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index 4f364fd9f..986cbd588 100644
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+          <link rel="alternate" href="/en/chapter_paperbook/" hreflang="en">
 
 
 
@@ -211,7 +211,7 @@
           </li>
 
           <li class="md-select__item">
-            <a href="/en/" hreflang="en" class="md-select__link">
+            <a href="/en/chapter_paperbook/" hreflang="en" class="md-select__link">
               English
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diff --git a/en/404.html b/en/404.html
index e5045f44d..585157fe8 100644
--- a/en/404.html
+++ b/en/404.html
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       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
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   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
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-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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-  Chapter 0. Preface
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-  Chapter 0. Preface
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-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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-  Chapter 1. Encounter with Algorithms
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-  1.1 Algorithms Are Everywhere
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
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   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
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-  2.2 Iteration and Recursion
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-  2.3 Time Complexity
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   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
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@@ -1258,7 +1258,7 @@
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-  4.1 Array
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1478,7 +1478,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1499,7 +1499,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1671,7 +1671,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1692,7 +1692,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1713,7 +1713,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1821,7 +1821,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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@@ -2066,7 +2066,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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@@ -2157,7 +2157,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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   <span class="md-ellipsis">
 
 
-  8.1 Heap
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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@@ -2280,7 +2280,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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@@ -2307,7 +2307,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2371,7 +2371,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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@@ -2392,7 +2392,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2413,7 +2413,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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@@ -2440,7 +2440,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2467,7 +2467,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2494,7 +2494,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2521,7 +2521,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2589,7 +2589,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2610,7 +2610,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2631,7 +2631,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2658,7 +2658,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2685,7 +2685,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2712,7 +2712,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2739,7 +2739,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2766,7 +2766,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2793,7 +2793,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2871,7 +2871,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2892,7 +2892,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2913,7 +2913,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2940,7 +2940,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2967,7 +2967,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -2994,7 +2994,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3021,7 +3021,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3048,7 +3048,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3075,7 +3075,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3102,7 +3102,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3129,7 +3129,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3156,7 +3156,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3183,7 +3183,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3210,7 +3210,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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@@ -3276,7 +3276,7 @@
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-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              16.2 &nbsp; Contributing Together
+              16.2 &nbsp; Contribute to the book
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Dark mode"  type="radio" name="__palette" id="__palette_0">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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@@ -177,9 +177,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3283,7 +3283,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3304,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3352,7 +3352,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3379,7 +3379,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3406,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3433,7 +3433,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3460,7 +3460,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3526,7 +3526,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3547,7 +3547,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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@@ -3568,7 +3568,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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@@ -3595,7 +3595,7 @@
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3773,7 +3773,7 @@
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+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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@@ -3815,7 +3815,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3842,7 +3842,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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@@ -3869,7 +3869,7 @@
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3896,7 +3896,7 @@
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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@@ -3923,7 +3923,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -3950,7 +3950,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -3977,7 +3977,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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@@ -4004,7 +4004,7 @@
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4070,7 +4070,7 @@
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4112,7 +4112,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4139,7 +4139,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4193,7 +4193,7 @@
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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@@ -4247,7 +4247,7 @@
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4309,7 +4309,7 @@
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4351,7 +4351,7 @@
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4387,7 +4387,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4404,7 +4404,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-minor-content-adjustments" class="md-nav__link">
+  <a href="#1-small-edits" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Minor Content Adjustments
+        1. &nbsp; Small edits
       </span>
     </span>
   </a>
@@ -4441,10 +4441,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#2-content-creation" class="md-nav__link">
+  <a href="#2-creating-content" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Content Creation
+        2. &nbsp; Creating content
       </span>
     </span>
   </a>
@@ -4452,10 +4452,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#3-docker-deployment" class="md-nav__link">
+  <a href="#3-deploying-with-docker" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Docker Deployment
+        3. &nbsp; Deploying with Docker
       </span>
     </span>
   </a>
@@ -4485,7 +4485,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4569,6 +4569,78 @@
 
 
 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item md-nav__item--nested">
+
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+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
+
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+          <div class="md-nav__link md-nav__container">
+            <a href="../../chapter_paperbook/" class="md-nav__link ">
+
+
+
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19 2-5 4.5v11l5-4.5zM6.5 5C4.55 5 2.45 5.4 1 6.5v14.66c0 .25.25.5.5.5.1 0 .15-.07.25-.07 1.35-.65 3.3-1.09 4.75-1.09 1.95 0 4.05.4 5.5 1.5 1.35-.85 3.8-1.5 5.5-1.5 1.65 0 3.35.31 4.75 1.06.1.05.15.03.25.03.25 0 .5-.25.5-.5V6.5c-.6-.45-1.25-.75-2-1V19c-1.1-.35-2.3-.5-3.5-.5-1.7 0-4.15.65-5.5 1.5V6.5C10.55 5.4 8.45 5 6.5 5"/></svg>
+
+  <span class="md-ellipsis">
+
+
+  Print edition (Chinese)
+
+
+  </span>
+
+
+
+
+
+    <span class="md-status md-status--new" title="Recently added">
+    </span>
+
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+            </a>
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
+            <span class="md-nav__icon md-icon"></span>
+
+
+  Print edition (Chinese)
+
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+
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+        </nav>
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 </nav>
                   </div>
@@ -4610,10 +4682,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-minor-content-adjustments" class="md-nav__link">
+  <a href="#1-small-edits" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Minor Content Adjustments
+        1. &nbsp; Small edits
       </span>
     </span>
   </a>
@@ -4621,10 +4693,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#2-content-creation" class="md-nav__link">
+  <a href="#2-creating-content" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Content Creation
+        2. &nbsp; Creating content
       </span>
     </span>
   </a>
@@ -4632,10 +4704,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#3-docker-deployment" class="md-nav__link">
+  <a href="#3-deploying-with-docker" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Docker Deployment
+        3. &nbsp; Deploying with Docker
       </span>
     </span>
   </a>
@@ -4679,36 +4751,36 @@
 
 
 <!-- Page content -->
-<h1 id="162-contributing-together">16.2 &nbsp; Contributing Together<a class="headerlink" href="#162-contributing-together" title="Permanent link">&para;</a></h1>
-<p>Due to limited capacity, there may be inevitable omissions and errors in this book. We appreciate your understanding and are grateful for your help in correcting them. If you discover typos, broken links, missing content, ambiguous wording, unclear explanations, or structural issues, please help us make corrections to provide readers with higher-quality learning resources.</p>
-<p>The GitHub IDs of all <a href="https://github.com/krahets/hello-algo/graphs/contributors">contributors</a> will be displayed on the homepage of the book repository, the web version, and the PDF version to acknowledge their selfless contributions to the open source community.</p>
+<h1 id="162-contribute-to-the-book">16.2 &nbsp; Contribute to the book<a class="headerlink" href="#162-contribute-to-the-book" title="Permanent link">&para;</a></h1>
+<p>Despite the author's best efforts, this book may contain omissions and errors. Thank you for your understanding. If you find typos, broken links, missing content, ambiguous wording, unclear explanations, or poorly organized passages, please help us correct them so that we can provide a better learning resource.</p>
+<p>To thank everyone for their generous contributions to the open-source community, the GitHub IDs of all <a href="https://github.com/krahets/hello-algo/graphs/contributors">contributors</a> are displayed on the home pages of the book's repository, website, and PDF edition.</p>
 <div class="admonition success">
-<p class="admonition-title">The Charm of Open Source</p>
-<p>The interval between two printings of a physical book is often quite long, making content updates very inconvenient.</p>
-<p>In this open source book, the time for content updates has been shortened to just days or even hours.</p>
+<p class="admonition-title">The beauty of open source</p>
+<p>Printed books often have long intervals between print runs, making their content difficult to update.</p>
+<p>In this open-source book, updates can be made within days or even hours.</p>
 </div>
-<h3 id="1-minor-content-adjustments">1. &nbsp; Minor Content Adjustments<a class="headerlink" href="#1-minor-content-adjustments" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 16-3, there is an "edit icon" in the top-right corner of each page. You can modify text or code by following these steps.</p>
+<h3 id="1-small-edits">1. &nbsp; Small edits<a class="headerlink" href="#1-small-edits" title="Permanent link">&para;</a></h3>
+<p>As shown in Figure 16-3, each page has an edit icon in its upper-right corner. Follow these steps to edit text or code.</p>
 <ol>
-<li>Click the "edit icon". If you encounter a prompt asking you to "Fork this repository", please approve the operation.</li>
-<li>Modify the content of the Markdown source file, verify the correctness of the content, and maintain consistent formatting as much as possible.</li>
-<li>Fill in a description of your changes at the bottom of the page, then click the "Propose file change" button. After the new page loads, click the "Create pull request" button to submit your pull request.</li>
+<li>Click the edit icon. If you are prompted to fork the repository, agree to do so.</li>
+<li>Edit the Markdown source file, check that the content is correct, and keep the formatting as consistent as possible.</li>
+<li>Enter a description of your changes at the bottom of the page, then click "Propose file change". On the next page, click "Create pull request" to submit your pull request.</li>
 </ol>
 <p><img alt="Page edit button" class="animation-figure" src="../contribution.assets/edit_markdown.png" /></p>
 <p align="center"> Figure 16-3 &nbsp; Page edit button </p>
 
-<p>Images cannot be directly modified. Please describe the issue by creating a new <a href="https://github.com/krahets/hello-algo/issues">Issue</a> or leaving a comment. We will promptly redraw and replace the images.</p>
-<h3 id="2-content-creation">2. &nbsp; Content Creation<a class="headerlink" href="#2-content-creation" title="Permanent link">&para;</a></h3>
-<p>If you are interested in contributing to this open source project, including translating code into other programming languages or expanding article content, you will need to follow the Pull Request workflow below.</p>
+<p>Images cannot be edited directly. Please describe the problem in a new <a href="https://github.com/krahets/hello-algo/issues">issue</a> or a comment, and we will redraw and replace the image as soon as possible.</p>
+<h3 id="2-creating-content">2. &nbsp; Creating content<a class="headerlink" href="#2-creating-content" title="Permanent link">&para;</a></h3>
+<p>If you would like to contribute to this open-source project, for example by translating code into another programming language or expanding an article, follow this pull request workflow.</p>
 <ol>
-<li>Log in to GitHub and Fork the book's <a href="https://github.com/krahets/hello-algo">code repository</a> to your personal account.</li>
-<li>Go to your forked repository page and use the <code>git clone</code> command to clone the repository to your local machine.</li>
-<li>Create content locally and conduct comprehensive tests to verify code correctness.</li>
-<li>Commit your local changes and push them to the remote repository.</li>
-<li>Refresh the repository webpage and click the "Create pull request" button to submit your pull request.</li>
+<li>Sign in to GitHub and fork the book's <a href="https://github.com/krahets/hello-algo">code repository</a> to your own account.</li>
+<li>Open your fork's repository page and use <code>git clone</code> to clone it to your computer.</li>
+<li>Create your content locally and test it thoroughly to verify that the code is correct.</li>
+<li>Commit your local changes, then push them to your remote repository.</li>
+<li>Refresh the repository page and click "Create pull request" to submit your pull request.</li>
 </ol>
-<h3 id="3-docker-deployment">3. &nbsp; Docker Deployment<a class="headerlink" href="#3-docker-deployment" title="Permanent link">&para;</a></h3>
-<p>From the root directory of <code>hello-algo</code>, run the following Docker command to access the project at <code>http://localhost:8000</code>:</p>
+<h3 id="3-deploying-with-docker">3. &nbsp; Deploying with Docker<a class="headerlink" href="#3-deploying-with-docker" title="Permanent link">&para;</a></h3>
+<p>Run the following Docker command in the <code>hello-algo</code> root directory to access the English edition at <code>http://localhost:8000/en/</code>:</p>
 <div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a>docker-compose<span class="w"> </span>up<span class="w"> </span>-d
 </code></pre></div>
 <p>Use the following command to remove the deployment:</p>
@@ -4736,7 +4808,7 @@ aria-label="Footer"
   <a
     href="../installation/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 16.1 Programming Environment Installation"
+    aria-label="Previous: 16.1   Setting up a programming environment"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4748,7 +4820,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        16.1 Programming Environment Installation
+        16.1 &nbsp; Setting up a programming environment
       </div>
     </div>
   </a>
@@ -4760,7 +4832,7 @@ aria-label="Footer"
   <a
     href="../terminology/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 16.3 Glossary"
+    aria-label="Next: 16.3   Glossary"
     rel="next"
   >
     <div class="md-footer__title">
@@ -4768,7 +4840,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        16.3 Glossary
+        16.3 &nbsp; Glossary
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -4785,7 +4857,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_appendix/index.html b/en/chapter_appendix/index.html
index d0bb8feb2..e406fbbe7 100644
--- a/en/chapter_appendix/index.html
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@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -119,7 +119,7 @@
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-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -167,9 +167,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
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-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
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@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
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   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
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-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
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   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
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@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
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   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
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@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3283,7 +3283,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3304,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3352,7 +3352,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3379,7 +3379,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3406,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3433,7 +3433,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3460,7 +3460,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3526,7 +3526,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3547,7 +3547,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3568,7 +3568,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3595,7 +3595,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3622,7 +3622,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3649,7 +3649,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3676,7 +3676,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3703,7 +3703,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3773,7 +3773,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3794,7 +3794,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3815,7 +3815,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3842,7 +3842,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3869,7 +3869,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3896,7 +3896,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -3923,7 +3923,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -3950,7 +3950,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -3977,7 +3977,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4004,7 +4004,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4070,7 +4070,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4091,7 +4091,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4112,7 +4112,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4139,7 +4139,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4166,7 +4166,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
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-  16.2 Contributing Together
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 <p><img alt="Appendix" class="cover-image" src="../assets/covers/chapter_appendix.jpg" /></p>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
-<li><a href="installation/">16.1 &nbsp; Programming Environment Installation</a></li>
-<li><a href="contribution/">16.2 &nbsp; Contributing Together</a></li>
+<li><a href="installation/">16.1 &nbsp; Setting up a programming environment</a></li>
+<li><a href="contribution/">16.2 &nbsp; Contribute to the book</a></li>
 <li><a href="terminology/">16.3 &nbsp; Glossary</a></li>
 </ul>
 
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   <a
     href="../chapter_greedy/exercises/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 15.6 Exercises"
+    aria-label="Previous: 15.6   Exercises"
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@@ -4619,7 +4691,7 @@ aria-label="Footer"
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       <div class="md-ellipsis">
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+        15.6 &nbsp; Exercises
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   <a
     href="./installation/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 16.1 Programming Environment Installation"
+    aria-label="Next: 16.1   Setting up a programming environment"
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+        16.1 &nbsp; Setting up a programming environment
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
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+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
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@@ -3568,7 +3568,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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@@ -3676,7 +3676,7 @@
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3773,7 +3773,7 @@
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3815,7 +3815,7 @@
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3869,7 +3869,7 @@
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4070,7 +4070,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4112,7 +4112,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4139,7 +4139,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4166,7 +4166,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4193,7 +4193,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4220,7 +4220,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4247,7 +4247,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4309,7 +4309,7 @@
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
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@@ -4360,7 +4360,7 @@
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4403,10 +4403,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1611-installing-ide" class="md-nav__link">
+  <a href="#1611-install-an-ide" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        16.1.1 &nbsp; Installing IDE
+        16.1.1 &nbsp; Install an IDE
       </span>
     </span>
   </a>
@@ -4414,22 +4414,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1612-installing-language-environments" class="md-nav__link">
+  <a href="#1612-install-a-language-environment" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        16.1.2 &nbsp; Installing Language Environments
+        16.1.2 &nbsp; Install a language environment
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="16.1.2 Installing Language Environments">
+    <nav class="md-nav" aria-label="16.1.2 Install a language environment">
       <ul class="md-nav__list">
 
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-  <a href="#1-python-environment" class="md-nav__link">
+  <a href="#1-python" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Python Environment
+        1. &nbsp; Python
       </span>
     </span>
   </a>
@@ -4437,10 +4437,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-cc-environment" class="md-nav__link">
+  <a href="#2-cc" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; C/C++ Environment
+        2. &nbsp; C/C++
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   </a>
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 </li>
 
           <li class="md-nav__item">
-  <a href="#3-java-environment" class="md-nav__link">
+  <a href="#3-java" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Java Environment
+        3. &nbsp; Java
       </span>
     </span>
   </a>
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 </li>
 
           <li class="md-nav__item">
-  <a href="#4-c-environment" class="md-nav__link">
+  <a href="#4-c" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4. &nbsp; C# Environment
+        4. &nbsp; C#
       </span>
     </span>
   </a>
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 </li>
 
           <li class="md-nav__item">
-  <a href="#5-go-environment" class="md-nav__link">
+  <a href="#5-go" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5. &nbsp; Go Environment
+        5. &nbsp; Go
       </span>
     </span>
   </a>
@@ -4481,10 +4481,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#6-swift-environment" class="md-nav__link">
+  <a href="#6-swift" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6. &nbsp; Swift Environment
+        6. &nbsp; Swift
       </span>
     </span>
   </a>
@@ -4492,10 +4492,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#7-javascript-environment" class="md-nav__link">
+  <a href="#7-javascript" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7. &nbsp; JavaScript Environment
+        7. &nbsp; JavaScript
       </span>
     </span>
   </a>
@@ -4503,10 +4503,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#8-typescript-environment" class="md-nav__link">
+  <a href="#8-typescript" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8. &nbsp; TypeScript Environment
+        8. &nbsp; TypeScript
       </span>
     </span>
   </a>
@@ -4514,10 +4514,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#9-dart-environment" class="md-nav__link">
+  <a href="#9-dart" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9. &nbsp; Dart Environment
+        9. &nbsp; Dart
       </span>
     </span>
   </a>
@@ -4525,10 +4525,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#10-rust-environment" class="md-nav__link">
+  <a href="#10-rust" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10. &nbsp; Rust Environment
+        10. &nbsp; Rust
       </span>
     </span>
   </a>
@@ -4563,7 +4563,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4590,7 +4590,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4674,6 +4674,78 @@
 
 
 
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-  <a href="#1611-installing-ide" class="md-nav__link">
+  <a href="#1611-install-an-ide" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        16.1.1 &nbsp; Installing IDE
+        16.1.1 &nbsp; Install an IDE
       </span>
     </span>
   </a>
@@ -4726,22 +4798,22 @@
 </li>
 
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-  <a href="#1612-installing-language-environments" class="md-nav__link">
+  <a href="#1612-install-a-language-environment" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        16.1.2 &nbsp; Installing Language Environments
+        16.1.2 &nbsp; Install a language environment
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="16.1.2 Installing Language Environments">
+    <nav class="md-nav" aria-label="16.1.2 Install a language environment">
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-  <a href="#1-python-environment" class="md-nav__link">
+  <a href="#1-python" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Python Environment
+        1. &nbsp; Python
       </span>
     </span>
   </a>
@@ -4749,10 +4821,10 @@
 </li>
 
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-  <a href="#2-cc-environment" class="md-nav__link">
+  <a href="#2-cc" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; C/C++ Environment
+        2. &nbsp; C/C++
       </span>
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   </a>
@@ -4760,10 +4832,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-java-environment" class="md-nav__link">
+  <a href="#3-java" class="md-nav__link">
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       <span class="md-typeset">
-        3. &nbsp; Java Environment
+        3. &nbsp; Java
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 </li>
 
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-  <a href="#4-c-environment" class="md-nav__link">
+  <a href="#4-c" class="md-nav__link">
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-        4. &nbsp; C# Environment
+        4. &nbsp; C#
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 </li>
 
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+  <a href="#5-go" class="md-nav__link">
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-        5. &nbsp; Go Environment
+        5. &nbsp; Go
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 </li>
 
           <li class="md-nav__item">
-  <a href="#6-swift-environment" class="md-nav__link">
+  <a href="#6-swift" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6. &nbsp; Swift Environment
+        6. &nbsp; Swift
       </span>
     </span>
   </a>
@@ -4804,10 +4876,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#7-javascript-environment" class="md-nav__link">
+  <a href="#7-javascript" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7. &nbsp; JavaScript Environment
+        7. &nbsp; JavaScript
       </span>
     </span>
   </a>
@@ -4815,10 +4887,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#8-typescript-environment" class="md-nav__link">
+  <a href="#8-typescript" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8. &nbsp; TypeScript Environment
+        8. &nbsp; TypeScript
       </span>
     </span>
   </a>
@@ -4826,10 +4898,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#9-dart-environment" class="md-nav__link">
+  <a href="#9-dart" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9. &nbsp; Dart Environment
+        9. &nbsp; Dart
       </span>
     </span>
   </a>
@@ -4837,10 +4909,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#10-rust-environment" class="md-nav__link">
+  <a href="#10-rust" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10. &nbsp; Rust Environment
+        10. &nbsp; Rust
       </span>
     </span>
   </a>
@@ -4889,71 +4961,71 @@
 
 
 <!-- Page content -->
-<h1 id="161-programming-environment-installation">16.1 &nbsp; Programming Environment Installation<a class="headerlink" href="#161-programming-environment-installation" title="Permanent link">&para;</a></h1>
-<h2 id="1611-installing-ide">16.1.1 &nbsp; Installing IDE<a class="headerlink" href="#1611-installing-ide" title="Permanent link">&para;</a></h2>
-<p>We recommend using the open-source and lightweight VS Code as the local integrated development environment (IDE). Visit the <a href="https://code.visualstudio.com/">VS Code official website</a>, and download and install the appropriate version of VS Code according to your operating system.</p>
-<p><img alt="Download VS Code from the Official Website" class="animation-figure" src="../installation.assets/vscode_installation.png" /></p>
-<p align="center"> Figure 16-1 &nbsp; Download VS Code from the Official Website </p>
+<h1 id="161-setting-up-a-programming-environment">16.1 &nbsp; Setting up a programming environment<a class="headerlink" href="#161-setting-up-a-programming-environment" title="Permanent link">&para;</a></h1>
+<h2 id="1611-install-an-ide">16.1.1 &nbsp; Install an IDE<a class="headerlink" href="#1611-install-an-ide" title="Permanent link">&para;</a></h2>
+<p>We recommend VS Code, a lightweight, open-source integrated development environment (IDE). Visit the <a href="https://code.visualstudio.com/">VS Code website</a>, then download and install the version for your operating system.</p>
+<p><img alt="Downloading VS Code from the official website" class="animation-figure" src="../installation.assets/vscode_installation.png" /></p>
+<p align="center"> Figure 16-1 &nbsp; Downloading VS Code from the official website </p>
 
-<p>VS Code has a powerful ecosystem of extensions that supports running and debugging most programming languages. For example, after installing the "Python Extension Pack" extension, you can debug Python code. The installation steps are shown in the following figure.</p>
-<p><img alt="Install VS Code Extensions" class="animation-figure" src="../installation.assets/vscode_extension_installation.png" /></p>
-<p align="center"> Figure 16-2 &nbsp; Install VS Code Extensions </p>
+<p>VS Code has an extensive ecosystem of extensions that support running and debugging code in most programming languages. For example, installing the Python Extension Pack lets you debug Python code. The installation steps are shown in Figure 16-2.</p>
+<p><img alt="Installing VS Code extensions" class="animation-figure" src="../installation.assets/vscode_extension_installation.png" /></p>
+<p align="center"> Figure 16-2 &nbsp; Installing VS Code extensions </p>
 
-<h2 id="1612-installing-language-environments">16.1.2 &nbsp; Installing Language Environments<a class="headerlink" href="#1612-installing-language-environments" title="Permanent link">&para;</a></h2>
-<h3 id="1-python-environment">1. &nbsp; Python Environment<a class="headerlink" href="#1-python-environment" title="Permanent link">&para;</a></h3>
+<h2 id="1612-install-a-language-environment">16.1.2 &nbsp; Install a language environment<a class="headerlink" href="#1612-install-a-language-environment" title="Permanent link">&para;</a></h2>
+<h3 id="1-python">1. &nbsp; Python<a class="headerlink" href="#1-python" title="Permanent link">&para;</a></h3>
 <ol>
-<li>Download and install <a href="https://docs.conda.io/en/latest/miniconda.html">Miniconda3</a> with Python 3.10 or later.</li>
-<li>Search for <code>python</code> in the VS Code extension marketplace and install the Python Extension Pack.</li>
-<li>(Optional) Enter <code>pip install black</code> on the command line to install the code formatter.</li>
+<li>Download and install <a href="https://docs.conda.io/en/latest/miniconda.html">Miniconda3</a>. Python 3.10 or later is required.</li>
+<li>Search for <code>python</code> in the VS Code Extensions Marketplace and install Python Extension Pack.</li>
+<li>(Optional) Run <code>pip install black</code> in a terminal to install a code formatter.</li>
 </ol>
-<h3 id="2-cc-environment">2. &nbsp; C/C++ Environment<a class="headerlink" href="#2-cc-environment" title="Permanent link">&para;</a></h3>
+<h3 id="2-cc">2. &nbsp; C/C++<a class="headerlink" href="#2-cc" title="Permanent link">&para;</a></h3>
 <ol>
-<li>Windows systems need to install <a href="https://sourceforge.net/projects/mingw-w64/files/">MinGW</a> (<a href="https://blog.csdn.net/qq_33698226/article/details/129031241">configuration tutorial</a>); macOS comes with Clang built-in and does not require installation.</li>
-<li>Search for <code>c++</code> in the VS Code extension marketplace and install the C/C++ Extension Pack.</li>
-<li>(Optional) Open the Settings page, search for the <code>Clang_format_fallback Style</code> code formatting option, and set it to <code>{ BasedOnStyle: Microsoft, BreakBeforeBraces: Attach }</code>.</li>
+<li>On Windows, install <a href="https://sourceforge.net/projects/mingw-w64/files/">MinGW</a> (<a href="https://blog.csdn.net/qq_33698226/article/details/129031241">setup guide</a>). macOS comes with Clang, so no installation is needed.</li>
+<li>Search for <code>c++</code> in the VS Code Extensions Marketplace and install C/C++ Extension Pack.</li>
+<li>(Optional) Open Settings, search for the <code>Clang_format_fallback Style</code> formatting option, and set it to <code>{ BasedOnStyle: Microsoft, BreakBeforeBraces: Attach }</code>.</li>
 </ol>
-<h3 id="3-java-environment">3. &nbsp; Java Environment<a class="headerlink" href="#3-java-environment" title="Permanent link">&para;</a></h3>
+<h3 id="3-java">3. &nbsp; Java<a class="headerlink" href="#3-java" title="Permanent link">&para;</a></h3>
 <ol>
-<li>Download and install <a href="https://jdk.java.net/18/">OpenJDK</a> (version 10 or later).</li>
-<li>Search for <code>java</code> in the VS Code extension marketplace and install the Extension Pack for Java.</li>
+<li>Download and install <a href="https://jdk.java.net/18/">OpenJDK</a> (a version later than JDK 9 is required).</li>
+<li>Search for <code>java</code> in the VS Code Extensions Marketplace and install Extension Pack for Java.</li>
 </ol>
-<h3 id="4-c-environment">4. &nbsp; C# Environment<a class="headerlink" href="#4-c-environment" title="Permanent link">&para;</a></h3>
+<h3 id="4-c">4. &nbsp; C#<a class="headerlink" href="#4-c" title="Permanent link">&para;</a></h3>
 <ol>
 <li>Download and install <a href="https://dotnet.microsoft.com/en-us/download">.NET 8.0</a>.</li>
-<li>Search for <code>C# Dev Kit</code> in the VS Code extension marketplace and install C# Dev Kit (<a href="https://code.visualstudio.com/docs/csharp/get-started">configuration tutorial</a>).</li>
-<li>You can also use Visual Studio (<a href="https://learn.microsoft.com/zh-cn/visualstudio/install/install-visual-studio?view=vs-2022">installation tutorial</a>).</li>
+<li>Search for <code>C# Dev Kit</code> in the VS Code Extensions Marketplace and install C# Dev Kit (<a href="https://code.visualstudio.com/docs/csharp/get-started">setup guide</a>).</li>
+<li>You can also use Visual Studio (<a href="https://learn.microsoft.com/en-us/visualstudio/install/install-visual-studio?view=vs-2022">installation guide</a>).</li>
 </ol>
-<h3 id="5-go-environment">5. &nbsp; Go Environment<a class="headerlink" href="#5-go-environment" title="Permanent link">&para;</a></h3>
+<h3 id="5-go">5. &nbsp; Go<a class="headerlink" href="#5-go" title="Permanent link">&para;</a></h3>
 <ol>
 <li>Download and install <a href="https://go.dev/dl/">Go</a>.</li>
-<li>Search for <code>go</code> in the VS Code extension marketplace and install Go.</li>
-<li>Press <code>Ctrl + Shift + P</code> to open the command palette, type <code>go</code>, select <code>Go: Install/Update Tools</code>, check all options and install.</li>
+<li>Search for <code>go</code> in the VS Code Extensions Marketplace and install Go.</li>
+<li>Press <code>Ctrl + Shift + P</code> to open the Command Palette, enter go, select <code>Go: Install/Update Tools</code>, then select and install all the tools.</li>
 </ol>
-<h3 id="6-swift-environment">6. &nbsp; Swift Environment<a class="headerlink" href="#6-swift-environment" title="Permanent link">&para;</a></h3>
+<h3 id="6-swift">6. &nbsp; Swift<a class="headerlink" href="#6-swift" title="Permanent link">&para;</a></h3>
 <ol>
 <li>Download and install <a href="https://www.swift.org/download/">Swift</a>.</li>
-<li>Search for <code>swift</code> in the VS Code extension marketplace and install <a href="https://marketplace.visualstudio.com/items?itemName=sswg.swift-lang">Swift for Visual Studio Code</a>.</li>
+<li>Search for <code>swift</code> in the VS Code Extensions Marketplace and install <a href="https://marketplace.visualstudio.com/items?itemName=sswg.swift-lang">Swift for Visual Studio Code</a>.</li>
 </ol>
-<h3 id="7-javascript-environment">7. &nbsp; JavaScript Environment<a class="headerlink" href="#7-javascript-environment" title="Permanent link">&para;</a></h3>
+<h3 id="7-javascript">7. &nbsp; JavaScript<a class="headerlink" href="#7-javascript" title="Permanent link">&para;</a></h3>
 <ol>
 <li>Download and install <a href="https://nodejs.org/en/">Node.js</a>.</li>
-<li>(Optional) Search for <code>Prettier</code> in the VS Code extension marketplace and install the code formatter.</li>
+<li>(Optional) Search for <code>Prettier</code> in the VS Code Extensions Marketplace and install this code formatter.</li>
 </ol>
-<h3 id="8-typescript-environment">8. &nbsp; TypeScript Environment<a class="headerlink" href="#8-typescript-environment" title="Permanent link">&para;</a></h3>
+<h3 id="8-typescript">8. &nbsp; TypeScript<a class="headerlink" href="#8-typescript" title="Permanent link">&para;</a></h3>
 <ol>
-<li>Follow the same installation steps as the JavaScript environment.</li>
+<li>Follow the JavaScript environment setup steps.</li>
 <li>Install <a href="https://github.com/privatenumber/tsx?tab=readme-ov-file#global-installation">TypeScript Execute (tsx)</a>.</li>
-<li>Search for <code>typescript</code> in the VS Code extension marketplace and install <a href="https://marketplace.visualstudio.com/items?itemName=yoavbls.pretty-ts-errors">Pretty TypeScript Errors</a>.</li>
+<li>Search for <code>typescript</code> in the VS Code Extensions Marketplace and install <a href="https://marketplace.visualstudio.com/items?itemName=yoavbls.pretty-ts-errors">Pretty TypeScript Errors</a>.</li>
 </ol>
-<h3 id="9-dart-environment">9. &nbsp; Dart Environment<a class="headerlink" href="#9-dart-environment" title="Permanent link">&para;</a></h3>
+<h3 id="9-dart">9. &nbsp; Dart<a class="headerlink" href="#9-dart" title="Permanent link">&para;</a></h3>
 <ol>
 <li>Download and install <a href="https://dart.dev/get-dart">Dart</a>.</li>
-<li>Search for <code>dart</code> in the VS Code extension marketplace and install <a href="https://marketplace.visualstudio.com/items?itemName=Dart-Code.dart-code">Dart</a>.</li>
+<li>Search for <code>dart</code> in the VS Code Extensions Marketplace and install <a href="https://marketplace.visualstudio.com/items?itemName=Dart-Code.dart-code">Dart</a>.</li>
 </ol>
-<h3 id="10-rust-environment">10. &nbsp; Rust Environment<a class="headerlink" href="#10-rust-environment" title="Permanent link">&para;</a></h3>
+<h3 id="10-rust">10. &nbsp; Rust<a class="headerlink" href="#10-rust" title="Permanent link">&para;</a></h3>
 <ol>
 <li>Download and install <a href="https://www.rust-lang.org/tools/install">Rust</a>.</li>
-<li>Search for <code>rust</code> in the VS Code extension marketplace and install <a href="https://marketplace.visualstudio.com/items?itemName=rust-lang.rust-analyzer">rust-analyzer</a>.</li>
+<li>Search for <code>rust</code> in the VS Code Extensions Marketplace and install <a href="https://marketplace.visualstudio.com/items?itemName=rust-lang.rust-analyzer">rust-analyzer</a>.</li>
 </ol>
 
 <!-- Source file information -->
@@ -4977,7 +5049,7 @@ aria-label="Footer"
   <a
     href="../"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: Chapter 16. &amp;nbsp; Appendix"
+    aria-label="Previous: Chapter 16.   Appendix"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -5001,7 +5073,7 @@ aria-label="Footer"
   <a
     href="../contribution/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 16.2 Contributing Together"
+    aria-label="Next: 16.2   Contribute to the book"
     rel="next"
   >
     <div class="md-footer__title">
@@ -5009,7 +5081,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        16.2 Contributing Together
+        16.2 &nbsp; Contribute to the book
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -5026,7 +5098,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_appendix/terminology/index.html b/en/chapter_appendix/terminology/index.html
index d0b838afe..05107d8c8 100644
--- a/en/chapter_appendix/terminology/index.html
+++ b/en/chapter_appendix/terminology/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -119,7 +119,7 @@
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-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -167,9 +167,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
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@@ -177,9 +177,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1265,7 +1265,7 @@
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-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
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@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3283,7 +3283,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3304,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
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@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3406,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3433,7 +3433,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3568,7 +3568,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
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@@ -3815,7 +3815,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4070,7 +4070,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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@@ -4112,7 +4112,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4193,7 +4193,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4309,7 +4309,7 @@
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4351,7 +4351,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4378,7 +4378,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4414,7 +4414,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4499,6 +4499,78 @@
 
 
 
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 <!-- Page content -->
 <h1 id="163-glossary">16.3 &nbsp; Glossary<a class="headerlink" href="#163-glossary" title="Permanent link">&para;</a></h1>
-<p>The following table lists important terms that appear in this book.</p>
-<p align="center"> Table 16-1 &nbsp; Important Terms in Data Structures and Algorithms </p>
+<p>Table 16-1 lists important terms used in this book. Familiarity with their English names will help when reading technical literature in English.</p>
+<p align="center"> Table 16-1 &nbsp; Key terms in data structures and algorithms </p>
 
 <div class="center-table">
 <table>
 <thead>
 <tr>
 <th>English</th>
+<th>Simplified Chinese</th>
 </tr>
 </thead>
 <tbody>
 <tr>
 <td>algorithm</td>
+<td>算法</td>
 </tr>
 <tr>
 <td>data structure</td>
+<td>数据结构</td>
 </tr>
 <tr>
 <td>code</td>
+<td>代码</td>
 </tr>
 <tr>
 <td>file</td>
+<td>文件</td>
 </tr>
 <tr>
 <td>function</td>
+<td>函数</td>
 </tr>
 <tr>
 <td>method</td>
+<td>方法</td>
 </tr>
 <tr>
 <td>variable</td>
+<td>变量</td>
 </tr>
 <tr>
 <td>asymptotic complexity analysis</td>
+<td>渐近复杂度分析</td>
 </tr>
 <tr>
 <td>time complexity</td>
+<td>时间复杂度</td>
 </tr>
 <tr>
 <td>space complexity</td>
+<td>空间复杂度</td>
 </tr>
 <tr>
 <td>loop</td>
+<td>循环</td>
 </tr>
 <tr>
 <td>iteration</td>
+<td>迭代</td>
 </tr>
 <tr>
 <td>recursion</td>
+<td>递归</td>
 </tr>
 <tr>
 <td>tail recursion</td>
+<td>尾递归</td>
 </tr>
 <tr>
 <td>recursion tree</td>
+<td>递归树</td>
 </tr>
 <tr>
 <td>big-<span class="arithmatex">\(O\)</span> notation</td>
+<td>大 <span class="arithmatex">\(O\)</span> 记号</td>
 </tr>
 <tr>
 <td>asymptotic upper bound</td>
+<td>渐近上界</td>
 </tr>
 <tr>
 <td>sign-magnitude</td>
+<td>原码</td>
 </tr>
 <tr>
-<td>1’s complement</td>
+<td>one's complement</td>
+<td>反码</td>
 </tr>
 <tr>
-<td>2’s complement</td>
+<td>two's complement</td>
+<td>补码</td>
 </tr>
 <tr>
 <td>array</td>
+<td>数组</td>
 </tr>
 <tr>
 <td>index</td>
+<td>索引</td>
 </tr>
 <tr>
 <td>linked list</td>
+<td>链表</td>
 </tr>
 <tr>
 <td>linked list node, list node</td>
+<td>链表节点</td>
 </tr>
 <tr>
 <td>head node</td>
+<td>头节点</td>
 </tr>
 <tr>
 <td>tail node</td>
+<td>尾节点</td>
 </tr>
 <tr>
 <td>list</td>
+<td>列表</td>
 </tr>
 <tr>
 <td>dynamic array</td>
+<td>动态数组</td>
 </tr>
 <tr>
 <td>hard disk</td>
+<td>硬盘</td>
 </tr>
 <tr>
 <td>random-access memory (RAM)</td>
+<td>内存</td>
 </tr>
 <tr>
 <td>cache memory</td>
+<td>缓存</td>
 </tr>
 <tr>
 <td>cache miss</td>
+<td>缓存未命中</td>
 </tr>
 <tr>
 <td>cache hit rate</td>
+<td>缓存命中率</td>
 </tr>
 <tr>
 <td>stack</td>
+<td>栈</td>
 </tr>
 <tr>
 <td>top of the stack</td>
+<td>栈顶</td>
 </tr>
 <tr>
 <td>bottom of the stack</td>
+<td>栈底</td>
 </tr>
 <tr>
 <td>queue</td>
+<td>队列</td>
 </tr>
 <tr>
 <td>double-ended queue</td>
+<td>双向队列</td>
 </tr>
 <tr>
 <td>front of the queue</td>
+<td>队首</td>
 </tr>
 <tr>
 <td>rear of the queue</td>
+<td>队尾</td>
 </tr>
 <tr>
 <td>hash table</td>
+<td>哈希表</td>
 </tr>
 <tr>
 <td>hash set</td>
+<td>哈希集合</td>
 </tr>
 <tr>
 <td>bucket</td>
+<td>桶</td>
 </tr>
 <tr>
 <td>hash function</td>
+<td>哈希函数</td>
 </tr>
 <tr>
 <td>hash collision</td>
+<td>哈希冲突</td>
 </tr>
 <tr>
 <td>load factor</td>
+<td>负载因子</td>
 </tr>
 <tr>
 <td>separate chaining</td>
+<td>链式地址</td>
 </tr>
 <tr>
 <td>open addressing</td>
+<td>开放寻址</td>
 </tr>
 <tr>
 <td>linear probing</td>
+<td>线性探测</td>
 </tr>
 <tr>
 <td>lazy deletion</td>
+<td>懒删除</td>
 </tr>
 <tr>
 <td>binary tree</td>
+<td>二叉树</td>
 </tr>
 <tr>
 <td>tree node</td>
+<td>树节点</td>
 </tr>
 <tr>
 <td>left-child node</td>
+<td>左子节点</td>
 </tr>
 <tr>
 <td>right-child node</td>
+<td>右子节点</td>
 </tr>
 <tr>
 <td>parent node</td>
+<td>父节点</td>
 </tr>
 <tr>
 <td>left subtree</td>
+<td>左子树</td>
 </tr>
 <tr>
 <td>right subtree</td>
+<td>右子树</td>
 </tr>
 <tr>
 <td>root node</td>
+<td>根节点</td>
 </tr>
 <tr>
 <td>leaf node</td>
+<td>叶节点</td>
 </tr>
 <tr>
 <td>edge</td>
+<td>边</td>
 </tr>
 <tr>
 <td>level</td>
+<td>层</td>
 </tr>
 <tr>
 <td>degree</td>
+<td>度</td>
 </tr>
 <tr>
 <td>height</td>
+<td>高度</td>
 </tr>
 <tr>
 <td>depth</td>
+<td>深度</td>
 </tr>
 <tr>
 <td>perfect binary tree</td>
+<td>完美二叉树</td>
 </tr>
 <tr>
 <td>complete binary tree</td>
+<td>完全二叉树</td>
 </tr>
 <tr>
 <td>full binary tree</td>
+<td>完满二叉树</td>
 </tr>
 <tr>
 <td>balanced binary tree</td>
+<td>平衡二叉树</td>
 </tr>
 <tr>
 <td>binary search tree</td>
+<td>二叉搜索树</td>
 </tr>
 <tr>
 <td>AVL tree</td>
+<td>AVL 树</td>
 </tr>
 <tr>
 <td>red-black tree</td>
+<td>红黑树</td>
 </tr>
 <tr>
 <td>level-order traversal</td>
+<td>层序遍历</td>
 </tr>
 <tr>
 <td>breadth-first traversal</td>
+<td>广度优先遍历</td>
 </tr>
 <tr>
 <td>depth-first traversal</td>
+<td>深度优先遍历</td>
 </tr>
 <tr>
-<td>pre-order traversal</td>
+<td>preorder traversal</td>
+<td>前序遍历</td>
 </tr>
 <tr>
-<td>in-order traversal</td>
+<td>inorder traversal</td>
+<td>中序遍历</td>
 </tr>
 <tr>
-<td>post-order traversal</td>
+<td>postorder traversal</td>
+<td>后序遍历</td>
 </tr>
 <tr>
 <td>balanced binary search tree</td>
+<td>平衡二叉搜索树</td>
 </tr>
 <tr>
 <td>balance factor</td>
+<td>平衡因子</td>
 </tr>
 <tr>
 <td>heap</td>
+<td>堆</td>
 </tr>
 <tr>
 <td>max heap</td>
+<td>大顶堆</td>
 </tr>
 <tr>
 <td>min heap</td>
+<td>小顶堆</td>
 </tr>
 <tr>
 <td>priority queue</td>
+<td>优先队列</td>
 </tr>
 <tr>
 <td>heapify</td>
+<td>堆化</td>
 </tr>
 <tr>
 <td>top-<span class="arithmatex">\(k\)</span> problem</td>
+<td>Top-<span class="arithmatex">\(k\)</span> 问题</td>
 </tr>
 <tr>
 <td>graph</td>
+<td>图</td>
 </tr>
 <tr>
 <td>vertex</td>
+<td>顶点</td>
 </tr>
 <tr>
 <td>undirected graph</td>
+<td>无向图</td>
 </tr>
 <tr>
 <td>directed graph</td>
+<td>有向图</td>
 </tr>
 <tr>
 <td>connected graph</td>
+<td>连通图</td>
 </tr>
 <tr>
 <td>disconnected graph</td>
+<td>非连通图</td>
 </tr>
 <tr>
 <td>weighted graph</td>
+<td>有权图</td>
 </tr>
 <tr>
 <td>adjacency</td>
+<td>邻接</td>
 </tr>
 <tr>
 <td>path</td>
+<td>路径</td>
 </tr>
 <tr>
 <td>in-degree</td>
+<td>入度</td>
 </tr>
 <tr>
 <td>out-degree</td>
+<td>出度</td>
 </tr>
 <tr>
 <td>adjacency matrix</td>
+<td>邻接矩阵</td>
 </tr>
 <tr>
 <td>adjacency list</td>
+<td>邻接表</td>
 </tr>
 <tr>
 <td>breadth-first search</td>
+<td>广度优先搜索</td>
 </tr>
 <tr>
 <td>depth-first search</td>
+<td>深度优先搜索</td>
 </tr>
 <tr>
 <td>binary search</td>
+<td>二分查找</td>
 </tr>
 <tr>
 <td>searching algorithm</td>
+<td>搜索算法</td>
 </tr>
 <tr>
 <td>sorting algorithm</td>
+<td>排序算法</td>
 </tr>
 <tr>
 <td>selection sort</td>
+<td>选择排序</td>
 </tr>
 <tr>
 <td>bubble sort</td>
+<td>冒泡排序</td>
 </tr>
 <tr>
 <td>insertion sort</td>
+<td>插入排序</td>
 </tr>
 <tr>
 <td>quick sort</td>
+<td>快速排序</td>
 </tr>
 <tr>
 <td>merge sort</td>
+<td>归并排序</td>
 </tr>
 <tr>
 <td>heap sort</td>
+<td>堆排序</td>
 </tr>
 <tr>
 <td>bucket sort</td>
+<td>桶排序</td>
 </tr>
 <tr>
 <td>counting sort</td>
+<td>计数排序</td>
 </tr>
 <tr>
 <td>radix sort</td>
+<td>基数排序</td>
 </tr>
 <tr>
 <td>divide and conquer</td>
+<td>分治</td>
 </tr>
 <tr>
 <td>Tower of Hanoi problem</td>
+<td>汉诺塔问题</td>
 </tr>
 <tr>
 <td>backtracking algorithm</td>
+<td>回溯算法</td>
 </tr>
 <tr>
 <td>constraint</td>
+<td>约束</td>
 </tr>
 <tr>
 <td>solution</td>
+<td>解</td>
 </tr>
 <tr>
 <td>state</td>
+<td>状态</td>
 </tr>
 <tr>
 <td>pruning</td>
+<td>剪枝</td>
 </tr>
 <tr>
 <td>permutations problem</td>
+<td>全排列问题</td>
 </tr>
 <tr>
 <td>subset-sum problem</td>
+<td>子集和问题</td>
 </tr>
 <tr>
 <td><span class="arithmatex">\(n\)</span>-queens problem</td>
+<td><span class="arithmatex">\(n\)</span> 皇后问题</td>
 </tr>
 <tr>
 <td>dynamic programming</td>
+<td>动态规划</td>
 </tr>
 <tr>
 <td>initial state</td>
+<td>初始状态</td>
 </tr>
 <tr>
-<td>state-transition equation</td>
+<td>state transition equation</td>
+<td>状态转移方程</td>
 </tr>
 <tr>
 <td>knapsack problem</td>
+<td>背包问题</td>
 </tr>
 <tr>
 <td>edit distance problem</td>
+<td>编辑距离问题</td>
 </tr>
 <tr>
 <td>greedy algorithm</td>
+<td>贪心算法</td>
 </tr>
 </tbody>
 </table>
@@ -4984,7 +5185,7 @@ aria-label="Footer"
   <a
     href="../contribution/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 16.2 Contributing Together"
+    aria-label="Previous: 16.2   Contribute to the book"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4996,7 +5197,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        16.2 Contributing Together
+        16.2 &nbsp; Contribute to the book
       </div>
     </div>
   </a>
@@ -5033,7 +5234,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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diff --git a/en/chapter_array_and_linkedlist/array/index.html b/en/chapter_array_and_linkedlist/array/index.html
index 3f79fe4c2..76b8eca92 100644
--- a/en/chapter_array_and_linkedlist/array/index.html
+++ b/en/chapter_array_and_linkedlist/array/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>4.1 Array - Hello Algo</title>
+        <title>4.1 Arrays - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
     <div data-md-component="skip">
 
 
-        <a href="#41-array" class="md-skip">
+        <a href="#41-arrays" class="md-skip">
           Skip to content
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@@ -119,7 +119,7 @@
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
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@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              4.1 &nbsp; Array
+              4.1 &nbsp; Arrays
 
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
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@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1225,7 +1225,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1246,7 +1246,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1276,7 +1276,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1293,7 +1293,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1322,19 +1322,19 @@
   <a href="#411-common-array-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4.1.1 &nbsp; Common Array Operations
+        4.1.1 &nbsp; Common array operations
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="4.1.1 Common Array Operations">
+    <nav class="md-nav" aria-label="4.1.1 Common array operations">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-initializing-arrays" class="md-nav__link">
+  <a href="#1-initializing-an-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Initializing Arrays
+        1. &nbsp; Initializing an array
       </span>
     </span>
   </a>
@@ -1342,10 +1342,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-accessing-elements" class="md-nav__link">
+  <a href="#2-accessing-an-element" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Accessing Elements
+        2. &nbsp; Accessing an element
       </span>
     </span>
   </a>
@@ -1353,10 +1353,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-inserting-elements" class="md-nav__link">
+  <a href="#3-inserting-an-element" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Inserting Elements
+        3. &nbsp; Inserting an element
       </span>
     </span>
   </a>
@@ -1364,10 +1364,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#4-removing-elements" class="md-nav__link">
+  <a href="#4-removing-an-element" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4. &nbsp; Removing Elements
+        4. &nbsp; Removing an element
       </span>
     </span>
   </a>
@@ -1375,10 +1375,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#5-traversing-arrays" class="md-nav__link">
+  <a href="#5-traversing-an-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5. &nbsp; Traversing Arrays
+        5. &nbsp; Traversing an array
       </span>
     </span>
   </a>
@@ -1386,10 +1386,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#6-finding-elements" class="md-nav__link">
+  <a href="#6-finding-an-element" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6. &nbsp; Finding Elements
+        6. &nbsp; Finding an element
       </span>
     </span>
   </a>
@@ -1397,10 +1397,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#7-expanding-arrays" class="md-nav__link">
+  <a href="#7-expanding-an-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7. &nbsp; Expanding Arrays
+        7. &nbsp; Expanding an array
       </span>
     </span>
   </a>
@@ -1413,10 +1413,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#412-advantages-and-limitations-of-arrays" class="md-nav__link">
+  <a href="#412-strengths-and-limitations-of-arrays" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4.1.2 &nbsp; Advantages and Limitations of Arrays
+        4.1.2 &nbsp; Strengths and limitations of arrays
       </span>
     </span>
   </a>
@@ -1427,7 +1427,7 @@
   <a href="#413-typical-applications-of-arrays" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4.1.3 &nbsp; Typical Applications of Arrays
+        4.1.3 &nbsp; Typical applications of arrays
       </span>
     </span>
   </a>
@@ -1457,7 +1457,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1484,7 +1484,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1511,7 +1511,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1538,7 +1538,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1565,7 +1565,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1629,7 +1629,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1650,7 +1650,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1671,7 +1671,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1698,7 +1698,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1725,7 +1725,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1752,7 +1752,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1779,7 +1779,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1843,7 +1843,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1864,7 +1864,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1885,7 +1885,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1912,7 +1912,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1939,7 +1939,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1966,7 +1966,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1993,7 +1993,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -2061,7 +2061,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -2082,7 +2082,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2103,7 +2103,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -2130,7 +2130,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2157,7 +2157,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2184,7 +2184,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2211,7 +2211,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2238,7 +2238,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2265,7 +2265,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2329,7 +2329,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2350,7 +2350,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2371,7 +2371,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2398,7 +2398,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2425,7 +2425,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2452,7 +2452,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2479,7 +2479,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2543,7 +2543,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2564,7 +2564,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2585,7 +2585,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2612,7 +2612,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2639,7 +2639,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2666,7 +2666,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2693,7 +2693,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2761,7 +2761,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2782,7 +2782,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2803,7 +2803,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2830,7 +2830,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2857,7 +2857,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2884,7 +2884,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2911,7 +2911,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2938,7 +2938,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2965,7 +2965,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -3043,7 +3043,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3064,7 +3064,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3085,7 +3085,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3112,7 +3112,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3139,7 +3139,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3166,7 +3166,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3193,7 +3193,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3220,7 +3220,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3247,7 +3247,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3274,7 +3274,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3301,7 +3301,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3328,7 +3328,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3355,7 +3355,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3382,7 +3382,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3448,7 +3448,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3469,7 +3469,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3490,7 +3490,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3517,7 +3517,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3544,7 +3544,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3571,7 +3571,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3598,7 +3598,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3625,7 +3625,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3691,7 +3691,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3712,7 +3712,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3733,7 +3733,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3760,7 +3760,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3787,7 +3787,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3814,7 +3814,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3841,7 +3841,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3868,7 +3868,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3938,7 +3938,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3959,7 +3959,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3980,7 +3980,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -4007,7 +4007,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -4034,7 +4034,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4061,7 +4061,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4088,7 +4088,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4115,7 +4115,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4142,7 +4142,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4169,7 +4169,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4235,7 +4235,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4256,7 +4256,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4277,7 +4277,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4304,7 +4304,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4331,7 +4331,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4358,7 +4358,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4385,7 +4385,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4412,7 +4412,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4472,7 +4472,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4493,7 +4493,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4514,7 +4514,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4541,7 +4541,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4568,7 +4568,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4652,6 +4652,78 @@
 
 
 
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@@ -4696,19 +4768,19 @@
   <a href="#411-common-array-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4.1.1 &nbsp; Common Array Operations
+        4.1.1 &nbsp; Common array operations
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="4.1.1 Common Array Operations">
+    <nav class="md-nav" aria-label="4.1.1 Common array operations">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-initializing-arrays" class="md-nav__link">
+  <a href="#1-initializing-an-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Initializing Arrays
+        1. &nbsp; Initializing an array
       </span>
     </span>
   </a>
@@ -4716,10 +4788,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-accessing-elements" class="md-nav__link">
+  <a href="#2-accessing-an-element" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Accessing Elements
+        2. &nbsp; Accessing an element
       </span>
     </span>
   </a>
@@ -4727,10 +4799,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-inserting-elements" class="md-nav__link">
+  <a href="#3-inserting-an-element" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Inserting Elements
+        3. &nbsp; Inserting an element
       </span>
     </span>
   </a>
@@ -4738,10 +4810,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#4-removing-elements" class="md-nav__link">
+  <a href="#4-removing-an-element" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4. &nbsp; Removing Elements
+        4. &nbsp; Removing an element
       </span>
     </span>
   </a>
@@ -4749,10 +4821,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#5-traversing-arrays" class="md-nav__link">
+  <a href="#5-traversing-an-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5. &nbsp; Traversing Arrays
+        5. &nbsp; Traversing an array
       </span>
     </span>
   </a>
@@ -4760,10 +4832,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#6-finding-elements" class="md-nav__link">
+  <a href="#6-finding-an-element" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6. &nbsp; Finding Elements
+        6. &nbsp; Finding an element
       </span>
     </span>
   </a>
@@ -4771,10 +4843,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#7-expanding-arrays" class="md-nav__link">
+  <a href="#7-expanding-an-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7. &nbsp; Expanding Arrays
+        7. &nbsp; Expanding an array
       </span>
     </span>
   </a>
@@ -4787,10 +4859,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#412-advantages-and-limitations-of-arrays" class="md-nav__link">
+  <a href="#412-strengths-and-limitations-of-arrays" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4.1.2 &nbsp; Advantages and Limitations of Arrays
+        4.1.2 &nbsp; Strengths and limitations of arrays
       </span>
     </span>
   </a>
@@ -4801,7 +4873,7 @@
   <a href="#413-typical-applications-of-arrays" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4.1.3 &nbsp; Typical Applications of Arrays
+        4.1.3 &nbsp; Typical applications of arrays
       </span>
     </span>
   </a>
@@ -4845,102 +4917,102 @@
 
 
 <!-- Page content -->
-<h1 id="41-array">4.1 &nbsp; Array<a class="headerlink" href="#41-array" title="Permanent link">&para;</a></h1>
-<p>An <u>array</u> is a linear data structure that stores elements of the same type in contiguous memory space. The position of an element in the array is called the element's <u>index</u>. Figure 4-1 illustrates the main concepts and storage method of arrays.</p>
-<p><img alt="Array definition and storage method" class="animation-figure" src="../array.assets/array_definition.png" /></p>
-<p align="center"> Figure 4-1 &nbsp; Array definition and storage method </p>
+<h1 id="41-arrays">4.1 &nbsp; Arrays<a class="headerlink" href="#41-arrays" title="Permanent link">&para;</a></h1>
+<p>An <u>array</u> is a linear data structure that stores elements of the same type in contiguous memory. An element's position in the array is its <u>index</u>. Figure 4-1 shows the main concepts and storage layout.</p>
+<p><img alt="Array definition and storage layout" class="animation-figure" src="../array.assets/array_definition.png" /></p>
+<p align="center"> Figure 4-1 &nbsp; Array definition and storage layout </p>
 
-<h2 id="411-common-array-operations">4.1.1 &nbsp; Common Array Operations<a class="headerlink" href="#411-common-array-operations" title="Permanent link">&para;</a></h2>
-<h3 id="1-initializing-arrays">1. &nbsp; Initializing Arrays<a class="headerlink" href="#1-initializing-arrays" title="Permanent link">&para;</a></h3>
-<p>We can choose between two array initialization methods based on our needs: with or without initial values. When no initial values are specified, most programming languages initialize array elements to <span class="arithmatex">\(0\)</span>:</p>
+<h2 id="411-common-array-operations">4.1.1 &nbsp; Common array operations<a class="headerlink" href="#411-common-array-operations" title="Permanent link">&para;</a></h2>
+<h3 id="1-initializing-an-array">1. &nbsp; Initializing an array<a class="headerlink" href="#1-initializing-an-array" title="Permanent link">&para;</a></h3>
+<p>We can initialize an array with or without explicit values. When values are not specified, most programming languages initialize the elements to <span class="arithmatex">\(0\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Initialize array</span>
+<div class="highlight"><span class="filename">array.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Initialize arrays</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="n">arr</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="mi">5</span>  <span class="c1"># [ 0, 0, 0, 0, 0 ]</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Initialize array */</span>
-<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="c1">// Stored on stack</span>
+<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Initialize arrays */</span>
+<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="c1">// Stored on the stack</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="kt">int</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{};</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="w"> </span><span class="p">};</span>
-<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="c1">// Stored on heap (requires manual memory release)</span>
+<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="c1">// Stored on the heap (memory must be freed manually)</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="kt">int</span><span class="o">*</span><span class="w"> </span><span class="n">arr1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="w"> </span><span class="p">{};</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="kt">int</span><span class="o">*</span><span class="w"> </span><span class="n">nums1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="w"> </span><span class="p">};</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Initialize array */</span>
+<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Initialize arrays */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="mi">5</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// { 0, 0, 0, 0, 0 }</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="w"> </span><span class="p">};</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Initialize array */</span>
+<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Initialize arrays */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="mi">5</span><span class="p">];</span><span class="w"> </span><span class="c1">// [ 0, 0, 0, 0, 0 ]</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">];</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Initialize array */</span>
+<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Initialize arrays */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">var</span><span class="w"> </span><span class="nx">arr</span><span class="w"> </span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="kt">int</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="c1">// In Go, specifying length ([5]int) creates an array; not specifying length ([]int) creates a slice</span>
-<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="c1">// Since Go's arrays are designed to have their length determined at compile time, only constants can be used to specify the length</span>
-<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="c1">// For convenience in implementing the extend() method, slices are treated as arrays below</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="c1">// In Go, [5]int is an array with a specified length; []int is a slice</span>
+<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="c1">// Go array lengths are determined at compile time and must be constants</span>
+<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="c1">// To make extend() easier to implement, we treat slices as arrays below</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="nx">nums</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Initialize array */</span>
+<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Initialize arrays */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">let</span><span class="w"> </span><span class="nv">arr</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="mi">5</span><span class="p">)</span><span class="w"> </span><span class="c1">// [0, 0, 0, 0, 0]</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="kd">let</span><span class="w"> </span><span class="nv">nums</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Initialize array */</span>
+<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Initialize arrays */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">var</span><span class="w"> </span><span class="nx">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">5</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="kd">var</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">2</span><span class="p">,</span><span class="w"> </span><span class="mf">5</span><span class="p">,</span><span class="w"> </span><span class="mf">4</span><span class="p">];</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Initialize array */</span>
+<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Initialize arrays */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">let</span><span class="w"> </span><span class="nx">arr</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">5</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="kd">let</span><span class="w"> </span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">2</span><span class="p">,</span><span class="w"> </span><span class="mf">5</span><span class="p">,</span><span class="w"> </span><span class="mf">4</span><span class="p">];</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Initialize array */</span>
+<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Initialize arrays */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">5</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// [0, 0, 0, 0, 0]</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="m">5</span><span class="p">,</span><span class="w"> </span><span class="m">4</span><span class="p">];</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Initialize array */</span>
+<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Initialize arrays */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="kd">let</span><span class="w"> </span><span class="n">arr</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">;</span><span class="w"> </span><span class="mi">5</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="mi">5</span><span class="p">];</span><span class="w"> </span><span class="c1">// [0, 0, 0, 0, 0]</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="kd">let</span><span class="w"> </span><span class="n">slice</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">&amp;</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="mi">5</span><span class="p">];</span>
-<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="c1">// In Rust, specifying length ([i32; 5]) creates an array; not specifying length (&amp;[i32]) creates a slice</span>
-<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="c1">// Since Rust's arrays are designed to have their length determined at compile time, only constants can be used to specify the length</span>
-<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="c1">// Vector is the type generally used as a dynamic array in Rust</span>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="c1">// For convenience in implementing the extend() method, vectors are treated as arrays below</span>
+<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="c1">// In Rust, [i32; 5] is an array with a specified length; &amp;[i32] is a slice</span>
+<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="c1">// Rust array lengths are determined at compile time and must be constants</span>
+<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="c1">// Vector is Rust's usual dynamic array type</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="c1">// To make extend() easier to implement, we treat a vector as an array below</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="kd">let</span><span class="w"> </span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">];</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Initialize array */</span>
+<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Initialize arrays */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">};</span><span class="w"> </span><span class="c1">// { 0, 0, 0, 0, 0 }</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="w"> </span><span class="p">};</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Initialize array */</span>
+<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Initialize arrays */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">var</span><span class="w"> </span><span class="nv">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="m">5</span><span class="p">)</span><span class="w"> </span><span class="c1">// { 0, 0, 0, 0, 0 }</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="kd">var</span><span class="w"> </span><span class="nv">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">intArrayOf</span><span class="p">(</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="m">5</span><span class="p">,</span><span class="w"> </span><span class="m">4</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Initialize array</span>
+<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Initialize arrays</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="o">]</span>
 </code></pre></div>
@@ -4948,349 +5020,359 @@
 </div>
 </div>
 <details class="pythontutor">
-<summary>Code Visualization</summary>
-<p><div style="height: 243px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E6%95%B0%E7%BB%84%0Aarr%20%3D%20%5B0%5D%20*%205%20%20%23%20%5B%200,%200,%200,%200,%200%20%5D%0Anums%20%3D%20%5B1,%203,%202,%205,%204%5D&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=0&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E6%95%B0%E7%BB%84%0Aarr%20%3D%20%5B0%5D%20*%205%20%20%23%20%5B%200,%200,%200,%200,%200%20%5D%0Anums%20%3D%20%5B1,%203,%202,%205,%204%5D&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=0&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 243px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=%23%20Initialize%20arrays%0Aarr%20%3D%20%5B0%5D%20%2A%205%20%20%23%20%5B%200%2C%200%2C%200%2C%200%2C%200%20%5D%0Anums%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=0&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%23%20Initialize%20arrays%0Aarr%20%3D%20%5B0%5D%20%2A%205%20%20%23%20%5B%200%2C%200%2C%200%2C%200%2C%200%20%5D%0Anums%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=0&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<h3 id="2-accessing-elements">2. &nbsp; Accessing Elements<a class="headerlink" href="#2-accessing-elements" title="Permanent link">&para;</a></h3>
-<p>Array elements are stored in contiguous memory space, which means calculating the memory address of array elements is very easy. Given the array's memory address (the memory address of the first element) and an element's index, we can use the formula shown in Figure 4-2 to calculate the element's memory address and directly access that element.</p>
-<p><img alt="Memory address calculation for array elements" class="animation-figure" src="../array.assets/array_memory_location_calculation.png" /></p>
-<p align="center"> Figure 4-2 &nbsp; Memory address calculation for array elements </p>
+<h3 id="2-accessing-an-element">2. &nbsp; Accessing an element<a class="headerlink" href="#2-accessing-an-element" title="Permanent link">&para;</a></h3>
+<p>Because array elements occupy contiguous memory, their addresses are easy to calculate. Given the array's address (the address of its first element) and an index, the formula in Figure 4-2 gives the element's address, allowing direct access.</p>
+<p><img alt="Calculating an array element's memory address" class="animation-figure" src="../array.assets/array_memory_location_calculation.png" /></p>
+<p align="center"> Figure 4-2 &nbsp; Calculating an array element's memory address </p>
 
-<p>Observing Figure 4-2, we find that the first element of an array has an index of <span class="arithmatex">\(0\)</span>, which may seem counterintuitive since counting from <span class="arithmatex">\(1\)</span> would be more natural. However, from the perspective of the address calculation formula, <strong>an index is essentially an offset from the memory address</strong>. The address offset of the first element is <span class="arithmatex">\(0\)</span>, so it is reasonable for its index to be <span class="arithmatex">\(0\)</span>.</p>
-<p>Accessing elements in an array is highly efficient; we can randomly access any element in the array in <span class="arithmatex">\(O(1)\)</span> time.</p>
+<p>The first element has index <span class="arithmatex">\(0\)</span>, which may seem unintuitive because counting from <span class="arithmatex">\(1\)</span> feels more natural. However, <strong>an index essentially describes an offset from the starting memory address</strong>. The first element's offset is <span class="arithmatex">\(0\)</span>, so index <span class="arithmatex">\(0\)</span> makes sense.</p>
+<p>Array access is very efficient: any element can be accessed directly in <span class="arithmatex">\(O(1)\)</span> time.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">array.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">random_access</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">])</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Random access to element"""</span>
-<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># Randomly select a number from the interval [0, len(nums)-1]</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Access an element at a random index"""</span>
+<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># Choose a random number in the range [0, len(nums)-1]</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="n">random_index</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Retrieve and return the random element</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Get and return the randomly selected element</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="n">random_num</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">random_index</span><span class="p">]</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>    <span class="k">return</span> <span class="n">random_num</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Random access to element */</span>
+<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Access an element at a random index */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">randomAccess</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// Randomly select a number from interval [0, size)</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// Choose a random number in [0, size)</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">randomIndex</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rand</span><span class="p">()</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">size</span><span class="p">;</span>
-<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="c1">// Retrieve and return the random element</span>
+<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="c1">// Get and return the randomly selected element</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">randomNum</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">randomIndex</span><span class="p">];</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">randomNum</span><span class="p">;</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Random access to element */</span>
+<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Access an element at a random index */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">randomAccess</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Randomly select a number in the interval [0, nums.length)</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Choose a random number in the interval [0, nums.length)</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">randomIndex</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ThreadLocalRandom</span><span class="p">.</span><span class="na">current</span><span class="p">().</span><span class="na">nextInt</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">);</span>
-<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="c1">// Retrieve and return the random element</span>
+<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="c1">// Get and return the randomly selected element</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">randomNum</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">randomIndex</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">randomNum</span><span class="p">;</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Random access to element */</span>
+<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Access an element at a random index */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">RandomAccess</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="n">Random</span><span class="w"> </span><span class="n">random</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">();</span>
-<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="c1">// Randomly select a number in interval [0, nums.Length)</span>
+<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="c1">// Randomly select a number from [0, nums.Length)</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">randomIndex</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">random</span><span class="p">.</span><span class="n">Next</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">);</span>
-<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="c1">// Retrieve and return the random element</span>
+<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="c1">// Get and return the randomly selected element</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">randomNum</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">randomIndex</span><span class="p">];</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">randomNum</span><span class="p">;</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Random access to element */</span>
+<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Access an element at a random index */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">randomAccess</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="nx">randomNum</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Randomly select a number in the interval [0, nums.length)</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Choose a random number in the interval [0, len(nums))</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="nx">randomIndex</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">rand</span><span class="p">.</span><span class="nx">Intn</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">))</span>
-<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="c1">// Retrieve and return the random element</span>
+<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="c1">// Get and return the randomly selected element</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="nx">randomNum</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">randomIndex</span><span class="p">]</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="k">return</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Random access to element */</span>
+<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Access an element at a random index */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">randomAccess</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Randomly select a number in interval [0, nums.count)</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Pick a random number in the interval [0, nums.count)</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">randomIndex</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="n">randomElement</span><span class="p">()</span><span class="o">!</span>
-<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="c1">// Retrieve and return the random element</span>
+<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="c1">// Get and return the randomly selected element</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">randomNum</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">randomIndex</span><span class="p">]</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">randomNum</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Random access to element */</span>
+<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Access an element at a random index */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">randomAccess</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Randomly select a number in the interval [0, nums.length)</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Choose a random index in the interval [0, nums.length)</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">random_index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nb">Math</span><span class="p">.</span><span class="nx">random</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">);</span>
-<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="c1">// Retrieve and return the random element</span>
+<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="c1">// Get and return the randomly selected element</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">random_num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">random_index</span><span class="p">];</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">random_num</span><span class="p">;</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Random access to element */</span>
+<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Access an element at a random index */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">randomAccess</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// Randomly select a number in the interval [0, nums.length)</span>
+<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// Choose a random index in the interval [0, nums.length)</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">random_index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nb">Math</span><span class="p">.</span><span class="nx">random</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">);</span>
-<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="c1">// Retrieve and return the random element</span>
+<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="c1">// Get and return the randomly selected element</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">random_num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">random_index</span><span class="p">];</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">random_num</span><span class="p">;</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Random access to element */</span>
+<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Access an element at a random index */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">randomAccess</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// Randomly select a number in the interval [0, nums.length)</span>
+<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// Choose a random index in the interval [0, nums.length)</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">randomIndex</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Random</span><span class="p">().</span><span class="n">nextInt</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">);</span>
-<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="c1">// Retrieve and return the random element</span>
+<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="c1">// Get and return the randomly selected element</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">randomNum</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">randomIndex</span><span class="p">];</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">randomNum</span><span class="p">;</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Random access to element */</span>
+<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Access an element at a random index */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">random_access</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Randomly select a number in interval [0, nums.len())</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Choose a random index in [0, nums.len())</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">random_index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rand</span><span class="p">::</span><span class="n">thread_rng</span><span class="p">().</span><span class="n">gen_range</span><span class="p">(</span><span class="mi">0</span><span class="o">..</span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">());</span>
-<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="c1">// Retrieve and return the random element</span>
+<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="c1">// Get and return the randomly selected element</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">random_num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">random_index</span><span class="p">];</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="n">random_num</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Random access to element */</span>
+<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Access an element at a random index */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">randomAccess</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Randomly select a number from interval [0, size)</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Choose a random number in [0, size)</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">randomIndex</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rand</span><span class="p">()</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">size</span><span class="p">;</span>
-<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="c1">// Retrieve and return the random element</span>
+<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="c1">// Get and return the randomly selected element</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">randomNum</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">randomIndex</span><span class="p">];</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">randomNum</span><span class="p">;</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Random access to element */</span>
+<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Access an element at a random index */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">randomAccess</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Randomly select a number in interval [0, nums.size)</span>
+<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Randomly select a number in [0, nums.size)</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">randomIndex</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ThreadLocalRandom</span><span class="p">.</span><span class="na">current</span><span class="p">().</span><span class="na">nextInt</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="p">)</span>
-<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="c1">// Retrieve and return the random element</span>
+<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="c1">// Get and return the randomly selected element</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">randomNum</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">randomIndex</span><span class="o">]</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">randomNum</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Random access element ###</span>
+<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Access an element at a random index ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">random_access</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Randomly select a number in the interval [0, nums.length)</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Choose a random index in the interval [0, nums.length)</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="n">random_index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">Random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">nums</span><span class="o">.</span><span class="n">length</span><span class="p">)</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a>
-<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="c1"># Retrieve and return the random element</span>
+<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="c1"># Get and return the randomly selected element</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="n">nums</span><span class="o">[</span><span class="n">random_index</span><span class="o">]</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h3 id="3-inserting-elements">3. &nbsp; Inserting Elements<a class="headerlink" href="#3-inserting-elements" title="Permanent link">&para;</a></h3>
-<p>Array elements are packed tightly together in memory, with no extra space between them for additional data. As shown in Figure 4-3, if we want to insert an element in the middle of an array, we need to shift all subsequent elements one position to the right and then assign the value at that index.</p>
-<p><img alt="Example of inserting an element into an array" class="animation-figure" src="../array.assets/array_insert_element.png" /></p>
-<p align="center"> Figure 4-3 &nbsp; Example of inserting an element into an array </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 531px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=import%20random%0A%0Adef%20random_access%28nums%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Access%20an%20element%20at%20a%20random%20index%22%22%22%0A%20%20%20%20%23%20Choose%20a%20random%20number%20in%20the%20range%20%5B0,%20len%28nums%29-1%5D%0A%20%20%20%20random_index%20%3D%20random.randint%280,%20len%28nums%29%20-%201%29%0A%20%20%20%20%23%20Get%20and%20return%20the%20randomly%20selected%20element%0A%20%20%20%20random_num%20%3D%20nums%5Brandom_index%5D%0A%20%20%20%20return%20random_num%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20array%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22Array%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20Random%20access%0A%20%20%20%20random_num%3A%20int%20%3D%20random_access%28nums%29%0A%20%20%20%20print%28%22Randomly%20selected%20element%20from%20nums%3A%22,%20random_num%29%0A&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=import%20random%0A%0Adef%20random_access%28nums%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Access%20an%20element%20at%20a%20random%20index%22%22%22%0A%20%20%20%20%23%20Choose%20a%20random%20number%20in%20the%20range%20%5B0,%20len%28nums%29-1%5D%0A%20%20%20%20random_index%20%3D%20random.randint%280,%20len%28nums%29%20-%201%29%0A%20%20%20%20%23%20Get%20and%20return%20the%20randomly%20selected%20element%0A%20%20%20%20random_num%20%3D%20nums%5Brandom_index%5D%0A%20%20%20%20return%20random_num%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20array%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22Array%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20Random%20access%0A%20%20%20%20random_num%3A%20int%20%3D%20random_access%28nums%29%0A%20%20%20%20print%28%22Randomly%20selected%20element%20from%20nums%3A%22,%20random_num%29%0A&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="3-inserting-an-element">3. &nbsp; Inserting an element<a class="headerlink" href="#3-inserting-an-element" title="Permanent link">&para;</a></h3>
+<p>Array elements sit next to one another, with no room between them for more data. As shown in Figure 4-3, inserting into the middle requires shifting every element at or after the insertion point one position toward the end, then assigning the new element at the desired index.</p>
+<p><img alt="Inserting an element into an array" class="animation-figure" src="../array.assets/array_insert_element.png" /></p>
+<p align="center"> Figure 4-3 &nbsp; Inserting an element into an array </p>
 
-<p>It is worth noting that since the length of an array is fixed, inserting an element will inevitably push the last element out of the array. We will leave the solution to this problem for discussion in the "List" chapter.</p>
+<p>Because an array's length is fixed, inserting an element causes the last element to be lost. We will discuss the solution in the section on lists.</p>
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 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">array.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">index</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Insert element num at index index in the array"""</span>
-<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="c1"># Move all elements at and after index index backward by one position</span>
+<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Insert element num at index `index` in the array"""</span>
+<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="c1"># Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">index</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>        <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="c1"># Assign num to the element at index index</span>
+<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="c1"># Assign num to the element at index `index`</span>
 <a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>    <span class="n">nums</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="n">num</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Insert element num at index index in the array */</span>
+<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Insert element num at index `index` in the array */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="c1">// Move all elements at and after index index backward by one position</span>
+<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index index</span>
+<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index `index`</span>
 <a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Insert element num at index index in the array */</span>
+<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Insert element num at index `index` in the array */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="c1">// Move all elements at and after index index backward by one position</span>
+<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index index</span>
+<a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index `index`</span>
 <a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Insert element num at index index in the array */</span>
+<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Insert element num at index `index` in the array */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Insert</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="c1">// Move all elements at and after index index backward by one position</span>
+<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index index</span>
+<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index `index`</span>
 <a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Insert element num at index index in the array */</span>
+<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Insert element num at index `index` in the array */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">insert</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="c1">// Move all elements at and after index index backward by one position</span>
+<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index index</span>
+<a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index `index`</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">    </span><span class="nx">nums</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span>
 <a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Insert element num at index index in the array */</span>
+<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Insert element num at index `index` in the array */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="c1">// Move all elements at and after index index backward by one position</span>
+<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="bp">stride</span><span class="p">(</span><span class="n">from</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">count</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">to</span><span class="p">:</span><span class="w"> </span><span class="n">index</span><span class="p">,</span><span class="w"> </span><span class="n">by</span><span class="p">:</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index index</span>
+<a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index `index`</span>
 <a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Insert element num at index index in the array */</span>
+<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Insert element num at index `index` in the array */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">insert</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="p">,</span><span class="w"> </span><span class="nx">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="c1">// Move all elements at and after index index backward by one position</span>
+<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index index</span>
+<a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index `index`</span>
 <a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="nx">nums</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Insert element num at index index in the array */</span>
+<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Insert element num at index `index` in the array */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">insert</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">index</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="c1">// Move all elements at and after index index backward by one position</span>
+<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index index</span>
+<a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index `index`</span>
 <a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">    </span><span class="nx">nums</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Insert element _num at array index index */</span>
+<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Insert element _num at index in the array */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">insert</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="c1">// Move all elements at and after index index backward by one position</span>
+<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
 <a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">  </span><span class="c1">// Assign _num to element at index</span>
+<a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">  </span><span class="c1">// Assign _num to the element at index</span>
 <a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">  </span><span class="n">nums</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span>
 <a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Insert element num at index index in the array */</span>
+<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Insert element num at index `index` in the array */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="c1">// Move all elements at and after index index backward by one position</span>
+<a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()).</span><span class="n">rev</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index index</span>
+<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index `index`</span>
 <a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Insert element num at index index in the array */</span>
+<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Insert element num at index `index` in the array */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="c1">// Move all elements at and after index index backward by one position</span>
+<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index index</span>
+<a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index `index`</span>
 <a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Insert element num at index index in the array */</span>
+<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Insert element num at index `index` in the array */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="c1">// Move all elements at and after index index backward by one position</span>
+<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="n">downTo</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span>
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index index</span>
+<a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="c1">// Assign num to the element at index `index`</span>
 <a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Insert element num at index in array ###</span>
+<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Insert element num at index `index` in the array ###</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="p">)</span>
-<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1"># Move all elements at and after index index backward by one position</span>
+<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1"># Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">downto</span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span>
 <a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a>
-<a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">  </span><span class="c1"># Assign num to the element at index index</span>
+<a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">  </span><span class="c1"># Assign num to the element at index `index`</span>
 <a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">  </span><span class="n">nums</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h3 id="4-removing-elements">4. &nbsp; Removing Elements<a class="headerlink" href="#4-removing-elements" title="Permanent link">&para;</a></h3>
-<p>Similarly, as shown in Figure 4-4, to delete the element at index <span class="arithmatex">\(i\)</span>, we need to shift all elements after index <span class="arithmatex">\(i\)</span> forward by one position.</p>
-<p><img alt="Example of removing an element from an array" class="animation-figure" src="../array.assets/array_remove_element.png" /></p>
-<p align="center"> Figure 4-4 &nbsp; Example of removing an element from an array </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 495px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20insert%28nums%3A%20list%5Bint%5D,%20num%3A%20int,%20index%3A%20int%29%3A%0A%20%20%20%20%22%22%22Insert%20element%20num%20at%20index%20%60index%60%20in%20the%20array%22%22%22%0A%20%20%20%20%23%20Shift%20all%20elements%20at%20and%20after%20index%20%60index%60%20one%20position%20to%20the%20right%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%20-%201,%20index,%20-1%29%3A%0A%20%20%20%20%20%20%20%20nums%5Bi%5D%20%3D%20nums%5Bi%20-%201%5D%0A%20%20%20%20%23%20Assign%20num%20to%20the%20element%20at%20index%20%60index%60%0A%20%20%20%20nums%5Bindex%5D%20%3D%20num%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20array%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22Array%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20Insert%20an%20element%0A%20%20%20%20insert%28nums,%206,%203%29%0A%20%20%20%20print%28%22Insert%20the%20number%206%20at%20index%203,%20giving%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20insert%28nums%3A%20list%5Bint%5D,%20num%3A%20int,%20index%3A%20int%29%3A%0A%20%20%20%20%22%22%22Insert%20element%20num%20at%20index%20%60index%60%20in%20the%20array%22%22%22%0A%20%20%20%20%23%20Shift%20all%20elements%20at%20and%20after%20index%20%60index%60%20one%20position%20to%20the%20right%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%20-%201,%20index,%20-1%29%3A%0A%20%20%20%20%20%20%20%20nums%5Bi%5D%20%3D%20nums%5Bi%20-%201%5D%0A%20%20%20%20%23%20Assign%20num%20to%20the%20element%20at%20index%20%60index%60%0A%20%20%20%20nums%5Bindex%5D%20%3D%20num%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20array%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22Array%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20Insert%20an%20element%0A%20%20%20%20insert%28nums,%206,%203%29%0A%20%20%20%20print%28%22Insert%20the%20number%206%20at%20index%203,%20giving%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="4-removing-an-element">4. &nbsp; Removing an element<a class="headerlink" href="#4-removing-an-element" title="Permanent link">&para;</a></h3>
+<p>Similarly, as shown in Figure 4-4, removing the element at index <span class="arithmatex">\(i\)</span> requires shifting every element after <span class="arithmatex">\(i\)</span> one position toward the beginning.</p>
+<p><img alt="Removing an element from an array" class="animation-figure" src="../array.assets/array_remove_element.png" /></p>
+<p align="center"> Figure 4-4 &nbsp; Removing an element from an array </p>
 
-<p>Note that after the deletion is complete, the original last element is no longer meaningful, so we do not need to modify it explicitly.</p>
+<p>After removal, the old final entry is no longer meaningful, so we do not need to change it explicitly.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:13"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">array.py</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">index</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Remove the element at index index"""</span>
-<a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="c1"># Move all elements after index index forward by one position</span>
+<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Remove the element at index `index`"""</span>
+<a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="c1"># Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">index</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>        <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Remove the element at index index */</span>
+<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Remove the element at index `index` */</span>
 <a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w">    </span><span class="p">}</span>
@@ -5298,9 +5380,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Remove the element at index index */</span>
+<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Remove the element at index `index` */</span>
 <a id="__codelineno-41-2" name="__codelineno-41-2" href="#__codelineno-41-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a><span class="w">    </span><span class="p">}</span>
@@ -5308,9 +5390,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Remove the element at index index */</span>
+<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Remove the element at index `index` */</span>
 <a id="__codelineno-42-2" name="__codelineno-42-2" href="#__codelineno-42-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Remove</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a><span class="w">    </span><span class="p">}</span>
@@ -5318,9 +5400,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Remove the element at index index */</span>
+<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Remove the element at index `index` */</span>
 <a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">remove</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a><span class="w">    </span><span class="p">}</span>
@@ -5328,9 +5410,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Remove the element at index index */</span>
+<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Remove the element at index `index` */</span>
 <a id="__codelineno-44-2" name="__codelineno-44-2" href="#__codelineno-44-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a><span class="w">    </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a><span class="w">    </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="bp">dropFirst</span><span class="p">(</span><span class="n">index</span><span class="p">).</span><span class="bp">dropLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></a><span class="w">    </span><span class="p">}</span>
@@ -5338,9 +5420,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Remove the element at index index */</span>
+<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Remove the element at index `index` */</span>
 <a id="__codelineno-45-2" name="__codelineno-45-2" href="#__codelineno-45-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">remove</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-45-6" name="__codelineno-45-6" href="#__codelineno-45-6"></a><span class="w">    </span><span class="p">}</span>
@@ -5348,9 +5430,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Remove the element at index index */</span>
+<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Remove the element at index `index` */</span>
 <a id="__codelineno-46-2" name="__codelineno-46-2" href="#__codelineno-46-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">remove</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">index</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="w">    </span><span class="p">}</span>
@@ -5358,9 +5440,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Remove the element at index index */</span>
+<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Remove the element at index `index` */</span>
 <a id="__codelineno-47-2" name="__codelineno-47-2" href="#__codelineno-47-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">remove</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
 <a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a><span class="w">  </span><span class="p">}</span>
@@ -5368,9 +5450,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Remove the element at index index */</span>
+<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Remove the element at index `index` */</span>
 <a id="__codelineno-48-2" name="__codelineno-48-2" href="#__codelineno-48-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">index</span><span class="o">..</span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a><span class="w">    </span><span class="p">}</span>
@@ -5378,10 +5460,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Remove the element at index index */</span>
-<a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="c1">// Note: stdio.h occupies the remove keyword</span>
+<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Remove the element at index `index` */</span>
+<a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="c1">// Note: stdio.h already declares a function named remove</span>
 <a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="kt">void</span><span class="w"> </span><span class="nf">removeItem</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">    </span><span class="p">}</span>
@@ -5389,9 +5471,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Remove the element at index index */</span>
+<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Remove the element at index `index` */</span>
 <a id="__codelineno-50-2" name="__codelineno-50-2" href="#__codelineno-50-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">index</span><span class="p">..</span><span class="o">&lt;</span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="o">]</span>
 <a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">    </span><span class="p">}</span>
@@ -5399,9 +5481,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Delete element at index ###</span>
+<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Remove the element at index `index` ###</span>
 <a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="p">)</span>
-<a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">  </span><span class="c1"># Move all elements after index index forward by one position</span>
+<a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">  </span><span class="c1"># Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">index</span><span class="o">...</span><span class="p">(</span><span class="n">nums</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span>
 <a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">  </span><span class="k">end</span>
@@ -5410,37 +5492,42 @@
 </div>
 </div>
 </div>
-<p>Overall, array insertion and deletion operations have the following drawbacks:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20remove%28nums%3A%20list%5Bint%5D,%20index%3A%20int%29%3A%0A%20%20%20%20%22%22%22Remove%20the%20element%20at%20index%20%60index%60%22%22%22%0A%20%20%20%20%23%20Shift%20all%20elements%20after%20index%20%60index%60%20one%20position%20to%20the%20left%0A%20%20%20%20for%20i%20in%20range%28index,%20len%28nums%29%20-%201%29%3A%0A%20%20%20%20%20%20%20%20nums%5Bi%5D%20%3D%20nums%5Bi%20%2B%201%5D%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20array%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22Array%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20Remove%20an%20element%0A%20%20%20%20remove%28nums,%202%29%0A%20%20%20%20print%28%22Remove%20the%20element%20at%20index%202,%20giving%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20remove%28nums%3A%20list%5Bint%5D,%20index%3A%20int%29%3A%0A%20%20%20%20%22%22%22Remove%20the%20element%20at%20index%20%60index%60%22%22%22%0A%20%20%20%20%23%20Shift%20all%20elements%20after%20index%20%60index%60%20one%20position%20to%20the%20left%0A%20%20%20%20for%20i%20in%20range%28index,%20len%28nums%29%20-%201%29%3A%0A%20%20%20%20%20%20%20%20nums%5Bi%5D%20%3D%20nums%5Bi%20%2B%201%5D%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20array%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22Array%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20Remove%20an%20element%0A%20%20%20%20remove%28nums,%202%29%0A%20%20%20%20print%28%22Remove%20the%20element%20at%20index%202,%20giving%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Overall, inserting and removing array elements have these drawbacks.</p>
 <ul>
-<li><strong>High time complexity</strong>: The average time complexity for both insertion and deletion in arrays is <span class="arithmatex">\(O(n)\)</span>, where <span class="arithmatex">\(n\)</span> is the length of the array.</li>
-<li><strong>Loss of elements</strong>: Since the length of an array is immutable, after inserting an element, elements that exceed the array's length will be lost.</li>
-<li><strong>Memory waste</strong>: We can initialize a relatively long array and use only the front portion, so that any overwritten tail elements are merely unused placeholders, but this wastes some memory space.</li>
+<li><strong>High time complexity</strong>: Both operations take <span class="arithmatex">\(O(n)\)</span> time on average, where <span class="arithmatex">\(n\)</span> is the array length.</li>
+<li><strong>Lost elements</strong>: Since the length is fixed, insertion loses any element pushed beyond the array's bounds.</li>
+<li><strong>Wasted memory</strong>: We can allocate a longer array and use only its beginning, so any final entries lost during insertion are unused. However, this leaves some memory wasted.</li>
 </ul>
-<h3 id="5-traversing-arrays">5. &nbsp; Traversing Arrays<a class="headerlink" href="#5-traversing-arrays" title="Permanent link">&para;</a></h3>
-<p>In most programming languages, we can traverse an array either by index or by directly iterating through each element in the array:</p>
+<h3 id="5-traversing-an-array">5. &nbsp; Traversing an array<a class="headerlink" href="#5-traversing-an-array" title="Permanent link">&para;</a></h3>
+<p>Most programming languages let us traverse an array either by index or by visiting its elements directly:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">array.py</span><pre><span></span><code><a id="__codelineno-52-1" name="__codelineno-52-1" href="#__codelineno-52-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">traverse</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]):</span>
-<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Traverse array"""</span>
+<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Traverse the array"""</span>
 <a id="__codelineno-52-3" name="__codelineno-52-3" href="#__codelineno-52-3"></a>    <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-52-4" name="__codelineno-52-4" href="#__codelineno-52-4"></a>    <span class="c1"># Traverse array by index</span>
+<a id="__codelineno-52-4" name="__codelineno-52-4" href="#__codelineno-52-4"></a>    <span class="c1"># Traverse the array by index</span>
 <a id="__codelineno-52-5" name="__codelineno-52-5" href="#__codelineno-52-5"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)):</span>
 <a id="__codelineno-52-6" name="__codelineno-52-6" href="#__codelineno-52-6"></a>        <span class="n">count</span> <span class="o">+=</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-<a id="__codelineno-52-7" name="__codelineno-52-7" href="#__codelineno-52-7"></a>    <span class="c1"># Direct traversal of array elements</span>
+<a id="__codelineno-52-7" name="__codelineno-52-7" href="#__codelineno-52-7"></a>    <span class="c1"># Iterate directly over the array elements</span>
 <a id="__codelineno-52-8" name="__codelineno-52-8" href="#__codelineno-52-8"></a>    <span class="k">for</span> <span class="n">num</span> <span class="ow">in</span> <span class="n">nums</span><span class="p">:</span>
 <a id="__codelineno-52-9" name="__codelineno-52-9" href="#__codelineno-52-9"></a>        <span class="n">count</span> <span class="o">+=</span> <span class="n">num</span>
-<a id="__codelineno-52-10" name="__codelineno-52-10" href="#__codelineno-52-10"></a>    <span class="c1"># Traverse simultaneously data index and elements</span>
+<a id="__codelineno-52-10" name="__codelineno-52-10" href="#__codelineno-52-10"></a>    <span class="c1"># Iterate over both the indices and the elements</span>
 <a id="__codelineno-52-11" name="__codelineno-52-11" href="#__codelineno-52-11"></a>    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">num</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">nums</span><span class="p">):</span>
 <a id="__codelineno-52-12" name="__codelineno-52-12" href="#__codelineno-52-12"></a>        <span class="n">count</span> <span class="o">+=</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-52-13" name="__codelineno-52-13" href="#__codelineno-52-13"></a>        <span class="n">count</span> <span class="o">+=</span> <span class="n">num</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Traverse array */</span>
+<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Traverse the array */</span>
 <a id="__codelineno-53-2" name="__codelineno-53-2" href="#__codelineno-53-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">traverse</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-53-4" name="__codelineno-53-4" href="#__codelineno-53-4"></a><span class="w">    </span><span class="c1">// Traverse array by index</span>
+<a id="__codelineno-53-4" name="__codelineno-53-4" href="#__codelineno-53-4"></a><span class="w">    </span><span class="c1">// Traverse the array by index</span>
 <a id="__codelineno-53-5" name="__codelineno-53-5" href="#__codelineno-53-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-6" name="__codelineno-53-6" href="#__codelineno-53-6"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-53-7" name="__codelineno-53-7" href="#__codelineno-53-7"></a><span class="w">    </span><span class="p">}</span>
@@ -5448,14 +5535,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Traverse array */</span>
+<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Traverse the array */</span>
 <a id="__codelineno-54-2" name="__codelineno-54-2" href="#__codelineno-54-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">traverse</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-54-4" name="__codelineno-54-4" href="#__codelineno-54-4"></a><span class="w">    </span><span class="c1">// Traverse array by index</span>
+<a id="__codelineno-54-4" name="__codelineno-54-4" href="#__codelineno-54-4"></a><span class="w">    </span><span class="c1">// Traverse the array by index</span>
 <a id="__codelineno-54-5" name="__codelineno-54-5" href="#__codelineno-54-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-6" name="__codelineno-54-6" href="#__codelineno-54-6"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-54-7" name="__codelineno-54-7" href="#__codelineno-54-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-54-8" name="__codelineno-54-8" href="#__codelineno-54-8"></a><span class="w">    </span><span class="c1">// Direct traversal of array elements</span>
+<a id="__codelineno-54-8" name="__codelineno-54-8" href="#__codelineno-54-8"></a><span class="w">    </span><span class="c1">// Iterate directly over the array elements</span>
 <a id="__codelineno-54-9" name="__codelineno-54-9" href="#__codelineno-54-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-10" name="__codelineno-54-10" href="#__codelineno-54-10"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-54-11" name="__codelineno-54-11" href="#__codelineno-54-11"></a><span class="w">    </span><span class="p">}</span>
@@ -5463,14 +5550,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Traverse array */</span>
+<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Traverse the array */</span>
 <a id="__codelineno-55-2" name="__codelineno-55-2" href="#__codelineno-55-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Traverse</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-55-4" name="__codelineno-55-4" href="#__codelineno-55-4"></a><span class="w">    </span><span class="c1">// Traverse array by index</span>
+<a id="__codelineno-55-4" name="__codelineno-55-4" href="#__codelineno-55-4"></a><span class="w">    </span><span class="c1">// Traverse the array by index</span>
 <a id="__codelineno-55-5" name="__codelineno-55-5" href="#__codelineno-55-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-6" name="__codelineno-55-6" href="#__codelineno-55-6"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-55-7" name="__codelineno-55-7" href="#__codelineno-55-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-55-8" name="__codelineno-55-8" href="#__codelineno-55-8"></a><span class="w">    </span><span class="c1">// Direct traversal of array elements</span>
+<a id="__codelineno-55-8" name="__codelineno-55-8" href="#__codelineno-55-8"></a><span class="w">    </span><span class="c1">// Iterate directly over the array elements</span>
 <a id="__codelineno-55-9" name="__codelineno-55-9" href="#__codelineno-55-9"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-10" name="__codelineno-55-10" href="#__codelineno-55-10"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-55-11" name="__codelineno-55-11" href="#__codelineno-55-11"></a><span class="w">    </span><span class="p">}</span>
@@ -5478,19 +5565,19 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Traverse array */</span>
+<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Traverse the array */</span>
 <a id="__codelineno-56-2" name="__codelineno-56-2" href="#__codelineno-56-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">traverse</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w">    </span><span class="nx">count</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a><span class="w">    </span><span class="c1">// Traverse array by index</span>
+<a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a><span class="w">    </span><span class="c1">// Traverse the array by index</span>
 <a id="__codelineno-56-5" name="__codelineno-56-5" href="#__codelineno-56-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-6" name="__codelineno-56-6" href="#__codelineno-56-6"></a><span class="w">        </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
 <a id="__codelineno-56-7" name="__codelineno-56-7" href="#__codelineno-56-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-56-8" name="__codelineno-56-8" href="#__codelineno-56-8"></a><span class="w">    </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-56-9" name="__codelineno-56-9" href="#__codelineno-56-9"></a><span class="w">    </span><span class="c1">// Direct traversal of array elements</span>
+<a id="__codelineno-56-9" name="__codelineno-56-9" href="#__codelineno-56-9"></a><span class="w">    </span><span class="c1">// Iterate directly over the array elements</span>
 <a id="__codelineno-56-10" name="__codelineno-56-10" href="#__codelineno-56-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-11" name="__codelineno-56-11" href="#__codelineno-56-11"></a><span class="w">        </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">num</span>
 <a id="__codelineno-56-12" name="__codelineno-56-12" href="#__codelineno-56-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-56-13" name="__codelineno-56-13" href="#__codelineno-56-13"></a><span class="w">    </span><span class="c1">// Traverse simultaneously data index and elements</span>
+<a id="__codelineno-56-13" name="__codelineno-56-13" href="#__codelineno-56-13"></a><span class="w">    </span><span class="c1">// Iterate over both the indices and the elements</span>
 <a id="__codelineno-56-14" name="__codelineno-56-14" href="#__codelineno-56-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-15" name="__codelineno-56-15" href="#__codelineno-56-15"></a><span class="w">        </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
 <a id="__codelineno-56-16" name="__codelineno-56-16" href="#__codelineno-56-16"></a><span class="w">        </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">num</span>
@@ -5499,18 +5586,18 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Traverse array */</span>
+<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Traverse the array */</span>
 <a id="__codelineno-57-2" name="__codelineno-57-2" href="#__codelineno-57-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">traverse</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="w">    </span><span class="c1">// Traverse array by index</span>
+<a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="w">    </span><span class="c1">// Traverse the array by index</span>
 <a id="__codelineno-57-5" name="__codelineno-57-5" href="#__codelineno-57-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-6" name="__codelineno-57-6" href="#__codelineno-57-6"></a><span class="w">        </span><span class="bp">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-57-7" name="__codelineno-57-7" href="#__codelineno-57-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-57-8" name="__codelineno-57-8" href="#__codelineno-57-8"></a><span class="w">    </span><span class="c1">// Direct traversal of array elements</span>
+<a id="__codelineno-57-8" name="__codelineno-57-8" href="#__codelineno-57-8"></a><span class="w">    </span><span class="c1">// Iterate directly over the array elements</span>
 <a id="__codelineno-57-9" name="__codelineno-57-9" href="#__codelineno-57-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-10" name="__codelineno-57-10" href="#__codelineno-57-10"></a><span class="w">        </span><span class="bp">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-57-11" name="__codelineno-57-11" href="#__codelineno-57-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-57-12" name="__codelineno-57-12" href="#__codelineno-57-12"></a><span class="w">    </span><span class="c1">// Traverse simultaneously data index and elements</span>
+<a id="__codelineno-57-12" name="__codelineno-57-12" href="#__codelineno-57-12"></a><span class="w">    </span><span class="c1">// Iterate over both the indices and the elements</span>
 <a id="__codelineno-57-13" name="__codelineno-57-13" href="#__codelineno-57-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">enumerated</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-14" name="__codelineno-57-14" href="#__codelineno-57-14"></a><span class="w">        </span><span class="bp">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-57-15" name="__codelineno-57-15" href="#__codelineno-57-15"></a><span class="w">        </span><span class="bp">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">num</span>
@@ -5519,14 +5606,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Traverse array */</span>
+<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Traverse the array */</span>
 <a id="__codelineno-58-2" name="__codelineno-58-2" href="#__codelineno-58-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">traverse</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-58-4" name="__codelineno-58-4" href="#__codelineno-58-4"></a><span class="w">    </span><span class="c1">// Traverse array by index</span>
+<a id="__codelineno-58-4" name="__codelineno-58-4" href="#__codelineno-58-4"></a><span class="w">    </span><span class="c1">// Traverse the array by index</span>
 <a id="__codelineno-58-5" name="__codelineno-58-5" href="#__codelineno-58-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-6" name="__codelineno-58-6" href="#__codelineno-58-6"></a><span class="w">        </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-58-7" name="__codelineno-58-7" href="#__codelineno-58-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-58-8" name="__codelineno-58-8" href="#__codelineno-58-8"></a><span class="w">    </span><span class="c1">// Direct traversal of array elements</span>
+<a id="__codelineno-58-8" name="__codelineno-58-8" href="#__codelineno-58-8"></a><span class="w">    </span><span class="c1">// Iterate directly over the array elements</span>
 <a id="__codelineno-58-9" name="__codelineno-58-9" href="#__codelineno-58-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-10" name="__codelineno-58-10" href="#__codelineno-58-10"></a><span class="w">        </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-58-11" name="__codelineno-58-11" href="#__codelineno-58-11"></a><span class="w">    </span><span class="p">}</span>
@@ -5534,14 +5621,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Traverse array */</span>
+<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Traverse the array */</span>
 <a id="__codelineno-59-2" name="__codelineno-59-2" href="#__codelineno-59-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">traverse</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-59-4" name="__codelineno-59-4" href="#__codelineno-59-4"></a><span class="w">    </span><span class="c1">// Traverse array by index</span>
+<a id="__codelineno-59-4" name="__codelineno-59-4" href="#__codelineno-59-4"></a><span class="w">    </span><span class="c1">// Traverse the array by index</span>
 <a id="__codelineno-59-5" name="__codelineno-59-5" href="#__codelineno-59-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-6" name="__codelineno-59-6" href="#__codelineno-59-6"></a><span class="w">        </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-59-7" name="__codelineno-59-7" href="#__codelineno-59-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-59-8" name="__codelineno-59-8" href="#__codelineno-59-8"></a><span class="w">    </span><span class="c1">// Direct traversal of array elements</span>
+<a id="__codelineno-59-8" name="__codelineno-59-8" href="#__codelineno-59-8"></a><span class="w">    </span><span class="c1">// Iterate directly over the array elements</span>
 <a id="__codelineno-59-9" name="__codelineno-59-9" href="#__codelineno-59-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-10" name="__codelineno-59-10" href="#__codelineno-59-10"></a><span class="w">        </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-59-11" name="__codelineno-59-11" href="#__codelineno-59-11"></a><span class="w">    </span><span class="p">}</span>
@@ -5549,18 +5636,18 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Traverse array elements */</span>
+<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Traverse the array elements */</span>
 <a id="__codelineno-60-2" name="__codelineno-60-2" href="#__codelineno-60-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">traverse</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-3" name="__codelineno-60-3" href="#__codelineno-60-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
-<a id="__codelineno-60-4" name="__codelineno-60-4" href="#__codelineno-60-4"></a><span class="w">  </span><span class="c1">// Traverse array by index</span>
+<a id="__codelineno-60-4" name="__codelineno-60-4" href="#__codelineno-60-4"></a><span class="w">  </span><span class="c1">// Traverse the array by index</span>
 <a id="__codelineno-60-5" name="__codelineno-60-5" href="#__codelineno-60-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-6" name="__codelineno-60-6" href="#__codelineno-60-6"></a><span class="w">    </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-60-7" name="__codelineno-60-7" href="#__codelineno-60-7"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-60-8" name="__codelineno-60-8" href="#__codelineno-60-8"></a><span class="w">  </span><span class="c1">// Direct traversal of array elements</span>
+<a id="__codelineno-60-8" name="__codelineno-60-8" href="#__codelineno-60-8"></a><span class="w">  </span><span class="c1">// Iterate directly over the array elements</span>
 <a id="__codelineno-60-9" name="__codelineno-60-9" href="#__codelineno-60-9"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-10" name="__codelineno-60-10" href="#__codelineno-60-10"></a><span class="w">    </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span>
 <a id="__codelineno-60-11" name="__codelineno-60-11" href="#__codelineno-60-11"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-60-12" name="__codelineno-60-12" href="#__codelineno-60-12"></a><span class="w">  </span><span class="c1">// Traverse array using forEach method</span>
+<a id="__codelineno-60-12" name="__codelineno-60-12" href="#__codelineno-60-12"></a><span class="w">  </span><span class="c1">// Traverse the array using forEach</span>
 <a id="__codelineno-60-13" name="__codelineno-60-13" href="#__codelineno-60-13"></a><span class="w">  </span><span class="n">nums</span><span class="p">.</span><span class="n">forEach</span><span class="p">((</span><span class="n">_num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-14" name="__codelineno-60-14" href="#__codelineno-60-14"></a><span class="w">    </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span>
 <a id="__codelineno-60-15" name="__codelineno-60-15" href="#__codelineno-60-15"></a><span class="w">  </span><span class="p">});</span>
@@ -5568,14 +5655,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Traverse array */</span>
+<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Traverse the array */</span>
 <a id="__codelineno-61-2" name="__codelineno-61-2" href="#__codelineno-61-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">traverse</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-3" name="__codelineno-61-3" href="#__codelineno-61-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">_count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-61-4" name="__codelineno-61-4" href="#__codelineno-61-4"></a><span class="w">    </span><span class="c1">// Traverse array by index</span>
+<a id="__codelineno-61-4" name="__codelineno-61-4" href="#__codelineno-61-4"></a><span class="w">    </span><span class="c1">// Traverse the array by index</span>
 <a id="__codelineno-61-5" name="__codelineno-61-5" href="#__codelineno-61-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-6" name="__codelineno-61-6" href="#__codelineno-61-6"></a><span class="w">        </span><span class="n">_count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-61-7" name="__codelineno-61-7" href="#__codelineno-61-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-61-8" name="__codelineno-61-8" href="#__codelineno-61-8"></a><span class="w">    </span><span class="c1">// Direct traversal of array elements</span>
+<a id="__codelineno-61-8" name="__codelineno-61-8" href="#__codelineno-61-8"></a><span class="w">    </span><span class="c1">// Iterate directly over the array elements</span>
 <a id="__codelineno-61-9" name="__codelineno-61-9" href="#__codelineno-61-9"></a><span class="w">    </span><span class="n">_count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-61-10" name="__codelineno-61-10" href="#__codelineno-61-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="o">&amp;</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-11" name="__codelineno-61-11" href="#__codelineno-61-11"></a><span class="w">        </span><span class="n">_count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
@@ -5584,10 +5671,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Traverse array */</span>
+<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Traverse the array */</span>
 <a id="__codelineno-62-2" name="__codelineno-62-2" href="#__codelineno-62-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">traverse</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-3" name="__codelineno-62-3" href="#__codelineno-62-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-62-4" name="__codelineno-62-4" href="#__codelineno-62-4"></a><span class="w">    </span><span class="c1">// Traverse array by index</span>
+<a id="__codelineno-62-4" name="__codelineno-62-4" href="#__codelineno-62-4"></a><span class="w">    </span><span class="c1">// Traverse the array by index</span>
 <a id="__codelineno-62-5" name="__codelineno-62-5" href="#__codelineno-62-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-6" name="__codelineno-62-6" href="#__codelineno-62-6"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-62-7" name="__codelineno-62-7" href="#__codelineno-62-7"></a><span class="w">    </span><span class="p">}</span>
@@ -5595,14 +5682,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Traverse array */</span>
+<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Traverse the array */</span>
 <a id="__codelineno-63-2" name="__codelineno-63-2" href="#__codelineno-63-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">traverse</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-3" name="__codelineno-63-3" href="#__codelineno-63-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-63-4" name="__codelineno-63-4" href="#__codelineno-63-4"></a><span class="w">    </span><span class="c1">// Traverse array by index</span>
+<a id="__codelineno-63-4" name="__codelineno-63-4" href="#__codelineno-63-4"></a><span class="w">    </span><span class="c1">// Traverse the array by index</span>
 <a id="__codelineno-63-5" name="__codelineno-63-5" href="#__codelineno-63-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">indices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-6" name="__codelineno-63-6" href="#__codelineno-63-6"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-63-7" name="__codelineno-63-7" href="#__codelineno-63-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-63-8" name="__codelineno-63-8" href="#__codelineno-63-8"></a><span class="w">    </span><span class="c1">// Direct traversal of array elements</span>
+<a id="__codelineno-63-8" name="__codelineno-63-8" href="#__codelineno-63-8"></a><span class="w">    </span><span class="c1">// Iterate directly over the array elements</span>
 <a id="__codelineno-63-9" name="__codelineno-63-9" href="#__codelineno-63-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-10" name="__codelineno-63-10" href="#__codelineno-63-10"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">j</span>
 <a id="__codelineno-63-11" name="__codelineno-63-11" href="#__codelineno-63-11"></a><span class="w">    </span><span class="p">}</span>
@@ -5610,16 +5697,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Traverse array ###</span>
+<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Traverse the array ###</span>
 <a id="__codelineno-64-2" name="__codelineno-64-2" href="#__codelineno-64-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">traverse</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
 <a id="__codelineno-64-3" name="__codelineno-64-3" href="#__codelineno-64-3"></a><span class="w">  </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-64-4" name="__codelineno-64-4" href="#__codelineno-64-4"></a>
-<a id="__codelineno-64-5" name="__codelineno-64-5" href="#__codelineno-64-5"></a><span class="w">  </span><span class="c1"># Traverse array by index</span>
+<a id="__codelineno-64-5" name="__codelineno-64-5" href="#__codelineno-64-5"></a><span class="w">  </span><span class="c1"># Traverse the array by index</span>
 <a id="__codelineno-64-6" name="__codelineno-64-6" href="#__codelineno-64-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">nums</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-64-7" name="__codelineno-64-7" href="#__codelineno-64-7"></a><span class="w">    </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-64-8" name="__codelineno-64-8" href="#__codelineno-64-8"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-64-9" name="__codelineno-64-9" href="#__codelineno-64-9"></a>
-<a id="__codelineno-64-10" name="__codelineno-64-10" href="#__codelineno-64-10"></a><span class="w">  </span><span class="c1"># Direct traversal of array elements</span>
+<a id="__codelineno-64-10" name="__codelineno-64-10" href="#__codelineno-64-10"></a><span class="w">  </span><span class="c1"># Iterate directly over the array elements</span>
 <a id="__codelineno-64-11" name="__codelineno-64-11" href="#__codelineno-64-11"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span>
 <a id="__codelineno-64-12" name="__codelineno-64-12" href="#__codelineno-64-12"></a><span class="w">    </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-64-13" name="__codelineno-64-13" href="#__codelineno-64-13"></a><span class="w">  </span><span class="k">end</span>
@@ -5628,14 +5715,19 @@
 </div>
 </div>
 </div>
-<h3 id="6-finding-elements">6. &nbsp; Finding Elements<a class="headerlink" href="#6-finding-elements" title="Permanent link">&para;</a></h3>
-<p>Finding a specified element in an array requires traversing the array and checking whether the element value matches in each iteration; if it matches, output the corresponding index.</p>
-<p>Since an array is a linear data structure, the above search operation is called a "linear search".</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20traverse%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Traverse%20the%20array%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20Traverse%20the%20array%20by%20index%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20nums%5Bi%5D%0A%20%20%20%20%23%20Iterate%20directly%20over%20the%20array%20elements%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20num%0A%20%20%20%20%23%20Iterate%20over%20both%20the%20indices%20and%20the%20elements%0A%20%20%20%20for%20i,%20num%20in%20enumerate%28nums%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20nums%5Bi%5D%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20num%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20array%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22Array%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20Traverse%20the%20array%0A%20%20%20%20traverse%28nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20traverse%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Traverse%20the%20array%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20Traverse%20the%20array%20by%20index%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20nums%5Bi%5D%0A%20%20%20%20%23%20Iterate%20directly%20over%20the%20array%20elements%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20num%0A%20%20%20%20%23%20Iterate%20over%20both%20the%20indices%20and%20the%20elements%0A%20%20%20%20for%20i,%20num%20in%20enumerate%28nums%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20nums%5Bi%5D%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20num%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20array%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22Array%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20Traverse%20the%20array%0A%20%20%20%20traverse%28nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="6-finding-an-element">6. &nbsp; Finding an element<a class="headerlink" href="#6-finding-an-element" title="Permanent link">&para;</a></h3>
+<p>To find a specified element, traverse the array and compare each value with the target. Return the index when a match is found.</p>
+<p>Because an array is a linear data structure, this is called linear search.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="6:13"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">array.py</span><pre><span></span><code><a id="__codelineno-65-1" name="__codelineno-65-1" href="#__codelineno-65-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">find</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-65-2" name="__codelineno-65-2" href="#__codelineno-65-2"></a><span class="w">    </span><span class="sd">"""Find the specified element in the array"""</span>
+<a id="__codelineno-65-2" name="__codelineno-65-2" href="#__codelineno-65-2"></a><span class="w">    </span><span class="sd">"""Find a specified element in the array"""</span>
 <a id="__codelineno-65-3" name="__codelineno-65-3" href="#__codelineno-65-3"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)):</span>
 <a id="__codelineno-65-4" name="__codelineno-65-4" href="#__codelineno-65-4"></a>        <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
 <a id="__codelineno-65-5" name="__codelineno-65-5" href="#__codelineno-65-5"></a>            <span class="k">return</span> <span class="n">i</span>
@@ -5643,7 +5735,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="cm">/* Find the specified element in the array */</span>
+<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="cm">/* Find a specified element in the array */</span>
 <a id="__codelineno-66-2" name="__codelineno-66-2" href="#__codelineno-66-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">find</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-66-3" name="__codelineno-66-3" href="#__codelineno-66-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-66-4" name="__codelineno-66-4" href="#__codelineno-66-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
@@ -5654,7 +5746,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="cm">/* Find the specified element in the array */</span>
+<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="cm">/* Find a specified element in the array */</span>
 <a id="__codelineno-67-2" name="__codelineno-67-2" href="#__codelineno-67-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">find</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-67-3" name="__codelineno-67-3" href="#__codelineno-67-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-67-4" name="__codelineno-67-4" href="#__codelineno-67-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
@@ -5665,7 +5757,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a><span class="cm">/* Find the specified element in the array */</span>
+<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a><span class="cm">/* Find a specified element in the array */</span>
 <a id="__codelineno-68-2" name="__codelineno-68-2" href="#__codelineno-68-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Find</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-68-3" name="__codelineno-68-3" href="#__codelineno-68-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-68-4" name="__codelineno-68-4" href="#__codelineno-68-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
@@ -5676,7 +5768,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a><span class="cm">/* Find the specified element in the array */</span>
+<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a><span class="cm">/* Find a specified element in the array */</span>
 <a id="__codelineno-69-2" name="__codelineno-69-2" href="#__codelineno-69-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">find</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-69-3" name="__codelineno-69-3" href="#__codelineno-69-3"></a><span class="w">    </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-69-4" name="__codelineno-69-4" href="#__codelineno-69-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
@@ -5690,7 +5782,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a><span class="cm">/* Find the specified element in the array */</span>
+<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a><span class="cm">/* Find a specified element in the array */</span>
 <a id="__codelineno-70-2" name="__codelineno-70-2" href="#__codelineno-70-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">find</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-70-3" name="__codelineno-70-3" href="#__codelineno-70-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-70-4" name="__codelineno-70-4" href="#__codelineno-70-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
@@ -5702,7 +5794,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a><span class="cm">/* Find the specified element in the array */</span>
+<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a><span class="cm">/* Find a specified element in the array */</span>
 <a id="__codelineno-71-2" name="__codelineno-71-2" href="#__codelineno-71-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">find</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-71-3" name="__codelineno-71-3" href="#__codelineno-71-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-71-4" name="__codelineno-71-4" href="#__codelineno-71-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
@@ -5712,7 +5804,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* Find the specified element in the array */</span>
+<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* Find a specified element in the array */</span>
 <a id="__codelineno-72-2" name="__codelineno-72-2" href="#__codelineno-72-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">find</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-72-3" name="__codelineno-72-3" href="#__codelineno-72-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-72-4" name="__codelineno-72-4" href="#__codelineno-72-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5724,7 +5816,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="cm">/* Find the specified element in the array */</span>
+<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="cm">/* Find a specified element in the array */</span>
 <a id="__codelineno-73-2" name="__codelineno-73-2" href="#__codelineno-73-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">find</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-73-3" name="__codelineno-73-3" href="#__codelineno-73-3"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-73-4" name="__codelineno-73-4" href="#__codelineno-73-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
@@ -5734,7 +5826,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="cm">/* Find the specified element in the array */</span>
+<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="cm">/* Find a specified element in the array */</span>
 <a id="__codelineno-74-2" name="__codelineno-74-2" href="#__codelineno-74-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">find</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="kt">usize</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-74-3" name="__codelineno-74-3" href="#__codelineno-74-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-74-4" name="__codelineno-74-4" href="#__codelineno-74-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
@@ -5746,7 +5838,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="cm">/* Find the specified element in the array */</span>
+<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="cm">/* Find a specified element in the array */</span>
 <a id="__codelineno-75-2" name="__codelineno-75-2" href="#__codelineno-75-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">find</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-75-3" name="__codelineno-75-3" href="#__codelineno-75-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-75-4" name="__codelineno-75-4" href="#__codelineno-75-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
@@ -5757,7 +5849,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a><span class="cm">/* Find the specified element in the array */</span>
+<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a><span class="cm">/* Find a specified element in the array */</span>
 <a id="__codelineno-76-2" name="__codelineno-76-2" href="#__codelineno-76-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">find</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-76-3" name="__codelineno-76-3" href="#__codelineno-76-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">indices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-76-4" name="__codelineno-76-4" href="#__codelineno-76-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
@@ -5768,7 +5860,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="c1">### Find specified element in array ###</span>
+<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="c1">### Find a specified element in the array ###</span>
 <a id="__codelineno-77-2" name="__codelineno-77-2" href="#__codelineno-77-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">find</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
 <a id="__codelineno-77-3" name="__codelineno-77-3" href="#__codelineno-77-3"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">nums</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-77-4" name="__codelineno-77-4" href="#__codelineno-77-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span>
@@ -5780,27 +5872,32 @@
 </div>
 </div>
 </div>
-<h3 id="7-expanding-arrays">7. &nbsp; Expanding Arrays<a class="headerlink" href="#7-expanding-arrays" title="Permanent link">&para;</a></h3>
-<p>In complex system environments, programs cannot guarantee that the memory space after an array is available, making it unsafe to expand the array's capacity. Therefore, in most programming languages, <strong>the length of an array is immutable</strong>.</p>
-<p>If we want to expand an array, we need to create a new, larger array and then copy the original array elements to the new array one by one. This is an <span class="arithmatex">\(O(n)\)</span> operation, which is very time-consuming when the array is large. The code is shown below:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 477px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20find%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20a%20specified%20element%20in%20the%20array%22%22%22%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20if%20nums%5Bi%5D%20%3D%3D%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20i%0A%20%20%20%20return%20-1%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20array%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22Array%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20Find%20an%20element%0A%20%20%20%20index%3A%20int%20%3D%20find%28nums,%203%29%0A%20%20%20%20print%28%22Find%20element%203%20in%20nums,%20giving%20index%20%3D%22,%20index%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20find%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20a%20specified%20element%20in%20the%20array%22%22%22%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20if%20nums%5Bi%5D%20%3D%3D%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20i%0A%20%20%20%20return%20-1%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20array%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22Array%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20Find%20an%20element%0A%20%20%20%20index%3A%20int%20%3D%20find%28nums,%203%29%0A%20%20%20%20print%28%22Find%20element%203%20in%20nums,%20giving%20index%20%3D%22,%20index%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="7-expanding-an-array">7. &nbsp; Expanding an array<a class="headerlink" href="#7-expanding-an-array" title="Permanent link">&para;</a></h3>
+<p>In a complex system, a program cannot ensure that memory immediately after an array is available, so extending it in place is unsafe. Therefore, in most programming languages, <strong>an array's length is fixed</strong>.</p>
+<p>To expand an array, create a larger one and copy every element into it. This takes <span class="arithmatex">\(O(n)\)</span> time and can be expensive for a large array. The code is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="7:13"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><input id="__tabbed_7_13" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Python</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Java</label><label for="__tabbed_7_4">C#</label><label for="__tabbed_7_5">Go</label><label for="__tabbed_7_6">Swift</label><label for="__tabbed_7_7">JS</label><label for="__tabbed_7_8">TS</label><label for="__tabbed_7_9">Dart</label><label for="__tabbed_7_10">Rust</label><label for="__tabbed_7_11">C</label><label for="__tabbed_7_12">Kotlin</label><label for="__tabbed_7_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">array.py</span><pre><span></span><code><a id="__codelineno-78-1" name="__codelineno-78-1" href="#__codelineno-78-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">extend</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">enlarge</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
-<a id="__codelineno-78-2" name="__codelineno-78-2" href="#__codelineno-78-2"></a><span class="w">    </span><span class="sd">"""Extend array length"""</span>
-<a id="__codelineno-78-3" name="__codelineno-78-3" href="#__codelineno-78-3"></a>    <span class="c1"># Initialize an array with extended length</span>
+<a id="__codelineno-78-2" name="__codelineno-78-2" href="#__codelineno-78-2"></a><span class="w">    </span><span class="sd">"""Extend the array length"""</span>
+<a id="__codelineno-78-3" name="__codelineno-78-3" href="#__codelineno-78-3"></a>    <span class="c1"># Initialize an array with the extended length</span>
 <a id="__codelineno-78-4" name="__codelineno-78-4" href="#__codelineno-78-4"></a>    <span class="n">res</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">+</span> <span class="n">enlarge</span><span class="p">)</span>
 <a id="__codelineno-78-5" name="__codelineno-78-5" href="#__codelineno-78-5"></a>    <span class="c1"># Copy all elements from the original array to the new array</span>
 <a id="__codelineno-78-6" name="__codelineno-78-6" href="#__codelineno-78-6"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)):</span>
 <a id="__codelineno-78-7" name="__codelineno-78-7" href="#__codelineno-78-7"></a>        <span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-<a id="__codelineno-78-8" name="__codelineno-78-8" href="#__codelineno-78-8"></a>    <span class="c1"># Return the extended new array</span>
+<a id="__codelineno-78-8" name="__codelineno-78-8" href="#__codelineno-78-8"></a>    <span class="c1"># Return the new, extended array</span>
 <a id="__codelineno-78-9" name="__codelineno-78-9" href="#__codelineno-78-9"></a>    <span class="k">return</span> <span class="n">res</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-79-1" name="__codelineno-79-1" href="#__codelineno-79-1"></a><span class="cm">/* Extend array length */</span>
+<div class="highlight"><span class="filename">array.cpp</span><pre><span></span><code><a id="__codelineno-79-1" name="__codelineno-79-1" href="#__codelineno-79-1"></a><span class="cm">/* Extend the array length */</span>
 <a id="__codelineno-79-2" name="__codelineno-79-2" href="#__codelineno-79-2"></a><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="nf">extend</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">enlarge</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-79-3" name="__codelineno-79-3" href="#__codelineno-79-3"></a><span class="w">    </span><span class="c1">// Initialize an array with extended length</span>
+<a id="__codelineno-79-3" name="__codelineno-79-3" href="#__codelineno-79-3"></a><span class="w">    </span><span class="c1">// Initialize an array with the extended length</span>
 <a id="__codelineno-79-4" name="__codelineno-79-4" href="#__codelineno-79-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">size</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">enlarge</span><span class="p">];</span>
 <a id="__codelineno-79-5" name="__codelineno-79-5" href="#__codelineno-79-5"></a><span class="w">    </span><span class="c1">// Copy all elements from the original array to the new array</span>
 <a id="__codelineno-79-6" name="__codelineno-79-6" href="#__codelineno-79-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5808,164 +5905,164 @@
 <a id="__codelineno-79-8" name="__codelineno-79-8" href="#__codelineno-79-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-79-9" name="__codelineno-79-9" href="#__codelineno-79-9"></a><span class="w">    </span><span class="c1">// Free memory</span>
 <a id="__codelineno-79-10" name="__codelineno-79-10" href="#__codelineno-79-10"></a><span class="w">    </span><span class="k">delete</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span>
-<a id="__codelineno-79-11" name="__codelineno-79-11" href="#__codelineno-79-11"></a><span class="w">    </span><span class="c1">// Return the extended new array</span>
+<a id="__codelineno-79-11" name="__codelineno-79-11" href="#__codelineno-79-11"></a><span class="w">    </span><span class="c1">// Return the new, extended array</span>
 <a id="__codelineno-79-12" name="__codelineno-79-12" href="#__codelineno-79-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-79-13" name="__codelineno-79-13" href="#__codelineno-79-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-80-1" name="__codelineno-80-1" href="#__codelineno-80-1"></a><span class="cm">/* Extend array length */</span>
+<div class="highlight"><span class="filename">array.java</span><pre><span></span><code><a id="__codelineno-80-1" name="__codelineno-80-1" href="#__codelineno-80-1"></a><span class="cm">/* Extend the array length */</span>
 <a id="__codelineno-80-2" name="__codelineno-80-2" href="#__codelineno-80-2"></a><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="nf">extend</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">enlarge</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-80-3" name="__codelineno-80-3" href="#__codelineno-80-3"></a><span class="w">    </span><span class="c1">// Initialize an array with extended length</span>
+<a id="__codelineno-80-3" name="__codelineno-80-3" href="#__codelineno-80-3"></a><span class="w">    </span><span class="c1">// Initialize an array with the extended length</span>
 <a id="__codelineno-80-4" name="__codelineno-80-4" href="#__codelineno-80-4"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">enlarge</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-80-5" name="__codelineno-80-5" href="#__codelineno-80-5"></a><span class="w">    </span><span class="c1">// Copy all elements from the original array to the new array</span>
 <a id="__codelineno-80-6" name="__codelineno-80-6" href="#__codelineno-80-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-7" name="__codelineno-80-7" href="#__codelineno-80-7"></a><span class="w">        </span><span class="n">res</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-80-8" name="__codelineno-80-8" href="#__codelineno-80-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-80-9" name="__codelineno-80-9" href="#__codelineno-80-9"></a><span class="w">    </span><span class="c1">// Return the extended new array</span>
+<a id="__codelineno-80-9" name="__codelineno-80-9" href="#__codelineno-80-9"></a><span class="w">    </span><span class="c1">// Return the new, extended array</span>
 <a id="__codelineno-80-10" name="__codelineno-80-10" href="#__codelineno-80-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-80-11" name="__codelineno-80-11" href="#__codelineno-80-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="cm">/* Extend array length */</span>
+<div class="highlight"><span class="filename">array.cs</span><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="cm">/* Extend the array length */</span>
 <a id="__codelineno-81-2" name="__codelineno-81-2" href="#__codelineno-81-2"></a><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="nf">Extend</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">enlarge</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-81-3" name="__codelineno-81-3" href="#__codelineno-81-3"></a><span class="w">    </span><span class="c1">// Initialize an array with extended length</span>
+<a id="__codelineno-81-3" name="__codelineno-81-3" href="#__codelineno-81-3"></a><span class="w">    </span><span class="c1">// Initialize an array with the extended length</span>
 <a id="__codelineno-81-4" name="__codelineno-81-4" href="#__codelineno-81-4"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">enlarge</span><span class="p">];</span>
 <a id="__codelineno-81-5" name="__codelineno-81-5" href="#__codelineno-81-5"></a><span class="w">    </span><span class="c1">// Copy all elements from the original array to the new array</span>
 <a id="__codelineno-81-6" name="__codelineno-81-6" href="#__codelineno-81-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-7" name="__codelineno-81-7" href="#__codelineno-81-7"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-81-8" name="__codelineno-81-8" href="#__codelineno-81-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-81-9" name="__codelineno-81-9" href="#__codelineno-81-9"></a><span class="w">    </span><span class="c1">// Return the extended new array</span>
+<a id="__codelineno-81-9" name="__codelineno-81-9" href="#__codelineno-81-9"></a><span class="w">    </span><span class="c1">// Return the new, extended array</span>
 <a id="__codelineno-81-10" name="__codelineno-81-10" href="#__codelineno-81-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-81-11" name="__codelineno-81-11" href="#__codelineno-81-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="cm">/* Extend array length */</span>
+<div class="highlight"><span class="filename">array.go</span><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="cm">/* Extend the array length */</span>
 <a id="__codelineno-82-2" name="__codelineno-82-2" href="#__codelineno-82-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">extend</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">enlarge</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-82-3" name="__codelineno-82-3" href="#__codelineno-82-3"></a><span class="w">    </span><span class="c1">// Initialize an array with extended length</span>
+<a id="__codelineno-82-3" name="__codelineno-82-3" href="#__codelineno-82-3"></a><span class="w">    </span><span class="c1">// Initialize an array with the extended length</span>
 <a id="__codelineno-82-4" name="__codelineno-82-4" href="#__codelineno-82-4"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="o">+</span><span class="nx">enlarge</span><span class="p">)</span>
 <a id="__codelineno-82-5" name="__codelineno-82-5" href="#__codelineno-82-5"></a><span class="w">    </span><span class="c1">// Copy all elements from the original array to the new array</span>
 <a id="__codelineno-82-6" name="__codelineno-82-6" href="#__codelineno-82-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-7" name="__codelineno-82-7" href="#__codelineno-82-7"></a><span class="w">        </span><span class="nx">res</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span>
 <a id="__codelineno-82-8" name="__codelineno-82-8" href="#__codelineno-82-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-82-9" name="__codelineno-82-9" href="#__codelineno-82-9"></a><span class="w">    </span><span class="c1">// Return the extended new array</span>
+<a id="__codelineno-82-9" name="__codelineno-82-9" href="#__codelineno-82-9"></a><span class="w">    </span><span class="c1">// Return the new, extended array</span>
 <a id="__codelineno-82-10" name="__codelineno-82-10" href="#__codelineno-82-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span>
 <a id="__codelineno-82-11" name="__codelineno-82-11" href="#__codelineno-82-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-83-1" name="__codelineno-83-1" href="#__codelineno-83-1"></a><span class="cm">/* Extend array length */</span>
+<div class="highlight"><span class="filename">array.swift</span><pre><span></span><code><a id="__codelineno-83-1" name="__codelineno-83-1" href="#__codelineno-83-1"></a><span class="cm">/* Extend the array length */</span>
 <a id="__codelineno-83-2" name="__codelineno-83-2" href="#__codelineno-83-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">extend</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">enlarge</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-83-3" name="__codelineno-83-3" href="#__codelineno-83-3"></a><span class="w">    </span><span class="c1">// Initialize an array with extended length</span>
+<a id="__codelineno-83-3" name="__codelineno-83-3" href="#__codelineno-83-3"></a><span class="w">    </span><span class="c1">// Initialize an array with the extended length</span>
 <a id="__codelineno-83-4" name="__codelineno-83-4" href="#__codelineno-83-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">count</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">enlarge</span><span class="p">)</span>
 <a id="__codelineno-83-5" name="__codelineno-83-5" href="#__codelineno-83-5"></a><span class="w">    </span><span class="c1">// Copy all elements from the original array to the new array</span>
 <a id="__codelineno-83-6" name="__codelineno-83-6" href="#__codelineno-83-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-7" name="__codelineno-83-7" href="#__codelineno-83-7"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-83-8" name="__codelineno-83-8" href="#__codelineno-83-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-83-9" name="__codelineno-83-9" href="#__codelineno-83-9"></a><span class="w">    </span><span class="c1">// Return the extended new array</span>
+<a id="__codelineno-83-9" name="__codelineno-83-9" href="#__codelineno-83-9"></a><span class="w">    </span><span class="c1">// Return the new, extended array</span>
 <a id="__codelineno-83-10" name="__codelineno-83-10" href="#__codelineno-83-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-83-11" name="__codelineno-83-11" href="#__codelineno-83-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-84-1" name="__codelineno-84-1" href="#__codelineno-84-1"></a><span class="cm">/* Extend array length */</span>
-<a id="__codelineno-84-2" name="__codelineno-84-2" href="#__codelineno-84-2"></a><span class="c1">// Note: JavaScript's Array is dynamic array, can be directly expanded</span>
-<a id="__codelineno-84-3" name="__codelineno-84-3" href="#__codelineno-84-3"></a><span class="c1">// For learning purposes, this function treats Array as fixed-length array</span>
+<div class="highlight"><span class="filename">array.js</span><pre><span></span><code><a id="__codelineno-84-1" name="__codelineno-84-1" href="#__codelineno-84-1"></a><span class="cm">/* Extend the array length */</span>
+<a id="__codelineno-84-2" name="__codelineno-84-2" href="#__codelineno-84-2"></a><span class="c1">// JavaScript Array is a dynamic array and can be extended directly</span>
+<a id="__codelineno-84-3" name="__codelineno-84-3" href="#__codelineno-84-3"></a><span class="c1">// For learning purposes, this function treats Array as a fixed-length array</span>
 <a id="__codelineno-84-4" name="__codelineno-84-4" href="#__codelineno-84-4"></a><span class="kd">function</span><span class="w"> </span><span class="nx">extend</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">enlarge</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-84-5" name="__codelineno-84-5" href="#__codelineno-84-5"></a><span class="w">    </span><span class="c1">// Initialize an array with extended length</span>
+<a id="__codelineno-84-5" name="__codelineno-84-5" href="#__codelineno-84-5"></a><span class="w">    </span><span class="c1">// Initialize an array with the extended length</span>
 <a id="__codelineno-84-6" name="__codelineno-84-6" href="#__codelineno-84-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">enlarge</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-84-7" name="__codelineno-84-7" href="#__codelineno-84-7"></a><span class="w">    </span><span class="c1">// Copy all elements from the original array to the new array</span>
 <a id="__codelineno-84-8" name="__codelineno-84-8" href="#__codelineno-84-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-9" name="__codelineno-84-9" href="#__codelineno-84-9"></a><span class="w">        </span><span class="nx">res</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-84-10" name="__codelineno-84-10" href="#__codelineno-84-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-84-11" name="__codelineno-84-11" href="#__codelineno-84-11"></a><span class="w">    </span><span class="c1">// Return the extended new array</span>
+<a id="__codelineno-84-11" name="__codelineno-84-11" href="#__codelineno-84-11"></a><span class="w">    </span><span class="c1">// Return the new, extended array</span>
 <a id="__codelineno-84-12" name="__codelineno-84-12" href="#__codelineno-84-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-84-13" name="__codelineno-84-13" href="#__codelineno-84-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-85-1" name="__codelineno-85-1" href="#__codelineno-85-1"></a><span class="cm">/* Extend array length */</span>
-<a id="__codelineno-85-2" name="__codelineno-85-2" href="#__codelineno-85-2"></a><span class="c1">// Note: TypeScript's Array is dynamic array, can be directly expanded</span>
-<a id="__codelineno-85-3" name="__codelineno-85-3" href="#__codelineno-85-3"></a><span class="c1">// For learning purposes, this function treats Array as fixed-length array</span>
+<div class="highlight"><span class="filename">array.ts</span><pre><span></span><code><a id="__codelineno-85-1" name="__codelineno-85-1" href="#__codelineno-85-1"></a><span class="cm">/* Extend the array length */</span>
+<a id="__codelineno-85-2" name="__codelineno-85-2" href="#__codelineno-85-2"></a><span class="c1">// Note: TypeScript Array is a dynamic array and can grow directly</span>
+<a id="__codelineno-85-3" name="__codelineno-85-3" href="#__codelineno-85-3"></a><span class="c1">// For learning purposes, this function treats Array as a fixed-length array</span>
 <a id="__codelineno-85-4" name="__codelineno-85-4" href="#__codelineno-85-4"></a><span class="kd">function</span><span class="w"> </span><span class="nx">extend</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">enlarge</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-85-5" name="__codelineno-85-5" href="#__codelineno-85-5"></a><span class="w">    </span><span class="c1">// Initialize an array with extended length</span>
+<a id="__codelineno-85-5" name="__codelineno-85-5" href="#__codelineno-85-5"></a><span class="w">    </span><span class="c1">// Initialize an array with the extended length</span>
 <a id="__codelineno-85-6" name="__codelineno-85-6" href="#__codelineno-85-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">enlarge</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-85-7" name="__codelineno-85-7" href="#__codelineno-85-7"></a><span class="w">    </span><span class="c1">// Copy all elements from the original array to the new array</span>
 <a id="__codelineno-85-8" name="__codelineno-85-8" href="#__codelineno-85-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-9" name="__codelineno-85-9" href="#__codelineno-85-9"></a><span class="w">        </span><span class="nx">res</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-85-10" name="__codelineno-85-10" href="#__codelineno-85-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-85-11" name="__codelineno-85-11" href="#__codelineno-85-11"></a><span class="w">    </span><span class="c1">// Return the extended new array</span>
+<a id="__codelineno-85-11" name="__codelineno-85-11" href="#__codelineno-85-11"></a><span class="w">    </span><span class="c1">// Return the new, extended array</span>
 <a id="__codelineno-85-12" name="__codelineno-85-12" href="#__codelineno-85-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-85-13" name="__codelineno-85-13" href="#__codelineno-85-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-86-1" name="__codelineno-86-1" href="#__codelineno-86-1"></a><span class="cm">/* Extend array length */</span>
+<div class="highlight"><span class="filename">array.dart</span><pre><span></span><code><a id="__codelineno-86-1" name="__codelineno-86-1" href="#__codelineno-86-1"></a><span class="cm">/* Extend the array length */</span>
 <a id="__codelineno-86-2" name="__codelineno-86-2" href="#__codelineno-86-2"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">extend</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">enlarge</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-86-3" name="__codelineno-86-3" href="#__codelineno-86-3"></a><span class="w">  </span><span class="c1">// Initialize an array with extended length</span>
+<a id="__codelineno-86-3" name="__codelineno-86-3" href="#__codelineno-86-3"></a><span class="w">  </span><span class="c1">// Initialize an array with the extended length</span>
 <a id="__codelineno-86-4" name="__codelineno-86-4" href="#__codelineno-86-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">enlarge</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-86-5" name="__codelineno-86-5" href="#__codelineno-86-5"></a><span class="w">  </span><span class="c1">// Copy all elements from the original array to the new array</span>
 <a id="__codelineno-86-6" name="__codelineno-86-6" href="#__codelineno-86-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-7" name="__codelineno-86-7" href="#__codelineno-86-7"></a><span class="w">    </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-86-8" name="__codelineno-86-8" href="#__codelineno-86-8"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-86-9" name="__codelineno-86-9" href="#__codelineno-86-9"></a><span class="w">  </span><span class="c1">// Return the extended new array</span>
+<a id="__codelineno-86-9" name="__codelineno-86-9" href="#__codelineno-86-9"></a><span class="w">  </span><span class="c1">// Return the new, extended array</span>
 <a id="__codelineno-86-10" name="__codelineno-86-10" href="#__codelineno-86-10"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-86-11" name="__codelineno-86-11" href="#__codelineno-86-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-87-1" name="__codelineno-87-1" href="#__codelineno-87-1"></a><span class="cm">/* Extend array length */</span>
+<div class="highlight"><span class="filename">array.rs</span><pre><span></span><code><a id="__codelineno-87-1" name="__codelineno-87-1" href="#__codelineno-87-1"></a><span class="cm">/* Extend the array length */</span>
 <a id="__codelineno-87-2" name="__codelineno-87-2" href="#__codelineno-87-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">extend</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">enlarge</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-87-3" name="__codelineno-87-3" href="#__codelineno-87-3"></a><span class="w">    </span><span class="c1">// Initialize an array with extended length</span>
+<a id="__codelineno-87-3" name="__codelineno-87-3" href="#__codelineno-87-3"></a><span class="w">    </span><span class="c1">// Initialize an array with the extended length</span>
 <a id="__codelineno-87-4" name="__codelineno-87-4" href="#__codelineno-87-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">enlarge</span><span class="p">];</span>
-<a id="__codelineno-87-5" name="__codelineno-87-5" href="#__codelineno-87-5"></a><span class="w">    </span><span class="c1">// Copy all elements from original array to new</span>
+<a id="__codelineno-87-5" name="__codelineno-87-5" href="#__codelineno-87-5"></a><span class="w">    </span><span class="c1">// Copy all elements from the original array to the new one</span>
 <a id="__codelineno-87-6" name="__codelineno-87-6" href="#__codelineno-87-6"></a><span class="w">    </span><span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="o">..</span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()].</span><span class="n">copy_from_slice</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span>
 <a id="__codelineno-87-7" name="__codelineno-87-7" href="#__codelineno-87-7"></a>
-<a id="__codelineno-87-8" name="__codelineno-87-8" href="#__codelineno-87-8"></a><span class="w">    </span><span class="c1">// Return the extended new array</span>
+<a id="__codelineno-87-8" name="__codelineno-87-8" href="#__codelineno-87-8"></a><span class="w">    </span><span class="c1">// Return the new, extended array</span>
 <a id="__codelineno-87-9" name="__codelineno-87-9" href="#__codelineno-87-9"></a><span class="w">    </span><span class="n">res</span>
 <a id="__codelineno-87-10" name="__codelineno-87-10" href="#__codelineno-87-10"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-88-1" name="__codelineno-88-1" href="#__codelineno-88-1"></a><span class="cm">/* Extend array length */</span>
+<div class="highlight"><span class="filename">array.c</span><pre><span></span><code><a id="__codelineno-88-1" name="__codelineno-88-1" href="#__codelineno-88-1"></a><span class="cm">/* Extend the array length */</span>
 <a id="__codelineno-88-2" name="__codelineno-88-2" href="#__codelineno-88-2"></a><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="nf">extend</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">enlarge</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-88-3" name="__codelineno-88-3" href="#__codelineno-88-3"></a><span class="w">    </span><span class="c1">// Initialize an array with extended length</span>
+<a id="__codelineno-88-3" name="__codelineno-88-3" href="#__codelineno-88-3"></a><span class="w">    </span><span class="c1">// Initialize an array with the extended length</span>
 <a id="__codelineno-88-4" name="__codelineno-88-4" href="#__codelineno-88-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">enlarge</span><span class="p">));</span>
 <a id="__codelineno-88-5" name="__codelineno-88-5" href="#__codelineno-88-5"></a><span class="w">    </span><span class="c1">// Copy all elements from the original array to the new array</span>
 <a id="__codelineno-88-6" name="__codelineno-88-6" href="#__codelineno-88-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-7" name="__codelineno-88-7" href="#__codelineno-88-7"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-88-8" name="__codelineno-88-8" href="#__codelineno-88-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-88-9" name="__codelineno-88-9" href="#__codelineno-88-9"></a><span class="w">    </span><span class="c1">// Initialize expanded space</span>
+<a id="__codelineno-88-9" name="__codelineno-88-9" href="#__codelineno-88-9"></a><span class="w">    </span><span class="c1">// Initialize the newly added space</span>
 <a id="__codelineno-88-10" name="__codelineno-88-10" href="#__codelineno-88-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">enlarge</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-11" name="__codelineno-88-11" href="#__codelineno-88-11"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-88-12" name="__codelineno-88-12" href="#__codelineno-88-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-88-13" name="__codelineno-88-13" href="#__codelineno-88-13"></a><span class="w">    </span><span class="c1">// Return the extended new array</span>
+<a id="__codelineno-88-13" name="__codelineno-88-13" href="#__codelineno-88-13"></a><span class="w">    </span><span class="c1">// Return the new, extended array</span>
 <a id="__codelineno-88-14" name="__codelineno-88-14" href="#__codelineno-88-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-88-15" name="__codelineno-88-15" href="#__codelineno-88-15"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-89-1" name="__codelineno-89-1" href="#__codelineno-89-1"></a><span class="cm">/* Extend array length */</span>
+<div class="highlight"><span class="filename">array.kt</span><pre><span></span><code><a id="__codelineno-89-1" name="__codelineno-89-1" href="#__codelineno-89-1"></a><span class="cm">/* Extend the array length */</span>
 <a id="__codelineno-89-2" name="__codelineno-89-2" href="#__codelineno-89-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">extend</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">enlarge</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="n">IntArray</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-89-3" name="__codelineno-89-3" href="#__codelineno-89-3"></a><span class="w">    </span><span class="c1">// Initialize an array with extended length</span>
+<a id="__codelineno-89-3" name="__codelineno-89-3" href="#__codelineno-89-3"></a><span class="w">    </span><span class="c1">// Initialize an array with the extended length</span>
 <a id="__codelineno-89-4" name="__codelineno-89-4" href="#__codelineno-89-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">enlarge</span><span class="p">)</span>
 <a id="__codelineno-89-5" name="__codelineno-89-5" href="#__codelineno-89-5"></a><span class="w">    </span><span class="c1">// Copy all elements from the original array to the new array</span>
 <a id="__codelineno-89-6" name="__codelineno-89-6" href="#__codelineno-89-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">indices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-89-7" name="__codelineno-89-7" href="#__codelineno-89-7"></a><span class="w">        </span><span class="n">res</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-89-8" name="__codelineno-89-8" href="#__codelineno-89-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-89-9" name="__codelineno-89-9" href="#__codelineno-89-9"></a><span class="w">    </span><span class="c1">// Return the extended new array</span>
+<a id="__codelineno-89-9" name="__codelineno-89-9" href="#__codelineno-89-9"></a><span class="w">    </span><span class="c1">// Return the new, extended array</span>
 <a id="__codelineno-89-10" name="__codelineno-89-10" href="#__codelineno-89-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-89-11" name="__codelineno-89-11" href="#__codelineno-89-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-90-1" name="__codelineno-90-1" href="#__codelineno-90-1"></a><span class="c1">### Extend array length ###</span>
-<a id="__codelineno-90-2" name="__codelineno-90-2" href="#__codelineno-90-2"></a><span class="c1"># Note: Ruby's Array is dynamic array, can be directly expanded</span>
-<a id="__codelineno-90-3" name="__codelineno-90-3" href="#__codelineno-90-3"></a><span class="c1"># For learning purposes, this function treats Array as fixed-length array</span>
+<div class="highlight"><span class="filename">array.rb</span><pre><span></span><code><a id="__codelineno-90-1" name="__codelineno-90-1" href="#__codelineno-90-1"></a><span class="c1">### Extend the array length ###</span>
+<a id="__codelineno-90-2" name="__codelineno-90-2" href="#__codelineno-90-2"></a><span class="c1"># Ruby Array is a dynamic array that can grow directly</span>
+<a id="__codelineno-90-3" name="__codelineno-90-3" href="#__codelineno-90-3"></a><span class="c1"># For learning purposes, this function treats Array as a fixed-length array</span>
 <a id="__codelineno-90-4" name="__codelineno-90-4" href="#__codelineno-90-4"></a><span class="k">def</span><span class="w"> </span><span class="nf">extend</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">enlarge</span><span class="p">)</span>
-<a id="__codelineno-90-5" name="__codelineno-90-5" href="#__codelineno-90-5"></a><span class="w">  </span><span class="c1"># Initialize an array with extended length</span>
+<a id="__codelineno-90-5" name="__codelineno-90-5" href="#__codelineno-90-5"></a><span class="w">  </span><span class="c1"># Initialize an array with the extended length</span>
 <a id="__codelineno-90-6" name="__codelineno-90-6" href="#__codelineno-90-6"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">nums</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">enlarge</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-90-7" name="__codelineno-90-7" href="#__codelineno-90-7"></a>
 <a id="__codelineno-90-8" name="__codelineno-90-8" href="#__codelineno-90-8"></a><span class="w">  </span><span class="c1"># Copy all elements from the original array to the new array</span>
@@ -5973,34 +6070,39 @@
 <a id="__codelineno-90-10" name="__codelineno-90-10" href="#__codelineno-90-10"></a><span class="w">    </span><span class="n">res</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-90-11" name="__codelineno-90-11" href="#__codelineno-90-11"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-90-12" name="__codelineno-90-12" href="#__codelineno-90-12"></a>
-<a id="__codelineno-90-13" name="__codelineno-90-13" href="#__codelineno-90-13"></a><span class="w">  </span><span class="c1"># Return the extended new array</span>
+<a id="__codelineno-90-13" name="__codelineno-90-13" href="#__codelineno-90-13"></a><span class="w">  </span><span class="c1"># Return the new, extended array</span>
 <a id="__codelineno-90-14" name="__codelineno-90-14" href="#__codelineno-90-14"></a><span class="w">  </span><span class="n">res</span>
 <a id="__codelineno-90-15" name="__codelineno-90-15" href="#__codelineno-90-15"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h2 id="412-advantages-and-limitations-of-arrays">4.1.2 &nbsp; Advantages and Limitations of Arrays<a class="headerlink" href="#412-advantages-and-limitations-of-arrays" title="Permanent link">&para;</a></h2>
-<p>Arrays are stored in contiguous memory space with elements of the same type. This approach contains rich prior information that the system can use to optimize the efficiency of data structure operations.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=%23%20Note%20that%20a%20Python%20list%20is%20a%20dynamic%20array%20and%20can%20be%20extended%20directly%0A%23%20For%20learning%20purposes,%20this%20function%20treats%20a%20list%20as%20a%20fixed-length%20array%0Adef%20extend%28nums%3A%20list%5Bint%5D,%20enlarge%3A%20int%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%22%22%22Extend%20the%20array%20length%22%22%22%0A%20%20%20%20%23%20Initialize%20an%20array%20with%20the%20extended%20length%0A%20%20%20%20res%20%3D%20%5B0%5D%20%2A%20%28len%28nums%29%20%2B%20enlarge%29%0A%20%20%20%20%23%20Copy%20all%20elements%20from%20the%20original%20array%20to%20the%20new%20array%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20res%5Bi%5D%20%3D%20nums%5Bi%5D%0A%20%20%20%20%23%20Return%20the%20new,%20extended%20array%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20array%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22Array%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20Extend%20the%20length%0A%20%20%20%20nums%20%3D%20extend%28nums,%203%29%0A%20%20%20%20print%28%22Extend%20the%20array%20length%20to%208,%20giving%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%23%20Note%20that%20a%20Python%20list%20is%20a%20dynamic%20array%20and%20can%20be%20extended%20directly%0A%23%20For%20learning%20purposes,%20this%20function%20treats%20a%20list%20as%20a%20fixed-length%20array%0Adef%20extend%28nums%3A%20list%5Bint%5D,%20enlarge%3A%20int%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%22%22%22Extend%20the%20array%20length%22%22%22%0A%20%20%20%20%23%20Initialize%20an%20array%20with%20the%20extended%20length%0A%20%20%20%20res%20%3D%20%5B0%5D%20%2A%20%28len%28nums%29%20%2B%20enlarge%29%0A%20%20%20%20%23%20Copy%20all%20elements%20from%20the%20original%20array%20to%20the%20new%20array%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20res%5Bi%5D%20%3D%20nums%5Bi%5D%0A%20%20%20%20%23%20Return%20the%20new,%20extended%20array%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20array%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20print%28%22Array%20nums%20%3D%22,%20nums%29%0A%0A%20%20%20%20%23%20Extend%20the%20length%0A%20%20%20%20nums%20%3D%20extend%28nums,%203%29%0A%20%20%20%20print%28%22Extend%20the%20array%20length%20to%208,%20giving%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="412-strengths-and-limitations-of-arrays">4.1.2 &nbsp; Strengths and limitations of arrays<a class="headerlink" href="#412-strengths-and-limitations-of-arrays" title="Permanent link">&para;</a></h2>
+<p>An array stores elements of one type in contiguous memory. These known properties give the system opportunities to optimize operations.</p>
 <ul>
-<li><strong>High space efficiency</strong>: Arrays allocate contiguous memory blocks for data without additional structural overhead.</li>
-<li><strong>Support for random access</strong>: Arrays allow accessing any element in <span class="arithmatex">\(O(1)\)</span> time.</li>
-<li><strong>Cache locality</strong>: When accessing array elements, the computer not only loads the element but also caches the surrounding data, thereby leveraging the cache to improve the execution speed of subsequent operations.</li>
+<li><strong>Space efficiency</strong>: Data occupies a contiguous block with no extra structural overhead.</li>
+<li><strong>Random access</strong>: Any element can be accessed in <span class="arithmatex">\(O(1)\)</span> time.</li>
+<li><strong>Cache locality</strong>: Accessing an element loads not only that element but also nearby data into the cache, speeding up later operations.</li>
 </ul>
-<p>Contiguous space storage is a double-edged sword with the following limitations:</p>
+<p>Contiguous storage also has limitations.</p>
 <ul>
-<li><strong>Low insertion and deletion efficiency</strong>: When an array has many elements, insertion and deletion operations require shifting a large number of elements.</li>
-<li><strong>Immutable length</strong>: After an array is initialized, its length is fixed. Expanding the array requires copying all data to a new array, which is very costly.</li>
-<li><strong>Space waste</strong>: If the allocated size of an array exceeds what is actually needed, the extra space is wasted.</li>
+<li><strong>Slow insertion and deletion</strong>: With many elements, these operations require substantial shifting.</li>
+<li><strong>Fixed length</strong>: Once initialized, an array cannot change length; expansion requires copying all data into a new array.</li>
+<li><strong>Wasted space</strong>: Any allocated space beyond actual needs is unused.</li>
 </ul>
-<h2 id="413-typical-applications-of-arrays">4.1.3 &nbsp; Typical Applications of Arrays<a class="headerlink" href="#413-typical-applications-of-arrays" title="Permanent link">&para;</a></h2>
-<p>Arrays are a fundamental and common data structure, frequently used in various algorithms and for implementing various complex data structures.</p>
+<h2 id="413-typical-applications-of-arrays">4.1.3 &nbsp; Typical applications of arrays<a class="headerlink" href="#413-typical-applications-of-arrays" title="Permanent link">&para;</a></h2>
+<p>Arrays are fundamental, widely used data structures. They appear throughout algorithms and underpin more complex structures.</p>
 <ul>
-<li><strong>Random access</strong>: If we want to randomly sample some items, we can use an array to store them and generate a random sequence to implement random sampling based on indices.</li>
-<li><strong>Sorting and searching</strong>: Arrays are the most commonly used data structure for sorting and searching algorithms. Quick sort, merge sort, binary search, and others are primarily performed on arrays.</li>
-<li><strong>Lookup tables</strong>: When we need to quickly find an element or its corresponding relationship, we can use an array as a lookup table. For example, if we want to implement a mapping from characters to ASCII codes, we can use the ASCII code value of a character as an index, with the corresponding element stored at that position in the array.</li>
-<li><strong>Machine learning</strong>: Neural networks make extensive use of linear algebra operations between vectors, matrices, and tensors, all of which are constructed in the form of arrays. Arrays are the most commonly used data structure in neural network programming.</li>
-<li><strong>Data structure implementation</strong>: Arrays can be used to implement stacks, queues, hash tables, heaps, graphs, and other data structures. For example, the adjacency matrix representation of a graph is essentially a two-dimensional array.</li>
+<li><strong>Random access</strong>: To draw random samples, store data in an array and generate random indices to select elements.</li>
+<li><strong>Sorting and searching</strong>: Arrays are the most common structures for these algorithms. Quick sort, merge sort, and binary search operate primarily on arrays.</li>
+<li><strong>Lookup tables</strong>: An array can provide fast access to an element or a mapping. For example, a character-to-ASCII mapping can use a character's ASCII value as an index and store its corresponding element there.</li>
+<li><strong>Machine learning</strong>: Neural networks make extensive use of linear algebra on vectors, matrices, and tensors, all built from arrays. Arrays are the most common data structure in neural network programming.</li>
+<li><strong>Implementing data structures</strong>: Arrays can implement stacks, queues, hash tables, heaps, graphs, and more. A graph's adjacency matrix, for example, is a two-dimensional array.</li>
 </ul>
 
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     href="../"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: Chapter 4. &amp;nbsp; Arrays and Linked Lists"
+    aria-label="Previous: Chapter 4.   Arrays and linked lists"
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+        Chapter 4. &nbsp; Arrays and linked lists
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   <a
     href="../linked_list/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 4.2 Linked List"
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+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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-  0.1 About This Book
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-  0.2 How to Use This Book
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-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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-  2.1 Algorithm Efficiency Evaluation
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+        4.6.1 &nbsp; Check your understanding
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-  <a href="#1-how-arrays-and-linked-lists-find-an-element" class="md-nav__link">
+  <a href="#1-finding-elements-in-arrays-and-linked-lists" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; How Arrays and Linked Lists Find an Element
+        1. &nbsp; Finding elements in arrays and linked lists
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 </li>
 
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-  <a href="#2-how-arrays-and-linked-lists-insert-an-element" class="md-nav__link">
+  <a href="#2-inserting-elements-into-arrays-and-linked-lists" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; How Arrays and Linked Lists Insert an Element
+        2. &nbsp; Inserting elements into arrays and linked lists
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 </li>
 
           <li class="md-nav__item">
-  <a href="#3-how-a-lists-capacity-grows" class="md-nav__link">
+  <a href="#3-how-does-a-lists-capacity-grow" class="md-nav__link">
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       <span class="md-typeset">
-        3. &nbsp; How a List's Capacity Grows
+        3. &nbsp; How does a list's capacity grow?
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   <a href="#462-programming-exercises" class="md-nav__link">
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       <span class="md-typeset">
-        4.6.2 &nbsp; Programming Exercises
+        4.6.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="4.6.2 Programming Exercises">
+    <nav class="md-nav" aria-label="4.6.2 Programming exercises">
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           <li class="md-nav__item">
-  <a href="#1-add-one-to-a-large-integer-stored-as-an-array" class="md-nav__link">
+  <a href="#1-adding-one-to-a-large-integer-stored-in-an-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Add One to a Large Integer Stored as an Array
+        1. &nbsp; Adding one to a large integer stored in an array
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 </li>
 
           <li class="md-nav__item">
-  <a href="#2-reverse-a-singly-linked-list" class="md-nav__link">
+  <a href="#2-reversing-a-singly-linked-list" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Reverse a Singly Linked List
+        2. &nbsp; Reversing a singly linked list
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     </span>
   </a>
@@ -1602,7 +1602,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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@@ -1644,7 +1644,7 @@
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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@@ -2076,7 +2076,7 @@
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-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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@@ -2558,7 +2558,7 @@
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-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -3016,7 +3016,7 @@
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
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@@ -3058,7 +3058,7 @@
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
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   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4208,7 +4208,7 @@
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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@@ -4250,7 +4250,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
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   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4625,6 +4625,78 @@
 
 
 
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+        4.6.1 &nbsp; Check your understanding
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       <span class="md-typeset">
-        1. &nbsp; How Arrays and Linked Lists Find an Element
+        1. &nbsp; Finding elements in arrays and linked lists
       </span>
     </span>
   </a>
@@ -4689,10 +4761,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-how-arrays-and-linked-lists-insert-an-element" class="md-nav__link">
+  <a href="#2-inserting-elements-into-arrays-and-linked-lists" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; How Arrays and Linked Lists Insert an Element
+        2. &nbsp; Inserting elements into arrays and linked lists
       </span>
     </span>
   </a>
@@ -4700,10 +4772,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-how-a-lists-capacity-grows" class="md-nav__link">
+  <a href="#3-how-does-a-lists-capacity-grow" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; How a List's Capacity Grows
+        3. &nbsp; How does a list's capacity grow?
       </span>
     </span>
   </a>
@@ -4719,19 +4791,19 @@
   <a href="#462-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4.6.2 &nbsp; Programming Exercises
+        4.6.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="4.6.2 Programming Exercises">
+    <nav class="md-nav" aria-label="4.6.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-add-one-to-a-large-integer-stored-as-an-array" class="md-nav__link">
+  <a href="#1-adding-one-to-a-large-integer-stored-in-an-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Add One to a Large Integer Stored as an Array
+        1. &nbsp; Adding one to a large integer stored in an array
       </span>
     </span>
   </a>
@@ -4739,10 +4811,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-reverse-a-singly-linked-list" class="md-nav__link">
+  <a href="#2-reversing-a-singly-linked-list" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Reverse a Singly Linked List
+        2. &nbsp; Reversing a singly linked list
       </span>
     </span>
   </a>
@@ -4792,117 +4864,117 @@
 
 <!-- Page content -->
 <h1 id="46-exercises">4.6 &nbsp; Exercises<a class="headerlink" href="#46-exercises" title="Permanent link">&para;</a></h1>
-<h2 id="461-concept-review">4.6.1 &nbsp; Concept Review<a class="headerlink" href="#461-concept-review" title="Permanent link">&para;</a></h2>
-<h3 id="1-how-arrays-and-linked-lists-find-an-element">1. &nbsp; How Arrays and Linked Lists Find an Element<a class="headerlink" href="#1-how-arrays-and-linked-lists-find-an-element" title="Permanent link">&para;</a></h3>
-<p>An array and a singly linked list both store <code>[A, B, C, D, E]</code> in order. You now need to access the fourth element, <code>D</code>:</p>
+<h2 id="461-check-your-understanding">4.6.1 &nbsp; Check your understanding<a class="headerlink" href="#461-check-your-understanding" title="Permanent link">&para;</a></h2>
+<h3 id="1-finding-elements-in-arrays-and-linked-lists">1. &nbsp; Finding elements in arrays and linked lists<a class="headerlink" href="#1-finding-elements-in-arrays-and-linked-lists" title="Permanent link">&para;</a></h3>
+<p>An array and a singly linked list both store <code>[A, B, C, D, E]</code> in order. We want to read the fourth element, <code>D</code>:</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Which index can the array use directly?</li>
-<li>Starting from the head node <code>A</code>, which nodes are visited in order as you follow <code>next</code>?</li>
-<li>As the requested element moves farther toward the end, how does the number of steps change for each structure? Which structure is better for repeated access by position, and why?</li>
+<li>Which index can we use to access it directly in the array?</li>
+<li>Starting at the head node <code>A</code>, which nodes must the singly linked list visit by following <code>next</code>?</li>
+<li>As the requested element moves farther toward the end, how does the number of steps change for each structure? Which is better suited to repeated access by position, and why?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>With zero-based indexing, the fourth element has index 3, so the array can access <code>arr[3]</code> directly.</p>
+<p>With zero-based indexing, the fourth element has index 3, so the array can read <code>arr[3]</code> directly.</p>
 </li>
 <li>
-<p>The singly linked list must start at the head. Its access path is <code>A → B → C → D</code>, requiring three moves through <code>next</code>.</p>
+<p>The singly linked list must start at the head and follow the path <code>A → B → C → D</code>, advancing along <code>next</code> three times.</p>
 </li>
 <li>
-<p>An array can locate an element directly from its starting address and index, so access by position has a time complexity of <span class="arithmatex">\(O(1)\)</span>.
-    To access the <span class="arithmatex">\(k\)</span>th node, a singly linked list must start at the head node and follow <code>next</code> <span class="arithmatex">\(k-1\)</span> times,
-    which takes <span class="arithmatex">\(O(n)\)</span> time in the worst case.</p>
-<p>This comparison concerns only access by position; it does not mean that linked lists are slower for every operation.</p>
+<p>An array can locate an element directly from its starting address and the index, giving time complexity <span class="arithmatex">\(O(1)\)</span> for access by position.
+    To access the <span class="arithmatex">\(k\)</span>-th node of a singly linked list, we must start at the head and follow <code>next</code> <span class="arithmatex">\(k-1\)</span> times,
+    taking <span class="arithmatex">\(O(n)\)</span> time in the worst case.</p>
+<p>This comparison concerns access by position only; it does not mean that linked lists are slower for every operation.</p>
 </li>
 </ol>
 </details>
-<h3 id="2-how-arrays-and-linked-lists-insert-an-element">2. &nbsp; How Arrays and Linked Lists Insert an Element<a class="headerlink" href="#2-how-arrays-and-linked-lists-insert-an-element" title="Permanent link">&para;</a></h3>
-<p>An array and a singly linked list both store <code>A, B, C, D</code>. You now need to insert <code>X</code> after <code>B</code>:</p>
+<h3 id="2-inserting-elements-into-arrays-and-linked-lists">2. &nbsp; Inserting elements into arrays and linked lists<a class="headerlink" href="#2-inserting-elements-into-arrays-and-linked-lists" title="Permanent link">&para;</a></h3>
+<p>An array and a singly linked list both store <code>A, B, C, D</code>. We want to insert <code>X</code> after <code>B</code>:</p>
 <ul>
-<li>The array has capacity 5 and is currently <code>[A, B, C, D, _]</code>.</li>
-<li>The linked list is <code>A → B → C → D</code>, and you already have a reference to node <code>B</code>.</li>
+<li>The array has capacity 5 and currently contains <code>[A, B, C, D, _]</code>.</li>
+<li>The linked list is <code>A → B → C → D</code>, and we already have a reference to node <code>B</code>.</li>
 </ul>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Which elements must the array move? Write the array after insertion.</li>
-<li>In what order should the linked list update <code>X.next</code> and <code>B.next</code>? Write the linked list after insertion.</li>
-<li>When comparing insertion efficiency, why is it important to state that "you already have a reference to node B"?</li>
+<li>Which elements must the array move? Write out the array after insertion.</li>
+<li>In what order should the linked list update <code>X.next</code> and <code>B.next</code>? Write out the resulting list.</li>
+<li>Why must a comparison of insertion efficiency specify that we already have a reference to node B?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>The array first moves <code>D</code> one position to the right, then moves <code>C</code> one position to the right, and finally places <code>X</code> at index 2,
-    producing <code>[A, B, X, C, D]</code>.</p>
+<p>First move <code>D</code> one position to the right, then move <code>C</code> one position to the right, and finally put <code>X</code> at index 2.
+    The result is <code>[A, B, X, C, D]</code>.</p>
 </li>
 <li>
-<p><code>B.next</code> originally points to <code>C</code>. First set <code>X.next = B.next</code>, making <code>X</code> point to <code>C</code>;
-    then set <code>B.next = X</code>. The result is <code>A → B → X → C → D</code>.
-    If <code>B.next</code> is overwritten first without saving the original link, <code>C</code> may become unreachable.</p>
+<p>Initially, <code>B.next</code> points to <code>C</code>. First set <code>X.next = B.next</code>, making <code>X</code> point to <code>C</code>.
+    Then set <code>B.next = X</code>. The result is <code>A → B → X → C → D</code>.
+    If we overwrite <code>B.next</code> first without saving the original link, we may lose access to <code>C</code>.</p>
 </li>
 <li>
-<p>Once the location of <code>B</code> is known, insertion into the linked list only changes two links and takes <span class="arithmatex">\(O(1)\)</span> time.
-    If <code>B</code> must first be found from the head, that search alone may take <span class="arithmatex">\(O(n)\)</span> time.</p>
+<p>Once the location of <code>B</code> is known, insertion into the linked list requires changing only two links and takes <span class="arithmatex">\(O(1)\)</span> time.
+    If we first need to search for <code>B</code> from the head, that search itself may take <span class="arithmatex">\(O(n)\)</span> time.</p>
 </li>
 </ol>
 </details>
-<h3 id="3-how-a-lists-capacity-grows">3. &nbsp; How a List's Capacity Grows<a class="headerlink" href="#3-how-a-lists-capacity-grows" title="Permanent link">&para;</a></h3>
+<h3 id="3-how-does-a-lists-capacity-grow">3. &nbsp; How does a list's capacity grow?<a class="headerlink" href="#3-how-does-a-lists-capacity-grow" title="Permanent link">&para;</a></h3>
 <p>An array-based list currently contains <code>[A, B, C]</code>, with length <code>size = 3</code> and capacity <code>capacity = 4</code>.
-When there is not enough capacity, a new array with twice the old capacity is created.</p>
+When capacity runs out, the new array is allocated with twice the previous capacity.</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>After appending <code>D</code>, what are the list's length and capacity? Is capacity expansion needed?</li>
+<li>After appending <code>D</code>, what are the list's length and capacity? Does it need to grow?</li>
 <li>When <code>E</code> is appended next, what does the capacity become? How many existing elements must be copied?</li>
-<li>The length of the underlying array cannot change. Why does the list's capacity nevertheless appear able to grow?</li>
+<li>If the length of the underlying array cannot change, why does the list's capacity appear to grow?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p><code>D</code> fits in the last empty position. The contents are now <code>[A, B, C, D]</code>,
-    with <code>size = 4</code> and <code>capacity = 4</code>, so no capacity expansion is needed.</p>
+<p><code>D</code> fits in the last empty position. The contents become <code>[A, B, C, D]</code>,
+    with <code>size = 4</code> and <code>capacity = 4</code>. No capacity increase is needed.</p>
 </li>
 <li>
-<p>There is no empty position when <code>E</code> is appended, so a new array with capacity 8 must be created.
-    The four existing elements <code>A, B, C, D</code> are copied into it before <code>E</code> is added.
-    The result is <code>size = 5</code> and <code>capacity = 8</code>.</p>
+<p>Appending <code>E</code> finds no empty position, so a new array of capacity 8 is created.
+    The four existing elements <code>A, B, C, D</code> are copied into it, and then <code>E</code> is added.
+    The list now has <code>size = 5</code> and <code>capacity = 8</code>.</p>
 </li>
 <li>
-<p>The original array itself does not grow. The list creates a larger new array, copies the existing elements,
-    and then uses the new array as its underlying storage. From the user's perspective, the capacity has grown.</p>
+<p>The original array itself does not become longer. The list creates a larger array, copies the existing elements,
+    and switches to using the new array for storage. From the user's perspective, its capacity has grown.</p>
 </li>
 </ol>
 </details>
-<h2 id="462-programming-exercises">4.6.2 &nbsp; Programming Exercises<a class="headerlink" href="#462-programming-exercises" title="Permanent link">&para;</a></h2>
-<h3 id="1-add-one-to-a-large-integer-stored-as-an-array">1. &nbsp; Add One to a Large Integer Stored as an Array<a class="headerlink" href="#1-add-one-to-a-large-integer-stored-as-an-array" title="Permanent link">&para;</a></h3>
-<p>The array <code>digits</code> stores the decimal digits of a non-negative integer from left to right. For example, <code>[3, 0, 8]</code> represents 308.
-The number 0 is represented by <code>[0]</code>; for every other input, the first digit is not 0.</p>
-<p>Simulate decimal column addition to add 1 to this integer, and return the result in the same array format.
-You may modify <code>digits</code> directly. If a new carry appears at the front, you may return a longer array.</p>
+<h2 id="462-programming-exercises">4.6.2 &nbsp; Programming exercises<a class="headerlink" href="#462-programming-exercises" title="Permanent link">&para;</a></h2>
+<h3 id="1-adding-one-to-a-large-integer-stored-in-an-array">1. &nbsp; Adding one to a large integer stored in an array<a class="headerlink" href="#1-adding-one-to-a-large-integer-stored-in-an-array" title="Permanent link">&para;</a></h3>
+<p>The array <code>digits</code> stores the digits of a nonnegative integer from left to right. For example, <code>[3, 0, 8]</code> represents 308.
+Zero is represented by <code>[0]</code>; all other inputs have a nonzero first digit.</p>
+<p>Simulate the usual digit-by-digit method for decimal addition to add 1 to this integer, and return the result in the same array format.
+You may modify <code>digits</code> directly. If a carry creates a new leading digit, you may return a longer array.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>Start with the last digit of the array, just as in ordinary column addition</li>
-<li>If the current digit is less than 9, add one and return immediately</li>
-<li>If the current digit is 9, change it to 0; if every digit is 9, place a 1 at the front</li>
+<li>Start at the last digit of the array, just as you would when adding by hand.</li>
+<li>If the current digit is less than 9, increment it and return immediately.</li>
+<li>If the current digit is 9, change it to 0. If all digits are 9, add a leading 1.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/plus-one/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
-<h3 id="2-reverse-a-singly-linked-list">2. &nbsp; Reverse a Singly Linked List<a class="headerlink" href="#2-reverse-a-singly-linked-list" title="Permanent link">&para;</a></h3>
+<h3 id="2-reversing-a-singly-linked-list">2. &nbsp; Reversing a singly linked list<a class="headerlink" href="#2-reversing-a-singly-linked-list" title="Permanent link">&para;</a></h3>
 <p>You are given the head node <code>head</code> of a singly linked list. Each node contains a value and a <code>next</code> reference to the following node.</p>
-<p>Use an iterative approach to reverse all links between the nodes, and return the new head node.
-Do not create any new linked list nodes.</p>
+<p>Use iteration to reverse every link between nodes, and return the new head.
+Do not create any new linked-list nodes.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>First draw three connected nodes and the two pointers prev and cur on paper</li>
-<li>Before changing cur.next, save the original next node in nxt</li>
-<li>After reversing cur.next, set prev = cur and then cur = nxt to continue with the next node in the original list</li>
+<li>First draw three connected nodes and the two pointers prev and cur on paper.</li>
+<li>Before changing cur.next, save the original next node in nxt.</li>
+<li>After reversing cur.next, set prev = cur and then cur = nxt to continue with the next node in the original list.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/reverse-linked-list/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
@@ -4928,7 +5000,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 4.5 Summary"
+    aria-label="Previous: 4.5   Summary"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4940,7 +5012,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        4.5 Summary
+        4.5 &nbsp; Summary
       </div>
     </div>
   </a>
@@ -4952,7 +5024,7 @@ aria-label="Footer"
   <a
     href="../../chapter_stack_and_queue/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: Chapter 5. &amp;nbsp; Stacks and Queues"
+    aria-label="Next: Chapter 5.   Stacks and queues"
     rel="next"
   >
     <div class="md-footer__title">
@@ -4960,7 +5032,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        Chapter 5. &nbsp; Stacks and Queues
+        Chapter 5. &nbsp; Stacks and queues
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -4977,7 +5049,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_array_and_linkedlist/index.html b/en/chapter_array_and_linkedlist/index.html
index 82f5b7423..01ff0e1e6 100644
--- a/en/chapter_array_and_linkedlist/index.html
+++ b/en/chapter_array_and_linkedlist/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>Chapter 4. Arrays and Linked Lists - Hello Algo</title>
+        <title>Chapter 4. Arrays and linked lists - Hello Algo</title>
 
 
 
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+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
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@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
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+              Chapter 4. &nbsp; Arrays and linked lists
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
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   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
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-  Before Starting
+  Foreword
 
           </label>
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
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   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
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-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
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-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
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-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
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-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
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-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
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   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2598,7 +2598,7 @@
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
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   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2694,7 +2694,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2721,7 +2721,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2748,7 +2748,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2775,7 +2775,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2880,7 +2880,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
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@@ -2922,7 +2922,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3057,7 +3057,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3111,7 +3111,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3138,7 +3138,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3165,7 +3165,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3192,7 +3192,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3285,7 +3285,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3306,7 +3306,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3327,7 +3327,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3381,7 +3381,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3408,7 +3408,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3435,7 +3435,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3462,7 +3462,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3528,7 +3528,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
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@@ -3570,7 +3570,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3597,7 +3597,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3624,7 +3624,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3651,7 +3651,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3678,7 +3678,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3705,7 +3705,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3775,7 +3775,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3817,7 +3817,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3844,7 +3844,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3871,7 +3871,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3898,7 +3898,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -3952,7 +3952,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -3979,7 +3979,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4006,7 +4006,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4072,7 +4072,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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@@ -4114,7 +4114,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
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 <!-- Page content -->
-<h1 id="chapter-4-arrays-and-linked-lists">Chapter 4. &nbsp; Arrays and Linked Lists<a class="headerlink" href="#chapter-4-arrays-and-linked-lists" title="Permanent link">&para;</a></h1>
-<p><img alt="Arrays and Linked Lists" class="cover-image" src="../assets/covers/chapter_array_and_linkedlist.jpg" /></p>
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+<p><img alt="Arrays and linked lists" class="cover-image" src="../assets/covers/chapter_array_and_linkedlist.jpg" /></p>
 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
 <p>The world of data structures is like a solid brick wall.</p>
-<p>The bricks of an array are neatly aligned, each pressed tightly against the next. The bricks of a linked list are scattered about, with connecting vines weaving freely through the gaps between them.</p>
+<p>In an array, the bricks line up neatly, each right beside the next. In a linked list, the bricks are scattered, with connecting vines weaving freely through the gaps.</p>
 </div>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
-<li><a href="array/">4.1 &nbsp; Array</a></li>
-<li><a href="linked_list/">4.2 &nbsp; Linked List</a></li>
-<li><a href="list/">4.3 &nbsp; List</a></li>
-<li><a href="ram_and_cache/">4.4 &nbsp; Random-Access Memory and Cache *</a></li>
+<li><a href="array/">4.1 &nbsp; Arrays</a></li>
+<li><a href="linked_list/">4.2 &nbsp; Linked lists</a></li>
+<li><a href="list/">4.3 &nbsp; Lists</a></li>
+<li><a href="ram_and_cache/">4.4 &nbsp; Memory and cache *</a></li>
 <li><a href="summary/">4.5 &nbsp; Summary</a></li>
 <li><a href="exercises/">4.6 &nbsp; Exercises</a></li>
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     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 3.6 Exercises"
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@@ -4627,7 +4699,7 @@ aria-label="Footer"
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+        3.6 &nbsp; Exercises
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   <a
     href="./array/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 4.1 Array"
+    aria-label="Next: 4.1   Arrays"
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+        4.1 &nbsp; Arrays
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
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-        <title>4.2 Linked List - Hello Algo</title>
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1660,7 +1660,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1832,7 +1832,7 @@
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1874,7 +1874,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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@@ -1901,7 +1901,7 @@
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1955,7 +1955,7 @@
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1982,7 +1982,7 @@
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2092,7 +2092,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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@@ -2119,7 +2119,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2146,7 +2146,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2173,7 +2173,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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@@ -2200,7 +2200,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2227,7 +2227,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2360,7 +2360,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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@@ -2441,7 +2441,7 @@
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2532,7 +2532,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2574,7 +2574,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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@@ -2601,7 +2601,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2750,7 +2750,7 @@
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2771,7 +2771,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2792,7 +2792,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2819,7 +2819,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2846,7 +2846,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2873,7 +2873,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2900,7 +2900,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2927,7 +2927,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2954,7 +2954,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -3032,7 +3032,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3053,7 +3053,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3074,7 +3074,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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@@ -3101,7 +3101,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3128,7 +3128,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3155,7 +3155,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3182,7 +3182,7 @@
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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@@ -3209,7 +3209,7 @@
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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@@ -3236,7 +3236,7 @@
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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@@ -3263,7 +3263,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3290,7 +3290,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3317,7 +3317,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3344,7 +3344,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3371,7 +3371,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3437,7 +3437,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3458,7 +3458,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3479,7 +3479,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3506,7 +3506,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3533,7 +3533,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3560,7 +3560,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3587,7 +3587,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3680,7 +3680,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3722,7 +3722,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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@@ -3803,7 +3803,7 @@
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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@@ -3830,7 +3830,7 @@
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3927,7 +3927,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3948,7 +3948,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3969,7 +3969,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -4023,7 +4023,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4050,7 +4050,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4104,7 +4104,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4131,7 +4131,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4158,7 +4158,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4224,7 +4224,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4245,7 +4245,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4266,7 +4266,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4293,7 +4293,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4347,7 +4347,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4374,7 +4374,7 @@
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4461,7 +4461,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4503,7 +4503,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4530,7 +4530,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4557,7 +4557,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4641,6 +4641,78 @@
 
 
 
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@@ -4685,19 +4757,19 @@
   <a href="#421-common-linked-list-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4.2.1 &nbsp; Common Linked List Operations
+        4.2.1 &nbsp; Common linked list operations
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="4.2.1 Common Linked List Operations">
+    <nav class="md-nav" aria-label="4.2.1 Common linked list operations">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-initializing-a-linked-list" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Initializing a Linked List
+        1. &nbsp; Initializing a linked list
       </span>
     </span>
   </a>
@@ -4708,7 +4780,7 @@
   <a href="#2-inserting-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Inserting a Node
+        2. &nbsp; Inserting a node
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   </a>
@@ -4719,7 +4791,7 @@
   <a href="#3-removing-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Removing a Node
+        3. &nbsp; Removing a node
       </span>
     </span>
   </a>
@@ -4730,7 +4802,7 @@
   <a href="#4-accessing-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4. &nbsp; Accessing a Node
+        4. &nbsp; Accessing a node
       </span>
     </span>
   </a>
@@ -4741,7 +4813,7 @@
   <a href="#5-finding-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5. &nbsp; Finding a Node
+        5. &nbsp; Finding a node
       </span>
     </span>
   </a>
@@ -4754,10 +4826,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#422-arrays-vs-linked-lists" class="md-nav__link">
+  <a href="#422-arrays-versus-linked-lists" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4.2.2 &nbsp; Arrays vs. Linked Lists
+        4.2.2 &nbsp; Arrays versus linked lists
       </span>
     </span>
   </a>
@@ -4768,7 +4840,7 @@
   <a href="#423-common-types-of-linked-lists" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4.2.3 &nbsp; Common Types of Linked Lists
+        4.2.3 &nbsp; Common types of linked lists
       </span>
     </span>
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   <a href="#424-typical-applications-of-linked-lists" class="md-nav__link">
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       <span class="md-typeset">
-        4.2.4 &nbsp; Typical Applications of Linked Lists
+        4.2.4 &nbsp; Typical applications of linked lists
       </span>
     </span>
   </a>
@@ -4823,20 +4895,20 @@
 
 
 <!-- Page content -->
-<h1 id="42-linked-list">4.2 &nbsp; Linked List<a class="headerlink" href="#42-linked-list" title="Permanent link">&para;</a></h1>
-<p>Memory is a shared resource for all programs. In a complex runtime environment, free memory may be scattered throughout the address space. We know that arrays require contiguous memory, and when an array is very large, the system may not be able to provide such a large contiguous block. This is where the flexibility of linked lists becomes apparent.</p>
-<p>A <u>linked list</u> is a linear data structure in which each element is a node object, and the nodes are connected through "references". A reference records the memory address of the next node, through which the next node can be accessed from the current node.</p>
-<p>This design allows linked-list nodes to be stored in different locations in memory, and their addresses do not need to be contiguous.</p>
-<p><img alt="Linked list definition and storage method" class="animation-figure" src="../linked_list.assets/linkedlist_definition.png" /></p>
-<p align="center"> Figure 4-5 &nbsp; Linked list definition and storage method </p>
+<h1 id="42-linked-lists">4.2 &nbsp; Linked lists<a class="headerlink" href="#42-linked-lists" title="Permanent link">&para;</a></h1>
+<p>Memory is a resource used by all programs, and in a complex system, free space may be scattered throughout it. An array requires contiguous storage, which may be unavailable for a very large array. This is where a linked list's flexibility helps.</p>
+<p>A <u>linked list</u> is a linear data structure whose elements are node objects connected by references. Each reference stores the next node's memory address, allowing us to reach it from the current node.</p>
+<p>This design lets nodes occupy scattered memory locations; their addresses need not be contiguous.</p>
+<p><img alt="Linked list definition and storage layout" class="animation-figure" src="../linked_list.assets/linkedlist_definition.png" /></p>
+<p align="center"> Figure 4-5 &nbsp; Linked list definition and storage layout </p>
 
-<p>Observing Figure 4-5, the basic unit of a linked list is a <u>node</u> object. Each node contains two pieces of data: the node's "value" and a "reference" to the next node.</p>
+<p>As shown above, the basic unit of a linked list is a <u>node</u> object. Each node holds a value and a reference to the next node.</p>
 <ul>
-<li>The first node of a linked list is called the "head node", and the last node is called the "tail node".</li>
-<li>The tail node points to "null", which is denoted as <code>null</code>, <code>nullptr</code>, and <code>None</code> in Java, C++, and Python, respectively.</li>
-<li>In languages that support pointers, such as C, C++, Go, and Rust, the aforementioned "reference" should be replaced with "pointer".</li>
+<li>The first node is called the head, and the last is called the tail.</li>
+<li>The tail points to a null value: <code>null</code> in Java, <code>nullptr</code> in C++, and <code>None</code> in Python.</li>
+<li>In languages supporting pointers, such as C, C++, Go, and Rust, these references are expressed as pointers.</li>
 </ul>
-<p>As shown in the following code, a linked list node <code>ListNode</code> contains not only a value but also an additional reference (pointer). Therefore, <strong>linked lists occupy more memory space than arrays when storing the same amount of data</strong>.</p>
+<p>As the code below shows, each <code>ListNode</code> stores a reference (pointer) in addition to its value. Thus, <strong>a linked list uses more memory than an array for the same amount of data</strong>.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -4880,7 +4952,7 @@
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">Next</span><span class="w"> </span><span class="o">*</span><span class="nx">ListNode</span><span class="w"> </span><span class="c1">// Pointer to the next node</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="p">}</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="c1">// NewListNode Constructor, creates a new linked list</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="c1">// NewListNode constructor: create a new linked list</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="kd">func</span><span class="w"> </span><span class="nx">NewListNode</span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">ListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">ListNode</span><span class="p">{</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">        </span><span class="nx">Val</span><span class="p">:</span><span class="w">  </span><span class="nx">val</span><span class="p">,</span>
@@ -4984,20 +5056,20 @@
 </div>
 </div>
 </div>
-<h2 id="421-common-linked-list-operations">4.2.1 &nbsp; Common Linked List Operations<a class="headerlink" href="#421-common-linked-list-operations" title="Permanent link">&para;</a></h2>
-<h3 id="1-initializing-a-linked-list">1. &nbsp; Initializing a Linked List<a class="headerlink" href="#1-initializing-a-linked-list" title="Permanent link">&para;</a></h3>
-<p>Building a linked list involves two steps: first, initializing each node object; second, constructing the reference relationships between nodes. Once initialization is complete, we can traverse all nodes starting from the head node of the linked list through the reference <code>next</code>.</p>
+<h2 id="421-common-linked-list-operations">4.2.1 &nbsp; Common linked list operations<a class="headerlink" href="#421-common-linked-list-operations" title="Permanent link">&para;</a></h2>
+<h3 id="1-initializing-a-linked-list">1. &nbsp; Initializing a linked list<a class="headerlink" href="#1-initializing-a-linked-list" title="Permanent link">&para;</a></h3>
+<p>Building a linked list takes two steps: initialize the node objects, then connect them with references. Afterward, start at the head and follow <code>next</code> references to visit every node in order.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="c1"># Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="c1"># Initialize each node</span>
+<div class="highlight"><span class="filename">linked_list.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="c1"># Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="c1"># Initialize the nodes</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="n">n0</span> <span class="o">=</span> <span class="n">ListNode</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="n">n1</span> <span class="o">=</span> <span class="n">ListNode</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="n">n2</span> <span class="o">=</span> <span class="n">ListNode</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a><span class="n">n3</span> <span class="o">=</span> <span class="n">ListNode</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a><span class="n">n4</span> <span class="o">=</span> <span class="n">ListNode</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a><span class="c1"># Build references between nodes</span>
+<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a><span class="c1"># Link the nodes with references</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a><span class="n">n0</span><span class="o">.</span><span class="n">next</span> <span class="o">=</span> <span class="n">n1</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a><span class="n">n1</span><span class="o">.</span><span class="n">next</span> <span class="o">=</span> <span class="n">n2</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a><span class="n">n2</span><span class="o">.</span><span class="n">next</span> <span class="o">=</span> <span class="n">n3</span>
@@ -5005,14 +5077,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
-<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="c1">// Initialize each node</span>
+<div class="highlight"><span class="filename">linked_list.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
+<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="c1">// Initialize the nodes</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
-<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="c1">// Build references between nodes</span>
+<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="c1">// Link the nodes with references</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="n">n0</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n1</span><span class="p">;</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="n">n1</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="n">n2</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n3</span><span class="p">;</span>
@@ -5020,14 +5092,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
-<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="c1">// Initialize each node</span>
+<div class="highlight"><span class="filename">linked_list.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
+<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="c1">// Initialize the nodes</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
-<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="c1">// Build references between nodes</span>
+<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="c1">// Link the nodes with references</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="n">n0</span><span class="p">.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n1</span><span class="p">;</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="n">n1</span><span class="p">.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="n">n2</span><span class="p">.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n3</span><span class="p">;</span>
@@ -5035,14 +5107,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
-<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="c1">// Initialize each node</span>
+<div class="highlight"><span class="filename">linked_list.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
+<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="c1">// Initialize the nodes</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
-<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="c1">// Build references between nodes</span>
+<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="c1">// Link the nodes with references</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="n">n0</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n1</span><span class="p">;</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="n">n1</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="n">n2</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n3</span><span class="p">;</span>
@@ -5050,14 +5122,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
-<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="c1">// Initialize each node</span>
+<div class="highlight"><span class="filename">linked_list.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
+<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="c1">// Initialize the nodes</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="nx">n0</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewListNode</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="nx">n1</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewListNode</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="nx">n2</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewListNode</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="nx">n3</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewListNode</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="nx">n4</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewListNode</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="c1">// Build references between nodes</span>
+<a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="c1">// Link the nodes with references</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="nx">n0</span><span class="p">.</span><span class="nx">Next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n1</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">Next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="nx">n2</span><span class="p">.</span><span class="nx">Next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n3</span>
@@ -5065,14 +5137,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
-<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="c1">// Initialize each node</span>
+<div class="highlight"><span class="filename">linked_list.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
+<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="c1">// Initialize the nodes</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="kd">let</span><span class="w"> </span><span class="nv">n0</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="kd">let</span><span class="w"> </span><span class="nv">n1</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="kd">let</span><span class="w"> </span><span class="nv">n2</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="kd">let</span><span class="w"> </span><span class="nv">n3</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="kd">let</span><span class="w"> </span><span class="nv">n4</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="mi">4</span><span class="p">)</span>
-<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="c1">// Build references between nodes</span>
+<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="c1">// Link the nodes with references</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="n">n0</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">n1</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="n">n1</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">n2</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="n">n2</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">n3</span>
@@ -5080,14 +5152,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
-<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="c1">// Initialize each node</span>
+<div class="highlight"><span class="filename">linked_list.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
+<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="c1">// Initialize the nodes</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="kd">const</span><span class="w"> </span><span class="nx">n0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="kd">const</span><span class="w"> </span><span class="nx">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="mf">3</span><span class="p">);</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="kd">const</span><span class="w"> </span><span class="nx">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="mf">2</span><span class="p">);</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="kd">const</span><span class="w"> </span><span class="nx">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="mf">5</span><span class="p">);</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="kd">const</span><span class="w"> </span><span class="nx">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="mf">4</span><span class="p">);</span>
-<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="c1">// Build references between nodes</span>
+<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="c1">// Link the nodes with references</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="nx">n0</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n1</span><span class="p">;</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span><span class="p">;</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="nx">n2</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n3</span><span class="p">;</span>
@@ -5095,14 +5167,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
-<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="c1">// Initialize each node</span>
+<div class="highlight"><span class="filename">linked_list.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
+<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="c1">// Initialize the nodes</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="kd">const</span><span class="w"> </span><span class="nx">n0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="kd">const</span><span class="w"> </span><span class="nx">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="mf">3</span><span class="p">);</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="kd">const</span><span class="w"> </span><span class="nx">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="mf">2</span><span class="p">);</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="kd">const</span><span class="w"> </span><span class="nx">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="mf">5</span><span class="p">);</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="kd">const</span><span class="w"> </span><span class="nx">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="mf">4</span><span class="p">);</span>
-<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="c1">// Build references between nodes</span>
+<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="c1">// Link the nodes with references</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="nx">n0</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n1</span><span class="p">;</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span><span class="p">;</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="nx">n2</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n3</span><span class="p">;</span>
@@ -5110,14 +5182,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span><span class="err">\</span>
-<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="c1">// Initialize each node</span>
+<div class="highlight"><span class="filename">linked_list.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span><span class="err">\</span>
+<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="c1">// Initialize the nodes</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">3</span><span class="p">);</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">2</span><span class="p">);</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">5</span><span class="p">);</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">4</span><span class="p">);</span>
-<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="c1">// Build references between nodes</span>
+<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="c1">// Link the nodes with references</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="n">n0</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n1</span><span class="p">;</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="n">n1</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="n">n2</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n3</span><span class="p">;</span>
@@ -5125,15 +5197,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
-<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="c1">// Initialize each node</span>
+<div class="highlight"><span class="filename">linked_list.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
+<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="c1">// Initialize the nodes</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="kd">let</span><span class="w"> </span><span class="n">n0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Rc</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">RefCell</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">ListNode</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">next</span><span class="p">:</span><span class="w"> </span><span class="nb">None</span><span class="w"> </span><span class="p">}));</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="kd">let</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Rc</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">RefCell</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">ListNode</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">next</span><span class="p">:</span><span class="w"> </span><span class="nb">None</span><span class="w"> </span><span class="p">}));</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="kd">let</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Rc</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">RefCell</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">ListNode</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="n">next</span><span class="p">:</span><span class="w"> </span><span class="nb">None</span><span class="w"> </span><span class="p">}));</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="kd">let</span><span class="w"> </span><span class="n">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Rc</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">RefCell</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">ListNode</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="n">next</span><span class="p">:</span><span class="w"> </span><span class="nb">None</span><span class="w"> </span><span class="p">}));</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="kd">let</span><span class="w"> </span><span class="n">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Rc</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">RefCell</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">ListNode</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="n">next</span><span class="p">:</span><span class="w"> </span><span class="nb">None</span><span class="w"> </span><span class="p">}));</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a>
-<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="c1">// Build references between nodes</span>
+<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="c1">// Link the nodes with references</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="n">n0</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">n1</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="n">n1</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">n2</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="n">n2</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">n3</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
@@ -5141,14 +5213,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
-<a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="c1">// Initialize each node</span>
+<div class="highlight"><span class="filename">linked_list.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
+<a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="c1">// Initialize the nodes</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newListNode</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newListNode</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newListNode</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newListNode</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="n">ListNode</span><span class="o">*</span><span class="w"> </span><span class="n">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newListNode</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
-<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="c1">// Build references between nodes</span>
+<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="c1">// Link the nodes with references</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="n">n0</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n1</span><span class="p">;</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="n">n1</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="n">n2</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n3</span><span class="p">;</span>
@@ -5156,14 +5228,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
-<a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="c1">// Initialize each node</span>
+<div class="highlight"><span class="filename">linked_list.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */</span>
+<a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="c1">// Initialize the nodes</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="kd">val</span><span class="w"> </span><span class="nv">n0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="kd">val</span><span class="w"> </span><span class="nv">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">3</span><span class="p">)</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="kd">val</span><span class="w"> </span><span class="nv">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">2</span><span class="p">)</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="kd">val</span><span class="w"> </span><span class="nv">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">5</span><span class="p">)</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="kd">val</span><span class="w"> </span><span class="nv">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">4</span><span class="p">)</span>
-<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="c1">// Build references between nodes</span>
+<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="c1">// Link the nodes with references</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="n">n0</span><span class="p">.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n1</span><span class="p">;</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="n">n1</span><span class="p">.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="n">n2</span><span class="p">.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n3</span><span class="p">;</span>
@@ -5171,14 +5243,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1"># Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4</span>
-<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="c1"># Initialize each node</span>
+<div class="highlight"><span class="filename">linked_list.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1"># Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4</span>
+<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="c1"># Initialize the nodes</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="n">n0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">ListNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">ListNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">ListNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="n">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">ListNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="n">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">ListNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="c1"># Build references between nodes</span>
+<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="c1"># Link the nodes with references</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="n">n0</span><span class="o">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n1</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="n">n1</span><span class="o">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="n">n2</span><span class="o">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n3</span>
@@ -5188,16 +5260,16 @@
 </div>
 </div>
 <details class="pythontutor">
-<summary>Code Visualization</summary>
-<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E9%93%BE%E8%A1%A8%201%20-%3E%203%20-%3E%202%20-%3E%205%20-%3E%204%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%90%84%E4%B8%AA%E8%8A%82%E7%82%B9%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E9%93%BE%E8%A1%A8%201%20-%3E%203%20-%3E%202%20-%3E%205%20-%3E%204%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%90%84%E4%B8%AA%E8%8A%82%E7%82%B9%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20successor%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20linked%20list%201%20-%3E%203%20-%3E%202%20-%3E%205%20-%3E%204%0A%20%20%20%20%23%20Initialize%20the%20nodes%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20Link%20the%20nodes%20with%20references%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20successor%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20linked%20list%201%20-%3E%203%20-%3E%202%20-%3E%205%20-%3E%204%0A%20%20%20%20%23%20Initialize%20the%20nodes%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20Link%20the%20nodes%20with%20references%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<p>An array is a single variable; for example, an array <code>nums</code> contains elements <code>nums[0]</code>, <code>nums[1]</code>, and so on. A linked list, by contrast, is composed of multiple independent node objects. <strong>We usually use the head node as a stand-in for the entire linked list</strong>; for example, the linked list in the above code can be referred to as linked list <code>n0</code>.</p>
-<h3 id="2-inserting-a-node">2. &nbsp; Inserting a Node<a class="headerlink" href="#2-inserting-a-node" title="Permanent link">&para;</a></h3>
-<p>Inserting a node in a linked list is very easy. As shown in Figure 4-6, suppose we want to insert a new node <code>P</code> between two adjacent nodes <code>n0</code> and <code>n1</code>. <strong>We only need to change two node references (pointers)</strong>, with a time complexity of <span class="arithmatex">\(O(1)\)</span>.</p>
-<p>In contrast, the time complexity of inserting an element in an array is <span class="arithmatex">\(O(n)\)</span>, which is inefficient when dealing with large amounts of data.</p>
-<p><img alt="Example of inserting a node into a linked list" class="animation-figure" src="../linked_list.assets/linkedlist_insert_node.png" /></p>
-<p align="center"> Figure 4-6 &nbsp; Example of inserting a node into a linked list </p>
+<p>An array is a single variable, such as <code>nums</code>, containing entries <code>nums[0]</code>, <code>nums[1]</code>, and so on. A linked list consists of separate node objects. <strong>We usually refer to the list by its head node</strong>; for example, the list above can be called <code>n0</code>.</p>
+<h3 id="2-inserting-a-node">2. &nbsp; Inserting a node<a class="headerlink" href="#2-inserting-a-node" title="Permanent link">&para;</a></h3>
+<p>Inserting a node is straightforward. As shown in Figure 4-6, inserting a new node <code>P</code> between adjacent nodes <code>n0</code> and <code>n1</code> <strong>requires changing only two references (pointers)</strong>, taking <span class="arithmatex">\(O(1)\)</span> time.</p>
+<p>In contrast, inserting an array element takes <span class="arithmatex">\(O(n)\)</span> time and is slower for large datasets.</p>
+<p><img alt="Inserting a node into a linked list" class="animation-figure" src="../linked_list.assets/linkedlist_insert_node.png" /></p>
+<p align="center"> Figure 4-6 &nbsp; Inserting a node into a linked list </p>
 
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
@@ -5301,7 +5373,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Insert node P after node n0 in the linked list */</span>
+<div class="highlight"><span class="filename">linked_list.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Insert node p after node n0 in the linked list */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">n0</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n0</span><span class="o">?.</span><span class="na">next</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="n">p</span><span class="o">?.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n1</span>
@@ -5310,8 +5382,8 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Insert node _p after node n0 in linked list ###</span>
-<a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="c1"># Ruby's `p` is a built-in function, `P` is a constant, so use `_p` instead</span>
+<div class="highlight"><span class="filename">linked_list.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Insert node _p after node n0 in the linked list ###</span>
+<a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="c1"># In Ruby, `p` is a built-in method and `P` denotes a constant, so use `_p` instead</span>
 <a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="k">def</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">n0</span><span class="p">,</span><span class="w"> </span><span class="n">_p</span><span class="p">)</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n0</span><span class="o">.</span><span class="n">next</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">  </span><span class="n">_p</span><span class="o">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n1</span>
@@ -5321,11 +5393,16 @@
 </div>
 </div>
 </div>
-<h3 id="3-removing-a-node">3. &nbsp; Removing a Node<a class="headerlink" href="#3-removing-a-node" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 4-7, removing a node in a linked list is also very convenient. <strong>We only need to change one node's reference (pointer)</strong>.</p>
-<p>Note that although node <code>P</code> still points to <code>n1</code> after the deletion operation is complete, the linked list can no longer access <code>P</code> when traversing, which means <code>P</code> no longer belongs to this linked list.</p>
-<p><img alt="Removing a node from a linked list" class="animation-figure" src="../linked_list.assets/linkedlist_remove_node.png" /></p>
-<p align="center"> Figure 4-7 &nbsp; Removing a node from a linked list </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0Adef%20insert%28n0%3A%20ListNode,%20P%3A%20ListNode%29%3A%0A%20%20%20%20%22%22%22Insert%20node%20P%20after%20node%20n0%20in%20the%20linked%20list%22%22%22%0A%20%20%20%20n1%20%3D%20n0.next%0A%20%20%20%20P.next%20%3D%20n1%0A%20%20%20%20n0.next%20%3D%20P%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20linked%20list%0A%20%20%20%20%23%20Initialize%20the%20individual%20nodes%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20Set%20up%20references%20between%20the%20nodes%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20Insert%20a%20node%0A%20%20%20%20p%20%3D%20ListNode%280%29%0A%20%20%20%20insert%28n0,%20p%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=39&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0Adef%20insert%28n0%3A%20ListNode,%20P%3A%20ListNode%29%3A%0A%20%20%20%20%22%22%22Insert%20node%20P%20after%20node%20n0%20in%20the%20linked%20list%22%22%22%0A%20%20%20%20n1%20%3D%20n0.next%0A%20%20%20%20P.next%20%3D%20n1%0A%20%20%20%20n0.next%20%3D%20P%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20linked%20list%0A%20%20%20%20%23%20Initialize%20the%20individual%20nodes%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20Set%20up%20references%20between%20the%20nodes%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20Insert%20a%20node%0A%20%20%20%20p%20%3D%20ListNode%280%29%0A%20%20%20%20insert%28n0,%20p%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=39&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="3-removing-a-node">3. &nbsp; Removing a node<a class="headerlink" href="#3-removing-a-node" title="Permanent link">&para;</a></h3>
+<p>Removing a node is also straightforward: as shown in Figure 4-7, <strong>only one reference (pointer) needs to change</strong>.</p>
+<p>Although <code>P</code> still points to <code>n1</code> after removal, traversing the list can no longer reach <code>P</code>. It therefore no longer belongs to the list.</p>
+<p><img alt="Removing a linked list node" class="animation-figure" src="../linked_list.assets/linkedlist_remove_node.png" /></p>
+<p align="center"> Figure 4-7 &nbsp; Removing a linked list node </p>
 
 <div class="tabbed-set tabbed-alternate" data-tabs="4:13"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label></div>
 <div class="tabbed-content">
@@ -5454,7 +5531,7 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">linked_list.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Remove the first node after node n0 in the linked list */</span>
-<a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="c1">// Note: stdio.h occupies the remove keyword</span>
+<a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="c1">// Note: stdio.h already declares a function named remove</span>
 <a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="kt">void</span><span class="w"> </span><span class="nf">removeItem</span><span class="p">(</span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">n0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">n0</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">)</span>
 <a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
@@ -5480,7 +5557,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Delete first node after node n0 in linked list ###</span>
+<div class="highlight"><span class="filename">linked_list.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Remove the first node after node n0 in the linked list ###</span>
 <a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">n0</span><span class="p">)</span>
 <a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n0</span><span class="o">.</span><span class="n">next</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a>
@@ -5493,13 +5570,18 @@
 </div>
 </div>
 </div>
-<h3 id="4-accessing-a-node">4. &nbsp; Accessing a Node<a class="headerlink" href="#4-accessing-a-node" title="Permanent link">&para;</a></h3>
-<p><strong>Accessing nodes in a linked list is less efficient</strong>. As mentioned in the previous section, we can access any element in an array in <span class="arithmatex">\(O(1)\)</span> time. This is not the case with linked lists. The program needs to start from the head node and traverse backward one by one until the target node is found. That is, accessing the <span class="arithmatex">\(i\)</span>-th node in a linked list requires <span class="arithmatex">\(i - 1\)</span> iterations, with a time complexity of <span class="arithmatex">\(O(n)\)</span>.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0Adef%20remove%28n0%3A%20ListNode%29%3A%0A%20%20%20%20%22%22%22Remove%20the%20first%20node%20after%20node%20n0%20in%20the%20linked%20list%22%22%22%0A%20%20%20%20if%20not%20n0.next%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20n0%20-%3E%20P%20-%3E%20n1%0A%20%20%20%20P%20%3D%20n0.next%0A%20%20%20%20n1%20%3D%20P.next%0A%20%20%20%20n0.next%20%3D%20n1%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20linked%20list%0A%20%20%20%20%23%20Initialize%20the%20individual%20nodes%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20Set%20up%20references%20between%20the%20nodes%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20Remove%20a%20node%0A%20%20%20%20remove%28n0%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=34&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0Adef%20remove%28n0%3A%20ListNode%29%3A%0A%20%20%20%20%22%22%22Remove%20the%20first%20node%20after%20node%20n0%20in%20the%20linked%20list%22%22%22%0A%20%20%20%20if%20not%20n0.next%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20n0%20-%3E%20P%20-%3E%20n1%0A%20%20%20%20P%20%3D%20n0.next%0A%20%20%20%20n1%20%3D%20P.next%0A%20%20%20%20n0.next%20%3D%20n1%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20linked%20list%0A%20%20%20%20%23%20Initialize%20the%20individual%20nodes%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20Set%20up%20references%20between%20the%20nodes%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20Remove%20a%20node%0A%20%20%20%20remove%28n0%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=34&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="4-accessing-a-node">4. &nbsp; Accessing a node<a class="headerlink" href="#4-accessing-a-node" title="Permanent link">&para;</a></h3>
+<p><strong>Accessing a node in a linked list is relatively slow</strong>. An array provides access to any element in <span class="arithmatex">\(O(1)\)</span> time, but a linked list must be traversed from its head until the desired node is reached. Accessing the <span class="arithmatex">\(i\)</span>-th node requires <span class="arithmatex">\(i - 1\)</span> iterations, giving <span class="arithmatex">\(O(n)\)</span> time complexity.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">linked_list.py</span><pre><span></span><code><a id="__codelineno-52-1" name="__codelineno-52-1" href="#__codelineno-52-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">access</span><span class="p">(</span><span class="n">head</span><span class="p">:</span> <span class="n">ListNode</span><span class="p">,</span> <span class="n">index</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Access the node at index index in the linked list"""</span>
+<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Access the node at index `index` in the linked list"""</span>
 <a id="__codelineno-52-3" name="__codelineno-52-3" href="#__codelineno-52-3"></a>    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">index</span><span class="p">):</span>
 <a id="__codelineno-52-4" name="__codelineno-52-4" href="#__codelineno-52-4"></a>        <span class="k">if</span> <span class="ow">not</span> <span class="n">head</span><span class="p">:</span>
 <a id="__codelineno-52-5" name="__codelineno-52-5" href="#__codelineno-52-5"></a>            <span class="k">return</span> <span class="kc">None</span>
@@ -5508,7 +5590,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Access the node at index index in the linked list */</span>
+<div class="highlight"><span class="filename">linked_list.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Access the node at index `index` in the linked list */</span>
 <a id="__codelineno-53-2" name="__codelineno-53-2" href="#__codelineno-53-2"></a><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="nf">access</span><span class="p">(</span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">head</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-4" name="__codelineno-53-4" href="#__codelineno-53-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">head</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span>
@@ -5520,7 +5602,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Access the node at index index in the linked list */</span>
+<div class="highlight"><span class="filename">linked_list.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Access the node at index `index` in the linked list */</span>
 <a id="__codelineno-54-2" name="__codelineno-54-2" href="#__codelineno-54-2"></a><span class="n">ListNode</span><span class="w"> </span><span class="nf">access</span><span class="p">(</span><span class="n">ListNode</span><span class="w"> </span><span class="n">head</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-4" name="__codelineno-54-4" href="#__codelineno-54-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">head</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
@@ -5532,7 +5614,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Access the node at index index in the linked list */</span>
+<div class="highlight"><span class="filename">linked_list.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Access the node at index `index` in the linked list */</span>
 <a id="__codelineno-55-2" name="__codelineno-55-2" href="#__codelineno-55-2"></a><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">Access</span><span class="p">(</span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">head</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-4" name="__codelineno-55-4" href="#__codelineno-55-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">head</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span>
@@ -5544,7 +5626,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Access the node at index index in the linked list */</span>
+<div class="highlight"><span class="filename">linked_list.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Access the node at index `index` in the linked list */</span>
 <a id="__codelineno-56-2" name="__codelineno-56-2" href="#__codelineno-56-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">access</span><span class="p">(</span><span class="nx">head</span><span class="w"> </span><span class="o">*</span><span class="nx">ListNode</span><span class="p">,</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">ListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">head</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
@@ -5557,7 +5639,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Access the node at index index in the linked list */</span>
+<div class="highlight"><span class="filename">linked_list.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Access the node at index `index` in the linked list */</span>
 <a id="__codelineno-57-2" name="__codelineno-57-2" href="#__codelineno-57-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">access</span><span class="p">(</span><span class="n">head</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">head</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">head</span>
 <a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="p">{</span>
@@ -5571,7 +5653,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Access the node at index index in the linked list */</span>
+<div class="highlight"><span class="filename">linked_list.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Access the node at index `index` in the linked list */</span>
 <a id="__codelineno-58-2" name="__codelineno-58-2" href="#__codelineno-58-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">access</span><span class="p">(</span><span class="nx">head</span><span class="p">,</span><span class="w"> </span><span class="nx">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-4" name="__codelineno-58-4" href="#__codelineno-58-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="nx">head</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5584,7 +5666,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Access the node at index index in the linked list */</span>
+<div class="highlight"><span class="filename">linked_list.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Access the node at index `index` in the linked list */</span>
 <a id="__codelineno-59-2" name="__codelineno-59-2" href="#__codelineno-59-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">access</span><span class="p">(</span><span class="nx">head</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="nx">index</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="nx">ListNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-4" name="__codelineno-59-4" href="#__codelineno-59-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="nx">head</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5597,7 +5679,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Access the node at index index in the linked list */</span>
+<div class="highlight"><span class="filename">linked_list.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Access the node at index `index` in the linked list */</span>
 <a id="__codelineno-60-2" name="__codelineno-60-2" href="#__codelineno-60-2"></a><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">access</span><span class="p">(</span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">head</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-3" name="__codelineno-60-3" href="#__codelineno-60-3"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-4" name="__codelineno-60-4" href="#__codelineno-60-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">head</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
@@ -5608,7 +5690,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Access the node at index index in the linked list */</span>
+<div class="highlight"><span class="filename">linked_list.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Access the node at index `index` in the linked list */</span>
 <a id="__codelineno-61-2" name="__codelineno-61-2" href="#__codelineno-61-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">access</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="p">(</span><span class="n">head</span><span class="p">:</span><span class="w"> </span><span class="nc">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-3" name="__codelineno-61-3" href="#__codelineno-61-3"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">dfs</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="p">(</span>
 <a id="__codelineno-61-4" name="__codelineno-61-4" href="#__codelineno-61-4"></a><span class="w">        </span><span class="n">head</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span>
@@ -5630,7 +5712,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Access the node at index index in the linked list */</span>
+<div class="highlight"><span class="filename">linked_list.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Access the node at index `index` in the linked list */</span>
 <a id="__codelineno-62-2" name="__codelineno-62-2" href="#__codelineno-62-2"></a><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="nf">access</span><span class="p">(</span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">head</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-3" name="__codelineno-62-3" href="#__codelineno-62-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-4" name="__codelineno-62-4" href="#__codelineno-62-4"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">head</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span>
@@ -5642,7 +5724,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Access the node at index index in the linked list */</span>
+<div class="highlight"><span class="filename">linked_list.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Access the node at index `index` in the linked list */</span>
 <a id="__codelineno-63-2" name="__codelineno-63-2" href="#__codelineno-63-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">access</span><span class="p">(</span><span class="n">head</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="n">ListNode?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-3" name="__codelineno-63-3" href="#__codelineno-63-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">h</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">head</span>
 <a id="__codelineno-63-4" name="__codelineno-63-4" href="#__codelineno-63-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5655,7 +5737,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Access node at index in linked list ###</span>
+<div class="highlight"><span class="filename">linked_list.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Access the node at index `index` in the linked list ###</span>
 <a id="__codelineno-64-2" name="__codelineno-64-2" href="#__codelineno-64-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">access</span><span class="p">(</span><span class="n">head</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="p">)</span>
 <a id="__codelineno-64-3" name="__codelineno-64-3" href="#__codelineno-64-3"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">index</span>
 <a id="__codelineno-64-4" name="__codelineno-64-4" href="#__codelineno-64-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="kp">nil</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">head</span><span class="o">.</span><span class="n">nil?</span>
@@ -5668,8 +5750,13 @@
 </div>
 </div>
 </div>
-<h3 id="5-finding-a-node">5. &nbsp; Finding a Node<a class="headerlink" href="#5-finding-a-node" title="Permanent link">&para;</a></h3>
-<p>Traverse the linked list to find a node with value <code>target</code>, and output the index of that node in the linked list. This process is also a linear search. The code is shown below:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0Adef%20access%28head%3A%20ListNode,%20index%3A%20int%29%20-%3E%20ListNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Access%20the%20node%20at%20index%20%60index%60%20in%20the%20linked%20list%22%22%22%0A%20%20%20%20for%20_%20in%20range%28index%29%3A%0A%20%20%20%20%20%20%20%20if%20not%20head%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%20%20%20%20head%20%3D%20head.next%0A%20%20%20%20return%20head%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20linked%20list%0A%20%20%20%20%23%20Initialize%20the%20individual%20nodes%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20Set%20up%20references%20between%20the%20nodes%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20Access%20a%20node%0A%20%20%20%20node%20%3D%20access%28n0,%203%29%0A%20%20%20%20print%28%22Value%20of%20the%20node%20at%20index%203%20in%20the%20linked%20list%20%3D%20%7B%7D%22.format%28node.val%29%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=34&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0Adef%20access%28head%3A%20ListNode,%20index%3A%20int%29%20-%3E%20ListNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Access%20the%20node%20at%20index%20%60index%60%20in%20the%20linked%20list%22%22%22%0A%20%20%20%20for%20_%20in%20range%28index%29%3A%0A%20%20%20%20%20%20%20%20if%20not%20head%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%20%20%20%20head%20%3D%20head.next%0A%20%20%20%20return%20head%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20linked%20list%0A%20%20%20%20%23%20Initialize%20the%20individual%20nodes%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20Set%20up%20references%20between%20the%20nodes%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20Access%20a%20node%0A%20%20%20%20node%20%3D%20access%28n0,%203%29%0A%20%20%20%20print%28%22Value%20of%20the%20node%20at%20index%203%20in%20the%20linked%20list%20%3D%20%7B%7D%22.format%28node.val%29%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=34&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="5-finding-a-node">5. &nbsp; Finding a node<a class="headerlink" href="#5-finding-a-node" title="Permanent link">&para;</a></h3>
+<p>Traverse the list to find a node with value <code>target</code>, then return its index. This is also linear search. The code is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="6:13"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -5850,7 +5937,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linked_list.rb</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="c1">### Find first node with value target in linked list ###</span>
+<div class="highlight"><span class="filename">linked_list.rb</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="c1">### Find the first node with value target in the linked list ###</span>
 <a id="__codelineno-77-2" name="__codelineno-77-2" href="#__codelineno-77-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">find</span><span class="p">(</span><span class="n">head</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
 <a id="__codelineno-77-3" name="__codelineno-77-3" href="#__codelineno-77-3"></a><span class="w">  </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-77-4" name="__codelineno-77-4" href="#__codelineno-77-4"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="n">head</span>
@@ -5865,9 +5952,14 @@
 </div>
 </div>
 </div>
-<h2 id="422-arrays-vs-linked-lists">4.2.2 &nbsp; Arrays vs. Linked Lists<a class="headerlink" href="#422-arrays-vs-linked-lists" title="Permanent link">&para;</a></h2>
-<p>Table 4-1 summarizes the characteristics of arrays and linked lists and compares their operational efficiencies. Since they employ two opposite storage strategies, their various properties and operational efficiencies also exhibit contrasting characteristics.</p>
-<p align="center"> Table 4-1 &nbsp; Comparison of array and linked list efficiencies </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0Adef%20find%28head%3A%20ListNode,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20the%20first%20node%20with%20value%20target%20in%20the%20linked%20list%22%22%22%0A%20%20%20%20index%20%3D%200%0A%20%20%20%20while%20head%3A%0A%20%20%20%20%20%20%20%20if%20head.val%20%3D%3D%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20index%0A%20%20%20%20%20%20%20%20head%20%3D%20head.next%0A%20%20%20%20%20%20%20%20index%20%2B%3D%201%0A%20%20%20%20return%20-1%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20linked%20list%0A%20%20%20%20%23%20Initialize%20the%20individual%20nodes%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20Set%20up%20references%20between%20the%20nodes%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20Find%20a%20node%0A%20%20%20%20index%20%3D%20find%28n0,%202%29%0A%20%20%20%20print%28%22Index%20of%20the%20node%20with%20value%202%20in%20the%20linked%20list%20%3D%20%7B%7D%22.format%28index%29%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=34&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0Adef%20find%28head%3A%20ListNode,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20the%20first%20node%20with%20value%20target%20in%20the%20linked%20list%22%22%22%0A%20%20%20%20index%20%3D%200%0A%20%20%20%20while%20head%3A%0A%20%20%20%20%20%20%20%20if%20head.val%20%3D%3D%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20index%0A%20%20%20%20%20%20%20%20head%20%3D%20head.next%0A%20%20%20%20%20%20%20%20index%20%2B%3D%201%0A%20%20%20%20return%20-1%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20linked%20list%0A%20%20%20%20%23%20Initialize%20the%20individual%20nodes%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20Set%20up%20references%20between%20the%20nodes%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20Find%20a%20node%0A%20%20%20%20index%20%3D%20find%28n0,%202%29%0A%20%20%20%20print%28%22Index%20of%20the%20node%20with%20value%202%20in%20the%20linked%20list%20%3D%20%7B%7D%22.format%28index%29%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=34&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="422-arrays-versus-linked-lists">4.2.2 &nbsp; Arrays versus linked lists<a class="headerlink" href="#422-arrays-versus-linked-lists" title="Permanent link">&para;</a></h2>
+<p>Table 4-1 compares their properties and operation costs. Their contrasting storage strategies give them contrasting strengths.</p>
+<p align="center"> Table 4-1 &nbsp; Comparing array and linked list efficiency </p>
 
 <div class="center-table">
 <table>
@@ -5875,49 +5967,49 @@
 <tr>
 <th></th>
 <th>Array</th>
-<th>Linked List</th>
+<th>Linked list</th>
 </tr>
 </thead>
 <tbody>
 <tr>
-<td>Storage method</td>
-<td>Contiguous memory space</td>
-<td>Scattered memory space</td>
+<td>Storage</td>
+<td>Contiguous memory</td>
+<td>Scattered memory</td>
 </tr>
 <tr>
-<td>Capacity expansion</td>
-<td>Immutable length</td>
+<td>Expansion</td>
+<td>Fixed length</td>
 <td>Flexible expansion</td>
 </tr>
 <tr>
 <td>Memory efficiency</td>
-<td>Elements occupy less memory, but space may be wasted</td>
-<td>Elements occupy more memory</td>
+<td>Less memory per element, but may waste space</td>
+<td>More memory per element</td>
 </tr>
 <tr>
-<td>Accessing an element</td>
+<td>Access an element</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 <td><span class="arithmatex">\(O(n)\)</span></td>
 </tr>
 <tr>
-<td>Adding an element</td>
+<td>Add an element</td>
 <td><span class="arithmatex">\(O(n)\)</span></td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
-<td>Removing an element</td>
+<td>Remove an element</td>
 <td><span class="arithmatex">\(O(n)\)</span></td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 </tbody>
 </table>
 </div>
-<h2 id="423-common-types-of-linked-lists">4.2.3 &nbsp; Common Types of Linked Lists<a class="headerlink" href="#423-common-types-of-linked-lists" title="Permanent link">&para;</a></h2>
-<p>As shown in Figure 4-8, there are three common types of linked lists:</p>
+<h2 id="423-common-types-of-linked-lists">4.2.3 &nbsp; Common types of linked lists<a class="headerlink" href="#423-common-types-of-linked-lists" title="Permanent link">&para;</a></h2>
+<p>Figure 4-8 shows three common types.</p>
 <ul>
-<li><strong>Singly linked list</strong>: This is the ordinary linked list introduced earlier. The nodes of a singly linked list contain a value and a reference to the next node. We call the first node the head node and the last node the tail node; the tail node points to <code>None</code>.</li>
-<li><strong>Circular linked list</strong>: If we make the tail node of a singly linked list point to the head node (connecting the tail to the head), we get a circular linked list. In a circular linked list, any node can be viewed as the head node.</li>
-<li><strong>Doubly linked list</strong>: Compared to a singly linked list, a doubly linked list records references in both directions. The node definition of a doubly linked list includes references to both the successor node (next node) and the predecessor node (previous node). Compared to a singly linked list, a doubly linked list is more flexible and can traverse the linked list in both directions, but it also requires more memory space.</li>
+<li><strong>Singly linked list</strong>: The ordinary linked list introduced above. Each node stores a value and a reference to the next node. The first and last nodes are the head and tail, and the tail points to <code>None</code>.</li>
+<li><strong>Circular linked list</strong>: Connecting a singly linked list's tail to its head forms a circular list. Any node can then be treated as the head.</li>
+<li><strong>Doubly linked list</strong>: Each node stores references (pointers) in both directions, to its successor and predecessor. This makes it more flexible and allows traversal in either direction, at the cost of additional memory.</li>
 </ul>
 <div class="tabbed-set tabbed-alternate" data-tabs="7:13"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><input id="__tabbed_7_13" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Python</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Java</label><label for="__tabbed_7_4">C#</label><label for="__tabbed_7_5">Go</label><label for="__tabbed_7_6">Swift</label><label for="__tabbed_7_7">JS</label><label for="__tabbed_7_8">TS</label><label for="__tabbed_7_9">Dart</label><label for="__tabbed_7_10">Rust</label><label for="__tabbed_7_11">C</label><label for="__tabbed_7_12">Kotlin</label><label for="__tabbed_7_13">Ruby</label></div>
 <div class="tabbed-content">
@@ -5926,16 +6018,16 @@
 <a id="__codelineno-78-2" name="__codelineno-78-2" href="#__codelineno-78-2"></a><span class="w">    </span><span class="sd">"""Doubly linked list node class"""</span>
 <a id="__codelineno-78-3" name="__codelineno-78-3" href="#__codelineno-78-3"></a>    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-78-4" name="__codelineno-78-4" href="#__codelineno-78-4"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">val</span>                <span class="c1"># Node value</span>
-<a id="__codelineno-78-5" name="__codelineno-78-5" href="#__codelineno-78-5"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span>  <span class="c1"># Reference to the successor node</span>
-<a id="__codelineno-78-6" name="__codelineno-78-6" href="#__codelineno-78-6"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">prev</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span>  <span class="c1"># Reference to the predecessor node</span>
+<a id="__codelineno-78-5" name="__codelineno-78-5" href="#__codelineno-78-5"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span>  <span class="c1"># Reference to the successor</span>
+<a id="__codelineno-78-6" name="__codelineno-78-6" href="#__codelineno-78-6"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">prev</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span>  <span class="c1"># Reference to the predecessor</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-79-1" name="__codelineno-79-1" href="#__codelineno-79-1"></a><span class="cm">/* Doubly linked list node structure */</span>
 <a id="__codelineno-79-2" name="__codelineno-79-2" href="#__codelineno-79-2"></a><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-3" name="__codelineno-79-3" href="#__codelineno-79-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w">         </span><span class="c1">// Node value</span>
-<a id="__codelineno-79-4" name="__codelineno-79-4" href="#__codelineno-79-4"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w">  </span><span class="c1">// Pointer to the successor node</span>
-<a id="__codelineno-79-5" name="__codelineno-79-5" href="#__codelineno-79-5"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">prev</span><span class="p">;</span><span class="w">  </span><span class="c1">// Pointer to the predecessor node</span>
+<a id="__codelineno-79-4" name="__codelineno-79-4" href="#__codelineno-79-4"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w">  </span><span class="c1">// Pointer to the successor</span>
+<a id="__codelineno-79-5" name="__codelineno-79-5" href="#__codelineno-79-5"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">prev</span><span class="p">;</span><span class="w">  </span><span class="c1">// Pointer to the predecessor</span>
 <a id="__codelineno-79-6" name="__codelineno-79-6" href="#__codelineno-79-6"></a><span class="w">    </span><span class="n">ListNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">val</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="w"> </span><span class="n">next</span><span class="p">(</span><span class="nb">nullptr</span><span class="p">),</span><span class="w"> </span><span class="n">prev</span><span class="p">(</span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{}</span><span class="w">  </span><span class="c1">// Constructor</span>
 <a id="__codelineno-79-7" name="__codelineno-79-7" href="#__codelineno-79-7"></a><span class="p">};</span>
 </code></pre></div>
@@ -5944,8 +6036,8 @@
 <div class="highlight"><pre><span></span><code><a id="__codelineno-80-1" name="__codelineno-80-1" href="#__codelineno-80-1"></a><span class="cm">/* Doubly linked list node class */</span>
 <a id="__codelineno-80-2" name="__codelineno-80-2" href="#__codelineno-80-2"></a><span class="kd">class</span> <span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-3" name="__codelineno-80-3" href="#__codelineno-80-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w">        </span><span class="c1">// Node value</span>
-<a id="__codelineno-80-4" name="__codelineno-80-4" href="#__codelineno-80-4"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the successor node</span>
-<a id="__codelineno-80-5" name="__codelineno-80-5" href="#__codelineno-80-5"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the predecessor node</span>
+<a id="__codelineno-80-4" name="__codelineno-80-4" href="#__codelineno-80-4"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the successor</span>
+<a id="__codelineno-80-5" name="__codelineno-80-5" href="#__codelineno-80-5"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the predecessor</span>
 <a id="__codelineno-80-6" name="__codelineno-80-6" href="#__codelineno-80-6"></a><span class="w">    </span><span class="n">ListNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"> </span><span class="p">}</span><span class="w">  </span><span class="c1">// Constructor</span>
 <a id="__codelineno-80-7" name="__codelineno-80-7" href="#__codelineno-80-7"></a><span class="p">}</span>
 </code></pre></div>
@@ -5954,8 +6046,8 @@
 <div class="highlight"><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="cm">/* Doubly linked list node class */</span>
 <a id="__codelineno-81-2" name="__codelineno-81-2" href="#__codelineno-81-2"></a><span class="k">class</span><span class="w"> </span><span class="nf">ListNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w">  </span><span class="c1">// Constructor</span>
 <a id="__codelineno-81-3" name="__codelineno-81-3" href="#__codelineno-81-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w">    </span><span class="c1">// Node value</span>
-<a id="__codelineno-81-4" name="__codelineno-81-4" href="#__codelineno-81-4"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the successor node</span>
-<a id="__codelineno-81-5" name="__codelineno-81-5" href="#__codelineno-81-5"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the predecessor node</span>
+<a id="__codelineno-81-4" name="__codelineno-81-4" href="#__codelineno-81-4"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the successor</span>
+<a id="__codelineno-81-5" name="__codelineno-81-5" href="#__codelineno-81-5"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the predecessor</span>
 <a id="__codelineno-81-6" name="__codelineno-81-6" href="#__codelineno-81-6"></a><span class="p">}</span>
 </code></pre></div>
 </div>
@@ -5963,11 +6055,11 @@
 <div class="highlight"><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="cm">/* Doubly linked list node structure */</span>
 <a id="__codelineno-82-2" name="__codelineno-82-2" href="#__codelineno-82-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">DoublyListNode</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-3" name="__codelineno-82-3" href="#__codelineno-82-3"></a><span class="w">    </span><span class="nx">Val</span><span class="w">  </span><span class="kt">int</span><span class="w">             </span><span class="c1">// Node value</span>
-<a id="__codelineno-82-4" name="__codelineno-82-4" href="#__codelineno-82-4"></a><span class="w">    </span><span class="nx">Next</span><span class="w"> </span><span class="o">*</span><span class="nx">DoublyListNode</span><span class="w"> </span><span class="c1">// Pointer to the successor node</span>
-<a id="__codelineno-82-5" name="__codelineno-82-5" href="#__codelineno-82-5"></a><span class="w">    </span><span class="nx">Prev</span><span class="w"> </span><span class="o">*</span><span class="nx">DoublyListNode</span><span class="w"> </span><span class="c1">// Pointer to the predecessor node</span>
+<a id="__codelineno-82-4" name="__codelineno-82-4" href="#__codelineno-82-4"></a><span class="w">    </span><span class="nx">Next</span><span class="w"> </span><span class="o">*</span><span class="nx">DoublyListNode</span><span class="w"> </span><span class="c1">// Pointer to the successor</span>
+<a id="__codelineno-82-5" name="__codelineno-82-5" href="#__codelineno-82-5"></a><span class="w">    </span><span class="nx">Prev</span><span class="w"> </span><span class="o">*</span><span class="nx">DoublyListNode</span><span class="w"> </span><span class="c1">// Pointer to the predecessor</span>
 <a id="__codelineno-82-6" name="__codelineno-82-6" href="#__codelineno-82-6"></a><span class="p">}</span>
 <a id="__codelineno-82-7" name="__codelineno-82-7" href="#__codelineno-82-7"></a>
-<a id="__codelineno-82-8" name="__codelineno-82-8" href="#__codelineno-82-8"></a><span class="c1">// NewDoublyListNode Initialization</span>
+<a id="__codelineno-82-8" name="__codelineno-82-8" href="#__codelineno-82-8"></a><span class="c1">// Initialize a node with NewDoublyListNode</span>
 <a id="__codelineno-82-9" name="__codelineno-82-9" href="#__codelineno-82-9"></a><span class="kd">func</span><span class="w"> </span><span class="nx">NewDoublyListNode</span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">DoublyListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-10" name="__codelineno-82-10" href="#__codelineno-82-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">DoublyListNode</span><span class="p">{</span>
 <a id="__codelineno-82-11" name="__codelineno-82-11" href="#__codelineno-82-11"></a><span class="w">        </span><span class="nx">Val</span><span class="p">:</span><span class="w">  </span><span class="nx">val</span><span class="p">,</span>
@@ -5981,8 +6073,8 @@
 <div class="highlight"><pre><span></span><code><a id="__codelineno-83-1" name="__codelineno-83-1" href="#__codelineno-83-1"></a><span class="cm">/* Doubly linked list node class */</span>
 <a id="__codelineno-83-2" name="__codelineno-83-2" href="#__codelineno-83-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-3" name="__codelineno-83-3" href="#__codelineno-83-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">val</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Node value</span>
-<a id="__codelineno-83-4" name="__codelineno-83-4" href="#__codelineno-83-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">next</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Reference to the successor node</span>
-<a id="__codelineno-83-5" name="__codelineno-83-5" href="#__codelineno-83-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">prev</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Reference to the predecessor node</span>
+<a id="__codelineno-83-4" name="__codelineno-83-4" href="#__codelineno-83-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">next</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Reference to the successor</span>
+<a id="__codelineno-83-5" name="__codelineno-83-5" href="#__codelineno-83-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">prev</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Reference to the predecessor</span>
 <a id="__codelineno-83-6" name="__codelineno-83-6" href="#__codelineno-83-6"></a>
 <a id="__codelineno-83-7" name="__codelineno-83-7" href="#__codelineno-83-7"></a><span class="w">    </span><span class="kd">init</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// Constructor</span>
 <a id="__codelineno-83-8" name="__codelineno-83-8" href="#__codelineno-83-8"></a><span class="w">        </span><span class="n">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">x</span>
@@ -5995,8 +6087,8 @@
 <a id="__codelineno-84-2" name="__codelineno-84-2" href="#__codelineno-84-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">ListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-3" name="__codelineno-84-3" href="#__codelineno-84-3"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">next</span><span class="p">,</span><span class="w"> </span><span class="nx">prev</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-4" name="__codelineno-84-4" href="#__codelineno-84-4"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span><span class="w">  </span><span class="o">===</span><span class="w">  </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">val</span><span class="p">;</span><span class="w">        </span><span class="c1">// Node value</span>
-<a id="__codelineno-84-5" name="__codelineno-84-5" href="#__codelineno-84-5"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">next</span><span class="w">  </span><span class="o">===</span><span class="w">  </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">next</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the successor node</span>
-<a id="__codelineno-84-6" name="__codelineno-84-6" href="#__codelineno-84-6"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">prev</span><span class="w">  </span><span class="o">===</span><span class="w">  </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">prev</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the predecessor node</span>
+<a id="__codelineno-84-5" name="__codelineno-84-5" href="#__codelineno-84-5"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">next</span><span class="w">  </span><span class="o">===</span><span class="w">  </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">next</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the successor</span>
+<a id="__codelineno-84-6" name="__codelineno-84-6" href="#__codelineno-84-6"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">prev</span><span class="w">  </span><span class="o">===</span><span class="w">  </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">prev</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the predecessor</span>
 <a id="__codelineno-84-7" name="__codelineno-84-7" href="#__codelineno-84-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-84-8" name="__codelineno-84-8" href="#__codelineno-84-8"></a><span class="p">}</span>
 </code></pre></div>
@@ -6009,8 +6101,8 @@
 <a id="__codelineno-85-5" name="__codelineno-85-5" href="#__codelineno-85-5"></a><span class="w">    </span><span class="nx">prev</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-85-6" name="__codelineno-85-6" href="#__codelineno-85-6"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">val?</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">next?</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="nx">prev?</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-7" name="__codelineno-85-7" href="#__codelineno-85-7"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span><span class="w">  </span><span class="o">===</span><span class="w">  </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">0</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">val</span><span class="p">;</span><span class="w">        </span><span class="c1">// Node value</span>
-<a id="__codelineno-85-8" name="__codelineno-85-8" href="#__codelineno-85-8"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">next</span><span class="w">  </span><span class="o">===</span><span class="w">  </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">next</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the successor node</span>
-<a id="__codelineno-85-9" name="__codelineno-85-9" href="#__codelineno-85-9"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">prev</span><span class="w">  </span><span class="o">===</span><span class="w">  </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">prev</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the predecessor node</span>
+<a id="__codelineno-85-8" name="__codelineno-85-8" href="#__codelineno-85-8"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">next</span><span class="w">  </span><span class="o">===</span><span class="w">  </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">next</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the successor</span>
+<a id="__codelineno-85-9" name="__codelineno-85-9" href="#__codelineno-85-9"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">prev</span><span class="w">  </span><span class="o">===</span><span class="w">  </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">prev</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the predecessor</span>
 <a id="__codelineno-85-10" name="__codelineno-85-10" href="#__codelineno-85-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-85-11" name="__codelineno-85-11" href="#__codelineno-85-11"></a><span class="p">}</span>
 </code></pre></div>
@@ -6019,8 +6111,8 @@
 <div class="highlight"><pre><span></span><code><a id="__codelineno-86-1" name="__codelineno-86-1" href="#__codelineno-86-1"></a><span class="cm">/* Doubly linked list node class */</span>
 <a id="__codelineno-86-2" name="__codelineno-86-2" href="#__codelineno-86-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-3" name="__codelineno-86-3" href="#__codelineno-86-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w">        </span><span class="c1">// Node value</span>
-<a id="__codelineno-86-4" name="__codelineno-86-4" href="#__codelineno-86-4"></a><span class="w">    </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the successor node</span>
-<a id="__codelineno-86-5" name="__codelineno-86-5" href="#__codelineno-86-5"></a><span class="w">    </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the predecessor node</span>
+<a id="__codelineno-86-4" name="__codelineno-86-4" href="#__codelineno-86-4"></a><span class="w">    </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the successor</span>
+<a id="__codelineno-86-5" name="__codelineno-86-5" href="#__codelineno-86-5"></a><span class="w">    </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to the predecessor</span>
 <a id="__codelineno-86-6" name="__codelineno-86-6" href="#__codelineno-86-6"></a><span class="w">    </span><span class="n">ListNode</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="n">next</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">prev</span><span class="p">]);</span><span class="w">  </span><span class="c1">// Constructor</span>
 <a id="__codelineno-86-7" name="__codelineno-86-7" href="#__codelineno-86-7"></a><span class="p">}</span>
 </code></pre></div>
@@ -6033,8 +6125,8 @@
 <a id="__codelineno-87-5" name="__codelineno-87-5" href="#__codelineno-87-5"></a><span class="cp">#[derive(Debug)]</span>
 <a id="__codelineno-87-6" name="__codelineno-87-6" href="#__codelineno-87-6"></a><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-7" name="__codelineno-87-7" href="#__codelineno-87-7"></a><span class="w">    </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="c1">// Node value</span>
-<a id="__codelineno-87-8" name="__codelineno-87-8" href="#__codelineno-87-8"></a><span class="w">    </span><span class="n">next</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="c1">// Pointer to the successor node</span>
-<a id="__codelineno-87-9" name="__codelineno-87-9" href="#__codelineno-87-9"></a><span class="w">    </span><span class="n">prev</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="c1">// Pointer to the predecessor node</span>
+<a id="__codelineno-87-8" name="__codelineno-87-8" href="#__codelineno-87-8"></a><span class="w">    </span><span class="n">next</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="c1">// Pointer to the successor</span>
+<a id="__codelineno-87-9" name="__codelineno-87-9" href="#__codelineno-87-9"></a><span class="w">    </span><span class="n">prev</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="c1">// Pointer to the predecessor</span>
 <a id="__codelineno-87-10" name="__codelineno-87-10" href="#__codelineno-87-10"></a><span class="p">}</span>
 <a id="__codelineno-87-11" name="__codelineno-87-11" href="#__codelineno-87-11"></a>
 <a id="__codelineno-87-12" name="__codelineno-87-12" href="#__codelineno-87-12"></a><span class="cm">/* Constructor */</span>
@@ -6053,8 +6145,8 @@
 <div class="highlight"><pre><span></span><code><a id="__codelineno-88-1" name="__codelineno-88-1" href="#__codelineno-88-1"></a><span class="cm">/* Doubly linked list node structure */</span>
 <a id="__codelineno-88-2" name="__codelineno-88-2" href="#__codelineno-88-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-3" name="__codelineno-88-3" href="#__codelineno-88-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w">               </span><span class="c1">// Node value</span>
-<a id="__codelineno-88-4" name="__codelineno-88-4" href="#__codelineno-88-4"></a><span class="w">    </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Pointer to the successor node</span>
-<a id="__codelineno-88-5" name="__codelineno-88-5" href="#__codelineno-88-5"></a><span class="w">    </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Pointer to the predecessor node</span>
+<a id="__codelineno-88-4" name="__codelineno-88-4" href="#__codelineno-88-4"></a><span class="w">    </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Pointer to the successor</span>
+<a id="__codelineno-88-5" name="__codelineno-88-5" href="#__codelineno-88-5"></a><span class="w">    </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Pointer to the predecessor</span>
 <a id="__codelineno-88-6" name="__codelineno-88-6" href="#__codelineno-88-6"></a><span class="p">}</span><span class="w"> </span><span class="n">ListNode</span><span class="p">;</span>
 <a id="__codelineno-88-7" name="__codelineno-88-7" href="#__codelineno-88-7"></a>
 <a id="__codelineno-88-8" name="__codelineno-88-8" href="#__codelineno-88-8"></a><span class="cm">/* Constructor */</span>
@@ -6073,8 +6165,8 @@
 <a id="__codelineno-89-2" name="__codelineno-89-2" href="#__codelineno-89-2"></a><span class="c1">// Constructor</span>
 <a id="__codelineno-89-3" name="__codelineno-89-3" href="#__codelineno-89-3"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ListNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-89-4" name="__codelineno-89-4" href="#__codelineno-89-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="w">           </span><span class="c1">// Node value</span>
-<a id="__codelineno-89-5" name="__codelineno-89-5" href="#__codelineno-89-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">next</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="w">  </span><span class="c1">// Reference to the successor node</span>
-<a id="__codelineno-89-6" name="__codelineno-89-6" href="#__codelineno-89-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">prev</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="w">  </span><span class="c1">// Reference to the predecessor node</span>
+<a id="__codelineno-89-5" name="__codelineno-89-5" href="#__codelineno-89-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">next</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="w">  </span><span class="c1">// Reference to the successor</span>
+<a id="__codelineno-89-6" name="__codelineno-89-6" href="#__codelineno-89-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">prev</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="w">  </span><span class="c1">// Reference to the predecessor</span>
 <a id="__codelineno-89-7" name="__codelineno-89-7" href="#__codelineno-89-7"></a><span class="p">}</span>
 </code></pre></div>
 </div>
@@ -6082,8 +6174,8 @@
 <div class="highlight"><pre><span></span><code><a id="__codelineno-90-1" name="__codelineno-90-1" href="#__codelineno-90-1"></a><span class="c1"># Doubly linked list node class</span>
 <a id="__codelineno-90-2" name="__codelineno-90-2" href="#__codelineno-90-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">ListNode</span>
 <a id="__codelineno-90-3" name="__codelineno-90-3" href="#__codelineno-90-3"></a><span class="w">  </span><span class="kp">attr_accessor</span><span class="w"> </span><span class="ss">:val</span><span class="w">    </span><span class="c1"># Node value</span>
-<a id="__codelineno-90-4" name="__codelineno-90-4" href="#__codelineno-90-4"></a><span class="w">  </span><span class="kp">attr_accessor</span><span class="w"> </span><span class="ss">:next</span><span class="w">   </span><span class="c1"># Reference to the successor node</span>
-<a id="__codelineno-90-5" name="__codelineno-90-5" href="#__codelineno-90-5"></a><span class="w">  </span><span class="kp">attr_accessor</span><span class="w"> </span><span class="ss">:prev</span><span class="w">   </span><span class="c1"># Reference to the predecessor node</span>
+<a id="__codelineno-90-4" name="__codelineno-90-4" href="#__codelineno-90-4"></a><span class="w">  </span><span class="kp">attr_accessor</span><span class="w"> </span><span class="ss">:next</span><span class="w">   </span><span class="c1"># Reference to the successor</span>
+<a id="__codelineno-90-5" name="__codelineno-90-5" href="#__codelineno-90-5"></a><span class="w">  </span><span class="kp">attr_accessor</span><span class="w"> </span><span class="ss">:prev</span><span class="w">   </span><span class="c1"># Reference to the predecessor</span>
 <a id="__codelineno-90-6" name="__codelineno-90-6" href="#__codelineno-90-6"></a>
 <a id="__codelineno-90-7" name="__codelineno-90-7" href="#__codelineno-90-7"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">initialize</span><span class="p">(</span><span class="n">val</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">next_node</span><span class="o">=</span><span class="kp">nil</span><span class="p">,</span><span class="w"> </span><span class="n">prev_node</span><span class="o">=</span><span class="kp">nil</span><span class="p">)</span>
 <a id="__codelineno-90-8" name="__codelineno-90-8" href="#__codelineno-90-8"></a><span class="w">    </span><span class="vi">@val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span>
@@ -6098,23 +6190,23 @@
 <p><img alt="Common types of linked lists" class="animation-figure" src="../linked_list.assets/linkedlist_common_types.png" /></p>
 <p align="center"> Figure 4-8 &nbsp; Common types of linked lists </p>
 
-<h2 id="424-typical-applications-of-linked-lists">4.2.4 &nbsp; Typical Applications of Linked Lists<a class="headerlink" href="#424-typical-applications-of-linked-lists" title="Permanent link">&para;</a></h2>
-<p>Singly linked lists are commonly used to implement stacks, queues, hash tables, and graphs.</p>
+<h2 id="424-typical-applications-of-linked-lists">4.2.4 &nbsp; Typical applications of linked lists<a class="headerlink" href="#424-typical-applications-of-linked-lists" title="Permanent link">&para;</a></h2>
+<p>Singly linked lists commonly implement stacks, queues, hash tables, and graphs.</p>
 <ul>
-<li><strong>Stacks and queues</strong>: When insertion and deletion operations both occur at one end of the linked list, it exhibits last-in-first-out characteristics, corresponding to a stack. When insertion operations occur at one end of the linked list and deletion operations occur at the other end, it exhibits first-in-first-out characteristics, corresponding to a queue.</li>
-<li><strong>Hash tables</strong>: Separate chaining is one of the mainstream solutions for resolving hash collisions. In this approach, all colliding elements are placed in a linked list.</li>
-<li><strong>Graphs</strong>: An adjacency list is a common way to represent a graph, where each vertex in the graph is associated with a linked list, and each element in the linked list represents another vertex connected to that vertex.</li>
+<li><strong>Stacks and queues</strong>: Inserting and removing at the same end gives last-in, first-out behavior, forming a stack. Inserting at one end and removing at the other gives first-in, first-out behavior, forming a queue.</li>
+<li><strong>Hash tables</strong>: Separate chaining is a common collision-handling strategy that places colliding elements in a linked list.</li>
+<li><strong>Graphs</strong>: An adjacency list associates each vertex with a linked list whose elements represent the other vertices connected to it.</li>
 </ul>
-<p>Doubly linked lists are commonly used in scenarios where quick access to the previous and next elements is needed.</p>
+<p>Doubly linked lists are useful when both the previous and next elements must be found quickly.</p>
 <ul>
-<li><strong>Advanced data structures</strong>: For example, in red-black trees and B-trees, we need to access the parent node of a node, which can be achieved by saving a reference to the parent node in the node, similar to a doubly linked list.</li>
-<li><strong>Browser history</strong>: In web browsers, when a user clicks the forward or backward button, the browser needs to know the previous and next web pages the user visited. The characteristics of doubly linked lists make this operation simple.</li>
-<li><strong>LRU algorithm</strong>: In cache eviction (LRU) algorithms, we need to quickly find the least recently used data and support quick addition and deletion of nodes. Using a doubly linked list is very suitable for this.</li>
+<li><strong>Advanced data structures</strong>: In structures such as red-black trees and B-trees, a node may need access to its parent. Storing a parent reference is similar to the backward link in a doubly linked list.</li>
+<li><strong>Browser history</strong>: The forward and back buttons require access to the next and previous visited pages. A doubly linked list makes this straightforward.</li>
+<li><strong>LRU caching</strong>: Least recently used eviction needs fast access to the least recently used data and efficient insertion and removal, making doubly linked lists a good fit.</li>
 </ul>
-<p>Circular linked lists are commonly used in scenarios that require periodic operations, such as operating system resource scheduling.</p>
+<p>Circular linked lists suit cyclic operations, such as operating-system resource scheduling.</p>
 <ul>
-<li><strong>Round-robin scheduling algorithm</strong>: In operating systems, round-robin scheduling is a common CPU scheduling algorithm that needs to cycle through a set of processes. Each process is assigned a time slice, and when the time slice expires, the CPU switches to the next process. This cyclic operation can be implemented using a circular linked list.</li>
-<li><strong>Data buffers</strong>: In some data buffer implementations, circular linked lists may also be used. For example, in audio and video players, the data stream may be divided into multiple buffer blocks and placed in a circular linked list to achieve seamless playback.</li>
+<li><strong>Round-robin scheduling</strong>: This common CPU scheduling algorithm cycles through processes, giving each a time slice before moving to the next. A circular linked list can implement this cycle.</li>
+<li><strong>Data buffers</strong>: Some buffers use circular linked lists. Audio and video players, for example, may divide a stream into buffer blocks arranged in a circular list for seamless playback.</li>
 </ul>
 
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@@ -6138,7 +6230,7 @@ aria-label="Footer"
   <a
     href="../array/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 4.1 Array"
+    aria-label="Previous: 4.1   Arrays"
     rel="prev"
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@@ -6150,7 +6242,7 @@ aria-label="Footer"
         Previous
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+        4.1 &nbsp; Arrays
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@@ -6162,7 +6254,7 @@ aria-label="Footer"
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     href="../list/"
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-    aria-label="Next: 4.3 List"
+    aria-label="Next: 4.3   Lists"
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@@ -6170,7 +6262,7 @@ aria-label="Footer"
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@@ -6187,7 +6279,7 @@ aria-label="Footer"
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+  8.1 &nbsp; Heaps
 
 
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+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
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-  8.5 Exercises
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+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
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+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
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+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
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-  10.4 Hash Optimization Strategy
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
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-  11.4 Insertion Sort
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-  11.7 Heap Sort
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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+  11.10 &nbsp; Radix sort
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
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+  12.3 &nbsp; Building a binary tree
 
 
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+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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+  13.3 &nbsp; Subset sum problem
 
 
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+  13.4 &nbsp; N-queens problem
 
 
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+  13.5 &nbsp; Summary
 
 
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+  13.6 &nbsp; Exercises
 
 
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+  Chapter 14. &nbsp; Dynamic programming
 
 
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+  Chapter 14. &nbsp; Dynamic programming
 
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+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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@@ -4630,6 +4630,78 @@
 
 
 
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   <a href="#431-common-list-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4.3.1 &nbsp; Common List Operations
+        4.3.1 &nbsp; Common list operations
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="4.3.1 Common List Operations">
+    <nav class="md-nav" aria-label="4.3.1 Common list operations">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-initialize-a-list" class="md-nav__link">
+  <a href="#1-initializing-a-list" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Initialize a List
+        1. &nbsp; Initializing a list
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+  <a href="#2-accessing-an-element" class="md-nav__link">
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+        2. &nbsp; Accessing an element
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       <span class="md-typeset">
-        3. &nbsp; Insert and Delete Elements
+        3. &nbsp; Inserting and removing elements
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 </li>
 
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       <span class="md-typeset">
-        4. &nbsp; Traverse a List
+        4. &nbsp; Traversing a list
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 </li>
 
           <li class="md-nav__item">
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+  <a href="#5-concatenating-lists" class="md-nav__link">
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+        5. &nbsp; Concatenating lists
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+  <a href="#6-sorting-a-list" class="md-nav__link">
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+        6. &nbsp; Sorting a list
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+  <a href="#432-implementing-a-list" class="md-nav__link">
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-        4.3.2 &nbsp; List Implementation
+        4.3.2 &nbsp; Implementing a list
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 <!-- Page content -->
-<h1 id="43-list">4.3 &nbsp; List<a class="headerlink" href="#43-list" title="Permanent link">&para;</a></h1>
-<p>A <u>list</u> is an abstract data structure concept that represents an ordered collection of elements, supporting operations such as element access, modification, insertion, deletion, and traversal, without requiring users to consider capacity limitations. Lists can be implemented based on linked lists or arrays.</p>
+<h1 id="43-lists">4.3 &nbsp; Lists<a class="headerlink" href="#43-lists" title="Permanent link">&para;</a></h1>
+<p>A <u>list</u> is an abstract data structure representing an ordered collection. It supports access, modification, insertion, removal, and traversal without requiring users to manage capacity limits. Lists can be implemented with linked lists or arrays.</p>
 <ul>
-<li>A linked list can naturally be viewed as a list: it supports insertion, deletion, search, and update, and can grow flexibly as needed.</li>
-<li>An array also supports insertion, deletion, search, and update, but because its length is fixed, it can only be regarded as a list with a capacity limit.</li>
+<li>A linked list naturally serves as a list, supporting these operations and flexible dynamic growth.</li>
+<li>An array also supports these operations, but its fixed length makes it a list with a length limit.</li>
 </ul>
-<p>When a list is implemented with an array, <strong>its fixed length makes it less practical</strong>. This is because we usually cannot determine in advance how much data we need to store, making it difficult to choose an appropriate capacity. If the capacity is too small, it may fail to meet our needs; if it is too large, memory space will be wasted.</p>
-<p>To solve this problem, we can use a <u>dynamic array</u> to implement a list. It inherits all the advantages of arrays while supporting dynamic resizing during program execution.</p>
-<p>In fact, <strong>the list types provided by the standard libraries of many programming languages are implemented with dynamic arrays</strong>, such as <code>list</code> in Python, <code>ArrayList</code> in Java, <code>vector</code> in C++, and <code>List</code> in C#. In the following discussion, we will treat "list" and "dynamic array" as equivalent concepts.</p>
-<h2 id="431-common-list-operations">4.3.1 &nbsp; Common List Operations<a class="headerlink" href="#431-common-list-operations" title="Permanent link">&para;</a></h2>
-<h3 id="1-initialize-a-list">1. &nbsp; Initialize a List<a class="headerlink" href="#1-initialize-a-list" title="Permanent link">&para;</a></h3>
-<p>We typically initialize a list in one of two ways: empty or with predefined values:</p>
+<p>When an array implements a list, <strong>its fixed length limits the list's usefulness</strong>. We often cannot predict how much data we will store. Too small an allocation may not meet our needs; too large an allocation wastes memory.</p>
+<p>A <u>dynamic array</u> solves this problem. It retains the advantages of an array while allowing capacity to grow during execution.</p>
+<p><strong>Many standard-library lists use dynamic arrays</strong>, including Python's <code>list</code>, Java's <code>ArrayList</code>, C++'s <code>vector</code>, and C#'s <code>List</code>. In the discussion that follows, we treat lists and dynamic arrays as equivalent concepts.</p>
+<h2 id="431-common-list-operations">4.3.1 &nbsp; Common list operations<a class="headerlink" href="#431-common-list-operations" title="Permanent link">&para;</a></h2>
+<h3 id="1-initializing-a-list">1. &nbsp; Initializing a list<a class="headerlink" href="#1-initializing-a-list" title="Permanent link">&para;</a></h3>
+<p>We usually initialize a list either without elements or with explicit initial values:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Initialize a list</span>
+<div class="highlight"><span class="filename">list.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Initialize lists</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="c1"># Without initial values</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="n">nums1</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="p">[]</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="c1"># With initial values</span>
@@ -4824,8 +4896,8 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Initialize a list */</span>
-<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="c1">// Note that vector in C++ is equivalent to nums as described in this article</span>
+<div class="highlight"><span class="filename">list.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Initialize lists */</span>
+<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="c1">// In C++, vector is the list type used for nums here</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="c1">// Without initial values</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums1</span><span class="p">;</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="c1">// With initial values</span>
@@ -4833,16 +4905,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Initialize a list */</span>
+<div class="highlight"><span class="filename">list.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Initialize lists */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="c1">// Without initial values</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
-<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="c1">// With initial values (note that array elements should use the wrapper class Integer[] instead of int[])</span>
+<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="c1">// With initial values (use Integer[], the boxed counterpart of int[])</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="n">Integer</span><span class="o">[]</span><span class="w"> </span><span class="n">numbers</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Integer</span><span class="o">[]</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="w"> </span><span class="p">};</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">(</span><span class="n">Arrays</span><span class="p">.</span><span class="na">asList</span><span class="p">(</span><span class="n">numbers</span><span class="p">));</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Initialize a list */</span>
+<div class="highlight"><span class="filename">list.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Initialize lists */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="c1">// Without initial values</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="c1">// With initial values</span>
@@ -4851,7 +4923,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list_test.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Initialize a list */</span>
+<div class="highlight"><span class="filename">list_test.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Initialize lists */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="c1">// Without initial values</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="nx">nums1</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">{}</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="c1">// With initial values</span>
@@ -4859,7 +4931,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Initialize a list */</span>
+<div class="highlight"><span class="filename">list.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Initialize lists */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Without initial values</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="kd">let</span><span class="w"> </span><span class="nv">nums1</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="c1">// With initial values</span>
@@ -4867,7 +4939,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Initialize a list */</span>
+<div class="highlight"><span class="filename">list.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Initialize lists */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="c1">// Without initial values</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="kd">const</span><span class="w"> </span><span class="nx">nums1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="c1">// With initial values</span>
@@ -4875,7 +4947,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Initialize a list */</span>
+<div class="highlight"><span class="filename">list.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Initialize lists */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="c1">// Without initial values</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="kd">const</span><span class="w"> </span><span class="nx">nums1</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="c1">// With initial values</span>
@@ -4883,7 +4955,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Initialize a list */</span>
+<div class="highlight"><span class="filename">list.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Initialize lists */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="c1">// Without initial values</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="c1">// With initial values</span>
@@ -4891,7 +4963,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Initialize a list */</span>
+<div class="highlight"><span class="filename">list.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Initialize lists */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="c1">// Without initial values</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="kd">let</span><span class="w"> </span><span class="n">nums1</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="c1">// With initial values</span>
@@ -4899,11 +4971,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// C does not provide built-in dynamic arrays</span>
+<div class="highlight"><span class="filename">list.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// C does not provide a built-in dynamic array</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Initialize a list */</span>
+<div class="highlight"><span class="filename">list.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Initialize lists */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="c1">// Without initial values</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="kd">var</span><span class="w"> </span><span class="nv">nums1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">listOf</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="c1">// With initial values</span>
@@ -4912,7 +4984,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Initialize a list</span>
+<div class="highlight"><span class="filename">list.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Initialize lists</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="c1"># Without initial values</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="n">nums1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="c1"># With initial values</span>
@@ -4922,127 +4994,127 @@
 </div>
 </div>
 <details class="pythontutor">
-<summary>Code Visualization</summary>
-<p><div style="height: 315px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%88%97%E8%A1%A8%0A%20%20%20%20%23%20%E6%97%A0%E5%88%9D%E5%A7%8B%E5%80%BC%0A%20%20%20%20nums1%20%3D%20%5B%5D%0A%20%20%20%20%23%20%E6%9C%89%E5%88%9D%E5%A7%8B%E5%80%BC%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%88%97%E8%A1%A8%0A%20%20%20%20%23%20%E6%97%A0%E5%88%9D%E5%A7%8B%E5%80%BC%0A%20%20%20%20nums1%20%3D%20%5B%5D%0A%20%20%20%20%23%20%E6%9C%89%E5%88%9D%E5%A7%8B%E5%80%BC%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 315px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20lists%0A%20%20%20%20%23%20Without%20initial%20values%0A%20%20%20%20nums1%20%3D%20%5B%5D%0A%20%20%20%20%23%20With%20initial%20values%0A%20%20%20%20nums%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20lists%0A%20%20%20%20%23%20Without%20initial%20values%0A%20%20%20%20nums1%20%3D%20%5B%5D%0A%20%20%20%20%23%20With%20initial%20values%0A%20%20%20%20nums%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<h3 id="2-access-elements">2. &nbsp; Access Elements<a class="headerlink" href="#2-access-elements" title="Permanent link">&para;</a></h3>
-<p>Since a list is essentially an array, we can access and update elements in <span class="arithmatex">\(O(1)\)</span> time complexity, which is very efficient.</p>
+<h3 id="2-accessing-an-element">2. &nbsp; Accessing an element<a class="headerlink" href="#2-accessing-an-element" title="Permanent link">&para;</a></h3>
+<p>Since the list is based on an array, accessing and updating an element are efficient <span class="arithmatex">\(O(1)\)</span> operations.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
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 <div class="highlight"><span class="filename">list.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="c1"># Access an element</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="n">num</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>  <span class="c1"># Access element at index 1</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="n">num</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>  <span class="c1"># Access the element at index 1</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="c1"># Update an element</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>    <span class="c1"># Update element at index 1 to 0</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>    <span class="c1"># Set the element at index 1 to 0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Access an element */</span>
-<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">];</span><span class="w">  </span><span class="c1">// Access element at index 1</span>
+<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">];</span><span class="w">  </span><span class="c1">// Access the element at index 1</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="cm">/* Update an element */</span>
-<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">  </span><span class="c1">// Update element at index 1 to 0</span>
+<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">  </span><span class="c1">// Set the element at index 1 to 0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Access an element */</span>
-<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Access element at index 1</span>
+<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Access the element at index 1</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="cm">/* Update an element */</span>
-<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="n">nums</span><span class="p">.</span><span class="na">set</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span><span class="w">  </span><span class="c1">// Update element at index 1 to 0</span>
+<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="n">nums</span><span class="p">.</span><span class="na">set</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span><span class="w">  </span><span class="c1">// Set the element at index 1 to 0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Access an element */</span>
-<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">];</span><span class="w">  </span><span class="c1">// Access element at index 1</span>
+<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">];</span><span class="w">  </span><span class="c1">// Access the element at index 1</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="cm">/* Update an element */</span>
-<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">  </span><span class="c1">// Update element at index 1 to 0</span>
+<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">  </span><span class="c1">// Set the element at index 1 to 0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list_test.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Access an element */</span>
-<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w">  </span><span class="c1">// Access element at index 1</span>
+<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w">  </span><span class="c1">// Access the element at index 1</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="cm">/* Update an element */</span>
-<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="nx">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w">     </span><span class="c1">// Update element at index 1 to 0</span>
+<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="nx">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w">     </span><span class="c1">// Set the element at index 1 to 0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Access an element */</span>
-<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">let</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="c1">// Access element at index 1</span>
+<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">let</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="c1">// Access the element at index 1</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="cm">/* Update an element */</span>
-<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1">// Update element at index 1 to 0</span>
+<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1">// Set the element at index 1 to 0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Access an element */</span>
-<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="mf">1</span><span class="p">];</span><span class="w">  </span><span class="c1">// Access element at index 1</span>
+<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="mf">1</span><span class="p">];</span><span class="w">  </span><span class="c1">// Access the element at index 1</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="cm">/* Update an element */</span>
-<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="nx">nums</span><span class="p">[</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w">  </span><span class="c1">// Update element at index 1 to 0</span>
+<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="nx">nums</span><span class="p">[</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w">  </span><span class="c1">// Set the element at index 1 to 0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Access an element */</span>
-<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="mf">1</span><span class="p">];</span><span class="w">  </span><span class="c1">// Access element at index 1</span>
+<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="mf">1</span><span class="p">];</span><span class="w">  </span><span class="c1">// Access the element at index 1</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="cm">/* Update an element */</span>
-<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="nx">nums</span><span class="p">[</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w">  </span><span class="c1">// Update element at index 1 to 0</span>
+<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="nx">nums</span><span class="p">[</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w">  </span><span class="c1">// Set the element at index 1 to 0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Access an element */</span>
-<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="kt">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="m">1</span><span class="p">];</span><span class="w">  </span><span class="c1">// Access element at index 1</span>
+<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="kt">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="m">1</span><span class="p">];</span><span class="w">  </span><span class="c1">// Access the element at index 1</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="cm">/* Update an element */</span>
-<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="n">nums</span><span class="p">[</span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w">  </span><span class="c1">// Update element at index 1 to 0</span>
+<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="n">nums</span><span class="p">[</span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w">  </span><span class="c1">// Set the element at index 1 to 0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Access an element */</span>
-<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="kd">let</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">];</span><span class="w">  </span><span class="c1">// Access element at index 1</span>
+<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="kd">let</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">];</span><span class="w">  </span><span class="c1">// Access the element at index 1</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="cm">/* Update an element */</span>
-<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">             </span><span class="c1">// Update element at index 1 to 0</span>
+<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="n">nums</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">             </span><span class="c1">// Set the element at index 1 to 0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="c1">// C does not provide built-in dynamic arrays</span>
+<div class="highlight"><span class="filename">list.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="c1">// C does not provide a built-in dynamic array</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Access an element */</span>
-<a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">val</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="m">1</span><span class="o">]</span><span class="w">       </span><span class="c1">// Access element at index 1</span>
+<a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">val</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="m">1</span><span class="o">]</span><span class="w">       </span><span class="c1">// Access the element at index 1</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="cm">/* Update an element */</span>
-<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="n">nums</span><span class="o">[</span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w">             </span><span class="c1">// Update element at index 1 to 0</span>
+<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="n">nums</span><span class="o">[</span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w">             </span><span class="c1">// Set the element at index 1 to 0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1"># Access an element</span>
-<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="c1"># Access element at index 1</span>
+<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="c1"># Access the element at index 1</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="c1"># Update an element</span>
-<a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="n">nums</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Update element at index 1 to 0</span>
+<a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="n">nums</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Set the element at index 1 to 0</span>
 </code></pre></div>
 </div>
 </div>
 </div>
 <details class="pythontutor">
-<summary>Code Visualization</summary>
-<p><div style="height: 369px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%88%97%E8%A1%A8%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%0A%20%20%20%20%23%20%E8%AE%BF%E9%97%AE%E5%85%83%E7%B4%A0%0A%20%20%20%20num%20%3D%20nums%5B1%5D%20%20%23%20%E8%AE%BF%E9%97%AE%E7%B4%A2%E5%BC%95%201%20%E5%A4%84%E7%9A%84%E5%85%83%E7%B4%A0%0A%0A%20%20%20%20%23%20%E6%9B%B4%E6%96%B0%E5%85%83%E7%B4%A0%0A%20%20%20%20nums%5B1%5D%20%3D%200%20%20%20%20%23%20%E5%B0%86%E7%B4%A2%E5%BC%95%201%20%E5%A4%84%E7%9A%84%E5%85%83%E7%B4%A0%E6%9B%B4%E6%96%B0%E4%B8%BA%200&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%88%97%E8%A1%A8%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%0A%20%20%20%20%23%20%E8%AE%BF%E9%97%AE%E5%85%83%E7%B4%A0%0A%20%20%20%20num%20%3D%20nums%5B1%5D%20%20%23%20%E8%AE%BF%E9%97%AE%E7%B4%A2%E5%BC%95%201%20%E5%A4%84%E7%9A%84%E5%85%83%E7%B4%A0%0A%0A%20%20%20%20%23%20%E6%9B%B4%E6%96%B0%E5%85%83%E7%B4%A0%0A%20%20%20%20nums%5B1%5D%20%3D%200%20%20%20%20%23%20%E5%B0%86%E7%B4%A2%E5%BC%95%201%20%E5%A4%84%E7%9A%84%E5%85%83%E7%B4%A0%E6%9B%B4%E6%96%B0%E4%B8%BA%200&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 369px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20lists%0A%20%20%20%20nums%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D%0A%0A%20%20%20%20%23%20Access%20an%20element%0A%20%20%20%20num%20%3D%20nums%5B1%5D%20%20%23%20Access%20the%20element%20at%20index%201%0A%0A%20%20%20%20%23%20Update%20an%20element%0A%20%20%20%20nums%5B1%5D%20%3D%200%20%20%20%20%23%20Set%20the%20element%20at%20index%201%20to%200&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20lists%0A%20%20%20%20nums%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D%0A%0A%20%20%20%20%23%20Access%20an%20element%0A%20%20%20%20num%20%3D%20nums%5B1%5D%20%20%23%20Access%20the%20element%20at%20index%201%0A%0A%20%20%20%20%23%20Update%20an%20element%0A%20%20%20%20nums%5B1%5D%20%3D%200%20%20%20%20%23%20Set%20the%20element%20at%20index%201%20to%200&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<h3 id="3-insert-and-delete-elements">3. &nbsp; Insert and Delete Elements<a class="headerlink" href="#3-insert-and-delete-elements" title="Permanent link">&para;</a></h3>
-<p>Compared to arrays, lists can freely add and delete elements. Adding an element at the end of a list has a time complexity of <span class="arithmatex">\(O(1)\)</span>, but inserting and deleting elements still have the same efficiency as arrays, with a time complexity of <span class="arithmatex">\(O(n)\)</span>.</p>
+<h3 id="3-inserting-and-removing-elements">3. &nbsp; Inserting and removing elements<a class="headerlink" href="#3-inserting-and-removing-elements" title="Permanent link">&para;</a></h3>
+<p>Unlike a fixed array, a list can freely add and remove elements. Appending takes <span class="arithmatex">\(O(1)\)</span> time, while insertion and removal retain the array's <span class="arithmatex">\(O(n)\)</span> time complexity.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="c1"># Clear the list</span>
 <a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="n">nums</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span>
 <a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>
-<a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a><span class="c1"># Add elements at the end</span>
+<a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a><span class="c1"># Append elements</span>
 <a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a><span class="n">nums</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a><span class="n">nums</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
 <a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a><span class="n">nums</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
@@ -5050,17 +5122,17 @@
 <a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a><span class="n">nums</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>
 <a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a><span class="c1"># Insert an element in the middle</span>
-<a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a><span class="n">nums</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>  <span class="c1"># Insert number 6 at index 3</span>
+<a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a><span class="n">nums</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>  <span class="c1"># Insert 6 at index 3</span>
 <a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>
-<a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a><span class="c1"># Delete an element</span>
-<a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a><span class="n">nums</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>        <span class="c1"># Delete element at index 3</span>
+<a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a><span class="c1"># Remove an element</span>
+<a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a><span class="n">nums</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>        <span class="c1"># Remove the element at index 3</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Clear the list */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="n">nums</span><span class="p">.</span><span class="n">clear</span><span class="p">();</span>
 <a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a>
-<a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="cm">/* Append elements */</span>
 <a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="n">nums</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="n">nums</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="n">nums</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
@@ -5068,17 +5140,17 @@
 <a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="n">nums</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a>
 <a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="cm">/* Insert an element in the middle */</span>
-<a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="n">nums</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">begin</span><span class="p">()</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">);</span><span class="w">  </span><span class="c1">// Insert number 6 at index 3</span>
+<a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="n">nums</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">begin</span><span class="p">()</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">);</span><span class="w">  </span><span class="c1">// Insert 6 at index 3</span>
 <a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a>
-<a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a><span class="cm">/* Delete an element */</span>
-<a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="n">nums</span><span class="p">.</span><span class="n">erase</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">begin</span><span class="p">()</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">3</span><span class="p">);</span><span class="w">      </span><span class="c1">// Delete element at index 3</span>
+<a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a><span class="cm">/* Remove an element */</span>
+<a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="n">nums</span><span class="p">.</span><span class="n">erase</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">begin</span><span class="p">()</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">3</span><span class="p">);</span><span class="w">      </span><span class="c1">// Remove the element at index 3</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Clear the list */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="n">nums</span><span class="p">.</span><span class="na">clear</span><span class="p">();</span>
 <a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a>
-<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="cm">/* Append elements */</span>
 <a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="n">nums</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="n">nums</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="n">nums</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
@@ -5086,17 +5158,17 @@
 <a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="n">nums</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a>
 <a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="cm">/* Insert an element in the middle */</span>
-<a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="n">nums</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">);</span><span class="w">  </span><span class="c1">// Insert number 6 at index 3</span>
+<a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="n">nums</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">);</span><span class="w">  </span><span class="c1">// Insert 6 at index 3</span>
 <a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a>
-<a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a><span class="cm">/* Delete an element */</span>
-<a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="n">nums</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span><span class="w">  </span><span class="c1">// Delete element at index 3</span>
+<a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a><span class="cm">/* Remove an element */</span>
+<a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="n">nums</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span><span class="w">  </span><span class="c1">// Remove the element at index 3</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Clear the list */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="n">nums</span><span class="p">.</span><span class="n">Clear</span><span class="p">();</span>
 <a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a>
-<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="cm">/* Append elements */</span>
 <a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="n">nums</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="n">nums</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="n">nums</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
@@ -5104,17 +5176,17 @@
 <a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="n">nums</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a>
 <a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="cm">/* Insert an element in the middle */</span>
-<a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="n">nums</span><span class="p">.</span><span class="n">Insert</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">);</span><span class="w">  </span><span class="c1">// Insert number 6 at index 3</span>
+<a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="n">nums</span><span class="p">.</span><span class="n">Insert</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">);</span><span class="w">  </span><span class="c1">// Insert 6 at index 3</span>
 <a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a>
-<a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="cm">/* Delete an element */</span>
-<a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="n">nums</span><span class="p">.</span><span class="n">RemoveAt</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span><span class="w">  </span><span class="c1">// Delete element at index 3</span>
+<a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="cm">/* Remove an element */</span>
+<a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="n">nums</span><span class="p">.</span><span class="n">RemoveAt</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span><span class="w">  </span><span class="c1">// Remove the element at index 3</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list_test.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Clear the list */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a>
-<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="cm">/* Append elements */</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span>
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
@@ -5122,17 +5194,17 @@
 <a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a>
 <a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="cm">/* Insert an element in the middle */</span>
-<a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">nums</span><span class="p">[:</span><span class="mi">3</span><span class="p">],</span><span class="w"> </span><span class="nb">append</span><span class="p">([]</span><span class="kt">int</span><span class="p">{</span><span class="mi">6</span><span class="p">},</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="mi">3</span><span class="p">:]...)...)</span><span class="w"> </span><span class="c1">// Insert number 6 at index 3</span>
+<a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">nums</span><span class="p">[:</span><span class="mi">3</span><span class="p">],</span><span class="w"> </span><span class="nb">append</span><span class="p">([]</span><span class="kt">int</span><span class="p">{</span><span class="mi">6</span><span class="p">},</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="mi">3</span><span class="p">:]...)...)</span><span class="w"> </span><span class="c1">// Insert 6 at index 3</span>
 <a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a>
-<a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="cm">/* Delete an element */</span>
-<a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">nums</span><span class="p">[:</span><span class="mi">3</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="mi">4</span><span class="p">:]...)</span><span class="w"> </span><span class="c1">// Delete element at index 3</span>
+<a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="cm">/* Remove an element */</span>
+<a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">nums</span><span class="p">[:</span><span class="mi">3</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="mi">4</span><span class="p">:]...)</span><span class="w"> </span><span class="c1">// Remove the element at index 3</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Clear the list */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="n">nums</span><span class="p">.</span><span class="bp">removeAll</span><span class="p">()</span>
 <a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a>
-<a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="cm">/* Append elements */</span>
 <a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="n">nums</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="n">nums</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
 <a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="n">nums</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
@@ -5140,17 +5212,17 @@
 <a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="n">nums</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a>
 <a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="cm">/* Insert an element in the middle */</span>
-<a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="n">nums</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="n">at</span><span class="p">:</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="c1">// Insert number 6 at index 3</span>
+<a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="n">nums</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="n">at</span><span class="p">:</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="c1">// Insert 6 at index 3</span>
 <a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a>
-<a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="cm">/* Delete an element */</span>
-<a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="n">nums</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">at</span><span class="p">:</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="c1">// Delete element at index 3</span>
+<a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="cm">/* Remove an element */</span>
+<a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="n">nums</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">at</span><span class="p">:</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="c1">// Remove the element at index 3</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Clear the list */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a>
-<a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="cm">/* Append elements */</span>
 <a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">3</span><span class="p">);</span>
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">2</span><span class="p">);</span>
@@ -5158,17 +5230,17 @@
 <a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">4</span><span class="p">);</span>
 <a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a>
 <a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="cm">/* Insert an element in the middle */</span>
-<a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="mf">6</span><span class="p">);</span><span class="w"> </span><span class="c1">// Insert number 6 at index 3</span>
+<a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="mf">6</span><span class="p">);</span><span class="w"> </span><span class="c1">// Insert 6 at index 3</span>
 <a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a>
-<a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="cm">/* Delete an element */</span>
-<a id="__codelineno-32-15" name="__codelineno-32-15" href="#__codelineno-32-15"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Delete element at index 3</span>
+<a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="cm">/* Remove an element */</span>
+<a id="__codelineno-32-15" name="__codelineno-32-15" href="#__codelineno-32-15"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Remove the element at index 3</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Clear the list */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a>
-<a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="cm">/* Append elements */</span>
 <a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">3</span><span class="p">);</span>
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">2</span><span class="p">);</span>
@@ -5176,17 +5248,17 @@
 <a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">4</span><span class="p">);</span>
 <a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a>
 <a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="cm">/* Insert an element in the middle */</span>
-<a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="mf">6</span><span class="p">);</span><span class="w"> </span><span class="c1">// Insert number 6 at index 3</span>
+<a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="mf">6</span><span class="p">);</span><span class="w"> </span><span class="c1">// Insert 6 at index 3</span>
 <a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a>
-<a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="cm">/* Delete an element */</span>
-<a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Delete element at index 3</span>
+<a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="cm">/* Remove an element */</span>
+<a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Remove the element at index 3</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Clear the list */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="n">nums</span><span class="p">.</span><span class="n">clear</span><span class="p">();</span>
 <a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a>
-<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="cm">/* Append elements */</span>
 <a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="n">nums</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="n">nums</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="m">3</span><span class="p">);</span>
 <a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="n">nums</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="m">2</span><span class="p">);</span>
@@ -5194,17 +5266,17 @@
 <a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="n">nums</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="m">4</span><span class="p">);</span>
 <a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a>
 <a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="cm">/* Insert an element in the middle */</span>
-<a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="n">nums</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">6</span><span class="p">);</span><span class="w"> </span><span class="c1">// Insert number 6 at index 3</span>
+<a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="n">nums</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">6</span><span class="p">);</span><span class="w"> </span><span class="c1">// Insert 6 at index 3</span>
 <a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a>
-<a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="cm">/* Delete an element */</span>
-<a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="n">nums</span><span class="p">.</span><span class="n">removeAt</span><span class="p">(</span><span class="m">3</span><span class="p">);</span><span class="w"> </span><span class="c1">// Delete element at index 3</span>
+<a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="cm">/* Remove an element */</span>
+<a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="n">nums</span><span class="p">.</span><span class="n">removeAt</span><span class="p">(</span><span class="m">3</span><span class="p">);</span><span class="w"> </span><span class="c1">// Remove the element at index 3</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Clear the list */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="n">nums</span><span class="p">.</span><span class="n">clear</span><span class="p">();</span>
 <a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a>
-<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="cm">/* Append elements */</span>
 <a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="n">nums</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="n">nums</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="n">nums</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
@@ -5212,21 +5284,21 @@
 <a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="n">nums</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a>
 <a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="cm">/* Insert an element in the middle */</span>
-<a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="n">nums</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">);</span><span class="w">  </span><span class="c1">// Insert number 6 at index 3</span>
+<a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="n">nums</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">);</span><span class="w">  </span><span class="c1">// Insert 6 at index 3</span>
 <a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a>
-<a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="cm">/* Delete an element */</span>
-<a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="n">nums</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span><span class="w">    </span><span class="c1">// Delete element at index 3</span>
+<a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="cm">/* Remove an element */</span>
+<a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="n">nums</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span><span class="w">    </span><span class="c1">// Remove the element at index 3</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="c1">// C does not provide built-in dynamic arrays</span>
+<div class="highlight"><span class="filename">list.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="c1">// C does not provide a built-in dynamic array</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Clear the list */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="n">nums</span><span class="p">.</span><span class="na">clear</span><span class="p">();</span>
 <a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a>
-<a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="cm">/* Append elements */</span>
 <a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="n">nums</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="n">nums</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="m">3</span><span class="p">);</span>
 <a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="n">nums</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="m">2</span><span class="p">);</span>
@@ -5234,17 +5306,17 @@
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="n">nums</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="m">4</span><span class="p">);</span>
 <a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a>
 <a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="cm">/* Insert an element in the middle */</span>
-<a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="n">nums</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">6</span><span class="p">);</span><span class="w">  </span><span class="c1">// Insert number 6 at index 3</span>
+<a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="n">nums</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">6</span><span class="p">);</span><span class="w">  </span><span class="c1">// Insert 6 at index 3</span>
 <a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a>
-<a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="cm">/* Delete an element */</span>
-<a id="__codelineno-37-15" name="__codelineno-37-15" href="#__codelineno-37-15"></a><span class="n">nums</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="m">3</span><span class="p">);</span><span class="w">  </span><span class="c1">// Delete element at index 3</span>
+<a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="cm">/* Remove an element */</span>
+<a id="__codelineno-37-15" name="__codelineno-37-15" href="#__codelineno-37-15"></a><span class="n">nums</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="m">3</span><span class="p">);</span><span class="w">  </span><span class="c1">// Remove the element at index 3</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1"># Clear the list</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="n">nums</span><span class="o">.</span><span class="n">clear</span>
 <a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a>
-<a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="c1"># Add elements at the end</span>
+<a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="c1"># Append elements</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="n">nums</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="n">nums</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">3</span>
 <a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="n">nums</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">2</span>
@@ -5252,21 +5324,21 @@
 <a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="n">nums</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">4</span>
 <a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a>
 <a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="c1"># Insert an element in the middle</span>
-<a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="n">nums</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">)</span><span class="w"> </span><span class="c1"># Insert number 6 at index 3</span>
+<a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="n">nums</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">)</span><span class="w"> </span><span class="c1"># Insert 6 at index 3</span>
 <a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a>
-<a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="c1"># Delete an element</span>
-<a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a><span class="n">nums</span><span class="o">.</span><span class="n">delete_at</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="c1"># Delete element at index 3</span>
+<a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="c1"># Remove an element</span>
+<a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a><span class="n">nums</span><span class="o">.</span><span class="n">delete_at</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="c1"># Remove the element at index 3</span>
 </code></pre></div>
 </div>
 </div>
 </div>
 <details class="pythontutor">
-<summary>Code Visualization</summary>
-<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E6%9C%89%E5%88%9D%E5%A7%8B%E5%80%BC%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%B8%85%E7%A9%BA%E5%88%97%E8%A1%A8%0A%20%20%20%20nums.clear%28%29%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%9C%A8%E5%B0%BE%E9%83%A8%E6%B7%BB%E5%8A%A0%E5%85%83%E7%B4%A0%0A%20%20%20%20nums.append%281%29%0A%20%20%20%20nums.append%283%29%0A%20%20%20%20nums.append%282%29%0A%20%20%20%20nums.append%285%29%0A%20%20%20%20nums.append%284%29%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%9C%A8%E4%B8%AD%E9%97%B4%E6%8F%92%E5%85%A5%E5%85%83%E7%B4%A0%0A%20%20%20%20nums.insert%283,%206%29%20%20%23%20%E5%9C%A8%E7%B4%A2%E5%BC%95%203%20%E5%A4%84%E6%8F%92%E5%85%A5%E6%95%B0%E5%AD%97%206%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E5%85%83%E7%B4%A0%0A%20%20%20%20nums.pop%283%29%20%20%20%20%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E7%B4%A2%E5%BC%95%203%20%E5%A4%84%E7%9A%84%E5%85%83%E7%B4%A0&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E6%9C%89%E5%88%9D%E5%A7%8B%E5%80%BC%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%B8%85%E7%A9%BA%E5%88%97%E8%A1%A8%0A%20%20%20%20nums.clear%28%29%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%9C%A8%E5%B0%BE%E9%83%A8%E6%B7%BB%E5%8A%A0%E5%85%83%E7%B4%A0%0A%20%20%20%20nums.append%281%29%0A%20%20%20%20nums.append%283%29%0A%20%20%20%20nums.append%282%29%0A%20%20%20%20nums.append%285%29%0A%20%20%20%20nums.append%284%29%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%9C%A8%E4%B8%AD%E9%97%B4%E6%8F%92%E5%85%A5%E5%85%83%E7%B4%A0%0A%20%20%20%20nums.insert%283,%206%29%20%20%23%20%E5%9C%A8%E7%B4%A2%E5%BC%95%203%20%E5%A4%84%E6%8F%92%E5%85%A5%E6%95%B0%E5%AD%97%206%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E5%85%83%E7%B4%A0%0A%20%20%20%20nums.pop%283%29%20%20%20%20%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E7%B4%A2%E5%BC%95%203%20%E5%A4%84%E7%9A%84%E5%85%83%E7%B4%A0&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20With%20initial%20values%0A%20%20%20%20nums%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20Clear%20the%20list%0A%20%20%20%20nums.clear%28%29%0A%20%20%20%20%0A%20%20%20%20%23%20Append%20elements%0A%20%20%20%20nums.append%281%29%0A%20%20%20%20nums.append%283%29%0A%20%20%20%20nums.append%282%29%0A%20%20%20%20nums.append%285%29%0A%20%20%20%20nums.append%284%29%0A%20%20%20%20%0A%20%20%20%20%23%20Insert%20an%20element%20in%20the%20middle%0A%20%20%20%20nums.insert%283%2C%206%29%20%20%23%20Insert%206%20at%20index%203%0A%20%20%20%20%0A%20%20%20%20%23%20Remove%20an%20element%0A%20%20%20%20nums.pop%283%29%20%20%20%20%20%20%20%20%23%20Remove%20the%20element%20at%20index%203&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20With%20initial%20values%0A%20%20%20%20nums%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20Clear%20the%20list%0A%20%20%20%20nums.clear%28%29%0A%20%20%20%20%0A%20%20%20%20%23%20Append%20elements%0A%20%20%20%20nums.append%281%29%0A%20%20%20%20nums.append%283%29%0A%20%20%20%20nums.append%282%29%0A%20%20%20%20nums.append%285%29%0A%20%20%20%20nums.append%284%29%0A%20%20%20%20%0A%20%20%20%20%23%20Insert%20an%20element%20in%20the%20middle%0A%20%20%20%20nums.insert%283%2C%206%29%20%20%23%20Insert%206%20at%20index%203%0A%20%20%20%20%0A%20%20%20%20%23%20Remove%20an%20element%0A%20%20%20%20nums.pop%283%29%20%20%20%20%20%20%20%20%23%20Remove%20the%20element%20at%20index%203&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<h3 id="4-traverse-a-list">4. &nbsp; Traverse a List<a class="headerlink" href="#4-traverse-a-list" title="Permanent link">&para;</a></h3>
-<p>Like arrays, lists can be traversed by index or by directly iterating through elements.</p>
+<h3 id="4-traversing-a-list">4. &nbsp; Traversing a list<a class="headerlink" href="#4-traversing-a-list" title="Permanent link">&para;</a></h3>
+<p>Like an array, a list can be traversed by index or by visiting its elements directly.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:13"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -5406,7 +5478,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="c1">// C does not provide built-in dynamic arrays</span>
+<div class="highlight"><span class="filename">list.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="c1">// C does not provide a built-in dynamic array</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
@@ -5439,67 +5511,67 @@
 </div>
 </div>
 <details class="pythontutor">
-<summary>Code Visualization</summary>
-<p><div style="height: 423px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%88%97%E8%A1%A8%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20%E9%80%9A%E8%BF%87%E7%B4%A2%E5%BC%95%E9%81%8D%E5%8E%86%E5%88%97%E8%A1%A8%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20nums%5Bi%5D%0A%0A%20%20%20%20%23%20%E7%9B%B4%E6%8E%A5%E9%81%8D%E5%8E%86%E5%88%97%E8%A1%A8%E5%85%83%E7%B4%A0%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20num&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%88%97%E8%A1%A8%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20%E9%80%9A%E8%BF%87%E7%B4%A2%E5%BC%95%E9%81%8D%E5%8E%86%E5%88%97%E8%A1%A8%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20nums%5Bi%5D%0A%0A%20%20%20%20%23%20%E7%9B%B4%E6%8E%A5%E9%81%8D%E5%8E%86%E5%88%97%E8%A1%A8%E5%85%83%E7%B4%A0%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20num&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 423px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20lists%0A%20%20%20%20nums%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20Traverse%20the%20list%20by%20index%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20nums%5Bi%5D%0A%0A%20%20%20%20%23%20Traverse%20list%20elements%20directly%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20num&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20lists%0A%20%20%20%20nums%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20Traverse%20the%20list%20by%20index%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20nums%5Bi%5D%0A%0A%20%20%20%20%23%20Traverse%20list%20elements%20directly%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20num&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<h3 id="5-concatenate-lists">5. &nbsp; Concatenate Lists<a class="headerlink" href="#5-concatenate-lists" title="Permanent link">&para;</a></h3>
-<p>Given a new list <code>nums1</code>, we can concatenate it to the end of the original list.</p>
+<h3 id="5-concatenating-lists">5. &nbsp; Concatenating lists<a class="headerlink" href="#5-concatenating-lists" title="Permanent link">&para;</a></h3>
+<p>Given another list <code>nums1</code>, we can append its elements to the end of the original list.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.py</span><pre><span></span><code><a id="__codelineno-52-1" name="__codelineno-52-1" href="#__codelineno-52-1"></a><span class="c1"># Concatenate two lists</span>
 <a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="n">nums1</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">6</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">9</span><span class="p">]</span>
-<a id="__codelineno-52-3" name="__codelineno-52-3" href="#__codelineno-52-3"></a><span class="n">nums</span> <span class="o">+=</span> <span class="n">nums1</span>  <span class="c1"># Concatenate list nums1 to the end of nums</span>
+<a id="__codelineno-52-3" name="__codelineno-52-3" href="#__codelineno-52-3"></a><span class="n">nums</span> <span class="o">+=</span> <span class="n">nums1</span>  <span class="c1"># Append nums1 to nums</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Concatenate two lists */</span>
 <a id="__codelineno-53-2" name="__codelineno-53-2" href="#__codelineno-53-2"></a><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">10</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="w"> </span><span class="p">};</span>
-<a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="c1">// Concatenate list nums1 to the end of nums</span>
+<a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="c1">// Append nums1 to nums</span>
 <a id="__codelineno-53-4" name="__codelineno-53-4" href="#__codelineno-53-4"></a><span class="n">nums</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">end</span><span class="p">(),</span><span class="w"> </span><span class="n">nums1</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span><span class="w"> </span><span class="n">nums1</span><span class="p">.</span><span class="n">end</span><span class="p">());</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Concatenate two lists */</span>
 <a id="__codelineno-54-2" name="__codelineno-54-2" href="#__codelineno-54-2"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">(</span><span class="n">Arrays</span><span class="p">.</span><span class="na">asList</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">Integer</span><span class="o">[]</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">10</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="w"> </span><span class="p">}));</span>
-<a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="n">nums</span><span class="p">.</span><span class="na">addAll</span><span class="p">(</span><span class="n">nums1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Concatenate list nums1 to the end of nums</span>
+<a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="n">nums</span><span class="p">.</span><span class="na">addAll</span><span class="p">(</span><span class="n">nums1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Append nums1 to nums</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Concatenate two lists */</span>
 <a id="__codelineno-55-2" name="__codelineno-55-2" href="#__codelineno-55-2"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">10</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">];</span>
-<a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="n">nums</span><span class="p">.</span><span class="n">AddRange</span><span class="p">(</span><span class="n">nums1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Concatenate list nums1 to the end of nums</span>
+<a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="n">nums</span><span class="p">.</span><span class="n">AddRange</span><span class="p">(</span><span class="n">nums1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Append nums1 to nums</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list_test.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Concatenate two lists */</span>
 <a id="__codelineno-56-2" name="__codelineno-56-2" href="#__codelineno-56-2"></a><span class="nx">nums1</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">{</span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">10</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">}</span>
-<a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">nums1</span><span class="p">...)</span><span class="w">  </span><span class="c1">// Concatenate list nums1 to the end of nums</span>
+<a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">nums1</span><span class="p">...)</span><span class="w">  </span><span class="c1">// Append nums1 to nums</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Concatenate two lists */</span>
 <a id="__codelineno-57-2" name="__codelineno-57-2" href="#__codelineno-57-2"></a><span class="kd">let</span><span class="w"> </span><span class="nv">nums1</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[</span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">10</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">]</span>
-<a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="n">nums</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">contentsOf</span><span class="p">:</span><span class="w"> </span><span class="n">nums1</span><span class="p">)</span><span class="w"> </span><span class="c1">// Concatenate list nums1 to the end of nums</span>
+<a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="n">nums</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">contentsOf</span><span class="p">:</span><span class="w"> </span><span class="n">nums1</span><span class="p">)</span><span class="w"> </span><span class="c1">// Append nums1 to nums</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Concatenate two lists */</span>
 <a id="__codelineno-58-2" name="__codelineno-58-2" href="#__codelineno-58-2"></a><span class="kd">const</span><span class="w"> </span><span class="nx">nums1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">6</span><span class="p">,</span><span class="w"> </span><span class="mf">8</span><span class="p">,</span><span class="w"> </span><span class="mf">7</span><span class="p">,</span><span class="w"> </span><span class="mf">10</span><span class="p">,</span><span class="w"> </span><span class="mf">9</span><span class="p">];</span>
-<a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">push</span><span class="p">(...</span><span class="nx">nums1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Concatenate list nums1 to the end of nums</span>
+<a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">push</span><span class="p">(...</span><span class="nx">nums1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Append nums1 to nums</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Concatenate two lists */</span>
 <a id="__codelineno-59-2" name="__codelineno-59-2" href="#__codelineno-59-2"></a><span class="kd">const</span><span class="w"> </span><span class="nx">nums1</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">6</span><span class="p">,</span><span class="w"> </span><span class="mf">8</span><span class="p">,</span><span class="w"> </span><span class="mf">7</span><span class="p">,</span><span class="w"> </span><span class="mf">10</span><span class="p">,</span><span class="w"> </span><span class="mf">9</span><span class="p">];</span>
-<a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">push</span><span class="p">(...</span><span class="nx">nums1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Concatenate list nums1 to the end of nums</span>
+<a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">push</span><span class="p">(...</span><span class="nx">nums1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Append nums1 to nums</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Concatenate two lists */</span>
 <a id="__codelineno-60-2" name="__codelineno-60-2" href="#__codelineno-60-2"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="m">6</span><span class="p">,</span><span class="w"> </span><span class="m">8</span><span class="p">,</span><span class="w"> </span><span class="m">7</span><span class="p">,</span><span class="w"> </span><span class="m">10</span><span class="p">,</span><span class="w"> </span><span class="m">9</span><span class="p">];</span>
-<a id="__codelineno-60-3" name="__codelineno-60-3" href="#__codelineno-60-3"></a><span class="n">nums</span><span class="p">.</span><span class="n">addAll</span><span class="p">(</span><span class="n">nums1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Concatenate list nums1 to the end of nums</span>
+<a id="__codelineno-60-3" name="__codelineno-60-3" href="#__codelineno-60-3"></a><span class="n">nums</span><span class="p">.</span><span class="n">addAll</span><span class="p">(</span><span class="n">nums1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Append nums1 to nums</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
@@ -5509,13 +5581,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="c1">// C does not provide built-in dynamic arrays</span>
+<div class="highlight"><span class="filename">list.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="c1">// C does not provide a built-in dynamic array</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">list.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Concatenate two lists */</span>
 <a id="__codelineno-63-2" name="__codelineno-63-2" href="#__codelineno-63-2"></a><span class="kd">val</span><span class="w"> </span><span class="nv">nums1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">intArrayOf</span><span class="p">(</span><span class="m">6</span><span class="p">,</span><span class="w"> </span><span class="m">8</span><span class="p">,</span><span class="w"> </span><span class="m">7</span><span class="p">,</span><span class="w"> </span><span class="m">10</span><span class="p">,</span><span class="w"> </span><span class="m">9</span><span class="p">).</span><span class="na">toMutableList</span><span class="p">()</span>
-<a id="__codelineno-63-3" name="__codelineno-63-3" href="#__codelineno-63-3"></a><span class="n">nums</span><span class="p">.</span><span class="na">addAll</span><span class="p">(</span><span class="n">nums1</span><span class="p">)</span><span class="w">  </span><span class="c1">// Concatenate list nums1 to the end of nums</span>
+<a id="__codelineno-63-3" name="__codelineno-63-3" href="#__codelineno-63-3"></a><span class="n">nums</span><span class="p">.</span><span class="na">addAll</span><span class="p">(</span><span class="n">nums1</span><span class="p">)</span><span class="w">  </span><span class="c1">// Append nums1 to nums</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
@@ -5527,92 +5599,92 @@
 </div>
 </div>
 <details class="pythontutor">
-<summary>Code Visualization</summary>
-<p><div style="height: 333px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%88%97%E8%A1%A8%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%8B%BC%E6%8E%A5%E4%B8%A4%E4%B8%AA%E5%88%97%E8%A1%A8%0A%20%20%20%20nums1%20%3D%20%5B6,%208,%207,%2010,%209%5D%0A%20%20%20%20nums%20%2B%3D%20nums1%20%20%23%20%E5%B0%86%E5%88%97%E8%A1%A8%20nums1%20%E6%8B%BC%E6%8E%A5%E5%88%B0%20nums%20%E4%B9%8B%E5%90%8E&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%88%97%E8%A1%A8%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%8B%BC%E6%8E%A5%E4%B8%A4%E4%B8%AA%E5%88%97%E8%A1%A8%0A%20%20%20%20nums1%20%3D%20%5B6,%208,%207,%2010,%209%5D%0A%20%20%20%20nums%20%2B%3D%20nums1%20%20%23%20%E5%B0%86%E5%88%97%E8%A1%A8%20nums1%20%E6%8B%BC%E6%8E%A5%E5%88%B0%20nums%20%E4%B9%8B%E5%90%8E&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 333px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20lists%0A%20%20%20%20nums%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20Concatenate%20two%20lists%0A%20%20%20%20nums1%20%3D%20%5B6%2C%208%2C%207%2C%2010%2C%209%5D%0A%20%20%20%20nums%20%2B%3D%20nums1%20%20%23%20Append%20nums1%20to%20nums&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20lists%0A%20%20%20%20nums%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20Concatenate%20two%20lists%0A%20%20%20%20nums1%20%3D%20%5B6%2C%208%2C%207%2C%2010%2C%209%5D%0A%20%20%20%20nums%20%2B%3D%20nums1%20%20%23%20Append%20nums1%20to%20nums&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<h3 id="6-sort-a-list">6. &nbsp; Sort a List<a class="headerlink" href="#6-sort-a-list" title="Permanent link">&para;</a></h3>
-<p>After sorting a list, we can use "binary search" and "two-pointer" algorithms, which are frequently tested in array algorithm problems.</p>
+<h3 id="6-sorting-a-list">6. &nbsp; Sorting a list<a class="headerlink" href="#6-sorting-a-list" title="Permanent link">&para;</a></h3>
+<p>Once the list is sorted, we can apply binary search and two-pointer algorithms, which often appear in array problems.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="6:13"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.py</span><pre><span></span><code><a id="__codelineno-65-1" name="__codelineno-65-1" href="#__codelineno-65-1"></a><span class="c1"># Sort a list</span>
-<a id="__codelineno-65-2" name="__codelineno-65-2" href="#__codelineno-65-2"></a><span class="n">nums</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>  <span class="c1"># After sorting, list elements are arranged from smallest to largest</span>
+<div class="highlight"><span class="filename">list.py</span><pre><span></span><code><a id="__codelineno-65-1" name="__codelineno-65-1" href="#__codelineno-65-1"></a><span class="c1"># Sort the list</span>
+<a id="__codelineno-65-2" name="__codelineno-65-2" href="#__codelineno-65-2"></a><span class="n">nums</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>  <span class="c1"># After sorting, elements are in ascending order</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.cpp</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="cm">/* Sort a list */</span>
-<a id="__codelineno-66-2" name="__codelineno-66-2" href="#__codelineno-66-2"></a><span class="n">sort</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">end</span><span class="p">());</span><span class="w">  </span><span class="c1">// After sorting, list elements are arranged from smallest to largest</span>
+<div class="highlight"><span class="filename">list.cpp</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="cm">/* Sort the list */</span>
+<a id="__codelineno-66-2" name="__codelineno-66-2" href="#__codelineno-66-2"></a><span class="n">sort</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">end</span><span class="p">());</span><span class="w">  </span><span class="c1">// After sorting, elements are in ascending order</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.java</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="cm">/* Sort a list */</span>
-<a id="__codelineno-67-2" name="__codelineno-67-2" href="#__codelineno-67-2"></a><span class="n">Collections</span><span class="p">.</span><span class="na">sort</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w">  </span><span class="c1">// After sorting, list elements are arranged from smallest to largest</span>
+<div class="highlight"><span class="filename">list.java</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="cm">/* Sort the list */</span>
+<a id="__codelineno-67-2" name="__codelineno-67-2" href="#__codelineno-67-2"></a><span class="n">Collections</span><span class="p">.</span><span class="na">sort</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w">  </span><span class="c1">// After sorting, elements are in ascending order</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.cs</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a><span class="cm">/* Sort a list */</span>
-<a id="__codelineno-68-2" name="__codelineno-68-2" href="#__codelineno-68-2"></a><span class="n">nums</span><span class="p">.</span><span class="n">Sort</span><span class="p">();</span><span class="w"> </span><span class="c1">// After sorting, list elements are arranged from smallest to largest</span>
+<div class="highlight"><span class="filename">list.cs</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a><span class="cm">/* Sort the list */</span>
+<a id="__codelineno-68-2" name="__codelineno-68-2" href="#__codelineno-68-2"></a><span class="n">nums</span><span class="p">.</span><span class="n">Sort</span><span class="p">();</span><span class="w"> </span><span class="c1">// After sorting, elements are in ascending order</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list_test.go</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a><span class="cm">/* Sort a list */</span>
-<a id="__codelineno-69-2" name="__codelineno-69-2" href="#__codelineno-69-2"></a><span class="nx">sort</span><span class="p">.</span><span class="nx">Ints</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w">  </span><span class="c1">// After sorting, list elements are arranged from smallest to largest</span>
+<div class="highlight"><span class="filename">list_test.go</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a><span class="cm">/* Sort the list */</span>
+<a id="__codelineno-69-2" name="__codelineno-69-2" href="#__codelineno-69-2"></a><span class="nx">sort</span><span class="p">.</span><span class="nx">Ints</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w">  </span><span class="c1">// After sorting, elements are in ascending order</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.swift</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a><span class="cm">/* Sort a list */</span>
-<a id="__codelineno-70-2" name="__codelineno-70-2" href="#__codelineno-70-2"></a><span class="n">nums</span><span class="p">.</span><span class="bp">sort</span><span class="p">()</span><span class="w"> </span><span class="c1">// After sorting, list elements are arranged from smallest to largest</span>
+<div class="highlight"><span class="filename">list.swift</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a><span class="cm">/* Sort the list */</span>
+<a id="__codelineno-70-2" name="__codelineno-70-2" href="#__codelineno-70-2"></a><span class="n">nums</span><span class="p">.</span><span class="bp">sort</span><span class="p">()</span><span class="w"> </span><span class="c1">// After sorting, elements are in ascending order</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.js</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a><span class="cm">/* Sort a list */</span>
-<a id="__codelineno-71-2" name="__codelineno-71-2" href="#__codelineno-71-2"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">sort</span><span class="p">((</span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">b</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">b</span><span class="p">);</span><span class="w">  </span><span class="c1">// After sorting, list elements are arranged from smallest to largest</span>
+<div class="highlight"><span class="filename">list.js</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a><span class="cm">/* Sort the list */</span>
+<a id="__codelineno-71-2" name="__codelineno-71-2" href="#__codelineno-71-2"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">sort</span><span class="p">((</span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">b</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">b</span><span class="p">);</span><span class="w">  </span><span class="c1">// After sorting, elements are in ascending order</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.ts</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* Sort a list */</span>
-<a id="__codelineno-72-2" name="__codelineno-72-2" href="#__codelineno-72-2"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">sort</span><span class="p">((</span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">b</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">b</span><span class="p">);</span><span class="w">  </span><span class="c1">// After sorting, list elements are arranged from smallest to largest</span>
+<div class="highlight"><span class="filename">list.ts</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* Sort the list */</span>
+<a id="__codelineno-72-2" name="__codelineno-72-2" href="#__codelineno-72-2"></a><span class="nx">nums</span><span class="p">.</span><span class="nx">sort</span><span class="p">((</span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">b</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">b</span><span class="p">);</span><span class="w">  </span><span class="c1">// After sorting, elements are in ascending order</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.dart</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="cm">/* Sort a list */</span>
-<a id="__codelineno-73-2" name="__codelineno-73-2" href="#__codelineno-73-2"></a><span class="n">nums</span><span class="p">.</span><span class="n">sort</span><span class="p">();</span><span class="w"> </span><span class="c1">// After sorting, list elements are arranged from smallest to largest</span>
+<div class="highlight"><span class="filename">list.dart</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="cm">/* Sort the list */</span>
+<a id="__codelineno-73-2" name="__codelineno-73-2" href="#__codelineno-73-2"></a><span class="n">nums</span><span class="p">.</span><span class="n">sort</span><span class="p">();</span><span class="w"> </span><span class="c1">// After sorting, elements are in ascending order</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.rs</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="cm">/* Sort a list */</span>
-<a id="__codelineno-74-2" name="__codelineno-74-2" href="#__codelineno-74-2"></a><span class="n">nums</span><span class="p">.</span><span class="n">sort</span><span class="p">();</span><span class="w"> </span><span class="c1">// After sorting, list elements are arranged from smallest to largest</span>
+<div class="highlight"><span class="filename">list.rs</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="cm">/* Sort the list */</span>
+<a id="__codelineno-74-2" name="__codelineno-74-2" href="#__codelineno-74-2"></a><span class="n">nums</span><span class="p">.</span><span class="n">sort</span><span class="p">();</span><span class="w"> </span><span class="c1">// After sorting, elements are in ascending order</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.c</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="c1">// C does not provide built-in dynamic arrays</span>
+<div class="highlight"><span class="filename">list.c</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="c1">// C does not provide a built-in dynamic array</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.kt</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a><span class="cm">/* Sort a list */</span>
-<a id="__codelineno-76-2" name="__codelineno-76-2" href="#__codelineno-76-2"></a><span class="n">nums</span><span class="p">.</span><span class="na">sort</span><span class="p">()</span><span class="w"> </span><span class="c1">// After sorting, list elements are arranged from smallest to largest</span>
+<div class="highlight"><span class="filename">list.kt</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a><span class="cm">/* Sort the list */</span>
+<a id="__codelineno-76-2" name="__codelineno-76-2" href="#__codelineno-76-2"></a><span class="n">nums</span><span class="p">.</span><span class="na">sort</span><span class="p">()</span><span class="w"> </span><span class="c1">// After sorting, elements are in ascending order</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">list.rb</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="c1"># Sort a list</span>
-<a id="__codelineno-77-2" name="__codelineno-77-2" href="#__codelineno-77-2"></a><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">sort</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="o">|</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=&gt;</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="c1"># After sorting, list elements are arranged from smallest to largest</span>
+<div class="highlight"><span class="filename">list.rb</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="c1"># Sort the list</span>
+<a id="__codelineno-77-2" name="__codelineno-77-2" href="#__codelineno-77-2"></a><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">sort</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="o">|</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=&gt;</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="c1"># After sorting, elements are in ascending order</span>
 </code></pre></div>
 </div>
 </div>
 </div>
 <details class="pythontutor">
-<summary>Code Visualization</summary>
-<p><div style="height: 315px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%88%97%E8%A1%A8%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%8E%92%E5%BA%8F%E5%88%97%E8%A1%A8%0A%20%20%20%20nums.sort%28%29%20%20%23%20%E6%8E%92%E5%BA%8F%E5%90%8E%EF%BC%8C%E5%88%97%E8%A1%A8%E5%85%83%E7%B4%A0%E4%BB%8E%E5%B0%8F%E5%88%B0%E5%A4%A7%E6%8E%92%E5%88%97&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%88%97%E8%A1%A8%0A%20%20%20%20nums%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%8E%92%E5%BA%8F%E5%88%97%E8%A1%A8%0A%20%20%20%20nums.sort%28%29%20%20%23%20%E6%8E%92%E5%BA%8F%E5%90%8E%EF%BC%8C%E5%88%97%E8%A1%A8%E5%85%83%E7%B4%A0%E4%BB%8E%E5%B0%8F%E5%88%B0%E5%A4%A7%E6%8E%92%E5%88%97&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 315px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20lists%0A%20%20%20%20nums%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20Sort%20the%20list%0A%20%20%20%20nums.sort%28%29%20%20%23%20After%20sorting%2C%20elements%20are%20in%20ascending%20order&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20lists%0A%20%20%20%20nums%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D%0A%20%20%20%20%0A%20%20%20%20%23%20Sort%20the%20list%0A%20%20%20%20nums.sort%28%29%20%20%23%20After%20sorting%2C%20elements%20are%20in%20ascending%20order&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<h2 id="432-list-implementation">4.3.2 &nbsp; List Implementation<a class="headerlink" href="#432-list-implementation" title="Permanent link">&para;</a></h2>
-<p>Many programming languages have built-in lists, such as Java, C++, and Python. Their implementations are quite complex, and the parameters are carefully considered, such as initial capacity, expansion multiples, and so on. Interested readers can consult the source code to learn more.</p>
-<p>To deepen our understanding of how lists work, we attempt to implement a simple list with three key design considerations:</p>
+<h2 id="432-implementing-a-list">4.3.2 &nbsp; Implementing a list<a class="headerlink" href="#432-implementing-a-list" title="Permanent link">&para;</a></h2>
+<p>Many languages, including Java, C++, and Python, provide lists. Their implementations are sophisticated, with carefully chosen parameters such as initial capacity and growth factor. Interested readers can study their source code.</p>
+<p>To understand how a list works, let's implement a simplified version with three key design choices.</p>
 <ul>
-<li><strong>Initial capacity</strong>: Select a reasonable initial capacity for the underlying array. In this example, we choose 10 as the initial capacity.</li>
-<li><strong>Size tracking</strong>: Declare a variable <code>size</code> to record the current number of elements in the list and update it in real-time as elements are inserted and deleted. Based on this variable, we can locate the end of the list and determine whether expansion is needed.</li>
-<li><strong>Expansion mechanism</strong>: When the list capacity is full upon inserting an element, we need to expand. We create a larger array based on the expansion multiple and then move all elements from the current array to the new array in order. In this example, we specify that the array should be expanded to 2 times its previous size each time.</li>
+<li><strong>Initial capacity</strong>: Choose a reasonable starting array capacity. We use 10 here.</li>
+<li><strong>Element count</strong>: Maintain <code>size</code>, the number of elements, updating it on insertion and removal. It identifies the list's end and tells us when expansion is needed.</li>
+<li><strong>Growth mechanism</strong>: If insertion finds the capacity full, allocate a larger array using a growth factor and copy all current elements into it. Here, capacity doubles each time.</li>
 </ul>
 <div class="tabbed-set tabbed-alternate" data-tabs="7:13"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><input id="__tabbed_7_13" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Python</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Java</label><label for="__tabbed_7_4">C#</label><label for="__tabbed_7_5">Go</label><label for="__tabbed_7_6">Swift</label><label for="__tabbed_7_7">JS</label><label for="__tabbed_7_8">TS</label><label for="__tabbed_7_9">Dart</label><label for="__tabbed_7_10">Rust</label><label for="__tabbed_7_11">C</label><label for="__tabbed_7_12">Kotlin</label><label for="__tabbed_7_13">Ruby</label></div>
 <div class="tabbed-content">
@@ -5623,47 +5695,47 @@
 <a id="__codelineno-78-4" name="__codelineno-78-4" href="#__codelineno-78-4"></a>    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
 <a id="__codelineno-78-5" name="__codelineno-78-5" href="#__codelineno-78-5"></a><span class="w">        </span><span class="sd">"""Constructor"""</span>
 <a id="__codelineno-78-6" name="__codelineno-78-6" href="#__codelineno-78-6"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_capacity</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">10</span>  <span class="c1"># List capacity</span>
-<a id="__codelineno-78-7" name="__codelineno-78-7" href="#__codelineno-78-7"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_arr</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">_capacity</span>  <span class="c1"># Array (stores list elements)</span>
+<a id="__codelineno-78-7" name="__codelineno-78-7" href="#__codelineno-78-7"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_arr</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">_capacity</span>  <span class="c1"># Array (stores the list elements)</span>
 <a id="__codelineno-78-8" name="__codelineno-78-8" href="#__codelineno-78-8"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># List length (current number of elements)</span>
-<a id="__codelineno-78-9" name="__codelineno-78-9" href="#__codelineno-78-9"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_extend_ratio</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">2</span>  <span class="c1"># Multiple by which the list capacity is extended each time</span>
+<a id="__codelineno-78-9" name="__codelineno-78-9" href="#__codelineno-78-9"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_extend_ratio</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">2</span>  <span class="c1"># Factor by which the list capacity grows each time</span>
 <a id="__codelineno-78-10" name="__codelineno-78-10" href="#__codelineno-78-10"></a>
 <a id="__codelineno-78-11" name="__codelineno-78-11" href="#__codelineno-78-11"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-78-12" name="__codelineno-78-12" href="#__codelineno-78-12"></a><span class="w">        </span><span class="sd">"""Get list length (current number of elements)"""</span>
+<a id="__codelineno-78-12" name="__codelineno-78-12" href="#__codelineno-78-12"></a><span class="w">        </span><span class="sd">"""Get the list length (current number of elements)"""</span>
 <a id="__codelineno-78-13" name="__codelineno-78-13" href="#__codelineno-78-13"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span>
 <a id="__codelineno-78-14" name="__codelineno-78-14" href="#__codelineno-78-14"></a>
 <a id="__codelineno-78-15" name="__codelineno-78-15" href="#__codelineno-78-15"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">capacity</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-78-16" name="__codelineno-78-16" href="#__codelineno-78-16"></a><span class="w">        </span><span class="sd">"""Get list capacity"""</span>
+<a id="__codelineno-78-16" name="__codelineno-78-16" href="#__codelineno-78-16"></a><span class="w">        </span><span class="sd">"""Get the list capacity"""</span>
 <a id="__codelineno-78-17" name="__codelineno-78-17" href="#__codelineno-78-17"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_capacity</span>
 <a id="__codelineno-78-18" name="__codelineno-78-18" href="#__codelineno-78-18"></a>
 <a id="__codelineno-78-19" name="__codelineno-78-19" href="#__codelineno-78-19"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">index</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-78-20" name="__codelineno-78-20" href="#__codelineno-78-20"></a><span class="w">        </span><span class="sd">"""Access element"""</span>
-<a id="__codelineno-78-21" name="__codelineno-78-21" href="#__codelineno-78-21"></a>        <span class="c1"># If the index is out of bounds, throw an exception, as below</span>
+<a id="__codelineno-78-20" name="__codelineno-78-20" href="#__codelineno-78-20"></a><span class="w">        </span><span class="sd">"""Access an element"""</span>
+<a id="__codelineno-78-21" name="__codelineno-78-21" href="#__codelineno-78-21"></a>        <span class="c1"># Raise an exception if the index is out of bounds; the same applies below</span>
 <a id="__codelineno-78-22" name="__codelineno-78-22" href="#__codelineno-78-22"></a>        <span class="k">if</span> <span class="n">index</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">index</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">:</span>
 <a id="__codelineno-78-23" name="__codelineno-78-23" href="#__codelineno-78-23"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-78-24" name="__codelineno-78-24" href="#__codelineno-78-24"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_arr</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
 <a id="__codelineno-78-25" name="__codelineno-78-25" href="#__codelineno-78-25"></a>
 <a id="__codelineno-78-26" name="__codelineno-78-26" href="#__codelineno-78-26"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">set</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">index</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-78-27" name="__codelineno-78-27" href="#__codelineno-78-27"></a><span class="w">        </span><span class="sd">"""Update element"""</span>
+<a id="__codelineno-78-27" name="__codelineno-78-27" href="#__codelineno-78-27"></a><span class="w">        </span><span class="sd">"""Update an element"""</span>
 <a id="__codelineno-78-28" name="__codelineno-78-28" href="#__codelineno-78-28"></a>        <span class="k">if</span> <span class="n">index</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">index</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">:</span>
 <a id="__codelineno-78-29" name="__codelineno-78-29" href="#__codelineno-78-29"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-78-30" name="__codelineno-78-30" href="#__codelineno-78-30"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_arr</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="n">num</span>
 <a id="__codelineno-78-31" name="__codelineno-78-31" href="#__codelineno-78-31"></a>
 <a id="__codelineno-78-32" name="__codelineno-78-32" href="#__codelineno-78-32"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-78-33" name="__codelineno-78-33" href="#__codelineno-78-33"></a><span class="w">        </span><span class="sd">"""Add element at the end"""</span>
-<a id="__codelineno-78-34" name="__codelineno-78-34" href="#__codelineno-78-34"></a>        <span class="c1"># When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-78-33" name="__codelineno-78-33" href="#__codelineno-78-33"></a><span class="w">        </span><span class="sd">"""Append an element at the end"""</span>
+<a id="__codelineno-78-34" name="__codelineno-78-34" href="#__codelineno-78-34"></a>        <span class="c1"># Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-78-35" name="__codelineno-78-35" href="#__codelineno-78-35"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">():</span>
 <a id="__codelineno-78-36" name="__codelineno-78-36" href="#__codelineno-78-36"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">extend_capacity</span><span class="p">()</span>
 <a id="__codelineno-78-37" name="__codelineno-78-37" href="#__codelineno-78-37"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_arr</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">]</span> <span class="o">=</span> <span class="n">num</span>
 <a id="__codelineno-78-38" name="__codelineno-78-38" href="#__codelineno-78-38"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">+=</span> <span class="mi">1</span>
 <a id="__codelineno-78-39" name="__codelineno-78-39" href="#__codelineno-78-39"></a>
 <a id="__codelineno-78-40" name="__codelineno-78-40" href="#__codelineno-78-40"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">index</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-78-41" name="__codelineno-78-41" href="#__codelineno-78-41"></a><span class="w">        </span><span class="sd">"""Insert element in the middle"""</span>
+<a id="__codelineno-78-41" name="__codelineno-78-41" href="#__codelineno-78-41"></a><span class="w">        </span><span class="sd">"""Insert an element in the middle"""</span>
 <a id="__codelineno-78-42" name="__codelineno-78-42" href="#__codelineno-78-42"></a>        <span class="k">if</span> <span class="n">index</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">index</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">:</span>
 <a id="__codelineno-78-43" name="__codelineno-78-43" href="#__codelineno-78-43"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Index out of bounds"</span><span class="p">)</span>
-<a id="__codelineno-78-44" name="__codelineno-78-44" href="#__codelineno-78-44"></a>        <span class="c1"># When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-78-44" name="__codelineno-78-44" href="#__codelineno-78-44"></a>        <span class="c1"># Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-78-45" name="__codelineno-78-45" href="#__codelineno-78-45"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">():</span>
 <a id="__codelineno-78-46" name="__codelineno-78-46" href="#__codelineno-78-46"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">extend_capacity</span><span class="p">()</span>
-<a id="__codelineno-78-47" name="__codelineno-78-47" href="#__codelineno-78-47"></a>        <span class="c1"># Move all elements at and after index index backward by one position</span>
+<a id="__codelineno-78-47" name="__codelineno-78-47" href="#__codelineno-78-47"></a>        <span class="c1"># Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-78-48" name="__codelineno-78-48" href="#__codelineno-78-48"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">index</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-78-49" name="__codelineno-78-49" href="#__codelineno-78-49"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_arr</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_arr</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-78-50" name="__codelineno-78-50" href="#__codelineno-78-50"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_arr</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="n">num</span>
@@ -5671,11 +5743,11 @@
 <a id="__codelineno-78-52" name="__codelineno-78-52" href="#__codelineno-78-52"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">+=</span> <span class="mi">1</span>
 <a id="__codelineno-78-53" name="__codelineno-78-53" href="#__codelineno-78-53"></a>
 <a id="__codelineno-78-54" name="__codelineno-78-54" href="#__codelineno-78-54"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">index</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-78-55" name="__codelineno-78-55" href="#__codelineno-78-55"></a><span class="w">        </span><span class="sd">"""Remove element"""</span>
+<a id="__codelineno-78-55" name="__codelineno-78-55" href="#__codelineno-78-55"></a><span class="w">        </span><span class="sd">"""Remove an element"""</span>
 <a id="__codelineno-78-56" name="__codelineno-78-56" href="#__codelineno-78-56"></a>        <span class="k">if</span> <span class="n">index</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">index</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">:</span>
 <a id="__codelineno-78-57" name="__codelineno-78-57" href="#__codelineno-78-57"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-78-58" name="__codelineno-78-58" href="#__codelineno-78-58"></a>        <span class="n">num</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_arr</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
-<a id="__codelineno-78-59" name="__codelineno-78-59" href="#__codelineno-78-59"></a>        <span class="c1"># Move all elements after index index forward by one position</span>
+<a id="__codelineno-78-59" name="__codelineno-78-59" href="#__codelineno-78-59"></a>        <span class="c1"># Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-78-60" name="__codelineno-78-60" href="#__codelineno-78-60"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">index</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-78-61" name="__codelineno-78-61" href="#__codelineno-78-61"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_arr</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_arr</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-78-62" name="__codelineno-78-62" href="#__codelineno-78-62"></a>        <span class="c1"># Update the number of elements</span>
@@ -5684,14 +5756,14 @@
 <a id="__codelineno-78-65" name="__codelineno-78-65" href="#__codelineno-78-65"></a>        <span class="k">return</span> <span class="n">num</span>
 <a id="__codelineno-78-66" name="__codelineno-78-66" href="#__codelineno-78-66"></a>
 <a id="__codelineno-78-67" name="__codelineno-78-67" href="#__codelineno-78-67"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">extend_capacity</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-<a id="__codelineno-78-68" name="__codelineno-78-68" href="#__codelineno-78-68"></a><span class="w">        </span><span class="sd">"""Extend list capacity"""</span>
-<a id="__codelineno-78-69" name="__codelineno-78-69" href="#__codelineno-78-69"></a>        <span class="c1"># Create a new array with length _extend_ratio times the original array, and copy the original array to the new array</span>
+<a id="__codelineno-78-68" name="__codelineno-78-68" href="#__codelineno-78-68"></a><span class="w">        </span><span class="sd">"""Grow the list capacity"""</span>
+<a id="__codelineno-78-69" name="__codelineno-78-69" href="#__codelineno-78-69"></a>        <span class="c1"># Create a new array with _extend_ratio times the original length and copy the original array into it</span>
 <a id="__codelineno-78-70" name="__codelineno-78-70" href="#__codelineno-78-70"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_arr</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_arr</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">()</span> <span class="o">*</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_extend_ratio</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-78-71" name="__codelineno-78-71" href="#__codelineno-78-71"></a>        <span class="c1"># Update list capacity</span>
+<a id="__codelineno-78-71" name="__codelineno-78-71" href="#__codelineno-78-71"></a>        <span class="c1"># Update the list capacity</span>
 <a id="__codelineno-78-72" name="__codelineno-78-72" href="#__codelineno-78-72"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_capacity</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_arr</span><span class="p">)</span>
 <a id="__codelineno-78-73" name="__codelineno-78-73" href="#__codelineno-78-73"></a>
 <a id="__codelineno-78-74" name="__codelineno-78-74" href="#__codelineno-78-74"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">to_array</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
-<a id="__codelineno-78-75" name="__codelineno-78-75" href="#__codelineno-78-75"></a><span class="w">        </span><span class="sd">"""Return list with valid length"""</span>
+<a id="__codelineno-78-75" name="__codelineno-78-75" href="#__codelineno-78-75"></a><span class="w">        </span><span class="sd">"""Return the portion of the list that contains elements"""</span>
 <a id="__codelineno-78-76" name="__codelineno-78-76" href="#__codelineno-78-76"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_arr</span><span class="p">[:</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">]</span>
 </code></pre></div>
 </div>
@@ -5699,10 +5771,10 @@
 <div class="highlight"><span class="filename">my_list.cpp</span><pre><span></span><code><a id="__codelineno-79-1" name="__codelineno-79-1" href="#__codelineno-79-1"></a><span class="cm">/* List class */</span>
 <a id="__codelineno-79-2" name="__codelineno-79-2" href="#__codelineno-79-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">MyList</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-3" name="__codelineno-79-3" href="#__codelineno-79-3"></a><span class="w">  </span><span class="k">private</span><span class="o">:</span>
-<a id="__codelineno-79-4" name="__codelineno-79-4" href="#__codelineno-79-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">arr</span><span class="p">;</span><span class="w">             </span><span class="c1">// Array (stores list elements)</span>
+<a id="__codelineno-79-4" name="__codelineno-79-4" href="#__codelineno-79-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">arr</span><span class="p">;</span><span class="w">             </span><span class="c1">// Array (stores the list elements)</span>
 <a id="__codelineno-79-5" name="__codelineno-79-5" href="#__codelineno-79-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">arrCapacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span><span class="w"> </span><span class="c1">// List capacity</span>
 <a id="__codelineno-79-6" name="__codelineno-79-6" href="#__codelineno-79-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">arrSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">      </span><span class="c1">// List length (current number of elements)</span>
-<a id="__codelineno-79-7" name="__codelineno-79-7" href="#__codelineno-79-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">   </span><span class="c1">// Multiple by which the list capacity is extended each time</span>
+<a id="__codelineno-79-7" name="__codelineno-79-7" href="#__codelineno-79-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">   </span><span class="c1">// Factor by which the list capacity grows each time</span>
 <a id="__codelineno-79-8" name="__codelineno-79-8" href="#__codelineno-79-8"></a>
 <a id="__codelineno-79-9" name="__codelineno-79-9" href="#__codelineno-79-9"></a><span class="w">  </span><span class="k">public</span><span class="o">:</span>
 <a id="__codelineno-79-10" name="__codelineno-79-10" href="#__codelineno-79-10"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -5715,34 +5787,34 @@
 <a id="__codelineno-79-17" name="__codelineno-79-17" href="#__codelineno-79-17"></a><span class="w">        </span><span class="k">delete</span><span class="p">[]</span><span class="w"> </span><span class="n">arr</span><span class="p">;</span>
 <a id="__codelineno-79-18" name="__codelineno-79-18" href="#__codelineno-79-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-79-19" name="__codelineno-79-19" href="#__codelineno-79-19"></a>
-<a id="__codelineno-79-20" name="__codelineno-79-20" href="#__codelineno-79-20"></a><span class="w">    </span><span class="cm">/* Get list length (current number of elements)*/</span>
+<a id="__codelineno-79-20" name="__codelineno-79-20" href="#__codelineno-79-20"></a><span class="w">    </span><span class="cm">/* Get the list length (current number of elements)*/</span>
 <a id="__codelineno-79-21" name="__codelineno-79-21" href="#__codelineno-79-21"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-22" name="__codelineno-79-22" href="#__codelineno-79-22"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">arrSize</span><span class="p">;</span>
 <a id="__codelineno-79-23" name="__codelineno-79-23" href="#__codelineno-79-23"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-79-24" name="__codelineno-79-24" href="#__codelineno-79-24"></a>
-<a id="__codelineno-79-25" name="__codelineno-79-25" href="#__codelineno-79-25"></a><span class="w">    </span><span class="cm">/* Get list capacity */</span>
+<a id="__codelineno-79-25" name="__codelineno-79-25" href="#__codelineno-79-25"></a><span class="w">    </span><span class="cm">/* Get the list capacity */</span>
 <a id="__codelineno-79-26" name="__codelineno-79-26" href="#__codelineno-79-26"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-27" name="__codelineno-79-27" href="#__codelineno-79-27"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">arrCapacity</span><span class="p">;</span>
 <a id="__codelineno-79-28" name="__codelineno-79-28" href="#__codelineno-79-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-79-29" name="__codelineno-79-29" href="#__codelineno-79-29"></a>
-<a id="__codelineno-79-30" name="__codelineno-79-30" href="#__codelineno-79-30"></a><span class="w">    </span><span class="cm">/* Update element */</span>
+<a id="__codelineno-79-30" name="__codelineno-79-30" href="#__codelineno-79-30"></a><span class="w">    </span><span class="cm">/* Access an element */</span>
 <a id="__codelineno-79-31" name="__codelineno-79-31" href="#__codelineno-79-31"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-79-32" name="__codelineno-79-32" href="#__codelineno-79-32"></a><span class="w">        </span><span class="c1">// If the index is out of bounds, throw an exception, as below</span>
+<a id="__codelineno-79-32" name="__codelineno-79-32" href="#__codelineno-79-32"></a><span class="w">        </span><span class="c1">// Raise an exception if the index is out of bounds; the same applies below</span>
 <a id="__codelineno-79-33" name="__codelineno-79-33" href="#__codelineno-79-33"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">())</span>
 <a id="__codelineno-79-34" name="__codelineno-79-34" href="#__codelineno-79-34"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">);</span>
 <a id="__codelineno-79-35" name="__codelineno-79-35" href="#__codelineno-79-35"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
 <a id="__codelineno-79-36" name="__codelineno-79-36" href="#__codelineno-79-36"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-79-37" name="__codelineno-79-37" href="#__codelineno-79-37"></a>
-<a id="__codelineno-79-38" name="__codelineno-79-38" href="#__codelineno-79-38"></a><span class="w">    </span><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-79-38" name="__codelineno-79-38" href="#__codelineno-79-38"></a><span class="w">    </span><span class="cm">/* Update an element */</span>
 <a id="__codelineno-79-39" name="__codelineno-79-39" href="#__codelineno-79-39"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">set</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-40" name="__codelineno-79-40" href="#__codelineno-79-40"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">())</span>
 <a id="__codelineno-79-41" name="__codelineno-79-41" href="#__codelineno-79-41"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">);</span>
 <a id="__codelineno-79-42" name="__codelineno-79-42" href="#__codelineno-79-42"></a><span class="w">        </span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-79-43" name="__codelineno-79-43" href="#__codelineno-79-43"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-79-44" name="__codelineno-79-44" href="#__codelineno-79-44"></a>
-<a id="__codelineno-79-45" name="__codelineno-79-45" href="#__codelineno-79-45"></a><span class="w">    </span><span class="cm">/* Direct traversal of list elements */</span>
+<a id="__codelineno-79-45" name="__codelineno-79-45" href="#__codelineno-79-45"></a><span class="w">    </span><span class="cm">/* Append an element at the end */</span>
 <a id="__codelineno-79-46" name="__codelineno-79-46" href="#__codelineno-79-46"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">add</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-79-47" name="__codelineno-79-47" href="#__codelineno-79-47"></a><span class="w">        </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-79-47" name="__codelineno-79-47" href="#__codelineno-79-47"></a><span class="w">        </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-79-48" name="__codelineno-79-48" href="#__codelineno-79-48"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span>
 <a id="__codelineno-79-49" name="__codelineno-79-49" href="#__codelineno-79-49"></a><span class="w">            </span><span class="n">extendCapacity</span><span class="p">();</span>
 <a id="__codelineno-79-50" name="__codelineno-79-50" href="#__codelineno-79-50"></a><span class="w">        </span><span class="n">arr</span><span class="p">[</span><span class="n">size</span><span class="p">()]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
@@ -5750,14 +5822,14 @@
 <a id="__codelineno-79-52" name="__codelineno-79-52" href="#__codelineno-79-52"></a><span class="w">        </span><span class="n">arrSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-79-53" name="__codelineno-79-53" href="#__codelineno-79-53"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-79-54" name="__codelineno-79-54" href="#__codelineno-79-54"></a>
-<a id="__codelineno-79-55" name="__codelineno-79-55" href="#__codelineno-79-55"></a><span class="w">    </span><span class="cm">/* Sort list */</span>
+<a id="__codelineno-79-55" name="__codelineno-79-55" href="#__codelineno-79-55"></a><span class="w">    </span><span class="cm">/* Insert an element in the middle */</span>
 <a id="__codelineno-79-56" name="__codelineno-79-56" href="#__codelineno-79-56"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-57" name="__codelineno-79-57" href="#__codelineno-79-57"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">())</span>
 <a id="__codelineno-79-58" name="__codelineno-79-58" href="#__codelineno-79-58"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">);</span>
-<a id="__codelineno-79-59" name="__codelineno-79-59" href="#__codelineno-79-59"></a><span class="w">        </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-79-59" name="__codelineno-79-59" href="#__codelineno-79-59"></a><span class="w">        </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-79-60" name="__codelineno-79-60" href="#__codelineno-79-60"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span>
 <a id="__codelineno-79-61" name="__codelineno-79-61" href="#__codelineno-79-61"></a><span class="w">            </span><span class="n">extendCapacity</span><span class="p">();</span>
-<a id="__codelineno-79-62" name="__codelineno-79-62" href="#__codelineno-79-62"></a><span class="w">        </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-79-62" name="__codelineno-79-62" href="#__codelineno-79-62"></a><span class="w">        </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-79-63" name="__codelineno-79-63" href="#__codelineno-79-63"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-64" name="__codelineno-79-64" href="#__codelineno-79-64"></a><span class="w">            </span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-79-65" name="__codelineno-79-65" href="#__codelineno-79-65"></a><span class="w">        </span><span class="p">}</span>
@@ -5766,12 +5838,12 @@
 <a id="__codelineno-79-68" name="__codelineno-79-68" href="#__codelineno-79-68"></a><span class="w">        </span><span class="n">arrSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-79-69" name="__codelineno-79-69" href="#__codelineno-79-69"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-79-70" name="__codelineno-79-70" href="#__codelineno-79-70"></a>
-<a id="__codelineno-79-71" name="__codelineno-79-71" href="#__codelineno-79-71"></a><span class="w">    </span><span class="cm">/* Remove element */</span>
+<a id="__codelineno-79-71" name="__codelineno-79-71" href="#__codelineno-79-71"></a><span class="w">    </span><span class="cm">/* Remove an element */</span>
 <a id="__codelineno-79-72" name="__codelineno-79-72" href="#__codelineno-79-72"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-73" name="__codelineno-79-73" href="#__codelineno-79-73"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">())</span>
 <a id="__codelineno-79-74" name="__codelineno-79-74" href="#__codelineno-79-74"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">);</span>
 <a id="__codelineno-79-75" name="__codelineno-79-75" href="#__codelineno-79-75"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
-<a id="__codelineno-79-76" name="__codelineno-79-76" href="#__codelineno-79-76"></a><span class="w">        </span><span class="c1">// Create a new array with length _extend_ratio times the original array, and copy the original array to the new array</span>
+<a id="__codelineno-79-76" name="__codelineno-79-76" href="#__codelineno-79-76"></a><span class="w">        </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-79-77" name="__codelineno-79-77" href="#__codelineno-79-77"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-78" name="__codelineno-79-78" href="#__codelineno-79-78"></a><span class="w">            </span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-79-79" name="__codelineno-79-79" href="#__codelineno-79-79"></a><span class="w">        </span><span class="p">}</span>
@@ -5781,9 +5853,9 @@
 <a id="__codelineno-79-83" name="__codelineno-79-83" href="#__codelineno-79-83"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-79-84" name="__codelineno-79-84" href="#__codelineno-79-84"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-79-85" name="__codelineno-79-85" href="#__codelineno-79-85"></a>
-<a id="__codelineno-79-86" name="__codelineno-79-86" href="#__codelineno-79-86"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-79-86" name="__codelineno-79-86" href="#__codelineno-79-86"></a><span class="w">    </span><span class="cm">/* Grow the list capacity */</span>
 <a id="__codelineno-79-87" name="__codelineno-79-87" href="#__codelineno-79-87"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">extendCapacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-79-88" name="__codelineno-79-88" href="#__codelineno-79-88"></a><span class="w">        </span><span class="c1">// Create a new array with length extendRatio times the original array</span>
+<a id="__codelineno-79-88" name="__codelineno-79-88" href="#__codelineno-79-88"></a><span class="w">        </span><span class="c1">// Create a new array with extendRatio times the original array's length</span>
 <a id="__codelineno-79-89" name="__codelineno-79-89" href="#__codelineno-79-89"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">newCapacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span>
 <a id="__codelineno-79-90" name="__codelineno-79-90" href="#__codelineno-79-90"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">;</span>
 <a id="__codelineno-79-91" name="__codelineno-79-91" href="#__codelineno-79-91"></a><span class="w">        </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">newCapacity</span><span class="p">];</span>
@@ -5796,9 +5868,9 @@
 <a id="__codelineno-79-98" name="__codelineno-79-98" href="#__codelineno-79-98"></a><span class="w">        </span><span class="n">arrCapacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newCapacity</span><span class="p">;</span>
 <a id="__codelineno-79-99" name="__codelineno-79-99" href="#__codelineno-79-99"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-79-100" name="__codelineno-79-100" href="#__codelineno-79-100"></a>
-<a id="__codelineno-79-101" name="__codelineno-79-101" href="#__codelineno-79-101"></a><span class="w">    </span><span class="cm">/* Convert list to Vector for printing */</span>
+<a id="__codelineno-79-101" name="__codelineno-79-101" href="#__codelineno-79-101"></a><span class="w">    </span><span class="cm">/* Convert the list to a vector for printing */</span>
 <a id="__codelineno-79-102" name="__codelineno-79-102" href="#__codelineno-79-102"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">toVector</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-79-103" name="__codelineno-79-103" href="#__codelineno-79-103"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-79-103" name="__codelineno-79-103" href="#__codelineno-79-103"></a><span class="w">        </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-79-104" name="__codelineno-79-104" href="#__codelineno-79-104"></a><span class="w">        </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">vec</span><span class="p">(</span><span class="n">size</span><span class="p">());</span>
 <a id="__codelineno-79-105" name="__codelineno-79-105" href="#__codelineno-79-105"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-106" name="__codelineno-79-106" href="#__codelineno-79-106"></a><span class="w">            </span><span class="n">vec</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
@@ -5811,44 +5883,44 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">my_list.java</span><pre><span></span><code><a id="__codelineno-80-1" name="__codelineno-80-1" href="#__codelineno-80-1"></a><span class="cm">/* List class */</span>
 <a id="__codelineno-80-2" name="__codelineno-80-2" href="#__codelineno-80-2"></a><span class="kd">class</span> <span class="nc">MyList</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-80-3" name="__codelineno-80-3" href="#__codelineno-80-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">arr</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array (stores list elements)</span>
+<a id="__codelineno-80-3" name="__codelineno-80-3" href="#__codelineno-80-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">arr</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array (stores the list elements)</span>
 <a id="__codelineno-80-4" name="__codelineno-80-4" href="#__codelineno-80-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span><span class="w"> </span><span class="c1">// List capacity</span>
 <a id="__codelineno-80-5" name="__codelineno-80-5" href="#__codelineno-80-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// List length (current number of elements)</span>
-<a id="__codelineno-80-6" name="__codelineno-80-6" href="#__codelineno-80-6"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Multiple by which the list capacity is extended each time</span>
+<a id="__codelineno-80-6" name="__codelineno-80-6" href="#__codelineno-80-6"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Factor by which the list capacity grows each time</span>
 <a id="__codelineno-80-7" name="__codelineno-80-7" href="#__codelineno-80-7"></a>
 <a id="__codelineno-80-8" name="__codelineno-80-8" href="#__codelineno-80-8"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-80-9" name="__codelineno-80-9" href="#__codelineno-80-9"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="nf">MyList</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-10" name="__codelineno-80-10" href="#__codelineno-80-10"></a><span class="w">        </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">capacity</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-80-11" name="__codelineno-80-11" href="#__codelineno-80-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-80-12" name="__codelineno-80-12" href="#__codelineno-80-12"></a>
-<a id="__codelineno-80-13" name="__codelineno-80-13" href="#__codelineno-80-13"></a><span class="w">    </span><span class="cm">/* Get list length (current number of elements) */</span>
+<a id="__codelineno-80-13" name="__codelineno-80-13" href="#__codelineno-80-13"></a><span class="w">    </span><span class="cm">/* Get the list length (current number of elements) */</span>
 <a id="__codelineno-80-14" name="__codelineno-80-14" href="#__codelineno-80-14"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-15" name="__codelineno-80-15" href="#__codelineno-80-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="p">;</span>
 <a id="__codelineno-80-16" name="__codelineno-80-16" href="#__codelineno-80-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-80-17" name="__codelineno-80-17" href="#__codelineno-80-17"></a>
-<a id="__codelineno-80-18" name="__codelineno-80-18" href="#__codelineno-80-18"></a><span class="w">    </span><span class="cm">/* Get list capacity */</span>
+<a id="__codelineno-80-18" name="__codelineno-80-18" href="#__codelineno-80-18"></a><span class="w">    </span><span class="cm">/* Get the list capacity */</span>
 <a id="__codelineno-80-19" name="__codelineno-80-19" href="#__codelineno-80-19"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-20" name="__codelineno-80-20" href="#__codelineno-80-20"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-80-21" name="__codelineno-80-21" href="#__codelineno-80-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-80-22" name="__codelineno-80-22" href="#__codelineno-80-22"></a>
-<a id="__codelineno-80-23" name="__codelineno-80-23" href="#__codelineno-80-23"></a><span class="w">    </span><span class="cm">/* Update element */</span>
+<a id="__codelineno-80-23" name="__codelineno-80-23" href="#__codelineno-80-23"></a><span class="w">    </span><span class="cm">/* Access an element */</span>
 <a id="__codelineno-80-24" name="__codelineno-80-24" href="#__codelineno-80-24"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-80-25" name="__codelineno-80-25" href="#__codelineno-80-25"></a><span class="w">        </span><span class="c1">// If the index is out of bounds, throw an exception, as below</span>
+<a id="__codelineno-80-25" name="__codelineno-80-25" href="#__codelineno-80-25"></a><span class="w">        </span><span class="c1">// Throw an exception if the index is out of bounds; the same applies below</span>
 <a id="__codelineno-80-26" name="__codelineno-80-26" href="#__codelineno-80-26"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">)</span>
 <a id="__codelineno-80-27" name="__codelineno-80-27" href="#__codelineno-80-27"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">);</span>
 <a id="__codelineno-80-28" name="__codelineno-80-28" href="#__codelineno-80-28"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">arr</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-80-29" name="__codelineno-80-29" href="#__codelineno-80-29"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-80-30" name="__codelineno-80-30" href="#__codelineno-80-30"></a>
-<a id="__codelineno-80-31" name="__codelineno-80-31" href="#__codelineno-80-31"></a><span class="w">    </span><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-80-31" name="__codelineno-80-31" href="#__codelineno-80-31"></a><span class="w">    </span><span class="cm">/* Update an element */</span>
 <a id="__codelineno-80-32" name="__codelineno-80-32" href="#__codelineno-80-32"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">set</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-33" name="__codelineno-80-33" href="#__codelineno-80-33"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">)</span>
 <a id="__codelineno-80-34" name="__codelineno-80-34" href="#__codelineno-80-34"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">);</span>
 <a id="__codelineno-80-35" name="__codelineno-80-35" href="#__codelineno-80-35"></a><span class="w">        </span><span class="n">arr</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-80-36" name="__codelineno-80-36" href="#__codelineno-80-36"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-80-37" name="__codelineno-80-37" href="#__codelineno-80-37"></a>
-<a id="__codelineno-80-38" name="__codelineno-80-38" href="#__codelineno-80-38"></a><span class="w">    </span><span class="cm">/* Direct traversal of list elements */</span>
+<a id="__codelineno-80-38" name="__codelineno-80-38" href="#__codelineno-80-38"></a><span class="w">    </span><span class="cm">/* Append at the end */</span>
 <a id="__codelineno-80-39" name="__codelineno-80-39" href="#__codelineno-80-39"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">add</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-80-40" name="__codelineno-80-40" href="#__codelineno-80-40"></a><span class="w">        </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-80-40" name="__codelineno-80-40" href="#__codelineno-80-40"></a><span class="w">        </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-80-41" name="__codelineno-80-41" href="#__codelineno-80-41"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span>
 <a id="__codelineno-80-42" name="__codelineno-80-42" href="#__codelineno-80-42"></a><span class="w">            </span><span class="n">extendCapacity</span><span class="p">();</span>
 <a id="__codelineno-80-43" name="__codelineno-80-43" href="#__codelineno-80-43"></a><span class="w">        </span><span class="n">arr</span><span class="o">[</span><span class="n">size</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
@@ -5856,14 +5928,14 @@
 <a id="__codelineno-80-45" name="__codelineno-80-45" href="#__codelineno-80-45"></a><span class="w">        </span><span class="n">size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-80-46" name="__codelineno-80-46" href="#__codelineno-80-46"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-80-47" name="__codelineno-80-47" href="#__codelineno-80-47"></a>
-<a id="__codelineno-80-48" name="__codelineno-80-48" href="#__codelineno-80-48"></a><span class="w">    </span><span class="cm">/* Sort list */</span>
+<a id="__codelineno-80-48" name="__codelineno-80-48" href="#__codelineno-80-48"></a><span class="w">    </span><span class="cm">/* Insert an element in the middle */</span>
 <a id="__codelineno-80-49" name="__codelineno-80-49" href="#__codelineno-80-49"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-50" name="__codelineno-80-50" href="#__codelineno-80-50"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">)</span>
 <a id="__codelineno-80-51" name="__codelineno-80-51" href="#__codelineno-80-51"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">);</span>
-<a id="__codelineno-80-52" name="__codelineno-80-52" href="#__codelineno-80-52"></a><span class="w">        </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-80-52" name="__codelineno-80-52" href="#__codelineno-80-52"></a><span class="w">        </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-80-53" name="__codelineno-80-53" href="#__codelineno-80-53"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span>
 <a id="__codelineno-80-54" name="__codelineno-80-54" href="#__codelineno-80-54"></a><span class="w">            </span><span class="n">extendCapacity</span><span class="p">();</span>
-<a id="__codelineno-80-55" name="__codelineno-80-55" href="#__codelineno-80-55"></a><span class="w">        </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-80-55" name="__codelineno-80-55" href="#__codelineno-80-55"></a><span class="w">        </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-80-56" name="__codelineno-80-56" href="#__codelineno-80-56"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-57" name="__codelineno-80-57" href="#__codelineno-80-57"></a><span class="w">            </span><span class="n">arr</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-80-58" name="__codelineno-80-58" href="#__codelineno-80-58"></a><span class="w">        </span><span class="p">}</span>
@@ -5872,12 +5944,12 @@
 <a id="__codelineno-80-61" name="__codelineno-80-61" href="#__codelineno-80-61"></a><span class="w">        </span><span class="n">size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-80-62" name="__codelineno-80-62" href="#__codelineno-80-62"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-80-63" name="__codelineno-80-63" href="#__codelineno-80-63"></a>
-<a id="__codelineno-80-64" name="__codelineno-80-64" href="#__codelineno-80-64"></a><span class="w">    </span><span class="cm">/* Remove element */</span>
+<a id="__codelineno-80-64" name="__codelineno-80-64" href="#__codelineno-80-64"></a><span class="w">    </span><span class="cm">/* Remove an element */</span>
 <a id="__codelineno-80-65" name="__codelineno-80-65" href="#__codelineno-80-65"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-66" name="__codelineno-80-66" href="#__codelineno-80-66"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">)</span>
 <a id="__codelineno-80-67" name="__codelineno-80-67" href="#__codelineno-80-67"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">);</span>
 <a id="__codelineno-80-68" name="__codelineno-80-68" href="#__codelineno-80-68"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-80-69" name="__codelineno-80-69" href="#__codelineno-80-69"></a><span class="w">        </span><span class="c1">// Move all elements after index forward by one position</span>
+<a id="__codelineno-80-69" name="__codelineno-80-69" href="#__codelineno-80-69"></a><span class="w">        </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-80-70" name="__codelineno-80-70" href="#__codelineno-80-70"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-71" name="__codelineno-80-71" href="#__codelineno-80-71"></a><span class="w">            </span><span class="n">arr</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-80-72" name="__codelineno-80-72" href="#__codelineno-80-72"></a><span class="w">        </span><span class="p">}</span>
@@ -5887,18 +5959,18 @@
 <a id="__codelineno-80-76" name="__codelineno-80-76" href="#__codelineno-80-76"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-80-77" name="__codelineno-80-77" href="#__codelineno-80-77"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-80-78" name="__codelineno-80-78" href="#__codelineno-80-78"></a>
-<a id="__codelineno-80-79" name="__codelineno-80-79" href="#__codelineno-80-79"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-80-79" name="__codelineno-80-79" href="#__codelineno-80-79"></a><span class="w">    </span><span class="cm">/* Grow the list capacity */</span>
 <a id="__codelineno-80-80" name="__codelineno-80-80" href="#__codelineno-80-80"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">extendCapacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-80-81" name="__codelineno-80-81" href="#__codelineno-80-81"></a><span class="w">        </span><span class="c1">// Create a new array with length extendRatio times the original array and copy the original array to the new array</span>
+<a id="__codelineno-80-81" name="__codelineno-80-81" href="#__codelineno-80-81"></a><span class="w">        </span><span class="c1">// Create an array extendRatio times as long as the original and copy the original array into it</span>
 <a id="__codelineno-80-82" name="__codelineno-80-82" href="#__codelineno-80-82"></a><span class="w">        </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Arrays</span><span class="p">.</span><span class="na">copyOf</span><span class="p">(</span><span class="n">arr</span><span class="p">,</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">);</span>
-<a id="__codelineno-80-83" name="__codelineno-80-83" href="#__codelineno-80-83"></a><span class="w">        </span><span class="c1">// Add elements at the end</span>
+<a id="__codelineno-80-83" name="__codelineno-80-83" href="#__codelineno-80-83"></a><span class="w">        </span><span class="c1">// Update the list capacity</span>
 <a id="__codelineno-80-84" name="__codelineno-80-84" href="#__codelineno-80-84"></a><span class="w">        </span><span class="n">capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
 <a id="__codelineno-80-85" name="__codelineno-80-85" href="#__codelineno-80-85"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-80-86" name="__codelineno-80-86" href="#__codelineno-80-86"></a>
-<a id="__codelineno-80-87" name="__codelineno-80-87" href="#__codelineno-80-87"></a><span class="w">    </span><span class="cm">/* Convert list to array */</span>
+<a id="__codelineno-80-87" name="__codelineno-80-87" href="#__codelineno-80-87"></a><span class="w">    </span><span class="cm">/* Convert the list to an array */</span>
 <a id="__codelineno-80-88" name="__codelineno-80-88" href="#__codelineno-80-88"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="nf">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-89" name="__codelineno-80-89" href="#__codelineno-80-89"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-80-90" name="__codelineno-80-90" href="#__codelineno-80-90"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-80-90" name="__codelineno-80-90" href="#__codelineno-80-90"></a><span class="w">        </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-80-91" name="__codelineno-80-91" href="#__codelineno-80-91"></a><span class="w">        </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">size</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-80-92" name="__codelineno-80-92" href="#__codelineno-80-92"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-93" name="__codelineno-80-93" href="#__codelineno-80-93"></a><span class="w">            </span><span class="n">arr</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">get</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
@@ -5911,44 +5983,44 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">my_list.cs</span><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="cm">/* List class */</span>
 <a id="__codelineno-81-2" name="__codelineno-81-2" href="#__codelineno-81-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">MyList</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-81-3" name="__codelineno-81-3" href="#__codelineno-81-3"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">arr</span><span class="p">;</span><span class="w">           </span><span class="c1">// Array (stores list elements)</span>
+<a id="__codelineno-81-3" name="__codelineno-81-3" href="#__codelineno-81-3"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">arr</span><span class="p">;</span><span class="w">           </span><span class="c1">// Array (stores the list elements)</span>
 <a id="__codelineno-81-4" name="__codelineno-81-4" href="#__codelineno-81-4"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">arrCapacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span><span class="w">    </span><span class="c1">// List capacity</span>
 <a id="__codelineno-81-5" name="__codelineno-81-5" href="#__codelineno-81-5"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">arrSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">         </span><span class="c1">// List length (current number of elements)</span>
-<a id="__codelineno-81-6" name="__codelineno-81-6" href="#__codelineno-81-6"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="k">readonly</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">  </span><span class="c1">// Multiple by which the list capacity is extended each time</span>
+<a id="__codelineno-81-6" name="__codelineno-81-6" href="#__codelineno-81-6"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="k">readonly</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">  </span><span class="c1">// Factor by which the list capacity grows each time</span>
 <a id="__codelineno-81-7" name="__codelineno-81-7" href="#__codelineno-81-7"></a>
 <a id="__codelineno-81-8" name="__codelineno-81-8" href="#__codelineno-81-8"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-81-9" name="__codelineno-81-9" href="#__codelineno-81-9"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nf">MyList</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-10" name="__codelineno-81-10" href="#__codelineno-81-10"></a><span class="w">        </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">arrCapacity</span><span class="p">];</span>
 <a id="__codelineno-81-11" name="__codelineno-81-11" href="#__codelineno-81-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-81-12" name="__codelineno-81-12" href="#__codelineno-81-12"></a>
-<a id="__codelineno-81-13" name="__codelineno-81-13" href="#__codelineno-81-13"></a><span class="w">    </span><span class="cm">/* Get list length (current number of elements) */</span>
+<a id="__codelineno-81-13" name="__codelineno-81-13" href="#__codelineno-81-13"></a><span class="w">    </span><span class="cm">/* Get the list length (current number of elements)*/</span>
 <a id="__codelineno-81-14" name="__codelineno-81-14" href="#__codelineno-81-14"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-15" name="__codelineno-81-15" href="#__codelineno-81-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">arrSize</span><span class="p">;</span>
 <a id="__codelineno-81-16" name="__codelineno-81-16" href="#__codelineno-81-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-81-17" name="__codelineno-81-17" href="#__codelineno-81-17"></a>
-<a id="__codelineno-81-18" name="__codelineno-81-18" href="#__codelineno-81-18"></a><span class="w">    </span><span class="cm">/* Get list capacity */</span>
+<a id="__codelineno-81-18" name="__codelineno-81-18" href="#__codelineno-81-18"></a><span class="w">    </span><span class="cm">/* Get the list capacity */</span>
 <a id="__codelineno-81-19" name="__codelineno-81-19" href="#__codelineno-81-19"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-20" name="__codelineno-81-20" href="#__codelineno-81-20"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">arrCapacity</span><span class="p">;</span>
 <a id="__codelineno-81-21" name="__codelineno-81-21" href="#__codelineno-81-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-81-22" name="__codelineno-81-22" href="#__codelineno-81-22"></a>
-<a id="__codelineno-81-23" name="__codelineno-81-23" href="#__codelineno-81-23"></a><span class="w">    </span><span class="cm">/* Update element */</span>
+<a id="__codelineno-81-23" name="__codelineno-81-23" href="#__codelineno-81-23"></a><span class="w">    </span><span class="cm">/* Access an element */</span>
 <a id="__codelineno-81-24" name="__codelineno-81-24" href="#__codelineno-81-24"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-81-25" name="__codelineno-81-25" href="#__codelineno-81-25"></a><span class="w">        </span><span class="c1">// If the index is out of bounds, throw an exception, as below</span>
+<a id="__codelineno-81-25" name="__codelineno-81-25" href="#__codelineno-81-25"></a><span class="w">        </span><span class="c1">// Throw an exception if the index is out of bounds; the same applies below</span>
 <a id="__codelineno-81-26" name="__codelineno-81-26" href="#__codelineno-81-26"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">arrSize</span><span class="p">)</span>
 <a id="__codelineno-81-27" name="__codelineno-81-27" href="#__codelineno-81-27"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">IndexOutOfRangeException</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">);</span>
 <a id="__codelineno-81-28" name="__codelineno-81-28" href="#__codelineno-81-28"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
 <a id="__codelineno-81-29" name="__codelineno-81-29" href="#__codelineno-81-29"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-81-30" name="__codelineno-81-30" href="#__codelineno-81-30"></a>
-<a id="__codelineno-81-31" name="__codelineno-81-31" href="#__codelineno-81-31"></a><span class="w">    </span><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-81-31" name="__codelineno-81-31" href="#__codelineno-81-31"></a><span class="w">    </span><span class="cm">/* Update an element */</span>
 <a id="__codelineno-81-32" name="__codelineno-81-32" href="#__codelineno-81-32"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">Set</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-33" name="__codelineno-81-33" href="#__codelineno-81-33"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">arrSize</span><span class="p">)</span>
 <a id="__codelineno-81-34" name="__codelineno-81-34" href="#__codelineno-81-34"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">IndexOutOfRangeException</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">);</span>
 <a id="__codelineno-81-35" name="__codelineno-81-35" href="#__codelineno-81-35"></a><span class="w">        </span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-81-36" name="__codelineno-81-36" href="#__codelineno-81-36"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-81-37" name="__codelineno-81-37" href="#__codelineno-81-37"></a>
-<a id="__codelineno-81-38" name="__codelineno-81-38" href="#__codelineno-81-38"></a><span class="w">    </span><span class="cm">/* Direct traversal of list elements */</span>
+<a id="__codelineno-81-38" name="__codelineno-81-38" href="#__codelineno-81-38"></a><span class="w">    </span><span class="cm">/* Append at the end */</span>
 <a id="__codelineno-81-39" name="__codelineno-81-39" href="#__codelineno-81-39"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">Add</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-81-40" name="__codelineno-81-40" href="#__codelineno-81-40"></a><span class="w">        </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-81-40" name="__codelineno-81-40" href="#__codelineno-81-40"></a><span class="w">        </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-81-41" name="__codelineno-81-41" href="#__codelineno-81-41"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">arrSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">arrCapacity</span><span class="p">)</span>
 <a id="__codelineno-81-42" name="__codelineno-81-42" href="#__codelineno-81-42"></a><span class="w">            </span><span class="n">ExtendCapacity</span><span class="p">();</span>
 <a id="__codelineno-81-43" name="__codelineno-81-43" href="#__codelineno-81-43"></a><span class="w">        </span><span class="n">arr</span><span class="p">[</span><span class="n">arrSize</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
@@ -5956,14 +6028,14 @@
 <a id="__codelineno-81-45" name="__codelineno-81-45" href="#__codelineno-81-45"></a><span class="w">        </span><span class="n">arrSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-81-46" name="__codelineno-81-46" href="#__codelineno-81-46"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-81-47" name="__codelineno-81-47" href="#__codelineno-81-47"></a>
-<a id="__codelineno-81-48" name="__codelineno-81-48" href="#__codelineno-81-48"></a><span class="w">    </span><span class="cm">/* Sort list */</span>
+<a id="__codelineno-81-48" name="__codelineno-81-48" href="#__codelineno-81-48"></a><span class="w">    </span><span class="cm">/* Insert an element in the middle */</span>
 <a id="__codelineno-81-49" name="__codelineno-81-49" href="#__codelineno-81-49"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">Insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-50" name="__codelineno-81-50" href="#__codelineno-81-50"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">arrSize</span><span class="p">)</span>
 <a id="__codelineno-81-51" name="__codelineno-81-51" href="#__codelineno-81-51"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">IndexOutOfRangeException</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">);</span>
-<a id="__codelineno-81-52" name="__codelineno-81-52" href="#__codelineno-81-52"></a><span class="w">        </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-81-52" name="__codelineno-81-52" href="#__codelineno-81-52"></a><span class="w">        </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-81-53" name="__codelineno-81-53" href="#__codelineno-81-53"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">arrSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">arrCapacity</span><span class="p">)</span>
 <a id="__codelineno-81-54" name="__codelineno-81-54" href="#__codelineno-81-54"></a><span class="w">            </span><span class="n">ExtendCapacity</span><span class="p">();</span>
-<a id="__codelineno-81-55" name="__codelineno-81-55" href="#__codelineno-81-55"></a><span class="w">        </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-81-55" name="__codelineno-81-55" href="#__codelineno-81-55"></a><span class="w">        </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-81-56" name="__codelineno-81-56" href="#__codelineno-81-56"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arrSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-57" name="__codelineno-81-57" href="#__codelineno-81-57"></a><span class="w">            </span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-81-58" name="__codelineno-81-58" href="#__codelineno-81-58"></a><span class="w">        </span><span class="p">}</span>
@@ -5972,12 +6044,12 @@
 <a id="__codelineno-81-61" name="__codelineno-81-61" href="#__codelineno-81-61"></a><span class="w">        </span><span class="n">arrSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-81-62" name="__codelineno-81-62" href="#__codelineno-81-62"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-81-63" name="__codelineno-81-63" href="#__codelineno-81-63"></a>
-<a id="__codelineno-81-64" name="__codelineno-81-64" href="#__codelineno-81-64"></a><span class="w">    </span><span class="cm">/* Remove element */</span>
+<a id="__codelineno-81-64" name="__codelineno-81-64" href="#__codelineno-81-64"></a><span class="w">    </span><span class="cm">/* Remove an element */</span>
 <a id="__codelineno-81-65" name="__codelineno-81-65" href="#__codelineno-81-65"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-66" name="__codelineno-81-66" href="#__codelineno-81-66"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">arrSize</span><span class="p">)</span>
 <a id="__codelineno-81-67" name="__codelineno-81-67" href="#__codelineno-81-67"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">IndexOutOfRangeException</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">);</span>
 <a id="__codelineno-81-68" name="__codelineno-81-68" href="#__codelineno-81-68"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
-<a id="__codelineno-81-69" name="__codelineno-81-69" href="#__codelineno-81-69"></a><span class="w">        </span><span class="c1">// Move all elements after index forward by one position</span>
+<a id="__codelineno-81-69" name="__codelineno-81-69" href="#__codelineno-81-69"></a><span class="w">        </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-81-70" name="__codelineno-81-70" href="#__codelineno-81-70"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">arrSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-71" name="__codelineno-81-71" href="#__codelineno-81-71"></a><span class="w">            </span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-81-72" name="__codelineno-81-72" href="#__codelineno-81-72"></a><span class="w">        </span><span class="p">}</span>
@@ -5987,17 +6059,17 @@
 <a id="__codelineno-81-76" name="__codelineno-81-76" href="#__codelineno-81-76"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-81-77" name="__codelineno-81-77" href="#__codelineno-81-77"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-81-78" name="__codelineno-81-78" href="#__codelineno-81-78"></a>
-<a id="__codelineno-81-79" name="__codelineno-81-79" href="#__codelineno-81-79"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-81-79" name="__codelineno-81-79" href="#__codelineno-81-79"></a><span class="w">    </span><span class="cm">/* Grow the list capacity */</span>
 <a id="__codelineno-81-80" name="__codelineno-81-80" href="#__codelineno-81-80"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">ExtendCapacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-81-81" name="__codelineno-81-81" href="#__codelineno-81-81"></a><span class="w">        </span><span class="c1">// Create new array of length arrCapacity * extendRatio and copy original array to new array</span>
+<a id="__codelineno-81-81" name="__codelineno-81-81" href="#__codelineno-81-81"></a><span class="w">        </span><span class="c1">// Create an array of length arrCapacity * extendRatio and copy the original array into it</span>
 <a id="__codelineno-81-82" name="__codelineno-81-82" href="#__codelineno-81-82"></a><span class="w">        </span><span class="n">Array</span><span class="p">.</span><span class="n">Resize</span><span class="p">(</span><span class="k">ref</span><span class="w"> </span><span class="n">arr</span><span class="p">,</span><span class="w"> </span><span class="n">arrCapacity</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">);</span>
-<a id="__codelineno-81-83" name="__codelineno-81-83" href="#__codelineno-81-83"></a><span class="w">        </span><span class="c1">// Add elements at the end</span>
+<a id="__codelineno-81-83" name="__codelineno-81-83" href="#__codelineno-81-83"></a><span class="w">        </span><span class="c1">// Update the list capacity</span>
 <a id="__codelineno-81-84" name="__codelineno-81-84" href="#__codelineno-81-84"></a><span class="w">        </span><span class="n">arrCapacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
 <a id="__codelineno-81-85" name="__codelineno-81-85" href="#__codelineno-81-85"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-81-86" name="__codelineno-81-86" href="#__codelineno-81-86"></a>
-<a id="__codelineno-81-87" name="__codelineno-81-87" href="#__codelineno-81-87"></a><span class="w">    </span><span class="cm">/* Convert list to array */</span>
+<a id="__codelineno-81-87" name="__codelineno-81-87" href="#__codelineno-81-87"></a><span class="w">    </span><span class="cm">/* Convert the list to an array */</span>
 <a id="__codelineno-81-88" name="__codelineno-81-88" href="#__codelineno-81-88"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="nf">ToArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-81-89" name="__codelineno-81-89" href="#__codelineno-81-89"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-81-89" name="__codelineno-81-89" href="#__codelineno-81-89"></a><span class="w">        </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-81-90" name="__codelineno-81-90" href="#__codelineno-81-90"></a><span class="w">        </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">arrSize</span><span class="p">];</span>
 <a id="__codelineno-81-91" name="__codelineno-81-91" href="#__codelineno-81-91"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">arrSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-92" name="__codelineno-81-92" href="#__codelineno-81-92"></a><span class="w">            </span><span class="n">arr</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Get</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
@@ -6008,7 +6080,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_list.go</span><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="cm">/* List class */</span>
+<div class="highlight"><span class="filename">my_list.go</span><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="cm">/* List type */</span>
 <a id="__codelineno-82-2" name="__codelineno-82-2" href="#__codelineno-82-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">myList</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-3" name="__codelineno-82-3" href="#__codelineno-82-3"></a><span class="w">    </span><span class="nx">arrCapacity</span><span class="w"> </span><span class="kt">int</span>
 <a id="__codelineno-82-4" name="__codelineno-82-4" href="#__codelineno-82-4"></a><span class="w">    </span><span class="nx">arr</span><span class="w">         </span><span class="p">[]</span><span class="kt">int</span>
@@ -6020,32 +6092,32 @@
 <a id="__codelineno-82-10" name="__codelineno-82-10" href="#__codelineno-82-10"></a><span class="kd">func</span><span class="w"> </span><span class="nx">newMyList</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="nx">myList</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-11" name="__codelineno-82-11" href="#__codelineno-82-11"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">myList</span><span class="p">{</span>
 <a id="__codelineno-82-12" name="__codelineno-82-12" href="#__codelineno-82-12"></a><span class="w">        </span><span class="nx">arrCapacity</span><span class="p">:</span><span class="w"> </span><span class="mi">10</span><span class="p">,</span><span class="w">              </span><span class="c1">// List capacity</span>
-<a id="__codelineno-82-13" name="__codelineno-82-13" href="#__codelineno-82-13"></a><span class="w">        </span><span class="nx">arr</span><span class="p">:</span><span class="w">         </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">10</span><span class="p">),</span><span class="w"> </span><span class="c1">// Array (stores list elements)</span>
+<a id="__codelineno-82-13" name="__codelineno-82-13" href="#__codelineno-82-13"></a><span class="w">        </span><span class="nx">arr</span><span class="p">:</span><span class="w">         </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">10</span><span class="p">),</span><span class="w"> </span><span class="c1">// Array (stores the list elements)</span>
 <a id="__codelineno-82-14" name="__codelineno-82-14" href="#__codelineno-82-14"></a><span class="w">        </span><span class="nx">arrSize</span><span class="p">:</span><span class="w">     </span><span class="mi">0</span><span class="p">,</span><span class="w">               </span><span class="c1">// List length (current number of elements)</span>
-<a id="__codelineno-82-15" name="__codelineno-82-15" href="#__codelineno-82-15"></a><span class="w">        </span><span class="nx">extendRatio</span><span class="p">:</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w">               </span><span class="c1">// Multiple by which the list capacity is extended each time</span>
+<a id="__codelineno-82-15" name="__codelineno-82-15" href="#__codelineno-82-15"></a><span class="w">        </span><span class="nx">extendRatio</span><span class="p">:</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w">               </span><span class="c1">// Factor by which the list capacity grows each time</span>
 <a id="__codelineno-82-16" name="__codelineno-82-16" href="#__codelineno-82-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-82-17" name="__codelineno-82-17" href="#__codelineno-82-17"></a><span class="p">}</span>
 <a id="__codelineno-82-18" name="__codelineno-82-18" href="#__codelineno-82-18"></a>
-<a id="__codelineno-82-19" name="__codelineno-82-19" href="#__codelineno-82-19"></a><span class="cm">/* Get list length (current number of elements) */</span>
+<a id="__codelineno-82-19" name="__codelineno-82-19" href="#__codelineno-82-19"></a><span class="cm">/* Get the list length (current number of elements) */</span>
 <a id="__codelineno-82-20" name="__codelineno-82-20" href="#__codelineno-82-20"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">l</span><span class="w"> </span><span class="o">*</span><span class="nx">myList</span><span class="p">)</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-21" name="__codelineno-82-21" href="#__codelineno-82-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrSize</span>
 <a id="__codelineno-82-22" name="__codelineno-82-22" href="#__codelineno-82-22"></a><span class="p">}</span>
 <a id="__codelineno-82-23" name="__codelineno-82-23" href="#__codelineno-82-23"></a>
-<a id="__codelineno-82-24" name="__codelineno-82-24" href="#__codelineno-82-24"></a><span class="cm">/* Get list capacity */</span>
+<a id="__codelineno-82-24" name="__codelineno-82-24" href="#__codelineno-82-24"></a><span class="cm">/*  Get the list capacity */</span>
 <a id="__codelineno-82-25" name="__codelineno-82-25" href="#__codelineno-82-25"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">l</span><span class="w"> </span><span class="o">*</span><span class="nx">myList</span><span class="p">)</span><span class="w"> </span><span class="nx">capacity</span><span class="p">()</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-26" name="__codelineno-82-26" href="#__codelineno-82-26"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrCapacity</span>
 <a id="__codelineno-82-27" name="__codelineno-82-27" href="#__codelineno-82-27"></a><span class="p">}</span>
 <a id="__codelineno-82-28" name="__codelineno-82-28" href="#__codelineno-82-28"></a>
-<a id="__codelineno-82-29" name="__codelineno-82-29" href="#__codelineno-82-29"></a><span class="cm">/* Update element */</span>
+<a id="__codelineno-82-29" name="__codelineno-82-29" href="#__codelineno-82-29"></a><span class="cm">/* Access an element */</span>
 <a id="__codelineno-82-30" name="__codelineno-82-30" href="#__codelineno-82-30"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">l</span><span class="w"> </span><span class="o">*</span><span class="nx">myList</span><span class="p">)</span><span class="w"> </span><span class="nx">get</span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-82-31" name="__codelineno-82-31" href="#__codelineno-82-31"></a><span class="w">    </span><span class="c1">// If the index is out of bounds, throw an exception, as below</span>
+<a id="__codelineno-82-31" name="__codelineno-82-31" href="#__codelineno-82-31"></a><span class="w">    </span><span class="c1">// Panic if the index is out of bounds; the same applies below</span>
 <a id="__codelineno-82-32" name="__codelineno-82-32" href="#__codelineno-82-32"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrSize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-33" name="__codelineno-82-33" href="#__codelineno-82-33"></a><span class="w">        </span><span class="nb">panic</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-82-34" name="__codelineno-82-34" href="#__codelineno-82-34"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-82-35" name="__codelineno-82-35" href="#__codelineno-82-35"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arr</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span>
 <a id="__codelineno-82-36" name="__codelineno-82-36" href="#__codelineno-82-36"></a><span class="p">}</span>
 <a id="__codelineno-82-37" name="__codelineno-82-37" href="#__codelineno-82-37"></a>
-<a id="__codelineno-82-38" name="__codelineno-82-38" href="#__codelineno-82-38"></a><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-82-38" name="__codelineno-82-38" href="#__codelineno-82-38"></a><span class="cm">/* Update an element */</span>
 <a id="__codelineno-82-39" name="__codelineno-82-39" href="#__codelineno-82-39"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">l</span><span class="w"> </span><span class="o">*</span><span class="nx">myList</span><span class="p">)</span><span class="w"> </span><span class="nx">set</span><span class="p">(</span><span class="nx">num</span><span class="p">,</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-40" name="__codelineno-82-40" href="#__codelineno-82-40"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrSize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-41" name="__codelineno-82-41" href="#__codelineno-82-41"></a><span class="w">        </span><span class="nb">panic</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
@@ -6053,9 +6125,9 @@
 <a id="__codelineno-82-43" name="__codelineno-82-43" href="#__codelineno-82-43"></a><span class="w">    </span><span class="nx">l</span><span class="p">.</span><span class="nx">arr</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span>
 <a id="__codelineno-82-44" name="__codelineno-82-44" href="#__codelineno-82-44"></a><span class="p">}</span>
 <a id="__codelineno-82-45" name="__codelineno-82-45" href="#__codelineno-82-45"></a>
-<a id="__codelineno-82-46" name="__codelineno-82-46" href="#__codelineno-82-46"></a><span class="cm">/* Direct traversal of list elements */</span>
+<a id="__codelineno-82-46" name="__codelineno-82-46" href="#__codelineno-82-46"></a><span class="cm">/* Append at the end */</span>
 <a id="__codelineno-82-47" name="__codelineno-82-47" href="#__codelineno-82-47"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">l</span><span class="w"> </span><span class="o">*</span><span class="nx">myList</span><span class="p">)</span><span class="w"> </span><span class="nx">add</span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-82-48" name="__codelineno-82-48" href="#__codelineno-82-48"></a><span class="w">    </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-82-48" name="__codelineno-82-48" href="#__codelineno-82-48"></a><span class="w">    </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-82-49" name="__codelineno-82-49" href="#__codelineno-82-49"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrCapacity</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-50" name="__codelineno-82-50" href="#__codelineno-82-50"></a><span class="w">        </span><span class="nx">l</span><span class="p">.</span><span class="nx">extendCapacity</span><span class="p">()</span>
 <a id="__codelineno-82-51" name="__codelineno-82-51" href="#__codelineno-82-51"></a><span class="w">    </span><span class="p">}</span>
@@ -6064,16 +6136,16 @@
 <a id="__codelineno-82-54" name="__codelineno-82-54" href="#__codelineno-82-54"></a><span class="w">    </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrSize</span><span class="o">++</span>
 <a id="__codelineno-82-55" name="__codelineno-82-55" href="#__codelineno-82-55"></a><span class="p">}</span>
 <a id="__codelineno-82-56" name="__codelineno-82-56" href="#__codelineno-82-56"></a>
-<a id="__codelineno-82-57" name="__codelineno-82-57" href="#__codelineno-82-57"></a><span class="cm">/* Sort list */</span>
+<a id="__codelineno-82-57" name="__codelineno-82-57" href="#__codelineno-82-57"></a><span class="cm">/* Insert an element in the middle */</span>
 <a id="__codelineno-82-58" name="__codelineno-82-58" href="#__codelineno-82-58"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">l</span><span class="w"> </span><span class="o">*</span><span class="nx">myList</span><span class="p">)</span><span class="w"> </span><span class="nx">insert</span><span class="p">(</span><span class="nx">num</span><span class="p">,</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-59" name="__codelineno-82-59" href="#__codelineno-82-59"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrSize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-60" name="__codelineno-82-60" href="#__codelineno-82-60"></a><span class="w">        </span><span class="nb">panic</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-82-61" name="__codelineno-82-61" href="#__codelineno-82-61"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-82-62" name="__codelineno-82-62" href="#__codelineno-82-62"></a><span class="w">    </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-82-62" name="__codelineno-82-62" href="#__codelineno-82-62"></a><span class="w">    </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-82-63" name="__codelineno-82-63" href="#__codelineno-82-63"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrCapacity</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-64" name="__codelineno-82-64" href="#__codelineno-82-64"></a><span class="w">        </span><span class="nx">l</span><span class="p">.</span><span class="nx">extendCapacity</span><span class="p">()</span>
 <a id="__codelineno-82-65" name="__codelineno-82-65" href="#__codelineno-82-65"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-82-66" name="__codelineno-82-66" href="#__codelineno-82-66"></a><span class="w">    </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-82-66" name="__codelineno-82-66" href="#__codelineno-82-66"></a><span class="w">    </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-82-67" name="__codelineno-82-67" href="#__codelineno-82-67"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">--</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-68" name="__codelineno-82-68" href="#__codelineno-82-68"></a><span class="w">        </span><span class="nx">l</span><span class="p">.</span><span class="nx">arr</span><span class="p">[</span><span class="nx">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arr</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span>
 <a id="__codelineno-82-69" name="__codelineno-82-69" href="#__codelineno-82-69"></a><span class="w">    </span><span class="p">}</span>
@@ -6082,13 +6154,13 @@
 <a id="__codelineno-82-72" name="__codelineno-82-72" href="#__codelineno-82-72"></a><span class="w">    </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrSize</span><span class="o">++</span>
 <a id="__codelineno-82-73" name="__codelineno-82-73" href="#__codelineno-82-73"></a><span class="p">}</span>
 <a id="__codelineno-82-74" name="__codelineno-82-74" href="#__codelineno-82-74"></a>
-<a id="__codelineno-82-75" name="__codelineno-82-75" href="#__codelineno-82-75"></a><span class="cm">/* Remove element */</span>
+<a id="__codelineno-82-75" name="__codelineno-82-75" href="#__codelineno-82-75"></a><span class="cm">/* Remove an element */</span>
 <a id="__codelineno-82-76" name="__codelineno-82-76" href="#__codelineno-82-76"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">l</span><span class="w"> </span><span class="o">*</span><span class="nx">myList</span><span class="p">)</span><span class="w"> </span><span class="nx">remove</span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-77" name="__codelineno-82-77" href="#__codelineno-82-77"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrSize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-78" name="__codelineno-82-78" href="#__codelineno-82-78"></a><span class="w">        </span><span class="nb">panic</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-82-79" name="__codelineno-82-79" href="#__codelineno-82-79"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-82-80" name="__codelineno-82-80" href="#__codelineno-82-80"></a><span class="w">    </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arr</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span>
-<a id="__codelineno-82-81" name="__codelineno-82-81" href="#__codelineno-82-81"></a><span class="w">    </span><span class="c1">// Create a new array with length _extend_ratio times the original array, and copy the original array to the new array</span>
+<a id="__codelineno-82-81" name="__codelineno-82-81" href="#__codelineno-82-81"></a><span class="w">    </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-82-82" name="__codelineno-82-82" href="#__codelineno-82-82"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrSize</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-83" name="__codelineno-82-83" href="#__codelineno-82-83"></a><span class="w">        </span><span class="nx">l</span><span class="p">.</span><span class="nx">arr</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arr</span><span class="p">[</span><span class="nx">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-82-84" name="__codelineno-82-84" href="#__codelineno-82-84"></a><span class="w">    </span><span class="p">}</span>
@@ -6098,17 +6170,17 @@
 <a id="__codelineno-82-88" name="__codelineno-82-88" href="#__codelineno-82-88"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span>
 <a id="__codelineno-82-89" name="__codelineno-82-89" href="#__codelineno-82-89"></a><span class="p">}</span>
 <a id="__codelineno-82-90" name="__codelineno-82-90" href="#__codelineno-82-90"></a>
-<a id="__codelineno-82-91" name="__codelineno-82-91" href="#__codelineno-82-91"></a><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-82-91" name="__codelineno-82-91" href="#__codelineno-82-91"></a><span class="cm">/* Grow the list capacity */</span>
 <a id="__codelineno-82-92" name="__codelineno-82-92" href="#__codelineno-82-92"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">l</span><span class="w"> </span><span class="o">*</span><span class="nx">myList</span><span class="p">)</span><span class="w"> </span><span class="nx">extendCapacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-82-93" name="__codelineno-82-93" href="#__codelineno-82-93"></a><span class="w">    </span><span class="c1">// Create a new array with length extendRatio times the original array and copy the original array to the new array</span>
+<a id="__codelineno-82-93" name="__codelineno-82-93" href="#__codelineno-82-93"></a><span class="w">    </span><span class="c1">// Extend the backing slice to extendRatio times its original length, preserving the existing elements</span>
 <a id="__codelineno-82-94" name="__codelineno-82-94" href="#__codelineno-82-94"></a><span class="w">    </span><span class="nx">l</span><span class="p">.</span><span class="nx">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">l</span><span class="p">.</span><span class="nx">arr</span><span class="p">,</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrCapacity</span><span class="o">*</span><span class="p">(</span><span class="nx">l</span><span class="p">.</span><span class="nx">extendRatio</span><span class="o">-</span><span class="mi">1</span><span class="p">))...)</span>
-<a id="__codelineno-82-95" name="__codelineno-82-95" href="#__codelineno-82-95"></a><span class="w">    </span><span class="c1">// Add elements at the end</span>
+<a id="__codelineno-82-95" name="__codelineno-82-95" href="#__codelineno-82-95"></a><span class="w">    </span><span class="c1">// Update the list capacity</span>
 <a id="__codelineno-82-96" name="__codelineno-82-96" href="#__codelineno-82-96"></a><span class="w">    </span><span class="nx">l</span><span class="p">.</span><span class="nx">arrCapacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">l</span><span class="p">.</span><span class="nx">arr</span><span class="p">)</span>
 <a id="__codelineno-82-97" name="__codelineno-82-97" href="#__codelineno-82-97"></a><span class="p">}</span>
 <a id="__codelineno-82-98" name="__codelineno-82-98" href="#__codelineno-82-98"></a>
-<a id="__codelineno-82-99" name="__codelineno-82-99" href="#__codelineno-82-99"></a><span class="cm">/* Return list with valid length */</span>
+<a id="__codelineno-82-99" name="__codelineno-82-99" href="#__codelineno-82-99"></a><span class="cm">/* Return the portion of the list that contains elements */</span>
 <a id="__codelineno-82-100" name="__codelineno-82-100" href="#__codelineno-82-100"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">l</span><span class="w"> </span><span class="o">*</span><span class="nx">myList</span><span class="p">)</span><span class="w"> </span><span class="nx">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-82-101" name="__codelineno-82-101" href="#__codelineno-82-101"></a><span class="w">    </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-82-101" name="__codelineno-82-101" href="#__codelineno-82-101"></a><span class="w">    </span><span class="c1">// Return a slice containing only the occupied elements</span>
 <a id="__codelineno-82-102" name="__codelineno-82-102" href="#__codelineno-82-102"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">l</span><span class="p">.</span><span class="nx">arr</span><span class="p">[:</span><span class="nx">l</span><span class="p">.</span><span class="nx">arrSize</span><span class="p">]</span>
 <a id="__codelineno-82-103" name="__codelineno-82-103" href="#__codelineno-82-103"></a><span class="p">}</span>
 </code></pre></div>
@@ -6116,10 +6188,10 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">my_list.swift</span><pre><span></span><code><a id="__codelineno-83-1" name="__codelineno-83-1" href="#__codelineno-83-1"></a><span class="cm">/* List class */</span>
 <a id="__codelineno-83-2" name="__codelineno-83-2" href="#__codelineno-83-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">MyList</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-83-3" name="__codelineno-83-3" href="#__codelineno-83-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">arr</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="c1">// Array (stores list elements)</span>
+<a id="__codelineno-83-3" name="__codelineno-83-3" href="#__codelineno-83-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">arr</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="c1">// Array (stores the list elements)</span>
 <a id="__codelineno-83-4" name="__codelineno-83-4" href="#__codelineno-83-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">_capacity</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// List capacity</span>
 <a id="__codelineno-83-5" name="__codelineno-83-5" href="#__codelineno-83-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">_size</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// List length (current number of elements)</span>
-<a id="__codelineno-83-6" name="__codelineno-83-6" href="#__codelineno-83-6"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nv">extendRatio</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Multiple by which the list capacity is extended each time</span>
+<a id="__codelineno-83-6" name="__codelineno-83-6" href="#__codelineno-83-6"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nv">extendRatio</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Factor by which the list capacity grows each time</span>
 <a id="__codelineno-83-7" name="__codelineno-83-7" href="#__codelineno-83-7"></a>
 <a id="__codelineno-83-8" name="__codelineno-83-8" href="#__codelineno-83-8"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-83-9" name="__codelineno-83-9" href="#__codelineno-83-9"></a><span class="w">    </span><span class="kd">init</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
@@ -6129,26 +6201,26 @@
 <a id="__codelineno-83-13" name="__codelineno-83-13" href="#__codelineno-83-13"></a><span class="w">        </span><span class="n">arr</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">_capacity</span><span class="p">)</span>
 <a id="__codelineno-83-14" name="__codelineno-83-14" href="#__codelineno-83-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-83-15" name="__codelineno-83-15" href="#__codelineno-83-15"></a>
-<a id="__codelineno-83-16" name="__codelineno-83-16" href="#__codelineno-83-16"></a><span class="w">    </span><span class="cm">/* Get list length (current number of elements) */</span>
+<a id="__codelineno-83-16" name="__codelineno-83-16" href="#__codelineno-83-16"></a><span class="w">    </span><span class="cm">/* Get the list length (current number of elements)*/</span>
 <a id="__codelineno-83-17" name="__codelineno-83-17" href="#__codelineno-83-17"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-18" name="__codelineno-83-18" href="#__codelineno-83-18"></a><span class="w">        </span><span class="n">_size</span>
 <a id="__codelineno-83-19" name="__codelineno-83-19" href="#__codelineno-83-19"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-83-20" name="__codelineno-83-20" href="#__codelineno-83-20"></a>
-<a id="__codelineno-83-21" name="__codelineno-83-21" href="#__codelineno-83-21"></a><span class="w">    </span><span class="cm">/* Get list capacity */</span>
+<a id="__codelineno-83-21" name="__codelineno-83-21" href="#__codelineno-83-21"></a><span class="w">    </span><span class="cm">/* Get the list capacity */</span>
 <a id="__codelineno-83-22" name="__codelineno-83-22" href="#__codelineno-83-22"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-23" name="__codelineno-83-23" href="#__codelineno-83-23"></a><span class="w">        </span><span class="n">_capacity</span>
 <a id="__codelineno-83-24" name="__codelineno-83-24" href="#__codelineno-83-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-83-25" name="__codelineno-83-25" href="#__codelineno-83-25"></a>
-<a id="__codelineno-83-26" name="__codelineno-83-26" href="#__codelineno-83-26"></a><span class="w">    </span><span class="cm">/* Update element */</span>
+<a id="__codelineno-83-26" name="__codelineno-83-26" href="#__codelineno-83-26"></a><span class="w">    </span><span class="cm">/* Access an element */</span>
 <a id="__codelineno-83-27" name="__codelineno-83-27" href="#__codelineno-83-27"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-83-28" name="__codelineno-83-28" href="#__codelineno-83-28"></a><span class="w">        </span><span class="c1">// Throw error if index out of bounds, same below</span>
+<a id="__codelineno-83-28" name="__codelineno-83-28" href="#__codelineno-83-28"></a><span class="w">        </span><span class="c1">// An out-of-bounds index triggers a fatal error; the same check is used below</span>
 <a id="__codelineno-83-29" name="__codelineno-83-29" href="#__codelineno-83-29"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-30" name="__codelineno-83-30" href="#__codelineno-83-30"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-83-31" name="__codelineno-83-31" href="#__codelineno-83-31"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-83-32" name="__codelineno-83-32" href="#__codelineno-83-32"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
 <a id="__codelineno-83-33" name="__codelineno-83-33" href="#__codelineno-83-33"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-83-34" name="__codelineno-83-34" href="#__codelineno-83-34"></a>
-<a id="__codelineno-83-35" name="__codelineno-83-35" href="#__codelineno-83-35"></a><span class="w">    </span><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-83-35" name="__codelineno-83-35" href="#__codelineno-83-35"></a><span class="w">    </span><span class="cm">/* Update an element */</span>
 <a id="__codelineno-83-36" name="__codelineno-83-36" href="#__codelineno-83-36"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">set</span><span class="p">(</span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-37" name="__codelineno-83-37" href="#__codelineno-83-37"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-38" name="__codelineno-83-38" href="#__codelineno-83-38"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
@@ -6156,9 +6228,9 @@
 <a id="__codelineno-83-40" name="__codelineno-83-40" href="#__codelineno-83-40"></a><span class="w">        </span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-83-41" name="__codelineno-83-41" href="#__codelineno-83-41"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-83-42" name="__codelineno-83-42" href="#__codelineno-83-42"></a>
-<a id="__codelineno-83-43" name="__codelineno-83-43" href="#__codelineno-83-43"></a><span class="w">    </span><span class="cm">/* Direct traversal of list elements */</span>
+<a id="__codelineno-83-43" name="__codelineno-83-43" href="#__codelineno-83-43"></a><span class="w">    </span><span class="cm">/* Append at the end */</span>
 <a id="__codelineno-83-44" name="__codelineno-83-44" href="#__codelineno-83-44"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">add</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-83-45" name="__codelineno-83-45" href="#__codelineno-83-45"></a><span class="w">        </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-83-45" name="__codelineno-83-45" href="#__codelineno-83-45"></a><span class="w">        </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-83-46" name="__codelineno-83-46" href="#__codelineno-83-46"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-47" name="__codelineno-83-47" href="#__codelineno-83-47"></a><span class="w">            </span><span class="n">extendCapacity</span><span class="p">()</span>
 <a id="__codelineno-83-48" name="__codelineno-83-48" href="#__codelineno-83-48"></a><span class="w">        </span><span class="p">}</span>
@@ -6167,16 +6239,16 @@
 <a id="__codelineno-83-51" name="__codelineno-83-51" href="#__codelineno-83-51"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-83-52" name="__codelineno-83-52" href="#__codelineno-83-52"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-83-53" name="__codelineno-83-53" href="#__codelineno-83-53"></a>
-<a id="__codelineno-83-54" name="__codelineno-83-54" href="#__codelineno-83-54"></a><span class="w">    </span><span class="cm">/* Sort list */</span>
+<a id="__codelineno-83-54" name="__codelineno-83-54" href="#__codelineno-83-54"></a><span class="w">    </span><span class="cm">/* Insert an element in the middle */</span>
 <a id="__codelineno-83-55" name="__codelineno-83-55" href="#__codelineno-83-55"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-56" name="__codelineno-83-56" href="#__codelineno-83-56"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-57" name="__codelineno-83-57" href="#__codelineno-83-57"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-83-58" name="__codelineno-83-58" href="#__codelineno-83-58"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-83-59" name="__codelineno-83-59" href="#__codelineno-83-59"></a><span class="w">        </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-83-59" name="__codelineno-83-59" href="#__codelineno-83-59"></a><span class="w">        </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-83-60" name="__codelineno-83-60" href="#__codelineno-83-60"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-61" name="__codelineno-83-61" href="#__codelineno-83-61"></a><span class="w">            </span><span class="n">extendCapacity</span><span class="p">()</span>
 <a id="__codelineno-83-62" name="__codelineno-83-62" href="#__codelineno-83-62"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-83-63" name="__codelineno-83-63" href="#__codelineno-83-63"></a><span class="w">        </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-83-63" name="__codelineno-83-63" href="#__codelineno-83-63"></a><span class="w">        </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-83-64" name="__codelineno-83-64" href="#__codelineno-83-64"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()).</span><span class="n">reversed</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-65" name="__codelineno-83-65" href="#__codelineno-83-65"></a><span class="w">            </span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-83-66" name="__codelineno-83-66" href="#__codelineno-83-66"></a><span class="w">        </span><span class="p">}</span>
@@ -6185,14 +6257,14 @@
 <a id="__codelineno-83-69" name="__codelineno-83-69" href="#__codelineno-83-69"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-83-70" name="__codelineno-83-70" href="#__codelineno-83-70"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-83-71" name="__codelineno-83-71" href="#__codelineno-83-71"></a>
-<a id="__codelineno-83-72" name="__codelineno-83-72" href="#__codelineno-83-72"></a><span class="w">    </span><span class="cm">/* Remove element */</span>
+<a id="__codelineno-83-72" name="__codelineno-83-72" href="#__codelineno-83-72"></a><span class="w">    </span><span class="cm">/* Remove an element */</span>
 <a id="__codelineno-83-73" name="__codelineno-83-73" href="#__codelineno-83-73"></a><span class="w">    </span><span class="p">@</span><span class="n">discardableResult</span>
 <a id="__codelineno-83-74" name="__codelineno-83-74" href="#__codelineno-83-74"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-75" name="__codelineno-83-75" href="#__codelineno-83-75"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-76" name="__codelineno-83-76" href="#__codelineno-83-76"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-83-77" name="__codelineno-83-77" href="#__codelineno-83-77"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-83-78" name="__codelineno-83-78" href="#__codelineno-83-78"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
-<a id="__codelineno-83-79" name="__codelineno-83-79" href="#__codelineno-83-79"></a><span class="w">        </span><span class="c1">// Move all elements after index forward by one position</span>
+<a id="__codelineno-83-79" name="__codelineno-83-79" href="#__codelineno-83-79"></a><span class="w">        </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-83-80" name="__codelineno-83-80" href="#__codelineno-83-80"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-81" name="__codelineno-83-81" href="#__codelineno-83-81"></a><span class="w">            </span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-83-82" name="__codelineno-83-82" href="#__codelineno-83-82"></a><span class="w">        </span><span class="p">}</span>
@@ -6202,15 +6274,15 @@
 <a id="__codelineno-83-86" name="__codelineno-83-86" href="#__codelineno-83-86"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-83-87" name="__codelineno-83-87" href="#__codelineno-83-87"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-83-88" name="__codelineno-83-88" href="#__codelineno-83-88"></a>
-<a id="__codelineno-83-89" name="__codelineno-83-89" href="#__codelineno-83-89"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-83-89" name="__codelineno-83-89" href="#__codelineno-83-89"></a><span class="w">    </span><span class="cm">/* Grow the list capacity */</span>
 <a id="__codelineno-83-90" name="__codelineno-83-90" href="#__codelineno-83-90"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">extendCapacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-83-91" name="__codelineno-83-91" href="#__codelineno-83-91"></a><span class="w">        </span><span class="c1">// Create a new array with length extendRatio times the original array and copy the original array to the new array</span>
+<a id="__codelineno-83-91" name="__codelineno-83-91" href="#__codelineno-83-91"></a><span class="w">        </span><span class="c1">// Create an array extendRatio times as long as the original and copy the original array into it</span>
 <a id="__codelineno-83-92" name="__codelineno-83-92" href="#__codelineno-83-92"></a><span class="w">        </span><span class="n">arr</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">extendRatio</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span>
-<a id="__codelineno-83-93" name="__codelineno-83-93" href="#__codelineno-83-93"></a><span class="w">        </span><span class="c1">// Add elements at the end</span>
+<a id="__codelineno-83-93" name="__codelineno-83-93" href="#__codelineno-83-93"></a><span class="w">        </span><span class="c1">// Update the list capacity</span>
 <a id="__codelineno-83-94" name="__codelineno-83-94" href="#__codelineno-83-94"></a><span class="w">        </span><span class="n">_capacity</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">arr</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-83-95" name="__codelineno-83-95" href="#__codelineno-83-95"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-83-96" name="__codelineno-83-96" href="#__codelineno-83-96"></a>
-<a id="__codelineno-83-97" name="__codelineno-83-97" href="#__codelineno-83-97"></a><span class="w">    </span><span class="cm">/* Convert list to array */</span>
+<a id="__codelineno-83-97" name="__codelineno-83-97" href="#__codelineno-83-97"></a><span class="w">    </span><span class="cm">/* Convert the list to an array */</span>
 <a id="__codelineno-83-98" name="__codelineno-83-98" href="#__codelineno-83-98"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-99" name="__codelineno-83-99" href="#__codelineno-83-99"></a><span class="w">        </span><span class="nb">Array</span><span class="p">(</span><span class="n">arr</span><span class="p">.</span><span class="kr">prefix</span><span class="p">(</span><span class="n">size</span><span class="p">()))</span>
 <a id="__codelineno-83-100" name="__codelineno-83-100" href="#__codelineno-83-100"></a><span class="w">    </span><span class="p">}</span>
@@ -6220,58 +6292,58 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">my_list.js</span><pre><span></span><code><a id="__codelineno-84-1" name="__codelineno-84-1" href="#__codelineno-84-1"></a><span class="cm">/* List class */</span>
 <a id="__codelineno-84-2" name="__codelineno-84-2" href="#__codelineno-84-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">MyList</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-84-3" name="__codelineno-84-3" href="#__codelineno-84-3"></a><span class="w">    </span><span class="n">#arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">();</span><span class="w"> </span><span class="c1">// Array (stores list elements)</span>
+<a id="__codelineno-84-3" name="__codelineno-84-3" href="#__codelineno-84-3"></a><span class="w">    </span><span class="n">#arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">();</span><span class="w"> </span><span class="c1">// Array (stores the list elements)</span>
 <a id="__codelineno-84-4" name="__codelineno-84-4" href="#__codelineno-84-4"></a><span class="w">    </span><span class="n">#capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">10</span><span class="p">;</span><span class="w"> </span><span class="c1">// List capacity</span>
 <a id="__codelineno-84-5" name="__codelineno-84-5" href="#__codelineno-84-5"></a><span class="w">    </span><span class="n">#size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// List length (current number of elements)</span>
-<a id="__codelineno-84-6" name="__codelineno-84-6" href="#__codelineno-84-6"></a><span class="w">    </span><span class="n">#extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Multiple by which the list capacity is extended each time</span>
+<a id="__codelineno-84-6" name="__codelineno-84-6" href="#__codelineno-84-6"></a><span class="w">    </span><span class="n">#extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Factor by which the list capacity grows each time</span>
 <a id="__codelineno-84-7" name="__codelineno-84-7" href="#__codelineno-84-7"></a>
 <a id="__codelineno-84-8" name="__codelineno-84-8" href="#__codelineno-84-8"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-84-9" name="__codelineno-84-9" href="#__codelineno-84-9"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-10" name="__codelineno-84-10" href="#__codelineno-84-10"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="p">);</span>
 <a id="__codelineno-84-11" name="__codelineno-84-11" href="#__codelineno-84-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-84-12" name="__codelineno-84-12" href="#__codelineno-84-12"></a>
-<a id="__codelineno-84-13" name="__codelineno-84-13" href="#__codelineno-84-13"></a><span class="w">    </span><span class="cm">/* Get list length (current number of elements) */</span>
+<a id="__codelineno-84-13" name="__codelineno-84-13" href="#__codelineno-84-13"></a><span class="w">    </span><span class="cm">/* Get the list length (current number of elements)*/</span>
 <a id="__codelineno-84-14" name="__codelineno-84-14" href="#__codelineno-84-14"></a><span class="w">    </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-15" name="__codelineno-84-15" href="#__codelineno-84-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="p">;</span>
 <a id="__codelineno-84-16" name="__codelineno-84-16" href="#__codelineno-84-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-84-17" name="__codelineno-84-17" href="#__codelineno-84-17"></a>
-<a id="__codelineno-84-18" name="__codelineno-84-18" href="#__codelineno-84-18"></a><span class="w">    </span><span class="cm">/* Get list capacity */</span>
+<a id="__codelineno-84-18" name="__codelineno-84-18" href="#__codelineno-84-18"></a><span class="w">    </span><span class="cm">/* Get the list capacity */</span>
 <a id="__codelineno-84-19" name="__codelineno-84-19" href="#__codelineno-84-19"></a><span class="w">    </span><span class="nx">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-20" name="__codelineno-84-20" href="#__codelineno-84-20"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="p">;</span>
 <a id="__codelineno-84-21" name="__codelineno-84-21" href="#__codelineno-84-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-84-22" name="__codelineno-84-22" href="#__codelineno-84-22"></a>
-<a id="__codelineno-84-23" name="__codelineno-84-23" href="#__codelineno-84-23"></a><span class="w">    </span><span class="cm">/* Update element */</span>
+<a id="__codelineno-84-23" name="__codelineno-84-23" href="#__codelineno-84-23"></a><span class="w">    </span><span class="cm">/* Access an element */</span>
 <a id="__codelineno-84-24" name="__codelineno-84-24" href="#__codelineno-84-24"></a><span class="w">    </span><span class="nx">get</span><span class="p">(</span><span class="nx">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-84-25" name="__codelineno-84-25" href="#__codelineno-84-25"></a><span class="w">        </span><span class="c1">// If the index is out of bounds, throw an exception, as below</span>
+<a id="__codelineno-84-25" name="__codelineno-84-25" href="#__codelineno-84-25"></a><span class="w">        </span><span class="c1">// Throw an exception if the index is out of bounds; the same applies below</span>
 <a id="__codelineno-84-26" name="__codelineno-84-26" href="#__codelineno-84-26"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
 <a id="__codelineno-84-27" name="__codelineno-84-27" href="#__codelineno-84-27"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#arr</span><span class="p">[</span><span class="nx">index</span><span class="p">];</span>
 <a id="__codelineno-84-28" name="__codelineno-84-28" href="#__codelineno-84-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-84-29" name="__codelineno-84-29" href="#__codelineno-84-29"></a>
-<a id="__codelineno-84-30" name="__codelineno-84-30" href="#__codelineno-84-30"></a><span class="w">    </span><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-84-30" name="__codelineno-84-30" href="#__codelineno-84-30"></a><span class="w">    </span><span class="cm">/* Update an element */</span>
 <a id="__codelineno-84-31" name="__codelineno-84-31" href="#__codelineno-84-31"></a><span class="w">    </span><span class="nx">set</span><span class="p">(</span><span class="nx">index</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-32" name="__codelineno-84-32" href="#__codelineno-84-32"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
 <a id="__codelineno-84-33" name="__codelineno-84-33" href="#__codelineno-84-33"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#arr</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-84-34" name="__codelineno-84-34" href="#__codelineno-84-34"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-84-35" name="__codelineno-84-35" href="#__codelineno-84-35"></a>
-<a id="__codelineno-84-36" name="__codelineno-84-36" href="#__codelineno-84-36"></a><span class="w">    </span><span class="cm">/* Direct traversal of list elements */</span>
+<a id="__codelineno-84-36" name="__codelineno-84-36" href="#__codelineno-84-36"></a><span class="w">    </span><span class="cm">/* Append an element at the end */</span>
 <a id="__codelineno-84-37" name="__codelineno-84-37" href="#__codelineno-84-37"></a><span class="w">    </span><span class="nx">add</span><span class="p">(</span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-84-38" name="__codelineno-84-38" href="#__codelineno-84-38"></a><span class="w">        </span><span class="c1">// If length equals capacity, need to expand</span>
+<a id="__codelineno-84-38" name="__codelineno-84-38" href="#__codelineno-84-38"></a><span class="w">        </span><span class="c1">// Grow the capacity if the length equals the capacity</span>
 <a id="__codelineno-84-39" name="__codelineno-84-39" href="#__codelineno-84-39"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-40" name="__codelineno-84-40" href="#__codelineno-84-40"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">extendCapacity</span><span class="p">();</span>
 <a id="__codelineno-84-41" name="__codelineno-84-41" href="#__codelineno-84-41"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-84-42" name="__codelineno-84-42" href="#__codelineno-84-42"></a><span class="w">        </span><span class="c1">// Add new element to end of list</span>
+<a id="__codelineno-84-42" name="__codelineno-84-42" href="#__codelineno-84-42"></a><span class="w">        </span><span class="c1">// Append the new element to the end of the list</span>
 <a id="__codelineno-84-43" name="__codelineno-84-43" href="#__codelineno-84-43"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#arr</span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-84-44" name="__codelineno-84-44" href="#__codelineno-84-44"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-84-45" name="__codelineno-84-45" href="#__codelineno-84-45"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-84-46" name="__codelineno-84-46" href="#__codelineno-84-46"></a>
-<a id="__codelineno-84-47" name="__codelineno-84-47" href="#__codelineno-84-47"></a><span class="w">    </span><span class="cm">/* Sort list */</span>
+<a id="__codelineno-84-47" name="__codelineno-84-47" href="#__codelineno-84-47"></a><span class="w">    </span><span class="cm">/* Insert an element in the middle */</span>
 <a id="__codelineno-84-48" name="__codelineno-84-48" href="#__codelineno-84-48"></a><span class="w">    </span><span class="nx">insert</span><span class="p">(</span><span class="nx">index</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-49" name="__codelineno-84-49" href="#__codelineno-84-49"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
-<a id="__codelineno-84-50" name="__codelineno-84-50" href="#__codelineno-84-50"></a><span class="w">        </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-84-50" name="__codelineno-84-50" href="#__codelineno-84-50"></a><span class="w">        </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-84-51" name="__codelineno-84-51" href="#__codelineno-84-51"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-52" name="__codelineno-84-52" href="#__codelineno-84-52"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">extendCapacity</span><span class="p">();</span>
 <a id="__codelineno-84-53" name="__codelineno-84-53" href="#__codelineno-84-53"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-84-54" name="__codelineno-84-54" href="#__codelineno-84-54"></a><span class="w">        </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-84-54" name="__codelineno-84-54" href="#__codelineno-84-54"></a><span class="w">        </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-84-55" name="__codelineno-84-55" href="#__codelineno-84-55"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-56" name="__codelineno-84-56" href="#__codelineno-84-56"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#arr</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#arr</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-84-57" name="__codelineno-84-57" href="#__codelineno-84-57"></a><span class="w">        </span><span class="p">}</span>
@@ -6280,11 +6352,11 @@
 <a id="__codelineno-84-60" name="__codelineno-84-60" href="#__codelineno-84-60"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-84-61" name="__codelineno-84-61" href="#__codelineno-84-61"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-84-62" name="__codelineno-84-62" href="#__codelineno-84-62"></a>
-<a id="__codelineno-84-63" name="__codelineno-84-63" href="#__codelineno-84-63"></a><span class="w">    </span><span class="cm">/* Remove element */</span>
+<a id="__codelineno-84-63" name="__codelineno-84-63" href="#__codelineno-84-63"></a><span class="w">    </span><span class="cm">/* Remove an element */</span>
 <a id="__codelineno-84-64" name="__codelineno-84-64" href="#__codelineno-84-64"></a><span class="w">    </span><span class="nx">remove</span><span class="p">(</span><span class="nx">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-65" name="__codelineno-84-65" href="#__codelineno-84-65"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
 <a id="__codelineno-84-66" name="__codelineno-84-66" href="#__codelineno-84-66"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#arr</span><span class="p">[</span><span class="nx">index</span><span class="p">];</span>
-<a id="__codelineno-84-67" name="__codelineno-84-67" href="#__codelineno-84-67"></a><span class="w">        </span><span class="c1">// Create a new array with length _extend_ratio times the original array, and copy the original array to the new array</span>
+<a id="__codelineno-84-67" name="__codelineno-84-67" href="#__codelineno-84-67"></a><span class="w">        </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-84-68" name="__codelineno-84-68" href="#__codelineno-84-68"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-69" name="__codelineno-84-69" href="#__codelineno-84-69"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#arr</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#arr</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-84-70" name="__codelineno-84-70" href="#__codelineno-84-70"></a><span class="w">        </span><span class="p">}</span>
@@ -6294,20 +6366,20 @@
 <a id="__codelineno-84-74" name="__codelineno-84-74" href="#__codelineno-84-74"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-84-75" name="__codelineno-84-75" href="#__codelineno-84-75"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-84-76" name="__codelineno-84-76" href="#__codelineno-84-76"></a>
-<a id="__codelineno-84-77" name="__codelineno-84-77" href="#__codelineno-84-77"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-84-77" name="__codelineno-84-77" href="#__codelineno-84-77"></a><span class="w">    </span><span class="cm">/* Grow the list capacity */</span>
 <a id="__codelineno-84-78" name="__codelineno-84-78" href="#__codelineno-84-78"></a><span class="w">    </span><span class="nx">extendCapacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-84-79" name="__codelineno-84-79" href="#__codelineno-84-79"></a><span class="w">        </span><span class="c1">// Create a new array with length extendRatio times the original array and copy the original array to the new array</span>
+<a id="__codelineno-84-79" name="__codelineno-84-79" href="#__codelineno-84-79"></a><span class="w">        </span><span class="c1">// Create an array extendRatio times the original length and copy the original array into it</span>
 <a id="__codelineno-84-80" name="__codelineno-84-80" href="#__codelineno-84-80"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#arr</span><span class="p">.</span><span class="nx">concat</span><span class="p">(</span>
 <a id="__codelineno-84-81" name="__codelineno-84-81" href="#__codelineno-84-81"></a><span class="w">            </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#extendRatio</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">))</span>
 <a id="__codelineno-84-82" name="__codelineno-84-82" href="#__codelineno-84-82"></a><span class="w">        </span><span class="p">);</span>
-<a id="__codelineno-84-83" name="__codelineno-84-83" href="#__codelineno-84-83"></a><span class="w">        </span><span class="c1">// Add elements at the end</span>
+<a id="__codelineno-84-83" name="__codelineno-84-83" href="#__codelineno-84-83"></a><span class="w">        </span><span class="c1">// Update the list capacity</span>
 <a id="__codelineno-84-84" name="__codelineno-84-84" href="#__codelineno-84-84"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#arr</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-84-85" name="__codelineno-84-85" href="#__codelineno-84-85"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-84-86" name="__codelineno-84-86" href="#__codelineno-84-86"></a>
-<a id="__codelineno-84-87" name="__codelineno-84-87" href="#__codelineno-84-87"></a><span class="w">    </span><span class="cm">/* Convert list to array */</span>
+<a id="__codelineno-84-87" name="__codelineno-84-87" href="#__codelineno-84-87"></a><span class="w">    </span><span class="cm">/* Convert the list to an array */</span>
 <a id="__codelineno-84-88" name="__codelineno-84-88" href="#__codelineno-84-88"></a><span class="w">    </span><span class="nx">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-89" name="__codelineno-84-89" href="#__codelineno-84-89"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">();</span>
-<a id="__codelineno-84-90" name="__codelineno-84-90" href="#__codelineno-84-90"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-84-90" name="__codelineno-84-90" href="#__codelineno-84-90"></a><span class="w">        </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-84-91" name="__codelineno-84-91" href="#__codelineno-84-91"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">size</span><span class="p">);</span>
 <a id="__codelineno-84-92" name="__codelineno-84-92" href="#__codelineno-84-92"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">size</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-93" name="__codelineno-84-93" href="#__codelineno-84-93"></a><span class="w">            </span><span class="nx">arr</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">i</span><span class="p">);</span>
@@ -6320,56 +6392,56 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">my_list.ts</span><pre><span></span><code><a id="__codelineno-85-1" name="__codelineno-85-1" href="#__codelineno-85-1"></a><span class="cm">/* List class */</span>
 <a id="__codelineno-85-2" name="__codelineno-85-2" href="#__codelineno-85-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">MyList</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-85-3" name="__codelineno-85-3" href="#__codelineno-85-3"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">arr</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array (stores list elements)</span>
+<a id="__codelineno-85-3" name="__codelineno-85-3" href="#__codelineno-85-3"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">arr</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array (stores the list elements)</span>
 <a id="__codelineno-85-4" name="__codelineno-85-4" href="#__codelineno-85-4"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">_capacity</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">10</span><span class="p">;</span><span class="w"> </span><span class="c1">// List capacity</span>
 <a id="__codelineno-85-5" name="__codelineno-85-5" href="#__codelineno-85-5"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">_size</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// List length (current number of elements)</span>
-<a id="__codelineno-85-6" name="__codelineno-85-6" href="#__codelineno-85-6"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">extendRatio</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Multiple by which the list capacity is extended each time</span>
+<a id="__codelineno-85-6" name="__codelineno-85-6" href="#__codelineno-85-6"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">extendRatio</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Factor by which the list capacity grows each time</span>
 <a id="__codelineno-85-7" name="__codelineno-85-7" href="#__codelineno-85-7"></a>
 <a id="__codelineno-85-8" name="__codelineno-85-8" href="#__codelineno-85-8"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-85-9" name="__codelineno-85-9" href="#__codelineno-85-9"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-10" name="__codelineno-85-10" href="#__codelineno-85-10"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">_capacity</span><span class="p">);</span>
 <a id="__codelineno-85-11" name="__codelineno-85-11" href="#__codelineno-85-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-85-12" name="__codelineno-85-12" href="#__codelineno-85-12"></a>
-<a id="__codelineno-85-13" name="__codelineno-85-13" href="#__codelineno-85-13"></a><span class="w">    </span><span class="cm">/* Get list length (current number of elements) */</span>
+<a id="__codelineno-85-13" name="__codelineno-85-13" href="#__codelineno-85-13"></a><span class="w">    </span><span class="cm">/* Get the list length (current number of elements)*/</span>
 <a id="__codelineno-85-14" name="__codelineno-85-14" href="#__codelineno-85-14"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-15" name="__codelineno-85-15" href="#__codelineno-85-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">_size</span><span class="p">;</span>
 <a id="__codelineno-85-16" name="__codelineno-85-16" href="#__codelineno-85-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-85-17" name="__codelineno-85-17" href="#__codelineno-85-17"></a>
-<a id="__codelineno-85-18" name="__codelineno-85-18" href="#__codelineno-85-18"></a><span class="w">    </span><span class="cm">/* Get list capacity */</span>
+<a id="__codelineno-85-18" name="__codelineno-85-18" href="#__codelineno-85-18"></a><span class="w">    </span><span class="cm">/* Get the list capacity */</span>
 <a id="__codelineno-85-19" name="__codelineno-85-19" href="#__codelineno-85-19"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nx">capacity</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-20" name="__codelineno-85-20" href="#__codelineno-85-20"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">_capacity</span><span class="p">;</span>
 <a id="__codelineno-85-21" name="__codelineno-85-21" href="#__codelineno-85-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-85-22" name="__codelineno-85-22" href="#__codelineno-85-22"></a>
-<a id="__codelineno-85-23" name="__codelineno-85-23" href="#__codelineno-85-23"></a><span class="w">    </span><span class="cm">/* Update element */</span>
+<a id="__codelineno-85-23" name="__codelineno-85-23" href="#__codelineno-85-23"></a><span class="w">    </span><span class="cm">/* Access an element */</span>
 <a id="__codelineno-85-24" name="__codelineno-85-24" href="#__codelineno-85-24"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nx">get</span><span class="p">(</span><span class="nx">index</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-85-25" name="__codelineno-85-25" href="#__codelineno-85-25"></a><span class="w">        </span><span class="c1">// If the index is out of bounds, throw an exception, as below</span>
+<a id="__codelineno-85-25" name="__codelineno-85-25" href="#__codelineno-85-25"></a><span class="w">        </span><span class="c1">// Throw an exception if the index is out of bounds; the same applies below</span>
 <a id="__codelineno-85-26" name="__codelineno-85-26" href="#__codelineno-85-26"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">_size</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
 <a id="__codelineno-85-27" name="__codelineno-85-27" href="#__codelineno-85-27"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">arr</span><span class="p">[</span><span class="nx">index</span><span class="p">];</span>
 <a id="__codelineno-85-28" name="__codelineno-85-28" href="#__codelineno-85-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-85-29" name="__codelineno-85-29" href="#__codelineno-85-29"></a>
-<a id="__codelineno-85-30" name="__codelineno-85-30" href="#__codelineno-85-30"></a><span class="w">    </span><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-85-30" name="__codelineno-85-30" href="#__codelineno-85-30"></a><span class="w">    </span><span class="cm">/* Update an element */</span>
 <a id="__codelineno-85-31" name="__codelineno-85-31" href="#__codelineno-85-31"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nx">set</span><span class="p">(</span><span class="nx">index</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-32" name="__codelineno-85-32" href="#__codelineno-85-32"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">_size</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
 <a id="__codelineno-85-33" name="__codelineno-85-33" href="#__codelineno-85-33"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">arr</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-85-34" name="__codelineno-85-34" href="#__codelineno-85-34"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-85-35" name="__codelineno-85-35" href="#__codelineno-85-35"></a>
-<a id="__codelineno-85-36" name="__codelineno-85-36" href="#__codelineno-85-36"></a><span class="w">    </span><span class="cm">/* Direct traversal of list elements */</span>
+<a id="__codelineno-85-36" name="__codelineno-85-36" href="#__codelineno-85-36"></a><span class="w">    </span><span class="cm">/* Append an element at the end */</span>
 <a id="__codelineno-85-37" name="__codelineno-85-37" href="#__codelineno-85-37"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nx">add</span><span class="p">(</span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-85-38" name="__codelineno-85-38" href="#__codelineno-85-38"></a><span class="w">        </span><span class="c1">// If length equals capacity, need to expand</span>
+<a id="__codelineno-85-38" name="__codelineno-85-38" href="#__codelineno-85-38"></a><span class="w">        </span><span class="c1">// Grow the capacity if the length equals the capacity</span>
 <a id="__codelineno-85-39" name="__codelineno-85-39" href="#__codelineno-85-39"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">_size</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">_capacity</span><span class="p">)</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">extendCapacity</span><span class="p">();</span>
-<a id="__codelineno-85-40" name="__codelineno-85-40" href="#__codelineno-85-40"></a><span class="w">        </span><span class="c1">// Add new element to end of list</span>
+<a id="__codelineno-85-40" name="__codelineno-85-40" href="#__codelineno-85-40"></a><span class="w">        </span><span class="c1">// Append the new element to the end of the list</span>
 <a id="__codelineno-85-41" name="__codelineno-85-41" href="#__codelineno-85-41"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">arr</span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="nx">_size</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-85-42" name="__codelineno-85-42" href="#__codelineno-85-42"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">_size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-85-43" name="__codelineno-85-43" href="#__codelineno-85-43"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-85-44" name="__codelineno-85-44" href="#__codelineno-85-44"></a>
-<a id="__codelineno-85-45" name="__codelineno-85-45" href="#__codelineno-85-45"></a><span class="w">    </span><span class="cm">/* Sort list */</span>
+<a id="__codelineno-85-45" name="__codelineno-85-45" href="#__codelineno-85-45"></a><span class="w">    </span><span class="cm">/* Insert an element in the middle */</span>
 <a id="__codelineno-85-46" name="__codelineno-85-46" href="#__codelineno-85-46"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nx">insert</span><span class="p">(</span><span class="nx">index</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-47" name="__codelineno-85-47" href="#__codelineno-85-47"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">_size</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
-<a id="__codelineno-85-48" name="__codelineno-85-48" href="#__codelineno-85-48"></a><span class="w">        </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-85-48" name="__codelineno-85-48" href="#__codelineno-85-48"></a><span class="w">        </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-85-49" name="__codelineno-85-49" href="#__codelineno-85-49"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">_size</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">_capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-50" name="__codelineno-85-50" href="#__codelineno-85-50"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">extendCapacity</span><span class="p">();</span>
 <a id="__codelineno-85-51" name="__codelineno-85-51" href="#__codelineno-85-51"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-85-52" name="__codelineno-85-52" href="#__codelineno-85-52"></a><span class="w">        </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-85-52" name="__codelineno-85-52" href="#__codelineno-85-52"></a><span class="w">        </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-85-53" name="__codelineno-85-53" href="#__codelineno-85-53"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">_size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-54" name="__codelineno-85-54" href="#__codelineno-85-54"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">arr</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">arr</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-85-55" name="__codelineno-85-55" href="#__codelineno-85-55"></a><span class="w">        </span><span class="p">}</span>
@@ -6378,11 +6450,11 @@
 <a id="__codelineno-85-58" name="__codelineno-85-58" href="#__codelineno-85-58"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">_size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-85-59" name="__codelineno-85-59" href="#__codelineno-85-59"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-85-60" name="__codelineno-85-60" href="#__codelineno-85-60"></a>
-<a id="__codelineno-85-61" name="__codelineno-85-61" href="#__codelineno-85-61"></a><span class="w">    </span><span class="cm">/* Remove element */</span>
+<a id="__codelineno-85-61" name="__codelineno-85-61" href="#__codelineno-85-61"></a><span class="w">    </span><span class="cm">/* Remove an element */</span>
 <a id="__codelineno-85-62" name="__codelineno-85-62" href="#__codelineno-85-62"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nx">remove</span><span class="p">(</span><span class="nx">index</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-63" name="__codelineno-85-63" href="#__codelineno-85-63"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">_size</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
 <a id="__codelineno-85-64" name="__codelineno-85-64" href="#__codelineno-85-64"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">arr</span><span class="p">[</span><span class="nx">index</span><span class="p">];</span>
-<a id="__codelineno-85-65" name="__codelineno-85-65" href="#__codelineno-85-65"></a><span class="w">        </span><span class="c1">// Move all elements after index forward by one position</span>
+<a id="__codelineno-85-65" name="__codelineno-85-65" href="#__codelineno-85-65"></a><span class="w">        </span><span class="c1">// Shift all elements after index one position to the left</span>
 <a id="__codelineno-85-66" name="__codelineno-85-66" href="#__codelineno-85-66"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">_size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-67" name="__codelineno-85-67" href="#__codelineno-85-67"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">arr</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">arr</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-85-68" name="__codelineno-85-68" href="#__codelineno-85-68"></a><span class="w">        </span><span class="p">}</span>
@@ -6392,20 +6464,20 @@
 <a id="__codelineno-85-72" name="__codelineno-85-72" href="#__codelineno-85-72"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-85-73" name="__codelineno-85-73" href="#__codelineno-85-73"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-85-74" name="__codelineno-85-74" href="#__codelineno-85-74"></a>
-<a id="__codelineno-85-75" name="__codelineno-85-75" href="#__codelineno-85-75"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-85-75" name="__codelineno-85-75" href="#__codelineno-85-75"></a><span class="w">    </span><span class="cm">/* Grow the list capacity */</span>
 <a id="__codelineno-85-76" name="__codelineno-85-76" href="#__codelineno-85-76"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nx">extendCapacity</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-85-77" name="__codelineno-85-77" href="#__codelineno-85-77"></a><span class="w">        </span><span class="c1">// Create new array of length size and copy original array to new array</span>
+<a id="__codelineno-85-77" name="__codelineno-85-77" href="#__codelineno-85-77"></a><span class="w">        </span><span class="c1">// Create a larger array by appending empty slots to a copy of the original array</span>
 <a id="__codelineno-85-78" name="__codelineno-85-78" href="#__codelineno-85-78"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">arr</span><span class="p">.</span><span class="nx">concat</span><span class="p">(</span>
 <a id="__codelineno-85-79" name="__codelineno-85-79" href="#__codelineno-85-79"></a><span class="w">            </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">extendRatio</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">))</span>
 <a id="__codelineno-85-80" name="__codelineno-85-80" href="#__codelineno-85-80"></a><span class="w">        </span><span class="p">);</span>
-<a id="__codelineno-85-81" name="__codelineno-85-81" href="#__codelineno-85-81"></a><span class="w">        </span><span class="c1">// Add elements at the end</span>
+<a id="__codelineno-85-81" name="__codelineno-85-81" href="#__codelineno-85-81"></a><span class="w">        </span><span class="c1">// Update the list capacity</span>
 <a id="__codelineno-85-82" name="__codelineno-85-82" href="#__codelineno-85-82"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">_capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">arr</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-85-83" name="__codelineno-85-83" href="#__codelineno-85-83"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-85-84" name="__codelineno-85-84" href="#__codelineno-85-84"></a>
-<a id="__codelineno-85-85" name="__codelineno-85-85" href="#__codelineno-85-85"></a><span class="w">    </span><span class="cm">/* Convert list to array */</span>
+<a id="__codelineno-85-85" name="__codelineno-85-85" href="#__codelineno-85-85"></a><span class="w">    </span><span class="cm">/* Convert the list to an array */</span>
 <a id="__codelineno-85-86" name="__codelineno-85-86" href="#__codelineno-85-86"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nx">toArray</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-87" name="__codelineno-85-87" href="#__codelineno-85-87"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">();</span>
-<a id="__codelineno-85-88" name="__codelineno-85-88" href="#__codelineno-85-88"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-85-88" name="__codelineno-85-88" href="#__codelineno-85-88"></a><span class="w">        </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-85-89" name="__codelineno-85-89" href="#__codelineno-85-89"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">size</span><span class="p">);</span>
 <a id="__codelineno-85-90" name="__codelineno-85-90" href="#__codelineno-85-90"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">size</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-91" name="__codelineno-85-91" href="#__codelineno-85-91"></a><span class="w">            </span><span class="nx">arr</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">i</span><span class="p">);</span>
@@ -6418,49 +6490,49 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">my_list.dart</span><pre><span></span><code><a id="__codelineno-86-1" name="__codelineno-86-1" href="#__codelineno-86-1"></a><span class="cm">/* List class */</span>
 <a id="__codelineno-86-2" name="__codelineno-86-2" href="#__codelineno-86-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">MyList</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-86-3" name="__codelineno-86-3" href="#__codelineno-86-3"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">_arr</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array (stores list elements)</span>
+<a id="__codelineno-86-3" name="__codelineno-86-3" href="#__codelineno-86-3"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">_arr</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array (stores the list elements)</span>
 <a id="__codelineno-86-4" name="__codelineno-86-4" href="#__codelineno-86-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">_capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">10</span><span class="p">;</span><span class="w"> </span><span class="c1">// List capacity</span>
 <a id="__codelineno-86-5" name="__codelineno-86-5" href="#__codelineno-86-5"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">_size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// List length (current number of elements)</span>
-<a id="__codelineno-86-6" name="__codelineno-86-6" href="#__codelineno-86-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">_extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Multiple by which the list capacity is extended each time</span>
+<a id="__codelineno-86-6" name="__codelineno-86-6" href="#__codelineno-86-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">_extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Factor by which the list capacity grows each time</span>
 <a id="__codelineno-86-7" name="__codelineno-86-7" href="#__codelineno-86-7"></a>
 <a id="__codelineno-86-8" name="__codelineno-86-8" href="#__codelineno-86-8"></a><span class="w">  </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-86-9" name="__codelineno-86-9" href="#__codelineno-86-9"></a><span class="w">  </span><span class="n">MyList</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-10" name="__codelineno-86-10" href="#__codelineno-86-10"></a><span class="w">    </span><span class="n">_arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">_capacity</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-86-11" name="__codelineno-86-11" href="#__codelineno-86-11"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-86-12" name="__codelineno-86-12" href="#__codelineno-86-12"></a>
-<a id="__codelineno-86-13" name="__codelineno-86-13" href="#__codelineno-86-13"></a><span class="w">  </span><span class="cm">/* Get list length (current number of elements) */</span>
+<a id="__codelineno-86-13" name="__codelineno-86-13" href="#__codelineno-86-13"></a><span class="w">  </span><span class="cm">/* Get the list length (current number of elements)*/</span>
 <a id="__codelineno-86-14" name="__codelineno-86-14" href="#__codelineno-86-14"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">_size</span><span class="p">;</span>
 <a id="__codelineno-86-15" name="__codelineno-86-15" href="#__codelineno-86-15"></a>
-<a id="__codelineno-86-16" name="__codelineno-86-16" href="#__codelineno-86-16"></a><span class="w">  </span><span class="cm">/* Get list capacity */</span>
+<a id="__codelineno-86-16" name="__codelineno-86-16" href="#__codelineno-86-16"></a><span class="w">  </span><span class="cm">/* Get the list capacity */</span>
 <a id="__codelineno-86-17" name="__codelineno-86-17" href="#__codelineno-86-17"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">_capacity</span><span class="p">;</span>
 <a id="__codelineno-86-18" name="__codelineno-86-18" href="#__codelineno-86-18"></a>
-<a id="__codelineno-86-19" name="__codelineno-86-19" href="#__codelineno-86-19"></a><span class="w">  </span><span class="cm">/* Update element */</span>
+<a id="__codelineno-86-19" name="__codelineno-86-19" href="#__codelineno-86-19"></a><span class="w">  </span><span class="cm">/* Access an element */</span>
 <a id="__codelineno-86-20" name="__codelineno-86-20" href="#__codelineno-86-20"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="kd">get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-21" name="__codelineno-86-21" href="#__codelineno-86-21"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">_size</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">RangeError</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
 <a id="__codelineno-86-22" name="__codelineno-86-22" href="#__codelineno-86-22"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_arr</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
 <a id="__codelineno-86-23" name="__codelineno-86-23" href="#__codelineno-86-23"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-86-24" name="__codelineno-86-24" href="#__codelineno-86-24"></a>
-<a id="__codelineno-86-25" name="__codelineno-86-25" href="#__codelineno-86-25"></a><span class="w">  </span><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-86-25" name="__codelineno-86-25" href="#__codelineno-86-25"></a><span class="w">  </span><span class="cm">/* Update an element */</span>
 <a id="__codelineno-86-26" name="__codelineno-86-26" href="#__codelineno-86-26"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="kd">set</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-27" name="__codelineno-86-27" href="#__codelineno-86-27"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">_size</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">RangeError</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
 <a id="__codelineno-86-28" name="__codelineno-86-28" href="#__codelineno-86-28"></a><span class="w">    </span><span class="n">_arr</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span>
 <a id="__codelineno-86-29" name="__codelineno-86-29" href="#__codelineno-86-29"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-86-30" name="__codelineno-86-30" href="#__codelineno-86-30"></a>
-<a id="__codelineno-86-31" name="__codelineno-86-31" href="#__codelineno-86-31"></a><span class="w">  </span><span class="cm">/* Direct traversal of list elements */</span>
+<a id="__codelineno-86-31" name="__codelineno-86-31" href="#__codelineno-86-31"></a><span class="w">  </span><span class="cm">/* Append an element at the end */</span>
 <a id="__codelineno-86-32" name="__codelineno-86-32" href="#__codelineno-86-32"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">add</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-86-33" name="__codelineno-86-33" href="#__codelineno-86-33"></a><span class="w">    </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-86-33" name="__codelineno-86-33" href="#__codelineno-86-33"></a><span class="w">    </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-86-34" name="__codelineno-86-34" href="#__codelineno-86-34"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">_capacity</span><span class="p">)</span><span class="w"> </span><span class="n">extendCapacity</span><span class="p">();</span>
 <a id="__codelineno-86-35" name="__codelineno-86-35" href="#__codelineno-86-35"></a><span class="w">    </span><span class="n">_arr</span><span class="p">[</span><span class="n">_size</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span>
 <a id="__codelineno-86-36" name="__codelineno-86-36" href="#__codelineno-86-36"></a><span class="w">    </span><span class="c1">// Update the number of elements</span>
 <a id="__codelineno-86-37" name="__codelineno-86-37" href="#__codelineno-86-37"></a><span class="w">    </span><span class="n">_size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-86-38" name="__codelineno-86-38" href="#__codelineno-86-38"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-86-39" name="__codelineno-86-39" href="#__codelineno-86-39"></a>
-<a id="__codelineno-86-40" name="__codelineno-86-40" href="#__codelineno-86-40"></a><span class="w">  </span><span class="cm">/* Sort list */</span>
+<a id="__codelineno-86-40" name="__codelineno-86-40" href="#__codelineno-86-40"></a><span class="w">  </span><span class="cm">/* Insert an element in the middle */</span>
 <a id="__codelineno-86-41" name="__codelineno-86-41" href="#__codelineno-86-41"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-42" name="__codelineno-86-42" href="#__codelineno-86-42"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">_size</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">RangeError</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
-<a id="__codelineno-86-43" name="__codelineno-86-43" href="#__codelineno-86-43"></a><span class="w">    </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-86-43" name="__codelineno-86-43" href="#__codelineno-86-43"></a><span class="w">    </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-86-44" name="__codelineno-86-44" href="#__codelineno-86-44"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">_capacity</span><span class="p">)</span><span class="w"> </span><span class="n">extendCapacity</span><span class="p">();</span>
-<a id="__codelineno-86-45" name="__codelineno-86-45" href="#__codelineno-86-45"></a><span class="w">    </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-86-45" name="__codelineno-86-45" href="#__codelineno-86-45"></a><span class="w">    </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-86-46" name="__codelineno-86-46" href="#__codelineno-86-46"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-47" name="__codelineno-86-47" href="#__codelineno-86-47"></a><span class="w">      </span><span class="n">_arr</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_arr</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-86-48" name="__codelineno-86-48" href="#__codelineno-86-48"></a><span class="w">    </span><span class="p">}</span>
@@ -6469,11 +6541,11 @@
 <a id="__codelineno-86-51" name="__codelineno-86-51" href="#__codelineno-86-51"></a><span class="w">    </span><span class="n">_size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-86-52" name="__codelineno-86-52" href="#__codelineno-86-52"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-86-53" name="__codelineno-86-53" href="#__codelineno-86-53"></a>
-<a id="__codelineno-86-54" name="__codelineno-86-54" href="#__codelineno-86-54"></a><span class="w">  </span><span class="cm">/* Remove element */</span>
+<a id="__codelineno-86-54" name="__codelineno-86-54" href="#__codelineno-86-54"></a><span class="w">  </span><span class="cm">/* Remove an element */</span>
 <a id="__codelineno-86-55" name="__codelineno-86-55" href="#__codelineno-86-55"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-56" name="__codelineno-86-56" href="#__codelineno-86-56"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">_size</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">RangeError</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
 <a id="__codelineno-86-57" name="__codelineno-86-57" href="#__codelineno-86-57"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_arr</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
-<a id="__codelineno-86-58" name="__codelineno-86-58" href="#__codelineno-86-58"></a><span class="w">    </span><span class="c1">// Move all elements after index forward by one position</span>
+<a id="__codelineno-86-58" name="__codelineno-86-58" href="#__codelineno-86-58"></a><span class="w">    </span><span class="c1">// Shift all elements after index one position to the left</span>
 <a id="__codelineno-86-59" name="__codelineno-86-59" href="#__codelineno-86-59"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">_size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-60" name="__codelineno-86-60" href="#__codelineno-86-60"></a><span class="w">      </span><span class="n">_arr</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_arr</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
 <a id="__codelineno-86-61" name="__codelineno-86-61" href="#__codelineno-86-61"></a><span class="w">    </span><span class="p">}</span>
@@ -6483,19 +6555,19 @@
 <a id="__codelineno-86-65" name="__codelineno-86-65" href="#__codelineno-86-65"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span>
 <a id="__codelineno-86-66" name="__codelineno-86-66" href="#__codelineno-86-66"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-86-67" name="__codelineno-86-67" href="#__codelineno-86-67"></a>
-<a id="__codelineno-86-68" name="__codelineno-86-68" href="#__codelineno-86-68"></a><span class="w">  </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-86-68" name="__codelineno-86-68" href="#__codelineno-86-68"></a><span class="w">  </span><span class="cm">/* Grow the list capacity */</span>
 <a id="__codelineno-86-69" name="__codelineno-86-69" href="#__codelineno-86-69"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">extendCapacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-86-70" name="__codelineno-86-70" href="#__codelineno-86-70"></a><span class="w">    </span><span class="c1">// Create new array with length _extendRatio times original array</span>
+<a id="__codelineno-86-70" name="__codelineno-86-70" href="#__codelineno-86-70"></a><span class="w">    </span><span class="c1">// Create a new array with _extendRatio times the length of the original array</span>
 <a id="__codelineno-86-71" name="__codelineno-86-71" href="#__codelineno-86-71"></a><span class="w">    </span><span class="kd">final</span><span class="w"> </span><span class="n">_newNums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">_capacity</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">_extendRatio</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
-<a id="__codelineno-86-72" name="__codelineno-86-72" href="#__codelineno-86-72"></a><span class="w">    </span><span class="c1">// Copy original array to new array</span>
+<a id="__codelineno-86-72" name="__codelineno-86-72" href="#__codelineno-86-72"></a><span class="w">    </span><span class="c1">// Copy the original array into the new array</span>
 <a id="__codelineno-86-73" name="__codelineno-86-73" href="#__codelineno-86-73"></a><span class="w">    </span><span class="n">List</span><span class="p">.</span><span class="n">copyRange</span><span class="p">(</span><span class="n">_newNums</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">_arr</span><span class="p">);</span>
-<a id="__codelineno-86-74" name="__codelineno-86-74" href="#__codelineno-86-74"></a><span class="w">    </span><span class="c1">// Update _arr reference</span>
+<a id="__codelineno-86-74" name="__codelineno-86-74" href="#__codelineno-86-74"></a><span class="w">    </span><span class="c1">// Update the reference _arr</span>
 <a id="__codelineno-86-75" name="__codelineno-86-75" href="#__codelineno-86-75"></a><span class="w">    </span><span class="n">_arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_newNums</span><span class="p">;</span>
-<a id="__codelineno-86-76" name="__codelineno-86-76" href="#__codelineno-86-76"></a><span class="w">    </span><span class="c1">// Add elements at the end</span>
+<a id="__codelineno-86-76" name="__codelineno-86-76" href="#__codelineno-86-76"></a><span class="w">    </span><span class="c1">// Update the list capacity</span>
 <a id="__codelineno-86-77" name="__codelineno-86-77" href="#__codelineno-86-77"></a><span class="w">    </span><span class="n">_capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_arr</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-86-78" name="__codelineno-86-78" href="#__codelineno-86-78"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-86-79" name="__codelineno-86-79" href="#__codelineno-86-79"></a>
-<a id="__codelineno-86-80" name="__codelineno-86-80" href="#__codelineno-86-80"></a><span class="w">  </span><span class="cm">/* Convert list to array */</span>
+<a id="__codelineno-86-80" name="__codelineno-86-80" href="#__codelineno-86-80"></a><span class="w">  </span><span class="cm">/* Convert the list to an array */</span>
 <a id="__codelineno-86-81" name="__codelineno-86-81" href="#__codelineno-86-81"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-82" name="__codelineno-86-82" href="#__codelineno-86-82"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-86-83" name="__codelineno-86-83" href="#__codelineno-86-83"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">_size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6507,13 +6579,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_list.rs</span><pre><span></span><code><a id="__codelineno-87-1" name="__codelineno-87-1" href="#__codelineno-87-1"></a><span class="cm">/* List class */</span>
+<div class="highlight"><span class="filename">my_list.rs</span><pre><span></span><code><a id="__codelineno-87-1" name="__codelineno-87-1" href="#__codelineno-87-1"></a><span class="cm">/* List type */</span>
 <a id="__codelineno-87-2" name="__codelineno-87-2" href="#__codelineno-87-2"></a><span class="cp">#[allow(dead_code)]</span>
 <a id="__codelineno-87-3" name="__codelineno-87-3" href="#__codelineno-87-3"></a><span class="k">struct</span><span class="w"> </span><span class="nc">MyList</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-87-4" name="__codelineno-87-4" href="#__codelineno-87-4"></a><span class="w">    </span><span class="n">arr</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">,</span><span class="w">       </span><span class="c1">// Array (stores list elements)</span>
+<a id="__codelineno-87-4" name="__codelineno-87-4" href="#__codelineno-87-4"></a><span class="w">    </span><span class="n">arr</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">,</span><span class="w">       </span><span class="c1">// Array (stores the list elements)</span>
 <a id="__codelineno-87-5" name="__codelineno-87-5" href="#__codelineno-87-5"></a><span class="w">    </span><span class="n">capacity</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w">     </span><span class="c1">// List capacity</span>
 <a id="__codelineno-87-6" name="__codelineno-87-6" href="#__codelineno-87-6"></a><span class="w">    </span><span class="n">size</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w">         </span><span class="c1">// List length (current number of elements)</span>
-<a id="__codelineno-87-7" name="__codelineno-87-7" href="#__codelineno-87-7"></a><span class="w">    </span><span class="n">extend_ratio</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="c1">// Multiple by which the list capacity is extended each time</span>
+<a id="__codelineno-87-7" name="__codelineno-87-7" href="#__codelineno-87-7"></a><span class="w">    </span><span class="n">extend_ratio</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="c1">// Factor by which the list capacity grows each time</span>
 <a id="__codelineno-87-8" name="__codelineno-87-8" href="#__codelineno-87-8"></a><span class="p">}</span>
 <a id="__codelineno-87-9" name="__codelineno-87-9" href="#__codelineno-87-9"></a>
 <a id="__codelineno-87-10" name="__codelineno-87-10" href="#__codelineno-87-10"></a><span class="cp">#[allow(unused, unused_comparisons)]</span>
@@ -6529,26 +6601,26 @@
 <a id="__codelineno-87-20" name="__codelineno-87-20" href="#__codelineno-87-20"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-87-21" name="__codelineno-87-21" href="#__codelineno-87-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-87-22" name="__codelineno-87-22" href="#__codelineno-87-22"></a>
-<a id="__codelineno-87-23" name="__codelineno-87-23" href="#__codelineno-87-23"></a><span class="w">    </span><span class="cm">/* Get list length (current number of elements) */</span>
+<a id="__codelineno-87-23" name="__codelineno-87-23" href="#__codelineno-87-23"></a><span class="w">    </span><span class="cm">/* Get the list length (current number of elements)*/</span>
 <a id="__codelineno-87-24" name="__codelineno-87-24" href="#__codelineno-87-24"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-25" name="__codelineno-87-25" href="#__codelineno-87-25"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">;</span>
 <a id="__codelineno-87-26" name="__codelineno-87-26" href="#__codelineno-87-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-87-27" name="__codelineno-87-27" href="#__codelineno-87-27"></a>
-<a id="__codelineno-87-28" name="__codelineno-87-28" href="#__codelineno-87-28"></a><span class="w">    </span><span class="cm">/* Get list capacity */</span>
+<a id="__codelineno-87-28" name="__codelineno-87-28" href="#__codelineno-87-28"></a><span class="w">    </span><span class="cm">/* Get the list capacity */</span>
 <a id="__codelineno-87-29" name="__codelineno-87-29" href="#__codelineno-87-29"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">capacity</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-30" name="__codelineno-87-30" href="#__codelineno-87-30"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-87-31" name="__codelineno-87-31" href="#__codelineno-87-31"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-87-32" name="__codelineno-87-32" href="#__codelineno-87-32"></a>
-<a id="__codelineno-87-33" name="__codelineno-87-33" href="#__codelineno-87-33"></a><span class="w">    </span><span class="cm">/* Update element */</span>
+<a id="__codelineno-87-33" name="__codelineno-87-33" href="#__codelineno-87-33"></a><span class="w">    </span><span class="cm">/* Access an element */</span>
 <a id="__codelineno-87-34" name="__codelineno-87-34" href="#__codelineno-87-34"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-87-35" name="__codelineno-87-35" href="#__codelineno-87-35"></a><span class="w">        </span><span class="c1">// If the index is out of bounds, throw an exception, as below</span>
+<a id="__codelineno-87-35" name="__codelineno-87-35" href="#__codelineno-87-35"></a><span class="w">        </span><span class="c1">// Panic if the index is out of bounds; the same applies below</span>
 <a id="__codelineno-87-36" name="__codelineno-87-36" href="#__codelineno-87-36"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-37" name="__codelineno-87-37" href="#__codelineno-87-37"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-87-38" name="__codelineno-87-38" href="#__codelineno-87-38"></a><span class="w">        </span><span class="p">};</span>
 <a id="__codelineno-87-39" name="__codelineno-87-39" href="#__codelineno-87-39"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
 <a id="__codelineno-87-40" name="__codelineno-87-40" href="#__codelineno-87-40"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-87-41" name="__codelineno-87-41" href="#__codelineno-87-41"></a>
-<a id="__codelineno-87-42" name="__codelineno-87-42" href="#__codelineno-87-42"></a><span class="w">    </span><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-87-42" name="__codelineno-87-42" href="#__codelineno-87-42"></a><span class="w">    </span><span class="cm">/* Update an element */</span>
 <a id="__codelineno-87-43" name="__codelineno-87-43" href="#__codelineno-87-43"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">set</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-44" name="__codelineno-87-44" href="#__codelineno-87-44"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-45" name="__codelineno-87-45" href="#__codelineno-87-45"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
@@ -6556,9 +6628,9 @@
 <a id="__codelineno-87-47" name="__codelineno-87-47" href="#__codelineno-87-47"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-87-48" name="__codelineno-87-48" href="#__codelineno-87-48"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-87-49" name="__codelineno-87-49" href="#__codelineno-87-49"></a>
-<a id="__codelineno-87-50" name="__codelineno-87-50" href="#__codelineno-87-50"></a><span class="w">    </span><span class="cm">/* Direct traversal of list elements */</span>
+<a id="__codelineno-87-50" name="__codelineno-87-50" href="#__codelineno-87-50"></a><span class="w">    </span><span class="cm">/* Append an element at the end */</span>
 <a id="__codelineno-87-51" name="__codelineno-87-51" href="#__codelineno-87-51"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">add</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-87-52" name="__codelineno-87-52" href="#__codelineno-87-52"></a><span class="w">        </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-87-52" name="__codelineno-87-52" href="#__codelineno-87-52"></a><span class="w">        </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-87-53" name="__codelineno-87-53" href="#__codelineno-87-53"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-54" name="__codelineno-87-54" href="#__codelineno-87-54"></a><span class="w">            </span><span class="bp">self</span><span class="p">.</span><span class="n">extend_capacity</span><span class="p">();</span>
 <a id="__codelineno-87-55" name="__codelineno-87-55" href="#__codelineno-87-55"></a><span class="w">        </span><span class="p">}</span>
@@ -6567,16 +6639,16 @@
 <a id="__codelineno-87-58" name="__codelineno-87-58" href="#__codelineno-87-58"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-87-59" name="__codelineno-87-59" href="#__codelineno-87-59"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-87-60" name="__codelineno-87-60" href="#__codelineno-87-60"></a>
-<a id="__codelineno-87-61" name="__codelineno-87-61" href="#__codelineno-87-61"></a><span class="w">    </span><span class="cm">/* Sort list */</span>
+<a id="__codelineno-87-61" name="__codelineno-87-61" href="#__codelineno-87-61"></a><span class="w">    </span><span class="cm">/* Insert an element in the middle */</span>
 <a id="__codelineno-87-62" name="__codelineno-87-62" href="#__codelineno-87-62"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-63" name="__codelineno-87-63" href="#__codelineno-87-63"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-64" name="__codelineno-87-64" href="#__codelineno-87-64"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-87-65" name="__codelineno-87-65" href="#__codelineno-87-65"></a><span class="w">        </span><span class="p">};</span>
-<a id="__codelineno-87-66" name="__codelineno-87-66" href="#__codelineno-87-66"></a><span class="w">        </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-87-66" name="__codelineno-87-66" href="#__codelineno-87-66"></a><span class="w">        </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-87-67" name="__codelineno-87-67" href="#__codelineno-87-67"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-68" name="__codelineno-87-68" href="#__codelineno-87-68"></a><span class="w">            </span><span class="bp">self</span><span class="p">.</span><span class="n">extend_capacity</span><span class="p">();</span>
 <a id="__codelineno-87-69" name="__codelineno-87-69" href="#__codelineno-87-69"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-87-70" name="__codelineno-87-70" href="#__codelineno-87-70"></a><span class="w">        </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-87-70" name="__codelineno-87-70" href="#__codelineno-87-70"></a><span class="w">        </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-87-71" name="__codelineno-87-71" href="#__codelineno-87-71"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="o">..</span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">).</span><span class="n">rev</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-72" name="__codelineno-87-72" href="#__codelineno-87-72"></a><span class="w">            </span><span class="bp">self</span><span class="p">.</span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-87-73" name="__codelineno-87-73" href="#__codelineno-87-73"></a><span class="w">        </span><span class="p">}</span>
@@ -6585,13 +6657,13 @@
 <a id="__codelineno-87-76" name="__codelineno-87-76" href="#__codelineno-87-76"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-87-77" name="__codelineno-87-77" href="#__codelineno-87-77"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-87-78" name="__codelineno-87-78" href="#__codelineno-87-78"></a>
-<a id="__codelineno-87-79" name="__codelineno-87-79" href="#__codelineno-87-79"></a><span class="w">    </span><span class="cm">/* Remove element */</span>
+<a id="__codelineno-87-79" name="__codelineno-87-79" href="#__codelineno-87-79"></a><span class="w">    </span><span class="cm">/* Remove an element */</span>
 <a id="__codelineno-87-80" name="__codelineno-87-80" href="#__codelineno-87-80"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-81" name="__codelineno-87-81" href="#__codelineno-87-81"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-82" name="__codelineno-87-82" href="#__codelineno-87-82"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-87-83" name="__codelineno-87-83" href="#__codelineno-87-83"></a><span class="w">        </span><span class="p">};</span>
 <a id="__codelineno-87-84" name="__codelineno-87-84" href="#__codelineno-87-84"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
-<a id="__codelineno-87-85" name="__codelineno-87-85" href="#__codelineno-87-85"></a><span class="w">        </span><span class="c1">// Create a new array with length _extend_ratio times the original array, and copy the original array to the new array</span>
+<a id="__codelineno-87-85" name="__codelineno-87-85" href="#__codelineno-87-85"></a><span class="w">        </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-87-86" name="__codelineno-87-86" href="#__codelineno-87-86"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">index</span><span class="o">..</span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-87" name="__codelineno-87-87" href="#__codelineno-87-87"></a><span class="w">            </span><span class="bp">self</span><span class="p">.</span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">arr</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-87-88" name="__codelineno-87-88" href="#__codelineno-87-88"></a><span class="w">        </span><span class="p">}</span>
@@ -6601,18 +6673,18 @@
 <a id="__codelineno-87-92" name="__codelineno-87-92" href="#__codelineno-87-92"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-87-93" name="__codelineno-87-93" href="#__codelineno-87-93"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-87-94" name="__codelineno-87-94" href="#__codelineno-87-94"></a>
-<a id="__codelineno-87-95" name="__codelineno-87-95" href="#__codelineno-87-95"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-87-95" name="__codelineno-87-95" href="#__codelineno-87-95"></a><span class="w">    </span><span class="cm">/* Grow the list capacity */</span>
 <a id="__codelineno-87-96" name="__codelineno-87-96" href="#__codelineno-87-96"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">extend_capacity</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-87-97" name="__codelineno-87-97" href="#__codelineno-87-97"></a><span class="w">        </span><span class="c1">// Create new array with length extend_ratio times original, copy original array to new array</span>
+<a id="__codelineno-87-97" name="__codelineno-87-97" href="#__codelineno-87-97"></a><span class="w">        </span><span class="c1">// Grow the backing vector to extend_ratio times its original length, preserving its elements</span>
 <a id="__codelineno-87-98" name="__codelineno-87-98" href="#__codelineno-87-98"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">new_capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">capacity</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">extend_ratio</span><span class="p">;</span>
 <a id="__codelineno-87-99" name="__codelineno-87-99" href="#__codelineno-87-99"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">arr</span><span class="p">.</span><span class="n">resize</span><span class="p">(</span><span class="n">new_capacity</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
-<a id="__codelineno-87-100" name="__codelineno-87-100" href="#__codelineno-87-100"></a><span class="w">        </span><span class="c1">// Add elements at the end</span>
+<a id="__codelineno-87-100" name="__codelineno-87-100" href="#__codelineno-87-100"></a><span class="w">        </span><span class="c1">// Update the list capacity</span>
 <a id="__codelineno-87-101" name="__codelineno-87-101" href="#__codelineno-87-101"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">new_capacity</span><span class="p">;</span>
 <a id="__codelineno-87-102" name="__codelineno-87-102" href="#__codelineno-87-102"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-87-103" name="__codelineno-87-103" href="#__codelineno-87-103"></a>
-<a id="__codelineno-87-104" name="__codelineno-87-104" href="#__codelineno-87-104"></a><span class="w">    </span><span class="cm">/* Convert list to array */</span>
+<a id="__codelineno-87-104" name="__codelineno-87-104" href="#__codelineno-87-104"></a><span class="w">    </span><span class="cm">/* Copy the list elements into a vector */</span>
 <a id="__codelineno-87-105" name="__codelineno-87-105" href="#__codelineno-87-105"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">to_array</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-87-106" name="__codelineno-87-106" href="#__codelineno-87-106"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-87-106" name="__codelineno-87-106" href="#__codelineno-87-106"></a><span class="w">        </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-87-107" name="__codelineno-87-107" href="#__codelineno-87-107"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-87-108" name="__codelineno-87-108" href="#__codelineno-87-108"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-109" name="__codelineno-87-109" href="#__codelineno-87-109"></a><span class="w">            </span><span class="n">arr</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">get</span><span class="p">(</span><span class="n">i</span><span class="p">));</span>
@@ -6623,12 +6695,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_list.c</span><pre><span></span><code><a id="__codelineno-88-1" name="__codelineno-88-1" href="#__codelineno-88-1"></a><span class="cm">/* List class */</span>
+<div class="highlight"><span class="filename">my_list.c</span><pre><span></span><code><a id="__codelineno-88-1" name="__codelineno-88-1" href="#__codelineno-88-1"></a><span class="cm">/* List structure */</span>
 <a id="__codelineno-88-2" name="__codelineno-88-2" href="#__codelineno-88-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-88-3" name="__codelineno-88-3" href="#__codelineno-88-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">arr</span><span class="p">;</span><span class="w">        </span><span class="c1">// Array (stores list elements)</span>
+<a id="__codelineno-88-3" name="__codelineno-88-3" href="#__codelineno-88-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">arr</span><span class="p">;</span><span class="w">        </span><span class="c1">// Array (stores the list elements)</span>
 <a id="__codelineno-88-4" name="__codelineno-88-4" href="#__codelineno-88-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span><span class="w">    </span><span class="c1">// List capacity</span>
 <a id="__codelineno-88-5" name="__codelineno-88-5" href="#__codelineno-88-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w">        </span><span class="c1">// List size</span>
-<a id="__codelineno-88-6" name="__codelineno-88-6" href="#__codelineno-88-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span><span class="w"> </span><span class="c1">// List expansion multiplier</span>
+<a id="__codelineno-88-6" name="__codelineno-88-6" href="#__codelineno-88-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span><span class="w"> </span><span class="c1">// List capacity growth factor</span>
 <a id="__codelineno-88-7" name="__codelineno-88-7" href="#__codelineno-88-7"></a><span class="p">}</span><span class="w"> </span><span class="n">MyList</span><span class="p">;</span>
 <a id="__codelineno-88-8" name="__codelineno-88-8" href="#__codelineno-88-8"></a>
 <a id="__codelineno-88-9" name="__codelineno-88-9" href="#__codelineno-88-9"></a><span class="cm">/* Constructor */</span>
@@ -6647,43 +6719,43 @@
 <a id="__codelineno-88-22" name="__codelineno-88-22" href="#__codelineno-88-22"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span>
 <a id="__codelineno-88-23" name="__codelineno-88-23" href="#__codelineno-88-23"></a><span class="p">}</span>
 <a id="__codelineno-88-24" name="__codelineno-88-24" href="#__codelineno-88-24"></a>
-<a id="__codelineno-88-25" name="__codelineno-88-25" href="#__codelineno-88-25"></a><span class="cm">/* Get list length */</span>
+<a id="__codelineno-88-25" name="__codelineno-88-25" href="#__codelineno-88-25"></a><span class="cm">/* Get the list size */</span>
 <a id="__codelineno-88-26" name="__codelineno-88-26" href="#__codelineno-88-26"></a><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="n">MyList</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-27" name="__codelineno-88-27" href="#__codelineno-88-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">;</span>
 <a id="__codelineno-88-28" name="__codelineno-88-28" href="#__codelineno-88-28"></a><span class="p">}</span>
 <a id="__codelineno-88-29" name="__codelineno-88-29" href="#__codelineno-88-29"></a>
-<a id="__codelineno-88-30" name="__codelineno-88-30" href="#__codelineno-88-30"></a><span class="cm">/* Get list capacity */</span>
+<a id="__codelineno-88-30" name="__codelineno-88-30" href="#__codelineno-88-30"></a><span class="cm">/* Get the list capacity */</span>
 <a id="__codelineno-88-31" name="__codelineno-88-31" href="#__codelineno-88-31"></a><span class="kt">int</span><span class="w"> </span><span class="nf">capacity</span><span class="p">(</span><span class="n">MyList</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-32" name="__codelineno-88-32" href="#__codelineno-88-32"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-88-33" name="__codelineno-88-33" href="#__codelineno-88-33"></a><span class="p">}</span>
 <a id="__codelineno-88-34" name="__codelineno-88-34" href="#__codelineno-88-34"></a>
-<a id="__codelineno-88-35" name="__codelineno-88-35" href="#__codelineno-88-35"></a><span class="cm">/* Update element */</span>
+<a id="__codelineno-88-35" name="__codelineno-88-35" href="#__codelineno-88-35"></a><span class="cm">/* Access an element */</span>
 <a id="__codelineno-88-36" name="__codelineno-88-36" href="#__codelineno-88-36"></a><span class="kt">int</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="n">MyList</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-37" name="__codelineno-88-37" href="#__codelineno-88-37"></a><span class="w">    </span><span class="n">assert</span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">);</span>
 <a id="__codelineno-88-38" name="__codelineno-88-38" href="#__codelineno-88-38"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
 <a id="__codelineno-88-39" name="__codelineno-88-39" href="#__codelineno-88-39"></a><span class="p">}</span>
 <a id="__codelineno-88-40" name="__codelineno-88-40" href="#__codelineno-88-40"></a>
-<a id="__codelineno-88-41" name="__codelineno-88-41" href="#__codelineno-88-41"></a><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-88-41" name="__codelineno-88-41" href="#__codelineno-88-41"></a><span class="cm">/* Update an element */</span>
 <a id="__codelineno-88-42" name="__codelineno-88-42" href="#__codelineno-88-42"></a><span class="kt">void</span><span class="w"> </span><span class="nf">set</span><span class="p">(</span><span class="n">MyList</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-43" name="__codelineno-88-43" href="#__codelineno-88-43"></a><span class="w">    </span><span class="n">assert</span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">);</span>
 <a id="__codelineno-88-44" name="__codelineno-88-44" href="#__codelineno-88-44"></a><span class="w">    </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-88-45" name="__codelineno-88-45" href="#__codelineno-88-45"></a><span class="p">}</span>
 <a id="__codelineno-88-46" name="__codelineno-88-46" href="#__codelineno-88-46"></a>
-<a id="__codelineno-88-47" name="__codelineno-88-47" href="#__codelineno-88-47"></a><span class="cm">/* Direct traversal of list elements */</span>
+<a id="__codelineno-88-47" name="__codelineno-88-47" href="#__codelineno-88-47"></a><span class="cm">/* Append an element at the end */</span>
 <a id="__codelineno-88-48" name="__codelineno-88-48" href="#__codelineno-88-48"></a><span class="kt">void</span><span class="w"> </span><span class="nf">add</span><span class="p">(</span><span class="n">MyList</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-49" name="__codelineno-88-49" href="#__codelineno-88-49"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">(</span><span class="n">nums</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-88-50" name="__codelineno-88-50" href="#__codelineno-88-50"></a><span class="w">        </span><span class="n">extendCapacity</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w"> </span><span class="c1">// Expand capacity</span>
+<a id="__codelineno-88-50" name="__codelineno-88-50" href="#__codelineno-88-50"></a><span class="w">        </span><span class="n">extendCapacity</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w"> </span><span class="c1">// Grow the capacity</span>
 <a id="__codelineno-88-51" name="__codelineno-88-51" href="#__codelineno-88-51"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-88-52" name="__codelineno-88-52" href="#__codelineno-88-52"></a><span class="w">    </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">arr</span><span class="p">[</span><span class="n">size</span><span class="p">(</span><span class="n">nums</span><span class="p">)]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-88-53" name="__codelineno-88-53" href="#__codelineno-88-53"></a><span class="w">    </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-88-54" name="__codelineno-88-54" href="#__codelineno-88-54"></a><span class="p">}</span>
 <a id="__codelineno-88-55" name="__codelineno-88-55" href="#__codelineno-88-55"></a>
-<a id="__codelineno-88-56" name="__codelineno-88-56" href="#__codelineno-88-56"></a><span class="cm">/* Sort list */</span>
+<a id="__codelineno-88-56" name="__codelineno-88-56" href="#__codelineno-88-56"></a><span class="cm">/* Insert an element in the middle */</span>
 <a id="__codelineno-88-57" name="__codelineno-88-57" href="#__codelineno-88-57"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">MyList</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-58" name="__codelineno-88-58" href="#__codelineno-88-58"></a><span class="w">    </span><span class="n">assert</span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">nums</span><span class="p">));</span>
-<a id="__codelineno-88-59" name="__codelineno-88-59" href="#__codelineno-88-59"></a><span class="w">    </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-88-59" name="__codelineno-88-59" href="#__codelineno-88-59"></a><span class="w">    </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-88-60" name="__codelineno-88-60" href="#__codelineno-88-60"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">(</span><span class="n">nums</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-88-61" name="__codelineno-88-61" href="#__codelineno-88-61"></a><span class="w">        </span><span class="n">extendCapacity</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w"> </span><span class="c1">// Expand capacity</span>
+<a id="__codelineno-88-61" name="__codelineno-88-61" href="#__codelineno-88-61"></a><span class="w">        </span><span class="n">extendCapacity</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w"> </span><span class="c1">// Grow the capacity</span>
 <a id="__codelineno-88-62" name="__codelineno-88-62" href="#__codelineno-88-62"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-88-63" name="__codelineno-88-63" href="#__codelineno-88-63"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="o">--</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-64" name="__codelineno-88-64" href="#__codelineno-88-64"></a><span class="w">        </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">arr</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">arr</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
@@ -6692,8 +6764,8 @@
 <a id="__codelineno-88-67" name="__codelineno-88-67" href="#__codelineno-88-67"></a><span class="w">    </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-88-68" name="__codelineno-88-68" href="#__codelineno-88-68"></a><span class="p">}</span>
 <a id="__codelineno-88-69" name="__codelineno-88-69" href="#__codelineno-88-69"></a>
-<a id="__codelineno-88-70" name="__codelineno-88-70" href="#__codelineno-88-70"></a><span class="cm">/* Remove element */</span>
-<a id="__codelineno-88-71" name="__codelineno-88-71" href="#__codelineno-88-71"></a><span class="c1">// Note: stdio.h occupies the remove keyword</span>
+<a id="__codelineno-88-70" name="__codelineno-88-70" href="#__codelineno-88-70"></a><span class="cm">/* Remove an element */</span>
+<a id="__codelineno-88-71" name="__codelineno-88-71" href="#__codelineno-88-71"></a><span class="c1">// Note: stdio.h already declares a function named remove</span>
 <a id="__codelineno-88-72" name="__codelineno-88-72" href="#__codelineno-88-72"></a><span class="kt">int</span><span class="w"> </span><span class="nf">removeItem</span><span class="p">(</span><span class="n">MyList</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-73" name="__codelineno-88-73" href="#__codelineno-88-73"></a><span class="w">    </span><span class="n">assert</span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">nums</span><span class="p">));</span>
 <a id="__codelineno-88-74" name="__codelineno-88-74" href="#__codelineno-88-74"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
@@ -6704,26 +6776,26 @@
 <a id="__codelineno-88-79" name="__codelineno-88-79" href="#__codelineno-88-79"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-88-80" name="__codelineno-88-80" href="#__codelineno-88-80"></a><span class="p">}</span>
 <a id="__codelineno-88-81" name="__codelineno-88-81" href="#__codelineno-88-81"></a>
-<a id="__codelineno-88-82" name="__codelineno-88-82" href="#__codelineno-88-82"></a><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-88-82" name="__codelineno-88-82" href="#__codelineno-88-82"></a><span class="cm">/* Grow the list capacity */</span>
 <a id="__codelineno-88-83" name="__codelineno-88-83" href="#__codelineno-88-83"></a><span class="kt">void</span><span class="w"> </span><span class="nf">extendCapacity</span><span class="p">(</span><span class="n">MyList</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-88-84" name="__codelineno-88-84" href="#__codelineno-88-84"></a><span class="w">    </span><span class="c1">// Allocate space first</span>
+<a id="__codelineno-88-84" name="__codelineno-88-84" href="#__codelineno-88-84"></a><span class="w">    </span><span class="c1">// Allocate new space first</span>
 <a id="__codelineno-88-85" name="__codelineno-88-85" href="#__codelineno-88-85"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">newCapacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">capacity</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">extendRatio</span><span class="p">;</span>
 <a id="__codelineno-88-86" name="__codelineno-88-86" href="#__codelineno-88-86"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">extend</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">newCapacity</span><span class="p">);</span>
 <a id="__codelineno-88-87" name="__codelineno-88-87" href="#__codelineno-88-87"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">arr</span><span class="p">;</span>
 <a id="__codelineno-88-88" name="__codelineno-88-88" href="#__codelineno-88-88"></a>
-<a id="__codelineno-88-89" name="__codelineno-88-89" href="#__codelineno-88-89"></a><span class="w">    </span><span class="c1">// Copy old data to new data</span>
+<a id="__codelineno-88-89" name="__codelineno-88-89" href="#__codelineno-88-89"></a><span class="w">    </span><span class="c1">// Copy the old data into the new array</span>
 <a id="__codelineno-88-90" name="__codelineno-88-90" href="#__codelineno-88-90"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span>
 <a id="__codelineno-88-91" name="__codelineno-88-91" href="#__codelineno-88-91"></a><span class="w">        </span><span class="n">extend</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">arr</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-88-92" name="__codelineno-88-92" href="#__codelineno-88-92"></a>
-<a id="__codelineno-88-93" name="__codelineno-88-93" href="#__codelineno-88-93"></a><span class="w">    </span><span class="c1">// Free old data</span>
+<a id="__codelineno-88-93" name="__codelineno-88-93" href="#__codelineno-88-93"></a><span class="w">    </span><span class="c1">// Free the old array</span>
 <a id="__codelineno-88-94" name="__codelineno-88-94" href="#__codelineno-88-94"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">temp</span><span class="p">);</span>
 <a id="__codelineno-88-95" name="__codelineno-88-95" href="#__codelineno-88-95"></a>
-<a id="__codelineno-88-96" name="__codelineno-88-96" href="#__codelineno-88-96"></a><span class="w">    </span><span class="c1">// Update new data</span>
+<a id="__codelineno-88-96" name="__codelineno-88-96" href="#__codelineno-88-96"></a><span class="w">    </span><span class="c1">// Update the array pointer and capacity</span>
 <a id="__codelineno-88-97" name="__codelineno-88-97" href="#__codelineno-88-97"></a><span class="w">    </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">extend</span><span class="p">;</span>
 <a id="__codelineno-88-98" name="__codelineno-88-98" href="#__codelineno-88-98"></a><span class="w">    </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newCapacity</span><span class="p">;</span>
 <a id="__codelineno-88-99" name="__codelineno-88-99" href="#__codelineno-88-99"></a><span class="p">}</span>
 <a id="__codelineno-88-100" name="__codelineno-88-100" href="#__codelineno-88-100"></a>
-<a id="__codelineno-88-101" name="__codelineno-88-101" href="#__codelineno-88-101"></a><span class="cm">/* Convert list to Array for printing */</span>
+<a id="__codelineno-88-101" name="__codelineno-88-101" href="#__codelineno-88-101"></a><span class="cm">/* Convert the list to an array for printing */</span>
 <a id="__codelineno-88-102" name="__codelineno-88-102" href="#__codelineno-88-102"></a><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="nf">toArray</span><span class="p">(</span><span class="n">MyList</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-103" name="__codelineno-88-103" href="#__codelineno-88-103"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="o">-&gt;</span><span class="n">arr</span><span class="p">;</span>
 <a id="__codelineno-88-104" name="__codelineno-88-104" href="#__codelineno-88-104"></a><span class="p">}</span>
@@ -6732,44 +6804,44 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">my_list.kt</span><pre><span></span><code><a id="__codelineno-89-1" name="__codelineno-89-1" href="#__codelineno-89-1"></a><span class="cm">/* List class */</span>
 <a id="__codelineno-89-2" name="__codelineno-89-2" href="#__codelineno-89-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">MyList</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-89-3" name="__codelineno-89-3" href="#__codelineno-89-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">arr</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">intArrayOf</span><span class="p">()</span><span class="w"> </span><span class="c1">// Array (stores list elements)</span>
+<a id="__codelineno-89-3" name="__codelineno-89-3" href="#__codelineno-89-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">arr</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">intArrayOf</span><span class="p">()</span><span class="w"> </span><span class="c1">// Array (stores the list elements)</span>
 <a id="__codelineno-89-4" name="__codelineno-89-4" href="#__codelineno-89-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">capacity</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">10</span><span class="w"> </span><span class="c1">// List capacity</span>
 <a id="__codelineno-89-5" name="__codelineno-89-5" href="#__codelineno-89-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">size</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// List length (current number of elements)</span>
-<a id="__codelineno-89-6" name="__codelineno-89-6" href="#__codelineno-89-6"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">extendRatio</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="c1">// Multiple by which the list capacity is extended each time</span>
+<a id="__codelineno-89-6" name="__codelineno-89-6" href="#__codelineno-89-6"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">extendRatio</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="c1">// Factor by which the list capacity grows each time</span>
 <a id="__codelineno-89-7" name="__codelineno-89-7" href="#__codelineno-89-7"></a>
 <a id="__codelineno-89-8" name="__codelineno-89-8" href="#__codelineno-89-8"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-89-9" name="__codelineno-89-9" href="#__codelineno-89-9"></a><span class="w">    </span><span class="k">init</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-89-10" name="__codelineno-89-10" href="#__codelineno-89-10"></a><span class="w">        </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">capacity</span><span class="p">)</span>
 <a id="__codelineno-89-11" name="__codelineno-89-11" href="#__codelineno-89-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-89-12" name="__codelineno-89-12" href="#__codelineno-89-12"></a>
-<a id="__codelineno-89-13" name="__codelineno-89-13" href="#__codelineno-89-13"></a><span class="w">    </span><span class="cm">/* Get list length (current number of elements) */</span>
+<a id="__codelineno-89-13" name="__codelineno-89-13" href="#__codelineno-89-13"></a><span class="w">    </span><span class="cm">/* Get the list length (current number of elements) */</span>
 <a id="__codelineno-89-14" name="__codelineno-89-14" href="#__codelineno-89-14"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">size</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-89-15" name="__codelineno-89-15" href="#__codelineno-89-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">size</span>
 <a id="__codelineno-89-16" name="__codelineno-89-16" href="#__codelineno-89-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-89-17" name="__codelineno-89-17" href="#__codelineno-89-17"></a>
-<a id="__codelineno-89-18" name="__codelineno-89-18" href="#__codelineno-89-18"></a><span class="w">    </span><span class="cm">/* Get list capacity */</span>
+<a id="__codelineno-89-18" name="__codelineno-89-18" href="#__codelineno-89-18"></a><span class="w">    </span><span class="cm">/* Get the list capacity */</span>
 <a id="__codelineno-89-19" name="__codelineno-89-19" href="#__codelineno-89-19"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">capacity</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-89-20" name="__codelineno-89-20" href="#__codelineno-89-20"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">capacity</span>
 <a id="__codelineno-89-21" name="__codelineno-89-21" href="#__codelineno-89-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-89-22" name="__codelineno-89-22" href="#__codelineno-89-22"></a>
-<a id="__codelineno-89-23" name="__codelineno-89-23" href="#__codelineno-89-23"></a><span class="w">    </span><span class="cm">/* Update element */</span>
+<a id="__codelineno-89-23" name="__codelineno-89-23" href="#__codelineno-89-23"></a><span class="w">    </span><span class="cm">/* Access an element */</span>
 <a id="__codelineno-89-24" name="__codelineno-89-24" href="#__codelineno-89-24"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-89-25" name="__codelineno-89-25" href="#__codelineno-89-25"></a><span class="w">        </span><span class="c1">// If the index is out of bounds, throw an exception, as below</span>
+<a id="__codelineno-89-25" name="__codelineno-89-25" href="#__codelineno-89-25"></a><span class="w">        </span><span class="c1">// Throw an exception if the index is out of bounds; the same applies below</span>
 <a id="__codelineno-89-26" name="__codelineno-89-26" href="#__codelineno-89-26"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">)</span>
 <a id="__codelineno-89-27" name="__codelineno-89-27" href="#__codelineno-89-27"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-89-28" name="__codelineno-89-28" href="#__codelineno-89-28"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">arr</span><span class="o">[</span><span class="n">index</span><span class="o">]</span>
 <a id="__codelineno-89-29" name="__codelineno-89-29" href="#__codelineno-89-29"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-89-30" name="__codelineno-89-30" href="#__codelineno-89-30"></a>
-<a id="__codelineno-89-31" name="__codelineno-89-31" href="#__codelineno-89-31"></a><span class="w">    </span><span class="cm">/* Add elements at the end */</span>
+<a id="__codelineno-89-31" name="__codelineno-89-31" href="#__codelineno-89-31"></a><span class="w">    </span><span class="cm">/* Update an element */</span>
 <a id="__codelineno-89-32" name="__codelineno-89-32" href="#__codelineno-89-32"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">set</span><span class="p">(</span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-89-33" name="__codelineno-89-33" href="#__codelineno-89-33"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">)</span>
 <a id="__codelineno-89-34" name="__codelineno-89-34" href="#__codelineno-89-34"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-89-35" name="__codelineno-89-35" href="#__codelineno-89-35"></a><span class="w">        </span><span class="n">arr</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-89-36" name="__codelineno-89-36" href="#__codelineno-89-36"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-89-37" name="__codelineno-89-37" href="#__codelineno-89-37"></a>
-<a id="__codelineno-89-38" name="__codelineno-89-38" href="#__codelineno-89-38"></a><span class="w">    </span><span class="cm">/* Direct traversal of list elements */</span>
+<a id="__codelineno-89-38" name="__codelineno-89-38" href="#__codelineno-89-38"></a><span class="w">    </span><span class="cm">/* Append at the end */</span>
 <a id="__codelineno-89-39" name="__codelineno-89-39" href="#__codelineno-89-39"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">add</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-89-40" name="__codelineno-89-40" href="#__codelineno-89-40"></a><span class="w">        </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-89-40" name="__codelineno-89-40" href="#__codelineno-89-40"></a><span class="w">        </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-89-41" name="__codelineno-89-41" href="#__codelineno-89-41"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span>
 <a id="__codelineno-89-42" name="__codelineno-89-42" href="#__codelineno-89-42"></a><span class="w">            </span><span class="n">extendCapacity</span><span class="p">()</span>
 <a id="__codelineno-89-43" name="__codelineno-89-43" href="#__codelineno-89-43"></a><span class="w">        </span><span class="n">arr</span><span class="o">[</span><span class="n">size</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span>
@@ -6777,14 +6849,14 @@
 <a id="__codelineno-89-45" name="__codelineno-89-45" href="#__codelineno-89-45"></a><span class="w">        </span><span class="n">size</span><span class="o">++</span>
 <a id="__codelineno-89-46" name="__codelineno-89-46" href="#__codelineno-89-46"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-89-47" name="__codelineno-89-47" href="#__codelineno-89-47"></a>
-<a id="__codelineno-89-48" name="__codelineno-89-48" href="#__codelineno-89-48"></a><span class="w">    </span><span class="cm">/* Sort list */</span>
+<a id="__codelineno-89-48" name="__codelineno-89-48" href="#__codelineno-89-48"></a><span class="w">    </span><span class="cm">/* Insert an element in the middle */</span>
 <a id="__codelineno-89-49" name="__codelineno-89-49" href="#__codelineno-89-49"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-89-50" name="__codelineno-89-50" href="#__codelineno-89-50"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">)</span>
 <a id="__codelineno-89-51" name="__codelineno-89-51" href="#__codelineno-89-51"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
-<a id="__codelineno-89-52" name="__codelineno-89-52" href="#__codelineno-89-52"></a><span class="w">        </span><span class="c1">// When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-89-52" name="__codelineno-89-52" href="#__codelineno-89-52"></a><span class="w">        </span><span class="c1">// Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-89-53" name="__codelineno-89-53" href="#__codelineno-89-53"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span>
 <a id="__codelineno-89-54" name="__codelineno-89-54" href="#__codelineno-89-54"></a><span class="w">            </span><span class="n">extendCapacity</span><span class="p">()</span>
-<a id="__codelineno-89-55" name="__codelineno-89-55" href="#__codelineno-89-55"></a><span class="w">        </span><span class="c1">// Move all elements after index index forward by one position</span>
+<a id="__codelineno-89-55" name="__codelineno-89-55" href="#__codelineno-89-55"></a><span class="w">        </span><span class="c1">// Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-89-56" name="__codelineno-89-56" href="#__codelineno-89-56"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="n">downTo</span><span class="w"> </span><span class="n">index</span><span class="p">)</span>
 <a id="__codelineno-89-57" name="__codelineno-89-57" href="#__codelineno-89-57"></a><span class="w">            </span><span class="n">arr</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="o">[</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-89-58" name="__codelineno-89-58" href="#__codelineno-89-58"></a><span class="w">        </span><span class="n">arr</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span>
@@ -6792,12 +6864,12 @@
 <a id="__codelineno-89-60" name="__codelineno-89-60" href="#__codelineno-89-60"></a><span class="w">        </span><span class="n">size</span><span class="o">++</span>
 <a id="__codelineno-89-61" name="__codelineno-89-61" href="#__codelineno-89-61"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-89-62" name="__codelineno-89-62" href="#__codelineno-89-62"></a>
-<a id="__codelineno-89-63" name="__codelineno-89-63" href="#__codelineno-89-63"></a><span class="w">    </span><span class="cm">/* Remove element */</span>
+<a id="__codelineno-89-63" name="__codelineno-89-63" href="#__codelineno-89-63"></a><span class="w">    </span><span class="cm">/* Remove an element */</span>
 <a id="__codelineno-89-64" name="__codelineno-89-64" href="#__codelineno-89-64"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-89-65" name="__codelineno-89-65" href="#__codelineno-89-65"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">)</span>
 <a id="__codelineno-89-66" name="__codelineno-89-66" href="#__codelineno-89-66"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-89-67" name="__codelineno-89-67" href="#__codelineno-89-67"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="o">[</span><span class="n">index</span><span class="o">]</span>
-<a id="__codelineno-89-68" name="__codelineno-89-68" href="#__codelineno-89-68"></a><span class="w">        </span><span class="c1">// Move all elements after index forward by one position</span>
+<a id="__codelineno-89-68" name="__codelineno-89-68" href="#__codelineno-89-68"></a><span class="w">        </span><span class="c1">// Shift all elements after index `index` one position to the left</span>
 <a id="__codelineno-89-69" name="__codelineno-89-69" href="#__codelineno-89-69"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">index</span><span class="p">..</span><span class="o">&lt;</span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-89-70" name="__codelineno-89-70" href="#__codelineno-89-70"></a><span class="w">            </span><span class="n">arr</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="o">]</span>
 <a id="__codelineno-89-71" name="__codelineno-89-71" href="#__codelineno-89-71"></a><span class="w">        </span><span class="c1">// Update the number of elements</span>
@@ -6806,18 +6878,18 @@
 <a id="__codelineno-89-74" name="__codelineno-89-74" href="#__codelineno-89-74"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-89-75" name="__codelineno-89-75" href="#__codelineno-89-75"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-89-76" name="__codelineno-89-76" href="#__codelineno-89-76"></a>
-<a id="__codelineno-89-77" name="__codelineno-89-77" href="#__codelineno-89-77"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-89-77" name="__codelineno-89-77" href="#__codelineno-89-77"></a><span class="w">    </span><span class="cm">/* Grow the list capacity */</span>
 <a id="__codelineno-89-78" name="__codelineno-89-78" href="#__codelineno-89-78"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">extendCapacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-89-79" name="__codelineno-89-79" href="#__codelineno-89-79"></a><span class="w">        </span><span class="c1">// Create a new array with length extendRatio times the original array and copy the original array to the new array</span>
+<a id="__codelineno-89-79" name="__codelineno-89-79" href="#__codelineno-89-79"></a><span class="w">        </span><span class="c1">// Create an array extendRatio times as long as the original and copy the original array into it</span>
 <a id="__codelineno-89-80" name="__codelineno-89-80" href="#__codelineno-89-80"></a><span class="w">        </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">.</span><span class="na">copyOf</span><span class="p">(</span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">)</span>
-<a id="__codelineno-89-81" name="__codelineno-89-81" href="#__codelineno-89-81"></a><span class="w">        </span><span class="c1">// Add elements at the end</span>
+<a id="__codelineno-89-81" name="__codelineno-89-81" href="#__codelineno-89-81"></a><span class="w">        </span><span class="c1">// Update the list capacity</span>
 <a id="__codelineno-89-82" name="__codelineno-89-82" href="#__codelineno-89-82"></a><span class="w">        </span><span class="n">capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-89-83" name="__codelineno-89-83" href="#__codelineno-89-83"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-89-84" name="__codelineno-89-84" href="#__codelineno-89-84"></a>
-<a id="__codelineno-89-85" name="__codelineno-89-85" href="#__codelineno-89-85"></a><span class="w">    </span><span class="cm">/* Convert list to array */</span>
+<a id="__codelineno-89-85" name="__codelineno-89-85" href="#__codelineno-89-85"></a><span class="w">    </span><span class="cm">/* Convert the list to an array */</span>
 <a id="__codelineno-89-86" name="__codelineno-89-86" href="#__codelineno-89-86"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">toArray</span><span class="p">():</span><span class="w"> </span><span class="n">IntArray</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-89-87" name="__codelineno-89-87" href="#__codelineno-89-87"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span>
-<a id="__codelineno-89-88" name="__codelineno-89-88" href="#__codelineno-89-88"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-89-88" name="__codelineno-89-88" href="#__codelineno-89-88"></a><span class="w">        </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-89-89" name="__codelineno-89-89" href="#__codelineno-89-89"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">size</span><span class="p">)</span>
 <a id="__codelineno-89-90" name="__codelineno-89-90" href="#__codelineno-89-90"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-89-91" name="__codelineno-89-91" href="#__codelineno-89-91"></a><span class="w">            </span><span class="n">arr</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">get</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
@@ -6830,8 +6902,8 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">my_list.rb</span><pre><span></span><code><a id="__codelineno-90-1" name="__codelineno-90-1" href="#__codelineno-90-1"></a><span class="c1">### List class ###</span>
 <a id="__codelineno-90-2" name="__codelineno-90-2" href="#__codelineno-90-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">MyList</span>
-<a id="__codelineno-90-3" name="__codelineno-90-3" href="#__codelineno-90-3"></a><span class="w">  </span><span class="kp">attr_reader</span><span class="w"> </span><span class="ss">:size</span><span class="w">       </span><span class="c1"># Get list length (current number of elements)</span>
-<a id="__codelineno-90-4" name="__codelineno-90-4" href="#__codelineno-90-4"></a><span class="w">  </span><span class="kp">attr_reader</span><span class="w"> </span><span class="ss">:capacity</span><span class="w">   </span><span class="c1"># Get list capacity</span>
+<a id="__codelineno-90-3" name="__codelineno-90-3" href="#__codelineno-90-3"></a><span class="w">  </span><span class="kp">attr_reader</span><span class="w"> </span><span class="ss">:size</span><span class="w">       </span><span class="c1"># Get the list length (current number of elements)</span>
+<a id="__codelineno-90-4" name="__codelineno-90-4" href="#__codelineno-90-4"></a><span class="w">  </span><span class="kp">attr_reader</span><span class="w"> </span><span class="ss">:capacity</span><span class="w">   </span><span class="c1"># Get the list capacity</span>
 <a id="__codelineno-90-5" name="__codelineno-90-5" href="#__codelineno-90-5"></a>
 <a id="__codelineno-90-6" name="__codelineno-90-6" href="#__codelineno-90-6"></a><span class="w">  </span><span class="c1">### Constructor ###</span>
 <a id="__codelineno-90-7" name="__codelineno-90-7" href="#__codelineno-90-7"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">initialize</span>
@@ -6841,22 +6913,22 @@
 <a id="__codelineno-90-11" name="__codelineno-90-11" href="#__codelineno-90-11"></a><span class="w">    </span><span class="vi">@arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">capacity</span><span class="p">)</span>
 <a id="__codelineno-90-12" name="__codelineno-90-12" href="#__codelineno-90-12"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-90-13" name="__codelineno-90-13" href="#__codelineno-90-13"></a>
-<a id="__codelineno-90-14" name="__codelineno-90-14" href="#__codelineno-90-14"></a><span class="w">  </span><span class="c1">### Access element ###</span>
+<a id="__codelineno-90-14" name="__codelineno-90-14" href="#__codelineno-90-14"></a><span class="w">  </span><span class="c1">### Access an element ###</span>
 <a id="__codelineno-90-15" name="__codelineno-90-15" href="#__codelineno-90-15"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
-<a id="__codelineno-90-16" name="__codelineno-90-16" href="#__codelineno-90-16"></a><span class="w">    </span><span class="c1"># If the index is out of bounds, throw an exception, as below</span>
+<a id="__codelineno-90-16" name="__codelineno-90-16" href="#__codelineno-90-16"></a><span class="w">    </span><span class="c1"># Raise an exception if the index is out of bounds; the same applies below</span>
 <a id="__codelineno-90-17" name="__codelineno-90-17" href="#__codelineno-90-17"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="p">,</span><span class="w"> </span><span class="s2">"Index out of bounds"</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span>
 <a id="__codelineno-90-18" name="__codelineno-90-18" href="#__codelineno-90-18"></a><span class="w">    </span><span class="vi">@arr</span><span class="o">[</span><span class="n">index</span><span class="o">]</span>
 <a id="__codelineno-90-19" name="__codelineno-90-19" href="#__codelineno-90-19"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-90-20" name="__codelineno-90-20" href="#__codelineno-90-20"></a>
-<a id="__codelineno-90-21" name="__codelineno-90-21" href="#__codelineno-90-21"></a><span class="w">  </span><span class="c1">### Access element ###</span>
+<a id="__codelineno-90-21" name="__codelineno-90-21" href="#__codelineno-90-21"></a><span class="w">  </span><span class="c1">### Update an element ###</span>
 <a id="__codelineno-90-22" name="__codelineno-90-22" href="#__codelineno-90-22"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">set</span><span class="p">(</span><span class="n">index</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-90-23" name="__codelineno-90-23" href="#__codelineno-90-23"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="p">,</span><span class="w"> </span><span class="s2">"Index out of bounds"</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span>
 <a id="__codelineno-90-24" name="__codelineno-90-24" href="#__codelineno-90-24"></a><span class="w">    </span><span class="vi">@arr</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-90-25" name="__codelineno-90-25" href="#__codelineno-90-25"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-90-26" name="__codelineno-90-26" href="#__codelineno-90-26"></a>
-<a id="__codelineno-90-27" name="__codelineno-90-27" href="#__codelineno-90-27"></a><span class="w">  </span><span class="c1">### Add element at end ###</span>
+<a id="__codelineno-90-27" name="__codelineno-90-27" href="#__codelineno-90-27"></a><span class="w">  </span><span class="c1">### Append an element at the end ###</span>
 <a id="__codelineno-90-28" name="__codelineno-90-28" href="#__codelineno-90-28"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">add</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
-<a id="__codelineno-90-29" name="__codelineno-90-29" href="#__codelineno-90-29"></a><span class="w">    </span><span class="c1"># When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-90-29" name="__codelineno-90-29" href="#__codelineno-90-29"></a><span class="w">    </span><span class="c1"># Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-90-30" name="__codelineno-90-30" href="#__codelineno-90-30"></a><span class="w">    </span><span class="n">extend_capacity</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span>
 <a id="__codelineno-90-31" name="__codelineno-90-31" href="#__codelineno-90-31"></a><span class="w">    </span><span class="vi">@arr</span><span class="o">[</span><span class="n">size</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-90-32" name="__codelineno-90-32" href="#__codelineno-90-32"></a>
@@ -6864,14 +6936,14 @@
 <a id="__codelineno-90-34" name="__codelineno-90-34" href="#__codelineno-90-34"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-90-35" name="__codelineno-90-35" href="#__codelineno-90-35"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-90-36" name="__codelineno-90-36" href="#__codelineno-90-36"></a>
-<a id="__codelineno-90-37" name="__codelineno-90-37" href="#__codelineno-90-37"></a><span class="w">  </span><span class="c1">### Insert element in middle ###</span>
+<a id="__codelineno-90-37" name="__codelineno-90-37" href="#__codelineno-90-37"></a><span class="w">  </span><span class="c1">### Insert an element in the middle ###</span>
 <a id="__codelineno-90-38" name="__codelineno-90-38" href="#__codelineno-90-38"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">index</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-90-39" name="__codelineno-90-39" href="#__codelineno-90-39"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="p">,</span><span class="w"> </span><span class="s2">"Index out of bounds"</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span>
 <a id="__codelineno-90-40" name="__codelineno-90-40" href="#__codelineno-90-40"></a>
-<a id="__codelineno-90-41" name="__codelineno-90-41" href="#__codelineno-90-41"></a><span class="w">    </span><span class="c1"># When the number of elements exceeds capacity, trigger the extension mechanism</span>
+<a id="__codelineno-90-41" name="__codelineno-90-41" href="#__codelineno-90-41"></a><span class="w">    </span><span class="c1"># Grow the capacity when adding an element would exceed it</span>
 <a id="__codelineno-90-42" name="__codelineno-90-42" href="#__codelineno-90-42"></a><span class="w">    </span><span class="n">extend_capacity</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span>
 <a id="__codelineno-90-43" name="__codelineno-90-43" href="#__codelineno-90-43"></a>
-<a id="__codelineno-90-44" name="__codelineno-90-44" href="#__codelineno-90-44"></a><span class="w">    </span><span class="c1"># Move all elements after index index forward by one position</span>
+<a id="__codelineno-90-44" name="__codelineno-90-44" href="#__codelineno-90-44"></a><span class="w">    </span><span class="c1"># Shift all elements at and after index `index` one position to the right</span>
 <a id="__codelineno-90-45" name="__codelineno-90-45" href="#__codelineno-90-45"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">downto</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
 <a id="__codelineno-90-46" name="__codelineno-90-46" href="#__codelineno-90-46"></a><span class="w">      </span><span class="vi">@arr</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@arr</span><span class="o">[</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-90-47" name="__codelineno-90-47" href="#__codelineno-90-47"></a><span class="w">    </span><span class="k">end</span>
@@ -6881,12 +6953,12 @@
 <a id="__codelineno-90-51" name="__codelineno-90-51" href="#__codelineno-90-51"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-90-52" name="__codelineno-90-52" href="#__codelineno-90-52"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-90-53" name="__codelineno-90-53" href="#__codelineno-90-53"></a>
-<a id="__codelineno-90-54" name="__codelineno-90-54" href="#__codelineno-90-54"></a><span class="w">  </span><span class="c1">### Delete element ###</span>
+<a id="__codelineno-90-54" name="__codelineno-90-54" href="#__codelineno-90-54"></a><span class="w">  </span><span class="c1">### Remove an element ###</span>
 <a id="__codelineno-90-55" name="__codelineno-90-55" href="#__codelineno-90-55"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
 <a id="__codelineno-90-56" name="__codelineno-90-56" href="#__codelineno-90-56"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="p">,</span><span class="w"> </span><span class="s2">"Index out of bounds"</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span>
 <a id="__codelineno-90-57" name="__codelineno-90-57" href="#__codelineno-90-57"></a><span class="w">    </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@arr</span><span class="o">[</span><span class="n">index</span><span class="o">]</span>
 <a id="__codelineno-90-58" name="__codelineno-90-58" href="#__codelineno-90-58"></a>
-<a id="__codelineno-90-59" name="__codelineno-90-59" href="#__codelineno-90-59"></a><span class="w">    </span><span class="c1"># Move all elements after index forward by one position</span>
+<a id="__codelineno-90-59" name="__codelineno-90-59" href="#__codelineno-90-59"></a><span class="w">    </span><span class="c1"># Shift all elements after index one position to the left</span>
 <a id="__codelineno-90-60" name="__codelineno-90-60" href="#__codelineno-90-60"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">index</span><span class="o">...</span><span class="n">size</span>
 <a id="__codelineno-90-61" name="__codelineno-90-61" href="#__codelineno-90-61"></a><span class="w">      </span><span class="vi">@arr</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@arr</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span>
 <a id="__codelineno-90-62" name="__codelineno-90-62" href="#__codelineno-90-62"></a><span class="w">    </span><span class="k">end</span>
@@ -6898,18 +6970,18 @@
 <a id="__codelineno-90-68" name="__codelineno-90-68" href="#__codelineno-90-68"></a><span class="w">    </span><span class="n">num</span>
 <a id="__codelineno-90-69" name="__codelineno-90-69" href="#__codelineno-90-69"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-90-70" name="__codelineno-90-70" href="#__codelineno-90-70"></a>
-<a id="__codelineno-90-71" name="__codelineno-90-71" href="#__codelineno-90-71"></a><span class="w">  </span><span class="c1">### Expand list capacity ###</span>
+<a id="__codelineno-90-71" name="__codelineno-90-71" href="#__codelineno-90-71"></a><span class="w">  </span><span class="c1">### Grow the list capacity ###</span>
 <a id="__codelineno-90-72" name="__codelineno-90-72" href="#__codelineno-90-72"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">extend_capacity</span>
-<a id="__codelineno-90-73" name="__codelineno-90-73" href="#__codelineno-90-73"></a><span class="w">    </span><span class="c1"># Create new array with length extend_ratio times original, copy original array to new array</span>
+<a id="__codelineno-90-73" name="__codelineno-90-73" href="#__codelineno-90-73"></a><span class="w">    </span><span class="c1"># Duplicate the original array and append capacity * (extend_ratio - 1) empty slots</span>
 <a id="__codelineno-90-74" name="__codelineno-90-74" href="#__codelineno-90-74"></a><span class="w">    </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@arr</span><span class="o">.</span><span class="n">dup</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">capacity</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="vi">@extend_ratio</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span>
-<a id="__codelineno-90-75" name="__codelineno-90-75" href="#__codelineno-90-75"></a><span class="w">    </span><span class="c1"># Add elements at the end</span>
+<a id="__codelineno-90-75" name="__codelineno-90-75" href="#__codelineno-90-75"></a><span class="w">    </span><span class="c1"># Update the list capacity</span>
 <a id="__codelineno-90-76" name="__codelineno-90-76" href="#__codelineno-90-76"></a><span class="w">    </span><span class="vi">@capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-90-77" name="__codelineno-90-77" href="#__codelineno-90-77"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-90-78" name="__codelineno-90-78" href="#__codelineno-90-78"></a>
-<a id="__codelineno-90-79" name="__codelineno-90-79" href="#__codelineno-90-79"></a><span class="w">  </span><span class="c1">### Convert list to array ###</span>
+<a id="__codelineno-90-79" name="__codelineno-90-79" href="#__codelineno-90-79"></a><span class="w">  </span><span class="c1">### Convert the list to an array ###</span>
 <a id="__codelineno-90-80" name="__codelineno-90-80" href="#__codelineno-90-80"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">to_array</span>
 <a id="__codelineno-90-81" name="__codelineno-90-81" href="#__codelineno-90-81"></a><span class="w">    </span><span class="n">sz</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span>
-<a id="__codelineno-90-82" name="__codelineno-90-82" href="#__codelineno-90-82"></a><span class="w">    </span><span class="c1"># Elements enqueue</span>
+<a id="__codelineno-90-82" name="__codelineno-90-82" href="#__codelineno-90-82"></a><span class="w">    </span><span class="c1"># Convert only the list elements within the occupied range</span>
 <a id="__codelineno-90-83" name="__codelineno-90-83" href="#__codelineno-90-83"></a><span class="w">    </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">sz</span><span class="p">)</span>
 <a id="__codelineno-90-84" name="__codelineno-90-84" href="#__codelineno-90-84"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">sz</span>
 <a id="__codelineno-90-85" name="__codelineno-90-85" href="#__codelineno-90-85"></a><span class="w">      </span><span class="n">arr</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">get</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
@@ -6921,6 +6993,11 @@
 </div>
 </div>
 </div>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
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target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
 
 <!-- Source file information -->
 
@@ -6943,7 +7020,7 @@ aria-label="Footer"
   <a
     href="../linked_list/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 4.2 Linked List"
+    aria-label="Previous: 4.2   Linked lists"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -6955,7 +7032,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        4.2 Linked List
+        4.2 &nbsp; Linked lists
       </div>
     </div>
   </a>
@@ -6967,7 +7044,7 @@ aria-label="Footer"
   <a
     href="../ram_and_cache/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 4.4 Random-Access Memory and Cache *"
+    aria-label="Next: 4.4   Memory and cache *"
     rel="next"
   >
     <div class="md-footer__title">
@@ -6975,7 +7052,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        4.4 Random-Access Memory and Cache *
+        4.4 &nbsp; Memory and cache *
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -6992,7 +7069,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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diff --git a/en/chapter_array_and_linkedlist/ram_and_cache.assets/computer_storage_devices.png b/en/chapter_array_and_linkedlist/ram_and_cache.assets/computer_storage_devices.png
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--- a/en/chapter_array_and_linkedlist/ram_and_cache/index.html
+++ b/en/chapter_array_and_linkedlist/ram_and_cache/index.html
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>4.4 Random-Access Memory and Cache * - Hello Algo</title>
+        <title>4.4 Memory and cache * - Hello Algo</title>
 
 
 
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         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
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     <div data-md-component="skip">
 
 
-        <a href="#44-random-access-memory-and-cache" class="md-skip">
+        <a href="#44-memory-and-cache" class="md-skip">
           Skip to content
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
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+              4.4 &nbsp; Memory and cache *
 
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-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
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-  Chapter 0. Preface
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   <span class="md-ellipsis">
 
 
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+  0.2 &nbsp; How to use this book
 
 
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-  0.3 Summary
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-  Chapter 1. Encounter with Algorithms
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-  Chapter 1. Encounter with Algorithms
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-  1.3 Summary
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-  Chapter 2. Complexity Analysis
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+  2.4 &nbsp; Space complexity
 
 
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  3.6 Exercises
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-  Chapter 4. Arrays and Linked Lists
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-  4.4 Random-Access Memory and Cache *
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   <a href="#441-computer-storage-devices" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4.4.1 &nbsp; Computer Storage Devices
+        4.4.1 &nbsp; Computer storage devices
       </span>
     </span>
   </a>
@@ -1414,7 +1414,7 @@
   <a href="#442-memory-efficiency-of-data-structures" class="md-nav__link">
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       <span class="md-typeset">
-        4.4.2 &nbsp; Memory Efficiency of Data Structures
+        4.4.2 &nbsp; Memory efficiency of data structures
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   <a href="#443-cache-efficiency-of-data-structures" class="md-nav__link">
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       <span class="md-typeset">
-        4.4.3 &nbsp; Cache Efficiency of Data Structures
+        4.4.3 &nbsp; Cache efficiency of data structures
       </span>
     </span>
   </a>
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   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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@@ -1802,7 +1802,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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@@ -2020,7 +2020,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2502,7 +2502,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2960,7 +2960,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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@@ -3002,7 +3002,7 @@
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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@@ -3272,7 +3272,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3299,7 +3299,7 @@
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4569,6 +4569,78 @@
 
 
 
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@@ -4613,7 +4685,7 @@
   <a href="#441-computer-storage-devices" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4.4.1 &nbsp; Computer Storage Devices
+        4.4.1 &nbsp; Computer storage devices
       </span>
     </span>
   </a>
@@ -4624,7 +4696,7 @@
   <a href="#442-memory-efficiency-of-data-structures" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4.4.2 &nbsp; Memory Efficiency of Data Structures
+        4.4.2 &nbsp; Memory efficiency of data structures
       </span>
     </span>
   </a>
@@ -4635,7 +4707,7 @@
   <a href="#443-cache-efficiency-of-data-structures" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4.4.3 &nbsp; Cache Efficiency of Data Structures
+        4.4.3 &nbsp; Cache efficiency of data structures
       </span>
     </span>
   </a>
@@ -4679,19 +4751,19 @@
 
 
 <!-- Page content -->
-<h1 id="44-random-access-memory-and-cache">4.4 &nbsp; Random-Access Memory and Cache *<a class="headerlink" href="#44-random-access-memory-and-cache" title="Permanent link">&para;</a></h1>
-<p>In the first two sections of this chapter, we explored arrays and linked lists, two fundamental and important data structures that represent two physical layouts: "contiguous storage" and "distributed storage", respectively.</p>
-<p>In fact, <strong>physical structure largely determines the efficiency with which programs utilize memory and cache</strong>, which in turn affects the overall performance of algorithmic programs.</p>
-<h2 id="441-computer-storage-devices">4.4.1 &nbsp; Computer Storage Devices<a class="headerlink" href="#441-computer-storage-devices" title="Permanent link">&para;</a></h2>
-<p>Computers include three types of storage devices: <u>hard disk</u>, <u>random-access memory (RAM)</u>, and <u>cache memory</u>. The following table shows their different roles and performance characteristics in a computer system.</p>
-<p align="center"> Table 4-2 &nbsp; Computer Storage Devices </p>
+<h1 id="44-memory-and-cache">4.4 &nbsp; Memory and cache *<a class="headerlink" href="#44-memory-and-cache" title="Permanent link">&para;</a></h1>
+<p>The first two sections of this chapter introduced arrays and linked lists, two fundamental structures representing contiguous and scattered storage, respectively.</p>
+<p><strong>Physical structure largely determines how efficiently a program uses memory and cache</strong>, affecting its overall performance.</p>
+<h2 id="441-computer-storage-devices">4.4.1 &nbsp; Computer storage devices<a class="headerlink" href="#441-computer-storage-devices" title="Permanent link">&para;</a></h2>
+<p>Computers have three types of storage: <u>disk storage</u>, <u>random-access memory (RAM)</u>, and <u>cache memory</u>. Table 4-2 compares their roles and performance characteristics.</p>
+<p align="center"> Table 4-2 &nbsp; Computer storage devices </p>
 
 <div class="center-table">
 <table>
 <thead>
 <tr>
 <th></th>
-<th>Hard Disk</th>
+<th>Disk storage</th>
 <th>RAM</th>
 <th>Cache</th>
 </tr>
@@ -4699,80 +4771,80 @@
 <tbody>
 <tr>
 <td>Purpose</td>
-<td>Long-term storage of data, including operating systems, programs, and files</td>
-<td>Temporary storage of currently running programs and data being processed</td>
-<td>Storage of frequently accessed data and instructions to reduce CPU's accesses to memory</td>
+<td>Long-term storage of the operating system, programs, files, and other data</td>
+<td>Temporary storage of running programs and data being processed</td>
+<td>Stores frequently accessed data and instructions to reduce CPU accesses to RAM</td>
 </tr>
 <tr>
 <td>Volatility</td>
-<td>Data is not lost after power-off</td>
-<td>Data is lost after power-off</td>
-<td>Data is lost after power-off</td>
+<td>Retains data without power</td>
+<td>Loses data without power</td>
+<td>Loses data without power</td>
 </tr>
 <tr>
 <td>Capacity</td>
-<td>Large, on the order of terabytes (TB)</td>
-<td>Small, on the order of gigabytes (GB)</td>
-<td>Very small, on the order of megabytes (MB)</td>
+<td>Large, on the TB scale</td>
+<td>Smaller, on the GB scale</td>
+<td>Very small, on the MB scale</td>
 </tr>
 <tr>
 <td>Speed</td>
-<td>Slow, hundreds to thousands of MB/s</td>
-<td>Fast, tens of GB/s</td>
+<td>Slower, hundreds to thousands of MB/s</td>
+<td>Faster, tens of GB/s</td>
 <td>Very fast, tens to hundreds of GB/s</td>
 </tr>
 <tr>
-<td>Cost (CNY/GB)</td>
-<td>Inexpensive, from a few tenths of a yuan to a few yuan per GB</td>
-<td>Expensive, from tens to hundreds of yuan per GB</td>
-<td>Very expensive, effectively bundled with the CPU package</td>
+<td>Price (Chinese yuan)</td>
+<td>Inexpensive, fractions of a yuan to several yuan per GB</td>
+<td>More expensive, tens to hundreds of yuan per GB</td>
+<td>Very expensive, priced as part of the CPU</td>
 </tr>
 </tbody>
 </table>
 </div>
-<p>We can imagine the computer storage system as a pyramid, as shown in the diagram below. Storage devices closer to the top are faster, have smaller capacity, and are more expensive. This multi-layered design is deliberate, the result of careful consideration by computer scientists and engineers.</p>
+<p>We can picture computer storage as the pyramid in Figure 4-9. Nearer the top, storage is faster, smaller, and more expensive. This hierarchy is a deliberate design by computer scientists and engineers.</p>
 <ul>
-<li><strong>Hard disk cannot be easily replaced by RAM</strong>. First, data in memory is lost after power-off, making it unsuitable for long-term data storage. Second, memory is tens of times more expensive than hard disk, which makes it difficult to popularize in the consumer market.</li>
-<li><strong>Cache cannot simultaneously achieve large capacity and high speed</strong>. As the capacity of L1, L2, and L3 caches increases, their physical size becomes larger, and the physical distance between them and the CPU core increases, resulting in longer data transmission time and higher element access latency. With current technology, the multi-layered cache structure represents the best balance point between capacity, speed, and cost.</li>
+<li><strong>RAM cannot readily replace disk storage</strong>. It loses data when power is removed, making it unsuitable for long-term storage. It also costs tens of times more than disk storage, making such a replacement impractical for consumers.</li>
+<li><strong>Cache capacity and speed are difficult to maximize together</strong>. From L1 to L2 to L3, increasing capacity means greater physical size and distance from the CPU core, increasing transfer time and access latency. With current technology, a multilevel cache offers the best balance of capacity, speed, and cost.</li>
 </ul>
-<p><img alt="Computer Storage System" class="animation-figure" src="../ram_and_cache.assets/storage_pyramid.png" /></p>
-<p align="center"> Figure 4-9 &nbsp; Computer Storage System </p>
+<p><img alt="Computer storage hierarchy" class="animation-figure" src="../ram_and_cache.assets/storage_pyramid.png" /></p>
+<p align="center"> Figure 4-9 &nbsp; Computer storage hierarchy </p>
 
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>The storage hierarchy of computers embodies a delicate balance among speed, capacity, and cost. In fact, such trade-offs are common across all industrial fields, requiring us to find the optimal balance point between different advantages and constraints.</p>
+<p>The storage hierarchy balances speed, capacity, and cost. Such tradeoffs occur throughout industry, requiring us to find the best balance among competing strengths and limitations.</p>
 </div>
-<p>In summary, <strong>hard disks are used for long-term storage of large amounts of data, RAM is used to temporarily store the data being processed during program execution, and cache is used to store frequently accessed data and instructions</strong>, thereby improving program execution efficiency. The three work together to keep the computer system running efficiently.</p>
-<p>As shown in the diagram below, during program execution, data is read from the hard disk into RAM for CPU computation. Cache can be viewed as part of the CPU. <strong>By intelligently loading data from RAM</strong>, it provides the CPU with high-speed access to data, significantly improving program execution efficiency and reducing reliance on slower RAM.</p>
-<p><img alt="Data Flow Among Hard Disk, RAM, and Cache" class="animation-figure" src="../ram_and_cache.assets/computer_storage_devices.png" /></p>
-<p align="center"> Figure 4-10 &nbsp; Data Flow Among Hard Disk, RAM, and Cache </p>
+<p>In general, <strong>disks store large amounts of data long term, RAM temporarily holds data being processed, and caches hold frequently accessed data and instructions</strong> to speed up execution. The three work together to keep the computer efficient.</p>
+<p>As shown in Figure 4-10, a running program reads data from disk into RAM for the CPU to process. The cache can be viewed as part of the CPU: <strong>it intelligently loads data from RAM</strong> and provides faster access, improving execution speed and reducing dependence on slower memory.</p>
+<p><img alt="Data flow between disk, RAM, and cache" class="animation-figure" src="../ram_and_cache.assets/computer_storage_devices.png" /></p>
+<p align="center"> Figure 4-10 &nbsp; Data flow between disk, RAM, and cache </p>
 
-<h2 id="442-memory-efficiency-of-data-structures">4.4.2 &nbsp; Memory Efficiency of Data Structures<a class="headerlink" href="#442-memory-efficiency-of-data-structures" title="Permanent link">&para;</a></h2>
-<p>In terms of memory space utilization, arrays and linked lists each have advantages and limitations.</p>
-<p>On one hand, <strong>memory is limited, and the same memory cannot be shared by multiple programs</strong>, so we hope data structures can utilize space as efficiently as possible. Array elements are tightly packed and do not require additional space to store references (pointers) between linked list nodes, thus having higher space efficiency. However, arrays need to allocate sufficient contiguous memory space at once, which may lead to memory waste, and array expansion requires additional time and space costs. In comparison, linked lists perform dynamic memory allocation and deallocation on a "node" basis, providing greater flexibility.</p>
-<p>On the other hand, during program execution, <strong>as memory is repeatedly allocated and freed, the degree of fragmentation of free memory becomes increasingly severe</strong>, leading to reduced memory utilization efficiency. Arrays, due to their contiguous storage approach, are relatively less prone to memory fragmentation. Conversely, linked list elements are distributed in storage, and frequent insertion and deletion operations are more likely to cause memory fragmentation.</p>
-<h2 id="443-cache-efficiency-of-data-structures">4.4.3 &nbsp; Cache Efficiency of Data Structures<a class="headerlink" href="#443-cache-efficiency-of-data-structures" title="Permanent link">&para;</a></h2>
-<p>Although cache has much smaller space capacity than memory, it is much faster than memory and plays a crucial role in program execution speed. Since cache capacity is limited and can only store a small portion of frequently accessed data, when the CPU attempts to access data that is not in the cache, a <u>cache miss</u> occurs, and the CPU must load the required data from the slower memory.</p>
-<p>Clearly, <strong>the fewer "cache misses," the higher the efficiency of CPU data reads and writes</strong>, and the better the program performance. We call the proportion of data that the CPU successfully obtains from the cache the <u>cache hit rate</u>, a metric typically used to measure cache efficiency.</p>
-<p>To achieve the highest efficiency possible, cache employs the following data loading mechanisms.</p>
+<h2 id="442-memory-efficiency-of-data-structures">4.4.2 &nbsp; Memory efficiency of data structures<a class="headerlink" href="#442-memory-efficiency-of-data-structures" title="Permanent link">&para;</a></h2>
+<p>Arrays and linked lists each have strengths and limitations in memory use.</p>
+<p>On one hand, <strong>memory is finite, and the same region cannot be shared by multiple programs</strong>, so data structures should use it efficiently. Arrays pack elements tightly without extra references (pointers) between nodes, making them more space-efficient. However, they require a sufficiently large contiguous allocation, which can waste memory, and expansion costs extra time and space. Linked lists instead allocate and reclaim memory node by node, providing greater flexibility.</p>
+<p>On the other hand, <strong>repeated allocation and deallocation increasingly fragment free memory</strong>, reducing its usability. Arrays' contiguous storage is less prone to causing fragmentation. Linked lists store elements in scattered locations, so frequent insertions and deletions are more likely to fragment memory.</p>
+<h2 id="443-cache-efficiency-of-data-structures">4.4.3 &nbsp; Cache efficiency of data structures<a class="headerlink" href="#443-cache-efficiency-of-data-structures" title="Permanent link">&para;</a></h2>
+<p>A cache is much smaller than RAM but far faster, making it crucial to execution speed. Its limited capacity holds only a small portion of frequently accessed data. If the CPU requests data not in the cache, a <u>cache miss</u> occurs, and the data must be fetched from slower RAM.</p>
+<p>Clearly, <strong>fewer cache misses mean more efficient CPU reads and writes</strong> and better program performance. The fraction of accesses successfully served by the cache is the <u>cache hit rate</u>, a common measure of cache efficiency.</p>
+<p>Caches use the following mechanisms to improve efficiency.</p>
 <ul>
-<li><strong>Cache lines</strong>: The cache does not store and load data on a byte-by-byte basis, but rather as cache lines. Compared to byte-by-byte transmission, cache line transmission is more efficient.</li>
-<li><strong>Prefetching mechanism</strong>: The processor attempts to predict data access patterns (e.g., sequential access, fixed-stride jumping access, etc.) and loads data into the cache according to specific patterns, thereby improving hit rate.</li>
-<li><strong>Spatial locality</strong>: If a piece of data is accessed, nearby data may also be accessed in the near future. Therefore, when the cache loads a particular piece of data, it also loads nearby data to improve hit rate.</li>
-<li><strong>Temporal locality</strong>: If a piece of data is accessed, it is likely to be accessed again in the near future. Cache leverages this principle by retaining recently accessed data to improve hit rate.</li>
+<li><strong>Cache lines</strong>: Data is stored and loaded in cache-line units rather than one byte at a time, making transfers more efficient.</li>
+<li><strong>Prefetching</strong>: The processor tries to predict access patterns, such as sequential access or fixed strides, and loads matching data into the cache to improve the hit rate.</li>
+<li><strong>Spatial locality</strong>: Accessing data suggests that nearby data may soon be needed, so the cache loads neighboring data as well.</li>
+<li><strong>Temporal locality</strong>: Recently accessed data is likely to be accessed again soon, so keeping it in the cache improves the hit rate.</li>
 </ul>
-<p>In fact, <strong>arrays and linked lists differ in how efficiently they utilize cache</strong>, mainly in the following respects.</p>
+<p><strong>Arrays and linked lists use the cache differently</strong>, mainly in the following ways.</p>
 <ul>
-<li><strong>Space occupied</strong>: Linked-list elements occupy more space than array elements, so less useful data can fit in the cache.</li>
-<li><strong>Cache lines</strong>: Linked list data are scattered throughout memory, while cache loads "by lines," so the proportion of invalid data loaded is higher.</li>
-<li><strong>Prefetching mechanism</strong>: Arrays have more "predictable" data access patterns than linked lists, making it easier for the system to guess which data will be loaded next.</li>
-<li><strong>Spatial locality</strong>: Arrays are stored in centralized memory space, so data near loaded data is more likely to be accessed soon.</li>
+<li><strong>Storage size</strong>: Linked list elements require more space, leaving less room for useful data in the cache.</li>
+<li><strong>Cache lines</strong>: Linked list data is scattered, so loading a whole cache line brings in a larger proportion of unneeded data.</li>
+<li><strong>Prefetching</strong>: Array access patterns are more predictable, making it easier to anticipate the next data needed.</li>
+<li><strong>Spatial locality</strong>: Array elements are stored together, so data near a loaded element is more likely to be accessed soon.</li>
 </ul>
-<p>Overall, <strong>arrays have higher cache hit rates, thus they usually outperform linked lists in operation efficiency</strong>. This makes data structures implemented based on arrays more popular when solving algorithmic problems.</p>
-<p>It is important to note that <strong>high cache efficiency does not mean arrays are superior to linked lists in all cases</strong>. In practical applications, which data structure to choose should be determined based on specific requirements. For example, both arrays and linked lists can implement the "stack" data structure (which will be discussed in detail in the next chapter), but they are suitable for different scenarios.</p>
+<p>Overall, <strong>arrays have higher cache hit rates and usually faster operations than linked lists</strong>. This makes array-based structures popular for algorithmic problems.</p>
+<p>However, <strong>better cache efficiency does not make arrays superior in every situation</strong>. Choose a structure based on the requirements. Both arrays and linked lists can implement a stack, discussed in the next chapter, but suit different situations.</p>
 <ul>
-<li>When solving algorithm problems, we tend to prefer stack implementations based on arrays, because they provide higher operation efficiency and the ability of random access, at the cost of needing to pre-allocate a certain amount of memory space for the array.</li>
-<li>If the data volume is very large, the dynamic nature is high, and the expected size of the stack is difficult to estimate, then a stack implementation based on linked lists is more suitable. Linked lists can distribute large amounts of data across different parts of memory and avoid the additional overhead produced by array expansion.</li>
+<li>For algorithm exercises, an array-based stack is often preferable because of faster operations and random access, at the cost of allocating some memory in advance.</li>
+<li>If the dataset is very large, changes frequently, and the stack's size is hard to predict, a linked-list-based stack may be more suitable. It can distribute data throughout memory and avoid the extra cost of array expansion.</li>
 </ul>
 
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-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
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-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
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-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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@@ -1791,7 +1791,7 @@
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2009,7 +2009,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2277,7 +2277,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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@@ -2449,7 +2449,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2491,7 +2491,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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@@ -2545,7 +2545,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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@@ -2572,7 +2572,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2667,7 +2667,7 @@
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2709,7 +2709,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2736,7 +2736,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2763,7 +2763,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2790,7 +2790,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2817,7 +2817,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2844,7 +2844,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2871,7 +2871,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2970,7 +2970,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2991,7 +2991,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3018,7 +3018,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3045,7 +3045,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3072,7 +3072,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3099,7 +3099,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3126,7 +3126,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3153,7 +3153,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3180,7 +3180,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3207,7 +3207,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3234,7 +3234,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3261,7 +3261,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3288,7 +3288,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3375,7 +3375,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3396,7 +3396,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3423,7 +3423,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3450,7 +3450,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3477,7 +3477,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3504,7 +3504,7 @@
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3597,7 +3597,7 @@
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3639,7 +3639,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3693,7 +3693,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3720,7 +3720,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3747,7 +3747,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3844,7 +3844,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3865,7 +3865,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3886,7 +3886,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3940,7 +3940,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3967,7 +3967,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -3994,7 +3994,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4021,7 +4021,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4048,7 +4048,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4075,7 +4075,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4141,7 +4141,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4162,7 +4162,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4183,7 +4183,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4210,7 +4210,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4237,7 +4237,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4264,7 +4264,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4291,7 +4291,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4378,7 +4378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4399,7 +4399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4420,7 +4420,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4447,7 +4447,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4474,7 +4474,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4558,6 +4558,78 @@
 
 
 
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+  <a href="#1-key-takeaways" class="md-nav__link">
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       <span class="md-typeset">
-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
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     </span>
   </a>
@@ -4658,62 +4730,62 @@
 
 <!-- Page content -->
 <h1 id="45-summary">4.5 &nbsp; Summary<a class="headerlink" href="#45-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
 <ul>
-<li>Arrays and linked lists are two fundamental data structures, representing two different ways data can be stored in computer memory: contiguous storage and scattered storage. Their strengths and weaknesses complement each other.</li>
-<li>Arrays support random access and use less memory; however, inserting and deleting elements is inefficient, and the length is immutable after initialization.</li>
-<li>Linked lists achieve efficient insertion and deletion of nodes by modifying references (pointers), and can flexibly adjust length; however, node access is inefficient and memory consumption is higher. Common linked list types include singly linked lists, circular linked lists, and doubly linked lists.</li>
-<li>A list is an ordered collection of elements that supports insertion, deletion, search, and modification, typically implemented based on dynamic arrays. It retains the advantages of arrays while allowing flexible adjustment of length.</li>
-<li>The emergence of lists has greatly improved the practicality of arrays, but it may also waste some memory space.</li>
-<li>During program execution, data is primarily stored in memory. Arrays provide higher memory space efficiency, while linked lists offer greater flexibility in memory usage.</li>
-<li>Caches provide fast data access to the CPU through mechanisms such as cache lines, prefetching, and spatial and temporal locality, significantly improving program execution efficiency.</li>
-<li>Because arrays have higher cache hit rates, they are generally more efficient than linked lists. When choosing a data structure, appropriate selection should be made based on specific requirements and scenarios.</li>
+<li>Arrays and linked lists are fundamental structures representing contiguous and scattered memory storage. Their strengths are complementary.</li>
+<li>Arrays support random access and use less memory, but insertion and deletion are slow, and their length is fixed after initialization.</li>
+<li>Linked lists efficiently insert and remove nodes by changing references (pointers), and their length is flexible. However, accessing nodes is slow and they use more memory. Common forms include singly linked, circular, and doubly linked lists.</li>
+<li>A list is an ordered collection supporting insertion, removal, access, and modification. It is usually implemented as a dynamic array, retaining array advantages while allowing its length to change.</li>
+<li>Lists make arrays much more practical, though they may waste some memory.</li>
+<li>During execution, data is mainly stored in RAM. Arrays use memory more efficiently, while linked lists offer more flexible allocation.</li>
+<li>Cache lines, prefetching, spatial locality, and temporal locality provide fast CPU access to data, substantially improving execution speed.</li>
+<li>Arrays usually outperform linked lists because of higher cache hit rates. Choose a structure according to the actual requirements and setting.</li>
 </ul>
 <h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
-<p><strong>Q</strong>: Does storing an array on the stack versus on the heap affect time efficiency and space efficiency?</p>
-<p>Arrays stored on the stack and on the heap are both stored in contiguous memory space, so data operation efficiency is basically the same. However, the stack and heap have their own characteristics, leading to the following differences.</p>
+<p><strong>Q</strong>: Does storing an array on the stack or heap affect time and space efficiency?</p>
+<p>Both use contiguous memory, so data operations have essentially the same efficiency. However, the stack and heap differ in these ways.</p>
 <ol>
-<li>Allocation and deallocation efficiency: The stack is a relatively small piece of memory, with allocation automatically handled by the compiler; the heap is relatively larger and can be dynamically allocated in code, more prone to fragmentation. Therefore, allocation and deallocation operations on the heap are usually slower than on the stack.</li>
-<li>Size limitations: Stack memory is relatively small, and the heap size is generally limited by available memory. Therefore, the heap is more suitable for storing large arrays.</li>
-<li>Flexibility: The size of an array on the stack must be determined at compile time, while the size of an array on the heap can be determined dynamically at runtime.</li>
+<li>Allocation and deallocation: Stack memory is relatively small and allocated automatically by the compiler. The heap is larger, supports dynamic allocation in code, and is more prone to fragmentation. Heap allocation and deallocation are therefore usually slower.</li>
+<li>Size limits: Stack space is relatively small, while the heap is generally limited by available memory, making it better suited to large arrays.</li>
+<li>Flexibility: Stack array sizes must be known at compile time, while heap array sizes can be determined at runtime.</li>
 </ol>
-<p><strong>Q</strong>: Why do arrays require elements of the same type, while linked lists do not emphasize this requirement?</p>
-<p>Linked lists are composed of nodes, with nodes connected through references (pointers), and each node can store different types of data, such as <code>int</code>, <code>double</code>, <code>string</code>, <code>object</code>, etc.</p>
-<p>In contrast, array elements must be of the same type so that their positions can be determined by calculating offsets. For example, if an array contains both <code>int</code> and <code>long</code> types, with individual elements occupying 4 bytes and 8 bytes respectively, then the following formula cannot be used to calculate the offset, because the array contains two different "element sizes".</p>
-<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># element address = array base address (address of the first element) + element size * element index</span>
+<p><strong>Q</strong>: Why must array elements have the same type, while this requirement is not emphasized for linked lists?</p>
+<p>Linked lists consist of nodes connected by references (pointers). Different nodes can hold different types, such as <code>int</code>, <code>double</code>, <code>string</code>, and <code>object</code>.</p>
+<p>Array elements must have the same type so that their positions can be calculated from offsets. If an array contained both <code>int</code> and <code>long</code>, occupying 4 and 8 bytes respectively, the following formula would not work because there would be two element sizes.</p>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Element address = array address (first element's address) + element size * element index</span>
 </code></pre></div>
-<p><strong>Q</strong>: After deleting node <code>P</code>, do we need to set <code>P.next</code> to <code>None</code>?</p>
-<p>It is not necessary to modify <code>P.next</code>. From the perspective of the linked list, traversing from the head node to the tail node will no longer encounter <code>P</code>. This means that node <code>P</code> has been removed from the linked list, and it doesn't matter where node <code>P</code> points to at this time—it won't affect the linked list.</p>
-<p>From an algorithms-and-problem-solving perspective, leaving the pointer connected is fine as long as the program logic is correct. From a standard-library implementation perspective, explicitly disconnecting it is safer and clearer. If it is not disconnected and the deleted node is not reclaimed properly, it may affect the reclamation of successor nodes.</p>
-<p><strong>Q</strong>: In a linked list, the time complexity of insertion and deletion operations is <span class="arithmatex">\(O(1)\)</span>. However, both insertion and deletion require <span class="arithmatex">\(O(n)\)</span> time to find the element; why isn't the time complexity <span class="arithmatex">\(O(n)\)</span>?</p>
-<p>If the element is first found and then deleted, the time complexity is indeed <span class="arithmatex">\(O(n)\)</span>. However, the advantage of <span class="arithmatex">\(O(1)\)</span> insertion and deletion in linked lists can be demonstrated in other applications. For example, a deque is well-suited for linked list implementation, where we maintain pointer variables always pointing to the head and tail nodes, with each insertion and deletion operation being <span class="arithmatex">\(O(1)\)</span>.</p>
-<p><strong>Q</strong>: In the diagram "Linked List Definition and Storage Methods", does the light blue pointer node occupy a single memory address, or does it share equally with the node value?</p>
-<p>This diagram is a qualitative representation; a quantitative representation requires analysis based on the specific situation.</p>
+<p><strong>Q</strong>: After removing node <code>P</code>, must we set <code>P.next</code> to <code>None</code>?</p>
+<p>Leaving <code>P.next</code> unchanged is acceptable. A traversal from the list's head to its tail no longer encounters <code>P</code>, so it has been removed. Whatever <code>P</code> points to no longer affects the list.</p>
+<p>For algorithm exercises, leaving it connected is fine as long as the logic is correct. For a standard library, disconnecting it is safer and clearer: if the removed node is not reclaimed properly, its reference may prevent its successor from being reclaimed.</p>
+<p><strong>Q</strong>: Insertion and deletion in a linked list take <span class="arithmatex">\(O(1)\)</span> time, but locating the element first takes <span class="arithmatex">\(O(n)\)</span>. Why is the overall complexity not <span class="arithmatex">\(O(n)\)</span>?</p>
+<p>Finding an element and then deleting it does take <span class="arithmatex">\(O(n)\)</span> time. However, <span class="arithmatex">\(O(1)\)</span> insertion and deletion are useful in other settings. A deque, for example, can use a linked list with pointers maintained at the head and tail, giving <span class="arithmatex">\(O(1)\)</span> operations.</p>
+<p><strong>Q</strong>: In the linked list storage diagram, does the light-blue pointer occupy a separate memory location, or share half the node's space with its value?</p>
+<p>The diagram is qualitative. Exact sizes depend on the circumstances.</p>
 <ul>
-<li>Different types of node values occupy different amounts of space, such as <code>int</code>, <code>long</code>, <code>double</code>, and instance objects, etc.</li>
-<li>The amount of memory space occupied by pointer variables depends on the operating system and compilation environment used, usually 8 bytes or 4 bytes.</li>
+<li>Node values of different types, such as <code>int</code>, <code>long</code>, <code>double</code>, and objects, occupy different amounts of space.</li>
+<li>A pointer's size depends on the operating system and compilation environment, and is usually 8 or 4 bytes.</li>
 </ul>
-<p><strong>Q</strong>: Is appending an element at the end of a list always <span class="arithmatex">\(O(1)\)</span>?</p>
-<p>If appending an element exceeds the list length, the list must first be expanded before adding. The system allocates a new block of memory and moves all elements from the original list to it, in which case the time complexity becomes <span class="arithmatex">\(O(n)\)</span>.</p>
-<p><strong>Q</strong>: "The emergence of lists has greatly improved the practicality of arrays, but may result in some wasted memory space"—does this space waste refer to the memory occupied by additional variables such as capacity, length, and expansion factor?</p>
-<p>This space waste mainly has two aspects: on one hand, lists typically set an initial length, which we may not need to fully utilize; on the other hand, to prevent frequent expansion, expansion generally multiplies by a coefficient, such as <span class="arithmatex">\(\times 1.5\)</span>. As a result, there will be many empty positions that we typically cannot completely fill.</p>
-<p><strong>Q</strong>: In Python, after initializing <code>n = [1, 2, 3]</code>, the addresses of these 3 elements are contiguous, but initializing <code>m = [2, 1, 3]</code> reveals that each element's id is not continuous; rather, they are the same as those in <code>n</code>. Since the addresses of these elements are not contiguous, is <code>m</code> still an array?</p>
-<p>If we replace list elements with linked list nodes <code>n = [n1, n2, n3, n4, n5]</code>, usually these 5 node objects are also scattered throughout memory. However, given a list index, we can still obtain the node memory address in <span class="arithmatex">\(O(1)\)</span> time, thereby accessing the corresponding node. This is because the array stores references to nodes, not the nodes themselves.</p>
-<p>Unlike many languages, numbers in Python are wrapped as objects, and lists store not the numbers themselves, but references to the numbers. Therefore, we find that the same numbers in two arrays have the same id, and the memory addresses of these numbers need not be contiguous.</p>
-<p><strong>Q</strong>: C++ STL has <code>std::list</code> which has already implemented a doubly linked list, but it seems that some algorithm books don't use it directly. Is there a limitation?</p>
-<p>On one hand, we often prefer to use arrays for implementing algorithms and only use linked lists when necessary, mainly for two reasons.</p>
+<p><strong>Q</strong>: Does appending to a list always take <span class="arithmatex">\(O(1)\)</span> time?</p>
+<p>If appending exceeds the available capacity, the list must first expand. The system allocates new memory and copies all existing elements, making that operation <span class="arithmatex">\(O(n)\)</span>.</p>
+<p><strong>Q</strong>: When lists waste memory, does this mean the extra variables for capacity, length, and growth factor?</p>
+<p>The main waste has two sources. First, the initial allocation may exceed what we need. Second, to avoid frequent resizing, capacity usually grows by a factor such as <span class="arithmatex">\(\times 1.5\)</span>, leaving empty slots that may never all be filled.</p>
+<p><strong>Q</strong>: In Python, after <code>n = [1, 2, 3]</code>, the three elements seem to have consecutive addresses. But in <code>m = [2, 1, 3]</code>, their IDs are not consecutive and match those in <code>n</code>. Is <code>m</code> still an array?</p>
+<p>Consider storing linked list nodes instead: <code>n = [n1, n2, n3, n4, n5]</code>. These five objects are usually scattered in memory, yet an index still gives a node's address in <span class="arithmatex">\(O(1)\)</span> time. The array stores references to the nodes, not the nodes themselves.</p>
+<p>Unlike many languages, Python also wraps numbers in objects. A list stores references to those objects rather than the numbers themselves. Thus, the same numbers in the two arrays can have the same IDs, and their addresses need not be contiguous.</p>
+<p><strong>Q</strong>: C++ STL's <code>std::list</code> already implements a doubly linked list. Why do algorithm books rarely use it directly? Does it have limitations?</p>
+<p>First, we often favor arrays and use linked lists only when needed, mainly for two reasons.</p>
 <ul>
-<li>Space overhead: Since each element requires two additional pointers (one for the previous element and one for the next element), <code>std::list</code> typically consumes more space than <code>std::vector</code>.</li>
-<li>Cache unfriendliness: Since data is not stored contiguously, <code>std::list</code> has lower cache utilization. In general, <code>std::vector</code> has better performance.</li>
+<li>Space overhead: Each element needs two extra pointers, to its predecessor and successor, so <code>std::list</code> usually consumes more space than <code>std::vector</code>.</li>
+<li>Poor cache locality: Its data is not contiguous, making cache use less efficient. In general, <code>std::vector</code> performs better.</li>
 </ul>
-<p>On the other hand, cases where linked lists are necessary mainly involve binary trees and graphs. Stacks and queues usually use the <code>stack</code> and <code>queue</code> provided by the programming language, rather than linked lists.</p>
-<p><strong>Q</strong>: Does the operation <code>res = [[0]] * n</code> create a 2D list where each <code>[0]</code> is independent?</p>
-<p>No, they are not independent. In this 2D list, all the <code>[0]</code> are actually references to the same object. If we modify one element, we will find that all corresponding elements change accordingly.</p>
-<p>If we want each <code>[0]</code> in the 2D list to be independent, we can use <code>res = [[0] for _ in range(n)]</code> to achieve this. The principle of this approach is to initialize <span class="arithmatex">\(n\)</span> independent <code>[0]</code> list objects.</p>
-<p><strong>Q</strong>: Does the operation <code>res = [0] * n</code> create a list where each integer 0 is independent?</p>
-<p>In this list, all the integer zeros reference the same object. This is because Python uses a caching mechanism for small integers (typically -5 to 256) to maximize object reuse and improve performance.</p>
-<p>Although they all reference the same object, we can still modify each element in the list independently. This is because Python integers are "immutable objects". When we modify an element, we actually switch that element to reference a different object, rather than changing the original object itself.</p>
-<p>However, when list elements are "mutable objects" (such as lists, dictionaries, or class instances), modifying an element directly changes the object itself, and all elements referencing that object will have the same change.</p>
+<p>Second, linked structures are mainly needed for binary trees and graphs. For stacks and queues, we usually use the language's provided <code>stack</code> and <code>queue</code> rather than a linked list directly.</p>
+<p><strong>Q</strong>: Does <code>res = [[0]] * n</code> create a two-dimensional list with independent <code>[0]</code> entries?</p>
+<p>No. Every <code>[0]</code> is a reference to the same object. Modifying an element through one reference changes the corresponding element seen through all the others.</p>
+<p>To create independent <code>[0]</code> lists, use <code>res = [[0] for _ in range(n)]</code>. This initializes <span class="arithmatex">\(n\)</span> separate list objects.</p>
+<p><strong>Q</strong>: Does <code>res = [0] * n</code> create independent integer zeros?</p>
+<p>All zeros in the list refer to the same object. Python caches small integers, usually -5 through 256, to maximize reuse and improve performance.</p>
+<p>Even so, each list entry can be changed independently because Python integers are immutable. Assigning a new value switches the entry to a different object reference instead of modifying the original object.</p>
+<p>When an element is mutable, such as a list, dictionary, or class instance, modifying the object itself is visible through every reference to it.</p>
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3283,7 +3283,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3304,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3352,7 +3352,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3379,7 +3379,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3406,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3433,7 +3433,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3460,7 +3460,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3528,7 +3528,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3549,7 +3549,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3579,7 +3579,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3596,7 +3596,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3622,10 +3622,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1311-attempt-and-backtrack" class="md-nav__link">
+  <a href="#1311-trying-choices-and-backtracking" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.1.1 &nbsp; Attempt and Backtrack
+        13.1.1 &nbsp; Trying choices and backtracking
       </span>
     </span>
   </a>
@@ -3644,10 +3644,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1313-framework-code" class="md-nav__link">
+  <a href="#1313-a-code-template" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.1.3 &nbsp; Framework Code
+        13.1.3 &nbsp; A code template
       </span>
     </span>
   </a>
@@ -3658,7 +3658,7 @@
   <a href="#1314-common-terminology" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.1.4 &nbsp; Common Terminology
+        13.1.4 &nbsp; Common terminology
       </span>
     </span>
   </a>
@@ -3666,10 +3666,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1315-advantages-and-limitations" class="md-nav__link">
+  <a href="#1315-strengths-and-limitations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.1.5 &nbsp; Advantages and Limitations
+        13.1.5 &nbsp; Strengths and limitations
       </span>
     </span>
   </a>
@@ -3677,10 +3677,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1316-typical-backtracking-examples" class="md-nav__link">
+  <a href="#1316-typical-backtracking-problems" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.1.6 &nbsp; Typical Backtracking Examples
+        13.1.6 &nbsp; Typical backtracking problems
       </span>
     </span>
   </a>
@@ -3710,7 +3710,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3737,7 +3737,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3764,7 +3764,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3791,7 +3791,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3818,7 +3818,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3888,7 +3888,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3909,7 +3909,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3930,7 +3930,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3957,7 +3957,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3984,7 +3984,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4011,7 +4011,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4038,7 +4038,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4065,7 +4065,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4092,7 +4092,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4119,7 +4119,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4185,7 +4185,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4206,7 +4206,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4227,7 +4227,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4254,7 +4254,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4281,7 +4281,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4308,7 +4308,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4335,7 +4335,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4362,7 +4362,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4422,7 +4422,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4443,7 +4443,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4464,7 +4464,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4491,7 +4491,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4518,7 +4518,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4602,6 +4602,78 @@
 
 
 
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         <li class="md-nav__item">
-  <a href="#1311-attempt-and-backtrack" class="md-nav__link">
+  <a href="#1311-trying-choices-and-backtracking" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.1.1 &nbsp; Attempt and Backtrack
+        13.1.1 &nbsp; Trying choices and backtracking
       </span>
     </span>
   </a>
@@ -4665,10 +4737,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1313-framework-code" class="md-nav__link">
+  <a href="#1313-a-code-template" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.1.3 &nbsp; Framework Code
+        13.1.3 &nbsp; A code template
       </span>
     </span>
   </a>
@@ -4679,7 +4751,7 @@
   <a href="#1314-common-terminology" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.1.4 &nbsp; Common Terminology
+        13.1.4 &nbsp; Common terminology
       </span>
     </span>
   </a>
@@ -4687,10 +4759,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1315-advantages-and-limitations" class="md-nav__link">
+  <a href="#1315-strengths-and-limitations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.1.5 &nbsp; Advantages and Limitations
+        13.1.5 &nbsp; Strengths and limitations
       </span>
     </span>
   </a>
@@ -4698,10 +4770,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1316-typical-backtracking-examples" class="md-nav__link">
+  <a href="#1316-typical-backtracking-problems" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.1.6 &nbsp; Typical Backtracking Examples
+        13.1.6 &nbsp; Typical backtracking problems
       </span>
     </span>
   </a>
@@ -4745,36 +4817,36 @@
 
 
 <!-- Page content -->
-<h1 id="131-backtracking-algorithm">13.1 &nbsp; Backtracking Algorithm<a class="headerlink" href="#131-backtracking-algorithm" title="Permanent link">&para;</a></h1>
-<p><u>The backtracking algorithm</u> is a method for solving problems through exhaustive search. Its core idea is to start from an initial state and exhaustively search all possible solutions. When a correct solution is found, it is recorded. This process continues until a solution is found or all possible choices have been tried without finding a solution.</p>
-<p>The backtracking algorithm typically employs "depth-first search" to traverse the solution space. In the "Binary Tree" chapter, we mentioned that preorder, inorder, and postorder traversals all belong to depth-first search. Next, we will construct a backtracking problem using preorder traversal to progressively understand how the backtracking algorithm works.</p>
+<h1 id="131-backtracking-algorithms">13.1 &nbsp; Backtracking algorithms<a class="headerlink" href="#131-backtracking-algorithms" title="Permanent link">&para;</a></h1>
+<p>A <u>backtracking algorithm</u> solves a problem through exhaustive search. Starting from an initial state, it explores possible solutions by brute force, recording valid solutions as it finds them, until a solution is found or every possible choice has been tried without success.</p>
+<p>Backtracking usually traverses the solution space through depth-first search. In the binary tree chapter, we learned that preorder, inorder, and postorder traversal are all forms of depth-first search. We will use preorder traversal to build a backtracking problem and explore the approach step by step.</p>
 <div class="admonition question">
 <p class="admonition-title">Example 1</p>
-<p>Given a binary tree, search and record all nodes with value <span class="arithmatex">\(7\)</span>, and return a list of these nodes.</p>
+<p>Given a binary tree, find and record every node with value <span class="arithmatex">\(7\)</span>. Return a list of these nodes.</p>
 </div>
-<p>For this problem, we perform a preorder traversal of the tree and check whether the current node's value is <span class="arithmatex">\(7\)</span>. If it is, we add the node to the result list <code>res</code>. The relevant implementation is shown in the following figure and code:</p>
+<p>Traverse the tree in preorder and check whether the current node's value is <span class="arithmatex">\(7\)</span>. If so, add that node's value to the result list <code>res</code>. The code and figure below show the process:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">preorder_traversal_i_compact.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">pre_order</span><span class="p">(</span><span class="n">root</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">):</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Preorder traversal: Example 1"""</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Preorder traversal: example 1"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="k">if</span> <span class="n">root</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>        <span class="k">return</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="k">if</span> <span class="n">root</span><span class="o">.</span><span class="n">val</span> <span class="o">==</span> <span class="mi">7</span><span class="p">:</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="c1"># Record solution</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="c1"># Record the solution</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="n">pre_order</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="n">pre_order</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_i_compact.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Preorder traversal: Example 1 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_i_compact.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Preorder traversal: example 1 */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">);</span>
@@ -4783,13 +4855,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_i_compact.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Preorder traversal: Example 1 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_i_compact.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Preorder traversal: example 1 */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">left</span><span class="p">);</span>
@@ -4798,13 +4870,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_i_compact.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Preorder traversal: Example 1 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_i_compact.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Preorder traversal: example 1 */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">PreOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="n">PreOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">left</span><span class="p">);</span>
@@ -4813,13 +4885,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_i_compact.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Preorder traversal: Example 1 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_i_compact.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Preorder traversal: example 1 */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">preOrderI</span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">Val</span><span class="p">).(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">        </span><span class="o">*</span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">root</span><span class="p">)</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="nx">preOrderI</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">Left</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span>
@@ -4828,13 +4900,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_i_compact.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Preorder traversal: Example 1 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_i_compact.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Preorder traversal: example 1 */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="k">guard</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nv">root</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">7</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="kr">left</span><span class="p">)</span>
@@ -4843,13 +4915,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_i_compact.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Preorder traversal: Example 1 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_i_compact.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Preorder traversal: example 1 */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">root</span><span class="p">);</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
@@ -4858,13 +4930,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_i_compact.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Preorder traversal: Example 1 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_i_compact.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Preorder traversal: example 1 */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">root</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">root</span><span class="p">);</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
@@ -4873,13 +4945,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_i_compact.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Preorder traversal: Example 1 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_i_compact.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Preorder traversal: example 1 */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">preOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">    </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">    </span><span class="c1">// Record solution</span>
+<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">    </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">  </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
@@ -4888,14 +4960,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_i_compact.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Preorder traversal: Example 1 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_i_compact.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Preorder traversal: example 1 */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">pre_order</span><span class="p">(</span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="n">is_none</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">            </span><span class="c1">// Record solution</span>
+<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">            </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">            </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="n">pre_order</span><span class="p">(</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">as_ref</span><span class="p">());</span>
@@ -4905,13 +4977,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_i_compact.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Preorder traversal: Example 1 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_i_compact.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Preorder traversal: example 1 */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">resSize</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">;</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">);</span>
@@ -4920,13 +4992,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_i_compact.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Preorder traversal: Example 1 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_i_compact.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Preorder traversal: example 1 */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">        </span><span class="n">res</span><span class="o">!!</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">left</span><span class="p">)</span>
@@ -4935,11 +5007,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_i_compact.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Pre-order traversal: example 1 ###</span>
+<div class="highlight"><span class="filename">preorder_traversal_i_compact.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Preorder traversal: example 1 ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">pre_order</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="n">root</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Record solution</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Record the solution</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="vg">$res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="n">pre_order</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
@@ -4949,29 +5021,34 @@
 </div>
 </div>
 </div>
-<p><img alt="Search for nodes in preorder traversal" class="animation-figure" src="../backtracking_algorithm.assets/preorder_find_nodes.png" /></p>
-<p align="center"> Figure 13-1 &nbsp; Search for nodes in preorder traversal </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20left%20child%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20right%20child%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%3A%20recursion%22%22%22%0A%20%20%20%20%23%20Return%20None%20if%20the%20index%20is%20out%20of%20bounds%20or%20the%20corresponding%20element%20is%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20Create%20the%20current%20node%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20Recursively%20build%20the%20left%20and%20right%20subtrees%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20pre_order%28root%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22Preorder%20traversal%3A%20example%201%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20if%20root.val%20%3D%3D%207%3A%0A%20%20%20%20%20%20%20%20%23%20Record%20the%20solution%0A%20%20%20%20%20%20%20%20res.append%28root%29%0A%20%20%20%20pre_order%28root.left%29%0A%20%20%20%20pre_order%28root.right%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20Preorder%20traversal%0A%20%20%20%20res%20%3D%20list%5BTreeNode%5D%28%29%0A%20%20%20%20pre_order%28root%29%0A%0A%20%20%20%20print%28%22%5CnOutput%20all%20nodes%20with%20value%207%22%29%0A%20%20%20%20print%28%5Bnode.val%20for%20node%20in%20res%5D%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=126&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20left%20child%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20right%20child%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%3A%20recursion%22%22%22%0A%20%20%20%20%23%20Return%20None%20if%20the%20index%20is%20out%20of%20bounds%20or%20the%20corresponding%20element%20is%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20Create%20the%20current%20node%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20Recursively%20build%20the%20left%20and%20right%20subtrees%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20pre_order%28root%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22Preorder%20traversal%3A%20example%201%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20if%20root.val%20%3D%3D%207%3A%0A%20%20%20%20%20%20%20%20%23%20Record%20the%20solution%0A%20%20%20%20%20%20%20%20res.append%28root%29%0A%20%20%20%20pre_order%28root.left%29%0A%20%20%20%20pre_order%28root.right%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20Preorder%20traversal%0A%20%20%20%20res%20%3D%20list%5BTreeNode%5D%28%29%0A%20%20%20%20pre_order%28root%29%0A%0A%20%20%20%20print%28%22%5CnOutput%20all%20nodes%20with%20value%207%22%29%0A%20%20%20%20print%28%5Bnode.val%20for%20node%20in%20res%5D%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=126&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p><img alt="Finding nodes during preorder traversal" class="animation-figure" src="../backtracking_algorithm.assets/preorder_find_nodes.png" /></p>
+<p align="center"> Figure 13-1 &nbsp; Finding nodes during preorder traversal </p>
 
-<h2 id="1311-attempt-and-backtrack">13.1.1 &nbsp; Attempt and Backtrack<a class="headerlink" href="#1311-attempt-and-backtrack" title="Permanent link">&para;</a></h2>
-<p><strong>The reason it is called a backtracking algorithm is that it employs "attempt" and "backtrack" strategies when searching the solution space</strong>. When the algorithm encounters a state where it cannot continue forward or cannot find a solution that satisfies the constraints, it will undo the previous choice, return to a previous state, and try other possible choices.</p>
-<p>For Example 1, visiting each node represents an "attempt", while skipping over a leaf node or the <code>return</code> that brings the traversal back to the parent node represents a "backtrack".</p>
-<p>It is worth noting that <strong>backtracking is not limited to function returns alone</strong>. To illustrate this, let's extend Example 1 slightly.</p>
+<h2 id="1311-trying-choices-and-backtracking">13.1.1 &nbsp; Trying choices and backtracking<a class="headerlink" href="#1311-trying-choices-and-backtracking" title="Permanent link">&para;</a></h2>
+<p><strong>Backtracking gets its name from its pattern of trying choices and undoing them while searching the solution space</strong>. When the search reaches a state that cannot advance or lead to a valid solution, it undoes the last choice, returns to the previous state, and tries other possibilities.</p>
+<p>In Example 1, visiting a node is a trial, while a <code>return</code> after passing a leaf or returning to a parent is backtracking.</p>
+<p><strong>Backtracking involves more than returning from a function</strong>. To see why, let's extend Example 1.</p>
 <div class="admonition question">
 <p class="admonition-title">Example 2</p>
-<p>In a binary tree, search all nodes with value <span class="arithmatex">\(7\)</span>, <strong>and return the paths from the root node to these nodes</strong>.</p>
+<p>Find every node with value <span class="arithmatex">\(7\)</span> in a binary tree, and <strong>return the paths from the root to those nodes</strong>.</p>
 </div>
-<p>Based on the code from Example 1, we need to use a list <code>path</code> to record the path of visited nodes. When we reach a node with value <span class="arithmatex">\(7\)</span>, we copy <code>path</code> and add it to the result list <code>res</code>. After traversal is complete, <code>res</code> contains all the solutions. The code is as follows:</p>
+<p>Extend the first solution with a list <code>path</code> recording the current path of visited nodes. When a node with value <span class="arithmatex">\(7\)</span> is reached, append a copy of <code>path</code> to the result list <code>res</code>. After traversal, <code>res</code> contains all solutions. The code is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">preorder_traversal_ii_compact.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">pre_order</span><span class="p">(</span><span class="n">root</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">):</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Preorder traversal: Example 2"""</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Preorder traversal: example 2"""</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="k">if</span> <span class="n">root</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>        <span class="k">return</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Attempt</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Try</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="n">path</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>    <span class="k">if</span> <span class="n">root</span><span class="o">.</span><span class="n">val</span> <span class="o">==</span> <span class="mi">7</span><span class="p">:</span>
-<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="c1"># Record solution</span>
+<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="c1"># Record the solution</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">path</span><span class="p">))</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>    <span class="n">pre_order</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>    <span class="n">pre_order</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
@@ -4980,15 +5057,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_ii_compact.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Preorder traversal: Example 2 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_ii_compact.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Preorder traversal: example 2 */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="n">path</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">path</span><span class="p">);</span>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">);</span>
@@ -4999,15 +5076,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_ii_compact.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Preorder traversal: Example 2 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_ii_compact.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Preorder traversal: example 2 */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="n">path</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">(</span><span class="n">path</span><span class="p">));</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">left</span><span class="p">);</span>
@@ -5018,15 +5095,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_ii_compact.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Preorder traversal: Example 2 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_ii_compact.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Preorder traversal: example 2 */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">PreOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="n">path</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="p">(</span><span class="n">path</span><span class="p">));</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">    </span><span class="n">PreOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">left</span><span class="p">);</span>
@@ -5037,15 +5114,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_ii_compact.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Preorder traversal: Example 2 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_ii_compact.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Preorder traversal: example 2 */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">preOrderII</span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[][]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">path</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="o">*</span><span class="nx">path</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">path</span><span class="p">,</span><span class="w"> </span><span class="nx">root</span><span class="p">)</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">root</span><span class="p">.</span><span class="nx">Val</span><span class="p">.(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">        </span><span class="o">*</span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nb">append</span><span class="p">([]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">{},</span><span class="w"> </span><span class="o">*</span><span class="nx">path</span><span class="p">...))</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">    </span><span class="nx">preOrderII</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">Left</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">path</span><span class="p">)</span>
@@ -5056,15 +5133,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_ii_compact.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Preorder traversal: Example 2 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_ii_compact.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Preorder traversal: example 2 */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="k">guard</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nv">root</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="n">path</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">7</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">path</span><span class="p">)</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="kr">left</span><span class="p">)</span>
@@ -5075,15 +5152,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_ii_compact.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Preorder traversal: Example 2 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_ii_compact.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Preorder traversal: example 2 */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">,</span><span class="w"> </span><span class="nx">path</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="nx">path</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">root</span><span class="p">);</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">path</span><span class="p">]);</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">    </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">path</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
@@ -5094,7 +5171,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_ii_compact.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Preorder traversal: Example 2 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_ii_compact.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Preorder traversal: example 2 */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">preOrder</span><span class="p">(</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="nx">root</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="nx">path</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">[],</span>
@@ -5103,10 +5180,10 @@
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="nx">path</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">root</span><span class="p">);</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">path</span><span class="p">]);</span>
 <a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">    </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">path</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
@@ -5117,7 +5194,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_ii_compact.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Preorder traversal: Example 2 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_ii_compact.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Preorder traversal: example 2 */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">preOrder</span><span class="p">(</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">root</span><span class="p">,</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">path</span><span class="p">,</span>
@@ -5127,10 +5204,10 @@
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">    </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a>
-<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">  </span><span class="c1">// Attempt</span>
+<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">  </span><span class="c1">// Try</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">  </span><span class="n">path</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">    </span><span class="c1">// Record solution</span>
+<a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">    </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">List</span><span class="p">.</span><span class="n">from</span><span class="p">(</span><span class="n">path</span><span class="p">));</span>
 <a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">  </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">path</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
@@ -5141,7 +5218,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_ii_compact.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Preorder traversal: Example 2 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_ii_compact.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Preorder traversal: example 2 */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">pre_order</span><span class="p">(</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="n">path</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span>
@@ -5151,10 +5228,10 @@
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">        </span><span class="c1">// Attempt</span>
+<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">        </span><span class="c1">// Try</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">        </span><span class="n">path</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">            </span><span class="c1">// Record solution</span>
+<a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">            </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">            </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">path</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">        </span><span class="n">pre_order</span><span class="p">(</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">path</span><span class="p">,</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">as_ref</span><span class="p">());</span>
@@ -5166,15 +5243,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_ii_compact.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Preorder traversal: Example 2 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_ii_compact.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Preorder traversal: example 2 */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="n">path</span><span class="p">[</span><span class="n">pathSize</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">;</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">pathSize</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">            </span><span class="n">res</span><span class="p">[</span><span class="n">resSize</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">path</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">        </span><span class="p">}</span>
@@ -5188,15 +5265,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_ii_compact.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Preorder traversal: Example 2 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_ii_compact.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Preorder traversal: example 2 */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="n">path</span><span class="o">!!</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">        </span><span class="n">res</span><span class="o">!!</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">path</span><span class="o">!!</span><span class="p">.</span><span class="na">toMutableList</span><span class="p">())</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">left</span><span class="p">)</span>
@@ -5207,14 +5284,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_ii_compact.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Pre-order traversal: example 2 ###</span>
+<div class="highlight"><span class="filename">preorder_traversal_ii_compact.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Preorder traversal: example 2 ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">pre_order</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="n">root</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a>
-<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># Attempt</span>
+<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># Try</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="vg">$path</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">root</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a>
-<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">  </span><span class="c1"># Record solution</span>
+<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">  </span><span class="c1"># Record the solution</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="vg">$res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="vg">$path</span><span class="o">.</span><span class="n">dup</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">  </span><span class="n">pre_order</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
@@ -5227,12 +5304,17 @@
 </div>
 </div>
 </div>
-<p>In each "attempt", we record the path by adding the current node to <code>path</code>; before "backtracking", we need to remove the node from <code>path</code>, <strong>to restore the state before this attempt</strong>.</p>
-<p>Observing the process shown in the following figure, <strong>we can understand attempt and backtrack as "advance" and "undo"</strong>, two operations that are the reverse of each other.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20left%20child%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20right%20child%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%3A%20recursion%22%22%22%0A%20%20%20%20%23%20Return%20None%20if%20the%20index%20is%20out%20of%20bounds%20or%20the%20corresponding%20element%20is%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20Create%20the%20current%20node%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20Recursively%20build%20the%20left%20and%20right%20subtrees%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20pre_order%28root%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22Preorder%20traversal%3A%20example%202%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Try%0A%20%20%20%20path.append%28root%29%0A%20%20%20%20if%20root.val%20%3D%3D%207%3A%0A%20%20%20%20%20%20%20%20%23%20Record%20the%20solution%0A%20%20%20%20%20%20%20%20res.append%28list%28path%29%29%0A%20%20%20%20pre_order%28root.left%29%0A%20%20%20%20pre_order%28root.right%29%0A%20%20%20%20%23%20Backtrack%0A%20%20%20%20path.pop%28%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20Preorder%20traversal%0A%20%20%20%20path%20%3D%20list%5BTreeNode%5D%28%29%0A%20%20%20%20res%20%3D%20list%5Blist%5BTreeNode%5D%5D%28%29%0A%20%20%20%20pre_order%28root%29%0A%0A%20%20%20%20print%28%22%5CnOutput%20all%20paths%20from%20the%20root%20to%20nodes%20with%20value%207%22%29%0A%20%20%20%20for%20path%20in%20res%3A%0A%20%20%20%20%20%20%20%20print%28%5Bnode.val%20for%20node%20in%20path%5D%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=126&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20left%20child%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20right%20child%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%3A%20recursion%22%22%22%0A%20%20%20%20%23%20Return%20None%20if%20the%20index%20is%20out%20of%20bounds%20or%20the%20corresponding%20element%20is%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20Create%20the%20current%20node%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20Recursively%20build%20the%20left%20and%20right%20subtrees%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20pre_order%28root%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22Preorder%20traversal%3A%20example%202%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Try%0A%20%20%20%20path.append%28root%29%0A%20%20%20%20if%20root.val%20%3D%3D%207%3A%0A%20%20%20%20%20%20%20%20%23%20Record%20the%20solution%0A%20%20%20%20%20%20%20%20res.append%28list%28path%29%29%0A%20%20%20%20pre_order%28root.left%29%0A%20%20%20%20pre_order%28root.right%29%0A%20%20%20%20%23%20Backtrack%0A%20%20%20%20path.pop%28%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20Preorder%20traversal%0A%20%20%20%20path%20%3D%20list%5BTreeNode%5D%28%29%0A%20%20%20%20res%20%3D%20list%5Blist%5BTreeNode%5D%5D%28%29%0A%20%20%20%20pre_order%28root%29%0A%0A%20%20%20%20print%28%22%5CnOutput%20all%20paths%20from%20the%20root%20to%20nodes%20with%20value%207%22%29%0A%20%20%20%20for%20path%20in%20res%3A%0A%20%20%20%20%20%20%20%20print%28%5Bnode.val%20for%20node%20in%20path%5D%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=126&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Each time we try a node, append it to <code>path</code>. Before backtracking, pop that node from <code>path</code> <strong>to restore the state from before the trial</strong>.</p>
+<p>As Figure 13-2 illustrates, <strong>trying a choice and backtracking are like moving forward and undoing that move</strong>: they are inverse operations.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:11"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">&lt;1&gt;</label><label for="__tabbed_3_2">&lt;2&gt;</label><label for="__tabbed_3_3">&lt;3&gt;</label><label for="__tabbed_3_4">&lt;4&gt;</label><label for="__tabbed_3_5">&lt;5&gt;</label><label for="__tabbed_3_6">&lt;6&gt;</label><label for="__tabbed_3_7">&lt;7&gt;</label><label for="__tabbed_3_8">&lt;8&gt;</label><label for="__tabbed_3_9">&lt;9&gt;</label><label for="__tabbed_3_10">&lt;10&gt;</label><label for="__tabbed_3_11">&lt;11&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Attempt and backtrack" class="animation-figure" src="../backtracking_algorithm.assets/preorder_find_paths_step1.png" /></p>
+<p><img alt="Trying choices and backtracking" class="animation-figure" src="../backtracking_algorithm.assets/preorder_find_paths_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="preorder_find_paths_step2" class="animation-figure" src="../backtracking_algorithm.assets/preorder_find_paths_step2.png" /></p>
@@ -5266,27 +5348,27 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 13-2 &nbsp; Attempt and backtrack </p>
+<p align="center"> Figure 13-2 &nbsp; Trying choices and backtracking </p>
 
 <h2 id="1312-pruning">13.1.2 &nbsp; Pruning<a class="headerlink" href="#1312-pruning" title="Permanent link">&para;</a></h2>
-<p>Complex backtracking problems usually contain one or more constraints. <strong>Constraints can typically be used for "pruning"</strong>.</p>
+<p>Complex backtracking problems usually have one or more constraints, <strong>which can often be used for pruning</strong>.</p>
 <div class="admonition question">
 <p class="admonition-title">Example 3</p>
-<p>In a binary tree, search all nodes with value <span class="arithmatex">\(7\)</span> and return the paths from the root node to these nodes, <strong>but require that the paths do not contain nodes with value <span class="arithmatex">\(3\)</span></strong>.</p>
+<p>Find every node with value <span class="arithmatex">\(7\)</span> in a binary tree and return the paths from the root to those nodes, <strong>requiring that no path contain a node with value <span class="arithmatex">\(3\)</span></strong>.</p>
 </div>
-<p>To satisfy the above constraints, <strong>we need to add pruning operations</strong>: during the search process, if we encounter a node with value <span class="arithmatex">\(3\)</span>, we return early and do not continue searching. The code is as follows:</p>
+<p>To enforce this constraint, <strong>add a pruning step</strong>: when a node with value <span class="arithmatex">\(3\)</span> is encountered, return immediately without searching farther. The code is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:13"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">preorder_traversal_iii_compact.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">pre_order</span><span class="p">(</span><span class="n">root</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">):</span>
-<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Preorder traversal: Example 3"""</span>
-<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="c1"># Pruning</span>
+<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Preorder traversal: example 3"""</span>
+<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="c1"># Prune</span>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="k">if</span> <span class="n">root</span> <span class="ow">is</span> <span class="kc">None</span> <span class="ow">or</span> <span class="n">root</span><span class="o">.</span><span class="n">val</span> <span class="o">==</span> <span class="mi">3</span><span class="p">:</span>
 <a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>        <span class="k">return</span>
-<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="c1"># Attempt</span>
+<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="c1"># Try</span>
 <a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>    <span class="n">path</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>    <span class="k">if</span> <span class="n">root</span><span class="o">.</span><span class="n">val</span> <span class="o">==</span> <span class="mi">7</span><span class="p">:</span>
-<a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>        <span class="c1"># Record solution</span>
+<a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>        <span class="c1"># Record the solution</span>
 <a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">path</span><span class="p">))</span>
 <a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a>    <span class="n">pre_order</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
 <a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a>    <span class="n">pre_order</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
@@ -5295,16 +5377,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_compact.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Preorder traversal: Example 3 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_compact.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Preorder traversal: example 3 */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="c1">// Pruning</span>
+<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="c1">// Prune</span>
 <a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">root</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">    </span><span class="n">path</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
 <a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">path</span><span class="p">);</span>
 <a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">);</span>
@@ -5315,16 +5397,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_compact.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Preorder traversal: Example 3 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_compact.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Preorder traversal: example 3 */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="c1">// Pruning</span>
+<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="c1">// Prune</span>
 <a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">    </span><span class="n">path</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
 <a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">(</span><span class="n">path</span><span class="p">));</span>
 <a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">left</span><span class="p">);</span>
@@ -5335,16 +5417,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_compact.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Preorder traversal: Example 3 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_compact.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Preorder traversal: example 3 */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">PreOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="c1">// Pruning</span>
+<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="c1">// Prune</span>
 <a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">    </span><span class="n">path</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
 <a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="p">(</span><span class="n">path</span><span class="p">));</span>
 <a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">    </span><span class="n">PreOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">left</span><span class="p">);</span>
@@ -5355,16 +5437,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_compact.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Preorder traversal: Example 3 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_compact.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Preorder traversal: example 3 */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">preOrderIII</span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[][]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">path</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="c1">// Pruning</span>
+<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="c1">// Prune</span>
 <a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">root</span><span class="p">.</span><span class="nx">Val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">    </span><span class="o">*</span><span class="nx">path</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">path</span><span class="p">,</span><span class="w"> </span><span class="nx">root</span><span class="p">)</span>
 <a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">root</span><span class="p">.</span><span class="nx">Val</span><span class="p">.(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">        </span><span class="o">*</span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nb">append</span><span class="p">([]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">{},</span><span class="w"> </span><span class="o">*</span><span class="nx">path</span><span class="p">...))</span>
 <a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="w">    </span><span class="nx">preOrderIII</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">Left</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">path</span><span class="p">)</span>
@@ -5375,16 +5457,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_compact.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Preorder traversal: Example 3 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_compact.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Preorder traversal: example 3 */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="c1">// Pruning</span>
+<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="c1">// Prune</span>
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="k">guard</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nv">root</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">    </span><span class="n">path</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">7</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">path</span><span class="p">)</span>
 <a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="kr">left</span><span class="p">)</span>
@@ -5395,16 +5477,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_compact.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Preorder traversal: Example 3 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_compact.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Preorder traversal: example 3 */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">,</span><span class="w"> </span><span class="nx">path</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="c1">// Pruning</span>
+<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="c1">// Prune</span>
 <a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">root</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">3</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="nx">path</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">root</span><span class="p">);</span>
 <a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">path</span><span class="p">]);</span>
 <a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="w">    </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">path</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
@@ -5415,20 +5497,20 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_compact.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Preorder traversal: Example 3 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_compact.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Preorder traversal: example 3 */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">preOrder</span><span class="p">(</span>
 <a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="nx">root</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="nx">path</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">[],</span>
 <a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">[][]</span>
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">    </span><span class="c1">// Pruning</span>
+<a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">    </span><span class="c1">// Prune</span>
 <a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">root</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">3</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="w">    </span><span class="nx">path</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">root</span><span class="p">);</span>
 <a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">path</span><span class="p">]);</span>
 <a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-17" name="__codelineno-33-17" href="#__codelineno-33-17"></a><span class="w">    </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">path</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
@@ -5439,7 +5521,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_compact.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Preorder traversal: Example 3 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_compact.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Preorder traversal: example 3 */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">preOrder</span><span class="p">(</span>
 <a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">root</span><span class="p">,</span>
 <a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">path</span><span class="p">,</span>
@@ -5449,10 +5531,10 @@
 <a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">    </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a>
-<a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w">  </span><span class="c1">// Attempt</span>
+<a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w">  </span><span class="c1">// Try</span>
 <a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="w">  </span><span class="n">path</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
 <a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">    </span><span class="c1">// Record solution</span>
+<a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">    </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">List</span><span class="p">.</span><span class="n">from</span><span class="p">(</span><span class="n">path</span><span class="p">));</span>
 <a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-17" name="__codelineno-34-17" href="#__codelineno-34-17"></a><span class="w">  </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">path</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
@@ -5463,21 +5545,21 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_compact.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Preorder traversal: Example 3 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_compact.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Preorder traversal: example 3 */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">pre_order</span><span class="p">(</span>
 <a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="n">path</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">    </span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="c1">// Pruning</span>
+<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="c1">// Prune</span>
 <a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="n">is_none</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="n">as_ref</span><span class="p">().</span><span class="n">unwrap</span><span class="p">().</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="w">        </span><span class="c1">// Attempt</span>
+<a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="w">        </span><span class="c1">// Try</span>
 <a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="w">        </span><span class="n">path</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">            </span><span class="c1">// Record solution</span>
+<a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">            </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-35-16" name="__codelineno-35-16" href="#__codelineno-35-16"></a><span class="w">            </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">path</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-35-17" name="__codelineno-35-17" href="#__codelineno-35-17"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-35-18" name="__codelineno-35-18" href="#__codelineno-35-18"></a><span class="w">        </span><span class="n">pre_order</span><span class="p">(</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">path</span><span class="p">,</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">as_ref</span><span class="p">());</span>
@@ -5489,16 +5571,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_compact.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Preorder traversal: Example 3 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_compact.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Preorder traversal: example 3 */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="c1">// Pruning</span>
+<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="c1">// Prune</span>
 <a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">root</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">    </span><span class="n">path</span><span class="p">[</span><span class="n">pathSize</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">;</span>
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">pathSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="w">            </span><span class="n">res</span><span class="p">[</span><span class="n">resSize</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">path</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">        </span><span class="p">}</span>
@@ -5512,16 +5594,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_compact.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Preorder traversal: Example 3 */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_compact.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Preorder traversal: example 3 */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="c1">// Pruning</span>
+<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="c1">// Prune</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">3</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="c1">// Attempt</span>
+<a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="c1">// Try</span>
 <a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">    </span><span class="n">path</span><span class="o">!!</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">7</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="w">        </span><span class="n">res</span><span class="o">!!</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">path</span><span class="o">!!</span><span class="p">.</span><span class="na">toMutableList</span><span class="p">())</span>
 <a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">left</span><span class="p">)</span>
@@ -5532,15 +5614,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_compact.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Pre-order traversal: example 3 ###</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_compact.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Preorder traversal: example 3 ###</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">pre_order</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
-<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1"># Pruning</span>
+<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1"># Prune</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">root</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">3</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a>
-<a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="c1"># Attempt</span>
+<a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="c1"># Try</span>
 <a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="w">  </span><span class="vg">$path</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a>
-<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">  </span><span class="c1"># Record solution</span>
+<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">  </span><span class="c1"># Record the solution</span>
 <a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">  </span><span class="vg">$res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="vg">$path</span><span class="o">.</span><span class="n">dup</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span>
 <a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a>
 <a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="w">  </span><span class="n">pre_order</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
@@ -5553,53 +5635,58 @@
 </div>
 </div>
 </div>
-<p>"Pruning" is a vivid term. As shown in the following figure, during the search process, <strong>we "prune" search branches that do not satisfy the constraints</strong>, avoiding many meaningless attempts and thus improving search efficiency.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20left%20child%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20right%20child%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%3A%20recursion%22%22%22%0A%20%20%20%20%23%20Return%20None%20if%20the%20index%20is%20out%20of%20bounds%20or%20the%20corresponding%20element%20is%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20Create%20the%20current%20node%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20Recursively%20build%20the%20left%20and%20right%20subtrees%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20pre_order%28root%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22Preorder%20traversal%3A%20example%203%22%22%22%0A%20%20%20%20%23%20Prune%0A%20%20%20%20if%20root%20is%20None%20or%20root.val%20%3D%3D%203%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Try%0A%20%20%20%20path.append%28root%29%0A%20%20%20%20if%20root.val%20%3D%3D%207%3A%0A%20%20%20%20%20%20%20%20%23%20Record%20the%20solution%0A%20%20%20%20%20%20%20%20res.append%28list%28path%29%29%0A%20%20%20%20pre_order%28root.left%29%0A%20%20%20%20pre_order%28root.right%29%0A%20%20%20%20%23%20Backtrack%0A%20%20%20%20path.pop%28%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20Preorder%20traversal%0A%20%20%20%20path%20%3D%20list%5BTreeNode%5D%28%29%0A%20%20%20%20res%20%3D%20list%5Blist%5BTreeNode%5D%5D%28%29%0A%20%20%20%20pre_order%28root%29%0A%0A%20%20%20%20print%28%22%5CnOutput%20all%20paths%20from%20the%20root%20to%20nodes%20with%20value%207%20that%20contain%20no%20node%20with%20value%203%22%29%0A%20%20%20%20for%20path%20in%20res%3A%0A%20%20%20%20%20%20%20%20print%28%5Bnode.val%20for%20node%20in%20path%5D%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=126&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20left%20child%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20right%20child%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%3A%20recursion%22%22%22%0A%20%20%20%20%23%20Return%20None%20if%20the%20index%20is%20out%20of%20bounds%20or%20the%20corresponding%20element%20is%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20Create%20the%20current%20node%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20Recursively%20build%20the%20left%20and%20right%20subtrees%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20pre_order%28root%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22Preorder%20traversal%3A%20example%203%22%22%22%0A%20%20%20%20%23%20Prune%0A%20%20%20%20if%20root%20is%20None%20or%20root.val%20%3D%3D%203%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Try%0A%20%20%20%20path.append%28root%29%0A%20%20%20%20if%20root.val%20%3D%3D%207%3A%0A%20%20%20%20%20%20%20%20%23%20Record%20the%20solution%0A%20%20%20%20%20%20%20%20res.append%28list%28path%29%29%0A%20%20%20%20pre_order%28root.left%29%0A%20%20%20%20pre_order%28root.right%29%0A%20%20%20%20%23%20Backtrack%0A%20%20%20%20path.pop%28%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20Preorder%20traversal%0A%20%20%20%20path%20%3D%20list%5BTreeNode%5D%28%29%0A%20%20%20%20res%20%3D%20list%5Blist%5BTreeNode%5D%5D%28%29%0A%20%20%20%20pre_order%28root%29%0A%0A%20%20%20%20print%28%22%5CnOutput%20all%20paths%20from%20the%20root%20to%20nodes%20with%20value%207%20that%20contain%20no%20node%20with%20value%203%22%29%0A%20%20%20%20for%20path%20in%20res%3A%0A%20%20%20%20%20%20%20%20print%28%5Bnode.val%20for%20node%20in%20path%5D%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=126&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Pruning is an apt name. As shown in Figure 13-3, <strong>we cut off search branches that violate the constraints</strong>, avoiding pointless trials and improving efficiency.</p>
 <p><img alt="Pruning according to constraints" class="animation-figure" src="../backtracking_algorithm.assets/preorder_find_constrained_paths.png" /></p>
 <p align="center"> Figure 13-3 &nbsp; Pruning according to constraints </p>
 
-<h2 id="1313-framework-code">13.1.3 &nbsp; Framework Code<a class="headerlink" href="#1313-framework-code" title="Permanent link">&para;</a></h2>
-<p>Next, we attempt to extract a general framework centered on backtracking's "attempt, backtrack, and pruning" to improve code generality.</p>
-<p>In the following framework code, <code>state</code> represents the current state of the problem, and <code>choices</code> represents the choices available in the current state:</p>
+<h2 id="1313-a-code-template">13.1.3 &nbsp; A code template<a class="headerlink" href="#1313-a-code-template" title="Permanent link">&para;</a></h2>
+<p>We will now extract the core pattern of trying, undoing, and pruning into a more general backtracking template.</p>
+<p>In the following template, <code>state</code> is the current problem state, and <code>choices</code> contains the choices available from that state:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="n">State</span><span class="p">,</span> <span class="n">choices</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="n">choice</span><span class="p">],</span> <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="n">state</span><span class="p">]):</span>
-<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Backtracking algorithm framework"""</span>
-<a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="c1"># Check if it is a solution</span>
+<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Backtracking template"""</span>
+<a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="c1"># Check whether the state is a solution</span>
 <a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="k">if</span> <span class="n">is_solution</span><span class="p">(</span><span class="n">state</span><span class="p">):</span>
 <a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>        <span class="c1"># Record the solution</span>
 <a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a>        <span class="n">record_solution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a>        <span class="c1"># Stop searching</span>
+<a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a>        <span class="c1"># Stop searching this branch</span>
 <a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a>        <span class="k">return</span>
-<a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a>    <span class="c1"># Traverse all choices</span>
+<a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a>    <span class="c1"># Try every choice</span>
 <a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a>    <span class="k">for</span> <span class="n">choice</span> <span class="ow">in</span> <span class="n">choices</span><span class="p">:</span>
-<a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a>        <span class="c1"># Pruning: check if the choice is valid</span>
+<a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a>        <span class="c1"># Prune: check whether the choice is valid</span>
 <a id="__codelineno-39-12" name="__codelineno-39-12" href="#__codelineno-39-12"></a>        <span class="k">if</span> <span class="n">is_valid</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">choice</span><span class="p">):</span>
-<a id="__codelineno-39-13" name="__codelineno-39-13" href="#__codelineno-39-13"></a>            <span class="c1"># Attempt: make a choice and update the state</span>
+<a id="__codelineno-39-13" name="__codelineno-39-13" href="#__codelineno-39-13"></a>            <span class="c1"># Try: make the choice and update the state</span>
 <a id="__codelineno-39-14" name="__codelineno-39-14" href="#__codelineno-39-14"></a>            <span class="n">make_choice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-39-15" name="__codelineno-39-15" href="#__codelineno-39-15"></a>            <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">choices</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-39-16" name="__codelineno-39-16" href="#__codelineno-39-16"></a>            <span class="c1"># Backtrack: undo the choice and restore to the previous state</span>
+<a id="__codelineno-39-16" name="__codelineno-39-16" href="#__codelineno-39-16"></a>            <span class="c1"># Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-39-17" name="__codelineno-39-17" href="#__codelineno-39-17"></a>            <span class="n">undo_choice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">choice</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Backtracking algorithm framework */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Backtracking template */</span>
 <a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">State</span><span class="w"> </span><span class="o">*</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">Choice</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">State</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="c1">// Check whether the state is a solution</span>
 <a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isSolution</span><span class="p">(</span><span class="n">state</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w">        </span><span class="n">recordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="w">        </span><span class="c1">// Stop searching</span>
+<a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="w">        </span><span class="c1">// Stop searching this branch</span>
 <a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-40-9" name="__codelineno-40-9" href="#__codelineno-40-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-40-10" name="__codelineno-40-10" href="#__codelineno-40-10"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-40-10" name="__codelineno-40-10" href="#__codelineno-40-10"></a><span class="w">    </span><span class="c1">// Try every choice</span>
 <a id="__codelineno-40-11" name="__codelineno-40-11" href="#__codelineno-40-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Choice</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-12" name="__codelineno-40-12" href="#__codelineno-40-12"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-40-12" name="__codelineno-40-12" href="#__codelineno-40-12"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-40-13" name="__codelineno-40-13" href="#__codelineno-40-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isValid</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-14" name="__codelineno-40-14" href="#__codelineno-40-14"></a><span class="w">            </span><span class="c1">// Attempt: make a choice and update the state</span>
+<a id="__codelineno-40-14" name="__codelineno-40-14" href="#__codelineno-40-14"></a><span class="w">            </span><span class="c1">// Try: make the choice and update the state</span>
 <a id="__codelineno-40-15" name="__codelineno-40-15" href="#__codelineno-40-15"></a><span class="w">            </span><span class="n">makeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-40-16" name="__codelineno-40-16" href="#__codelineno-40-16"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-40-17" name="__codelineno-40-17" href="#__codelineno-40-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore to the previous state</span>
+<a id="__codelineno-40-17" name="__codelineno-40-17" href="#__codelineno-40-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-40-18" name="__codelineno-40-18" href="#__codelineno-40-18"></a><span class="w">            </span><span class="n">undoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-40-19" name="__codelineno-40-19" href="#__codelineno-40-19"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-40-20" name="__codelineno-40-20" href="#__codelineno-40-20"></a><span class="w">    </span><span class="p">}</span>
@@ -5607,23 +5694,23 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Backtracking algorithm framework */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Backtracking template */</span>
 <a id="__codelineno-41-2" name="__codelineno-41-2" href="#__codelineno-41-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">State</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Choice</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">State</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="c1">// Check whether the state is a solution</span>
 <a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isSolution</span><span class="p">(</span><span class="n">state</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a><span class="w">        </span><span class="n">recordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="w">        </span><span class="c1">// Stop searching</span>
+<a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="w">        </span><span class="c1">// Stop searching this branch</span>
 <a id="__codelineno-41-8" name="__codelineno-41-8" href="#__codelineno-41-8"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-41-9" name="__codelineno-41-9" href="#__codelineno-41-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-41-10" name="__codelineno-41-10" href="#__codelineno-41-10"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-41-10" name="__codelineno-41-10" href="#__codelineno-41-10"></a><span class="w">    </span><span class="c1">// Try every choice</span>
 <a id="__codelineno-41-11" name="__codelineno-41-11" href="#__codelineno-41-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Choice</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-12" name="__codelineno-41-12" href="#__codelineno-41-12"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-41-12" name="__codelineno-41-12" href="#__codelineno-41-12"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-41-13" name="__codelineno-41-13" href="#__codelineno-41-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isValid</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-14" name="__codelineno-41-14" href="#__codelineno-41-14"></a><span class="w">            </span><span class="c1">// Attempt: make a choice and update the state</span>
+<a id="__codelineno-41-14" name="__codelineno-41-14" href="#__codelineno-41-14"></a><span class="w">            </span><span class="c1">// Try: make the choice and update the state</span>
 <a id="__codelineno-41-15" name="__codelineno-41-15" href="#__codelineno-41-15"></a><span class="w">            </span><span class="n">makeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-41-16" name="__codelineno-41-16" href="#__codelineno-41-16"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-41-17" name="__codelineno-41-17" href="#__codelineno-41-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore to the previous state</span>
+<a id="__codelineno-41-17" name="__codelineno-41-17" href="#__codelineno-41-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-41-18" name="__codelineno-41-18" href="#__codelineno-41-18"></a><span class="w">            </span><span class="n">undoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-41-19" name="__codelineno-41-19" href="#__codelineno-41-19"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-41-20" name="__codelineno-41-20" href="#__codelineno-41-20"></a><span class="w">    </span><span class="p">}</span>
@@ -5631,23 +5718,23 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Backtracking algorithm framework */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Backtracking template */</span>
 <a id="__codelineno-42-2" name="__codelineno-42-2" href="#__codelineno-42-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Backtrack</span><span class="p">(</span><span class="n">State</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Choice</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">State</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="c1">// Check whether the state is a solution</span>
 <a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">IsSolution</span><span class="p">(</span><span class="n">state</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a><span class="w">        </span><span class="n">RecordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="w">        </span><span class="c1">// Stop searching</span>
+<a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="w">        </span><span class="c1">// Stop searching this branch</span>
 <a id="__codelineno-42-8" name="__codelineno-42-8" href="#__codelineno-42-8"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-42-9" name="__codelineno-42-9" href="#__codelineno-42-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-42-10" name="__codelineno-42-10" href="#__codelineno-42-10"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-42-10" name="__codelineno-42-10" href="#__codelineno-42-10"></a><span class="w">    </span><span class="c1">// Try every choice</span>
 <a id="__codelineno-42-11" name="__codelineno-42-11" href="#__codelineno-42-11"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">Choice</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-12" name="__codelineno-42-12" href="#__codelineno-42-12"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-42-12" name="__codelineno-42-12" href="#__codelineno-42-12"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-42-13" name="__codelineno-42-13" href="#__codelineno-42-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">IsValid</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-14" name="__codelineno-42-14" href="#__codelineno-42-14"></a><span class="w">            </span><span class="c1">// Attempt: make a choice and update the state</span>
+<a id="__codelineno-42-14" name="__codelineno-42-14" href="#__codelineno-42-14"></a><span class="w">            </span><span class="c1">// Try: make the choice and update the state</span>
 <a id="__codelineno-42-15" name="__codelineno-42-15" href="#__codelineno-42-15"></a><span class="w">            </span><span class="n">MakeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-42-16" name="__codelineno-42-16" href="#__codelineno-42-16"></a><span class="w">            </span><span class="n">Backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-42-17" name="__codelineno-42-17" href="#__codelineno-42-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore to the previous state</span>
+<a id="__codelineno-42-17" name="__codelineno-42-17" href="#__codelineno-42-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-42-18" name="__codelineno-42-18" href="#__codelineno-42-18"></a><span class="w">            </span><span class="n">UndoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-42-19" name="__codelineno-42-19" href="#__codelineno-42-19"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-42-20" name="__codelineno-42-20" href="#__codelineno-42-20"></a><span class="w">    </span><span class="p">}</span>
@@ -5655,23 +5742,23 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Backtracking algorithm framework */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Backtracking template */</span>
 <a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">*</span><span class="nx">State</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="w"> </span><span class="p">[]</span><span class="nx">Choice</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="nx">State</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="c1">// Check whether the state is a solution</span>
 <a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">isSolution</span><span class="p">(</span><span class="nx">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a><span class="w">        </span><span class="nx">recordSolution</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span>
-<a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="w">        </span><span class="c1">// Stop searching</span>
+<a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="w">        </span><span class="c1">// Stop searching this branch</span>
 <a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-43-9" name="__codelineno-43-9" href="#__codelineno-43-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></a><span class="w">    </span><span class="c1">// Try every choice</span>
 <a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">choices</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-12" name="__codelineno-43-12" href="#__codelineno-43-12"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-43-12" name="__codelineno-43-12" href="#__codelineno-43-12"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-43-13" name="__codelineno-43-13" href="#__codelineno-43-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">isValid</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-14" name="__codelineno-43-14" href="#__codelineno-43-14"></a><span class="w">            </span><span class="c1">// Attempt: make a choice and update the state</span>
+<a id="__codelineno-43-14" name="__codelineno-43-14" href="#__codelineno-43-14"></a><span class="w">            </span><span class="c1">// Try: make the choice and update the state</span>
 <a id="__codelineno-43-15" name="__codelineno-43-15" href="#__codelineno-43-15"></a><span class="w">            </span><span class="nx">makeChoice</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">)</span>
 <a id="__codelineno-43-16" name="__codelineno-43-16" href="#__codelineno-43-16"></a><span class="w">            </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span>
-<a id="__codelineno-43-17" name="__codelineno-43-17" href="#__codelineno-43-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore to the previous state</span>
+<a id="__codelineno-43-17" name="__codelineno-43-17" href="#__codelineno-43-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-43-18" name="__codelineno-43-18" href="#__codelineno-43-18"></a><span class="w">            </span><span class="nx">undoChoice</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">)</span>
 <a id="__codelineno-43-19" name="__codelineno-43-19" href="#__codelineno-43-19"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-43-20" name="__codelineno-43-20" href="#__codelineno-43-20"></a><span class="w">    </span><span class="p">}</span>
@@ -5679,23 +5766,23 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Backtracking algorithm framework */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Backtracking template */</span>
 <a id="__codelineno-44-2" name="__codelineno-44-2" href="#__codelineno-44-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="n">State</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="n">Choice</span><span class="p">],</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="n">State</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a><span class="w">    </span><span class="c1">// Check whether the state is a solution</span>
 <a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">isSolution</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></a><span class="w">        </span><span class="n">recordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-44-7" name="__codelineno-44-7" href="#__codelineno-44-7"></a><span class="w">        </span><span class="c1">// Stop searching</span>
+<a id="__codelineno-44-7" name="__codelineno-44-7" href="#__codelineno-44-7"></a><span class="w">        </span><span class="c1">// Stop searching this branch</span>
 <a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-44-9" name="__codelineno-44-9" href="#__codelineno-44-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-44-10" name="__codelineno-44-10" href="#__codelineno-44-10"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-44-10" name="__codelineno-44-10" href="#__codelineno-44-10"></a><span class="w">    </span><span class="c1">// Try every choice</span>
 <a id="__codelineno-44-11" name="__codelineno-44-11" href="#__codelineno-44-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-44-12" name="__codelineno-44-12" href="#__codelineno-44-12"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-44-12" name="__codelineno-44-12" href="#__codelineno-44-12"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-44-13" name="__codelineno-44-13" href="#__codelineno-44-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">isValid</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">:</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-44-14" name="__codelineno-44-14" href="#__codelineno-44-14"></a><span class="w">            </span><span class="c1">// Attempt: make a choice and update the state</span>
+<a id="__codelineno-44-14" name="__codelineno-44-14" href="#__codelineno-44-14"></a><span class="w">            </span><span class="c1">// Try: make the choice and update the state</span>
 <a id="__codelineno-44-15" name="__codelineno-44-15" href="#__codelineno-44-15"></a><span class="w">            </span><span class="n">makeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">:</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-44-16" name="__codelineno-44-16" href="#__codelineno-44-16"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-44-17" name="__codelineno-44-17" href="#__codelineno-44-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore to the previous state</span>
+<a id="__codelineno-44-17" name="__codelineno-44-17" href="#__codelineno-44-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-44-18" name="__codelineno-44-18" href="#__codelineno-44-18"></a><span class="w">            </span><span class="n">undoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">:</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-44-19" name="__codelineno-44-19" href="#__codelineno-44-19"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-44-20" name="__codelineno-44-20" href="#__codelineno-44-20"></a><span class="w">    </span><span class="p">}</span>
@@ -5703,23 +5790,23 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Backtracking algorithm framework */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Backtracking template */</span>
 <a id="__codelineno-45-2" name="__codelineno-45-2" href="#__codelineno-45-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="c1">// Check whether the state is a solution</span>
 <a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">isSolution</span><span class="p">(</span><span class="nx">state</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-45-6" name="__codelineno-45-6" href="#__codelineno-45-6"></a><span class="w">        </span><span class="nx">recordSolution</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-45-7" name="__codelineno-45-7" href="#__codelineno-45-7"></a><span class="w">        </span><span class="c1">// Stop searching</span>
+<a id="__codelineno-45-7" name="__codelineno-45-7" href="#__codelineno-45-7"></a><span class="w">        </span><span class="c1">// Stop searching this branch</span>
 <a id="__codelineno-45-8" name="__codelineno-45-8" href="#__codelineno-45-8"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-45-9" name="__codelineno-45-9" href="#__codelineno-45-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-45-10" name="__codelineno-45-10" href="#__codelineno-45-10"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-45-10" name="__codelineno-45-10" href="#__codelineno-45-10"></a><span class="w">    </span><span class="c1">// Try every choice</span>
 <a id="__codelineno-45-11" name="__codelineno-45-11" href="#__codelineno-45-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-12" name="__codelineno-45-12" href="#__codelineno-45-12"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-45-12" name="__codelineno-45-12" href="#__codelineno-45-12"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-45-13" name="__codelineno-45-13" href="#__codelineno-45-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">isValid</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-14" name="__codelineno-45-14" href="#__codelineno-45-14"></a><span class="w">            </span><span class="c1">// Attempt: make a choice and update the state</span>
+<a id="__codelineno-45-14" name="__codelineno-45-14" href="#__codelineno-45-14"></a><span class="w">            </span><span class="c1">// Try: make the choice and update the state</span>
 <a id="__codelineno-45-15" name="__codelineno-45-15" href="#__codelineno-45-15"></a><span class="w">            </span><span class="nx">makeChoice</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">);</span>
 <a id="__codelineno-45-16" name="__codelineno-45-16" href="#__codelineno-45-16"></a><span class="w">            </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-45-17" name="__codelineno-45-17" href="#__codelineno-45-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore to the previous state</span>
+<a id="__codelineno-45-17" name="__codelineno-45-17" href="#__codelineno-45-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-45-18" name="__codelineno-45-18" href="#__codelineno-45-18"></a><span class="w">            </span><span class="nx">undoChoice</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">);</span>
 <a id="__codelineno-45-19" name="__codelineno-45-19" href="#__codelineno-45-19"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-45-20" name="__codelineno-45-20" href="#__codelineno-45-20"></a><span class="w">    </span><span class="p">}</span>
@@ -5727,23 +5814,23 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Backtracking algorithm framework */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Backtracking template */</span>
 <a id="__codelineno-46-2" name="__codelineno-46-2" href="#__codelineno-46-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="o">:</span><span class="w"> </span><span class="kt">State</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="o">:</span><span class="w"> </span><span class="kt">Choice</span><span class="p">[],</span><span class="w"> </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">State</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="c1">// Check whether the state is a solution</span>
 <a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">isSolution</span><span class="p">(</span><span class="nx">state</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="w">        </span><span class="nx">recordSolution</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="w">        </span><span class="c1">// Stop searching</span>
+<a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="w">        </span><span class="c1">// Stop searching this branch</span>
 <a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-46-10" name="__codelineno-46-10" href="#__codelineno-46-10"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-46-10" name="__codelineno-46-10" href="#__codelineno-46-10"></a><span class="w">    </span><span class="c1">// Try every choice</span>
 <a id="__codelineno-46-11" name="__codelineno-46-11" href="#__codelineno-46-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-46-13" name="__codelineno-46-13" href="#__codelineno-46-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">isValid</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-14" name="__codelineno-46-14" href="#__codelineno-46-14"></a><span class="w">            </span><span class="c1">// Attempt: make a choice and update the state</span>
+<a id="__codelineno-46-14" name="__codelineno-46-14" href="#__codelineno-46-14"></a><span class="w">            </span><span class="c1">// Try: make the choice and update the state</span>
 <a id="__codelineno-46-15" name="__codelineno-46-15" href="#__codelineno-46-15"></a><span class="w">            </span><span class="nx">makeChoice</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">);</span>
 <a id="__codelineno-46-16" name="__codelineno-46-16" href="#__codelineno-46-16"></a><span class="w">            </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-46-17" name="__codelineno-46-17" href="#__codelineno-46-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore to the previous state</span>
+<a id="__codelineno-46-17" name="__codelineno-46-17" href="#__codelineno-46-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-46-18" name="__codelineno-46-18" href="#__codelineno-46-18"></a><span class="w">            </span><span class="nx">undoChoice</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">);</span>
 <a id="__codelineno-46-19" name="__codelineno-46-19" href="#__codelineno-46-19"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-46-20" name="__codelineno-46-20" href="#__codelineno-46-20"></a><span class="w">    </span><span class="p">}</span>
@@ -5751,23 +5838,23 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Backtracking algorithm framework */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Backtracking template */</span>
 <a id="__codelineno-47-2" name="__codelineno-47-2" href="#__codelineno-47-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">State</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Choice</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">State</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="c1">// Check whether the state is a solution</span>
 <a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isSolution</span><span class="p">(</span><span class="n">state</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w">    </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a><span class="w">    </span><span class="n">recordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-47-7" name="__codelineno-47-7" href="#__codelineno-47-7"></a><span class="w">    </span><span class="c1">// Stop searching</span>
+<a id="__codelineno-47-7" name="__codelineno-47-7" href="#__codelineno-47-7"></a><span class="w">    </span><span class="c1">// Stop searching this branch</span>
 <a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a><span class="w">    </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-47-9" name="__codelineno-47-9" href="#__codelineno-47-9"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-47-10" name="__codelineno-47-10" href="#__codelineno-47-10"></a><span class="w">  </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-47-10" name="__codelineno-47-10" href="#__codelineno-47-10"></a><span class="w">  </span><span class="c1">// Try every choice</span>
 <a id="__codelineno-47-11" name="__codelineno-47-11" href="#__codelineno-47-11"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Choice</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-12" name="__codelineno-47-12" href="#__codelineno-47-12"></a><span class="w">    </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-47-12" name="__codelineno-47-12" href="#__codelineno-47-12"></a><span class="w">    </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-47-13" name="__codelineno-47-13" href="#__codelineno-47-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isValid</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-14" name="__codelineno-47-14" href="#__codelineno-47-14"></a><span class="w">      </span><span class="c1">// Attempt: make a choice and update the state</span>
+<a id="__codelineno-47-14" name="__codelineno-47-14" href="#__codelineno-47-14"></a><span class="w">      </span><span class="c1">// Try: make the choice and update the state</span>
 <a id="__codelineno-47-15" name="__codelineno-47-15" href="#__codelineno-47-15"></a><span class="w">      </span><span class="n">makeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-47-16" name="__codelineno-47-16" href="#__codelineno-47-16"></a><span class="w">      </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-47-17" name="__codelineno-47-17" href="#__codelineno-47-17"></a><span class="w">      </span><span class="c1">// Backtrack: undo the choice and restore to the previous state</span>
+<a id="__codelineno-47-17" name="__codelineno-47-17" href="#__codelineno-47-17"></a><span class="w">      </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-47-18" name="__codelineno-47-18" href="#__codelineno-47-18"></a><span class="w">      </span><span class="n">undoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-47-19" name="__codelineno-47-19" href="#__codelineno-47-19"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-47-20" name="__codelineno-47-20" href="#__codelineno-47-20"></a><span class="w">  </span><span class="p">}</span>
@@ -5775,23 +5862,23 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Backtracking algorithm framework */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Backtracking template */</span>
 <a id="__codelineno-48-2" name="__codelineno-48-2" href="#__codelineno-48-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="n">State</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Choice</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">State</span><span class="o">&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="c1">// Check whether the state is a solution</span>
 <a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">is_solution</span><span class="p">(</span><span class="n">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a><span class="w">        </span><span class="n">record_solution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="w">        </span><span class="c1">// Stop searching</span>
+<a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="w">        </span><span class="c1">// Stop searching this branch</span>
 <a id="__codelineno-48-8" name="__codelineno-48-8" href="#__codelineno-48-8"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="w">    </span><span class="c1">// Try every choice</span>
 <a id="__codelineno-48-11" name="__codelineno-48-11" href="#__codelineno-48-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-12" name="__codelineno-48-12" href="#__codelineno-48-12"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-48-12" name="__codelineno-48-12" href="#__codelineno-48-12"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-48-13" name="__codelineno-48-13" href="#__codelineno-48-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">is_valid</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-14" name="__codelineno-48-14" href="#__codelineno-48-14"></a><span class="w">            </span><span class="c1">// Attempt: make a choice and update the state</span>
+<a id="__codelineno-48-14" name="__codelineno-48-14" href="#__codelineno-48-14"></a><span class="w">            </span><span class="c1">// Try: make the choice and update the state</span>
 <a id="__codelineno-48-15" name="__codelineno-48-15" href="#__codelineno-48-15"></a><span class="w">            </span><span class="n">make_choice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-48-16" name="__codelineno-48-16" href="#__codelineno-48-16"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-48-17" name="__codelineno-48-17" href="#__codelineno-48-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore to the previous state</span>
+<a id="__codelineno-48-17" name="__codelineno-48-17" href="#__codelineno-48-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-48-18" name="__codelineno-48-18" href="#__codelineno-48-18"></a><span class="w">            </span><span class="n">undo_choice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-48-19" name="__codelineno-48-19" href="#__codelineno-48-19"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-48-20" name="__codelineno-48-20" href="#__codelineno-48-20"></a><span class="w">    </span><span class="p">}</span>
@@ -5799,23 +5886,23 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Backtracking algorithm framework */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Backtracking template */</span>
 <a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">State</span><span class="w"> </span><span class="o">*</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">Choice</span><span class="w"> </span><span class="o">*</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">numChoices</span><span class="p">,</span><span class="w"> </span><span class="n">State</span><span class="w"> </span><span class="o">*</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">numRes</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="c1">// Check whether the state is a solution</span>
 <a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isSolution</span><span class="p">(</span><span class="n">state</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w">        </span><span class="n">recordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">numRes</span><span class="p">);</span>
-<a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">        </span><span class="c1">// Stop searching</span>
+<a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">        </span><span class="c1">// Stop searching this branch</span>
 <a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-49-9" name="__codelineno-49-9" href="#__codelineno-49-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-49-10" name="__codelineno-49-10" href="#__codelineno-49-10"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-49-10" name="__codelineno-49-10" href="#__codelineno-49-10"></a><span class="w">    </span><span class="c1">// Try every choice</span>
 <a id="__codelineno-49-11" name="__codelineno-49-11" href="#__codelineno-49-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">numChoices</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-12" name="__codelineno-49-12" href="#__codelineno-49-12"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-49-12" name="__codelineno-49-12" href="#__codelineno-49-12"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-49-13" name="__codelineno-49-13" href="#__codelineno-49-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isValid</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-14" name="__codelineno-49-14" href="#__codelineno-49-14"></a><span class="w">            </span><span class="c1">// Attempt: make a choice and update the state</span>
+<a id="__codelineno-49-14" name="__codelineno-49-14" href="#__codelineno-49-14"></a><span class="w">            </span><span class="c1">// Try: make the choice and update the state</span>
 <a id="__codelineno-49-15" name="__codelineno-49-15" href="#__codelineno-49-15"></a><span class="w">            </span><span class="n">makeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
 <a id="__codelineno-49-16" name="__codelineno-49-16" href="#__codelineno-49-16"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">numChoices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">numRes</span><span class="p">);</span>
-<a id="__codelineno-49-17" name="__codelineno-49-17" href="#__codelineno-49-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore to the previous state</span>
+<a id="__codelineno-49-17" name="__codelineno-49-17" href="#__codelineno-49-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-49-18" name="__codelineno-49-18" href="#__codelineno-49-18"></a><span class="w">            </span><span class="n">undoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
 <a id="__codelineno-49-19" name="__codelineno-49-19" href="#__codelineno-49-19"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-49-20" name="__codelineno-49-20" href="#__codelineno-49-20"></a><span class="w">    </span><span class="p">}</span>
@@ -5823,23 +5910,23 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Backtracking algorithm framework */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Backtracking template */</span>
 <a id="__codelineno-50-2" name="__codelineno-50-2" href="#__codelineno-50-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">State?</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Choice?</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">State?</span><span class="o">&gt;?</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="c1">// Check whether the state is a solution</span>
 <a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isSolution</span><span class="p">(</span><span class="n">state</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">        </span><span class="n">recordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="w">        </span><span class="c1">// Stop searching</span>
+<a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="w">        </span><span class="c1">// Stop searching this branch</span>
 <a id="__codelineno-50-8" name="__codelineno-50-8" href="#__codelineno-50-8"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-50-10" name="__codelineno-50-10" href="#__codelineno-50-10"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-50-10" name="__codelineno-50-10" href="#__codelineno-50-10"></a><span class="w">    </span><span class="c1">// Try every choice</span>
 <a id="__codelineno-50-11" name="__codelineno-50-11" href="#__codelineno-50-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-12" name="__codelineno-50-12" href="#__codelineno-50-12"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-50-12" name="__codelineno-50-12" href="#__codelineno-50-12"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-50-13" name="__codelineno-50-13" href="#__codelineno-50-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isValid</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-14" name="__codelineno-50-14" href="#__codelineno-50-14"></a><span class="w">            </span><span class="c1">// Attempt: make a choice and update the state</span>
+<a id="__codelineno-50-14" name="__codelineno-50-14" href="#__codelineno-50-14"></a><span class="w">            </span><span class="c1">// Try: make the choice and update the state</span>
 <a id="__codelineno-50-15" name="__codelineno-50-15" href="#__codelineno-50-15"></a><span class="w">            </span><span class="n">makeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-50-16" name="__codelineno-50-16" href="#__codelineno-50-16"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-50-17" name="__codelineno-50-17" href="#__codelineno-50-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore to the previous state</span>
+<a id="__codelineno-50-17" name="__codelineno-50-17" href="#__codelineno-50-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-50-18" name="__codelineno-50-18" href="#__codelineno-50-18"></a><span class="w">            </span><span class="n">undoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-50-19" name="__codelineno-50-19" href="#__codelineno-50-19"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-50-20" name="__codelineno-50-20" href="#__codelineno-50-20"></a><span class="w">    </span><span class="p">}</span>
@@ -5847,23 +5934,23 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Backtracking algorithm framework ###</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Backtracking template ###</span>
 <a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">    </span><span class="c1"># Check if it is a solution</span>
+<a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">    </span><span class="c1"># Check whether the state is a solution</span>
 <a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">is_solution?</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
 <a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">        </span><span class="c1"># Record the solution</span>
 <a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">        </span><span class="n">record_solution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a>
-<a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a><span class="w">    </span><span class="c1"># Traverse all choices</span>
+<a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a><span class="w">    </span><span class="c1"># Try every choice</span>
 <a id="__codelineno-51-11" name="__codelineno-51-11" href="#__codelineno-51-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span>
-<a id="__codelineno-51-12" name="__codelineno-51-12" href="#__codelineno-51-12"></a><span class="w">        </span><span class="c1"># Pruning: check if the choice is valid</span>
+<a id="__codelineno-51-12" name="__codelineno-51-12" href="#__codelineno-51-12"></a><span class="w">        </span><span class="c1"># Prune: check whether the choice is valid</span>
 <a id="__codelineno-51-13" name="__codelineno-51-13" href="#__codelineno-51-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">is_valid?</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
-<a id="__codelineno-51-14" name="__codelineno-51-14" href="#__codelineno-51-14"></a><span class="w">            </span><span class="c1"># Attempt: make a choice and update the state</span>
+<a id="__codelineno-51-14" name="__codelineno-51-14" href="#__codelineno-51-14"></a><span class="w">            </span><span class="c1"># Try: make the choice and update the state</span>
 <a id="__codelineno-51-15" name="__codelineno-51-15" href="#__codelineno-51-15"></a><span class="w">            </span><span class="n">make_choice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-51-16" name="__codelineno-51-16" href="#__codelineno-51-16"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-51-17" name="__codelineno-51-17" href="#__codelineno-51-17"></a><span class="w">            </span><span class="c1"># Backtrack: undo the choice and restore to the previous state</span>
+<a id="__codelineno-51-17" name="__codelineno-51-17" href="#__codelineno-51-17"></a><span class="w">            </span><span class="c1"># Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-51-18" name="__codelineno-51-18" href="#__codelineno-51-18"></a><span class="w">            </span><span class="n">undo_choice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-51-19" name="__codelineno-51-19" href="#__codelineno-51-19"></a><span class="w">        </span><span class="k">end</span>
 <a id="__codelineno-51-20" name="__codelineno-51-20" href="#__codelineno-51-20"></a><span class="w">    </span><span class="k">end</span>
@@ -5872,93 +5959,93 @@
 </div>
 </div>
 </div>
-<p>Next, we solve Example 3 based on the framework code. The state <code>state</code> is the node traversal path, the choices <code>choices</code> are the left and right child nodes of the current node, and the result <code>res</code> is a list of paths:</p>
+<p>Now apply this template to Example 3. The <code>state</code> is the current path of visited nodes, <code>choices</code> consists of the current node's left and right children, and <code>res</code> is the list of paths:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="6:13"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">preorder_traversal_iii_template.py</span><pre><span></span><code><a id="__codelineno-52-1" name="__codelineno-52-1" href="#__codelineno-52-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">is_solution</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">])</span> <span class="o">-&gt;</span> <span class="nb">bool</span><span class="p">:</span>
-<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Check if the current state is a solution"""</span>
+<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Check whether the current state is a solution"""</span>
 <a id="__codelineno-52-3" name="__codelineno-52-3" href="#__codelineno-52-3"></a>    <span class="k">return</span> <span class="n">state</span> <span class="ow">and</span> <span class="n">state</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">val</span> <span class="o">==</span> <span class="mi">7</span>
 <a id="__codelineno-52-4" name="__codelineno-52-4" href="#__codelineno-52-4"></a>
 <a id="__codelineno-52-5" name="__codelineno-52-5" href="#__codelineno-52-5"></a><span class="k">def</span><span class="w"> </span><span class="nf">record_solution</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span> <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">]]):</span>
-<a id="__codelineno-52-6" name="__codelineno-52-6" href="#__codelineno-52-6"></a><span class="w">    </span><span class="sd">"""Record solution"""</span>
+<a id="__codelineno-52-6" name="__codelineno-52-6" href="#__codelineno-52-6"></a><span class="w">    </span><span class="sd">"""Record the solution"""</span>
 <a id="__codelineno-52-7" name="__codelineno-52-7" href="#__codelineno-52-7"></a>    <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">state</span><span class="p">))</span>
 <a id="__codelineno-52-8" name="__codelineno-52-8" href="#__codelineno-52-8"></a>
 <a id="__codelineno-52-9" name="__codelineno-52-9" href="#__codelineno-52-9"></a><span class="k">def</span><span class="w"> </span><span class="nf">is_valid</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span> <span class="n">choice</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">bool</span><span class="p">:</span>
-<a id="__codelineno-52-10" name="__codelineno-52-10" href="#__codelineno-52-10"></a><span class="w">    </span><span class="sd">"""Check if the choice is valid under the current state"""</span>
+<a id="__codelineno-52-10" name="__codelineno-52-10" href="#__codelineno-52-10"></a><span class="w">    </span><span class="sd">"""Check whether the choice is valid in the current state"""</span>
 <a id="__codelineno-52-11" name="__codelineno-52-11" href="#__codelineno-52-11"></a>    <span class="k">return</span> <span class="n">choice</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="ow">and</span> <span class="n">choice</span><span class="o">.</span><span class="n">val</span> <span class="o">!=</span> <span class="mi">3</span>
 <a id="__codelineno-52-12" name="__codelineno-52-12" href="#__codelineno-52-12"></a>
 <a id="__codelineno-52-13" name="__codelineno-52-13" href="#__codelineno-52-13"></a><span class="k">def</span><span class="w"> </span><span class="nf">make_choice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span> <span class="n">choice</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">):</span>
-<a id="__codelineno-52-14" name="__codelineno-52-14" href="#__codelineno-52-14"></a><span class="w">    </span><span class="sd">"""Update state"""</span>
+<a id="__codelineno-52-14" name="__codelineno-52-14" href="#__codelineno-52-14"></a><span class="w">    </span><span class="sd">"""Update the state"""</span>
 <a id="__codelineno-52-15" name="__codelineno-52-15" href="#__codelineno-52-15"></a>    <span class="n">state</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-52-16" name="__codelineno-52-16" href="#__codelineno-52-16"></a>
 <a id="__codelineno-52-17" name="__codelineno-52-17" href="#__codelineno-52-17"></a><span class="k">def</span><span class="w"> </span><span class="nf">undo_choice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span> <span class="n">choice</span><span class="p">:</span> <span class="n">TreeNode</span><span class="p">):</span>
-<a id="__codelineno-52-18" name="__codelineno-52-18" href="#__codelineno-52-18"></a><span class="w">    </span><span class="sd">"""Restore state"""</span>
+<a id="__codelineno-52-18" name="__codelineno-52-18" href="#__codelineno-52-18"></a><span class="w">    </span><span class="sd">"""Restore the state"""</span>
 <a id="__codelineno-52-19" name="__codelineno-52-19" href="#__codelineno-52-19"></a>    <span class="n">state</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
 <a id="__codelineno-52-20" name="__codelineno-52-20" href="#__codelineno-52-20"></a>
 <a id="__codelineno-52-21" name="__codelineno-52-21" href="#__codelineno-52-21"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-52-22" name="__codelineno-52-22" href="#__codelineno-52-22"></a>    <span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span> <span class="n">choices</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span> <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">]]</span>
 <a id="__codelineno-52-23" name="__codelineno-52-23" href="#__codelineno-52-23"></a><span class="p">):</span>
-<a id="__codelineno-52-24" name="__codelineno-52-24" href="#__codelineno-52-24"></a><span class="w">    </span><span class="sd">"""Backtracking algorithm: Example 3"""</span>
-<a id="__codelineno-52-25" name="__codelineno-52-25" href="#__codelineno-52-25"></a>    <span class="c1"># Check if it is a solution</span>
+<a id="__codelineno-52-24" name="__codelineno-52-24" href="#__codelineno-52-24"></a><span class="w">    </span><span class="sd">"""Backtracking: example 3"""</span>
+<a id="__codelineno-52-25" name="__codelineno-52-25" href="#__codelineno-52-25"></a>    <span class="c1"># Check whether this is a solution</span>
 <a id="__codelineno-52-26" name="__codelineno-52-26" href="#__codelineno-52-26"></a>    <span class="k">if</span> <span class="n">is_solution</span><span class="p">(</span><span class="n">state</span><span class="p">):</span>
-<a id="__codelineno-52-27" name="__codelineno-52-27" href="#__codelineno-52-27"></a>        <span class="c1"># Record solution</span>
+<a id="__codelineno-52-27" name="__codelineno-52-27" href="#__codelineno-52-27"></a>        <span class="c1"># Record the solution</span>
 <a id="__codelineno-52-28" name="__codelineno-52-28" href="#__codelineno-52-28"></a>        <span class="n">record_solution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-52-29" name="__codelineno-52-29" href="#__codelineno-52-29"></a>    <span class="c1"># Traverse all choices</span>
+<a id="__codelineno-52-29" name="__codelineno-52-29" href="#__codelineno-52-29"></a>    <span class="c1"># Iterate over all choices</span>
 <a id="__codelineno-52-30" name="__codelineno-52-30" href="#__codelineno-52-30"></a>    <span class="k">for</span> <span class="n">choice</span> <span class="ow">in</span> <span class="n">choices</span><span class="p">:</span>
-<a id="__codelineno-52-31" name="__codelineno-52-31" href="#__codelineno-52-31"></a>        <span class="c1"># Pruning: check if the choice is valid</span>
+<a id="__codelineno-52-31" name="__codelineno-52-31" href="#__codelineno-52-31"></a>        <span class="c1"># Prune: check whether the choice is valid</span>
 <a id="__codelineno-52-32" name="__codelineno-52-32" href="#__codelineno-52-32"></a>        <span class="k">if</span> <span class="n">is_valid</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">choice</span><span class="p">):</span>
-<a id="__codelineno-52-33" name="__codelineno-52-33" href="#__codelineno-52-33"></a>            <span class="c1"># Attempt: make choice, update state</span>
+<a id="__codelineno-52-33" name="__codelineno-52-33" href="#__codelineno-52-33"></a>            <span class="c1"># Try: make a choice and update the state</span>
 <a id="__codelineno-52-34" name="__codelineno-52-34" href="#__codelineno-52-34"></a>            <span class="n">make_choice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">choice</span><span class="p">)</span>
-<a id="__codelineno-52-35" name="__codelineno-52-35" href="#__codelineno-52-35"></a>            <span class="c1"># Proceed to the next round of selection</span>
+<a id="__codelineno-52-35" name="__codelineno-52-35" href="#__codelineno-52-35"></a>            <span class="c1"># Proceed to the next choice</span>
 <a id="__codelineno-52-36" name="__codelineno-52-36" href="#__codelineno-52-36"></a>            <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="p">[</span><span class="n">choice</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="n">choice</span><span class="o">.</span><span class="n">right</span><span class="p">],</span> <span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-52-37" name="__codelineno-52-37" href="#__codelineno-52-37"></a>            <span class="c1"># Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-52-37" name="__codelineno-52-37" href="#__codelineno-52-37"></a>            <span class="c1"># Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-52-38" name="__codelineno-52-38" href="#__codelineno-52-38"></a>            <span class="n">undo_choice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">choice</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_template.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Check if the current state is a solution */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_template.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Check whether the current state is a solution */</span>
 <a id="__codelineno-53-2" name="__codelineno-53-2" href="#__codelineno-53-2"></a><span class="kt">bool</span><span class="w"> </span><span class="nf">isSolution</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">!</span><span class="n">state</span><span class="p">.</span><span class="n">empty</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">back</span><span class="p">()</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">;</span>
 <a id="__codelineno-53-4" name="__codelineno-53-4" href="#__codelineno-53-4"></a><span class="p">}</span>
 <a id="__codelineno-53-5" name="__codelineno-53-5" href="#__codelineno-53-5"></a>
-<a id="__codelineno-53-6" name="__codelineno-53-6" href="#__codelineno-53-6"></a><span class="cm">/* Record solution */</span>
+<a id="__codelineno-53-6" name="__codelineno-53-6" href="#__codelineno-53-6"></a><span class="cm">/* Record the solution */</span>
 <a id="__codelineno-53-7" name="__codelineno-53-7" href="#__codelineno-53-7"></a><span class="kt">void</span><span class="w"> </span><span class="nf">recordSolution</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-8" name="__codelineno-53-8" href="#__codelineno-53-8"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">state</span><span class="p">);</span>
 <a id="__codelineno-53-9" name="__codelineno-53-9" href="#__codelineno-53-9"></a><span class="p">}</span>
 <a id="__codelineno-53-10" name="__codelineno-53-10" href="#__codelineno-53-10"></a>
-<a id="__codelineno-53-11" name="__codelineno-53-11" href="#__codelineno-53-11"></a><span class="cm">/* Check if the choice is valid under the current state */</span>
+<a id="__codelineno-53-11" name="__codelineno-53-11" href="#__codelineno-53-11"></a><span class="cm">/* Check whether the choice is valid in the current state */</span>
 <a id="__codelineno-53-12" name="__codelineno-53-12" href="#__codelineno-53-12"></a><span class="kt">bool</span><span class="w"> </span><span class="nf">isValid</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-13" name="__codelineno-53-13" href="#__codelineno-53-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choice</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-53-14" name="__codelineno-53-14" href="#__codelineno-53-14"></a><span class="p">}</span>
 <a id="__codelineno-53-15" name="__codelineno-53-15" href="#__codelineno-53-15"></a>
-<a id="__codelineno-53-16" name="__codelineno-53-16" href="#__codelineno-53-16"></a><span class="cm">/* Update state */</span>
+<a id="__codelineno-53-16" name="__codelineno-53-16" href="#__codelineno-53-16"></a><span class="cm">/* Update the state */</span>
 <a id="__codelineno-53-17" name="__codelineno-53-17" href="#__codelineno-53-17"></a><span class="kt">void</span><span class="w"> </span><span class="nf">makeChoice</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-18" name="__codelineno-53-18" href="#__codelineno-53-18"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-53-19" name="__codelineno-53-19" href="#__codelineno-53-19"></a><span class="p">}</span>
 <a id="__codelineno-53-20" name="__codelineno-53-20" href="#__codelineno-53-20"></a>
-<a id="__codelineno-53-21" name="__codelineno-53-21" href="#__codelineno-53-21"></a><span class="cm">/* Restore state */</span>
+<a id="__codelineno-53-21" name="__codelineno-53-21" href="#__codelineno-53-21"></a><span class="cm">/* Restore the state */</span>
 <a id="__codelineno-53-22" name="__codelineno-53-22" href="#__codelineno-53-22"></a><span class="kt">void</span><span class="w"> </span><span class="nf">undoChoice</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-23" name="__codelineno-53-23" href="#__codelineno-53-23"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span>
 <a id="__codelineno-53-24" name="__codelineno-53-24" href="#__codelineno-53-24"></a><span class="p">}</span>
 <a id="__codelineno-53-25" name="__codelineno-53-25" href="#__codelineno-53-25"></a>
-<a id="__codelineno-53-26" name="__codelineno-53-26" href="#__codelineno-53-26"></a><span class="cm">/* Backtracking algorithm: Example 3 */</span>
+<a id="__codelineno-53-26" name="__codelineno-53-26" href="#__codelineno-53-26"></a><span class="cm">/* Backtracking: example 3 */</span>
 <a id="__codelineno-53-27" name="__codelineno-53-27" href="#__codelineno-53-27"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-53-28" name="__codelineno-53-28" href="#__codelineno-53-28"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-53-28" name="__codelineno-53-28" href="#__codelineno-53-28"></a><span class="w">    </span><span class="c1">// Check whether this is a solution</span>
 <a id="__codelineno-53-29" name="__codelineno-53-29" href="#__codelineno-53-29"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isSolution</span><span class="p">(</span><span class="n">state</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-53-30" name="__codelineno-53-30" href="#__codelineno-53-30"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-53-30" name="__codelineno-53-30" href="#__codelineno-53-30"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-53-31" name="__codelineno-53-31" href="#__codelineno-53-31"></a><span class="w">        </span><span class="n">recordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-53-32" name="__codelineno-53-32" href="#__codelineno-53-32"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-53-33" name="__codelineno-53-33" href="#__codelineno-53-33"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-53-33" name="__codelineno-53-33" href="#__codelineno-53-33"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-53-34" name="__codelineno-53-34" href="#__codelineno-53-34"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">choice</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-53-35" name="__codelineno-53-35" href="#__codelineno-53-35"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-53-35" name="__codelineno-53-35" href="#__codelineno-53-35"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-53-36" name="__codelineno-53-36" href="#__codelineno-53-36"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isValid</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-53-37" name="__codelineno-53-37" href="#__codelineno-53-37"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-53-37" name="__codelineno-53-37" href="#__codelineno-53-37"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-53-38" name="__codelineno-53-38" href="#__codelineno-53-38"></a><span class="w">            </span><span class="n">makeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
-<a id="__codelineno-53-39" name="__codelineno-53-39" href="#__codelineno-53-39"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-53-39" name="__codelineno-53-39" href="#__codelineno-53-39"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-53-40" name="__codelineno-53-40" href="#__codelineno-53-40"></a><span class="w">            </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="n">nextChoices</span><span class="p">{</span><span class="n">choice</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">};</span>
 <a id="__codelineno-53-41" name="__codelineno-53-41" href="#__codelineno-53-41"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">nextChoices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-53-42" name="__codelineno-53-42" href="#__codelineno-53-42"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-53-42" name="__codelineno-53-42" href="#__codelineno-53-42"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-53-43" name="__codelineno-53-43" href="#__codelineno-53-43"></a><span class="w">            </span><span class="n">undoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-53-44" name="__codelineno-53-44" href="#__codelineno-53-44"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-53-45" name="__codelineno-53-45" href="#__codelineno-53-45"></a><span class="w">    </span><span class="p">}</span>
@@ -5966,47 +6053,47 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_template.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Check if the current state is a solution */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_template.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Check whether the current state is a solution */</span>
 <a id="__codelineno-54-2" name="__codelineno-54-2" href="#__codelineno-54-2"></a><span class="kt">boolean</span><span class="w"> </span><span class="nf">isSolution</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">!</span><span class="n">state</span><span class="p">.</span><span class="na">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">).</span><span class="na">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">;</span>
 <a id="__codelineno-54-4" name="__codelineno-54-4" href="#__codelineno-54-4"></a><span class="p">}</span>
 <a id="__codelineno-54-5" name="__codelineno-54-5" href="#__codelineno-54-5"></a>
-<a id="__codelineno-54-6" name="__codelineno-54-6" href="#__codelineno-54-6"></a><span class="cm">/* Record solution */</span>
+<a id="__codelineno-54-6" name="__codelineno-54-6" href="#__codelineno-54-6"></a><span class="cm">/* Record the solution */</span>
 <a id="__codelineno-54-7" name="__codelineno-54-7" href="#__codelineno-54-7"></a><span class="kt">void</span><span class="w"> </span><span class="nf">recordSolution</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-8" name="__codelineno-54-8" href="#__codelineno-54-8"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-54-9" name="__codelineno-54-9" href="#__codelineno-54-9"></a><span class="p">}</span>
 <a id="__codelineno-54-10" name="__codelineno-54-10" href="#__codelineno-54-10"></a>
-<a id="__codelineno-54-11" name="__codelineno-54-11" href="#__codelineno-54-11"></a><span class="cm">/* Check if the choice is valid under the current state */</span>
+<a id="__codelineno-54-11" name="__codelineno-54-11" href="#__codelineno-54-11"></a><span class="cm">/* Check whether the choice is valid in the current state */</span>
 <a id="__codelineno-54-12" name="__codelineno-54-12" href="#__codelineno-54-12"></a><span class="kt">boolean</span><span class="w"> </span><span class="nf">isValid</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-13" name="__codelineno-54-13" href="#__codelineno-54-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choice</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-54-14" name="__codelineno-54-14" href="#__codelineno-54-14"></a><span class="p">}</span>
 <a id="__codelineno-54-15" name="__codelineno-54-15" href="#__codelineno-54-15"></a>
-<a id="__codelineno-54-16" name="__codelineno-54-16" href="#__codelineno-54-16"></a><span class="cm">/* Update state */</span>
+<a id="__codelineno-54-16" name="__codelineno-54-16" href="#__codelineno-54-16"></a><span class="cm">/* Update the state */</span>
 <a id="__codelineno-54-17" name="__codelineno-54-17" href="#__codelineno-54-17"></a><span class="kt">void</span><span class="w"> </span><span class="nf">makeChoice</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-18" name="__codelineno-54-18" href="#__codelineno-54-18"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-54-19" name="__codelineno-54-19" href="#__codelineno-54-19"></a><span class="p">}</span>
 <a id="__codelineno-54-20" name="__codelineno-54-20" href="#__codelineno-54-20"></a>
-<a id="__codelineno-54-21" name="__codelineno-54-21" href="#__codelineno-54-21"></a><span class="cm">/* Restore state */</span>
+<a id="__codelineno-54-21" name="__codelineno-54-21" href="#__codelineno-54-21"></a><span class="cm">/* Restore the state */</span>
 <a id="__codelineno-54-22" name="__codelineno-54-22" href="#__codelineno-54-22"></a><span class="kt">void</span><span class="w"> </span><span class="nf">undoChoice</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-23" name="__codelineno-54-23" href="#__codelineno-54-23"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-54-24" name="__codelineno-54-24" href="#__codelineno-54-24"></a><span class="p">}</span>
 <a id="__codelineno-54-25" name="__codelineno-54-25" href="#__codelineno-54-25"></a>
-<a id="__codelineno-54-26" name="__codelineno-54-26" href="#__codelineno-54-26"></a><span class="cm">/* Backtracking algorithm: Example 3 */</span>
+<a id="__codelineno-54-26" name="__codelineno-54-26" href="#__codelineno-54-26"></a><span class="cm">/* Backtracking: example 3 */</span>
 <a id="__codelineno-54-27" name="__codelineno-54-27" href="#__codelineno-54-27"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-54-28" name="__codelineno-54-28" href="#__codelineno-54-28"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-54-28" name="__codelineno-54-28" href="#__codelineno-54-28"></a><span class="w">    </span><span class="c1">// Check whether this is a solution</span>
 <a id="__codelineno-54-29" name="__codelineno-54-29" href="#__codelineno-54-29"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isSolution</span><span class="p">(</span><span class="n">state</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-54-30" name="__codelineno-54-30" href="#__codelineno-54-30"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-54-30" name="__codelineno-54-30" href="#__codelineno-54-30"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-54-31" name="__codelineno-54-31" href="#__codelineno-54-31"></a><span class="w">        </span><span class="n">recordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-54-32" name="__codelineno-54-32" href="#__codelineno-54-32"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-54-33" name="__codelineno-54-33" href="#__codelineno-54-33"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-54-33" name="__codelineno-54-33" href="#__codelineno-54-33"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-54-34" name="__codelineno-54-34" href="#__codelineno-54-34"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-54-35" name="__codelineno-54-35" href="#__codelineno-54-35"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-54-35" name="__codelineno-54-35" href="#__codelineno-54-35"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-54-36" name="__codelineno-54-36" href="#__codelineno-54-36"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isValid</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-54-37" name="__codelineno-54-37" href="#__codelineno-54-37"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-54-37" name="__codelineno-54-37" href="#__codelineno-54-37"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-54-38" name="__codelineno-54-38" href="#__codelineno-54-38"></a><span class="w">            </span><span class="n">makeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
-<a id="__codelineno-54-39" name="__codelineno-54-39" href="#__codelineno-54-39"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-54-39" name="__codelineno-54-39" href="#__codelineno-54-39"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-54-40" name="__codelineno-54-40" href="#__codelineno-54-40"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">Arrays</span><span class="p">.</span><span class="na">asList</span><span class="p">(</span><span class="n">choice</span><span class="p">.</span><span class="na">left</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">.</span><span class="na">right</span><span class="p">),</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-54-41" name="__codelineno-54-41" href="#__codelineno-54-41"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-54-41" name="__codelineno-54-41" href="#__codelineno-54-41"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-54-42" name="__codelineno-54-42" href="#__codelineno-54-42"></a><span class="w">            </span><span class="n">undoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-54-43" name="__codelineno-54-43" href="#__codelineno-54-43"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-54-44" name="__codelineno-54-44" href="#__codelineno-54-44"></a><span class="w">    </span><span class="p">}</span>
@@ -6014,47 +6101,47 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_template.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Check if the current state is a solution */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_template.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Check whether the current state is a solution */</span>
 <a id="__codelineno-55-2" name="__codelineno-55-2" href="#__codelineno-55-2"></a><span class="kt">bool</span><span class="w"> </span><span class="nf">IsSolution</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="o">^</span><span class="mi">1</span><span class="p">].</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">;</span>
 <a id="__codelineno-55-4" name="__codelineno-55-4" href="#__codelineno-55-4"></a><span class="p">}</span>
 <a id="__codelineno-55-5" name="__codelineno-55-5" href="#__codelineno-55-5"></a>
-<a id="__codelineno-55-6" name="__codelineno-55-6" href="#__codelineno-55-6"></a><span class="cm">/* Record solution */</span>
+<a id="__codelineno-55-6" name="__codelineno-55-6" href="#__codelineno-55-6"></a><span class="cm">/* Record the solution */</span>
 <a id="__codelineno-55-7" name="__codelineno-55-7" href="#__codelineno-55-7"></a><span class="k">void</span><span class="w"> </span><span class="nf">RecordSolution</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-8" name="__codelineno-55-8" href="#__codelineno-55-8"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-55-9" name="__codelineno-55-9" href="#__codelineno-55-9"></a><span class="p">}</span>
 <a id="__codelineno-55-10" name="__codelineno-55-10" href="#__codelineno-55-10"></a>
-<a id="__codelineno-55-11" name="__codelineno-55-11" href="#__codelineno-55-11"></a><span class="cm">/* Check if the choice is valid under the current state */</span>
+<a id="__codelineno-55-11" name="__codelineno-55-11" href="#__codelineno-55-11"></a><span class="cm">/* Check whether the choice is valid in the current state */</span>
 <a id="__codelineno-55-12" name="__codelineno-55-12" href="#__codelineno-55-12"></a><span class="kt">bool</span><span class="w"> </span><span class="nf">IsValid</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-13" name="__codelineno-55-13" href="#__codelineno-55-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choice</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-55-14" name="__codelineno-55-14" href="#__codelineno-55-14"></a><span class="p">}</span>
 <a id="__codelineno-55-15" name="__codelineno-55-15" href="#__codelineno-55-15"></a>
-<a id="__codelineno-55-16" name="__codelineno-55-16" href="#__codelineno-55-16"></a><span class="cm">/* Update state */</span>
+<a id="__codelineno-55-16" name="__codelineno-55-16" href="#__codelineno-55-16"></a><span class="cm">/* Update the state */</span>
 <a id="__codelineno-55-17" name="__codelineno-55-17" href="#__codelineno-55-17"></a><span class="k">void</span><span class="w"> </span><span class="nf">MakeChoice</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-18" name="__codelineno-55-18" href="#__codelineno-55-18"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-55-19" name="__codelineno-55-19" href="#__codelineno-55-19"></a><span class="p">}</span>
 <a id="__codelineno-55-20" name="__codelineno-55-20" href="#__codelineno-55-20"></a>
-<a id="__codelineno-55-21" name="__codelineno-55-21" href="#__codelineno-55-21"></a><span class="cm">/* Restore state */</span>
+<a id="__codelineno-55-21" name="__codelineno-55-21" href="#__codelineno-55-21"></a><span class="cm">/* Restore the state */</span>
 <a id="__codelineno-55-22" name="__codelineno-55-22" href="#__codelineno-55-22"></a><span class="k">void</span><span class="w"> </span><span class="nf">UndoChoice</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-23" name="__codelineno-55-23" href="#__codelineno-55-23"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="n">RemoveAt</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-55-24" name="__codelineno-55-24" href="#__codelineno-55-24"></a><span class="p">}</span>
 <a id="__codelineno-55-25" name="__codelineno-55-25" href="#__codelineno-55-25"></a>
-<a id="__codelineno-55-26" name="__codelineno-55-26" href="#__codelineno-55-26"></a><span class="cm">/* Backtracking algorithm: Example 3 */</span>
+<a id="__codelineno-55-26" name="__codelineno-55-26" href="#__codelineno-55-26"></a><span class="cm">/* Backtracking: example 3 */</span>
 <a id="__codelineno-55-27" name="__codelineno-55-27" href="#__codelineno-55-27"></a><span class="k">void</span><span class="w"> </span><span class="nf">Backtrack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-55-28" name="__codelineno-55-28" href="#__codelineno-55-28"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-55-28" name="__codelineno-55-28" href="#__codelineno-55-28"></a><span class="w">    </span><span class="c1">// Check whether this is a solution</span>
 <a id="__codelineno-55-29" name="__codelineno-55-29" href="#__codelineno-55-29"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">IsSolution</span><span class="p">(</span><span class="n">state</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-55-30" name="__codelineno-55-30" href="#__codelineno-55-30"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-55-30" name="__codelineno-55-30" href="#__codelineno-55-30"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-55-31" name="__codelineno-55-31" href="#__codelineno-55-31"></a><span class="w">        </span><span class="n">RecordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-55-32" name="__codelineno-55-32" href="#__codelineno-55-32"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-55-33" name="__codelineno-55-33" href="#__codelineno-55-33"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-55-33" name="__codelineno-55-33" href="#__codelineno-55-33"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-55-34" name="__codelineno-55-34" href="#__codelineno-55-34"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-55-35" name="__codelineno-55-35" href="#__codelineno-55-35"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-55-35" name="__codelineno-55-35" href="#__codelineno-55-35"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-55-36" name="__codelineno-55-36" href="#__codelineno-55-36"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">IsValid</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-55-37" name="__codelineno-55-37" href="#__codelineno-55-37"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-55-37" name="__codelineno-55-37" href="#__codelineno-55-37"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-55-38" name="__codelineno-55-38" href="#__codelineno-55-38"></a><span class="w">            </span><span class="n">MakeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
-<a id="__codelineno-55-39" name="__codelineno-55-39" href="#__codelineno-55-39"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-55-39" name="__codelineno-55-39" href="#__codelineno-55-39"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-55-40" name="__codelineno-55-40" href="#__codelineno-55-40"></a><span class="w">            </span><span class="n">Backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="p">[</span><span class="n">choice</span><span class="p">.</span><span class="n">left</span><span class="o">!</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">.</span><span class="n">right</span><span class="o">!</span><span class="p">],</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-55-41" name="__codelineno-55-41" href="#__codelineno-55-41"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-55-41" name="__codelineno-55-41" href="#__codelineno-55-41"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-55-42" name="__codelineno-55-42" href="#__codelineno-55-42"></a><span class="w">            </span><span class="n">UndoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-55-43" name="__codelineno-55-43" href="#__codelineno-55-43"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-55-44" name="__codelineno-55-44" href="#__codelineno-55-44"></a><span class="w">    </span><span class="p">}</span>
@@ -6062,49 +6149,49 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_template.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Check if the current state is a solution */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_template.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Check whether the current state is a solution */</span>
 <a id="__codelineno-56-2" name="__codelineno-56-2" href="#__codelineno-56-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">isSolution</span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[</span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">].</span><span class="nx">Val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span>
 <a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a><span class="p">}</span>
 <a id="__codelineno-56-5" name="__codelineno-56-5" href="#__codelineno-56-5"></a>
-<a id="__codelineno-56-6" name="__codelineno-56-6" href="#__codelineno-56-6"></a><span class="cm">/* Record solution */</span>
+<a id="__codelineno-56-6" name="__codelineno-56-6" href="#__codelineno-56-6"></a><span class="cm">/* Record the solution */</span>
 <a id="__codelineno-56-7" name="__codelineno-56-7" href="#__codelineno-56-7"></a><span class="kd">func</span><span class="w"> </span><span class="nx">recordSolution</span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[][]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-8" name="__codelineno-56-8" href="#__codelineno-56-8"></a><span class="w">    </span><span class="o">*</span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nb">append</span><span class="p">([]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">{},</span><span class="w"> </span><span class="o">*</span><span class="nx">state</span><span class="p">...))</span>
 <a id="__codelineno-56-9" name="__codelineno-56-9" href="#__codelineno-56-9"></a><span class="p">}</span>
 <a id="__codelineno-56-10" name="__codelineno-56-10" href="#__codelineno-56-10"></a>
-<a id="__codelineno-56-11" name="__codelineno-56-11" href="#__codelineno-56-11"></a><span class="cm">/* Check if the choice is valid under the current state */</span>
+<a id="__codelineno-56-11" name="__codelineno-56-11" href="#__codelineno-56-11"></a><span class="cm">/* Check whether the choice is valid in the current state */</span>
 <a id="__codelineno-56-12" name="__codelineno-56-12" href="#__codelineno-56-12"></a><span class="kd">func</span><span class="w"> </span><span class="nx">isValid</span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-13" name="__codelineno-56-13" href="#__codelineno-56-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">choice</span><span class="p">.</span><span class="nx">Val</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">3</span>
 <a id="__codelineno-56-14" name="__codelineno-56-14" href="#__codelineno-56-14"></a><span class="p">}</span>
 <a id="__codelineno-56-15" name="__codelineno-56-15" href="#__codelineno-56-15"></a>
-<a id="__codelineno-56-16" name="__codelineno-56-16" href="#__codelineno-56-16"></a><span class="cm">/* Update state */</span>
+<a id="__codelineno-56-16" name="__codelineno-56-16" href="#__codelineno-56-16"></a><span class="cm">/* Update the state */</span>
 <a id="__codelineno-56-17" name="__codelineno-56-17" href="#__codelineno-56-17"></a><span class="kd">func</span><span class="w"> </span><span class="nx">makeChoice</span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-18" name="__codelineno-56-18" href="#__codelineno-56-18"></a><span class="w">    </span><span class="o">*</span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">)</span>
 <a id="__codelineno-56-19" name="__codelineno-56-19" href="#__codelineno-56-19"></a><span class="p">}</span>
 <a id="__codelineno-56-20" name="__codelineno-56-20" href="#__codelineno-56-20"></a>
-<a id="__codelineno-56-21" name="__codelineno-56-21" href="#__codelineno-56-21"></a><span class="cm">/* Restore state */</span>
+<a id="__codelineno-56-21" name="__codelineno-56-21" href="#__codelineno-56-21"></a><span class="cm">/* Restore the state */</span>
 <a id="__codelineno-56-22" name="__codelineno-56-22" href="#__codelineno-56-22"></a><span class="kd">func</span><span class="w"> </span><span class="nx">undoChoice</span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-23" name="__codelineno-56-23" href="#__codelineno-56-23"></a><span class="w">    </span><span class="o">*</span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[:</span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-56-24" name="__codelineno-56-24" href="#__codelineno-56-24"></a><span class="p">}</span>
 <a id="__codelineno-56-25" name="__codelineno-56-25" href="#__codelineno-56-25"></a>
-<a id="__codelineno-56-26" name="__codelineno-56-26" href="#__codelineno-56-26"></a><span class="cm">/* Backtracking algorithm: Example 3 */</span>
+<a id="__codelineno-56-26" name="__codelineno-56-26" href="#__codelineno-56-26"></a><span class="cm">/* Backtracking: example 3 */</span>
 <a id="__codelineno-56-27" name="__codelineno-56-27" href="#__codelineno-56-27"></a><span class="kd">func</span><span class="w"> </span><span class="nx">backtrackIII</span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[][]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-56-28" name="__codelineno-56-28" href="#__codelineno-56-28"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-56-28" name="__codelineno-56-28" href="#__codelineno-56-28"></a><span class="w">    </span><span class="c1">// Check whether this is a solution</span>
 <a id="__codelineno-56-29" name="__codelineno-56-29" href="#__codelineno-56-29"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">isSolution</span><span class="p">(</span><span class="nx">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-56-30" name="__codelineno-56-30" href="#__codelineno-56-30"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-56-30" name="__codelineno-56-30" href="#__codelineno-56-30"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-56-31" name="__codelineno-56-31" href="#__codelineno-56-31"></a><span class="w">        </span><span class="nx">recordSolution</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span>
 <a id="__codelineno-56-32" name="__codelineno-56-32" href="#__codelineno-56-32"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-56-33" name="__codelineno-56-33" href="#__codelineno-56-33"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-56-33" name="__codelineno-56-33" href="#__codelineno-56-33"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-56-34" name="__codelineno-56-34" href="#__codelineno-56-34"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="o">*</span><span class="nx">choices</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-56-35" name="__codelineno-56-35" href="#__codelineno-56-35"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-56-35" name="__codelineno-56-35" href="#__codelineno-56-35"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-56-36" name="__codelineno-56-36" href="#__codelineno-56-36"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">isValid</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-56-37" name="__codelineno-56-37" href="#__codelineno-56-37"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-56-37" name="__codelineno-56-37" href="#__codelineno-56-37"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-56-38" name="__codelineno-56-38" href="#__codelineno-56-38"></a><span class="w">            </span><span class="nx">makeChoice</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">)</span>
-<a id="__codelineno-56-39" name="__codelineno-56-39" href="#__codelineno-56-39"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-56-39" name="__codelineno-56-39" href="#__codelineno-56-39"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-56-40" name="__codelineno-56-40" href="#__codelineno-56-40"></a><span class="w">            </span><span class="nx">temp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-56-41" name="__codelineno-56-41" href="#__codelineno-56-41"></a><span class="w">            </span><span class="nx">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">temp</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">.</span><span class="nx">Left</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">.</span><span class="nx">Right</span><span class="p">)</span>
 <a id="__codelineno-56-42" name="__codelineno-56-42" href="#__codelineno-56-42"></a><span class="w">            </span><span class="nx">backtrackIII</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">temp</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span>
-<a id="__codelineno-56-43" name="__codelineno-56-43" href="#__codelineno-56-43"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-56-43" name="__codelineno-56-43" href="#__codelineno-56-43"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-56-44" name="__codelineno-56-44" href="#__codelineno-56-44"></a><span class="w">            </span><span class="nx">undoChoice</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">)</span>
 <a id="__codelineno-56-45" name="__codelineno-56-45" href="#__codelineno-56-45"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-56-46" name="__codelineno-56-46" href="#__codelineno-56-46"></a><span class="w">    </span><span class="p">}</span>
@@ -6112,46 +6199,46 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_template.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Check if the current state is a solution */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_template.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Check whether the current state is a solution */</span>
 <a id="__codelineno-57-2" name="__codelineno-57-2" href="#__codelineno-57-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">isSolution</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="n">TreeNode</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="w">    </span><span class="o">!</span><span class="n">state</span><span class="p">.</span><span class="bp">isEmpty</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="bp">last</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">7</span>
 <a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="p">}</span>
 <a id="__codelineno-57-5" name="__codelineno-57-5" href="#__codelineno-57-5"></a>
-<a id="__codelineno-57-6" name="__codelineno-57-6" href="#__codelineno-57-6"></a><span class="cm">/* Record solution */</span>
+<a id="__codelineno-57-6" name="__codelineno-57-6" href="#__codelineno-57-6"></a><span class="cm">/* Record the solution */</span>
 <a id="__codelineno-57-7" name="__codelineno-57-7" href="#__codelineno-57-7"></a><span class="kd">func</span><span class="w"> </span><span class="nf">recordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[[</span><span class="n">TreeNode</span><span class="p">]])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-8" name="__codelineno-57-8" href="#__codelineno-57-8"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
 <a id="__codelineno-57-9" name="__codelineno-57-9" href="#__codelineno-57-9"></a><span class="p">}</span>
 <a id="__codelineno-57-10" name="__codelineno-57-10" href="#__codelineno-57-10"></a>
-<a id="__codelineno-57-11" name="__codelineno-57-11" href="#__codelineno-57-11"></a><span class="cm">/* Check if the choice is valid under the current state */</span>
+<a id="__codelineno-57-11" name="__codelineno-57-11" href="#__codelineno-57-11"></a><span class="cm">/* Check whether the choice is valid in the current state */</span>
 <a id="__codelineno-57-12" name="__codelineno-57-12" href="#__codelineno-57-12"></a><span class="kd">func</span><span class="w"> </span><span class="nf">isValid</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span><span class="w"> </span><span class="n">choice</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-13" name="__codelineno-57-13" href="#__codelineno-57-13"></a><span class="w">    </span><span class="n">choice</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choice</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">3</span>
 <a id="__codelineno-57-14" name="__codelineno-57-14" href="#__codelineno-57-14"></a><span class="p">}</span>
 <a id="__codelineno-57-15" name="__codelineno-57-15" href="#__codelineno-57-15"></a>
-<a id="__codelineno-57-16" name="__codelineno-57-16" href="#__codelineno-57-16"></a><span class="cm">/* Update state */</span>
+<a id="__codelineno-57-16" name="__codelineno-57-16" href="#__codelineno-57-16"></a><span class="cm">/* Update the state */</span>
 <a id="__codelineno-57-17" name="__codelineno-57-17" href="#__codelineno-57-17"></a><span class="kd">func</span><span class="w"> </span><span class="nf">makeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span><span class="w"> </span><span class="n">choice</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-18" name="__codelineno-57-18" href="#__codelineno-57-18"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-57-19" name="__codelineno-57-19" href="#__codelineno-57-19"></a><span class="p">}</span>
 <a id="__codelineno-57-20" name="__codelineno-57-20" href="#__codelineno-57-20"></a>
-<a id="__codelineno-57-21" name="__codelineno-57-21" href="#__codelineno-57-21"></a><span class="cm">/* Restore state */</span>
+<a id="__codelineno-57-21" name="__codelineno-57-21" href="#__codelineno-57-21"></a><span class="cm">/* Restore the state */</span>
 <a id="__codelineno-57-22" name="__codelineno-57-22" href="#__codelineno-57-22"></a><span class="kd">func</span><span class="w"> </span><span class="nf">undoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span><span class="w"> </span><span class="n">choice</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-23" name="__codelineno-57-23" href="#__codelineno-57-23"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span>
 <a id="__codelineno-57-24" name="__codelineno-57-24" href="#__codelineno-57-24"></a><span class="p">}</span>
 <a id="__codelineno-57-25" name="__codelineno-57-25" href="#__codelineno-57-25"></a>
-<a id="__codelineno-57-26" name="__codelineno-57-26" href="#__codelineno-57-26"></a><span class="cm">/* Backtracking algorithm: Example 3 */</span>
+<a id="__codelineno-57-26" name="__codelineno-57-26" href="#__codelineno-57-26"></a><span class="cm">/* Backtracking: example 3 */</span>
 <a id="__codelineno-57-27" name="__codelineno-57-27" href="#__codelineno-57-27"></a><span class="kd">func</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="n">TreeNode</span><span class="p">],</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[[</span><span class="n">TreeNode</span><span class="p">]])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-57-28" name="__codelineno-57-28" href="#__codelineno-57-28"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-57-28" name="__codelineno-57-28" href="#__codelineno-57-28"></a><span class="w">    </span><span class="c1">// Check whether this is a solution</span>
 <a id="__codelineno-57-29" name="__codelineno-57-29" href="#__codelineno-57-29"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">isSolution</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-30" name="__codelineno-57-30" href="#__codelineno-57-30"></a><span class="w">        </span><span class="n">recordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-57-31" name="__codelineno-57-31" href="#__codelineno-57-31"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-57-32" name="__codelineno-57-32" href="#__codelineno-57-32"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-57-32" name="__codelineno-57-32" href="#__codelineno-57-32"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-57-33" name="__codelineno-57-33" href="#__codelineno-57-33"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-57-34" name="__codelineno-57-34" href="#__codelineno-57-34"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-57-34" name="__codelineno-57-34" href="#__codelineno-57-34"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-57-35" name="__codelineno-57-35" href="#__codelineno-57-35"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">isValid</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">:</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-57-36" name="__codelineno-57-36" href="#__codelineno-57-36"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-57-36" name="__codelineno-57-36" href="#__codelineno-57-36"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-57-37" name="__codelineno-57-37" href="#__codelineno-57-37"></a><span class="w">            </span><span class="n">makeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">:</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
-<a id="__codelineno-57-38" name="__codelineno-57-38" href="#__codelineno-57-38"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-57-38" name="__codelineno-57-38" href="#__codelineno-57-38"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-57-39" name="__codelineno-57-39" href="#__codelineno-57-39"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="n">choice</span><span class="p">.</span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">.</span><span class="kr">right</span><span class="p">].</span><span class="n">compactMap</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nv">$0</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-57-40" name="__codelineno-57-40" href="#__codelineno-57-40"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-57-40" name="__codelineno-57-40" href="#__codelineno-57-40"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-57-41" name="__codelineno-57-41" href="#__codelineno-57-41"></a><span class="w">            </span><span class="n">undoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">:</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-57-42" name="__codelineno-57-42" href="#__codelineno-57-42"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-57-43" name="__codelineno-57-43" href="#__codelineno-57-43"></a><span class="w">    </span><span class="p">}</span>
@@ -6159,47 +6246,47 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_template.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Check if the current state is a solution */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_template.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Check whether the current state is a solution */</span>
 <a id="__codelineno-58-2" name="__codelineno-58-2" href="#__codelineno-58-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">isSolution</span><span class="p">(</span><span class="nx">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">state</span><span class="p">[</span><span class="nx">state</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="o">?</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">7</span><span class="p">;</span>
 <a id="__codelineno-58-4" name="__codelineno-58-4" href="#__codelineno-58-4"></a><span class="p">}</span>
 <a id="__codelineno-58-5" name="__codelineno-58-5" href="#__codelineno-58-5"></a>
-<a id="__codelineno-58-6" name="__codelineno-58-6" href="#__codelineno-58-6"></a><span class="cm">/* Record solution */</span>
+<a id="__codelineno-58-6" name="__codelineno-58-6" href="#__codelineno-58-6"></a><span class="cm">/* Record the solution */</span>
 <a id="__codelineno-58-7" name="__codelineno-58-7" href="#__codelineno-58-7"></a><span class="kd">function</span><span class="w"> </span><span class="nx">recordSolution</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-8" name="__codelineno-58-8" href="#__codelineno-58-8"></a><span class="w">    </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">state</span><span class="p">]);</span>
 <a id="__codelineno-58-9" name="__codelineno-58-9" href="#__codelineno-58-9"></a><span class="p">}</span>
 <a id="__codelineno-58-10" name="__codelineno-58-10" href="#__codelineno-58-10"></a>
-<a id="__codelineno-58-11" name="__codelineno-58-11" href="#__codelineno-58-11"></a><span class="cm">/* Check if the choice is valid under the current state */</span>
+<a id="__codelineno-58-11" name="__codelineno-58-11" href="#__codelineno-58-11"></a><span class="cm">/* Check whether the choice is valid in the current state */</span>
 <a id="__codelineno-58-12" name="__codelineno-58-12" href="#__codelineno-58-12"></a><span class="kd">function</span><span class="w"> </span><span class="nx">isValid</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-13" name="__codelineno-58-13" href="#__codelineno-58-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">choice</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="mf">3</span><span class="p">;</span>
 <a id="__codelineno-58-14" name="__codelineno-58-14" href="#__codelineno-58-14"></a><span class="p">}</span>
 <a id="__codelineno-58-15" name="__codelineno-58-15" href="#__codelineno-58-15"></a>
-<a id="__codelineno-58-16" name="__codelineno-58-16" href="#__codelineno-58-16"></a><span class="cm">/* Update state */</span>
+<a id="__codelineno-58-16" name="__codelineno-58-16" href="#__codelineno-58-16"></a><span class="cm">/* Update the state */</span>
 <a id="__codelineno-58-17" name="__codelineno-58-17" href="#__codelineno-58-17"></a><span class="kd">function</span><span class="w"> </span><span class="nx">makeChoice</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-18" name="__codelineno-58-18" href="#__codelineno-58-18"></a><span class="w">    </span><span class="nx">state</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">choice</span><span class="p">);</span>
 <a id="__codelineno-58-19" name="__codelineno-58-19" href="#__codelineno-58-19"></a><span class="p">}</span>
 <a id="__codelineno-58-20" name="__codelineno-58-20" href="#__codelineno-58-20"></a>
-<a id="__codelineno-58-21" name="__codelineno-58-21" href="#__codelineno-58-21"></a><span class="cm">/* Restore state */</span>
+<a id="__codelineno-58-21" name="__codelineno-58-21" href="#__codelineno-58-21"></a><span class="cm">/* Restore the state */</span>
 <a id="__codelineno-58-22" name="__codelineno-58-22" href="#__codelineno-58-22"></a><span class="kd">function</span><span class="w"> </span><span class="nx">undoChoice</span><span class="p">(</span><span class="nx">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-23" name="__codelineno-58-23" href="#__codelineno-58-23"></a><span class="w">    </span><span class="nx">state</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-58-24" name="__codelineno-58-24" href="#__codelineno-58-24"></a><span class="p">}</span>
 <a id="__codelineno-58-25" name="__codelineno-58-25" href="#__codelineno-58-25"></a>
-<a id="__codelineno-58-26" name="__codelineno-58-26" href="#__codelineno-58-26"></a><span class="cm">/* Backtracking algorithm: Example 3 */</span>
+<a id="__codelineno-58-26" name="__codelineno-58-26" href="#__codelineno-58-26"></a><span class="cm">/* Backtracking: example 3 */</span>
 <a id="__codelineno-58-27" name="__codelineno-58-27" href="#__codelineno-58-27"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-58-28" name="__codelineno-58-28" href="#__codelineno-58-28"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-58-28" name="__codelineno-58-28" href="#__codelineno-58-28"></a><span class="w">    </span><span class="c1">// Check whether this is a solution</span>
 <a id="__codelineno-58-29" name="__codelineno-58-29" href="#__codelineno-58-29"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">isSolution</span><span class="p">(</span><span class="nx">state</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-58-30" name="__codelineno-58-30" href="#__codelineno-58-30"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-58-30" name="__codelineno-58-30" href="#__codelineno-58-30"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-58-31" name="__codelineno-58-31" href="#__codelineno-58-31"></a><span class="w">        </span><span class="nx">recordSolution</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
 <a id="__codelineno-58-32" name="__codelineno-58-32" href="#__codelineno-58-32"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-58-33" name="__codelineno-58-33" href="#__codelineno-58-33"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-58-33" name="__codelineno-58-33" href="#__codelineno-58-33"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-58-34" name="__codelineno-58-34" href="#__codelineno-58-34"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-58-35" name="__codelineno-58-35" href="#__codelineno-58-35"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-58-35" name="__codelineno-58-35" href="#__codelineno-58-35"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-58-36" name="__codelineno-58-36" href="#__codelineno-58-36"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">isValid</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-58-37" name="__codelineno-58-37" href="#__codelineno-58-37"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-58-37" name="__codelineno-58-37" href="#__codelineno-58-37"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-58-38" name="__codelineno-58-38" href="#__codelineno-58-38"></a><span class="w">            </span><span class="nx">makeChoice</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">);</span>
-<a id="__codelineno-58-39" name="__codelineno-58-39" href="#__codelineno-58-39"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-58-39" name="__codelineno-58-39" href="#__codelineno-58-39"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-58-40" name="__codelineno-58-40" href="#__codelineno-58-40"></a><span class="w">            </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="p">[</span><span class="nx">choice</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">.</span><span class="nx">right</span><span class="p">],</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-58-41" name="__codelineno-58-41" href="#__codelineno-58-41"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-58-41" name="__codelineno-58-41" href="#__codelineno-58-41"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-58-42" name="__codelineno-58-42" href="#__codelineno-58-42"></a><span class="w">            </span><span class="nx">undoChoice</span><span class="p">(</span><span class="nx">state</span><span class="p">);</span>
 <a id="__codelineno-58-43" name="__codelineno-58-43" href="#__codelineno-58-43"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-58-44" name="__codelineno-58-44" href="#__codelineno-58-44"></a><span class="w">    </span><span class="p">}</span>
@@ -6207,51 +6294,51 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_template.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Check if the current state is a solution */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_template.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Check whether the current state is a solution */</span>
 <a id="__codelineno-59-2" name="__codelineno-59-2" href="#__codelineno-59-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">isSolution</span><span class="p">(</span><span class="nx">state</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">state</span><span class="p">[</span><span class="nx">state</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="o">?</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">7</span><span class="p">;</span>
 <a id="__codelineno-59-4" name="__codelineno-59-4" href="#__codelineno-59-4"></a><span class="p">}</span>
 <a id="__codelineno-59-5" name="__codelineno-59-5" href="#__codelineno-59-5"></a>
-<a id="__codelineno-59-6" name="__codelineno-59-6" href="#__codelineno-59-6"></a><span class="cm">/* Record solution */</span>
+<a id="__codelineno-59-6" name="__codelineno-59-6" href="#__codelineno-59-6"></a><span class="cm">/* Record the solution */</span>
 <a id="__codelineno-59-7" name="__codelineno-59-7" href="#__codelineno-59-7"></a><span class="kd">function</span><span class="w"> </span><span class="nx">recordSolution</span><span class="p">(</span><span class="nx">state</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">[],</span><span class="w"> </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">[][])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-8" name="__codelineno-59-8" href="#__codelineno-59-8"></a><span class="w">    </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">state</span><span class="p">]);</span>
 <a id="__codelineno-59-9" name="__codelineno-59-9" href="#__codelineno-59-9"></a><span class="p">}</span>
 <a id="__codelineno-59-10" name="__codelineno-59-10" href="#__codelineno-59-10"></a>
-<a id="__codelineno-59-11" name="__codelineno-59-11" href="#__codelineno-59-11"></a><span class="cm">/* Check if the choice is valid under the current state */</span>
+<a id="__codelineno-59-11" name="__codelineno-59-11" href="#__codelineno-59-11"></a><span class="cm">/* Check whether the choice is valid in the current state */</span>
 <a id="__codelineno-59-12" name="__codelineno-59-12" href="#__codelineno-59-12"></a><span class="kd">function</span><span class="w"> </span><span class="nx">isValid</span><span class="p">(</span><span class="nx">state</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">[],</span><span class="w"> </span><span class="nx">choice</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-13" name="__codelineno-59-13" href="#__codelineno-59-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">choice</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="mf">3</span><span class="p">;</span>
 <a id="__codelineno-59-14" name="__codelineno-59-14" href="#__codelineno-59-14"></a><span class="p">}</span>
 <a id="__codelineno-59-15" name="__codelineno-59-15" href="#__codelineno-59-15"></a>
-<a id="__codelineno-59-16" name="__codelineno-59-16" href="#__codelineno-59-16"></a><span class="cm">/* Update state */</span>
+<a id="__codelineno-59-16" name="__codelineno-59-16" href="#__codelineno-59-16"></a><span class="cm">/* Update the state */</span>
 <a id="__codelineno-59-17" name="__codelineno-59-17" href="#__codelineno-59-17"></a><span class="kd">function</span><span class="w"> </span><span class="nx">makeChoice</span><span class="p">(</span><span class="nx">state</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">[],</span><span class="w"> </span><span class="nx">choice</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-18" name="__codelineno-59-18" href="#__codelineno-59-18"></a><span class="w">    </span><span class="nx">state</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">choice</span><span class="p">);</span>
 <a id="__codelineno-59-19" name="__codelineno-59-19" href="#__codelineno-59-19"></a><span class="p">}</span>
 <a id="__codelineno-59-20" name="__codelineno-59-20" href="#__codelineno-59-20"></a>
-<a id="__codelineno-59-21" name="__codelineno-59-21" href="#__codelineno-59-21"></a><span class="cm">/* Restore state */</span>
+<a id="__codelineno-59-21" name="__codelineno-59-21" href="#__codelineno-59-21"></a><span class="cm">/* Restore the state */</span>
 <a id="__codelineno-59-22" name="__codelineno-59-22" href="#__codelineno-59-22"></a><span class="kd">function</span><span class="w"> </span><span class="nx">undoChoice</span><span class="p">(</span><span class="nx">state</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-23" name="__codelineno-59-23" href="#__codelineno-59-23"></a><span class="w">    </span><span class="nx">state</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-59-24" name="__codelineno-59-24" href="#__codelineno-59-24"></a><span class="p">}</span>
 <a id="__codelineno-59-25" name="__codelineno-59-25" href="#__codelineno-59-25"></a>
-<a id="__codelineno-59-26" name="__codelineno-59-26" href="#__codelineno-59-26"></a><span class="cm">/* Backtracking algorithm: Example 3 */</span>
+<a id="__codelineno-59-26" name="__codelineno-59-26" href="#__codelineno-59-26"></a><span class="cm">/* Backtracking: example 3 */</span>
 <a id="__codelineno-59-27" name="__codelineno-59-27" href="#__codelineno-59-27"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span>
 <a id="__codelineno-59-28" name="__codelineno-59-28" href="#__codelineno-59-28"></a><span class="w">    </span><span class="nx">state</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">[],</span>
 <a id="__codelineno-59-29" name="__codelineno-59-29" href="#__codelineno-59-29"></a><span class="w">    </span><span class="nx">choices</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">[],</span>
 <a id="__codelineno-59-30" name="__codelineno-59-30" href="#__codelineno-59-30"></a><span class="w">    </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">[][]</span>
 <a id="__codelineno-59-31" name="__codelineno-59-31" href="#__codelineno-59-31"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-59-32" name="__codelineno-59-32" href="#__codelineno-59-32"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-59-32" name="__codelineno-59-32" href="#__codelineno-59-32"></a><span class="w">    </span><span class="c1">// Check whether this is a solution</span>
 <a id="__codelineno-59-33" name="__codelineno-59-33" href="#__codelineno-59-33"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">isSolution</span><span class="p">(</span><span class="nx">state</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-59-34" name="__codelineno-59-34" href="#__codelineno-59-34"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-59-34" name="__codelineno-59-34" href="#__codelineno-59-34"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-59-35" name="__codelineno-59-35" href="#__codelineno-59-35"></a><span class="w">        </span><span class="nx">recordSolution</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
 <a id="__codelineno-59-36" name="__codelineno-59-36" href="#__codelineno-59-36"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-59-37" name="__codelineno-59-37" href="#__codelineno-59-37"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-59-37" name="__codelineno-59-37" href="#__codelineno-59-37"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-59-38" name="__codelineno-59-38" href="#__codelineno-59-38"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-59-39" name="__codelineno-59-39" href="#__codelineno-59-39"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-59-39" name="__codelineno-59-39" href="#__codelineno-59-39"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-59-40" name="__codelineno-59-40" href="#__codelineno-59-40"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">isValid</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-59-41" name="__codelineno-59-41" href="#__codelineno-59-41"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-59-41" name="__codelineno-59-41" href="#__codelineno-59-41"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-59-42" name="__codelineno-59-42" href="#__codelineno-59-42"></a><span class="w">            </span><span class="nx">makeChoice</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">);</span>
-<a id="__codelineno-59-43" name="__codelineno-59-43" href="#__codelineno-59-43"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-59-43" name="__codelineno-59-43" href="#__codelineno-59-43"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-59-44" name="__codelineno-59-44" href="#__codelineno-59-44"></a><span class="w">            </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="p">[</span><span class="nx">choice</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">.</span><span class="nx">right</span><span class="p">],</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-59-45" name="__codelineno-59-45" href="#__codelineno-59-45"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-59-45" name="__codelineno-59-45" href="#__codelineno-59-45"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-59-46" name="__codelineno-59-46" href="#__codelineno-59-46"></a><span class="w">            </span><span class="nx">undoChoice</span><span class="p">(</span><span class="nx">state</span><span class="p">);</span>
 <a id="__codelineno-59-47" name="__codelineno-59-47" href="#__codelineno-59-47"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-59-48" name="__codelineno-59-48" href="#__codelineno-59-48"></a><span class="w">    </span><span class="p">}</span>
@@ -6259,51 +6346,51 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_template.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Check if the current state is a solution */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_template.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Check whether the current state is a solution */</span>
 <a id="__codelineno-60-2" name="__codelineno-60-2" href="#__codelineno-60-2"></a><span class="kt">bool</span><span class="w"> </span><span class="n">isSolution</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-3" name="__codelineno-60-3" href="#__codelineno-60-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">isNotEmpty</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">last</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">7</span><span class="p">;</span>
 <a id="__codelineno-60-4" name="__codelineno-60-4" href="#__codelineno-60-4"></a><span class="p">}</span>
 <a id="__codelineno-60-5" name="__codelineno-60-5" href="#__codelineno-60-5"></a>
-<a id="__codelineno-60-6" name="__codelineno-60-6" href="#__codelineno-60-6"></a><span class="cm">/* Record solution */</span>
+<a id="__codelineno-60-6" name="__codelineno-60-6" href="#__codelineno-60-6"></a><span class="cm">/* Record the solution */</span>
 <a id="__codelineno-60-7" name="__codelineno-60-7" href="#__codelineno-60-7"></a><span class="kt">void</span><span class="w"> </span><span class="n">recordSolution</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-8" name="__codelineno-60-8" href="#__codelineno-60-8"></a><span class="w">  </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">List</span><span class="p">.</span><span class="n">from</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-60-9" name="__codelineno-60-9" href="#__codelineno-60-9"></a><span class="p">}</span>
 <a id="__codelineno-60-10" name="__codelineno-60-10" href="#__codelineno-60-10"></a>
-<a id="__codelineno-60-11" name="__codelineno-60-11" href="#__codelineno-60-11"></a><span class="cm">/* Check if the choice is valid under the current state */</span>
+<a id="__codelineno-60-11" name="__codelineno-60-11" href="#__codelineno-60-11"></a><span class="cm">/* Check whether the choice is valid in the current state */</span>
 <a id="__codelineno-60-12" name="__codelineno-60-12" href="#__codelineno-60-12"></a><span class="kt">bool</span><span class="w"> </span><span class="n">isValid</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-13" name="__codelineno-60-13" href="#__codelineno-60-13"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choice</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="m">3</span><span class="p">;</span>
 <a id="__codelineno-60-14" name="__codelineno-60-14" href="#__codelineno-60-14"></a><span class="p">}</span>
 <a id="__codelineno-60-15" name="__codelineno-60-15" href="#__codelineno-60-15"></a>
-<a id="__codelineno-60-16" name="__codelineno-60-16" href="#__codelineno-60-16"></a><span class="cm">/* Update state */</span>
+<a id="__codelineno-60-16" name="__codelineno-60-16" href="#__codelineno-60-16"></a><span class="cm">/* Update the state */</span>
 <a id="__codelineno-60-17" name="__codelineno-60-17" href="#__codelineno-60-17"></a><span class="kt">void</span><span class="w"> </span><span class="n">makeChoice</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-18" name="__codelineno-60-18" href="#__codelineno-60-18"></a><span class="w">  </span><span class="n">state</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">choice</span><span class="o">!</span><span class="p">);</span>
 <a id="__codelineno-60-19" name="__codelineno-60-19" href="#__codelineno-60-19"></a><span class="p">}</span>
 <a id="__codelineno-60-20" name="__codelineno-60-20" href="#__codelineno-60-20"></a>
-<a id="__codelineno-60-21" name="__codelineno-60-21" href="#__codelineno-60-21"></a><span class="cm">/* Restore state */</span>
+<a id="__codelineno-60-21" name="__codelineno-60-21" href="#__codelineno-60-21"></a><span class="cm">/* Restore the state */</span>
 <a id="__codelineno-60-22" name="__codelineno-60-22" href="#__codelineno-60-22"></a><span class="kt">void</span><span class="w"> </span><span class="n">undoChoice</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-23" name="__codelineno-60-23" href="#__codelineno-60-23"></a><span class="w">  </span><span class="n">state</span><span class="p">.</span><span class="n">removeLast</span><span class="p">();</span>
 <a id="__codelineno-60-24" name="__codelineno-60-24" href="#__codelineno-60-24"></a><span class="p">}</span>
 <a id="__codelineno-60-25" name="__codelineno-60-25" href="#__codelineno-60-25"></a>
-<a id="__codelineno-60-26" name="__codelineno-60-26" href="#__codelineno-60-26"></a><span class="cm">/* Backtracking algorithm: Example 3 */</span>
+<a id="__codelineno-60-26" name="__codelineno-60-26" href="#__codelineno-60-26"></a><span class="cm">/* Backtracking: example 3 */</span>
 <a id="__codelineno-60-27" name="__codelineno-60-27" href="#__codelineno-60-27"></a><span class="kt">void</span><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span>
 <a id="__codelineno-60-28" name="__codelineno-60-28" href="#__codelineno-60-28"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span>
 <a id="__codelineno-60-29" name="__codelineno-60-29" href="#__codelineno-60-29"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">?&gt;</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span>
 <a id="__codelineno-60-30" name="__codelineno-60-30" href="#__codelineno-60-30"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">,</span>
 <a id="__codelineno-60-31" name="__codelineno-60-31" href="#__codelineno-60-31"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-60-32" name="__codelineno-60-32" href="#__codelineno-60-32"></a><span class="w">  </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-60-32" name="__codelineno-60-32" href="#__codelineno-60-32"></a><span class="w">  </span><span class="c1">// Check whether this is a solution</span>
 <a id="__codelineno-60-33" name="__codelineno-60-33" href="#__codelineno-60-33"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isSolution</span><span class="p">(</span><span class="n">state</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-60-34" name="__codelineno-60-34" href="#__codelineno-60-34"></a><span class="w">    </span><span class="c1">// Record solution</span>
+<a id="__codelineno-60-34" name="__codelineno-60-34" href="#__codelineno-60-34"></a><span class="w">    </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-60-35" name="__codelineno-60-35" href="#__codelineno-60-35"></a><span class="w">    </span><span class="n">recordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-60-36" name="__codelineno-60-36" href="#__codelineno-60-36"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-60-37" name="__codelineno-60-37" href="#__codelineno-60-37"></a><span class="w">  </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-60-37" name="__codelineno-60-37" href="#__codelineno-60-37"></a><span class="w">  </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-60-38" name="__codelineno-60-38" href="#__codelineno-60-38"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-60-39" name="__codelineno-60-39" href="#__codelineno-60-39"></a><span class="w">    </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-60-39" name="__codelineno-60-39" href="#__codelineno-60-39"></a><span class="w">    </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-60-40" name="__codelineno-60-40" href="#__codelineno-60-40"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isValid</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-60-41" name="__codelineno-60-41" href="#__codelineno-60-41"></a><span class="w">      </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-60-41" name="__codelineno-60-41" href="#__codelineno-60-41"></a><span class="w">      </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-60-42" name="__codelineno-60-42" href="#__codelineno-60-42"></a><span class="w">      </span><span class="n">makeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
-<a id="__codelineno-60-43" name="__codelineno-60-43" href="#__codelineno-60-43"></a><span class="w">      </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-60-43" name="__codelineno-60-43" href="#__codelineno-60-43"></a><span class="w">      </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-60-44" name="__codelineno-60-44" href="#__codelineno-60-44"></a><span class="w">      </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="p">[</span><span class="n">choice</span><span class="o">!</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">.</span><span class="n">right</span><span class="p">],</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-60-45" name="__codelineno-60-45" href="#__codelineno-60-45"></a><span class="w">      </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-60-45" name="__codelineno-60-45" href="#__codelineno-60-45"></a><span class="w">      </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-60-46" name="__codelineno-60-46" href="#__codelineno-60-46"></a><span class="w">      </span><span class="n">undoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-60-47" name="__codelineno-60-47" href="#__codelineno-60-47"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-60-48" name="__codelineno-60-48" href="#__codelineno-60-48"></a><span class="w">  </span><span class="p">}</span>
@@ -6311,12 +6398,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_template.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Check if the current state is a solution */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_template.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Check whether the current state is a solution */</span>
 <a id="__codelineno-61-2" name="__codelineno-61-2" href="#__codelineno-61-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">is_solution</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-3" name="__codelineno-61-3" href="#__codelineno-61-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">!</span><span class="n">state</span><span class="p">.</span><span class="n">is_empty</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">last</span><span class="p">().</span><span class="n">unwrap</span><span class="p">().</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">;</span>
 <a id="__codelineno-61-4" name="__codelineno-61-4" href="#__codelineno-61-4"></a><span class="p">}</span>
 <a id="__codelineno-61-5" name="__codelineno-61-5" href="#__codelineno-61-5"></a>
-<a id="__codelineno-61-6" name="__codelineno-61-6" href="#__codelineno-61-6"></a><span class="cm">/* Record solution */</span>
+<a id="__codelineno-61-6" name="__codelineno-61-6" href="#__codelineno-61-6"></a><span class="cm">/* Record the solution */</span>
 <a id="__codelineno-61-7" name="__codelineno-61-7" href="#__codelineno-61-7"></a><span class="k">fn</span><span class="w"> </span><span class="nf">record_solution</span><span class="p">(</span>
 <a id="__codelineno-61-8" name="__codelineno-61-8" href="#__codelineno-61-8"></a><span class="w">    </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-61-9" name="__codelineno-61-9" href="#__codelineno-61-9"></a><span class="w">    </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span>
@@ -6324,39 +6411,39 @@
 <a id="__codelineno-61-11" name="__codelineno-61-11" href="#__codelineno-61-11"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-61-12" name="__codelineno-61-12" href="#__codelineno-61-12"></a><span class="p">}</span>
 <a id="__codelineno-61-13" name="__codelineno-61-13" href="#__codelineno-61-13"></a>
-<a id="__codelineno-61-14" name="__codelineno-61-14" href="#__codelineno-61-14"></a><span class="cm">/* Check if the choice is valid under the current state */</span>
+<a id="__codelineno-61-14" name="__codelineno-61-14" href="#__codelineno-61-14"></a><span class="cm">/* Check whether the choice is valid in the current state */</span>
 <a id="__codelineno-61-15" name="__codelineno-61-15" href="#__codelineno-61-15"></a><span class="k">fn</span><span class="w"> </span><span class="nf">is_valid</span><span class="p">(</span><span class="n">_</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-16" name="__codelineno-61-16" href="#__codelineno-61-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">choice</span><span class="p">.</span><span class="n">is_some</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choice</span><span class="p">.</span><span class="n">unwrap</span><span class="p">().</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-61-17" name="__codelineno-61-17" href="#__codelineno-61-17"></a><span class="p">}</span>
 <a id="__codelineno-61-18" name="__codelineno-61-18" href="#__codelineno-61-18"></a>
-<a id="__codelineno-61-19" name="__codelineno-61-19" href="#__codelineno-61-19"></a><span class="cm">/* Update state */</span>
+<a id="__codelineno-61-19" name="__codelineno-61-19" href="#__codelineno-61-19"></a><span class="cm">/* Update the state */</span>
 <a id="__codelineno-61-20" name="__codelineno-61-20" href="#__codelineno-61-20"></a><span class="k">fn</span><span class="w"> </span><span class="nf">make_choice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">:</span><span class="w"> </span><span class="nc">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-21" name="__codelineno-61-21" href="#__codelineno-61-21"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
 <a id="__codelineno-61-22" name="__codelineno-61-22" href="#__codelineno-61-22"></a><span class="p">}</span>
 <a id="__codelineno-61-23" name="__codelineno-61-23" href="#__codelineno-61-23"></a>
-<a id="__codelineno-61-24" name="__codelineno-61-24" href="#__codelineno-61-24"></a><span class="cm">/* Restore state */</span>
+<a id="__codelineno-61-24" name="__codelineno-61-24" href="#__codelineno-61-24"></a><span class="cm">/* Restore the state */</span>
 <a id="__codelineno-61-25" name="__codelineno-61-25" href="#__codelineno-61-25"></a><span class="k">fn</span><span class="w"> </span><span class="nf">undo_choice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">_</span><span class="p">:</span><span class="w"> </span><span class="nc">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-26" name="__codelineno-61-26" href="#__codelineno-61-26"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span>
 <a id="__codelineno-61-27" name="__codelineno-61-27" href="#__codelineno-61-27"></a><span class="p">}</span>
 <a id="__codelineno-61-28" name="__codelineno-61-28" href="#__codelineno-61-28"></a>
-<a id="__codelineno-61-29" name="__codelineno-61-29" href="#__codelineno-61-29"></a><span class="cm">/* Backtracking algorithm: Example 3 */</span>
+<a id="__codelineno-61-29" name="__codelineno-61-29" href="#__codelineno-61-29"></a><span class="cm">/* Backtracking: example 3 */</span>
 <a id="__codelineno-61-30" name="__codelineno-61-30" href="#__codelineno-61-30"></a><span class="k">fn</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-61-31" name="__codelineno-61-31" href="#__codelineno-61-31"></a><span class="w">    </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-61-32" name="__codelineno-61-32" href="#__codelineno-61-32"></a><span class="w">    </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-61-33" name="__codelineno-61-33" href="#__codelineno-61-33"></a><span class="w">    </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-61-34" name="__codelineno-61-34" href="#__codelineno-61-34"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-61-35" name="__codelineno-61-35" href="#__codelineno-61-35"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-61-35" name="__codelineno-61-35" href="#__codelineno-61-35"></a><span class="w">    </span><span class="c1">// Check whether this is a solution</span>
 <a id="__codelineno-61-36" name="__codelineno-61-36" href="#__codelineno-61-36"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">is_solution</span><span class="p">(</span><span class="n">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-61-37" name="__codelineno-61-37" href="#__codelineno-61-37"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-61-37" name="__codelineno-61-37" href="#__codelineno-61-37"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-61-38" name="__codelineno-61-38" href="#__codelineno-61-38"></a><span class="w">        </span><span class="n">record_solution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-61-39" name="__codelineno-61-39" href="#__codelineno-61-39"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-61-40" name="__codelineno-61-40" href="#__codelineno-61-40"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-61-40" name="__codelineno-61-40" href="#__codelineno-61-40"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-61-41" name="__codelineno-61-41" href="#__codelineno-61-41"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="o">&amp;</span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">iter</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-61-42" name="__codelineno-61-42" href="#__codelineno-61-42"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-61-42" name="__codelineno-61-42" href="#__codelineno-61-42"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-61-43" name="__codelineno-61-43" href="#__codelineno-61-43"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">is_valid</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-61-44" name="__codelineno-61-44" href="#__codelineno-61-44"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-61-44" name="__codelineno-61-44" href="#__codelineno-61-44"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-61-45" name="__codelineno-61-45" href="#__codelineno-61-45"></a><span class="w">            </span><span class="n">make_choice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">.</span><span class="n">unwrap</span><span class="p">().</span><span class="n">clone</span><span class="p">());</span>
-<a id="__codelineno-61-46" name="__codelineno-61-46" href="#__codelineno-61-46"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-61-46" name="__codelineno-61-46" href="#__codelineno-61-46"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-61-47" name="__codelineno-61-47" href="#__codelineno-61-47"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span>
 <a id="__codelineno-61-48" name="__codelineno-61-48" href="#__codelineno-61-48"></a><span class="w">                </span><span class="n">state</span><span class="p">,</span>
 <a id="__codelineno-61-49" name="__codelineno-61-49" href="#__codelineno-61-49"></a><span class="w">                </span><span class="o">&amp;</span><span class="fm">vec!</span><span class="p">[</span>
@@ -6365,7 +6452,7 @@
 <a id="__codelineno-61-52" name="__codelineno-61-52" href="#__codelineno-61-52"></a><span class="w">                </span><span class="p">],</span>
 <a id="__codelineno-61-53" name="__codelineno-61-53" href="#__codelineno-61-53"></a><span class="w">                </span><span class="n">res</span><span class="p">,</span>
 <a id="__codelineno-61-54" name="__codelineno-61-54" href="#__codelineno-61-54"></a><span class="w">            </span><span class="p">);</span>
-<a id="__codelineno-61-55" name="__codelineno-61-55" href="#__codelineno-61-55"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-61-55" name="__codelineno-61-55" href="#__codelineno-61-55"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-61-56" name="__codelineno-61-56" href="#__codelineno-61-56"></a><span class="w">            </span><span class="n">undo_choice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">.</span><span class="n">unwrap</span><span class="p">().</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-61-57" name="__codelineno-61-57" href="#__codelineno-61-57"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-61-58" name="__codelineno-61-58" href="#__codelineno-61-58"></a><span class="w">    </span><span class="p">}</span>
@@ -6373,12 +6460,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_template.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Check if the current state is a solution */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_template.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Check whether the current state is a solution */</span>
 <a id="__codelineno-62-2" name="__codelineno-62-2" href="#__codelineno-62-2"></a><span class="kt">bool</span><span class="w"> </span><span class="nf">isSolution</span><span class="p">(</span><span class="kt">void</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-3" name="__codelineno-62-3" href="#__codelineno-62-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">pathSize</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">path</span><span class="p">[</span><span class="n">pathSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span><span class="p">;</span>
 <a id="__codelineno-62-4" name="__codelineno-62-4" href="#__codelineno-62-4"></a><span class="p">}</span>
 <a id="__codelineno-62-5" name="__codelineno-62-5" href="#__codelineno-62-5"></a>
-<a id="__codelineno-62-6" name="__codelineno-62-6" href="#__codelineno-62-6"></a><span class="cm">/* Record solution */</span>
+<a id="__codelineno-62-6" name="__codelineno-62-6" href="#__codelineno-62-6"></a><span class="cm">/* Record the solution */</span>
 <a id="__codelineno-62-7" name="__codelineno-62-7" href="#__codelineno-62-7"></a><span class="kt">void</span><span class="w"> </span><span class="nf">recordSolution</span><span class="p">(</span><span class="kt">void</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-8" name="__codelineno-62-8" href="#__codelineno-62-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">pathSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-9" name="__codelineno-62-9" href="#__codelineno-62-9"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">resSize</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">path</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
@@ -6386,39 +6473,39 @@
 <a id="__codelineno-62-11" name="__codelineno-62-11" href="#__codelineno-62-11"></a><span class="w">    </span><span class="n">resSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-62-12" name="__codelineno-62-12" href="#__codelineno-62-12"></a><span class="p">}</span>
 <a id="__codelineno-62-13" name="__codelineno-62-13" href="#__codelineno-62-13"></a>
-<a id="__codelineno-62-14" name="__codelineno-62-14" href="#__codelineno-62-14"></a><span class="cm">/* Check if the choice is valid under the current state */</span>
+<a id="__codelineno-62-14" name="__codelineno-62-14" href="#__codelineno-62-14"></a><span class="cm">/* Check whether the choice is valid in the current state */</span>
 <a id="__codelineno-62-15" name="__codelineno-62-15" href="#__codelineno-62-15"></a><span class="kt">bool</span><span class="w"> </span><span class="nf">isValid</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-16" name="__codelineno-62-16" href="#__codelineno-62-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choice</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-62-17" name="__codelineno-62-17" href="#__codelineno-62-17"></a><span class="p">}</span>
 <a id="__codelineno-62-18" name="__codelineno-62-18" href="#__codelineno-62-18"></a>
-<a id="__codelineno-62-19" name="__codelineno-62-19" href="#__codelineno-62-19"></a><span class="cm">/* Update state */</span>
+<a id="__codelineno-62-19" name="__codelineno-62-19" href="#__codelineno-62-19"></a><span class="cm">/* Update the state */</span>
 <a id="__codelineno-62-20" name="__codelineno-62-20" href="#__codelineno-62-20"></a><span class="kt">void</span><span class="w"> </span><span class="nf">makeChoice</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-21" name="__codelineno-62-21" href="#__codelineno-62-21"></a><span class="w">    </span><span class="n">path</span><span class="p">[</span><span class="n">pathSize</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choice</span><span class="p">;</span>
 <a id="__codelineno-62-22" name="__codelineno-62-22" href="#__codelineno-62-22"></a><span class="p">}</span>
 <a id="__codelineno-62-23" name="__codelineno-62-23" href="#__codelineno-62-23"></a>
-<a id="__codelineno-62-24" name="__codelineno-62-24" href="#__codelineno-62-24"></a><span class="cm">/* Restore state */</span>
+<a id="__codelineno-62-24" name="__codelineno-62-24" href="#__codelineno-62-24"></a><span class="cm">/* Restore the state */</span>
 <a id="__codelineno-62-25" name="__codelineno-62-25" href="#__codelineno-62-25"></a><span class="kt">void</span><span class="w"> </span><span class="nf">undoChoice</span><span class="p">(</span><span class="kt">void</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-26" name="__codelineno-62-26" href="#__codelineno-62-26"></a><span class="w">    </span><span class="n">pathSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-62-27" name="__codelineno-62-27" href="#__codelineno-62-27"></a><span class="p">}</span>
 <a id="__codelineno-62-28" name="__codelineno-62-28" href="#__codelineno-62-28"></a>
-<a id="__codelineno-62-29" name="__codelineno-62-29" href="#__codelineno-62-29"></a><span class="cm">/* Backtracking algorithm: Example 3 */</span>
+<a id="__codelineno-62-29" name="__codelineno-62-29" href="#__codelineno-62-29"></a><span class="cm">/* Backtracking: example 3 */</span>
 <a id="__codelineno-62-30" name="__codelineno-62-30" href="#__codelineno-62-30"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">choices</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-62-31" name="__codelineno-62-31" href="#__codelineno-62-31"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-62-31" name="__codelineno-62-31" href="#__codelineno-62-31"></a><span class="w">    </span><span class="c1">// Check whether this is a solution</span>
 <a id="__codelineno-62-32" name="__codelineno-62-32" href="#__codelineno-62-32"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isSolution</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-62-33" name="__codelineno-62-33" href="#__codelineno-62-33"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-62-33" name="__codelineno-62-33" href="#__codelineno-62-33"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-62-34" name="__codelineno-62-34" href="#__codelineno-62-34"></a><span class="w">        </span><span class="n">recordSolution</span><span class="p">();</span>
 <a id="__codelineno-62-35" name="__codelineno-62-35" href="#__codelineno-62-35"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-62-36" name="__codelineno-62-36" href="#__codelineno-62-36"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-62-36" name="__codelineno-62-36" href="#__codelineno-62-36"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-62-37" name="__codelineno-62-37" href="#__codelineno-62-37"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-38" name="__codelineno-62-38" href="#__codelineno-62-38"></a><span class="w">        </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
-<a id="__codelineno-62-39" name="__codelineno-62-39" href="#__codelineno-62-39"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-62-39" name="__codelineno-62-39" href="#__codelineno-62-39"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-62-40" name="__codelineno-62-40" href="#__codelineno-62-40"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isValid</span><span class="p">(</span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-62-41" name="__codelineno-62-41" href="#__codelineno-62-41"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-62-41" name="__codelineno-62-41" href="#__codelineno-62-41"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-62-42" name="__codelineno-62-42" href="#__codelineno-62-42"></a><span class="w">            </span><span class="n">makeChoice</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
-<a id="__codelineno-62-43" name="__codelineno-62-43" href="#__codelineno-62-43"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-62-43" name="__codelineno-62-43" href="#__codelineno-62-43"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-62-44" name="__codelineno-62-44" href="#__codelineno-62-44"></a><span class="w">            </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">nextChoices</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="n">choice</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">};</span>
 <a id="__codelineno-62-45" name="__codelineno-62-45" href="#__codelineno-62-45"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">nextChoices</span><span class="p">);</span>
-<a id="__codelineno-62-46" name="__codelineno-62-46" href="#__codelineno-62-46"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-62-46" name="__codelineno-62-46" href="#__codelineno-62-46"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-62-47" name="__codelineno-62-47" href="#__codelineno-62-47"></a><span class="w">            </span><span class="n">undoChoice</span><span class="p">();</span>
 <a id="__codelineno-62-48" name="__codelineno-62-48" href="#__codelineno-62-48"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-62-49" name="__codelineno-62-49" href="#__codelineno-62-49"></a><span class="w">    </span><span class="p">}</span>
@@ -6426,51 +6513,51 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_template.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Check if the current state is a solution */</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_template.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Check whether the current state is a solution */</span>
 <a id="__codelineno-63-2" name="__codelineno-63-2" href="#__codelineno-63-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">isSolution</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">TreeNode?</span><span class="o">&gt;</span><span class="p">):</span><span class="w"> </span><span class="kt">Boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-3" name="__codelineno-63-3" href="#__codelineno-63-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="na">isNotEmpty</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">state</span><span class="o">[</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]?.</span><span class="na">_val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">7</span>
 <a id="__codelineno-63-4" name="__codelineno-63-4" href="#__codelineno-63-4"></a><span class="p">}</span>
 <a id="__codelineno-63-5" name="__codelineno-63-5" href="#__codelineno-63-5"></a>
-<a id="__codelineno-63-6" name="__codelineno-63-6" href="#__codelineno-63-6"></a><span class="cm">/* Record solution */</span>
+<a id="__codelineno-63-6" name="__codelineno-63-6" href="#__codelineno-63-6"></a><span class="cm">/* Record the solution */</span>
 <a id="__codelineno-63-7" name="__codelineno-63-7" href="#__codelineno-63-7"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">recordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">TreeNode?</span><span class="o">&gt;?</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">TreeNode?</span><span class="o">&gt;?&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-8" name="__codelineno-63-8" href="#__codelineno-63-8"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">state</span><span class="o">!!</span><span class="p">.</span><span class="na">toMutableList</span><span class="p">())</span>
 <a id="__codelineno-63-9" name="__codelineno-63-9" href="#__codelineno-63-9"></a><span class="p">}</span>
 <a id="__codelineno-63-10" name="__codelineno-63-10" href="#__codelineno-63-10"></a>
-<a id="__codelineno-63-11" name="__codelineno-63-11" href="#__codelineno-63-11"></a><span class="cm">/* Check if the choice is valid under the current state */</span>
+<a id="__codelineno-63-11" name="__codelineno-63-11" href="#__codelineno-63-11"></a><span class="cm">/* Check whether the choice is valid in the current state */</span>
 <a id="__codelineno-63-12" name="__codelineno-63-12" href="#__codelineno-63-12"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">isValid</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">TreeNode?</span><span class="o">&gt;?</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">):</span><span class="w"> </span><span class="kt">Boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-13" name="__codelineno-63-13" href="#__codelineno-63-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choice</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="m">3</span>
 <a id="__codelineno-63-14" name="__codelineno-63-14" href="#__codelineno-63-14"></a><span class="p">}</span>
 <a id="__codelineno-63-15" name="__codelineno-63-15" href="#__codelineno-63-15"></a>
-<a id="__codelineno-63-16" name="__codelineno-63-16" href="#__codelineno-63-16"></a><span class="cm">/* Update state */</span>
+<a id="__codelineno-63-16" name="__codelineno-63-16" href="#__codelineno-63-16"></a><span class="cm">/* Update the state */</span>
 <a id="__codelineno-63-17" name="__codelineno-63-17" href="#__codelineno-63-17"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">makeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">TreeNode?</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-18" name="__codelineno-63-18" href="#__codelineno-63-18"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-63-19" name="__codelineno-63-19" href="#__codelineno-63-19"></a><span class="p">}</span>
 <a id="__codelineno-63-20" name="__codelineno-63-20" href="#__codelineno-63-20"></a>
-<a id="__codelineno-63-21" name="__codelineno-63-21" href="#__codelineno-63-21"></a><span class="cm">/* Restore state */</span>
+<a id="__codelineno-63-21" name="__codelineno-63-21" href="#__codelineno-63-21"></a><span class="cm">/* Restore the state */</span>
 <a id="__codelineno-63-22" name="__codelineno-63-22" href="#__codelineno-63-22"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">undoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">TreeNode?</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-23" name="__codelineno-63-23" href="#__codelineno-63-23"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="na">removeLast</span><span class="p">()</span>
 <a id="__codelineno-63-24" name="__codelineno-63-24" href="#__codelineno-63-24"></a><span class="p">}</span>
 <a id="__codelineno-63-25" name="__codelineno-63-25" href="#__codelineno-63-25"></a>
-<a id="__codelineno-63-26" name="__codelineno-63-26" href="#__codelineno-63-26"></a><span class="cm">/* Backtracking algorithm: Example 3 */</span>
+<a id="__codelineno-63-26" name="__codelineno-63-26" href="#__codelineno-63-26"></a><span class="cm">/* Backtracking: example 3 */</span>
 <a id="__codelineno-63-27" name="__codelineno-63-27" href="#__codelineno-63-27"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-63-28" name="__codelineno-63-28" href="#__codelineno-63-28"></a><span class="w">    </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">TreeNode?</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-63-29" name="__codelineno-63-29" href="#__codelineno-63-29"></a><span class="w">    </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">TreeNode?</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-63-30" name="__codelineno-63-30" href="#__codelineno-63-30"></a><span class="w">    </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">TreeNode?</span><span class="o">&gt;?&gt;</span>
 <a id="__codelineno-63-31" name="__codelineno-63-31" href="#__codelineno-63-31"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-63-32" name="__codelineno-63-32" href="#__codelineno-63-32"></a><span class="w">    </span><span class="c1">// Check if it is a solution</span>
+<a id="__codelineno-63-32" name="__codelineno-63-32" href="#__codelineno-63-32"></a><span class="w">    </span><span class="c1">// Check whether this is a solution</span>
 <a id="__codelineno-63-33" name="__codelineno-63-33" href="#__codelineno-63-33"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isSolution</span><span class="p">(</span><span class="n">state</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-63-34" name="__codelineno-63-34" href="#__codelineno-63-34"></a><span class="w">        </span><span class="c1">// Record solution</span>
+<a id="__codelineno-63-34" name="__codelineno-63-34" href="#__codelineno-63-34"></a><span class="w">        </span><span class="c1">// Record the solution</span>
 <a id="__codelineno-63-35" name="__codelineno-63-35" href="#__codelineno-63-35"></a><span class="w">        </span><span class="n">recordSolution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-63-36" name="__codelineno-63-36" href="#__codelineno-63-36"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-63-37" name="__codelineno-63-37" href="#__codelineno-63-37"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-63-37" name="__codelineno-63-37" href="#__codelineno-63-37"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-63-38" name="__codelineno-63-38" href="#__codelineno-63-38"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-63-39" name="__codelineno-63-39" href="#__codelineno-63-39"></a><span class="w">        </span><span class="c1">// Pruning: check if the choice is valid</span>
+<a id="__codelineno-63-39" name="__codelineno-63-39" href="#__codelineno-63-39"></a><span class="w">        </span><span class="c1">// Prune: check whether the choice is valid</span>
 <a id="__codelineno-63-40" name="__codelineno-63-40" href="#__codelineno-63-40"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isValid</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-63-41" name="__codelineno-63-41" href="#__codelineno-63-41"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-63-41" name="__codelineno-63-41" href="#__codelineno-63-41"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-63-42" name="__codelineno-63-42" href="#__codelineno-63-42"></a><span class="w">            </span><span class="n">makeChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
-<a id="__codelineno-63-43" name="__codelineno-63-43" href="#__codelineno-63-43"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-63-43" name="__codelineno-63-43" href="#__codelineno-63-43"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-63-44" name="__codelineno-63-44" href="#__codelineno-63-44"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">mutableListOf</span><span class="p">(</span><span class="n">choice</span><span class="o">!!</span><span class="p">.</span><span class="na">left</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">.</span><span class="na">right</span><span class="p">),</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-63-45" name="__codelineno-63-45" href="#__codelineno-63-45"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-63-45" name="__codelineno-63-45" href="#__codelineno-63-45"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-63-46" name="__codelineno-63-46" href="#__codelineno-63-46"></a><span class="w">            </span><span class="n">undoChoice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-63-47" name="__codelineno-63-47" href="#__codelineno-63-47"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-63-48" name="__codelineno-63-48" href="#__codelineno-63-48"></a><span class="w">    </span><span class="p">}</span>
@@ -6478,45 +6565,45 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">preorder_traversal_iii_template.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Check if current state is solution ###</span>
+<div class="highlight"><span class="filename">preorder_traversal_iii_template.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Check whether the current state is a solution ###</span>
 <a id="__codelineno-64-2" name="__codelineno-64-2" href="#__codelineno-64-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">is_solution?</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
 <a id="__codelineno-64-3" name="__codelineno-64-3" href="#__codelineno-64-3"></a><span class="w">  </span><span class="o">!</span><span class="n">state</span><span class="o">.</span><span class="n">empty?</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">state</span><span class="o">.</span><span class="n">last</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">7</span>
 <a id="__codelineno-64-4" name="__codelineno-64-4" href="#__codelineno-64-4"></a><span class="k">end</span>
 <a id="__codelineno-64-5" name="__codelineno-64-5" href="#__codelineno-64-5"></a>
-<a id="__codelineno-64-6" name="__codelineno-64-6" href="#__codelineno-64-6"></a><span class="c1">### Record solution ###</span>
+<a id="__codelineno-64-6" name="__codelineno-64-6" href="#__codelineno-64-6"></a><span class="c1">### Record the solution ###</span>
 <a id="__codelineno-64-7" name="__codelineno-64-7" href="#__codelineno-64-7"></a><span class="k">def</span><span class="w"> </span><span class="nf">record_solution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-64-8" name="__codelineno-64-8" href="#__codelineno-64-8"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">state</span><span class="o">.</span><span class="n">dup</span>
 <a id="__codelineno-64-9" name="__codelineno-64-9" href="#__codelineno-64-9"></a><span class="k">end</span>
 <a id="__codelineno-64-10" name="__codelineno-64-10" href="#__codelineno-64-10"></a>
-<a id="__codelineno-64-11" name="__codelineno-64-11" href="#__codelineno-64-11"></a><span class="c1">### Check if choice is valid in current state ###</span>
+<a id="__codelineno-64-11" name="__codelineno-64-11" href="#__codelineno-64-11"></a><span class="c1">### Check whether the choice is valid in the current state ###</span>
 <a id="__codelineno-64-12" name="__codelineno-64-12" href="#__codelineno-64-12"></a><span class="k">def</span><span class="w"> </span><span class="nf">is_valid?</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-64-13" name="__codelineno-64-13" href="#__codelineno-64-13"></a><span class="w">  </span><span class="n">choice</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choice</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">3</span>
 <a id="__codelineno-64-14" name="__codelineno-64-14" href="#__codelineno-64-14"></a><span class="k">end</span>
 <a id="__codelineno-64-15" name="__codelineno-64-15" href="#__codelineno-64-15"></a>
-<a id="__codelineno-64-16" name="__codelineno-64-16" href="#__codelineno-64-16"></a><span class="c1">### Update state ###</span>
+<a id="__codelineno-64-16" name="__codelineno-64-16" href="#__codelineno-64-16"></a><span class="c1">### Update the state ###</span>
 <a id="__codelineno-64-17" name="__codelineno-64-17" href="#__codelineno-64-17"></a><span class="k">def</span><span class="w"> </span><span class="nf">make_choice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-64-18" name="__codelineno-64-18" href="#__codelineno-64-18"></a><span class="w">  </span><span class="n">state</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">choice</span>
 <a id="__codelineno-64-19" name="__codelineno-64-19" href="#__codelineno-64-19"></a><span class="k">end</span>
 <a id="__codelineno-64-20" name="__codelineno-64-20" href="#__codelineno-64-20"></a>
-<a id="__codelineno-64-21" name="__codelineno-64-21" href="#__codelineno-64-21"></a><span class="c1">### Restore state ###</span>
+<a id="__codelineno-64-21" name="__codelineno-64-21" href="#__codelineno-64-21"></a><span class="c1">### Restore the state ###</span>
 <a id="__codelineno-64-22" name="__codelineno-64-22" href="#__codelineno-64-22"></a><span class="k">def</span><span class="w"> </span><span class="nf">undo_choice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-64-23" name="__codelineno-64-23" href="#__codelineno-64-23"></a><span class="w">  </span><span class="n">state</span><span class="o">.</span><span class="n">pop</span>
 <a id="__codelineno-64-24" name="__codelineno-64-24" href="#__codelineno-64-24"></a><span class="k">end</span>
 <a id="__codelineno-64-25" name="__codelineno-64-25" href="#__codelineno-64-25"></a>
 <a id="__codelineno-64-26" name="__codelineno-64-26" href="#__codelineno-64-26"></a><span class="c1">### Backtracking: example 3 ###</span>
 <a id="__codelineno-64-27" name="__codelineno-64-27" href="#__codelineno-64-27"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-64-28" name="__codelineno-64-28" href="#__codelineno-64-28"></a><span class="w">  </span><span class="c1"># Check if it is a solution</span>
+<a id="__codelineno-64-28" name="__codelineno-64-28" href="#__codelineno-64-28"></a><span class="w">  </span><span class="c1"># Check whether this is a solution</span>
 <a id="__codelineno-64-29" name="__codelineno-64-29" href="#__codelineno-64-29"></a><span class="w">  </span><span class="n">record_solution</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">is_solution?</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
 <a id="__codelineno-64-30" name="__codelineno-64-30" href="#__codelineno-64-30"></a>
-<a id="__codelineno-64-31" name="__codelineno-64-31" href="#__codelineno-64-31"></a><span class="w">  </span><span class="c1"># Traverse all choices</span>
+<a id="__codelineno-64-31" name="__codelineno-64-31" href="#__codelineno-64-31"></a><span class="w">  </span><span class="c1"># Iterate over all choices</span>
 <a id="__codelineno-64-32" name="__codelineno-64-32" href="#__codelineno-64-32"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span>
-<a id="__codelineno-64-33" name="__codelineno-64-33" href="#__codelineno-64-33"></a><span class="w">    </span><span class="c1"># Pruning: check if the choice is valid</span>
+<a id="__codelineno-64-33" name="__codelineno-64-33" href="#__codelineno-64-33"></a><span class="w">    </span><span class="c1"># Prune: check whether the choice is valid</span>
 <a id="__codelineno-64-34" name="__codelineno-64-34" href="#__codelineno-64-34"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">is_valid?</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
-<a id="__codelineno-64-35" name="__codelineno-64-35" href="#__codelineno-64-35"></a><span class="w">      </span><span class="c1"># Attempt: make choice, update state</span>
+<a id="__codelineno-64-35" name="__codelineno-64-35" href="#__codelineno-64-35"></a><span class="w">      </span><span class="c1"># Try: make a choice and update the state</span>
 <a id="__codelineno-64-36" name="__codelineno-64-36" href="#__codelineno-64-36"></a><span class="w">      </span><span class="n">make_choice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
-<a id="__codelineno-64-37" name="__codelineno-64-37" href="#__codelineno-64-37"></a><span class="w">      </span><span class="c1"># Proceed to the next round of selection</span>
+<a id="__codelineno-64-37" name="__codelineno-64-37" href="#__codelineno-64-37"></a><span class="w">      </span><span class="c1"># Proceed to the next choice</span>
 <a id="__codelineno-64-38" name="__codelineno-64-38" href="#__codelineno-64-38"></a><span class="w">      </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="o">[</span><span class="n">choice</span><span class="o">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="o">.</span><span class="n">right</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-64-39" name="__codelineno-64-39" href="#__codelineno-64-39"></a><span class="w">      </span><span class="c1"># Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-64-39" name="__codelineno-64-39" href="#__codelineno-64-39"></a><span class="w">      </span><span class="c1"># Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-64-40" name="__codelineno-64-40" href="#__codelineno-64-40"></a><span class="w">      </span><span class="n">undo_choice</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-64-41" name="__codelineno-64-41" href="#__codelineno-64-41"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-64-42" name="__codelineno-64-42" href="#__codelineno-64-42"></a><span class="w">  </span><span class="k">end</span>
@@ -6525,14 +6612,19 @@
 </div>
 </div>
 </div>
-<p>As per the problem statement, we should continue searching after finding a node with value <span class="arithmatex">\(7\)</span>. <strong>Therefore, we need to remove the <code>return</code> statement after recording the solution</strong>. The following figure compares the search process with and without the <code>return</code> statement.</p>
-<p><img alt="Comparison of search process with and without return statement" class="animation-figure" src="../backtracking_algorithm.assets/backtrack_remove_return_or_not.png" /></p>
-<p align="center"> Figure 13-4 &nbsp; Comparison of search process with and without return statement </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20left%20child%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20right%20child%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%3A%20recursion%22%22%22%0A%20%20%20%20%23%20Return%20None%20if%20the%20index%20is%20out%20of%20bounds%20or%20the%20corresponding%20element%20is%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20Create%20the%20current%20node%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20Recursively%20build%20the%20left%20and%20right%20subtrees%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20is_solution%28state%3A%20list%5BTreeNode%5D%29%20-%3E%20bool%3A%0A%20%20%20%20%22%22%22Check%20whether%20the%20current%20state%20is%20a%20solution%22%22%22%0A%20%20%20%20return%20state%20and%20state%5B-1%5D.val%20%3D%3D%207%0A%0Adef%20record_solution%28state%3A%20list%5BTreeNode%5D,%20res%3A%20list%5Blist%5BTreeNode%5D%5D%29%3A%0A%20%20%20%20%22%22%22Record%20the%20solution%22%22%22%0A%20%20%20%20res.append%28list%28state%29%29%0A%0Adef%20is_valid%28state%3A%20list%5BTreeNode%5D,%20choice%3A%20TreeNode%29%20-%3E%20bool%3A%0A%20%20%20%20%22%22%22Check%20whether%20the%20choice%20is%20valid%20in%20the%20current%20state%22%22%22%0A%20%20%20%20return%20choice%20is%20not%20None%20and%20choice.val%20%21%3D%203%0A%0Adef%20make_choice%28state%3A%20list%5BTreeNode%5D,%20choice%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22Update%20the%20state%22%22%22%0A%20%20%20%20state.append%28choice%29%0A%0Adef%20undo_choice%28state%3A%20list%5BTreeNode%5D,%20choice%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22Restore%20the%20state%22%22%22%0A%20%20%20%20state.pop%28%29%0A%0Adef%20backtrack%28%0A%20%20%20%20state%3A%20list%5BTreeNode%5D,%20choices%3A%20list%5BTreeNode%5D,%20res%3A%20list%5Blist%5BTreeNode%5D%5D%0A%29%3A%0A%20%20%20%20%22%22%22Backtracking%3A%20example%203%22%22%22%0A%20%20%20%20%23%20Check%20whether%20this%20is%20a%20solution%0A%20%20%20%20if%20is_solution%28state%29%3A%0A%20%20%20%20%20%20%20%20%23%20Record%20the%20solution%0A%20%20%20%20%20%20%20%20record_solution%28state,%20res%29%0A%20%20%20%20%23%20Iterate%20over%20all%20choices%0A%20%20%20%20for%20choice%20in%20choices%3A%0A%20%20%20%20%20%20%20%20%23%20Prune%3A%20check%20whether%20the%20choice%20is%20valid%0A%20%20%20%20%20%20%20%20if%20is_valid%28state,%20choice%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Try%3A%20make%20a%20choice%20and%20update%20the%20state%0A%20%20%20%20%20%20%20%20%20%20%20%20make_choice%28state,%20choice%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Proceed%20to%20the%20next%20choice%0A%20%20%20%20%20%20%20%20%20%20%20%20backtrack%28state,%20%5Bchoice.left,%20choice.right%5D,%20res%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Backtrack%3A%20undo%20the%20choice%20and%20restore%20the%20previous%20state%0A%20%20%20%20%20%20%20%20%20%20%20%20undo_choice%28state,%20choice%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20Backtracking%20algorithm%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20backtrack%28state%3D%5B%5D,%20choices%3D%5Broot%5D,%20res%3Dres%29%0A%20%20%20%20print%28%22%5CnOutput%20all%20paths%22%29%0A%20%20%20%20for%20path%20in%20res%3A%0A%20%20%20%20%20%20%20%20print%28%5Bnode.val%20for%20node%20in%20path%5D%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=138&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20left%20child%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20right%20child%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%3A%20recursion%22%22%22%0A%20%20%20%20%23%20Return%20None%20if%20the%20index%20is%20out%20of%20bounds%20or%20the%20corresponding%20element%20is%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20Create%20the%20current%20node%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20Recursively%20build%20the%20left%20and%20right%20subtrees%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20is_solution%28state%3A%20list%5BTreeNode%5D%29%20-%3E%20bool%3A%0A%20%20%20%20%22%22%22Check%20whether%20the%20current%20state%20is%20a%20solution%22%22%22%0A%20%20%20%20return%20state%20and%20state%5B-1%5D.val%20%3D%3D%207%0A%0Adef%20record_solution%28state%3A%20list%5BTreeNode%5D,%20res%3A%20list%5Blist%5BTreeNode%5D%5D%29%3A%0A%20%20%20%20%22%22%22Record%20the%20solution%22%22%22%0A%20%20%20%20res.append%28list%28state%29%29%0A%0Adef%20is_valid%28state%3A%20list%5BTreeNode%5D,%20choice%3A%20TreeNode%29%20-%3E%20bool%3A%0A%20%20%20%20%22%22%22Check%20whether%20the%20choice%20is%20valid%20in%20the%20current%20state%22%22%22%0A%20%20%20%20return%20choice%20is%20not%20None%20and%20choice.val%20%21%3D%203%0A%0Adef%20make_choice%28state%3A%20list%5BTreeNode%5D,%20choice%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22Update%20the%20state%22%22%22%0A%20%20%20%20state.append%28choice%29%0A%0Adef%20undo_choice%28state%3A%20list%5BTreeNode%5D,%20choice%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22Restore%20the%20state%22%22%22%0A%20%20%20%20state.pop%28%29%0A%0Adef%20backtrack%28%0A%20%20%20%20state%3A%20list%5BTreeNode%5D,%20choices%3A%20list%5BTreeNode%5D,%20res%3A%20list%5Blist%5BTreeNode%5D%5D%0A%29%3A%0A%20%20%20%20%22%22%22Backtracking%3A%20example%203%22%22%22%0A%20%20%20%20%23%20Check%20whether%20this%20is%20a%20solution%0A%20%20%20%20if%20is_solution%28state%29%3A%0A%20%20%20%20%20%20%20%20%23%20Record%20the%20solution%0A%20%20%20%20%20%20%20%20record_solution%28state,%20res%29%0A%20%20%20%20%23%20Iterate%20over%20all%20choices%0A%20%20%20%20for%20choice%20in%20choices%3A%0A%20%20%20%20%20%20%20%20%23%20Prune%3A%20check%20whether%20the%20choice%20is%20valid%0A%20%20%20%20%20%20%20%20if%20is_valid%28state,%20choice%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Try%3A%20make%20a%20choice%20and%20update%20the%20state%0A%20%20%20%20%20%20%20%20%20%20%20%20make_choice%28state,%20choice%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Proceed%20to%20the%20next%20choice%0A%20%20%20%20%20%20%20%20%20%20%20%20backtrack%28state,%20%5Bchoice.left,%20choice.right%5D,%20res%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Backtrack%3A%20undo%20the%20choice%20and%20restore%20the%20previous%20state%0A%20%20%20%20%20%20%20%20%20%20%20%20undo_choice%28state,%20choice%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20Backtracking%20algorithm%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20backtrack%28state%3D%5B%5D,%20choices%3D%5Broot%5D,%20res%3Dres%29%0A%20%20%20%20print%28%22%5CnOutput%20all%20paths%22%29%0A%20%20%20%20for%20path%20in%20res%3A%0A%20%20%20%20%20%20%20%20print%28%5Bnode.val%20for%20node%20in%20path%5D%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=138&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>The problem requires us to keep searching after finding a node with value <span class="arithmatex">\(7\)</span>, <strong>so remove the <code>return</code> after recording a solution</strong>. Figure 13-4 compares the searches with and without that <code>return</code>.</p>
+<p><img alt="Comparing searches with and without return" class="animation-figure" src="../backtracking_algorithm.assets/backtrack_remove_return_or_not.png" /></p>
+<p align="center"> Figure 13-4 &nbsp; Comparing searches with and without return </p>
 
-<p>Compared to code based on preorder traversal, code based on the backtracking algorithm framework appears more verbose, but is more general. In fact, <strong>many backtracking problems can be solved within this framework</strong>. We only need to define <code>state</code> and <code>choices</code> for the specific problem and implement each method in the framework.</p>
-<h2 id="1314-common-terminology">13.1.4 &nbsp; Common Terminology<a class="headerlink" href="#1314-common-terminology" title="Permanent link">&para;</a></h2>
-<p>To analyze algorithmic problems more clearly, we summarize the meanings of common terminology used in backtracking algorithms and provide corresponding examples from Example 3, as shown in the following table.</p>
-<p align="center"> Table 13-1 &nbsp; Common Backtracking Algorithm Terminology </p>
+<p>The template-based implementation is more verbose than the preorder version, but more general. <strong>Many backtracking problems can be solved within this template</strong>: define <code>state</code> and <code>choices</code> for the problem, then implement each helper method.</p>
+<h2 id="1314-common-terminology">13.1.4 &nbsp; Common terminology<a class="headerlink" href="#1314-common-terminology" title="Permanent link">&para;</a></h2>
+<p>Table 13-1 summarizes common backtracking terms and illustrates them with Example 3, helping us analyze problems more clearly.</p>
+<p align="center"> Table 13-1 &nbsp; Common backtracking terminology </p>
 
 <div class="center-table">
 <table>
@@ -6545,79 +6637,79 @@
 </thead>
 <tbody>
 <tr>
-<td>Solution (solution)</td>
-<td>A solution is an answer that satisfies the specific conditions of a problem; there may be one or more solutions</td>
-<td>All paths from root to nodes with value <span class="arithmatex">\(7\)</span> that satisfy the constraint</td>
+<td>Solution</td>
+<td>An answer satisfying the problem's conditions; there may be one or more</td>
+<td>All paths from the root to a node valued <span class="arithmatex">\(7\)</span> that satisfy the constraint</td>
 </tr>
 <tr>
-<td>Constraint (constraint)</td>
-<td>A constraint is a condition in the problem that limits the feasibility of solutions, typically used for pruning</td>
-<td>Paths do not contain nodes with value <span class="arithmatex">\(3\)</span></td>
+<td>Constraint</td>
+<td>A condition restricting which solutions are feasible, often used for pruning</td>
+<td>The path contains no node valued <span class="arithmatex">\(3\)</span></td>
 </tr>
 <tr>
-<td>State (state)</td>
-<td>State represents the situation of a problem at a certain moment, including the choices already made</td>
-<td>The currently visited node path, i.e., the <code>path</code> list of nodes</td>
+<td>State</td>
+<td>The situation at a particular point, including the choices already made</td>
+<td>The current path of visited nodes, stored in <code>path</code></td>
 </tr>
 <tr>
-<td>Attempt (attempt)</td>
-<td>An attempt is the process of exploring the solution space according to available choices, including making choices, updating state, and checking if it is a solution</td>
-<td>Recursively visit left (right) child nodes, add nodes to <code>path</code>, check if node value is <span class="arithmatex">\(7\)</span></td>
+<td>Trial</td>
+<td>Exploring the solution space by making a choice, updating the state, and checking for a solution</td>
+<td>Recursively visit a left or right child, append it to <code>path</code>, and check whether its value is <span class="arithmatex">\(7\)</span></td>
 </tr>
 <tr>
-<td>Backtrack (backtracking)</td>
-<td>Backtracking refers to undoing previous choices and returning to a previous state when encountering a state that does not satisfy constraints</td>
-<td>Stop searching when passing over leaf nodes, ending node visits, or encountering nodes with value <span class="arithmatex">\(3\)</span>; function returns</td>
+<td>Backtracking</td>
+<td>Undoing an earlier choice and returning to the previous state when a constraint cannot be met</td>
+<td>Stop searching and return after passing a leaf, finishing a node, or encountering value <span class="arithmatex">\(3\)</span></td>
 </tr>
 <tr>
-<td>Pruning (pruning)</td>
-<td>Pruning is a method of avoiding meaningless search paths according to problem characteristics and constraints, which can improve search efficiency</td>
-<td>When encountering a node with value <span class="arithmatex">\(3\)</span>, do not continue searching</td>
+<td>Pruning</td>
+<td>Using problem properties and constraints to avoid pointless search paths and improve efficiency</td>
+<td>Stop searching when a node valued <span class="arithmatex">\(3\)</span> is encountered</td>
 </tr>
 </tbody>
 </table>
 </div>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>The concepts of problem, solution, state, etc. are universal and appear in divide-and-conquer, backtracking, dynamic programming, greedy algorithms, and others.</p>
+<p>Concepts such as problems, solutions, and states are shared by divide and conquer, backtracking, dynamic programming, greedy algorithms, and other approaches.</p>
 </div>
-<h2 id="1315-advantages-and-limitations">13.1.5 &nbsp; Advantages and Limitations<a class="headerlink" href="#1315-advantages-and-limitations" title="Permanent link">&para;</a></h2>
-<p>The backtracking algorithm is essentially a depth-first search algorithm that tries all possible solutions until it finds one that satisfies the conditions. The advantage of this approach is that it can find all possible solutions, and with reasonable pruning operations, it achieves high efficiency.</p>
-<p>However, when dealing with large-scale or complex problems, <strong>the running efficiency of the backtracking algorithm may be unacceptable</strong>.</p>
+<h2 id="1315-strengths-and-limitations">13.1.5 &nbsp; Strengths and limitations<a class="headerlink" href="#1315-strengths-and-limitations" title="Permanent link">&para;</a></h2>
+<p>Backtracking is fundamentally a depth-first search that tries possible solutions until it finds ones meeting the conditions. It can find all possible solutions and can be highly efficient with effective pruning.</p>
+<p>However, for large or complex problems, <strong>backtracking may be unacceptably slow</strong>.</p>
 <ul>
-<li><strong>Time</strong>: The backtracking algorithm usually needs to traverse all possibilities in the state space, and the time complexity can reach exponential or factorial order.</li>
-<li><strong>Space</strong>: During recursive calls, the current state needs to be saved (such as paths, auxiliary variables used for pruning, etc.), and when the depth is large, the space requirement can become very large.</li>
+<li><strong>Time</strong>: Backtracking often explores every possibility in the state space, leading to exponential or factorial time complexity.</li>
+<li><strong>Space</strong>: Recursive calls must preserve state, such as the path and auxiliary pruning variables. Deep recursion can therefore require substantial space.</li>
 </ul>
-<p>Nevertheless, <strong>the backtracking algorithm is still the best solution for certain search problems and constraint satisfaction problems</strong>. For these problems, since we cannot predict which choices will generate valid solutions, we must traverse all possible choices. In this case, <strong>the key is how to optimize efficiency</strong>. There are two common efficiency optimization methods.</p>
+<p>Even so, <strong>backtracking remains the best approach for some search and constraint satisfaction problems</strong>. When we cannot predict which choices lead to valid solutions, we must explore every possibility. <strong>The key is then to improve efficiency</strong>, commonly through two techniques.</p>
 <ul>
-<li><strong>Pruning</strong>: Avoid searching paths that are guaranteed not to produce solutions, thereby saving time and space.</li>
-<li><strong>Heuristic search</strong>: Introduce certain strategies or estimation values during the search process to prioritize searching paths that are most likely to produce valid solutions.</li>
+<li><strong>Pruning</strong>: Avoid paths that cannot produce a solution, saving time and space.</li>
+<li><strong>Heuristic search</strong>: Use strategies or estimates to explore paths most likely to yield valid solutions first.</li>
 </ul>
-<h2 id="1316-typical-backtracking-examples">13.1.6 &nbsp; Typical Backtracking Examples<a class="headerlink" href="#1316-typical-backtracking-examples" title="Permanent link">&para;</a></h2>
-<p>The backtracking algorithm can be used to solve many search problems, constraint satisfaction problems, and combinatorial optimization problems.</p>
-<p><strong>Search problems</strong>: The goal of these problems is to find solutions that satisfy specific conditions.</p>
+<h2 id="1316-typical-backtracking-problems">13.1.6 &nbsp; Typical backtracking problems<a class="headerlink" href="#1316-typical-backtracking-problems" title="Permanent link">&para;</a></h2>
+<p>Backtracking can solve many search, constraint satisfaction, and combinatorial optimization problems.</p>
+<p><strong>Search problems</strong> seek solutions satisfying specified conditions.</p>
 <ul>
-<li>Permutation problem: Given a set, find all possible permutations and combinations.</li>
-<li>Subset sum problem: Given a set and a target sum, find all subsets in the set whose elements sum to the target.</li>
-<li>Tower of Hanoi: Given three pegs and a series of disks of different sizes, move all disks from one peg to another, moving only one disk at a time, and never placing a larger disk on a smaller disk.</li>
+<li>Permutations: Given a collection, find every possible ordering of its elements.</li>
+<li>Subset sum: Given a collection and a target sum, find every subset whose sum equals the target.</li>
+<li>Tower of Hanoi: Given three rods and disks of different sizes, move all disks from one rod to another, moving only one at a time and never placing a larger disk on a smaller one.</li>
 </ul>
-<p><strong>Constraint satisfaction problems</strong>: The goal of these problems is to find solutions that satisfy all constraints.</p>
+<p><strong>Constraint satisfaction problems</strong> seek solutions meeting all constraints.</p>
 <ul>
-<li>N-Queens: Place <span class="arithmatex">\(n\)</span> queens on an <span class="arithmatex">\(n \times n\)</span> chessboard such that they do not attack each other.</li>
-<li>Sudoku: Fill numbers <span class="arithmatex">\(1\)</span> to <span class="arithmatex">\(9\)</span> in a <span class="arithmatex">\(9 \times 9\)</span> grid such that each row, column, and <span class="arithmatex">\(3 \times 3\)</span> subgrid contains no repeated digits.</li>
-<li>Graph coloring: Given an undirected graph, color each vertex with the minimum number of colors such that adjacent vertices have different colors.</li>
+<li><span class="arithmatex">\(n\)</span>-queens: Place <span class="arithmatex">\(n\)</span> queens on an <span class="arithmatex">\(n \times n\)</span> chessboard so that no two attack each other.</li>
+<li>Sudoku: Fill a <span class="arithmatex">\(9 \times 9\)</span> grid with digits <span class="arithmatex">\(1\)</span> ~ <span class="arithmatex">\(9\)</span> so that no digit repeats in any row, column, or <span class="arithmatex">\(3 \times 3\)</span> subgrid.</li>
+<li>Graph coloring: Color an undirected graph's vertices with as few colors as possible, ensuring adjacent vertices have different colors.</li>
 </ul>
-<p><strong>Combinatorial optimization problems</strong>: The goal of these problems is to find an optimal solution that satisfies certain conditions in a combinatorial space.</p>
+<p><strong>Combinatorial optimization problems</strong> seek an optimal solution satisfying conditions within a space of combinations.</p>
 <ul>
-<li>0-1 Knapsack: Given a set of items and a knapsack, each item has a value and weight. Under the knapsack capacity constraint, select items to maximize total value.</li>
-<li>Traveling Salesman Problem: Starting from a point in a graph, visit all other points exactly once and return to the starting point, finding the shortest path.</li>
-<li>Maximum Clique: Given an undirected graph, find the largest complete subgraph, i.e., a subgraph where any two vertices are connected by an edge.</li>
+<li>0-1 knapsack: Given items with values and weights and a knapsack, choose items to maximize total value within the capacity limit.</li>
+<li>Traveling salesperson: Starting at a vertex in a graph, visit every other vertex exactly once and return to the start along the shortest possible route.</li>
+<li>Maximum clique: Given an undirected graph, find its largest complete subgraph, in which every pair of vertices is connected by an edge.</li>
 </ul>
-<p>Note that for many combinatorial optimization problems, backtracking is not the optimal solution.</p>
+<p>For many combinatorial optimization problems, backtracking is not the best approach.</p>
 <ul>
-<li>The 0-1 Knapsack problem is usually solved using dynamic programming to achieve higher time efficiency.</li>
-<li>The Traveling Salesman Problem is a famous NP-Hard problem; common solutions include genetic algorithms and ant colony algorithms.</li>
-<li>The Maximum Clique problem is a classical problem in graph theory and can be solved using heuristic algorithms such as greedy algorithms.</li>
+<li>0-1 knapsack is usually solved more efficiently with dynamic programming.</li>
+<li>The traveling salesperson problem is a well-known NP-hard problem. Common approaches include genetic algorithms and ant colony algorithms.</li>
+<li>Maximum clique is a classic graph theory problem that can be approached with heuristics such as greedy algorithms.</li>
 </ul>
 
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   <a
     href="../"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: Chapter 13. &amp;nbsp; Backtracking"
+    aria-label="Previous: Chapter 13.   Backtracking"
     rel="prev"
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@@ -6665,7 +6757,7 @@ aria-label="Footer"
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     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 13.2 Permutations Problem"
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         Next
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+        13.2 &nbsp; Permutations problem
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@@ -6690,7 +6782,7 @@ aria-label="Footer"
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-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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-  0.2 How to Use This Book
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-  0.3 Summary
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-  Chapter 1. Encounter with Algorithms
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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-  Chapter 2. Complexity Analysis
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-  2.6 Exercises
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-  Chapter 3. Data Structures
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-  3.1 Classification of Data Structures
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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+  10.1 &nbsp; Binary search
 
 
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@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3283,7 +3283,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3304,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3352,7 +3352,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3379,7 +3379,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3406,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3433,7 +3433,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3460,7 +3460,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3528,7 +3528,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3549,7 +3549,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3570,7 +3570,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3597,7 +3597,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3624,7 +3624,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3651,7 +3651,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3678,7 +3678,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3714,7 +3714,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3731,7 +3731,7 @@
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3757,22 +3757,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1361-concept-review" class="md-nav__link">
+  <a href="#1361-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.6.1 &nbsp; Concept Review
+        13.6.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="13.6.1 Concept Review">
+    <nav class="md-nav" aria-label="13.6.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-will-this-permutation-algorithm-miss-results" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Will This Permutation Algorithm Miss Results?
+        1. &nbsp; Will this permutation algorithm miss results?
       </span>
     </span>
   </a>
@@ -3783,7 +3783,7 @@
   <a href="#2-does-the-order-of-choosing-numbers-matter" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Does the Order of Choosing Numbers Matter?
+        2. &nbsp; Does the order of choosing numbers matter?
       </span>
     </span>
   </a>
@@ -3794,7 +3794,7 @@
   <a href="#3-where-can-the-next-queen-be-placed" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Where Can the Next Queen Be Placed?
+        3. &nbsp; Where can the next queen be placed?
       </span>
     </span>
   </a>
@@ -3810,19 +3810,19 @@
   <a href="#1362-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.6.2 &nbsp; Programming Exercises
+        13.6.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="13.6.2 Programming Exercises">
+    <nav class="md-nav" aria-label="13.6.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-permutations-of-distinct-elements" class="md-nav__link">
+  <a href="#1-all-permutations-of-distinct-elements" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Permutations of Distinct Elements
+        1. &nbsp; All permutations of distinct elements
       </span>
     </span>
   </a>
@@ -3900,7 +3900,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3921,7 +3921,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3942,7 +3942,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3969,7 +3969,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3996,7 +3996,7 @@
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4023,7 +4023,7 @@
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4050,7 +4050,7 @@
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4077,7 +4077,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4104,7 +4104,7 @@
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4131,7 +4131,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4197,7 +4197,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4218,7 +4218,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4239,7 +4239,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4266,7 +4266,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4293,7 +4293,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4320,7 +4320,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4347,7 +4347,7 @@
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4374,7 +4374,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4434,7 +4434,7 @@
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4455,7 +4455,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4476,7 +4476,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4503,7 +4503,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4530,7 +4530,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4614,6 +4614,78 @@
 
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1361-concept-review" class="md-nav__link">
+  <a href="#1361-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.6.1 &nbsp; Concept Review
+        13.6.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="13.6.1 Concept Review">
+    <nav class="md-nav" aria-label="13.6.1 Check your understanding">
       <ul class="md-nav__list">
 
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   <a href="#1-will-this-permutation-algorithm-miss-results" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Will This Permutation Algorithm Miss Results?
+        1. &nbsp; Will this permutation algorithm miss results?
       </span>
     </span>
   </a>
@@ -4681,7 +4753,7 @@
   <a href="#2-does-the-order-of-choosing-numbers-matter" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Does the Order of Choosing Numbers Matter?
+        2. &nbsp; Does the order of choosing numbers matter?
       </span>
     </span>
   </a>
@@ -4692,7 +4764,7 @@
   <a href="#3-where-can-the-next-queen-be-placed" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Where Can the Next Queen Be Placed?
+        3. &nbsp; Where can the next queen be placed?
       </span>
     </span>
   </a>
@@ -4708,19 +4780,19 @@
   <a href="#1362-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.6.2 &nbsp; Programming Exercises
+        13.6.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="13.6.2 Programming Exercises">
+    <nav class="md-nav" aria-label="13.6.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-permutations-of-distinct-elements" class="md-nav__link">
+  <a href="#1-all-permutations-of-distinct-elements" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Permutations of Distinct Elements
+        1. &nbsp; All permutations of distinct elements
       </span>
     </span>
   </a>
@@ -4770,99 +4842,99 @@
 
 <!-- Page content -->
 <h1 id="136-exercises">13.6 &nbsp; Exercises<a class="headerlink" href="#136-exercises" title="Permanent link">&para;</a></h1>
-<h2 id="1361-concept-review">13.6.1 &nbsp; Concept Review<a class="headerlink" href="#1361-concept-review" title="Permanent link">&para;</a></h2>
-<h3 id="1-will-this-permutation-algorithm-miss-results">1. &nbsp; Will This Permutation Algorithm Miss Results?<a class="headerlink" href="#1-will-this-permutation-algorithm-miss-results" title="Permanent link">&para;</a></h3>
-<p>A backtracking algorithm tries to generate all permutations using <code>1, 2, 3</code> in that order. Each time it chooses a number <code>x</code>, it:</p>
+<h2 id="1361-check-your-understanding">13.6.1 &nbsp; Check your understanding<a class="headerlink" href="#1361-check-your-understanding" title="Permanent link">&para;</a></h2>
+<h3 id="1-will-this-permutation-algorithm-miss-results">1. &nbsp; Will this permutation algorithm miss results?<a class="headerlink" href="#1-will-this-permutation-algorithm-miss-results" title="Permanent link">&para;</a></h3>
+<p>A backtracking algorithm tries the numbers <code>1, 2, 3</code> in order to generate all permutations. Whenever it chooses a number <code>x</code>, it:</p>
 <ol>
 <li>Appends <code>x</code> to the current path.</li>
-<li>Marks <code>x</code> as "used."</li>
+<li>Marks <code>x</code> as used.</li>
 <li>Recursively fills the next position.</li>
 </ol>
-<p>When the recursive call returns, the student removes only <code>x</code> from the end of the path and then tries the next number.</p>
+<p>After the recursive call returns, the student removes only <code>x</code> from the end of the path, then continues trying the next number.</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Which permutation does the algorithm generate first? Can it still generate all 6 permutations?</li>
-<li>Before returning to the previous level, is removing only the last number from the path enough? If not, what else must be done? Explain why.</li>
+<li>Which permutation does the algorithm produce first? Can it still produce all six permutations?</li>
+<li>Before returning to the previous recursion level, is removing the last number from the path enough? If not, what else is needed? Explain.</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>It first generates <code>[1, 2, 3]</code>, but it cannot generate all the permutations. Although the path becomes shorter when calls return,
-    the markers for 1, 2, and 3 all remain "used," leaving no available number for later branches.</p>
+<p>It first produces <code>[1, 2, 3]</code>, but cannot generate every permutation. Although the path becomes shorter as calls return,
+    the numbers 1, 2, and 3 remain marked as used, leaving later branches with no available numbers.</p>
 </li>
 <li>
-<p>It is not enough. After removing <code>x</code> from the end of the path, the algorithm must also mark <code>x</code> as "unused" again.
-    The current path and the used markers together describe the search state. A choice changes both of them, so backtracking must restore both
-    before another branch can choose <code>x</code> again.</p>
+<p>No. After removing <code>x</code> from the end of the path, we must also mark <code>x</code> as unused again.
+    The current path and the used markers together describe the search state. Choosing a number changes both, so backtracking must restore both
+    to allow other branches to choose <code>x</code> again.</p>
 </li>
 </ol>
 </details>
-<h3 id="2-does-the-order-of-choosing-numbers-matter">2. &nbsp; Does the Order of Choosing Numbers Matter?<a class="headerlink" href="#2-does-the-order-of-choosing-numbers-matter" title="Permanent link">&para;</a></h3>
-<p>You are given the sorted array <code>[2, 3, 5]</code> and target value 5. Each number may be chosen repeatedly.
-The algorithm requires the numbers along each search path to appear only in nondecreasing order.</p>
+<h3 id="2-does-the-order-of-choosing-numbers-matter">2. &nbsp; Does the order of choosing numbers matter?<a class="headerlink" href="#2-does-the-order-of-choosing-numbers-matter" title="Permanent link">&para;</a></h3>
+<p>Given the sorted array <code>[2, 3, 5]</code> and target value 5, each number may be chosen repeatedly.
+The algorithm requires the numbers along each search path to appear in nondecreasing order.</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Which distinct combinations can be found?</li>
-<li>Why is there no need to search for the same group of numbers in different orders? What does the nondecreasing-order restriction accomplish?</li>
-<li>Suppose the current path is <code>[3]</code>, the remaining amount is 2, and the next candidate is 3. Why can the algorithm stop checking all later candidates at this level?</li>
+<li>Which distinct combinations can be obtained?</li>
+<li>Why is it unnecessary to search the same set of numbers in different orders? What does the nondecreasing-order restriction achieve?</li>
+<li>The current path is <code>[3]</code>, the remaining sum is 2, and the next candidate is 3. Why can we stop checking all further candidates at this level?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
 <p>The distinct combinations are <code>[2, 3]</code> and <code>[5]</code>.</p>
 </li>
 <li>
-<p>This exercise treats <code>[2, 3]</code> and <code>[3, 2]</code> as the same combination; the order in which numbers are selected does not create a different answer.
-    Requiring numbers in a path to appear in nondecreasing order lets the search skip duplicate arrangements such as <code>[3, 2]</code>.</p>
+<p>This problem treats <code>[2, 3]</code> and <code>[3, 2]</code> as the same combination: selection order does not distinguish answers.
+    Requiring the path's numbers to appear in nondecreasing order lets the search skip duplicate combinations such as <code>[3, 2]</code> directly.</p>
 </li>
 <li>
-<p>The remaining amount is 2, while candidate 3 is already greater than 2. Because the array is sorted,
-    all later candidates are even larger and cannot be added to the current combination, so the algorithm can stop checking this level immediately.</p>
+<p>The remaining sum is 2, and candidate 3 already exceeds it. Since the array is sorted,
+    all candidates after 3 are larger still and cannot join the current combination. We can therefore stop checking this level.</p>
 </li>
 </ol>
 </details>
-<h3 id="3-where-can-the-next-queen-be-placed">3. &nbsp; Where Can the Next Queen Be Placed?<a class="headerlink" href="#3-where-can-the-next-queen-be-placed" title="Permanent link">&para;</a></h3>
-<p>Place queens row by row on a <code>4 × 4</code> chessboard, with both row and column indices starting at 0.
-Queens have already been placed at <code>(0, 1)</code> and <code>(1, 3)</code>. The next queen must be placed in row 2.</p>
+<h3 id="3-where-can-the-next-queen-be-placed">3. &nbsp; Where can the next queen be placed?<a class="headerlink" href="#3-where-can-the-next-queen-be-placed" title="Permanent link">&para;</a></h3>
+<p>Place queens row by row on a <code>4 × 4</code> board, with both row and column indices starting at 0.
+Queens are already at <code>(0, 1)</code> and <code>(1, 3)</code>. The next queen must be placed in row 2.</p>
 <!-- numbered-subquestions -->
 
 <ol>
 <li>Which columns are ruled out because they already contain a queen?</li>
-<li>Among the remaining columns, which positions are ruled out because they share a diagonal with an existing queen?</li>
+<li>Among the remaining columns, which positions are ruled out by sharing a diagonal with a queen?</li>
 <li>Which positions in row 2 remain available to try?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>Columns 1 and 3 already contain queens, so positions <code>(2, 1)</code> and <code>(2, 3)</code> are ruled out.</p>
+<p>Columns 1 and 3 already contain queens, ruling out positions <code>(2, 1)</code> and <code>(2, 3)</code>.</p>
 </li>
 <li>
-<p>Among the remaining positions, <code>(2, 2)</code> lies on the same diagonal as <code>(1, 3)</code>, so it is also ruled out.
+<p>Of the remaining positions, <code>(2, 2)</code> shares a diagonal with <code>(1, 3)</code>, so it is also ruled out.
     Position <code>(2, 0)</code> shares neither a column nor a diagonal with either existing queen.</p>
 </li>
 <li>
 <p>The only position to try in row 2 is <code>(2, 0)</code>.</p>
-<p>This step shows only that the current placement is valid. If the board cannot be completed later, the algorithm must still backtrack and try an earlier alternative.</p>
+<p>This step establishes only that the current placement is valid. If the board cannot be completed later, we must still backtrack and try other earlier choices.</p>
 </li>
 </ol>
 </details>
-<h2 id="1362-programming-exercises">13.6.2 &nbsp; Programming Exercises<a class="headerlink" href="#1362-programming-exercises" title="Permanent link">&para;</a></h2>
-<h3 id="1-permutations-of-distinct-elements">1. &nbsp; Permutations of Distinct Elements<a class="headerlink" href="#1-permutations-of-distinct-elements" title="Permanent link">&para;</a></h3>
+<h2 id="1362-programming-exercises">13.6.2 &nbsp; Programming exercises<a class="headerlink" href="#1362-programming-exercises" title="Permanent link">&para;</a></h2>
+<h3 id="1-all-permutations-of-distinct-elements">1. &nbsp; All permutations of distinct elements<a class="headerlink" href="#1-all-permutations-of-distinct-elements" title="Permanent link">&para;</a></h3>
 <p>The integer array <code>nums</code> contains at least one element, and all its elements are distinct.
-List every possible order formed by using each element exactly once, and return each order as an array.
+List every possible ordering that uses each element exactly once, returning each ordering as an array.
 The permutations may appear in any order in the result.
-Use backtracking, with a Boolean array recording whether the element at each position has already been selected for the current permutation.</p>
+Use backtracking and a Boolean array to record whether the element at each position is already included in the current permutation.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>The recursion depth indicates which position of the permutation is currently being filled</li>
-<li>At each level, try only elements that have not yet been used</li>
-<li>When the path's length equals the length of <code>nums</code>, add a copy of the path to the answer</li>
+<li>The recursion depth indicates which position in the permutation is being filled.</li>
+<li>At each level, try only unused elements.</li>
+<li>When the path length reaches the length of <code>nums</code>, add a copy of the path to the answer.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/permutations/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
@@ -4888,7 +4960,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 13.5 Summary"
+    aria-label="Previous: 13.5   Summary"
     rel="prev"
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@@ -4900,7 +4972,7 @@ aria-label="Footer"
         Previous
       </span>
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+        13.5 &nbsp; Summary
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@@ -4912,7 +4984,7 @@ aria-label="Footer"
   <a
     href="../../chapter_dynamic_programming/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: Chapter 14. &amp;nbsp; Dynamic Programming"
+    aria-label="Next: Chapter 14.   Dynamic programming"
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@@ -4920,7 +4992,7 @@ aria-label="Footer"
         Next
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+        Chapter 14. &nbsp; Dynamic programming
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@@ -4937,7 +5009,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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diff --git a/en/chapter_backtracking/index.html b/en/chapter_backtracking/index.html
index 81afc0208..2b2725570 100644
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@@ -6,7 +6,7 @@
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+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
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@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
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@@ -1506,7 +1506,7 @@
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
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   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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+  Chapter 9. &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
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-  9.4 Summary
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
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-  10.1 Binary Search
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   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
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+  12.3 &nbsp; Building a binary tree
 
 
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-  12.5 Summary
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-  12.6 Exercises
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-  Chapter 13. Backtracking
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-  13.1 Backtracking Algorithm
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.5 Summary
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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@@ -3775,7 +3775,7 @@
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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@@ -3817,7 +3817,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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@@ -3925,7 +3925,7 @@
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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@@ -3952,7 +3952,7 @@
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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@@ -4072,7 +4072,7 @@
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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@@ -4114,7 +4114,7 @@
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
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   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
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   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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+
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+
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+  Print edition (Chinese)
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+    <span class="md-status md-status--new" title="Recently added">
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 </nav>
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@@ -4581,15 +4653,15 @@
 <p><img alt="Backtracking" class="cover-image" src="../assets/covers/chapter_backtracking.jpg" /></p>
 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
-<p>We are like explorers in a maze, and may encounter difficulties on the path forward.</p>
-<p>The power of backtracking allows us to start over, keep trying, and eventually find the exit leading to light.</p>
+<p>Like explorers in a maze, we may encounter obstacles as we move forward.</p>
+<p>Backtracking lets us start again, try new paths, and eventually find our way out into the light.</p>
 </div>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
-<li><a href="backtracking_algorithm/">13.1 &nbsp; Backtracking Algorithm</a></li>
-<li><a href="permutations_problem/">13.2 &nbsp; Permutations Problem</a></li>
-<li><a href="subset_sum_problem/">13.3 &nbsp; Subset-Sum Problem</a></li>
-<li><a href="n_queens_problem/">13.4 &nbsp; N-Queens Problem</a></li>
+<li><a href="backtracking_algorithm/">13.1 &nbsp; Backtracking algorithms</a></li>
+<li><a href="permutations_problem/">13.2 &nbsp; Permutations problem</a></li>
+<li><a href="subset_sum_problem/">13.3 &nbsp; Subset sum problem</a></li>
+<li><a href="n_queens_problem/">13.4 &nbsp; N-queens problem</a></li>
 <li><a href="summary/">13.5 &nbsp; Summary</a></li>
 <li><a href="exercises/">13.6 &nbsp; Exercises</a></li>
 </ul>
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   <a
     href="../chapter_divide_and_conquer/exercises/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 12.6 Exercises"
+    aria-label="Previous: 12.6   Exercises"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4627,7 +4699,7 @@ aria-label="Footer"
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       <div class="md-ellipsis">
-        12.6 Exercises
+        12.6 &nbsp; Exercises
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   <a
     href="./backtracking_algorithm/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 13.1 Backtracking Algorithm"
+    aria-label="Next: 13.1   Backtracking algorithms"
     rel="next"
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     <div class="md-footer__title">
@@ -4647,7 +4719,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        13.1 Backtracking Algorithm
+        13.1 &nbsp; Backtracking algorithms
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     <div class="md-footer__button md-icon">
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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-        <title>13.4 N-Queens Problem - Hello Algo</title>
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-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
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         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3283,7 +3283,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3304,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3352,7 +3352,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3379,7 +3379,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3406,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3433,7 +3433,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3460,7 +3460,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3528,7 +3528,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3549,7 +3549,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3570,7 +3570,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3597,7 +3597,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3624,7 +3624,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3660,7 +3660,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3677,7 +3677,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3703,10 +3703,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-row-by-row-placement-strategy" class="md-nav__link">
+  <a href="#1-placing-queens-row-by-row" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Row-By-Row Placement Strategy
+        1. &nbsp; Placing queens row by row
       </span>
     </span>
   </a>
@@ -3714,10 +3714,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#2-column-and-diagonal-pruning" class="md-nav__link">
+  <a href="#2-pruning-by-columns-and-diagonals" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Column and Diagonal Pruning
+        2. &nbsp; Pruning by columns and diagonals
       </span>
     </span>
   </a>
@@ -3728,7 +3728,7 @@
   <a href="#3-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Code Implementation
+        3. &nbsp; Code implementation
       </span>
     </span>
   </a>
@@ -3758,7 +3758,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3785,7 +3785,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3855,7 +3855,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3876,7 +3876,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3897,7 +3897,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3924,7 +3924,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3951,7 +3951,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3978,7 +3978,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4005,7 +4005,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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@@ -4389,7 +4389,7 @@
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-row-by-row-placement-strategy" class="md-nav__link">
+  <a href="#1-placing-queens-row-by-row" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Row-By-Row Placement Strategy
+        1. &nbsp; Placing queens row by row
       </span>
     </span>
   </a>
@@ -4621,10 +4693,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#2-column-and-diagonal-pruning" class="md-nav__link">
+  <a href="#2-pruning-by-columns-and-diagonals" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Column and Diagonal Pruning
+        2. &nbsp; Pruning by columns and diagonals
       </span>
     </span>
   </a>
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   <a href="#3-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
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-        3. &nbsp; Code Implementation
+        3. &nbsp; Code implementation
       </span>
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@@ -4679,41 +4751,41 @@
 
 
 <!-- Page content -->
-<h1 id="134-n-queens-problem">13.4 &nbsp; N-Queens Problem<a class="headerlink" href="#134-n-queens-problem" title="Permanent link">&para;</a></h1>
+<h1 id="134-n-queens-problem">13.4 &nbsp; N-queens problem<a class="headerlink" href="#134-n-queens-problem" title="Permanent link">&para;</a></h1>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>According to the rules of chess, a queen can attack any piece in the same row, column, or diagonal. Given <span class="arithmatex">\(n\)</span> queens and an <span class="arithmatex">\(n \times n\)</span> chessboard, find an arrangement such that no two queens can attack each other.</p>
+<p>In chess, a queen can attack any piece in the same row, column, or diagonal. Given <span class="arithmatex">\(n\)</span> queens and an <span class="arithmatex">\(n \times n\)</span> chessboard, find all arrangements in which no two queens can attack each other.</p>
 </div>
-<p>As shown in Figure 13-15, when <span class="arithmatex">\(n = 4\)</span>, there are two solutions that can be found. From the perspective of the backtracking algorithm, an <span class="arithmatex">\(n \times n\)</span> chessboard has <span class="arithmatex">\(n^2\)</span> squares, which provide all the choices <code>choices</code>. During the process of placing queens one by one, the chessboard state changes continuously, and the chessboard at each moment represents the state <code>state</code>.</p>
-<p><img alt="Solution to the 4-queens problem" class="animation-figure" src="../n_queens_problem.assets/solution_4_queens.png" /></p>
-<p align="center"> Figure 13-15 &nbsp; Solution to the 4-queens problem </p>
+<p>As shown in Figure 13-15, there are two solutions when <span class="arithmatex">\(n = 4\)</span>. From a backtracking perspective, the <span class="arithmatex">\(n \times n\)</span> board has <span class="arithmatex">\(n^2\)</span> squares, which provide all possible <code>choices</code>. As we place queens one by one, the board changes; its arrangement at any moment is the current <code>state</code>.</p>
+<p><img alt="Solutions to the 4-queens problem" class="animation-figure" src="../n_queens_problem.assets/solution_4_queens.png" /></p>
+<p align="center"> Figure 13-15 &nbsp; Solutions to the 4-queens problem </p>
 
-<p>Figure 13-16 illustrates the three constraints of this problem: <strong>multiple queens cannot be in the same row, the same column, or on the same diagonal</strong>. It is worth noting that diagonals are divided into two types: the main diagonal <code>\</code> and the anti-diagonal <code>/</code>.</p>
+<p>Figure 13-16 shows the three constraints: <strong>no two queens may share a row, column, or diagonal</strong>. Diagonals run in two directions: the main-diagonal direction <code>\</code> and the antidiagonal direction <code>/</code>.</p>
 <p><img alt="Constraints of the n-queens problem" class="animation-figure" src="../n_queens_problem.assets/n_queens_constraints.png" /></p>
 <p align="center"> Figure 13-16 &nbsp; Constraints of the n-queens problem </p>
 
-<h3 id="1-row-by-row-placement-strategy">1. &nbsp; Row-By-Row Placement Strategy<a class="headerlink" href="#1-row-by-row-placement-strategy" title="Permanent link">&para;</a></h3>
-<p>Since both the number of queens and the number of rows on the chessboard are <span class="arithmatex">\(n\)</span>, we can easily derive a conclusion: <strong>each row of the chessboard allows one and only one queen to be placed</strong>.</p>
-<p>This means we can adopt a row-by-row placement strategy: starting from the first row, place one queen in each row until the last row is completed.</p>
-<p>Figure 13-17 shows the row-by-row placement process for the 4-queens problem. Due to space limitations, the figure only expands one search branch of the first row, and all schemes that violate the column or diagonal constraints are pruned.</p>
-<p><img alt="Row-by-row placement strategy" class="animation-figure" src="../n_queens_problem.assets/n_queens_placing.png" /></p>
-<p align="center"> Figure 13-17 &nbsp; Row-by-row placement strategy </p>
+<h3 id="1-placing-queens-row-by-row">1. &nbsp; Placing queens row by row<a class="headerlink" href="#1-placing-queens-row-by-row" title="Permanent link">&para;</a></h3>
+<p>There are as many queens as rows, namely <span class="arithmatex">\(n\)</span>, so we can immediately conclude that <strong>each row must contain exactly one queen</strong>.</p>
+<p>We can therefore place queens row by row: start at the first row and place one queen in each row through the last.</p>
+<p>Figure 13-17 illustrates this process for 4 queens. To fit the illustration, only one branch from the first row is expanded, and all arrangements violating the column or diagonal constraints are pruned.</p>
+<p><img alt="Placing queens row by row" class="animation-figure" src="../n_queens_problem.assets/n_queens_placing.png" /></p>
+<p align="center"> Figure 13-17 &nbsp; Placing queens row by row </p>
 
-<p>Essentially, <strong>the row-by-row placement strategy serves a pruning function</strong>, as it avoids all search branches where multiple queens appear in the same row.</p>
-<h3 id="2-column-and-diagonal-pruning">2. &nbsp; Column and Diagonal Pruning<a class="headerlink" href="#2-column-and-diagonal-pruning" title="Permanent link">&para;</a></h3>
-<p>To satisfy the column constraint, we can use a boolean array <code>cols</code> of length <span class="arithmatex">\(n\)</span> to record whether each column has a queen. Before each placement decision, we use <code>cols</code> to prune columns that already have queens, and dynamically update the state of <code>cols</code> during backtracking.</p>
+<p>In essence, <strong>placing queens row by row is itself a form of pruning</strong>: it eliminates every search branch that places multiple queens in the same row.</p>
+<h3 id="2-pruning-by-columns-and-diagonals">2. &nbsp; Pruning by columns and diagonals<a class="headerlink" href="#2-pruning-by-columns-and-diagonals" title="Permanent link">&para;</a></h3>
+<p>To enforce the column constraint, use a Boolean array <code>cols</code> of length <span class="arithmatex">\(n\)</span> to record whether each column contains a queen. Before placing a queen, consult <code>cols</code> to prune columns already occupied, and update <code>cols</code> as the search proceeds and backtracks.</p>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>Please note that the origin of the matrix is located in the upper-left corner, where the row index increases from top to bottom, and the column index increases from left to right.</p>
+<p>The matrix starts at its top-left corner. Row indices increase downward, and column indices increase to the right.</p>
 </div>
-<p>So how do we handle diagonal constraints? Consider a square on the chessboard with row and column indices <span class="arithmatex">\((row, col)\)</span>. If we select a specific main diagonal in the matrix, we find that all squares on that diagonal have the same difference between their row and column indices, <strong>meaning that <span class="arithmatex">\(row - col\)</span> is a constant value for all squares on the main diagonal</strong>.</p>
-<p>In other words, if two squares satisfy <span class="arithmatex">\(row_1 - col_1 = row_2 - col_2\)</span>, they must be on the same main diagonal. Using this pattern, we can use the array <code>diags1</code> shown in Figure 13-18 to record whether there is a queen on each main diagonal.</p>
-<p>Similarly, <strong>for all squares on an anti-diagonal, the sum <span class="arithmatex">\(row + col\)</span> is a constant value</strong>. We can likewise use the array <code>diags2</code> to handle anti-diagonal constraints.</p>
-<p><img alt="Handling column and diagonal constraints" class="animation-figure" src="../n_queens_problem.assets/n_queens_cols_diagonals.png" /></p>
-<p align="center"> Figure 13-18 &nbsp; Handling column and diagonal constraints </p>
+<p>How do we enforce the diagonal constraints? Let <span class="arithmatex">\((row, col)\)</span> be the row and column indices of a square. Along any diagonal in the main-diagonal direction, the row index minus the column index is the same: <strong><span class="arithmatex">\(row - col\)</span> is constant along that diagonal</strong>.</p>
+<p>Thus, if two squares satisfy <span class="arithmatex">\(row_1 - col_1 = row_2 - col_2\)</span>, they lie on the same main-direction diagonal. Using this property, the array <code>diags1</code> in Figure 13-18 records whether each such diagonal contains a queen.</p>
+<p>Similarly, <strong><span class="arithmatex">\(row + col\)</span> is constant along each antidiagonal</strong>. We can use an array <code>diags2</code> to enforce the antidiagonal constraint.</p>
+<p><img alt="Enforcing column and diagonal constraints" class="animation-figure" src="../n_queens_problem.assets/n_queens_cols_diagonals.png" /></p>
+<p align="center"> Figure 13-18 &nbsp; Enforcing column and diagonal constraints </p>
 
-<h3 id="3-code-implementation">3. &nbsp; Code Implementation<a class="headerlink" href="#3-code-implementation" title="Permanent link">&para;</a></h3>
-<p>Please note that in an <span class="arithmatex">\(n \times n\)</span> square matrix, the range of <span class="arithmatex">\(row - col\)</span> is <span class="arithmatex">\([-n + 1, n - 1]\)</span>, and the range of <span class="arithmatex">\(row + col\)</span> is <span class="arithmatex">\([0, 2n - 2]\)</span>. Therefore, the number of both main diagonals and anti-diagonals is <span class="arithmatex">\(2n - 1\)</span>, meaning the length of both arrays <code>diags1</code> and <code>diags2</code> is <span class="arithmatex">\(2n - 1\)</span>.</p>
+<h3 id="3-code-implementation">3. &nbsp; Code implementation<a class="headerlink" href="#3-code-implementation" title="Permanent link">&para;</a></h3>
+<p>In a square matrix of order <span class="arithmatex">\(n\)</span>, <span class="arithmatex">\(row - col\)</span> ranges over <span class="arithmatex">\([-n + 1, n - 1]\)</span>, and <span class="arithmatex">\(row + col\)</span> ranges over <span class="arithmatex">\([0, 2n - 2]\)</span>. There are therefore <span class="arithmatex">\(2n - 1\)</span> diagonals in each direction, so both <code>diags1</code> and <code>diags2</code> have length <span class="arithmatex">\(2n - 1\)</span>.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -4726,34 +4798,34 @@
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="n">diags1</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="n">diags2</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="p">):</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="w">    </span><span class="sd">"""Backtracking algorithm: N queens"""</span>
-<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="c1"># When all rows are placed, record the solution</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="w">    </span><span class="sd">"""Backtracking: n-queens"""</span>
+<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="c1"># Record a solution when a queen has been placed in every row</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="k">if</span> <span class="n">row</span> <span class="o">==</span> <span class="n">n</span><span class="p">:</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">([</span><span class="nb">list</span><span class="p">(</span><span class="n">row</span><span class="p">)</span> <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">state</span><span class="p">])</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>        <span class="k">return</span>
-<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="c1"># Traverse all columns</span>
+<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="c1"># Iterate over all columns</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>    <span class="k">for</span> <span class="n">col</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>        <span class="c1"># Calculate the main diagonal and anti-diagonal corresponding to this cell</span>
+<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>        <span class="c1"># Calculate the main-diagonal and anti-diagonal indices for this square</span>
 <a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>        <span class="n">diag1</span> <span class="o">=</span> <span class="n">row</span> <span class="o">-</span> <span class="n">col</span> <span class="o">+</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
 <a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>        <span class="n">diag2</span> <span class="o">=</span> <span class="n">row</span> <span class="o">+</span> <span class="n">col</span>
-<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>        <span class="c1"># Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell</span>
+<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>        <span class="c1"># Prune: the square must have no queen in its column, main diagonal, or anti-diagonal</span>
 <a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a>        <span class="k">if</span> <span class="ow">not</span> <span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span> <span class="ow">and</span> <span class="ow">not</span> <span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]:</span>
-<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>            <span class="c1"># Attempt: place the queen in this cell</span>
+<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>            <span class="c1"># Try: place a queen on this square</span>
 <a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a>            <span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"Q"</span>
 <a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a>            <span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span> <span class="o">=</span> <span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span> <span class="o">=</span> <span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
-<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a>            <span class="c1"># Place the next row</span>
+<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a>            <span class="c1"># Place a queen in the next row</span>
 <a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a>            <span class="n">backtrack</span><span class="p">(</span><span class="n">row</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="n">res</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">diags1</span><span class="p">,</span> <span class="n">diags2</span><span class="p">)</span>
-<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a>            <span class="c1"># Backtrack: restore this cell to an empty cell</span>
+<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a>            <span class="c1"># Backtrack: restore this square to empty</span>
 <a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a>            <span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"#"</span>
 <a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a>            <span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span> <span class="o">=</span> <span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span> <span class="o">=</span> <span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span> <span class="o">=</span> <span class="kc">False</span>
 <a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a>
 <a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a><span class="k">def</span><span class="w"> </span><span class="nf">n_queens</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">str</span><span class="p">]]]:</span>
-<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a><span class="w">    </span><span class="sd">"""Solve N queens"""</span>
-<a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a>    <span class="c1"># Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell</span>
+<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a><span class="w">    </span><span class="sd">"""Solve the n-queens problem"""</span>
+<a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a>    <span class="c1"># Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square</span>
 <a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a>    <span class="n">state</span> <span class="o">=</span> <span class="p">[[</span><span class="s2">"#"</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span>
-<a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a>    <span class="n">cols</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span>  <span class="c1"># Record whether there is a queen in the column</span>
-<a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a>    <span class="n">diags1</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>  <span class="c1"># Record whether there is a queen on the main diagonal</span>
-<a id="__codelineno-0-37" name="__codelineno-0-37" href="#__codelineno-0-37"></a>    <span class="n">diags2</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>  <span class="c1"># Record whether there is a queen on the anti-diagonal</span>
+<a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a>    <span class="n">cols</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span>  <span class="c1"># Track whether a column contains a queen</span>
+<a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a>    <span class="n">diags1</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>  <span class="c1"># Track whether a main diagonal contains a queen</span>
+<a id="__codelineno-0-37" name="__codelineno-0-37" href="#__codelineno-0-37"></a>    <span class="n">diags2</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>  <span class="c1"># Track whether an anti-diagonal contains a queen</span>
 <a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></a>    <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
 <a id="__codelineno-0-39" name="__codelineno-0-39" href="#__codelineno-0-39"></a>    <span class="n">backtrack</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="n">res</span><span class="p">,</span> <span class="n">cols</span><span class="p">,</span> <span class="n">diags1</span><span class="p">,</span> <span class="n">diags2</span><span class="p">)</span>
 <a id="__codelineno-0-40" name="__codelineno-0-40" href="#__codelineno-0-40"></a>
@@ -4761,40 +4833,40 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">n_queens.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Backtracking algorithm: N queens */</span>
+<div class="highlight"><span class="filename">n_queens.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Backtracking: n-queens */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">row</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">string</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">string</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">cols</span><span class="p">,</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">               </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">diags2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// When all rows are placed, record the solution</span>
+<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// Record a solution when a queen has been placed in every row</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">state</span><span class="p">);</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="c1">// Traverse all columns</span>
+<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="c1">// Iterate over all columns</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="c1">// Calculate the main diagonal and anti-diagonal corresponding to this cell</span>
+<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="c1">// Calculate the main-diagonal and anti-diagonal indices for this square</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">col</span><span class="p">;</span>
-<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">        </span><span class="c1">// Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell</span>
+<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">        </span><span class="c1">// Prune: the square must have no queen in its column, main diagonal, or anti-diagonal</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">            </span><span class="c1">// Attempt: place the queen in this cell</span>
+<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">            </span><span class="c1">// Try: place a queen on this square</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">            </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"Q"</span><span class="p">;</span>
 <a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">            </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span>
-<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">            </span><span class="c1">// Place the next row</span>
+<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">            </span><span class="c1">// Place a queen in the next row</span>
 <a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
-<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w">            </span><span class="c1">// Backtrack: restore this cell to an empty cell</span>
+<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w">            </span><span class="c1">// Backtrack: restore this square to empty</span>
 <a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w">            </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"#"</span><span class="p">;</span>
 <a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="w">            </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">false</span><span class="p">;</span>
 <a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="p">}</span>
 <a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a>
-<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="cm">/* Solve N queens */</span>
+<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="cm">/* Solve the n-queens problem */</span>
 <a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">string</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="n">nQueens</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="w">    </span><span class="c1">// Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell</span>
+<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="w">    </span><span class="c1">// Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square</span>
 <a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">string</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">string</span><span class="o">&gt;</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="s">"#"</span><span class="p">));</span>
-<a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">cols</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span><span class="w">           </span><span class="c1">// Record whether there is a queen in the column</span>
-<a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">diags1</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the main diagonal</span>
-<a id="__codelineno-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">diags2</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the anti-diagonal</span>
+<a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">cols</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span><span class="w">           </span><span class="c1">// Track whether a column contains a queen</span>
+<a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">diags1</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Track whether a main diagonal contains a queen</span>
+<a id="__codelineno-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">diags2</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Track whether an anti-diagonal contains a queen</span>
 <a id="__codelineno-1-35" name="__codelineno-1-35" href="#__codelineno-1-35"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">string</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-1-36" name="__codelineno-1-36" href="#__codelineno-1-36"></a>
 <a id="__codelineno-1-37" name="__codelineno-1-37" href="#__codelineno-1-37"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
@@ -4804,10 +4876,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">n_queens.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Backtracking algorithm: N queens */</span>
+<div class="highlight"><span class="filename">n_queens.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Backtracking: n-queens */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">row</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">,</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">        </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// When all rows are placed, record the solution</span>
+<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// Record a solution when a queen has been placed in every row</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">        </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">copyState</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;</span><span class="w"> </span><span class="n">sRow</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4816,28 +4888,28 @@
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">copyState</span><span class="p">);</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">    </span><span class="c1">// Traverse all columns</span>
+<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">    </span><span class="c1">// Iterate over all columns</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="c1">// Calculate the main diagonal and anti-diagonal corresponding to this cell</span>
+<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="c1">// Calculate the main-diagonal and anti-diagonal indices for this square</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">col</span><span class="p">;</span>
-<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">        </span><span class="c1">// Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell</span>
+<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">        </span><span class="c1">// Prune: the square must have no queen in its column, main diagonal, or anti-diagonal</span>
 <a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">            </span><span class="c1">// Attempt: place the queen in this cell</span>
+<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">            </span><span class="c1">// Try: place a queen on this square</span>
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">row</span><span class="p">).</span><span class="na">set</span><span class="p">(</span><span class="n">col</span><span class="p">,</span><span class="w"> </span><span class="s">"Q"</span><span class="p">);</span>
 <a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="w">            </span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
-<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="w">            </span><span class="c1">// Place the next row</span>
+<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="w">            </span><span class="c1">// Place a queen in the next row</span>
 <a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
-<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="w">            </span><span class="c1">// Backtrack: restore this cell to an empty cell</span>
+<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="w">            </span><span class="c1">// Backtrack: restore this square to empty</span>
 <a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">row</span><span class="p">).</span><span class="na">set</span><span class="p">(</span><span class="n">col</span><span class="p">,</span><span class="w"> </span><span class="s">"#"</span><span class="p">);</span>
 <a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a><span class="w">            </span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
 <a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a><span class="p">}</span>
 <a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a>
-<a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="cm">/* Solve N queens */</span>
+<a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="cm">/* Solve the n-queens problem */</span>
 <a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="nf">nQueens</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a><span class="w">    </span><span class="c1">// Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell</span>
+<a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a><span class="w">    </span><span class="c1">// Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square</span>
 <a id="__codelineno-2-35" name="__codelineno-2-35" href="#__codelineno-2-35"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-37" name="__codelineno-2-37" href="#__codelineno-2-37"></a><span class="w">        </span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
@@ -4846,9 +4918,9 @@
 <a id="__codelineno-2-40" name="__codelineno-2-40" href="#__codelineno-2-40"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-2-41" name="__codelineno-2-41" href="#__codelineno-2-41"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">row</span><span class="p">);</span>
 <a id="__codelineno-2-42" name="__codelineno-2-42" href="#__codelineno-2-42"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-43" name="__codelineno-2-43" href="#__codelineno-2-43"></a><span class="w">    </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record whether there is a queen in the column</span>
-<a id="__codelineno-2-44" name="__codelineno-2-44" href="#__codelineno-2-44"></a><span class="w">    </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the main diagonal</span>
-<a id="__codelineno-2-45" name="__codelineno-2-45" href="#__codelineno-2-45"></a><span class="w">    </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the anti-diagonal</span>
+<a id="__codelineno-2-43" name="__codelineno-2-43" href="#__codelineno-2-43"></a><span class="w">    </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// Track whether a column contains a queen</span>
+<a id="__codelineno-2-44" name="__codelineno-2-44" href="#__codelineno-2-44"></a><span class="w">    </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// Track whether a main diagonal contains a queen</span>
+<a id="__codelineno-2-45" name="__codelineno-2-45" href="#__codelineno-2-45"></a><span class="w">    </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// Track whether an anti-diagonal contains a queen</span>
 <a id="__codelineno-2-46" name="__codelineno-2-46" href="#__codelineno-2-46"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-2-47" name="__codelineno-2-47" href="#__codelineno-2-47"></a>
 <a id="__codelineno-2-48" name="__codelineno-2-48" href="#__codelineno-2-48"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
@@ -4858,10 +4930,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">n_queens.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Backtracking algorithm: N queens */</span>
+<div class="highlight"><span class="filename">n_queens.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Backtracking: n-queens */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">row</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">string</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">string</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">,</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">        </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">diags2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// When all rows are placed, record the solution</span>
+<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Record a solution when a queen has been placed in every row</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">        </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">string</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">copyState</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">string</span><span class="o">&gt;</span><span class="w"> </span><span class="n">sRow</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4870,28 +4942,28 @@
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">copyState</span><span class="p">);</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">    </span><span class="c1">// Traverse all columns</span>
+<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">    </span><span class="c1">// Iterate over all columns</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">        </span><span class="c1">// Calculate the main diagonal and anti-diagonal corresponding to this cell</span>
+<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">        </span><span class="c1">// Calculate the main-diagonal and anti-diagonal indices for this square</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">col</span><span class="p">;</span>
-<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">        </span><span class="c1">// Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell</span>
+<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">        </span><span class="c1">// Prune: the square must have no queen in its column, main diagonal, or anti-diagonal</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">            </span><span class="c1">// Attempt: place the queen in this cell</span>
+<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">            </span><span class="c1">// Try: place a queen on this square</span>
 <a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w">            </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"Q"</span><span class="p">;</span>
 <a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w">            </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">true</span><span class="p">;</span>
-<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="w">            </span><span class="c1">// Place the next row</span>
+<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="w">            </span><span class="c1">// Place a queen in the next row</span>
 <a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w">            </span><span class="n">Backtrack</span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
-<a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="w">            </span><span class="c1">// Backtrack: restore this cell to an empty cell</span>
+<a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="w">            </span><span class="c1">// Backtrack: restore this square to empty</span>
 <a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="w">            </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"#"</span><span class="p">;</span>
 <a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a><span class="w">            </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">false</span><span class="p">;</span>
 <a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a><span class="p">}</span>
 <a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a>
-<a id="__codelineno-3-32" name="__codelineno-3-32" href="#__codelineno-3-32"></a><span class="cm">/* Solve N queens */</span>
+<a id="__codelineno-3-32" name="__codelineno-3-32" href="#__codelineno-3-32"></a><span class="cm">/* Solve the n-queens problem */</span>
 <a id="__codelineno-3-33" name="__codelineno-3-33" href="#__codelineno-3-33"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">string</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="n">NQueens</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-34" name="__codelineno-3-34" href="#__codelineno-3-34"></a><span class="w">    </span><span class="c1">// Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell</span>
+<a id="__codelineno-3-34" name="__codelineno-3-34" href="#__codelineno-3-34"></a><span class="w">    </span><span class="c1">// Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square</span>
 <a id="__codelineno-3-35" name="__codelineno-3-35" href="#__codelineno-3-35"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">string</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-3-36" name="__codelineno-3-36" href="#__codelineno-3-36"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-37" name="__codelineno-3-37" href="#__codelineno-3-37"></a><span class="w">        </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">string</span><span class="o">&gt;</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
@@ -4900,9 +4972,9 @@
 <a id="__codelineno-3-40" name="__codelineno-3-40" href="#__codelineno-3-40"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-3-41" name="__codelineno-3-41" href="#__codelineno-3-41"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">row</span><span class="p">);</span>
 <a id="__codelineno-3-42" name="__codelineno-3-42" href="#__codelineno-3-42"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-43" name="__codelineno-3-43" href="#__codelineno-3-43"></a><span class="w">    </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">bool</span><span class="p">[</span><span class="n">n</span><span class="p">];</span><span class="w"> </span><span class="c1">// Record whether there is a queen in the column</span>
-<a id="__codelineno-3-44" name="__codelineno-3-44" href="#__codelineno-3-44"></a><span class="w">    </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">bool</span><span class="p">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the main diagonal</span>
-<a id="__codelineno-3-45" name="__codelineno-3-45" href="#__codelineno-3-45"></a><span class="w">    </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">bool</span><span class="p">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the anti-diagonal</span>
+<a id="__codelineno-3-43" name="__codelineno-3-43" href="#__codelineno-3-43"></a><span class="w">    </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">bool</span><span class="p">[</span><span class="n">n</span><span class="p">];</span><span class="w"> </span><span class="c1">// Track whether a column contains a queen</span>
+<a id="__codelineno-3-44" name="__codelineno-3-44" href="#__codelineno-3-44"></a><span class="w">    </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">bool</span><span class="p">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="c1">// Track whether a main diagonal contains a queen</span>
+<a id="__codelineno-3-45" name="__codelineno-3-45" href="#__codelineno-3-45"></a><span class="w">    </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">bool</span><span class="p">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="c1">// Track whether an anti-diagonal contains a queen</span>
 <a id="__codelineno-3-46" name="__codelineno-3-46" href="#__codelineno-3-46"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">string</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-3-47" name="__codelineno-3-47" href="#__codelineno-3-47"></a>
 <a id="__codelineno-3-48" name="__codelineno-3-48" href="#__codelineno-3-48"></a><span class="w">    </span><span class="n">Backtrack</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
@@ -4912,9 +4984,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">n_queens.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Backtracking algorithm: N queens */</span>
+<div class="highlight"><span class="filename">n_queens.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Backtracking: n-queens */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">row</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">*</span><span class="p">[][]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[][][]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="kt">bool</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// When all rows are placed, record the solution</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Record a solution when a queen has been placed in every row</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">        </span><span class="nx">newState</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">))</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">_</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">newState</span><span class="w"> </span><span class="p">{</span>
@@ -4925,28 +4997,28 @@
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="o">*</span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">newState</span><span class="p">)</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="c1">// Traverse all columns</span>
+<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="c1">// Iterate over all columns</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">col</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">        </span><span class="c1">// Calculate the main diagonal and anti-diagonal corresponding to this cell</span>
+<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">        </span><span class="c1">// Calculate the main-diagonal and anti-diagonal indices for this square</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">        </span><span class="nx">diag1</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">        </span><span class="nx">diag2</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">col</span>
-<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">        </span><span class="c1">// Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell</span>
+<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">        </span><span class="c1">// Prune: the square must have no queen in its column, main diagonal, or anti-diagonal</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="p">(</span><span class="o">*</span><span class="nx">cols</span><span class="p">)[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="p">(</span><span class="o">*</span><span class="nx">diags1</span><span class="p">)[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="p">(</span><span class="o">*</span><span class="nx">diags2</span><span class="p">)[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">            </span><span class="c1">// Attempt: place the queen in this cell</span>
+<a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">            </span><span class="c1">// Try: place a queen on this square</span>
 <a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">            </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"Q"</span>
 <a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w">            </span><span class="p">(</span><span class="o">*</span><span class="nx">cols</span><span class="p">)[</span><span class="nx">col</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags1</span><span class="p">)[</span><span class="nx">diag1</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags2</span><span class="p">)[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">,</span><span class="w"> </span><span class="kc">true</span><span class="p">,</span><span class="w"> </span><span class="kc">true</span>
-<a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="w">            </span><span class="c1">// Place the next row</span>
+<a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="w">            </span><span class="c1">// Place a queen in the next row</span>
 <a id="__codelineno-4-25" name="__codelineno-4-25" href="#__codelineno-4-25"></a><span class="w">            </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">row</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="p">)</span>
-<a id="__codelineno-4-26" name="__codelineno-4-26" href="#__codelineno-4-26"></a><span class="w">            </span><span class="c1">// Backtrack: restore this cell to an empty cell</span>
+<a id="__codelineno-4-26" name="__codelineno-4-26" href="#__codelineno-4-26"></a><span class="w">            </span><span class="c1">// Backtrack: restore this square to empty</span>
 <a id="__codelineno-4-27" name="__codelineno-4-27" href="#__codelineno-4-27"></a><span class="w">            </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"#"</span>
 <a id="__codelineno-4-28" name="__codelineno-4-28" href="#__codelineno-4-28"></a><span class="w">            </span><span class="p">(</span><span class="o">*</span><span class="nx">cols</span><span class="p">)[</span><span class="nx">col</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags1</span><span class="p">)[</span><span class="nx">diag1</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">diags2</span><span class="p">)[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span>
 <a id="__codelineno-4-29" name="__codelineno-4-29" href="#__codelineno-4-29"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-4-30" name="__codelineno-4-30" href="#__codelineno-4-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-31" name="__codelineno-4-31" href="#__codelineno-4-31"></a><span class="p">}</span>
 <a id="__codelineno-4-32" name="__codelineno-4-32" href="#__codelineno-4-32"></a>
-<a id="__codelineno-4-33" name="__codelineno-4-33" href="#__codelineno-4-33"></a><span class="cm">/* Solve N queens */</span>
+<a id="__codelineno-4-33" name="__codelineno-4-33" href="#__codelineno-4-33"></a><span class="cm">/* Solve the n-queens problem */</span>
 <a id="__codelineno-4-34" name="__codelineno-4-34" href="#__codelineno-4-34"></a><span class="kd">func</span><span class="w"> </span><span class="nx">nQueens</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">[][][]</span><span class="kt">string</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-35" name="__codelineno-4-35" href="#__codelineno-4-35"></a><span class="w">    </span><span class="c1">// Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell</span>
+<a id="__codelineno-4-35" name="__codelineno-4-35" href="#__codelineno-4-35"></a><span class="w">    </span><span class="c1">// Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square</span>
 <a id="__codelineno-4-36" name="__codelineno-4-36" href="#__codelineno-4-36"></a><span class="w">    </span><span class="nx">state</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
 <a id="__codelineno-4-37" name="__codelineno-4-37" href="#__codelineno-4-37"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-38" name="__codelineno-4-38" href="#__codelineno-4-38"></a><span class="w">        </span><span class="nx">row</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
@@ -4955,7 +5027,7 @@
 <a id="__codelineno-4-41" name="__codelineno-4-41" href="#__codelineno-4-41"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-4-42" name="__codelineno-4-42" href="#__codelineno-4-42"></a><span class="w">        </span><span class="nx">state</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">row</span>
 <a id="__codelineno-4-43" name="__codelineno-4-43" href="#__codelineno-4-43"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-44" name="__codelineno-4-44" href="#__codelineno-4-44"></a><span class="w">    </span><span class="c1">// Record whether there is a queen in the column</span>
+<a id="__codelineno-4-44" name="__codelineno-4-44" href="#__codelineno-4-44"></a><span class="w">    </span><span class="c1">// Track whether a column contains a queen</span>
 <a id="__codelineno-4-45" name="__codelineno-4-45" href="#__codelineno-4-45"></a><span class="w">    </span><span class="nx">cols</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">bool</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
 <a id="__codelineno-4-46" name="__codelineno-4-46" href="#__codelineno-4-46"></a><span class="w">    </span><span class="nx">diags1</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">bool</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="o">*</span><span class="nx">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-4-47" name="__codelineno-4-47" href="#__codelineno-4-47"></a><span class="w">    </span><span class="nx">diags2</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">bool</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="o">*</span><span class="nx">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
@@ -4966,28 +5038,28 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">n_queens.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Backtracking algorithm: N queens */</span>
+<div class="highlight"><span class="filename">n_queens.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Backtracking: n-queens */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">row</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[[</span><span class="nb">String</span><span class="p">]],</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[[[</span><span class="nb">String</span><span class="p">]]],</span><span class="w"> </span><span class="n">cols</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Bool</span><span class="p">],</span><span class="w"> </span><span class="n">diags1</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Bool</span><span class="p">],</span><span class="w"> </span><span class="n">diags2</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Bool</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// When all rows are placed, record the solution</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Record a solution when a queen has been placed in every row</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="c1">// Traverse all columns</span>
+<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="c1">// Iterate over all columns</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="c1">// Calculate the main diagonal and anti-diagonal corresponding to this cell</span>
+<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="c1">// Calculate the main-diagonal and anti-diagonal indices for this square</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">diag1</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">diag2</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">col</span>
-<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="c1">// Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell</span>
+<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="c1">// Prune: the square must have no queen in its column, main diagonal, or anti-diagonal</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">            </span><span class="c1">// Attempt: place the queen in this cell</span>
+<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">            </span><span class="c1">// Try: place a queen on this square</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">            </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"Q"</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">            </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">true</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">            </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">true</span>
 <a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">            </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">true</span>
-<a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w">            </span><span class="c1">// Place the next row</span>
+<a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w">            </span><span class="c1">// Place a queen in the next row</span>
 <a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">row</span><span class="p">:</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">diags2</span><span class="p">)</span>
-<a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="w">            </span><span class="c1">// Backtrack: restore this cell to an empty cell</span>
+<a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="w">            </span><span class="c1">// Backtrack: restore this square to empty</span>
 <a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="w">            </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"#"</span>
 <a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="w">            </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">false</span>
 <a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a><span class="w">            </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">false</span>
@@ -4996,13 +5068,13 @@
 <a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-29" name="__codelineno-5-29" href="#__codelineno-5-29"></a><span class="p">}</span>
 <a id="__codelineno-5-30" name="__codelineno-5-30" href="#__codelineno-5-30"></a>
-<a id="__codelineno-5-31" name="__codelineno-5-31" href="#__codelineno-5-31"></a><span class="cm">/* Solve N queens */</span>
+<a id="__codelineno-5-31" name="__codelineno-5-31" href="#__codelineno-5-31"></a><span class="cm">/* Solve the n-queens problem */</span>
 <a id="__codelineno-5-32" name="__codelineno-5-32" href="#__codelineno-5-32"></a><span class="kd">func</span><span class="w"> </span><span class="nf">nQueens</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[[[</span><span class="nb">String</span><span class="p">]]]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-33" name="__codelineno-5-33" href="#__codelineno-5-33"></a><span class="w">    </span><span class="c1">// Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell</span>
+<a id="__codelineno-5-33" name="__codelineno-5-33" href="#__codelineno-5-33"></a><span class="w">    </span><span class="c1">// Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square</span>
 <a id="__codelineno-5-34" name="__codelineno-5-34" href="#__codelineno-5-34"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">state</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="s">"#"</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">),</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-5-35" name="__codelineno-5-35" href="#__codelineno-5-35"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">cols</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="c1">// Record whether there is a queen in the column</span>
-<a id="__codelineno-5-36" name="__codelineno-5-36" href="#__codelineno-5-36"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">diags1</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the main diagonal</span>
-<a id="__codelineno-5-37" name="__codelineno-5-37" href="#__codelineno-5-37"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">diags2</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the anti-diagonal</span>
+<a id="__codelineno-5-35" name="__codelineno-5-35" href="#__codelineno-5-35"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">cols</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="c1">// Track whether a column contains a queen</span>
+<a id="__codelineno-5-36" name="__codelineno-5-36" href="#__codelineno-5-36"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">diags1</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="c1">// Track whether a main diagonal contains a queen</span>
+<a id="__codelineno-5-37" name="__codelineno-5-37" href="#__codelineno-5-37"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">diags2</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="c1">// Track whether an anti-diagonal contains a queen</span>
 <a id="__codelineno-5-38" name="__codelineno-5-38" href="#__codelineno-5-38"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="p">:</span><span class="w"> </span><span class="p">[[[</span><span class="nb">String</span><span class="p">]]]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
 <a id="__codelineno-5-39" name="__codelineno-5-39" href="#__codelineno-5-39"></a>
 <a id="__codelineno-5-40" name="__codelineno-5-40" href="#__codelineno-5-40"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">row</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">diags2</span><span class="p">)</span>
@@ -5012,39 +5084,39 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">n_queens.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Backtracking algorithm: N queens */</span>
+<div class="highlight"><span class="filename">n_queens.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Backtracking: n-queens */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">row</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// When all rows are placed, record the solution</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Record a solution when a queen has been placed in every row</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">row</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">state</span><span class="p">.</span><span class="nx">map</span><span class="p">((</span><span class="nx">row</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">row</span><span class="p">.</span><span class="nx">slice</span><span class="p">()));</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="c1">// Traverse all columns</span>
+<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="c1">// Iterate over all columns</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">col</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="c1">// Calculate the main diagonal and anti-diagonal corresponding to this cell</span>
+<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="c1">// Calculate the main-diagonal and anti-diagonal indices for this square</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">col</span><span class="p">;</span>
-<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="c1">// Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell</span>
+<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="c1">// Prune: the square must have no queen in its column, main diagonal, or anti-diagonal</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">            </span><span class="c1">// Attempt: place the queen in this cell</span>
+<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">            </span><span class="c1">// Try: place a queen on this square</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">            </span><span class="nx">state</span><span class="p">[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s1">'Q'</span><span class="p">;</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">            </span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
-<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">            </span><span class="c1">// Place the next row</span>
+<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">            </span><span class="c1">// Place a queen in the next row</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">            </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="p">);</span>
-<a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">            </span><span class="c1">// Backtrack: restore this cell to an empty cell</span>
+<a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">            </span><span class="c1">// Backtrack: restore this square to empty</span>
 <a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">            </span><span class="nx">state</span><span class="p">[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s1">'#'</span><span class="p">;</span>
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="w">            </span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
 <a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="p">}</span>
 <a id="__codelineno-6-26" name="__codelineno-6-26" href="#__codelineno-6-26"></a>
-<a id="__codelineno-6-27" name="__codelineno-6-27" href="#__codelineno-6-27"></a><span class="cm">/* Solve N queens */</span>
+<a id="__codelineno-6-27" name="__codelineno-6-27" href="#__codelineno-6-27"></a><span class="cm">/* Solve the n-queens problem */</span>
 <a id="__codelineno-6-28" name="__codelineno-6-28" href="#__codelineno-6-28"></a><span class="kd">function</span><span class="w"> </span><span class="nx">nQueens</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-29" name="__codelineno-6-29" href="#__codelineno-6-29"></a><span class="w">    </span><span class="c1">// Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell</span>
+<a id="__codelineno-6-29" name="__codelineno-6-29" href="#__codelineno-6-29"></a><span class="w">    </span><span class="c1">// Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square</span>
 <a id="__codelineno-6-30" name="__codelineno-6-30" href="#__codelineno-6-30"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="s1">'#'</span><span class="p">));</span>
-<a id="__codelineno-6-31" name="__codelineno-6-31" href="#__codelineno-6-31"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record whether there is a queen in the column</span>
-<a id="__codelineno-6-32" name="__codelineno-6-32" href="#__codelineno-6-32"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the main diagonal</span>
-<a id="__codelineno-6-33" name="__codelineno-6-33" href="#__codelineno-6-33"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the anti-diagonal</span>
+<a id="__codelineno-6-31" name="__codelineno-6-31" href="#__codelineno-6-31"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Track whether a column contains a queen</span>
+<a id="__codelineno-6-32" name="__codelineno-6-32" href="#__codelineno-6-32"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Track whether a main diagonal contains a queen</span>
+<a id="__codelineno-6-33" name="__codelineno-6-33" href="#__codelineno-6-33"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Track whether an anti-diagonal contains a queen</span>
 <a id="__codelineno-6-34" name="__codelineno-6-34" href="#__codelineno-6-34"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-6-35" name="__codelineno-6-35" href="#__codelineno-6-35"></a>
 <a id="__codelineno-6-36" name="__codelineno-6-36" href="#__codelineno-6-36"></a><span class="w">    </span><span class="nx">backtrack</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="p">);</span>
@@ -5053,7 +5125,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">n_queens.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Backtracking algorithm: N queens */</span>
+<div class="highlight"><span class="filename">n_queens.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Backtracking: n-queens */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="nx">row</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
@@ -5063,37 +5135,37 @@
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="nx">diags1</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="p">[],</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="nx">diags2</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="p">[]</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="c1">// When all rows are placed, record the solution</span>
+<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="c1">// Record a solution when a queen has been placed in every row</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">row</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">state</span><span class="p">.</span><span class="nx">map</span><span class="p">((</span><span class="nx">row</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">row</span><span class="p">.</span><span class="nx">slice</span><span class="p">()));</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="c1">// Traverse all columns</span>
+<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="c1">// Iterate over all columns</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">col</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">        </span><span class="c1">// Calculate the main diagonal and anti-diagonal corresponding to this cell</span>
+<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">        </span><span class="c1">// Calculate the main-diagonal and anti-diagonal indices for this square</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">col</span><span class="p">;</span>
-<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">        </span><span class="c1">// Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell</span>
+<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">        </span><span class="c1">// Prune: the square must have no queen in its column, main diagonal, or anti-diagonal</span>
 <a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">            </span><span class="c1">// Attempt: place the queen in this cell</span>
+<a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">            </span><span class="c1">// Try: place a queen on this square</span>
 <a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">            </span><span class="nx">state</span><span class="p">[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s1">'Q'</span><span class="p">;</span>
 <a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">            </span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
-<a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="w">            </span><span class="c1">// Place the next row</span>
+<a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="w">            </span><span class="c1">// Place a queen in the next row</span>
 <a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="w">            </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="p">);</span>
-<a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w">            </span><span class="c1">// Backtrack: restore this cell to an empty cell</span>
+<a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w">            </span><span class="c1">// Backtrack: restore this square to empty</span>
 <a id="__codelineno-7-29" name="__codelineno-7-29" href="#__codelineno-7-29"></a><span class="w">            </span><span class="nx">state</span><span class="p">[</span><span class="nx">row</span><span class="p">][</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s1">'#'</span><span class="p">;</span>
 <a id="__codelineno-7-30" name="__codelineno-7-30" href="#__codelineno-7-30"></a><span class="w">            </span><span class="nx">cols</span><span class="p">[</span><span class="nx">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags1</span><span class="p">[</span><span class="nx">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">diags2</span><span class="p">[</span><span class="nx">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
 <a id="__codelineno-7-31" name="__codelineno-7-31" href="#__codelineno-7-31"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-7-32" name="__codelineno-7-32" href="#__codelineno-7-32"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-33" name="__codelineno-7-33" href="#__codelineno-7-33"></a><span class="p">}</span>
 <a id="__codelineno-7-34" name="__codelineno-7-34" href="#__codelineno-7-34"></a>
-<a id="__codelineno-7-35" name="__codelineno-7-35" href="#__codelineno-7-35"></a><span class="cm">/* Solve N queens */</span>
+<a id="__codelineno-7-35" name="__codelineno-7-35" href="#__codelineno-7-35"></a><span class="cm">/* Solve the n-queens problem */</span>
 <a id="__codelineno-7-36" name="__codelineno-7-36" href="#__codelineno-7-36"></a><span class="kd">function</span><span class="w"> </span><span class="nx">nQueens</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">[][][]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-37" name="__codelineno-7-37" href="#__codelineno-7-37"></a><span class="w">    </span><span class="c1">// Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell</span>
+<a id="__codelineno-7-37" name="__codelineno-7-37" href="#__codelineno-7-37"></a><span class="w">    </span><span class="c1">// Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square</span>
 <a id="__codelineno-7-38" name="__codelineno-7-38" href="#__codelineno-7-38"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">n</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="s1">'#'</span><span class="p">));</span>
-<a id="__codelineno-7-39" name="__codelineno-7-39" href="#__codelineno-7-39"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record whether there is a queen in the column</span>
-<a id="__codelineno-7-40" name="__codelineno-7-40" href="#__codelineno-7-40"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the main diagonal</span>
-<a id="__codelineno-7-41" name="__codelineno-7-41" href="#__codelineno-7-41"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the anti-diagonal</span>
+<a id="__codelineno-7-39" name="__codelineno-7-39" href="#__codelineno-7-39"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Track whether a column contains a queen</span>
+<a id="__codelineno-7-40" name="__codelineno-7-40" href="#__codelineno-7-40"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Track whether a main diagonal contains a queen</span>
+<a id="__codelineno-7-41" name="__codelineno-7-41" href="#__codelineno-7-41"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Track whether an anti-diagonal contains a queen</span>
 <a id="__codelineno-7-42" name="__codelineno-7-42" href="#__codelineno-7-42"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">[][][]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-7-43" name="__codelineno-7-43" href="#__codelineno-7-43"></a>
 <a id="__codelineno-7-44" name="__codelineno-7-44" href="#__codelineno-7-44"></a><span class="w">    </span><span class="nx">backtrack</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">cols</span><span class="p">,</span><span class="w"> </span><span class="nx">diags1</span><span class="p">,</span><span class="w"> </span><span class="nx">diags2</span><span class="p">);</span>
@@ -5102,7 +5174,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">n_queens.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Backtracking algorithm: N queens */</span>
+<div class="highlight"><span class="filename">n_queens.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Backtracking: n-queens */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">row</span><span class="p">,</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span>
@@ -5112,7 +5184,7 @@
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">diags2</span><span class="p">,</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="c1">// When all rows are placed, record the solution</span>
+<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="c1">// Record a solution when a queen has been placed in every row</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">String</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">copyState</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">String</span><span class="o">&gt;</span><span class="w"> </span><span class="n">sRow</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5121,21 +5193,21 @@
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">copyState</span><span class="p">);</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">    </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">  </span><span class="c1">// Traverse all columns</span>
+<a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">  </span><span class="c1">// Iterate over all columns</span>
 <a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="w">    </span><span class="c1">// Calculate the main diagonal and anti-diagonal corresponding to this cell</span>
+<a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="w">    </span><span class="c1">// Calculate the main-diagonal and anti-diagonal indices for this square</span>
 <a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">col</span><span class="p">;</span>
-<a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="w">    </span><span class="c1">// Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell</span>
+<a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="w">    </span><span class="c1">// Prune: the square must have no queen in its column, main diagonal, or anti-diagonal</span>
 <a id="__codelineno-8-26" name="__codelineno-8-26" href="#__codelineno-8-26"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a><span class="w">      </span><span class="c1">// Attempt: place the queen in this cell</span>
+<a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a><span class="w">      </span><span class="c1">// Try: place a queen on this square</span>
 <a id="__codelineno-8-28" name="__codelineno-8-28" href="#__codelineno-8-28"></a><span class="w">      </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"Q"</span><span class="p">;</span>
 <a id="__codelineno-8-29" name="__codelineno-8-29" href="#__codelineno-8-29"></a><span class="w">      </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
 <a id="__codelineno-8-30" name="__codelineno-8-30" href="#__codelineno-8-30"></a><span class="w">      </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
 <a id="__codelineno-8-31" name="__codelineno-8-31" href="#__codelineno-8-31"></a><span class="w">      </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
-<a id="__codelineno-8-32" name="__codelineno-8-32" href="#__codelineno-8-32"></a><span class="w">      </span><span class="c1">// Place the next row</span>
+<a id="__codelineno-8-32" name="__codelineno-8-32" href="#__codelineno-8-32"></a><span class="w">      </span><span class="c1">// Place a queen in the next row</span>
 <a id="__codelineno-8-33" name="__codelineno-8-33" href="#__codelineno-8-33"></a><span class="w">      </span><span class="n">backtrack</span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
-<a id="__codelineno-8-34" name="__codelineno-8-34" href="#__codelineno-8-34"></a><span class="w">      </span><span class="c1">// Backtrack: restore this cell to an empty cell</span>
+<a id="__codelineno-8-34" name="__codelineno-8-34" href="#__codelineno-8-34"></a><span class="w">      </span><span class="c1">// Backtrack: restore this square to empty</span>
 <a id="__codelineno-8-35" name="__codelineno-8-35" href="#__codelineno-8-35"></a><span class="w">      </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"#"</span><span class="p">;</span>
 <a id="__codelineno-8-36" name="__codelineno-8-36" href="#__codelineno-8-36"></a><span class="w">      </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
 <a id="__codelineno-8-37" name="__codelineno-8-37" href="#__codelineno-8-37"></a><span class="w">      </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
@@ -5144,13 +5216,13 @@
 <a id="__codelineno-8-40" name="__codelineno-8-40" href="#__codelineno-8-40"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-41" name="__codelineno-8-41" href="#__codelineno-8-41"></a><span class="p">}</span>
 <a id="__codelineno-8-42" name="__codelineno-8-42" href="#__codelineno-8-42"></a>
-<a id="__codelineno-8-43" name="__codelineno-8-43" href="#__codelineno-8-43"></a><span class="cm">/* Solve N queens */</span>
+<a id="__codelineno-8-43" name="__codelineno-8-43" href="#__codelineno-8-43"></a><span class="cm">/* Solve the n-queens problem */</span>
 <a id="__codelineno-8-44" name="__codelineno-8-44" href="#__codelineno-8-44"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">String</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="n">nQueens</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-45" name="__codelineno-8-45" href="#__codelineno-8-45"></a><span class="w">  </span><span class="c1">// Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell</span>
+<a id="__codelineno-8-45" name="__codelineno-8-45" href="#__codelineno-8-45"></a><span class="w">  </span><span class="c1">// Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square</span>
 <a id="__codelineno-8-46" name="__codelineno-8-46" href="#__codelineno-8-46"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">String</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="s2">"#"</span><span class="p">));</span>
-<a id="__codelineno-8-47" name="__codelineno-8-47" href="#__codelineno-8-47"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record whether there is a queen in the column</span>
-<a id="__codelineno-8-48" name="__codelineno-8-48" href="#__codelineno-8-48"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the main diagonal</span>
-<a id="__codelineno-8-49" name="__codelineno-8-49" href="#__codelineno-8-49"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the anti-diagonal</span>
+<a id="__codelineno-8-47" name="__codelineno-8-47" href="#__codelineno-8-47"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Track whether a column contains a queen</span>
+<a id="__codelineno-8-48" name="__codelineno-8-48" href="#__codelineno-8-48"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Track whether a main diagonal contains a queen</span>
+<a id="__codelineno-8-49" name="__codelineno-8-49" href="#__codelineno-8-49"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">);</span><span class="w"> </span><span class="c1">// Track whether an anti-diagonal contains a queen</span>
 <a id="__codelineno-8-50" name="__codelineno-8-50" href="#__codelineno-8-50"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">String</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-8-51" name="__codelineno-8-51" href="#__codelineno-8-51"></a>
 <a id="__codelineno-8-52" name="__codelineno-8-52" href="#__codelineno-8-52"></a><span class="w">  </span><span class="n">backtrack</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
@@ -5160,7 +5232,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">n_queens.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Backtracking algorithm: N queens */</span>
+<div class="highlight"><span class="filename">n_queens.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Backtracking: n-queens */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="n">row</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span>
@@ -5170,37 +5242,37 @@
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="n">diags1</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">bool</span><span class="p">],</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="n">diags2</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">bool</span><span class="p">],</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="c1">// When all rows are placed, record the solution</span>
+<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="c1">// Record a solution when a queen has been placed in every row</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">    </span><span class="c1">// Traverse all columns</span>
+<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">    </span><span class="c1">// Iterate over all columns</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">        </span><span class="c1">// Calculate the main diagonal and anti-diagonal corresponding to this cell</span>
+<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">        </span><span class="c1">// Calculate the main-diagonal and anti-diagonal indices for this square</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">col</span><span class="p">;</span>
 <a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">col</span><span class="p">;</span>
-<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">        </span><span class="c1">// Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell</span>
+<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">        </span><span class="c1">// Prune: the square must have no queen in its column, main diagonal, or anti-diagonal</span>
 <a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="w">            </span><span class="c1">// Attempt: place the queen in this cell</span>
+<a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="w">            </span><span class="c1">// Try: place a queen on this square</span>
 <a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a><span class="w">            </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"Q"</span><span class="p">.</span><span class="n">into</span><span class="p">();</span>
 <a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a><span class="w">            </span><span class="p">(</span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">],</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">],</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">])</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">,</span><span class="w"> </span><span class="kc">true</span><span class="p">,</span><span class="w"> </span><span class="kc">true</span><span class="p">);</span>
-<a id="__codelineno-9-26" name="__codelineno-9-26" href="#__codelineno-9-26"></a><span class="w">            </span><span class="c1">// Place the next row</span>
+<a id="__codelineno-9-26" name="__codelineno-9-26" href="#__codelineno-9-26"></a><span class="w">            </span><span class="c1">// Place a queen in the next row</span>
 <a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
-<a id="__codelineno-9-28" name="__codelineno-9-28" href="#__codelineno-9-28"></a><span class="w">            </span><span class="c1">// Backtrack: restore this cell to an empty cell</span>
+<a id="__codelineno-9-28" name="__codelineno-9-28" href="#__codelineno-9-28"></a><span class="w">            </span><span class="c1">// Backtrack: restore this square to empty</span>
 <a id="__codelineno-9-29" name="__codelineno-9-29" href="#__codelineno-9-29"></a><span class="w">            </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"#"</span><span class="p">.</span><span class="n">into</span><span class="p">();</span>
 <a id="__codelineno-9-30" name="__codelineno-9-30" href="#__codelineno-9-30"></a><span class="w">            </span><span class="p">(</span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">],</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">],</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">])</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">);</span>
 <a id="__codelineno-9-31" name="__codelineno-9-31" href="#__codelineno-9-31"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-32" name="__codelineno-9-32" href="#__codelineno-9-32"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-33" name="__codelineno-9-33" href="#__codelineno-9-33"></a><span class="p">}</span>
 <a id="__codelineno-9-34" name="__codelineno-9-34" href="#__codelineno-9-34"></a>
-<a id="__codelineno-9-35" name="__codelineno-9-35" href="#__codelineno-9-35"></a><span class="cm">/* Solve N queens */</span>
+<a id="__codelineno-9-35" name="__codelineno-9-35" href="#__codelineno-9-35"></a><span class="cm">/* Solve the n-queens problem */</span>
 <a id="__codelineno-9-36" name="__codelineno-9-36" href="#__codelineno-9-36"></a><span class="k">fn</span><span class="w"> </span><span class="nf">n_queens</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">String</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-37" name="__codelineno-9-37" href="#__codelineno-9-37"></a><span class="w">    </span><span class="c1">// Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell</span>
+<a id="__codelineno-9-37" name="__codelineno-9-37" href="#__codelineno-9-37"></a><span class="w">    </span><span class="c1">// Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square</span>
 <a id="__codelineno-9-38" name="__codelineno-9-38" href="#__codelineno-9-38"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">String</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="fm">vec!</span><span class="p">[</span><span class="s">"#"</span><span class="p">.</span><span class="n">to_string</span><span class="p">();</span><span class="w"> </span><span class="n">n</span><span class="p">];</span><span class="w"> </span><span class="n">n</span><span class="p">];</span>
-<a id="__codelineno-9-39" name="__codelineno-9-39" href="#__codelineno-9-39"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="p">];</span><span class="w"> </span><span class="c1">// Record whether there is a queen in the column</span>
-<a id="__codelineno-9-40" name="__codelineno-9-40" href="#__codelineno-9-40"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the main diagonal</span>
-<a id="__codelineno-9-41" name="__codelineno-9-41" href="#__codelineno-9-41"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the anti-diagonal</span>
+<a id="__codelineno-9-39" name="__codelineno-9-39" href="#__codelineno-9-39"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="p">];</span><span class="w"> </span><span class="c1">// Track whether a column contains a queen</span>
+<a id="__codelineno-9-40" name="__codelineno-9-40" href="#__codelineno-9-40"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="c1">// Track whether a main diagonal contains a queen</span>
+<a id="__codelineno-9-41" name="__codelineno-9-41" href="#__codelineno-9-41"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="c1">// Track whether an anti-diagonal contains a queen</span>
 <a id="__codelineno-9-42" name="__codelineno-9-42" href="#__codelineno-9-42"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">String</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-9-43" name="__codelineno-9-43" href="#__codelineno-9-43"></a>
 <a id="__codelineno-9-44" name="__codelineno-9-44" href="#__codelineno-9-44"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span>
@@ -5218,10 +5290,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">n_queens.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Backtracking algorithm: N queens */</span>
+<div class="highlight"><span class="filename">n_queens.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Backtracking: n-queens */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">row</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">char</span><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">][</span><span class="n">MAX_SIZE</span><span class="p">],</span><span class="w"> </span><span class="kt">char</span><span class="w"> </span><span class="o">***</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">resSize</span><span class="p">,</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="n">cols</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">],</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">               </span><span class="kt">bool</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="c1">// When all rows are placed, record the solution</span>
+<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="c1">// Record a solution when a queen has been placed in every row</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="o">*</span><span class="n">resSize</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">char</span><span class="w"> </span><span class="o">**</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5231,38 +5303,38 @@
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">        </span><span class="p">(</span><span class="o">*</span><span class="n">resSize</span><span class="p">)</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="c1">// Traverse all columns</span>
+<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="c1">// Iterate over all columns</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">col</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="c1">// Calculate the main diagonal and anti-diagonal corresponding to this cell</span>
+<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="c1">// Calculate the main-diagonal and anti-diagonal indices for this square</span>
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">col</span><span class="p">;</span>
-<a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">        </span><span class="c1">// Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell</span>
+<a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">        </span><span class="c1">// Prune: the square must have no queen in its column, main diagonal, or anti-diagonal</span>
 <a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">            </span><span class="c1">// Attempt: place the queen in this cell</span>
+<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">            </span><span class="c1">// Try: place a queen on this square</span>
 <a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">            </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="sc">'Q'</span><span class="p">;</span>
 <a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">            </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span>
-<a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">            </span><span class="c1">// Place the next row</span>
+<a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">            </span><span class="c1">// Place a queen in the next row</span>
 <a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">resSize</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">);</span>
-<a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="w">            </span><span class="c1">// Backtrack: restore this cell to an empty cell</span>
+<a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="w">            </span><span class="c1">// Backtrack: restore this square to empty</span>
 <a id="__codelineno-10-27" name="__codelineno-10-27" href="#__codelineno-10-27"></a><span class="w">            </span><span class="n">state</span><span class="p">[</span><span class="n">row</span><span class="p">][</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="sc">'#'</span><span class="p">;</span>
 <a id="__codelineno-10-28" name="__codelineno-10-28" href="#__codelineno-10-28"></a><span class="w">            </span><span class="n">cols</span><span class="p">[</span><span class="n">col</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="n">diag1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="n">diag2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">false</span><span class="p">;</span>
 <a id="__codelineno-10-29" name="__codelineno-10-29" href="#__codelineno-10-29"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-10-30" name="__codelineno-10-30" href="#__codelineno-10-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-31" name="__codelineno-10-31" href="#__codelineno-10-31"></a><span class="p">}</span>
 <a id="__codelineno-10-32" name="__codelineno-10-32" href="#__codelineno-10-32"></a>
-<a id="__codelineno-10-33" name="__codelineno-10-33" href="#__codelineno-10-33"></a><span class="cm">/* Solve N queens */</span>
+<a id="__codelineno-10-33" name="__codelineno-10-33" href="#__codelineno-10-33"></a><span class="cm">/* Solve the n-queens problem */</span>
 <a id="__codelineno-10-34" name="__codelineno-10-34" href="#__codelineno-10-34"></a><span class="kt">char</span><span class="w"> </span><span class="o">***</span><span class="nf">nQueens</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">returnSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-35" name="__codelineno-10-35" href="#__codelineno-10-35"></a><span class="w">    </span><span class="kt">char</span><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">][</span><span class="n">MAX_SIZE</span><span class="p">];</span>
-<a id="__codelineno-10-36" name="__codelineno-10-36" href="#__codelineno-10-36"></a><span class="w">    </span><span class="c1">// Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell</span>
+<a id="__codelineno-10-36" name="__codelineno-10-36" href="#__codelineno-10-36"></a><span class="w">    </span><span class="c1">// Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square</span>
 <a id="__codelineno-10-37" name="__codelineno-10-37" href="#__codelineno-10-37"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-38" name="__codelineno-10-38" href="#__codelineno-10-38"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-39" name="__codelineno-10-39" href="#__codelineno-10-39"></a><span class="w">            </span><span class="n">state</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="sc">'#'</span><span class="p">;</span>
 <a id="__codelineno-10-40" name="__codelineno-10-40" href="#__codelineno-10-40"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-10-41" name="__codelineno-10-41" href="#__codelineno-10-41"></a><span class="w">        </span><span class="n">state</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">n</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="sc">'\0'</span><span class="p">;</span>
 <a id="__codelineno-10-42" name="__codelineno-10-42" href="#__codelineno-10-42"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-43" name="__codelineno-10-43" href="#__codelineno-10-43"></a><span class="w">    </span><span class="kt">bool</span><span class="w"> </span><span class="n">cols</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="nb">false</span><span class="p">};</span><span class="w">           </span><span class="c1">// Record whether there is a queen in the column</span>
-<a id="__codelineno-10-44" name="__codelineno-10-44" href="#__codelineno-10-44"></a><span class="w">    </span><span class="kt">bool</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="nb">false</span><span class="p">};</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the main diagonal</span>
-<a id="__codelineno-10-45" name="__codelineno-10-45" href="#__codelineno-10-45"></a><span class="w">    </span><span class="kt">bool</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="nb">false</span><span class="p">};</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the anti-diagonal</span>
+<a id="__codelineno-10-43" name="__codelineno-10-43" href="#__codelineno-10-43"></a><span class="w">    </span><span class="kt">bool</span><span class="w"> </span><span class="n">cols</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="nb">false</span><span class="p">};</span><span class="w">           </span><span class="c1">// Track whether a column contains a queen</span>
+<a id="__codelineno-10-44" name="__codelineno-10-44" href="#__codelineno-10-44"></a><span class="w">    </span><span class="kt">bool</span><span class="w"> </span><span class="n">diags1</span><span class="p">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="nb">false</span><span class="p">};</span><span class="w"> </span><span class="c1">// Track whether a main diagonal contains a queen</span>
+<a id="__codelineno-10-45" name="__codelineno-10-45" href="#__codelineno-10-45"></a><span class="w">    </span><span class="kt">bool</span><span class="w"> </span><span class="n">diags2</span><span class="p">[</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="nb">false</span><span class="p">};</span><span class="w"> </span><span class="c1">// Track whether an anti-diagonal contains a queen</span>
 <a id="__codelineno-10-46" name="__codelineno-10-46" href="#__codelineno-10-46"></a>
 <a id="__codelineno-10-47" name="__codelineno-10-47" href="#__codelineno-10-47"></a><span class="w">    </span><span class="kt">char</span><span class="w"> </span><span class="o">***</span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">char</span><span class="w"> </span><span class="o">***</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">char</span><span class="w"> </span><span class="o">**</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="p">);</span>
 <a id="__codelineno-10-48" name="__codelineno-10-48" href="#__codelineno-10-48"></a><span class="w">    </span><span class="o">*</span><span class="n">returnSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
@@ -5272,7 +5344,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">n_queens.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Backtracking algorithm: N queens */</span>
+<div class="highlight"><span class="filename">n_queens.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Backtracking: n-queens */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="n">row</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span>
@@ -5282,7 +5354,7 @@
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="n">diags1</span><span class="p">:</span><span class="w"> </span><span class="n">BooleanArray</span><span class="p">,</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="n">diags2</span><span class="p">:</span><span class="w"> </span><span class="n">BooleanArray</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="c1">// When all rows are placed, record the solution</span>
+<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="c1">// Record a solution when a queen has been placed in every row</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">copyState</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">String</span><span class="o">&gt;&gt;</span><span class="p">()</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">sRow</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">state</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5291,21 +5363,21 @@
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">copyState</span><span class="p">)</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">    </span><span class="c1">// Traverse all columns</span>
+<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">    </span><span class="c1">// Iterate over all columns</span>
 <a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">col</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">        </span><span class="c1">// Calculate the main diagonal and anti-diagonal corresponding to this cell</span>
+<a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">        </span><span class="c1">// Calculate the main-diagonal and anti-diagonal indices for this square</span>
 <a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">col</span>
-<a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w">        </span><span class="c1">// Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell</span>
+<a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w">        </span><span class="c1">// Prune: the square must have no queen in its column, main diagonal, or anti-diagonal</span>
 <a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w">            </span><span class="c1">// Attempt: place the queen in this cell</span>
+<a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w">            </span><span class="c1">// Try: place a queen on this square</span>
 <a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="w">            </span><span class="n">state</span><span class="o">[</span><span class="n">row</span><span class="o">][</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"Q"</span>
 <a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="w">            </span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span>
 <a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a><span class="w">            </span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span>
 <a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a><span class="w">            </span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span>
-<a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="w">            </span><span class="c1">// Place the next row</span>
+<a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="w">            </span><span class="c1">// Place a queen in the next row</span>
 <a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">)</span>
-<a id="__codelineno-11-34" name="__codelineno-11-34" href="#__codelineno-11-34"></a><span class="w">            </span><span class="c1">// Backtrack: restore this cell to an empty cell</span>
+<a id="__codelineno-11-34" name="__codelineno-11-34" href="#__codelineno-11-34"></a><span class="w">            </span><span class="c1">// Backtrack: restore this square to empty</span>
 <a id="__codelineno-11-35" name="__codelineno-11-35" href="#__codelineno-11-35"></a><span class="w">            </span><span class="n">state</span><span class="o">[</span><span class="n">row</span><span class="o">][</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"#"</span>
 <a id="__codelineno-11-36" name="__codelineno-11-36" href="#__codelineno-11-36"></a><span class="w">            </span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span>
 <a id="__codelineno-11-37" name="__codelineno-11-37" href="#__codelineno-11-37"></a><span class="w">            </span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span>
@@ -5314,9 +5386,9 @@
 <a id="__codelineno-11-40" name="__codelineno-11-40" href="#__codelineno-11-40"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-41" name="__codelineno-11-41" href="#__codelineno-11-41"></a><span class="p">}</span>
 <a id="__codelineno-11-42" name="__codelineno-11-42" href="#__codelineno-11-42"></a>
-<a id="__codelineno-11-43" name="__codelineno-11-43" href="#__codelineno-11-43"></a><span class="cm">/* Solve N queens */</span>
+<a id="__codelineno-11-43" name="__codelineno-11-43" href="#__codelineno-11-43"></a><span class="cm">/* Solve the n-queens problem */</span>
 <a id="__codelineno-11-44" name="__codelineno-11-44" href="#__codelineno-11-44"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">nQueens</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">String</span><span class="o">&gt;&gt;?&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-45" name="__codelineno-11-45" href="#__codelineno-11-45"></a><span class="w">    </span><span class="c1">// Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell</span>
+<a id="__codelineno-11-45" name="__codelineno-11-45" href="#__codelineno-11-45"></a><span class="w">    </span><span class="c1">// Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square</span>
 <a id="__codelineno-11-46" name="__codelineno-11-46" href="#__codelineno-11-46"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">String</span><span class="o">&gt;&gt;</span><span class="p">()</span>
 <a id="__codelineno-11-47" name="__codelineno-11-47" href="#__codelineno-11-47"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-48" name="__codelineno-11-48" href="#__codelineno-11-48"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">row</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="kt">String</span><span class="o">&gt;</span><span class="p">()</span>
@@ -5325,9 +5397,9 @@
 <a id="__codelineno-11-51" name="__codelineno-11-51" href="#__codelineno-11-51"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-11-52" name="__codelineno-11-52" href="#__codelineno-11-52"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">row</span><span class="p">)</span>
 <a id="__codelineno-11-53" name="__codelineno-11-53" href="#__codelineno-11-53"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-54" name="__codelineno-11-54" href="#__codelineno-11-54"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">BooleanArray</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="c1">// Record whether there is a queen in the column</span>
-<a id="__codelineno-11-55" name="__codelineno-11-55" href="#__codelineno-11-55"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">BooleanArray</span><span class="p">(</span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the main diagonal</span>
-<a id="__codelineno-11-56" name="__codelineno-11-56" href="#__codelineno-11-56"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">BooleanArray</span><span class="p">(</span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="c1">// Record whether there is a queen on the anti-diagonal</span>
+<a id="__codelineno-11-54" name="__codelineno-11-54" href="#__codelineno-11-54"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">BooleanArray</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="c1">// Track whether a column contains a queen</span>
+<a id="__codelineno-11-55" name="__codelineno-11-55" href="#__codelineno-11-55"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">BooleanArray</span><span class="p">(</span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="c1">// Track whether a main diagonal contains a queen</span>
+<a id="__codelineno-11-56" name="__codelineno-11-56" href="#__codelineno-11-56"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">BooleanArray</span><span class="p">(</span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="c1">// Track whether an anti-diagonal contains a queen</span>
 <a id="__codelineno-11-57" name="__codelineno-11-57" href="#__codelineno-11-57"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">String</span><span class="o">&gt;&gt;?&gt;</span><span class="p">()</span>
 <a id="__codelineno-11-58" name="__codelineno-11-58" href="#__codelineno-11-58"></a>
 <a id="__codelineno-11-59" name="__codelineno-11-59" href="#__codelineno-11-59"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">)</span>
@@ -5337,40 +5409,40 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">n_queens.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Backtracking: n queens ###</span>
+<div class="highlight"><span class="filename">n_queens.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Backtracking: n-queens ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">row</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># When all rows are placed, record the solution</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Record a solution when a queen has been placed in every row</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">state</span><span class="o">.</span><span class="n">map</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">row</span><span class="o">|</span><span class="w"> </span><span class="n">row</span><span class="o">.</span><span class="n">dup</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">    </span><span class="k">return</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a>
-<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="c1"># Traverse all columns</span>
+<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="c1"># Iterate over all columns</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">n</span>
-<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="c1"># Calculate the main diagonal and anti-diagonal corresponding to this cell</span>
+<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="c1"># Calculate the main-diagonal and anti-diagonal indices for this square</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">    </span><span class="n">diag1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="n">diag2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">col</span>
-<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="c1"># Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell</span>
+<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="c1"># Prune: the square must have no queen in its column, main diagonal, or anti-diagonal</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span>
-<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">      </span><span class="c1"># Attempt: place the queen in this cell</span>
+<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">      </span><span class="c1"># Try: place a queen on this square</span>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">      </span><span class="n">state</span><span class="o">[</span><span class="n">row</span><span class="o">][</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"Q"</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">      </span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">true</span>
-<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">      </span><span class="c1"># Place the next row</span>
+<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">      </span><span class="c1"># Place a queen in the next row</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="w">      </span><span class="n">backtrack</span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">)</span>
-<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">      </span><span class="c1"># Backtrack: restore this cell to an empty cell</span>
+<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">      </span><span class="c1"># Backtrack: restore this square to empty</span>
 <a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w">      </span><span class="n">state</span><span class="o">[</span><span class="n">row</span><span class="o">][</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"#"</span>
 <a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="w">      </span><span class="n">cols</span><span class="o">[</span><span class="n">col</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags1</span><span class="o">[</span><span class="n">diag1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">diags2</span><span class="o">[</span><span class="n">diag2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">false</span>
 <a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></a><span class="k">end</span>
 <a id="__codelineno-12-27" name="__codelineno-12-27" href="#__codelineno-12-27"></a>
-<a id="__codelineno-12-28" name="__codelineno-12-28" href="#__codelineno-12-28"></a><span class="c1">### Solve n queens ###</span>
+<a id="__codelineno-12-28" name="__codelineno-12-28" href="#__codelineno-12-28"></a><span class="c1">### Solve the n-queens problem ###</span>
 <a id="__codelineno-12-29" name="__codelineno-12-29" href="#__codelineno-12-29"></a><span class="k">def</span><span class="w"> </span><span class="nf">n_queens</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-12-30" name="__codelineno-12-30" href="#__codelineno-12-30"></a><span class="w">  </span><span class="c1"># Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell</span>
+<a id="__codelineno-12-30" name="__codelineno-12-30" href="#__codelineno-12-30"></a><span class="w">  </span><span class="c1"># Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square</span>
 <a id="__codelineno-12-31" name="__codelineno-12-31" href="#__codelineno-12-31"></a><span class="w">  </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="s2">"#"</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-12-32" name="__codelineno-12-32" href="#__codelineno-12-32"></a><span class="w">  </span><span class="n">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kp">false</span><span class="p">)</span><span class="w"> </span><span class="c1"># Record whether there is a queen in the column</span>
-<a id="__codelineno-12-33" name="__codelineno-12-33" href="#__codelineno-12-33"></a><span class="w">  </span><span class="n">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="kp">false</span><span class="p">)</span><span class="w"> </span><span class="c1"># Record whether there is a queen on the main diagonal</span>
-<a id="__codelineno-12-34" name="__codelineno-12-34" href="#__codelineno-12-34"></a><span class="w">  </span><span class="n">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="kp">false</span><span class="p">)</span><span class="w"> </span><span class="c1"># Record whether there is a queen on the anti-diagonal</span>
+<a id="__codelineno-12-32" name="__codelineno-12-32" href="#__codelineno-12-32"></a><span class="w">  </span><span class="n">cols</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kp">false</span><span class="p">)</span><span class="w"> </span><span class="c1"># Track whether a column contains a queen</span>
+<a id="__codelineno-12-33" name="__codelineno-12-33" href="#__codelineno-12-33"></a><span class="w">  </span><span class="n">diags1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="kp">false</span><span class="p">)</span><span class="w"> </span><span class="c1"># Track whether a main diagonal contains a queen</span>
+<a id="__codelineno-12-34" name="__codelineno-12-34" href="#__codelineno-12-34"></a><span class="w">  </span><span class="n">diags2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="kp">false</span><span class="p">)</span><span class="w"> </span><span class="c1"># Track whether an anti-diagonal contains a queen</span>
 <a id="__codelineno-12-35" name="__codelineno-12-35" href="#__codelineno-12-35"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-12-36" name="__codelineno-12-36" href="#__codelineno-12-36"></a><span class="w">  </span><span class="n">backtrack</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cols</span><span class="p">,</span><span class="w"> </span><span class="n">diags1</span><span class="p">,</span><span class="w"> </span><span class="n">diags2</span><span class="p">)</span>
 <a id="__codelineno-12-37" name="__codelineno-12-37" href="#__codelineno-12-37"></a>
@@ -5380,8 +5452,13 @@
 </div>
 </div>
 </div>
-<p>Placing <span class="arithmatex">\(n\)</span> queens row by row, considering the column constraint, from the first row to the last row there are <span class="arithmatex">\(n\)</span>, <span class="arithmatex">\(n-1\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, <span class="arithmatex">\(1\)</span> choices, using <span class="arithmatex">\(O(n!)\)</span> time. When recording a solution, it is necessary to copy the matrix <code>state</code> and add it to <code>res</code>, and the copy operation uses <span class="arithmatex">\(O(n^2)\)</span> time. Therefore, <strong>the overall time complexity is <span class="arithmatex">\(O(n! \cdot n^2)\)</span></strong>. In practice, pruning based on diagonal constraints can also significantly reduce the search space, so the search efficiency is often better than the time complexity mentioned above.</p>
-<p>The array <code>state</code> uses <span class="arithmatex">\(O(n^2)\)</span> space, and the arrays <code>cols</code>, <code>diags1</code>, and <code>diags2</code> each use <span class="arithmatex">\(O(n)\)</span> space. The maximum recursion depth is <span class="arithmatex">\(n\)</span>, using <span class="arithmatex">\(O(n)\)</span> stack frame space. Therefore, <strong>the space complexity is <span class="arithmatex">\(O(n^2)\)</span></strong>.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20backtrack%28%0A%20%20%20%20row%3A%20int,%0A%20%20%20%20n%3A%20int,%0A%20%20%20%20state%3A%20list%5Blist%5Bstr%5D%5D,%0A%20%20%20%20res%3A%20list%5Blist%5Blist%5Bstr%5D%5D%5D,%0A%20%20%20%20cols%3A%20list%5Bbool%5D,%0A%20%20%20%20diags1%3A%20list%5Bbool%5D,%0A%20%20%20%20diags2%3A%20list%5Bbool%5D,%0A%29%3A%0A%20%20%20%20%22%22%22Backtracking%3A%20N-queens%22%22%22%0A%20%20%20%20%23%20Record%20a%20solution%20when%20a%20queen%20has%20been%20placed%20in%20every%20row%0A%20%20%20%20if%20row%20%3D%3D%20n%3A%0A%20%20%20%20%20%20%20%20res.append%28%5Blist%28row%29%20for%20row%20in%20state%5D%29%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Iterate%20over%20all%20columns%0A%20%20%20%20for%20col%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20%23%20Calculate%20the%20main-diagonal%20and%20anti-diagonal%20indices%20for%20this%20square%0A%20%20%20%20%20%20%20%20diag1%20%3D%20row%20-%20col%20%2B%20n%20-%201%0A%20%20%20%20%20%20%20%20diag2%20%3D%20row%20%2B%20col%0A%20%20%20%20%20%20%20%20%23%20Prune%3A%20the%20square%20must%20have%20no%20queen%20in%20its%20column,%20main%20diagonal,%20or%20anti-diagonal%0A%20%20%20%20%20%20%20%20if%20not%20cols%5Bcol%5D%20and%20not%20diags1%5Bdiag1%5D%20and%20not%20diags2%5Bdiag2%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Try%3A%20place%20a%20queen%20on%20this%20square%0A%20%20%20%20%20%20%20%20%20%20%20%20state%5Brow%5D%5Bcol%5D%20%3D%20%22Q%22%0A%20%20%20%20%20%20%20%20%20%20%20%20cols%5Bcol%5D%20%3D%20diags1%5Bdiag1%5D%20%3D%20diags2%5Bdiag2%5D%20%3D%20True%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Place%20a%20queen%20in%20the%20next%20row%0A%20%20%20%20%20%20%20%20%20%20%20%20backtrack%28row%20%2B%201,%20n,%20state,%20res,%20cols,%20diags1,%20diags2%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Backtrack%3A%20restore%20this%20square%20to%20empty%0A%20%20%20%20%20%20%20%20%20%20%20%20state%5Brow%5D%5Bcol%5D%20%3D%20%22%23%22%0A%20%20%20%20%20%20%20%20%20%20%20%20cols%5Bcol%5D%20%3D%20diags1%5Bdiag1%5D%20%3D%20diags2%5Bdiag2%5D%20%3D%20False%0A%0A%0Adef%20n_queens%28n%3A%20int%29%20-%3E%20list%5Blist%5Blist%5Bstr%5D%5D%5D%3A%0A%20%20%20%20%22%22%22Solve%20the%20N-queens%20problem%22%22%22%0A%20%20%20%20%23%20Initialize%20an%20n%2An%20board,%20where%20%27Q%27%20represents%20a%20queen%20and%20%27%23%27%20represents%20an%20empty%20square%0A%20%20%20%20state%20%3D%20%5B%5B%22%23%22%20for%20_%20in%20range%28n%29%5D%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20cols%20%3D%20%5BFalse%5D%20%2A%20n%20%20%23%20Track%20whether%20a%20column%20contains%20a%20queen%0A%20%20%20%20diags1%20%3D%20%5BFalse%5D%20%2A%20%282%20%2A%20n%20-%201%29%20%20%23%20Track%20whether%20a%20main%20diagonal%20contains%20a%20queen%0A%20%20%20%20diags2%20%3D%20%5BFalse%5D%20%2A%20%282%20%2A%20n%20-%201%29%20%20%23%20Track%20whether%20an%20anti-diagonal%20contains%20a%20queen%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20backtrack%280,%20n,%20state,%20res,%20cols,%20diags1,%20diags2%29%0A%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%204%0A%20%20%20%20res%20%3D%20n_queens%28n%29%0A%0A%20%20%20%20print%28f%22The%20input%20board%20has%20width%20and%20height%20%7Bn%7D%22%29%0A%20%20%20%20print%28f%22Number%20of%20ways%20to%20place%20the%20queens%3A%20%7Blen%28res%29%7D%22%29%0A%20%20%20%20for%20state%20in%20res%3A%0A%20%20%20%20%20%20%20%20print%28%22--------------------%22%29%0A%20%20%20%20%20%20%20%20for%20row%20in%20state%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20print%28row%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=61&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20backtrack%28%0A%20%20%20%20row%3A%20int,%0A%20%20%20%20n%3A%20int,%0A%20%20%20%20state%3A%20list%5Blist%5Bstr%5D%5D,%0A%20%20%20%20res%3A%20list%5Blist%5Blist%5Bstr%5D%5D%5D,%0A%20%20%20%20cols%3A%20list%5Bbool%5D,%0A%20%20%20%20diags1%3A%20list%5Bbool%5D,%0A%20%20%20%20diags2%3A%20list%5Bbool%5D,%0A%29%3A%0A%20%20%20%20%22%22%22Backtracking%3A%20N-queens%22%22%22%0A%20%20%20%20%23%20Record%20a%20solution%20when%20a%20queen%20has%20been%20placed%20in%20every%20row%0A%20%20%20%20if%20row%20%3D%3D%20n%3A%0A%20%20%20%20%20%20%20%20res.append%28%5Blist%28row%29%20for%20row%20in%20state%5D%29%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Iterate%20over%20all%20columns%0A%20%20%20%20for%20col%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20%23%20Calculate%20the%20main-diagonal%20and%20anti-diagonal%20indices%20for%20this%20square%0A%20%20%20%20%20%20%20%20diag1%20%3D%20row%20-%20col%20%2B%20n%20-%201%0A%20%20%20%20%20%20%20%20diag2%20%3D%20row%20%2B%20col%0A%20%20%20%20%20%20%20%20%23%20Prune%3A%20the%20square%20must%20have%20no%20queen%20in%20its%20column,%20main%20diagonal,%20or%20anti-diagonal%0A%20%20%20%20%20%20%20%20if%20not%20cols%5Bcol%5D%20and%20not%20diags1%5Bdiag1%5D%20and%20not%20diags2%5Bdiag2%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Try%3A%20place%20a%20queen%20on%20this%20square%0A%20%20%20%20%20%20%20%20%20%20%20%20state%5Brow%5D%5Bcol%5D%20%3D%20%22Q%22%0A%20%20%20%20%20%20%20%20%20%20%20%20cols%5Bcol%5D%20%3D%20diags1%5Bdiag1%5D%20%3D%20diags2%5Bdiag2%5D%20%3D%20True%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Place%20a%20queen%20in%20the%20next%20row%0A%20%20%20%20%20%20%20%20%20%20%20%20backtrack%28row%20%2B%201,%20n,%20state,%20res,%20cols,%20diags1,%20diags2%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Backtrack%3A%20restore%20this%20square%20to%20empty%0A%20%20%20%20%20%20%20%20%20%20%20%20state%5Brow%5D%5Bcol%5D%20%3D%20%22%23%22%0A%20%20%20%20%20%20%20%20%20%20%20%20cols%5Bcol%5D%20%3D%20diags1%5Bdiag1%5D%20%3D%20diags2%5Bdiag2%5D%20%3D%20False%0A%0A%0Adef%20n_queens%28n%3A%20int%29%20-%3E%20list%5Blist%5Blist%5Bstr%5D%5D%5D%3A%0A%20%20%20%20%22%22%22Solve%20the%20N-queens%20problem%22%22%22%0A%20%20%20%20%23%20Initialize%20an%20n%2An%20board,%20where%20%27Q%27%20represents%20a%20queen%20and%20%27%23%27%20represents%20an%20empty%20square%0A%20%20%20%20state%20%3D%20%5B%5B%22%23%22%20for%20_%20in%20range%28n%29%5D%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20cols%20%3D%20%5BFalse%5D%20%2A%20n%20%20%23%20Track%20whether%20a%20column%20contains%20a%20queen%0A%20%20%20%20diags1%20%3D%20%5BFalse%5D%20%2A%20%282%20%2A%20n%20-%201%29%20%20%23%20Track%20whether%20a%20main%20diagonal%20contains%20a%20queen%0A%20%20%20%20diags2%20%3D%20%5BFalse%5D%20%2A%20%282%20%2A%20n%20-%201%29%20%20%23%20Track%20whether%20an%20anti-diagonal%20contains%20a%20queen%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20backtrack%280,%20n,%20state,%20res,%20cols,%20diags1,%20diags2%29%0A%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%204%0A%20%20%20%20res%20%3D%20n_queens%28n%29%0A%0A%20%20%20%20print%28f%22The%20input%20board%20has%20width%20and%20height%20%7Bn%7D%22%29%0A%20%20%20%20print%28f%22Number%20of%20ways%20to%20place%20the%20queens%3A%20%7Blen%28res%29%7D%22%29%0A%20%20%20%20for%20state%20in%20res%3A%0A%20%20%20%20%20%20%20%20print%28%22--------------------%22%29%0A%20%20%20%20%20%20%20%20for%20row%20in%20state%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20print%28row%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=61&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>We place a queen in each of <span class="arithmatex">\(n\)</span> rows. Considering the column constraint, the successive rows have <span class="arithmatex">\(n\)</span>, <span class="arithmatex">\(n-1\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, and <span class="arithmatex">\(1\)</span> choices, giving <span class="arithmatex">\(O(n!)\)</span> time. Recording a solution requires copying the matrix <code>state</code> and appending the copy to <code>res</code>, which takes <span class="arithmatex">\(O(n^2)\)</span> time. <strong>The overall time complexity is therefore <span class="arithmatex">\(O(n! \cdot n^2)\)</span></strong>. In practice, diagonal pruning greatly reduces the search space, so the search is often faster than this bound suggests.</p>
+<p>The array <code>state</code> uses <span class="arithmatex">\(O(n^2)\)</span> space, while <code>cols</code>, <code>diags1</code>, and <code>diags2</code> each use <span class="arithmatex">\(O(n)\)</span> space. The maximum recursion depth is <span class="arithmatex">\(n\)</span>, requiring <span class="arithmatex">\(O(n)\)</span> space for stack frames. <strong>The space complexity is therefore <span class="arithmatex">\(O(n^2)\)</span></strong>.</p>
 
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   <a
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@@ -5416,7 +5493,7 @@ aria-label="Footer"
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-        13.3 Subset-Sum Problem
+        13.3 &nbsp; Subset sum problem
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@@ -5428,7 +5505,7 @@ aria-label="Footer"
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-    aria-label="Next: 13.5 Summary"
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@@ -5453,7 +5530,7 @@ aria-label="Footer"
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@@ -41,7 +41,7 @@
 
 
 
-        <title>13.2 Permutations Problem - Hello Algo</title>
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@@ -67,8 +67,8 @@
 
 
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-  Before Starting
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-  Before Starting
+  Foreword
 
           </label>
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
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-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3283,7 +3283,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3304,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3352,7 +3352,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3379,7 +3379,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3406,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3433,7 +3433,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3460,7 +3460,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3528,7 +3528,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3549,7 +3549,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3570,7 +3570,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3606,7 +3606,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3623,7 +3623,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3649,22 +3649,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1321-case-with-distinct-elements" class="md-nav__link">
+  <a href="#1321-distinct-elements" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.2.1 &nbsp; Case with Distinct Elements
+        13.2.1 &nbsp; Distinct elements
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="13.2.1 Case with Distinct Elements">
+    <nav class="md-nav" aria-label="13.2.1 Distinct elements">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-pruning-duplicate-choices" class="md-nav__link">
+  <a href="#1-pruning-repeated-selections" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Pruning Duplicate Choices
+        1. &nbsp; Pruning repeated selections
       </span>
     </span>
   </a>
@@ -3675,7 +3675,7 @@
   <a href="#2-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Code Implementation
+        2. &nbsp; Code implementation
       </span>
     </span>
   </a>
@@ -3688,22 +3688,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1322-case-with-duplicate-elements" class="md-nav__link">
+  <a href="#1322-duplicate-elements" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.2.2 &nbsp; Case with Duplicate Elements
+        13.2.2 &nbsp; Duplicate elements
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="13.2.2 Case with Duplicate Elements">
+    <nav class="md-nav" aria-label="13.2.2 Duplicate elements">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-pruning-equal-elements" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Pruning Equal Elements
+        1. &nbsp; Pruning equal elements
       </span>
     </span>
   </a>
@@ -3714,7 +3714,7 @@
   <a href="#2-code-implementation_1" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Code Implementation
+        2. &nbsp; Code implementation
       </span>
     </span>
   </a>
@@ -3722,10 +3722,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-comparison-of-two-pruning-methods" class="md-nav__link">
+  <a href="#3-comparing-the-two-pruning-methods" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Comparison of Two Pruning Methods
+        3. &nbsp; Comparing the two pruning methods
       </span>
     </span>
   </a>
@@ -3760,7 +3760,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3787,7 +3787,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3814,7 +3814,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3841,7 +3841,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3911,7 +3911,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3932,7 +3932,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3953,7 +3953,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3980,7 +3980,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -4007,7 +4007,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4034,7 +4034,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4061,7 +4061,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4088,7 +4088,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4115,7 +4115,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4142,7 +4142,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4208,7 +4208,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4229,7 +4229,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4277,7 +4277,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1321-case-with-distinct-elements" class="md-nav__link">
+  <a href="#1321-distinct-elements" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.2.1 &nbsp; Case with Distinct Elements
+        13.2.1 &nbsp; Distinct elements
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="13.2.1 Case with Distinct Elements">
+    <nav class="md-nav" aria-label="13.2.1 Distinct elements">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-pruning-duplicate-choices" class="md-nav__link">
+  <a href="#1-pruning-repeated-selections" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Pruning Duplicate Choices
+        1. &nbsp; Pruning repeated selections
       </span>
     </span>
   </a>
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   <a href="#2-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
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-        2. &nbsp; Code Implementation
+        2. &nbsp; Code implementation
       </span>
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 </li>
 
         <li class="md-nav__item">
-  <a href="#1322-case-with-duplicate-elements" class="md-nav__link">
+  <a href="#1322-duplicate-elements" class="md-nav__link">
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       <span class="md-typeset">
-        13.2.2 &nbsp; Case with Duplicate Elements
+        13.2.2 &nbsp; Duplicate elements
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="13.2.2 Case with Duplicate Elements">
+    <nav class="md-nav" aria-label="13.2.2 Duplicate elements">
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           <li class="md-nav__item">
   <a href="#1-pruning-equal-elements" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Pruning Equal Elements
+        1. &nbsp; Pruning equal elements
       </span>
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   <a href="#2-code-implementation_1" class="md-nav__link">
     <span class="md-ellipsis">
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-        2. &nbsp; Code Implementation
+        2. &nbsp; Code implementation
       </span>
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 </li>
 
           <li class="md-nav__item">
-  <a href="#3-comparison-of-two-pruning-methods" class="md-nav__link">
+  <a href="#3-comparing-the-two-pruning-methods" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Comparison of Two Pruning Methods
+        3. &nbsp; Comparing the two pruning methods
       </span>
     </span>
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 <!-- Page content -->
-<h1 id="132-permutations-problem">13.2 &nbsp; Permutations Problem<a class="headerlink" href="#132-permutations-problem" title="Permanent link">&para;</a></h1>
-<p>The permutations problem is a classic application of backtracking algorithms. It is defined as finding all possible arrangements of elements in a given collection (such as an array or string).</p>
-<p>Table 13-2 shows several example datasets, including input arrays and their corresponding permutations.</p>
-<p align="center"> Table 13-2 &nbsp; Permutations Examples </p>
+<h1 id="132-permutations-problem">13.2 &nbsp; Permutations problem<a class="headerlink" href="#132-permutations-problem" title="Permanent link">&para;</a></h1>
+<p>Generating permutations is a classic application of backtracking. Given a collection, such as an array or a string, the task is to find every possible ordering of its elements.</p>
+<p>Table 13-2 gives several input arrays and all their permutations.</p>
+<p align="center"> Table 13-2 &nbsp; Examples of permutations </p>
 
 <div class="center-table">
 <table>
 <thead>
 <tr>
-<th style="text-align: left;">Input Array</th>
-<th style="text-align: left;">All Permutations</th>
+<th style="text-align: left;">Input array</th>
+<th style="text-align: left;">All permutations</th>
 </tr>
 </thead>
 <tbody>
@@ -4820,51 +4892,51 @@
 </tbody>
 </table>
 </div>
-<h2 id="1321-case-with-distinct-elements">13.2.1 &nbsp; Case with Distinct Elements<a class="headerlink" href="#1321-case-with-distinct-elements" title="Permanent link">&para;</a></h2>
+<h2 id="1321-distinct-elements">13.2.1 &nbsp; Distinct elements<a class="headerlink" href="#1321-distinct-elements" title="Permanent link">&para;</a></h2>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
 <p>Given an integer array with no duplicate elements, return all possible permutations.</p>
 </div>
-<p>From the perspective of backtracking algorithms, <strong>we can imagine the process of generating permutations as the result of a series of choices</strong>. Suppose the input array is <span class="arithmatex">\([1, 2, 3]\)</span>. If we first choose <span class="arithmatex">\(1\)</span>, then choose <span class="arithmatex">\(3\)</span>, and finally choose <span class="arithmatex">\(2\)</span>, we obtain the permutation <span class="arithmatex">\([1, 3, 2]\)</span>. Backtracking means undoing a choice and then trying other choices.</p>
-<p>From the perspective of backtracking code, the candidate set <code>choices</code> consists of all elements in the input array, and the state <code>state</code> is the elements that have been chosen so far. Note that each element can only be chosen once, <strong>therefore all elements in <code>state</code> should be unique</strong>.</p>
-<p>As shown in Figure 13-5, we can unfold the search process into a recursion tree, where each node in the tree represents the current state <code>state</code>. Starting from the root node, after three rounds of choices, we reach a leaf node, and each leaf node corresponds to a permutation.</p>
-<p><img alt="Recursion tree of permutations" class="animation-figure" src="../permutations_problem.assets/permutations_i.png" /></p>
-<p align="center"> Figure 13-5 &nbsp; Recursion tree of permutations </p>
+<p>From a backtracking perspective, <strong>we can view a permutation as the result of a sequence of choices</strong>. For input <span class="arithmatex">\([1, 2, 3]\)</span>, choosing <span class="arithmatex">\(1\)</span>, then <span class="arithmatex">\(3\)</span>, then <span class="arithmatex">\(2\)</span> produces <span class="arithmatex">\([1, 3, 2]\)</span>. Backtracking undoes a choice so that other choices can be tried.</p>
+<p>In the code, <code>choices</code> contains all the input elements, and <code>state</code> contains the elements chosen so far. Each element may be chosen only once, <strong>so every element in <code>state</code> must be unique</strong>.</p>
+<p>As shown in Figure 13-5, we can expand the search into a recursion tree, where each node represents the current <code>state</code>. Starting from the root, three rounds of choices lead to a leaf, and each leaf corresponds to one permutation.</p>
+<p><img alt="Recursion tree for permutations" class="animation-figure" src="../permutations_problem.assets/permutations_i.png" /></p>
+<p align="center"> Figure 13-5 &nbsp; Recursion tree for permutations </p>
 
-<h3 id="1-pruning-duplicate-choices">1. &nbsp; Pruning Duplicate Choices<a class="headerlink" href="#1-pruning-duplicate-choices" title="Permanent link">&para;</a></h3>
-<p>To ensure that each element is chosen only once, we consider introducing a boolean array <code>selected</code>, where <code>selected[i]</code> indicates whether <code>choices[i]</code> has been chosen. We implement the following pruning operation based on it.</p>
+<h3 id="1-pruning-repeated-selections">1. &nbsp; Pruning repeated selections<a class="headerlink" href="#1-pruning-repeated-selections" title="Permanent link">&para;</a></h3>
+<p>To ensure that each element is chosen only once, introduce a Boolean array <code>selected</code>, where <code>selected[i]</code> indicates whether <code>choices[i]</code> has been chosen. Use it for the following pruning steps.</p>
 <ul>
-<li>After making a choice <code>choices[i]</code>, we set <code>selected[i]</code> to <span class="arithmatex">\(\text{True}\)</span>, indicating that it has been chosen.</li>
-<li>When traversing the candidate list <code>choices</code>, we skip all nodes that have been chosen, which is pruning.</li>
+<li>After choosing <code>choices[i]</code>, set <code>selected[i]</code> to <span class="arithmatex">\(\text{True}\)</span> to mark it as selected.</li>
+<li>When iterating over <code>choices</code>, skip all selected elements, pruning their branches.</li>
 </ul>
-<p>As shown in Figure 13-6, suppose we choose <span class="arithmatex">\(1\)</span> in the first round, <span class="arithmatex">\(3\)</span> in the second round, and <span class="arithmatex">\(2\)</span> in the third round. Then we need to prune the branch of element <span class="arithmatex">\(1\)</span> in the second round and prune the branches of elements <span class="arithmatex">\(1\)</span> and <span class="arithmatex">\(3\)</span> in the third round.</p>
-<p><img alt="Pruning example of permutations" class="animation-figure" src="../permutations_problem.assets/permutations_i_pruning.png" /></p>
-<p align="center"> Figure 13-6 &nbsp; Pruning example of permutations </p>
+<p>As shown in Figure 13-6, if we choose 1 first, 3 second, and 2 third, we must prune the branch for 1 in the second round and the branches for 1 and 3 in the third round.</p>
+<p><img alt="Example of pruning permutations" class="animation-figure" src="../permutations_problem.assets/permutations_i_pruning.png" /></p>
+<p align="center"> Figure 13-6 &nbsp; Example of pruning permutations </p>
 
-<p>Observing the above figure, we find that this pruning operation reduces the search space size from <span class="arithmatex">\(O(n^n)\)</span> to <span class="arithmatex">\(O(n!)\)</span>.</p>
-<h3 id="2-code-implementation">2. &nbsp; Code Implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">&para;</a></h3>
-<p>After understanding the above information, we can fill in the blanks in the template code. To shorten the overall code, we do not implement each function in the template separately, but instead unfold them in the <code>backtrack()</code> function:</p>
+<p>This pruning reduces the search space from <span class="arithmatex">\(O(n^n)\)</span> to <span class="arithmatex">\(O(n!)\)</span>.</p>
+<h3 id="2-code-implementation">2. &nbsp; Code implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">&para;</a></h3>
+<p>With these details established, we can fill in the backtracking template. To keep the code short, we inline the template's helper functions into <code>backtrack()</code>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">permutations_i.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>    <span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">choices</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">selected</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span> <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]]</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="p">):</span>
-<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w">    </span><span class="sd">"""Backtracking algorithm: Permutations I"""</span>
-<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w">    </span><span class="sd">"""Backtracking: permutations I"""</span>
+<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">state</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">choices</span><span class="p">):</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">state</span><span class="p">))</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="k">return</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="c1"># Traverse all choices</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="c1"># Iterate over all choices</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">choice</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">choices</span><span class="p">):</span>
-<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="c1"># Pruning: do not allow repeated selection of elements</span>
+<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="c1"># Prune: do not choose an element more than once</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="k">if</span> <span class="ow">not</span> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]:</span>
-<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>            <span class="c1"># Attempt: make choice, update state</span>
+<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>            <span class="c1"># Try: make a choice and update the state</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>            <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>            <span class="n">state</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
-<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>            <span class="c1"># Proceed to the next round of selection</span>
+<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>            <span class="c1"># Proceed to the next choice</span>
 <a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>            <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">choices</span><span class="p">,</span> <span class="n">selected</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>            <span class="c1"># Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>            <span class="c1"># Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>            <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="kc">False</span>
 <a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>            <span class="n">state</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
 <a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a>
@@ -4876,24 +4948,24 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_i.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Backtracking algorithm: Permutations I */</span>
+<div class="highlight"><span class="filename">permutations_i.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Backtracking: permutations I */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">size</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">state</span><span class="p">);</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
-<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements</span>
+<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="c1">// Prune: do not choose an element more than once</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
-<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">false</span><span class="p">;</span>
 <a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span>
 <a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w">        </span><span class="p">}</span>
@@ -4911,24 +4983,24 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_i.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Backtracking algorithm: Permutations I */</span>
+<div class="highlight"><span class="filename">permutations_i.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Backtracking: permutations I */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements</span>
+<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="c1">// Prune: do not choose an element more than once</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">            </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
-<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">            </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
 <a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">        </span><span class="p">}</span>
@@ -4944,24 +5016,24 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_i.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Backtracking algorithm: Permutations I */</span>
+<div class="highlight"><span class="filename">permutations_i.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Backtracking: permutations I */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Backtrack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">Length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
-<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements</span>
+<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="c1">// Prune: do not choose an element more than once</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">true</span><span class="p">;</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
-<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">            </span><span class="n">Backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">false</span><span class="p">;</span>
 <a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="n">RemoveAt</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w">        </span><span class="p">}</span>
@@ -4977,24 +5049,24 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_i.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Backtracking algorithm: Permutations I */</span>
+<div class="highlight"><span class="filename">permutations_i.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Backtracking: permutations I */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">backtrackI</span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">selected</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="kt">bool</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[][]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">        </span><span class="nx">newState</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">append</span><span class="p">([]</span><span class="kt">int</span><span class="p">{},</span><span class="w"> </span><span class="o">*</span><span class="nx">state</span><span class="p">...)</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="o">*</span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">newState</span><span class="p">)</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">        </span><span class="nx">choice</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">)[</span><span class="nx">i</span><span class="p">]</span>
-<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements</span>
+<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="c1">// Prune: do not choose an element more than once</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="p">(</span><span class="o">*</span><span class="nx">selected</span><span class="p">)[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">            </span><span class="p">(</span><span class="o">*</span><span class="nx">selected</span><span class="p">)[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">            </span><span class="o">*</span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">)</span>
-<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">            </span><span class="nx">backtrackI</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">selected</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span>
-<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">            </span><span class="p">(</span><span class="o">*</span><span class="nx">selected</span><span class="p">)[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">            </span><span class="o">*</span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[:</span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">        </span><span class="p">}</span>
@@ -5012,23 +5084,23 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_i.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Backtracking algorithm: Permutations I */</span>
+<div class="highlight"><span class="filename">permutations_i.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Backtracking: permutations I */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">selected</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Bool</span><span class="p">],</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="bp">count</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="bp">count</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">enumerated</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements</span>
+<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="c1">// Prune: do not choose an element more than once</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">true</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
-<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">false</span>
 <a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w">        </span><span class="p">}</span>
@@ -5046,23 +5118,23 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_i.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Backtracking algorithm: Permutations I */</span>
+<div class="highlight"><span class="filename">permutations_i.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Backtracking: permutations I */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">selected</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">state</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">choices</span><span class="p">.</span><span class="nx">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">state</span><span class="p">]);</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="nx">choices</span><span class="p">.</span><span class="nx">forEach</span><span class="p">((</span><span class="nx">choice</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements</span>
+<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="c1">// Prune: do not choose an element more than once</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="nx">selected</span><span class="p">[</span><span class="nx">i</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">            </span><span class="nx">selected</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">            </span><span class="nx">state</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">choice</span><span class="p">);</span>
-<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">            </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">selected</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">            </span><span class="nx">selected</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">            </span><span class="nx">state</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">        </span><span class="p">}</span>
@@ -5078,28 +5150,28 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_i.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Backtracking algorithm: Permutations I */</span>
+<div class="highlight"><span class="filename">permutations_i.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Backtracking: permutations I */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="nx">state</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="nx">choices</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="nx">selected</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="p">[],</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[][]</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">state</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">choices</span><span class="p">.</span><span class="nx">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">state</span><span class="p">]);</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">    </span><span class="nx">choices</span><span class="p">.</span><span class="nx">forEach</span><span class="p">((</span><span class="nx">choice</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements</span>
+<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">        </span><span class="c1">// Prune: do not choose an element more than once</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="nx">selected</span><span class="p">[</span><span class="nx">i</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">            </span><span class="nx">selected</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">            </span><span class="nx">state</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">choice</span><span class="p">);</span>
-<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">            </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">selected</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">            </span><span class="nx">selected</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
 <a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">            </span><span class="nx">state</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">        </span><span class="p">}</span>
@@ -5115,29 +5187,29 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_i.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Backtracking algorithm: Permutations I */</span>
+<div class="highlight"><span class="filename">permutations_i.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Backtracking: permutations I */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">,</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">List</span><span class="p">.</span><span class="n">from</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
-<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">    </span><span class="c1">// Pruning: do not allow repeated selection of elements</span>
+<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">    </span><span class="c1">// Prune: do not choose an element more than once</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">      </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">      </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">      </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
 <a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">      </span><span class="n">state</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
-<a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="w">      </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="w">      </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="w">      </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="w">      </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="w">      </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="w">      </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
 <a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="w">      </span><span class="n">state</span><span class="p">.</span><span class="n">removeLast</span><span class="p">();</span>
 <a id="__codelineno-8-26" name="__codelineno-8-26" href="#__codelineno-8-26"></a><span class="w">    </span><span class="p">}</span>
@@ -5153,24 +5225,24 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_i.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Backtracking algorithm: Permutations I */</span>
+<div class="highlight"><span class="filename">permutations_i.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Backtracking: permutations I */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">selected</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">bool</span><span class="p">],</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">state</span><span class="p">);</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">choices</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
-<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements</span>
+<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="c1">// Prune: do not choose an element more than once</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
-<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">clone</span><span class="p">(),</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
 <a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span>
 <a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">        </span><span class="p">}</span>
@@ -5179,16 +5251,16 @@
 <a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a>
 <a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a><span class="cm">/* Permutations I */</span>
 <a id="__codelineno-9-26" name="__codelineno-9-26" href="#__codelineno-9-26"></a><span class="k">fn</span><span class="w"> </span><span class="nf">permutations_i</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span><span class="w"> </span><span class="c1">// State (subset)</span>
+<a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span><span class="w"> </span><span class="c1">// Result list (permutations)</span>
 <a id="__codelineno-9-28" name="__codelineno-9-28" href="#__codelineno-9-28"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()],</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-9-29" name="__codelineno-9-29" href="#__codelineno-9-29"></a><span class="w">    </span><span class="n">res</span>
 <a id="__codelineno-9-30" name="__codelineno-9-30" href="#__codelineno-9-30"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_i.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Backtracking algorithm: Permutations I */</span>
+<div class="highlight"><span class="filename">permutations_i.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Backtracking: permutations I */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">stateSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">,</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="o">*</span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">**</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">resSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">stateSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="o">*</span><span class="n">resSize</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="n">choicesSize</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5197,17 +5269,17 @@
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">        </span><span class="p">(</span><span class="o">*</span><span class="n">resSize</span><span class="p">)</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
-<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements</span>
+<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">        </span><span class="c1">// Prune: do not choose an element more than once</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span>
 <a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">            </span><span class="n">state</span><span class="p">[</span><span class="n">stateSize</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choice</span><span class="p">;</span>
-<a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">stateSize</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">resSize</span><span class="p">);</span>
-<a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">false</span><span class="p">;</span>
 <a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="w">    </span><span class="p">}</span>
@@ -5233,29 +5305,29 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_i.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Backtracking algorithm: Permutations I */</span>
+<div class="highlight"><span class="filename">permutations_i.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Backtracking: permutations I */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="n">selected</span><span class="p">:</span><span class="w"> </span><span class="n">BooleanArray</span><span class="p">,</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?&gt;</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">toMutableList</span><span class="p">())</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">indices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
-<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements</span>
+<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">        </span><span class="c1">// Prune: do not choose an element more than once</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">            </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span>
 <a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
-<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">            </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span>
 <a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="na">removeAt</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w">        </span><span class="p">}</span>
@@ -5273,22 +5345,22 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">permutations_i.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Backtracking: permutations I ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">state</span><span class="o">.</span><span class="n">dup</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">    </span><span class="k">return</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a>
-<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="c1"># Traverse all choices</span>
+<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="c1"># Iterate over all choices</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="n">choices</span><span class="o">.</span><span class="n">each_with_index</span><span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="o">|</span><span class="n">choice</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="o">|</span>
-<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="c1"># Pruning: do not allow repeated selection of elements</span>
+<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="c1"># Prune: do not choose an element more than once</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">    </span><span class="k">unless</span><span class="w"> </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
-<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">      </span><span class="c1"># Attempt: make choice, update state</span>
+<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">      </span><span class="c1"># Try: make a choice and update the state</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">      </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">true</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">      </span><span class="n">state</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">choice</span>
-<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">      </span><span class="c1"># Proceed to the next round of selection</span>
+<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">      </span><span class="c1"># Proceed to the next choice</span>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">      </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">      </span><span class="c1"># Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">      </span><span class="c1"># Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">      </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">false</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="w">      </span><span class="n">state</span><span class="o">.</span><span class="n">pop</span>
 <a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">    </span><span class="k">end</span>
@@ -5305,49 +5377,54 @@
 </div>
 </div>
 </div>
-<h2 id="1322-case-with-duplicate-elements">13.2.2 &nbsp; Case with Duplicate Elements<a class="headerlink" href="#1322-case-with-duplicate-elements" title="Permanent link">&para;</a></h2>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20backtrack%28%0A%20%20%20%20state%3A%20list%5Bint%5D,%20choices%3A%20list%5Bint%5D,%20selected%3A%20list%5Bbool%5D,%20res%3A%20list%5Blist%5Bint%5D%5D%0A%29%3A%0A%20%20%20%20%22%22%22Backtracking%3A%20permutations%20I%22%22%22%0A%20%20%20%20%23%20Record%20a%20solution%20when%20the%20state%20length%20equals%20the%20number%20of%20elements%0A%20%20%20%20if%20len%28state%29%20%3D%3D%20len%28choices%29%3A%0A%20%20%20%20%20%20%20%20res.append%28list%28state%29%29%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Iterate%20over%20all%20choices%0A%20%20%20%20for%20i,%20choice%20in%20enumerate%28choices%29%3A%0A%20%20%20%20%20%20%20%20%23%20Prune%3A%20do%20not%20choose%20an%20element%20more%20than%20once%0A%20%20%20%20%20%20%20%20if%20not%20selected%5Bi%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Try%3A%20make%20a%20choice%20and%20update%20the%20state%0A%20%20%20%20%20%20%20%20%20%20%20%20selected%5Bi%5D%20%3D%20True%0A%20%20%20%20%20%20%20%20%20%20%20%20state.append%28choice%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Proceed%20to%20the%20next%20choice%0A%20%20%20%20%20%20%20%20%20%20%20%20backtrack%28state,%20choices,%20selected,%20res%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Backtrack%3A%20undo%20the%20choice%20and%20restore%20the%20previous%20state%0A%20%20%20%20%20%20%20%20%20%20%20%20selected%5Bi%5D%20%3D%20False%0A%20%20%20%20%20%20%20%20%20%20%20%20state.pop%28%29%0A%0A%0Adef%20permutations_i%28nums%3A%20list%5Bint%5D%29%20-%3E%20list%5Blist%5Bint%5D%5D%3A%0A%20%20%20%20%22%22%22Permutations%20I%22%22%22%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20backtrack%28state%3D%5B%5D,%20choices%3Dnums,%20selected%3D%5BFalse%5D%20%2A%20len%28nums%29,%20res%3Dres%29%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B1,%202,%203%5D%0A%0A%20%20%20%20res%20%3D%20permutations_i%28nums%29%0A%0A%20%20%20%20print%28f%22Input%20array%20nums%20%3D%20%7Bnums%7D%22%29%0A%20%20%20%20print%28f%22All%20permutations%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=13&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20backtrack%28%0A%20%20%20%20state%3A%20list%5Bint%5D,%20choices%3A%20list%5Bint%5D,%20selected%3A%20list%5Bbool%5D,%20res%3A%20list%5Blist%5Bint%5D%5D%0A%29%3A%0A%20%20%20%20%22%22%22Backtracking%3A%20permutations%20I%22%22%22%0A%20%20%20%20%23%20Record%20a%20solution%20when%20the%20state%20length%20equals%20the%20number%20of%20elements%0A%20%20%20%20if%20len%28state%29%20%3D%3D%20len%28choices%29%3A%0A%20%20%20%20%20%20%20%20res.append%28list%28state%29%29%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Iterate%20over%20all%20choices%0A%20%20%20%20for%20i,%20choice%20in%20enumerate%28choices%29%3A%0A%20%20%20%20%20%20%20%20%23%20Prune%3A%20do%20not%20choose%20an%20element%20more%20than%20once%0A%20%20%20%20%20%20%20%20if%20not%20selected%5Bi%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Try%3A%20make%20a%20choice%20and%20update%20the%20state%0A%20%20%20%20%20%20%20%20%20%20%20%20selected%5Bi%5D%20%3D%20True%0A%20%20%20%20%20%20%20%20%20%20%20%20state.append%28choice%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Proceed%20to%20the%20next%20choice%0A%20%20%20%20%20%20%20%20%20%20%20%20backtrack%28state,%20choices,%20selected,%20res%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Backtrack%3A%20undo%20the%20choice%20and%20restore%20the%20previous%20state%0A%20%20%20%20%20%20%20%20%20%20%20%20selected%5Bi%5D%20%3D%20False%0A%20%20%20%20%20%20%20%20%20%20%20%20state.pop%28%29%0A%0A%0Adef%20permutations_i%28nums%3A%20list%5Bint%5D%29%20-%3E%20list%5Blist%5Bint%5D%5D%3A%0A%20%20%20%20%22%22%22Permutations%20I%22%22%22%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20backtrack%28state%3D%5B%5D,%20choices%3Dnums,%20selected%3D%5BFalse%5D%20%2A%20len%28nums%29,%20res%3Dres%29%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B1,%202,%203%5D%0A%0A%20%20%20%20res%20%3D%20permutations_i%28nums%29%0A%0A%20%20%20%20print%28f%22Input%20array%20nums%20%3D%20%7Bnums%7D%22%29%0A%20%20%20%20print%28f%22All%20permutations%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=13&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1322-duplicate-elements">13.2.2 &nbsp; Duplicate elements<a class="headerlink" href="#1322-duplicate-elements" title="Permanent link">&para;</a></h2>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given an integer array that <strong>may contain duplicate elements</strong>, return all unique permutations.</p>
+<p>Given an integer array that <strong>may contain duplicate elements</strong>, return all distinct permutations.</p>
 </div>
-<p>Suppose the input array is <span class="arithmatex">\([1, 1, 2]\)</span>. To distinguish the two duplicate elements <span class="arithmatex">\(1\)</span>, we denote the second <span class="arithmatex">\(1\)</span> as <span class="arithmatex">\(\hat{1}\)</span>.</p>
-<p>As shown in Figure 13-7, half of the permutations generated by the above method are duplicates.</p>
+<p>Suppose the input is <span class="arithmatex">\([1, 1, 2]\)</span>. To distinguish the two occurrences of <span class="arithmatex">\(1\)</span>, denote the second <span class="arithmatex">\(1\)</span> by <span class="arithmatex">\(\hat{1}\)</span>.</p>
+<p>As shown in Figure 13-7, half the permutations generated by the previous method are duplicates.</p>
 <p><img alt="Duplicate permutations" class="animation-figure" src="../permutations_problem.assets/permutations_ii.png" /></p>
 <p align="center"> Figure 13-7 &nbsp; Duplicate permutations </p>
 
-<p>So how do we remove duplicate permutations? The most direct approach is to use a hash set to directly deduplicate the permutation results. However, this is not elegant because <strong>the search branches that generate duplicate permutations are unnecessary and should be identified and pruned early</strong>, which can further improve algorithm efficiency.</p>
-<h3 id="1-pruning-equal-elements">1. &nbsp; Pruning Equal Elements<a class="headerlink" href="#1-pruning-equal-elements" title="Permanent link">&para;</a></h3>
-<p>Observe Figure 13-8. In the first round, choosing <span class="arithmatex">\(1\)</span> or choosing <span class="arithmatex">\(\hat{1}\)</span> is equivalent. All permutations generated under these two choices are duplicates. Therefore, we should prune <span class="arithmatex">\(\hat{1}\)</span>.</p>
-<p>Similarly, after choosing <span class="arithmatex">\(2\)</span> in the first round, the <span class="arithmatex">\(1\)</span> and <span class="arithmatex">\(\hat{1}\)</span> in the second round also produce duplicate branches, so the second round's <span class="arithmatex">\(\hat{1}\)</span> should also be pruned.</p>
-<p>Essentially, <strong>our goal is to ensure that multiple equal elements are chosen only once in a certain round of choices</strong>.</p>
+<p>How can we remove these duplicates? A straightforward approach is to use a hash set to deduplicate the results. However, this is inelegant: <strong>branches that generate duplicate permutations are unnecessary and should be identified and pruned in advance</strong>, improving efficiency.</p>
+<h3 id="1-pruning-equal-elements">1. &nbsp; Pruning equal elements<a class="headerlink" href="#1-pruning-equal-elements" title="Permanent link">&para;</a></h3>
+<p>In Figure 13-8, choosing <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(\hat{1}\)</span> in the first round is equivalent: every permutation produced under one choice is duplicated under the other. We should therefore prune <span class="arithmatex">\(\hat{1}\)</span>.</p>
+<p>Similarly, after choosing <span class="arithmatex">\(2\)</span> in the first round, choosing either <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(\hat{1}\)</span> in the second produces duplicate branches. Prune the second-round choice of <span class="arithmatex">\(\hat{1}\)</span> as well.</p>
+<p>In essence, <strong>within any one round of choices, we want to try only one of the elements with the same value</strong>.</p>
 <p><img alt="Pruning duplicate permutations" class="animation-figure" src="../permutations_problem.assets/permutations_ii_pruning.png" /></p>
 <p align="center"> Figure 13-8 &nbsp; Pruning duplicate permutations </p>
 
-<h3 id="2-code-implementation_1">2. &nbsp; Code Implementation<a class="headerlink" href="#2-code-implementation_1" title="Permanent link">&para;</a></h3>
-<p>Building on the code from the previous problem, we initialize a hash set <code>duplicated</code> in each round of choices to record which elements have already been tried in that round, and prune equal elements:</p>
+<h3 id="2-code-implementation_1">2. &nbsp; Code implementation<a class="headerlink" href="#2-code-implementation_1" title="Permanent link">&para;</a></h3>
+<p>Building on the previous solution, create a hash set <code>duplicated</code> in each round to record the values already tried in that round and prune repeated values:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">permutations_ii.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a>    <span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">choices</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">selected</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">],</span> <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]]</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="p">):</span>
-<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="w">    </span><span class="sd">"""Backtracking algorithm: Permutations II"""</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="w">    </span><span class="sd">"""Backtracking: permutations II"""</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">state</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">choices</span><span class="p">):</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">state</span><span class="p">))</span>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="k">return</span>
-<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>    <span class="c1"># Traverse all choices</span>
+<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>    <span class="c1"># Iterate over all choices</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>    <span class="n">duplicated</span> <span class="o">=</span> <span class="nb">set</span><span class="p">[</span><span class="nb">int</span><span class="p">]()</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>    <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">choice</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">choices</span><span class="p">):</span>
-<a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>        <span class="c1"># Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
+<a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>        <span class="c1"># Prune: do not reuse an element or choose the same value again at this depth</span>
 <a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>        <span class="k">if</span> <span class="ow">not</span> <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="ow">and</span> <span class="n">choice</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">duplicated</span><span class="p">:</span>
-<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>            <span class="c1"># Attempt: make choice, update state</span>
-<a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>            <span class="n">duplicated</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>  <span class="c1"># Record the selected element value</span>
+<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>            <span class="c1"># Try: make a choice and update the state</span>
+<a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>            <span class="n">duplicated</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>  <span class="c1"># Record the chosen value</span>
 <a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>            <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="kc">True</span>
 <a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>            <span class="n">state</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
-<a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a>            <span class="c1"># Proceed to the next round of selection</span>
+<a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a>            <span class="c1"># Proceed to the next choice</span>
 <a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a>            <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">choices</span><span class="p">,</span> <span class="n">selected</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a>            <span class="c1"># Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a>            <span class="c1"># Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-13-21" name="__codelineno-13-21" href="#__codelineno-13-21"></a>            <span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="kc">False</span>
 <a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a>            <span class="n">state</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
 <a id="__codelineno-13-23" name="__codelineno-13-23" href="#__codelineno-13-23"></a>
@@ -5359,26 +5436,26 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_ii.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Backtracking algorithm: Permutations II */</span>
+<div class="highlight"><span class="filename">permutations_ii.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Backtracking: permutations II */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">size</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">state</span><span class="p">);</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="n">unordered_set</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">duplicated</span><span class="p">;</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
-<a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
+<a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">        </span><span class="c1">// Prune: do not reuse an element or choose the same value again at this depth</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">duplicated</span><span class="p">.</span><span class="n">find</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">duplicated</span><span class="p">.</span><span class="n">end</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
-<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">            </span><span class="n">duplicated</span><span class="p">.</span><span class="n">emplace</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the selected element value</span>
+<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
+<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">            </span><span class="n">duplicated</span><span class="p">.</span><span class="n">emplace</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the chosen value</span>
 <a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
-<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">false</span><span class="p">;</span>
 <a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span>
 <a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a><span class="w">        </span><span class="p">}</span>
@@ -5396,26 +5473,26 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_ii.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Backtracking algorithm: Permutations II */</span>
+<div class="highlight"><span class="filename">permutations_ii.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Backtracking: permutations II */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">boolean</span><span class="o">[]</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="n">Set</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">duplicated</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">HashSet</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="p">();</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
+<a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">        </span><span class="c1">// Prune: do not reuse an element or choose the same value again at this depth</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">duplicated</span><span class="p">.</span><span class="na">contains</span><span class="p">(</span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
-<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">            </span><span class="n">duplicated</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the selected element value</span>
+<a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
+<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">            </span><span class="n">duplicated</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the chosen value</span>
 <a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">            </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
 <a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
-<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="w">            </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
 <a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w">        </span><span class="p">}</span>
@@ -5431,26 +5508,26 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_ii.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Backtracking algorithm: Permutations II */</span>
+<div class="highlight"><span class="filename">permutations_ii.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Backtracking: permutations II */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Backtrack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">bool</span><span class="p">[]</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">Length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="n">HashSet</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">duplicated</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
-<a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
+<a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">        </span><span class="c1">// Prune: do not reuse an element or choose the same value again at this depth</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">duplicated</span><span class="p">.</span><span class="n">Contains</span><span class="p">(</span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
-<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">            </span><span class="n">duplicated</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the selected element value</span>
+<a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
+<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">            </span><span class="n">duplicated</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the chosen value</span>
 <a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">true</span><span class="p">;</span>
 <a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
-<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w">            </span><span class="n">Backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">false</span><span class="p">;</span>
 <a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="n">RemoveAt</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">        </span><span class="p">}</span>
@@ -5466,27 +5543,27 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_ii.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Backtracking algorithm: Permutations II */</span>
+<div class="highlight"><span class="filename">permutations_ii.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Backtracking: permutations II */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">backtrackII</span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">selected</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="kt">bool</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[][]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">        </span><span class="nx">newState</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">append</span><span class="p">([]</span><span class="kt">int</span><span class="p">{},</span><span class="w"> </span><span class="o">*</span><span class="nx">state</span><span class="p">...)</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">        </span><span class="o">*</span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">newState</span><span class="p">)</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">    </span><span class="nx">duplicated</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">(</span><span class="kt">map</span><span class="p">[</span><span class="kt">int</span><span class="p">]</span><span class="kd">struct</span><span class="p">{},</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">        </span><span class="nx">choice</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">)[</span><span class="nx">i</span><span class="p">]</span>
-<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
+<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">        </span><span class="c1">// Prune: do not reuse an element or choose the same value again at this depth</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">ok</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">duplicated</span><span class="p">[</span><span class="nx">choice</span><span class="p">];</span><span class="w"> </span><span class="o">!</span><span class="nx">ok</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="p">(</span><span class="o">*</span><span class="nx">selected</span><span class="p">)[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
-<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">            </span><span class="c1">// Record the selected element value</span>
+<a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
+<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">            </span><span class="c1">// Record the chosen value</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">            </span><span class="nx">duplicated</span><span class="p">[</span><span class="nx">choice</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kd">struct</span><span class="p">{}{}</span>
 <a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">            </span><span class="p">(</span><span class="o">*</span><span class="nx">selected</span><span class="p">)[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span>
 <a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="w">            </span><span class="o">*</span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="p">)</span>
-<a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a><span class="w">            </span><span class="nx">backtrackII</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">selected</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span>
-<a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-17-22" name="__codelineno-17-22" href="#__codelineno-17-22"></a><span class="w">            </span><span class="p">(</span><span class="o">*</span><span class="nx">selected</span><span class="p">)[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span>
 <a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a><span class="w">            </span><span class="o">*</span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[:</span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-17-24" name="__codelineno-17-24" href="#__codelineno-17-24"></a><span class="w">        </span><span class="p">}</span>
@@ -5504,25 +5581,25 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_ii.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Backtracking algorithm: Permutations II */</span>
+<div class="highlight"><span class="filename">permutations_ii.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Backtracking: permutations II */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">selected</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Bool</span><span class="p">],</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="bp">count</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="bp">count</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">duplicated</span><span class="p">:</span><span class="w"> </span><span class="n">Set</span><span class="p">&lt;</span><span class="nb">Int</span><span class="p">&gt;</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">enumerated</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
+<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">        </span><span class="c1">// Prune: do not reuse an element or choose the same value again at this depth</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="o">!</span><span class="n">duplicated</span><span class="p">.</span><span class="bp">contains</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
-<a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">            </span><span class="n">duplicated</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="c1">// Record the selected element value</span>
+<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
+<a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">            </span><span class="n">duplicated</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="c1">// Record the chosen value</span>
 <a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">true</span>
 <a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
-<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">false</span>
 <a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span>
 <a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a><span class="w">        </span><span class="p">}</span>
@@ -5540,25 +5617,25 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_ii.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Backtracking algorithm: Permutations II */</span>
+<div class="highlight"><span class="filename">permutations_ii.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Backtracking: permutations II */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">selected</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">state</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">choices</span><span class="p">.</span><span class="nx">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">state</span><span class="p">]);</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">duplicated</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Set</span><span class="p">();</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="nx">choices</span><span class="p">.</span><span class="nx">forEach</span><span class="p">((</span><span class="nx">choice</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
+<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">        </span><span class="c1">// Prune: do not reuse an element or choose the same value again at this depth</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="nx">selected</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="nx">duplicated</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
-<a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">            </span><span class="nx">duplicated</span><span class="p">.</span><span class="nx">add</span><span class="p">(</span><span class="nx">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the selected element value</span>
+<a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
+<a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">            </span><span class="nx">duplicated</span><span class="p">.</span><span class="nx">add</span><span class="p">(</span><span class="nx">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the chosen value</span>
 <a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">            </span><span class="nx">selected</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">            </span><span class="nx">state</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">choice</span><span class="p">);</span>
-<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">            </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">selected</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">            </span><span class="nx">selected</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
 <a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="w">            </span><span class="nx">state</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="w">        </span><span class="p">}</span>
@@ -5574,30 +5651,30 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_ii.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Backtracking algorithm: Permutations II */</span>
+<div class="highlight"><span class="filename">permutations_ii.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Backtracking: permutations II */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="nx">state</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="nx">choices</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="nx">selected</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="p">[],</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[][]</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">state</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">choices</span><span class="p">.</span><span class="nx">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">state</span><span class="p">]);</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">duplicated</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Set</span><span class="p">();</span>
 <a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">    </span><span class="nx">choices</span><span class="p">.</span><span class="nx">forEach</span><span class="p">((</span><span class="nx">choice</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
+<a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">        </span><span class="c1">// Prune: do not reuse an element or choose the same value again at this depth</span>
 <a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="nx">selected</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="nx">duplicated</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
-<a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">            </span><span class="nx">duplicated</span><span class="p">.</span><span class="nx">add</span><span class="p">(</span><span class="nx">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the selected element value</span>
+<a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
+<a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">            </span><span class="nx">duplicated</span><span class="p">.</span><span class="nx">add</span><span class="p">(</span><span class="nx">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the chosen value</span>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">            </span><span class="nx">selected</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
 <a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="w">            </span><span class="nx">state</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">choice</span><span class="p">);</span>
-<a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-20-23" name="__codelineno-20-23" href="#__codelineno-20-23"></a><span class="w">            </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">selected</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a><span class="w">            </span><span class="nx">selected</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
 <a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="w">            </span><span class="nx">state</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-20-27" name="__codelineno-20-27" href="#__codelineno-20-27"></a><span class="w">        </span><span class="p">}</span>
@@ -5613,31 +5690,31 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_ii.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Backtracking algorithm: Permutations II */</span>
+<div class="highlight"><span class="filename">permutations_ii.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Backtracking: permutations II */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">,</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">List</span><span class="p">.</span><span class="n">from</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">    </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">  </span><span class="n">Set</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">duplicated</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{};</span>
 <a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
-<a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">    </span><span class="c1">// Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
+<a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">    </span><span class="c1">// Prune: do not reuse an element or choose the same value again at this depth</span>
 <a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">duplicated</span><span class="p">.</span><span class="n">contains</span><span class="p">(</span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="w">      </span><span class="c1">// Attempt: make choice, update state</span>
-<a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="w">      </span><span class="n">duplicated</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the selected element value</span>
+<a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="w">      </span><span class="c1">// Try: make a choice and update the state</span>
+<a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="w">      </span><span class="n">duplicated</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the chosen value</span>
 <a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="w">      </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
 <a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">      </span><span class="n">state</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
-<a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">      </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">      </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">      </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-21-25" name="__codelineno-21-25" href="#__codelineno-21-25"></a><span class="w">      </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-21-25" name="__codelineno-21-25" href="#__codelineno-21-25"></a><span class="w">      </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a><span class="w">      </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
 <a id="__codelineno-21-27" name="__codelineno-21-27" href="#__codelineno-21-27"></a><span class="w">      </span><span class="n">state</span><span class="p">.</span><span class="n">removeLast</span><span class="p">();</span>
 <a id="__codelineno-21-28" name="__codelineno-21-28" href="#__codelineno-21-28"></a><span class="w">    </span><span class="p">}</span>
@@ -5653,26 +5730,26 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_ii.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Backtracking algorithm: Permutations II */</span>
+<div class="highlight"><span class="filename">permutations_ii.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Backtracking: permutations II */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">selected</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">bool</span><span class="p">],</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">state</span><span class="p">);</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">duplicated</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashSet</span><span class="p">::</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">choices</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
-<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
+<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">        </span><span class="c1">// Prune: do not reuse an element or choose the same value again at this depth</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">duplicated</span><span class="p">.</span><span class="n">contains</span><span class="p">(</span><span class="o">&amp;</span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
-<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">            </span><span class="n">duplicated</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the selected element value</span>
+<a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
+<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">            </span><span class="n">duplicated</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the chosen value</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
 <a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">choice</span><span class="p">);</span>
-<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">clone</span><span class="p">(),</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span>
 <a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span>
 <a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w">        </span><span class="p">}</span>
@@ -5688,9 +5765,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_ii.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Backtracking algorithm: Permutations II */</span>
+<div class="highlight"><span class="filename">permutations_ii.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Backtracking: permutations II */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">stateSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">,</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="o">*</span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">**</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">resSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">stateSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="o">*</span><span class="n">resSize</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="n">choicesSize</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5699,19 +5776,19 @@
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">        </span><span class="p">(</span><span class="o">*</span><span class="n">resSize</span><span class="p">)</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">    </span><span class="kt">bool</span><span class="w"> </span><span class="n">duplicated</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="nb">false</span><span class="p">};</span>
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
-<a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
+<a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w">        </span><span class="c1">// Prune: do not reuse an element or choose the same value again at this depth</span>
 <a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">duplicated</span><span class="p">[</span><span class="n">choice</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
-<a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w">            </span><span class="n">duplicated</span><span class="p">[</span><span class="n">choice</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record the selected element value</span>
+<a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
+<a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w">            </span><span class="n">duplicated</span><span class="p">[</span><span class="n">choice</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record the chosen value</span>
 <a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span>
 <a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="w">            </span><span class="n">state</span><span class="p">[</span><span class="n">stateSize</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choice</span><span class="p">;</span>
-<a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">stateSize</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">resSize</span><span class="p">);</span>
-<a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="w">            </span><span class="n">selected</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">false</span><span class="p">;</span>
 <a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a><span class="w">    </span><span class="p">}</span>
@@ -5737,31 +5814,31 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">permutations_ii.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Backtracking algorithm: Permutations II */</span>
+<div class="highlight"><span class="filename">permutations_ii.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Backtracking: permutations II */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="n">selected</span><span class="p">:</span><span class="w"> </span><span class="n">BooleanArray</span><span class="p">,</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?&gt;</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="c1">// When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="c1">// Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">toMutableList</span><span class="p">())</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">duplicated</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashSet</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">indices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
-<a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">        </span><span class="c1">// Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
+<a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">        </span><span class="c1">// Prune: do not reuse an element or choose the same value again at this depth</span>
 <a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">duplicated</span><span class="p">.</span><span class="na">contains</span><span class="p">(</span><span class="n">choice</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="w">            </span><span class="c1">// Attempt: make choice, update state</span>
-<a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w">            </span><span class="n">duplicated</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="c1">// Record the selected element value</span>
+<a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="w">            </span><span class="c1">// Try: make a choice and update the state</span>
+<a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w">            </span><span class="n">duplicated</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span><span class="w"> </span><span class="c1">// Record the chosen value</span>
 <a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="w">            </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span>
 <a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
-<a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">            </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">            </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a><span class="w">            </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-24-25" name="__codelineno-24-25" href="#__codelineno-24-25"></a><span class="w">            </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-24-25" name="__codelineno-24-25" href="#__codelineno-24-25"></a><span class="w">            </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-24-26" name="__codelineno-24-26" href="#__codelineno-24-26"></a><span class="w">            </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span>
 <a id="__codelineno-24-27" name="__codelineno-24-27" href="#__codelineno-24-27"></a><span class="w">            </span><span class="n">state</span><span class="p">.</span><span class="na">removeAt</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-24-28" name="__codelineno-24-28" href="#__codelineno-24-28"></a><span class="w">        </span><span class="p">}</span>
@@ -5779,24 +5856,24 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">permutations_ii.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Backtracking: permutations II ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># When the state length equals the number of elements, record the solution</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Record a solution when the state length equals the number of elements</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">state</span><span class="o">.</span><span class="n">dup</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">    </span><span class="k">return</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a>
-<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="c1"># Traverse all choices</span>
+<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="c1"># Iterate over all choices</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="n">duplicated</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">Set</span><span class="o">.</span><span class="n">new</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">  </span><span class="n">choices</span><span class="o">.</span><span class="n">each_with_index</span><span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="o">|</span><span class="n">choice</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="o">|</span>
-<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">    </span><span class="c1"># Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements</span>
+<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">    </span><span class="c1"># Prune: do not reuse an element or choose the same value again at this depth</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="o">!</span><span class="n">duplicated</span><span class="o">.</span><span class="n">include?</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
-<a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">      </span><span class="c1"># Attempt: make choice, update state</span>
+<a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">      </span><span class="c1"># Try: make a choice and update the state</span>
 <a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">      </span><span class="n">duplicated</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">choice</span><span class="p">)</span>
 <a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">      </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">true</span>
 <a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w">      </span><span class="n">state</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">choice</span>
-<a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w">      </span><span class="c1"># Proceed to the next round of selection</span>
+<a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w">      </span><span class="c1"># Proceed to the next choice</span>
 <a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="w">      </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">selected</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a><span class="w">      </span><span class="c1"># Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a><span class="w">      </span><span class="c1"># Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="w">      </span><span class="n">selected</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">false</span>
 <a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="w">      </span><span class="n">state</span><span class="o">.</span><span class="n">pop</span>
 <a id="__codelineno-25-23" name="__codelineno-25-23" href="#__codelineno-25-23"></a><span class="w">    </span><span class="k">end</span>
@@ -5813,17 +5890,22 @@
 </div>
 </div>
 </div>
-<p>Assuming elements are pairwise distinct, there are <span class="arithmatex">\(n!\)</span> (factorial) permutations of <span class="arithmatex">\(n\)</span> elements. When recording results, we need to copy a list of length <span class="arithmatex">\(n\)</span>, using <span class="arithmatex">\(O(n)\)</span> time. <strong>Therefore, the time complexity is <span class="arithmatex">\(O(n! \cdot n)\)</span></strong>.</p>
-<p>The maximum recursion depth is <span class="arithmatex">\(n\)</span>, using <span class="arithmatex">\(O(n)\)</span> stack frame space. <code>selected</code> uses <span class="arithmatex">\(O(n)\)</span> space. At most <span class="arithmatex">\(n\)</span> <code>duplicated</code> sets exist simultaneously, using <span class="arithmatex">\(O(n^2)\)</span> space. <strong>Therefore, the space complexity is <span class="arithmatex">\(O(n^2)\)</span></strong>.</p>
-<h3 id="3-comparison-of-two-pruning-methods">3. &nbsp; Comparison of Two Pruning Methods<a class="headerlink" href="#3-comparison-of-two-pruning-methods" title="Permanent link">&para;</a></h3>
-<p>Note that although both <code>selected</code> and <code>duplicated</code> are used for pruning, they have different objectives.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20backtrack%28%0A%20%20%20%20state%3A%20list%5Bint%5D,%20choices%3A%20list%5Bint%5D,%20selected%3A%20list%5Bbool%5D,%20res%3A%20list%5Blist%5Bint%5D%5D%0A%29%3A%0A%20%20%20%20%22%22%22Backtracking%3A%20permutations%20II%22%22%22%0A%20%20%20%20%23%20Record%20a%20solution%20when%20the%20state%20length%20equals%20the%20number%20of%20elements%0A%20%20%20%20if%20len%28state%29%20%3D%3D%20len%28choices%29%3A%0A%20%20%20%20%20%20%20%20res.append%28list%28state%29%29%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Iterate%20over%20all%20choices%0A%20%20%20%20duplicated%20%3D%20set%5Bint%5D%28%29%0A%20%20%20%20for%20i,%20choice%20in%20enumerate%28choices%29%3A%0A%20%20%20%20%20%20%20%20%23%20Prune%3A%20do%20not%20reuse%20an%20element%20or%20choose%20the%20same%20value%20again%20at%20this%20depth%0A%20%20%20%20%20%20%20%20if%20not%20selected%5Bi%5D%20and%20choice%20not%20in%20duplicated%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Try%3A%20make%20a%20choice%20and%20update%20the%20state%0A%20%20%20%20%20%20%20%20%20%20%20%20duplicated.add%28choice%29%20%20%23%20Record%20the%20chosen%20value%0A%20%20%20%20%20%20%20%20%20%20%20%20selected%5Bi%5D%20%3D%20True%0A%20%20%20%20%20%20%20%20%20%20%20%20state.append%28choice%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Proceed%20to%20the%20next%20choice%0A%20%20%20%20%20%20%20%20%20%20%20%20backtrack%28state,%20choices,%20selected,%20res%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Backtrack%3A%20undo%20the%20choice%20and%20restore%20the%20previous%20state%0A%20%20%20%20%20%20%20%20%20%20%20%20selected%5Bi%5D%20%3D%20False%0A%20%20%20%20%20%20%20%20%20%20%20%20state.pop%28%29%0A%0A%0Adef%20permutations_ii%28nums%3A%20list%5Bint%5D%29%20-%3E%20list%5Blist%5Bint%5D%5D%3A%0A%20%20%20%20%22%22%22Permutations%20II%22%22%22%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20backtrack%28state%3D%5B%5D,%20choices%3Dnums,%20selected%3D%5BFalse%5D%20%2A%20len%28nums%29,%20res%3Dres%29%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B1,%202,%202%5D%0A%0A%20%20%20%20res%20%3D%20permutations_ii%28nums%29%0A%0A%20%20%20%20print%28f%22Input%20array%20nums%20%3D%20%7Bnums%7D%22%29%0A%20%20%20%20print%28f%22All%20permutations%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=13&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20backtrack%28%0A%20%20%20%20state%3A%20list%5Bint%5D,%20choices%3A%20list%5Bint%5D,%20selected%3A%20list%5Bbool%5D,%20res%3A%20list%5Blist%5Bint%5D%5D%0A%29%3A%0A%20%20%20%20%22%22%22Backtracking%3A%20permutations%20II%22%22%22%0A%20%20%20%20%23%20Record%20a%20solution%20when%20the%20state%20length%20equals%20the%20number%20of%20elements%0A%20%20%20%20if%20len%28state%29%20%3D%3D%20len%28choices%29%3A%0A%20%20%20%20%20%20%20%20res.append%28list%28state%29%29%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Iterate%20over%20all%20choices%0A%20%20%20%20duplicated%20%3D%20set%5Bint%5D%28%29%0A%20%20%20%20for%20i,%20choice%20in%20enumerate%28choices%29%3A%0A%20%20%20%20%20%20%20%20%23%20Prune%3A%20do%20not%20reuse%20an%20element%20or%20choose%20the%20same%20value%20again%20at%20this%20depth%0A%20%20%20%20%20%20%20%20if%20not%20selected%5Bi%5D%20and%20choice%20not%20in%20duplicated%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Try%3A%20make%20a%20choice%20and%20update%20the%20state%0A%20%20%20%20%20%20%20%20%20%20%20%20duplicated.add%28choice%29%20%20%23%20Record%20the%20chosen%20value%0A%20%20%20%20%20%20%20%20%20%20%20%20selected%5Bi%5D%20%3D%20True%0A%20%20%20%20%20%20%20%20%20%20%20%20state.append%28choice%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Proceed%20to%20the%20next%20choice%0A%20%20%20%20%20%20%20%20%20%20%20%20backtrack%28state,%20choices,%20selected,%20res%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Backtrack%3A%20undo%20the%20choice%20and%20restore%20the%20previous%20state%0A%20%20%20%20%20%20%20%20%20%20%20%20selected%5Bi%5D%20%3D%20False%0A%20%20%20%20%20%20%20%20%20%20%20%20state.pop%28%29%0A%0A%0Adef%20permutations_ii%28nums%3A%20list%5Bint%5D%29%20-%3E%20list%5Blist%5Bint%5D%5D%3A%0A%20%20%20%20%22%22%22Permutations%20II%22%22%22%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20backtrack%28state%3D%5B%5D,%20choices%3Dnums,%20selected%3D%5BFalse%5D%20%2A%20len%28nums%29,%20res%3Dres%29%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B1,%202,%202%5D%0A%0A%20%20%20%20res%20%3D%20permutations_ii%28nums%29%0A%0A%20%20%20%20print%28f%22Input%20array%20nums%20%3D%20%7Bnums%7D%22%29%0A%20%20%20%20print%28f%22All%20permutations%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=13&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>If all elements are distinct, <span class="arithmatex">\(n\)</span> elements have <span class="arithmatex">\(n!\)</span> permutations, where the exclamation mark denotes factorial. Recording each result requires copying a list of length <span class="arithmatex">\(n\)</span>, taking <span class="arithmatex">\(O(n)\)</span> time. <strong>The time complexity is therefore <span class="arithmatex">\(O(n!n)\)</span></strong>.</p>
+<p>The maximum recursion depth is <span class="arithmatex">\(n\)</span>, requiring <span class="arithmatex">\(O(n)\)</span> space for stack frames. The array <code>selected</code> uses <span class="arithmatex">\(O(n)\)</span> space. At most <span class="arithmatex">\(n\)</span> <code>duplicated</code> sets exist at once, using <span class="arithmatex">\(O(n^2)\)</span> space. <strong>The space complexity is therefore <span class="arithmatex">\(O(n^2)\)</span></strong>.</p>
+<h3 id="3-comparing-the-two-pruning-methods">3. &nbsp; Comparing the two pruning methods<a class="headerlink" href="#3-comparing-the-two-pruning-methods" title="Permanent link">&para;</a></h3>
+<p>Although both <code>selected</code> and <code>duplicated</code> are used for pruning, they serve different purposes.</p>
 <ul>
-<li><strong>Pruning duplicate choices</strong>: There is only one <code>selected</code> throughout the entire search process. It records which elements are included in the current state, and its purpose is to prevent an element from appearing repeatedly in <code>state</code>.</li>
-<li><strong>Pruning equal elements</strong>: Each round of choices (each <code>backtrack</code> function call) contains a <code>duplicated</code> set. It records which elements have been chosen in this round's iteration (the <code>for</code> loop), and its purpose is to ensure that equal elements are chosen only once.</li>
+<li><strong>Pruning repeated selections</strong>: There is only one <code>selected</code> array throughout the search. It records which elements are in the current state, preventing the same element from appearing twice in <code>state</code>.</li>
+<li><strong>Pruning equal elements</strong>: Each round of choices (each call to <code>backtrack</code>) has its own <code>duplicated</code> set. It records the values already chosen in that round's <code>for</code> loop, ensuring that equal elements are tried only once in that round.</li>
 </ul>
-<p>Figure 13-9 shows the effective scope of the two pruning conditions. Note that each node in the tree represents a choice, and the nodes on the path from the root to a leaf node form a permutation.</p>
-<p><img alt="Effective scope of two pruning conditions" class="animation-figure" src="../permutations_problem.assets/permutations_ii_pruning_summary.png" /></p>
-<p align="center"> Figure 13-9 &nbsp; Effective scope of two pruning conditions </p>
+<p>Figure 13-9 shows the scope of each pruning condition. Each node represents a choice, and the nodes along a path from root to leaf form a permutation.</p>
+<p><img alt="Scope of the two pruning conditions" class="animation-figure" src="../permutations_problem.assets/permutations_ii_pruning_summary.png" /></p>
+<p align="center"> Figure 13-9 &nbsp; Scope of the two pruning conditions </p>
 
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   <a
     href="../backtracking_algorithm/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 13.1 Backtracking Algorithm"
+    aria-label="Previous: 13.1   Backtracking algorithms"
     rel="prev"
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@@ -5858,7 +5940,7 @@ aria-label="Footer"
         Previous
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-        13.1 Backtracking Algorithm
+        13.1 &nbsp; Backtracking algorithms
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   <a
     href="../subset_sum_problem/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 13.3 Subset-Sum Problem"
+    aria-label="Next: 13.3   Subset sum problem"
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+        13.3 &nbsp; Subset sum problem
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@@ -5895,7 +5977,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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@@ -41,7 +41,7 @@
 
 
 
-        <title>13.3 Subset-Sum Problem - Hello Algo</title>
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-  Before Starting
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-  Before Starting
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-  0.1 About This Book
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-  0.2 How to Use This Book
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-  0.3 Summary
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-  Chapter 1. Encounter with Algorithms
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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-  1.3 Summary
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-  2.2 Iteration and Recursion
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-  2.6 Exercises
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
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-  3.1 Classification of Data Structures
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-  3.2 Basic Data Types
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+  6.2 &nbsp; Hash collisions
 
 
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+  7.5 &nbsp; AVL trees *
 
 
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+  Chapter 9. &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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+  Chapter 10. &nbsp; Searching
 
 
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+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3283,7 +3283,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3304,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3352,7 +3352,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3379,7 +3379,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3406,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3433,7 +3433,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3460,7 +3460,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3528,7 +3528,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3549,7 +3549,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3570,7 +3570,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3597,7 +3597,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3633,7 +3633,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3650,7 +3650,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3676,22 +3676,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1331-without-duplicate-elements" class="md-nav__link">
+  <a href="#1331-distinct-elements" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.3.1 &nbsp; Without Duplicate Elements
+        13.3.1 &nbsp; Distinct elements
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="13.3.1 Without Duplicate Elements">
+    <nav class="md-nav" aria-label="13.3.1 Distinct elements">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-using-the-permutation-solution-as-a-reference" class="md-nav__link">
+  <a href="#1-starting-from-the-permutations-solution" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Using the Permutation Solution as a Reference
+        1. &nbsp; Starting from the permutations solution
       </span>
     </span>
   </a>
@@ -3702,7 +3702,7 @@
   <a href="#2-pruning-duplicate-subsets" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Pruning Duplicate Subsets
+        2. &nbsp; Pruning duplicate subsets
       </span>
     </span>
   </a>
@@ -3713,7 +3713,7 @@
   <a href="#3-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Code Implementation
+        3. &nbsp; Code implementation
       </span>
     </span>
   </a>
@@ -3726,22 +3726,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1332-with-duplicate-elements-in-array" class="md-nav__link">
+  <a href="#1332-duplicate-elements" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.3.2 &nbsp; With Duplicate Elements in Array
+        13.3.2 &nbsp; Duplicate elements
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="13.3.2 With Duplicate Elements in Array">
+    <nav class="md-nav" aria-label="13.3.2 Duplicate elements">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-pruning-equal-elements" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Pruning Equal Elements
+        1. &nbsp; Pruning equal elements
       </span>
     </span>
   </a>
@@ -3752,7 +3752,7 @@
   <a href="#2-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Code Implementation
+        2. &nbsp; Code implementation
       </span>
     </span>
   </a>
@@ -3787,7 +3787,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3814,7 +3814,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3841,7 +3841,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3911,7 +3911,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3932,7 +3932,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3953,7 +3953,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3980,7 +3980,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -4007,7 +4007,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4034,7 +4034,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4061,7 +4061,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4088,7 +4088,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4115,7 +4115,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4142,7 +4142,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4208,7 +4208,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4229,7 +4229,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4250,7 +4250,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4277,7 +4277,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4304,7 +4304,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4331,7 +4331,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4358,7 +4358,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4385,7 +4385,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4445,7 +4445,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4466,7 +4466,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4487,7 +4487,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4514,7 +4514,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4541,7 +4541,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4625,6 +4625,78 @@
 
 
 
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@@ -4666,22 +4738,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1331-without-duplicate-elements" class="md-nav__link">
+  <a href="#1331-distinct-elements" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.3.1 &nbsp; Without Duplicate Elements
+        13.3.1 &nbsp; Distinct elements
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="13.3.1 Without Duplicate Elements">
+    <nav class="md-nav" aria-label="13.3.1 Distinct elements">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-using-the-permutation-solution-as-a-reference" class="md-nav__link">
+  <a href="#1-starting-from-the-permutations-solution" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Using the Permutation Solution as a Reference
+        1. &nbsp; Starting from the permutations solution
       </span>
     </span>
   </a>
@@ -4692,7 +4764,7 @@
   <a href="#2-pruning-duplicate-subsets" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Pruning Duplicate Subsets
+        2. &nbsp; Pruning duplicate subsets
       </span>
     </span>
   </a>
@@ -4703,7 +4775,7 @@
   <a href="#3-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Code Implementation
+        3. &nbsp; Code implementation
       </span>
     </span>
   </a>
@@ -4716,22 +4788,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1332-with-duplicate-elements-in-array" class="md-nav__link">
+  <a href="#1332-duplicate-elements" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        13.3.2 &nbsp; With Duplicate Elements in Array
+        13.3.2 &nbsp; Duplicate elements
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="13.3.2 With Duplicate Elements in Array">
+    <nav class="md-nav" aria-label="13.3.2 Duplicate elements">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-pruning-equal-elements" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Pruning Equal Elements
+        1. &nbsp; Pruning equal elements
       </span>
     </span>
   </a>
@@ -4742,7 +4814,7 @@
   <a href="#2-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Code Implementation
+        2. &nbsp; Code implementation
       </span>
     </span>
   </a>
@@ -4791,20 +4863,20 @@
 
 
 <!-- Page content -->
-<h1 id="133-subset-sum-problem">13.3 &nbsp; Subset-Sum Problem<a class="headerlink" href="#133-subset-sum-problem" title="Permanent link">&para;</a></h1>
-<h2 id="1331-without-duplicate-elements">13.3.1 &nbsp; Without Duplicate Elements<a class="headerlink" href="#1331-without-duplicate-elements" title="Permanent link">&para;</a></h2>
+<h1 id="133-subset-sum-problem">13.3 &nbsp; Subset sum problem<a class="headerlink" href="#133-subset-sum-problem" title="Permanent link">&para;</a></h1>
+<h2 id="1331-distinct-elements">13.3.1 &nbsp; Distinct elements<a class="headerlink" href="#1331-distinct-elements" title="Permanent link">&para;</a></h2>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given a positive integer array <code>nums</code> and a target positive integer <code>target</code>, find all possible combinations where the sum of elements in the combination equals <code>target</code>. The given array has no duplicate elements, and each element can be selected multiple times. Return these combinations in list form, where the list should not contain duplicate combinations.</p>
+<p>Given an array of positive integers <code>nums</code> and a positive target integer <code>target</code>, find all combinations whose elements sum to <code>target</code>. The array has no duplicate elements, and each element may be chosen multiple times. Return the combinations as a list containing no duplicates.</p>
 </div>
-<p>For example, given the set <span class="arithmatex">\(\{3, 4, 5\}\)</span> and target integer <span class="arithmatex">\(9\)</span>, the solutions are <span class="arithmatex">\(\{3, 3, 3\}, \{4, 5\}\)</span>. Note the following two points:</p>
+<p>For example, with input <span class="arithmatex">\(\{3, 4, 5\}\)</span> and target <span class="arithmatex">\(9\)</span>, the solutions are <span class="arithmatex">\(\{3, 3, 3\}, \{4, 5\}\)</span>. Keep two points in mind.</p>
 <ul>
-<li>Elements in the input set can be selected repeatedly without limit.</li>
-<li>Subsets do not distinguish element order; for example, <span class="arithmatex">\(\{4, 5\}\)</span> and <span class="arithmatex">\(\{5, 4\}\)</span> are the same subset.</li>
+<li>Each input element may be chosen an unlimited number of times.</li>
+<li>Element order does not matter: <span class="arithmatex">\(\{4, 5\}\)</span> and <span class="arithmatex">\(\{5, 4\}\)</span> represent the same subset.</li>
 </ul>
-<h3 id="1-using-the-permutation-solution-as-a-reference">1. &nbsp; Using the Permutation Solution as a Reference<a class="headerlink" href="#1-using-the-permutation-solution-as-a-reference" title="Permanent link">&para;</a></h3>
-<p>Similar to the permutation problem, we can view the process of generating subsets as the result of a series of choices and update the running sum during the selection process. When the sum equals <code>target</code>, we record the subset in the result list.</p>
-<p>Unlike the permutation problem, <strong>elements in this problem can be selected any number of times</strong>, so we do not need to use a <code>selected</code> boolean list to track whether an element has already been selected. With a few small changes to the permutation code, we obtain an initial solution:</p>
+<h3 id="1-starting-from-the-permutations-solution">1. &nbsp; Starting from the permutations solution<a class="headerlink" href="#1-starting-from-the-permutations-solution" title="Permanent link">&para;</a></h3>
+<p>As with permutations, we can view generating a subset as a sequence of choices. Update the sum as we make choices, and record the subset when the sum equals <code>target</code>.</p>
+<p>Unlike permutations, <strong>each element may be chosen an unlimited number of times</strong>, so we do not need a Boolean <code>selected</code> list. A small modification to the permutations code gives an initial solution:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -4815,233 +4887,233 @@
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="n">choices</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]],</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="p">):</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="w">    </span><span class="sd">"""Backtracking algorithm: Subset sum I"""</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="c1"># When the subset sum equals target, record the solution</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="w">    </span><span class="sd">"""Backtracking: subset sum I"""</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="c1"># Record a solution when the subset sum equals target</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="k">if</span> <span class="n">total</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">state</span><span class="p">))</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="k">return</span>
-<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>    <span class="c1"># Traverse all choices</span>
+<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>    <span class="c1"># Iterate over all choices</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">choices</span><span class="p">)):</span>
-<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>        <span class="c1"># Pruning: if the subset sum exceeds target, skip this choice</span>
+<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>        <span class="c1"># Prune: skip this choice if the subset sum would exceed target</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>        <span class="k">if</span> <span class="n">total</span> <span class="o">+</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">target</span><span class="p">:</span>
 <a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>            <span class="k">continue</span>
-<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>        <span class="c1"># Attempt: make choice, update element sum total</span>
+<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>        <span class="c1"># Try: make a choice and update the sum total</span>
 <a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>        <span class="n">state</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>        <span class="c1"># Proceed to the next round of selection</span>
+<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>        <span class="c1"># Proceed to the next choice</span>
 <a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a>        <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">total</span> <span class="o">+</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">choices</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>        <span class="c1"># Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>        <span class="c1"># Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a>        <span class="n">state</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
 <a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a>
 <a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a><span class="k">def</span><span class="w"> </span><span class="nf">subset_sum_i_naive</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]]:</span>
-<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a><span class="w">    </span><span class="sd">"""Solve subset sum I (including duplicate subsets)"""</span>
+<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a><span class="w">    </span><span class="sd">"""Solve subset sum I (with duplicate subsets)"""</span>
 <a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a>    <span class="n">state</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c1"># State (subset)</span>
 <a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a>    <span class="n">total</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Subset sum</span>
-<a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a>    <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c1"># Result list (subset list)</span>
+<a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a>    <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c1"># Result list (list of subsets)</span>
 <a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a>    <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">total</span><span class="p">,</span> <span class="n">nums</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a>    <span class="k">return</span> <span class="n">res</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i_naive.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i_naive.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">total</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Record a solution when the subset sum equals target</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">total</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">state</span><span class="p">);</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">size_t</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="c1">// Pruning: if the subset sum exceeds target, skip this choice</span>
+<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="c1">// Prune: skip this choice if the subset sum would exceed target</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update element sum total</span>
+<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the sum total</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span>
 <a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="p">}</span>
 <a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a>
-<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="cm">/* Solve subset sum I (including duplicate subsets) */</span>
+<a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="cm">/* Solve subset sum I (with duplicate subsets) */</span>
 <a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">subsetSumINaive</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">;</span><span class="w">       </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">           </span><span class="c1">// Subset sum</span>
-<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">;</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">;</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i_naive.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i_naive.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">total</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Record a solution when the subset sum equals target</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">total</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="c1">// Pruning: if the subset sum exceeds target, skip this choice</span>
+<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="c1">// Prune: skip this choice if the subset sum would exceed target</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update element sum total</span>
+<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the sum total</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">);</span>
-<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="p">}</span>
 <a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a>
-<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="cm">/* Solve subset sum I (including duplicate subsets) */</span>
+<a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="cm">/* Solve subset sum I (with duplicate subsets) */</span>
 <a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="nf">subsetSumINaive</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Subset sum</span>
-<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i_naive.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i_naive.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Backtrack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">total</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Record a solution when the subset sum equals target</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">total</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="c1">// Pruning: if the subset sum exceeds target, skip this choice</span>
+<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="c1">// Prune: skip this choice if the subset sum would exceed target</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update element sum total</span>
+<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the sum total</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">        </span><span class="n">Backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">RemoveAt</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="p">}</span>
 <a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a>
-<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="cm">/* Solve subset sum I (including duplicate subsets) */</span>
+<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="cm">/* Solve subset sum I (with duplicate subsets) */</span>
 <a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">SubsetSumINaive</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Subset sum</span>
-<a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="w">    </span><span class="n">Backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i_naive.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i_naive.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">backtrackSubsetSumINaive</span><span class="p">(</span><span class="nx">total</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[][]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Record a solution when the subset sum equals target</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">total</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">        </span><span class="nx">newState</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">append</span><span class="p">([]</span><span class="kt">int</span><span class="p">{},</span><span class="w"> </span><span class="o">*</span><span class="nx">state</span><span class="p">...)</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="o">*</span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">newState</span><span class="p">)</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="c1">// Pruning: if the subset sum exceeds target, skip this choice</span>
+<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="c1">// Prune: skip this choice if the subset sum would exceed target</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">total</span><span class="o">+</span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">)[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">            </span><span class="k">continue</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update element sum total</span>
+<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the sum total</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">        </span><span class="o">*</span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">)[</span><span class="nx">i</span><span class="p">])</span>
-<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">        </span><span class="nx">backtrackSubsetSumINaive</span><span class="p">(</span><span class="nx">total</span><span class="o">+</span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">)[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span>
-<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">        </span><span class="o">*</span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[:</span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="p">}</span>
 <a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a>
-<a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="cm">/* Solve subset sum I (including duplicate subsets) */</span>
+<a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="cm">/* Solve subset sum I (with duplicate subsets) */</span>
 <a id="__codelineno-4-25" name="__codelineno-4-25" href="#__codelineno-4-25"></a><span class="kd">func</span><span class="w"> </span><span class="nx">subsetSumINaive</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">[][]</span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-26" name="__codelineno-4-26" href="#__codelineno-4-26"></a><span class="w">    </span><span class="nx">state</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-4-27" name="__codelineno-4-27" href="#__codelineno-4-27"></a><span class="w">    </span><span class="nx">total</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="w">              </span><span class="c1">// Subset sum</span>
-<a id="__codelineno-4-28" name="__codelineno-4-28" href="#__codelineno-4-28"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-4-28" name="__codelineno-4-28" href="#__codelineno-4-28"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-4-29" name="__codelineno-4-29" href="#__codelineno-4-29"></a><span class="w">    </span><span class="nx">backtrackSubsetSumINaive</span><span class="p">(</span><span class="nx">total</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">res</span><span class="p">)</span>
 <a id="__codelineno-4-30" name="__codelineno-4-30" href="#__codelineno-4-30"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span>
 <a id="__codelineno-4-31" name="__codelineno-4-31" href="#__codelineno-4-31"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i_naive.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i_naive.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Record a solution when the subset sum equals target</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="bp">indices</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="c1">// Pruning: if the subset sum exceeds target, skip this choice</span>
+<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="c1">// Prune: skip this choice if the subset sum would exceed target</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">            </span><span class="k">continue</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update element sum total</span>
+<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the sum total</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="p">:</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="p">}</span>
 <a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a>
-<a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="cm">/* Solve subset sum I (including duplicate subsets) */</span>
+<a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="cm">/* Solve subset sum I (with duplicate subsets) */</span>
 <a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="kd">func</span><span class="w"> </span><span class="nf">subsetSumINaive</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">state</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-5-26" name="__codelineno-5-26" href="#__codelineno-5-26"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">total</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1">// Subset sum</span>
-<a id="__codelineno-5-27" name="__codelineno-5-27" href="#__codelineno-5-27"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="p">:</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-5-27" name="__codelineno-5-27" href="#__codelineno-5-27"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="p">:</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="p">:</span><span class="w"> </span><span class="n">total</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-5-29" name="__codelineno-5-29" href="#__codelineno-5-29"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-5-30" name="__codelineno-5-30" href="#__codelineno-5-30"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i_naive.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i_naive.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">total</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Record a solution when the subset sum equals target</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">total</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">state</span><span class="p">]);</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">choices</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="c1">// Pruning: if the subset sum exceeds target, skip this choice</span>
+<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="c1">// Prune: skip this choice if the subset sum would exceed target</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update element sum total</span>
+<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the sum total</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">        </span><span class="nx">state</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">]);</span>
-<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">        </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">        </span><span class="nx">state</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="p">}</span>
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a>
-<a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="cm">/* Solve subset sum I (including duplicate subsets) */</span>
+<a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="cm">/* Solve subset sum I (with duplicate subsets) */</span>
 <a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a><span class="kd">function</span><span class="w"> </span><span class="nx">subsetSumINaive</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-6-26" name="__codelineno-6-26" href="#__codelineno-6-26"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">total</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Subset sum</span>
-<a id="__codelineno-6-27" name="__codelineno-6-27" href="#__codelineno-6-27"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-6-27" name="__codelineno-6-27" href="#__codelineno-6-27"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-6-28" name="__codelineno-6-28" href="#__codelineno-6-28"></a><span class="w">    </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">total</span><span class="p">,</span><span class="w"> </span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
 <a id="__codelineno-6-29" name="__codelineno-6-29" href="#__codelineno-6-29"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-6-30" name="__codelineno-6-30" href="#__codelineno-6-30"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i_naive.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i_naive.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="nx">state</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
@@ -5049,38 +5121,38 @@
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="nx">choices</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[][]</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// Record a solution when the subset sum equals target</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">total</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">state</span><span class="p">]);</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">choices</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">        </span><span class="c1">// Pruning: if the subset sum exceeds target, skip this choice</span>
+<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">        </span><span class="c1">// Prune: skip this choice if the subset sum would exceed target</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update element sum total</span>
+<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the sum total</span>
 <a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">        </span><span class="nx">state</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">]);</span>
-<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">        </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">        </span><span class="nx">state</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="p">}</span>
 <a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a>
-<a id="__codelineno-7-29" name="__codelineno-7-29" href="#__codelineno-7-29"></a><span class="cm">/* Solve subset sum I (including duplicate subsets) */</span>
+<a id="__codelineno-7-29" name="__codelineno-7-29" href="#__codelineno-7-29"></a><span class="cm">/* Solve subset sum I (with duplicate subsets) */</span>
 <a id="__codelineno-7-30" name="__codelineno-7-30" href="#__codelineno-7-30"></a><span class="kd">function</span><span class="w"> </span><span class="nx">subsetSumINaive</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[][]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-31" name="__codelineno-7-31" href="#__codelineno-7-31"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-7-32" name="__codelineno-7-32" href="#__codelineno-7-32"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">total</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Subset sum</span>
-<a id="__codelineno-7-33" name="__codelineno-7-33" href="#__codelineno-7-33"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-7-33" name="__codelineno-7-33" href="#__codelineno-7-33"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-7-34" name="__codelineno-7-34" href="#__codelineno-7-34"></a><span class="w">    </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">total</span><span class="p">,</span><span class="w"> </span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
 <a id="__codelineno-7-35" name="__codelineno-7-35" href="#__codelineno-7-35"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-7-36" name="__codelineno-7-36" href="#__codelineno-7-36"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i_naive.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i_naive.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span>
@@ -5088,38 +5160,38 @@
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">,</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="c1">// Record a solution when the subset sum equals target</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">total</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">List</span><span class="p">.</span><span class="n">from</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">  </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">  </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">    </span><span class="c1">// Pruning: if the subset sum exceeds target, skip this choice</span>
+<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">    </span><span class="c1">// Prune: skip this choice if the subset sum would exceed target</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">      </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">    </span><span class="c1">// Attempt: make choice, update element sum total</span>
+<a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">    </span><span class="c1">// Try: make a choice and update the sum total</span>
 <a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="w">    </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="w">    </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="w">    </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="w">    </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="n">removeLast</span><span class="p">();</span>
 <a id="__codelineno-8-26" name="__codelineno-8-26" href="#__codelineno-8-26"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a><span class="p">}</span>
 <a id="__codelineno-8-28" name="__codelineno-8-28" href="#__codelineno-8-28"></a>
-<a id="__codelineno-8-29" name="__codelineno-8-29" href="#__codelineno-8-29"></a><span class="cm">/* Solve subset sum I (including duplicate subsets) */</span>
+<a id="__codelineno-8-29" name="__codelineno-8-29" href="#__codelineno-8-29"></a><span class="cm">/* Solve subset sum I (with duplicate subsets) */</span>
 <a id="__codelineno-8-30" name="__codelineno-8-30" href="#__codelineno-8-30"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">subsetSumINaive</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-31" name="__codelineno-8-31" href="#__codelineno-8-31"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// State (subset)</span>
-<a id="__codelineno-8-32" name="__codelineno-8-32" href="#__codelineno-8-32"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Sum of elements</span>
-<a id="__codelineno-8-33" name="__codelineno-8-33" href="#__codelineno-8-33"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-8-32" name="__codelineno-8-32" href="#__codelineno-8-32"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Sum of the elements</span>
+<a id="__codelineno-8-33" name="__codelineno-8-33" href="#__codelineno-8-33"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-8-34" name="__codelineno-8-34" href="#__codelineno-8-34"></a><span class="w">  </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-8-35" name="__codelineno-8-35" href="#__codelineno-8-35"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-8-36" name="__codelineno-8-36" href="#__codelineno-8-36"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i_naive.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i_naive.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span>
@@ -5127,40 +5199,40 @@
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// Record a solution when the subset sum equals target</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">choices</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">        </span><span class="c1">// Pruning: if the subset sum exceeds target, skip this choice</span>
+<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">        </span><span class="c1">// Prune: skip this choice if the subset sum would exceed target</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update element sum total</span>
+<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the sum total</span>
 <a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span>
 <a id="__codelineno-9-26" name="__codelineno-9-26" href="#__codelineno-9-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="p">}</span>
 <a id="__codelineno-9-28" name="__codelineno-9-28" href="#__codelineno-9-28"></a>
-<a id="__codelineno-9-29" name="__codelineno-9-29" href="#__codelineno-9-29"></a><span class="cm">/* Solve subset sum I (including duplicate subsets) */</span>
+<a id="__codelineno-9-29" name="__codelineno-9-29" href="#__codelineno-9-29"></a><span class="cm">/* Solve subset sum I (with duplicate subsets) */</span>
 <a id="__codelineno-9-30" name="__codelineno-9-30" href="#__codelineno-9-30"></a><span class="k">fn</span><span class="w"> </span><span class="nf">subset_sum_i_naive</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-31" name="__codelineno-9-31" href="#__codelineno-9-31"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-9-32" name="__codelineno-9-32" href="#__codelineno-9-32"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Subset sum</span>
-<a id="__codelineno-9-33" name="__codelineno-9-33" href="#__codelineno-9-33"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-9-33" name="__codelineno-9-33" href="#__codelineno-9-33"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-9-34" name="__codelineno-9-34" href="#__codelineno-9-34"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-9-35" name="__codelineno-9-35" href="#__codelineno-9-35"></a><span class="w">    </span><span class="n">res</span>
 <a id="__codelineno-9-36" name="__codelineno-9-36" href="#__codelineno-9-36"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i_naive.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i_naive.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">total</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Record a solution when the subset sum equals target</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">total</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">stateSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">            </span><span class="n">res</span><span class="p">[</span><span class="n">resSize</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
@@ -5168,30 +5240,30 @@
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">        </span><span class="n">resColSizes</span><span class="p">[</span><span class="n">resSize</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stateSize</span><span class="p">;</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="c1">// Pruning: if the subset sum exceeds target, skip this choice</span>
+<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="c1">// Prune: skip this choice if the subset sum would exceed target</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update element sum total</span>
+<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the sum total</span>
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">        </span><span class="n">state</span><span class="p">[</span><span class="n">stateSize</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
-<a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">);</span>
-<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">        </span><span class="n">stateSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="p">}</span>
 <a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a>
-<a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="cm">/* Solve subset sum I (including duplicate subsets) */</span>
+<a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="cm">/* Solve subset sum I (with duplicate subsets) */</span>
 <a id="__codelineno-10-27" name="__codelineno-10-27" href="#__codelineno-10-27"></a><span class="kt">void</span><span class="w"> </span><span class="nf">subsetSumINaive</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">numsSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-28" name="__codelineno-10-28" href="#__codelineno-10-28"></a><span class="w">    </span><span class="n">resSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize solution count to 0</span>
+<a id="__codelineno-10-28" name="__codelineno-10-28" href="#__codelineno-10-28"></a><span class="w">    </span><span class="n">resSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the solution count to 0</span>
 <a id="__codelineno-10-29" name="__codelineno-10-29" href="#__codelineno-10-29"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">numsSize</span><span class="p">);</span>
 <a id="__codelineno-10-30" name="__codelineno-10-30" href="#__codelineno-10-30"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i_naive.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i_naive.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span>
@@ -5199,31 +5271,31 @@
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?&gt;</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// Record a solution when the subset sum equals target</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">total</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">toMutableList</span><span class="p">())</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">indices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">        </span><span class="c1">// Pruning: if the subset sum exceeds target, skip this choice</span>
+<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">        </span><span class="c1">// Prune: skip this choice if the subset sum would exceed target</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">            </span><span class="k">continue</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update element sum total</span>
+<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the sum total</span>
 <a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">)</span>
-<a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="na">removeAt</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="p">}</span>
 <a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a>
-<a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="cm">/* Solve subset sum I (including duplicate subsets) */</span>
+<a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="cm">/* Solve subset sum I (with duplicate subsets) */</span>
 <a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">subsetSumINaive</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">total</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Subset sum</span>
-<a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?&gt;</span><span class="p">()</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?&gt;</span><span class="p">()</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-11-34" name="__codelineno-11-34" href="#__codelineno-11-34"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-11-35" name="__codelineno-11-35" href="#__codelineno-11-35"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-11-36" name="__codelineno-11-36" href="#__codelineno-11-36"></a><span class="p">}</span>
@@ -5232,30 +5304,30 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">subset_sum_i_naive.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Backtracking: subset sum I ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># When the subset sum equals target, record the solution</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Record a solution when the subset sum equals target</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">state</span><span class="o">.</span><span class="n">dup</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">    </span><span class="k">return</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a>
-<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="c1"># Traverse all choices</span>
+<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="c1"># Iterate over all choices</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">choices</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="c1"># Pruning: if the subset sum exceeds target, skip this choice</span>
+<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="c1"># Prune: skip this choice if the subset sum would exceed target</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">    </span><span class="k">next</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span>
-<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="c1"># Attempt: make choice, update element sum total</span>
+<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="c1"># Try: make a choice and update the sum total</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="n">state</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
-<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">    </span><span class="c1"># Proceed to the next round of selection</span>
+<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">    </span><span class="c1"># Proceed to the next choice</span>
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">    </span><span class="c1"># Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">    </span><span class="c1"># Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">    </span><span class="n">state</span><span class="o">.</span><span class="n">pop</span>
 <a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="k">end</span>
 <a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a>
-<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="c1">### Solve subset sum I (with duplicate subsets) ###</span>
+<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="c1">### Solve subset sum I (with duplicate subsets)###</span>
 <a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="k">def</span><span class="w"> </span><span class="nf">subset_sum_i_naive</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
 <a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a><span class="w">  </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span><span class="w"> </span><span class="c1"># State (subset)</span>
 <a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="w">  </span><span class="n">total</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Subset sum</span>
-<a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span><span class="w"> </span><span class="c1"># Result list (subset list)</span>
+<a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span><span class="w"> </span><span class="c1"># Result list (list of subsets)</span>
 <a id="__codelineno-12-27" name="__codelineno-12-27" href="#__codelineno-12-27"></a><span class="w">  </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">total</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-12-28" name="__codelineno-12-28" href="#__codelineno-12-28"></a><span class="w">  </span><span class="n">res</span>
 <a id="__codelineno-12-29" name="__codelineno-12-29" href="#__codelineno-12-29"></a><span class="k">end</span>
@@ -5263,38 +5335,43 @@
 </div>
 </div>
 </div>
-<p>Running the above code on array <span class="arithmatex">\([3, 4, 5]\)</span> with target value <span class="arithmatex">\(9\)</span> produces <span class="arithmatex">\([3, 3, 3], [4, 5], [5, 4]\)</span>. <strong>Although we successfully found all subsets that sum to <span class="arithmatex">\(9\)</span>, there are duplicate subsets <span class="arithmatex">\([4, 5]\)</span> and <span class="arithmatex">\([5, 4]\)</span></strong>.</p>
-<p>This is because the search process distinguishes the order of selections, but subsets do not distinguish selection order. As shown in Figure 13-10, selecting 4 first and then 5 versus selecting 5 first and then 4 are different branches, but they correspond to the same subset.</p>
-<p><img alt="Subset search and boundary pruning" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_i_naive.png" /></p>
-<p align="center"> Figure 13-10 &nbsp; Subset search and boundary pruning </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20backtrack%28%0A%20%20%20%20state%3A%20list%5Bint%5D,%0A%20%20%20%20target%3A%20int,%0A%20%20%20%20total%3A%20int,%0A%20%20%20%20choices%3A%20list%5Bint%5D,%0A%20%20%20%20res%3A%20list%5Blist%5Bint%5D%5D,%0A%29%3A%0A%20%20%20%20%22%22%22Backtracking%3A%20subset%20sum%20I%22%22%22%0A%20%20%20%20%23%20Record%20a%20solution%20when%20the%20subset%20sum%20equals%20target%0A%20%20%20%20if%20total%20%3D%3D%20target%3A%0A%20%20%20%20%20%20%20%20res.append%28list%28state%29%29%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Iterate%20over%20all%20choices%0A%20%20%20%20for%20i%20in%20range%28len%28choices%29%29%3A%0A%20%20%20%20%20%20%20%20%23%20Prune%3A%20skip%20this%20choice%20if%20the%20subset%20sum%20would%20exceed%20target%0A%20%20%20%20%20%20%20%20if%20total%20%2B%20choices%5Bi%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20continue%0A%20%20%20%20%20%20%20%20%23%20Try%3A%20make%20a%20choice%20and%20update%20the%20sum%20total%0A%20%20%20%20%20%20%20%20state.append%28choices%5Bi%5D%29%0A%20%20%20%20%20%20%20%20%23%20Proceed%20to%20the%20next%20choice%0A%20%20%20%20%20%20%20%20backtrack%28state,%20target,%20total%20%2B%20choices%5Bi%5D,%20choices,%20res%29%0A%20%20%20%20%20%20%20%20%23%20Backtrack%3A%20undo%20the%20choice%20and%20restore%20the%20previous%20state%0A%20%20%20%20%20%20%20%20state.pop%28%29%0A%0A%0Adef%20subset_sum_i_naive%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20list%5Blist%5Bint%5D%5D%3A%0A%20%20%20%20%22%22%22Solve%20subset%20sum%20I%20%28with%20duplicate%20subsets%29%22%22%22%0A%20%20%20%20state%20%3D%20%5B%5D%20%20%23%20State%20%28subset%29%0A%20%20%20%20total%20%3D%200%20%20%23%20Subset%20sum%0A%20%20%20%20res%20%3D%20%5B%5D%20%20%23%20Result%20list%20%28list%20of%20subsets%29%0A%20%20%20%20backtrack%28state,%20target,%20total,%20nums,%20res%29%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B3,%204,%205%5D%0A%20%20%20%20target%20%3D%209%0A%20%20%20%20res%20%3D%20subset_sum_i_naive%28nums,%20target%29%0A%0A%20%20%20%20print%28f%22Input%20array%20nums%20%3D%20%7Bnums%7D,%20target%20%3D%20%7Btarget%7D%22%29%0A%20%20%20%20print%28f%22All%20subsets%20with%20sum%20%7Btarget%7D%3A%20res%20%3D%20%7Bres%7D%22%29%0A%20%20%20%20print%28f%22Note%20that%20this%20method%20produces%20duplicate%20subsets%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20backtrack%28%0A%20%20%20%20state%3A%20list%5Bint%5D,%0A%20%20%20%20target%3A%20int,%0A%20%20%20%20total%3A%20int,%0A%20%20%20%20choices%3A%20list%5Bint%5D,%0A%20%20%20%20res%3A%20list%5Blist%5Bint%5D%5D,%0A%29%3A%0A%20%20%20%20%22%22%22Backtracking%3A%20subset%20sum%20I%22%22%22%0A%20%20%20%20%23%20Record%20a%20solution%20when%20the%20subset%20sum%20equals%20target%0A%20%20%20%20if%20total%20%3D%3D%20target%3A%0A%20%20%20%20%20%20%20%20res.append%28list%28state%29%29%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Iterate%20over%20all%20choices%0A%20%20%20%20for%20i%20in%20range%28len%28choices%29%29%3A%0A%20%20%20%20%20%20%20%20%23%20Prune%3A%20skip%20this%20choice%20if%20the%20subset%20sum%20would%20exceed%20target%0A%20%20%20%20%20%20%20%20if%20total%20%2B%20choices%5Bi%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20continue%0A%20%20%20%20%20%20%20%20%23%20Try%3A%20make%20a%20choice%20and%20update%20the%20sum%20total%0A%20%20%20%20%20%20%20%20state.append%28choices%5Bi%5D%29%0A%20%20%20%20%20%20%20%20%23%20Proceed%20to%20the%20next%20choice%0A%20%20%20%20%20%20%20%20backtrack%28state,%20target,%20total%20%2B%20choices%5Bi%5D,%20choices,%20res%29%0A%20%20%20%20%20%20%20%20%23%20Backtrack%3A%20undo%20the%20choice%20and%20restore%20the%20previous%20state%0A%20%20%20%20%20%20%20%20state.pop%28%29%0A%0A%0Adef%20subset_sum_i_naive%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20list%5Blist%5Bint%5D%5D%3A%0A%20%20%20%20%22%22%22Solve%20subset%20sum%20I%20%28with%20duplicate%20subsets%29%22%22%22%0A%20%20%20%20state%20%3D%20%5B%5D%20%20%23%20State%20%28subset%29%0A%20%20%20%20total%20%3D%200%20%20%23%20Subset%20sum%0A%20%20%20%20res%20%3D%20%5B%5D%20%20%23%20Result%20list%20%28list%20of%20subsets%29%0A%20%20%20%20backtrack%28state,%20target,%20total,%20nums,%20res%29%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B3,%204,%205%5D%0A%20%20%20%20target%20%3D%209%0A%20%20%20%20res%20%3D%20subset_sum_i_naive%28nums,%20target%29%0A%0A%20%20%20%20print%28f%22Input%20array%20nums%20%3D%20%7Bnums%7D,%20target%20%3D%20%7Btarget%7D%22%29%0A%20%20%20%20print%28f%22All%20subsets%20with%20sum%20%7Btarget%7D%3A%20res%20%3D%20%7Bres%7D%22%29%0A%20%20%20%20print%28f%22Note%20that%20this%20method%20produces%20duplicate%20subsets%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>For input array <span class="arithmatex">\([3, 4, 5]\)</span> and target <span class="arithmatex">\(9\)</span>, this code returns <span class="arithmatex">\([3, 3, 3], [4, 5], [5, 4]\)</span>. <strong>It finds all subsets summing to <span class="arithmatex">\(9\)</span>, but includes the duplicate subsets <span class="arithmatex">\([4, 5]\)</span> and <span class="arithmatex">\([5, 4]\)</span></strong>.</p>
+<p>The search distinguishes the order of choices, whereas subsets do not. As shown in Figure 13-10, choosing <span class="arithmatex">\(4\)</span> then <span class="arithmatex">\(5\)</span> and choosing <span class="arithmatex">\(5\)</span> then <span class="arithmatex">\(4\)</span> follow different branches but produce the same subset.</p>
+<p><img alt="Subset search and pruning when the sum exceeds the target" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_i_naive.png" /></p>
+<p align="center"> Figure 13-10 &nbsp; Subset search and pruning when the sum exceeds the target </p>
 
-<p>To eliminate duplicate subsets, <strong>one straightforward idea is to deduplicate the result list</strong>. However, this approach is very inefficient for two reasons:</p>
+<p><strong>One straightforward way to remove duplicate subsets is to deduplicate the result list</strong>. However, this is inefficient for two reasons.</p>
 <ul>
-<li>When there are many array elements, especially when <code>target</code> is large, the search process generates many duplicate subsets.</li>
-<li>Comparing subsets (arrays) is very time-consuming, requiring sorting the arrays first, then comparing each element in them.</li>
+<li>With many array elements, especially when <code>target</code> is large, the search generates many duplicate subsets.</li>
+<li>Comparing subsets (arrays) is expensive: we must sort them first, then compare their elements one by one.</li>
 </ul>
-<h3 id="2-pruning-duplicate-subsets">2. &nbsp; Pruning Duplicate Subsets<a class="headerlink" href="#2-pruning-duplicate-subsets" title="Permanent link">&para;</a></h3>
-<p><strong>We consider deduplication through pruning during the search process</strong>. Observing Figure 13-11, duplicate subsets occur when array elements are selected in different orders, as in the following cases:</p>
+<h3 id="2-pruning-duplicate-subsets">2. &nbsp; Pruning duplicate subsets<a class="headerlink" href="#2-pruning-duplicate-subsets" title="Permanent link">&para;</a></h3>
+<p><strong>Instead, we can eliminate duplicates by pruning during the search</strong>. As shown in Figure 13-11, duplicates arise when array elements are chosen in different orders, as in the following example.</p>
 <ol>
-<li>When the first and second rounds select <span class="arithmatex">\(3\)</span> and <span class="arithmatex">\(4\)</span> respectively, all subsets containing these two elements are generated, denoted as <span class="arithmatex">\([3, 4, \dots]\)</span>.</li>
-<li>Afterward, when the first round selects <span class="arithmatex">\(4\)</span>, <strong>the second round should skip <span class="arithmatex">\(3\)</span></strong>, because the subset <span class="arithmatex">\([4, 3, \dots]\)</span> generated by this choice is an exact duplicate of the subset generated in step <code>1.</code></li>
+<li>Choosing <span class="arithmatex">\(3\)</span> and then <span class="arithmatex">\(4\)</span> in the first two rounds generates all subsets beginning with these choices, denoted by <span class="arithmatex">\([3, 4, \dots]\)</span>.</li>
+<li>Later, after choosing <span class="arithmatex">\(4\)</span> first, <strong>we should skip <span class="arithmatex">\(3\)</span> in the second round</strong>, because subsets <span class="arithmatex">\([4, 3, \dots]\)</span> duplicate those generated in step <code>1.</code>.</li>
 </ol>
-<p>In the search process, each level's choices are tried from left to right, so the rightmost branches are pruned more.</p>
+<p>At each level of the search, choices are tried from left to right, so branches farther to the right have more choices pruned.</p>
 <ol>
-<li>The first two rounds select <span class="arithmatex">\(3\)</span> and <span class="arithmatex">\(5\)</span>, generating subset <span class="arithmatex">\([3, 5, \dots]\)</span>.</li>
-<li>The first two rounds select <span class="arithmatex">\(4\)</span> and <span class="arithmatex">\(5\)</span>, generating subset <span class="arithmatex">\([4, 5, \dots]\)</span>.</li>
-<li>If the first round selects <span class="arithmatex">\(5\)</span>, <strong>the second round should skip <span class="arithmatex">\(3\)</span> and <span class="arithmatex">\(4\)</span></strong>, because subsets <span class="arithmatex">\([5, 3, \dots]\)</span> and <span class="arithmatex">\([5, 4, \dots]\)</span> are exact duplicates of the subsets described in steps <code>1.</code> and <code>2.</code></li>
+<li>Choosing <span class="arithmatex">\(3\)</span> and then <span class="arithmatex">\(5\)</span> generates subsets <span class="arithmatex">\([3, 5, \dots]\)</span>.</li>
+<li>Choosing <span class="arithmatex">\(4\)</span> and then <span class="arithmatex">\(5\)</span> generates subsets <span class="arithmatex">\([4, 5, \dots]\)</span>.</li>
+<li>If <span class="arithmatex">\(5\)</span> is chosen first, <strong>skip both <span class="arithmatex">\(3\)</span> and <span class="arithmatex">\(4\)</span> in the second round</strong>, because <span class="arithmatex">\([5, 3, \dots]\)</span> and <span class="arithmatex">\([5, 4, \dots]\)</span> duplicate the subsets described in steps <code>1.</code> and <code>2.</code>.</li>
 </ol>
-<p><img alt="Different selection orders leading to duplicate subsets" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_i_pruning.png" /></p>
-<p align="center"> Figure 13-11 &nbsp; Different selection orders leading to duplicate subsets </p>
+<p><img alt="Duplicate subsets caused by different choice orders" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_i_pruning.png" /></p>
+<p align="center"> Figure 13-11 &nbsp; Duplicate subsets caused by different choice orders </p>
 
-<p>In summary, given an input array <span class="arithmatex">\([x_1, x_2, \dots, x_n]\)</span>, let the selection sequence in the search process be <span class="arithmatex">\([x_{i_1}, x_{i_2}, \dots, x_{i_m}]\)</span>. This selection sequence must satisfy <span class="arithmatex">\(i_1 \leq i_2 \leq \dots \leq i_m\)</span>; <strong>any selection sequence that does not satisfy this condition will cause duplicates and should be pruned</strong>.</p>
-<h3 id="3-code-implementation">3. &nbsp; Code Implementation<a class="headerlink" href="#3-code-implementation" title="Permanent link">&para;</a></h3>
-<p>To implement this pruning, we initialize a variable <code>start</code> to indicate the starting point of traversal. <strong>After making choice <span class="arithmatex">\(x_{i}\)</span>, set the next round to start traversal from index <span class="arithmatex">\(i\)</span></strong>. This ensures that the selection sequence satisfies <span class="arithmatex">\(i_1 \leq i_2 \leq \dots \leq i_m\)</span>, guaranteeing subset uniqueness.</p>
-<p>In addition, we have made the following two optimizations to the code:</p>
+<p>In general, for input <span class="arithmatex">\([x_1, x_2, \dots, x_n]\)</span>, a sequence of choices <span class="arithmatex">\([x_{i_1}, x_{i_2}, \dots, x_{i_m}]\)</span> must satisfy <span class="arithmatex">\(i_1 \leq i_2 \leq \dots \leq i_m\)</span>. <strong>Any sequence violating this condition creates a duplicate and should be pruned</strong>.</p>
+<h3 id="3-code-implementation">3. &nbsp; Code implementation<a class="headerlink" href="#3-code-implementation" title="Permanent link">&para;</a></h3>
+<p>To implement this pruning, introduce <code>start</code> to specify where iteration begins. <strong>After choosing <span class="arithmatex">\(x_{i}\)</span>, begin the next round at index <span class="arithmatex">\(i\)</span></strong>. This enforces <span class="arithmatex">\(i_1 \leq i_2 \leq \dots \leq i_m\)</span> and ensures that each subset is unique.</p>
+<p>We also make two further improvements.</p>
 <ul>
-<li>Before starting the search, first sort the array <code>nums</code>. When traversing all choices, <strong>end the loop immediately when the subset sum exceeds <code>target</code></strong>, because subsequent elements are larger, and their subset sums must exceed <code>target</code>.</li>
-<li>Omit the element sum variable <code>total</code> and <strong>use subtraction on <code>target</code> to track the sum of elements</strong>. Record the solution when <code>target</code> equals <span class="arithmatex">\(0\)</span>.</li>
+<li>Sort <code>nums</code> before starting the search. While iterating over choices, <strong>stop the loop as soon as the subset sum exceeds <code>target</code></strong>, because all later elements are larger and would also exceed it.</li>
+<li>Remove the sum variable <code>total</code> and <strong>track the sum by subtracting from <code>target</code></strong>. Record a solution when <code>target</code> reaches <span class="arithmatex">\(0\)</span>.</li>
 </ul>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
@@ -5302,56 +5379,56 @@
 <div class="highlight"><span class="filename">subset_sum_i.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a>    <span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">choices</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">start</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]]</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="p">):</span>
-<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="w">    </span><span class="sd">"""Backtracking algorithm: Subset sum I"""</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># When the subset sum equals target, record the solution</span>
+<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="w">    </span><span class="sd">"""Backtracking: subset sum I"""</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Record a solution when the remaining target is zero</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="k">if</span> <span class="n">target</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">state</span><span class="p">))</span>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="k">return</span>
-<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>    <span class="c1"># Traverse all choices</span>
-<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>    <span class="c1"># Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>    <span class="c1"># Iterate over all choices</span>
+<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>    <span class="c1"># Pruning 2: iterate from start to avoid generating duplicate subsets</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">choices</span><span class="p">)):</span>
-<a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>        <span class="c1"># Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>        <span class="c1"># This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>        <span class="c1"># Pruning 1: exit the loop if this choice exceeds the remaining target</span>
+<a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>        <span class="c1"># The array is sorted, so later elements are at least as large and also exceed the remaining target</span>
 <a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>        <span class="k">if</span> <span class="n">target</span> <span class="o">-</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>            <span class="k">break</span>
-<a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>        <span class="c1"># Attempt: make choice, update target, start</span>
+<a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>        <span class="c1"># Try: make a choice and update target and start</span>
 <a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>        <span class="n">state</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-<a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a>        <span class="c1"># Proceed to the next round of selection</span>
+<a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a>        <span class="c1"># Proceed to the next choice</span>
 <a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a>        <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">target</span> <span class="o">-</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">choices</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a>        <span class="c1"># Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a>        <span class="c1"># Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-13-21" name="__codelineno-13-21" href="#__codelineno-13-21"></a>        <span class="n">state</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
 <a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a>
 <a id="__codelineno-13-23" name="__codelineno-13-23" href="#__codelineno-13-23"></a><span class="k">def</span><span class="w"> </span><span class="nf">subset_sum_i</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]]:</span>
 <a id="__codelineno-13-24" name="__codelineno-13-24" href="#__codelineno-13-24"></a><span class="w">    </span><span class="sd">"""Solve subset sum I"""</span>
 <a id="__codelineno-13-25" name="__codelineno-13-25" href="#__codelineno-13-25"></a>    <span class="n">state</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c1"># State (subset)</span>
 <a id="__codelineno-13-26" name="__codelineno-13-26" href="#__codelineno-13-26"></a>    <span class="n">nums</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>  <span class="c1"># Sort nums</span>
-<a id="__codelineno-13-27" name="__codelineno-13-27" href="#__codelineno-13-27"></a>    <span class="n">start</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Start point for traversal</span>
-<a id="__codelineno-13-28" name="__codelineno-13-28" href="#__codelineno-13-28"></a>    <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c1"># Result list (subset list)</span>
+<a id="__codelineno-13-27" name="__codelineno-13-27" href="#__codelineno-13-27"></a>    <span class="n">start</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Starting index for iteration</span>
+<a id="__codelineno-13-28" name="__codelineno-13-28" href="#__codelineno-13-28"></a>    <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c1"># Result list (list of subsets)</span>
 <a id="__codelineno-13-29" name="__codelineno-13-29" href="#__codelineno-13-29"></a>    <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">nums</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-13-30" name="__codelineno-13-30" href="#__codelineno-13-30"></a>    <span class="k">return</span> <span class="n">res</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// Record a solution when the remaining target reaches 0</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">state</span><span class="p">);</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">start</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if this choice exceeds the remaining target</span>
+<a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and also exceed the remaining target</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span>
 <a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a><span class="p">}</span>
@@ -5360,34 +5437,34 @@
 <a id="__codelineno-14-26" name="__codelineno-14-26" href="#__codelineno-14-26"></a><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">subsetSumI</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-27" name="__codelineno-14-27" href="#__codelineno-14-27"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">;</span><span class="w">              </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-14-28" name="__codelineno-14-28" href="#__codelineno-14-28"></a><span class="w">    </span><span class="n">sort</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">end</span><span class="p">());</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-14-29" name="__codelineno-14-29" href="#__codelineno-14-29"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">                  </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-14-30" name="__codelineno-14-30" href="#__codelineno-14-30"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">;</span><span class="w">        </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-14-29" name="__codelineno-14-29" href="#__codelineno-14-29"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">                  </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-14-30" name="__codelineno-14-30" href="#__codelineno-14-30"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">;</span><span class="w">        </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-14-31" name="__codelineno-14-31" href="#__codelineno-14-31"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-14-32" name="__codelineno-14-32" href="#__codelineno-14-32"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-14-33" name="__codelineno-14-33" href="#__codelineno-14-33"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Record a solution when the remaining target is zero, so the subset sums to the original target</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">start</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if the next element exceeds the remaining target</span>
+<a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and also exceed the remaining target</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">);</span>
-<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="p">}</span>
@@ -5396,34 +5473,34 @@
 <a id="__codelineno-15-26" name="__codelineno-15-26" href="#__codelineno-15-26"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="nf">subsetSumI</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-27" name="__codelineno-15-27" href="#__codelineno-15-27"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-15-28" name="__codelineno-15-28" href="#__codelineno-15-28"></a><span class="w">    </span><span class="n">Arrays</span><span class="p">.</span><span class="na">sort</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-15-29" name="__codelineno-15-29" href="#__codelineno-15-29"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-15-30" name="__codelineno-15-30" href="#__codelineno-15-30"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-15-29" name="__codelineno-15-29" href="#__codelineno-15-29"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-15-30" name="__codelineno-15-30" href="#__codelineno-15-30"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-15-31" name="__codelineno-15-31" href="#__codelineno-15-31"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-15-32" name="__codelineno-15-32" href="#__codelineno-15-32"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-15-33" name="__codelineno-15-33" href="#__codelineno-15-33"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Backtrack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// Record a solution when the subset reaches the original target (remaining target is 0)</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">start</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if this choice would exceed the remaining target</span>
+<a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and would also exceed the remaining target</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w">        </span><span class="n">Backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">RemoveAt</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="p">}</span>
@@ -5432,35 +5509,35 @@
 <a id="__codelineno-16-26" name="__codelineno-16-26" href="#__codelineno-16-26"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">SubsetSumI</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-27" name="__codelineno-16-27" href="#__codelineno-16-27"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-16-28" name="__codelineno-16-28" href="#__codelineno-16-28"></a><span class="w">    </span><span class="n">Array</span><span class="p">.</span><span class="n">Sort</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-16-29" name="__codelineno-16-29" href="#__codelineno-16-29"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-16-30" name="__codelineno-16-30" href="#__codelineno-16-30"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-16-29" name="__codelineno-16-29" href="#__codelineno-16-29"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-16-30" name="__codelineno-16-30" href="#__codelineno-16-30"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-16-31" name="__codelineno-16-31" href="#__codelineno-16-31"></a><span class="w">    </span><span class="n">Backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-16-32" name="__codelineno-16-32" href="#__codelineno-16-32"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-16-33" name="__codelineno-16-33" href="#__codelineno-16-33"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">backtrackSubsetSumI</span><span class="p">(</span><span class="nx">start</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[][]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Record a solution when the subset reaches the original target sum (remaining target is zero)</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">        </span><span class="nx">newState</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">append</span><span class="p">([]</span><span class="kt">int</span><span class="p">{},</span><span class="w"> </span><span class="o">*</span><span class="nx">state</span><span class="p">...)</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">        </span><span class="o">*</span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">newState</span><span class="p">)</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">start</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if this choice exceeds the remaining target</span>
+<a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and also exceed the remaining target</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">target</span><span class="o">-</span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">)[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="w">        </span><span class="o">*</span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">)[</span><span class="nx">i</span><span class="p">])</span>
-<a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a><span class="w">        </span><span class="nx">backtrackSubsetSumI</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="o">-</span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">)[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span>
-<a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-17-22" name="__codelineno-17-22" href="#__codelineno-17-22"></a><span class="w">        </span><span class="o">*</span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[:</span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-24" name="__codelineno-17-24" href="#__codelineno-17-24"></a><span class="p">}</span>
@@ -5469,34 +5546,34 @@
 <a id="__codelineno-17-27" name="__codelineno-17-27" href="#__codelineno-17-27"></a><span class="kd">func</span><span class="w"> </span><span class="nx">subsetSumI</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">[][]</span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-28" name="__codelineno-17-28" href="#__codelineno-17-28"></a><span class="w">    </span><span class="nx">state</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-17-29" name="__codelineno-17-29" href="#__codelineno-17-29"></a><span class="w">    </span><span class="nx">sort</span><span class="p">.</span><span class="nx">Ints</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w">         </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-17-30" name="__codelineno-17-30" href="#__codelineno-17-30"></a><span class="w">    </span><span class="nx">start</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="w">              </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-17-31" name="__codelineno-17-31" href="#__codelineno-17-31"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-17-30" name="__codelineno-17-30" href="#__codelineno-17-30"></a><span class="w">    </span><span class="nx">start</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="w">              </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-17-31" name="__codelineno-17-31" href="#__codelineno-17-31"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-17-32" name="__codelineno-17-32" href="#__codelineno-17-32"></a><span class="w">    </span><span class="nx">backtrackSubsetSumI</span><span class="p">(</span><span class="nx">start</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">res</span><span class="p">)</span>
 <a id="__codelineno-17-33" name="__codelineno-17-33" href="#__codelineno-17-33"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span>
 <a id="__codelineno-17-34" name="__codelineno-17-34" href="#__codelineno-17-34"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">start</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Record a solution when the remaining target reaches zero</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="bp">dropFirst</span><span class="p">(</span><span class="n">start</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if this choice exceeds the remaining target</span>
+<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and also exceed the remaining target</span>
 <a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-<a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span>
 <a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-23" name="__codelineno-18-23" href="#__codelineno-18-23"></a><span class="p">}</span>
@@ -5505,34 +5582,34 @@
 <a id="__codelineno-18-26" name="__codelineno-18-26" href="#__codelineno-18-26"></a><span class="kd">func</span><span class="w"> </span><span class="nf">subsetSumI</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-27" name="__codelineno-18-27" href="#__codelineno-18-27"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">state</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-18-28" name="__codelineno-18-28" href="#__codelineno-18-28"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">nums</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">sorted</span><span class="p">()</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-18-29" name="__codelineno-18-29" href="#__codelineno-18-29"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">start</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-18-30" name="__codelineno-18-30" href="#__codelineno-18-30"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="p">:</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-18-29" name="__codelineno-18-29" href="#__codelineno-18-29"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">start</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-18-30" name="__codelineno-18-30" href="#__codelineno-18-30"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="p">:</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-18-31" name="__codelineno-18-31" href="#__codelineno-18-31"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">:</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-18-32" name="__codelineno-18-32" href="#__codelineno-18-32"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-18-33" name="__codelineno-18-33" href="#__codelineno-18-33"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">start</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Record a solution when the remaining target is zero, so the subset sum equals the original target</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">target</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">state</span><span class="p">]);</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">start</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">choices</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if the next choice exceeds the remaining target</span>
+<a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and would also exceed the remaining target</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">        </span><span class="nx">state</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">]);</span>
-<a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">        </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="w">        </span><span class="nx">state</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-23" name="__codelineno-19-23" href="#__codelineno-19-23"></a><span class="p">}</span>
@@ -5541,15 +5618,15 @@
 <a id="__codelineno-19-26" name="__codelineno-19-26" href="#__codelineno-19-26"></a><span class="kd">function</span><span class="w"> </span><span class="nx">subsetSumI</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-27" name="__codelineno-19-27" href="#__codelineno-19-27"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-19-28" name="__codelineno-19-28" href="#__codelineno-19-28"></a><span class="w">    </span><span class="nx">nums</span><span class="p">.</span><span class="nx">sort</span><span class="p">((</span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">b</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">b</span><span class="p">);</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-19-29" name="__codelineno-19-29" href="#__codelineno-19-29"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-19-30" name="__codelineno-19-30" href="#__codelineno-19-30"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-19-29" name="__codelineno-19-29" href="#__codelineno-19-29"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-19-30" name="__codelineno-19-30" href="#__codelineno-19-30"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-19-31" name="__codelineno-19-31" href="#__codelineno-19-31"></a><span class="w">    </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">start</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
 <a id="__codelineno-19-32" name="__codelineno-19-32" href="#__codelineno-19-32"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-19-33" name="__codelineno-19-33" href="#__codelineno-19-33"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="nx">state</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
@@ -5557,24 +5634,24 @@
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="nx">start</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[][]</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="c1">// Record a solution when the subset reaches the original target (remaining target is 0)</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">target</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">state</span><span class="p">]);</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">start</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">choices</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if this choice would exceed the remaining target</span>
+<a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and would also exceed the remaining target</span>
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-20-23" name="__codelineno-20-23" href="#__codelineno-20-23"></a><span class="w">        </span><span class="nx">state</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">]);</span>
-<a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a><span class="w">        </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-20-27" name="__codelineno-20-27" href="#__codelineno-20-27"></a><span class="w">        </span><span class="nx">state</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-20-28" name="__codelineno-20-28" href="#__codelineno-20-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-29" name="__codelineno-20-29" href="#__codelineno-20-29"></a><span class="p">}</span>
@@ -5583,15 +5660,15 @@
 <a id="__codelineno-20-32" name="__codelineno-20-32" href="#__codelineno-20-32"></a><span class="kd">function</span><span class="w"> </span><span class="nx">subsetSumI</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[][]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-33" name="__codelineno-20-33" href="#__codelineno-20-33"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-20-34" name="__codelineno-20-34" href="#__codelineno-20-34"></a><span class="w">    </span><span class="nx">nums</span><span class="p">.</span><span class="nx">sort</span><span class="p">((</span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">b</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">b</span><span class="p">);</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-20-35" name="__codelineno-20-35" href="#__codelineno-20-35"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-20-36" name="__codelineno-20-36" href="#__codelineno-20-36"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-20-35" name="__codelineno-20-35" href="#__codelineno-20-35"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-20-36" name="__codelineno-20-36" href="#__codelineno-20-36"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-20-37" name="__codelineno-20-37" href="#__codelineno-20-37"></a><span class="w">    </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">start</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
 <a id="__codelineno-20-38" name="__codelineno-20-38" href="#__codelineno-20-38"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-20-39" name="__codelineno-20-39" href="#__codelineno-20-39"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span>
@@ -5599,24 +5676,24 @@
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="p">,</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">,</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="c1">// Record a solution when the subset sum reaches the original target (the remaining target is 0)</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">List</span><span class="p">.</span><span class="n">from</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">    </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">  </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">  </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">  </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">  </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
 <a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">start</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">    </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">    </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">    </span><span class="c1">// Pruning 1: exit the loop if this choice exceeds the remaining target</span>
+<a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">    </span><span class="c1">// The array is sorted, so later elements are at least as large and also exceed the remaining target</span>
 <a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="w">      </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">    </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">    </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">    </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">    </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-21-25" name="__codelineno-21-25" href="#__codelineno-21-25"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a><span class="w">    </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a><span class="w">    </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-21-27" name="__codelineno-21-27" href="#__codelineno-21-27"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="n">removeLast</span><span class="p">();</span>
 <a id="__codelineno-21-28" name="__codelineno-21-28" href="#__codelineno-21-28"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-29" name="__codelineno-21-29" href="#__codelineno-21-29"></a><span class="p">}</span>
@@ -5625,15 +5702,15 @@
 <a id="__codelineno-21-32" name="__codelineno-21-32" href="#__codelineno-21-32"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">subsetSumI</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-33" name="__codelineno-21-33" href="#__codelineno-21-33"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-21-34" name="__codelineno-21-34" href="#__codelineno-21-34"></a><span class="w">  </span><span class="n">nums</span><span class="p">.</span><span class="n">sort</span><span class="p">();</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-21-35" name="__codelineno-21-35" href="#__codelineno-21-35"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-21-36" name="__codelineno-21-36" href="#__codelineno-21-36"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-21-35" name="__codelineno-21-35" href="#__codelineno-21-35"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-21-36" name="__codelineno-21-36" href="#__codelineno-21-36"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-21-37" name="__codelineno-21-37" href="#__codelineno-21-37"></a><span class="w">  </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-21-38" name="__codelineno-21-38" href="#__codelineno-21-38"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-21-39" name="__codelineno-21-39" href="#__codelineno-21-39"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span>
@@ -5641,24 +5718,24 @@
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="n">start</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">    </span><span class="c1">// Record a solution when the remaining target is zero</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">start</span><span class="o">..</span><span class="n">choices</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if the next element exceeds the remaining target</span>
+<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and also exceed the remaining target</span>
 <a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-22-25" name="__codelineno-22-25" href="#__codelineno-22-25"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-22-26" name="__codelineno-22-26" href="#__codelineno-22-26"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-22-26" name="__codelineno-22-26" href="#__codelineno-22-26"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-22-27" name="__codelineno-22-27" href="#__codelineno-22-27"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span>
 <a id="__codelineno-22-28" name="__codelineno-22-28" href="#__codelineno-22-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-29" name="__codelineno-22-29" href="#__codelineno-22-29"></a><span class="p">}</span>
@@ -5667,17 +5744,17 @@
 <a id="__codelineno-22-32" name="__codelineno-22-32" href="#__codelineno-22-32"></a><span class="k">fn</span><span class="w"> </span><span class="nf">subset_sum_i</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-33" name="__codelineno-22-33" href="#__codelineno-22-33"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-22-34" name="__codelineno-22-34" href="#__codelineno-22-34"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="n">sort</span><span class="p">();</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-22-35" name="__codelineno-22-35" href="#__codelineno-22-35"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-22-36" name="__codelineno-22-36" href="#__codelineno-22-36"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-22-35" name="__codelineno-22-35" href="#__codelineno-22-35"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-22-36" name="__codelineno-22-36" href="#__codelineno-22-36"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-22-37" name="__codelineno-22-37" href="#__codelineno-22-37"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-22-38" name="__codelineno-22-38" href="#__codelineno-22-38"></a><span class="w">    </span><span class="n">res</span>
 <a id="__codelineno-22-39" name="__codelineno-22-39" href="#__codelineno-22-39"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Record a solution when the remaining target is 0</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">stateSize</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">            </span><span class="n">res</span><span class="p">[</span><span class="n">resSize</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
@@ -5685,20 +5762,20 @@
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">        </span><span class="n">resColSizes</span><span class="p">[</span><span class="n">resSize</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stateSize</span><span class="p">;</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">start</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if the choice exceeds the remaining target</span>
+<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and also exceed the remaining target</span>
 <a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="w">        </span><span class="n">state</span><span class="p">[</span><span class="n">stateSize</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="w">        </span><span class="n">stateSize</span><span class="o">++</span><span class="p">;</span>
-<a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
-<a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="w">        </span><span class="n">stateSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a><span class="p">}</span>
@@ -5706,13 +5783,13 @@
 <a id="__codelineno-23-29" name="__codelineno-23-29" href="#__codelineno-23-29"></a><span class="cm">/* Solve subset sum I */</span>
 <a id="__codelineno-23-30" name="__codelineno-23-30" href="#__codelineno-23-30"></a><span class="kt">void</span><span class="w"> </span><span class="nf">subsetSumI</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">numsSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-31" name="__codelineno-23-31" href="#__codelineno-23-31"></a><span class="w">    </span><span class="n">qsort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">numsSize</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">),</span><span class="w"> </span><span class="n">cmp</span><span class="p">);</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-23-32" name="__codelineno-23-32" href="#__codelineno-23-32"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">                           </span><span class="c1">// Start point for traversal</span>
+<a id="__codelineno-23-32" name="__codelineno-23-32" href="#__codelineno-23-32"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">                           </span><span class="c1">// Starting index for iteration</span>
 <a id="__codelineno-23-33" name="__codelineno-23-33" href="#__codelineno-23-33"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">numsSize</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">);</span>
 <a id="__codelineno-23-34" name="__codelineno-23-34" href="#__codelineno-23-34"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_i.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Backtracking algorithm: Subset sum I */</span>
+<div class="highlight"><span class="filename">subset_sum_i.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Backtracking: subset sum I */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span>
@@ -5720,24 +5797,24 @@
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="n">start</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?&gt;</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="c1">// Record a solution when the subset reaches the original target (remaining target is 0)</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">toMutableList</span><span class="p">())</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
 <a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">start</span><span class="p">..</span><span class="o">&lt;</span><span class="n">choices</span><span class="p">.</span><span class="na">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if this choice would exceed the remaining target</span>
+<a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and would also exceed the remaining target</span>
 <a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">)</span>
-<a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-24-25" name="__codelineno-24-25" href="#__codelineno-24-25"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-24-26" name="__codelineno-24-26" href="#__codelineno-24-26"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-24-26" name="__codelineno-24-26" href="#__codelineno-24-26"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-24-27" name="__codelineno-24-27" href="#__codelineno-24-27"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="na">removeAt</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-24-28" name="__codelineno-24-28" href="#__codelineno-24-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-29" name="__codelineno-24-29" href="#__codelineno-24-29"></a><span class="p">}</span>
@@ -5746,8 +5823,8 @@
 <a id="__codelineno-24-32" name="__codelineno-24-32" href="#__codelineno-24-32"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">subsetSumI</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-33" name="__codelineno-24-33" href="#__codelineno-24-33"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-24-34" name="__codelineno-24-34" href="#__codelineno-24-34"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="na">sort</span><span class="p">()</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-24-35" name="__codelineno-24-35" href="#__codelineno-24-35"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-24-36" name="__codelineno-24-36" href="#__codelineno-24-36"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?&gt;</span><span class="p">()</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-24-35" name="__codelineno-24-35" href="#__codelineno-24-35"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-24-36" name="__codelineno-24-36" href="#__codelineno-24-36"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?&gt;</span><span class="p">()</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-24-37" name="__codelineno-24-37" href="#__codelineno-24-37"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-24-38" name="__codelineno-24-38" href="#__codelineno-24-38"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-24-39" name="__codelineno-24-39" href="#__codelineno-24-39"></a><span class="p">}</span>
@@ -5756,22 +5833,22 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">subset_sum_i.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Backtracking: subset sum I ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># When the subset sum equals target, record the solution</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Record a solution when the remaining target reaches zero</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="n">target</span><span class="o">.</span><span class="n">zero?</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">state</span><span class="o">.</span><span class="n">dup</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">    </span><span class="k">return</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="k">end</span>
-<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">  </span><span class="c1"># Traverse all choices</span>
-<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="c1"># Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">  </span><span class="c1"># Iterate over all choices</span>
+<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="c1"># Pruning 2: iterate from start to avoid generating duplicate subsets</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">start</span><span class="o">...</span><span class="n">choices</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">    </span><span class="c1"># Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">    </span><span class="c1"># This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">    </span><span class="c1"># Pruning 1: exit the loop if this choice would exceed the remaining target</span>
+<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">    </span><span class="c1"># The array is sorted, so later elements are at least as large and would also exceed the remaining target</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">    </span><span class="k">break</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">    </span><span class="c1"># Attempt: make choice, update target, start</span>
+<a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">    </span><span class="c1"># Try: make a choice and update target and start</span>
 <a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">    </span><span class="n">state</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
-<a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">    </span><span class="c1"># Proceed to the next round of selection</span>
+<a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">    </span><span class="c1"># Proceed to the next choice</span>
 <a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w">    </span><span class="c1"># Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w">    </span><span class="c1"># Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="w">    </span><span class="n">state</span><span class="o">.</span><span class="n">pop</span>
 <a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="k">end</span>
@@ -5780,8 +5857,8 @@
 <a id="__codelineno-25-24" name="__codelineno-25-24" href="#__codelineno-25-24"></a><span class="k">def</span><span class="w"> </span><span class="nf">subset_sum_i</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
 <a id="__codelineno-25-25" name="__codelineno-25-25" href="#__codelineno-25-25"></a><span class="w">  </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span><span class="w"> </span><span class="c1"># State (subset)</span>
 <a id="__codelineno-25-26" name="__codelineno-25-26" href="#__codelineno-25-26"></a><span class="w">  </span><span class="n">nums</span><span class="o">.</span><span class="n">sort!</span><span class="w"> </span><span class="c1"># Sort nums</span>
-<a id="__codelineno-25-27" name="__codelineno-25-27" href="#__codelineno-25-27"></a><span class="w">  </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Start point for traversal</span>
-<a id="__codelineno-25-28" name="__codelineno-25-28" href="#__codelineno-25-28"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span><span class="w"> </span><span class="c1"># Result list (subset list)</span>
+<a id="__codelineno-25-27" name="__codelineno-25-27" href="#__codelineno-25-27"></a><span class="w">  </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Starting index for iteration</span>
+<a id="__codelineno-25-28" name="__codelineno-25-28" href="#__codelineno-25-28"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span><span class="w"> </span><span class="c1"># Result list (list of subsets)</span>
 <a id="__codelineno-25-29" name="__codelineno-25-29" href="#__codelineno-25-29"></a><span class="w">  </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-25-30" name="__codelineno-25-30" href="#__codelineno-25-30"></a><span class="w">  </span><span class="n">res</span>
 <a id="__codelineno-25-31" name="__codelineno-25-31" href="#__codelineno-25-31"></a><span class="k">end</span>
@@ -5789,89 +5866,94 @@
 </div>
 </div>
 </div>
-<p>Figure 13-12 shows the complete backtracking process produced by running the above code on array <span class="arithmatex">\([3, 4, 5]\)</span> with target value <span class="arithmatex">\(9\)</span>.</p>
-<p><img alt="Subset-sum I backtracking process" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_i.png" /></p>
-<p align="center"> Figure 13-12 &nbsp; Subset-sum I backtracking process </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20backtrack%28%0A%20%20%20%20state%3A%20list%5Bint%5D,%20target%3A%20int,%20choices%3A%20list%5Bint%5D,%20start%3A%20int,%20res%3A%20list%5Blist%5Bint%5D%5D%0A%29%3A%0A%20%20%20%20%22%22%22Backtracking%3A%20subset%20sum%20I%22%22%22%0A%20%20%20%20%23%20Record%20a%20solution%20when%20the%20remaining%20target%20is%20zero%0A%20%20%20%20if%20target%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20res.append%28list%28state%29%29%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Iterate%20over%20all%20choices%0A%20%20%20%20%23%20Pruning%202%3A%20iterate%20from%20start%20to%20avoid%20generating%20duplicate%20subsets%0A%20%20%20%20for%20i%20in%20range%28start,%20len%28choices%29%29%3A%0A%20%20%20%20%20%20%20%20%23%20Pruning%201%3A%20exit%20the%20loop%20if%20the%20chosen%20value%20exceeds%20the%20remaining%20target%0A%20%20%20%20%20%20%20%20%23%20The%20array%20is%20sorted,%20so%20later%20elements%20are%20at%20least%20as%20large%20and%20also%20exceed%20the%20remaining%20target%0A%20%20%20%20%20%20%20%20if%20target%20-%20choices%5Bi%5D%20%3C%200%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20%23%20Try%3A%20make%20a%20choice%20and%20update%20target%20and%20start%0A%20%20%20%20%20%20%20%20state.append%28choices%5Bi%5D%29%0A%20%20%20%20%20%20%20%20%23%20Proceed%20to%20the%20next%20choice%0A%20%20%20%20%20%20%20%20backtrack%28state,%20target%20-%20choices%5Bi%5D,%20choices,%20i,%20res%29%0A%20%20%20%20%20%20%20%20%23%20Backtrack%3A%20undo%20the%20choice%20and%20restore%20the%20previous%20state%0A%20%20%20%20%20%20%20%20state.pop%28%29%0A%0A%0Adef%20subset_sum_i%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20list%5Blist%5Bint%5D%5D%3A%0A%20%20%20%20%22%22%22Solve%20subset%20sum%20I%22%22%22%0A%20%20%20%20state%20%3D%20%5B%5D%20%20%23%20State%20%28subset%29%0A%20%20%20%20nums.sort%28%29%20%20%23%20Sort%20nums%0A%20%20%20%20start%20%3D%200%20%20%23%20Starting%20index%20for%20iteration%0A%20%20%20%20res%20%3D%20%5B%5D%20%20%23%20Result%20list%20%28list%20of%20subsets%29%0A%20%20%20%20backtrack%28state,%20target,%20nums,%20start,%20res%29%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B3,%204,%205%5D%0A%20%20%20%20target%20%3D%209%0A%20%20%20%20res%20%3D%20subset_sum_i%28nums,%20target%29%0A%0A%20%20%20%20print%28f%22Input%20array%20nums%20%3D%20%7Bnums%7D,%20target%20%3D%20%7Btarget%7D%22%29%0A%20%20%20%20print%28f%22All%20subsets%20with%20sum%20%7Btarget%7D%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20backtrack%28%0A%20%20%20%20state%3A%20list%5Bint%5D,%20target%3A%20int,%20choices%3A%20list%5Bint%5D,%20start%3A%20int,%20res%3A%20list%5Blist%5Bint%5D%5D%0A%29%3A%0A%20%20%20%20%22%22%22Backtracking%3A%20subset%20sum%20I%22%22%22%0A%20%20%20%20%23%20Record%20a%20solution%20when%20the%20remaining%20target%20is%20zero%0A%20%20%20%20if%20target%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20res.append%28list%28state%29%29%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Iterate%20over%20all%20choices%0A%20%20%20%20%23%20Pruning%202%3A%20iterate%20from%20start%20to%20avoid%20generating%20duplicate%20subsets%0A%20%20%20%20for%20i%20in%20range%28start,%20len%28choices%29%29%3A%0A%20%20%20%20%20%20%20%20%23%20Pruning%201%3A%20exit%20the%20loop%20if%20the%20chosen%20value%20exceeds%20the%20remaining%20target%0A%20%20%20%20%20%20%20%20%23%20The%20array%20is%20sorted,%20so%20later%20elements%20are%20at%20least%20as%20large%20and%20also%20exceed%20the%20remaining%20target%0A%20%20%20%20%20%20%20%20if%20target%20-%20choices%5Bi%5D%20%3C%200%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20%23%20Try%3A%20make%20a%20choice%20and%20update%20target%20and%20start%0A%20%20%20%20%20%20%20%20state.append%28choices%5Bi%5D%29%0A%20%20%20%20%20%20%20%20%23%20Proceed%20to%20the%20next%20choice%0A%20%20%20%20%20%20%20%20backtrack%28state,%20target%20-%20choices%5Bi%5D,%20choices,%20i,%20res%29%0A%20%20%20%20%20%20%20%20%23%20Backtrack%3A%20undo%20the%20choice%20and%20restore%20the%20previous%20state%0A%20%20%20%20%20%20%20%20state.pop%28%29%0A%0A%0Adef%20subset_sum_i%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20list%5Blist%5Bint%5D%5D%3A%0A%20%20%20%20%22%22%22Solve%20subset%20sum%20I%22%22%22%0A%20%20%20%20state%20%3D%20%5B%5D%20%20%23%20State%20%28subset%29%0A%20%20%20%20nums.sort%28%29%20%20%23%20Sort%20nums%0A%20%20%20%20start%20%3D%200%20%20%23%20Starting%20index%20for%20iteration%0A%20%20%20%20res%20%3D%20%5B%5D%20%20%23%20Result%20list%20%28list%20of%20subsets%29%0A%20%20%20%20backtrack%28state,%20target,%20nums,%20start,%20res%29%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B3,%204,%205%5D%0A%20%20%20%20target%20%3D%209%0A%20%20%20%20res%20%3D%20subset_sum_i%28nums,%20target%29%0A%0A%20%20%20%20print%28f%22Input%20array%20nums%20%3D%20%7Bnums%7D,%20target%20%3D%20%7Btarget%7D%22%29%0A%20%20%20%20print%28f%22All%20subsets%20with%20sum%20%7Btarget%7D%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 13-12 shows the complete backtracking process for input <span class="arithmatex">\([3, 4, 5]\)</span> and target <span class="arithmatex">\(9\)</span>.</p>
+<p><img alt="Backtracking for subset sum I" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_i.png" /></p>
+<p align="center"> Figure 13-12 &nbsp; Backtracking for subset sum I </p>
 
-<h2 id="1332-with-duplicate-elements-in-array">13.3.2 &nbsp; With Duplicate Elements in Array<a class="headerlink" href="#1332-with-duplicate-elements-in-array" title="Permanent link">&para;</a></h2>
+<h2 id="1332-duplicate-elements">13.3.2 &nbsp; Duplicate elements<a class="headerlink" href="#1332-duplicate-elements" title="Permanent link">&para;</a></h2>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given a positive integer array <code>nums</code> and a target positive integer <code>target</code>, find all possible combinations where the sum of elements in the combination equals <code>target</code>. <strong>The given array may contain duplicate elements, and each element can be selected at most once</strong>. Return these combinations in list form, where the list should not contain duplicate combinations.</p>
+<p>Given an array of positive integers <code>nums</code> and a positive target integer <code>target</code>, find all combinations whose elements sum to <code>target</code>. <strong>The array may contain duplicate elements, and each element may be chosen only once</strong>. Return the combinations as a list containing no duplicates.</p>
 </div>
-<p>Compared to the previous problem, <strong>the input array in this problem may contain duplicate elements</strong>, which introduces a new issue. For example, given array <span class="arithmatex">\([4, \hat{4}, 5]\)</span> and target value <span class="arithmatex">\(9\)</span>, the output of the existing code is <span class="arithmatex">\([4, 5], [\hat{4}, 5]\)</span>, which contains duplicate subsets.</p>
-<p><strong>The reason for this duplication is that equal elements are selected multiple times in a certain round</strong>. In Figure 13-13, the first round has three choices, two of which are <span class="arithmatex">\(4\)</span>, creating two duplicate search branches that output duplicate subsets. Similarly, the two <span class="arithmatex">\(4\)</span>'s in the second round also produce duplicate subsets.</p>
+<p>Unlike the previous problem, <strong>the input may contain duplicate elements</strong>, which introduces a new issue. For array <span class="arithmatex">\([4, \hat{4}, 5]\)</span> and target <span class="arithmatex">\(9\)</span>, for example, the existing code produces <span class="arithmatex">\([4, 5], [\hat{4}, 5]\)</span>: duplicate subsets.</p>
+<p><strong>These duplicates occur because equal elements are tried more than once in a single round</strong>. In Figure 13-13, the first round has three choices, two of which are <span class="arithmatex">\(4\)</span>. They generate duplicate branches and therefore duplicate subsets. The two occurrences of <span class="arithmatex">\(4\)</span> in the second round cause the same problem.</p>
 <p><img alt="Duplicate subsets caused by equal elements" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_ii_repeat.png" /></p>
 <p align="center"> Figure 13-13 &nbsp; Duplicate subsets caused by equal elements </p>
 
-<h3 id="1-pruning-equal-elements">1. &nbsp; Pruning Equal Elements<a class="headerlink" href="#1-pruning-equal-elements" title="Permanent link">&para;</a></h3>
-<p>To solve this problem, <strong>we need to limit equal elements to be selected only once in each round</strong>. The implementation is quite clever: since the array is already sorted, equal elements are adjacent. This means that in a given round of selection, if the current element equals the element to its left, then the same value has already been chosen in this round, so we skip the current element directly.</p>
-<p>At the same time, <strong>this problem specifies that each array element can only be selected once</strong>. Fortunately, we can also use the variable <code>start</code> to satisfy this constraint: after making choice <span class="arithmatex">\(x_{i}\)</span>, set the next round to start traversal from index <span class="arithmatex">\(i + 1\)</span> onwards. This both eliminates duplicate subsets and avoids selecting elements multiple times.</p>
-<h3 id="2-code-implementation">2. &nbsp; Code Implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">&para;</a></h3>
+<h3 id="1-pruning-equal-elements">1. &nbsp; Pruning equal elements<a class="headerlink" href="#1-pruning-equal-elements" title="Permanent link">&para;</a></h3>
+<p>To solve this, <strong>try each value only once in each round</strong>. The implementation uses the fact that equal elements are adjacent in a sorted array. Within a round, if an element equals the preceding element, that value has already been tried, so skip the current element.</p>
+<p>In addition, <strong>each array element may be chosen only once</strong>. Fortunately, <code>start</code> can enforce this constraint too: after choosing <span class="arithmatex">\(x_{i}\)</span>, begin the next round at index <span class="arithmatex">\(i + 1\)</span>. This both eliminates duplicate subsets and prevents reusing an element.</p>
+<h3 id="2-code-implementation">2. &nbsp; Code implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">&para;</a></h3>
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 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">subset_sum_ii.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a>    <span class="n">state</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">choices</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">start</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]]</span>
 <a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a><span class="p">):</span>
-<a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a><span class="w">    </span><span class="sd">"""Backtracking algorithm: Subset sum II"""</span>
-<a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>    <span class="c1"># When the subset sum equals target, record the solution</span>
+<a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a><span class="w">    </span><span class="sd">"""Backtracking: subset sum II"""</span>
+<a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>    <span class="c1"># Record a solution when the remaining target is zero</span>
 <a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="k">if</span> <span class="n">target</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">state</span><span class="p">))</span>
 <a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>        <span class="k">return</span>
-<a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>    <span class="c1"># Traverse all choices</span>
-<a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>    <span class="c1"># Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
-<a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a>    <span class="c1"># Pruning 3: start traversing from start to avoid repeatedly selecting the same element</span>
+<a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>    <span class="c1"># Iterate over all choices</span>
+<a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>    <span class="c1"># Pruning 2: iterate from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a>    <span class="c1"># Pruning 3: start after the previously chosen index to avoid choosing the same element again</span>
 <a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">choices</span><span class="p">)):</span>
-<a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>        <span class="c1"># Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a>        <span class="c1"># This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>        <span class="c1"># Pruning 1: exit the loop if this choice exceeds the remaining target</span>
+<a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a>        <span class="c1"># The array is sorted, so later elements are at least as large and also exceed the remaining target</span>
 <a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a>        <span class="k">if</span> <span class="n">target</span> <span class="o">-</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-26-16" name="__codelineno-26-16" href="#__codelineno-26-16"></a>            <span class="k">break</span>
-<a id="__codelineno-26-17" name="__codelineno-26-17" href="#__codelineno-26-17"></a>        <span class="c1"># Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly</span>
+<a id="__codelineno-26-17" name="__codelineno-26-17" href="#__codelineno-26-17"></a>        <span class="c1"># Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it</span>
 <a id="__codelineno-26-18" name="__codelineno-26-18" href="#__codelineno-26-18"></a>        <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="n">start</span> <span class="ow">and</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]:</span>
 <a id="__codelineno-26-19" name="__codelineno-26-19" href="#__codelineno-26-19"></a>            <span class="k">continue</span>
-<a id="__codelineno-26-20" name="__codelineno-26-20" href="#__codelineno-26-20"></a>        <span class="c1"># Attempt: make choice, update target, start</span>
+<a id="__codelineno-26-20" name="__codelineno-26-20" href="#__codelineno-26-20"></a>        <span class="c1"># Try: make a choice and update target and start</span>
 <a id="__codelineno-26-21" name="__codelineno-26-21" href="#__codelineno-26-21"></a>        <span class="n">state</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-<a id="__codelineno-26-22" name="__codelineno-26-22" href="#__codelineno-26-22"></a>        <span class="c1"># Proceed to the next round of selection</span>
+<a id="__codelineno-26-22" name="__codelineno-26-22" href="#__codelineno-26-22"></a>        <span class="c1"># Proceed to the next choice</span>
 <a id="__codelineno-26-23" name="__codelineno-26-23" href="#__codelineno-26-23"></a>        <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">target</span> <span class="o">-</span> <span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">choices</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-26-24" name="__codelineno-26-24" href="#__codelineno-26-24"></a>        <span class="c1"># Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-26-24" name="__codelineno-26-24" href="#__codelineno-26-24"></a>        <span class="c1"># Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-26-25" name="__codelineno-26-25" href="#__codelineno-26-25"></a>        <span class="n">state</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
 <a id="__codelineno-26-26" name="__codelineno-26-26" href="#__codelineno-26-26"></a>
 <a id="__codelineno-26-27" name="__codelineno-26-27" href="#__codelineno-26-27"></a><span class="k">def</span><span class="w"> </span><span class="nf">subset_sum_ii</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]]:</span>
 <a id="__codelineno-26-28" name="__codelineno-26-28" href="#__codelineno-26-28"></a><span class="w">    </span><span class="sd">"""Solve subset sum II"""</span>
 <a id="__codelineno-26-29" name="__codelineno-26-29" href="#__codelineno-26-29"></a>    <span class="n">state</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c1"># State (subset)</span>
 <a id="__codelineno-26-30" name="__codelineno-26-30" href="#__codelineno-26-30"></a>    <span class="n">nums</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>  <span class="c1"># Sort nums</span>
-<a id="__codelineno-26-31" name="__codelineno-26-31" href="#__codelineno-26-31"></a>    <span class="n">start</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Start point for traversal</span>
-<a id="__codelineno-26-32" name="__codelineno-26-32" href="#__codelineno-26-32"></a>    <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c1"># Result list (subset list)</span>
+<a id="__codelineno-26-31" name="__codelineno-26-31" href="#__codelineno-26-31"></a>    <span class="n">start</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Starting index for iteration</span>
+<a id="__codelineno-26-32" name="__codelineno-26-32" href="#__codelineno-26-32"></a>    <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>  <span class="c1"># Result list (list of subsets)</span>
 <a id="__codelineno-26-33" name="__codelineno-26-33" href="#__codelineno-26-33"></a>    <span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">nums</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-26-34" name="__codelineno-26-34" href="#__codelineno-26-34"></a>    <span class="k">return</span> <span class="n">res</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_ii.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Backtracking algorithm: Subset sum II */</span>
+<div class="highlight"><span class="filename">subset_sum_ii.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Backtracking: subset sum II */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="c1">// Record a solution when the remaining target reaches 0</span>
 <a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">state</span><span class="p">);</span>
 <a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
-<a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a><span class="w">    </span><span class="c1">// Pruning 3: start traversing from start to avoid repeatedly selecting the same element</span>
+<a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a><span class="w">    </span><span class="c1">// Pruning 3: iterate from start to avoid choosing the same element again</span>
 <a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">start</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if this choice exceeds the remaining target</span>
+<a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and also exceed the remaining target</span>
 <a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-27-16" name="__codelineno-27-16" href="#__codelineno-27-16"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-27-17" name="__codelineno-27-17" href="#__codelineno-27-17"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly</span>
+<a id="__codelineno-27-17" name="__codelineno-27-17" href="#__codelineno-27-17"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it</span>
 <a id="__codelineno-27-18" name="__codelineno-27-18" href="#__codelineno-27-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-19" name="__codelineno-27-19" href="#__codelineno-27-19"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-27-20" name="__codelineno-27-20" href="#__codelineno-27-20"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-27-21" name="__codelineno-27-21" href="#__codelineno-27-21"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-27-21" name="__codelineno-27-21" href="#__codelineno-27-21"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-27-22" name="__codelineno-27-22" href="#__codelineno-27-22"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-27-23" name="__codelineno-27-23" href="#__codelineno-27-23"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-27-23" name="__codelineno-27-23" href="#__codelineno-27-23"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-27-24" name="__codelineno-27-24" href="#__codelineno-27-24"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-27-25" name="__codelineno-27-25" href="#__codelineno-27-25"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-27-25" name="__codelineno-27-25" href="#__codelineno-27-25"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-27-26" name="__codelineno-27-26" href="#__codelineno-27-26"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span>
 <a id="__codelineno-27-27" name="__codelineno-27-27" href="#__codelineno-27-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-28" name="__codelineno-27-28" href="#__codelineno-27-28"></a><span class="p">}</span>
@@ -5880,39 +5962,39 @@
 <a id="__codelineno-27-31" name="__codelineno-27-31" href="#__codelineno-27-31"></a><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">subsetSumII</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-32" name="__codelineno-27-32" href="#__codelineno-27-32"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">;</span><span class="w">              </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-27-33" name="__codelineno-27-33" href="#__codelineno-27-33"></a><span class="w">    </span><span class="n">sort</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">end</span><span class="p">());</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-27-34" name="__codelineno-27-34" href="#__codelineno-27-34"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">                  </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-27-35" name="__codelineno-27-35" href="#__codelineno-27-35"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">;</span><span class="w">        </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-27-34" name="__codelineno-27-34" href="#__codelineno-27-34"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">                  </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-27-35" name="__codelineno-27-35" href="#__codelineno-27-35"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">;</span><span class="w">        </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-27-36" name="__codelineno-27-36" href="#__codelineno-27-36"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-27-37" name="__codelineno-27-37" href="#__codelineno-27-37"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-27-38" name="__codelineno-27-38" href="#__codelineno-27-38"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_ii.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Backtracking algorithm: Subset sum II */</span>
+<div class="highlight"><span class="filename">subset_sum_ii.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Backtracking: subset sum II */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="c1">// Record a solution when the remaining target is zero, so the subset sums to the original target</span>
 <a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
-<a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="w">    </span><span class="c1">// Pruning 3: start traversing from start to avoid repeatedly selecting the same element</span>
+<a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="w">    </span><span class="c1">// Pruning 3: iterate from start to avoid choosing the same element again</span>
 <a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">start</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if the next element exceeds the remaining target</span>
+<a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and also exceed the remaining target</span>
 <a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly</span>
+<a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it</span>
 <a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-19" name="__codelineno-28-19" href="#__codelineno-28-19"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-28-20" name="__codelineno-28-20" href="#__codelineno-28-20"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-28-21" name="__codelineno-28-21" href="#__codelineno-28-21"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-28-21" name="__codelineno-28-21" href="#__codelineno-28-21"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-28-22" name="__codelineno-28-22" href="#__codelineno-28-22"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">);</span>
-<a id="__codelineno-28-23" name="__codelineno-28-23" href="#__codelineno-28-23"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-28-23" name="__codelineno-28-23" href="#__codelineno-28-23"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-28-24" name="__codelineno-28-24" href="#__codelineno-28-24"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-28-25" name="__codelineno-28-25" href="#__codelineno-28-25"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-28-25" name="__codelineno-28-25" href="#__codelineno-28-25"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-28-26" name="__codelineno-28-26" href="#__codelineno-28-26"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-28-27" name="__codelineno-28-27" href="#__codelineno-28-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-28" name="__codelineno-28-28" href="#__codelineno-28-28"></a><span class="p">}</span>
@@ -5921,39 +6003,39 @@
 <a id="__codelineno-28-31" name="__codelineno-28-31" href="#__codelineno-28-31"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="nf">subsetSumII</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-32" name="__codelineno-28-32" href="#__codelineno-28-32"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-28-33" name="__codelineno-28-33" href="#__codelineno-28-33"></a><span class="w">    </span><span class="n">Arrays</span><span class="p">.</span><span class="na">sort</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-28-34" name="__codelineno-28-34" href="#__codelineno-28-34"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-28-35" name="__codelineno-28-35" href="#__codelineno-28-35"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-28-34" name="__codelineno-28-34" href="#__codelineno-28-34"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-28-35" name="__codelineno-28-35" href="#__codelineno-28-35"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-28-36" name="__codelineno-28-36" href="#__codelineno-28-36"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-28-37" name="__codelineno-28-37" href="#__codelineno-28-37"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-28-38" name="__codelineno-28-38" href="#__codelineno-28-38"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_ii.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Backtracking algorithm: Subset sum II */</span>
+<div class="highlight"><span class="filename">subset_sum_ii.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Backtracking: subset sum II */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Backtrack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="c1">// Record a solution when the subset reaches the original target (remaining target is 0)</span>
 <a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
-<a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w">    </span><span class="c1">// Pruning 3: start traversing from start to avoid repeatedly selecting the same element</span>
+<a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w">    </span><span class="c1">// Pruning 3: start after the previously chosen index to avoid choosing the same element again</span>
 <a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">start</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if this choice would exceed the remaining target</span>
+<a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and would also exceed the remaining target</span>
 <a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly</span>
+<a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it</span>
 <a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-19" name="__codelineno-29-19" href="#__codelineno-29-19"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-29-20" name="__codelineno-29-20" href="#__codelineno-29-20"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-29-22" name="__codelineno-29-22" href="#__codelineno-29-22"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-29-23" name="__codelineno-29-23" href="#__codelineno-29-23"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-29-23" name="__codelineno-29-23" href="#__codelineno-29-23"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-29-24" name="__codelineno-29-24" href="#__codelineno-29-24"></a><span class="w">        </span><span class="n">Backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-29-25" name="__codelineno-29-25" href="#__codelineno-29-25"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-29-25" name="__codelineno-29-25" href="#__codelineno-29-25"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-29-26" name="__codelineno-29-26" href="#__codelineno-29-26"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">RemoveAt</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-29-27" name="__codelineno-29-27" href="#__codelineno-29-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-28" name="__codelineno-29-28" href="#__codelineno-29-28"></a><span class="p">}</span>
@@ -5962,40 +6044,40 @@
 <a id="__codelineno-29-31" name="__codelineno-29-31" href="#__codelineno-29-31"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">SubsetSumII</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-32" name="__codelineno-29-32" href="#__codelineno-29-32"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-29-33" name="__codelineno-29-33" href="#__codelineno-29-33"></a><span class="w">    </span><span class="n">Array</span><span class="p">.</span><span class="n">Sort</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-29-34" name="__codelineno-29-34" href="#__codelineno-29-34"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-29-35" name="__codelineno-29-35" href="#__codelineno-29-35"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-29-34" name="__codelineno-29-34" href="#__codelineno-29-34"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-29-35" name="__codelineno-29-35" href="#__codelineno-29-35"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-29-36" name="__codelineno-29-36" href="#__codelineno-29-36"></a><span class="w">    </span><span class="n">Backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-29-37" name="__codelineno-29-37" href="#__codelineno-29-37"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-29-38" name="__codelineno-29-38" href="#__codelineno-29-38"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_ii.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Backtracking algorithm: Subset sum II */</span>
+<div class="highlight"><span class="filename">subset_sum_ii.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Backtracking: subset sum II */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">backtrackSubsetSumII</span><span class="p">(</span><span class="nx">start</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[][]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="c1">// Record a solution when the subset reaches the original target sum (remaining target is zero)</span>
 <a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">        </span><span class="nx">newState</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">append</span><span class="p">([]</span><span class="kt">int</span><span class="p">{},</span><span class="w"> </span><span class="o">*</span><span class="nx">state</span><span class="p">...)</span>
 <a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">        </span><span class="o">*</span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">newState</span><span class="p">)</span>
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
-<a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">    </span><span class="c1">// Pruning 3: start traversing from start to avoid repeatedly selecting the same element</span>
+<a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">    </span><span class="c1">// Pruning 3: start after the previously chosen index to avoid choosing the same element again</span>
 <a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">start</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if this choice exceeds the remaining target</span>
+<a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and also exceed the remaining target</span>
 <a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">target</span><span class="o">-</span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">)[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly</span>
+<a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it</span>
 <a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">start</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">)[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">)[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-20" name="__codelineno-30-20" href="#__codelineno-30-20"></a><span class="w">            </span><span class="k">continue</span>
 <a id="__codelineno-30-21" name="__codelineno-30-21" href="#__codelineno-30-21"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-30-22" name="__codelineno-30-22" href="#__codelineno-30-22"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-30-22" name="__codelineno-30-22" href="#__codelineno-30-22"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-30-23" name="__codelineno-30-23" href="#__codelineno-30-23"></a><span class="w">        </span><span class="o">*</span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">)[</span><span class="nx">i</span><span class="p">])</span>
-<a id="__codelineno-30-24" name="__codelineno-30-24" href="#__codelineno-30-24"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-30-24" name="__codelineno-30-24" href="#__codelineno-30-24"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-30-25" name="__codelineno-30-25" href="#__codelineno-30-25"></a><span class="w">        </span><span class="nx">backtrackSubsetSumII</span><span class="p">(</span><span class="nx">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="o">-</span><span class="p">(</span><span class="o">*</span><span class="nx">choices</span><span class="p">)[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span>
-<a id="__codelineno-30-26" name="__codelineno-30-26" href="#__codelineno-30-26"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-30-26" name="__codelineno-30-26" href="#__codelineno-30-26"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-30-27" name="__codelineno-30-27" href="#__codelineno-30-27"></a><span class="w">        </span><span class="o">*</span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)[:</span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">state</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-30-28" name="__codelineno-30-28" href="#__codelineno-30-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-29" name="__codelineno-30-29" href="#__codelineno-30-29"></a><span class="p">}</span>
@@ -6004,39 +6086,39 @@
 <a id="__codelineno-30-32" name="__codelineno-30-32" href="#__codelineno-30-32"></a><span class="kd">func</span><span class="w"> </span><span class="nx">subsetSumII</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">[][]</span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-33" name="__codelineno-30-33" href="#__codelineno-30-33"></a><span class="w">    </span><span class="nx">state</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-30-34" name="__codelineno-30-34" href="#__codelineno-30-34"></a><span class="w">    </span><span class="nx">sort</span><span class="p">.</span><span class="nx">Ints</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w">         </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-30-35" name="__codelineno-30-35" href="#__codelineno-30-35"></a><span class="w">    </span><span class="nx">start</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="w">              </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-30-36" name="__codelineno-30-36" href="#__codelineno-30-36"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-30-35" name="__codelineno-30-35" href="#__codelineno-30-35"></a><span class="w">    </span><span class="nx">start</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="w">              </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-30-36" name="__codelineno-30-36" href="#__codelineno-30-36"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-30-37" name="__codelineno-30-37" href="#__codelineno-30-37"></a><span class="w">    </span><span class="nx">backtrackSubsetSumII</span><span class="p">(</span><span class="nx">start</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">res</span><span class="p">)</span>
 <a id="__codelineno-30-38" name="__codelineno-30-38" href="#__codelineno-30-38"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span>
 <a id="__codelineno-30-39" name="__codelineno-30-39" href="#__codelineno-30-39"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_ii.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Backtracking algorithm: Subset sum II */</span>
+<div class="highlight"><span class="filename">subset_sum_ii.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Backtracking: subset sum II */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">start</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="c1">// Record a solution when the remaining target reaches zero</span>
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
-<a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">    </span><span class="c1">// Pruning 3: start traversing from start to avoid repeatedly selecting the same element</span>
+<a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">    </span><span class="c1">// Pruning 3: start after the previously chosen index to avoid choosing the same element again</span>
 <a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="bp">dropFirst</span><span class="p">(</span><span class="n">start</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if this choice exceeds the remaining target</span>
+<a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and also exceed the remaining target</span>
 <a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-31-17" name="__codelineno-31-17" href="#__codelineno-31-17"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly</span>
+<a id="__codelineno-31-17" name="__codelineno-31-17" href="#__codelineno-31-17"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it</span>
 <a id="__codelineno-31-18" name="__codelineno-31-18" href="#__codelineno-31-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-19" name="__codelineno-31-19" href="#__codelineno-31-19"></a><span class="w">            </span><span class="k">continue</span>
 <a id="__codelineno-31-20" name="__codelineno-31-20" href="#__codelineno-31-20"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-31-21" name="__codelineno-31-21" href="#__codelineno-31-21"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-31-21" name="__codelineno-31-21" href="#__codelineno-31-21"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-31-22" name="__codelineno-31-22" href="#__codelineno-31-22"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-<a id="__codelineno-31-23" name="__codelineno-31-23" href="#__codelineno-31-23"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-31-23" name="__codelineno-31-23" href="#__codelineno-31-23"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-31-24" name="__codelineno-31-24" href="#__codelineno-31-24"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-31-25" name="__codelineno-31-25" href="#__codelineno-31-25"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-31-25" name="__codelineno-31-25" href="#__codelineno-31-25"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-31-26" name="__codelineno-31-26" href="#__codelineno-31-26"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span>
 <a id="__codelineno-31-27" name="__codelineno-31-27" href="#__codelineno-31-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-28" name="__codelineno-31-28" href="#__codelineno-31-28"></a><span class="p">}</span>
@@ -6045,39 +6127,39 @@
 <a id="__codelineno-31-31" name="__codelineno-31-31" href="#__codelineno-31-31"></a><span class="kd">func</span><span class="w"> </span><span class="nf">subsetSumII</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-32" name="__codelineno-31-32" href="#__codelineno-31-32"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">state</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-31-33" name="__codelineno-31-33" href="#__codelineno-31-33"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">nums</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">sorted</span><span class="p">()</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-31-34" name="__codelineno-31-34" href="#__codelineno-31-34"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">start</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-31-35" name="__codelineno-31-35" href="#__codelineno-31-35"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="p">:</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-31-34" name="__codelineno-31-34" href="#__codelineno-31-34"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">start</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-31-35" name="__codelineno-31-35" href="#__codelineno-31-35"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="p">:</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-31-36" name="__codelineno-31-36" href="#__codelineno-31-36"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">:</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-31-37" name="__codelineno-31-37" href="#__codelineno-31-37"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-31-38" name="__codelineno-31-38" href="#__codelineno-31-38"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_ii.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Backtracking algorithm: Subset sum II */</span>
+<div class="highlight"><span class="filename">subset_sum_ii.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Backtracking: subset sum II */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">start</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="c1">// Record a solution when the remaining target is zero, so the subset sum equals the original target</span>
 <a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">target</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">state</span><span class="p">]);</span>
 <a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
-<a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a><span class="w">    </span><span class="c1">// Pruning 3: start traversing from start to avoid repeatedly selecting the same element</span>
+<a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a><span class="w">    </span><span class="c1">// Pruning 3: start after the previously chosen index to avoid choosing the same element again</span>
 <a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">start</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">choices</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if the next choice exceeds the remaining target</span>
+<a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and would also exceed the remaining target</span>
 <a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-15" name="__codelineno-32-15" href="#__codelineno-32-15"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-32-16" name="__codelineno-32-16" href="#__codelineno-32-16"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-32-17" name="__codelineno-32-17" href="#__codelineno-32-17"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly</span>
+<a id="__codelineno-32-17" name="__codelineno-32-17" href="#__codelineno-32-17"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it</span>
 <a id="__codelineno-32-18" name="__codelineno-32-18" href="#__codelineno-32-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">start</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-19" name="__codelineno-32-19" href="#__codelineno-32-19"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-32-20" name="__codelineno-32-20" href="#__codelineno-32-20"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-32-21" name="__codelineno-32-21" href="#__codelineno-32-21"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-32-21" name="__codelineno-32-21" href="#__codelineno-32-21"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-32-22" name="__codelineno-32-22" href="#__codelineno-32-22"></a><span class="w">        </span><span class="nx">state</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">]);</span>
-<a id="__codelineno-32-23" name="__codelineno-32-23" href="#__codelineno-32-23"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-32-23" name="__codelineno-32-23" href="#__codelineno-32-23"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-32-24" name="__codelineno-32-24" href="#__codelineno-32-24"></a><span class="w">        </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-32-25" name="__codelineno-32-25" href="#__codelineno-32-25"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-32-25" name="__codelineno-32-25" href="#__codelineno-32-25"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-32-26" name="__codelineno-32-26" href="#__codelineno-32-26"></a><span class="w">        </span><span class="nx">state</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-32-27" name="__codelineno-32-27" href="#__codelineno-32-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-28" name="__codelineno-32-28" href="#__codelineno-32-28"></a><span class="p">}</span>
@@ -6086,15 +6168,15 @@
 <a id="__codelineno-32-31" name="__codelineno-32-31" href="#__codelineno-32-31"></a><span class="kd">function</span><span class="w"> </span><span class="nx">subsetSumII</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-32" name="__codelineno-32-32" href="#__codelineno-32-32"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-32-33" name="__codelineno-32-33" href="#__codelineno-32-33"></a><span class="w">    </span><span class="nx">nums</span><span class="p">.</span><span class="nx">sort</span><span class="p">((</span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">b</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">b</span><span class="p">);</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-32-34" name="__codelineno-32-34" href="#__codelineno-32-34"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-32-35" name="__codelineno-32-35" href="#__codelineno-32-35"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-32-34" name="__codelineno-32-34" href="#__codelineno-32-34"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-32-35" name="__codelineno-32-35" href="#__codelineno-32-35"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-32-36" name="__codelineno-32-36" href="#__codelineno-32-36"></a><span class="w">    </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">start</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
 <a id="__codelineno-32-37" name="__codelineno-32-37" href="#__codelineno-32-37"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-32-38" name="__codelineno-32-38" href="#__codelineno-32-38"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_ii.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Backtracking algorithm: Subset sum II */</span>
+<div class="highlight"><span class="filename">subset_sum_ii.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Backtracking: subset sum II */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span>
 <a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="nx">state</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
@@ -6102,29 +6184,29 @@
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="w">    </span><span class="nx">start</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">    </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[][]</span>
 <a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">    </span><span class="c1">// Record a solution when the subset reaches the original target (remaining target is 0)</span>
 <a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">target</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">([...</span><span class="nx">state</span><span class="p">]);</span>
 <a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
-<a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a><span class="w">    </span><span class="c1">// Pruning 3: start traversing from start to avoid repeatedly selecting the same element</span>
+<a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a><span class="w">    </span><span class="c1">// Pruning 3: start after the previously chosen index to avoid choosing the same element again</span>
 <a id="__codelineno-33-17" name="__codelineno-33-17" href="#__codelineno-33-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">start</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">choices</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-18" name="__codelineno-33-18" href="#__codelineno-33-18"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-33-19" name="__codelineno-33-19" href="#__codelineno-33-19"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-33-18" name="__codelineno-33-18" href="#__codelineno-33-18"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if this choice would exceed the remaining target</span>
+<a id="__codelineno-33-19" name="__codelineno-33-19" href="#__codelineno-33-19"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and would also exceed the remaining target</span>
 <a id="__codelineno-33-20" name="__codelineno-33-20" href="#__codelineno-33-20"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-21" name="__codelineno-33-21" href="#__codelineno-33-21"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-33-22" name="__codelineno-33-22" href="#__codelineno-33-22"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-33-23" name="__codelineno-33-23" href="#__codelineno-33-23"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly</span>
+<a id="__codelineno-33-23" name="__codelineno-33-23" href="#__codelineno-33-23"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it</span>
 <a id="__codelineno-33-24" name="__codelineno-33-24" href="#__codelineno-33-24"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">start</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-25" name="__codelineno-33-25" href="#__codelineno-33-25"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-33-26" name="__codelineno-33-26" href="#__codelineno-33-26"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-33-27" name="__codelineno-33-27" href="#__codelineno-33-27"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-33-27" name="__codelineno-33-27" href="#__codelineno-33-27"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-33-28" name="__codelineno-33-28" href="#__codelineno-33-28"></a><span class="w">        </span><span class="nx">state</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">]);</span>
-<a id="__codelineno-33-29" name="__codelineno-33-29" href="#__codelineno-33-29"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-33-29" name="__codelineno-33-29" href="#__codelineno-33-29"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-33-30" name="__codelineno-33-30" href="#__codelineno-33-30"></a><span class="w">        </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">choices</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
-<a id="__codelineno-33-31" name="__codelineno-33-31" href="#__codelineno-33-31"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-33-31" name="__codelineno-33-31" href="#__codelineno-33-31"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-33-32" name="__codelineno-33-32" href="#__codelineno-33-32"></a><span class="w">        </span><span class="nx">state</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-33-33" name="__codelineno-33-33" href="#__codelineno-33-33"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-34" name="__codelineno-33-34" href="#__codelineno-33-34"></a><span class="p">}</span>
@@ -6133,15 +6215,15 @@
 <a id="__codelineno-33-37" name="__codelineno-33-37" href="#__codelineno-33-37"></a><span class="kd">function</span><span class="w"> </span><span class="nx">subsetSumII</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[][]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-38" name="__codelineno-33-38" href="#__codelineno-33-38"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-33-39" name="__codelineno-33-39" href="#__codelineno-33-39"></a><span class="w">    </span><span class="nx">nums</span><span class="p">.</span><span class="nx">sort</span><span class="p">((</span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">b</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">b</span><span class="p">);</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-33-40" name="__codelineno-33-40" href="#__codelineno-33-40"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-33-41" name="__codelineno-33-41" href="#__codelineno-33-41"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-33-40" name="__codelineno-33-40" href="#__codelineno-33-40"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-33-41" name="__codelineno-33-41" href="#__codelineno-33-41"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-33-42" name="__codelineno-33-42" href="#__codelineno-33-42"></a><span class="w">    </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">start</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
 <a id="__codelineno-33-43" name="__codelineno-33-43" href="#__codelineno-33-43"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-33-44" name="__codelineno-33-44" href="#__codelineno-33-44"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_ii.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Backtracking algorithm: Subset sum II */</span>
+<div class="highlight"><span class="filename">subset_sum_ii.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Backtracking: subset sum II */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span>
 <a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="p">,</span>
 <a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span>
@@ -6149,29 +6231,29 @@
 <a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="p">,</span>
 <a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">,</span>
 <a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w">  </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w">  </span><span class="c1">// Record a solution when the subset sum reaches the original target (the remaining target is 0)</span>
 <a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">List</span><span class="p">.</span><span class="n">from</span><span class="p">(</span><span class="n">state</span><span class="p">));</span>
 <a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="w">    </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">  </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">  </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
-<a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w">  </span><span class="c1">// Pruning 3: start traversing from start to avoid repeatedly selecting the same element</span>
+<a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">  </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">  </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w">  </span><span class="c1">// Pruning 3: start after the previously chosen index to avoid choosing the same element again</span>
 <a id="__codelineno-34-17" name="__codelineno-34-17" href="#__codelineno-34-17"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">start</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choices</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-18" name="__codelineno-34-18" href="#__codelineno-34-18"></a><span class="w">    </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-34-19" name="__codelineno-34-19" href="#__codelineno-34-19"></a><span class="w">    </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-34-18" name="__codelineno-34-18" href="#__codelineno-34-18"></a><span class="w">    </span><span class="c1">// Pruning 1: exit the loop if this choice exceeds the remaining target</span>
+<a id="__codelineno-34-19" name="__codelineno-34-19" href="#__codelineno-34-19"></a><span class="w">    </span><span class="c1">// The array is sorted, so later elements are at least as large and also exceed the remaining target</span>
 <a id="__codelineno-34-20" name="__codelineno-34-20" href="#__codelineno-34-20"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-21" name="__codelineno-34-21" href="#__codelineno-34-21"></a><span class="w">      </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-34-22" name="__codelineno-34-22" href="#__codelineno-34-22"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-34-23" name="__codelineno-34-23" href="#__codelineno-34-23"></a><span class="w">    </span><span class="c1">// Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly</span>
+<a id="__codelineno-34-23" name="__codelineno-34-23" href="#__codelineno-34-23"></a><span class="w">    </span><span class="c1">// Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it</span>
 <a id="__codelineno-34-24" name="__codelineno-34-24" href="#__codelineno-34-24"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-25" name="__codelineno-34-25" href="#__codelineno-34-25"></a><span class="w">      </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-34-26" name="__codelineno-34-26" href="#__codelineno-34-26"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-34-27" name="__codelineno-34-27" href="#__codelineno-34-27"></a><span class="w">    </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-34-27" name="__codelineno-34-27" href="#__codelineno-34-27"></a><span class="w">    </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-34-28" name="__codelineno-34-28" href="#__codelineno-34-28"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-34-29" name="__codelineno-34-29" href="#__codelineno-34-29"></a><span class="w">    </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-34-29" name="__codelineno-34-29" href="#__codelineno-34-29"></a><span class="w">    </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-34-30" name="__codelineno-34-30" href="#__codelineno-34-30"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-34-31" name="__codelineno-34-31" href="#__codelineno-34-31"></a><span class="w">    </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-34-31" name="__codelineno-34-31" href="#__codelineno-34-31"></a><span class="w">    </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-34-32" name="__codelineno-34-32" href="#__codelineno-34-32"></a><span class="w">    </span><span class="n">state</span><span class="p">.</span><span class="n">removeLast</span><span class="p">();</span>
 <a id="__codelineno-34-33" name="__codelineno-34-33" href="#__codelineno-34-33"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-34" name="__codelineno-34-34" href="#__codelineno-34-34"></a><span class="p">}</span>
@@ -6180,15 +6262,15 @@
 <a id="__codelineno-34-37" name="__codelineno-34-37" href="#__codelineno-34-37"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">subsetSumII</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-38" name="__codelineno-34-38" href="#__codelineno-34-38"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-34-39" name="__codelineno-34-39" href="#__codelineno-34-39"></a><span class="w">  </span><span class="n">nums</span><span class="p">.</span><span class="n">sort</span><span class="p">();</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-34-40" name="__codelineno-34-40" href="#__codelineno-34-40"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-34-41" name="__codelineno-34-41" href="#__codelineno-34-41"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-34-40" name="__codelineno-34-40" href="#__codelineno-34-40"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-34-41" name="__codelineno-34-41" href="#__codelineno-34-41"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-34-42" name="__codelineno-34-42" href="#__codelineno-34-42"></a><span class="w">  </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-34-43" name="__codelineno-34-43" href="#__codelineno-34-43"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-34-44" name="__codelineno-34-44" href="#__codelineno-34-44"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_ii.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Backtracking algorithm: Subset sum II */</span>
+<div class="highlight"><span class="filename">subset_sum_ii.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Backtracking: subset sum II */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span>
@@ -6196,29 +6278,29 @@
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="n">start</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span>
 <a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">    </span><span class="c1">// Record a solution when the remaining target is zero</span>
 <a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
-<a id="__codelineno-35-16" name="__codelineno-35-16" href="#__codelineno-35-16"></a><span class="w">    </span><span class="c1">// Pruning 3: start traversing from start to avoid repeatedly selecting the same element</span>
+<a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-35-16" name="__codelineno-35-16" href="#__codelineno-35-16"></a><span class="w">    </span><span class="c1">// Pruning 3: iterate from start to avoid choosing the same element again</span>
 <a id="__codelineno-35-17" name="__codelineno-35-17" href="#__codelineno-35-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">start</span><span class="o">..</span><span class="n">choices</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-18" name="__codelineno-35-18" href="#__codelineno-35-18"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-35-19" name="__codelineno-35-19" href="#__codelineno-35-19"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-35-18" name="__codelineno-35-18" href="#__codelineno-35-18"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if the next element exceeds the remaining target</span>
+<a id="__codelineno-35-19" name="__codelineno-35-19" href="#__codelineno-35-19"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and also exceed the remaining target</span>
 <a id="__codelineno-35-20" name="__codelineno-35-20" href="#__codelineno-35-20"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-21" name="__codelineno-35-21" href="#__codelineno-35-21"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-35-22" name="__codelineno-35-22" href="#__codelineno-35-22"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-35-23" name="__codelineno-35-23" href="#__codelineno-35-23"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly</span>
+<a id="__codelineno-35-23" name="__codelineno-35-23" href="#__codelineno-35-23"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it</span>
 <a id="__codelineno-35-24" name="__codelineno-35-24" href="#__codelineno-35-24"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-25" name="__codelineno-35-25" href="#__codelineno-35-25"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-35-26" name="__codelineno-35-26" href="#__codelineno-35-26"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-35-27" name="__codelineno-35-27" href="#__codelineno-35-27"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-35-27" name="__codelineno-35-27" href="#__codelineno-35-27"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-35-28" name="__codelineno-35-28" href="#__codelineno-35-28"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-35-29" name="__codelineno-35-29" href="#__codelineno-35-29"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-35-29" name="__codelineno-35-29" href="#__codelineno-35-29"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-35-30" name="__codelineno-35-30" href="#__codelineno-35-30"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
-<a id="__codelineno-35-31" name="__codelineno-35-31" href="#__codelineno-35-31"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-35-31" name="__codelineno-35-31" href="#__codelineno-35-31"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-35-32" name="__codelineno-35-32" href="#__codelineno-35-32"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span>
 <a id="__codelineno-35-33" name="__codelineno-35-33" href="#__codelineno-35-33"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-34" name="__codelineno-35-34" href="#__codelineno-35-34"></a><span class="p">}</span>
@@ -6227,17 +6309,17 @@
 <a id="__codelineno-35-37" name="__codelineno-35-37" href="#__codelineno-35-37"></a><span class="k">fn</span><span class="w"> </span><span class="nf">subset_sum_ii</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-38" name="__codelineno-35-38" href="#__codelineno-35-38"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-35-39" name="__codelineno-35-39" href="#__codelineno-35-39"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="n">sort</span><span class="p">();</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-35-40" name="__codelineno-35-40" href="#__codelineno-35-40"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-35-41" name="__codelineno-35-41" href="#__codelineno-35-41"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-35-40" name="__codelineno-35-40" href="#__codelineno-35-40"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-35-41" name="__codelineno-35-41" href="#__codelineno-35-41"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-35-42" name="__codelineno-35-42" href="#__codelineno-35-42"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-35-43" name="__codelineno-35-43" href="#__codelineno-35-43"></a><span class="w">    </span><span class="n">res</span>
 <a id="__codelineno-35-44" name="__codelineno-35-44" href="#__codelineno-35-44"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_ii.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Backtracking algorithm: Subset sum II */</span>
+<div class="highlight"><span class="filename">subset_sum_ii.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Backtracking: subset sum II */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">start</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="c1">// Record a solution when the remaining target is 0</span>
 <a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">stateSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">            </span><span class="n">res</span><span class="p">[</span><span class="n">resSize</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
@@ -6245,24 +6327,24 @@
 <a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">        </span><span class="n">resColSizes</span><span class="p">[</span><span class="n">resSize</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stateSize</span><span class="p">;</span>
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
-<a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">    </span><span class="c1">// Pruning 3: start traversing from start to avoid repeatedly selecting the same element</span>
+<a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">    </span><span class="c1">// Pruning 3: iterate from start to avoid choosing the same element again</span>
 <a id="__codelineno-36-14" name="__codelineno-36-14" href="#__codelineno-36-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">start</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a><span class="w">        </span><span class="c1">// Pruning 1: Skip if subset sum exceeds target</span>
+<a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a><span class="w">        </span><span class="c1">// Pruning 1: skip this choice if it exceeds the remaining target</span>
 <a id="__codelineno-36-16" name="__codelineno-36-16" href="#__codelineno-36-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-17" name="__codelineno-36-17" href="#__codelineno-36-17"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-36-18" name="__codelineno-36-18" href="#__codelineno-36-18"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-36-19" name="__codelineno-36-19" href="#__codelineno-36-19"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly</span>
+<a id="__codelineno-36-19" name="__codelineno-36-19" href="#__codelineno-36-19"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it</span>
 <a id="__codelineno-36-20" name="__codelineno-36-20" href="#__codelineno-36-20"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-21" name="__codelineno-36-21" href="#__codelineno-36-21"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-36-22" name="__codelineno-36-22" href="#__codelineno-36-22"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-36-23" name="__codelineno-36-23" href="#__codelineno-36-23"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-36-23" name="__codelineno-36-23" href="#__codelineno-36-23"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-36-24" name="__codelineno-36-24" href="#__codelineno-36-24"></a><span class="w">        </span><span class="n">state</span><span class="p">[</span><span class="n">stateSize</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-36-25" name="__codelineno-36-25" href="#__codelineno-36-25"></a><span class="w">        </span><span class="n">stateSize</span><span class="o">++</span><span class="p">;</span>
-<a id="__codelineno-36-26" name="__codelineno-36-26" href="#__codelineno-36-26"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-36-26" name="__codelineno-36-26" href="#__codelineno-36-26"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-36-27" name="__codelineno-36-27" href="#__codelineno-36-27"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">choicesSize</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-36-28" name="__codelineno-36-28" href="#__codelineno-36-28"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-36-28" name="__codelineno-36-28" href="#__codelineno-36-28"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-36-29" name="__codelineno-36-29" href="#__codelineno-36-29"></a><span class="w">        </span><span class="n">stateSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-36-30" name="__codelineno-36-30" href="#__codelineno-36-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-36-31" name="__codelineno-36-31" href="#__codelineno-36-31"></a><span class="p">}</span>
@@ -6277,7 +6359,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">subset_sum_ii.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Backtracking algorithm: Subset sum II */</span>
+<div class="highlight"><span class="filename">subset_sum_ii.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Backtracking: subset sum II */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span>
 <a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span>
@@ -6285,29 +6367,29 @@
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="n">start</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span>
 <a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?&gt;</span>
 <a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">    </span><span class="c1">// When the subset sum equals target, record the solution</span>
+<a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">    </span><span class="c1">// Record a solution when the subset reaches the original target (remaining target is 0)</span>
 <a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">toMutableList</span><span class="p">())</span>
 <a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
-<a id="__codelineno-37-15" name="__codelineno-37-15" href="#__codelineno-37-15"></a><span class="w">    </span><span class="c1">// Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
-<a id="__codelineno-37-16" name="__codelineno-37-16" href="#__codelineno-37-16"></a><span class="w">    </span><span class="c1">// Pruning 3: start traversing from start to avoid repeatedly selecting the same element</span>
+<a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
+<a id="__codelineno-37-15" name="__codelineno-37-15" href="#__codelineno-37-15"></a><span class="w">    </span><span class="c1">// Pruning 2: iterate from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-37-16" name="__codelineno-37-16" href="#__codelineno-37-16"></a><span class="w">    </span><span class="c1">// Pruning 3: start after the previously chosen index to avoid choosing the same element again</span>
 <a id="__codelineno-37-17" name="__codelineno-37-17" href="#__codelineno-37-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">start</span><span class="p">..</span><span class="o">&lt;</span><span class="n">choices</span><span class="p">.</span><span class="na">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-18" name="__codelineno-37-18" href="#__codelineno-37-18"></a><span class="w">        </span><span class="c1">// Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-37-19" name="__codelineno-37-19" href="#__codelineno-37-19"></a><span class="w">        </span><span class="c1">// This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-37-18" name="__codelineno-37-18" href="#__codelineno-37-18"></a><span class="w">        </span><span class="c1">// Pruning 1: exit the loop if this choice would exceed the remaining target</span>
+<a id="__codelineno-37-19" name="__codelineno-37-19" href="#__codelineno-37-19"></a><span class="w">        </span><span class="c1">// The array is sorted, so later elements are at least as large and would also exceed the remaining target</span>
 <a id="__codelineno-37-20" name="__codelineno-37-20" href="#__codelineno-37-20"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-21" name="__codelineno-37-21" href="#__codelineno-37-21"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-37-22" name="__codelineno-37-22" href="#__codelineno-37-22"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-37-23" name="__codelineno-37-23" href="#__codelineno-37-23"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly</span>
+<a id="__codelineno-37-23" name="__codelineno-37-23" href="#__codelineno-37-23"></a><span class="w">        </span><span class="c1">// Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it</span>
 <a id="__codelineno-37-24" name="__codelineno-37-24" href="#__codelineno-37-24"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-25" name="__codelineno-37-25" href="#__codelineno-37-25"></a><span class="w">            </span><span class="k">continue</span>
 <a id="__codelineno-37-26" name="__codelineno-37-26" href="#__codelineno-37-26"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-37-27" name="__codelineno-37-27" href="#__codelineno-37-27"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update target, start</span>
+<a id="__codelineno-37-27" name="__codelineno-37-27" href="#__codelineno-37-27"></a><span class="w">        </span><span class="c1">// Try: make a choice and update target and start</span>
 <a id="__codelineno-37-28" name="__codelineno-37-28" href="#__codelineno-37-28"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">)</span>
-<a id="__codelineno-37-29" name="__codelineno-37-29" href="#__codelineno-37-29"></a><span class="w">        </span><span class="c1">// Proceed to the next round of selection</span>
+<a id="__codelineno-37-29" name="__codelineno-37-29" href="#__codelineno-37-29"></a><span class="w">        </span><span class="c1">// Proceed to the next choice</span>
 <a id="__codelineno-37-30" name="__codelineno-37-30" href="#__codelineno-37-30"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-37-31" name="__codelineno-37-31" href="#__codelineno-37-31"></a><span class="w">        </span><span class="c1">// Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-37-31" name="__codelineno-37-31" href="#__codelineno-37-31"></a><span class="w">        </span><span class="c1">// Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-37-32" name="__codelineno-37-32" href="#__codelineno-37-32"></a><span class="w">        </span><span class="n">state</span><span class="p">.</span><span class="na">removeAt</span><span class="p">(</span><span class="n">state</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-37-33" name="__codelineno-37-33" href="#__codelineno-37-33"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-34" name="__codelineno-37-34" href="#__codelineno-37-34"></a><span class="p">}</span>
@@ -6316,8 +6398,8 @@
 <a id="__codelineno-37-37" name="__codelineno-37-37" href="#__codelineno-37-37"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">subsetSumII</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-38" name="__codelineno-37-38" href="#__codelineno-37-38"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span><span class="w"> </span><span class="c1">// State (subset)</span>
 <a id="__codelineno-37-39" name="__codelineno-37-39" href="#__codelineno-37-39"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="na">sort</span><span class="p">()</span><span class="w"> </span><span class="c1">// Sort nums</span>
-<a id="__codelineno-37-40" name="__codelineno-37-40" href="#__codelineno-37-40"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Start point for traversal</span>
-<a id="__codelineno-37-41" name="__codelineno-37-41" href="#__codelineno-37-41"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?&gt;</span><span class="p">()</span><span class="w"> </span><span class="c1">// Result list (subset list)</span>
+<a id="__codelineno-37-40" name="__codelineno-37-40" href="#__codelineno-37-40"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Starting index for iteration</span>
+<a id="__codelineno-37-41" name="__codelineno-37-41" href="#__codelineno-37-41"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?&gt;</span><span class="p">()</span><span class="w"> </span><span class="c1">// Result list (list of subsets)</span>
 <a id="__codelineno-37-42" name="__codelineno-37-42" href="#__codelineno-37-42"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-37-43" name="__codelineno-37-43" href="#__codelineno-37-43"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-37-44" name="__codelineno-37-44" href="#__codelineno-37-44"></a><span class="p">}</span>
@@ -6326,26 +6408,26 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">subset_sum_ii.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Backtracking: subset sum II ###</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1"># When the subset sum equals target, record the solution</span>
+<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1"># Record a solution when the remaining target reaches zero</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="n">target</span><span class="o">.</span><span class="n">zero?</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">state</span><span class="o">.</span><span class="n">dup</span>
 <a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">    </span><span class="k">return</span>
 <a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a>
-<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">  </span><span class="c1"># Traverse all choices</span>
-<a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">  </span><span class="c1"># Pruning 2: start traversing from start to avoid generating duplicate subsets</span>
-<a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="w">  </span><span class="c1"># Pruning 3: start traversing from start to avoid repeatedly selecting the same element</span>
+<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">  </span><span class="c1"># Iterate over all choices</span>
+<a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">  </span><span class="c1"># Pruning 2: iterate from start to avoid generating duplicate subsets</span>
+<a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="w">  </span><span class="c1"># Pruning 3: start after the previously chosen index to avoid choosing the same element again</span>
 <a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">start</span><span class="o">...</span><span class="n">choices</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">    </span><span class="c1"># Pruning 1: if the subset sum exceeds target, end the loop directly</span>
-<a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="w">    </span><span class="c1"># This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target</span>
+<a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">    </span><span class="c1"># Pruning 1: exit the loop if this choice would exceed the remaining target</span>
+<a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="w">    </span><span class="c1"># The array is sorted, so later elements are at least as large and would also exceed the remaining target</span>
 <a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a><span class="w">    </span><span class="k">break</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w">    </span><span class="c1"># Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly</span>
+<a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w">    </span><span class="c1"># Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it</span>
 <a id="__codelineno-38-17" name="__codelineno-38-17" href="#__codelineno-38-17"></a><span class="w">    </span><span class="k">next</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span>
-<a id="__codelineno-38-18" name="__codelineno-38-18" href="#__codelineno-38-18"></a><span class="w">    </span><span class="c1"># Attempt: make choice, update target, start</span>
+<a id="__codelineno-38-18" name="__codelineno-38-18" href="#__codelineno-38-18"></a><span class="w">    </span><span class="c1"># Try: make a choice and update target and start</span>
 <a id="__codelineno-38-19" name="__codelineno-38-19" href="#__codelineno-38-19"></a><span class="w">    </span><span class="n">state</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
-<a id="__codelineno-38-20" name="__codelineno-38-20" href="#__codelineno-38-20"></a><span class="w">    </span><span class="c1"># Proceed to the next round of selection</span>
+<a id="__codelineno-38-20" name="__codelineno-38-20" href="#__codelineno-38-20"></a><span class="w">    </span><span class="c1"># Proceed to the next choice</span>
 <a id="__codelineno-38-21" name="__codelineno-38-21" href="#__codelineno-38-21"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">choices</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-38-22" name="__codelineno-38-22" href="#__codelineno-38-22"></a><span class="w">    </span><span class="c1"># Backtrack: undo choice, restore to previous state</span>
+<a id="__codelineno-38-22" name="__codelineno-38-22" href="#__codelineno-38-22"></a><span class="w">    </span><span class="c1"># Backtrack: undo the choice and restore the previous state</span>
 <a id="__codelineno-38-23" name="__codelineno-38-23" href="#__codelineno-38-23"></a><span class="w">    </span><span class="n">state</span><span class="o">.</span><span class="n">pop</span>
 <a id="__codelineno-38-24" name="__codelineno-38-24" href="#__codelineno-38-24"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-25" name="__codelineno-38-25" href="#__codelineno-38-25"></a><span class="k">end</span>
@@ -6354,8 +6436,8 @@
 <a id="__codelineno-38-28" name="__codelineno-38-28" href="#__codelineno-38-28"></a><span class="k">def</span><span class="w"> </span><span class="nf">subset_sum_ii</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
 <a id="__codelineno-38-29" name="__codelineno-38-29" href="#__codelineno-38-29"></a><span class="w">  </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span><span class="w"> </span><span class="c1"># State (subset)</span>
 <a id="__codelineno-38-30" name="__codelineno-38-30" href="#__codelineno-38-30"></a><span class="w">  </span><span class="n">nums</span><span class="o">.</span><span class="n">sort!</span><span class="w"> </span><span class="c1"># Sort nums</span>
-<a id="__codelineno-38-31" name="__codelineno-38-31" href="#__codelineno-38-31"></a><span class="w">  </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Start point for traversal</span>
-<a id="__codelineno-38-32" name="__codelineno-38-32" href="#__codelineno-38-32"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span><span class="w"> </span><span class="c1"># Result list (subset list)</span>
+<a id="__codelineno-38-31" name="__codelineno-38-31" href="#__codelineno-38-31"></a><span class="w">  </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Starting index for iteration</span>
+<a id="__codelineno-38-32" name="__codelineno-38-32" href="#__codelineno-38-32"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span><span class="w"> </span><span class="c1"># Result list (list of subsets)</span>
 <a id="__codelineno-38-33" name="__codelineno-38-33" href="#__codelineno-38-33"></a><span class="w">  </span><span class="n">backtrack</span><span class="p">(</span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-38-34" name="__codelineno-38-34" href="#__codelineno-38-34"></a><span class="w">  </span><span class="n">res</span>
 <a id="__codelineno-38-35" name="__codelineno-38-35" href="#__codelineno-38-35"></a><span class="k">end</span>
@@ -6363,9 +6445,14 @@
 </div>
 </div>
 </div>
-<p>Figure 13-14 shows the backtracking process for array <span class="arithmatex">\([4, 4, 5]\)</span> with target value <span class="arithmatex">\(9\)</span>, which includes four types of pruning operations. Combine the illustration with the code comments to understand the entire search process and how each pruning operation works.</p>
-<p><img alt="Subset-sum II backtracking process" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_ii.png" /></p>
-<p align="center"> Figure 13-14 &nbsp; Subset-sum II backtracking process </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20backtrack%28%0A%20%20%20%20state%3A%20list%5Bint%5D,%20target%3A%20int,%20choices%3A%20list%5Bint%5D,%20start%3A%20int,%20res%3A%20list%5Blist%5Bint%5D%5D%0A%29%3A%0A%20%20%20%20%22%22%22Backtracking%3A%20subset%20sum%20II%22%22%22%0A%20%20%20%20%23%20Record%20a%20solution%20when%20the%20remaining%20target%20is%20zero%0A%20%20%20%20if%20target%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20res.append%28list%28state%29%29%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Iterate%20over%20all%20choices%0A%20%20%20%20%23%20Pruning%202%3A%20iterate%20from%20start%20to%20avoid%20generating%20duplicate%20subsets%0A%20%20%20%20%23%20Pruning%203%3A%20start%20after%20the%20previously%20chosen%20index%20to%20avoid%20choosing%20the%20same%20element%20again%0A%20%20%20%20for%20i%20in%20range%28start,%20len%28choices%29%29%3A%0A%20%20%20%20%20%20%20%20%23%20Pruning%201%3A%20exit%20the%20loop%20if%20the%20chosen%20value%20exceeds%20the%20remaining%20target%0A%20%20%20%20%20%20%20%20%23%20The%20array%20is%20sorted,%20so%20later%20elements%20are%20at%20least%20as%20large%20and%20also%20exceed%20the%20remaining%20target%0A%20%20%20%20%20%20%20%20if%20target%20-%20choices%5Bi%5D%20%3C%200%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20%23%20Pruning%204%3A%20if%20this%20element%20equals%20its%20left%20neighbor%20at%20the%20same%20search%20depth,%20the%20branch%20is%20a%20duplicate%3B%20skip%20it%0A%20%20%20%20%20%20%20%20if%20i%20%3E%20start%20and%20choices%5Bi%5D%20%3D%3D%20choices%5Bi%20-%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20continue%0A%20%20%20%20%20%20%20%20%23%20Try%3A%20make%20a%20choice%20and%20update%20target%20and%20start%0A%20%20%20%20%20%20%20%20state.append%28choices%5Bi%5D%29%0A%20%20%20%20%20%20%20%20%23%20Proceed%20to%20the%20next%20choice%0A%20%20%20%20%20%20%20%20backtrack%28state,%20target%20-%20choices%5Bi%5D,%20choices,%20i%20%2B%201,%20res%29%0A%20%20%20%20%20%20%20%20%23%20Backtrack%3A%20undo%20the%20choice%20and%20restore%20the%20previous%20state%0A%20%20%20%20%20%20%20%20state.pop%28%29%0A%0A%0Adef%20subset_sum_ii%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20list%5Blist%5Bint%5D%5D%3A%0A%20%20%20%20%22%22%22Solve%20subset%20sum%20II%22%22%22%0A%20%20%20%20state%20%3D%20%5B%5D%20%20%23%20State%20%28subset%29%0A%20%20%20%20nums.sort%28%29%20%20%23%20Sort%20nums%0A%20%20%20%20start%20%3D%200%20%20%23%20Starting%20index%20for%20iteration%0A%20%20%20%20res%20%3D%20%5B%5D%20%20%23%20Result%20list%20%28list%20of%20subsets%29%0A%20%20%20%20backtrack%28state,%20target,%20nums,%20start,%20res%29%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B4,%204,%205%5D%0A%20%20%20%20target%20%3D%209%0A%20%20%20%20res%20%3D%20subset_sum_ii%28nums,%20target%29%0A%0A%20%20%20%20print%28f%22Input%20array%20nums%20%3D%20%7Bnums%7D,%20target%20%3D%20%7Btarget%7D%22%29%0A%20%20%20%20print%28f%22All%20subsets%20with%20sum%20%7Btarget%7D%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20backtrack%28%0A%20%20%20%20state%3A%20list%5Bint%5D,%20target%3A%20int,%20choices%3A%20list%5Bint%5D,%20start%3A%20int,%20res%3A%20list%5Blist%5Bint%5D%5D%0A%29%3A%0A%20%20%20%20%22%22%22Backtracking%3A%20subset%20sum%20II%22%22%22%0A%20%20%20%20%23%20Record%20a%20solution%20when%20the%20remaining%20target%20is%20zero%0A%20%20%20%20if%20target%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20res.append%28list%28state%29%29%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Iterate%20over%20all%20choices%0A%20%20%20%20%23%20Pruning%202%3A%20iterate%20from%20start%20to%20avoid%20generating%20duplicate%20subsets%0A%20%20%20%20%23%20Pruning%203%3A%20start%20after%20the%20previously%20chosen%20index%20to%20avoid%20choosing%20the%20same%20element%20again%0A%20%20%20%20for%20i%20in%20range%28start,%20len%28choices%29%29%3A%0A%20%20%20%20%20%20%20%20%23%20Pruning%201%3A%20exit%20the%20loop%20if%20the%20chosen%20value%20exceeds%20the%20remaining%20target%0A%20%20%20%20%20%20%20%20%23%20The%20array%20is%20sorted,%20so%20later%20elements%20are%20at%20least%20as%20large%20and%20also%20exceed%20the%20remaining%20target%0A%20%20%20%20%20%20%20%20if%20target%20-%20choices%5Bi%5D%20%3C%200%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20%23%20Pruning%204%3A%20if%20this%20element%20equals%20its%20left%20neighbor%20at%20the%20same%20search%20depth,%20the%20branch%20is%20a%20duplicate%3B%20skip%20it%0A%20%20%20%20%20%20%20%20if%20i%20%3E%20start%20and%20choices%5Bi%5D%20%3D%3D%20choices%5Bi%20-%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20continue%0A%20%20%20%20%20%20%20%20%23%20Try%3A%20make%20a%20choice%20and%20update%20target%20and%20start%0A%20%20%20%20%20%20%20%20state.append%28choices%5Bi%5D%29%0A%20%20%20%20%20%20%20%20%23%20Proceed%20to%20the%20next%20choice%0A%20%20%20%20%20%20%20%20backtrack%28state,%20target%20-%20choices%5Bi%5D,%20choices,%20i%20%2B%201,%20res%29%0A%20%20%20%20%20%20%20%20%23%20Backtrack%3A%20undo%20the%20choice%20and%20restore%20the%20previous%20state%0A%20%20%20%20%20%20%20%20state.pop%28%29%0A%0A%0Adef%20subset_sum_ii%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20list%5Blist%5Bint%5D%5D%3A%0A%20%20%20%20%22%22%22Solve%20subset%20sum%20II%22%22%22%0A%20%20%20%20state%20%3D%20%5B%5D%20%20%23%20State%20%28subset%29%0A%20%20%20%20nums.sort%28%29%20%20%23%20Sort%20nums%0A%20%20%20%20start%20%3D%200%20%20%23%20Starting%20index%20for%20iteration%0A%20%20%20%20res%20%3D%20%5B%5D%20%20%23%20Result%20list%20%28list%20of%20subsets%29%0A%20%20%20%20backtrack%28state,%20target,%20nums,%20start,%20res%29%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B4,%204,%205%5D%0A%20%20%20%20target%20%3D%209%0A%20%20%20%20res%20%3D%20subset_sum_ii%28nums,%20target%29%0A%0A%20%20%20%20print%28f%22Input%20array%20nums%20%3D%20%7Bnums%7D,%20target%20%3D%20%7Btarget%7D%22%29%0A%20%20%20%20print%28f%22All%20subsets%20with%20sum%20%7Btarget%7D%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 13-14 shows backtracking for array <span class="arithmatex">\([4, 4, 5]\)</span> and target <span class="arithmatex">\(9\)</span>, including four types of pruning. Follow the illustration alongside the code comments to understand the full search and how each pruning rule works.</p>
+<p><img alt="Backtracking for subset sum II" class="animation-figure" src="../subset_sum_problem.assets/subset_sum_ii.png" /></p>
+<p align="center"> Figure 13-14 &nbsp; Backtracking for subset sum II </p>
 
 <!-- Source file information -->
 
@@ -6388,7 +6475,7 @@ aria-label="Footer"
   <a
     href="../permutations_problem/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 13.2 Permutations Problem"
+    aria-label="Previous: 13.2   Permutations problem"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -6400,7 +6487,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        13.2 Permutations Problem
+        13.2 &nbsp; Permutations problem
       </div>
     </div>
   </a>
@@ -6412,7 +6499,7 @@ aria-label="Footer"
   <a
     href="../n_queens_problem/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 13.4 N-Queens Problem"
+    aria-label="Next: 13.4   N-queens problem"
     rel="next"
   >
     <div class="md-footer__title">
@@ -6420,7 +6507,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        13.4 N-Queens Problem
+        13.4 &nbsp; N-queens problem
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -6437,7 +6524,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
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+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-  Before Starting
+  Foreword
 
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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-  Chapter 0. Preface
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+  0.1 &nbsp; About this book
 
 
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-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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-  Chapter 1. Encounter with Algorithms
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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-  Chapter 2. Complexity Analysis
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-  2.2 Iteration and Recursion
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-  2.6 Exercises
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
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-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
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-  3.3 Number Encoding *
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-  3.4 Character Encoding *
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-  3.5 Summary
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-  3.6 Exercises
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
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-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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@@ -2420,7 +2420,7 @@
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-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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@@ -2638,7 +2638,7 @@
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-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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@@ -2692,7 +2692,7 @@
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
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@@ -2920,7 +2920,7 @@
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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@@ -3190,7 +3190,7 @@
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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@@ -3217,7 +3217,7 @@
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4558,6 +4558,78 @@
 
 
 
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+  <a href="#1-key-takeaways" class="md-nav__link">
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-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
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     </span>
   </a>
@@ -4658,25 +4730,25 @@
 
 <!-- Page content -->
 <h1 id="135-summary">13.5 &nbsp; Summary<a class="headerlink" href="#135-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
 <ul>
-<li>The backtracking algorithm is fundamentally an exhaustive search method. It finds solutions that meet specified conditions by performing a depth-first traversal of the solution space. During the search process, when a solution satisfying the conditions is found, it is recorded. The search ends either after finding all solutions or when the traversal is complete.</li>
-<li>The backtracking algorithm search process consists of two parts: attempting and backtracking. It tries various choices through depth-first search. When encountering situations that violate constraints, it reverts the previous choice, returns to the previous state, and continues exploring other options. Attempting and backtracking are operations in opposite directions.</li>
-<li>Backtracking problems typically contain multiple constraints, which can be utilized to implement pruning operations. Pruning can terminate unnecessary search branches early, significantly improving search efficiency.</li>
-<li>The backtracking algorithm is primarily used to solve search problems and constraint satisfaction problems. While combinatorial optimization problems can be solved with backtracking, there are often more efficient or better-performing solutions available.</li>
-<li>The permutation problem aims to find all possible permutations of elements in a given set. We use an array to record whether each element has been selected, thereby pruning search branches that attempt to select the same element repeatedly, ensuring each element is selected exactly once.</li>
-<li>In the permutation problem, if the set contains duplicate elements, the final result will contain duplicate permutations. We need to impose a constraint so that equal elements can only be selected once per round, which is typically achieved using a hash set.</li>
-<li>The subset-sum problem aims to find all subsets of a given set that sum to a target value. Since the set is unordered but the search process outputs results in all orders, duplicate subsets are generated. We sort the data before backtracking and use a variable to indicate the starting point of each round's traversal, thereby pruning search branches that generate duplicate subsets.</li>
-<li>For the subset-sum problem, equal elements in the array produce duplicate subsets. We leverage the precondition that the array is sorted by checking whether adjacent elements are equal to implement pruning, ensuring that equal elements can only be selected once per round.</li>
-<li>The <span class="arithmatex">\(n\)</span> queens problem aims to find placements of <span class="arithmatex">\(n\)</span> queens on an <span class="arithmatex">\(n \times n\)</span> chessboard such that no two queens can attack each other. The constraints of this problem include row constraints, column constraints, and main and anti-diagonal constraints. To satisfy row constraints, we adopt a row-by-row placement strategy, ensuring exactly one queen is placed in each row.</li>
-<li>The handling of column constraints and diagonal constraints is similar. For column constraints, we use an array to record whether each column has a queen, thereby indicating whether a selected cell is valid. For diagonal constraints, we use two arrays to separately record whether queens exist on each main or anti-diagonal. The challenge lies in finding the row-column index pattern that characterizes cells on the same main (anti-)diagonal.</li>
+<li>Backtracking is fundamentally exhaustive search. It traverses the solution space depth first to find solutions meeting the requirements, records them when found, and continues until all solutions are found or traversal is complete.</li>
+<li>Backtracking alternates between trying a choice and undoing it. It explores choices through depth-first search; when a constraint is violated, it undoes the last choice, restores the previous state, and tries another choice. Trying and undoing are opposite operations.</li>
+<li>Backtracking problems usually have multiple constraints that enable pruning. Pruning stops unnecessary branches early and can greatly improve search efficiency.</li>
+<li>Backtracking is mainly used for search and constraint satisfaction problems. It can also solve combinatorial optimization problems, though more efficient or more effective approaches often exist.</li>
+<li>The permutations problem asks for every possible ordering of a collection's elements. An array tracks whether each element has been selected, pruning branches that select the same element again and ensuring each is used only once.</li>
+<li>If the input contains equal elements, permutation generation produces duplicate results. We must try each value only once per round, usually by using a hash set.</li>
+<li>The subset sum problem asks for all subsets of a collection whose sum equals a target. Subsets disregard order, but the search generates all orderings, producing duplicates. Sorting the data before backtracking and using a variable to mark each round's starting index allows us to prune duplicate-generating branches.</li>
+<li>Equal elements also produce duplicate subsets. Because the array is sorted, we can prune by comparing adjacent elements, ensuring that each value is tried only once per round.</li>
+<li>The <span class="arithmatex">\(n\)</span>-queens problem asks for arrangements of <span class="arithmatex">\(n\)</span> queens on an <span class="arithmatex">\(n \times n\)</span> chessboard such that no two queens can attack each other. It imposes row, column, and both diagonal constraints. Placing queens row by row ensures exactly one queen per row.</li>
+<li>Column and diagonal constraints are handled similarly. An array records whether each column contains a queen, indicating whether a chosen square is valid. Two more arrays track queens on diagonals in each direction. The main challenge is identifying the relationship between row and column indices for squares on the same diagonal.</li>
 </ul>
 <h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
-<p><strong>Q</strong>: How can we understand the relationship between backtracking and recursion?</p>
-<p>Overall, backtracking is an algorithmic strategy, while recursion is better viewed as a tool.</p>
+<p><strong>Q</strong>: How are backtracking and recursion related?</p>
+<p>Broadly speaking, backtracking is an algorithmic strategy, while recursion is a tool.</p>
 <ul>
-<li>Backtracking is typically implemented with recursion. However, backtracking is only one application of recursion, specifically its use in search problems.</li>
-<li>The structure of recursion reflects a problem-solving paradigm based on decomposing a problem into subproblems, and it is commonly used in divide-and-conquer, backtracking, and dynamic programming (memoized recursion).</li>
+<li>Backtracking is usually implemented recursively. It is one application of recursion, specifically to search problems.</li>
+<li>Recursion expresses the idea of decomposing a problem into subproblems. It is commonly used in divide and conquer, backtracking, and dynamic programming (memoized recursion).</li>
 </ul>
 
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-    aria-label="Previous: 13.4 N-Queens Problem"
+    aria-label="Previous: 13.4   N-queens problem"
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-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
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-  1.2 What Is an Algorithm
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         <li class="md-nav__item">
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+  <a href="#261-check-your-understanding" class="md-nav__link">
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-        2.6.1 &nbsp; Concept Review
+        2.6.1 &nbsp; Check your understanding
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+    <nav class="md-nav" aria-label="2.6.1 Check your understanding">
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+  6.5 &nbsp; Exercises
 
 
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@@ -2023,7 +2023,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -2044,7 +2044,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2065,7 +2065,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -2092,7 +2092,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2119,7 +2119,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2146,7 +2146,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2173,7 +2173,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2200,7 +2200,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2227,7 +2227,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2291,7 +2291,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2312,7 +2312,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2333,7 +2333,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2360,7 +2360,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2387,7 +2387,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2414,7 +2414,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2441,7 +2441,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2505,7 +2505,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2526,7 +2526,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2547,7 +2547,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2574,7 +2574,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2601,7 +2601,7 @@
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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@@ -2628,7 +2628,7 @@
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2723,7 +2723,7 @@
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2744,7 +2744,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2765,7 +2765,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2792,7 +2792,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2819,7 +2819,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2846,7 +2846,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2873,7 +2873,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2900,7 +2900,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2927,7 +2927,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -3005,7 +3005,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3026,7 +3026,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3047,7 +3047,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3074,7 +3074,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3101,7 +3101,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3128,7 +3128,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3155,7 +3155,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3182,7 +3182,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3209,7 +3209,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3236,7 +3236,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3263,7 +3263,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3290,7 +3290,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3317,7 +3317,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3344,7 +3344,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3410,7 +3410,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3431,7 +3431,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3452,7 +3452,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3479,7 +3479,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3506,7 +3506,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3533,7 +3533,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3560,7 +3560,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3587,7 +3587,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3653,7 +3653,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3674,7 +3674,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3695,7 +3695,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3722,7 +3722,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3749,7 +3749,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3776,7 +3776,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3803,7 +3803,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3830,7 +3830,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3900,7 +3900,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3921,7 +3921,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3942,7 +3942,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3969,7 +3969,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3996,7 +3996,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4023,7 +4023,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4050,7 +4050,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4077,7 +4077,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4104,7 +4104,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4131,7 +4131,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4197,7 +4197,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4218,7 +4218,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4239,7 +4239,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4266,7 +4266,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4293,7 +4293,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4320,7 +4320,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4347,7 +4347,7 @@
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4374,7 +4374,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4434,7 +4434,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4455,7 +4455,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4476,7 +4476,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4503,7 +4503,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4530,7 +4530,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4614,6 +4614,78 @@
 
 
 
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@@ -4655,22 +4727,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#261-concept-review" class="md-nav__link">
+  <a href="#261-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.6.1 &nbsp; Concept Review
+        2.6.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="2.6.1 Concept Review">
+    <nav class="md-nav" aria-label="2.6.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-time-and-space-complexity-of-iteration-and-recursion" class="md-nav__link">
+  <a href="#1-time-and-space-in-iteration-and-recursion" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Time and Space Complexity of Iteration and Recursion
+        1. &nbsp; Time and space in iteration and recursion
       </span>
     </span>
   </a>
@@ -4678,10 +4750,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-time-complexity-of-three-code-fragments" class="md-nav__link">
+  <a href="#2-time-complexity-of-three-code-snippets" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Time Complexity of Three Code Fragments
+        2. &nbsp; Time complexity of three code snippets
       </span>
     </span>
   </a>
@@ -4689,10 +4761,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-which-reversal-uses-less-space" class="md-nav__link">
+  <a href="#3-which-reversal-method-uses-less-space" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Which Reversal Uses Less Space?
+        3. &nbsp; Which reversal method uses less space?
       </span>
     </span>
   </a>
@@ -4708,19 +4780,19 @@
   <a href="#262-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.6.2 &nbsp; Programming Exercises
+        2.6.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="2.6.2 Programming Exercises">
+    <nav class="md-nav" aria-label="2.6.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-fibonacci-number" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Fibonacci Number
+        1. &nbsp; Fibonacci number
       </span>
     </span>
   </a>
@@ -4770,29 +4842,29 @@
 
 <!-- Page content -->
 <h1 id="26-exercises">2.6 &nbsp; Exercises<a class="headerlink" href="#26-exercises" title="Permanent link">&para;</a></h1>
-<h2 id="261-concept-review">2.6.1 &nbsp; Concept Review<a class="headerlink" href="#261-concept-review" title="Permanent link">&para;</a></h2>
-<h3 id="1-time-and-space-complexity-of-iteration-and-recursion">1. &nbsp; Time and Space Complexity of Iteration and Recursion<a class="headerlink" href="#1-time-and-space-complexity-of-iteration-and-recursion" title="Permanent link">&para;</a></h3>
-<p>The two functions below both calculate <span class="arithmatex">\(1 + 2 + \dots + n\)</span> (assume <span class="arithmatex">\(n \ge 1\)</span>). Set <code>n</code> to 4,
-answer the questions by following the program's actual execution order, and then compare the efficiency of the two approaches.</p>
+<h2 id="261-check-your-understanding">2.6.1 &nbsp; Check your understanding<a class="headerlink" href="#261-check-your-understanding" title="Permanent link">&para;</a></h2>
+<h3 id="1-time-and-space-in-iteration-and-recursion">1. &nbsp; Time and space in iteration and recursion<a class="headerlink" href="#1-time-and-space-in-iteration-and-recursion" title="Permanent link">&para;</a></h3>
+<p>Both functions below compute <span class="arithmatex">\(1 + 2 + \dots + n\)</span> (assume <span class="arithmatex">\(n \ge 1\)</span>). Set <code>n</code> to 4,
+follow the program's execution order to answer the questions, and then compare the efficiency of the two approaches.</p>
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 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">complexity_exercises.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">sum_iter</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Iterative summation"""</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Sum using iteration"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="n">res</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="n">res</span> <span class="o">+=</span> <span class="n">i</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="k">return</span> <span class="n">res</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="k">def</span><span class="w"> </span><span class="nf">sum_recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="w">    </span><span class="sd">"""Recursive summation"""</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="w">    </span><span class="sd">"""Sum using recursion"""</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="k">return</span> <span class="mi">1</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="k">return</span> <span class="n">n</span> <span class="o">+</span> <span class="n">sum_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Iterative summation */</span>
+<div class="highlight"><span class="filename">complexity_exercises.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Sum using iteration */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">sumIter</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4801,7 +4873,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="p">}</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a>
-<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="cm">/* Recursive summation */</span>
+<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="cm">/* Sum using recursion */</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="kt">int</span><span class="w"> </span><span class="nf">sumRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -4811,7 +4883,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Iterative summation */</span>
+<div class="highlight"><span class="filename">complexity_exercises.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Sum using iteration */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">sumIter</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4820,7 +4892,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="p">}</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a>
-<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="cm">/* Recursive summation */</span>
+<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="cm">/* Sum using recursion */</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="kt">int</span><span class="w"> </span><span class="nf">sumRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -4830,7 +4902,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Iterative summation */</span>
+<div class="highlight"><span class="filename">complexity_exercises.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Sum using iteration */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">SumIter</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4839,7 +4911,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="p">}</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a>
-<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="cm">/* Recursive summation */</span>
+<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="cm">/* Sum using recursion */</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="kt">int</span><span class="w"> </span><span class="nf">SumRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -4849,7 +4921,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Iterative summation */</span>
+<div class="highlight"><span class="filename">complexity_exercises.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Sum using iteration */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">sumIter</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
@@ -4858,7 +4930,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="p">}</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a>
-<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="cm">/* Recursive summation */</span>
+<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="cm">/* Sum using recursion */</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="kd">func</span><span class="w"> </span><span class="nx">sumRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span>
@@ -4868,7 +4940,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Iterative summation */</span>
+<div class="highlight"><span class="filename">complexity_exercises.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Sum using iteration */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">sumIter</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
@@ -4877,7 +4949,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="p">}</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a>
-<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="cm">/* Recursive summation */</span>
+<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="cm">/* Sum using recursion */</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="kd">func</span><span class="w"> </span><span class="nf">sumRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span>
@@ -4887,7 +4959,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Iterative summation */</span>
+<div class="highlight"><span class="filename">complexity_exercises.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Sum using iteration */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">sumIter</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4896,7 +4968,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="p">}</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a>
-<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="cm">/* Recursive summation */</span>
+<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="cm">/* Sum using recursion */</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="kd">function</span><span class="w"> </span><span class="nx">sumRecur</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
@@ -4906,7 +4978,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Iterative summation */</span>
+<div class="highlight"><span class="filename">complexity_exercises.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Sum using iteration */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">sumIter</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4915,7 +4987,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="p">}</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a>
-<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="cm">/* Recursive summation */</span>
+<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="cm">/* Sum using recursion */</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="kd">function</span><span class="w"> </span><span class="nx">sumRecur</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
@@ -4925,7 +4997,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Iterative summation */</span>
+<div class="highlight"><span class="filename">complexity_exercises.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Sum using iteration */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">sumIter</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4934,7 +5006,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="p">}</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a>
-<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="cm">/* Recursive summation */</span>
+<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="cm">/* Sum using recursion */</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="kt">int</span><span class="w"> </span><span class="n">sumRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
@@ -4944,7 +5016,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Iterative summation */</span>
+<div class="highlight"><span class="filename">complexity_exercises.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Sum using iteration */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">sum_iter</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
@@ -4953,7 +5025,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="n">res</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="p">}</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a>
-<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="cm">/* Recursive summation */</span>
+<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="cm">/* Sum using recursion */</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="k">fn</span><span class="w"> </span><span class="nf">sum_recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -4963,7 +5035,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Iterative summation */</span>
+<div class="highlight"><span class="filename">complexity_exercises.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Sum using iteration */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">sumIter</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4972,7 +5044,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="p">}</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a>
-<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="cm">/* Recursive summation */</span>
+<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="cm">/* Sum using recursion */</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="kt">int</span><span class="w"> </span><span class="nf">sumRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -4982,7 +5054,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Iterative summation */</span>
+<div class="highlight"><span class="filename">complexity_exercises.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Sum using iteration */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">sumIter</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4991,7 +5063,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="p">}</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a>
-<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="cm">/* Recursive summation */</span>
+<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="cm">/* Sum using recursion */</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">sumRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">1</span>
@@ -5001,7 +5073,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Iterative summation ###</span>
+<div class="highlight"><span class="filename">complexity_exercises.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Sum using iteration ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">sum_iter</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">n</span>
@@ -5010,7 +5082,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="n">res</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="k">end</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a>
-<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="c1">### Recursive summation ###</span>
+<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="c1">### Sum using recursion ###</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="k">def</span><span class="w"> </span><span class="nf">sum_recur</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a>
@@ -5023,45 +5095,45 @@ answer the questions by following the program's actual execution order, and then
 <!-- numbered-subquestions -->
 
 <ol>
-<li>When the iterative function runs with <code>n = 4</code>, what is the value of the accumulator <code>res</code> after each loop iteration?</li>
-<li>When the recursive function runs with <code>n = 4</code>, which values does the argument <code>n</code> take in order? As the calls return from the deepest level, how is the result obtained?</li>
-<li>What are the time and space complexities of the two approaches? Explain your reasoning using the execution processes from Questions 1 and 2.</li>
+<li>When the iterative function runs with <code>n = 4</code>, what is the value of the accumulator <code>res</code> at the end of each iteration?</li>
+<li>When the recursive function runs with <code>n = 4</code>, what values does the parameter <code>n</code> take, in order? How is the result computed as the calls return, starting from the deepest call?</li>
+<li>What are the time and space complexities of each approach? Explain using the execution steps from questions 1 and 2.</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>The loop variable <code>i</code> takes the values <code>1, 2, 3, 4</code>. After each iteration, <code>res</code> becomes
+<p>The loop variable <code>i</code> takes the values <code>1, 2, 3, 4</code> in order. At the end of each iteration, <code>res</code> becomes
     <code>1, 3, 6, 10</code>, respectively, so the iterative function returns 10.</p>
 </li>
 <li>
-<p>The argument <code>n</code> takes the values <code>4 → 3 → 2 → 1</code>.
-    The deepest call returns 1. The remaining calls then obtain <code>2 + 1 = 3</code>, <code>3 + 3 = 6</code>, and <code>4 + 6 = 10</code>, in that order.
-    At the deepest point, all four function calls are still unfinished.</p>
+<p>The parameter <code>n</code> follows the sequence <code>4 → 3 → 2 → 1</code>.
+    The deepest call returns 1. The remaining calls then compute <code>2 + 1 = 3</code>, <code>3 + 3 = 6</code>, and <code>4 + 6 = 10</code>, in order.
+    At the deepest point, all four function calls are still active.</p>
 </li>
 <li>
-<p>Both functions perform a number of loop iterations or calls proportional to <span class="arithmatex">\(n\)</span>, so both have a time complexity of <span class="arithmatex">\(O(n)\)</span>.
-    Their space complexities differ. The iterative version uses only a constant number of variables, so its space complexity is <span class="arithmatex">\(O(1)\)</span>.
-    In the recursive version, earlier calls must wait for a result before returning, so the call stack holds up to <span class="arithmatex">\(n\)</span> calls at the same time.
-    Its space complexity is <span class="arithmatex">\(O(n)\)</span>.</p>
-<p>When analyzing space complexity, remember to include the space used by recursive calls as well as the variables written in the code.</p>
+<p>Both functions perform a number of iterations or calls proportional to <span class="arithmatex">\(n\)</span>, so both have time complexity <span class="arithmatex">\(O(n)\)</span>.
+    Their space complexities differ: the iterative version uses only a constant number of variables, giving <span class="arithmatex">\(O(1)\)</span>.
+    In the recursive version, earlier calls must wait for their results until the base case is reached. The call stack therefore holds up to <span class="arithmatex">\(n\)</span> calls at once,
+    giving space complexity <span class="arithmatex">\(O(n)\)</span>.</p>
+<p>When analyzing space complexity, consider the space used by recursive calls as well as the variables in the code.</p>
 </li>
 </ol>
 </details>
-<h3 id="2-time-complexity-of-three-code-fragments">2. &nbsp; Time Complexity of Three Code Fragments<a class="headerlink" href="#2-time-complexity-of-three-code-fragments" title="Permanent link">&para;</a></h3>
-<p>Each of the following code fragments takes a positive integer <span class="arithmatex">\(n\)</span> as input. Order them from lowest to highest time complexity, and give the complexity of each one.</p>
+<h3 id="2-time-complexity-of-three-code-snippets">2. &nbsp; Time complexity of three code snippets<a class="headerlink" href="#2-time-complexity-of-three-code-snippets" title="Permanent link">&para;</a></h3>
+<p>Each of the following three code snippets takes a positive integer <span class="arithmatex">\(n\)</span> as input. Rank them from lowest to highest time complexity, and state each complexity.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">complexity_exercises.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">linear_loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Linear loop"""</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Linear-time loop"""</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="n">res</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>        <span class="n">res</span> <span class="o">+=</span> <span class="n">i</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="k">return</span> <span class="n">res</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a><span class="k">def</span><span class="w"> </span><span class="nf">quadratic_loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a><span class="w">    </span><span class="sd">"""Quadratic loop"""</span>
+<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a><span class="w">    </span><span class="sd">"""Quadratic-time loop"""</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>    <span class="n">res</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
@@ -5069,14 +5141,14 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>    <span class="k">return</span> <span class="n">res</span>
 <a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>
 <a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a><span class="k">def</span><span class="w"> </span><span class="nf">logarithmic_loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a><span class="w">    </span><span class="sd">"""Logarithmic loop"""</span>
+<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a><span class="w">    </span><span class="sd">"""Logarithmic-time loop"""</span>
 <a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a>    <span class="k">while</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a>        <span class="n">n</span> <span class="o">//=</span> <span class="mi">2</span>
 <a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a>    <span class="k">return</span> <span class="n">n</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Linear loop */</span>
+<div class="highlight"><span class="filename">complexity_exercises.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Linear-time loop */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">linearLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5085,7 +5157,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="p">}</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a>
-<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="cm">/* Quadratic loop */</span>
+<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="cm">/* Quadratic-time loop */</span>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="kt">int</span><span class="w"> </span><span class="nf">quadraticLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5096,7 +5168,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="p">}</span>
 <a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a>
-<a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="cm">/* Logarithmic loop */</span>
+<a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="cm">/* Logarithmic-time loop */</span>
 <a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a><span class="kt">int</span><span class="w"> </span><span class="nf">logarithmicLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-24" name="__codelineno-14-24" href="#__codelineno-14-24"></a><span class="w">        </span><span class="n">n</span><span class="w"> </span><span class="o">/=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
@@ -5106,7 +5178,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Linear loop */</span>
+<div class="highlight"><span class="filename">complexity_exercises.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Linear-time loop */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">linearLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5115,7 +5187,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="p">}</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a>
-<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="cm">/* Quadratic loop */</span>
+<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="cm">/* Quadratic-time loop */</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="kt">int</span><span class="w"> </span><span class="nf">quadraticLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5126,7 +5198,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="p">}</span>
 <a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a>
-<a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="cm">/* Logarithmic loop */</span>
+<a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="cm">/* Logarithmic-time loop */</span>
 <a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="kt">int</span><span class="w"> </span><span class="nf">logarithmicLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-24" name="__codelineno-15-24" href="#__codelineno-15-24"></a><span class="w">        </span><span class="n">n</span><span class="w"> </span><span class="o">/=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
@@ -5136,7 +5208,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Linear loop */</span>
+<div class="highlight"><span class="filename">complexity_exercises.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Linear-time loop */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">LinearLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5145,7 +5217,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="p">}</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a>
-<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="cm">/* Quadratic loop */</span>
+<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="cm">/* Quadratic-time loop */</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="kt">int</span><span class="w"> </span><span class="nf">QuadraticLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5156,7 +5228,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="p">}</span>
 <a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a>
-<a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="cm">/* Logarithmic loop */</span>
+<a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="cm">/* Logarithmic-time loop */</span>
 <a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a><span class="kt">int</span><span class="w"> </span><span class="nf">LogarithmicLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-24" name="__codelineno-16-24" href="#__codelineno-16-24"></a><span class="w">        </span><span class="n">n</span><span class="w"> </span><span class="o">/=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
@@ -5166,7 +5238,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Linear loop */</span>
+<div class="highlight"><span class="filename">complexity_exercises.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Linear-time loop */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">linearLoop</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
@@ -5175,7 +5247,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="p">}</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a>
-<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="cm">/* Quadratic loop */</span>
+<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="cm">/* Quadratic-time loop */</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="kd">func</span><span class="w"> </span><span class="nx">quadraticLoop</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
@@ -5186,7 +5258,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span>
 <a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="p">}</span>
 <a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a>
-<a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="cm">/* Logarithmic loop */</span>
+<a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="cm">/* Logarithmic-time loop */</span>
 <a id="__codelineno-17-22" name="__codelineno-17-22" href="#__codelineno-17-22"></a><span class="kd">func</span><span class="w"> </span><span class="nx">logarithmicLoop</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-24" name="__codelineno-17-24" href="#__codelineno-17-24"></a><span class="w">        </span><span class="nx">n</span><span class="w"> </span><span class="o">/=</span><span class="w"> </span><span class="mi">2</span>
@@ -5196,7 +5268,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Linear loop */</span>
+<div class="highlight"><span class="filename">complexity_exercises.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Linear-time loop */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">linearLoop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
@@ -5205,7 +5277,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="p">}</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a>
-<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="cm">/* Quadratic loop */</span>
+<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="cm">/* Quadratic-time loop */</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="kd">func</span><span class="w"> </span><span class="nf">quadraticLoop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
@@ -5216,7 +5288,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="p">}</span>
 <a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a>
-<a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a><span class="cm">/* Logarithmic loop */</span>
+<a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a><span class="cm">/* Logarithmic-time loop */</span>
 <a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a><span class="kd">func</span><span class="w"> </span><span class="nf">logarithmicLoop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-23" name="__codelineno-18-23" href="#__codelineno-18-23"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">n</span>
 <a id="__codelineno-18-24" name="__codelineno-18-24" href="#__codelineno-18-24"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
@@ -5227,7 +5299,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Linear loop */</span>
+<div class="highlight"><span class="filename">complexity_exercises.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Linear-time loop */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">linearLoop</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5236,7 +5308,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="p">}</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a>
-<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="cm">/* Quadratic loop */</span>
+<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="cm">/* Quadratic-time loop */</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="kd">function</span><span class="w"> </span><span class="nx">quadraticLoop</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5247,7 +5319,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="p">}</span>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a>
-<a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="cm">/* Logarithmic loop */</span>
+<a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="cm">/* Logarithmic-time loop */</span>
 <a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="kd">function</span><span class="w"> </span><span class="nx">logarithmicLoop</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-23" name="__codelineno-19-23" href="#__codelineno-19-23"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-24" name="__codelineno-19-24" href="#__codelineno-19-24"></a><span class="w">        </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span>
@@ -5257,7 +5329,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Linear loop */</span>
+<div class="highlight"><span class="filename">complexity_exercises.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Linear-time loop */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">linearLoop</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5266,7 +5338,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="p">}</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a>
-<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="cm">/* Quadratic loop */</span>
+<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="cm">/* Quadratic-time loop */</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="kd">function</span><span class="w"> </span><span class="nx">quadraticLoop</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5277,7 +5349,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="p">}</span>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a>
-<a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="cm">/* Logarithmic loop */</span>
+<a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="cm">/* Logarithmic-time loop */</span>
 <a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="kd">function</span><span class="w"> </span><span class="nx">logarithmicLoop</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-23" name="__codelineno-20-23" href="#__codelineno-20-23"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">        </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span>
@@ -5287,7 +5359,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Linear loop */</span>
+<div class="highlight"><span class="filename">complexity_exercises.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Linear-time loop */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">linearLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5296,7 +5368,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="p">}</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a>
-<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="cm">/* Quadratic loop */</span>
+<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="cm">/* Quadratic-time loop */</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="kt">int</span><span class="w"> </span><span class="n">quadraticLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5307,7 +5379,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="p">}</span>
 <a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a>
-<a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="cm">/* Logarithmic loop */</span>
+<a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="cm">/* Logarithmic-time loop */</span>
 <a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="kt">int</span><span class="w"> </span><span class="n">logarithmicLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">    </span><span class="n">n</span><span class="w"> </span><span class="o">~/=</span><span class="w"> </span><span class="m">2</span><span class="p">;</span>
@@ -5317,7 +5389,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Linear loop */</span>
+<div class="highlight"><span class="filename">complexity_exercises.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Linear-time loop */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">linear_loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
@@ -5326,7 +5398,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="n">res</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="p">}</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a>
-<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="cm">/* Quadratic loop */</span>
+<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="cm">/* Quadratic-time loop */</span>
 <a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="k">fn</span><span class="w"> </span><span class="nf">quadratic_loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
@@ -5337,7 +5409,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">    </span><span class="n">res</span>
 <a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="p">}</span>
 <a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a>
-<a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="cm">/* Logarithmic loop */</span>
+<a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="cm">/* Logarithmic-time loop */</span>
 <a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="k">fn</span><span class="w"> </span><span class="nf">logarithmic_loop</span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w">        </span><span class="n">n</span><span class="w"> </span><span class="o">/=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
@@ -5347,7 +5419,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Linear loop */</span>
+<div class="highlight"><span class="filename">complexity_exercises.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Linear-time loop */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">linearLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5356,7 +5428,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="p">}</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a>
-<a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="cm">/* Quadratic loop */</span>
+<a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="cm">/* Quadratic-time loop */</span>
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="kt">int</span><span class="w"> </span><span class="nf">quadraticLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5367,7 +5439,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="p">}</span>
 <a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a>
-<a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="cm">/* Logarithmic loop */</span>
+<a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="cm">/* Logarithmic-time loop */</span>
 <a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="kt">int</span><span class="w"> </span><span class="nf">logarithmicLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">        </span><span class="n">n</span><span class="w"> </span><span class="o">/=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
@@ -5377,7 +5449,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Linear loop */</span>
+<div class="highlight"><span class="filename">complexity_exercises.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Linear-time loop */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">linearLoop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="n">until</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5386,7 +5458,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="p">}</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a>
-<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="cm">/* Quadratic loop */</span>
+<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="cm">/* Quadratic-time loop */</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">quadraticLoop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="n">until</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5397,7 +5469,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="p">}</span>
 <a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a>
-<a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="cm">/* Logarithmic loop */</span>
+<a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="cm">/* Logarithmic-time loop */</span>
 <a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">logarithmicLoop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">value</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span>
 <a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">value</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5408,7 +5480,7 @@ answer the questions by following the program's actual execution order, and then
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">complexity_exercises.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Linear loop ###</span>
+<div class="highlight"><span class="filename">complexity_exercises.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Linear-time loop ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">linear_loop</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">n</span>
@@ -5417,7 +5489,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="n">res</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="k">end</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a>
-<a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="c1">### Quadratic loop ###</span>
+<a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="c1">### Quadratic-time loop ###</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="k">def</span><span class="w"> </span><span class="nf">quadratic_loop</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">n</span>
@@ -5428,7 +5500,7 @@ answer the questions by following the program's actual execution order, and then
 <a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w">  </span><span class="n">res</span>
 <a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="k">end</span>
 <a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a>
-<a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="c1">### Logarithmic loop ###</span>
+<a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="c1">### Logarithmic-time loop ###</span>
 <a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="k">def</span><span class="w"> </span><span class="nf">logarithmic_loop</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-25-23" name="__codelineno-25-23" href="#__codelineno-25-23"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">/=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-25-24" name="__codelineno-25-24" href="#__codelineno-25-24"></a><span class="w">  </span><span class="n">n</span>
@@ -5438,48 +5510,48 @@ answer the questions by following the program's actual execution order, and then
 </div>
 </div>
 <details class="success">
-<summary>Answer</summary>
-<p>From lowest to highest, the order is Fragment 3 with <span class="arithmatex">\(O(\log n)\)</span>, Fragment 1 with <span class="arithmatex">\(O(n)\)</span>, and Fragment 2 with <span class="arithmatex">\(O(n^2)\)</span>.
-Fragment 3 halves <span class="arithmatex">\(n\)</span> in each iteration, so it runs about <span class="arithmatex">\(\log_2 n\)</span> times.
-The loop in Fragment 1 runs exactly <span class="arithmatex">\(n\)</span> times. The inner loop in Fragment 2 runs
-<span class="arithmatex">\(n,n-1,\dots,1\)</span> times, for a total of <span class="arithmatex">\(n(n+1)/2\)</span>, so its time complexity is quadratic.</p>
+<summary>Reference answer</summary>
+<p>From lowest to highest: snippet 3, <span class="arithmatex">\(O(\log n)\)</span>; snippet 1, <span class="arithmatex">\(O(n)\)</span>; snippet 2, <span class="arithmatex">\(O(n^2)\)</span>.
+Snippet 3 halves <span class="arithmatex">\(n\)</span> at each iteration, so it runs approximately <span class="arithmatex">\(\log_2 n\)</span> times.
+Snippet 1 runs its loop exactly <span class="arithmatex">\(n\)</span> times. In snippet 2, the inner loop runs
+<span class="arithmatex">\(n,n-1,\dots,1\)</span> times in successive iterations, for a total of <span class="arithmatex">\(n(n+1)/2\)</span>, giving quadratic time complexity.</p>
 </details>
-<h3 id="3-which-reversal-uses-less-space">3. &nbsp; Which Reversal Uses Less Space?<a class="headerlink" href="#3-which-reversal-uses-less-space" title="Permanent link">&para;</a></h3>
-<p>There are two ways to reverse all the elements in the array <code>nums</code>:</p>
+<h3 id="3-which-reversal-method-uses-less-space">3. &nbsp; Which reversal method uses less space?<a class="headerlink" href="#3-which-reversal-method-uses-less-space" title="Permanent link">&para;</a></h3>
+<p>There are two ways to reverse all the elements in an array <code>nums</code>:</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Create a new array <code>res</code> of the same length, copy the elements into it in reverse order, and return it.</li>
+<li>Create an array <code>res</code> of the same length, copy the elements into it in reverse order, and return it.</li>
 <li>
-<p>Move two indices <code>i</code> and <code>j</code> inward from the beginning and end, swapping <code>nums[i]</code> and <code>nums[j]</code> at each step.</p>
-<p>What is the space complexity of each approach? Which one is an "in-place" operation?</p>
+<p>Move two indices, <code>i</code> and <code>j</code>, inward from the beginning and end, swapping <code>nums[i]</code> with <code>nums[j]</code> in pairs.</p>
+<p>What is the space complexity of each approach? Which operation is in-place?</p>
 </li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>This approach needs an auxiliary array with the same length as the input, so its space complexity is <span class="arithmatex">\(O(n)\)</span>.</p>
+<p>An auxiliary array as long as the input is needed, giving space complexity <span class="arithmatex">\(O(n)\)</span>.</p>
 </li>
 <li>
-<p>This approach uses only two index variables,
-    so its space complexity is <span class="arithmatex">\(O(1)\)</span>. It is an in-place operation.</p>
-<p>Note that an in-place reversal changes the input array,
-so it should be preferred only when modifying the input is allowed. If the original array must be kept, the copying cost of the first approach is unavoidable.</p>
+<p>Only two index variables are used,
+    giving space complexity <span class="arithmatex">\(O(1)\)</span>. This is an in-place operation.</p>
+<p>Keep in mind that in-place reversal modifies the input array.
+Prefer it only when modifying the input is allowed. If the original array must be preserved, the copying cost of approach 1 is unavoidable.</p>
 </li>
 </ol>
 </details>
-<h2 id="262-programming-exercises">2.6.2 &nbsp; Programming Exercises<a class="headerlink" href="#262-programming-exercises" title="Permanent link">&para;</a></h2>
-<h3 id="1-fibonacci-number">1. &nbsp; Fibonacci Number<a class="headerlink" href="#1-fibonacci-number" title="Permanent link">&para;</a></h3>
-<p>The Fibonacci sequence is defined by <span class="arithmatex">\(F(0)=0\)</span>, <span class="arithmatex">\(F(1)=1\)</span>, and, for <span class="arithmatex">\(n\ge2\)</span>,
+<h2 id="262-programming-exercises">2.6.2 &nbsp; Programming exercises<a class="headerlink" href="#262-programming-exercises" title="Permanent link">&para;</a></h2>
+<h3 id="1-fibonacci-number">1. &nbsp; Fibonacci number<a class="headerlink" href="#1-fibonacci-number" title="Permanent link">&para;</a></h3>
+<p>The Fibonacci sequence satisfies <span class="arithmatex">\(F(0)=0\)</span>, <span class="arithmatex">\(F(1)=1\)</span>, and, for <span class="arithmatex">\(n\ge2\)</span>,
 <span class="arithmatex">\(F(n)=F(n-1)+F(n-2)\)</span>.</p>
-<p>Given a non-negative integer <code>n</code>, use a loop to calculate and return <span class="arithmatex">\(F(n)\)</span>. Do not use recursion.</p>
+<p>Given a nonnegative integer <code>n</code>, use a loop to compute and return <span class="arithmatex">\(F(n)\)</span> without recursion.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>Handle the cases where n is 0 or 1 separately</li>
-<li>Only the previous two terms are needed to calculate the next term; there is no need to store the entire sequence</li>
-<li>When updating the two variables, take care not to overwrite an old value before it is used</li>
+<li>Handle n equal to 0 or 1 separately first.</li>
+<li>Computing the next term requires only the previous two terms; there is no need to store the entire sequence.</li>
+<li>When updating the two variables, take care not to overwrite an old value before you have finished using it.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/fibonacci-number/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
@@ -5505,7 +5577,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 2.5 Summary"
+    aria-label="Previous: 2.5   Summary"
     rel="prev"
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     <div class="md-footer__button md-icon">
@@ -5517,7 +5589,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        2.5 Summary
+        2.5 &nbsp; Summary
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@@ -5529,7 +5601,7 @@ aria-label="Footer"
   <a
     href="../../chapter_data_structure/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: Chapter 3. &amp;nbsp; Data Structures"
+    aria-label="Next: Chapter 3.   Data structures"
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@@ -5537,7 +5609,7 @@ aria-label="Footer"
         Next
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+        Chapter 3. &nbsp; Data structures
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@@ -5554,7 +5626,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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index d42898a55..766c995e7 100644
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@@ -6,7 +6,7 @@
       <meta charset="utf-8">
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+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
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@@ -41,7 +41,7 @@
 
 
 
-        <title>Chapter 2. Complexity Analysis - Hello Algo</title>
+        <title>Chapter 2. Complexity analysis - Hello Algo</title>
 
 
 
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -739,7 +739,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -760,7 +760,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -781,7 +781,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -982,7 +982,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1003,7 +1003,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1267,7 +1267,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
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@@ -1508,7 +1508,7 @@
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1701,7 +1701,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1722,7 +1722,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1919,7 +1919,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1940,7 +1940,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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@@ -1967,7 +1967,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  10.2 Binary Search Insertion Point
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
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-  11.5 Quick Sort
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
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-  11.12 Exercises
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-  Chapter 12. Divide and Conquer
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
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-  13.6 Exercises
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-  Chapter 14. Dynamic Programming
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-  Chapter 14. Dynamic Programming
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-  14.1 Introduction to Dynamic Programming
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  Chapter 16. Appendix
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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@@ -4577,19 +4649,19 @@
 
 
 <!-- Page content -->
-<h1 id="chapter-2-complexity-analysis">Chapter 2. &nbsp; Complexity Analysis<a class="headerlink" href="#chapter-2-complexity-analysis" title="Permanent link">&para;</a></h1>
+<h1 id="chapter-2-complexity-analysis">Chapter 2. &nbsp; Complexity analysis<a class="headerlink" href="#chapter-2-complexity-analysis" title="Permanent link">&para;</a></h1>
 <p><img alt="Complexity analysis" class="cover-image" src="../assets/covers/chapter_complexity_analysis.jpg" /></p>
 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
-<p>Complexity analysis is like a space-time guide in the vast universe of algorithms.</p>
-<p>It leads us to explore deeply within the two dimensions of time and space, seeking more elegant solutions.</p>
+<p>Complexity analysis is our guide to time and space in the vast universe of algorithms.</p>
+<p>It leads us to explore these two dimensions in depth as we search for more elegant solutions.</p>
 </div>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
-<li><a href="performance_evaluation/">2.1 &nbsp; Algorithm Efficiency Evaluation</a></li>
-<li><a href="iteration_and_recursion/">2.2 &nbsp; Iteration and Recursion</a></li>
-<li><a href="time_complexity/">2.3 &nbsp; Time Complexity</a></li>
-<li><a href="space_complexity/">2.4 &nbsp; Space Complexity</a></li>
+<li><a href="performance_evaluation/">2.1 &nbsp; Evaluating algorithm efficiency</a></li>
+<li><a href="iteration_and_recursion/">2.2 &nbsp; Iteration and recursion</a></li>
+<li><a href="time_complexity/">2.3 &nbsp; Time complexity</a></li>
+<li><a href="space_complexity/">2.4 &nbsp; Space complexity</a></li>
 <li><a href="summary/">2.5 &nbsp; Summary</a></li>
 <li><a href="exercises/">2.6 &nbsp; Exercises</a></li>
 </ul>
@@ -4615,7 +4687,7 @@ aria-label="Footer"
   <a
     href="../chapter_introduction/summary/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 1.3 Summary"
+    aria-label="Previous: 1.3   Summary"
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@@ -4627,7 +4699,7 @@ aria-label="Footer"
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+        1.3 &nbsp; Summary
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   <a
     href="./performance_evaluation/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 2.1 Algorithm Efficiency Evaluation"
+    aria-label="Next: 2.1   Evaluating algorithm efficiency"
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@@ -4647,7 +4719,7 @@ aria-label="Footer"
         Next
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       <div class="md-ellipsis">
-        2.1 Algorithm Efficiency Evaluation
+        2.1 &nbsp; Evaluating algorithm efficiency
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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@@ -177,9 +177,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Light mode"  type="radio" name="__palette" id="__palette_1">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -739,7 +739,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -760,7 +760,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -781,7 +781,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -817,7 +817,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -834,7 +834,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -872,10 +872,10 @@
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-for-loop" class="md-nav__link">
+  <a href="#1-for-loops" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; For Loop
+        1. &nbsp; For loops
       </span>
     </span>
   </a>
@@ -883,10 +883,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-while-loop" class="md-nav__link">
+  <a href="#2-while-loops" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; While Loop
+        2. &nbsp; While loops
       </span>
     </span>
   </a>
@@ -897,7 +897,7 @@
   <a href="#3-nested-loops" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Nested Loops
+        3. &nbsp; Nested loops
       </span>
     </span>
   </a>
@@ -922,10 +922,10 @@
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-call-stack" class="md-nav__link">
+  <a href="#1-the-call-stack" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Call Stack
+        1. &nbsp; The call stack
       </span>
     </span>
   </a>
@@ -936,7 +936,7 @@
   <a href="#2-tail-recursion" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Tail Recursion
+        2. &nbsp; Tail recursion
       </span>
     </span>
   </a>
@@ -944,10 +944,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-recursion-tree" class="md-nav__link">
+  <a href="#3-recursion-trees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Recursion Tree
+        3. &nbsp; Recursion trees
       </span>
     </span>
   </a>
@@ -960,10 +960,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#223-comparison-of-the-two" class="md-nav__link">
+  <a href="#223-comparing-iteration-and-recursion" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.2.3 &nbsp; Comparison of the Two
+        2.2.3 &nbsp; Comparing iteration and recursion
       </span>
     </span>
   </a>
@@ -993,7 +993,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -1020,7 +1020,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -1047,7 +1047,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -1074,7 +1074,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -1140,7 +1140,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1161,7 +1161,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1182,7 +1182,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1209,7 +1209,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1236,7 +1236,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1263,7 +1263,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1290,7 +1290,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1317,7 +1317,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1383,7 +1383,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1404,7 +1404,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1425,7 +1425,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1452,7 +1452,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1479,7 +1479,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1624,7 +1624,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1645,7 +1645,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1666,7 +1666,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1693,7 +1693,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1838,7 +1838,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1859,7 +1859,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1880,7 +1880,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1907,7 +1907,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1934,7 +1934,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1961,7 +1961,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1988,7 +1988,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -2056,7 +2056,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -2077,7 +2077,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2098,7 +2098,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -2125,7 +2125,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2152,7 +2152,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2179,7 +2179,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2324,7 +2324,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2345,7 +2345,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2366,7 +2366,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2393,7 +2393,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2538,7 +2538,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2559,7 +2559,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2580,7 +2580,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2607,7 +2607,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2634,7 +2634,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2661,7 +2661,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2688,7 +2688,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2756,7 +2756,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2777,7 +2777,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2798,7 +2798,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2825,7 +2825,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2852,7 +2852,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2879,7 +2879,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2906,7 +2906,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2933,7 +2933,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2960,7 +2960,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -3038,7 +3038,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3059,7 +3059,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3080,7 +3080,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3107,7 +3107,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3134,7 +3134,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3161,7 +3161,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3188,7 +3188,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3215,7 +3215,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3242,7 +3242,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3269,7 +3269,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3296,7 +3296,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3323,7 +3323,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3350,7 +3350,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3377,7 +3377,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3443,7 +3443,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3464,7 +3464,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3485,7 +3485,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3512,7 +3512,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3539,7 +3539,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3566,7 +3566,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3593,7 +3593,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3620,7 +3620,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3686,7 +3686,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3707,7 +3707,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3728,7 +3728,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3755,7 +3755,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3782,7 +3782,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3809,7 +3809,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3836,7 +3836,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3933,7 +3933,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3954,7 +3954,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3975,7 +3975,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -4029,7 +4029,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4230,7 +4230,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4272,7 +4272,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4647,6 +4647,78 @@
 
 
 
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@@ -4700,10 +4772,10 @@
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-for-loop" class="md-nav__link">
+  <a href="#1-for-loops" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; For Loop
+        1. &nbsp; For loops
       </span>
     </span>
   </a>
@@ -4711,10 +4783,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-while-loop" class="md-nav__link">
+  <a href="#2-while-loops" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; While Loop
+        2. &nbsp; While loops
       </span>
     </span>
   </a>
@@ -4725,7 +4797,7 @@
   <a href="#3-nested-loops" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Nested Loops
+        3. &nbsp; Nested loops
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     </span>
   </a>
@@ -4750,10 +4822,10 @@
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-call-stack" class="md-nav__link">
+  <a href="#1-the-call-stack" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Call Stack
+        1. &nbsp; The call stack
       </span>
     </span>
   </a>
@@ -4764,7 +4836,7 @@
   <a href="#2-tail-recursion" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Tail Recursion
+        2. &nbsp; Tail recursion
       </span>
     </span>
   </a>
@@ -4772,10 +4844,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-recursion-tree" class="md-nav__link">
+  <a href="#3-recursion-trees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Recursion Tree
+        3. &nbsp; Recursion trees
       </span>
     </span>
   </a>
@@ -4788,10 +4860,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#223-comparison-of-the-two" class="md-nav__link">
+  <a href="#223-comparing-iteration-and-recursion" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.2.3 &nbsp; Comparison of the Two
+        2.2.3 &nbsp; Comparing iteration and recursion
       </span>
     </span>
   </a>
@@ -4835,20 +4907,20 @@
 
 
 <!-- Page content -->
-<h1 id="22-iteration-and-recursion">2.2 &nbsp; Iteration and Recursion<a class="headerlink" href="#22-iteration-and-recursion" title="Permanent link">&para;</a></h1>
-<p>In algorithms, repeatedly executing a task is very common and closely related to complexity analysis. Therefore, before introducing time complexity and space complexity, let's first understand how to implement repeated task execution in programs, namely the two basic program control structures: iteration and recursion.</p>
+<h1 id="22-iteration-and-recursion">2.2 &nbsp; Iteration and recursion<a class="headerlink" href="#22-iteration-and-recursion" title="Permanent link">&para;</a></h1>
+<p>Algorithms often repeat tasks, and repetition is closely tied to complexity analysis. Before introducing time and space complexity, let's examine two basic ways to repeat work in a program: iteration and recursion.</p>
 <h2 id="221-iteration">2.2.1 &nbsp; Iteration<a class="headerlink" href="#221-iteration" title="Permanent link">&para;</a></h2>
-<p><u>Iteration</u> is a control structure for repeatedly executing a task. In iteration, a program repeatedly executes a segment of code under certain conditions until those conditions are no longer satisfied.</p>
-<h3 id="1-for-loop">1. &nbsp; For Loop<a class="headerlink" href="#1-for-loop" title="Permanent link">&para;</a></h3>
-<p>The <code>for</code> loop is one of the most common forms of iteration, <strong>suitable for use when the number of iterations is known in advance</strong>.</p>
-<p>The following function implements the summation <span class="arithmatex">\(1 + 2 + \dots + n\)</span> using a <code>for</code> loop, with the result stored in the variable <code>res</code>. Note that in Python, <code>range(a, b)</code> corresponds to a "left-closed, right-open" interval, with the traversal range being <span class="arithmatex">\(a, a + 1, \dots, b-1\)</span>:</p>
+<p><u>Iteration</u> is a control structure for repeating a task. A program repeatedly executes a block of code while a condition holds, stopping when it no longer does.</p>
+<h3 id="1-for-loops">1. &nbsp; For loops<a class="headerlink" href="#1-for-loops" title="Permanent link">&para;</a></h3>
+<p>A <code>for</code> loop is one of the most common forms of iteration, <strong>especially when the number of iterations is known in advance</strong>.</p>
+<p>The following function uses a <code>for</code> loop to compute <span class="arithmatex">\(1 + 2 + \dots + n\)</span>, storing the result in <code>res</code>. In Python, <code>range(a, b)</code> includes the left endpoint but excludes the right, so it visits <span class="arithmatex">\(a, a + 1, \dots, b-1\)</span>:</p>
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 <div class="highlight"><span class="filename">iteration.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">for_loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""for loop"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="n">res</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="c1"># Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="c1"># Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="n">res</span> <span class="o">+=</span> <span class="n">i</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="k">return</span> <span class="n">res</span>
@@ -4858,7 +4930,7 @@
 <div class="highlight"><span class="filename">iteration.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* for loop */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">forLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="p">}</span>
@@ -4870,7 +4942,7 @@
 <div class="highlight"><span class="filename">iteration.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* for loop */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">forLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="p">}</span>
@@ -4882,7 +4954,7 @@
 <div class="highlight"><span class="filename">iteration.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* for loop */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">ForLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="p">}</span>
@@ -4894,7 +4966,7 @@
 <div class="highlight"><span class="filename">iteration.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* for loop */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">forLoop</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">i</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="p">}</span>
@@ -4906,7 +4978,7 @@
 <div class="highlight"><span class="filename">iteration.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* for loop */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">forLoop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="p">}</span>
@@ -4918,7 +4990,7 @@
 <div class="highlight"><span class="filename">iteration.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* for loop */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">forLoop</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="p">}</span>
@@ -4930,7 +5002,7 @@
 <div class="highlight"><span class="filename">iteration.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* for loop */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">forLoop</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="p">}</span>
@@ -4942,7 +5014,7 @@
 <div class="highlight"><span class="filename">iteration.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* for loop */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">forLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
-<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="p">}</span>
@@ -4954,7 +5026,7 @@
 <div class="highlight"><span class="filename">iteration.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* for loop */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">for_loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="p">}</span>
@@ -4966,7 +5038,7 @@
 <div class="highlight"><span class="filename">iteration.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* for loop */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">forLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="p">}</span>
@@ -4978,7 +5050,7 @@
 <div class="highlight"><span class="filename">iteration.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* for loop */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">forLoop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="p">}</span>
@@ -4991,7 +5063,7 @@
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">for_loop</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">n</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="k">end</span>
@@ -5002,25 +5074,30 @@
 </div>
 </div>
 </div>
-<p>Figure 2-1 shows the flowchart of this summation function.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 423px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20for_loop%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22for%20loop%22%22%22%0A%20%20%20%20res%20%3D%200%0A%20%20%20%20%23%20Sum%201,%202,%20...,%20n-1,%20n%20in%20a%20loop%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20res%20%2B%3D%20i%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20for_loop%28n%29%0A%20%20%20%20print%28f%22%5CnSum%20from%20the%20for%20loop%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20for_loop%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22for%20loop%22%22%22%0A%20%20%20%20res%20%3D%200%0A%20%20%20%20%23%20Sum%201,%202,%20...,%20n-1,%20n%20in%20a%20loop%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20res%20%2B%3D%20i%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20for_loop%28n%29%0A%20%20%20%20print%28f%22%5CnSum%20from%20the%20for%20loop%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 2-1 is a flowchart of this summation function.</p>
 <p><img alt="Flowchart of the summation function" class="animation-figure" src="../iteration_and_recursion.assets/iteration.png" /></p>
 <p align="center"> Figure 2-1 &nbsp; Flowchart of the summation function </p>
 
-<p>The number of operations in this summation function is proportional to the input data size <span class="arithmatex">\(n\)</span>, or has a "linear relationship". In fact, <strong>time complexity describes precisely this "linear relationship"</strong>. Related content will be introduced in detail in the next section.</p>
-<h3 id="2-while-loop">2. &nbsp; While Loop<a class="headerlink" href="#2-while-loop" title="Permanent link">&para;</a></h3>
-<p>Similar to the <code>for</code> loop, the <code>while</code> loop is also a method for implementing iteration. In a <code>while</code> loop, the program first checks the condition in each round; if the condition is true, it continues execution, otherwise it ends the loop.</p>
-<p>Below we use a <code>while</code> loop to implement the summation <span class="arithmatex">\(1 + 2 + \dots + n\)</span>:</p>
+<p>Its operation count is proportional to the input size <span class="arithmatex">\(n\)</span>: the relationship is linear. <strong>Time complexity describes this relationship</strong>, as the next section will explain.</p>
+<h3 id="2-while-loops">2. &nbsp; While loops<a class="headerlink" href="#2-while-loops" title="Permanent link">&para;</a></h3>
+<p>Like <code>for</code>, a <code>while</code> loop implements iteration. Before each iteration, it checks a condition: if true, execution continues; otherwise, the loop ends.</p>
+<p>Here is a <code>while</code> loop computing <span class="arithmatex">\(1 + 2 + \dots + n\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
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 <div class="highlight"><span class="filename">iteration.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">while_loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""while loop"""</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="n">res</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="n">i</span> <span class="o">=</span> <span class="mi">1</span>  <span class="c1"># Initialize condition variable</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="n">i</span> <span class="o">=</span> <span class="mi">1</span>  <span class="c1"># Initialize the loop control variable</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="n">n</span><span class="p">:</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="n">res</span> <span class="o">+=</span> <span class="n">i</span>
-<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>  <span class="c1"># Update condition variable</span>
+<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>  <span class="c1"># Update the loop control variable</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>    <span class="k">return</span> <span class="n">res</span>
 </code></pre></div>
 </div>
@@ -5028,11 +5105,11 @@
 <div class="highlight"><span class="filename">iteration.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* while loop */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">whileLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">        </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">        </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="p">}</span>
@@ -5042,11 +5119,11 @@
 <div class="highlight"><span class="filename">iteration.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* while loop */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">whileLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">        </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">        </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="p">}</span>
@@ -5056,11 +5133,11 @@
 <div class="highlight"><span class="filename">iteration.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* while loop */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">WhileLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="p">}</span>
@@ -5070,12 +5147,12 @@
 <div class="highlight"><span class="filename">iteration.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* while loop */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">whileLoop</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="c1">// Initialize condition variable</span>
+<a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="c1">// Initialize the loop control variable</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">i</span>
-<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">        </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">        </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">        </span><span class="nx">i</span><span class="o">++</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span>
@@ -5086,11 +5163,11 @@
 <div class="highlight"><span class="filename">iteration.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* while loop */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">whileLoop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span>
-<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="p">}</span>
@@ -5100,11 +5177,11 @@
 <div class="highlight"><span class="filename">iteration.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* while loop */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">whileLoop</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
-<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">        </span><span class="nx">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">        </span><span class="nx">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="p">}</span>
@@ -5114,11 +5191,11 @@
 <div class="highlight"><span class="filename">iteration.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* while loop */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">whileLoop</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
-<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">        </span><span class="nx">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">        </span><span class="nx">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="p">}</span>
@@ -5128,11 +5205,11 @@
 <div class="highlight"><span class="filename">iteration.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* while loop */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">whileLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
-<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">    </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">    </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="p">}</span>
@@ -5142,12 +5219,12 @@
 <div class="highlight"><span class="filename">iteration.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* while loop */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">while_loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
+<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a>
-<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">    </span><span class="n">res</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="p">}</span>
@@ -5157,11 +5234,11 @@
 <div class="highlight"><span class="filename">iteration.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* while loop */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">whileLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">        </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">        </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="p">}</span>
@@ -5171,11 +5248,11 @@
 <div class="highlight"><span class="filename">iteration.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* while loop */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">whileLoop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="c1">// Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span>
-<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">        </span><span class="n">i</span><span class="o">++</span><span class="w"> </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">        </span><span class="n">i</span><span class="o">++</span><span class="w"> </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="p">}</span>
@@ -5185,12 +5262,12 @@
 <div class="highlight"><span class="filename">iteration.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### while loop ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">while_loop</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Initialize condition variable</span>
+<a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Initialize the loop control variable</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a>
-<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="c1"># Sum 1, 2, ..., n-1, n</span>
+<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="c1"># Sum 1, 2, ..., n-1, n in a loop</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span>
-<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Update condition variable</span>
+<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Update the loop control variable</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">  </span><span class="n">res</span>
@@ -5199,19 +5276,24 @@
 </div>
 </div>
 </div>
-<p><strong>The <code>while</code> loop has greater flexibility than the <code>for</code> loop</strong>. In a <code>while</code> loop, we can freely design the initialization and update steps of the condition variable.</p>
-<p>For example, in the following code, the condition variable <span class="arithmatex">\(i\)</span> is updated twice per round, which is not convenient to implement using a <code>for</code> loop:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20while_loop%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22while%20loop%22%22%22%0A%20%20%20%20res%20%3D%200%0A%20%20%20%20i%20%3D%201%20%20%23%20Initialize%20the%20loop%20control%20variable%0A%20%20%20%20%23%20Sum%201,%202,%20...,%20n-1,%20n%20in%20a%20loop%0A%20%20%20%20while%20i%20%3C%3D%20n%3A%0A%20%20%20%20%20%20%20%20res%20%2B%3D%20i%0A%20%20%20%20%20%20%20%20i%20%2B%3D%201%20%20%23%20Update%20the%20loop%20control%20variable%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20while_loop%28n%29%0A%20%20%20%20print%28f%22%5CnSum%20from%20the%20while%20loop%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20while_loop%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22while%20loop%22%22%22%0A%20%20%20%20res%20%3D%200%0A%20%20%20%20i%20%3D%201%20%20%23%20Initialize%20the%20loop%20control%20variable%0A%20%20%20%20%23%20Sum%201,%202,%20...,%20n-1,%20n%20in%20a%20loop%0A%20%20%20%20while%20i%20%3C%3D%20n%3A%0A%20%20%20%20%20%20%20%20res%20%2B%3D%20i%0A%20%20%20%20%20%20%20%20i%20%2B%3D%201%20%20%23%20Update%20the%20loop%20control%20variable%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20while_loop%28n%29%0A%20%20%20%20print%28f%22%5CnSum%20from%20the%20while%20loop%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p><strong>A <code>while</code> loop offers more flexibility than a <code>for</code> loop</strong> in how its condition variables are initialized and updated.</p>
+<p>For example, the following code updates the condition variable <span class="arithmatex">\(i\)</span> twice per iteration, which is less convenient to express with a <code>for</code> loop:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
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 <div class="tabbed-block">
 <div class="highlight"><span class="filename">iteration.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">while_loop_ii</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""while loop (two updates)"""</span>
 <a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="n">res</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="n">i</span> <span class="o">=</span> <span class="mi">1</span>  <span class="c1"># Initialize condition variable</span>
-<a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>    <span class="c1"># Sum 1, 4, 10, ...</span>
+<a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="n">i</span> <span class="o">=</span> <span class="mi">1</span>  <span class="c1"># Initialize the loop control variable</span>
+<a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>    <span class="c1"># Sum 1, 4, 10, ... in a loop</span>
 <a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="n">n</span><span class="p">:</span>
 <a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>        <span class="n">res</span> <span class="o">+=</span> <span class="n">i</span>
-<a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>        <span class="c1"># Update condition variable</span>
+<a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>        <span class="c1"># Update the loop control variable</span>
 <a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>        <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
 <a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>        <span class="n">i</span> <span class="o">*=</span> <span class="mi">2</span>
 <a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a>    <span class="k">return</span> <span class="n">res</span>
@@ -5221,11 +5303,11 @@
 <div class="highlight"><span class="filename">iteration.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* while loop (two updates) */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">whileLoopII</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ...</span>
+<a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ... in a loop</span>
 <a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">        </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">        </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">        </span><span class="n">i</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="w">    </span><span class="p">}</span>
@@ -5237,11 +5319,11 @@
 <div class="highlight"><span class="filename">iteration.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* while loop (two updates) */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">whileLoopII</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ...</span>
+<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ... in a loop</span>
 <a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">        </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">        </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="w">        </span><span class="n">i</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="w">    </span><span class="p">}</span>
@@ -5253,11 +5335,11 @@
 <div class="highlight"><span class="filename">iteration.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* while loop (two updates) */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">WhileLoopII</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ...</span>
+<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ... in a loop</span>
 <a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">        </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">        </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span>
 <a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="w">    </span><span class="p">}</span>
@@ -5269,12 +5351,12 @@
 <div class="highlight"><span class="filename">iteration.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* while loop (two updates) */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">whileLoopII</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="c1">// Initialize condition variable</span>
+<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="c1">// Initialize the loop control variable</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">    </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ...</span>
+<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ... in a loop</span>
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">i</span>
-<a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">        </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">        </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">        </span><span class="nx">i</span><span class="o">++</span>
 <a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">        </span><span class="nx">i</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="w">    </span><span class="p">}</span>
@@ -5286,11 +5368,11 @@
 <div class="highlight"><span class="filename">iteration.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* while loop (two updates) */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">whileLoopII</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ...</span>
+<a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ... in a loop</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span>
-<a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">        </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">        </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">    </span><span class="p">}</span>
@@ -5302,11 +5384,11 @@
 <div class="highlight"><span class="filename">iteration.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* while loop (two updates) */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">whileLoopII</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ...</span>
+<a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ... in a loop</span>
 <a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
-<a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">        </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">        </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w">        </span><span class="nx">i</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a><span class="w">        </span><span class="nx">i</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
 <a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="w">    </span><span class="p">}</span>
@@ -5318,11 +5400,11 @@
 <div class="highlight"><span class="filename">iteration.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* while loop (two updates) */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">whileLoopII</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ...</span>
+<a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ... in a loop</span>
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
-<a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">        </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">        </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">        </span><span class="nx">i</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="w">        </span><span class="nx">i</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
 <a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">    </span><span class="p">}</span>
@@ -5334,11 +5416,11 @@
 <div class="highlight"><span class="filename">iteration.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* while loop (two updates) */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">whileLoopII</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
-<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="c1">// Sum 1, 4, 10, ...</span>
+<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="c1">// Sum 1, 4, 10, ... in a loop</span>
 <a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">    </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">    </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w">    </span><span class="n">i</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="m">2</span><span class="p">;</span>
 <a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w">  </span><span class="p">}</span>
@@ -5350,12 +5432,12 @@
 <div class="highlight"><span class="filename">iteration.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* while loop (two updates) */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">while_loop_ii</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
+<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
 <a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a>
-<a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ...</span>
+<a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ... in a loop</span>
 <a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">        </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">        </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="w">    </span><span class="p">}</span>
@@ -5367,11 +5449,11 @@
 <div class="highlight"><span class="filename">iteration.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* while loop (two updates) */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">whileLoopII</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ...</span>
+<a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ... in a loop</span>
 <a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">        </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">        </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="w">        </span><span class="n">i</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="w">    </span><span class="p">}</span>
@@ -5383,11 +5465,11 @@
 <div class="highlight"><span class="filename">iteration.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* while loop (two updates) */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">whileLoopII</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Initialize condition variable</span>
-<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ...</span>
+<a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Initialize the loop control variable</span>
+<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="c1">// Sum 1, 4, 10, ... in a loop</span>
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span>
-<a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">        </span><span class="c1">// Update condition variable</span>
+<a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">        </span><span class="c1">// Update the loop control variable</span>
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">        </span><span class="n">i</span><span class="o">++</span>
 <a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="m">2</span>
 <a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="w">    </span><span class="p">}</span>
@@ -5396,15 +5478,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">iteration.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### while loop (two updates) ###</span>
+<div class="highlight"><span class="filename">iteration.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### while loop (two updates)###</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">while_loop_ii</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Initialize condition variable</span>
+<a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Initialize the loop control variable</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a>
-<a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="c1"># Sum 1, 4, 10, ...</span>
+<a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="c1"># Sum 1, 4, 10, ... in a loop</span>
 <a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span>
 <a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">i</span>
-<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">    </span><span class="c1"># Update condition variable</span>
+<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">    </span><span class="c1"># Update the loop control variable</span>
 <a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="w">  </span><span class="k">end</span>
@@ -5415,30 +5497,35 @@
 </div>
 </div>
 </div>
-<p>Overall, <strong><code>for</code> loops have more compact code, while <code>while</code> loops are more flexible</strong>; both can implement iterative structures. The choice of which to use should be determined based on the requirements of the specific problem.</p>
-<h3 id="3-nested-loops">3. &nbsp; Nested Loops<a class="headerlink" href="#3-nested-loops" title="Permanent link">&para;</a></h3>
-<p>We can nest one loop structure inside another. Below is an example using <code>for</code> loops:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 495px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20while_loop_ii%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22while%20loop%20%28two%20updates%29%22%22%22%0A%20%20%20%20res%20%3D%200%0A%20%20%20%20i%20%3D%201%20%20%23%20Initialize%20the%20loop%20control%20variable%0A%20%20%20%20%23%20Sum%201,%204,%2010,%20...%20in%20a%20loop%0A%20%20%20%20while%20i%20%3C%3D%20n%3A%0A%20%20%20%20%20%20%20%20res%20%2B%3D%20i%0A%20%20%20%20%20%20%20%20%23%20Update%20the%20loop%20control%20variable%0A%20%20%20%20%20%20%20%20i%20%2B%3D%201%0A%20%20%20%20%20%20%20%20i%20%2A%3D%202%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20while_loop_ii%28n%29%0A%20%20%20%20print%28f%22%5CnSum%20from%20the%20while%20loop%20%28two%20updates%29%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20while_loop_ii%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22while%20loop%20%28two%20updates%29%22%22%22%0A%20%20%20%20res%20%3D%200%0A%20%20%20%20i%20%3D%201%20%20%23%20Initialize%20the%20loop%20control%20variable%0A%20%20%20%20%23%20Sum%201,%204,%2010,%20...%20in%20a%20loop%0A%20%20%20%20while%20i%20%3C%3D%20n%3A%0A%20%20%20%20%20%20%20%20res%20%2B%3D%20i%0A%20%20%20%20%20%20%20%20%23%20Update%20the%20loop%20control%20variable%0A%20%20%20%20%20%20%20%20i%20%2B%3D%201%0A%20%20%20%20%20%20%20%20i%20%2A%3D%202%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20while_loop_ii%28n%29%0A%20%20%20%20print%28f%22%5CnSum%20from%20the%20while%20loop%20%28two%20updates%29%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>In general, <strong><code>for</code> loops are more compact, while <code>while</code> loops are more flexible</strong>. Both implement iteration; choose according to the problem's needs.</p>
+<h3 id="3-nested-loops">3. &nbsp; Nested loops<a class="headerlink" href="#3-nested-loops" title="Permanent link">&para;</a></h3>
+<p>We can place one loop inside another, as in the following <code>for</code>-loop example:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:13"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">iteration.py</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">nested_for_loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">str</span><span class="p">:</span>
-<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Nested for loop"""</span>
+<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Nested for loops"""</span>
 <a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="n">res</span> <span class="o">=</span> <span class="s2">""</span>
-<a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="c1"># Loop i = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="c1"># Loop over i = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-<a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a>        <span class="c1"># Loop j = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a>        <span class="c1"># Loop over j = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a>            <span class="n">res</span> <span class="o">+=</span> <span class="sa">f</span><span class="s2">"(</span><span class="si">{</span><span class="n">i</span><span class="si">}</span><span class="s2">, </span><span class="si">{</span><span class="n">j</span><span class="si">}</span><span class="s2">), "</span>
 <a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a>    <span class="k">return</span> <span class="n">res</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">iteration.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Nested for loop */</span>
+<div class="highlight"><span class="filename">iteration.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Nested for loops */</span>
 <a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="n">string</span><span class="w"> </span><span class="nf">nestedForLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="n">ostringstream</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
-<a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="c1">// Loop i = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="c1">// Loop over i = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w">        </span><span class="c1">// Loop j = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w">        </span><span class="c1">// Loop over j = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="w">            </span><span class="n">res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">"("</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">", "</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">"), "</span><span class="p">;</span>
 <a id="__codelineno-40-9" name="__codelineno-40-9" href="#__codelineno-40-9"></a><span class="w">        </span><span class="p">}</span>
@@ -5448,12 +5535,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">iteration.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Nested for loop */</span>
+<div class="highlight"><span class="filename">iteration.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Nested for loops */</span>
 <a id="__codelineno-41-2" name="__codelineno-41-2" href="#__codelineno-41-2"></a><span class="n">String</span><span class="w"> </span><span class="nf">nestedForLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="n">StringBuilder</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">StringBuilder</span><span class="p">();</span>
-<a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">    </span><span class="c1">// Loop i = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">    </span><span class="c1">// Loop over i = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a><span class="w">        </span><span class="c1">// Loop j = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a><span class="w">        </span><span class="c1">// Loop over j = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-8" name="__codelineno-41-8" href="#__codelineno-41-8"></a><span class="w">            </span><span class="n">res</span><span class="p">.</span><span class="na">append</span><span class="p">(</span><span class="s">"("</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="s">", "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="s">"), "</span><span class="p">);</span>
 <a id="__codelineno-41-9" name="__codelineno-41-9" href="#__codelineno-41-9"></a><span class="w">        </span><span class="p">}</span>
@@ -5463,12 +5550,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">iteration.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Nested for loop */</span>
+<div class="highlight"><span class="filename">iteration.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Nested for loops */</span>
 <a id="__codelineno-42-2" name="__codelineno-42-2" href="#__codelineno-42-2"></a><span class="kt">string</span><span class="w"> </span><span class="nf">NestedForLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="n">StringBuilder</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">();</span>
-<a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">    </span><span class="c1">// Loop i = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">    </span><span class="c1">// Loop over i = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a><span class="w">        </span><span class="c1">// Loop j = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a><span class="w">        </span><span class="c1">// Loop over j = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-8" name="__codelineno-42-8" href="#__codelineno-42-8"></a><span class="w">            </span><span class="n">res</span><span class="p">.</span><span class="n">Append</span><span class="p">(</span><span class="s">$"({i}, {j}), "</span><span class="p">);</span>
 <a id="__codelineno-42-9" name="__codelineno-42-9" href="#__codelineno-42-9"></a><span class="w">        </span><span class="p">}</span>
@@ -5478,13 +5565,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">iteration.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Nested for loop */</span>
+<div class="highlight"><span class="filename">iteration.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Nested for loops */</span>
 <a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">nestedForLoop</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="s">""</span>
-<a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="c1">// Loop i = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="c1">// Loop over i = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="w">            </span><span class="c1">// Loop j = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="w">            </span><span class="c1">// Loop over j = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a><span class="w">            </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Sprintf</span><span class="p">(</span><span class="s">"(%d, %d), "</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span>
 <a id="__codelineno-43-9" name="__codelineno-43-9" href="#__codelineno-43-9"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></a><span class="w">    </span><span class="p">}</span>
@@ -5493,12 +5580,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">iteration.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Nested for loop */</span>
+<div class="highlight"><span class="filename">iteration.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Nested for loops */</span>
 <a id="__codelineno-44-2" name="__codelineno-44-2" href="#__codelineno-44-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">nestedForLoop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">String</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">""</span>
-<a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a><span class="w">    </span><span class="c1">// Loop i = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a><span class="w">    </span><span class="c1">// Loop over i = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></a><span class="w">        </span><span class="c1">// Loop j = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></a><span class="w">        </span><span class="c1">// Loop over j = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-44-7" name="__codelineno-44-7" href="#__codelineno-44-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="w">            </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="s">"(</span><span class="si">\(</span><span class="n">i</span><span class="si">)</span><span class="s">, </span><span class="si">\(</span><span class="n">j</span><span class="si">)</span><span class="s">), "</span><span class="p">)</span>
 <a id="__codelineno-44-9" name="__codelineno-44-9" href="#__codelineno-44-9"></a><span class="w">        </span><span class="p">}</span>
@@ -5508,12 +5595,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">iteration.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Nested for loop */</span>
+<div class="highlight"><span class="filename">iteration.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Nested for loops */</span>
 <a id="__codelineno-45-2" name="__codelineno-45-2" href="#__codelineno-45-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">nestedForLoop</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s1">''</span><span class="p">;</span>
-<a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="c1">// Loop i = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="c1">// Loop over i = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-6" name="__codelineno-45-6" href="#__codelineno-45-6"></a><span class="w">        </span><span class="c1">// Loop j = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-45-6" name="__codelineno-45-6" href="#__codelineno-45-6"></a><span class="w">        </span><span class="c1">// Loop over j = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-45-7" name="__codelineno-45-7" href="#__codelineno-45-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-8" name="__codelineno-45-8" href="#__codelineno-45-8"></a><span class="w">            </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="sb">`(</span><span class="si">${</span><span class="nx">i</span><span class="si">}</span><span class="sb">, </span><span class="si">${</span><span class="nx">j</span><span class="si">}</span><span class="sb">), `</span><span class="p">;</span>
 <a id="__codelineno-45-9" name="__codelineno-45-9" href="#__codelineno-45-9"></a><span class="w">        </span><span class="p">}</span>
@@ -5523,12 +5610,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">iteration.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Nested for loop */</span>
+<div class="highlight"><span class="filename">iteration.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Nested for loops */</span>
 <a id="__codelineno-46-2" name="__codelineno-46-2" href="#__codelineno-46-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">nestedForLoop</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s1">''</span><span class="p">;</span>
-<a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">    </span><span class="c1">// Loop i = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">    </span><span class="c1">// Loop over i = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="w">        </span><span class="c1">// Loop j = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="w">        </span><span class="c1">// Loop over j = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a><span class="w">            </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="sb">`(</span><span class="si">${</span><span class="nx">i</span><span class="si">}</span><span class="sb">, </span><span class="si">${</span><span class="nx">j</span><span class="si">}</span><span class="sb">), `</span><span class="p">;</span>
 <a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="w">        </span><span class="p">}</span>
@@ -5538,12 +5625,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">iteration.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Nested for loop */</span>
+<div class="highlight"><span class="filename">iteration.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Nested for loops */</span>
 <a id="__codelineno-47-2" name="__codelineno-47-2" href="#__codelineno-47-2"></a><span class="kt">String</span><span class="w"> </span><span class="n">nestedForLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="kt">String</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">""</span><span class="p">;</span>
-<a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="c1">// Loop i = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="c1">// Loop over i = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a><span class="w">    </span><span class="c1">// Loop j = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a><span class="w">    </span><span class="c1">// Loop over j = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-47-7" name="__codelineno-47-7" href="#__codelineno-47-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a><span class="w">      </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="s2">"(</span><span class="si">$</span><span class="n">i</span><span class="s2">, </span><span class="si">$</span><span class="n">j</span><span class="s2">), "</span><span class="p">;</span>
 <a id="__codelineno-47-9" name="__codelineno-47-9" href="#__codelineno-47-9"></a><span class="w">    </span><span class="p">}</span>
@@ -5553,12 +5640,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">iteration.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Nested for loop */</span>
+<div class="highlight"><span class="filename">iteration.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Nested for loops */</span>
 <a id="__codelineno-48-2" name="__codelineno-48-2" href="#__codelineno-48-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">nested_for_loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">String</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[];</span>
-<a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w">    </span><span class="c1">// Loop i = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w">    </span><span class="c1">// Loop over i = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a><span class="w">        </span><span class="c1">// Loop j = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a><span class="w">        </span><span class="c1">// Loop over j = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-8" name="__codelineno-48-8" href="#__codelineno-48-8"></a><span class="w">            </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="fm">format!</span><span class="p">(</span><span class="s">"({}, {}), "</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">));</span>
 <a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a><span class="w">        </span><span class="p">}</span>
@@ -5568,14 +5655,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">iteration.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Nested for loop */</span>
+<div class="highlight"><span class="filename">iteration.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Nested for loops */</span>
 <a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="nf">nestedForLoop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="c1">// n * n is the number of points, "(i, j), " string max length is 6+10*2, plus extra space for null character \0</span>
+<a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="c1">// There are n * n points; each "(i, j), " takes at most 6+10*2 characters, plus space for the final null character \0</span>
 <a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">26</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">    </span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">char</span><span class="p">));</span>
-<a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w">    </span><span class="c1">// Loop i = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w">    </span><span class="c1">// Loop over i = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w">        </span><span class="c1">// Loop j = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w">        </span><span class="c1">// Loop over j = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-49-9" name="__codelineno-49-9" href="#__codelineno-49-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-10" name="__codelineno-49-10" href="#__codelineno-49-10"></a><span class="w">            </span><span class="kt">char</span><span class="w"> </span><span class="n">tmp</span><span class="p">[</span><span class="mi">26</span><span class="p">];</span>
 <a id="__codelineno-49-11" name="__codelineno-49-11" href="#__codelineno-49-11"></a><span class="w">            </span><span class="n">snprintf</span><span class="p">(</span><span class="n">tmp</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="n">tmp</span><span class="p">),</span><span class="w"> </span><span class="s">"(%d, %d), "</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
@@ -5587,12 +5674,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">iteration.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Nested for loop */</span>
+<div class="highlight"><span class="filename">iteration.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Nested for loops */</span>
 <a id="__codelineno-50-2" name="__codelineno-50-2" href="#__codelineno-50-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">nestedForLoop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">String</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">StringBuilder</span><span class="p">()</span>
-<a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="c1">// Loop i = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="c1">// Loop over i = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">        </span><span class="c1">// Loop j = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">        </span><span class="c1">// Loop over j = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-8" name="__codelineno-50-8" href="#__codelineno-50-8"></a><span class="w">            </span><span class="n">res</span><span class="p">.</span><span class="na">append</span><span class="p">(</span><span class="s">" (</span><span class="si">$</span><span class="n">i</span><span class="s">, </span><span class="si">$</span><span class="n">j</span><span class="s">), "</span><span class="p">)</span>
 <a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="w">        </span><span class="p">}</span>
@@ -5602,13 +5689,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">iteration.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Nested for loop ###</span>
+<div class="highlight"><span class="filename">iteration.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Nested for loops ###</span>
 <a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">nested_for_loop</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">""</span>
 <a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a>
-<a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">  </span><span class="c1"># Loop i = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">  </span><span class="c1"># Loop over i = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">n</span>
-<a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="w">    </span><span class="c1"># Loop j = 1, 2, ..., n-1, n</span>
+<a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="w">    </span><span class="c1"># Loop over j = 1, 2, ..., n-1, n</span>
 <a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">n</span>
 <a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w">      </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="s2">"(</span><span class="si">#{</span><span class="n">i</span><span class="si">}</span><span class="s2">, </span><span class="si">#{</span><span class="n">j</span><span class="si">}</span><span class="s2">), "</span>
 <a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a><span class="w">    </span><span class="k">end</span>
@@ -5620,48 +5707,53 @@
 </div>
 </div>
 </div>
-<p>Figure 2-2 shows the flowchart of this nested loop.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20nested_for_loop%28n%3A%20int%29%20-%3E%20str%3A%0A%20%20%20%20%22%22%22Nested%20for%20loops%22%22%22%0A%20%20%20%20res%20%3D%20%22%22%0A%20%20%20%20%23%20Loop%20over%20i%20%3D%201,%202,%20...,%20n-1,%20n%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%23%20Loop%20over%20j%20%3D%201,%202,%20...,%20n-1,%20n%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20res%20%2B%3D%20f%22%28%7Bi%7D,%20%7Bj%7D%29,%20%22%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20nested_for_loop%28n%29%0A%20%20%20%20print%28f%22%5CnTraversal%20from%20the%20nested%20for%20loops%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20nested_for_loop%28n%3A%20int%29%20-%3E%20str%3A%0A%20%20%20%20%22%22%22Nested%20for%20loops%22%22%22%0A%20%20%20%20res%20%3D%20%22%22%0A%20%20%20%20%23%20Loop%20over%20i%20%3D%201,%202,%20...,%20n-1,%20n%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%23%20Loop%20over%20j%20%3D%201,%202,%20...,%20n-1,%20n%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20res%20%2B%3D%20f%22%28%7Bi%7D,%20%7Bj%7D%29,%20%22%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20nested_for_loop%28n%29%0A%20%20%20%20print%28f%22%5CnTraversal%20from%20the%20nested%20for%20loops%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 2-2 shows the nested loop's flowchart.</p>
 <p><img alt="Flowchart of nested loops" class="animation-figure" src="../iteration_and_recursion.assets/nested_iteration.png" /></p>
 <p align="center"> Figure 2-2 &nbsp; Flowchart of nested loops </p>
 
-<p>In this case, the number of operations of the function is proportional to <span class="arithmatex">\(n^2\)</span>, or the algorithm's running time has a "quadratic relationship" with the input data size <span class="arithmatex">\(n\)</span>.</p>
-<p>We can continue adding nested loops, where each additional level of nesting can be viewed as an increase in dimensionality, raising the time complexity to a "cubic relationship", a "quartic relationship", and so on.</p>
+<p>Here, the operation count is proportional to <span class="arithmatex">\(n^2\)</span>: runtime has a quadratic relationship with input size <span class="arithmatex">\(n\)</span>.</p>
+<p>Adding further nesting increases the dimension each time, producing cubic, fourth-power, and higher-order time complexity.</p>
 <h2 id="222-recursion">2.2.2 &nbsp; Recursion<a class="headerlink" href="#222-recursion" title="Permanent link">&para;</a></h2>
-<p><u>Recursion</u> is an algorithmic strategy that solves problems by having a function call itself. It mainly consists of two phases.</p>
+<p><u>Recursion</u> is an algorithmic strategy in which a function calls itself to solve a problem. It has two main phases.</p>
 <ol>
-<li><strong>Descend</strong>: The program continuously calls itself deeper, usually passing in smaller or more simplified parameters, until reaching a "termination condition".</li>
-<li><strong>Ascend</strong>: After triggering the "termination condition", the program returns layer by layer from the deepest recursive function, aggregating the result of each layer.</li>
+<li><strong>Descent</strong>: The function calls itself more deeply, usually with smaller or simpler arguments, until a base case is reached.</li>
+<li><strong>Return</strong>: Once the base case is reached, calls return from the deepest level upward, combining results along the way.</li>
 </ol>
-<p>From an implementation perspective, recursive code mainly consists of three elements.</p>
+<p>In implementation, recursive code has three main elements.</p>
 <ol>
-<li><strong>Termination condition</strong>: Used to determine when to switch from "descending" to "ascending".</li>
-<li><strong>Recursive call</strong>: Corresponds to "descending", where the function calls itself, usually with smaller or more simplified parameters.</li>
-<li><strong>Return result</strong>: Corresponds to "ascending", returning the result of the current recursion level to the previous layer.</li>
+<li><strong>Base case</strong>: Determines when descent stops and returning begins.</li>
+<li><strong>Recursive call</strong>: The function calls itself, usually with smaller or simpler arguments.</li>
+<li><strong>Returned result</strong>: The current level returns its result to the previous level.</li>
 </ol>
-<p>Observe the following code. We only need to call the function <code>recur(n)</code> to complete the calculation of <span class="arithmatex">\(1 + 2 + \dots + n\)</span>:</p>
+<p>In the following code, calling <code>recur(n)</code> computes <span class="arithmatex">\(1 + 2 + \dots + n\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.py</span><pre><span></span><code><a id="__codelineno-52-1" name="__codelineno-52-1" href="#__codelineno-52-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Recursion"""</span>
-<a id="__codelineno-52-3" name="__codelineno-52-3" href="#__codelineno-52-3"></a>    <span class="c1"># Termination condition</span>
+<a id="__codelineno-52-3" name="__codelineno-52-3" href="#__codelineno-52-3"></a>    <span class="c1"># Base case</span>
 <a id="__codelineno-52-4" name="__codelineno-52-4" href="#__codelineno-52-4"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-52-5" name="__codelineno-52-5" href="#__codelineno-52-5"></a>        <span class="k">return</span> <span class="mi">1</span>
-<a id="__codelineno-52-6" name="__codelineno-52-6" href="#__codelineno-52-6"></a>    <span class="c1"># Recurse: recursive call</span>
+<a id="__codelineno-52-6" name="__codelineno-52-6" href="#__codelineno-52-6"></a>    <span class="c1"># Recurse: make a recursive call</span>
 <a id="__codelineno-52-7" name="__codelineno-52-7" href="#__codelineno-52-7"></a>    <span class="n">res</span> <span class="o">=</span> <span class="n">recur</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-52-8" name="__codelineno-52-8" href="#__codelineno-52-8"></a>    <span class="c1"># Return: return result</span>
+<a id="__codelineno-52-8" name="__codelineno-52-8" href="#__codelineno-52-8"></a>    <span class="c1"># Unwind: return the result</span>
 <a id="__codelineno-52-9" name="__codelineno-52-9" href="#__codelineno-52-9"></a>    <span class="k">return</span> <span class="n">n</span> <span class="o">+</span> <span class="n">res</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Recursion */</span>
 <a id="__codelineno-53-2" name="__codelineno-53-2" href="#__codelineno-53-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-53-4" name="__codelineno-53-4" href="#__codelineno-53-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-53-5" name="__codelineno-53-5" href="#__codelineno-53-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-53-6" name="__codelineno-53-6" href="#__codelineno-53-6"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-53-6" name="__codelineno-53-6" href="#__codelineno-53-6"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-53-7" name="__codelineno-53-7" href="#__codelineno-53-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-53-8" name="__codelineno-53-8" href="#__codelineno-53-8"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-53-8" name="__codelineno-53-8" href="#__codelineno-53-8"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-53-9" name="__codelineno-53-9" href="#__codelineno-53-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-53-10" name="__codelineno-53-10" href="#__codelineno-53-10"></a><span class="p">}</span>
 </code></pre></div>
@@ -5669,12 +5761,12 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Recursion */</span>
 <a id="__codelineno-54-2" name="__codelineno-54-2" href="#__codelineno-54-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-54-4" name="__codelineno-54-4" href="#__codelineno-54-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-54-5" name="__codelineno-54-5" href="#__codelineno-54-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-54-6" name="__codelineno-54-6" href="#__codelineno-54-6"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-54-6" name="__codelineno-54-6" href="#__codelineno-54-6"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-54-7" name="__codelineno-54-7" href="#__codelineno-54-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-54-8" name="__codelineno-54-8" href="#__codelineno-54-8"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-54-8" name="__codelineno-54-8" href="#__codelineno-54-8"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-54-9" name="__codelineno-54-9" href="#__codelineno-54-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-54-10" name="__codelineno-54-10" href="#__codelineno-54-10"></a><span class="p">}</span>
 </code></pre></div>
@@ -5682,12 +5774,12 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Recursion */</span>
 <a id="__codelineno-55-2" name="__codelineno-55-2" href="#__codelineno-55-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Recur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-55-4" name="__codelineno-55-4" href="#__codelineno-55-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-55-5" name="__codelineno-55-5" href="#__codelineno-55-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-55-6" name="__codelineno-55-6" href="#__codelineno-55-6"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-55-6" name="__codelineno-55-6" href="#__codelineno-55-6"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-55-7" name="__codelineno-55-7" href="#__codelineno-55-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-55-8" name="__codelineno-55-8" href="#__codelineno-55-8"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-55-8" name="__codelineno-55-8" href="#__codelineno-55-8"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-55-9" name="__codelineno-55-9" href="#__codelineno-55-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-55-10" name="__codelineno-55-10" href="#__codelineno-55-10"></a><span class="p">}</span>
 </code></pre></div>
@@ -5695,13 +5787,13 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Recursion */</span>
 <a id="__codelineno-56-2" name="__codelineno-56-2" href="#__codelineno-56-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">recur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-5" name="__codelineno-56-5" href="#__codelineno-56-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-56-6" name="__codelineno-56-6" href="#__codelineno-56-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-56-7" name="__codelineno-56-7" href="#__codelineno-56-7"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-56-7" name="__codelineno-56-7" href="#__codelineno-56-7"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-56-8" name="__codelineno-56-8" href="#__codelineno-56-8"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">recur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-56-9" name="__codelineno-56-9" href="#__codelineno-56-9"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-56-9" name="__codelineno-56-9" href="#__codelineno-56-9"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-56-10" name="__codelineno-56-10" href="#__codelineno-56-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">res</span>
 <a id="__codelineno-56-11" name="__codelineno-56-11" href="#__codelineno-56-11"></a><span class="p">}</span>
 </code></pre></div>
@@ -5709,13 +5801,13 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Recursion */</span>
 <a id="__codelineno-57-2" name="__codelineno-57-2" href="#__codelineno-57-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-5" name="__codelineno-57-5" href="#__codelineno-57-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-57-6" name="__codelineno-57-6" href="#__codelineno-57-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-57-7" name="__codelineno-57-7" href="#__codelineno-57-7"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-57-7" name="__codelineno-57-7" href="#__codelineno-57-7"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-57-8" name="__codelineno-57-8" href="#__codelineno-57-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-57-9" name="__codelineno-57-9" href="#__codelineno-57-9"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-57-9" name="__codelineno-57-9" href="#__codelineno-57-9"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-57-10" name="__codelineno-57-10" href="#__codelineno-57-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-57-11" name="__codelineno-57-11" href="#__codelineno-57-11"></a><span class="p">}</span>
 </code></pre></div>
@@ -5723,11 +5815,11 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Recursion */</span>
 <a id="__codelineno-58-2" name="__codelineno-58-2" href="#__codelineno-58-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">recur</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-58-4" name="__codelineno-58-4" href="#__codelineno-58-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-58-5" name="__codelineno-58-5" href="#__codelineno-58-5"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-58-5" name="__codelineno-58-5" href="#__codelineno-58-5"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-58-6" name="__codelineno-58-6" href="#__codelineno-58-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">recur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-58-7" name="__codelineno-58-7" href="#__codelineno-58-7"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-58-7" name="__codelineno-58-7" href="#__codelineno-58-7"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-58-8" name="__codelineno-58-8" href="#__codelineno-58-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-58-9" name="__codelineno-58-9" href="#__codelineno-58-9"></a><span class="p">}</span>
 </code></pre></div>
@@ -5735,11 +5827,11 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Recursion */</span>
 <a id="__codelineno-59-2" name="__codelineno-59-2" href="#__codelineno-59-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">recur</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-59-4" name="__codelineno-59-4" href="#__codelineno-59-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-59-5" name="__codelineno-59-5" href="#__codelineno-59-5"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-59-5" name="__codelineno-59-5" href="#__codelineno-59-5"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-59-6" name="__codelineno-59-6" href="#__codelineno-59-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">recur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-59-7" name="__codelineno-59-7" href="#__codelineno-59-7"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-59-7" name="__codelineno-59-7" href="#__codelineno-59-7"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-59-8" name="__codelineno-59-8" href="#__codelineno-59-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-59-9" name="__codelineno-59-9" href="#__codelineno-59-9"></a><span class="p">}</span>
 </code></pre></div>
@@ -5747,11 +5839,11 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Recursion */</span>
 <a id="__codelineno-60-2" name="__codelineno-60-2" href="#__codelineno-60-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">recur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-60-3" name="__codelineno-60-3" href="#__codelineno-60-3"></a><span class="w">  </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-60-3" name="__codelineno-60-3" href="#__codelineno-60-3"></a><span class="w">  </span><span class="c1">// Base case</span>
 <a id="__codelineno-60-4" name="__codelineno-60-4" href="#__codelineno-60-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
-<a id="__codelineno-60-5" name="__codelineno-60-5" href="#__codelineno-60-5"></a><span class="w">  </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-60-5" name="__codelineno-60-5" href="#__codelineno-60-5"></a><span class="w">  </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-60-6" name="__codelineno-60-6" href="#__codelineno-60-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
-<a id="__codelineno-60-7" name="__codelineno-60-7" href="#__codelineno-60-7"></a><span class="w">  </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-60-7" name="__codelineno-60-7" href="#__codelineno-60-7"></a><span class="w">  </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-60-8" name="__codelineno-60-8" href="#__codelineno-60-8"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-60-9" name="__codelineno-60-9" href="#__codelineno-60-9"></a><span class="p">}</span>
 </code></pre></div>
@@ -5759,13 +5851,13 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Recursion */</span>
 <a id="__codelineno-61-2" name="__codelineno-61-2" href="#__codelineno-61-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-61-3" name="__codelineno-61-3" href="#__codelineno-61-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-61-3" name="__codelineno-61-3" href="#__codelineno-61-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-61-4" name="__codelineno-61-4" href="#__codelineno-61-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-5" name="__codelineno-61-5" href="#__codelineno-61-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-61-6" name="__codelineno-61-6" href="#__codelineno-61-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-61-7" name="__codelineno-61-7" href="#__codelineno-61-7"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-61-7" name="__codelineno-61-7" href="#__codelineno-61-7"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-61-8" name="__codelineno-61-8" href="#__codelineno-61-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-61-9" name="__codelineno-61-9" href="#__codelineno-61-9"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-61-9" name="__codelineno-61-9" href="#__codelineno-61-9"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-61-10" name="__codelineno-61-10" href="#__codelineno-61-10"></a><span class="w">    </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-61-11" name="__codelineno-61-11" href="#__codelineno-61-11"></a><span class="p">}</span>
 </code></pre></div>
@@ -5773,12 +5865,12 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Recursion */</span>
 <a id="__codelineno-62-2" name="__codelineno-62-2" href="#__codelineno-62-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-62-3" name="__codelineno-62-3" href="#__codelineno-62-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-62-3" name="__codelineno-62-3" href="#__codelineno-62-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-62-4" name="__codelineno-62-4" href="#__codelineno-62-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-62-5" name="__codelineno-62-5" href="#__codelineno-62-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-62-6" name="__codelineno-62-6" href="#__codelineno-62-6"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-62-6" name="__codelineno-62-6" href="#__codelineno-62-6"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-62-7" name="__codelineno-62-7" href="#__codelineno-62-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-62-8" name="__codelineno-62-8" href="#__codelineno-62-8"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-62-8" name="__codelineno-62-8" href="#__codelineno-62-8"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-62-9" name="__codelineno-62-9" href="#__codelineno-62-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-62-10" name="__codelineno-62-10" href="#__codelineno-62-10"></a><span class="p">}</span>
 </code></pre></div>
@@ -5786,12 +5878,12 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Recursion */</span>
 <a id="__codelineno-63-2" name="__codelineno-63-2" href="#__codelineno-63-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-63-3" name="__codelineno-63-3" href="#__codelineno-63-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-63-3" name="__codelineno-63-3" href="#__codelineno-63-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-63-4" name="__codelineno-63-4" href="#__codelineno-63-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-63-5" name="__codelineno-63-5" href="#__codelineno-63-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-63-6" name="__codelineno-63-6" href="#__codelineno-63-6"></a><span class="w">    </span><span class="c1">// Descend: recursive call</span>
+<a id="__codelineno-63-6" name="__codelineno-63-6" href="#__codelineno-63-6"></a><span class="w">    </span><span class="c1">// Descending: recursive calls</span>
 <a id="__codelineno-63-7" name="__codelineno-63-7" href="#__codelineno-63-7"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
-<a id="__codelineno-63-8" name="__codelineno-63-8" href="#__codelineno-63-8"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-63-8" name="__codelineno-63-8" href="#__codelineno-63-8"></a><span class="w">    </span><span class="c1">// Unwinding: return the result</span>
 <a id="__codelineno-63-9" name="__codelineno-63-9" href="#__codelineno-63-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-63-10" name="__codelineno-63-10" href="#__codelineno-63-10"></a><span class="p">}</span>
 </code></pre></div>
@@ -5799,68 +5891,73 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Recursion ###</span>
 <a id="__codelineno-64-2" name="__codelineno-64-2" href="#__codelineno-64-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-64-3" name="__codelineno-64-3" href="#__codelineno-64-3"></a><span class="w">  </span><span class="c1"># Termination condition</span>
+<a id="__codelineno-64-3" name="__codelineno-64-3" href="#__codelineno-64-3"></a><span class="w">  </span><span class="c1"># Base case</span>
 <a id="__codelineno-64-4" name="__codelineno-64-4" href="#__codelineno-64-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-64-5" name="__codelineno-64-5" href="#__codelineno-64-5"></a><span class="w">  </span><span class="c1"># Recurse: recursive call</span>
+<a id="__codelineno-64-5" name="__codelineno-64-5" href="#__codelineno-64-5"></a><span class="w">  </span><span class="c1"># Recurse: make a recursive call</span>
 <a id="__codelineno-64-6" name="__codelineno-64-6" href="#__codelineno-64-6"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-64-7" name="__codelineno-64-7" href="#__codelineno-64-7"></a><span class="w">  </span><span class="c1"># Return: return result</span>
+<a id="__codelineno-64-7" name="__codelineno-64-7" href="#__codelineno-64-7"></a><span class="w">  </span><span class="c1"># Unwind: return the result</span>
 <a id="__codelineno-64-8" name="__codelineno-64-8" href="#__codelineno-64-8"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-64-9" name="__codelineno-64-9" href="#__codelineno-64-9"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<p>Figure 2-3 shows the recursive process of this function.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Recursion%22%22%22%0A%20%20%20%20%23%20Base%20case%0A%20%20%20%20if%20n%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20%23%20Recursive-call%20phase%0A%20%20%20%20res%20%3D%20recur%28n%20-%201%29%0A%20%20%20%20%23%20Unwinding%20phase%3A%20compute%20the%20result%0A%20%20%20%20return%20n%20%2B%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20recur%28n%29%0A%20%20%20%20print%28f%22%5CnSum%20from%20the%20recursive%20function%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Recursion%22%22%22%0A%20%20%20%20%23%20Base%20case%0A%20%20%20%20if%20n%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20%23%20Recursive-call%20phase%0A%20%20%20%20res%20%3D%20recur%28n%20-%201%29%0A%20%20%20%20%23%20Unwinding%20phase%3A%20compute%20the%20result%0A%20%20%20%20return%20n%20%2B%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20recur%28n%29%0A%20%20%20%20print%28f%22%5CnSum%20from%20the%20recursive%20function%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 2-3 illustrates the recursive process.</p>
 <p><img alt="Recursive process of the summation function" class="animation-figure" src="../iteration_and_recursion.assets/recursion_sum.png" /></p>
 <p align="center"> Figure 2-3 &nbsp; Recursive process of the summation function </p>
 
-<p>Although from a computational perspective, iteration and recursion can achieve the same results, <strong>they represent two completely different paradigms for thinking about and solving problems</strong>.</p>
+<p>Iteration and recursion can produce the same answer, <strong>but represent two different ways of thinking about and solving a problem</strong>.</p>
 <ul>
-<li><strong>Iteration</strong>: Solves problems "bottom-up". Starting from the most basic steps, these steps are then repeatedly executed or accumulated until the task is complete.</li>
-<li><strong>Recursion</strong>: Solves problems "top-down". The original problem is decomposed into smaller subproblems that have the same form as the original problem. These subproblems continue to be decomposed into even smaller subproblems until reaching the base case (where the solution is known).</li>
+<li><strong>Iteration</strong> works bottom up, starting with basic steps and repeatedly performing or accumulating them until the task is complete.</li>
+<li><strong>Recursion</strong> works top down, dividing the original problem into smaller subproblems of the same form, then repeating the division until reaching base cases with known solutions.</li>
 </ul>
-<p>Taking the above summation function as an example, let the problem be <span class="arithmatex">\(f(n) = 1 + 2 + \dots + n\)</span>.</p>
+<p>For the summation example, let <span class="arithmatex">\(f(n) = 1 + 2 + \dots + n\)</span>.</p>
 <ul>
-<li><strong>Iteration</strong>: Simulates the summation process in a loop, traversing from <span class="arithmatex">\(1\)</span> to <span class="arithmatex">\(n\)</span>, performing the summation operation in each round to obtain <span class="arithmatex">\(f(n)\)</span>.</li>
-<li><strong>Recursion</strong>: Decomposes the problem into the subproblem <span class="arithmatex">\(f(n) = n + f(n-1)\)</span>, continuously decomposing (recursively) until terminating at the base case <span class="arithmatex">\(f(1) = 1\)</span>.</li>
+<li><strong>Iteration</strong>: Loop from <span class="arithmatex">\(1\)</span> to <span class="arithmatex">\(n\)</span>, accumulating the sum at each step to obtain <span class="arithmatex">\(f(n)\)</span>.</li>
+<li><strong>Recursion</strong>: Decompose the problem as <span class="arithmatex">\(f(n) = n + f(n-1)\)</span> and keep doing so recursively until the base case <span class="arithmatex">\(f(1) = 1\)</span>.</li>
 </ul>
-<h3 id="1-call-stack">1. &nbsp; Call Stack<a class="headerlink" href="#1-call-stack" title="Permanent link">&para;</a></h3>
-<p>Each time a recursive function calls itself, the system allocates memory for the newly invoked function to store local variables, call addresses, and other information. This leads to two consequences.</p>
+<h3 id="1-the-call-stack">1. &nbsp; The call stack<a class="headerlink" href="#1-the-call-stack" title="Permanent link">&para;</a></h3>
+<p>Each time a recursive function calls itself, the system allocates memory for the new call's local variables, return address, and other information. This has two consequences.</p>
 <ul>
-<li>The function's context data is stored in a memory area called "stack frame space", which is not released until the function returns. Therefore, <strong>recursion usually consumes more memory space than iteration</strong>.</li>
-<li>Recursive function calls incur additional overhead. <strong>Therefore, recursion is usually less time-efficient than loops</strong>.</li>
+<li>The call's context is stored in a stack frame until it returns. <strong>Recursion therefore usually uses more memory than iteration</strong>.</li>
+<li>Each recursive call adds overhead. <strong>Recursion is therefore usually slower than a loop</strong>.</li>
 </ul>
-<p>As shown in Figure 2-4, before the termination condition is triggered, there are <span class="arithmatex">\(n\)</span> unreturned recursive functions existing simultaneously, with a <strong>recursion depth of <span class="arithmatex">\(n\)</span></strong>.</p>
-<p><img alt="Recursion call depth" class="animation-figure" src="../iteration_and_recursion.assets/recursion_sum_depth.png" /></p>
-<p align="center"> Figure 2-4 &nbsp; Recursion call depth </p>
+<p>As shown in Figure 2-4, before the base case is reached, <span class="arithmatex">\(n\)</span> recursive calls remain active at once: <strong>the recursion depth is <span class="arithmatex">\(n\)</span></strong>.</p>
+<p><img alt="Depth of recursive calls" class="animation-figure" src="../iteration_and_recursion.assets/recursion_sum_depth.png" /></p>
+<p align="center"> Figure 2-4 &nbsp; Depth of recursive calls </p>
 
-<p>In practice, the recursion depth allowed by programming languages is usually limited, and excessively deep recursion may lead to stack overflow errors.</p>
-<h3 id="2-tail-recursion">2. &nbsp; Tail Recursion<a class="headerlink" href="#2-tail-recursion" title="Permanent link">&para;</a></h3>
-<p>Interestingly, <strong>if a function makes the recursive call as the very last step before returning</strong>, the compiler or interpreter may optimize it so that its space efficiency is comparable to iteration. This case is called <u>tail recursion</u>.</p>
+<p>Programming languages usually limit recursion depth, and excessively deep recursion may cause a stack overflow.</p>
+<h3 id="2-tail-recursion">2. &nbsp; Tail recursion<a class="headerlink" href="#2-tail-recursion" title="Permanent link">&para;</a></h3>
+<p>Interestingly, <strong>if the recursive call is the last operation before a function returns</strong>, a compiler or interpreter can optimize it to use as little space as iteration. This is called <u>tail recursion</u>.</p>
 <ul>
-<li><strong>Regular recursion</strong>: When a function returns to the previous level, it needs to continue executing code, so the system needs to save the context of the previous layer's call.</li>
-<li><strong>Tail recursion</strong>: The recursive call is the last operation before the function returns, meaning that after returning to the previous level, there is no need to continue executing other operations, so the system does not need to save the context of the previous layer's function.</li>
+<li><strong>Ordinary recursion</strong>: After a call returns, its caller must continue executing code, so the caller's context must be preserved.</li>
+<li><strong>Tail recursion</strong>: Nothing remains to be done after the recursive call returns, so the caller's context need not be retained.</li>
 </ul>
-<p>Taking the calculation of <span class="arithmatex">\(1 + 2 + \dots + n\)</span> as an example, we can set the result variable <code>res</code> as a function parameter to implement tail recursion:</p>
+<p>For <span class="arithmatex">\(1 + 2 + \dots + n\)</span>, passing the accumulated result <code>res</code> as a parameter gives a tail-recursive solution:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="6:13"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.py</span><pre><span></span><code><a id="__codelineno-65-1" name="__codelineno-65-1" href="#__codelineno-65-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">tail_recur</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">res</span><span class="p">):</span>
 <a id="__codelineno-65-2" name="__codelineno-65-2" href="#__codelineno-65-2"></a><span class="w">    </span><span class="sd">"""Tail recursion"""</span>
-<a id="__codelineno-65-3" name="__codelineno-65-3" href="#__codelineno-65-3"></a>    <span class="c1"># Termination condition</span>
+<a id="__codelineno-65-3" name="__codelineno-65-3" href="#__codelineno-65-3"></a>    <span class="c1"># Base case</span>
 <a id="__codelineno-65-4" name="__codelineno-65-4" href="#__codelineno-65-4"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-65-5" name="__codelineno-65-5" href="#__codelineno-65-5"></a>        <span class="k">return</span> <span class="n">res</span>
-<a id="__codelineno-65-6" name="__codelineno-65-6" href="#__codelineno-65-6"></a>    <span class="c1"># Tail recursive call</span>
+<a id="__codelineno-65-6" name="__codelineno-65-6" href="#__codelineno-65-6"></a>    <span class="c1"># Tail-recursive call</span>
 <a id="__codelineno-65-7" name="__codelineno-65-7" href="#__codelineno-65-7"></a>    <span class="k">return</span> <span class="n">tail_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">res</span> <span class="o">+</span> <span class="n">n</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.cpp</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="cm">/* Tail recursion */</span>
 <a id="__codelineno-66-2" name="__codelineno-66-2" href="#__codelineno-66-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">tailRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-66-3" name="__codelineno-66-3" href="#__codelineno-66-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-66-3" name="__codelineno-66-3" href="#__codelineno-66-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-66-4" name="__codelineno-66-4" href="#__codelineno-66-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-66-5" name="__codelineno-66-5" href="#__codelineno-66-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
-<a id="__codelineno-66-6" name="__codelineno-66-6" href="#__codelineno-66-6"></a><span class="w">    </span><span class="c1">// Tail recursive call</span>
+<a id="__codelineno-66-6" name="__codelineno-66-6" href="#__codelineno-66-6"></a><span class="w">    </span><span class="c1">// Tail-recursive call</span>
 <a id="__codelineno-66-7" name="__codelineno-66-7" href="#__codelineno-66-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">tailRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-66-8" name="__codelineno-66-8" href="#__codelineno-66-8"></a><span class="p">}</span>
 </code></pre></div>
@@ -5868,10 +5965,10 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.java</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="cm">/* Tail recursion */</span>
 <a id="__codelineno-67-2" name="__codelineno-67-2" href="#__codelineno-67-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">tailRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-67-3" name="__codelineno-67-3" href="#__codelineno-67-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-67-3" name="__codelineno-67-3" href="#__codelineno-67-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-67-4" name="__codelineno-67-4" href="#__codelineno-67-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-67-5" name="__codelineno-67-5" href="#__codelineno-67-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
-<a id="__codelineno-67-6" name="__codelineno-67-6" href="#__codelineno-67-6"></a><span class="w">    </span><span class="c1">// Tail recursive call</span>
+<a id="__codelineno-67-6" name="__codelineno-67-6" href="#__codelineno-67-6"></a><span class="w">    </span><span class="c1">// Tail-recursive call</span>
 <a id="__codelineno-67-7" name="__codelineno-67-7" href="#__codelineno-67-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">tailRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-67-8" name="__codelineno-67-8" href="#__codelineno-67-8"></a><span class="p">}</span>
 </code></pre></div>
@@ -5879,10 +5976,10 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.cs</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a><span class="cm">/* Tail recursion */</span>
 <a id="__codelineno-68-2" name="__codelineno-68-2" href="#__codelineno-68-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">TailRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-68-3" name="__codelineno-68-3" href="#__codelineno-68-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-68-3" name="__codelineno-68-3" href="#__codelineno-68-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-68-4" name="__codelineno-68-4" href="#__codelineno-68-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-68-5" name="__codelineno-68-5" href="#__codelineno-68-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
-<a id="__codelineno-68-6" name="__codelineno-68-6" href="#__codelineno-68-6"></a><span class="w">    </span><span class="c1">// Tail recursive call</span>
+<a id="__codelineno-68-6" name="__codelineno-68-6" href="#__codelineno-68-6"></a><span class="w">    </span><span class="c1">// Tail-recursive call</span>
 <a id="__codelineno-68-7" name="__codelineno-68-7" href="#__codelineno-68-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nf">TailRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-68-8" name="__codelineno-68-8" href="#__codelineno-68-8"></a><span class="p">}</span>
 </code></pre></div>
@@ -5890,11 +5987,11 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.go</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a><span class="cm">/* Tail recursion */</span>
 <a id="__codelineno-69-2" name="__codelineno-69-2" href="#__codelineno-69-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">tailRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-69-3" name="__codelineno-69-3" href="#__codelineno-69-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-69-3" name="__codelineno-69-3" href="#__codelineno-69-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-69-4" name="__codelineno-69-4" href="#__codelineno-69-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-69-5" name="__codelineno-69-5" href="#__codelineno-69-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span>
 <a id="__codelineno-69-6" name="__codelineno-69-6" href="#__codelineno-69-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-69-7" name="__codelineno-69-7" href="#__codelineno-69-7"></a><span class="w">    </span><span class="c1">// Tail recursive call</span>
+<a id="__codelineno-69-7" name="__codelineno-69-7" href="#__codelineno-69-7"></a><span class="w">    </span><span class="c1">// Tail-recursive call</span>
 <a id="__codelineno-69-8" name="__codelineno-69-8" href="#__codelineno-69-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">tailRecur</span><span class="p">(</span><span class="nx">n</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="o">+</span><span class="nx">n</span><span class="p">)</span>
 <a id="__codelineno-69-9" name="__codelineno-69-9" href="#__codelineno-69-9"></a><span class="p">}</span>
 </code></pre></div>
@@ -5902,11 +5999,11 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.swift</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a><span class="cm">/* Tail recursion */</span>
 <a id="__codelineno-70-2" name="__codelineno-70-2" href="#__codelineno-70-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">tailRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-70-3" name="__codelineno-70-3" href="#__codelineno-70-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-70-3" name="__codelineno-70-3" href="#__codelineno-70-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-70-4" name="__codelineno-70-4" href="#__codelineno-70-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-70-5" name="__codelineno-70-5" href="#__codelineno-70-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-70-6" name="__codelineno-70-6" href="#__codelineno-70-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-70-7" name="__codelineno-70-7" href="#__codelineno-70-7"></a><span class="w">    </span><span class="c1">// Tail recursive call</span>
+<a id="__codelineno-70-7" name="__codelineno-70-7" href="#__codelineno-70-7"></a><span class="w">    </span><span class="c1">// Tail-recursive call</span>
 <a id="__codelineno-70-8" name="__codelineno-70-8" href="#__codelineno-70-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">tailRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-70-9" name="__codelineno-70-9" href="#__codelineno-70-9"></a><span class="p">}</span>
 </code></pre></div>
@@ -5914,9 +6011,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.js</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a><span class="cm">/* Tail recursion */</span>
 <a id="__codelineno-71-2" name="__codelineno-71-2" href="#__codelineno-71-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">tailRecur</span><span class="p">(</span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-71-3" name="__codelineno-71-3" href="#__codelineno-71-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-71-3" name="__codelineno-71-3" href="#__codelineno-71-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-71-4" name="__codelineno-71-4" href="#__codelineno-71-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
-<a id="__codelineno-71-5" name="__codelineno-71-5" href="#__codelineno-71-5"></a><span class="w">    </span><span class="c1">// Tail recursive call</span>
+<a id="__codelineno-71-5" name="__codelineno-71-5" href="#__codelineno-71-5"></a><span class="w">    </span><span class="c1">// Tail-recursive call</span>
 <a id="__codelineno-71-6" name="__codelineno-71-6" href="#__codelineno-71-6"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">tailRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="p">);</span>
 <a id="__codelineno-71-7" name="__codelineno-71-7" href="#__codelineno-71-7"></a><span class="p">}</span>
 </code></pre></div>
@@ -5924,9 +6021,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.ts</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* Tail recursion */</span>
 <a id="__codelineno-72-2" name="__codelineno-72-2" href="#__codelineno-72-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">tailRecur</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-72-3" name="__codelineno-72-3" href="#__codelineno-72-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-72-3" name="__codelineno-72-3" href="#__codelineno-72-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-72-4" name="__codelineno-72-4" href="#__codelineno-72-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
-<a id="__codelineno-72-5" name="__codelineno-72-5" href="#__codelineno-72-5"></a><span class="w">    </span><span class="c1">// Tail recursive call</span>
+<a id="__codelineno-72-5" name="__codelineno-72-5" href="#__codelineno-72-5"></a><span class="w">    </span><span class="c1">// Tail-recursive call</span>
 <a id="__codelineno-72-6" name="__codelineno-72-6" href="#__codelineno-72-6"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">tailRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="p">);</span>
 <a id="__codelineno-72-7" name="__codelineno-72-7" href="#__codelineno-72-7"></a><span class="p">}</span>
 </code></pre></div>
@@ -5934,9 +6031,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.dart</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="cm">/* Tail recursion */</span>
 <a id="__codelineno-73-2" name="__codelineno-73-2" href="#__codelineno-73-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">tailRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-73-3" name="__codelineno-73-3" href="#__codelineno-73-3"></a><span class="w">  </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-73-3" name="__codelineno-73-3" href="#__codelineno-73-3"></a><span class="w">  </span><span class="c1">// Base case</span>
 <a id="__codelineno-73-4" name="__codelineno-73-4" href="#__codelineno-73-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
-<a id="__codelineno-73-5" name="__codelineno-73-5" href="#__codelineno-73-5"></a><span class="w">  </span><span class="c1">// Tail recursive call</span>
+<a id="__codelineno-73-5" name="__codelineno-73-5" href="#__codelineno-73-5"></a><span class="w">  </span><span class="c1">// Tail-recursive call</span>
 <a id="__codelineno-73-6" name="__codelineno-73-6" href="#__codelineno-73-6"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">tailRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-73-7" name="__codelineno-73-7" href="#__codelineno-73-7"></a><span class="p">}</span>
 </code></pre></div>
@@ -5944,11 +6041,11 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.rs</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="cm">/* Tail recursion */</span>
 <a id="__codelineno-74-2" name="__codelineno-74-2" href="#__codelineno-74-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">tail_recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-74-3" name="__codelineno-74-3" href="#__codelineno-74-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-74-3" name="__codelineno-74-3" href="#__codelineno-74-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-74-4" name="__codelineno-74-4" href="#__codelineno-74-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-74-5" name="__codelineno-74-5" href="#__codelineno-74-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-74-6" name="__codelineno-74-6" href="#__codelineno-74-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-74-7" name="__codelineno-74-7" href="#__codelineno-74-7"></a><span class="w">    </span><span class="c1">// Tail recursive call</span>
+<a id="__codelineno-74-7" name="__codelineno-74-7" href="#__codelineno-74-7"></a><span class="w">    </span><span class="c1">// Tail-recursive call</span>
 <a id="__codelineno-74-8" name="__codelineno-74-8" href="#__codelineno-74-8"></a><span class="w">    </span><span class="n">tail_recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-74-9" name="__codelineno-74-9" href="#__codelineno-74-9"></a><span class="p">}</span>
 </code></pre></div>
@@ -5956,10 +6053,10 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.c</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="cm">/* Tail recursion */</span>
 <a id="__codelineno-75-2" name="__codelineno-75-2" href="#__codelineno-75-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">tailRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-75-3" name="__codelineno-75-3" href="#__codelineno-75-3"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-75-3" name="__codelineno-75-3" href="#__codelineno-75-3"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-75-4" name="__codelineno-75-4" href="#__codelineno-75-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-75-5" name="__codelineno-75-5" href="#__codelineno-75-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
-<a id="__codelineno-75-6" name="__codelineno-75-6" href="#__codelineno-75-6"></a><span class="w">    </span><span class="c1">// Tail recursive call</span>
+<a id="__codelineno-75-6" name="__codelineno-75-6" href="#__codelineno-75-6"></a><span class="w">    </span><span class="c1">// Tail-recursive call</span>
 <a id="__codelineno-75-7" name="__codelineno-75-7" href="#__codelineno-75-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">tailRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-75-8" name="__codelineno-75-8" href="#__codelineno-75-8"></a><span class="p">}</span>
 </code></pre></div>
@@ -5967,11 +6064,11 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.kt</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a><span class="cm">/* Tail recursion */</span>
 <a id="__codelineno-76-2" name="__codelineno-76-2" href="#__codelineno-76-2"></a><span class="kd">tailrec</span><span class="w"> </span><span class="kd">fun</span><span class="w"> </span><span class="nf">tailRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-76-3" name="__codelineno-76-3" href="#__codelineno-76-3"></a><span class="w">    </span><span class="c1">// Add tailrec keyword to enable tail recursion optimization</span>
-<a id="__codelineno-76-4" name="__codelineno-76-4" href="#__codelineno-76-4"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-76-3" name="__codelineno-76-3" href="#__codelineno-76-3"></a><span class="w">    </span><span class="c1">// Add the tailrec modifier to enable tail-call optimization</span>
+<a id="__codelineno-76-4" name="__codelineno-76-4" href="#__codelineno-76-4"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-76-5" name="__codelineno-76-5" href="#__codelineno-76-5"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span>
 <a id="__codelineno-76-6" name="__codelineno-76-6" href="#__codelineno-76-6"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
-<a id="__codelineno-76-7" name="__codelineno-76-7" href="#__codelineno-76-7"></a><span class="w">    </span><span class="c1">// Tail recursive call</span>
+<a id="__codelineno-76-7" name="__codelineno-76-7" href="#__codelineno-76-7"></a><span class="w">    </span><span class="c1">// Tail-recursive call</span>
 <a id="__codelineno-76-8" name="__codelineno-76-8" href="#__codelineno-76-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">tailRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-76-9" name="__codelineno-76-9" href="#__codelineno-76-9"></a><span class="p">}</span>
 </code></pre></div>
@@ -5979,62 +6076,67 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.rb</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="c1">### Tail recursion ###</span>
 <a id="__codelineno-77-2" name="__codelineno-77-2" href="#__codelineno-77-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">tail_recur</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-77-3" name="__codelineno-77-3" href="#__codelineno-77-3"></a><span class="w">  </span><span class="c1"># Termination condition</span>
+<a id="__codelineno-77-3" name="__codelineno-77-3" href="#__codelineno-77-3"></a><span class="w">  </span><span class="c1"># Base case</span>
 <a id="__codelineno-77-4" name="__codelineno-77-4" href="#__codelineno-77-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-77-5" name="__codelineno-77-5" href="#__codelineno-77-5"></a><span class="w">  </span><span class="c1"># Tail recursive call</span>
+<a id="__codelineno-77-5" name="__codelineno-77-5" href="#__codelineno-77-5"></a><span class="w">  </span><span class="c1"># Tail-recursive call</span>
 <a id="__codelineno-77-6" name="__codelineno-77-6" href="#__codelineno-77-6"></a><span class="w">  </span><span class="n">tail_recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-77-7" name="__codelineno-77-7" href="#__codelineno-77-7"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<p>The execution process of tail recursion is shown in Figure 2-5. Comparing regular recursion and tail recursion, the summation operation is performed at different points.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 423px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20tail_recur%28n,%20res%29%3A%0A%20%20%20%20%22%22%22Tail%20recursion%22%22%22%0A%20%20%20%20%23%20Base%20case%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20res%0A%20%20%20%20%23%20Tail-recursive%20call%0A%20%20%20%20return%20tail_recur%28n%20-%201,%20res%20%2B%20n%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20tail_recur%28n,%200%29%0A%20%20%20%20print%28f%22%5CnSum%20from%20the%20tail-recursive%20function%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20tail_recur%28n,%20res%29%3A%0A%20%20%20%20%22%22%22Tail%20recursion%22%22%22%0A%20%20%20%20%23%20Base%20case%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20res%0A%20%20%20%20%23%20Tail-recursive%20call%0A%20%20%20%20return%20tail_recur%28n%20-%201,%20res%20%2B%20n%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20tail_recur%28n,%200%29%0A%20%20%20%20print%28f%22%5CnSum%20from%20the%20tail-recursive%20function%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 2-5 illustrates tail recursion. Ordinary and tail recursion perform their additions at different points.</p>
 <ul>
-<li><strong>Regular recursion</strong>: The summation operation is performed during the "ascending" process, requiring an additional summation operation after each layer returns.</li>
-<li><strong>Tail recursion</strong>: The summation operation is performed during the "descending" process; the "ascending" process only needs to return layer by layer.</li>
+<li><strong>Ordinary recursion</strong>: Addition occurs during the return phase, after each deeper call returns.</li>
+<li><strong>Tail recursion</strong>: Addition occurs during descent; the return phase simply returns through the levels.</li>
 </ul>
-<p><img alt="Tail recursion process" class="animation-figure" src="../iteration_and_recursion.assets/tail_recursion_sum.png" /></p>
-<p align="center"> Figure 2-5 &nbsp; Tail recursion process </p>
+<p><img alt="Tail-recursive summation" class="animation-figure" src="../iteration_and_recursion.assets/tail_recursion_sum.png" /></p>
+<p align="center"> Figure 2-5 &nbsp; Tail-recursive summation </p>
 
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>Please note that many compilers or interpreters do not support tail recursion optimization. For example, Python does not support tail recursion optimization by default, so even if a function is in tail recursive form, it may still encounter stack overflow issues.</p>
+<p>Many compilers and interpreters do not support tail-recursion optimization. Python, for example, does not optimize it by default, so even a tail-recursive function can run into a stack overflow.</p>
 </div>
-<h3 id="3-recursion-tree">3. &nbsp; Recursion Tree<a class="headerlink" href="#3-recursion-tree" title="Permanent link">&para;</a></h3>
-<p>When dealing with algorithmic problems related to "divide and conquer", recursion often provides a more intuitive approach and more readable code than iteration. Taking the "Fibonacci sequence" as an example.</p>
+<h3 id="3-recursion-trees">3. &nbsp; Recursion trees<a class="headerlink" href="#3-recursion-trees" title="Permanent link">&para;</a></h3>
+<p>For divide-and-conquer problems, recursion often provides a more intuitive approach and more readable code than iteration. Consider the Fibonacci sequence.</p>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given a Fibonacci sequence <span class="arithmatex">\(0, 1, 1, 2, 3, 5, 8, 13, \dots\)</span>, find the <span class="arithmatex">\(n\)</span>-th number in the sequence.</p>
+<p>Given the Fibonacci sequence <span class="arithmatex">\(0, 1, 1, 2, 3, 5, 8, 13, \dots\)</span>, find its <span class="arithmatex">\(n\)</span>-th number.</p>
 </div>
-<p>Let the <span class="arithmatex">\(n\)</span>-th number of the Fibonacci sequence be <span class="arithmatex">\(f(n)\)</span>. Two conclusions can be easily obtained.</p>
+<p>Let <span class="arithmatex">\(f(n)\)</span> be its <span class="arithmatex">\(n\)</span>-th number. Two facts follow.</p>
 <ul>
-<li>The first two numbers of the sequence are <span class="arithmatex">\(f(1) = 0\)</span> and <span class="arithmatex">\(f(2) = 1\)</span>.</li>
-<li>Each number in the sequence is the sum of the previous two numbers, i.e., <span class="arithmatex">\(f(n) = f(n - 1) + f(n - 2)\)</span>.</li>
+<li>The first two numbers are <span class="arithmatex">\(f(1) = 0\)</span> and <span class="arithmatex">\(f(2) = 1\)</span>.</li>
+<li>Every later number is the sum of the two preceding numbers: <span class="arithmatex">\(f(n) = f(n - 1) + f(n - 2)\)</span>.</li>
 </ul>
-<p>Following the recurrence relation to make recursive calls, with the first two numbers as termination conditions, we can write the recursive code. Calling <code>fib(n)</code> will give us the <span class="arithmatex">\(n\)</span>-th number of the Fibonacci sequence:</p>
+<p>Use the recurrence for recursive calls and the first two numbers as base cases. Calling <code>fib(n)</code> then returns the <span class="arithmatex">\(n\)</span>-th Fibonacci number:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="7:13"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><input id="__tabbed_7_13" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Python</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Java</label><label for="__tabbed_7_4">C#</label><label for="__tabbed_7_5">Go</label><label for="__tabbed_7_6">Swift</label><label for="__tabbed_7_7">JS</label><label for="__tabbed_7_8">TS</label><label for="__tabbed_7_9">Dart</label><label for="__tabbed_7_10">Rust</label><label for="__tabbed_7_11">C</label><label for="__tabbed_7_12">Kotlin</label><label for="__tabbed_7_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.py</span><pre><span></span><code><a id="__codelineno-78-1" name="__codelineno-78-1" href="#__codelineno-78-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">fib</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-78-2" name="__codelineno-78-2" href="#__codelineno-78-2"></a><span class="w">    </span><span class="sd">"""Fibonacci sequence: recursion"""</span>
-<a id="__codelineno-78-3" name="__codelineno-78-3" href="#__codelineno-78-3"></a>    <span class="c1"># Termination condition f(1) = 0, f(2) = 1</span>
+<a id="__codelineno-78-3" name="__codelineno-78-3" href="#__codelineno-78-3"></a>    <span class="c1"># Base cases: f(1) = 0, f(2) = 1</span>
 <a id="__codelineno-78-4" name="__codelineno-78-4" href="#__codelineno-78-4"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
 <a id="__codelineno-78-5" name="__codelineno-78-5" href="#__codelineno-78-5"></a>        <span class="k">return</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span>
-<a id="__codelineno-78-6" name="__codelineno-78-6" href="#__codelineno-78-6"></a>    <span class="c1"># Recursive call f(n) = f(n-1) + f(n-2)</span>
+<a id="__codelineno-78-6" name="__codelineno-78-6" href="#__codelineno-78-6"></a>    <span class="c1"># Recursive calls: f(n) = f(n-1) + f(n-2)</span>
 <a id="__codelineno-78-7" name="__codelineno-78-7" href="#__codelineno-78-7"></a>    <span class="n">res</span> <span class="o">=</span> <span class="n">fib</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">fib</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span>
-<a id="__codelineno-78-8" name="__codelineno-78-8" href="#__codelineno-78-8"></a>    <span class="c1"># Return result f(n)</span>
+<a id="__codelineno-78-8" name="__codelineno-78-8" href="#__codelineno-78-8"></a>    <span class="c1"># Return the result f(n)</span>
 <a id="__codelineno-78-9" name="__codelineno-78-9" href="#__codelineno-78-9"></a>    <span class="k">return</span> <span class="n">res</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.cpp</span><pre><span></span><code><a id="__codelineno-79-1" name="__codelineno-79-1" href="#__codelineno-79-1"></a><span class="cm">/* Fibonacci sequence: recursion */</span>
 <a id="__codelineno-79-2" name="__codelineno-79-2" href="#__codelineno-79-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">fib</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-79-3" name="__codelineno-79-3" href="#__codelineno-79-3"></a><span class="w">    </span><span class="c1">// Termination condition f(1) = 0, f(2) = 1</span>
+<a id="__codelineno-79-3" name="__codelineno-79-3" href="#__codelineno-79-3"></a><span class="w">    </span><span class="c1">// Base cases: f(1) = 0, f(2) = 1</span>
 <a id="__codelineno-79-4" name="__codelineno-79-4" href="#__codelineno-79-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-79-5" name="__codelineno-79-5" href="#__codelineno-79-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-79-6" name="__codelineno-79-6" href="#__codelineno-79-6"></a><span class="w">    </span><span class="c1">// Recursive call f(n) = f(n-1) + f(n-2)</span>
+<a id="__codelineno-79-6" name="__codelineno-79-6" href="#__codelineno-79-6"></a><span class="w">    </span><span class="c1">// Recursive calls: f(n) = f(n-1) + f(n-2)</span>
 <a id="__codelineno-79-7" name="__codelineno-79-7" href="#__codelineno-79-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">);</span>
-<a id="__codelineno-79-8" name="__codelineno-79-8" href="#__codelineno-79-8"></a><span class="w">    </span><span class="c1">// Return result f(n)</span>
+<a id="__codelineno-79-8" name="__codelineno-79-8" href="#__codelineno-79-8"></a><span class="w">    </span><span class="c1">// Return the result f(n)</span>
 <a id="__codelineno-79-9" name="__codelineno-79-9" href="#__codelineno-79-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-79-10" name="__codelineno-79-10" href="#__codelineno-79-10"></a><span class="p">}</span>
 </code></pre></div>
@@ -6042,12 +6144,12 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.java</span><pre><span></span><code><a id="__codelineno-80-1" name="__codelineno-80-1" href="#__codelineno-80-1"></a><span class="cm">/* Fibonacci sequence: recursion */</span>
 <a id="__codelineno-80-2" name="__codelineno-80-2" href="#__codelineno-80-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">fib</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-80-3" name="__codelineno-80-3" href="#__codelineno-80-3"></a><span class="w">    </span><span class="c1">// Termination condition f(1) = 0, f(2) = 1</span>
+<a id="__codelineno-80-3" name="__codelineno-80-3" href="#__codelineno-80-3"></a><span class="w">    </span><span class="c1">// Base cases: f(1) = 0, f(2) = 1</span>
 <a id="__codelineno-80-4" name="__codelineno-80-4" href="#__codelineno-80-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-80-5" name="__codelineno-80-5" href="#__codelineno-80-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-80-6" name="__codelineno-80-6" href="#__codelineno-80-6"></a><span class="w">    </span><span class="c1">// Recursive call f(n) = f(n-1) + f(n-2)</span>
+<a id="__codelineno-80-6" name="__codelineno-80-6" href="#__codelineno-80-6"></a><span class="w">    </span><span class="c1">// Recursive calls: f(n) = f(n-1) + f(n-2)</span>
 <a id="__codelineno-80-7" name="__codelineno-80-7" href="#__codelineno-80-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">);</span>
-<a id="__codelineno-80-8" name="__codelineno-80-8" href="#__codelineno-80-8"></a><span class="w">    </span><span class="c1">// Return result f(n)</span>
+<a id="__codelineno-80-8" name="__codelineno-80-8" href="#__codelineno-80-8"></a><span class="w">    </span><span class="c1">// Return the result f(n)</span>
 <a id="__codelineno-80-9" name="__codelineno-80-9" href="#__codelineno-80-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-80-10" name="__codelineno-80-10" href="#__codelineno-80-10"></a><span class="p">}</span>
 </code></pre></div>
@@ -6055,12 +6157,12 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.cs</span><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="cm">/* Fibonacci sequence: recursion */</span>
 <a id="__codelineno-81-2" name="__codelineno-81-2" href="#__codelineno-81-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Fib</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-81-3" name="__codelineno-81-3" href="#__codelineno-81-3"></a><span class="w">    </span><span class="c1">// Termination condition f(1) = 0, f(2) = 1</span>
+<a id="__codelineno-81-3" name="__codelineno-81-3" href="#__codelineno-81-3"></a><span class="w">    </span><span class="c1">// Base cases: f(1) = 0, f(2) = 1</span>
 <a id="__codelineno-81-4" name="__codelineno-81-4" href="#__codelineno-81-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-81-5" name="__codelineno-81-5" href="#__codelineno-81-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-81-6" name="__codelineno-81-6" href="#__codelineno-81-6"></a><span class="w">    </span><span class="c1">// Recursive call f(n) = f(n-1) + f(n-2)</span>
+<a id="__codelineno-81-6" name="__codelineno-81-6" href="#__codelineno-81-6"></a><span class="w">    </span><span class="c1">// Recursive calls: f(n) = f(n-1) + f(n-2)</span>
 <a id="__codelineno-81-7" name="__codelineno-81-7" href="#__codelineno-81-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">Fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">);</span>
-<a id="__codelineno-81-8" name="__codelineno-81-8" href="#__codelineno-81-8"></a><span class="w">    </span><span class="c1">// Return result f(n)</span>
+<a id="__codelineno-81-8" name="__codelineno-81-8" href="#__codelineno-81-8"></a><span class="w">    </span><span class="c1">// Return the result f(n)</span>
 <a id="__codelineno-81-9" name="__codelineno-81-9" href="#__codelineno-81-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-81-10" name="__codelineno-81-10" href="#__codelineno-81-10"></a><span class="p">}</span>
 </code></pre></div>
@@ -6068,13 +6170,13 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.go</span><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="cm">/* Fibonacci sequence: recursion */</span>
 <a id="__codelineno-82-2" name="__codelineno-82-2" href="#__codelineno-82-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">fib</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-82-3" name="__codelineno-82-3" href="#__codelineno-82-3"></a><span class="w">    </span><span class="c1">// Termination condition f(1) = 0, f(2) = 1</span>
+<a id="__codelineno-82-3" name="__codelineno-82-3" href="#__codelineno-82-3"></a><span class="w">    </span><span class="c1">// Base cases: f(1) = 0, f(2) = 1</span>
 <a id="__codelineno-82-4" name="__codelineno-82-4" href="#__codelineno-82-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-5" name="__codelineno-82-5" href="#__codelineno-82-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-82-6" name="__codelineno-82-6" href="#__codelineno-82-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-82-7" name="__codelineno-82-7" href="#__codelineno-82-7"></a><span class="w">    </span><span class="c1">// Recursive call f(n) = f(n-1) + f(n-2)</span>
+<a id="__codelineno-82-7" name="__codelineno-82-7" href="#__codelineno-82-7"></a><span class="w">    </span><span class="c1">// Recursive calls: f(n) = f(n-1) + f(n-2)</span>
 <a id="__codelineno-82-8" name="__codelineno-82-8" href="#__codelineno-82-8"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">fib</span><span class="p">(</span><span class="nx">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">fib</span><span class="p">(</span><span class="nx">n</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
-<a id="__codelineno-82-9" name="__codelineno-82-9" href="#__codelineno-82-9"></a><span class="w">    </span><span class="c1">// Return result f(n)</span>
+<a id="__codelineno-82-9" name="__codelineno-82-9" href="#__codelineno-82-9"></a><span class="w">    </span><span class="c1">// Return the result f(n)</span>
 <a id="__codelineno-82-10" name="__codelineno-82-10" href="#__codelineno-82-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span>
 <a id="__codelineno-82-11" name="__codelineno-82-11" href="#__codelineno-82-11"></a><span class="p">}</span>
 </code></pre></div>
@@ -6082,13 +6184,13 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.swift</span><pre><span></span><code><a id="__codelineno-83-1" name="__codelineno-83-1" href="#__codelineno-83-1"></a><span class="cm">/* Fibonacci sequence: recursion */</span>
 <a id="__codelineno-83-2" name="__codelineno-83-2" href="#__codelineno-83-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">fib</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-83-3" name="__codelineno-83-3" href="#__codelineno-83-3"></a><span class="w">    </span><span class="c1">// Termination condition f(1) = 0, f(2) = 1</span>
+<a id="__codelineno-83-3" name="__codelineno-83-3" href="#__codelineno-83-3"></a><span class="w">    </span><span class="c1">// Base cases: f(1) = 0, f(2) = 1</span>
 <a id="__codelineno-83-4" name="__codelineno-83-4" href="#__codelineno-83-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-5" name="__codelineno-83-5" href="#__codelineno-83-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-83-6" name="__codelineno-83-6" href="#__codelineno-83-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-83-7" name="__codelineno-83-7" href="#__codelineno-83-7"></a><span class="w">    </span><span class="c1">// Recursive call f(n) = f(n-1) + f(n-2)</span>
+<a id="__codelineno-83-7" name="__codelineno-83-7" href="#__codelineno-83-7"></a><span class="w">    </span><span class="c1">// Recursive calls: f(n) = f(n-1) + f(n-2)</span>
 <a id="__codelineno-83-8" name="__codelineno-83-8" href="#__codelineno-83-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
-<a id="__codelineno-83-9" name="__codelineno-83-9" href="#__codelineno-83-9"></a><span class="w">    </span><span class="c1">// Return result f(n)</span>
+<a id="__codelineno-83-9" name="__codelineno-83-9" href="#__codelineno-83-9"></a><span class="w">    </span><span class="c1">// Return the result f(n)</span>
 <a id="__codelineno-83-10" name="__codelineno-83-10" href="#__codelineno-83-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-83-11" name="__codelineno-83-11" href="#__codelineno-83-11"></a><span class="p">}</span>
 </code></pre></div>
@@ -6096,11 +6198,11 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.js</span><pre><span></span><code><a id="__codelineno-84-1" name="__codelineno-84-1" href="#__codelineno-84-1"></a><span class="cm">/* Fibonacci sequence: recursion */</span>
 <a id="__codelineno-84-2" name="__codelineno-84-2" href="#__codelineno-84-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">fib</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-84-3" name="__codelineno-84-3" href="#__codelineno-84-3"></a><span class="w">    </span><span class="c1">// Termination condition f(1) = 0, f(2) = 1</span>
+<a id="__codelineno-84-3" name="__codelineno-84-3" href="#__codelineno-84-3"></a><span class="w">    </span><span class="c1">// Base cases: f(1) = 0, f(2) = 1</span>
 <a id="__codelineno-84-4" name="__codelineno-84-4" href="#__codelineno-84-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-84-5" name="__codelineno-84-5" href="#__codelineno-84-5"></a><span class="w">    </span><span class="c1">// Recursive call f(n) = f(n-1) + f(n-2)</span>
+<a id="__codelineno-84-5" name="__codelineno-84-5" href="#__codelineno-84-5"></a><span class="w">    </span><span class="c1">// Recursive calls: f(n) = f(n-1) + f(n-2)</span>
 <a id="__codelineno-84-6" name="__codelineno-84-6" href="#__codelineno-84-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">fib</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">fib</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span>
-<a id="__codelineno-84-7" name="__codelineno-84-7" href="#__codelineno-84-7"></a><span class="w">    </span><span class="c1">// Return result f(n)</span>
+<a id="__codelineno-84-7" name="__codelineno-84-7" href="#__codelineno-84-7"></a><span class="w">    </span><span class="c1">// Return the result f(n)</span>
 <a id="__codelineno-84-8" name="__codelineno-84-8" href="#__codelineno-84-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-84-9" name="__codelineno-84-9" href="#__codelineno-84-9"></a><span class="p">}</span>
 </code></pre></div>
@@ -6108,11 +6210,11 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.ts</span><pre><span></span><code><a id="__codelineno-85-1" name="__codelineno-85-1" href="#__codelineno-85-1"></a><span class="cm">/* Fibonacci sequence: recursion */</span>
 <a id="__codelineno-85-2" name="__codelineno-85-2" href="#__codelineno-85-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">fib</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-85-3" name="__codelineno-85-3" href="#__codelineno-85-3"></a><span class="w">    </span><span class="c1">// Termination condition f(1) = 0, f(2) = 1</span>
+<a id="__codelineno-85-3" name="__codelineno-85-3" href="#__codelineno-85-3"></a><span class="w">    </span><span class="c1">// Base cases: f(1) = 0, f(2) = 1</span>
 <a id="__codelineno-85-4" name="__codelineno-85-4" href="#__codelineno-85-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-85-5" name="__codelineno-85-5" href="#__codelineno-85-5"></a><span class="w">    </span><span class="c1">// Recursive call f(n) = f(n-1) + f(n-2)</span>
+<a id="__codelineno-85-5" name="__codelineno-85-5" href="#__codelineno-85-5"></a><span class="w">    </span><span class="c1">// Recursive calls: f(n) = f(n-1) + f(n-2)</span>
 <a id="__codelineno-85-6" name="__codelineno-85-6" href="#__codelineno-85-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">fib</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">fib</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span>
-<a id="__codelineno-85-7" name="__codelineno-85-7" href="#__codelineno-85-7"></a><span class="w">    </span><span class="c1">// Return result f(n)</span>
+<a id="__codelineno-85-7" name="__codelineno-85-7" href="#__codelineno-85-7"></a><span class="w">    </span><span class="c1">// Return the result f(n)</span>
 <a id="__codelineno-85-8" name="__codelineno-85-8" href="#__codelineno-85-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-85-9" name="__codelineno-85-9" href="#__codelineno-85-9"></a><span class="p">}</span>
 </code></pre></div>
@@ -6120,11 +6222,11 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.dart</span><pre><span></span><code><a id="__codelineno-86-1" name="__codelineno-86-1" href="#__codelineno-86-1"></a><span class="cm">/* Fibonacci sequence: recursion */</span>
 <a id="__codelineno-86-2" name="__codelineno-86-2" href="#__codelineno-86-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-86-3" name="__codelineno-86-3" href="#__codelineno-86-3"></a><span class="w">  </span><span class="c1">// Termination condition f(1) = 0, f(2) = 1</span>
+<a id="__codelineno-86-3" name="__codelineno-86-3" href="#__codelineno-86-3"></a><span class="w">  </span><span class="c1">// Base cases: f(1) = 0, f(2) = 1</span>
 <a id="__codelineno-86-4" name="__codelineno-86-4" href="#__codelineno-86-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
-<a id="__codelineno-86-5" name="__codelineno-86-5" href="#__codelineno-86-5"></a><span class="w">  </span><span class="c1">// Recursive call f(n) = f(n-1) + f(n-2)</span>
+<a id="__codelineno-86-5" name="__codelineno-86-5" href="#__codelineno-86-5"></a><span class="w">  </span><span class="c1">// Recursive calls: f(n) = f(n-1) + f(n-2)</span>
 <a id="__codelineno-86-6" name="__codelineno-86-6" href="#__codelineno-86-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">2</span><span class="p">);</span>
-<a id="__codelineno-86-7" name="__codelineno-86-7" href="#__codelineno-86-7"></a><span class="w">  </span><span class="c1">// Return result f(n)</span>
+<a id="__codelineno-86-7" name="__codelineno-86-7" href="#__codelineno-86-7"></a><span class="w">  </span><span class="c1">// Return the result f(n)</span>
 <a id="__codelineno-86-8" name="__codelineno-86-8" href="#__codelineno-86-8"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-86-9" name="__codelineno-86-9" href="#__codelineno-86-9"></a><span class="p">}</span>
 </code></pre></div>
@@ -6132,13 +6234,13 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.rs</span><pre><span></span><code><a id="__codelineno-87-1" name="__codelineno-87-1" href="#__codelineno-87-1"></a><span class="cm">/* Fibonacci sequence: recursion */</span>
 <a id="__codelineno-87-2" name="__codelineno-87-2" href="#__codelineno-87-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">fib</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-87-3" name="__codelineno-87-3" href="#__codelineno-87-3"></a><span class="w">    </span><span class="c1">// Termination condition f(1) = 0, f(2) = 1</span>
+<a id="__codelineno-87-3" name="__codelineno-87-3" href="#__codelineno-87-3"></a><span class="w">    </span><span class="c1">// Base cases: f(1) = 0, f(2) = 1</span>
 <a id="__codelineno-87-4" name="__codelineno-87-4" href="#__codelineno-87-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-5" name="__codelineno-87-5" href="#__codelineno-87-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-87-6" name="__codelineno-87-6" href="#__codelineno-87-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-87-7" name="__codelineno-87-7" href="#__codelineno-87-7"></a><span class="w">    </span><span class="c1">// Recursive call f(n) = f(n-1) + f(n-2)</span>
+<a id="__codelineno-87-7" name="__codelineno-87-7" href="#__codelineno-87-7"></a><span class="w">    </span><span class="c1">// Recursive calls: f(n) = f(n-1) + f(n-2)</span>
 <a id="__codelineno-87-8" name="__codelineno-87-8" href="#__codelineno-87-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">);</span>
-<a id="__codelineno-87-9" name="__codelineno-87-9" href="#__codelineno-87-9"></a><span class="w">    </span><span class="c1">// Return result</span>
+<a id="__codelineno-87-9" name="__codelineno-87-9" href="#__codelineno-87-9"></a><span class="w">    </span><span class="c1">// Return the result</span>
 <a id="__codelineno-87-10" name="__codelineno-87-10" href="#__codelineno-87-10"></a><span class="w">    </span><span class="n">res</span>
 <a id="__codelineno-87-11" name="__codelineno-87-11" href="#__codelineno-87-11"></a><span class="p">}</span>
 </code></pre></div>
@@ -6146,12 +6248,12 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.c</span><pre><span></span><code><a id="__codelineno-88-1" name="__codelineno-88-1" href="#__codelineno-88-1"></a><span class="cm">/* Fibonacci sequence: recursion */</span>
 <a id="__codelineno-88-2" name="__codelineno-88-2" href="#__codelineno-88-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">fib</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-88-3" name="__codelineno-88-3" href="#__codelineno-88-3"></a><span class="w">    </span><span class="c1">// Termination condition f(1) = 0, f(2) = 1</span>
+<a id="__codelineno-88-3" name="__codelineno-88-3" href="#__codelineno-88-3"></a><span class="w">    </span><span class="c1">// Base cases: f(1) = 0, f(2) = 1</span>
 <a id="__codelineno-88-4" name="__codelineno-88-4" href="#__codelineno-88-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-88-5" name="__codelineno-88-5" href="#__codelineno-88-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-88-6" name="__codelineno-88-6" href="#__codelineno-88-6"></a><span class="w">    </span><span class="c1">// Recursive call f(n) = f(n-1) + f(n-2)</span>
+<a id="__codelineno-88-6" name="__codelineno-88-6" href="#__codelineno-88-6"></a><span class="w">    </span><span class="c1">// Recursive calls: f(n) = f(n-1) + f(n-2)</span>
 <a id="__codelineno-88-7" name="__codelineno-88-7" href="#__codelineno-88-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">);</span>
-<a id="__codelineno-88-8" name="__codelineno-88-8" href="#__codelineno-88-8"></a><span class="w">    </span><span class="c1">// Return result f(n)</span>
+<a id="__codelineno-88-8" name="__codelineno-88-8" href="#__codelineno-88-8"></a><span class="w">    </span><span class="c1">// Return the result f(n)</span>
 <a id="__codelineno-88-9" name="__codelineno-88-9" href="#__codelineno-88-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-88-10" name="__codelineno-88-10" href="#__codelineno-88-10"></a><span class="p">}</span>
 </code></pre></div>
@@ -6159,12 +6261,12 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.kt</span><pre><span></span><code><a id="__codelineno-89-1" name="__codelineno-89-1" href="#__codelineno-89-1"></a><span class="cm">/* Fibonacci sequence: recursion */</span>
 <a id="__codelineno-89-2" name="__codelineno-89-2" href="#__codelineno-89-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">fib</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-89-3" name="__codelineno-89-3" href="#__codelineno-89-3"></a><span class="w">    </span><span class="c1">// Termination condition f(1) = 0, f(2) = 1</span>
+<a id="__codelineno-89-3" name="__codelineno-89-3" href="#__codelineno-89-3"></a><span class="w">    </span><span class="c1">// Base cases: f(1) = 0, f(2) = 1</span>
 <a id="__codelineno-89-4" name="__codelineno-89-4" href="#__codelineno-89-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span>
 <a id="__codelineno-89-5" name="__codelineno-89-5" href="#__codelineno-89-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-89-6" name="__codelineno-89-6" href="#__codelineno-89-6"></a><span class="w">    </span><span class="c1">// Recursive call f(n) = f(n-1) + f(n-2)</span>
+<a id="__codelineno-89-6" name="__codelineno-89-6" href="#__codelineno-89-6"></a><span class="w">    </span><span class="c1">// Recursive calls: f(n) = f(n-1) + f(n-2)</span>
 <a id="__codelineno-89-7" name="__codelineno-89-7" href="#__codelineno-89-7"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">2</span><span class="p">)</span>
-<a id="__codelineno-89-8" name="__codelineno-89-8" href="#__codelineno-89-8"></a><span class="w">    </span><span class="c1">// Return result f(n)</span>
+<a id="__codelineno-89-8" name="__codelineno-89-8" href="#__codelineno-89-8"></a><span class="w">    </span><span class="c1">// Return the result f(n)</span>
 <a id="__codelineno-89-9" name="__codelineno-89-9" href="#__codelineno-89-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-89-10" name="__codelineno-89-10" href="#__codelineno-89-10"></a><span class="p">}</span>
 </code></pre></div>
@@ -6172,29 +6274,34 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.rb</span><pre><span></span><code><a id="__codelineno-90-1" name="__codelineno-90-1" href="#__codelineno-90-1"></a><span class="c1">### Fibonacci sequence: recursion ###</span>
 <a id="__codelineno-90-2" name="__codelineno-90-2" href="#__codelineno-90-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">fib</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-90-3" name="__codelineno-90-3" href="#__codelineno-90-3"></a><span class="w">  </span><span class="c1"># Termination condition f(1) = 0, f(2) = 1</span>
+<a id="__codelineno-90-3" name="__codelineno-90-3" href="#__codelineno-90-3"></a><span class="w">  </span><span class="c1"># Base cases: f(1) = 0, f(2) = 1</span>
 <a id="__codelineno-90-4" name="__codelineno-90-4" href="#__codelineno-90-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span>
-<a id="__codelineno-90-5" name="__codelineno-90-5" href="#__codelineno-90-5"></a><span class="w">  </span><span class="c1"># Recursive call f(n) = f(n-1) + f(n-2)</span>
+<a id="__codelineno-90-5" name="__codelineno-90-5" href="#__codelineno-90-5"></a><span class="w">  </span><span class="c1"># Recursive calls: f(n) = f(n-1) + f(n-2)</span>
 <a id="__codelineno-90-6" name="__codelineno-90-6" href="#__codelineno-90-6"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">fib</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
-<a id="__codelineno-90-7" name="__codelineno-90-7" href="#__codelineno-90-7"></a><span class="w">  </span><span class="c1"># Return result f(n)</span>
+<a id="__codelineno-90-7" name="__codelineno-90-7" href="#__codelineno-90-7"></a><span class="w">  </span><span class="c1"># Return the result f(n)</span>
 <a id="__codelineno-90-8" name="__codelineno-90-8" href="#__codelineno-90-8"></a><span class="w">  </span><span class="n">res</span>
 <a id="__codelineno-90-9" name="__codelineno-90-9" href="#__codelineno-90-9"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<p>Observing the above code, we make two recursive calls within the function, <strong>meaning that one call produces two call branches</strong>. As shown in Figure 2-6, this repeated recursive calling eventually produces a <u>recursion tree</u> with <span class="arithmatex">\(n\)</span> levels.</p>
-<p><img alt="Recursion tree of the Fibonacci sequence" class="animation-figure" src="../iteration_and_recursion.assets/recursion_tree.png" /></p>
-<p align="center"> Figure 2-6 &nbsp; Recursion tree of the Fibonacci sequence </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20fib%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Fibonacci%20sequence%3A%20recursion%22%22%22%0A%20%20%20%20%23%20Base%20cases%3A%20f%281%29%20%3D%200,%20f%282%29%20%3D%201%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20n%20-%201%0A%20%20%20%20%23%20Recursive%20calls%3A%20f%28n%29%20%3D%20f%28n-1%29%20%2B%20f%28n-2%29%0A%20%20%20%20res%20%3D%20fib%28n%20-%201%29%20%2B%20fib%28n%20-%202%29%0A%20%20%20%20%23%20Return%20the%20result%20f%28n%29%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20fib%28n%29%0A%20%20%20%20print%28f%22%5CnTerm%20%7Bn%7D%20of%20the%20Fibonacci%20sequence%20is%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20fib%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Fibonacci%20sequence%3A%20recursion%22%22%22%0A%20%20%20%20%23%20Base%20cases%3A%20f%281%29%20%3D%200,%20f%282%29%20%3D%201%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20n%20-%201%0A%20%20%20%20%23%20Recursive%20calls%3A%20f%28n%29%20%3D%20f%28n-1%29%20%2B%20f%28n-2%29%0A%20%20%20%20res%20%3D%20fib%28n%20-%201%29%20%2B%20fib%28n%20-%202%29%0A%20%20%20%20%23%20Return%20the%20result%20f%28n%29%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20fib%28n%29%0A%20%20%20%20print%28f%22%5CnTerm%20%7Bn%7D%20of%20the%20Fibonacci%20sequence%20is%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>The function makes two recursive calls, <strong>so each call creates two branches</strong>. Continuing this process produces a <u>recursion tree</u> with <span class="arithmatex">\(n\)</span> levels, as shown in Figure 2-6.</p>
+<p><img alt="Recursion tree for the Fibonacci sequence" class="animation-figure" src="../iteration_and_recursion.assets/recursion_tree.png" /></p>
+<p align="center"> Figure 2-6 &nbsp; Recursion tree for the Fibonacci sequence </p>
 
-<p>Fundamentally, recursion embodies the paradigm of "decomposing a problem into smaller subproblems", and this divide-and-conquer strategy is crucial.</p>
+<p>Recursion embodies the idea of breaking a problem into smaller subproblems, a crucial divide-and-conquer strategy.</p>
 <ul>
-<li>From an algorithmic perspective, many important algorithmic strategies such as searching, sorting, backtracking, divide and conquer, and dynamic programming directly or indirectly apply this way of thinking.</li>
-<li>From a data structure perspective, recursion is naturally suited for handling problems related to linked lists, trees, and graphs, because they are well-suited for analysis using divide-and-conquer thinking.</li>
+<li>In algorithms, searching, sorting, backtracking, divide and conquer, dynamic programming, and many other techniques use this idea directly or indirectly.</li>
+<li>In data structures, recursion naturally handles linked lists, trees, and graphs because divide-and-conquer reasoning suits them well.</li>
 </ul>
-<h2 id="223-comparison-of-the-two">2.2.3 &nbsp; Comparison of the Two<a class="headerlink" href="#223-comparison-of-the-two" title="Permanent link">&para;</a></h2>
-<p>Summarizing the above content, as shown in Table 2-1, iteration and recursion differ in implementation, performance, and applicability.</p>
-<p align="center"> Table 2-1 &nbsp; Comparison of iteration and recursion characteristics </p>
+<h2 id="223-comparing-iteration-and-recursion">2.2.3 &nbsp; Comparing iteration and recursion<a class="headerlink" href="#223-comparing-iteration-and-recursion" title="Permanent link">&para;</a></h2>
+<p>Table 2-1 summarizes their differences in implementation, performance, and suitability.</p>
+<p align="center"> Table 2-1 &nbsp; Comparing iteration and recursion </p>
 
 <div class="center-table">
 <table>
@@ -6208,38 +6315,38 @@
 <tbody>
 <tr>
 <td>Implementation</td>
-<td>Loop structure</td>
-<td>Function calls itself</td>
+<td>Loops</td>
+<td>A function calls itself</td>
 </tr>
 <tr>
 <td>Time efficiency</td>
-<td>Generally more efficient, no function call overhead</td>
-<td>Each function call incurs overhead</td>
+<td>Usually high, with no function-call overhead</td>
+<td>Each call adds overhead</td>
 </tr>
 <tr>
-<td>Memory usage</td>
-<td>Usually uses a fixed amount of memory space</td>
-<td>Accumulated function calls may use a large amount of stack frame space</td>
+<td>Memory use</td>
+<td>Usually a fixed amount of memory</td>
+<td>Accumulated calls may require many stack frames</td>
 </tr>
 <tr>
 <td>Suitable problems</td>
-<td>Suitable for simple loop tasks, with intuitive and readable code</td>
-<td>Suitable for subproblem decomposition, such as trees, graphs, divide and conquer, backtracking, etc., with concise and clear code structure</td>
+<td>Simple repetitive tasks, with straightforward, readable code</td>
+<td>Subproblem decomposition, including trees, graphs, divide and conquer, and backtracking; concise, clear structure</td>
 </tr>
 </tbody>
 </table>
 </div>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>If you find the following content difficult to understand, you can review it after reading the "Stack" chapter.</p>
+<p>If the following discussion feels difficult, return to it after reading the section on stacks.</p>
 </div>
-<p>What is the intrinsic relationship between iteration and recursion? Taking the above recursive function as an example, the summation operation is performed during the "ascending" phase of recursion. This means that the function called first actually completes its summation operation last, <strong>and this working mechanism is similar to the "last-in, first-out" principle of stacks</strong>.</p>
-<p>In fact, recursive terminology such as "call stack" and "stack frame space" already hints at the close relationship between recursion and stacks.</p>
+<p>How are iteration and recursion connected? In the summation example, additions occur during the return phase. The first function called is the last to finish its addition, <strong>matching a stack's last-in, first-out behavior</strong>.</p>
+<p>The terms call stack and stack frame already suggest this close relationship.</p>
 <ol>
-<li><strong>Descend</strong>: When a function is called, the system allocates a new stack frame on the "call stack" for that function to store the function's local variables, parameters, return address, and other data.</li>
-<li><strong>Ascend</strong>: When the function completes execution and returns, the corresponding stack frame is removed from the "call stack", restoring the execution environment of the previous function.</li>
+<li><strong>Descent</strong>: A function call adds a new stack frame to the call stack, holding local variables, parameters, the return address, and other data.</li>
+<li><strong>Return</strong>: When the function finishes, its frame is removed, restoring the caller's execution context.</li>
 </ol>
-<p>Therefore, <strong>we can use an explicit stack to simulate the behavior of the call stack</strong>, thus transforming recursion into iterative form:</p>
+<p>Thus, <strong>an explicit stack can simulate the call stack</strong>, converting recursion into iteration:</p>
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@@ -6248,13 +6355,13 @@
 <a id="__codelineno-91-3" name="__codelineno-91-3" href="#__codelineno-91-3"></a>    <span class="c1"># Use an explicit stack to simulate the system call stack</span>
 <a id="__codelineno-91-4" name="__codelineno-91-4" href="#__codelineno-91-4"></a>    <span class="n">stack</span> <span class="o">=</span> <span class="p">[]</span>
 <a id="__codelineno-91-5" name="__codelineno-91-5" href="#__codelineno-91-5"></a>    <span class="n">res</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-91-6" name="__codelineno-91-6" href="#__codelineno-91-6"></a>    <span class="c1"># Recurse: recursive call</span>
+<a id="__codelineno-91-6" name="__codelineno-91-6" href="#__codelineno-91-6"></a>    <span class="c1"># Simulate the recursive-call phase</span>
 <a id="__codelineno-91-7" name="__codelineno-91-7" href="#__codelineno-91-7"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-<a id="__codelineno-91-8" name="__codelineno-91-8" href="#__codelineno-91-8"></a>        <span class="c1"># Simulate "recurse" with "push"</span>
+<a id="__codelineno-91-8" name="__codelineno-91-8" href="#__codelineno-91-8"></a>        <span class="c1"># Simulate recursive calls by pushing onto the stack</span>
 <a id="__codelineno-91-9" name="__codelineno-91-9" href="#__codelineno-91-9"></a>        <span class="n">stack</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-<a id="__codelineno-91-10" name="__codelineno-91-10" href="#__codelineno-91-10"></a>    <span class="c1"># Return: return result</span>
+<a id="__codelineno-91-10" name="__codelineno-91-10" href="#__codelineno-91-10"></a>    <span class="c1"># Simulate unwinding and accumulating the result</span>
 <a id="__codelineno-91-11" name="__codelineno-91-11" href="#__codelineno-91-11"></a>    <span class="k">while</span> <span class="n">stack</span><span class="p">:</span>
-<a id="__codelineno-91-12" name="__codelineno-91-12" href="#__codelineno-91-12"></a>        <span class="c1"># Simulate "return" with "pop"</span>
+<a id="__codelineno-91-12" name="__codelineno-91-12" href="#__codelineno-91-12"></a>        <span class="c1"># Simulate unwinding by popping from the stack</span>
 <a id="__codelineno-91-13" name="__codelineno-91-13" href="#__codelineno-91-13"></a>        <span class="n">res</span> <span class="o">+=</span> <span class="n">stack</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
 <a id="__codelineno-91-14" name="__codelineno-91-14" href="#__codelineno-91-14"></a>    <span class="c1"># res = 1+2+3+...+n</span>
 <a id="__codelineno-91-15" name="__codelineno-91-15" href="#__codelineno-91-15"></a>    <span class="k">return</span> <span class="n">res</span>
@@ -6266,14 +6373,14 @@
 <a id="__codelineno-92-3" name="__codelineno-92-3" href="#__codelineno-92-3"></a><span class="w">    </span><span class="c1">// Use an explicit stack to simulate the system call stack</span>
 <a id="__codelineno-92-4" name="__codelineno-92-4" href="#__codelineno-92-4"></a><span class="w">    </span><span class="n">stack</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">stack</span><span class="p">;</span>
 <a id="__codelineno-92-5" name="__codelineno-92-5" href="#__codelineno-92-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-92-6" name="__codelineno-92-6" href="#__codelineno-92-6"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-92-6" name="__codelineno-92-6" href="#__codelineno-92-6"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-92-7" name="__codelineno-92-7" href="#__codelineno-92-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-92-8" name="__codelineno-92-8" href="#__codelineno-92-8"></a><span class="w">        </span><span class="c1">// Simulate "recurse" with "push"</span>
+<a id="__codelineno-92-8" name="__codelineno-92-8" href="#__codelineno-92-8"></a><span class="w">        </span><span class="c1">// Simulate recursive calls by pushing onto the stack</span>
 <a id="__codelineno-92-9" name="__codelineno-92-9" href="#__codelineno-92-9"></a><span class="w">        </span><span class="n">stack</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-92-10" name="__codelineno-92-10" href="#__codelineno-92-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-92-11" name="__codelineno-92-11" href="#__codelineno-92-11"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-92-11" name="__codelineno-92-11" href="#__codelineno-92-11"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-92-12" name="__codelineno-92-12" href="#__codelineno-92-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">stack</span><span class="p">.</span><span class="n">empty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-92-13" name="__codelineno-92-13" href="#__codelineno-92-13"></a><span class="w">        </span><span class="c1">// Simulate "return" with "pop"</span>
+<a id="__codelineno-92-13" name="__codelineno-92-13" href="#__codelineno-92-13"></a><span class="w">        </span><span class="c1">// Simulate unwinding by popping from the stack</span>
 <a id="__codelineno-92-14" name="__codelineno-92-14" href="#__codelineno-92-14"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">top</span><span class="p">();</span>
 <a id="__codelineno-92-15" name="__codelineno-92-15" href="#__codelineno-92-15"></a><span class="w">        </span><span class="n">stack</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span>
 <a id="__codelineno-92-16" name="__codelineno-92-16" href="#__codelineno-92-16"></a><span class="w">    </span><span class="p">}</span>
@@ -6288,14 +6395,14 @@
 <a id="__codelineno-93-3" name="__codelineno-93-3" href="#__codelineno-93-3"></a><span class="w">    </span><span class="c1">// Use an explicit stack to simulate the system call stack</span>
 <a id="__codelineno-93-4" name="__codelineno-93-4" href="#__codelineno-93-4"></a><span class="w">    </span><span class="n">Stack</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Stack</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-93-5" name="__codelineno-93-5" href="#__codelineno-93-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-93-6" name="__codelineno-93-6" href="#__codelineno-93-6"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-93-6" name="__codelineno-93-6" href="#__codelineno-93-6"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-93-7" name="__codelineno-93-7" href="#__codelineno-93-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-93-8" name="__codelineno-93-8" href="#__codelineno-93-8"></a><span class="w">        </span><span class="c1">// Simulate "recurse" with "push"</span>
+<a id="__codelineno-93-8" name="__codelineno-93-8" href="#__codelineno-93-8"></a><span class="w">        </span><span class="c1">// Simulate recursive calls by pushing onto the stack</span>
 <a id="__codelineno-93-9" name="__codelineno-93-9" href="#__codelineno-93-9"></a><span class="w">        </span><span class="n">stack</span><span class="p">.</span><span class="na">push</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-93-10" name="__codelineno-93-10" href="#__codelineno-93-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-93-11" name="__codelineno-93-11" href="#__codelineno-93-11"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-93-11" name="__codelineno-93-11" href="#__codelineno-93-11"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-93-12" name="__codelineno-93-12" href="#__codelineno-93-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">stack</span><span class="p">.</span><span class="na">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-93-13" name="__codelineno-93-13" href="#__codelineno-93-13"></a><span class="w">        </span><span class="c1">// Simulate "return" with "pop"</span>
+<a id="__codelineno-93-13" name="__codelineno-93-13" href="#__codelineno-93-13"></a><span class="w">        </span><span class="c1">// Simulate unwinding by popping from the stack</span>
 <a id="__codelineno-93-14" name="__codelineno-93-14" href="#__codelineno-93-14"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">pop</span><span class="p">();</span>
 <a id="__codelineno-93-15" name="__codelineno-93-15" href="#__codelineno-93-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-93-16" name="__codelineno-93-16" href="#__codelineno-93-16"></a><span class="w">    </span><span class="c1">// res = 1+2+3+...+n</span>
@@ -6309,14 +6416,14 @@
 <a id="__codelineno-94-3" name="__codelineno-94-3" href="#__codelineno-94-3"></a><span class="w">    </span><span class="c1">// Use an explicit stack to simulate the system call stack</span>
 <a id="__codelineno-94-4" name="__codelineno-94-4" href="#__codelineno-94-4"></a><span class="w">    </span><span class="n">Stack</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">();</span>
 <a id="__codelineno-94-5" name="__codelineno-94-5" href="#__codelineno-94-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-94-6" name="__codelineno-94-6" href="#__codelineno-94-6"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-94-6" name="__codelineno-94-6" href="#__codelineno-94-6"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-94-7" name="__codelineno-94-7" href="#__codelineno-94-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-94-8" name="__codelineno-94-8" href="#__codelineno-94-8"></a><span class="w">        </span><span class="c1">// Simulate "recurse" with "push"</span>
+<a id="__codelineno-94-8" name="__codelineno-94-8" href="#__codelineno-94-8"></a><span class="w">        </span><span class="c1">// Simulate recursive calls by pushing onto the stack</span>
 <a id="__codelineno-94-9" name="__codelineno-94-9" href="#__codelineno-94-9"></a><span class="w">        </span><span class="n">stack</span><span class="p">.</span><span class="n">Push</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-94-10" name="__codelineno-94-10" href="#__codelineno-94-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-94-11" name="__codelineno-94-11" href="#__codelineno-94-11"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-94-11" name="__codelineno-94-11" href="#__codelineno-94-11"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-94-12" name="__codelineno-94-12" href="#__codelineno-94-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">stack</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-94-13" name="__codelineno-94-13" href="#__codelineno-94-13"></a><span class="w">        </span><span class="c1">// Simulate "return" with "pop"</span>
+<a id="__codelineno-94-13" name="__codelineno-94-13" href="#__codelineno-94-13"></a><span class="w">        </span><span class="c1">// Simulate unwinding by popping from the stack</span>
 <a id="__codelineno-94-14" name="__codelineno-94-14" href="#__codelineno-94-14"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">Pop</span><span class="p">();</span>
 <a id="__codelineno-94-15" name="__codelineno-94-15" href="#__codelineno-94-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-94-16" name="__codelineno-94-16" href="#__codelineno-94-16"></a><span class="w">    </span><span class="c1">// res = 1+2+3+...+n</span>
@@ -6330,14 +6437,14 @@
 <a id="__codelineno-95-3" name="__codelineno-95-3" href="#__codelineno-95-3"></a><span class="w">    </span><span class="c1">// Use an explicit stack to simulate the system call stack</span>
 <a id="__codelineno-95-4" name="__codelineno-95-4" href="#__codelineno-95-4"></a><span class="w">    </span><span class="nx">stack</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">list</span><span class="p">.</span><span class="nx">New</span><span class="p">()</span>
 <a id="__codelineno-95-5" name="__codelineno-95-5" href="#__codelineno-95-5"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-95-6" name="__codelineno-95-6" href="#__codelineno-95-6"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-95-6" name="__codelineno-95-6" href="#__codelineno-95-6"></a><span class="w">    </span><span class="c1">// Descending: simulate recursive calls</span>
 <a id="__codelineno-95-7" name="__codelineno-95-7" href="#__codelineno-95-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-95-8" name="__codelineno-95-8" href="#__codelineno-95-8"></a><span class="w">        </span><span class="c1">// Simulate "recurse" with "push"</span>
+<a id="__codelineno-95-8" name="__codelineno-95-8" href="#__codelineno-95-8"></a><span class="w">        </span><span class="c1">// Simulate recursive calls by pushing onto the stack</span>
 <a id="__codelineno-95-9" name="__codelineno-95-9" href="#__codelineno-95-9"></a><span class="w">        </span><span class="nx">stack</span><span class="p">.</span><span class="nx">PushBack</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span>
 <a id="__codelineno-95-10" name="__codelineno-95-10" href="#__codelineno-95-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-95-11" name="__codelineno-95-11" href="#__codelineno-95-11"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-95-11" name="__codelineno-95-11" href="#__codelineno-95-11"></a><span class="w">    </span><span class="c1">// Unwinding: simulate returning results</span>
 <a id="__codelineno-95-12" name="__codelineno-95-12" href="#__codelineno-95-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">stack</span><span class="p">.</span><span class="nx">Len</span><span class="p">()</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-95-13" name="__codelineno-95-13" href="#__codelineno-95-13"></a><span class="w">        </span><span class="c1">// Simulate "return" with "pop"</span>
+<a id="__codelineno-95-13" name="__codelineno-95-13" href="#__codelineno-95-13"></a><span class="w">        </span><span class="c1">// Simulate unwinding by popping from the stack</span>
 <a id="__codelineno-95-14" name="__codelineno-95-14" href="#__codelineno-95-14"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">stack</span><span class="p">.</span><span class="nx">Back</span><span class="p">().</span><span class="nx">Value</span><span class="p">.(</span><span class="kt">int</span><span class="p">)</span>
 <a id="__codelineno-95-15" name="__codelineno-95-15" href="#__codelineno-95-15"></a><span class="w">        </span><span class="nx">stack</span><span class="p">.</span><span class="nx">Remove</span><span class="p">(</span><span class="nx">stack</span><span class="p">.</span><span class="nx">Back</span><span class="p">())</span>
 <a id="__codelineno-95-16" name="__codelineno-95-16" href="#__codelineno-95-16"></a><span class="w">    </span><span class="p">}</span>
@@ -6352,14 +6459,14 @@
 <a id="__codelineno-96-3" name="__codelineno-96-3" href="#__codelineno-96-3"></a><span class="w">    </span><span class="c1">// Use an explicit stack to simulate the system call stack</span>
 <a id="__codelineno-96-4" name="__codelineno-96-4" href="#__codelineno-96-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">stack</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
 <a id="__codelineno-96-5" name="__codelineno-96-5" href="#__codelineno-96-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-96-6" name="__codelineno-96-6" href="#__codelineno-96-6"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-96-6" name="__codelineno-96-6" href="#__codelineno-96-6"></a><span class="w">    </span><span class="c1">// Descending: simulate recursive calls</span>
 <a id="__codelineno-96-7" name="__codelineno-96-7" href="#__codelineno-96-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="p">).</span><span class="n">reversed</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-96-8" name="__codelineno-96-8" href="#__codelineno-96-8"></a><span class="w">        </span><span class="c1">// Simulate "recurse" with "push"</span>
+<a id="__codelineno-96-8" name="__codelineno-96-8" href="#__codelineno-96-8"></a><span class="w">        </span><span class="c1">// Simulate recursive calls by pushing onto the stack</span>
 <a id="__codelineno-96-9" name="__codelineno-96-9" href="#__codelineno-96-9"></a><span class="w">        </span><span class="n">stack</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-96-10" name="__codelineno-96-10" href="#__codelineno-96-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-96-11" name="__codelineno-96-11" href="#__codelineno-96-11"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-96-11" name="__codelineno-96-11" href="#__codelineno-96-11"></a><span class="w">    </span><span class="c1">// Unwinding: simulate returning results</span>
 <a id="__codelineno-96-12" name="__codelineno-96-12" href="#__codelineno-96-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="o">!</span><span class="n">stack</span><span class="p">.</span><span class="bp">isEmpty</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-96-13" name="__codelineno-96-13" href="#__codelineno-96-13"></a><span class="w">        </span><span class="c1">// Simulate "return" with "pop"</span>
+<a id="__codelineno-96-13" name="__codelineno-96-13" href="#__codelineno-96-13"></a><span class="w">        </span><span class="c1">// Simulate unwinding by popping from the stack</span>
 <a id="__codelineno-96-14" name="__codelineno-96-14" href="#__codelineno-96-14"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span>
 <a id="__codelineno-96-15" name="__codelineno-96-15" href="#__codelineno-96-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-96-16" name="__codelineno-96-16" href="#__codelineno-96-16"></a><span class="w">    </span><span class="c1">// res = 1+2+3+...+n</span>
@@ -6373,14 +6480,14 @@
 <a id="__codelineno-97-3" name="__codelineno-97-3" href="#__codelineno-97-3"></a><span class="w">    </span><span class="c1">// Use an explicit stack to simulate the system call stack</span>
 <a id="__codelineno-97-4" name="__codelineno-97-4" href="#__codelineno-97-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-97-5" name="__codelineno-97-5" href="#__codelineno-97-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-97-6" name="__codelineno-97-6" href="#__codelineno-97-6"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-97-6" name="__codelineno-97-6" href="#__codelineno-97-6"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-97-7" name="__codelineno-97-7" href="#__codelineno-97-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-97-8" name="__codelineno-97-8" href="#__codelineno-97-8"></a><span class="w">        </span><span class="c1">// Simulate "recurse" with "push"</span>
+<a id="__codelineno-97-8" name="__codelineno-97-8" href="#__codelineno-97-8"></a><span class="w">        </span><span class="c1">// Simulate recursive calls by pushing onto the stack</span>
 <a id="__codelineno-97-9" name="__codelineno-97-9" href="#__codelineno-97-9"></a><span class="w">        </span><span class="nx">stack</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">i</span><span class="p">);</span>
 <a id="__codelineno-97-10" name="__codelineno-97-10" href="#__codelineno-97-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-97-11" name="__codelineno-97-11" href="#__codelineno-97-11"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-97-11" name="__codelineno-97-11" href="#__codelineno-97-11"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-97-12" name="__codelineno-97-12" href="#__codelineno-97-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">stack</span><span class="p">.</span><span class="nx">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-97-13" name="__codelineno-97-13" href="#__codelineno-97-13"></a><span class="w">        </span><span class="c1">// Simulate "return" with "pop"</span>
+<a id="__codelineno-97-13" name="__codelineno-97-13" href="#__codelineno-97-13"></a><span class="w">        </span><span class="c1">// Simulate unwinding by popping from the stack</span>
 <a id="__codelineno-97-14" name="__codelineno-97-14" href="#__codelineno-97-14"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">stack</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-97-15" name="__codelineno-97-15" href="#__codelineno-97-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-97-16" name="__codelineno-97-16" href="#__codelineno-97-16"></a><span class="w">    </span><span class="c1">// res = 1+2+3+...+n</span>
@@ -6391,17 +6498,17 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.ts</span><pre><span></span><code><a id="__codelineno-98-1" name="__codelineno-98-1" href="#__codelineno-98-1"></a><span class="cm">/* Simulate recursion using iteration */</span>
 <a id="__codelineno-98-2" name="__codelineno-98-2" href="#__codelineno-98-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">forLoopRecur</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-98-3" name="__codelineno-98-3" href="#__codelineno-98-3"></a><span class="w">    </span><span class="c1">// Use an explicit stack to simulate the system call stack</span>
+<a id="__codelineno-98-3" name="__codelineno-98-3" href="#__codelineno-98-3"></a><span class="w">    </span><span class="c1">// Use an explicit stack to simulate the system call stack </span>
 <a id="__codelineno-98-4" name="__codelineno-98-4" href="#__codelineno-98-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">stack</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-98-5" name="__codelineno-98-5" href="#__codelineno-98-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-98-6" name="__codelineno-98-6" href="#__codelineno-98-6"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-98-6" name="__codelineno-98-6" href="#__codelineno-98-6"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-98-7" name="__codelineno-98-7" href="#__codelineno-98-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-98-8" name="__codelineno-98-8" href="#__codelineno-98-8"></a><span class="w">        </span><span class="c1">// Simulate "recurse" with "push"</span>
+<a id="__codelineno-98-8" name="__codelineno-98-8" href="#__codelineno-98-8"></a><span class="w">        </span><span class="c1">// Simulate recursive calls by pushing onto the stack</span>
 <a id="__codelineno-98-9" name="__codelineno-98-9" href="#__codelineno-98-9"></a><span class="w">        </span><span class="nx">stack</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">i</span><span class="p">);</span>
 <a id="__codelineno-98-10" name="__codelineno-98-10" href="#__codelineno-98-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-98-11" name="__codelineno-98-11" href="#__codelineno-98-11"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-98-11" name="__codelineno-98-11" href="#__codelineno-98-11"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-98-12" name="__codelineno-98-12" href="#__codelineno-98-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">stack</span><span class="p">.</span><span class="nx">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-98-13" name="__codelineno-98-13" href="#__codelineno-98-13"></a><span class="w">        </span><span class="c1">// Simulate "return" with "pop"</span>
+<a id="__codelineno-98-13" name="__codelineno-98-13" href="#__codelineno-98-13"></a><span class="w">        </span><span class="c1">// Simulate unwinding by popping from the stack</span>
 <a id="__codelineno-98-14" name="__codelineno-98-14" href="#__codelineno-98-14"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">stack</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-98-15" name="__codelineno-98-15" href="#__codelineno-98-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-98-16" name="__codelineno-98-16" href="#__codelineno-98-16"></a><span class="w">    </span><span class="c1">// res = 1+2+3+...+n</span>
@@ -6415,14 +6522,14 @@
 <a id="__codelineno-99-3" name="__codelineno-99-3" href="#__codelineno-99-3"></a><span class="w">  </span><span class="c1">// Use an explicit stack to simulate the system call stack</span>
 <a id="__codelineno-99-4" name="__codelineno-99-4" href="#__codelineno-99-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-99-5" name="__codelineno-99-5" href="#__codelineno-99-5"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
-<a id="__codelineno-99-6" name="__codelineno-99-6" href="#__codelineno-99-6"></a><span class="w">  </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-99-6" name="__codelineno-99-6" href="#__codelineno-99-6"></a><span class="w">  </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-99-7" name="__codelineno-99-7" href="#__codelineno-99-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-99-8" name="__codelineno-99-8" href="#__codelineno-99-8"></a><span class="w">    </span><span class="c1">// Simulate "recurse" with "push"</span>
+<a id="__codelineno-99-8" name="__codelineno-99-8" href="#__codelineno-99-8"></a><span class="w">    </span><span class="c1">// Simulate recursive calls by pushing onto the stack</span>
 <a id="__codelineno-99-9" name="__codelineno-99-9" href="#__codelineno-99-9"></a><span class="w">    </span><span class="n">stack</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-99-10" name="__codelineno-99-10" href="#__codelineno-99-10"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-99-11" name="__codelineno-99-11" href="#__codelineno-99-11"></a><span class="w">  </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-99-11" name="__codelineno-99-11" href="#__codelineno-99-11"></a><span class="w">  </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-99-12" name="__codelineno-99-12" href="#__codelineno-99-12"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">stack</span><span class="p">.</span><span class="n">isEmpty</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-99-13" name="__codelineno-99-13" href="#__codelineno-99-13"></a><span class="w">    </span><span class="c1">// Simulate "return" with "pop"</span>
+<a id="__codelineno-99-13" name="__codelineno-99-13" href="#__codelineno-99-13"></a><span class="w">    </span><span class="c1">// Simulate unwinding by popping from the stack</span>
 <a id="__codelineno-99-14" name="__codelineno-99-14" href="#__codelineno-99-14"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">removeLast</span><span class="p">();</span>
 <a id="__codelineno-99-15" name="__codelineno-99-15" href="#__codelineno-99-15"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-99-16" name="__codelineno-99-16" href="#__codelineno-99-16"></a><span class="w">  </span><span class="c1">// res = 1+2+3+...+n</span>
@@ -6436,14 +6543,14 @@
 <a id="__codelineno-100-3" name="__codelineno-100-3" href="#__codelineno-100-3"></a><span class="w">    </span><span class="c1">// Use an explicit stack to simulate the system call stack</span>
 <a id="__codelineno-100-4" name="__codelineno-100-4" href="#__codelineno-100-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-100-5" name="__codelineno-100-5" href="#__codelineno-100-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-100-6" name="__codelineno-100-6" href="#__codelineno-100-6"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-100-6" name="__codelineno-100-6" href="#__codelineno-100-6"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-100-7" name="__codelineno-100-7" href="#__codelineno-100-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="p">).</span><span class="n">rev</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-100-8" name="__codelineno-100-8" href="#__codelineno-100-8"></a><span class="w">        </span><span class="c1">// Simulate "recurse" with "push"</span>
+<a id="__codelineno-100-8" name="__codelineno-100-8" href="#__codelineno-100-8"></a><span class="w">        </span><span class="c1">// Simulate recursive calls by pushing onto the stack</span>
 <a id="__codelineno-100-9" name="__codelineno-100-9" href="#__codelineno-100-9"></a><span class="w">        </span><span class="n">stack</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-100-10" name="__codelineno-100-10" href="#__codelineno-100-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-100-11" name="__codelineno-100-11" href="#__codelineno-100-11"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-100-11" name="__codelineno-100-11" href="#__codelineno-100-11"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-100-12" name="__codelineno-100-12" href="#__codelineno-100-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="o">!</span><span class="n">stack</span><span class="p">.</span><span class="n">is_empty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-100-13" name="__codelineno-100-13" href="#__codelineno-100-13"></a><span class="w">        </span><span class="c1">// Simulate "return" with "pop"</span>
+<a id="__codelineno-100-13" name="__codelineno-100-13" href="#__codelineno-100-13"></a><span class="w">        </span><span class="c1">// Simulate unwinding by popping from the stack</span>
 <a id="__codelineno-100-14" name="__codelineno-100-14" href="#__codelineno-100-14"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">pop</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span>
 <a id="__codelineno-100-15" name="__codelineno-100-15" href="#__codelineno-100-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-100-16" name="__codelineno-100-16" href="#__codelineno-100-16"></a><span class="w">    </span><span class="c1">// res = 1+2+3+...+n</span>
@@ -6454,17 +6561,17 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">recursion.c</span><pre><span></span><code><a id="__codelineno-101-1" name="__codelineno-101-1" href="#__codelineno-101-1"></a><span class="cm">/* Simulate recursion using iteration */</span>
 <a id="__codelineno-101-2" name="__codelineno-101-2" href="#__codelineno-101-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">forLoopRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-101-3" name="__codelineno-101-3" href="#__codelineno-101-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">stack</span><span class="p">[</span><span class="mi">1000</span><span class="p">];</span><span class="w"> </span><span class="c1">// Use a large array to simulate stack</span>
-<a id="__codelineno-101-4" name="__codelineno-101-4" href="#__codelineno-101-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">top</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span><span class="w">    </span><span class="c1">// Stack top index</span>
+<a id="__codelineno-101-3" name="__codelineno-101-3" href="#__codelineno-101-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">stack</span><span class="p">[</span><span class="mi">1000</span><span class="p">];</span><span class="w"> </span><span class="c1">// Use a large array to simulate a stack</span>
+<a id="__codelineno-101-4" name="__codelineno-101-4" href="#__codelineno-101-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">top</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span><span class="w">    </span><span class="c1">// Top-of-stack index</span>
 <a id="__codelineno-101-5" name="__codelineno-101-5" href="#__codelineno-101-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-101-6" name="__codelineno-101-6" href="#__codelineno-101-6"></a><span class="w">    </span><span class="c1">// Recurse: recursive call</span>
+<a id="__codelineno-101-6" name="__codelineno-101-6" href="#__codelineno-101-6"></a><span class="w">    </span><span class="c1">// Recurse: make a recursive call</span>
 <a id="__codelineno-101-7" name="__codelineno-101-7" href="#__codelineno-101-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-101-8" name="__codelineno-101-8" href="#__codelineno-101-8"></a><span class="w">        </span><span class="c1">// Simulate "recurse" with "push"</span>
+<a id="__codelineno-101-8" name="__codelineno-101-8" href="#__codelineno-101-8"></a><span class="w">        </span><span class="c1">// Simulate recursive calls by pushing onto the stack</span>
 <a id="__codelineno-101-9" name="__codelineno-101-9" href="#__codelineno-101-9"></a><span class="w">        </span><span class="n">stack</span><span class="p">[</span><span class="mi">1</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">top</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-101-10" name="__codelineno-101-10" href="#__codelineno-101-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-101-11" name="__codelineno-101-11" href="#__codelineno-101-11"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-101-11" name="__codelineno-101-11" href="#__codelineno-101-11"></a><span class="w">    </span><span class="c1">// Unwind: return the result</span>
 <a id="__codelineno-101-12" name="__codelineno-101-12" href="#__codelineno-101-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">top</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-101-13" name="__codelineno-101-13" href="#__codelineno-101-13"></a><span class="w">        </span><span class="c1">// Simulate "return" with "pop"</span>
+<a id="__codelineno-101-13" name="__codelineno-101-13" href="#__codelineno-101-13"></a><span class="w">        </span><span class="c1">// Simulate unwinding by popping from the stack</span>
 <a id="__codelineno-101-14" name="__codelineno-101-14" href="#__codelineno-101-14"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">stack</span><span class="p">[</span><span class="n">top</span><span class="o">--</span><span class="p">];</span>
 <a id="__codelineno-101-15" name="__codelineno-101-15" href="#__codelineno-101-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-101-16" name="__codelineno-101-16" href="#__codelineno-101-16"></a><span class="w">    </span><span class="c1">// res = 1+2+3+...+n</span>
@@ -6478,14 +6585,14 @@
 <a id="__codelineno-102-3" name="__codelineno-102-3" href="#__codelineno-102-3"></a><span class="w">    </span><span class="c1">// Use an explicit stack to simulate the system call stack</span>
 <a id="__codelineno-102-4" name="__codelineno-102-4" href="#__codelineno-102-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Stack</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-102-5" name="__codelineno-102-5" href="#__codelineno-102-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-102-6" name="__codelineno-102-6" href="#__codelineno-102-6"></a><span class="w">    </span><span class="c1">// Descend: recursive call</span>
+<a id="__codelineno-102-6" name="__codelineno-102-6" href="#__codelineno-102-6"></a><span class="w">    </span><span class="c1">// Descending: recursive calls</span>
 <a id="__codelineno-102-7" name="__codelineno-102-7" href="#__codelineno-102-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="n">downTo</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-102-8" name="__codelineno-102-8" href="#__codelineno-102-8"></a><span class="w">        </span><span class="c1">// Simulate "recurse" with "push"</span>
+<a id="__codelineno-102-8" name="__codelineno-102-8" href="#__codelineno-102-8"></a><span class="w">        </span><span class="c1">// Simulate recursive calls by pushing onto the stack</span>
 <a id="__codelineno-102-9" name="__codelineno-102-9" href="#__codelineno-102-9"></a><span class="w">        </span><span class="n">stack</span><span class="p">.</span><span class="na">push</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-102-10" name="__codelineno-102-10" href="#__codelineno-102-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-102-11" name="__codelineno-102-11" href="#__codelineno-102-11"></a><span class="w">    </span><span class="c1">// Return: return result</span>
+<a id="__codelineno-102-11" name="__codelineno-102-11" href="#__codelineno-102-11"></a><span class="w">    </span><span class="c1">// Unwinding: return the result</span>
 <a id="__codelineno-102-12" name="__codelineno-102-12" href="#__codelineno-102-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">stack</span><span class="p">.</span><span class="na">isNotEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-102-13" name="__codelineno-102-13" href="#__codelineno-102-13"></a><span class="w">        </span><span class="c1">// Simulate "return" with "pop"</span>
+<a id="__codelineno-102-13" name="__codelineno-102-13" href="#__codelineno-102-13"></a><span class="w">        </span><span class="c1">// Simulate unwinding by popping from the stack</span>
 <a id="__codelineno-102-14" name="__codelineno-102-14" href="#__codelineno-102-14"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">pop</span><span class="p">()</span>
 <a id="__codelineno-102-15" name="__codelineno-102-15" href="#__codelineno-102-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-102-16" name="__codelineno-102-16" href="#__codelineno-102-16"></a><span class="w">    </span><span class="c1">// res = 1+2+3+...+n</span>
@@ -6494,18 +6601,18 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">recursion.rb</span><pre><span></span><code><a id="__codelineno-103-1" name="__codelineno-103-1" href="#__codelineno-103-1"></a><span class="c1">### Use iteration to simulate recursion ###</span>
+<div class="highlight"><span class="filename">recursion.rb</span><pre><span></span><code><a id="__codelineno-103-1" name="__codelineno-103-1" href="#__codelineno-103-1"></a><span class="c1">### Simulate recursion using iteration ###</span>
 <a id="__codelineno-103-2" name="__codelineno-103-2" href="#__codelineno-103-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">for_loop_recur</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-103-3" name="__codelineno-103-3" href="#__codelineno-103-3"></a><span class="w">  </span><span class="c1"># Use an explicit stack to simulate the system call stack</span>
 <a id="__codelineno-103-4" name="__codelineno-103-4" href="#__codelineno-103-4"></a><span class="w">  </span><span class="n">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-103-5" name="__codelineno-103-5" href="#__codelineno-103-5"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-103-6" name="__codelineno-103-6" href="#__codelineno-103-6"></a>
-<a id="__codelineno-103-7" name="__codelineno-103-7" href="#__codelineno-103-7"></a><span class="w">  </span><span class="c1"># Recurse: recursive call</span>
+<a id="__codelineno-103-7" name="__codelineno-103-7" href="#__codelineno-103-7"></a><span class="w">  </span><span class="c1"># Recurse: make a recursive call</span>
 <a id="__codelineno-103-8" name="__codelineno-103-8" href="#__codelineno-103-8"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">n</span><span class="o">.</span><span class="n">downto</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-103-9" name="__codelineno-103-9" href="#__codelineno-103-9"></a><span class="w">    </span><span class="c1"># Simulate "recurse" with "push"</span>
+<a id="__codelineno-103-9" name="__codelineno-103-9" href="#__codelineno-103-9"></a><span class="w">    </span><span class="c1"># Simulate recursive calls by pushing onto the stack</span>
 <a id="__codelineno-103-10" name="__codelineno-103-10" href="#__codelineno-103-10"></a><span class="w">    </span><span class="n">stack</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-103-11" name="__codelineno-103-11" href="#__codelineno-103-11"></a><span class="w">  </span><span class="k">end</span>
-<a id="__codelineno-103-12" name="__codelineno-103-12" href="#__codelineno-103-12"></a><span class="w">  </span><span class="c1"># Return: return result</span>
+<a id="__codelineno-103-12" name="__codelineno-103-12" href="#__codelineno-103-12"></a><span class="w">  </span><span class="c1"># Unwind: return the result</span>
 <a id="__codelineno-103-13" name="__codelineno-103-13" href="#__codelineno-103-13"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="o">!</span><span class="n">stack</span><span class="o">.</span><span class="n">empty?</span>
 <a id="__codelineno-103-14" name="__codelineno-103-14" href="#__codelineno-103-14"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">stack</span><span class="o">.</span><span class="n">pop</span>
 <a id="__codelineno-103-15" name="__codelineno-103-15" href="#__codelineno-103-15"></a><span class="w">  </span><span class="k">end</span>
@@ -6517,12 +6624,17 @@
 </div>
 </div>
 </div>
-<p>Observing the above code, when recursion is transformed into iteration, the code becomes more complex. Although iteration and recursion can be converted into each other in many cases, it may not be worthwhile to do so for the following two reasons.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20for_loop_recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Simulate%20recursion%20using%20iteration%22%22%22%0A%20%20%20%20%23%20Use%20an%20explicit%20stack%20to%20simulate%20the%20system%20call%20stack%0A%20%20%20%20stack%20%3D%20%5B%5D%0A%20%20%20%20res%20%3D%200%0A%20%20%20%20%23%20Recursive-call%20phase%0A%20%20%20%20for%20i%20in%20range%28n,%200,%20-1%29%3A%0A%20%20%20%20%20%20%20%20%23%20Simulate%20recursive%20calls%20by%20pushing%20onto%20the%20stack%0A%20%20%20%20%20%20%20%20stack.append%28i%29%0A%20%20%20%20%23%20Unwinding%20phase%3A%20compute%20the%20result%0A%20%20%20%20while%20stack%3A%0A%20%20%20%20%20%20%20%20%23%20Simulate%20unwinding%20by%20popping%20from%20the%20stack%0A%20%20%20%20%20%20%20%20res%20%2B%3D%20stack.pop%28%29%0A%20%20%20%20%23%20res%20%3D%201%2B2%2B3%2B...%2Bn%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20for_loop_recur%28n%29%0A%20%20%20%20print%28f%22%5CnSum%20from%20simulating%20recursion%20using%20iteration%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20for_loop_recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Simulate%20recursion%20using%20iteration%22%22%22%0A%20%20%20%20%23%20Use%20an%20explicit%20stack%20to%20simulate%20the%20system%20call%20stack%0A%20%20%20%20stack%20%3D%20%5B%5D%0A%20%20%20%20res%20%3D%200%0A%20%20%20%20%23%20Recursive-call%20phase%0A%20%20%20%20for%20i%20in%20range%28n,%200,%20-1%29%3A%0A%20%20%20%20%20%20%20%20%23%20Simulate%20recursive%20calls%20by%20pushing%20onto%20the%20stack%0A%20%20%20%20%20%20%20%20stack.append%28i%29%0A%20%20%20%20%23%20Unwinding%20phase%3A%20compute%20the%20result%0A%20%20%20%20while%20stack%3A%0A%20%20%20%20%20%20%20%20%23%20Simulate%20unwinding%20by%20popping%20from%20the%20stack%0A%20%20%20%20%20%20%20%20res%20%2B%3D%20stack.pop%28%29%0A%20%20%20%20%23%20res%20%3D%201%2B2%2B3%2B...%2Bn%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20res%20%3D%20for_loop_recur%28n%29%0A%20%20%20%20print%28f%22%5CnSum%20from%20simulating%20recursion%20using%20iteration%3A%20res%20%3D%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>The resulting iterative code is more complicated. Although iteration and recursion can often be converted into each other, doing so is not always worthwhile.</p>
 <ul>
-<li>The transformed code may be more difficult to understand and less readable.</li>
-<li>For some complex problems, simulating the behavior of the system call stack can be very difficult.</li>
+<li>The converted code may be harder to understand and less readable.</li>
+<li>Simulating the call stack can be very difficult for complex problems.</li>
 </ul>
-<p>In summary, <strong>choosing between iteration and recursion depends on the nature of the specific problem</strong>. In programming practice, it is crucial to weigh the pros and cons of both and choose the appropriate method based on the context.</p>
+<p><strong>The choice between iteration and recursion depends on the problem</strong>. In practice, weigh their strengths and weaknesses and choose the approach that fits the situation.</p>
 
 <!-- Source file information -->
 
@@ -6545,7 +6657,7 @@ aria-label="Footer"
   <a
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     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 2.1 Algorithm Efficiency Evaluation"
+    aria-label="Previous: 2.1   Evaluating algorithm efficiency"
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@@ -6557,7 +6669,7 @@ aria-label="Footer"
         Previous
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+        2.1 &nbsp; Evaluating algorithm efficiency
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@@ -6569,7 +6681,7 @@ aria-label="Footer"
   <a
     href="../time_complexity/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 2.3 Time Complexity"
+    aria-label="Next: 2.3   Time complexity"
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@@ -6577,7 +6689,7 @@ aria-label="Footer"
         Next
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+        2.3 &nbsp; Time complexity
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     <div class="md-footer__button md-icon">
@@ -6594,7 +6706,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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diff --git a/en/chapter_computational_complexity/performance_evaluation/index.html b/en/chapter_computational_complexity/performance_evaluation/index.html
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+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>2.1 Algorithm Efficiency Evaluation - Hello Algo</title>
+        <title>2.1 Evaluating algorithm efficiency - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
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-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
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@@ -101,7 +101,7 @@
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-        <a href="#21-algorithm-efficiency-evaluation" class="md-skip">
+        <a href="#21-evaluating-algorithm-efficiency" class="md-skip">
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
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-  Before Starting
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-  Before Starting
+  Foreword
 
           </label>
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-  Chapter 0. Preface
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-  Chapter 0. Preface
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-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
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-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
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-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
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-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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         <li class="md-nav__item">
-  <a href="#211-actual-testing" class="md-nav__link">
+  <a href="#211-practical-testing" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.1.1 &nbsp; Actual Testing
+        2.1.1 &nbsp; Practical testing
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+        2.1.2 &nbsp; Theoretical estimation
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     </span>
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-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
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+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
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-  3.5 Summary
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
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+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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+  6.4 &nbsp; Summary
 
 
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+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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+  Chapter 7. &nbsp; Trees
 
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-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  11.9 Counting Sort
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+  11.10 &nbsp; Radix sort
 
 
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+  12.3 &nbsp; Building a binary tree
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 15. Greedy
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+  15.3 &nbsp; Maximum capacity problem
 
 
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+  15.6 &nbsp; Exercises
 
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#211-actual-testing" class="md-nav__link">
+  <a href="#211-practical-testing" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.1.1 &nbsp; Actual Testing
+        2.1.1 &nbsp; Practical testing
       </span>
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-        2.1.2 &nbsp; Theoretical Estimation
+        2.1.2 &nbsp; Theoretical estimation
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 <!-- Page content -->
-<h1 id="21-algorithm-efficiency-evaluation">2.1 &nbsp; Algorithm Efficiency Evaluation<a class="headerlink" href="#21-algorithm-efficiency-evaluation" title="Permanent link">&para;</a></h1>
-<p>In algorithm design, we pursue the following two levels of objectives sequentially.</p>
+<h1 id="21-evaluating-algorithm-efficiency">2.1 &nbsp; Evaluating algorithm efficiency<a class="headerlink" href="#21-evaluating-algorithm-efficiency" title="Permanent link">&para;</a></h1>
+<p>Algorithm design pursues two goals in sequence.</p>
 <ol>
-<li><strong>Finding a solution to the problem</strong>: The algorithm must reliably obtain the correct solution within the specified input range.</li>
-<li><strong>Seeking the optimal solution</strong>: Multiple solutions may exist for the same problem, and we hope to find an algorithm that is as efficient as possible.</li>
+<li><strong>Find a solution</strong>: The algorithm must reliably produce correct results across the specified input range.</li>
+<li><strong>Find the best solution</strong>: When several approaches exist, we want the most efficient algorithm possible.</li>
 </ol>
-<p>In other words, under the premise of being able to solve the problem, algorithm efficiency has become the primary evaluation criterion for measuring the quality of algorithms. It includes the following two dimensions.</p>
+<p>Once correctness is established, efficiency becomes a primary measure of an algorithm's quality. It has two dimensions.</p>
 <ul>
-<li><strong>Time efficiency</strong>: The length of time the algorithm runs.</li>
-<li><strong>Space efficiency</strong>: The size of memory space the algorithm occupies.</li>
+<li><strong>Time efficiency</strong>: How long the algorithm takes to run.</li>
+<li><strong>Space efficiency</strong>: How much memory it uses.</li>
 </ul>
-<p>In short, <strong>our goal is to design data structures and algorithms that are "both fast and memory-efficient"</strong>. Effectively evaluating algorithm efficiency is crucial, because only in this way can we compare various algorithms and guide the algorithm design and optimization process.</p>
-<p>Efficiency evaluation methods are mainly divided into two types: actual testing and theoretical estimation.</p>
-<h2 id="211-actual-testing">2.1.1 &nbsp; Actual Testing<a class="headerlink" href="#211-actual-testing" title="Permanent link">&para;</a></h2>
-<p>Suppose we now have algorithm <code>A</code> and algorithm <code>B</code>, both of which can solve the same problem, and we need to compare their efficiency. The most direct method is to run them on a computer and measure their running time and memory usage. This evaluation approach can reflect real-world behavior, but it also has considerable limitations.</p>
-<p>On one hand, <strong>it is difficult to eliminate interference factors from the testing environment</strong>. Hardware configuration affects algorithmic performance. For example, if an algorithm has a high degree of parallelism, it is more suitable for running on multi-core CPUs; if an algorithm performs memory-intensive operations, it will benefit more from high-performance memory. In other words, the test results of an algorithm on different machines may be inconsistent. This means we need to test on various machines and calculate average efficiency, which is impractical.</p>
-<p>On the other hand, <strong>conducting complete testing is very resource-intensive</strong>. As the input data volume changes, the algorithm will exhibit different efficiencies. For example, when the input data volume is small, the running time of algorithm <code>A</code> is shorter than algorithm <code>B</code>; but when the input data volume is large, the test results may be exactly the opposite. Therefore, to obtain convincing conclusions, we need to test input data of various scales, which requires a large amount of computational resources.</p>
-<h2 id="212-theoretical-estimation">2.1.2 &nbsp; Theoretical Estimation<a class="headerlink" href="#212-theoretical-estimation" title="Permanent link">&para;</a></h2>
-<p>Since actual testing has considerable limitations, we can consider evaluating algorithm efficiency through theoretical calculation. This estimation method is called <u>asymptotic complexity analysis</u>, or <u>complexity analysis</u> for short.</p>
-<p>Complexity analysis can reflect the relationship between the time and space resources required for algorithm execution and the input data scale. <strong>It describes the growth trend of the time and space required for algorithm execution as the input data scale increases</strong>. This definition is a bit cumbersome, so we can break it down into three key points to understand.</p>
+<p><strong>Our goal is to design data structures and algorithms that are both fast and economical with memory</strong>. Evaluating efficiency effectively lets us compare algorithms and guide their design and optimization.</p>
+<p>There are two main evaluation methods: practical testing and theoretical estimation.</p>
+<h2 id="211-practical-testing">2.1.1 &nbsp; Practical testing<a class="headerlink" href="#211-practical-testing" title="Permanent link">&para;</a></h2>
+<p>Suppose algorithms <code>A</code> and <code>B</code> solve the same problem. The most direct comparison is to run both on a computer and measure runtime and memory use. This reflects actual behavior, but has substantial limitations.</p>
+<p>First, <strong>it is difficult to remove the influence of the test environment</strong>. Hardware affects performance: a highly parallel algorithm benefits from a multicore CPU, while a memory-intensive algorithm benefits from faster RAM. Results may differ across machines, so a fair average would require testing on many kinds of hardware, which is impractical.</p>
+<p>Second, <strong>comprehensive testing consumes substantial resources</strong>. Efficiency changes with input size. Algorithm <code>A</code> may be faster than <code>B</code> for small inputs, while the reverse holds for large inputs. A convincing conclusion requires testing many input sizes, consuming considerable computing resources.</p>
+<h2 id="212-theoretical-estimation">2.1.2 &nbsp; Theoretical estimation<a class="headerlink" href="#212-theoretical-estimation" title="Permanent link">&para;</a></h2>
+<p>Given these limitations, we can estimate efficiency through calculation alone. This is called <u>asymptotic complexity analysis</u>, or simply <u>complexity analysis</u>.</p>
+<p>Complexity analysis relates an algorithm's time and space requirements to its input size. <strong>It describes how runtime and memory use grow as the input gets larger</strong>. We can unpack this definition into three points.</p>
 <ul>
-<li>"Time and space resources" correspond to <u>time complexity</u> and <u>space complexity</u>, respectively.</li>
-<li>"As the input data scale increases" means that complexity reflects the relationship between algorithm running efficiency and input data scale.</li>
-<li>"Growth trend of time and space" indicates that complexity analysis focuses not on the specific values of running time or occupied space, but on how "fast" time or space grows.</li>
+<li>Time and space resources correspond to <u>time complexity</u> and <u>space complexity</u>.</li>
+<li>Growth in input size means complexity relates efficiency to the amount of input data.</li>
+<li>Growth trends mean we study how quickly time or space increases, rather than its exact value.</li>
 </ul>
-<p><strong>Complexity analysis overcomes the drawbacks of the actual testing method</strong>, reflected in the following aspects.</p>
+<p><strong>Complexity analysis overcomes limitations of practical testing</strong> in several ways.</p>
 <ul>
-<li>It does not need to actually run the code, making it more environmentally friendly and energy-efficient.</li>
-<li>It is independent of the testing environment, and the analysis results are applicable to all running platforms.</li>
-<li>It can reflect algorithm efficiency at different data volumes, especially algorithm performance at large data volumes.</li>
+<li>It does not require running code, saving energy and computing resources.</li>
+<li>It is independent of the test environment, so its conclusions apply across platforms.</li>
+<li>It reveals efficiency at different input sizes, especially for large inputs.</li>
 </ul>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>If you are still confused about the concept of complexity, don't worry—we will introduce it in detail in subsequent chapters.</p>
+<p>If complexity still feels unclear, do not worry. The following sections will explain it in detail.</p>
 </div>
-<p>Complexity analysis provides us with a "ruler" for evaluating algorithm efficiency, allowing us to measure the time and space resources required to execute a certain algorithm and compare the efficiency between different algorithms.</p>
-<p>Complexity is a mathematical concept that may feel abstract and challenging for beginners. From this perspective, complexity analysis may not be the most suitable topic to introduce first. However, when we discuss the characteristics of a certain data structure or algorithm, it is difficult to avoid analyzing its running speed and space usage.</p>
-<p>In summary, it is recommended that before diving deep into data structures and algorithms, <strong>you first establish a preliminary understanding of complexity analysis so that you can analyze the complexity of simple algorithms</strong>.</p>
+<p>Complexity analysis gives us a yardstick for evaluating the time and space an algorithm needs and comparing different algorithms.</p>
+<p>As a mathematical concept, complexity can be abstract and challenging for beginners. This makes it a difficult topic to introduce first. Yet discussing a data structure or algorithm's properties inevitably involves runtime and memory use.</p>
+<p>Before studying data structures and algorithms in depth, <strong>build an initial understanding of complexity analysis so that you can analyze simple algorithms</strong>.</p>
 
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@@ -4717,7 +4789,7 @@ aria-label="Footer"
   <a
     href="../"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: Chapter 2. &amp;nbsp; Complexity Analysis"
+    aria-label="Previous: Chapter 2.   Complexity analysis"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4729,7 +4801,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        Chapter 2. &nbsp; Complexity Analysis
+        Chapter 2. &nbsp; Complexity analysis
       </div>
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   </a>
@@ -4741,7 +4813,7 @@ aria-label="Footer"
   <a
     href="../iteration_and_recursion/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 2.2 Iteration and Recursion"
+    aria-label="Next: 2.2   Iteration and recursion"
     rel="next"
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@@ -4749,7 +4821,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        2.2 Iteration and Recursion
+        2.2 &nbsp; Iteration and recursion
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@@ -4766,7 +4838,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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-        <title>2.4 Space Complexity - Hello Algo</title>
+        <title>2.4 Space complexity - Hello Algo</title>
 
 
 
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-  Before Starting
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-  Chapter 0. Preface
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   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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-  Chapter 1. Encounter with Algorithms
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   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
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-  2.1 Algorithm Efficiency Evaluation
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-  2.2 Iteration and Recursion
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#241-algorithm-related-space" class="md-nav__link">
+  <a href="#241-space-used-by-an-algorithm" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
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   </a>
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 </li>
 
         <li class="md-nav__item">
-  <a href="#242-calculation-method" class="md-nav__link">
+  <a href="#242-deriving-space-complexity" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.4.2 &nbsp; Calculation Method
+        2.4.2 &nbsp; Deriving space complexity
       </span>
     </span>
   </a>
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 </li>
 
         <li class="md-nav__item">
-  <a href="#243-common-types" class="md-nav__link">
+  <a href="#243-common-complexity-classes" class="md-nav__link">
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       <span class="md-typeset">
-        2.4.3 &nbsp; Common Types
+        2.4.3 &nbsp; Common complexity classes
       </span>
     </span>
   </a>
 
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+    <nav class="md-nav" aria-label="2.4.3 Common complexity classes">
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-  <a href="#1-constant-order-o1" class="md-nav__link">
+  <a href="#1-constant-space-o1" class="md-nav__link">
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-        1. &nbsp; Constant Order
+        1. &nbsp; Constant space
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-  <a href="#2-linear-order-on" class="md-nav__link">
+  <a href="#2-linear-space-on" class="md-nav__link">
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-        2. &nbsp; Linear Order
+        2. &nbsp; Linear space
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-  <a href="#3-quadratic-order-on2" class="md-nav__link">
+  <a href="#3-quadratic-space-on2" class="md-nav__link">
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-        3. &nbsp; Quadratic Order
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-  <a href="#4-exponential-order-o2n" class="md-nav__link">
+  <a href="#4-exponential-space-o2n" class="md-nav__link">
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-        4. &nbsp; Exponential Order
+        4. &nbsp; Exponential space
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-  <a href="#5-logarithmic-order-olog-n" class="md-nav__link">
+  <a href="#5-logarithmic-space-olog-n" class="md-nav__link">
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-        5. &nbsp; Logarithmic Order
+        5. &nbsp; Logarithmic space
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+  <a href="#244-balancing-time-and-space" class="md-nav__link">
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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+  4.2 &nbsp; Linked lists
 
 
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+  4.3 &nbsp; Lists
 
 
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+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
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+  5.2 &nbsp; Queues
 
 
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+  5.3 &nbsp; Deques
 
 
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+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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@@ -4682,10 +4754,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#241-algorithm-related-space" class="md-nav__link">
+  <a href="#241-space-used-by-an-algorithm" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.4.1 &nbsp; Algorithm-Related Space
+        2.4.1 &nbsp; Space used by an algorithm
       </span>
     </span>
   </a>
@@ -4693,10 +4765,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#242-calculation-method" class="md-nav__link">
+  <a href="#242-deriving-space-complexity" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.4.2 &nbsp; Calculation Method
+        2.4.2 &nbsp; Deriving space complexity
       </span>
     </span>
   </a>
@@ -4704,22 +4776,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#243-common-types" class="md-nav__link">
+  <a href="#243-common-complexity-classes" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.4.3 &nbsp; Common Types
+        2.4.3 &nbsp; Common complexity classes
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="2.4.3 Common Types">
+    <nav class="md-nav" aria-label="2.4.3 Common complexity classes">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-constant-order-o1" class="md-nav__link">
+  <a href="#1-constant-space-o1" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Constant Order
+        1. &nbsp; Constant space
       </span>
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   </a>
@@ -4727,10 +4799,10 @@
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           <li class="md-nav__item">
-  <a href="#2-linear-order-on" class="md-nav__link">
+  <a href="#2-linear-space-on" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Linear Order
+        2. &nbsp; Linear space
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   </a>
@@ -4738,10 +4810,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-quadratic-order-on2" class="md-nav__link">
+  <a href="#3-quadratic-space-on2" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Quadratic Order
+        3. &nbsp; Quadratic space
       </span>
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@@ -4749,10 +4821,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#4-exponential-order-o2n" class="md-nav__link">
+  <a href="#4-exponential-space-o2n" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4. &nbsp; Exponential Order
+        4. &nbsp; Exponential space
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@@ -4760,10 +4832,10 @@
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           <li class="md-nav__item">
-  <a href="#5-logarithmic-order-olog-n" class="md-nav__link">
+  <a href="#5-logarithmic-space-olog-n" class="md-nav__link">
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       <span class="md-typeset">
-        5. &nbsp; Logarithmic Order
+        5. &nbsp; Logarithmic space
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@@ -4776,10 +4848,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#244-trading-time-for-space" class="md-nav__link">
+  <a href="#244-balancing-time-and-space" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.4.4 &nbsp; Trading Time for Space
+        2.4.4 &nbsp; Balancing time and space
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@@ -4823,27 +4895,27 @@
 
 
 <!-- Page content -->
-<h1 id="24-space-complexity">2.4 &nbsp; Space Complexity<a class="headerlink" href="#24-space-complexity" title="Permanent link">&para;</a></h1>
-<p><u>Space complexity</u> measures the growth trend of memory space occupied by an algorithm as the data size increases. This concept is very similar to time complexity, except that "running time" is replaced with "occupied memory space".</p>
-<h2 id="241-algorithm-related-space">2.4.1 &nbsp; Algorithm-Related Space<a class="headerlink" href="#241-algorithm-related-space" title="Permanent link">&para;</a></h2>
-<p>The memory space used by an algorithm during execution mainly includes the following types.</p>
+<h1 id="24-space-complexity">2.4 &nbsp; Space complexity<a class="headerlink" href="#24-space-complexity" title="Permanent link">&para;</a></h1>
+<p><u>Space complexity</u> measures how an algorithm's memory use grows as the input gets larger. It is much like time complexity, with memory usage replacing runtime.</p>
+<h2 id="241-space-used-by-an-algorithm">2.4.1 &nbsp; Space used by an algorithm<a class="headerlink" href="#241-space-used-by-an-algorithm" title="Permanent link">&para;</a></h2>
+<p>The memory an algorithm uses during execution falls into these main categories.</p>
 <ul>
-<li><strong>Input space</strong>: Used to store the input data of the algorithm.</li>
-<li><strong>Temporary space</strong>: Used to store variables, objects, function contexts, and other data during the algorithm's execution.</li>
-<li><strong>Output space</strong>: Used to store the output data of the algorithm.</li>
+<li><strong>Input space</strong>: Stores the input data.</li>
+<li><strong>Temporary space</strong>: Stores variables, objects, function contexts, and other working data.</li>
+<li><strong>Output space</strong>: Stores the output data.</li>
 </ul>
-<p>In general, the scope of space complexity statistics is "temporary space" plus "output space".</p>
-<p>Temporary space can be further divided into three parts.</p>
+<p>Space complexity generally counts temporary space plus output space.</p>
+<p>Temporary space can be divided further into three parts.</p>
 <ul>
-<li><strong>Temporary data</strong>: Used to save various constants, variables, objects, etc., during the algorithm's execution.</li>
-<li><strong>Stack frame space</strong>: Used to save the context data of called functions. The system creates a stack frame at the top of the stack each time a function is called, and the stack frame space is released after the function returns.</li>
-<li><strong>Instruction space</strong>: Used to save compiled program instructions, which are usually ignored in actual statistics.</li>
+<li><strong>Temporary data</strong>: Constants, variables, objects, and other data used during execution.</li>
+<li><strong>Stack frames</strong>: Context data for function calls. Each call creates a frame at the top of the stack, which is released when the function returns.</li>
+<li><strong>Instruction space</strong>: Compiled program instructions, usually ignored in the analysis.</li>
 </ul>
-<p>When analyzing the space complexity of a program, <strong>we usually consider three parts: temporary data, stack frame space, and output data</strong>, as shown in the following figure.</p>
-<p><img alt="Algorithm-related space" class="animation-figure" src="../space_complexity.assets/space_types.png" /></p>
-<p align="center"> Figure 2-15 &nbsp; Algorithm-related space </p>
+<p><strong>We usually count temporary data, stack frames, and output data</strong> when analyzing space complexity, as shown in Figure 2-15.</p>
+<p><img alt="Space used by an algorithm" class="animation-figure" src="../space_complexity.assets/space_types.png" /></p>
+<p align="center"> Figure 2-15 &nbsp; Space used by an algorithm </p>
 
-<p>The related code is as follows:</p>
+<p>The corresponding code is:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -4859,10 +4931,10 @@
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="k">return</span> <span class="mi">0</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="k">def</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>  <span class="c1"># Input data</span>
-<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>    <span class="n">A</span> <span class="o">=</span> <span class="mi">0</span>                 <span class="c1"># Temporary data (constant, usually represented by uppercase letters)</span>
+<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>    <span class="n">A</span> <span class="o">=</span> <span class="mi">0</span>                 <span class="c1"># Temporary data (constant, conventionally named in uppercase)</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>    <span class="n">b</span> <span class="o">=</span> <span class="mi">0</span>                 <span class="c1"># Temporary data (variable)</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="n">node</span> <span class="o">=</span> <span class="n">Node</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>        <span class="c1"># Temporary data (object)</span>
-<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>    <span class="n">c</span> <span class="o">=</span> <span class="n">function</span><span class="p">()</span>        <span class="c1"># Stack frame space (function call)</span>
+<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>    <span class="n">c</span> <span class="o">=</span> <span class="n">function</span><span class="p">()</span>        <span class="c1"># Stack frame (function call)</span>
 <a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>    <span class="k">return</span> <span class="n">A</span> <span class="o">+</span> <span class="n">b</span> <span class="o">+</span> <span class="n">c</span>      <span class="c1"># Output data</span>
 </code></pre></div>
 </div>
@@ -4884,7 +4956,7 @@
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">    </span><span class="k">const</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">          </span><span class="c1">// Temporary data (constant)</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">                </span><span class="c1">// Temporary data (variable)</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">    </span><span class="n">Node</span><span class="o">*</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Node</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// Temporary data (object)</span>
-<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">func</span><span class="p">();</span><span class="w">           </span><span class="c1">// Stack frame space (function call)</span>
+<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">func</span><span class="p">();</span><span class="w">           </span><span class="c1">// Stack frame (function call)</span>
 <a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">c</span><span class="p">;</span><span class="w">         </span><span class="c1">// Output data</span>
 <a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="p">}</span>
 </code></pre></div>
@@ -4907,7 +4979,7 @@
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">          </span><span class="c1">// Temporary data (constant)</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">                </span><span class="c1">// Temporary data (variable)</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">    </span><span class="n">Node</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Node</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w">  </span><span class="c1">// Temporary data (object)</span>
-<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">function</span><span class="p">();</span><span class="w">       </span><span class="c1">// Stack frame space (function call)</span>
+<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">function</span><span class="p">();</span><span class="w">       </span><span class="c1">// Stack frame (function call)</span>
 <a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">c</span><span class="p">;</span><span class="w">         </span><span class="c1">// Output data</span>
 <a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="p">}</span>
 </code></pre></div>
@@ -4929,7 +5001,7 @@
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="k">const</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">          </span><span class="c1">// Temporary data (constant)</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">                </span><span class="c1">// Temporary data (variable)</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">    </span><span class="n">Node</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w">       </span><span class="c1">// Temporary data (object)</span>
-<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Function</span><span class="p">();</span><span class="w">       </span><span class="c1">// Stack frame space (function call)</span>
+<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Function</span><span class="p">();</span><span class="w">       </span><span class="c1">// Stack frame (function call)</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">c</span><span class="p">;</span><span class="w">         </span><span class="c1">// Output data</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="p">}</span>
 </code></pre></div>
@@ -4941,7 +5013,7 @@
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">next</span><span class="w"> </span><span class="o">*</span><span class="nx">node</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="p">}</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="cm">/* Create node structure */</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="cm">/* Create a node structure  */</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="kd">func</span><span class="w"> </span><span class="nx">newNode</span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">node</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">node</span><span class="p">{</span><span class="nx">val</span><span class="p">:</span><span class="w"> </span><span class="nx">val</span><span class="p">}</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="p">}</span>
@@ -4956,7 +5028,7 @@
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w">             </span><span class="c1">// Temporary data (constant)</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">    </span><span class="nx">b</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="w">                  </span><span class="c1">// Temporary data (variable)</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">    </span><span class="nx">newNode</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="w">              </span><span class="c1">// Temporary data (object)</span>
-<a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">    </span><span class="nx">c</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">function</span><span class="p">()</span><span class="w">         </span><span class="c1">// Stack frame space (function call)</span>
+<a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">    </span><span class="nx">c</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">function</span><span class="p">()</span><span class="w">         </span><span class="c1">// Stack frame (function call)</span>
 <a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">b</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">c</span><span class="w">        </span><span class="c1">// Output data</span>
 <a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="p">}</span>
 </code></pre></div>
@@ -4982,7 +5054,7 @@
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span><span class="w">             </span><span class="c1">// Temporary data (constant)</span>
 <a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">b</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span><span class="w">             </span><span class="c1">// Temporary data (variable)</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Node</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// Temporary data (object)</span>
-<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">c</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">function</span><span class="p">()</span><span class="w">    </span><span class="c1">// Stack frame space (function call)</span>
+<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">c</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">function</span><span class="p">()</span><span class="w">    </span><span class="c1">// Stack frame (function call)</span>
 <a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">c</span><span class="w">      </span><span class="c1">// Output data</span>
 <a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="p">}</span>
 </code></pre></div>
@@ -5008,7 +5080,7 @@
 <a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w">              </span><span class="c1">// Temporary data (constant)</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w">                </span><span class="c1">// Temporary data (variable)</span>
 <a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Node</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// Temporary data (object)</span>
-<a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">constFunc</span><span class="p">();</span><span class="w">    </span><span class="c1">// Stack frame space (function call)</span>
+<a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">constFunc</span><span class="p">();</span><span class="w">    </span><span class="c1">// Stack frame (function call)</span>
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">b</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">c</span><span class="p">;</span><span class="w">         </span><span class="c1">// Output data</span>
 <a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="p">}</span>
 </code></pre></div>
@@ -5034,7 +5106,7 @@
 <a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w">                        </span><span class="c1">// Temporary data (constant)</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w">                          </span><span class="c1">// Temporary data (variable)</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Node</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span><span class="w">           </span><span class="c1">// Temporary data (object)</span>
-<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">constFunc</span><span class="p">();</span><span class="w">              </span><span class="c1">// Stack frame space (function call)</span>
+<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">constFunc</span><span class="p">();</span><span class="w">              </span><span class="c1">// Stack frame (function call)</span>
 <a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">b</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">c</span><span class="p">;</span><span class="w">                   </span><span class="c1">// Output data</span>
 <a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="p">}</span>
 </code></pre></div>
@@ -5057,7 +5129,7 @@
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="k">const</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w">      </span><span class="c1">// Temporary data (constant)</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w">            </span><span class="c1">// Temporary data (variable)</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">  </span><span class="n">Node</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Node</span><span class="p">(</span><span class="m">0</span><span class="p">);</span><span class="w">  </span><span class="c1">// Temporary data (object)</span>
-<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">function</span><span class="p">();</span><span class="w">   </span><span class="c1">// Stack frame space (function call)</span>
+<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">function</span><span class="p">();</span><span class="w">   </span><span class="c1">// Stack frame (function call)</span>
 <a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">c</span><span class="p">;</span><span class="w">     </span><span class="c1">// Output data</span>
 <a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="p">}</span>
 </code></pre></div>
@@ -5072,7 +5144,7 @@
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="n">next</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">Node</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="p">}</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a>
-<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="cm">/* Create Node structure */</span>
+<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="cm">/* Create a Node structure */</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="k">impl</span><span class="w"> </span><span class="n">Node</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">new</span><span class="p">(</span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">Self</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">        </span><span class="bp">Self</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="nc">val</span><span class="p">,</span><span class="w"> </span><span class="n">next</span><span class="p">:</span><span class="w"> </span><span class="nb">None</span><span class="w"> </span><span class="p">}</span>
@@ -5080,7 +5152,7 @@
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="p">}</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="cm">/* Function */</span>
-<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="k">fn</span><span class="w"> </span><span class="nf">function</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
+<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="k">fn</span><span class="w"> </span><span class="nf">function</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span><span class="w">      </span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">    </span><span class="c1">// Perform some operations...</span>
 <a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="p">}</span>
@@ -5089,7 +5161,7 @@
 <a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a><span class="w">    </span><span class="k">const</span><span class="w"> </span><span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">               </span><span class="c1">// Temporary data (constant)</span>
 <a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">                  </span><span class="c1">// Temporary data (variable)</span>
 <a id="__codelineno-9-26" name="__codelineno-9-26" href="#__codelineno-9-26"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Node</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w">        </span><span class="c1">// Temporary data (object)</span>
-<a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">function</span><span class="p">();</span><span class="w">             </span><span class="c1">// Stack frame space (function call)</span>
+<a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">function</span><span class="p">();</span><span class="w">             </span><span class="c1">// Stack frame (function call)</span>
 <a id="__codelineno-9-28" name="__codelineno-9-28" href="#__codelineno-9-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">c</span><span class="p">;</span><span class="w">               </span><span class="c1">// Output data</span>
 <a id="__codelineno-9-29" name="__codelineno-9-29" href="#__codelineno-9-29"></a><span class="p">}</span>
 </code></pre></div>
@@ -5104,7 +5176,7 @@
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="kt">int</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// Input data</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="k">const</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">   </span><span class="c1">// Temporary data (constant)</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">         </span><span class="c1">// Temporary data (variable)</span>
-<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">func</span><span class="p">();</span><span class="w">    </span><span class="c1">// Stack frame space (function call)</span>
+<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">func</span><span class="p">();</span><span class="w">    </span><span class="c1">// Stack frame (function call)</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">c</span><span class="p">;</span><span class="w">  </span><span class="c1">// Output data</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="p">}</span>
 </code></pre></div>
@@ -5125,7 +5197,7 @@
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w">                </span><span class="c1">// Temporary data (constant)</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w">                </span><span class="c1">// Temporary data (variable)</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Node</span><span class="p">(</span><span class="m">0</span><span class="p">)</span><span class="w">       </span><span class="c1">// Temporary data (object)</span>
-<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">function</span><span class="p">()</span><span class="w">       </span><span class="c1">// Stack frame space (function call)</span>
+<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">function</span><span class="p">()</span><span class="w">       </span><span class="c1">// Stack frame (function call)</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">c</span><span class="w">         </span><span class="c1">// Output data</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="p">}</span>
 </code></pre></div>
@@ -5152,20 +5224,20 @@
 <a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w">               </span><span class="c1"># Temporary data (constant)</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="w">    </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w">               </span><span class="c1"># Temporary data (variable)</span>
 <a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">Node</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="w">  </span><span class="c1"># Temporary data (object)</span>
-<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w">    </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">function</span><span class="w">        </span><span class="c1"># Stack frame space (function call)</span>
+<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w">    </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">function</span><span class="w">        </span><span class="c1"># Stack frame (function call)</span>
 <a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">c</span><span class="w">           </span><span class="c1"># Output data</span>
 <a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h2 id="242-calculation-method">2.4.2 &nbsp; Calculation Method<a class="headerlink" href="#242-calculation-method" title="Permanent link">&para;</a></h2>
-<p>The calculation method for space complexity is roughly the same as for time complexity, except that what we measure changes from the "number of operations" to the "amount of space used".</p>
-<p>Unlike time complexity, <strong>we usually only focus on the worst-case space complexity</strong>. This is because memory space is a hard requirement, and we must ensure that sufficient memory space is reserved for all input data.</p>
-<p>Observe the following code. Here, "worst case" in worst-case space complexity has two meanings.</p>
+<h2 id="242-deriving-space-complexity">2.4.2 &nbsp; Deriving space complexity<a class="headerlink" href="#242-deriving-space-complexity" title="Permanent link">&para;</a></h2>
+<p>The approach is much like time complexity analysis, except that we count memory used rather than operations performed.</p>
+<p>Unlike time complexity, <strong>we usually focus only on worst-case space complexity</strong>. Memory is a hard requirement: enough must be available for every valid input.</p>
+<p>In the following code, "worst case" has two meanings.</p>
 <ol>
-<li><strong>Based on the worst input data</strong>: When <span class="arithmatex">\(n &lt; 10\)</span>, the space complexity is <span class="arithmatex">\(O(1)\)</span>; but when <span class="arithmatex">\(n &gt; 10\)</span>, the initialized array <code>nums</code> occupies <span class="arithmatex">\(O(n)\)</span> space, so the worst-case space complexity is <span class="arithmatex">\(O(n)\)</span>.</li>
-<li><strong>Based on the peak memory during algorithm execution</strong>: For example, before executing the last line, the program occupies <span class="arithmatex">\(O(1)\)</span> space; when initializing the array <code>nums</code>, the program occupies <span class="arithmatex">\(O(n)\)</span> space, so the worst-case space complexity is <span class="arithmatex">\(O(n)\)</span>.</li>
+<li><strong>The worst input</strong>: When <span class="arithmatex">\(n &lt; 10\)</span>, space complexity is <span class="arithmatex">\(O(1)\)</span>. When <span class="arithmatex">\(n &gt; 10\)</span>, initializing <code>nums</code> requires <span class="arithmatex">\(O(n)\)</span> space, so worst-case space complexity is <span class="arithmatex">\(O(n)\)</span>.</li>
+<li><strong>Peak memory during execution</strong>: Before the final line, the program uses <span class="arithmatex">\(O(1)\)</span> space. Initializing <code>nums</code> raises usage to <span class="arithmatex">\(O(n)\)</span>, so worst-case space complexity is <span class="arithmatex">\(O(n)\)</span>.</li>
 </ol>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
@@ -5296,7 +5368,7 @@
 </div>
 </div>
 </div>
-<p><strong>In recursive functions, it is necessary to count the stack frame space</strong>. Observe the following code:</p>
+<p><strong>For recursive functions, remember to count stack frames</strong>. Consider the following code:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -5305,12 +5377,12 @@
 <a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="k">return</span> <span class="mi">0</span>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>
 <a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a><span class="k">def</span><span class="w"> </span><span class="nf">loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a><span class="w">    </span><span class="sd">"""Loop has space complexity of O(1)"""</span>
+<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a><span class="w">    </span><span class="sd">"""The loop uses O(1) space"""</span>
 <a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>        <span class="n">function</span><span class="p">()</span>
 <a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>
 <a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a><span class="k">def</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a><span class="w">    </span><span class="sd">"""Recursion has space complexity of O(n)"""</span>
+<a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a><span class="w">    </span><span class="sd">"""The recursion uses O(n) space"""</span>
 <a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>        <span class="k">return</span>
 <a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a>    <span class="k">return</span> <span class="n">recur</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
@@ -5321,13 +5393,13 @@
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="w">    </span><span class="c1">// Perform some operations</span>
 <a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="p">}</span>
-<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="cm">/* Loop has space complexity of O(1) */</span>
+<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="cm">/* The loop uses O(1) space */</span>
 <a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="kt">void</span><span class="w"> </span><span class="nf">loop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">        </span><span class="n">func</span><span class="p">();</span>
 <a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a><span class="p">}</span>
-<a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="cm">/* Recursion has space complexity of O(n) */</span>
+<a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="cm">/* The recursion uses O(n) space */</span>
 <a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="kt">void</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a><span class="w">    </span><span class="n">recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
@@ -5339,13 +5411,13 @@
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="w">    </span><span class="c1">// Perform some operations</span>
 <a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="p">}</span>
-<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="cm">/* Loop has space complexity of O(1) */</span>
+<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="cm">/* The loop uses O(1) space */</span>
 <a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="kt">void</span><span class="w"> </span><span class="nf">loop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">        </span><span class="n">function</span><span class="p">();</span>
 <a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="p">}</span>
-<a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="cm">/* Recursion has space complexity of O(n) */</span>
+<a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="cm">/* The recursion uses O(n) space */</span>
 <a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="kt">void</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a><span class="w">    </span><span class="n">recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
@@ -5357,13 +5429,13 @@
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="w">    </span><span class="c1">// Perform some operations</span>
 <a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="p">}</span>
-<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="cm">/* Loop has space complexity of O(1) */</span>
+<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="cm">/* The loop uses O(1) space */</span>
 <a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="k">void</span><span class="w"> </span><span class="nf">Loop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">        </span><span class="n">Function</span><span class="p">();</span>
 <a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="p">}</span>
-<a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="cm">/* Recursion has space complexity of O(n) */</span>
+<a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="cm">/* The recursion uses O(n) space */</span>
 <a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Recur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nf">Recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
@@ -5376,14 +5448,14 @@
 <a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="p">}</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a>
-<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="cm">/* Loop has space complexity of O(1) */</span>
+<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="cm">/* The loop uses O(1) space */</span>
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="kd">func</span><span class="w"> </span><span class="nx">loop</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">        </span><span class="nx">function</span><span class="p">()</span>
 <a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="p">}</span>
 <a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a>
-<a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="cm">/* Recursion has space complexity of O(n) */</span>
+<a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="cm">/* The recursion uses O(n) space */</span>
 <a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="kd">func</span><span class="w"> </span><span class="nx">recur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="w">        </span><span class="k">return</span>
@@ -5399,14 +5471,14 @@
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="p">}</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a>
-<a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="cm">/* Loop has space complexity of O(1) */</span>
+<a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="cm">/* The loop uses O(1) space */</span>
 <a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="kd">func</span><span class="w"> </span><span class="nf">loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">        </span><span class="n">function</span><span class="p">()</span>
 <a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="p">}</span>
 <a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a>
-<a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="cm">/* Recursion has space complexity of O(n) */</span>
+<a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="cm">/* The recursion uses O(n) space */</span>
 <a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="kd">func</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-17" name="__codelineno-31-17" href="#__codelineno-31-17"></a><span class="w">        </span><span class="k">return</span>
@@ -5420,13 +5492,13 @@
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="w">    </span><span class="c1">// Perform some operations</span>
 <a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="p">}</span>
-<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="cm">/* Loop has space complexity of O(1) */</span>
+<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="cm">/* The loop uses O(1) space */</span>
 <a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="kd">function</span><span class="w"> </span><span class="nx">loop</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">        </span><span class="nx">constFunc</span><span class="p">();</span>
 <a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a><span class="p">}</span>
-<a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="cm">/* Recursion has space complexity of O(n) */</span>
+<a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="cm">/* The recursion uses O(n) space */</span>
 <a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="kd">function</span><span class="w"> </span><span class="nx">recur</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">recur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
@@ -5438,13 +5510,13 @@
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="w">    </span><span class="c1">// Perform some operations</span>
 <a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="p">}</span>
-<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="cm">/* Loop has space complexity of O(1) */</span>
+<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="cm">/* The loop uses O(1) space */</span>
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="kd">function</span><span class="w"> </span><span class="nx">loop</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">        </span><span class="nx">constFunc</span><span class="p">();</span>
 <a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="p">}</span>
-<a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="cm">/* Recursion has space complexity of O(n) */</span>
+<a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="cm">/* The recursion uses O(n) space */</span>
 <a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="kd">function</span><span class="w"> </span><span class="nx">recur</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">recur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
@@ -5456,13 +5528,13 @@
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="w">  </span><span class="c1">// Perform some operations</span>
 <a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="p">}</span>
-<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="cm">/* Loop has space complexity of O(1) */</span>
+<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="cm">/* The loop uses O(1) space */</span>
 <a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="kt">void</span><span class="w"> </span><span class="n">loop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">    </span><span class="n">function</span><span class="p">();</span>
 <a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="p">}</span>
-<a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="cm">/* Recursion has space complexity of O(n) */</span>
+<a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="cm">/* The recursion uses O(n) space */</span>
 <a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="kt">void</span><span class="w"> </span><span class="n">recur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">  </span><span class="n">recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
@@ -5474,13 +5546,13 @@
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="w">    </span><span class="c1">// Perform some operations</span>
 <a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="p">}</span>
-<a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="cm">/* Loop has space complexity of O(1) */</span>
+<a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="cm">/* The loop uses O(1) space */</span>
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="k">fn</span><span class="w"> </span><span class="nf">loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">        </span><span class="n">function</span><span class="p">();</span>
 <a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="p">}</span>
-<a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="cm">/* Recursion has space complexity of O(n) */</span>
+<a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="cm">/* The recursion uses O(n) space */</span>
 <a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="k">fn</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
@@ -5494,13 +5566,13 @@
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="w">    </span><span class="c1">// Perform some operations</span>
 <a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="p">}</span>
-<a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="cm">/* Loop has space complexity of O(1) */</span>
+<a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="cm">/* The loop uses O(1) space */</span>
 <a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="kt">void</span><span class="w"> </span><span class="nf">loop</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">        </span><span class="n">func</span><span class="p">();</span>
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="p">}</span>
-<a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="cm">/* Recursion has space complexity of O(n) */</span>
+<a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="cm">/* The recursion uses O(n) space */</span>
 <a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="kt">void</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-36-14" name="__codelineno-36-14" href="#__codelineno-36-14"></a><span class="w">    </span><span class="n">recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
@@ -5512,13 +5584,13 @@
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="w">    </span><span class="c1">// Perform some operations</span>
 <a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="p">}</span>
-<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="cm">/* Loop has space complexity of O(1) */</span>
+<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="cm">/* The loop uses O(1) space */</span>
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">loop</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">        </span><span class="n">function</span><span class="p">()</span>
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="p">}</span>
-<a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="cm">/* Recursion has space complexity of O(n) */</span>
+<a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="cm">/* The recursion uses O(n) space */</span>
 <a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span>
 <a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
@@ -5531,12 +5603,12 @@
 <a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">    </span><span class="mi">0</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="k">end</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a>
-<a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="c1">### Loop has space complexity of O(1) ###</span>
+<a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="c1">### The loop uses O(1) space ###</span>
 <a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="k">def</span><span class="w"> </span><span class="nf">loop</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">    </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">function</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="k">end</span>
 <a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a>
-<a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="c1">### Recursion has space complexity of O(n) ###</span>
+<a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="c1">### The recursion uses O(n) space ###</span>
 <a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="k">def</span><span class="w"> </span><span class="nf">recur</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="w">    </span><span class="n">recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
@@ -5545,25 +5617,25 @@
 </div>
 </div>
 </div>
-<p>The time complexity of both functions <code>loop()</code> and <code>recur()</code> is <span class="arithmatex">\(O(n)\)</span>, but their space complexities are different.</p>
+<p>Both <code>loop()</code> and <code>recur()</code> take <span class="arithmatex">\(O(n)\)</span> time, but their space complexities differ.</p>
 <ul>
-<li>The function <code>loop()</code> calls <code>function()</code> <span class="arithmatex">\(n\)</span> times in a loop. In each iteration, <code>function()</code> returns and releases its stack frame space, so the space complexity remains <span class="arithmatex">\(O(1)\)</span>.</li>
-<li>The recursive function <code>recur()</code> has <span class="arithmatex">\(n\)</span> unreturned <code>recur()</code> instances existing simultaneously during execution, thus occupying <span class="arithmatex">\(O(n)\)</span> stack frame space.</li>
+<li><code>loop()</code> calls <code>function()</code> <span class="arithmatex">\(n\)</span> times. Each call returns and releases its stack frame before the next iteration, so space complexity remains <span class="arithmatex">\(O(1)\)</span>.</li>
+<li>During <code>recur()</code>, <span class="arithmatex">\(n\)</span> calls can remain active at once, using <span class="arithmatex">\(O(n)\)</span> space for stack frames.</li>
 </ul>
-<h2 id="243-common-types">2.4.3 &nbsp; Common Types<a class="headerlink" href="#243-common-types" title="Permanent link">&para;</a></h2>
-<p>Let the input data size be <span class="arithmatex">\(n\)</span>. The following figure shows common types of space complexity (arranged from low to high).</p>
+<h2 id="243-common-complexity-classes">2.4.3 &nbsp; Common complexity classes<a class="headerlink" href="#243-common-complexity-classes" title="Permanent link">&para;</a></h2>
+<p>For input size <span class="arithmatex">\(n\)</span>, Figure 2-16 shows common space complexities in increasing order of growth.</p>
 <div class="arithmatex">\[
 \begin{aligned}
 &amp; O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
 &amp; \text{Constant} &lt; \text{Logarithmic} &lt; \text{Linear} &lt; \text{Quadratic} &lt; \text{Exponential}
 \end{aligned}
 \]</div>
-<p><img alt="Common types of space complexity" class="animation-figure" src="../space_complexity.assets/space_complexity_common_types.png" /></p>
-<p align="center"> Figure 2-16 &nbsp; Common types of space complexity </p>
+<p><img alt="Common space complexity classes" class="animation-figure" src="../space_complexity.assets/space_complexity_common_types.png" /></p>
+<p align="center"> Figure 2-16 &nbsp; Common space complexity classes </p>
 
-<h3 id="1-constant-order-o1">1. &nbsp; Constant Order <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#1-constant-order-o1" title="Permanent link">&para;</a></h3>
-<p>Constant order is common for constants, variables, and objects whose number is independent of the input data size <span class="arithmatex">\(n\)</span>.</p>
-<p>It should be noted that memory occupied by initializing variables or calling functions in a loop is released when entering the next iteration, so it does not accumulate space, and the space complexity remains <span class="arithmatex">\(O(1)\)</span>:</p>
+<h3 id="1-constant-space-o1">1. &nbsp; Constant space <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#1-constant-space-o1" title="Permanent link">&para;</a></h3>
+<p>Constant space commonly occurs when the number of constants, variables, and objects is independent of input size <span class="arithmatex">\(n\)</span>.</p>
+<p>Memory for variables initialized or functions called within a loop is released by the next iteration, so it does not accumulate. Space complexity remains <span class="arithmatex">\(O(1)\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:13"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -5573,15 +5645,15 @@
 <a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="k">return</span> <span class="mi">0</span>
 <a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>
 <a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a><span class="k">def</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a><span class="w">    </span><span class="sd">"""Constant order"""</span>
-<a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a>    <span class="c1"># Constants, variables, objects occupy O(1) space</span>
+<a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a><span class="w">    </span><span class="sd">"""Constant complexity"""</span>
+<a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a>    <span class="c1"># Constants, variables, and objects occupy O(1) space</span>
 <a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a>    <span class="n">a</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a>    <span class="n">nums</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="mi">10000</span>
 <a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a>    <span class="n">node</span> <span class="o">=</span> <span class="n">ListNode</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-39-12" name="__codelineno-39-12" href="#__codelineno-39-12"></a>    <span class="c1"># Variables in the loop occupy O(1) space</span>
 <a id="__codelineno-39-13" name="__codelineno-39-13" href="#__codelineno-39-13"></a>    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-39-14" name="__codelineno-39-14" href="#__codelineno-39-14"></a>        <span class="n">c</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-39-15" name="__codelineno-39-15" href="#__codelineno-39-15"></a>    <span class="c1"># Functions in the loop occupy O(1) space</span>
+<a id="__codelineno-39-15" name="__codelineno-39-15" href="#__codelineno-39-15"></a>    <span class="c1"># Function calls in the loop occupy O(1) space</span>
 <a id="__codelineno-39-16" name="__codelineno-39-16" href="#__codelineno-39-16"></a>    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-39-17" name="__codelineno-39-17" href="#__codelineno-39-17"></a>        <span class="n">function</span><span class="p">()</span>
 </code></pre></div>
@@ -5593,9 +5665,9 @@
 <a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="p">}</span>
 <a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a>
-<a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="cm">/* Constant order */</span>
+<a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="kt">void</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-9" name="__codelineno-40-9" href="#__codelineno-40-9"></a><span class="w">    </span><span class="c1">// Constants, variables, objects occupy O(1) space</span>
+<a id="__codelineno-40-9" name="__codelineno-40-9" href="#__codelineno-40-9"></a><span class="w">    </span><span class="c1">// Constants, variables, and objects occupy O(1) space</span>
 <a id="__codelineno-40-10" name="__codelineno-40-10" href="#__codelineno-40-10"></a><span class="w">    </span><span class="k">const</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-40-11" name="__codelineno-40-11" href="#__codelineno-40-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-40-12" name="__codelineno-40-12" href="#__codelineno-40-12"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">(</span><span class="mi">10000</span><span class="p">);</span>
@@ -5604,7 +5676,7 @@
 <a id="__codelineno-40-15" name="__codelineno-40-15" href="#__codelineno-40-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-16" name="__codelineno-40-16" href="#__codelineno-40-16"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-40-17" name="__codelineno-40-17" href="#__codelineno-40-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-40-18" name="__codelineno-40-18" href="#__codelineno-40-18"></a><span class="w">    </span><span class="c1">// Functions in the loop occupy O(1) space</span>
+<a id="__codelineno-40-18" name="__codelineno-40-18" href="#__codelineno-40-18"></a><span class="w">    </span><span class="c1">// Function calls in the loop occupy O(1) space</span>
 <a id="__codelineno-40-19" name="__codelineno-40-19" href="#__codelineno-40-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-20" name="__codelineno-40-20" href="#__codelineno-40-20"></a><span class="w">        </span><span class="n">func</span><span class="p">();</span>
 <a id="__codelineno-40-21" name="__codelineno-40-21" href="#__codelineno-40-21"></a><span class="w">    </span><span class="p">}</span>
@@ -5618,9 +5690,9 @@
 <a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="p">}</span>
 <a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a>
-<a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="cm">/* Constant order */</span>
+<a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-41-8" name="__codelineno-41-8" href="#__codelineno-41-8"></a><span class="kt">void</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-9" name="__codelineno-41-9" href="#__codelineno-41-9"></a><span class="w">    </span><span class="c1">// Constants, variables, objects occupy O(1) space</span>
+<a id="__codelineno-41-9" name="__codelineno-41-9" href="#__codelineno-41-9"></a><span class="w">    </span><span class="c1">// Constants, variables, and objects occupy O(1) space</span>
 <a id="__codelineno-41-10" name="__codelineno-41-10" href="#__codelineno-41-10"></a><span class="w">    </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-41-11" name="__codelineno-41-11" href="#__codelineno-41-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-41-12" name="__codelineno-41-12" href="#__codelineno-41-12"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="mi">10000</span><span class="o">]</span><span class="p">;</span>
@@ -5629,7 +5701,7 @@
 <a id="__codelineno-41-15" name="__codelineno-41-15" href="#__codelineno-41-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-16" name="__codelineno-41-16" href="#__codelineno-41-16"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-41-17" name="__codelineno-41-17" href="#__codelineno-41-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-41-18" name="__codelineno-41-18" href="#__codelineno-41-18"></a><span class="w">    </span><span class="c1">// Functions in the loop occupy O(1) space</span>
+<a id="__codelineno-41-18" name="__codelineno-41-18" href="#__codelineno-41-18"></a><span class="w">    </span><span class="c1">// Function calls in the loop occupy O(1) space</span>
 <a id="__codelineno-41-19" name="__codelineno-41-19" href="#__codelineno-41-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-20" name="__codelineno-41-20" href="#__codelineno-41-20"></a><span class="w">        </span><span class="n">function</span><span class="p">();</span>
 <a id="__codelineno-41-21" name="__codelineno-41-21" href="#__codelineno-41-21"></a><span class="w">    </span><span class="p">}</span>
@@ -5643,9 +5715,9 @@
 <a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="p">}</span>
 <a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a>
-<a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="cm">/* Constant order */</span>
+<a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-42-8" name="__codelineno-42-8" href="#__codelineno-42-8"></a><span class="k">void</span><span class="w"> </span><span class="nf">Constant</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-9" name="__codelineno-42-9" href="#__codelineno-42-9"></a><span class="w">    </span><span class="c1">// Constants, variables, objects occupy O(1) space</span>
+<a id="__codelineno-42-9" name="__codelineno-42-9" href="#__codelineno-42-9"></a><span class="w">    </span><span class="c1">// Constants, variables, and objects occupy O(1) space</span>
 <a id="__codelineno-42-10" name="__codelineno-42-10" href="#__codelineno-42-10"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-42-11" name="__codelineno-42-11" href="#__codelineno-42-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-42-12" name="__codelineno-42-12" href="#__codelineno-42-12"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="mi">10000</span><span class="p">];</span>
@@ -5654,7 +5726,7 @@
 <a id="__codelineno-42-15" name="__codelineno-42-15" href="#__codelineno-42-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-16" name="__codelineno-42-16" href="#__codelineno-42-16"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-42-17" name="__codelineno-42-17" href="#__codelineno-42-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-42-18" name="__codelineno-42-18" href="#__codelineno-42-18"></a><span class="w">    </span><span class="c1">// Functions in the loop occupy O(1) space</span>
+<a id="__codelineno-42-18" name="__codelineno-42-18" href="#__codelineno-42-18"></a><span class="w">    </span><span class="c1">// Function calls in the loop occupy O(1) space</span>
 <a id="__codelineno-42-19" name="__codelineno-42-19" href="#__codelineno-42-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-20" name="__codelineno-42-20" href="#__codelineno-42-20"></a><span class="w">        </span><span class="n">Function</span><span class="p">();</span>
 <a id="__codelineno-42-21" name="__codelineno-42-21" href="#__codelineno-42-21"></a><span class="w">    </span><span class="p">}</span>
@@ -5668,9 +5740,9 @@
 <a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="p">}</span>
 <a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a>
-<a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="cm">/* Constant order */</span>
+<a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a><span class="kd">func</span><span class="w"> </span><span class="nx">spaceConstant</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-9" name="__codelineno-43-9" href="#__codelineno-43-9"></a><span class="w">    </span><span class="c1">// Constants, variables, objects occupy O(1) space</span>
+<a id="__codelineno-43-9" name="__codelineno-43-9" href="#__codelineno-43-9"></a><span class="w">    </span><span class="c1">// Constants, variables, and objects occupy O(1) space</span>
 <a id="__codelineno-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w">    </span><span class="nx">b</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-43-12" name="__codelineno-43-12" href="#__codelineno-43-12"></a><span class="w">    </span><span class="nx">nums</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">10000</span><span class="p">)</span>
@@ -5680,7 +5752,7 @@
 <a id="__codelineno-43-16" name="__codelineno-43-16" href="#__codelineno-43-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-17" name="__codelineno-43-17" href="#__codelineno-43-17"></a><span class="w">        </span><span class="nx">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-43-18" name="__codelineno-43-18" href="#__codelineno-43-18"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-43-19" name="__codelineno-43-19" href="#__codelineno-43-19"></a><span class="w">    </span><span class="c1">// Functions in the loop occupy O(1) space</span>
+<a id="__codelineno-43-19" name="__codelineno-43-19" href="#__codelineno-43-19"></a><span class="w">    </span><span class="c1">// Function calls in the loop occupy O(1) space</span>
 <a id="__codelineno-43-20" name="__codelineno-43-20" href="#__codelineno-43-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-21" name="__codelineno-43-21" href="#__codelineno-43-21"></a><span class="w">        </span><span class="nx">function</span><span class="p">()</span>
 <a id="__codelineno-43-22" name="__codelineno-43-22" href="#__codelineno-43-22"></a><span class="w">    </span><span class="p">}</span>
@@ -5699,9 +5771,9 @@
 <a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></a><span class="p">}</span>
 <a id="__codelineno-44-7" name="__codelineno-44-7" href="#__codelineno-44-7"></a>
-<a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="cm">/* Constant order */</span>
+<a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-44-9" name="__codelineno-44-9" href="#__codelineno-44-9"></a><span class="kd">func</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-44-10" name="__codelineno-44-10" href="#__codelineno-44-10"></a><span class="w">    </span><span class="c1">// Constants, variables, objects occupy O(1) space</span>
+<a id="__codelineno-44-10" name="__codelineno-44-10" href="#__codelineno-44-10"></a><span class="w">    </span><span class="c1">// Constants, variables, and objects occupy O(1) space</span>
 <a id="__codelineno-44-11" name="__codelineno-44-11" href="#__codelineno-44-11"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-44-12" name="__codelineno-44-12" href="#__codelineno-44-12"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">b</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-44-13" name="__codelineno-44-13" href="#__codelineno-44-13"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">nums</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="mi">10000</span><span class="p">)</span>
@@ -5710,7 +5782,7 @@
 <a id="__codelineno-44-16" name="__codelineno-44-16" href="#__codelineno-44-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-17" name="__codelineno-44-17" href="#__codelineno-44-17"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">c</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-44-18" name="__codelineno-44-18" href="#__codelineno-44-18"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-44-19" name="__codelineno-44-19" href="#__codelineno-44-19"></a><span class="w">    </span><span class="c1">// Functions in the loop occupy O(1) space</span>
+<a id="__codelineno-44-19" name="__codelineno-44-19" href="#__codelineno-44-19"></a><span class="w">    </span><span class="c1">// Function calls in the loop occupy O(1) space</span>
 <a id="__codelineno-44-20" name="__codelineno-44-20" href="#__codelineno-44-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-21" name="__codelineno-44-21" href="#__codelineno-44-21"></a><span class="w">        </span><span class="n">function</span><span class="p">()</span>
 <a id="__codelineno-44-22" name="__codelineno-44-22" href="#__codelineno-44-22"></a><span class="w">    </span><span class="p">}</span>
@@ -5724,9 +5796,9 @@
 <a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="p">}</span>
 <a id="__codelineno-45-6" name="__codelineno-45-6" href="#__codelineno-45-6"></a>
-<a id="__codelineno-45-7" name="__codelineno-45-7" href="#__codelineno-45-7"></a><span class="cm">/* Constant order */</span>
+<a id="__codelineno-45-7" name="__codelineno-45-7" href="#__codelineno-45-7"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-45-8" name="__codelineno-45-8" href="#__codelineno-45-8"></a><span class="kd">function</span><span class="w"> </span><span class="nx">constant</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-9" name="__codelineno-45-9" href="#__codelineno-45-9"></a><span class="w">    </span><span class="c1">// Constants, variables, objects occupy O(1) space</span>
+<a id="__codelineno-45-9" name="__codelineno-45-9" href="#__codelineno-45-9"></a><span class="w">    </span><span class="c1">// Constants, variables, and objects occupy O(1) space</span>
 <a id="__codelineno-45-10" name="__codelineno-45-10" href="#__codelineno-45-10"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-45-11" name="__codelineno-45-11" href="#__codelineno-45-11"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-45-12" name="__codelineno-45-12" href="#__codelineno-45-12"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">10000</span><span class="p">);</span>
@@ -5735,7 +5807,7 @@
 <a id="__codelineno-45-15" name="__codelineno-45-15" href="#__codelineno-45-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-16" name="__codelineno-45-16" href="#__codelineno-45-16"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-45-17" name="__codelineno-45-17" href="#__codelineno-45-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-45-18" name="__codelineno-45-18" href="#__codelineno-45-18"></a><span class="w">    </span><span class="c1">// Functions in the loop occupy O(1) space</span>
+<a id="__codelineno-45-18" name="__codelineno-45-18" href="#__codelineno-45-18"></a><span class="w">    </span><span class="c1">// Function calls in the loop occupy O(1) space</span>
 <a id="__codelineno-45-19" name="__codelineno-45-19" href="#__codelineno-45-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-20" name="__codelineno-45-20" href="#__codelineno-45-20"></a><span class="w">        </span><span class="nx">constFunc</span><span class="p">();</span>
 <a id="__codelineno-45-21" name="__codelineno-45-21" href="#__codelineno-45-21"></a><span class="w">    </span><span class="p">}</span>
@@ -5749,9 +5821,9 @@
 <a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="p">}</span>
 <a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a>
-<a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="cm">/* Constant order */</span>
+<a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a><span class="kd">function</span><span class="w"> </span><span class="nx">constant</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="w">    </span><span class="c1">// Constants, variables, objects occupy O(1) space</span>
+<a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="w">    </span><span class="c1">// Constants, variables, and objects occupy O(1) space</span>
 <a id="__codelineno-46-10" name="__codelineno-46-10" href="#__codelineno-46-10"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-46-11" name="__codelineno-46-11" href="#__codelineno-46-11"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">10000</span><span class="p">);</span>
@@ -5760,7 +5832,7 @@
 <a id="__codelineno-46-15" name="__codelineno-46-15" href="#__codelineno-46-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-16" name="__codelineno-46-16" href="#__codelineno-46-16"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-46-17" name="__codelineno-46-17" href="#__codelineno-46-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-46-18" name="__codelineno-46-18" href="#__codelineno-46-18"></a><span class="w">    </span><span class="c1">// Functions in the loop occupy O(1) space</span>
+<a id="__codelineno-46-18" name="__codelineno-46-18" href="#__codelineno-46-18"></a><span class="w">    </span><span class="c1">// Function calls in the loop occupy O(1) space</span>
 <a id="__codelineno-46-19" name="__codelineno-46-19" href="#__codelineno-46-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-20" name="__codelineno-46-20" href="#__codelineno-46-20"></a><span class="w">        </span><span class="nx">constFunc</span><span class="p">();</span>
 <a id="__codelineno-46-21" name="__codelineno-46-21" href="#__codelineno-46-21"></a><span class="w">    </span><span class="p">}</span>
@@ -5774,9 +5846,9 @@
 <a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="p">}</span>
 <a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a>
-<a id="__codelineno-47-7" name="__codelineno-47-7" href="#__codelineno-47-7"></a><span class="cm">/* Constant order */</span>
+<a id="__codelineno-47-7" name="__codelineno-47-7" href="#__codelineno-47-7"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a><span class="kt">void</span><span class="w"> </span><span class="n">constant</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-9" name="__codelineno-47-9" href="#__codelineno-47-9"></a><span class="w">  </span><span class="c1">// Constants, variables, objects occupy O(1) space</span>
+<a id="__codelineno-47-9" name="__codelineno-47-9" href="#__codelineno-47-9"></a><span class="w">  </span><span class="c1">// Constants, variables, and objects occupy O(1) space</span>
 <a id="__codelineno-47-10" name="__codelineno-47-10" href="#__codelineno-47-10"></a><span class="w">  </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-47-11" name="__codelineno-47-11" href="#__codelineno-47-11"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-47-12" name="__codelineno-47-12" href="#__codelineno-47-12"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">10000</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
@@ -5785,7 +5857,7 @@
 <a id="__codelineno-47-15" name="__codelineno-47-15" href="#__codelineno-47-15"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-16" name="__codelineno-47-16" href="#__codelineno-47-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-47-17" name="__codelineno-47-17" href="#__codelineno-47-17"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-47-18" name="__codelineno-47-18" href="#__codelineno-47-18"></a><span class="w">  </span><span class="c1">// Functions in the loop occupy O(1) space</span>
+<a id="__codelineno-47-18" name="__codelineno-47-18" href="#__codelineno-47-18"></a><span class="w">  </span><span class="c1">// Function calls in the loop occupy O(1) space</span>
 <a id="__codelineno-47-19" name="__codelineno-47-19" href="#__codelineno-47-19"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-20" name="__codelineno-47-20" href="#__codelineno-47-20"></a><span class="w">    </span><span class="n">function</span><span class="p">();</span>
 <a id="__codelineno-47-21" name="__codelineno-47-21" href="#__codelineno-47-21"></a><span class="w">  </span><span class="p">}</span>
@@ -5799,10 +5871,10 @@
 <a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="p">}</span>
 <a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a>
-<a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="cm">/* Constant order */</span>
+<a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-48-8" name="__codelineno-48-8" href="#__codelineno-48-8"></a><span class="cp">#[allow(unused)]</span>
 <a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a><span class="k">fn</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="w">    </span><span class="c1">// Constants, variables, objects occupy O(1) space</span>
+<a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="w">    </span><span class="c1">// Constants, variables, and objects occupy O(1) space</span>
 <a id="__codelineno-48-11" name="__codelineno-48-11" href="#__codelineno-48-11"></a><span class="w">    </span><span class="k">const</span><span class="w"> </span><span class="n">A</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-48-12" name="__codelineno-48-12" href="#__codelineno-48-12"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-48-13" name="__codelineno-48-13" href="#__codelineno-48-13"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="mi">10000</span><span class="p">];</span>
@@ -5811,7 +5883,7 @@
 <a id="__codelineno-48-16" name="__codelineno-48-16" href="#__codelineno-48-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-17" name="__codelineno-48-17" href="#__codelineno-48-17"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-48-18" name="__codelineno-48-18" href="#__codelineno-48-18"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-48-19" name="__codelineno-48-19" href="#__codelineno-48-19"></a><span class="w">    </span><span class="c1">// Functions in the loop occupy O(1) space</span>
+<a id="__codelineno-48-19" name="__codelineno-48-19" href="#__codelineno-48-19"></a><span class="w">    </span><span class="c1">// Function calls in the loop occupy O(1) space</span>
 <a id="__codelineno-48-20" name="__codelineno-48-20" href="#__codelineno-48-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-21" name="__codelineno-48-21" href="#__codelineno-48-21"></a><span class="w">        </span><span class="n">function</span><span class="p">();</span>
 <a id="__codelineno-48-22" name="__codelineno-48-22" href="#__codelineno-48-22"></a><span class="w">    </span><span class="p">}</span>
@@ -5825,9 +5897,9 @@
 <a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="p">}</span>
 <a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a>
-<a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="cm">/* Constant order */</span>
+<a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="kt">void</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-9" name="__codelineno-49-9" href="#__codelineno-49-9"></a><span class="w">    </span><span class="c1">// Constants, variables, objects occupy O(1) space</span>
+<a id="__codelineno-49-9" name="__codelineno-49-9" href="#__codelineno-49-9"></a><span class="w">    </span><span class="c1">// Constants, variables, and objects occupy O(1) space</span>
 <a id="__codelineno-49-10" name="__codelineno-49-10" href="#__codelineno-49-10"></a><span class="w">    </span><span class="k">const</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-49-11" name="__codelineno-49-11" href="#__codelineno-49-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-49-12" name="__codelineno-49-12" href="#__codelineno-49-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="mi">1000</span><span class="p">];</span>
@@ -5837,7 +5909,7 @@
 <a id="__codelineno-49-16" name="__codelineno-49-16" href="#__codelineno-49-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-17" name="__codelineno-49-17" href="#__codelineno-49-17"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-49-18" name="__codelineno-49-18" href="#__codelineno-49-18"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-49-19" name="__codelineno-49-19" href="#__codelineno-49-19"></a><span class="w">    </span><span class="c1">// Functions in the loop occupy O(1) space</span>
+<a id="__codelineno-49-19" name="__codelineno-49-19" href="#__codelineno-49-19"></a><span class="w">    </span><span class="c1">// Function calls in the loop occupy O(1) space</span>
 <a id="__codelineno-49-20" name="__codelineno-49-20" href="#__codelineno-49-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-21" name="__codelineno-49-21" href="#__codelineno-49-21"></a><span class="w">        </span><span class="n">func</span><span class="p">();</span>
 <a id="__codelineno-49-22" name="__codelineno-49-22" href="#__codelineno-49-22"></a><span class="w">    </span><span class="p">}</span>
@@ -5851,9 +5923,9 @@
 <a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="p">}</span>
 <a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a>
-<a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="cm">/* Constant order */</span>
+<a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-50-8" name="__codelineno-50-8" href="#__codelineno-50-8"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="w">    </span><span class="c1">// Constants, variables, objects occupy O(1) space</span>
+<a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="w">    </span><span class="c1">// Constants, variables, and objects occupy O(1) space</span>
 <a id="__codelineno-50-10" name="__codelineno-50-10" href="#__codelineno-50-10"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-50-11" name="__codelineno-50-11" href="#__codelineno-50-11"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-50-12" name="__codelineno-50-12" href="#__codelineno-50-12"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Array</span><span class="p">(</span><span class="m">10000</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="p">}</span>
@@ -5862,7 +5934,7 @@
 <a id="__codelineno-50-15" name="__codelineno-50-15" href="#__codelineno-50-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-16" name="__codelineno-50-16" href="#__codelineno-50-16"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-50-17" name="__codelineno-50-17" href="#__codelineno-50-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-50-18" name="__codelineno-50-18" href="#__codelineno-50-18"></a><span class="w">    </span><span class="c1">// Functions in the loop occupy O(1) space</span>
+<a id="__codelineno-50-18" name="__codelineno-50-18" href="#__codelineno-50-18"></a><span class="w">    </span><span class="c1">// Function calls in the loop occupy O(1) space</span>
 <a id="__codelineno-50-19" name="__codelineno-50-19" href="#__codelineno-50-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-20" name="__codelineno-50-20" href="#__codelineno-50-20"></a><span class="w">        </span><span class="n">function</span><span class="p">()</span>
 <a id="__codelineno-50-21" name="__codelineno-50-21" href="#__codelineno-50-21"></a><span class="w">    </span><span class="p">}</span>
@@ -5876,48 +5948,53 @@
 <a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="w">  </span><span class="mi">0</span>
 <a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="k">end</span>
 <a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a>
-<a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="c1">### Constant time ###</span>
+<a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="c1">### Constant complexity ###</span>
 <a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="k">def</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w">  </span><span class="c1"># Constants, variables, objects occupy O(1) space</span>
+<a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w">  </span><span class="c1"># Constants, variables, and objects occupy O(1) space</span>
 <a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a><span class="w">  </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-51-11" name="__codelineno-51-11" href="#__codelineno-51-11"></a><span class="w">  </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">10000</span>
 <a id="__codelineno-51-12" name="__codelineno-51-12" href="#__codelineno-51-12"></a><span class="w">  </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">ListNode</span><span class="o">.</span><span class="n">new</span>
 <a id="__codelineno-51-13" name="__codelineno-51-13" href="#__codelineno-51-13"></a>
 <a id="__codelineno-51-14" name="__codelineno-51-14" href="#__codelineno-51-14"></a><span class="w">  </span><span class="c1"># Variables in the loop occupy O(1) space</span>
 <a id="__codelineno-51-15" name="__codelineno-51-15" href="#__codelineno-51-15"></a><span class="w">  </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-51-16" name="__codelineno-51-16" href="#__codelineno-51-16"></a><span class="w">  </span><span class="c1"># Functions in the loop occupy O(1) space</span>
+<a id="__codelineno-51-16" name="__codelineno-51-16" href="#__codelineno-51-16"></a><span class="w">  </span><span class="c1"># Function calls in the loop occupy O(1) space</span>
 <a id="__codelineno-51-17" name="__codelineno-51-17" href="#__codelineno-51-17"></a><span class="w">  </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">function</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-51-18" name="__codelineno-51-18" href="#__codelineno-51-18"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h3 id="2-linear-order-on">2. &nbsp; Linear Order <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#2-linear-order-on" title="Permanent link">&para;</a></h3>
-<p>Linear order is common in arrays, linked lists, stacks, queues, etc., where the number of elements is proportional to <span class="arithmatex">\(n\)</span>:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0Adef%20function%28%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Function%22%22%22%0A%20%20%20%20%23%20Perform%20some%20operations%0A%20%20%20%20return%200%0A%0Adef%20constant%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22Constant%20complexity%22%22%22%0A%20%20%20%20%23%20Constants,%20variables,%20and%20objects%20occupy%20O%281%29%20space%0A%20%20%20%20a%20%3D%200%0A%20%20%20%20nums%20%3D%20%5B0%5D%20%2A%2010%0A%20%20%20%20node%20%3D%20ListNode%280%29%0A%20%20%20%20%23%20Variables%20in%20the%20loop%20occupy%20O%281%29%20space%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20c%20%3D%200%0A%20%20%20%20%23%20Function%20calls%20in%20the%20loop%20occupy%20O%281%29%20space%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20function%28%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20Constant%20complexity%0A%20%20%20%20constant%28n%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0Adef%20function%28%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Function%22%22%22%0A%20%20%20%20%23%20Perform%20some%20operations%0A%20%20%20%20return%200%0A%0Adef%20constant%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22Constant%20complexity%22%22%22%0A%20%20%20%20%23%20Constants,%20variables,%20and%20objects%20occupy%20O%281%29%20space%0A%20%20%20%20a%20%3D%200%0A%20%20%20%20nums%20%3D%20%5B0%5D%20%2A%2010%0A%20%20%20%20node%20%3D%20ListNode%280%29%0A%20%20%20%20%23%20Variables%20in%20the%20loop%20occupy%20O%281%29%20space%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20c%20%3D%200%0A%20%20%20%20%23%20Function%20calls%20in%20the%20loop%20occupy%20O%281%29%20space%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20function%28%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20Constant%20complexity%0A%20%20%20%20constant%28n%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="2-linear-space-on">2. &nbsp; Linear space <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#2-linear-space-on" title="Permanent link">&para;</a></h3>
+<p>Linear space commonly occurs in arrays, linked lists, stacks, and queues whose element counts are proportional to <span class="arithmatex">\(n\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">space_complexity.py</span><pre><span></span><code><a id="__codelineno-52-1" name="__codelineno-52-1" href="#__codelineno-52-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Linear order"""</span>
+<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Linear complexity"""</span>
 <a id="__codelineno-52-3" name="__codelineno-52-3" href="#__codelineno-52-3"></a>    <span class="c1"># A list of length n occupies O(n) space</span>
 <a id="__codelineno-52-4" name="__codelineno-52-4" href="#__codelineno-52-4"></a>    <span class="n">nums</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span>
-<a id="__codelineno-52-5" name="__codelineno-52-5" href="#__codelineno-52-5"></a>    <span class="c1"># A hash table of length n occupies O(n) space</span>
+<a id="__codelineno-52-5" name="__codelineno-52-5" href="#__codelineno-52-5"></a>    <span class="c1"># A hash table with n entries occupies O(n) space</span>
 <a id="__codelineno-52-6" name="__codelineno-52-6" href="#__codelineno-52-6"></a>    <span class="n">hmap</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">[</span><span class="nb">int</span><span class="p">,</span> <span class="nb">str</span><span class="p">]()</span>
 <a id="__codelineno-52-7" name="__codelineno-52-7" href="#__codelineno-52-7"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-52-8" name="__codelineno-52-8" href="#__codelineno-52-8"></a>        <span class="n">hmap</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">space_complexity.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-53-2" name="__codelineno-53-2" href="#__codelineno-53-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="w">    </span><span class="c1">// Array of length n uses O(n) space</span>
+<a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="w">    </span><span class="c1">// An array of length n uses O(n) space</span>
 <a id="__codelineno-53-4" name="__codelineno-53-4" href="#__codelineno-53-4"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">(</span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-53-5" name="__codelineno-53-5" href="#__codelineno-53-5"></a><span class="w">    </span><span class="c1">// A list of length n occupies O(n) space</span>
 <a id="__codelineno-53-6" name="__codelineno-53-6" href="#__codelineno-53-6"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nodes</span><span class="p">;</span>
 <a id="__codelineno-53-7" name="__codelineno-53-7" href="#__codelineno-53-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-8" name="__codelineno-53-8" href="#__codelineno-53-8"></a><span class="w">        </span><span class="n">nodes</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">ListNode</span><span class="p">(</span><span class="n">i</span><span class="p">));</span>
 <a id="__codelineno-53-9" name="__codelineno-53-9" href="#__codelineno-53-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-53-10" name="__codelineno-53-10" href="#__codelineno-53-10"></a><span class="w">    </span><span class="c1">// A hash table of length n occupies O(n) space</span>
+<a id="__codelineno-53-10" name="__codelineno-53-10" href="#__codelineno-53-10"></a><span class="w">    </span><span class="c1">// A hash table with n entries occupies O(n) space</span>
 <a id="__codelineno-53-11" name="__codelineno-53-11" href="#__codelineno-53-11"></a><span class="w">    </span><span class="n">unordered_map</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="n">string</span><span class="o">&gt;</span><span class="w"> </span><span class="n">map</span><span class="p">;</span>
 <a id="__codelineno-53-12" name="__codelineno-53-12" href="#__codelineno-53-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-13" name="__codelineno-53-13" href="#__codelineno-53-13"></a><span class="w">        </span><span class="n">map</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">to_string</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
@@ -5926,16 +6003,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-54-2" name="__codelineno-54-2" href="#__codelineno-54-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="w">    </span><span class="c1">// Array of length n uses O(n) space</span>
+<a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="w">    </span><span class="c1">// An array of length n uses O(n) space</span>
 <a id="__codelineno-54-4" name="__codelineno-54-4" href="#__codelineno-54-4"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-54-5" name="__codelineno-54-5" href="#__codelineno-54-5"></a><span class="w">    </span><span class="c1">// A list of length n occupies O(n) space</span>
 <a id="__codelineno-54-6" name="__codelineno-54-6" href="#__codelineno-54-6"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nodes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-54-7" name="__codelineno-54-7" href="#__codelineno-54-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-8" name="__codelineno-54-8" href="#__codelineno-54-8"></a><span class="w">        </span><span class="n">nodes</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="n">i</span><span class="p">));</span>
 <a id="__codelineno-54-9" name="__codelineno-54-9" href="#__codelineno-54-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-54-10" name="__codelineno-54-10" href="#__codelineno-54-10"></a><span class="w">    </span><span class="c1">// A hash table of length n occupies O(n) space</span>
+<a id="__codelineno-54-10" name="__codelineno-54-10" href="#__codelineno-54-10"></a><span class="w">    </span><span class="c1">// A hash table with n entries occupies O(n) space</span>
 <a id="__codelineno-54-11" name="__codelineno-54-11" href="#__codelineno-54-11"></a><span class="w">    </span><span class="n">Map</span><span class="o">&lt;</span><span class="n">Integer</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="o">&gt;</span><span class="w"> </span><span class="n">map</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">HashMap</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-54-12" name="__codelineno-54-12" href="#__codelineno-54-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-13" name="__codelineno-54-13" href="#__codelineno-54-13"></a><span class="w">        </span><span class="n">map</span><span class="p">.</span><span class="na">put</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="p">.</span><span class="na">valueOf</span><span class="p">(</span><span class="n">i</span><span class="p">));</span>
@@ -5944,16 +6021,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">space_complexity.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-55-2" name="__codelineno-55-2" href="#__codelineno-55-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Linear</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="w">    </span><span class="c1">// Array of length n uses O(n) space</span>
+<a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="w">    </span><span class="c1">// An array of length n uses O(n) space</span>
 <a id="__codelineno-55-4" name="__codelineno-55-4" href="#__codelineno-55-4"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="p">];</span>
 <a id="__codelineno-55-5" name="__codelineno-55-5" href="#__codelineno-55-5"></a><span class="w">    </span><span class="c1">// A list of length n occupies O(n) space</span>
 <a id="__codelineno-55-6" name="__codelineno-55-6" href="#__codelineno-55-6"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nodes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-55-7" name="__codelineno-55-7" href="#__codelineno-55-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-8" name="__codelineno-55-8" href="#__codelineno-55-8"></a><span class="w">        </span><span class="n">nodes</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="n">i</span><span class="p">));</span>
 <a id="__codelineno-55-9" name="__codelineno-55-9" href="#__codelineno-55-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-55-10" name="__codelineno-55-10" href="#__codelineno-55-10"></a><span class="w">    </span><span class="c1">// A hash table of length n occupies O(n) space</span>
+<a id="__codelineno-55-10" name="__codelineno-55-10" href="#__codelineno-55-10"></a><span class="w">    </span><span class="c1">// A hash table with n entries occupies O(n) space</span>
 <a id="__codelineno-55-11" name="__codelineno-55-11" href="#__codelineno-55-11"></a><span class="w">    </span><span class="n">Dictionary</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">string</span><span class="o">&gt;</span><span class="w"> </span><span class="n">map</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-55-12" name="__codelineno-55-12" href="#__codelineno-55-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-13" name="__codelineno-55-13" href="#__codelineno-55-13"></a><span class="w">        </span><span class="n">map</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">.</span><span class="n">ToString</span><span class="p">());</span>
@@ -5962,16 +6039,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">space_complexity.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-56-2" name="__codelineno-56-2" href="#__codelineno-56-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">spaceLinear</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w">    </span><span class="c1">// Array of length n uses O(n) space</span>
+<a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w">    </span><span class="c1">// An array of length n uses O(n) space</span>
 <a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a><span class="w">    </span><span class="nx">_</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
 <a id="__codelineno-56-5" name="__codelineno-56-5" href="#__codelineno-56-5"></a><span class="w">    </span><span class="c1">// A list of length n occupies O(n) space</span>
 <a id="__codelineno-56-6" name="__codelineno-56-6" href="#__codelineno-56-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nx">nodes</span><span class="w"> </span><span class="p">[]</span><span class="o">*</span><span class="nx">node</span>
 <a id="__codelineno-56-7" name="__codelineno-56-7" href="#__codelineno-56-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-8" name="__codelineno-56-8" href="#__codelineno-56-8"></a><span class="w">        </span><span class="nx">nodes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">nodes</span><span class="p">,</span><span class="w"> </span><span class="nx">newNode</span><span class="p">(</span><span class="nx">i</span><span class="p">))</span>
 <a id="__codelineno-56-9" name="__codelineno-56-9" href="#__codelineno-56-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-56-10" name="__codelineno-56-10" href="#__codelineno-56-10"></a><span class="w">    </span><span class="c1">// A hash table of length n occupies O(n) space</span>
+<a id="__codelineno-56-10" name="__codelineno-56-10" href="#__codelineno-56-10"></a><span class="w">    </span><span class="c1">// A hash table with n entries occupies O(n) space</span>
 <a id="__codelineno-56-11" name="__codelineno-56-11" href="#__codelineno-56-11"></a><span class="w">    </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">(</span><span class="kt">map</span><span class="p">[</span><span class="kt">int</span><span class="p">]</span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
 <a id="__codelineno-56-12" name="__codelineno-56-12" href="#__codelineno-56-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-13" name="__codelineno-56-13" href="#__codelineno-56-13"></a><span class="w">        </span><span class="nx">m</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">strconv</span><span class="p">.</span><span class="nx">Itoa</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span>
@@ -5980,28 +6057,28 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">space_complexity.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-57-2" name="__codelineno-57-2" href="#__codelineno-57-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="w">    </span><span class="c1">// Array of length n uses O(n) space</span>
+<a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="w">    </span><span class="c1">// An array of length n uses O(n) space</span>
 <a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">nums</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-57-5" name="__codelineno-57-5" href="#__codelineno-57-5"></a><span class="w">    </span><span class="c1">// A list of length n occupies O(n) space</span>
 <a id="__codelineno-57-6" name="__codelineno-57-6" href="#__codelineno-57-6"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">nodes</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">).</span><span class="bp">map</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="nv">$0</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-57-7" name="__codelineno-57-7" href="#__codelineno-57-7"></a><span class="w">    </span><span class="c1">// A hash table of length n occupies O(n) space</span>
+<a id="__codelineno-57-7" name="__codelineno-57-7" href="#__codelineno-57-7"></a><span class="w">    </span><span class="c1">// A hash table with n entries occupies O(n) space</span>
 <a id="__codelineno-57-8" name="__codelineno-57-8" href="#__codelineno-57-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">map</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Dictionary</span><span class="p">(</span><span class="n">uniqueKeysWithValues</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">).</span><span class="bp">map</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="p">(</span><span class="nv">$0</span><span class="p">,</span><span class="w"> </span><span class="s">"</span><span class="si">\(</span><span class="nv">$0</span><span class="si">)</span><span class="s">"</span><span class="p">)</span><span class="w"> </span><span class="p">})</span>
 <a id="__codelineno-57-9" name="__codelineno-57-9" href="#__codelineno-57-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">space_complexity.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-58-2" name="__codelineno-58-2" href="#__codelineno-58-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">linear</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a><span class="w">    </span><span class="c1">// Array of length n uses O(n) space</span>
+<a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a><span class="w">    </span><span class="c1">// An array of length n uses O(n) space</span>
 <a id="__codelineno-58-4" name="__codelineno-58-4" href="#__codelineno-58-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">);</span>
 <a id="__codelineno-58-5" name="__codelineno-58-5" href="#__codelineno-58-5"></a><span class="w">    </span><span class="c1">// A list of length n occupies O(n) space</span>
 <a id="__codelineno-58-6" name="__codelineno-58-6" href="#__codelineno-58-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">nodes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-58-7" name="__codelineno-58-7" href="#__codelineno-58-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-8" name="__codelineno-58-8" href="#__codelineno-58-8"></a><span class="w">        </span><span class="nx">nodes</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="nx">i</span><span class="p">));</span>
 <a id="__codelineno-58-9" name="__codelineno-58-9" href="#__codelineno-58-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-58-10" name="__codelineno-58-10" href="#__codelineno-58-10"></a><span class="w">    </span><span class="c1">// A hash table of length n occupies O(n) space</span>
+<a id="__codelineno-58-10" name="__codelineno-58-10" href="#__codelineno-58-10"></a><span class="w">    </span><span class="c1">// A hash table with n entries occupies O(n) space</span>
 <a id="__codelineno-58-11" name="__codelineno-58-11" href="#__codelineno-58-11"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">map</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Map</span><span class="p">();</span>
 <a id="__codelineno-58-12" name="__codelineno-58-12" href="#__codelineno-58-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-13" name="__codelineno-58-13" href="#__codelineno-58-13"></a><span class="w">        </span><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">.</span><span class="nx">toString</span><span class="p">());</span>
@@ -6010,16 +6087,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">space_complexity.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-59-2" name="__codelineno-59-2" href="#__codelineno-59-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">linear</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="w">    </span><span class="c1">// Array of length n uses O(n) space</span>
+<a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="w">    </span><span class="c1">// An array of length n uses O(n) space</span>
 <a id="__codelineno-59-4" name="__codelineno-59-4" href="#__codelineno-59-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">);</span>
 <a id="__codelineno-59-5" name="__codelineno-59-5" href="#__codelineno-59-5"></a><span class="w">    </span><span class="c1">// A list of length n occupies O(n) space</span>
 <a id="__codelineno-59-6" name="__codelineno-59-6" href="#__codelineno-59-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">nodes</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-59-7" name="__codelineno-59-7" href="#__codelineno-59-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-8" name="__codelineno-59-8" href="#__codelineno-59-8"></a><span class="w">        </span><span class="nx">nodes</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="nx">i</span><span class="p">));</span>
 <a id="__codelineno-59-9" name="__codelineno-59-9" href="#__codelineno-59-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-59-10" name="__codelineno-59-10" href="#__codelineno-59-10"></a><span class="w">    </span><span class="c1">// A hash table of length n occupies O(n) space</span>
+<a id="__codelineno-59-10" name="__codelineno-59-10" href="#__codelineno-59-10"></a><span class="w">    </span><span class="c1">// A hash table with n entries occupies O(n) space</span>
 <a id="__codelineno-59-11" name="__codelineno-59-11" href="#__codelineno-59-11"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">map</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Map</span><span class="p">();</span>
 <a id="__codelineno-59-12" name="__codelineno-59-12" href="#__codelineno-59-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-13" name="__codelineno-59-13" href="#__codelineno-59-13"></a><span class="w">        </span><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">.</span><span class="nx">toString</span><span class="p">());</span>
@@ -6028,16 +6105,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">space_complexity.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-60-2" name="__codelineno-60-2" href="#__codelineno-60-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">linear</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-60-3" name="__codelineno-60-3" href="#__codelineno-60-3"></a><span class="w">  </span><span class="c1">// Array of length n uses O(n) space</span>
+<a id="__codelineno-60-3" name="__codelineno-60-3" href="#__codelineno-60-3"></a><span class="w">  </span><span class="c1">// An array of length n uses O(n) space</span>
 <a id="__codelineno-60-4" name="__codelineno-60-4" href="#__codelineno-60-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-60-5" name="__codelineno-60-5" href="#__codelineno-60-5"></a><span class="w">  </span><span class="c1">// A list of length n occupies O(n) space</span>
 <a id="__codelineno-60-6" name="__codelineno-60-6" href="#__codelineno-60-6"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nodes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-60-7" name="__codelineno-60-7" href="#__codelineno-60-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-8" name="__codelineno-60-8" href="#__codelineno-60-8"></a><span class="w">    </span><span class="n">nodes</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">ListNode</span><span class="p">(</span><span class="n">i</span><span class="p">));</span>
 <a id="__codelineno-60-9" name="__codelineno-60-9" href="#__codelineno-60-9"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-60-10" name="__codelineno-60-10" href="#__codelineno-60-10"></a><span class="w">  </span><span class="c1">// A hash table of length n occupies O(n) space</span>
+<a id="__codelineno-60-10" name="__codelineno-60-10" href="#__codelineno-60-10"></a><span class="w">  </span><span class="c1">// A hash table with n entries occupies O(n) space</span>
 <a id="__codelineno-60-11" name="__codelineno-60-11" href="#__codelineno-60-11"></a><span class="w">  </span><span class="n">Map</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">String</span><span class="o">&gt;</span><span class="w"> </span><span class="n">map</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashMap</span><span class="p">();</span>
 <a id="__codelineno-60-12" name="__codelineno-60-12" href="#__codelineno-60-12"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-13" name="__codelineno-60-13" href="#__codelineno-60-13"></a><span class="w">    </span><span class="n">map</span><span class="p">.</span><span class="n">putIfAbsent</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">i</span><span class="p">.</span><span class="n">toString</span><span class="p">());</span>
@@ -6046,17 +6123,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">space_complexity.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-61-2" name="__codelineno-61-2" href="#__codelineno-61-2"></a><span class="cp">#[allow(unused)]</span>
 <a id="__codelineno-61-3" name="__codelineno-61-3" href="#__codelineno-61-3"></a><span class="k">fn</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-61-4" name="__codelineno-61-4" href="#__codelineno-61-4"></a><span class="w">    </span><span class="c1">// Array of length n uses O(n) space</span>
+<a id="__codelineno-61-4" name="__codelineno-61-4" href="#__codelineno-61-4"></a><span class="w">    </span><span class="c1">// An array of length n uses O(n) space</span>
 <a id="__codelineno-61-5" name="__codelineno-61-5" href="#__codelineno-61-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">];</span>
 <a id="__codelineno-61-6" name="__codelineno-61-6" href="#__codelineno-61-6"></a><span class="w">    </span><span class="c1">// A list of length n occupies O(n) space</span>
 <a id="__codelineno-61-7" name="__codelineno-61-7" href="#__codelineno-61-7"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">nodes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-61-8" name="__codelineno-61-8" href="#__codelineno-61-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-9" name="__codelineno-61-9" href="#__codelineno-61-9"></a><span class="w">        </span><span class="n">nodes</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">ListNode</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
 <a id="__codelineno-61-10" name="__codelineno-61-10" href="#__codelineno-61-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-61-11" name="__codelineno-61-11" href="#__codelineno-61-11"></a><span class="w">    </span><span class="c1">// A hash table of length n occupies O(n) space</span>
+<a id="__codelineno-61-11" name="__codelineno-61-11" href="#__codelineno-61-11"></a><span class="w">    </span><span class="c1">// A hash table with n entries occupies O(n) space</span>
 <a id="__codelineno-61-12" name="__codelineno-61-12" href="#__codelineno-61-12"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">map</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashMap</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-61-13" name="__codelineno-61-13" href="#__codelineno-61-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-14" name="__codelineno-61-14" href="#__codelineno-61-14"></a><span class="w">        </span><span class="n">map</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">.</span><span class="n">to_string</span><span class="p">());</span>
@@ -6072,9 +6149,9 @@
 <a id="__codelineno-62-5" name="__codelineno-62-5" href="#__codelineno-62-5"></a><span class="w">    </span><span class="n">UT_hash_handle</span><span class="w"> </span><span class="n">hh</span><span class="p">;</span><span class="w"> </span><span class="c1">// Implemented using uthash.h</span>
 <a id="__codelineno-62-6" name="__codelineno-62-6" href="#__codelineno-62-6"></a><span class="p">}</span><span class="w"> </span><span class="n">HashTable</span><span class="p">;</span>
 <a id="__codelineno-62-7" name="__codelineno-62-7" href="#__codelineno-62-7"></a>
-<a id="__codelineno-62-8" name="__codelineno-62-8" href="#__codelineno-62-8"></a><span class="cm">/* Linear order */</span>
+<a id="__codelineno-62-8" name="__codelineno-62-8" href="#__codelineno-62-8"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-62-9" name="__codelineno-62-9" href="#__codelineno-62-9"></a><span class="kt">void</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-62-10" name="__codelineno-62-10" href="#__codelineno-62-10"></a><span class="w">    </span><span class="c1">// Array of length n uses O(n) space</span>
+<a id="__codelineno-62-10" name="__codelineno-62-10" href="#__codelineno-62-10"></a><span class="w">    </span><span class="c1">// An array of length n uses O(n) space</span>
 <a id="__codelineno-62-11" name="__codelineno-62-11" href="#__codelineno-62-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-62-12" name="__codelineno-62-12" href="#__codelineno-62-12"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span>
 <a id="__codelineno-62-13" name="__codelineno-62-13" href="#__codelineno-62-13"></a>
@@ -6083,13 +6160,13 @@
 <a id="__codelineno-62-16" name="__codelineno-62-16" href="#__codelineno-62-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-17" name="__codelineno-62-17" href="#__codelineno-62-17"></a><span class="w">        </span><span class="n">nodes</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newListNode</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-62-18" name="__codelineno-62-18" href="#__codelineno-62-18"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-62-19" name="__codelineno-62-19" href="#__codelineno-62-19"></a><span class="w">    </span><span class="c1">// Memory release</span>
+<a id="__codelineno-62-19" name="__codelineno-62-19" href="#__codelineno-62-19"></a><span class="w">    </span><span class="c1">// Free memory</span>
 <a id="__codelineno-62-20" name="__codelineno-62-20" href="#__codelineno-62-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-21" name="__codelineno-62-21" href="#__codelineno-62-21"></a><span class="w">        </span><span class="n">free</span><span class="p">(</span><span class="n">nodes</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
 <a id="__codelineno-62-22" name="__codelineno-62-22" href="#__codelineno-62-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-62-23" name="__codelineno-62-23" href="#__codelineno-62-23"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">nodes</span><span class="p">);</span>
 <a id="__codelineno-62-24" name="__codelineno-62-24" href="#__codelineno-62-24"></a>
-<a id="__codelineno-62-25" name="__codelineno-62-25" href="#__codelineno-62-25"></a><span class="w">    </span><span class="c1">// A hash table of length n occupies O(n) space</span>
+<a id="__codelineno-62-25" name="__codelineno-62-25" href="#__codelineno-62-25"></a><span class="w">    </span><span class="c1">// A hash table with n entries occupies O(n) space</span>
 <a id="__codelineno-62-26" name="__codelineno-62-26" href="#__codelineno-62-26"></a><span class="w">    </span><span class="n">HashTable</span><span class="w"> </span><span class="o">*</span><span class="n">h</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-62-27" name="__codelineno-62-27" href="#__codelineno-62-27"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-28" name="__codelineno-62-28" href="#__codelineno-62-28"></a><span class="w">        </span><span class="n">HashTable</span><span class="w"> </span><span class="o">*</span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">HashTable</span><span class="p">));</span>
@@ -6098,7 +6175,7 @@
 <a id="__codelineno-62-31" name="__codelineno-62-31" href="#__codelineno-62-31"></a><span class="w">        </span><span class="n">HASH_ADD_INT</span><span class="p">(</span><span class="n">h</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">tmp</span><span class="p">);</span>
 <a id="__codelineno-62-32" name="__codelineno-62-32" href="#__codelineno-62-32"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-62-33" name="__codelineno-62-33" href="#__codelineno-62-33"></a>
-<a id="__codelineno-62-34" name="__codelineno-62-34" href="#__codelineno-62-34"></a><span class="w">    </span><span class="c1">// Memory release</span>
+<a id="__codelineno-62-34" name="__codelineno-62-34" href="#__codelineno-62-34"></a><span class="w">    </span><span class="c1">// Free memory</span>
 <a id="__codelineno-62-35" name="__codelineno-62-35" href="#__codelineno-62-35"></a><span class="w">    </span><span class="n">HashTable</span><span class="w"> </span><span class="o">*</span><span class="n">curr</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">tmp</span><span class="p">;</span>
 <a id="__codelineno-62-36" name="__codelineno-62-36" href="#__codelineno-62-36"></a><span class="w">    </span><span class="n">HASH_ITER</span><span class="p">(</span><span class="n">hh</span><span class="p">,</span><span class="w"> </span><span class="n">h</span><span class="p">,</span><span class="w"> </span><span class="n">curr</span><span class="p">,</span><span class="w"> </span><span class="n">tmp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-37" name="__codelineno-62-37" href="#__codelineno-62-37"></a><span class="w">        </span><span class="n">HASH_DEL</span><span class="p">(</span><span class="n">h</span><span class="p">,</span><span class="w"> </span><span class="n">curr</span><span class="p">);</span>
@@ -6108,16 +6185,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">space_complexity.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-63-2" name="__codelineno-63-2" href="#__codelineno-63-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-63-3" name="__codelineno-63-3" href="#__codelineno-63-3"></a><span class="w">    </span><span class="c1">// Array of length n uses O(n) space</span>
+<a id="__codelineno-63-3" name="__codelineno-63-3" href="#__codelineno-63-3"></a><span class="w">    </span><span class="c1">// An array of length n uses O(n) space</span>
 <a id="__codelineno-63-4" name="__codelineno-63-4" href="#__codelineno-63-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Array</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-63-5" name="__codelineno-63-5" href="#__codelineno-63-5"></a><span class="w">    </span><span class="c1">// A list of length n occupies O(n) space</span>
 <a id="__codelineno-63-6" name="__codelineno-63-6" href="#__codelineno-63-6"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">nodes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-63-7" name="__codelineno-63-7" href="#__codelineno-63-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-8" name="__codelineno-63-8" href="#__codelineno-63-8"></a><span class="w">        </span><span class="n">nodes</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">ListNode</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
 <a id="__codelineno-63-9" name="__codelineno-63-9" href="#__codelineno-63-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-63-10" name="__codelineno-63-10" href="#__codelineno-63-10"></a><span class="w">    </span><span class="c1">// A hash table of length n occupies O(n) space</span>
+<a id="__codelineno-63-10" name="__codelineno-63-10" href="#__codelineno-63-10"></a><span class="w">    </span><span class="c1">// A hash table with n entries occupies O(n) space</span>
 <a id="__codelineno-63-11" name="__codelineno-63-11" href="#__codelineno-63-11"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">map</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableMapOf</span><span class="o">&lt;</span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="kt">String</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-63-12" name="__codelineno-63-12" href="#__codelineno-63-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-13" name="__codelineno-63-13" href="#__codelineno-63-13"></a><span class="w">        </span><span class="n">map</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">.</span><span class="na">toString</span><span class="p">()</span>
@@ -6126,12 +6203,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Linear time ###</span>
+<div class="highlight"><span class="filename">space_complexity.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Linear complexity ###</span>
 <a id="__codelineno-64-2" name="__codelineno-64-2" href="#__codelineno-64-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-64-3" name="__codelineno-64-3" href="#__codelineno-64-3"></a><span class="w">  </span><span class="c1"># A list of length n occupies O(n) space</span>
 <a id="__codelineno-64-4" name="__codelineno-64-4" href="#__codelineno-64-4"></a><span class="w">  </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-64-5" name="__codelineno-64-5" href="#__codelineno-64-5"></a>
-<a id="__codelineno-64-6" name="__codelineno-64-6" href="#__codelineno-64-6"></a><span class="w">  </span><span class="c1"># A hash table of length n occupies O(n) space</span>
+<a id="__codelineno-64-6" name="__codelineno-64-6" href="#__codelineno-64-6"></a><span class="w">  </span><span class="c1"># A hash table with n entries occupies O(n) space</span>
 <a id="__codelineno-64-7" name="__codelineno-64-7" href="#__codelineno-64-7"></a><span class="w">  </span><span class="n">hmap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{}</span>
 <a id="__codelineno-64-8" name="__codelineno-64-8" href="#__codelineno-64-8"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">n</span>
 <a id="__codelineno-64-9" name="__codelineno-64-9" href="#__codelineno-64-9"></a><span class="w">    </span><span class="n">hmap</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="o">.</span><span class="n">to_s</span>
@@ -6141,22 +6218,27 @@
 </div>
 </div>
 </div>
-<p>As shown in the following figure, the recursion depth of this function is <span class="arithmatex">\(n\)</span>, meaning that there are <span class="arithmatex">\(n\)</span> unreturned <code>linear_recur()</code> functions existing simultaneously, using <span class="arithmatex">\(O(n)\)</span> stack frame space:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 477px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20linear%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22Linear%20complexity%22%22%22%0A%20%20%20%20%23%20A%20list%20of%20length%20n%20occupies%20O%28n%29%20space%0A%20%20%20%20nums%20%3D%20%5B0%5D%20%2A%20n%0A%20%20%20%20%23%20A%20hash%20table%20with%20n%20entries%20occupies%20O%28n%29%20space%0A%20%20%20%20hmap%20%3D%20dict%5Bint,%20str%5D%28%29%0A%20%20%20%20for%20i%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20hmap%5Bi%5D%20%3D%20str%28i%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20Linear%20complexity%0A%20%20%20%20linear%28n%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=20&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20linear%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22Linear%20complexity%22%22%22%0A%20%20%20%20%23%20A%20list%20of%20length%20n%20occupies%20O%28n%29%20space%0A%20%20%20%20nums%20%3D%20%5B0%5D%20%2A%20n%0A%20%20%20%20%23%20A%20hash%20table%20with%20n%20entries%20occupies%20O%28n%29%20space%0A%20%20%20%20hmap%20%3D%20dict%5Bint,%20str%5D%28%29%0A%20%20%20%20for%20i%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20hmap%5Bi%5D%20%3D%20str%28i%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20Linear%20complexity%0A%20%20%20%20linear%28n%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=20&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>As shown in Figure 2-17, this function has recursion depth <span class="arithmatex">\(n\)</span>, with <span class="arithmatex">\(n\)</span> active calls to <code>linear_recur()</code> at once, using <span class="arithmatex">\(O(n)\)</span> space for stack frames:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="6:13"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Ruby</label></div>
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 <div class="tabbed-block">
 <div class="highlight"><span class="filename">space_complexity.py</span><pre><span></span><code><a id="__codelineno-65-1" name="__codelineno-65-1" href="#__codelineno-65-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">linear_recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-65-2" name="__codelineno-65-2" href="#__codelineno-65-2"></a><span class="w">    </span><span class="sd">"""Linear order (recursive implementation)"""</span>
-<a id="__codelineno-65-3" name="__codelineno-65-3" href="#__codelineno-65-3"></a>    <span class="nb">print</span><span class="p">(</span><span class="s2">"Recursion n ="</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
+<a id="__codelineno-65-2" name="__codelineno-65-2" href="#__codelineno-65-2"></a><span class="w">    </span><span class="sd">"""Linear complexity (recursive implementation)"""</span>
+<a id="__codelineno-65-3" name="__codelineno-65-3" href="#__codelineno-65-3"></a>    <span class="nb">print</span><span class="p">(</span><span class="s2">"Recursion: n ="</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-65-4" name="__codelineno-65-4" href="#__codelineno-65-4"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-65-5" name="__codelineno-65-5" href="#__codelineno-65-5"></a>        <span class="k">return</span>
 <a id="__codelineno-65-6" name="__codelineno-65-6" href="#__codelineno-65-6"></a>    <span class="n">linear_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.cpp</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="cm">/* Linear order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.cpp</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="cm">/* Linear complexity (recursive implementation) */</span>
 <a id="__codelineno-66-2" name="__codelineno-66-2" href="#__codelineno-66-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">linearRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-66-3" name="__codelineno-66-3" href="#__codelineno-66-3"></a><span class="w">    </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">"Recursion n = "</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
+<a id="__codelineno-66-3" name="__codelineno-66-3" href="#__codelineno-66-3"></a><span class="w">    </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">"Recursion: n = "</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
 <a id="__codelineno-66-4" name="__codelineno-66-4" href="#__codelineno-66-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-66-5" name="__codelineno-66-5" href="#__codelineno-66-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-66-6" name="__codelineno-66-6" href="#__codelineno-66-6"></a><span class="w">    </span><span class="n">linearRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
@@ -6164,9 +6246,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="cm">/* Linear order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="cm">/* Linear complexity (recursive implementation) */</span>
 <a id="__codelineno-67-2" name="__codelineno-67-2" href="#__codelineno-67-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">linearRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-67-3" name="__codelineno-67-3" href="#__codelineno-67-3"></a><span class="w">    </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="s">"Recursion n = "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
+<a id="__codelineno-67-3" name="__codelineno-67-3" href="#__codelineno-67-3"></a><span class="w">    </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="s">"Recursive call: n = "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-67-4" name="__codelineno-67-4" href="#__codelineno-67-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-67-5" name="__codelineno-67-5" href="#__codelineno-67-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-67-6" name="__codelineno-67-6" href="#__codelineno-67-6"></a><span class="w">    </span><span class="n">linearRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
@@ -6174,18 +6256,18 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.cs</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a><span class="cm">/* Linear order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.cs</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a><span class="cm">/* Linear complexity (recursive implementation) */</span>
 <a id="__codelineno-68-2" name="__codelineno-68-2" href="#__codelineno-68-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">LinearRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-68-3" name="__codelineno-68-3" href="#__codelineno-68-3"></a><span class="w">    </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="s">"Recursion n = "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
+<a id="__codelineno-68-3" name="__codelineno-68-3" href="#__codelineno-68-3"></a><span class="w">    </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="s">"Recursive call: n = "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-68-4" name="__codelineno-68-4" href="#__codelineno-68-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-68-5" name="__codelineno-68-5" href="#__codelineno-68-5"></a><span class="w">    </span><span class="n">LinearRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-68-6" name="__codelineno-68-6" href="#__codelineno-68-6"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.go</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a><span class="cm">/* Linear order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.go</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a><span class="cm">/* Linear complexity (recursive implementation) */</span>
 <a id="__codelineno-69-2" name="__codelineno-69-2" href="#__codelineno-69-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">spaceLinearRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-69-3" name="__codelineno-69-3" href="#__codelineno-69-3"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="s">"Recursion n ="</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
+<a id="__codelineno-69-3" name="__codelineno-69-3" href="#__codelineno-69-3"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="s">"Recursive call: n ="</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
 <a id="__codelineno-69-4" name="__codelineno-69-4" href="#__codelineno-69-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-69-5" name="__codelineno-69-5" href="#__codelineno-69-5"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-69-6" name="__codelineno-69-6" href="#__codelineno-69-6"></a><span class="w">    </span><span class="p">}</span>
@@ -6194,9 +6276,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.swift</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a><span class="cm">/* Linear order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.swift</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a><span class="cm">/* Linear complexity (recursive implementation) */</span>
 <a id="__codelineno-70-2" name="__codelineno-70-2" href="#__codelineno-70-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">linearRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-70-3" name="__codelineno-70-3" href="#__codelineno-70-3"></a><span class="w">    </span><span class="bp">print</span><span class="p">(</span><span class="s">"Recursion n = </span><span class="si">\(</span><span class="n">n</span><span class="si">)</span><span class="s">"</span><span class="p">)</span>
+<a id="__codelineno-70-3" name="__codelineno-70-3" href="#__codelineno-70-3"></a><span class="w">    </span><span class="bp">print</span><span class="p">(</span><span class="s">"Recursion: n = </span><span class="si">\(</span><span class="n">n</span><span class="si">)</span><span class="s">"</span><span class="p">)</span>
 <a id="__codelineno-70-4" name="__codelineno-70-4" href="#__codelineno-70-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-70-5" name="__codelineno-70-5" href="#__codelineno-70-5"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-70-6" name="__codelineno-70-6" href="#__codelineno-70-6"></a><span class="w">    </span><span class="p">}</span>
@@ -6205,36 +6287,36 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.js</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a><span class="cm">/* Linear order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.js</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a><span class="cm">/* Linear complexity (recursive implementation) */</span>
 <a id="__codelineno-71-2" name="__codelineno-71-2" href="#__codelineno-71-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">linearRecur</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-71-3" name="__codelineno-71-3" href="#__codelineno-71-3"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="sb">`Recursion n = </span><span class="si">${</span><span class="nx">n</span><span class="si">}</span><span class="sb">`</span><span class="p">);</span>
+<a id="__codelineno-71-3" name="__codelineno-71-3" href="#__codelineno-71-3"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="sb">`Recursive call with n = </span><span class="si">${</span><span class="nx">n</span><span class="si">}</span><span class="sb">`</span><span class="p">);</span>
 <a id="__codelineno-71-4" name="__codelineno-71-4" href="#__codelineno-71-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-71-5" name="__codelineno-71-5" href="#__codelineno-71-5"></a><span class="w">    </span><span class="nx">linearRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-71-6" name="__codelineno-71-6" href="#__codelineno-71-6"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.ts</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* Linear order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.ts</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* Linear complexity (recursive implementation) */</span>
 <a id="__codelineno-72-2" name="__codelineno-72-2" href="#__codelineno-72-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">linearRecur</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-72-3" name="__codelineno-72-3" href="#__codelineno-72-3"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="sb">`Recursion n = </span><span class="si">${</span><span class="nx">n</span><span class="si">}</span><span class="sb">`</span><span class="p">);</span>
+<a id="__codelineno-72-3" name="__codelineno-72-3" href="#__codelineno-72-3"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="sb">`Recursive call with n = </span><span class="si">${</span><span class="nx">n</span><span class="si">}</span><span class="sb">`</span><span class="p">);</span>
 <a id="__codelineno-72-4" name="__codelineno-72-4" href="#__codelineno-72-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-72-5" name="__codelineno-72-5" href="#__codelineno-72-5"></a><span class="w">    </span><span class="nx">linearRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-72-6" name="__codelineno-72-6" href="#__codelineno-72-6"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.dart</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="cm">/* Linear order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.dart</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="cm">/* Linear complexity (recursive implementation) */</span>
 <a id="__codelineno-73-2" name="__codelineno-73-2" href="#__codelineno-73-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">linearRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-73-3" name="__codelineno-73-3" href="#__codelineno-73-3"></a><span class="w">  </span><span class="n">print</span><span class="p">(</span><span class="s1">'Recursion n = </span><span class="si">$</span><span class="n">n</span><span class="s1">'</span><span class="p">);</span>
+<a id="__codelineno-73-3" name="__codelineno-73-3" href="#__codelineno-73-3"></a><span class="w">  </span><span class="n">print</span><span class="p">(</span><span class="s1">'Recursive call: n = </span><span class="si">$</span><span class="n">n</span><span class="s1">'</span><span class="p">);</span>
 <a id="__codelineno-73-4" name="__codelineno-73-4" href="#__codelineno-73-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-73-5" name="__codelineno-73-5" href="#__codelineno-73-5"></a><span class="w">  </span><span class="n">linearRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-73-6" name="__codelineno-73-6" href="#__codelineno-73-6"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.rs</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="cm">/* Linear order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.rs</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="cm">/* Linear complexity (recursive implementation) */</span>
 <a id="__codelineno-74-2" name="__codelineno-74-2" href="#__codelineno-74-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">linear_recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-74-3" name="__codelineno-74-3" href="#__codelineno-74-3"></a><span class="w">    </span><span class="fm">println!</span><span class="p">(</span><span class="s">"Recursion n = {}"</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
+<a id="__codelineno-74-3" name="__codelineno-74-3" href="#__codelineno-74-3"></a><span class="w">    </span><span class="fm">println!</span><span class="p">(</span><span class="s">"Recursive call n = {}"</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-74-4" name="__codelineno-74-4" href="#__codelineno-74-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-74-5" name="__codelineno-74-5" href="#__codelineno-74-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-74-6" name="__codelineno-74-6" href="#__codelineno-74-6"></a><span class="w">    </span><span class="p">};</span>
@@ -6243,9 +6325,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.c</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="cm">/* Linear order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.c</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="cm">/* Linear complexity (recursive implementation) */</span>
 <a id="__codelineno-75-2" name="__codelineno-75-2" href="#__codelineno-75-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">linearRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-75-3" name="__codelineno-75-3" href="#__codelineno-75-3"></a><span class="w">    </span><span class="n">printf</span><span class="p">(</span><span class="s">"Recursion n = %d</span><span class="se">\r\n</span><span class="s">"</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
+<a id="__codelineno-75-3" name="__codelineno-75-3" href="#__codelineno-75-3"></a><span class="w">    </span><span class="n">printf</span><span class="p">(</span><span class="s">"Recursive call n = %d</span><span class="se">\r\n</span><span class="s">"</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-75-4" name="__codelineno-75-4" href="#__codelineno-75-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-75-5" name="__codelineno-75-5" href="#__codelineno-75-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-75-6" name="__codelineno-75-6" href="#__codelineno-75-6"></a><span class="w">    </span><span class="n">linearRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
@@ -6253,9 +6335,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.kt</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a><span class="cm">/* Linear order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.kt</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a><span class="cm">/* Linear complexity (recursive implementation) */</span>
 <a id="__codelineno-76-2" name="__codelineno-76-2" href="#__codelineno-76-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">linearRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-76-3" name="__codelineno-76-3" href="#__codelineno-76-3"></a><span class="w">    </span><span class="n">println</span><span class="p">(</span><span class="s">"Recursion n = </span><span class="si">$</span><span class="n">n</span><span class="s">"</span><span class="p">)</span>
+<a id="__codelineno-76-3" name="__codelineno-76-3" href="#__codelineno-76-3"></a><span class="w">    </span><span class="n">println</span><span class="p">(</span><span class="s">"Recursive call: n = </span><span class="si">$</span><span class="n">n</span><span class="s">"</span><span class="p">)</span>
 <a id="__codelineno-76-4" name="__codelineno-76-4" href="#__codelineno-76-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-76-5" name="__codelineno-76-5" href="#__codelineno-76-5"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-76-6" name="__codelineno-76-6" href="#__codelineno-76-6"></a><span class="w">    </span><span class="n">linearRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
@@ -6263,9 +6345,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.rb</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="c1">### Linear space (recursive) ###</span>
+<div class="highlight"><span class="filename">space_complexity.rb</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="c1">### Linear complexity (recursive implementation)###</span>
 <a id="__codelineno-77-2" name="__codelineno-77-2" href="#__codelineno-77-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">linear_recur</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-77-3" name="__codelineno-77-3" href="#__codelineno-77-3"></a><span class="w">  </span><span class="nb">puts</span><span class="w"> </span><span class="s2">"Recursion n = </span><span class="si">#{</span><span class="n">n</span><span class="si">}</span><span class="s2">"</span>
+<a id="__codelineno-77-3" name="__codelineno-77-3" href="#__codelineno-77-3"></a><span class="w">  </span><span class="nb">puts</span><span class="w"> </span><span class="s2">"Recursion: n = </span><span class="si">#{</span><span class="n">n</span><span class="si">}</span><span class="s2">"</span>
 <a id="__codelineno-77-4" name="__codelineno-77-4" href="#__codelineno-77-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-77-5" name="__codelineno-77-5" href="#__codelineno-77-5"></a><span class="w">  </span><span class="n">linear_recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-77-6" name="__codelineno-77-6" href="#__codelineno-77-6"></a><span class="k">end</span>
@@ -6273,24 +6355,29 @@
 </div>
 </div>
 </div>
-<p><img alt="Linear order space complexity generated by recursive function" class="animation-figure" src="../space_complexity.assets/space_complexity_recursive_linear.png" /></p>
-<p align="center"> Figure 2-17 &nbsp; Linear order space complexity generated by recursive function </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 441px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20linear_recur%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22Linear%20complexity%20%28recursive%20implementation%29%22%22%22%0A%20%20%20%20print%28%22Recursion%3A%20n%20%3D%22,%20n%29%0A%20%20%20%20if%20n%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20linear_recur%28n%20-%201%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20Linear%20complexity%0A%20%20%20%20linear_recur%28n%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=25&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20linear_recur%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22Linear%20complexity%20%28recursive%20implementation%29%22%22%22%0A%20%20%20%20print%28%22Recursion%3A%20n%20%3D%22,%20n%29%0A%20%20%20%20if%20n%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20linear_recur%28n%20-%201%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20Linear%20complexity%0A%20%20%20%20linear_recur%28n%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=25&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p><img alt="Linear space complexity from recursion" class="animation-figure" src="../space_complexity.assets/space_complexity_recursive_linear.png" /></p>
+<p align="center"> Figure 2-17 &nbsp; Linear space complexity from recursion </p>
 
-<h3 id="3-quadratic-order-on2">3. &nbsp; Quadratic Order <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#3-quadratic-order-on2" title="Permanent link">&para;</a></h3>
-<p>Quadratic order is common in matrices and graphs, where the number of elements is quadratically related to <span class="arithmatex">\(n\)</span>:</p>
+<h3 id="3-quadratic-space-on2">3. &nbsp; Quadratic space <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#3-quadratic-space-on2" title="Permanent link">&para;</a></h3>
+<p>Quadratic space commonly occurs in matrices and graphs whose element counts grow with the square of <span class="arithmatex">\(n\)</span>:</p>
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 <div class="highlight"><span class="filename">space_complexity.py</span><pre><span></span><code><a id="__codelineno-78-1" name="__codelineno-78-1" href="#__codelineno-78-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-78-2" name="__codelineno-78-2" href="#__codelineno-78-2"></a><span class="w">    </span><span class="sd">"""Quadratic order"""</span>
-<a id="__codelineno-78-3" name="__codelineno-78-3" href="#__codelineno-78-3"></a>    <span class="c1"># A 2D list occupies O(n^2) space</span>
+<a id="__codelineno-78-2" name="__codelineno-78-2" href="#__codelineno-78-2"></a><span class="w">    </span><span class="sd">"""Quadratic complexity"""</span>
+<a id="__codelineno-78-3" name="__codelineno-78-3" href="#__codelineno-78-3"></a>    <span class="c1"># The two-dimensional list occupies O(n^2) space</span>
 <a id="__codelineno-78-4" name="__codelineno-78-4" href="#__codelineno-78-4"></a>    <span class="n">num_matrix</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.cpp</span><pre><span></span><code><a id="__codelineno-79-1" name="__codelineno-79-1" href="#__codelineno-79-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">space_complexity.cpp</span><pre><span></span><code><a id="__codelineno-79-1" name="__codelineno-79-1" href="#__codelineno-79-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-79-2" name="__codelineno-79-2" href="#__codelineno-79-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-79-3" name="__codelineno-79-3" href="#__codelineno-79-3"></a><span class="w">    </span><span class="c1">// 2D list uses O(n^2) space</span>
+<a id="__codelineno-79-3" name="__codelineno-79-3" href="#__codelineno-79-3"></a><span class="w">    </span><span class="c1">// The two-dimensional list occupies O(n^2) space</span>
 <a id="__codelineno-79-4" name="__codelineno-79-4" href="#__codelineno-79-4"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">numMatrix</span><span class="p">;</span>
 <a id="__codelineno-79-5" name="__codelineno-79-5" href="#__codelineno-79-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-6" name="__codelineno-79-6" href="#__codelineno-79-6"></a><span class="w">        </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
@@ -6303,11 +6390,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-80-1" name="__codelineno-80-1" href="#__codelineno-80-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-80-1" name="__codelineno-80-1" href="#__codelineno-80-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-80-2" name="__codelineno-80-2" href="#__codelineno-80-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-80-3" name="__codelineno-80-3" href="#__codelineno-80-3"></a><span class="w">    </span><span class="c1">// Matrix uses O(n^2) space</span>
+<a id="__codelineno-80-3" name="__codelineno-80-3" href="#__codelineno-80-3"></a><span class="w">    </span><span class="c1">// The matrix uses O(n^2) space</span>
 <a id="__codelineno-80-4" name="__codelineno-80-4" href="#__codelineno-80-4"></a><span class="w">    </span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">numMatrix</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="o">][</span><span class="n">n</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-80-5" name="__codelineno-80-5" href="#__codelineno-80-5"></a><span class="w">    </span><span class="c1">// 2D list uses O(n^2) space</span>
+<a id="__codelineno-80-5" name="__codelineno-80-5" href="#__codelineno-80-5"></a><span class="w">    </span><span class="c1">// The two-dimensional list occupies O(n^2) space</span>
 <a id="__codelineno-80-6" name="__codelineno-80-6" href="#__codelineno-80-6"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">numList</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-80-7" name="__codelineno-80-7" href="#__codelineno-80-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-8" name="__codelineno-80-8" href="#__codelineno-80-8"></a><span class="w">        </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
@@ -6320,11 +6407,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.cs</span><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">space_complexity.cs</span><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-81-2" name="__codelineno-81-2" href="#__codelineno-81-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Quadratic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-81-3" name="__codelineno-81-3" href="#__codelineno-81-3"></a><span class="w">    </span><span class="c1">// Matrix uses O(n^2) space</span>
+<a id="__codelineno-81-3" name="__codelineno-81-3" href="#__codelineno-81-3"></a><span class="w">    </span><span class="c1">// The matrix uses O(n^2) space</span>
 <a id="__codelineno-81-4" name="__codelineno-81-4" href="#__codelineno-81-4"></a><span class="w">    </span><span class="kt">int</span><span class="p">[,]</span><span class="w"> </span><span class="n">numMatrix</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">];</span>
-<a id="__codelineno-81-5" name="__codelineno-81-5" href="#__codelineno-81-5"></a><span class="w">    </span><span class="c1">// 2D list uses O(n^2) space</span>
+<a id="__codelineno-81-5" name="__codelineno-81-5" href="#__codelineno-81-5"></a><span class="w">    </span><span class="c1">// The two-dimensional list occupies O(n^2) space</span>
 <a id="__codelineno-81-6" name="__codelineno-81-6" href="#__codelineno-81-6"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">numList</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-81-7" name="__codelineno-81-7" href="#__codelineno-81-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-8" name="__codelineno-81-8" href="#__codelineno-81-8"></a><span class="w">        </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
@@ -6337,9 +6424,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.go</span><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">space_complexity.go</span><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-82-2" name="__codelineno-82-2" href="#__codelineno-82-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">spaceQuadratic</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-82-3" name="__codelineno-82-3" href="#__codelineno-82-3"></a><span class="w">    </span><span class="c1">// Matrix uses O(n^2) space</span>
+<a id="__codelineno-82-3" name="__codelineno-82-3" href="#__codelineno-82-3"></a><span class="w">    </span><span class="c1">// The matrix uses O(n^2) space</span>
 <a id="__codelineno-82-4" name="__codelineno-82-4" href="#__codelineno-82-4"></a><span class="w">    </span><span class="nx">numMatrix</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
 <a id="__codelineno-82-5" name="__codelineno-82-5" href="#__codelineno-82-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-6" name="__codelineno-82-6" href="#__codelineno-82-6"></a><span class="w">        </span><span class="nx">numMatrix</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
@@ -6348,21 +6435,21 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.swift</span><pre><span></span><code><a id="__codelineno-83-1" name="__codelineno-83-1" href="#__codelineno-83-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">space_complexity.swift</span><pre><span></span><code><a id="__codelineno-83-1" name="__codelineno-83-1" href="#__codelineno-83-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-83-2" name="__codelineno-83-2" href="#__codelineno-83-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-83-3" name="__codelineno-83-3" href="#__codelineno-83-3"></a><span class="w">    </span><span class="c1">// 2D list uses O(n^2) space</span>
+<a id="__codelineno-83-3" name="__codelineno-83-3" href="#__codelineno-83-3"></a><span class="w">    </span><span class="c1">// A fully materialized two-dimensional list uses O(n^2) space; Swift may share repeated row storage</span>
 <a id="__codelineno-83-4" name="__codelineno-83-4" href="#__codelineno-83-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">numList</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">),</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-83-5" name="__codelineno-83-5" href="#__codelineno-83-5"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.js</span><pre><span></span><code><a id="__codelineno-84-1" name="__codelineno-84-1" href="#__codelineno-84-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">space_complexity.js</span><pre><span></span><code><a id="__codelineno-84-1" name="__codelineno-84-1" href="#__codelineno-84-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-84-2" name="__codelineno-84-2" href="#__codelineno-84-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">quadratic</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-84-3" name="__codelineno-84-3" href="#__codelineno-84-3"></a><span class="w">    </span><span class="c1">// Matrix uses O(n^2) space</span>
+<a id="__codelineno-84-3" name="__codelineno-84-3" href="#__codelineno-84-3"></a><span class="w">    </span><span class="c1">// The matrix uses O(n^2) space</span>
 <a id="__codelineno-84-4" name="__codelineno-84-4" href="#__codelineno-84-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">numMatrix</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span>
 <a id="__codelineno-84-5" name="__codelineno-84-5" href="#__codelineno-84-5"></a><span class="w">        </span><span class="p">.</span><span class="nx">fill</span><span class="p">(</span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-84-6" name="__codelineno-84-6" href="#__codelineno-84-6"></a><span class="w">        </span><span class="p">.</span><span class="nx">map</span><span class="p">(()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">null</span><span class="p">));</span>
-<a id="__codelineno-84-7" name="__codelineno-84-7" href="#__codelineno-84-7"></a><span class="w">    </span><span class="c1">// 2D list uses O(n^2) space</span>
+<a id="__codelineno-84-7" name="__codelineno-84-7" href="#__codelineno-84-7"></a><span class="w">    </span><span class="c1">// The two-dimensional list occupies O(n^2) space</span>
 <a id="__codelineno-84-8" name="__codelineno-84-8" href="#__codelineno-84-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">numList</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-84-9" name="__codelineno-84-9" href="#__codelineno-84-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-10" name="__codelineno-84-10" href="#__codelineno-84-10"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
@@ -6375,13 +6462,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.ts</span><pre><span></span><code><a id="__codelineno-85-1" name="__codelineno-85-1" href="#__codelineno-85-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">space_complexity.ts</span><pre><span></span><code><a id="__codelineno-85-1" name="__codelineno-85-1" href="#__codelineno-85-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-85-2" name="__codelineno-85-2" href="#__codelineno-85-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">quadratic</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-85-3" name="__codelineno-85-3" href="#__codelineno-85-3"></a><span class="w">    </span><span class="c1">// Matrix uses O(n^2) space</span>
+<a id="__codelineno-85-3" name="__codelineno-85-3" href="#__codelineno-85-3"></a><span class="w">    </span><span class="c1">// The matrix uses O(n^2) space</span>
 <a id="__codelineno-85-4" name="__codelineno-85-4" href="#__codelineno-85-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">numMatrix</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span>
 <a id="__codelineno-85-5" name="__codelineno-85-5" href="#__codelineno-85-5"></a><span class="w">        </span><span class="p">.</span><span class="nx">fill</span><span class="p">(</span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-85-6" name="__codelineno-85-6" href="#__codelineno-85-6"></a><span class="w">        </span><span class="p">.</span><span class="nx">map</span><span class="p">(()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">null</span><span class="p">));</span>
-<a id="__codelineno-85-7" name="__codelineno-85-7" href="#__codelineno-85-7"></a><span class="w">    </span><span class="c1">// 2D list uses O(n^2) space</span>
+<a id="__codelineno-85-7" name="__codelineno-85-7" href="#__codelineno-85-7"></a><span class="w">    </span><span class="c1">// The two-dimensional list occupies O(n^2) space</span>
 <a id="__codelineno-85-8" name="__codelineno-85-8" href="#__codelineno-85-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">numList</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-85-9" name="__codelineno-85-9" href="#__codelineno-85-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-10" name="__codelineno-85-10" href="#__codelineno-85-10"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
@@ -6394,11 +6481,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.dart</span><pre><span></span><code><a id="__codelineno-86-1" name="__codelineno-86-1" href="#__codelineno-86-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">space_complexity.dart</span><pre><span></span><code><a id="__codelineno-86-1" name="__codelineno-86-1" href="#__codelineno-86-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-86-2" name="__codelineno-86-2" href="#__codelineno-86-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">quadratic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-86-3" name="__codelineno-86-3" href="#__codelineno-86-3"></a><span class="w">  </span><span class="c1">// Matrix uses O(n^2) space</span>
+<a id="__codelineno-86-3" name="__codelineno-86-3" href="#__codelineno-86-3"></a><span class="w">  </span><span class="c1">// The matrix uses O(n^2) space</span>
 <a id="__codelineno-86-4" name="__codelineno-86-4" href="#__codelineno-86-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">numMatrix</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">_</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">));</span>
-<a id="__codelineno-86-5" name="__codelineno-86-5" href="#__codelineno-86-5"></a><span class="w">  </span><span class="c1">// 2D list uses O(n^2) space</span>
+<a id="__codelineno-86-5" name="__codelineno-86-5" href="#__codelineno-86-5"></a><span class="w">  </span><span class="c1">// The two-dimensional list occupies O(n^2) space</span>
 <a id="__codelineno-86-6" name="__codelineno-86-6" href="#__codelineno-86-6"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">numList</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-86-7" name="__codelineno-86-7" href="#__codelineno-86-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-8" name="__codelineno-86-8" href="#__codelineno-86-8"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
@@ -6411,12 +6498,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.rs</span><pre><span></span><code><a id="__codelineno-87-1" name="__codelineno-87-1" href="#__codelineno-87-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">space_complexity.rs</span><pre><span></span><code><a id="__codelineno-87-1" name="__codelineno-87-1" href="#__codelineno-87-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-87-2" name="__codelineno-87-2" href="#__codelineno-87-2"></a><span class="cp">#[allow(unused)]</span>
 <a id="__codelineno-87-3" name="__codelineno-87-3" href="#__codelineno-87-3"></a><span class="k">fn</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-87-4" name="__codelineno-87-4" href="#__codelineno-87-4"></a><span class="w">    </span><span class="c1">// Matrix uses O(n^2) space</span>
+<a id="__codelineno-87-4" name="__codelineno-87-4" href="#__codelineno-87-4"></a><span class="w">    </span><span class="c1">// The matrix uses O(n^2) space</span>
 <a id="__codelineno-87-5" name="__codelineno-87-5" href="#__codelineno-87-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">num_matrix</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">];</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">];</span>
-<a id="__codelineno-87-6" name="__codelineno-87-6" href="#__codelineno-87-6"></a><span class="w">    </span><span class="c1">// 2D list uses O(n^2) space</span>
+<a id="__codelineno-87-6" name="__codelineno-87-6" href="#__codelineno-87-6"></a><span class="w">    </span><span class="c1">// The two-dimensional list occupies O(n^2) space</span>
 <a id="__codelineno-87-7" name="__codelineno-87-7" href="#__codelineno-87-7"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">num_list</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-87-8" name="__codelineno-87-8" href="#__codelineno-87-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-9" name="__codelineno-87-9" href="#__codelineno-87-9"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
@@ -6429,9 +6516,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.c</span><pre><span></span><code><a id="__codelineno-88-1" name="__codelineno-88-1" href="#__codelineno-88-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">space_complexity.c</span><pre><span></span><code><a id="__codelineno-88-1" name="__codelineno-88-1" href="#__codelineno-88-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-88-2" name="__codelineno-88-2" href="#__codelineno-88-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-88-3" name="__codelineno-88-3" href="#__codelineno-88-3"></a><span class="w">    </span><span class="c1">// 2D list uses O(n^2) space</span>
+<a id="__codelineno-88-3" name="__codelineno-88-3" href="#__codelineno-88-3"></a><span class="w">    </span><span class="c1">// The two-dimensional list occupies O(n^2) space</span>
 <a id="__codelineno-88-4" name="__codelineno-88-4" href="#__codelineno-88-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">**</span><span class="n">numMatrix</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-88-5" name="__codelineno-88-5" href="#__codelineno-88-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-6" name="__codelineno-88-6" href="#__codelineno-88-6"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
@@ -6441,7 +6528,7 @@
 <a id="__codelineno-88-10" name="__codelineno-88-10" href="#__codelineno-88-10"></a><span class="w">        </span><span class="n">numMatrix</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
 <a id="__codelineno-88-11" name="__codelineno-88-11" href="#__codelineno-88-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-88-12" name="__codelineno-88-12" href="#__codelineno-88-12"></a>
-<a id="__codelineno-88-13" name="__codelineno-88-13" href="#__codelineno-88-13"></a><span class="w">    </span><span class="c1">// Memory release</span>
+<a id="__codelineno-88-13" name="__codelineno-88-13" href="#__codelineno-88-13"></a><span class="w">    </span><span class="c1">// Free memory</span>
 <a id="__codelineno-88-14" name="__codelineno-88-14" href="#__codelineno-88-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-15" name="__codelineno-88-15" href="#__codelineno-88-15"></a><span class="w">        </span><span class="n">free</span><span class="p">(</span><span class="n">numMatrix</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
 <a id="__codelineno-88-16" name="__codelineno-88-16" href="#__codelineno-88-16"></a><span class="w">    </span><span class="p">}</span>
@@ -6450,11 +6537,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.kt</span><pre><span></span><code><a id="__codelineno-89-1" name="__codelineno-89-1" href="#__codelineno-89-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">space_complexity.kt</span><pre><span></span><code><a id="__codelineno-89-1" name="__codelineno-89-1" href="#__codelineno-89-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-89-2" name="__codelineno-89-2" href="#__codelineno-89-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-89-3" name="__codelineno-89-3" href="#__codelineno-89-3"></a><span class="w">    </span><span class="c1">// Matrix uses O(n^2) space</span>
+<a id="__codelineno-89-3" name="__codelineno-89-3" href="#__codelineno-89-3"></a><span class="w">    </span><span class="c1">// A full n*n matrix uses O(n^2) space; only its outer array is allocated here</span>
 <a id="__codelineno-89-4" name="__codelineno-89-4" href="#__codelineno-89-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">numMatrix</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arrayOfNulls</span><span class="o">&lt;</span><span class="n">Array</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?&gt;</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-89-5" name="__codelineno-89-5" href="#__codelineno-89-5"></a><span class="w">    </span><span class="c1">// 2D list uses O(n^2) space</span>
+<a id="__codelineno-89-5" name="__codelineno-89-5" href="#__codelineno-89-5"></a><span class="w">    </span><span class="c1">// The two-dimensional list occupies O(n^2) space</span>
 <a id="__codelineno-89-6" name="__codelineno-89-6" href="#__codelineno-89-6"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">numList</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;&gt;</span><span class="p">()</span>
 <a id="__codelineno-89-7" name="__codelineno-89-7" href="#__codelineno-89-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-89-8" name="__codelineno-89-8" href="#__codelineno-89-8"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span>
@@ -6467,137 +6554,142 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.rb</span><pre><span></span><code><a id="__codelineno-90-1" name="__codelineno-90-1" href="#__codelineno-90-1"></a><span class="c1">### Quadratic time ###</span>
+<div class="highlight"><span class="filename">space_complexity.rb</span><pre><span></span><code><a id="__codelineno-90-1" name="__codelineno-90-1" href="#__codelineno-90-1"></a><span class="c1">### Quadratic complexity ###</span>
 <a id="__codelineno-90-2" name="__codelineno-90-2" href="#__codelineno-90-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-90-3" name="__codelineno-90-3" href="#__codelineno-90-3"></a><span class="w">  </span><span class="c1"># 2D list uses O(n^2) space</span>
+<a id="__codelineno-90-3" name="__codelineno-90-3" href="#__codelineno-90-3"></a><span class="w">  </span><span class="c1"># The two-dimensional list occupies O(n^2) space</span>
 <a id="__codelineno-90-4" name="__codelineno-90-4" href="#__codelineno-90-4"></a><span class="w">  </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-90-5" name="__codelineno-90-5" href="#__codelineno-90-5"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<p>As shown in the following figure, the recursion depth of this function is <span class="arithmatex">\(n\)</span>, and an array is initialized in each recursive function with lengths of <span class="arithmatex">\(n\)</span>, <span class="arithmatex">\(n-1\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, <span class="arithmatex">\(1\)</span>, with an average length of <span class="arithmatex">\(n / 2\)</span>, thus occupying <span class="arithmatex">\(O(n^2)\)</span> space overall:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 405px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20quadratic%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22Quadratic%20complexity%22%22%22%0A%20%20%20%20%23%20The%20two-dimensional%20list%20occupies%20O%28n%5E2%29%20space%0A%20%20%20%20num_matrix%20%3D%20%5B%5B0%5D%20%2A%20n%20for%20_%20in%20range%28n%29%5D%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20Quadratic%20complexity%0A%20%20%20%20quadratic%28n%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20quadratic%28n%3A%20int%29%3A%0A%20%20%20%20%22%22%22Quadratic%20complexity%22%22%22%0A%20%20%20%20%23%20The%20two-dimensional%20list%20occupies%20O%28n%5E2%29%20space%0A%20%20%20%20num_matrix%20%3D%20%5B%5B0%5D%20%2A%20n%20for%20_%20in%20range%28n%29%5D%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20Quadratic%20complexity%0A%20%20%20%20quadratic%28n%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>As shown in Figure 2-18, the recursion depth is <span class="arithmatex">\(n\)</span>, and each call initializes an array. Their lengths are <span class="arithmatex">\(n\)</span>, <span class="arithmatex">\(n-1\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, and <span class="arithmatex">\(1\)</span>, averaging <span class="arithmatex">\(n / 2\)</span>, so total space is <span class="arithmatex">\(O(n^2)\)</span>:</p>
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 <div class="highlight"><span class="filename">space_complexity.py</span><pre><span></span><code><a id="__codelineno-91-1" name="__codelineno-91-1" href="#__codelineno-91-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">quadratic_recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-91-2" name="__codelineno-91-2" href="#__codelineno-91-2"></a><span class="w">    </span><span class="sd">"""Quadratic order (recursive implementation)"""</span>
+<a id="__codelineno-91-2" name="__codelineno-91-2" href="#__codelineno-91-2"></a><span class="w">    </span><span class="sd">"""Quadratic complexity (recursive implementation)"""</span>
 <a id="__codelineno-91-3" name="__codelineno-91-3" href="#__codelineno-91-3"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-91-4" name="__codelineno-91-4" href="#__codelineno-91-4"></a>        <span class="k">return</span> <span class="mi">0</span>
-<a id="__codelineno-91-5" name="__codelineno-91-5" href="#__codelineno-91-5"></a>    <span class="c1"># Array nums length is n, n-1, ..., 2, 1</span>
+<a id="__codelineno-91-5" name="__codelineno-91-5" href="#__codelineno-91-5"></a>    <span class="c1"># The lengths of array nums are n, n-1, ..., 2, 1</span>
 <a id="__codelineno-91-6" name="__codelineno-91-6" href="#__codelineno-91-6"></a>    <span class="n">nums</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span>
 <a id="__codelineno-91-7" name="__codelineno-91-7" href="#__codelineno-91-7"></a>    <span class="k">return</span> <span class="n">quadratic_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.cpp</span><pre><span></span><code><a id="__codelineno-92-1" name="__codelineno-92-1" href="#__codelineno-92-1"></a><span class="cm">/* Quadratic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.cpp</span><pre><span></span><code><a id="__codelineno-92-1" name="__codelineno-92-1" href="#__codelineno-92-1"></a><span class="cm">/* Quadratic complexity (recursive implementation) */</span>
 <a id="__codelineno-92-2" name="__codelineno-92-2" href="#__codelineno-92-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">quadraticRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-92-3" name="__codelineno-92-3" href="#__codelineno-92-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-92-4" name="__codelineno-92-4" href="#__codelineno-92-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-92-5" name="__codelineno-92-5" href="#__codelineno-92-5"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">(</span><span class="n">n</span><span class="p">);</span>
-<a id="__codelineno-92-6" name="__codelineno-92-6" href="#__codelineno-92-6"></a><span class="w">    </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">"In recursion n = "</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">", nums length = "</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
+<a id="__codelineno-92-6" name="__codelineno-92-6" href="#__codelineno-92-6"></a><span class="w">    </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">"Recursion: n = "</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">" , length of nums = "</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
 <a id="__codelineno-92-7" name="__codelineno-92-7" href="#__codelineno-92-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">quadraticRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-92-8" name="__codelineno-92-8" href="#__codelineno-92-8"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-93-1" name="__codelineno-93-1" href="#__codelineno-93-1"></a><span class="cm">/* Quadratic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-93-1" name="__codelineno-93-1" href="#__codelineno-93-1"></a><span class="cm">/* Quadratic complexity (recursive implementation) */</span>
 <a id="__codelineno-93-2" name="__codelineno-93-2" href="#__codelineno-93-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">quadraticRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-93-3" name="__codelineno-93-3" href="#__codelineno-93-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-93-4" name="__codelineno-93-4" href="#__codelineno-93-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-93-5" name="__codelineno-93-5" href="#__codelineno-93-5"></a><span class="w">    </span><span class="c1">// Array nums has length n, n-1, ..., 2, 1</span>
+<a id="__codelineno-93-5" name="__codelineno-93-5" href="#__codelineno-93-5"></a><span class="w">    </span><span class="c1">// The lengths of array nums are n, n-1, ..., 2, 1</span>
 <a id="__codelineno-93-6" name="__codelineno-93-6" href="#__codelineno-93-6"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-93-7" name="__codelineno-93-7" href="#__codelineno-93-7"></a><span class="w">    </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="s">"In recursion n = "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="s">", nums length = "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">);</span>
+<a id="__codelineno-93-7" name="__codelineno-93-7" href="#__codelineno-93-7"></a><span class="w">    </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="s">"Recursive call: n = "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="s">" with nums length = "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">);</span>
 <a id="__codelineno-93-8" name="__codelineno-93-8" href="#__codelineno-93-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">quadraticRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-93-9" name="__codelineno-93-9" href="#__codelineno-93-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.cs</span><pre><span></span><code><a id="__codelineno-94-1" name="__codelineno-94-1" href="#__codelineno-94-1"></a><span class="cm">/* Quadratic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.cs</span><pre><span></span><code><a id="__codelineno-94-1" name="__codelineno-94-1" href="#__codelineno-94-1"></a><span class="cm">/* Quadratic complexity (recursive implementation) */</span>
 <a id="__codelineno-94-2" name="__codelineno-94-2" href="#__codelineno-94-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">QuadraticRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-94-3" name="__codelineno-94-3" href="#__codelineno-94-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-94-4" name="__codelineno-94-4" href="#__codelineno-94-4"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="p">];</span>
-<a id="__codelineno-94-5" name="__codelineno-94-5" href="#__codelineno-94-5"></a><span class="w">    </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="s">"Recursion n = "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="s">", nums length = "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">);</span>
+<a id="__codelineno-94-5" name="__codelineno-94-5" href="#__codelineno-94-5"></a><span class="w">    </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="s">"Recursive call: n = "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="s">" with nums length = "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">);</span>
 <a id="__codelineno-94-6" name="__codelineno-94-6" href="#__codelineno-94-6"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nf">QuadraticRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-94-7" name="__codelineno-94-7" href="#__codelineno-94-7"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.go</span><pre><span></span><code><a id="__codelineno-95-1" name="__codelineno-95-1" href="#__codelineno-95-1"></a><span class="cm">/* Quadratic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.go</span><pre><span></span><code><a id="__codelineno-95-1" name="__codelineno-95-1" href="#__codelineno-95-1"></a><span class="cm">/* Quadratic complexity (recursive implementation) */</span>
 <a id="__codelineno-95-2" name="__codelineno-95-2" href="#__codelineno-95-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">spaceQuadraticRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-95-3" name="__codelineno-95-3" href="#__codelineno-95-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-95-4" name="__codelineno-95-4" href="#__codelineno-95-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-95-5" name="__codelineno-95-5" href="#__codelineno-95-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-95-6" name="__codelineno-95-6" href="#__codelineno-95-6"></a><span class="w">    </span><span class="nx">nums</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
-<a id="__codelineno-95-7" name="__codelineno-95-7" href="#__codelineno-95-7"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Printf</span><span class="p">(</span><span class="s">"In recursion n = %d, nums length = %d \n"</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">))</span>
+<a id="__codelineno-95-7" name="__codelineno-95-7" href="#__codelineno-95-7"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Printf</span><span class="p">(</span><span class="s">"Recursive call n = %d: length of nums = %d \n"</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">))</span>
 <a id="__codelineno-95-8" name="__codelineno-95-8" href="#__codelineno-95-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">spaceQuadraticRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-95-9" name="__codelineno-95-9" href="#__codelineno-95-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.swift</span><pre><span></span><code><a id="__codelineno-96-1" name="__codelineno-96-1" href="#__codelineno-96-1"></a><span class="cm">/* Quadratic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.swift</span><pre><span></span><code><a id="__codelineno-96-1" name="__codelineno-96-1" href="#__codelineno-96-1"></a><span class="cm">/* Quadratic complexity (recursive implementation) */</span>
 <a id="__codelineno-96-2" name="__codelineno-96-2" href="#__codelineno-96-2"></a><span class="p">@</span><span class="n">discardableResult</span>
 <a id="__codelineno-96-3" name="__codelineno-96-3" href="#__codelineno-96-3"></a><span class="kd">func</span><span class="w"> </span><span class="nf">quadraticRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-96-4" name="__codelineno-96-4" href="#__codelineno-96-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-96-5" name="__codelineno-96-5" href="#__codelineno-96-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-96-6" name="__codelineno-96-6" href="#__codelineno-96-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-96-7" name="__codelineno-96-7" href="#__codelineno-96-7"></a><span class="w">    </span><span class="c1">// Array nums has length n, n-1, ..., 2, 1</span>
+<a id="__codelineno-96-7" name="__codelineno-96-7" href="#__codelineno-96-7"></a><span class="w">    </span><span class="c1">// The lengths of array nums are n, n-1, ..., 2, 1</span>
 <a id="__codelineno-96-8" name="__codelineno-96-8" href="#__codelineno-96-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">nums</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-96-9" name="__codelineno-96-9" href="#__codelineno-96-9"></a><span class="w">    </span><span class="bp">print</span><span class="p">(</span><span class="s">"In recursion n = </span><span class="si">\(</span><span class="n">n</span><span class="si">)</span><span class="s">, nums length = </span><span class="si">\(</span><span class="n">nums</span><span class="p">.</span><span class="bp">count</span><span class="si">)</span><span class="s">"</span><span class="p">)</span>
+<a id="__codelineno-96-9" name="__codelineno-96-9" href="#__codelineno-96-9"></a><span class="w">    </span><span class="bp">print</span><span class="p">(</span><span class="s">"At recursion n = </span><span class="si">\(</span><span class="n">n</span><span class="si">)</span><span class="s">, nums has length </span><span class="si">\(</span><span class="n">nums</span><span class="p">.</span><span class="bp">count</span><span class="si">)</span><span class="s">"</span><span class="p">)</span>
 <a id="__codelineno-96-10" name="__codelineno-96-10" href="#__codelineno-96-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">quadraticRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-96-11" name="__codelineno-96-11" href="#__codelineno-96-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.js</span><pre><span></span><code><a id="__codelineno-97-1" name="__codelineno-97-1" href="#__codelineno-97-1"></a><span class="cm">/* Quadratic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.js</span><pre><span></span><code><a id="__codelineno-97-1" name="__codelineno-97-1" href="#__codelineno-97-1"></a><span class="cm">/* Quadratic complexity (recursive implementation) */</span>
 <a id="__codelineno-97-2" name="__codelineno-97-2" href="#__codelineno-97-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">quadraticRecur</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-97-3" name="__codelineno-97-3" href="#__codelineno-97-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-97-4" name="__codelineno-97-4" href="#__codelineno-97-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">);</span>
-<a id="__codelineno-97-5" name="__codelineno-97-5" href="#__codelineno-97-5"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="sb">`In recursion n = </span><span class="si">${</span><span class="nx">n</span><span class="si">}</span><span class="sb">, nums length = </span><span class="si">${</span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="si">}</span><span class="sb">`</span><span class="p">);</span>
+<a id="__codelineno-97-5" name="__codelineno-97-5" href="#__codelineno-97-5"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="sb">`In the recursive call with n = </span><span class="si">${</span><span class="nx">n</span><span class="si">}</span><span class="sb">, nums has length </span><span class="si">${</span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="si">}</span><span class="sb">`</span><span class="p">);</span>
 <a id="__codelineno-97-6" name="__codelineno-97-6" href="#__codelineno-97-6"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">quadraticRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-97-7" name="__codelineno-97-7" href="#__codelineno-97-7"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.ts</span><pre><span></span><code><a id="__codelineno-98-1" name="__codelineno-98-1" href="#__codelineno-98-1"></a><span class="cm">/* Quadratic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.ts</span><pre><span></span><code><a id="__codelineno-98-1" name="__codelineno-98-1" href="#__codelineno-98-1"></a><span class="cm">/* Quadratic complexity (recursive implementation) */</span>
 <a id="__codelineno-98-2" name="__codelineno-98-2" href="#__codelineno-98-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">quadraticRecur</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-98-3" name="__codelineno-98-3" href="#__codelineno-98-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-98-4" name="__codelineno-98-4" href="#__codelineno-98-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">);</span>
-<a id="__codelineno-98-5" name="__codelineno-98-5" href="#__codelineno-98-5"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="sb">`In recursion n = </span><span class="si">${</span><span class="nx">n</span><span class="si">}</span><span class="sb">, nums length = </span><span class="si">${</span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="si">}</span><span class="sb">`</span><span class="p">);</span>
+<a id="__codelineno-98-5" name="__codelineno-98-5" href="#__codelineno-98-5"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="sb">`In the recursive call with n = </span><span class="si">${</span><span class="nx">n</span><span class="si">}</span><span class="sb">, nums has length </span><span class="si">${</span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="si">}</span><span class="sb">`</span><span class="p">);</span>
 <a id="__codelineno-98-6" name="__codelineno-98-6" href="#__codelineno-98-6"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">quadraticRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-98-7" name="__codelineno-98-7" href="#__codelineno-98-7"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.dart</span><pre><span></span><code><a id="__codelineno-99-1" name="__codelineno-99-1" href="#__codelineno-99-1"></a><span class="cm">/* Quadratic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.dart</span><pre><span></span><code><a id="__codelineno-99-1" name="__codelineno-99-1" href="#__codelineno-99-1"></a><span class="cm">/* Quadratic complexity (recursive implementation) */</span>
 <a id="__codelineno-99-2" name="__codelineno-99-2" href="#__codelineno-99-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">quadraticRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-99-3" name="__codelineno-99-3" href="#__codelineno-99-3"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-99-4" name="__codelineno-99-4" href="#__codelineno-99-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
-<a id="__codelineno-99-5" name="__codelineno-99-5" href="#__codelineno-99-5"></a><span class="w">  </span><span class="n">print</span><span class="p">(</span><span class="s1">'In recursion n = </span><span class="si">$</span><span class="n">n</span><span class="s1">, nums length = </span><span class="si">${</span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="si">}</span><span class="s1">'</span><span class="p">);</span>
+<a id="__codelineno-99-5" name="__codelineno-99-5" href="#__codelineno-99-5"></a><span class="w">  </span><span class="n">print</span><span class="p">(</span><span class="s1">'In the recursive call with n = </span><span class="si">$</span><span class="n">n</span><span class="s1">, length of nums = </span><span class="si">${</span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="si">}</span><span class="s1">'</span><span class="p">);</span>
 <a id="__codelineno-99-6" name="__codelineno-99-6" href="#__codelineno-99-6"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">quadraticRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-99-7" name="__codelineno-99-7" href="#__codelineno-99-7"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.rs</span><pre><span></span><code><a id="__codelineno-100-1" name="__codelineno-100-1" href="#__codelineno-100-1"></a><span class="cm">/* Quadratic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.rs</span><pre><span></span><code><a id="__codelineno-100-1" name="__codelineno-100-1" href="#__codelineno-100-1"></a><span class="cm">/* Quadratic complexity (recursive implementation) */</span>
 <a id="__codelineno-100-2" name="__codelineno-100-2" href="#__codelineno-100-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">quadratic_recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-100-3" name="__codelineno-100-3" href="#__codelineno-100-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-100-4" name="__codelineno-100-4" href="#__codelineno-100-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-100-5" name="__codelineno-100-5" href="#__codelineno-100-5"></a><span class="w">    </span><span class="p">};</span>
-<a id="__codelineno-100-6" name="__codelineno-100-6" href="#__codelineno-100-6"></a><span class="w">    </span><span class="c1">// Array nums has length n, n-1, ..., 2, 1</span>
+<a id="__codelineno-100-6" name="__codelineno-100-6" href="#__codelineno-100-6"></a><span class="w">    </span><span class="c1">// The lengths of array nums are n, n-1, ..., 2, 1</span>
 <a id="__codelineno-100-7" name="__codelineno-100-7" href="#__codelineno-100-7"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">];</span>
-<a id="__codelineno-100-8" name="__codelineno-100-8" href="#__codelineno-100-8"></a><span class="w">    </span><span class="fm">println!</span><span class="p">(</span><span class="s">"In recursion n = {}, nums length = {}"</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">());</span>
+<a id="__codelineno-100-8" name="__codelineno-100-8" href="#__codelineno-100-8"></a><span class="w">    </span><span class="fm">println!</span><span class="p">(</span><span class="s">"In recursive call n = {}, nums has length {}"</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">());</span>
 <a id="__codelineno-100-9" name="__codelineno-100-9" href="#__codelineno-100-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">quadratic_recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-100-10" name="__codelineno-100-10" href="#__codelineno-100-10"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.c</span><pre><span></span><code><a id="__codelineno-101-1" name="__codelineno-101-1" href="#__codelineno-101-1"></a><span class="cm">/* Quadratic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.c</span><pre><span></span><code><a id="__codelineno-101-1" name="__codelineno-101-1" href="#__codelineno-101-1"></a><span class="cm">/* Quadratic complexity (recursive implementation) */</span>
 <a id="__codelineno-101-2" name="__codelineno-101-2" href="#__codelineno-101-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">quadraticRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-101-3" name="__codelineno-101-3" href="#__codelineno-101-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-101-4" name="__codelineno-101-4" href="#__codelineno-101-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-101-5" name="__codelineno-101-5" href="#__codelineno-101-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
-<a id="__codelineno-101-6" name="__codelineno-101-6" href="#__codelineno-101-6"></a><span class="w">    </span><span class="n">printf</span><span class="p">(</span><span class="s">"In recursion n = %d, nums length = %d</span><span class="se">\r\n</span><span class="s">"</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
+<a id="__codelineno-101-6" name="__codelineno-101-6" href="#__codelineno-101-6"></a><span class="w">    </span><span class="n">printf</span><span class="p">(</span><span class="s">"In recursive call n = %d, nums has length = %d</span><span class="se">\r\n</span><span class="s">"</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-101-7" name="__codelineno-101-7" href="#__codelineno-101-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">quadraticRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-101-8" name="__codelineno-101-8" href="#__codelineno-101-8"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span>
 <a id="__codelineno-101-9" name="__codelineno-101-9" href="#__codelineno-101-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
@@ -6605,23 +6697,23 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.kt</span><pre><span></span><code><a id="__codelineno-102-1" name="__codelineno-102-1" href="#__codelineno-102-1"></a><span class="cm">/* Quadratic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">space_complexity.kt</span><pre><span></span><code><a id="__codelineno-102-1" name="__codelineno-102-1" href="#__codelineno-102-1"></a><span class="cm">/* Quadratic allocation total (recursive example; tailrec removes recursive stack growth) */</span>
 <a id="__codelineno-102-2" name="__codelineno-102-2" href="#__codelineno-102-2"></a><span class="kd">tailrec</span><span class="w"> </span><span class="kd">fun</span><span class="w"> </span><span class="nf">quadraticRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-102-3" name="__codelineno-102-3" href="#__codelineno-102-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="m">0</span><span class="p">)</span>
 <a id="__codelineno-102-4" name="__codelineno-102-4" href="#__codelineno-102-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-102-5" name="__codelineno-102-5" href="#__codelineno-102-5"></a><span class="w">    </span><span class="c1">// Array nums has length n, n-1, ..., 2, 1</span>
+<a id="__codelineno-102-5" name="__codelineno-102-5" href="#__codelineno-102-5"></a><span class="w">    </span><span class="c1">// The lengths of array nums are n, n-1, ..., 2, 1</span>
 <a id="__codelineno-102-6" name="__codelineno-102-6" href="#__codelineno-102-6"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Array</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-102-7" name="__codelineno-102-7" href="#__codelineno-102-7"></a><span class="w">    </span><span class="n">println</span><span class="p">(</span><span class="s">"In recursion n = </span><span class="si">$</span><span class="n">n</span><span class="s">, nums length = </span><span class="si">${</span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="si">}</span><span class="s">"</span><span class="p">)</span>
+<a id="__codelineno-102-7" name="__codelineno-102-7" href="#__codelineno-102-7"></a><span class="w">    </span><span class="n">println</span><span class="p">(</span><span class="s">"In the recursive call with n = </span><span class="si">$</span><span class="n">n</span><span class="s">, nums has length </span><span class="si">${</span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="si">}</span><span class="s">"</span><span class="p">)</span>
 <a id="__codelineno-102-8" name="__codelineno-102-8" href="#__codelineno-102-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">quadraticRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-102-9" name="__codelineno-102-9" href="#__codelineno-102-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.rb</span><pre><span></span><code><a id="__codelineno-103-1" name="__codelineno-103-1" href="#__codelineno-103-1"></a><span class="c1">### Quadratic space (recursive) ###</span>
+<div class="highlight"><span class="filename">space_complexity.rb</span><pre><span></span><code><a id="__codelineno-103-1" name="__codelineno-103-1" href="#__codelineno-103-1"></a><span class="c1">### Quadratic complexity (recursive implementation)###</span>
 <a id="__codelineno-103-2" name="__codelineno-103-2" href="#__codelineno-103-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">quadratic_recur</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-103-3" name="__codelineno-103-3" href="#__codelineno-103-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-103-4" name="__codelineno-103-4" href="#__codelineno-103-4"></a>
-<a id="__codelineno-103-5" name="__codelineno-103-5" href="#__codelineno-103-5"></a><span class="w">  </span><span class="c1"># Array nums has length n, n-1, ..., 2, 1</span>
+<a id="__codelineno-103-5" name="__codelineno-103-5" href="#__codelineno-103-5"></a><span class="w">  </span><span class="c1"># The lengths of array nums are n, n-1, ..., 2, 1</span>
 <a id="__codelineno-103-6" name="__codelineno-103-6" href="#__codelineno-103-6"></a><span class="w">  </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-103-7" name="__codelineno-103-7" href="#__codelineno-103-7"></a><span class="w">  </span><span class="n">quadratic_recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-103-8" name="__codelineno-103-8" href="#__codelineno-103-8"></a><span class="k">end</span>
@@ -6629,16 +6721,21 @@
 </div>
 </div>
 </div>
-<p><img alt="Quadratic order space complexity generated by recursive function" class="animation-figure" src="../space_complexity.assets/space_complexity_recursive_quadratic.png" /></p>
-<p align="center"> Figure 2-18 &nbsp; Quadratic order space complexity generated by recursive function </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20quadratic_recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Quadratic%20complexity%20%28recursive%20implementation%29%22%22%22%0A%20%20%20%20if%20n%20%3C%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20The%20lengths%20of%20array%20nums%20are%20n,%20n-1,%20...,%202,%201%0A%20%20%20%20nums%20%3D%20%5B0%5D%20%2A%20n%0A%20%20%20%20return%20quadratic_recur%28n%20-%201%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20Quadratic%20complexity%0A%20%20%20%20quadratic_recur%28n%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=28&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20quadratic_recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Quadratic%20complexity%20%28recursive%20implementation%29%22%22%22%0A%20%20%20%20if%20n%20%3C%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20The%20lengths%20of%20array%20nums%20are%20n,%20n-1,%20...,%202,%201%0A%20%20%20%20nums%20%3D%20%5B0%5D%20%2A%20n%0A%20%20%20%20return%20quadratic_recur%28n%20-%201%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20Quadratic%20complexity%0A%20%20%20%20quadratic_recur%28n%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=28&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p><img alt="Quadratic space complexity from recursion" class="animation-figure" src="../space_complexity.assets/space_complexity_recursive_quadratic.png" /></p>
+<p align="center"> Figure 2-18 &nbsp; Quadratic space complexity from recursion </p>
 
-<h3 id="4-exponential-order-o2n">4. &nbsp; Exponential Order <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#4-exponential-order-o2n" title="Permanent link">&para;</a></h3>
-<p>Exponential order is common in binary trees. Observe the following figure: a "full binary tree" with <span class="arithmatex">\(n\)</span> levels has <span class="arithmatex">\(2^n - 1\)</span> nodes, occupying <span class="arithmatex">\(O(2^n)\)</span> space:</p>
+<h3 id="4-exponential-space-o2n">4. &nbsp; Exponential space <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#4-exponential-space-o2n" title="Permanent link">&para;</a></h3>
+<p>Exponential space commonly occurs in binary trees. In Figure 2-19, a perfect binary tree with <span class="arithmatex">\(n\)</span> levels contains <span class="arithmatex">\(2^n - 1\)</span> nodes and uses <span class="arithmatex">\(O(2^n)\)</span> space:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="9:13"><input checked="checked" id="__tabbed_9_1" name="__tabbed_9" type="radio" /><input id="__tabbed_9_2" name="__tabbed_9" type="radio" /><input id="__tabbed_9_3" name="__tabbed_9" type="radio" /><input id="__tabbed_9_4" name="__tabbed_9" type="radio" /><input id="__tabbed_9_5" name="__tabbed_9" type="radio" /><input id="__tabbed_9_6" name="__tabbed_9" type="radio" /><input id="__tabbed_9_7" name="__tabbed_9" type="radio" /><input id="__tabbed_9_8" name="__tabbed_9" type="radio" /><input id="__tabbed_9_9" name="__tabbed_9" type="radio" /><input id="__tabbed_9_10" name="__tabbed_9" type="radio" /><input id="__tabbed_9_11" name="__tabbed_9" type="radio" /><input id="__tabbed_9_12" name="__tabbed_9" type="radio" /><input id="__tabbed_9_13" name="__tabbed_9" type="radio" /><div class="tabbed-labels"><label for="__tabbed_9_1">Python</label><label for="__tabbed_9_2">C++</label><label for="__tabbed_9_3">Java</label><label for="__tabbed_9_4">C#</label><label for="__tabbed_9_5">Go</label><label for="__tabbed_9_6">Swift</label><label for="__tabbed_9_7">JS</label><label for="__tabbed_9_8">TS</label><label for="__tabbed_9_9">Dart</label><label for="__tabbed_9_10">Rust</label><label for="__tabbed_9_11">C</label><label for="__tabbed_9_12">Kotlin</label><label for="__tabbed_9_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">space_complexity.py</span><pre><span></span><code><a id="__codelineno-104-1" name="__codelineno-104-1" href="#__codelineno-104-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">build_tree</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-104-2" name="__codelineno-104-2" href="#__codelineno-104-2"></a><span class="w">    </span><span class="sd">"""Exponential order (build full binary tree)"""</span>
+<a id="__codelineno-104-2" name="__codelineno-104-2" href="#__codelineno-104-2"></a><span class="w">    </span><span class="sd">"""Exponential complexity (building a perfect binary tree)"""</span>
 <a id="__codelineno-104-3" name="__codelineno-104-3" href="#__codelineno-104-3"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-104-4" name="__codelineno-104-4" href="#__codelineno-104-4"></a>        <span class="k">return</span> <span class="kc">None</span>
 <a id="__codelineno-104-5" name="__codelineno-104-5" href="#__codelineno-104-5"></a>    <span class="n">root</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
@@ -6648,7 +6745,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.cpp</span><pre><span></span><code><a id="__codelineno-105-1" name="__codelineno-105-1" href="#__codelineno-105-1"></a><span class="cm">/* Driver Code */</span>
+<div class="highlight"><span class="filename">space_complexity.cpp</span><pre><span></span><code><a id="__codelineno-105-1" name="__codelineno-105-1" href="#__codelineno-105-1"></a><span class="cm">/* Exponential complexity (building a perfect binary tree) */</span>
 <a id="__codelineno-105-2" name="__codelineno-105-2" href="#__codelineno-105-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">buildTree</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-105-3" name="__codelineno-105-3" href="#__codelineno-105-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-105-4" name="__codelineno-105-4" href="#__codelineno-105-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">;</span>
@@ -6660,7 +6757,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-106-1" name="__codelineno-106-1" href="#__codelineno-106-1"></a><span class="cm">/* Driver Code */</span>
+<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-106-1" name="__codelineno-106-1" href="#__codelineno-106-1"></a><span class="cm">/* Exponential complexity (building a perfect binary tree) */</span>
 <a id="__codelineno-106-2" name="__codelineno-106-2" href="#__codelineno-106-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">buildTree</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-106-3" name="__codelineno-106-3" href="#__codelineno-106-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-106-4" name="__codelineno-106-4" href="#__codelineno-106-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
@@ -6672,7 +6769,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.cs</span><pre><span></span><code><a id="__codelineno-107-1" name="__codelineno-107-1" href="#__codelineno-107-1"></a><span class="cm">/* Driver Code */</span>
+<div class="highlight"><span class="filename">space_complexity.cs</span><pre><span></span><code><a id="__codelineno-107-1" name="__codelineno-107-1" href="#__codelineno-107-1"></a><span class="cm">/* Exponential complexity (building a perfect binary tree) */</span>
 <a id="__codelineno-107-2" name="__codelineno-107-2" href="#__codelineno-107-2"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">BuildTree</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-107-3" name="__codelineno-107-3" href="#__codelineno-107-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
 <a id="__codelineno-107-4" name="__codelineno-107-4" href="#__codelineno-107-4"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6684,7 +6781,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.go</span><pre><span></span><code><a id="__codelineno-108-1" name="__codelineno-108-1" href="#__codelineno-108-1"></a><span class="cm">/* Driver Code */</span>
+<div class="highlight"><span class="filename">space_complexity.go</span><pre><span></span><code><a id="__codelineno-108-1" name="__codelineno-108-1" href="#__codelineno-108-1"></a><span class="cm">/* Exponential complexity (building a perfect binary tree) */</span>
 <a id="__codelineno-108-2" name="__codelineno-108-2" href="#__codelineno-108-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">buildTree</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-108-3" name="__codelineno-108-3" href="#__codelineno-108-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-108-4" name="__codelineno-108-4" href="#__codelineno-108-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
@@ -6697,7 +6794,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.swift</span><pre><span></span><code><a id="__codelineno-109-1" name="__codelineno-109-1" href="#__codelineno-109-1"></a><span class="cm">/* Driver Code */</span>
+<div class="highlight"><span class="filename">space_complexity.swift</span><pre><span></span><code><a id="__codelineno-109-1" name="__codelineno-109-1" href="#__codelineno-109-1"></a><span class="cm">/* Exponential complexity (building a perfect binary tree) */</span>
 <a id="__codelineno-109-2" name="__codelineno-109-2" href="#__codelineno-109-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">buildTree</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-109-3" name="__codelineno-109-3" href="#__codelineno-109-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-109-4" name="__codelineno-109-4" href="#__codelineno-109-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
@@ -6710,7 +6807,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.js</span><pre><span></span><code><a id="__codelineno-110-1" name="__codelineno-110-1" href="#__codelineno-110-1"></a><span class="cm">/* Driver Code */</span>
+<div class="highlight"><span class="filename">space_complexity.js</span><pre><span></span><code><a id="__codelineno-110-1" name="__codelineno-110-1" href="#__codelineno-110-1"></a><span class="cm">/* Exponential complexity (building a perfect binary tree) */</span>
 <a id="__codelineno-110-2" name="__codelineno-110-2" href="#__codelineno-110-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">buildTree</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-110-3" name="__codelineno-110-3" href="#__codelineno-110-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-110-4" name="__codelineno-110-4" href="#__codelineno-110-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
@@ -6721,7 +6818,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.ts</span><pre><span></span><code><a id="__codelineno-111-1" name="__codelineno-111-1" href="#__codelineno-111-1"></a><span class="cm">/* Driver Code */</span>
+<div class="highlight"><span class="filename">space_complexity.ts</span><pre><span></span><code><a id="__codelineno-111-1" name="__codelineno-111-1" href="#__codelineno-111-1"></a><span class="cm">/* Exponential complexity (building a perfect binary tree) */</span>
 <a id="__codelineno-111-2" name="__codelineno-111-2" href="#__codelineno-111-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">buildTree</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-111-3" name="__codelineno-111-3" href="#__codelineno-111-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-111-4" name="__codelineno-111-4" href="#__codelineno-111-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
@@ -6732,7 +6829,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.dart</span><pre><span></span><code><a id="__codelineno-112-1" name="__codelineno-112-1" href="#__codelineno-112-1"></a><span class="cm">/* Driver Code */</span>
+<div class="highlight"><span class="filename">space_complexity.dart</span><pre><span></span><code><a id="__codelineno-112-1" name="__codelineno-112-1" href="#__codelineno-112-1"></a><span class="cm">/* Exponential complexity (building a perfect binary tree) */</span>
 <a id="__codelineno-112-2" name="__codelineno-112-2" href="#__codelineno-112-2"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">buildTree</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-112-3" name="__codelineno-112-3" href="#__codelineno-112-3"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-112-4" name="__codelineno-112-4" href="#__codelineno-112-4"></a><span class="w">  </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="m">0</span><span class="p">);</span>
@@ -6743,7 +6840,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.rs</span><pre><span></span><code><a id="__codelineno-113-1" name="__codelineno-113-1" href="#__codelineno-113-1"></a><span class="cm">/* Driver Code */</span>
+<div class="highlight"><span class="filename">space_complexity.rs</span><pre><span></span><code><a id="__codelineno-113-1" name="__codelineno-113-1" href="#__codelineno-113-1"></a><span class="cm">/* Exponential complexity (building a perfect binary tree) */</span>
 <a id="__codelineno-113-2" name="__codelineno-113-2" href="#__codelineno-113-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">build_tree</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-113-3" name="__codelineno-113-3" href="#__codelineno-113-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-113-4" name="__codelineno-113-4" href="#__codelineno-113-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nb">None</span><span class="p">;</span>
@@ -6756,7 +6853,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.c</span><pre><span></span><code><a id="__codelineno-114-1" name="__codelineno-114-1" href="#__codelineno-114-1"></a><span class="cm">/* Driver Code */</span>
+<div class="highlight"><span class="filename">space_complexity.c</span><pre><span></span><code><a id="__codelineno-114-1" name="__codelineno-114-1" href="#__codelineno-114-1"></a><span class="cm">/* Exponential complexity (building a perfect binary tree) */</span>
 <a id="__codelineno-114-2" name="__codelineno-114-2" href="#__codelineno-114-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">buildTree</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-114-3" name="__codelineno-114-3" href="#__codelineno-114-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-114-4" name="__codelineno-114-4" href="#__codelineno-114-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
@@ -6768,7 +6865,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.kt</span><pre><span></span><code><a id="__codelineno-115-1" name="__codelineno-115-1" href="#__codelineno-115-1"></a><span class="cm">/* Driver Code */</span>
+<div class="highlight"><span class="filename">space_complexity.kt</span><pre><span></span><code><a id="__codelineno-115-1" name="__codelineno-115-1" href="#__codelineno-115-1"></a><span class="cm">/* Exponential complexity (building a perfect binary tree) */</span>
 <a id="__codelineno-115-2" name="__codelineno-115-2" href="#__codelineno-115-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">buildTree</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="n">TreeNode?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-115-3" name="__codelineno-115-3" href="#__codelineno-115-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span>
 <a id="__codelineno-115-4" name="__codelineno-115-4" href="#__codelineno-115-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span>
@@ -6780,7 +6877,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">space_complexity.rb</span><pre><span></span><code><a id="__codelineno-116-1" name="__codelineno-116-1" href="#__codelineno-116-1"></a><span class="c1">### Exponential space (build full binary tree) ###</span>
+<div class="highlight"><span class="filename">space_complexity.rb</span><pre><span></span><code><a id="__codelineno-116-1" name="__codelineno-116-1" href="#__codelineno-116-1"></a><span class="c1">### Exponential complexity (building a perfect binary tree)###</span>
 <a id="__codelineno-116-2" name="__codelineno-116-2" href="#__codelineno-116-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">build_tree</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-116-3" name="__codelineno-116-3" href="#__codelineno-116-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-116-4" name="__codelineno-116-4" href="#__codelineno-116-4"></a>
@@ -6793,16 +6890,21 @@
 </div>
 </div>
 </div>
-<p><img alt="Exponential order space complexity generated by full binary tree" class="animation-figure" src="../space_complexity.assets/space_complexity_exponential.png" /></p>
-<p align="center"> Figure 2-19 &nbsp; Exponential order space complexity generated by full binary tree </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20left%20child%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20right%20child%0A%0Adef%20build_tree%28n%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Exponential%20complexity%20%28building%20a%20perfect%20binary%20tree%29%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20root%20%3D%20TreeNode%280%29%0A%20%20%20%20root.left%20%3D%20build_tree%28n%20-%201%29%0A%20%20%20%20root.right%20%3D%20build_tree%28n%20-%201%29%0A%20%20%20%20return%20root%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20Exponential%20complexity%0A%20%20%20%20root%20%3D%20build_tree%28n%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=507&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20left%20child%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20right%20child%0A%0Adef%20build_tree%28n%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Exponential%20complexity%20%28building%20a%20perfect%20binary%20tree%29%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20root%20%3D%20TreeNode%280%29%0A%20%20%20%20root.left%20%3D%20build_tree%28n%20-%201%29%0A%20%20%20%20root.right%20%3D%20build_tree%28n%20-%201%29%0A%20%20%20%20return%20root%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%205%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20%23%20Exponential%20complexity%0A%20%20%20%20root%20%3D%20build_tree%28n%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=507&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p><img alt="Exponential space used by a perfect binary tree" class="animation-figure" src="../space_complexity.assets/space_complexity_exponential.png" /></p>
+<p align="center"> Figure 2-19 &nbsp; Exponential space used by a perfect binary tree </p>
 
-<h3 id="5-logarithmic-order-olog-n">5. &nbsp; Logarithmic Order <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#5-logarithmic-order-olog-n" title="Permanent link">&para;</a></h3>
-<p>Logarithmic order is common in divide-and-conquer algorithms. For example, merge sort: given an input array of length <span class="arithmatex">\(n\)</span>, each recursion divides the array in half from the midpoint, forming a recursion tree of height <span class="arithmatex">\(\log n\)</span>, using <span class="arithmatex">\(O(\log n)\)</span> stack frame space.</p>
-<p>Another example is converting a number to a string. Given a positive integer <span class="arithmatex">\(n\)</span>, it has <span class="arithmatex">\(\lfloor \log_{10} n \rfloor + 1\)</span> digits, i.e., the corresponding string length is <span class="arithmatex">\(\lfloor \log_{10} n \rfloor + 1\)</span>, so the space complexity is <span class="arithmatex">\(O(\log_{10} n + 1) = O(\log n)\)</span>.</p>
-<h2 id="244-trading-time-for-space">2.4.4 &nbsp; Trading Time for Space<a class="headerlink" href="#244-trading-time-for-space" title="Permanent link">&para;</a></h2>
-<p>Ideally, we hope that both the time complexity and space complexity of an algorithm can reach optimal. However, in practice, optimizing both time complexity and space complexity simultaneously is usually very difficult.</p>
-<p><strong>Reducing time complexity usually comes at the cost of increasing space complexity, and vice versa</strong>. Sacrificing memory space to improve execution speed is called "trading space for time"; the reverse is called "trading time for space".</p>
-<p>The choice of which approach depends on which aspect we value more. In most cases, time is more precious than space, so "trading space for time" is usually the more common strategy. Of course, when the data volume is very large, controlling space complexity is also very important.</p>
+<h3 id="5-logarithmic-space-olog-n">5. &nbsp; Logarithmic space <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#5-logarithmic-space-olog-n" title="Permanent link">&para;</a></h3>
+<p>Logarithmic space often appears in divide-and-conquer algorithms. For example, merge sort splits an array of length <span class="arithmatex">\(n\)</span> in half at each recursive step, producing a recursion tree of height <span class="arithmatex">\(\log n\)</span> and using <span class="arithmatex">\(O(\log n)\)</span> space for stack frames.</p>
+<p>Another example is converting a number to a string. A positive integer <span class="arithmatex">\(n\)</span> has <span class="arithmatex">\(\lfloor \log_{10} n \rfloor + 1\)</span> digits, giving a string of length <span class="arithmatex">\(\lfloor \log_{10} n \rfloor + 1\)</span>. The space complexity is therefore <span class="arithmatex">\(O(\log_{10} n + 1) = O(\log n)\)</span>.</p>
+<h2 id="244-balancing-time-and-space">2.4.4 &nbsp; Balancing time and space<a class="headerlink" href="#244-balancing-time-and-space" title="Permanent link">&para;</a></h2>
+<p>Ideally, we would minimize both time and space complexity. In practice, optimizing both simultaneously is usually difficult.</p>
+<p><strong>Reducing time complexity often increases space complexity, and vice versa</strong>. Using more memory to run faster is called trading space for time; accepting slower execution to use less memory is trading time for space.</p>
+<p>The choice depends on which resource matters more. Time is usually more valuable than space, so trading space for time is common. For very large datasets, however, keeping space complexity under control is also crucial.</p>
 
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-  Before Starting
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  2.6 Exercises
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-  3.2 Basic Data Types
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+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
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@@ -2009,7 +2009,7 @@
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-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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+  8.5 &nbsp; Exercises
 
 
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+  Chapter 9. &nbsp; Graphs
 
 
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+  Chapter 9. &nbsp; Graphs
 
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+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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+  9.3 &nbsp; Graph traversal
 
 
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+  9.4 &nbsp; Summary
 
 
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+  9.5 &nbsp; Exercises
 
 
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+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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+  Chapter 13. &nbsp; Backtracking
 
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4558,6 +4558,78 @@
 
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
       </span>
     </span>
   </a>
@@ -4658,47 +4730,47 @@
 
 <!-- Page content -->
 <h1 id="25-summary">2.5 &nbsp; Summary<a class="headerlink" href="#25-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
-<p><strong>Algorithm Efficiency Assessment</strong></p>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
+<p><strong>Evaluating algorithm efficiency</strong></p>
 <ul>
-<li>Time efficiency and space efficiency are the two primary evaluation metrics for measuring algorithm performance.</li>
-<li>We can evaluate algorithm efficiency through actual testing, but it is difficult to eliminate the influence of the testing environment, and it consumes substantial computational resources.</li>
-<li>Complexity analysis can overcome the limitations of actual testing. Its results apply across running platforms, and it can reveal algorithm efficiency under different data scales.</li>
+<li>Time and space efficiency are the two main measures of algorithm quality.</li>
+<li>Practical testing measures efficiency, but cannot easily remove environmental effects and consumes substantial computing resources.</li>
+<li>Complexity analysis avoids those limitations, applies across platforms, and reveals efficiency at different input sizes.</li>
 </ul>
-<p><strong>Time Complexity</strong></p>
+<p><strong>Time complexity</strong></p>
 <ul>
-<li>Time complexity is used to measure the trend of algorithm runtime as data volume increases. It can effectively evaluate algorithm efficiency, but it may be less informative in certain situations, such as when the input data volume is small or when time complexities are identical, making it impossible to precisely compare algorithm efficiency.</li>
-<li>Worst-case time complexity is represented using Big <span class="arithmatex">\(O\)</span> notation, corresponding to the asymptotic upper bound of a function, reflecting the growth level of the number of operations <span class="arithmatex">\(T(n)\)</span> as <span class="arithmatex">\(n\)</span> approaches positive infinity.</li>
-<li>Deriving time complexity involves two steps: first, counting the number of operations, then determining the asymptotic upper bound.</li>
-<li>Common time complexities arranged from low to high include <span class="arithmatex">\(O(1)\)</span>, <span class="arithmatex">\(O(\log n)\)</span>, <span class="arithmatex">\(O(n)\)</span>, <span class="arithmatex">\(O(n \log n)\)</span>, <span class="arithmatex">\(O(n^2)\)</span>, <span class="arithmatex">\(O(2^n)\)</span>, and <span class="arithmatex">\(O(n!)\)</span>.</li>
-<li>The time complexity of some algorithms is not fixed, but rather depends on the distribution of input data. Time complexity is divided into worst-case, best-case, and average-case time complexity. Best-case time complexity is rarely used because input data generally needs to satisfy strict conditions to achieve the best case.</li>
-<li>Average time complexity reflects the algorithm's runtime efficiency under random data input, and is closest to the algorithm's performance in practical applications. Calculating average time complexity requires analyzing the input data distribution and the resulting mathematical expectation.</li>
+<li>Time complexity describes how runtime grows with input size. It is useful for evaluating efficiency, but may not distinguish algorithms accurately for small inputs or equal complexity classes.</li>
+<li>Worst-case time complexity is expressed with big-<span class="arithmatex">\(O\)</span> notation, an asymptotic upper bound describing the growth of the operation count <span class="arithmatex">\(T(n)\)</span> as <span class="arithmatex">\(n\)</span> tends to positive infinity.</li>
+<li>Deriving time complexity takes two steps: count operations, then determine the asymptotic upper bound.</li>
+<li>Common complexities, from slower to faster growth, include <span class="arithmatex">\(O(1)\)</span>, <span class="arithmatex">\(O(\log n)\)</span>, <span class="arithmatex">\(O(n)\)</span>, <span class="arithmatex">\(O(n \log n)\)</span>, <span class="arithmatex">\(O(n^2)\)</span>, <span class="arithmatex">\(O(2^n)\)</span>, and <span class="arithmatex">\(O(n!)\)</span>.</li>
+<li>Some algorithms' complexity depends on the input's arrangement. We distinguish worst-case, best-case, and average-case complexity. Best-case complexity is rarely used because reaching it generally requires strict input conditions.</li>
+<li>Average-case complexity describes performance on random inputs and is closest to typical practical behavior. Calculating it requires an input distribution and its mathematical expectation.</li>
 </ul>
-<p><strong>Space Complexity</strong></p>
+<p><strong>Space complexity</strong></p>
 <ul>
-<li>Space complexity serves a similar purpose to time complexity, used to measure the trend of algorithm memory usage as data volume increases.</li>
-<li>The memory space related to algorithm execution can be divided into input space, temporary space, and output space. Typically, input space is not included in space complexity calculations. Temporary space can be divided into temporary data, stack frame space, and instruction space, where stack frame space usually affects space complexity only in recursive functions.</li>
-<li>We typically only focus on worst-case space complexity, which is the space complexity of an algorithm under worst-case input data and worst-case runtime.</li>
-<li>Common space complexities arranged from low to high include <span class="arithmatex">\(O(1)\)</span>, <span class="arithmatex">\(O(\log n)\)</span>, <span class="arithmatex">\(O(n)\)</span>, <span class="arithmatex">\(O(n^2)\)</span>, and <span class="arithmatex">\(O(2^n)\)</span>.</li>
+<li>Like time complexity, space complexity describes a growth trend: how memory usage increases with input size.</li>
+<li>Memory used during execution includes input, temporary, and output space. Input space is usually excluded from space complexity. Temporary space includes temporary data, stack frames, and instructions; stack frames usually affect complexity only in recursive functions.</li>
+<li>We usually consider worst-case space complexity, measuring memory under the worst input and at the moment of maximum usage.</li>
+<li>Common space complexities, from slower to faster growth, include <span class="arithmatex">\(O(1)\)</span>, <span class="arithmatex">\(O(\log n)\)</span>, <span class="arithmatex">\(O(n)\)</span>, <span class="arithmatex">\(O(n^2)\)</span>, and <span class="arithmatex">\(O(2^n)\)</span>.</li>
 </ul>
 <h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
-<p><strong>Q</strong>: Is the space complexity of tail recursion <span class="arithmatex">\(O(1)\)</span>?</p>
-<p>Theoretically, the space complexity of tail recursive functions can be optimized to <span class="arithmatex">\(O(1)\)</span>. However, most programming languages (such as Java, Python, C++, Go, C#, etc.) do not support automatic tail recursion optimization, so the space complexity is generally considered to be <span class="arithmatex">\(O(n)\)</span>.</p>
-<p><strong>Q</strong>: What is the difference between the terms function and method?</p>
-<p>A <u>function</u> can be executed independently, with all parameters passed explicitly. A <u>method</u> is associated with an object, is implicitly bound to the object that invokes it, and can operate on data contained in class instances.</p>
-<p>The following examples use several common programming languages for illustration.</p>
+<p><strong>Q</strong>: Does tail recursion have <span class="arithmatex">\(O(1)\)</span> space complexity?</p>
+<p>In theory, tail recursion can be optimized to <span class="arithmatex">\(O(1)\)</span> space. However, most languages, including Java, Python, C++, Go, and C#, do not automatically optimize tail recursion, so its space complexity is usually treated as <span class="arithmatex">\(O(n)\)</span>.</p>
+<p><strong>Q</strong>: What is the difference between a function and a method?</p>
+<p>A <u>function</u> can run independently, with all arguments passed explicitly. A <u>method</u> is associated with an object, receives that object implicitly when called, and can operate on the instance's data.</p>
+<p>Here are examples from several common languages.</p>
 <ul>
-<li>C is a procedural programming language without object-oriented concepts, so it only has functions. However, we can simulate object-oriented programming by creating structures (struct), and functions associated with structures are equivalent to methods in other programming languages.</li>
-<li>Java and C# are object-oriented programming languages where code blocks (methods) are typically part of a class. Static methods behave like functions because they are bound to the class and cannot access specific instance variables.</li>
-<li>C++ and Python support both procedural programming (functions) and object-oriented programming (methods).</li>
+<li>C is procedural and has no built-in object-oriented model, so it has functions only. We can simulate object-oriented programming with structures (<code>struct</code>); functions associated with a structure then act like methods in other languages.</li>
+<li>Java and C# are object-oriented, and executable blocks (methods) are usually part of a class. Static methods behave like functions: they belong to the class and cannot access a particular instance's variables directly.</li>
+<li>C++ and Python support both procedural programming with functions and object-oriented programming with methods.</li>
 </ul>
-<p><strong>Q</strong>: Does the diagram for "common space complexity types" reflect the absolute size of occupied space?</p>
-<p>No, the diagram shows space complexity, which reflects growth trends rather than the absolute size of occupied space.</p>
-<p>Assuming <span class="arithmatex">\(n = 8\)</span>, you might find that the values of each curve do not correspond to the functions. This is because each curve contains a constant term used to compress the value range into a visually comfortable range.</p>
-<p>In practice, because we generally do not know the "constant-term" cost of each method, we usually cannot choose the optimal solution for cases like <span class="arithmatex">\(n = 8\)</span> based on complexity alone. But for <span class="arithmatex">\(n = 8^5\)</span>, the choice is straightforward, because the growth trend already dominates.</p>
-<p><strong>Q</strong>: Are there situations where algorithms are designed to sacrifice time (or space) based on actual use cases?</p>
-<p>In practical applications, most situations choose to sacrifice space for time. For example, with database indexes, we typically choose to build B+ trees or hash indexes, occupying substantial memory space in exchange for efficient queries of <span class="arithmatex">\(O(\log n)\)</span> or even <span class="arithmatex">\(O(1)\)</span>.</p>
-<p>In scenarios where space resources are precious, time may be sacrificed for space. For example, in embedded development, device memory is precious, and engineers may forgo using hash tables and choose to use array sequential search to save memory usage, at the cost of slower searches.</p>
+<p><strong>Q</strong>: Does the common space complexity chart show absolute memory usage?</p>
+<p>No. It illustrates growth trends rather than absolute amounts of memory.</p>
+<p>For <span class="arithmatex">\(n = 8\)</span>, the plotted values may not match the corresponding functions exactly. Each curve includes a constant term that adjusts it to a visually convenient range.</p>
+<p>In practice, these constant terms are usually unknown, so complexity alone cannot generally identify the best solution for <span class="arithmatex">\(n = 8\)</span>. For <span class="arithmatex">\(n = 8^5\)</span>, the choice is easier because growth rates dominate.</p>
+<p><strong>Q</strong>: Do algorithms sometimes trade time for space, or space for time, depending on the application?</p>
+<p>Most practical applications trade space for time. Database indexes, for example, often use B+ trees or hash indexes, consuming substantial memory to achieve <span class="arithmatex">\(O(\log n)\)</span> or even <span class="arithmatex">\(O(1)\)</span> query time.</p>
+<p>When memory is scarce, the reverse tradeoff may be useful. In embedded development, an engineer might replace a hash table with sequential array search to save memory, accepting slower lookups.</p>
 
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-    aria-label="Previous: 2.4 Space Complexity"
+    aria-label="Previous: 2.4   Space complexity"
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-  Chapter 1. Encounter with Algorithms
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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-  Chapter 2. Complexity Analysis
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-        2.3.1 &nbsp; Counting Time Growth Trends
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-        2.3.3 &nbsp; Derivation Method
+        2.3.3 &nbsp; How to derive time complexity
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+        2.3.4 &nbsp; Common complexity classes
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-  <a href="#235-worst-best-and-average-time-complexities" class="md-nav__link">
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-        2.3.5 &nbsp; Worst, Best, and Average Time Complexities
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
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-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1480,7 +1480,7 @@
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-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1935,7 +1935,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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@@ -2016,7 +2016,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2153,7 +2153,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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@@ -2180,7 +2180,7 @@
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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@@ -2207,7 +2207,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2261,7 +2261,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2288,7 +2288,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2315,7 +2315,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2379,7 +2379,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2421,7 +2421,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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@@ -2448,7 +2448,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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@@ -2475,7 +2475,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2502,7 +2502,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2529,7 +2529,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2593,7 +2593,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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@@ -2614,7 +2614,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2635,7 +2635,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2662,7 +2662,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2689,7 +2689,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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@@ -2716,7 +2716,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2743,7 +2743,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2811,7 +2811,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2832,7 +2832,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2853,7 +2853,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2880,7 +2880,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2907,7 +2907,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2934,7 +2934,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2961,7 +2961,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2988,7 +2988,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -3015,7 +3015,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -3093,7 +3093,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3114,7 +3114,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3135,7 +3135,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3162,7 +3162,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3189,7 +3189,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3216,7 +3216,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3243,7 +3243,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3270,7 +3270,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3297,7 +3297,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3324,7 +3324,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3351,7 +3351,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3378,7 +3378,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3405,7 +3405,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3432,7 +3432,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3498,7 +3498,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3519,7 +3519,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3540,7 +3540,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3567,7 +3567,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3594,7 +3594,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3621,7 +3621,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3648,7 +3648,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3741,7 +3741,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3783,7 +3783,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3837,7 +3837,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3864,7 +3864,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3891,7 +3891,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3988,7 +3988,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -4009,7 +4009,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4030,7 +4030,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -4084,7 +4084,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4111,7 +4111,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4138,7 +4138,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4165,7 +4165,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4192,7 +4192,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4219,7 +4219,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4285,7 +4285,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4306,7 +4306,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4327,7 +4327,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4354,7 +4354,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4381,7 +4381,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4408,7 +4408,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4435,7 +4435,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4462,7 +4462,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4522,7 +4522,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4543,7 +4543,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4564,7 +4564,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4591,7 +4591,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4618,7 +4618,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4702,6 +4702,78 @@
 
 
 
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         <li class="md-nav__item">
-  <a href="#231-counting-time-growth-trends" class="md-nav__link">
+  <a href="#231-measuring-growth-in-runtime" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.3.1 &nbsp; Counting Time Growth Trends
+        2.3.1 &nbsp; Measuring growth in runtime
       </span>
     </span>
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@@ -4754,10 +4826,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#232-asymptotic-upper-bound-of-functions" class="md-nav__link">
+  <a href="#232-asymptotic-upper-bounds" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.3.2 &nbsp; Asymptotic Upper Bound of Functions
+        2.3.2 &nbsp; Asymptotic upper bounds
       </span>
     </span>
   </a>
@@ -4765,22 +4837,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#233-derivation-method" class="md-nav__link">
+  <a href="#233-how-to-derive-time-complexity" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.3.3 &nbsp; Derivation Method
+        2.3.3 &nbsp; How to derive time complexity
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="2.3.3 Derivation Method">
+    <nav class="md-nav" aria-label="2.3.3 How to derive time complexity">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-step-1-count-the-number-of-operations" class="md-nav__link">
+  <a href="#1-step-1-count-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Step 1: Count the Number of Operations
+        1. &nbsp; Step 1: Count operations
       </span>
     </span>
   </a>
@@ -4791,7 +4863,7 @@
   <a href="#2-step-2-determine-the-asymptotic-upper-bound" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Step 2: Determine the Asymptotic Upper Bound
+        2. &nbsp; Step 2: Determine the asymptotic upper bound
       </span>
     </span>
   </a>
@@ -4804,22 +4876,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#234-common-types" class="md-nav__link">
+  <a href="#234-common-complexity-classes" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.3.4 &nbsp; Common Types
+        2.3.4 &nbsp; Common complexity classes
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="2.3.4 Common Types">
+    <nav class="md-nav" aria-label="2.3.4 Common complexity classes">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-constant-order-o1" class="md-nav__link">
+  <a href="#1-constant-time-o1" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Constant Order
+        1. &nbsp; Constant time
       </span>
     </span>
   </a>
@@ -4827,10 +4899,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-linear-order-on" class="md-nav__link">
+  <a href="#2-linear-time-on" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Linear Order
+        2. &nbsp; Linear time
       </span>
     </span>
   </a>
@@ -4838,10 +4910,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-quadratic-order-on2" class="md-nav__link">
+  <a href="#3-quadratic-time-on2" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Quadratic Order
+        3. &nbsp; Quadratic time
       </span>
     </span>
   </a>
@@ -4849,10 +4921,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#4-exponential-order-o2n" class="md-nav__link">
+  <a href="#4-exponential-time-o2n" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4. &nbsp; Exponential Order
+        4. &nbsp; Exponential time
       </span>
     </span>
   </a>
@@ -4860,10 +4932,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#5-logarithmic-order-olog-n" class="md-nav__link">
+  <a href="#5-logarithmic-time-olog-n" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5. &nbsp; Logarithmic Order
+        5. &nbsp; Logarithmic time
       </span>
     </span>
   </a>
@@ -4871,10 +4943,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#6-linearithmic-order-on-log-n" class="md-nav__link">
+  <a href="#6-linearithmic-time-on-log-n" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6. &nbsp; Linearithmic Order
+        6. &nbsp; Linearithmic time
       </span>
     </span>
   </a>
@@ -4882,10 +4954,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#7-factorial-order-on" class="md-nav__link">
+  <a href="#7-factorial-time-on" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7. &nbsp; Factorial Order
+        7. &nbsp; Factorial time
       </span>
     </span>
   </a>
@@ -4898,10 +4970,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#235-worst-best-and-average-time-complexities" class="md-nav__link">
+  <a href="#235-worst-case-best-case-and-average-case-time-complexity" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2.3.5 &nbsp; Worst, Best, and Average Time Complexities
+        2.3.5 &nbsp; Worst-case, best-case, and average-case time complexity
       </span>
     </span>
   </a>
@@ -4945,34 +5017,34 @@
 
 
 <!-- Page content -->
-<h1 id="23-time-complexity">2.3 &nbsp; Time Complexity<a class="headerlink" href="#23-time-complexity" title="Permanent link">&para;</a></h1>
-<p>Runtime can intuitively and accurately reflect the efficiency of an algorithm. If we want to accurately estimate the runtime of a piece of code, how should we proceed?</p>
+<h1 id="23-time-complexity">2.3 &nbsp; Time complexity<a class="headerlink" href="#23-time-complexity" title="Permanent link">&para;</a></h1>
+<p>Runtime gives a direct, accurate picture of an algorithm's efficiency. How might we predict the runtime of a piece of code precisely?</p>
 <ol>
-<li><strong>Determine the running platform</strong>, including hardware configuration, programming language, system environment, etc., as these factors all affect code execution efficiency.</li>
-<li><strong>Evaluate the runtime required for various computational operations</strong>, for example, an addition operation <code>+</code> requires 1 ns, a multiplication operation <code>*</code> requires 10 ns, a print operation <code>print()</code> requires 5 ns, etc.</li>
-<li><strong>Count all computational operations in the code</strong>, and sum the execution times of all operations to obtain the runtime.</li>
+<li><strong>Specify the execution platform</strong>, including hardware, programming language, and system environment, all of which affect performance.</li>
+<li><strong>Determine the time required by each operation</strong>: for example, addition <code>+</code> might take 1 ns, multiplication <code>*</code> 10 ns, and <code>print()</code> 5 ns.</li>
+<li><strong>Count every operation in the code</strong> and sum their execution times to obtain the total runtime.</li>
 </ol>
-<p>For example, in the following code, the input data size is <span class="arithmatex">\(n\)</span>:</p>
+<p>For example, consider the following code with input size <span class="arithmatex">\(n\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># On a certain running platform</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># On a particular execution platform</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="n">a</span> <span class="o">=</span> <span class="mi">2</span>      <span class="c1"># 1 ns</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c1"># 1 ns</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">*</span> <span class="mi">2</span>  <span class="c1"># 10 ns</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Loop n times</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Repeat n times</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>  <span class="c1"># 1 ns</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="nb">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>        <span class="c1"># 5 ns</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="c1">// On a certain running platform</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="c1">// On a particular execution platform</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">  </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">  </span><span class="c1">// 10 ns</span>
-<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Loop n times</span>
+<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w">  </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span><span class="w">         </span><span class="c1">// 5 ns</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="p">}</span>
@@ -4980,12 +5052,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="c1">// On a certain running platform</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="c1">// On a particular execution platform</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">  </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">  </span><span class="c1">// 10 ns</span>
-<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// Loop n times</span>
+<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w">  </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">        </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w">     </span><span class="c1">// 5 ns</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="p">}</span>
@@ -4993,12 +5065,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="c1">// On a certain running platform</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="c1">// On a particular execution platform</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">  </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">  </span><span class="c1">// 10 ns</span>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Loop n times</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w">  </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">        </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w">      </span><span class="c1">// 5 ns</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="p">}</span>
@@ -5006,12 +5078,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="c1">// On a certain running platform</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="c1">// On a particular execution platform</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">algorithm</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">2</span><span class="w">     </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w">  </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="w">  </span><span class="c1">// 10 ns</span>
-<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="c1">// Loop n times</span>
+<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span><span class="w">  </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">        </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="nx">a</span><span class="p">)</span><span class="w">        </span><span class="c1">// 5 ns</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="p">}</span>
@@ -5019,12 +5091,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="c1">// On a certain running platform</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="c1">// On a particular execution platform</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// 10 ns</span>
-<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// Loop n times</span>
+<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">        </span><span class="bp">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// 5 ns</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="p">}</span>
@@ -5032,12 +5104,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="c1">// On a certain running platform</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="c1">// On a particular execution platform</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">algorithm</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 10 ns</span>
-<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// Loop n times</span>
+<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="k">for</span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// 5 ns</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="p">}</span>
@@ -5045,12 +5117,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="c1">// On a certain running platform</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="c1">// On a particular execution platform</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">algorithm</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nx">a</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 10 ns</span>
-<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="c1">// Loop n times</span>
+<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="k">for</span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// 5 ns</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="p">}</span>
@@ -5058,12 +5130,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="c1">// On a certain running platform</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="c1">// On a particular execution platform</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// 10 ns</span>
-<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="c1">// Loop n times</span>
+<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="c1">// Repeat n times</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">    </span><span class="n">print</span><span class="p">(</span><span class="m">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// 5 ns</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="p">}</span>
@@ -5071,12 +5143,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="c1">// On a certain running platform</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="c1">// On a particular execution platform</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">      </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">          </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">          </span><span class="c1">// 10 ns</span>
-<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// Loop n times</span>
+<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span><span class="w">     </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">        </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{}"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span><span class="w">  </span><span class="c1">// 5 ns</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="p">}</span>
@@ -5084,12 +5156,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// On a certain running platform</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// On a particular execution platform</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">  </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">  </span><span class="c1">// 10 ns</span>
-<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="c1">// Loop n times</span>
+<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w">   </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">        </span><span class="n">printf</span><span class="p">(</span><span class="s">"%d"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span><span class="w">            </span><span class="c1">// 5 ns</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="p">}</span>
@@ -5097,12 +5169,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="c1">// On a certain running platform</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="c1">// On a particular execution platform</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="c1">// 10 ns</span>
-<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// Loop n times</span>
+<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w">  </span><span class="c1">// 1 ns</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">        </span><span class="n">println</span><span class="p">(</span><span class="m">0</span><span class="p">)</span><span class="w">      </span><span class="c1">// 5 ns</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="p">}</span>
@@ -5110,12 +5182,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># On a certain running platform</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># On a particular execution platform</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w">       </span><span class="c1"># 1 ns</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w">   </span><span class="c1"># 1 ns</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="w">   </span><span class="c1"># 10 ns</span>
-<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">    </span><span class="c1"># Loop n times</span>
+<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">    </span><span class="c1"># Repeat n times</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">    </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="c1"># 1 ns</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">        </span><span class="nb">puts</span><span class="w"> </span><span class="mi">0</span><span class="w">      </span><span class="c1"># 5 ns</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">    </span><span class="k">end</span>
@@ -5124,42 +5196,42 @@
 </div>
 </div>
 </div>
-<p>According to the above method, the algorithm's runtime can be obtained as <span class="arithmatex">\((6n + 12)\)</span> ns:</p>
+<p>This gives a runtime of <span class="arithmatex">\((6n + 12)\)</span> ns:</p>
 <div class="arithmatex">\[
 1 + 1 + 10 + (1 + 5) \times n = 6n + 12
 \]</div>
-<p>In reality, however, <strong>trying to count an algorithm's exact runtime is neither practical nor realistic</strong>. First, we do not want to tie the estimated time to the running platform, because algorithms need to run on many different platforms. Second, it is difficult to know the runtime of each type of operation, which makes the estimation process extremely difficult.</p>
-<h2 id="231-counting-time-growth-trends">2.3.1 &nbsp; Counting Time Growth Trends<a class="headerlink" href="#231-counting-time-growth-trends" title="Permanent link">&para;</a></h2>
-<p>Time complexity analysis does not count the algorithm's runtime, <strong>but rather counts the growth trend of the algorithm's runtime as the data volume increases</strong>.</p>
-<p>The concept of "time growth trend" is rather abstract; let us understand it through an example. Suppose the input data size is <span class="arithmatex">\(n\)</span>, and given three algorithms <code>A</code>, <code>B</code>, and <code>C</code>:</p>
+<p>In practice, <strong>calculating runtime this way is neither suitable nor realistic</strong>. We do not want estimates tied to one platform, since algorithms run on many platforms. Also, the time taken by each operation is difficult to determine, making accurate prediction very hard.</p>
+<h2 id="231-measuring-growth-in-runtime">2.3.1 &nbsp; Measuring growth in runtime<a class="headerlink" href="#231-measuring-growth-in-runtime" title="Permanent link">&para;</a></h2>
+<p>Time complexity analysis measures <strong>how runtime grows as the input gets larger</strong>, rather than the runtime itself.</p>
+<p>To make this abstract idea concrete, consider three algorithms, <code>A</code>, <code>B</code>, and <code>C</code>, with input size <span class="arithmatex">\(n\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="c1"># Time complexity of algorithm A: constant order</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="c1"># Algorithm A: constant time complexity</span>
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">algorithm_A</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="nb">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="c1"># Time complexity of algorithm B: linear order</span>
+<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="c1"># Algorithm B: linear time complexity</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="k">def</span><span class="w"> </span><span class="nf">algorithm_B</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="nb">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a><span class="c1"># Time complexity of algorithm C: constant order</span>
+<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a><span class="c1"># Algorithm C: constant time complexity</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a><span class="k">def</span><span class="w"> </span><span class="nf">algorithm_C</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000000</span><span class="p">):</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>        <span class="nb">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="c1">// Time complexity of algorithm A: constant order</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="c1">// Algorithm A: constant time complexity</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm_A</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="p">}</span>
-<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="c1">// Time complexity of algorithm B: linear order</span>
+<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="c1">// Algorithm B: linear time complexity</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm_B</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">        </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="p">}</span>
-<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="c1">// Time complexity of algorithm C: constant order</span>
+<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="c1">// Algorithm C: constant time complexity</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm_C</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">1000000</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">        </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
@@ -5168,17 +5240,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="c1">// Time complexity of algorithm A: constant order</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="c1">// Algorithm A: constant time complexity</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm_A</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="p">}</span>
-<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="c1">// Time complexity of algorithm B: linear order</span>
+<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="c1">// Algorithm B: linear time complexity</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm_B</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">        </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="p">}</span>
-<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="c1">// Time complexity of algorithm C: constant order</span>
+<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="c1">// Algorithm C: constant time complexity</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm_C</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">1000000</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">        </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
@@ -5187,17 +5259,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="c1">// Time complexity of algorithm A: constant order</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="c1">// Algorithm A: constant time complexity</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">AlgorithmA</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="p">}</span>
-<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="c1">// Time complexity of algorithm B: linear order</span>
+<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="c1">// Algorithm B: linear time complexity</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="k">void</span><span class="w"> </span><span class="nf">AlgorithmB</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">        </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="p">}</span>
-<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="c1">// Time complexity of algorithm C: constant order</span>
+<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="c1">// Algorithm C: constant time complexity</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="k">void</span><span class="w"> </span><span class="nf">AlgorithmC</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">1000000</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">        </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
@@ -5206,17 +5278,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="c1">// Time complexity of algorithm A: constant order</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="c1">// Algorithm A: constant time complexity</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">algorithm_A</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="p">}</span>
-<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="c1">// Time complexity of algorithm B: linear order</span>
+<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="c1">// Algorithm B: linear time complexity</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="kd">func</span><span class="w"> </span><span class="nx">algorithm_B</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">        </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="p">}</span>
-<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="c1">// Time complexity of algorithm C: constant order</span>
+<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="c1">// Algorithm C: constant time complexity</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="kd">func</span><span class="w"> </span><span class="nx">algorithm_C</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">1000000</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">        </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
@@ -5225,19 +5297,19 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="c1">// Time complexity of algorithm A: constant order</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="c1">// Algorithm A: constant time complexity</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">algorithmA</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="bp">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="p">}</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a>
-<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="c1">// Time complexity of algorithm B: linear order</span>
+<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="c1">// Algorithm B: linear time complexity</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="kd">func</span><span class="w"> </span><span class="nf">algorithmB</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">        </span><span class="bp">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="p">}</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a>
-<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="c1">// Time complexity of algorithm C: constant order</span>
+<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="c1">// Algorithm C: constant time complexity</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="kd">func</span><span class="w"> </span><span class="nf">algorithmC</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="mi">1_000_000</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">        </span><span class="bp">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
@@ -5246,17 +5318,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="c1">// Time complexity of algorithm A: constant order</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="c1">// Algorithm A: constant time complexity</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">algorithm_A</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="p">}</span>
-<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="c1">// Time complexity of algorithm B: linear order</span>
+<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="c1">// Algorithm B: linear time complexity</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="kd">function</span><span class="w"> </span><span class="nx">algorithm_B</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="p">}</span>
-<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="c1">// Time complexity of algorithm C: constant order</span>
+<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="c1">// Algorithm C: constant time complexity</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="kd">function</span><span class="w"> </span><span class="nx">algorithm_C</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">1000000</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
@@ -5265,17 +5337,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="c1">// Time complexity of algorithm A: constant order</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="c1">// Algorithm A: constant time complexity</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">algorithm_A</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="p">}</span>
-<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="c1">// Time complexity of algorithm B: linear order</span>
+<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="c1">// Algorithm B: linear time complexity</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="kd">function</span><span class="w"> </span><span class="nx">algorithm_B</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="p">}</span>
-<a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="c1">// Time complexity of algorithm C: constant order</span>
+<a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="c1">// Algorithm C: constant time complexity</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="kd">function</span><span class="w"> </span><span class="nx">algorithm_C</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">1000000</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
@@ -5284,17 +5356,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="c1">// Time complexity of algorithm A: constant order</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="c1">// Algorithm A: constant time complexity</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">algorithmA</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="n">print</span><span class="p">(</span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="p">}</span>
-<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="c1">// Time complexity of algorithm B: linear order</span>
+<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="c1">// Algorithm B: linear time complexity</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="kt">void</span><span class="w"> </span><span class="n">algorithmB</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">    </span><span class="n">print</span><span class="p">(</span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="p">}</span>
-<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="c1">// Time complexity of algorithm C: constant order</span>
+<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="c1">// Algorithm C: constant time complexity</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="kt">void</span><span class="w"> </span><span class="n">algorithmC</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">1000000</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">    </span><span class="n">print</span><span class="p">(</span><span class="m">0</span><span class="p">);</span>
@@ -5303,17 +5375,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="c1">// Time complexity of algorithm A: constant order</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="c1">// Algorithm A: constant time complexity</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">algorithm_A</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{}"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="p">}</span>
-<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="c1">// Time complexity of algorithm B: linear order</span>
+<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="c1">// Algorithm B: linear time complexity</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="k">fn</span><span class="w"> </span><span class="nf">algorithm_B</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">        </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{}"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="p">}</span>
-<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="c1">// Time complexity of algorithm C: constant order</span>
+<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="c1">// Algorithm C: constant time complexity</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="k">fn</span><span class="w"> </span><span class="nf">algorithm_C</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="mi">1000000</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">        </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{}"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
@@ -5322,17 +5394,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="c1">// Time complexity of algorithm A: constant order</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="c1">// Algorithm A: constant time complexity</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm_A</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="n">printf</span><span class="p">(</span><span class="s">"%d"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="p">}</span>
-<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="c1">// Time complexity of algorithm B: linear order</span>
+<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="c1">// Algorithm B: linear time complexity</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm_B</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">        </span><span class="n">printf</span><span class="p">(</span><span class="s">"%d"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="p">}</span>
-<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="c1">// Time complexity of algorithm C: constant order</span>
+<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="c1">// Algorithm C: constant time complexity</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm_C</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">1000000</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">        </span><span class="n">printf</span><span class="p">(</span><span class="s">"%d"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
@@ -5341,17 +5413,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="c1">// Time complexity of algorithm A: constant order</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="c1">// Algorithm A: constant time complexity</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">algoritm_A</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="n">println</span><span class="p">(</span><span class="m">0</span><span class="p">)</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="p">}</span>
-<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="c1">// Time complexity of algorithm B: linear order</span>
+<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="c1">// Algorithm B: linear time complexity</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">algorithm_B</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">){</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">        </span><span class="n">println</span><span class="p">(</span><span class="m">0</span><span class="p">)</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="p">}</span>
-<a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="c1">// Time complexity of algorithm C: constant order</span>
+<a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="c1">// Algorithm C: constant time complexity</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">algorithm_C</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="m">1000000</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">        </span><span class="n">println</span><span class="p">(</span><span class="m">0</span><span class="p">)</span>
@@ -5360,17 +5432,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1"># Time complexity of algorithm A: constant order</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1"># Algorithm A: constant time complexity</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">algorithm_A</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">    </span><span class="nb">puts</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="k">end</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a>
-<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="c1"># Time complexity of algorithm B: linear order</span>
+<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="c1"># Algorithm B: linear time complexity</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="k">def</span><span class="w"> </span><span class="nf">algorithm_B</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">    </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nb">puts</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="k">end</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a>
-<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="c1"># Time complexity of algorithm C: constant order</span>
+<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="c1"># Algorithm C: constant time complexity</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="k">def</span><span class="w"> </span><span class="nf">algorithm_C</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">    </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="mi">1_000_000</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nb">puts</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="k">end</span>
@@ -5378,23 +5450,23 @@
 </div>
 </div>
 </div>
-<p>Figure 2-7 shows the time complexity of the above three algorithm functions.</p>
+<p>Figure 2-7 shows their time complexities.</p>
 <ul>
-<li>Algorithm <code>A</code> has only <span class="arithmatex">\(1\)</span> print operation, and the algorithm's runtime does not grow as <span class="arithmatex">\(n\)</span> increases. We call the time complexity of this algorithm "constant order".</li>
-<li>In algorithm <code>B</code>, the print operation needs to loop <span class="arithmatex">\(n\)</span> times, and the algorithm's runtime grows linearly as <span class="arithmatex">\(n\)</span> increases. The time complexity of this algorithm is called "linear order".</li>
-<li>In algorithm <code>C</code>, the print operation needs to loop <span class="arithmatex">\(1000000\)</span> times. Although the runtime is very long, it is independent of the input data size <span class="arithmatex">\(n\)</span>. Therefore, the time complexity of <code>C</code> is the same as <code>A</code>, still "constant order".</li>
+<li>Algorithm <code>A</code> performs <span class="arithmatex">\(1\)</span> print operation. Its runtime does not grow with <span class="arithmatex">\(n\)</span>, so it has constant time complexity.</li>
+<li>Algorithm <code>B</code> prints <span class="arithmatex">\(n\)</span> times. Its runtime grows linearly with <span class="arithmatex">\(n\)</span>, so it has linear time complexity.</li>
+<li>Algorithm <code>C</code> prints <span class="arithmatex">\(1000000\)</span> times. It may take a long time, but that time is independent of input size <span class="arithmatex">\(n\)</span>. Its complexity is therefore constant, like <code>A</code>.</li>
 </ul>
-<p><img alt="Time growth trends of algorithms A, B, and C" class="animation-figure" src="../time_complexity.assets/time_complexity_simple_example.png" /></p>
-<p align="center"> Figure 2-7 &nbsp; Time growth trends of algorithms A, B, and C </p>
+<p><img alt="Runtime growth of algorithms A, B, and C" class="animation-figure" src="../time_complexity.assets/time_complexity_simple_example.png" /></p>
+<p align="center"> Figure 2-7 &nbsp; Runtime growth of algorithms A, B, and C </p>
 
-<p>Compared to directly counting the algorithm's runtime, what are the characteristics of time complexity analysis?</p>
+<p>How does time complexity analysis differ from calculating runtime directly?</p>
 <ul>
-<li><strong>Time complexity can effectively evaluate algorithm efficiency</strong>. For example, the runtime of algorithm <code>B</code> grows linearly; when <span class="arithmatex">\(n &gt; 1\)</span> it is slower than algorithm <code>A</code>, and when <span class="arithmatex">\(n &gt; 1000000\)</span> it is slower than algorithm <code>C</code>. In fact, as long as the input data size <span class="arithmatex">\(n\)</span> is sufficiently large, an algorithm with "constant order" complexity will always be superior to one with "linear order" complexity, which is precisely the meaning of time growth trend.</li>
-<li><strong>The derivation method for time complexity is simpler</strong>. Obviously, the running platform and the types of computational operations are both unrelated to the growth trend of the algorithm's runtime. Therefore, in time complexity analysis, we can simply treat the execution time of all computational operations as the same "unit time", reducing "tracking the runtime of each operation" to "counting the number of operations", which greatly reduces the difficulty of estimation.</li>
-<li><strong>Time complexity also has certain limitations</strong>. For example, although algorithms <code>A</code> and <code>C</code> have the same time complexity, their actual runtimes differ significantly. Similarly, although algorithm <code>B</code> has a higher time complexity than <code>C</code>, when the input data size <span class="arithmatex">\(n\)</span> is small, algorithm <code>B</code> is clearly superior to algorithm <code>C</code>. In such cases, it is often difficult to judge the efficiency of algorithms based solely on time complexity. Of course, despite the above issues, complexity analysis remains the most effective and commonly used method for evaluating algorithm efficiency.</li>
+<li><strong>Time complexity evaluates efficiency effectively</strong>. Algorithm <code>B</code> grows linearly, becoming slower than <code>A</code> when <span class="arithmatex">\(n &gt; 1\)</span> and slower than <code>C</code> when <span class="arithmatex">\(n &gt; 1000000\)</span>. With sufficiently large input size <span class="arithmatex">\(n\)</span>, a constant-time algorithm will outperform a linear-time one. This is what a growth trend tells us.</li>
+<li><strong>Time complexity is easier to derive</strong>. The execution platform and kinds of elementary operations do not determine the growth trend. We can treat each elementary operation as taking the same unit of time, reducing runtime estimation to operation counting and greatly simplifying analysis.</li>
+<li><strong>Time complexity has limitations</strong>. Algorithms <code>A</code> and <code>C</code> have the same complexity but very different runtimes. Although <code>B</code> has higher complexity than <code>C</code>, it is much faster for small <span class="arithmatex">\(n\)</span>. Complexity alone may not settle such comparisons. Even so, it remains the most effective and widely used way to evaluate algorithm efficiency.</li>
 </ul>
-<h2 id="232-asymptotic-upper-bound-of-functions">2.3.2 &nbsp; Asymptotic Upper Bound of Functions<a class="headerlink" href="#232-asymptotic-upper-bound-of-functions" title="Permanent link">&para;</a></h2>
-<p>Given a function with input size <span class="arithmatex">\(n\)</span>:</p>
+<h2 id="232-asymptotic-upper-bounds">2.3.2 &nbsp; Asymptotic upper bounds<a class="headerlink" href="#232-asymptotic-upper-bounds" title="Permanent link">&para;</a></h2>
+<p>Consider this function with input size <span class="arithmatex">\(n\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -5402,7 +5474,7 @@
 <a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a>    <span class="n">a</span> <span class="o">=</span> <span class="mi">1</span>      <span class="c1"># +1</span>
 <a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c1"># +1</span>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">*</span> <span class="mi">2</span>  <span class="c1"># +1</span>
-<a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>    <span class="c1"># Loop n times</span>
+<a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>    <span class="c1"># Repeat n times</span>
 <a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>  <span class="c1"># +1</span>
 <a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>        <span class="nb">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>        <span class="c1"># +1</span>
 </code></pre></div>
@@ -5412,8 +5484,8 @@
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +1</span>
 <a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +1</span>
 <a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">  </span><span class="c1">// +1</span>
-<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">    </span><span class="c1">// Loop n times</span>
-<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// +1 (i++ is executed each round)</span>
+<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
+<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// +1 (i ++ executes each iteration)</span>
 <a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">        </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span><span class="w">    </span><span class="c1">// +1</span>
 <a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="p">}</span>
@@ -5424,8 +5496,8 @@
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +1</span>
 <a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +1</span>
 <a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">  </span><span class="c1">// +1</span>
-<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">    </span><span class="c1">// Loop n times</span>
-<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// +1 (i++ is executed each round)</span>
+<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
+<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// +1 (i ++ executes each iteration)</span>
 <a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">        </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w">    </span><span class="c1">// +1</span>
 <a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="p">}</span>
@@ -5436,8 +5508,8 @@
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +1</span>
 <a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +1</span>
 <a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">  </span><span class="c1">// +1</span>
-<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">    </span><span class="c1">// Loop n times</span>
-<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w">   </span><span class="c1">// +1 (i++ is executed each round)</span>
+<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
+<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w">   </span><span class="c1">// +1 (i ++ executes each iteration)</span>
 <a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">        </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w">   </span><span class="c1">// +1</span>
 <a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="p">}</span>
@@ -5448,7 +5520,7 @@
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="w">      </span><span class="c1">// +1</span>
 <a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w">   </span><span class="c1">// +1</span>
 <a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="w">   </span><span class="c1">// +1</span>
-<a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">    </span><span class="c1">// Loop n times</span>
+<a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
 <a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span><span class="w">   </span><span class="c1">// +1</span>
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">        </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="nx">a</span><span class="p">)</span><span class="w">         </span><span class="c1">// +1</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">    </span><span class="p">}</span>
@@ -5460,7 +5532,7 @@
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// +1</span>
-<a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="c1">// Loop n times</span>
+<a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">        </span><span class="bp">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">    </span><span class="p">}</span>
@@ -5472,8 +5544,8 @@
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
-<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="c1">// Loop n times</span>
-<a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">    </span><span class="k">for</span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">){</span><span class="w"> </span><span class="c1">// +1 (i++ is executed each round)</span>
+<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
+<a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">    </span><span class="k">for</span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">){</span><span class="w"> </span><span class="c1">// +1 (i ++ executes each iteration)</span>
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="p">}</span>
@@ -5484,8 +5556,8 @@
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nx">a</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
-<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="c1">// Loop n times</span>
-<a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="w">    </span><span class="k">for</span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">){</span><span class="w"> </span><span class="c1">// +1 (i++ is executed each round)</span>
+<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
+<a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="w">    </span><span class="k">for</span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">){</span><span class="w"> </span><span class="c1">// +1 (i ++ executes each iteration)</span>
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="p">}</span>
@@ -5496,8 +5568,8 @@
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// +1</span>
-<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="c1">// Loop n times</span>
-<a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// +1 (i++ is executed each round)</span>
+<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="c1">// Repeat n times</span>
+<a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// +1 (i ++ executes each iteration)</span>
 <a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">    </span><span class="n">print</span><span class="p">(</span><span class="m">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="p">}</span>
@@ -5509,8 +5581,8 @@
 <a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">      </span><span class="c1">// +1</span>
 <a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">      </span><span class="c1">// +1</span>
 <a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a>
-<a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="c1">// Loop n times</span>
-<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// +1 (i++ is executed each round)</span>
+<a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
+<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// +1 (i ++ executes each iteration)</span>
 <a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">        </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{}"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="p">}</span>
@@ -5521,8 +5593,8 @@
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +1</span>
 <a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +1</span>
 <a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">  </span><span class="c1">// +1</span>
-<a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">    </span><span class="c1">// Loop n times</span>
-<a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w">   </span><span class="c1">// +1 (i++ is executed each round)</span>
+<a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
+<a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w">   </span><span class="c1">// +1 (i ++ executes each iteration)</span>
 <a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="w">        </span><span class="n">printf</span><span class="p">(</span><span class="s">"%d"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span><span class="w">            </span><span class="c1">// +1</span>
 <a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="p">}</span>
@@ -5533,8 +5605,8 @@
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="c1">// +1</span>
-<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="c1">// Loop n times</span>
-<a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// +1 (i++ is executed each round)</span>
+<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="c1">// Repeat n times</span>
+<a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// +1 (i ++ executes each iteration)</span>
 <a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">        </span><span class="n">println</span><span class="p">(</span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// +1</span>
 <a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="p">}</span>
@@ -5545,7 +5617,7 @@
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="w">       </span><span class="c1"># +1</span>
 <a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w">   </span><span class="c1"># +1</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="w">   </span><span class="c1"># +1</span>
-<a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">    </span><span class="c1"># Loop n times</span>
+<a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">    </span><span class="c1"># Repeat n times</span>
 <a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">    </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="c1"># +1</span>
 <a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="w">        </span><span class="nb">puts</span><span class="w"> </span><span class="mi">0</span><span class="w">      </span><span class="c1"># +1</span>
 <a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">    </span><span class="k">end</span>
@@ -5554,42 +5626,42 @@
 </div>
 </div>
 </div>
-<p>Let the number of operations of the algorithm be a function of the input data size <span class="arithmatex">\(n\)</span>, denoted as <span class="arithmatex">\(T(n)\)</span>. Then the number of operations of the above function is:</p>
+<p>Let <span class="arithmatex">\(T(n)\)</span> be the operation count as a function of input size <span class="arithmatex">\(n\)</span>. For the function above:</p>
 <div class="arithmatex">\[
 T(n) = 3 + 2n
 \]</div>
-<p><span class="arithmatex">\(T(n)\)</span> is a linear function, indicating that its runtime growth trend is linear, and therefore its time complexity is linear order.</p>
-<p>We denote the time complexity of linear order as <span class="arithmatex">\(O(n)\)</span>. This mathematical symbol is called <u>big-<span class="arithmatex">\(O\)</span> notation</u>, representing the <u>asymptotic upper bound</u> of the function <span class="arithmatex">\(T(n)\)</span>.</p>
-<p>Time complexity analysis essentially calculates the asymptotic upper bound of "the number of operations <span class="arithmatex">\(T(n)\)</span>", which has a clear mathematical definition.</p>
+<p>Since <span class="arithmatex">\(T(n)\)</span> is a first-degree function, runtime grows linearly, giving linear time complexity.</p>
+<p>We write linear time complexity as <span class="arithmatex">\(O(n)\)</span>. This is called <u>big-<span class="arithmatex">\(O\)</span> notation</u>, and it gives an <u>asymptotic upper bound</u> on <span class="arithmatex">\(T(n)\)</span>.</p>
+<p>Time complexity analysis essentially finds an asymptotic upper bound on the operation count <span class="arithmatex">\(T(n)\)</span>. It has a precise mathematical definition.</p>
 <div class="admonition note">
-<p class="admonition-title">Asymptotic upper bound of functions</p>
-<p>If there exist positive real numbers <span class="arithmatex">\(c\)</span> and <span class="arithmatex">\(n_0\)</span> such that for all <span class="arithmatex">\(n &gt; n_0\)</span>, we have <span class="arithmatex">\(T(n) \leq c \cdot f(n)\)</span>, then <span class="arithmatex">\(f(n)\)</span> can be considered as an asymptotic upper bound of <span class="arithmatex">\(T(n)\)</span>, denoted as <span class="arithmatex">\(T(n) = O(f(n))\)</span>.</p>
+<p class="admonition-title">Asymptotic upper bound</p>
+<p>If a positive real constant <span class="arithmatex">\(c\)</span> and a real number <span class="arithmatex">\(n_0\)</span> exist such that <span class="arithmatex">\(T(n) \leq c \cdot f(n)\)</span> for all <span class="arithmatex">\(n &gt; n_0\)</span>, then <span class="arithmatex">\(f(n)\)</span> is an asymptotic upper bound on <span class="arithmatex">\(T(n)\)</span>, written <span class="arithmatex">\(T(n) = O(f(n))\)</span>.</p>
 </div>
-<p>As shown in Figure 2-8, calculating the asymptotic upper bound is to find a function <span class="arithmatex">\(f(n)\)</span> such that when <span class="arithmatex">\(n\)</span> tends to infinity, <span class="arithmatex">\(T(n)\)</span> and <span class="arithmatex">\(f(n)\)</span> are at the same growth level, differing only by a constant coefficient <span class="arithmatex">\(c\)</span>.</p>
+<p>As shown in Figure 2-8, we seek a function <span class="arithmatex">\(f(n)\)</span> such that, as <span class="arithmatex">\(n\)</span> tends to infinity, <span class="arithmatex">\(T(n)\)</span> and <span class="arithmatex">\(f(n)\)</span> have the same order of growth, differing only by a constant factor <span class="arithmatex">\(c\)</span>.</p>
 <p><img alt="Asymptotic upper bound of a function" class="animation-figure" src="../time_complexity.assets/asymptotic_upper_bound.png" /></p>
 <p align="center"> Figure 2-8 &nbsp; Asymptotic upper bound of a function </p>
 
-<h2 id="233-derivation-method">2.3.3 &nbsp; Derivation Method<a class="headerlink" href="#233-derivation-method" title="Permanent link">&para;</a></h2>
-<p>The idea of an asymptotic upper bound is somewhat mathematical. If you feel you haven't fully understood it, don't worry. We can first master the derivation method, and gradually grasp its mathematical meaning through continuous practice.</p>
-<p>According to the definition, after determining <span class="arithmatex">\(f(n)\)</span>, we can obtain the time complexity <span class="arithmatex">\(O(f(n))\)</span>. So how do we determine the asymptotic upper bound <span class="arithmatex">\(f(n)\)</span>? Overall, it is divided into two steps: first count the number of operations, then determine the asymptotic upper bound.</p>
-<h3 id="1-step-1-count-the-number-of-operations">1. &nbsp; Step 1: Count the Number of Operations<a class="headerlink" href="#1-step-1-count-the-number-of-operations" title="Permanent link">&para;</a></h3>
-<p>For code, count from top to bottom line by line. However, since the constant coefficient <span class="arithmatex">\(c\)</span> in <span class="arithmatex">\(c \cdot f(n)\)</span> above can be of any size, <strong>coefficients and constant terms in the number of operations <span class="arithmatex">\(T(n)\)</span> can all be ignored</strong>. According to this principle, the following counting simplification techniques can be summarized.</p>
+<h2 id="233-how-to-derive-time-complexity">2.3.3 &nbsp; How to derive time complexity<a class="headerlink" href="#233-how-to-derive-time-complexity" title="Permanent link">&para;</a></h2>
+<p>The mathematics of asymptotic bounds may feel unfamiliar. Do not worry if it is not fully clear yet: learn the practical method first, and its meaning will become clearer with practice.</p>
+<p>Once we identify <span class="arithmatex">\(f(n)\)</span>, the time complexity is <span class="arithmatex">\(O(f(n))\)</span>. Finding the asymptotic upper bound <span class="arithmatex">\(f(n)\)</span> takes two steps: count operations, then determine the bound.</p>
+<h3 id="1-step-1-count-operations">1. &nbsp; Step 1: Count operations<a class="headerlink" href="#1-step-1-count-operations" title="Permanent link">&para;</a></h3>
+<p>We can count code operations line by line. Since the constant factor <span class="arithmatex">\(c\)</span> in <span class="arithmatex">\(c \cdot f(n)\)</span> may be arbitrarily large, <strong>coefficients and constant terms in <span class="arithmatex">\(T(n)\)</span> can be ignored</strong>. This gives three useful shortcuts.</p>
 <ol>
-<li><strong>Ignore constants in <span class="arithmatex">\(T(n)\)</span></strong>. Because they are all independent of <span class="arithmatex">\(n\)</span>, they do not affect time complexity.</li>
-<li><strong>Omit all coefficients</strong>. For example, looping <span class="arithmatex">\(2n\)</span> times, <span class="arithmatex">\(5n + 1\)</span> times, etc., can all be simplified as <span class="arithmatex">\(n\)</span> times, because the coefficient before <span class="arithmatex">\(n\)</span> does not affect time complexity.</li>
-<li><strong>Use multiplication for nested loops</strong>. The total number of operations equals the product of the number of operations in the outer and inner loops, with each layer of loop still able to apply techniques <code>1.</code> and <code>2.</code> separately.</li>
+<li><strong>Ignore constant terms in <span class="arithmatex">\(T(n)\)</span></strong>. They are independent of <span class="arithmatex">\(n\)</span> and do not affect complexity.</li>
+<li><strong>Drop coefficients</strong>. For example, <span class="arithmatex">\(2n\)</span> or <span class="arithmatex">\(5n + 1\)</span> iterations can be counted simply as <span class="arithmatex">\(n\)</span>, because the coefficient of <span class="arithmatex">\(n\)</span> does not affect complexity.</li>
+<li><strong>Multiply for nested loops</strong>. Multiply the outer and inner iteration counts, applying shortcuts <code>1.</code> and <code>2.</code> to each level.</li>
 </ol>
-<p>Given a function, we can use the above techniques to count the number of operations:</p>
+<p>We can apply these shortcuts to the following function:</p>
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 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a>    <span class="n">a</span> <span class="o">=</span> <span class="mi">1</span>      <span class="c1"># +0 (Technique 1)</span>
-<a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">n</span>  <span class="c1"># +0 (Technique 1)</span>
-<a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="c1"># +n (Technique 2)</span>
+<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a>    <span class="n">a</span> <span class="o">=</span> <span class="mi">1</span>      <span class="c1"># +0 (shortcut 1)</span>
+<a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">n</span>  <span class="c1"># +0 (shortcut 1)</span>
+<a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="c1"># +n (shortcut 2)</span>
 <a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span> <span class="o">*</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a>        <span class="nb">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a>    <span class="c1"># +n*n (Technique 3)</span>
+<a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a>    <span class="c1"># +n*n (shortcut 3)</span>
 <a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a>            <span class="nb">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
@@ -5597,13 +5669,13 @@ T(n) = 3 + 2n
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="c1">// +n (Technique 2)</span>
+<a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="c1">// +n (shortcut 2)</span>
 <a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w">        </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
 <a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="w">    </span><span class="c1">// +n*n (Technique 3)</span>
+<a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="w">    </span><span class="c1">// +n*n (shortcut 3)</span>
 <a id="__codelineno-40-9" name="__codelineno-40-9" href="#__codelineno-40-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-10" name="__codelineno-40-10" href="#__codelineno-40-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-11" name="__codelineno-40-11" href="#__codelineno-40-11"></a><span class="w">            </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
@@ -5614,13 +5686,13 @@ T(n) = 3 + 2n
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-2" name="__codelineno-41-2" href="#__codelineno-41-2"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">    </span><span class="c1">// +n (Technique 2)</span>
+<a id="__codelineno-41-2" name="__codelineno-41-2" href="#__codelineno-41-2"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">    </span><span class="c1">// +n (shortcut 2)</span>
 <a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a><span class="w">        </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-41-8" name="__codelineno-41-8" href="#__codelineno-41-8"></a><span class="w">    </span><span class="c1">// +n*n (Technique 3)</span>
+<a id="__codelineno-41-8" name="__codelineno-41-8" href="#__codelineno-41-8"></a><span class="w">    </span><span class="c1">// +n*n (shortcut 3)</span>
 <a id="__codelineno-41-9" name="__codelineno-41-9" href="#__codelineno-41-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-10" name="__codelineno-41-10" href="#__codelineno-41-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-11" name="__codelineno-41-11" href="#__codelineno-41-11"></a><span class="w">            </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
@@ -5631,13 +5703,13 @@ T(n) = 3 + 2n
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="k">void</span><span class="w"> </span><span class="nf">Algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-2" name="__codelineno-42-2" href="#__codelineno-42-2"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">    </span><span class="c1">// +n (Technique 2)</span>
+<a id="__codelineno-42-2" name="__codelineno-42-2" href="#__codelineno-42-2"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">    </span><span class="c1">// +n (shortcut 2)</span>
 <a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a><span class="w">        </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-42-8" name="__codelineno-42-8" href="#__codelineno-42-8"></a><span class="w">    </span><span class="c1">// +n*n (Technique 3)</span>
+<a id="__codelineno-42-8" name="__codelineno-42-8" href="#__codelineno-42-8"></a><span class="w">    </span><span class="c1">// +n*n (shortcut 3)</span>
 <a id="__codelineno-42-9" name="__codelineno-42-9" href="#__codelineno-42-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-10" name="__codelineno-42-10" href="#__codelineno-42-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-11" name="__codelineno-42-11" href="#__codelineno-42-11"></a><span class="w">            </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
@@ -5648,13 +5720,13 @@ T(n) = 3 + 2n
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="kd">func</span><span class="w"> </span><span class="nx">algorithm</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="w">     </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="w">  </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="c1">// +n (Technique 2)</span>
+<a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="w">     </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="w">  </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="c1">// +n (shortcut 2)</span>
 <a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a><span class="w">        </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a><span class="w">    </span><span class="c1">// +n*n (Technique 3)</span>
+<a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a><span class="w">    </span><span class="c1">// +n*n (shortcut 3)</span>
 <a id="__codelineno-43-9" name="__codelineno-43-9" href="#__codelineno-43-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w">            </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
@@ -5665,13 +5737,13 @@ T(n) = 3 + 2n
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="kd">func</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-44-2" name="__codelineno-44-2" href="#__codelineno-44-2"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a><span class="w">    </span><span class="c1">// +n (Technique 2)</span>
+<a id="__codelineno-44-2" name="__codelineno-44-2" href="#__codelineno-44-2"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a><span class="w">    </span><span class="c1">// +n (shortcut 2)</span>
 <a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="p">(</span><span class="mi">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></a><span class="w">        </span><span class="bp">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-44-7" name="__codelineno-44-7" href="#__codelineno-44-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="w">    </span><span class="c1">// +n*n (Technique 3)</span>
+<a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="w">    </span><span class="c1">// +n*n (shortcut 3)</span>
 <a id="__codelineno-44-9" name="__codelineno-44-9" href="#__codelineno-44-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-10" name="__codelineno-44-10" href="#__codelineno-44-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-11" name="__codelineno-44-11" href="#__codelineno-44-11"></a><span class="w">            </span><span class="bp">print</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
@@ -5682,13 +5754,13 @@ T(n) = 3 + 2n
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="kd">function</span><span class="w"> </span><span class="nx">algorithm</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-2" name="__codelineno-45-2" href="#__codelineno-45-2"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="c1">// +n (Technique 2)</span>
+<a id="__codelineno-45-2" name="__codelineno-45-2" href="#__codelineno-45-2"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="c1">// +n (shortcut 2)</span>
 <a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-6" name="__codelineno-45-6" href="#__codelineno-45-6"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-45-7" name="__codelineno-45-7" href="#__codelineno-45-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-45-8" name="__codelineno-45-8" href="#__codelineno-45-8"></a><span class="w">    </span><span class="c1">// +n*n (Technique 3)</span>
+<a id="__codelineno-45-8" name="__codelineno-45-8" href="#__codelineno-45-8"></a><span class="w">    </span><span class="c1">// +n*n (shortcut 3)</span>
 <a id="__codelineno-45-9" name="__codelineno-45-9" href="#__codelineno-45-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-10" name="__codelineno-45-10" href="#__codelineno-45-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-11" name="__codelineno-45-11" href="#__codelineno-45-11"></a><span class="w">            </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
@@ -5699,13 +5771,13 @@ T(n) = 3 + 2n
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="kd">function</span><span class="w"> </span><span class="nx">algorithm</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-2" name="__codelineno-46-2" href="#__codelineno-46-2"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">    </span><span class="c1">// +n (Technique 2)</span>
+<a id="__codelineno-46-2" name="__codelineno-46-2" href="#__codelineno-46-2"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">    </span><span class="c1">// +n (shortcut 2)</span>
 <a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a><span class="w">    </span><span class="c1">// +n*n (Technique 3)</span>
+<a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a><span class="w">    </span><span class="c1">// +n*n (shortcut 3)</span>
 <a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-10" name="__codelineno-46-10" href="#__codelineno-46-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-11" name="__codelineno-46-11" href="#__codelineno-46-11"></a><span class="w">            </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
@@ -5716,13 +5788,13 @@ T(n) = 3 + 2n
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="kt">void</span><span class="w"> </span><span class="n">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-2" name="__codelineno-47-2" href="#__codelineno-47-2"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="c1">// +n (Technique 2)</span>
+<a id="__codelineno-47-2" name="__codelineno-47-2" href="#__codelineno-47-2"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="c1">// +n (shortcut 2)</span>
 <a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a><span class="w">    </span><span class="n">print</span><span class="p">(</span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-47-7" name="__codelineno-47-7" href="#__codelineno-47-7"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a><span class="w">  </span><span class="c1">// +n*n (Technique 3)</span>
+<a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a><span class="w">  </span><span class="c1">// +n*n (shortcut 3)</span>
 <a id="__codelineno-47-9" name="__codelineno-47-9" href="#__codelineno-47-9"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-10" name="__codelineno-47-10" href="#__codelineno-47-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-11" name="__codelineno-47-11" href="#__codelineno-47-11"></a><span class="w">      </span><span class="n">print</span><span class="p">(</span><span class="m">0</span><span class="p">);</span>
@@ -5733,15 +5805,15 @@ T(n) = 3 + 2n
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="k">fn</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-2" name="__codelineno-48-2" href="#__codelineno-48-2"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">     </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w">        </span><span class="c1">// +0 (Technique 1)</span>
+<a id="__codelineno-48-2" name="__codelineno-48-2" href="#__codelineno-48-2"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">     </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w">        </span><span class="c1">// +0 (shortcut 1)</span>
 <a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a>
-<a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w">    </span><span class="c1">// +n (Technique 2)</span>
+<a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w">    </span><span class="c1">// +n (shortcut 2)</span>
 <a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="p">(</span><span class="mi">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="w">        </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{}"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-48-8" name="__codelineno-48-8" href="#__codelineno-48-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a>
-<a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="w">    </span><span class="c1">// +n*n (Technique 3)</span>
+<a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="w">    </span><span class="c1">// +n*n (shortcut 3)</span>
 <a id="__codelineno-48-11" name="__codelineno-48-11" href="#__codelineno-48-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-12" name="__codelineno-48-12" href="#__codelineno-48-12"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-13" name="__codelineno-48-13" href="#__codelineno-48-13"></a><span class="w">            </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{}"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
@@ -5752,13 +5824,13 @@ T(n) = 3 + 2n
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="c1">// +n (Technique 2)</span>
+<a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w">  </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="c1">// +n (shortcut 2)</span>
 <a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w">        </span><span class="n">printf</span><span class="p">(</span><span class="s">"%d"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w">    </span><span class="c1">// +n*n (Technique 3)</span>
+<a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w">    </span><span class="c1">// +n*n (shortcut 3)</span>
 <a id="__codelineno-49-9" name="__codelineno-49-9" href="#__codelineno-49-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-10" name="__codelineno-49-10" href="#__codelineno-49-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-11" name="__codelineno-49-11" href="#__codelineno-49-11"></a><span class="w">            </span><span class="n">printf</span><span class="p">(</span><span class="s">"%d"</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
@@ -5769,13 +5841,13 @@ T(n) = 3 + 2n
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-2" name="__codelineno-50-2" href="#__codelineno-50-2"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="w">   </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w">   </span><span class="c1">// +0 (Technique 1)</span>
-<a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="c1">// +n (Technique 2)</span>
+<a id="__codelineno-50-2" name="__codelineno-50-2" href="#__codelineno-50-2"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="w">   </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w">   </span><span class="c1">// +0 (shortcut 1)</span>
+<a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="c1">// +n (shortcut 2)</span>
 <a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="m">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">        </span><span class="n">println</span><span class="p">(</span><span class="m">0</span><span class="p">)</span>
 <a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-50-8" name="__codelineno-50-8" href="#__codelineno-50-8"></a><span class="w">    </span><span class="c1">// +n*n (Technique 3)</span>
+<a id="__codelineno-50-8" name="__codelineno-50-8" href="#__codelineno-50-8"></a><span class="w">    </span><span class="c1">// +n*n (shortcut 3)</span>
 <a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-10" name="__codelineno-50-10" href="#__codelineno-50-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-11" name="__codelineno-50-11" href="#__codelineno-50-11"></a><span class="w">            </span><span class="n">println</span><span class="p">(</span><span class="m">0</span><span class="p">)</span>
@@ -5786,11 +5858,11 @@ T(n) = 3 + 2n
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="w">       </span><span class="c1"># +0 (Technique 1)</span>
-<a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w">   </span><span class="c1"># +0 (Technique 1)</span>
-<a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="w">    </span><span class="c1"># +n (Technique 2)</span>
+<a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="w">       </span><span class="c1"># +0 (shortcut 1)</span>
+<a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">    </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w">   </span><span class="c1"># +0 (shortcut 1)</span>
+<a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="w">    </span><span class="c1"># +n (shortcut 2)</span>
 <a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">    </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="p">(</span><span class="mi">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nb">puts</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">    </span><span class="c1"># +n*n (Technique 3)</span>
+<a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">    </span><span class="c1"># +n*n (shortcut 3)</span>
 <a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="w">    </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">n</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="k">do</span>
 <a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="w">        </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nb">puts</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w">    </span><span class="k">end</span>
@@ -5799,25 +5871,25 @@ T(n) = 3 + 2n
 </div>
 </div>
 </div>
-<p>The following formula shows the counting results before and after using the above techniques; both derive a time complexity of <span class="arithmatex">\(O(n^2)\)</span>.</p>
+<p>The formulas below compare full and simplified counts. Both give time complexity <span class="arithmatex">\(O(n^2)\)</span>.</p>
 <div class="arithmatex">\[
 \begin{aligned}
-T(n) &amp; = 2n(n + 1) + (5n + 1) + 2 &amp; \text{Complete count (-.-|||)} \newline
+T(n) &amp; = 2n(n + 1) + (5n + 1) + 2 &amp; \text{Full count (-.-|||)} \newline
 &amp; = 2n^2 + 7n + 3 \newline
-T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
+T(n) &amp; = n^2 + n &amp; \text{Shortcut count (o.O)}
 \end{aligned}
 \]</div>
-<h3 id="2-step-2-determine-the-asymptotic-upper-bound">2. &nbsp; Step 2: Determine the Asymptotic Upper Bound<a class="headerlink" href="#2-step-2-determine-the-asymptotic-upper-bound" title="Permanent link">&para;</a></h3>
-<p><strong>Time complexity is determined by the highest-order term in <span class="arithmatex">\(T(n)\)</span></strong>. This is because as <span class="arithmatex">\(n\)</span> tends to infinity, the highest-order term will play a dominant role, and the influence of other terms can be ignored.</p>
-<p>Table 2-2 shows some examples, where some exaggerated values are used to emphasize the conclusion that "coefficients cannot shake the order". When <span class="arithmatex">\(n\)</span> tends to infinity, these constants become insignificant.</p>
-<p align="center"> Table 2-2 &nbsp; Time complexities corresponding to different numbers of operations </p>
+<h3 id="2-step-2-determine-the-asymptotic-upper-bound">2. &nbsp; Step 2: Determine the asymptotic upper bound<a class="headerlink" href="#2-step-2-determine-the-asymptotic-upper-bound" title="Permanent link">&para;</a></h3>
+<p><strong>The highest-order term in <span class="arithmatex">\(T(n)\)</span> determines time complexity</strong>. As <span class="arithmatex">\(n\)</span> tends to infinity, that term dominates, and the others become negligible.</p>
+<p>Table 2-2 includes deliberately extreme coefficients to emphasize that coefficients cannot overcome growth order. As <span class="arithmatex">\(n\)</span> tends to infinity, those constants become insignificant.</p>
+<p align="center"> Table 2-2 &nbsp; Operation counts and their time complexities </p>
 
 <div class="center-table">
 <table>
 <thead>
 <tr>
-<th>Number of Operations <span class="arithmatex">\(T(n)\)</span></th>
-<th>Time Complexity <span class="arithmatex">\(O(f(n))\)</span></th>
+<th>Operation count <span class="arithmatex">\(T(n)\)</span></th>
+<th>Time complexity <span class="arithmatex">\(O(f(n))\)</span></th>
 </tr>
 </thead>
 <tbody>
@@ -5844,25 +5916,25 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </tbody>
 </table>
 </div>
-<h2 id="234-common-types">2.3.4 &nbsp; Common Types<a class="headerlink" href="#234-common-types" title="Permanent link">&para;</a></h2>
-<p>Let the input data size be <span class="arithmatex">\(n\)</span>. Common time complexity types are shown in Figure 2-9 (arranged in order from low to high).</p>
+<h2 id="234-common-complexity-classes">2.3.4 &nbsp; Common complexity classes<a class="headerlink" href="#234-common-complexity-classes" title="Permanent link">&para;</a></h2>
+<p>For input size <span class="arithmatex">\(n\)</span>, common time complexities are shown below in increasing order of growth.</p>
 <div class="arithmatex">\[
 \begin{aligned}
 &amp; O(1) &lt; O(\log n) &lt; O(n) &lt; O(n \log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!) \newline
 &amp; \text{Constant} &lt; \text{Logarithmic} &lt; \text{Linear} &lt; \text{Linearithmic} &lt; \text{Quadratic} &lt; \text{Exponential} &lt; \text{Factorial}
 \end{aligned}
 \]</div>
-<p><img alt="Common time complexity types" class="animation-figure" src="../time_complexity.assets/time_complexity_common_types.png" /></p>
-<p align="center"> Figure 2-9 &nbsp; Common time complexity types </p>
+<p><img alt="Common time complexity classes" class="animation-figure" src="../time_complexity.assets/time_complexity_common_types.png" /></p>
+<p align="center"> Figure 2-9 &nbsp; Common time complexity classes </p>
 
-<h3 id="1-constant-order-o1">1. &nbsp; Constant Order <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#1-constant-order-o1" title="Permanent link">&para;</a></h3>
-<p>The number of operations in constant order is independent of the input data size <span class="arithmatex">\(n\)</span>, meaning it does not change as <span class="arithmatex">\(n\)</span> changes.</p>
-<p>In the following function, although the value of <code>size</code> may be large, it is independent of the input data size <span class="arithmatex">\(n\)</span>, so the time complexity remains <span class="arithmatex">\(O(1)\)</span>:</p>
+<h3 id="1-constant-time-o1">1. &nbsp; Constant time <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#1-constant-time-o1" title="Permanent link">&para;</a></h3>
+<p>A constant operation count is independent of input size <span class="arithmatex">\(n\)</span>: changing <span class="arithmatex">\(n\)</span> does not change the count.</p>
+<p>In the following function, <code>size</code> may be large, but it is independent of input size <span class="arithmatex">\(n\)</span>, so the time complexity remains <span class="arithmatex">\(O(1)\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">time_complexity.py</span><pre><span></span><code><a id="__codelineno-52-1" name="__codelineno-52-1" href="#__codelineno-52-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Constant order"""</span>
+<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Constant complexity"""</span>
 <a id="__codelineno-52-3" name="__codelineno-52-3" href="#__codelineno-52-3"></a>    <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-52-4" name="__codelineno-52-4" href="#__codelineno-52-4"></a>    <span class="n">size</span> <span class="o">=</span> <span class="mi">100000</span>
 <a id="__codelineno-52-5" name="__codelineno-52-5" href="#__codelineno-52-5"></a>    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">size</span><span class="p">):</span>
@@ -5871,7 +5943,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Constant order */</span>
+<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-53-2" name="__codelineno-53-2" href="#__codelineno-53-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-53-4" name="__codelineno-53-4" href="#__codelineno-53-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">100000</span><span class="p">;</span>
@@ -5882,7 +5954,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Constant order */</span>
+<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-54-2" name="__codelineno-54-2" href="#__codelineno-54-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-54-4" name="__codelineno-54-4" href="#__codelineno-54-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">100000</span><span class="p">;</span>
@@ -5893,7 +5965,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Constant order */</span>
+<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-55-2" name="__codelineno-55-2" href="#__codelineno-55-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Constant</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-55-4" name="__codelineno-55-4" href="#__codelineno-55-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">100000</span><span class="p">;</span>
@@ -5904,7 +5976,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Constant order */</span>
+<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-56-2" name="__codelineno-56-2" href="#__codelineno-56-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">constant</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w">    </span><span class="nx">count</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a><span class="w">    </span><span class="nx">size</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">100000</span>
@@ -5916,7 +5988,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Constant order */</span>
+<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-57-2" name="__codelineno-57-2" href="#__codelineno-57-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">size</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">100_000</span>
@@ -5928,7 +6000,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Constant order */</span>
+<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-58-2" name="__codelineno-58-2" href="#__codelineno-58-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">constant</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-58-4" name="__codelineno-58-4" href="#__codelineno-58-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">100000</span><span class="p">;</span>
@@ -5938,7 +6010,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Constant order */</span>
+<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-59-2" name="__codelineno-59-2" href="#__codelineno-59-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">constant</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-59-4" name="__codelineno-59-4" href="#__codelineno-59-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">100000</span><span class="p">;</span>
@@ -5948,7 +6020,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Constant order */</span>
+<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-60-2" name="__codelineno-60-2" href="#__codelineno-60-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">constant</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-3" name="__codelineno-60-3" href="#__codelineno-60-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-60-4" name="__codelineno-60-4" href="#__codelineno-60-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">100000</span><span class="p">;</span>
@@ -5960,7 +6032,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Constant order */</span>
+<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-61-2" name="__codelineno-61-2" href="#__codelineno-61-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-3" name="__codelineno-61-3" href="#__codelineno-61-3"></a><span class="w">    </span><span class="n">_</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span>
 <a id="__codelineno-61-4" name="__codelineno-61-4" href="#__codelineno-61-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
@@ -5973,7 +6045,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Constant order */</span>
+<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-62-2" name="__codelineno-62-2" href="#__codelineno-62-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-3" name="__codelineno-62-3" href="#__codelineno-62-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-62-4" name="__codelineno-62-4" href="#__codelineno-62-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">100000</span><span class="p">;</span>
@@ -5985,7 +6057,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Constant order */</span>
+<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Constant complexity */</span>
 <a id="__codelineno-63-2" name="__codelineno-63-2" href="#__codelineno-63-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-3" name="__codelineno-63-3" href="#__codelineno-63-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-63-4" name="__codelineno-63-4" href="#__codelineno-63-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">100000</span>
@@ -5996,7 +6068,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Constant time ###</span>
+<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Constant complexity ###</span>
 <a id="__codelineno-64-2" name="__codelineno-64-2" href="#__codelineno-64-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">constant</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-64-3" name="__codelineno-64-3" href="#__codelineno-64-3"></a><span class="w">  </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-64-4" name="__codelineno-64-4" href="#__codelineno-64-4"></a><span class="w">  </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">100000</span>
@@ -6009,13 +6081,18 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </div>
 </div>
 </div>
-<h3 id="2-linear-order-on">2. &nbsp; Linear Order <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#2-linear-order-on" title="Permanent link">&para;</a></h3>
-<p>The number of operations in linear order grows linearly relative to the input data size <span class="arithmatex">\(n\)</span>. Linear order typically appears in single-layer loops:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20constant%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Constant%20complexity%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20size%20%3D%2010%0A%20%20%20%20for%20_%20in%20range%28size%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20constant%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20constant%20complexity%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20constant%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Constant%20complexity%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20size%20%3D%2010%0A%20%20%20%20for%20_%20in%20range%28size%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20constant%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20constant%20complexity%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="2-linear-time-on">2. &nbsp; Linear time <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#2-linear-time-on" title="Permanent link">&para;</a></h3>
+<p>A linear operation count grows proportionally to input size <span class="arithmatex">\(n\)</span>. It commonly appears in a single loop:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="6:13"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">time_complexity.py</span><pre><span></span><code><a id="__codelineno-65-1" name="__codelineno-65-1" href="#__codelineno-65-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-65-2" name="__codelineno-65-2" href="#__codelineno-65-2"></a><span class="w">    </span><span class="sd">"""Linear order"""</span>
+<a id="__codelineno-65-2" name="__codelineno-65-2" href="#__codelineno-65-2"></a><span class="w">    </span><span class="sd">"""Linear complexity"""</span>
 <a id="__codelineno-65-3" name="__codelineno-65-3" href="#__codelineno-65-3"></a>    <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-65-4" name="__codelineno-65-4" href="#__codelineno-65-4"></a>    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-65-5" name="__codelineno-65-5" href="#__codelineno-65-5"></a>        <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
@@ -6023,7 +6100,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-66-2" name="__codelineno-66-2" href="#__codelineno-66-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-66-3" name="__codelineno-66-3" href="#__codelineno-66-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-66-4" name="__codelineno-66-4" href="#__codelineno-66-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span>
@@ -6033,7 +6110,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-67-2" name="__codelineno-67-2" href="#__codelineno-67-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-67-3" name="__codelineno-67-3" href="#__codelineno-67-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-67-4" name="__codelineno-67-4" href="#__codelineno-67-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span>
@@ -6043,7 +6120,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-68-2" name="__codelineno-68-2" href="#__codelineno-68-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Linear</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-68-3" name="__codelineno-68-3" href="#__codelineno-68-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-68-4" name="__codelineno-68-4" href="#__codelineno-68-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span>
@@ -6053,7 +6130,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-69-2" name="__codelineno-69-2" href="#__codelineno-69-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">linear</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-69-3" name="__codelineno-69-3" href="#__codelineno-69-3"></a><span class="w">    </span><span class="nx">count</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-69-4" name="__codelineno-69-4" href="#__codelineno-69-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
@@ -6064,7 +6141,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-70-2" name="__codelineno-70-2" href="#__codelineno-70-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-70-3" name="__codelineno-70-3" href="#__codelineno-70-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-70-4" name="__codelineno-70-4" href="#__codelineno-70-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
@@ -6075,7 +6152,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-71-2" name="__codelineno-71-2" href="#__codelineno-71-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">linear</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-71-3" name="__codelineno-71-3" href="#__codelineno-71-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-71-4" name="__codelineno-71-4" href="#__codelineno-71-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="nx">count</span><span class="o">++</span><span class="p">;</span>
@@ -6084,7 +6161,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-72-2" name="__codelineno-72-2" href="#__codelineno-72-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">linear</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-72-3" name="__codelineno-72-3" href="#__codelineno-72-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-72-4" name="__codelineno-72-4" href="#__codelineno-72-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="nx">count</span><span class="o">++</span><span class="p">;</span>
@@ -6093,7 +6170,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-73-2" name="__codelineno-73-2" href="#__codelineno-73-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">linear</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-73-3" name="__codelineno-73-3" href="#__codelineno-73-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-73-4" name="__codelineno-73-4" href="#__codelineno-73-4"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6104,7 +6181,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-74-2" name="__codelineno-74-2" href="#__codelineno-74-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-74-3" name="__codelineno-74-3" href="#__codelineno-74-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-74-4" name="__codelineno-74-4" href="#__codelineno-74-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
@@ -6115,7 +6192,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-75-2" name="__codelineno-75-2" href="#__codelineno-75-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-75-3" name="__codelineno-75-3" href="#__codelineno-75-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-75-4" name="__codelineno-75-4" href="#__codelineno-75-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6126,7 +6203,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a><span class="cm">/* Linear order */</span>
+<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a><span class="cm">/* Linear complexity */</span>
 <a id="__codelineno-76-2" name="__codelineno-76-2" href="#__codelineno-76-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-76-3" name="__codelineno-76-3" href="#__codelineno-76-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-76-4" name="__codelineno-76-4" href="#__codelineno-76-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span>
@@ -6136,7 +6213,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="c1">### Linear time ###</span>
+<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="c1">### Linear complexity ###</span>
 <a id="__codelineno-77-2" name="__codelineno-77-2" href="#__codelineno-77-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">linear</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-77-3" name="__codelineno-77-3" href="#__codelineno-77-3"></a><span class="w">  </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-77-4" name="__codelineno-77-4" href="#__codelineno-77-4"></a><span class="w">  </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">}</span>
@@ -6146,24 +6223,29 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </div>
 </div>
 </div>
-<p>Operations such as traversing arrays and traversing linked lists have a time complexity of <span class="arithmatex">\(O(n)\)</span>, where <span class="arithmatex">\(n\)</span> is the length of the array or linked list:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 441px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20linear%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Linear%20complexity%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20linear%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20linear%20complexity%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20linear%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Linear%20complexity%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20linear%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20linear%20complexity%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Traversing an array or linked list takes <span class="arithmatex">\(O(n)\)</span> time, where <span class="arithmatex">\(n\)</span> is its length:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="7:13"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><input id="__tabbed_7_13" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Python</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Java</label><label for="__tabbed_7_4">C#</label><label for="__tabbed_7_5">Go</label><label for="__tabbed_7_6">Swift</label><label for="__tabbed_7_7">JS</label><label for="__tabbed_7_8">TS</label><label for="__tabbed_7_9">Dart</label><label for="__tabbed_7_10">Rust</label><label for="__tabbed_7_11">C</label><label for="__tabbed_7_12">Kotlin</label><label for="__tabbed_7_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">time_complexity.py</span><pre><span></span><code><a id="__codelineno-78-1" name="__codelineno-78-1" href="#__codelineno-78-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">array_traversal</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">])</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-78-2" name="__codelineno-78-2" href="#__codelineno-78-2"></a><span class="w">    </span><span class="sd">"""Linear order (traversing array)"""</span>
+<a id="__codelineno-78-2" name="__codelineno-78-2" href="#__codelineno-78-2"></a><span class="w">    </span><span class="sd">"""Linear complexity (array traversal)"""</span>
 <a id="__codelineno-78-3" name="__codelineno-78-3" href="#__codelineno-78-3"></a>    <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-78-4" name="__codelineno-78-4" href="#__codelineno-78-4"></a>    <span class="c1"># Number of iterations is proportional to the array length</span>
+<a id="__codelineno-78-4" name="__codelineno-78-4" href="#__codelineno-78-4"></a>    <span class="c1"># The number of iterations is proportional to the array length</span>
 <a id="__codelineno-78-5" name="__codelineno-78-5" href="#__codelineno-78-5"></a>    <span class="k">for</span> <span class="n">num</span> <span class="ow">in</span> <span class="n">nums</span><span class="p">:</span>
 <a id="__codelineno-78-6" name="__codelineno-78-6" href="#__codelineno-78-6"></a>        <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
 <a id="__codelineno-78-7" name="__codelineno-78-7" href="#__codelineno-78-7"></a>    <span class="k">return</span> <span class="n">count</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-79-1" name="__codelineno-79-1" href="#__codelineno-79-1"></a><span class="cm">/* Linear order (traversing array) */</span>
+<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-79-1" name="__codelineno-79-1" href="#__codelineno-79-1"></a><span class="cm">/* Linear complexity (array traversal) */</span>
 <a id="__codelineno-79-2" name="__codelineno-79-2" href="#__codelineno-79-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">arrayTraversal</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-3" name="__codelineno-79-3" href="#__codelineno-79-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-79-4" name="__codelineno-79-4" href="#__codelineno-79-4"></a><span class="w">    </span><span class="c1">// Number of iterations is proportional to the array length</span>
+<a id="__codelineno-79-4" name="__codelineno-79-4" href="#__codelineno-79-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the array length</span>
 <a id="__codelineno-79-5" name="__codelineno-79-5" href="#__codelineno-79-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-6" name="__codelineno-79-6" href="#__codelineno-79-6"></a><span class="w">        </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-79-7" name="__codelineno-79-7" href="#__codelineno-79-7"></a><span class="w">    </span><span class="p">}</span>
@@ -6172,10 +6254,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-80-1" name="__codelineno-80-1" href="#__codelineno-80-1"></a><span class="cm">/* Linear order (traversing array) */</span>
+<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-80-1" name="__codelineno-80-1" href="#__codelineno-80-1"></a><span class="cm">/* Linear complexity (array traversal) */</span>
 <a id="__codelineno-80-2" name="__codelineno-80-2" href="#__codelineno-80-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">arrayTraversal</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-3" name="__codelineno-80-3" href="#__codelineno-80-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-80-4" name="__codelineno-80-4" href="#__codelineno-80-4"></a><span class="w">    </span><span class="c1">// Number of iterations is proportional to the array length</span>
+<a id="__codelineno-80-4" name="__codelineno-80-4" href="#__codelineno-80-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the array length</span>
 <a id="__codelineno-80-5" name="__codelineno-80-5" href="#__codelineno-80-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-6" name="__codelineno-80-6" href="#__codelineno-80-6"></a><span class="w">        </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-80-7" name="__codelineno-80-7" href="#__codelineno-80-7"></a><span class="w">    </span><span class="p">}</span>
@@ -6184,10 +6266,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="cm">/* Linear order (traversing array) */</span>
+<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="cm">/* Linear complexity (array traversal) */</span>
 <a id="__codelineno-81-2" name="__codelineno-81-2" href="#__codelineno-81-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">ArrayTraversal</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-3" name="__codelineno-81-3" href="#__codelineno-81-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-81-4" name="__codelineno-81-4" href="#__codelineno-81-4"></a><span class="w">    </span><span class="c1">// Number of iterations is proportional to the array length</span>
+<a id="__codelineno-81-4" name="__codelineno-81-4" href="#__codelineno-81-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the array length</span>
 <a id="__codelineno-81-5" name="__codelineno-81-5" href="#__codelineno-81-5"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-6" name="__codelineno-81-6" href="#__codelineno-81-6"></a><span class="w">        </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-81-7" name="__codelineno-81-7" href="#__codelineno-81-7"></a><span class="w">    </span><span class="p">}</span>
@@ -6196,10 +6278,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="cm">/* Linear order (traversing array) */</span>
+<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="cm">/* Linear complexity (array traversal) */</span>
 <a id="__codelineno-82-2" name="__codelineno-82-2" href="#__codelineno-82-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">arrayTraversal</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-3" name="__codelineno-82-3" href="#__codelineno-82-3"></a><span class="w">    </span><span class="nx">count</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-82-4" name="__codelineno-82-4" href="#__codelineno-82-4"></a><span class="w">    </span><span class="c1">// Number of iterations is proportional to the array length</span>
+<a id="__codelineno-82-4" name="__codelineno-82-4" href="#__codelineno-82-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the array length</span>
 <a id="__codelineno-82-5" name="__codelineno-82-5" href="#__codelineno-82-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-6" name="__codelineno-82-6" href="#__codelineno-82-6"></a><span class="w">        </span><span class="nx">count</span><span class="o">++</span>
 <a id="__codelineno-82-7" name="__codelineno-82-7" href="#__codelineno-82-7"></a><span class="w">    </span><span class="p">}</span>
@@ -6208,10 +6290,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-83-1" name="__codelineno-83-1" href="#__codelineno-83-1"></a><span class="cm">/* Linear order (traversing array) */</span>
+<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-83-1" name="__codelineno-83-1" href="#__codelineno-83-1"></a><span class="cm">/* Linear complexity (array traversal) */</span>
 <a id="__codelineno-83-2" name="__codelineno-83-2" href="#__codelineno-83-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">arrayTraversal</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-3" name="__codelineno-83-3" href="#__codelineno-83-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-83-4" name="__codelineno-83-4" href="#__codelineno-83-4"></a><span class="w">    </span><span class="c1">// Number of iterations is proportional to the array length</span>
+<a id="__codelineno-83-4" name="__codelineno-83-4" href="#__codelineno-83-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the array length</span>
 <a id="__codelineno-83-5" name="__codelineno-83-5" href="#__codelineno-83-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-6" name="__codelineno-83-6" href="#__codelineno-83-6"></a><span class="w">        </span><span class="bp">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-83-7" name="__codelineno-83-7" href="#__codelineno-83-7"></a><span class="w">    </span><span class="p">}</span>
@@ -6220,10 +6302,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-84-1" name="__codelineno-84-1" href="#__codelineno-84-1"></a><span class="cm">/* Linear order (traversing array) */</span>
+<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-84-1" name="__codelineno-84-1" href="#__codelineno-84-1"></a><span class="cm">/* Linear complexity (array traversal) */</span>
 <a id="__codelineno-84-2" name="__codelineno-84-2" href="#__codelineno-84-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">arrayTraversal</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-3" name="__codelineno-84-3" href="#__codelineno-84-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-84-4" name="__codelineno-84-4" href="#__codelineno-84-4"></a><span class="w">    </span><span class="c1">// Number of iterations is proportional to the array length</span>
+<a id="__codelineno-84-4" name="__codelineno-84-4" href="#__codelineno-84-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the array length</span>
 <a id="__codelineno-84-5" name="__codelineno-84-5" href="#__codelineno-84-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-6" name="__codelineno-84-6" href="#__codelineno-84-6"></a><span class="w">        </span><span class="nx">count</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-84-7" name="__codelineno-84-7" href="#__codelineno-84-7"></a><span class="w">    </span><span class="p">}</span>
@@ -6232,10 +6314,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-85-1" name="__codelineno-85-1" href="#__codelineno-85-1"></a><span class="cm">/* Linear order (traversing array) */</span>
+<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-85-1" name="__codelineno-85-1" href="#__codelineno-85-1"></a><span class="cm">/* Linear complexity (array traversal) */</span>
 <a id="__codelineno-85-2" name="__codelineno-85-2" href="#__codelineno-85-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">arrayTraversal</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-3" name="__codelineno-85-3" href="#__codelineno-85-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-85-4" name="__codelineno-85-4" href="#__codelineno-85-4"></a><span class="w">    </span><span class="c1">// Number of iterations is proportional to the array length</span>
+<a id="__codelineno-85-4" name="__codelineno-85-4" href="#__codelineno-85-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the array length</span>
 <a id="__codelineno-85-5" name="__codelineno-85-5" href="#__codelineno-85-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-6" name="__codelineno-85-6" href="#__codelineno-85-6"></a><span class="w">        </span><span class="nx">count</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-85-7" name="__codelineno-85-7" href="#__codelineno-85-7"></a><span class="w">    </span><span class="p">}</span>
@@ -6244,10 +6326,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-86-1" name="__codelineno-86-1" href="#__codelineno-86-1"></a><span class="cm">/* Linear order (traversing array) */</span>
+<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-86-1" name="__codelineno-86-1" href="#__codelineno-86-1"></a><span class="cm">/* Linear complexity (array traversal) */</span>
 <a id="__codelineno-86-2" name="__codelineno-86-2" href="#__codelineno-86-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">arrayTraversal</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-3" name="__codelineno-86-3" href="#__codelineno-86-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
-<a id="__codelineno-86-4" name="__codelineno-86-4" href="#__codelineno-86-4"></a><span class="w">  </span><span class="c1">// Number of iterations is proportional to the array length</span>
+<a id="__codelineno-86-4" name="__codelineno-86-4" href="#__codelineno-86-4"></a><span class="w">  </span><span class="c1">// The number of iterations is proportional to the array length</span>
 <a id="__codelineno-86-5" name="__codelineno-86-5" href="#__codelineno-86-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-6" name="__codelineno-86-6" href="#__codelineno-86-6"></a><span class="w">    </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-86-7" name="__codelineno-86-7" href="#__codelineno-86-7"></a><span class="w">  </span><span class="p">}</span>
@@ -6256,10 +6338,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-87-1" name="__codelineno-87-1" href="#__codelineno-87-1"></a><span class="cm">/* Linear order (traversing array) */</span>
+<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-87-1" name="__codelineno-87-1" href="#__codelineno-87-1"></a><span class="cm">/* Linear complexity (array traversal) */</span>
 <a id="__codelineno-87-2" name="__codelineno-87-2" href="#__codelineno-87-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">array_traversal</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-3" name="__codelineno-87-3" href="#__codelineno-87-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-87-4" name="__codelineno-87-4" href="#__codelineno-87-4"></a><span class="w">    </span><span class="c1">// Number of iterations is proportional to the array length</span>
+<a id="__codelineno-87-4" name="__codelineno-87-4" href="#__codelineno-87-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the array length</span>
 <a id="__codelineno-87-5" name="__codelineno-87-5" href="#__codelineno-87-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-6" name="__codelineno-87-6" href="#__codelineno-87-6"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-87-7" name="__codelineno-87-7" href="#__codelineno-87-7"></a><span class="w">    </span><span class="p">}</span>
@@ -6268,10 +6350,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-88-1" name="__codelineno-88-1" href="#__codelineno-88-1"></a><span class="cm">/* Linear order (traversing array) */</span>
+<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-88-1" name="__codelineno-88-1" href="#__codelineno-88-1"></a><span class="cm">/* Linear complexity (array traversal) */</span>
 <a id="__codelineno-88-2" name="__codelineno-88-2" href="#__codelineno-88-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">arrayTraversal</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-3" name="__codelineno-88-3" href="#__codelineno-88-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-88-4" name="__codelineno-88-4" href="#__codelineno-88-4"></a><span class="w">    </span><span class="c1">// Number of iterations is proportional to the array length</span>
+<a id="__codelineno-88-4" name="__codelineno-88-4" href="#__codelineno-88-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the array length</span>
 <a id="__codelineno-88-5" name="__codelineno-88-5" href="#__codelineno-88-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-6" name="__codelineno-88-6" href="#__codelineno-88-6"></a><span class="w">        </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-88-7" name="__codelineno-88-7" href="#__codelineno-88-7"></a><span class="w">    </span><span class="p">}</span>
@@ -6280,10 +6362,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-89-1" name="__codelineno-89-1" href="#__codelineno-89-1"></a><span class="cm">/* Linear order (traversing array) */</span>
+<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-89-1" name="__codelineno-89-1" href="#__codelineno-89-1"></a><span class="cm">/* Linear complexity (array traversal) */</span>
 <a id="__codelineno-89-2" name="__codelineno-89-2" href="#__codelineno-89-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">arrayTraversal</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-89-3" name="__codelineno-89-3" href="#__codelineno-89-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-89-4" name="__codelineno-89-4" href="#__codelineno-89-4"></a><span class="w">    </span><span class="c1">// Number of iterations is proportional to the array length</span>
+<a id="__codelineno-89-4" name="__codelineno-89-4" href="#__codelineno-89-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the array length</span>
 <a id="__codelineno-89-5" name="__codelineno-89-5" href="#__codelineno-89-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-89-6" name="__codelineno-89-6" href="#__codelineno-89-6"></a><span class="w">        </span><span class="n">count</span><span class="o">++</span>
 <a id="__codelineno-89-7" name="__codelineno-89-7" href="#__codelineno-89-7"></a><span class="w">    </span><span class="p">}</span>
@@ -6292,11 +6374,11 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-90-1" name="__codelineno-90-1" href="#__codelineno-90-1"></a><span class="c1">### Linear time (array traversal) ###</span>
+<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-90-1" name="__codelineno-90-1" href="#__codelineno-90-1"></a><span class="c1">### Linear complexity (array traversal)###</span>
 <a id="__codelineno-90-2" name="__codelineno-90-2" href="#__codelineno-90-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">array_traversal</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
 <a id="__codelineno-90-3" name="__codelineno-90-3" href="#__codelineno-90-3"></a><span class="w">  </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-90-4" name="__codelineno-90-4" href="#__codelineno-90-4"></a>
-<a id="__codelineno-90-5" name="__codelineno-90-5" href="#__codelineno-90-5"></a><span class="w">  </span><span class="c1"># Number of iterations is proportional to the array length</span>
+<a id="__codelineno-90-5" name="__codelineno-90-5" href="#__codelineno-90-5"></a><span class="w">  </span><span class="c1"># The number of iterations is proportional to the array length</span>
 <a id="__codelineno-90-6" name="__codelineno-90-6" href="#__codelineno-90-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span>
 <a id="__codelineno-90-7" name="__codelineno-90-7" href="#__codelineno-90-7"></a><span class="w">    </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-90-8" name="__codelineno-90-8" href="#__codelineno-90-8"></a><span class="w">  </span><span class="k">end</span>
@@ -6307,16 +6389,21 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </div>
 </div>
 </div>
-<p>It is worth noting that <strong>the input data size <span class="arithmatex">\(n\)</span> should be determined according to the type of input data</strong>. For example, in the first example, the variable <span class="arithmatex">\(n\)</span> is the input data size; in the second example, the array length <span class="arithmatex">\(n\)</span> is the data size.</p>
-<h3 id="3-quadratic-order-on2">3. &nbsp; Quadratic Order <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#3-quadratic-order-on2" title="Permanent link">&para;</a></h3>
-<p>The number of operations in quadratic order grows quadratically relative to the input data size <span class="arithmatex">\(n\)</span>. Quadratic order typically appears in nested loops, where both the outer and inner loops have a time complexity of <span class="arithmatex">\(O(n)\)</span>, resulting in an overall time complexity of <span class="arithmatex">\(O(n^2)\)</span>:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20array_traversal%28nums%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Linear%20complexity%20%28array%20traversal%29%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20The%20number%20of%20iterations%20is%20proportional%20to%20the%20array%20length%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20array_traversal%28%5B0%5D%20%2A%20n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20linear%20complexity%20%28array%20traversal%29%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20array_traversal%28nums%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Linear%20complexity%20%28array%20traversal%29%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20The%20number%20of%20iterations%20is%20proportional%20to%20the%20array%20length%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20array_traversal%28%5B0%5D%20%2A%20n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20linear%20complexity%20%28array%20traversal%29%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p><strong>The meaning of input size <span class="arithmatex">\(n\)</span> depends on the input type</strong>. In the first example, it is the variable <span class="arithmatex">\(n\)</span>; in the second, it is the array length <span class="arithmatex">\(n\)</span>.</p>
+<h3 id="3-quadratic-time-on2">3. &nbsp; Quadratic time <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#3-quadratic-time-on2" title="Permanent link">&para;</a></h3>
+<p>A quadratic operation count grows with the square of input size <span class="arithmatex">\(n\)</span>. It commonly appears in nested loops where both levels take <span class="arithmatex">\(O(n)\)</span> time, giving <span class="arithmatex">\(O(n^2)\)</span> overall:</p>
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 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">time_complexity.py</span><pre><span></span><code><a id="__codelineno-91-1" name="__codelineno-91-1" href="#__codelineno-91-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-91-2" name="__codelineno-91-2" href="#__codelineno-91-2"></a><span class="w">    </span><span class="sd">"""Quadratic order"""</span>
+<a id="__codelineno-91-2" name="__codelineno-91-2" href="#__codelineno-91-2"></a><span class="w">    </span><span class="sd">"""Quadratic complexity"""</span>
 <a id="__codelineno-91-3" name="__codelineno-91-3" href="#__codelineno-91-3"></a>    <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-91-4" name="__codelineno-91-4" href="#__codelineno-91-4"></a>    <span class="c1"># Number of iterations is quadratically related to the data size n</span>
+<a id="__codelineno-91-4" name="__codelineno-91-4" href="#__codelineno-91-4"></a>    <span class="c1"># The number of iterations is proportional to the square of the input size n</span>
 <a id="__codelineno-91-5" name="__codelineno-91-5" href="#__codelineno-91-5"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-91-6" name="__codelineno-91-6" href="#__codelineno-91-6"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-91-7" name="__codelineno-91-7" href="#__codelineno-91-7"></a>            <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
@@ -6324,10 +6411,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-92-1" name="__codelineno-92-1" href="#__codelineno-92-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-92-1" name="__codelineno-92-1" href="#__codelineno-92-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-92-2" name="__codelineno-92-2" href="#__codelineno-92-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-92-3" name="__codelineno-92-3" href="#__codelineno-92-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-92-4" name="__codelineno-92-4" href="#__codelineno-92-4"></a><span class="w">    </span><span class="c1">// Number of iterations is quadratically related to the data size n</span>
+<a id="__codelineno-92-4" name="__codelineno-92-4" href="#__codelineno-92-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the square of the input size n</span>
 <a id="__codelineno-92-5" name="__codelineno-92-5" href="#__codelineno-92-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-92-6" name="__codelineno-92-6" href="#__codelineno-92-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-92-7" name="__codelineno-92-7" href="#__codelineno-92-7"></a><span class="w">            </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
@@ -6338,10 +6425,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-93-1" name="__codelineno-93-1" href="#__codelineno-93-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-93-1" name="__codelineno-93-1" href="#__codelineno-93-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-93-2" name="__codelineno-93-2" href="#__codelineno-93-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-93-3" name="__codelineno-93-3" href="#__codelineno-93-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-93-4" name="__codelineno-93-4" href="#__codelineno-93-4"></a><span class="w">    </span><span class="c1">// Number of iterations is quadratically related to the data size n</span>
+<a id="__codelineno-93-4" name="__codelineno-93-4" href="#__codelineno-93-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the square of the input size n</span>
 <a id="__codelineno-93-5" name="__codelineno-93-5" href="#__codelineno-93-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-93-6" name="__codelineno-93-6" href="#__codelineno-93-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-93-7" name="__codelineno-93-7" href="#__codelineno-93-7"></a><span class="w">            </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
@@ -6352,10 +6439,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-94-1" name="__codelineno-94-1" href="#__codelineno-94-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-94-1" name="__codelineno-94-1" href="#__codelineno-94-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-94-2" name="__codelineno-94-2" href="#__codelineno-94-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Quadratic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-94-3" name="__codelineno-94-3" href="#__codelineno-94-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-94-4" name="__codelineno-94-4" href="#__codelineno-94-4"></a><span class="w">    </span><span class="c1">// Number of iterations is quadratically related to the data size n</span>
+<a id="__codelineno-94-4" name="__codelineno-94-4" href="#__codelineno-94-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the square of the input size n</span>
 <a id="__codelineno-94-5" name="__codelineno-94-5" href="#__codelineno-94-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-94-6" name="__codelineno-94-6" href="#__codelineno-94-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-94-7" name="__codelineno-94-7" href="#__codelineno-94-7"></a><span class="w">            </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
@@ -6366,10 +6453,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-95-1" name="__codelineno-95-1" href="#__codelineno-95-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-95-1" name="__codelineno-95-1" href="#__codelineno-95-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-95-2" name="__codelineno-95-2" href="#__codelineno-95-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">quadratic</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-95-3" name="__codelineno-95-3" href="#__codelineno-95-3"></a><span class="w">    </span><span class="nx">count</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-95-4" name="__codelineno-95-4" href="#__codelineno-95-4"></a><span class="w">    </span><span class="c1">// Number of iterations is quadratically related to the data size n</span>
+<a id="__codelineno-95-4" name="__codelineno-95-4" href="#__codelineno-95-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the square of the input size n</span>
 <a id="__codelineno-95-5" name="__codelineno-95-5" href="#__codelineno-95-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-95-6" name="__codelineno-95-6" href="#__codelineno-95-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-95-7" name="__codelineno-95-7" href="#__codelineno-95-7"></a><span class="w">            </span><span class="nx">count</span><span class="o">++</span>
@@ -6380,10 +6467,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-96-1" name="__codelineno-96-1" href="#__codelineno-96-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-96-1" name="__codelineno-96-1" href="#__codelineno-96-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-96-2" name="__codelineno-96-2" href="#__codelineno-96-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-96-3" name="__codelineno-96-3" href="#__codelineno-96-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-96-4" name="__codelineno-96-4" href="#__codelineno-96-4"></a><span class="w">    </span><span class="c1">// Number of iterations is quadratically related to the data size n</span>
+<a id="__codelineno-96-4" name="__codelineno-96-4" href="#__codelineno-96-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the square of the input size n</span>
 <a id="__codelineno-96-5" name="__codelineno-96-5" href="#__codelineno-96-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-96-6" name="__codelineno-96-6" href="#__codelineno-96-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-96-7" name="__codelineno-96-7" href="#__codelineno-96-7"></a><span class="w">            </span><span class="bp">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
@@ -6394,10 +6481,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-97-1" name="__codelineno-97-1" href="#__codelineno-97-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-97-1" name="__codelineno-97-1" href="#__codelineno-97-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-97-2" name="__codelineno-97-2" href="#__codelineno-97-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">quadratic</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-97-3" name="__codelineno-97-3" href="#__codelineno-97-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-97-4" name="__codelineno-97-4" href="#__codelineno-97-4"></a><span class="w">    </span><span class="c1">// Number of iterations is quadratically related to the data size n</span>
+<a id="__codelineno-97-4" name="__codelineno-97-4" href="#__codelineno-97-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the square of the input size n</span>
 <a id="__codelineno-97-5" name="__codelineno-97-5" href="#__codelineno-97-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-97-6" name="__codelineno-97-6" href="#__codelineno-97-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-97-7" name="__codelineno-97-7" href="#__codelineno-97-7"></a><span class="w">            </span><span class="nx">count</span><span class="o">++</span><span class="p">;</span>
@@ -6408,10 +6495,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-98-1" name="__codelineno-98-1" href="#__codelineno-98-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-98-1" name="__codelineno-98-1" href="#__codelineno-98-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-98-2" name="__codelineno-98-2" href="#__codelineno-98-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">quadratic</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-98-3" name="__codelineno-98-3" href="#__codelineno-98-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-98-4" name="__codelineno-98-4" href="#__codelineno-98-4"></a><span class="w">    </span><span class="c1">// Number of iterations is quadratically related to the data size n</span>
+<a id="__codelineno-98-4" name="__codelineno-98-4" href="#__codelineno-98-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the square of the input size n</span>
 <a id="__codelineno-98-5" name="__codelineno-98-5" href="#__codelineno-98-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-98-6" name="__codelineno-98-6" href="#__codelineno-98-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-98-7" name="__codelineno-98-7" href="#__codelineno-98-7"></a><span class="w">            </span><span class="nx">count</span><span class="o">++</span><span class="p">;</span>
@@ -6422,10 +6509,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-99-1" name="__codelineno-99-1" href="#__codelineno-99-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-99-1" name="__codelineno-99-1" href="#__codelineno-99-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-99-2" name="__codelineno-99-2" href="#__codelineno-99-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">quadratic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-99-3" name="__codelineno-99-3" href="#__codelineno-99-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
-<a id="__codelineno-99-4" name="__codelineno-99-4" href="#__codelineno-99-4"></a><span class="w">  </span><span class="c1">// Number of iterations is quadratically related to the data size n</span>
+<a id="__codelineno-99-4" name="__codelineno-99-4" href="#__codelineno-99-4"></a><span class="w">  </span><span class="c1">// The number of iterations is proportional to the square of the input size n</span>
 <a id="__codelineno-99-5" name="__codelineno-99-5" href="#__codelineno-99-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-99-6" name="__codelineno-99-6" href="#__codelineno-99-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-99-7" name="__codelineno-99-7" href="#__codelineno-99-7"></a><span class="w">      </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
@@ -6436,10 +6523,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-100-1" name="__codelineno-100-1" href="#__codelineno-100-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-100-1" name="__codelineno-100-1" href="#__codelineno-100-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-100-2" name="__codelineno-100-2" href="#__codelineno-100-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-100-3" name="__codelineno-100-3" href="#__codelineno-100-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-100-4" name="__codelineno-100-4" href="#__codelineno-100-4"></a><span class="w">    </span><span class="c1">// Number of iterations is quadratically related to the data size n</span>
+<a id="__codelineno-100-4" name="__codelineno-100-4" href="#__codelineno-100-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the square of the input size n</span>
 <a id="__codelineno-100-5" name="__codelineno-100-5" href="#__codelineno-100-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-100-6" name="__codelineno-100-6" href="#__codelineno-100-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-100-7" name="__codelineno-100-7" href="#__codelineno-100-7"></a><span class="w">            </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -6450,10 +6537,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-101-1" name="__codelineno-101-1" href="#__codelineno-101-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-101-1" name="__codelineno-101-1" href="#__codelineno-101-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-101-2" name="__codelineno-101-2" href="#__codelineno-101-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-101-3" name="__codelineno-101-3" href="#__codelineno-101-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-101-4" name="__codelineno-101-4" href="#__codelineno-101-4"></a><span class="w">    </span><span class="c1">// Number of iterations is quadratically related to the data size n</span>
+<a id="__codelineno-101-4" name="__codelineno-101-4" href="#__codelineno-101-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the square of the input size n</span>
 <a id="__codelineno-101-5" name="__codelineno-101-5" href="#__codelineno-101-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-101-6" name="__codelineno-101-6" href="#__codelineno-101-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-101-7" name="__codelineno-101-7" href="#__codelineno-101-7"></a><span class="w">            </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
@@ -6464,10 +6551,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-102-1" name="__codelineno-102-1" href="#__codelineno-102-1"></a><span class="cm">/* Exponential order */</span>
+<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-102-1" name="__codelineno-102-1" href="#__codelineno-102-1"></a><span class="cm">/* Quadratic complexity */</span>
 <a id="__codelineno-102-2" name="__codelineno-102-2" href="#__codelineno-102-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-102-3" name="__codelineno-102-3" href="#__codelineno-102-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-102-4" name="__codelineno-102-4" href="#__codelineno-102-4"></a><span class="w">    </span><span class="c1">// Number of iterations is quadratically related to the data size n</span>
+<a id="__codelineno-102-4" name="__codelineno-102-4" href="#__codelineno-102-4"></a><span class="w">    </span><span class="c1">// The number of iterations is proportional to the square of the input size n</span>
 <a id="__codelineno-102-5" name="__codelineno-102-5" href="#__codelineno-102-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-102-6" name="__codelineno-102-6" href="#__codelineno-102-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-102-7" name="__codelineno-102-7" href="#__codelineno-102-7"></a><span class="w">            </span><span class="n">count</span><span class="o">++</span>
@@ -6478,11 +6565,11 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-103-1" name="__codelineno-103-1" href="#__codelineno-103-1"></a><span class="c1">### Quadratic time ###</span>
+<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-103-1" name="__codelineno-103-1" href="#__codelineno-103-1"></a><span class="c1">### Quadratic complexity ###</span>
 <a id="__codelineno-103-2" name="__codelineno-103-2" href="#__codelineno-103-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">quadratic</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-103-3" name="__codelineno-103-3" href="#__codelineno-103-3"></a><span class="w">  </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-103-4" name="__codelineno-103-4" href="#__codelineno-103-4"></a>
-<a id="__codelineno-103-5" name="__codelineno-103-5" href="#__codelineno-103-5"></a><span class="w">  </span><span class="c1"># Number of iterations is quadratically related to the data size n</span>
+<a id="__codelineno-103-5" name="__codelineno-103-5" href="#__codelineno-103-5"></a><span class="w">  </span><span class="c1"># The number of iterations is proportional to the square of the input size n</span>
 <a id="__codelineno-103-6" name="__codelineno-103-6" href="#__codelineno-103-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">n</span>
 <a id="__codelineno-103-7" name="__codelineno-103-7" href="#__codelineno-103-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">n</span>
 <a id="__codelineno-103-8" name="__codelineno-103-8" href="#__codelineno-103-8"></a><span class="w">      </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
@@ -6495,44 +6582,49 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </div>
 </div>
 </div>
-<p>Figure 2-10 compares constant order, linear order, and quadratic order time complexities.</p>
-<p><img alt="Time complexities of constant, linear, and quadratic orders" class="animation-figure" src="../time_complexity.assets/time_complexity_constant_linear_quadratic.png" /></p>
-<p align="center"> Figure 2-10 &nbsp; Time complexities of constant, linear, and quadratic orders </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 477px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20quadratic%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Quadratic%20complexity%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20The%20number%20of%20iterations%20is%20proportional%20to%20the%20square%20of%20the%20input%20size%20n%0A%20%20%20%20for%20i%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20quadratic%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20quadratic%20complexity%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20quadratic%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Quadratic%20complexity%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20The%20number%20of%20iterations%20is%20proportional%20to%20the%20square%20of%20the%20input%20size%20n%0A%20%20%20%20for%20i%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20quadratic%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20quadratic%20complexity%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 2-10 compares constant, linear, and quadratic time complexity.</p>
+<p><img alt="Constant, linear, and quadratic time complexity" class="animation-figure" src="../time_complexity.assets/time_complexity_constant_linear_quadratic.png" /></p>
+<p align="center"> Figure 2-10 &nbsp; Constant, linear, and quadratic time complexity </p>
 
-<p>Taking bubble sort as an example, the outer loop executes <span class="arithmatex">\(n - 1\)</span> times, and the inner loop executes <span class="arithmatex">\(n-1\)</span>, <span class="arithmatex">\(n-2\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, <span class="arithmatex">\(1\)</span> times, averaging <span class="arithmatex">\(n / 2\)</span> times, resulting in a time complexity of <span class="arithmatex">\(O((n - 1) n / 2) = O(n^2)\)</span>:</p>
+<p>In bubble sort, for example, the outer loop runs <span class="arithmatex">\(n - 1\)</span> times, while the inner loop runs <span class="arithmatex">\(n-1\)</span>, <span class="arithmatex">\(n-2\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, and <span class="arithmatex">\(1\)</span> times, averaging <span class="arithmatex">\(n / 2\)</span>. The resulting complexity is <span class="arithmatex">\(O((n - 1) n / 2) = O(n^2)\)</span>:</p>
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 <div class="highlight"><span class="filename">time_complexity.py</span><pre><span></span><code><a id="__codelineno-104-1" name="__codelineno-104-1" href="#__codelineno-104-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">bubble_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">])</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-104-2" name="__codelineno-104-2" href="#__codelineno-104-2"></a><span class="w">    </span><span class="sd">"""Quadratic order (bubble sort)"""</span>
+<a id="__codelineno-104-2" name="__codelineno-104-2" href="#__codelineno-104-2"></a><span class="w">    </span><span class="sd">"""Quadratic complexity (bubble sort)"""</span>
 <a id="__codelineno-104-3" name="__codelineno-104-3" href="#__codelineno-104-3"></a>    <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Counter</span>
-<a id="__codelineno-104-4" name="__codelineno-104-4" href="#__codelineno-104-4"></a>    <span class="c1"># Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-104-4" name="__codelineno-104-4" href="#__codelineno-104-4"></a>    <span class="c1"># Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-104-5" name="__codelineno-104-5" href="#__codelineno-104-5"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-<a id="__codelineno-104-6" name="__codelineno-104-6" href="#__codelineno-104-6"></a>        <span class="c1"># Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-104-6" name="__codelineno-104-6" href="#__codelineno-104-6"></a>        <span class="c1"># Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-104-7" name="__codelineno-104-7" href="#__codelineno-104-7"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
 <a id="__codelineno-104-8" name="__codelineno-104-8" href="#__codelineno-104-8"></a>            <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]:</span>
 <a id="__codelineno-104-9" name="__codelineno-104-9" href="#__codelineno-104-9"></a>                <span class="c1"># Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-104-10" name="__codelineno-104-10" href="#__codelineno-104-10"></a>                <span class="n">tmp</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-104-11" name="__codelineno-104-11" href="#__codelineno-104-11"></a>                <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-104-12" name="__codelineno-104-12" href="#__codelineno-104-12"></a>                <span class="n">nums</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">tmp</span>
-<a id="__codelineno-104-13" name="__codelineno-104-13" href="#__codelineno-104-13"></a>                <span class="n">count</span> <span class="o">+=</span> <span class="mi">3</span>  <span class="c1"># Element swap includes 3 unit operations</span>
+<a id="__codelineno-104-13" name="__codelineno-104-13" href="#__codelineno-104-13"></a>                <span class="n">count</span> <span class="o">+=</span> <span class="mi">3</span>  <span class="c1"># Swapping two elements takes 3 elementary operations</span>
 <a id="__codelineno-104-14" name="__codelineno-104-14" href="#__codelineno-104-14"></a>    <span class="k">return</span> <span class="n">count</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-105-1" name="__codelineno-105-1" href="#__codelineno-105-1"></a><span class="cm">/* Quadratic order (bubble sort) */</span>
+<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-105-1" name="__codelineno-105-1" href="#__codelineno-105-1"></a><span class="cm">/* Quadratic complexity (bubble sort) */</span>
 <a id="__codelineno-105-2" name="__codelineno-105-2" href="#__codelineno-105-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">bubbleSort</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-105-3" name="__codelineno-105-3" href="#__codelineno-105-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Counter</span>
-<a id="__codelineno-105-4" name="__codelineno-105-4" href="#__codelineno-105-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-105-4" name="__codelineno-105-4" href="#__codelineno-105-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-105-5" name="__codelineno-105-5" href="#__codelineno-105-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-105-6" name="__codelineno-105-6" href="#__codelineno-105-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-105-6" name="__codelineno-105-6" href="#__codelineno-105-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-105-7" name="__codelineno-105-7" href="#__codelineno-105-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-105-8" name="__codelineno-105-8" href="#__codelineno-105-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-105-9" name="__codelineno-105-9" href="#__codelineno-105-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-105-10" name="__codelineno-105-10" href="#__codelineno-105-10"></a><span class="w">                </span><span class="kt">int</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-105-11" name="__codelineno-105-11" href="#__codelineno-105-11"></a><span class="w">                </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-105-12" name="__codelineno-105-12" href="#__codelineno-105-12"></a><span class="w">                </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
-<a id="__codelineno-105-13" name="__codelineno-105-13" href="#__codelineno-105-13"></a><span class="w">                </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// Element swap includes 3 unit operations</span>
+<a id="__codelineno-105-13" name="__codelineno-105-13" href="#__codelineno-105-13"></a><span class="w">                </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// Swapping two elements takes 3 elementary operations</span>
 <a id="__codelineno-105-14" name="__codelineno-105-14" href="#__codelineno-105-14"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-105-15" name="__codelineno-105-15" href="#__codelineno-105-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-105-16" name="__codelineno-105-16" href="#__codelineno-105-16"></a><span class="w">    </span><span class="p">}</span>
@@ -6541,19 +6633,19 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-106-1" name="__codelineno-106-1" href="#__codelineno-106-1"></a><span class="cm">/* Quadratic order (bubble sort) */</span>
+<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-106-1" name="__codelineno-106-1" href="#__codelineno-106-1"></a><span class="cm">/* Quadratic complexity (bubble sort) */</span>
 <a id="__codelineno-106-2" name="__codelineno-106-2" href="#__codelineno-106-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">bubbleSort</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-106-3" name="__codelineno-106-3" href="#__codelineno-106-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Counter</span>
-<a id="__codelineno-106-4" name="__codelineno-106-4" href="#__codelineno-106-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-106-4" name="__codelineno-106-4" href="#__codelineno-106-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-106-5" name="__codelineno-106-5" href="#__codelineno-106-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-106-6" name="__codelineno-106-6" href="#__codelineno-106-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-106-6" name="__codelineno-106-6" href="#__codelineno-106-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-106-7" name="__codelineno-106-7" href="#__codelineno-106-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-106-8" name="__codelineno-106-8" href="#__codelineno-106-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-106-9" name="__codelineno-106-9" href="#__codelineno-106-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-106-10" name="__codelineno-106-10" href="#__codelineno-106-10"></a><span class="w">                </span><span class="kt">int</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-106-11" name="__codelineno-106-11" href="#__codelineno-106-11"></a><span class="w">                </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-106-12" name="__codelineno-106-12" href="#__codelineno-106-12"></a><span class="w">                </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
-<a id="__codelineno-106-13" name="__codelineno-106-13" href="#__codelineno-106-13"></a><span class="w">                </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// Element swap includes 3 unit operations</span>
+<a id="__codelineno-106-13" name="__codelineno-106-13" href="#__codelineno-106-13"></a><span class="w">                </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// Swapping two elements takes 3 elementary operations</span>
 <a id="__codelineno-106-14" name="__codelineno-106-14" href="#__codelineno-106-14"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-106-15" name="__codelineno-106-15" href="#__codelineno-106-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-106-16" name="__codelineno-106-16" href="#__codelineno-106-16"></a><span class="w">    </span><span class="p">}</span>
@@ -6562,17 +6654,17 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-107-1" name="__codelineno-107-1" href="#__codelineno-107-1"></a><span class="cm">/* Quadratic order (bubble sort) */</span>
+<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-107-1" name="__codelineno-107-1" href="#__codelineno-107-1"></a><span class="cm">/* Quadratic complexity (bubble sort) */</span>
 <a id="__codelineno-107-2" name="__codelineno-107-2" href="#__codelineno-107-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">BubbleSort</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-107-3" name="__codelineno-107-3" href="#__codelineno-107-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">  </span><span class="c1">// Counter</span>
-<a id="__codelineno-107-4" name="__codelineno-107-4" href="#__codelineno-107-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-107-4" name="__codelineno-107-4" href="#__codelineno-107-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-107-5" name="__codelineno-107-5" href="#__codelineno-107-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-107-6" name="__codelineno-107-6" href="#__codelineno-107-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-107-6" name="__codelineno-107-6" href="#__codelineno-107-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-107-7" name="__codelineno-107-7" href="#__codelineno-107-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-107-8" name="__codelineno-107-8" href="#__codelineno-107-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-107-9" name="__codelineno-107-9" href="#__codelineno-107-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-107-10" name="__codelineno-107-10" href="#__codelineno-107-10"></a><span class="w">                </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
-<a id="__codelineno-107-11" name="__codelineno-107-11" href="#__codelineno-107-11"></a><span class="w">                </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w">  </span><span class="c1">// Element swap includes 3 unit operations</span>
+<a id="__codelineno-107-11" name="__codelineno-107-11" href="#__codelineno-107-11"></a><span class="w">                </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w">  </span><span class="c1">// Swapping two elements takes 3 elementary operations</span>
 <a id="__codelineno-107-12" name="__codelineno-107-12" href="#__codelineno-107-12"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-107-13" name="__codelineno-107-13" href="#__codelineno-107-13"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-107-14" name="__codelineno-107-14" href="#__codelineno-107-14"></a><span class="w">    </span><span class="p">}</span>
@@ -6581,19 +6673,19 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-108-1" name="__codelineno-108-1" href="#__codelineno-108-1"></a><span class="cm">/* Quadratic order (bubble sort) */</span>
+<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-108-1" name="__codelineno-108-1" href="#__codelineno-108-1"></a><span class="cm">/* Quadratic complexity (bubble sort) */</span>
 <a id="__codelineno-108-2" name="__codelineno-108-2" href="#__codelineno-108-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">bubbleSort</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-108-3" name="__codelineno-108-3" href="#__codelineno-108-3"></a><span class="w">    </span><span class="nx">count</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1">// Counter</span>
-<a id="__codelineno-108-4" name="__codelineno-108-4" href="#__codelineno-108-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-108-4" name="__codelineno-108-4" href="#__codelineno-108-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-108-5" name="__codelineno-108-5" href="#__codelineno-108-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-108-6" name="__codelineno-108-6" href="#__codelineno-108-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-108-6" name="__codelineno-108-6" href="#__codelineno-108-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-108-7" name="__codelineno-108-7" href="#__codelineno-108-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-108-8" name="__codelineno-108-8" href="#__codelineno-108-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-108-9" name="__codelineno-108-9" href="#__codelineno-108-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-108-10" name="__codelineno-108-10" href="#__codelineno-108-10"></a><span class="w">                </span><span class="nx">tmp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span>
 <a id="__codelineno-108-11" name="__codelineno-108-11" href="#__codelineno-108-11"></a><span class="w">                </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-108-12" name="__codelineno-108-12" href="#__codelineno-108-12"></a><span class="w">                </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span>
-<a id="__codelineno-108-13" name="__codelineno-108-13" href="#__codelineno-108-13"></a><span class="w">                </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="c1">// Element swap includes 3 unit operations</span>
+<a id="__codelineno-108-13" name="__codelineno-108-13" href="#__codelineno-108-13"></a><span class="w">                </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="c1">// Swapping two elements takes 3 elementary operations</span>
 <a id="__codelineno-108-14" name="__codelineno-108-14" href="#__codelineno-108-14"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-108-15" name="__codelineno-108-15" href="#__codelineno-108-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-108-16" name="__codelineno-108-16" href="#__codelineno-108-16"></a><span class="w">    </span><span class="p">}</span>
@@ -6602,19 +6694,19 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-109-1" name="__codelineno-109-1" href="#__codelineno-109-1"></a><span class="cm">/* Quadratic order (bubble sort) */</span>
+<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-109-1" name="__codelineno-109-1" href="#__codelineno-109-1"></a><span class="cm">/* Quadratic complexity (bubble sort) */</span>
 <a id="__codelineno-109-2" name="__codelineno-109-2" href="#__codelineno-109-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">bubbleSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-109-3" name="__codelineno-109-3" href="#__codelineno-109-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1">// Counter</span>
-<a id="__codelineno-109-4" name="__codelineno-109-4" href="#__codelineno-109-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-109-4" name="__codelineno-109-4" href="#__codelineno-109-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-109-5" name="__codelineno-109-5" href="#__codelineno-109-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="bp">dropFirst</span><span class="p">().</span><span class="n">reversed</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-109-6" name="__codelineno-109-6" href="#__codelineno-109-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-109-6" name="__codelineno-109-6" href="#__codelineno-109-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-109-7" name="__codelineno-109-7" href="#__codelineno-109-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-109-8" name="__codelineno-109-8" href="#__codelineno-109-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-109-9" name="__codelineno-109-9" href="#__codelineno-109-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-109-10" name="__codelineno-109-10" href="#__codelineno-109-10"></a><span class="w">                </span><span class="kd">let</span><span class="w"> </span><span class="nv">tmp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-109-11" name="__codelineno-109-11" href="#__codelineno-109-11"></a><span class="w">                </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-109-12" name="__codelineno-109-12" href="#__codelineno-109-12"></a><span class="w">                </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">tmp</span>
-<a id="__codelineno-109-13" name="__codelineno-109-13" href="#__codelineno-109-13"></a><span class="w">                </span><span class="bp">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="c1">// Element swap includes 3 unit operations</span>
+<a id="__codelineno-109-13" name="__codelineno-109-13" href="#__codelineno-109-13"></a><span class="w">                </span><span class="bp">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="c1">// Swapping two elements takes 3 elementary operations</span>
 <a id="__codelineno-109-14" name="__codelineno-109-14" href="#__codelineno-109-14"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-109-15" name="__codelineno-109-15" href="#__codelineno-109-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-109-16" name="__codelineno-109-16" href="#__codelineno-109-16"></a><span class="w">    </span><span class="p">}</span>
@@ -6623,19 +6715,19 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-110-1" name="__codelineno-110-1" href="#__codelineno-110-1"></a><span class="cm">/* Quadratic order (bubble sort) */</span>
+<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-110-1" name="__codelineno-110-1" href="#__codelineno-110-1"></a><span class="cm">/* Quadratic complexity (bubble sort) */</span>
 <a id="__codelineno-110-2" name="__codelineno-110-2" href="#__codelineno-110-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">bubbleSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-110-3" name="__codelineno-110-3" href="#__codelineno-110-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Counter</span>
-<a id="__codelineno-110-4" name="__codelineno-110-4" href="#__codelineno-110-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-110-4" name="__codelineno-110-4" href="#__codelineno-110-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-110-5" name="__codelineno-110-5" href="#__codelineno-110-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-110-6" name="__codelineno-110-6" href="#__codelineno-110-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-110-6" name="__codelineno-110-6" href="#__codelineno-110-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-110-7" name="__codelineno-110-7" href="#__codelineno-110-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-110-8" name="__codelineno-110-8" href="#__codelineno-110-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-110-9" name="__codelineno-110-9" href="#__codelineno-110-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-110-10" name="__codelineno-110-10" href="#__codelineno-110-10"></a><span class="w">                </span><span class="kd">let</span><span class="w"> </span><span class="nx">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-110-11" name="__codelineno-110-11" href="#__codelineno-110-11"></a><span class="w">                </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-110-12" name="__codelineno-110-12" href="#__codelineno-110-12"></a><span class="w">                </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span><span class="p">;</span>
-<a id="__codelineno-110-13" name="__codelineno-110-13" href="#__codelineno-110-13"></a><span class="w">                </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mf">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// Element swap includes 3 unit operations</span>
+<a id="__codelineno-110-13" name="__codelineno-110-13" href="#__codelineno-110-13"></a><span class="w">                </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mf">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// Swapping two elements takes 3 elementary operations</span>
 <a id="__codelineno-110-14" name="__codelineno-110-14" href="#__codelineno-110-14"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-110-15" name="__codelineno-110-15" href="#__codelineno-110-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-110-16" name="__codelineno-110-16" href="#__codelineno-110-16"></a><span class="w">    </span><span class="p">}</span>
@@ -6644,19 +6736,19 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-111-1" name="__codelineno-111-1" href="#__codelineno-111-1"></a><span class="cm">/* Quadratic order (bubble sort) */</span>
+<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-111-1" name="__codelineno-111-1" href="#__codelineno-111-1"></a><span class="cm">/* Quadratic complexity (bubble sort) */</span>
 <a id="__codelineno-111-2" name="__codelineno-111-2" href="#__codelineno-111-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">bubbleSort</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-111-3" name="__codelineno-111-3" href="#__codelineno-111-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Counter</span>
-<a id="__codelineno-111-4" name="__codelineno-111-4" href="#__codelineno-111-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-111-4" name="__codelineno-111-4" href="#__codelineno-111-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-111-5" name="__codelineno-111-5" href="#__codelineno-111-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-111-6" name="__codelineno-111-6" href="#__codelineno-111-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-111-6" name="__codelineno-111-6" href="#__codelineno-111-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-111-7" name="__codelineno-111-7" href="#__codelineno-111-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-111-8" name="__codelineno-111-8" href="#__codelineno-111-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-111-9" name="__codelineno-111-9" href="#__codelineno-111-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-111-10" name="__codelineno-111-10" href="#__codelineno-111-10"></a><span class="w">                </span><span class="kd">let</span><span class="w"> </span><span class="nx">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-111-11" name="__codelineno-111-11" href="#__codelineno-111-11"></a><span class="w">                </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-111-12" name="__codelineno-111-12" href="#__codelineno-111-12"></a><span class="w">                </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span><span class="p">;</span>
-<a id="__codelineno-111-13" name="__codelineno-111-13" href="#__codelineno-111-13"></a><span class="w">                </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mf">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// Element swap includes 3 unit operations</span>
+<a id="__codelineno-111-13" name="__codelineno-111-13" href="#__codelineno-111-13"></a><span class="w">                </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mf">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// Swapping two elements takes 3 elementary operations</span>
 <a id="__codelineno-111-14" name="__codelineno-111-14" href="#__codelineno-111-14"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-111-15" name="__codelineno-111-15" href="#__codelineno-111-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-111-16" name="__codelineno-111-16" href="#__codelineno-111-16"></a><span class="w">    </span><span class="p">}</span>
@@ -6665,19 +6757,19 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-112-1" name="__codelineno-112-1" href="#__codelineno-112-1"></a><span class="cm">/* Quadratic order (bubble sort) */</span>
+<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-112-1" name="__codelineno-112-1" href="#__codelineno-112-1"></a><span class="cm">/* Quadratic complexity (bubble sort) */</span>
 <a id="__codelineno-112-2" name="__codelineno-112-2" href="#__codelineno-112-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">bubbleSort</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-112-3" name="__codelineno-112-3" href="#__codelineno-112-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Counter</span>
-<a id="__codelineno-112-4" name="__codelineno-112-4" href="#__codelineno-112-4"></a><span class="w">  </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-112-4" name="__codelineno-112-4" href="#__codelineno-112-4"></a><span class="w">  </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-112-5" name="__codelineno-112-5" href="#__codelineno-112-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-112-6" name="__codelineno-112-6" href="#__codelineno-112-6"></a><span class="w">    </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-112-6" name="__codelineno-112-6" href="#__codelineno-112-6"></a><span class="w">    </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-112-7" name="__codelineno-112-7" href="#__codelineno-112-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-112-8" name="__codelineno-112-8" href="#__codelineno-112-8"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-112-9" name="__codelineno-112-9" href="#__codelineno-112-9"></a><span class="w">        </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-112-10" name="__codelineno-112-10" href="#__codelineno-112-10"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-112-11" name="__codelineno-112-11" href="#__codelineno-112-11"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
 <a id="__codelineno-112-12" name="__codelineno-112-12" href="#__codelineno-112-12"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
-<a id="__codelineno-112-13" name="__codelineno-112-13" href="#__codelineno-112-13"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="m">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// Element swap includes 3 unit operations</span>
+<a id="__codelineno-112-13" name="__codelineno-112-13" href="#__codelineno-112-13"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="m">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// Swapping two elements takes 3 elementary operations</span>
 <a id="__codelineno-112-14" name="__codelineno-112-14" href="#__codelineno-112-14"></a><span class="w">      </span><span class="p">}</span>
 <a id="__codelineno-112-15" name="__codelineno-112-15" href="#__codelineno-112-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-112-16" name="__codelineno-112-16" href="#__codelineno-112-16"></a><span class="w">  </span><span class="p">}</span>
@@ -6686,20 +6778,20 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-113-1" name="__codelineno-113-1" href="#__codelineno-113-1"></a><span class="cm">/* Quadratic order (bubble sort) */</span>
+<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-113-1" name="__codelineno-113-1" href="#__codelineno-113-1"></a><span class="cm">/* Quadratic complexity (bubble sort) */</span>
 <a id="__codelineno-113-2" name="__codelineno-113-2" href="#__codelineno-113-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">bubble_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-113-3" name="__codelineno-113-3" href="#__codelineno-113-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Counter</span>
 <a id="__codelineno-113-4" name="__codelineno-113-4" href="#__codelineno-113-4"></a>
-<a id="__codelineno-113-5" name="__codelineno-113-5" href="#__codelineno-113-5"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-113-5" name="__codelineno-113-5" href="#__codelineno-113-5"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-113-6" name="__codelineno-113-6" href="#__codelineno-113-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="o">..</span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()).</span><span class="n">rev</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-113-7" name="__codelineno-113-7" href="#__codelineno-113-7"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-113-7" name="__codelineno-113-7" href="#__codelineno-113-7"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-113-8" name="__codelineno-113-8" href="#__codelineno-113-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">i</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-113-9" name="__codelineno-113-9" href="#__codelineno-113-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-113-10" name="__codelineno-113-10" href="#__codelineno-113-10"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-113-11" name="__codelineno-113-11" href="#__codelineno-113-11"></a><span class="w">                </span><span class="kd">let</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-113-12" name="__codelineno-113-12" href="#__codelineno-113-12"></a><span class="w">                </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-113-13" name="__codelineno-113-13" href="#__codelineno-113-13"></a><span class="w">                </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
-<a id="__codelineno-113-14" name="__codelineno-113-14" href="#__codelineno-113-14"></a><span class="w">                </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// Element swap includes 3 unit operations</span>
+<a id="__codelineno-113-14" name="__codelineno-113-14" href="#__codelineno-113-14"></a><span class="w">                </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// Swapping two elements takes 3 elementary operations</span>
 <a id="__codelineno-113-15" name="__codelineno-113-15" href="#__codelineno-113-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-113-16" name="__codelineno-113-16" href="#__codelineno-113-16"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-113-17" name="__codelineno-113-17" href="#__codelineno-113-17"></a><span class="w">    </span><span class="p">}</span>
@@ -6708,19 +6800,19 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-114-1" name="__codelineno-114-1" href="#__codelineno-114-1"></a><span class="cm">/* Quadratic order (bubble sort) */</span>
+<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-114-1" name="__codelineno-114-1" href="#__codelineno-114-1"></a><span class="cm">/* Quadratic complexity (bubble sort) */</span>
 <a id="__codelineno-114-2" name="__codelineno-114-2" href="#__codelineno-114-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">bubbleSort</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-114-3" name="__codelineno-114-3" href="#__codelineno-114-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Counter</span>
-<a id="__codelineno-114-4" name="__codelineno-114-4" href="#__codelineno-114-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-114-4" name="__codelineno-114-4" href="#__codelineno-114-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-114-5" name="__codelineno-114-5" href="#__codelineno-114-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-114-6" name="__codelineno-114-6" href="#__codelineno-114-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-114-6" name="__codelineno-114-6" href="#__codelineno-114-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-114-7" name="__codelineno-114-7" href="#__codelineno-114-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-114-8" name="__codelineno-114-8" href="#__codelineno-114-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-114-9" name="__codelineno-114-9" href="#__codelineno-114-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-114-10" name="__codelineno-114-10" href="#__codelineno-114-10"></a><span class="w">                </span><span class="kt">int</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-114-11" name="__codelineno-114-11" href="#__codelineno-114-11"></a><span class="w">                </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-114-12" name="__codelineno-114-12" href="#__codelineno-114-12"></a><span class="w">                </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
-<a id="__codelineno-114-13" name="__codelineno-114-13" href="#__codelineno-114-13"></a><span class="w">                </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// Element swap includes 3 unit operations</span>
+<a id="__codelineno-114-13" name="__codelineno-114-13" href="#__codelineno-114-13"></a><span class="w">                </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="c1">// Swapping two elements takes 3 elementary operations</span>
 <a id="__codelineno-114-14" name="__codelineno-114-14" href="#__codelineno-114-14"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-114-15" name="__codelineno-114-15" href="#__codelineno-114-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-114-16" name="__codelineno-114-16" href="#__codelineno-114-16"></a><span class="w">    </span><span class="p">}</span>
@@ -6729,19 +6821,19 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-115-1" name="__codelineno-115-1" href="#__codelineno-115-1"></a><span class="cm">/* Quadratic order (bubble sort) */</span>
+<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-115-1" name="__codelineno-115-1" href="#__codelineno-115-1"></a><span class="cm">/* Quadratic complexity (bubble sort) */</span>
 <a id="__codelineno-115-2" name="__codelineno-115-2" href="#__codelineno-115-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">bubbleSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-115-3" name="__codelineno-115-3" href="#__codelineno-115-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Counter</span>
-<a id="__codelineno-115-4" name="__codelineno-115-4" href="#__codelineno-115-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-115-4" name="__codelineno-115-4" href="#__codelineno-115-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-115-5" name="__codelineno-115-5" href="#__codelineno-115-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="n">downTo</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-115-6" name="__codelineno-115-6" href="#__codelineno-115-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-115-6" name="__codelineno-115-6" href="#__codelineno-115-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-115-7" name="__codelineno-115-7" href="#__codelineno-115-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-115-8" name="__codelineno-115-8" href="#__codelineno-115-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-115-9" name="__codelineno-115-9" href="#__codelineno-115-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-115-10" name="__codelineno-115-10" href="#__codelineno-115-10"></a><span class="w">                </span><span class="kd">val</span><span class="w"> </span><span class="nv">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-115-11" name="__codelineno-115-11" href="#__codelineno-115-11"></a><span class="w">                </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="o">]</span>
 <a id="__codelineno-115-12" name="__codelineno-115-12" href="#__codelineno-115-12"></a><span class="w">                </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span>
-<a id="__codelineno-115-13" name="__codelineno-115-13" href="#__codelineno-115-13"></a><span class="w">                </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="m">3</span><span class="w"> </span><span class="c1">// Element swap includes 3 unit operations</span>
+<a id="__codelineno-115-13" name="__codelineno-115-13" href="#__codelineno-115-13"></a><span class="w">                </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="m">3</span><span class="w"> </span><span class="c1">// Swapping two elements takes 3 elementary operations</span>
 <a id="__codelineno-115-14" name="__codelineno-115-14" href="#__codelineno-115-14"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-115-15" name="__codelineno-115-15" href="#__codelineno-115-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-115-16" name="__codelineno-115-16" href="#__codelineno-115-16"></a><span class="w">    </span><span class="p">}</span>
@@ -6750,20 +6842,20 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-116-1" name="__codelineno-116-1" href="#__codelineno-116-1"></a><span class="c1">### Quadratic time (bubble sort) ###</span>
+<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-116-1" name="__codelineno-116-1" href="#__codelineno-116-1"></a><span class="c1">### Quadratic complexity (bubble sort)###</span>
 <a id="__codelineno-116-2" name="__codelineno-116-2" href="#__codelineno-116-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">bubble_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
 <a id="__codelineno-116-3" name="__codelineno-116-3" href="#__codelineno-116-3"></a><span class="w">  </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w">  </span><span class="c1"># Counter</span>
 <a id="__codelineno-116-4" name="__codelineno-116-4" href="#__codelineno-116-4"></a>
-<a id="__codelineno-116-5" name="__codelineno-116-5" href="#__codelineno-116-5"></a><span class="w">  </span><span class="c1"># Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-116-5" name="__codelineno-116-5" href="#__codelineno-116-5"></a><span class="w">  </span><span class="c1"># Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-116-6" name="__codelineno-116-6" href="#__codelineno-116-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">downto</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-116-7" name="__codelineno-116-7" href="#__codelineno-116-7"></a><span class="w">    </span><span class="c1"># Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-116-7" name="__codelineno-116-7" href="#__codelineno-116-7"></a><span class="w">    </span><span class="c1"># Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-116-8" name="__codelineno-116-8" href="#__codelineno-116-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">i</span>
 <a id="__codelineno-116-9" name="__codelineno-116-9" href="#__codelineno-116-9"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span>
 <a id="__codelineno-116-10" name="__codelineno-116-10" href="#__codelineno-116-10"></a><span class="w">        </span><span class="c1"># Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-116-11" name="__codelineno-116-11" href="#__codelineno-116-11"></a><span class="w">        </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-116-12" name="__codelineno-116-12" href="#__codelineno-116-12"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span>
 <a id="__codelineno-116-13" name="__codelineno-116-13" href="#__codelineno-116-13"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span>
-<a id="__codelineno-116-14" name="__codelineno-116-14" href="#__codelineno-116-14"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="c1"># Element swap includes 3 unit operations</span>
+<a id="__codelineno-116-14" name="__codelineno-116-14" href="#__codelineno-116-14"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="c1"># Swapping two elements takes 3 elementary operations</span>
 <a id="__codelineno-116-15" name="__codelineno-116-15" href="#__codelineno-116-15"></a><span class="w">      </span><span class="k">end</span>
 <a id="__codelineno-116-16" name="__codelineno-116-16" href="#__codelineno-116-16"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-116-17" name="__codelineno-116-17" href="#__codelineno-116-17"></a><span class="w">  </span><span class="k">end</span>
@@ -6774,17 +6866,22 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </div>
 </div>
 </div>
-<h3 id="4-exponential-order-o2n">4. &nbsp; Exponential Order <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#4-exponential-order-o2n" title="Permanent link">&para;</a></h3>
-<p>Biological "cell division" is a typical example of exponential order growth: the initial state is <span class="arithmatex">\(1\)</span> cell, after one round of division it becomes <span class="arithmatex">\(2\)</span>, after two rounds it becomes <span class="arithmatex">\(4\)</span>, and so on; after <span class="arithmatex">\(n\)</span> rounds of division there are <span class="arithmatex">\(2^n\)</span> cells.</p>
-<p>Figure 2-11 and the following code simulate the cell division process, with a time complexity of <span class="arithmatex">\(O(2^n)\)</span>. Note that the input <span class="arithmatex">\(n\)</span> represents the number of division rounds, and the return value <code>count</code> represents the total number of divisions.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20bubble_sort%28nums%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Quadratic%20complexity%20%28bubble%20sort%29%22%22%22%0A%20%20%20%20count%20%3D%200%20%20%23%20Counter%0A%20%20%20%20%23%20Outer%20loop%3A%20the%20unsorted%20range%20is%20%5B0,%20i%5D%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%20-%201,%200,%20-1%29%3A%0A%20%20%20%20%20%20%20%20%23%20Inner%20loop%3A%20move%20the%20largest%20element%20in%20the%20unsorted%20range%20%5B0,%20i%5D%20to%20its%20right%20end%20through%20swaps%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28i%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20nums%5Bj%5D%20%3E%20nums%5Bj%20%2B%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Swap%20nums%5Bj%5D%20and%20nums%5Bj%20%2B%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20tmp%20%3D%20nums%5Bj%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bj%5D%20%3D%20nums%5Bj%20%2B%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bj%20%2B%201%5D%20%3D%20tmp%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20count%20%2B%3D%203%20%20%23%20Swapping%20two%20elements%20takes%203%20elementary%20operations%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20nums%20%3D%20%5Bi%20for%20i%20in%20range%28n,%200,%20-1%29%5D%20%20%23%20%5Bn,%20n-1,%20...,%202,%201%5D%0A%20%20%20%20count%20%3D%20bubble_sort%28nums%29%0A%20%20%20%20print%28%22Operation%20count%20for%20quadratic%20complexity%20%28bubble%20sort%29%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20bubble_sort%28nums%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Quadratic%20complexity%20%28bubble%20sort%29%22%22%22%0A%20%20%20%20count%20%3D%200%20%20%23%20Counter%0A%20%20%20%20%23%20Outer%20loop%3A%20the%20unsorted%20range%20is%20%5B0,%20i%5D%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%20-%201,%200,%20-1%29%3A%0A%20%20%20%20%20%20%20%20%23%20Inner%20loop%3A%20move%20the%20largest%20element%20in%20the%20unsorted%20range%20%5B0,%20i%5D%20to%20its%20right%20end%20through%20swaps%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28i%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20nums%5Bj%5D%20%3E%20nums%5Bj%20%2B%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Swap%20nums%5Bj%5D%20and%20nums%5Bj%20%2B%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20tmp%20%3D%20nums%5Bj%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bj%5D%20%3D%20nums%5Bj%20%2B%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bj%20%2B%201%5D%20%3D%20tmp%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20count%20%2B%3D%203%20%20%23%20Swapping%20two%20elements%20takes%203%20elementary%20operations%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20nums%20%3D%20%5Bi%20for%20i%20in%20range%28n,%200,%20-1%29%5D%20%20%23%20%5Bn,%20n-1,%20...,%202,%201%5D%0A%20%20%20%20count%20%3D%20bubble_sort%28nums%29%0A%20%20%20%20print%28%22Operation%20count%20for%20quadratic%20complexity%20%28bubble%20sort%29%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="4-exponential-time-o2n">4. &nbsp; Exponential time <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#4-exponential-time-o2n" title="Permanent link">&para;</a></h3>
+<p>Cell division is a classic example of exponential growth. Starting with <span class="arithmatex">\(1\)</span> cell, one round produces <span class="arithmatex">\(2\)</span>, two rounds produce <span class="arithmatex">\(4\)</span>, and after <span class="arithmatex">\(n\)</span> rounds there are <span class="arithmatex">\(2^n\)</span> cells.</p>
+<p>The figure and code below simulate cell division in <span class="arithmatex">\(O(2^n)\)</span> time. The input <span class="arithmatex">\(n\)</span> is the number of rounds, while the returned <code>count</code> is the total number of individual divisions.</p>
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 <div class="highlight"><span class="filename">time_complexity.py</span><pre><span></span><code><a id="__codelineno-117-1" name="__codelineno-117-1" href="#__codelineno-117-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">exponential</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-117-2" name="__codelineno-117-2" href="#__codelineno-117-2"></a><span class="w">    </span><span class="sd">"""Exponential order (loop implementation)"""</span>
+<a id="__codelineno-117-2" name="__codelineno-117-2" href="#__codelineno-117-2"></a><span class="w">    </span><span class="sd">"""Exponential complexity (iterative implementation)"""</span>
 <a id="__codelineno-117-3" name="__codelineno-117-3" href="#__codelineno-117-3"></a>    <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-117-4" name="__codelineno-117-4" href="#__codelineno-117-4"></a>    <span class="n">base</span> <span class="o">=</span> <span class="mi">1</span>
-<a id="__codelineno-117-5" name="__codelineno-117-5" href="#__codelineno-117-5"></a>    <span class="c1"># Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
+<a id="__codelineno-117-5" name="__codelineno-117-5" href="#__codelineno-117-5"></a>    <span class="c1"># Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
 <a id="__codelineno-117-6" name="__codelineno-117-6" href="#__codelineno-117-6"></a>    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-117-7" name="__codelineno-117-7" href="#__codelineno-117-7"></a>        <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">base</span><span class="p">):</span>
 <a id="__codelineno-117-8" name="__codelineno-117-8" href="#__codelineno-117-8"></a>            <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
@@ -6794,10 +6891,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-118-1" name="__codelineno-118-1" href="#__codelineno-118-1"></a><span class="cm">/* Exponential order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-118-1" name="__codelineno-118-1" href="#__codelineno-118-1"></a><span class="cm">/* Exponential complexity (iterative implementation) */</span>
 <a id="__codelineno-118-2" name="__codelineno-118-2" href="#__codelineno-118-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">exponential</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-118-3" name="__codelineno-118-3" href="#__codelineno-118-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-118-4" name="__codelineno-118-4" href="#__codelineno-118-4"></a><span class="w">    </span><span class="c1">// Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
+<a id="__codelineno-118-4" name="__codelineno-118-4" href="#__codelineno-118-4"></a><span class="w">    </span><span class="c1">// Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
 <a id="__codelineno-118-5" name="__codelineno-118-5" href="#__codelineno-118-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-118-6" name="__codelineno-118-6" href="#__codelineno-118-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-118-7" name="__codelineno-118-7" href="#__codelineno-118-7"></a><span class="w">            </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
@@ -6810,10 +6907,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-119-1" name="__codelineno-119-1" href="#__codelineno-119-1"></a><span class="cm">/* Exponential order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-119-1" name="__codelineno-119-1" href="#__codelineno-119-1"></a><span class="cm">/* Exponential complexity (iterative implementation) */</span>
 <a id="__codelineno-119-2" name="__codelineno-119-2" href="#__codelineno-119-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">exponential</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-119-3" name="__codelineno-119-3" href="#__codelineno-119-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-119-4" name="__codelineno-119-4" href="#__codelineno-119-4"></a><span class="w">    </span><span class="c1">// Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
+<a id="__codelineno-119-4" name="__codelineno-119-4" href="#__codelineno-119-4"></a><span class="w">    </span><span class="c1">// Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
 <a id="__codelineno-119-5" name="__codelineno-119-5" href="#__codelineno-119-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-119-6" name="__codelineno-119-6" href="#__codelineno-119-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-119-7" name="__codelineno-119-7" href="#__codelineno-119-7"></a><span class="w">            </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
@@ -6826,10 +6923,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-120-1" name="__codelineno-120-1" href="#__codelineno-120-1"></a><span class="cm">/* Exponential order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-120-1" name="__codelineno-120-1" href="#__codelineno-120-1"></a><span class="cm">/* Exponential complexity (iterative implementation) */</span>
 <a id="__codelineno-120-2" name="__codelineno-120-2" href="#__codelineno-120-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Exponential</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-120-3" name="__codelineno-120-3" href="#__codelineno-120-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">bas</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-120-4" name="__codelineno-120-4" href="#__codelineno-120-4"></a><span class="w">    </span><span class="c1">// Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
+<a id="__codelineno-120-4" name="__codelineno-120-4" href="#__codelineno-120-4"></a><span class="w">    </span><span class="c1">// Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
 <a id="__codelineno-120-5" name="__codelineno-120-5" href="#__codelineno-120-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-120-6" name="__codelineno-120-6" href="#__codelineno-120-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">bas</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-120-7" name="__codelineno-120-7" href="#__codelineno-120-7"></a><span class="w">            </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
@@ -6842,10 +6939,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-121-1" name="__codelineno-121-1" href="#__codelineno-121-1"></a><span class="cm">/* Exponential order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-121-1" name="__codelineno-121-1" href="#__codelineno-121-1"></a><span class="cm">/* Exponential complexity (iterative implementation)*/</span>
 <a id="__codelineno-121-2" name="__codelineno-121-2" href="#__codelineno-121-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">exponential</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-121-3" name="__codelineno-121-3" href="#__codelineno-121-3"></a><span class="w">    </span><span class="nx">count</span><span class="p">,</span><span class="w"> </span><span class="nx">base</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-121-4" name="__codelineno-121-4" href="#__codelineno-121-4"></a><span class="w">    </span><span class="c1">// Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
+<a id="__codelineno-121-4" name="__codelineno-121-4" href="#__codelineno-121-4"></a><span class="w">    </span><span class="c1">// Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
 <a id="__codelineno-121-5" name="__codelineno-121-5" href="#__codelineno-121-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-121-6" name="__codelineno-121-6" href="#__codelineno-121-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">base</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-121-7" name="__codelineno-121-7" href="#__codelineno-121-7"></a><span class="w">            </span><span class="nx">count</span><span class="o">++</span>
@@ -6858,11 +6955,11 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-122-1" name="__codelineno-122-1" href="#__codelineno-122-1"></a><span class="cm">/* Exponential order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-122-1" name="__codelineno-122-1" href="#__codelineno-122-1"></a><span class="cm">/* Exponential complexity (iterative implementation) */</span>
 <a id="__codelineno-122-2" name="__codelineno-122-2" href="#__codelineno-122-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">exponential</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-122-3" name="__codelineno-122-3" href="#__codelineno-122-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-122-4" name="__codelineno-122-4" href="#__codelineno-122-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">base</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-122-5" name="__codelineno-122-5" href="#__codelineno-122-5"></a><span class="w">    </span><span class="c1">// Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
+<a id="__codelineno-122-5" name="__codelineno-122-5" href="#__codelineno-122-5"></a><span class="w">    </span><span class="c1">// Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
 <a id="__codelineno-122-6" name="__codelineno-122-6" href="#__codelineno-122-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-122-7" name="__codelineno-122-7" href="#__codelineno-122-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-122-8" name="__codelineno-122-8" href="#__codelineno-122-8"></a><span class="w">            </span><span class="bp">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
@@ -6875,11 +6972,11 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-123-1" name="__codelineno-123-1" href="#__codelineno-123-1"></a><span class="cm">/* Exponential order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-123-1" name="__codelineno-123-1" href="#__codelineno-123-1"></a><span class="cm">/* Exponential complexity (iterative implementation) */</span>
 <a id="__codelineno-123-2" name="__codelineno-123-2" href="#__codelineno-123-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">exponential</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-123-3" name="__codelineno-123-3" href="#__codelineno-123-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
 <a id="__codelineno-123-4" name="__codelineno-123-4" href="#__codelineno-123-4"></a><span class="w">        </span><span class="nx">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-123-5" name="__codelineno-123-5" href="#__codelineno-123-5"></a><span class="w">    </span><span class="c1">// Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
+<a id="__codelineno-123-5" name="__codelineno-123-5" href="#__codelineno-123-5"></a><span class="w">    </span><span class="c1">// Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
 <a id="__codelineno-123-6" name="__codelineno-123-6" href="#__codelineno-123-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-123-7" name="__codelineno-123-7" href="#__codelineno-123-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">base</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-123-8" name="__codelineno-123-8" href="#__codelineno-123-8"></a><span class="w">            </span><span class="nx">count</span><span class="o">++</span><span class="p">;</span>
@@ -6892,11 +6989,11 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-124-1" name="__codelineno-124-1" href="#__codelineno-124-1"></a><span class="cm">/* Exponential order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-124-1" name="__codelineno-124-1" href="#__codelineno-124-1"></a><span class="cm">/* Exponential complexity (iterative implementation) */</span>
 <a id="__codelineno-124-2" name="__codelineno-124-2" href="#__codelineno-124-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">exponential</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-124-3" name="__codelineno-124-3" href="#__codelineno-124-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
 <a id="__codelineno-124-4" name="__codelineno-124-4" href="#__codelineno-124-4"></a><span class="w">        </span><span class="nx">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-124-5" name="__codelineno-124-5" href="#__codelineno-124-5"></a><span class="w">    </span><span class="c1">// Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
+<a id="__codelineno-124-5" name="__codelineno-124-5" href="#__codelineno-124-5"></a><span class="w">    </span><span class="c1">// Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
 <a id="__codelineno-124-6" name="__codelineno-124-6" href="#__codelineno-124-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-124-7" name="__codelineno-124-7" href="#__codelineno-124-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">base</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-124-8" name="__codelineno-124-8" href="#__codelineno-124-8"></a><span class="w">            </span><span class="nx">count</span><span class="o">++</span><span class="p">;</span>
@@ -6909,10 +7006,10 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-125-1" name="__codelineno-125-1" href="#__codelineno-125-1"></a><span class="cm">/* Exponential order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-125-1" name="__codelineno-125-1" href="#__codelineno-125-1"></a><span class="cm">/* Exponential complexity (iterative implementation) */</span>
 <a id="__codelineno-125-2" name="__codelineno-125-2" href="#__codelineno-125-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">exponential</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-125-3" name="__codelineno-125-3" href="#__codelineno-125-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
-<a id="__codelineno-125-4" name="__codelineno-125-4" href="#__codelineno-125-4"></a><span class="w">  </span><span class="c1">// Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
+<a id="__codelineno-125-4" name="__codelineno-125-4" href="#__codelineno-125-4"></a><span class="w">  </span><span class="c1">// Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
 <a id="__codelineno-125-5" name="__codelineno-125-5" href="#__codelineno-125-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-125-6" name="__codelineno-125-6" href="#__codelineno-125-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-125-7" name="__codelineno-125-7" href="#__codelineno-125-7"></a><span class="w">      </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
@@ -6925,11 +7022,11 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-126-1" name="__codelineno-126-1" href="#__codelineno-126-1"></a><span class="cm">/* Exponential order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-126-1" name="__codelineno-126-1" href="#__codelineno-126-1"></a><span class="cm">/* Exponential complexity (iterative implementation) */</span>
 <a id="__codelineno-126-2" name="__codelineno-126-2" href="#__codelineno-126-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">exponential</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-126-3" name="__codelineno-126-3" href="#__codelineno-126-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-126-4" name="__codelineno-126-4" href="#__codelineno-126-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-126-5" name="__codelineno-126-5" href="#__codelineno-126-5"></a><span class="w">    </span><span class="c1">// Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
+<a id="__codelineno-126-5" name="__codelineno-126-5" href="#__codelineno-126-5"></a><span class="w">    </span><span class="c1">// Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
 <a id="__codelineno-126-6" name="__codelineno-126-6" href="#__codelineno-126-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-126-7" name="__codelineno-126-7" href="#__codelineno-126-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">base</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-126-8" name="__codelineno-126-8" href="#__codelineno-126-8"></a><span class="w">            </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
@@ -6942,11 +7039,11 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-127-1" name="__codelineno-127-1" href="#__codelineno-127-1"></a><span class="cm">/* Exponential order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-127-1" name="__codelineno-127-1" href="#__codelineno-127-1"></a><span class="cm">/* Exponential complexity (iterative implementation) */</span>
 <a id="__codelineno-127-2" name="__codelineno-127-2" href="#__codelineno-127-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">exponential</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-127-3" name="__codelineno-127-3" href="#__codelineno-127-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-127-4" name="__codelineno-127-4" href="#__codelineno-127-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">bas</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-127-5" name="__codelineno-127-5" href="#__codelineno-127-5"></a><span class="w">    </span><span class="c1">// Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
+<a id="__codelineno-127-5" name="__codelineno-127-5" href="#__codelineno-127-5"></a><span class="w">    </span><span class="c1">// Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
 <a id="__codelineno-127-6" name="__codelineno-127-6" href="#__codelineno-127-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-127-7" name="__codelineno-127-7" href="#__codelineno-127-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">bas</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-127-8" name="__codelineno-127-8" href="#__codelineno-127-8"></a><span class="w">            </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
@@ -6959,11 +7056,11 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-128-1" name="__codelineno-128-1" href="#__codelineno-128-1"></a><span class="cm">/* Exponential order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-128-1" name="__codelineno-128-1" href="#__codelineno-128-1"></a><span class="cm">/* Exponential complexity (iterative implementation) */</span>
 <a id="__codelineno-128-2" name="__codelineno-128-2" href="#__codelineno-128-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">exponential</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-128-3" name="__codelineno-128-3" href="#__codelineno-128-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-128-4" name="__codelineno-128-4" href="#__codelineno-128-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-128-5" name="__codelineno-128-5" href="#__codelineno-128-5"></a><span class="w">    </span><span class="c1">// Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
+<a id="__codelineno-128-5" name="__codelineno-128-5" href="#__codelineno-128-5"></a><span class="w">    </span><span class="c1">// Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
 <a id="__codelineno-128-6" name="__codelineno-128-6" href="#__codelineno-128-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-128-7" name="__codelineno-128-7" href="#__codelineno-128-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">base</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-128-8" name="__codelineno-128-8" href="#__codelineno-128-8"></a><span class="w">            </span><span class="n">count</span><span class="o">++</span>
@@ -6976,11 +7073,11 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-129-1" name="__codelineno-129-1" href="#__codelineno-129-1"></a><span class="c1">### Exponential time (iterative) ###</span>
+<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-129-1" name="__codelineno-129-1" href="#__codelineno-129-1"></a><span class="c1">### Exponential complexity (iterative implementation)###</span>
 <a id="__codelineno-129-2" name="__codelineno-129-2" href="#__codelineno-129-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">exponential</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-129-3" name="__codelineno-129-3" href="#__codelineno-129-3"></a><span class="w">  </span><span class="n">count</span><span class="p">,</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-129-4" name="__codelineno-129-4" href="#__codelineno-129-4"></a>
-<a id="__codelineno-129-5" name="__codelineno-129-5" href="#__codelineno-129-5"></a><span class="w">  </span><span class="c1"># Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
+<a id="__codelineno-129-5" name="__codelineno-129-5" href="#__codelineno-129-5"></a><span class="w">  </span><span class="c1"># Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)</span>
 <a id="__codelineno-129-6" name="__codelineno-129-6" href="#__codelineno-129-6"></a><span class="w">  </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="k">do</span>
 <a id="__codelineno-129-7" name="__codelineno-129-7" href="#__codelineno-129-7"></a><span class="w">    </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">base</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-129-8" name="__codelineno-129-8" href="#__codelineno-129-8"></a><span class="w">    </span><span class="n">base</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">2</span>
@@ -6993,22 +7090,27 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </div>
 </div>
 </div>
-<p><img alt="Time complexity of exponential order" class="animation-figure" src="../time_complexity.assets/time_complexity_exponential.png" /></p>
-<p align="center"> Figure 2-11 &nbsp; Time complexity of exponential order </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 531px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20exponential%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Exponential%20complexity%20%28iterative%20implementation%29%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20base%20%3D%201%0A%20%20%20%20%23%20Each%20cell%20divides%20into%20two%20per%20round,%20giving%20the%20sequence%201,%202,%204,%208,%20...,%202%5E%28n-1%29%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20for%20_%20in%20range%28base%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20%20%20%20%20base%20%2A%3D%202%0A%20%20%20%20%23%20count%20%3D%201%20%2B%202%20%2B%204%20%2B%208%20%2B%20..%20%2B%202%5E%28n-1%29%20%3D%202%5En%20-%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20exponential%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20exponential%20complexity%20%28iterative%20implementation%29%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
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+</details>
+<p><img alt="Exponential time complexity" class="animation-figure" src="../time_complexity.assets/time_complexity_exponential.png" /></p>
+<p align="center"> Figure 2-11 &nbsp; Exponential time complexity </p>
 
-<p>In actual algorithms, exponential order often appears in recursive functions. For example, in the following code, it recursively splits in two, stopping after <span class="arithmatex">\(n\)</span> splits:</p>
+<p>Exponential time often appears in recursive algorithms. The following function branches into two calls at each level, stopping after <span class="arithmatex">\(n\)</span> levels:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="11:13"><input checked="checked" id="__tabbed_11_1" name="__tabbed_11" type="radio" /><input id="__tabbed_11_2" name="__tabbed_11" type="radio" /><input id="__tabbed_11_3" name="__tabbed_11" type="radio" /><input id="__tabbed_11_4" name="__tabbed_11" type="radio" /><input id="__tabbed_11_5" name="__tabbed_11" type="radio" /><input id="__tabbed_11_6" name="__tabbed_11" type="radio" /><input id="__tabbed_11_7" name="__tabbed_11" type="radio" /><input id="__tabbed_11_8" name="__tabbed_11" type="radio" /><input id="__tabbed_11_9" name="__tabbed_11" type="radio" /><input id="__tabbed_11_10" name="__tabbed_11" type="radio" /><input id="__tabbed_11_11" name="__tabbed_11" type="radio" /><input id="__tabbed_11_12" name="__tabbed_11" type="radio" /><input id="__tabbed_11_13" name="__tabbed_11" type="radio" /><div class="tabbed-labels"><label for="__tabbed_11_1">Python</label><label for="__tabbed_11_2">C++</label><label for="__tabbed_11_3">Java</label><label for="__tabbed_11_4">C#</label><label for="__tabbed_11_5">Go</label><label for="__tabbed_11_6">Swift</label><label for="__tabbed_11_7">JS</label><label for="__tabbed_11_8">TS</label><label for="__tabbed_11_9">Dart</label><label for="__tabbed_11_10">Rust</label><label for="__tabbed_11_11">C</label><label for="__tabbed_11_12">Kotlin</label><label for="__tabbed_11_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">time_complexity.py</span><pre><span></span><code><a id="__codelineno-130-1" name="__codelineno-130-1" href="#__codelineno-130-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">exp_recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-130-2" name="__codelineno-130-2" href="#__codelineno-130-2"></a><span class="w">    </span><span class="sd">"""Exponential order (recursive implementation)"""</span>
+<a id="__codelineno-130-2" name="__codelineno-130-2" href="#__codelineno-130-2"></a><span class="w">    </span><span class="sd">"""Exponential complexity (recursive implementation)"""</span>
 <a id="__codelineno-130-3" name="__codelineno-130-3" href="#__codelineno-130-3"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-130-4" name="__codelineno-130-4" href="#__codelineno-130-4"></a>        <span class="k">return</span> <span class="mi">1</span>
 <a id="__codelineno-130-5" name="__codelineno-130-5" href="#__codelineno-130-5"></a>    <span class="k">return</span> <span class="n">exp_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">exp_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-131-1" name="__codelineno-131-1" href="#__codelineno-131-1"></a><span class="cm">/* Exponential order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-131-1" name="__codelineno-131-1" href="#__codelineno-131-1"></a><span class="cm">/* Exponential complexity (recursive implementation) */</span>
 <a id="__codelineno-131-2" name="__codelineno-131-2" href="#__codelineno-131-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">expRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-131-3" name="__codelineno-131-3" href="#__codelineno-131-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-131-4" name="__codelineno-131-4" href="#__codelineno-131-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -7017,7 +7119,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-132-1" name="__codelineno-132-1" href="#__codelineno-132-1"></a><span class="cm">/* Exponential order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-132-1" name="__codelineno-132-1" href="#__codelineno-132-1"></a><span class="cm">/* Exponential complexity (recursive implementation) */</span>
 <a id="__codelineno-132-2" name="__codelineno-132-2" href="#__codelineno-132-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">expRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-132-3" name="__codelineno-132-3" href="#__codelineno-132-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-132-4" name="__codelineno-132-4" href="#__codelineno-132-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -7026,7 +7128,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-133-1" name="__codelineno-133-1" href="#__codelineno-133-1"></a><span class="cm">/* Exponential order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-133-1" name="__codelineno-133-1" href="#__codelineno-133-1"></a><span class="cm">/* Exponential complexity (recursive implementation) */</span>
 <a id="__codelineno-133-2" name="__codelineno-133-2" href="#__codelineno-133-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">ExpRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-133-3" name="__codelineno-133-3" href="#__codelineno-133-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-133-4" name="__codelineno-133-4" href="#__codelineno-133-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nf">ExpRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">ExpRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -7034,7 +7136,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-134-1" name="__codelineno-134-1" href="#__codelineno-134-1"></a><span class="cm">/* Exponential order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-134-1" name="__codelineno-134-1" href="#__codelineno-134-1"></a><span class="cm">/* Exponential complexity (recursive implementation)*/</span>
 <a id="__codelineno-134-2" name="__codelineno-134-2" href="#__codelineno-134-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">expRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-134-3" name="__codelineno-134-3" href="#__codelineno-134-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-134-4" name="__codelineno-134-4" href="#__codelineno-134-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span>
@@ -7044,7 +7146,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-135-1" name="__codelineno-135-1" href="#__codelineno-135-1"></a><span class="cm">/* Exponential order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-135-1" name="__codelineno-135-1" href="#__codelineno-135-1"></a><span class="cm">/* Exponential complexity (recursive implementation) */</span>
 <a id="__codelineno-135-2" name="__codelineno-135-2" href="#__codelineno-135-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">expRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-135-3" name="__codelineno-135-3" href="#__codelineno-135-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-135-4" name="__codelineno-135-4" href="#__codelineno-135-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span>
@@ -7054,7 +7156,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-136-1" name="__codelineno-136-1" href="#__codelineno-136-1"></a><span class="cm">/* Exponential order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-136-1" name="__codelineno-136-1" href="#__codelineno-136-1"></a><span class="cm">/* Exponential complexity (recursive implementation) */</span>
 <a id="__codelineno-136-2" name="__codelineno-136-2" href="#__codelineno-136-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">expRecur</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-136-3" name="__codelineno-136-3" href="#__codelineno-136-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-136-4" name="__codelineno-136-4" href="#__codelineno-136-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">expRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">expRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
@@ -7062,7 +7164,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-137-1" name="__codelineno-137-1" href="#__codelineno-137-1"></a><span class="cm">/* Exponential order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-137-1" name="__codelineno-137-1" href="#__codelineno-137-1"></a><span class="cm">/* Exponential complexity (recursive implementation) */</span>
 <a id="__codelineno-137-2" name="__codelineno-137-2" href="#__codelineno-137-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">expRecur</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-137-3" name="__codelineno-137-3" href="#__codelineno-137-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-137-4" name="__codelineno-137-4" href="#__codelineno-137-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">expRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">expRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
@@ -7070,7 +7172,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-138-1" name="__codelineno-138-1" href="#__codelineno-138-1"></a><span class="cm">/* Exponential order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-138-1" name="__codelineno-138-1" href="#__codelineno-138-1"></a><span class="cm">/* Exponential complexity (recursive implementation) */</span>
 <a id="__codelineno-138-2" name="__codelineno-138-2" href="#__codelineno-138-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">expRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-138-3" name="__codelineno-138-3" href="#__codelineno-138-3"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-138-4" name="__codelineno-138-4" href="#__codelineno-138-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">expRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">expRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
@@ -7078,7 +7180,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-139-1" name="__codelineno-139-1" href="#__codelineno-139-1"></a><span class="cm">/* Exponential order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-139-1" name="__codelineno-139-1" href="#__codelineno-139-1"></a><span class="cm">/* Exponential complexity (recursive implementation) */</span>
 <a id="__codelineno-139-2" name="__codelineno-139-2" href="#__codelineno-139-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">exp_recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-139-3" name="__codelineno-139-3" href="#__codelineno-139-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-139-4" name="__codelineno-139-4" href="#__codelineno-139-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -7088,7 +7190,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-140-1" name="__codelineno-140-1" href="#__codelineno-140-1"></a><span class="cm">/* Exponential order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-140-1" name="__codelineno-140-1" href="#__codelineno-140-1"></a><span class="cm">/* Exponential complexity (recursive implementation) */</span>
 <a id="__codelineno-140-2" name="__codelineno-140-2" href="#__codelineno-140-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">expRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-140-3" name="__codelineno-140-3" href="#__codelineno-140-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-140-4" name="__codelineno-140-4" href="#__codelineno-140-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -7097,7 +7199,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-141-1" name="__codelineno-141-1" href="#__codelineno-141-1"></a><span class="cm">/* Exponential order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-141-1" name="__codelineno-141-1" href="#__codelineno-141-1"></a><span class="cm">/* Exponential complexity (recursive implementation) */</span>
 <a id="__codelineno-141-2" name="__codelineno-141-2" href="#__codelineno-141-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">expRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-141-3" name="__codelineno-141-3" href="#__codelineno-141-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-141-4" name="__codelineno-141-4" href="#__codelineno-141-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">1</span>
@@ -7107,7 +7209,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-142-1" name="__codelineno-142-1" href="#__codelineno-142-1"></a><span class="c1">### Exponential time (recursive) ###</span>
+<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-142-1" name="__codelineno-142-1" href="#__codelineno-142-1"></a><span class="c1">### Exponential complexity (recursive implementation)###</span>
 <a id="__codelineno-142-2" name="__codelineno-142-2" href="#__codelineno-142-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">exp_recur</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-142-3" name="__codelineno-142-3" href="#__codelineno-142-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-142-4" name="__codelineno-142-4" href="#__codelineno-142-4"></a><span class="w">  </span><span class="n">exp_recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">exp_recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
@@ -7116,15 +7218,20 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </div>
 </div>
 </div>
-<p>Exponential order growth is very rapid and is common in exhaustive methods (brute force search, backtracking, etc.). For problems with large data scales, exponential order is unacceptable and typically requires dynamic programming or greedy algorithms to solve.</p>
-<h3 id="5-logarithmic-order-olog-n">5. &nbsp; Logarithmic Order <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#5-logarithmic-order-olog-n" title="Permanent link">&para;</a></h3>
-<p>In contrast to exponential order, logarithmic order reflects the situation of "reducing to half each round". Let the input data size be <span class="arithmatex">\(n\)</span>. Since it is reduced to half each round, the number of loops is <span class="arithmatex">\(\log_2 n\)</span>, which is the inverse function of <span class="arithmatex">\(2^n\)</span>.</p>
-<p>Figure 2-12 and the following code simulate the process of "reducing to half each round", with a time complexity of <span class="arithmatex">\(O(\log_2 n)\)</span>, abbreviated as <span class="arithmatex">\(O(\log n)\)</span>:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 423px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20exp_recur%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Exponential%20complexity%20%28recursive%20implementation%29%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20return%20exp_recur%28n%20-%201%29%20%2B%20exp_recur%28n%20-%201%29%20%2B%201%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%207%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20exp_recur%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20exponential%20complexity%20%28recursive%20implementation%29%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20exp_recur%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Exponential%20complexity%20%28recursive%20implementation%29%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20return%20exp_recur%28n%20-%201%29%20%2B%20exp_recur%28n%20-%201%29%20%2B%201%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%207%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20exp_recur%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20exponential%20complexity%20%28recursive%20implementation%29%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Exponential growth is very rapid and common in exhaustive approaches such as brute-force search and backtracking. It is unacceptable for large inputs, which usually require approaches such as dynamic programming or greedy algorithms.</p>
+<h3 id="5-logarithmic-time-olog-n">5. &nbsp; Logarithmic time <span class="arithmatex">\(O(\log n)\)</span><a class="headerlink" href="#5-logarithmic-time-olog-n" title="Permanent link">&para;</a></h3>
+<p>Logarithmic growth corresponds to repeatedly halving a quantity, the reverse of exponential growth. Starting with input size <span class="arithmatex">\(n\)</span>, halving each round gives <span class="arithmatex">\(\log_2 n\)</span> iterations, the inverse of <span class="arithmatex">\(2^n\)</span>.</p>
+<p>The figure and code below simulate repeated halving, taking <span class="arithmatex">\(O(\log_2 n)\)</span> time, usually written <span class="arithmatex">\(O(\log n)\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="12:13"><input checked="checked" id="__tabbed_12_1" name="__tabbed_12" type="radio" /><input id="__tabbed_12_2" name="__tabbed_12" type="radio" /><input id="__tabbed_12_3" name="__tabbed_12" type="radio" /><input id="__tabbed_12_4" name="__tabbed_12" type="radio" /><input id="__tabbed_12_5" name="__tabbed_12" type="radio" /><input id="__tabbed_12_6" name="__tabbed_12" type="radio" /><input id="__tabbed_12_7" name="__tabbed_12" type="radio" /><input id="__tabbed_12_8" name="__tabbed_12" type="radio" /><input id="__tabbed_12_9" name="__tabbed_12" type="radio" /><input id="__tabbed_12_10" name="__tabbed_12" type="radio" /><input id="__tabbed_12_11" name="__tabbed_12" type="radio" /><input id="__tabbed_12_12" name="__tabbed_12" type="radio" /><input id="__tabbed_12_13" name="__tabbed_12" type="radio" /><div class="tabbed-labels"><label for="__tabbed_12_1">Python</label><label for="__tabbed_12_2">C++</label><label for="__tabbed_12_3">Java</label><label for="__tabbed_12_4">C#</label><label for="__tabbed_12_5">Go</label><label for="__tabbed_12_6">Swift</label><label for="__tabbed_12_7">JS</label><label for="__tabbed_12_8">TS</label><label for="__tabbed_12_9">Dart</label><label for="__tabbed_12_10">Rust</label><label for="__tabbed_12_11">C</label><label for="__tabbed_12_12">Kotlin</label><label for="__tabbed_12_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">time_complexity.py</span><pre><span></span><code><a id="__codelineno-143-1" name="__codelineno-143-1" href="#__codelineno-143-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">logarithmic</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-143-2" name="__codelineno-143-2" href="#__codelineno-143-2"></a><span class="w">    </span><span class="sd">"""Logarithmic order (loop implementation)"""</span>
+<a id="__codelineno-143-2" name="__codelineno-143-2" href="#__codelineno-143-2"></a><span class="w">    </span><span class="sd">"""Logarithmic complexity (iterative implementation)"""</span>
 <a id="__codelineno-143-3" name="__codelineno-143-3" href="#__codelineno-143-3"></a>    <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-143-4" name="__codelineno-143-4" href="#__codelineno-143-4"></a>    <span class="k">while</span> <span class="n">n</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-143-5" name="__codelineno-143-5" href="#__codelineno-143-5"></a>        <span class="n">n</span> <span class="o">=</span> <span class="n">n</span> <span class="o">/</span> <span class="mi">2</span>
@@ -7133,7 +7240,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-144-1" name="__codelineno-144-1" href="#__codelineno-144-1"></a><span class="cm">/* Logarithmic order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-144-1" name="__codelineno-144-1" href="#__codelineno-144-1"></a><span class="cm">/* Logarithmic complexity (iterative implementation) */</span>
 <a id="__codelineno-144-2" name="__codelineno-144-2" href="#__codelineno-144-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">logarithmic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-144-3" name="__codelineno-144-3" href="#__codelineno-144-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-144-4" name="__codelineno-144-4" href="#__codelineno-144-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7145,7 +7252,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-145-1" name="__codelineno-145-1" href="#__codelineno-145-1"></a><span class="cm">/* Logarithmic order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-145-1" name="__codelineno-145-1" href="#__codelineno-145-1"></a><span class="cm">/* Logarithmic complexity (iterative implementation) */</span>
 <a id="__codelineno-145-2" name="__codelineno-145-2" href="#__codelineno-145-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">logarithmic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-145-3" name="__codelineno-145-3" href="#__codelineno-145-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-145-4" name="__codelineno-145-4" href="#__codelineno-145-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7157,7 +7264,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-146-1" name="__codelineno-146-1" href="#__codelineno-146-1"></a><span class="cm">/* Logarithmic order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-146-1" name="__codelineno-146-1" href="#__codelineno-146-1"></a><span class="cm">/* Logarithmic complexity (iterative implementation) */</span>
 <a id="__codelineno-146-2" name="__codelineno-146-2" href="#__codelineno-146-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Logarithmic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-146-3" name="__codelineno-146-3" href="#__codelineno-146-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-146-4" name="__codelineno-146-4" href="#__codelineno-146-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7169,7 +7276,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-147-1" name="__codelineno-147-1" href="#__codelineno-147-1"></a><span class="cm">/* Logarithmic order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-147-1" name="__codelineno-147-1" href="#__codelineno-147-1"></a><span class="cm">/* Logarithmic complexity (iterative implementation)*/</span>
 <a id="__codelineno-147-2" name="__codelineno-147-2" href="#__codelineno-147-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">logarithmic</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-147-3" name="__codelineno-147-3" href="#__codelineno-147-3"></a><span class="w">    </span><span class="nx">count</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-147-4" name="__codelineno-147-4" href="#__codelineno-147-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
@@ -7181,7 +7288,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-148-1" name="__codelineno-148-1" href="#__codelineno-148-1"></a><span class="cm">/* Logarithmic order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-148-1" name="__codelineno-148-1" href="#__codelineno-148-1"></a><span class="cm">/* Logarithmic complexity (iterative implementation) */</span>
 <a id="__codelineno-148-2" name="__codelineno-148-2" href="#__codelineno-148-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">logarithmic</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-148-3" name="__codelineno-148-3" href="#__codelineno-148-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-148-4" name="__codelineno-148-4" href="#__codelineno-148-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">n</span>
@@ -7194,7 +7301,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-149-1" name="__codelineno-149-1" href="#__codelineno-149-1"></a><span class="cm">/* Logarithmic order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-149-1" name="__codelineno-149-1" href="#__codelineno-149-1"></a><span class="cm">/* Logarithmic complexity (iterative implementation) */</span>
 <a id="__codelineno-149-2" name="__codelineno-149-2" href="#__codelineno-149-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">logarithmic</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-149-3" name="__codelineno-149-3" href="#__codelineno-149-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-149-4" name="__codelineno-149-4" href="#__codelineno-149-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7206,7 +7313,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-150-1" name="__codelineno-150-1" href="#__codelineno-150-1"></a><span class="cm">/* Logarithmic order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-150-1" name="__codelineno-150-1" href="#__codelineno-150-1"></a><span class="cm">/* Logarithmic complexity (iterative implementation) */</span>
 <a id="__codelineno-150-2" name="__codelineno-150-2" href="#__codelineno-150-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">logarithmic</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-150-3" name="__codelineno-150-3" href="#__codelineno-150-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-150-4" name="__codelineno-150-4" href="#__codelineno-150-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7218,7 +7325,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-151-1" name="__codelineno-151-1" href="#__codelineno-151-1"></a><span class="cm">/* Logarithmic order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-151-1" name="__codelineno-151-1" href="#__codelineno-151-1"></a><span class="cm">/* Logarithmic complexity (iterative implementation) */</span>
 <a id="__codelineno-151-2" name="__codelineno-151-2" href="#__codelineno-151-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">logarithmic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-151-3" name="__codelineno-151-3" href="#__codelineno-151-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-151-4" name="__codelineno-151-4" href="#__codelineno-151-4"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7230,7 +7337,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-152-1" name="__codelineno-152-1" href="#__codelineno-152-1"></a><span class="cm">/* Logarithmic order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-152-1" name="__codelineno-152-1" href="#__codelineno-152-1"></a><span class="cm">/* Logarithmic complexity (iterative implementation) */</span>
 <a id="__codelineno-152-2" name="__codelineno-152-2" href="#__codelineno-152-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">logarithmic</span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-152-3" name="__codelineno-152-3" href="#__codelineno-152-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-152-4" name="__codelineno-152-4" href="#__codelineno-152-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
@@ -7242,7 +7349,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-153-1" name="__codelineno-153-1" href="#__codelineno-153-1"></a><span class="cm">/* Logarithmic order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-153-1" name="__codelineno-153-1" href="#__codelineno-153-1"></a><span class="cm">/* Logarithmic complexity (iterative implementation) */</span>
 <a id="__codelineno-153-2" name="__codelineno-153-2" href="#__codelineno-153-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">logarithmic</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-153-3" name="__codelineno-153-3" href="#__codelineno-153-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-153-4" name="__codelineno-153-4" href="#__codelineno-153-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7254,7 +7361,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-154-1" name="__codelineno-154-1" href="#__codelineno-154-1"></a><span class="cm">/* Logarithmic order (loop implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-154-1" name="__codelineno-154-1" href="#__codelineno-154-1"></a><span class="cm">/* Logarithmic complexity (iterative implementation) */</span>
 <a id="__codelineno-154-2" name="__codelineno-154-2" href="#__codelineno-154-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">logarithmic</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-154-3" name="__codelineno-154-3" href="#__codelineno-154-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span>
 <a id="__codelineno-154-4" name="__codelineno-154-4" href="#__codelineno-154-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
@@ -7267,7 +7374,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-155-1" name="__codelineno-155-1" href="#__codelineno-155-1"></a><span class="c1">### Logarithmic time (iterative) ###</span>
+<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-155-1" name="__codelineno-155-1" href="#__codelineno-155-1"></a><span class="c1">### Logarithmic complexity (iterative implementation)###</span>
 <a id="__codelineno-155-2" name="__codelineno-155-2" href="#__codelineno-155-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">logarithmic</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-155-3" name="__codelineno-155-3" href="#__codelineno-155-3"></a><span class="w">  </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-155-4" name="__codelineno-155-4" href="#__codelineno-155-4"></a>
@@ -7282,22 +7389,27 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </div>
 </div>
 </div>
-<p><img alt="Time complexity of logarithmic order" class="animation-figure" src="../time_complexity.assets/time_complexity_logarithmic.png" /></p>
-<p align="center"> Figure 2-12 &nbsp; Time complexity of logarithmic order </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 459px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20logarithmic%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Logarithmic%20complexity%20%28iterative%20implementation%29%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20while%20n%20%3E%201%3A%0A%20%20%20%20%20%20%20%20n%20%3D%20n%20/%202%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20logarithmic%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20logarithmic%20complexity%20%28iterative%20implementation%29%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20logarithmic%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Logarithmic%20complexity%20%28iterative%20implementation%29%22%22%22%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20while%20n%20%3E%201%3A%0A%20%20%20%20%20%20%20%20n%20%3D%20n%20/%202%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20logarithmic%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20logarithmic%20complexity%20%28iterative%20implementation%29%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p><img alt="Logarithmic time complexity" class="animation-figure" src="../time_complexity.assets/time_complexity_logarithmic.png" /></p>
+<p align="center"> Figure 2-12 &nbsp; Logarithmic time complexity </p>
 
-<p>Like exponential order, logarithmic order also commonly appears in recursive functions. The following code forms a recursion tree of height <span class="arithmatex">\(\log_2 n\)</span>:</p>
+<p>Like exponential growth, logarithmic growth often occurs in recursion. The following code produces a recursion tree of height <span class="arithmatex">\(\log_2 n\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="13:13"><input checked="checked" id="__tabbed_13_1" name="__tabbed_13" type="radio" /><input id="__tabbed_13_2" name="__tabbed_13" type="radio" /><input id="__tabbed_13_3" name="__tabbed_13" type="radio" /><input id="__tabbed_13_4" name="__tabbed_13" type="radio" /><input id="__tabbed_13_5" name="__tabbed_13" type="radio" /><input id="__tabbed_13_6" name="__tabbed_13" type="radio" /><input id="__tabbed_13_7" name="__tabbed_13" type="radio" /><input id="__tabbed_13_8" name="__tabbed_13" type="radio" /><input id="__tabbed_13_9" name="__tabbed_13" type="radio" /><input id="__tabbed_13_10" name="__tabbed_13" type="radio" /><input id="__tabbed_13_11" name="__tabbed_13" type="radio" /><input id="__tabbed_13_12" name="__tabbed_13" type="radio" /><input id="__tabbed_13_13" name="__tabbed_13" type="radio" /><div class="tabbed-labels"><label for="__tabbed_13_1">Python</label><label for="__tabbed_13_2">C++</label><label for="__tabbed_13_3">Java</label><label for="__tabbed_13_4">C#</label><label for="__tabbed_13_5">Go</label><label for="__tabbed_13_6">Swift</label><label for="__tabbed_13_7">JS</label><label for="__tabbed_13_8">TS</label><label for="__tabbed_13_9">Dart</label><label for="__tabbed_13_10">Rust</label><label for="__tabbed_13_11">C</label><label for="__tabbed_13_12">Kotlin</label><label for="__tabbed_13_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">time_complexity.py</span><pre><span></span><code><a id="__codelineno-156-1" name="__codelineno-156-1" href="#__codelineno-156-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">log_recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-156-2" name="__codelineno-156-2" href="#__codelineno-156-2"></a><span class="w">    </span><span class="sd">"""Logarithmic order (recursive implementation)"""</span>
+<a id="__codelineno-156-2" name="__codelineno-156-2" href="#__codelineno-156-2"></a><span class="w">    </span><span class="sd">"""Logarithmic complexity (recursive implementation)"""</span>
 <a id="__codelineno-156-3" name="__codelineno-156-3" href="#__codelineno-156-3"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-156-4" name="__codelineno-156-4" href="#__codelineno-156-4"></a>        <span class="k">return</span> <span class="mi">0</span>
 <a id="__codelineno-156-5" name="__codelineno-156-5" href="#__codelineno-156-5"></a>    <span class="k">return</span> <span class="n">log_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-157-1" name="__codelineno-157-1" href="#__codelineno-157-1"></a><span class="cm">/* Logarithmic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-157-1" name="__codelineno-157-1" href="#__codelineno-157-1"></a><span class="cm">/* Logarithmic complexity (recursive implementation) */</span>
 <a id="__codelineno-157-2" name="__codelineno-157-2" href="#__codelineno-157-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">logRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-157-3" name="__codelineno-157-3" href="#__codelineno-157-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-157-4" name="__codelineno-157-4" href="#__codelineno-157-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
@@ -7306,7 +7418,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-158-1" name="__codelineno-158-1" href="#__codelineno-158-1"></a><span class="cm">/* Logarithmic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-158-1" name="__codelineno-158-1" href="#__codelineno-158-1"></a><span class="cm">/* Logarithmic complexity (recursive implementation) */</span>
 <a id="__codelineno-158-2" name="__codelineno-158-2" href="#__codelineno-158-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">logRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-158-3" name="__codelineno-158-3" href="#__codelineno-158-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-158-4" name="__codelineno-158-4" href="#__codelineno-158-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
@@ -7315,7 +7427,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-159-1" name="__codelineno-159-1" href="#__codelineno-159-1"></a><span class="cm">/* Logarithmic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-159-1" name="__codelineno-159-1" href="#__codelineno-159-1"></a><span class="cm">/* Logarithmic complexity (recursive implementation) */</span>
 <a id="__codelineno-159-2" name="__codelineno-159-2" href="#__codelineno-159-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">LogRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-159-3" name="__codelineno-159-3" href="#__codelineno-159-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-159-4" name="__codelineno-159-4" href="#__codelineno-159-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nf">LogRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -7323,7 +7435,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-160-1" name="__codelineno-160-1" href="#__codelineno-160-1"></a><span class="cm">/* Logarithmic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-160-1" name="__codelineno-160-1" href="#__codelineno-160-1"></a><span class="cm">/* Logarithmic complexity (recursive implementation)*/</span>
 <a id="__codelineno-160-2" name="__codelineno-160-2" href="#__codelineno-160-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">logRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-160-3" name="__codelineno-160-3" href="#__codelineno-160-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-160-4" name="__codelineno-160-4" href="#__codelineno-160-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span>
@@ -7333,7 +7445,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-161-1" name="__codelineno-161-1" href="#__codelineno-161-1"></a><span class="cm">/* Logarithmic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-161-1" name="__codelineno-161-1" href="#__codelineno-161-1"></a><span class="cm">/* Logarithmic complexity (recursive implementation) */</span>
 <a id="__codelineno-161-2" name="__codelineno-161-2" href="#__codelineno-161-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">logRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-161-3" name="__codelineno-161-3" href="#__codelineno-161-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-161-4" name="__codelineno-161-4" href="#__codelineno-161-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span>
@@ -7343,7 +7455,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-162-1" name="__codelineno-162-1" href="#__codelineno-162-1"></a><span class="cm">/* Logarithmic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-162-1" name="__codelineno-162-1" href="#__codelineno-162-1"></a><span class="cm">/* Logarithmic complexity (recursive implementation) */</span>
 <a id="__codelineno-162-2" name="__codelineno-162-2" href="#__codelineno-162-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">logRecur</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-162-3" name="__codelineno-162-3" href="#__codelineno-162-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-162-4" name="__codelineno-162-4" href="#__codelineno-162-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">logRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
@@ -7351,7 +7463,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-163-1" name="__codelineno-163-1" href="#__codelineno-163-1"></a><span class="cm">/* Logarithmic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-163-1" name="__codelineno-163-1" href="#__codelineno-163-1"></a><span class="cm">/* Logarithmic complexity (recursive implementation) */</span>
 <a id="__codelineno-163-2" name="__codelineno-163-2" href="#__codelineno-163-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">logRecur</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-163-3" name="__codelineno-163-3" href="#__codelineno-163-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-163-4" name="__codelineno-163-4" href="#__codelineno-163-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">logRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
@@ -7359,7 +7471,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-164-1" name="__codelineno-164-1" href="#__codelineno-164-1"></a><span class="cm">/* Logarithmic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-164-1" name="__codelineno-164-1" href="#__codelineno-164-1"></a><span class="cm">/* Logarithmic complexity (recursive implementation) */</span>
 <a id="__codelineno-164-2" name="__codelineno-164-2" href="#__codelineno-164-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">logRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-164-3" name="__codelineno-164-3" href="#__codelineno-164-3"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-164-4" name="__codelineno-164-4" href="#__codelineno-164-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">logRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
@@ -7367,7 +7479,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-165-1" name="__codelineno-165-1" href="#__codelineno-165-1"></a><span class="cm">/* Logarithmic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-165-1" name="__codelineno-165-1" href="#__codelineno-165-1"></a><span class="cm">/* Logarithmic complexity (recursive implementation) */</span>
 <a id="__codelineno-165-2" name="__codelineno-165-2" href="#__codelineno-165-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">log_recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-165-3" name="__codelineno-165-3" href="#__codelineno-165-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-165-4" name="__codelineno-165-4" href="#__codelineno-165-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
@@ -7377,7 +7489,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-166-1" name="__codelineno-166-1" href="#__codelineno-166-1"></a><span class="cm">/* Logarithmic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-166-1" name="__codelineno-166-1" href="#__codelineno-166-1"></a><span class="cm">/* Logarithmic complexity (recursive implementation) */</span>
 <a id="__codelineno-166-2" name="__codelineno-166-2" href="#__codelineno-166-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">logRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-166-3" name="__codelineno-166-3" href="#__codelineno-166-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-166-4" name="__codelineno-166-4" href="#__codelineno-166-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
@@ -7386,7 +7498,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-167-1" name="__codelineno-167-1" href="#__codelineno-167-1"></a><span class="cm">/* Logarithmic order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-167-1" name="__codelineno-167-1" href="#__codelineno-167-1"></a><span class="cm">/* Logarithmic complexity (recursive implementation) */</span>
 <a id="__codelineno-167-2" name="__codelineno-167-2" href="#__codelineno-167-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">logRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-167-3" name="__codelineno-167-3" href="#__codelineno-167-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-167-4" name="__codelineno-167-4" href="#__codelineno-167-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">0</span>
@@ -7395,7 +7507,7 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-168-1" name="__codelineno-168-1" href="#__codelineno-168-1"></a><span class="c1">### Logarithmic time (recursive) ###</span>
+<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-168-1" name="__codelineno-168-1" href="#__codelineno-168-1"></a><span class="c1">### Logarithmic complexity (recursive implementation)###</span>
 <a id="__codelineno-168-2" name="__codelineno-168-2" href="#__codelineno-168-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">log_recur</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-168-3" name="__codelineno-168-3" href="#__codelineno-168-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-168-4" name="__codelineno-168-4" href="#__codelineno-168-4"></a><span class="w">  </span><span class="n">log_recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
@@ -7404,34 +7516,39 @@ T(n) &amp; = n^2 + n &amp; \text{Simplified count (o.O)}
 </div>
 </div>
 </div>
-<p>Logarithmic order commonly appears in algorithms based on the divide-and-conquer strategy, reflecting the idea of repeatedly splitting a problem and simplifying it. It grows slowly and is the ideal time complexity second only to constant order.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 423px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20log_recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Logarithmic%20complexity%20%28recursive%20implementation%29%22%22%22%0A%20%20%20%20if%20n%20%3C%3D%201%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20return%20log_recur%28n%20/%202%29%20%2B%201%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20log_recur%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20logarithmic%20complexity%20%28recursive%20implementation%29%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20log_recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Logarithmic%20complexity%20%28recursive%20implementation%29%22%22%22%0A%20%20%20%20if%20n%20%3C%3D%201%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20return%20log_recur%28n%20/%202%29%20%2B%201%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20log_recur%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20logarithmic%20complexity%20%28recursive%20implementation%29%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Logarithmic time often arises in divide-and-conquer algorithms, reflecting the idea of dividing a problem into smaller, simpler parts. It grows slowly and is highly desirable, second only to constant time among these classes.</p>
 <div class="admonition tip">
 <p class="admonition-title">What is the base of <span class="arithmatex">\(O(\log n)\)</span>?</p>
-<p>To be precise, "dividing into <span class="arithmatex">\(m\)</span>" corresponds to a time complexity of <span class="arithmatex">\(O(\log_m n)\)</span>. And through the logarithmic base change formula, we can obtain time complexities with different bases that are equal:</p>
+<p>More precisely, reducing the size by a factor of <span class="arithmatex">\(m\)</span> gives <span class="arithmatex">\(O(\log_m n)\)</span>. The change-of-base formula shows that different fixed bases yield the same complexity:</p>
 <div class="arithmatex">\[
 O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
 \]</div>
-<p>That is to say, the base <span class="arithmatex">\(m\)</span> can be converted without affecting the complexity. Therefore, we usually omit the base <span class="arithmatex">\(m\)</span> and denote logarithmic order simply as <span class="arithmatex">\(O(\log n)\)</span>.</p>
+<p>Thus, changing the base <span class="arithmatex">\(m\)</span> does not affect the complexity. We usually omit <span class="arithmatex">\(m\)</span> and simply write <span class="arithmatex">\(O(\log n)\)</span>.</p>
 </div>
-<h3 id="6-linearithmic-order-on-log-n">6. &nbsp; Linearithmic Order <span class="arithmatex">\(O(n \log n)\)</span><a class="headerlink" href="#6-linearithmic-order-on-log-n" title="Permanent link">&para;</a></h3>
-<p>Linearithmic order commonly appears in nested loops, where the time complexities of the two layers of loops are <span class="arithmatex">\(O(\log n)\)</span> and <span class="arithmatex">\(O(n)\)</span> respectively. The relevant code is as follows:</p>
+<h3 id="6-linearithmic-time-on-log-n">6. &nbsp; Linearithmic time <span class="arithmatex">\(O(n \log n)\)</span><a class="headerlink" href="#6-linearithmic-time-on-log-n" title="Permanent link">&para;</a></h3>
+<p>Linearithmic time often appears when one level of a nested process takes <span class="arithmatex">\(O(\log n)\)</span> and another takes <span class="arithmatex">\(O(n)\)</span>. An example follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="14:13"><input checked="checked" id="__tabbed_14_1" name="__tabbed_14" type="radio" /><input id="__tabbed_14_2" name="__tabbed_14" type="radio" /><input id="__tabbed_14_3" name="__tabbed_14" type="radio" /><input id="__tabbed_14_4" name="__tabbed_14" type="radio" /><input id="__tabbed_14_5" name="__tabbed_14" type="radio" /><input id="__tabbed_14_6" name="__tabbed_14" type="radio" /><input id="__tabbed_14_7" name="__tabbed_14" type="radio" /><input id="__tabbed_14_8" name="__tabbed_14" type="radio" /><input id="__tabbed_14_9" name="__tabbed_14" type="radio" /><input id="__tabbed_14_10" name="__tabbed_14" type="radio" /><input id="__tabbed_14_11" name="__tabbed_14" type="radio" /><input id="__tabbed_14_12" name="__tabbed_14" type="radio" /><input id="__tabbed_14_13" name="__tabbed_14" type="radio" /><div class="tabbed-labels"><label for="__tabbed_14_1">Python</label><label for="__tabbed_14_2">C++</label><label for="__tabbed_14_3">Java</label><label for="__tabbed_14_4">C#</label><label for="__tabbed_14_5">Go</label><label for="__tabbed_14_6">Swift</label><label for="__tabbed_14_7">JS</label><label for="__tabbed_14_8">TS</label><label for="__tabbed_14_9">Dart</label><label for="__tabbed_14_10">Rust</label><label for="__tabbed_14_11">C</label><label for="__tabbed_14_12">Kotlin</label><label for="__tabbed_14_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">time_complexity.py</span><pre><span></span><code><a id="__codelineno-169-1" name="__codelineno-169-1" href="#__codelineno-169-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">linear_log_recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-169-2" name="__codelineno-169-2" href="#__codelineno-169-2"></a><span class="w">    </span><span class="sd">"""Linearithmic order"""</span>
+<a id="__codelineno-169-2" name="__codelineno-169-2" href="#__codelineno-169-2"></a><span class="w">    </span><span class="sd">"""Linearithmic complexity"""</span>
 <a id="__codelineno-169-3" name="__codelineno-169-3" href="#__codelineno-169-3"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-169-4" name="__codelineno-169-4" href="#__codelineno-169-4"></a>        <span class="k">return</span> <span class="mi">1</span>
-<a id="__codelineno-169-5" name="__codelineno-169-5" href="#__codelineno-169-5"></a>    <span class="c1"># Divide into two, the scale of subproblems is reduced by half</span>
+<a id="__codelineno-169-5" name="__codelineno-169-5" href="#__codelineno-169-5"></a>    <span class="c1"># Split into two subproblems, each half the original size</span>
 <a id="__codelineno-169-6" name="__codelineno-169-6" href="#__codelineno-169-6"></a>    <span class="n">count</span> <span class="o">=</span> <span class="n">linear_log_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">linear_log_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)</span>
-<a id="__codelineno-169-7" name="__codelineno-169-7" href="#__codelineno-169-7"></a>    <span class="c1"># Current subproblem contains n operations</span>
+<a id="__codelineno-169-7" name="__codelineno-169-7" href="#__codelineno-169-7"></a>    <span class="c1"># The current subproblem includes n operations</span>
 <a id="__codelineno-169-8" name="__codelineno-169-8" href="#__codelineno-169-8"></a>    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-169-9" name="__codelineno-169-9" href="#__codelineno-169-9"></a>        <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
 <a id="__codelineno-169-10" name="__codelineno-169-10" href="#__codelineno-169-10"></a>    <span class="k">return</span> <span class="n">count</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-170-1" name="__codelineno-170-1" href="#__codelineno-170-1"></a><span class="cm">/* Linearithmic order */</span>
+<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-170-1" name="__codelineno-170-1" href="#__codelineno-170-1"></a><span class="cm">/* Linearithmic complexity */</span>
 <a id="__codelineno-170-2" name="__codelineno-170-2" href="#__codelineno-170-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">linearLogRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-170-3" name="__codelineno-170-3" href="#__codelineno-170-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-170-4" name="__codelineno-170-4" href="#__codelineno-170-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -7444,7 +7561,7 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-171-1" name="__codelineno-171-1" href="#__codelineno-171-1"></a><span class="cm">/* Linearithmic order */</span>
+<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-171-1" name="__codelineno-171-1" href="#__codelineno-171-1"></a><span class="cm">/* Linearithmic complexity */</span>
 <a id="__codelineno-171-2" name="__codelineno-171-2" href="#__codelineno-171-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">linearLogRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-171-3" name="__codelineno-171-3" href="#__codelineno-171-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-171-4" name="__codelineno-171-4" href="#__codelineno-171-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -7457,7 +7574,7 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-172-1" name="__codelineno-172-1" href="#__codelineno-172-1"></a><span class="cm">/* Linearithmic order */</span>
+<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-172-1" name="__codelineno-172-1" href="#__codelineno-172-1"></a><span class="cm">/* Linearithmic complexity */</span>
 <a id="__codelineno-172-2" name="__codelineno-172-2" href="#__codelineno-172-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">LinearLogRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-172-3" name="__codelineno-172-3" href="#__codelineno-172-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-172-4" name="__codelineno-172-4" href="#__codelineno-172-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">LinearLogRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">LinearLogRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">);</span>
@@ -7469,7 +7586,7 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-173-1" name="__codelineno-173-1" href="#__codelineno-173-1"></a><span class="cm">/* Linearithmic order */</span>
+<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-173-1" name="__codelineno-173-1" href="#__codelineno-173-1"></a><span class="cm">/* Linearithmic complexity */</span>
 <a id="__codelineno-173-2" name="__codelineno-173-2" href="#__codelineno-173-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">linearLogRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-173-3" name="__codelineno-173-3" href="#__codelineno-173-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-173-4" name="__codelineno-173-4" href="#__codelineno-173-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span>
@@ -7483,7 +7600,7 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-174-1" name="__codelineno-174-1" href="#__codelineno-174-1"></a><span class="cm">/* Linearithmic order */</span>
+<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-174-1" name="__codelineno-174-1" href="#__codelineno-174-1"></a><span class="cm">/* Linearithmic complexity */</span>
 <a id="__codelineno-174-2" name="__codelineno-174-2" href="#__codelineno-174-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">linearLogRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-174-3" name="__codelineno-174-3" href="#__codelineno-174-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-174-4" name="__codelineno-174-4" href="#__codelineno-174-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span>
@@ -7497,7 +7614,7 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-175-1" name="__codelineno-175-1" href="#__codelineno-175-1"></a><span class="cm">/* Linearithmic order */</span>
+<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-175-1" name="__codelineno-175-1" href="#__codelineno-175-1"></a><span class="cm">/* Linearithmic complexity */</span>
 <a id="__codelineno-175-2" name="__codelineno-175-2" href="#__codelineno-175-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">linearLogRecur</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-175-3" name="__codelineno-175-3" href="#__codelineno-175-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-175-4" name="__codelineno-175-4" href="#__codelineno-175-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">linearLogRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">linearLogRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span>
@@ -7509,7 +7626,7 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-176-1" name="__codelineno-176-1" href="#__codelineno-176-1"></a><span class="cm">/* Linearithmic order */</span>
+<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-176-1" name="__codelineno-176-1" href="#__codelineno-176-1"></a><span class="cm">/* Linearithmic complexity */</span>
 <a id="__codelineno-176-2" name="__codelineno-176-2" href="#__codelineno-176-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">linearLogRecur</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-176-3" name="__codelineno-176-3" href="#__codelineno-176-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-176-4" name="__codelineno-176-4" href="#__codelineno-176-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">linearLogRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">linearLogRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span>
@@ -7521,7 +7638,7 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-177-1" name="__codelineno-177-1" href="#__codelineno-177-1"></a><span class="cm">/* Linearithmic order */</span>
+<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-177-1" name="__codelineno-177-1" href="#__codelineno-177-1"></a><span class="cm">/* Linearithmic complexity */</span>
 <a id="__codelineno-177-2" name="__codelineno-177-2" href="#__codelineno-177-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">linearLogRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-177-3" name="__codelineno-177-3" href="#__codelineno-177-3"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-177-4" name="__codelineno-177-4" href="#__codelineno-177-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">linearLogRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">linearLogRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">);</span>
@@ -7533,7 +7650,7 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-178-1" name="__codelineno-178-1" href="#__codelineno-178-1"></a><span class="cm">/* Linearithmic order */</span>
+<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-178-1" name="__codelineno-178-1" href="#__codelineno-178-1"></a><span class="cm">/* Linearithmic complexity */</span>
 <a id="__codelineno-178-2" name="__codelineno-178-2" href="#__codelineno-178-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">linear_log_recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-178-3" name="__codelineno-178-3" href="#__codelineno-178-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-178-4" name="__codelineno-178-4" href="#__codelineno-178-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -7547,7 +7664,7 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-179-1" name="__codelineno-179-1" href="#__codelineno-179-1"></a><span class="cm">/* Linearithmic order */</span>
+<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-179-1" name="__codelineno-179-1" href="#__codelineno-179-1"></a><span class="cm">/* Linearithmic complexity */</span>
 <a id="__codelineno-179-2" name="__codelineno-179-2" href="#__codelineno-179-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">linearLogRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-179-3" name="__codelineno-179-3" href="#__codelineno-179-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-179-4" name="__codelineno-179-4" href="#__codelineno-179-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -7560,7 +7677,7 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-180-1" name="__codelineno-180-1" href="#__codelineno-180-1"></a><span class="cm">/* Linearithmic order */</span>
+<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-180-1" name="__codelineno-180-1" href="#__codelineno-180-1"></a><span class="cm">/* Linearithmic complexity */</span>
 <a id="__codelineno-180-2" name="__codelineno-180-2" href="#__codelineno-180-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">linearLogRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-180-3" name="__codelineno-180-3" href="#__codelineno-180-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-180-4" name="__codelineno-180-4" href="#__codelineno-180-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">1</span>
@@ -7573,7 +7690,7 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-181-1" name="__codelineno-181-1" href="#__codelineno-181-1"></a><span class="c1">### Linearithmic time ###</span>
+<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-181-1" name="__codelineno-181-1" href="#__codelineno-181-1"></a><span class="c1">### Linearithmic complexity ###</span>
 <a id="__codelineno-181-2" name="__codelineno-181-2" href="#__codelineno-181-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">linear_log_recur</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-181-3" name="__codelineno-181-3" href="#__codelineno-181-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-181-4" name="__codelineno-181-4" href="#__codelineno-181-4"></a>
@@ -7586,38 +7703,43 @@ O(\log_m n) = O(\log_k n / \log_k m) = O(\log_k n)
 </div>
 </div>
 </div>
-<p>Figure 2-13 shows how linearithmic order is generated. Each level of the binary tree has a total of <span class="arithmatex">\(n\)</span> operations, and the tree has <span class="arithmatex">\(\log_2 n + 1\)</span> levels, resulting in a time complexity of <span class="arithmatex">\(O(n \log n)\)</span>.</p>
-<p><img alt="Time complexity of linearithmic order" class="animation-figure" src="../time_complexity.assets/time_complexity_logarithmic_linear.png" /></p>
-<p align="center"> Figure 2-13 &nbsp; Time complexity of linearithmic order </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 477px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20linear_log_recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Linearithmic%20complexity%22%22%22%0A%20%20%20%20if%20n%20%3C%3D%201%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20count%20%3D%20linear_log_recur%28n%20//%202%29%20%2B%20linear_log_recur%28n%20//%202%29%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20linear_log_recur%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20linearithmic%20complexity%20%28recursive%20implementation%29%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20linear_log_recur%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Linearithmic%20complexity%22%22%22%0A%20%20%20%20if%20n%20%3C%3D%201%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20count%20%3D%20linear_log_recur%28n%20//%202%29%20%2B%20linear_log_recur%28n%20//%202%29%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%208%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20linear_log_recur%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20linearithmic%20complexity%20%28recursive%20implementation%29%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 2-13 shows how linearithmic growth arises. Each level of the binary tree performs <span class="arithmatex">\(n\)</span> operations, and there are <span class="arithmatex">\(\log_2 n + 1\)</span> levels, giving <span class="arithmatex">\(O(n \log n)\)</span> time.</p>
+<p><img alt="Linearithmic time complexity" class="animation-figure" src="../time_complexity.assets/time_complexity_logarithmic_linear.png" /></p>
+<p align="center"> Figure 2-13 &nbsp; Linearithmic time complexity </p>
 
-<p>Mainstream sorting algorithms typically have a time complexity of <span class="arithmatex">\(O(n \log n)\)</span>, such as quicksort, merge sort, and heap sort.</p>
-<h3 id="7-factorial-order-on">7. &nbsp; Factorial Order <span class="arithmatex">\(O(n!)\)</span><a class="headerlink" href="#7-factorial-order-on" title="Permanent link">&para;</a></h3>
-<p>Factorial order corresponds to the mathematical "permutation" problem. Given <span class="arithmatex">\(n\)</span> distinct elements, find all possible permutation schemes; the number of schemes is:</p>
+<p>Common sorting algorithms, including quick sort, merge sort, and heap sort, typically take <span class="arithmatex">\(O(n \log n)\)</span> time.</p>
+<h3 id="7-factorial-time-on">7. &nbsp; Factorial time <span class="arithmatex">\(O(n!)\)</span><a class="headerlink" href="#7-factorial-time-on" title="Permanent link">&para;</a></h3>
+<p>Factorial growth arises when generating every permutation of <span class="arithmatex">\(n\)</span> distinct elements. The number of permutations is:</p>
 <div class="arithmatex">\[
 n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 \]</div>
-<p>Factorials are typically implemented using recursion. As shown in Figure 2-14 and the following code, the first level splits into <span class="arithmatex">\(n\)</span> branches, the second level splits into <span class="arithmatex">\(n - 1\)</span> branches, and so on, until the <span class="arithmatex">\(n\)</span>-th level when splitting stops:</p>
+<p>Algorithms with factorial time complexity are usually implemented recursively. As the figure and code below show, each call creates <span class="arithmatex">\(n\)</span> branches at the first level, <span class="arithmatex">\(n - 1\)</span> at the second, and so on until level <span class="arithmatex">\(n\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="15:13"><input checked="checked" id="__tabbed_15_1" name="__tabbed_15" type="radio" /><input id="__tabbed_15_2" name="__tabbed_15" type="radio" /><input id="__tabbed_15_3" name="__tabbed_15" type="radio" /><input id="__tabbed_15_4" name="__tabbed_15" type="radio" /><input id="__tabbed_15_5" name="__tabbed_15" type="radio" /><input id="__tabbed_15_6" name="__tabbed_15" type="radio" /><input id="__tabbed_15_7" name="__tabbed_15" type="radio" /><input id="__tabbed_15_8" name="__tabbed_15" type="radio" /><input id="__tabbed_15_9" name="__tabbed_15" type="radio" /><input id="__tabbed_15_10" name="__tabbed_15" type="radio" /><input id="__tabbed_15_11" name="__tabbed_15" type="radio" /><input id="__tabbed_15_12" name="__tabbed_15" type="radio" /><input id="__tabbed_15_13" name="__tabbed_15" type="radio" /><div class="tabbed-labels"><label for="__tabbed_15_1">Python</label><label for="__tabbed_15_2">C++</label><label for="__tabbed_15_3">Java</label><label for="__tabbed_15_4">C#</label><label for="__tabbed_15_5">Go</label><label for="__tabbed_15_6">Swift</label><label for="__tabbed_15_7">JS</label><label for="__tabbed_15_8">TS</label><label for="__tabbed_15_9">Dart</label><label for="__tabbed_15_10">Rust</label><label for="__tabbed_15_11">C</label><label for="__tabbed_15_12">Kotlin</label><label for="__tabbed_15_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">time_complexity.py</span><pre><span></span><code><a id="__codelineno-182-1" name="__codelineno-182-1" href="#__codelineno-182-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">factorial_recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-182-2" name="__codelineno-182-2" href="#__codelineno-182-2"></a><span class="w">    </span><span class="sd">"""Factorial order (recursive implementation)"""</span>
+<a id="__codelineno-182-2" name="__codelineno-182-2" href="#__codelineno-182-2"></a><span class="w">    </span><span class="sd">"""Factorial complexity (recursive implementation)"""</span>
 <a id="__codelineno-182-3" name="__codelineno-182-3" href="#__codelineno-182-3"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-182-4" name="__codelineno-182-4" href="#__codelineno-182-4"></a>        <span class="k">return</span> <span class="mi">1</span>
 <a id="__codelineno-182-5" name="__codelineno-182-5" href="#__codelineno-182-5"></a>    <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-182-6" name="__codelineno-182-6" href="#__codelineno-182-6"></a>    <span class="c1"># Split from 1 into n</span>
+<a id="__codelineno-182-6" name="__codelineno-182-6" href="#__codelineno-182-6"></a>    <span class="c1"># Split one problem into n subproblems</span>
 <a id="__codelineno-182-7" name="__codelineno-182-7" href="#__codelineno-182-7"></a>    <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-182-8" name="__codelineno-182-8" href="#__codelineno-182-8"></a>        <span class="n">count</span> <span class="o">+=</span> <span class="n">factorial_recur</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-182-9" name="__codelineno-182-9" href="#__codelineno-182-9"></a>    <span class="k">return</span> <span class="n">count</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-183-1" name="__codelineno-183-1" href="#__codelineno-183-1"></a><span class="cm">/* Factorial order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-183-1" name="__codelineno-183-1" href="#__codelineno-183-1"></a><span class="cm">/* Factorial complexity (recursive implementation) */</span>
 <a id="__codelineno-183-2" name="__codelineno-183-2" href="#__codelineno-183-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">factorialRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-183-3" name="__codelineno-183-3" href="#__codelineno-183-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-183-4" name="__codelineno-183-4" href="#__codelineno-183-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-183-5" name="__codelineno-183-5" href="#__codelineno-183-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-183-6" name="__codelineno-183-6" href="#__codelineno-183-6"></a><span class="w">    </span><span class="c1">// Split from 1 into n</span>
+<a id="__codelineno-183-6" name="__codelineno-183-6" href="#__codelineno-183-6"></a><span class="w">    </span><span class="c1">// Split one problem into n subproblems</span>
 <a id="__codelineno-183-7" name="__codelineno-183-7" href="#__codelineno-183-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-183-8" name="__codelineno-183-8" href="#__codelineno-183-8"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">factorialRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-183-9" name="__codelineno-183-9" href="#__codelineno-183-9"></a><span class="w">    </span><span class="p">}</span>
@@ -7626,12 +7748,12 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-184-1" name="__codelineno-184-1" href="#__codelineno-184-1"></a><span class="cm">/* Factorial order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-184-1" name="__codelineno-184-1" href="#__codelineno-184-1"></a><span class="cm">/* Factorial complexity (recursive implementation) */</span>
 <a id="__codelineno-184-2" name="__codelineno-184-2" href="#__codelineno-184-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">factorialRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-184-3" name="__codelineno-184-3" href="#__codelineno-184-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-184-4" name="__codelineno-184-4" href="#__codelineno-184-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-184-5" name="__codelineno-184-5" href="#__codelineno-184-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-184-6" name="__codelineno-184-6" href="#__codelineno-184-6"></a><span class="w">    </span><span class="c1">// Split from 1 into n</span>
+<a id="__codelineno-184-6" name="__codelineno-184-6" href="#__codelineno-184-6"></a><span class="w">    </span><span class="c1">// Split one problem into n subproblems</span>
 <a id="__codelineno-184-7" name="__codelineno-184-7" href="#__codelineno-184-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-184-8" name="__codelineno-184-8" href="#__codelineno-184-8"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">factorialRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-184-9" name="__codelineno-184-9" href="#__codelineno-184-9"></a><span class="w">    </span><span class="p">}</span>
@@ -7640,11 +7762,11 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-185-1" name="__codelineno-185-1" href="#__codelineno-185-1"></a><span class="cm">/* Factorial order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.cs</span><pre><span></span><code><a id="__codelineno-185-1" name="__codelineno-185-1" href="#__codelineno-185-1"></a><span class="cm">/* Factorial complexity (recursive implementation) */</span>
 <a id="__codelineno-185-2" name="__codelineno-185-2" href="#__codelineno-185-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">FactorialRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-185-3" name="__codelineno-185-3" href="#__codelineno-185-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-185-4" name="__codelineno-185-4" href="#__codelineno-185-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-185-5" name="__codelineno-185-5" href="#__codelineno-185-5"></a><span class="w">    </span><span class="c1">// Split from 1 into n</span>
+<a id="__codelineno-185-5" name="__codelineno-185-5" href="#__codelineno-185-5"></a><span class="w">    </span><span class="c1">// Split one problem into n subproblems</span>
 <a id="__codelineno-185-6" name="__codelineno-185-6" href="#__codelineno-185-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-185-7" name="__codelineno-185-7" href="#__codelineno-185-7"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">FactorialRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-185-8" name="__codelineno-185-8" href="#__codelineno-185-8"></a><span class="w">    </span><span class="p">}</span>
@@ -7653,13 +7775,13 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-186-1" name="__codelineno-186-1" href="#__codelineno-186-1"></a><span class="cm">/* Factorial order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.go</span><pre><span></span><code><a id="__codelineno-186-1" name="__codelineno-186-1" href="#__codelineno-186-1"></a><span class="cm">/* Factorial complexity (recursive implementation) */</span>
 <a id="__codelineno-186-2" name="__codelineno-186-2" href="#__codelineno-186-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">factorialRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-186-3" name="__codelineno-186-3" href="#__codelineno-186-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-186-4" name="__codelineno-186-4" href="#__codelineno-186-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-186-5" name="__codelineno-186-5" href="#__codelineno-186-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-186-6" name="__codelineno-186-6" href="#__codelineno-186-6"></a><span class="w">    </span><span class="nx">count</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-186-7" name="__codelineno-186-7" href="#__codelineno-186-7"></a><span class="w">    </span><span class="c1">// Split from 1 into n</span>
+<a id="__codelineno-186-7" name="__codelineno-186-7" href="#__codelineno-186-7"></a><span class="w">    </span><span class="c1">// Split one problem into n subproblems</span>
 <a id="__codelineno-186-8" name="__codelineno-186-8" href="#__codelineno-186-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-186-9" name="__codelineno-186-9" href="#__codelineno-186-9"></a><span class="w">        </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">factorialRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-186-10" name="__codelineno-186-10" href="#__codelineno-186-10"></a><span class="w">    </span><span class="p">}</span>
@@ -7668,13 +7790,13 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-187-1" name="__codelineno-187-1" href="#__codelineno-187-1"></a><span class="cm">/* Factorial order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.swift</span><pre><span></span><code><a id="__codelineno-187-1" name="__codelineno-187-1" href="#__codelineno-187-1"></a><span class="cm">/* Factorial complexity (recursive implementation) */</span>
 <a id="__codelineno-187-2" name="__codelineno-187-2" href="#__codelineno-187-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">factorialRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-187-3" name="__codelineno-187-3" href="#__codelineno-187-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-187-4" name="__codelineno-187-4" href="#__codelineno-187-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-187-5" name="__codelineno-187-5" href="#__codelineno-187-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-187-6" name="__codelineno-187-6" href="#__codelineno-187-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-187-7" name="__codelineno-187-7" href="#__codelineno-187-7"></a><span class="w">    </span><span class="c1">// Split from 1 into n</span>
+<a id="__codelineno-187-7" name="__codelineno-187-7" href="#__codelineno-187-7"></a><span class="w">    </span><span class="c1">// Split one problem into n subproblems</span>
 <a id="__codelineno-187-8" name="__codelineno-187-8" href="#__codelineno-187-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-187-9" name="__codelineno-187-9" href="#__codelineno-187-9"></a><span class="w">        </span><span class="bp">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">factorialRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-187-10" name="__codelineno-187-10" href="#__codelineno-187-10"></a><span class="w">    </span><span class="p">}</span>
@@ -7683,11 +7805,11 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-188-1" name="__codelineno-188-1" href="#__codelineno-188-1"></a><span class="cm">/* Factorial order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.js</span><pre><span></span><code><a id="__codelineno-188-1" name="__codelineno-188-1" href="#__codelineno-188-1"></a><span class="cm">/* Factorial complexity (recursive implementation) */</span>
 <a id="__codelineno-188-2" name="__codelineno-188-2" href="#__codelineno-188-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">factorialRecur</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-188-3" name="__codelineno-188-3" href="#__codelineno-188-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-188-4" name="__codelineno-188-4" href="#__codelineno-188-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-188-5" name="__codelineno-188-5" href="#__codelineno-188-5"></a><span class="w">    </span><span class="c1">// Split from 1 into n</span>
+<a id="__codelineno-188-5" name="__codelineno-188-5" href="#__codelineno-188-5"></a><span class="w">    </span><span class="c1">// Split one problem into n subproblems</span>
 <a id="__codelineno-188-6" name="__codelineno-188-6" href="#__codelineno-188-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-188-7" name="__codelineno-188-7" href="#__codelineno-188-7"></a><span class="w">        </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">factorialRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-188-8" name="__codelineno-188-8" href="#__codelineno-188-8"></a><span class="w">    </span><span class="p">}</span>
@@ -7696,11 +7818,11 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-189-1" name="__codelineno-189-1" href="#__codelineno-189-1"></a><span class="cm">/* Factorial order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.ts</span><pre><span></span><code><a id="__codelineno-189-1" name="__codelineno-189-1" href="#__codelineno-189-1"></a><span class="cm">/* Factorial complexity (recursive implementation) */</span>
 <a id="__codelineno-189-2" name="__codelineno-189-2" href="#__codelineno-189-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">factorialRecur</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-189-3" name="__codelineno-189-3" href="#__codelineno-189-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-189-4" name="__codelineno-189-4" href="#__codelineno-189-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-189-5" name="__codelineno-189-5" href="#__codelineno-189-5"></a><span class="w">    </span><span class="c1">// Split from 1 into n</span>
+<a id="__codelineno-189-5" name="__codelineno-189-5" href="#__codelineno-189-5"></a><span class="w">    </span><span class="c1">// Split one problem into n subproblems</span>
 <a id="__codelineno-189-6" name="__codelineno-189-6" href="#__codelineno-189-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-189-7" name="__codelineno-189-7" href="#__codelineno-189-7"></a><span class="w">        </span><span class="nx">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">factorialRecur</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-189-8" name="__codelineno-189-8" href="#__codelineno-189-8"></a><span class="w">    </span><span class="p">}</span>
@@ -7709,11 +7831,11 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-190-1" name="__codelineno-190-1" href="#__codelineno-190-1"></a><span class="cm">/* Factorial order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.dart</span><pre><span></span><code><a id="__codelineno-190-1" name="__codelineno-190-1" href="#__codelineno-190-1"></a><span class="cm">/* Factorial complexity (recursive implementation) */</span>
 <a id="__codelineno-190-2" name="__codelineno-190-2" href="#__codelineno-190-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">factorialRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-190-3" name="__codelineno-190-3" href="#__codelineno-190-3"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-190-4" name="__codelineno-190-4" href="#__codelineno-190-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
-<a id="__codelineno-190-5" name="__codelineno-190-5" href="#__codelineno-190-5"></a><span class="w">  </span><span class="c1">// Split from 1 into n</span>
+<a id="__codelineno-190-5" name="__codelineno-190-5" href="#__codelineno-190-5"></a><span class="w">  </span><span class="c1">// Split one problem into n subproblems</span>
 <a id="__codelineno-190-6" name="__codelineno-190-6" href="#__codelineno-190-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-190-7" name="__codelineno-190-7" href="#__codelineno-190-7"></a><span class="w">    </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">factorialRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-190-8" name="__codelineno-190-8" href="#__codelineno-190-8"></a><span class="w">  </span><span class="p">}</span>
@@ -7722,13 +7844,13 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-191-1" name="__codelineno-191-1" href="#__codelineno-191-1"></a><span class="cm">/* Factorial order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.rs</span><pre><span></span><code><a id="__codelineno-191-1" name="__codelineno-191-1" href="#__codelineno-191-1"></a><span class="cm">/* Factorial complexity (recursive implementation) */</span>
 <a id="__codelineno-191-2" name="__codelineno-191-2" href="#__codelineno-191-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">factorial_recur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-191-3" name="__codelineno-191-3" href="#__codelineno-191-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-191-4" name="__codelineno-191-4" href="#__codelineno-191-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-191-5" name="__codelineno-191-5" href="#__codelineno-191-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-191-6" name="__codelineno-191-6" href="#__codelineno-191-6"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-191-7" name="__codelineno-191-7" href="#__codelineno-191-7"></a><span class="w">    </span><span class="c1">// Split from 1 into n</span>
+<a id="__codelineno-191-7" name="__codelineno-191-7" href="#__codelineno-191-7"></a><span class="w">    </span><span class="c1">// Split one problem into n subproblems</span>
 <a id="__codelineno-191-8" name="__codelineno-191-8" href="#__codelineno-191-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-191-9" name="__codelineno-191-9" href="#__codelineno-191-9"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">factorial_recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-191-10" name="__codelineno-191-10" href="#__codelineno-191-10"></a><span class="w">    </span><span class="p">}</span>
@@ -7737,7 +7859,7 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-192-1" name="__codelineno-192-1" href="#__codelineno-192-1"></a><span class="cm">/* Factorial order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.c</span><pre><span></span><code><a id="__codelineno-192-1" name="__codelineno-192-1" href="#__codelineno-192-1"></a><span class="cm">/* Factorial complexity (recursive implementation) */</span>
 <a id="__codelineno-192-2" name="__codelineno-192-2" href="#__codelineno-192-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">factorialRecur</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-192-3" name="__codelineno-192-3" href="#__codelineno-192-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-192-4" name="__codelineno-192-4" href="#__codelineno-192-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -7750,12 +7872,12 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-193-1" name="__codelineno-193-1" href="#__codelineno-193-1"></a><span class="cm">/* Factorial order (recursive implementation) */</span>
+<div class="highlight"><span class="filename">time_complexity.kt</span><pre><span></span><code><a id="__codelineno-193-1" name="__codelineno-193-1" href="#__codelineno-193-1"></a><span class="cm">/* Factorial complexity (recursive implementation) */</span>
 <a id="__codelineno-193-2" name="__codelineno-193-2" href="#__codelineno-193-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">factorialRecur</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-193-3" name="__codelineno-193-3" href="#__codelineno-193-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span>
 <a id="__codelineno-193-4" name="__codelineno-193-4" href="#__codelineno-193-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-193-5" name="__codelineno-193-5" href="#__codelineno-193-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-193-6" name="__codelineno-193-6" href="#__codelineno-193-6"></a><span class="w">    </span><span class="c1">// Split from 1 into n</span>
+<a id="__codelineno-193-6" name="__codelineno-193-6" href="#__codelineno-193-6"></a><span class="w">    </span><span class="c1">// Split one problem into n subproblems</span>
 <a id="__codelineno-193-7" name="__codelineno-193-7" href="#__codelineno-193-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-193-8" name="__codelineno-193-8" href="#__codelineno-193-8"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">factorialRecur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-193-9" name="__codelineno-193-9" href="#__codelineno-193-9"></a><span class="w">    </span><span class="p">}</span>
@@ -7764,12 +7886,12 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-194-1" name="__codelineno-194-1" href="#__codelineno-194-1"></a><span class="c1">### Factorial time (recursive) ###</span>
+<div class="highlight"><span class="filename">time_complexity.rb</span><pre><span></span><code><a id="__codelineno-194-1" name="__codelineno-194-1" href="#__codelineno-194-1"></a><span class="c1">### Factorial complexity (recursive implementation)###</span>
 <a id="__codelineno-194-2" name="__codelineno-194-2" href="#__codelineno-194-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">factorial_recur</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-194-3" name="__codelineno-194-3" href="#__codelineno-194-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-194-4" name="__codelineno-194-4" href="#__codelineno-194-4"></a>
 <a id="__codelineno-194-5" name="__codelineno-194-5" href="#__codelineno-194-5"></a><span class="w">  </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-194-6" name="__codelineno-194-6" href="#__codelineno-194-6"></a><span class="w">  </span><span class="c1"># Split from 1 into n</span>
+<a id="__codelineno-194-6" name="__codelineno-194-6" href="#__codelineno-194-6"></a><span class="w">  </span><span class="c1"># Split one problem into n subproblems</span>
 <a id="__codelineno-194-7" name="__codelineno-194-7" href="#__codelineno-194-7"></a><span class="w">  </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">factorial_recur</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-194-8" name="__codelineno-194-8" href="#__codelineno-194-8"></a>
 <a id="__codelineno-194-9" name="__codelineno-194-9" href="#__codelineno-194-9"></a><span class="w">  </span><span class="n">count</span>
@@ -7778,58 +7900,63 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </div>
 </div>
 </div>
-<p><img alt="Time complexity of factorial order" class="animation-figure" src="../time_complexity.assets/time_complexity_factorial.png" /></p>
-<p align="center"> Figure 2-14 &nbsp; Time complexity of factorial order </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 495px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20factorial_recur%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Factorial%20complexity%20%28recursive%20implementation%29%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20Split%20one%20problem%20into%20n%20subproblems%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20factorial_recur%28n%20-%201%29%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%204%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20factorial_recur%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20factorial%20complexity%20%28recursive%20implementation%29%20%3D%22,%20count%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20factorial_recur%28n%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Factorial%20complexity%20%28recursive%20implementation%29%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20Split%20one%20problem%20into%20n%20subproblems%0A%20%20%20%20for%20_%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20count%20%2B%3D%20factorial_recur%28n%20-%201%29%0A%20%20%20%20return%20count%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%204%0A%20%20%20%20print%28%22Input%20size%20n%20%3D%22,%20n%29%0A%0A%20%20%20%20count%20%3D%20factorial_recur%28n%29%0A%20%20%20%20print%28%22Operation%20count%20for%20factorial%20complexity%20%28recursive%20implementation%29%20%3D%22,%20count%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p><img alt="Factorial time complexity" class="animation-figure" src="../time_complexity.assets/time_complexity_factorial.png" /></p>
+<p align="center"> Figure 2-14 &nbsp; Factorial time complexity </p>
 
-<p>Note that because when <span class="arithmatex">\(n \geq 4\)</span> we always have <span class="arithmatex">\(n! &gt; 2^n\)</span>, factorial order grows faster than exponential order, and is also unacceptable for large <span class="arithmatex">\(n\)</span>.</p>
-<h2 id="235-worst-best-and-average-time-complexities">2.3.5 &nbsp; Worst, Best, and Average Time Complexities<a class="headerlink" href="#235-worst-best-and-average-time-complexities" title="Permanent link">&para;</a></h2>
-<p><strong>The time efficiency of an algorithm is often not fixed, but is related to the distribution of the input data</strong>. Suppose we input an array <code>nums</code> of length <span class="arithmatex">\(n\)</span>, where <code>nums</code> consists of numbers from <span class="arithmatex">\(1\)</span> to <span class="arithmatex">\(n\)</span>, with each number appearing only once, but the element order is randomly shuffled. The task is to return the index of element <span class="arithmatex">\(1\)</span>. We can draw the following conclusions.</p>
+<p>Since <span class="arithmatex">\(n! &gt; 2^n\)</span> whenever <span class="arithmatex">\(n \geq 4\)</span>, factorial growth is faster than exponential growth and is also unacceptable for large <span class="arithmatex">\(n\)</span>.</p>
+<h2 id="235-worst-case-best-case-and-average-case-time-complexity">2.3.5 &nbsp; Worst-case, best-case, and average-case time complexity<a class="headerlink" href="#235-worst-case-best-case-and-average-case-time-complexity" title="Permanent link">&para;</a></h2>
+<p><strong>An algorithm's efficiency often depends on how its input data is arranged</strong>. Suppose <code>nums</code> is an array of length <span class="arithmatex">\(n\)</span> containing every integer from <span class="arithmatex">\(1\)</span> through <span class="arithmatex">\(n\)</span> exactly once in a random order. The task is to return the index of <span class="arithmatex">\(1\)</span>. We can conclude the following.</p>
 <ul>
-<li>When <code>nums = [?, ?, ..., 1]</code>, i.e., when the last element is <span class="arithmatex">\(1\)</span>, it requires a complete traversal of the array, <strong>reaching worst-case time complexity <span class="arithmatex">\(O(n)\)</span></strong>.</li>
-<li>When <code>nums = [1, ?, ?, ...]</code>, i.e., when the first element is <span class="arithmatex">\(1\)</span>, no matter how long the array is, there is no need to continue traversing, <strong>reaching best-case time complexity <span class="arithmatex">\(\Omega(1)\)</span></strong>.</li>
+<li>If <code>nums = [?, ?, ..., 1]</code>, with <span class="arithmatex">\(1\)</span> last, we must traverse the whole array, <strong>giving worst-case time complexity <span class="arithmatex">\(O(n)\)</span></strong>.</li>
+<li>If <code>nums = [1, ?, ?, ...]</code>, with <span class="arithmatex">\(1\)</span> first, no further traversal is needed regardless of array length, <strong>giving best-case time complexity <span class="arithmatex">\(\Omega(1)\)</span></strong>.</li>
 </ul>
-<p>The "worst-case time complexity" corresponds to the function's asymptotic upper bound, denoted using big-<span class="arithmatex">\(O\)</span> notation. Correspondingly, the "best-case time complexity" corresponds to the function's asymptotic lower bound, denoted using <span class="arithmatex">\(\Omega\)</span> notation:</p>
+<p>Worst-case time complexity corresponds to an asymptotic upper bound, expressed with big-<span class="arithmatex">\(O\)</span> notation. Best-case time complexity corresponds to an asymptotic lower bound, expressed with <span class="arithmatex">\(\Omega\)</span>:</p>
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 <div class="tabbed-block">
 <div class="highlight"><span class="filename">worst_best_time_complexity.py</span><pre><span></span><code><a id="__codelineno-195-1" name="__codelineno-195-1" href="#__codelineno-195-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">random_numbers</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
-<a id="__codelineno-195-2" name="__codelineno-195-2" href="#__codelineno-195-2"></a><span class="w">    </span><span class="sd">"""Generate an array with elements: 1, 2, ..., n, shuffled in order"""</span>
-<a id="__codelineno-195-3" name="__codelineno-195-3" href="#__codelineno-195-3"></a>    <span class="c1"># Generate array nums =: 1, 2, 3, ..., n</span>
+<a id="__codelineno-195-2" name="__codelineno-195-2" href="#__codelineno-195-2"></a><span class="w">    </span><span class="sd">"""Generate an array containing 1, 2, ..., n in shuffled order"""</span>
+<a id="__codelineno-195-3" name="__codelineno-195-3" href="#__codelineno-195-3"></a>    <span class="c1"># Generate the array nums = 1, 2, 3, ..., n</span>
 <a id="__codelineno-195-4" name="__codelineno-195-4" href="#__codelineno-195-4"></a>    <span class="n">nums</span> <span class="o">=</span> <span class="p">[</span><span class="n">i</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-<a id="__codelineno-195-5" name="__codelineno-195-5" href="#__codelineno-195-5"></a>    <span class="c1"># Randomly shuffle array elements</span>
+<a id="__codelineno-195-5" name="__codelineno-195-5" href="#__codelineno-195-5"></a>    <span class="c1"># Randomly shuffle the array elements</span>
 <a id="__codelineno-195-6" name="__codelineno-195-6" href="#__codelineno-195-6"></a>    <span class="n">random</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
 <a id="__codelineno-195-7" name="__codelineno-195-7" href="#__codelineno-195-7"></a>    <span class="k">return</span> <span class="n">nums</span>
 <a id="__codelineno-195-8" name="__codelineno-195-8" href="#__codelineno-195-8"></a>
 <a id="__codelineno-195-9" name="__codelineno-195-9" href="#__codelineno-195-9"></a><span class="k">def</span><span class="w"> </span><span class="nf">find_one</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">])</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-195-10" name="__codelineno-195-10" href="#__codelineno-195-10"></a><span class="w">    </span><span class="sd">"""Find the index of number 1 in array nums"""</span>
+<a id="__codelineno-195-10" name="__codelineno-195-10" href="#__codelineno-195-10"></a><span class="w">    </span><span class="sd">"""Find the index of the number 1 in array nums"""</span>
 <a id="__codelineno-195-11" name="__codelineno-195-11" href="#__codelineno-195-11"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)):</span>
-<a id="__codelineno-195-12" name="__codelineno-195-12" href="#__codelineno-195-12"></a>        <span class="c1"># When element 1 is at the head of the array, best time complexity O(1) is achieved</span>
-<a id="__codelineno-195-13" name="__codelineno-195-13" href="#__codelineno-195-13"></a>        <span class="c1"># When element 1 is at the tail of the array, worst time complexity O(n) is achieved</span>
+<a id="__codelineno-195-12" name="__codelineno-195-12" href="#__codelineno-195-12"></a>        <span class="c1"># When 1 is at the start of the array, the best-case time complexity is O(1)</span>
+<a id="__codelineno-195-13" name="__codelineno-195-13" href="#__codelineno-195-13"></a>        <span class="c1"># When 1 is at the end of the array, the worst-case time complexity is O(n)</span>
 <a id="__codelineno-195-14" name="__codelineno-195-14" href="#__codelineno-195-14"></a>        <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-195-15" name="__codelineno-195-15" href="#__codelineno-195-15"></a>            <span class="k">return</span> <span class="n">i</span>
 <a id="__codelineno-195-16" name="__codelineno-195-16" href="#__codelineno-195-16"></a>    <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">worst_best_time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-196-1" name="__codelineno-196-1" href="#__codelineno-196-1"></a><span class="cm">/* Generate an array with elements { 1, 2, ..., n }, order shuffled */</span>
+<div class="highlight"><span class="filename">worst_best_time_complexity.cpp</span><pre><span></span><code><a id="__codelineno-196-1" name="__codelineno-196-1" href="#__codelineno-196-1"></a><span class="cm">/* Generate an array containing { 1, 2, ..., n } in shuffled order */</span>
 <a id="__codelineno-196-2" name="__codelineno-196-2" href="#__codelineno-196-2"></a><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">randomNumbers</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-196-3" name="__codelineno-196-3" href="#__codelineno-196-3"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">(</span><span class="n">n</span><span class="p">);</span>
-<a id="__codelineno-196-4" name="__codelineno-196-4" href="#__codelineno-196-4"></a><span class="w">    </span><span class="c1">// Generate array nums = { 1, 2, 3, ..., n }</span>
+<a id="__codelineno-196-4" name="__codelineno-196-4" href="#__codelineno-196-4"></a><span class="w">    </span><span class="c1">// Generate the array nums = { 1, 2, 3, ..., n }</span>
 <a id="__codelineno-196-5" name="__codelineno-196-5" href="#__codelineno-196-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-196-6" name="__codelineno-196-6" href="#__codelineno-196-6"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-196-7" name="__codelineno-196-7" href="#__codelineno-196-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-196-8" name="__codelineno-196-8" href="#__codelineno-196-8"></a><span class="w">    </span><span class="c1">// Use system time to generate random seed</span>
+<a id="__codelineno-196-8" name="__codelineno-196-8" href="#__codelineno-196-8"></a><span class="w">    </span><span class="c1">// Use the system time to generate a random seed</span>
 <a id="__codelineno-196-9" name="__codelineno-196-9" href="#__codelineno-196-9"></a><span class="w">    </span><span class="kt">unsigned</span><span class="w"> </span><span class="n">seed</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">chrono</span><span class="o">::</span><span class="n">system_clock</span><span class="o">::</span><span class="n">now</span><span class="p">().</span><span class="n">time_since_epoch</span><span class="p">().</span><span class="n">count</span><span class="p">();</span>
-<a id="__codelineno-196-10" name="__codelineno-196-10" href="#__codelineno-196-10"></a><span class="w">    </span><span class="c1">// Randomly shuffle array elements</span>
+<a id="__codelineno-196-10" name="__codelineno-196-10" href="#__codelineno-196-10"></a><span class="w">    </span><span class="c1">// Randomly shuffle the array elements</span>
 <a id="__codelineno-196-11" name="__codelineno-196-11" href="#__codelineno-196-11"></a><span class="w">    </span><span class="n">shuffle</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">end</span><span class="p">(),</span><span class="w"> </span><span class="n">default_random_engine</span><span class="p">(</span><span class="n">seed</span><span class="p">));</span>
 <a id="__codelineno-196-12" name="__codelineno-196-12" href="#__codelineno-196-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span>
 <a id="__codelineno-196-13" name="__codelineno-196-13" href="#__codelineno-196-13"></a><span class="p">}</span>
 <a id="__codelineno-196-14" name="__codelineno-196-14" href="#__codelineno-196-14"></a>
-<a id="__codelineno-196-15" name="__codelineno-196-15" href="#__codelineno-196-15"></a><span class="cm">/* Find the index of number 1 in array nums */</span>
+<a id="__codelineno-196-15" name="__codelineno-196-15" href="#__codelineno-196-15"></a><span class="cm">/* Find the index of the number 1 in array nums */</span>
 <a id="__codelineno-196-16" name="__codelineno-196-16" href="#__codelineno-196-16"></a><span class="kt">int</span><span class="w"> </span><span class="n">findOne</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-196-17" name="__codelineno-196-17" href="#__codelineno-196-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-196-18" name="__codelineno-196-18" href="#__codelineno-196-18"></a><span class="w">        </span><span class="c1">// When element 1 is at the head of the array, best time complexity O(1) is achieved</span>
-<a id="__codelineno-196-19" name="__codelineno-196-19" href="#__codelineno-196-19"></a><span class="w">        </span><span class="c1">// When element 1 is at the tail of the array, worst time complexity O(n) is achieved</span>
+<a id="__codelineno-196-18" name="__codelineno-196-18" href="#__codelineno-196-18"></a><span class="w">        </span><span class="c1">// When 1 is at the start of the array, the best-case time complexity is O(1)</span>
+<a id="__codelineno-196-19" name="__codelineno-196-19" href="#__codelineno-196-19"></a><span class="w">        </span><span class="c1">// When 1 is at the end of the array, the worst-case time complexity is O(n)</span>
 <a id="__codelineno-196-20" name="__codelineno-196-20" href="#__codelineno-196-20"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-196-21" name="__codelineno-196-21" href="#__codelineno-196-21"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-196-22" name="__codelineno-196-22" href="#__codelineno-196-22"></a><span class="w">    </span><span class="p">}</span>
@@ -7838,14 +7965,14 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">worst_best_time_complexity.java</span><pre><span></span><code><a id="__codelineno-197-1" name="__codelineno-197-1" href="#__codelineno-197-1"></a><span class="cm">/* Generate an array with elements { 1, 2, ..., n }, order shuffled */</span>
+<div class="highlight"><span class="filename">worst_best_time_complexity.java</span><pre><span></span><code><a id="__codelineno-197-1" name="__codelineno-197-1" href="#__codelineno-197-1"></a><span class="cm">/* Generate an array containing { 1, 2, ..., n } in shuffled order */</span>
 <a id="__codelineno-197-2" name="__codelineno-197-2" href="#__codelineno-197-2"></a><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="nf">randomNumbers</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-197-3" name="__codelineno-197-3" href="#__codelineno-197-3"></a><span class="w">    </span><span class="n">Integer</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Integer</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-197-4" name="__codelineno-197-4" href="#__codelineno-197-4"></a><span class="w">    </span><span class="c1">// Generate array nums = { 1, 2, 3, ..., n }</span>
+<a id="__codelineno-197-4" name="__codelineno-197-4" href="#__codelineno-197-4"></a><span class="w">    </span><span class="c1">// Generate the array nums = { 1, 2, 3, ..., n }</span>
 <a id="__codelineno-197-5" name="__codelineno-197-5" href="#__codelineno-197-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-197-6" name="__codelineno-197-6" href="#__codelineno-197-6"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-197-7" name="__codelineno-197-7" href="#__codelineno-197-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-197-8" name="__codelineno-197-8" href="#__codelineno-197-8"></a><span class="w">    </span><span class="c1">// Randomly shuffle array elements</span>
+<a id="__codelineno-197-8" name="__codelineno-197-8" href="#__codelineno-197-8"></a><span class="w">    </span><span class="c1">// Randomly shuffle the array elements</span>
 <a id="__codelineno-197-9" name="__codelineno-197-9" href="#__codelineno-197-9"></a><span class="w">    </span><span class="n">Collections</span><span class="p">.</span><span class="na">shuffle</span><span class="p">(</span><span class="n">Arrays</span><span class="p">.</span><span class="na">asList</span><span class="p">(</span><span class="n">nums</span><span class="p">));</span>
 <a id="__codelineno-197-10" name="__codelineno-197-10" href="#__codelineno-197-10"></a><span class="w">    </span><span class="c1">// Integer[] -&gt; int[]</span>
 <a id="__codelineno-197-11" name="__codelineno-197-11" href="#__codelineno-197-11"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="p">;</span>
@@ -7855,11 +7982,11 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 <a id="__codelineno-197-15" name="__codelineno-197-15" href="#__codelineno-197-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-197-16" name="__codelineno-197-16" href="#__codelineno-197-16"></a><span class="p">}</span>
 <a id="__codelineno-197-17" name="__codelineno-197-17" href="#__codelineno-197-17"></a>
-<a id="__codelineno-197-18" name="__codelineno-197-18" href="#__codelineno-197-18"></a><span class="cm">/* Find the index of number 1 in array nums */</span>
+<a id="__codelineno-197-18" name="__codelineno-197-18" href="#__codelineno-197-18"></a><span class="cm">/* Find the index of the number 1 in array nums */</span>
 <a id="__codelineno-197-19" name="__codelineno-197-19" href="#__codelineno-197-19"></a><span class="kt">int</span><span class="w"> </span><span class="nf">findOne</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-197-20" name="__codelineno-197-20" href="#__codelineno-197-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-197-21" name="__codelineno-197-21" href="#__codelineno-197-21"></a><span class="w">        </span><span class="c1">// When element 1 is at the head of the array, best time complexity O(1) is achieved</span>
-<a id="__codelineno-197-22" name="__codelineno-197-22" href="#__codelineno-197-22"></a><span class="w">        </span><span class="c1">// When element 1 is at the tail of the array, worst time complexity O(n) is achieved</span>
+<a id="__codelineno-197-21" name="__codelineno-197-21" href="#__codelineno-197-21"></a><span class="w">        </span><span class="c1">// When 1 is at the start of the array, the best-case time complexity is O(1)</span>
+<a id="__codelineno-197-22" name="__codelineno-197-22" href="#__codelineno-197-22"></a><span class="w">        </span><span class="c1">// When 1 is at the end of the array, the worst-case time complexity is O(n)</span>
 <a id="__codelineno-197-23" name="__codelineno-197-23" href="#__codelineno-197-23"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-197-24" name="__codelineno-197-24" href="#__codelineno-197-24"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-197-25" name="__codelineno-197-25" href="#__codelineno-197-25"></a><span class="w">    </span><span class="p">}</span>
@@ -7868,15 +7995,15 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">worst_best_time_complexity.cs</span><pre><span></span><code><a id="__codelineno-198-1" name="__codelineno-198-1" href="#__codelineno-198-1"></a><span class="cm">/* Generate an array with elements { 1, 2, ..., n }, order shuffled */</span>
+<div class="highlight"><span class="filename">worst_best_time_complexity.cs</span><pre><span></span><code><a id="__codelineno-198-1" name="__codelineno-198-1" href="#__codelineno-198-1"></a><span class="cm">/* Generate an array containing { 1, 2, ..., n } in shuffled order */</span>
 <a id="__codelineno-198-2" name="__codelineno-198-2" href="#__codelineno-198-2"></a><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="nf">RandomNumbers</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-198-3" name="__codelineno-198-3" href="#__codelineno-198-3"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="p">];</span>
-<a id="__codelineno-198-4" name="__codelineno-198-4" href="#__codelineno-198-4"></a><span class="w">    </span><span class="c1">// Generate array nums = { 1, 2, 3, ..., n }</span>
+<a id="__codelineno-198-4" name="__codelineno-198-4" href="#__codelineno-198-4"></a><span class="w">    </span><span class="c1">// Generate the array nums = { 1, 2, 3, ..., n }</span>
 <a id="__codelineno-198-5" name="__codelineno-198-5" href="#__codelineno-198-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-198-6" name="__codelineno-198-6" href="#__codelineno-198-6"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-198-7" name="__codelineno-198-7" href="#__codelineno-198-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-198-8" name="__codelineno-198-8" href="#__codelineno-198-8"></a>
-<a id="__codelineno-198-9" name="__codelineno-198-9" href="#__codelineno-198-9"></a><span class="w">    </span><span class="c1">// Randomly shuffle array elements</span>
+<a id="__codelineno-198-9" name="__codelineno-198-9" href="#__codelineno-198-9"></a><span class="w">    </span><span class="c1">// Randomly shuffle the array elements</span>
 <a id="__codelineno-198-10" name="__codelineno-198-10" href="#__codelineno-198-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-198-11" name="__codelineno-198-11" href="#__codelineno-198-11"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Random</span><span class="p">().</span><span class="n">Next</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">);</span>
 <a id="__codelineno-198-12" name="__codelineno-198-12" href="#__codelineno-198-12"></a><span class="w">        </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">index</span><span class="p">])</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">index</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
@@ -7884,11 +8011,11 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 <a id="__codelineno-198-14" name="__codelineno-198-14" href="#__codelineno-198-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span>
 <a id="__codelineno-198-15" name="__codelineno-198-15" href="#__codelineno-198-15"></a><span class="p">}</span>
 <a id="__codelineno-198-16" name="__codelineno-198-16" href="#__codelineno-198-16"></a>
-<a id="__codelineno-198-17" name="__codelineno-198-17" href="#__codelineno-198-17"></a><span class="cm">/* Find the index of number 1 in array nums */</span>
+<a id="__codelineno-198-17" name="__codelineno-198-17" href="#__codelineno-198-17"></a><span class="cm">/* Find the index of the number 1 in array nums */</span>
 <a id="__codelineno-198-18" name="__codelineno-198-18" href="#__codelineno-198-18"></a><span class="kt">int</span><span class="w"> </span><span class="nf">FindOne</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-198-19" name="__codelineno-198-19" href="#__codelineno-198-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-198-20" name="__codelineno-198-20" href="#__codelineno-198-20"></a><span class="w">        </span><span class="c1">// When element 1 is at the head of the array, best time complexity O(1) is achieved</span>
-<a id="__codelineno-198-21" name="__codelineno-198-21" href="#__codelineno-198-21"></a><span class="w">        </span><span class="c1">// When element 1 is at the tail of the array, worst time complexity O(n) is achieved</span>
+<a id="__codelineno-198-20" name="__codelineno-198-20" href="#__codelineno-198-20"></a><span class="w">        </span><span class="c1">// When 1 is at the start of the array, the best-case time complexity is O(1)</span>
+<a id="__codelineno-198-21" name="__codelineno-198-21" href="#__codelineno-198-21"></a><span class="w">        </span><span class="c1">// When 1 is at the end of the array, the worst-case time complexity is O(n)</span>
 <a id="__codelineno-198-22" name="__codelineno-198-22" href="#__codelineno-198-22"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-198-23" name="__codelineno-198-23" href="#__codelineno-198-23"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-198-24" name="__codelineno-198-24" href="#__codelineno-198-24"></a><span class="w">    </span><span class="p">}</span>
@@ -7897,25 +8024,25 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">worst_best_time_complexity.go</span><pre><span></span><code><a id="__codelineno-199-1" name="__codelineno-199-1" href="#__codelineno-199-1"></a><span class="cm">/* Generate an array with elements { 1, 2, ..., n }, order shuffled */</span>
+<div class="highlight"><span class="filename">worst_best_time_complexity.go</span><pre><span></span><code><a id="__codelineno-199-1" name="__codelineno-199-1" href="#__codelineno-199-1"></a><span class="cm">/* Generate an array containing { 1, 2, ..., n } in shuffled order */</span>
 <a id="__codelineno-199-2" name="__codelineno-199-2" href="#__codelineno-199-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">randomNumbers</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-199-3" name="__codelineno-199-3" href="#__codelineno-199-3"></a><span class="w">    </span><span class="nx">nums</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
-<a id="__codelineno-199-4" name="__codelineno-199-4" href="#__codelineno-199-4"></a><span class="w">    </span><span class="c1">// Generate array nums = { 1, 2, 3, ..., n }</span>
+<a id="__codelineno-199-4" name="__codelineno-199-4" href="#__codelineno-199-4"></a><span class="w">    </span><span class="c1">// Generate the array nums = { 1, 2, 3, ..., n }</span>
 <a id="__codelineno-199-5" name="__codelineno-199-5" href="#__codelineno-199-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-199-6" name="__codelineno-199-6" href="#__codelineno-199-6"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-199-7" name="__codelineno-199-7" href="#__codelineno-199-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-199-8" name="__codelineno-199-8" href="#__codelineno-199-8"></a><span class="w">    </span><span class="c1">// Randomly shuffle array elements</span>
+<a id="__codelineno-199-8" name="__codelineno-199-8" href="#__codelineno-199-8"></a><span class="w">    </span><span class="c1">// Randomly shuffle the array elements</span>
 <a id="__codelineno-199-9" name="__codelineno-199-9" href="#__codelineno-199-9"></a><span class="w">    </span><span class="nx">rand</span><span class="p">.</span><span class="nx">Shuffle</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">),</span><span class="w"> </span><span class="kd">func</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-199-10" name="__codelineno-199-10" href="#__codelineno-199-10"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
 <a id="__codelineno-199-11" name="__codelineno-199-11" href="#__codelineno-199-11"></a><span class="w">    </span><span class="p">})</span>
 <a id="__codelineno-199-12" name="__codelineno-199-12" href="#__codelineno-199-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">nums</span>
 <a id="__codelineno-199-13" name="__codelineno-199-13" href="#__codelineno-199-13"></a><span class="p">}</span>
 <a id="__codelineno-199-14" name="__codelineno-199-14" href="#__codelineno-199-14"></a>
-<a id="__codelineno-199-15" name="__codelineno-199-15" href="#__codelineno-199-15"></a><span class="cm">/* Find the index of number 1 in array nums */</span>
+<a id="__codelineno-199-15" name="__codelineno-199-15" href="#__codelineno-199-15"></a><span class="cm">/* Find the index of the number 1 in array nums */</span>
 <a id="__codelineno-199-16" name="__codelineno-199-16" href="#__codelineno-199-16"></a><span class="kd">func</span><span class="w"> </span><span class="nx">findOne</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-199-17" name="__codelineno-199-17" href="#__codelineno-199-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-199-18" name="__codelineno-199-18" href="#__codelineno-199-18"></a><span class="w">        </span><span class="c1">// When element 1 is at the head of the array, best time complexity O(1) is achieved</span>
-<a id="__codelineno-199-19" name="__codelineno-199-19" href="#__codelineno-199-19"></a><span class="w">        </span><span class="c1">// When element 1 is at the tail of the array, worst time complexity O(n) is achieved</span>
+<a id="__codelineno-199-18" name="__codelineno-199-18" href="#__codelineno-199-18"></a><span class="w">        </span><span class="c1">// When 1 is at the start of the array, the best-case time complexity is O(1)</span>
+<a id="__codelineno-199-19" name="__codelineno-199-19" href="#__codelineno-199-19"></a><span class="w">        </span><span class="c1">// When 1 is at the end of the array, the worst-case time complexity is O(n)</span>
 <a id="__codelineno-199-20" name="__codelineno-199-20" href="#__codelineno-199-20"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-199-21" name="__codelineno-199-21" href="#__codelineno-199-21"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span>
 <a id="__codelineno-199-22" name="__codelineno-199-22" href="#__codelineno-199-22"></a><span class="w">        </span><span class="p">}</span>
@@ -7925,20 +8052,20 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">worst_best_time_complexity.swift</span><pre><span></span><code><a id="__codelineno-200-1" name="__codelineno-200-1" href="#__codelineno-200-1"></a><span class="cm">/* Generate an array with elements { 1, 2, ..., n }, order shuffled */</span>
+<div class="highlight"><span class="filename">worst_best_time_complexity.swift</span><pre><span></span><code><a id="__codelineno-200-1" name="__codelineno-200-1" href="#__codelineno-200-1"></a><span class="cm">/* Generate an array containing { 1, 2, ..., n } in shuffled order */</span>
 <a id="__codelineno-200-2" name="__codelineno-200-2" href="#__codelineno-200-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">randomNumbers</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-200-3" name="__codelineno-200-3" href="#__codelineno-200-3"></a><span class="w">    </span><span class="c1">// Generate array nums = { 1, 2, 3, ..., n }</span>
+<a id="__codelineno-200-3" name="__codelineno-200-3" href="#__codelineno-200-3"></a><span class="w">    </span><span class="c1">// Generate the array nums = { 1, 2, 3, ..., n }</span>
 <a id="__codelineno-200-4" name="__codelineno-200-4" href="#__codelineno-200-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">nums</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-200-5" name="__codelineno-200-5" href="#__codelineno-200-5"></a><span class="w">    </span><span class="c1">// Randomly shuffle array elements</span>
+<a id="__codelineno-200-5" name="__codelineno-200-5" href="#__codelineno-200-5"></a><span class="w">    </span><span class="c1">// Randomly shuffle the array elements</span>
 <a id="__codelineno-200-6" name="__codelineno-200-6" href="#__codelineno-200-6"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="n">shuffle</span><span class="p">()</span>
 <a id="__codelineno-200-7" name="__codelineno-200-7" href="#__codelineno-200-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span>
 <a id="__codelineno-200-8" name="__codelineno-200-8" href="#__codelineno-200-8"></a><span class="p">}</span>
 <a id="__codelineno-200-9" name="__codelineno-200-9" href="#__codelineno-200-9"></a>
-<a id="__codelineno-200-10" name="__codelineno-200-10" href="#__codelineno-200-10"></a><span class="cm">/* Find the index of number 1 in array nums */</span>
+<a id="__codelineno-200-10" name="__codelineno-200-10" href="#__codelineno-200-10"></a><span class="cm">/* Find the index of the number 1 in array nums */</span>
 <a id="__codelineno-200-11" name="__codelineno-200-11" href="#__codelineno-200-11"></a><span class="kd">func</span><span class="w"> </span><span class="nf">findOne</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-200-12" name="__codelineno-200-12" href="#__codelineno-200-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-200-13" name="__codelineno-200-13" href="#__codelineno-200-13"></a><span class="w">        </span><span class="c1">// When element 1 is at the head of the array, best time complexity O(1) is achieved</span>
-<a id="__codelineno-200-14" name="__codelineno-200-14" href="#__codelineno-200-14"></a><span class="w">        </span><span class="c1">// When element 1 is at the tail of the array, worst time complexity O(n) is achieved</span>
+<a id="__codelineno-200-13" name="__codelineno-200-13" href="#__codelineno-200-13"></a><span class="w">        </span><span class="c1">// When 1 is at the start of the array, the best-case time complexity is O(1)</span>
+<a id="__codelineno-200-14" name="__codelineno-200-14" href="#__codelineno-200-14"></a><span class="w">        </span><span class="c1">// When 1 is at the end of the array, the worst-case time complexity is O(n)</span>
 <a id="__codelineno-200-15" name="__codelineno-200-15" href="#__codelineno-200-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-200-16" name="__codelineno-200-16" href="#__codelineno-200-16"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-200-17" name="__codelineno-200-17" href="#__codelineno-200-17"></a><span class="w">        </span><span class="p">}</span>
@@ -7948,14 +8075,14 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">worst_best_time_complexity.js</span><pre><span></span><code><a id="__codelineno-201-1" name="__codelineno-201-1" href="#__codelineno-201-1"></a><span class="cm">/* Generate an array with elements { 1, 2, ..., n }, order shuffled */</span>
+<div class="highlight"><span class="filename">worst_best_time_complexity.js</span><pre><span></span><code><a id="__codelineno-201-1" name="__codelineno-201-1" href="#__codelineno-201-1"></a><span class="cm">/* Generate an array containing { 1, 2, ..., n } in shuffled order */</span>
 <a id="__codelineno-201-2" name="__codelineno-201-2" href="#__codelineno-201-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">randomNumbers</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-201-3" name="__codelineno-201-3" href="#__codelineno-201-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">);</span>
-<a id="__codelineno-201-4" name="__codelineno-201-4" href="#__codelineno-201-4"></a><span class="w">    </span><span class="c1">// Generate array nums = { 1, 2, 3, ..., n }</span>
+<a id="__codelineno-201-4" name="__codelineno-201-4" href="#__codelineno-201-4"></a><span class="w">    </span><span class="c1">// Generate the array nums = { 1, 2, 3, ..., n }</span>
 <a id="__codelineno-201-5" name="__codelineno-201-5" href="#__codelineno-201-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-201-6" name="__codelineno-201-6" href="#__codelineno-201-6"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-201-7" name="__codelineno-201-7" href="#__codelineno-201-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-201-8" name="__codelineno-201-8" href="#__codelineno-201-8"></a><span class="w">    </span><span class="c1">// Randomly shuffle array elements</span>
+<a id="__codelineno-201-8" name="__codelineno-201-8" href="#__codelineno-201-8"></a><span class="w">    </span><span class="c1">// Randomly shuffle the array elements</span>
 <a id="__codelineno-201-9" name="__codelineno-201-9" href="#__codelineno-201-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-201-10" name="__codelineno-201-10" href="#__codelineno-201-10"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nb">Math</span><span class="p">.</span><span class="nx">random</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">));</span>
 <a id="__codelineno-201-11" name="__codelineno-201-11" href="#__codelineno-201-11"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
@@ -7965,11 +8092,11 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 <a id="__codelineno-201-15" name="__codelineno-201-15" href="#__codelineno-201-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">nums</span><span class="p">;</span>
 <a id="__codelineno-201-16" name="__codelineno-201-16" href="#__codelineno-201-16"></a><span class="p">}</span>
 <a id="__codelineno-201-17" name="__codelineno-201-17" href="#__codelineno-201-17"></a>
-<a id="__codelineno-201-18" name="__codelineno-201-18" href="#__codelineno-201-18"></a><span class="cm">/* Find the index of number 1 in array nums */</span>
+<a id="__codelineno-201-18" name="__codelineno-201-18" href="#__codelineno-201-18"></a><span class="cm">/* Find the index of the number 1 in array nums */</span>
 <a id="__codelineno-201-19" name="__codelineno-201-19" href="#__codelineno-201-19"></a><span class="kd">function</span><span class="w"> </span><span class="nx">findOne</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-201-20" name="__codelineno-201-20" href="#__codelineno-201-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-201-21" name="__codelineno-201-21" href="#__codelineno-201-21"></a><span class="w">        </span><span class="c1">// When element 1 is at the head of the array, best time complexity O(1) is achieved</span>
-<a id="__codelineno-201-22" name="__codelineno-201-22" href="#__codelineno-201-22"></a><span class="w">        </span><span class="c1">// When element 1 is at the tail of the array, worst time complexity O(n) is achieved</span>
+<a id="__codelineno-201-21" name="__codelineno-201-21" href="#__codelineno-201-21"></a><span class="w">        </span><span class="c1">// When 1 is at the start of the array, the best-case time complexity is O(1)</span>
+<a id="__codelineno-201-22" name="__codelineno-201-22" href="#__codelineno-201-22"></a><span class="w">        </span><span class="c1">// When 1 is at the end of the array, the worst-case time complexity is O(n)</span>
 <a id="__codelineno-201-23" name="__codelineno-201-23" href="#__codelineno-201-23"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-201-24" name="__codelineno-201-24" href="#__codelineno-201-24"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-201-25" name="__codelineno-201-25" href="#__codelineno-201-25"></a><span class="w">        </span><span class="p">}</span>
@@ -7979,14 +8106,14 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">worst_best_time_complexity.ts</span><pre><span></span><code><a id="__codelineno-202-1" name="__codelineno-202-1" href="#__codelineno-202-1"></a><span class="cm">/* Generate an array with elements { 1, 2, ..., n }, order shuffled */</span>
+<div class="highlight"><span class="filename">worst_best_time_complexity.ts</span><pre><span></span><code><a id="__codelineno-202-1" name="__codelineno-202-1" href="#__codelineno-202-1"></a><span class="cm">/* Generate an array containing { 1, 2, ..., n } in shuffled order */</span>
 <a id="__codelineno-202-2" name="__codelineno-202-2" href="#__codelineno-202-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">randomNumbers</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-202-3" name="__codelineno-202-3" href="#__codelineno-202-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">);</span>
-<a id="__codelineno-202-4" name="__codelineno-202-4" href="#__codelineno-202-4"></a><span class="w">    </span><span class="c1">// Generate array nums = { 1, 2, 3, ..., n }</span>
+<a id="__codelineno-202-4" name="__codelineno-202-4" href="#__codelineno-202-4"></a><span class="w">    </span><span class="c1">// Generate the array nums = { 1, 2, 3, ..., n }</span>
 <a id="__codelineno-202-5" name="__codelineno-202-5" href="#__codelineno-202-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-202-6" name="__codelineno-202-6" href="#__codelineno-202-6"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-202-7" name="__codelineno-202-7" href="#__codelineno-202-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-202-8" name="__codelineno-202-8" href="#__codelineno-202-8"></a><span class="w">    </span><span class="c1">// Randomly shuffle array elements</span>
+<a id="__codelineno-202-8" name="__codelineno-202-8" href="#__codelineno-202-8"></a><span class="w">    </span><span class="c1">// Randomly shuffle the array elements</span>
 <a id="__codelineno-202-9" name="__codelineno-202-9" href="#__codelineno-202-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-202-10" name="__codelineno-202-10" href="#__codelineno-202-10"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nb">Math</span><span class="p">.</span><span class="nx">random</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">));</span>
 <a id="__codelineno-202-11" name="__codelineno-202-11" href="#__codelineno-202-11"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
@@ -7996,11 +8123,11 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 <a id="__codelineno-202-15" name="__codelineno-202-15" href="#__codelineno-202-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">nums</span><span class="p">;</span>
 <a id="__codelineno-202-16" name="__codelineno-202-16" href="#__codelineno-202-16"></a><span class="p">}</span>
 <a id="__codelineno-202-17" name="__codelineno-202-17" href="#__codelineno-202-17"></a>
-<a id="__codelineno-202-18" name="__codelineno-202-18" href="#__codelineno-202-18"></a><span class="cm">/* Find the index of number 1 in array nums */</span>
+<a id="__codelineno-202-18" name="__codelineno-202-18" href="#__codelineno-202-18"></a><span class="cm">/* Find the index of the number 1 in array nums */</span>
 <a id="__codelineno-202-19" name="__codelineno-202-19" href="#__codelineno-202-19"></a><span class="kd">function</span><span class="w"> </span><span class="nx">findOne</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-202-20" name="__codelineno-202-20" href="#__codelineno-202-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-202-21" name="__codelineno-202-21" href="#__codelineno-202-21"></a><span class="w">        </span><span class="c1">// When element 1 is at the head of the array, best time complexity O(1) is achieved</span>
-<a id="__codelineno-202-22" name="__codelineno-202-22" href="#__codelineno-202-22"></a><span class="w">        </span><span class="c1">// When element 1 is at the tail of the array, worst time complexity O(n) is achieved</span>
+<a id="__codelineno-202-21" name="__codelineno-202-21" href="#__codelineno-202-21"></a><span class="w">        </span><span class="c1">// When 1 is at the start of the array, the best-case time complexity is O(1)</span>
+<a id="__codelineno-202-22" name="__codelineno-202-22" href="#__codelineno-202-22"></a><span class="w">        </span><span class="c1">// When 1 is at the end of the array, the worst-case time complexity is O(n)</span>
 <a id="__codelineno-202-23" name="__codelineno-202-23" href="#__codelineno-202-23"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-202-24" name="__codelineno-202-24" href="#__codelineno-202-24"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-202-25" name="__codelineno-202-25" href="#__codelineno-202-25"></a><span class="w">        </span><span class="p">}</span>
@@ -8010,24 +8137,24 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">worst_best_time_complexity.dart</span><pre><span></span><code><a id="__codelineno-203-1" name="__codelineno-203-1" href="#__codelineno-203-1"></a><span class="cm">/* Generate an array with elements { 1, 2, ..., n }, order shuffled */</span>
+<div class="highlight"><span class="filename">worst_best_time_complexity.dart</span><pre><span></span><code><a id="__codelineno-203-1" name="__codelineno-203-1" href="#__codelineno-203-1"></a><span class="cm">/* Generate an array containing { 1, 2, ..., n } in shuffled order */</span>
 <a id="__codelineno-203-2" name="__codelineno-203-2" href="#__codelineno-203-2"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">randomNumbers</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-203-3" name="__codelineno-203-3" href="#__codelineno-203-3"></a><span class="w">  </span><span class="kd">final</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
-<a id="__codelineno-203-4" name="__codelineno-203-4" href="#__codelineno-203-4"></a><span class="w">  </span><span class="c1">// Generate array nums = { 1, 2, 3, ..., n }</span>
+<a id="__codelineno-203-4" name="__codelineno-203-4" href="#__codelineno-203-4"></a><span class="w">  </span><span class="c1">// Generate the array nums = { 1, 2, 3, ..., n }</span>
 <a id="__codelineno-203-5" name="__codelineno-203-5" href="#__codelineno-203-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-203-6" name="__codelineno-203-6" href="#__codelineno-203-6"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-203-7" name="__codelineno-203-7" href="#__codelineno-203-7"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-203-8" name="__codelineno-203-8" href="#__codelineno-203-8"></a><span class="w">  </span><span class="c1">// Randomly shuffle array elements</span>
+<a id="__codelineno-203-8" name="__codelineno-203-8" href="#__codelineno-203-8"></a><span class="w">  </span><span class="c1">// Randomly shuffle the array elements</span>
 <a id="__codelineno-203-9" name="__codelineno-203-9" href="#__codelineno-203-9"></a><span class="w">  </span><span class="n">nums</span><span class="p">.</span><span class="n">shuffle</span><span class="p">();</span>
 <a id="__codelineno-203-10" name="__codelineno-203-10" href="#__codelineno-203-10"></a>
 <a id="__codelineno-203-11" name="__codelineno-203-11" href="#__codelineno-203-11"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span>
 <a id="__codelineno-203-12" name="__codelineno-203-12" href="#__codelineno-203-12"></a><span class="p">}</span>
 <a id="__codelineno-203-13" name="__codelineno-203-13" href="#__codelineno-203-13"></a>
-<a id="__codelineno-203-14" name="__codelineno-203-14" href="#__codelineno-203-14"></a><span class="cm">/* Find the index of number 1 in array nums */</span>
+<a id="__codelineno-203-14" name="__codelineno-203-14" href="#__codelineno-203-14"></a><span class="cm">/* Find the index of the number 1 in array nums */</span>
 <a id="__codelineno-203-15" name="__codelineno-203-15" href="#__codelineno-203-15"></a><span class="kt">int</span><span class="w"> </span><span class="n">findOne</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-203-16" name="__codelineno-203-16" href="#__codelineno-203-16"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-203-17" name="__codelineno-203-17" href="#__codelineno-203-17"></a><span class="w">    </span><span class="c1">// When element 1 is at the head of the array, best time complexity O(1) is achieved</span>
-<a id="__codelineno-203-18" name="__codelineno-203-18" href="#__codelineno-203-18"></a><span class="w">    </span><span class="c1">// When element 1 is at the tail of the array, worst time complexity O(n) is achieved</span>
+<a id="__codelineno-203-17" name="__codelineno-203-17" href="#__codelineno-203-17"></a><span class="w">    </span><span class="c1">// When 1 is at the start of the array, the best-case time complexity is O(1)</span>
+<a id="__codelineno-203-18" name="__codelineno-203-18" href="#__codelineno-203-18"></a><span class="w">    </span><span class="c1">// When 1 is at the end of the array, the worst-case time complexity is O(n)</span>
 <a id="__codelineno-203-19" name="__codelineno-203-19" href="#__codelineno-203-19"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-203-20" name="__codelineno-203-20" href="#__codelineno-203-20"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-203-21" name="__codelineno-203-21" href="#__codelineno-203-21"></a>
@@ -8036,20 +8163,20 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">worst_best_time_complexity.rs</span><pre><span></span><code><a id="__codelineno-204-1" name="__codelineno-204-1" href="#__codelineno-204-1"></a><span class="cm">/* Generate an array with elements { 1, 2, ..., n }, order shuffled */</span>
+<div class="highlight"><span class="filename">worst_best_time_complexity.rs</span><pre><span></span><code><a id="__codelineno-204-1" name="__codelineno-204-1" href="#__codelineno-204-1"></a><span class="cm">/* Generate an array containing { 1, 2, ..., n } in shuffled order */</span>
 <a id="__codelineno-204-2" name="__codelineno-204-2" href="#__codelineno-204-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">random_numbers</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-204-3" name="__codelineno-204-3" href="#__codelineno-204-3"></a><span class="w">    </span><span class="c1">// Generate array nums = { 1, 2, 3, ..., n }</span>
+<a id="__codelineno-204-3" name="__codelineno-204-3" href="#__codelineno-204-3"></a><span class="w">    </span><span class="c1">// Generate the array nums = { 1, 2, 3, ..., n }</span>
 <a id="__codelineno-204-4" name="__codelineno-204-4" href="#__codelineno-204-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="p">).</span><span class="n">collect</span><span class="p">::</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">();</span>
-<a id="__codelineno-204-5" name="__codelineno-204-5" href="#__codelineno-204-5"></a><span class="w">    </span><span class="c1">// Randomly shuffle array elements</span>
+<a id="__codelineno-204-5" name="__codelineno-204-5" href="#__codelineno-204-5"></a><span class="w">    </span><span class="c1">// Randomly shuffle the array elements</span>
 <a id="__codelineno-204-6" name="__codelineno-204-6" href="#__codelineno-204-6"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="n">shuffle</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">thread_rng</span><span class="p">());</span>
 <a id="__codelineno-204-7" name="__codelineno-204-7" href="#__codelineno-204-7"></a><span class="w">    </span><span class="n">nums</span>
 <a id="__codelineno-204-8" name="__codelineno-204-8" href="#__codelineno-204-8"></a><span class="p">}</span>
 <a id="__codelineno-204-9" name="__codelineno-204-9" href="#__codelineno-204-9"></a>
-<a id="__codelineno-204-10" name="__codelineno-204-10" href="#__codelineno-204-10"></a><span class="cm">/* Find the index of number 1 in array nums */</span>
+<a id="__codelineno-204-10" name="__codelineno-204-10" href="#__codelineno-204-10"></a><span class="cm">/* Find the index of the number 1 in array nums */</span>
 <a id="__codelineno-204-11" name="__codelineno-204-11" href="#__codelineno-204-11"></a><span class="k">fn</span><span class="w"> </span><span class="nf">find_one</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="kt">usize</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-204-12" name="__codelineno-204-12" href="#__codelineno-204-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-204-13" name="__codelineno-204-13" href="#__codelineno-204-13"></a><span class="w">        </span><span class="c1">// When element 1 is at the head of the array, best time complexity O(1) is achieved</span>
-<a id="__codelineno-204-14" name="__codelineno-204-14" href="#__codelineno-204-14"></a><span class="w">        </span><span class="c1">// When element 1 is at the tail of the array, worst time complexity O(n) is achieved</span>
+<a id="__codelineno-204-13" name="__codelineno-204-13" href="#__codelineno-204-13"></a><span class="w">        </span><span class="c1">// When 1 is at the start of the array, the best-case time complexity is O(1)</span>
+<a id="__codelineno-204-14" name="__codelineno-204-14" href="#__codelineno-204-14"></a><span class="w">        </span><span class="c1">// When 1 is at the end of the array, the worst-case time complexity is O(n)</span>
 <a id="__codelineno-204-15" name="__codelineno-204-15" href="#__codelineno-204-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-204-16" name="__codelineno-204-16" href="#__codelineno-204-16"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-204-17" name="__codelineno-204-17" href="#__codelineno-204-17"></a><span class="w">        </span><span class="p">}</span>
@@ -8059,15 +8186,15 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">worst_best_time_complexity.c</span><pre><span></span><code><a id="__codelineno-205-1" name="__codelineno-205-1" href="#__codelineno-205-1"></a><span class="cm">/* Generate an array with elements { 1, 2, ..., n }, order shuffled */</span>
+<div class="highlight"><span class="filename">worst_best_time_complexity.c</span><pre><span></span><code><a id="__codelineno-205-1" name="__codelineno-205-1" href="#__codelineno-205-1"></a><span class="cm">/* Generate an array containing { 1, 2, ..., n } in shuffled order */</span>
 <a id="__codelineno-205-2" name="__codelineno-205-2" href="#__codelineno-205-2"></a><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="nf">randomNumbers</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-205-3" name="__codelineno-205-3" href="#__codelineno-205-3"></a><span class="w">    </span><span class="c1">// Allocate heap memory (create 1D variable-length array: n elements of type int)</span>
+<a id="__codelineno-205-3" name="__codelineno-205-3" href="#__codelineno-205-3"></a><span class="w">    </span><span class="c1">// Allocate heap memory for a one-dimensional array of n int elements</span>
 <a id="__codelineno-205-4" name="__codelineno-205-4" href="#__codelineno-205-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
-<a id="__codelineno-205-5" name="__codelineno-205-5" href="#__codelineno-205-5"></a><span class="w">    </span><span class="c1">// Generate array nums = { 1, 2, 3, ..., n }</span>
+<a id="__codelineno-205-5" name="__codelineno-205-5" href="#__codelineno-205-5"></a><span class="w">    </span><span class="c1">// Generate the array nums = { 1, 2, 3, ..., n }</span>
 <a id="__codelineno-205-6" name="__codelineno-205-6" href="#__codelineno-205-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-205-7" name="__codelineno-205-7" href="#__codelineno-205-7"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-205-8" name="__codelineno-205-8" href="#__codelineno-205-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-205-9" name="__codelineno-205-9" href="#__codelineno-205-9"></a><span class="w">    </span><span class="c1">// Randomly shuffle array elements</span>
+<a id="__codelineno-205-9" name="__codelineno-205-9" href="#__codelineno-205-9"></a><span class="w">    </span><span class="c1">// Randomly shuffle the array elements</span>
 <a id="__codelineno-205-10" name="__codelineno-205-10" href="#__codelineno-205-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-205-11" name="__codelineno-205-11" href="#__codelineno-205-11"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rand</span><span class="p">()</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-205-12" name="__codelineno-205-12" href="#__codelineno-205-12"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
@@ -8077,11 +8204,11 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 <a id="__codelineno-205-16" name="__codelineno-205-16" href="#__codelineno-205-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span>
 <a id="__codelineno-205-17" name="__codelineno-205-17" href="#__codelineno-205-17"></a><span class="p">}</span>
 <a id="__codelineno-205-18" name="__codelineno-205-18" href="#__codelineno-205-18"></a>
-<a id="__codelineno-205-19" name="__codelineno-205-19" href="#__codelineno-205-19"></a><span class="cm">/* Find the index of number 1 in array nums */</span>
+<a id="__codelineno-205-19" name="__codelineno-205-19" href="#__codelineno-205-19"></a><span class="cm">/* Find the index of the number 1 in array nums */</span>
 <a id="__codelineno-205-20" name="__codelineno-205-20" href="#__codelineno-205-20"></a><span class="kt">int</span><span class="w"> </span><span class="nf">findOne</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-205-21" name="__codelineno-205-21" href="#__codelineno-205-21"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-205-22" name="__codelineno-205-22" href="#__codelineno-205-22"></a><span class="w">        </span><span class="c1">// When element 1 is at the head of the array, best time complexity O(1) is achieved</span>
-<a id="__codelineno-205-23" name="__codelineno-205-23" href="#__codelineno-205-23"></a><span class="w">        </span><span class="c1">// When element 1 is at the tail of the array, worst time complexity O(n) is achieved</span>
+<a id="__codelineno-205-22" name="__codelineno-205-22" href="#__codelineno-205-22"></a><span class="w">        </span><span class="c1">// When 1 is at the start of the array, the best-case time complexity is O(1)</span>
+<a id="__codelineno-205-23" name="__codelineno-205-23" href="#__codelineno-205-23"></a><span class="w">        </span><span class="c1">// When 1 is at the end of the array, the worst-case time complexity is O(n)</span>
 <a id="__codelineno-205-24" name="__codelineno-205-24" href="#__codelineno-205-24"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-205-25" name="__codelineno-205-25" href="#__codelineno-205-25"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-205-26" name="__codelineno-205-26" href="#__codelineno-205-26"></a><span class="w">    </span><span class="p">}</span>
@@ -8090,14 +8217,14 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">worst_best_time_complexity.kt</span><pre><span></span><code><a id="__codelineno-206-1" name="__codelineno-206-1" href="#__codelineno-206-1"></a><span class="cm">/* Generate an array with elements { 1, 2, ..., n }, order shuffled */</span>
+<div class="highlight"><span class="filename">worst_best_time_complexity.kt</span><pre><span></span><code><a id="__codelineno-206-1" name="__codelineno-206-1" href="#__codelineno-206-1"></a><span class="cm">/* Generate an array containing { 1, 2, ..., n } in shuffled order */</span>
 <a id="__codelineno-206-2" name="__codelineno-206-2" href="#__codelineno-206-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">randomNumbers</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="n">Array</span><span class="o">&lt;</span><span class="kt">Int?</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-206-3" name="__codelineno-206-3" href="#__codelineno-206-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-206-4" name="__codelineno-206-4" href="#__codelineno-206-4"></a><span class="w">    </span><span class="c1">// Generate array nums = { 1, 2, 3, ..., n }</span>
+<a id="__codelineno-206-4" name="__codelineno-206-4" href="#__codelineno-206-4"></a><span class="w">    </span><span class="c1">// Generate the array nums = { 1, 2, 3, ..., n }</span>
 <a id="__codelineno-206-5" name="__codelineno-206-5" href="#__codelineno-206-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-206-6" name="__codelineno-206-6" href="#__codelineno-206-6"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-206-7" name="__codelineno-206-7" href="#__codelineno-206-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-206-8" name="__codelineno-206-8" href="#__codelineno-206-8"></a><span class="w">    </span><span class="c1">// Randomly shuffle array elements</span>
+<a id="__codelineno-206-8" name="__codelineno-206-8" href="#__codelineno-206-8"></a><span class="w">    </span><span class="c1">// Randomly shuffle the array elements</span>
 <a id="__codelineno-206-9" name="__codelineno-206-9" href="#__codelineno-206-9"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="na">shuffle</span><span class="p">()</span>
 <a id="__codelineno-206-10" name="__codelineno-206-10" href="#__codelineno-206-10"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arrayOfNulls</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-206-11" name="__codelineno-206-11" href="#__codelineno-206-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -8106,11 +8233,11 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 <a id="__codelineno-206-14" name="__codelineno-206-14" href="#__codelineno-206-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-206-15" name="__codelineno-206-15" href="#__codelineno-206-15"></a><span class="p">}</span>
 <a id="__codelineno-206-16" name="__codelineno-206-16" href="#__codelineno-206-16"></a>
-<a id="__codelineno-206-17" name="__codelineno-206-17" href="#__codelineno-206-17"></a><span class="cm">/* Find the index of number 1 in array nums */</span>
+<a id="__codelineno-206-17" name="__codelineno-206-17" href="#__codelineno-206-17"></a><span class="cm">/* Find the index of the number 1 in array nums */</span>
 <a id="__codelineno-206-18" name="__codelineno-206-18" href="#__codelineno-206-18"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">findOne</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">Array</span><span class="o">&lt;</span><span class="kt">Int?</span><span class="o">&gt;</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-206-19" name="__codelineno-206-19" href="#__codelineno-206-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">indices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-206-20" name="__codelineno-206-20" href="#__codelineno-206-20"></a><span class="w">        </span><span class="c1">// When element 1 is at the head of the array, best time complexity O(1) is achieved</span>
-<a id="__codelineno-206-21" name="__codelineno-206-21" href="#__codelineno-206-21"></a><span class="w">        </span><span class="c1">// When element 1 is at the tail of the array, worst time complexity O(n) is achieved</span>
+<a id="__codelineno-206-20" name="__codelineno-206-20" href="#__codelineno-206-20"></a><span class="w">        </span><span class="c1">// When 1 is at the start of the array, the best-case time complexity is O(1)</span>
+<a id="__codelineno-206-21" name="__codelineno-206-21" href="#__codelineno-206-21"></a><span class="w">        </span><span class="c1">// When 1 is at the end of the array, the worst-case time complexity is O(n)</span>
 <a id="__codelineno-206-22" name="__codelineno-206-22" href="#__codelineno-206-22"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-206-23" name="__codelineno-206-23" href="#__codelineno-206-23"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-206-24" name="__codelineno-206-24" href="#__codelineno-206-24"></a><span class="w">    </span><span class="p">}</span>
@@ -8119,19 +8246,19 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">worst_best_time_complexity.rb</span><pre><span></span><code><a id="__codelineno-207-1" name="__codelineno-207-1" href="#__codelineno-207-1"></a><span class="c1">### Generate array with elements: 1, 2, ..., n, shuffled ###</span>
+<div class="highlight"><span class="filename">worst_best_time_complexity.rb</span><pre><span></span><code><a id="__codelineno-207-1" name="__codelineno-207-1" href="#__codelineno-207-1"></a><span class="c1">### Generate an array containing 1, 2, ..., n in shuffled order ###</span>
 <a id="__codelineno-207-2" name="__codelineno-207-2" href="#__codelineno-207-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">random_numbers</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-207-3" name="__codelineno-207-3" href="#__codelineno-207-3"></a><span class="w">  </span><span class="c1"># Generate array nums =: 1, 2, 3, ..., n</span>
+<a id="__codelineno-207-3" name="__codelineno-207-3" href="#__codelineno-207-3"></a><span class="w">  </span><span class="c1"># Generate the array nums = 1, 2, 3, ..., n</span>
 <a id="__codelineno-207-4" name="__codelineno-207-4" href="#__codelineno-207-4"></a><span class="w">  </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-207-5" name="__codelineno-207-5" href="#__codelineno-207-5"></a><span class="w">  </span><span class="c1"># Randomly shuffle array elements</span>
+<a id="__codelineno-207-5" name="__codelineno-207-5" href="#__codelineno-207-5"></a><span class="w">  </span><span class="c1"># Randomly shuffle the array elements</span>
 <a id="__codelineno-207-6" name="__codelineno-207-6" href="#__codelineno-207-6"></a><span class="w">  </span><span class="n">nums</span><span class="o">.</span><span class="n">shuffle!</span>
 <a id="__codelineno-207-7" name="__codelineno-207-7" href="#__codelineno-207-7"></a><span class="k">end</span>
 <a id="__codelineno-207-8" name="__codelineno-207-8" href="#__codelineno-207-8"></a>
-<a id="__codelineno-207-9" name="__codelineno-207-9" href="#__codelineno-207-9"></a><span class="c1">### Find index of number 1 in array nums ###</span>
+<a id="__codelineno-207-9" name="__codelineno-207-9" href="#__codelineno-207-9"></a><span class="c1">### Find the index of the number 1 in array nums ###</span>
 <a id="__codelineno-207-10" name="__codelineno-207-10" href="#__codelineno-207-10"></a><span class="k">def</span><span class="w"> </span><span class="nf">find_one</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
 <a id="__codelineno-207-11" name="__codelineno-207-11" href="#__codelineno-207-11"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">nums</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-207-12" name="__codelineno-207-12" href="#__codelineno-207-12"></a><span class="w">    </span><span class="c1"># When element 1 is at the head of the array, best time complexity O(1) is achieved</span>
-<a id="__codelineno-207-13" name="__codelineno-207-13" href="#__codelineno-207-13"></a><span class="w">    </span><span class="c1"># When element 1 is at the tail of the array, worst time complexity O(n) is achieved</span>
+<a id="__codelineno-207-12" name="__codelineno-207-12" href="#__codelineno-207-12"></a><span class="w">    </span><span class="c1"># When 1 is at the start of the array, the best-case time complexity is O(1)</span>
+<a id="__codelineno-207-13" name="__codelineno-207-13" href="#__codelineno-207-13"></a><span class="w">    </span><span class="c1"># When 1 is at the end of the array, the worst-case time complexity is O(n)</span>
 <a id="__codelineno-207-14" name="__codelineno-207-14" href="#__codelineno-207-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-207-15" name="__codelineno-207-15" href="#__codelineno-207-15"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-207-16" name="__codelineno-207-16" href="#__codelineno-207-16"></a>
@@ -8141,13 +8268,18 @@ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1
 </div>
 </div>
 </div>
-<p>It is worth noting that we rarely use best-case time complexity in practice, because it can usually only be achieved with a very small probability and may be somewhat misleading. <strong>The worst-case time complexity is more practical because it gives a safety value for efficiency</strong>, allowing us to use the algorithm with confidence.</p>
-<p>From the above example, we can see that both worst-case and best-case time complexities arise only under particular input distributions, which may occur with very low probability and may not truly reflect the algorithm's running efficiency. In contrast, <strong>average time complexity can reflect the algorithm's running efficiency under random input data</strong>, denoted using the <span class="arithmatex">\(\Theta\)</span> notation.</p>
-<p>For some algorithms, we can simply derive the average case under random data distribution. For example, in the above example, since the input array is shuffled, the probability of element <span class="arithmatex">\(1\)</span> appearing at any index is equal, so the algorithm's average number of loops is half the array length <span class="arithmatex">\(n / 2\)</span>, giving an average time complexity of <span class="arithmatex">\(\Theta(n / 2) = \Theta(n)\)</span>.</p>
-<p>But for more complex algorithms, calculating average time complexity is often quite difficult, because it is hard to analyze the overall mathematical expectation under data distribution. In this case, we usually use worst-case time complexity as the criterion for judging algorithm efficiency.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=import%20random%0A%0Adef%20random_numbers%28n%3A%20int%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%22%22%22Generate%20an%20array%20containing%201,%202,%20...,%20n%20in%20shuffled%20order%22%22%22%0A%20%20%20%20%23%20Generate%20the%20array%20nums%20%3D%201,%202,%203,%20...,%20n%0A%20%20%20%20nums%20%3D%20%5Bi%20for%20i%20in%20range%281,%20n%20%2B%201%29%5D%0A%20%20%20%20%23%20Randomly%20shuffle%20the%20array%20elements%0A%20%20%20%20random.shuffle%28nums%29%0A%20%20%20%20return%20nums%0A%0Adef%20find_one%28nums%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20the%20index%20of%20the%20number%201%20in%20array%20nums%22%22%22%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20%23%20When%201%20is%20at%20the%20start%20of%20the%20array,%20the%20best-case%20time%20complexity%20is%20O%281%29%0A%20%20%20%20%20%20%20%20%23%20When%201%20is%20at%20the%20end%20of%20the%20array,%20the%20worst-case%20time%20complexity%20is%20O%28n%29%0A%20%20%20%20%20%20%20%20if%20nums%5Bi%5D%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20i%0A%20%20%20%20return%20-1%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%2010%0A%20%20%20%20nums%20%3D%20random_numbers%28n%29%0A%20%20%20%20index%20%3D%20find_one%28nums%29%0A%20%20%20%20print%28%22%5CnArray%20%5B%201,%202,%20...,%20n%20%5D%20after%20shuffling%20%3D%22,%20nums%29%0A%20%20%20%20print%28%22The%20index%20of%20the%20number%201%20is%22,%20index%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=25&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=import%20random%0A%0Adef%20random_numbers%28n%3A%20int%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%22%22%22Generate%20an%20array%20containing%201,%202,%20...,%20n%20in%20shuffled%20order%22%22%22%0A%20%20%20%20%23%20Generate%20the%20array%20nums%20%3D%201,%202,%203,%20...,%20n%0A%20%20%20%20nums%20%3D%20%5Bi%20for%20i%20in%20range%281,%20n%20%2B%201%29%5D%0A%20%20%20%20%23%20Randomly%20shuffle%20the%20array%20elements%0A%20%20%20%20random.shuffle%28nums%29%0A%20%20%20%20return%20nums%0A%0Adef%20find_one%28nums%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20the%20index%20of%20the%20number%201%20in%20array%20nums%22%22%22%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20%23%20When%201%20is%20at%20the%20start%20of%20the%20array,%20the%20best-case%20time%20complexity%20is%20O%281%29%0A%20%20%20%20%20%20%20%20%23%20When%201%20is%20at%20the%20end%20of%20the%20array,%20the%20worst-case%20time%20complexity%20is%20O%28n%29%0A%20%20%20%20%20%20%20%20if%20nums%5Bi%5D%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20i%0A%20%20%20%20return%20-1%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%2010%0A%20%20%20%20nums%20%3D%20random_numbers%28n%29%0A%20%20%20%20index%20%3D%20find_one%28nums%29%0A%20%20%20%20print%28%22%5CnArray%20%5B%201,%202,%20...,%20n%20%5D%20after%20shuffling%20%3D%22,%20nums%29%0A%20%20%20%20print%28%22The%20index%20of%20the%20number%201%20is%22,%20index%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=25&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Best-case complexity is rarely used in practice because it may occur only with low probability and can be misleading. <strong>Worst-case complexity is more useful because it gives a safe performance bound</strong>, allowing us to use the algorithm with confidence.</p>
+<p>In this example, both the worst and best cases occur only for particular arrangements, perhaps with low probability, and may not reflect ordinary performance. <strong>Average-case complexity describes performance on random inputs</strong> and is expressed with <span class="arithmatex">\(\Theta\)</span>.</p>
+<p>For some algorithms, the average case is easy to derive. In the example above, <span class="arithmatex">\(1\)</span> is equally likely to occur at any index in the shuffled array. The average number of iterations is half the array length, <span class="arithmatex">\(n / 2\)</span>, giving <span class="arithmatex">\(\Theta(n / 2) = \Theta(n)\)</span>.</p>
+<p>For more complex algorithms, average-case complexity can be difficult to calculate because the overall expectation under the input distribution is hard to derive. In such cases, we usually judge efficiency by worst-case complexity.</p>
 <div class="admonition question">
-<p class="admonition-title">Why is the <span class="arithmatex">\(\Theta\)</span> symbol rarely seen?</p>
-<p>This may be because the <span class="arithmatex">\(O\)</span> symbol is too catchy, so we often use it to represent average time complexity. But strictly speaking, this practice is not standard. In this book and other materials, if you encounter expressions like "average time complexity <span class="arithmatex">\(O(n)\)</span>", please understand it directly as <span class="arithmatex">\(\Theta(n)\)</span>.</p>
+<p class="admonition-title">Why is <span class="arithmatex">\(\Theta\)</span> rarely seen?</p>
+<p>Perhaps because <span class="arithmatex">\(O\)</span> is so familiar, it is often used for average-case complexity too. Strictly speaking, this usage is not standard. In this book and other resources, read a statement such as "average-case time complexity <span class="arithmatex">\(O(n)\)</span>" as <span class="arithmatex">\(\Theta(n)\)</span>.</p>
 </div>
 
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+    aria-label="Previous: 2.2   Iteration and recursion"
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
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-        <title>3.2 Basic Data Types - Hello Algo</title>
+        <title>3.2 Basic data types - Hello Algo</title>
 
 
 
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+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
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-  1.1 Algorithms Are Everywhere
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+  Chapter 6. &nbsp; Hash tables
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -1732,7 +1732,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1759,7 +1759,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1786,7 +1786,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1813,7 +1813,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1840,7 +1840,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1908,7 +1908,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1929,7 +1929,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1950,7 +1950,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1977,7 +1977,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2004,7 +2004,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2031,7 +2031,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2058,7 +2058,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2085,7 +2085,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2176,7 +2176,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2218,7 +2218,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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@@ -2272,7 +2272,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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@@ -2299,7 +2299,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2326,7 +2326,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2390,7 +2390,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2432,7 +2432,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2459,7 +2459,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2486,7 +2486,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2513,7 +2513,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2608,7 +2608,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2629,7 +2629,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2650,7 +2650,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2677,7 +2677,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2704,7 +2704,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2731,7 +2731,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2758,7 +2758,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2785,7 +2785,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2812,7 +2812,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2890,7 +2890,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2911,7 +2911,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2932,7 +2932,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2959,7 +2959,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2986,7 +2986,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3013,7 +3013,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3040,7 +3040,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3067,7 +3067,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3094,7 +3094,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3121,7 +3121,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3148,7 +3148,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3175,7 +3175,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3202,7 +3202,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3229,7 +3229,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3295,7 +3295,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3316,7 +3316,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
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@@ -3337,7 +3337,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3364,7 +3364,7 @@
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3391,7 +3391,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3418,7 +3418,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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@@ -3445,7 +3445,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3472,7 +3472,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3538,7 +3538,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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@@ -3559,7 +3559,7 @@
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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@@ -3580,7 +3580,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3607,7 +3607,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3634,7 +3634,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3661,7 +3661,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3688,7 +3688,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3715,7 +3715,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3785,7 +3785,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3806,7 +3806,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3827,7 +3827,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3854,7 +3854,7 @@
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3881,7 +3881,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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@@ -3935,7 +3935,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -3962,7 +3962,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -3989,7 +3989,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4016,7 +4016,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4082,7 +4082,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4103,7 +4103,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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@@ -4124,7 +4124,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4151,7 +4151,7 @@
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4178,7 +4178,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4205,7 +4205,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4232,7 +4232,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4259,7 +4259,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4319,7 +4319,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4340,7 +4340,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4361,7 +4361,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4388,7 +4388,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4415,7 +4415,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4499,6 +4499,78 @@
 
 
 
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@@ -4563,34 +4635,34 @@
 
 
 <!-- Page content -->
-<h1 id="32-basic-data-types">3.2 &nbsp; Basic Data Types<a class="headerlink" href="#32-basic-data-types" title="Permanent link">&para;</a></h1>
-<p>When we talk about data stored in computers, we think of various forms such as text, images, videos, audio, 3D models, and more. Although these kinds of data are organized in different ways, they are all composed of various basic data types.</p>
-<p><strong>Basic data types are types that the CPU can directly operate on</strong>, and they are directly used in algorithms, mainly including the following.</p>
+<h1 id="32-basic-data-types">3.2 &nbsp; Basic data types<a class="headerlink" href="#32-basic-data-types" title="Permanent link">&para;</a></h1>
+<p>Computer data comes in many forms: text, images, video, audio, 3D models, and more. Although organized differently, all these forms are built from basic data types.</p>
+<p><strong>Basic data types are types on which the CPU can operate directly</strong> and which algorithms use directly. The main types include the following.</p>
 <ul>
-<li>Integer types <code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>.</li>
-<li>Floating-point types <code>float</code>, <code>double</code>, used to represent decimal numbers.</li>
-<li>Character type <code>char</code>, used to represent letters, punctuation marks, and even emojis in various languages.</li>
-<li>Boolean type <code>bool</code>, used to represent "yes" and "no" judgments.</li>
+<li>Integer types: <code>byte</code>, <code>short</code>, <code>int</code>, and <code>long</code>.</li>
+<li>Floating-point types: <code>float</code> and <code>double</code>, used to represent fractional numbers.</li>
+<li>The character type <code>char</code>, used for letters in different languages, punctuation, and even emoji.</li>
+<li>The Boolean type <code>bool</code>, used for yes-or-no values.</li>
 </ul>
-<p><strong>Basic data types are stored in binary form in computers</strong>. A binary digit is one bit. In most modern operating systems, <span class="arithmatex">\(1\)</span> byte consists of <span class="arithmatex">\(8\)</span> bits.</p>
-<p>The range of values for basic data types depends on the size of the space they occupy. Below is an example using Java.</p>
+<p><strong>Basic data types are stored in binary form</strong>. One binary digit is <span class="arithmatex">\(1\)</span> bit. In the vast majority of modern operating systems, <span class="arithmatex">\(1\)</span> byte consists of <span class="arithmatex">\(8\)</span> bits.</p>
+<p>A basic data type's value range depends on the space it occupies. Taking Java as an example:</p>
 <ul>
-<li>Integer type <code>byte</code> occupies <span class="arithmatex">\(1\)</span> byte = <span class="arithmatex">\(8\)</span> bits, and can represent <span class="arithmatex">\(2^{8}\)</span> numbers.</li>
-<li>Integer type <code>int</code> occupies <span class="arithmatex">\(4\)</span> bytes = <span class="arithmatex">\(32\)</span> bits, and can represent <span class="arithmatex">\(2^{32}\)</span> numbers.</li>
+<li>An integer <code>byte</code> occupies <span class="arithmatex">\(1\)</span> byte = <span class="arithmatex">\(8\)</span> bits and can represent <span class="arithmatex">\(2^{8}\)</span> distinct values.</li>
+<li>An integer <code>int</code> occupies <span class="arithmatex">\(4\)</span> bytes = <span class="arithmatex">\(32\)</span> bits and can represent <span class="arithmatex">\(2^{32}\)</span> distinct values.</li>
 </ul>
-<p>The following table lists the space occupied, value ranges, and default values of various basic data types in Java. You don't need to memorize this table; a general understanding is sufficient, and you can refer to it when needed.</p>
-<p align="center"> Table 3-1 &nbsp; Space occupied and value ranges of basic data types </p>
+<p>Table 3-1 lists storage sizes, value ranges, and default values for Java's basic data types. There is no need to memorize it; understand the general picture and refer back when needed.</p>
+<p align="center"> Table 3-1 &nbsp; Storage sizes and value ranges of basic data types </p>
 
 <div class="center-table">
 <table>
 <thead>
 <tr>
 <th>Type</th>
-<th>Symbol</th>
-<th>Space Occupied</th>
-<th>Minimum Value</th>
-<th>Maximum Value</th>
-<th>Default Value</th>
+<th>Name</th>
+<th>Storage size</th>
+<th>Minimum value</th>
+<th>Maximum value</th>
+<th>Default value</th>
 </tr>
 </thead>
 <tbody>
@@ -4627,7 +4699,7 @@
 <td><span class="arithmatex">\(0\)</span></td>
 </tr>
 <tr>
-<td>Float</td>
+<td>Floating-point</td>
 <td><code>float</code></td>
 <td>4 bytes</td>
 <td><span class="arithmatex">\(1.175 \times 10^{-38}\)</span></td>
@@ -4661,31 +4733,31 @@
 </tbody>
 </table>
 </div>
-<p>Please note that Table 3-1 applies specifically to Java's basic data types. Each programming language has its own type definitions, and their space usage, value ranges, and default values may vary.</p>
+<p>Table 3-1 describes Java's basic data types. Each programming language defines its own types, whose sizes, value ranges, and default values may differ.</p>
 <ul>
-<li>In Python, the integer type <code>int</code> can be of any size, limited only by available memory; the floating-point type <code>float</code> is double-precision 64-bit; there is no <code>char</code> type, a single character is actually a string <code>str</code> of length 1.</li>
-<li>C and C++ do not explicitly specify the size of basic data types, which varies by implementation and platform. The above table follows the LP64 <a href="https://en.cppreference.com/w/cpp/language/types#Properties">data model</a>, which is used in Unix 64-bit operating systems including Linux and macOS.</li>
-<li>The size of character <code>char</code> is 1 byte in C and C++, and in most programming languages it depends on the specific character encoding method, as detailed in the "Character Encoding" section.</li>
-<li>Even though representing a boolean value requires only 1 bit (<span class="arithmatex">\(0\)</span> or <span class="arithmatex">\(1\)</span>), it is usually stored as 1 byte in memory. This is because modern computer CPUs typically use 1 byte as the minimum addressable memory unit.</li>
+<li>In Python, an <code>int</code> can be arbitrarily large, limited only by available memory. A <code>float</code> is a 64-bit double-precision number. There is no <code>char</code> type: a single character is a <code>str</code> of length 1.</li>
+<li>C and C++ do not fix the sizes of basic data types; they vary by implementation and platform. Table 3-1 follows the LP64 <a href="https://en.cppreference.com/w/cpp/language/types#Properties">data model</a>, used by 64-bit Unix operating systems including Linux and macOS.</li>
+<li>A <code>char</code> occupies 1 byte in C and C++. In most programming languages, character size depends on the encoding used; see the character encoding section.</li>
+<li>Although a Boolean needs only 1 bit (<span class="arithmatex">\(0\)</span> or <span class="arithmatex">\(1\)</span>), it usually occupies 1 byte in memory because modern CPUs typically use a byte as the smallest addressable memory unit.</li>
 </ul>
-<p>So, what is the relationship between basic data types and data structures? We know that data structures are ways of organizing and storing data in computers. Here, the emphasis is on the "structure", not the "data".</p>
-<p>If we want to represent "a row of numbers", we naturally think of using an array. This is because the linear structure of an array can represent the adjacency and order relationships of numbers, but whether the stored content is integer <code>int</code>, floating-point <code>float</code>, or character <code>char</code> is unrelated to the "data structure".</p>
-<p>In other words, <strong>basic data types provide the "content type" of data, while data structures provide the "organization method" of data</strong>. For example, in the following code, we use the same data structure (array) to store and represent different basic data types, including <code>int</code>, <code>float</code>, <code>char</code>, <code>bool</code>, etc.</p>
+<p>How do basic data types relate to data structures? A data structure is a way to organize and store data in a computer. The emphasis is on structure, rather than the data itself.</p>
+<p>To represent a row of numbers, an array comes naturally to mind. Its linear structure captures adjacency and ordering, regardless of whether the stored values are integers (<code>int</code>), fractional numbers (<code>float</code>), or characters (<code>char</code>).</p>
+<p>In other words, <strong>basic data types specify what kind of data is stored, while data structures specify how it is organized</strong>. The following code uses the same data structure, an array, for different basic types, including <code>int</code>, <code>float</code>, <code>char</code>, and <code>bool</code>.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Initialize arrays using various basic data types</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Initialize arrays with different basic data types</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="n">numbers</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="mi">5</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="n">decimals</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">float</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">]</span> <span class="o">*</span> <span class="mi">5</span>
-<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="c1"># In Python, characters are actually strings of length 1</span>
+<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="c1"># A Python character is a string of length 1</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="n">characters</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">str</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'0'</span><span class="p">]</span> <span class="o">*</span> <span class="mi">5</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="n">bools</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">bool</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">]</span> <span class="o">*</span> <span class="mi">5</span>
-<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="c1"># Python lists can freely store various basic data types and object references</span>
+<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="c1"># Python lists can hold different basic data types and object references</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="n">data</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="s1">'a'</span><span class="p">,</span> <span class="kc">False</span><span class="p">,</span> <span class="n">ListNode</span><span class="p">(</span><span class="mi">0</span><span class="p">)]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="c1">// Initialize arrays using various basic data types</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="c1">// Initialize arrays with different basic data types</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">numbers</span><span class="p">[</span><span class="mi">5</span><span class="p">];</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="kt">float</span><span class="w"> </span><span class="n">decimals</span><span class="p">[</span><span class="mi">5</span><span class="p">];</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="kt">char</span><span class="w"> </span><span class="n">characters</span><span class="p">[</span><span class="mi">5</span><span class="p">];</span>
@@ -4693,7 +4765,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="c1">// Initialize arrays using various basic data types</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="c1">// Initialize arrays with different basic data types</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">numbers</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="mi">5</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="kt">float</span><span class="o">[]</span><span class="w"> </span><span class="n">decimals</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">float</span><span class="o">[</span><span class="mi">5</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="kt">char</span><span class="o">[]</span><span class="w"> </span><span class="n">characters</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">char</span><span class="o">[</span><span class="mi">5</span><span class="o">]</span><span class="p">;</span>
@@ -4701,7 +4773,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="c1">// Initialize arrays using various basic data types</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="c1">// Initialize arrays with different basic data types</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">numbers</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="mi">5</span><span class="p">];</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="kt">float</span><span class="p">[]</span><span class="w"> </span><span class="n">decimals</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">float</span><span class="p">[</span><span class="mi">5</span><span class="p">];</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="kt">char</span><span class="p">[]</span><span class="w"> </span><span class="n">characters</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">char</span><span class="p">[</span><span class="mi">5</span><span class="p">];</span>
@@ -4709,7 +4781,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="c1">// Initialize arrays using various basic data types</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="c1">// Initialize arrays with different basic data types</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">var</span><span class="w"> </span><span class="nx">numbers</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="kt">int</span><span class="p">{}</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="kd">var</span><span class="w"> </span><span class="nx">decimals</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="kt">float64</span><span class="p">{}</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="kd">var</span><span class="w"> </span><span class="nx">characters</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="kt">byte</span><span class="p">{}</span>
@@ -4717,7 +4789,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="c1">// Initialize arrays using various basic data types</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="c1">// Initialize arrays with different basic data types</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">let</span><span class="w"> </span><span class="nv">numbers</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="kd">let</span><span class="w"> </span><span class="nv">decimals</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mf">0.0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="kd">let</span><span class="w"> </span><span class="nv">characters</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Character</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="s">"a"</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="mi">5</span><span class="p">)</span>
@@ -4725,19 +4797,19 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="c1">// JavaScript arrays can freely store various basic data types and objects</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="c1">// JavaScript arrays can hold different basic data types and objects</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">const</span><span class="w"> </span><span class="nx">array</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="mf">0.0</span><span class="p">,</span><span class="w"> </span><span class="s1">'a'</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">];</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="c1">// Initialize arrays using various basic data types</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="c1">// Initialize arrays with different basic data types</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">const</span><span class="w"> </span><span class="nx">numbers</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="kd">const</span><span class="w"> </span><span class="nx">characters</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="kd">const</span><span class="w"> </span><span class="nx">bools</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="c1">// Initialize arrays using various basic data types</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="c1">// Initialize arrays with different basic data types</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">numbers</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">5</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">double</span><span class="o">&gt;</span><span class="w"> </span><span class="n">decimals</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">5</span><span class="p">,</span><span class="w"> </span><span class="m">0.0</span><span class="p">);</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">String</span><span class="o">&gt;</span><span class="w"> </span><span class="n">characters</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">5</span><span class="p">,</span><span class="w"> </span><span class="s1">'a'</span><span class="p">);</span>
@@ -4745,7 +4817,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="c1">// Initialize arrays using various basic data types</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="c1">// Initialize arrays with different basic data types</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="kd">let</span><span class="w"> </span><span class="n">numbers</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="mi">5</span><span class="p">];</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="kd">let</span><span class="w"> </span><span class="n">decimals</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">f32</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mf">0.0</span><span class="p">;</span><span class="w"> </span><span class="mi">5</span><span class="p">];</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="kd">let</span><span class="w"> </span><span class="n">characters</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">char</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="sc">'0'</span><span class="p">;</span><span class="w"> </span><span class="mi">5</span><span class="p">];</span>
@@ -4753,7 +4825,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// Initialize arrays using various basic data types</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// Initialize arrays with different basic data types</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">numbers</span><span class="p">[</span><span class="mi">10</span><span class="p">];</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="kt">float</span><span class="w"> </span><span class="n">decimals</span><span class="p">[</span><span class="mi">10</span><span class="p">];</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="kt">char</span><span class="w"> </span><span class="n">characters</span><span class="p">[</span><span class="mi">10</span><span class="p">];</span>
@@ -4761,7 +4833,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="c1">// Initialize arrays using various basic data types</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="c1">// Initialize arrays with different basic data types</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">val</span><span class="w"> </span><span class="nv">numbers</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="m">5</span><span class="p">)</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="kd">val</span><span class="w"> </span><span class="nv">decinals</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">FloatArray</span><span class="p">(</span><span class="m">5</span><span class="p">)</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="kd">val</span><span class="w"> </span><span class="nv">characters</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">CharArray</span><span class="p">(</span><span class="m">5</span><span class="p">)</span>
@@ -4769,16 +4841,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Ruby lists can freely store various basic data types and object references</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Ruby arrays can hold different basic data types and object references</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="n">data</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="o">.</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="s1">'a'</span><span class="p">,</span><span class="w"> </span><span class="kp">false</span><span class="p">,</span><span class="w"> </span><span class="no">ListNode</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">]</span>
 </code></pre></div>
 </div>
 </div>
 </div>
 <details class="pythontutor">
-<summary>Visualized Execution</summary>
-<p><div style="height: 477px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E4%BD%BF%E7%94%A8%E5%A4%9A%E7%A7%8D%E5%9F%BA%E6%9C%AC%E6%95%B0%E6%8D%AE%E7%B1%BB%E5%9E%8B%E6%9D%A5%E5%88%9D%E5%A7%8B%E5%8C%96%E6%95%B0%E7%BB%84%0A%20%20%20%20numbers%20%3D%20%5B0%5D%20*%205%0A%20%20%20%20decimals%20%3D%20%5B0.0%5D%20*%205%0A%20%20%20%20%23%20Python%20%E7%9A%84%E5%AD%97%E7%AC%A6%E5%AE%9E%E9%99%85%E4%B8%8A%E6%98%AF%E9%95%BF%E5%BA%A6%E4%B8%BA%201%20%E7%9A%84%E5%AD%97%E7%AC%A6%E4%B8%B2%0A%20%20%20%20characters%20%3D%20%5B'0'%5D%20*%205%0A%20%20%20%20bools%20%3D%20%5BFalse%5D%20*%205%0A%20%20%20%20%23%20Python%20%E7%9A%84%E5%88%97%E8%A1%A8%E5%8F%AF%E4%BB%A5%E8%87%AA%E7%94%B1%E5%AD%98%E5%82%A8%E5%90%84%E7%A7%8D%E5%9F%BA%E6%9C%AC%E6%95%B0%E6%8D%AE%E7%B1%BB%E5%9E%8B%E5%92%8C%E5%AF%B9%E8%B1%A1%E5%BC%95%E7%94%A8%0A%20%20%20%20data%20%3D%20%5B0,%200.0,%20'a',%20False,%20ListNode%280%29%5D&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=12&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E4%BD%BF%E7%94%A8%E5%A4%9A%E7%A7%8D%E5%9F%BA%E6%9C%AC%E6%95%B0%E6%8D%AE%E7%B1%BB%E5%9E%8B%E6%9D%A5%E5%88%9D%E5%A7%8B%E5%8C%96%E6%95%B0%E7%BB%84%0A%20%20%20%20numbers%20%3D%20%5B0%5D%20*%205%0A%20%20%20%20decimals%20%3D%20%5B0.0%5D%20*%205%0A%20%20%20%20%23%20Python%20%E7%9A%84%E5%AD%97%E7%AC%A6%E5%AE%9E%E9%99%85%E4%B8%8A%E6%98%AF%E9%95%BF%E5%BA%A6%E4%B8%BA%201%20%E7%9A%84%E5%AD%97%E7%AC%A6%E4%B8%B2%0A%20%20%20%20characters%20%3D%20%5B'0'%5D%20*%205%0A%20%20%20%20bools%20%3D%20%5BFalse%5D%20*%205%0A%20%20%20%20%23%20Python%20%E7%9A%84%E5%88%97%E8%A1%A8%E5%8F%AF%E4%BB%A5%E8%87%AA%E7%94%B1%E5%AD%98%E5%82%A8%E5%90%84%E7%A7%8D%E5%9F%BA%E6%9C%AC%E6%95%B0%E6%8D%AE%E7%B1%BB%E5%9E%8B%E5%92%8C%E5%AF%B9%E8%B1%A1%E5%BC%95%E7%94%A8%0A%20%20%20%20data%20%3D%20%5B0,%200.0,%20'a',%20False,%20ListNode%280%29%5D&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=12&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 477px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20successor%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20arrays%20with%20different%20basic%20data%20types%0A%20%20%20%20numbers%20%3D%20%5B0%5D%20%2A%205%0A%20%20%20%20decimals%20%3D%20%5B0.0%5D%20%2A%205%0A%20%20%20%20%23%20A%20Python%20character%20is%20a%20string%20of%20length%201%0A%20%20%20%20characters%20%3D%20%5B%270%27%5D%20%2A%205%0A%20%20%20%20bools%20%3D%20%5BFalse%5D%20%2A%205%0A%20%20%20%20%23%20Python%20lists%20can%20hold%20different%20basic%20data%20types%20and%20object%20references%0A%20%20%20%20data%20%3D%20%5B0%2C%200.0%2C%20%27a%27%2C%20False%2C%20ListNode%280%29%5D&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=12&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20successor%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20arrays%20with%20different%20basic%20data%20types%0A%20%20%20%20numbers%20%3D%20%5B0%5D%20%2A%205%0A%20%20%20%20decimals%20%3D%20%5B0.0%5D%20%2A%205%0A%20%20%20%20%23%20A%20Python%20character%20is%20a%20string%20of%20length%201%0A%20%20%20%20characters%20%3D%20%5B%270%27%5D%20%2A%205%0A%20%20%20%20bools%20%3D%20%5BFalse%5D%20%2A%205%0A%20%20%20%20%23%20Python%20lists%20can%20hold%20different%20basic%20data%20types%20and%20object%20references%0A%20%20%20%20data%20%3D%20%5B0%2C%200.0%2C%20%27a%27%2C%20False%2C%20ListNode%280%29%5D&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=12&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
 
 <!-- Source file information -->
@@ -4802,7 +4874,7 @@ aria-label="Footer"
   <a
     href="../classification_of_data_structure/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 3.1 Classification of Data Structures"
+    aria-label="Previous: 3.1   Classifying data structures"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4814,7 +4886,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        3.1 Classification of Data Structures
+        3.1 &nbsp; Classifying data structures
       </div>
     </div>
   </a>
@@ -4826,7 +4898,7 @@ aria-label="Footer"
   <a
     href="../number_encoding/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 3.3 Number Encoding *"
+    aria-label="Next: 3.3   Number encoding *"
     rel="next"
   >
     <div class="md-footer__title">
@@ -4834,7 +4906,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        3.3 Number Encoding *
+        3.3 &nbsp; Number encoding *
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -4851,7 +4923,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_data_structure/character_encoding/index.html b/en/chapter_data_structure/character_encoding/index.html
index ff1ecd912..ddf64936f 100644
--- a/en/chapter_data_structure/character_encoding/index.html
+++ b/en/chapter_data_structure/character_encoding/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>3.4 Character Encoding * - Hello Algo</title>
+        <title>3.4 Character encoding * - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -119,7 +119,7 @@
         <path
             d="M480 32c0-12.9-7.8-24.6-19.8-29.6s-25.7-2.2-34.9 6.9L381.7 53c-48 48-113.1 75-181 75H192 160 64c-35.3 0-64 28.7-64 64v96c0 35.3 28.7 64 64 64l0 128c0 17.7 14.3 32 32 32h64c17.7 0 32-14.3 32-32V352l8.7 0c67.9 0 133 27 181 75l43.6 43.6c9.2 9.2 22.9 11.9 34.9 6.9s19.8-16.6 19.8-29.6V300.4c18.6-8.8 32-32.5 32-60.4s-13.4-51.6-32-60.4V32zm-64 76.7V240 371.3C357.2 317.8 280.5 288 200.7 288H192V192h8.7c79.8 0 156.5-29.8 215.3-83.3z" />
     </svg>
-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              3.4 &nbsp; Character Encoding *
+              3.4 &nbsp; Character encoding *
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
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@@ -177,9 +177,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -982,7 +982,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1024,7 +1024,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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   <a href="#341-ascii-character-set" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3.4.1 &nbsp; ASCII Character Set
+        3.4.1 &nbsp; ASCII character set
       </span>
     </span>
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@@ -1171,7 +1171,7 @@
   <a href="#342-gbk-character-set" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3.4.2 &nbsp; GBK Character Set
+        3.4.2 &nbsp; GBK character set
       </span>
     </span>
   </a>
@@ -1182,7 +1182,7 @@
   <a href="#343-unicode-character-set" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3.4.3 &nbsp; Unicode Character Set
+        3.4.3 &nbsp; Unicode character set
       </span>
     </span>
   </a>
@@ -1193,7 +1193,7 @@
   <a href="#344-utf-8-encoding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3.4.4 &nbsp; UTF-8 Encoding
+        3.4.4 &nbsp; UTF-8 encoding
       </span>
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   </a>
@@ -1204,7 +1204,7 @@
   <a href="#345-character-encoding-in-programming-languages" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3.4.5 &nbsp; Character Encoding in Programming Languages
+        3.4.5 &nbsp; Character encoding in programming languages
       </span>
     </span>
   </a>
@@ -1234,7 +1234,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1327,7 +1327,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1369,7 +1369,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1477,7 +1477,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1504,7 +1504,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1568,7 +1568,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1589,7 +1589,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1610,7 +1610,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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@@ -1664,7 +1664,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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@@ -1691,7 +1691,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1782,7 +1782,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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@@ -1824,7 +1824,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1932,7 +1932,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -2000,7 +2000,7 @@
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
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@@ -2042,7 +2042,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2096,7 +2096,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2123,7 +2123,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2150,7 +2150,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2177,7 +2177,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2204,7 +2204,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2268,7 +2268,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2310,7 +2310,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2337,7 +2337,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2364,7 +2364,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2391,7 +2391,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2418,7 +2418,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2482,7 +2482,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2524,7 +2524,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2578,7 +2578,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2605,7 +2605,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2632,7 +2632,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2700,7 +2700,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2721,7 +2721,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2742,7 +2742,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2769,7 +2769,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2796,7 +2796,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2823,7 +2823,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2850,7 +2850,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2877,7 +2877,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2904,7 +2904,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2982,7 +2982,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3003,7 +3003,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
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@@ -3024,7 +3024,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3051,7 +3051,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3078,7 +3078,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3105,7 +3105,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3132,7 +3132,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3159,7 +3159,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3186,7 +3186,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3213,7 +3213,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3240,7 +3240,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3267,7 +3267,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3294,7 +3294,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3321,7 +3321,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3387,7 +3387,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3408,7 +3408,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3429,7 +3429,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3456,7 +3456,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3483,7 +3483,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3510,7 +3510,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3537,7 +3537,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3564,7 +3564,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3630,7 +3630,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
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@@ -3672,7 +3672,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3699,7 +3699,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3726,7 +3726,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3753,7 +3753,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3780,7 +3780,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3807,7 +3807,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3877,7 +3877,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3898,7 +3898,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
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-  14.3 Dynamic Programming Problem-Solving Approach
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-  14.5 Unbounded Knapsack Problem
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-  14.6 Edit Distance Problem
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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+        3.4.5 &nbsp; Character encoding in programming languages
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 <!-- Page content -->
-<h1 id="34-character-encoding">3.4 &nbsp; Character Encoding *<a class="headerlink" href="#34-character-encoding" title="Permanent link">&para;</a></h1>
-<p>In computers, all data is stored in binary form, and character <code>char</code> is no exception. To represent characters, we need to establish a "character set" that defines a one-to-one correspondence between each character and binary numbers. With a character set, computers can convert binary numbers to characters by looking up the table.</p>
-<h2 id="341-ascii-character-set">3.4.1 &nbsp; ASCII Character Set<a class="headerlink" href="#341-ascii-character-set" title="Permanent link">&para;</a></h2>
-<p><u>ASCII code</u> is the earliest character set, with the full name American Standard Code for Information Interchange. It uses 7 binary bits (the lower 7 bits of one byte) to represent a character, and can represent a maximum of 128 different characters. As shown in Figure 3-6, ASCII code includes uppercase and lowercase English letters, numbers 0 ~ 9, some punctuation marks, and some control characters (such as newline and tab).</p>
-<p><img alt="ASCII code" class="animation-figure" src="../character_encoding.assets/ascii_table.png" /></p>
-<p align="center"> Figure 3-6 &nbsp; ASCII code </p>
+<h1 id="34-character-encoding">3.4 &nbsp; Character encoding *<a class="headerlink" href="#34-character-encoding" title="Permanent link">&para;</a></h1>
+<p>All computer data is stored as binary numbers, including characters of type <code>char</code>. To represent characters, we need a character set that assigns each character a unique binary number. The computer can then convert numbers to characters by looking them up.</p>
+<h2 id="341-ascii-character-set">3.4.1 &nbsp; ASCII character set<a class="headerlink" href="#341-ascii-character-set" title="Permanent link">&para;</a></h2>
+<p><u>ASCII</u>, short for American Standard Code for Information Interchange, was the earliest character set. It uses 7 bits (the lower 7 bits of a byte) per character, allowing up to 128 characters. As shown in Figure 3-6, ASCII includes uppercase and lowercase English letters, digits 0 ~ 9, punctuation, and control characters such as newlines and tabs.</p>
+<p><img alt="ASCII codes" class="animation-figure" src="../character_encoding.assets/ascii_table.png" /></p>
+<p align="center"> Figure 3-6 &nbsp; ASCII codes </p>
 
-<p>However, <strong>ASCII code can only represent English</strong>. With the globalization of computers, a character set called <u>EASCII</u> that can represent more languages emerged. It expands from the 7-bit basis of ASCII to 8 bits, and can represent 256 different characters.</p>
-<p>Worldwide, a batch of EASCII character sets suitable for different regions have appeared successively. The first 128 characters of these character sets are unified as ASCII code, and the last 128 characters are defined differently to adapt to the needs of different languages.</p>
-<h2 id="342-gbk-character-set">3.4.2 &nbsp; GBK Character Set<a class="headerlink" href="#342-gbk-character-set" title="Permanent link">&para;</a></h2>
-<p>Later, people found that <strong>EASCII still could not provide enough characters for many languages</strong>. For example, there are nearly one hundred thousand Chinese characters, and several thousand are used in everyday life. In 1980, the China National Standardization Administration released the <u>GB2312</u> character set, which included 6,763 Chinese characters, basically meeting the needs of computer processing for Chinese.</p>
-<p>However, GB2312 cannot handle some rare characters and traditional Chinese characters. The <u>GBK</u> character set is an extension based on GB2312, which includes a total of 21,886 Chinese characters. In the GBK encoding scheme, ASCII characters are represented using one byte, and Chinese characters are represented using two bytes.</p>
-<h2 id="343-unicode-character-set">3.4.3 &nbsp; Unicode Character Set<a class="headerlink" href="#343-unicode-character-set" title="Permanent link">&para;</a></h2>
-<p>With the vigorous development of computer technology, character sets and encoding standards flourished, which brought many problems. On the one hand, these character sets generally only define characters for specific languages and cannot work normally in multilingual environments. On the other hand, multiple character set standards exist for the same language, and if two computers use different encoding standards, garbled characters will appear during information transmission.</p>
-<p>Researchers of that era thought: <strong>If a sufficiently complete character set were released to include all languages and symbols in the world, wouldn't that solve problems in cross-language environments and eliminate garbled text</strong>? Driven by this idea, a large and comprehensive character set, Unicode, was born.</p>
-<p><u>Unicode</u>, or Unified Code, can theoretically accommodate over one million characters. It is committed to including characters from around the world into a unified character set, providing a universal character set to handle and display various language texts, reducing garbled character problems caused by different encoding standards. Since its release in 1991, Unicode has continuously expanded to include new languages and characters. As of September 2022, Unicode has included 149,186 characters, including characters, symbols, and even emojis from various languages.</p>
-<p>As a universal character set, Unicode essentially assigns each character a unique "code point" (character identifier), whose range is U+0000 to U+10FFFF, forming a unified character numbering space. However, <strong>Unicode does not specify how to store these character code points in computers</strong>. We can't help but ask: when Unicode code points of multiple lengths appear simultaneously in a text, how does the system parse the characters? For example, given an encoding with a length of 2 bytes, how does the system determine whether it is one 2-byte character or two 1-byte characters?</p>
-<p>For the above problem, <strong>a straightforward solution is to store all characters as equal-length encodings</strong>. As shown in Figure 3-7, each character in "Hello" occupies 1 byte, and each character in "算法" (algorithm) occupies 2 bytes. We can encode all characters in "Hello 算法" as 2 bytes in length by padding the high bits with 0. In this way, the system can parse one character every 2 bytes and restore the content of this phrase.</p>
-<p><img alt="Unicode encoding example" class="animation-figure" src="../character_encoding.assets/unicode_hello_algo.png" /></p>
-<p align="center"> Figure 3-7 &nbsp; Unicode encoding example </p>
+<p>However, <strong>ASCII can represent only English text</strong>. As computers spread worldwide, <u>EASCII</u> character sets emerged to support more languages. Extending ASCII from 7 bits to 8 allowed 256 distinct characters.</p>
+<p>Different regions developed their own EASCII character sets. The first 128 characters remained identical to ASCII, while the remaining 128 varied to meet different languages' needs.</p>
+<h2 id="342-gbk-character-set">3.4.2 &nbsp; GBK character set<a class="headerlink" href="#342-gbk-character-set" title="Permanent link">&para;</a></h2>
+<p>Eventually, <strong>even EASCII proved too small for many languages</strong>. Chinese, for example, has nearly 100,000 characters, with thousands used daily. In 1980, China's National Bureau of Standards released <u>GB2312</u>, containing 6,763 Chinese characters and meeting most basic needs for processing Chinese text.</p>
+<p>GB2312 could not handle some rare and traditional characters. <u>GBK</u> extended it to 21,886 Chinese characters. In GBK encoding, ASCII characters use one byte, and Chinese characters use two.</p>
+<h2 id="343-unicode-character-set">3.4.3 &nbsp; Unicode character set<a class="headerlink" href="#343-unicode-character-set" title="Permanent link">&para;</a></h2>
+<p>As computing grew, character sets and encoding standards proliferated, creating problems. Most covered only particular languages and did not work well in multilingual settings. Multiple standards also existed for the same language, so exchanging text between computers using different encodings could produce garbled text.</p>
+<p>Researchers asked: <strong>could a sufficiently comprehensive character set, containing all the world's languages and symbols, solve both multilingual compatibility and garbled-text problems</strong>? This idea led to Unicode.</p>
+<p><u>Unicode</u>, whose Chinese name means "unified code", can theoretically accommodate over a million characters. It aims to bring the world's characters into one universal set for processing and displaying different languages, reducing garbled text caused by incompatible standards. Since its release in 1991, it has continued adding languages and characters. As of September 2022, Unicode contained 149,186 characters, including letters, symbols, and emoji.</p>
+<p>As a universal character set, Unicode assigns each character a unique code point, or character number, from U+0000 to U+10FFFF. <strong>It does not, by itself, specify how these code points are stored in a computer</strong>. How can a system parse text containing code points of different lengths? For example, how does it distinguish one two-byte character from two one-byte characters?</p>
+<p><strong>One straightforward solution is to store every character in a fixed-width encoding</strong>. In Figure 3-7, each character in "Hello" occupies 1 byte, while each in "算法" occupies 2 bytes. Padding the high bits with zeros lets us encode every character in "Hello 算法" with 2 bytes. The system can then read one character every 2 bytes to recover the phrase.</p>
+<p><img alt="Example of Unicode encoding" class="animation-figure" src="../character_encoding.assets/unicode_hello_algo.png" /></p>
+<p align="center"> Figure 3-7 &nbsp; Example of Unicode encoding </p>
 
-<p>However, ASCII code has already proven to us that encoding English only requires 1 byte. If the above scheme is adopted, the size of English text will be twice that under ASCII encoding, which is very wasteful of memory space. Therefore, we need a more efficient Unicode encoding method.</p>
-<h2 id="344-utf-8-encoding">3.4.4 &nbsp; UTF-8 Encoding<a class="headerlink" href="#344-utf-8-encoding" title="Permanent link">&para;</a></h2>
-<p>Currently, UTF-8 has become the most widely used Unicode encoding method internationally. <strong>It is a variable-length encoding</strong> that uses 1 to 4 bytes to represent a character, depending on the complexity of the character. ASCII characters only require 1 byte, Latin and Greek letters require 2 bytes, commonly used Chinese characters require 3 bytes, and some other rare characters require 4 bytes.</p>
-<p>The encoding rules of UTF-8 are not complicated and can be divided into the following two cases.</p>
+<p>ASCII has already shown that English text needs only 1 byte per character. This scheme would double its storage compared with ASCII, wasting memory. We therefore need a more efficient Unicode encoding.</p>
+<h2 id="344-utf-8-encoding">3.4.4 &nbsp; UTF-8 encoding<a class="headerlink" href="#344-utf-8-encoding" title="Permanent link">&para;</a></h2>
+<p>UTF-8 is now the most widely used Unicode encoding worldwide. <strong>It is variable-length</strong>, using 1 to 4 bytes per character depending on the character. ASCII characters need 1 byte, Latin and Greek letters need 2, common Chinese characters need 3, and some less common characters need 4.</p>
+<p>UTF-8's rules have two cases.</p>
 <ul>
-<li>For 1-byte characters, set the highest bit to <span class="arithmatex">\(0\)</span>, and set the remaining 7 bits to the Unicode code point. It is worth noting that ASCII characters occupy the first 128 code points in the Unicode character set. That is to say, <strong>UTF-8 encoding is backward compatible with ASCII code</strong>. This means we can use UTF-8 to parse very old ASCII code text.</li>
-<li>For characters with a length of <span class="arithmatex">\(n\)</span> bytes (where <span class="arithmatex">\(n &gt; 1\)</span>), set the highest <span class="arithmatex">\(n\)</span> bits of the first byte to <span class="arithmatex">\(1\)</span>, and set the <span class="arithmatex">\((n + 1)\)</span>-th bit to <span class="arithmatex">\(0\)</span>; starting from the second byte, set the highest 2 bits of each byte to <span class="arithmatex">\(10\)</span>; use all remaining bits to fill in the Unicode code point of the character.</li>
+<li>For a one-byte character, set the highest bit to <span class="arithmatex">\(0\)</span> and the remaining 7 bits to its Unicode code point. ASCII occupies the first 128 Unicode code points, so <strong>UTF-8 is backward-compatible with ASCII</strong>. UTF-8 can therefore decode old ASCII text.</li>
+<li>For an <span class="arithmatex">\(n\)</span>-byte character, where <span class="arithmatex">\(n &gt; 1\)</span>, set the first byte's leading <span class="arithmatex">\(n\)</span> bits to <span class="arithmatex">\(1\)</span> and the next bit, <span class="arithmatex">\(n + 1\)</span>, to <span class="arithmatex">\(0\)</span>. Starting with the second byte, set the leading 2 bits of every byte to <span class="arithmatex">\(10\)</span>. Fill the remaining bits with the character's Unicode code point.</li>
 </ul>
-<p>Figure 3-8 shows the UTF-8 encoding corresponding to "Hello 算法". It can be observed that since the highest <span class="arithmatex">\(n\)</span> bits are all set to <span class="arithmatex">\(1\)</span>, the system can determine that the character length is <span class="arithmatex">\(n\)</span> by counting the leading <span class="arithmatex">\(1\)</span> bits.</p>
-<p>But why set the highest 2 bits of all other bytes to <span class="arithmatex">\(10\)</span>? In fact, this <span class="arithmatex">\(10\)</span> can serve as a check symbol. Assuming the system starts parsing text from an incorrect byte, the <span class="arithmatex">\(10\)</span> at the beginning of the byte can help the system quickly determine an anomaly.</p>
-<p>The reason for using <span class="arithmatex">\(10\)</span> as a check symbol is that under UTF-8 encoding rules, it is impossible for a character's highest two bits to be <span class="arithmatex">\(10\)</span>. This conclusion can be proven by contradiction: assuming the highest two bits of a character are <span class="arithmatex">\(10\)</span>, it means the length of the character is <span class="arithmatex">\(1\)</span>, corresponding to ASCII code. However, the highest bit of ASCII code should be <span class="arithmatex">\(0\)</span>, which contradicts the assumption.</p>
-<p><img alt="UTF-8 encoding example" class="animation-figure" src="../character_encoding.assets/utf-8_hello_algo.png" /></p>
-<p align="center"> Figure 3-8 &nbsp; UTF-8 encoding example </p>
+<p>Figure 3-8 shows UTF-8 for "Hello算法". Since the leading <span class="arithmatex">\(n\)</span> bits are <span class="arithmatex">\(1\)</span>, the system can determine the character length <span class="arithmatex">\(n\)</span> by counting the leading <span class="arithmatex">\(1\)</span> bits.</p>
+<p>Why do all remaining bytes begin with <span class="arithmatex">\(10\)</span>? This <span class="arithmatex">\(10\)</span> acts as a check marker. If decoding starts at the wrong byte, the leading <span class="arithmatex">\(10\)</span> helps the system quickly detect the problem.</p>
+<p>The prefix <span class="arithmatex">\(10\)</span> works as a marker because no character's first byte can begin with <span class="arithmatex">\(10\)</span> under UTF-8's rules. To see this by contradiction, suppose it did begin with <span class="arithmatex">\(10\)</span>. The single leading one would imply a length of <span class="arithmatex">\(1\)</span> byte, corresponding to ASCII, but an ASCII byte must begin with <span class="arithmatex">\(0\)</span>, a contradiction.</p>
+<p><img alt="Example of UTF-8 encoding" class="animation-figure" src="../character_encoding.assets/utf-8_hello_algo.png" /></p>
+<p align="center"> Figure 3-8 &nbsp; Example of UTF-8 encoding </p>
 
-<p>In addition to UTF-8, common encoding methods also include the following two.</p>
+<p>Two other common encodings are:</p>
 <ul>
-<li><strong>UTF-16 encoding</strong>: Uses 2 or 4 bytes to represent a character. All ASCII characters and commonly used non-English characters are represented with 2 bytes; a few characters need to use 4 bytes. For 2-byte characters, UTF-16 encoding is equal to the Unicode code point.</li>
-<li><strong>UTF-32 encoding</strong>: Every character uses 4 bytes. This means that UTF-32 takes up more space than UTF-8 and UTF-16, especially for text with a high proportion of ASCII characters.</li>
+<li><strong>UTF-16</strong>: Uses 2 or 4 bytes per character. ASCII and common non-English characters use 2 bytes; some characters require 4. For a two-byte character, its UTF-16 encoding equals its Unicode code point.</li>
+<li><strong>UTF-32</strong>: Uses 4 bytes for every character. It therefore takes more space than UTF-8 or UTF-16, especially for text containing mostly ASCII characters.</li>
 </ul>
-<p>From the perspective of storage space occupation, using UTF-8 to represent English characters is very efficient because it only requires 1 byte; using UTF-16 encoding for some non-English characters (such as Chinese) will be more efficient because it only requires 2 bytes, while UTF-8 may require 3 bytes.</p>
-<p>From a compatibility perspective, UTF-8 has the best universality, and many tools and libraries support UTF-8 first.</p>
-<h2 id="345-character-encoding-in-programming-languages">3.4.5 &nbsp; Character Encoding in Programming Languages<a class="headerlink" href="#345-character-encoding-in-programming-languages" title="Permanent link">&para;</a></h2>
-<p>For many programming languages in the past, strings during program execution used internal encodings such as UTF-16 or UTF-32. Under these representations, we can often treat strings like arrays during processing, and this approach has the following advantages.</p>
+<p>In terms of storage, UTF-8 is efficient for English characters because each uses only 1 byte. UTF-16 is more efficient for some non-English characters, such as Chinese, using 2 bytes where UTF-8 may require 3.</p>
+<p>In terms of compatibility, UTF-8 is the most widely supported, and many tools and libraries prioritize it.</p>
+<h2 id="345-character-encoding-in-programming-languages">3.4.5 &nbsp; Character encoding in programming languages<a class="headerlink" href="#345-character-encoding-in-programming-languages" title="Permanent link">&para;</a></h2>
+<p>Historically, most programming languages have stored strings at runtime using fixed-width encodings such as UTF-16 or UTF-32. Fixed-width encoding lets us treat a string as an array, with the following advantages.</p>
 <ul>
-<li><strong>Random access</strong>: UTF-16 encoded strings can be easily accessed randomly. UTF-8 is a variable-length encoding. To find the <span class="arithmatex">\(i\)</span>-th character, we need to traverse from the beginning of the string to the <span class="arithmatex">\(i\)</span>-th character, which requires <span class="arithmatex">\(O(n)\)</span> time.</li>
-<li><strong>Character counting</strong>: Similar to random access, calculating the length of a UTF-16 encoded string is also an <span class="arithmatex">\(O(1)\)</span> operation. However, calculating the length of a UTF-8 encoded string requires traversing the entire string.</li>
-<li><strong>String operations</strong>: Many string operations (such as splitting, joining, inserting, deleting, etc.) on UTF-16 encoded strings are easier to perform. Performing these operations on UTF-8 encoded strings usually requires additional calculations to ensure that invalid UTF-8 encoding is not generated.</li>
+<li><strong>Random access</strong>: UTF-16 strings support straightforward random access. With variable-length UTF-8, finding the <span class="arithmatex">\(i\)</span>-th character requires traversing from the beginning to the <span class="arithmatex">\(i\)</span>-th character, taking <span class="arithmatex">\(O(n)\)</span> time.</li>
+<li><strong>Character counting</strong>: Similarly, finding the length of a UTF-16 string is an <span class="arithmatex">\(O(1)\)</span> operation, while finding the character count of a UTF-8 string requires traversing the whole string.</li>
+<li><strong>String operations</strong>: Operations such as splitting, concatenation, insertion, and deletion are easier on UTF-16 strings. With UTF-8, they usually require extra work to avoid producing invalid encodings.</li>
 </ul>
-<p>In fact, the design of character encoding schemes for programming languages is a very interesting topic involving many factors.</p>
+<p>The design of string representations is an interesting subject involving many factors.</p>
 <ul>
-<li>Java's <code>String</code> type uses UTF-16 encoding, with each character occupying 2 bytes. This is because at the beginning of Java language design, people believed that 16 bits were sufficient to represent all possible characters. However, this was an incorrect judgment. Later, the Unicode specification expanded beyond 16 bits, so characters in Java may now be represented by a pair of 16-bit values (called "surrogate pairs").</li>
-<li>The strings of JavaScript and TypeScript use UTF-16 encoding for reasons similar to Java. When Netscape first introduced the JavaScript language in 1995, Unicode was still in its early stages of development, and at that time, using 16-bit encoding was sufficient to represent all Unicode characters.</li>
-<li>C# uses UTF-16 encoding mainly because the .NET platform was designed by Microsoft, and many of Microsoft's technologies (including the Windows operating system) extensively use UTF-16 encoding.</li>
+<li>Java's <code>String</code> uses UTF-16, with 2 bytes per character. When Java was designed, 16 bits were thought sufficient for every character. This turned out to be incorrect: Unicode later expanded beyond 16 bits, so a Java character may now require a pair of 16-bit values called a surrogate pair.</li>
+<li>JavaScript and TypeScript use UTF-16 for similar historical reasons. When Netscape introduced JavaScript in 1995, Unicode was still young, and 16 bits could represent all its characters at the time.</li>
+<li>C# uses UTF-16 largely because Microsoft designed .NET, and many Microsoft technologies, including Windows, use UTF-16 extensively.</li>
 </ul>
-<p>Due to the underestimation of character quantities by the above programming languages, they had to adopt the "surrogate pair" method to represent Unicode characters with lengths exceeding 16 bits. This is a reluctant compromise. On the one hand, in strings containing surrogate pairs, one character may occupy 2 bytes or 4 bytes, thus losing the advantage of fixed-length encoding. On the other hand, handling surrogate pairs requires additional code, which increases the complexity and difficulty of debugging in programming.</p>
-<p>For the above reasons, some programming languages have proposed different encoding schemes.</p>
+<p>Having underestimated the number of characters, these languages had to adopt surrogate pairs for Unicode characters beyond 16 bits. This was a compromise. A character in such a string may occupy 2 or 4 bytes, losing the benefit of fixed width, and handling surrogate pairs requires additional code, making programming and debugging more complex.</p>
+<p>For these reasons, some languages use different representations.</p>
 <ul>
-<li>Python's <code>str</code> uses Unicode encoding and adopts a flexible string representation where the stored character length depends on the largest Unicode code point in the string. If all characters in the string are ASCII characters, each character occupies 1 byte; if there are characters exceeding the ASCII range but all within the Basic Multilingual Plane (BMP), each character occupies 2 bytes; if there are characters exceeding the BMP, each character occupies 4 bytes.</li>
-<li>Go language's <code>string</code> type uses UTF-8 encoding internally. Go language also provides the <code>rune</code> type, which is used to represent a single Unicode code point.</li>
-<li>Rust language's <code>str</code> and <code>String</code> types use UTF-8 encoding internally. Rust also provides the <code>char</code> type for representing a single Unicode code point.</li>
+<li>Python's <code>str</code> uses Unicode with a flexible representation whose character width depends on the largest code point in the string. If all characters are ASCII, each uses 1 byte. If some exceed ASCII but all lie within the Basic Multilingual Plane (BMP), each uses 2 bytes. If any lie outside the BMP, each uses 4 bytes.</li>
+<li>Go's <code>string</code> uses UTF-8 internally. Go also provides <code>rune</code> to represent a single Unicode code point.</li>
+<li>Rust's <code>str</code> and <code>String</code> use UTF-8 internally, while <code>char</code> represents a single Unicode code point.</li>
 </ul>
-<p>It should be noted that the above discussion is about how strings are stored in programming languages, <strong>which is different from how strings are stored in files or transmitted over networks</strong>. In file storage or network transmission, we usually encode strings into UTF-8 format to achieve optimal compatibility and space efficiency.</p>
+<p>These descriptions concern string storage within programming languages. <strong>How strings are stored in files or transmitted over a network is a separate question</strong>. For file storage and network transmission, strings are usually encoded as UTF-8 for compatibility and space efficiency.</p>
 
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@@ -4808,7 +4880,7 @@ aria-label="Footer"
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-    aria-label="Previous: 3.3 Number Encoding *"
+    aria-label="Previous: 3.3   Number encoding *"
     rel="prev"
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@@ -4820,7 +4892,7 @@ aria-label="Footer"
         Previous
       </span>
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+        3.3 &nbsp; Number encoding *
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   <a
     href="../summary/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 3.5 Summary"
+    aria-label="Next: 3.5   Summary"
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-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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@@ -41,7 +41,7 @@
 
 
 
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-  0.2 How to Use This Book
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-  0.3 Summary
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-  Chapter 1. Encounter with Algorithms
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-  1.2 What Is an Algorithm
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-  2.6 Exercises
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-  Chapter 3. Data Structures
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-  3.1 Classification of Data Structures
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+  <a href="#311-logical-structure-linear-and-nonlinear" class="md-nav__link">
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+  6.2 &nbsp; Hash collisions
 
 
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+  7.4 &nbsp; Binary search trees
 
 
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+  7.5 &nbsp; AVL trees *
 
 
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+  16.2 &nbsp; Contribute to the book
 
 
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-  <a href="#311-logical-structure-linear-and-non-linear" class="md-nav__link">
+  <a href="#311-logical-structure-linear-and-nonlinear" class="md-nav__link">
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-        3.1.1 &nbsp; Logical Structure: Linear and Non-Linear
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-  <a href="#312-physical-structure-contiguous-and-dispersed" class="md-nav__link">
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       <span class="md-typeset">
-        3.1.2 &nbsp; Physical Structure: Contiguous and Dispersed
+        3.1.2 &nbsp; Physical structure: contiguous and scattered storage
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 <!-- Page content -->
-<h1 id="31-classification-of-data-structures">3.1 &nbsp; Classification of Data Structures<a class="headerlink" href="#31-classification-of-data-structures" title="Permanent link">&para;</a></h1>
-<p>Common data structures include arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graphs. They can be classified from two dimensions: "logical structure" and "physical structure".</p>
-<h2 id="311-logical-structure-linear-and-non-linear">3.1.1 &nbsp; Logical Structure: Linear and Non-Linear<a class="headerlink" href="#311-logical-structure-linear-and-non-linear" title="Permanent link">&para;</a></h2>
-<p><strong>Logical structure reveals the logical relationships between data elements</strong>. In arrays and linked lists, data is arranged in a certain order, embodying linear relationships between elements; while in trees, data is arranged hierarchically from top to bottom, showing parent-descendant relationships; graphs are composed of nodes and edges, reflecting complex network relationships.</p>
-<p>As shown in Figure 3-1, logical structures can be divided into two major categories: "linear" and "non-linear". Linear structures are more intuitive, indicating that data is linearly arranged in logical relationships; non-linear structures are the opposite, arranged non-linearly.</p>
+<h1 id="31-classifying-data-structures">3.1 &nbsp; Classifying data structures<a class="headerlink" href="#31-classifying-data-structures" title="Permanent link">&para;</a></h1>
+<p>Common data structures include arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graphs. They can be classified by their logical structure and physical structure.</p>
+<h2 id="311-logical-structure-linear-and-nonlinear">3.1.1 &nbsp; Logical structure: linear and nonlinear<a class="headerlink" href="#311-logical-structure-linear-and-nonlinear" title="Permanent link">&para;</a></h2>
+<p><strong>Logical structure describes the relationships between data elements</strong>. Arrays and linked lists arrange data in sequence, expressing linear relationships. Trees arrange data hierarchically from top to bottom, expressing ancestry and descent. Graphs consist of vertices and edges that express complex network relationships.</p>
+<p>As shown in Figure 3-1, logical structures fall into two broad categories: linear and nonlinear. A linear structure arranges data in a logical sequence; a nonlinear structure does not.</p>
 <ul>
-<li><strong>Linear data structures</strong>: Arrays, linked lists, stacks, queues, hash tables, where elements have a one-to-one sequential relationship.</li>
-<li><strong>Non-linear data structures</strong>: Trees, heaps, graphs, hash tables.</li>
+<li><strong>Linear data structures</strong>: Arrays, linked lists, stacks, queues, and hash tables. Elements have one-to-one sequential relationships.</li>
+<li><strong>Nonlinear data structures</strong>: Trees, heaps, graphs, and hash tables.</li>
 </ul>
-<p>Non-linear data structures can be further divided into tree structures and network structures.</p>
+<p>Nonlinear data structures can be divided further into tree and network structures.</p>
 <ul>
-<li><strong>Tree structures</strong>: Trees, heaps, hash tables, where elements have a one-to-many relationship.</li>
-<li><strong>Network structures</strong>: Graphs, where elements have a many-to-many relationship.</li>
+<li><strong>Tree structures</strong>: Trees, heaps, and hash tables, with one-to-many relationships.</li>
+<li><strong>Network structures</strong>: Graphs, with many-to-many relationships.</li>
 </ul>
-<p><img alt="Linear and non-linear data structures" class="animation-figure" src="../classification_of_data_structure.assets/classification_logic_structure.png" /></p>
-<p align="center"> Figure 3-1 &nbsp; Linear and non-linear data structures </p>
+<p><img alt="Linear and nonlinear data structures" class="animation-figure" src="../classification_of_data_structure.assets/classification_logic_structure.png" /></p>
+<p align="center"> Figure 3-1 &nbsp; Linear and nonlinear data structures </p>
 
-<h2 id="312-physical-structure-contiguous-and-dispersed">3.1.2 &nbsp; Physical Structure: Contiguous and Dispersed<a class="headerlink" href="#312-physical-structure-contiguous-and-dispersed" title="Permanent link">&para;</a></h2>
-<p><strong>When an algorithm program runs, the data being processed is mainly stored in memory</strong>. Figure 3-2 shows a computer memory stick, where each black square contains a memory space. We can imagine memory as a huge Excel spreadsheet, where each cell can store a certain amount of data.</p>
-<p><strong>The system accesses data at the target location through memory addresses</strong>. As shown in Figure 3-2, the computer assigns a number to each cell in the spreadsheet according to specific rules, ensuring that each memory space has a unique memory address. With these addresses, the program can access data in memory.</p>
-<p><img alt="Memory stick, memory space, memory address" class="animation-figure" src="../classification_of_data_structure.assets/computer_memory_location.png" /></p>
-<p align="center"> Figure 3-2 &nbsp; Memory stick, memory space, memory address </p>
+<h2 id="312-physical-structure-contiguous-and-scattered-storage">3.1.2 &nbsp; Physical structure: contiguous and scattered storage<a class="headerlink" href="#312-physical-structure-contiguous-and-scattered-storage" title="Permanent link">&para;</a></h2>
+<p><strong>While an algorithm runs, the data it processes is mainly stored in memory</strong>. Figure 3-2 shows a memory module, with each black chip containing storage space. We can imagine memory as a huge Excel spreadsheet, where each cell holds a certain amount of data.</p>
+<p><strong>The system uses memory addresses to access data at particular locations</strong>. As shown in Figure 3-2, the computer assigns each cell a number according to specific rules, giving every memory location a unique address. Programs use these addresses to access data.</p>
+<p><img alt="Memory module, storage space, and memory addresses" class="animation-figure" src="../classification_of_data_structure.assets/computer_memory_location.png" /></p>
+<p align="center"> Figure 3-2 &nbsp; Memory module, storage space, and memory addresses </p>
 
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>It should be noted that comparing memory to an Excel spreadsheet is only a simplified analogy. The actual workings of memory are much more complex, involving concepts such as address space, memory management, cache mechanisms, virtual memory, and physical memory.</p>
+<p>The Excel spreadsheet analogy is a simplification. Real memory is more complex and involves address spaces, memory management, caching, virtual memory, and physical memory.</p>
 </div>
-<p>Memory is a shared resource for all programs. When a block of memory is occupied by a program, it usually cannot be used by other programs at the same time. <strong>Therefore, in the design of data structures and algorithms, memory resources are an important consideration</strong>. For example, the peak memory occupied by an algorithm should not exceed the remaining free memory of the system; if there is a lack of contiguous large memory blocks, then the data structure chosen must be able to be stored in dispersed memory spaces.</p>
-<p>As shown in Figure 3-3, <strong>physical structure reflects the way data is stored in computer memory</strong>. It can be divided into contiguous-space storage (arrays) and dispersed-space storage (linked lists). At a low level, physical structure determines how data is accessed, updated, inserted, and deleted. These two physical structures exhibit complementary characteristics in terms of time efficiency and space efficiency.</p>
-<p><img alt="Contiguous space storage and dispersed space storage" class="animation-figure" src="../classification_of_data_structure.assets/classification_phisical_structure.png" /></p>
-<p align="center"> Figure 3-3 &nbsp; Contiguous space storage and dispersed space storage </p>
+<p>Memory is a resource used by all programs. When a program occupies a region of memory, other programs usually cannot use that region at the same time. <strong>Memory resources are therefore an important consideration when designing data structures and algorithms</strong>. For example, an algorithm's peak memory usage should not exceed the system's available memory. If no large contiguous region is available, the chosen data structure must support scattered storage.</p>
+<p>As shown in Figure 3-3, <strong>physical structure describes how data is stored in computer memory</strong>: either contiguously (arrays) or in scattered locations (linked lists). It determines how data can be accessed, updated, inserted, and deleted. The two storage forms have complementary strengths in time and space efficiency.</p>
+<p><img alt="Contiguous and scattered storage" class="animation-figure" src="../classification_of_data_structure.assets/classification_phisical_structure.png" /></p>
+<p align="center"> Figure 3-3 &nbsp; Contiguous and scattered storage </p>
 
-<p>It is worth noting that <strong>all data structures are implemented based on arrays, linked lists, or a combination of both</strong>. For example, stacks and queues can be implemented using either arrays or linked lists; while the implementation of hash tables may include both arrays and linked lists.</p>
+<p><strong>All data structures are implemented using arrays, linked lists, or a combination of the two</strong>. For example, stacks and queues can use either arrays or linked lists, while hash tables may use both.</p>
 <ul>
-<li><strong>Can be implemented based on arrays</strong>: Stacks, queues, hash tables, trees, heaps, graphs, matrices, tensors (arrays with dimensions <span class="arithmatex">\(\geq 3\)</span>), etc.</li>
-<li><strong>Can be implemented based on linked lists</strong>: Stacks, queues, hash tables, trees, heaps, graphs, etc.</li>
+<li><strong>Array-based implementations</strong>: Stacks, queues, hash tables, trees, heaps, graphs, matrices, tensors (arrays with dimension <span class="arithmatex">\(\geq 3\)</span>), and more.</li>
+<li><strong>Linked-list-based implementations</strong>: Stacks, queues, hash tables, trees, heaps, graphs, and more.</li>
 </ul>
-<p>After initialization, linked lists can still adjust their length during program execution, so they are also called "dynamic data structures". After initialization, the length of arrays cannot be changed, so they are also called "static data structures". It is worth noting that arrays can change length by reallocating memory, thus retaining a limited degree of flexibility.</p>
+<p>A linked list can change length during execution after initialization, so it is called a dynamic data structure. An array's length is fixed after initialization, so it is called a static data structure. However, reallocating memory can change an array's length, providing a degree of dynamic behavior.</p>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>If you find it difficult to understand physical structure, it is recommended to read the next chapter first, and then review this section.</p>
+<p>If physical structure feels difficult to understand, read the next chapter first and then return to this section.</p>
 </div>
 
 <!-- Source file information -->
@@ -4721,7 +4793,7 @@ aria-label="Footer"
   <a
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     class="md-footer__link md-footer__link--prev"
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+    aria-label="Previous: Chapter 3.   Data structures"
     rel="prev"
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@@ -4733,7 +4805,7 @@ aria-label="Footer"
         Previous
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@@ -4745,7 +4817,7 @@ aria-label="Footer"
   <a
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     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 3.2 Basic Data Types"
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@@ -4770,7 +4842,7 @@ aria-label="Footer"
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+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
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-  3.2 Basic Data Types
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-  3.3 Number Encoding *
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
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+    <nav class="md-nav" aria-label="3.6.1 Check your understanding">
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-  <a href="#1-data-relationships-in-everyday-situations" class="md-nav__link">
+  <a href="#1-relationships-between-data-in-everyday-situations" class="md-nav__link">
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-  <a href="#2-storing-a-logical-order-in-memory" class="md-nav__link">
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-        2. &nbsp; Storing a Logical Order in Memory
+        2. &nbsp; Storing a logical sequence in memory
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-        3. &nbsp; Data Types and Structures in Homework Records
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+    <nav class="md-nav" aria-label="3.6.2 Programming exercises">
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-  <a href="#1-count-the-1s-in-a-binary-representation" class="md-nav__link">
+  <a href="#1-counting-ones-in-a-binary-representation" class="md-nav__link">
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+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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+  4.6 &nbsp; Exercises
 
 
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+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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+  Chapter 6. &nbsp; Hash tables
 
 
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+  6.2 &nbsp; Hash collisions
 
 
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+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
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+  7.3 &nbsp; Array representation of a binary tree
 
 
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+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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+  7.6 &nbsp; Summary
 
 
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+  7.7 &nbsp; Exercises
 
 
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+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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@@ -2547,7 +2547,7 @@
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-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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@@ -2765,7 +2765,7 @@
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-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
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+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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+  Chapter 13. &nbsp; Backtracking
 
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
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+  13.6 &nbsp; Exercises
 
 
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+  Chapter 14. &nbsp; Dynamic programming
 
 
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+  Chapter 14. &nbsp; Dynamic programming
 
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+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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+  Chapter 15. &nbsp; Greedy algorithms
 
 
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+  Chapter 15. &nbsp; Greedy algorithms
 
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+  15.1 &nbsp; Greedy algorithms
 
 
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+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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@@ -4614,6 +4614,78 @@
 
 
 
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@@ -4655,22 +4727,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#361-concept-review" class="md-nav__link">
+  <a href="#361-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3.6.1 &nbsp; Concept Review
+        3.6.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="3.6.1 Concept Review">
+    <nav class="md-nav" aria-label="3.6.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-data-relationships-in-everyday-situations" class="md-nav__link">
+  <a href="#1-relationships-between-data-in-everyday-situations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Data Relationships in Everyday Situations
+        1. &nbsp; Relationships between data in everyday situations
       </span>
     </span>
   </a>
@@ -4678,10 +4750,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-storing-a-logical-order-in-memory" class="md-nav__link">
+  <a href="#2-storing-a-logical-sequence-in-memory" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Storing a Logical Order in Memory
+        2. &nbsp; Storing a logical sequence in memory
       </span>
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   <a href="#3-data-types-and-structures-in-homework-records" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Data Types and Structures in Homework Records
+        3. &nbsp; Data types and structures in homework records
       </span>
     </span>
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@@ -4708,19 +4780,19 @@
   <a href="#362-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3.6.2 &nbsp; Programming Exercises
+        3.6.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="3.6.2 Programming Exercises">
+    <nav class="md-nav" aria-label="3.6.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-count-the-1s-in-a-binary-representation" class="md-nav__link">
+  <a href="#1-counting-ones-in-a-binary-representation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Count the 1s in a Binary Representation
+        1. &nbsp; Counting ones in a binary representation
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@@ -4770,96 +4842,96 @@
 
 <!-- Page content -->
 <h1 id="36-exercises">3.6 &nbsp; Exercises<a class="headerlink" href="#36-exercises" title="Permanent link">&para;</a></h1>
-<h2 id="361-concept-review">3.6.1 &nbsp; Concept Review<a class="headerlink" href="#361-concept-review" title="Permanent link">&para;</a></h2>
-<h3 id="1-data-relationships-in-everyday-situations">1. &nbsp; Data Relationships in Everyday Situations<a class="headerlink" href="#1-data-relationships-in-everyday-situations" title="Permanent link">&para;</a></h3>
-<p>Based on the relationships among the data, choose one of "linear structure," "tree structure," and "network structure" for each situation below, and explain why:</p>
+<h2 id="361-check-your-understanding">3.6.1 &nbsp; Check your understanding<a class="headerlink" href="#361-check-your-understanding" title="Permanent link">&para;</a></h2>
+<h3 id="1-relationships-between-data-in-everyday-situations">1. &nbsp; Relationships between data in everyday situations<a class="headerlink" href="#1-relationships-between-data-in-everyday-situations" title="Permanent link">&para;</a></h3>
+<p>For each situation below, choose a linear, tree, or network structure based on the relationships between the data, and explain your choice:</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Students stand in a line; we consider only each student's immediate neighbors in front of and behind them.</li>
-<li>A school is organized in levels: "school → grade → class."</li>
-<li>City roads connect many intersections. One intersection may lead to several others, and the roads may form cycles.</li>
+<li>Students stand in a line, each concerned only with the people immediately in front of and behind them.</li>
+<li>A school is organized in a hierarchy: school → grade → class.</li>
+<li>City roads connect intersections. An intersection can lead to several others, and the roads may form cycles.</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>This is a linear structure. Except for the first and last students, each person is adjacent to exactly one person in front and one behind, so the relationships extend along a single line.</p>
+<p>A linear structure. Apart from the first and last people, each person has exactly one neighbor in front and one behind, forming a line of relationships.</p>
 </li>
 <li>
-<p>This is a tree structure. Each class belongs to one grade, and each grade belongs to the school, so the relationships form levels from top to bottom.</p>
+<p>A tree structure. Each class belongs to a grade, and each grade belongs to the school, forming a hierarchy from top to bottom.</p>
 </li>
 <li>
-<p>This is a network structure. An intersection can connect to several other intersections, and the routes can form cycles, so they cannot be arranged in a single order or a strict hierarchy.</p>
-<p>To identify a structure, first examine the relationships among the elements rather than how much space it uses in memory.</p>
+<p>A network structure. An intersection can connect to several others, and routes can form cycles. These relationships cannot be arranged in a single sequence or a strict hierarchy.</p>
+<p>To identify a structure, first examine the relationships between its elements, rather than how much memory it occupies.</p>
 </li>
 </ol>
 </details>
-<h3 id="2-storing-a-logical-order-in-memory">2. &nbsp; Storing a Logical Order in Memory<a class="headerlink" href="#2-storing-a-logical-order-in-memory" title="Permanent link">&para;</a></h3>
-<p>Consider two simplified memory layouts for storing the logical order <code>A → B → C</code>:</p>
+<h3 id="2-storing-a-logical-sequence-in-memory">2. &nbsp; Storing a logical sequence in memory<a class="headerlink" href="#2-storing-a-logical-sequence-in-memory" title="Permanent link">&para;</a></h3>
+<p>Consider two simplified memory layouts for storing the logical sequence <code>A → B → C</code>:</p>
 <ul>
 <li>Layout A: <code>A, B, C</code> are stored in memory cells numbered <code>20, 21, 22</code>, respectively.</li>
-<li>Layout B: <code>A, B, C</code> are stored in memory cells numbered <code>20, 7, 31</code>, respectively. <code>A</code> records the location of <code>B</code>, and <code>B</code> records the location of <code>C</code>.</li>
+<li>Layout B: <code>A, B, C</code> are stored in cells numbered <code>20, 7, 31</code>, respectively. <code>A</code> records the location of <code>B</code>, and <code>B</code> records the location of <code>C</code>.</li>
 </ul>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Which layout uses contiguous-space storage, and which uses dispersed-space storage?</li>
+<li>Which layout uses contiguous storage, and which uses noncontiguous storage?</li>
 <li>Which layout is more like an array, and which is more like a linked list?</li>
-<li>The memory-cell numbers in Layout B are not in increasing order. Why can it still represent the logical order <code>A → B → C</code>?</li>
+<li>The cell numbers in layout B are not in ascending order. Why can it still represent the logical sequence <code>A → B → C</code>?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>Layout A uses consecutive memory cells, so it uses contiguous-space storage. The nodes in Layout B are scattered across different locations, so it uses dispersed-space storage.</p>
+<p>Layout A uses consecutive memory cells, so its storage is contiguous. In layout B, nodes occupy separate locations, so its storage is noncontiguous.</p>
 </li>
 <li>
-<p>Layout A is more like an array, and Layout B is more like a linked list.</p>
+<p>Layout A is more like an array; layout B is more like a linked list.</p>
 </li>
 <li>
-<p>The logical order is determined by the links recorded between nodes, not by the numerical order of the memory-cell numbers.
-    The location stored by <code>A</code> leads to <code>B</code>, and the location stored by <code>B</code> then leads to <code>C</code>, so <code>A, B, C</code> can still be visited in order.</p>
-<p>This also shows that logical structure and physical structure are two different ways of viewing the same data.</p>
+<p>The logical order is determined by the links recorded between nodes, not by the numerical order of their memory cell numbers.
+    The location stored in <code>A</code> leads to <code>B</code>, and the location stored in <code>B</code> leads to <code>C</code>, so we can still visit <code>A, B, C</code> in sequence.</p>
+<p>This also shows that logical and physical structure are two ways of looking at the same data.</p>
 </li>
 </ol>
 </details>
-<h3 id="3-data-types-and-structures-in-homework-records">3. &nbsp; Data Types and Structures in Homework Records<a class="headerlink" href="#3-data-types-and-structures-in-homework-records" title="Permanent link">&para;</a></h3>
-<p>A study group records, in seating order, whether each of four students submitted their homework:</p>
+<h3 id="3-data-types-and-structures-in-homework-records">3. &nbsp; Data types and structures in homework records<a class="headerlink" href="#3-data-types-and-structures-in-homework-records" title="Permanent link">&para;</a></h3>
+<p>A study group records whether four students have submitted their homework, in seating order:</p>
 <p><code>[true, false, true, true]</code></p>
 <!-- numbered-subquestions -->
 
 <ol>
 <li>Which basic data type is suitable for each element?</li>
-<li>The four elements are arranged in a line by seating order. What logical structure does this use?</li>
-<li>Suppose the group later records each student's homework score as <code>[90, 0, 85, 100]</code>. Does this change the data's "content type" or its "organization"? Explain why.</li>
+<li>These four elements form a sequence in seating order. What logical structure does this use?</li>
+<li>If the group later records homework scores instead, as <code>[90, 0, 85, 100]</code>, has the type of content changed, or the way it is organized? Explain.</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>Each element represents only "yes" or "no," so the Boolean type <code>bool</code> is suitable.</p>
+<p>Each element represents only yes or no, so the Boolean type <code>bool</code> is appropriate.</p>
 </li>
 <li>
-<p>The elements are arranged in seating order, forming a linear structure that can be stored in an array.</p>
+<p>The elements form a linear structure in seating order and can be stored in an array.</p>
 </li>
 <li>
-<p>The content type changes: the elements change from Boolean values to integers. The organization does not change.
-    The data is still arranged in a line by seating order and can still be stored in an array, which is a linear structure.</p>
-<p>A basic data type describes "what is stored," while a data structure describes "how the data is organized."</p>
+<p>The content type has changed: the elements are now integers instead of Boolean values. Their organization has not changed.
+    The data still forms a sequence in seating order and can still use the linear structure of an array.</p>
+<p>A basic data type describes what is stored; a data structure describes how the data is organized.</p>
 </li>
 </ol>
 </details>
-<h2 id="362-programming-exercises">3.6.2 &nbsp; Programming Exercises<a class="headerlink" href="#362-programming-exercises" title="Permanent link">&para;</a></h2>
-<h3 id="1-count-the-1s-in-a-binary-representation">1. &nbsp; Count the 1s in a Binary Representation<a class="headerlink" href="#1-count-the-1s-in-a-binary-representation" title="Permanent link">&para;</a></h3>
-<p>Given a non-negative integer <code>n</code>, count the number of 1s in its binary representation.</p>
-<p>Use bitwise operations. Do not convert the binary representation to a string or use a built-in function that directly counts 1s.</p>
+<h2 id="362-programming-exercises">3.6.2 &nbsp; Programming exercises<a class="headerlink" href="#362-programming-exercises" title="Permanent link">&para;</a></h2>
+<h3 id="1-counting-ones-in-a-binary-representation">1. &nbsp; Counting ones in a binary representation<a class="headerlink" href="#1-counting-ones-in-a-binary-representation" title="Permanent link">&para;</a></h3>
+<p>Given a nonnegative integer <code>n</code>, count the ones in its binary representation.</p>
+<p>Use bitwise operations. Do not convert the binary representation to a string or use a built-in function that counts the ones directly.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>n &amp; 1 extracts the rightmost bit of n, which tells you whether that bit is 1</li>
-<li>Shifting right by one discards the current rightmost bit; most languages use the operator &gt;&gt;</li>
-<li>After implementing the method that checks and shifts one bit at a time, observe that n &amp; (n - 1) turns the rightmost 1 in n into 0</li>
+<li>n &amp; 1 extracts the rightmost bit of n, allowing you to check whether it is 1.</li>
+<li>Shifting right by one bit discards the current rightmost bit; most languages use the &gt;&gt; operator.</li>
+<li>After implementing the method that checks each bit and shifts right, observe that n &amp; (n - 1) changes the rightmost 1 in n to 0.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/number-of-1-bits/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
@@ -4885,7 +4957,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 3.5 Summary"
+    aria-label="Previous: 3.5   Summary"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4897,7 +4969,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        3.5 Summary
+        3.5 &nbsp; Summary
       </div>
     </div>
   </a>
@@ -4909,7 +4981,7 @@ aria-label="Footer"
   <a
     href="../../chapter_array_and_linkedlist/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: Chapter 4. &amp;nbsp; Arrays and Linked Lists"
+    aria-label="Next: Chapter 4.   Arrays and linked lists"
     rel="next"
   >
     <div class="md-footer__title">
@@ -4917,7 +4989,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        Chapter 4. &nbsp; Arrays and Linked Lists
+        Chapter 4. &nbsp; Arrays and linked lists
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -4934,7 +5006,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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diff --git a/en/chapter_data_structure/index.html b/en/chapter_data_structure/index.html
index 465207dfe..57e4dc905 100644
--- a/en/chapter_data_structure/index.html
+++ b/en/chapter_data_structure/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>Chapter 3. Data Structures - Hello Algo</title>
+        <title>Chapter 3. Data structures - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
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@@ -154,7 +154,7 @@
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+              Chapter 3. &nbsp; Data structures
 
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -982,7 +982,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1003,7 +1003,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1024,7 +1024,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1051,7 +1051,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1078,7 +1078,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1105,7 +1105,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1132,7 +1132,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1159,7 +1159,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1225,7 +1225,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1246,7 +1246,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1267,7 +1267,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1294,7 +1294,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1375,7 +1375,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1466,7 +1466,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1487,7 +1487,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1508,7 +1508,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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@@ -1535,7 +1535,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1680,7 +1680,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1701,7 +1701,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1722,7 +1722,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1749,7 +1749,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1776,7 +1776,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1803,7 +1803,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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@@ -1830,7 +1830,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1898,7 +1898,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1919,7 +1919,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1940,7 +1940,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1967,7 +1967,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1994,7 +1994,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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@@ -2021,7 +2021,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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@@ -2048,7 +2048,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2075,7 +2075,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2102,7 +2102,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2166,7 +2166,7 @@
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2187,7 +2187,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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@@ -2208,7 +2208,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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@@ -2235,7 +2235,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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@@ -2262,7 +2262,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.5 Exercises
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-  Chapter 10. Searching
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
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-  11.2 Selection Sort
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-  11.3 Bubble Sort
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-  11.4 Insertion Sort
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   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
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+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
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-  12.6 Exercises
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-  Chapter 13. Backtracking
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@@ -3570,7 +3570,7 @@
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-  13.1 Backtracking Algorithm
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
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@@ -3817,7 +3817,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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@@ -3871,7 +3871,7 @@
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -3952,7 +3952,7 @@
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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@@ -3979,7 +3979,7 @@
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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@@ -4072,7 +4072,7 @@
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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@@ -4114,7 +4114,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
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   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
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   </span>
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   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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@@ -4577,19 +4649,19 @@
 
 
 <!-- Page content -->
-<h1 id="chapter-3-data-structures">Chapter 3. &nbsp; Data Structures<a class="headerlink" href="#chapter-3-data-structures" title="Permanent link">&para;</a></h1>
+<h1 id="chapter-3-data-structures">Chapter 3. &nbsp; Data structures<a class="headerlink" href="#chapter-3-data-structures" title="Permanent link">&para;</a></h1>
 <p><img alt="Data structures" class="cover-image" src="../assets/covers/chapter_data_structure.jpg" /></p>
 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
-<p>Data structures are like a sturdy and diverse framework.</p>
-<p>It provides a blueprint for the orderly organization of data, upon which algorithms come to life.</p>
+<p>Data structures provide a framework that is both sturdy and versatile.</p>
+<p>They offer a blueprint for organizing data, a foundation on which algorithms come to life.</p>
 </div>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
-<li><a href="classification_of_data_structure/">3.1 &nbsp; Classification of Data Structures</a></li>
-<li><a href="basic_data_types/">3.2 &nbsp; Basic Data Types</a></li>
-<li><a href="number_encoding/">3.3 &nbsp; Number Encoding *</a></li>
-<li><a href="character_encoding/">3.4 &nbsp; Character Encoding *</a></li>
+<li><a href="classification_of_data_structure/">3.1 &nbsp; Classifying data structures</a></li>
+<li><a href="basic_data_types/">3.2 &nbsp; Basic data types</a></li>
+<li><a href="number_encoding/">3.3 &nbsp; Number encoding *</a></li>
+<li><a href="character_encoding/">3.4 &nbsp; Character encoding *</a></li>
 <li><a href="summary/">3.5 &nbsp; Summary</a></li>
 <li><a href="exercises/">3.6 &nbsp; Exercises</a></li>
 </ul>
@@ -4615,7 +4687,7 @@ aria-label="Footer"
   <a
     href="../chapter_computational_complexity/exercises/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 2.6 Exercises"
+    aria-label="Previous: 2.6   Exercises"
     rel="prev"
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@@ -4627,7 +4699,7 @@ aria-label="Footer"
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       </span>
       <div class="md-ellipsis">
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+        2.6 &nbsp; Exercises
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   <a
     href="./classification_of_data_structure/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 3.1 Classification of Data Structures"
+    aria-label="Next: 3.1   Classifying data structures"
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@@ -4647,7 +4719,7 @@ aria-label="Footer"
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       <div class="md-ellipsis">
-        3.1 Classification of Data Structures
+        3.1 &nbsp; Classifying data structures
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@@ -4664,7 +4736,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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-        <title>3.3 Number Encoding * - Hello Algo</title>
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -982,7 +982,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
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@@ -1024,7 +1024,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1051,7 +1051,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1087,7 +1087,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#331-sign-magnitude-1s-complement-and-2s-complement" class="md-nav__link">
+  <a href="#331-sign-magnitude-ones-complement-and-twos-complement" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3.3.1 &nbsp; Sign-Magnitude, 1's Complement, and 2's Complement
+        3.3.1 &nbsp; Sign-magnitude, one's complement, and two's complement
       </span>
     </span>
   </a>
@@ -1141,10 +1141,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#332-floating-point-number-encoding" class="md-nav__link">
+  <a href="#332-floating-point-encoding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3.3.2 &nbsp; Floating-Point Number Encoding
+        3.3.2 &nbsp; Floating-point encoding
       </span>
     </span>
   </a>
@@ -1174,7 +1174,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1201,7 +1201,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1228,7 +1228,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1294,7 +1294,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1336,7 +1336,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1390,7 +1390,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1417,7 +1417,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1444,7 +1444,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1535,7 +1535,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1577,7 +1577,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1604,7 +1604,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1631,7 +1631,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1658,7 +1658,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1685,7 +1685,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1749,7 +1749,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1791,7 +1791,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1818,7 +1818,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1845,7 +1845,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1872,7 +1872,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1899,7 +1899,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1967,7 +1967,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2009,7 +2009,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -2036,7 +2036,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2063,7 +2063,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2117,7 +2117,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2144,7 +2144,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2171,7 +2171,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2235,7 +2235,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2277,7 +2277,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2304,7 +2304,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2331,7 +2331,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2358,7 +2358,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2385,7 +2385,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2449,7 +2449,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2491,7 +2491,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2518,7 +2518,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2545,7 +2545,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2572,7 +2572,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2599,7 +2599,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2688,7 +2688,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2709,7 +2709,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2736,7 +2736,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2763,7 +2763,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2790,7 +2790,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2817,7 +2817,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2844,7 +2844,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2871,7 +2871,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2970,7 +2970,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2991,7 +2991,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3018,7 +3018,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3045,7 +3045,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3072,7 +3072,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3099,7 +3099,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3126,7 +3126,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3153,7 +3153,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3180,7 +3180,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3207,7 +3207,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3234,7 +3234,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3261,7 +3261,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3288,7 +3288,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3375,7 +3375,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3396,7 +3396,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3423,7 +3423,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3450,7 +3450,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3477,7 +3477,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3504,7 +3504,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3531,7 +3531,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3597,7 +3597,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3618,7 +3618,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3639,7 +3639,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3666,7 +3666,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3693,7 +3693,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3720,7 +3720,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3747,7 +3747,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3774,7 +3774,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3844,7 +3844,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3865,7 +3865,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3886,7 +3886,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3913,7 +3913,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3940,7 +3940,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3967,7 +3967,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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-  <a href="#331-sign-magnitude-1s-complement-and-2s-complement" class="md-nav__link">
+  <a href="#331-sign-magnitude-ones-complement-and-twos-complement" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3.3.1 &nbsp; Sign-Magnitude, 1's Complement, and 2's Complement
+        3.3.1 &nbsp; Sign-magnitude, one's complement, and two's complement
       </span>
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   </a>
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 </li>
 
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+  <a href="#332-floating-point-encoding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3.3.2 &nbsp; Floating-Point Number Encoding
+        3.3.2 &nbsp; Floating-point encoding
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 <!-- Page content -->
-<h1 id="33-number-encoding">3.3 &nbsp; Number Encoding *<a class="headerlink" href="#33-number-encoding" title="Permanent link">&para;</a></h1>
+<h1 id="33-number-encoding">3.3 &nbsp; Number encoding *<a class="headerlink" href="#33-number-encoding" title="Permanent link">&para;</a></h1>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>In this book, chapters marked with an asterisk * are optional readings. If you are short on time or find them challenging, you may skip these initially and return to them after completing the essential chapters.</p>
+<p>Sections marked with * in their titles are optional. If you are short on time or find them difficult, you can skip them initially and return after completing the required sections.</p>
 </div>
-<h2 id="331-sign-magnitude-1s-complement-and-2s-complement">3.3.1 &nbsp; Sign-Magnitude, 1's Complement, and 2's Complement<a class="headerlink" href="#331-sign-magnitude-1s-complement-and-2s-complement" title="Permanent link">&para;</a></h2>
-<p>In the table from the previous section, we found that all integer types can represent one more negative number than positive numbers. For example, the <code>byte</code> range is <span class="arithmatex">\([-128, 127]\)</span>. This phenomenon is counterintuitive, and its underlying cause lies in sign-magnitude, 1's complement, and 2's complement representations.</p>
-<p>First, it should be noted that <strong>numbers are stored in computers in the form of "2's complement"</strong>. Before analyzing the reasons for this, let's first define these three concepts.</p>
+<h2 id="331-sign-magnitude-ones-complement-and-twos-complement">3.3.1 &nbsp; Sign-magnitude, one's complement, and two's complement<a class="headerlink" href="#331-sign-magnitude-ones-complement-and-twos-complement" title="Permanent link">&para;</a></h2>
+<p>The previous table showed that every integer type represents one more negative value than positive value. For example, <code>byte</code> ranges over <span class="arithmatex">\([-128, 127]\)</span>. Understanding this counterintuitive fact requires sign-magnitude, one's complement, and two's complement.</p>
+<p>First, <strong>integers are stored in two's complement form</strong>. Before explaining why, let's define the three representations.</p>
 <ul>
-<li><strong>Sign-magnitude</strong>: We treat the highest bit of the binary representation of a number as the sign bit, where <span class="arithmatex">\(0\)</span> represents a positive number and <span class="arithmatex">\(1\)</span> represents a negative number, and the remaining bits represent the value of the number.</li>
-<li><strong>1's complement</strong>: The 1's complement of a positive number is the same as its sign-magnitude. For a negative number, the 1's complement is obtained by inverting all bits except the sign bit of its sign-magnitude.</li>
-<li><strong>2's complement</strong>: The 2's complement of a positive number is the same as its sign-magnitude. For a negative number, the 2's complement is obtained by adding <span class="arithmatex">\(1\)</span> to its 1's complement.</li>
+<li><strong>Sign-magnitude</strong>: The highest bit is the sign: <span class="arithmatex">\(0\)</span> for positive and <span class="arithmatex">\(1\)</span> for negative. The remaining bits represent the magnitude.</li>
+<li><strong>One's complement</strong>: A positive number has the same representation as sign-magnitude. For a negative number, invert every bit except the sign bit of its sign-magnitude representation.</li>
+<li><strong>Two's complement</strong>: A positive number again has the same representation as sign-magnitude. For a negative number, add <span class="arithmatex">\(1\)</span> to its one's complement.</li>
 </ul>
-<p>Figure 3-4 shows the conversion methods among sign-magnitude, 1's complement, and 2's complement.</p>
-<p><img alt="Conversions among sign-magnitude, 1's complement, and 2's complement" class="animation-figure" src="../number_encoding.assets/1s_2s_complement.png" /></p>
-<p align="center"> Figure 3-4 &nbsp; Conversions among sign-magnitude, 1's complement, and 2's complement </p>
+<p>Figure 3-4 shows how to convert among these representations.</p>
+<p><img alt="Converting between sign-magnitude, one's complement, and two's complement" class="animation-figure" src="../number_encoding.assets/1s_2s_complement.png" /></p>
+<p align="center"> Figure 3-4 &nbsp; Converting between sign-magnitude, one's complement, and two's complement </p>
 
-<p><u>Sign-magnitude</u>, although the most intuitive, has some limitations. On one hand, <strong>the sign-magnitude of negative numbers cannot be directly used in operations</strong>. For example, calculating <span class="arithmatex">\(1 + (-2)\)</span> in sign-magnitude yields <span class="arithmatex">\(-3\)</span>, which is clearly incorrect.</p>
+<p>Although <u>sign-magnitude</u> is intuitive, it has limitations. First, <strong>negative sign-magnitude values cannot be used directly in ordinary binary arithmetic</strong>. For example, directly adding <span class="arithmatex">\(1 + (-2)\)</span> in this representation gives <span class="arithmatex">\(-3\)</span>, which is clearly wrong.</p>
 <div class="arithmatex">\[
 \begin{aligned}
 &amp; 1 + (-2) \newline
@@ -4683,109 +4755,109 @@
 &amp; \rightarrow -3
 \end{aligned}
 \]</div>
-<p>To solve this problem, computers introduced <u>1's complement</u>. If we first convert sign-magnitude to 1's complement and calculate <span class="arithmatex">\(1 + (-2)\)</span> in 1's complement, then convert the result back to sign-magnitude, we can obtain the correct result of <span class="arithmatex">\(-1\)</span>.</p>
+<p>To address this, computers introduced <u>one's complement</u>. Convert the operands to one's complement, calculate <span class="arithmatex">\(1 + (-2)\)</span>, and convert the result back to sign-magnitude to obtain the correct answer, <span class="arithmatex">\(-1\)</span>.</p>
 <div class="arithmatex">\[
 \begin{aligned}
 &amp; 1 + (-2) \newline
-&amp; \rightarrow 0000 \; 0001 \; \text{(Sign-magnitude)} + 1000 \; 0010 \; \text{(Sign-magnitude)} \newline
-&amp; = 0000 \; 0001 \; \text{(1's complement)} + 1111  \; 1101 \; \text{(1's complement)} \newline
-&amp; = 1111 \; 1110 \; \text{(1's complement)} \newline
-&amp; = 1000 \; 0001 \; \text{(Sign-magnitude)} \newline
+&amp; \rightarrow 0000 \; 0001 \; \text{(sign-magnitude)} + 1000 \; 0010 \; \text{(sign-magnitude)} \newline
+&amp; = 0000 \; 0001 \; \text{(one's complement)} + 1111  \; 1101 \; \text{(one's complement)} \newline
+&amp; = 1111 \; 1110 \; \text{(one's complement)} \newline
+&amp; = 1000 \; 0001 \; \text{(sign-magnitude)} \newline
 &amp; \rightarrow -1
 \end{aligned}
 \]</div>
-<p>On the other hand, <strong>the sign-magnitude of the number zero has two representations, <span class="arithmatex">\(+0\)</span> and <span class="arithmatex">\(-0\)</span></strong>. This means that the number zero corresponds to two different binary encodings, which may cause ambiguity. For example, in conditional judgments, if we don't distinguish between positive zero and negative zero, it may lead to incorrect judgment results. If we want to handle the ambiguity of positive and negative zero, we need to introduce additional judgment operations, which may reduce the computational efficiency of the computer.</p>
+<p>Second, <strong>sign-magnitude has two representations of zero, <span class="arithmatex">\(+0\)</span> and <span class="arithmatex">\(-0\)</span></strong>. Two binary encodings for zero can create ambiguity. Conditional checks may be wrong if they do not handle both forms, while special handling requires extra checks that can reduce efficiency.</p>
 <div class="arithmatex">\[
 \begin{aligned}
 +0 &amp; \rightarrow 0000 \; 0000 \newline
 -0 &amp; \rightarrow 1000 \; 0000
 \end{aligned}
 \]</div>
-<p>Like sign-magnitude, 1's complement also has the problem of positive and negative zero ambiguity. Therefore, computers further introduced <u>2's complement</u>. Let's first observe the conversion process of negative zero from sign-magnitude to 1's complement to 2's complement:</p>
+<p>One's complement also has positive and negative zero, so computers introduced <u>two's complement</u>. Consider converting negative zero through all three representations:</p>
 <div class="arithmatex">\[
 \begin{aligned}
--0 \rightarrow \; &amp; 1000 \; 0000 \; \text{(Sign-magnitude)} \newline
-= \; &amp; 1111 \; 1111 \; \text{(1's complement)} \newline
-= 1 \; &amp; 0000 \; 0000 \; \text{(2's complement)} \newline
+-0 \rightarrow \; &amp; 1000 \; 0000 \; \text{(sign-magnitude)} \newline
+= \; &amp; 1111 \; 1111 \; \text{(one's complement)} \newline
+= 1 \; &amp; 0000 \; 0000 \; \text{(two's complement)} \newline
 \end{aligned}
 \]</div>
-<p>Adding <span class="arithmatex">\(1\)</span> to the 1's complement of negative zero produces a carry, but since the <code>byte</code> type has a length of only 8 bits, the <span class="arithmatex">\(1\)</span> that overflows to the 9<sup>th</sup> bit is discarded. That is to say, <strong>the 2's complement of negative zero is <span class="arithmatex">\(0000 \; 0000\)</span>, which is the same as the 2's complement of positive zero</strong>. This means that in 2's complement representation, there is only one zero, and the positive and negative zero ambiguity is thus resolved.</p>
-<p>One last question remains: the range of the <code>byte</code> type is <span class="arithmatex">\([-128, 127]\)</span>, so where does the extra negative number <span class="arithmatex">\(-128\)</span> come from? We notice that all integers in the interval <span class="arithmatex">\([-127, +127]\)</span> have corresponding sign-magnitude, 1's complement, and 2's complement, and sign-magnitude and 2's complement can be converted to each other.</p>
-<p>However, <strong>the 2's complement <span class="arithmatex">\(1000 \; 0000\)</span> is an exception, and it does not have a corresponding sign-magnitude</strong>. According to the conversion method, we get that the sign-magnitude of this 2's complement is <span class="arithmatex">\(0000 \; 0000\)</span>. This is clearly contradictory because this sign-magnitude represents the number <span class="arithmatex">\(0\)</span>, and its 2's complement should be itself. The computer specifies that this special 2's complement <span class="arithmatex">\(1000 \; 0000\)</span> represents <span class="arithmatex">\(-128\)</span>. In fact, the result of calculating <span class="arithmatex">\((-1) + (-127)\)</span> in 2's complement is <span class="arithmatex">\(-128\)</span>.</p>
+<p>Adding <span class="arithmatex">\(1\)</span> to the one's complement of negative zero produces a carry. A <code>byte</code> has only 8 bits, so the <span class="arithmatex">\(1\)</span> carried into the ninth bit is discarded. <strong>Negative zero's two's complement is therefore <span class="arithmatex">\(0000 \; 0000\)</span>, the same as positive zero</strong>. Two's complement has only one zero, eliminating the ambiguity.</p>
+<p>One question remains: if <code>byte</code> ranges over <span class="arithmatex">\([-128, 127]\)</span>, where does the extra negative value <span class="arithmatex">\(-128\)</span> come from? Every integer in <span class="arithmatex">\([-127, +127]\)</span> has sign-magnitude, one's complement, and two's complement representations, and its sign-magnitude and two's complement forms can be converted into each other.</p>
+<p>However, <strong>the two's complement value <span class="arithmatex">\(1000 \; 0000\)</span> is an exception with no corresponding sign-magnitude form</strong>. Applying the conversion gives <span class="arithmatex">\(0000 \; 0000\)</span>, but that represents <span class="arithmatex">\(0\)</span>, whose two's complement should be itself. The special pattern <span class="arithmatex">\(1000 \; 0000\)</span> is defined to represent <span class="arithmatex">\(-128\)</span>. Indeed, two's complement addition of <span class="arithmatex">\((-1) + (-127)\)</span> gives <span class="arithmatex">\(-128\)</span>.</p>
 <div class="arithmatex">\[
 \begin{aligned}
 &amp; (-127) + (-1) \newline
-&amp; \rightarrow 1111 \; 1111 \; \text{(Sign-magnitude)} + 1000 \; 0001 \; \text{(Sign-magnitude)} \newline
-&amp; = 1000 \; 0000 \; \text{(1's complement)} + 1111  \; 1110 \; \text{(1's complement)} \newline
-&amp; = 1000 \; 0001 \; \text{(2's complement)} + 1111  \; 1111 \; \text{(2's complement)} \newline
-&amp; = 1000 \; 0000 \; \text{(2's complement)} \newline
+&amp; \rightarrow 1111 \; 1111 \; \text{(sign-magnitude)} + 1000 \; 0001 \; \text{(sign-magnitude)} \newline
+&amp; = 1000 \; 0000 \; \text{(one's complement)} + 1111  \; 1110 \; \text{(one's complement)} \newline
+&amp; = 1000 \; 0001 \; \text{(two's complement)} + 1111  \; 1111 \; \text{(two's complement)} \newline
+&amp; = 1000 \; 0000 \; \text{(two's complement)} \newline
 &amp; \rightarrow -128
 \end{aligned}
 \]</div>
-<p>You may have noticed that all the above calculations are addition operations. This hints at an important fact: <strong>the hardware circuits inside computers are mainly designed based on addition operations</strong>. This is because addition operations are simpler to implement in hardware compared to other operations (such as multiplication, division, and subtraction), easier to parallelize, and have faster operation speeds.</p>
-<p>Please note that this does not mean that computers can only perform addition. <strong>By combining addition with some basic logical operations, computers can implement various other mathematical operations</strong>. For example, calculating the subtraction <span class="arithmatex">\(a - b\)</span> can be converted to calculating the addition <span class="arithmatex">\(a + (-b)\)</span>; calculating multiplication and division can be converted to calculating multiple additions or subtractions.</p>
-<p>We can now summarize why computers use 2's complement: with 2's complement representation, computers can use the same circuits and operations to handle the addition of positive and negative numbers, without designing special hardware circuits for subtraction or separately handling the ambiguity of positive and negative zero. This greatly simplifies hardware design and improves efficiency.</p>
-<p>The design of 2's complement is very ingenious. Due to space limitations, we will stop here. Interested readers are encouraged to explore further.</p>
-<h2 id="332-floating-point-number-encoding">3.3.2 &nbsp; Floating-Point Number Encoding<a class="headerlink" href="#332-floating-point-number-encoding" title="Permanent link">&para;</a></h2>
-<p>Careful readers may have noticed: <code>int</code> and <code>float</code> have the same length, both are 4 bytes, but why does <code>float</code> have a much larger range than <code>int</code>? This is very counterintuitive because it stands to reason that <code>float</code> needs to represent decimals, so the range should be smaller.</p>
-<p>In fact, <strong>this is because floating-point number <code>float</code> uses a different representation method</strong>. Let's denote a 32-bit binary number as:</p>
+<p>All the calculations above use addition. This points to an important fact: <strong>computer arithmetic circuits are largely built around addition</strong>. Compared with operations such as multiplication, division, and subtraction, addition is simpler to implement in hardware, easier to parallelize, and faster.</p>
+<p>This does not mean computers can only add. <strong>Combining addition with basic logical operations makes other arithmetic possible</strong>. For example, subtraction <span class="arithmatex">\(a - b\)</span> becomes addition <span class="arithmatex">\(a + (-b)\)</span>, while multiplication and division can be expressed as repeated additions or subtractions.</p>
+<p>We can now summarize why computers use two's complement. It lets the same circuitry and operations handle both positive and negative addition, without dedicated subtraction circuitry or special handling for positive and negative zero. This simplifies hardware and improves efficiency.</p>
+<p>Two's complement is an elegant design. Space limits our discussion here, but interested readers are encouraged to explore it further.</p>
+<h2 id="332-floating-point-encoding">3.3.2 &nbsp; Floating-point encoding<a class="headerlink" href="#332-floating-point-encoding" title="Permanent link">&para;</a></h2>
+<p>Both <code>int</code> and <code>float</code> occupy 4 bytes, yet <code>float</code> has a much wider range. This seems surprising: since <code>float</code> also represents fractions, should its range not be smaller?</p>
+<p><strong>The reason is that <code>float</code> uses a different representation</strong>. Write a 32-bit binary number as:</p>
 <div class="arithmatex">\[
 b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0
 \]</div>
-<p>According to the IEEE 754 standard, a 32-bit <code>float</code> consists of the following three parts.</p>
+<p>Under IEEE 754, a 32-bit <code>float</code> has three parts.</p>
 <ul>
-<li>Sign bit <span class="arithmatex">\(\mathrm{S}\)</span>: occupies 1 bit, corresponding to <span class="arithmatex">\(b_{31}\)</span>.</li>
-<li>Exponent bit <span class="arithmatex">\(\mathrm{E}\)</span>: occupies 8 bits, corresponding to <span class="arithmatex">\(b_{30} b_{29} \ldots b_{23}\)</span>.</li>
-<li>Fraction bit <span class="arithmatex">\(\mathrm{N}\)</span>: occupies 23 bits, corresponding to <span class="arithmatex">\(b_{22} b_{21} \ldots b_0\)</span>.</li>
+<li>Sign <span class="arithmatex">\(\mathrm{S}\)</span>: 1 bit, corresponding to <span class="arithmatex">\(b_{31}\)</span>.</li>
+<li>Exponent <span class="arithmatex">\(\mathrm{E}\)</span>: 8 bits, corresponding to <span class="arithmatex">\(b_{30} b_{29} \ldots b_{23}\)</span>.</li>
+<li>Fraction <span class="arithmatex">\(\mathrm{N}\)</span>: 23 bits, corresponding to <span class="arithmatex">\(b_{22} b_{21} \ldots b_0\)</span>.</li>
 </ul>
-<p>The calculation method for the value corresponding to the binary <code>float</code> is:</p>
+<p>The value represented by the binary <code>float</code> is:</p>
 <div class="arithmatex">\[
 \text {val} = (-1)^{b_{31}} \times 2^{\left(b_{30} b_{29} \ldots b_{23}\right)_2-127} \times\left(1 . b_{22} b_{21} \ldots b_0\right)_2
 \]</div>
-<p>Converted to decimal, the calculation formula is:</p>
+<p>In decimal notation, this becomes:</p>
 <div class="arithmatex">\[
 \text {val}=(-1)^{\mathrm{S}} \times 2^{\mathrm{E} -127} \times (1 + \mathrm{N})
 \]</div>
-<p>The range of each component is:</p>
+<p>The components have these ranges:</p>
 <div class="arithmatex">\[
 \begin{aligned}
 \mathrm{S} \in &amp; \{ 0, 1\}, \quad \mathrm{E} \in \{ 1, 2, \dots, 254 \} \newline
 (1 + \mathrm{N}) = &amp; (1 + \sum_{i=1}^{23} b_{23-i} 2^{-i}) \subset [1, 2 - 2^{-23}]
 \end{aligned}
 \]</div>
-<p><img alt="Calculation example of float under IEEE 754 standard" class="animation-figure" src="../number_encoding.assets/ieee_754_float.png" /></p>
-<p align="center"> Figure 3-5 &nbsp; Calculation example of float under IEEE 754 standard </p>
+<p><img alt="Example of an IEEE 754 float calculation" class="animation-figure" src="../number_encoding.assets/ieee_754_float.png" /></p>
+<p align="center"> Figure 3-5 &nbsp; Example of an IEEE 754 float calculation </p>
 
-<p>Observing Figure 3-5, given example data <span class="arithmatex">\(\mathrm{S} = 0\)</span>, <span class="arithmatex">\(\mathrm{E} = 124\)</span>, <span class="arithmatex">\(\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\)</span>, we have:</p>
+<p>In Figure 3-5, <span class="arithmatex">\(\mathrm{S} = 0\)</span>, <span class="arithmatex">\(\mathrm{E} = 124\)</span>, and <span class="arithmatex">\(\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\)</span>, giving:</p>
 <div class="arithmatex">\[
 \text { val } = (-1)^0 \times 2^{124 - 127} \times (1 + 0.375) = 0.171875
 \]</div>
-<p>Now we can answer the initial question: <strong>the representation of <code>float</code> includes an exponent bit, resulting in a range far greater than <code>int</code></strong>. According to the above calculation, the maximum positive number that <code>float</code> can represent is <span class="arithmatex">\(2^{254 - 127} \times (2 - 2^{-23}) \approx 3.4 \times 10^{38}\)</span>, and the minimum negative number can be obtained by switching the sign bit.</p>
-<p><strong>Although floating-point number <code>float</code> expands the range, its side effect is sacrificing precision</strong>. The integer type <code>int</code> uses all 32 bits to represent numbers, and the numbers are evenly distributed; however, due to the existence of the exponent bit, the larger the value of floating-point number <code>float</code>, the larger the difference between two adjacent numbers tends to be.</p>
-<p>As shown in Table 3-2, exponent bits <span class="arithmatex">\(\mathrm{E} = 0\)</span> and <span class="arithmatex">\(\mathrm{E} = 255\)</span> have special meanings, <strong>used to represent zero, infinity, <span class="arithmatex">\(\mathrm{NaN}\)</span>, etc.</strong></p>
-<p align="center"> Table 3-2 &nbsp; Meaning of exponent bits </p>
+<p>We can now answer the original question: <strong>the exponent gives <code>float</code> a much wider range than <code>int</code></strong>. The largest positive <code>float</code> is <span class="arithmatex">\(2^{254 - 127} \times (2 - 2^{-23}) \approx 3.4 \times 10^{38}\)</span>; changing the sign bit gives the most negative value.</p>
+<p><strong>The wider range of <code>float</code> comes at the cost of precision</strong>. An <code>int</code> uses all 32 bits to represent evenly spaced integer values. Because <code>float</code> includes an exponent, the gap between adjacent representable values tends to grow as their magnitude increases.</p>
+<p>As shown in Table 3-2, exponent values <span class="arithmatex">\(\mathrm{E} = 0\)</span> and <span class="arithmatex">\(\mathrm{E} = 255\)</span> have special meanings, <strong>representing zero, infinity, and <span class="arithmatex">\(\mathrm{NaN}\)</span>, among other values</strong>.</p>
+<p align="center"> Table 3-2 &nbsp; Meanings of exponent values </p>
 
 <div class="center-table">
 <table>
 <thead>
 <tr>
-<th>Exponent Bit E</th>
-<th>Fraction Bit <span class="arithmatex">\(\mathrm{N} = 0\)</span></th>
-<th>Fraction Bit <span class="arithmatex">\(\mathrm{N} \ne 0\)</span></th>
-<th>Calculation Formula</th>
+<th>Exponent E</th>
+<th>Fraction <span class="arithmatex">\(\mathrm{N} = 0\)</span></th>
+<th>Fraction <span class="arithmatex">\(\mathrm{N} \ne 0\)</span></th>
+<th>Formula</th>
 </tr>
 </thead>
 <tbody>
 <tr>
 <td><span class="arithmatex">\(0\)</span></td>
 <td><span class="arithmatex">\(\pm 0\)</span></td>
-<td>Subnormal Number</td>
+<td>Subnormal number</td>
 <td><span class="arithmatex">\((-1)^{\mathrm{S}} \times 2^{-126} \times (0.\mathrm{N})\)</span></td>
 </tr>
 <tr>
 <td><span class="arithmatex">\(1, 2, \dots, 254\)</span></td>
-<td>Normal Number</td>
-<td>Normal Number</td>
+<td>Normal number</td>
+<td>Normal number</td>
 <td><span class="arithmatex">\((-1)^{\mathrm{S}} \times 2^{(\mathrm{E} -127)} \times (1.\mathrm{N})\)</span></td>
 </tr>
 <tr>
@@ -4797,8 +4869,8 @@ b_{31} b_{30} b_{29} \ldots b_2 b_1 b_0
 </tbody>
 </table>
 </div>
-<p>It is worth noting that subnormal numbers significantly improve the precision of floating-point numbers. The smallest positive normal number is <span class="arithmatex">\(2^{-126}\)</span>, and the smallest positive subnormal number is <span class="arithmatex">\(2^{-126} \times 2^{-23}\)</span>.</p>
-<p>Double-precision <code>double</code> also uses a representation method similar to <code>float</code>, which will not be elaborated here.</p>
+<p>Subnormal numbers substantially improve precision near zero. The smallest positive normal number is <span class="arithmatex">\(2^{-126}\)</span>, and the smallest positive subnormal number is <span class="arithmatex">\(2^{-126} \times 2^{-23}\)</span>.</p>
+<p>Double-precision <code>double</code> uses a similar representation, which we will not detail here.</p>
 
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   <a
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     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 3.2 Basic Data Types"
+    aria-label="Previous: 3.2   Basic data types"
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@@ -4833,7 +4905,7 @@ aria-label="Footer"
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+        3.2 &nbsp; Basic data types
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@@ -4845,7 +4917,7 @@ aria-label="Footer"
   <a
     href="../character_encoding/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 3.4 Character Encoding *"
+    aria-label="Next: 3.4   Character encoding *"
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@@ -4853,7 +4925,7 @@ aria-label="Footer"
         Next
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+        3.4 &nbsp; Character encoding *
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@@ -4870,7 +4942,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -982,7 +982,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
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-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
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           </label>
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-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
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   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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+  8.1 &nbsp; Heaps
 
 
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+  16.2 &nbsp; Contribute to the book
 
 
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-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
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-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
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 <!-- Page content -->
 <h1 id="35-summary">3.5 &nbsp; Summary<a class="headerlink" href="#35-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
 <ul>
-<li>Data structures can be classified from two perspectives: logical structure and physical structure. Logical structure describes the logical relationships between data elements, while physical structure describes how data is stored in computer memory.</li>
-<li>Common logical structures include linear, tree-like, and network structures. We typically classify data structures as linear (arrays, linked lists, stacks, queues) and non-linear (trees, graphs, heaps) based on their logical structure. The implementation of hash tables may involve both linear and non-linear data structures.</li>
-<li>When a program runs, data is stored in computer memory. Each memory space has a corresponding memory address, and the program accesses data through these memory addresses.</li>
-<li>Physical structures are primarily divided into contiguous space storage (arrays) and dispersed space storage (linked lists). All data structures are implemented using arrays, linked lists, or a combination of both.</li>
-<li>Basic data types in computers include integers <code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>, floating-point numbers <code>float</code>, <code>double</code>, characters <code>char</code>, and booleans <code>bool</code>. Their value ranges depend on the size of space they occupy and their representation method.</li>
-<li>Sign-magnitude, 1's complement, and 2's complement are three methods for encoding numbers in computers, and they can be converted into each other. The most significant bit of sign-magnitude is the sign bit, and the remaining bits represent the value of the number.</li>
-<li>Integers are stored in computers in 2's complement form. Under 2's complement representation, computers can treat the addition of positive and negative numbers uniformly, without needing to design special hardware circuits for subtraction, and there is no ambiguity of positive and negative zero.</li>
-<li>The encoding of floating-point numbers consists of 1 sign bit, 8 exponent bits, and 23 fraction bits. Due to the exponent bits, the range of floating-point numbers is much larger than that of integers, at the cost of sacrificing precision.</li>
-<li>ASCII is the earliest English character set, with a length of 1 byte, containing a total of 128 characters. GBK is a commonly used Chinese character set, containing over 20,000 Chinese characters. Unicode is committed to providing a complete character set standard, collecting characters from various languages around the world, thereby solving the garbled text problem caused by inconsistent character encoding methods.</li>
-<li>UTF-8 is the most popular Unicode encoding method and has excellent compatibility. It is a variable-length encoding method with good scalability, effectively improving storage space efficiency. UTF-16 and UTF-32 are common Unicode encoding methods. When encoding Chinese characters, UTF-16 occupies less space than UTF-8. Programming languages such as Java and C# use UTF-16 encoding by default.</li>
+<li>Data structures can be classified by logical and physical structure. Logical structure describes relationships between data elements, while physical structure describes how data is stored in memory.</li>
+<li>Common logical structures are linear, tree, and network structures. We usually classify data structures as linear (arrays, linked lists, stacks, queues) or nonlinear (trees, graphs, heaps). A hash table implementation may contain both linear and nonlinear structures.</li>
+<li>During execution, data resides in computer memory. Each location has a memory address, which programs use to access its data.</li>
+<li>The two main physical structures are contiguous storage (arrays) and scattered storage (linked lists). All data structures are implemented using arrays, linked lists, or both.</li>
+<li>Basic data types include integers (<code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>), floating-point numbers (<code>float</code>, <code>double</code>), characters (<code>char</code>), and Booleans (<code>bool</code>). Their value ranges depend on storage size and representation.</li>
+<li>Sign-magnitude, one's complement, and two's complement are three ways to encode numbers, and can be converted into one another. In sign-magnitude representation, the highest bit is the sign and the remaining bits give the magnitude.</li>
+<li>Integers are stored in two's complement. This lets the computer handle positive and negative addition uniformly, avoids separate hardware circuitry for subtraction, and eliminates the ambiguity of positive and negative zero.</li>
+<li>A floating-point encoding consists of 1 sign bit, 8 exponent bits, and 23 fraction bits. The exponent provides a much wider range than integers, at the cost of precision.</li>
+<li>ASCII was the earliest English character set. It uses 1 byte and contains 128 characters. GBK is a common Chinese character set with over 20,000 Chinese characters. Unicode aims to provide a comprehensive character standard for the world's languages, avoiding garbled text caused by incompatible encodings.</li>
+<li>UTF-8 is the most popular Unicode encoding and is widely compatible. Its variable length makes it extensible and space-efficient. UTF-16 and UTF-32 use fixed lengths. For Chinese text, UTF-16 uses less space than UTF-8. Languages such as Java and C# use UTF-16 by default.</li>
 </ul>
 <h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
-<p><strong>Q</strong>: Why do hash tables contain both linear and non-linear data structures?</p>
-<p>The underlying structure of a hash table is an array. To resolve hash collisions, we may use "chaining" (discussed in the subsequent "Hash Collision" section): each bucket in the array points to a linked list, which may be converted to a tree (usually a red-black tree) when the list length exceeds a certain threshold.</p>
-<p>From a storage perspective, the underlying structure of a hash table is an array, where each bucket slot may contain a value, a linked list, or a tree. Therefore, hash tables may contain both linear data structures (arrays, linked lists) and non-linear data structures (trees).</p>
-<p><strong>Q</strong>: Is the length of the <code>char</code> type 1 byte?</p>
-<p>The length of the <code>char</code> type is determined by the encoding method used by the programming language. For example, Java, JavaScript, TypeScript, and C# all use UTF-16 encoding (to store Unicode code points), so the <code>char</code> type has a length of 2 bytes.</p>
-<p><strong>Q</strong>: Is there ambiguity in referring to array-based data structures as "static data structures"? Stacks can also perform "dynamic" operations such as push and pop.</p>
-<p>Stacks can indeed implement dynamic data operations, but the data structure is still "static" (fixed length). Although array-based data structures can dynamically add or remove elements, their capacity is fixed. If the data volume exceeds the pre-allocated size, a new larger array needs to be created, and the contents of the old array must be copied to the new array.</p>
-<p><strong>Q</strong>: When constructing a stack (queue), its size is not specified. Why are they "static data structures"?</p>
-<p>In high-level programming languages, we do not need to manually specify the initial capacity of a stack (queue); the class handles this automatically. For example, the initial capacity of Java's <code>ArrayList</code> is typically 10. Additionally, the expansion operation is also automatically implemented. See the subsequent "List" section for details.</p>
-<p><strong>Q</strong>: The method of converting sign-magnitude to 2's complement is "first negate then add 1". So converting 2's complement to sign-magnitude should be the inverse operation "first subtract 1 then negate". However, 2's complement can also be converted to sign-magnitude through "first negate then add 1". Why is this?</p>
-<p>This is because the mutual conversion between sign-magnitude and 2's complement is actually the process of computing the "complement". Let us first define the complement: assuming <span class="arithmatex">\(a + b = c\)</span>, then we say that <span class="arithmatex">\(a\)</span> is the complement of <span class="arithmatex">\(b\)</span> to <span class="arithmatex">\(c\)</span>, and conversely, <span class="arithmatex">\(b\)</span> is the complement of <span class="arithmatex">\(a\)</span> to <span class="arithmatex">\(c\)</span>.</p>
-<p>Given an <span class="arithmatex">\(n = 4\)</span> bit binary number <span class="arithmatex">\(0010\)</span>, if we treat this number as sign-magnitude (ignoring the sign bit), then its 2's complement can be obtained through "first negate then add 1":</p>
+<p><strong>Q</strong>: Why can a hash table contain both linear and nonlinear data structures?</p>
+<p>A hash table is built on an array. To handle hash collisions, it may use separate chaining, discussed later in the Hash collisions section: each array bucket points to a linked list, which may be converted to a tree (usually a red-black tree) if it grows beyond a threshold.</p>
+<p>From a storage perspective, each bucket in the underlying array may contain a value, a linked list, or a tree. A hash table may therefore contain both linear structures (arrays and linked lists) and nonlinear structures (trees).</p>
+<p><strong>Q</strong>: Does <code>char</code> occupy 1 byte?</p>
+<p>The size of <code>char</code> depends on the programming language's encoding. For example, Java, JavaScript, TypeScript, and C# use UTF-16 to store Unicode code points, so their <code>char</code> type occupies 2 bytes.</p>
+<p><strong>Q</strong>: Is calling array-based structures static misleading? A stack supports dynamic operations such as pushing and popping.</p>
+<p>A stack does support dynamic data operations, but its underlying structure remains static in length. Although array-based structures can add and remove elements dynamically, their capacity is fixed. When the data exceeds the allocated capacity, a larger array must be created and the old contents copied into it.</p>
+<p><strong>Q</strong>: When we create a stack or queue without specifying its size, why is it still called a static data structure?</p>
+<p>In high-level languages, the class automatically chooses an initial capacity for a stack or queue. For example, Java's <code>ArrayList</code> typically starts with a capacity of 10. Resizing is also automatic; see the later section on lists.</p>
+<p><strong>Q</strong>: Converting sign-magnitude to two's complement uses "invert, then add 1". Its inverse should be "subtract 1, then invert", so why does "invert, then add 1" also work in the reverse direction?</p>
+<p>These conversions are really computations of a complement. If <span class="arithmatex">\(a + b = c\)</span>, we call <span class="arithmatex">\(a\)</span> the complement of <span class="arithmatex">\(b\)</span> with respect to <span class="arithmatex">\(c\)</span>, and conversely <span class="arithmatex">\(b\)</span> the complement of <span class="arithmatex">\(a\)</span> with respect to <span class="arithmatex">\(c\)</span>.</p>
+<p>Consider the <span class="arithmatex">\(n = 4\)</span>-bit binary number <span class="arithmatex">\(0010\)</span>. Treating it as sign-magnitude and disregarding the sign bit, we obtain its two's complement by inverting and adding 1:</p>
 <div class="arithmatex">\[
 0010 \rightarrow 1101 \rightarrow 1110
 \]</div>
-<p>We find that the sum of sign-magnitude and 2's complement is <span class="arithmatex">\(0010 + 1110 = 10000\)</span>, which means the 2's complement <span class="arithmatex">\(1110\)</span> is the "complement" of sign-magnitude <span class="arithmatex">\(0010\)</span> to <span class="arithmatex">\(10000\)</span>. <strong>This means the above "first negate then add 1" is actually the process of computing the complement to <span class="arithmatex">\(10000\)</span></strong>.</p>
-<p>So, what is the "complement" of 2's complement <span class="arithmatex">\(1110\)</span> to <span class="arithmatex">\(10000\)</span>? We can still use "first negate then add 1" to obtain it:</p>
+<p>The two representations sum to <span class="arithmatex">\(0010 + 1110 = 10000\)</span>. Thus, <span class="arithmatex">\(1110\)</span> is the complement of <span class="arithmatex">\(0010\)</span> with respect to <span class="arithmatex">\(10000\)</span>. <strong>The "invert, then add 1" operation computes the complement with respect to <span class="arithmatex">\(10000\)</span></strong>.</p>
+<p>What is the complement of <span class="arithmatex">\(1110\)</span> with respect to <span class="arithmatex">\(10000\)</span>? We can again invert and add 1:</p>
 <div class="arithmatex">\[
 1110 \rightarrow 0001 \rightarrow 0010
 \]</div>
-<p>In other words, sign-magnitude and 2's complement are each other's "complement" to <span class="arithmatex">\(10000\)</span>, so "sign-magnitude to 2's complement" and "2's complement to sign-magnitude" can be implemented using the same operation (first negate then add 1).</p>
-<p>Of course, we can also use the inverse operation to find the sign-magnitude of 2's complement <span class="arithmatex">\(1110\)</span>, that is, "first subtract 1 then negate":</p>
+<p>The two representations are complements of each other with respect to <span class="arithmatex">\(10000\)</span>, so conversion in either direction can use the same operation: invert, then add 1.</p>
+<p>Of course, we can also recover the sign-magnitude form of <span class="arithmatex">\(1110\)</span> by the inverse operation: subtract 1, then invert.</p>
 <div class="arithmatex">\[
 1110 \rightarrow 1101 \rightarrow 0010
 \]</div>
-<p>In summary, both "first negate then add 1" and "first subtract 1 then negate" are computing the complement to <span class="arithmatex">\(10000\)</span>, and they are equivalent.</p>
-<p>Essentially, the "negate" operation is actually finding the complement to <span class="arithmatex">\(1111\)</span> (because "sign-magnitude + 1's complement = 1111" always holds); and adding 1 to the 1's complement yields the 2's complement, which is the complement to <span class="arithmatex">\(10000\)</span>.</p>
-<p>The above uses <span class="arithmatex">\(n = 4\)</span> as an example, and it can be generalized to binary numbers of any number of bits.</p>
+<p>Both "invert, then add 1" and "subtract 1, then invert" compute the complement with respect to <span class="arithmatex">\(10000\)</span>, so they are equivalent.</p>
+<p>In essence, inverting computes the complement with respect to <span class="arithmatex">\(1111\)</span>, since <code>sign-magnitude + one's complement = 1111</code>. Adding 1 to the one's complement then gives the complement with respect to <span class="arithmatex">\(10000\)</span>.</p>
+<p>The example uses <span class="arithmatex">\(n = 4\)</span>, but the reasoning extends to binary numbers of any bit width.</p>
 
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-              12.2 &nbsp; Divide and Conquer Search Strategy
+              12.2 &nbsp; Divide-and-conquer search
 
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-  Before Starting
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+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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+  8.5 &nbsp; Exercises
 
 
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+  Chapter 9. &nbsp; Graphs
 
 
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+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
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+  10.1 &nbsp; Binary search
 
 
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+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
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+  Chapter 11. &nbsp; Sorting
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  Chapter 12. Divide and Conquer
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-  12.2 Divide and Conquer Search Strategy
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-  <a href="#1-implementing-binary-search-based-on-divide-and-conquer" class="md-nav__link">
+  <a href="#1-implementing-binary-search-with-divide-and-conquer" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Implementing Binary Search Based on Divide and Conquer
+        1. &nbsp; Implementing binary search with divide and conquer
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-  12.3 Building a Binary Tree Problem
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-  12.6 Exercises
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+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
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-  14.5 Unbounded Knapsack Problem
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+  14.6 &nbsp; Edit distance problem
 
 
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-  14.8 Exercises
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-  Chapter 15. Greedy
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+  15.3 &nbsp; Maximum capacity problem
 
 
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-  Chapter 16. Appendix
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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-  <a href="#1-implementing-binary-search-based-on-divide-and-conquer" class="md-nav__link">
+  <a href="#1-implementing-binary-search-with-divide-and-conquer" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Implementing Binary Search Based on Divide and Conquer
+        1. &nbsp; Implementing binary search with divide and conquer
       </span>
     </span>
   </a>
@@ -4635,86 +4707,86 @@
 
 
 <!-- Page content -->
-<h1 id="122-divide-and-conquer-search-strategy">12.2 &nbsp; Divide and Conquer Search Strategy<a class="headerlink" href="#122-divide-and-conquer-search-strategy" title="Permanent link">&para;</a></h1>
-<p>We have already learned that search algorithms are divided into two major categories.</p>
+<h1 id="122-divide-and-conquer-search">12.2 &nbsp; Divide-and-conquer search<a class="headerlink" href="#122-divide-and-conquer-search" title="Permanent link">&para;</a></h1>
+<p>We have learned that search algorithms fall into two broad categories.</p>
 <ul>
-<li><strong>Brute-force search</strong>: Implemented by traversing the data structure, with a time complexity of <span class="arithmatex">\(O(n)\)</span>.</li>
-<li><strong>Adaptive search</strong>: Leverages specific data organization or prior information, with time complexity reaching <span class="arithmatex">\(O(\log n)\)</span> or even <span class="arithmatex">\(O(1)\)</span>.</li>
+<li><strong>Brute-force search</strong>: Traverses a data structure, taking <span class="arithmatex">\(O(n)\)</span> time.</li>
+<li><strong>Adaptive search</strong>: Uses a particular organization of data or prior information to achieve <span class="arithmatex">\(O(\log n)\)</span> or even <span class="arithmatex">\(O(1)\)</span> time.</li>
 </ul>
-<p>In fact, <strong>search algorithms with time complexity of <span class="arithmatex">\(O(\log n)\)</span> are typically implemented based on the divide and conquer strategy</strong>, such as binary search and trees.</p>
+<p>In fact, <strong>search algorithms that run in <span class="arithmatex">\(O(\log n)\)</span> time are usually based on divide and conquer</strong>, as seen in binary search and trees.</p>
 <ul>
-<li>Each step of binary search divides the problem (searching for a target element in an array) into a smaller problem (searching for the target element in half of the array), continuing until the array is empty or the target element is found.</li>
-<li>Trees are representative of the divide and conquer idea. In data structures such as binary search trees, AVL trees, and heaps, the time complexity of various operations is <span class="arithmatex">\(O(\log n)\)</span>.</li>
+<li>Each step of binary search reduces the problem of finding a target in an array to a smaller problem of finding it in half of the array. This continues until the array is empty or the target is found.</li>
+<li>Trees exemplify divide and conquer. In data structures such as binary search trees, AVL trees, and heaps, various operations take <span class="arithmatex">\(O(\log n)\)</span> time.</li>
 </ul>
-<p>The divide and conquer strategy of binary search is as follows.</p>
+<p>Binary search uses divide and conquer in the following way.</p>
 <ul>
-<li><strong>The problem can be decomposed</strong>: Binary search recursively decomposes the original problem (searching in an array) into subproblems (searching in half of the array), achieved by comparing the middle element with the target element.</li>
-<li><strong>Subproblems are independent</strong>: In binary search, each round only processes one subproblem, which is not affected by other subproblems.</li>
-<li><strong>Solutions of subproblems do not need to be merged</strong>: Binary search aims to find a specific element, so there is no need to merge the solutions of subproblems. When a subproblem is solved, the original problem is also solved.</li>
+<li><strong>The problem can be divided</strong>: By comparing the middle element with the target, binary search recursively reduces the original problem of searching an array to the subproblem of searching half of that array.</li>
+<li><strong>The subproblems are independent</strong>: Binary search handles only one subproblem at each step, unaffected by other subproblems.</li>
+<li><strong>The subproblem solutions need not be combined</strong>: Binary search seeks a particular element, so there is no need to combine solutions. Solving the subproblem also solves the original problem.</li>
 </ul>
-<p>Divide and conquer can improve search efficiency because brute-force search can only eliminate one option per round, <strong>while divide and conquer search can eliminate half of the options per round</strong>.</p>
-<h3 id="1-implementing-binary-search-based-on-divide-and-conquer">1. &nbsp; Implementing Binary Search Based on Divide and Conquer<a class="headerlink" href="#1-implementing-binary-search-based-on-divide-and-conquer" title="Permanent link">&para;</a></h3>
-<p>In previous sections, binary search was implemented based on iteration. Now we implement it based on divide and conquer (recursion).</p>
+<p>Divide and conquer makes searching faster because brute-force search eliminates only one option per step, <strong>whereas divide-and-conquer search can eliminate half the options</strong>.</p>
+<h3 id="1-implementing-binary-search-with-divide-and-conquer">1. &nbsp; Implementing binary search with divide and conquer<a class="headerlink" href="#1-implementing-binary-search-with-divide-and-conquer" title="Permanent link">&para;</a></h3>
+<p>Earlier chapters implemented binary search iteratively. We will now implement it recursively using divide and conquer.</p>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given a sorted array <code>nums</code> of length <span class="arithmatex">\(n\)</span>, where all elements are unique, find <code>target</code>.</p>
+<p>Given a sorted array <code>nums</code> of length <span class="arithmatex">\(n\)</span> with distinct elements, find the element <code>target</code>.</p>
 </div>
-<p>From a divide and conquer perspective, we denote the subproblem corresponding to the search interval <span class="arithmatex">\([i, j]\)</span> as <span class="arithmatex">\(f(i, j)\)</span>.</p>
-<p>Starting from the original problem <span class="arithmatex">\(f(0, n-1)\)</span>, perform binary search through the following steps.</p>
+<p>From a divide-and-conquer perspective, let <span class="arithmatex">\(f(i, j)\)</span> denote the subproblem for the search interval <span class="arithmatex">\([i, j]\)</span>.</p>
+<p>Starting with the original problem <span class="arithmatex">\(f(0, n-1)\)</span>, perform binary search as follows.</p>
 <ol>
-<li>Calculate the midpoint <span class="arithmatex">\(m\)</span> of the search interval <span class="arithmatex">\([i, j]\)</span>, and use it to eliminate half of the search interval.</li>
-<li>Recursively solve the subproblem reduced by half in size, which could be <span class="arithmatex">\(f(i, m-1)\)</span> or <span class="arithmatex">\(f(m+1, j)\)</span>.</li>
-<li>Repeat steps <code>1.</code> and <code>2.</code> until <code>target</code> is found, or return when the interval is empty.</li>
+<li>Calculate the midpoint <span class="arithmatex">\(m\)</span> of the search interval <span class="arithmatex">\([i, j]\)</span> and use it to eliminate half the interval.</li>
+<li>Recursively solve the subproblem of half the size, either <span class="arithmatex">\(f(i, m-1)\)</span> or <span class="arithmatex">\(f(m+1, j)\)</span>.</li>
+<li>Repeat steps <code>1.</code> and <code>2.</code>, returning when <code>target</code> is found or the interval is empty.</li>
 </ol>
-<p>Figure 12-4 shows the divide and conquer process of binary search for element <span class="arithmatex">\(6\)</span> in an array.</p>
-<p><img alt="Divide and conquer process of binary search" class="animation-figure" src="../binary_search_recur.assets/binary_search_recur.png" /></p>
-<p align="center"> Figure 12-4 &nbsp; Divide and conquer process of binary search </p>
+<p>Figure 12-4 shows the divide-and-conquer process for finding <span class="arithmatex">\(6\)</span> in an array using binary search.</p>
+<p><img alt="Divide-and-conquer process for binary search" class="animation-figure" src="../binary_search_recur.assets/binary_search_recur.png" /></p>
+<p align="center"> Figure 12-4 &nbsp; Divide-and-conquer process for binary search </p>
 
-<p>In the implementation code, we declare a recursive function <code>dfs()</code> to solve the problem <span class="arithmatex">\(f(i, j)\)</span>:</p>
+<p>In the implementation, we define a recursive function <code>dfs()</code> to solve <span class="arithmatex">\(f(i, j)\)</span>:</p>
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 <div class="highlight"><span class="filename">binary_search_recur.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Binary search: problem f(i, j)"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># If the interval is empty, it means there is no target element, return -1</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># If the interval is empty, the target is absent; return -1</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="k">if</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="n">j</span><span class="p">:</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Calculate the midpoint index m</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">target</span><span class="p">:</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="c1"># Recursion subproblem f(m+1, j)</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="c1"># Recurse on subproblem f(m+1, j)</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="k">return</span> <span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="k">elif</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">target</span><span class="p">:</span>
-<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="c1"># Recursion subproblem f(i, m-1)</span>
+<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="c1"># Recurse on subproblem f(i, m-1)</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="k">return</span> <span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>    <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>        <span class="c1"># Found the target element, return its index</span>
+<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>        <span class="c1"># The target element was found; return its index</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>        <span class="k">return</span> <span class="n">m</span>
 <a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>
 <a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="k">def</span><span class="w"> </span><span class="nf">binary_search</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="w">    </span><span class="sd">"""Binary search"""</span>
 <a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
-<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a>    <span class="c1"># Solve the problem f(0, n-1)</span>
+<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a>    <span class="c1"># Solve problem f(0, n-1)</span>
 <a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>    <span class="k">return</span> <span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_recur.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Binary search: problem f(i, j) */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, it means there is no target element, return -1</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, the target is absent; return -1</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(m+1, j)</span>
+<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(m+1, j)</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(i, m-1)</span>
+<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(i, m-1)</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">        </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">        </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="p">}</span>
@@ -4722,7 +4794,7 @@
 <a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="cm">/* Binary search */</span>
 <a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearch</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="w">    </span><span class="c1">// Solve the problem f(0, n-1)</span>
+<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="w">    </span><span class="c1">// Solve problem f(0, n-1)</span>
 <a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="p">}</span>
 </code></pre></div>
@@ -4730,20 +4802,20 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_recur.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Binary search: problem f(i, j) */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, it means there is no target element, return -1</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, the target is absent; return -1</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(m+1, j)</span>
+<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(m+1, j)</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(i, m-1)</span>
+<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(i, m-1)</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">        </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">        </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="p">}</span>
@@ -4751,7 +4823,7 @@
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="cm">/* Binary search */</span>
 <a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearch</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
-<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="w">    </span><span class="c1">// Solve the problem f(0, n-1)</span>
+<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="w">    </span><span class="c1">// Solve problem f(0, n-1)</span>
 <a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="p">}</span>
 </code></pre></div>
@@ -4759,20 +4831,20 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_recur.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Binary search: problem f(i, j) */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">DFS</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, it means there is no target element, return -1</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, the target is absent; return -1</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(m+1, j)</span>
+<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(m+1, j)</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nf">DFS</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(i, m-1)</span>
+<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(i, m-1)</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nf">DFS</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">        </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">        </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="p">}</span>
@@ -4780,7 +4852,7 @@
 <a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="cm">/* Binary search */</span>
 <a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="kt">int</span><span class="w"> </span><span class="nf">BinarySearch</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
-<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w">    </span><span class="c1">// Solve the problem f(0, n-1)</span>
+<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w">    </span><span class="c1">// Solve problem f(0, n-1)</span>
 <a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nf">DFS</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="p">}</span>
 </code></pre></div>
@@ -4788,23 +4860,23 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_recur.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Binary search: problem f(i, j) */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// If interval is empty, indicating no target element, return -1</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, the target is absent; return -1</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// Calculate midpoint index</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">//    Calculate the midpoint index</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">((</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// Compare midpoint with target element</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">//Compare the midpoint element with the target</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="c1">// If smaller, recurse on right half of array</span>
-<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(m+1, j)</span>
+<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="c1">// If it is smaller, recurse on the right half</span>
+<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(m+1, j)</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="c1">// If larger, recurse on left half of array</span>
-<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(i, m-1)</span>
+<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="c1">// If it is larger, recurse on the left half</span>
+<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(i, m-1)</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">        </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">        </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="p">}</span>
@@ -4819,27 +4891,27 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_recur.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Binary search: problem f(i, j) */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, it means there is no target element, return -1</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, the target is absent; return -1</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(m+1, j)</span>
+<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(m+1, j)</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(i, m-1)</span>
+<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(i, m-1)</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">        </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">        </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">m</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="p">}</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a>
 <a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="cm">/* Binary search */</span>
 <a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="kd">func</span><span class="w"> </span><span class="nf">binarySearch</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="w">    </span><span class="c1">// Solve the problem f(0, n-1)</span>
+<a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="w">    </span><span class="c1">// Solve problem f(0, n-1)</span>
 <a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">startIndex</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">endIndex</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a><span class="p">}</span>
 </code></pre></div>
@@ -4847,20 +4919,20 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_recur.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Binary search: problem f(i, j) */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, it means there is no target element, return -1</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, the target is absent; return -1</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">((</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(m+1, j)</span>
+<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(m+1, j)</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">);</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(i, m-1)</span>
+<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(i, m-1)</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">        </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">        </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span>
 <a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="p">}</span>
@@ -4868,7 +4940,7 @@
 <a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="cm">/* Binary search */</span>
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearch</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a><span class="w">    </span><span class="c1">// Solve the problem f(0, n-1)</span>
+<a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a><span class="w">    </span><span class="c1">// Solve problem f(0, n-1)</span>
 <a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-6-26" name="__codelineno-6-26" href="#__codelineno-6-26"></a><span class="p">}</span>
 </code></pre></div>
@@ -4876,20 +4948,20 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_recur.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Binary search: problem f(i, j) */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, it means there is no target element, return -1</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, the target is absent; return -1</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">((</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(m+1, j)</span>
+<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(m+1, j)</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">);</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(i, m-1)</span>
+<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(i, m-1)</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">        </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">        </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span>
 <a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="p">}</span>
@@ -4897,7 +4969,7 @@
 <a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="cm">/* Binary search */</span>
 <a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearch</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">    </span><span class="c1">// Solve the problem f(0, n-1)</span>
+<a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">    </span><span class="c1">// Solve problem f(0, n-1)</span>
 <a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">,</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="p">}</span>
 </code></pre></div>
@@ -4905,20 +4977,20 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_recur.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Binary search: problem f(i, j) */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// If the interval is empty, it means there is no target element, return -1</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// If the interval is empty, the target is absent; return -1</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">;</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="c1">// Recursion subproblem f(m+1, j)</span>
+<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="c1">// Recurse on subproblem f(m+1, j)</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">    </span><span class="c1">// Recursion subproblem f(i, m-1)</span>
+<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">    </span><span class="c1">// Recurse on subproblem f(i, m-1)</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">    </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">    </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="p">}</span>
@@ -4926,7 +4998,7 @@
 <a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="cm">/* Binary search */</span>
 <a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="kt">int</span><span class="w"> </span><span class="n">binarySearch</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
-<a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="w">  </span><span class="c1">// Solve the problem f(0, n-1)</span>
+<a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="w">  </span><span class="c1">// Solve problem f(0, n-1)</span>
 <a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-8-26" name="__codelineno-8-26" href="#__codelineno-8-26"></a><span class="p">}</span>
 </code></pre></div>
@@ -4934,19 +5006,19 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_recur.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Binary search: problem f(i, j) */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, it means there is no target element, return -1</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, the target is absent; return -1</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">m</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(m+1, j)</span>
+<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(m+1, j)</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(i, m-1)</span>
+<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(i, m-1)</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">        </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">        </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="p">}</span>
@@ -4954,7 +5026,7 @@
 <a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="cm">/* Binary search */</span>
 <a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="k">fn</span><span class="w"> </span><span class="nf">binary_search</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
-<a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="w">    </span><span class="c1">// Solve the problem f(0, n-1)</span>
+<a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="w">    </span><span class="c1">// Solve problem f(0, n-1)</span>
 <a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a><span class="p">}</span>
 </code></pre></div>
@@ -4962,20 +5034,20 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_recur.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Binary search: problem f(i, j) */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, it means there is no target element, return -1</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// If the interval is empty, the target is absent; return -1</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(m+1, j)</span>
+<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(m+1, j)</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(i, m-1)</span>
+<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(i, m-1)</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="p">}</span>
@@ -4983,7 +5055,7 @@
 <a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="cm">/* Binary search */</span>
 <a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearch</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">numsSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">numsSize</span><span class="p">;</span>
-<a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">    </span><span class="c1">// Solve the problem f(0, n-1)</span>
+<a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">    </span><span class="c1">// Solve problem f(0, n-1)</span>
 <a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="p">}</span>
 </code></pre></div>
@@ -4996,20 +5068,20 @@
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="c1">// If the interval is empty, it means there is no target element, return -1</span>
+<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="c1">// If the interval is empty, the target is absent; return -1</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="m">1</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">2</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(m+1, j)</span>
+<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(m+1, j)</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">        </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">        </span><span class="c1">// Recursion subproblem f(i, m-1)</span>
+<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">        </span><span class="c1">// Recurse on subproblem f(i, m-1)</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">        </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">        </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">        </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">        </span><span class="n">m</span>
 <a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="p">}</span>
@@ -5017,7 +5089,7 @@
 <a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="cm">/* Binary search */</span>
 <a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">binarySearch</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span>
-<a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="w">    </span><span class="c1">// Solve the problem f(0, n-1)</span>
+<a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="w">    </span><span class="c1">// Solve problem f(0, n-1)</span>
 <a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a><span class="p">}</span>
 </code></pre></div>
@@ -5025,20 +5097,20 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_recur.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Binary search: problem f(i, j) ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># If the interval is empty, it means there is no target element, return -1</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># If the interval is empty, the target is absent; return -1</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">j</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="c1"># Calculate the midpoint index m</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span>
-<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="c1"># Recursion subproblem f(m+1, j)</span>
+<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="c1"># Recurse on subproblem f(m+1, j)</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">  </span><span class="k">elsif</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span>
-<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="c1"># Recursion subproblem f(i, m-1)</span>
+<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="c1"># Recurse on subproblem f(i, m-1)</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">  </span><span class="k">else</span>
-<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">    </span><span class="c1"># Found the target element, return its index</span>
+<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">    </span><span class="c1"># The target element was found; return its index</span>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">m</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="k">end</span>
@@ -5046,13 +5118,18 @@
 <a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="c1">### Binary search ###</span>
 <a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="k">def</span><span class="w"> </span><span class="nf">binary_search</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
 <a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a><span class="w">  </span><span class="c1"># Solve the problem f(0, n-1)</span>
+<a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a><span class="w">  </span><span class="c1"># Solve problem f(0, n-1)</span>
 <a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="w">  </span><span class="n">dfs</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20dfs%28nums%3A%20list%5Bint%5D,%20target%3A%20int,%20i%3A%20int,%20j%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Binary%20search%3A%20problem%20f%28i,%20j%29%22%22%22%0A%20%20%20%20%23%20If%20the%20interval%20is%20empty,%20the%20target%20is%20absent%3B%20return%20-1%0A%20%20%20%20if%20i%20%3E%20j%3A%0A%20%20%20%20%20%20%20%20return%20-1%0A%20%20%20%20%23%20Calculate%20the%20midpoint%20index%20m%0A%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%0A%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%23%20Recurse%20on%20subproblem%20f%28m%2B1,%20j%29%0A%20%20%20%20%20%20%20%20return%20dfs%28nums,%20target,%20m%20%2B%201,%20j%29%0A%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%23%20Recurse%20on%20subproblem%20f%28i,%20m-1%29%0A%20%20%20%20%20%20%20%20return%20dfs%28nums,%20target,%20i,%20m%20-%201%29%0A%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%23%20The%20target%20element%20was%20found%3B%20return%20its%20index%0A%20%20%20%20%20%20%20%20return%20m%0A%0Adef%20binary_search%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Binary%20search%22%22%22%0A%20%20%20%20n%20%3D%20len%28nums%29%0A%20%20%20%20%23%20Solve%20problem%20f%280,%20n-1%29%0A%20%20%20%20return%20dfs%28nums,%20target,%200,%20n%20-%201%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%208,%2012,%2015,%2023,%2026,%2031,%2035%5D%0A%0A%20%20%20%20%23%20Binary%20search%20%28closed%20interval%29%0A%20%20%20%20index%20%3D%20binary_search%28nums,%20target%29%0A%20%20%20%20print%28%22Index%20of%20target%20element%206%20%3D%20%22,%20index%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20dfs%28nums%3A%20list%5Bint%5D,%20target%3A%20int,%20i%3A%20int,%20j%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Binary%20search%3A%20problem%20f%28i,%20j%29%22%22%22%0A%20%20%20%20%23%20If%20the%20interval%20is%20empty,%20the%20target%20is%20absent%3B%20return%20-1%0A%20%20%20%20if%20i%20%3E%20j%3A%0A%20%20%20%20%20%20%20%20return%20-1%0A%20%20%20%20%23%20Calculate%20the%20midpoint%20index%20m%0A%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%0A%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%23%20Recurse%20on%20subproblem%20f%28m%2B1,%20j%29%0A%20%20%20%20%20%20%20%20return%20dfs%28nums,%20target,%20m%20%2B%201,%20j%29%0A%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%23%20Recurse%20on%20subproblem%20f%28i,%20m-1%29%0A%20%20%20%20%20%20%20%20return%20dfs%28nums,%20target,%20i,%20m%20-%201%29%0A%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%23%20The%20target%20element%20was%20found%3B%20return%20its%20index%0A%20%20%20%20%20%20%20%20return%20m%0A%0Adef%20binary_search%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Binary%20search%22%22%22%0A%20%20%20%20n%20%3D%20len%28nums%29%0A%20%20%20%20%23%20Solve%20problem%20f%280,%20n-1%29%0A%20%20%20%20return%20dfs%28nums,%20target,%200,%20n%20-%201%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%208,%2012,%2015,%2023,%2026,%2031,%2035%5D%0A%0A%20%20%20%20%23%20Binary%20search%20%28closed%20interval%29%0A%20%20%20%20index%20%3D%20binary_search%28nums,%20target%29%0A%20%20%20%20print%28%22Index%20of%20target%20element%206%20%3D%20%22,%20index%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
 
 <!-- Source file information -->
 
@@ -5075,7 +5152,7 @@ aria-label="Footer"
   <a
     href="../divide_and_conquer/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 12.1 Divide and Conquer Algorithms"
+    aria-label="Previous: 12.1   Divide and conquer"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -5087,7 +5164,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        12.1 Divide and Conquer Algorithms
+        12.1 &nbsp; Divide and conquer
       </div>
     </div>
   </a>
@@ -5099,7 +5176,7 @@ aria-label="Footer"
   <a
     href="../build_binary_tree_problem/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 12.3 Building a Binary Tree Problem"
+    aria-label="Next: 12.3   Building a binary tree"
     rel="next"
   >
     <div class="md-footer__title">
@@ -5107,7 +5184,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        12.3 Building a Binary Tree Problem
+        12.3 &nbsp; Building a binary tree
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -5124,7 +5201,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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diff --git a/en/chapter_divide_and_conquer/build_binary_tree_problem/index.html b/en/chapter_divide_and_conquer/build_binary_tree_problem/index.html
index 61444a38b..6d44996c4 100644
--- a/en/chapter_divide_and_conquer/build_binary_tree_problem/index.html
+++ b/en/chapter_divide_and_conquer/build_binary_tree_problem/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>12.3 Building a Binary Tree Problem - Hello Algo</title>
+        <title>12.3 Building a binary tree - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
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+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
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+        <a href="#123-building-a-binary-tree" class="md-skip">
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@@ -119,7 +119,7 @@
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         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              12.3 &nbsp; Building a Binary Tree Problem
+              12.3 &nbsp; Building a binary tree
 
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@@ -167,9 +167,9 @@
 
 
 
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
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   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
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   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
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   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
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@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
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@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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@@ -3327,7 +3327,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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@@ -3433,10 +3433,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-determining-if-it-is-a-divide-and-conquer-problem" class="md-nav__link">
+  <a href="#1-identifying-a-divide-and-conquer-problem" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Determining If It Is a Divide and Conquer Problem
+        1. &nbsp; Identifying a divide-and-conquer problem
       </span>
     </span>
   </a>
@@ -3444,10 +3444,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#2-how-to-divide-subtrees" class="md-nav__link">
+  <a href="#2-dividing-the-subtrees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; How to Divide Subtrees
+        2. &nbsp; Dividing the subtrees
       </span>
     </span>
   </a>
@@ -3455,10 +3455,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#3-describing-subtree-intervals-based-on-variables" class="md-nav__link">
+  <a href="#3-describing-subtree-intervals-with-variables" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Describing Subtree Intervals Based on Variables
+        3. &nbsp; Describing subtree intervals with variables
       </span>
     </span>
   </a>
@@ -3469,7 +3469,7 @@
   <a href="#4-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4. &nbsp; Code Implementation
+        4. &nbsp; Code implementation
       </span>
     </span>
   </a>
@@ -3499,7 +3499,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3526,7 +3526,7 @@
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3619,7 +3619,7 @@
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
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@@ -3661,7 +3661,7 @@
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4163,7 +4163,7 @@
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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@@ -4580,6 +4580,78 @@
 
 
 
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@@ -4621,10 +4693,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-determining-if-it-is-a-divide-and-conquer-problem" class="md-nav__link">
+  <a href="#1-identifying-a-divide-and-conquer-problem" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Determining If It Is a Divide and Conquer Problem
+        1. &nbsp; Identifying a divide-and-conquer problem
       </span>
     </span>
   </a>
@@ -4632,10 +4704,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#2-how-to-divide-subtrees" class="md-nav__link">
+  <a href="#2-dividing-the-subtrees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; How to Divide Subtrees
+        2. &nbsp; Dividing the subtrees
       </span>
     </span>
   </a>
@@ -4643,10 +4715,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#3-describing-subtree-intervals-based-on-variables" class="md-nav__link">
+  <a href="#3-describing-subtree-intervals-with-variables" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Describing Subtree Intervals Based on Variables
+        3. &nbsp; Describing subtree intervals with variables
       </span>
     </span>
   </a>
@@ -4657,7 +4729,7 @@
   <a href="#4-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4. &nbsp; Code Implementation
+        4. &nbsp; Code implementation
       </span>
     </span>
   </a>
@@ -4701,54 +4773,54 @@
 
 
 <!-- Page content -->
-<h1 id="123-building-a-binary-tree-problem">12.3 &nbsp; Building a Binary Tree Problem<a class="headerlink" href="#123-building-a-binary-tree-problem" title="Permanent link">&para;</a></h1>
+<h1 id="123-building-a-binary-tree">12.3 &nbsp; Building a binary tree<a class="headerlink" href="#123-building-a-binary-tree" title="Permanent link">&para;</a></h1>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given the preorder traversal <code>preorder</code> and inorder traversal <code>inorder</code> of a binary tree, construct the binary tree and return the root node of the binary tree. Assume there are no duplicate node values in the binary tree (as shown in Figure 12-5).</p>
+<p>Given a binary tree's preorder traversal <code>preorder</code> and inorder traversal <code>inorder</code>, reconstruct the tree and return its root node. Assume that all node values are distinct, as shown in Figure 12-5.</p>
 </div>
-<p><img alt="Example data for building a binary tree" class="animation-figure" src="../build_binary_tree_problem.assets/build_tree_example.png" /></p>
-<p align="center"> Figure 12-5 &nbsp; Example data for building a binary tree </p>
+<p><img alt="Example input for reconstructing a binary tree" class="animation-figure" src="../build_binary_tree_problem.assets/build_tree_example.png" /></p>
+<p align="center"> Figure 12-5 &nbsp; Example input for reconstructing a binary tree </p>
 
-<h3 id="1-determining-if-it-is-a-divide-and-conquer-problem">1. &nbsp; Determining If It Is a Divide and Conquer Problem<a class="headerlink" href="#1-determining-if-it-is-a-divide-and-conquer-problem" title="Permanent link">&para;</a></h3>
-<p>The original problem is defined as constructing a binary tree from <code>preorder</code> and <code>inorder</code>, which is a typical divide and conquer problem.</p>
+<h3 id="1-identifying-a-divide-and-conquer-problem">1. &nbsp; Identifying a divide-and-conquer problem<a class="headerlink" href="#1-identifying-a-divide-and-conquer-problem" title="Permanent link">&para;</a></h3>
+<p>Reconstructing a binary tree from <code>preorder</code> and <code>inorder</code> is a typical divide-and-conquer problem.</p>
 <ul>
-<li><strong>The problem can be decomposed</strong>: From a divide and conquer perspective, we can divide the original problem into two subproblems: constructing the left subtree and constructing the right subtree, plus one operation: initializing the root node. For each subtree (subproblem), we can still reuse the above division method, dividing it into smaller subtrees (subproblems) until the smallest subproblem (empty subtree) is reached.</li>
-<li><strong>Subproblems are independent</strong>: The left and right subtrees are independent of each other; there is no overlap between them. When constructing the left subtree, we only need to focus on the parts of the inorder and preorder traversals corresponding to the left subtree. The same applies to the right subtree.</li>
-<li><strong>Solutions of subproblems can be merged</strong>: Once we have the left and right subtrees (solutions of subproblems), we can link them to the root node to obtain the solution to the original problem.</li>
+<li><strong>The problem can be divided</strong>: We can split the original problem into two subproblems, building the left subtree and building the right subtree, plus one operation: initializing the root node. Each subtree can be divided in the same way into smaller subtrees, until we reach the smallest subproblem, an empty subtree.</li>
+<li><strong>The subproblems are independent</strong>: The left and right subtrees are independent and do not overlap. To build the left subtree, we need only the portions of the preorder and inorder traversals corresponding to it. The same applies to the right subtree.</li>
+<li><strong>The subproblem solutions can be combined</strong>: Once we have built the left and right subtrees, we can attach them to the root to obtain the solution to the original problem.</li>
 </ul>
-<h3 id="2-how-to-divide-subtrees">2. &nbsp; How to Divide Subtrees<a class="headerlink" href="#2-how-to-divide-subtrees" title="Permanent link">&para;</a></h3>
-<p>Based on the above analysis, this problem can be solved using divide and conquer, <strong>but how do we divide the left and right subtrees through the preorder traversal <code>preorder</code> and inorder traversal <code>inorder</code></strong>?</p>
-<p>According to the definition, both <code>preorder</code> and <code>inorder</code> can be divided into three parts.</p>
+<h3 id="2-dividing-the-subtrees">2. &nbsp; Dividing the subtrees<a class="headerlink" href="#2-dividing-the-subtrees" title="Permanent link">&para;</a></h3>
+<p>The analysis above shows that divide and conquer applies, <strong>but how do we use <code>preorder</code> and <code>inorder</code> to separate the left and right subtrees</strong>?</p>
+<p>By definition, both <code>preorder</code> and <code>inorder</code> consist of three parts.</p>
 <ul>
-<li>Preorder traversal: <code>[ Root Node | Left Subtree | Right Subtree ]</code>, for example, the tree in Figure 12-5 corresponds to <code>[ 3 | 9 | 2 1 7 ]</code>.</li>
-<li>Inorder traversal: <code>[ Left Subtree | Root Node | Right Subtree ]</code>, for example, the tree in Figure 12-5 corresponds to <code>[ 9 | 3 | 1 2 7 ]</code>.</li>
+<li>Preorder: <code>[ root | left subtree | right subtree ]</code>. For the tree above, this is <code>[ 3 | 9 | 2 1 7 ]</code>.</li>
+<li>Inorder: <code>[ left subtree | root | right subtree ]</code>. For the tree above, this is <code>[ 9 | 3 | 1 2 7 ]</code>.</li>
 </ul>
-<p>Using the data from the figure above as an example, we can obtain the division results through the steps shown in Figure 12-6.</p>
+<p>Using the example above, we can divide the traversals with the steps shown in Figure 12-6.</p>
 <ol>
-<li>The first element 3 in the preorder traversal is the value of the root node.</li>
-<li>Find the index of root node 3 in <code>inorder</code>, and use this index to divide <code>inorder</code> into <code>[ 9 | 3 | 1 2 7 ]</code>.</li>
-<li>Based on the division result of <code>inorder</code>, it is easy to determine that the left and right subtrees have 1 and 3 nodes respectively, allowing us to divide <code>preorder</code> into <code>[ 3 | 9 | 2 1 7 ]</code>.</li>
+<li>The first element in the preorder traversal, 3, is the root value.</li>
+<li>Find the index of root value 3 in <code>inorder</code>, then use it to divide <code>inorder</code> into <code>[ 9 | 3 | 1 2 7 ]</code>.</li>
+<li>This division of <code>inorder</code> tells us that the left and right subtrees have 1 and 3 nodes, respectively, allowing us to divide <code>preorder</code> into <code>[ 3 | 9 | 2 1 7 ]</code>.</li>
 </ol>
-<p><img alt="Dividing subtrees in preorder and inorder traversals" class="animation-figure" src="../build_binary_tree_problem.assets/build_tree_preorder_inorder_division.png" /></p>
-<p align="center"> Figure 12-6 &nbsp; Dividing subtrees in preorder and inorder traversals </p>
+<p><img alt="Separating subtrees in preorder and inorder traversals" class="animation-figure" src="../build_binary_tree_problem.assets/build_tree_preorder_inorder_division.png" /></p>
+<p align="center"> Figure 12-6 &nbsp; Separating subtrees in preorder and inorder traversals </p>
 
-<h3 id="3-describing-subtree-intervals-based-on-variables">3. &nbsp; Describing Subtree Intervals Based on Variables<a class="headerlink" href="#3-describing-subtree-intervals-based-on-variables" title="Permanent link">&para;</a></h3>
-<p>Based on the above division method, <strong>we have obtained the index intervals of the root node, left subtree, and right subtree in <code>preorder</code> and <code>inorder</code></strong>. To describe these index intervals, we need to use several index variables.</p>
+<h3 id="3-describing-subtree-intervals-with-variables">3. &nbsp; Describing subtree intervals with variables<a class="headerlink" href="#3-describing-subtree-intervals-with-variables" title="Permanent link">&para;</a></h3>
+<p>Using this method, <strong>we have identified the index intervals for the root, left subtree, and right subtree in <code>preorder</code> and <code>inorder</code></strong>. We need a few pointer variables to describe these intervals.</p>
 <ul>
-<li>Denote the index of the current tree's root node in <code>preorder</code> as <span class="arithmatex">\(i\)</span>.</li>
-<li>Denote the index of the current tree's root node in <code>inorder</code> as <span class="arithmatex">\(m\)</span>.</li>
-<li>Denote the index interval of the current tree in <code>inorder</code> as <span class="arithmatex">\([l, r]\)</span>.</li>
+<li>Let <span class="arithmatex">\(i\)</span> be the index of the current tree's root in <code>preorder</code>.</li>
+<li>Let <span class="arithmatex">\(m\)</span> be the index of the current tree's root in <code>inorder</code>.</li>
+<li>Let <span class="arithmatex">\([l, r]\)</span> be the interval occupied by the current tree in <code>inorder</code>.</li>
 </ul>
-<p>As shown in Table 12-1, through these variables we can represent the index of the root node in <code>preorder</code> and the index intervals of the subtrees in <code>inorder</code>.</p>
-<p align="center"> Table 12-1 &nbsp; Indices of root node and subtrees in preorder and inorder traversals </p>
+<p>As shown in Table 12-1, these variables describe the root index in <code>preorder</code> and each subtree's interval in <code>inorder</code>.</p>
+<p align="center"> Table 12-1 &nbsp; Root and subtree indices in preorder and inorder traversals </p>
 
 <div class="center-table">
 <table>
 <thead>
 <tr>
 <th></th>
-<th>Root node index in <code>preorder</code></th>
-<th>Subtree index interval in <code>inorder</code></th>
+<th>Root index in <code>preorder</code></th>
+<th>Subtree interval in <code>inorder</code></th>
 </tr>
 </thead>
 <tbody>
@@ -4770,12 +4842,12 @@
 </tbody>
 </table>
 </div>
-<p>Please note that <span class="arithmatex">\((m-l)\)</span> in the right subtree root node index means "the number of nodes in the left subtree". It is recommended to understand this in conjunction with Figure 12-7.</p>
-<p><img alt="Index interval representation of root node and left and right subtrees" class="animation-figure" src="../build_binary_tree_problem.assets/build_tree_division_pointers.png" /></p>
-<p align="center"> Figure 12-7 &nbsp; Index interval representation of root node and left and right subtrees </p>
+<p>The term <span class="arithmatex">\((m-l)\)</span> in the right subtree's root index is the number of nodes in the left subtree. Figure 12-7 helps illustrate this.</p>
+<p><img alt="Index intervals for the root and the left and right subtrees" class="animation-figure" src="../build_binary_tree_problem.assets/build_tree_division_pointers.png" /></p>
+<p align="center"> Figure 12-7 &nbsp; Index intervals for the root and the left and right subtrees </p>
 
-<h3 id="4-code-implementation">4. &nbsp; Code Implementation<a class="headerlink" href="#4-code-implementation" title="Permanent link">&para;</a></h3>
-<p>To improve the efficiency of querying <span class="arithmatex">\(m\)</span>, we use a hash table <code>hmap</code> to store the mapping from elements in the <code>inorder</code> array to their indices:</p>
+<h3 id="4-code-implementation">4. &nbsp; Code implementation<a class="headerlink" href="#4-code-implementation" title="Permanent link">&para;</a></h3>
+<p>To find <span class="arithmatex">\(m\)</span> efficiently, we use a hash table <code>hmap</code> mapping each element of <code>inorder</code> to its index:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -4786,13 +4858,13 @@
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="n">l</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="n">r</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="w">    </span><span class="sd">"""Build binary tree: divide and conquer"""</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="c1"># Terminate when the subtree interval is empty</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="w">    </span><span class="sd">"""Build a binary tree: divide and conquer"""</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="c1"># Stop when the subtree interval is empty</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="k">if</span> <span class="n">r</span> <span class="o">-</span> <span class="n">l</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="k">return</span> <span class="kc">None</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="c1"># Initialize the root node</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>    <span class="n">root</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">preorder</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>    <span class="c1"># Query m to divide the left and right subtrees</span>
+<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>    <span class="c1"># Look up m to separate the left and right subtrees</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="n">m</span> <span class="o">=</span> <span class="n">inorder_map</span><span class="p">[</span><span class="n">preorder</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>    <span class="c1"># Subproblem: build the left subtree</span>
 <a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>    <span class="n">root</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">dfs</span><span class="p">(</span><span class="n">preorder</span><span class="p">,</span> <span class="n">inorder_map</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
@@ -4802,22 +4874,22 @@
 <a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a>    <span class="k">return</span> <span class="n">root</span>
 <a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>
 <a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="k">def</span><span class="w"> </span><span class="nf">build_tree</span><span class="p">(</span><span class="n">preorder</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">inorder</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">])</span> <span class="o">-&gt;</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a><span class="w">    </span><span class="sd">"""Build binary tree"""</span>
-<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a>    <span class="c1"># Initialize hash map, storing the mapping from inorder elements to indices</span>
+<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a><span class="w">    </span><span class="sd">"""Build a binary tree"""</span>
+<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a>    <span class="c1"># Initialize a hash table mapping each element in inorder to its index</span>
 <a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a>    <span class="n">inorder_map</span> <span class="o">=</span> <span class="p">{</span><span class="n">val</span><span class="p">:</span> <span class="n">i</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">val</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">inorder</span><span class="p">)}</span>
 <a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a>    <span class="n">root</span> <span class="o">=</span> <span class="n">dfs</span><span class="p">(</span><span class="n">preorder</span><span class="p">,</span> <span class="n">inorder_map</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">inorder</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a>    <span class="k">return</span> <span class="n">root</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">build_tree.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Build binary tree: divide and conquer */</span>
+<div class="highlight"><span class="filename">build_tree.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Build a binary tree: divide and conquer */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">dfs</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">unordered_map</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Terminate when the subtree interval is empty</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Stop when the subtree interval is empty</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Initialize the root node</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">preorder</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="c1">// Query m to divide the left and right subtrees</span>
+<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="c1">// Look up m to separate the left and right subtrees</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">[</span><span class="n">preorder</span><span class="p">[</span><span class="n">i</span><span class="p">]];</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">    </span><span class="c1">// Subproblem: build the left subtree</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">    </span><span class="n">root</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
@@ -4827,9 +4899,9 @@
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">root</span><span class="p">;</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="p">}</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a>
-<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="cm">/* Build binary tree */</span>
+<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="cm">/* Build a binary tree */</span>
 <a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">buildTree</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">inorder</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">    </span><span class="c1">// Initialize hash map, storing the mapping from inorder elements to indices</span>
+<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">    </span><span class="c1">// Initialize a hash table mapping each element in inorder to its index</span>
 <a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w">    </span><span class="n">unordered_map</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">;</span>
 <a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">inorder</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="w">        </span><span class="n">inorderMap</span><span class="p">[</span><span class="n">inorder</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
@@ -4840,14 +4912,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">build_tree.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Build binary tree: divide and conquer */</span>
+<div class="highlight"><span class="filename">build_tree.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Build a binary tree: divide and conquer */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">Map</span><span class="o">&lt;</span><span class="n">Integer</span><span class="p">,</span><span class="w"> </span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Terminate when the subtree interval is empty</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Stop when the subtree interval is empty</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// Initialize the root node</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">preorder</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">);</span>
-<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="c1">// Query m to divide the left and right subtrees</span>
+<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="c1">// Look up m to separate the left and right subtrees</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">preorder</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">);</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="c1">// Subproblem: build the left subtree</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="n">root</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
@@ -4857,9 +4929,9 @@
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">root</span><span class="p">;</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="p">}</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a>
-<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="cm">/* Build binary tree */</span>
+<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="cm">/* Build a binary tree */</span>
 <a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">buildTree</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">inorder</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">    </span><span class="c1">// Initialize hash map, storing the mapping from inorder elements to indices</span>
+<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">    </span><span class="c1">// Initialize a hash table mapping each element in inorder to its index</span>
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">    </span><span class="n">Map</span><span class="o">&lt;</span><span class="n">Integer</span><span class="p">,</span><span class="w"> </span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">inorderMap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">HashMap</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">inorder</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="w">        </span><span class="n">inorderMap</span><span class="p">.</span><span class="na">put</span><span class="p">(</span><span class="n">inorder</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
@@ -4870,14 +4942,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">build_tree.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Build binary tree: divide and conquer */</span>
+<div class="highlight"><span class="filename">build_tree.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Build a binary tree: divide and conquer */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">DFS</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">Dictionary</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Terminate when the subtree interval is empty</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Stop when the subtree interval is empty</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Initialize the root node</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="n">preorder</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="c1">// Query m to divide the left and right subtrees</span>
+<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="c1">// Look up m to separate the left and right subtrees</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">[</span><span class="n">preorder</span><span class="p">[</span><span class="n">i</span><span class="p">]];</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="c1">// Subproblem: build the left subtree</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">    </span><span class="n">root</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">DFS</span><span class="p">(</span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
@@ -4887,9 +4959,9 @@
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">root</span><span class="p">;</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="p">}</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a>
-<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="cm">/* Build binary tree */</span>
+<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="cm">/* Build a binary tree */</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">BuildTree</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">inorder</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">    </span><span class="c1">// Initialize hash map, storing the mapping from inorder elements to indices</span>
+<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">    </span><span class="c1">// Initialize a hash table mapping each element in inorder to its index</span>
 <a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w">    </span><span class="n">Dictionary</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">inorderMap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">inorder</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="w">        </span><span class="n">inorderMap</span><span class="p">.</span><span class="n">TryAdd</span><span class="p">(</span><span class="n">inorder</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
@@ -4900,15 +4972,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">build_tree.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Build binary tree: divide and conquer */</span>
+<div class="highlight"><span class="filename">build_tree.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Build a binary tree: divide and conquer */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">dfsBuildTree</span><span class="p">(</span><span class="nx">preorder</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">inorderMap</span><span class="w"> </span><span class="kt">map</span><span class="p">[</span><span class="kt">int</span><span class="p">]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">l</span><span class="p">,</span><span class="w"> </span><span class="nx">r</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Terminate when the subtree interval is empty</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Stop when the subtree interval is empty</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">r</span><span class="o">-</span><span class="nx">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// Initialize the root node</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="nx">root</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="nx">preorder</span><span class="p">[</span><span class="nx">i</span><span class="p">])</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// Query m to divide the left and right subtrees</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// Look up m to separate the left and right subtrees</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">inorderMap</span><span class="p">[</span><span class="nx">preorder</span><span class="p">[</span><span class="nx">i</span><span class="p">]]</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">    </span><span class="c1">// Subproblem: build the left subtree</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="nx">root</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dfsBuildTree</span><span class="p">(</span><span class="nx">preorder</span><span class="p">,</span><span class="w"> </span><span class="nx">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">l</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
@@ -4918,9 +4990,9 @@
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">root</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="p">}</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a>
-<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="cm">/* Build binary tree */</span>
+<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="cm">/* Build a binary tree */</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="kd">func</span><span class="w"> </span><span class="nx">buildTree</span><span class="p">(</span><span class="nx">preorder</span><span class="p">,</span><span class="w"> </span><span class="nx">inorder</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">    </span><span class="c1">// Initialize hash map, storing the mapping from inorder elements to indices</span>
+<a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">    </span><span class="c1">// Initialize a hash table mapping each element in inorder to its index</span>
 <a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">    </span><span class="nx">inorderMap</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">(</span><span class="kt">map</span><span class="p">[</span><span class="kt">int</span><span class="p">]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">inorder</span><span class="p">))</span>
 <a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">inorder</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="w">        </span><span class="nx">inorderMap</span><span class="p">[</span><span class="nx">inorder</span><span class="p">[</span><span class="nx">i</span><span class="p">]]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span>
@@ -4932,15 +5004,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">build_tree.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Build binary tree: divide and conquer */</span>
+<div class="highlight"><span class="filename">build_tree.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Build a binary tree: divide and conquer */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">preorder</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">r</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Terminate when the subtree interval is empty</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Stop when the subtree interval is empty</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// Initialize the root node</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">root</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="n">preorder</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="c1">// Query m to divide the left and right subtrees</span>
+<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="c1">// Look up m to separate the left and right subtrees</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">[</span><span class="n">preorder</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="o">!</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">    </span><span class="c1">// Subproblem: build the left subtree</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">    </span><span class="n">root</span><span class="p">.</span><span class="kr">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">preorder</span><span class="p">:</span><span class="w"> </span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">:</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">:</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">r</span><span class="p">:</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
@@ -4950,22 +5022,22 @@
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">root</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="p">}</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a>
-<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="cm">/* Build binary tree */</span>
+<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="cm">/* Build a binary tree */</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="kd">func</span><span class="w"> </span><span class="nf">buildTree</span><span class="p">(</span><span class="n">preorder</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">inorder</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w">    </span><span class="c1">// Initialize hash map, storing the mapping from inorder elements to indices</span>
+<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w">    </span><span class="c1">// Initialize a hash table mapping each element in inorder to its index</span>
 <a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">inorderMap</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">inorder</span><span class="p">.</span><span class="n">enumerated</span><span class="p">().</span><span class="bp">reduce</span><span class="p">(</span><span class="n">into</span><span class="p">:</span><span class="w"> </span><span class="p">[:])</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nv">$0</span><span class="p">[</span><span class="nv">$1</span><span class="p">.</span><span class="n">element</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nv">$1</span><span class="p">.</span><span class="n">offset</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">preorder</span><span class="p">:</span><span class="w"> </span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">:</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">inorder</span><span class="p">.</span><span class="n">startIndex</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">:</span><span class="w"> </span><span class="n">inorder</span><span class="p">.</span><span class="n">startIndex</span><span class="p">,</span><span class="w"> </span><span class="n">r</span><span class="p">:</span><span class="w"> </span><span class="n">inorder</span><span class="p">.</span><span class="n">endIndex</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">build_tree.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Build binary tree: divide and conquer */</span>
+<div class="highlight"><span class="filename">build_tree.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Build a binary tree: divide and conquer */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">preorder</span><span class="p">,</span><span class="w"> </span><span class="nx">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">l</span><span class="p">,</span><span class="w"> </span><span class="nx">r</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Terminate when the subtree interval is empty</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Stop when the subtree interval is empty</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">r</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="c1">// Initialize the root node</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="nx">preorder</span><span class="p">[</span><span class="nx">i</span><span class="p">]);</span>
-<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="c1">// Query m to divide the left and right subtrees</span>
+<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="c1">// Look up m to separate the left and right subtrees</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">inorderMap</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">preorder</span><span class="p">[</span><span class="nx">i</span><span class="p">]);</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="c1">// Subproblem: build the left subtree</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="nx">root</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">preorder</span><span class="p">,</span><span class="w"> </span><span class="nx">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">l</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
@@ -4975,9 +5047,9 @@
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">root</span><span class="p">;</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="p">}</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a>
-<a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="cm">/* Build binary tree */</span>
+<a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="cm">/* Build a binary tree */</span>
 <a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="kd">function</span><span class="w"> </span><span class="nx">buildTree</span><span class="p">(</span><span class="nx">preorder</span><span class="p">,</span><span class="w"> </span><span class="nx">inorder</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">    </span><span class="c1">// Initialize hash map, storing the mapping from inorder elements to indices</span>
+<a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">    </span><span class="c1">// Initialize a hash table mapping each element in inorder to its index</span>
 <a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">inorderMap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Map</span><span class="p">();</span>
 <a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">inorder</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="w">        </span><span class="nx">inorderMap</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="nx">inorder</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">i</span><span class="p">);</span>
@@ -4988,7 +5060,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">build_tree.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Build binary tree: divide and conquer */</span>
+<div class="highlight"><span class="filename">build_tree.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Build a binary tree: divide and conquer */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="nx">preorder</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="nx">inorderMap</span><span class="o">:</span><span class="w"> </span><span class="kt">Map</span><span class="o">&lt;</span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span>
@@ -4996,11 +5068,11 @@
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="nx">l</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="nx">r</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// Terminate when the subtree interval is empty</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// Stop when the subtree interval is empty</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">r</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="c1">// Initialize the root node</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">root</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="nx">preorder</span><span class="p">[</span><span class="nx">i</span><span class="p">]);</span>
-<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="c1">// Query m to divide the left and right subtrees</span>
+<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="c1">// Look up m to separate the left and right subtrees</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">inorderMap</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">preorder</span><span class="p">[</span><span class="nx">i</span><span class="p">]);</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="c1">// Subproblem: build the left subtree</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="nx">root</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">preorder</span><span class="p">,</span><span class="w"> </span><span class="nx">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">l</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
@@ -5010,9 +5082,9 @@
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">root</span><span class="p">;</span>
 <a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="p">}</span>
 <a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a>
-<a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="cm">/* Build binary tree */</span>
+<a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="cm">/* Build a binary tree */</span>
 <a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="kd">function</span><span class="w"> </span><span class="nx">buildTree</span><span class="p">(</span><span class="nx">preorder</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">inorder</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">    </span><span class="c1">// Initialize hash map, storing the mapping from inorder elements to indices</span>
+<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">    </span><span class="c1">// Initialize a hash table mapping each element in inorder to its index</span>
 <a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">inorderMap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Map</span><span class="o">&lt;</span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="kt">number</span><span class="o">&gt;</span><span class="p">();</span>
 <a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">inorder</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w">        </span><span class="nx">inorderMap</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="nx">inorder</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">i</span><span class="p">);</span>
@@ -5023,7 +5095,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">build_tree.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Build binary tree: divide and conquer */</span>
+<div class="highlight"><span class="filename">build_tree.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Build a binary tree: divide and conquer */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">preorder</span><span class="p">,</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="n">Map</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">,</span>
@@ -5031,13 +5103,13 @@
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="p">,</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">r</span><span class="p">,</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="c1">// Terminate when the subtree interval is empty</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="c1">// Stop when the subtree interval is empty</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="c1">// Initialize the root node</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">preorder</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="c1">// Query m to divide the left and right subtrees</span>
+<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="c1">// Look up m to separate the left and right subtrees</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">[</span><span class="n">preorder</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="o">!</span><span class="p">;</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">  </span><span class="c1">// Subproblem: build the left subtree</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">  </span><span class="n">root</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
@@ -5047,9 +5119,9 @@
 <a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">root</span><span class="p">;</span>
 <a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="p">}</span>
 <a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a>
-<a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="cm">/* Build binary tree */</span>
+<a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="cm">/* Build a binary tree */</span>
 <a id="__codelineno-8-26" name="__codelineno-8-26" href="#__codelineno-8-26"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">buildTree</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">inorder</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a><span class="w">  </span><span class="c1">// Initialize hash map, storing the mapping from inorder elements to indices</span>
+<a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a><span class="w">  </span><span class="c1">// Initialize a hash table mapping each element in inorder to its index</span>
 <a id="__codelineno-8-28" name="__codelineno-8-28" href="#__codelineno-8-28"></a><span class="w">  </span><span class="n">Map</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">inorderMap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{};</span>
 <a id="__codelineno-8-29" name="__codelineno-8-29" href="#__codelineno-8-29"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">inorder</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-30" name="__codelineno-8-30" href="#__codelineno-8-30"></a><span class="w">    </span><span class="n">inorderMap</span><span class="p">[</span><span class="n">inorder</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
@@ -5060,7 +5132,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">build_tree.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Build binary tree: divide and conquer */</span>
+<div class="highlight"><span class="filename">build_tree.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Build a binary tree: divide and conquer */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="n">preorder</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="n">inorder_map</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">HashMap</span><span class="o">&lt;</span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">,</span>
@@ -5068,13 +5140,13 @@
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="n">l</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="n">r</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// Terminate when the subtree interval is empty</span>
+<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// Stop when the subtree interval is empty</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nb">None</span><span class="p">;</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">    </span><span class="c1">// Initialize the root node</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">preorder</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]);</span>
-<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">    </span><span class="c1">// Query m to divide the left and right subtrees</span>
+<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">    </span><span class="c1">// Look up m to separate the left and right subtrees</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">inorder_map</span><span class="p">.</span><span class="n">get</span><span class="p">(</span><span class="o">&amp;</span><span class="n">preorder</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]).</span><span class="n">unwrap</span><span class="p">();</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">    </span><span class="c1">// Subproblem: build the left subtree</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">    </span><span class="n">root</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">inorder_map</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
@@ -5084,9 +5156,9 @@
 <a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="w">    </span><span class="nb">Some</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="p">}</span>
 <a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a>
-<a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a><span class="cm">/* Build binary tree */</span>
+<a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a><span class="cm">/* Build a binary tree */</span>
 <a id="__codelineno-9-26" name="__codelineno-9-26" href="#__codelineno-9-26"></a><span class="k">fn</span><span class="w"> </span><span class="nf">build_tree</span><span class="p">(</span><span class="n">preorder</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">inorder</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="w">    </span><span class="c1">// Initialize hash map, storing the mapping from inorder elements to indices</span>
+<a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="w">    </span><span class="c1">// Initialize a hash table mapping each element in inorder to its index</span>
 <a id="__codelineno-9-28" name="__codelineno-9-28" href="#__codelineno-9-28"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">inorder_map</span><span class="p">:</span><span class="w"> </span><span class="nc">HashMap</span><span class="o">&lt;</span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="kt">i32</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashMap</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-9-29" name="__codelineno-9-29" href="#__codelineno-9-29"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">inorder</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-30" name="__codelineno-9-30" href="#__codelineno-9-30"></a><span class="w">        </span><span class="n">inorder_map</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="n">inorder</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">);</span>
@@ -5097,9 +5169,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">build_tree.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Build binary tree: divide and conquer */</span>
+<div class="highlight"><span class="filename">build_tree.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Build a binary tree: divide and conquer */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">r</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Terminate when the subtree interval is empty</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Stop when the subtree interval is empty</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="c1">// Initialize the root node</span>
@@ -5107,7 +5179,7 @@
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="n">root</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">preorder</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="n">root</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="n">root</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
-<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// Query m to divide the left and right subtrees</span>
+<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// Look up m to separate the left and right subtrees</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">[</span><span class="n">preorder</span><span class="p">[</span><span class="n">i</span><span class="p">]];</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">    </span><span class="c1">// Subproblem: build the left subtree</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="n">root</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">);</span>
@@ -5117,9 +5189,9 @@
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">root</span><span class="p">;</span>
 <a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="p">}</span>
 <a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a>
-<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="cm">/* Build binary tree */</span>
+<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="cm">/* Build a binary tree */</span>
 <a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">buildTree</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">preorderSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">inorder</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">inorderSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">    </span><span class="c1">// Initialize hash map, storing the mapping from inorder elements to indices</span>
+<a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">    </span><span class="c1">// Initialize a lookup array mapping inorder values to their indices</span>
 <a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">inorderMap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="p">);</span>
 <a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">inorderSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="w">        </span><span class="n">inorderMap</span><span class="p">[</span><span class="n">inorder</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
@@ -5131,7 +5203,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">build_tree.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Build binary tree: divide and conquer */</span>
+<div class="highlight"><span class="filename">build_tree.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Build a binary tree: divide and conquer */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="n">preorder</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="n">inorderMap</span><span class="p">:</span><span class="w"> </span><span class="n">Map</span><span class="o">&lt;</span><span class="kt">Int?</span><span class="p">,</span><span class="w"> </span><span class="kt">Int?</span><span class="o">&gt;</span><span class="p">,</span>
@@ -5139,11 +5211,11 @@
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="n">l</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="n">r</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="p">):</span><span class="w"> </span><span class="n">TreeNode?</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// Terminate when the subtree interval is empty</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// Stop when the subtree interval is empty</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="c1">// Initialize the root node</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">preorder</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">)</span>
-<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="c1">// Query m to divide the left and right subtrees</span>
+<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="c1">// Look up m to separate the left and right subtrees</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">inorderMap</span><span class="o">[</span><span class="n">preorder</span><span class="o">[</span><span class="n">i</span><span class="o">]]!!</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="c1">// Subproblem: build the left subtree</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="n">root</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">inorderMap</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
@@ -5153,9 +5225,9 @@
 <a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">root</span>
 <a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="p">}</span>
 <a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a>
-<a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="cm">/* Build binary tree */</span>
+<a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="cm">/* Build a binary tree */</span>
 <a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">buildTree</span><span class="p">(</span><span class="n">preorder</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">inorder</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">):</span><span class="w"> </span><span class="n">TreeNode?</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w">    </span><span class="c1">// Initialize hash map, storing the mapping from inorder elements to indices</span>
+<a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w">    </span><span class="c1">// Initialize a hash table mapping each element in inorder to its index</span>
 <a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">inorderMap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashMap</span><span class="o">&lt;</span><span class="kt">Int?</span><span class="p">,</span><span class="w"> </span><span class="kt">Int?</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">inorder</span><span class="p">.</span><span class="na">indices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="w">        </span><span class="n">inorderMap</span><span class="o">[</span><span class="n">inorder</span><span class="o">[</span><span class="n">i</span><span class="o">]]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span>
@@ -5166,14 +5238,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">build_tree.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Build binary tree: divide and conquer ###</span>
+<div class="highlight"><span class="filename">build_tree.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Build a binary tree: divide and conquer ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">inorder_map</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">r</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Terminate when the subtree interval is empty</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Stop when the subtree interval is empty</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="c1"># Initialize the root node</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">TreeNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">preorder</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">)</span>
-<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="c1"># Query m to divide the left and right subtrees</span>
+<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="c1"># Look up m to separate the left and right subtrees</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">inorder_map</span><span class="o">[</span><span class="n">preorder</span><span class="o">[</span><span class="n">i</span><span class="o">]]</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="c1"># Subproblem: build the left subtree</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">  </span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">inorder_map</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
@@ -5184,9 +5256,9 @@
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">  </span><span class="n">root</span>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="k">end</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a>
-<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="c1">### Build binary tree ###</span>
+<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="c1">### Build a binary tree ###</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="k">def</span><span class="w"> </span><span class="nf">build_tree</span><span class="p">(</span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">inorder</span><span class="p">)</span>
-<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">  </span><span class="c1"># Initialize hash map, storing the mapping from inorder elements to indices</span>
+<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">  </span><span class="c1"># Initialize a hash table mapping each element in inorder to its index</span>
 <a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w">  </span><span class="n">inorder_map</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{}</span>
 <a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="w">  </span><span class="n">inorder</span><span class="o">.</span><span class="n">each_with_index</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="n">inorder_map</span><span class="o">[</span><span class="n">val</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a><span class="w">  </span><span class="n">dfs</span><span class="p">(</span><span class="n">preorder</span><span class="p">,</span><span class="w"> </span><span class="n">inorder_map</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">inorder</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
@@ -5195,11 +5267,16 @@
 </div>
 </div>
 </div>
-<p>Figure 12-8 shows the recursive process of building the binary tree. Each node is established during the downward "recursion" process, while each edge (reference) is established during the upward "return" process.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20left%20child%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20right%20child%0A%0Adef%20dfs%28%0A%20%20%20%20preorder%3A%20list%5Bint%5D,%0A%20%20%20%20inorder_map%3A%20dict%5Bint,%20int%5D,%0A%20%20%20%20i%3A%20int,%0A%20%20%20%20l%3A%20int,%0A%20%20%20%20r%3A%20int,%0A%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Build%20a%20binary%20tree%3A%20divide%20and%20conquer%22%22%22%0A%20%20%20%20%23%20Stop%20when%20the%20subtree%20interval%20is%20empty%0A%20%20%20%20if%20r%20-%20l%20%3C%200%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20Initialize%20the%20root%20node%0A%20%20%20%20root%20%3D%20TreeNode%28preorder%5Bi%5D%29%0A%20%20%20%20%23%20Look%20up%20m%20to%20separate%20the%20left%20and%20right%20subtrees%0A%20%20%20%20m%20%3D%20inorder_map%5Bpreorder%5Bi%5D%5D%0A%20%20%20%20%23%20Subproblem%3A%20build%20the%20left%20subtree%0A%20%20%20%20root.left%20%3D%20dfs%28preorder,%20inorder_map,%20i%20%2B%201,%20l,%20m%20-%201%29%0A%20%20%20%20%23%20Subproblem%3A%20build%20the%20right%20subtree%0A%20%20%20%20root.right%20%3D%20dfs%28preorder,%20inorder_map,%20i%20%2B%201%20%2B%20m%20-%20l,%20m%20%2B%201,%20r%29%0A%20%20%20%20%23%20Return%20the%20root%20node%0A%20%20%20%20return%20root%0A%0A%0Adef%20build_tree%28preorder%3A%20list%5Bint%5D,%20inorder%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Build%20a%20binary%20tree%22%22%22%0A%20%20%20%20%23%20Initialize%20a%20hash%20table%20mapping%20each%20element%20in%20inorder%20to%20its%20index%0A%20%20%20%20inorder_map%20%3D%20%7Bval%3A%20i%20for%20i,%20val%20in%20enumerate%28inorder%29%7D%0A%20%20%20%20root%20%3D%20dfs%28preorder,%20inorder_map,%200,%200,%20len%28inorder%29%20-%201%29%0A%20%20%20%20return%20root%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20preorder%20%3D%20%5B3,%209,%202,%201,%207%5D%0A%20%20%20%20inorder%20%3D%20%5B9,%203,%201,%202,%207%5D%0A%20%20%20%20print%28f%22Preorder%20traversal%20%3D%20%7Bpreorder%7D%22%29%0A%20%20%20%20print%28f%22Inorder%20traversal%20%3D%20%7Binorder%7D%22%29%0A%20%20%20%20root%20%3D%20build_tree%28preorder,%20inorder%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=21&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20left%20child%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20right%20child%0A%0Adef%20dfs%28%0A%20%20%20%20preorder%3A%20list%5Bint%5D,%0A%20%20%20%20inorder_map%3A%20dict%5Bint,%20int%5D,%0A%20%20%20%20i%3A%20int,%0A%20%20%20%20l%3A%20int,%0A%20%20%20%20r%3A%20int,%0A%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Build%20a%20binary%20tree%3A%20divide%20and%20conquer%22%22%22%0A%20%20%20%20%23%20Stop%20when%20the%20subtree%20interval%20is%20empty%0A%20%20%20%20if%20r%20-%20l%20%3C%200%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20Initialize%20the%20root%20node%0A%20%20%20%20root%20%3D%20TreeNode%28preorder%5Bi%5D%29%0A%20%20%20%20%23%20Look%20up%20m%20to%20separate%20the%20left%20and%20right%20subtrees%0A%20%20%20%20m%20%3D%20inorder_map%5Bpreorder%5Bi%5D%5D%0A%20%20%20%20%23%20Subproblem%3A%20build%20the%20left%20subtree%0A%20%20%20%20root.left%20%3D%20dfs%28preorder,%20inorder_map,%20i%20%2B%201,%20l,%20m%20-%201%29%0A%20%20%20%20%23%20Subproblem%3A%20build%20the%20right%20subtree%0A%20%20%20%20root.right%20%3D%20dfs%28preorder,%20inorder_map,%20i%20%2B%201%20%2B%20m%20-%20l,%20m%20%2B%201,%20r%29%0A%20%20%20%20%23%20Return%20the%20root%20node%0A%20%20%20%20return%20root%0A%0A%0Adef%20build_tree%28preorder%3A%20list%5Bint%5D,%20inorder%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Build%20a%20binary%20tree%22%22%22%0A%20%20%20%20%23%20Initialize%20a%20hash%20table%20mapping%20each%20element%20in%20inorder%20to%20its%20index%0A%20%20%20%20inorder_map%20%3D%20%7Bval%3A%20i%20for%20i,%20val%20in%20enumerate%28inorder%29%7D%0A%20%20%20%20root%20%3D%20dfs%28preorder,%20inorder_map,%200,%200,%20len%28inorder%29%20-%201%29%0A%20%20%20%20return%20root%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20preorder%20%3D%20%5B3,%209,%202,%201,%207%5D%0A%20%20%20%20inorder%20%3D%20%5B9,%203,%201,%202,%207%5D%0A%20%20%20%20print%28f%22Preorder%20traversal%20%3D%20%7Bpreorder%7D%22%29%0A%20%20%20%20print%28f%22Inorder%20traversal%20%3D%20%7Binorder%7D%22%29%0A%20%20%20%20root%20%3D%20build_tree%28preorder,%20inorder%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=21&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 12-8 shows the recursive reconstruction process. Nodes are created during recursive descent, while edges (references) are established as the recursive calls return.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:9"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">&lt;1&gt;</label><label for="__tabbed_2_2">&lt;2&gt;</label><label for="__tabbed_2_3">&lt;3&gt;</label><label for="__tabbed_2_4">&lt;4&gt;</label><label for="__tabbed_2_5">&lt;5&gt;</label><label for="__tabbed_2_6">&lt;6&gt;</label><label for="__tabbed_2_7">&lt;7&gt;</label><label for="__tabbed_2_8">&lt;8&gt;</label><label for="__tabbed_2_9">&lt;9&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Recursive process of building a binary tree" class="animation-figure" src="../build_binary_tree_problem.assets/built_tree_step1.png" /></p>
+<p><img alt="Recursive process for building a binary tree" class="animation-figure" src="../build_binary_tree_problem.assets/built_tree_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="built_tree_step2" class="animation-figure" src="../build_binary_tree_problem.assets/built_tree_step2.png" /></p>
@@ -5227,14 +5304,14 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 12-8 &nbsp; Recursive process of building a binary tree </p>
+<p align="center"> Figure 12-8 &nbsp; Recursive process for building a binary tree </p>
 
-<p>The division results of the preorder traversal <code>preorder</code> and inorder traversal <code>inorder</code> within each recursive function are shown in Figure 12-9.</p>
-<p><img alt="Division results in each recursive function" class="animation-figure" src="../build_binary_tree_problem.assets/built_tree_overall.png" /></p>
-<p align="center"> Figure 12-9 &nbsp; Division results in each recursive function </p>
+<p>Figure 12-9 shows how <code>preorder</code> and <code>inorder</code> are divided in each recursive call.</p>
+<p><img alt="Division within each recursive call" class="animation-figure" src="../build_binary_tree_problem.assets/built_tree_overall.png" /></p>
+<p align="center"> Figure 12-9 &nbsp; Division within each recursive call </p>
 
-<p>Let the number of nodes in the tree be <span class="arithmatex">\(n\)</span>. Initializing each node (executing one recursive function <code>dfs()</code>) takes <span class="arithmatex">\(O(1)\)</span> time. <strong>Therefore, the overall time complexity is <span class="arithmatex">\(O(n)\)</span></strong>.</p>
-<p>The hash table stores the mapping from <code>inorder</code> elements to their indices, with a space complexity of <span class="arithmatex">\(O(n)\)</span>. In the worst case, when the binary tree degenerates into a linked list, the recursion depth reaches <span class="arithmatex">\(n\)</span>, using <span class="arithmatex">\(O(n)\)</span> stack frame space. <strong>Therefore, the overall space complexity is <span class="arithmatex">\(O(n)\)</span></strong>.</p>
+<p>For a tree with <span class="arithmatex">\(n\)</span> nodes, initializing each node (executing one call to <code>dfs()</code>) takes <span class="arithmatex">\(O(1)\)</span> time. <strong>The overall time complexity is therefore <span class="arithmatex">\(O(n)\)</span></strong>.</p>
+<p>The hash table maps <code>inorder</code> elements to indices, requiring <span class="arithmatex">\(O(n)\)</span> space. In the worst case, the binary tree degenerates into a linked list, and the recursion depth reaches <span class="arithmatex">\(n\)</span>, using <span class="arithmatex">\(O(n)\)</span> space for stack frames. <strong>The overall space complexity is therefore <span class="arithmatex">\(O(n)\)</span></strong>.</p>
 
 <!-- Source file information -->
 
@@ -5257,7 +5334,7 @@ aria-label="Footer"
   <a
     href="../binary_search_recur/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 12.2 Divide and Conquer Search Strategy"
+    aria-label="Previous: 12.2   Divide-and-conquer search"
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@@ -5269,7 +5346,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        12.2 Divide and Conquer Search Strategy
+        12.2 &nbsp; Divide-and-conquer search
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     </div>
   </a>
@@ -5281,7 +5358,7 @@ aria-label="Footer"
   <a
     href="../hanota_problem/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 12.4 Tower of Hanoi Problem"
+    aria-label="Next: 12.4   Tower of Hanoi"
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@@ -5289,7 +5366,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        12.4 Tower of Hanoi Problem
+        12.4 &nbsp; Tower of Hanoi
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     <div class="md-footer__button md-icon">
@@ -5306,7 +5383,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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diff --git a/en/chapter_divide_and_conquer/divide_and_conquer/index.html b/en/chapter_divide_and_conquer/divide_and_conquer/index.html
index 56d7448b4..021d1746a 100644
--- a/en/chapter_divide_and_conquer/divide_and_conquer/index.html
+++ b/en/chapter_divide_and_conquer/divide_and_conquer/index.html
@@ -6,7 +6,7 @@
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>12.1 Divide and Conquer Algorithms - Hello Algo</title>
+        <title>12.1 Divide and conquer - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
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+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
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-        <a href="#121-divide-and-conquer-algorithms" class="md-skip">
+        <a href="#121-divide-and-conquer" class="md-skip">
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@@ -119,7 +119,7 @@
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 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              12.1 &nbsp; Divide and Conquer Algorithms
+              12.1 &nbsp; Divide and conquer
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
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   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
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   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
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@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3306,7 +3306,7 @@
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
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@@ -3336,7 +3336,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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         <li class="md-nav__item">
-  <a href="#1211-how-to-determine-divide-and-conquer-problems" class="md-nav__link">
+  <a href="#1211-recognizing-divide-and-conquer-problems" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        12.1.1 &nbsp; How to Determine Divide and Conquer Problems
+        12.1.1 &nbsp; Recognizing divide-and-conquer problems
       </span>
     </span>
   </a>
@@ -3390,22 +3390,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1212-improving-efficiency-through-divide-and-conquer" class="md-nav__link">
+  <a href="#1212-improving-efficiency-with-divide-and-conquer" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        12.1.2 &nbsp; Improving Efficiency Through Divide and Conquer
+        12.1.2 &nbsp; Improving efficiency with divide and conquer
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="12.1.2 Improving Efficiency Through Divide and Conquer">
+    <nav class="md-nav" aria-label="12.1.2 Improving efficiency with divide and conquer">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-operation-count-optimization" class="md-nav__link">
+  <a href="#1-reducing-the-number-of-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Operation Count Optimization
+        1. &nbsp; Reducing the number of operations
       </span>
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   </a>
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 </li>
 
           <li class="md-nav__item">
-  <a href="#2-parallel-computation-optimization" class="md-nav__link">
+  <a href="#2-using-parallel-computation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Parallel Computation Optimization
+        2. &nbsp; Using parallel computation
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-        12.1.3 &nbsp; Common Applications of Divide and Conquer
+        12.1.3 &nbsp; Common applications of divide and conquer
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
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-  14.1 Introduction to Dynamic Programming
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-  14.2 Characteristics of Dynamic Programming Problems
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-  14.3 Dynamic Programming Problem-Solving Approach
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1211-how-to-determine-divide-and-conquer-problems" class="md-nav__link">
+  <a href="#1211-recognizing-divide-and-conquer-problems" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        12.1.1 &nbsp; How to Determine Divide and Conquer Problems
+        12.1.1 &nbsp; Recognizing divide-and-conquer problems
       </span>
     </span>
   </a>
@@ -4649,22 +4721,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1212-improving-efficiency-through-divide-and-conquer" class="md-nav__link">
+  <a href="#1212-improving-efficiency-with-divide-and-conquer" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        12.1.2 &nbsp; Improving Efficiency Through Divide and Conquer
+        12.1.2 &nbsp; Improving efficiency with divide and conquer
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="12.1.2 Improving Efficiency Through Divide and Conquer">
+    <nav class="md-nav" aria-label="12.1.2 Improving efficiency with divide and conquer">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-operation-count-optimization" class="md-nav__link">
+  <a href="#1-reducing-the-number-of-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Operation Count Optimization
+        1. &nbsp; Reducing the number of operations
       </span>
     </span>
   </a>
@@ -4672,10 +4744,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-parallel-computation-optimization" class="md-nav__link">
+  <a href="#2-using-parallel-computation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Parallel Computation Optimization
+        2. &nbsp; Using parallel computation
       </span>
     </span>
   </a>
@@ -4691,7 +4763,7 @@
   <a href="#1213-common-applications-of-divide-and-conquer" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        12.1.3 &nbsp; Common Applications of Divide and Conquer
+        12.1.3 &nbsp; Common applications of divide and conquer
       </span>
     </span>
   </a>
@@ -4735,45 +4807,45 @@
 
 
 <!-- Page content -->
-<h1 id="121-divide-and-conquer-algorithms">12.1 &nbsp; Divide and Conquer Algorithms<a class="headerlink" href="#121-divide-and-conquer-algorithms" title="Permanent link">&para;</a></h1>
-<p><u>Divide and conquer</u> is a very important and common algorithmic strategy. Divide and conquer is typically implemented based on recursion, consisting of two steps: "divide" and "conquer".</p>
+<h1 id="121-divide-and-conquer">12.1 &nbsp; Divide and conquer<a class="headerlink" href="#121-divide-and-conquer" title="Permanent link">&para;</a></h1>
+<p><u>Divide and conquer</u> is a fundamental and widely used algorithmic strategy. Usually implemented recursively, it has two stages: dividing the problem and combining its solutions.</p>
 <ol>
-<li><strong>Divide (partition phase)</strong>: Recursively divide the original problem into two or more subproblems until the smallest subproblem is reached.</li>
-<li><strong>Conquer (merge phase)</strong>: Starting from the smallest subproblems with known solutions, merge the solutions of subproblems from bottom to top to construct the solution to the original problem.</li>
+<li><strong>Divide (the partitioning stage)</strong>: Recursively split the original problem into two or more subproblems until the smallest subproblems are reached.</li>
+<li><strong>Conquer (the combining stage)</strong>: Starting with the known solutions to the smallest subproblems, combine solutions from the bottom up to construct the solution to the original problem.</li>
 </ol>
-<p>As shown in Figure 12-1, "merge sort" is one of the typical applications of the divide and conquer strategy.</p>
+<p>As shown in Figure 12-1, merge sort is a classic application of divide and conquer.</p>
 <ol>
-<li><strong>Divide</strong>: Recursively divide the original array (original problem) into two subarrays (subproblems) until the subarray has only one element (smallest subproblem).</li>
-<li><strong>Conquer</strong>: Merge the sorted subarrays (solutions to subproblems) from bottom to top to obtain a sorted original array (solution to the original problem).</li>
+<li><strong>Divide</strong>: Recursively split the original array (the original problem) into two subarrays (subproblems) until each subarray contains just one element (the smallest subproblem).</li>
+<li><strong>Conquer</strong>: Merge the sorted subarrays (subproblem solutions) from the bottom up to obtain the sorted original array (the solution to the original problem).</li>
 </ol>
-<p><img alt="Divide and conquer strategy of merge sort" class="animation-figure" src="../divide_and_conquer.assets/divide_and_conquer_merge_sort.png" /></p>
-<p align="center"> Figure 12-1 &nbsp; Divide and conquer strategy of merge sort </p>
+<p><img alt="Divide-and-conquer strategy in merge sort" class="animation-figure" src="../divide_and_conquer.assets/divide_and_conquer_merge_sort.png" /></p>
+<p align="center"> Figure 12-1 &nbsp; Divide-and-conquer strategy in merge sort </p>
 
-<h2 id="1211-how-to-determine-divide-and-conquer-problems">12.1.1 &nbsp; How to Determine Divide and Conquer Problems<a class="headerlink" href="#1211-how-to-determine-divide-and-conquer-problems" title="Permanent link">&para;</a></h2>
-<p>Whether a problem is suitable for solving with divide and conquer can usually be determined based on the following criteria.</p>
+<h2 id="1211-recognizing-divide-and-conquer-problems">12.1.1 &nbsp; Recognizing divide-and-conquer problems<a class="headerlink" href="#1211-recognizing-divide-and-conquer-problems" title="Permanent link">&para;</a></h2>
+<p>The following criteria usually help determine whether divide and conquer is suitable for a problem.</p>
 <ol>
-<li><strong>The problem can be decomposed</strong>: The original problem can be divided into smaller, similar subproblems, and can be recursively divided in the same way.</li>
-<li><strong>Subproblems are independent</strong>: There is no overlap between subproblems, they are independent of each other and can be solved independently.</li>
-<li><strong>Solutions of subproblems can be merged</strong>: The solution to the original problem is obtained by merging the solutions of subproblems.</li>
+<li><strong>The problem can be divided</strong>: The original problem can be split into smaller, similar subproblems, which can in turn be divided recursively in the same way.</li>
+<li><strong>The subproblems are independent</strong>: They neither overlap nor depend on one another, so they can be solved independently.</li>
+<li><strong>The subproblem solutions can be combined</strong>: Combining them yields the solution to the original problem.</li>
 </ol>
-<p>Clearly, merge sort satisfies these three criteria.</p>
+<p>Merge sort clearly satisfies all three criteria.</p>
 <ol>
-<li><strong>The problem can be decomposed</strong>: Recursively divide the array (original problem) into two subarrays (subproblems).</li>
-<li><strong>Subproblems are independent</strong>: Each subarray can be sorted independently (subproblems can be solved independently).</li>
-<li><strong>Solutions of subproblems can be merged</strong>: Two sorted subarrays (solutions of subproblems) can be merged into one sorted array (solution of the original problem).</li>
+<li><strong>The problem can be divided</strong>: Recursively split an array (the original problem) into two subarrays (subproblems).</li>
+<li><strong>The subproblems are independent</strong>: Each subarray can be sorted independently.</li>
+<li><strong>The subproblem solutions can be combined</strong>: Merge the two sorted subarrays (subproblem solutions) into a single sorted array (the solution to the original problem).</li>
 </ol>
-<h2 id="1212-improving-efficiency-through-divide-and-conquer">12.1.2 &nbsp; Improving Efficiency Through Divide and Conquer<a class="headerlink" href="#1212-improving-efficiency-through-divide-and-conquer" title="Permanent link">&para;</a></h2>
-<p><strong>Divide and conquer can not only effectively solve algorithmic problems, but can often also improve algorithmic efficiency</strong>. In sorting algorithms, quick sort, merge sort, and heap sort are faster than selection, bubble, and insertion sort because they apply the divide and conquer strategy.</p>
-<p>This raises the question: <strong>Why can divide and conquer improve algorithm efficiency, and what is the underlying logic</strong>? In other words, why is dividing a large problem into multiple subproblems, solving the subproblems, and merging their solutions more efficient than directly solving the original problem? This question can be discussed from two aspects: operation count and parallel computation.</p>
-<h3 id="1-operation-count-optimization">1. &nbsp; Operation Count Optimization<a class="headerlink" href="#1-operation-count-optimization" title="Permanent link">&para;</a></h3>
-<p>Taking "bubble sort" as an example, processing an array of length <span class="arithmatex">\(n\)</span> requires <span class="arithmatex">\(O(n^2)\)</span> time. Suppose we divide the array at the midpoint into two subarrays, as shown in Figure 12-2. The division requires <span class="arithmatex">\(O(n)\)</span> time, sorting each subarray requires <span class="arithmatex">\(O((n / 2)^2)\)</span> time, and merging the two subarrays requires <span class="arithmatex">\(O(n)\)</span> time, resulting in an overall time complexity of:</p>
+<h2 id="1212-improving-efficiency-with-divide-and-conquer">12.1.2 &nbsp; Improving efficiency with divide and conquer<a class="headerlink" href="#1212-improving-efficiency-with-divide-and-conquer" title="Permanent link">&para;</a></h2>
+<p><strong>Divide and conquer not only solves algorithmic problems effectively, but often improves efficiency as well</strong>. Quick sort, merge sort, and heap sort outperform selection sort, bubble sort, and insertion sort because they use this strategy.</p>
+<p>This raises a question: <strong>why does divide and conquer improve efficiency, and what is the underlying reason</strong>? Why is dividing a large problem, solving its subproblems, and combining their solutions faster than solving it directly? We can examine this from two perspectives: the number of operations and parallel computation.</p>
+<h3 id="1-reducing-the-number-of-operations">1. &nbsp; Reducing the number of operations<a class="headerlink" href="#1-reducing-the-number-of-operations" title="Permanent link">&para;</a></h3>
+<p>Consider bubble sort, which takes <span class="arithmatex">\(O(n^2)\)</span> time to sort an array of length <span class="arithmatex">\(n\)</span>. Suppose we split the array at its midpoint, as shown in Figure 12-2. Splitting takes <span class="arithmatex">\(O(n)\)</span> time, sorting each half takes <span class="arithmatex">\(O((n / 2)^2)\)</span> time, and merging the halves takes <span class="arithmatex">\(O(n)\)</span> time. The total time complexity is:</p>
 <div class="arithmatex">\[
 O(n + (\frac{n}{2})^2 \times 2 + n) = O(\frac{n^2}{2} + 2n)
 \]</div>
-<p><img alt="Bubble sort before and after array division" class="animation-figure" src="../divide_and_conquer.assets/divide_and_conquer_bubble_sort.png" /></p>
-<p align="center"> Figure 12-2 &nbsp; Bubble sort before and after array division </p>
+<p><img alt="Bubble sort before and after dividing the array" class="animation-figure" src="../divide_and_conquer.assets/divide_and_conquer_bubble_sort.png" /></p>
+<p align="center"> Figure 12-2 &nbsp; Bubble sort before and after dividing the array </p>
 
-<p>Next, we compute the following inequality, where the left and right sides represent the total number of operations before and after division, respectively:</p>
+<p>Now consider the following inequality, whose left and right sides represent the total operation counts before and after division:</p>
 <div class="arithmatex">\[
 \begin{aligned}
 n^2 &amp; &gt; \frac{n^2}{2} + 2n \newline
@@ -4781,36 +4853,36 @@ n^2 - \frac{n^2}{2} - 2n &amp; &gt; 0 \newline
 n(n - 4) &amp; &gt; 0
 \end{aligned}
 \]</div>
-<p><strong>This means that when <span class="arithmatex">\(n &gt; 4\)</span>, the number of operations after division is smaller, and sorting efficiency should be higher</strong>. Note that the time complexity after division is still quadratic <span class="arithmatex">\(O(n^2)\)</span>, but the constant term in the complexity has become smaller.</p>
-<p>Going further, <strong>what if we continuously divide the subarrays from their midpoints into two subarrays</strong> until the subarrays have only one element? This approach is actually "merge sort", with a time complexity of <span class="arithmatex">\(O(n \log n)\)</span>.</p>
-<p>Thinking further, <strong>what if we set multiple division points</strong> and evenly divide the original array into <span class="arithmatex">\(k\)</span> subarrays? This situation is very similar to "bucket sort", which is well-suited for sorting massive amounts of data, with a theoretical time complexity of <span class="arithmatex">\(O(n + k)\)</span>.</p>
-<h3 id="2-parallel-computation-optimization">2. &nbsp; Parallel Computation Optimization<a class="headerlink" href="#2-parallel-computation-optimization" title="Permanent link">&para;</a></h3>
-<p>We know that the subproblems generated by divide and conquer are independent of each other, <strong>so they can typically be solved in parallel</strong>. This means divide and conquer can not only reduce the time complexity of algorithms, <strong>but is also amenable to parallel optimization by the operating system</strong>.</p>
-<p>Parallel optimization is particularly effective in multi-core or multi-processor environments, as the system can simultaneously handle multiple subproblems, making fuller use of computing resources and significantly reducing overall runtime.</p>
-<p>For example, in the "bucket sort" shown in Figure 12-3, we evenly distribute massive data into various buckets, and the sorting tasks for all buckets can be distributed to various computing units. After completion, the results are merged.</p>
+<p><strong>When <span class="arithmatex">\(n &gt; 4\)</span>, dividing the array reduces the number of operations and should make sorting faster</strong>. The time complexity is still quadratic, <span class="arithmatex">\(O(n^2)\)</span>; only the constant factor has decreased.</p>
+<p>What if <strong>we keep splitting each subarray into two at its midpoint</strong> until every subarray contains just one element? This is precisely the idea behind merge sort, which takes <span class="arithmatex">\(O(n \log n)\)</span> time.</p>
+<p>What if <strong>we introduce more split points</strong>, dividing the original array evenly into <span class="arithmatex">\(k\)</span> subarrays? This resembles bucket sort, which is well suited to sorting large amounts of data and can theoretically achieve <span class="arithmatex">\(O(n + k)\)</span> time.</p>
+<h3 id="2-using-parallel-computation">2. &nbsp; Using parallel computation<a class="headerlink" href="#2-using-parallel-computation" title="Permanent link">&para;</a></h3>
+<p>The subproblems produced by divide and conquer are independent, <strong>so they can usually be solved in parallel</strong>. Thus, divide and conquer can both reduce time complexity and <strong>help the operating system parallelize the work</strong>.</p>
+<p>Parallelization is especially effective on multicore or multiprocessor systems. Multiple subproblems can be processed simultaneously, making fuller use of computing resources and substantially reducing total runtime.</p>
+<p>For example, in the bucket sort shown in Figure 12-3, a large dataset is evenly distributed among buckets. The sorting tasks for different buckets can then be assigned to different computing units, and their results combined afterward.</p>
 <p><img alt="Parallel computation in bucket sort" class="animation-figure" src="../divide_and_conquer.assets/divide_and_conquer_parallel_computing.png" /></p>
 <p align="center"> Figure 12-3 &nbsp; Parallel computation in bucket sort </p>
 
-<h2 id="1213-common-applications-of-divide-and-conquer">12.1.3 &nbsp; Common Applications of Divide and Conquer<a class="headerlink" href="#1213-common-applications-of-divide-and-conquer" title="Permanent link">&para;</a></h2>
-<p>On the one hand, divide and conquer can be used to solve many classic algorithmic problems.</p>
+<h2 id="1213-common-applications-of-divide-and-conquer">12.1.3 &nbsp; Common applications of divide and conquer<a class="headerlink" href="#1213-common-applications-of-divide-and-conquer" title="Permanent link">&para;</a></h2>
+<p>Divide and conquer solves many classic algorithmic problems.</p>
 <ul>
-<li><strong>Finding the closest pair of points</strong>: This algorithm first divides the point set into two parts, then finds the closest pair of points in each part separately, and finally finds the closest pair of points that spans both parts.</li>
-<li><strong>Large integer multiplication</strong>: For example, the Karatsuba algorithm, which decomposes large integer multiplication into several smaller integer multiplications and additions.</li>
-<li><strong>Matrix multiplication</strong>: For example, the Strassen algorithm, which decomposes large matrix multiplication into multiple small matrix multiplications and additions.</li>
-<li><strong>Tower of Hanoi problem</strong>: The Tower of Hanoi problem can be solved through recursion, which is a typical application of the divide and conquer strategy.</li>
-<li><strong>Solving inversion pairs</strong>: In a sequence, if a preceding number is greater than a following number, these two numbers form an inversion pair. Solving the inversion pair problem can utilize the divide and conquer approach with the help of merge sort.</li>
+<li><strong>Closest pair of points</strong>: Divide a set of points into two parts, find the closest pair within each part, then find the closest pair spanning the two parts.</li>
+<li><strong>Large integer multiplication</strong>: Karatsuba's algorithm, for example, breaks multiplication of large integers into several smaller integer multiplications and additions.</li>
+<li><strong>Matrix multiplication</strong>: Strassen's algorithm, for example, breaks multiplication of large matrices into multiple smaller matrix multiplications and additions.</li>
+<li><strong>Tower of Hanoi</strong>: This problem can be solved recursively and is a classic application of divide and conquer.</li>
+<li><strong>Counting inversions</strong>: In a sequence, an earlier number and a later number form an inversion if the earlier one is larger. Inversions can be counted using divide and conquer with merge sort.</li>
 </ul>
-<p>On the other hand, divide and conquer is widely applied in the design of algorithms and data structures.</p>
+<p>Divide and conquer is also widely used in designing algorithms and data structures.</p>
 <ul>
-<li><strong>Binary search</strong>: Binary search divides a sorted array into two parts from the midpoint index, then decides which half to eliminate based on the comparison result between the target value and the middle element value, and performs the same binary-search step on the remaining interval.</li>
-<li><strong>Merge sort</strong>: Already introduced at the beginning of this section, no further elaboration needed.</li>
-<li><strong>Quick sort</strong>: Quick sort selects a pivot value, then divides the array into two subarrays, one with elements smaller than the pivot and the other with elements larger than the pivot, then performs the same division operation on these two parts until the subarrays have only one element.</li>
-<li><strong>Bucket sort</strong>: The basic idea of bucket sort is to scatter data into multiple buckets, then sort the elements within each bucket, and finally extract the elements from each bucket in sequence to obtain a sorted array.</li>
-<li><strong>Trees</strong>: For example, binary search trees, AVL trees, red-black trees, B-trees, B+ trees, etc. Their search, insertion, and deletion operations can all be viewed as applications of the divide and conquer strategy.</li>
-<li><strong>Heaps</strong>: A heap is a special complete binary tree, and its various operations, such as insertion, deletion, and heapify, actually imply the divide and conquer idea.</li>
-<li><strong>Hash tables</strong>: Although hash tables do not directly apply divide and conquer, some methods for resolving hash collisions indirectly apply the divide and conquer strategy. For example, long linked lists in chaining may be converted to red-black trees to improve lookup efficiency.</li>
+<li><strong>Binary search</strong>: Split a sorted array at its middle index, compare the middle element with the target to decide which half to discard, then repeat the same operation on the remaining interval.</li>
+<li><strong>Merge sort</strong>: This was described at the start of the section.</li>
+<li><strong>Quick sort</strong>: Choose a pivot and divide the array into two subarrays, one containing elements smaller than the pivot and the other containing larger elements. Repeat the division on both parts until each subarray contains just one element.</li>
+<li><strong>Bucket sort</strong>: Distribute data among multiple buckets, sort the elements in each bucket, and then take the elements from the buckets in order to obtain a sorted array.</li>
+<li><strong>Trees</strong>: Operations such as search, insertion, and deletion in binary search trees, AVL trees, red-black trees, B-trees, and B+ trees can be viewed as applications of divide and conquer.</li>
+<li><strong>Heaps</strong>: A heap is a special kind of complete binary tree. Its operations, such as insertion, deletion, and heapification, implicitly use divide and conquer.</li>
+<li><strong>Hash tables</strong>: Although hash tables do not directly use divide and conquer, some collision-handling strategies use it indirectly. For example, long linked lists in separate chaining may be converted into red-black trees to speed up lookups.</li>
 </ul>
-<p>It can be seen that <strong>divide and conquer is a "quietly pervasive" algorithmic idea</strong>, embedded in various algorithms and data structures.</p>
+<p>As we can see, <strong>divide and conquer quietly underlies many algorithms and data structures</strong>, often without being explicitly named.</p>
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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@@ -2019,7 +2019,7 @@
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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@@ -2378,7 +2378,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
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-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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@@ -2665,7 +2665,7 @@
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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@@ -2692,7 +2692,7 @@
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3285,7 +3285,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3306,7 +3306,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3327,7 +3327,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3381,7 +3381,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3408,7 +3408,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3435,7 +3435,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3514,22 +3514,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1261-concept-review" class="md-nav__link">
+  <a href="#1261-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        12.6.1 &nbsp; Concept Review
+        12.6.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="12.6.1 Concept Review">
+    <nav class="md-nav" aria-label="12.6.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-which-tasks-are-suitable-for-divide-and-conquer" class="md-nav__link">
+  <a href="#1-which-tasks-suit-divide-and-conquer" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Which Tasks Are Suitable for Divide and Conquer?
+        1. &nbsp; Which tasks suit divide and conquer?
       </span>
     </span>
   </a>
@@ -3537,10 +3537,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-how-exponentiation-by-squaring-reduces-computation" class="md-nav__link">
+  <a href="#2-how-fast-exponentiation-reduces-computation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; How Exponentiation by Squaring Reduces Computation
+        2. &nbsp; How fast exponentiation reduces computation
       </span>
     </span>
   </a>
@@ -3548,10 +3548,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-split-traversal-sequences-into-left-and-right-subtrees" class="md-nav__link">
+  <a href="#3-splitting-traversal-sequences-into-left-and-right-subtrees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Split Traversal Sequences into Left and Right Subtrees
+        3. &nbsp; Splitting traversal sequences into left and right subtrees
       </span>
     </span>
   </a>
@@ -3567,19 +3567,19 @@
   <a href="#1262-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        12.6.2 &nbsp; Programming Exercises
+        12.6.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="12.6.2 Programming Exercises">
+    <nav class="md-nav" aria-label="12.6.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-exponentiation-by-squaring" class="md-nav__link">
+  <a href="#1-fast-exponentiation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Exponentiation by Squaring
+        1. &nbsp; Fast exponentiation
       </span>
     </span>
   </a>
@@ -3653,7 +3653,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3695,7 +3695,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3776,7 +3776,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3803,7 +3803,7 @@
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3900,7 +3900,7 @@
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3942,7 +3942,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3996,7 +3996,7 @@
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4077,7 +4077,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4104,7 +4104,7 @@
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4197,7 +4197,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4218,7 +4218,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4239,7 +4239,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4266,7 +4266,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4434,7 +4434,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4476,7 +4476,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4503,7 +4503,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4530,7 +4530,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4614,6 +4614,78 @@
 
 
 
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-  <a href="#1261-concept-review" class="md-nav__link">
+  <a href="#1261-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        12.6.1 &nbsp; Concept Review
+        12.6.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="12.6.1 Concept Review">
+    <nav class="md-nav" aria-label="12.6.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-which-tasks-are-suitable-for-divide-and-conquer" class="md-nav__link">
+  <a href="#1-which-tasks-suit-divide-and-conquer" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Which Tasks Are Suitable for Divide and Conquer?
+        1. &nbsp; Which tasks suit divide and conquer?
       </span>
     </span>
   </a>
@@ -4678,10 +4750,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-how-exponentiation-by-squaring-reduces-computation" class="md-nav__link">
+  <a href="#2-how-fast-exponentiation-reduces-computation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; How Exponentiation by Squaring Reduces Computation
+        2. &nbsp; How fast exponentiation reduces computation
       </span>
     </span>
   </a>
@@ -4689,10 +4761,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-split-traversal-sequences-into-left-and-right-subtrees" class="md-nav__link">
+  <a href="#3-splitting-traversal-sequences-into-left-and-right-subtrees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Split Traversal Sequences into Left and Right Subtrees
+        3. &nbsp; Splitting traversal sequences into left and right subtrees
       </span>
     </span>
   </a>
@@ -4708,19 +4780,19 @@
   <a href="#1262-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        12.6.2 &nbsp; Programming Exercises
+        12.6.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="12.6.2 Programming Exercises">
+    <nav class="md-nav" aria-label="12.6.2 Programming exercises">
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-  <a href="#1-exponentiation-by-squaring" class="md-nav__link">
+  <a href="#1-fast-exponentiation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Exponentiation by Squaring
+        1. &nbsp; Fast exponentiation
       </span>
     </span>
   </a>
@@ -4770,34 +4842,34 @@
 
 <!-- Page content -->
 <h1 id="126-exercises">12.6 &nbsp; Exercises<a class="headerlink" href="#126-exercises" title="Permanent link">&para;</a></h1>
-<h2 id="1261-concept-review">12.6.1 &nbsp; Concept Review<a class="headerlink" href="#1261-concept-review" title="Permanent link">&para;</a></h2>
-<h3 id="1-which-tasks-are-suitable-for-divide-and-conquer">1. &nbsp; Which Tasks Are Suitable for Divide and Conquer?<a class="headerlink" href="#1-which-tasks-are-suitable-for-divide-and-conquer" title="Permanent link">&para;</a></h3>
-<p>A student wants to solve each task below by "dividing it into two halves, solving each half separately, and then combining the results."
-Classify each task as "suitable for divide and conquer," "can use divide and conquer, but it will not reduce the total work," or "the two halves cannot be solved independently," and explain why.</p>
+<h2 id="1261-check-your-understanding">12.6.1 &nbsp; Check your understanding<a class="headerlink" href="#1261-check-your-understanding" title="Permanent link">&para;</a></h2>
+<h3 id="1-which-tasks-suit-divide-and-conquer">1. &nbsp; Which tasks suit divide and conquer?<a class="headerlink" href="#1-which-tasks-suit-divide-and-conquer" title="Permanent link">&para;</a></h3>
+<p>A student wants to solve the following tasks by splitting each in half, solving the halves separately, and combining the results.
+For each task, decide whether divide and conquer is suitable, is possible but does not reduce the total work, or is unsuitable because the halves cannot be solved independently. Explain.</p>
 <!-- numbered-subquestions -->
 
 <ol>
 <li>Sort an unsorted array.</li>
 <li>Find the maximum value in an array.</li>
-<li>Execute a sequence of <code>push(x)</code> and <code>pop()</code> stack operations in order, and output the element returned by each <code>pop()</code>.</li>
+<li>Execute a sequence of stack operations <code>push(x)</code> and <code>pop()</code> in order, outputting the element returned by each <code>pop()</code>.</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
-<li>Suitable: divide the array in half, sort each half independently, and merge them in <span class="arithmatex">\(O(n)\)</span> time. This is exactly merge sort.</li>
-<li>Divide and conquer can be used, but it does not reduce the total work. The two halves still require examining all <span class="arithmatex">\(n\)</span> elements in total,
-    so the time complexity remains <span class="arithmatex">\(O(n)\)</span>, just like a direct scan.</li>
-<li>The two halves cannot be solved independently. The stack's contents at the beginning of the second half depend on the result of executing the first half,
-    so the two halves cannot be completed without knowing each other's results.</li>
+<li>Suitable: split in half, sort the halves independently, and merge in <span class="arithmatex">\(O(n)\)</span> time. This is exactly merge sort.</li>
+<li>Possible, but without reducing the total work: the two halves must still examine all <span class="arithmatex">\(n\)</span> elements between them,
+    taking <span class="arithmatex">\(O(n)\)</span> time, just like a direct scan.</li>
+<li>The halves cannot be solved independently: the stack contents at the start of the second half depend on the results of the first half.
+    The second half therefore cannot run independently of the first half's result.</li>
 </ol>
 </details>
-<h3 id="2-how-exponentiation-by-squaring-reduces-computation">2. &nbsp; How Exponentiation by Squaring Reduces Computation<a class="headerlink" href="#2-how-exponentiation-by-squaring-reduces-computation" title="Permanent link">&para;</a></h3>
-<p>The recursive function below uses divide and conquer to calculate <span class="arithmatex">\(x^n\)</span>:</p>
+<h3 id="2-how-fast-exponentiation-reduces-computation">2. &nbsp; How fast exponentiation reduces computation<a class="headerlink" href="#2-how-fast-exponentiation-reduces-computation" title="Permanent link">&para;</a></h3>
+<p>The following recursive function uses divide and conquer to compute <span class="arithmatex">\(x^n\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">fast_power.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">fast_pow</span><span class="p">(</span><span class="n">x</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Exponentiation by squaring"""</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Fast exponentiation"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>        <span class="k">return</span> <span class="mi">1</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="n">half</span> <span class="o">=</span> <span class="n">fast_pow</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">)</span>
@@ -4807,7 +4879,7 @@ Classify each task as "suitable for divide and conquer," "can use divide and con
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">fast_power.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Exponentiation by squaring */</span>
+<div class="highlight"><span class="filename">fast_power.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Fast exponentiation */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">fastPow</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -4821,7 +4893,7 @@ Classify each task as "suitable for divide and conquer," "can use divide and con
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">fast_power.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Exponentiation by squaring */</span>
+<div class="highlight"><span class="filename">fast_power.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Fast exponentiation */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">fastPow</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -4835,7 +4907,7 @@ Classify each task as "suitable for divide and conquer," "can use divide and con
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">fast_power.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Exponentiation by squaring */</span>
+<div class="highlight"><span class="filename">fast_power.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Fast exponentiation */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">FastPow</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -4849,7 +4921,7 @@ Classify each task as "suitable for divide and conquer," "can use divide and con
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">fast_power.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Exponentiation by squaring */</span>
+<div class="highlight"><span class="filename">fast_power.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Fast exponentiation */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">fastPow</span><span class="p">(</span><span class="nx">x</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span>
@@ -4863,7 +4935,7 @@ Classify each task as "suitable for divide and conquer," "can use divide and con
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">fast_power.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Exponentiation by squaring */</span>
+<div class="highlight"><span class="filename">fast_power.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Fast exponentiation */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">fastPow</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span>
@@ -4877,7 +4949,7 @@ Classify each task as "suitable for divide and conquer," "can use divide and con
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">fast_power.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Exponentiation by squaring */</span>
+<div class="highlight"><span class="filename">fast_power.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Fast exponentiation */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">fastPow</span><span class="p">(</span><span class="nx">x</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
@@ -4891,7 +4963,7 @@ Classify each task as "suitable for divide and conquer," "can use divide and con
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">fast_power.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Exponentiation by squaring */</span>
+<div class="highlight"><span class="filename">fast_power.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Fast exponentiation */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">fastPow</span><span class="p">(</span><span class="nx">x</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
@@ -4905,7 +4977,7 @@ Classify each task as "suitable for divide and conquer," "can use divide and con
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">fast_power.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Exponentiation by squaring */</span>
+<div class="highlight"><span class="filename">fast_power.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Fast exponentiation */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">fastPow</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
@@ -4919,7 +4991,7 @@ Classify each task as "suitable for divide and conquer," "can use divide and con
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">fast_power.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Exponentiation by squaring */</span>
+<div class="highlight"><span class="filename">fast_power.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Fast exponentiation */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">fast_pow</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -4933,7 +5005,7 @@ Classify each task as "suitable for divide and conquer," "can use divide and con
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">fast_power.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Exponentiation by squaring */</span>
+<div class="highlight"><span class="filename">fast_power.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Fast exponentiation */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">fastPow</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -4947,7 +5019,7 @@ Classify each task as "suitable for divide and conquer," "can use divide and con
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">fast_power.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Exponentiation by squaring */</span>
+<div class="highlight"><span class="filename">fast_power.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Fast exponentiation */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">fastPow</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">1</span>
@@ -4961,7 +5033,7 @@ Classify each task as "suitable for divide and conquer," "can use divide and con
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">fast_power.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Exponentiation by squaring ###</span>
+<div class="highlight"><span class="filename">fast_power.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Fast exponentiation ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">fast_pow</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a>
@@ -4974,72 +5046,72 @@ Classify each task as "suitable for divide and conquer," "can use divide and con
 </div>
 </div>
 </div>
-<p>Set <code>x = 3</code> and <code>n = 5</code>, and use this function to calculate the result:</p>
+<p>Run it with <code>x = 3</code> and <code>n = 5</code>:</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>As the recursive calls proceed, which values does the argument <code>n</code> take in order?</li>
-<li>Starting from the deepest call, what value does each level return?</li>
-<li>Why should the recursive result be stored in <code>half</code> instead of calling the same subproblem once on each side of the multiplication?</li>
+<li>What values does <code>n</code> take in successive recursive calls?</li>
+<li>Starting with the deepest call, what value does each call return?</li>
+<li>Why should the recursive result be saved in <code>half</code> instead of making the same recursive call on each side of the multiplication?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>The argument takes the values <code>5 → 2 → 1 → 0</code>. The exponent is halved at each step until the base case is reached.</p>
+<p>The parameter follows <code>5 → 2 → 1 → 0</code>, halving the exponent each time until the base case is reached.</p>
 </li>
 <li>
-<p>When <code>n = 0</code>, the function returns 1. When <code>n = 1</code>, it returns <span class="arithmatex">\(1×1×3=3\)</span>.
-    When <code>n = 2</code>, it returns <span class="arithmatex">\(3×3=9\)</span>. When <code>n = 5</code>, it returns <span class="arithmatex">\(9×9×3=243\)</span>.</p>
+<p>When <code>n = 0</code>, the return value is 1. When <code>n = 1</code>, it is <span class="arithmatex">\(1×1×3=3\)</span>.
+    When <code>n = 2</code>, it is <span class="arithmatex">\(3×3=9\)</span>; when <code>n = 5</code>, it is <span class="arithmatex">\(9×9×3=243\)</span>.</p>
 </li>
 <li>
-<p>If the same subproblem were called once on each side of the multiplication, the two recursive calls would perform exactly the same calculation.
-    Storing the result in <code>half</code> means that each level makes only one recursive call, so the recursion depth is about <span class="arithmatex">\(\log n\)</span>.
-    Making two calls would cause a great deal of repeated computation.</p>
+<p>Calling the same subproblem on both sides of the multiplication repeats exactly the same computation.
+    Saving the result in <code>half</code> means each level makes only one recursive call, with recursion depth approximately <span class="arithmatex">\(\log n\)</span>.
+    Making two calls would cause substantial repeated computation.</p>
 </li>
 </ol>
 </details>
-<h3 id="3-split-traversal-sequences-into-left-and-right-subtrees">3. &nbsp; Split Traversal Sequences into Left and Right Subtrees<a class="headerlink" href="#3-split-traversal-sequences-into-left-and-right-subtrees" title="Permanent link">&para;</a></h3>
-<p>A binary tree has no duplicate nodes. Its preorder and inorder traversals are:</p>
+<h3 id="3-splitting-traversal-sequences-into-left-and-right-subtrees">3. &nbsp; Splitting traversal sequences into left and right subtrees<a class="headerlink" href="#3-splitting-traversal-sequences-into-left-and-right-subtrees" title="Permanent link">&para;</a></h3>
+<p>A binary tree with distinct node values has these preorder and inorder traversals:</p>
 <ul>
 <li>Preorder: <code>[A, B, D, E, C]</code></li>
 <li>Inorder: <code>[D, B, E, A, C]</code></li>
 </ul>
-<p>Split the sequences only once, at the root. You do not need to continue recursively or draw the whole tree:</p>
+<p>Split only at the root level; there is no need to recurse further or draw the entire tree.</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Which node is the root?</li>
-<li>Which subsequences of the inorder traversal correspond to the left and right subtrees?</li>
-<li>Which subsequences of the preorder traversal correspond to the left and right subtrees? Which nodes are the root's direct children?</li>
+<li>What is the root node?</li>
+<li>Which parts of the inorder sequence belong to the left and right subtrees?</li>
+<li>Which parts of the preorder sequence belong to the left and right subtrees? What are the root's immediate children?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>The first node in a preorder traversal is the root, so the root is <code>A</code>.</p>
+<p>The first node in preorder is the root, so the root is <code>A</code>.</p>
 </li>
 <li>
-<p><code>A</code> divides the inorder traversal into two parts: <code>[D, B, E]</code> for the left subtree and <code>[C]</code> for the right subtree.</p>
+<p><code>A</code> splits the inorder sequence into two parts: <code>[D, B, E]</code> for the left subtree and <code>[C]</code> for the right subtree.</p>
 </li>
 <li>
-<p>The left subtree contains 3 nodes, so the 3 preorder elements after the root <code>A</code> belong to the left subtree,
-    giving <code>[B, D, E]</code>. The remaining <code>[C]</code> belongs to the right subtree.
-    Therefore, the root's left child is <code>B</code>, and its right child is <code>C</code>.</p>
+<p>The left subtree has three nodes, so the three preorder elements after root <code>A</code> belong to the left subtree:
+    <code>[B, D, E]</code>. The remaining <code>[C]</code> belongs to the right subtree.
+    Thus, the root's left child is <code>B</code>, and its right child is <code>C</code>.</p>
 </li>
 </ol>
 </details>
-<h2 id="1262-programming-exercises">12.6.2 &nbsp; Programming Exercises<a class="headerlink" href="#1262-programming-exercises" title="Permanent link">&para;</a></h2>
-<h3 id="1-exponentiation-by-squaring">1. &nbsp; Exponentiation by Squaring<a class="headerlink" href="#1-exponentiation-by-squaring" title="Permanent link">&para;</a></h3>
-<p>Given a real number <code>x</code> and an integer <code>n</code>, calculate <span class="arithmatex">\(x^n\)</span> without calling the language's built-in power function.
-Use recursive divide and conquer: halve the exponent at each step and reuse the result of the subproblem already calculated.
-This exercise defines <span class="arithmatex">\(x^0=1\)</span>, including when <code>x = 0</code>. When <code>n &lt; 0</code>, <code>x != 0</code> is guaranteed, and the answer can be transformed into <span class="arithmatex">\((1/x)^{-n}\)</span>.</p>
+<h2 id="1262-programming-exercises">12.6.2 &nbsp; Programming exercises<a class="headerlink" href="#1262-programming-exercises" title="Permanent link">&para;</a></h2>
+<h3 id="1-fast-exponentiation">1. &nbsp; Fast exponentiation<a class="headerlink" href="#1-fast-exponentiation" title="Permanent link">&para;</a></h3>
+<p>Given a real number <code>x</code> and an integer <code>n</code>, compute <span class="arithmatex">\(x^n\)</span> without calling a built-in power function.
+Use recursive divide and conquer: halve the exponent at each step and reuse the computed subproblem result.
+For this problem, <span class="arithmatex">\(x^0=1\)</span>, including when <code>x = 0</code>. When <code>n &lt; 0</code>, <code>x != 0</code> is guaranteed, and the answer can be expressed as <span class="arithmatex">\((1/x)^{-n}\)</span>.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>When n is 0, the answer is 1</li>
-<li>After calculating x to the power n // 2, store the result in half rather than making the recursive call a second time</li>
-<li>When n &lt; 0, first change x to 1 / x and then change n to -n; in C++ or Java, first convert n to a 64-bit integer to avoid overflow when negating the smallest 32-bit integer</li>
+<li>When n is 0, the answer is 1.</li>
+<li>After computing x to the power n // 2, save it in half instead of making the recursive call a second time.</li>
+<li>When n &lt; 0, first replace x with 1 / x and n with -n. In C++ or Java, you can convert n to a 64-bit integer first to avoid overflow when negating the smallest 32-bit integer.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/powx-n/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
@@ -5065,7 +5137,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 12.5 Summary"
+    aria-label="Previous: 12.5   Summary"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -5077,7 +5149,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        12.5 Summary
+        12.5 &nbsp; Summary
       </div>
     </div>
   </a>
@@ -5089,7 +5161,7 @@ aria-label="Footer"
   <a
     href="../../chapter_backtracking/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: Chapter 13. &amp;nbsp; Backtracking"
+    aria-label="Next: Chapter 13.   Backtracking"
     rel="next"
   >
     <div class="md-footer__title">
@@ -5114,7 +5186,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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diff --git a/en/chapter_divide_and_conquer/hanota_problem/index.html b/en/chapter_divide_and_conquer/hanota_problem/index.html
index 1834305f1..8fa5a3293 100644
--- a/en/chapter_divide_and_conquer/hanota_problem/index.html
+++ b/en/chapter_divide_and_conquer/hanota_problem/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>12.4 Tower of Hanoi Problem - Hello Algo</title>
+        <title>12.4 Tower of Hanoi - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
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+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
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+        <a href="#124-tower-of-hanoi" class="md-skip">
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@@ -119,7 +119,7 @@
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             d="M480 32c0-12.9-7.8-24.6-19.8-29.6s-25.7-2.2-34.9 6.9L381.7 53c-48 48-113.1 75-181 75H192 160 64c-35.3 0-64 28.7-64 64v96c0 35.3 28.7 64 64 64l0 128c0 17.7 14.3 32 32 32h64c17.7 0 32-14.3 32-32V352l8.7 0c67.9 0 133 27 181 75l43.6 43.6c9.2 9.2 22.9 11.9 34.9 6.9s19.8-16.6 19.8-29.6V300.4c18.6-8.8 32-32.5 32-60.4s-13.4-51.6-32-60.4V32zm-64 76.7V240 371.3C357.2 317.8 280.5 288 200.7 288H192V192h8.7c79.8 0 156.5-29.8 215.3-83.3z" />
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
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@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              12.4 &nbsp; Tower of Hanoi Problem
+              12.4 &nbsp; Tower of Hanoi
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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@@ -440,7 +440,7 @@
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
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@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3285,7 +3285,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3306,7 +3306,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3327,7 +3327,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3381,7 +3381,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3417,7 +3417,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3434,7 +3434,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3463,7 +3463,7 @@
   <a href="#1-considering-the-base-cases" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Considering the Base Cases
+        1. &nbsp; Considering the base cases
       </span>
     </span>
   </a>
@@ -3471,10 +3471,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#2-subproblem-decomposition" class="md-nav__link">
+  <a href="#2-dividing-the-problem" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Subproblem Decomposition
+        2. &nbsp; Dividing the problem
       </span>
     </span>
   </a>
@@ -3485,7 +3485,7 @@
   <a href="#3-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Code Implementation
+        3. &nbsp; Code implementation
       </span>
     </span>
   </a>
@@ -3515,7 +3515,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3542,7 +3542,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3608,7 +3608,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3629,7 +3629,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3650,7 +3650,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3677,7 +3677,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3704,7 +3704,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3731,7 +3731,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3758,7 +3758,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3785,7 +3785,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3855,7 +3855,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3876,7 +3876,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3897,7 +3897,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3924,7 +3924,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3951,7 +3951,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3978,7 +3978,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4005,7 +4005,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4032,7 +4032,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4059,7 +4059,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4086,7 +4086,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4152,7 +4152,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4173,7 +4173,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4194,7 +4194,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4221,7 +4221,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4248,7 +4248,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4275,7 +4275,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4302,7 +4302,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4329,7 +4329,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4389,7 +4389,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4410,7 +4410,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4431,7 +4431,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4458,7 +4458,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
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-        1. &nbsp; Considering the Base Cases
+        1. &nbsp; Considering the base cases
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 </li>
 
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-  <a href="#2-subproblem-decomposition" class="md-nav__link">
+  <a href="#2-dividing-the-problem" class="md-nav__link">
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-        2. &nbsp; Subproblem Decomposition
+        2. &nbsp; Dividing the problem
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-        3. &nbsp; Code Implementation
+        3. &nbsp; Code implementation
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 <!-- Page content -->
-<h1 id="124-tower-of-hanoi-problem">12.4 &nbsp; Tower of Hanoi Problem<a class="headerlink" href="#124-tower-of-hanoi-problem" title="Permanent link">&para;</a></h1>
-<p>In merge sort and building binary trees, we decompose the original problem into two subproblems, each half the size of the original problem. However, for the Tower of Hanoi problem, we adopt a different decomposition strategy.</p>
+<h1 id="124-tower-of-hanoi">12.4 &nbsp; Tower of Hanoi<a class="headerlink" href="#124-tower-of-hanoi" title="Permanent link">&para;</a></h1>
+<p>In merge sort and binary tree reconstruction, we divided the original problem into two subproblems, each half its size. The Tower of Hanoi uses a different division strategy.</p>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given three pillars, denoted as <code>A</code>, <code>B</code>, and <code>C</code>. Initially, pillar <code>A</code> has <span class="arithmatex">\(n\)</span> discs stacked on it, arranged from top to bottom in ascending order of size. Our task is to move these <span class="arithmatex">\(n\)</span> discs to pillar <code>C</code> while maintaining their original order (as shown in Figure 12-10). The following rules must be followed when moving the discs.</p>
+<p>Given three rods called <code>A</code>, <code>B</code>, and <code>C</code>, initially rod <code>A</code> holds <span class="arithmatex">\(n\)</span> disks arranged from smallest at the top to largest at the bottom. Move all <span class="arithmatex">\(n\)</span> disks to rod <code>C</code> while preserving their order, as shown in Figure 12-10. The following rules apply.</p>
 <ol>
-<li>A disc can only be taken from the top of one pillar and placed on top of another pillar.</li>
-<li>Only one disc can be moved at a time.</li>
-<li>A smaller disc must always be on top of a larger disc.</li>
+<li>A disk can only be removed from the top of one rod and placed on top of another.</li>
+<li>Only one disk may be moved at a time.</li>
+<li>A smaller disk must always be above a larger disk.</li>
 </ol>
 </div>
-<p><img alt="Example of the Tower of Hanoi problem" class="animation-figure" src="../hanota_problem.assets/hanota_example.png" /></p>
-<p align="center"> Figure 12-10 &nbsp; Example of the Tower of Hanoi problem </p>
+<p><img alt="Example of the Tower of Hanoi" class="animation-figure" src="../hanota_problem.assets/hanota_example.png" /></p>
+<p align="center"> Figure 12-10 &nbsp; Example of the Tower of Hanoi </p>
 
-<p><strong>We denote the Tower of Hanoi problem of size <span class="arithmatex">\(i\)</span> as <span class="arithmatex">\(f(i)\)</span></strong>. For example, <span class="arithmatex">\(f(3)\)</span> represents moving <span class="arithmatex">\(3\)</span> discs from <code>A</code> to <code>C</code>.</p>
-<h3 id="1-considering-the-base-cases">1. &nbsp; Considering the Base Cases<a class="headerlink" href="#1-considering-the-base-cases" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 12-11, for problem <span class="arithmatex">\(f(1)\)</span>, when there is only one disc, we can move it directly from <code>A</code> to <code>C</code>.</p>
+<p><strong>Let <span class="arithmatex">\(f(i)\)</span> denote a Tower of Hanoi problem of size <span class="arithmatex">\(i\)</span></strong>. For example, <span class="arithmatex">\(f(3)\)</span> means moving <span class="arithmatex">\(3\)</span> disks from <code>A</code> to <code>C</code>.</p>
+<h3 id="1-considering-the-base-cases">1. &nbsp; Considering the base cases<a class="headerlink" href="#1-considering-the-base-cases" title="Permanent link">&para;</a></h3>
+<p>For <span class="arithmatex">\(f(1)\)</span>, with just one disk, simply move it directly from <code>A</code> to <code>C</code>, as shown in Figure 12-11.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:2"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -4708,11 +4780,11 @@
 </div>
 <p align="center"> Figure 12-11 &nbsp; Solution for a problem of size 1 </p>
 
-<p>As shown in Figure 12-12, for problem <span class="arithmatex">\(f(2)\)</span>, when there are two discs, <strong>since we must always keep the smaller disc on top of the larger disc, we need to use <code>B</code> to assist in the move</strong>.</p>
+<p>For <span class="arithmatex">\(f(2)\)</span>, with two disks, <strong>we need to use <code>B</code> because the smaller disk must remain above the larger one</strong>, as shown in Figure 12-12.</p>
 <ol>
-<li>First, move the smaller disc from <code>A</code> to <code>B</code>.</li>
-<li>Then move the larger disc from <code>A</code> to <code>C</code>.</li>
-<li>Finally, move the smaller disc from <code>B</code> to <code>C</code>.</li>
+<li>Move the smaller top disk from <code>A</code> to <code>B</code>.</li>
+<li>Move the larger disk from <code>A</code> to <code>C</code>.</li>
+<li>Move the smaller disk from <code>B</code> to <code>C</code>.</li>
 </ol>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:4"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">&lt;1&gt;</label><label for="__tabbed_2_2">&lt;2&gt;</label><label for="__tabbed_2_3">&lt;3&gt;</label><label for="__tabbed_2_4">&lt;4&gt;</label></div>
 <div class="tabbed-content">
@@ -4732,14 +4804,14 @@
 </div>
 <p align="center"> Figure 12-12 &nbsp; Solution for a problem of size 2 </p>
 
-<p>The process of solving problem <span class="arithmatex">\(f(2)\)</span> can be summarized as: <strong>moving two discs from <code>A</code> to <code>C</code> with the help of <code>B</code></strong>. Here, <code>C</code> is called the target pillar, and <code>B</code> is called the buffer pillar.</p>
-<h3 id="2-subproblem-decomposition">2. &nbsp; Subproblem Decomposition<a class="headerlink" href="#2-subproblem-decomposition" title="Permanent link">&para;</a></h3>
-<p>For problem <span class="arithmatex">\(f(3)\)</span>, when there are three discs, the situation becomes slightly more complex.</p>
-<p>Since we already know the solutions to <span class="arithmatex">\(f(1)\)</span> and <span class="arithmatex">\(f(2)\)</span>, we can think from a divide and conquer perspective, <strong>treating the top two discs on <code>A</code> as a whole</strong>, and execute the steps shown in Figure 12-13. This successfully moves the three discs from <code>A</code> to <code>C</code>.</p>
+<p>We can summarize the solution to <span class="arithmatex">\(f(2)\)</span> as <strong>moving two disks from <code>A</code> to <code>C</code> using <code>B</code></strong>. Here, <code>C</code> is the destination rod and <code>B</code> is the auxiliary rod.</p>
+<h3 id="2-dividing-the-problem">2. &nbsp; Dividing the problem<a class="headerlink" href="#2-dividing-the-problem" title="Permanent link">&para;</a></h3>
+<p>For <span class="arithmatex">\(f(3)\)</span>, with three disks, the situation is a little more complicated.</p>
+<p>Since we know how to solve <span class="arithmatex">\(f(1)\)</span> and <span class="arithmatex">\(f(2)\)</span>, we can use divide and conquer: <strong>treat the top two disks on <code>A</code> as a group</strong> and follow the steps in Figure 12-13. This moves all three disks from <code>A</code> to <code>C</code>.</p>
 <ol>
-<li>Let <code>B</code> be the target pillar and <code>C</code> be the buffer pillar, and move two discs from <code>A</code> to <code>B</code>.</li>
-<li>Move the remaining disc from <code>A</code> directly to <code>C</code>.</li>
-<li>Let <code>C</code> be the target pillar and <code>A</code> be the buffer pillar, and move two discs from <code>B</code> to <code>C</code>.</li>
+<li>Use <code>B</code> as the destination and <code>C</code> as the auxiliary rod to move two disks from <code>A</code> to <code>B</code>.</li>
+<li>Move the remaining disk directly from <code>A</code> to <code>C</code>.</li>
+<li>Use <code>C</code> as the destination and <code>A</code> as the auxiliary rod to move two disks from <code>B</code> to <code>C</code>.</li>
 </ol>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:4"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">&lt;1&gt;</label><label for="__tabbed_3_2">&lt;2&gt;</label><label for="__tabbed_3_3">&lt;3&gt;</label><label for="__tabbed_3_4">&lt;4&gt;</label></div>
 <div class="tabbed-content">
@@ -4759,38 +4831,38 @@
 </div>
 <p align="center"> Figure 12-13 &nbsp; Solution for a problem of size 3 </p>
 
-<p>Essentially, <strong>we divide problem <span class="arithmatex">\(f(3)\)</span> into two subproblems <span class="arithmatex">\(f(2)\)</span> and one subproblem <span class="arithmatex">\(f(1)\)</span></strong>. By solving these three subproblems in order, the original problem is solved. This shows that the subproblems are independent and their solutions can be merged.</p>
-<p>From this, we can summarize the divide and conquer strategy for solving the Tower of Hanoi problem shown in Figure 12-14: divide the original problem <span class="arithmatex">\(f(n)\)</span> into two subproblems <span class="arithmatex">\(f(n-1)\)</span> and one subproblem <span class="arithmatex">\(f(1)\)</span>, and solve these three subproblems in the following order.</p>
+<p>In essence, <strong>we have divided <span class="arithmatex">\(f(3)\)</span> into two <span class="arithmatex">\(f(2)\)</span> subproblems and one <span class="arithmatex">\(f(1)\)</span> subproblem</strong>. Solving these three subproblems in order solves the original problem. This shows that the subproblems are independent and their solutions can be combined.</p>
+<p>We can now state the divide-and-conquer strategy for the Tower of Hanoi, illustrated in Figure 12-14: divide <span class="arithmatex">\(f(n)\)</span> into two subproblems <span class="arithmatex">\(f(n-1)\)</span> and one subproblem <span class="arithmatex">\(f(1)\)</span>, then solve them in this order.</p>
 <ol>
-<li>Move <span class="arithmatex">\(n-1\)</span> discs from <code>A</code> to <code>B</code> with the help of <code>C</code>.</li>
-<li>Move the remaining <span class="arithmatex">\(1\)</span> disc directly from <code>A</code> to <code>C</code>.</li>
-<li>Move <span class="arithmatex">\(n-1\)</span> discs from <code>B</code> to <code>C</code> with the help of <code>A</code>.</li>
+<li>Move <span class="arithmatex">\(n-1\)</span> disks from <code>A</code> to <code>B</code> using <code>C</code>.</li>
+<li>Move the remaining <span class="arithmatex">\(1\)</span> disk directly from <code>A</code> to <code>C</code>.</li>
+<li>Move <span class="arithmatex">\(n-1\)</span> disks from <code>B</code> to <code>C</code> using <code>A</code>.</li>
 </ol>
-<p>For these two subproblems <span class="arithmatex">\(f(n-1)\)</span>, <strong>we can recursively divide them in the same way</strong> until reaching the smallest subproblem <span class="arithmatex">\(f(1)\)</span>. The solution to <span class="arithmatex">\(f(1)\)</span> is known and requires only one move operation.</p>
-<p><img alt="Divide and conquer strategy for solving the Tower of Hanoi problem" class="animation-figure" src="../hanota_problem.assets/hanota_divide_and_conquer.png" /></p>
-<p align="center"> Figure 12-14 &nbsp; Divide and conquer strategy for solving the Tower of Hanoi problem </p>
+<p>Each of the two <span class="arithmatex">\(f(n-1)\)</span> subproblems <strong>can be divided recursively in the same way</strong> until we reach the smallest subproblem, <span class="arithmatex">\(f(1)\)</span>. The solution to <span class="arithmatex">\(f(1)\)</span> is a single move.</p>
+<p><img alt="Divide-and-conquer strategy for the Tower of Hanoi" class="animation-figure" src="../hanota_problem.assets/hanota_divide_and_conquer.png" /></p>
+<p align="center"> Figure 12-14 &nbsp; Divide-and-conquer strategy for the Tower of Hanoi </p>
 
-<h3 id="3-code-implementation">3. &nbsp; Code Implementation<a class="headerlink" href="#3-code-implementation" title="Permanent link">&para;</a></h3>
-<p>In the code, we declare a recursive function <code>dfs(i, src, buf, tar)</code>, whose purpose is to move the top <span class="arithmatex">\(i\)</span> discs from pillar <code>src</code> to target pillar <code>tar</code> with the help of buffer pillar <code>buf</code>:</p>
+<h3 id="3-code-implementation">3. &nbsp; Code implementation<a class="headerlink" href="#3-code-implementation" title="Permanent link">&para;</a></h3>
+<p>We define a recursive function <code>dfs(i, src, buf, tar)</code> that moves the top <span class="arithmatex">\(i\)</span> disks from rod <code>src</code> to destination rod <code>tar</code> using auxiliary rod <code>buf</code>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:13"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">hanota.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">move</span><span class="p">(</span><span class="n">src</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">tar</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]):</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Move a disk"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Take out a disk from the top of src</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Move one disk"""</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Take a disk from the top of src</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="n">pan</span> <span class="o">=</span> <span class="n">src</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># Place the disk on top of tar</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="n">tar</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pan</span><span class="p">)</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">src</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">buf</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">tar</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]):</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="w">    </span><span class="sd">"""Solve the Tower of Hanoi problem f(i)"""</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="c1"># If there is only one disk left in src, move it directly to tar</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="c1"># If this subproblem has only one disk to move, move it directly from src to tar</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span> <span class="n">tar</span><span class="p">)</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="k">return</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>    <span class="c1"># Subproblem f(i-1): move the top i-1 disks from src to buf using tar</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="n">dfs</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">src</span><span class="p">,</span> <span class="n">tar</span><span class="p">,</span> <span class="n">buf</span><span class="p">)</span>
-<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>    <span class="c1"># Subproblem f(1): move the remaining disk from src to tar</span>
+<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>    <span class="c1"># Subproblem f(1): move the remaining disk of this subproblem from src to tar</span>
 <a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>    <span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span> <span class="n">tar</span><span class="p">)</span>
 <a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>    <span class="c1"># Subproblem f(i-1): move the top i-1 disks from buf to tar using src</span>
 <a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>    <span class="n">dfs</span><span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">buf</span><span class="p">,</span> <span class="n">src</span><span class="p">,</span> <span class="n">tar</span><span class="p">)</span>
@@ -4803,9 +4875,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hanota.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Move a disk */</span>
+<div class="highlight"><span class="filename">hanota.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Move one disk */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">move</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">tar</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Take out a disk from the top of src</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Take a disk from the top of src</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">pan</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">src</span><span class="p">.</span><span class="n">back</span><span class="p">();</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="n">src</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Place the disk on top of tar</span>
@@ -4814,14 +4886,14 @@
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="cm">/* Solve the Tower of Hanoi problem f(i) */</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="kt">void</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">tar</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="c1">// If there is only one disk left in src, move it directly to tar</span>
+<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="c1">// If only one disk remains to be moved from src, move it directly to tar</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">        </span><span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">);</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from src to buf using tar</span>
 <a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">);</span>
-<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk from src to tar</span>
+<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk of this subproblem from src to tar</span>
 <a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">    </span><span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">);</span>
 <a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from buf to tar using src</span>
 <a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">);</span>
@@ -4836,9 +4908,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hanota.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Move a disk */</span>
+<div class="highlight"><span class="filename">hanota.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Move one disk */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">move</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">tar</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Take out a disk from the top of src</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Take a disk from the top of src</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="n">Integer</span><span class="w"> </span><span class="n">pan</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">src</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">src</span><span class="p">.</span><span class="na">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="c1">// Place the disk on top of tar</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="n">tar</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">pan</span><span class="p">);</span>
@@ -4846,14 +4918,14 @@
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="cm">/* Solve the Tower of Hanoi problem f(i) */</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="kt">void</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">tar</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="c1">// If there is only one disk left in src, move it directly to tar</span>
+<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="c1">// If only one disk remains to be moved from src, move it directly to tar</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">);</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from src to buf using tar</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">);</span>
-<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk from src to tar</span>
+<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk of this subproblem from src to tar</span>
 <a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">    </span><span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">);</span>
 <a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from buf to tar using src</span>
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">);</span>
@@ -4868,9 +4940,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hanota.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Move a disk */</span>
+<div class="highlight"><span class="filename">hanota.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Move one disk */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Move</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">tar</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Take out a disk from the top of src</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Take a disk from the top of src</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">pan</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">src</span><span class="p">[</span><span class="o">^</span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="n">src</span><span class="p">.</span><span class="n">RemoveAt</span><span class="p">(</span><span class="n">src</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Place the disk on top of tar</span>
@@ -4879,14 +4951,14 @@
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="cm">/* Solve the Tower of Hanoi problem f(i) */</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="k">void</span><span class="w"> </span><span class="nf">DFS</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">tar</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="c1">// If there is only one disk left in src, move it directly to tar</span>
+<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="c1">// If only one disk in this subproblem remains on src, move it directly to tar</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">        </span><span class="n">Move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">);</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from src to buf using tar</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">    </span><span class="n">DFS</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">);</span>
-<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk from src to tar</span>
+<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk of this subproblem from src to tar</span>
 <a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">    </span><span class="n">Move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">);</span>
 <a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from buf to tar using src</span>
 <a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w">    </span><span class="n">DFS</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">);</span>
@@ -4901,26 +4973,26 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hanota.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Move a disk */</span>
+<div class="highlight"><span class="filename">hanota.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Move one disk */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">move</span><span class="p">(</span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="w"> </span><span class="o">*</span><span class="nx">list</span><span class="p">.</span><span class="nx">List</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Take out a disk from the top of src</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Take a disk from the top of src</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">pan</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">src</span><span class="p">.</span><span class="nx">Back</span><span class="p">()</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="c1">// Place the disk on top of tar</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="nx">tar</span><span class="p">.</span><span class="nx">PushBack</span><span class="p">(</span><span class="nx">pan</span><span class="p">.</span><span class="nx">Value</span><span class="p">)</span>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// Remove top disk from src</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// Remove the disk from the top of src</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="nx">src</span><span class="p">.</span><span class="nx">Remove</span><span class="p">(</span><span class="nx">pan</span><span class="p">)</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="p">}</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="cm">/* Solve the Tower of Hanoi problem f(i) */</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="kd">func</span><span class="w"> </span><span class="nx">dfsHanota</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">buf</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="w"> </span><span class="o">*</span><span class="nx">list</span><span class="p">.</span><span class="nx">List</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="c1">// If there is only one disk left in src, move it directly to tar</span>
+<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="c1">// If only one disk in this subproblem remains on src, move it directly to tar</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="nx">move</span><span class="p">(</span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="p">)</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from src to buf using tar</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">    </span><span class="nx">dfsHanota</span><span class="p">(</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="p">,</span><span class="w"> </span><span class="nx">buf</span><span class="p">)</span>
-<a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk from src to tar</span>
+<a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk of this subproblem from src to tar</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">    </span><span class="nx">move</span><span class="p">(</span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="p">)</span>
 <a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from buf to tar using src</span>
 <a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w">    </span><span class="nx">dfsHanota</span><span class="p">(</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">buf</span><span class="p">,</span><span class="w"> </span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="p">)</span>
@@ -4935,9 +5007,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hanota.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Move a disk */</span>
+<div class="highlight"><span class="filename">hanota.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Move one disk */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">move</span><span class="p">(</span><span class="n">src</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">tar</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Take out a disk from the top of src</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Take a disk from the top of src</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">pan</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">src</span><span class="p">.</span><span class="n">popLast</span><span class="p">()</span><span class="o">!</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="c1">// Place the disk on top of tar</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="n">tar</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">pan</span><span class="p">)</span>
@@ -4945,14 +5017,14 @@
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="cm">/* Solve the Tower of Hanoi problem f(i) */</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="kd">func</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">buf</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">tar</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">    </span><span class="c1">// If there is only one disk left in src, move it directly to tar</span>
+<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">    </span><span class="c1">// If only one disk in this subproblem remains on src, move it directly to tar</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">tar</span><span class="p">)</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from src to buf using tar</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">tar</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">buf</span><span class="p">)</span>
-<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk from src to tar</span>
+<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk of this subproblem from src to tar</span>
 <a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">    </span><span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">tar</span><span class="p">)</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from buf to tar using src</span>
 <a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">tar</span><span class="p">)</span>
@@ -4961,16 +5033,16 @@
 <a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="cm">/* Solve the Tower of Hanoi problem */</span>
 <a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a><span class="kd">func</span><span class="w"> </span><span class="nf">solveHanota</span><span class="p">(</span><span class="n">A</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">B</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">C</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-26" name="__codelineno-5-26" href="#__codelineno-5-26"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">A</span><span class="p">.</span><span class="bp">count</span>
-<a id="__codelineno-5-27" name="__codelineno-5-27" href="#__codelineno-5-27"></a><span class="w">    </span><span class="c1">// The tail of the list is the top of the rod</span>
-<a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="w">    </span><span class="c1">// Move top n disks from src to C using B</span>
+<a id="__codelineno-5-27" name="__codelineno-5-27" href="#__codelineno-5-27"></a><span class="w">    </span><span class="c1">// The end of each list represents the top of the peg</span>
+<a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="w">    </span><span class="c1">// Move the top n disks from A to C using B</span>
 <a id="__codelineno-5-29" name="__codelineno-5-29" href="#__codelineno-5-29"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">A</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">B</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">C</span><span class="p">)</span>
 <a id="__codelineno-5-30" name="__codelineno-5-30" href="#__codelineno-5-30"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hanota.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Move a disk */</span>
+<div class="highlight"><span class="filename">hanota.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Move one disk */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">move</span><span class="p">(</span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Take out a disk from the top of src</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Take a disk from the top of src</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">pan</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">src</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="c1">// Place the disk on top of tar</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="nx">tar</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">pan</span><span class="p">);</span>
@@ -4978,14 +5050,14 @@
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="cm">/* Solve the Tower of Hanoi problem f(i) */</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="kd">function</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">buf</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">    </span><span class="c1">// If there is only one disk left in src, move it directly to tar</span>
+<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">    </span><span class="c1">// If only one disk remains to be moved from src, move it directly to tar</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="nx">move</span><span class="p">(</span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="p">);</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from src to buf using tar</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">    </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="p">,</span><span class="w"> </span><span class="nx">buf</span><span class="p">);</span>
-<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk from src to tar</span>
+<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk of this subproblem from src to tar</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">    </span><span class="nx">move</span><span class="p">(</span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="p">);</span>
 <a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from buf to tar using src</span>
 <a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">    </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">buf</span><span class="p">,</span><span class="w"> </span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="p">);</span>
@@ -5000,9 +5072,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hanota.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Move a disk */</span>
+<div class="highlight"><span class="filename">hanota.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Move one disk */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">move</span><span class="p">(</span><span class="nx">src</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">tar</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Take out a disk from the top of src</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Take a disk from the top of src</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">pan</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">src</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="c1">// Place the disk on top of tar</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="nx">tar</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">pan</span><span class="p">);</span>
@@ -5010,14 +5082,14 @@
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="cm">/* Solve the Tower of Hanoi problem f(i) */</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="kd">function</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">src</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">buf</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">tar</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="c1">// If there is only one disk left in src, move it directly to tar</span>
+<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="c1">// If only one disk remains to be moved from src, move it directly to tar</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="nx">move</span><span class="p">(</span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="p">);</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from src to buf using tar</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">    </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="p">,</span><span class="w"> </span><span class="nx">buf</span><span class="p">);</span>
-<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk from src to tar</span>
+<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk of this subproblem from src to tar</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">    </span><span class="nx">move</span><span class="p">(</span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="p">);</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from buf to tar using src</span>
 <a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">    </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">buf</span><span class="p">,</span><span class="w"> </span><span class="nx">src</span><span class="p">,</span><span class="w"> </span><span class="nx">tar</span><span class="p">);</span>
@@ -5032,9 +5104,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hanota.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Move a disk */</span>
+<div class="highlight"><span class="filename">hanota.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Move one disk */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">move</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">tar</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Take out a disk from the top of src</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Take a disk from the top of src</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">pan</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">src</span><span class="p">.</span><span class="n">removeLast</span><span class="p">();</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// Place the disk on top of tar</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="n">tar</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">pan</span><span class="p">);</span>
@@ -5042,14 +5114,14 @@
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="cm">/* Solve the Tower of Hanoi problem f(i) */</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="kt">void</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">tar</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="c1">// If there is only one disk left in src, move it directly to tar</span>
+<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="c1">// If only one disk remains to be moved from src, move it directly to tar</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">    </span><span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">);</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">  </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from src to buf using tar</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">  </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">);</span>
-<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">  </span><span class="c1">// Subproblem f(1): move the remaining disk from src to tar</span>
+<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">  </span><span class="c1">// Subproblem f(1): move the remaining disk of this subproblem from src to tar</span>
 <a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">  </span><span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">);</span>
 <a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">  </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from buf to tar using src</span>
 <a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="w">  </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">);</span>
@@ -5064,9 +5136,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hanota.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Move a disk */</span>
+<div class="highlight"><span class="filename">hanota.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Move one disk */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">move_pan</span><span class="p">(</span><span class="n">src</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Take out a disk from the top of src</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Take a disk from the top of src</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">pan</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">src</span><span class="p">.</span><span class="n">pop</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="c1">// Place the disk on top of tar</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="n">tar</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">pan</span><span class="p">);</span>
@@ -5074,14 +5146,14 @@
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="cm">/* Solve the Tower of Hanoi problem f(i) */</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="k">fn</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="c1">// If there is only one disk left in src, move it directly to tar</span>
+<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="c1">// If only one disk remains to be moved from src, move it directly to tar</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">        </span><span class="n">move_pan</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">);</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from src to buf using tar</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">);</span>
-<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk from src to tar</span>
+<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk of this subproblem from src to tar</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">    </span><span class="n">move_pan</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">);</span>
 <a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from buf to tar using src</span>
 <a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">);</span>
@@ -5096,9 +5168,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hanota.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Move a disk */</span>
+<div class="highlight"><span class="filename">hanota.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Move one disk */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">move</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">srcSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">tar</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">tarSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Take out a disk from the top of src</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Take a disk from the top of src</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">pan</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">src</span><span class="p">[</span><span class="o">*</span><span class="n">srcSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="n">src</span><span class="p">[</span><span class="o">*</span><span class="n">srcSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="p">(</span><span class="o">*</span><span class="n">srcSize</span><span class="p">)</span><span class="o">--</span><span class="p">;</span>
@@ -5109,14 +5181,14 @@
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="cm">/* Solve the Tower of Hanoi problem f(i) */</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="kt">void</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">srcSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">bufSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">tar</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">tarSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="c1">// If there is only one disk left in src, move it directly to tar</span>
+<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="c1">// If only one disk remains to be moved from src, move it directly to tar</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">srcSize</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">,</span><span class="w"> </span><span class="n">tarSize</span><span class="p">);</span>
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from src to buf using tar</span>
 <a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">srcSize</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">,</span><span class="w"> </span><span class="n">tarSize</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="n">bufSize</span><span class="p">);</span>
-<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk from src to tar</span>
+<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk of this subproblem from src to tar</span>
 <a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">    </span><span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">srcSize</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">,</span><span class="w"> </span><span class="n">tarSize</span><span class="p">);</span>
 <a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from buf to tar using src</span>
 <a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="n">bufSize</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">srcSize</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">,</span><span class="w"> </span><span class="n">tarSize</span><span class="p">);</span>
@@ -5130,9 +5202,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hanota.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Move a disk */</span>
+<div class="highlight"><span class="filename">hanota.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Move one disk */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">move</span><span class="p">(</span><span class="n">src</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Take out a disk from the top of src</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Take a disk from the top of src</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">pan</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">src</span><span class="p">.</span><span class="na">removeAt</span><span class="p">(</span><span class="n">src</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="c1">// Place the disk on top of tar</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="n">tar</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">pan</span><span class="p">)</span>
@@ -5140,14 +5212,14 @@
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="cm">/* Solve the Tower of Hanoi problem f(i) */</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="c1">// If there is only one disk left in src, move it directly to tar</span>
+<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="c1">// If only one disk in this subproblem remains on src, move it directly to tar</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">)</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from src to buf using tar</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">)</span>
-<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk from src to tar</span>
+<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">    </span><span class="c1">// Subproblem f(1): move the remaining disk of this subproblem from src to tar</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">    </span><span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">)</span>
 <a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">    </span><span class="c1">// Subproblem f(i-1): move the top i-1 disks from buf to tar using src</span>
 <a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">)</span>
@@ -5164,15 +5236,15 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">hanota.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Move one disk ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Take out a disk from the top of src</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Take a disk from the top of src</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="n">pan</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">src</span><span class="o">.</span><span class="n">pop</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Place the disk on top of tar</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="n">tar</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">pan</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="k">end</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a>
-<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="c1">### Solve Tower of Hanoi f(i) ###</span>
+<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="c1">### Solve the Tower of Hanoi problem f(i) ###</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">)</span>
-<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">  </span><span class="c1"># If there is only one disk left in src, move it directly to tar</span>
+<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">  </span><span class="c1"># If only one disk remains to be moved from src, move it directly to tar</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">)</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="k">return</span>
@@ -5180,13 +5252,13 @@
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">  </span><span class="c1"># Subproblem f(i-1): move the top i-1 disks from src to buf using tar</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">  </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">)</span>
-<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">  </span><span class="c1"># Subproblem f(1): move the remaining disk from src to tar</span>
+<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">  </span><span class="c1"># Subproblem f(1): move the remaining disk in the current subproblem from src to tar</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="w">  </span><span class="n">move</span><span class="p">(</span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">)</span>
 <a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">  </span><span class="c1"># Subproblem f(i-1): move the top i-1 disks from buf to tar using src</span>
 <a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w">  </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">buf</span><span class="p">,</span><span class="w"> </span><span class="n">src</span><span class="p">,</span><span class="w"> </span><span class="n">tar</span><span class="p">)</span>
 <a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="k">end</span>
 <a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a>
-<a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="c1">### Solve Tower of Hanoi ###</span>
+<a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="c1">### Solve the Tower of Hanoi problem ###</span>
 <a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></a><span class="k">def</span><span class="w"> </span><span class="nf">solve_hanota</span><span class="p">(</span><span class="n">_A</span><span class="p">,</span><span class="w"> </span><span class="n">_B</span><span class="p">,</span><span class="w"> </span><span class="n">_C</span><span class="p">)</span>
 <a id="__codelineno-12-27" name="__codelineno-12-27" href="#__codelineno-12-27"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_A</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-12-28" name="__codelineno-12-28" href="#__codelineno-12-28"></a><span class="w">  </span><span class="c1"># Move the top n disks from A to C using B</span>
@@ -5196,14 +5268,19 @@
 </div>
 </div>
 </div>
-<p>As shown in Figure 12-15, the Tower of Hanoi problem forms a recursion tree of height <span class="arithmatex">\(n\)</span>, where each node represents a subproblem corresponding to an invocation of the <code>dfs()</code> function, <strong>therefore the time complexity is <span class="arithmatex">\(O(2^n)\)</span> and the space complexity is <span class="arithmatex">\(O(n)\)</span></strong>.</p>
-<p><img alt="Recursion tree of the Tower of Hanoi problem" class="animation-figure" src="../hanota_problem.assets/hanota_recursive_tree.png" /></p>
-<p align="center"> Figure 12-15 &nbsp; Recursion tree of the Tower of Hanoi problem </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20move%28src%3A%20list%5Bint%5D,%20tar%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Move%20one%20disk%22%22%22%0A%20%20%20%20%23%20Take%20a%20disk%20from%20the%20top%20of%20src%0A%20%20%20%20pan%20%3D%20src.pop%28%29%0A%20%20%20%20%23%20Place%20the%20disk%20on%20top%20of%20tar%0A%20%20%20%20tar.append%28pan%29%0A%0A%0Adef%20dfs%28i%3A%20int,%20src%3A%20list%5Bint%5D,%20buf%3A%20list%5Bint%5D,%20tar%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Solve%20the%20Tower%20of%20Hanoi%20problem%20f%28i%29%22%22%22%0A%20%20%20%20%23%20If%20the%20current%20subproblem%20has%20only%20one%20disk,%20move%20it%20from%20src%20directly%20to%20tar%0A%20%20%20%20if%20i%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20move%28src,%20tar%29%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Subproblem%20f%28i-1%29%3A%20move%20the%20top%20i-1%20disks%20from%20src%20to%20buf%20using%20tar%0A%20%20%20%20dfs%28i%20-%201,%20src,%20tar,%20buf%29%0A%20%20%20%20%23%20Subproblem%20f%281%29%3A%20move%20the%20remaining%20disk%20of%20this%20subproblem%20from%20src%20to%20tar%0A%20%20%20%20move%28src,%20tar%29%0A%20%20%20%20%23%20Subproblem%20f%28i-1%29%3A%20move%20the%20top%20i-1%20disks%20from%20buf%20to%20tar%20using%20src%0A%20%20%20%20dfs%28i%20-%201,%20buf,%20src,%20tar%29%0A%0A%0Adef%20solve_hanota%28A%3A%20list%5Bint%5D,%20B%3A%20list%5Bint%5D,%20C%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Solve%20the%20Tower%20of%20Hanoi%20problem%22%22%22%0A%20%20%20%20n%20%3D%20len%28A%29%0A%20%20%20%20%23%20Move%20the%20top%20n%20disks%20from%20A%20to%20C%20using%20B%0A%20%20%20%20dfs%28n,%20A,%20B,%20C%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20The%20end%20of%20each%20list%20represents%20the%20top%20of%20the%20peg%0A%20%20%20%20A%20%3D%20%5B5,%204,%203,%202,%201%5D%0A%20%20%20%20B%20%3D%20%5B%5D%0A%20%20%20%20C%20%3D%20%5B%5D%0A%20%20%20%20print%28%22Initial%20state%3A%22%29%0A%20%20%20%20print%28f%22A%20%3D%20%7BA%7D%22%29%0A%20%20%20%20print%28f%22B%20%3D%20%7BB%7D%22%29%0A%20%20%20%20print%28f%22C%20%3D%20%7BC%7D%22%29%0A%0A%20%20%20%20solve_hanota%28A,%20B,%20C%29%0A%0A%20%20%20%20print%28%22After%20moving%20the%20disks%3A%22%29%0A%20%20%20%20print%28f%22A%20%3D%20%7BA%7D%22%29%0A%20%20%20%20print%28f%22B%20%3D%20%7BB%7D%22%29%0A%20%20%20%20print%28f%22C%20%3D%20%7BC%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=12&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20move%28src%3A%20list%5Bint%5D,%20tar%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Move%20one%20disk%22%22%22%0A%20%20%20%20%23%20Take%20a%20disk%20from%20the%20top%20of%20src%0A%20%20%20%20pan%20%3D%20src.pop%28%29%0A%20%20%20%20%23%20Place%20the%20disk%20on%20top%20of%20tar%0A%20%20%20%20tar.append%28pan%29%0A%0A%0Adef%20dfs%28i%3A%20int,%20src%3A%20list%5Bint%5D,%20buf%3A%20list%5Bint%5D,%20tar%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Solve%20the%20Tower%20of%20Hanoi%20problem%20f%28i%29%22%22%22%0A%20%20%20%20%23%20If%20the%20current%20subproblem%20has%20only%20one%20disk,%20move%20it%20from%20src%20directly%20to%20tar%0A%20%20%20%20if%20i%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20move%28src,%20tar%29%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Subproblem%20f%28i-1%29%3A%20move%20the%20top%20i-1%20disks%20from%20src%20to%20buf%20using%20tar%0A%20%20%20%20dfs%28i%20-%201,%20src,%20tar,%20buf%29%0A%20%20%20%20%23%20Subproblem%20f%281%29%3A%20move%20the%20remaining%20disk%20of%20this%20subproblem%20from%20src%20to%20tar%0A%20%20%20%20move%28src,%20tar%29%0A%20%20%20%20%23%20Subproblem%20f%28i-1%29%3A%20move%20the%20top%20i-1%20disks%20from%20buf%20to%20tar%20using%20src%0A%20%20%20%20dfs%28i%20-%201,%20buf,%20src,%20tar%29%0A%0A%0Adef%20solve_hanota%28A%3A%20list%5Bint%5D,%20B%3A%20list%5Bint%5D,%20C%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Solve%20the%20Tower%20of%20Hanoi%20problem%22%22%22%0A%20%20%20%20n%20%3D%20len%28A%29%0A%20%20%20%20%23%20Move%20the%20top%20n%20disks%20from%20A%20to%20C%20using%20B%0A%20%20%20%20dfs%28n,%20A,%20B,%20C%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20The%20end%20of%20each%20list%20represents%20the%20top%20of%20the%20peg%0A%20%20%20%20A%20%3D%20%5B5,%204,%203,%202,%201%5D%0A%20%20%20%20B%20%3D%20%5B%5D%0A%20%20%20%20C%20%3D%20%5B%5D%0A%20%20%20%20print%28%22Initial%20state%3A%22%29%0A%20%20%20%20print%28f%22A%20%3D%20%7BA%7D%22%29%0A%20%20%20%20print%28f%22B%20%3D%20%7BB%7D%22%29%0A%20%20%20%20print%28f%22C%20%3D%20%7BC%7D%22%29%0A%0A%20%20%20%20solve_hanota%28A,%20B,%20C%29%0A%0A%20%20%20%20print%28%22After%20moving%20the%20disks%3A%22%29%0A%20%20%20%20print%28f%22A%20%3D%20%7BA%7D%22%29%0A%20%20%20%20print%28f%22B%20%3D%20%7BB%7D%22%29%0A%20%20%20%20print%28f%22C%20%3D%20%7BC%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=12&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>As shown in Figure 12-15, the Tower of Hanoi forms a recursion tree of height <span class="arithmatex">\(n\)</span>. Each node represents a subproblem and corresponds to a call to <code>dfs()</code>. <strong>The time complexity is therefore <span class="arithmatex">\(O(2^n)\)</span>, and the space complexity is <span class="arithmatex">\(O(n)\)</span></strong>.</p>
+<p><img alt="Recursion tree for the Tower of Hanoi" class="animation-figure" src="../hanota_problem.assets/hanota_recursive_tree.png" /></p>
+<p align="center"> Figure 12-15 &nbsp; Recursion tree for the Tower of Hanoi </p>
 
 <div class="admonition quote">
 <p class="admonition-title">Quote</p>
-<p>The Tower of Hanoi problem originates from an ancient legend. In a temple in ancient India, monks had three tall diamond pillars and <span class="arithmatex">\(64\)</span> golden discs of different sizes. The monks continuously moved the discs, believing that when the last disc was correctly placed, the world would come to an end.</p>
-<p>However, even if the monks moved one disc per second, it would take approximately <span class="arithmatex">\(2^{64} \approx 1.84×10^{19}\)</span> seconds, which is about <span class="arithmatex">\(585\)</span> billion years, far exceeding current estimates of the age of the universe. Therefore, if this legend is true, we should not need to worry about the end of the world.</p>
+<p>The Tower of Hanoi comes from an ancient legend. In a temple in ancient India, monks had three tall diamond rods and <span class="arithmatex">\(64\)</span> gold disks of different sizes. They continually moved the disks, believing that the world would end when the last disk was placed in its correct position.</p>
+<p>Even at one move per second, however, this would take about <span class="arithmatex">\(2^{64} \approx 1.84×10^{19}\)</span> seconds, or roughly <span class="arithmatex">\(585\)</span> billion years, far longer than current estimates of the age of the universe. If the legend is true, we should have little reason to worry about the end of the world.</p>
 </div>
 
 <!-- Source file information -->
@@ -5227,7 +5304,7 @@ aria-label="Footer"
   <a
     href="../build_binary_tree_problem/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 12.3 Building a Binary Tree Problem"
+    aria-label="Previous: 12.3   Building a binary tree"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -5239,7 +5316,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        12.3 Building a Binary Tree Problem
+        12.3 &nbsp; Building a binary tree
       </div>
     </div>
   </a>
@@ -5251,7 +5328,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 12.5 Summary"
+    aria-label="Next: 12.5   Summary"
     rel="next"
   >
     <div class="md-footer__title">
@@ -5259,7 +5336,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        12.5 Summary
+        12.5 &nbsp; Summary
       </div>
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     <div class="md-footer__button md-icon">
@@ -5276,7 +5353,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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diff --git a/en/chapter_divide_and_conquer/index.html b/en/chapter_divide_and_conquer/index.html
index 280c76e94..1d6051808 100644
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>Chapter 12. Divide and Conquer - Hello Algo</title>
+        <title>Chapter 12. Divide and conquer - Hello Algo</title>
 
 
 
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-              Chapter 12. &nbsp; Divide and Conquer
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-  Before Starting
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-  Before Starting
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
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-  Chapter 1. Encounter with Algorithms
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
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-  1.3 Summary
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-  2.2 Iteration and Recursion
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-  2.3 Time Complexity
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.5 Summary
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
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-  Chapter 3. Data Structures
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   <span class="md-ellipsis">
 
 
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-  3.2 Basic Data Types
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-  3.3 Number Encoding *
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
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-  4.5 Summary
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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@@ -1678,7 +1678,7 @@
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
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-  Chapter 6. Hashing
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   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
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   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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@@ -2378,7 +2378,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
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-  14.8 Exercises
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 <!-- Page content -->
-<h1 id="chapter-12-divide-and-conquer">Chapter 12. &nbsp; Divide and Conquer<a class="headerlink" href="#chapter-12-divide-and-conquer" title="Permanent link">&para;</a></h1>
+<h1 id="chapter-12-divide-and-conquer">Chapter 12. &nbsp; Divide and conquer<a class="headerlink" href="#chapter-12-divide-and-conquer" title="Permanent link">&para;</a></h1>
 <p><img alt="Divide and conquer" class="cover-image" src="../assets/covers/chapter_divide_and_conquer.jpg" /></p>
 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
-<p>Difficult problems are decomposed layer by layer, with each decomposition making them simpler.</p>
-<p>Divide and conquer reveals an important truth: start with what is simple, and nothing remains complex.</p>
+<p>A difficult problem is broken down layer by layer, becoming simpler with each division.</p>
+<p>Divide and conquer reveals an important truth: start with the simple parts, and complexity becomes manageable.</p>
 </div>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
-<li><a href="divide_and_conquer/">12.1 &nbsp; Divide and Conquer Algorithms</a></li>
-<li><a href="binary_search_recur/">12.2 &nbsp; Divide and Conquer Search Strategy</a></li>
-<li><a href="build_binary_tree_problem/">12.3 &nbsp; Building a Binary Tree Problem</a></li>
-<li><a href="hanota_problem/">12.4 &nbsp; Tower of Hanoi Problem</a></li>
+<li><a href="divide_and_conquer/">12.1 &nbsp; Divide and conquer</a></li>
+<li><a href="binary_search_recur/">12.2 &nbsp; Divide-and-conquer search</a></li>
+<li><a href="build_binary_tree_problem/">12.3 &nbsp; Building a binary tree</a></li>
+<li><a href="hanota_problem/">12.4 &nbsp; Tower of Hanoi</a></li>
 <li><a href="summary/">12.5 &nbsp; Summary</a></li>
 <li><a href="exercises/">12.6 &nbsp; Exercises</a></li>
 </ul>
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   <a
     href="../chapter_sorting/exercises/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 11.12 Exercises"
+    aria-label="Previous: 11.12   Exercises"
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@@ -4627,7 +4699,7 @@ aria-label="Footer"
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+        11.12 &nbsp; Exercises
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     href="./divide_and_conquer/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 12.1 Divide and Conquer Algorithms"
+    aria-label="Next: 12.1   Divide and conquer"
     rel="next"
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@@ -4647,7 +4719,7 @@ aria-label="Footer"
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       </span>
       <div class="md-ellipsis">
-        12.1 Divide and Conquer Algorithms
+        12.1 &nbsp; Divide and conquer
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     <div class="md-footer__button md-icon">
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
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-  2.6 Exercises
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   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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@@ -2287,7 +2287,7 @@
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3285,7 +3285,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3306,7 +3306,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3327,7 +3327,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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+  12.5 &nbsp; Summary
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4547,6 +4547,78 @@
 
 
 
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+  <a href="#1-key-takeaways" class="md-nav__link">
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-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
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   </a>
@@ -4636,17 +4708,17 @@
 
 <!-- Page content -->
 <h1 id="125-summary">12.5 &nbsp; Summary<a class="headerlink" href="#125-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
 <ul>
-<li>Divide and conquer is a common algorithm design strategy consisting of two phases, divide (partition) and conquer (merge), and is typically implemented recursively.</li>
-<li>The criteria for determining whether a problem is a divide and conquer problem include: whether the problem can be decomposed, whether subproblems are independent, and whether subproblems can be merged.</li>
-<li>Merge sort is a typical application of the divide and conquer strategy. It recursively divides an array into two equal-length subarrays until only one element remains, then merges them layer by layer to complete the sorting.</li>
-<li>Introducing the divide and conquer strategy can often improve algorithm efficiency. On one hand, it reduces the number of operations; on the other hand, it makes parallel optimization by the system easier.</li>
-<li>Divide and conquer can solve many algorithmic problems and is also widely used in data structures and algorithm design, making it ubiquitous.</li>
-<li>Compared to brute-force search, adaptive search is more efficient. Search algorithms with time complexity of <span class="arithmatex">\(O(\log n)\)</span> are typically implemented based on the divide and conquer strategy.</li>
-<li>Binary search is another typical application of divide and conquer. It does not include the step of merging solutions of subproblems. We can implement binary search through recursive divide and conquer.</li>
-<li>In the problem of building a binary tree, building the tree (original problem) can be divided into building the left subtree and right subtree (subproblems), which can be achieved by dividing the index intervals of the preorder and inorder traversals.</li>
-<li>In the Tower of Hanoi problem, a problem of size <span class="arithmatex">\(n\)</span> can be divided into two subproblems of size <span class="arithmatex">\(n-1\)</span> and one subproblem of size <span class="arithmatex">\(1\)</span>. After solving these three subproblems in order, the original problem is solved.</li>
+<li>Divide and conquer is a common algorithm design strategy with two stages: dividing the problem and combining solutions. It is usually implemented recursively.</li>
+<li>To recognize a divide-and-conquer problem, ask whether it can be divided, whether the subproblems are independent, and whether their solutions can be combined.</li>
+<li>Merge sort is a classic example. It recursively divides an array into two equally sized subarrays until each contains one element, then merges them level by level to complete the sort.</li>
+<li>Divide and conquer often improves efficiency by reducing the number of operations and making it easier to parallelize the work.</li>
+<li>Divide and conquer solves many algorithmic problems and appears widely in the design of data structures and algorithms.</li>
+<li>Adaptive search is more efficient than brute-force search. Search algorithms with <span class="arithmatex">\(O(\log n)\)</span> time complexity are usually based on divide and conquer.</li>
+<li>Binary search is another classic example, with no step for combining subproblem solutions. It can be implemented through recursive divide and conquer.</li>
+<li>Reconstructing a binary tree can be divided into building its left and right subtrees. This is achieved by partitioning index intervals in the preorder and inorder traversals.</li>
+<li>A Tower of Hanoi problem of size <span class="arithmatex">\(n\)</span> can be divided into two subproblems of size <span class="arithmatex">\(n-1\)</span> and one of size <span class="arithmatex">\(1\)</span>. Solving these three subproblems in order solves the original problem.</li>
 </ul>
 
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-  1.2 What Is an Algorithm
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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+  Chapter 10. &nbsp; Searching
 
 
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+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.12 Exercises
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-  12.2 Divide and Conquer Search Strategy
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+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
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-  Chapter 15. Greedy
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-  Chapter 16. Appendix
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-  16.2 Contributing Together
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-  16.3 Glossary
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     <span class="md-ellipsis">
       <span class="md-typeset">
-        14.2.1 &nbsp; Optimal Substructure
+        14.2.1 &nbsp; Optimal substructure
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 </li>
 
         <li class="md-nav__item">
-  <a href="#1422-no-aftereffects" class="md-nav__link">
+  <a href="#1422-no-aftereffect-property" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        14.2.2 &nbsp; No Aftereffects
+        14.2.2 &nbsp; No-aftereffect property
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 <!-- Page content -->
-<h1 id="142-characteristics-of-dynamic-programming-problems">14.2 &nbsp; Characteristics of Dynamic Programming Problems<a class="headerlink" href="#142-characteristics-of-dynamic-programming-problems" title="Permanent link">&para;</a></h1>
-<p>In the previous section, we learned how dynamic programming solves the original problem by decomposing it into subproblems. In fact, subproblem decomposition is a general algorithmic approach, with different emphases in divide and conquer, dynamic programming, and backtracking.</p>
+<h1 id="142-properties-of-dynamic-programming-problems">14.2 &nbsp; Properties of dynamic programming problems<a class="headerlink" href="#142-properties-of-dynamic-programming-problems" title="Permanent link">&para;</a></h1>
+<p>The previous section showed how dynamic programming solves a problem by decomposing it into subproblems. Decomposition is a general algorithmic idea, but divide and conquer, dynamic programming, and backtracking use it differently.</p>
 <ul>
-<li>Divide and conquer algorithms recursively divide the original problem into multiple independent subproblems until the smallest subproblems are reached, and merge the solutions to the subproblems during backtracking to ultimately obtain the solution to the original problem.</li>
-<li>Dynamic programming also recursively decomposes problems, but the main difference from divide and conquer algorithms is that subproblems in dynamic programming are interdependent, and many overlapping subproblems appear during the decomposition process.</li>
-<li>Backtracking algorithms enumerate all possible solutions through trial and error, and avoid unnecessary search branches through pruning. The solution to the original problem consists of a series of decision steps, and we can regard the subsequence before each decision step as a subproblem.</li>
+<li>Divide and conquer recursively splits a problem into independent subproblems until the smallest ones are reached, then combines their solutions as the recursion returns to solve the original problem.</li>
+<li>Dynamic programming also decomposes problems recursively. Its main difference from divide and conquer is that its subproblems depend on one another, and many overlapping subproblems appear during decomposition.</li>
+<li>Backtracking enumerates possible solutions by trying choices and undoing them, using pruning to avoid unnecessary branches. A solution consists of a sequence of decisions, and the prefix before each decision can be viewed as a subproblem.</li>
 </ul>
-<p>In fact, dynamic programming is commonly used to solve optimization problems, which not only contain overlapping subproblems but also have two other major characteristics: optimal substructure and no aftereffects.</p>
-<h2 id="1421-optimal-substructure">14.2.1 &nbsp; Optimal Substructure<a class="headerlink" href="#1421-optimal-substructure" title="Permanent link">&para;</a></h2>
-<p>We make a slight modification to the stair climbing problem to make it more suitable for demonstrating the concept of optimal substructure.</p>
+<p>Dynamic programming is often used for optimization problems. Besides overlapping subproblems, these problems have two key properties: optimal substructure and the no-aftereffect property.</p>
+<h2 id="1421-optimal-substructure">14.2.1 &nbsp; Optimal substructure<a class="headerlink" href="#1421-optimal-substructure" title="Permanent link">&para;</a></h2>
+<p>Let's modify the climbing stairs problem slightly to illustrate optimal substructure more clearly.</p>
 <div class="admonition question">
-<p class="admonition-title">Climbing stairs with minimum cost</p>
-<p>Given a staircase, you can climb <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(2\)</span> steps at a time, and each step is labeled with a non-negative integer representing the cost of stepping on it. Given a non-negative integer array <span class="arithmatex">\(cost\)</span>, where <span class="arithmatex">\(cost[i]\)</span> represents the cost of the <span class="arithmatex">\(i\)</span>-th step and <span class="arithmatex">\(cost[0]\)</span> is the ground (starting point), what is the minimum cost required to reach the top?</p>
+<p class="admonition-title">Minimum cost to climb stairs</p>
+<p>Given a staircase, you can climb <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(2\)</span> steps at a time. Each step has a nonnegative integer indicating the cost paid on that step. A nonnegative integer array <span class="arithmatex">\(cost\)</span> gives the costs: <span class="arithmatex">\(cost[i]\)</span> is the cost of step <span class="arithmatex">\(i\)</span>, and <span class="arithmatex">\(cost[0]\)</span> refers to the ground (the starting point). What is the minimum cost of reaching the top?</p>
 </div>
-<p>As shown in Figure 14-6, if the costs of the <span class="arithmatex">\(1\)</span>st, <span class="arithmatex">\(2\)</span>nd, and <span class="arithmatex">\(3\)</span>rd steps are <span class="arithmatex">\(1\)</span>, <span class="arithmatex">\(10\)</span>, and <span class="arithmatex">\(1\)</span> respectively, then climbing from the ground to the <span class="arithmatex">\(3\)</span>rd step requires a minimum cost of <span class="arithmatex">\(2\)</span>.</p>
-<p><img alt="Minimum cost to climb to the 3rd step" class="animation-figure" src="../dp_problem_features.assets/min_cost_cs_example.png" /></p>
-<p align="center"> Figure 14-6 &nbsp; Minimum cost to climb to the 3rd step </p>
+<p>As shown in Figure 14-6, if steps <span class="arithmatex">\(1\)</span>, <span class="arithmatex">\(2\)</span>, and <span class="arithmatex">\(3\)</span> cost <span class="arithmatex">\(1\)</span>, <span class="arithmatex">\(10\)</span>, and <span class="arithmatex">\(1\)</span>, respectively, the minimum cost of reaching step <span class="arithmatex">\(3\)</span> from the ground is <span class="arithmatex">\(2\)</span>.</p>
+<p><img alt="Minimum cost of reaching step 3" class="animation-figure" src="../dp_problem_features.assets/min_cost_cs_example.png" /></p>
+<p align="center"> Figure 14-6 &nbsp; Minimum cost of reaching step 3 </p>
 
-<p>Let <span class="arithmatex">\(dp[i]\)</span> be the accumulated cost of climbing to the <span class="arithmatex">\(i\)</span>-th step. Since the <span class="arithmatex">\(i\)</span>-th step can only come from the <span class="arithmatex">\(i-1\)</span>-th or <span class="arithmatex">\(i-2\)</span>-th step, <span class="arithmatex">\(dp[i]\)</span> can only equal <span class="arithmatex">\(dp[i-1] + cost[i]\)</span> or <span class="arithmatex">\(dp[i-2] + cost[i]\)</span>. To minimize the cost, we should choose the smaller of the two:</p>
+<p>Let <span class="arithmatex">\(dp[i]\)</span> be the total cost of reaching step <span class="arithmatex">\(i\)</span>. Since step <span class="arithmatex">\(i\)</span> can only be reached from step <span class="arithmatex">\(i - 1\)</span> or <span class="arithmatex">\(i - 2\)</span>, <span class="arithmatex">\(dp[i]\)</span> must be either <span class="arithmatex">\(dp[i - 1] + cost[i]\)</span> or <span class="arithmatex">\(dp[i - 2] + cost[i]\)</span>. To minimize the cost, choose the smaller of the two:</p>
 <div class="arithmatex">\[
 dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 \]</div>
-<p>This leads us to the meaning of optimal substructure: <strong>the optimal solution to the original problem is constructed from the optimal solutions to the subproblems</strong>.</p>
-<p>This problem clearly has optimal substructure: we select the better one from the optimal solutions to the two subproblems <span class="arithmatex">\(dp[i-1]\)</span> and <span class="arithmatex">\(dp[i-2]\)</span>, and use it to construct the optimal solution to the original problem <span class="arithmatex">\(dp[i]\)</span>.</p>
-<p>So, does the stair climbing problem from the previous section have optimal substructure? Its goal is to find the number of ways, which seems to be a counting problem, but if we change the question: "Find the maximum number of ways". We surprisingly discover that <strong>although the problem before and after modification are equivalent, the optimal substructure has emerged</strong>: the maximum number of ways for the <span class="arithmatex">\(n\)</span>-th step equals the sum of the maximum number of ways for the <span class="arithmatex">\(n-1\)</span>-th and <span class="arithmatex">\(n-2\)</span>-th steps. Therefore, the interpretation of optimal substructure is quite flexible and will have different meanings in different problems.</p>
-<p>According to the state transition equation and the initial states <span class="arithmatex">\(dp[1] = cost[1]\)</span> and <span class="arithmatex">\(dp[2] = cost[2]\)</span>, we can obtain the dynamic programming code:</p>
+<p>This introduces the meaning of optimal substructure: <strong>an optimal solution to the original problem is constructed from optimal solutions to its subproblems</strong>.</p>
+<p>This problem clearly has optimal substructure: choose the better of <span class="arithmatex">\(dp[i-1]\)</span> and <span class="arithmatex">\(dp[i-2]\)</span> and use it to construct the optimal solution <span class="arithmatex">\(dp[i]\)</span>.</p>
+<p>Does the climbing stairs problem from the previous section also have optimal substructure? Its goal is to count solutions, so it seems to be a counting problem. But rephrase the goal as finding the maximum number of solutions, and <strong>although the problem remains equivalent, optimal substructure becomes apparent</strong>: the maximum number of ways to reach step <span class="arithmatex">\(n\)</span> is the sum of the maximum numbers for steps <span class="arithmatex">\(n-1\)</span> and <span class="arithmatex">\(n-2\)</span>. The interpretation of optimal substructure is therefore flexible and can vary by problem.</p>
+<p>The state transition equation, together with the initial states <span class="arithmatex">\(dp[1] = cost[1]\)</span> and <span class="arithmatex">\(dp[2] = cost[2]\)</span>, gives the following dynamic programming code:</p>
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 <div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">min_cost_climbing_stairs_dp</span><span class="p">(</span><span class="n">cost</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">])</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Minimum cost climbing stairs: Dynamic programming"""</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Minimum cost to climb stairs: dynamic programming"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">cost</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="k">return</span> <span class="n">cost</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="c1"># Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="c1"># Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">cost</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">cost</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="c1"># State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="c1"># State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">2</span><span class="p">])</span> <span class="o">+</span> <span class="n">cost</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Minimum cost climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Minimum cost to climb stairs: dynamic programming */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minCostClimbingStairsDP</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">cost</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="n">n</span><span class="p">];</span>
-<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="mi">2</span><span class="p">];</span>
-<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">    </span><span class="p">}</span>
@@ -4721,17 +4793,17 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Minimum cost climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Minimum cost to climb stairs: dynamic programming */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minCostClimbingStairsDP</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">cost</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="mi">2</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">    </span><span class="p">}</span>
@@ -4740,17 +4812,17 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Minimum cost climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Minimum cost to climb stairs: dynamic programming */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">MinCostClimbingStairsDP</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">cost</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="n">n</span><span class="p">];</span>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="mi">2</span><span class="p">];</span>
-<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="p">}</span>
@@ -4759,7 +4831,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Minimum cost climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Minimum cost to climb stairs: dynamic programming */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">minCostClimbingStairsDP</span><span class="p">(</span><span class="nx">cost</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">cost</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
@@ -4771,12 +4843,12 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">b</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cost</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cost</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">2</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">cost</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">    </span><span class="p">}</span>
@@ -4785,18 +4857,18 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Minimum cost climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Minimum cost to climb stairs: dynamic programming */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">minCostClimbingStairsDP</span><span class="p">(</span><span class="n">cost</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">cost</span><span class="p">.</span><span class="bp">count</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">    </span><span class="p">}</span>
@@ -4805,18 +4877,18 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Minimum cost climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Minimum cost to climb stairs: dynamic programming */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">minCostClimbingStairsDP</span><span class="p">(</span><span class="nx">cost</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cost</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">cost</span><span class="p">[</span><span class="nx">n</span><span class="p">];</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cost</span><span class="p">[</span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cost</span><span class="p">[</span><span class="mf">2</span><span class="p">];</span>
-<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">3</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">2</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">cost</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">    </span><span class="p">}</span>
@@ -4825,18 +4897,18 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Minimum cost climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Minimum cost to climb stairs: dynamic programming */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">minCostClimbingStairsDP</span><span class="p">(</span><span class="nx">cost</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cost</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">cost</span><span class="p">[</span><span class="nx">n</span><span class="p">];</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cost</span><span class="p">[</span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cost</span><span class="p">[</span><span class="mf">2</span><span class="p">];</span>
-<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">3</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">2</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">cost</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="p">}</span>
@@ -4845,16 +4917,16 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Minimum cost climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Minimum cost to climb stairs: dynamic programming */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">minCostClimbingStairsDP</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">cost</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="n">n</span><span class="p">];</span>
-<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
-<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="n">dp</span><span class="p">[</span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="m">1</span><span class="p">];</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="n">dp</span><span class="p">[</span><span class="m">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="m">2</span><span class="p">];</span>
-<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">  </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">  </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">3</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">2</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="p">}</span>
@@ -4863,18 +4935,18 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Minimum cost climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Minimum cost to climb stairs: dynamic programming */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">min_cost_climbing_stairs_dp</span><span class="p">(</span><span class="n">cost</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="n">n</span><span class="p">];</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="mi">2</span><span class="p">];</span>
-<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">3</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cmp</span><span class="p">::</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">    </span><span class="p">}</span>
@@ -4883,17 +4955,17 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Minimum cost climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Minimum cost to climb stairs: dynamic programming */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minCostClimbingStairsDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">cost</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">costSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">costSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="n">n</span><span class="p">];</span>
-<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
-<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="mi">2</span><span class="p">];</span>
-<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="p">}</span>
@@ -4905,16 +4977,16 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Minimum cost climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Minimum cost to climb stairs: dynamic programming */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">minCostClimbingStairsDP</span><span class="p">(</span><span class="n">cost</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="n">n</span><span class="o">]</span>
-<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
-<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="m">1</span><span class="o">]</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="m">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="m">2</span><span class="o">]</span>
-<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">3.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">2</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="p">}</span>
@@ -4923,15 +4995,15 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Minimum cost climbing stairs: DP ###</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Minimum cost to climb stairs: dynamic programming ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">min_cost_climbing_stairs_dp</span><span class="p">(</span><span class="n">cost</span><span class="p">)</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="c1"># Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="c1"># Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="n">dp</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="mi">2</span><span class="o">]</span>
-<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="c1"># State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="c1"># State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="p">(</span><span class="mi">3</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="o">]].</span><span class="n">min</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">  </span><span class="n">dp</span><span class="o">[</span><span class="n">n</span><span class="o">]</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="k">end</span>
@@ -4939,16 +5011,21 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </div>
 </div>
 </div>
-<p>Figure 14-7 shows the dynamic programming process for the above code.</p>
-<p><img alt="Dynamic programming process for climbing stairs with minimum cost" class="animation-figure" src="../dp_problem_features.assets/min_cost_cs_dp.png" /></p>
-<p align="center"> Figure 14-7 &nbsp; Dynamic programming process for climbing stairs with minimum cost </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20min_cost_climbing_stairs_dp%28cost%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Minimum%20cost%20to%20climb%20stairs%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28cost%29%20-%201%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20cost%5Bn%5D%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%20to%20store%20subproblem%20solutions%0A%20%20%20%20dp%20%3D%20%5B0%5D%20%2A%20%28n%20%2B%201%29%0A%20%20%20%20%23%20Initial%20states%3A%20set%20the%20solutions%20to%20the%20smallest%20subproblems%0A%20%20%20%20dp%5B1%5D,%20dp%5B2%5D%20%3D%20cost%5B1%5D,%20cost%5B2%5D%0A%20%20%20%20%23%20State%20transitions%3A%20solve%20progressively%20larger%20subproblems%20using%20smaller%20ones%0A%20%20%20%20for%20i%20in%20range%283,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%20%3D%20min%28dp%5Bi%20-%201%5D,%20dp%5Bi%20-%202%5D%29%20%2B%20cost%5Bi%5D%0A%20%20%20%20return%20dp%5Bn%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20cost%20%3D%20%5B0,%201,%2010,%201,%201,%201,%2010,%201,%201,%2010,%201%5D%0A%20%20%20%20print%28f%22Input%20list%20of%20stair%20costs%3A%20%7Bcost%7D%22%29%0A%0A%20%20%20%20res%20%3D%20min_cost_climbing_stairs_dp%28cost%29%0A%20%20%20%20print%28f%22Minimum%20cost%20to%20finish%20climbing%20the%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20min_cost_climbing_stairs_dp%28cost%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Minimum%20cost%20to%20climb%20stairs%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28cost%29%20-%201%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20cost%5Bn%5D%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%20to%20store%20subproblem%20solutions%0A%20%20%20%20dp%20%3D%20%5B0%5D%20%2A%20%28n%20%2B%201%29%0A%20%20%20%20%23%20Initial%20states%3A%20set%20the%20solutions%20to%20the%20smallest%20subproblems%0A%20%20%20%20dp%5B1%5D,%20dp%5B2%5D%20%3D%20cost%5B1%5D,%20cost%5B2%5D%0A%20%20%20%20%23%20State%20transitions%3A%20solve%20progressively%20larger%20subproblems%20using%20smaller%20ones%0A%20%20%20%20for%20i%20in%20range%283,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%20%3D%20min%28dp%5Bi%20-%201%5D,%20dp%5Bi%20-%202%5D%29%20%2B%20cost%5Bi%5D%0A%20%20%20%20return%20dp%5Bn%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20cost%20%3D%20%5B0,%201,%2010,%201,%201,%201,%2010,%201,%201,%2010,%201%5D%0A%20%20%20%20print%28f%22Input%20list%20of%20stair%20costs%3A%20%7Bcost%7D%22%29%0A%0A%20%20%20%20res%20%3D%20min_cost_climbing_stairs_dp%28cost%29%0A%20%20%20%20print%28f%22Minimum%20cost%20to%20finish%20climbing%20the%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 14-7 illustrates this dynamic programming process.</p>
+<p><img alt="Dynamic programming for the minimum climbing cost" class="animation-figure" src="../dp_problem_features.assets/min_cost_cs_dp.png" /></p>
+<p align="center"> Figure 14-7 &nbsp; Dynamic programming for the minimum climbing cost </p>
 
-<p>This problem can also be space-optimized, compressing from one dimension to zero, reducing the space complexity from <span class="arithmatex">\(O(n)\)</span> to <span class="arithmatex">\(O(1)\)</span>:</p>
+<p>We can also reduce the state storage from a one-dimensional array to scalar variables, lowering the space complexity from <span class="arithmatex">\(O(n)\)</span> to <span class="arithmatex">\(O(1)\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">min_cost_climbing_stairs_dp_comp</span><span class="p">(</span><span class="n">cost</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">])</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Minimum cost climbing stairs: Space-optimized dynamic programming"""</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Minimum cost to climb stairs: space-optimized dynamic programming"""</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">cost</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>        <span class="k">return</span> <span class="n">cost</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
@@ -4959,7 +5036,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Minimum cost climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Minimum cost to climb stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minCostClimbingStairsDPComp</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">cost</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
@@ -4975,7 +5052,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Minimum cost climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Minimum cost to climb stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minCostClimbingStairsDPComp</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">cost</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
@@ -4991,7 +5068,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Minimum cost climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Minimum cost to climb stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">MinCostClimbingStairsDPComp</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">cost</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
@@ -5007,7 +5084,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Minimum cost climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Minimum cost to climb stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">minCostClimbingStairsDPComp</span><span class="p">(</span><span class="nx">cost</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">cost</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
@@ -5019,9 +5096,9 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">b</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">    </span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">b</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">cost</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="nx">cost</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
-<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">        </span><span class="nx">tmp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">b</span>
 <a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="w">        </span><span class="nx">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">min</span><span class="p">(</span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">tmp</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">cost</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
@@ -5032,7 +5109,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Minimum cost climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Minimum cost to climb stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">minCostClimbingStairsDPComp</span><span class="p">(</span><span class="n">cost</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">cost</span><span class="p">.</span><span class="bp">count</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
@@ -5047,7 +5124,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Minimum cost climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Minimum cost to climb stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">minCostClimbingStairsDPComp</span><span class="p">(</span><span class="nx">cost</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cost</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5065,7 +5142,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Minimum cost climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Minimum cost to climb stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">minCostClimbingStairsDPComp</span><span class="p">(</span><span class="nx">cost</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cost</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5083,7 +5160,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Minimum cost climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Minimum cost to climb stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">minCostClimbingStairsDPComp</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">cost</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">cost</span><span class="p">[</span><span class="n">n</span><span class="p">];</span>
@@ -5098,7 +5175,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Minimum cost climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Minimum cost to climb stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">min_cost_climbing_stairs_dp_comp</span><span class="p">(</span><span class="n">cost</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
@@ -5115,7 +5192,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Minimum cost climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Minimum cost to climb stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minCostClimbingStairsDPComp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">cost</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">costSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">costSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
@@ -5131,7 +5208,7 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Minimum cost climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Minimum cost to climb stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">minCostClimbingStairsDPComp</span><span class="p">(</span><span class="n">cost</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="n">n</span><span class="o">]</span>
@@ -5147,20 +5224,20 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Minimum cost climbing stairs: DP ###</span>
+<div class="highlight"><span class="filename">min_cost_climbing_stairs_dp.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Minimum cost to climb stairs: dynamic programming ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">min_cost_climbing_stairs_dp</span><span class="p">(</span><span class="n">cost</span><span class="p">)</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span>
-<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="c1"># Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="c1"># Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">  </span><span class="n">dp</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="mi">2</span><span class="o">]</span>
-<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="c1"># State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="c1"># State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="p">(</span><span class="mi">3</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="o">]].</span><span class="n">min</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">  </span><span class="n">dp</span><span class="o">[</span><span class="n">n</span><span class="o">]</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="k">end</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a>
-<a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="c1"># Minimum cost climbing stairs: Space-optimized dynamic programming</span>
+<a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="c1"># Minimum cost to climb stairs: space-optimized dynamic programming</span>
 <a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="k">def</span><span class="w"> </span><span class="nf">min_cost_climbing_stairs_dp_comp</span><span class="p">(</span><span class="n">cost</span><span class="p">)</span>
 <a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cost</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">cost</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span>
@@ -5172,49 +5249,54 @@ dp[i] = \min(dp[i-1], dp[i-2]) + cost[i]
 </div>
 </div>
 </div>
-<h2 id="1422-no-aftereffects">14.2.2 &nbsp; No Aftereffects<a class="headerlink" href="#1422-no-aftereffects" title="Permanent link">&para;</a></h2>
-<p>No aftereffects is one of the important characteristics that enable dynamic programming to solve problems effectively. Its definition is: <strong>given a certain state, its future development is only related to the current state and has nothing to do with all past states</strong>.</p>
-<p>Taking the stair climbing problem as an example, given state <span class="arithmatex">\(i\)</span>, it will develop into states <span class="arithmatex">\(i+1\)</span> and <span class="arithmatex">\(i+2\)</span>, corresponding to jumping <span class="arithmatex">\(1\)</span> step and jumping <span class="arithmatex">\(2\)</span> steps, respectively. When making these two choices, we do not need to consider the states before state <span class="arithmatex">\(i\)</span>, as they have no effect on the future of state <span class="arithmatex">\(i\)</span>.</p>
-<p>However, if we add a constraint to the stair climbing problem, the situation changes.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 513px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20min_cost_climbing_stairs_dp_comp%28cost%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Minimum%20cost%20to%20climb%20stairs%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28cost%29%20-%201%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20cost%5Bn%5D%0A%20%20%20%20a,%20b%20%3D%20cost%5B1%5D,%20cost%5B2%5D%0A%20%20%20%20for%20i%20in%20range%283,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20a,%20b%20%3D%20b,%20min%28a,%20b%29%20%2B%20cost%5Bi%5D%0A%20%20%20%20return%20b%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20cost%20%3D%20%5B0,%201,%2010,%201,%201,%201,%2010,%201,%201,%2010,%201%5D%0A%20%20%20%20print%28f%22Input%20list%20of%20stair%20costs%3A%20%7Bcost%7D%22%29%0A%0A%20%20%20%20res%20%3D%20min_cost_climbing_stairs_dp_comp%28cost%29%0A%20%20%20%20print%28f%22Minimum%20cost%20to%20finish%20climbing%20the%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20min_cost_climbing_stairs_dp_comp%28cost%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Minimum%20cost%20to%20climb%20stairs%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28cost%29%20-%201%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20cost%5Bn%5D%0A%20%20%20%20a,%20b%20%3D%20cost%5B1%5D,%20cost%5B2%5D%0A%20%20%20%20for%20i%20in%20range%283,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20a,%20b%20%3D%20b,%20min%28a,%20b%29%20%2B%20cost%5Bi%5D%0A%20%20%20%20return%20b%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20cost%20%3D%20%5B0,%201,%2010,%201,%201,%201,%2010,%201,%201,%2010,%201%5D%0A%20%20%20%20print%28f%22Input%20list%20of%20stair%20costs%3A%20%7Bcost%7D%22%29%0A%0A%20%20%20%20res%20%3D%20min_cost_climbing_stairs_dp_comp%28cost%29%0A%20%20%20%20print%28f%22Minimum%20cost%20to%20finish%20climbing%20the%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1422-no-aftereffect-property">14.2.2 &nbsp; No-aftereffect property<a class="headerlink" href="#1422-no-aftereffect-property" title="Permanent link">&para;</a></h2>
+<p>The no-aftereffect property is central to effective dynamic programming: <strong>given a fully specified state, its future depends only on that state, not on the sequence of states that led to it</strong>.</p>
+<p>In the climbing stairs problem, state <span class="arithmatex">\(i\)</span> can lead to <span class="arithmatex">\(i+1\)</span> or <span class="arithmatex">\(i+2\)</span>, corresponding to moves of <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(2\)</span> steps. Neither choice requires us to consider states before <span class="arithmatex">\(i\)</span>: they do not affect the future of state <span class="arithmatex">\(i\)</span>.</p>
+<p>Adding a constraint changes the situation.</p>
 <div class="admonition question">
-<p class="admonition-title">Climbing stairs with constraint</p>
-<p>Given a staircase with <span class="arithmatex">\(n\)</span> steps, where you can climb <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(2\)</span> steps at a time, <strong>but you cannot jump <span class="arithmatex">\(1\)</span> step in two consecutive rounds</strong>. How many ways are there to climb to the top?</p>
+<p class="admonition-title">Climbing stairs with a constraint</p>
+<p>Given a staircase with <span class="arithmatex">\(n\)</span> steps, you may climb <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(2\)</span> steps at a time, <strong>but may not make two consecutive moves of <span class="arithmatex">\(1\)</span> step</strong>. How many ways are there to reach the top?</p>
 </div>
-<p>As shown in Figure 14-8, there are only <span class="arithmatex">\(2\)</span> feasible ways to climb to the <span class="arithmatex">\(3\)</span>rd step. The path with three consecutive <span class="arithmatex">\(1\)</span>-step jumps does not satisfy the constraint and is therefore discarded.</p>
-<p><img alt="Number of ways to climb to the 3rd step with constraint" class="animation-figure" src="../dp_problem_features.assets/climbing_stairs_constraint_example.png" /></p>
-<p align="center"> Figure 14-8 &nbsp; Number of ways to climb to the 3rd step with constraint </p>
+<p>As shown in Figure 14-8, only <span class="arithmatex">\(2\)</span> ways remain to reach step <span class="arithmatex">\(3\)</span>: taking <span class="arithmatex">\(1\)</span> step three times in a row violates the constraint and is discarded.</p>
+<p><img alt="Number of ways to reach step 3 with the constraint" class="animation-figure" src="../dp_problem_features.assets/climbing_stairs_constraint_example.png" /></p>
+<p align="center"> Figure 14-8 &nbsp; Number of ways to reach step 3 with the constraint </p>
 
-<p>In this problem, if the previous round was a jump of <span class="arithmatex">\(1\)</span> step, then the next round must jump <span class="arithmatex">\(2\)</span> steps. This means that <strong>the next choice cannot be determined solely by the current state (current stair step number), but also depends on the previous state (the stair step number from the previous round)</strong>.</p>
-<p>It is not difficult to see that this problem no longer satisfies no aftereffects, and the state transition equation <span class="arithmatex">\(dp[i] = dp[i-1] + dp[i-2]\)</span> also fails, because <span class="arithmatex">\(dp[i-1]\)</span> represents jumping <span class="arithmatex">\(1\)</span> step in this round, but it includes many solutions where "the previous round was a jump of <span class="arithmatex">\(1\)</span> step", which cannot be directly counted in <span class="arithmatex">\(dp[i]\)</span> to satisfy the constraint.</p>
-<p>For this reason, we need to expand the state definition: <strong>state <span class="arithmatex">\([i, j]\)</span> represents being on the <span class="arithmatex">\(i\)</span>-th step with the previous round having jumped <span class="arithmatex">\(j\)</span> steps</strong>, where <span class="arithmatex">\(j \in \{1, 2\}\)</span>. This state definition effectively distinguishes whether the previous round was a jump of <span class="arithmatex">\(1\)</span> step or <span class="arithmatex">\(2\)</span> steps, allowing us to determine where the current state came from.</p>
+<p>If the previous move climbed <span class="arithmatex">\(1\)</span> step, the next must climb <span class="arithmatex">\(2\)</span> steps. Thus, <strong>the next choice cannot be determined by the current state (the current step number) alone; it also depends on the preceding state (the previous step number)</strong>.</p>
+<p>The no-aftereffect property no longer holds with this state definition, and <span class="arithmatex">\(dp[i] = dp[i-1] + dp[i-2]\)</span> is no longer valid. Using <span class="arithmatex">\(dp[i-1]\)</span> means taking <span class="arithmatex">\(1\)</span> step now, but it includes many ways in which the preceding move also took <span class="arithmatex">\(1\)</span> step. We cannot simply add all of <span class="arithmatex">\(dp[i-1]\)</span> to <span class="arithmatex">\(dp[i]\)</span> without violating the constraint.</p>
+<p>We therefore expand the state definition: <strong>state <span class="arithmatex">\([i, j]\)</span> means we are on step <span class="arithmatex">\(i\)</span> and the previous move climbed <span class="arithmatex">\(j\)</span> steps</strong>, where <span class="arithmatex">\(j \in \{1, 2\}\)</span>. This distinguishes whether the preceding move climbed <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(2\)</span> steps, allowing us to determine how the current state could have been reached.</p>
 <ul>
-<li>When the previous round jumped <span class="arithmatex">\(1\)</span> step, the round before that could only choose to jump <span class="arithmatex">\(2\)</span> steps, i.e., <span class="arithmatex">\(dp[i, 1]\)</span> can only transition from <span class="arithmatex">\(dp[i-1, 2]\)</span>.</li>
-<li>When the previous round jumped <span class="arithmatex">\(2\)</span> steps, the round before that could choose to jump <span class="arithmatex">\(1\)</span> step or <span class="arithmatex">\(2\)</span> steps, i.e., <span class="arithmatex">\(dp[i, 2]\)</span> can transition from <span class="arithmatex">\(dp[i-2, 1]\)</span> or <span class="arithmatex">\(dp[i-2, 2]\)</span>.</li>
+<li>If the previous move climbed <span class="arithmatex">\(1\)</span> step, the move before it must have climbed <span class="arithmatex">\(2\)</span> steps. Thus, <span class="arithmatex">\(dp[i, 1]\)</span> can only come from <span class="arithmatex">\(dp[i-1, 2]\)</span>.</li>
+<li>If the previous move climbed <span class="arithmatex">\(2\)</span> steps, the move before it could have climbed <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(2\)</span> steps. Thus, <span class="arithmatex">\(dp[i, 2]\)</span> can come from <span class="arithmatex">\(dp[i-2, 1]\)</span> or <span class="arithmatex">\(dp[i-2, 2]\)</span>.</li>
 </ul>
-<p>As shown in Figure 14-9, under this definition, <span class="arithmatex">\(dp[i, j]\)</span> represents the number of ways for state <span class="arithmatex">\([i, j]\)</span>. The state transition equation is then:</p>
+<p>As shown in Figure 14-9, <span class="arithmatex">\(dp[i, j]\)</span> now counts the ways to reach state <span class="arithmatex">\([i, j]\)</span>. The state transition equations are:</p>
 <div class="arithmatex">\[
 \begin{cases}
 dp[i, 1] = dp[i-1, 2] \\
 dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
 \end{cases}
 \]</div>
-<p><img alt="Recurrence relation considering constraints" class="animation-figure" src="../dp_problem_features.assets/climbing_stairs_constraint_state_transfer.png" /></p>
-<p align="center"> Figure 14-9 &nbsp; Recurrence relation considering constraints </p>
+<p><img alt="Recurrence with the constraint" class="animation-figure" src="../dp_problem_features.assets/climbing_stairs_constraint_state_transfer.png" /></p>
+<p align="center"> Figure 14-9 &nbsp; Recurrence with the constraint </p>
 
-<p>Finally, return <span class="arithmatex">\(dp[n, 1] + dp[n, 2]\)</span>, where the sum of the two represents the total number of ways to climb to the <span class="arithmatex">\(n\)</span>-th step:</p>
+<p>Finally, return <span class="arithmatex">\(dp[n, 1] + dp[n, 2]\)</span>, the total number of ways to reach step <span class="arithmatex">\(n\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
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 <div class="highlight"><span class="filename">climbing_stairs_constraint_dp.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_constraint_dp</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Climbing stairs with constraint: Dynamic programming"""</span>
+<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Climbing stairs with constraints: dynamic programming"""</span>
 <a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>        <span class="k">return</span> <span class="mi">1</span>
-<a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>    <span class="c1"># Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>    <span class="c1"># Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="mi">3</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>    <span class="c1"># Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>    <span class="c1"># Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>    <span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span>
 <a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>    <span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span>
-<a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>    <span class="c1"># State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>    <span class="c1"># State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a>        <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span>
 <a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>        <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">2</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span>
@@ -5222,19 +5304,19 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Climbing stairs with constraint: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Climbing stairs with constraints: dynamic programming */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsConstraintDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">));</span>
-<a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">];</span>
 <a id="__codelineno-27-16" name="__codelineno-27-16" href="#__codelineno-27-16"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">];</span>
@@ -5244,19 +5326,19 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Climbing stairs with constraint: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Climbing stairs with constraints: dynamic programming */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsConstraintDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="mi">3</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="mi">1</span><span class="o">][</span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="mi">1</span><span class="o">][</span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="mi">2</span><span class="o">][</span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="mi">2</span><span class="o">][</span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="mi">2</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="o">][</span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="o">][</span><span class="mi">2</span><span class="o">]</span><span class="p">;</span>
@@ -5266,19 +5348,19 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Climbing stairs with constraint: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Climbing stairs with constraints: dynamic programming */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">ClimbingStairsConstraintDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="kt">int</span><span class="p">[,]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">];</span>
-<a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">];</span>
 <a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">];</span>
@@ -5288,19 +5370,19 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Climbing stairs with constraint: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Climbing stairs with constraints: dynamic programming */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">climbingStairsConstraintDP</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][</span><span class="mi">3</span><span class="p">]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span>
 <a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">2</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span>
@@ -5310,19 +5392,19 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Climbing stairs with constraint: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Climbing stairs with constraints: dynamic programming */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">climbingStairsConstraintDP</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="mi">3</span><span class="p">),</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span>
 <a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span>
@@ -5332,19 +5414,19 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Climbing stairs with constraint: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Climbing stairs with constraints: dynamic programming */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">climbingStairsConstraintDP</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">(</span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">),</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">3</span><span class="p">));</span>
-<a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">1</span><span class="p">][</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">1</span><span class="p">][</span><span class="mf">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">2</span><span class="p">][</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">2</span><span class="p">][</span><span class="mf">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">3</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-15" name="__codelineno-32-15" href="#__codelineno-32-15"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="mf">2</span><span class="p">];</span>
 <a id="__codelineno-32-16" name="__codelineno-32-16" href="#__codelineno-32-16"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">2</span><span class="p">][</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">2</span><span class="p">][</span><span class="mf">2</span><span class="p">];</span>
@@ -5354,19 +5436,19 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Climbing stairs with constraint: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Climbing stairs with constraints: dynamic programming */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">climbingStairsConstraintDP</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">3</span><span class="p">));</span>
-<a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">1</span><span class="p">][</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">1</span><span class="p">][</span><span class="mf">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">2</span><span class="p">][</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">2</span><span class="p">][</span><span class="mf">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">3</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="mf">2</span><span class="p">];</span>
 <a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">2</span><span class="p">][</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">2</span><span class="p">][</span><span class="mf">2</span><span class="p">];</span>
@@ -5376,19 +5458,19 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Climbing stairs with constraint: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Climbing stairs with constraints: dynamic programming */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">climbingStairsConstraintDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">));</span>
-<a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">  </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">  </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w">  </span><span class="n">dp</span><span class="p">[</span><span class="m">1</span><span class="p">][</span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="w">  </span><span class="n">dp</span><span class="p">[</span><span class="m">1</span><span class="p">][</span><span class="m">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w">  </span><span class="n">dp</span><span class="p">[</span><span class="m">2</span><span class="p">][</span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="w">  </span><span class="n">dp</span><span class="p">[</span><span class="m">2</span><span class="p">][</span><span class="m">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
-<a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">  </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">  </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">3</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="m">2</span><span class="p">];</span>
 <a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="m">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">2</span><span class="p">][</span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">2</span><span class="p">][</span><span class="m">2</span><span class="p">];</span>
@@ -5398,19 +5480,19 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Climbing stairs with constraint: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Climbing stairs with constraints: dynamic programming */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">climbing_stairs_constraint_dp</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">    </span><span class="p">};</span>
-<a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="fm">vec!</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="mi">3</span><span class="p">];</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">3</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">];</span>
 <a id="__codelineno-35-16" name="__codelineno-35-16" href="#__codelineno-35-16"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">];</span>
@@ -5420,22 +5502,22 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Climbing stairs with constraint: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Climbing stairs with constraints: dynamic programming */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsConstraintDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">**</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">((</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">));</span>
 <a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-36-14" name="__codelineno-36-14" href="#__codelineno-36-14"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-36-16" name="__codelineno-36-16" href="#__codelineno-36-16"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-36-16" name="__codelineno-36-16" href="#__codelineno-36-16"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-36-17" name="__codelineno-36-17" href="#__codelineno-36-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-18" name="__codelineno-36-18" href="#__codelineno-36-18"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">];</span>
 <a id="__codelineno-36-19" name="__codelineno-36-19" href="#__codelineno-36-19"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">][</span><span class="mi">2</span><span class="p">];</span>
@@ -5451,19 +5533,19 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Climbing stairs with constraint: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Climbing stairs with constraints: dynamic programming */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">climbingStairsConstraintDP</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Array</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="m">3</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="m">1</span><span class="o">][</span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="m">1</span><span class="o">][</span><span class="m">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="m">2</span><span class="o">][</span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="m">2</span><span class="o">][</span><span class="m">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">3.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-15" name="__codelineno-37-15" href="#__codelineno-37-15"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">][</span><span class="m">2</span><span class="o">]</span>
 <a id="__codelineno-37-16" name="__codelineno-37-16" href="#__codelineno-37-16"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="m">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">2</span><span class="o">][</span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">2</span><span class="o">][</span><span class="m">2</span><span class="o">]</span>
@@ -5473,16 +5555,16 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Climbing stairs with constraint: DP ###</span>
+<div class="highlight"><span class="filename">climbing_stairs_constraint_dp.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Climbing stairs with constraints: dynamic programming ###</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_constraint_dp</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a>
-<a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">  </span><span class="c1"># Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">  </span><span class="c1"># Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="w">  </span><span class="c1"># Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="w">  </span><span class="c1"># Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">  </span><span class="n">dp</span><span class="o">[</span><span class="mi">1</span><span class="o">][</span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">1</span><span class="o">][</span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">  </span><span class="n">dp</span><span class="o">[</span><span class="mi">2</span><span class="o">][</span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">2</span><span class="o">][</span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">  </span><span class="c1"># State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">  </span><span class="c1"># State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">3</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="mi">2</span><span class="o">]</span>
 <a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="o">][</span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="o">][</span><span class="mi">2</span><span class="o">]</span>
@@ -5494,13 +5576,18 @@ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2]
 </div>
 </div>
 </div>
-<p>In the above case, since we only need to consider one more preceding state, we can still make the problem satisfy no aftereffects by expanding the state definition. However, some problems have very severe "aftereffects".</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20climbing_stairs_constraint_dp%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Climbing%20stairs%20with%20constraints%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%20to%20store%20subproblem%20solutions%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%203%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20Initial%20states%3A%20set%20the%20solutions%20to%20the%20smallest%20subproblems%0A%20%20%20%20dp%5B1%5D%5B1%5D,%20dp%5B1%5D%5B2%5D%20%3D%201,%200%0A%20%20%20%20dp%5B2%5D%5B1%5D,%20dp%5B2%5D%5B2%5D%20%3D%200,%201%0A%20%20%20%20%23%20State%20transitions%3A%20solve%20progressively%20larger%20subproblems%20using%20smaller%20ones%0A%20%20%20%20for%20i%20in%20range%283,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B1%5D%20%3D%20dp%5Bi%20-%201%5D%5B2%5D%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B2%5D%20%3D%20dp%5Bi%20-%202%5D%5B1%5D%20%2B%20dp%5Bi%20-%202%5D%5B2%5D%0A%20%20%20%20return%20dp%5Bn%5D%5B1%5D%20%2B%20dp%5Bn%5D%5B2%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_constraint_dp%28n%29%0A%20%20%20%20print%28f%22Number%20of%20ways%20to%20climb%20%7Bn%7D%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20climbing_stairs_constraint_dp%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Climbing%20stairs%20with%20constraints%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%20to%20store%20subproblem%20solutions%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%203%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20Initial%20states%3A%20set%20the%20solutions%20to%20the%20smallest%20subproblems%0A%20%20%20%20dp%5B1%5D%5B1%5D,%20dp%5B1%5D%5B2%5D%20%3D%201,%200%0A%20%20%20%20dp%5B2%5D%5B1%5D,%20dp%5B2%5D%5B2%5D%20%3D%200,%201%0A%20%20%20%20%23%20State%20transitions%3A%20solve%20progressively%20larger%20subproblems%20using%20smaller%20ones%0A%20%20%20%20for%20i%20in%20range%283,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B1%5D%20%3D%20dp%5Bi%20-%201%5D%5B2%5D%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B2%5D%20%3D%20dp%5Bi%20-%202%5D%5B1%5D%20%2B%20dp%5Bi%20-%202%5D%5B2%5D%0A%20%20%20%20return%20dp%5Bn%5D%5B1%5D%20%2B%20dp%5Bn%5D%5B2%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_constraint_dp%28n%29%0A%20%20%20%20print%28f%22Number%20of%20ways%20to%20climb%20%7Bn%7D%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>In this example, only one preceding state mattered, so expanding the state definition restored the no-aftereffect property. Some problems, however, depend much more strongly on their history.</p>
 <div class="admonition question">
-<p class="admonition-title">Climbing stairs with obstacle generation</p>
-<p>Given a staircase with <span class="arithmatex">\(n\)</span> steps, where you can climb <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(2\)</span> steps at a time. <strong>Whenever you reach the <span class="arithmatex">\(i\)</span>-th step, the system automatically places an obstacle on the <span class="arithmatex">\(2i\)</span>-th step, and no subsequent round is allowed to jump to the <span class="arithmatex">\(2i\)</span>-th step</strong>. For example, if the first two rounds jump to the <span class="arithmatex">\(2\)</span>nd and <span class="arithmatex">\(3\)</span>rd steps, then afterwards you cannot jump to the <span class="arithmatex">\(4\)</span>th and <span class="arithmatex">\(6\)</span>th steps. How many ways are there to climb to the top?</p>
+<p class="admonition-title">Climbing stairs that generate obstacles</p>
+<p>Given a staircase with <span class="arithmatex">\(n\)</span> steps, you may climb <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(2\)</span> steps at a time. <strong>Whenever you reach step <span class="arithmatex">\(i\)</span>, the system places an obstacle on step <span class="arithmatex">\(2i\)</span>, and you may never land on step <span class="arithmatex">\(2i\)</span> afterward</strong>. For example, if your first two moves land on steps <span class="arithmatex">\(2\)</span> and <span class="arithmatex">\(3\)</span>, then steps <span class="arithmatex">\(4\)</span> and <span class="arithmatex">\(6\)</span> are blocked for all later moves. How many ways are there to reach the top?</p>
 </div>
-<p>In this problem, the next jump depends on all past states, because each jump places obstacles on higher steps, affecting future jumps. For such problems, dynamic programming is often difficult to solve.</p>
-<p>In fact, many complex combinatorial optimization problems (such as the traveling salesman problem) do not satisfy no aftereffects. For such problems, we usually use other methods, such as heuristic search, genetic algorithms, and reinforcement learning, to obtain usable locally optimal solutions within a limited time.</p>
+<p>Here, each next move depends on all previous states, because every move places an obstacle higher up and affects future moves. Such problems are often difficult to solve with dynamic programming.</p>
+<p>Many complex combinatorial optimization problems, such as the traveling salesperson problem, do not satisfy the no-aftereffect property. For such problems, we often choose other methods, such as heuristic search, genetic algorithms, or reinforcement learning, to obtain useful locally optimal solutions within a limited time.</p>
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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@@ -177,9 +177,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3283,7 +3283,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3304,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3352,7 +3352,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3379,7 +3379,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3406,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3433,7 +3433,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3460,7 +3460,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3526,7 +3526,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3547,7 +3547,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3568,7 +3568,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3595,7 +3595,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3622,7 +3622,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3649,7 +3649,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3676,7 +3676,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3703,7 +3703,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3775,7 +3775,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3796,7 +3796,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
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-  14.2 Characteristics of Dynamic Programming Problems
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 <!-- Page content -->
-<h1 id="143-dynamic-programming-problem-solving-approach">14.3 &nbsp; Dynamic Programming Problem-Solving Approach<a class="headerlink" href="#143-dynamic-programming-problem-solving-approach" title="Permanent link">&para;</a></h1>
-<p>The previous two sections introduced the main characteristics of dynamic programming problems. Next, let us explore two more practical issues together.</p>
+<h1 id="143-solving-dynamic-programming-problems">14.3 &nbsp; Solving dynamic programming problems<a class="headerlink" href="#143-solving-dynamic-programming-problems" title="Permanent link">&para;</a></h1>
+<p>The previous two sections introduced the main properties of dynamic programming problems. We will now explore two practical questions.</p>
 <ol>
-<li>How to determine whether a problem is a dynamic programming problem?</li>
-<li>What is the complete process for solving a dynamic programming problem, and where should we start?</li>
+<li>How can we recognize a dynamic programming problem?</li>
+<li>Where should we start, and what are the steps for solving one?</li>
 </ol>
-<h2 id="1431-problem-identification">14.3.1 &nbsp; Problem Identification<a class="headerlink" href="#1431-problem-identification" title="Permanent link">&para;</a></h2>
-<p>Generally speaking, if a problem contains overlapping subproblems, optimal substructure, and satisfies no aftereffects, then it is usually suitable for solving with dynamic programming. However, it is difficult to directly extract these characteristics from the problem description. Therefore, we usually relax the conditions and <strong>first observe whether the problem is suitable for solving with backtracking (exhaustive search)</strong>.</p>
-<p><strong>Problems suitable for solving with backtracking usually satisfy the "decision tree model"</strong>, which means the problem can be described using a tree structure, where each node represents a decision and each path represents a sequence of decisions.</p>
-<p>In other words, if a problem contains an explicit concept of decisions, and the solution is generated through a series of decisions, then it satisfies the decision tree model and can usually be solved using backtracking.</p>
-<p>On this basis, dynamic programming problems also have some positive indicators.</p>
+<h2 id="1431-recognizing-the-problem">14.3.1 &nbsp; Recognizing the problem<a class="headerlink" href="#1431-recognizing-the-problem" title="Permanent link">&para;</a></h2>
+<p>In general, a problem is suitable for dynamic programming if it has overlapping subproblems, optimal substructure, and the no-aftereffect property. These can be difficult to identify directly from a problem statement, so we usually start with a broader test: <strong>could backtracking (exhaustive search) solve it</strong>?</p>
+<p><strong>Problems suited to backtracking usually fit a decision tree model</strong>. They can be represented as a tree in which each node is a decision and each path is a sequence of decisions.</p>
+<p>In other words, if a problem has clear choices and a solution is produced through a sequence of them, it fits the decision tree model and can usually be solved by backtracking.</p>
+<p>Beyond this, some signs make dynamic programming more promising.</p>
 <ul>
-<li>The problem contains descriptions such as maximum (minimum) or most (least), indicating optimization.</li>
-<li>The problem's state can be represented using a list, multi-dimensional matrix, or tree, and a state has a recurrence relation with its surrounding states.</li>
+<li>The problem asks for an optimum, such as a maximum, minimum, greatest number, or smallest number.</li>
+<li>States can be represented by a list, a multidimensional matrix, or a tree, and a recurrence relates each state to nearby states.</li>
 </ul>
-<p>Correspondingly, there are also some negative indicators.</p>
+<p>Other signs make it less promising.</p>
 <ul>
-<li>The goal of the problem is to find all possible solutions, rather than finding the optimal solution.</li>
-<li>The problem description has obvious permutation and combination characteristics, requiring the return of specific multiple solutions.</li>
+<li>The goal is to find all possible solutions rather than an optimal one.</li>
+<li>The problem explicitly involves permutations or combinations and asks for multiple concrete arrangements.</li>
 </ul>
-<p>If a problem satisfies the decision tree model and has relatively obvious positive indicators, we can assume it is a dynamic programming problem and verify that assumption during the solving process.</p>
-<h2 id="1432-problem-solving-steps">14.3.2 &nbsp; Problem-Solving Steps<a class="headerlink" href="#1432-problem-solving-steps" title="Permanent link">&para;</a></h2>
-<p>The problem-solving process for dynamic programming varies depending on the nature and difficulty of the problem, but generally follows these steps: describe decisions, define states, establish the <span class="arithmatex">\(dp\)</span> table, derive state transition equations, determine boundary conditions, etc.</p>
-<p>To illustrate the problem-solving steps more vividly, we use a classic problem "minimum path sum" as an example.</p>
+<p>If a problem fits the decision tree model and has clear positive signs, we can tentatively treat it as a dynamic programming problem and verify that assumption while solving it.</p>
+<h2 id="1432-steps-for-solving-the-problem">14.3.2 &nbsp; Steps for solving the problem<a class="headerlink" href="#1432-steps-for-solving-the-problem" title="Permanent link">&para;</a></h2>
+<p>The process varies with the problem's nature and difficulty, but usually includes describing the decisions, defining states, creating a <span class="arithmatex">\(dp\)</span> table, deriving a state transition equation, and identifying boundary conditions.</p>
+<p>To make these steps concrete, we will use the classic minimum path sum problem.</p>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given an <span class="arithmatex">\(n \times m\)</span> two-dimensional grid <code>grid</code> in which each cell contains a non-negative integer representing its cost, a robot starts from the top-left cell and can only move down or right at each step until reaching the bottom-right cell. Return the minimum path sum from the top-left to the bottom-right.</p>
+<p>Given an <span class="arithmatex">\(n \times m\)</span> grid <code>grid</code>, each cell contains a nonnegative integer representing its cost. A robot starts at the top-left cell and can move one cell down or right at a time until it reaches the bottom-right cell. Return the minimum path sum from the top-left to the bottom-right.</p>
 </div>
-<p>Figure 14-10 shows an example where the minimum path sum for the given grid is <span class="arithmatex">\(13\)</span>.</p>
-<p><img alt="Minimum path sum example data" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_example.png" /></p>
-<p align="center"> Figure 14-10 &nbsp; Minimum path sum example data </p>
+<p>In the example shown in Figure 14-10, the minimum path sum is <span class="arithmatex">\(13\)</span>.</p>
+<p><img alt="Example input for minimum path sum" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_example.png" /></p>
+<p align="center"> Figure 14-10 &nbsp; Example input for minimum path sum </p>
 
-<p><strong>Step 1: Think about the decisions in each round, define the state, and thus obtain the <span class="arithmatex">\(dp\)</span> table</strong></p>
-<p>The decision in each round of this problem is to move one step down or right from the current cell. Let the row and column indices of the current cell be <span class="arithmatex">\([i, j]\)</span>. After moving down or right, the indices become <span class="arithmatex">\([i+1, j]\)</span> or <span class="arithmatex">\([i, j+1]\)</span>. Therefore, the state should include two variables, the row index and column index, denoted as <span class="arithmatex">\([i, j]\)</span>.</p>
-<p>State <span class="arithmatex">\([i, j]\)</span> corresponds to the subproblem: <strong>the minimum path sum from the starting point <span class="arithmatex">\([0, 0]\)</span> to <span class="arithmatex">\([i, j]\)</span></strong>, denoted as <span class="arithmatex">\(dp[i, j]\)</span>.</p>
-<p>From this, we obtain the two-dimensional <span class="arithmatex">\(dp\)</span> matrix shown in Figure 14-11, whose size is the same as the input grid <span class="arithmatex">\(grid\)</span>.</p>
+<p><strong>Step 1: Identify each decision, define the state, and establish the <span class="arithmatex">\(dp\)</span> table</strong></p>
+<p>Each decision moves one cell down or right. If the current cell has indices <span class="arithmatex">\([i, j]\)</span>, the next indices are <span class="arithmatex">\([i+1, j]\)</span> or <span class="arithmatex">\([i, j+1]\)</span>. The state therefore contains two variables, the row and column indices, and is denoted by <span class="arithmatex">\([i, j]\)</span>.</p>
+<p>State <span class="arithmatex">\([i, j]\)</span> corresponds to the subproblem of finding the minimum path sum from the starting point <span class="arithmatex">\([0, 0]\)</span> to <span class="arithmatex">\([i, j]\)</span>. Its solution is <span class="arithmatex">\(dp[i, j]\)</span>.</p>
+<p>This gives the two-dimensional <span class="arithmatex">\(dp\)</span> matrix shown in Figure 14-11, with the same dimensions as the input <span class="arithmatex">\(grid\)</span>.</p>
 <p><img alt="State definition and dp table" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_solution_state_definition.png" /></p>
 <p align="center"> Figure 14-11 &nbsp; State definition and dp table </p>
 
 <div class="admonition note">
 <p class="admonition-title">Note</p>
-<p>The dynamic programming and backtracking processes can be described as a sequence of decisions, and the state consists of all decision variables. It should contain all variables describing the progress of problem-solving, and should contain sufficient information to derive the next state.</p>
-<p>Each state corresponds to a subproblem, and we define a <span class="arithmatex">\(dp\)</span> table to store the solutions to all subproblems. Each independent variable of the state is a dimension of the <span class="arithmatex">\(dp\)</span> table. Essentially, the <span class="arithmatex">\(dp\)</span> table is a mapping between states and solutions to subproblems.</p>
+<p>Dynamic programming and backtracking can be described as sequences of decisions. A state consists of the decision variables and should contain every variable needed to describe progress, with enough information to derive the next state.</p>
+<p>Each state corresponds to a subproblem. A <span class="arithmatex">\(dp\)</span> table stores the solutions to all subproblems, with one dimension of the <span class="arithmatex">\(dp\)</span> table for each independent state variable. In essence, the <span class="arithmatex">\(dp\)</span> table maps states to subproblem solutions.</p>
 </div>
-<p><strong>Step 2: Identify the optimal substructure, and then derive the state transition equation</strong></p>
-<p>For state <span class="arithmatex">\([i, j]\)</span>, it can only transition from the cell above <span class="arithmatex">\([i-1, j]\)</span> or the cell to the left <span class="arithmatex">\([i, j-1]\)</span>. Therefore, the optimal substructure is: the minimum path sum to reach <span class="arithmatex">\([i, j]\)</span> is determined by the smaller of the minimum path sums of <span class="arithmatex">\([i, j-1]\)</span> and <span class="arithmatex">\([i-1, j]\)</span>.</p>
-<p>Based on the above analysis, the state transition equation shown in Figure 14-12 can be derived:</p>
+<p><strong>Step 2: Identify optimal substructure and derive the state transition equation</strong></p>
+<p>State <span class="arithmatex">\([i, j]\)</span> can only be reached from the cell above, <span class="arithmatex">\([i-1, j]\)</span>, or the cell to the left, <span class="arithmatex">\([i, j-1]\)</span>. The optimal substructure is therefore that the minimum path sum to <span class="arithmatex">\([i, j]\)</span> is determined by the smaller of the minimum path sums to <span class="arithmatex">\([i, j-1]\)</span> and <span class="arithmatex">\([i-1, j]\)</span>.</p>
+<p>This gives the state transition equation illustrated in Figure 14-12:</p>
 <div class="arithmatex">\[
 dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 \]</div>
@@ -4812,627 +4884,637 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 
 <div class="admonition note">
 <p class="admonition-title">Note</p>
-<p>Based on the defined <span class="arithmatex">\(dp\)</span> table, think about the relationship between the original problem and subproblems, and find the method to construct the optimal solution to the original problem from the optimal solutions to the subproblems, which is the optimal substructure.</p>
-<p>Once we identify the optimal substructure, we can use it to construct the state transition equation.</p>
+<p>Using the defined <span class="arithmatex">\(dp\)</span> table, examine the relationship between the original problem and its subproblems. Find how to construct an optimal solution from optimal subproblem solutions: this is the optimal substructure.</p>
+<p>Once we identify optimal substructure, we can use it to build the state transition equation.</p>
 </div>
-<p><strong>Step 3: Determine boundary conditions and state transition order</strong></p>
-<p>In this problem, states in the first row can only come from the state to their left, and states in the first column can only come from the state above them. Therefore, the first row <span class="arithmatex">\(i = 0\)</span> and first column <span class="arithmatex">\(j = 0\)</span> are boundary conditions.</p>
-<p>As shown in Figure 14-13, since each cell transitions from the cell to its left and the cell above it, we use loops to traverse the matrix, with the outer loop traversing rows and the inner loop traversing columns.</p>
-<p><img alt="Boundary conditions and state transition order" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_solution_initial_state.png" /></p>
-<p align="center"> Figure 14-13 &nbsp; Boundary conditions and state transition order </p>
+<p><strong>Step 3: Determine the boundary conditions and evaluation order</strong></p>
+<p>A state in the first row can only come from its left, and a state in the first column can only come from above. Thus, the first row <span class="arithmatex">\(i = 0\)</span> and first column <span class="arithmatex">\(j = 0\)</span> define the boundary conditions.</p>
+<p>As shown in Figure 14-13, each cell depends on the cells to its left and above. We therefore traverse the matrix with an outer loop over rows and an inner loop over columns.</p>
+<p><img alt="Boundary conditions and evaluation order" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_solution_initial_state.png" /></p>
+<p align="center"> Figure 14-13 &nbsp; Boundary conditions and evaluation order </p>
 
 <div class="admonition note">
 <p class="admonition-title">Note</p>
-<p>Boundary conditions in dynamic programming are used to initialize the <span class="arithmatex">\(dp\)</span> table, while in search they are used for pruning.</p>
-<p>The core of state transition order is to ensure that when computing the solution to the current problem, all the smaller subproblems it depends on have already been computed correctly.</p>
+<p>Boundary conditions initialize the <span class="arithmatex">\(dp\)</span> table in dynamic programming and support pruning in search.</p>
+<p>The key to choosing an evaluation order is ensuring that all smaller subproblems needed by the current state have already been solved correctly.</p>
 </div>
-<p>Based on the above analysis, we can directly write the dynamic programming code. However, subproblem decomposition is a top-down approach, so implementing in the order "brute force search <span class="arithmatex">\(\rightarrow\)</span> memoization <span class="arithmatex">\(\rightarrow\)</span> dynamic programming" is more aligned with thinking habits.</p>
-<h3 id="1-method-1-brute-force-search">1. &nbsp; Method 1: Brute Force Search<a class="headerlink" href="#1-method-1-brute-force-search" title="Permanent link">&para;</a></h3>
-<p>Starting from state <span class="arithmatex">\([i, j]\)</span>, we continuously decompose it into smaller states <span class="arithmatex">\([i-1, j]\)</span> and <span class="arithmatex">\([i, j-1]\)</span>. The recursive function includes the following elements.</p>
+<p>We could now write the dynamic programming solution directly. However, decomposing a problem is a top-down way of thinking, so implementing "brute-force search <span class="arithmatex">\(\rightarrow\)</span> memoized search <span class="arithmatex">\(\rightarrow\)</span> dynamic programming" often follows our reasoning more naturally.</p>
+<h3 id="1-method-1-brute-force-search">1. &nbsp; Method 1: Brute-force search<a class="headerlink" href="#1-method-1-brute-force-search" title="Permanent link">&para;</a></h3>
+<p>Starting from <span class="arithmatex">\([i, j]\)</span>, recursively decompose the problem into smaller states <span class="arithmatex">\([i-1, j]\)</span> and <span class="arithmatex">\([i, j-1]\)</span>. The recursive function has the following elements.</p>
 <ul>
-<li><strong>Recursive parameters</strong>: state <span class="arithmatex">\([i, j]\)</span>.</li>
-<li><strong>Return value</strong>: minimum path sum from <span class="arithmatex">\([0, 0]\)</span> to <span class="arithmatex">\([i, j]\)</span>, which is <span class="arithmatex">\(dp[i, j]\)</span>.</li>
-<li><strong>Termination condition</strong>: when <span class="arithmatex">\(i = 0\)</span> and <span class="arithmatex">\(j = 0\)</span>, return cost <span class="arithmatex">\(grid[0, 0]\)</span>.</li>
-<li><strong>Pruning</strong>: when <span class="arithmatex">\(i &lt; 0\)</span> or <span class="arithmatex">\(j &lt; 0\)</span>, the index is out of bounds, return cost <span class="arithmatex">\(+\infty\)</span>, representing infeasibility.</li>
+<li><strong>Parameters</strong>: The state <span class="arithmatex">\([i, j]\)</span>.</li>
+<li><strong>Return value</strong>: The minimum path sum <span class="arithmatex">\(dp[i, j]\)</span> from <span class="arithmatex">\([0, 0]\)</span> to <span class="arithmatex">\([i, j]\)</span>.</li>
+<li><strong>Base case</strong>: When <span class="arithmatex">\(i = 0\)</span> and <span class="arithmatex">\(j = 0\)</span>, return <span class="arithmatex">\(grid[0, 0]\)</span>.</li>
+<li><strong>Pruning</strong>: If <span class="arithmatex">\(i &lt; 0\)</span> or <span class="arithmatex">\(j &lt; 0\)</span>, the indices are out of bounds. Return cost <span class="arithmatex">\(+\infty\)</span> to mark the path as infeasible.</li>
 </ul>
-<p>The implementation code is as follows:</p>
+<p>The implementation is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">min_path_sum.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">min_path_sum_dfs</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]],</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Minimum path sum: Brute-force search"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Minimum path sum: brute-force search"""</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Stop searching at the top-left cell</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">j</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="k">return</span> <span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="k">return</span> <span class="n">inf</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="c1"># Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="c1"># Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="n">up</span> <span class="o">=</span> <span class="n">min_path_sum_dfs</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="n">left</span> <span class="o">=</span> <span class="n">min_path_sum_dfs</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="c1"># Return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="c1"># Return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>    <span class="k">return</span> <span class="nb">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span> <span class="n">up</span><span class="p">)</span> <span class="o">+</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Minimum path sum: Brute-force search */</span>
+<div class="highlight"><span class="filename">min_path_sum.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Minimum path sum: brute-force search */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDFS</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)</span>
+<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">INT_MAX</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Minimum path sum: Brute-force search */</span>
+<div class="highlight"><span class="filename">min_path_sum.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Minimum path sum: brute-force search */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDFS</span><span class="p">(</span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">Integer</span><span class="p">.</span><span class="na">MAX_VALUE</span><span class="p">;</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)</span>
+<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Minimum path sum: Brute-force search */</span>
+<div class="highlight"><span class="filename">min_path_sum.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Minimum path sum: brute-force search */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">MinPathSumDFS</span><span class="p">(</span><span class="kt">int</span><span class="p">[][]</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kt">int</span><span class="p">.</span><span class="n">MaxValue</span><span class="p">;</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)</span>
+<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MinPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MinPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Minimum path sum: Brute-force search */</span>
+<div class="highlight"><span class="filename">min_path_sum.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Minimum path sum: brute-force search */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">minPathSumDFS</span><span class="p">(</span><span class="nx">grid</span><span class="w"> </span><span class="p">[][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return math.MaxInt to represent an infinite cost</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">math</span><span class="p">.</span><span class="nx">MaxInt</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)</span>
+<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="nx">up</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">minPathSumDFS</span><span class="p">(</span><span class="nx">grid</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="nx">left</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">minPathSumDFS</span><span class="p">(</span><span class="nx">grid</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Min</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">left</span><span class="p">),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">up</span><span class="p">)))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Minimum path sum: Brute-force search */</span>
+<div class="highlight"><span class="filename">min_path_sum.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Minimum path sum: brute-force search */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]],</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return Int.max to represent an infinite cost</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">.</span><span class="bp">max</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)</span>
+<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">up</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="bp">min</span><span class="p">(</span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Minimum path sum: Brute-force search */</span>
+<div class="highlight"><span class="filename">min_path_sum.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Minimum path sum: brute-force search */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">minPathSumDFS</span><span class="p">(</span><span class="nx">grid</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="mf">0</span><span class="p">];</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">Infinity</span><span class="p">;</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)</span>
+<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">minPathSumDFS</span><span class="p">(</span><span class="nx">grid</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">);</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">minPathSumDFS</span><span class="p">(</span><span class="nx">grid</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Minimum path sum: Brute-force search */</span>
+<div class="highlight"><span class="filename">min_path_sum.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Minimum path sum: brute-force search */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">minPathSumDFS</span><span class="p">(</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="nx">grid</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="nb">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="nx">j</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="mf">0</span><span class="p">];</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">Infinity</span><span class="p">;</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="c1">// Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)</span>
+<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="c1">// Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">minPathSumDFS</span><span class="p">(</span><span class="nx">grid</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">);</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">minPathSumDFS</span><span class="p">(</span><span class="nx">grid</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Minimum path sum: Brute-force search */</span>
+<div class="highlight"><span class="filename">min_path_sum.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Minimum path sum: brute-force search */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="m">0</span><span class="p">][</span><span class="m">0</span><span class="p">];</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// If a row or column index is out of bounds, return a large cost representing +∞</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="c1">// In Dart, int type is fixed-range integer, no value representing "infinity"</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="c1">// Dart int has no infinity value; use 2^31 as a sentinel for this example</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">BigInt</span><span class="p">.</span><span class="n">from</span><span class="p">(</span><span class="m">2</span><span class="p">).</span><span class="n">pow</span><span class="p">(</span><span class="m">31</span><span class="p">).</span><span class="n">toInt</span><span class="p">();</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="c1">// Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)</span>
+<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="c1">// Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
-<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="c1">// Return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="c1">// Return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Minimum path sum: Brute-force search */</span>
+<div class="highlight"><span class="filename">min_path_sum.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Minimum path sum: brute-force search */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">min_path_sum_dfs</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return i32::MAX to represent infinite cost</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kt">i32</span><span class="p">::</span><span class="n">MAX</span><span class="p">;</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)</span>
+<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min_path_sum_dfs</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min_path_sum_dfs</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">    </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Minimum path sum: Brute-force search */</span>
+<div class="highlight"><span class="filename">min_path_sum.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Minimum path sum: brute-force search */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDFS</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">][</span><span class="n">MAX_SIZE</span><span class="p">],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)</span>
+<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">INT_MAX</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Minimum path sum: Brute-force search */</span>
+<div class="highlight"><span class="filename">min_path_sum.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Minimum path sum: brute-force search */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="n">Array</span><span class="o">&lt;</span><span class="n">IntArray</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="m">0</span><span class="o">][</span><span class="m">0</span><span class="o">]</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return Int.MAX_VALUE as an infinite-cost sentinel</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kt">Int</span><span class="p">.</span><span class="na">MAX_VALUE</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)</span>
+<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="c1">// Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFS</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
-<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">    </span><span class="c1">// Return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Minimum path sum: brute force search ###</span>
+<div class="highlight"><span class="filename">min_path_sum.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Minimum path sum: brute-force search ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">min_path_sum_dfs</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Stop searching at the top-left cell</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="nb">Float</span><span class="o">::</span><span class="no">INFINITY</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="c1"># Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)</span>
+<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="c1"># Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min_path_sum_dfs</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min_path_sum_dfs</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="c1"># Return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="c1"># Return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">  </span><span class="o">[</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="o">].</span><span class="n">min</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<p>Figure 14-14 shows the recursion tree rooted at <span class="arithmatex">\(dp[2, 1]\)</span>, which includes some overlapping subproblems whose number will increase sharply as the size of grid <code>grid</code> grows.</p>
-<p>Essentially, the reason for overlapping subproblems is: <strong>there are multiple paths from the top-left corner to reach a certain cell</strong>.</p>
-<p><img alt="Brute force search recursion tree" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_dfs.png" /></p>
-<p align="center"> Figure 14-14 &nbsp; Brute force search recursion tree </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dfs%28grid%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20j%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Minimum%20path%20sum%3A%20brute-force%20search%22%22%22%0A%20%20%20%20%23%20Stop%20searching%20at%20the%20top-left%20cell%0A%20%20%20%20if%20i%20%3D%3D%200%20and%20j%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20If%20a%20row%20or%20column%20index%20is%20out%20of%20bounds,%20return%20a%20cost%20of%20%2B%E2%88%9E%0A%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%3A%0A%20%20%20%20%20%20%20%20return%20inf%0A%20%20%20%20%23%20Calculate%20the%20minimum%20path%20costs%20from%20the%20top-left%20corner%20to%20%28i-1,%20j%29%20and%20%28i,%20j-1%29%0A%20%20%20%20up%20%3D%20min_path_sum_dfs%28grid,%20i%20-%201,%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs%28grid,%20i,%20j%20-%201%29%0A%20%20%20%20%23%20Return%20the%20minimum%20path%20cost%20from%20the%20top-left%20corner%20to%20%28i,%20j%29%0A%20%20%20%20return%20min%28left,%20up%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20Brute-force%20search%0A%20%20%20%20res%20%3D%20min_path_sum_dfs%28grid,%20n%20-%201,%20m%20-%201%29%0A%20%20%20%20print%28f%22Minimum%20path%20sum%20from%20the%20top-left%20to%20the%20bottom-right%20corner%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dfs%28grid%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20j%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Minimum%20path%20sum%3A%20brute-force%20search%22%22%22%0A%20%20%20%20%23%20Stop%20searching%20at%20the%20top-left%20cell%0A%20%20%20%20if%20i%20%3D%3D%200%20and%20j%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20If%20a%20row%20or%20column%20index%20is%20out%20of%20bounds,%20return%20a%20cost%20of%20%2B%E2%88%9E%0A%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%3A%0A%20%20%20%20%20%20%20%20return%20inf%0A%20%20%20%20%23%20Calculate%20the%20minimum%20path%20costs%20from%20the%20top-left%20corner%20to%20%28i-1,%20j%29%20and%20%28i,%20j-1%29%0A%20%20%20%20up%20%3D%20min_path_sum_dfs%28grid,%20i%20-%201,%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs%28grid,%20i,%20j%20-%201%29%0A%20%20%20%20%23%20Return%20the%20minimum%20path%20cost%20from%20the%20top-left%20corner%20to%20%28i,%20j%29%0A%20%20%20%20return%20min%28left,%20up%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20Brute-force%20search%0A%20%20%20%20res%20%3D%20min_path_sum_dfs%28grid,%20n%20-%201,%20m%20-%201%29%0A%20%20%20%20print%28f%22Minimum%20path%20sum%20from%20the%20top-left%20to%20the%20bottom-right%20corner%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 14-14 shows the recursion tree rooted at <span class="arithmatex">\(dp[2, 1]\)</span>. It includes overlapping subproblems, whose number grows rapidly with the size of <code>grid</code>.</p>
+<p>The underlying reason for overlapping subproblems is that <strong>multiple paths can reach the same cell from the top-left corner</strong>.</p>
+<p><img alt="Recursion tree for brute-force search" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_dfs.png" /></p>
+<p align="center"> Figure 14-14 &nbsp; Recursion tree for brute-force search </p>
 
-<p>Each state has two choices, down and right, so the total number of steps from the top-left corner to the bottom-right corner is <span class="arithmatex">\(m + n - 2\)</span>, giving a worst-case time complexity of <span class="arithmatex">\(O(2^{m + n})\)</span>, where <span class="arithmatex">\(n\)</span> and <span class="arithmatex">\(m\)</span> are the number of rows and columns of the grid, respectively. Note that this calculation does not account for situations near the grid boundaries, where only one choice remains when reaching the grid boundary, so the actual number of paths will be somewhat less.</p>
-<h3 id="2-method-2-memoization">2. &nbsp; Method 2: Memoization<a class="headerlink" href="#2-method-2-memoization" title="Permanent link">&para;</a></h3>
-<p>We introduce a memo list <code>mem</code> of the same size as grid <code>grid</code> to record the solutions to subproblems and prune overlapping subproblems:</p>
+<p>Each state offers two choices, down and right, and reaching the bottom-right corner takes <span class="arithmatex">\(m + n - 2\)</span> moves. The worst-case time complexity is therefore <span class="arithmatex">\(O(2^{m + n})\)</span>, where <span class="arithmatex">\(n\)</span> and <span class="arithmatex">\(m\)</span> are the numbers of rows and columns. This estimate ignores the grid boundaries, where only one choice remains, so the actual number of paths is smaller.</p>
+<h3 id="2-method-2-memoized-search">2. &nbsp; Method 2: Memoized search<a class="headerlink" href="#2-method-2-memoized-search" title="Permanent link">&para;</a></h3>
+<p>Introduce a memoization table <code>mem</code> with the same dimensions as <code>grid</code> to store subproblem solutions and prune repeated subproblems:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">min_path_sum.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">min_path_sum_dfs_mem</span><span class="p">(</span>
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a>    <span class="n">grid</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]],</span> <span class="n">mem</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]],</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="nb">int</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="w">    </span><span class="sd">"""Minimum path sum: Memoization search"""</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="w">    </span><span class="sd">"""Minimum path sum: memoized search"""</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Stop searching at the top-left cell</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">j</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="k">return</span> <span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>    <span class="c1"># If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>    <span class="c1"># If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>    <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>        <span class="k">return</span> <span class="n">inf</span>
-<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>    <span class="c1"># If there's a record, return it directly</span>
+<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>    <span class="c1"># If a result has already been recorded, return it directly</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>    <span class="k">if</span> <span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>        <span class="k">return</span> <span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
-<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>    <span class="c1"># Minimum path cost for left and upper cells</span>
+<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>    <span class="c1"># Minimum path costs to the cells on the left and above</span>
 <a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>    <span class="n">up</span> <span class="o">=</span> <span class="n">min_path_sum_dfs_mem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span> <span class="n">mem</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>    <span class="n">left</span> <span class="o">=</span> <span class="n">min_path_sum_dfs_mem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span> <span class="n">mem</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>    <span class="c1"># Record and return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>    <span class="c1"># Record and return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a>    <span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span> <span class="n">up</span><span class="p">)</span> <span class="o">+</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a>    <span class="k">return</span> <span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Minimum path sum: Memoization search */</span>
+<div class="highlight"><span class="filename">min_path_sum.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Minimum path sum: memoized search */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDFSMem</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">-1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">    </span><span class="c1">// Minimum path cost for left and upper cells</span>
+<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">    </span><span class="c1">// Minimum path costs to the cells on the left and above</span>
 <a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="w">    </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">INT_MAX</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
 <a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Minimum path sum: Memoization search */</span>
+<div class="highlight"><span class="filename">min_path_sum.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Minimum path sum: memoized search */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">Integer</span><span class="p">.</span><span class="na">MAX_VALUE</span><span class="p">;</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="c1">// Minimum path cost for left and upper cells</span>
+<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="c1">// Minimum path costs to the cells on the left and above</span>
 <a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w">    </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Minimum path sum: Memoization search */</span>
+<div class="highlight"><span class="filename">min_path_sum.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Minimum path sum: memoized search */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">MinPathSumDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="p">[][]</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[][]</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kt">int</span><span class="p">.</span><span class="n">MaxValue</span><span class="p">;</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">    </span><span class="c1">// Minimum path cost for left and upper cells</span>
+<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">    </span><span class="c1">// Minimum path costs to the cells on the left and above</span>
 <a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MinPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MinPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w">    </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Minimum path sum: Memoization search */</span>
+<div class="highlight"><span class="filename">min_path_sum.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Minimum path sum: memoized search */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">minPathSumDFSMem</span><span class="p">(</span><span class="nx">grid</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="w"> </span><span class="p">[][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return math.MaxInt to represent an infinite cost</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">math</span><span class="p">.</span><span class="nx">MaxInt</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">    </span><span class="c1">// Minimum path cost for left and upper cells</span>
+<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">    </span><span class="c1">// Minimum path costs to the cells on the left and above</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">    </span><span class="nx">up</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">minPathSumDFSMem</span><span class="p">(</span><span class="nx">grid</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span>
 <a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">    </span><span class="nx">left</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">minPathSumDFSMem</span><span class="p">(</span><span class="nx">grid</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="w">    </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Min</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">left</span><span class="p">),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">up</span><span class="p">)))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span>
 <a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span>
 <a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Minimum path sum: Memoization search */</span>
+<div class="highlight"><span class="filename">min_path_sum.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Minimum path sum: memoized search */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">minPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]],</span><span class="w"> </span><span class="n">mem</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]],</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return Int.max to represent an infinite cost</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">.</span><span class="bp">max</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">    </span><span class="c1">// Minimum path cost for left and upper cells</span>
+<a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">    </span><span class="c1">// Minimum path costs to the cells on the left and above</span>
 <a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">up</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">    </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">min</span><span class="p">(</span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Minimum path sum: Memoization search */</span>
+<div class="highlight"><span class="filename">min_path_sum.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Minimum path sum: memoized search */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">minPathSumDFSMem</span><span class="p">(</span><span class="nx">grid</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="mf">0</span><span class="p">];</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">Infinity</span><span class="p">;</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">    </span><span class="c1">// Minimum path cost for left and upper cells</span>
+<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">    </span><span class="c1">// Minimum path costs to the cells on the left and above</span>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">minPathSumDFSMem</span><span class="p">(</span><span class="nx">grid</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">);</span>
 <a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">minPathSumDFSMem</span><span class="p">(</span><span class="nx">grid</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">    </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Minimum path sum: Memoization search */</span>
+<div class="highlight"><span class="filename">min_path_sum.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Minimum path sum: memoized search */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">minPathSumDFSMem</span><span class="p">(</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="nx">grid</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="nb">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="nx">mem</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="nb">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="nx">j</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="mf">0</span><span class="p">];</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">Infinity</span><span class="p">;</span>
 <a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">    </span><span class="c1">// Minimum path cost for left and upper cells</span>
+<a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">    </span><span class="c1">// Minimum path costs to the cells on the left and above</span>
 <a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">minPathSumDFSMem</span><span class="p">(</span><span class="nx">grid</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">);</span>
 <a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">minPathSumDFSMem</span><span class="p">(</span><span class="nx">grid</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-20-23" name="__codelineno-20-23" href="#__codelineno-20-23"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-20-23" name="__codelineno-20-23" href="#__codelineno-20-23"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">    </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Minimum path sum: Memoization search */</span>
+<div class="highlight"><span class="filename">min_path_sum.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Minimum path sum: memoized search */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="m">0</span><span class="p">][</span><span class="m">0</span><span class="p">];</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="c1">// If a row or column index is out of bounds, return a large cost representing +∞</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">    </span><span class="c1">// In Dart, int type is fixed-range integer, no value representing "infinity"</span>
+<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">    </span><span class="c1">// Dart int has no infinity value; use 2^31 as a sentinel for this example</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">BigInt</span><span class="p">.</span><span class="n">from</span><span class="p">(</span><span class="m">2</span><span class="p">).</span><span class="n">pow</span><span class="p">(</span><span class="m">31</span><span class="p">).</span><span class="n">toInt</span><span class="p">();</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">  </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">  </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">  </span><span class="c1">// Minimum path cost for left and upper cells</span>
+<a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">  </span><span class="c1">// Minimum path costs to the cells on the left and above</span>
 <a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
-<a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="w">  </span><span class="c1">// Record and return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="w">  </span><span class="c1">// Record and return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="w">  </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Minimum path sum: Memoization search */</span>
+<div class="highlight"><span class="filename">min_path_sum.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Minimum path sum: memoized search */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">min_path_sum_dfs_mem</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return i32::MAX to represent infinite cost</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kt">i32</span><span class="p">::</span><span class="n">MAX</span><span class="p">;</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">];</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">    </span><span class="c1">// Minimum path cost for left and upper cells</span>
+<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">    </span><span class="c1">// Minimum path costs to the cells on the left and above</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min_path_sum_dfs_mem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min_path_sum_dfs_mem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w">    </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">];</span>
 <a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w">    </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span>
 <a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Minimum path sum: Memoization search */</span>
+<div class="highlight"><span class="filename">min_path_sum.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Minimum path sum: memoized search */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">][</span><span class="n">MAX_SIZE</span><span class="p">],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">][</span><span class="n">MAX_SIZE</span><span class="p">],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">-1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">    </span><span class="c1">// Minimum path cost for left and upper cells</span>
+<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">    </span><span class="c1">// Minimum path costs to the cells on the left and above</span>
 <a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w">    </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">INT_MAX</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
 <a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Minimum path sum: Memoization search */</span>
+<div class="highlight"><span class="filename">min_path_sum.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Minimum path sum: memoized search */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">minPathSumDFSMem</span><span class="p">(</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="n">Array</span><span class="o">&lt;</span><span class="n">IntArray</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="n">mem</span><span class="p">:</span><span class="w"> </span><span class="n">Array</span><span class="o">&lt;</span><span class="n">IntArray</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="c1">// If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="c1">// Stop searching at the top-left cell</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="m">0</span><span class="o">][</span><span class="m">0</span><span class="o">]</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="c1">// If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="c1">// If a row or column index is out of bounds, return Int.MAX_VALUE as an infinite-cost sentinel</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kt">Int</span><span class="p">.</span><span class="na">MAX_VALUE</span>
 <a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w">    </span><span class="c1">// Minimum path cost for left and upper cells</span>
+<a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w">    </span><span class="c1">// Minimum path costs to the cells on the left and above</span>
 <a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">minPathSumDFSMem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
-<a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">    </span><span class="c1">// Record and return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a><span class="w">    </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-24-25" name="__codelineno-24-25" href="#__codelineno-24-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-24-26" name="__codelineno-24-26" href="#__codelineno-24-26"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Minimum path sum: memoization search ###</span>
+<div class="highlight"><span class="filename">min_path_sum.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Minimum path sum: memoized search ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">min_path_sum_dfs_mem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># If it's the top-left cell, terminate the search</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Stop searching at the top-left cell</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># If row or column index is out of bounds, return +∞ cost</span>
+<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># If a row or column index is out of bounds, return a cost of +∞</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="nb">Float</span><span class="o">::</span><span class="no">INFINITY</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="c1"># If there's a record, return it directly</span>
+<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="c1"># If a result has already been recorded, return it directly</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
-<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="c1"># Minimum path cost for left and upper cells</span>
+<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="c1"># Minimum path costs to the cells on the left and above</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="n">up</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min_path_sum_dfs_mem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">  </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min_path_sum_dfs_mem</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">  </span><span class="c1"># Record and return the minimum path cost from top-left to (i, j)</span>
+<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">  </span><span class="c1"># Record and return the minimum path cost from the top-left corner to (i, j)</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">  </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">up</span><span class="o">].</span><span class="n">min</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<p>As shown in Figure 14-15, after introducing memoization, all subproblem solutions only need to be computed once, so the time complexity depends on the total number of states, which is the grid size <span class="arithmatex">\(O(nm)\)</span>.</p>
-<p><img alt="Memoization recursion tree" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_dfs_mem.png" /></p>
-<p align="center"> Figure 14-15 &nbsp; Memoization recursion tree </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dfs_mem%28%0A%20%20%20%20grid%3A%20list%5Blist%5Bint%5D%5D,%20mem%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20j%3A%20int%0A%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Minimum%20path%20sum%3A%20memoized%20search%22%22%22%0A%20%20%20%20%23%20Stop%20searching%20at%20the%20top-left%20cell%0A%20%20%20%20if%20i%20%3D%3D%200%20and%20j%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20If%20a%20row%20or%20column%20index%20is%20out%20of%20bounds,%20return%20a%20cost%20of%20%2B%E2%88%9E%0A%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%3A%0A%20%20%20%20%20%20%20%20return%20inf%0A%20%20%20%20%23%20If%20a%20result%20has%20already%20been%20recorded,%20return%20it%20directly%0A%20%20%20%20if%20mem%5Bi%5D%5Bj%5D%20%21%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%5Bj%5D%0A%20%20%20%20%23%20Minimum%20path%20costs%20to%20the%20cells%20on%20the%20left%20and%20above%0A%20%20%20%20up%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20i%20-%201,%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20i,%20j%20-%201%29%0A%20%20%20%20%23%20Record%20and%20return%20the%20minimum%20path%20cost%20from%20the%20top-left%20corner%20to%20%28i,%20j%29%0A%20%20%20%20mem%5Bi%5D%5Bj%5D%20%3D%20min%28left,%20up%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20mem%5Bi%5D%5Bj%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%23%20Memoized%20search%0A%20%20%20%20mem%20%3D%20%5B%5B-1%5D%20%2A%20m%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20res%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20n%20-%201,%20m%20-%201%29%0A%20%20%20%20print%28f%22Minimum%20path%20sum%20from%20the%20top-left%20to%20the%20bottom-right%20corner%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dfs_mem%28%0A%20%20%20%20grid%3A%20list%5Blist%5Bint%5D%5D,%20mem%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20j%3A%20int%0A%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Minimum%20path%20sum%3A%20memoized%20search%22%22%22%0A%20%20%20%20%23%20Stop%20searching%20at%20the%20top-left%20cell%0A%20%20%20%20if%20i%20%3D%3D%200%20and%20j%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20If%20a%20row%20or%20column%20index%20is%20out%20of%20bounds,%20return%20a%20cost%20of%20%2B%E2%88%9E%0A%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%3A%0A%20%20%20%20%20%20%20%20return%20inf%0A%20%20%20%20%23%20If%20a%20result%20has%20already%20been%20recorded,%20return%20it%20directly%0A%20%20%20%20if%20mem%5Bi%5D%5Bj%5D%20%21%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%5Bj%5D%0A%20%20%20%20%23%20Minimum%20path%20costs%20to%20the%20cells%20on%20the%20left%20and%20above%0A%20%20%20%20up%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20i%20-%201,%20j%29%0A%20%20%20%20left%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20i,%20j%20-%201%29%0A%20%20%20%20%23%20Record%20and%20return%20the%20minimum%20path%20cost%20from%20the%20top-left%20corner%20to%20%28i,%20j%29%0A%20%20%20%20mem%5Bi%5D%5Bj%5D%20%3D%20min%28left,%20up%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20mem%5Bi%5D%5Bj%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%23%20Memoized%20search%0A%20%20%20%20mem%20%3D%20%5B%5B-1%5D%20%2A%20m%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20res%20%3D%20min_path_sum_dfs_mem%28grid,%20mem,%20n%20-%201,%20m%20-%201%29%0A%20%20%20%20print%28f%22Minimum%20path%20sum%20from%20the%20top-left%20to%20the%20bottom-right%20corner%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=16&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>As shown in Figure 14-15, memoization ensures that every subproblem is solved only once. The time complexity is therefore determined by the number of states, the grid size <span class="arithmatex">\(O(nm)\)</span>.</p>
+<p><img alt="Recursion tree for memoized search" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_dfs_mem.png" /></p>
+<p align="center"> Figure 14-15 &nbsp; Recursion tree for memoized search </p>
 
-<h3 id="3-method-3-dynamic-programming">3. &nbsp; Method 3: Dynamic Programming<a class="headerlink" href="#3-method-3-dynamic-programming" title="Permanent link">&para;</a></h3>
-<p>Implement the dynamic programming solution based on iteration, as shown in the code below:</p>
+<h3 id="3-method-3-dynamic-programming">3. &nbsp; Method 3: Dynamic programming<a class="headerlink" href="#3-method-3-dynamic-programming" title="Permanent link">&para;</a></h3>
+<p>The following code implements dynamic programming iteratively:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">min_path_sum.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">min_path_sum_dp</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]])</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Minimum path sum: Dynamic programming"""</span>
+<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Minimum path sum: dynamic programming"""</span>
 <a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">grid</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-<a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="c1"># Initialize dp table</span>
+<a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">m</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span>
 <a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>    <span class="c1"># State transition: first row</span>
+<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>    <span class="c1"># State transitions: first row</span>
 <a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
 <a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>        <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
-<a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>    <span class="c1"># State transition: first column</span>
+<a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>    <span class="c1"># State transitions: first column</span>
 <a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a>        <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-<a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>    <span class="c1"># State transition: rest of the rows and columns</span>
+<a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>    <span class="c1"># State transitions: remaining rows and columns</span>
 <a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
 <a id="__codelineno-26-16" name="__codelineno-26-16" href="#__codelineno-26-16"></a>            <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">])</span> <span class="o">+</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
@@ -5440,21 +5522,21 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Minimum path sum: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Minimum path sum: dynamic programming */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDP</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">grid</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">.</span><span class="n">size</span><span class="p">(),</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">].</span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">m</span><span class="p">));</span>
 <a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
-<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="w">    </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="w">    </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-27-16" name="__codelineno-27-16" href="#__codelineno-27-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-17" name="__codelineno-27-17" href="#__codelineno-27-17"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-18" name="__codelineno-27-18" href="#__codelineno-27-18"></a><span class="w">            </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
@@ -5465,21 +5547,21 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Minimum path sum: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Minimum path sum: dynamic programming */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDP</span><span class="p">(</span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">grid</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">.</span><span class="na">length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
-<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">    </span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="o">][</span><span class="n">m</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="w">    </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="w">    </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a><span class="w">            </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="p">;</span>
@@ -5490,21 +5572,21 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Minimum path sum: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Minimum path sum: dynamic programming */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">MinPathSumDP</span><span class="p">(</span><span class="kt">int</span><span class="p">[][]</span><span class="w"> </span><span class="n">grid</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">.</span><span class="n">Length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">].</span><span class="n">Length</span><span class="p">;</span>
-<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">    </span><span class="kt">int</span><span class="p">[,]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="p">];</span>
 <a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
-<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="w">    </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="w">    </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a><span class="w">            </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
@@ -5515,24 +5597,24 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Minimum path sum: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Minimum path sum: dynamic programming */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">minPathSumDP</span><span class="p">(</span><span class="nx">grid</span><span class="w"> </span><span class="p">[][]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">grid</span><span class="p">),</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
 <a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="p">)</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-<a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span>
 <a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="w">    </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="w">    </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-20" name="__codelineno-30-20" href="#__codelineno-30-20"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-21" name="__codelineno-30-21" href="#__codelineno-30-21"></a><span class="w">            </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Min</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">j</span><span class="p">])))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span>
@@ -5543,22 +5625,22 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Minimum path sum: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Minimum path sum: dynamic programming */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">minPathSumDP</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">grid</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">].</span><span class="bp">count</span>
-<a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">m</span><span class="p">),</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-<a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="w">    </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="w">    </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-31-17" name="__codelineno-31-17" href="#__codelineno-31-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-18" name="__codelineno-31-18" href="#__codelineno-31-18"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-19" name="__codelineno-31-19" href="#__codelineno-31-19"></a><span class="w">            </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
@@ -5569,24 +5651,24 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Minimum path sum: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Minimum path sum: dynamic programming */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">minPathSumDP</span><span class="p">(</span><span class="nx">grid</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grid</span><span class="p">.</span><span class="nx">length</span><span class="p">,</span>
 <a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">].</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span>
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">        </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span>
 <a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="p">);</span>
 <a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="mf">0</span><span class="p">];</span>
-<a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="w">    </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="w">    </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-32-15" name="__codelineno-32-15" href="#__codelineno-32-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-16" name="__codelineno-32-16" href="#__codelineno-32-16"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">0</span><span class="p">];</span>
 <a id="__codelineno-32-17" name="__codelineno-32-17" href="#__codelineno-32-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-32-18" name="__codelineno-32-18" href="#__codelineno-32-18"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-32-18" name="__codelineno-32-18" href="#__codelineno-32-18"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-32-19" name="__codelineno-32-19" href="#__codelineno-32-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-20" name="__codelineno-32-20" href="#__codelineno-32-20"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-21" name="__codelineno-32-21" href="#__codelineno-32-21"></a><span class="w">            </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
@@ -5597,24 +5679,24 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Minimum path sum: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Minimum path sum: dynamic programming */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">minPathSumDP</span><span class="p">(</span><span class="nx">grid</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="nb">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;&gt;</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grid</span><span class="p">.</span><span class="nx">length</span><span class="p">,</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">].</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">n</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span>
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">        </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">m</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span>
 <a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">    </span><span class="p">);</span>
 <a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="mf">0</span><span class="p">];</span>
-<a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">    </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">    </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">0</span><span class="p">];</span>
 <a id="__codelineno-33-17" name="__codelineno-33-17" href="#__codelineno-33-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-33-18" name="__codelineno-33-18" href="#__codelineno-33-18"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-33-18" name="__codelineno-33-18" href="#__codelineno-33-18"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-33-19" name="__codelineno-33-19" href="#__codelineno-33-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-20" name="__codelineno-33-20" href="#__codelineno-33-20"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-21" name="__codelineno-33-21" href="#__codelineno-33-21"></a><span class="w">            </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
@@ -5625,21 +5707,21 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Minimum path sum: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Minimum path sum: dynamic programming */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">minPathSumDP</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">grid</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">.</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="m">0</span><span class="p">].</span><span class="n">length</span><span class="p">;</span>
-<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">));</span>
 <a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">][</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="m">0</span><span class="p">][</span><span class="m">0</span><span class="p">];</span>
-<a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">  </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">  </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="m">0</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w">  </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w">  </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="m">0</span><span class="p">];</span>
 <a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">  </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">  </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-17" name="__codelineno-34-17" href="#__codelineno-34-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-18" name="__codelineno-34-18" href="#__codelineno-34-18"></a><span class="w">      </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
@@ -5650,21 +5732,21 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Minimum path sum: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Minimum path sum: dynamic programming */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">min_path_sum_dp</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">grid</span><span class="p">.</span><span class="n">len</span><span class="p">(),</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">].</span><span class="n">len</span><span class="p">());</span>
-<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">m</span><span class="p">];</span><span class="w"> </span><span class="n">n</span><span class="p">];</span>
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
-<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="w">    </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="w">    </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-35-16" name="__codelineno-35-16" href="#__codelineno-35-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-17" name="__codelineno-35-17" href="#__codelineno-35-17"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-18" name="__codelineno-35-18" href="#__codelineno-35-18"></a><span class="w">            </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
@@ -5675,23 +5757,23 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Minimum path sum: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Minimum path sum: dynamic programming */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">][</span><span class="n">MAX_SIZE</span><span class="p">],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">**</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">));</span>
 <a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
-<a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">    </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">    </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-36-14" name="__codelineno-36-14" href="#__codelineno-36-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-36-16" name="__codelineno-36-16" href="#__codelineno-36-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-36-17" name="__codelineno-36-17" href="#__codelineno-36-17"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-36-17" name="__codelineno-36-17" href="#__codelineno-36-17"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-36-18" name="__codelineno-36-18" href="#__codelineno-36-18"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-19" name="__codelineno-36-19" href="#__codelineno-36-19"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-20" name="__codelineno-36-20" href="#__codelineno-36-20"></a><span class="w">            </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
@@ -5707,22 +5789,22 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Minimum path sum: Dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Minimum path sum: dynamic programming */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">minPathSumDP</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="n">Array</span><span class="o">&lt;</span><span class="n">IntArray</span><span class="o">&gt;</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="m">0</span><span class="o">]</span><span class="p">.</span><span class="na">size</span>
-<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Array</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="m">0</span><span class="o">][</span><span class="m">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="m">0</span><span class="o">][</span><span class="m">0</span><span class="o">]</span>
-<a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="m">0</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="m">0</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="m">0</span><span class="o">][</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="w">    </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="w">    </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="m">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">][</span><span class="m">0</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="m">0</span><span class="o">]</span>
 <a id="__codelineno-37-15" name="__codelineno-37-15" href="#__codelineno-37-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-37-16" name="__codelineno-37-16" href="#__codelineno-37-16"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-37-16" name="__codelineno-37-16" href="#__codelineno-37-16"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-37-17" name="__codelineno-37-17" href="#__codelineno-37-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-18" name="__codelineno-37-18" href="#__codelineno-37-18"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-19" name="__codelineno-37-19" href="#__codelineno-37-19"></a><span class="w">            </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span>
@@ -5736,14 +5818,14 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 <div class="highlight"><span class="filename">min_path_sum.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Minimum path sum: dynamic programming ###</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">min_path_sum_dp</span><span class="p">(</span><span class="n">grid</span><span class="p">)</span>
 <a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="o">.</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">grid</span><span class="o">.</span><span class="n">first</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="c1"># Initialize dp table</span>
+<a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span>
-<a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="w">  </span><span class="c1"># State transition: first row</span>
+<a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="w">  </span><span class="c1"># State transitions: first row</span>
 <a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">  </span><span class="p">(</span><span class="mi">1</span><span class="o">...</span><span class="n">m</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">j</span><span class="o">|</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">  </span><span class="c1"># State transition: first column</span>
+<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">  </span><span class="c1"># State transitions: first column</span>
 <a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">  </span><span class="p">(</span><span class="mi">1</span><span class="o">...</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="w">  </span><span class="c1"># State transition: rest of the rows and columns</span>
+<a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="w">  </span><span class="c1"># State transitions: remaining rows and columns</span>
 <a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="n">n</span>
 <a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="n">m</span>
 <a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="w">      </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">j</span><span class="o">]].</span><span class="n">min</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span>
@@ -5755,12 +5837,17 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </div>
 </div>
 </div>
-<p>Figure 14-16 shows the state transition process for minimum path sum, which traverses the entire grid, <strong>thus the time complexity is <span class="arithmatex">\(O(nm)\)</span></strong>.</p>
-<p>The array <code>dp</code> has size <span class="arithmatex">\(n \times m\)</span>, <strong>thus the space complexity is <span class="arithmatex">\(O(nm)\)</span></strong>.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dp%28grid%3A%20list%5Blist%5Bint%5D%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Minimum%20path%20sum%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%20m%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20dp%5B0%5D%5B0%5D%20%3D%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20State%20transitions%3A%20first%20row%0A%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Bj%5D%20%3D%20dp%5B0%5D%5Bj%20-%201%5D%20%2B%20grid%5B0%5D%5Bj%5D%0A%20%20%20%20%23%20State%20transitions%3A%20first%20column%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B0%5D%20%3D%20dp%5Bi%20-%201%5D%5B0%5D%20%2B%20grid%5Bi%5D%5B0%5D%0A%20%20%20%20%23%20State%20transitions%3A%20remaining%20rows%20and%20columns%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20min%28dp%5Bi%5D%5Bj%20-%201%5D,%20dp%5Bi%20-%201%5D%5Bj%5D%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20dp%5Bn%20-%201%5D%5Bm%20-%201%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20Dynamic%20programming%0A%20%20%20%20res%20%3D%20min_path_sum_dp%28grid%29%0A%20%20%20%20print%28f%22Minimum%20path%20sum%20from%20the%20top-left%20to%20the%20bottom-right%20corner%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dp%28grid%3A%20list%5Blist%5Bint%5D%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Minimum%20path%20sum%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%20m%20for%20_%20in%20range%28n%29%5D%0A%20%20%20%20dp%5B0%5D%5B0%5D%20%3D%20grid%5B0%5D%5B0%5D%0A%20%20%20%20%23%20State%20transitions%3A%20first%20row%0A%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Bj%5D%20%3D%20dp%5B0%5D%5Bj%20-%201%5D%20%2B%20grid%5B0%5D%5Bj%5D%0A%20%20%20%20%23%20State%20transitions%3A%20first%20column%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B0%5D%20%3D%20dp%5Bi%20-%201%5D%5B0%5D%20%2B%20grid%5Bi%5D%5B0%5D%0A%20%20%20%20%23%20State%20transitions%3A%20remaining%20rows%20and%20columns%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20min%28dp%5Bi%5D%5Bj%20-%201%5D,%20dp%5Bi%20-%201%5D%5Bj%5D%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20dp%5Bn%20-%201%5D%5Bm%20-%201%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20Dynamic%20programming%0A%20%20%20%20res%20%3D%20min_path_sum_dp%28grid%29%0A%20%20%20%20print%28f%22Minimum%20path%20sum%20from%20the%20top-left%20to%20the%20bottom-right%20corner%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 14-16 shows the state transitions for minimum path sum. They visit the entire grid, <strong>so the time complexity is <span class="arithmatex">\(O(nm)\)</span></strong>.</p>
+<p>The array <code>dp</code> has dimensions <span class="arithmatex">\(n \times m\)</span>, <strong>so the space complexity is <span class="arithmatex">\(O(nm)\)</span></strong>.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:12"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">&lt;1&gt;</label><label for="__tabbed_4_2">&lt;2&gt;</label><label for="__tabbed_4_3">&lt;3&gt;</label><label for="__tabbed_4_4">&lt;4&gt;</label><label for="__tabbed_4_5">&lt;5&gt;</label><label for="__tabbed_4_6">&lt;6&gt;</label><label for="__tabbed_4_7">&lt;7&gt;</label><label for="__tabbed_4_8">&lt;8&gt;</label><label for="__tabbed_4_9">&lt;9&gt;</label><label for="__tabbed_4_10">&lt;10&gt;</label><label for="__tabbed_4_11">&lt;11&gt;</label><label for="__tabbed_4_12">&lt;12&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Dynamic programming process for minimum path sum" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_dp_step1.png" /></p>
+<p><img alt="Dynamic programming for minimum path sum" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_dp_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="min_path_sum_dp_step2" class="animation-figure" src="../dp_solution_pipeline.assets/min_path_sum_dp_step2.png" /></p>
@@ -5797,49 +5884,49 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </div>
 </div>
 </div>
-<p align="center"> Figure 14-16 &nbsp; Dynamic programming process for minimum path sum </p>
+<p align="center"> Figure 14-16 &nbsp; Dynamic programming for minimum path sum </p>
 
-<h3 id="4-space-optimization">4. &nbsp; Space Optimization<a class="headerlink" href="#4-space-optimization" title="Permanent link">&para;</a></h3>
-<p>Since each cell is only related to the cell to its left and the cell above it, we can use a single-row array to implement the <span class="arithmatex">\(dp\)</span> table.</p>
-<p>Note that since the array <code>dp</code> can only represent the state of one row, we cannot initialize the first column state in advance, but rather update it when traversing each row:</p>
+<h3 id="4-space-optimization">4. &nbsp; Space optimization<a class="headerlink" href="#4-space-optimization" title="Permanent link">&para;</a></h3>
+<p>Each cell depends only on the cells to its left and above, so a single-row array is enough to represent the <span class="arithmatex">\(dp\)</span> table.</p>
+<p>Because <code>dp</code> stores only one row of states, we cannot initialize the entire first column in advance. Instead, update its entry when processing each row:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">min_path_sum.py</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">min_path_sum_dp_comp</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]])</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Minimum path sum: Space-optimized dynamic programming"""</span>
+<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Minimum path sum: space-optimized dynamic programming"""</span>
 <a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">grid</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-<a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="c1"># Initialize dp table</span>
+<a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">m</span>
-<a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a>    <span class="c1"># State transition: first row</span>
+<a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a>    <span class="c1"># State transitions: first row</span>
 <a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a>    <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a>    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
 <a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a>        <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
-<a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a>    <span class="c1"># State transition: rest of the rows</span>
+<a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a>    <span class="c1"># State transitions: remaining rows</span>
 <a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
-<a id="__codelineno-39-12" name="__codelineno-39-12" href="#__codelineno-39-12"></a>        <span class="c1"># State transition: first column</span>
+<a id="__codelineno-39-12" name="__codelineno-39-12" href="#__codelineno-39-12"></a>        <span class="c1"># State transitions: first column</span>
 <a id="__codelineno-39-13" name="__codelineno-39-13" href="#__codelineno-39-13"></a>        <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-<a id="__codelineno-39-14" name="__codelineno-39-14" href="#__codelineno-39-14"></a>        <span class="c1"># State transition: rest of the columns</span>
+<a id="__codelineno-39-14" name="__codelineno-39-14" href="#__codelineno-39-14"></a>        <span class="c1"># State transitions: remaining columns</span>
 <a id="__codelineno-39-15" name="__codelineno-39-15" href="#__codelineno-39-15"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">):</span>
 <a id="__codelineno-39-16" name="__codelineno-39-16" href="#__codelineno-39-16"></a>            <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">])</span> <span class="o">+</span> <span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-39-17" name="__codelineno-39-17" href="#__codelineno-39-17"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Minimum path sum: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Minimum path sum: space-optimized dynamic programming */</span>
 <a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDPComp</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">grid</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">.</span><span class="n">size</span><span class="p">(),</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">].</span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">m</span><span class="p">);</span>
-<a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-9" name="__codelineno-40-9" href="#__codelineno-40-9"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-40-10" name="__codelineno-40-10" href="#__codelineno-40-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-40-11" name="__codelineno-40-11" href="#__codelineno-40-11"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-40-11" name="__codelineno-40-11" href="#__codelineno-40-11"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-40-12" name="__codelineno-40-12" href="#__codelineno-40-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-13" name="__codelineno-40-13" href="#__codelineno-40-13"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-40-13" name="__codelineno-40-13" href="#__codelineno-40-13"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-40-14" name="__codelineno-40-14" href="#__codelineno-40-14"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
-<a id="__codelineno-40-15" name="__codelineno-40-15" href="#__codelineno-40-15"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-40-15" name="__codelineno-40-15" href="#__codelineno-40-15"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-40-16" name="__codelineno-40-16" href="#__codelineno-40-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-17" name="__codelineno-40-17" href="#__codelineno-40-17"></a><span class="w">            </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-40-18" name="__codelineno-40-18" href="#__codelineno-40-18"></a><span class="w">        </span><span class="p">}</span>
@@ -5849,21 +5936,21 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Minimum path sum: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Minimum path sum: space-optimized dynamic programming */</span>
 <a id="__codelineno-41-2" name="__codelineno-41-2" href="#__codelineno-41-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDPComp</span><span class="p">(</span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">grid</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">.</span><span class="na">length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
-<a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-41-8" name="__codelineno-41-8" href="#__codelineno-41-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-9" name="__codelineno-41-9" href="#__codelineno-41-9"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-41-10" name="__codelineno-41-10" href="#__codelineno-41-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-41-11" name="__codelineno-41-11" href="#__codelineno-41-11"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-41-11" name="__codelineno-41-11" href="#__codelineno-41-11"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-41-12" name="__codelineno-41-12" href="#__codelineno-41-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-13" name="__codelineno-41-13" href="#__codelineno-41-13"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-41-13" name="__codelineno-41-13" href="#__codelineno-41-13"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-41-14" name="__codelineno-41-14" href="#__codelineno-41-14"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-41-15" name="__codelineno-41-15" href="#__codelineno-41-15"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-41-15" name="__codelineno-41-15" href="#__codelineno-41-15"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-41-16" name="__codelineno-41-16" href="#__codelineno-41-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-17" name="__codelineno-41-17" href="#__codelineno-41-17"></a><span class="w">            </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-41-18" name="__codelineno-41-18" href="#__codelineno-41-18"></a><span class="w">        </span><span class="p">}</span>
@@ -5873,21 +5960,21 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Minimum path sum: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Minimum path sum: space-optimized dynamic programming */</span>
 <a id="__codelineno-42-2" name="__codelineno-42-2" href="#__codelineno-42-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">MinPathSumDPComp</span><span class="p">(</span><span class="kt">int</span><span class="p">[][]</span><span class="w"> </span><span class="n">grid</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">.</span><span class="n">Length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">].</span><span class="n">Length</span><span class="p">;</span>
-<a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">m</span><span class="p">];</span>
 <a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
-<a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-42-8" name="__codelineno-42-8" href="#__codelineno-42-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-9" name="__codelineno-42-9" href="#__codelineno-42-9"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-42-10" name="__codelineno-42-10" href="#__codelineno-42-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-42-11" name="__codelineno-42-11" href="#__codelineno-42-11"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-42-11" name="__codelineno-42-11" href="#__codelineno-42-11"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-42-12" name="__codelineno-42-12" href="#__codelineno-42-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-13" name="__codelineno-42-13" href="#__codelineno-42-13"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-42-13" name="__codelineno-42-13" href="#__codelineno-42-13"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-42-14" name="__codelineno-42-14" href="#__codelineno-42-14"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
-<a id="__codelineno-42-15" name="__codelineno-42-15" href="#__codelineno-42-15"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-42-15" name="__codelineno-42-15" href="#__codelineno-42-15"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-42-16" name="__codelineno-42-16" href="#__codelineno-42-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-17" name="__codelineno-42-17" href="#__codelineno-42-17"></a><span class="w">            </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-42-18" name="__codelineno-42-18" href="#__codelineno-42-18"></a><span class="w">        </span><span class="p">}</span>
@@ -5897,21 +5984,21 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Minimum path sum: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Minimum path sum: space-optimized dynamic programming */</span>
 <a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">minPathSumDPComp</span><span class="p">(</span><span class="nx">grid</span><span class="w"> </span><span class="p">[][]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">grid</span><span class="p">),</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
-<a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="p">)</span>
-<a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-9" name="__codelineno-43-9" href="#__codelineno-43-9"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span>
 <a id="__codelineno-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-43-12" name="__codelineno-43-12" href="#__codelineno-43-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-13" name="__codelineno-43-13" href="#__codelineno-43-13"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-43-13" name="__codelineno-43-13" href="#__codelineno-43-13"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-43-14" name="__codelineno-43-14" href="#__codelineno-43-14"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-<a id="__codelineno-43-15" name="__codelineno-43-15" href="#__codelineno-43-15"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-43-15" name="__codelineno-43-15" href="#__codelineno-43-15"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-43-16" name="__codelineno-43-16" href="#__codelineno-43-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-17" name="__codelineno-43-17" href="#__codelineno-43-17"></a><span class="w">            </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Min</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">])))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span>
 <a id="__codelineno-43-18" name="__codelineno-43-18" href="#__codelineno-43-18"></a><span class="w">        </span><span class="p">}</span>
@@ -5921,22 +6008,22 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Minimum path sum: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Minimum path sum: space-optimized dynamic programming */</span>
 <a id="__codelineno-44-2" name="__codelineno-44-2" href="#__codelineno-44-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">minPathSumDPComp</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">grid</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">].</span><span class="bp">count</span>
-<a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">m</span><span class="p">)</span>
-<a id="__codelineno-44-7" name="__codelineno-44-7" href="#__codelineno-44-7"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-44-7" name="__codelineno-44-7" href="#__codelineno-44-7"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-44-9" name="__codelineno-44-9" href="#__codelineno-44-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-10" name="__codelineno-44-10" href="#__codelineno-44-10"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-44-11" name="__codelineno-44-11" href="#__codelineno-44-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-44-12" name="__codelineno-44-12" href="#__codelineno-44-12"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-44-12" name="__codelineno-44-12" href="#__codelineno-44-12"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-44-13" name="__codelineno-44-13" href="#__codelineno-44-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-44-14" name="__codelineno-44-14" href="#__codelineno-44-14"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-44-14" name="__codelineno-44-14" href="#__codelineno-44-14"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-44-15" name="__codelineno-44-15" href="#__codelineno-44-15"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
-<a id="__codelineno-44-16" name="__codelineno-44-16" href="#__codelineno-44-16"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-44-16" name="__codelineno-44-16" href="#__codelineno-44-16"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-44-17" name="__codelineno-44-17" href="#__codelineno-44-17"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-18" name="__codelineno-44-18" href="#__codelineno-44-18"></a><span class="w">            </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-44-19" name="__codelineno-44-19" href="#__codelineno-44-19"></a><span class="w">        </span><span class="p">}</span>
@@ -5946,22 +6033,22 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Minimum path sum: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Minimum path sum: space-optimized dynamic programming */</span>
 <a id="__codelineno-45-2" name="__codelineno-45-2" href="#__codelineno-45-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">minPathSumDPComp</span><span class="p">(</span><span class="nx">grid</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grid</span><span class="p">.</span><span class="nx">length</span><span class="p">,</span>
 <a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">].</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-45-6" name="__codelineno-45-6" href="#__codelineno-45-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">m</span><span class="p">);</span>
-<a id="__codelineno-45-7" name="__codelineno-45-7" href="#__codelineno-45-7"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-45-7" name="__codelineno-45-7" href="#__codelineno-45-7"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-45-8" name="__codelineno-45-8" href="#__codelineno-45-8"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="mf">0</span><span class="p">];</span>
 <a id="__codelineno-45-9" name="__codelineno-45-9" href="#__codelineno-45-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-10" name="__codelineno-45-10" href="#__codelineno-45-10"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-45-11" name="__codelineno-45-11" href="#__codelineno-45-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-45-12" name="__codelineno-45-12" href="#__codelineno-45-12"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-45-12" name="__codelineno-45-12" href="#__codelineno-45-12"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-45-13" name="__codelineno-45-13" href="#__codelineno-45-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-14" name="__codelineno-45-14" href="#__codelineno-45-14"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-45-14" name="__codelineno-45-14" href="#__codelineno-45-14"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-45-15" name="__codelineno-45-15" href="#__codelineno-45-15"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">0</span><span class="p">];</span>
-<a id="__codelineno-45-16" name="__codelineno-45-16" href="#__codelineno-45-16"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-45-16" name="__codelineno-45-16" href="#__codelineno-45-16"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-45-17" name="__codelineno-45-17" href="#__codelineno-45-17"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-18" name="__codelineno-45-18" href="#__codelineno-45-18"></a><span class="w">            </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-45-19" name="__codelineno-45-19" href="#__codelineno-45-19"></a><span class="w">        </span><span class="p">}</span>
@@ -5971,22 +6058,22 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Minimum path sum: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Minimum path sum: space-optimized dynamic programming */</span>
 <a id="__codelineno-46-2" name="__codelineno-46-2" href="#__codelineno-46-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">minPathSumDPComp</span><span class="p">(</span><span class="nx">grid</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="nb">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;&gt;</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grid</span><span class="p">.</span><span class="nx">length</span><span class="p">,</span>
 <a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">].</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">m</span><span class="p">);</span>
-<a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="mf">0</span><span class="p">];</span>
 <a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-10" name="__codelineno-46-10" href="#__codelineno-46-10"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-46-11" name="__codelineno-46-11" href="#__codelineno-46-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-46-13" name="__codelineno-46-13" href="#__codelineno-46-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-14" name="__codelineno-46-14" href="#__codelineno-46-14"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-46-14" name="__codelineno-46-14" href="#__codelineno-46-14"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-46-15" name="__codelineno-46-15" href="#__codelineno-46-15"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">0</span><span class="p">];</span>
-<a id="__codelineno-46-16" name="__codelineno-46-16" href="#__codelineno-46-16"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-46-16" name="__codelineno-46-16" href="#__codelineno-46-16"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-46-17" name="__codelineno-46-17" href="#__codelineno-46-17"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-18" name="__codelineno-46-18" href="#__codelineno-46-18"></a><span class="w">            </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">grid</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-46-19" name="__codelineno-46-19" href="#__codelineno-46-19"></a><span class="w">        </span><span class="p">}</span>
@@ -5996,20 +6083,20 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Minimum path sum: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Minimum path sum: space-optimized dynamic programming */</span>
 <a id="__codelineno-47-2" name="__codelineno-47-2" href="#__codelineno-47-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">minPathSumDPComp</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">grid</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">.</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="m">0</span><span class="p">].</span><span class="n">length</span><span class="p">;</span>
-<a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a><span class="w">  </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="m">0</span><span class="p">][</span><span class="m">0</span><span class="p">];</span>
 <a id="__codelineno-47-7" name="__codelineno-47-7" href="#__codelineno-47-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="m">0</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-47-9" name="__codelineno-47-9" href="#__codelineno-47-9"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-47-10" name="__codelineno-47-10" href="#__codelineno-47-10"></a><span class="w">  </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-47-10" name="__codelineno-47-10" href="#__codelineno-47-10"></a><span class="w">  </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-47-11" name="__codelineno-47-11" href="#__codelineno-47-11"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-12" name="__codelineno-47-12" href="#__codelineno-47-12"></a><span class="w">    </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-47-12" name="__codelineno-47-12" href="#__codelineno-47-12"></a><span class="w">    </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-47-13" name="__codelineno-47-13" href="#__codelineno-47-13"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="m">0</span><span class="p">];</span>
-<a id="__codelineno-47-14" name="__codelineno-47-14" href="#__codelineno-47-14"></a><span class="w">    </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-47-14" name="__codelineno-47-14" href="#__codelineno-47-14"></a><span class="w">    </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-47-15" name="__codelineno-47-15" href="#__codelineno-47-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-16" name="__codelineno-47-16" href="#__codelineno-47-16"></a><span class="w">      </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-47-17" name="__codelineno-47-17" href="#__codelineno-47-17"></a><span class="w">    </span><span class="p">}</span>
@@ -6019,21 +6106,21 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Minimum path sum: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Minimum path sum: space-optimized dynamic programming */</span>
 <a id="__codelineno-48-2" name="__codelineno-48-2" href="#__codelineno-48-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">min_path_sum_dp_comp</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">grid</span><span class="p">.</span><span class="n">len</span><span class="p">(),</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">].</span><span class="n">len</span><span class="p">());</span>
-<a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">m</span><span class="p">];</span>
-<a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-48-8" name="__codelineno-48-8" href="#__codelineno-48-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-48-11" name="__codelineno-48-11" href="#__codelineno-48-11"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-48-11" name="__codelineno-48-11" href="#__codelineno-48-11"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-48-12" name="__codelineno-48-12" href="#__codelineno-48-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-13" name="__codelineno-48-13" href="#__codelineno-48-13"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-48-13" name="__codelineno-48-13" href="#__codelineno-48-13"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-48-14" name="__codelineno-48-14" href="#__codelineno-48-14"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
-<a id="__codelineno-48-15" name="__codelineno-48-15" href="#__codelineno-48-15"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-48-15" name="__codelineno-48-15" href="#__codelineno-48-15"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-48-16" name="__codelineno-48-16" href="#__codelineno-48-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-17" name="__codelineno-48-17" href="#__codelineno-48-17"></a><span class="w">            </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-48-18" name="__codelineno-48-18" href="#__codelineno-48-18"></a><span class="w">        </span><span class="p">}</span>
@@ -6043,20 +6130,20 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Minimum path sum: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Minimum path sum: space-optimized dynamic programming */</span>
 <a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">minPathSumDPComp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">][</span><span class="n">MAX_SIZE</span><span class="p">],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
-<a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-49-9" name="__codelineno-49-9" href="#__codelineno-49-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-49-10" name="__codelineno-49-10" href="#__codelineno-49-10"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-49-10" name="__codelineno-49-10" href="#__codelineno-49-10"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-49-11" name="__codelineno-49-11" href="#__codelineno-49-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-12" name="__codelineno-49-12" href="#__codelineno-49-12"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-49-12" name="__codelineno-49-12" href="#__codelineno-49-12"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-49-13" name="__codelineno-49-13" href="#__codelineno-49-13"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">];</span>
-<a id="__codelineno-49-14" name="__codelineno-49-14" href="#__codelineno-49-14"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-49-14" name="__codelineno-49-14" href="#__codelineno-49-14"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-49-15" name="__codelineno-49-15" href="#__codelineno-49-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-16" name="__codelineno-49-16" href="#__codelineno-49-16"></a><span class="w">            </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-49-17" name="__codelineno-49-17" href="#__codelineno-49-17"></a><span class="w">        </span><span class="p">}</span>
@@ -6069,22 +6156,22 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Minimum path sum: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">min_path_sum.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Minimum path sum: space-optimized dynamic programming */</span>
 <a id="__codelineno-50-2" name="__codelineno-50-2" href="#__codelineno-50-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">minPathSumDPComp</span><span class="p">(</span><span class="n">grid</span><span class="p">:</span><span class="w"> </span><span class="n">Array</span><span class="o">&lt;</span><span class="n">IntArray</span><span class="o">&gt;</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="m">0</span><span class="o">]</span><span class="p">.</span><span class="na">size</span>
-<a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
-<a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-50-8" name="__codelineno-50-8" href="#__codelineno-50-8"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="m">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="m">0</span><span class="o">][</span><span class="m">0</span><span class="o">]</span>
 <a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-10" name="__codelineno-50-10" href="#__codelineno-50-10"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="m">0</span><span class="o">][</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-50-11" name="__codelineno-50-11" href="#__codelineno-50-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-50-12" name="__codelineno-50-12" href="#__codelineno-50-12"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-50-12" name="__codelineno-50-12" href="#__codelineno-50-12"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-50-13" name="__codelineno-50-13" href="#__codelineno-50-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-14" name="__codelineno-50-14" href="#__codelineno-50-14"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-50-14" name="__codelineno-50-14" href="#__codelineno-50-14"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-50-15" name="__codelineno-50-15" href="#__codelineno-50-15"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="m">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="m">0</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="m">0</span><span class="o">]</span>
-<a id="__codelineno-50-16" name="__codelineno-50-16" href="#__codelineno-50-16"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-50-16" name="__codelineno-50-16" href="#__codelineno-50-16"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-50-17" name="__codelineno-50-17" href="#__codelineno-50-17"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-18" name="__codelineno-50-18" href="#__codelineno-50-18"></a><span class="w">            </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-50-19" name="__codelineno-50-19" href="#__codelineno-50-19"></a><span class="w">        </span><span class="p">}</span>
@@ -6094,19 +6181,19 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">min_path_sum.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Minimum path sum: space-optimized DP ###</span>
+<div class="highlight"><span class="filename">min_path_sum.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Minimum path sum: space-optimized dynamic programming ###</span>
 <a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">min_path_sum_dp_comp</span><span class="p">(</span><span class="n">grid</span><span class="p">)</span>
 <a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">  </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="o">.</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">grid</span><span class="o">.</span><span class="n">first</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="w">  </span><span class="c1"># Initialize dp table</span>
+<a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="w">  </span><span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">  </span><span class="c1"># State transition: first row</span>
+<a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">  </span><span class="c1"># State transitions: first row</span>
 <a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="w">  </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span>
 <a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="w">  </span><span class="p">(</span><span class="mi">1</span><span class="o">...</span><span class="n">m</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">j</span><span class="o">|</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w">  </span><span class="c1"># State transition: rest of the rows</span>
+<a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w">  </span><span class="c1"># State transitions: remaining rows</span>
 <a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="n">n</span>
-<a id="__codelineno-51-11" name="__codelineno-51-11" href="#__codelineno-51-11"></a><span class="w">    </span><span class="c1"># State transition: first column</span>
+<a id="__codelineno-51-11" name="__codelineno-51-11" href="#__codelineno-51-11"></a><span class="w">    </span><span class="c1"># State transitions: first column</span>
 <a id="__codelineno-51-12" name="__codelineno-51-12" href="#__codelineno-51-12"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span>
-<a id="__codelineno-51-13" name="__codelineno-51-13" href="#__codelineno-51-13"></a><span class="w">    </span><span class="c1"># State transition: rest of the columns</span>
+<a id="__codelineno-51-13" name="__codelineno-51-13" href="#__codelineno-51-13"></a><span class="w">    </span><span class="c1"># State transitions: remaining columns</span>
 <a id="__codelineno-51-14" name="__codelineno-51-14" href="#__codelineno-51-14"></a><span class="w">    </span><span class="p">(</span><span class="mi">1</span><span class="o">...</span><span class="n">m</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">j</span><span class="o">|</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]].</span><span class="n">min</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">grid</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-51-15" name="__codelineno-51-15" href="#__codelineno-51-15"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-51-16" name="__codelineno-51-16" href="#__codelineno-51-16"></a><span class="w">  </span><span class="n">dp</span><span class="o">[</span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span>
@@ -6115,6 +6202,11 @@ dp[i, j] = \min(dp[i-1, j], dp[i, j-1]) + grid[i, j]
 </div>
 </div>
 </div>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dp_comp%28grid%3A%20list%5Blist%5Bint%5D%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Minimum%20path%20sum%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B0%5D%20%2A%20m%0A%20%20%20%20%23%20State%20transitions%3A%20first%20row%0A%20%20%20%20dp%5B0%5D%20%3D%20grid%5B0%5D%5B0%5D%0A%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20dp%5Bj%20-%201%5D%20%2B%20grid%5B0%5D%5Bj%5D%0A%20%20%20%20%23%20State%20transitions%3A%20remaining%20rows%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20%23%20State%20transitions%3A%20first%20column%0A%20%20%20%20%20%20%20%20dp%5B0%5D%20%3D%20dp%5B0%5D%20%2B%20grid%5Bi%5D%5B0%5D%0A%20%20%20%20%20%20%20%20%23%20State%20transitions%3A%20remaining%20columns%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20min%28dp%5Bj%20-%201%5D,%20dp%5Bj%5D%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20dp%5Bm%20-%201%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20Space-optimized%20dynamic%20programming%0A%20%20%20%20res%20%3D%20min_path_sum_dp_comp%28grid%29%0A%20%20%20%20print%28f%22Minimum%20path%20sum%20from%20the%20top-left%20to%20the%20bottom-right%20corner%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20math%20import%20inf%0A%0Adef%20min_path_sum_dp_comp%28grid%3A%20list%5Blist%5Bint%5D%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Minimum%20path%20sum%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B0%5D%20%2A%20m%0A%20%20%20%20%23%20State%20transitions%3A%20first%20row%0A%20%20%20%20dp%5B0%5D%20%3D%20grid%5B0%5D%5B0%5D%0A%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20dp%5Bj%20-%201%5D%20%2B%20grid%5B0%5D%5Bj%5D%0A%20%20%20%20%23%20State%20transitions%3A%20remaining%20rows%0A%20%20%20%20for%20i%20in%20range%281,%20n%29%3A%0A%20%20%20%20%20%20%20%20%23%20State%20transitions%3A%20first%20column%0A%20%20%20%20%20%20%20%20dp%5B0%5D%20%3D%20dp%5B0%5D%20%2B%20grid%5Bi%5D%5B0%5D%0A%20%20%20%20%20%20%20%20%23%20State%20transitions%3A%20remaining%20columns%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20min%28dp%5Bj%20-%201%5D,%20dp%5Bj%5D%29%20%2B%20grid%5Bi%5D%5Bj%5D%0A%20%20%20%20return%20dp%5Bm%20-%201%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20grid%20%3D%20%5B%5B1,%203,%201,%205%5D,%20%5B2,%202,%204,%202%5D,%20%5B5,%203,%202,%201%5D,%20%5B4,%203,%205,%202%5D%5D%0A%20%20%20%20n,%20m%20%3D%20len%28grid%29,%20len%28grid%5B0%5D%29%0A%0A%20%20%20%20%23%20Space-optimized%20dynamic%20programming%0A%20%20%20%20res%20%3D%20min_path_sum_dp_comp%28grid%29%0A%20%20%20%20print%28f%22Minimum%20path%20sum%20from%20the%20top-left%20to%20the%20bottom-right%20corner%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
 
 <!-- Source file information -->
 
@@ -6137,7 +6229,7 @@ aria-label="Footer"
   <a
     href="../dp_problem_features/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 14.2 Characteristics of Dynamic Programming Problems"
+    aria-label="Previous: 14.2   Properties of dynamic programming problems"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -6149,7 +6241,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        14.2 Characteristics of Dynamic Programming Problems
+        14.2 &nbsp; Properties of dynamic programming problems
       </div>
     </div>
   </a>
@@ -6161,7 +6253,7 @@ aria-label="Footer"
   <a
     href="../knapsack_problem/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 14.4 0-1 Knapsack Problem"
+    aria-label="Next: 14.4   0-1 knapsack problem"
     rel="next"
   >
     <div class="md-footer__title">
@@ -6169,7 +6261,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        14.4 0-1 Knapsack Problem
+        14.4 &nbsp; 0-1 knapsack problem
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -6186,7 +6278,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>14.6 Edit Distance Problem - Hello Algo</title>
+        <title>14.6 Edit distance problem - Hello Algo</title>
 
 
 
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+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-  Before Starting
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-  0.1 About This Book
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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-  1.3 Summary
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-  2.2 Iteration and Recursion
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-  2.4 Space Complexity
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-  2.5 Summary
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
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-  4.5 Summary
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-  4.6 Exercises
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
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+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
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+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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+  Chapter 7. &nbsp; Trees
 
 
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-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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+  7.6 &nbsp; Summary
 
 
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+  7.7 &nbsp; Exercises
 
 
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+  Chapter 8. &nbsp; Heaps
 
 
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+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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+  8.3 &nbsp; The top-k problem
 
 
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+  11.3 &nbsp; Bubble sort
 
 
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+  11.4 &nbsp; Insertion sort
 
 
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+  11.10 &nbsp; Radix sort
 
 
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-  Chapter 15. Greedy
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-  Chapter 16. Appendix
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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+        1. &nbsp; Dynamic programming approach
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 <!-- Page content -->
-<h1 id="146-edit-distance-problem">14.6 &nbsp; Edit Distance Problem<a class="headerlink" href="#146-edit-distance-problem" title="Permanent link">&para;</a></h1>
-<p>Edit distance, also known as Levenshtein distance, refers to the minimum number of edits required to transform one string into another, commonly used in information retrieval and natural language processing to measure the similarity between two sequences.</p>
+<h1 id="146-edit-distance-problem">14.6 &nbsp; Edit distance problem<a class="headerlink" href="#146-edit-distance-problem" title="Permanent link">&para;</a></h1>
+<p>Edit distance, also called Levenshtein distance, is the minimum number of edits needed to transform one string into another. It is commonly used in information retrieval and natural language processing to measure similarity between sequences.</p>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given two strings <span class="arithmatex">\(s\)</span> and <span class="arithmatex">\(t\)</span>, return the minimum number of edits required to transform <span class="arithmatex">\(s\)</span> into <span class="arithmatex">\(t\)</span>.</p>
-<p>You can perform three types of edit operations on a string: insert a character, delete a character, or replace a character with any other character.</p>
+<p>Given two strings <span class="arithmatex">\(s\)</span> and <span class="arithmatex">\(t\)</span>, return the minimum number of edits needed to transform <span class="arithmatex">\(s\)</span> into <span class="arithmatex">\(t\)</span>.</p>
+<p>Three edits are allowed: insert a character, delete a character, or replace a character with any other character.</p>
 </div>
-<p>As shown in Figure 14-27, transforming <code>kitten</code> into <code>sitting</code> requires 3 edits, including 2 replacements and 1 insertion; transforming <code>hello</code> into <code>algo</code> requires 3 steps, including 2 replacements and 1 deletion.</p>
-<p><img alt="Example data for edit distance" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_example.png" /></p>
-<p align="center"> Figure 14-27 &nbsp; Example data for edit distance </p>
+<p>As shown in Figure 14-27, transforming <code>kitten</code> into <code>sitting</code> takes 3 edits: 2 replacements and 1 insertion. Transforming <code>hello</code> into <code>algo</code> also takes 3 edits: 2 replacements and 1 deletion.</p>
+<p><img alt="Examples of edit distance" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_example.png" /></p>
+<p align="center"> Figure 14-27 &nbsp; Examples of edit distance </p>
 
-<p><strong>The edit distance problem can be naturally explained using the decision tree model</strong>. Strings correspond to tree nodes, and each edit operation corresponds to an edge in the tree.</p>
-<p>As shown in Figure 14-28, without restricting operations, each node can branch into many edges, with each edge corresponding to one operation, meaning there are many possible paths to transform <code>hello</code> into <code>algo</code>.</p>
-<p>From the perspective of the decision tree, the goal of this problem is to find the shortest path between node <code>hello</code> and node <code>algo</code>.</p>
-<p><img alt="Representing edit distance problem based on decision tree model" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_decision_tree.png" /></p>
-<p align="center"> Figure 14-28 &nbsp; Representing edit distance problem based on decision tree model </p>
+<p><strong>Edit distance fits naturally into a decision tree model</strong>. Strings are nodes, and each decision (an edit) is an edge.</p>
+<p>As shown in Figure 14-28, without restrictions on the edits, each node can have many outgoing edges, one for each possible operation. Thus, many paths can transform <code>hello</code> into <code>algo</code>.</p>
+<p>From the decision tree perspective, the goal is to find the shortest path between nodes <code>hello</code> and <code>algo</code>.</p>
+<p><img alt="Edit distance as a decision tree" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_decision_tree.png" /></p>
+<p align="center"> Figure 14-28 &nbsp; Edit distance as a decision tree </p>
 
-<h3 id="1-dynamic-programming-approach">1. &nbsp; Dynamic Programming Approach<a class="headerlink" href="#1-dynamic-programming-approach" title="Permanent link">&para;</a></h3>
-<p><strong>Step 1: Think about the decisions in each round, define the state, and thus obtain the <span class="arithmatex">\(dp\)</span> table</strong></p>
-<p>Each round of decision involves performing one edit operation on string <span class="arithmatex">\(s\)</span>.</p>
-<p>We want the problem size to gradually decrease during the editing process so that we can construct subproblems. Let the lengths of strings <span class="arithmatex">\(s\)</span> and <span class="arithmatex">\(t\)</span> be <span class="arithmatex">\(n\)</span> and <span class="arithmatex">\(m\)</span> respectively. We first consider the tail characters of the two strings, <span class="arithmatex">\(s[n-1]\)</span> and <span class="arithmatex">\(t[m-1]\)</span>.</p>
+<h3 id="1-dynamic-programming-approach">1. &nbsp; Dynamic programming approach<a class="headerlink" href="#1-dynamic-programming-approach" title="Permanent link">&para;</a></h3>
+<p><strong>Step 1: Identify each decision, define the state, and establish the <span class="arithmatex">\(dp\)</span> table</strong></p>
+<p>Each decision applies one edit to string <span class="arithmatex">\(s\)</span>.</p>
+<p>We want each edit to reduce the problem's size so that we can form subproblems. Let the lengths of <span class="arithmatex">\(s\)</span> and <span class="arithmatex">\(t\)</span> be <span class="arithmatex">\(n\)</span> and <span class="arithmatex">\(m\)</span>. First consider their last characters, <span class="arithmatex">\(s[n-1]\)</span> and <span class="arithmatex">\(t[m-1]\)</span>.</p>
 <ul>
-<li>If <span class="arithmatex">\(s[n-1]\)</span> and <span class="arithmatex">\(t[m-1]\)</span> are the same, we can skip them and directly consider <span class="arithmatex">\(s[n-2]\)</span> and <span class="arithmatex">\(t[m-2]\)</span>.</li>
-<li>If <span class="arithmatex">\(s[n-1]\)</span> and <span class="arithmatex">\(t[m-1]\)</span> are different, we need to perform one edit on <span class="arithmatex">\(s\)</span> (insert, delete, or replace) to make the tail characters of the two strings the same, allowing us to skip them and consider a smaller-scale problem.</li>
+<li>If <span class="arithmatex">\(s[n-1]\)</span> and <span class="arithmatex">\(t[m-1]\)</span> match, skip them and consider <span class="arithmatex">\(s[n-2]\)</span> and <span class="arithmatex">\(t[m-2]\)</span>.</li>
+<li>If <span class="arithmatex">\(s[n-1]\)</span> and <span class="arithmatex">\(t[m-1]\)</span> differ, apply an edit to <span class="arithmatex">\(s\)</span> (insertion, deletion, or replacement) to match the endings, allowing us to move on to a smaller problem.</li>
 </ul>
-<p>In other words, each round of decision (edit operation) we make on string <span class="arithmatex">\(s\)</span> will change the remaining characters to be matched in <span class="arithmatex">\(s\)</span> and <span class="arithmatex">\(t\)</span>. Therefore, the state is the <span class="arithmatex">\(i\)</span>-th and <span class="arithmatex">\(j\)</span>-th characters currently being considered in <span class="arithmatex">\(s\)</span> and <span class="arithmatex">\(t\)</span>, denoted as <span class="arithmatex">\([i, j]\)</span>.</p>
-<p>State <span class="arithmatex">\([i, j]\)</span> corresponds to the subproblem: <strong>the minimum number of edits required to change the first <span class="arithmatex">\(i\)</span> characters of <span class="arithmatex">\(s\)</span> into the first <span class="arithmatex">\(j\)</span> characters of <span class="arithmatex">\(t\)</span></strong>.</p>
-<p>From this, we obtain a two-dimensional <span class="arithmatex">\(dp\)</span> table of size <span class="arithmatex">\((n+1) \times (m+1)\)</span>.</p>
-<p><strong>Step 2: Identify the optimal substructure, and then derive the state transition equation</strong></p>
-<p>Consider subproblem <span class="arithmatex">\(dp[i, j]\)</span>, where the tail characters of the corresponding two strings are <span class="arithmatex">\(s[i-1]\)</span> and <span class="arithmatex">\(t[j-1]\)</span>, which can be divided into the three cases shown in Figure 14-29 based on different edit operations.</p>
+<p>Each decision (edit) in <span class="arithmatex">\(s\)</span> changes the characters still to be matched in <span class="arithmatex">\(s\)</span> and <span class="arithmatex">\(t\)</span>. The state therefore specifies the <span class="arithmatex">\(i\)</span>-th and <span class="arithmatex">\(j\)</span>-th characters currently under consideration in <span class="arithmatex">\(s\)</span> and <span class="arithmatex">\(t\)</span>, denoted by <span class="arithmatex">\([i, j]\)</span>.</p>
+<p>State <span class="arithmatex">\([i, j]\)</span> represents the subproblem of finding <strong>the minimum number of edits needed to transform the first <span class="arithmatex">\(i\)</span> characters of <span class="arithmatex">\(s\)</span> into the first <span class="arithmatex">\(j\)</span> characters of <span class="arithmatex">\(t\)</span></strong>.</p>
+<p>This gives a two-dimensional <span class="arithmatex">\(dp\)</span> table of size <span class="arithmatex">\((n+1) \times (m+1)\)</span>.</p>
+<p><strong>Step 2: Identify optimal substructure and derive the state transition equation</strong></p>
+<p>For subproblem <span class="arithmatex">\(dp[i, j]\)</span>, the final characters are <span class="arithmatex">\(s[i-1]\)</span> and <span class="arithmatex">\(t[j-1]\)</span>. The three possible edits give the cases shown in Figure 14-29.</p>
 <ol>
-<li>Insert <span class="arithmatex">\(t[j-1]\)</span> after <span class="arithmatex">\(s[i-1]\)</span>, then the remaining subproblem is <span class="arithmatex">\(dp[i, j-1]\)</span>.</li>
-<li>Delete <span class="arithmatex">\(s[i-1]\)</span>, then the remaining subproblem is <span class="arithmatex">\(dp[i-1, j]\)</span>.</li>
-<li>Replace <span class="arithmatex">\(s[i-1]\)</span> with <span class="arithmatex">\(t[j-1]\)</span>, then the remaining subproblem is <span class="arithmatex">\(dp[i-1, j-1]\)</span>.</li>
+<li>Insert <span class="arithmatex">\(t[j-1]\)</span> after <span class="arithmatex">\(s[i-1]\)</span>, leaving subproblem <span class="arithmatex">\(dp[i, j-1]\)</span>.</li>
+<li>Delete <span class="arithmatex">\(s[i-1]\)</span>, leaving subproblem <span class="arithmatex">\(dp[i-1, j]\)</span>.</li>
+<li>Replace <span class="arithmatex">\(s[i-1]\)</span> with <span class="arithmatex">\(t[j-1]\)</span>, leaving subproblem <span class="arithmatex">\(dp[i-1, j-1]\)</span>.</li>
 </ol>
-<p><img alt="State transition for edit distance" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_state_transfer.png" /></p>
-<p align="center"> Figure 14-29 &nbsp; State transition for edit distance </p>
+<p><img alt="State transitions for edit distance" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_state_transfer.png" /></p>
+<p align="center"> Figure 14-29 &nbsp; State transitions for edit distance </p>
 
-<p>Based on the above analysis, we obtain the optimal substructure: the minimum number of edits for <span class="arithmatex">\(dp[i, j]\)</span> equals the minimum of <span class="arithmatex">\(dp[i, j-1]\)</span>, <span class="arithmatex">\(dp[i-1, j]\)</span>, and <span class="arithmatex">\(dp[i-1, j-1]\)</span>, plus the current edit cost of <span class="arithmatex">\(1\)</span>. The corresponding state transition equation is:</p>
+<p>The optimal substructure follows: the minimum edit count <span class="arithmatex">\(dp[i, j]\)</span> is the smallest of <span class="arithmatex">\(dp[i, j-1]\)</span>, <span class="arithmatex">\(dp[i-1, j]\)</span>, and <span class="arithmatex">\(dp[i-1, j-1]\)</span>, plus <span class="arithmatex">\(1\)</span> for the current edit. The state transition equation is:</p>
 <div class="arithmatex">\[
 dp[i, j] = \min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1
 \]</div>
-<p>Please note that <strong>when <span class="arithmatex">\(s[i-1]\)</span> and <span class="arithmatex">\(t[j-1]\)</span> are the same, no edit is required for the current character</strong>, in which case the state transition equation is:</p>
+<p>When <strong><span class="arithmatex">\(s[i-1]\)</span> and <span class="arithmatex">\(t[j-1]\)</span> match, no edit is needed for the current characters</strong>, and the state transition equation becomes:</p>
 <div class="arithmatex">\[
 dp[i, j] = dp[i-1, j-1]
 \]</div>
-<p><strong>Step 3: Determine boundary conditions and state transition order</strong></p>
-<p>When both strings are empty, the number of edit steps is <span class="arithmatex">\(0\)</span>, i.e., <span class="arithmatex">\(dp[0, 0] = 0\)</span>. When <span class="arithmatex">\(s\)</span> is empty but <span class="arithmatex">\(t\)</span> is not, the minimum number of edit steps equals the length of <span class="arithmatex">\(t\)</span>, i.e., the first row <span class="arithmatex">\(dp[0, j] = j\)</span>. When <span class="arithmatex">\(s\)</span> is not empty but <span class="arithmatex">\(t\)</span> is empty, the minimum number of edit steps equals the length of <span class="arithmatex">\(s\)</span>, i.e., the first column <span class="arithmatex">\(dp[i, 0] = i\)</span>.</p>
-<p>Observing the state transition equation, the solution <span class="arithmatex">\(dp[i, j]\)</span> depends on solutions to the left, above, and upper-left, so the entire <span class="arithmatex">\(dp\)</span> table can be traversed in order through two nested loops.</p>
-<h3 id="2-code-implementation">2. &nbsp; Code Implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">&para;</a></h3>
+<p><strong>Step 3: Determine the boundary conditions and evaluation order</strong></p>
+<p>If both strings are empty, <span class="arithmatex">\(0\)</span> edits are needed, so <span class="arithmatex">\(dp[0, 0] = 0\)</span>. If <span class="arithmatex">\(s\)</span> is empty but <span class="arithmatex">\(t\)</span> is not, the minimum edit count is the length of <span class="arithmatex">\(t\)</span>, giving the first row <span class="arithmatex">\(dp[0, j] = j\)</span>. If <span class="arithmatex">\(s\)</span> is nonempty but <span class="arithmatex">\(t\)</span> is empty, the count is the length of <span class="arithmatex">\(s\)</span>, giving the first column <span class="arithmatex">\(dp[i, 0] = i\)</span>.</p>
+<p>The solution <span class="arithmatex">\(dp[i, j]\)</span> depends on entries to its left, above, and upper left, so two nested loops can traverse the <span class="arithmatex">\(dp\)</span> table in increasing index order.</p>
+<h3 id="2-code-implementation">2. &nbsp; Code implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">&para;</a></h3>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">edit_distance.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">edit_distance_dp</span><span class="p">(</span><span class="n">s</span><span class="p">:</span> <span class="nb">str</span><span class="p">,</span> <span class="n">t</span><span class="p">:</span> <span class="nb">str</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Edit distance: Dynamic programming"""</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Edit distance: dynamic programming"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># State transition: first row and first column</span>
+<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># State transitions: first row and first column</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">i</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">j</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="c1"># State transition: rest of the rows and columns</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="c1"># State transitions: remaining rows and columns</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>            <span class="k">if</span> <span class="n">s</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">t</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]:</span>
-<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>                <span class="c1"># If two characters are equal, skip both characters</span>
+<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>                <span class="c1"># If the two characters are equal, skip both of them</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>            <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>                <span class="c1"># Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>                <span class="c1"># Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span> <span class="o">+</span> <span class="mi">1</span>
 <a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">m</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Edit distance: Dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Edit distance: dynamic programming */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">editDistanceDP</span><span class="p">(</span><span class="n">string</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">length</span><span class="p">(),</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">length</span><span class="p">();</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">));</span>
-<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w">        </span><span class="p">}</span>
@@ -4782,25 +4854,25 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Edit distance: Dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Edit distance: dynamic programming */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">editDistanceDP</span><span class="p">(</span><span class="n">String</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="na">length</span><span class="p">(),</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="na">length</span><span class="p">();</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">.</span><span class="na">charAt</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="na">charAt</span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="p">),</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="w">        </span><span class="p">}</span>
@@ -4810,25 +4882,25 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Edit distance: Dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Edit distance: dynamic programming */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">EditDistanceDP</span><span class="p">(</span><span class="kt">string</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">Length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="kt">int</span><span class="p">[,]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w">        </span><span class="p">}</span>
@@ -4838,7 +4910,7 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Edit distance: Dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Edit distance: dynamic programming */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">editDistanceDP</span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">t</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">s</span><span class="p">)</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">t</span><span class="p">)</span>
@@ -4846,21 +4918,21 @@ dp[i, j] = dp[i-1, j-1]
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">j</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">s</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">t</span><span class="p">[</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">                </span><span class="c1">// If the two bytes are equal, skip both (each byte is one character for ASCII input)</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">MinInt</span><span class="p">(</span><span class="nx">MinInt</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">j</span><span class="p">]),</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-4-25" name="__codelineno-4-25" href="#__codelineno-4-25"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-4-26" name="__codelineno-4-26" href="#__codelineno-4-26"></a><span class="w">        </span><span class="p">}</span>
@@ -4870,26 +4942,26 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Edit distance: Dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Edit distance: dynamic programming */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">editDistanceDP</span><span class="p">(</span><span class="n">s</span><span class="p">:</span><span class="w"> </span><span class="nb">String</span><span class="p">,</span><span class="w"> </span><span class="n">t</span><span class="p">:</span><span class="w"> </span><span class="nb">String</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">),</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">j</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">                </span><span class="c1">// If the two current UTF-8 bytes match, skip both</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">min</span><span class="p">(</span><span class="bp">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="w">        </span><span class="p">}</span>
@@ -4899,26 +4971,26 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Edit distance: Dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Edit distance: dynamic programming */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">editDistanceDP</span><span class="p">(</span><span class="nx">s</span><span class="p">,</span><span class="w"> </span><span class="nx">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">length</span><span class="p">,</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">));</span>
-<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">j</span><span class="p">;</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span>
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="w">                    </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">j</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="w">            </span><span class="p">}</span>
@@ -4929,28 +5001,28 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Edit distance: Dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Edit distance: dynamic programming */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">editDistanceDP</span><span class="p">(</span><span class="nx">s</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">t</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">length</span><span class="p">,</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">        </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="p">);</span>
-<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">j</span><span class="p">;</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span>
 <a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">                    </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">j</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">            </span><span class="p">}</span>
@@ -4961,25 +5033,25 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Edit distance: Dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Edit distance: dynamic programming */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">editDistanceDP</span><span class="p">(</span><span class="kt">String</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="kt">String</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">_</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">));</span>
-<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">        </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">        </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">      </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">        </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">        </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="w">      </span><span class="p">}</span>
 <a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="w">    </span><span class="p">}</span>
@@ -4989,25 +5061,25 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Edit distance: Dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Edit distance: dynamic programming */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">edit_distance_dp</span><span class="p">(</span><span class="n">s</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="kt">str</span><span class="p">,</span><span class="w"> </span><span class="n">t</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="kt">str</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">.</span><span class="n">len</span><span class="p">(),</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">len</span><span class="p">());</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">chars</span><span class="p">().</span><span class="n">nth</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">chars</span><span class="p">().</span><span class="n">nth</span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span>
 <a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">                    </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">min</span><span class="p">(</span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="w">            </span><span class="p">}</span>
@@ -5018,27 +5090,27 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Edit distance: Dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Edit distance: dynamic programming */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">editDistanceDP</span><span class="p">(</span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">**</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">((</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">));</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">myMin</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">        </span><span class="p">}</span>
@@ -5053,26 +5125,26 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Edit distance: Dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Edit distance: dynamic programming */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">editDistanceDP</span><span class="p">(</span><span class="n">s</span><span class="p">:</span><span class="w"> </span><span class="kt">String</span><span class="p">,</span><span class="w"> </span><span class="n">t</span><span class="p">:</span><span class="w"> </span><span class="kt">String</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="na">length</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="na">length</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Array</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="m">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="m">0</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="p">),</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w">        </span><span class="p">}</span>
@@ -5086,17 +5158,17 @@ dp[i, j] = dp[i-1, j-1]
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">edit_distance_dp</span><span class="p">(</span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="n">t</span><span class="p">)</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="o">.</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">t</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># State transition: first row and first column</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># State transitions: first row and first column</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="p">(</span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="p">(</span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">j</span><span class="o">|</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="c1"># State transition: rest of the rows and columns</span>
+<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="c1"># State transitions: remaining rows and columns</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">s</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span>
-<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">        </span><span class="c1"># If two characters are equal, skip both characters</span>
+<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">        </span><span class="c1"># If the two characters are equal, skip both of them</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">      </span><span class="k">else</span>
-<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">        </span><span class="c1"># Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">        </span><span class="c1"># Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]].</span><span class="n">min</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">      </span><span class="k">end</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">    </span><span class="k">end</span>
@@ -5107,11 +5179,16 @@ dp[i, j] = dp[i-1, j-1]
 </div>
 </div>
 </div>
-<p>As shown in Figure 14-30, the state transition process for the edit distance problem is very similar to that of the knapsack problem; both can be viewed as the process of filling a two-dimensional grid.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20edit_distance_dp%28s%3A%20str,%20t%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Edit%20distance%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%20%28m%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20State%20transitions%3A%20first%20row%20and%20first%20column%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B0%5D%20%3D%20i%0A%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Bj%5D%20%3D%20j%0A%20%20%20%20%23%20State%20transitions%3A%20remaining%20rows%20and%20columns%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20s%5Bi%20-%201%5D%20%3D%3D%20t%5Bj%20-%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20the%20two%20characters%20are%20equal,%20skip%20both%20of%20them%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20dp%5Bi%20-%201%5D%5Bj%20-%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Minimum%20edits%20%3D%20the%20minimum%20among%20the%20insertion,%20deletion,%20and%20replacement%20subproblems%20%2B%201%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20min%28dp%5Bi%5D%5Bj%20-%201%5D,%20dp%5Bi%20-%201%5D%5Bj%5D,%20dp%5Bi%20-%201%5D%5Bj%20-%201%5D%29%20%2B%201%0A%20%20%20%20return%20dp%5Bn%5D%5Bm%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20s%20%3D%20%22bag%22%0A%20%20%20%20t%20%3D%20%22pack%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%0A%20%20%20%20%23%20Dynamic%20programming%0A%20%20%20%20res%20%3D%20edit_distance_dp%28s,%20t%29%0A%20%20%20%20print%28f%22Changing%20%7Bs%7D%20to%20%7Bt%7D%20requires%20at%20least%20%7Bres%7D%20edits%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20edit_distance_dp%28s%3A%20str,%20t%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Edit%20distance%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%20%28m%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20State%20transitions%3A%20first%20row%20and%20first%20column%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B0%5D%20%3D%20i%0A%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Bj%5D%20%3D%20j%0A%20%20%20%20%23%20State%20transitions%3A%20remaining%20rows%20and%20columns%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20s%5Bi%20-%201%5D%20%3D%3D%20t%5Bj%20-%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20the%20two%20characters%20are%20equal,%20skip%20both%20of%20them%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20dp%5Bi%20-%201%5D%5Bj%20-%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Minimum%20edits%20%3D%20the%20minimum%20among%20the%20insertion,%20deletion,%20and%20replacement%20subproblems%20%2B%201%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bj%5D%20%3D%20min%28dp%5Bi%5D%5Bj%20-%201%5D,%20dp%5Bi%20-%201%5D%5Bj%5D,%20dp%5Bi%20-%201%5D%5Bj%20-%201%5D%29%20%2B%201%0A%20%20%20%20return%20dp%5Bn%5D%5Bm%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20s%20%3D%20%22bag%22%0A%20%20%20%20t%20%3D%20%22pack%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%0A%20%20%20%20%23%20Dynamic%20programming%0A%20%20%20%20res%20%3D%20edit_distance_dp%28s,%20t%29%0A%20%20%20%20print%28f%22Changing%20%7Bs%7D%20to%20%7Bt%7D%20requires%20at%20least%20%7Bres%7D%20edits%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>As shown in Figure 14-30, the state transition process resembles that of the knapsack problem: both fill a two-dimensional grid.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:15"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><input id="__tabbed_2_14" name="__tabbed_2" type="radio" /><input id="__tabbed_2_15" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">&lt;1&gt;</label><label for="__tabbed_2_2">&lt;2&gt;</label><label for="__tabbed_2_3">&lt;3&gt;</label><label for="__tabbed_2_4">&lt;4&gt;</label><label for="__tabbed_2_5">&lt;5&gt;</label><label for="__tabbed_2_6">&lt;6&gt;</label><label for="__tabbed_2_7">&lt;7&gt;</label><label for="__tabbed_2_8">&lt;8&gt;</label><label for="__tabbed_2_9">&lt;9&gt;</label><label for="__tabbed_2_10">&lt;10&gt;</label><label for="__tabbed_2_11">&lt;11&gt;</label><label for="__tabbed_2_12">&lt;12&gt;</label><label for="__tabbed_2_13">&lt;13&gt;</label><label for="__tabbed_2_14">&lt;14&gt;</label><label for="__tabbed_2_15">&lt;15&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Dynamic programming process for edit distance" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step1.png" /></p>
+<p><img alt="Dynamic programming for edit distance" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="edit_distance_dp_step2" class="animation-figure" src="../edit_distance_problem.assets/edit_distance_dp_step2.png" /></p>
@@ -5157,64 +5234,64 @@ dp[i, j] = dp[i-1, j-1]
 </div>
 </div>
 </div>
-<p align="center"> Figure 14-30 &nbsp; Dynamic programming process for edit distance </p>
+<p align="center"> Figure 14-30 &nbsp; Dynamic programming for edit distance </p>
 
-<h3 id="3-space-optimization">3. &nbsp; Space Optimization<a class="headerlink" href="#3-space-optimization" title="Permanent link">&para;</a></h3>
-<p>Since <span class="arithmatex">\(dp[i, j]\)</span> depends on the states above <span class="arithmatex">\(dp[i-1, j]\)</span>, to the left <span class="arithmatex">\(dp[i, j-1]\)</span>, and at the upper-left <span class="arithmatex">\(dp[i-1, j-1]\)</span>, forward traversal will lose the upper-left state <span class="arithmatex">\(dp[i-1, j-1]\)</span>, while reverse traversal cannot construct <span class="arithmatex">\(dp[i, j-1]\)</span> in advance, so neither traversal order is suitable.</p>
-<p>For this reason, we can use a variable <code>leftup</code> to temporarily store the upper-left solution <span class="arithmatex">\(dp[i-1, j-1]\)</span>, so we only need to consider the solutions to the left and above. This situation is the same as in the unbounded knapsack problem, so we can use forward traversal. The code is as follows:</p>
+<h3 id="3-space-optimization">3. &nbsp; Space optimization<a class="headerlink" href="#3-space-optimization" title="Permanent link">&para;</a></h3>
+<p>The state <span class="arithmatex">\(dp[i,j]\)</span> depends on <span class="arithmatex">\(dp[i-1, j]\)</span> above, <span class="arithmatex">\(dp[i, j-1]\)</span> to the left, and <span class="arithmatex">\(dp[i-1, j-1]\)</span> at the upper left. Forward iteration loses <span class="arithmatex">\(dp[i-1, j-1]\)</span>, while backward iteration cannot compute <span class="arithmatex">\(dp[i, j-1]\)</span> in advance. Neither order alone is sufficient.</p>
+<p>We can use a variable <code>leftup</code> to preserve the upper-left solution <span class="arithmatex">\(dp[i-1, j-1]\)</span>, leaving only the left and upper entries to consider. The situation then matches unbounded knapsack, and forward iteration works. The code is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">edit_distance.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">edit_distance_dp_comp</span><span class="p">(</span><span class="n">s</span><span class="p">:</span> <span class="nb">str</span><span class="p">,</span> <span class="n">t</span><span class="p">:</span> <span class="nb">str</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Edit distance: Space-optimized dynamic programming"""</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Edit distance: space-optimized dynamic programming"""</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="n">n</span><span class="p">,</span> <span class="n">m</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">s</span><span class="p">),</span> <span class="nb">len</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># State transition: first row</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># State transitions: first row</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">j</span>
-<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>    <span class="c1"># State transition: rest of the rows</span>
+<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>    <span class="c1"># State transitions: remaining rows</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>        <span class="c1"># State transition: first column</span>
+<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>        <span class="c1"># State transitions: first column</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>        <span class="n">leftup</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>  <span class="c1"># Temporarily store dp[i-1, j-1]</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>        <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-<a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>        <span class="c1"># State transition: rest of the columns</span>
+<a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>        <span class="c1"># State transitions: remaining columns</span>
 <a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>            <span class="n">temp</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>            <span class="k">if</span> <span class="n">s</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">t</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]:</span>
-<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>                <span class="c1"># If two characters are equal, skip both characters</span>
+<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>                <span class="c1"># If the two characters are equal, skip both of them</span>
 <a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">leftup</span>
 <a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a>            <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a>                <span class="c1"># Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a>                <span class="c1"># Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-13-21" name="__codelineno-13-21" href="#__codelineno-13-21"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span> <span class="o">-</span> <span class="mi">1</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">leftup</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
-<a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a>            <span class="n">leftup</span> <span class="o">=</span> <span class="n">temp</span>  <span class="c1"># Update for next round's dp[i-1, j-1]</span>
+<a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a>            <span class="n">leftup</span> <span class="o">=</span> <span class="n">temp</span>  <span class="c1"># Update to dp[i-1, j-1] for the next iteration</span>
 <a id="__codelineno-13-23" name="__codelineno-13-23" href="#__codelineno-13-23"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Edit distance: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Edit distance: space-optimized dynamic programming */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">editDistanceDPComp</span><span class="p">(</span><span class="n">string</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">length</span><span class="p">(),</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">length</span><span class="p">();</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
-<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// Temporarily store dp[i-1, j-1]</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">            </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span><span class="p">;</span>
 <a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-14-24" name="__codelineno-14-24" href="#__codelineno-14-24"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update for next round's dp[i-1, j-1]</span>
+<a id="__codelineno-14-24" name="__codelineno-14-24" href="#__codelineno-14-24"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update to dp[i-1, j-1] for the next iteration</span>
 <a id="__codelineno-14-25" name="__codelineno-14-25" href="#__codelineno-14-25"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-14-26" name="__codelineno-14-26" href="#__codelineno-14-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-27" name="__codelineno-14-27" href="#__codelineno-14-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">];</span>
@@ -5222,30 +5299,30 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Edit distance: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Edit distance: space-optimized dynamic programming */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">editDistanceDPComp</span><span class="p">(</span><span class="n">String</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="na">length</span><span class="p">(),</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="na">length</span><span class="p">();</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// Temporarily store dp[i-1, j-1]</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">            </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">.</span><span class="na">charAt</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="na">charAt</span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span><span class="p">;</span>
 <a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-15-24" name="__codelineno-15-24" href="#__codelineno-15-24"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update for next round's dp[i-1, j-1]</span>
+<a id="__codelineno-15-24" name="__codelineno-15-24" href="#__codelineno-15-24"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update to dp[i-1, j-1] for the next iteration</span>
 <a id="__codelineno-15-25" name="__codelineno-15-25" href="#__codelineno-15-25"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-15-26" name="__codelineno-15-26" href="#__codelineno-15-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-27" name="__codelineno-15-27" href="#__codelineno-15-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="p">;</span>
@@ -5253,30 +5330,30 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Edit distance: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Edit distance: space-optimized dynamic programming */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">EditDistanceDPComp</span><span class="p">(</span><span class="kt">string</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">Length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// Temporarily store dp[i-1, j-1]</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">            </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span><span class="p">;</span>
 <a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-16-24" name="__codelineno-16-24" href="#__codelineno-16-24"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update for next round's dp[i-1, j-1]</span>
+<a id="__codelineno-16-24" name="__codelineno-16-24" href="#__codelineno-16-24"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update to dp[i-1, j-1] for the next iteration</span>
 <a id="__codelineno-16-25" name="__codelineno-16-25" href="#__codelineno-16-25"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-16-26" name="__codelineno-16-26" href="#__codelineno-16-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-27" name="__codelineno-16-27" href="#__codelineno-16-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">];</span>
@@ -5284,31 +5361,31 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Edit distance: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Edit distance: space-optimized dynamic programming */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">editDistanceDPComp</span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">t</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">s</span><span class="p">)</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">t</span><span class="p">)</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">j</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">        </span><span class="nx">leftUp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="c1">// Temporarily store dp[i-1, j-1]</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span>
-<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">            </span><span class="nx">temp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span>
 <a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">s</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">t</span><span class="p">[</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="w">                </span><span class="c1">// If the two bytes are equal, skip both (each byte is one character for ASCII input)</span>
 <a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">leftUp</span>
 <a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-22" name="__codelineno-17-22" href="#__codelineno-17-22"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-17-22" name="__codelineno-17-22" href="#__codelineno-17-22"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">MinInt</span><span class="p">(</span><span class="nx">MinInt</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]),</span><span class="w"> </span><span class="nx">leftUp</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-17-24" name="__codelineno-17-24" href="#__codelineno-17-24"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-17-25" name="__codelineno-17-25" href="#__codelineno-17-25"></a><span class="w">            </span><span class="nx">leftUp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="w"> </span><span class="c1">// Update for next round's dp[i-1, j-1]</span>
+<a id="__codelineno-17-25" name="__codelineno-17-25" href="#__codelineno-17-25"></a><span class="w">            </span><span class="nx">leftUp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="w"> </span><span class="c1">// Update to dp[i-1, j-1] for the next iteration</span>
 <a id="__codelineno-17-26" name="__codelineno-17-26" href="#__codelineno-17-26"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-17-27" name="__codelineno-17-27" href="#__codelineno-17-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-28" name="__codelineno-17-28" href="#__codelineno-17-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span>
@@ -5316,31 +5393,31 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Edit distance: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Edit distance: space-optimized dynamic programming */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">editDistanceDPComp</span><span class="p">(</span><span class="n">s</span><span class="p">:</span><span class="w"> </span><span class="nb">String</span><span class="p">,</span><span class="w"> </span><span class="n">t</span><span class="p">:</span><span class="w"> </span><span class="nb">String</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">j</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">leftup</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="c1">// Temporarily store dp[i-1, j-1]</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i</span>
-<a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="nv">temp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">utf8CString</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">                </span><span class="c1">// If the two current UTF-8 bytes match, skip both</span>
 <a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">leftup</span>
 <a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-18-23" name="__codelineno-18-23" href="#__codelineno-18-23"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">min</span><span class="p">(</span><span class="bp">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-18-24" name="__codelineno-18-24" href="#__codelineno-18-24"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-18-25" name="__codelineno-18-25" href="#__codelineno-18-25"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="c1">// Update for next round's dp[i-1, j-1]</span>
+<a id="__codelineno-18-25" name="__codelineno-18-25" href="#__codelineno-18-25"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="c1">// Update to dp[i-1, j-1] for the next iteration</span>
 <a id="__codelineno-18-26" name="__codelineno-18-26" href="#__codelineno-18-26"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-18-27" name="__codelineno-18-27" href="#__codelineno-18-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-28" name="__codelineno-18-28" href="#__codelineno-18-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">]</span>
@@ -5348,31 +5425,31 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Edit distance: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Edit distance: space-optimized dynamic programming */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">editDistanceDPComp</span><span class="p">(</span><span class="nx">s</span><span class="p">,</span><span class="w"> </span><span class="nx">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">length</span><span class="p">,</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
-<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">j</span><span class="p">;</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// Temporarily store dp[i-1, j-1]</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
-<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">            </span><span class="kd">const</span><span class="w"> </span><span class="nx">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">leftup</span><span class="p">;</span>
 <a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-19-23" name="__codelineno-19-23" href="#__codelineno-19-23"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">],</span><span class="w"> </span><span class="nx">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-19-24" name="__codelineno-19-24" href="#__codelineno-19-24"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-19-25" name="__codelineno-19-25" href="#__codelineno-19-25"></a><span class="w">            </span><span class="nx">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update for next round's dp[i-1, j-1]</span>
+<a id="__codelineno-19-25" name="__codelineno-19-25" href="#__codelineno-19-25"></a><span class="w">            </span><span class="nx">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update to dp[i-1, j-1] for the next iteration</span>
 <a id="__codelineno-19-26" name="__codelineno-19-26" href="#__codelineno-19-26"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-19-27" name="__codelineno-19-27" href="#__codelineno-19-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-28" name="__codelineno-19-28" href="#__codelineno-19-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">m</span><span class="p">];</span>
@@ -5380,31 +5457,31 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Edit distance: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Edit distance: space-optimized dynamic programming */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">editDistanceDPComp</span><span class="p">(</span><span class="nx">s</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">t</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">length</span><span class="p">,</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
-<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">j</span><span class="p">;</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// Temporarily store dp[i-1, j-1]</span>
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
-<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">            </span><span class="kd">const</span><span class="w"> </span><span class="nx">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">charAt</span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">leftup</span><span class="p">;</span>
 <a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-20-23" name="__codelineno-20-23" href="#__codelineno-20-23"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">j</span><span class="p">],</span><span class="w"> </span><span class="nx">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a><span class="w">            </span><span class="nx">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update for next round's dp[i-1, j-1]</span>
+<a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a><span class="w">            </span><span class="nx">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update to dp[i-1, j-1] for the next iteration</span>
 <a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-20-27" name="__codelineno-20-27" href="#__codelineno-20-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-28" name="__codelineno-20-28" href="#__codelineno-20-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">m</span><span class="p">];</span>
@@ -5412,30 +5489,30 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Edit distance: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Edit distance: space-optimized dynamic programming */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">editDistanceDPComp</span><span class="p">(</span><span class="kt">String</span><span class="w"> </span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="kt">String</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
-<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">    </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">    </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// Temporarily store dp[i-1, j-1]</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">    </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">    </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">      </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">        </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">        </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span><span class="p">;</span>
 <a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="w">      </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="w">        </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="w">        </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">      </span><span class="p">}</span>
-<a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">      </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update for next round's dp[i-1, j-1]</span>
+<a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">      </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update to dp[i-1, j-1] for the next iteration</span>
 <a id="__codelineno-21-25" name="__codelineno-21-25" href="#__codelineno-21-25"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-27" name="__codelineno-21-27" href="#__codelineno-21-27"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">];</span>
@@ -5443,30 +5520,30 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Edit distance: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Edit distance: space-optimized dynamic programming */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">edit_distance_dp_comp</span><span class="p">(</span><span class="n">s</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="kt">str</span><span class="p">,</span><span class="w"> </span><span class="n">t</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="kt">str</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">.</span><span class="n">len</span><span class="p">(),</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">len</span><span class="p">());</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// Temporarily store dp[i-1, j-1]</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
-<a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="n">chars</span><span class="p">().</span><span class="n">nth</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="n">chars</span><span class="p">().</span><span class="n">nth</span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span><span class="p">;</span>
 <a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">min</span><span class="p">(</span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update for next round's dp[i-1, j-1]</span>
+<a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update to dp[i-1, j-1] for the next iteration</span>
 <a id="__codelineno-22-25" name="__codelineno-22-25" href="#__codelineno-22-25"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-26" name="__codelineno-22-26" href="#__codelineno-22-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-27" name="__codelineno-22-27" href="#__codelineno-22-27"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">]</span>
@@ -5474,29 +5551,29 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Edit distance: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Edit distance: space-optimized dynamic programming */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">editDistanceDPComp</span><span class="p">(</span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
-<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// Temporarily store dp[i-1, j-1]</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">            </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span><span class="p">;</span>
 <a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">myMin</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update for next round's dp[i-1, j-1]</span>
+<a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update to dp[i-1, j-1] for the next iteration</span>
 <a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">m</span><span class="p">];</span>
@@ -5507,31 +5584,31 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Edit distance: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">edit_distance.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Edit distance: space-optimized dynamic programming */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">editDistanceDPComp</span><span class="p">(</span><span class="n">s</span><span class="p">:</span><span class="w"> </span><span class="kt">String</span><span class="p">,</span><span class="w"> </span><span class="n">t</span><span class="p">:</span><span class="w"> </span><span class="kt">String</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="p">.</span><span class="na">length</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">.</span><span class="na">length</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
-<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="c1">// State transition: first row</span>
+<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="c1">// State transitions: first row</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows</span>
+<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">        </span><span class="c1">// State transition: first column</span>
+<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">        </span><span class="c1">// State transitions: first column</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="m">0</span><span class="o">]</span><span class="w"> </span><span class="c1">// Temporarily store dp[i-1, j-1]</span>
 <a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="m">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span>
-<a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">        </span><span class="c1">// State transition: rest of the columns</span>
+<a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">        </span><span class="c1">// State transitions: remaining columns</span>
 <a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">            </span><span class="kd">val</span><span class="w"> </span><span class="nv">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="w">                </span><span class="c1">// If two characters are equal, skip both characters</span>
+<a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="w">                </span><span class="c1">// If the two characters are equal, skip both of them</span>
 <a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span>
 <a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">                </span><span class="c1">// Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">                </span><span class="c1">// Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">),</span><span class="w"> </span><span class="n">leftup</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-24-25" name="__codelineno-24-25" href="#__codelineno-24-25"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="c1">// Update for next round's dp[i-1, j-1]</span>
+<a id="__codelineno-24-25" name="__codelineno-24-25" href="#__codelineno-24-25"></a><span class="w">            </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="c1">// Update to dp[i-1, j-1] for the next iteration</span>
 <a id="__codelineno-24-26" name="__codelineno-24-26" href="#__codelineno-24-26"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-24-27" name="__codelineno-24-27" href="#__codelineno-24-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-28" name="__codelineno-24-28" href="#__codelineno-24-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">m</span><span class="o">]</span>
@@ -5539,28 +5616,28 @@ dp[i, j] = dp[i-1, j-1]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">edit_distance.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Edit distance: space-optimized DP ###</span>
+<div class="highlight"><span class="filename">edit_distance.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Edit distance: space-optimized dynamic programming ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">edit_distance_dp_comp</span><span class="p">(</span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="n">t</span><span class="p">)</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="o">.</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">t</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># State transition: first row</span>
+<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># State transitions: first row</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="p">(</span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">j</span><span class="o">|</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="c1"># State transition: rest of the rows</span>
+<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="c1"># State transitions: remaining rows</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">    </span><span class="c1"># State transition: first column</span>
+<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">    </span><span class="c1"># State transitions: first column</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">    </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">.</span><span class="n">first</span><span class="w"> </span><span class="c1"># Temporarily store dp[i-1, j-1]</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">    </span><span class="c1"># State transition: rest of the columns</span>
+<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">    </span><span class="c1"># State transitions: remaining columns</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">      </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">s</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">t</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span>
-<a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">        </span><span class="c1"># If two characters are equal, skip both characters</span>
+<a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">        </span><span class="c1"># If the two characters are equal, skip both of them</span>
 <a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftup</span>
 <a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w">      </span><span class="k">else</span>
-<a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="w">        </span><span class="c1"># Minimum edit steps = minimum edit steps of insert, delete, replace + 1</span>
+<a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="w">        </span><span class="c1"># Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1</span>
 <a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">leftup</span><span class="o">].</span><span class="n">min</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="w">      </span><span class="k">end</span>
-<a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="w">      </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="c1"># Update for next round's dp[i-1, j-1]</span>
+<a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="w">      </span><span class="n">leftup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="c1"># Update to dp[i-1, j-1] for the next iteration</span>
 <a id="__codelineno-25-23" name="__codelineno-25-23" href="#__codelineno-25-23"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-25-24" name="__codelineno-25-24" href="#__codelineno-25-24"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-25" name="__codelineno-25-25" href="#__codelineno-25-25"></a><span class="w">  </span><span class="n">dp</span><span class="o">[</span><span class="n">m</span><span class="o">]</span>
@@ -5569,6 +5646,11 @@ dp[i, j] = dp[i-1, j-1]
 </div>
 </div>
 </div>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20edit_distance_dp_comp%28s%3A%20str,%20t%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Edit%20distance%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%20%20%20%20dp%20%3D%20%5B0%5D%20%2A%20%28m%20%2B%201%29%0A%20%20%20%20%23%20State%20transitions%3A%20first%20row%0A%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20j%0A%20%20%20%20%23%20State%20transitions%3A%20remaining%20rows%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%23%20State%20transitions%3A%20first%20column%0A%20%20%20%20%20%20%20%20leftup%20%3D%20dp%5B0%5D%20%20%23%20Temporarily%20store%20dp%5Bi-1,%20j-1%5D%0A%20%20%20%20%20%20%20%20dp%5B0%5D%20%2B%3D%201%0A%20%20%20%20%20%20%20%20%23%20State%20transitions%3A%20remaining%20columns%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20temp%20%3D%20dp%5Bj%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20s%5Bi%20-%201%5D%20%3D%3D%20t%5Bj%20-%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20the%20two%20characters%20are%20equal,%20skip%20both%20of%20them%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20leftup%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Minimum%20edits%20%3D%20the%20minimum%20among%20the%20insertion,%20deletion,%20and%20replacement%20subproblems%20%2B%201%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20min%28dp%5Bj%20-%201%5D,%20dp%5Bj%5D,%20leftup%29%20%2B%201%0A%20%20%20%20%20%20%20%20%20%20%20%20leftup%20%3D%20temp%20%20%23%20Update%20to%20dp%5Bi-1,%20j-1%5D%20for%20the%20next%20iteration%0A%20%20%20%20return%20dp%5Bm%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20s%20%3D%20%22bag%22%0A%20%20%20%20t%20%3D%20%22pack%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%0A%20%20%20%20%23%20Space-optimized%20dynamic%20programming%0A%20%20%20%20res%20%3D%20edit_distance_dp_comp%28s,%20t%29%0A%20%20%20%20print%28f%22Changing%20%7Bs%7D%20to%20%7Bt%7D%20requires%20at%20least%20%7Bres%7D%20edits%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20edit_distance_dp_comp%28s%3A%20str,%20t%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Edit%20distance%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%20%20%20%20dp%20%3D%20%5B0%5D%20%2A%20%28m%20%2B%201%29%0A%20%20%20%20%23%20State%20transitions%3A%20first%20row%0A%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20j%0A%20%20%20%20%23%20State%20transitions%3A%20remaining%20rows%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%23%20State%20transitions%3A%20first%20column%0A%20%20%20%20%20%20%20%20leftup%20%3D%20dp%5B0%5D%20%20%23%20Temporarily%20store%20dp%5Bi-1,%20j-1%5D%0A%20%20%20%20%20%20%20%20dp%5B0%5D%20%2B%3D%201%0A%20%20%20%20%20%20%20%20%23%20State%20transitions%3A%20remaining%20columns%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%281,%20m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20temp%20%3D%20dp%5Bj%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20s%5Bi%20-%201%5D%20%3D%3D%20t%5Bj%20-%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20the%20two%20characters%20are%20equal,%20skip%20both%20of%20them%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20leftup%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Minimum%20edits%20%3D%20the%20minimum%20among%20the%20insertion,%20deletion,%20and%20replacement%20subproblems%20%2B%201%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bj%5D%20%3D%20min%28dp%5Bj%20-%201%5D,%20dp%5Bj%5D,%20leftup%29%20%2B%201%0A%20%20%20%20%20%20%20%20%20%20%20%20leftup%20%3D%20temp%20%20%23%20Update%20to%20dp%5Bi-1,%20j-1%5D%20for%20the%20next%20iteration%0A%20%20%20%20return%20dp%5Bm%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20s%20%3D%20%22bag%22%0A%20%20%20%20t%20%3D%20%22pack%22%0A%20%20%20%20n,%20m%20%3D%20len%28s%29,%20len%28t%29%0A%0A%20%20%20%20%23%20Space-optimized%20dynamic%20programming%0A%20%20%20%20res%20%3D%20edit_distance_dp_comp%28s,%20t%29%0A%20%20%20%20print%28f%22Changing%20%7Bs%7D%20to%20%7Bt%7D%20requires%20at%20least%20%7Bres%7D%20edits%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
 
 <!-- Source file information -->
 
@@ -5591,7 +5673,7 @@ aria-label="Footer"
   <a
     href="../unbounded_knapsack_problem/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 14.5 Unbounded Knapsack Problem"
+    aria-label="Previous: 14.5   Unbounded knapsack problem"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -5603,7 +5685,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        14.5 Unbounded Knapsack Problem
+        14.5 &nbsp; Unbounded knapsack problem
       </div>
     </div>
   </a>
@@ -5615,7 +5697,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 14.7 Summary"
+    aria-label="Next: 14.7   Summary"
     rel="next"
   >
     <div class="md-footer__title">
@@ -5623,7 +5705,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        14.7 Summary
+        14.7 &nbsp; Summary
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -5640,7 +5722,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_dynamic_programming/exercises/index.html b/en/chapter_dynamic_programming/exercises/index.html
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
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-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
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-  Chapter 0. Preface
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
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-  0.3 Summary
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-  Chapter 1. Encounter with Algorithms
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-  Chapter 1. Encounter with Algorithms
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-  2.6 Exercises
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-  Chapter 3. Data Structures
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
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-  3.6 Exercises
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
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-  4.4 Random-Access Memory and Cache *
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-  4.5 Summary
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-  4.6 Exercises
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-  Chapter 5. Stacks and Queues
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-  Chapter 5. Stacks and Queues
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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@@ -1720,7 +1720,7 @@
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
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-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
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@@ -2638,7 +2638,7 @@
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-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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@@ -2878,7 +2878,7 @@
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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@@ -3028,7 +3028,7 @@
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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@@ -3055,7 +3055,7 @@
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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@@ -3082,7 +3082,7 @@
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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@@ -3109,7 +3109,7 @@
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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@@ -3136,7 +3136,7 @@
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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@@ -3190,7 +3190,7 @@
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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@@ -3283,7 +3283,7 @@
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3352,7 +3352,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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@@ -3379,7 +3379,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3406,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3433,7 +3433,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3460,7 +3460,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3526,7 +3526,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3568,7 +3568,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3595,7 +3595,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3622,7 +3622,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3649,7 +3649,7 @@
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3676,7 +3676,7 @@
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3775,7 +3775,7 @@
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3817,7 +3817,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3844,7 +3844,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3871,7 +3871,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3898,7 +3898,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1481-concept-review" class="md-nav__link">
+  <a href="#1481-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        14.8.1 &nbsp; Concept Review
+        14.8.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="14.8.1 Concept Review">
+    <nav class="md-nav" aria-label="14.8.1 Check your understanding">
       <ul class="md-nav__list">
 
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-  <a href="#1-when-is-dynamic-programming-appropriate" class="md-nav__link">
+  <a href="#1-when-is-dynamic-programming-a-good-fit" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; When Is Dynamic Programming Appropriate?
+        1. &nbsp; When is dynamic programming a good fit?
       </span>
     </span>
   </a>
@@ -4081,10 +4081,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-calculating-one-cell-in-a-knapsack-table" class="md-nav__link">
+  <a href="#2-how-do-you-calculate-a-cell-in-the-knapsack-table" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Calculating One Cell in a Knapsack Table
+        2. &nbsp; How do you calculate a cell in the knapsack table?
       </span>
     </span>
   </a>
@@ -4092,10 +4092,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-in-which-order-should-knapsack-capacities-be-updated" class="md-nav__link">
+  <a href="#3-in-what-order-should-knapsack-capacities-be-updated" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; In Which Order Should Knapsack Capacities Be Updated?
+        3. &nbsp; In what order should knapsack capacities be updated?
       </span>
     </span>
   </a>
@@ -4111,19 +4111,19 @@
   <a href="#1482-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        14.8.2 &nbsp; Programming Exercises
+        14.8.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="14.8.2 Programming Exercises">
+    <nav class="md-nav" aria-label="14.8.2 Programming exercises">
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-  <a href="#1-number-of-ways-to-climb-stairs" class="md-nav__link">
+  <a href="#1-counting-ways-to-climb-stairs" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Number of Ways to Climb Stairs
+        1. &nbsp; Counting ways to climb stairs
       </span>
     </span>
   </a>
@@ -4134,7 +4134,7 @@
   <a href="#2-0-1-knapsack" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; 0-1 Knapsack
+        2. &nbsp; 0-1 knapsack
       </span>
     </span>
   </a>
@@ -4208,7 +4208,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4229,7 +4229,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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@@ -4250,7 +4250,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4625,6 +4625,78 @@
 
 
 
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+  <a href="#1481-check-your-understanding" class="md-nav__link">
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       <span class="md-typeset">
-        14.8.1 &nbsp; Concept Review
+        14.8.1 &nbsp; Check your understanding
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     </span>
   </a>
 
-    <nav class="md-nav" aria-label="14.8.1 Concept Review">
+    <nav class="md-nav" aria-label="14.8.1 Check your understanding">
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+  <a href="#1-when-is-dynamic-programming-a-good-fit" class="md-nav__link">
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       <span class="md-typeset">
-        1. &nbsp; When Is Dynamic Programming Appropriate?
+        1. &nbsp; When is dynamic programming a good fit?
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 </li>
 
           <li class="md-nav__item">
-  <a href="#2-calculating-one-cell-in-a-knapsack-table" class="md-nav__link">
+  <a href="#2-how-do-you-calculate-a-cell-in-the-knapsack-table" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Calculating One Cell in a Knapsack Table
+        2. &nbsp; How do you calculate a cell in the knapsack table?
       </span>
     </span>
   </a>
@@ -4700,10 +4772,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-in-which-order-should-knapsack-capacities-be-updated" class="md-nav__link">
+  <a href="#3-in-what-order-should-knapsack-capacities-be-updated" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; In Which Order Should Knapsack Capacities Be Updated?
+        3. &nbsp; In what order should knapsack capacities be updated?
       </span>
     </span>
   </a>
@@ -4719,19 +4791,19 @@
   <a href="#1482-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        14.8.2 &nbsp; Programming Exercises
+        14.8.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="14.8.2 Programming Exercises">
+    <nav class="md-nav" aria-label="14.8.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-number-of-ways-to-climb-stairs" class="md-nav__link">
+  <a href="#1-counting-ways-to-climb-stairs" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Number of Ways to Climb Stairs
+        1. &nbsp; Counting ways to climb stairs
       </span>
     </span>
   </a>
@@ -4742,7 +4814,7 @@
   <a href="#2-0-1-knapsack" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; 0-1 Knapsack
+        2. &nbsp; 0-1 knapsack
       </span>
     </span>
   </a>
@@ -4792,125 +4864,125 @@
 
 <!-- Page content -->
 <h1 id="148-exercises">14.8 &nbsp; Exercises<a class="headerlink" href="#148-exercises" title="Permanent link">&para;</a></h1>
-<h2 id="1481-concept-review">14.8.1 &nbsp; Concept Review<a class="headerlink" href="#1481-concept-review" title="Permanent link">&para;</a></h2>
-<h3 id="1-when-is-dynamic-programming-appropriate">1. &nbsp; When Is Dynamic Programming Appropriate?<a class="headerlink" href="#1-when-is-dynamic-programming-appropriate" title="Permanent link">&para;</a></h3>
-<p>A student says, "Whenever a recurrence can be written, dynamic programming should be used."
-For each task below, decide whether dynamic programming, backtracking, or a loop or mathematical formula without a <code>dp</code> table is more appropriate. Give one key reason.</p>
+<h2 id="1481-check-your-understanding">14.8.1 &nbsp; Check your understanding<a class="headerlink" href="#1481-check-your-understanding" title="Permanent link">&para;</a></h2>
+<h3 id="1-when-is-dynamic-programming-a-good-fit">1. &nbsp; When is dynamic programming a good fit?<a class="headerlink" href="#1-when-is-dynamic-programming-a-good-fit" title="Permanent link">&para;</a></h3>
+<p>A student says, “Whenever you can write a recurrence relation, you should use dynamic programming.”
+For each task below, decide whether dynamic programming, backtracking, or a loop or mathematical formula without a <code>dp</code> table is the best fit. Give one key reason for each choice.</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Using coin denominations <code>[1, 3, 4]</code>, make an amount of 6 with the fewest coins. Each denomination may be used repeatedly.</li>
+<li>Find the minimum number of coins needed to make an amount of 6 using denominations <code>[1, 3, 4]</code>. Each denomination can be used any number of times.</li>
 <li>Output all permutations of <code>[1, 2, 3]</code>.</li>
 <li>
 <p>Calculate <span class="arithmatex">\(1 + 2 + \dots + n\)</span>.</p>
-<p>For the task you consider suitable for dynamic programming, also state what <code>dp[i]</code> represents.</p>
+<p>For the task you consider suitable for dynamic programming, also explain what <code>dp[i]</code> represents.</p>
 </li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>Dynamic programming is suitable. Let <code>dp[i]</code> be the minimum number of coins needed to make amount <code>i</code>.
-    For every coin <code>c</code> that does not exceed <code>i</code>, <code>dp[i-c] + 1</code> is a candidate answer,
-    and the minimum of these candidates is chosen. Different choices repeatedly encounter the same amounts, and an optimal solution for a larger amount can be built from optimal solutions for smaller amounts.
-    The answer for amount 6 is 2, using <code>3 + 3</code>.</p>
+<p>Dynamic programming is a good fit. Let <code>dp[i]</code> be the minimum number of coins needed to make an amount of <code>i</code>.
+    For each coin <code>c</code> whose denomination does not exceed <code>i</code>, consider <code>dp[i-c] + 1</code> as a candidate answer,
+    then take the minimum of these candidates. Different choices repeatedly lead to the same amounts, and an optimal solution for a larger amount can be built from optimal solutions for smaller amounts.
+    The answer for an amount of 6 is 2: <code>3 + 3</code>.</p>
 </li>
 <li>
-<p>Backtracking is suitable. The task requires generating all 6 permutations one by one. Backtracking can systematically make a choice, continue searching,
-    undo the choice, and then try another branch. Regardless of the method, actually outputting every permutation requires enumerating them.</p>
+<p>Use backtracking. The task requires generating all 6 permutations individually. Backtracking systematically makes a choice, continues the search,
+    then undoes the choice and explores other branches. Whatever method you use, outputting all the permutations requires enumerating them.</p>
 </li>
 <li>
-<p>A loop or the arithmetic-series formula is sufficient. Although the recurrence <code>S(i) = S(i-1) + i</code> can be written, calculating <code>S(i)</code> depends on only one smaller value, <code>S(i-1)</code>.
-    Each partial sum needs to be calculated only once, so there are no repeated subproblems and no need for a <code>dp</code> table. "A recurrence can be written" does not mean "dynamic programming is needed."</p>
+<p>A loop or the formula for an arithmetic series is sufficient. Although you can write <code>S(i) = S(i-1) + i</code>, calculating <code>S(i)</code> depends on just one smaller value, <code>S(i-1)</code>.
+    Each partial sum needs to be calculated only once, so there are no overlapping subproblems and no need for a <code>dp</code> table. Being able to write a recurrence does not mean you need dynamic programming.</p>
 </li>
 </ol>
 </details>
-<h3 id="2-calculating-one-cell-in-a-knapsack-table">2. &nbsp; Calculating One Cell in a Knapsack Table<a class="headerlink" href="#2-calculating-one-cell-in-a-knapsack-table" title="Permanent link">&para;</a></h3>
-<p>Consider this 0-1 knapsack problem: item weights <code>wgt = [1, 2, 3]</code>, values <code>val = [5, 11, 15]</code>, and knapsack capacity 4.
-<code>dp[i][c]</code> is the maximum value obtainable using only the first <span class="arithmatex">\(i\)</span> items with a knapsack capacity limit of <span class="arithmatex">\(c\)</span>;
-the knapsack does not have to be filled exactly.</p>
-<p>Calculate only the state <code>dp[3][4]</code>. You are given <code>dp[2][4] = 16</code> and <code>dp[2][1] = 5</code>:</p>
+<h3 id="2-how-do-you-calculate-a-cell-in-the-knapsack-table">2. &nbsp; How do you calculate a cell in the knapsack table?<a class="headerlink" href="#2-how-do-you-calculate-a-cell-in-the-knapsack-table" title="Permanent link">&para;</a></h3>
+<p>Consider a 0-1 knapsack problem with item weights <code>wgt = [1, 2, 3]</code>, values <code>val = [5, 11, 15]</code>, and a knapsack capacity of 4.
+Here, <code>dp[i][c]</code> is the maximum value obtainable using only the first <span class="arithmatex">\(i\)</span> items with a capacity limit of <span class="arithmatex">\(c\)</span>.
+The knapsack does not have to be filled exactly.</p>
+<p>Calculate just the state <code>dp[3][4]</code>. You are given <code>dp[2][4] = 16</code> and <code>dp[2][1] = 5</code>.</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>If the third item is not selected, what is the candidate value?</li>
-<li>If the third item is selected, how much capacity remains, and what is the candidate value?</li>
-<li>What should <code>dp[3][4]</code> be? Which items does this value correspond to selecting?</li>
+<li>What is the candidate value if you do not take item 3?</li>
+<li>If you take item 3, how much capacity remains, and what is the candidate value?</li>
+<li>What should <code>dp[3][4]</code> be? Which items does this solution take?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>If the third item is not selected, keep the result from the first two items. The candidate value is <code>dp[2][4] = 16</code>.</p>
+<p>If you do not take item 3, use the result for the first two items. The candidate value is <code>dp[2][4] = 16</code>.</p>
 </li>
 <li>
-<p>The third item has weight 3, so capacity <span class="arithmatex">\(4-3=1\)</span> remains after it is placed in the knapsack. The candidate value is
+<p>Item 3 weighs 3, so taking it leaves a capacity of <span class="arithmatex">\(4-3=1\)</span>. The candidate value is
     <code>dp[2][1] + 15 = 5 + 15 = 20</code>.</p>
 </li>
 <li>
-<p>Comparing 16 and 20 gives <code>dp[3][4] = 20</code>. This corresponds to selecting the first and third items,
-    whose total weight is <span class="arithmatex">\(1+3=4\)</span> and total value is <span class="arithmatex">\(5+15=20\)</span>.</p>
-<p>This state calculation demonstrates one "select or do not select" comparison in the 0-1 knapsack problem.</p>
+<p>Comparing 16 and 20 gives <code>dp[3][4] = 20</code>. This solution takes items 1 and 3,
+    with a total weight of <span class="arithmatex">\(1+3=4\)</span> and a total value of <span class="arithmatex">\(5+15=20\)</span>.</p>
+<p>Calculating this state illustrates a single “take it or leave it” comparison in the 0-1 knapsack problem.</p>
 </li>
 </ol>
 </details>
-<h3 id="3-in-which-order-should-knapsack-capacities-be-updated">3. &nbsp; In Which Order Should Knapsack Capacities Be Updated?<a class="headerlink" href="#3-in-which-order-should-knapsack-capacities-be-updated" title="Permanent link">&para;</a></h3>
-<p>A 0-1 knapsack problem has only one item, with weight 2 and value 5, and the knapsack has capacity 4.
-The item can be selected at most once. The initial one-dimensional array is <code>dp = [0, 0, 0, 0, 0]</code>.</p>
-<p>A student processes the item by updating capacities from 2 to 4:</p>
+<h3 id="3-in-what-order-should-knapsack-capacities-be-updated">3. &nbsp; In what order should knapsack capacities be updated?<a class="headerlink" href="#3-in-what-order-should-knapsack-capacities-be-updated" title="Permanent link">&para;</a></h3>
+<p>A 0-1 knapsack problem has just one item, with weight 2 and value 5. The knapsack capacity is 4.
+Each item can be taken at most once, and the initial one-dimensional array is <code>dp = [0, 0, 0, 0, 0]</code>.</p>
+<p>A student processes this item by updating capacities from 2 to 4:</p>
 <ul>
-<li>After updating <code>dp[2]</code>, its value is 5.</li>
-<li>After updating <code>dp[3]</code>, its value is also 5.</li>
-<li>When updating <code>dp[4]</code>, the student uses the newly obtained <code>dp[2]</code>, producing <code>dp[4] = 10</code>.</li>
+<li>Updating <code>dp[2]</code> gives 5.</li>
+<li>Updating <code>dp[3]</code> also gives 5.</li>
+<li>When updating <code>dp[4]</code>, the student uses the newly computed <code>dp[2]</code>, obtaining <code>dp[4] = 10</code>.</li>
 </ul>
 <!-- numbered-subquestions -->
 
 <ol>
 <li>Is <code>dp[4] = 10</code> correct? Why or why not?</li>
-<li>Given that each item may be selected at most once, what should <code>dp[4]</code> be?</li>
-<li>When processing each item, should capacities be updated from largest to smallest or from smallest to largest? What problem does this avoid?</li>
+<li>Given that each item can be taken at most once, what should <code>dp[4]</code> be?</li>
+<li>When processing each item, should capacities be updated from largest to smallest or from smallest to largest? What problem does this prevent?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>The result is incorrect. A value of 10 is equivalent to placing the item with value 5 into the knapsack twice,
-    violating the condition that each item may be selected at most once.</p>
+<p>The result is incorrect. A value of 10 means taking the item worth 5 twice,
+    which violates the condition that each item can be taken at most once.</p>
 </li>
 <li>
-<p>The knapsack can contain at most this one item, so the correct value of <code>dp[4]</code> is 5.</p>
+<p>The knapsack can contain only this one item, so the correct value of <code>dp[4]</code> is 5.</p>
 </li>
 <li>
-<p>Capacities should be updated from largest to smallest, in the order 4, 3, 2.
-    Then, when calculating <code>dp[c]</code>, the value read from <code>dp[c-2]</code> still comes from before the current item was processed,
-    preventing the current item from being reused during the same round.</p>
+<p>Update capacities from largest to smallest: 4, then 3, then 2.
+    This ensures that when calculating <code>dp[c]</code>, the value read from <code>dp[c-2]</code> is still the result from before the current item was processed.
+    The current item therefore cannot be used more than once in the same round of updates.</p>
 </li>
 </ol>
 </details>
-<h2 id="1482-programming-exercises">14.8.2 &nbsp; Programming Exercises<a class="headerlink" href="#1482-programming-exercises" title="Permanent link">&para;</a></h2>
-<h3 id="1-number-of-ways-to-climb-stairs">1. &nbsp; Number of Ways to Climb Stairs<a class="headerlink" href="#1-number-of-ways-to-climb-stairs" title="Permanent link">&para;</a></h3>
-<p>A staircase has <code>n</code> steps. Each move climbs either 1 or 2 steps, and you must land exactly on step <code>n</code>.
-Calculate the number of distinct ways to reach the top. Assume <code>n &gt;= 1</code>; ways are distinguished only by their sequence of 1-step and 2-step moves.
-Use a one-dimensional dynamic programming array. For now, do not use the space optimization that keeps only two states.</p>
+<h2 id="1482-programming-exercises">14.8.2 &nbsp; Programming exercises<a class="headerlink" href="#1482-programming-exercises" title="Permanent link">&para;</a></h2>
+<h3 id="1-counting-ways-to-climb-stairs">1. &nbsp; Counting ways to climb stairs<a class="headerlink" href="#1-counting-ways-to-climb-stairs" title="Permanent link">&para;</a></h3>
+<p>A staircase has <code>n</code> steps. You can climb either 1 or 2 steps at a time and must land exactly on step <code>n</code>.
+Calculate the number of different ways to climb the staircase. Assume <code>n &gt;= 1</code>; two ways differ only in their sequence of 1-step and 2-step moves.
+Use a one-dimensional dynamic programming array. For now, do not use the space optimization that retains only two states.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>The last move to step i can cover only 1 or 2 steps</li>
-<li>Therefore, dp[i] = dp[i-1] + dp[i-2]</li>
-<li>Handle the cases where n is 1 or 2 first, then fill the table starting from step 3</li>
+<li>The final move to step i must be either a 1-step or a 2-step move.</li>
+<li>This gives dp[i] = dp[i-1] + dp[i-2].</li>
+<li>Handle n = 1 and n = 2 first, then fill the table starting at step 3.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/climbing-stairs/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
-<h3 id="2-0-1-knapsack">2. &nbsp; 0-1 Knapsack<a class="headerlink" href="#2-0-1-knapsack" title="Permanent link">&para;</a></h3>
-<p>You are given equal-length arrays <code>wgt</code> and <code>val</code>. Item <code>i</code> has positive integer weight <code>wgt[i]</code> and non-negative integer value <code>val[i]</code>.
-The knapsack capacity <code>cap</code> is a non-negative integer. Each item may be selected at most once. Find the maximum total value that can be placed in the knapsack
-without exceeding <code>cap</code>. Use one-dimensional dynamic programming.</p>
+<h3 id="2-0-1-knapsack">2. &nbsp; 0-1 knapsack<a class="headerlink" href="#2-0-1-knapsack" title="Permanent link">&para;</a></h3>
+<p>You are given arrays <code>wgt</code> and <code>val</code> of equal length. Item <code>i</code> has a positive integer weight <code>wgt[i]</code> and a nonnegative integer value <code>val[i]</code>.
+The knapsack capacity <code>cap</code> is a nonnegative integer. Each item can be taken at most once. Find the maximum total value
+that fits in the knapsack without exceeding <code>cap</code> in total weight. Use one-dimensional dynamic programming.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>Initialize an array dp of length cap + 1, where dp[c] is the maximum value for a capacity limit of c</li>
-<li>When processing item i, compare dp[c], which does not select it, with dp[c-wgt[i]] + val[i], which does</li>
-<li>Update capacities from largest to smallest to avoid selecting the current item repeatedly in the same round</li>
+<li>Initialize an array dp of length cap + 1, where dp[c] is the maximum value obtainable with a capacity limit of c.</li>
+<li>When processing item i, compare dp[c] for leaving it out with dp[c-wgt[i]] + val[i] for taking it.</li>
+<li>Update capacities from largest to smallest to avoid taking the current item more than once in the same round.</li>
 </ol>
 </details>
 
@@ -4935,7 +5007,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 14.7 Summary"
+    aria-label="Previous: 14.7   Summary"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4947,7 +5019,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        14.7 Summary
+        14.7 &nbsp; Summary
       </div>
     </div>
   </a>
@@ -4959,7 +5031,7 @@ aria-label="Footer"
   <a
     href="../../chapter_greedy/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: Chapter 15. &amp;nbsp; Greedy"
+    aria-label="Next: Chapter 15.   Greedy algorithms"
     rel="next"
   >
     <div class="md-footer__title">
@@ -4967,7 +5039,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        Chapter 15. &nbsp; Greedy
+        Chapter 15. &nbsp; Greedy algorithms
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -4984,7 +5056,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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diff --git a/en/chapter_dynamic_programming/index.html b/en/chapter_dynamic_programming/index.html
index 99b87efc8..ac19adc1e 100644
--- a/en/chapter_dynamic_programming/index.html
+++ b/en/chapter_dynamic_programming/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>Chapter 14. Dynamic Programming - Hello Algo</title>
+        <title>Chapter 14. Dynamic programming - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
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-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
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 </div>
 
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@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              Chapter 14. &nbsp; Dynamic Programming
+              Chapter 14. &nbsp; Dynamic programming
 
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@@ -167,9 +167,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
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   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
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@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
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@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3283,7 +3283,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3304,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3352,7 +3352,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3379,7 +3379,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3406,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3433,7 +3433,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3460,7 +3460,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3526,7 +3526,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3547,7 +3547,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3568,7 +3568,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3595,7 +3595,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3622,7 +3622,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3649,7 +3649,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3676,7 +3676,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3703,7 +3703,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3775,7 +3775,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3796,7 +3796,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3817,7 +3817,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3844,7 +3844,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3871,7 +3871,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  16.2 Contributing Together
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 <!-- Page content -->
-<h1 id="chapter-14-dynamic-programming">Chapter 14. &nbsp; Dynamic Programming<a class="headerlink" href="#chapter-14-dynamic-programming" title="Permanent link">&para;</a></h1>
+<h1 id="chapter-14-dynamic-programming">Chapter 14. &nbsp; Dynamic programming<a class="headerlink" href="#chapter-14-dynamic-programming" title="Permanent link">&para;</a></h1>
 <p><img alt="Dynamic programming" class="cover-image" src="../assets/covers/chapter_dynamic_programming.jpg" /></p>
 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
-<p>Streams flow into rivers, rivers flow into the sea.</p>
-<p>Dynamic programming combines solutions to small problems into the answer to a large problem, leading us step by step to the other shore of problem-solving.</p>
+<p>Streams flow into rivers, and rivers flow into the sea.</p>
+<p>Dynamic programming brings together solutions to small problems to answer a larger one, guiding us step by step to the other shore.</p>
 </div>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
-<li><a href="intro_to_dynamic_programming/">14.1 &nbsp; Introduction to Dynamic Programming</a></li>
-<li><a href="dp_problem_features/">14.2 &nbsp; Characteristics of Dynamic Programming Problems</a></li>
-<li><a href="dp_solution_pipeline/">14.3 &nbsp; Dynamic Programming Problem-Solving Approach</a></li>
-<li><a href="knapsack_problem/">14.4 &nbsp; 0-1 Knapsack Problem</a></li>
-<li><a href="unbounded_knapsack_problem/">14.5 &nbsp; Unbounded Knapsack Problem</a></li>
-<li><a href="edit_distance_problem/">14.6 &nbsp; Edit Distance Problem</a></li>
+<li><a href="intro_to_dynamic_programming/">14.1 &nbsp; Introduction to dynamic programming</a></li>
+<li><a href="dp_problem_features/">14.2 &nbsp; Properties of dynamic programming problems</a></li>
+<li><a href="dp_solution_pipeline/">14.3 &nbsp; Solving dynamic programming problems</a></li>
+<li><a href="knapsack_problem/">14.4 &nbsp; 0-1 knapsack problem</a></li>
+<li><a href="unbounded_knapsack_problem/">14.5 &nbsp; Unbounded knapsack problem</a></li>
+<li><a href="edit_distance_problem/">14.6 &nbsp; Edit distance problem</a></li>
 <li><a href="summary/">14.7 &nbsp; Summary</a></li>
 <li><a href="exercises/">14.8 &nbsp; Exercises</a></li>
 </ul>
@@ -4617,7 +4689,7 @@ aria-label="Footer"
   <a
     href="../chapter_backtracking/exercises/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 13.6 Exercises"
+    aria-label="Previous: 13.6   Exercises"
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@@ -4629,7 +4701,7 @@ aria-label="Footer"
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       </span>
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-        13.6 Exercises
+        13.6 &nbsp; Exercises
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   </a>
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   <a
     href="./intro_to_dynamic_programming/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 14.1 Introduction to Dynamic Programming"
+    aria-label="Next: 14.1   Introduction to dynamic programming"
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@@ -4649,7 +4721,7 @@ aria-label="Footer"
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-        14.1 Introduction to Dynamic Programming
+        14.1 &nbsp; Introduction to dynamic programming
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
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+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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+  Chapter 10. &nbsp; Searching
 
 
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+  10.1 &nbsp; Binary search
 
 
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+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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@@ -3136,7 +3136,7 @@
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
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-  12.2 Divide and Conquer Search Strategy
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+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
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+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-        14.1.1 &nbsp; Method 1: Brute Force Search
+        14.1.1 &nbsp; Method 1: Brute-force search
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 </li>
 
         <li class="md-nav__item">
-  <a href="#1412-method-2-memoization" class="md-nav__link">
+  <a href="#1412-method-2-memoized-search" class="md-nav__link">
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       <span class="md-typeset">
-        14.1.2 &nbsp; Method 2: Memoization
+        14.1.2 &nbsp; Method 2: Memoized search
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-        14.1.3 &nbsp; Method 3: Dynamic Programming
+        14.1.3 &nbsp; Method 3: Dynamic programming
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-  14.2 Characteristics of Dynamic Programming Problems
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-  14.3 Dynamic Programming Problem-Solving Approach
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+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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   <a href="#1411-method-1-brute-force-search" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        14.1.1 &nbsp; Method 1: Brute Force Search
+        14.1.1 &nbsp; Method 1: Brute-force search
       </span>
     </span>
   </a>
@@ -4632,10 +4704,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1412-method-2-memoization" class="md-nav__link">
+  <a href="#1412-method-2-memoized-search" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        14.1.2 &nbsp; Method 2: Memoization
+        14.1.2 &nbsp; Method 2: Memoized search
       </span>
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@@ -4646,7 +4718,7 @@
   <a href="#1413-method-3-dynamic-programming" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        14.1.3 &nbsp; Method 3: Dynamic Programming
+        14.1.3 &nbsp; Method 3: Dynamic programming
       </span>
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@@ -4657,7 +4729,7 @@
   <a href="#1414-space-optimization" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        14.1.4 &nbsp; Space Optimization
+        14.1.4 &nbsp; Space optimization
       </span>
     </span>
   </a>
@@ -4701,40 +4773,40 @@
 
 
 <!-- Page content -->
-<h1 id="141-introduction-to-dynamic-programming">14.1 &nbsp; Introduction to Dynamic Programming<a class="headerlink" href="#141-introduction-to-dynamic-programming" title="Permanent link">&para;</a></h1>
-<p><u>Dynamic programming</u> is an important algorithmic paradigm that decomposes a problem into a series of smaller subproblems and avoids redundant computation by storing the solutions to subproblems, thereby significantly improving time efficiency.</p>
-<p>In this section, we start with a classic example, first presenting its brute force backtracking solution, observing the overlapping subproblems within it, and then gradually deriving a more efficient dynamic programming solution.</p>
+<h1 id="141-introduction-to-dynamic-programming">14.1 &nbsp; Introduction to dynamic programming<a class="headerlink" href="#141-introduction-to-dynamic-programming" title="Permanent link">&para;</a></h1>
+<p><u>Dynamic programming</u> is an important algorithmic paradigm. It breaks a problem into smaller subproblems and stores their solutions to avoid repeated computation, greatly improving time efficiency.</p>
+<p>We will start with a classic example, give a brute-force backtracking solution, identify its overlapping subproblems, and gradually derive a more efficient dynamic programming solution.</p>
 <div class="admonition question">
 <p class="admonition-title">Climbing stairs</p>
-<p>Given a staircase with <span class="arithmatex">\(n\)</span> steps, where you can climb <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(2\)</span> steps at a time, how many different ways are there to reach the top?</p>
+<p>Given a staircase with <span class="arithmatex">\(n\)</span> steps, you may climb <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(2\)</span> steps at a time. How many ways are there to reach the top?</p>
 </div>
-<p>As shown in Figure 14-1, for a <span class="arithmatex">\(3\)</span>-step staircase, there are <span class="arithmatex">\(3\)</span> different ways to reach the top.</p>
-<p><img alt="Number of ways to reach the 3rd step" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_example.png" /></p>
-<p align="center"> Figure 14-1 &nbsp; Number of ways to reach the 3rd step </p>
+<p>As shown in Figure 14-1, a staircase with <span class="arithmatex">\(3\)</span> steps has <span class="arithmatex">\(3\)</span> possible ways to reach the top.</p>
+<p><img alt="Number of ways to reach step 3" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_example.png" /></p>
+<p align="center"> Figure 14-1 &nbsp; Number of ways to reach step 3 </p>
 
-<p>The goal of this problem is to determine the number of ways, so <strong>we can consider using backtracking to enumerate all possibilities</strong>. Specifically, imagine climbing stairs as a multi-round selection process: starting from the ground, choosing to go up <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(2\)</span> steps in each round, incrementing the count by <span class="arithmatex">\(1\)</span> whenever the top of the stairs is reached, and pruning when exceeding the top. The code is as follows:</p>
+<p>The goal is to count solutions, so <strong>we can use backtracking to enumerate every possibility</strong>. View climbing as a sequence of choices: starting from the ground, choose to climb <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(2\)</span> steps each time. Add <span class="arithmatex">\(1\)</span> to the count whenever we reach the top, and prune any branch that goes beyond it. The code is as follows:</p>
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 <div class="highlight"><span class="filename">climbing_stairs_backtrack.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">state</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">])</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Backtracking"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># When climbing to the n-th stair, add 1 to the solution count</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># When step n is reached, increase the number of ways by 1</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="k">if</span> <span class="n">state</span> <span class="o">==</span> <span class="n">n</span><span class="p">:</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Traverse all choices</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Iterate over all choices</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="k">for</span> <span class="n">choice</span> <span class="ow">in</span> <span class="n">choices</span><span class="p">:</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="c1"># Pruning: not allowed to go beyond the n-th stair</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="c1"># Prune: do not go beyond step n</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="k">if</span> <span class="n">state</span> <span class="o">+</span> <span class="n">choice</span> <span class="o">&gt;</span> <span class="n">n</span><span class="p">:</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>            <span class="k">continue</span>
-<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="c1"># Attempt: make a choice, update state</span>
+<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="c1"># Try: make a choice and update the state</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span> <span class="n">state</span> <span class="o">+</span> <span class="n">choice</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="c1"># Backtrack</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_backtrack</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="w">    </span><span class="sd">"""Climbing stairs: Backtracking"""</span>
-<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>    <span class="n">choices</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>  <span class="c1"># Can choose to climb up 1 or 2 stairs</span>
-<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>    <span class="n">state</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Start climbing from the 0-th stair</span>
-<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>    <span class="n">res</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span>  <span class="c1"># Use res[0] to record the solution count</span>
+<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="w">    </span><span class="sd">"""Climbing stairs: backtracking"""</span>
+<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>    <span class="n">choices</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>  <span class="c1"># Choose to climb 1 or 2 steps</span>
+<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>    <span class="n">state</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Start climbing from step 0</span>
+<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>    <span class="n">res</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span>  <span class="c1"># Use res[0] to record the number of ways</span>
 <a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>    <span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span> <span class="n">state</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a>    <span class="k">return</span> <span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
 </code></pre></div>
@@ -4742,25 +4814,25 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_backtrack.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Backtracking */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// When climbing to the n-th stair, add 1 to the solution count</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// When step n is reached, increase the number of ways by 1</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">++</span><span class="p">;</span>
-<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="k">auto</span><span class="w"> </span><span class="o">&amp;</span><span class="n">choice</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="c1">// Pruning: not allowed to go beyond the n-th stair</span>
+<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="c1">// Prune: do not go beyond step n</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
-<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="c1">// Backtrack</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="p">}</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a>
-<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="cm">/* Climbing stairs: Backtracking */</span>
+<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="cm">/* Climbing stairs: backtracking */</span>
 <a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsBacktrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">};</span><span class="w"> </span><span class="c1">// Can choose to climb up 1 or 2 stairs</span>
-<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">                </span><span class="c1">// Start climbing from the 0-th stair</span>
-<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="mi">0</span><span class="p">};</span><span class="w">        </span><span class="c1">// Use res[0] to record the solution count</span>
+<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">};</span><span class="w"> </span><span class="c1">// Choose to climb 1 or 2 steps</span>
+<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">                </span><span class="c1">// Start climbing from step 0</span>
+<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="mi">0</span><span class="p">};</span><span class="w">        </span><span class="c1">// Use res[0] to record the number of ways</span>
 <a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="p">}</span>
@@ -4769,26 +4841,26 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_backtrack.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Backtracking */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// When climbing to the n-th stair, add 1 to the solution count</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// When step n is reached, increase the number of ways by 1</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">set</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Integer</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">        </span><span class="c1">// Pruning: not allowed to go beyond the n-th stair</span>
+<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">        </span><span class="c1">// Prune: do not go beyond step n</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
-<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="c1">// Backtrack</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="p">}</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a>
-<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="cm">/* Climbing stairs: Backtracking */</span>
+<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="cm">/* Climbing stairs: backtracking */</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsBacktrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Arrays</span><span class="p">.</span><span class="na">asList</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Can choose to climb up 1 or 2 stairs</span>
-<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start climbing from the 0-th stair</span>
+<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Arrays</span><span class="p">.</span><span class="na">asList</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Choose to climb 1 or 2 steps</span>
+<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start climbing from step 0</span>
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
-<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// Use res[0] to record the solution count</span>
+<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// Use res[0] to record the number of ways</span>
 <a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="p">}</span>
@@ -4797,25 +4869,25 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_backtrack.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Backtracking */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Backtrack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// When climbing to the n-th stair, add 1 to the solution count</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// When step n is reached, increase the number of ways by 1</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">++</span><span class="p">;</span>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">        </span><span class="c1">// Pruning: not allowed to go beyond the n-th stair</span>
+<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">        </span><span class="c1">// Prune: do not go beyond step n</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
-<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="n">Backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="c1">// Backtrack</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="p">}</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a>
-<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="cm">/* Climbing stairs: Backtracking */</span>
+<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="cm">/* Climbing stairs: backtracking */</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="kt">int</span><span class="w"> </span><span class="nf">ClimbingStairsBacktrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">];</span><span class="w"> </span><span class="c1">// Can choose to climb up 1 or 2 stairs</span>
-<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start climbing from the 0-th stair</span>
-<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// Use res[0] to record the solution count</span>
+<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">];</span><span class="w"> </span><span class="c1">// Choose to climb 1 or 2 steps</span>
+<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start climbing from step 0</span>
+<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">0</span><span class="p">];</span><span class="w"> </span><span class="c1">// Use res[0] to record the number of ways</span>
 <a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w">    </span><span class="n">Backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="p">}</span>
@@ -4824,30 +4896,30 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_backtrack.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Backtracking */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">choices</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// When climbing to the n-th stair, add 1 to the solution count</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// When step n is reached, increase the number of ways by 1</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">        </span><span class="nx">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">choices</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">        </span><span class="c1">// Pruning: not allowed to go beyond the n-th stair</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">        </span><span class="c1">// Prune: do not go beyond step n</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">state</span><span class="o">+</span><span class="nx">choice</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">            </span><span class="k">continue</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="o">+</span><span class="nx">choice</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="c1">// Backtrack</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="p">}</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a>
-<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="cm">/* Climbing stairs: Backtracking */</span>
+<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="cm">/* Climbing stairs: backtracking */</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="kd">func</span><span class="w"> </span><span class="nx">climbingStairsBacktrack</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">    </span><span class="c1">// Can choose to climb up 1 or 2 stairs</span>
+<a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">    </span><span class="c1">// Choose to climb 1 or 2 steps</span>
 <a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">    </span><span class="nx">choices</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">}</span>
-<a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w">    </span><span class="c1">// Start climbing from the 0-th stair</span>
+<a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w">    </span><span class="c1">// Start climbing from step 0</span>
 <a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="w">    </span><span class="nx">state</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-4-25" name="__codelineno-4-25" href="#__codelineno-4-25"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-4-26" name="__codelineno-4-26" href="#__codelineno-4-26"></a><span class="w">    </span><span class="c1">// Use res[0] to record the solution count</span>
+<a id="__codelineno-4-26" name="__codelineno-4-26" href="#__codelineno-4-26"></a><span class="w">    </span><span class="c1">// Use res[0] to record the number of ways</span>
 <a id="__codelineno-4-27" name="__codelineno-4-27" href="#__codelineno-4-27"></a><span class="w">    </span><span class="nx">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-4-28" name="__codelineno-4-28" href="#__codelineno-4-28"></a><span class="w">    </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span>
 <a id="__codelineno-4-29" name="__codelineno-4-29" href="#__codelineno-4-29"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
@@ -4857,28 +4929,28 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_backtrack.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Backtracking */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// When climbing to the n-th stair, add 1 to the solution count</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// When step n is reached, increase the number of ways by 1</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="c1">// Pruning: not allowed to go beyond the n-th stair</span>
+<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="c1">// Prune: do not go beyond step n</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">            </span><span class="k">continue</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="c1">// Backtrack</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="p">}</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a>
-<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="cm">/* Climbing stairs: Backtracking */</span>
+<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="cm">/* Climbing stairs: backtracking */</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="kd">func</span><span class="w"> </span><span class="nf">climbingStairsBacktrack</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">choices</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="c1">// Can choose to climb up 1 or 2 stairs</span>
-<a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">state</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1">// Start climbing from the 0-th stair</span>
+<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">choices</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="c1">// Choose to climb 1 or 2 steps</span>
+<a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">state</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1">// Start climbing from step 0</span>
 <a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
-<a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// Use res[0] to record the solution count</span>
+<a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// Use res[0] to record the number of ways</span>
 <a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-5-26" name="__codelineno-5-26" href="#__codelineno-5-26"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-5-27" name="__codelineno-5-27" href="#__codelineno-5-27"></a><span class="p">}</span>
@@ -4887,24 +4959,24 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_backtrack.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Backtracking */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// When climbing to the n-th stair, add 1 to the solution count</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// When step n is reached, increase the number of ways by 1</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">        </span><span class="c1">// Pruning: not allowed to go beyond the n-th stair</span>
+<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">        </span><span class="c1">// Prune: do not go beyond step n</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="k">continue</span><span class="p">;</span>
-<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">choice</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="c1">// Backtrack</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="p">}</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a>
-<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="cm">/* Climbing stairs: Backtracking */</span>
+<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="cm">/* Climbing stairs: backtracking */</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="kd">function</span><span class="w"> </span><span class="nx">climbingStairsBacktrack</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="mf">2</span><span class="p">];</span><span class="w"> </span><span class="c1">// Can choose to climb up 1 or 2 stairs</span>
-<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start climbing from the 0-th stair</span>
+<a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="mf">2</span><span class="p">];</span><span class="w"> </span><span class="c1">// Choose to climb 1 or 2 steps</span>
+<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start climbing from step 0</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Map</span><span class="p">();</span>
-<a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">    </span><span class="nx">res</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="mf">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// Use res[0] to record the solution count</span>
+<a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">    </span><span class="nx">res</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="mf">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// Store the number of ways at key 0 in res</span>
 <a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">    </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="p">}</span>
@@ -4918,24 +4990,24 @@
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">Map</span><span class="o">&lt;</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="nx">any</span><span class="o">&gt;</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// When climbing to the n-th stair, add 1 to the solution count</span>
+<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// When step n is reached, increase the number of ways by 1</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">        </span><span class="c1">// Pruning: not allowed to go beyond the n-th stair</span>
+<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">        </span><span class="c1">// Prune: do not go beyond step n</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">choice</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="k">continue</span><span class="p">;</span>
-<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">        </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">choice</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">        </span><span class="c1">// Backtrack</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="p">}</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a>
-<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="cm">/* Climbing stairs: Backtracking */</span>
+<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="cm">/* Climbing stairs: backtracking */</span>
 <a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="kd">function</span><span class="w"> </span><span class="nx">climbingStairsBacktrack</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="mf">2</span><span class="p">];</span><span class="w"> </span><span class="c1">// Can choose to climb up 1 or 2 stairs</span>
-<a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start climbing from the 0-th stair</span>
+<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="mf">2</span><span class="p">];</span><span class="w"> </span><span class="c1">// Choose to climb 1 or 2 steps</span>
+<a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start climbing from step 0</span>
 <a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Map</span><span class="p">();</span>
-<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">    </span><span class="nx">res</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="mf">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// Use res[0] to record the solution count</span>
+<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">    </span><span class="nx">res</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="mf">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// Store the number of ways at key 0 in res</span>
 <a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="w">    </span><span class="nx">backtrack</span><span class="p">(</span><span class="nx">choices</span><span class="p">,</span><span class="w"> </span><span class="nx">state</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">);</span>
 <a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="p">}</span>
@@ -4944,26 +5016,26 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_backtrack.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Backtracking */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">backtrack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// When climbing to the n-th stair, add 1 to the solution count</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// When step n is reached, increase the number of ways by 1</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">    </span><span class="n">res</span><span class="p">[</span><span class="m">0</span><span class="p">]</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="c1">// Pruning: not allowed to go beyond the n-th stair</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="c1">// Prune: do not go beyond step n</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="k">continue</span><span class="p">;</span>
-<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">    </span><span class="c1">// Backtrack</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="p">}</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a>
-<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="cm">/* Climbing stairs: Backtracking */</span>
+<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="cm">/* Climbing stairs: backtracking */</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="kt">int</span><span class="w"> </span><span class="n">climbingStairsBacktrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">];</span><span class="w"> </span><span class="c1">// Can choose to climb up 1 or 2 stairs</span>
-<a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start climbing from the 0-th stair</span>
+<a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">];</span><span class="w"> </span><span class="c1">// Choose to climb 1 or 2 steps</span>
+<a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start climbing from step 0</span>
 <a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="w">  </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="m">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// Use res[0] to record the solution count</span>
+<a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="w">  </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="m">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// Use res[0] to record the number of ways</span>
 <a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="w">  </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">[</span><span class="m">0</span><span class="p">];</span>
 <a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="p">}</span>
@@ -4972,28 +5044,28 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_backtrack.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Backtracking */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">state</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// When climbing to the n-th stair, add 1 to the solution count</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// When step n is reached, increase the number of ways by 1</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="o">&amp;</span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">        </span><span class="c1">// Pruning: not allowed to go beyond the n-th stair</span>
+<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">        </span><span class="c1">// Prune: do not go beyond step n</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">        </span><span class="c1">// Backtrack</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="p">}</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a>
-<a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="cm">/* Climbing stairs: Backtracking */</span>
+<a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="cm">/* Climbing stairs: backtracking */</span>
 <a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="k">fn</span><span class="w"> </span><span class="nf">climbing_stairs_backtrack</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">];</span><span class="w"> </span><span class="c1">// Can choose to climb up 1 or 2 stairs</span>
-<a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start climbing from the 0-th stair</span>
+<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">];</span><span class="w"> </span><span class="c1">// Choose to climb 1 or 2 steps</span>
+<a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Start climbing from step 0</span>
 <a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
-<a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// Use res[0] to record the solution count</span>
+<a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w"> </span><span class="c1">// Use res[0] to record the number of ways</span>
 <a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="o">&amp;</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-9-26" name="__codelineno-9-26" href="#__codelineno-9-26"></a><span class="w">    </span><span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="p">}</span>
@@ -5002,27 +5074,27 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_backtrack.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Backtracking */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">len</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// When climbing to the n-th stair, add 1 to the solution count</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// When step n is reached, increase the number of ways by 1</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">++</span><span class="p">;</span>
-<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">len</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
-<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">        </span><span class="c1">// Pruning: not allowed to go beyond the n-th stair</span>
+<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">        </span><span class="c1">// Prune: do not go beyond step n</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span>
-<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">len</span><span class="p">);</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">        </span><span class="c1">// Backtrack</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="p">}</span>
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a>
-<a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="cm">/* Climbing stairs: Backtracking */</span>
+<a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="cm">/* Climbing stairs: backtracking */</span>
 <a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsBacktrack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">};</span><span class="w"> </span><span class="c1">// Can choose to climb up 1 or 2 stairs</span>
-<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">           </span><span class="c1">// Start climbing from the 0-th stair</span>
+<a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">choices</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">};</span><span class="w"> </span><span class="c1">// Choose to climb 1 or 2 steps</span>
+<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">           </span><span class="c1">// Start climbing from step 0</span>
 <a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
-<a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">    </span><span class="o">*</span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Use res[0] to record the solution count</span>
+<a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">    </span><span class="o">*</span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Use res[0] to record the number of ways</span>
 <a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">len</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">);</span>
 <a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">len</span><span class="p">);</span>
 <a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">result</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">*</span><span class="n">res</span><span class="p">;</span>
@@ -5039,25 +5111,25 @@
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="c1">// When climbing to the n-th stair, add 1 to the solution count</span>
+<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="c1">// When step n is reached, increase the number of ways by 1</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">        </span><span class="n">res</span><span class="o">[</span><span class="m">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">res</span><span class="o">[</span><span class="m">0</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="c1">// Traverse all choices</span>
+<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="c1">// Iterate over all choices</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="c1">// Pruning: not allowed to go beyond the n-th stair</span>
+<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="c1">// Prune: do not go beyond step n</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="k">continue</span>
-<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">        </span><span class="c1">// Attempt: make choice, update state</span>
+<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">        </span><span class="c1">// Try: make a choice and update the state</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">        </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">        </span><span class="c1">// Backtrack</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="p">}</span>
 <a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a>
-<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="cm">/* Climbing stairs: Backtracking */</span>
+<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="cm">/* Climbing stairs: backtracking */</span>
 <a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">climbingStairsBacktrack</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="p">(</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="c1">// Can choose to climb up 1 or 2 stairs</span>
-<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Start climbing from the 0-th stair</span>
+<a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="p">(</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="c1">// Choose to climb 1 or 2 steps</span>
+<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Start climbing from step 0</span>
 <a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span>
-<a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// Use res[0] to record the solution count</span>
+<a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// Use res[0] to record the number of ways</span>
 <a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="o">[</span><span class="m">0</span><span class="o">]</span>
 <a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="p">}</span>
@@ -5066,14 +5138,14 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_backtrack.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Backtracking ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># When climbing to the n-th stair, add 1 to the solution count</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># When step n is reached, increase the number of ways by 1</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="n">res</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">n</span>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Traverse all choices</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Iterate over all choices</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">choices</span>
-<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">    </span><span class="c1"># Pruning: not allowed to go beyond the n-th stair</span>
+<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">    </span><span class="c1"># Prune: do not go beyond step n</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="k">next</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">n</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a>
-<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="c1"># Attempt: make choice, update state</span>
+<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="c1"># Try: make a choice and update the state</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">choice</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">  </span><span class="c1"># Backtrack</span>
@@ -5081,9 +5153,9 @@
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a>
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="c1">### Climbing stairs: backtracking ###</span>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_backtrack</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">  </span><span class="n">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="c1"># Can choose to climb up 1 or 2 stairs</span>
-<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">  </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Start climbing from the 0-th stair</span>
-<a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="c1"># Use res[0] to record the solution count</span>
+<a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">  </span><span class="n">choices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="c1"># Choose to climb 1 or 2 steps</span>
+<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">  </span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Start climbing from step 0</span>
+<a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="c1"># Use res[0] to record the number of ways</span>
 <a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">  </span><span class="n">backtrack</span><span class="p">(</span><span class="n">choices</span><span class="p">,</span><span class="w"> </span><span class="n">state</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w">  </span><span class="n">res</span><span class="o">.</span><span class="n">first</span>
 <a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="k">end</span>
@@ -5091,29 +5163,34 @@
 </div>
 </div>
 </div>
-<h2 id="1411-method-1-brute-force-search">14.1.1 &nbsp; Method 1: Brute Force Search<a class="headerlink" href="#1411-method-1-brute-force-search" title="Permanent link">&para;</a></h2>
-<p>Backtracking algorithms typically do not explicitly decompose problems, but rather treat solving the problem as a series of decision steps, searching for all possible solutions through trial and pruning.</p>
-<p>We can try to analyze this problem from the perspective of problem decomposition. Let the number of ways to climb to the <span class="arithmatex">\(i\)</span>-th step be <span class="arithmatex">\(dp[i]\)</span>, then <span class="arithmatex">\(dp[i]\)</span> is the original problem, and its subproblems include:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20backtrack%28choices%3A%20list%5Bint%5D,%20state%3A%20int,%20n%3A%20int,%20res%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Backtracking%22%22%22%0A%20%20%20%20%23%20When%20step%20n%20is%20reached,%20increase%20the%20number%20of%20ways%20by%201%0A%20%20%20%20if%20state%20%3D%3D%20n%3A%0A%20%20%20%20%20%20%20%20res%5B0%5D%20%2B%3D%201%0A%20%20%20%20%23%20Iterate%20over%20all%20choices%0A%20%20%20%20for%20choice%20in%20choices%3A%0A%20%20%20%20%20%20%20%20%23%20Prune%3A%20do%20not%20go%20beyond%20step%20n%0A%20%20%20%20%20%20%20%20if%20state%20%2B%20choice%20%3E%20n%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20continue%0A%20%20%20%20%20%20%20%20%23%20Try%3A%20make%20a%20choice%20and%20update%20the%20state%0A%20%20%20%20%20%20%20%20backtrack%28choices,%20state%20%2B%20choice,%20n,%20res%29%0A%20%20%20%20%20%20%20%20%23%20Backtrack%0A%0A%0Adef%20climbing_stairs_backtrack%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Climbing%20stairs%3A%20backtracking%22%22%22%0A%20%20%20%20choices%20%3D%20%5B1,%202%5D%20%20%23%20Choose%20to%20climb%201%20or%202%20steps%0A%20%20%20%20state%20%3D%200%20%20%23%20Start%20climbing%20from%20step%200%0A%20%20%20%20res%20%3D%20%5B0%5D%20%20%23%20Use%20res%5B0%5D%20to%20record%20the%20number%20of%20ways%0A%20%20%20%20backtrack%28choices,%20state,%20n,%20res%29%0A%20%20%20%20return%20res%5B0%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%204%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_backtrack%28n%29%0A%20%20%20%20print%28f%22Number%20of%20ways%20to%20climb%20%7Bn%7D%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20backtrack%28choices%3A%20list%5Bint%5D,%20state%3A%20int,%20n%3A%20int,%20res%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Backtracking%22%22%22%0A%20%20%20%20%23%20When%20step%20n%20is%20reached,%20increase%20the%20number%20of%20ways%20by%201%0A%20%20%20%20if%20state%20%3D%3D%20n%3A%0A%20%20%20%20%20%20%20%20res%5B0%5D%20%2B%3D%201%0A%20%20%20%20%23%20Iterate%20over%20all%20choices%0A%20%20%20%20for%20choice%20in%20choices%3A%0A%20%20%20%20%20%20%20%20%23%20Prune%3A%20do%20not%20go%20beyond%20step%20n%0A%20%20%20%20%20%20%20%20if%20state%20%2B%20choice%20%3E%20n%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20continue%0A%20%20%20%20%20%20%20%20%23%20Try%3A%20make%20a%20choice%20and%20update%20the%20state%0A%20%20%20%20%20%20%20%20backtrack%28choices,%20state%20%2B%20choice,%20n,%20res%29%0A%20%20%20%20%20%20%20%20%23%20Backtrack%0A%0A%0Adef%20climbing_stairs_backtrack%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Climbing%20stairs%3A%20backtracking%22%22%22%0A%20%20%20%20choices%20%3D%20%5B1,%202%5D%20%20%23%20Choose%20to%20climb%201%20or%202%20steps%0A%20%20%20%20state%20%3D%200%20%20%23%20Start%20climbing%20from%20step%200%0A%20%20%20%20res%20%3D%20%5B0%5D%20%20%23%20Use%20res%5B0%5D%20to%20record%20the%20number%20of%20ways%0A%20%20%20%20backtrack%28choices,%20state,%20n,%20res%29%0A%20%20%20%20return%20res%5B0%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%204%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_backtrack%28n%29%0A%20%20%20%20print%28f%22Number%20of%20ways%20to%20climb%20%7Bn%7D%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1411-method-1-brute-force-search">14.1.1 &nbsp; Method 1: Brute-force search<a class="headerlink" href="#1411-method-1-brute-force-search" title="Permanent link">&para;</a></h2>
+<p>Backtracking usually does not explicitly decompose the problem. Instead, it treats a solution as a sequence of decisions and searches all possible solutions through trial and pruning.</p>
+<p>Let's analyze this problem through decomposition. Let <span class="arithmatex">\(dp[i]\)</span> be the number of ways to reach step <span class="arithmatex">\(i\)</span>. The original problem is <span class="arithmatex">\(dp[i]\)</span>, and its subproblems include:</p>
 <div class="arithmatex">\[
 dp[i-1], dp[i-2], \dots, dp[2], dp[1]
 \]</div>
-<p>Since we can only go up <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(2\)</span> steps in each round, when we stand on the <span class="arithmatex">\(i\)</span>-th step, we could only have been on the <span class="arithmatex">\(i-1\)</span>-th or <span class="arithmatex">\(i-2\)</span>-th step in the previous round. In other words, we can only reach the <span class="arithmatex">\(i\)</span>-th step from the <span class="arithmatex">\(i-1\)</span>-th or <span class="arithmatex">\(i-2\)</span>-th step.</p>
-<p>This leads to an important conclusion: <strong>the number of ways to climb to the <span class="arithmatex">\(i-1\)</span>-th step plus the number of ways to climb to the <span class="arithmatex">\(i-2\)</span>-th step equals the number of ways to climb to the <span class="arithmatex">\(i\)</span>-th step</strong>. The formula is as follows:</p>
+<p>Since each move climbs <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(2\)</span> steps, when we stand on step <span class="arithmatex">\(i\)</span>, we must previously have been on step <span class="arithmatex">\(i - 1\)</span> or <span class="arithmatex">\(i - 2\)</span>. In other words, step <span class="arithmatex">\(i\)</span> can only be reached from step <span class="arithmatex">\(i -1\)</span> or <span class="arithmatex">\(i - 2\)</span>.</p>
+<p>This gives an important result: <strong>the number of ways to reach step <span class="arithmatex">\(i - 1\)</span> plus the number of ways to reach step <span class="arithmatex">\(i - 2\)</span> equals the number of ways to reach step <span class="arithmatex">\(i\)</span></strong>. In symbols:</p>
 <div class="arithmatex">\[
 dp[i] = dp[i-1] + dp[i-2]
 \]</div>
-<p>This means that in the stair climbing problem, there exists a recurrence relation among the subproblems, and <strong>the solution to the original problem can be constructed from the solutions to the subproblems</strong>. Figure 14-2 illustrates this recurrence relation.</p>
-<p><img alt="Recurrence relation for the number of ways" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_state_transfer.png" /></p>
-<p align="center"> Figure 14-2 &nbsp; Recurrence relation for the number of ways </p>
+<p>Thus, the subproblems are linked by a recurrence: <strong>the solution to the original problem can be constructed from subproblem solutions</strong>. Figure 14-2 illustrates this relationship.</p>
+<p><img alt="Recurrence for the number of ways" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_state_transfer.png" /></p>
+<p align="center"> Figure 14-2 &nbsp; Recurrence for the number of ways </p>
 
-<p>We can obtain a brute force search solution based on the recurrence formula. Starting from <span class="arithmatex">\(dp[n]\)</span>, <strong>recursively decompose a larger problem into the sum of two smaller problems</strong>, until reaching the smallest subproblems <span class="arithmatex">\(dp[1]\)</span> and <span class="arithmatex">\(dp[2]\)</span> and returning. Among them, the solutions to the smallest subproblems are known, namely <span class="arithmatex">\(dp[1] = 1\)</span> and <span class="arithmatex">\(dp[2] = 2\)</span>, representing <span class="arithmatex">\(1\)</span> and <span class="arithmatex">\(2\)</span> ways to climb to the <span class="arithmatex">\(1\)</span>st and <span class="arithmatex">\(2\)</span>nd steps, respectively.</p>
-<p>Observe the following code: like standard backtracking code, it also uses depth-first search but is more concise:</p>
+<p>The recurrence gives a brute-force search solution. Starting at <span class="arithmatex">\(dp[n]\)</span>, <strong>recursively express a larger problem as the sum of two smaller ones</strong> until reaching <span class="arithmatex">\(dp[1]\)</span> or <span class="arithmatex">\(dp[2]\)</span>. Their solutions are known: <span class="arithmatex">\(dp[1] = 1\)</span> and <span class="arithmatex">\(dp[2] = 2\)</span>, meaning steps <span class="arithmatex">\(1\)</span> and <span class="arithmatex">\(2\)</span> can be reached in <span class="arithmatex">\(1\)</span> and <span class="arithmatex">\(2\)</span> ways, respectively.</p>
+<p>Like the standard backtracking code, the following code performs depth-first search, but is more concise:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dfs.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Search"""</span>
-<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># The values of dp[1] and dp[2] are known; return the corresponding value</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>        <span class="k">return</span> <span class="n">i</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="c1"># dp[i] = dp[i-1] + dp[i-2]</span>
@@ -5121,14 +5198,14 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>    <span class="k">return</span> <span class="n">count</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dfs</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a><span class="w">    </span><span class="sd">"""Climbing stairs: Search"""</span>
+<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a><span class="w">    </span><span class="sd">"""Climbing stairs: search"""</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>    <span class="k">return</span> <span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dfs.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Search */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
@@ -5136,7 +5213,7 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="p">}</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a>
-<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="cm">/* Climbing stairs: Search */</span>
+<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="cm">/* Climbing stairs: search */</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsDFS</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="p">}</span>
@@ -5145,7 +5222,7 @@ dp[i] = dp[i-1] + dp[i-2]
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dfs.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Search */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
@@ -5153,7 +5230,7 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="p">}</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a>
-<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="cm">/* Climbing stairs: Search */</span>
+<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="cm">/* Climbing stairs: search */</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsDFS</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="p">}</span>
@@ -5162,7 +5239,7 @@ dp[i] = dp[i-1] + dp[i-2]
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dfs.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Search */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">DFS</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
@@ -5170,7 +5247,7 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="p">}</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a>
-<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="cm">/* Climbing stairs: Search */</span>
+<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="cm">/* Climbing stairs: search */</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="kt">int</span><span class="w"> </span><span class="nf">ClimbingStairsDFS</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nf">DFS</span><span class="p">(</span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="p">}</span>
@@ -5179,7 +5256,7 @@ dp[i] = dp[i-1] + dp[i-2]
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dfs.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Search */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="p">}</span>
@@ -5188,7 +5265,7 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">count</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="p">}</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a>
-<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="cm">/* Climbing stairs: Search */</span>
+<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="cm">/* Climbing stairs: search */</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="kd">func</span><span class="w"> </span><span class="nx">climbingStairsDFS</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span>
 <a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="p">}</span>
@@ -5197,7 +5274,7 @@ dp[i] = dp[i-1] + dp[i-2]
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dfs.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Search */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="p">}</span>
@@ -5206,7 +5283,7 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="bp">count</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="p">}</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a>
-<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="cm">/* Climbing stairs: Search */</span>
+<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="cm">/* Climbing stairs: search */</span>
 <a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="kd">func</span><span class="w"> </span><span class="nf">climbingStairsDFS</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="p">}</span>
@@ -5215,14 +5292,14 @@ dp[i] = dp[i-1] + dp[i-2]
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dfs.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Search */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">count</span><span class="p">;</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="p">}</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a>
-<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="cm">/* Climbing stairs: Search */</span>
+<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="cm">/* Climbing stairs: search */</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="kd">function</span><span class="w"> </span><span class="nx">climbingStairsDFS</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">n</span><span class="p">);</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="p">}</span>
@@ -5231,14 +5308,14 @@ dp[i] = dp[i-1] + dp[i-2]
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dfs.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Search */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">count</span><span class="p">;</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="p">}</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a>
-<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="cm">/* Climbing stairs: Search */</span>
+<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="cm">/* Climbing stairs: search */</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="kd">function</span><span class="w"> </span><span class="nx">climbingStairsDFS</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">n</span><span class="p">);</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="p">}</span>
@@ -5247,14 +5324,14 @@ dp[i] = dp[i-1] + dp[i-2]
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dfs.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Search */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">2</span><span class="p">);</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="p">}</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a>
-<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="cm">/* Climbing stairs: Search */</span>
+<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="cm">/* Climbing stairs: search */</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="kt">int</span><span class="w"> </span><span class="n">climbingStairsDFS</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="p">}</span>
@@ -5263,7 +5340,7 @@ dp[i] = dp[i-1] + dp[i-2]
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dfs.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Search */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="p">}</span>
@@ -5272,7 +5349,7 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">    </span><span class="n">count</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="p">}</span>
 <a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a>
-<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="cm">/* Climbing stairs: Search */</span>
+<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="cm">/* Climbing stairs: search */</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="k">fn</span><span class="w"> </span><span class="nf">climbing_stairs_dfs</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="p">}</span>
@@ -5281,7 +5358,7 @@ dp[i] = dp[i-1] + dp[i-2]
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dfs.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Search */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
@@ -5289,7 +5366,7 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="p">}</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a>
-<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="cm">/* Climbing stairs: Search */</span>
+<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="cm">/* Climbing stairs: search */</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsDFS</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="p">}</span>
@@ -5298,14 +5375,14 @@ dp[i] = dp[i-1] + dp[i-2]
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dfs.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Search */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">2</span><span class="p">)</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">count</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="p">}</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a>
-<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="cm">/* Climbing stairs: Search */</span>
+<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="cm">/* Climbing stairs: search */</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">climbingStairsDFS</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="p">}</span>
@@ -5314,7 +5391,7 @@ dp[i] = dp[i-1] + dp[i-2]
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dfs.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Search ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># dp[i] = dp[i-1] + dp[i-2]</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
@@ -5328,28 +5405,33 @@ dp[i] = dp[i-1] + dp[i-2]
 </div>
 </div>
 </div>
-<p>Figure 14-3 shows the recursion tree formed by brute force search. For the problem <span class="arithmatex">\(dp[n]\)</span>, the depth of its recursion tree is <span class="arithmatex">\(n\)</span>, with a time complexity of <span class="arithmatex">\(O(2^n)\)</span>. Exponential growth is explosive; if we input a relatively large <span class="arithmatex">\(n\)</span>, the wait can be very long.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20dfs%28i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Search%22%22%22%0A%20%20%20%20%23%20The%20values%20of%20dp%5B1%5D%20and%20dp%5B2%5D%20are%20known%3B%20return%20the%20corresponding%20value%0A%20%20%20%20if%20i%20%3D%3D%201%20or%20i%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20i%0A%20%20%20%20%23%20dp%5Bi%5D%20%3D%20dp%5Bi-1%5D%20%2B%20dp%5Bi-2%5D%0A%20%20%20%20count%20%3D%20dfs%28i%20-%201%29%20%2B%20dfs%28i%20-%202%29%0A%20%20%20%20return%20count%0A%0A%0Adef%20climbing_stairs_dfs%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Climbing%20stairs%3A%20search%22%22%22%0A%20%20%20%20return%20dfs%28n%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dfs%28n%29%0A%20%20%20%20print%28f%22Number%20of%20ways%20to%20climb%20%7Bn%7D%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20dfs%28i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Search%22%22%22%0A%20%20%20%20%23%20The%20values%20of%20dp%5B1%5D%20and%20dp%5B2%5D%20are%20known%3B%20return%20the%20corresponding%20value%0A%20%20%20%20if%20i%20%3D%3D%201%20or%20i%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20i%0A%20%20%20%20%23%20dp%5Bi%5D%20%3D%20dp%5Bi-1%5D%20%2B%20dp%5Bi-2%5D%0A%20%20%20%20count%20%3D%20dfs%28i%20-%201%29%20%2B%20dfs%28i%20-%202%29%0A%20%20%20%20return%20count%0A%0A%0Adef%20climbing_stairs_dfs%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Climbing%20stairs%3A%20search%22%22%22%0A%20%20%20%20return%20dfs%28n%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dfs%28n%29%0A%20%20%20%20print%28f%22Number%20of%20ways%20to%20climb%20%7Bn%7D%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 14-3 shows the recursion tree for brute-force search. For <span class="arithmatex">\(dp[n]\)</span>, its depth is <span class="arithmatex">\(n\)</span>, and the time complexity is <span class="arithmatex">\(O(2^n)\)</span>. Exponential growth is explosive, so a relatively large <span class="arithmatex">\(n\)</span> leads to a very long wait.</p>
 <p><img alt="Recursion tree for climbing stairs" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_dfs_tree.png" /></p>
 <p align="center"> Figure 14-3 &nbsp; Recursion tree for climbing stairs </p>
 
-<p>Observing the above figure, <strong>the exponential time complexity is caused by "overlapping subproblems"</strong>. For example, <span class="arithmatex">\(dp[9]\)</span> is decomposed into <span class="arithmatex">\(dp[8]\)</span> and <span class="arithmatex">\(dp[7]\)</span>, and <span class="arithmatex">\(dp[8]\)</span> is decomposed into <span class="arithmatex">\(dp[7]\)</span> and <span class="arithmatex">\(dp[6]\)</span>, both of which contain the subproblem <span class="arithmatex">\(dp[7]\)</span>.</p>
-<p>And so on, subproblems contain smaller overlapping subproblems, ad infinitum. The vast majority of computational resources are wasted on these overlapping subproblems.</p>
-<h2 id="1412-method-2-memoization">14.1.2 &nbsp; Method 2: Memoization<a class="headerlink" href="#1412-method-2-memoization" title="Permanent link">&para;</a></h2>
-<p>To improve algorithm efficiency, <strong>we want all overlapping subproblems to be computed only once</strong>. For this purpose, we declare an array <code>mem</code> to record the solution to each subproblem and prune overlapping subproblems during the search process.</p>
+<p>As Figure 14-3 shows, <strong>overlapping subproblems cause the exponential time complexity</strong>. For example, <span class="arithmatex">\(dp[9]\)</span> splits into <span class="arithmatex">\(dp[8]\)</span> and <span class="arithmatex">\(dp[7]\)</span>, while <span class="arithmatex">\(dp[8]\)</span> splits into <span class="arithmatex">\(dp[7]\)</span> and <span class="arithmatex">\(dp[6]\)</span>. Both contain <span class="arithmatex">\(dp[7]\)</span>.</p>
+<p>These subproblems in turn contain smaller overlapping subproblems, and the pattern repeats at deeper levels of the recursion tree. Most of the computation is wasted on solving the same subproblems repeatedly.</p>
+<h2 id="1412-method-2-memoized-search">14.1.2 &nbsp; Method 2: Memoized search<a class="headerlink" href="#1412-method-2-memoized-search" title="Permanent link">&para;</a></h2>
+<p>To improve efficiency, <strong>we want every overlapping subproblem to be solved only once</strong>. Introduce an array <code>mem</code> to record each solution and prune repeated subproblems during the search.</p>
 <ol>
-<li>When computing <span class="arithmatex">\(dp[i]\)</span> for the first time, we record it in <code>mem[i]</code> for later use.</li>
-<li>When we need to compute <span class="arithmatex">\(dp[i]\)</span> again, we can directly retrieve the result from <code>mem[i]</code>, thereby avoiding redundant computation of that subproblem.</li>
+<li>The first time we calculate <span class="arithmatex">\(dp[i]\)</span>, store it in <code>mem[i]</code> for later use.</li>
+<li>When <span class="arithmatex">\(dp[i]\)</span> is needed again, retrieve it directly from <code>mem[i]</code> instead of recomputing it.</li>
 </ol>
 <p>The code is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dfs_mem.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">mem</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">])</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Memoization search"""</span>
-<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="c1"># Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Memoized search"""</span>
+<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="c1"># The values of dp[1] and dp[2] are known; return the corresponding value</span>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
 <a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>        <span class="k">return</span> <span class="n">i</span>
-<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="c1"># If record dp[i] exists, return it directly</span>
+<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="c1"># If dp[i] has been recorded, return it directly</span>
 <a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>    <span class="k">if</span> <span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>        <span class="k">return</span> <span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>    <span class="c1"># dp[i] = dp[i-1] + dp[i-2]</span>
@@ -5359,19 +5441,19 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>    <span class="k">return</span> <span class="n">count</span>
 <a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a>
 <a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dfs_mem</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-16" name="__codelineno-26-16" href="#__codelineno-26-16"></a><span class="w">    </span><span class="sd">"""Climbing stairs: Memoization search"""</span>
-<a id="__codelineno-26-17" name="__codelineno-26-17" href="#__codelineno-26-17"></a>    <span class="c1"># mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record</span>
+<a id="__codelineno-26-16" name="__codelineno-26-16" href="#__codelineno-26-16"></a><span class="w">    </span><span class="sd">"""Climbing stairs: memoized search"""</span>
+<a id="__codelineno-26-17" name="__codelineno-26-17" href="#__codelineno-26-17"></a>    <span class="c1"># mem[i] stores the number of ways to reach step i; -1 means no result has been recorded</span>
 <a id="__codelineno-26-18" name="__codelineno-26-18" href="#__codelineno-26-18"></a>    <span class="n">mem</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-26-19" name="__codelineno-26-19" href="#__codelineno-26-19"></a>    <span class="k">return</span> <span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">mem</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Memoization search */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Memoized search */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">mem</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="c1">// If record dp[i] exists, return it directly</span>
+<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="c1">// If dp[i] has been recorded, return it directly</span>
 <a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">-1</span><span class="p">)</span>
 <a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">    </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
@@ -5381,21 +5463,21 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span>
 <a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a><span class="p">}</span>
 <a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a>
-<a id="__codelineno-27-16" name="__codelineno-27-16" href="#__codelineno-27-16"></a><span class="cm">/* Climbing stairs: Memoization search */</span>
+<a id="__codelineno-27-16" name="__codelineno-27-16" href="#__codelineno-27-16"></a><span class="cm">/* Climbing stairs: memoized search */</span>
 <a id="__codelineno-27-17" name="__codelineno-27-17" href="#__codelineno-27-17"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-18" name="__codelineno-27-18" href="#__codelineno-27-18"></a><span class="w">    </span><span class="c1">// mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record</span>
+<a id="__codelineno-27-18" name="__codelineno-27-18" href="#__codelineno-27-18"></a><span class="w">    </span><span class="c1">// mem[i] stores the number of ways to reach step i; -1 means no result has been recorded</span>
 <a id="__codelineno-27-19" name="__codelineno-27-19" href="#__codelineno-27-19"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">mem</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">-1</span><span class="p">);</span>
 <a id="__codelineno-27-20" name="__codelineno-27-20" href="#__codelineno-27-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">);</span>
 <a id="__codelineno-27-21" name="__codelineno-27-21" href="#__codelineno-27-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Memoization search */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Memoized search */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="c1">// If record dp[i] exists, return it directly</span>
+<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="c1">// If dp[i] has been recorded, return it directly</span>
 <a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="w">    </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
@@ -5405,9 +5487,9 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span>
 <a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a><span class="p">}</span>
 <a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a>
-<a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a><span class="cm">/* Climbing stairs: Memoization search */</span>
+<a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a><span class="cm">/* Climbing stairs: memoized search */</span>
 <a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a><span class="w">    </span><span class="c1">// mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record</span>
+<a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a><span class="w">    </span><span class="c1">// mem[i] stores the number of ways to reach step i; -1 means no result has been recorded</span>
 <a id="__codelineno-28-19" name="__codelineno-28-19" href="#__codelineno-28-19"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">mem</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-20" name="__codelineno-28-20" href="#__codelineno-28-20"></a><span class="w">    </span><span class="n">Arrays</span><span class="p">.</span><span class="na">fill</span><span class="p">(</span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-28-21" name="__codelineno-28-21" href="#__codelineno-28-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">);</span>
@@ -5415,12 +5497,12 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Memoization search */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Memoized search */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">DFS</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="c1">// If record dp[i] exists, return it directly</span>
+<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="c1">// If dp[i] has been recorded, return it directly</span>
 <a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">    </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
@@ -5430,9 +5512,9 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span>
 <a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="p">}</span>
 <a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a>
-<a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="cm">/* Climbing stairs: Memoization search */</span>
+<a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="cm">/* Climbing stairs: memoized search */</span>
 <a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></a><span class="kt">int</span><span class="w"> </span><span class="nf">ClimbingStairsDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a><span class="w">    </span><span class="c1">// mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record</span>
+<a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a><span class="w">    </span><span class="c1">// mem[i] stores the number of ways to reach step i; -1 means no result has been recorded</span>
 <a id="__codelineno-29-19" name="__codelineno-29-19" href="#__codelineno-29-19"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">mem</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-29-20" name="__codelineno-29-20" href="#__codelineno-29-20"></a><span class="w">    </span><span class="n">Array</span><span class="p">.</span><span class="n">Fill</span><span class="p">(</span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nf">DFS</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">);</span>
@@ -5440,13 +5522,13 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Memoization search */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Memoized search */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">dfsMem</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span>
 <a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">    </span><span class="c1">// If record dp[i] exists, return it directly</span>
+<a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">    </span><span class="c1">// If dp[i] has been recorded, return it directly</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
 <a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">    </span><span class="p">}</span>
@@ -5457,9 +5539,9 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">count</span>
 <a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="p">}</span>
 <a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></a>
-<a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a><span class="cm">/* Climbing stairs: Memoization search */</span>
+<a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a><span class="cm">/* Climbing stairs: memoized search */</span>
 <a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></a><span class="kd">func</span><span class="w"> </span><span class="nx">climbingStairsDFSMem</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-20" name="__codelineno-30-20" href="#__codelineno-30-20"></a><span class="w">    </span><span class="c1">// mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record</span>
+<a id="__codelineno-30-20" name="__codelineno-30-20" href="#__codelineno-30-20"></a><span class="w">    </span><span class="c1">// mem[i] stores the number of ways to reach step i; -1 means no result has been recorded</span>
 <a id="__codelineno-30-21" name="__codelineno-30-21" href="#__codelineno-30-21"></a><span class="w">    </span><span class="nx">mem</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-30-22" name="__codelineno-30-22" href="#__codelineno-30-22"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">mem</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-23" name="__codelineno-30-23" href="#__codelineno-30-23"></a><span class="w">        </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
@@ -5469,13 +5551,13 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Memoization search */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Memoized search */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">    </span><span class="c1">// If record dp[i] exists, return it directly</span>
+<a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">    </span><span class="c1">// If dp[i] has been recorded, return it directly</span>
 <a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">    </span><span class="p">}</span>
@@ -5486,20 +5568,20 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="bp">count</span>
 <a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="p">}</span>
 <a id="__codelineno-31-17" name="__codelineno-31-17" href="#__codelineno-31-17"></a>
-<a id="__codelineno-31-18" name="__codelineno-31-18" href="#__codelineno-31-18"></a><span class="cm">/* Climbing stairs: Memoization search */</span>
+<a id="__codelineno-31-18" name="__codelineno-31-18" href="#__codelineno-31-18"></a><span class="cm">/* Climbing stairs: memoized search */</span>
 <a id="__codelineno-31-19" name="__codelineno-31-19" href="#__codelineno-31-19"></a><span class="kd">func</span><span class="w"> </span><span class="nf">climbingStairsDFSMem</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-20" name="__codelineno-31-20" href="#__codelineno-31-20"></a><span class="w">    </span><span class="c1">// mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record</span>
+<a id="__codelineno-31-20" name="__codelineno-31-20" href="#__codelineno-31-20"></a><span class="w">    </span><span class="c1">// mem[i] stores the number of ways to reach step i; -1 means no result has been recorded</span>
 <a id="__codelineno-31-21" name="__codelineno-31-21" href="#__codelineno-31-21"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">mem</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-31-22" name="__codelineno-31-22" href="#__codelineno-31-22"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">mem</span><span class="p">)</span>
 <a id="__codelineno-31-23" name="__codelineno-31-23" href="#__codelineno-31-23"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Memoization search */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Memoized search */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
-<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="c1">// If record dp[i] exists, return it directly</span>
+<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="c1">// If dp[i] has been recorded, return it directly</span>
 <a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">    </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
 <a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">2</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">);</span>
@@ -5508,20 +5590,20 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">count</span><span class="p">;</span>
 <a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="p">}</span>
 <a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a>
-<a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="cm">/* Climbing stairs: Memoization search */</span>
+<a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="cm">/* Climbing stairs: memoized search */</span>
 <a id="__codelineno-32-15" name="__codelineno-32-15" href="#__codelineno-32-15"></a><span class="kd">function</span><span class="w"> </span><span class="nx">climbingStairsDFSMem</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-16" name="__codelineno-32-16" href="#__codelineno-32-16"></a><span class="w">    </span><span class="c1">// mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record</span>
+<a id="__codelineno-32-16" name="__codelineno-32-16" href="#__codelineno-32-16"></a><span class="w">    </span><span class="c1">// mem[i] stores the number of ways to reach step i; -1 means no result has been recorded</span>
 <a id="__codelineno-32-17" name="__codelineno-32-17" href="#__codelineno-32-17"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">mem</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="o">-</span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-32-18" name="__codelineno-32-18" href="#__codelineno-32-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">);</span>
 <a id="__codelineno-32-19" name="__codelineno-32-19" href="#__codelineno-32-19"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Memoization search */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Memoized search */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
-<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="c1">// If record dp[i] exists, return it directly</span>
+<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="c1">// If dp[i] has been recorded, return it directly</span>
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">    </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
 <a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">2</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">);</span>
@@ -5530,20 +5612,20 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">count</span><span class="p">;</span>
 <a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="p">}</span>
 <a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a>
-<a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="cm">/* Climbing stairs: Memoization search */</span>
+<a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="cm">/* Climbing stairs: memoized search */</span>
 <a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="kd">function</span><span class="w"> </span><span class="nx">climbingStairsDFSMem</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a><span class="w">    </span><span class="c1">// mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record</span>
+<a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a><span class="w">    </span><span class="c1">// mem[i] stores the number of ways to reach step i; -1 means no result has been recorded</span>
 <a id="__codelineno-33-17" name="__codelineno-33-17" href="#__codelineno-33-17"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">mem</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="o">-</span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-33-18" name="__codelineno-33-18" href="#__codelineno-33-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">);</span>
 <a id="__codelineno-33-19" name="__codelineno-33-19" href="#__codelineno-33-19"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Memoization search */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Memoized search */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="c1">// If record dp[i] exists, return it directly</span>
+<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="c1">// If dp[i] has been recorded, return it directly</span>
 <a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">  </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
 <a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">);</span>
@@ -5552,22 +5634,22 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span>
 <a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="p">}</span>
 <a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a>
-<a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="cm">/* Climbing stairs: Memoization search */</span>
+<a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="cm">/* Climbing stairs: memoized search */</span>
 <a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="kt">int</span><span class="w"> </span><span class="n">climbingStairsDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w">  </span><span class="c1">// mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record</span>
+<a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w">  </span><span class="c1">// mem[i] stores the number of ways to reach step i; -1 means no result has been recorded</span>
 <a id="__codelineno-34-17" name="__codelineno-34-17" href="#__codelineno-34-17"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">mem</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-34-18" name="__codelineno-34-18" href="#__codelineno-34-18"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">);</span>
 <a id="__codelineno-34-19" name="__codelineno-34-19" href="#__codelineno-34-19"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Memoization search */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Memoized search */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="c1">// If record dp[i] exists, return it directly</span>
+<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="c1">// If dp[i] has been recorded, return it directly</span>
 <a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w">    </span><span class="p">}</span>
@@ -5578,21 +5660,21 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">    </span><span class="n">count</span>
 <a id="__codelineno-35-16" name="__codelineno-35-16" href="#__codelineno-35-16"></a><span class="p">}</span>
 <a id="__codelineno-35-17" name="__codelineno-35-17" href="#__codelineno-35-17"></a>
-<a id="__codelineno-35-18" name="__codelineno-35-18" href="#__codelineno-35-18"></a><span class="cm">/* Climbing stairs: Memoization search */</span>
+<a id="__codelineno-35-18" name="__codelineno-35-18" href="#__codelineno-35-18"></a><span class="cm">/* Climbing stairs: memoized search */</span>
 <a id="__codelineno-35-19" name="__codelineno-35-19" href="#__codelineno-35-19"></a><span class="k">fn</span><span class="w"> </span><span class="nf">climbing_stairs_dfs_mem</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-20" name="__codelineno-35-20" href="#__codelineno-35-20"></a><span class="w">    </span><span class="c1">// mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record</span>
+<a id="__codelineno-35-20" name="__codelineno-35-20" href="#__codelineno-35-20"></a><span class="w">    </span><span class="c1">// mem[i] stores the number of ways to reach step i; -1 means no result has been recorded</span>
 <a id="__codelineno-35-21" name="__codelineno-35-21" href="#__codelineno-35-21"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">mem</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-35-22" name="__codelineno-35-22" href="#__codelineno-35-22"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span>
 <a id="__codelineno-35-23" name="__codelineno-35-23" href="#__codelineno-35-23"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Memoization search */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Memoized search */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">mem</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
-<a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">    </span><span class="c1">// If record dp[i] exists, return it directly</span>
+<a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">    </span><span class="c1">// If dp[i] has been recorded, return it directly</span>
 <a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">-1</span><span class="p">)</span>
 <a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="w">    </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
@@ -5602,9 +5684,9 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">count</span><span class="p">;</span>
 <a id="__codelineno-36-14" name="__codelineno-36-14" href="#__codelineno-36-14"></a><span class="p">}</span>
 <a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a>
-<a id="__codelineno-36-16" name="__codelineno-36-16" href="#__codelineno-36-16"></a><span class="cm">/* Climbing stairs: Memoization search */</span>
+<a id="__codelineno-36-16" name="__codelineno-36-16" href="#__codelineno-36-16"></a><span class="cm">/* Climbing stairs: memoized search */</span>
 <a id="__codelineno-36-17" name="__codelineno-36-17" href="#__codelineno-36-17"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-18" name="__codelineno-36-18" href="#__codelineno-36-18"></a><span class="w">    </span><span class="c1">// mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record</span>
+<a id="__codelineno-36-18" name="__codelineno-36-18" href="#__codelineno-36-18"></a><span class="w">    </span><span class="c1">// mem[i] stores the number of ways to reach step i; -1 means no result has been recorded</span>
 <a id="__codelineno-36-19" name="__codelineno-36-19" href="#__codelineno-36-19"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">mem</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">((</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-36-20" name="__codelineno-36-20" href="#__codelineno-36-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-21" name="__codelineno-36-21" href="#__codelineno-36-21"></a><span class="w">        </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
@@ -5616,11 +5698,11 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Memoization search */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Memoized search */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">i</span>
-<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="c1">// If record dp[i] exists, return it directly</span>
+<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="c1">// If dp[i] has been recorded, return it directly</span>
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="c1">// dp[i] = dp[i-1] + dp[i-2]</span>
 <a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span>
@@ -5629,9 +5711,9 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">count</span>
 <a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="p">}</span>
 <a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a>
-<a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="cm">/* Climbing stairs: Memoization search */</span>
+<a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="cm">/* Climbing stairs: memoized search */</span>
 <a id="__codelineno-37-15" name="__codelineno-37-15" href="#__codelineno-37-15"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">climbingStairsDFSMem</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-16" name="__codelineno-37-16" href="#__codelineno-37-16"></a><span class="w">    </span><span class="c1">// mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record</span>
+<a id="__codelineno-37-16" name="__codelineno-37-16" href="#__codelineno-37-16"></a><span class="w">    </span><span class="c1">// mem[i] stores the number of ways to reach step i; -1 means no result has been recorded</span>
 <a id="__codelineno-37-17" name="__codelineno-37-17" href="#__codelineno-37-17"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">mem</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-37-18" name="__codelineno-37-18" href="#__codelineno-37-18"></a><span class="w">    </span><span class="n">mem</span><span class="p">.</span><span class="na">fill</span><span class="p">(</span><span class="o">-</span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-37-19" name="__codelineno-37-19" href="#__codelineno-37-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span>
@@ -5639,11 +5721,11 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Memoization search ###</span>
+<div class="highlight"><span class="filename">climbing_stairs_dfs_mem.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Memoized search ###</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span>
-<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1"># Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1"># The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span>
-<a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">  </span><span class="c1"># If record dp[i] exists, return it directly</span>
+<a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">  </span><span class="c1"># If dp[i] has been recorded, return it directly</span>
 <a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a>
 <a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">  </span><span class="c1"># dp[i] = dp[i-1] + dp[i-2]</span>
@@ -5652,9 +5734,9 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="w">  </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">count</span>
 <a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="k">end</span>
 <a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a>
-<a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="c1">### Climbing stairs: memoization search ###</span>
+<a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="c1">### Climbing stairs: memoized search ###</span>
 <a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dfs_mem</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w">  </span><span class="c1"># mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record</span>
+<a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w">  </span><span class="c1"># mem[i] stores the number of ways to reach step i; -1 means no result has been recorded</span>
 <a id="__codelineno-38-17" name="__codelineno-38-17" href="#__codelineno-38-17"></a><span class="w">  </span><span class="n">mem</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-38-18" name="__codelineno-38-18" href="#__codelineno-38-18"></a><span class="w">  </span><span class="n">dfs</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">)</span>
 <a id="__codelineno-38-19" name="__codelineno-38-19" href="#__codelineno-38-19"></a><span class="k">end</span>
@@ -5662,42 +5744,47 @@ dp[i] = dp[i-1] + dp[i-2]
 </div>
 </div>
 </div>
-<p>Observe Figure 14-4: <strong>after memoization, all overlapping subproblems need to be computed only once, reducing the time complexity to <span class="arithmatex">\(O(n)\)</span></strong>, which is a tremendous leap.</p>
-<p><img alt="Recursion tree with memoization" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_dfs_memo_tree.png" /></p>
-<p align="center"> Figure 14-4 &nbsp; Recursion tree with memoization </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20dfs%28i%3A%20int,%20mem%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Memoized%20search%22%22%22%0A%20%20%20%20%23%20The%20values%20of%20dp%5B1%5D%20and%20dp%5B2%5D%20are%20known%3B%20return%20the%20corresponding%20value%0A%20%20%20%20if%20i%20%3D%3D%201%20or%20i%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20i%0A%20%20%20%20%23%20If%20dp%5Bi%5D%20has%20been%20recorded,%20return%20it%20directly%0A%20%20%20%20if%20mem%5Bi%5D%20%21%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%0A%20%20%20%20%23%20dp%5Bi%5D%20%3D%20dp%5Bi-1%5D%20%2B%20dp%5Bi-2%5D%0A%20%20%20%20count%20%3D%20dfs%28i%20-%201,%20mem%29%20%2B%20dfs%28i%20-%202,%20mem%29%0A%20%20%20%20%23%20Record%20dp%5Bi%5D%0A%20%20%20%20mem%5Bi%5D%20%3D%20count%0A%20%20%20%20return%20count%0A%0A%0Adef%20climbing_stairs_dfs_mem%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Climbing%20stairs%3A%20memoized%20search%22%22%22%0A%20%20%20%20%23%20mem%5Bi%5D%20stores%20the%20number%20of%20ways%20to%20reach%20step%20i%3B%20-1%20means%20no%20result%20has%20been%20recorded%0A%20%20%20%20mem%20%3D%20%5B-1%5D%20%2A%20%28n%20%2B%201%29%0A%20%20%20%20return%20dfs%28n,%20mem%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dfs_mem%28n%29%0A%20%20%20%20print%28f%22Number%20of%20ways%20to%20climb%20%7Bn%7D%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20dfs%28i%3A%20int,%20mem%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Memoized%20search%22%22%22%0A%20%20%20%20%23%20The%20values%20of%20dp%5B1%5D%20and%20dp%5B2%5D%20are%20known%3B%20return%20the%20corresponding%20value%0A%20%20%20%20if%20i%20%3D%3D%201%20or%20i%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20i%0A%20%20%20%20%23%20If%20dp%5Bi%5D%20has%20been%20recorded,%20return%20it%20directly%0A%20%20%20%20if%20mem%5Bi%5D%20%21%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%0A%20%20%20%20%23%20dp%5Bi%5D%20%3D%20dp%5Bi-1%5D%20%2B%20dp%5Bi-2%5D%0A%20%20%20%20count%20%3D%20dfs%28i%20-%201,%20mem%29%20%2B%20dfs%28i%20-%202,%20mem%29%0A%20%20%20%20%23%20Record%20dp%5Bi%5D%0A%20%20%20%20mem%5Bi%5D%20%3D%20count%0A%20%20%20%20return%20count%0A%0A%0Adef%20climbing_stairs_dfs_mem%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Climbing%20stairs%3A%20memoized%20search%22%22%22%0A%20%20%20%20%23%20mem%5Bi%5D%20stores%20the%20number%20of%20ways%20to%20reach%20step%20i%3B%20-1%20means%20no%20result%20has%20been%20recorded%0A%20%20%20%20mem%20%3D%20%5B-1%5D%20%2A%20%28n%20%2B%201%29%0A%20%20%20%20return%20dfs%28n,%20mem%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dfs_mem%28n%29%0A%20%20%20%20print%28f%22Number%20of%20ways%20to%20climb%20%7Bn%7D%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>As shown in Figure 14-4, <strong>memoization makes every overlapping subproblem require only one computation, reducing the time complexity to <span class="arithmatex">\(O(n)\)</span></strong>: a major improvement.</p>
+<p><img alt="Recursion tree for memoized search" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_dfs_memo_tree.png" /></p>
+<p align="center"> Figure 14-4 &nbsp; Recursion tree for memoized search </p>
 
-<h2 id="1413-method-3-dynamic-programming">14.1.3 &nbsp; Method 3: Dynamic Programming<a class="headerlink" href="#1413-method-3-dynamic-programming" title="Permanent link">&para;</a></h2>
-<p><strong>Memoization is a "top-down" method</strong>: we start from the original problem (root node), recursively decompose larger subproblems into smaller ones, until reaching the smallest known subproblems (leaf nodes). Afterward, by backtracking, we collect the solutions to the subproblems layer by layer to construct the solution to the original problem.</p>
-<p>In contrast, <strong>dynamic programming is a "bottom-up" method</strong>: starting from the solutions to the smallest subproblems, iteratively constructing solutions to larger subproblems until obtaining the solution to the original problem.</p>
-<p>Since dynamic programming does not include a backtracking process, it only requires loop iteration for implementation and does not need recursion. In the following code, we initialize an array <code>dp</code> to store the solutions to subproblems, which serves the same recording function as the array <code>mem</code> in memoization:</p>
+<h2 id="1413-method-3-dynamic-programming">14.1.3 &nbsp; Method 3: Dynamic programming<a class="headerlink" href="#1413-method-3-dynamic-programming" title="Permanent link">&para;</a></h2>
+<p><strong>Memoized search is a top-down approach</strong>: start with the original problem (the root), recursively divide larger subproblems into smaller ones until reaching the smallest subproblems with known solutions (the leaves), then collect their solutions as recursion returns to construct the solution to the original problem.</p>
+<p>In contrast, <strong>dynamic programming is a bottom-up approach</strong>: start with the smallest subproblem solutions and iteratively build solutions to larger subproblems until the original problem is solved.</p>
+<p>Because this dynamic programming process does not backtrack, it needs only iterative loops, not recursion. In the following code, the array <code>dp</code> stores subproblem solutions, serving the same role as <code>mem</code> in memoized search:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:13"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dp.py</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dp</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Climbing stairs: Dynamic programming"""</span>
+<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Climbing stairs: dynamic programming"""</span>
 <a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
 <a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>        <span class="k">return</span> <span class="n">n</span>
-<a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>    <span class="c1"># Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>    <span class="c1"># Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a>    <span class="c1"># Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a>    <span class="c1"># Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a>    <span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span>
-<a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a>    <span class="c1"># State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a>    <span class="c1"># State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a>        <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">2</span><span class="p">]</span>
 <a id="__codelineno-39-12" name="__codelineno-39-12" href="#__codelineno-39-12"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Climbing stairs: dynamic programming */</span>
 <a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="p">;</span>
-<a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-40-9" name="__codelineno-40-9" href="#__codelineno-40-9"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
-<a id="__codelineno-40-10" name="__codelineno-40-10" href="#__codelineno-40-10"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-40-10" name="__codelineno-40-10" href="#__codelineno-40-10"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-40-11" name="__codelineno-40-11" href="#__codelineno-40-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-12" name="__codelineno-40-12" href="#__codelineno-40-12"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">];</span>
 <a id="__codelineno-40-13" name="__codelineno-40-13" href="#__codelineno-40-13"></a><span class="w">    </span><span class="p">}</span>
@@ -5706,16 +5793,16 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Climbing stairs: dynamic programming */</span>
 <a id="__codelineno-41-2" name="__codelineno-41-2" href="#__codelineno-41-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="p">;</span>
-<a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-41-8" name="__codelineno-41-8" href="#__codelineno-41-8"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-41-9" name="__codelineno-41-9" href="#__codelineno-41-9"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
-<a id="__codelineno-41-10" name="__codelineno-41-10" href="#__codelineno-41-10"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-41-10" name="__codelineno-41-10" href="#__codelineno-41-10"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-41-11" name="__codelineno-41-11" href="#__codelineno-41-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-12" name="__codelineno-41-12" href="#__codelineno-41-12"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-41-13" name="__codelineno-41-13" href="#__codelineno-41-13"></a><span class="w">    </span><span class="p">}</span>
@@ -5724,16 +5811,16 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Climbing stairs: dynamic programming */</span>
 <a id="__codelineno-42-2" name="__codelineno-42-2" href="#__codelineno-42-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">ClimbingStairsDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="p">;</span>
-<a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-42-8" name="__codelineno-42-8" href="#__codelineno-42-8"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-42-9" name="__codelineno-42-9" href="#__codelineno-42-9"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
-<a id="__codelineno-42-10" name="__codelineno-42-10" href="#__codelineno-42-10"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-42-10" name="__codelineno-42-10" href="#__codelineno-42-10"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-42-11" name="__codelineno-42-11" href="#__codelineno-42-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-12" name="__codelineno-42-12" href="#__codelineno-42-12"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">];</span>
 <a id="__codelineno-42-13" name="__codelineno-42-13" href="#__codelineno-42-13"></a><span class="w">    </span><span class="p">}</span>
@@ -5742,17 +5829,17 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Climbing stairs: dynamic programming */</span>
 <a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">climbingStairsDP</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">n</span>
 <a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-43-9" name="__codelineno-43-9" href="#__codelineno-43-9"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span>
-<a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-43-12" name="__codelineno-43-12" href="#__codelineno-43-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-13" name="__codelineno-43-13" href="#__codelineno-43-13"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span>
 <a id="__codelineno-43-14" name="__codelineno-43-14" href="#__codelineno-43-14"></a><span class="w">    </span><span class="p">}</span>
@@ -5761,17 +5848,17 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Climbing stairs: dynamic programming */</span>
 <a id="__codelineno-44-2" name="__codelineno-44-2" href="#__codelineno-44-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">climbingStairsDP</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span>
 <a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-44-7" name="__codelineno-44-7" href="#__codelineno-44-7"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-44-9" name="__codelineno-44-9" href="#__codelineno-44-9"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-44-10" name="__codelineno-44-10" href="#__codelineno-44-10"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">2</span>
-<a id="__codelineno-44-11" name="__codelineno-44-11" href="#__codelineno-44-11"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-44-11" name="__codelineno-44-11" href="#__codelineno-44-11"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-44-12" name="__codelineno-44-12" href="#__codelineno-44-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-13" name="__codelineno-44-13" href="#__codelineno-44-13"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">]</span>
 <a id="__codelineno-44-14" name="__codelineno-44-14" href="#__codelineno-44-14"></a><span class="w">    </span><span class="p">}</span>
@@ -5780,15 +5867,15 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Climbing stairs: dynamic programming */</span>
 <a id="__codelineno-45-2" name="__codelineno-45-2" href="#__codelineno-45-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">climbingStairsDP</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span>
-<a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="o">-</span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-45-6" name="__codelineno-45-6" href="#__codelineno-45-6"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-45-6" name="__codelineno-45-6" href="#__codelineno-45-6"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-45-7" name="__codelineno-45-7" href="#__codelineno-45-7"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-45-8" name="__codelineno-45-8" href="#__codelineno-45-8"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
-<a id="__codelineno-45-9" name="__codelineno-45-9" href="#__codelineno-45-9"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-45-9" name="__codelineno-45-9" href="#__codelineno-45-9"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-45-10" name="__codelineno-45-10" href="#__codelineno-45-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">3</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-11" name="__codelineno-45-11" href="#__codelineno-45-11"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">2</span><span class="p">];</span>
 <a id="__codelineno-45-12" name="__codelineno-45-12" href="#__codelineno-45-12"></a><span class="w">    </span><span class="p">}</span>
@@ -5797,15 +5884,15 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Climbing stairs: dynamic programming */</span>
 <a id="__codelineno-46-2" name="__codelineno-46-2" href="#__codelineno-46-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">climbingStairsDP</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span>
-<a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="o">-</span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
-<a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-46-10" name="__codelineno-46-10" href="#__codelineno-46-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">3</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-11" name="__codelineno-46-11" href="#__codelineno-46-11"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">2</span><span class="p">];</span>
 <a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w">    </span><span class="p">}</span>
@@ -5814,15 +5901,15 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Climbing stairs: dynamic programming */</span>
 <a id="__codelineno-47-2" name="__codelineno-47-2" href="#__codelineno-47-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">climbingStairsDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="p">;</span>
-<a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
-<a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a><span class="w">  </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a><span class="w">  </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-47-7" name="__codelineno-47-7" href="#__codelineno-47-7"></a><span class="w">  </span><span class="n">dp</span><span class="p">[</span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a><span class="w">  </span><span class="n">dp</span><span class="p">[</span><span class="m">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2</span><span class="p">;</span>
-<a id="__codelineno-47-9" name="__codelineno-47-9" href="#__codelineno-47-9"></a><span class="w">  </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-47-9" name="__codelineno-47-9" href="#__codelineno-47-9"></a><span class="w">  </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-47-10" name="__codelineno-47-10" href="#__codelineno-47-10"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">3</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-11" name="__codelineno-47-11" href="#__codelineno-47-11"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">2</span><span class="p">];</span>
 <a id="__codelineno-47-12" name="__codelineno-47-12" href="#__codelineno-47-12"></a><span class="w">  </span><span class="p">}</span>
@@ -5831,18 +5918,18 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Climbing stairs: dynamic programming */</span>
 <a id="__codelineno-48-2" name="__codelineno-48-2" href="#__codelineno-48-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">climbing_stairs_dp</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="c1">// Known dp[1] and dp[2], return them</span>
+<a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="c1">// The values of dp[1] and dp[2] are known; return them</span>
 <a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
 <a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-48-8" name="__codelineno-48-8" href="#__codelineno-48-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-48-11" name="__codelineno-48-11" href="#__codelineno-48-11"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
-<a id="__codelineno-48-12" name="__codelineno-48-12" href="#__codelineno-48-12"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-48-12" name="__codelineno-48-12" href="#__codelineno-48-12"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-48-13" name="__codelineno-48-13" href="#__codelineno-48-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">3</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-14" name="__codelineno-48-14" href="#__codelineno-48-14"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">];</span>
 <a id="__codelineno-48-15" name="__codelineno-48-15" href="#__codelineno-48-15"></a><span class="w">    </span><span class="p">}</span>
@@ -5851,16 +5938,16 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Climbing stairs: dynamic programming */</span>
 <a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="p">;</span>
-<a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">((</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
-<a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-49-9" name="__codelineno-49-9" href="#__codelineno-49-9"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
-<a id="__codelineno-49-10" name="__codelineno-49-10" href="#__codelineno-49-10"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-49-10" name="__codelineno-49-10" href="#__codelineno-49-10"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-49-11" name="__codelineno-49-11" href="#__codelineno-49-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-12" name="__codelineno-49-12" href="#__codelineno-49-12"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">];</span>
 <a id="__codelineno-49-13" name="__codelineno-49-13" href="#__codelineno-49-13"></a><span class="w">    </span><span class="p">}</span>
@@ -5871,15 +5958,15 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Climbing stairs: Dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Climbing stairs: dynamic programming */</span>
 <a id="__codelineno-50-2" name="__codelineno-50-2" href="#__codelineno-50-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">climbingStairsDP</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">n</span>
-<a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="c1">// Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
-<a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">    </span><span class="c1">// Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">    </span><span class="c1">// Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-50-8" name="__codelineno-50-8" href="#__codelineno-50-8"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="m">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2</span>
-<a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-50-10" name="__codelineno-50-10" href="#__codelineno-50-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">3.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-11" name="__codelineno-50-11" href="#__codelineno-50-11"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">2</span><span class="o">]</span>
 <a id="__codelineno-50-12" name="__codelineno-50-12" href="#__codelineno-50-12"></a><span class="w">    </span><span class="p">}</span>
@@ -5892,11 +5979,11 @@ dp[i] = dp[i-1] + dp[i-2]
 <a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dp</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a>
-<a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">  </span><span class="c1"># Initialize dp table, used to store solutions to subproblems</span>
+<a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">  </span><span class="c1"># Initialize the dp table to store subproblem solutions</span>
 <a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="w">  </span><span class="c1"># Initial state: preset the solution to the smallest subproblem</span>
+<a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="w">  </span><span class="c1"># Initial states: set the solutions to the smallest subproblems</span>
 <a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="w">  </span><span class="n">dp</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span>
-<a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w">  </span><span class="c1"># State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w">  </span><span class="c1"># State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a><span class="w">  </span><span class="p">(</span><span class="mi">3</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="o">]</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-51-11" name="__codelineno-51-11" href="#__codelineno-51-11"></a>
 <a id="__codelineno-51-12" name="__codelineno-51-12" href="#__codelineno-51-12"></a><span class="w">  </span><span class="n">dp</span><span class="o">[</span><span class="n">n</span><span class="o">]</span>
@@ -5905,24 +5992,29 @@ dp[i] = dp[i-1] + dp[i-2]
 </div>
 </div>
 </div>
-<p>Figure 14-5 simulates the execution process of the above code.</p>
-<p><img alt="Dynamic programming process for climbing stairs" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_dp.png" /></p>
-<p align="center"> Figure 14-5 &nbsp; Dynamic programming process for climbing stairs </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20climbing_stairs_dp%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Climbing%20stairs%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20n%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%20to%20store%20subproblem%20solutions%0A%20%20%20%20dp%20%3D%20%5B0%5D%20%2A%20%28n%20%2B%201%29%0A%20%20%20%20%23%20Initial%20states%3A%20set%20the%20solutions%20to%20the%20smallest%20subproblems%0A%20%20%20%20dp%5B1%5D,%20dp%5B2%5D%20%3D%201,%202%0A%20%20%20%20%23%20State%20transitions%3A%20solve%20progressively%20larger%20subproblems%20using%20smaller%20ones%0A%20%20%20%20for%20i%20in%20range%283,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%20%3D%20dp%5Bi%20-%201%5D%20%2B%20dp%5Bi%20-%202%5D%0A%20%20%20%20return%20dp%5Bn%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dp%28n%29%0A%20%20%20%20print%28f%22Number%20of%20ways%20to%20climb%20%7Bn%7D%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20climbing_stairs_dp%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Climbing%20stairs%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20n%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%20to%20store%20subproblem%20solutions%0A%20%20%20%20dp%20%3D%20%5B0%5D%20%2A%20%28n%20%2B%201%29%0A%20%20%20%20%23%20Initial%20states%3A%20set%20the%20solutions%20to%20the%20smallest%20subproblems%0A%20%20%20%20dp%5B1%5D,%20dp%5B2%5D%20%3D%201,%202%0A%20%20%20%20%23%20State%20transitions%3A%20solve%20progressively%20larger%20subproblems%20using%20smaller%20ones%0A%20%20%20%20for%20i%20in%20range%283,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%20%3D%20dp%5Bi%20-%201%5D%20%2B%20dp%5Bi%20-%202%5D%0A%20%20%20%20return%20dp%5Bn%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dp%28n%29%0A%20%20%20%20print%28f%22Number%20of%20ways%20to%20climb%20%7Bn%7D%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 14-5 traces the code's execution.</p>
+<p><img alt="Dynamic programming for climbing stairs" class="animation-figure" src="../intro_to_dynamic_programming.assets/climbing_stairs_dp.png" /></p>
+<p align="center"> Figure 14-5 &nbsp; Dynamic programming for climbing stairs </p>
 
-<p>Like backtracking algorithms, dynamic programming also uses the "state" concept to represent specific stages of problem solving, with each state corresponding to a subproblem and its corresponding local optimal solution. For example, the state in the stair climbing problem is defined as the current stair step number <span class="arithmatex">\(i\)</span>.</p>
-<p>Based on the above content, we can summarize the commonly used terminology in dynamic programming.</p>
+<p>Like backtracking, dynamic programming uses states to represent stages of solving a problem. Each state corresponds to a subproblem and its locally optimal solution. For climbing stairs, the state is the current step number <span class="arithmatex">\(i\)</span>.</p>
+<p>We can now summarize some common dynamic programming terminology.</p>
 <ul>
-<li>The array <code>dp</code> is called the <u>dp table</u>, where <span class="arithmatex">\(dp[i]\)</span> represents the solution to the subproblem corresponding to state <span class="arithmatex">\(i\)</span>.</li>
-<li>The states corresponding to the smallest subproblems (the <span class="arithmatex">\(1\)</span>st and <span class="arithmatex">\(2\)</span>nd steps) are called <u>initial states</u>.</li>
-<li>The recurrence formula <span class="arithmatex">\(dp[i] = dp[i-1] + dp[i-2]\)</span> is called the <u>state transition equation</u>.</li>
+<li>The array <code>dp</code> is called the <u>dp table</u>, and <span class="arithmatex">\(dp[i]\)</span> stores the solution to the subproblem for state <span class="arithmatex">\(i\)</span>.</li>
+<li>The states for the smallest subproblems (steps <span class="arithmatex">\(1\)</span> and <span class="arithmatex">\(2\)</span>) are called the <u>initial states</u>.</li>
+<li>The recurrence <span class="arithmatex">\(dp[i] = dp[i-1] + dp[i-2]\)</span> is called the <u>state transition equation</u>.</li>
 </ul>
-<h2 id="1414-space-optimization">14.1.4 &nbsp; Space Optimization<a class="headerlink" href="#1414-space-optimization" title="Permanent link">&para;</a></h2>
-<p>Observant readers may have noticed that <strong>since <span class="arithmatex">\(dp[i]\)</span> is only related to <span class="arithmatex">\(dp[i-1]\)</span> and <span class="arithmatex">\(dp[i-2]\)</span>, we do not need to use an array <code>dp</code> to store the solutions to all subproblems</strong>, and can instead use two variables that roll forward. The code is as follows:</p>
+<h2 id="1414-space-optimization">14.1.4 &nbsp; Space optimization<a class="headerlink" href="#1414-space-optimization" title="Permanent link">&para;</a></h2>
+<p>You may have noticed that <strong><span class="arithmatex">\(dp[i]\)</span> depends only on <span class="arithmatex">\(dp[i-1]\)</span> and <span class="arithmatex">\(dp[i-2]\)</span>, so we do not need an entire <code>dp</code> array storing every subproblem solution</strong>. Two variables updated as we proceed are enough. The code is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">climbing_stairs_dp.py</span><pre><span></span><code><a id="__codelineno-52-1" name="__codelineno-52-1" href="#__codelineno-52-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dp_comp</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Climbing stairs: Space-optimized dynamic programming"""</span>
+<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Climbing stairs: space-optimized dynamic programming"""</span>
 <a id="__codelineno-52-3" name="__codelineno-52-3" href="#__codelineno-52-3"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">1</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
 <a id="__codelineno-52-4" name="__codelineno-52-4" href="#__codelineno-52-4"></a>        <span class="k">return</span> <span class="n">n</span>
 <a id="__codelineno-52-5" name="__codelineno-52-5" href="#__codelineno-52-5"></a>    <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span>
@@ -5932,7 +6024,7 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Climbing stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-53-2" name="__codelineno-53-2" href="#__codelineno-53-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsDPComp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-53-4" name="__codelineno-53-4" href="#__codelineno-53-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="p">;</span>
@@ -5947,7 +6039,7 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Climbing stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-54-2" name="__codelineno-54-2" href="#__codelineno-54-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsDPComp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-54-4" name="__codelineno-54-4" href="#__codelineno-54-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="p">;</span>
@@ -5962,7 +6054,7 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Climbing stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-55-2" name="__codelineno-55-2" href="#__codelineno-55-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">ClimbingStairsDPComp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-55-4" name="__codelineno-55-4" href="#__codelineno-55-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="p">;</span>
@@ -5977,13 +6069,13 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Climbing stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-56-2" name="__codelineno-56-2" href="#__codelineno-56-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">climbingStairsDPComp</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">n</span>
 <a id="__codelineno-56-5" name="__codelineno-56-5" href="#__codelineno-56-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-56-6" name="__codelineno-56-6" href="#__codelineno-56-6"></a><span class="w">    </span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">b</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span>
-<a id="__codelineno-56-7" name="__codelineno-56-7" href="#__codelineno-56-7"></a><span class="w">    </span><span class="c1">// State transition: gradually solve larger subproblems from smaller ones</span>
+<a id="__codelineno-56-7" name="__codelineno-56-7" href="#__codelineno-56-7"></a><span class="w">    </span><span class="c1">// State transitions: solve progressively larger subproblems using smaller ones</span>
 <a id="__codelineno-56-8" name="__codelineno-56-8" href="#__codelineno-56-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-9" name="__codelineno-56-9" href="#__codelineno-56-9"></a><span class="w">        </span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">b</span><span class="p">,</span><span class="w"> </span><span class="nx">a</span><span class="o">+</span><span class="nx">b</span>
 <a id="__codelineno-56-10" name="__codelineno-56-10" href="#__codelineno-56-10"></a><span class="w">    </span><span class="p">}</span>
@@ -5992,7 +6084,7 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Climbing stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-57-2" name="__codelineno-57-2" href="#__codelineno-57-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">climbingStairsDPComp</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span>
@@ -6007,7 +6099,7 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Climbing stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-58-2" name="__codelineno-58-2" href="#__codelineno-58-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">climbingStairsDPComp</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span>
 <a id="__codelineno-58-4" name="__codelineno-58-4" href="#__codelineno-58-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span>
@@ -6022,7 +6114,7 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Climbing stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-59-2" name="__codelineno-59-2" href="#__codelineno-59-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">climbingStairsDPComp</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span>
 <a id="__codelineno-59-4" name="__codelineno-59-4" href="#__codelineno-59-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span>
@@ -6037,7 +6129,7 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Climbing stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-60-2" name="__codelineno-60-2" href="#__codelineno-60-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">climbingStairsDPComp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-3" name="__codelineno-60-3" href="#__codelineno-60-3"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="p">;</span>
 <a id="__codelineno-60-4" name="__codelineno-60-4" href="#__codelineno-60-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2</span><span class="p">;</span>
@@ -6051,7 +6143,7 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Climbing stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-61-2" name="__codelineno-61-2" href="#__codelineno-61-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">climbing_stairs_dp_comp</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-3" name="__codelineno-61-3" href="#__codelineno-61-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-4" name="__codelineno-61-4" href="#__codelineno-61-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
@@ -6067,7 +6159,7 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Climbing stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-62-2" name="__codelineno-62-2" href="#__codelineno-62-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">climbingStairsDPComp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-3" name="__codelineno-62-3" href="#__codelineno-62-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-62-4" name="__codelineno-62-4" href="#__codelineno-62-4"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="p">;</span>
@@ -6082,7 +6174,7 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Climbing stairs: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Climbing stairs: space-optimized dynamic programming */</span>
 <a id="__codelineno-63-2" name="__codelineno-63-2" href="#__codelineno-63-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">climbingStairsDPComp</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-3" name="__codelineno-63-3" href="#__codelineno-63-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">n</span>
 <a id="__codelineno-63-4" name="__codelineno-63-4" href="#__codelineno-63-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span>
@@ -6097,7 +6189,7 @@ dp[i] = dp[i-1] + dp[i-2]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">climbing_stairs_dp.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Climbing stairs: space-optimized DP ###</span>
+<div class="highlight"><span class="filename">climbing_stairs_dp.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Climbing stairs: space-optimized dynamic programming ###</span>
 <a id="__codelineno-64-2" name="__codelineno-64-2" href="#__codelineno-64-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">climbing_stairs_dp_comp</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-64-3" name="__codelineno-64-3" href="#__codelineno-64-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-64-4" name="__codelineno-64-4" href="#__codelineno-64-4"></a>
@@ -6110,8 +6202,13 @@ dp[i] = dp[i-1] + dp[i-2]
 </div>
 </div>
 </div>
-<p>As the above code shows, by eliminating the space occupied by the array <code>dp</code>, the space complexity is reduced from <span class="arithmatex">\(O(n)\)</span> to <span class="arithmatex">\(O(1)\)</span>.</p>
-<p>In dynamic programming problems, the current state often depends only on a limited number of preceding states, allowing us to retain only the necessary states and save memory space through "dimension reduction". <strong>This space optimization technique is called "rolling variable" or "rolling array"</strong>.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 477px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20climbing_stairs_dp_comp%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Climbing%20stairs%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20n%0A%20%20%20%20a,%20b%20%3D%201,%202%0A%20%20%20%20for%20_%20in%20range%283,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20a,%20b%20%3D%20b,%20a%20%2B%20b%0A%20%20%20%20return%20b%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dp_comp%28n%29%0A%20%20%20%20print%28f%22Number%20of%20ways%20to%20climb%20%7Bn%7D%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20climbing_stairs_dp_comp%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Climbing%20stairs%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20if%20n%20%3D%3D%201%20or%20n%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20return%20n%0A%20%20%20%20a,%20b%20%3D%201,%202%0A%20%20%20%20for%20_%20in%20range%283,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20a,%20b%20%3D%20b,%20a%20%2B%20b%0A%20%20%20%20return%20b%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%209%0A%0A%20%20%20%20res%20%3D%20climbing_stairs_dp_comp%28n%29%0A%20%20%20%20print%28f%22Number%20of%20ways%20to%20climb%20%7Bn%7D%20stairs%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Removing the <code>dp</code> array reduces space complexity from <span class="arithmatex">\(O(n)\)</span> to <span class="arithmatex">\(O(1)\)</span>.</p>
+<p>In dynamic programming, a state often depends on only a limited number of preceding states. We can keep just those needed, reducing the storage dimensions to save memory. <strong>This technique is called using rolling variables or a rolling array</strong>.</p>
 
 <!-- Source file information -->
 
@@ -6134,7 +6231,7 @@ aria-label="Footer"
   <a
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     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: Chapter 14. &amp;nbsp; Dynamic Programming"
+    aria-label="Previous: Chapter 14.   Dynamic programming"
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@@ -6146,7 +6243,7 @@ aria-label="Footer"
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@@ -6158,7 +6255,7 @@ aria-label="Footer"
   <a
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     class="md-footer__link md-footer__link--next"
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@@ -6166,7 +6263,7 @@ aria-label="Footer"
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+        14.2 &nbsp; Properties of dynamic programming problems
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@@ -6183,7 +6280,7 @@ aria-label="Footer"
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+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
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@@ -41,7 +41,7 @@
 
 
 
-        <title>14.4 0-1 Knapsack Problem - Hello Algo</title>
+        <title>14.4 0-1 knapsack problem - Hello Algo</title>
 
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
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-  2.2 Iteration and Recursion
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-  2.3 Time Complexity
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-  2.4 Space Complexity
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
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-  Chapter 3. Data Structures
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-  3.2 Basic Data Types
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-  3.3 Number Encoding *
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-  3.6 Exercises
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  5.5 Exercises
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-  Chapter 6. Hashing
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-  Chapter 6. Hashing
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
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-  7.1 Binary Tree
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
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-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
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+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
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         <li class="md-nav__item">
-  <a href="#2-method-2-memoization" class="md-nav__link">
+  <a href="#2-method-2-memoized-search" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Method 2: Memoization
+        2. &nbsp; Method 2: Memoized search
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Method 1: Brute Force Search
+        1. &nbsp; Method 1: Brute-force search
       </span>
     </span>
   </a>
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 </li>
 
         <li class="md-nav__item">
-  <a href="#2-method-2-memoization" class="md-nav__link">
+  <a href="#2-method-2-memoized-search" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Method 2: Memoization
+        2. &nbsp; Method 2: Memoized search
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     <span class="md-ellipsis">
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-        3. &nbsp; Method 3: Dynamic Programming
+        3. &nbsp; Method 3: Dynamic programming
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-        4. &nbsp; Space Optimization
+        4. &nbsp; Space optimization
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@@ -4701,479 +4773,484 @@
 
 
 <!-- Page content -->
-<h1 id="144-0-1-knapsack-problem">14.4 &nbsp; 0-1 Knapsack Problem<a class="headerlink" href="#144-0-1-knapsack-problem" title="Permanent link">&para;</a></h1>
-<p>The knapsack problem is an excellent introductory problem for dynamic programming and is one of the most common problem forms in dynamic programming. It has many variants, such as the 0-1 knapsack problem, the unbounded knapsack problem, and the multiple knapsack problem.</p>
-<p>In this section, we will first solve the most common 0-1 knapsack problem.</p>
+<h1 id="144-0-1-knapsack-problem">14.4 &nbsp; 0-1 knapsack problem<a class="headerlink" href="#144-0-1-knapsack-problem" title="Permanent link">&para;</a></h1>
+<p>The knapsack problem is an excellent introduction to dynamic programming and one of its most common problem families. Variants include 0-1 knapsack, unbounded knapsack, and bounded knapsack.</p>
+<p>We will begin with the most common variant, 0-1 knapsack.</p>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given <span class="arithmatex">\(n\)</span> items and a knapsack with capacity <span class="arithmatex">\(cap\)</span>, where the weight and value of the <span class="arithmatex">\(i\)</span>-th item are <span class="arithmatex">\(wgt[i-1]\)</span> and <span class="arithmatex">\(val[i-1]\)</span>, respectively. Each item can be selected at most once. What is the maximum value that can fit in the knapsack under the capacity limit?</p>
+<p>Given <span class="arithmatex">\(n\)</span> items, where item <span class="arithmatex">\(i\)</span> has weight <span class="arithmatex">\(wgt[i-1]\)</span> and value <span class="arithmatex">\(val[i-1]\)</span>, and a knapsack of capacity <span class="arithmatex">\(cap\)</span>, each item may be chosen only once. What is the maximum total value that fits within the capacity?</p>
 </div>
-<p>Observe Figure 14-17. Since item number <span class="arithmatex">\(i\)</span> starts counting from <span class="arithmatex">\(1\)</span> and array indices start from <span class="arithmatex">\(0\)</span>, item <span class="arithmatex">\(i\)</span> corresponds to weight <span class="arithmatex">\(wgt[i-1]\)</span> and value <span class="arithmatex">\(val[i-1]\)</span>.</p>
-<p><img alt="Example data for 0-1 knapsack" class="animation-figure" src="../knapsack_problem.assets/knapsack_example.png" /></p>
-<p align="center"> Figure 14-17 &nbsp; Example data for 0-1 knapsack </p>
+<p>As shown in Figure 14-17, item numbers <span class="arithmatex">\(i\)</span> start at <span class="arithmatex">\(1\)</span>, while array indices start at <span class="arithmatex">\(0\)</span>. Item <span class="arithmatex">\(i\)</span> therefore has weight <span class="arithmatex">\(wgt[i-1]\)</span> and value <span class="arithmatex">\(val[i-1]\)</span>.</p>
+<p><img alt="Example input for 0-1 knapsack" class="animation-figure" src="../knapsack_problem.assets/knapsack_example.png" /></p>
+<p align="center"> Figure 14-17 &nbsp; Example input for 0-1 knapsack </p>
 
-<p>We can view the 0-1 knapsack problem as a process consisting of <span class="arithmatex">\(n\)</span> rounds of decisions, where for each item there are two decisions: not putting it in and putting it in, thus the problem satisfies the decision tree model.</p>
-<p>The goal of this problem is to find "the maximum value that can be placed in the knapsack within the capacity limit", so it is more likely to be a dynamic programming problem.</p>
-<p><strong>Step 1: Think about the decisions in each round, define the state, and thus obtain the <span class="arithmatex">\(dp\)</span> table</strong></p>
-<p>For each item, if not placed in the knapsack, the knapsack capacity remains unchanged; if placed in, the knapsack capacity decreases. From this, we can derive the state definition: current item number <span class="arithmatex">\(i\)</span> and knapsack capacity <span class="arithmatex">\(c\)</span>, denoted as <span class="arithmatex">\([i, c]\)</span>.</p>
-<p>State <span class="arithmatex">\([i, c]\)</span> corresponds to the subproblem: <strong>the maximum value among the first <span class="arithmatex">\(i\)</span> items in a knapsack of capacity <span class="arithmatex">\(c\)</span></strong>, denoted as <span class="arithmatex">\(dp[i, c]\)</span>.</p>
-<p>What we need to find is <span class="arithmatex">\(dp[n, cap]\)</span>, so we need a two-dimensional <span class="arithmatex">\(dp\)</span> table of size <span class="arithmatex">\((n+1) \times (cap+1)\)</span>.</p>
-<p><strong>Step 2: Identify the optimal substructure, and then derive the state transition equation</strong></p>
-<p>After making the decision for item <span class="arithmatex">\(i\)</span>, what remains is the subproblem of the first <span class="arithmatex">\(i-1\)</span> items, which can be divided into the following two cases.</p>
+<p>We can view 0-1 knapsack as <span class="arithmatex">\(n\)</span> rounds of decisions, with two choices for each item: leave it out or put it in. It therefore fits the decision tree model.</p>
+<p>The objective is to maximize value within a capacity limit, making dynamic programming a promising approach.</p>
+<p><strong>Step 1: Identify each decision, define the state, and establish the <span class="arithmatex">\(dp\)</span> table</strong></p>
+<p>Leaving an item out keeps the available capacity unchanged; putting it in reduces the capacity. The state therefore consists of the current item number <span class="arithmatex">\(i\)</span> and capacity <span class="arithmatex">\(c\)</span>, denoted by <span class="arithmatex">\([i, c]\)</span>.</p>
+<p>State <span class="arithmatex">\([i, c]\)</span> represents the subproblem of finding <strong>the maximum value obtainable from the first <span class="arithmatex">\(i\)</span> items with capacity <span class="arithmatex">\(c\)</span></strong>, denoted by <span class="arithmatex">\(dp[i, c]\)</span>.</p>
+<p>We seek <span class="arithmatex">\(dp[n, cap]\)</span>, so we need a two-dimensional <span class="arithmatex">\(dp\)</span> table of size <span class="arithmatex">\((n+1) \times (cap+1)\)</span>.</p>
+<p><strong>Step 2: Identify optimal substructure and derive the state transition equation</strong></p>
+<p>After deciding what to do with item <span class="arithmatex">\(i\)</span>, the remaining subproblem concerns the first <span class="arithmatex">\(i-1\)</span> items. There are two cases.</p>
 <ul>
-<li><strong>Not putting item <span class="arithmatex">\(i\)</span></strong>: The knapsack capacity remains unchanged, and the state changes to <span class="arithmatex">\([i-1, c]\)</span>.</li>
-<li><strong>Putting item <span class="arithmatex">\(i\)</span></strong>: The knapsack capacity decreases by <span class="arithmatex">\(wgt[i-1]\)</span>, the value increases by <span class="arithmatex">\(val[i-1]\)</span>, and the state changes to <span class="arithmatex">\([i-1, c-wgt[i-1]]\)</span>.</li>
+<li><strong>Leave item <span class="arithmatex">\(i\)</span> out</strong>: Capacity stays unchanged, and the remaining state is <span class="arithmatex">\([i-1, c]\)</span>.</li>
+<li><strong>Put item <span class="arithmatex">\(i\)</span> in</strong>: Capacity decreases by <span class="arithmatex">\(wgt[i-1]\)</span>, value increases by <span class="arithmatex">\(val[i-1]\)</span>, and the remaining state is <span class="arithmatex">\([i-1, c-wgt[i-1]]\)</span>.</li>
 </ul>
-<p>The above analysis reveals the optimal substructure of this problem: <strong>the maximum value <span class="arithmatex">\(dp[i, c]\)</span> equals the greater of the values obtained by not putting item <span class="arithmatex">\(i\)</span> into the knapsack and by putting it into the knapsack</strong>. From this, the state transition equation can be derived:</p>
+<p>This reveals the optimal substructure: <strong>the maximum value <span class="arithmatex">\(dp[i, c]\)</span> is the larger value obtained by leaving item <span class="arithmatex">\(i\)</span> out or putting item <span class="arithmatex">\(i\)</span> in</strong>. The state transition equation is:</p>
 <div class="arithmatex">\[
 dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 \]</div>
-<p>Note that if the weight of the current item <span class="arithmatex">\(wgt[i - 1]\)</span> exceeds the remaining knapsack capacity <span class="arithmatex">\(c\)</span>, then the only option is not to put it in the knapsack.</p>
-<p><strong>Step 3: Determine boundary conditions and state transition order</strong></p>
-<p>When there are no items or the knapsack capacity is <span class="arithmatex">\(0\)</span>, the maximum value is <span class="arithmatex">\(0\)</span>, i.e., the first column <span class="arithmatex">\(dp[i, 0]\)</span> and the first row <span class="arithmatex">\(dp[0, c]\)</span> are both equal to <span class="arithmatex">\(0\)</span>.</p>
-<p>The current state <span class="arithmatex">\([i, c]\)</span> transitions from the state above <span class="arithmatex">\([i-1, c]\)</span> and the upper-left state <span class="arithmatex">\([i-1, c-wgt[i-1]]\)</span>, so we can traverse the entire <span class="arithmatex">\(dp\)</span> table in forward order using two nested loops.</p>
-<p>Based on the above analysis, we will next implement the brute force search, memoization, and dynamic programming solutions in order.</p>
-<h3 id="1-method-1-brute-force-search">1. &nbsp; Method 1: Brute Force Search<a class="headerlink" href="#1-method-1-brute-force-search" title="Permanent link">&para;</a></h3>
-<p>The search code includes the following elements.</p>
+<p>If the current item's weight <span class="arithmatex">\(wgt[i - 1]\)</span> exceeds the remaining capacity <span class="arithmatex">\(c\)</span>, the only option is to leave it out.</p>
+<p><strong>Step 3: Determine the boundary conditions and evaluation order</strong></p>
+<p>With no items or capacity <span class="arithmatex">\(0\)</span>, the maximum value is <span class="arithmatex">\(0\)</span>. Thus, the first column <span class="arithmatex">\(dp[i, 0]\)</span> and first row <span class="arithmatex">\(dp[0, c]\)</span> are both <span class="arithmatex">\(0\)</span>.</p>
+<p>The current state <span class="arithmatex">\([i, c]\)</span> depends on <span class="arithmatex">\([i-1, c]\)</span> above and <span class="arithmatex">\([i-1, c-wgt[i-1]]\)</span> to the upper left, so two nested loops can traverse the <span class="arithmatex">\(dp\)</span> table in increasing index order.</p>
+<p>We will now implement brute-force search, memoized search, and dynamic programming in that order.</p>
+<h3 id="1-method-1-brute-force-search">1. &nbsp; Method 1: Brute-force search<a class="headerlink" href="#1-method-1-brute-force-search" title="Permanent link">&para;</a></h3>
+<p>The search has the following elements.</p>
 <ul>
-<li><strong>Recursive parameters</strong>: state <span class="arithmatex">\([i, c]\)</span>.</li>
-<li><strong>Return value</strong>: solution to the subproblem <span class="arithmatex">\(dp[i, c]\)</span>.</li>
-<li><strong>Termination condition</strong>: when there are no items left (<span class="arithmatex">\(i = 0\)</span>) or the remaining knapsack capacity is <span class="arithmatex">\(0\)</span>, terminate the recursion and return value <span class="arithmatex">\(0\)</span>.</li>
-<li><strong>Pruning</strong>: if the weight of the current item exceeds the remaining knapsack capacity, only the option of not putting it in is available.</li>
+<li><strong>Parameters</strong>: The state <span class="arithmatex">\([i, c]\)</span>.</li>
+<li><strong>Return value</strong>: The subproblem solution <span class="arithmatex">\(dp[i, c]\)</span>.</li>
+<li><strong>Base case</strong>: If there are no items left, <span class="arithmatex">\(i = 0\)</span>, or the remaining capacity is <span class="arithmatex">\(0\)</span>, stop recursion and return value <span class="arithmatex">\(0\)</span>.</li>
+<li><strong>Pruning</strong>: If the current item is heavier than the remaining capacity, only the choice to leave it out is allowed.</li>
 </ul>
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 <div class="highlight"><span class="filename">knapsack.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">val</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">c</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""0-1 knapsack: Brute-force search"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""0-1 knapsack: brute-force search"""</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">c</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="k">return</span> <span class="mi">0</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="k">if</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">c</span><span class="p">:</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="k">return</span> <span class="n">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="c1"># Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="c1"># Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="n">no</span> <span class="o">=</span> <span class="n">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="n">yes</span> <span class="o">=</span> <span class="n">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">c</span> <span class="o">-</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span> <span class="o">+</span> <span class="n">val</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="c1"># Return the larger value of the two options</span>
+<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="c1"># Return the larger value from the two choices</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>    <span class="k">return</span> <span class="nb">max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span> <span class="n">yes</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* 0-1 knapsack: Brute-force search */</span>
+<div class="highlight"><span class="filename">knapsack.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* 0-1 knapsack: brute-force search */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDFS</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">    </span><span class="c1">// Return the larger value of the two options</span>
+<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">    </span><span class="c1">// Return the larger value from the two choices</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">);</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* 0-1 knapsack: Brute-force search */</span>
+<div class="highlight"><span class="filename">knapsack.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* 0-1 knapsack: brute-force search */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDFS</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">    </span><span class="c1">// Return the larger value of the two options</span>
+<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">    </span><span class="c1">// Return the larger value from the two choices</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">);</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* 0-1 knapsack: Brute-force search */</span>
+<div class="highlight"><span class="filename">knapsack.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* 0-1 knapsack: brute-force search */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">KnapsackDFS</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">weight</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">weight</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nf">KnapsackDFS</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">KnapsackDFS</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">KnapsackDFS</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">weight</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="c1">// Return the larger value of the two options</span>
+<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="c1">// Return the larger value from the two choices</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">);</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* 0-1 knapsack: Brute-force search */</span>
+<div class="highlight"><span class="filename">knapsack.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* 0-1 knapsack: brute-force search */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="nx">no</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="nx">yes</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="o">-</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="c1">// Return the larger value of the two options</span>
+<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="c1">// Return the larger value from the two choices</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Max</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">no</span><span class="p">),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">yes</span><span class="p">)))</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* 0-1 knapsack: Brute-force search */</span>
+<div class="highlight"><span class="filename">knapsack.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* 0-1 knapsack: brute-force search */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">:</span><span class="w"> </span><span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">no</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">:</span><span class="w"> </span><span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">yes</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">:</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span>
-<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">    </span><span class="c1">// Return the larger value of the two options</span>
+<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">    </span><span class="c1">// Return the larger value from the two choices</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="bp">max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">)</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* 0-1 knapsack: Brute-force search */</span>
+<div class="highlight"><span class="filename">knapsack.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* 0-1 knapsack: brute-force search */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">);</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">);</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
-<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">    </span><span class="c1">// Return the larger value of the two options</span>
+<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">    </span><span class="c1">// Return the larger value from the two choices</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span><span class="nx">no</span><span class="p">,</span><span class="w"> </span><span class="nx">yes</span><span class="p">);</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* 0-1 knapsack: Brute-force search */</span>
+<div class="highlight"><span class="filename">knapsack.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* 0-1 knapsack: brute-force search */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="nx">wgt</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="nx">c</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">);</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">);</span>
 <a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">knapsackDFS</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
-<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">    </span><span class="c1">// Return the larger value of the two options</span>
+<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">    </span><span class="c1">// Return the larger value from the two choices</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span><span class="nx">no</span><span class="p">,</span><span class="w"> </span><span class="nx">yes</span><span class="p">);</span>
 <a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 0-1 knapsack: Brute-force search */</span>
+<div class="highlight"><span class="filename">knapsack.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* 0-1 knapsack: brute-force search */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
-<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">  </span><span class="c1">// Return the larger value of the two options</span>
+<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">  </span><span class="c1">// Return the larger value from the two choices</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">);</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* 0-1 knapsack: Brute-force search */</span>
+<div class="highlight"><span class="filename">knapsack.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* 0-1 knapsack: brute-force search */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">    </span><span class="c1">// Return the larger value of the two options</span>
+<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">    </span><span class="c1">// Return the larger value from the two choices</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">    </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">)</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 0-1 knapsack: Brute-force search */</span>
+<div class="highlight"><span class="filename">knapsack.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* 0-1 knapsack: brute-force search */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDFS</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">wgt</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="c1">// Return the larger value of the two options</span>
+<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="c1">// Return the larger value from the two choices</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">myMax</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">);</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* 0-1 knapsack: Brute-force search */</span>
+<div class="highlight"><span class="filename">knapsack.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* 0-1 knapsack: brute-force search */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">knapsackDFS</span><span class="p">(</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="n">c</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFS</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">_val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span>
-<a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">    </span><span class="c1">// Return the larger value of the two options</span>
+<a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">    </span><span class="c1">// Return the larger value from the two choices</span>
 <a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">)</span>
 <a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### 0-1 knapsack: brute force search ###</span>
+<div class="highlight"><span class="filename">knapsack.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### 0-1 knapsack: brute-force search ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span>
-<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="c1"># Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="c1"># Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="n">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="n">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsack_dfs</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span>
-<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="c1"># Return the larger value of the two options</span>
+<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="c1"># Return the larger value from the two choices</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">  </span><span class="o">[</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="o">].</span><span class="n">max</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<p>As shown in Figure 14-18, since each item generates two search branches, excluding it and including it, the time complexity is <span class="arithmatex">\(O(2^n)\)</span>.</p>
-<p>Observing the recursion tree, it is easy to see overlapping subproblems, such as <span class="arithmatex">\(dp[1, 10]\)</span>. When there are many items, large knapsack capacity, and especially many items with the same weight, the number of overlapping subproblems will increase significantly.</p>
-<p><img alt="Brute force search recursion tree for 0-1 knapsack problem" class="animation-figure" src="../knapsack_problem.assets/knapsack_dfs.png" /></p>
-<p align="center"> Figure 14-18 &nbsp; Brute force search recursion tree for 0-1 knapsack problem </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dfs%28wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20i%3A%20int,%20c%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20knapsack%3A%20brute-force%20search%22%22%22%0A%20%20%20%20%23%20If%20all%20items%20have%20been%20considered%20or%20no%20capacity%20remains,%20return%20value%200%0A%20%20%20%20if%20i%20%3D%3D%200%20or%20c%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20If%20the%20item%20exceeds%20the%20remaining%20capacity,%20it%20must%20be%20excluded%0A%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20return%20knapsack_dfs%28wgt,%20val,%20i%20-%201,%20c%29%0A%20%20%20%20%23%20Calculate%20the%20maximum%20values%20obtained%20by%20excluding%20and%20including%20item%20i%0A%20%20%20%20no%20%3D%20knapsack_dfs%28wgt,%20val,%20i%20-%201,%20c%29%0A%20%20%20%20yes%20%3D%20knapsack_dfs%28wgt,%20val,%20i%20-%201,%20c%20-%20wgt%5Bi%20-%201%5D%29%20%2B%20val%5Bi%20-%201%5D%0A%20%20%20%20%23%20Return%20the%20larger%20value%20from%20the%20two%20choices%0A%20%20%20%20return%20max%28no,%20yes%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10,%2020,%2030,%2040,%2050%5D%0A%20%20%20%20val%20%3D%20%5B50,%20120,%20150,%20210,%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20Brute-force%20search%0A%20%20%20%20res%20%3D%20knapsack_dfs%28wgt,%20val,%20n,%20cap%29%0A%20%20%20%20print%28f%22Maximum%20total%20value%20within%20the%20knapsack%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dfs%28wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20i%3A%20int,%20c%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20knapsack%3A%20brute-force%20search%22%22%22%0A%20%20%20%20%23%20If%20all%20items%20have%20been%20considered%20or%20no%20capacity%20remains,%20return%20value%200%0A%20%20%20%20if%20i%20%3D%3D%200%20or%20c%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20If%20the%20item%20exceeds%20the%20remaining%20capacity,%20it%20must%20be%20excluded%0A%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20return%20knapsack_dfs%28wgt,%20val,%20i%20-%201,%20c%29%0A%20%20%20%20%23%20Calculate%20the%20maximum%20values%20obtained%20by%20excluding%20and%20including%20item%20i%0A%20%20%20%20no%20%3D%20knapsack_dfs%28wgt,%20val,%20i%20-%201,%20c%29%0A%20%20%20%20yes%20%3D%20knapsack_dfs%28wgt,%20val,%20i%20-%201,%20c%20-%20wgt%5Bi%20-%201%5D%29%20%2B%20val%5Bi%20-%201%5D%0A%20%20%20%20%23%20Return%20the%20larger%20value%20from%20the%20two%20choices%0A%20%20%20%20return%20max%28no,%20yes%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10,%2020,%2030,%2040,%2050%5D%0A%20%20%20%20val%20%3D%20%5B50,%20120,%20150,%20210,%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20Brute-force%20search%0A%20%20%20%20res%20%3D%20knapsack_dfs%28wgt,%20val,%20n,%20cap%29%0A%20%20%20%20print%28f%22Maximum%20total%20value%20within%20the%20knapsack%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>As shown in Figure 14-18, each item creates two branches, excluding it or including it, so the time complexity is <span class="arithmatex">\(O(2^n)\)</span>.</p>
+<p>The recursion tree contains overlapping subproblems, such as <span class="arithmatex">\(dp[1, 10]\)</span>. Their number grows substantially with more items and larger capacity, especially when many items share the same weight.</p>
+<p><img alt="Brute-force recursion tree for 0-1 knapsack" class="animation-figure" src="../knapsack_problem.assets/knapsack_dfs.png" /></p>
+<p align="center"> Figure 14-18 &nbsp; Brute-force recursion tree for 0-1 knapsack </p>
 
-<h3 id="2-method-2-memoization">2. &nbsp; Method 2: Memoization<a class="headerlink" href="#2-method-2-memoization" title="Permanent link">&para;</a></h3>
-<p>To ensure that overlapping subproblems are only computed once, we use a memo list <code>mem</code> to record the solutions to subproblems, where <code>mem[i][c]</code> corresponds to <span class="arithmatex">\(dp[i, c]\)</span>.</p>
-<p>After introducing memoization, <strong>the time complexity depends on the number of subproblems</strong>, which is <span class="arithmatex">\(O(n \times cap)\)</span>. The implementation code is as follows:</p>
+<h3 id="2-method-2-memoized-search">2. &nbsp; Method 2: Memoized search<a class="headerlink" href="#2-method-2-memoized-search" title="Permanent link">&para;</a></h3>
+<p>To solve each overlapping subproblem only once, use a memoization table <code>mem</code>, where <code>mem[i][c]</code> stores <span class="arithmatex">\(dp[i, c]\)</span>.</p>
+<p>With memoization, <strong>the time complexity depends on the number of subproblems</strong>, giving <span class="arithmatex">\(O(n \times cap)\)</span>. The code is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">knapsack_dfs_mem</span><span class="p">(</span>
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a>    <span class="n">wgt</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">val</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">mem</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]],</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">c</span><span class="p">:</span> <span class="nb">int</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="w">    </span><span class="sd">"""0-1 knapsack: Memoization search"""</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="w">    </span><span class="sd">"""0-1 knapsack: memoized search"""</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">c</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="k">return</span> <span class="mi">0</span>
-<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>    <span class="c1"># If there's a record, return it directly</span>
+<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>    <span class="c1"># If a result has already been recorded, return it directly</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>    <span class="k">if</span> <span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>        <span class="k">return</span> <span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span>
-<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>    <span class="c1"># If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>    <span class="c1"># If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>    <span class="k">if</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">c</span><span class="p">:</span>
 <a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>        <span class="k">return</span> <span class="n">knapsack_dfs_mem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">mem</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
-<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>    <span class="c1"># Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>    <span class="c1"># Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>    <span class="n">no</span> <span class="o">=</span> <span class="n">knapsack_dfs_mem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">mem</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>    <span class="n">yes</span> <span class="o">=</span> <span class="n">knapsack_dfs_mem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span> <span class="n">val</span><span class="p">,</span> <span class="n">mem</span><span class="p">,</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">c</span> <span class="o">-</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span> <span class="o">+</span> <span class="n">val</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>    <span class="c1"># Record and return the larger value of the two options</span>
+<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>    <span class="c1"># Record and return the larger value from the two choices</span>
 <a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a>    <span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span> <span class="n">yes</span><span class="p">)</span>
 <a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a>    <span class="k">return</span> <span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* 0-1 knapsack: Memoization search */</span>
+<div class="highlight"><span class="filename">knapsack.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* 0-1 knapsack: memoized search */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDFSMem</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">-1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">    </span><span class="c1">// Record and return the larger value of the two options</span>
+<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">    </span><span class="c1">// Record and return the larger value from the two choices</span>
 <a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="w">    </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">);</span>
 <a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* 0-1 knapsack: Memoization search */</span>
+<div class="highlight"><span class="filename">knapsack.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* 0-1 knapsack: memoized search */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">    </span><span class="c1">// Record and return the larger value of the two options</span>
+<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">    </span><span class="c1">// Record and return the larger value from the two choices</span>
 <a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w">    </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">);</span>
 <a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* 0-1 knapsack: Memoization search */</span>
+<div class="highlight"><span class="filename">knapsack.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* 0-1 knapsack: memoized search */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">KnapsackDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">weight</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[][]</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">weight</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nf">KnapsackDFSMem</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">KnapsackDFSMem</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">KnapsackDFSMem</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">weight</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">    </span><span class="c1">// Record and return the larger value of the two options</span>
+<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">    </span><span class="c1">// Record and return the larger value from the two choices</span>
 <a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w">    </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">);</span>
 <a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* 0-1 knapsack: Memoization search */</span>
+<div class="highlight"><span class="filename">knapsack.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* 0-1 knapsack: memoized search */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="w"> </span><span class="p">[][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">    </span><span class="nx">no</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span>
 <a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">    </span><span class="nx">yes</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="o">-</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-<a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="w">    </span><span class="c1">// Return the larger value of the two options</span>
+<a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="w">    </span><span class="c1">// Return the larger value from the two choices</span>
 <a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="w">    </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Max</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">no</span><span class="p">),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">yes</span><span class="p">)))</span>
 <a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span>
 <a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* 0-1 knapsack: Memoization search */</span>
+<div class="highlight"><span class="filename">knapsack.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* 0-1 knapsack: memoized search */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">mem</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]],</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">:</span><span class="w"> </span><span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">no</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">:</span><span class="w"> </span><span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">yes</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">:</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span>
-<a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">    </span><span class="c1">// Record and return the larger value of the two options</span>
+<a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">    </span><span class="c1">// Record and return the larger value from the two choices</span>
 <a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">    </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">)</span>
 <a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span>
 <a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* 0-1 knapsack: Memoization search */</span>
+<div class="highlight"><span class="filename">knapsack.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* 0-1 knapsack: memoized search */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">];</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">);</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">);</span>
 <a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">yes</span><span class="w"> </span><span class="o">=</span>
 <a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">        </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
-<a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">    </span><span class="c1">// Record and return the larger value of the two options</span>
+<a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">    </span><span class="c1">// Record and return the larger value from the two choices</span>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">    </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span><span class="nx">no</span><span class="p">,</span><span class="w"> </span><span class="nx">yes</span><span class="p">);</span>
 <a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">];</span>
 <a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* 0-1 knapsack: Memoization search */</span>
+<div class="highlight"><span class="filename">knapsack.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* 0-1 knapsack: memoized search */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">knapsackDFSMem</span><span class="p">(</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="nx">wgt</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span>
@@ -5181,30 +5258,30 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="nx">c</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">];</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">);</span>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="p">);</span>
 <a id="__codelineno-20-23" name="__codelineno-20-23" href="#__codelineno-20-23"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">yes</span><span class="w"> </span><span class="o">=</span>
 <a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">        </span><span class="nx">knapsackDFSMem</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">mem</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
-<a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a><span class="w">    </span><span class="c1">// Record and return the larger value of the two options</span>
+<a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a><span class="w">    </span><span class="c1">// Record and return the larger value from the two choices</span>
 <a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="w">    </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span><span class="nx">no</span><span class="p">,</span><span class="w"> </span><span class="nx">yes</span><span class="p">);</span>
 <a id="__codelineno-20-27" name="__codelineno-20-27" href="#__codelineno-20-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">mem</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">];</span>
 <a id="__codelineno-20-28" name="__codelineno-20-28" href="#__codelineno-20-28"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* 0-1 knapsack: Memoization search */</span>
+<div class="highlight"><span class="filename">knapsack.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* 0-1 knapsack: memoized search */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">val</span><span class="p">,</span>
@@ -5212,77 +5289,77 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">,</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">  </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">  </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="w">  </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="w">  </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
-<a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">  </span><span class="c1">// Record and return the larger value of the two options</span>
+<a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">  </span><span class="c1">// Record and return the larger value from the two choices</span>
 <a id="__codelineno-21-25" name="__codelineno-21-25" href="#__codelineno-21-25"></a><span class="w">  </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">);</span>
 <a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-21-27" name="__codelineno-21-27" href="#__codelineno-21-27"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* 0-1 knapsack: Memoization search */</span>
+<div class="highlight"><span class="filename">knapsack.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* 0-1 knapsack: memoized search */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">knapsack_dfs_mem</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">mem</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsack_dfs_mem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsack_dfs_mem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsack_dfs_mem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">    </span><span class="c1">// Record and return the larger value of the two options</span>
+<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">    </span><span class="c1">// Record and return the larger value from the two choices</span>
 <a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w">    </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">);</span>
 <a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w">    </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span>
 <a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* 0-1 knapsack: Memoization search */</span>
+<div class="highlight"><span class="filename">knapsack.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* 0-1 knapsack: memoized search */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDFSMem</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">wgt</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">memCols</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">**</span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">-1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">memCols</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">memCols</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">);</span>
 <a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">memCols</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">    </span><span class="c1">// Record and return the larger value of the two options</span>
+<a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">    </span><span class="c1">// Record and return the larger value from the two choices</span>
 <a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w">    </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMax</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">);</span>
 <a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* 0-1 knapsack: Memoization search */</span>
+<div class="highlight"><span class="filename">knapsack.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* 0-1 knapsack: memoized search */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">knapsackDFSMem</span><span class="p">(</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span>
@@ -5290,86 +5367,91 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="n">c</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="c1">// If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="c1">// If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">    </span><span class="c1">// If there's a record, return it directly</span>
+<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">    </span><span class="c1">// If a result has already been recorded, return it directly</span>
 <a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span>
 <a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">    </span><span class="c1">// If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">    </span><span class="c1">// If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="w">    </span><span class="c1">// Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="w">    </span><span class="c1">// Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsackDFSMem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">_val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span>
-<a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a><span class="w">    </span><span class="c1">// Record and return the larger value of the two options</span>
+<a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a><span class="w">    </span><span class="c1">// Record and return the larger value from the two choices</span>
 <a id="__codelineno-24-25" name="__codelineno-24-25" href="#__codelineno-24-25"></a><span class="w">    </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="p">)</span>
 <a id="__codelineno-24-26" name="__codelineno-24-26" href="#__codelineno-24-26"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span>
 <a id="__codelineno-24-27" name="__codelineno-24-27" href="#__codelineno-24-27"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### 0-1 knapsack: memoization search ###</span>
+<div class="highlight"><span class="filename">knapsack.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### 0-1 knapsack: memoized search ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">knapsack_dfs_mem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">)</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># If all items have been selected or knapsack has no remaining capacity, return value 0</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># If all items have been considered or no capacity remains, return value 0</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># If there's a record, return it directly</span>
+<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># If a result has already been recorded, return it directly</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
-<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="c1"># If exceeds knapsack capacity, can only choose not to put it in</span>
+<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="c1"># If the item exceeds the remaining capacity, it must be excluded</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">knapsack_dfs_mem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span>
-<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="c1"># Calculate the maximum value of not putting in and putting in item i</span>
+<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="c1"># Calculate the maximum values obtained by excluding and including item i</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="n">no</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsack_dfs_mem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">)</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">  </span><span class="n">yes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">knapsack_dfs_mem</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">mem</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span>
-<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">  </span><span class="c1"># Record and return the larger value of the two options</span>
+<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">  </span><span class="c1"># Record and return the larger value from the two choices</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">  </span><span class="n">mem</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">no</span><span class="p">,</span><span class="w"> </span><span class="n">yes</span><span class="o">].</span><span class="n">max</span>
 <a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<p>Figure 14-19 shows the search branches pruned in memoization.</p>
-<p><img alt="Memoization recursion tree for 0-1 knapsack problem" class="animation-figure" src="../knapsack_problem.assets/knapsack_dfs_mem.png" /></p>
-<p align="center"> Figure 14-19 &nbsp; Memoization recursion tree for 0-1 knapsack problem </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dfs_mem%28%0A%20%20%20%20wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20mem%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20c%3A%20int%0A%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20knapsack%3A%20memoized%20search%22%22%22%0A%20%20%20%20%23%20If%20all%20items%20have%20been%20considered%20or%20no%20capacity%20remains,%20return%20value%200%0A%20%20%20%20if%20i%20%3D%3D%200%20or%20c%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20If%20a%20result%20has%20already%20been%20recorded,%20return%20it%20directly%0A%20%20%20%20if%20mem%5Bi%5D%5Bc%5D%20%21%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%5Bc%5D%0A%20%20%20%20%23%20If%20the%20item%20exceeds%20the%20remaining%20capacity,%20it%20must%20be%20excluded%0A%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20return%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20i%20-%201,%20c%29%0A%20%20%20%20%23%20Calculate%20the%20maximum%20values%20obtained%20by%20excluding%20and%20including%20item%20i%0A%20%20%20%20no%20%3D%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20i%20-%201,%20c%29%0A%20%20%20%20yes%20%3D%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20i%20-%201,%20c%20-%20wgt%5Bi%20-%201%5D%29%20%2B%20val%5Bi%20-%201%5D%0A%20%20%20%20%23%20Record%20and%20return%20the%20larger%20value%20from%20the%20two%20choices%0A%20%20%20%20mem%5Bi%5D%5Bc%5D%20%3D%20max%28no,%20yes%29%0A%20%20%20%20return%20mem%5Bi%5D%5Bc%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10,%2020,%2030,%2040,%2050%5D%0A%20%20%20%20val%20%3D%20%5B50,%20120,%20150,%20210,%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20Memoized%20search%0A%20%20%20%20mem%20%3D%20%5B%5B-1%5D%20%2A%20%28cap%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20res%20%3D%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20n,%20cap%29%0A%20%20%20%20print%28f%22Maximum%20total%20value%20within%20the%20knapsack%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=20&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dfs_mem%28%0A%20%20%20%20wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20mem%3A%20list%5Blist%5Bint%5D%5D,%20i%3A%20int,%20c%3A%20int%0A%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20knapsack%3A%20memoized%20search%22%22%22%0A%20%20%20%20%23%20If%20all%20items%20have%20been%20considered%20or%20no%20capacity%20remains,%20return%20value%200%0A%20%20%20%20if%20i%20%3D%3D%200%20or%20c%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%200%0A%20%20%20%20%23%20If%20a%20result%20has%20already%20been%20recorded,%20return%20it%20directly%0A%20%20%20%20if%20mem%5Bi%5D%5Bc%5D%20%21%3D%20-1%3A%0A%20%20%20%20%20%20%20%20return%20mem%5Bi%5D%5Bc%5D%0A%20%20%20%20%23%20If%20the%20item%20exceeds%20the%20remaining%20capacity,%20it%20must%20be%20excluded%0A%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20return%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20i%20-%201,%20c%29%0A%20%20%20%20%23%20Calculate%20the%20maximum%20values%20obtained%20by%20excluding%20and%20including%20item%20i%0A%20%20%20%20no%20%3D%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20i%20-%201,%20c%29%0A%20%20%20%20yes%20%3D%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20i%20-%201,%20c%20-%20wgt%5Bi%20-%201%5D%29%20%2B%20val%5Bi%20-%201%5D%0A%20%20%20%20%23%20Record%20and%20return%20the%20larger%20value%20from%20the%20two%20choices%0A%20%20%20%20mem%5Bi%5D%5Bc%5D%20%3D%20max%28no,%20yes%29%0A%20%20%20%20return%20mem%5Bi%5D%5Bc%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10,%2020,%2030,%2040,%2050%5D%0A%20%20%20%20val%20%3D%20%5B50,%20120,%20150,%20210,%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20Memoized%20search%0A%20%20%20%20mem%20%3D%20%5B%5B-1%5D%20%2A%20%28cap%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20res%20%3D%20knapsack_dfs_mem%28wgt,%20val,%20mem,%20n,%20cap%29%0A%20%20%20%20print%28f%22Maximum%20total%20value%20within%20the%20knapsack%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=20&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 14-19 shows the branches pruned by memoized search.</p>
+<p><img alt="Memoized recursion tree for 0-1 knapsack" class="animation-figure" src="../knapsack_problem.assets/knapsack_dfs_mem.png" /></p>
+<p align="center"> Figure 14-19 &nbsp; Memoized recursion tree for 0-1 knapsack </p>
 
-<h3 id="3-method-3-dynamic-programming">3. &nbsp; Method 3: Dynamic Programming<a class="headerlink" href="#3-method-3-dynamic-programming" title="Permanent link">&para;</a></h3>
-<p>Dynamic programming is essentially the process of filling the <span class="arithmatex">\(dp\)</span> table during state transitions. The code is as follows:</p>
+<h3 id="3-method-3-dynamic-programming">3. &nbsp; Method 3: Dynamic programming<a class="headerlink" href="#3-method-3-dynamic-programming" title="Permanent link">&para;</a></h3>
+<p>Dynamic programming essentially fills the <span class="arithmatex">\(dp\)</span> table using state transitions, as shown in the code below:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">knapsack_dp</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">val</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">cap</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""0-1 knapsack: Dynamic programming"""</span>
+<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""0-1 knapsack: dynamic programming"""</span>
 <a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">wgt</span><span class="p">)</span>
-<a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="c1"># Initialize dp table</span>
+<a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">cap</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="c1"># State transition</span>
+<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="c1"># State transitions</span>
 <a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">cap</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>            <span class="k">if</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">c</span><span class="p">:</span>
-<a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>                <span class="c1"># If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>                <span class="c1"># If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">]</span>
 <a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a>            <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>                <span class="c1"># The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>                <span class="c1"># Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">c</span> <span class="o">-</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]]</span> <span class="o">+</span> <span class="n">val</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
 <a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">cap</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* 0-1 knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* 0-1 knapsack: dynamic programming */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDP</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">));</span>
-<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-27-16" name="__codelineno-27-16" href="#__codelineno-27-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5379,19 +5461,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* 0-1 knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* 0-1 knapsack: dynamic programming */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDP</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
-<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">    </span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">max</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">);</span>
 <a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5401,19 +5483,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* 0-1 knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* 0-1 knapsack: dynamic programming */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">KnapsackDP</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">weight</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">weight</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
-<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">    </span><span class="kt">int</span><span class="p">[,]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">weight</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">weight</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">]);</span>
 <a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5423,22 +5505,22 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* 0-1 knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* 0-1 knapsack: dynamic programming */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">knapsackDP</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">wgt</span><span class="p">)</span>
-<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span>
 <a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Max</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">c</span><span class="p">]),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">c</span><span class="o">-</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span><span class="o">+</span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">])))</span>
 <a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></a><span class="w">        </span><span class="p">}</span>
@@ -5448,19 +5530,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* 0-1 knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* 0-1 knapsack: dynamic programming */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">knapsackDP</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">cap</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="bp">count</span>
-<a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">),</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">]</span>
 <a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span>
 <a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5470,21 +5552,21 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* 0-1 knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* 0-1 knapsack: dynamic programming */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">knapsackDP</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">wgt</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span>
 <a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">        </span><span class="p">.</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">)</span>
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">        </span><span class="p">.</span><span class="nx">map</span><span class="p">(()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">));</span>
-<a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">c</span><span class="p">];</span>
 <a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-15" name="__codelineno-32-15" href="#__codelineno-32-15"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-32-15" name="__codelineno-32-15" href="#__codelineno-32-15"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-32-16" name="__codelineno-32-16" href="#__codelineno-32-16"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span>
 <a id="__codelineno-32-17" name="__codelineno-32-17" href="#__codelineno-32-17"></a><span class="w">                    </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">c</span><span class="p">],</span>
 <a id="__codelineno-32-18" name="__codelineno-32-18" href="#__codelineno-32-18"></a><span class="w">                    </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span>
@@ -5497,25 +5579,25 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* 0-1 knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* 0-1 knapsack: dynamic programming */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">knapsackDP</span><span class="p">(</span>
 <a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="nx">wgt</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="nx">cap</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span>
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">wgt</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span>
 <a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="w">        </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span>
 <a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">    </span><span class="p">);</span>
-<a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-33-17" name="__codelineno-33-17" href="#__codelineno-33-17"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">c</span><span class="p">];</span>
 <a id="__codelineno-33-18" name="__codelineno-33-18" href="#__codelineno-33-18"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-19" name="__codelineno-33-19" href="#__codelineno-33-19"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-33-19" name="__codelineno-33-19" href="#__codelineno-33-19"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-33-20" name="__codelineno-33-20" href="#__codelineno-33-20"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span>
 <a id="__codelineno-33-21" name="__codelineno-33-21" href="#__codelineno-33-21"></a><span class="w">                    </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">c</span><span class="p">],</span>
 <a id="__codelineno-33-22" name="__codelineno-33-22" href="#__codelineno-33-22"></a><span class="w">                    </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span>
@@ -5528,19 +5610,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* 0-1 knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* 0-1 knapsack: dynamic programming */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">knapsackDP</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
-<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">));</span>
-<a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="c1">// State transition</span>
+<a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="c1">// State transitions</span>
 <a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="w">        </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="w">        </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="w">      </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">        </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">        </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]);</span>
 <a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">      </span><span class="p">}</span>
 <a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w">    </span><span class="p">}</span>
@@ -5550,19 +5632,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* 0-1 knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* 0-1 knapsack: dynamic programming */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">knapsack_dp</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">cap</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
-<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">cap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">max</span><span class="p">(</span>
 <a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">                    </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">],</span>
 <a id="__codelineno-35-16" name="__codelineno-35-16" href="#__codelineno-35-16"></a><span class="w">                    </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span>
@@ -5575,22 +5657,22 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* 0-1 knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* 0-1 knapsack: dynamic programming */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">wgt</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">wgtSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgtSize</span><span class="p">;</span>
-<a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">**</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">((</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">));</span>
 <a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-36-14" name="__codelineno-36-14" href="#__codelineno-36-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-16" name="__codelineno-36-16" href="#__codelineno-36-16"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-36-16" name="__codelineno-36-16" href="#__codelineno-36-16"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-36-17" name="__codelineno-36-17" href="#__codelineno-36-17"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMax</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-36-18" name="__codelineno-36-18" href="#__codelineno-36-18"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-36-19" name="__codelineno-36-19" href="#__codelineno-36-19"></a><span class="w">        </span><span class="p">}</span>
@@ -5605,19 +5687,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* 0-1 knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* 0-1 knapsack: dynamic programming */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">knapsackDP</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="na">size</span>
-<a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Array</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">][</span><span class="n">c</span><span class="o">]</span>
 <a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">_val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-37-15" name="__codelineno-37-15" href="#__codelineno-37-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-37-16" name="__codelineno-37-16" href="#__codelineno-37-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5630,16 +5712,16 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 <div class="highlight"><span class="filename">knapsack.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### 0-1 knapsack: dynamic programming ###</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">knapsack_dp</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span>
 <a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="c1"># Initialize dp table</span>
+<a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="c1"># State transition</span>
+<a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="c1"># State transitions</span>
 <a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span>
-<a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">        </span><span class="c1"># If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">        </span><span class="c1"># If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">c</span><span class="o">]</span>
 <a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="w">      </span><span class="k">else</span>
-<a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">        </span><span class="c1"># The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">        </span><span class="c1"># Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]].</span><span class="n">max</span>
 <a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a><span class="w">      </span><span class="k">end</span>
 <a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w">    </span><span class="k">end</span>
@@ -5650,11 +5732,16 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </div>
 </div>
 </div>
-<p>As shown in Figure 14-20, both time complexity and space complexity are determined by the size of the array <code>dp</code>, which is <span class="arithmatex">\(O(n \times cap)\)</span>.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dp%28wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20cap%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20knapsack%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%20%28cap%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20State%20transitions%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20c%20in%20range%281,%20cap%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20item%20i%20exceeds%20the%20remaining%20capacity,%20do%20not%20choose%20it%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bc%5D%20%3D%20dp%5Bi%20-%201%5D%5Bc%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Take%20the%20larger%20value%20obtained%20by%20excluding%20and%20including%20item%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bc%5D%20%3D%20max%28dp%5Bi%20-%201%5D%5Bc%5D,%20dp%5Bi%20-%201%5D%5Bc%20-%20wgt%5Bi%20-%201%5D%5D%20%2B%20val%5Bi%20-%201%5D%29%0A%20%20%20%20return%20dp%5Bn%5D%5Bcap%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10,%2020,%2030,%2040,%2050%5D%0A%20%20%20%20val%20%3D%20%5B50,%20120,%20150,%20210,%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20Dynamic%20programming%0A%20%20%20%20res%20%3D%20knapsack_dp%28wgt,%20val,%20cap%29%0A%20%20%20%20print%28f%22Maximum%20total%20value%20within%20the%20knapsack%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dp%28wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20cap%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20knapsack%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%20%28cap%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20State%20transitions%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20c%20in%20range%281,%20cap%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20item%20i%20exceeds%20the%20remaining%20capacity,%20do%20not%20choose%20it%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bc%5D%20%3D%20dp%5Bi%20-%201%5D%5Bc%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Take%20the%20larger%20value%20obtained%20by%20excluding%20and%20including%20item%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bc%5D%20%3D%20max%28dp%5Bi%20-%201%5D%5Bc%5D,%20dp%5Bi%20-%201%5D%5Bc%20-%20wgt%5Bi%20-%201%5D%5D%20%2B%20val%5Bi%20-%201%5D%29%0A%20%20%20%20return%20dp%5Bn%5D%5Bcap%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10,%2020,%2030,%2040,%2050%5D%0A%20%20%20%20val%20%3D%20%5B50,%20120,%20150,%20210,%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20Dynamic%20programming%0A%20%20%20%20res%20%3D%20knapsack_dp%28wgt,%20val,%20cap%29%0A%20%20%20%20print%28f%22Maximum%20total%20value%20within%20the%20knapsack%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>As shown in Figure 14-20, both time and space complexity are determined by the size of <code>dp</code>: <span class="arithmatex">\(O(n \times cap)\)</span>.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:14"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><input id="__tabbed_4_14" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">&lt;1&gt;</label><label for="__tabbed_4_2">&lt;2&gt;</label><label for="__tabbed_4_3">&lt;3&gt;</label><label for="__tabbed_4_4">&lt;4&gt;</label><label for="__tabbed_4_5">&lt;5&gt;</label><label for="__tabbed_4_6">&lt;6&gt;</label><label for="__tabbed_4_7">&lt;7&gt;</label><label for="__tabbed_4_8">&lt;8&gt;</label><label for="__tabbed_4_9">&lt;9&gt;</label><label for="__tabbed_4_10">&lt;10&gt;</label><label for="__tabbed_4_11">&lt;11&gt;</label><label for="__tabbed_4_12">&lt;12&gt;</label><label for="__tabbed_4_13">&lt;13&gt;</label><label for="__tabbed_4_14">&lt;14&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Dynamic programming process for 0-1 knapsack problem" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step1.png" /></p>
+<p><img alt="Dynamic programming for 0-1 knapsack" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="knapsack_dp_step2" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_step2.png" /></p>
@@ -5697,20 +5784,20 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </div>
 </div>
 </div>
-<p align="center"> Figure 14-20 &nbsp; Dynamic programming process for 0-1 knapsack problem </p>
+<p align="center"> Figure 14-20 &nbsp; Dynamic programming for 0-1 knapsack </p>
 
-<h3 id="4-space-optimization">4. &nbsp; Space Optimization<a class="headerlink" href="#4-space-optimization" title="Permanent link">&para;</a></h3>
-<p>Since each state is only related to the state in the row above it, we can use two arrays rolling forward to reduce the space complexity from <span class="arithmatex">\(O(n \times cap)\)</span> to <span class="arithmatex">\(O(cap)\)</span>.</p>
-<p>Further thinking, can we achieve space optimization using just one array? Observing, we can see that each state is transferred from the cell directly above or the cell in the upper-left. If there is only one array, when we start traversing row <span class="arithmatex">\(i\)</span>, that array still stores the state of row <span class="arithmatex">\(i-1\)</span>.</p>
+<h3 id="4-space-optimization">4. &nbsp; Space optimization<a class="headerlink" href="#4-space-optimization" title="Permanent link">&para;</a></h3>
+<p>Each state depends only on the previous row, so two rolling arrays reduce space complexity from <span class="arithmatex">\(O(n \times cap)\)</span> to <span class="arithmatex">\(O(cap)\)</span>.</p>
+<p>Can we go further and use just one array? Each state depends on entries directly above or to the upper left. With one array, when we begin processing row <span class="arithmatex">\(i\)</span>, it still holds the states from row <span class="arithmatex">\(i-1\)</span>.</p>
 <ul>
-<li>If using forward traversal, then when traversing to <span class="arithmatex">\(dp[i, j]\)</span>, the values in the upper-left <span class="arithmatex">\(dp[i-1, 1]\)</span> ~ <span class="arithmatex">\(dp[i-1, j-1]\)</span> may have already been overwritten, thus preventing correct state transition.</li>
-<li>If using reverse traversal, there will be no overwriting issue, and state transition can proceed correctly.</li>
+<li>With forward iteration, by the time we reach <span class="arithmatex">\(dp[i, j]\)</span>, the upper-left values <span class="arithmatex">\(dp[i-1, 1]\)</span> ~ <span class="arithmatex">\(dp[i-1, j-1]\)</span> may have been overwritten, making the transition incorrect.</li>
+<li>With backward iteration, these values are not overwritten before they are needed, so transitions remain correct.</li>
 </ul>
-<p>Figure 14-21 shows the transition process from row <span class="arithmatex">\(i = 1\)</span> to row <span class="arithmatex">\(i = 2\)</span> using a single array. Please consider the difference between forward and reverse traversal.</p>
+<p>Figure 14-21 shows one array changing from row <span class="arithmatex">\(i = 1\)</span> to row <span class="arithmatex">\(i = 2\)</span>. Consider how forward and backward iteration differ.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:6"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">&lt;1&gt;</label><label for="__tabbed_5_2">&lt;2&gt;</label><label for="__tabbed_5_3">&lt;3&gt;</label><label for="__tabbed_5_4">&lt;4&gt;</label><label for="__tabbed_5_5">&lt;5&gt;</label><label for="__tabbed_5_6">&lt;6&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Space-optimized dynamic programming process for 0-1 knapsack" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_comp_step1.png" /></p>
+<p><img alt="Space-optimized dynamic programming for 0-1 knapsack" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_comp_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="knapsack_dp_comp_step2" class="animation-figure" src="../knapsack_problem.assets/knapsack_dp_comp_step2.png" /></p>
@@ -5729,42 +5816,42 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </div>
 </div>
 </div>
-<p align="center"> Figure 14-21 &nbsp; Space-optimized dynamic programming process for 0-1 knapsack </p>
+<p align="center"> Figure 14-21 &nbsp; Space-optimized dynamic programming for 0-1 knapsack </p>
 
-<p>In the code implementation, we simply need to delete the first dimension <span class="arithmatex">\(i\)</span> of the array <code>dp</code> and change the inner loop to reverse traversal:</p>
+<p>In the implementation, simply remove the first dimension <span class="arithmatex">\(i\)</span> from <code>dp</code> and reverse the inner loop:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="6:13"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">knapsack.py</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">knapsack_dp_comp</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">val</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">cap</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""0-1 knapsack: Space-optimized dynamic programming"""</span>
+<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""0-1 knapsack: space-optimized dynamic programming"""</span>
 <a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">wgt</span><span class="p">)</span>
-<a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="c1"># Initialize dp table</span>
+<a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">cap</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a>    <span class="c1"># State transition</span>
+<a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a>    <span class="c1"># State transitions</span>
 <a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-<a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a>        <span class="c1"># Traverse in reverse order</span>
+<a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a>        <span class="c1"># Traverse in decreasing order</span>
 <a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a>        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">cap</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a>            <span class="k">if</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">c</span><span class="p">:</span>
-<a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a>                <span class="c1"># If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a>                <span class="c1"># If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-39-12" name="__codelineno-39-12" href="#__codelineno-39-12"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span>
 <a id="__codelineno-39-13" name="__codelineno-39-13" href="#__codelineno-39-13"></a>            <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-39-14" name="__codelineno-39-14" href="#__codelineno-39-14"></a>                <span class="c1"># The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-39-14" name="__codelineno-39-14" href="#__codelineno-39-14"></a>                <span class="c1"># Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-39-15" name="__codelineno-39-15" href="#__codelineno-39-15"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">c</span> <span class="o">-</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]]</span> <span class="o">+</span> <span class="n">val</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
 <a id="__codelineno-39-16" name="__codelineno-39-16" href="#__codelineno-39-16"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">cap</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* 0-1 knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* 0-1 knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDPComp</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
-<a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="w">        </span><span class="c1">// Traverse in reverse order</span>
+<a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="w">        </span><span class="c1">// Traverse in decreasing order</span>
 <a id="__codelineno-40-9" name="__codelineno-40-9" href="#__codelineno-40-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-10" name="__codelineno-40-10" href="#__codelineno-40-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-11" name="__codelineno-40-11" href="#__codelineno-40-11"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-40-11" name="__codelineno-40-11" href="#__codelineno-40-11"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-40-12" name="__codelineno-40-12" href="#__codelineno-40-12"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-40-13" name="__codelineno-40-13" href="#__codelineno-40-13"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-40-14" name="__codelineno-40-14" href="#__codelineno-40-14"></a><span class="w">        </span><span class="p">}</span>
@@ -5774,17 +5861,17 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* 0-1 knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* 0-1 knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-41-2" name="__codelineno-41-2" href="#__codelineno-41-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDPComp</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
-<a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-8" name="__codelineno-41-8" href="#__codelineno-41-8"></a><span class="w">        </span><span class="c1">// Traverse in reverse order</span>
+<a id="__codelineno-41-8" name="__codelineno-41-8" href="#__codelineno-41-8"></a><span class="w">        </span><span class="c1">// Traverse in decreasing order</span>
 <a id="__codelineno-41-9" name="__codelineno-41-9" href="#__codelineno-41-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-10" name="__codelineno-41-10" href="#__codelineno-41-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-11" name="__codelineno-41-11" href="#__codelineno-41-11"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-41-11" name="__codelineno-41-11" href="#__codelineno-41-11"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-41-12" name="__codelineno-41-12" href="#__codelineno-41-12"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">max</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">);</span>
 <a id="__codelineno-41-13" name="__codelineno-41-13" href="#__codelineno-41-13"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-41-14" name="__codelineno-41-14" href="#__codelineno-41-14"></a><span class="w">        </span><span class="p">}</span>
@@ -5794,20 +5881,20 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* 0-1 knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* 0-1 knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-42-2" name="__codelineno-42-2" href="#__codelineno-42-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">KnapsackDPComp</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">weight</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">weight</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
-<a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-8" name="__codelineno-42-8" href="#__codelineno-42-8"></a><span class="w">        </span><span class="c1">// Traverse in reverse order</span>
+<a id="__codelineno-42-8" name="__codelineno-42-8" href="#__codelineno-42-8"></a><span class="w">        </span><span class="c1">// Traverse in decreasing order</span>
 <a id="__codelineno-42-9" name="__codelineno-42-9" href="#__codelineno-42-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-10" name="__codelineno-42-10" href="#__codelineno-42-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">weight</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-11" name="__codelineno-42-11" href="#__codelineno-42-11"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-42-11" name="__codelineno-42-11" href="#__codelineno-42-11"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-42-12" name="__codelineno-42-12" href="#__codelineno-42-12"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-42-13" name="__codelineno-42-13" href="#__codelineno-42-13"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-14" name="__codelineno-42-14" href="#__codelineno-42-14"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-42-14" name="__codelineno-42-14" href="#__codelineno-42-14"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-42-15" name="__codelineno-42-15" href="#__codelineno-42-15"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">weight</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-42-16" name="__codelineno-42-16" href="#__codelineno-42-16"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-42-17" name="__codelineno-42-17" href="#__codelineno-42-17"></a><span class="w">        </span><span class="p">}</span>
@@ -5817,17 +5904,17 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* 0-1 knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* 0-1 knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">knapsackDPComp</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">wgt</span><span class="p">)</span>
-<a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a><span class="w">        </span><span class="c1">// Traverse in reverse order</span>
+<a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a><span class="w">        </span><span class="c1">// Traverse in decreasing order</span>
 <a id="__codelineno-43-9" name="__codelineno-43-9" href="#__codelineno-43-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="o">--</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-43-12" name="__codelineno-43-12" href="#__codelineno-43-12"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Max</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">]),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="o">-</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span><span class="o">+</span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">])))</span>
 <a id="__codelineno-43-13" name="__codelineno-43-13" href="#__codelineno-43-13"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-43-14" name="__codelineno-43-14" href="#__codelineno-43-14"></a><span class="w">        </span><span class="p">}</span>
@@ -5837,17 +5924,17 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* 0-1 knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* 0-1 knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-44-2" name="__codelineno-44-2" href="#__codelineno-44-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">knapsackDPComp</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">cap</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="bp">count</span>
-<a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-44-7" name="__codelineno-44-7" href="#__codelineno-44-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="w">        </span><span class="c1">// Traverse in reverse order</span>
+<a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="w">        </span><span class="c1">// Traverse in decreasing order</span>
 <a id="__codelineno-44-9" name="__codelineno-44-9" href="#__codelineno-44-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">cap</span><span class="p">).</span><span class="n">reversed</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-10" name="__codelineno-44-10" href="#__codelineno-44-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-44-11" name="__codelineno-44-11" href="#__codelineno-44-11"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-44-11" name="__codelineno-44-11" href="#__codelineno-44-11"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-44-12" name="__codelineno-44-12" href="#__codelineno-44-12"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span>
 <a id="__codelineno-44-13" name="__codelineno-44-13" href="#__codelineno-44-13"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-44-14" name="__codelineno-44-14" href="#__codelineno-44-14"></a><span class="w">        </span><span class="p">}</span>
@@ -5857,17 +5944,17 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* 0-1 knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* 0-1 knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-45-2" name="__codelineno-45-2" href="#__codelineno-45-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">knapsackDPComp</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">wgt</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
-<a id="__codelineno-45-6" name="__codelineno-45-6" href="#__codelineno-45-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-45-6" name="__codelineno-45-6" href="#__codelineno-45-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-45-7" name="__codelineno-45-7" href="#__codelineno-45-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-8" name="__codelineno-45-8" href="#__codelineno-45-8"></a><span class="w">        </span><span class="c1">// Traverse in reverse order</span>
+<a id="__codelineno-45-8" name="__codelineno-45-8" href="#__codelineno-45-8"></a><span class="w">        </span><span class="c1">// Traverse in decreasing order</span>
 <a id="__codelineno-45-9" name="__codelineno-45-9" href="#__codelineno-45-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-10" name="__codelineno-45-10" href="#__codelineno-45-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-11" name="__codelineno-45-11" href="#__codelineno-45-11"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-45-11" name="__codelineno-45-11" href="#__codelineno-45-11"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-45-12" name="__codelineno-45-12" href="#__codelineno-45-12"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]);</span>
 <a id="__codelineno-45-13" name="__codelineno-45-13" href="#__codelineno-45-13"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-45-14" name="__codelineno-45-14" href="#__codelineno-45-14"></a><span class="w">        </span><span class="p">}</span>
@@ -5877,21 +5964,21 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* 0-1 knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* 0-1 knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-46-2" name="__codelineno-46-2" href="#__codelineno-46-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">knapsackDPComp</span><span class="p">(</span>
 <a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="nx">wgt</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">    </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">    </span><span class="nx">cap</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span>
 <a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">wgt</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
-<a id="__codelineno-46-10" name="__codelineno-46-10" href="#__codelineno-46-10"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-46-10" name="__codelineno-46-10" href="#__codelineno-46-10"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-46-11" name="__codelineno-46-11" href="#__codelineno-46-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w">        </span><span class="c1">// Traverse in reverse order</span>
+<a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w">        </span><span class="c1">// Traverse in decreasing order</span>
 <a id="__codelineno-46-13" name="__codelineno-46-13" href="#__codelineno-46-13"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-14" name="__codelineno-46-14" href="#__codelineno-46-14"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-15" name="__codelineno-46-15" href="#__codelineno-46-15"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-46-15" name="__codelineno-46-15" href="#__codelineno-46-15"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-46-16" name="__codelineno-46-16" href="#__codelineno-46-16"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]);</span>
 <a id="__codelineno-46-17" name="__codelineno-46-17" href="#__codelineno-46-17"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-46-18" name="__codelineno-46-18" href="#__codelineno-46-18"></a><span class="w">        </span><span class="p">}</span>
@@ -5901,17 +5988,17 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* 0-1 knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* 0-1 knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-47-2" name="__codelineno-47-2" href="#__codelineno-47-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">knapsackDPComp</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
-<a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
-<a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a><span class="w">  </span><span class="c1">// State transition</span>
+<a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a><span class="w">  </span><span class="c1">// State transitions</span>
 <a id="__codelineno-47-7" name="__codelineno-47-7" href="#__codelineno-47-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a><span class="w">    </span><span class="c1">// Traverse in reverse order</span>
+<a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a><span class="w">    </span><span class="c1">// Traverse in decreasing order</span>
 <a id="__codelineno-47-9" name="__codelineno-47-9" href="#__codelineno-47-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-10" name="__codelineno-47-10" href="#__codelineno-47-10"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-11" name="__codelineno-47-11" href="#__codelineno-47-11"></a><span class="w">        </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-47-11" name="__codelineno-47-11" href="#__codelineno-47-11"></a><span class="w">        </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-47-12" name="__codelineno-47-12" href="#__codelineno-47-12"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]);</span>
 <a id="__codelineno-47-13" name="__codelineno-47-13" href="#__codelineno-47-13"></a><span class="w">      </span><span class="p">}</span>
 <a id="__codelineno-47-14" name="__codelineno-47-14" href="#__codelineno-47-14"></a><span class="w">    </span><span class="p">}</span>
@@ -5921,17 +6008,17 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* 0-1 knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* 0-1 knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-48-2" name="__codelineno-48-2" href="#__codelineno-48-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">knapsack_dp_comp</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">cap</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
-<a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-8" name="__codelineno-48-8" href="#__codelineno-48-8"></a><span class="w">        </span><span class="c1">// Traverse in reverse order</span>
+<a id="__codelineno-48-8" name="__codelineno-48-8" href="#__codelineno-48-8"></a><span class="w">        </span><span class="c1">// Traverse in decreasing order</span>
 <a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="o">..=</span><span class="n">cap</span><span class="p">).</span><span class="n">rev</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-11" name="__codelineno-48-11" href="#__codelineno-48-11"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-48-11" name="__codelineno-48-11" href="#__codelineno-48-11"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-48-12" name="__codelineno-48-12" href="#__codelineno-48-12"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-48-13" name="__codelineno-48-13" href="#__codelineno-48-13"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-48-14" name="__codelineno-48-14" href="#__codelineno-48-14"></a><span class="w">        </span><span class="p">}</span>
@@ -5941,17 +6028,17 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* 0-1 knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* 0-1 knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">knapsackDPComp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">wgt</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">wgtSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgtSize</span><span class="p">;</span>
-<a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
-<a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w">        </span><span class="c1">// Traverse in reverse order</span>
+<a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w">        </span><span class="c1">// Traverse in decreasing order</span>
 <a id="__codelineno-49-9" name="__codelineno-49-9" href="#__codelineno-49-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-10" name="__codelineno-49-10" href="#__codelineno-49-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-11" name="__codelineno-49-11" href="#__codelineno-49-11"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-49-11" name="__codelineno-49-11" href="#__codelineno-49-11"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-49-12" name="__codelineno-49-12" href="#__codelineno-49-12"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMax</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-49-13" name="__codelineno-49-13" href="#__codelineno-49-13"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-49-14" name="__codelineno-49-14" href="#__codelineno-49-14"></a><span class="w">        </span><span class="p">}</span>
@@ -5964,17 +6051,17 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* 0-1 knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">knapsack.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* 0-1 knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-50-2" name="__codelineno-50-2" href="#__codelineno-50-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">knapsackDPComp</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="na">size</span>
-<a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
-<a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-8" name="__codelineno-50-8" href="#__codelineno-50-8"></a><span class="w">        </span><span class="c1">// Traverse in reverse order</span>
+<a id="__codelineno-50-8" name="__codelineno-50-8" href="#__codelineno-50-8"></a><span class="w">        </span><span class="c1">// Traverse in decreasing order</span>
 <a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="n">downTo</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-10" name="__codelineno-50-10" href="#__codelineno-50-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-11" name="__codelineno-50-11" href="#__codelineno-50-11"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-50-11" name="__codelineno-50-11" href="#__codelineno-50-11"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-50-12" name="__codelineno-50-12" href="#__codelineno-50-12"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">_val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-50-13" name="__codelineno-50-13" href="#__codelineno-50-13"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-50-14" name="__codelineno-50-14" href="#__codelineno-50-14"></a><span class="w">        </span><span class="p">}</span>
@@ -5984,20 +6071,20 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">knapsack.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### 0-1 knapsack: space-optimized DP ###</span>
+<div class="highlight"><span class="filename">knapsack.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### 0-1 knapsack: space-optimized dynamic programming ###</span>
 <a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">knapsack_dp_comp</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span>
 <a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="w">  </span><span class="c1"># Initialize dp table</span>
+<a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="w">  </span><span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">  </span><span class="c1"># State transition</span>
+<a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">  </span><span class="c1"># State transitions</span>
 <a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="w">    </span><span class="c1"># Traverse in reverse order</span>
+<a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="w">    </span><span class="c1"># Traverse in decreasing order</span>
 <a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">cap</span><span class="o">.</span><span class="n">downto</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span>
-<a id="__codelineno-51-11" name="__codelineno-51-11" href="#__codelineno-51-11"></a><span class="w">        </span><span class="c1"># If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-51-11" name="__codelineno-51-11" href="#__codelineno-51-11"></a><span class="w">        </span><span class="c1"># If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-51-12" name="__codelineno-51-12" href="#__codelineno-51-12"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span>
 <a id="__codelineno-51-13" name="__codelineno-51-13" href="#__codelineno-51-13"></a><span class="w">      </span><span class="k">else</span>
-<a id="__codelineno-51-14" name="__codelineno-51-14" href="#__codelineno-51-14"></a><span class="w">        </span><span class="c1"># The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-51-14" name="__codelineno-51-14" href="#__codelineno-51-14"></a><span class="w">        </span><span class="c1"># Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-51-15" name="__codelineno-51-15" href="#__codelineno-51-15"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]].</span><span class="n">max</span>
 <a id="__codelineno-51-16" name="__codelineno-51-16" href="#__codelineno-51-16"></a><span class="w">      </span><span class="k">end</span>
 <a id="__codelineno-51-17" name="__codelineno-51-17" href="#__codelineno-51-17"></a><span class="w">    </span><span class="k">end</span>
@@ -6008,6 +6095,11 @@ dp[i, c] = \max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1])
 </div>
 </div>
 </div>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dp_comp%28wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20cap%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20knapsack%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B0%5D%20%2A%20%28cap%20%2B%201%29%0A%20%20%20%20%23%20State%20transitions%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%23%20Traverse%20in%20decreasing%20order%0A%20%20%20%20%20%20%20%20for%20c%20in%20range%28cap,%200,%20-1%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20item%20i%20exceeds%20the%20remaining%20capacity,%20do%20not%20choose%20it%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bc%5D%20%3D%20dp%5Bc%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Take%20the%20larger%20value%20obtained%20by%20excluding%20and%20including%20item%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bc%5D%20%3D%20max%28dp%5Bc%5D,%20dp%5Bc%20-%20wgt%5Bi%20-%201%5D%5D%20%2B%20val%5Bi%20-%201%5D%29%0A%20%20%20%20return%20dp%5Bcap%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10,%2020,%2030,%2040,%2050%5D%0A%20%20%20%20val%20%3D%20%5B50,%20120,%20150,%20210,%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20Space-optimized%20dynamic%20programming%0A%20%20%20%20res%20%3D%20knapsack_dp_comp%28wgt,%20val,%20cap%29%0A%20%20%20%20print%28f%22Maximum%20total%20value%20within%20the%20knapsack%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20knapsack_dp_comp%28wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20cap%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%220-1%20knapsack%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B0%5D%20%2A%20%28cap%20%2B%201%29%0A%20%20%20%20%23%20State%20transitions%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%23%20Traverse%20in%20decreasing%20order%0A%20%20%20%20%20%20%20%20for%20c%20in%20range%28cap,%200,%20-1%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20item%20i%20exceeds%20the%20remaining%20capacity,%20do%20not%20choose%20it%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bc%5D%20%3D%20dp%5Bc%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Take%20the%20larger%20value%20obtained%20by%20excluding%20and%20including%20item%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bc%5D%20%3D%20max%28dp%5Bc%5D,%20dp%5Bc%20-%20wgt%5Bi%20-%201%5D%5D%20%2B%20val%5Bi%20-%201%5D%29%0A%20%20%20%20return%20dp%5Bcap%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10,%2020,%2030,%2040,%2050%5D%0A%20%20%20%20val%20%3D%20%5B50,%20120,%20150,%20210,%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20Space-optimized%20dynamic%20programming%0A%20%20%20%20res%20%3D%20knapsack_dp_comp%28wgt,%20val,%20cap%29%0A%20%20%20%20print%28f%22Maximum%20total%20value%20within%20the%20knapsack%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
 
 <!-- Source file information -->
 
@@ -6030,7 +6122,7 @@ aria-label="Footer"
   <a
     href="../dp_solution_pipeline/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 14.3 Dynamic Programming Problem-Solving Approach"
+    aria-label="Previous: 14.3   Solving dynamic programming problems"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -6042,7 +6134,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        14.3 Dynamic Programming Problem-Solving Approach
+        14.3 &nbsp; Solving dynamic programming problems
       </div>
     </div>
   </a>
@@ -6054,7 +6146,7 @@ aria-label="Footer"
   <a
     href="../unbounded_knapsack_problem/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 14.5 Unbounded Knapsack Problem"
+    aria-label="Next: 14.5   Unbounded knapsack problem"
     rel="next"
   >
     <div class="md-footer__title">
@@ -6062,7 +6154,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        14.5 Unbounded Knapsack Problem
+        14.5 &nbsp; Unbounded knapsack problem
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -6079,7 +6171,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_dynamic_programming/summary/index.html b/en/chapter_dynamic_programming/summary/index.html
index a053786e4..d75eedbb2 100644
--- a/en/chapter_dynamic_programming/summary/index.html
+++ b/en/chapter_dynamic_programming/summary/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
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@@ -167,9 +167,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
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@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
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-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
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-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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           <ul class="md-nav__list" data-md-scrollfix>
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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@@ -2287,7 +2287,7 @@
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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@@ -2378,7 +2378,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
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@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3283,7 +3283,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3304,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3352,7 +3352,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3379,7 +3379,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3406,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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@@ -3433,7 +3433,7 @@
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3526,7 +3526,7 @@
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3568,7 +3568,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3775,7 +3775,7 @@
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3817,7 +3817,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-key-points" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Key Points
+        1. &nbsp; Key takeaways
       </span>
     </span>
   </a>
@@ -4064,7 +4064,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4130,7 +4130,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4172,7 +4172,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4253,7 +4253,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4280,7 +4280,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4367,7 +4367,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4409,7 +4409,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4436,7 +4436,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4463,7 +4463,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4547,6 +4547,78 @@
 
 
 
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+        1. &nbsp; Key takeaways
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 <!-- Page content -->
 <h1 id="147-summary">14.7 &nbsp; Summary<a class="headerlink" href="#147-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-points">1. &nbsp; Key Points<a class="headerlink" href="#1-key-points" title="Permanent link">&para;</a></h3>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
 <ul>
-<li>Dynamic programming decomposes problems and avoids redundant computation by storing the solutions to subproblems, thereby significantly improving computational efficiency.</li>
-<li>Without considering time constraints, all dynamic programming problems can be solved using backtracking (brute force search), but the recursion tree contains a large number of overlapping subproblems, resulting in extremely low efficiency. By introducing a memo list, we can store the solutions to all computed subproblems, ensuring that overlapping subproblems are only computed once.</li>
-<li>Memoization is a top-down recursive solution, while the corresponding dynamic programming is a bottom-up iterative solution, similar to "filling in a table". Since the current state only depends on certain local states, we can eliminate one dimension of the <span class="arithmatex">\(dp\)</span> table to reduce space complexity.</li>
-<li>Subproblem decomposition is a general algorithmic approach, with different properties in divide and conquer, dynamic programming, and backtracking.</li>
-<li>Dynamic programming problems have three major characteristics: overlapping subproblems, optimal substructure, and no aftereffects.</li>
-<li>If the optimal solution to the original problem can be constructed from the optimal solutions to the subproblems, then it has optimal substructure.</li>
-<li>No aftereffects means that for a given state, its future development is only related to that state and has nothing to do with all past states. Many combinatorial optimization problems do not satisfy this property and cannot be solved efficiently using dynamic programming.</li>
+<li>Dynamic programming decomposes problems and stores subproblem solutions to avoid repeated computation and improve efficiency.</li>
+<li>Without a time limit, every dynamic programming problem can be solved by backtracking (brute-force search). However, the recursion tree contains many overlapping subproblems, making it very inefficient. Memoization stores computed solutions so that each overlapping subproblem is solved only once.</li>
+<li>Memoized search is a top-down recursive approach, while the corresponding dynamic programming solution works bottom up, like filling a table. Since the current state depends only on certain nearby states, removing a dimension from the <span class="arithmatex">\(dp\)</span> table can reduce space complexity.</li>
+<li>Decomposing a problem into subproblems is a general algorithmic idea with different characteristics in divide and conquer, dynamic programming, and backtracking.</li>
+<li>Dynamic programming problems have three key properties: overlapping subproblems, optimal substructure, and the no-aftereffect property.</li>
+<li>A problem has optimal substructure if its optimal solution can be constructed from optimal solutions to subproblems.</li>
+<li>The no-aftereffect property means a state's future depends only on that state, not on the history leading to it. Many combinatorial optimization problems lack this property and cannot be solved quickly with dynamic programming.</li>
 </ul>
-<p><strong>Knapsack problem</strong></p>
+<p><strong>Knapsack problems</strong></p>
 <ul>
-<li>The knapsack problem is one of the most typical dynamic programming problems, with variants such as the 0-1 knapsack, unbounded knapsack, and multiple knapsack.</li>
-<li>The state definition for the 0-1 knapsack is the maximum value achievable using the first <span class="arithmatex">\(i\)</span> items with a knapsack capacity of <span class="arithmatex">\(c\)</span>. Based on the two decisions of not putting an item in the knapsack and putting it in, the optimal substructure can be identified and the state transition equation constructed. In space optimization, since each state depends on the state directly above and to the upper-left, the list needs to be traversed in reverse order to avoid overwriting the upper-left state.</li>
-<li>The unbounded knapsack problem has no limit on the selection quantity of each type of item, so the state transition for choosing to put in an item differs from the 0-1 knapsack problem. Since the state depends on the state directly above and directly to the left, space optimization should use forward traversal.</li>
-<li>The coin change problem is a variant of the unbounded knapsack problem. It changes from seeking the "maximum" value to seeking the "minimum" number of coins, so <span class="arithmatex">\(\max()\)</span> in the state transition equation should be changed to <span class="arithmatex">\(\min()\)</span>. It changes from seeking "not exceeding" the knapsack capacity to seeking "exactly" making up the target amount, so <span class="arithmatex">\(amt + 1\)</span> is used to represent the invalid solution of "unable to make up the target amount".</li>
-<li>Coin change problem II changes from seeking the "minimum number of coins" to seeking the "number of coin combinations", so the state transition equation correspondingly changes from <span class="arithmatex">\(\min()\)</span> to a summation operator.</li>
+<li>Knapsack is one of the most typical dynamic programming problem families, with variants including 0-1, unbounded, and bounded knapsack.</li>
+<li>In 0-1 knapsack, the state value is the maximum value obtainable from the first <span class="arithmatex">\(i\)</span> items within capacity <span class="arithmatex">\(c\)</span>. The choices to exclude or include an item reveal optimal substructure and give the state transition equation. After space optimization, backward iteration is necessary because each state depends on entries above and to the upper left, which must not be overwritten prematurely.</li>
+<li>Unbounded knapsack allows unlimited copies of each item, changing the transition for including an item. Because a state depends on entries directly above and to the left, space optimization requires forward iteration.</li>
+<li>Coin change is a variant of unbounded knapsack. Minimizing the number of coins instead of maximizing value changes <span class="arithmatex">\(\max()\)</span> to <span class="arithmatex">\(\min()\)</span> in the transition. Requiring the target amount exactly, rather than staying within a capacity, calls for using <span class="arithmatex">\(amt + 1\)</span> to represent an infeasible solution.</li>
+<li>Coin change II counts combinations instead of minimizing coin count, so the transition replaces <span class="arithmatex">\(\min()\)</span> with addition.</li>
 </ul>
-<p><strong>Edit distance problem</strong></p>
+<p><strong>Edit distance</strong></p>
 <ul>
-<li>Edit distance (Levenshtein distance) is used to measure the similarity between two strings, defined as the minimum number of edit steps from one string to another, with edit operations including insert, delete, and replace.</li>
-<li>The state definition for the edit distance problem is the minimum number of edit steps required to change the first <span class="arithmatex">\(i\)</span> characters of <span class="arithmatex">\(s\)</span> into the first <span class="arithmatex">\(j\)</span> characters of <span class="arithmatex">\(t\)</span>. When <span class="arithmatex">\(s[i-1] \ne t[j-1]\)</span>, there are three decisions: insert, delete, replace, each with corresponding remaining subproblems. From this, the optimal substructure can be identified and the state transition equation constructed. When <span class="arithmatex">\(s[i-1] = t[j-1]\)</span>, no edit is required for the current character.</li>
-<li>In edit distance, the state depends on the state directly above, directly to the left, and to the upper-left, so after space optimization, neither forward nor reverse traversal can correctly perform state transitions. For this reason, we use a variable to temporarily store the upper-left state, thus transforming to a situation equivalent to the unbounded knapsack problem, allowing for forward traversal after space optimization.</li>
+<li>Edit distance (Levenshtein distance) measures similarity between strings. It is the minimum number of edits needed to turn one string into another, allowing insertion, deletion, and replacement.</li>
+<li>The edit distance state is the minimum number of edits needed to transform the first <span class="arithmatex">\(i\)</span> characters of <span class="arithmatex">\(s\)</span> into the first <span class="arithmatex">\(j\)</span> characters of <span class="arithmatex">\(t\)</span>. When <span class="arithmatex">\(s[i-1] \ne t[j-1]\)</span>, insertion, deletion, and replacement each leave a corresponding subproblem, revealing optimal substructure and the state transition equation. When <span class="arithmatex">\(s[i-1] = t[j-1]\)</span>, no edit is needed for those characters.</li>
+<li>Edit distance depends on states above, to the left, and to the upper left, so neither forward nor backward iteration alone works after space optimization. A temporary variable preserves the upper-left state, reducing the situation to one like unbounded knapsack and allowing forward iteration.</li>
 </ul>
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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@@ -3406,7 +3406,7 @@
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3526,7 +3526,7 @@
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3568,7 +3568,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3622,7 +3622,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3649,7 +3649,7 @@
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3676,7 +3676,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3775,7 +3775,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3817,7 +3817,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -3951,7 +3951,7 @@
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1451-unbounded-knapsack-problem" class="md-nav__link">
+  <a href="#1451-unbounded-knapsack" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        14.5.1 &nbsp; Unbounded Knapsack Problem
+        14.5.1 &nbsp; Unbounded knapsack
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="14.5.1 Unbounded Knapsack Problem">
+    <nav class="md-nav" aria-label="14.5.1 Unbounded knapsack">
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           <li class="md-nav__item">
   <a href="#1-dynamic-programming-approach" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Dynamic Programming Approach
+        1. &nbsp; Dynamic programming approach
       </span>
     </span>
   </a>
@@ -4003,7 +4003,7 @@
   <a href="#2-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Code Implementation
+        2. &nbsp; Code implementation
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   <a href="#3-space-optimization" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Space Optimization
+        3. &nbsp; Space optimization
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   <a href="#1452-coin-change-problem" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        14.5.2 &nbsp; Coin Change Problem
+        14.5.2 &nbsp; Coin change problem
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="14.5.2 Coin Change Problem">
+    <nav class="md-nav" aria-label="14.5.2 Coin change problem">
       <ul class="md-nav__list">
 
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   <a href="#1-dynamic-programming-approach_1" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Dynamic Programming Approach
+        1. &nbsp; Dynamic programming approach
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     <span class="md-ellipsis">
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+        2. &nbsp; Code implementation
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     <span class="md-ellipsis">
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-        3. &nbsp; Space Optimization
+        3. &nbsp; Space optimization
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   <a href="#1453-coin-change-problem-ii" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        14.5.3 &nbsp; Coin Change Problem II
+        14.5.3 &nbsp; Coin change problem II
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="14.5.3 Coin Change Problem II">
+    <nav class="md-nav" aria-label="14.5.3 Coin change problem II">
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     <span class="md-ellipsis">
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-        1. &nbsp; Dynamic Programming Approach
+        1. &nbsp; Dynamic programming approach
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     <span class="md-ellipsis">
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+        3. &nbsp; Space optimization
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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@@ -4311,7 +4311,7 @@
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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@@ -4686,6 +4686,78 @@
 
 
 
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+  <a href="#1451-unbounded-knapsack" class="md-nav__link">
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       <span class="md-typeset">
-        14.5.1 &nbsp; Unbounded Knapsack Problem
+        14.5.1 &nbsp; Unbounded knapsack
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="14.5.1 Unbounded Knapsack Problem">
+    <nav class="md-nav" aria-label="14.5.1 Unbounded knapsack">
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-        1. &nbsp; Dynamic Programming Approach
+        1. &nbsp; Dynamic programming approach
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   <a href="#2-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Code Implementation
+        2. &nbsp; Code implementation
       </span>
     </span>
   </a>
@@ -4764,7 +4836,7 @@
   <a href="#3-space-optimization" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Space Optimization
+        3. &nbsp; Space optimization
       </span>
     </span>
   </a>
@@ -4780,19 +4852,19 @@
   <a href="#1452-coin-change-problem" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        14.5.2 &nbsp; Coin Change Problem
+        14.5.2 &nbsp; Coin change problem
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="14.5.2 Coin Change Problem">
+    <nav class="md-nav" aria-label="14.5.2 Coin change problem">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-dynamic-programming-approach_1" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Dynamic Programming Approach
+        1. &nbsp; Dynamic programming approach
       </span>
     </span>
   </a>
@@ -4803,7 +4875,7 @@
   <a href="#2-code-implementation_1" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Code Implementation
+        2. &nbsp; Code implementation
       </span>
     </span>
   </a>
@@ -4814,7 +4886,7 @@
   <a href="#3-space-optimization_1" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Space Optimization
+        3. &nbsp; Space optimization
       </span>
     </span>
   </a>
@@ -4830,19 +4902,19 @@
   <a href="#1453-coin-change-problem-ii" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        14.5.3 &nbsp; Coin Change Problem II
+        14.5.3 &nbsp; Coin change problem II
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="14.5.3 Coin Change Problem II">
+    <nav class="md-nav" aria-label="14.5.3 Coin change problem II">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-dynamic-programming-approach_2" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Dynamic Programming Approach
+        1. &nbsp; Dynamic programming approach
       </span>
     </span>
   </a>
@@ -4853,7 +4925,7 @@
   <a href="#2-code-implementation_2" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Code Implementation
+        2. &nbsp; Code implementation
       </span>
     </span>
   </a>
@@ -4864,7 +4936,7 @@
   <a href="#3-space-optimization_2" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Space Optimization
+        3. &nbsp; Space optimization
       </span>
     </span>
   </a>
@@ -4913,67 +4985,67 @@
 
 
 <!-- Page content -->
-<h1 id="145-unbounded-knapsack-problem">14.5 &nbsp; Unbounded Knapsack Problem<a class="headerlink" href="#145-unbounded-knapsack-problem" title="Permanent link">&para;</a></h1>
-<p>In this section, we first solve another common knapsack problem: the unbounded knapsack, and then explore a special case of it: the coin change problem.</p>
-<h2 id="1451-unbounded-knapsack-problem">14.5.1 &nbsp; Unbounded Knapsack Problem<a class="headerlink" href="#1451-unbounded-knapsack-problem" title="Permanent link">&para;</a></h2>
+<h1 id="145-unbounded-knapsack-problem">14.5 &nbsp; Unbounded knapsack problem<a class="headerlink" href="#145-unbounded-knapsack-problem" title="Permanent link">&para;</a></h1>
+<p>We will now solve another common knapsack variant, unbounded knapsack, and then examine a special case: coin change.</p>
+<h2 id="1451-unbounded-knapsack">14.5.1 &nbsp; Unbounded knapsack<a class="headerlink" href="#1451-unbounded-knapsack" title="Permanent link">&para;</a></h2>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given <span class="arithmatex">\(n\)</span> items, where the weight of the <span class="arithmatex">\(i\)</span>-th item is <span class="arithmatex">\(wgt[i-1]\)</span> and its value is <span class="arithmatex">\(val[i-1]\)</span>, and a knapsack with capacity <span class="arithmatex">\(cap\)</span>. <strong>Each item can be selected multiple times</strong>. What is the maximum value that can be placed in the knapsack within the capacity limit? An example is shown in Figure 14-22.</p>
+<p>Given <span class="arithmatex">\(n\)</span> items, where item <span class="arithmatex">\(i\)</span> has weight <span class="arithmatex">\(wgt[i-1]\)</span> and value <span class="arithmatex">\(val[i-1]\)</span>, and a knapsack of capacity <span class="arithmatex">\(cap\)</span>, <strong>each item may be chosen repeatedly</strong>. What is the maximum value that fits within the capacity? An example is shown in Figure 14-22.</p>
 </div>
-<p><img alt="Example data for unbounded knapsack problem" class="animation-figure" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_example.png" /></p>
-<p align="center"> Figure 14-22 &nbsp; Example data for unbounded knapsack problem </p>
+<p><img alt="Example input for unbounded knapsack" class="animation-figure" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_example.png" /></p>
+<p align="center"> Figure 14-22 &nbsp; Example input for unbounded knapsack </p>
 
-<h3 id="1-dynamic-programming-approach">1. &nbsp; Dynamic Programming Approach<a class="headerlink" href="#1-dynamic-programming-approach" title="Permanent link">&para;</a></h3>
-<p>The unbounded knapsack problem is very similar to the 0-1 knapsack problem, <strong>differing only in that there is no limit on the number of times an item can be selected</strong>.</p>
+<h3 id="1-dynamic-programming-approach">1. &nbsp; Dynamic programming approach<a class="headerlink" href="#1-dynamic-programming-approach" title="Permanent link">&para;</a></h3>
+<p>Unbounded knapsack is very similar to 0-1 knapsack. <strong>The only difference is that there is no limit on how many times an item may be chosen</strong>.</p>
 <ul>
-<li>In the 0-1 knapsack problem, there is only one of each type of item, so after placing item <span class="arithmatex">\(i\)</span> in the knapsack, we can only choose from the first <span class="arithmatex">\(i-1\)</span> items.</li>
-<li>In the unbounded knapsack problem, the quantity of each type of item is unlimited, so after placing item <span class="arithmatex">\(i\)</span> in the knapsack, <strong>we can still choose from the first <span class="arithmatex">\(i\)</span> items</strong>.</li>
+<li>In 0-1 knapsack, there is only one of each item, so after including item <span class="arithmatex">\(i\)</span>, we can choose only from the first <span class="arithmatex">\(i-1\)</span> items.</li>
+<li>In unbounded knapsack, each item has an unlimited supply, so after including item <span class="arithmatex">\(i\)</span>, <strong>we can still choose from the first <span class="arithmatex">\(i\)</span> items</strong>.</li>
 </ul>
-<p>Under the rules of the unbounded knapsack problem, the changes in state <span class="arithmatex">\([i, c]\)</span> are divided into two cases.</p>
+<p>For unbounded knapsack, state <span class="arithmatex">\([i, c]\)</span> has two cases.</p>
 <ul>
-<li><strong>Not putting item <span class="arithmatex">\(i\)</span></strong>: Same as the 0-1 knapsack problem, transfer to <span class="arithmatex">\([i-1, c]\)</span>.</li>
-<li><strong>Putting item <span class="arithmatex">\(i\)</span></strong>: Different from the 0-1 knapsack problem, transfer to <span class="arithmatex">\([i, c-wgt[i-1]]\)</span>.</li>
+<li><strong>Leave item <span class="arithmatex">\(i\)</span> out</strong>: As in 0-1 knapsack, the remaining state is <span class="arithmatex">\([i-1, c]\)</span>.</li>
+<li><strong>Put item <span class="arithmatex">\(i\)</span> in</strong>: Unlike 0-1 knapsack, the remaining state is <span class="arithmatex">\([i, c-wgt[i-1]]\)</span>.</li>
 </ul>
-<p>Thus, the state transition equation becomes:</p>
+<p>The state transition equation becomes:</p>
 <div class="arithmatex">\[
 dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 \]</div>
-<h3 id="2-code-implementation">2. &nbsp; Code Implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">&para;</a></h3>
-<p>Comparing the code for the two problems, there is one change in state transition from <span class="arithmatex">\(i-1\)</span> to <span class="arithmatex">\(i\)</span>, with everything else identical:</p>
+<h3 id="2-code-implementation">2. &nbsp; Code implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">&para;</a></h3>
+<p>Compared with 0-1 knapsack, only one <span class="arithmatex">\(i-1\)</span> in the transition changes to <span class="arithmatex">\(i\)</span>; the rest of the code is identical:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">unbounded_knapsack.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">unbounded_knapsack_dp</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">val</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">cap</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Unbounded knapsack: Dynamic programming"""</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Unbounded knapsack: dynamic programming"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">wgt</span><span class="p">)</span>
-<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="c1"># Initialize dp table</span>
+<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">cap</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># State transition</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># State transitions</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">cap</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>            <span class="k">if</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">c</span><span class="p">:</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>                <span class="c1"># If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>                <span class="c1"># If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">]</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>            <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>                <span class="c1"># The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>                <span class="c1"># Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span> <span class="o">-</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]]</span> <span class="o">+</span> <span class="n">val</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">cap</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Unbounded knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Unbounded knapsack: dynamic programming */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">unboundedKnapsackDP</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">));</span>
-<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">        </span><span class="p">}</span>
@@ -4983,19 +5055,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Unbounded knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Unbounded knapsack: dynamic programming */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">unboundedKnapsackDP</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
-<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">max</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">);</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5005,19 +5077,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Unbounded knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Unbounded knapsack: dynamic programming */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">UnboundedKnapsackDP</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
-<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="kt">int</span><span class="p">[,]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5027,22 +5099,22 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Unbounded knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Unbounded knapsack: dynamic programming */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">unboundedKnapsackDP</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">wgt</span><span class="p">)</span>
-<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Max</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">c</span><span class="p">]),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="o">-</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span><span class="o">+</span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">])))</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">        </span><span class="p">}</span>
@@ -5052,19 +5124,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Unbounded knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Unbounded knapsack: dynamic programming */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">unboundedKnapsackDP</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">cap</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="bp">count</span>
-<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">),</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">]</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5074,21 +5146,21 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Unbounded knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Unbounded knapsack: dynamic programming */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">unboundedKnapsackDP</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">wgt</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">        </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="nx">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="p">);</span>
-<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">c</span><span class="p">];</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">                    </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">c</span><span class="p">],</span>
 <a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">                    </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span>
@@ -5101,25 +5173,25 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Unbounded knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Unbounded knapsack: dynamic programming */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">unboundedKnapsackDP</span><span class="p">(</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="nx">wgt</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="nx">cap</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">wgt</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="p">);</span>
-<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">c</span><span class="p">];</span>
 <a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span>
 <a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">                    </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">c</span><span class="p">],</span>
 <a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">                    </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span>
@@ -5132,19 +5204,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Unbounded knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Unbounded knapsack: dynamic programming */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">unboundedKnapsackDP</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
-<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">));</span>
-<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="c1">// State transition</span>
+<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="c1">// State transitions</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">        </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">        </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">      </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">        </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">        </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]);</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">      </span><span class="p">}</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">    </span><span class="p">}</span>
@@ -5154,19 +5226,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Unbounded knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Unbounded knapsack: dynamic programming */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">unbounded_knapsack_dp</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">cap</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
-<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">cap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5176,22 +5248,22 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Unbounded knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Unbounded knapsack: dynamic programming */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">unboundedKnapsackDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">wgt</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">wgtSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgtSize</span><span class="p">;</span>
-<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">**</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">((</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">));</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMax</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">        </span><span class="p">}</span>
@@ -5206,19 +5278,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Unbounded knapsack: Dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Unbounded knapsack: dynamic programming */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">unboundedKnapsackDP</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="na">size</span>
-<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Array</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">][</span><span class="n">c</span><span class="o">]</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">_val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5231,16 +5303,16 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 <div class="highlight"><span class="filename">unbounded_knapsack.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Unbounded knapsack: dynamic programming ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">unbounded_knapsack_dp</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="c1"># Initialize dp table</span>
+<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="c1"># State transition</span>
+<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="c1"># State transitions</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span>
-<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">        </span><span class="c1"># If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">        </span><span class="c1"># If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">c</span><span class="o">]</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">      </span><span class="k">else</span>
-<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">        </span><span class="c1"># The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">        </span><span class="c1"># Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]].</span><span class="n">max</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">      </span><span class="k">end</span>
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">    </span><span class="k">end</span>
@@ -5251,13 +5323,18 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </div>
 </div>
 </div>
-<h3 id="3-space-optimization">3. &nbsp; Space Optimization<a class="headerlink" href="#3-space-optimization" title="Permanent link">&para;</a></h3>
-<p>Since the current state is transferred from states on the left and above, <strong>after space optimization, each row in the <span class="arithmatex">\(dp\)</span> table should be traversed in forward order</strong>.</p>
-<p>This traversal order is exactly opposite to the 0-1 knapsack. Please refer to Figure 14-23 to understand the difference between the two.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20unbounded_knapsack_dp%28wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20cap%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Unbounded%20knapsack%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%20%28cap%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20State%20transitions%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20c%20in%20range%281,%20cap%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20item%20i%20exceeds%20the%20remaining%20capacity,%20do%20not%20choose%20it%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bc%5D%20%3D%20dp%5Bi%20-%201%5D%5Bc%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Take%20the%20larger%20value%20obtained%20by%20excluding%20and%20including%20item%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bc%5D%20%3D%20max%28dp%5Bi%20-%201%5D%5Bc%5D,%20dp%5Bi%5D%5Bc%20-%20wgt%5Bi%20-%201%5D%5D%20%2B%20val%5Bi%20-%201%5D%29%0A%20%20%20%20return%20dp%5Bn%5D%5Bcap%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B1,%202,%203%5D%0A%20%20%20%20val%20%3D%20%5B5,%2011,%2015%5D%0A%20%20%20%20cap%20%3D%204%0A%0A%20%20%20%20%23%20Dynamic%20programming%0A%20%20%20%20res%20%3D%20unbounded_knapsack_dp%28wgt,%20val,%20cap%29%0A%20%20%20%20print%28f%22Maximum%20total%20value%20within%20the%20knapsack%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20unbounded_knapsack_dp%28wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20cap%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Unbounded%20knapsack%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%20%28cap%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20State%20transitions%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20c%20in%20range%281,%20cap%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20item%20i%20exceeds%20the%20remaining%20capacity,%20do%20not%20choose%20it%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bc%5D%20%3D%20dp%5Bi%20-%201%5D%5Bc%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Take%20the%20larger%20value%20obtained%20by%20excluding%20and%20including%20item%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Bc%5D%20%3D%20max%28dp%5Bi%20-%201%5D%5Bc%5D,%20dp%5Bi%5D%5Bc%20-%20wgt%5Bi%20-%201%5D%5D%20%2B%20val%5Bi%20-%201%5D%29%0A%20%20%20%20return%20dp%5Bn%5D%5Bcap%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B1,%202,%203%5D%0A%20%20%20%20val%20%3D%20%5B5,%2011,%2015%5D%0A%20%20%20%20cap%20%3D%204%0A%0A%20%20%20%20%23%20Dynamic%20programming%0A%20%20%20%20res%20%3D%20unbounded_knapsack_dp%28wgt,%20val,%20cap%29%0A%20%20%20%20print%28f%22Maximum%20total%20value%20within%20the%20knapsack%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="3-space-optimization">3. &nbsp; Space optimization<a class="headerlink" href="#3-space-optimization" title="Permanent link">&para;</a></h3>
+<p>The current state depends on states to its left and above, <strong>so after space optimization, each row of the <span class="arithmatex">\(dp\)</span> table must be processed in forward order</strong>.</p>
+<p>This is the opposite of 0-1 knapsack. Use Figure 14-23 to understand the difference.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:6"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">&lt;1&gt;</label><label for="__tabbed_2_2">&lt;2&gt;</label><label for="__tabbed_2_3">&lt;3&gt;</label><label for="__tabbed_2_4">&lt;4&gt;</label><label for="__tabbed_2_5">&lt;5&gt;</label><label for="__tabbed_2_6">&lt;6&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Space-optimized dynamic programming process for unbounded knapsack problem" class="animation-figure" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step1.png" /></p>
+<p><img alt="Space-optimized dynamic programming for unbounded knapsack" class="animation-figure" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="unbounded_knapsack_dp_comp_step2" class="animation-figure" src="../unbounded_knapsack_problem.assets/unbounded_knapsack_dp_comp_step2.png" /></p>
@@ -5276,44 +5353,44 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </div>
 </div>
 </div>
-<p align="center"> Figure 14-23 &nbsp; Space-optimized dynamic programming process for unbounded knapsack problem </p>
+<p align="center"> Figure 14-23 &nbsp; Space-optimized dynamic programming for unbounded knapsack </p>
 
-<p>The code implementation is relatively simple, just delete the first dimension of the array <code>dp</code>:</p>
+<p>The implementation simply removes the first dimension of <code>dp</code>:</p>
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 <div class="highlight"><span class="filename">unbounded_knapsack.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">unbounded_knapsack_dp_comp</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">val</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">cap</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Unbounded knapsack: Space-optimized dynamic programming"""</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Unbounded knapsack: space-optimized dynamic programming"""</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">wgt</span><span class="p">)</span>
-<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="c1"># Initialize dp table</span>
+<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">cap</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="c1"># State transition</span>
+<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="c1"># State transitions</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="c1"># Traverse in forward order</span>
+<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="c1"># Traverse in increasing order</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>        <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">cap</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>            <span class="k">if</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">c</span><span class="p">:</span>
-<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>                <span class="c1"># If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>                <span class="c1"># If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span>
 <a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>            <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>                <span class="c1"># The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>                <span class="c1"># Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">c</span> <span class="o">-</span> <span class="n">wgt</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]]</span> <span class="o">+</span> <span class="n">val</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">])</span>
 <a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">cap</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Unbounded knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Unbounded knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">unboundedKnapsackDPComp</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
-<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5323,19 +5400,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Unbounded knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Unbounded knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">unboundedKnapsackDPComp</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
-<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">max</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">);</span>
 <a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5345,19 +5422,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Unbounded knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Unbounded knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">UnboundedKnapsackDPComp</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
-<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5367,19 +5444,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Unbounded knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Unbounded knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">unboundedKnapsackDPComp</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">wgt</span><span class="p">)</span>
-<a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">]</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Max</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">]),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="o">-</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span><span class="o">+</span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">])))</span>
 <a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5389,19 +5466,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Unbounded knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Unbounded knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">unboundedKnapsackDPComp</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">cap</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="bp">count</span>
-<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span>
 <a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5411,19 +5488,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Unbounded knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Unbounded knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">unboundedKnapsackDPComp</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">wgt</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="nx">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">);</span>
-<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">];</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]);</span>
 <a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5433,23 +5510,23 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Unbounded knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Unbounded knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">unboundedKnapsackDPComp</span><span class="p">(</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="nx">wgt</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="nx">cap</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">wgt</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">);</span>
-<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span><span class="w"> </span><span class="nx">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">];</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]);</span>
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">        </span><span class="p">}</span>
@@ -5459,19 +5536,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Unbounded knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Unbounded knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">unboundedKnapsackDPComp</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
-<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
-<a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="c1">// State transition</span>
+<a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="c1">// State transitions</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">        </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">        </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">      </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">        </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">        </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]);</span>
 <a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">      </span><span class="p">}</span>
 <a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">    </span><span class="p">}</span>
@@ -5481,19 +5558,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Unbounded knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Unbounded knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">unbounded_knapsack_dp_comp</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">cap</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
-<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">cap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5503,19 +5580,19 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Unbounded knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Unbounded knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">unboundedKnapsackDPComp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">wgt</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">wgtSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgtSize</span><span class="p">;</span>
-<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
-<a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span><span class="w"> </span><span class="n">c</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">];</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMax</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5528,23 +5605,23 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Unbounded knapsack: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Unbounded knapsack: space-optimized dynamic programming */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">unboundedKnapsackDPComp</span><span class="p">(</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="n">cap</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="na">size</span>
-<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
-<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">                </span><span class="c1">// If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">                </span><span class="c1">// If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span>
 <a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">                </span><span class="c1">// The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">                </span><span class="c1">// Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">_val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w">        </span><span class="p">}</span>
@@ -5554,20 +5631,20 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">unbounded_knapsack.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Unbounded knapsack: space-optimized DP ###</span>
+<div class="highlight"><span class="filename">unbounded_knapsack.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Unbounded knapsack: space-optimized dynamic programming ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">unbounded_knapsack_dp_comp</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="c1"># Initialize dp table</span>
+<a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="c1"># State transition</span>
+<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="c1"># State transitions</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">    </span><span class="c1"># Traverse in forward order</span>
+<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">    </span><span class="c1"># Traverse in increasing order</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">cap</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">c</span>
-<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">        </span><span class="c1"># If exceeds knapsack capacity, don't select item i</span>
+<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">        </span><span class="c1"># If item i exceeds the remaining capacity, do not choose it</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">      </span><span class="k">else</span>
-<a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">        </span><span class="c1"># The larger value between not selecting and selecting item i</span>
+<a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">        </span><span class="c1"># Take the larger value obtained by excluding and including item i</span>
 <a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">c</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">val</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]].</span><span class="n">max</span>
 <a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">      </span><span class="k">end</span>
 <a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w">    </span><span class="k">end</span>
@@ -5578,83 +5655,88 @@ dp[i, c] = \max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1])
 </div>
 </div>
 </div>
-<h2 id="1452-coin-change-problem">14.5.2 &nbsp; Coin Change Problem<a class="headerlink" href="#1452-coin-change-problem" title="Permanent link">&para;</a></h2>
-<p>The knapsack problem represents a large class of dynamic programming problems and has many variants, such as the coin change problem.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20unbounded_knapsack_dp_comp%28wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20cap%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Unbounded%20knapsack%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B0%5D%20%2A%20%28cap%20%2B%201%29%0A%20%20%20%20%23%20State%20transitions%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%23%20Traverse%20in%20increasing%20order%0A%20%20%20%20%20%20%20%20for%20c%20in%20range%281,%20cap%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20item%20i%20exceeds%20the%20remaining%20capacity,%20do%20not%20choose%20it%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bc%5D%20%3D%20dp%5Bc%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Take%20the%20larger%20value%20obtained%20by%20excluding%20and%20including%20item%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bc%5D%20%3D%20max%28dp%5Bc%5D,%20dp%5Bc%20-%20wgt%5Bi%20-%201%5D%5D%20%2B%20val%5Bi%20-%201%5D%29%0A%20%20%20%20return%20dp%5Bcap%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B1,%202,%203%5D%0A%20%20%20%20val%20%3D%20%5B5,%2011,%2015%5D%0A%20%20%20%20cap%20%3D%204%0A%0A%20%20%20%20%23%20Space-optimized%20dynamic%20programming%0A%20%20%20%20res%20%3D%20unbounded_knapsack_dp_comp%28wgt,%20val,%20cap%29%0A%20%20%20%20print%28f%22Maximum%20total%20value%20within%20the%20knapsack%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20unbounded_knapsack_dp_comp%28wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20cap%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Unbounded%20knapsack%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B0%5D%20%2A%20%28cap%20%2B%201%29%0A%20%20%20%20%23%20State%20transitions%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%23%20Traverse%20in%20increasing%20order%0A%20%20%20%20%20%20%20%20for%20c%20in%20range%281,%20cap%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20wgt%5Bi%20-%201%5D%20%3E%20c%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20item%20i%20exceeds%20the%20remaining%20capacity,%20do%20not%20choose%20it%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bc%5D%20%3D%20dp%5Bc%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Take%20the%20larger%20value%20obtained%20by%20excluding%20and%20including%20item%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bc%5D%20%3D%20max%28dp%5Bc%5D,%20dp%5Bc%20-%20wgt%5Bi%20-%201%5D%5D%20%2B%20val%5Bi%20-%201%5D%29%0A%20%20%20%20return%20dp%5Bcap%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B1,%202,%203%5D%0A%20%20%20%20val%20%3D%20%5B5,%2011,%2015%5D%0A%20%20%20%20cap%20%3D%204%0A%0A%20%20%20%20%23%20Space-optimized%20dynamic%20programming%0A%20%20%20%20res%20%3D%20unbounded_knapsack_dp_comp%28wgt,%20val,%20cap%29%0A%20%20%20%20print%28f%22Maximum%20total%20value%20within%20the%20knapsack%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1452-coin-change-problem">14.5.2 &nbsp; Coin change problem<a class="headerlink" href="#1452-coin-change-problem" title="Permanent link">&para;</a></h2>
+<p>Knapsack represents a broad family of dynamic programming problems, with many variants such as coin change.</p>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given <span class="arithmatex">\(n\)</span> types of coins, where the denomination of the <span class="arithmatex">\(i\)</span>-th type of coin is <span class="arithmatex">\(coins[i - 1]\)</span>, and the target amount is <span class="arithmatex">\(amt\)</span>. <strong>Each type of coin can be selected multiple times</strong>. What is the minimum number of coins needed to make up the target amount? If it is impossible to make up the target amount, return <span class="arithmatex">\(-1\)</span>. An example is shown in Figure 14-24.</p>
+<p>Given <span class="arithmatex">\(n\)</span> coin denominations, where denomination <span class="arithmatex">\(i\)</span> has value <span class="arithmatex">\(coins[i - 1]\)</span>, and a target amount <span class="arithmatex">\(amt\)</span>, <strong>each denomination may be used repeatedly</strong>. What is the minimum number of coins needed to make the target? Return <span class="arithmatex">\(-1\)</span> if it cannot be made. An example is shown in Figure 14-24.</p>
 </div>
-<p><img alt="Example data for coin change problem" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_example.png" /></p>
-<p align="center"> Figure 14-24 &nbsp; Example data for coin change problem </p>
+<p><img alt="Example input for coin change" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_example.png" /></p>
+<p align="center"> Figure 14-24 &nbsp; Example input for coin change </p>
 
-<h3 id="1-dynamic-programming-approach_1">1. &nbsp; Dynamic Programming Approach<a class="headerlink" href="#1-dynamic-programming-approach_1" title="Permanent link">&para;</a></h3>
-<p><strong>The coin change problem can be viewed as a special case of the unbounded knapsack problem</strong>, with the following connections and differences.</p>
+<h3 id="1-dynamic-programming-approach_1">1. &nbsp; Dynamic programming approach<a class="headerlink" href="#1-dynamic-programming-approach_1" title="Permanent link">&para;</a></h3>
+<p><strong>Coin change can be viewed as a special case of unbounded knapsack</strong>, with the following connections and differences.</p>
 <ul>
-<li>The two problems can be converted to each other: "item" corresponds to "coin", "item weight" corresponds to "coin denomination", and "knapsack capacity" corresponds to "target amount".</li>
-<li>The optimization goals are opposite: the unbounded knapsack problem aims to maximize item value, while the coin change problem aims to minimize the number of coins.</li>
-<li>The unbounded knapsack problem seeks solutions "not exceeding" the knapsack capacity, while the coin change problem seeks solutions that "exactly" make up the target amount.</li>
+<li>The two problems can be translated into one another: items correspond to coins, item weights to coin denominations, and knapsack capacity to the target amount.</li>
+<li>The optimization objectives are opposite: unbounded knapsack maximizes item value, while coin change minimizes the number of coins.</li>
+<li>Unbounded knapsack must stay within a capacity limit, whereas coin change must reach the target amount exactly.</li>
 </ul>
-<p><strong>Step 1: Think about the decisions in each round, define the state, and thus obtain the <span class="arithmatex">\(dp\)</span> table</strong></p>
-<p>State <span class="arithmatex">\([i, a]\)</span> corresponds to the subproblem: <strong>the minimum number of coins among the first <span class="arithmatex">\(i\)</span> types of coins that can make up amount <span class="arithmatex">\(a\)</span></strong>, denoted as <span class="arithmatex">\(dp[i, a]\)</span>.</p>
+<p><strong>Step 1: Identify each decision, define the state, and establish the <span class="arithmatex">\(dp\)</span> table</strong></p>
+<p>State <span class="arithmatex">\([i, a]\)</span> represents the subproblem of finding <strong>the minimum number of coins needed to make amount <span class="arithmatex">\(a\)</span> using the first <span class="arithmatex">\(i\)</span> denominations</strong>, denoted by <span class="arithmatex">\(dp[i, a]\)</span>.</p>
 <p>The two-dimensional <span class="arithmatex">\(dp\)</span> table has size <span class="arithmatex">\((n+1) \times (amt+1)\)</span>.</p>
-<p><strong>Step 2: Identify the optimal substructure, and then derive the state transition equation</strong></p>
-<p>This problem differs from the unbounded knapsack problem in the following two aspects regarding the state transition equation.</p>
+<p><strong>Step 2: Identify optimal substructure and derive the state transition equation</strong></p>
+<p>The state transition differs from unbounded knapsack in two ways.</p>
 <ul>
-<li>This problem seeks the minimum value, so the operator <span class="arithmatex">\(\max()\)</span> needs to be changed to <span class="arithmatex">\(\min()\)</span>.</li>
-<li>The optimization target is the number of coins rather than item value, so when a coin is selected, simply add <span class="arithmatex">\(1\)</span>.</li>
+<li>We want a minimum, so replace <span class="arithmatex">\(\max()\)</span> with <span class="arithmatex">\(\min()\)</span>.</li>
+<li>We optimize the coin count rather than item value, so choosing a coin adds <span class="arithmatex">\(+1\)</span>.</li>
 </ul>
 <div class="arithmatex">\[
 dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 \]</div>
-<p><strong>Step 3: Determine boundary conditions and state transition order</strong></p>
-<p>When the target amount is <span class="arithmatex">\(0\)</span>, the minimum number of coins needed to make it up is <span class="arithmatex">\(0\)</span>, so all <span class="arithmatex">\(dp[i, 0]\)</span> in the first column equal <span class="arithmatex">\(0\)</span>.</p>
-<p>When there are no coins, <strong>it is impossible to make up any amount <span class="arithmatex">\(&gt; 0\)</span></strong>, which is an invalid solution. To enable the <span class="arithmatex">\(\min()\)</span> function in the state transition equation to identify and filter out invalid solutions, we consider using <span class="arithmatex">\(+ \infty\)</span> to represent them, i.e., set all <span class="arithmatex">\(dp[0, a]\)</span> in the first row to <span class="arithmatex">\(+ \infty\)</span>.</p>
-<h3 id="2-code-implementation_1">2. &nbsp; Code Implementation<a class="headerlink" href="#2-code-implementation_1" title="Permanent link">&para;</a></h3>
-<p>Most programming languages do not provide a <span class="arithmatex">\(+ \infty\)</span> variable, and can only use the maximum value of integer type <code>int</code> as a substitute. However, this can lead to integer overflow: the <span class="arithmatex">\(+ 1\)</span> operation in the state transition equation may cause overflow.</p>
-<p>For this reason, we use the number <span class="arithmatex">\(amt + 1\)</span> to represent invalid solutions, because the maximum number of coins needed to make up <span class="arithmatex">\(amt\)</span> is at most <span class="arithmatex">\(amt\)</span>. Before returning, check whether <span class="arithmatex">\(dp[n, amt]\)</span> equals <span class="arithmatex">\(amt + 1\)</span>; if so, return <span class="arithmatex">\(-1\)</span>, indicating that the target amount cannot be made up. The code is as follows:</p>
+<p><strong>Step 3: Determine the boundary conditions and evaluation order</strong></p>
+<p>A target amount of <span class="arithmatex">\(0\)</span> requires <span class="arithmatex">\(0\)</span> coins, so every entry <span class="arithmatex">\(dp[i, 0]\)</span> in the first column is <span class="arithmatex">\(0\)</span>.</p>
+<p>With no coins, <strong>any target amount <span class="arithmatex">\(&gt; 0\)</span> is impossible</strong>. To let <span class="arithmatex">\(\min()\)</span> recognize and discard these infeasible solutions, represent them by <span class="arithmatex">\(+ \infty\)</span>: set the first-row entries <span class="arithmatex">\(dp[0, a]\)</span> to <span class="arithmatex">\(+ \infty\)</span>.</p>
+<h3 id="2-code-implementation_1">2. &nbsp; Code implementation<a class="headerlink" href="#2-code-implementation_1" title="Permanent link">&para;</a></h3>
+<p>Most programming languages do not provide an integer value for <span class="arithmatex">\(+ \infty\)</span>, so the maximum <code>int</code> value is used instead. However, the <span class="arithmatex">\(+ 1\)</span> in the transition may then overflow.</p>
+<p>Instead, use <span class="arithmatex">\(amt + 1\)</span> to represent an infeasible solution, since making <span class="arithmatex">\(amt\)</span> requires at most <span class="arithmatex">\(amt\)</span> coins. Before returning, check whether <span class="arithmatex">\(dp[n, amt]\)</span> equals <span class="arithmatex">\(amt + 1\)</span>. If so, return <span class="arithmatex">\(-1\)</span> to indicate that the target cannot be made. The code is as follows:</p>
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 <div class="highlight"><span class="filename">coin_change.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">coin_change_dp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">amt</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Coin change: Dynamic programming"""</span>
+<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Coin change: dynamic programming"""</span>
 <a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coins</span><span class="p">)</span>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="n">MAX</span> <span class="o">=</span> <span class="n">amt</span> <span class="o">+</span> <span class="mi">1</span>
-<a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>    <span class="c1"># Initialize dp table</span>
+<a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>    <span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>    <span class="c1"># State transition: first row and first column</span>
+<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>    <span class="c1"># State transitions: first row and first column</span>
 <a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>    <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>        <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">MAX</span>
-<a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>    <span class="c1"># State transition: rest of the rows and columns</span>
+<a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>    <span class="c1"># State transitions: remaining rows and columns</span>
 <a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a>        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>            <span class="k">if</span> <span class="n">coins</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">a</span><span class="p">:</span>
-<a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a>                <span class="c1"># If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a>                <span class="c1"># If the coin value exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">]</span>
 <a id="__codelineno-26-16" name="__codelineno-26-16" href="#__codelineno-26-16"></a>            <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-26-17" name="__codelineno-26-17" href="#__codelineno-26-17"></a>                <span class="c1"># The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-26-17" name="__codelineno-26-17" href="#__codelineno-26-17"></a>                <span class="c1"># Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-26-18" name="__codelineno-26-18" href="#__codelineno-26-18"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span> <span class="o">-</span> <span class="n">coins</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-26-19" name="__codelineno-26-19" href="#__codelineno-26-19"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">amt</span><span class="p">]</span> <span class="k">if</span> <span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">amt</span><span class="p">]</span> <span class="o">!=</span> <span class="n">MAX</span> <span class="k">else</span> <span class="o">-</span><span class="mi">1</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Coin change: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Coin change: dynamic programming */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeDP</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
 <a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">));</span>
-<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MAX</span><span class="p">;</span>
 <a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-27-16" name="__codelineno-27-16" href="#__codelineno-27-16"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-27-17" name="__codelineno-27-17" href="#__codelineno-27-17"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-18" name="__codelineno-27-18" href="#__codelineno-27-18"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-27-18" name="__codelineno-27-18" href="#__codelineno-27-18"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-27-19" name="__codelineno-27-19" href="#__codelineno-27-19"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-27-20" name="__codelineno-27-20" href="#__codelineno-27-20"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-27-21" name="__codelineno-27-21" href="#__codelineno-27-21"></a><span class="w">        </span><span class="p">}</span>
@@ -5664,24 +5746,24 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Coin change: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Coin change: dynamic programming */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeDP</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
 <a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MAX</span><span class="p">;</span>
 <a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-28-19" name="__codelineno-28-19" href="#__codelineno-28-19"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-28-20" name="__codelineno-28-20" href="#__codelineno-28-20"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-28-21" name="__codelineno-28-21" href="#__codelineno-28-21"></a><span class="w">        </span><span class="p">}</span>
@@ -5691,24 +5773,24 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Coin change: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Coin change: dynamic programming */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">CoinChangeDP</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
 <a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="kt">int</span><span class="p">[,]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MAX</span><span class="p">;</span>
 <a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-29-19" name="__codelineno-29-19" href="#__codelineno-29-19"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-29-20" name="__codelineno-29-20" href="#__codelineno-29-20"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a><span class="w">        </span><span class="p">}</span>
@@ -5718,27 +5800,27 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Coin change: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Coin change: dynamic programming */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">coinChangeDP</span><span class="p">(</span><span class="nx">coins</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">coins</span><span class="p">)</span>
 <a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="nx">max</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">max</span>
 <a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a><span class="w">                </span><span class="c1">// If the coin exceeds the current target amount a, do not choose coin i</span>
 <a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span>
 <a id="__codelineno-30-20" name="__codelineno-30-20" href="#__codelineno-30-20"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-21" name="__codelineno-30-21" href="#__codelineno-30-21"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-30-21" name="__codelineno-30-21" href="#__codelineno-30-21"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-30-22" name="__codelineno-30-22" href="#__codelineno-30-22"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Min</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">a</span><span class="p">]),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="o">-</span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span><span class="o">+</span><span class="mi">1</span><span class="p">)))</span>
 <a id="__codelineno-30-23" name="__codelineno-30-23" href="#__codelineno-30-23"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-30-24" name="__codelineno-30-24" href="#__codelineno-30-24"></a><span class="w">        </span><span class="p">}</span>
@@ -5751,24 +5833,24 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Coin change: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Coin change: dynamic programming */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">coinChangeDP</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">amt</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">MAX</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">),</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">MAX</span>
 <a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="w">                </span><span class="c1">// If the coin exceeds the current target amount a, do not choose coin i</span>
 <a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">]</span>
 <a id="__codelineno-31-17" name="__codelineno-31-17" href="#__codelineno-31-17"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-18" name="__codelineno-31-18" href="#__codelineno-31-18"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-31-18" name="__codelineno-31-18" href="#__codelineno-31-18"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-31-19" name="__codelineno-31-19" href="#__codelineno-31-19"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-31-20" name="__codelineno-31-20" href="#__codelineno-31-20"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-31-21" name="__codelineno-31-21" href="#__codelineno-31-21"></a><span class="w">        </span><span class="p">}</span>
@@ -5778,26 +5860,26 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Coin change: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Coin change: dynamic programming */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">coinChangeDP</span><span class="p">(</span><span class="nx">coins</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">coins</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span>
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">        </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span>
 <a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="p">);</span>
-<a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">MAX</span><span class="p">;</span>
 <a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-15" name="__codelineno-32-15" href="#__codelineno-32-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-16" name="__codelineno-32-16" href="#__codelineno-32-16"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-17" name="__codelineno-32-17" href="#__codelineno-32-17"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-32-17" name="__codelineno-32-17" href="#__codelineno-32-17"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-32-18" name="__codelineno-32-18" href="#__codelineno-32-18"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">a</span><span class="p">];</span>
 <a id="__codelineno-32-19" name="__codelineno-32-19" href="#__codelineno-32-19"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-20" name="__codelineno-32-20" href="#__codelineno-32-20"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-32-20" name="__codelineno-32-20" href="#__codelineno-32-20"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-32-21" name="__codelineno-32-21" href="#__codelineno-32-21"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">a</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-32-22" name="__codelineno-32-22" href="#__codelineno-32-22"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-32-23" name="__codelineno-32-23" href="#__codelineno-32-23"></a><span class="w">        </span><span class="p">}</span>
@@ -5807,26 +5889,26 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Coin change: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Coin change: dynamic programming */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">coinChangeDP</span><span class="p">(</span><span class="nx">coins</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">coins</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span>
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">        </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span>
 <a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">    </span><span class="p">);</span>
-<a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">MAX</span><span class="p">;</span>
 <a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-17" name="__codelineno-33-17" href="#__codelineno-33-17"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-33-17" name="__codelineno-33-17" href="#__codelineno-33-17"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-33-18" name="__codelineno-33-18" href="#__codelineno-33-18"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">a</span><span class="p">];</span>
 <a id="__codelineno-33-19" name="__codelineno-33-19" href="#__codelineno-33-19"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-20" name="__codelineno-33-20" href="#__codelineno-33-20"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-33-20" name="__codelineno-33-20" href="#__codelineno-33-20"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-33-21" name="__codelineno-33-21" href="#__codelineno-33-21"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">a</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-33-22" name="__codelineno-33-22" href="#__codelineno-33-22"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-33-23" name="__codelineno-33-23" href="#__codelineno-33-23"></a><span class="w">        </span><span class="p">}</span>
@@ -5836,24 +5918,24 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Coin change: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Coin change: dynamic programming */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">coinChangeDP</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
-<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">));</span>
-<a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">  </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">  </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MAX</span><span class="p">;</span>
 <a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w">  </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w">  </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">        </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">        </span><span class="c1">// If the coin exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-34-17" name="__codelineno-34-17" href="#__codelineno-34-17"></a><span class="w">      </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-18" name="__codelineno-34-18" href="#__codelineno-34-18"></a><span class="w">        </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-34-18" name="__codelineno-34-18" href="#__codelineno-34-18"></a><span class="w">        </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-34-19" name="__codelineno-34-19" href="#__codelineno-34-19"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">a</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-34-20" name="__codelineno-34-20" href="#__codelineno-34-20"></a><span class="w">      </span><span class="p">}</span>
 <a id="__codelineno-34-21" name="__codelineno-34-21" href="#__codelineno-34-21"></a><span class="w">    </span><span class="p">}</span>
@@ -5863,24 +5945,24 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Coin change: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Coin change: dynamic programming */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">coin_change_dp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">amt</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
 <a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">amt</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">;</span>
 <a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">amt</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-35-16" name="__codelineno-35-16" href="#__codelineno-35-16"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-35-17" name="__codelineno-35-17" href="#__codelineno-35-17"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-18" name="__codelineno-35-18" href="#__codelineno-35-18"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-35-18" name="__codelineno-35-18" href="#__codelineno-35-18"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-35-19" name="__codelineno-35-19" href="#__codelineno-35-19"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-35-20" name="__codelineno-35-20" href="#__codelineno-35-20"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-35-21" name="__codelineno-35-21" href="#__codelineno-35-21"></a><span class="w">        </span><span class="p">}</span>
@@ -5894,27 +5976,27 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Coin change: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Coin change: dynamic programming */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">coins</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">coinsSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coinsSize</span><span class="p">;</span>
 <a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">**</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">((</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">));</span>
 <a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MAX</span><span class="p">;</span>
 <a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-36-14" name="__codelineno-36-14" href="#__codelineno-36-14"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-36-14" name="__codelineno-36-14" href="#__codelineno-36-14"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-16" name="__codelineno-36-16" href="#__codelineno-36-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-17" name="__codelineno-36-17" href="#__codelineno-36-17"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-18" name="__codelineno-36-18" href="#__codelineno-36-18"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-36-18" name="__codelineno-36-18" href="#__codelineno-36-18"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-36-19" name="__codelineno-36-19" href="#__codelineno-36-19"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-36-20" name="__codelineno-36-20" href="#__codelineno-36-20"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-21" name="__codelineno-36-21" href="#__codelineno-36-21"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-36-21" name="__codelineno-36-21" href="#__codelineno-36-21"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-36-22" name="__codelineno-36-22" href="#__codelineno-36-22"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-36-23" name="__codelineno-36-23" href="#__codelineno-36-23"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-36-24" name="__codelineno-36-24" href="#__codelineno-36-24"></a><span class="w">        </span><span class="p">}</span>
@@ -5930,24 +6012,24 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Coin change: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Coin change: dynamic programming */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">coinChangeDP</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">amt</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Array</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="c1">// State transition: first row and first column</span>
+<a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="c1">// State transitions: first row and first column</span>
 <a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="m">0</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MAX</span>
 <a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-15" name="__codelineno-37-15" href="#__codelineno-37-15"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-37-15" name="__codelineno-37-15" href="#__codelineno-37-15"></a><span class="w">                </span><span class="c1">// If the coin value exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-37-16" name="__codelineno-37-16" href="#__codelineno-37-16"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">][</span><span class="n">a</span><span class="o">]</span>
 <a id="__codelineno-37-17" name="__codelineno-37-17" href="#__codelineno-37-17"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-18" name="__codelineno-37-18" href="#__codelineno-37-18"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-37-18" name="__codelineno-37-18" href="#__codelineno-37-18"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-37-19" name="__codelineno-37-19" href="#__codelineno-37-19"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-37-20" name="__codelineno-37-20" href="#__codelineno-37-20"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-37-21" name="__codelineno-37-21" href="#__codelineno-37-21"></a><span class="w">        </span><span class="p">}</span>
@@ -5961,18 +6043,18 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">coin_change_dp</span><span class="p">(</span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span>
 <a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="n">_MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">  </span><span class="c1"># Initialize dp table</span>
+<a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">  </span><span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="w">  </span><span class="c1"># State transition: first row and first column</span>
+<a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="w">  </span><span class="c1"># State transitions: first row and first column</span>
 <a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">  </span><span class="p">(</span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">a</span><span class="o">|</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_MAX</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">  </span><span class="c1"># State transition: rest of the rows and columns</span>
+<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">  </span><span class="c1"># State transitions: remaining rows and columns</span>
 <a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span>
-<a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">        </span><span class="c1"># If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">        </span><span class="c1"># If the coin exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">a</span><span class="o">]</span>
 <a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a><span class="w">      </span><span class="k">else</span>
-<a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w">        </span><span class="c1"># The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w">        </span><span class="c1"># Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-38-17" name="__codelineno-38-17" href="#__codelineno-38-17"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">].</span><span class="n">min</span>
 <a id="__codelineno-38-18" name="__codelineno-38-18" href="#__codelineno-38-18"></a><span class="w">      </span><span class="k">end</span>
 <a id="__codelineno-38-19" name="__codelineno-38-19" href="#__codelineno-38-19"></a><span class="w">    </span><span class="k">end</span>
@@ -5983,11 +6065,16 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </div>
 </div>
 </div>
-<p>Figure 14-25 shows the dynamic programming process for coin change, which is very similar to the unbounded knapsack problem.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20coin_change_dp%28coins%3A%20list%5Bint%5D,%20amt%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Coin%20change%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28coins%29%0A%20%20%20%20MAX%20%3D%20amt%20%2B%201%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%20%28amt%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20State%20transitions%3A%20first%20row%20and%20first%20column%0A%20%20%20%20for%20a%20in%20range%281,%20amt%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Ba%5D%20%3D%20MAX%0A%20%20%20%20%23%20State%20transitions%3A%20remaining%20rows%20and%20columns%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20a%20in%20range%281,%20amt%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20coins%5Bi%20-%201%5D%20%3E%20a%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20the%20coin%20value%20exceeds%20the%20current%20amount%20a,%20do%20not%20choose%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Ba%5D%20%3D%20dp%5Bi%20-%201%5D%5Ba%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Take%20the%20smaller%20result%20of%20not%20choosing%20and%20choosing%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Ba%5D%20%3D%20min%28dp%5Bi%20-%201%5D%5Ba%5D,%20dp%5Bi%5D%5Ba%20-%20coins%5Bi%20-%201%5D%5D%20%2B%201%29%0A%20%20%20%20return%20dp%5Bn%5D%5Bamt%5D%20if%20dp%5Bn%5D%5Bamt%5D%20%21%3D%20MAX%20else%20-1%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20coins%20%3D%20%5B1,%202,%205%5D%0A%20%20%20%20amt%20%3D%204%0A%0A%20%20%20%20%23%20Dynamic%20programming%0A%20%20%20%20res%20%3D%20coin_change_dp%28coins,%20amt%29%0A%20%20%20%20print%28f%22Minimum%20number%20of%20coins%20needed%20to%20make%20the%20target%20amount%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20coin_change_dp%28coins%3A%20list%5Bint%5D,%20amt%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Coin%20change%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28coins%29%0A%20%20%20%20MAX%20%3D%20amt%20%2B%201%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%20%28amt%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20State%20transitions%3A%20first%20row%20and%20first%20column%0A%20%20%20%20for%20a%20in%20range%281,%20amt%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5B0%5D%5Ba%5D%20%3D%20MAX%0A%20%20%20%20%23%20State%20transitions%3A%20remaining%20rows%20and%20columns%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20a%20in%20range%281,%20amt%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20coins%5Bi%20-%201%5D%20%3E%20a%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20the%20coin%20value%20exceeds%20the%20current%20amount%20a,%20do%20not%20choose%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Ba%5D%20%3D%20dp%5Bi%20-%201%5D%5Ba%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Take%20the%20smaller%20result%20of%20not%20choosing%20and%20choosing%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Ba%5D%20%3D%20min%28dp%5Bi%20-%201%5D%5Ba%5D,%20dp%5Bi%5D%5Ba%20-%20coins%5Bi%20-%201%5D%5D%20%2B%201%29%0A%20%20%20%20return%20dp%5Bn%5D%5Bamt%5D%20if%20dp%5Bn%5D%5Bamt%5D%20%21%3D%20MAX%20else%20-1%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20coins%20%3D%20%5B1,%202,%205%5D%0A%20%20%20%20amt%20%3D%204%0A%0A%20%20%20%20%23%20Dynamic%20programming%0A%20%20%20%20res%20%3D%20coin_change_dp%28coins,%20amt%29%0A%20%20%20%20print%28f%22Minimum%20number%20of%20coins%20needed%20to%20make%20the%20target%20amount%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 14-25 shows dynamic programming for coin change, which closely resembles unbounded knapsack.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:15"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><input id="__tabbed_5_14" name="__tabbed_5" type="radio" /><input id="__tabbed_5_15" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">&lt;1&gt;</label><label for="__tabbed_5_2">&lt;2&gt;</label><label for="__tabbed_5_3">&lt;3&gt;</label><label for="__tabbed_5_4">&lt;4&gt;</label><label for="__tabbed_5_5">&lt;5&gt;</label><label for="__tabbed_5_6">&lt;6&gt;</label><label for="__tabbed_5_7">&lt;7&gt;</label><label for="__tabbed_5_8">&lt;8&gt;</label><label for="__tabbed_5_9">&lt;9&gt;</label><label for="__tabbed_5_10">&lt;10&gt;</label><label for="__tabbed_5_11">&lt;11&gt;</label><label for="__tabbed_5_12">&lt;12&gt;</label><label for="__tabbed_5_13">&lt;13&gt;</label><label for="__tabbed_5_14">&lt;14&gt;</label><label for="__tabbed_5_15">&lt;15&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Dynamic programming process for coin change problem" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step1.png" /></p>
+<p><img alt="Dynamic programming for coin change" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="coin_change_dp_step2" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_dp_step2.png" /></p>
@@ -6033,49 +6120,49 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </div>
 </div>
 </div>
-<p align="center"> Figure 14-25 &nbsp; Dynamic programming process for coin change problem </p>
+<p align="center"> Figure 14-25 &nbsp; Dynamic programming for coin change </p>
 
-<h3 id="3-space-optimization_1">3. &nbsp; Space Optimization<a class="headerlink" href="#3-space-optimization_1" title="Permanent link">&para;</a></h3>
-<p>The space optimization for the coin change problem is handled in the same way as the unbounded knapsack problem:</p>
+<h3 id="3-space-optimization_1">3. &nbsp; Space optimization<a class="headerlink" href="#3-space-optimization_1" title="Permanent link">&para;</a></h3>
+<p>Space optimization is the same as for unbounded knapsack:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="6:13"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">coin_change.py</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">coin_change_dp_comp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">amt</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Coin change: Space-optimized dynamic programming"""</span>
+<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Coin change: space-optimized dynamic programming"""</span>
 <a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coins</span><span class="p">)</span>
 <a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="n">MAX</span> <span class="o">=</span> <span class="n">amt</span> <span class="o">+</span> <span class="mi">1</span>
-<a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>    <span class="c1"># Initialize dp table</span>
+<a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>    <span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[</span><span class="n">MAX</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a>    <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a>    <span class="c1"># State transition</span>
+<a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a>    <span class="c1"># State transitions</span>
 <a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-<a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a>        <span class="c1"># Traverse in forward order</span>
+<a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a>        <span class="c1"># Traverse in increasing order</span>
 <a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a>        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-39-12" name="__codelineno-39-12" href="#__codelineno-39-12"></a>            <span class="k">if</span> <span class="n">coins</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">a</span><span class="p">:</span>
-<a id="__codelineno-39-13" name="__codelineno-39-13" href="#__codelineno-39-13"></a>                <span class="c1"># If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-39-13" name="__codelineno-39-13" href="#__codelineno-39-13"></a>                <span class="c1"># If the coin value exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-39-14" name="__codelineno-39-14" href="#__codelineno-39-14"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span>
 <a id="__codelineno-39-15" name="__codelineno-39-15" href="#__codelineno-39-15"></a>            <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-39-16" name="__codelineno-39-16" href="#__codelineno-39-16"></a>                <span class="c1"># The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-39-16" name="__codelineno-39-16" href="#__codelineno-39-16"></a>                <span class="c1"># Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-39-17" name="__codelineno-39-17" href="#__codelineno-39-17"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">],</span> <span class="n">dp</span><span class="p">[</span><span class="n">a</span> <span class="o">-</span> <span class="n">coins</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-39-18" name="__codelineno-39-18" href="#__codelineno-39-18"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">amt</span><span class="p">]</span> <span class="k">if</span> <span class="n">dp</span><span class="p">[</span><span class="n">amt</span><span class="p">]</span> <span class="o">!=</span> <span class="n">MAX</span> <span class="k">else</span> <span class="o">-</span><span class="mi">1</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Coin change: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Coin change: space-optimized dynamic programming */</span>
 <a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeDPComp</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
 <a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">MAX</span><span class="p">);</span>
 <a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-40-9" name="__codelineno-40-9" href="#__codelineno-40-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-10" name="__codelineno-40-10" href="#__codelineno-40-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-11" name="__codelineno-40-11" href="#__codelineno-40-11"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-12" name="__codelineno-40-12" href="#__codelineno-40-12"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-40-12" name="__codelineno-40-12" href="#__codelineno-40-12"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-40-13" name="__codelineno-40-13" href="#__codelineno-40-13"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-40-14" name="__codelineno-40-14" href="#__codelineno-40-14"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-15" name="__codelineno-40-15" href="#__codelineno-40-15"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-40-15" name="__codelineno-40-15" href="#__codelineno-40-15"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-40-16" name="__codelineno-40-16" href="#__codelineno-40-16"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-40-17" name="__codelineno-40-17" href="#__codelineno-40-17"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-40-18" name="__codelineno-40-18" href="#__codelineno-40-18"></a><span class="w">        </span><span class="p">}</span>
@@ -6085,22 +6172,22 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Coin change: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Coin change: space-optimized dynamic programming */</span>
 <a id="__codelineno-41-2" name="__codelineno-41-2" href="#__codelineno-41-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeDPComp</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
 <a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="w">    </span><span class="n">Arrays</span><span class="p">.</span><span class="na">fill</span><span class="p">(</span><span class="n">dp</span><span class="p">,</span><span class="w"> </span><span class="n">MAX</span><span class="p">);</span>
 <a id="__codelineno-41-8" name="__codelineno-41-8" href="#__codelineno-41-8"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-41-9" name="__codelineno-41-9" href="#__codelineno-41-9"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-41-9" name="__codelineno-41-9" href="#__codelineno-41-9"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-41-10" name="__codelineno-41-10" href="#__codelineno-41-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-11" name="__codelineno-41-11" href="#__codelineno-41-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-12" name="__codelineno-41-12" href="#__codelineno-41-12"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-13" name="__codelineno-41-13" href="#__codelineno-41-13"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-41-13" name="__codelineno-41-13" href="#__codelineno-41-13"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-41-14" name="__codelineno-41-14" href="#__codelineno-41-14"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-41-15" name="__codelineno-41-15" href="#__codelineno-41-15"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-16" name="__codelineno-41-16" href="#__codelineno-41-16"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-41-16" name="__codelineno-41-16" href="#__codelineno-41-16"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-41-17" name="__codelineno-41-17" href="#__codelineno-41-17"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-41-18" name="__codelineno-41-18" href="#__codelineno-41-18"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-41-19" name="__codelineno-41-19" href="#__codelineno-41-19"></a><span class="w">        </span><span class="p">}</span>
@@ -6110,22 +6197,22 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Coin change: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Coin change: space-optimized dynamic programming */</span>
 <a id="__codelineno-42-2" name="__codelineno-42-2" href="#__codelineno-42-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">CoinChangeDPComp</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
 <a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="w">    </span><span class="n">Array</span><span class="p">.</span><span class="n">Fill</span><span class="p">(</span><span class="n">dp</span><span class="p">,</span><span class="w"> </span><span class="n">MAX</span><span class="p">);</span>
 <a id="__codelineno-42-8" name="__codelineno-42-8" href="#__codelineno-42-8"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-42-9" name="__codelineno-42-9" href="#__codelineno-42-9"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-42-9" name="__codelineno-42-9" href="#__codelineno-42-9"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-42-10" name="__codelineno-42-10" href="#__codelineno-42-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-11" name="__codelineno-42-11" href="#__codelineno-42-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-12" name="__codelineno-42-12" href="#__codelineno-42-12"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-13" name="__codelineno-42-13" href="#__codelineno-42-13"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-42-13" name="__codelineno-42-13" href="#__codelineno-42-13"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-42-14" name="__codelineno-42-14" href="#__codelineno-42-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-42-15" name="__codelineno-42-15" href="#__codelineno-42-15"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-16" name="__codelineno-42-16" href="#__codelineno-42-16"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-42-16" name="__codelineno-42-16" href="#__codelineno-42-16"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-42-17" name="__codelineno-42-17" href="#__codelineno-42-17"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-42-18" name="__codelineno-42-18" href="#__codelineno-42-18"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-42-19" name="__codelineno-42-19" href="#__codelineno-42-19"></a><span class="w">        </span><span class="p">}</span>
@@ -6135,24 +6222,24 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Coin change: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Coin change: dynamic programming */</span>
 <a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">coinChangeDPComp</span><span class="p">(</span><span class="nx">coins</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">coins</span><span class="p">)</span>
 <a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="nx">max</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">max</span>
 <a id="__codelineno-43-9" name="__codelineno-43-9" href="#__codelineno-43-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-12" name="__codelineno-43-12" href="#__codelineno-43-12"></a><span class="w">        </span><span class="c1">// Traverse in forward order</span>
+<a id="__codelineno-43-12" name="__codelineno-43-12" href="#__codelineno-43-12"></a><span class="w">        </span><span class="c1">// Traverse in increasing order</span>
 <a id="__codelineno-43-13" name="__codelineno-43-13" href="#__codelineno-43-13"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-14" name="__codelineno-43-14" href="#__codelineno-43-14"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-15" name="__codelineno-43-15" href="#__codelineno-43-15"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-43-15" name="__codelineno-43-15" href="#__codelineno-43-15"></a><span class="w">                </span><span class="c1">// If the coin exceeds the current target amount a, do not choose coin i</span>
 <a id="__codelineno-43-16" name="__codelineno-43-16" href="#__codelineno-43-16"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span>
 <a id="__codelineno-43-17" name="__codelineno-43-17" href="#__codelineno-43-17"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-18" name="__codelineno-43-18" href="#__codelineno-43-18"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-43-18" name="__codelineno-43-18" href="#__codelineno-43-18"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-43-19" name="__codelineno-43-19" href="#__codelineno-43-19"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Min</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="o">-</span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span><span class="o">+</span><span class="mi">1</span><span class="p">)))</span>
 <a id="__codelineno-43-20" name="__codelineno-43-20" href="#__codelineno-43-20"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-43-21" name="__codelineno-43-21" href="#__codelineno-43-21"></a><span class="w">        </span><span class="p">}</span>
@@ -6165,21 +6252,21 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Coin change: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Coin change: space-optimized dynamic programming */</span>
 <a id="__codelineno-44-2" name="__codelineno-44-2" href="#__codelineno-44-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">coinChangeDPComp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">amt</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">MAX</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="n">MAX</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-44-7" name="__codelineno-44-7" href="#__codelineno-44-7"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-44-9" name="__codelineno-44-9" href="#__codelineno-44-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-10" name="__codelineno-44-10" href="#__codelineno-44-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-11" name="__codelineno-44-11" href="#__codelineno-44-11"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-44-12" name="__codelineno-44-12" href="#__codelineno-44-12"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-44-12" name="__codelineno-44-12" href="#__codelineno-44-12"></a><span class="w">                </span><span class="c1">// If the coin exceeds the current target amount a, do not choose coin i</span>
 <a id="__codelineno-44-13" name="__codelineno-44-13" href="#__codelineno-44-13"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span>
 <a id="__codelineno-44-14" name="__codelineno-44-14" href="#__codelineno-44-14"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-44-15" name="__codelineno-44-15" href="#__codelineno-44-15"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-44-15" name="__codelineno-44-15" href="#__codelineno-44-15"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-44-16" name="__codelineno-44-16" href="#__codelineno-44-16"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-44-17" name="__codelineno-44-17" href="#__codelineno-44-17"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-44-18" name="__codelineno-44-18" href="#__codelineno-44-18"></a><span class="w">        </span><span class="p">}</span>
@@ -6189,21 +6276,21 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Coin change: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Coin change: space-optimized dynamic programming */</span>
 <a id="__codelineno-45-2" name="__codelineno-45-2" href="#__codelineno-45-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">coinChangeDPComp</span><span class="p">(</span><span class="nx">coins</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">coins</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-45-6" name="__codelineno-45-6" href="#__codelineno-45-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">MAX</span><span class="p">);</span>
 <a id="__codelineno-45-7" name="__codelineno-45-7" href="#__codelineno-45-7"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-45-8" name="__codelineno-45-8" href="#__codelineno-45-8"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-45-8" name="__codelineno-45-8" href="#__codelineno-45-8"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-45-9" name="__codelineno-45-9" href="#__codelineno-45-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-10" name="__codelineno-45-10" href="#__codelineno-45-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-11" name="__codelineno-45-11" href="#__codelineno-45-11"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-12" name="__codelineno-45-12" href="#__codelineno-45-12"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-45-12" name="__codelineno-45-12" href="#__codelineno-45-12"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-45-13" name="__codelineno-45-13" href="#__codelineno-45-13"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">];</span>
 <a id="__codelineno-45-14" name="__codelineno-45-14" href="#__codelineno-45-14"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-15" name="__codelineno-45-15" href="#__codelineno-45-15"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-45-15" name="__codelineno-45-15" href="#__codelineno-45-15"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-45-16" name="__codelineno-45-16" href="#__codelineno-45-16"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-45-17" name="__codelineno-45-17" href="#__codelineno-45-17"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-45-18" name="__codelineno-45-18" href="#__codelineno-45-18"></a><span class="w">        </span><span class="p">}</span>
@@ -6213,21 +6300,21 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Coin change: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Coin change: space-optimized dynamic programming */</span>
 <a id="__codelineno-46-2" name="__codelineno-46-2" href="#__codelineno-46-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">coinChangeDPComp</span><span class="p">(</span><span class="nx">coins</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">coins</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">MAX</span><span class="p">);</span>
 <a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-10" name="__codelineno-46-10" href="#__codelineno-46-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-11" name="__codelineno-46-11" href="#__codelineno-46-11"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-46-13" name="__codelineno-46-13" href="#__codelineno-46-13"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">];</span>
 <a id="__codelineno-46-14" name="__codelineno-46-14" href="#__codelineno-46-14"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-15" name="__codelineno-46-15" href="#__codelineno-46-15"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-46-15" name="__codelineno-46-15" href="#__codelineno-46-15"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-46-16" name="__codelineno-46-16" href="#__codelineno-46-16"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">],</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-46-17" name="__codelineno-46-17" href="#__codelineno-46-17"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-46-18" name="__codelineno-46-18" href="#__codelineno-46-18"></a><span class="w">        </span><span class="p">}</span>
@@ -6237,21 +6324,21 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Coin change: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Coin change: space-optimized dynamic programming */</span>
 <a id="__codelineno-47-2" name="__codelineno-47-2" href="#__codelineno-47-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">coinChangeDPComp</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
-<a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w">  </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w">  </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">MAX</span><span class="p">);</span>
 <a id="__codelineno-47-7" name="__codelineno-47-7" href="#__codelineno-47-7"></a><span class="w">  </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
-<a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a><span class="w">  </span><span class="c1">// State transition</span>
+<a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a><span class="w">  </span><span class="c1">// State transitions</span>
 <a id="__codelineno-47-9" name="__codelineno-47-9" href="#__codelineno-47-9"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-10" name="__codelineno-47-10" href="#__codelineno-47-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-11" name="__codelineno-47-11" href="#__codelineno-47-11"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-12" name="__codelineno-47-12" href="#__codelineno-47-12"></a><span class="w">        </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-47-12" name="__codelineno-47-12" href="#__codelineno-47-12"></a><span class="w">        </span><span class="c1">// If the coin exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-47-13" name="__codelineno-47-13" href="#__codelineno-47-13"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-47-14" name="__codelineno-47-14" href="#__codelineno-47-14"></a><span class="w">      </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-15" name="__codelineno-47-15" href="#__codelineno-47-15"></a><span class="w">        </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-47-15" name="__codelineno-47-15" href="#__codelineno-47-15"></a><span class="w">        </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-47-16" name="__codelineno-47-16" href="#__codelineno-47-16"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-47-17" name="__codelineno-47-17" href="#__codelineno-47-17"></a><span class="w">      </span><span class="p">}</span>
 <a id="__codelineno-47-18" name="__codelineno-47-18" href="#__codelineno-47-18"></a><span class="w">    </span><span class="p">}</span>
@@ -6261,22 +6348,22 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Coin change: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Coin change: space-optimized dynamic programming */</span>
 <a id="__codelineno-48-2" name="__codelineno-48-2" href="#__codelineno-48-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">coin_change_dp_comp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">amt</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
 <a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="w">    </span><span class="n">dp</span><span class="p">.</span><span class="n">fill</span><span class="p">(</span><span class="n">max</span><span class="p">);</span>
 <a id="__codelineno-48-8" name="__codelineno-48-8" href="#__codelineno-48-8"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-11" name="__codelineno-48-11" href="#__codelineno-48-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">amt</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-12" name="__codelineno-48-12" href="#__codelineno-48-12"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-13" name="__codelineno-48-13" href="#__codelineno-48-13"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-48-13" name="__codelineno-48-13" href="#__codelineno-48-13"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-48-14" name="__codelineno-48-14" href="#__codelineno-48-14"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-48-15" name="__codelineno-48-15" href="#__codelineno-48-15"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-16" name="__codelineno-48-16" href="#__codelineno-48-16"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-48-16" name="__codelineno-48-16" href="#__codelineno-48-16"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-48-17" name="__codelineno-48-17" href="#__codelineno-48-17"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-48-18" name="__codelineno-48-18" href="#__codelineno-48-18"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-48-19" name="__codelineno-48-19" href="#__codelineno-48-19"></a><span class="w">        </span><span class="p">}</span>
@@ -6290,25 +6377,25 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Coin change: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Coin change: space-optimized dynamic programming */</span>
 <a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeDPComp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">coins</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">coinsSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coinsSize</span><span class="p">;</span>
 <a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">((</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MAX</span><span class="p">;</span>
 <a id="__codelineno-49-9" name="__codelineno-49-9" href="#__codelineno-49-9"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span>
 <a id="__codelineno-49-10" name="__codelineno-49-10" href="#__codelineno-49-10"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-49-11" name="__codelineno-49-11" href="#__codelineno-49-11"></a>
-<a id="__codelineno-49-12" name="__codelineno-49-12" href="#__codelineno-49-12"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-49-12" name="__codelineno-49-12" href="#__codelineno-49-12"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-49-13" name="__codelineno-49-13" href="#__codelineno-49-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-14" name="__codelineno-49-14" href="#__codelineno-49-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-15" name="__codelineno-49-15" href="#__codelineno-49-15"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-16" name="__codelineno-49-16" href="#__codelineno-49-16"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-49-16" name="__codelineno-49-16" href="#__codelineno-49-16"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-49-17" name="__codelineno-49-17" href="#__codelineno-49-17"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-49-18" name="__codelineno-49-18" href="#__codelineno-49-18"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-19" name="__codelineno-49-19" href="#__codelineno-49-19"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-49-19" name="__codelineno-49-19" href="#__codelineno-49-19"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-49-20" name="__codelineno-49-20" href="#__codelineno-49-20"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">],</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-49-21" name="__codelineno-49-21" href="#__codelineno-49-21"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-49-22" name="__codelineno-49-22" href="#__codelineno-49-22"></a><span class="w">        </span><span class="p">}</span>
@@ -6321,22 +6408,22 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Coin change: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Coin change: space-optimized dynamic programming */</span>
 <a id="__codelineno-50-2" name="__codelineno-50-2" href="#__codelineno-50-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">coinChangeDPComp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">amt</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="w">    </span><span class="n">dp</span><span class="p">.</span><span class="na">fill</span><span class="p">(</span><span class="n">MAX</span><span class="p">)</span>
 <a id="__codelineno-50-8" name="__codelineno-50-8" href="#__codelineno-50-8"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="m">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-50-10" name="__codelineno-50-10" href="#__codelineno-50-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-11" name="__codelineno-50-11" href="#__codelineno-50-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-12" name="__codelineno-50-12" href="#__codelineno-50-12"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-13" name="__codelineno-50-13" href="#__codelineno-50-13"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-50-13" name="__codelineno-50-13" href="#__codelineno-50-13"></a><span class="w">                </span><span class="c1">// If the coin value exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-50-14" name="__codelineno-50-14" href="#__codelineno-50-14"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span>
 <a id="__codelineno-50-15" name="__codelineno-50-15" href="#__codelineno-50-15"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-16" name="__codelineno-50-16" href="#__codelineno-50-16"></a><span class="w">                </span><span class="c1">// The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-50-16" name="__codelineno-50-16" href="#__codelineno-50-16"></a><span class="w">                </span><span class="c1">// Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-50-17" name="__codelineno-50-17" href="#__codelineno-50-17"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-50-18" name="__codelineno-50-18" href="#__codelineno-50-18"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-50-19" name="__codelineno-50-19" href="#__codelineno-50-19"></a><span class="w">        </span><span class="p">}</span>
@@ -6346,22 +6433,22 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Coin change: space-optimized DP ###</span>
+<div class="highlight"><span class="filename">coin_change.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Coin change: space-optimized dynamic programming ###</span>
 <a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">coin_change_dp_comp</span><span class="p">(</span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span>
 <a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="w">  </span><span class="n">_MAX</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">  </span><span class="c1"># Initialize dp table</span>
+<a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">  </span><span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">_MAX</span><span class="p">)</span>
 <a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="w">  </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="w">  </span><span class="c1"># State transition</span>
+<a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="w">  </span><span class="c1"># State transitions</span>
 <a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a><span class="w">    </span><span class="c1"># Traverse in forward order</span>
+<a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a><span class="w">    </span><span class="c1"># Traverse in increasing order</span>
 <a id="__codelineno-51-11" name="__codelineno-51-11" href="#__codelineno-51-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-51-12" name="__codelineno-51-12" href="#__codelineno-51-12"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span>
-<a id="__codelineno-51-13" name="__codelineno-51-13" href="#__codelineno-51-13"></a><span class="w">        </span><span class="c1"># If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-51-13" name="__codelineno-51-13" href="#__codelineno-51-13"></a><span class="w">        </span><span class="c1"># If the coin exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-51-14" name="__codelineno-51-14" href="#__codelineno-51-14"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span>
 <a id="__codelineno-51-15" name="__codelineno-51-15" href="#__codelineno-51-15"></a><span class="w">      </span><span class="k">else</span>
-<a id="__codelineno-51-16" name="__codelineno-51-16" href="#__codelineno-51-16"></a><span class="w">        </span><span class="c1"># The smaller value between not selecting and selecting coin i</span>
+<a id="__codelineno-51-16" name="__codelineno-51-16" href="#__codelineno-51-16"></a><span class="w">        </span><span class="c1"># Take the smaller result of not choosing and choosing coin i</span>
 <a id="__codelineno-51-17" name="__codelineno-51-17" href="#__codelineno-51-17"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">].</span><span class="n">min</span>
 <a id="__codelineno-51-18" name="__codelineno-51-18" href="#__codelineno-51-18"></a><span class="w">      </span><span class="k">end</span>
 <a id="__codelineno-51-19" name="__codelineno-51-19" href="#__codelineno-51-19"></a><span class="w">    </span><span class="k">end</span>
@@ -6372,63 +6459,68 @@ dp[i, a] = \min(dp[i-1, a], dp[i, a - coins[i-1]] + 1)
 </div>
 </div>
 </div>
-<h2 id="1453-coin-change-problem-ii">14.5.3 &nbsp; Coin Change Problem II<a class="headerlink" href="#1453-coin-change-problem-ii" title="Permanent link">&para;</a></h2>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20coin_change_dp_comp%28coins%3A%20list%5Bint%5D,%20amt%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Coin%20change%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28coins%29%0A%20%20%20%20MAX%20%3D%20amt%20%2B%201%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5BMAX%5D%20%2A%20%28amt%20%2B%201%29%0A%20%20%20%20dp%5B0%5D%20%3D%200%0A%20%20%20%20%23%20State%20transitions%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%23%20Traverse%20in%20increasing%20order%0A%20%20%20%20%20%20%20%20for%20a%20in%20range%281,%20amt%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20coins%5Bi%20-%201%5D%20%3E%20a%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20the%20coin%20value%20exceeds%20the%20current%20amount%20a,%20do%20not%20choose%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Ba%5D%20%3D%20dp%5Ba%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Take%20the%20smaller%20result%20of%20not%20choosing%20and%20choosing%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Ba%5D%20%3D%20min%28dp%5Ba%5D,%20dp%5Ba%20-%20coins%5Bi%20-%201%5D%5D%20%2B%201%29%0A%20%20%20%20return%20dp%5Bamt%5D%20if%20dp%5Bamt%5D%20%21%3D%20MAX%20else%20-1%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20coins%20%3D%20%5B1,%202,%205%5D%0A%20%20%20%20amt%20%3D%204%0A%0A%20%20%20%20%23%20Space-optimized%20dynamic%20programming%0A%20%20%20%20res%20%3D%20coin_change_dp_comp%28coins,%20amt%29%0A%20%20%20%20print%28f%22Minimum%20number%20of%20coins%20needed%20to%20make%20the%20target%20amount%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20coin_change_dp_comp%28coins%3A%20list%5Bint%5D,%20amt%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Coin%20change%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28coins%29%0A%20%20%20%20MAX%20%3D%20amt%20%2B%201%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5BMAX%5D%20%2A%20%28amt%20%2B%201%29%0A%20%20%20%20dp%5B0%5D%20%3D%200%0A%20%20%20%20%23%20State%20transitions%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%23%20Traverse%20in%20increasing%20order%0A%20%20%20%20%20%20%20%20for%20a%20in%20range%281,%20amt%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20coins%5Bi%20-%201%5D%20%3E%20a%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20the%20coin%20value%20exceeds%20the%20current%20amount%20a,%20do%20not%20choose%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Ba%5D%20%3D%20dp%5Ba%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Take%20the%20smaller%20result%20of%20not%20choosing%20and%20choosing%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Ba%5D%20%3D%20min%28dp%5Ba%5D,%20dp%5Ba%20-%20coins%5Bi%20-%201%5D%5D%20%2B%201%29%0A%20%20%20%20return%20dp%5Bamt%5D%20if%20dp%5Bamt%5D%20%21%3D%20MAX%20else%20-1%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20coins%20%3D%20%5B1,%202,%205%5D%0A%20%20%20%20amt%20%3D%204%0A%0A%20%20%20%20%23%20Space-optimized%20dynamic%20programming%0A%20%20%20%20res%20%3D%20coin_change_dp_comp%28coins,%20amt%29%0A%20%20%20%20print%28f%22Minimum%20number%20of%20coins%20needed%20to%20make%20the%20target%20amount%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1453-coin-change-problem-ii">14.5.3 &nbsp; Coin change problem II<a class="headerlink" href="#1453-coin-change-problem-ii" title="Permanent link">&para;</a></h2>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given <span class="arithmatex">\(n\)</span> types of coins, where the denomination of the <span class="arithmatex">\(i\)</span>-th type of coin is <span class="arithmatex">\(coins[i - 1]\)</span>, and the target amount is <span class="arithmatex">\(amt\)</span>. Each type of coin can be selected multiple times. <strong>What is the number of coin combinations that can make up the target amount?</strong> An example is shown in Figure 14-26.</p>
+<p>Given <span class="arithmatex">\(n\)</span> coin denominations, where denomination <span class="arithmatex">\(i\)</span> has value <span class="arithmatex">\(coins[i - 1]\)</span>, and a target amount <span class="arithmatex">\(amt\)</span>, each denomination may be used repeatedly. <strong>How many combinations of coins make the target amount</strong>? An example is shown in Figure 14-26.</p>
 </div>
-<p><img alt="Example data for coin change problem II" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_ii_example.png" /></p>
-<p align="center"> Figure 14-26 &nbsp; Example data for coin change problem II </p>
+<p><img alt="Example input for coin change II" class="animation-figure" src="../unbounded_knapsack_problem.assets/coin_change_ii_example.png" /></p>
+<p align="center"> Figure 14-26 &nbsp; Example input for coin change II </p>
 
-<h3 id="1-dynamic-programming-approach_2">1. &nbsp; Dynamic Programming Approach<a class="headerlink" href="#1-dynamic-programming-approach_2" title="Permanent link">&para;</a></h3>
-<p>Compared to the previous problem, this problem's goal is to find the number of combinations, so the subproblem becomes: <strong>the number of combinations among the first <span class="arithmatex">\(i\)</span> types of coins that can make up amount <span class="arithmatex">\(a\)</span></strong>. The <span class="arithmatex">\(dp\)</span> table remains a two-dimensional matrix of size <span class="arithmatex">\((n+1) \times (amt + 1)\)</span>.</p>
-<p>The number of combinations for the current state equals the sum of the combinations from not selecting the current coin and selecting the current coin. The state transition equation is:</p>
+<h3 id="1-dynamic-programming-approach_2">1. &nbsp; Dynamic programming approach<a class="headerlink" href="#1-dynamic-programming-approach_2" title="Permanent link">&para;</a></h3>
+<p>This time, we count combinations, so the subproblem becomes <strong>the number of combinations that make amount <span class="arithmatex">\(a\)</span> using the first <span class="arithmatex">\(i\)</span> denominations</strong>. The <span class="arithmatex">\(dp\)</span> table remains a two-dimensional matrix of size <span class="arithmatex">\((n+1) \times (amt + 1)\)</span>.</p>
+<p>The number of combinations for the current state is the sum of those that exclude the current coin and those that include it. The state transition equation is:</p>
 <div class="arithmatex">\[
 dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 \]</div>
-<p>When the target amount is <span class="arithmatex">\(0\)</span>, no coins need to be selected to make up the target amount, so all <span class="arithmatex">\(dp[i, 0]\)</span> in the first column should be initialized to <span class="arithmatex">\(1\)</span>. When there are no coins, it is impossible to make up any amount <span class="arithmatex">\(&gt;0\)</span>, so all <span class="arithmatex">\(dp[0, a]\)</span> in the first row equal <span class="arithmatex">\(0\)</span>.</p>
-<h3 id="2-code-implementation_2">2. &nbsp; Code Implementation<a class="headerlink" href="#2-code-implementation_2" title="Permanent link">&para;</a></h3>
+<p>A target of <span class="arithmatex">\(0\)</span> can be made by choosing no coins, so initialize every first-column entry <span class="arithmatex">\(dp[i, 0]\)</span> to <span class="arithmatex">\(1\)</span>. With no coins, any target <span class="arithmatex">\(&gt;0\)</span> is impossible, so the first-row entries <span class="arithmatex">\(dp[0, a]\)</span> are <span class="arithmatex">\(0\)</span>.</p>
+<h3 id="2-code-implementation_2">2. &nbsp; Code implementation<a class="headerlink" href="#2-code-implementation_2" title="Permanent link">&para;</a></h3>
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 <div class="highlight"><span class="filename">coin_change_ii.py</span><pre><span></span><code><a id="__codelineno-52-1" name="__codelineno-52-1" href="#__codelineno-52-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">coin_change_ii_dp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">amt</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Coin change II: Dynamic programming"""</span>
+<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Coin change II: dynamic programming"""</span>
 <a id="__codelineno-52-3" name="__codelineno-52-3" href="#__codelineno-52-3"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coins</span><span class="p">)</span>
-<a id="__codelineno-52-4" name="__codelineno-52-4" href="#__codelineno-52-4"></a>    <span class="c1"># Initialize dp table</span>
+<a id="__codelineno-52-4" name="__codelineno-52-4" href="#__codelineno-52-4"></a>    <span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-52-5" name="__codelineno-52-5" href="#__codelineno-52-5"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
-<a id="__codelineno-52-6" name="__codelineno-52-6" href="#__codelineno-52-6"></a>    <span class="c1"># Initialize first column</span>
+<a id="__codelineno-52-6" name="__codelineno-52-6" href="#__codelineno-52-6"></a>    <span class="c1"># Initialize the first column</span>
 <a id="__codelineno-52-7" name="__codelineno-52-7" href="#__codelineno-52-7"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-52-8" name="__codelineno-52-8" href="#__codelineno-52-8"></a>        <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-<a id="__codelineno-52-9" name="__codelineno-52-9" href="#__codelineno-52-9"></a>    <span class="c1"># State transition</span>
+<a id="__codelineno-52-9" name="__codelineno-52-9" href="#__codelineno-52-9"></a>    <span class="c1"># State transitions</span>
 <a id="__codelineno-52-10" name="__codelineno-52-10" href="#__codelineno-52-10"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-52-11" name="__codelineno-52-11" href="#__codelineno-52-11"></a>        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-52-12" name="__codelineno-52-12" href="#__codelineno-52-12"></a>            <span class="k">if</span> <span class="n">coins</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">a</span><span class="p">:</span>
-<a id="__codelineno-52-13" name="__codelineno-52-13" href="#__codelineno-52-13"></a>                <span class="c1"># If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-52-13" name="__codelineno-52-13" href="#__codelineno-52-13"></a>                <span class="c1"># If the coin value exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-52-14" name="__codelineno-52-14" href="#__codelineno-52-14"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">]</span>
 <a id="__codelineno-52-15" name="__codelineno-52-15" href="#__codelineno-52-15"></a>            <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-52-16" name="__codelineno-52-16" href="#__codelineno-52-16"></a>                <span class="c1"># Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-52-16" name="__codelineno-52-16" href="#__codelineno-52-16"></a>                <span class="c1"># Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-52-17" name="__codelineno-52-17" href="#__codelineno-52-17"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">]</span> <span class="o">+</span> <span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span> <span class="o">-</span> <span class="n">coins</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]]</span>
 <a id="__codelineno-52-18" name="__codelineno-52-18" href="#__codelineno-52-18"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">n</span><span class="p">][</span><span class="n">amt</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Coin change II: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Coin change II: dynamic programming */</span>
 <a id="__codelineno-53-2" name="__codelineno-53-2" href="#__codelineno-53-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeIIDP</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-53-4" name="__codelineno-53-4" href="#__codelineno-53-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-53-4" name="__codelineno-53-4" href="#__codelineno-53-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-53-5" name="__codelineno-53-5" href="#__codelineno-53-5"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">));</span>
-<a id="__codelineno-53-6" name="__codelineno-53-6" href="#__codelineno-53-6"></a><span class="w">    </span><span class="c1">// Initialize first column</span>
+<a id="__codelineno-53-6" name="__codelineno-53-6" href="#__codelineno-53-6"></a><span class="w">    </span><span class="c1">// Initialize the first column</span>
 <a id="__codelineno-53-7" name="__codelineno-53-7" href="#__codelineno-53-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-8" name="__codelineno-53-8" href="#__codelineno-53-8"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-53-9" name="__codelineno-53-9" href="#__codelineno-53-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-53-10" name="__codelineno-53-10" href="#__codelineno-53-10"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-53-10" name="__codelineno-53-10" href="#__codelineno-53-10"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-53-11" name="__codelineno-53-11" href="#__codelineno-53-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-12" name="__codelineno-53-12" href="#__codelineno-53-12"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-13" name="__codelineno-53-13" href="#__codelineno-53-13"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-53-14" name="__codelineno-53-14" href="#__codelineno-53-14"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-53-14" name="__codelineno-53-14" href="#__codelineno-53-14"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-53-15" name="__codelineno-53-15" href="#__codelineno-53-15"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-53-16" name="__codelineno-53-16" href="#__codelineno-53-16"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-53-17" name="__codelineno-53-17" href="#__codelineno-53-17"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-53-17" name="__codelineno-53-17" href="#__codelineno-53-17"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-53-18" name="__codelineno-53-18" href="#__codelineno-53-18"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]];</span>
 <a id="__codelineno-53-19" name="__codelineno-53-19" href="#__codelineno-53-19"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-53-20" name="__codelineno-53-20" href="#__codelineno-53-20"></a><span class="w">        </span><span class="p">}</span>
@@ -6438,23 +6530,23 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Coin change II: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Coin change II: dynamic programming */</span>
 <a id="__codelineno-54-2" name="__codelineno-54-2" href="#__codelineno-54-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeIIDP</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
-<a id="__codelineno-54-4" name="__codelineno-54-4" href="#__codelineno-54-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-54-4" name="__codelineno-54-4" href="#__codelineno-54-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-54-5" name="__codelineno-54-5" href="#__codelineno-54-5"></a><span class="w">    </span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-54-6" name="__codelineno-54-6" href="#__codelineno-54-6"></a><span class="w">    </span><span class="c1">// Initialize first column</span>
+<a id="__codelineno-54-6" name="__codelineno-54-6" href="#__codelineno-54-6"></a><span class="w">    </span><span class="c1">// Initialize the first column</span>
 <a id="__codelineno-54-7" name="__codelineno-54-7" href="#__codelineno-54-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-8" name="__codelineno-54-8" href="#__codelineno-54-8"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-54-9" name="__codelineno-54-9" href="#__codelineno-54-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-54-10" name="__codelineno-54-10" href="#__codelineno-54-10"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-54-10" name="__codelineno-54-10" href="#__codelineno-54-10"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-54-11" name="__codelineno-54-11" href="#__codelineno-54-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-12" name="__codelineno-54-12" href="#__codelineno-54-12"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-13" name="__codelineno-54-13" href="#__codelineno-54-13"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-54-14" name="__codelineno-54-14" href="#__codelineno-54-14"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-54-14" name="__codelineno-54-14" href="#__codelineno-54-14"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-54-15" name="__codelineno-54-15" href="#__codelineno-54-15"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-54-16" name="__codelineno-54-16" href="#__codelineno-54-16"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-54-17" name="__codelineno-54-17" href="#__codelineno-54-17"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-54-17" name="__codelineno-54-17" href="#__codelineno-54-17"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-54-18" name="__codelineno-54-18" href="#__codelineno-54-18"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span><span class="p">;</span>
 <a id="__codelineno-54-19" name="__codelineno-54-19" href="#__codelineno-54-19"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-54-20" name="__codelineno-54-20" href="#__codelineno-54-20"></a><span class="w">        </span><span class="p">}</span>
@@ -6464,23 +6556,23 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Coin change II: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Coin change II: dynamic programming */</span>
 <a id="__codelineno-55-2" name="__codelineno-55-2" href="#__codelineno-55-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">CoinChangeIIDP</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
-<a id="__codelineno-55-4" name="__codelineno-55-4" href="#__codelineno-55-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-55-4" name="__codelineno-55-4" href="#__codelineno-55-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-55-5" name="__codelineno-55-5" href="#__codelineno-55-5"></a><span class="w">    </span><span class="kt">int</span><span class="p">[,]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-55-6" name="__codelineno-55-6" href="#__codelineno-55-6"></a><span class="w">    </span><span class="c1">// Initialize first column</span>
+<a id="__codelineno-55-6" name="__codelineno-55-6" href="#__codelineno-55-6"></a><span class="w">    </span><span class="c1">// Initialize the first column</span>
 <a id="__codelineno-55-7" name="__codelineno-55-7" href="#__codelineno-55-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-8" name="__codelineno-55-8" href="#__codelineno-55-8"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-55-9" name="__codelineno-55-9" href="#__codelineno-55-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-55-10" name="__codelineno-55-10" href="#__codelineno-55-10"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-55-10" name="__codelineno-55-10" href="#__codelineno-55-10"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-55-11" name="__codelineno-55-11" href="#__codelineno-55-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-12" name="__codelineno-55-12" href="#__codelineno-55-12"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-13" name="__codelineno-55-13" href="#__codelineno-55-13"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-55-14" name="__codelineno-55-14" href="#__codelineno-55-14"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-55-14" name="__codelineno-55-14" href="#__codelineno-55-14"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-55-15" name="__codelineno-55-15" href="#__codelineno-55-15"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-55-16" name="__codelineno-55-16" href="#__codelineno-55-16"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-55-17" name="__codelineno-55-17" href="#__codelineno-55-17"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-55-17" name="__codelineno-55-17" href="#__codelineno-55-17"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-55-18" name="__codelineno-55-18" href="#__codelineno-55-18"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]];</span>
 <a id="__codelineno-55-19" name="__codelineno-55-19" href="#__codelineno-55-19"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-55-20" name="__codelineno-55-20" href="#__codelineno-55-20"></a><span class="w">        </span><span class="p">}</span>
@@ -6490,26 +6582,26 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Coin change II: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Coin change II: dynamic programming */</span>
 <a id="__codelineno-56-2" name="__codelineno-56-2" href="#__codelineno-56-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">coinChangeIIDP</span><span class="p">(</span><span class="nx">coins</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">coins</span><span class="p">)</span>
-<a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-56-5" name="__codelineno-56-5" href="#__codelineno-56-5"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-56-6" name="__codelineno-56-6" href="#__codelineno-56-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-7" name="__codelineno-56-7" href="#__codelineno-56-7"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-56-8" name="__codelineno-56-8" href="#__codelineno-56-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-56-9" name="__codelineno-56-9" href="#__codelineno-56-9"></a><span class="w">    </span><span class="c1">// Initialize first column</span>
+<a id="__codelineno-56-9" name="__codelineno-56-9" href="#__codelineno-56-9"></a><span class="w">    </span><span class="c1">// Initialize the first column</span>
 <a id="__codelineno-56-10" name="__codelineno-56-10" href="#__codelineno-56-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-11" name="__codelineno-56-11" href="#__codelineno-56-11"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-56-12" name="__codelineno-56-12" href="#__codelineno-56-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-56-13" name="__codelineno-56-13" href="#__codelineno-56-13"></a><span class="w">    </span><span class="c1">// State transition: rest of the rows and columns</span>
+<a id="__codelineno-56-13" name="__codelineno-56-13" href="#__codelineno-56-13"></a><span class="w">    </span><span class="c1">// State transitions: remaining rows and columns</span>
 <a id="__codelineno-56-14" name="__codelineno-56-14" href="#__codelineno-56-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-15" name="__codelineno-56-15" href="#__codelineno-56-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-16" name="__codelineno-56-16" href="#__codelineno-56-16"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-56-17" name="__codelineno-56-17" href="#__codelineno-56-17"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-56-17" name="__codelineno-56-17" href="#__codelineno-56-17"></a><span class="w">                </span><span class="c1">// If the coin exceeds the current target amount a, do not choose coin i</span>
 <a id="__codelineno-56-18" name="__codelineno-56-18" href="#__codelineno-56-18"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span>
 <a id="__codelineno-56-19" name="__codelineno-56-19" href="#__codelineno-56-19"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-56-20" name="__codelineno-56-20" href="#__codelineno-56-20"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-56-20" name="__codelineno-56-20" href="#__codelineno-56-20"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-56-21" name="__codelineno-56-21" href="#__codelineno-56-21"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="o">-</span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span>
 <a id="__codelineno-56-22" name="__codelineno-56-22" href="#__codelineno-56-22"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-56-23" name="__codelineno-56-23" href="#__codelineno-56-23"></a><span class="w">        </span><span class="p">}</span>
@@ -6519,23 +6611,23 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Coin change II: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Coin change II: dynamic programming */</span>
 <a id="__codelineno-57-2" name="__codelineno-57-2" href="#__codelineno-57-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">coinChangeIIDP</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">amt</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="bp">count</span>
-<a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-57-5" name="__codelineno-57-5" href="#__codelineno-57-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">),</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-57-6" name="__codelineno-57-6" href="#__codelineno-57-6"></a><span class="w">    </span><span class="c1">// Initialize first column</span>
+<a id="__codelineno-57-6" name="__codelineno-57-6" href="#__codelineno-57-6"></a><span class="w">    </span><span class="c1">// Initialize the first column</span>
 <a id="__codelineno-57-7" name="__codelineno-57-7" href="#__codelineno-57-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-8" name="__codelineno-57-8" href="#__codelineno-57-8"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-57-9" name="__codelineno-57-9" href="#__codelineno-57-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-57-10" name="__codelineno-57-10" href="#__codelineno-57-10"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-57-10" name="__codelineno-57-10" href="#__codelineno-57-10"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-57-11" name="__codelineno-57-11" href="#__codelineno-57-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-12" name="__codelineno-57-12" href="#__codelineno-57-12"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-13" name="__codelineno-57-13" href="#__codelineno-57-13"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-57-14" name="__codelineno-57-14" href="#__codelineno-57-14"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-57-14" name="__codelineno-57-14" href="#__codelineno-57-14"></a><span class="w">                </span><span class="c1">// If the coin exceeds the current target amount a, do not choose coin i</span>
 <a id="__codelineno-57-15" name="__codelineno-57-15" href="#__codelineno-57-15"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">]</span>
 <a id="__codelineno-57-16" name="__codelineno-57-16" href="#__codelineno-57-16"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-57-17" name="__codelineno-57-17" href="#__codelineno-57-17"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-57-17" name="__codelineno-57-17" href="#__codelineno-57-17"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-57-18" name="__codelineno-57-18" href="#__codelineno-57-18"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span>
 <a id="__codelineno-57-19" name="__codelineno-57-19" href="#__codelineno-57-19"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-57-20" name="__codelineno-57-20" href="#__codelineno-57-20"></a><span class="w">        </span><span class="p">}</span>
@@ -6545,25 +6637,25 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Coin change II: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Coin change II: dynamic programming */</span>
 <a id="__codelineno-58-2" name="__codelineno-58-2" href="#__codelineno-58-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">coinChangeIIDP</span><span class="p">(</span><span class="nx">coins</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">coins</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-58-4" name="__codelineno-58-4" href="#__codelineno-58-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-58-4" name="__codelineno-58-4" href="#__codelineno-58-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-58-5" name="__codelineno-58-5" href="#__codelineno-58-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span>
 <a id="__codelineno-58-6" name="__codelineno-58-6" href="#__codelineno-58-6"></a><span class="w">        </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span>
 <a id="__codelineno-58-7" name="__codelineno-58-7" href="#__codelineno-58-7"></a><span class="w">    </span><span class="p">);</span>
-<a id="__codelineno-58-8" name="__codelineno-58-8" href="#__codelineno-58-8"></a><span class="w">    </span><span class="c1">// Initialize first column</span>
+<a id="__codelineno-58-8" name="__codelineno-58-8" href="#__codelineno-58-8"></a><span class="w">    </span><span class="c1">// Initialize the first column</span>
 <a id="__codelineno-58-9" name="__codelineno-58-9" href="#__codelineno-58-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-10" name="__codelineno-58-10" href="#__codelineno-58-10"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-58-11" name="__codelineno-58-11" href="#__codelineno-58-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-58-12" name="__codelineno-58-12" href="#__codelineno-58-12"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-58-12" name="__codelineno-58-12" href="#__codelineno-58-12"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-58-13" name="__codelineno-58-13" href="#__codelineno-58-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-14" name="__codelineno-58-14" href="#__codelineno-58-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-15" name="__codelineno-58-15" href="#__codelineno-58-15"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-58-16" name="__codelineno-58-16" href="#__codelineno-58-16"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-58-16" name="__codelineno-58-16" href="#__codelineno-58-16"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-58-17" name="__codelineno-58-17" href="#__codelineno-58-17"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">a</span><span class="p">];</span>
 <a id="__codelineno-58-18" name="__codelineno-58-18" href="#__codelineno-58-18"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-58-19" name="__codelineno-58-19" href="#__codelineno-58-19"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-58-19" name="__codelineno-58-19" href="#__codelineno-58-19"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-58-20" name="__codelineno-58-20" href="#__codelineno-58-20"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]];</span>
 <a id="__codelineno-58-21" name="__codelineno-58-21" href="#__codelineno-58-21"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-58-22" name="__codelineno-58-22" href="#__codelineno-58-22"></a><span class="w">        </span><span class="p">}</span>
@@ -6573,25 +6665,25 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Coin change II: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Coin change II: dynamic programming */</span>
 <a id="__codelineno-59-2" name="__codelineno-59-2" href="#__codelineno-59-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">coinChangeIIDP</span><span class="p">(</span><span class="nx">coins</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">coins</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-59-4" name="__codelineno-59-4" href="#__codelineno-59-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-59-4" name="__codelineno-59-4" href="#__codelineno-59-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-59-5" name="__codelineno-59-5" href="#__codelineno-59-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span>
 <a id="__codelineno-59-6" name="__codelineno-59-6" href="#__codelineno-59-6"></a><span class="w">        </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span>
 <a id="__codelineno-59-7" name="__codelineno-59-7" href="#__codelineno-59-7"></a><span class="w">    </span><span class="p">);</span>
-<a id="__codelineno-59-8" name="__codelineno-59-8" href="#__codelineno-59-8"></a><span class="w">    </span><span class="c1">// Initialize first column</span>
+<a id="__codelineno-59-8" name="__codelineno-59-8" href="#__codelineno-59-8"></a><span class="w">    </span><span class="c1">// Initialize the first column</span>
 <a id="__codelineno-59-9" name="__codelineno-59-9" href="#__codelineno-59-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-10" name="__codelineno-59-10" href="#__codelineno-59-10"></a><span class="w">        </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-59-11" name="__codelineno-59-11" href="#__codelineno-59-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-59-12" name="__codelineno-59-12" href="#__codelineno-59-12"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-59-12" name="__codelineno-59-12" href="#__codelineno-59-12"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-59-13" name="__codelineno-59-13" href="#__codelineno-59-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-14" name="__codelineno-59-14" href="#__codelineno-59-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-15" name="__codelineno-59-15" href="#__codelineno-59-15"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-59-16" name="__codelineno-59-16" href="#__codelineno-59-16"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-59-16" name="__codelineno-59-16" href="#__codelineno-59-16"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-59-17" name="__codelineno-59-17" href="#__codelineno-59-17"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">a</span><span class="p">];</span>
 <a id="__codelineno-59-18" name="__codelineno-59-18" href="#__codelineno-59-18"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-59-19" name="__codelineno-59-19" href="#__codelineno-59-19"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-59-19" name="__codelineno-59-19" href="#__codelineno-59-19"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-59-20" name="__codelineno-59-20" href="#__codelineno-59-20"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">][</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]];</span>
 <a id="__codelineno-59-21" name="__codelineno-59-21" href="#__codelineno-59-21"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-59-22" name="__codelineno-59-22" href="#__codelineno-59-22"></a><span class="w">        </span><span class="p">}</span>
@@ -6601,23 +6693,23 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Coin change II: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Coin change II: dynamic programming */</span>
 <a id="__codelineno-60-2" name="__codelineno-60-2" href="#__codelineno-60-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">coinChangeIIDP</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-3" name="__codelineno-60-3" href="#__codelineno-60-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
-<a id="__codelineno-60-4" name="__codelineno-60-4" href="#__codelineno-60-4"></a><span class="w">  </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-60-4" name="__codelineno-60-4" href="#__codelineno-60-4"></a><span class="w">  </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-60-5" name="__codelineno-60-5" href="#__codelineno-60-5"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">));</span>
-<a id="__codelineno-60-6" name="__codelineno-60-6" href="#__codelineno-60-6"></a><span class="w">  </span><span class="c1">// Initialize first column</span>
+<a id="__codelineno-60-6" name="__codelineno-60-6" href="#__codelineno-60-6"></a><span class="w">  </span><span class="c1">// Initialize the first column</span>
 <a id="__codelineno-60-7" name="__codelineno-60-7" href="#__codelineno-60-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-8" name="__codelineno-60-8" href="#__codelineno-60-8"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-60-9" name="__codelineno-60-9" href="#__codelineno-60-9"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-60-10" name="__codelineno-60-10" href="#__codelineno-60-10"></a><span class="w">  </span><span class="c1">// State transition</span>
+<a id="__codelineno-60-10" name="__codelineno-60-10" href="#__codelineno-60-10"></a><span class="w">  </span><span class="c1">// State transitions</span>
 <a id="__codelineno-60-11" name="__codelineno-60-11" href="#__codelineno-60-11"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-12" name="__codelineno-60-12" href="#__codelineno-60-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-13" name="__codelineno-60-13" href="#__codelineno-60-13"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-60-14" name="__codelineno-60-14" href="#__codelineno-60-14"></a><span class="w">        </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-60-14" name="__codelineno-60-14" href="#__codelineno-60-14"></a><span class="w">        </span><span class="c1">// If the coin exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-60-15" name="__codelineno-60-15" href="#__codelineno-60-15"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-60-16" name="__codelineno-60-16" href="#__codelineno-60-16"></a><span class="w">      </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-60-17" name="__codelineno-60-17" href="#__codelineno-60-17"></a><span class="w">        </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-60-17" name="__codelineno-60-17" href="#__codelineno-60-17"></a><span class="w">        </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-60-18" name="__codelineno-60-18" href="#__codelineno-60-18"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]];</span>
 <a id="__codelineno-60-19" name="__codelineno-60-19" href="#__codelineno-60-19"></a><span class="w">      </span><span class="p">}</span>
 <a id="__codelineno-60-20" name="__codelineno-60-20" href="#__codelineno-60-20"></a><span class="w">    </span><span class="p">}</span>
@@ -6627,23 +6719,23 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Coin change II: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Coin change II: dynamic programming */</span>
 <a id="__codelineno-61-2" name="__codelineno-61-2" href="#__codelineno-61-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">coin_change_ii_dp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">amt</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-3" name="__codelineno-61-3" href="#__codelineno-61-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
-<a id="__codelineno-61-4" name="__codelineno-61-4" href="#__codelineno-61-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-61-4" name="__codelineno-61-4" href="#__codelineno-61-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-61-5" name="__codelineno-61-5" href="#__codelineno-61-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-61-6" name="__codelineno-61-6" href="#__codelineno-61-6"></a><span class="w">    </span><span class="c1">// Initialize first column</span>
+<a id="__codelineno-61-6" name="__codelineno-61-6" href="#__codelineno-61-6"></a><span class="w">    </span><span class="c1">// Initialize the first column</span>
 <a id="__codelineno-61-7" name="__codelineno-61-7" href="#__codelineno-61-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-8" name="__codelineno-61-8" href="#__codelineno-61-8"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-61-9" name="__codelineno-61-9" href="#__codelineno-61-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-61-10" name="__codelineno-61-10" href="#__codelineno-61-10"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-61-10" name="__codelineno-61-10" href="#__codelineno-61-10"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-61-11" name="__codelineno-61-11" href="#__codelineno-61-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-12" name="__codelineno-61-12" href="#__codelineno-61-12"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">amt</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-13" name="__codelineno-61-13" href="#__codelineno-61-13"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-61-14" name="__codelineno-61-14" href="#__codelineno-61-14"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-61-14" name="__codelineno-61-14" href="#__codelineno-61-14"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-61-15" name="__codelineno-61-15" href="#__codelineno-61-15"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-61-16" name="__codelineno-61-16" href="#__codelineno-61-16"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-61-17" name="__codelineno-61-17" href="#__codelineno-61-17"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-61-17" name="__codelineno-61-17" href="#__codelineno-61-17"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-61-18" name="__codelineno-61-18" href="#__codelineno-61-18"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">];</span>
 <a id="__codelineno-61-19" name="__codelineno-61-19" href="#__codelineno-61-19"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-61-20" name="__codelineno-61-20" href="#__codelineno-61-20"></a><span class="w">        </span><span class="p">}</span>
@@ -6653,26 +6745,26 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Coin change II: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Coin change II: dynamic programming */</span>
 <a id="__codelineno-62-2" name="__codelineno-62-2" href="#__codelineno-62-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeIIDP</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">coins</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">coinsSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-3" name="__codelineno-62-3" href="#__codelineno-62-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coinsSize</span><span class="p">;</span>
-<a id="__codelineno-62-4" name="__codelineno-62-4" href="#__codelineno-62-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-62-4" name="__codelineno-62-4" href="#__codelineno-62-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-62-5" name="__codelineno-62-5" href="#__codelineno-62-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">**</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">((</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">));</span>
 <a id="__codelineno-62-6" name="__codelineno-62-6" href="#__codelineno-62-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-7" name="__codelineno-62-7" href="#__codelineno-62-7"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-62-8" name="__codelineno-62-8" href="#__codelineno-62-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-62-9" name="__codelineno-62-9" href="#__codelineno-62-9"></a><span class="w">    </span><span class="c1">// Initialize first column</span>
+<a id="__codelineno-62-9" name="__codelineno-62-9" href="#__codelineno-62-9"></a><span class="w">    </span><span class="c1">// Initialize the first column</span>
 <a id="__codelineno-62-10" name="__codelineno-62-10" href="#__codelineno-62-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-11" name="__codelineno-62-11" href="#__codelineno-62-11"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-62-12" name="__codelineno-62-12" href="#__codelineno-62-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-62-13" name="__codelineno-62-13" href="#__codelineno-62-13"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-62-13" name="__codelineno-62-13" href="#__codelineno-62-13"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-62-14" name="__codelineno-62-14" href="#__codelineno-62-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-15" name="__codelineno-62-15" href="#__codelineno-62-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-16" name="__codelineno-62-16" href="#__codelineno-62-16"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-62-17" name="__codelineno-62-17" href="#__codelineno-62-17"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-62-17" name="__codelineno-62-17" href="#__codelineno-62-17"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-62-18" name="__codelineno-62-18" href="#__codelineno-62-18"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-62-19" name="__codelineno-62-19" href="#__codelineno-62-19"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-62-20" name="__codelineno-62-20" href="#__codelineno-62-20"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-62-20" name="__codelineno-62-20" href="#__codelineno-62-20"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-62-21" name="__codelineno-62-21" href="#__codelineno-62-21"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]];</span>
 <a id="__codelineno-62-22" name="__codelineno-62-22" href="#__codelineno-62-22"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-62-23" name="__codelineno-62-23" href="#__codelineno-62-23"></a><span class="w">        </span><span class="p">}</span>
@@ -6688,23 +6780,23 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Coin change II: Dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Coin change II: dynamic programming */</span>
 <a id="__codelineno-63-2" name="__codelineno-63-2" href="#__codelineno-63-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">coinChangeIIDP</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">amt</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-3" name="__codelineno-63-3" href="#__codelineno-63-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="na">size</span>
-<a id="__codelineno-63-4" name="__codelineno-63-4" href="#__codelineno-63-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-63-4" name="__codelineno-63-4" href="#__codelineno-63-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-63-5" name="__codelineno-63-5" href="#__codelineno-63-5"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Array</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-63-6" name="__codelineno-63-6" href="#__codelineno-63-6"></a><span class="w">    </span><span class="c1">// Initialize first column</span>
+<a id="__codelineno-63-6" name="__codelineno-63-6" href="#__codelineno-63-6"></a><span class="w">    </span><span class="c1">// Initialize the first column</span>
 <a id="__codelineno-63-7" name="__codelineno-63-7" href="#__codelineno-63-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-8" name="__codelineno-63-8" href="#__codelineno-63-8"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="m">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-63-9" name="__codelineno-63-9" href="#__codelineno-63-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-63-10" name="__codelineno-63-10" href="#__codelineno-63-10"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-63-10" name="__codelineno-63-10" href="#__codelineno-63-10"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-63-11" name="__codelineno-63-11" href="#__codelineno-63-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-12" name="__codelineno-63-12" href="#__codelineno-63-12"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-13" name="__codelineno-63-13" href="#__codelineno-63-13"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-63-14" name="__codelineno-63-14" href="#__codelineno-63-14"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-63-14" name="__codelineno-63-14" href="#__codelineno-63-14"></a><span class="w">                </span><span class="c1">// If the coin value exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-63-15" name="__codelineno-63-15" href="#__codelineno-63-15"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">][</span><span class="n">a</span><span class="o">]</span>
 <a id="__codelineno-63-16" name="__codelineno-63-16" href="#__codelineno-63-16"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-63-17" name="__codelineno-63-17" href="#__codelineno-63-17"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-63-17" name="__codelineno-63-17" href="#__codelineno-63-17"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-63-18" name="__codelineno-63-18" href="#__codelineno-63-18"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]]</span>
 <a id="__codelineno-63-19" name="__codelineno-63-19" href="#__codelineno-63-19"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-63-20" name="__codelineno-63-20" href="#__codelineno-63-20"></a><span class="w">        </span><span class="p">}</span>
@@ -6717,18 +6809,18 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 <div class="highlight"><span class="filename">coin_change_ii.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Coin change II: dynamic programming ###</span>
 <a id="__codelineno-64-2" name="__codelineno-64-2" href="#__codelineno-64-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">coin_change_ii_dp</span><span class="p">(</span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span>
 <a id="__codelineno-64-3" name="__codelineno-64-3" href="#__codelineno-64-3"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-64-4" name="__codelineno-64-4" href="#__codelineno-64-4"></a><span class="w">  </span><span class="c1"># Initialize dp table</span>
+<a id="__codelineno-64-4" name="__codelineno-64-4" href="#__codelineno-64-4"></a><span class="w">  </span><span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-64-5" name="__codelineno-64-5" href="#__codelineno-64-5"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-64-6" name="__codelineno-64-6" href="#__codelineno-64-6"></a><span class="w">  </span><span class="c1"># Initialize first column</span>
+<a id="__codelineno-64-6" name="__codelineno-64-6" href="#__codelineno-64-6"></a><span class="w">  </span><span class="c1"># Initialize the first column</span>
 <a id="__codelineno-64-7" name="__codelineno-64-7" href="#__codelineno-64-7"></a><span class="w">  </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-64-8" name="__codelineno-64-8" href="#__codelineno-64-8"></a><span class="w">  </span><span class="c1"># State transition</span>
+<a id="__codelineno-64-8" name="__codelineno-64-8" href="#__codelineno-64-8"></a><span class="w">  </span><span class="c1"># State transitions</span>
 <a id="__codelineno-64-9" name="__codelineno-64-9" href="#__codelineno-64-9"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-64-10" name="__codelineno-64-10" href="#__codelineno-64-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-64-11" name="__codelineno-64-11" href="#__codelineno-64-11"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span>
-<a id="__codelineno-64-12" name="__codelineno-64-12" href="#__codelineno-64-12"></a><span class="w">        </span><span class="c1"># If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-64-12" name="__codelineno-64-12" href="#__codelineno-64-12"></a><span class="w">        </span><span class="c1"># If the coin exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-64-13" name="__codelineno-64-13" href="#__codelineno-64-13"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">a</span><span class="o">]</span>
 <a id="__codelineno-64-14" name="__codelineno-64-14" href="#__codelineno-64-14"></a><span class="w">      </span><span class="k">else</span>
-<a id="__codelineno-64-15" name="__codelineno-64-15" href="#__codelineno-64-15"></a><span class="w">        </span><span class="c1"># Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-64-15" name="__codelineno-64-15" href="#__codelineno-64-15"></a><span class="w">        </span><span class="c1"># Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-64-16" name="__codelineno-64-16" href="#__codelineno-64-16"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">][</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span>
 <a id="__codelineno-64-17" name="__codelineno-64-17" href="#__codelineno-64-17"></a><span class="w">      </span><span class="k">end</span>
 <a id="__codelineno-64-18" name="__codelineno-64-18" href="#__codelineno-64-18"></a><span class="w">    </span><span class="k">end</span>
@@ -6739,45 +6831,50 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </div>
 </div>
 </div>
-<h3 id="3-space-optimization_2">3. &nbsp; Space Optimization<a class="headerlink" href="#3-space-optimization_2" title="Permanent link">&para;</a></h3>
-<p>The space optimization is handled in the same way, just delete the coin dimension:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20coin_change_ii_dp%28coins%3A%20list%5Bint%5D,%20amt%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Coin%20change%20II%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28coins%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%20%28amt%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20Initialize%20the%20first%20column%0A%20%20%20%20for%20i%20in%20range%28n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B0%5D%20%3D%201%0A%20%20%20%20%23%20State%20transitions%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20a%20in%20range%281,%20amt%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20coins%5Bi%20-%201%5D%20%3E%20a%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20the%20coin%20value%20exceeds%20the%20current%20amount%20a,%20do%20not%20choose%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Ba%5D%20%3D%20dp%5Bi%20-%201%5D%5Ba%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Add%20the%20numbers%20of%20ways%20without%20and%20with%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Ba%5D%20%3D%20dp%5Bi%20-%201%5D%5Ba%5D%20%2B%20dp%5Bi%5D%5Ba%20-%20coins%5Bi%20-%201%5D%5D%0A%20%20%20%20return%20dp%5Bn%5D%5Bamt%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20coins%20%3D%20%5B1,%202,%205%5D%0A%20%20%20%20amt%20%3D%205%0A%0A%20%20%20%20%23%20Dynamic%20programming%0A%20%20%20%20res%20%3D%20coin_change_ii_dp%28coins,%20amt%29%0A%20%20%20%20print%28f%22Number%20of%20coin%20combinations%20that%20make%20the%20target%20amount%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20coin_change_ii_dp%28coins%3A%20list%5Bint%5D,%20amt%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Coin%20change%20II%3A%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28coins%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B%5B0%5D%20%2A%20%28amt%20%2B%201%29%20for%20_%20in%20range%28n%20%2B%201%29%5D%0A%20%20%20%20%23%20Initialize%20the%20first%20column%0A%20%20%20%20for%20i%20in%20range%28n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20dp%5Bi%5D%5B0%5D%20%3D%201%0A%20%20%20%20%23%20State%20transitions%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20a%20in%20range%281,%20amt%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20coins%5Bi%20-%201%5D%20%3E%20a%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20the%20coin%20value%20exceeds%20the%20current%20amount%20a,%20do%20not%20choose%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Ba%5D%20%3D%20dp%5Bi%20-%201%5D%5Ba%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Add%20the%20numbers%20of%20ways%20without%20and%20with%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Bi%5D%5Ba%5D%20%3D%20dp%5Bi%20-%201%5D%5Ba%5D%20%2B%20dp%5Bi%5D%5Ba%20-%20coins%5Bi%20-%201%5D%5D%0A%20%20%20%20return%20dp%5Bn%5D%5Bamt%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20coins%20%3D%20%5B1,%202,%205%5D%0A%20%20%20%20amt%20%3D%205%0A%0A%20%20%20%20%23%20Dynamic%20programming%0A%20%20%20%20res%20%3D%20coin_change_ii_dp%28coins,%20amt%29%0A%20%20%20%20print%28f%22Number%20of%20coin%20combinations%20that%20make%20the%20target%20amount%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="3-space-optimization_2">3. &nbsp; Space optimization<a class="headerlink" href="#3-space-optimization_2" title="Permanent link">&para;</a></h3>
+<p>The space optimization is the same: remove the coin dimension:</p>
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 <div class="highlight"><span class="filename">coin_change_ii.py</span><pre><span></span><code><a id="__codelineno-65-1" name="__codelineno-65-1" href="#__codelineno-65-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">coin_change_ii_dp_comp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">amt</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-65-2" name="__codelineno-65-2" href="#__codelineno-65-2"></a><span class="w">    </span><span class="sd">"""Coin change II: Space-optimized dynamic programming"""</span>
+<a id="__codelineno-65-2" name="__codelineno-65-2" href="#__codelineno-65-2"></a><span class="w">    </span><span class="sd">"""Coin change II: space-optimized dynamic programming"""</span>
 <a id="__codelineno-65-3" name="__codelineno-65-3" href="#__codelineno-65-3"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coins</span><span class="p">)</span>
-<a id="__codelineno-65-4" name="__codelineno-65-4" href="#__codelineno-65-4"></a>    <span class="c1"># Initialize dp table</span>
+<a id="__codelineno-65-4" name="__codelineno-65-4" href="#__codelineno-65-4"></a>    <span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-65-5" name="__codelineno-65-5" href="#__codelineno-65-5"></a>    <span class="n">dp</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-65-6" name="__codelineno-65-6" href="#__codelineno-65-6"></a>    <span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
-<a id="__codelineno-65-7" name="__codelineno-65-7" href="#__codelineno-65-7"></a>    <span class="c1"># State transition</span>
+<a id="__codelineno-65-7" name="__codelineno-65-7" href="#__codelineno-65-7"></a>    <span class="c1"># State transitions</span>
 <a id="__codelineno-65-8" name="__codelineno-65-8" href="#__codelineno-65-8"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-<a id="__codelineno-65-9" name="__codelineno-65-9" href="#__codelineno-65-9"></a>        <span class="c1"># Traverse in forward order</span>
+<a id="__codelineno-65-9" name="__codelineno-65-9" href="#__codelineno-65-9"></a>        <span class="c1"># Traverse in increasing order</span>
 <a id="__codelineno-65-10" name="__codelineno-65-10" href="#__codelineno-65-10"></a>        <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">amt</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-65-11" name="__codelineno-65-11" href="#__codelineno-65-11"></a>            <span class="k">if</span> <span class="n">coins</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">a</span><span class="p">:</span>
-<a id="__codelineno-65-12" name="__codelineno-65-12" href="#__codelineno-65-12"></a>                <span class="c1"># If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-65-12" name="__codelineno-65-12" href="#__codelineno-65-12"></a>                <span class="c1"># If the coin value exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-65-13" name="__codelineno-65-13" href="#__codelineno-65-13"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span>
 <a id="__codelineno-65-14" name="__codelineno-65-14" href="#__codelineno-65-14"></a>            <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-65-15" name="__codelineno-65-15" href="#__codelineno-65-15"></a>                <span class="c1"># Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-65-15" name="__codelineno-65-15" href="#__codelineno-65-15"></a>                <span class="c1"># Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-65-16" name="__codelineno-65-16" href="#__codelineno-65-16"></a>                <span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">+</span> <span class="n">dp</span><span class="p">[</span><span class="n">a</span> <span class="o">-</span> <span class="n">coins</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]]</span>
 <a id="__codelineno-65-17" name="__codelineno-65-17" href="#__codelineno-65-17"></a>    <span class="k">return</span> <span class="n">dp</span><span class="p">[</span><span class="n">amt</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.cpp</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="cm">/* Coin change II: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.cpp</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="cm">/* Coin change II: space-optimized dynamic programming */</span>
 <a id="__codelineno-66-2" name="__codelineno-66-2" href="#__codelineno-66-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeIIDPComp</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-66-3" name="__codelineno-66-3" href="#__codelineno-66-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-66-4" name="__codelineno-66-4" href="#__codelineno-66-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-66-4" name="__codelineno-66-4" href="#__codelineno-66-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-66-5" name="__codelineno-66-5" href="#__codelineno-66-5"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-66-6" name="__codelineno-66-6" href="#__codelineno-66-6"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-66-7" name="__codelineno-66-7" href="#__codelineno-66-7"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-66-7" name="__codelineno-66-7" href="#__codelineno-66-7"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-66-8" name="__codelineno-66-8" href="#__codelineno-66-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-66-9" name="__codelineno-66-9" href="#__codelineno-66-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-66-10" name="__codelineno-66-10" href="#__codelineno-66-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-66-11" name="__codelineno-66-11" href="#__codelineno-66-11"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-66-11" name="__codelineno-66-11" href="#__codelineno-66-11"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-66-12" name="__codelineno-66-12" href="#__codelineno-66-12"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-66-13" name="__codelineno-66-13" href="#__codelineno-66-13"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-66-14" name="__codelineno-66-14" href="#__codelineno-66-14"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-66-14" name="__codelineno-66-14" href="#__codelineno-66-14"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-66-15" name="__codelineno-66-15" href="#__codelineno-66-15"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]];</span>
 <a id="__codelineno-66-16" name="__codelineno-66-16" href="#__codelineno-66-16"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-66-17" name="__codelineno-66-17" href="#__codelineno-66-17"></a><span class="w">        </span><span class="p">}</span>
@@ -6787,20 +6884,20 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.java</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="cm">/* Coin change II: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.java</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="cm">/* Coin change II: space-optimized dynamic programming */</span>
 <a id="__codelineno-67-2" name="__codelineno-67-2" href="#__codelineno-67-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeIIDPComp</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-67-3" name="__codelineno-67-3" href="#__codelineno-67-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
-<a id="__codelineno-67-4" name="__codelineno-67-4" href="#__codelineno-67-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-67-4" name="__codelineno-67-4" href="#__codelineno-67-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-67-5" name="__codelineno-67-5" href="#__codelineno-67-5"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-67-6" name="__codelineno-67-6" href="#__codelineno-67-6"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-67-7" name="__codelineno-67-7" href="#__codelineno-67-7"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-67-7" name="__codelineno-67-7" href="#__codelineno-67-7"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-67-8" name="__codelineno-67-8" href="#__codelineno-67-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-67-9" name="__codelineno-67-9" href="#__codelineno-67-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-67-10" name="__codelineno-67-10" href="#__codelineno-67-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-67-11" name="__codelineno-67-11" href="#__codelineno-67-11"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-67-11" name="__codelineno-67-11" href="#__codelineno-67-11"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-67-12" name="__codelineno-67-12" href="#__codelineno-67-12"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-67-13" name="__codelineno-67-13" href="#__codelineno-67-13"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-67-14" name="__codelineno-67-14" href="#__codelineno-67-14"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-67-14" name="__codelineno-67-14" href="#__codelineno-67-14"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-67-15" name="__codelineno-67-15" href="#__codelineno-67-15"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span><span class="p">;</span>
 <a id="__codelineno-67-16" name="__codelineno-67-16" href="#__codelineno-67-16"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-67-17" name="__codelineno-67-17" href="#__codelineno-67-17"></a><span class="w">        </span><span class="p">}</span>
@@ -6810,20 +6907,20 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.cs</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a><span class="cm">/* Coin change II: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.cs</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a><span class="cm">/* Coin change II: space-optimized dynamic programming */</span>
 <a id="__codelineno-68-2" name="__codelineno-68-2" href="#__codelineno-68-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">CoinChangeIIDPComp</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-68-3" name="__codelineno-68-3" href="#__codelineno-68-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
-<a id="__codelineno-68-4" name="__codelineno-68-4" href="#__codelineno-68-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-68-4" name="__codelineno-68-4" href="#__codelineno-68-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-68-5" name="__codelineno-68-5" href="#__codelineno-68-5"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-68-6" name="__codelineno-68-6" href="#__codelineno-68-6"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-68-7" name="__codelineno-68-7" href="#__codelineno-68-7"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-68-7" name="__codelineno-68-7" href="#__codelineno-68-7"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-68-8" name="__codelineno-68-8" href="#__codelineno-68-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-68-9" name="__codelineno-68-9" href="#__codelineno-68-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-68-10" name="__codelineno-68-10" href="#__codelineno-68-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-68-11" name="__codelineno-68-11" href="#__codelineno-68-11"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-68-11" name="__codelineno-68-11" href="#__codelineno-68-11"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-68-12" name="__codelineno-68-12" href="#__codelineno-68-12"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-68-13" name="__codelineno-68-13" href="#__codelineno-68-13"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-68-14" name="__codelineno-68-14" href="#__codelineno-68-14"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-68-14" name="__codelineno-68-14" href="#__codelineno-68-14"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-68-15" name="__codelineno-68-15" href="#__codelineno-68-15"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]];</span>
 <a id="__codelineno-68-16" name="__codelineno-68-16" href="#__codelineno-68-16"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-68-17" name="__codelineno-68-17" href="#__codelineno-68-17"></a><span class="w">        </span><span class="p">}</span>
@@ -6833,21 +6930,21 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.go</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a><span class="cm">/* Coin change II: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.go</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a><span class="cm">/* Coin change II: space-optimized dynamic programming */</span>
 <a id="__codelineno-69-2" name="__codelineno-69-2" href="#__codelineno-69-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">coinChangeIIDPComp</span><span class="p">(</span><span class="nx">coins</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-69-3" name="__codelineno-69-3" href="#__codelineno-69-3"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">coins</span><span class="p">)</span>
-<a id="__codelineno-69-4" name="__codelineno-69-4" href="#__codelineno-69-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-69-4" name="__codelineno-69-4" href="#__codelineno-69-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-69-5" name="__codelineno-69-5" href="#__codelineno-69-5"></a><span class="w">    </span><span class="nx">dp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-69-6" name="__codelineno-69-6" href="#__codelineno-69-6"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-69-7" name="__codelineno-69-7" href="#__codelineno-69-7"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-69-7" name="__codelineno-69-7" href="#__codelineno-69-7"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-69-8" name="__codelineno-69-8" href="#__codelineno-69-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-69-9" name="__codelineno-69-9" href="#__codelineno-69-9"></a><span class="w">        </span><span class="c1">// Traverse in forward order</span>
+<a id="__codelineno-69-9" name="__codelineno-69-9" href="#__codelineno-69-9"></a><span class="w">        </span><span class="c1">// Traverse in increasing order</span>
 <a id="__codelineno-69-10" name="__codelineno-69-10" href="#__codelineno-69-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-69-11" name="__codelineno-69-11" href="#__codelineno-69-11"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-69-12" name="__codelineno-69-12" href="#__codelineno-69-12"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-69-12" name="__codelineno-69-12" href="#__codelineno-69-12"></a><span class="w">                </span><span class="c1">// If the coin exceeds the current target amount a, do not choose coin i</span>
 <a id="__codelineno-69-13" name="__codelineno-69-13" href="#__codelineno-69-13"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span>
 <a id="__codelineno-69-14" name="__codelineno-69-14" href="#__codelineno-69-14"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-69-15" name="__codelineno-69-15" href="#__codelineno-69-15"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-69-15" name="__codelineno-69-15" href="#__codelineno-69-15"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-69-16" name="__codelineno-69-16" href="#__codelineno-69-16"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="o">-</span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span>
 <a id="__codelineno-69-17" name="__codelineno-69-17" href="#__codelineno-69-17"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-69-18" name="__codelineno-69-18" href="#__codelineno-69-18"></a><span class="w">        </span><span class="p">}</span>
@@ -6857,20 +6954,20 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.swift</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a><span class="cm">/* Coin change II: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.swift</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a><span class="cm">/* Coin change II: space-optimized dynamic programming */</span>
 <a id="__codelineno-70-2" name="__codelineno-70-2" href="#__codelineno-70-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">coinChangeIIDPComp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">amt</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-70-3" name="__codelineno-70-3" href="#__codelineno-70-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="bp">count</span>
-<a id="__codelineno-70-4" name="__codelineno-70-4" href="#__codelineno-70-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-70-4" name="__codelineno-70-4" href="#__codelineno-70-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-70-5" name="__codelineno-70-5" href="#__codelineno-70-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-70-6" name="__codelineno-70-6" href="#__codelineno-70-6"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-70-7" name="__codelineno-70-7" href="#__codelineno-70-7"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-70-7" name="__codelineno-70-7" href="#__codelineno-70-7"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-70-8" name="__codelineno-70-8" href="#__codelineno-70-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-70-9" name="__codelineno-70-9" href="#__codelineno-70-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-70-10" name="__codelineno-70-10" href="#__codelineno-70-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-70-11" name="__codelineno-70-11" href="#__codelineno-70-11"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-70-11" name="__codelineno-70-11" href="#__codelineno-70-11"></a><span class="w">                </span><span class="c1">// If the coin exceeds the current target amount a, do not choose coin i</span>
 <a id="__codelineno-70-12" name="__codelineno-70-12" href="#__codelineno-70-12"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span>
 <a id="__codelineno-70-13" name="__codelineno-70-13" href="#__codelineno-70-13"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-70-14" name="__codelineno-70-14" href="#__codelineno-70-14"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-70-14" name="__codelineno-70-14" href="#__codelineno-70-14"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-70-15" name="__codelineno-70-15" href="#__codelineno-70-15"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]]</span>
 <a id="__codelineno-70-16" name="__codelineno-70-16" href="#__codelineno-70-16"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-70-17" name="__codelineno-70-17" href="#__codelineno-70-17"></a><span class="w">        </span><span class="p">}</span>
@@ -6880,20 +6977,20 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.js</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a><span class="cm">/* Coin change II: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.js</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a><span class="cm">/* Coin change II: space-optimized dynamic programming */</span>
 <a id="__codelineno-71-2" name="__codelineno-71-2" href="#__codelineno-71-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">coinChangeIIDPComp</span><span class="p">(</span><span class="nx">coins</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-71-3" name="__codelineno-71-3" href="#__codelineno-71-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">coins</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-71-4" name="__codelineno-71-4" href="#__codelineno-71-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-71-4" name="__codelineno-71-4" href="#__codelineno-71-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-71-5" name="__codelineno-71-5" href="#__codelineno-71-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-71-6" name="__codelineno-71-6" href="#__codelineno-71-6"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-71-7" name="__codelineno-71-7" href="#__codelineno-71-7"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-71-7" name="__codelineno-71-7" href="#__codelineno-71-7"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-71-8" name="__codelineno-71-8" href="#__codelineno-71-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-71-9" name="__codelineno-71-9" href="#__codelineno-71-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-71-10" name="__codelineno-71-10" href="#__codelineno-71-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-71-11" name="__codelineno-71-11" href="#__codelineno-71-11"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-71-11" name="__codelineno-71-11" href="#__codelineno-71-11"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-71-12" name="__codelineno-71-12" href="#__codelineno-71-12"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">];</span>
 <a id="__codelineno-71-13" name="__codelineno-71-13" href="#__codelineno-71-13"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-71-14" name="__codelineno-71-14" href="#__codelineno-71-14"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-71-14" name="__codelineno-71-14" href="#__codelineno-71-14"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-71-15" name="__codelineno-71-15" href="#__codelineno-71-15"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]];</span>
 <a id="__codelineno-71-16" name="__codelineno-71-16" href="#__codelineno-71-16"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-71-17" name="__codelineno-71-17" href="#__codelineno-71-17"></a><span class="w">        </span><span class="p">}</span>
@@ -6903,20 +7000,20 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.ts</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* Coin change II: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.ts</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* Coin change II: space-optimized dynamic programming */</span>
 <a id="__codelineno-72-2" name="__codelineno-72-2" href="#__codelineno-72-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">coinChangeIIDPComp</span><span class="p">(</span><span class="nx">coins</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-72-3" name="__codelineno-72-3" href="#__codelineno-72-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">coins</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-72-4" name="__codelineno-72-4" href="#__codelineno-72-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-72-4" name="__codelineno-72-4" href="#__codelineno-72-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-72-5" name="__codelineno-72-5" href="#__codelineno-72-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">.</span><span class="kr">from</span><span class="p">({</span><span class="w"> </span><span class="nx">length</span><span class="o">:</span><span class="w"> </span><span class="kt">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-72-6" name="__codelineno-72-6" href="#__codelineno-72-6"></a><span class="w">    </span><span class="nx">dp</span><span class="p">[</span><span class="mf">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-72-7" name="__codelineno-72-7" href="#__codelineno-72-7"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-72-7" name="__codelineno-72-7" href="#__codelineno-72-7"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-72-8" name="__codelineno-72-8" href="#__codelineno-72-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-72-9" name="__codelineno-72-9" href="#__codelineno-72-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">amt</span><span class="p">;</span><span class="w"> </span><span class="nx">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-72-10" name="__codelineno-72-10" href="#__codelineno-72-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-72-11" name="__codelineno-72-11" href="#__codelineno-72-11"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-72-11" name="__codelineno-72-11" href="#__codelineno-72-11"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-72-12" name="__codelineno-72-12" href="#__codelineno-72-12"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">];</span>
 <a id="__codelineno-72-13" name="__codelineno-72-13" href="#__codelineno-72-13"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-72-14" name="__codelineno-72-14" href="#__codelineno-72-14"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-72-14" name="__codelineno-72-14" href="#__codelineno-72-14"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-72-15" name="__codelineno-72-15" href="#__codelineno-72-15"></a><span class="w">                </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">dp</span><span class="p">[</span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]];</span>
 <a id="__codelineno-72-16" name="__codelineno-72-16" href="#__codelineno-72-16"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-72-17" name="__codelineno-72-17" href="#__codelineno-72-17"></a><span class="w">        </span><span class="p">}</span>
@@ -6926,20 +7023,20 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.dart</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="cm">/* Coin change II: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.dart</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="cm">/* Coin change II: space-optimized dynamic programming */</span>
 <a id="__codelineno-73-2" name="__codelineno-73-2" href="#__codelineno-73-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">coinChangeIIDPComp</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-73-3" name="__codelineno-73-3" href="#__codelineno-73-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
-<a id="__codelineno-73-4" name="__codelineno-73-4" href="#__codelineno-73-4"></a><span class="w">  </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-73-4" name="__codelineno-73-4" href="#__codelineno-73-4"></a><span class="w">  </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-73-5" name="__codelineno-73-5" href="#__codelineno-73-5"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-73-6" name="__codelineno-73-6" href="#__codelineno-73-6"></a><span class="w">  </span><span class="n">dp</span><span class="p">[</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
-<a id="__codelineno-73-7" name="__codelineno-73-7" href="#__codelineno-73-7"></a><span class="w">  </span><span class="c1">// State transition</span>
+<a id="__codelineno-73-7" name="__codelineno-73-7" href="#__codelineno-73-7"></a><span class="w">  </span><span class="c1">// State transitions</span>
 <a id="__codelineno-73-8" name="__codelineno-73-8" href="#__codelineno-73-8"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-73-9" name="__codelineno-73-9" href="#__codelineno-73-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-73-10" name="__codelineno-73-10" href="#__codelineno-73-10"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-73-11" name="__codelineno-73-11" href="#__codelineno-73-11"></a><span class="w">        </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-73-11" name="__codelineno-73-11" href="#__codelineno-73-11"></a><span class="w">        </span><span class="c1">// If the coin exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-73-12" name="__codelineno-73-12" href="#__codelineno-73-12"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-73-13" name="__codelineno-73-13" href="#__codelineno-73-13"></a><span class="w">      </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-73-14" name="__codelineno-73-14" href="#__codelineno-73-14"></a><span class="w">        </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-73-14" name="__codelineno-73-14" href="#__codelineno-73-14"></a><span class="w">        </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-73-15" name="__codelineno-73-15" href="#__codelineno-73-15"></a><span class="w">        </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]];</span>
 <a id="__codelineno-73-16" name="__codelineno-73-16" href="#__codelineno-73-16"></a><span class="w">      </span><span class="p">}</span>
 <a id="__codelineno-73-17" name="__codelineno-73-17" href="#__codelineno-73-17"></a><span class="w">    </span><span class="p">}</span>
@@ -6949,20 +7046,20 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.rs</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="cm">/* Coin change II: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.rs</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="cm">/* Coin change II: space-optimized dynamic programming */</span>
 <a id="__codelineno-74-2" name="__codelineno-74-2" href="#__codelineno-74-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">coin_change_ii_dp_comp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">amt</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-74-3" name="__codelineno-74-3" href="#__codelineno-74-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
-<a id="__codelineno-74-4" name="__codelineno-74-4" href="#__codelineno-74-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-74-4" name="__codelineno-74-4" href="#__codelineno-74-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-74-5" name="__codelineno-74-5" href="#__codelineno-74-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-74-6" name="__codelineno-74-6" href="#__codelineno-74-6"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-74-7" name="__codelineno-74-7" href="#__codelineno-74-7"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-74-7" name="__codelineno-74-7" href="#__codelineno-74-7"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-74-8" name="__codelineno-74-8" href="#__codelineno-74-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-74-9" name="__codelineno-74-9" href="#__codelineno-74-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..=</span><span class="n">amt</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-74-10" name="__codelineno-74-10" href="#__codelineno-74-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-74-11" name="__codelineno-74-11" href="#__codelineno-74-11"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-74-11" name="__codelineno-74-11" href="#__codelineno-74-11"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-74-12" name="__codelineno-74-12" href="#__codelineno-74-12"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-74-13" name="__codelineno-74-13" href="#__codelineno-74-13"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-74-14" name="__codelineno-74-14" href="#__codelineno-74-14"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-74-14" name="__codelineno-74-14" href="#__codelineno-74-14"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-74-15" name="__codelineno-74-15" href="#__codelineno-74-15"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">];</span>
 <a id="__codelineno-74-16" name="__codelineno-74-16" href="#__codelineno-74-16"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-74-17" name="__codelineno-74-17" href="#__codelineno-74-17"></a><span class="w">        </span><span class="p">}</span>
@@ -6972,20 +7069,20 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.c</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="cm">/* Coin change II: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.c</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="cm">/* Coin change II: space-optimized dynamic programming */</span>
 <a id="__codelineno-75-2" name="__codelineno-75-2" href="#__codelineno-75-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeIIDPComp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">coins</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">coinsSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-75-3" name="__codelineno-75-3" href="#__codelineno-75-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coinsSize</span><span class="p">;</span>
-<a id="__codelineno-75-4" name="__codelineno-75-4" href="#__codelineno-75-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-75-4" name="__codelineno-75-4" href="#__codelineno-75-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-75-5" name="__codelineno-75-5" href="#__codelineno-75-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-75-6" name="__codelineno-75-6" href="#__codelineno-75-6"></a><span class="w">    </span><span class="n">dp</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-75-7" name="__codelineno-75-7" href="#__codelineno-75-7"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-75-7" name="__codelineno-75-7" href="#__codelineno-75-7"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-75-8" name="__codelineno-75-8" href="#__codelineno-75-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-75-9" name="__codelineno-75-9" href="#__codelineno-75-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">amt</span><span class="p">;</span><span class="w"> </span><span class="n">a</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-75-10" name="__codelineno-75-10" href="#__codelineno-75-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-75-11" name="__codelineno-75-11" href="#__codelineno-75-11"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-75-11" name="__codelineno-75-11" href="#__codelineno-75-11"></a><span class="w">                </span><span class="c1">// If the coin exceeds the target amount, do not choose coin i</span>
 <a id="__codelineno-75-12" name="__codelineno-75-12" href="#__codelineno-75-12"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">];</span>
 <a id="__codelineno-75-13" name="__codelineno-75-13" href="#__codelineno-75-13"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-75-14" name="__codelineno-75-14" href="#__codelineno-75-14"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-75-14" name="__codelineno-75-14" href="#__codelineno-75-14"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-75-15" name="__codelineno-75-15" href="#__codelineno-75-15"></a><span class="w">                </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="p">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]];</span>
 <a id="__codelineno-75-16" name="__codelineno-75-16" href="#__codelineno-75-16"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-75-17" name="__codelineno-75-17" href="#__codelineno-75-17"></a><span class="w">        </span><span class="p">}</span>
@@ -6998,20 +7095,20 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.kt</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a><span class="cm">/* Coin change II: Space-optimized dynamic programming */</span>
+<div class="highlight"><span class="filename">coin_change_ii.kt</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a><span class="cm">/* Coin change II: space-optimized dynamic programming */</span>
 <a id="__codelineno-76-2" name="__codelineno-76-2" href="#__codelineno-76-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">coinChangeIIDPComp</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">amt</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-76-3" name="__codelineno-76-3" href="#__codelineno-76-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="na">size</span>
-<a id="__codelineno-76-4" name="__codelineno-76-4" href="#__codelineno-76-4"></a><span class="w">    </span><span class="c1">// Initialize dp table</span>
+<a id="__codelineno-76-4" name="__codelineno-76-4" href="#__codelineno-76-4"></a><span class="w">    </span><span class="c1">// Initialize the dp table</span>
 <a id="__codelineno-76-5" name="__codelineno-76-5" href="#__codelineno-76-5"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-76-6" name="__codelineno-76-6" href="#__codelineno-76-6"></a><span class="w">    </span><span class="n">dp</span><span class="o">[</span><span class="m">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-76-7" name="__codelineno-76-7" href="#__codelineno-76-7"></a><span class="w">    </span><span class="c1">// State transition</span>
+<a id="__codelineno-76-7" name="__codelineno-76-7" href="#__codelineno-76-7"></a><span class="w">    </span><span class="c1">// State transitions</span>
 <a id="__codelineno-76-8" name="__codelineno-76-8" href="#__codelineno-76-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-76-9" name="__codelineno-76-9" href="#__codelineno-76-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="na">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-76-10" name="__codelineno-76-10" href="#__codelineno-76-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-76-11" name="__codelineno-76-11" href="#__codelineno-76-11"></a><span class="w">                </span><span class="c1">// If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-76-11" name="__codelineno-76-11" href="#__codelineno-76-11"></a><span class="w">                </span><span class="c1">// If the coin value exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-76-12" name="__codelineno-76-12" href="#__codelineno-76-12"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span>
 <a id="__codelineno-76-13" name="__codelineno-76-13" href="#__codelineno-76-13"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-76-14" name="__codelineno-76-14" href="#__codelineno-76-14"></a><span class="w">                </span><span class="c1">// Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-76-14" name="__codelineno-76-14" href="#__codelineno-76-14"></a><span class="w">                </span><span class="c1">// Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-76-15" name="__codelineno-76-15" href="#__codelineno-76-15"></a><span class="w">                </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]]</span>
 <a id="__codelineno-76-16" name="__codelineno-76-16" href="#__codelineno-76-16"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-76-17" name="__codelineno-76-17" href="#__codelineno-76-17"></a><span class="w">        </span><span class="p">}</span>
@@ -7021,21 +7118,21 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_ii.rb</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="c1">### Coin change II: space-optimized DP ###</span>
+<div class="highlight"><span class="filename">coin_change_ii.rb</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="c1">### Coin change II: space-optimized dynamic programming ###</span>
 <a id="__codelineno-77-2" name="__codelineno-77-2" href="#__codelineno-77-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">coin_change_ii_dp_comp</span><span class="p">(</span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span>
 <a id="__codelineno-77-3" name="__codelineno-77-3" href="#__codelineno-77-3"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-77-4" name="__codelineno-77-4" href="#__codelineno-77-4"></a><span class="w">  </span><span class="c1"># Initialize dp table</span>
+<a id="__codelineno-77-4" name="__codelineno-77-4" href="#__codelineno-77-4"></a><span class="w">  </span><span class="c1"># Initialize the dp table</span>
 <a id="__codelineno-77-5" name="__codelineno-77-5" href="#__codelineno-77-5"></a><span class="w">  </span><span class="n">dp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-77-6" name="__codelineno-77-6" href="#__codelineno-77-6"></a><span class="w">  </span><span class="n">dp</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-77-7" name="__codelineno-77-7" href="#__codelineno-77-7"></a><span class="w">  </span><span class="c1"># State transition</span>
+<a id="__codelineno-77-7" name="__codelineno-77-7" href="#__codelineno-77-7"></a><span class="w">  </span><span class="c1"># State transitions</span>
 <a id="__codelineno-77-8" name="__codelineno-77-8" href="#__codelineno-77-8"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-77-9" name="__codelineno-77-9" href="#__codelineno-77-9"></a><span class="w">    </span><span class="c1"># Traverse in forward order</span>
+<a id="__codelineno-77-9" name="__codelineno-77-9" href="#__codelineno-77-9"></a><span class="w">    </span><span class="c1"># Traverse in increasing order</span>
 <a id="__codelineno-77-10" name="__codelineno-77-10" href="#__codelineno-77-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-77-11" name="__codelineno-77-11" href="#__codelineno-77-11"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">a</span>
-<a id="__codelineno-77-12" name="__codelineno-77-12" href="#__codelineno-77-12"></a><span class="w">        </span><span class="c1"># If exceeds target amount, don't select coin i</span>
+<a id="__codelineno-77-12" name="__codelineno-77-12" href="#__codelineno-77-12"></a><span class="w">        </span><span class="c1"># If the coin exceeds the current amount a, do not choose coin i</span>
 <a id="__codelineno-77-13" name="__codelineno-77-13" href="#__codelineno-77-13"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span>
 <a id="__codelineno-77-14" name="__codelineno-77-14" href="#__codelineno-77-14"></a><span class="w">      </span><span class="k">else</span>
-<a id="__codelineno-77-15" name="__codelineno-77-15" href="#__codelineno-77-15"></a><span class="w">        </span><span class="c1"># Sum of the two options: not selecting and selecting coin i</span>
+<a id="__codelineno-77-15" name="__codelineno-77-15" href="#__codelineno-77-15"></a><span class="w">        </span><span class="c1"># Add the numbers of ways without and with coin i</span>
 <a id="__codelineno-77-16" name="__codelineno-77-16" href="#__codelineno-77-16"></a><span class="w">        </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">dp</span><span class="o">[</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]]</span>
 <a id="__codelineno-77-17" name="__codelineno-77-17" href="#__codelineno-77-17"></a><span class="w">      </span><span class="k">end</span>
 <a id="__codelineno-77-18" name="__codelineno-77-18" href="#__codelineno-77-18"></a><span class="w">    </span><span class="k">end</span>
@@ -7046,6 +7143,11 @@ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]]
 </div>
 </div>
 </div>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20coin_change_ii_dp_comp%28coins%3A%20list%5Bint%5D,%20amt%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Coin%20change%20II%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28coins%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B0%5D%20%2A%20%28amt%20%2B%201%29%0A%20%20%20%20dp%5B0%5D%20%3D%201%0A%20%20%20%20%23%20State%20transitions%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%23%20Traverse%20in%20increasing%20order%0A%20%20%20%20%20%20%20%20for%20a%20in%20range%281,%20amt%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20coins%5Bi%20-%201%5D%20%3E%20a%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20the%20coin%20value%20exceeds%20the%20current%20amount%20a,%20do%20not%20choose%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Ba%5D%20%3D%20dp%5Ba%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Add%20the%20numbers%20of%20ways%20without%20and%20with%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Ba%5D%20%3D%20dp%5Ba%5D%20%2B%20dp%5Ba%20-%20coins%5Bi%20-%201%5D%5D%0A%20%20%20%20return%20dp%5Bamt%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20coins%20%3D%20%5B1,%202,%205%5D%0A%20%20%20%20amt%20%3D%205%0A%0A%20%20%20%20%23%20Space-optimized%20dynamic%20programming%0A%20%20%20%20res%20%3D%20coin_change_ii_dp_comp%28coins,%20amt%29%0A%20%20%20%20print%28f%22Number%20of%20coin%20combinations%20that%20make%20the%20target%20amount%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20coin_change_ii_dp_comp%28coins%3A%20list%5Bint%5D,%20amt%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Coin%20change%20II%3A%20space-optimized%20dynamic%20programming%22%22%22%0A%20%20%20%20n%20%3D%20len%28coins%29%0A%20%20%20%20%23%20Initialize%20the%20dp%20table%0A%20%20%20%20dp%20%3D%20%5B0%5D%20%2A%20%28amt%20%2B%201%29%0A%20%20%20%20dp%5B0%5D%20%3D%201%0A%20%20%20%20%23%20State%20transitions%0A%20%20%20%20for%20i%20in%20range%281,%20n%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%23%20Traverse%20in%20increasing%20order%0A%20%20%20%20%20%20%20%20for%20a%20in%20range%281,%20amt%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20coins%5Bi%20-%201%5D%20%3E%20a%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20the%20coin%20value%20exceeds%20the%20current%20amount%20a,%20do%20not%20choose%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Ba%5D%20%3D%20dp%5Ba%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Add%20the%20numbers%20of%20ways%20without%20and%20with%20coin%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20dp%5Ba%5D%20%3D%20dp%5Ba%5D%20%2B%20dp%5Ba%20-%20coins%5Bi%20-%201%5D%5D%0A%20%20%20%20return%20dp%5Bamt%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20coins%20%3D%20%5B1,%202,%205%5D%0A%20%20%20%20amt%20%3D%205%0A%0A%20%20%20%20%23%20Space-optimized%20dynamic%20programming%0A%20%20%20%20res%20%3D%20coin_change_ii_dp_comp%28coins,%20amt%29%0A%20%20%20%20print%28f%22Number%20of%20coin%20combinations%20that%20make%20the%20target%20amount%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
 
 <!-- Source file information -->
 
@@ -7068,7 +7170,7 @@ aria-label="Footer"
   <a
     href="../knapsack_problem/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 14.4 0-1 Knapsack Problem"
+    aria-label="Previous: 14.4   0-1 knapsack problem"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -7080,7 +7182,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        14.4 0-1 Knapsack Problem
+        14.4 &nbsp; 0-1 knapsack problem
       </div>
     </div>
   </a>
@@ -7092,7 +7194,7 @@ aria-label="Footer"
   <a
     href="../edit_distance_problem/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 14.6 Edit Distance Problem"
+    aria-label="Next: 14.6   Edit distance problem"
     rel="next"
   >
     <div class="md-footer__title">
@@ -7100,7 +7202,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        14.6 Edit Distance Problem
+        14.6 &nbsp; Edit distance problem
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -7117,7 +7219,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_graph/exercises/index.html b/en/chapter_graph/exercises/index.html
index b6edaf35a..c38346861 100644
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@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
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-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
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-  Before Starting
+  Foreword
 
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           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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-  Chapter 0. Preface
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-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
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@@ -617,7 +617,7 @@
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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-  2.3 Time Complexity
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
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-  3.2 Basic Data Types
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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@@ -1103,7 +1103,7 @@
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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@@ -1130,7 +1130,7 @@
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2380,7 +2380,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2401,7 +2401,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2422,7 +2422,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2449,7 +2449,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2476,7 +2476,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2503,7 +2503,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2539,7 +2539,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2556,7 +2556,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2582,22 +2582,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#951-concept-review" class="md-nav__link">
+  <a href="#951-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.5.1 &nbsp; Concept Review
+        9.5.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="9.5.1 Concept Review">
+    <nav class="md-nav" aria-label="9.5.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-represent-the-same-graph-in-two-ways" class="md-nav__link">
+  <a href="#1-representing-the-same-graph-in-two-ways" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Represent the Same Graph in Two Ways
+        1. &nbsp; Representing the same graph in two ways
       </span>
     </span>
   </a>
@@ -2608,7 +2608,7 @@
   <a href="#2-breadth-first-and-depth-first-traversal-orders" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Breadth-First and Depth-First Traversal Orders
+        2. &nbsp; Breadth-first and depth-first traversal orders
       </span>
     </span>
   </a>
@@ -2619,7 +2619,7 @@
   <a href="#3-can-one-bfs-visit-the-entire-graph" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Can One BFS Visit the Entire Graph?
+        3. &nbsp; Can one BFS visit the entire graph?
       </span>
     </span>
   </a>
@@ -2635,19 +2635,19 @@
   <a href="#952-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.5.2 &nbsp; Programming Exercises
+        9.5.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="9.5.2 Programming Exercises">
+    <nav class="md-nav" aria-label="9.5.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-determine-whether-a-path-exists-in-an-undirected-graph" class="md-nav__link">
+  <a href="#1-checking-for-a-path-in-an-undirected-graph" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Determine Whether a Path Exists in an Undirected Graph
+        1. &nbsp; Checking for a path in an undirected graph
       </span>
     </span>
   </a>
@@ -2723,7 +2723,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2744,7 +2744,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2765,7 +2765,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2792,7 +2792,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2819,7 +2819,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2846,7 +2846,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2873,7 +2873,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2900,7 +2900,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2927,7 +2927,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -3005,7 +3005,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3026,7 +3026,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3047,7 +3047,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3101,7 +3101,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3128,7 +3128,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3155,7 +3155,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3209,7 +3209,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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@@ -3236,7 +3236,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3263,7 +3263,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3290,7 +3290,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3317,7 +3317,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3344,7 +3344,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3410,7 +3410,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3431,7 +3431,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3452,7 +3452,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3506,7 +3506,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3533,7 +3533,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3560,7 +3560,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3653,7 +3653,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
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@@ -3695,7 +3695,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3749,7 +3749,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3776,7 +3776,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3803,7 +3803,7 @@
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3900,7 +3900,7 @@
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
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@@ -3942,7 +3942,7 @@
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4077,7 +4077,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4104,7 +4104,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4131,7 +4131,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4197,7 +4197,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4239,7 +4239,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4347,7 +4347,7 @@
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4434,7 +4434,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4476,7 +4476,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4503,7 +4503,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4530,7 +4530,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4614,6 +4614,78 @@
 
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#951-concept-review" class="md-nav__link">
+  <a href="#951-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.5.1 &nbsp; Concept Review
+        9.5.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="9.5.1 Concept Review">
+    <nav class="md-nav" aria-label="9.5.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-represent-the-same-graph-in-two-ways" class="md-nav__link">
+  <a href="#1-representing-the-same-graph-in-two-ways" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Represent the Same Graph in Two Ways
+        1. &nbsp; Representing the same graph in two ways
       </span>
     </span>
   </a>
@@ -4681,7 +4753,7 @@
   <a href="#2-breadth-first-and-depth-first-traversal-orders" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Breadth-First and Depth-First Traversal Orders
+        2. &nbsp; Breadth-first and depth-first traversal orders
       </span>
     </span>
   </a>
@@ -4692,7 +4764,7 @@
   <a href="#3-can-one-bfs-visit-the-entire-graph" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Can One BFS Visit the Entire Graph?
+        3. &nbsp; Can one BFS visit the entire graph?
       </span>
     </span>
   </a>
@@ -4708,19 +4780,19 @@
   <a href="#952-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.5.2 &nbsp; Programming Exercises
+        9.5.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="9.5.2 Programming Exercises">
+    <nav class="md-nav" aria-label="9.5.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-determine-whether-a-path-exists-in-an-undirected-graph" class="md-nav__link">
+  <a href="#1-checking-for-a-path-in-an-undirected-graph" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Determine Whether a Path Exists in an Undirected Graph
+        1. &nbsp; Checking for a path in an undirected graph
       </span>
     </span>
   </a>
@@ -4770,20 +4842,20 @@
 
 <!-- Page content -->
 <h1 id="95-exercises">9.5 &nbsp; Exercises<a class="headerlink" href="#95-exercises" title="Permanent link">&para;</a></h1>
-<h2 id="951-concept-review">9.5.1 &nbsp; Concept Review<a class="headerlink" href="#951-concept-review" title="Permanent link">&para;</a></h2>
-<h3 id="1-represent-the-same-graph-in-two-ways">1. &nbsp; Represent the Same Graph in Two Ways<a class="headerlink" href="#1-represent-the-same-graph-in-two-ways" title="Permanent link">&para;</a></h3>
-<p>An undirected graph has four vertices, <code>A, B, C, D</code>, and the edges
+<h2 id="951-check-your-understanding">9.5.1 &nbsp; Check your understanding<a class="headerlink" href="#951-check-your-understanding" title="Permanent link">&para;</a></h2>
+<h3 id="1-representing-the-same-graph-in-two-ways">1. &nbsp; Representing the same graph in two ways<a class="headerlink" href="#1-representing-the-same-graph-in-two-ways" title="Permanent link">&para;</a></h3>
+<p>An undirected graph has four vertices <code>A, B, C, D</code> and edges
 <code>A-B, A-C, B-C, C-D</code>.</p>
 <!-- numbered-subquestions -->
 
 <ol>
 <li>Write its adjacency list.</li>
-<li>Fill in its adjacency matrix using only 0s and 1s.</li>
-<li>To determine whether <code>A</code> and <code>D</code> are directly connected, which graph representation requires checking only one stored entry?</li>
+<li>Fill in its adjacency matrix using only 0 and 1.</li>
+<li>Which representation lets us determine whether <code>A</code> and <code>D</code> are directly connected by inspecting just one stored entry?</li>
 <li>If a graph has many vertices but few edges, which representation usually uses less space?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
 <p>The adjacency list is:</p>
@@ -4838,80 +4910,80 @@
 </table>
 </li>
 <li>
-<p>In an adjacency matrix, you can directly check row <code>A</code>, column <code>D</code>, making it well suited to determining whether any two vertices are directly connected.</p>
+<p>In the adjacency matrix, we can inspect row <code>A</code>, column <code>D</code> directly. This makes it well suited to checking whether any two vertices are directly connected.</p>
 </li>
 <li>
-<p>When a graph has many vertices but few edges, an adjacency list records only the edges that actually exist. It usually uses less space than an adjacency matrix, which reserves a position for every pair of vertices.</p>
+<p>When there are many vertices but few edges, an adjacency list records only existing edges. It usually uses less space than an adjacency matrix, which reserves an entry for every pair of vertices.</p>
 </li>
 </ol>
 </details>
-<h3 id="2-breadth-first-and-depth-first-traversal-orders">2. &nbsp; Breadth-First and Depth-First Traversal Orders<a class="headerlink" href="#2-breadth-first-and-depth-first-traversal-orders" title="Permanent link">&para;</a></h3>
+<h3 id="2-breadth-first-and-depth-first-traversal-orders">2. &nbsp; Breadth-first and depth-first traversal orders<a class="headerlink" href="#2-breadth-first-and-depth-first-traversal-orders" title="Permanent link">&para;</a></h3>
 <p>An undirected graph has vertices <code>A, B, C, D, E</code> and edges
 <code>A-B, A-C, B-D, C-D, D-E</code>.</p>
-<p>Start at A. Whenever there are several unvisited adjacent vertices, choose them in alphabetical order:</p>
+<p>Start at A. Whenever several unvisited neighbors are available, choose them in alphabetical order:</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Write the visit order of breadth-first traversal (BFS).</li>
-<li>Write the visit order of recursive depth-first traversal (DFS).</li>
-<li>Why must both traversals record which vertices have already been visited?</li>
+<li>Write the breadth-first traversal (BFS) order.</li>
+<li>Write the recursive depth-first traversal (DFS) order.</li>
+<li>Why must both traversals keep track of vertices already visited?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>The BFS visit order is <code>A, B, C, D, E</code>. It first visits B and C, which are one edge away from A,
-    and then visits the more distant D and E.</p>
+<p>The BFS order is <code>A, B, C, D, E</code>. It first visits B and C, which are one edge away from A,
+    and then the more distant D and E.</p>
 </li>
 <li>
-<p>The DFS visit order is <code>A, B, D, C, E</code>. It repeatedly enters an unvisited adjacent vertex,
-    first following <code>A → B → D → C</code>. When C has no new adjacent vertex, it returns to D and then visits E.</p>
+<p>The DFS order is <code>A, B, D, C, E</code>. It proceeds into an unvisited neighbor of the current vertex,
+    first following <code>A → B → D → C</code>. When C has no new neighbors to visit, it returns to D and then visits E.</p>
 </li>
 <li>
-<p>The graph contains a cycle, such as <code>A-B-D-C-A</code>. Without recording visited vertices,
-    a traversal could repeatedly visit the same vertices around the cycle and fail to terminate normally.</p>
+<p>The graph contains cycles, such as <code>A-B-D-C-A</code>. Without recording visited vertices,
+    traversal could repeatedly follow a cycle through the same vertices and fail to terminate.</p>
 </li>
 </ol>
 </details>
-<h3 id="3-can-one-bfs-visit-the-entire-graph">3. &nbsp; Can One BFS Visit the Entire Graph?<a class="headerlink" href="#3-can-one-bfs-visit-the-entire-graph" title="Permanent link">&para;</a></h3>
+<h3 id="3-can-one-bfs-visit-the-entire-graph">3. &nbsp; Can one BFS visit the entire graph?<a class="headerlink" href="#3-can-one-bfs-visit-the-entire-graph" title="Permanent link">&para;</a></h3>
 <p>An undirected graph has vertices <code>A, B, C, D, E, F</code> and only the edges
 <code>A-B, B-C, D-E</code>.</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Which vertices can one BFS starting at A visit?</li>
-<li>Based on Question 1, has this BFS visited every vertex in the graph? Why or why not?</li>
-<li>Suppose you scan all vertices in alphabetical order and start a new BFS whenever you reach an unvisited vertex.
-    What is the starting vertex of each BFS? Into how many mutually disconnected parts (connected components) is the graph divided?</li>
+<li>Which vertices can a single BFS starting from A visit?</li>
+<li>Based on your answer, has this BFS visited every vertex in the graph? Why?</li>
+<li>Suppose we scan all vertices alphabetically and start a new BFS whenever we encounter an unvisited vertex.
+    What are the starting vertices of these searches? How many separate connected parts, or connected components, does the graph have?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>Starting from A, the traversal can visit only <code>A, B, C</code>.</p>
+<p>Starting from A reaches only <code>A, B, C</code>.</p>
 </li>
 <li>
-<p>It has not visited every vertex. <code>D, E</code> form another connected part, while F is an isolated vertex.
-    None of them has a path to A, so they cannot be reached from A.</p>
+<p>Not every vertex is visited. <code>D, E</code> form another connected part, and F is an isolated vertex.
+    None has a path to A, so they cannot be reached from A.</p>
 </li>
 <li>
-<p>The three BFS traversals start at <code>A, D, F</code>, and visit
-    <code>{A, B, C}</code>, <code>{D, E}</code>, and <code>{F}</code>, respectively. Therefore, the graph has 3 connected components.</p>
+<p>The three BFS starting vertices are <code>A, D, F</code>, in order, visiting
+    <code>{A, B, C}</code>, <code>{D, E}</code>, and <code>{F}</code>, respectively. Thus, the graph has three connected components.</p>
 </li>
 </ol>
 </details>
-<h2 id="952-programming-exercises">9.5.2 &nbsp; Programming Exercises<a class="headerlink" href="#952-programming-exercises" title="Permanent link">&para;</a></h2>
-<h3 id="1-determine-whether-a-path-exists-in-an-undirected-graph">1. &nbsp; Determine Whether a Path Exists in an Undirected Graph<a class="headerlink" href="#1-determine-whether-a-path-exists-in-an-undirected-graph" title="Permanent link">&para;</a></h3>
+<h2 id="952-programming-exercises">9.5.2 &nbsp; Programming exercises<a class="headerlink" href="#952-programming-exercises" title="Permanent link">&para;</a></h2>
+<h3 id="1-checking-for-a-path-in-an-undirected-graph">1. &nbsp; Checking for a path in an undirected graph<a class="headerlink" href="#1-checking-for-a-path-in-an-undirected-graph" title="Permanent link">&para;</a></h3>
 <p>You are given an undirected graph with <span class="arithmatex">\(n\)</span> vertices numbered from <span class="arithmatex">\(0\)</span> to <span class="arithmatex">\(n-1\)</span>. Each entry <code>[u, v]</code> in the array <code>edges</code> represents an undirected edge between vertices <code>u</code> and <code>v</code>.</p>
-<p>You are also given a starting vertex <code>source</code> and a destination vertex <code>destination</code>. First build an adjacency list from <code>edges</code>, then use BFS or DFS
-to determine whether a path exists from <code>source</code> to <code>destination</code>. Return <code>true</code> if one exists and <code>false</code> otherwise.
+<p>You are also given a starting vertex <code>source</code> and an ending vertex <code>destination</code>. First build an adjacency list from <code>edges</code>, then use BFS or DFS
+to determine whether a path exists from <code>source</code> to <code>destination</code>. Return <code>true</code> if it does, or <code>false</code> otherwise.
 The graph may contain cycles and may be disconnected.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>Add every undirected edge in both directions</li>
-<li>The graph may contain cycles, so you must record which vertices have already been visited</li>
-<li>Starting from source, return true if you encounter destination; if the traversal ends without reaching it, return false</li>
+<li>Add each undirected edge in both directions.</li>
+<li>The graph may contain cycles, so keep track of vertices already visited.</li>
+<li>Starting from source, return true if you reach destination. Return false if traversal ends without reaching it.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/find-if-path-exists-in-graph/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
@@ -4937,7 +5009,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 9.4 Summary"
+    aria-label="Previous: 9.4   Summary"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4949,7 +5021,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        9.4 Summary
+        9.4 &nbsp; Summary
       </div>
     </div>
   </a>
@@ -4961,7 +5033,7 @@ aria-label="Footer"
   <a
     href="../../chapter_searching/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: Chapter 10. &amp;nbsp; Searching"
+    aria-label="Next: Chapter 10.   Searching"
     rel="next"
   >
     <div class="md-footer__title">
@@ -4986,7 +5058,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_graph/graph/index.html b/en/chapter_graph/graph/index.html
index 4933e6ecc..ed813099c 100644
--- a/en/chapter_graph/graph/index.html
+++ b/en/chapter_graph/graph/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>9.1 Graph - Hello Algo</title>
+        <title>9.1 Graphs - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
     <div data-md-component="skip">
 
 
-        <a href="#91-graph" class="md-skip">
+        <a href="#91-graphs" class="md-skip">
           Skip to content
         </a>
 
@@ -119,7 +119,7 @@
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             d="M480 32c0-12.9-7.8-24.6-19.8-29.6s-25.7-2.2-34.9 6.9L381.7 53c-48 48-113.1 75-181 75H192 160 64c-35.3 0-64 28.7-64 64v96c0 35.3 28.7 64 64 64l0 128c0 17.7 14.3 32 32 32h64c17.7 0 32-14.3 32-32V352l8.7 0c67.9 0 133 27 181 75l43.6 43.6c9.2 9.2 22.9 11.9 34.9 6.9s19.8-16.6 19.8-29.6V300.4c18.6-8.8 32-32.5 32-60.4s-13.4-51.6-32-60.4V32zm-64 76.7V240 371.3C357.2 317.8 280.5 288 200.7 288H192V192h8.7c79.8 0 156.5-29.8 215.3-83.3z" />
     </svg>
-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              9.1 &nbsp; Graph
+              9.1 &nbsp; Graphs
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2380,7 +2380,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2401,7 +2401,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2431,7 +2431,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2448,7 +2448,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2474,10 +2474,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#911-common-types-and-terminology-of-graphs" class="md-nav__link">
+  <a href="#911-common-graph-types-and-terminology" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.1.1 &nbsp; Common Types and Terminology of Graphs
+        9.1.1 &nbsp; Common graph types and terminology
       </span>
     </span>
   </a>
@@ -2485,22 +2485,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#912-representation-of-graphs" class="md-nav__link">
+  <a href="#912-representing-a-graph" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.1.2 &nbsp; Representation of Graphs
+        9.1.2 &nbsp; Representing a graph
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="9.1.2 Representation of Graphs">
+    <nav class="md-nav" aria-label="9.1.2 Representing a graph">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-adjacency-matrix" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Adjacency Matrix
+        1. &nbsp; Adjacency matrix
       </span>
     </span>
   </a>
@@ -2511,7 +2511,7 @@
   <a href="#2-adjacency-list" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Adjacency List
+        2. &nbsp; Adjacency list
       </span>
     </span>
   </a>
@@ -2527,7 +2527,7 @@
   <a href="#913-common-applications-of-graphs" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.1.3 &nbsp; Common Applications of Graphs
+        9.1.3 &nbsp; Common applications of graphs
       </span>
     </span>
   </a>
@@ -2557,7 +2557,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2584,7 +2584,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2611,7 +2611,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2706,7 +2706,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2727,7 +2727,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2748,7 +2748,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2775,7 +2775,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2802,7 +2802,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2829,7 +2829,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2856,7 +2856,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2883,7 +2883,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2910,7 +2910,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2988,7 +2988,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3009,7 +3009,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3030,7 +3030,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3057,7 +3057,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3084,7 +3084,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3111,7 +3111,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3138,7 +3138,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3165,7 +3165,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3192,7 +3192,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3219,7 +3219,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3246,7 +3246,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3273,7 +3273,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3300,7 +3300,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3327,7 +3327,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3393,7 +3393,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3414,7 +3414,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3435,7 +3435,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3462,7 +3462,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3489,7 +3489,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3516,7 +3516,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3543,7 +3543,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3570,7 +3570,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3636,7 +3636,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3657,7 +3657,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3678,7 +3678,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3705,7 +3705,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3732,7 +3732,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3759,7 +3759,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3786,7 +3786,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3813,7 +3813,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3883,7 +3883,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3904,7 +3904,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3925,7 +3925,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3952,7 +3952,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3979,7 +3979,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4006,7 +4006,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4033,7 +4033,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4060,7 +4060,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4087,7 +4087,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4114,7 +4114,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4180,7 +4180,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4201,7 +4201,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4222,7 +4222,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4249,7 +4249,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4276,7 +4276,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4303,7 +4303,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4330,7 +4330,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4357,7 +4357,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4417,7 +4417,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4438,7 +4438,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4459,7 +4459,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4486,7 +4486,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4513,7 +4513,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4597,6 +4597,78 @@
 
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#911-common-types-and-terminology-of-graphs" class="md-nav__link">
+  <a href="#911-common-graph-types-and-terminology" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.1.1 &nbsp; Common Types and Terminology of Graphs
+        9.1.1 &nbsp; Common graph types and terminology
       </span>
     </span>
   </a>
@@ -4649,22 +4721,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#912-representation-of-graphs" class="md-nav__link">
+  <a href="#912-representing-a-graph" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.1.2 &nbsp; Representation of Graphs
+        9.1.2 &nbsp; Representing a graph
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="9.1.2 Representation of Graphs">
+    <nav class="md-nav" aria-label="9.1.2 Representing a graph">
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           <li class="md-nav__item">
   <a href="#1-adjacency-matrix" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Adjacency Matrix
+        1. &nbsp; Adjacency matrix
       </span>
     </span>
   </a>
@@ -4675,7 +4747,7 @@
   <a href="#2-adjacency-list" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Adjacency List
+        2. &nbsp; Adjacency list
       </span>
     </span>
   </a>
@@ -4691,7 +4763,7 @@
   <a href="#913-common-applications-of-graphs" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.1.3 &nbsp; Common Applications of Graphs
+        9.1.3 &nbsp; Common applications of graphs
       </span>
     </span>
   </a>
@@ -4735,8 +4807,8 @@
 
 
 <!-- Page content -->
-<h1 id="91-graph">9.1 &nbsp; Graph<a class="headerlink" href="#91-graph" title="Permanent link">&para;</a></h1>
-<p>A <u>graph</u> is a nonlinear data structure consisting of <u>vertices</u> and <u>edges</u>. We can abstractly represent a graph <span class="arithmatex">\(G\)</span> as a set of vertices <span class="arithmatex">\(V\)</span> and a set of edges <span class="arithmatex">\(E\)</span>. The following example shows a graph containing 5 vertices and 7 edges.</p>
+<h1 id="91-graphs">9.1 &nbsp; Graphs<a class="headerlink" href="#91-graphs" title="Permanent link">&para;</a></h1>
+<p>A <u>graph</u> is a nonlinear data structure made up of <u>vertices</u> and <u>edges</u>. We can represent a graph <span class="arithmatex">\(G\)</span> abstractly using a set of vertices <span class="arithmatex">\(V\)</span> and a set of edges <span class="arithmatex">\(E\)</span>. The following example describes a graph with 5 vertices and 7 edges.</p>
 <div class="arithmatex">\[
 \begin{aligned}
 V &amp; = \{ 1, 2, 3, 4, 5 \} \newline
@@ -4744,62 +4816,62 @@ E &amp; = \{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \} \newline
 G &amp; = \{ V, E \} \newline
 \end{aligned}
 \]</div>
-<p>If we view vertices as nodes and edges as references (pointers) connecting them, we can regard a graph as an extension of the linked list data structure. As shown in Figure 9-1, <strong>compared to linear relationships (linked lists) and divide-and-conquer relationships (trees), network relationships (graphs) have a higher degree of freedom and are therefore more complex</strong>.</p>
-<p><img alt="Relationships among linked lists, trees, and graphs" class="animation-figure" src="../graph.assets/linkedlist_tree_graph.png" /></p>
-<p align="center"> Figure 9-1 &nbsp; Relationships among linked lists, trees, and graphs </p>
+<p>If we think of vertices as nodes and edges as references (pointers) connecting those nodes, a graph can be viewed as an extension of a linked list. As Figure 9-1 shows, <strong>network relationships (graphs) allow more freedom than linear relationships (linked lists) or divide-and-conquer relationships (trees)</strong>, making them more complex.</p>
+<p><img alt="Relationships between linked lists, trees, and graphs" class="animation-figure" src="../graph.assets/linkedlist_tree_graph.png" /></p>
+<p align="center"> Figure 9-1 &nbsp; Relationships between linked lists, trees, and graphs </p>
 
-<h2 id="911-common-types-and-terminology-of-graphs">9.1.1 &nbsp; Common Types and Terminology of Graphs<a class="headerlink" href="#911-common-types-and-terminology-of-graphs" title="Permanent link">&para;</a></h2>
-<p>Graphs can be divided into <u>undirected graphs</u> and <u>directed graphs</u> based on whether edges have direction, as shown in Figure 9-2.</p>
+<h2 id="911-common-graph-types-and-terminology">9.1.1 &nbsp; Common graph types and terminology<a class="headerlink" href="#911-common-graph-types-and-terminology" title="Permanent link">&para;</a></h2>
+<p>Depending on whether their edges have a direction, graphs are classified as <u>undirected graphs</u> or <u>directed graphs</u>, as shown in Figure 9-2.</p>
 <ul>
-<li>In undirected graphs, edges represent a "bidirectional" connection between two vertices, such as friendships on WeChat or QQ.</li>
-<li>In directed graphs, edges have directionality, meaning edges <span class="arithmatex">\(A \rightarrow B\)</span> and <span class="arithmatex">\(A \leftarrow B\)</span> are independent of each other, such as following and follower relationships on Weibo or TikTok.</li>
+<li>In an undirected graph, an edge represents a two-way connection between two vertices, such as a friendship on WeChat or QQ.</li>
+<li>In a directed graph, edges have a direction: the edges <span class="arithmatex">\(A \rightarrow B\)</span> and <span class="arithmatex">\(A \leftarrow B\)</span> are independent of each other. Examples include following and being followed on Weibo or Douyin.</li>
 </ul>
 <p><img alt="Directed and undirected graphs" class="animation-figure" src="../graph.assets/directed_graph.png" /></p>
 <p align="center"> Figure 9-2 &nbsp; Directed and undirected graphs </p>
 
-<p>Graphs can be divided into <u>connected graphs</u> and <u>disconnected graphs</u> based on whether all vertices are connected, as shown in Figure 9-3.</p>
+<p>Depending on whether all vertices are connected, graphs are classified as <u>connected graphs</u> or <u>disconnected graphs</u>, as shown in Figure 9-3.</p>
 <ul>
-<li>For connected graphs, starting from any vertex, all other vertices can be reached.</li>
-<li>For disconnected graphs, starting from a certain vertex, at least one vertex cannot be reached.</li>
+<li>In a connected graph, every other vertex can be reached from any starting vertex.</li>
+<li>In a disconnected graph, at least one vertex cannot be reached from a given starting vertex.</li>
 </ul>
 <p><img alt="Connected and disconnected graphs" class="animation-figure" src="../graph.assets/connected_graph.png" /></p>
 <p align="center"> Figure 9-3 &nbsp; Connected and disconnected graphs </p>
 
-<p>We can also add a "weight" variable to edges, resulting in <u>weighted graphs</u> as shown in Figure 9-4. For example, in mobile games like "Honor of Kings", the system calculates the "intimacy" between players based on how long they have played together, and such intimacy networks can be represented using weighted graphs.</p>
+<p>We can also assign a weight to each edge to obtain a <u>weighted graph</u>, as shown in Figure 9-4. For example, mobile games such as Honor of Kings calculate an affinity score between players based on the time they spend playing together. A weighted graph can represent this network of affinities.</p>
 <p><img alt="Weighted and unweighted graphs" class="animation-figure" src="../graph.assets/weighted_graph.png" /></p>
 <p align="center"> Figure 9-4 &nbsp; Weighted and unweighted graphs </p>
 
-<p>Graph data structures include the following commonly used terms.</p>
+<p>The following terms are commonly used when discussing graphs.</p>
 <ul>
-<li><u>Adjacency</u>: When two vertices are connected by an edge, these two vertices are said to be "adjacent". In Figure 9-4, the adjacent vertices of vertex 1 are vertices 2, 3, and 5.</li>
-<li><u>Path</u>: The sequence of edges from vertex A to vertex B is called a "path" from A to B. In Figure 9-4, the edge sequence 1-5-2-4 is a path from vertex 1 to vertex 4.</li>
-<li><u>Degree</u>: The number of edges a vertex has. For directed graphs, <u>in-degree</u> indicates how many edges point to the vertex, and <u>out-degree</u> indicates how many edges leave the vertex.</li>
+<li><u>Adjacency</u>: Two vertices are adjacent if an edge connects them. In Figure 9-4, the vertices adjacent to vertex 1 are 2, 3, and 5.</li>
+<li><u>Path</u>: A sequence of edges followed from vertex A to vertex B is called a path from A to B. In Figure 9-4, the edge sequence 1-5-2-4 forms a path from vertex 1 to vertex 4.</li>
+<li><u>Degree</u>: The number of edges incident to a vertex. In a directed graph, the <u>in-degree</u> is the number of edges pointing to the vertex, and the <u>out-degree</u> is the number of edges pointing away from it.</li>
 </ul>
-<h2 id="912-representation-of-graphs">9.1.2 &nbsp; Representation of Graphs<a class="headerlink" href="#912-representation-of-graphs" title="Permanent link">&para;</a></h2>
-<p>Common representations of graphs include "adjacency matrices" and "adjacency lists". The following uses undirected graphs as examples.</p>
-<h3 id="1-adjacency-matrix">1. &nbsp; Adjacency Matrix<a class="headerlink" href="#1-adjacency-matrix" title="Permanent link">&para;</a></h3>
-<p>Given a graph with <span class="arithmatex">\(n\)</span> vertices, an <u>adjacency matrix</u> uses an <span class="arithmatex">\(n \times n\)</span> matrix to represent the graph, where each row (column) represents a vertex, and matrix elements represent edges, using <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(0\)</span> to indicate whether an edge exists between two vertices.</p>
-<p>As shown in Figure 9-5, let the adjacency matrix be <span class="arithmatex">\(M\)</span> and the vertex list be <span class="arithmatex">\(V\)</span>. Then matrix element <span class="arithmatex">\(M[i, j] = 1\)</span> indicates that an edge exists between vertex <span class="arithmatex">\(V[i]\)</span> and vertex <span class="arithmatex">\(V[j]\)</span>, whereas <span class="arithmatex">\(M[i, j] = 0\)</span> indicates no edge between the two vertices.</p>
-<p><img alt="Adjacency matrix representation of a graph" class="animation-figure" src="../graph.assets/adjacency_matrix.png" /></p>
-<p align="center"> Figure 9-5 &nbsp; Adjacency matrix representation of a graph </p>
+<h2 id="912-representing-a-graph">9.1.2 &nbsp; Representing a graph<a class="headerlink" href="#912-representing-a-graph" title="Permanent link">&para;</a></h2>
+<p>Two common ways to represent a graph are an adjacency matrix and an adjacency list. We will use undirected graphs in the following examples.</p>
+<h3 id="1-adjacency-matrix">1. &nbsp; Adjacency matrix<a class="headerlink" href="#1-adjacency-matrix" title="Permanent link">&para;</a></h3>
+<p>For a graph with <span class="arithmatex">\(n\)</span> vertices, an <u>adjacency matrix</u> represents the graph using an <span class="arithmatex">\(n \times n\)</span> matrix. Each row (or column) represents a vertex, and the matrix entries represent edges: <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(0\)</span> indicates whether an edge exists between two vertices.</p>
+<p>As shown in Figure 9-5, let <span class="arithmatex">\(M\)</span> be the adjacency matrix and <span class="arithmatex">\(V\)</span> the vertex list. The entry <span class="arithmatex">\(M[i, j] = 1\)</span> means that an edge connects vertices <span class="arithmatex">\(V[i]\)</span> and <span class="arithmatex">\(V[j]\)</span>, whereas <span class="arithmatex">\(M[i, j] = 0\)</span> means that no edge connects them.</p>
+<p><img alt="Representing a graph with an adjacency matrix" class="animation-figure" src="../graph.assets/adjacency_matrix.png" /></p>
+<p align="center"> Figure 9-5 &nbsp; Representing a graph with an adjacency matrix </p>
 
-<p>Adjacency matrices have the following properties.</p>
+<p>An adjacency matrix has the following properties.</p>
 <ul>
-<li>In simple graphs, vertices cannot connect to themselves, so the elements on the main diagonal of the adjacency matrix are meaningless.</li>
-<li>For undirected graphs, edges in both directions are equivalent, so the adjacency matrix is symmetric about the main diagonal.</li>
-<li>Replacing the <span class="arithmatex">\(1\)</span> and <span class="arithmatex">\(0\)</span> entries in the adjacency matrix with weights allows it to represent weighted graphs.</li>
+<li>In a simple graph, a vertex cannot connect to itself, so the entries on the main diagonal do not represent meaningful connections.</li>
+<li>In an undirected graph, edges in the two directions are equivalent, so the adjacency matrix is symmetric about its main diagonal.</li>
+<li>Replacing the <span class="arithmatex">\(1\)</span> and <span class="arithmatex">\(0\)</span> entries with weights allows the matrix to represent a weighted graph.</li>
 </ul>
-<p>When using adjacency matrices to represent graphs, we can directly access matrix elements to obtain edges, resulting in highly efficient addition, deletion, lookup, and modification operations, all with a time complexity of <span class="arithmatex">\(O(1)\)</span>. However, the space complexity of the matrix is <span class="arithmatex">\(O(n^2)\)</span>, which consumes significant memory.</p>
-<h3 id="2-adjacency-list">2. &nbsp; Adjacency List<a class="headerlink" href="#2-adjacency-list" title="Permanent link">&para;</a></h3>
-<p>An <u>adjacency list</u> uses <span class="arithmatex">\(n\)</span> linked lists to represent a graph, with linked list nodes representing vertices. The <span class="arithmatex">\(i\)</span>-th linked list corresponds to vertex <span class="arithmatex">\(i\)</span> and stores all adjacent vertices of that vertex (vertices connected to that vertex). Figure 9-6 shows an example of a graph stored using an adjacency list.</p>
-<p><img alt="Adjacency list representation of a graph" class="animation-figure" src="../graph.assets/adjacency_list.png" /></p>
-<p align="center"> Figure 9-6 &nbsp; Adjacency list representation of a graph </p>
+<p>With an adjacency matrix, we can access a matrix entry directly to retrieve an edge. Adding, removing, looking up, and modifying edges are therefore efficient, each taking <span class="arithmatex">\(O(1)\)</span> time. However, the matrix requires <span class="arithmatex">\(O(n^2)\)</span> space and uses a substantial amount of memory.</p>
+<h3 id="2-adjacency-list">2. &nbsp; Adjacency list<a class="headerlink" href="#2-adjacency-list" title="Permanent link">&para;</a></h3>
+<p>An <u>adjacency list</u> represents a graph using <span class="arithmatex">\(n\)</span> linked lists whose nodes represent vertices. The <span class="arithmatex">\(i\)</span>-th linked list corresponds to vertex <span class="arithmatex">\(i\)</span> and stores all of its adjacent vertices (the vertices connected to it). Figure 9-6 shows an example of a graph stored as an adjacency list.</p>
+<p><img alt="Representing a graph with an adjacency list" class="animation-figure" src="../graph.assets/adjacency_list.png" /></p>
+<p align="center"> Figure 9-6 &nbsp; Representing a graph with an adjacency list </p>
 
-<p>Adjacency lists only store edges that actually exist, and the total number of edges is typically much less than <span class="arithmatex">\(n^2\)</span>, making them more space-efficient. However, finding edges in an adjacency list requires traversing the linked list, so it is less time-efficient than an adjacency matrix.</p>
-<p>As shown in Figure 9-6, <strong>the structure of adjacency lists is very similar to separate chaining in hash tables, so we can use similar methods to improve efficiency</strong>. For example, when a linked list becomes long, it can be converted into an AVL tree or red-black tree, improving the time complexity from <span class="arithmatex">\(O(n)\)</span> to <span class="arithmatex">\(O(\log n)\)</span>; it can also be converted into a hash table, reducing the time complexity to <span class="arithmatex">\(O(1)\)</span>.</p>
-<h2 id="913-common-applications-of-graphs">9.1.3 &nbsp; Common Applications of Graphs<a class="headerlink" href="#913-common-applications-of-graphs" title="Permanent link">&para;</a></h2>
-<p>As shown in Table 9-1, many real-world systems can be modeled using graphs, and corresponding problems can be reduced to graph computation problems.</p>
-<p align="center"> Table 9-1 &nbsp; Common graphs in real life </p>
+<p>An adjacency list stores only edges that actually exist. Since the total number of edges is usually much smaller than <span class="arithmatex">\(n^2\)</span>, it saves space. However, finding an edge requires traversing a linked list, so lookups are slower than with an adjacency matrix.</p>
+<p>As Figure 9-6 shows, <strong>an adjacency list closely resembles separate chaining in a hash table, so similar techniques can improve its efficiency</strong>. For example, a long linked list can be replaced with an AVL tree or a red-black tree to improve lookup time from <span class="arithmatex">\(O(n)\)</span> to <span class="arithmatex">\(O(\log n)\)</span>. Replacing it with a hash table can further reduce the time complexity to <span class="arithmatex">\(O(1)\)</span>.</p>
+<h2 id="913-common-applications-of-graphs">9.1.3 &nbsp; Common applications of graphs<a class="headerlink" href="#913-common-applications-of-graphs" title="Permanent link">&para;</a></h2>
+<p>As shown in Table 9-1, many real-world systems can be modeled as graphs, and their associated problems can be reduced to graph computation problems.</p>
+<p align="center"> Table 9-1 &nbsp; Common graphs in everyday life </p>
 
 <div class="center-table">
 <table>
@@ -4808,27 +4880,27 @@ G &amp; = \{ V, E \} \newline
 <th></th>
 <th>Vertices</th>
 <th>Edges</th>
-<th>Graph Computation Problem</th>
+<th>Graph computation problem</th>
 </tr>
 </thead>
 <tbody>
 <tr>
 <td>Social network</td>
 <td>Users</td>
-<td>Friend relationships</td>
-<td>Potential friend recommendation</td>
+<td>Friendships</td>
+<td>Recommending potential friends</td>
 </tr>
 <tr>
-<td>Subway lines</td>
+<td>Subway network</td>
 <td>Stations</td>
-<td>Connectivity between stations</td>
-<td>Shortest route recommendation</td>
+<td>Connections between stations</td>
+<td>Recommending the shortest route</td>
 </tr>
 <tr>
 <td>Solar system</td>
 <td>Celestial bodies</td>
-<td>Gravitational forces between celestial bodies</td>
-<td>Planetary orbit calculation</td>
+<td>Gravitational interactions between bodies</td>
+<td>Calculating planetary orbits</td>
 </tr>
 </tbody>
 </table>
@@ -4855,7 +4927,7 @@ aria-label="Footer"
   <a
     href="../"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: Chapter 9. &amp;nbsp; Graph"
+    aria-label="Previous: Chapter 9.   Graphs"
     rel="prev"
   >
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@@ -4867,7 +4939,7 @@ aria-label="Footer"
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       </span>
       <div class="md-ellipsis">
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+        Chapter 9. &nbsp; Graphs
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@@ -4879,7 +4951,7 @@ aria-label="Footer"
   <a
     href="../graph_operations/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 9.2 Basic Operations on Graphs"
+    aria-label="Next: 9.2   Basic graph operations"
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@@ -4887,7 +4959,7 @@ aria-label="Footer"
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+        9.2 &nbsp; Basic graph operations
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@@ -4904,7 +4976,7 @@ aria-label="Footer"
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@@ -6,7 +6,7 @@
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+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
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@@ -41,7 +41,7 @@
 
 
 
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+        <title>9.2 Basic graph operations - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
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@@ -101,7 +101,7 @@
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+        <a href="#92-basic-graph-operations" class="md-skip">
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+              9.2 &nbsp; Basic graph operations
 
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-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2380,7 +2380,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2401,7 +2401,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2422,7 +2422,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2458,7 +2458,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2475,7 +2475,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2501,10 +2501,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#921-implementation-based-on-adjacency-matrix" class="md-nav__link">
+  <a href="#921-adjacency-matrix-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.2.1 &nbsp; Implementation Based on Adjacency Matrix
+        9.2.1 &nbsp; Adjacency matrix implementation
       </span>
     </span>
   </a>
@@ -2512,10 +2512,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#922-implementation-based-on-adjacency-list" class="md-nav__link">
+  <a href="#922-adjacency-list-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.2.2 &nbsp; Implementation Based on Adjacency List
+        9.2.2 &nbsp; Adjacency list implementation
       </span>
     </span>
   </a>
@@ -2523,10 +2523,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#923-efficiency-comparison" class="md-nav__link">
+  <a href="#923-comparing-efficiency" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.2.3 &nbsp; Efficiency Comparison
+        9.2.3 &nbsp; Comparing efficiency
       </span>
     </span>
   </a>
@@ -2556,7 +2556,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2583,7 +2583,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2610,7 +2610,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2678,7 +2678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2699,7 +2699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2720,7 +2720,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2747,7 +2747,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2774,7 +2774,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2801,7 +2801,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2828,7 +2828,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2855,7 +2855,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2882,7 +2882,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2960,7 +2960,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2981,7 +2981,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3002,7 +3002,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3029,7 +3029,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3056,7 +3056,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3083,7 +3083,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3110,7 +3110,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3137,7 +3137,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3164,7 +3164,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3191,7 +3191,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3218,7 +3218,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3245,7 +3245,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3272,7 +3272,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3299,7 +3299,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3365,7 +3365,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3386,7 +3386,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3407,7 +3407,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3434,7 +3434,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3461,7 +3461,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3488,7 +3488,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3515,7 +3515,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3542,7 +3542,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3608,7 +3608,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3629,7 +3629,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3650,7 +3650,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3677,7 +3677,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3704,7 +3704,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3731,7 +3731,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3758,7 +3758,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3785,7 +3785,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3855,7 +3855,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3876,7 +3876,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3897,7 +3897,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3924,7 +3924,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3951,7 +3951,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3978,7 +3978,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4005,7 +4005,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4032,7 +4032,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4059,7 +4059,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4086,7 +4086,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4152,7 +4152,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4173,7 +4173,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4194,7 +4194,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4221,7 +4221,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4248,7 +4248,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4275,7 +4275,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4302,7 +4302,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4329,7 +4329,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4389,7 +4389,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4410,7 +4410,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4431,7 +4431,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4458,7 +4458,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4485,7 +4485,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4569,6 +4569,78 @@
 
 
 
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@@ -4610,10 +4682,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#921-implementation-based-on-adjacency-matrix" class="md-nav__link">
+  <a href="#921-adjacency-matrix-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.2.1 &nbsp; Implementation Based on Adjacency Matrix
+        9.2.1 &nbsp; Adjacency matrix implementation
       </span>
     </span>
   </a>
@@ -4621,10 +4693,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#922-implementation-based-on-adjacency-list" class="md-nav__link">
+  <a href="#922-adjacency-list-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.2.2 &nbsp; Implementation Based on Adjacency List
+        9.2.2 &nbsp; Adjacency list implementation
       </span>
     </span>
   </a>
@@ -4632,10 +4704,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#923-efficiency-comparison" class="md-nav__link">
+  <a href="#923-comparing-efficiency" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.2.3 &nbsp; Efficiency Comparison
+        9.2.3 &nbsp; Comparing efficiency
       </span>
     </span>
   </a>
@@ -4679,20 +4751,20 @@
 
 
 <!-- Page content -->
-<h1 id="92-basic-operations-on-graphs">9.2 &nbsp; Basic Operations on Graphs<a class="headerlink" href="#92-basic-operations-on-graphs" title="Permanent link">&para;</a></h1>
-<p>Basic operations on graphs can be divided into operations on "edges" and operations on "vertices". Their implementations differ depending on whether the graph is represented as an "adjacency matrix" or an "adjacency list".</p>
-<h2 id="921-implementation-based-on-adjacency-matrix">9.2.1 &nbsp; Implementation Based on Adjacency Matrix<a class="headerlink" href="#921-implementation-based-on-adjacency-matrix" title="Permanent link">&para;</a></h2>
-<p>Given an undirected graph with <span class="arithmatex">\(n\)</span> vertices, the various operations are implemented as shown in Figure 9-7.</p>
+<h1 id="92-basic-graph-operations">9.2 &nbsp; Basic graph operations<a class="headerlink" href="#92-basic-graph-operations" title="Permanent link">&para;</a></h1>
+<p>Basic graph operations fall into two categories: operations on edges and operations on vertices. Their implementations differ depending on whether the graph is represented by an adjacency matrix or an adjacency list.</p>
+<h2 id="921-adjacency-matrix-implementation">9.2.1 &nbsp; Adjacency matrix implementation<a class="headerlink" href="#921-adjacency-matrix-implementation" title="Permanent link">&para;</a></h2>
+<p>For an undirected graph with <span class="arithmatex">\(n\)</span> vertices, the operations are implemented as shown in Figure 9-7.</p>
 <ul>
-<li><strong>Adding or removing an edge</strong>: Directly modify the specified edge in the adjacency matrix, using <span class="arithmatex">\(O(1)\)</span> time. Since it is an undirected graph, both directions of the edge need to be updated simultaneously.</li>
-<li><strong>Adding a vertex</strong>: Add a row and a column at the end of the adjacency matrix and fill them all with <span class="arithmatex">\(0\)</span>s, using <span class="arithmatex">\(O(n)\)</span> time.</li>
-<li><strong>Removing a vertex</strong>: Delete a row and a column in the adjacency matrix. The worst case occurs when removing the first row and column, requiring <span class="arithmatex">\((n-1)^2\)</span> elements to be "moved up and to the left", thus using <span class="arithmatex">\(O(n^2)\)</span> time.</li>
-<li><strong>Initialization</strong>: Given <span class="arithmatex">\(n\)</span> vertices, initialize a vertex list <code>vertices</code> of length <span class="arithmatex">\(n\)</span>, using <span class="arithmatex">\(O(n)\)</span> time; initialize an adjacency matrix <code>adjMat</code> of size <span class="arithmatex">\(n \times n\)</span>, using <span class="arithmatex">\(O(n^2)\)</span> time.</li>
+<li><strong>Add or remove an edge</strong>: Modify the corresponding entry in the adjacency matrix directly, taking <span class="arithmatex">\(O(1)\)</span> time. Since the graph is undirected, update the edge in both directions.</li>
+<li><strong>Add a vertex</strong>: Append one row and one column to the adjacency matrix and fill them with <span class="arithmatex">\(0\)</span>, taking <span class="arithmatex">\(O(n)\)</span> time.</li>
+<li><strong>Remove a vertex</strong>: Delete the corresponding row and column from the adjacency matrix. The worst case occurs when deleting the first row and column: <span class="arithmatex">\((n-1)^2\)</span> entries must be shifted up and to the left, taking <span class="arithmatex">\(O(n^2)\)</span> time.</li>
+<li><strong>Initialize the graph</strong>: Given <span class="arithmatex">\(n\)</span> vertices, initialize the vertex list <code>vertices</code> of length <span class="arithmatex">\(n\)</span> in <span class="arithmatex">\(O(n)\)</span> time, then initialize the <span class="arithmatex">\(n \times n\)</span> adjacency matrix <code>adjMat</code> in <span class="arithmatex">\(O(n^2)\)</span> time.</li>
 </ul>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:5"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Initialization, adding and removing edges, adding and removing vertices in adjacency matrix" class="animation-figure" src="../graph_operations.assets/adjacency_matrix_step1_initialization.png" /></p>
+<p><img alt="Initializing an adjacency matrix and adding or removing edges and vertices" class="animation-figure" src="../graph_operations.assets/adjacency_matrix_step1_initialization.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="adjacency_matrix_add_edge" class="animation-figure" src="../graph_operations.assets/adjacency_matrix_step2_add_edge.png" /></p>
@@ -4708,26 +4780,26 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 9-7 &nbsp; Initialization, adding and removing edges, adding and removing vertices in adjacency matrix </p>
+<p align="center"> Figure 9-7 &nbsp; Initializing an adjacency matrix and adding or removing edges and vertices </p>
 
-<p>The following is the implementation code for graphs represented using an adjacency matrix:</p>
+<p>The following code implements a graph using an adjacency matrix:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_adjacency_matrix.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">GraphAdjMat</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Undirected graph class based on adjacency matrix"""</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Undirected graph class implemented with an adjacency matrix"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">vertices</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">edges</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]]):</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w">        </span><span class="sd">"""Constructor"""</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="c1"># Vertex list, where the element represents the "vertex value" and the index represents the "vertex index"</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="c1"># Vertex list: elements are vertex values, and indices are vertex indices</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="p">[]</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="c1"># Adjacency matrix, where the row and column indices correspond to the "vertex index"</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="c1"># Adjacency matrix: row and column indices correspond to vertex indices</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">adj_mat</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]]</span> <span class="o">=</span> <span class="p">[]</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="c1"># Add vertices</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="c1"># Add a vertex</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="k">for</span> <span class="n">val</span> <span class="ow">in</span> <span class="n">vertices</span><span class="p">:</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">add_vertex</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
-<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="c1"># Add edges</span>
-<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>        <span class="c1"># Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices</span>
+<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="c1"># Add an edge</span>
+<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>        <span class="c1"># Note that the entries in edges are vertex indices, corresponding to element indices in vertices</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>        <span class="k">for</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">edges</span><span class="p">:</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">e</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">e</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
 <a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>
@@ -4736,9 +4808,9 @@
 <a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">vertices</span><span class="p">)</span>
 <a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a>
 <a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">add_vertex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="w">        </span><span class="sd">"""Add vertex"""</span>
+<a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="w">        </span><span class="sd">"""Add a vertex"""</span>
 <a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a>        <span class="n">n</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span>
-<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a>        <span class="c1"># Add the value of the new vertex to the vertex list</span>
+<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a>        <span class="c1"># Append the new vertex value to the vertex list</span>
 <a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a>        <span class="c1"># Add a row to the adjacency matrix</span>
 <a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a>        <span class="n">new_row</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span>
@@ -4748,58 +4820,58 @@
 <a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a>            <span class="n">row</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a>
 <a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">remove_vertex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">index</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a><span class="w">        </span><span class="sd">"""Remove vertex"""</span>
+<a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a><span class="w">        </span><span class="sd">"""Remove a vertex"""</span>
 <a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a>        <span class="k">if</span> <span class="n">index</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">():</span>
 <a id="__codelineno-0-37" name="__codelineno-0-37" href="#__codelineno-0-37"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">()</span>
-<a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></a>        <span class="c1"># Remove the vertex at index from the vertex list</span>
+<a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></a>        <span class="c1"># Remove the vertex at index `index` from the vertex list</span>
 <a id="__codelineno-0-39" name="__codelineno-0-39" href="#__codelineno-0-39"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
-<a id="__codelineno-0-40" name="__codelineno-0-40" href="#__codelineno-0-40"></a>        <span class="c1"># Remove the row at index from the adjacency matrix</span>
+<a id="__codelineno-0-40" name="__codelineno-0-40" href="#__codelineno-0-40"></a>        <span class="c1"># Remove row `index` from the adjacency matrix</span>
 <a id="__codelineno-0-41" name="__codelineno-0-41" href="#__codelineno-0-41"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">adj_mat</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
-<a id="__codelineno-0-42" name="__codelineno-0-42" href="#__codelineno-0-42"></a>        <span class="c1"># Remove the column at index from the adjacency matrix</span>
+<a id="__codelineno-0-42" name="__codelineno-0-42" href="#__codelineno-0-42"></a>        <span class="c1"># Remove column `index` from the adjacency matrix</span>
 <a id="__codelineno-0-43" name="__codelineno-0-43" href="#__codelineno-0-43"></a>        <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">adj_mat</span><span class="p">:</span>
 <a id="__codelineno-0-44" name="__codelineno-0-44" href="#__codelineno-0-44"></a>            <span class="n">row</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
 <a id="__codelineno-0-45" name="__codelineno-0-45" href="#__codelineno-0-45"></a>
 <a id="__codelineno-0-46" name="__codelineno-0-46" href="#__codelineno-0-46"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">add_edge</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-0-47" name="__codelineno-0-47" href="#__codelineno-0-47"></a><span class="w">        </span><span class="sd">"""Add edge"""</span>
-<a id="__codelineno-0-48" name="__codelineno-0-48" href="#__codelineno-0-48"></a>        <span class="c1"># Parameters i, j correspond to the vertices element indices</span>
-<a id="__codelineno-0-49" name="__codelineno-0-49" href="#__codelineno-0-49"></a>        <span class="c1"># Handle index out of bounds and equality</span>
+<a id="__codelineno-0-47" name="__codelineno-0-47" href="#__codelineno-0-47"></a><span class="w">        </span><span class="sd">"""Add an edge"""</span>
+<a id="__codelineno-0-48" name="__codelineno-0-48" href="#__codelineno-0-48"></a>        <span class="c1"># Parameters i and j are element indices in vertices</span>
+<a id="__codelineno-0-49" name="__codelineno-0-49" href="#__codelineno-0-49"></a>        <span class="c1"># Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-0-50" name="__codelineno-0-50" href="#__codelineno-0-50"></a>        <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">i</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span> <span class="ow">or</span> <span class="n">j</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span> <span class="ow">or</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
 <a id="__codelineno-0-51" name="__codelineno-0-51" href="#__codelineno-0-51"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">()</span>
-<a id="__codelineno-0-52" name="__codelineno-0-52" href="#__codelineno-0-52"></a>        <span class="c1"># In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)</span>
+<a id="__codelineno-0-52" name="__codelineno-0-52" href="#__codelineno-0-52"></a>        <span class="c1"># For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal</span>
 <a id="__codelineno-0-53" name="__codelineno-0-53" href="#__codelineno-0-53"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">adj_mat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
 <a id="__codelineno-0-54" name="__codelineno-0-54" href="#__codelineno-0-54"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">adj_mat</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
 <a id="__codelineno-0-55" name="__codelineno-0-55" href="#__codelineno-0-55"></a>
 <a id="__codelineno-0-56" name="__codelineno-0-56" href="#__codelineno-0-56"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">remove_edge</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">j</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-0-57" name="__codelineno-0-57" href="#__codelineno-0-57"></a><span class="w">        </span><span class="sd">"""Remove edge"""</span>
-<a id="__codelineno-0-58" name="__codelineno-0-58" href="#__codelineno-0-58"></a>        <span class="c1"># Parameters i, j correspond to the vertices element indices</span>
-<a id="__codelineno-0-59" name="__codelineno-0-59" href="#__codelineno-0-59"></a>        <span class="c1"># Handle index out of bounds and equality</span>
+<a id="__codelineno-0-57" name="__codelineno-0-57" href="#__codelineno-0-57"></a><span class="w">        </span><span class="sd">"""Remove an edge"""</span>
+<a id="__codelineno-0-58" name="__codelineno-0-58" href="#__codelineno-0-58"></a>        <span class="c1"># Parameters i and j are element indices in vertices</span>
+<a id="__codelineno-0-59" name="__codelineno-0-59" href="#__codelineno-0-59"></a>        <span class="c1"># Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-0-60" name="__codelineno-0-60" href="#__codelineno-0-60"></a>        <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">j</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">i</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span> <span class="ow">or</span> <span class="n">j</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span> <span class="ow">or</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
 <a id="__codelineno-0-61" name="__codelineno-0-61" href="#__codelineno-0-61"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">()</span>
 <a id="__codelineno-0-62" name="__codelineno-0-62" href="#__codelineno-0-62"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">adj_mat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-0-63" name="__codelineno-0-63" href="#__codelineno-0-63"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">adj_mat</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-0-64" name="__codelineno-0-64" href="#__codelineno-0-64"></a>
 <a id="__codelineno-0-65" name="__codelineno-0-65" href="#__codelineno-0-65"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">print</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-<a id="__codelineno-0-66" name="__codelineno-0-66" href="#__codelineno-0-66"></a><span class="w">        </span><span class="sd">"""Print adjacency matrix"""</span>
+<a id="__codelineno-0-66" name="__codelineno-0-66" href="#__codelineno-0-66"></a><span class="w">        </span><span class="sd">"""Print the adjacency matrix"""</span>
 <a id="__codelineno-0-67" name="__codelineno-0-67" href="#__codelineno-0-67"></a>        <span class="nb">print</span><span class="p">(</span><span class="s2">"Vertex list ="</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">vertices</span><span class="p">)</span>
 <a id="__codelineno-0-68" name="__codelineno-0-68" href="#__codelineno-0-68"></a>        <span class="nb">print</span><span class="p">(</span><span class="s2">"Adjacency matrix ="</span><span class="p">)</span>
 <a id="__codelineno-0-69" name="__codelineno-0-69" href="#__codelineno-0-69"></a>        <span class="n">print_matrix</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">adj_mat</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_matrix.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Undirected graph class based on adjacency matrix */</span>
+<div class="highlight"><span class="filename">graph_adjacency_matrix.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency matrix */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">GraphAdjMat</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">vertices</span><span class="p">;</span><span class="w">       </span><span class="c1">// Vertex list, where the element represents the "vertex value" and the index represents the "vertex index"</span>
-<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">adjMat</span><span class="p">;</span><span class="w"> </span><span class="c1">// Adjacency matrix, where the row and column indices correspond to the "vertex index"</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">vertices</span><span class="p">;</span><span class="w">       </span><span class="c1">// Vertex list: elements are vertex values, and indices are vertex indices</span>
+<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">adjMat</span><span class="p">;</span><span class="w"> </span><span class="c1">// Adjacency matrix: row and column indices correspond to vertex indices</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">  </span><span class="k">public</span><span class="o">:</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="n">GraphAdjMat</span><span class="p">(</span><span class="k">const</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">vertices</span><span class="p">,</span><span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">edges</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">        </span><span class="c1">// Add vertex</span>
+<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">        </span><span class="c1">// Add a vertex</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">vertices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">            </span><span class="n">addVertex</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="c1">// Add edge</span>
-<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">        </span><span class="c1">// Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices</span>
+<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="c1">// Add an edge</span>
+<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">        </span><span class="c1">// Note that the entries in edges are vertex indices, corresponding to element indices in vertices</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="k">const</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">edge</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">edges</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">            </span><span class="n">addEdge</span><span class="p">(</span><span class="n">edge</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="w"> </span><span class="n">edge</span><span class="p">[</span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">        </span><span class="p">}</span>
@@ -4810,10 +4882,10 @@
 <a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">vertices</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
 <a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a>
-<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">addVertex</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="w">        </span><span class="c1">// Add the value of the new vertex to the vertex list</span>
+<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="w">        </span><span class="c1">// Append the new vertex value to the vertex list</span>
 <a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="w">        </span><span class="n">vertices</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="w">        </span><span class="c1">// Add a row to the adjacency matrix</span>
 <a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">.</span><span class="n">emplace_back</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">));</span>
@@ -4823,45 +4895,45 @@
 <a id="__codelineno-1-35" name="__codelineno-1-35" href="#__codelineno-1-35"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-1-36" name="__codelineno-1-36" href="#__codelineno-1-36"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-37" name="__codelineno-1-37" href="#__codelineno-1-37"></a>
-<a id="__codelineno-1-38" name="__codelineno-1-38" href="#__codelineno-1-38"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-1-38" name="__codelineno-1-38" href="#__codelineno-1-38"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-1-39" name="__codelineno-1-39" href="#__codelineno-1-39"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">removeVertex</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-40" name="__codelineno-1-40" href="#__codelineno-1-40"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-41" name="__codelineno-1-41" href="#__codelineno-1-41"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Vertex does not exist"</span><span class="p">);</span>
+<a id="__codelineno-1-41" name="__codelineno-1-41" href="#__codelineno-1-41"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Invalid vertex"</span><span class="p">);</span>
 <a id="__codelineno-1-42" name="__codelineno-1-42" href="#__codelineno-1-42"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-1-43" name="__codelineno-1-43" href="#__codelineno-1-43"></a><span class="w">        </span><span class="c1">// Remove the vertex at index from the vertex list</span>
+<a id="__codelineno-1-43" name="__codelineno-1-43" href="#__codelineno-1-43"></a><span class="w">        </span><span class="c1">// Remove the vertex at index `index` from the vertex list</span>
 <a id="__codelineno-1-44" name="__codelineno-1-44" href="#__codelineno-1-44"></a><span class="w">        </span><span class="n">vertices</span><span class="p">.</span><span class="n">erase</span><span class="p">(</span><span class="n">vertices</span><span class="p">.</span><span class="n">begin</span><span class="p">()</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">index</span><span class="p">);</span>
-<a id="__codelineno-1-45" name="__codelineno-1-45" href="#__codelineno-1-45"></a><span class="w">        </span><span class="c1">// Remove the row at index from the adjacency matrix</span>
+<a id="__codelineno-1-45" name="__codelineno-1-45" href="#__codelineno-1-45"></a><span class="w">        </span><span class="c1">// Remove row `index` from the adjacency matrix</span>
 <a id="__codelineno-1-46" name="__codelineno-1-46" href="#__codelineno-1-46"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">.</span><span class="n">erase</span><span class="p">(</span><span class="n">adjMat</span><span class="p">.</span><span class="n">begin</span><span class="p">()</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">index</span><span class="p">);</span>
-<a id="__codelineno-1-47" name="__codelineno-1-47" href="#__codelineno-1-47"></a><span class="w">        </span><span class="c1">// Remove the column at index from the adjacency matrix</span>
+<a id="__codelineno-1-47" name="__codelineno-1-47" href="#__codelineno-1-47"></a><span class="w">        </span><span class="c1">// Remove column `index` from the adjacency matrix</span>
 <a id="__codelineno-1-48" name="__codelineno-1-48" href="#__codelineno-1-48"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">row</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">adjMat</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-49" name="__codelineno-1-49" href="#__codelineno-1-49"></a><span class="w">            </span><span class="n">row</span><span class="p">.</span><span class="n">erase</span><span class="p">(</span><span class="n">row</span><span class="p">.</span><span class="n">begin</span><span class="p">()</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">index</span><span class="p">);</span>
 <a id="__codelineno-1-50" name="__codelineno-1-50" href="#__codelineno-1-50"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-1-51" name="__codelineno-1-51" href="#__codelineno-1-51"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-52" name="__codelineno-1-52" href="#__codelineno-1-52"></a>
-<a id="__codelineno-1-53" name="__codelineno-1-53" href="#__codelineno-1-53"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
-<a id="__codelineno-1-54" name="__codelineno-1-54" href="#__codelineno-1-54"></a><span class="w">    </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-1-53" name="__codelineno-1-53" href="#__codelineno-1-53"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
+<a id="__codelineno-1-54" name="__codelineno-1-54" href="#__codelineno-1-54"></a><span class="w">    </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-1-55" name="__codelineno-1-55" href="#__codelineno-1-55"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">addEdge</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-56" name="__codelineno-1-56" href="#__codelineno-1-56"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-1-56" name="__codelineno-1-56" href="#__codelineno-1-56"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-1-57" name="__codelineno-1-57" href="#__codelineno-1-57"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-58" name="__codelineno-1-58" href="#__codelineno-1-58"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Vertex does not exist"</span><span class="p">);</span>
+<a id="__codelineno-1-58" name="__codelineno-1-58" href="#__codelineno-1-58"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Invalid vertex"</span><span class="p">);</span>
 <a id="__codelineno-1-59" name="__codelineno-1-59" href="#__codelineno-1-59"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-1-60" name="__codelineno-1-60" href="#__codelineno-1-60"></a><span class="w">        </span><span class="c1">// In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)</span>
+<a id="__codelineno-1-60" name="__codelineno-1-60" href="#__codelineno-1-60"></a><span class="w">        </span><span class="c1">// For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal</span>
 <a id="__codelineno-1-61" name="__codelineno-1-61" href="#__codelineno-1-61"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-1-62" name="__codelineno-1-62" href="#__codelineno-1-62"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-1-63" name="__codelineno-1-63" href="#__codelineno-1-63"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-64" name="__codelineno-1-64" href="#__codelineno-1-64"></a>
-<a id="__codelineno-1-65" name="__codelineno-1-65" href="#__codelineno-1-65"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
-<a id="__codelineno-1-66" name="__codelineno-1-66" href="#__codelineno-1-66"></a><span class="w">    </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-1-65" name="__codelineno-1-65" href="#__codelineno-1-65"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
+<a id="__codelineno-1-66" name="__codelineno-1-66" href="#__codelineno-1-66"></a><span class="w">    </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-1-67" name="__codelineno-1-67" href="#__codelineno-1-67"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">removeEdge</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-68" name="__codelineno-1-68" href="#__codelineno-1-68"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-1-68" name="__codelineno-1-68" href="#__codelineno-1-68"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-1-69" name="__codelineno-1-69" href="#__codelineno-1-69"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-70" name="__codelineno-1-70" href="#__codelineno-1-70"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Vertex does not exist"</span><span class="p">);</span>
+<a id="__codelineno-1-70" name="__codelineno-1-70" href="#__codelineno-1-70"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Invalid vertex"</span><span class="p">);</span>
 <a id="__codelineno-1-71" name="__codelineno-1-71" href="#__codelineno-1-71"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-1-72" name="__codelineno-1-72" href="#__codelineno-1-72"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-1-73" name="__codelineno-1-73" href="#__codelineno-1-73"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-1-74" name="__codelineno-1-74" href="#__codelineno-1-74"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-75" name="__codelineno-1-75" href="#__codelineno-1-75"></a>
-<a id="__codelineno-1-76" name="__codelineno-1-76" href="#__codelineno-1-76"></a><span class="w">    </span><span class="cm">/* Print adjacency matrix */</span>
+<a id="__codelineno-1-76" name="__codelineno-1-76" href="#__codelineno-1-76"></a><span class="w">    </span><span class="cm">/* Print the adjacency matrix */</span>
 <a id="__codelineno-1-77" name="__codelineno-1-77" href="#__codelineno-1-77"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-78" name="__codelineno-1-78" href="#__codelineno-1-78"></a><span class="w">        </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">"Vertex list = "</span><span class="p">;</span>
 <a id="__codelineno-1-79" name="__codelineno-1-79" href="#__codelineno-1-79"></a><span class="w">        </span><span class="n">printVector</span><span class="p">(</span><span class="n">vertices</span><span class="p">);</span>
@@ -4872,21 +4944,21 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_matrix.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Undirected graph class based on adjacency matrix */</span>
+<div class="highlight"><span class="filename">graph_adjacency_matrix.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency matrix */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kd">class</span> <span class="nc">GraphAdjMat</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">vertices</span><span class="p">;</span><span class="w"> </span><span class="c1">// Vertex list, where the element represents the "vertex value" and the index represents the "vertex index"</span>
-<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">adjMat</span><span class="p">;</span><span class="w"> </span><span class="c1">// Adjacency matrix, where the row and column indices correspond to the "vertex index"</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">vertices</span><span class="p">;</span><span class="w"> </span><span class="c1">// Vertex list: elements are vertex values, and indices are vertex indices</span>
+<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">adjMat</span><span class="p">;</span><span class="w"> </span><span class="c1">// Adjacency matrix: row and column indices correspond to vertex indices</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="nf">GraphAdjMat</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">vertices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[][]</span><span class="w"> </span><span class="n">edges</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="na">vertices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="na">adjMat</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
-<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="c1">// Add vertex</span>
+<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="c1">// Add a vertex</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">vertices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">            </span><span class="n">addVertex</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">        </span><span class="c1">// Add edge</span>
-<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="c1">// Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices</span>
+<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">        </span><span class="c1">// Add an edge</span>
+<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="c1">// Note that the entries in edges are vertex indices, corresponding to element indices in vertices</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">e</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">edges</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">            </span><span class="n">addEdge</span><span class="p">(</span><span class="n">e</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">e</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="p">);</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">        </span><span class="p">}</span>
@@ -4897,10 +4969,10 @@
 <a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">vertices</span><span class="p">.</span><span class="na">size</span><span class="p">();</span>
 <a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a>
-<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">addVertex</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="w">        </span><span class="c1">// Add the value of the new vertex to the vertex list</span>
+<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="w">        </span><span class="c1">// Append the new vertex value to the vertex list</span>
 <a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a><span class="w">        </span><span class="n">vertices</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="w">        </span><span class="c1">// Add a row to the adjacency matrix</span>
 <a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="w">        </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">newRow</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">(</span><span class="n">n</span><span class="p">);</span>
@@ -4914,42 +4986,42 @@
 <a id="__codelineno-2-40" name="__codelineno-2-40" href="#__codelineno-2-40"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-2-41" name="__codelineno-2-41" href="#__codelineno-2-41"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-42" name="__codelineno-2-42" href="#__codelineno-2-42"></a>
-<a id="__codelineno-2-43" name="__codelineno-2-43" href="#__codelineno-2-43"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-2-43" name="__codelineno-2-43" href="#__codelineno-2-43"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-2-44" name="__codelineno-2-44" href="#__codelineno-2-44"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">removeVertex</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-45" name="__codelineno-2-45" href="#__codelineno-2-45"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">())</span>
 <a id="__codelineno-2-46" name="__codelineno-2-46" href="#__codelineno-2-46"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">();</span>
-<a id="__codelineno-2-47" name="__codelineno-2-47" href="#__codelineno-2-47"></a><span class="w">        </span><span class="c1">// Remove the vertex at index from the vertex list</span>
+<a id="__codelineno-2-47" name="__codelineno-2-47" href="#__codelineno-2-47"></a><span class="w">        </span><span class="c1">// Remove the vertex at index `index` from the vertex list</span>
 <a id="__codelineno-2-48" name="__codelineno-2-48" href="#__codelineno-2-48"></a><span class="w">        </span><span class="n">vertices</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
-<a id="__codelineno-2-49" name="__codelineno-2-49" href="#__codelineno-2-49"></a><span class="w">        </span><span class="c1">// Remove the row at index from the adjacency matrix</span>
+<a id="__codelineno-2-49" name="__codelineno-2-49" href="#__codelineno-2-49"></a><span class="w">        </span><span class="c1">// Remove row `index` from the adjacency matrix</span>
 <a id="__codelineno-2-50" name="__codelineno-2-50" href="#__codelineno-2-50"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
-<a id="__codelineno-2-51" name="__codelineno-2-51" href="#__codelineno-2-51"></a><span class="w">        </span><span class="c1">// Remove the column at index from the adjacency matrix</span>
+<a id="__codelineno-2-51" name="__codelineno-2-51" href="#__codelineno-2-51"></a><span class="w">        </span><span class="c1">// Remove column `index` from the adjacency matrix</span>
 <a id="__codelineno-2-52" name="__codelineno-2-52" href="#__codelineno-2-52"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">adjMat</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-53" name="__codelineno-2-53" href="#__codelineno-2-53"></a><span class="w">            </span><span class="n">row</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
 <a id="__codelineno-2-54" name="__codelineno-2-54" href="#__codelineno-2-54"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-2-55" name="__codelineno-2-55" href="#__codelineno-2-55"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-56" name="__codelineno-2-56" href="#__codelineno-2-56"></a>
-<a id="__codelineno-2-57" name="__codelineno-2-57" href="#__codelineno-2-57"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
-<a id="__codelineno-2-58" name="__codelineno-2-58" href="#__codelineno-2-58"></a><span class="w">    </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-2-57" name="__codelineno-2-57" href="#__codelineno-2-57"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
+<a id="__codelineno-2-58" name="__codelineno-2-58" href="#__codelineno-2-58"></a><span class="w">    </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-2-59" name="__codelineno-2-59" href="#__codelineno-2-59"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">addEdge</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-60" name="__codelineno-2-60" href="#__codelineno-2-60"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-2-60" name="__codelineno-2-60" href="#__codelineno-2-60"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-2-61" name="__codelineno-2-61" href="#__codelineno-2-61"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-2-62" name="__codelineno-2-62" href="#__codelineno-2-62"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">();</span>
-<a id="__codelineno-2-63" name="__codelineno-2-63" href="#__codelineno-2-63"></a><span class="w">        </span><span class="c1">// In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)</span>
+<a id="__codelineno-2-63" name="__codelineno-2-63" href="#__codelineno-2-63"></a><span class="w">        </span><span class="c1">// For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal</span>
 <a id="__codelineno-2-64" name="__codelineno-2-64" href="#__codelineno-2-64"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">i</span><span class="p">).</span><span class="na">set</span><span class="p">(</span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-2-65" name="__codelineno-2-65" href="#__codelineno-2-65"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">j</span><span class="p">).</span><span class="na">set</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-2-66" name="__codelineno-2-66" href="#__codelineno-2-66"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-67" name="__codelineno-2-67" href="#__codelineno-2-67"></a>
-<a id="__codelineno-2-68" name="__codelineno-2-68" href="#__codelineno-2-68"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
-<a id="__codelineno-2-69" name="__codelineno-2-69" href="#__codelineno-2-69"></a><span class="w">    </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-2-68" name="__codelineno-2-68" href="#__codelineno-2-68"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
+<a id="__codelineno-2-69" name="__codelineno-2-69" href="#__codelineno-2-69"></a><span class="w">    </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-2-70" name="__codelineno-2-70" href="#__codelineno-2-70"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">removeEdge</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-71" name="__codelineno-2-71" href="#__codelineno-2-71"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-2-71" name="__codelineno-2-71" href="#__codelineno-2-71"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-2-72" name="__codelineno-2-72" href="#__codelineno-2-72"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-2-73" name="__codelineno-2-73" href="#__codelineno-2-73"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">();</span>
 <a id="__codelineno-2-74" name="__codelineno-2-74" href="#__codelineno-2-74"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">i</span><span class="p">).</span><span class="na">set</span><span class="p">(</span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-2-75" name="__codelineno-2-75" href="#__codelineno-2-75"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">j</span><span class="p">).</span><span class="na">set</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-2-76" name="__codelineno-2-76" href="#__codelineno-2-76"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-77" name="__codelineno-2-77" href="#__codelineno-2-77"></a>
-<a id="__codelineno-2-78" name="__codelineno-2-78" href="#__codelineno-2-78"></a><span class="w">    </span><span class="cm">/* Print adjacency matrix */</span>
+<a id="__codelineno-2-78" name="__codelineno-2-78" href="#__codelineno-2-78"></a><span class="w">    </span><span class="cm">/* Print the adjacency matrix */</span>
 <a id="__codelineno-2-79" name="__codelineno-2-79" href="#__codelineno-2-79"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-80" name="__codelineno-2-80" href="#__codelineno-2-80"></a><span class="w">        </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">print</span><span class="p">(</span><span class="s">"Vertex list = "</span><span class="p">);</span>
 <a id="__codelineno-2-81" name="__codelineno-2-81" href="#__codelineno-2-81"></a><span class="w">        </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="n">vertices</span><span class="p">);</span>
@@ -4960,21 +5032,21 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_matrix.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Undirected graph class based on adjacency matrix */</span>
+<div class="highlight"><span class="filename">graph_adjacency_matrix.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency matrix */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">GraphAdjMat</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">vertices</span><span class="p">;</span><span class="w">     </span><span class="c1">// Vertex list, where the element represents the "vertex value" and the index represents the "vertex index"</span>
-<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">adjMat</span><span class="p">;</span><span class="w"> </span><span class="c1">// Adjacency matrix, where the row and column indices correspond to the "vertex index"</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">vertices</span><span class="p">;</span><span class="w">     </span><span class="c1">// Vertex list: elements are vertex values, and indices are vertex indices</span>
+<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">adjMat</span><span class="p">;</span><span class="w"> </span><span class="c1">// Adjacency matrix: row and column indices correspond to vertex indices</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nf">GraphAdjMat</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">vertices</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[][]</span><span class="w"> </span><span class="n">edges</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">vertices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">adjMat</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="c1">// Add vertex</span>
+<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="c1">// Add a vertex</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">vertices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">            </span><span class="n">AddVertex</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">        </span><span class="c1">// Add edge</span>
-<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">        </span><span class="c1">// Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices</span>
+<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">        </span><span class="c1">// Add an edge</span>
+<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">        </span><span class="c1">// Note that the entries in edges are vertex indices, corresponding to element indices in vertices</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">e</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">edges</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">            </span><span class="n">AddEdge</span><span class="p">(</span><span class="n">e</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="w"> </span><span class="n">e</span><span class="p">[</span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">        </span><span class="p">}</span>
@@ -4985,10 +5057,10 @@
 <a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">vertices</span><span class="p">.</span><span class="n">Count</span><span class="p">;</span>
 <a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a>
-<a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">AddVertex</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Size</span><span class="p">();</span>
-<a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="w">        </span><span class="c1">// Add the value of the new vertex to the vertex list</span>
+<a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="w">        </span><span class="c1">// Append the new vertex value to the vertex list</span>
 <a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a><span class="w">        </span><span class="n">vertices</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a><span class="w">        </span><span class="c1">// Add a row to the adjacency matrix</span>
 <a id="__codelineno-3-32" name="__codelineno-3-32" href="#__codelineno-3-32"></a><span class="w">        </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">newRow</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="n">n</span><span class="p">);</span>
@@ -5002,42 +5074,42 @@
 <a id="__codelineno-3-40" name="__codelineno-3-40" href="#__codelineno-3-40"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-3-41" name="__codelineno-3-41" href="#__codelineno-3-41"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-42" name="__codelineno-3-42" href="#__codelineno-3-42"></a>
-<a id="__codelineno-3-43" name="__codelineno-3-43" href="#__codelineno-3-43"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-3-43" name="__codelineno-3-43" href="#__codelineno-3-43"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-3-44" name="__codelineno-3-44" href="#__codelineno-3-44"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">RemoveVertex</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-45" name="__codelineno-3-45" href="#__codelineno-3-45"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">Size</span><span class="p">())</span>
 <a id="__codelineno-3-46" name="__codelineno-3-46" href="#__codelineno-3-46"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">IndexOutOfRangeException</span><span class="p">();</span>
-<a id="__codelineno-3-47" name="__codelineno-3-47" href="#__codelineno-3-47"></a><span class="w">        </span><span class="c1">// Remove the vertex at index from the vertex list</span>
+<a id="__codelineno-3-47" name="__codelineno-3-47" href="#__codelineno-3-47"></a><span class="w">        </span><span class="c1">// Remove the vertex at index `index` from the vertex list</span>
 <a id="__codelineno-3-48" name="__codelineno-3-48" href="#__codelineno-3-48"></a><span class="w">        </span><span class="n">vertices</span><span class="p">.</span><span class="n">RemoveAt</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
-<a id="__codelineno-3-49" name="__codelineno-3-49" href="#__codelineno-3-49"></a><span class="w">        </span><span class="c1">// Remove the row at index from the adjacency matrix</span>
+<a id="__codelineno-3-49" name="__codelineno-3-49" href="#__codelineno-3-49"></a><span class="w">        </span><span class="c1">// Remove row `index` from the adjacency matrix</span>
 <a id="__codelineno-3-50" name="__codelineno-3-50" href="#__codelineno-3-50"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">.</span><span class="n">RemoveAt</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
-<a id="__codelineno-3-51" name="__codelineno-3-51" href="#__codelineno-3-51"></a><span class="w">        </span><span class="c1">// Remove the column at index from the adjacency matrix</span>
+<a id="__codelineno-3-51" name="__codelineno-3-51" href="#__codelineno-3-51"></a><span class="w">        </span><span class="c1">// Remove column `index` from the adjacency matrix</span>
 <a id="__codelineno-3-52" name="__codelineno-3-52" href="#__codelineno-3-52"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">adjMat</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-53" name="__codelineno-3-53" href="#__codelineno-3-53"></a><span class="w">            </span><span class="n">row</span><span class="p">.</span><span class="n">RemoveAt</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
 <a id="__codelineno-3-54" name="__codelineno-3-54" href="#__codelineno-3-54"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-3-55" name="__codelineno-3-55" href="#__codelineno-3-55"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-56" name="__codelineno-3-56" href="#__codelineno-3-56"></a>
-<a id="__codelineno-3-57" name="__codelineno-3-57" href="#__codelineno-3-57"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
-<a id="__codelineno-3-58" name="__codelineno-3-58" href="#__codelineno-3-58"></a><span class="w">    </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-3-57" name="__codelineno-3-57" href="#__codelineno-3-57"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
+<a id="__codelineno-3-58" name="__codelineno-3-58" href="#__codelineno-3-58"></a><span class="w">    </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-3-59" name="__codelineno-3-59" href="#__codelineno-3-59"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">AddEdge</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-60" name="__codelineno-3-60" href="#__codelineno-3-60"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-3-60" name="__codelineno-3-60" href="#__codelineno-3-60"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-3-61" name="__codelineno-3-61" href="#__codelineno-3-61"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">Size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">Size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-3-62" name="__codelineno-3-62" href="#__codelineno-3-62"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">IndexOutOfRangeException</span><span class="p">();</span>
-<a id="__codelineno-3-63" name="__codelineno-3-63" href="#__codelineno-3-63"></a><span class="w">        </span><span class="c1">// In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)</span>
+<a id="__codelineno-3-63" name="__codelineno-3-63" href="#__codelineno-3-63"></a><span class="w">        </span><span class="c1">// For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal</span>
 <a id="__codelineno-3-64" name="__codelineno-3-64" href="#__codelineno-3-64"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-3-65" name="__codelineno-3-65" href="#__codelineno-3-65"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-3-66" name="__codelineno-3-66" href="#__codelineno-3-66"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-67" name="__codelineno-3-67" href="#__codelineno-3-67"></a>
-<a id="__codelineno-3-68" name="__codelineno-3-68" href="#__codelineno-3-68"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
-<a id="__codelineno-3-69" name="__codelineno-3-69" href="#__codelineno-3-69"></a><span class="w">    </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-3-68" name="__codelineno-3-68" href="#__codelineno-3-68"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
+<a id="__codelineno-3-69" name="__codelineno-3-69" href="#__codelineno-3-69"></a><span class="w">    </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-3-70" name="__codelineno-3-70" href="#__codelineno-3-70"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">RemoveEdge</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-71" name="__codelineno-3-71" href="#__codelineno-3-71"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-3-71" name="__codelineno-3-71" href="#__codelineno-3-71"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-3-72" name="__codelineno-3-72" href="#__codelineno-3-72"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">Size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">Size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-3-73" name="__codelineno-3-73" href="#__codelineno-3-73"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">IndexOutOfRangeException</span><span class="p">();</span>
 <a id="__codelineno-3-74" name="__codelineno-3-74" href="#__codelineno-3-74"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-3-75" name="__codelineno-3-75" href="#__codelineno-3-75"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-3-76" name="__codelineno-3-76" href="#__codelineno-3-76"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-77" name="__codelineno-3-77" href="#__codelineno-3-77"></a>
-<a id="__codelineno-3-78" name="__codelineno-3-78" href="#__codelineno-3-78"></a><span class="w">    </span><span class="cm">/* Print adjacency matrix */</span>
+<a id="__codelineno-3-78" name="__codelineno-3-78" href="#__codelineno-3-78"></a><span class="w">    </span><span class="cm">/* Print the adjacency matrix */</span>
 <a id="__codelineno-3-79" name="__codelineno-3-79" href="#__codelineno-3-79"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">Print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-80" name="__codelineno-3-80" href="#__codelineno-3-80"></a><span class="w">        </span><span class="n">Console</span><span class="p">.</span><span class="n">Write</span><span class="p">(</span><span class="s">"Vertex list = "</span><span class="p">);</span>
 <a id="__codelineno-3-81" name="__codelineno-3-81" href="#__codelineno-3-81"></a><span class="w">        </span><span class="n">PrintUtil</span><span class="p">.</span><span class="n">PrintList</span><span class="p">(</span><span class="n">vertices</span><span class="p">);</span>
@@ -5048,29 +5120,29 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_matrix.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Undirected graph class based on adjacency matrix */</span>
+<div class="highlight"><span class="filename">graph_adjacency_matrix.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Undirected graph type implemented with an adjacency matrix */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">graphAdjMat</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Vertex list, where the element represents the "vertex value" and the index represents the "vertex index"</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Vertex list: elements are vertex values, and indices are vertex indices</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">vertices</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span>
-<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="c1">// Adjacency matrix, where the row and column indices correspond to the "vertex index"</span>
+<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="c1">// Adjacency matrix: row and column indices correspond to vertex indices</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="nx">adjMat</span><span class="w"> </span><span class="p">[][]</span><span class="kt">int</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="p">}</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="cm">/* Constructor */</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="kd">func</span><span class="w"> </span><span class="nx">newGraphAdjMat</span><span class="p">(</span><span class="nx">vertices</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">edges</span><span class="w"> </span><span class="p">[][]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">graphAdjMat</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">    </span><span class="c1">// Add vertex</span>
+<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">    </span><span class="c1">// Allocate the adjacency matrix for all vertices</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">vertices</span><span class="p">)</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="nx">adjMat</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">adjMat</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="c1">// Initialize graph</span>
+<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="c1">// Initialize the graph</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">    </span><span class="nx">g</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">graphAdjMat</span><span class="p">{</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">        </span><span class="nx">vertices</span><span class="p">:</span><span class="w"> </span><span class="nx">vertices</span><span class="p">,</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">        </span><span class="nx">adjMat</span><span class="p">:</span><span class="w">   </span><span class="nx">adjMat</span><span class="p">,</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">    </span><span class="c1">// Add edge</span>
-<a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w">    </span><span class="c1">// Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices</span>
+<a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">    </span><span class="c1">// Add all edges</span>
+<a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w">    </span><span class="c1">// Note that the entries in edges are vertex indices, corresponding to element indices in vertices</span>
 <a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">edges</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-25" name="__codelineno-4-25" href="#__codelineno-4-25"></a><span class="w">        </span><span class="nx">g</span><span class="p">.</span><span class="nx">addEdge</span><span class="p">(</span><span class="nx">edges</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mi">0</span><span class="p">],</span><span class="w"> </span><span class="nx">edges</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="mi">1</span><span class="p">])</span>
 <a id="__codelineno-4-26" name="__codelineno-4-26" href="#__codelineno-4-26"></a><span class="w">    </span><span class="p">}</span>
@@ -5082,10 +5154,10 @@
 <a id="__codelineno-4-32" name="__codelineno-4-32" href="#__codelineno-4-32"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">g</span><span class="p">.</span><span class="nx">vertices</span><span class="p">)</span>
 <a id="__codelineno-4-33" name="__codelineno-4-33" href="#__codelineno-4-33"></a><span class="p">}</span>
 <a id="__codelineno-4-34" name="__codelineno-4-34" href="#__codelineno-4-34"></a>
-<a id="__codelineno-4-35" name="__codelineno-4-35" href="#__codelineno-4-35"></a><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-4-35" name="__codelineno-4-35" href="#__codelineno-4-35"></a><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-4-36" name="__codelineno-4-36" href="#__codelineno-4-36"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">g</span><span class="w"> </span><span class="o">*</span><span class="nx">graphAdjMat</span><span class="p">)</span><span class="w"> </span><span class="nx">addVertex</span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-37" name="__codelineno-4-37" href="#__codelineno-4-37"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span>
-<a id="__codelineno-4-38" name="__codelineno-4-38" href="#__codelineno-4-38"></a><span class="w">    </span><span class="c1">// Add the value of the new vertex to the vertex list</span>
+<a id="__codelineno-4-38" name="__codelineno-4-38" href="#__codelineno-4-38"></a><span class="w">    </span><span class="c1">// Append the new vertex value to the vertex list</span>
 <a id="__codelineno-4-39" name="__codelineno-4-39" href="#__codelineno-4-39"></a><span class="w">    </span><span class="nx">g</span><span class="p">.</span><span class="nx">vertices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">g</span><span class="p">.</span><span class="nx">vertices</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span>
 <a id="__codelineno-4-40" name="__codelineno-4-40" href="#__codelineno-4-40"></a><span class="w">    </span><span class="c1">// Add a row to the adjacency matrix</span>
 <a id="__codelineno-4-41" name="__codelineno-4-41" href="#__codelineno-4-41"></a><span class="w">    </span><span class="nx">newRow</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
@@ -5096,45 +5168,45 @@
 <a id="__codelineno-4-46" name="__codelineno-4-46" href="#__codelineno-4-46"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-47" name="__codelineno-4-47" href="#__codelineno-4-47"></a><span class="p">}</span>
 <a id="__codelineno-4-48" name="__codelineno-4-48" href="#__codelineno-4-48"></a>
-<a id="__codelineno-4-49" name="__codelineno-4-49" href="#__codelineno-4-49"></a><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-4-49" name="__codelineno-4-49" href="#__codelineno-4-49"></a><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-4-50" name="__codelineno-4-50" href="#__codelineno-4-50"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">g</span><span class="w"> </span><span class="o">*</span><span class="nx">graphAdjMat</span><span class="p">)</span><span class="w"> </span><span class="nx">removeVertex</span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-51" name="__codelineno-4-51" href="#__codelineno-4-51"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-52" name="__codelineno-4-52" href="#__codelineno-4-52"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-4-53" name="__codelineno-4-53" href="#__codelineno-4-53"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-54" name="__codelineno-4-54" href="#__codelineno-4-54"></a><span class="w">    </span><span class="c1">// Remove the vertex at index from the vertex list</span>
+<a id="__codelineno-4-54" name="__codelineno-4-54" href="#__codelineno-4-54"></a><span class="w">    </span><span class="c1">// Remove the vertex at index `index` from the vertex list</span>
 <a id="__codelineno-4-55" name="__codelineno-4-55" href="#__codelineno-4-55"></a><span class="w">    </span><span class="nx">g</span><span class="p">.</span><span class="nx">vertices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">g</span><span class="p">.</span><span class="nx">vertices</span><span class="p">[:</span><span class="nx">index</span><span class="p">],</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">vertices</span><span class="p">[</span><span class="nx">index</span><span class="o">+</span><span class="mi">1</span><span class="p">:]...)</span>
-<a id="__codelineno-4-56" name="__codelineno-4-56" href="#__codelineno-4-56"></a><span class="w">    </span><span class="c1">// Remove the row at index from the adjacency matrix</span>
+<a id="__codelineno-4-56" name="__codelineno-4-56" href="#__codelineno-4-56"></a><span class="w">    </span><span class="c1">// Remove row `index` from the adjacency matrix</span>
 <a id="__codelineno-4-57" name="__codelineno-4-57" href="#__codelineno-4-57"></a><span class="w">    </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjMat</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">g</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[:</span><span class="nx">index</span><span class="p">],</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">index</span><span class="o">+</span><span class="mi">1</span><span class="p">:]...)</span>
-<a id="__codelineno-4-58" name="__codelineno-4-58" href="#__codelineno-4-58"></a><span class="w">    </span><span class="c1">// Remove the column at index from the adjacency matrix</span>
+<a id="__codelineno-4-58" name="__codelineno-4-58" href="#__codelineno-4-58"></a><span class="w">    </span><span class="c1">// Remove column `index` from the adjacency matrix</span>
 <a id="__codelineno-4-59" name="__codelineno-4-59" href="#__codelineno-4-59"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjMat</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-60" name="__codelineno-4-60" href="#__codelineno-4-60"></a><span class="w">        </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">g</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">i</span><span class="p">][:</span><span class="nx">index</span><span class="p">],</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">index</span><span class="o">+</span><span class="mi">1</span><span class="p">:]...)</span>
 <a id="__codelineno-4-61" name="__codelineno-4-61" href="#__codelineno-4-61"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-62" name="__codelineno-4-62" href="#__codelineno-4-62"></a><span class="p">}</span>
 <a id="__codelineno-4-63" name="__codelineno-4-63" href="#__codelineno-4-63"></a>
-<a id="__codelineno-4-64" name="__codelineno-4-64" href="#__codelineno-4-64"></a><span class="cm">/* Add edge */</span>
-<a id="__codelineno-4-65" name="__codelineno-4-65" href="#__codelineno-4-65"></a><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-4-64" name="__codelineno-4-64" href="#__codelineno-4-64"></a><span class="cm">/* Add an edge */</span>
+<a id="__codelineno-4-65" name="__codelineno-4-65" href="#__codelineno-4-65"></a><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-4-66" name="__codelineno-4-66" href="#__codelineno-4-66"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">g</span><span class="w"> </span><span class="o">*</span><span class="nx">graphAdjMat</span><span class="p">)</span><span class="w"> </span><span class="nx">addEdge</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-67" name="__codelineno-4-67" href="#__codelineno-4-67"></a><span class="w">    </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-4-67" name="__codelineno-4-67" href="#__codelineno-4-67"></a><span class="w">    </span><span class="c1">// Check for out-of-bounds or equal indices (the constructed error is not returned)</span>
 <a id="__codelineno-4-68" name="__codelineno-4-68" href="#__codelineno-4-68"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-69" name="__codelineno-4-69" href="#__codelineno-4-69"></a><span class="w">        </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Errorf</span><span class="p">(</span><span class="s">"%s"</span><span class="p">,</span><span class="w"> </span><span class="s">"Index Out Of Bounds Exception"</span><span class="p">)</span>
+<a id="__codelineno-4-69" name="__codelineno-4-69" href="#__codelineno-4-69"></a><span class="w">        </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Errorf</span><span class="p">(</span><span class="s">"%s"</span><span class="p">,</span><span class="w"> </span><span class="s">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-4-70" name="__codelineno-4-70" href="#__codelineno-4-70"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-71" name="__codelineno-4-71" href="#__codelineno-4-71"></a><span class="w">    </span><span class="c1">// In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)</span>
+<a id="__codelineno-4-71" name="__codelineno-4-71" href="#__codelineno-4-71"></a><span class="w">    </span><span class="c1">// For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal</span>
 <a id="__codelineno-4-72" name="__codelineno-4-72" href="#__codelineno-4-72"></a><span class="w">    </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-4-73" name="__codelineno-4-73" href="#__codelineno-4-73"></a><span class="w">    </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">j</span><span class="p">][</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-4-74" name="__codelineno-4-74" href="#__codelineno-4-74"></a><span class="p">}</span>
 <a id="__codelineno-4-75" name="__codelineno-4-75" href="#__codelineno-4-75"></a>
-<a id="__codelineno-4-76" name="__codelineno-4-76" href="#__codelineno-4-76"></a><span class="cm">/* Remove edge */</span>
-<a id="__codelineno-4-77" name="__codelineno-4-77" href="#__codelineno-4-77"></a><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-4-76" name="__codelineno-4-76" href="#__codelineno-4-76"></a><span class="cm">/* Remove an edge */</span>
+<a id="__codelineno-4-77" name="__codelineno-4-77" href="#__codelineno-4-77"></a><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-4-78" name="__codelineno-4-78" href="#__codelineno-4-78"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">g</span><span class="w"> </span><span class="o">*</span><span class="nx">graphAdjMat</span><span class="p">)</span><span class="w"> </span><span class="nx">removeEdge</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-79" name="__codelineno-4-79" href="#__codelineno-4-79"></a><span class="w">    </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-4-79" name="__codelineno-4-79" href="#__codelineno-4-79"></a><span class="w">    </span><span class="c1">// Check for out-of-bounds or equal indices (the constructed error is not returned)</span>
 <a id="__codelineno-4-80" name="__codelineno-4-80" href="#__codelineno-4-80"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-81" name="__codelineno-4-81" href="#__codelineno-4-81"></a><span class="w">        </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Errorf</span><span class="p">(</span><span class="s">"%s"</span><span class="p">,</span><span class="w"> </span><span class="s">"Index Out Of Bounds Exception"</span><span class="p">)</span>
+<a id="__codelineno-4-81" name="__codelineno-4-81" href="#__codelineno-4-81"></a><span class="w">        </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Errorf</span><span class="p">(</span><span class="s">"%s"</span><span class="p">,</span><span class="w"> </span><span class="s">"Index out of bounds"</span><span class="p">)</span>
 <a id="__codelineno-4-82" name="__codelineno-4-82" href="#__codelineno-4-82"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-83" name="__codelineno-4-83" href="#__codelineno-4-83"></a><span class="w">    </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-4-84" name="__codelineno-4-84" href="#__codelineno-4-84"></a><span class="w">    </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">j</span><span class="p">][</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-4-85" name="__codelineno-4-85" href="#__codelineno-4-85"></a><span class="p">}</span>
 <a id="__codelineno-4-86" name="__codelineno-4-86" href="#__codelineno-4-86"></a>
-<a id="__codelineno-4-87" name="__codelineno-4-87" href="#__codelineno-4-87"></a><span class="cm">/* Print adjacency matrix */</span>
+<a id="__codelineno-4-87" name="__codelineno-4-87" href="#__codelineno-4-87"></a><span class="cm">/* Print the adjacency matrix */</span>
 <a id="__codelineno-4-88" name="__codelineno-4-88" href="#__codelineno-4-88"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">g</span><span class="w"> </span><span class="o">*</span><span class="nx">graphAdjMat</span><span class="p">)</span><span class="w"> </span><span class="nb">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-89" name="__codelineno-4-89" href="#__codelineno-4-89"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Printf</span><span class="p">(</span><span class="s">"\tVertex list = %v\n"</span><span class="p">,</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">vertices</span><span class="p">)</span>
 <a id="__codelineno-4-90" name="__codelineno-4-90" href="#__codelineno-4-90"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Printf</span><span class="p">(</span><span class="s">"\tAdjacency matrix = \n"</span><span class="p">)</span>
@@ -5145,21 +5217,21 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_matrix.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Undirected graph class based on adjacency matrix */</span>
+<div class="highlight"><span class="filename">graph_adjacency_matrix.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency matrix */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">GraphAdjMat</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">vertices</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="c1">// Vertex list, where the element represents the "vertex value" and the index represents the "vertex index"</span>
-<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">adjMat</span><span class="p">:</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]]</span><span class="w"> </span><span class="c1">// Adjacency matrix, where the row and column indices correspond to the "vertex index"</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">vertices</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="c1">// Vertex list: elements are vertex values, and indices are vertex indices</span>
+<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">adjMat</span><span class="p">:</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]]</span><span class="w"> </span><span class="c1">// Adjacency matrix: row and column indices correspond to vertex indices</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="kd">init</span><span class="p">(</span><span class="n">vertices</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">edges</span><span class="p">:</span><span class="w"> </span><span class="p">[[</span><span class="nb">Int</span><span class="p">]])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">        </span><span class="kc">self</span><span class="p">.</span><span class="n">vertices</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="n">adjMat</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
-<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="c1">// Add vertex</span>
+<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="c1">// Add all vertices</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">vertices</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">            </span><span class="n">addVertex</span><span class="p">(</span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="c1">// Add edge</span>
-<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="c1">// Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices</span>
+<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="c1">// Add all edges</span>
+<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="c1">// Note that the entries in edges are vertex indices, corresponding to element indices in vertices</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">e</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">edges</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">            </span><span class="n">addEdge</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">e</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="n">e</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">        </span><span class="p">}</span>
@@ -5170,10 +5242,10 @@
 <a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="w">        </span><span class="n">vertices</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a>
-<a id="__codelineno-5-26" name="__codelineno-5-26" href="#__codelineno-5-26"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-5-26" name="__codelineno-5-26" href="#__codelineno-5-26"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-5-27" name="__codelineno-5-27" href="#__codelineno-5-27"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">addVertex</span><span class="p">(</span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span>
-<a id="__codelineno-5-29" name="__codelineno-5-29" href="#__codelineno-5-29"></a><span class="w">        </span><span class="c1">// Add the value of the new vertex to the vertex list</span>
+<a id="__codelineno-5-29" name="__codelineno-5-29" href="#__codelineno-5-29"></a><span class="w">        </span><span class="c1">// Append the new vertex value to the vertex list</span>
 <a id="__codelineno-5-30" name="__codelineno-5-30" href="#__codelineno-5-30"></a><span class="w">        </span><span class="n">vertices</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-5-31" name="__codelineno-5-31" href="#__codelineno-5-31"></a><span class="w">        </span><span class="c1">// Add a row to the adjacency matrix</span>
 <a id="__codelineno-5-32" name="__codelineno-5-32" href="#__codelineno-5-32"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">newRow</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">n</span><span class="p">)</span>
@@ -5184,45 +5256,45 @@
 <a id="__codelineno-5-37" name="__codelineno-5-37" href="#__codelineno-5-37"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-38" name="__codelineno-5-38" href="#__codelineno-5-38"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-39" name="__codelineno-5-39" href="#__codelineno-5-39"></a>
-<a id="__codelineno-5-40" name="__codelineno-5-40" href="#__codelineno-5-40"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-5-40" name="__codelineno-5-40" href="#__codelineno-5-40"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-5-41" name="__codelineno-5-41" href="#__codelineno-5-41"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">removeVertex</span><span class="p">(</span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-42" name="__codelineno-5-42" href="#__codelineno-5-42"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-43" name="__codelineno-5-43" href="#__codelineno-5-43"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Out of bounds"</span><span class="p">)</span>
+<a id="__codelineno-5-43" name="__codelineno-5-43" href="#__codelineno-5-43"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Invalid index"</span><span class="p">)</span>
 <a id="__codelineno-5-44" name="__codelineno-5-44" href="#__codelineno-5-44"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-5-45" name="__codelineno-5-45" href="#__codelineno-5-45"></a><span class="w">        </span><span class="c1">// Remove the vertex at index from the vertex list</span>
+<a id="__codelineno-5-45" name="__codelineno-5-45" href="#__codelineno-5-45"></a><span class="w">        </span><span class="c1">// Remove the vertex at index `index` from the vertex list</span>
 <a id="__codelineno-5-46" name="__codelineno-5-46" href="#__codelineno-5-46"></a><span class="w">        </span><span class="n">vertices</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">at</span><span class="p">:</span><span class="w"> </span><span class="n">index</span><span class="p">)</span>
-<a id="__codelineno-5-47" name="__codelineno-5-47" href="#__codelineno-5-47"></a><span class="w">        </span><span class="c1">// Remove the row at index from the adjacency matrix</span>
+<a id="__codelineno-5-47" name="__codelineno-5-47" href="#__codelineno-5-47"></a><span class="w">        </span><span class="c1">// Remove row `index` from the adjacency matrix</span>
 <a id="__codelineno-5-48" name="__codelineno-5-48" href="#__codelineno-5-48"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">at</span><span class="p">:</span><span class="w"> </span><span class="n">index</span><span class="p">)</span>
-<a id="__codelineno-5-49" name="__codelineno-5-49" href="#__codelineno-5-49"></a><span class="w">        </span><span class="c1">// Remove the column at index from the adjacency matrix</span>
+<a id="__codelineno-5-49" name="__codelineno-5-49" href="#__codelineno-5-49"></a><span class="w">        </span><span class="c1">// Remove column `index` from the adjacency matrix</span>
 <a id="__codelineno-5-50" name="__codelineno-5-50" href="#__codelineno-5-50"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">adjMat</span><span class="p">.</span><span class="bp">indices</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-51" name="__codelineno-5-51" href="#__codelineno-5-51"></a><span class="w">            </span><span class="n">adjMat</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">remove</span><span class="p">(</span><span class="n">at</span><span class="p">:</span><span class="w"> </span><span class="n">index</span><span class="p">)</span>
 <a id="__codelineno-5-52" name="__codelineno-5-52" href="#__codelineno-5-52"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-53" name="__codelineno-5-53" href="#__codelineno-5-53"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-54" name="__codelineno-5-54" href="#__codelineno-5-54"></a>
-<a id="__codelineno-5-55" name="__codelineno-5-55" href="#__codelineno-5-55"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
-<a id="__codelineno-5-56" name="__codelineno-5-56" href="#__codelineno-5-56"></a><span class="w">    </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-5-55" name="__codelineno-5-55" href="#__codelineno-5-55"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
+<a id="__codelineno-5-56" name="__codelineno-5-56" href="#__codelineno-5-56"></a><span class="w">    </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-5-57" name="__codelineno-5-57" href="#__codelineno-5-57"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">addEdge</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-58" name="__codelineno-5-58" href="#__codelineno-5-58"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-5-58" name="__codelineno-5-58" href="#__codelineno-5-58"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-5-59" name="__codelineno-5-59" href="#__codelineno-5-59"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-60" name="__codelineno-5-60" href="#__codelineno-5-60"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Out of bounds"</span><span class="p">)</span>
+<a id="__codelineno-5-60" name="__codelineno-5-60" href="#__codelineno-5-60"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Invalid index"</span><span class="p">)</span>
 <a id="__codelineno-5-61" name="__codelineno-5-61" href="#__codelineno-5-61"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-5-62" name="__codelineno-5-62" href="#__codelineno-5-62"></a><span class="w">        </span><span class="c1">// In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)</span>
+<a id="__codelineno-5-62" name="__codelineno-5-62" href="#__codelineno-5-62"></a><span class="w">        </span><span class="c1">// For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal</span>
 <a id="__codelineno-5-63" name="__codelineno-5-63" href="#__codelineno-5-63"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-64" name="__codelineno-5-64" href="#__codelineno-5-64"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-65" name="__codelineno-5-65" href="#__codelineno-5-65"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-66" name="__codelineno-5-66" href="#__codelineno-5-66"></a>
-<a id="__codelineno-5-67" name="__codelineno-5-67" href="#__codelineno-5-67"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
-<a id="__codelineno-5-68" name="__codelineno-5-68" href="#__codelineno-5-68"></a><span class="w">    </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-5-67" name="__codelineno-5-67" href="#__codelineno-5-67"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
+<a id="__codelineno-5-68" name="__codelineno-5-68" href="#__codelineno-5-68"></a><span class="w">    </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-5-69" name="__codelineno-5-69" href="#__codelineno-5-69"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">removeEdge</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-70" name="__codelineno-5-70" href="#__codelineno-5-70"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-5-70" name="__codelineno-5-70" href="#__codelineno-5-70"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-5-71" name="__codelineno-5-71" href="#__codelineno-5-71"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-72" name="__codelineno-5-72" href="#__codelineno-5-72"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Out of bounds"</span><span class="p">)</span>
+<a id="__codelineno-5-72" name="__codelineno-5-72" href="#__codelineno-5-72"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Invalid index"</span><span class="p">)</span>
 <a id="__codelineno-5-73" name="__codelineno-5-73" href="#__codelineno-5-73"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-74" name="__codelineno-5-74" href="#__codelineno-5-74"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-5-75" name="__codelineno-5-75" href="#__codelineno-5-75"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-5-76" name="__codelineno-5-76" href="#__codelineno-5-76"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-77" name="__codelineno-5-77" href="#__codelineno-5-77"></a>
-<a id="__codelineno-5-78" name="__codelineno-5-78" href="#__codelineno-5-78"></a><span class="w">    </span><span class="cm">/* Print adjacency matrix */</span>
+<a id="__codelineno-5-78" name="__codelineno-5-78" href="#__codelineno-5-78"></a><span class="w">    </span><span class="cm">/* Print the adjacency matrix */</span>
 <a id="__codelineno-5-79" name="__codelineno-5-79" href="#__codelineno-5-79"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-80" name="__codelineno-5-80" href="#__codelineno-5-80"></a><span class="w">        </span><span class="n">Swift</span><span class="p">.</span><span class="bp">print</span><span class="p">(</span><span class="s">"Vertex list = "</span><span class="p">,</span><span class="w"> </span><span class="n">terminator</span><span class="p">:</span><span class="w"> </span><span class="s">""</span><span class="p">)</span>
 <a id="__codelineno-5-81" name="__codelineno-5-81" href="#__codelineno-5-81"></a><span class="w">        </span><span class="n">Swift</span><span class="p">.</span><span class="bp">print</span><span class="p">(</span><span class="n">vertices</span><span class="p">)</span>
@@ -5233,21 +5305,21 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_matrix.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Undirected graph class based on adjacency matrix */</span>
+<div class="highlight"><span class="filename">graph_adjacency_matrix.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency matrix */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">GraphAdjMat</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="nx">vertices</span><span class="p">;</span><span class="w"> </span><span class="c1">// Vertex list, where the element represents the "vertex value" and the index represents the "vertex index"</span>
-<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="nx">adjMat</span><span class="p">;</span><span class="w"> </span><span class="c1">// Adjacency matrix, where the row and column indices correspond to the "vertex index"</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="nx">vertices</span><span class="p">;</span><span class="w"> </span><span class="c1">// Vertex list: elements are vertex values, and indices are vertex indices</span>
+<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="nx">adjMat</span><span class="p">;</span><span class="w"> </span><span class="c1">// Adjacency matrix: row and column indices correspond to vertex indices</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">vertices</span><span class="p">,</span><span class="w"> </span><span class="nx">edges</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">vertices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="c1">// Add vertex</span>
+<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="c1">// Add a vertex</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">vertices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">addVertex</span><span class="p">(</span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="c1">// Add edge</span>
-<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">        </span><span class="c1">// Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices</span>
+<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="c1">// Add an edge</span>
+<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">        </span><span class="c1">// Note that the entries in edges are vertex indices, corresponding to element indices in vertices</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">e</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">edges</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">addEdge</span><span class="p">(</span><span class="nx">e</span><span class="p">[</span><span class="mf">0</span><span class="p">],</span><span class="w"> </span><span class="nx">e</span><span class="p">[</span><span class="mf">1</span><span class="p">]);</span>
 <a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">        </span><span class="p">}</span>
@@ -5258,10 +5330,10 @@
 <a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">vertices</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a>
-<a id="__codelineno-6-26" name="__codelineno-6-26" href="#__codelineno-6-26"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-6-26" name="__codelineno-6-26" href="#__codelineno-6-26"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-6-27" name="__codelineno-6-27" href="#__codelineno-6-27"></a><span class="w">    </span><span class="nx">addVertex</span><span class="p">(</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-28" name="__codelineno-6-28" href="#__codelineno-6-28"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">();</span>
-<a id="__codelineno-6-29" name="__codelineno-6-29" href="#__codelineno-6-29"></a><span class="w">        </span><span class="c1">// Add the value of the new vertex to the vertex list</span>
+<a id="__codelineno-6-29" name="__codelineno-6-29" href="#__codelineno-6-29"></a><span class="w">        </span><span class="c1">// Append the new vertex value to the vertex list</span>
 <a id="__codelineno-6-30" name="__codelineno-6-30" href="#__codelineno-6-30"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">vertices</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-6-31" name="__codelineno-6-31" href="#__codelineno-6-31"></a><span class="w">        </span><span class="c1">// Add a row to the adjacency matrix</span>
 <a id="__codelineno-6-32" name="__codelineno-6-32" href="#__codelineno-6-32"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">newRow</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
@@ -5275,46 +5347,46 @@
 <a id="__codelineno-6-40" name="__codelineno-6-40" href="#__codelineno-6-40"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-6-41" name="__codelineno-6-41" href="#__codelineno-6-41"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-42" name="__codelineno-6-42" href="#__codelineno-6-42"></a>
-<a id="__codelineno-6-43" name="__codelineno-6-43" href="#__codelineno-6-43"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-6-43" name="__codelineno-6-43" href="#__codelineno-6-43"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-6-44" name="__codelineno-6-44" href="#__codelineno-6-44"></a><span class="w">    </span><span class="nx">removeVertex</span><span class="p">(</span><span class="nx">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-45" name="__codelineno-6-45" href="#__codelineno-6-45"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-46" name="__codelineno-6-46" href="#__codelineno-6-46"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">RangeError</span><span class="p">(</span><span class="s1">'Index Out Of Bounds Exception'</span><span class="p">);</span>
+<a id="__codelineno-6-46" name="__codelineno-6-46" href="#__codelineno-6-46"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">RangeError</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
 <a id="__codelineno-6-47" name="__codelineno-6-47" href="#__codelineno-6-47"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-6-48" name="__codelineno-6-48" href="#__codelineno-6-48"></a><span class="w">        </span><span class="c1">// Remove the vertex at index from the vertex list</span>
+<a id="__codelineno-6-48" name="__codelineno-6-48" href="#__codelineno-6-48"></a><span class="w">        </span><span class="c1">// Remove the vertex at index `index` from the vertex list</span>
 <a id="__codelineno-6-49" name="__codelineno-6-49" href="#__codelineno-6-49"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">vertices</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="nx">index</span><span class="p">,</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-6-50" name="__codelineno-6-50" href="#__codelineno-6-50"></a>
-<a id="__codelineno-6-51" name="__codelineno-6-51" href="#__codelineno-6-51"></a><span class="w">        </span><span class="c1">// Remove the row at index from the adjacency matrix</span>
+<a id="__codelineno-6-51" name="__codelineno-6-51" href="#__codelineno-6-51"></a><span class="w">        </span><span class="c1">// Remove row `index` from the adjacency matrix</span>
 <a id="__codelineno-6-52" name="__codelineno-6-52" href="#__codelineno-6-52"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="nx">index</span><span class="p">,</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-6-53" name="__codelineno-6-53" href="#__codelineno-6-53"></a><span class="w">        </span><span class="c1">// Remove the column at index from the adjacency matrix</span>
+<a id="__codelineno-6-53" name="__codelineno-6-53" href="#__codelineno-6-53"></a><span class="w">        </span><span class="c1">// Remove column `index` from the adjacency matrix</span>
 <a id="__codelineno-6-54" name="__codelineno-6-54" href="#__codelineno-6-54"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-55" name="__codelineno-6-55" href="#__codelineno-6-55"></a><span class="w">            </span><span class="nx">row</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="nx">index</span><span class="p">,</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-6-56" name="__codelineno-6-56" href="#__codelineno-6-56"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-6-57" name="__codelineno-6-57" href="#__codelineno-6-57"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-58" name="__codelineno-6-58" href="#__codelineno-6-58"></a>
-<a id="__codelineno-6-59" name="__codelineno-6-59" href="#__codelineno-6-59"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
-<a id="__codelineno-6-60" name="__codelineno-6-60" href="#__codelineno-6-60"></a><span class="w">    </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-6-59" name="__codelineno-6-59" href="#__codelineno-6-59"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
+<a id="__codelineno-6-60" name="__codelineno-6-60" href="#__codelineno-6-60"></a><span class="w">    </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-6-61" name="__codelineno-6-61" href="#__codelineno-6-61"></a><span class="w">    </span><span class="nx">addEdge</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-62" name="__codelineno-6-62" href="#__codelineno-6-62"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-6-62" name="__codelineno-6-62" href="#__codelineno-6-62"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-6-63" name="__codelineno-6-63" href="#__codelineno-6-63"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-64" name="__codelineno-6-64" href="#__codelineno-6-64"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">RangeError</span><span class="p">(</span><span class="s1">'Index Out Of Bounds Exception'</span><span class="p">);</span>
+<a id="__codelineno-6-64" name="__codelineno-6-64" href="#__codelineno-6-64"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">RangeError</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
 <a id="__codelineno-6-65" name="__codelineno-6-65" href="#__codelineno-6-65"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-6-66" name="__codelineno-6-66" href="#__codelineno-6-66"></a><span class="w">        </span><span class="c1">// In undirected graph, adjacency matrix is symmetric about main diagonal, i.e., satisfies (i, j) === (j, i)</span>
+<a id="__codelineno-6-66" name="__codelineno-6-66" href="#__codelineno-6-66"></a><span class="w">        </span><span class="c1">// In an undirected graph, the adjacency matrix is symmetric about its main diagonal: (i, j) === (j, i)</span>
 <a id="__codelineno-6-67" name="__codelineno-6-67" href="#__codelineno-6-67"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-6-68" name="__codelineno-6-68" href="#__codelineno-6-68"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">j</span><span class="p">][</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-6-69" name="__codelineno-6-69" href="#__codelineno-6-69"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-70" name="__codelineno-6-70" href="#__codelineno-6-70"></a>
-<a id="__codelineno-6-71" name="__codelineno-6-71" href="#__codelineno-6-71"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
-<a id="__codelineno-6-72" name="__codelineno-6-72" href="#__codelineno-6-72"></a><span class="w">    </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-6-71" name="__codelineno-6-71" href="#__codelineno-6-71"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
+<a id="__codelineno-6-72" name="__codelineno-6-72" href="#__codelineno-6-72"></a><span class="w">    </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-6-73" name="__codelineno-6-73" href="#__codelineno-6-73"></a><span class="w">    </span><span class="nx">removeEdge</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-74" name="__codelineno-6-74" href="#__codelineno-6-74"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-6-74" name="__codelineno-6-74" href="#__codelineno-6-74"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-6-75" name="__codelineno-6-75" href="#__codelineno-6-75"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-76" name="__codelineno-6-76" href="#__codelineno-6-76"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">RangeError</span><span class="p">(</span><span class="s1">'Index Out Of Bounds Exception'</span><span class="p">);</span>
+<a id="__codelineno-6-76" name="__codelineno-6-76" href="#__codelineno-6-76"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">RangeError</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
 <a id="__codelineno-6-77" name="__codelineno-6-77" href="#__codelineno-6-77"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-6-78" name="__codelineno-6-78" href="#__codelineno-6-78"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-6-79" name="__codelineno-6-79" href="#__codelineno-6-79"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">j</span><span class="p">][</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-6-80" name="__codelineno-6-80" href="#__codelineno-6-80"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-81" name="__codelineno-6-81" href="#__codelineno-6-81"></a>
-<a id="__codelineno-6-82" name="__codelineno-6-82" href="#__codelineno-6-82"></a><span class="w">    </span><span class="cm">/* Print adjacency matrix */</span>
+<a id="__codelineno-6-82" name="__codelineno-6-82" href="#__codelineno-6-82"></a><span class="w">    </span><span class="cm">/* Print the adjacency matrix */</span>
 <a id="__codelineno-6-83" name="__codelineno-6-83" href="#__codelineno-6-83"></a><span class="w">    </span><span class="nx">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-84" name="__codelineno-6-84" href="#__codelineno-6-84"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Vertex list = '</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">vertices</span><span class="p">);</span>
 <a id="__codelineno-6-85" name="__codelineno-6-85" href="#__codelineno-6-85"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Adjacency matrix ='</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">);</span>
@@ -5323,21 +5395,21 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_matrix.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Undirected graph class based on adjacency matrix */</span>
+<div class="highlight"><span class="filename">graph_adjacency_matrix.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency matrix */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">GraphAdjMat</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="nx">vertices</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[];</span><span class="w"> </span><span class="c1">// Vertex list, where the element represents the "vertex value" and the index represents the "vertex index"</span>
-<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="nx">adjMat</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[][];</span><span class="w"> </span><span class="c1">// Adjacency matrix, where the row and column indices correspond to the "vertex index"</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="nx">vertices</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[];</span><span class="w"> </span><span class="c1">// Vertex list: elements are vertex values, and indices are vertex indices</span>
+<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="nx">adjMat</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[][];</span><span class="w"> </span><span class="c1">// Adjacency matrix: row and column indices correspond to vertex indices</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">vertices</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">edges</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[][])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">vertices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="c1">// Add vertex</span>
+<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="c1">// Add a vertex</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">vertices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">addVertex</span><span class="p">(</span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="c1">// Add edge</span>
-<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">        </span><span class="c1">// Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices</span>
+<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="c1">// Add an edge</span>
+<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">        </span><span class="c1">// Note that the entries in edges are vertex indices, corresponding to element indices in vertices</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">e</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">edges</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">addEdge</span><span class="p">(</span><span class="nx">e</span><span class="p">[</span><span class="mf">0</span><span class="p">],</span><span class="w"> </span><span class="nx">e</span><span class="p">[</span><span class="mf">1</span><span class="p">]);</span>
 <a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">        </span><span class="p">}</span>
@@ -5348,10 +5420,10 @@
 <a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">vertices</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a>
-<a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="w">    </span><span class="nx">addVertex</span><span class="p">(</span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">();</span>
-<a id="__codelineno-7-29" name="__codelineno-7-29" href="#__codelineno-7-29"></a><span class="w">        </span><span class="c1">// Add the value of the new vertex to the vertex list</span>
+<a id="__codelineno-7-29" name="__codelineno-7-29" href="#__codelineno-7-29"></a><span class="w">        </span><span class="c1">// Append the new vertex value to the vertex list</span>
 <a id="__codelineno-7-30" name="__codelineno-7-30" href="#__codelineno-7-30"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">vertices</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-7-31" name="__codelineno-7-31" href="#__codelineno-7-31"></a><span class="w">        </span><span class="c1">// Add a row to the adjacency matrix</span>
 <a id="__codelineno-7-32" name="__codelineno-7-32" href="#__codelineno-7-32"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">newRow</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
@@ -5365,46 +5437,46 @@
 <a id="__codelineno-7-40" name="__codelineno-7-40" href="#__codelineno-7-40"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-7-41" name="__codelineno-7-41" href="#__codelineno-7-41"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-42" name="__codelineno-7-42" href="#__codelineno-7-42"></a>
-<a id="__codelineno-7-43" name="__codelineno-7-43" href="#__codelineno-7-43"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-7-43" name="__codelineno-7-43" href="#__codelineno-7-43"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-7-44" name="__codelineno-7-44" href="#__codelineno-7-44"></a><span class="w">    </span><span class="nx">removeVertex</span><span class="p">(</span><span class="nx">index</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-45" name="__codelineno-7-45" href="#__codelineno-7-45"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-46" name="__codelineno-7-46" href="#__codelineno-7-46"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">RangeError</span><span class="p">(</span><span class="s1">'Index Out Of Bounds Exception'</span><span class="p">);</span>
+<a id="__codelineno-7-46" name="__codelineno-7-46" href="#__codelineno-7-46"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">RangeError</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
 <a id="__codelineno-7-47" name="__codelineno-7-47" href="#__codelineno-7-47"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-7-48" name="__codelineno-7-48" href="#__codelineno-7-48"></a><span class="w">        </span><span class="c1">// Remove the vertex at index from the vertex list</span>
+<a id="__codelineno-7-48" name="__codelineno-7-48" href="#__codelineno-7-48"></a><span class="w">        </span><span class="c1">// Remove the vertex at index `index` from the vertex list</span>
 <a id="__codelineno-7-49" name="__codelineno-7-49" href="#__codelineno-7-49"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">vertices</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="nx">index</span><span class="p">,</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-7-50" name="__codelineno-7-50" href="#__codelineno-7-50"></a>
-<a id="__codelineno-7-51" name="__codelineno-7-51" href="#__codelineno-7-51"></a><span class="w">        </span><span class="c1">// Remove the row at index from the adjacency matrix</span>
+<a id="__codelineno-7-51" name="__codelineno-7-51" href="#__codelineno-7-51"></a><span class="w">        </span><span class="c1">// Remove row `index` from the adjacency matrix</span>
 <a id="__codelineno-7-52" name="__codelineno-7-52" href="#__codelineno-7-52"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="nx">index</span><span class="p">,</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-7-53" name="__codelineno-7-53" href="#__codelineno-7-53"></a><span class="w">        </span><span class="c1">// Remove the column at index from the adjacency matrix</span>
+<a id="__codelineno-7-53" name="__codelineno-7-53" href="#__codelineno-7-53"></a><span class="w">        </span><span class="c1">// Remove column `index` from the adjacency matrix</span>
 <a id="__codelineno-7-54" name="__codelineno-7-54" href="#__codelineno-7-54"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">row</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-55" name="__codelineno-7-55" href="#__codelineno-7-55"></a><span class="w">            </span><span class="nx">row</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="nx">index</span><span class="p">,</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-7-56" name="__codelineno-7-56" href="#__codelineno-7-56"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-7-57" name="__codelineno-7-57" href="#__codelineno-7-57"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-58" name="__codelineno-7-58" href="#__codelineno-7-58"></a>
-<a id="__codelineno-7-59" name="__codelineno-7-59" href="#__codelineno-7-59"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
-<a id="__codelineno-7-60" name="__codelineno-7-60" href="#__codelineno-7-60"></a><span class="w">    </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-7-59" name="__codelineno-7-59" href="#__codelineno-7-59"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
+<a id="__codelineno-7-60" name="__codelineno-7-60" href="#__codelineno-7-60"></a><span class="w">    </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-7-61" name="__codelineno-7-61" href="#__codelineno-7-61"></a><span class="w">    </span><span class="nx">addEdge</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-62" name="__codelineno-7-62" href="#__codelineno-7-62"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-7-62" name="__codelineno-7-62" href="#__codelineno-7-62"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-7-63" name="__codelineno-7-63" href="#__codelineno-7-63"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-64" name="__codelineno-7-64" href="#__codelineno-7-64"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">RangeError</span><span class="p">(</span><span class="s1">'Index Out Of Bounds Exception'</span><span class="p">);</span>
+<a id="__codelineno-7-64" name="__codelineno-7-64" href="#__codelineno-7-64"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">RangeError</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
 <a id="__codelineno-7-65" name="__codelineno-7-65" href="#__codelineno-7-65"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-7-66" name="__codelineno-7-66" href="#__codelineno-7-66"></a><span class="w">        </span><span class="c1">// In undirected graph, adjacency matrix is symmetric about main diagonal, i.e., satisfies (i, j) === (j, i)</span>
+<a id="__codelineno-7-66" name="__codelineno-7-66" href="#__codelineno-7-66"></a><span class="w">        </span><span class="c1">// In an undirected graph, the adjacency matrix is symmetric about its main diagonal: (i, j) === (j, i)</span>
 <a id="__codelineno-7-67" name="__codelineno-7-67" href="#__codelineno-7-67"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-7-68" name="__codelineno-7-68" href="#__codelineno-7-68"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">j</span><span class="p">][</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-7-69" name="__codelineno-7-69" href="#__codelineno-7-69"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-70" name="__codelineno-7-70" href="#__codelineno-7-70"></a>
-<a id="__codelineno-7-71" name="__codelineno-7-71" href="#__codelineno-7-71"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
-<a id="__codelineno-7-72" name="__codelineno-7-72" href="#__codelineno-7-72"></a><span class="w">    </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-7-71" name="__codelineno-7-71" href="#__codelineno-7-71"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
+<a id="__codelineno-7-72" name="__codelineno-7-72" href="#__codelineno-7-72"></a><span class="w">    </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-7-73" name="__codelineno-7-73" href="#__codelineno-7-73"></a><span class="w">    </span><span class="nx">removeEdge</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-74" name="__codelineno-7-74" href="#__codelineno-7-74"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-7-74" name="__codelineno-7-74" href="#__codelineno-7-74"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-7-75" name="__codelineno-7-75" href="#__codelineno-7-75"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-76" name="__codelineno-7-76" href="#__codelineno-7-76"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">RangeError</span><span class="p">(</span><span class="s1">'Index Out Of Bounds Exception'</span><span class="p">);</span>
+<a id="__codelineno-7-76" name="__codelineno-7-76" href="#__codelineno-7-76"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">RangeError</span><span class="p">(</span><span class="s1">'Index out of bounds'</span><span class="p">);</span>
 <a id="__codelineno-7-77" name="__codelineno-7-77" href="#__codelineno-7-77"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-7-78" name="__codelineno-7-78" href="#__codelineno-7-78"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">i</span><span class="p">][</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-7-79" name="__codelineno-7-79" href="#__codelineno-7-79"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">[</span><span class="nx">j</span><span class="p">][</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-7-80" name="__codelineno-7-80" href="#__codelineno-7-80"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-81" name="__codelineno-7-81" href="#__codelineno-7-81"></a>
-<a id="__codelineno-7-82" name="__codelineno-7-82" href="#__codelineno-7-82"></a><span class="w">    </span><span class="cm">/* Print adjacency matrix */</span>
+<a id="__codelineno-7-82" name="__codelineno-7-82" href="#__codelineno-7-82"></a><span class="w">    </span><span class="cm">/* Print the adjacency matrix */</span>
 <a id="__codelineno-7-83" name="__codelineno-7-83" href="#__codelineno-7-83"></a><span class="w">    </span><span class="nx">print</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-84" name="__codelineno-7-84" href="#__codelineno-7-84"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Vertex list = '</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">vertices</span><span class="p">);</span>
 <a id="__codelineno-7-85" name="__codelineno-7-85" href="#__codelineno-7-85"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Adjacency matrix ='</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjMat</span><span class="p">);</span>
@@ -5413,21 +5485,21 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_matrix.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Undirected graph class based on adjacency matrix */</span>
+<div class="highlight"><span class="filename">graph_adjacency_matrix.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency matrix */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">GraphAdjMat</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">vertices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Vertex elements, elements represent "vertex values", indices represent "vertex indices"</span>
-<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">adjMat</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Adjacency matrix, where the row and column indices correspond to the "vertex index"</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">vertices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">// Vertex entries: each element is a vertex value, and its index is the vertex index</span>
+<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">adjMat</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span><span class="w"> </span><span class="c1">//Adjacency matrix: row and column indices correspond to vertex indices</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="n">GraphAdjMat</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">vertices</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">edges</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">vertices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">adjMat</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="c1">// Add vertex</span>
+<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="c1">// Add a vertex</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">vertices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">      </span><span class="n">addVertex</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="c1">// Add edge</span>
-<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">    </span><span class="c1">// Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices</span>
+<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="c1">// Add an edge</span>
+<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">    </span><span class="c1">// Note that the entries in edges are vertex indices, corresponding to element indices in vertices</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">e</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">edges</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">      </span><span class="n">addEdge</span><span class="p">(</span><span class="n">e</span><span class="p">[</span><span class="m">0</span><span class="p">],</span><span class="w"> </span><span class="n">e</span><span class="p">[</span><span class="m">1</span><span class="p">]);</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">    </span><span class="p">}</span>
@@ -5438,10 +5510,10 @@
 <a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">vertices</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a>
-<a id="__codelineno-8-26" name="__codelineno-8-26" href="#__codelineno-8-26"></a><span class="w">  </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-8-26" name="__codelineno-8-26" href="#__codelineno-8-26"></a><span class="w">  </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">addVertex</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-28" name="__codelineno-8-28" href="#__codelineno-8-28"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-8-29" name="__codelineno-8-29" href="#__codelineno-8-29"></a><span class="w">    </span><span class="c1">// Add the value of the new vertex to the vertex list</span>
+<a id="__codelineno-8-29" name="__codelineno-8-29" href="#__codelineno-8-29"></a><span class="w">    </span><span class="c1">// Append the new vertex value to the vertex list</span>
 <a id="__codelineno-8-30" name="__codelineno-8-30" href="#__codelineno-8-30"></a><span class="w">    </span><span class="n">vertices</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-8-31" name="__codelineno-8-31" href="#__codelineno-8-31"></a><span class="w">    </span><span class="c1">// Add a row to the adjacency matrix</span>
 <a id="__codelineno-8-32" name="__codelineno-8-32" href="#__codelineno-8-32"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">newRow</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="nl">growable:</span><span class="w"> </span><span class="kc">true</span><span class="p">);</span>
@@ -5452,37 +5524,37 @@
 <a id="__codelineno-8-37" name="__codelineno-8-37" href="#__codelineno-8-37"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-8-38" name="__codelineno-8-38" href="#__codelineno-8-38"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-39" name="__codelineno-8-39" href="#__codelineno-8-39"></a>
-<a id="__codelineno-8-40" name="__codelineno-8-40" href="#__codelineno-8-40"></a><span class="w">  </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-8-40" name="__codelineno-8-40" href="#__codelineno-8-40"></a><span class="w">  </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-8-41" name="__codelineno-8-41" href="#__codelineno-8-41"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">removeVertex</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-42" name="__codelineno-8-42" href="#__codelineno-8-42"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-43" name="__codelineno-8-43" href="#__codelineno-8-43"></a><span class="w">      </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexError</span><span class="p">;</span>
 <a id="__codelineno-8-44" name="__codelineno-8-44" href="#__codelineno-8-44"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-8-45" name="__codelineno-8-45" href="#__codelineno-8-45"></a><span class="w">    </span><span class="c1">// Remove the vertex at index from the vertex list</span>
+<a id="__codelineno-8-45" name="__codelineno-8-45" href="#__codelineno-8-45"></a><span class="w">    </span><span class="c1">// Remove the vertex at index `index` from the vertex list</span>
 <a id="__codelineno-8-46" name="__codelineno-8-46" href="#__codelineno-8-46"></a><span class="w">    </span><span class="n">vertices</span><span class="p">.</span><span class="n">removeAt</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
-<a id="__codelineno-8-47" name="__codelineno-8-47" href="#__codelineno-8-47"></a><span class="w">    </span><span class="c1">// Remove the row at index from the adjacency matrix</span>
+<a id="__codelineno-8-47" name="__codelineno-8-47" href="#__codelineno-8-47"></a><span class="w">    </span><span class="c1">// Remove row `index` from the adjacency matrix</span>
 <a id="__codelineno-8-48" name="__codelineno-8-48" href="#__codelineno-8-48"></a><span class="w">    </span><span class="n">adjMat</span><span class="p">.</span><span class="n">removeAt</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
-<a id="__codelineno-8-49" name="__codelineno-8-49" href="#__codelineno-8-49"></a><span class="w">    </span><span class="c1">// Remove the column at index from the adjacency matrix</span>
+<a id="__codelineno-8-49" name="__codelineno-8-49" href="#__codelineno-8-49"></a><span class="w">    </span><span class="c1">// Remove column `index` from the adjacency matrix</span>
 <a id="__codelineno-8-50" name="__codelineno-8-50" href="#__codelineno-8-50"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">adjMat</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-51" name="__codelineno-8-51" href="#__codelineno-8-51"></a><span class="w">      </span><span class="n">row</span><span class="p">.</span><span class="n">removeAt</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
 <a id="__codelineno-8-52" name="__codelineno-8-52" href="#__codelineno-8-52"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-8-53" name="__codelineno-8-53" href="#__codelineno-8-53"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-54" name="__codelineno-8-54" href="#__codelineno-8-54"></a>
-<a id="__codelineno-8-55" name="__codelineno-8-55" href="#__codelineno-8-55"></a><span class="w">  </span><span class="cm">/* Add edge */</span>
-<a id="__codelineno-8-56" name="__codelineno-8-56" href="#__codelineno-8-56"></a><span class="w">  </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-8-55" name="__codelineno-8-55" href="#__codelineno-8-55"></a><span class="w">  </span><span class="cm">/* Add an edge */</span>
+<a id="__codelineno-8-56" name="__codelineno-8-56" href="#__codelineno-8-56"></a><span class="w">  </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-8-57" name="__codelineno-8-57" href="#__codelineno-8-57"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">addEdge</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-58" name="__codelineno-8-58" href="#__codelineno-8-58"></a><span class="w">    </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-8-58" name="__codelineno-8-58" href="#__codelineno-8-58"></a><span class="w">    </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-8-59" name="__codelineno-8-59" href="#__codelineno-8-59"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-60" name="__codelineno-8-60" href="#__codelineno-8-60"></a><span class="w">      </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexError</span><span class="p">;</span>
 <a id="__codelineno-8-61" name="__codelineno-8-61" href="#__codelineno-8-61"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-8-62" name="__codelineno-8-62" href="#__codelineno-8-62"></a><span class="w">    </span><span class="c1">// In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)</span>
+<a id="__codelineno-8-62" name="__codelineno-8-62" href="#__codelineno-8-62"></a><span class="w">    </span><span class="c1">// For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal</span>
 <a id="__codelineno-8-63" name="__codelineno-8-63" href="#__codelineno-8-63"></a><span class="w">    </span><span class="n">adjMat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-8-64" name="__codelineno-8-64" href="#__codelineno-8-64"></a><span class="w">    </span><span class="n">adjMat</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-8-65" name="__codelineno-8-65" href="#__codelineno-8-65"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-66" name="__codelineno-8-66" href="#__codelineno-8-66"></a>
-<a id="__codelineno-8-67" name="__codelineno-8-67" href="#__codelineno-8-67"></a><span class="w">  </span><span class="cm">/* Remove edge */</span>
-<a id="__codelineno-8-68" name="__codelineno-8-68" href="#__codelineno-8-68"></a><span class="w">  </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-8-67" name="__codelineno-8-67" href="#__codelineno-8-67"></a><span class="w">  </span><span class="cm">/* Remove an edge */</span>
+<a id="__codelineno-8-68" name="__codelineno-8-68" href="#__codelineno-8-68"></a><span class="w">  </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-8-69" name="__codelineno-8-69" href="#__codelineno-8-69"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">removeEdge</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-70" name="__codelineno-8-70" href="#__codelineno-8-70"></a><span class="w">    </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-8-70" name="__codelineno-8-70" href="#__codelineno-8-70"></a><span class="w">    </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-8-71" name="__codelineno-8-71" href="#__codelineno-8-71"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-72" name="__codelineno-8-72" href="#__codelineno-8-72"></a><span class="w">      </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexError</span><span class="p">;</span>
 <a id="__codelineno-8-73" name="__codelineno-8-73" href="#__codelineno-8-73"></a><span class="w">    </span><span class="p">}</span>
@@ -5490,7 +5562,7 @@
 <a id="__codelineno-8-75" name="__codelineno-8-75" href="#__codelineno-8-75"></a><span class="w">    </span><span class="n">adjMat</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-8-76" name="__codelineno-8-76" href="#__codelineno-8-76"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-77" name="__codelineno-8-77" href="#__codelineno-8-77"></a>
-<a id="__codelineno-8-78" name="__codelineno-8-78" href="#__codelineno-8-78"></a><span class="w">  </span><span class="cm">/* Print adjacency matrix */</span>
+<a id="__codelineno-8-78" name="__codelineno-8-78" href="#__codelineno-8-78"></a><span class="w">  </span><span class="cm">/* Print the adjacency matrix */</span>
 <a id="__codelineno-8-79" name="__codelineno-8-79" href="#__codelineno-8-79"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">printAdjMat</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-80" name="__codelineno-8-80" href="#__codelineno-8-80"></a><span class="w">    </span><span class="n">print</span><span class="p">(</span><span class="s2">"Vertex list = </span><span class="si">$</span><span class="n">vertices</span><span class="s2">"</span><span class="p">);</span>
 <a id="__codelineno-8-81" name="__codelineno-8-81" href="#__codelineno-8-81"></a><span class="w">    </span><span class="n">print</span><span class="p">(</span><span class="s2">"Adjacency matrix = "</span><span class="p">);</span>
@@ -5500,11 +5572,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_matrix.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Undirected graph type based on adjacency matrix */</span>
+<div class="highlight"><span class="filename">graph_adjacency_matrix.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Undirected graph type implemented with an adjacency matrix */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">GraphAdjMat</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Vertex list, where the element represents the "vertex value" and the index represents the "vertex index"</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Vertex list: elements are vertex values, and indices are vertex indices</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="n">vertices</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">,</span>
-<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="c1">// Adjacency matrix, where the row and column indices correspond to the "vertex index"</span>
+<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="c1">// Adjacency matrix: row and column indices correspond to vertex indices</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="n">adj_mat</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="p">}</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a>
@@ -5515,12 +5587,12 @@
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">            </span><span class="n">vertices</span><span class="p">:</span><span class="w"> </span><span class="nc">vec</span><span class="o">!</span><span class="p">[],</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">            </span><span class="n">adj_mat</span><span class="p">:</span><span class="w"> </span><span class="nc">vec</span><span class="o">!</span><span class="p">[],</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">        </span><span class="p">};</span>
-<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">        </span><span class="c1">// Add vertex</span>
+<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">        </span><span class="c1">// Add a vertex</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">vertices</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">            </span><span class="n">graph</span><span class="p">.</span><span class="n">add_vertex</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">        </span><span class="c1">// Add edge</span>
-<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">        </span><span class="c1">// Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices</span>
+<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">        </span><span class="c1">// Add an edge</span>
+<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">        </span><span class="c1">// Note that the entries in edges are vertex indices, corresponding to element indices in vertices</span>
 <a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">edge</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">edges</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="w">            </span><span class="n">graph</span><span class="p">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">edge</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="w"> </span><span class="n">edge</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
 <a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a><span class="w">        </span><span class="p">}</span>
@@ -5533,10 +5605,10 @@
 <a id="__codelineno-9-31" name="__codelineno-9-31" href="#__codelineno-9-31"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">vertices</span><span class="p">.</span><span class="n">len</span><span class="p">()</span>
 <a id="__codelineno-9-32" name="__codelineno-9-32" href="#__codelineno-9-32"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-33" name="__codelineno-9-33" href="#__codelineno-9-33"></a>
-<a id="__codelineno-9-34" name="__codelineno-9-34" href="#__codelineno-9-34"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-9-34" name="__codelineno-9-34" href="#__codelineno-9-34"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-9-35" name="__codelineno-9-35" href="#__codelineno-9-35"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">add_vertex</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-36" name="__codelineno-9-36" href="#__codelineno-9-36"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-9-37" name="__codelineno-9-37" href="#__codelineno-9-37"></a><span class="w">        </span><span class="c1">// Add the value of the new vertex to the vertex list</span>
+<a id="__codelineno-9-37" name="__codelineno-9-37" href="#__codelineno-9-37"></a><span class="w">        </span><span class="c1">// Append the new vertex value to the vertex list</span>
 <a id="__codelineno-9-38" name="__codelineno-9-38" href="#__codelineno-9-38"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">vertices</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-9-39" name="__codelineno-9-39" href="#__codelineno-9-39"></a><span class="w">        </span><span class="c1">// Add a row to the adjacency matrix</span>
 <a id="__codelineno-9-40" name="__codelineno-9-40" href="#__codelineno-9-40"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">adj_mat</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="p">]);</span>
@@ -5546,46 +5618,46 @@
 <a id="__codelineno-9-44" name="__codelineno-9-44" href="#__codelineno-9-44"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-45" name="__codelineno-9-45" href="#__codelineno-9-45"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-46" name="__codelineno-9-46" href="#__codelineno-9-46"></a>
-<a id="__codelineno-9-47" name="__codelineno-9-47" href="#__codelineno-9-47"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-9-47" name="__codelineno-9-47" href="#__codelineno-9-47"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-9-48" name="__codelineno-9-48" href="#__codelineno-9-48"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">remove_vertex</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-49" name="__codelineno-9-49" href="#__codelineno-9-49"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-50" name="__codelineno-9-50" href="#__codelineno-9-50"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"index error"</span><span class="p">)</span>
+<a id="__codelineno-9-50" name="__codelineno-9-50" href="#__codelineno-9-50"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"Invalid vertex index"</span><span class="p">)</span>
 <a id="__codelineno-9-51" name="__codelineno-9-51" href="#__codelineno-9-51"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-9-52" name="__codelineno-9-52" href="#__codelineno-9-52"></a><span class="w">        </span><span class="c1">// Remove the vertex at index from the vertex list</span>
+<a id="__codelineno-9-52" name="__codelineno-9-52" href="#__codelineno-9-52"></a><span class="w">        </span><span class="c1">// Remove the vertex at index `index` from the vertex list</span>
 <a id="__codelineno-9-53" name="__codelineno-9-53" href="#__codelineno-9-53"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">vertices</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
-<a id="__codelineno-9-54" name="__codelineno-9-54" href="#__codelineno-9-54"></a><span class="w">        </span><span class="c1">// Remove the row at index from the adjacency matrix</span>
+<a id="__codelineno-9-54" name="__codelineno-9-54" href="#__codelineno-9-54"></a><span class="w">        </span><span class="c1">// Remove row `index` from the adjacency matrix</span>
 <a id="__codelineno-9-55" name="__codelineno-9-55" href="#__codelineno-9-55"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">adj_mat</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
-<a id="__codelineno-9-56" name="__codelineno-9-56" href="#__codelineno-9-56"></a><span class="w">        </span><span class="c1">// Remove the column at index from the adjacency matrix</span>
+<a id="__codelineno-9-56" name="__codelineno-9-56" href="#__codelineno-9-56"></a><span class="w">        </span><span class="c1">// Remove column `index` from the adjacency matrix</span>
 <a id="__codelineno-9-57" name="__codelineno-9-57" href="#__codelineno-9-57"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">adj_mat</span><span class="p">.</span><span class="n">iter_mut</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-58" name="__codelineno-9-58" href="#__codelineno-9-58"></a><span class="w">            </span><span class="n">row</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
 <a id="__codelineno-9-59" name="__codelineno-9-59" href="#__codelineno-9-59"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-60" name="__codelineno-9-60" href="#__codelineno-9-60"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-61" name="__codelineno-9-61" href="#__codelineno-9-61"></a>
-<a id="__codelineno-9-62" name="__codelineno-9-62" href="#__codelineno-9-62"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
+<a id="__codelineno-9-62" name="__codelineno-9-62" href="#__codelineno-9-62"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
 <a id="__codelineno-9-63" name="__codelineno-9-63" href="#__codelineno-9-63"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">add_edge</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-64" name="__codelineno-9-64" href="#__codelineno-9-64"></a><span class="w">        </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
-<a id="__codelineno-9-65" name="__codelineno-9-65" href="#__codelineno-9-65"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-9-64" name="__codelineno-9-64" href="#__codelineno-9-64"></a><span class="w">        </span><span class="c1">// Parameters i and j are element indices in vertices</span>
+<a id="__codelineno-9-65" name="__codelineno-9-65" href="#__codelineno-9-65"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-9-66" name="__codelineno-9-66" href="#__codelineno-9-66"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-67" name="__codelineno-9-67" href="#__codelineno-9-67"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"index error"</span><span class="p">)</span>
+<a id="__codelineno-9-67" name="__codelineno-9-67" href="#__codelineno-9-67"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"Invalid vertex index"</span><span class="p">)</span>
 <a id="__codelineno-9-68" name="__codelineno-9-68" href="#__codelineno-9-68"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-9-69" name="__codelineno-9-69" href="#__codelineno-9-69"></a><span class="w">        </span><span class="c1">// In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)</span>
+<a id="__codelineno-9-69" name="__codelineno-9-69" href="#__codelineno-9-69"></a><span class="w">        </span><span class="c1">// For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal</span>
 <a id="__codelineno-9-70" name="__codelineno-9-70" href="#__codelineno-9-70"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">adj_mat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-71" name="__codelineno-9-71" href="#__codelineno-9-71"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">adj_mat</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-72" name="__codelineno-9-72" href="#__codelineno-9-72"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-73" name="__codelineno-9-73" href="#__codelineno-9-73"></a>
-<a id="__codelineno-9-74" name="__codelineno-9-74" href="#__codelineno-9-74"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
-<a id="__codelineno-9-75" name="__codelineno-9-75" href="#__codelineno-9-75"></a><span class="w">    </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-9-74" name="__codelineno-9-74" href="#__codelineno-9-74"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
+<a id="__codelineno-9-75" name="__codelineno-9-75" href="#__codelineno-9-75"></a><span class="w">    </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-9-76" name="__codelineno-9-76" href="#__codelineno-9-76"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">remove_edge</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-77" name="__codelineno-9-77" href="#__codelineno-9-77"></a><span class="w">        </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
-<a id="__codelineno-9-78" name="__codelineno-9-78" href="#__codelineno-9-78"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-9-77" name="__codelineno-9-77" href="#__codelineno-9-77"></a><span class="w">        </span><span class="c1">// Parameters i and j are element indices in vertices</span>
+<a id="__codelineno-9-78" name="__codelineno-9-78" href="#__codelineno-9-78"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-9-79" name="__codelineno-9-79" href="#__codelineno-9-79"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-80" name="__codelineno-9-80" href="#__codelineno-9-80"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"index error"</span><span class="p">)</span>
+<a id="__codelineno-9-80" name="__codelineno-9-80" href="#__codelineno-9-80"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"Invalid vertex index"</span><span class="p">)</span>
 <a id="__codelineno-9-81" name="__codelineno-9-81" href="#__codelineno-9-81"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-82" name="__codelineno-9-82" href="#__codelineno-9-82"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">adj_mat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-9-83" name="__codelineno-9-83" href="#__codelineno-9-83"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">adj_mat</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-9-84" name="__codelineno-9-84" href="#__codelineno-9-84"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-85" name="__codelineno-9-85" href="#__codelineno-9-85"></a>
-<a id="__codelineno-9-86" name="__codelineno-9-86" href="#__codelineno-9-86"></a><span class="w">    </span><span class="cm">/* Print adjacency matrix */</span>
+<a id="__codelineno-9-86" name="__codelineno-9-86" href="#__codelineno-9-86"></a><span class="w">    </span><span class="cm">/* Print the adjacency matrix */</span>
 <a id="__codelineno-9-87" name="__codelineno-9-87" href="#__codelineno-9-87"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">print</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-88" name="__codelineno-9-88" href="#__codelineno-9-88"></a><span class="w">        </span><span class="fm">println!</span><span class="p">(</span><span class="s">"Vertex list = {:?}"</span><span class="p">,</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">vertices</span><span class="p">);</span>
 <a id="__codelineno-9-89" name="__codelineno-9-89" href="#__codelineno-9-89"></a><span class="w">        </span><span class="fm">println!</span><span class="p">(</span><span class="s">"Adjacency matrix ="</span><span class="p">);</span>
@@ -5599,7 +5671,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_matrix.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Undirected graph structure based on adjacency matrix */</span>
+<div class="highlight"><span class="filename">graph_adjacency_matrix.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Undirected graph structure implemented with an adjacency matrix */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">vertices</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">];</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">adjMat</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">][</span><span class="n">MAX_SIZE</span><span class="p">];</span>
@@ -5623,13 +5695,13 @@
 <a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">graph</span><span class="p">);</span>
 <a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="p">}</span>
 <a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a>
-<a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="kt">void</span><span class="w"> </span><span class="nf">addVertex</span><span class="p">(</span><span class="n">GraphAdjMat</span><span class="w"> </span><span class="o">*</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-27" name="__codelineno-10-27" href="#__codelineno-10-27"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-28" name="__codelineno-10-28" href="#__codelineno-10-28"></a><span class="w">        </span><span class="n">fprintf</span><span class="p">(</span><span class="n">stderr</span><span class="p">,</span><span class="w"> </span><span class="s">"Graph vertex count has reached maximum</span><span class="se">\n</span><span class="s">"</span><span class="p">);</span>
+<a id="__codelineno-10-28" name="__codelineno-10-28" href="#__codelineno-10-28"></a><span class="w">        </span><span class="n">fprintf</span><span class="p">(</span><span class="n">stderr</span><span class="p">,</span><span class="w"> </span><span class="s">"The graph has reached its maximum number of vertices</span><span class="se">\n</span><span class="s">"</span><span class="p">);</span>
 <a id="__codelineno-10-29" name="__codelineno-10-29" href="#__codelineno-10-29"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-10-30" name="__codelineno-10-30" href="#__codelineno-10-30"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-31" name="__codelineno-10-31" href="#__codelineno-10-31"></a><span class="w">    </span><span class="c1">// Add nth vertex and zero nth row and column</span>
+<a id="__codelineno-10-31" name="__codelineno-10-31" href="#__codelineno-10-31"></a><span class="w">    </span><span class="c1">// Add the vertex at index n and set row n and column n to zero</span>
 <a id="__codelineno-10-32" name="__codelineno-10-32" href="#__codelineno-10-32"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">;</span>
 <a id="__codelineno-10-33" name="__codelineno-10-33" href="#__codelineno-10-33"></a><span class="w">    </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">vertices</span><span class="p">[</span><span class="n">n</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-10-34" name="__codelineno-10-34" href="#__codelineno-10-34"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5638,23 +5710,23 @@
 <a id="__codelineno-10-37" name="__codelineno-10-37" href="#__codelineno-10-37"></a><span class="w">    </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-10-38" name="__codelineno-10-38" href="#__codelineno-10-38"></a><span class="p">}</span>
 <a id="__codelineno-10-39" name="__codelineno-10-39" href="#__codelineno-10-39"></a>
-<a id="__codelineno-10-40" name="__codelineno-10-40" href="#__codelineno-10-40"></a><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-10-40" name="__codelineno-10-40" href="#__codelineno-10-40"></a><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-10-41" name="__codelineno-10-41" href="#__codelineno-10-41"></a><span class="kt">void</span><span class="w"> </span><span class="nf">removeVertex</span><span class="p">(</span><span class="n">GraphAdjMat</span><span class="w"> </span><span class="o">*</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-42" name="__codelineno-10-42" href="#__codelineno-10-42"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-43" name="__codelineno-10-43" href="#__codelineno-10-43"></a><span class="w">        </span><span class="n">fprintf</span><span class="p">(</span><span class="n">stderr</span><span class="p">,</span><span class="w"> </span><span class="s">"Vertex index out of bounds</span><span class="se">\n</span><span class="s">"</span><span class="p">);</span>
 <a id="__codelineno-10-44" name="__codelineno-10-44" href="#__codelineno-10-44"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-10-45" name="__codelineno-10-45" href="#__codelineno-10-45"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-46" name="__codelineno-10-46" href="#__codelineno-10-46"></a><span class="w">    </span><span class="c1">// Remove the vertex at index from the vertex list</span>
+<a id="__codelineno-10-46" name="__codelineno-10-46" href="#__codelineno-10-46"></a><span class="w">    </span><span class="c1">// Remove the vertex at index `index` from the vertex list</span>
 <a id="__codelineno-10-47" name="__codelineno-10-47" href="#__codelineno-10-47"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-48" name="__codelineno-10-48" href="#__codelineno-10-48"></a><span class="w">        </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">vertices</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">vertices</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-10-49" name="__codelineno-10-49" href="#__codelineno-10-49"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-50" name="__codelineno-10-50" href="#__codelineno-10-50"></a><span class="w">    </span><span class="c1">// Remove the row at index from the adjacency matrix</span>
+<a id="__codelineno-10-50" name="__codelineno-10-50" href="#__codelineno-10-50"></a><span class="w">    </span><span class="c1">// Remove row `index` from the adjacency matrix</span>
 <a id="__codelineno-10-51" name="__codelineno-10-51" href="#__codelineno-10-51"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-52" name="__codelineno-10-52" href="#__codelineno-10-52"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-53" name="__codelineno-10-53" href="#__codelineno-10-53"></a><span class="w">            </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">adjMat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">adjMat</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">][</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-10-54" name="__codelineno-10-54" href="#__codelineno-10-54"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-10-55" name="__codelineno-10-55" href="#__codelineno-10-55"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-56" name="__codelineno-10-56" href="#__codelineno-10-56"></a><span class="w">    </span><span class="c1">// Remove the column at index from the adjacency matrix</span>
+<a id="__codelineno-10-56" name="__codelineno-10-56" href="#__codelineno-10-56"></a><span class="w">    </span><span class="c1">// Remove column `index` from the adjacency matrix</span>
 <a id="__codelineno-10-57" name="__codelineno-10-57" href="#__codelineno-10-57"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-58" name="__codelineno-10-58" href="#__codelineno-10-58"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-59" name="__codelineno-10-59" href="#__codelineno-10-59"></a><span class="w">            </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">adjMat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">adjMat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
@@ -5663,29 +5735,29 @@
 <a id="__codelineno-10-62" name="__codelineno-10-62" href="#__codelineno-10-62"></a><span class="w">    </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-10-63" name="__codelineno-10-63" href="#__codelineno-10-63"></a><span class="p">}</span>
 <a id="__codelineno-10-64" name="__codelineno-10-64" href="#__codelineno-10-64"></a>
-<a id="__codelineno-10-65" name="__codelineno-10-65" href="#__codelineno-10-65"></a><span class="cm">/* Add edge */</span>
-<a id="__codelineno-10-66" name="__codelineno-10-66" href="#__codelineno-10-66"></a><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-10-65" name="__codelineno-10-65" href="#__codelineno-10-65"></a><span class="cm">/* Add an edge */</span>
+<a id="__codelineno-10-66" name="__codelineno-10-66" href="#__codelineno-10-66"></a><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-10-67" name="__codelineno-10-67" href="#__codelineno-10-67"></a><span class="kt">void</span><span class="w"> </span><span class="nf">addEdge</span><span class="p">(</span><span class="n">GraphAdjMat</span><span class="w"> </span><span class="o">*</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-68" name="__codelineno-10-68" href="#__codelineno-10-68"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-69" name="__codelineno-10-69" href="#__codelineno-10-69"></a><span class="w">        </span><span class="n">fprintf</span><span class="p">(</span><span class="n">stderr</span><span class="p">,</span><span class="w"> </span><span class="s">"Edge index out of bounds or equal</span><span class="se">\n</span><span class="s">"</span><span class="p">);</span>
+<a id="__codelineno-10-69" name="__codelineno-10-69" href="#__codelineno-10-69"></a><span class="w">        </span><span class="n">fprintf</span><span class="p">(</span><span class="n">stderr</span><span class="p">,</span><span class="w"> </span><span class="s">"Edge endpoint indices are out of bounds or equal</span><span class="se">\n</span><span class="s">"</span><span class="p">);</span>
 <a id="__codelineno-10-70" name="__codelineno-10-70" href="#__codelineno-10-70"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-10-71" name="__codelineno-10-71" href="#__codelineno-10-71"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-72" name="__codelineno-10-72" href="#__codelineno-10-72"></a><span class="w">    </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">adjMat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-10-73" name="__codelineno-10-73" href="#__codelineno-10-73"></a><span class="w">    </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">adjMat</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-10-74" name="__codelineno-10-74" href="#__codelineno-10-74"></a><span class="p">}</span>
 <a id="__codelineno-10-75" name="__codelineno-10-75" href="#__codelineno-10-75"></a>
-<a id="__codelineno-10-76" name="__codelineno-10-76" href="#__codelineno-10-76"></a><span class="cm">/* Remove edge */</span>
-<a id="__codelineno-10-77" name="__codelineno-10-77" href="#__codelineno-10-77"></a><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-10-76" name="__codelineno-10-76" href="#__codelineno-10-76"></a><span class="cm">/* Remove an edge */</span>
+<a id="__codelineno-10-77" name="__codelineno-10-77" href="#__codelineno-10-77"></a><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-10-78" name="__codelineno-10-78" href="#__codelineno-10-78"></a><span class="kt">void</span><span class="w"> </span><span class="nf">removeEdge</span><span class="p">(</span><span class="n">GraphAdjMat</span><span class="w"> </span><span class="o">*</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-79" name="__codelineno-10-79" href="#__codelineno-10-79"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-80" name="__codelineno-10-80" href="#__codelineno-10-80"></a><span class="w">        </span><span class="n">fprintf</span><span class="p">(</span><span class="n">stderr</span><span class="p">,</span><span class="w"> </span><span class="s">"Edge index out of bounds or equal</span><span class="se">\n</span><span class="s">"</span><span class="p">);</span>
+<a id="__codelineno-10-80" name="__codelineno-10-80" href="#__codelineno-10-80"></a><span class="w">        </span><span class="n">fprintf</span><span class="p">(</span><span class="n">stderr</span><span class="p">,</span><span class="w"> </span><span class="s">"Edge endpoint indices are out of bounds or equal</span><span class="se">\n</span><span class="s">"</span><span class="p">);</span>
 <a id="__codelineno-10-81" name="__codelineno-10-81" href="#__codelineno-10-81"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-10-82" name="__codelineno-10-82" href="#__codelineno-10-82"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-83" name="__codelineno-10-83" href="#__codelineno-10-83"></a><span class="w">    </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">adjMat</span><span class="p">[</span><span class="n">i</span><span class="p">][</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-10-84" name="__codelineno-10-84" href="#__codelineno-10-84"></a><span class="w">    </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">adjMat</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-10-85" name="__codelineno-10-85" href="#__codelineno-10-85"></a><span class="p">}</span>
 <a id="__codelineno-10-86" name="__codelineno-10-86" href="#__codelineno-10-86"></a>
-<a id="__codelineno-10-87" name="__codelineno-10-87" href="#__codelineno-10-87"></a><span class="cm">/* Print adjacency matrix */</span>
+<a id="__codelineno-10-87" name="__codelineno-10-87" href="#__codelineno-10-87"></a><span class="cm">/* Print the adjacency matrix */</span>
 <a id="__codelineno-10-88" name="__codelineno-10-88" href="#__codelineno-10-88"></a><span class="kt">void</span><span class="w"> </span><span class="nf">printGraphAdjMat</span><span class="p">(</span><span class="n">GraphAdjMat</span><span class="w"> </span><span class="o">*</span><span class="n">graph</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-89" name="__codelineno-10-89" href="#__codelineno-10-89"></a><span class="w">    </span><span class="n">printf</span><span class="p">(</span><span class="s">"Vertex list = "</span><span class="p">);</span>
 <a id="__codelineno-10-90" name="__codelineno-10-90" href="#__codelineno-10-90"></a><span class="w">    </span><span class="n">printArray</span><span class="p">(</span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">vertices</span><span class="p">,</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">);</span>
@@ -5697,19 +5769,19 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_matrix.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Undirected graph class based on adjacency matrix */</span>
+<div class="highlight"><span class="filename">graph_adjacency_matrix.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency matrix */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">GraphAdjMat</span><span class="p">(</span><span class="n">vertices</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">edges</span><span class="p">:</span><span class="w"> </span><span class="n">Array</span><span class="o">&lt;</span><span class="n">IntArray</span><span class="o">&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">vertices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span><span class="w"> </span><span class="c1">// Vertex list, where the element represents the "vertex value" and the index represents the "vertex index"</span>
-<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">adjMat</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;&gt;</span><span class="p">()</span><span class="w"> </span><span class="c1">// Adjacency matrix, where the row and column indices correspond to the "vertex index"</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">vertices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span><span class="w"> </span><span class="c1">// Vertex list: elements are vertex values, and indices are vertex indices</span>
+<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">adjMat</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;&gt;</span><span class="p">()</span><span class="w"> </span><span class="c1">// Adjacency matrix: row and column indices correspond to vertex indices</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="k">init</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">        </span><span class="c1">// Add vertex</span>
+<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">        </span><span class="c1">// Add a vertex</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">vertex</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">vertices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">            </span><span class="n">addVertex</span><span class="p">(</span><span class="n">vertex</span><span class="p">)</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="c1">// Add edge</span>
-<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="c1">// Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices</span>
+<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="c1">// Add an edge</span>
+<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="c1">// Note that the entries in edges are vertex indices, corresponding to element indices in vertices</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">edge</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">edges</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">            </span><span class="n">addEdge</span><span class="p">(</span><span class="n">edge</span><span class="o">[</span><span class="m">0</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">edge</span><span class="o">[</span><span class="m">1</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">        </span><span class="p">}</span>
@@ -5720,10 +5792,10 @@
 <a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">vertices</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a>
-<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">addVertex</span><span class="p">(</span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span>
-<a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w">        </span><span class="c1">// Add the value of the new vertex to the vertex list</span>
+<a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w">        </span><span class="c1">// Append the new vertex value to the vertex list</span>
 <a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="w">        </span><span class="n">vertices</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">_val</span><span class="p">)</span>
 <a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="w">        </span><span class="c1">// Add a row to the adjacency matrix</span>
 <a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">newRow</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span>
@@ -5737,42 +5809,42 @@
 <a id="__codelineno-11-38" name="__codelineno-11-38" href="#__codelineno-11-38"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-11-39" name="__codelineno-11-39" href="#__codelineno-11-39"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-40" name="__codelineno-11-40" href="#__codelineno-11-40"></a>
-<a id="__codelineno-11-41" name="__codelineno-11-41" href="#__codelineno-11-41"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-11-41" name="__codelineno-11-41" href="#__codelineno-11-41"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-11-42" name="__codelineno-11-42" href="#__codelineno-11-42"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">removeVertex</span><span class="p">(</span><span class="n">index</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-43" name="__codelineno-11-43" href="#__codelineno-11-43"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">())</span>
 <a id="__codelineno-11-44" name="__codelineno-11-44" href="#__codelineno-11-44"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">()</span>
-<a id="__codelineno-11-45" name="__codelineno-11-45" href="#__codelineno-11-45"></a><span class="w">        </span><span class="c1">// Remove the vertex at index from the vertex list</span>
+<a id="__codelineno-11-45" name="__codelineno-11-45" href="#__codelineno-11-45"></a><span class="w">        </span><span class="c1">// Remove the vertex at index `index` from the vertex list</span>
 <a id="__codelineno-11-46" name="__codelineno-11-46" href="#__codelineno-11-46"></a><span class="w">        </span><span class="n">vertices</span><span class="p">.</span><span class="na">removeAt</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
-<a id="__codelineno-11-47" name="__codelineno-11-47" href="#__codelineno-11-47"></a><span class="w">        </span><span class="c1">// Remove the row at index from the adjacency matrix</span>
+<a id="__codelineno-11-47" name="__codelineno-11-47" href="#__codelineno-11-47"></a><span class="w">        </span><span class="c1">// Remove row `index` from the adjacency matrix</span>
 <a id="__codelineno-11-48" name="__codelineno-11-48" href="#__codelineno-11-48"></a><span class="w">        </span><span class="n">adjMat</span><span class="p">.</span><span class="na">removeAt</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
-<a id="__codelineno-11-49" name="__codelineno-11-49" href="#__codelineno-11-49"></a><span class="w">        </span><span class="c1">// Remove the column at index from the adjacency matrix</span>
+<a id="__codelineno-11-49" name="__codelineno-11-49" href="#__codelineno-11-49"></a><span class="w">        </span><span class="c1">// Remove column `index` from the adjacency matrix</span>
 <a id="__codelineno-11-50" name="__codelineno-11-50" href="#__codelineno-11-50"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">row</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">adjMat</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-51" name="__codelineno-11-51" href="#__codelineno-11-51"></a><span class="w">            </span><span class="n">row</span><span class="p">.</span><span class="na">removeAt</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
 <a id="__codelineno-11-52" name="__codelineno-11-52" href="#__codelineno-11-52"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-11-53" name="__codelineno-11-53" href="#__codelineno-11-53"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-54" name="__codelineno-11-54" href="#__codelineno-11-54"></a>
-<a id="__codelineno-11-55" name="__codelineno-11-55" href="#__codelineno-11-55"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
-<a id="__codelineno-11-56" name="__codelineno-11-56" href="#__codelineno-11-56"></a><span class="w">    </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-11-55" name="__codelineno-11-55" href="#__codelineno-11-55"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
+<a id="__codelineno-11-56" name="__codelineno-11-56" href="#__codelineno-11-56"></a><span class="w">    </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-11-57" name="__codelineno-11-57" href="#__codelineno-11-57"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">addEdge</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-58" name="__codelineno-11-58" href="#__codelineno-11-58"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-11-58" name="__codelineno-11-58" href="#__codelineno-11-58"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-11-59" name="__codelineno-11-59" href="#__codelineno-11-59"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-11-60" name="__codelineno-11-60" href="#__codelineno-11-60"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">()</span>
-<a id="__codelineno-11-61" name="__codelineno-11-61" href="#__codelineno-11-61"></a><span class="w">        </span><span class="c1">// In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)</span>
+<a id="__codelineno-11-61" name="__codelineno-11-61" href="#__codelineno-11-61"></a><span class="w">        </span><span class="c1">// For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal</span>
 <a id="__codelineno-11-62" name="__codelineno-11-62" href="#__codelineno-11-62"></a><span class="w">        </span><span class="n">adjMat</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-11-63" name="__codelineno-11-63" href="#__codelineno-11-63"></a><span class="w">        </span><span class="n">adjMat</span><span class="o">[</span><span class="n">j</span><span class="o">][</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-11-64" name="__codelineno-11-64" href="#__codelineno-11-64"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-65" name="__codelineno-11-65" href="#__codelineno-11-65"></a>
-<a id="__codelineno-11-66" name="__codelineno-11-66" href="#__codelineno-11-66"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
-<a id="__codelineno-11-67" name="__codelineno-11-67" href="#__codelineno-11-67"></a><span class="w">    </span><span class="c1">// Parameters i, j correspond to the vertices element indices</span>
+<a id="__codelineno-11-66" name="__codelineno-11-66" href="#__codelineno-11-66"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
+<a id="__codelineno-11-67" name="__codelineno-11-67" href="#__codelineno-11-67"></a><span class="w">    </span><span class="c1">// Parameters i and j are element indices in vertices</span>
 <a id="__codelineno-11-68" name="__codelineno-11-68" href="#__codelineno-11-68"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">removeEdge</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-69" name="__codelineno-11-69" href="#__codelineno-11-69"></a><span class="w">        </span><span class="c1">// Handle index out of bounds and equality</span>
+<a id="__codelineno-11-69" name="__codelineno-11-69" href="#__codelineno-11-69"></a><span class="w">        </span><span class="c1">// Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-11-70" name="__codelineno-11-70" href="#__codelineno-11-70"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
 <a id="__codelineno-11-71" name="__codelineno-11-71" href="#__codelineno-11-71"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">()</span>
 <a id="__codelineno-11-72" name="__codelineno-11-72" href="#__codelineno-11-72"></a><span class="w">        </span><span class="n">adjMat</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-11-73" name="__codelineno-11-73" href="#__codelineno-11-73"></a><span class="w">        </span><span class="n">adjMat</span><span class="o">[</span><span class="n">j</span><span class="o">][</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-11-74" name="__codelineno-11-74" href="#__codelineno-11-74"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-75" name="__codelineno-11-75" href="#__codelineno-11-75"></a>
-<a id="__codelineno-11-76" name="__codelineno-11-76" href="#__codelineno-11-76"></a><span class="w">    </span><span class="cm">/* Print adjacency matrix */</span>
+<a id="__codelineno-11-76" name="__codelineno-11-76" href="#__codelineno-11-76"></a><span class="w">    </span><span class="cm">/* Print the adjacency matrix */</span>
 <a id="__codelineno-11-77" name="__codelineno-11-77" href="#__codelineno-11-77"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-78" name="__codelineno-11-78" href="#__codelineno-11-78"></a><span class="w">        </span><span class="n">print</span><span class="p">(</span><span class="s">"Vertex list = "</span><span class="p">)</span>
 <a id="__codelineno-11-79" name="__codelineno-11-79" href="#__codelineno-11-79"></a><span class="w">        </span><span class="n">println</span><span class="p">(</span><span class="n">vertices</span><span class="p">)</span>
@@ -5783,30 +5855,30 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_matrix.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Undirected graph class based on adjacency matrix ###</span>
+<div class="highlight"><span class="filename">graph_adjacency_matrix.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Undirected graph class implemented with an adjacency matrix ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">GraphAdjMat</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">initialize</span><span class="p">(</span><span class="n">vertices</span><span class="p">,</span><span class="w"> </span><span class="n">edges</span><span class="p">)</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">    </span><span class="c1">### Constructor ###</span>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">    </span><span class="c1"># Vertex list, where the element represents the "vertex value" and the index represents the "vertex index"</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">    </span><span class="c1"># Vertex list: elements are vertex values, and indices are vertex indices</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">    </span><span class="vi">@vertices</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
-<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">    </span><span class="c1"># Adjacency matrix, where the row and column indices correspond to the "vertex index"</span>
+<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">    </span><span class="c1"># Adjacency matrix: row and column indices correspond to vertex indices</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="vi">@adj_mat</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
-<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">    </span><span class="c1"># Add vertex</span>
+<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">    </span><span class="c1"># Add a vertex</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="n">vertices</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">val</span><span class="o">|</span><span class="w"> </span><span class="n">add_vertex</span><span class="p">(</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="c1"># Add edge</span>
-<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">    </span><span class="c1"># Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices</span>
+<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="c1"># Add an edge</span>
+<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">    </span><span class="c1"># Note that the entries in edges are vertex indices, corresponding to element indices in vertices</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="n">edges</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">e</span><span class="o">|</span><span class="w"> </span><span class="n">add_edge</span><span class="p">(</span><span class="n">e</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">e</span><span class="o">[</span><span class="mi">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a>
-<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">  </span><span class="c1">### Get number of vertices ###</span>
+<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">  </span><span class="c1">### Get the number of vertices ###</span>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">size</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">    </span><span class="vi">@vertices</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a>
-<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">  </span><span class="c1">### Add vertex ###</span>
+<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">  </span><span class="c1">### Add a vertex ###</span>
 <a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">add_vertex</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="w">    </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span>
-<a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a><span class="w">    </span><span class="c1"># Add the value of the new vertex to the vertex list</span>
+<a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a><span class="w">    </span><span class="c1"># Append the new vertex value to the vertex list</span>
 <a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="w">    </span><span class="vi">@vertices</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">val</span>
 <a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></a><span class="w">    </span><span class="c1"># Add a row to the adjacency matrix</span>
 <a id="__codelineno-12-27" name="__codelineno-12-27" href="#__codelineno-12-27"></a><span class="w">    </span><span class="n">new_row</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
@@ -5815,34 +5887,34 @@
 <a id="__codelineno-12-30" name="__codelineno-12-30" href="#__codelineno-12-30"></a><span class="w">    </span><span class="vi">@adj_mat</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">row</span><span class="o">|</span><span class="w"> </span><span class="n">row</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-12-31" name="__codelineno-12-31" href="#__codelineno-12-31"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-32" name="__codelineno-12-32" href="#__codelineno-12-32"></a>
-<a id="__codelineno-12-33" name="__codelineno-12-33" href="#__codelineno-12-33"></a><span class="w">  </span><span class="c1">### Delete vertex ###</span>
+<a id="__codelineno-12-33" name="__codelineno-12-33" href="#__codelineno-12-33"></a><span class="w">  </span><span class="c1">### Remove a vertex ###</span>
 <a id="__codelineno-12-34" name="__codelineno-12-34" href="#__codelineno-12-34"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">remove_vertex</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
 <a id="__codelineno-12-35" name="__codelineno-12-35" href="#__codelineno-12-35"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span>
 <a id="__codelineno-12-36" name="__codelineno-12-36" href="#__codelineno-12-36"></a>
-<a id="__codelineno-12-37" name="__codelineno-12-37" href="#__codelineno-12-37"></a><span class="w">    </span><span class="c1"># Remove the vertex at index from the vertex list</span>
+<a id="__codelineno-12-37" name="__codelineno-12-37" href="#__codelineno-12-37"></a><span class="w">    </span><span class="c1"># Remove the vertex at index `index` from the vertex list</span>
 <a id="__codelineno-12-38" name="__codelineno-12-38" href="#__codelineno-12-38"></a><span class="w">    </span><span class="vi">@vertices</span><span class="o">.</span><span class="n">delete_at</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
-<a id="__codelineno-12-39" name="__codelineno-12-39" href="#__codelineno-12-39"></a><span class="w">    </span><span class="c1"># Remove the row at index from the adjacency matrix</span>
+<a id="__codelineno-12-39" name="__codelineno-12-39" href="#__codelineno-12-39"></a><span class="w">    </span><span class="c1"># Remove row `index` from the adjacency matrix</span>
 <a id="__codelineno-12-40" name="__codelineno-12-40" href="#__codelineno-12-40"></a><span class="w">    </span><span class="vi">@adj_mat</span><span class="o">.</span><span class="n">delete_at</span><span class="p">(</span><span class="n">index</span><span class="p">)</span>
-<a id="__codelineno-12-41" name="__codelineno-12-41" href="#__codelineno-12-41"></a><span class="w">    </span><span class="c1"># Remove the column at index from the adjacency matrix</span>
+<a id="__codelineno-12-41" name="__codelineno-12-41" href="#__codelineno-12-41"></a><span class="w">    </span><span class="c1"># Remove column `index` from the adjacency matrix</span>
 <a id="__codelineno-12-42" name="__codelineno-12-42" href="#__codelineno-12-42"></a><span class="w">    </span><span class="vi">@adj_mat</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">row</span><span class="o">|</span><span class="w"> </span><span class="n">row</span><span class="o">.</span><span class="n">delete_at</span><span class="p">(</span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-12-43" name="__codelineno-12-43" href="#__codelineno-12-43"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-44" name="__codelineno-12-44" href="#__codelineno-12-44"></a>
-<a id="__codelineno-12-45" name="__codelineno-12-45" href="#__codelineno-12-45"></a><span class="w">  </span><span class="c1">### Add edge ###</span>
+<a id="__codelineno-12-45" name="__codelineno-12-45" href="#__codelineno-12-45"></a><span class="w">  </span><span class="c1">### Add an edge ###</span>
 <a id="__codelineno-12-46" name="__codelineno-12-46" href="#__codelineno-12-46"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">add_edge</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
-<a id="__codelineno-12-47" name="__codelineno-12-47" href="#__codelineno-12-47"></a><span class="w">    </span><span class="c1"># Parameters i, j correspond to the vertices element indices</span>
-<a id="__codelineno-12-48" name="__codelineno-12-48" href="#__codelineno-12-48"></a><span class="w">    </span><span class="c1"># Handle index out of bounds and equality</span>
+<a id="__codelineno-12-47" name="__codelineno-12-47" href="#__codelineno-12-47"></a><span class="w">    </span><span class="c1"># Parameters i and j are element indices in vertices</span>
+<a id="__codelineno-12-48" name="__codelineno-12-48" href="#__codelineno-12-48"></a><span class="w">    </span><span class="c1"># Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-12-49" name="__codelineno-12-49" href="#__codelineno-12-49"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span>
 <a id="__codelineno-12-50" name="__codelineno-12-50" href="#__codelineno-12-50"></a><span class="w">      </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span>
 <a id="__codelineno-12-51" name="__codelineno-12-51" href="#__codelineno-12-51"></a><span class="w">    </span><span class="k">end</span>
-<a id="__codelineno-12-52" name="__codelineno-12-52" href="#__codelineno-12-52"></a><span class="w">    </span><span class="c1"># In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)</span>
+<a id="__codelineno-12-52" name="__codelineno-12-52" href="#__codelineno-12-52"></a><span class="w">    </span><span class="c1"># For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal</span>
 <a id="__codelineno-12-53" name="__codelineno-12-53" href="#__codelineno-12-53"></a><span class="w">    </span><span class="vi">@adj_mat</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-54" name="__codelineno-12-54" href="#__codelineno-12-54"></a><span class="w">    </span><span class="vi">@adj_mat</span><span class="o">[</span><span class="n">j</span><span class="o">][</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-55" name="__codelineno-12-55" href="#__codelineno-12-55"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-56" name="__codelineno-12-56" href="#__codelineno-12-56"></a>
-<a id="__codelineno-12-57" name="__codelineno-12-57" href="#__codelineno-12-57"></a><span class="w">  </span><span class="c1">### Delete edge ###</span>
+<a id="__codelineno-12-57" name="__codelineno-12-57" href="#__codelineno-12-57"></a><span class="w">  </span><span class="c1">### Remove an edge ###</span>
 <a id="__codelineno-12-58" name="__codelineno-12-58" href="#__codelineno-12-58"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">remove_edge</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
-<a id="__codelineno-12-59" name="__codelineno-12-59" href="#__codelineno-12-59"></a><span class="w">    </span><span class="c1"># Parameters i, j correspond to the vertices element indices</span>
-<a id="__codelineno-12-60" name="__codelineno-12-60" href="#__codelineno-12-60"></a><span class="w">    </span><span class="c1"># Handle index out of bounds and equality</span>
+<a id="__codelineno-12-59" name="__codelineno-12-59" href="#__codelineno-12-59"></a><span class="w">    </span><span class="c1"># Parameters i and j are element indices in vertices</span>
+<a id="__codelineno-12-60" name="__codelineno-12-60" href="#__codelineno-12-60"></a><span class="w">    </span><span class="c1"># Handle out-of-bounds or equal indices</span>
 <a id="__codelineno-12-61" name="__codelineno-12-61" href="#__codelineno-12-61"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">j</span>
 <a id="__codelineno-12-62" name="__codelineno-12-62" href="#__codelineno-12-62"></a><span class="w">      </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span>
 <a id="__codelineno-12-63" name="__codelineno-12-63" href="#__codelineno-12-63"></a><span class="w">    </span><span class="k">end</span>
@@ -5850,7 +5922,7 @@
 <a id="__codelineno-12-65" name="__codelineno-12-65" href="#__codelineno-12-65"></a><span class="w">    </span><span class="vi">@adj_mat</span><span class="o">[</span><span class="n">j</span><span class="o">][</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-12-66" name="__codelineno-12-66" href="#__codelineno-12-66"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-67" name="__codelineno-12-67" href="#__codelineno-12-67"></a>
-<a id="__codelineno-12-68" name="__codelineno-12-68" href="#__codelineno-12-68"></a><span class="w">  </span><span class="c1">### Print adjacency matrix ###</span>
+<a id="__codelineno-12-68" name="__codelineno-12-68" href="#__codelineno-12-68"></a><span class="w">  </span><span class="c1">### Print the adjacency matrix ###</span>
 <a id="__codelineno-12-69" name="__codelineno-12-69" href="#__codelineno-12-69"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="n">__print__</span>
 <a id="__codelineno-12-70" name="__codelineno-12-70" href="#__codelineno-12-70"></a><span class="w">    </span><span class="nb">puts</span><span class="w"> </span><span class="s2">"Vertex list = </span><span class="si">#{</span><span class="vi">@vertices</span><span class="si">}</span><span class="s2">"</span>
 <a id="__codelineno-12-71" name="__codelineno-12-71" href="#__codelineno-12-71"></a><span class="w">    </span><span class="nb">puts</span><span class="w"> </span><span class="s1">'Adjacency matrix ='</span>
@@ -5861,19 +5933,24 @@
 </div>
 </div>
 </div>
-<h2 id="922-implementation-based-on-adjacency-list">9.2.2 &nbsp; Implementation Based on Adjacency List<a class="headerlink" href="#922-implementation-based-on-adjacency-list" title="Permanent link">&para;</a></h2>
-<p>Given an undirected graph with a total of <span class="arithmatex">\(n\)</span> vertices and <span class="arithmatex">\(m\)</span> edges, the various operations can be implemented as shown in Figure 9-8.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20GraphAdjMat%3A%0A%20%20%20%20%22%22%22Undirected%20graph%20class%20implemented%20with%20an%20adjacency%20matrix%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20vertices%3A%20list%5Bint%5D,%20edges%3A%20list%5Blist%5Bint%5D%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self.vertices%3A%20list%5Bint%5D%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20self.adj_mat%3A%20list%5Blist%5Bint%5D%5D%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20%23%20Add%20a%20vertex%0A%20%20%20%20%20%20%20%20for%20val%20in%20vertices%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_vertex%28val%29%0A%20%20%20%20%20%20%20%20%23%20Add%20an%20edge%0A%20%20%20%20%20%20%20%20for%20e%20in%20edges%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_edge%28e%5B0%5D,%20e%5B1%5D%29%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20number%20of%20vertices%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self.vertices%29%0A%0A%20%20%20%20def%20add_vertex%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Add%20a%20vertex%22%22%22%0A%20%20%20%20%20%20%20%20n%20%3D%20self.size%28%29%0A%20%20%20%20%20%20%20%20%23%20Append%20the%20new%20vertex%20value%20to%20the%20vertex%20list%0A%20%20%20%20%20%20%20%20self.vertices.append%28val%29%0A%20%20%20%20%20%20%20%20%23%20Add%20a%20row%20to%20the%20adjacency%20matrix%0A%20%20%20%20%20%20%20%20new_row%20%3D%20%5B0%5D%20%2A%20n%0A%20%20%20%20%20%20%20%20self.adj_mat.append%28new_row%29%0A%20%20%20%20%20%20%20%20%23%20Add%20a%20column%20to%20the%20adjacency%20matrix%0A%20%20%20%20%20%20%20%20for%20row%20in%20self.adj_mat%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20row.append%280%29%0A%0A%20%20%20%20def%20remove_vertex%28self,%20index%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Remove%20a%20vertex%22%22%22%0A%20%20%20%20%20%20%20%20if%20index%20%3E%3D%20self.size%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%29%0A%20%20%20%20%20%20%20%20%23%20Remove%20the%20vertex%20at%20index%20%60index%60%20from%20the%20vertex%20list%0A%20%20%20%20%20%20%20%20self.vertices.pop%28index%29%0A%20%20%20%20%20%20%20%20%23%20Remove%20row%20%60index%60%20from%20the%20adjacency%20matrix%0A%20%20%20%20%20%20%20%20self.adj_mat.pop%28index%29%0A%20%20%20%20%20%20%20%20%23%20Remove%20column%20%60index%60%20from%20the%20adjacency%20matrix%0A%20%20%20%20%20%20%20%20for%20row%20in%20self.adj_mat%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20row.pop%28index%29%0A%0A%20%20%20%20def%20add_edge%28self,%20i%3A%20int,%20j%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Add%20an%20edge%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Handle%20out-of-bounds%20or%20equal%20indices%0A%20%20%20%20%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%20or%20i%20%3E%3D%20self.size%28%29%20or%20j%20%3E%3D%20self.size%28%29%20or%20i%20%3D%3D%20j%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%29%0A%20%20%20%20%20%20%20%20self.adj_mat%5Bi%5D%5Bj%5D%20%3D%201%0A%20%20%20%20%20%20%20%20self.adj_mat%5Bj%5D%5Bi%5D%20%3D%201%0A%0A%20%20%20%20def%20remove_edge%28self,%20i%3A%20int,%20j%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Remove%20an%20edge%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Handle%20out-of-bounds%20or%20equal%20indices%0A%20%20%20%20%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%20or%20i%20%3E%3D%20self.size%28%29%20or%20j%20%3E%3D%20self.size%28%29%20or%20i%20%3D%3D%20j%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%29%0A%20%20%20%20%20%20%20%20self.adj_mat%5Bi%5D%5Bj%5D%20%3D%200%0A%20%20%20%20%20%20%20%20self.adj_mat%5Bj%5D%5Bi%5D%20%3D%200%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20undirected%20graph%0A%20%20%20%20vertices%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20edges%20%3D%20%5B%5B0,%201%5D,%20%5B0,%203%5D,%20%5B1,%202%5D,%20%5B2,%203%5D,%20%5B2,%204%5D,%20%5B3,%204%5D%5D%0A%20%20%20%20graph%20%3D%20GraphAdjMat%28vertices,%20edges%29%0A%0A%20%20%20%20%23%20Add%20an%20edge%0A%20%20%20%20%23%20Vertices%201%20and%202%20have%20indices%200%20and%202,%20respectively%0A%20%20%20%20graph.add_edge%280,%202%29%0A%0A%20%20%20%20%23%20Remove%20an%20edge%0A%20%20%20%20%23%20Vertices%201%20and%203%20have%20indices%200%20and%201,%20respectively%0A%20%20%20%20graph.remove_edge%280,%201%29%0A%0A%20%20%20%20%23%20Add%20a%20vertex%0A%20%20%20%20graph.add_vertex%286%29%0A%0A%20%20%20%20%23%20Remove%20a%20vertex%0A%20%20%20%20%23%20Vertex%203%20has%20index%201%0A%20%20%20%20graph.remove_vertex%281%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20GraphAdjMat%3A%0A%20%20%20%20%22%22%22Undirected%20graph%20class%20implemented%20with%20an%20adjacency%20matrix%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20vertices%3A%20list%5Bint%5D,%20edges%3A%20list%5Blist%5Bint%5D%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self.vertices%3A%20list%5Bint%5D%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20self.adj_mat%3A%20list%5Blist%5Bint%5D%5D%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20%23%20Add%20a%20vertex%0A%20%20%20%20%20%20%20%20for%20val%20in%20vertices%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_vertex%28val%29%0A%20%20%20%20%20%20%20%20%23%20Add%20an%20edge%0A%20%20%20%20%20%20%20%20for%20e%20in%20edges%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_edge%28e%5B0%5D,%20e%5B1%5D%29%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20number%20of%20vertices%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self.vertices%29%0A%0A%20%20%20%20def%20add_vertex%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Add%20a%20vertex%22%22%22%0A%20%20%20%20%20%20%20%20n%20%3D%20self.size%28%29%0A%20%20%20%20%20%20%20%20%23%20Append%20the%20new%20vertex%20value%20to%20the%20vertex%20list%0A%20%20%20%20%20%20%20%20self.vertices.append%28val%29%0A%20%20%20%20%20%20%20%20%23%20Add%20a%20row%20to%20the%20adjacency%20matrix%0A%20%20%20%20%20%20%20%20new_row%20%3D%20%5B0%5D%20%2A%20n%0A%20%20%20%20%20%20%20%20self.adj_mat.append%28new_row%29%0A%20%20%20%20%20%20%20%20%23%20Add%20a%20column%20to%20the%20adjacency%20matrix%0A%20%20%20%20%20%20%20%20for%20row%20in%20self.adj_mat%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20row.append%280%29%0A%0A%20%20%20%20def%20remove_vertex%28self,%20index%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Remove%20a%20vertex%22%22%22%0A%20%20%20%20%20%20%20%20if%20index%20%3E%3D%20self.size%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%29%0A%20%20%20%20%20%20%20%20%23%20Remove%20the%20vertex%20at%20index%20%60index%60%20from%20the%20vertex%20list%0A%20%20%20%20%20%20%20%20self.vertices.pop%28index%29%0A%20%20%20%20%20%20%20%20%23%20Remove%20row%20%60index%60%20from%20the%20adjacency%20matrix%0A%20%20%20%20%20%20%20%20self.adj_mat.pop%28index%29%0A%20%20%20%20%20%20%20%20%23%20Remove%20column%20%60index%60%20from%20the%20adjacency%20matrix%0A%20%20%20%20%20%20%20%20for%20row%20in%20self.adj_mat%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20row.pop%28index%29%0A%0A%20%20%20%20def%20add_edge%28self,%20i%3A%20int,%20j%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Add%20an%20edge%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Handle%20out-of-bounds%20or%20equal%20indices%0A%20%20%20%20%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%20or%20i%20%3E%3D%20self.size%28%29%20or%20j%20%3E%3D%20self.size%28%29%20or%20i%20%3D%3D%20j%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%29%0A%20%20%20%20%20%20%20%20self.adj_mat%5Bi%5D%5Bj%5D%20%3D%201%0A%20%20%20%20%20%20%20%20self.adj_mat%5Bj%5D%5Bi%5D%20%3D%201%0A%0A%20%20%20%20def%20remove_edge%28self,%20i%3A%20int,%20j%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Remove%20an%20edge%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Handle%20out-of-bounds%20or%20equal%20indices%0A%20%20%20%20%20%20%20%20if%20i%20%3C%200%20or%20j%20%3C%200%20or%20i%20%3E%3D%20self.size%28%29%20or%20j%20%3E%3D%20self.size%28%29%20or%20i%20%3D%3D%20j%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%29%0A%20%20%20%20%20%20%20%20self.adj_mat%5Bi%5D%5Bj%5D%20%3D%200%0A%20%20%20%20%20%20%20%20self.adj_mat%5Bj%5D%5Bi%5D%20%3D%200%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20undirected%20graph%0A%20%20%20%20vertices%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20edges%20%3D%20%5B%5B0,%201%5D,%20%5B0,%203%5D,%20%5B1,%202%5D,%20%5B2,%203%5D,%20%5B2,%204%5D,%20%5B3,%204%5D%5D%0A%20%20%20%20graph%20%3D%20GraphAdjMat%28vertices,%20edges%29%0A%0A%20%20%20%20%23%20Add%20an%20edge%0A%20%20%20%20%23%20Vertices%201%20and%202%20have%20indices%200%20and%202,%20respectively%0A%20%20%20%20graph.add_edge%280,%202%29%0A%0A%20%20%20%20%23%20Remove%20an%20edge%0A%20%20%20%20%23%20Vertices%201%20and%203%20have%20indices%200%20and%201,%20respectively%0A%20%20%20%20graph.remove_edge%280,%201%29%0A%0A%20%20%20%20%23%20Add%20a%20vertex%0A%20%20%20%20graph.add_vertex%286%29%0A%0A%20%20%20%20%23%20Remove%20a%20vertex%0A%20%20%20%20%23%20Vertex%203%20has%20index%201%0A%20%20%20%20graph.remove_vertex%281%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="922-adjacency-list-implementation">9.2.2 &nbsp; Adjacency list implementation<a class="headerlink" href="#922-adjacency-list-implementation" title="Permanent link">&para;</a></h2>
+<p>For an undirected graph with <span class="arithmatex">\(n\)</span> vertices and <span class="arithmatex">\(m\)</span> edges, the operations can be implemented as shown in Figure 9-8.</p>
 <ul>
-<li><strong>Adding an edge</strong>: Add the edge at the end of the corresponding vertex's linked list, using <span class="arithmatex">\(O(1)\)</span> time. Since it is an undirected graph, edges in both directions need to be added simultaneously.</li>
-<li><strong>Removing an edge</strong>: Find and remove the specified edge in the corresponding vertex's linked list, using <span class="arithmatex">\(O(m)\)</span> time. In an undirected graph, edges in both directions need to be removed simultaneously.</li>
-<li><strong>Adding a vertex</strong>: Add a linked list to the adjacency list, with the new vertex as the head node, using <span class="arithmatex">\(O(1)\)</span> time.</li>
-<li><strong>Removing a vertex</strong>: Traverse the entire adjacency list and remove all edges containing the specified vertex, using <span class="arithmatex">\(O(n + m)\)</span> time.</li>
-<li><strong>Initialization</strong>: Create <span class="arithmatex">\(n\)</span> vertices and <span class="arithmatex">\(2m\)</span> edges in the adjacency list, using <span class="arithmatex">\(O(n + m)\)</span> time.</li>
+<li><strong>Add an edge</strong>: Append the edge to the linked list of the corresponding vertex, taking <span class="arithmatex">\(O(1)\)</span> time. Since the graph is undirected, add the edge in both directions.</li>
+<li><strong>Remove an edge</strong>: Find and remove the specified edge in the corresponding vertex's linked list, taking <span class="arithmatex">\(O(m)\)</span> time. For an undirected graph, remove the edge in both directions.</li>
+<li><strong>Add a vertex</strong>: Add a linked list to the adjacency list with the new vertex as its head node, taking <span class="arithmatex">\(O(1)\)</span> time.</li>
+<li><strong>Remove a vertex</strong>: Traverse the entire adjacency list and remove every edge involving the specified vertex, taking <span class="arithmatex">\(O(n + m)\)</span> time.</li>
+<li><strong>Initialize the graph</strong>: Create <span class="arithmatex">\(n\)</span> vertices and <span class="arithmatex">\(2m\)</span> edge entries in the adjacency list, taking <span class="arithmatex">\(O(n + m)\)</span> time.</li>
 </ul>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:5"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">&lt;1&gt;</label><label for="__tabbed_3_2">&lt;2&gt;</label><label for="__tabbed_3_3">&lt;3&gt;</label><label for="__tabbed_3_4">&lt;4&gt;</label><label for="__tabbed_3_5">&lt;5&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Initialization, adding and removing edges, adding and removing vertices in adjacency list" class="animation-figure" src="../graph_operations.assets/adjacency_list_step1_initialization.png" /></p>
+<p><img alt="Initializing an adjacency list and adding or removing edges and vertices" class="animation-figure" src="../graph_operations.assets/adjacency_list_step1_initialization.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="adjacency_list_add_edge" class="animation-figure" src="../graph_operations.assets/adjacency_list_step2_add_edge.png" /></p>
@@ -5889,23 +5966,23 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 9-8 &nbsp; Initialization, adding and removing edges, adding and removing vertices in adjacency list </p>
+<p align="center"> Figure 9-8 &nbsp; Initializing an adjacency list and adding or removing edges and vertices </p>
 
-<p>The following code shows the adjacency list implementation. Compared with Figure 9-8, the actual code differs in the following ways.</p>
+<p>The following code implements an adjacency list, with these differences from Figure 9-8.</p>
 <ul>
-<li>For convenience in adding and removing vertices, and to simplify the code, we use lists (dynamic arrays) instead of linked lists.</li>
-<li>A hash table is used to store the adjacency list, where <code>key</code> is the vertex instance and <code>value</code> is the list (linked list) of adjacent vertices for that vertex.</li>
+<li>To make adding and removing vertices easier and simplify the code, we use lists (dynamic arrays) in place of linked lists.</li>
+<li>A hash table stores the adjacency list. Each <code>key</code> is a vertex instance, and its <code>value</code> is the list (linked list) of adjacent vertices.</li>
 </ul>
-<p>Additionally, we use the <code>Vertex</code> class to represent vertices in the adjacency list for the following reason: if we used list indices to distinguish different vertices, as with adjacency matrices, then to delete the vertex at index <span class="arithmatex">\(i\)</span>, we would need to traverse the entire adjacency list and decrement all indices greater than <span class="arithmatex">\(i\)</span> by <span class="arithmatex">\(1\)</span>, which is very inefficient. However, if each vertex is a unique <code>Vertex</code> instance, deleting one vertex does not require modifying the others.</p>
+<p>We also use the <code>Vertex</code> class to represent vertices in the adjacency list. If we identified vertices by list indices, as in the adjacency matrix, deleting the vertex at index <span class="arithmatex">\(i\)</span> would require traversing the entire adjacency list and subtracting <span class="arithmatex">\(1\)</span> from every index greater than <span class="arithmatex">\(i\)</span>. This would be inefficient. When each vertex is a unique <code>Vertex</code> instance, deleting one vertex requires no changes to the other vertices.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:13"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_adjacency_list.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">GraphAdjList</span><span class="p">:</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Undirected graph class based on adjacency list"""</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Undirected graph class implemented with an adjacency list"""</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">edges</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">list</span><span class="p">[</span><span class="n">Vertex</span><span class="p">]]):</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="w">        </span><span class="sd">"""Constructor"""</span>
-<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>        <span class="c1"># Adjacency list, key: vertex, value: all adjacent vertices of that vertex</span>
+<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>        <span class="c1"># Adjacency list: key is a vertex, value is the list of all its neighbors</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">[</span><span class="n">Vertex</span><span class="p">,</span> <span class="nb">list</span><span class="p">[</span><span class="n">Vertex</span><span class="p">]]()</span>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="c1"># Add all vertices and edges</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>        <span class="k">for</span> <span class="n">edge</span> <span class="ow">in</span> <span class="n">edges</span><span class="p">:</span>
@@ -5918,7 +5995,7 @@
 <a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span><span class="p">)</span>
 <a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>
 <a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">add_edge</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">vet1</span><span class="p">:</span> <span class="n">Vertex</span><span class="p">,</span> <span class="n">vet2</span><span class="p">:</span> <span class="n">Vertex</span><span class="p">):</span>
-<a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a><span class="w">        </span><span class="sd">"""Add edge"""</span>
+<a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a><span class="w">        </span><span class="sd">"""Add an edge"""</span>
 <a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a>        <span class="k">if</span> <span class="n">vet1</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span> <span class="ow">or</span> <span class="n">vet2</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span> <span class="ow">or</span> <span class="n">vet1</span> <span class="o">==</span> <span class="n">vet2</span><span class="p">:</span>
 <a id="__codelineno-13-21" name="__codelineno-13-21" href="#__codelineno-13-21"></a>            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">()</span>
 <a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a>        <span class="c1"># Add edge vet1 - vet2</span>
@@ -5926,7 +6003,7 @@
 <a id="__codelineno-13-24" name="__codelineno-13-24" href="#__codelineno-13-24"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span><span class="p">[</span><span class="n">vet2</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span>
 <a id="__codelineno-13-25" name="__codelineno-13-25" href="#__codelineno-13-25"></a>
 <a id="__codelineno-13-26" name="__codelineno-13-26" href="#__codelineno-13-26"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">remove_edge</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">vet1</span><span class="p">:</span> <span class="n">Vertex</span><span class="p">,</span> <span class="n">vet2</span><span class="p">:</span> <span class="n">Vertex</span><span class="p">):</span>
-<a id="__codelineno-13-27" name="__codelineno-13-27" href="#__codelineno-13-27"></a><span class="w">        </span><span class="sd">"""Remove edge"""</span>
+<a id="__codelineno-13-27" name="__codelineno-13-27" href="#__codelineno-13-27"></a><span class="w">        </span><span class="sd">"""Remove an edge"""</span>
 <a id="__codelineno-13-28" name="__codelineno-13-28" href="#__codelineno-13-28"></a>        <span class="k">if</span> <span class="n">vet1</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span> <span class="ow">or</span> <span class="n">vet2</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span> <span class="ow">or</span> <span class="n">vet1</span> <span class="o">==</span> <span class="n">vet2</span><span class="p">:</span>
 <a id="__codelineno-13-29" name="__codelineno-13-29" href="#__codelineno-13-29"></a>            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">()</span>
 <a id="__codelineno-13-30" name="__codelineno-13-30" href="#__codelineno-13-30"></a>        <span class="c1"># Remove edge vet1 - vet2</span>
@@ -5934,25 +6011,25 @@
 <a id="__codelineno-13-32" name="__codelineno-13-32" href="#__codelineno-13-32"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span><span class="p">[</span><span class="n">vet2</span><span class="p">]</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span>
 <a id="__codelineno-13-33" name="__codelineno-13-33" href="#__codelineno-13-33"></a>
 <a id="__codelineno-13-34" name="__codelineno-13-34" href="#__codelineno-13-34"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">add_vertex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">vet</span><span class="p">:</span> <span class="n">Vertex</span><span class="p">):</span>
-<a id="__codelineno-13-35" name="__codelineno-13-35" href="#__codelineno-13-35"></a><span class="w">        </span><span class="sd">"""Add vertex"""</span>
+<a id="__codelineno-13-35" name="__codelineno-13-35" href="#__codelineno-13-35"></a><span class="w">        </span><span class="sd">"""Add a vertex"""</span>
 <a id="__codelineno-13-36" name="__codelineno-13-36" href="#__codelineno-13-36"></a>        <span class="k">if</span> <span class="n">vet</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span><span class="p">:</span>
 <a id="__codelineno-13-37" name="__codelineno-13-37" href="#__codelineno-13-37"></a>            <span class="k">return</span>
-<a id="__codelineno-13-38" name="__codelineno-13-38" href="#__codelineno-13-38"></a>        <span class="c1"># Add a new linked list in the adjacency list</span>
+<a id="__codelineno-13-38" name="__codelineno-13-38" href="#__codelineno-13-38"></a>        <span class="c1"># Add a new list to the adjacency list</span>
 <a id="__codelineno-13-39" name="__codelineno-13-39" href="#__codelineno-13-39"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span><span class="p">[</span><span class="n">vet</span><span class="p">]</span> <span class="o">=</span> <span class="p">[]</span>
 <a id="__codelineno-13-40" name="__codelineno-13-40" href="#__codelineno-13-40"></a>
 <a id="__codelineno-13-41" name="__codelineno-13-41" href="#__codelineno-13-41"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">remove_vertex</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">vet</span><span class="p">:</span> <span class="n">Vertex</span><span class="p">):</span>
-<a id="__codelineno-13-42" name="__codelineno-13-42" href="#__codelineno-13-42"></a><span class="w">        </span><span class="sd">"""Remove vertex"""</span>
+<a id="__codelineno-13-42" name="__codelineno-13-42" href="#__codelineno-13-42"></a><span class="w">        </span><span class="sd">"""Remove a vertex"""</span>
 <a id="__codelineno-13-43" name="__codelineno-13-43" href="#__codelineno-13-43"></a>        <span class="k">if</span> <span class="n">vet</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span><span class="p">:</span>
 <a id="__codelineno-13-44" name="__codelineno-13-44" href="#__codelineno-13-44"></a>            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">()</span>
-<a id="__codelineno-13-45" name="__codelineno-13-45" href="#__codelineno-13-45"></a>        <span class="c1"># Remove the linked list corresponding to vertex vet in the adjacency list</span>
+<a id="__codelineno-13-45" name="__codelineno-13-45" href="#__codelineno-13-45"></a>        <span class="c1"># Remove the list corresponding to vertex vet from the adjacency list</span>
 <a id="__codelineno-13-46" name="__codelineno-13-46" href="#__codelineno-13-46"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span>
-<a id="__codelineno-13-47" name="__codelineno-13-47" href="#__codelineno-13-47"></a>        <span class="c1"># Traverse the linked lists of other vertices and remove all edges containing vet</span>
+<a id="__codelineno-13-47" name="__codelineno-13-47" href="#__codelineno-13-47"></a>        <span class="c1"># Traverse the lists of the other vertices and remove all edges incident to vet</span>
 <a id="__codelineno-13-48" name="__codelineno-13-48" href="#__codelineno-13-48"></a>        <span class="k">for</span> <span class="n">vertex</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span><span class="p">:</span>
 <a id="__codelineno-13-49" name="__codelineno-13-49" href="#__codelineno-13-49"></a>            <span class="k">if</span> <span class="n">vet</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span><span class="p">[</span><span class="n">vertex</span><span class="p">]:</span>
 <a id="__codelineno-13-50" name="__codelineno-13-50" href="#__codelineno-13-50"></a>                <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span><span class="p">[</span><span class="n">vertex</span><span class="p">]</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span>
 <a id="__codelineno-13-51" name="__codelineno-13-51" href="#__codelineno-13-51"></a>
 <a id="__codelineno-13-52" name="__codelineno-13-52" href="#__codelineno-13-52"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">print</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-<a id="__codelineno-13-53" name="__codelineno-13-53" href="#__codelineno-13-53"></a><span class="w">        </span><span class="sd">"""Print adjacency list"""</span>
+<a id="__codelineno-13-53" name="__codelineno-13-53" href="#__codelineno-13-53"></a><span class="w">        </span><span class="sd">"""Print the adjacency list"""</span>
 <a id="__codelineno-13-54" name="__codelineno-13-54" href="#__codelineno-13-54"></a>        <span class="nb">print</span><span class="p">(</span><span class="s2">"Adjacency list ="</span><span class="p">)</span>
 <a id="__codelineno-13-55" name="__codelineno-13-55" href="#__codelineno-13-55"></a>        <span class="k">for</span> <span class="n">vertex</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span><span class="p">:</span>
 <a id="__codelineno-13-56" name="__codelineno-13-56" href="#__codelineno-13-56"></a>            <span class="n">tmp</span> <span class="o">=</span> <span class="p">[</span><span class="n">v</span><span class="o">.</span><span class="n">val</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">adj_list</span><span class="p">[</span><span class="n">vertex</span><span class="p">]]</span>
@@ -5960,13 +6037,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_list.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Undirected graph class based on adjacency list */</span>
+<div class="highlight"><span class="filename">graph_adjacency_list.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency list */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">GraphAdjList</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">  </span><span class="k">public</span><span class="o">:</span>
-<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="c1">// Adjacency list, key: vertex, value: all adjacent vertices of that vertex</span>
+<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="c1">// Adjacency list: key is a vertex, value is the list of all its neighbors</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="n">unordered_map</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*&gt;&gt;</span><span class="w"> </span><span class="n">adjList</span><span class="p">;</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a>
-<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="cm">/* Remove specified node from vector */</span>
+<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="cm">/* Remove the specified vertex from the vector */</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">vec</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">vec</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">vec</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5991,45 +6068,45 @@
 <a id="__codelineno-14-29" name="__codelineno-14-29" href="#__codelineno-14-29"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">adjList</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
 <a id="__codelineno-14-30" name="__codelineno-14-30" href="#__codelineno-14-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-31" name="__codelineno-14-31" href="#__codelineno-14-31"></a>
-<a id="__codelineno-14-32" name="__codelineno-14-32" href="#__codelineno-14-32"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
+<a id="__codelineno-14-32" name="__codelineno-14-32" href="#__codelineno-14-32"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
 <a id="__codelineno-14-33" name="__codelineno-14-33" href="#__codelineno-14-33"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">addEdge</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet1</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-34" name="__codelineno-14-34" href="#__codelineno-14-34"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="n">count</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="n">count</span><span class="p">(</span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">vet1</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span>
-<a id="__codelineno-14-35" name="__codelineno-14-35" href="#__codelineno-14-35"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">invalid_argument</span><span class="p">(</span><span class="s">"Vertex does not exist"</span><span class="p">);</span>
+<a id="__codelineno-14-35" name="__codelineno-14-35" href="#__codelineno-14-35"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">invalid_argument</span><span class="p">(</span><span class="s">"Invalid vertex"</span><span class="p">);</span>
 <a id="__codelineno-14-36" name="__codelineno-14-36" href="#__codelineno-14-36"></a><span class="w">        </span><span class="c1">// Add edge vet1 - vet2</span>
 <a id="__codelineno-14-37" name="__codelineno-14-37" href="#__codelineno-14-37"></a><span class="w">        </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet1</span><span class="p">].</span><span class="n">push_back</span><span class="p">(</span><span class="n">vet2</span><span class="p">);</span>
 <a id="__codelineno-14-38" name="__codelineno-14-38" href="#__codelineno-14-38"></a><span class="w">        </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet2</span><span class="p">].</span><span class="n">push_back</span><span class="p">(</span><span class="n">vet1</span><span class="p">);</span>
 <a id="__codelineno-14-39" name="__codelineno-14-39" href="#__codelineno-14-39"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-40" name="__codelineno-14-40" href="#__codelineno-14-40"></a>
-<a id="__codelineno-14-41" name="__codelineno-14-41" href="#__codelineno-14-41"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
+<a id="__codelineno-14-41" name="__codelineno-14-41" href="#__codelineno-14-41"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
 <a id="__codelineno-14-42" name="__codelineno-14-42" href="#__codelineno-14-42"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">removeEdge</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet1</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-43" name="__codelineno-14-43" href="#__codelineno-14-43"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="n">count</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="n">count</span><span class="p">(</span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">vet1</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span>
-<a id="__codelineno-14-44" name="__codelineno-14-44" href="#__codelineno-14-44"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">invalid_argument</span><span class="p">(</span><span class="s">"Vertex does not exist"</span><span class="p">);</span>
+<a id="__codelineno-14-44" name="__codelineno-14-44" href="#__codelineno-14-44"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">invalid_argument</span><span class="p">(</span><span class="s">"Invalid vertex"</span><span class="p">);</span>
 <a id="__codelineno-14-45" name="__codelineno-14-45" href="#__codelineno-14-45"></a><span class="w">        </span><span class="c1">// Remove edge vet1 - vet2</span>
 <a id="__codelineno-14-46" name="__codelineno-14-46" href="#__codelineno-14-46"></a><span class="w">        </span><span class="n">remove</span><span class="p">(</span><span class="n">adjList</span><span class="p">[</span><span class="n">vet1</span><span class="p">],</span><span class="w"> </span><span class="n">vet2</span><span class="p">);</span>
 <a id="__codelineno-14-47" name="__codelineno-14-47" href="#__codelineno-14-47"></a><span class="w">        </span><span class="n">remove</span><span class="p">(</span><span class="n">adjList</span><span class="p">[</span><span class="n">vet2</span><span class="p">],</span><span class="w"> </span><span class="n">vet1</span><span class="p">);</span>
 <a id="__codelineno-14-48" name="__codelineno-14-48" href="#__codelineno-14-48"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-49" name="__codelineno-14-49" href="#__codelineno-14-49"></a>
-<a id="__codelineno-14-50" name="__codelineno-14-50" href="#__codelineno-14-50"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-14-50" name="__codelineno-14-50" href="#__codelineno-14-50"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-14-51" name="__codelineno-14-51" href="#__codelineno-14-51"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">addVertex</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-52" name="__codelineno-14-52" href="#__codelineno-14-52"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">adjList</span><span class="p">.</span><span class="n">count</span><span class="p">(</span><span class="n">vet</span><span class="p">))</span>
 <a id="__codelineno-14-53" name="__codelineno-14-53" href="#__codelineno-14-53"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-14-54" name="__codelineno-14-54" href="#__codelineno-14-54"></a><span class="w">        </span><span class="c1">// Add a new linked list in the adjacency list</span>
+<a id="__codelineno-14-54" name="__codelineno-14-54" href="#__codelineno-14-54"></a><span class="w">        </span><span class="c1">// Add a new list to the adjacency list</span>
 <a id="__codelineno-14-55" name="__codelineno-14-55" href="#__codelineno-14-55"></a><span class="w">        </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*&gt;</span><span class="p">();</span>
 <a id="__codelineno-14-56" name="__codelineno-14-56" href="#__codelineno-14-56"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-57" name="__codelineno-14-57" href="#__codelineno-14-57"></a>
-<a id="__codelineno-14-58" name="__codelineno-14-58" href="#__codelineno-14-58"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-14-58" name="__codelineno-14-58" href="#__codelineno-14-58"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-14-59" name="__codelineno-14-59" href="#__codelineno-14-59"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">removeVertex</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-60" name="__codelineno-14-60" href="#__codelineno-14-60"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="n">count</span><span class="p">(</span><span class="n">vet</span><span class="p">))</span>
-<a id="__codelineno-14-61" name="__codelineno-14-61" href="#__codelineno-14-61"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">invalid_argument</span><span class="p">(</span><span class="s">"Vertex does not exist"</span><span class="p">);</span>
-<a id="__codelineno-14-62" name="__codelineno-14-62" href="#__codelineno-14-62"></a><span class="w">        </span><span class="c1">// Remove the linked list corresponding to vertex vet in the adjacency list</span>
+<a id="__codelineno-14-61" name="__codelineno-14-61" href="#__codelineno-14-61"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">invalid_argument</span><span class="p">(</span><span class="s">"Invalid vertex"</span><span class="p">);</span>
+<a id="__codelineno-14-62" name="__codelineno-14-62" href="#__codelineno-14-62"></a><span class="w">        </span><span class="c1">// Remove the list corresponding to vertex vet from the adjacency list</span>
 <a id="__codelineno-14-63" name="__codelineno-14-63" href="#__codelineno-14-63"></a><span class="w">        </span><span class="n">adjList</span><span class="p">.</span><span class="n">erase</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span>
-<a id="__codelineno-14-64" name="__codelineno-14-64" href="#__codelineno-14-64"></a><span class="w">        </span><span class="c1">// Traverse the linked lists of other vertices and remove all edges containing vet</span>
+<a id="__codelineno-14-64" name="__codelineno-14-64" href="#__codelineno-14-64"></a><span class="w">        </span><span class="c1">// Traverse the lists of the other vertices and remove all edges incident to vet</span>
 <a id="__codelineno-14-65" name="__codelineno-14-65" href="#__codelineno-14-65"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="k">auto</span><span class="w"> </span><span class="o">&amp;</span><span class="n">adj</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">adjList</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-66" name="__codelineno-14-66" href="#__codelineno-14-66"></a><span class="w">            </span><span class="n">remove</span><span class="p">(</span><span class="n">adj</span><span class="p">.</span><span class="n">second</span><span class="p">,</span><span class="w"> </span><span class="n">vet</span><span class="p">);</span>
 <a id="__codelineno-14-67" name="__codelineno-14-67" href="#__codelineno-14-67"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-14-68" name="__codelineno-14-68" href="#__codelineno-14-68"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-69" name="__codelineno-14-69" href="#__codelineno-14-69"></a>
-<a id="__codelineno-14-70" name="__codelineno-14-70" href="#__codelineno-14-70"></a><span class="w">    </span><span class="cm">/* Print adjacency list */</span>
+<a id="__codelineno-14-70" name="__codelineno-14-70" href="#__codelineno-14-70"></a><span class="w">    </span><span class="cm">/* Print the adjacency list */</span>
 <a id="__codelineno-14-71" name="__codelineno-14-71" href="#__codelineno-14-71"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-72" name="__codelineno-14-72" href="#__codelineno-14-72"></a><span class="w">        </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">"Adjacency list ="</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
 <a id="__codelineno-14-73" name="__codelineno-14-73" href="#__codelineno-14-73"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="k">auto</span><span class="w"> </span><span class="o">&amp;</span><span class="n">adj</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">adjList</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6043,9 +6120,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_list.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Undirected graph class based on adjacency list */</span>
+<div class="highlight"><span class="filename">graph_adjacency_list.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency list */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kd">class</span> <span class="nc">GraphAdjList</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Adjacency list, key: vertex, value: all adjacent vertices of that vertex</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Adjacency list: key is a vertex, value is the list of all its neighbors</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="n">Map</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">adjList</span><span class="p">;</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -6064,7 +6141,7 @@
 <a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">adjList</span><span class="p">.</span><span class="na">size</span><span class="p">();</span>
 <a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a>
-<a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
+<a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
 <a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">addEdge</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet1</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-24" name="__codelineno-15-24" href="#__codelineno-15-24"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="na">containsKey</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="na">containsKey</span><span class="p">(</span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">vet1</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span>
 <a id="__codelineno-15-25" name="__codelineno-15-25" href="#__codelineno-15-25"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IllegalArgumentException</span><span class="p">();</span>
@@ -6073,7 +6150,7 @@
 <a id="__codelineno-15-28" name="__codelineno-15-28" href="#__codelineno-15-28"></a><span class="w">        </span><span class="n">adjList</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">vet2</span><span class="p">).</span><span class="na">add</span><span class="p">(</span><span class="n">vet1</span><span class="p">);</span>
 <a id="__codelineno-15-29" name="__codelineno-15-29" href="#__codelineno-15-29"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-30" name="__codelineno-15-30" href="#__codelineno-15-30"></a>
-<a id="__codelineno-15-31" name="__codelineno-15-31" href="#__codelineno-15-31"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
+<a id="__codelineno-15-31" name="__codelineno-15-31" href="#__codelineno-15-31"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
 <a id="__codelineno-15-32" name="__codelineno-15-32" href="#__codelineno-15-32"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">removeEdge</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet1</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-33" name="__codelineno-15-33" href="#__codelineno-15-33"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="na">containsKey</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="na">containsKey</span><span class="p">(</span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">vet1</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span>
 <a id="__codelineno-15-34" name="__codelineno-15-34" href="#__codelineno-15-34"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IllegalArgumentException</span><span class="p">();</span>
@@ -6082,27 +6159,27 @@
 <a id="__codelineno-15-37" name="__codelineno-15-37" href="#__codelineno-15-37"></a><span class="w">        </span><span class="n">adjList</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">vet2</span><span class="p">).</span><span class="na">remove</span><span class="p">(</span><span class="n">vet1</span><span class="p">);</span>
 <a id="__codelineno-15-38" name="__codelineno-15-38" href="#__codelineno-15-38"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-39" name="__codelineno-15-39" href="#__codelineno-15-39"></a>
-<a id="__codelineno-15-40" name="__codelineno-15-40" href="#__codelineno-15-40"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-15-40" name="__codelineno-15-40" href="#__codelineno-15-40"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-15-41" name="__codelineno-15-41" href="#__codelineno-15-41"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">addVertex</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-42" name="__codelineno-15-42" href="#__codelineno-15-42"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">adjList</span><span class="p">.</span><span class="na">containsKey</span><span class="p">(</span><span class="n">vet</span><span class="p">))</span>
 <a id="__codelineno-15-43" name="__codelineno-15-43" href="#__codelineno-15-43"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-15-44" name="__codelineno-15-44" href="#__codelineno-15-44"></a><span class="w">        </span><span class="c1">// Add a new linked list in the adjacency list</span>
+<a id="__codelineno-15-44" name="__codelineno-15-44" href="#__codelineno-15-44"></a><span class="w">        </span><span class="c1">// Add a new list to the adjacency list</span>
 <a id="__codelineno-15-45" name="__codelineno-15-45" href="#__codelineno-15-45"></a><span class="w">        </span><span class="n">adjList</span><span class="p">.</span><span class="na">put</span><span class="p">(</span><span class="n">vet</span><span class="p">,</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">());</span>
 <a id="__codelineno-15-46" name="__codelineno-15-46" href="#__codelineno-15-46"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-47" name="__codelineno-15-47" href="#__codelineno-15-47"></a>
-<a id="__codelineno-15-48" name="__codelineno-15-48" href="#__codelineno-15-48"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-15-48" name="__codelineno-15-48" href="#__codelineno-15-48"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-15-49" name="__codelineno-15-49" href="#__codelineno-15-49"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">removeVertex</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-50" name="__codelineno-15-50" href="#__codelineno-15-50"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="na">containsKey</span><span class="p">(</span><span class="n">vet</span><span class="p">))</span>
 <a id="__codelineno-15-51" name="__codelineno-15-51" href="#__codelineno-15-51"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IllegalArgumentException</span><span class="p">();</span>
-<a id="__codelineno-15-52" name="__codelineno-15-52" href="#__codelineno-15-52"></a><span class="w">        </span><span class="c1">// Remove the linked list corresponding to vertex vet in the adjacency list</span>
+<a id="__codelineno-15-52" name="__codelineno-15-52" href="#__codelineno-15-52"></a><span class="w">        </span><span class="c1">// Remove the list corresponding to vertex vet from the adjacency list</span>
 <a id="__codelineno-15-53" name="__codelineno-15-53" href="#__codelineno-15-53"></a><span class="w">        </span><span class="n">adjList</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span>
-<a id="__codelineno-15-54" name="__codelineno-15-54" href="#__codelineno-15-54"></a><span class="w">        </span><span class="c1">// Traverse the linked lists of other vertices and remove all edges containing vet</span>
+<a id="__codelineno-15-54" name="__codelineno-15-54" href="#__codelineno-15-54"></a><span class="w">        </span><span class="c1">// Traverse the lists of the other vertices and remove all edges incident to vet</span>
 <a id="__codelineno-15-55" name="__codelineno-15-55" href="#__codelineno-15-55"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">list</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">adjList</span><span class="p">.</span><span class="na">values</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-56" name="__codelineno-15-56" href="#__codelineno-15-56"></a><span class="w">            </span><span class="n">list</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span>
 <a id="__codelineno-15-57" name="__codelineno-15-57" href="#__codelineno-15-57"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-15-58" name="__codelineno-15-58" href="#__codelineno-15-58"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-59" name="__codelineno-15-59" href="#__codelineno-15-59"></a>
-<a id="__codelineno-15-60" name="__codelineno-15-60" href="#__codelineno-15-60"></a><span class="w">    </span><span class="cm">/* Print adjacency list */</span>
+<a id="__codelineno-15-60" name="__codelineno-15-60" href="#__codelineno-15-60"></a><span class="w">    </span><span class="cm">/* Print the adjacency list */</span>
 <a id="__codelineno-15-61" name="__codelineno-15-61" href="#__codelineno-15-61"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-62" name="__codelineno-15-62" href="#__codelineno-15-62"></a><span class="w">        </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="s">"Adjacency list ="</span><span class="p">);</span>
 <a id="__codelineno-15-63" name="__codelineno-15-63" href="#__codelineno-15-63"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Map</span><span class="p">.</span><span class="na">Entry</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">adjList</span><span class="p">.</span><span class="na">entrySet</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
@@ -6116,9 +6193,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_list.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Undirected graph class based on adjacency list */</span>
+<div class="highlight"><span class="filename">graph_adjacency_list.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency list */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">GraphAdjList</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// Adjacency list, key: vertex, value: all adjacent vertices of that vertex</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// Adjacency list: key is a vertex, value is the list of all its neighbors</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="n">Dictionary</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">adjList</span><span class="p">;</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -6137,7 +6214,7 @@
 <a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">adjList</span><span class="p">.</span><span class="n">Count</span><span class="p">;</span>
 <a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a>
-<a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
+<a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
 <a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">AddEdge</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet1</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-24" name="__codelineno-16-24" href="#__codelineno-16-24"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="n">ContainsKey</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="n">ContainsKey</span><span class="p">(</span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">vet1</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span>
 <a id="__codelineno-16-25" name="__codelineno-16-25" href="#__codelineno-16-25"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">InvalidOperationException</span><span class="p">();</span>
@@ -6146,7 +6223,7 @@
 <a id="__codelineno-16-28" name="__codelineno-16-28" href="#__codelineno-16-28"></a><span class="w">        </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet2</span><span class="p">].</span><span class="n">Add</span><span class="p">(</span><span class="n">vet1</span><span class="p">);</span>
 <a id="__codelineno-16-29" name="__codelineno-16-29" href="#__codelineno-16-29"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-30" name="__codelineno-16-30" href="#__codelineno-16-30"></a>
-<a id="__codelineno-16-31" name="__codelineno-16-31" href="#__codelineno-16-31"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
+<a id="__codelineno-16-31" name="__codelineno-16-31" href="#__codelineno-16-31"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
 <a id="__codelineno-16-32" name="__codelineno-16-32" href="#__codelineno-16-32"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">RemoveEdge</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet1</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-33" name="__codelineno-16-33" href="#__codelineno-16-33"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="n">ContainsKey</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="n">ContainsKey</span><span class="p">(</span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">vet1</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span>
 <a id="__codelineno-16-34" name="__codelineno-16-34" href="#__codelineno-16-34"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">InvalidOperationException</span><span class="p">();</span>
@@ -6155,27 +6232,27 @@
 <a id="__codelineno-16-37" name="__codelineno-16-37" href="#__codelineno-16-37"></a><span class="w">        </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet2</span><span class="p">].</span><span class="n">Remove</span><span class="p">(</span><span class="n">vet1</span><span class="p">);</span>
 <a id="__codelineno-16-38" name="__codelineno-16-38" href="#__codelineno-16-38"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-39" name="__codelineno-16-39" href="#__codelineno-16-39"></a>
-<a id="__codelineno-16-40" name="__codelineno-16-40" href="#__codelineno-16-40"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-16-40" name="__codelineno-16-40" href="#__codelineno-16-40"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-16-41" name="__codelineno-16-41" href="#__codelineno-16-41"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">AddVertex</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-42" name="__codelineno-16-42" href="#__codelineno-16-42"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">adjList</span><span class="p">.</span><span class="n">ContainsKey</span><span class="p">(</span><span class="n">vet</span><span class="p">))</span>
 <a id="__codelineno-16-43" name="__codelineno-16-43" href="#__codelineno-16-43"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-16-44" name="__codelineno-16-44" href="#__codelineno-16-44"></a><span class="w">        </span><span class="c1">// Add a new linked list in the adjacency list</span>
+<a id="__codelineno-16-44" name="__codelineno-16-44" href="#__codelineno-16-44"></a><span class="w">        </span><span class="c1">// Add a new list to the adjacency list</span>
 <a id="__codelineno-16-45" name="__codelineno-16-45" href="#__codelineno-16-45"></a><span class="w">        </span><span class="n">adjList</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">vet</span><span class="p">,</span><span class="w"> </span><span class="p">[]);</span>
 <a id="__codelineno-16-46" name="__codelineno-16-46" href="#__codelineno-16-46"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-47" name="__codelineno-16-47" href="#__codelineno-16-47"></a>
-<a id="__codelineno-16-48" name="__codelineno-16-48" href="#__codelineno-16-48"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-16-48" name="__codelineno-16-48" href="#__codelineno-16-48"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-16-49" name="__codelineno-16-49" href="#__codelineno-16-49"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">RemoveVertex</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-50" name="__codelineno-16-50" href="#__codelineno-16-50"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="n">ContainsKey</span><span class="p">(</span><span class="n">vet</span><span class="p">))</span>
 <a id="__codelineno-16-51" name="__codelineno-16-51" href="#__codelineno-16-51"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">InvalidOperationException</span><span class="p">();</span>
-<a id="__codelineno-16-52" name="__codelineno-16-52" href="#__codelineno-16-52"></a><span class="w">        </span><span class="c1">// Remove the linked list corresponding to vertex vet in the adjacency list</span>
+<a id="__codelineno-16-52" name="__codelineno-16-52" href="#__codelineno-16-52"></a><span class="w">        </span><span class="c1">// Remove the list corresponding to vertex vet from the adjacency list</span>
 <a id="__codelineno-16-53" name="__codelineno-16-53" href="#__codelineno-16-53"></a><span class="w">        </span><span class="n">adjList</span><span class="p">.</span><span class="n">Remove</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span>
-<a id="__codelineno-16-54" name="__codelineno-16-54" href="#__codelineno-16-54"></a><span class="w">        </span><span class="c1">// Traverse the linked lists of other vertices and remove all edges containing vet</span>
+<a id="__codelineno-16-54" name="__codelineno-16-54" href="#__codelineno-16-54"></a><span class="w">        </span><span class="c1">// Traverse the lists of the other vertices and remove all edges incident to vet</span>
 <a id="__codelineno-16-55" name="__codelineno-16-55" href="#__codelineno-16-55"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">list</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">adjList</span><span class="p">.</span><span class="n">Values</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-56" name="__codelineno-16-56" href="#__codelineno-16-56"></a><span class="w">            </span><span class="n">list</span><span class="p">.</span><span class="n">Remove</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span>
 <a id="__codelineno-16-57" name="__codelineno-16-57" href="#__codelineno-16-57"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-16-58" name="__codelineno-16-58" href="#__codelineno-16-58"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-59" name="__codelineno-16-59" href="#__codelineno-16-59"></a>
-<a id="__codelineno-16-60" name="__codelineno-16-60" href="#__codelineno-16-60"></a><span class="w">    </span><span class="cm">/* Print adjacency list */</span>
+<a id="__codelineno-16-60" name="__codelineno-16-60" href="#__codelineno-16-60"></a><span class="w">    </span><span class="cm">/* Print the adjacency list */</span>
 <a id="__codelineno-16-61" name="__codelineno-16-61" href="#__codelineno-16-61"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">Print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-62" name="__codelineno-16-62" href="#__codelineno-16-62"></a><span class="w">        </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="s">"Adjacency list ="</span><span class="p">);</span>
 <a id="__codelineno-16-63" name="__codelineno-16-63" href="#__codelineno-16-63"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">KeyValuePair</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">adjList</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6189,9 +6266,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_list.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Undirected graph class based on adjacency list */</span>
+<div class="highlight"><span class="filename">graph_adjacency_list.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Undirected graph type implemented with an adjacency list */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">graphAdjList</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Adjacency list, key: vertex, value: all adjacent vertices of that vertex</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Adjacency list: key is a vertex, value is the list of all its neighbors</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="nx">adjList</span><span class="w"> </span><span class="kt">map</span><span class="p">[</span><span class="nx">Vertex</span><span class="p">][]</span><span class="nx">Vertex</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="p">}</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a>
@@ -6214,19 +6291,19 @@
 <a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">)</span>
 <a id="__codelineno-17-24" name="__codelineno-17-24" href="#__codelineno-17-24"></a><span class="p">}</span>
 <a id="__codelineno-17-25" name="__codelineno-17-25" href="#__codelineno-17-25"></a>
-<a id="__codelineno-17-26" name="__codelineno-17-26" href="#__codelineno-17-26"></a><span class="cm">/* Add edge */</span>
+<a id="__codelineno-17-26" name="__codelineno-17-26" href="#__codelineno-17-26"></a><span class="cm">/* Add an edge */</span>
 <a id="__codelineno-17-27" name="__codelineno-17-27" href="#__codelineno-17-27"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">g</span><span class="w"> </span><span class="o">*</span><span class="nx">graphAdjList</span><span class="p">)</span><span class="w"> </span><span class="nx">addEdge</span><span class="p">(</span><span class="nx">vet1</span><span class="w"> </span><span class="nx">Vertex</span><span class="p">,</span><span class="w"> </span><span class="nx">vet2</span><span class="w"> </span><span class="nx">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-28" name="__codelineno-17-28" href="#__codelineno-17-28"></a><span class="w">    </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">ok1</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">vet1</span><span class="p">]</span>
 <a id="__codelineno-17-29" name="__codelineno-17-29" href="#__codelineno-17-29"></a><span class="w">    </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">ok2</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">vet2</span><span class="p">]</span>
 <a id="__codelineno-17-30" name="__codelineno-17-30" href="#__codelineno-17-30"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="nx">ok1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="o">!</span><span class="nx">ok2</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">vet1</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">vet2</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-31" name="__codelineno-17-31" href="#__codelineno-17-31"></a><span class="w">        </span><span class="nb">panic</span><span class="p">(</span><span class="s">"error"</span><span class="p">)</span>
 <a id="__codelineno-17-32" name="__codelineno-17-32" href="#__codelineno-17-32"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-33" name="__codelineno-17-33" href="#__codelineno-17-33"></a><span class="w">    </span><span class="c1">// Add edge vet1 - vet2, add anonymous struct{},</span>
+<a id="__codelineno-17-33" name="__codelineno-17-33" href="#__codelineno-17-33"></a><span class="w">    </span><span class="c1">// Add edge vet1 - vet2 by appending each vertex to the other vertex's neighbor list</span>
 <a id="__codelineno-17-34" name="__codelineno-17-34" href="#__codelineno-17-34"></a><span class="w">    </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">vet1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">vet1</span><span class="p">],</span><span class="w"> </span><span class="nx">vet2</span><span class="p">)</span>
 <a id="__codelineno-17-35" name="__codelineno-17-35" href="#__codelineno-17-35"></a><span class="w">    </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">vet2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">vet2</span><span class="p">],</span><span class="w"> </span><span class="nx">vet1</span><span class="p">)</span>
 <a id="__codelineno-17-36" name="__codelineno-17-36" href="#__codelineno-17-36"></a><span class="p">}</span>
 <a id="__codelineno-17-37" name="__codelineno-17-37" href="#__codelineno-17-37"></a>
-<a id="__codelineno-17-38" name="__codelineno-17-38" href="#__codelineno-17-38"></a><span class="cm">/* Remove edge */</span>
+<a id="__codelineno-17-38" name="__codelineno-17-38" href="#__codelineno-17-38"></a><span class="cm">/* Remove an edge */</span>
 <a id="__codelineno-17-39" name="__codelineno-17-39" href="#__codelineno-17-39"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">g</span><span class="w"> </span><span class="o">*</span><span class="nx">graphAdjList</span><span class="p">)</span><span class="w"> </span><span class="nx">removeEdge</span><span class="p">(</span><span class="nx">vet1</span><span class="w"> </span><span class="nx">Vertex</span><span class="p">,</span><span class="w"> </span><span class="nx">vet2</span><span class="w"> </span><span class="nx">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-40" name="__codelineno-17-40" href="#__codelineno-17-40"></a><span class="w">    </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">ok1</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">vet1</span><span class="p">]</span>
 <a id="__codelineno-17-41" name="__codelineno-17-41" href="#__codelineno-17-41"></a><span class="w">    </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">ok2</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">vet2</span><span class="p">]</span>
@@ -6238,31 +6315,31 @@
 <a id="__codelineno-17-47" name="__codelineno-17-47" href="#__codelineno-17-47"></a><span class="w">    </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">vet2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">DeleteSliceElms</span><span class="p">(</span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">vet2</span><span class="p">],</span><span class="w"> </span><span class="nx">vet1</span><span class="p">)</span>
 <a id="__codelineno-17-48" name="__codelineno-17-48" href="#__codelineno-17-48"></a><span class="p">}</span>
 <a id="__codelineno-17-49" name="__codelineno-17-49" href="#__codelineno-17-49"></a>
-<a id="__codelineno-17-50" name="__codelineno-17-50" href="#__codelineno-17-50"></a><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-17-50" name="__codelineno-17-50" href="#__codelineno-17-50"></a><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-17-51" name="__codelineno-17-51" href="#__codelineno-17-51"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">g</span><span class="w"> </span><span class="o">*</span><span class="nx">graphAdjList</span><span class="p">)</span><span class="w"> </span><span class="nx">addVertex</span><span class="p">(</span><span class="nx">vet</span><span class="w"> </span><span class="nx">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-52" name="__codelineno-17-52" href="#__codelineno-17-52"></a><span class="w">    </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">ok</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">vet</span><span class="p">]</span>
 <a id="__codelineno-17-53" name="__codelineno-17-53" href="#__codelineno-17-53"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">ok</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-54" name="__codelineno-17-54" href="#__codelineno-17-54"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-17-55" name="__codelineno-17-55" href="#__codelineno-17-55"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-56" name="__codelineno-17-56" href="#__codelineno-17-56"></a><span class="w">    </span><span class="c1">// Add a new linked list in the adjacency list</span>
+<a id="__codelineno-17-56" name="__codelineno-17-56" href="#__codelineno-17-56"></a><span class="w">    </span><span class="c1">// Add a new list to the adjacency list</span>
 <a id="__codelineno-17-57" name="__codelineno-17-57" href="#__codelineno-17-57"></a><span class="w">    </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">vet</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="nx">Vertex</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-17-58" name="__codelineno-17-58" href="#__codelineno-17-58"></a><span class="p">}</span>
 <a id="__codelineno-17-59" name="__codelineno-17-59" href="#__codelineno-17-59"></a>
-<a id="__codelineno-17-60" name="__codelineno-17-60" href="#__codelineno-17-60"></a><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-17-60" name="__codelineno-17-60" href="#__codelineno-17-60"></a><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-17-61" name="__codelineno-17-61" href="#__codelineno-17-61"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">g</span><span class="w"> </span><span class="o">*</span><span class="nx">graphAdjList</span><span class="p">)</span><span class="w"> </span><span class="nx">removeVertex</span><span class="p">(</span><span class="nx">vet</span><span class="w"> </span><span class="nx">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-62" name="__codelineno-17-62" href="#__codelineno-17-62"></a><span class="w">    </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">ok</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">vet</span><span class="p">]</span>
 <a id="__codelineno-17-63" name="__codelineno-17-63" href="#__codelineno-17-63"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="nx">ok</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-64" name="__codelineno-17-64" href="#__codelineno-17-64"></a><span class="w">        </span><span class="nb">panic</span><span class="p">(</span><span class="s">"error"</span><span class="p">)</span>
 <a id="__codelineno-17-65" name="__codelineno-17-65" href="#__codelineno-17-65"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-66" name="__codelineno-17-66" href="#__codelineno-17-66"></a><span class="w">    </span><span class="c1">// Remove the linked list corresponding to vertex vet in the adjacency list</span>
+<a id="__codelineno-17-66" name="__codelineno-17-66" href="#__codelineno-17-66"></a><span class="w">    </span><span class="c1">// Remove the list corresponding to vertex vet from the adjacency list</span>
 <a id="__codelineno-17-67" name="__codelineno-17-67" href="#__codelineno-17-67"></a><span class="w">    </span><span class="nb">delete</span><span class="p">(</span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">,</span><span class="w"> </span><span class="nx">vet</span><span class="p">)</span>
-<a id="__codelineno-17-68" name="__codelineno-17-68" href="#__codelineno-17-68"></a><span class="w">    </span><span class="c1">// Traverse the linked lists of other vertices and remove all edges containing vet</span>
+<a id="__codelineno-17-68" name="__codelineno-17-68" href="#__codelineno-17-68"></a><span class="w">    </span><span class="c1">// Traverse the lists of the other vertices and remove all edges incident to vet</span>
 <a id="__codelineno-17-69" name="__codelineno-17-69" href="#__codelineno-17-69"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">v</span><span class="p">,</span><span class="w"> </span><span class="nx">list</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-70" name="__codelineno-17-70" href="#__codelineno-17-70"></a><span class="w">        </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">v</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">DeleteSliceElms</span><span class="p">(</span><span class="nx">list</span><span class="p">,</span><span class="w"> </span><span class="nx">vet</span><span class="p">)</span>
 <a id="__codelineno-17-71" name="__codelineno-17-71" href="#__codelineno-17-71"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-72" name="__codelineno-17-72" href="#__codelineno-17-72"></a><span class="p">}</span>
 <a id="__codelineno-17-73" name="__codelineno-17-73" href="#__codelineno-17-73"></a>
-<a id="__codelineno-17-74" name="__codelineno-17-74" href="#__codelineno-17-74"></a><span class="cm">/* Print adjacency list */</span>
+<a id="__codelineno-17-74" name="__codelineno-17-74" href="#__codelineno-17-74"></a><span class="cm">/* Print the adjacency list */</span>
 <a id="__codelineno-17-75" name="__codelineno-17-75" href="#__codelineno-17-75"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">g</span><span class="w"> </span><span class="o">*</span><span class="nx">graphAdjList</span><span class="p">)</span><span class="w"> </span><span class="nb">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-76" name="__codelineno-17-76" href="#__codelineno-17-76"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nx">builder</span><span class="w"> </span><span class="nx">strings</span><span class="p">.</span><span class="nx">Builder</span>
 <a id="__codelineno-17-77" name="__codelineno-17-77" href="#__codelineno-17-77"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Printf</span><span class="p">(</span><span class="s">"Adjacency list = \n"</span><span class="p">)</span>
@@ -6278,9 +6355,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_list.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Undirected graph class based on adjacency list */</span>
+<div class="highlight"><span class="filename">graph_adjacency_list.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency list */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">GraphAdjList</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Adjacency list, key: vertex, value: all adjacent vertices of that vertex</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Adjacency list: key is a vertex, value is the list of all its neighbors</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kd">private</span><span class="p">(</span><span class="kr">set</span><span class="p">)</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">adjList</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="n">Vertex</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="n">Vertex</span><span class="p">]]</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -6299,49 +6376,49 @@
 <a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">        </span><span class="n">adjList</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a>
-<a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
+<a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
 <a id="__codelineno-18-23" name="__codelineno-18-23" href="#__codelineno-18-23"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kd">func</span><span class="w"> </span><span class="nf">addEdge</span><span class="p">(</span><span class="n">vet1</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">,</span><span class="w"> </span><span class="n">vet2</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-24" name="__codelineno-18-24" href="#__codelineno-18-24"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet1</span><span class="p">]</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet2</span><span class="p">]</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">vet1</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">vet2</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-25" name="__codelineno-18-25" href="#__codelineno-18-25"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Invalid parameter"</span><span class="p">)</span>
+<a id="__codelineno-18-25" name="__codelineno-18-25" href="#__codelineno-18-25"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Invalid arguments"</span><span class="p">)</span>
 <a id="__codelineno-18-26" name="__codelineno-18-26" href="#__codelineno-18-26"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-18-27" name="__codelineno-18-27" href="#__codelineno-18-27"></a><span class="w">        </span><span class="c1">// Add edge vet1 - vet2</span>
 <a id="__codelineno-18-28" name="__codelineno-18-28" href="#__codelineno-18-28"></a><span class="w">        </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet1</span><span class="p">]?.</span><span class="n">append</span><span class="p">(</span><span class="n">vet2</span><span class="p">)</span>
 <a id="__codelineno-18-29" name="__codelineno-18-29" href="#__codelineno-18-29"></a><span class="w">        </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet2</span><span class="p">]?.</span><span class="n">append</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span>
 <a id="__codelineno-18-30" name="__codelineno-18-30" href="#__codelineno-18-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-31" name="__codelineno-18-31" href="#__codelineno-18-31"></a>
-<a id="__codelineno-18-32" name="__codelineno-18-32" href="#__codelineno-18-32"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
+<a id="__codelineno-18-32" name="__codelineno-18-32" href="#__codelineno-18-32"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
 <a id="__codelineno-18-33" name="__codelineno-18-33" href="#__codelineno-18-33"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kd">func</span><span class="w"> </span><span class="nf">removeEdge</span><span class="p">(</span><span class="n">vet1</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">,</span><span class="w"> </span><span class="n">vet2</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-34" name="__codelineno-18-34" href="#__codelineno-18-34"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet1</span><span class="p">]</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet2</span><span class="p">]</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">vet1</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">vet2</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-35" name="__codelineno-18-35" href="#__codelineno-18-35"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Invalid parameter"</span><span class="p">)</span>
+<a id="__codelineno-18-35" name="__codelineno-18-35" href="#__codelineno-18-35"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Invalid arguments"</span><span class="p">)</span>
 <a id="__codelineno-18-36" name="__codelineno-18-36" href="#__codelineno-18-36"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-18-37" name="__codelineno-18-37" href="#__codelineno-18-37"></a><span class="w">        </span><span class="c1">// Remove edge vet1 - vet2</span>
 <a id="__codelineno-18-38" name="__codelineno-18-38" href="#__codelineno-18-38"></a><span class="w">        </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet1</span><span class="p">]?.</span><span class="bp">removeAll</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nv">$0</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">vet2</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-18-39" name="__codelineno-18-39" href="#__codelineno-18-39"></a><span class="w">        </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet2</span><span class="p">]?.</span><span class="bp">removeAll</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nv">$0</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">vet1</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-18-40" name="__codelineno-18-40" href="#__codelineno-18-40"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-41" name="__codelineno-18-41" href="#__codelineno-18-41"></a>
-<a id="__codelineno-18-42" name="__codelineno-18-42" href="#__codelineno-18-42"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-18-42" name="__codelineno-18-42" href="#__codelineno-18-42"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-18-43" name="__codelineno-18-43" href="#__codelineno-18-43"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kd">func</span><span class="w"> </span><span class="nf">addVertex</span><span class="p">(</span><span class="n">vet</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-44" name="__codelineno-18-44" href="#__codelineno-18-44"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-45" name="__codelineno-18-45" href="#__codelineno-18-45"></a><span class="w">            </span><span class="k">return</span>
 <a id="__codelineno-18-46" name="__codelineno-18-46" href="#__codelineno-18-46"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-47" name="__codelineno-18-47" href="#__codelineno-18-47"></a><span class="w">        </span><span class="c1">// Add a new linked list in the adjacency list</span>
+<a id="__codelineno-18-47" name="__codelineno-18-47" href="#__codelineno-18-47"></a><span class="w">        </span><span class="c1">// Add a new list to the adjacency list</span>
 <a id="__codelineno-18-48" name="__codelineno-18-48" href="#__codelineno-18-48"></a><span class="w">        </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
 <a id="__codelineno-18-49" name="__codelineno-18-49" href="#__codelineno-18-49"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-50" name="__codelineno-18-50" href="#__codelineno-18-50"></a>
-<a id="__codelineno-18-51" name="__codelineno-18-51" href="#__codelineno-18-51"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-18-51" name="__codelineno-18-51" href="#__codelineno-18-51"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-18-52" name="__codelineno-18-52" href="#__codelineno-18-52"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kd">func</span><span class="w"> </span><span class="nf">removeVertex</span><span class="p">(</span><span class="n">vet</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-53" name="__codelineno-18-53" href="#__codelineno-18-53"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet</span><span class="p">]</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-54" name="__codelineno-18-54" href="#__codelineno-18-54"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Invalid parameter"</span><span class="p">)</span>
+<a id="__codelineno-18-54" name="__codelineno-18-54" href="#__codelineno-18-54"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Invalid arguments"</span><span class="p">)</span>
 <a id="__codelineno-18-55" name="__codelineno-18-55" href="#__codelineno-18-55"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-56" name="__codelineno-18-56" href="#__codelineno-18-56"></a><span class="w">        </span><span class="c1">// Remove the linked list corresponding to vertex vet in the adjacency list</span>
+<a id="__codelineno-18-56" name="__codelineno-18-56" href="#__codelineno-18-56"></a><span class="w">        </span><span class="c1">// Remove the list corresponding to vertex vet from the adjacency list</span>
 <a id="__codelineno-18-57" name="__codelineno-18-57" href="#__codelineno-18-57"></a><span class="w">        </span><span class="n">adjList</span><span class="p">.</span><span class="n">removeValue</span><span class="p">(</span><span class="n">forKey</span><span class="p">:</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span>
-<a id="__codelineno-18-58" name="__codelineno-18-58" href="#__codelineno-18-58"></a><span class="w">        </span><span class="c1">// Traverse the linked lists of other vertices and remove all edges containing vet</span>
+<a id="__codelineno-18-58" name="__codelineno-18-58" href="#__codelineno-18-58"></a><span class="w">        </span><span class="c1">// Traverse the lists of the other vertices and remove all edges incident to vet</span>
 <a id="__codelineno-18-59" name="__codelineno-18-59" href="#__codelineno-18-59"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">key</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">adjList</span><span class="p">.</span><span class="n">keys</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-60" name="__codelineno-18-60" href="#__codelineno-18-60"></a><span class="w">            </span><span class="n">adjList</span><span class="p">[</span><span class="n">key</span><span class="p">]?.</span><span class="bp">removeAll</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nv">$0</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">vet</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-18-61" name="__codelineno-18-61" href="#__codelineno-18-61"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-18-62" name="__codelineno-18-62" href="#__codelineno-18-62"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-63" name="__codelineno-18-63" href="#__codelineno-18-63"></a>
-<a id="__codelineno-18-64" name="__codelineno-18-64" href="#__codelineno-18-64"></a><span class="w">    </span><span class="cm">/* Print adjacency list */</span>
+<a id="__codelineno-18-64" name="__codelineno-18-64" href="#__codelineno-18-64"></a><span class="w">    </span><span class="cm">/* Print the adjacency list */</span>
 <a id="__codelineno-18-65" name="__codelineno-18-65" href="#__codelineno-18-65"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kd">func</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-66" name="__codelineno-18-66" href="#__codelineno-18-66"></a><span class="w">        </span><span class="n">Swift</span><span class="p">.</span><span class="bp">print</span><span class="p">(</span><span class="s">"Adjacency list ="</span><span class="p">)</span>
 <a id="__codelineno-18-67" name="__codelineno-18-67" href="#__codelineno-18-67"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">vertex</span><span class="p">,</span><span class="w"> </span><span class="n">list</span><span class="p">)</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">adjList</span><span class="w"> </span><span class="p">{</span>
@@ -6353,9 +6430,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_list.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Undirected graph class based on adjacency list */</span>
+<div class="highlight"><span class="filename">graph_adjacency_list.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency list */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">GraphAdjList</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Adjacency list, key: vertex, value: all adjacent vertices of that vertex</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Adjacency list: key is a vertex, value is the list of all its neighbors</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="nx">adjList</span><span class="p">;</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -6374,21 +6451,21 @@
 <a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">size</span><span class="p">;</span>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a>
-<a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
+<a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
 <a id="__codelineno-19-23" name="__codelineno-19-23" href="#__codelineno-19-23"></a><span class="w">    </span><span class="nx">addEdge</span><span class="p">(</span><span class="nx">vet1</span><span class="p">,</span><span class="w"> </span><span class="nx">vet2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-24" name="__codelineno-19-24" href="#__codelineno-19-24"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span>
 <a id="__codelineno-19-25" name="__codelineno-19-25" href="#__codelineno-19-25"></a><span class="w">            </span><span class="o">!</span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span>
 <a id="__codelineno-19-26" name="__codelineno-19-26" href="#__codelineno-19-26"></a><span class="w">            </span><span class="o">!</span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">vet2</span><span class="p">)</span><span class="w"> </span><span class="o">||</span>
 <a id="__codelineno-19-27" name="__codelineno-19-27" href="#__codelineno-19-27"></a><span class="w">            </span><span class="nx">vet1</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">vet2</span>
 <a id="__codelineno-19-28" name="__codelineno-19-28" href="#__codelineno-19-28"></a><span class="w">        </span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-29" name="__codelineno-19-29" href="#__codelineno-19-29"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Illegal Argument Exception'</span><span class="p">);</span>
+<a id="__codelineno-19-29" name="__codelineno-19-29" href="#__codelineno-19-29"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Invalid argument'</span><span class="p">);</span>
 <a id="__codelineno-19-30" name="__codelineno-19-30" href="#__codelineno-19-30"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-19-31" name="__codelineno-19-31" href="#__codelineno-19-31"></a><span class="w">        </span><span class="c1">// Add edge vet1 - vet2</span>
 <a id="__codelineno-19-32" name="__codelineno-19-32" href="#__codelineno-19-32"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet1</span><span class="p">).</span><span class="nx">push</span><span class="p">(</span><span class="nx">vet2</span><span class="p">);</span>
 <a id="__codelineno-19-33" name="__codelineno-19-33" href="#__codelineno-19-33"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet2</span><span class="p">).</span><span class="nx">push</span><span class="p">(</span><span class="nx">vet1</span><span class="p">);</span>
 <a id="__codelineno-19-34" name="__codelineno-19-34" href="#__codelineno-19-34"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-35" name="__codelineno-19-35" href="#__codelineno-19-35"></a>
-<a id="__codelineno-19-36" name="__codelineno-19-36" href="#__codelineno-19-36"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
+<a id="__codelineno-19-36" name="__codelineno-19-36" href="#__codelineno-19-36"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
 <a id="__codelineno-19-37" name="__codelineno-19-37" href="#__codelineno-19-37"></a><span class="w">    </span><span class="nx">removeEdge</span><span class="p">(</span><span class="nx">vet1</span><span class="p">,</span><span class="w"> </span><span class="nx">vet2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-38" name="__codelineno-19-38" href="#__codelineno-19-38"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span>
 <a id="__codelineno-19-39" name="__codelineno-19-39" href="#__codelineno-19-39"></a><span class="w">            </span><span class="o">!</span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span>
@@ -6396,28 +6473,28 @@
 <a id="__codelineno-19-41" name="__codelineno-19-41" href="#__codelineno-19-41"></a><span class="w">            </span><span class="nx">vet1</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">vet2</span><span class="w"> </span><span class="o">||</span>
 <a id="__codelineno-19-42" name="__codelineno-19-42" href="#__codelineno-19-42"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet1</span><span class="p">).</span><span class="nx">indexOf</span><span class="p">(</span><span class="nx">vet2</span><span class="p">)</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span>
 <a id="__codelineno-19-43" name="__codelineno-19-43" href="#__codelineno-19-43"></a><span class="w">        </span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-44" name="__codelineno-19-44" href="#__codelineno-19-44"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Illegal Argument Exception'</span><span class="p">);</span>
+<a id="__codelineno-19-44" name="__codelineno-19-44" href="#__codelineno-19-44"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Invalid argument'</span><span class="p">);</span>
 <a id="__codelineno-19-45" name="__codelineno-19-45" href="#__codelineno-19-45"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-19-46" name="__codelineno-19-46" href="#__codelineno-19-46"></a><span class="w">        </span><span class="c1">// Remove edge vet1 - vet2</span>
 <a id="__codelineno-19-47" name="__codelineno-19-47" href="#__codelineno-19-47"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet1</span><span class="p">).</span><span class="nx">splice</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet1</span><span class="p">).</span><span class="nx">indexOf</span><span class="p">(</span><span class="nx">vet2</span><span class="p">),</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-19-48" name="__codelineno-19-48" href="#__codelineno-19-48"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet2</span><span class="p">).</span><span class="nx">splice</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet2</span><span class="p">).</span><span class="nx">indexOf</span><span class="p">(</span><span class="nx">vet1</span><span class="p">),</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-19-49" name="__codelineno-19-49" href="#__codelineno-19-49"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-50" name="__codelineno-19-50" href="#__codelineno-19-50"></a>
-<a id="__codelineno-19-51" name="__codelineno-19-51" href="#__codelineno-19-51"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-19-51" name="__codelineno-19-51" href="#__codelineno-19-51"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-19-52" name="__codelineno-19-52" href="#__codelineno-19-52"></a><span class="w">    </span><span class="nx">addVertex</span><span class="p">(</span><span class="nx">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-53" name="__codelineno-19-53" href="#__codelineno-19-53"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">vet</span><span class="p">))</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-19-54" name="__codelineno-19-54" href="#__codelineno-19-54"></a><span class="w">        </span><span class="c1">// Add a new linked list in the adjacency list</span>
+<a id="__codelineno-19-54" name="__codelineno-19-54" href="#__codelineno-19-54"></a><span class="w">        </span><span class="c1">// Add a new list to the adjacency list</span>
 <a id="__codelineno-19-55" name="__codelineno-19-55" href="#__codelineno-19-55"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="nx">vet</span><span class="p">,</span><span class="w"> </span><span class="p">[]);</span>
 <a id="__codelineno-19-56" name="__codelineno-19-56" href="#__codelineno-19-56"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-57" name="__codelineno-19-57" href="#__codelineno-19-57"></a>
-<a id="__codelineno-19-58" name="__codelineno-19-58" href="#__codelineno-19-58"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-19-58" name="__codelineno-19-58" href="#__codelineno-19-58"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-19-59" name="__codelineno-19-59" href="#__codelineno-19-59"></a><span class="w">    </span><span class="nx">removeVertex</span><span class="p">(</span><span class="nx">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-60" name="__codelineno-19-60" href="#__codelineno-19-60"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">vet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-61" name="__codelineno-19-61" href="#__codelineno-19-61"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Illegal Argument Exception'</span><span class="p">);</span>
+<a id="__codelineno-19-61" name="__codelineno-19-61" href="#__codelineno-19-61"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Invalid argument'</span><span class="p">);</span>
 <a id="__codelineno-19-62" name="__codelineno-19-62" href="#__codelineno-19-62"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-19-63" name="__codelineno-19-63" href="#__codelineno-19-63"></a><span class="w">        </span><span class="c1">// Remove the linked list corresponding to vertex vet in the adjacency list</span>
+<a id="__codelineno-19-63" name="__codelineno-19-63" href="#__codelineno-19-63"></a><span class="w">        </span><span class="c1">// Remove the list corresponding to vertex vet from the adjacency list</span>
 <a id="__codelineno-19-64" name="__codelineno-19-64" href="#__codelineno-19-64"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="ow">delete</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span>
-<a id="__codelineno-19-65" name="__codelineno-19-65" href="#__codelineno-19-65"></a><span class="w">        </span><span class="c1">// Traverse the linked lists of other vertices and remove all edges containing vet</span>
+<a id="__codelineno-19-65" name="__codelineno-19-65" href="#__codelineno-19-65"></a><span class="w">        </span><span class="c1">// Traverse the lists of the other vertices and remove all edges incident to vet</span>
 <a id="__codelineno-19-66" name="__codelineno-19-66" href="#__codelineno-19-66"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">set</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">values</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-67" name="__codelineno-19-67" href="#__codelineno-19-67"></a><span class="w">            </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">set</span><span class="p">.</span><span class="nx">indexOf</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span>
 <a id="__codelineno-19-68" name="__codelineno-19-68" href="#__codelineno-19-68"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6426,7 +6503,7 @@
 <a id="__codelineno-19-71" name="__codelineno-19-71" href="#__codelineno-19-71"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-19-72" name="__codelineno-19-72" href="#__codelineno-19-72"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-73" name="__codelineno-19-73" href="#__codelineno-19-73"></a>
-<a id="__codelineno-19-74" name="__codelineno-19-74" href="#__codelineno-19-74"></a><span class="w">    </span><span class="cm">/* Print adjacency list */</span>
+<a id="__codelineno-19-74" name="__codelineno-19-74" href="#__codelineno-19-74"></a><span class="w">    </span><span class="cm">/* Print the adjacency list */</span>
 <a id="__codelineno-19-75" name="__codelineno-19-75" href="#__codelineno-19-75"></a><span class="w">    </span><span class="nx">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-76" name="__codelineno-19-76" href="#__codelineno-19-76"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Adjacency list ='</span><span class="p">);</span>
 <a id="__codelineno-19-77" name="__codelineno-19-77" href="#__codelineno-19-77"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="p">[</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">value</span><span class="p">]</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6441,9 +6518,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_list.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Undirected graph class based on adjacency list */</span>
+<div class="highlight"><span class="filename">graph_adjacency_list.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency list */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">GraphAdjList</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// Adjacency list, key: vertex, value: all adjacent vertices of that vertex</span>
+<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// Adjacency list: key is a vertex, value is the list of all its neighbors</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="nx">adjList</span><span class="o">:</span><span class="w"> </span><span class="kt">Map</span><span class="o">&lt;</span><span class="nx">Vertex</span><span class="p">,</span><span class="w"> </span><span class="nx">Vertex</span><span class="p">[]</span><span class="o">&gt;</span><span class="p">;</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -6462,21 +6539,21 @@
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">size</span><span class="p">;</span>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a>
-<a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
+<a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
 <a id="__codelineno-20-23" name="__codelineno-20-23" href="#__codelineno-20-23"></a><span class="w">    </span><span class="nx">addEdge</span><span class="p">(</span><span class="nx">vet1</span><span class="o">:</span><span class="w"> </span><span class="kt">Vertex</span><span class="p">,</span><span class="w"> </span><span class="nx">vet2</span><span class="o">:</span><span class="w"> </span><span class="kt">Vertex</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span>
 <a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a><span class="w">            </span><span class="o">!</span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span>
 <a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="w">            </span><span class="o">!</span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">vet2</span><span class="p">)</span><span class="w"> </span><span class="o">||</span>
 <a id="__codelineno-20-27" name="__codelineno-20-27" href="#__codelineno-20-27"></a><span class="w">            </span><span class="nx">vet1</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">vet2</span>
 <a id="__codelineno-20-28" name="__codelineno-20-28" href="#__codelineno-20-28"></a><span class="w">        </span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-29" name="__codelineno-20-29" href="#__codelineno-20-29"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Illegal Argument Exception'</span><span class="p">);</span>
+<a id="__codelineno-20-29" name="__codelineno-20-29" href="#__codelineno-20-29"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Invalid argument'</span><span class="p">);</span>
 <a id="__codelineno-20-30" name="__codelineno-20-30" href="#__codelineno-20-30"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-20-31" name="__codelineno-20-31" href="#__codelineno-20-31"></a><span class="w">        </span><span class="c1">// Add edge vet1 - vet2</span>
 <a id="__codelineno-20-32" name="__codelineno-20-32" href="#__codelineno-20-32"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet1</span><span class="p">).</span><span class="nx">push</span><span class="p">(</span><span class="nx">vet2</span><span class="p">);</span>
 <a id="__codelineno-20-33" name="__codelineno-20-33" href="#__codelineno-20-33"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet2</span><span class="p">).</span><span class="nx">push</span><span class="p">(</span><span class="nx">vet1</span><span class="p">);</span>
 <a id="__codelineno-20-34" name="__codelineno-20-34" href="#__codelineno-20-34"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-35" name="__codelineno-20-35" href="#__codelineno-20-35"></a>
-<a id="__codelineno-20-36" name="__codelineno-20-36" href="#__codelineno-20-36"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
+<a id="__codelineno-20-36" name="__codelineno-20-36" href="#__codelineno-20-36"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
 <a id="__codelineno-20-37" name="__codelineno-20-37" href="#__codelineno-20-37"></a><span class="w">    </span><span class="nx">removeEdge</span><span class="p">(</span><span class="nx">vet1</span><span class="o">:</span><span class="w"> </span><span class="kt">Vertex</span><span class="p">,</span><span class="w"> </span><span class="nx">vet2</span><span class="o">:</span><span class="w"> </span><span class="kt">Vertex</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-38" name="__codelineno-20-38" href="#__codelineno-20-38"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span>
 <a id="__codelineno-20-39" name="__codelineno-20-39" href="#__codelineno-20-39"></a><span class="w">            </span><span class="o">!</span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span>
@@ -6484,28 +6561,28 @@
 <a id="__codelineno-20-41" name="__codelineno-20-41" href="#__codelineno-20-41"></a><span class="w">            </span><span class="nx">vet1</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">vet2</span><span class="w"> </span><span class="o">||</span>
 <a id="__codelineno-20-42" name="__codelineno-20-42" href="#__codelineno-20-42"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet1</span><span class="p">).</span><span class="nx">indexOf</span><span class="p">(</span><span class="nx">vet2</span><span class="p">)</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span>
 <a id="__codelineno-20-43" name="__codelineno-20-43" href="#__codelineno-20-43"></a><span class="w">        </span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-44" name="__codelineno-20-44" href="#__codelineno-20-44"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Illegal Argument Exception'</span><span class="p">);</span>
+<a id="__codelineno-20-44" name="__codelineno-20-44" href="#__codelineno-20-44"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Invalid argument'</span><span class="p">);</span>
 <a id="__codelineno-20-45" name="__codelineno-20-45" href="#__codelineno-20-45"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-20-46" name="__codelineno-20-46" href="#__codelineno-20-46"></a><span class="w">        </span><span class="c1">// Remove edge vet1 - vet2</span>
 <a id="__codelineno-20-47" name="__codelineno-20-47" href="#__codelineno-20-47"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet1</span><span class="p">).</span><span class="nx">splice</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet1</span><span class="p">).</span><span class="nx">indexOf</span><span class="p">(</span><span class="nx">vet2</span><span class="p">),</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-20-48" name="__codelineno-20-48" href="#__codelineno-20-48"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet2</span><span class="p">).</span><span class="nx">splice</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet2</span><span class="p">).</span><span class="nx">indexOf</span><span class="p">(</span><span class="nx">vet1</span><span class="p">),</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-20-49" name="__codelineno-20-49" href="#__codelineno-20-49"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-50" name="__codelineno-20-50" href="#__codelineno-20-50"></a>
-<a id="__codelineno-20-51" name="__codelineno-20-51" href="#__codelineno-20-51"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-20-51" name="__codelineno-20-51" href="#__codelineno-20-51"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-20-52" name="__codelineno-20-52" href="#__codelineno-20-52"></a><span class="w">    </span><span class="nx">addVertex</span><span class="p">(</span><span class="nx">vet</span><span class="o">:</span><span class="w"> </span><span class="kt">Vertex</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-53" name="__codelineno-20-53" href="#__codelineno-20-53"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">vet</span><span class="p">))</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-20-54" name="__codelineno-20-54" href="#__codelineno-20-54"></a><span class="w">        </span><span class="c1">// Add a new linked list in the adjacency list</span>
+<a id="__codelineno-20-54" name="__codelineno-20-54" href="#__codelineno-20-54"></a><span class="w">        </span><span class="c1">// Add a new list to the adjacency list</span>
 <a id="__codelineno-20-55" name="__codelineno-20-55" href="#__codelineno-20-55"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="nx">vet</span><span class="p">,</span><span class="w"> </span><span class="p">[]);</span>
 <a id="__codelineno-20-56" name="__codelineno-20-56" href="#__codelineno-20-56"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-57" name="__codelineno-20-57" href="#__codelineno-20-57"></a>
-<a id="__codelineno-20-58" name="__codelineno-20-58" href="#__codelineno-20-58"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-20-58" name="__codelineno-20-58" href="#__codelineno-20-58"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-20-59" name="__codelineno-20-59" href="#__codelineno-20-59"></a><span class="w">    </span><span class="nx">removeVertex</span><span class="p">(</span><span class="nx">vet</span><span class="o">:</span><span class="w"> </span><span class="kt">Vertex</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-60" name="__codelineno-20-60" href="#__codelineno-20-60"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">vet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-61" name="__codelineno-20-61" href="#__codelineno-20-61"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Illegal Argument Exception'</span><span class="p">);</span>
+<a id="__codelineno-20-61" name="__codelineno-20-61" href="#__codelineno-20-61"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Invalid argument'</span><span class="p">);</span>
 <a id="__codelineno-20-62" name="__codelineno-20-62" href="#__codelineno-20-62"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-20-63" name="__codelineno-20-63" href="#__codelineno-20-63"></a><span class="w">        </span><span class="c1">// Remove the linked list corresponding to vertex vet in the adjacency list</span>
+<a id="__codelineno-20-63" name="__codelineno-20-63" href="#__codelineno-20-63"></a><span class="w">        </span><span class="c1">// Remove the list corresponding to vertex vet from the adjacency list</span>
 <a id="__codelineno-20-64" name="__codelineno-20-64" href="#__codelineno-20-64"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="ow">delete</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span>
-<a id="__codelineno-20-65" name="__codelineno-20-65" href="#__codelineno-20-65"></a><span class="w">        </span><span class="c1">// Traverse the linked lists of other vertices and remove all edges containing vet</span>
+<a id="__codelineno-20-65" name="__codelineno-20-65" href="#__codelineno-20-65"></a><span class="w">        </span><span class="c1">// Traverse the lists of the other vertices and remove all edges incident to vet</span>
 <a id="__codelineno-20-66" name="__codelineno-20-66" href="#__codelineno-20-66"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">set</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">values</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-67" name="__codelineno-20-67" href="#__codelineno-20-67"></a><span class="w">            </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">set</span><span class="p">.</span><span class="nx">indexOf</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span>
 <a id="__codelineno-20-68" name="__codelineno-20-68" href="#__codelineno-20-68"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6514,7 +6591,7 @@
 <a id="__codelineno-20-71" name="__codelineno-20-71" href="#__codelineno-20-71"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-20-72" name="__codelineno-20-72" href="#__codelineno-20-72"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-73" name="__codelineno-20-73" href="#__codelineno-20-73"></a>
-<a id="__codelineno-20-74" name="__codelineno-20-74" href="#__codelineno-20-74"></a><span class="w">    </span><span class="cm">/* Print adjacency list */</span>
+<a id="__codelineno-20-74" name="__codelineno-20-74" href="#__codelineno-20-74"></a><span class="w">    </span><span class="cm">/* Print the adjacency list */</span>
 <a id="__codelineno-20-75" name="__codelineno-20-75" href="#__codelineno-20-75"></a><span class="w">    </span><span class="nx">print</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-76" name="__codelineno-20-76" href="#__codelineno-20-76"></a><span class="w">        </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Adjacency list ='</span><span class="p">);</span>
 <a id="__codelineno-20-77" name="__codelineno-20-77" href="#__codelineno-20-77"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="p">[</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">value</span><span class="p">]</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">entries</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
@@ -6529,9 +6606,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_list.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Undirected graph class based on adjacency list */</span>
+<div class="highlight"><span class="filename">graph_adjacency_list.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency list */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">GraphAdjList</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// Adjacency list, key: vertex, value: all adjacent vertices of that vertex</span>
+<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// Adjacency list: key is a vertex, value is the list of all its neighbors</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="n">Map</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">adjList</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{};</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="cm">/* Constructor */</span>
@@ -6548,7 +6625,7 @@
 <a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">adjList</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a>
-<a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="w">  </span><span class="cm">/* Add edge */</span>
+<a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="w">  </span><span class="cm">/* Add an edge */</span>
 <a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">addEdge</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet1</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="n">containsKey</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span>
 <a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">        </span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="n">containsKey</span><span class="p">(</span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="o">||</span>
@@ -6560,7 +6637,7 @@
 <a id="__codelineno-21-29" name="__codelineno-21-29" href="#__codelineno-21-29"></a><span class="w">    </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet2</span><span class="p">]</span><span class="o">!</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">vet1</span><span class="p">);</span>
 <a id="__codelineno-21-30" name="__codelineno-21-30" href="#__codelineno-21-30"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-31" name="__codelineno-21-31" href="#__codelineno-21-31"></a>
-<a id="__codelineno-21-32" name="__codelineno-21-32" href="#__codelineno-21-32"></a><span class="w">  </span><span class="cm">/* Remove edge */</span>
+<a id="__codelineno-21-32" name="__codelineno-21-32" href="#__codelineno-21-32"></a><span class="w">  </span><span class="cm">/* Remove an edge */</span>
 <a id="__codelineno-21-33" name="__codelineno-21-33" href="#__codelineno-21-33"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">removeEdge</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet1</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-34" name="__codelineno-21-34" href="#__codelineno-21-34"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="n">containsKey</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span>
 <a id="__codelineno-21-35" name="__codelineno-21-35" href="#__codelineno-21-35"></a><span class="w">        </span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="n">containsKey</span><span class="p">(</span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="o">||</span>
@@ -6572,27 +6649,27 @@
 <a id="__codelineno-21-41" name="__codelineno-21-41" href="#__codelineno-21-41"></a><span class="w">    </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet2</span><span class="p">]</span><span class="o">!</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">vet1</span><span class="p">);</span>
 <a id="__codelineno-21-42" name="__codelineno-21-42" href="#__codelineno-21-42"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-43" name="__codelineno-21-43" href="#__codelineno-21-43"></a>
-<a id="__codelineno-21-44" name="__codelineno-21-44" href="#__codelineno-21-44"></a><span class="w">  </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-21-44" name="__codelineno-21-44" href="#__codelineno-21-44"></a><span class="w">  </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-21-45" name="__codelineno-21-45" href="#__codelineno-21-45"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">addVertex</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-46" name="__codelineno-21-46" href="#__codelineno-21-46"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">adjList</span><span class="p">.</span><span class="n">containsKey</span><span class="p">(</span><span class="n">vet</span><span class="p">))</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-21-47" name="__codelineno-21-47" href="#__codelineno-21-47"></a><span class="w">    </span><span class="c1">// Add a new linked list in the adjacency list</span>
+<a id="__codelineno-21-47" name="__codelineno-21-47" href="#__codelineno-21-47"></a><span class="w">    </span><span class="c1">// Add a new list to the adjacency list</span>
 <a id="__codelineno-21-48" name="__codelineno-21-48" href="#__codelineno-21-48"></a><span class="w">    </span><span class="n">adjList</span><span class="p">[</span><span class="n">vet</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-21-49" name="__codelineno-21-49" href="#__codelineno-21-49"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-50" name="__codelineno-21-50" href="#__codelineno-21-50"></a>
-<a id="__codelineno-21-51" name="__codelineno-21-51" href="#__codelineno-21-51"></a><span class="w">  </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-21-51" name="__codelineno-21-51" href="#__codelineno-21-51"></a><span class="w">  </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-21-52" name="__codelineno-21-52" href="#__codelineno-21-52"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">removeVertex</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-53" name="__codelineno-21-53" href="#__codelineno-21-53"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="n">containsKey</span><span class="p">(</span><span class="n">vet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-54" name="__codelineno-21-54" href="#__codelineno-21-54"></a><span class="w">      </span><span class="k">throw</span><span class="w"> </span><span class="n">ArgumentError</span><span class="p">;</span>
 <a id="__codelineno-21-55" name="__codelineno-21-55" href="#__codelineno-21-55"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-21-56" name="__codelineno-21-56" href="#__codelineno-21-56"></a><span class="w">    </span><span class="c1">// Remove the linked list corresponding to vertex vet in the adjacency list</span>
+<a id="__codelineno-21-56" name="__codelineno-21-56" href="#__codelineno-21-56"></a><span class="w">    </span><span class="c1">// Remove the list corresponding to vertex vet from the adjacency list</span>
 <a id="__codelineno-21-57" name="__codelineno-21-57" href="#__codelineno-21-57"></a><span class="w">    </span><span class="n">adjList</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span>
-<a id="__codelineno-21-58" name="__codelineno-21-58" href="#__codelineno-21-58"></a><span class="w">    </span><span class="c1">// Traverse the linked lists of other vertices and remove all edges containing vet</span>
+<a id="__codelineno-21-58" name="__codelineno-21-58" href="#__codelineno-21-58"></a><span class="w">    </span><span class="c1">// Traverse the lists of the other vertices and remove all edges incident to vet</span>
 <a id="__codelineno-21-59" name="__codelineno-21-59" href="#__codelineno-21-59"></a><span class="w">    </span><span class="n">adjList</span><span class="p">.</span><span class="n">forEach</span><span class="p">((</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">value</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-60" name="__codelineno-21-60" href="#__codelineno-21-60"></a><span class="w">      </span><span class="n">value</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span>
 <a id="__codelineno-21-61" name="__codelineno-21-61" href="#__codelineno-21-61"></a><span class="w">    </span><span class="p">});</span>
 <a id="__codelineno-21-62" name="__codelineno-21-62" href="#__codelineno-21-62"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-63" name="__codelineno-21-63" href="#__codelineno-21-63"></a>
-<a id="__codelineno-21-64" name="__codelineno-21-64" href="#__codelineno-21-64"></a><span class="w">  </span><span class="cm">/* Print adjacency list */</span>
+<a id="__codelineno-21-64" name="__codelineno-21-64" href="#__codelineno-21-64"></a><span class="w">  </span><span class="cm">/* Print the adjacency list */</span>
 <a id="__codelineno-21-65" name="__codelineno-21-65" href="#__codelineno-21-65"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">printAdjList</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-66" name="__codelineno-21-66" href="#__codelineno-21-66"></a><span class="w">    </span><span class="n">print</span><span class="p">(</span><span class="s2">"Adjacency list ="</span><span class="p">);</span>
 <a id="__codelineno-21-67" name="__codelineno-21-67" href="#__codelineno-21-67"></a><span class="w">    </span><span class="n">adjList</span><span class="p">.</span><span class="n">forEach</span><span class="p">((</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">value</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6607,10 +6684,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_list.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Undirected graph type based on adjacency list */</span>
+<div class="highlight"><span class="filename">graph_adjacency_list.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Undirected graph type implemented with an adjacency list */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">GraphAdjList</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Adjacency list, key: vertex, value: all adjacent vertices of that vertex</span>
-<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="n">adj_list</span><span class="p">:</span><span class="w"> </span><span class="nc">HashMap</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="p">,</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="c1">// maybe HashSet&lt;Vertex&gt; for value part is better?</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Adjacency list: key is a vertex, value is the list of all its neighbors</span>
+<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="n">adj_list</span><span class="p">:</span><span class="w"> </span><span class="nc">HashMap</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="p">,</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="c1">// A HashSet&lt;Vertex&gt; may be a better choice for the neighbor collection</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="p">}</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="k">impl</span><span class="w"> </span><span class="n">GraphAdjList</span><span class="w"> </span><span class="p">{</span>
@@ -6635,21 +6712,21 @@
 <a id="__codelineno-22-26" name="__codelineno-22-26" href="#__codelineno-22-26"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">adj_list</span><span class="p">.</span><span class="n">len</span><span class="p">()</span>
 <a id="__codelineno-22-27" name="__codelineno-22-27" href="#__codelineno-22-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-28" name="__codelineno-22-28" href="#__codelineno-22-28"></a>
-<a id="__codelineno-22-29" name="__codelineno-22-29" href="#__codelineno-22-29"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
+<a id="__codelineno-22-29" name="__codelineno-22-29" href="#__codelineno-22-29"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
 <a id="__codelineno-22-30" name="__codelineno-22-30" href="#__codelineno-22-30"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">add_edge</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">vet1</span><span class="p">:</span><span class="w"> </span><span class="nc">Vertex</span><span class="p">,</span><span class="w"> </span><span class="n">vet2</span><span class="p">:</span><span class="w"> </span><span class="nc">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-31" name="__codelineno-22-31" href="#__codelineno-22-31"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">vet1</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">vet2</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-32" name="__codelineno-22-32" href="#__codelineno-22-32"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"value error"</span><span class="p">);</span>
+<a id="__codelineno-22-32" name="__codelineno-22-32" href="#__codelineno-22-32"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"Vertices must be distinct"</span><span class="p">);</span>
 <a id="__codelineno-22-33" name="__codelineno-22-33" href="#__codelineno-22-33"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-34" name="__codelineno-22-34" href="#__codelineno-22-34"></a><span class="w">        </span><span class="c1">// Add edge vet1 - vet2</span>
 <a id="__codelineno-22-35" name="__codelineno-22-35" href="#__codelineno-22-35"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">adj_list</span><span class="p">.</span><span class="n">entry</span><span class="p">(</span><span class="n">vet1</span><span class="p">).</span><span class="n">or_default</span><span class="p">().</span><span class="n">push</span><span class="p">(</span><span class="n">vet2</span><span class="p">);</span>
 <a id="__codelineno-22-36" name="__codelineno-22-36" href="#__codelineno-22-36"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">adj_list</span><span class="p">.</span><span class="n">entry</span><span class="p">(</span><span class="n">vet2</span><span class="p">).</span><span class="n">or_default</span><span class="p">().</span><span class="n">push</span><span class="p">(</span><span class="n">vet1</span><span class="p">);</span>
 <a id="__codelineno-22-37" name="__codelineno-22-37" href="#__codelineno-22-37"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-38" name="__codelineno-22-38" href="#__codelineno-22-38"></a>
-<a id="__codelineno-22-39" name="__codelineno-22-39" href="#__codelineno-22-39"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
+<a id="__codelineno-22-39" name="__codelineno-22-39" href="#__codelineno-22-39"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
 <a id="__codelineno-22-40" name="__codelineno-22-40" href="#__codelineno-22-40"></a><span class="w">    </span><span class="cp">#[allow(unused)]</span>
 <a id="__codelineno-22-41" name="__codelineno-22-41" href="#__codelineno-22-41"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">remove_edge</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">vet1</span><span class="p">:</span><span class="w"> </span><span class="nc">Vertex</span><span class="p">,</span><span class="w"> </span><span class="n">vet2</span><span class="p">:</span><span class="w"> </span><span class="nc">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-42" name="__codelineno-22-42" href="#__codelineno-22-42"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">vet1</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">vet2</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-43" name="__codelineno-22-43" href="#__codelineno-22-43"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"value error"</span><span class="p">);</span>
+<a id="__codelineno-22-43" name="__codelineno-22-43" href="#__codelineno-22-43"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"Vertices must be distinct"</span><span class="p">);</span>
 <a id="__codelineno-22-44" name="__codelineno-22-44" href="#__codelineno-22-44"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-45" name="__codelineno-22-45" href="#__codelineno-22-45"></a><span class="w">        </span><span class="c1">// Remove edge vet1 - vet2</span>
 <a id="__codelineno-22-46" name="__codelineno-22-46" href="#__codelineno-22-46"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">adj_list</span>
@@ -6660,27 +6737,27 @@
 <a id="__codelineno-22-51" name="__codelineno-22-51" href="#__codelineno-22-51"></a><span class="w">            </span><span class="p">.</span><span class="n">and_modify</span><span class="p">(</span><span class="o">|</span><span class="n">v</span><span class="o">|</span><span class="w"> </span><span class="n">v</span><span class="p">.</span><span class="n">retain</span><span class="p">(</span><span class="o">|&amp;</span><span class="n">e</span><span class="o">|</span><span class="w"> </span><span class="n">e</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">vet1</span><span class="p">));</span>
 <a id="__codelineno-22-52" name="__codelineno-22-52" href="#__codelineno-22-52"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-53" name="__codelineno-22-53" href="#__codelineno-22-53"></a>
-<a id="__codelineno-22-54" name="__codelineno-22-54" href="#__codelineno-22-54"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-22-54" name="__codelineno-22-54" href="#__codelineno-22-54"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-22-55" name="__codelineno-22-55" href="#__codelineno-22-55"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">add_vertex</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">vet</span><span class="p">:</span><span class="w"> </span><span class="nc">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-56" name="__codelineno-22-56" href="#__codelineno-22-56"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">adj_list</span><span class="p">.</span><span class="n">contains_key</span><span class="p">(</span><span class="o">&amp;</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-57" name="__codelineno-22-57" href="#__codelineno-22-57"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-22-58" name="__codelineno-22-58" href="#__codelineno-22-58"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-22-59" name="__codelineno-22-59" href="#__codelineno-22-59"></a><span class="w">        </span><span class="c1">// Add a new linked list in the adjacency list</span>
+<a id="__codelineno-22-59" name="__codelineno-22-59" href="#__codelineno-22-59"></a><span class="w">        </span><span class="c1">// Add a new neighbor list to the adjacency list</span>
 <a id="__codelineno-22-60" name="__codelineno-22-60" href="#__codelineno-22-60"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">adj_list</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="n">vet</span><span class="p">,</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[]);</span>
 <a id="__codelineno-22-61" name="__codelineno-22-61" href="#__codelineno-22-61"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-62" name="__codelineno-22-62" href="#__codelineno-22-62"></a>
-<a id="__codelineno-22-63" name="__codelineno-22-63" href="#__codelineno-22-63"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-22-63" name="__codelineno-22-63" href="#__codelineno-22-63"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-22-64" name="__codelineno-22-64" href="#__codelineno-22-64"></a><span class="w">    </span><span class="cp">#[allow(unused)]</span>
 <a id="__codelineno-22-65" name="__codelineno-22-65" href="#__codelineno-22-65"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">remove_vertex</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">vet</span><span class="p">:</span><span class="w"> </span><span class="nc">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-66" name="__codelineno-22-66" href="#__codelineno-22-66"></a><span class="w">        </span><span class="c1">// Remove the linked list corresponding to vertex vet in the adjacency list</span>
+<a id="__codelineno-22-66" name="__codelineno-22-66" href="#__codelineno-22-66"></a><span class="w">        </span><span class="c1">// Remove the neighbor list for vertex vet from the adjacency list</span>
 <a id="__codelineno-22-67" name="__codelineno-22-67" href="#__codelineno-22-67"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">adj_list</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="o">&amp;</span><span class="n">vet</span><span class="p">);</span>
-<a id="__codelineno-22-68" name="__codelineno-22-68" href="#__codelineno-22-68"></a><span class="w">        </span><span class="c1">// Traverse the linked lists of other vertices and remove all edges containing vet</span>
+<a id="__codelineno-22-68" name="__codelineno-22-68" href="#__codelineno-22-68"></a><span class="w">        </span><span class="c1">// Traverse the other neighbor lists and remove every edge incident to vet</span>
 <a id="__codelineno-22-69" name="__codelineno-22-69" href="#__codelineno-22-69"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">list</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">adj_list</span><span class="p">.</span><span class="n">values_mut</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-70" name="__codelineno-22-70" href="#__codelineno-22-70"></a><span class="w">            </span><span class="n">list</span><span class="p">.</span><span class="n">retain</span><span class="p">(</span><span class="o">|&amp;</span><span class="n">v</span><span class="o">|</span><span class="w"> </span><span class="n">v</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">vet</span><span class="p">);</span>
 <a id="__codelineno-22-71" name="__codelineno-22-71" href="#__codelineno-22-71"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-72" name="__codelineno-22-72" href="#__codelineno-22-72"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-73" name="__codelineno-22-73" href="#__codelineno-22-73"></a>
-<a id="__codelineno-22-74" name="__codelineno-22-74" href="#__codelineno-22-74"></a><span class="w">    </span><span class="cm">/* Print adjacency list */</span>
+<a id="__codelineno-22-74" name="__codelineno-22-74" href="#__codelineno-22-74"></a><span class="w">    </span><span class="cm">/* Print the adjacency list */</span>
 <a id="__codelineno-22-75" name="__codelineno-22-75" href="#__codelineno-22-75"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">print</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-76" name="__codelineno-22-76" href="#__codelineno-22-76"></a><span class="w">        </span><span class="fm">println!</span><span class="p">(</span><span class="s">"Adjacency list ="</span><span class="p">);</span>
 <a id="__codelineno-22-77" name="__codelineno-22-77" href="#__codelineno-22-77"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">vertex</span><span class="p">,</span><span class="w"> </span><span class="n">list</span><span class="p">)</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="o">&amp;</span><span class="bp">self</span><span class="p">.</span><span class="n">adj_list</span><span class="w"> </span><span class="p">{</span>
@@ -6692,13 +6769,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_list.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Node structure */</span>
+<div class="highlight"><span class="filename">graph_adjacency_list.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Adjacency-list node structure */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">AdjListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vertex</span><span class="p">;</span><span class="w">           </span><span class="c1">// Vertex</span>
-<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">struct</span><span class="w"> </span><span class="nc">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Successor node</span>
+<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">struct</span><span class="w"> </span><span class="nc">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Next node</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="p">}</span><span class="w"> </span><span class="n">AdjListNode</span><span class="p">;</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a>
-<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="cm">/* Find node corresponding to vertex */</span>
+<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="cm">/* Find the list node corresponding to a vertex */</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="nf">findNode</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="o">*</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">heads</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">-&gt;</span><span class="n">vertex</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6708,36 +6785,36 @@
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="p">}</span>
 <a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a>
-<a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="cm">/* Add edge helper function */</span>
+<a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="cm">/* Helper for adding an edge */</span>
 <a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="kt">void</span><span class="w"> </span><span class="nf">addEdgeHelper</span><span class="p">(</span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">head</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w">    </span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">AdjListNode</span><span class="p">));</span>
 <a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="w">    </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">vertex</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">vet</span><span class="p">;</span>
-<a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="w">    </span><span class="c1">// Head insertion</span>
+<a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="w">    </span><span class="c1">// Insert at the front of the neighbor list</span>
 <a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="w">    </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">head</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="w">    </span><span class="n">head</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="p">}</span>
 <a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a>
-<a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="cm">/* Remove edge helper function */</span>
+<a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="cm">/* Helper for removing an edge */</span>
 <a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a><span class="kt">void</span><span class="w"> </span><span class="nf">removeEdgeHelper</span><span class="p">(</span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">head</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-28" name="__codelineno-23-28" href="#__codelineno-23-28"></a><span class="w">    </span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">head</span><span class="p">;</span>
 <a id="__codelineno-23-29" name="__codelineno-23-29" href="#__codelineno-23-29"></a><span class="w">    </span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">head</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
-<a id="__codelineno-23-30" name="__codelineno-23-30" href="#__codelineno-23-30"></a><span class="w">    </span><span class="c1">// Search for node corresponding to vet in list</span>
+<a id="__codelineno-23-30" name="__codelineno-23-30" href="#__codelineno-23-30"></a><span class="w">    </span><span class="c1">// Search the linked list for the node corresponding to vet</span>
 <a id="__codelineno-23-31" name="__codelineno-23-31" href="#__codelineno-23-31"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">vertex</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-32" name="__codelineno-23-32" href="#__codelineno-23-32"></a><span class="w">        </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
 <a id="__codelineno-23-33" name="__codelineno-23-33" href="#__codelineno-23-33"></a><span class="w">        </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-23-34" name="__codelineno-23-34" href="#__codelineno-23-34"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-23-35" name="__codelineno-23-35" href="#__codelineno-23-35"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span>
 <a id="__codelineno-23-36" name="__codelineno-23-36" href="#__codelineno-23-36"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-23-37" name="__codelineno-23-37" href="#__codelineno-23-37"></a><span class="w">    </span><span class="c1">// Remove node corresponding to vet from list</span>
+<a id="__codelineno-23-37" name="__codelineno-23-37" href="#__codelineno-23-37"></a><span class="w">    </span><span class="c1">// Remove the node corresponding to vet from the linked list</span>
 <a id="__codelineno-23-38" name="__codelineno-23-38" href="#__codelineno-23-38"></a><span class="w">    </span><span class="n">pre</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-23-39" name="__codelineno-23-39" href="#__codelineno-23-39"></a><span class="w">    </span><span class="c1">// Free memory</span>
 <a id="__codelineno-23-40" name="__codelineno-23-40" href="#__codelineno-23-40"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">cur</span><span class="p">);</span>
 <a id="__codelineno-23-41" name="__codelineno-23-41" href="#__codelineno-23-41"></a><span class="p">}</span>
 <a id="__codelineno-23-42" name="__codelineno-23-42" href="#__codelineno-23-42"></a>
-<a id="__codelineno-23-43" name="__codelineno-23-43" href="#__codelineno-23-43"></a><span class="cm">/* Undirected graph class based on adjacency list */</span>
+<a id="__codelineno-23-43" name="__codelineno-23-43" href="#__codelineno-23-43"></a><span class="cm">/* Undirected graph structure implemented with an adjacency list */</span>
 <a id="__codelineno-23-44" name="__codelineno-23-44" href="#__codelineno-23-44"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-45" name="__codelineno-23-45" href="#__codelineno-23-45"></a><span class="w">    </span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">heads</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">];</span><span class="w"> </span><span class="c1">// Node array</span>
-<a id="__codelineno-23-46" name="__codelineno-23-46" href="#__codelineno-23-46"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w">                     </span><span class="c1">// Node count</span>
+<a id="__codelineno-23-45" name="__codelineno-23-45" href="#__codelineno-23-45"></a><span class="w">    </span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">heads</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">];</span><span class="w"> </span><span class="c1">// Array of list heads</span>
+<a id="__codelineno-23-46" name="__codelineno-23-46" href="#__codelineno-23-46"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w">                     </span><span class="c1">// Number of vertices</span>
 <a id="__codelineno-23-47" name="__codelineno-23-47" href="#__codelineno-23-47"></a><span class="p">}</span><span class="w"> </span><span class="n">GraphAdjList</span><span class="p">;</span>
 <a id="__codelineno-23-48" name="__codelineno-23-48" href="#__codelineno-23-48"></a>
 <a id="__codelineno-23-49" name="__codelineno-23-49" href="#__codelineno-23-49"></a><span class="cm">/* Constructor */</span>
@@ -6770,7 +6847,7 @@
 <a id="__codelineno-23-76" name="__codelineno-23-76" href="#__codelineno-23-76"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">graph</span><span class="p">);</span>
 <a id="__codelineno-23-77" name="__codelineno-23-77" href="#__codelineno-23-77"></a><span class="p">}</span>
 <a id="__codelineno-23-78" name="__codelineno-23-78" href="#__codelineno-23-78"></a>
-<a id="__codelineno-23-79" name="__codelineno-23-79" href="#__codelineno-23-79"></a><span class="cm">/* Find node corresponding to vertex */</span>
+<a id="__codelineno-23-79" name="__codelineno-23-79" href="#__codelineno-23-79"></a><span class="cm">/* Find the list node corresponding to a vertex */</span>
 <a id="__codelineno-23-80" name="__codelineno-23-80" href="#__codelineno-23-80"></a><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="nf">findNode</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="o">*</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-81" name="__codelineno-23-81" href="#__codelineno-23-81"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-82" name="__codelineno-23-82" href="#__codelineno-23-82"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">heads</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">-&gt;</span><span class="n">vertex</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6780,7 +6857,7 @@
 <a id="__codelineno-23-86" name="__codelineno-23-86" href="#__codelineno-23-86"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-23-87" name="__codelineno-23-87" href="#__codelineno-23-87"></a><span class="p">}</span>
 <a id="__codelineno-23-88" name="__codelineno-23-88" href="#__codelineno-23-88"></a>
-<a id="__codelineno-23-89" name="__codelineno-23-89" href="#__codelineno-23-89"></a><span class="cm">/* Add edge */</span>
+<a id="__codelineno-23-89" name="__codelineno-23-89" href="#__codelineno-23-89"></a><span class="cm">/* Add an edge */</span>
 <a id="__codelineno-23-90" name="__codelineno-23-90" href="#__codelineno-23-90"></a><span class="kt">void</span><span class="w"> </span><span class="nf">addEdge</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="o">*</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet1</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-91" name="__codelineno-23-91" href="#__codelineno-23-91"></a><span class="w">    </span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">head1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findNode</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">vet1</span><span class="p">);</span>
 <a id="__codelineno-23-92" name="__codelineno-23-92" href="#__codelineno-23-92"></a><span class="w">    </span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">head2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findNode</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">vet2</span><span class="p">);</span>
@@ -6790,7 +6867,7 @@
 <a id="__codelineno-23-96" name="__codelineno-23-96" href="#__codelineno-23-96"></a><span class="w">    </span><span class="n">addEdgeHelper</span><span class="p">(</span><span class="n">head2</span><span class="p">,</span><span class="w"> </span><span class="n">vet1</span><span class="p">);</span>
 <a id="__codelineno-23-97" name="__codelineno-23-97" href="#__codelineno-23-97"></a><span class="p">}</span>
 <a id="__codelineno-23-98" name="__codelineno-23-98" href="#__codelineno-23-98"></a>
-<a id="__codelineno-23-99" name="__codelineno-23-99" href="#__codelineno-23-99"></a><span class="cm">/* Remove edge */</span>
+<a id="__codelineno-23-99" name="__codelineno-23-99" href="#__codelineno-23-99"></a><span class="cm">/* Remove an edge */</span>
 <a id="__codelineno-23-100" name="__codelineno-23-100" href="#__codelineno-23-100"></a><span class="kt">void</span><span class="w"> </span><span class="nf">removeEdge</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="o">*</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet1</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-101" name="__codelineno-23-101" href="#__codelineno-23-101"></a><span class="w">    </span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">head1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findNode</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">vet1</span><span class="p">);</span>
 <a id="__codelineno-23-102" name="__codelineno-23-102" href="#__codelineno-23-102"></a><span class="w">    </span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">head2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findNode</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">vet2</span><span class="p">);</span>
@@ -6800,28 +6877,28 @@
 <a id="__codelineno-23-106" name="__codelineno-23-106" href="#__codelineno-23-106"></a><span class="w">    </span><span class="n">removeEdgeHelper</span><span class="p">(</span><span class="n">head2</span><span class="p">,</span><span class="w"> </span><span class="n">head1</span><span class="o">-&gt;</span><span class="n">vertex</span><span class="p">);</span>
 <a id="__codelineno-23-107" name="__codelineno-23-107" href="#__codelineno-23-107"></a><span class="p">}</span>
 <a id="__codelineno-23-108" name="__codelineno-23-108" href="#__codelineno-23-108"></a>
-<a id="__codelineno-23-109" name="__codelineno-23-109" href="#__codelineno-23-109"></a><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-23-109" name="__codelineno-23-109" href="#__codelineno-23-109"></a><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-23-110" name="__codelineno-23-110" href="#__codelineno-23-110"></a><span class="kt">void</span><span class="w"> </span><span class="nf">addVertex</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="o">*</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-111" name="__codelineno-23-111" href="#__codelineno-23-111"></a><span class="w">    </span><span class="n">assert</span><span class="p">(</span><span class="n">graph</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="p">);</span>
 <a id="__codelineno-23-112" name="__codelineno-23-112" href="#__codelineno-23-112"></a><span class="w">    </span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">head</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">AdjListNode</span><span class="p">));</span>
 <a id="__codelineno-23-113" name="__codelineno-23-113" href="#__codelineno-23-113"></a><span class="w">    </span><span class="n">head</span><span class="o">-&gt;</span><span class="n">vertex</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">vet</span><span class="p">;</span>
 <a id="__codelineno-23-114" name="__codelineno-23-114" href="#__codelineno-23-114"></a><span class="w">    </span><span class="n">head</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
-<a id="__codelineno-23-115" name="__codelineno-23-115" href="#__codelineno-23-115"></a><span class="w">    </span><span class="c1">// Add a new linked list in the adjacency list</span>
+<a id="__codelineno-23-115" name="__codelineno-23-115" href="#__codelineno-23-115"></a><span class="w">    </span><span class="c1">// Add a new list to the adjacency list</span>
 <a id="__codelineno-23-116" name="__codelineno-23-116" href="#__codelineno-23-116"></a><span class="w">    </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">heads</span><span class="p">[</span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">head</span><span class="p">;</span>
 <a id="__codelineno-23-117" name="__codelineno-23-117" href="#__codelineno-23-117"></a><span class="p">}</span>
 <a id="__codelineno-23-118" name="__codelineno-23-118" href="#__codelineno-23-118"></a>
-<a id="__codelineno-23-119" name="__codelineno-23-119" href="#__codelineno-23-119"></a><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-23-119" name="__codelineno-23-119" href="#__codelineno-23-119"></a><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-23-120" name="__codelineno-23-120" href="#__codelineno-23-120"></a><span class="kt">void</span><span class="w"> </span><span class="nf">removeVertex</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="o">*</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-121" name="__codelineno-23-121" href="#__codelineno-23-121"></a><span class="w">    </span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findNode</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">vet</span><span class="p">);</span>
 <a id="__codelineno-23-122" name="__codelineno-23-122" href="#__codelineno-23-122"></a><span class="w">    </span><span class="n">assert</span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">);</span>
-<a id="__codelineno-23-123" name="__codelineno-23-123" href="#__codelineno-23-123"></a><span class="w">    </span><span class="c1">// Remove the linked list corresponding to vertex vet in the adjacency list</span>
+<a id="__codelineno-23-123" name="__codelineno-23-123" href="#__codelineno-23-123"></a><span class="w">    </span><span class="c1">// Remove the list corresponding to vertex vet from the adjacency list</span>
 <a id="__codelineno-23-124" name="__codelineno-23-124" href="#__codelineno-23-124"></a><span class="w">    </span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-23-125" name="__codelineno-23-125" href="#__codelineno-23-125"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-126" name="__codelineno-23-126" href="#__codelineno-23-126"></a><span class="w">        </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
 <a id="__codelineno-23-127" name="__codelineno-23-127" href="#__codelineno-23-127"></a><span class="w">        </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-23-128" name="__codelineno-23-128" href="#__codelineno-23-128"></a><span class="w">        </span><span class="n">free</span><span class="p">(</span><span class="n">pre</span><span class="p">);</span>
 <a id="__codelineno-23-129" name="__codelineno-23-129" href="#__codelineno-23-129"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-130" name="__codelineno-23-130" href="#__codelineno-23-130"></a><span class="w">    </span><span class="c1">// Traverse the linked lists of other vertices and remove all edges containing vet</span>
+<a id="__codelineno-23-130" name="__codelineno-23-130" href="#__codelineno-23-130"></a><span class="w">    </span><span class="c1">// Traverse the lists of the other vertices and remove all edges incident to vet</span>
 <a id="__codelineno-23-131" name="__codelineno-23-131" href="#__codelineno-23-131"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-132" name="__codelineno-23-132" href="#__codelineno-23-132"></a><span class="w">        </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">heads</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-23-133" name="__codelineno-23-133" href="#__codelineno-23-133"></a><span class="w">        </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
@@ -6835,7 +6912,7 @@
 <a id="__codelineno-23-141" name="__codelineno-23-141" href="#__codelineno-23-141"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-23-142" name="__codelineno-23-142" href="#__codelineno-23-142"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-23-143" name="__codelineno-23-143" href="#__codelineno-23-143"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-144" name="__codelineno-23-144" href="#__codelineno-23-144"></a><span class="w">    </span><span class="c1">// Move vertices after this vertex forward to fill gap</span>
+<a id="__codelineno-23-144" name="__codelineno-23-144" href="#__codelineno-23-144"></a><span class="w">    </span><span class="c1">// Shift subsequent vertices forward to fill the gap</span>
 <a id="__codelineno-23-145" name="__codelineno-23-145" href="#__codelineno-23-145"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-23-146" name="__codelineno-23-146" href="#__codelineno-23-146"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-147" name="__codelineno-23-147" href="#__codelineno-23-147"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">graph</span><span class="o">-&gt;</span><span class="n">heads</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">node</span><span class="p">)</span>
@@ -6850,9 +6927,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_list.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Undirected graph class based on adjacency list */</span>
+<div class="highlight"><span class="filename">graph_adjacency_list.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Undirected graph class implemented with an adjacency list */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">GraphAdjList</span><span class="p">(</span><span class="n">edges</span><span class="p">:</span><span class="w"> </span><span class="n">Array</span><span class="o">&lt;</span><span class="n">Array</span><span class="o">&lt;</span><span class="n">Vertex?</span><span class="o">&gt;&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Adjacency list, key: vertex, value: all adjacent vertices of that vertex</span>
+<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Adjacency list: key is a vertex, value is the list of all its neighbors</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">adjList</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashMap</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="p">,</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;&gt;</span><span class="p">()</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -6870,7 +6947,7 @@
 <a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">adjList</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a>
-<a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="w">    </span><span class="cm">/* Add edge */</span>
+<a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="w">    </span><span class="cm">/* Add an edge */</span>
 <a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">addEdge</span><span class="p">(</span><span class="n">vet1</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">,</span><span class="w"> </span><span class="n">vet2</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="na">containsKey</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="na">containsKey</span><span class="p">(</span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">vet1</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span>
 <a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">IllegalArgumentException</span><span class="p">()</span>
@@ -6879,7 +6956,7 @@
 <a id="__codelineno-24-27" name="__codelineno-24-27" href="#__codelineno-24-27"></a><span class="w">        </span><span class="n">adjList</span><span class="o">[</span><span class="n">vet2</span><span class="o">]?.</span><span class="na">add</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span>
 <a id="__codelineno-24-28" name="__codelineno-24-28" href="#__codelineno-24-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-29" name="__codelineno-24-29" href="#__codelineno-24-29"></a>
-<a id="__codelineno-24-30" name="__codelineno-24-30" href="#__codelineno-24-30"></a><span class="w">    </span><span class="cm">/* Remove edge */</span>
+<a id="__codelineno-24-30" name="__codelineno-24-30" href="#__codelineno-24-30"></a><span class="w">    </span><span class="cm">/* Remove an edge */</span>
 <a id="__codelineno-24-31" name="__codelineno-24-31" href="#__codelineno-24-31"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">removeEdge</span><span class="p">(</span><span class="n">vet1</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">,</span><span class="w"> </span><span class="n">vet2</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-32" name="__codelineno-24-32" href="#__codelineno-24-32"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="na">containsKey</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="na">containsKey</span><span class="p">(</span><span class="n">vet2</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">vet1</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span>
 <a id="__codelineno-24-33" name="__codelineno-24-33" href="#__codelineno-24-33"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">IllegalArgumentException</span><span class="p">()</span>
@@ -6888,27 +6965,27 @@
 <a id="__codelineno-24-36" name="__codelineno-24-36" href="#__codelineno-24-36"></a><span class="w">        </span><span class="n">adjList</span><span class="o">[</span><span class="n">vet2</span><span class="o">]?.</span><span class="na">remove</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span>
 <a id="__codelineno-24-37" name="__codelineno-24-37" href="#__codelineno-24-37"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-38" name="__codelineno-24-38" href="#__codelineno-24-38"></a>
-<a id="__codelineno-24-39" name="__codelineno-24-39" href="#__codelineno-24-39"></a><span class="w">    </span><span class="cm">/* Add vertex */</span>
+<a id="__codelineno-24-39" name="__codelineno-24-39" href="#__codelineno-24-39"></a><span class="w">    </span><span class="cm">/* Add a vertex */</span>
 <a id="__codelineno-24-40" name="__codelineno-24-40" href="#__codelineno-24-40"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">addVertex</span><span class="p">(</span><span class="n">vet</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-41" name="__codelineno-24-41" href="#__codelineno-24-41"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">adjList</span><span class="p">.</span><span class="na">containsKey</span><span class="p">(</span><span class="n">vet</span><span class="p">))</span>
 <a id="__codelineno-24-42" name="__codelineno-24-42" href="#__codelineno-24-42"></a><span class="w">            </span><span class="k">return</span>
-<a id="__codelineno-24-43" name="__codelineno-24-43" href="#__codelineno-24-43"></a><span class="w">        </span><span class="c1">// Add a new linked list in the adjacency list</span>
+<a id="__codelineno-24-43" name="__codelineno-24-43" href="#__codelineno-24-43"></a><span class="w">        </span><span class="c1">// Add a new list to the adjacency list</span>
 <a id="__codelineno-24-44" name="__codelineno-24-44" href="#__codelineno-24-44"></a><span class="w">        </span><span class="n">adjList</span><span class="o">[</span><span class="n">vet</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="p">()</span>
 <a id="__codelineno-24-45" name="__codelineno-24-45" href="#__codelineno-24-45"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-46" name="__codelineno-24-46" href="#__codelineno-24-46"></a>
-<a id="__codelineno-24-47" name="__codelineno-24-47" href="#__codelineno-24-47"></a><span class="w">    </span><span class="cm">/* Remove vertex */</span>
+<a id="__codelineno-24-47" name="__codelineno-24-47" href="#__codelineno-24-47"></a><span class="w">    </span><span class="cm">/* Remove a vertex */</span>
 <a id="__codelineno-24-48" name="__codelineno-24-48" href="#__codelineno-24-48"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">removeVertex</span><span class="p">(</span><span class="n">vet</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-49" name="__codelineno-24-49" href="#__codelineno-24-49"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">adjList</span><span class="p">.</span><span class="na">containsKey</span><span class="p">(</span><span class="n">vet</span><span class="p">))</span>
 <a id="__codelineno-24-50" name="__codelineno-24-50" href="#__codelineno-24-50"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">IllegalArgumentException</span><span class="p">()</span>
-<a id="__codelineno-24-51" name="__codelineno-24-51" href="#__codelineno-24-51"></a><span class="w">        </span><span class="c1">// Remove the linked list corresponding to vertex vet in the adjacency list</span>
+<a id="__codelineno-24-51" name="__codelineno-24-51" href="#__codelineno-24-51"></a><span class="w">        </span><span class="c1">// Remove the list corresponding to vertex vet from the adjacency list</span>
 <a id="__codelineno-24-52" name="__codelineno-24-52" href="#__codelineno-24-52"></a><span class="w">        </span><span class="n">adjList</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span>
-<a id="__codelineno-24-53" name="__codelineno-24-53" href="#__codelineno-24-53"></a><span class="w">        </span><span class="c1">// Traverse the linked lists of other vertices and remove all edges containing vet</span>
+<a id="__codelineno-24-53" name="__codelineno-24-53" href="#__codelineno-24-53"></a><span class="w">        </span><span class="c1">// Traverse the lists of the other vertices and remove all edges incident to vet</span>
 <a id="__codelineno-24-54" name="__codelineno-24-54" href="#__codelineno-24-54"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">list</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">adjList</span><span class="p">.</span><span class="na">values</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-55" name="__codelineno-24-55" href="#__codelineno-24-55"></a><span class="w">            </span><span class="n">list</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span>
 <a id="__codelineno-24-56" name="__codelineno-24-56" href="#__codelineno-24-56"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-24-57" name="__codelineno-24-57" href="#__codelineno-24-57"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-58" name="__codelineno-24-58" href="#__codelineno-24-58"></a>
-<a id="__codelineno-24-59" name="__codelineno-24-59" href="#__codelineno-24-59"></a><span class="w">    </span><span class="cm">/* Print adjacency list */</span>
+<a id="__codelineno-24-59" name="__codelineno-24-59" href="#__codelineno-24-59"></a><span class="w">    </span><span class="cm">/* Print the adjacency list */</span>
 <a id="__codelineno-24-60" name="__codelineno-24-60" href="#__codelineno-24-60"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-61" name="__codelineno-24-61" href="#__codelineno-24-61"></a><span class="w">        </span><span class="n">println</span><span class="p">(</span><span class="s">"Adjacency list ="</span><span class="p">)</span>
 <a id="__codelineno-24-62" name="__codelineno-24-62" href="#__codelineno-24-62"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">adjList</span><span class="p">.</span><span class="na">entries</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6923,13 +7000,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_adjacency_list.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Undirected graph class based on adjacency list ###</span>
+<div class="highlight"><span class="filename">graph_adjacency_list.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Undirected graph class implemented with an adjacency list ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">GraphAdjList</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="kp">attr_reader</span><span class="w"> </span><span class="ss">:adj_list</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1">### Constructor ###</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">initialize</span><span class="p">(</span><span class="n">edges</span><span class="p">)</span>
-<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">    </span><span class="c1"># Adjacency list, key: vertex, value: all adjacent vertices of that vertex</span>
+<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">    </span><span class="c1"># Adjacency list: key is a vertex, value is the list of all its neighbors</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">    </span><span class="vi">@adj_list</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{}</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">    </span><span class="c1"># Add all vertices and edges</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">edge</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">edges</span>
@@ -6939,12 +7016,12 @@
 <a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a>
-<a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w">  </span><span class="c1">### Get number of vertices ###</span>
+<a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w">  </span><span class="c1">### Get the number of vertices ###</span>
 <a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">size</span>
 <a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="w">    </span><span class="vi">@adj_list</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a>
-<a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="w">  </span><span class="c1">### Add edge ###</span>
+<a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="w">  </span><span class="c1">### Add an edge ###</span>
 <a id="__codelineno-25-23" name="__codelineno-25-23" href="#__codelineno-25-23"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">add_edge</span><span class="p">(</span><span class="n">vet1</span><span class="p">,</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span>
 <a id="__codelineno-25-24" name="__codelineno-25-24" href="#__codelineno-25-24"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">ArgumentError</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="vi">@adj_list</span><span class="o">.</span><span class="n">include?</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="o">!</span><span class="vi">@adj_list</span><span class="o">.</span><span class="n">include?</span><span class="p">(</span><span class="n">vet2</span><span class="p">)</span>
 <a id="__codelineno-25-25" name="__codelineno-25-25" href="#__codelineno-25-25"></a>
@@ -6952,7 +7029,7 @@
 <a id="__codelineno-25-27" name="__codelineno-25-27" href="#__codelineno-25-27"></a><span class="w">    </span><span class="vi">@adj_list</span><span class="o">[</span><span class="n">vet2</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">vet1</span>
 <a id="__codelineno-25-28" name="__codelineno-25-28" href="#__codelineno-25-28"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-29" name="__codelineno-25-29" href="#__codelineno-25-29"></a>
-<a id="__codelineno-25-30" name="__codelineno-25-30" href="#__codelineno-25-30"></a><span class="w">  </span><span class="c1">### Delete edge ###</span>
+<a id="__codelineno-25-30" name="__codelineno-25-30" href="#__codelineno-25-30"></a><span class="w">  </span><span class="c1">### Remove an edge ###</span>
 <a id="__codelineno-25-31" name="__codelineno-25-31" href="#__codelineno-25-31"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">remove_edge</span><span class="p">(</span><span class="n">vet1</span><span class="p">,</span><span class="w"> </span><span class="n">vet2</span><span class="p">)</span>
 <a id="__codelineno-25-32" name="__codelineno-25-32" href="#__codelineno-25-32"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">ArgumentError</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="vi">@adj_list</span><span class="o">.</span><span class="n">include?</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="o">!</span><span class="vi">@adj_list</span><span class="o">.</span><span class="n">include?</span><span class="p">(</span><span class="n">vet2</span><span class="p">)</span>
 <a id="__codelineno-25-33" name="__codelineno-25-33" href="#__codelineno-25-33"></a>
@@ -6961,27 +7038,27 @@
 <a id="__codelineno-25-36" name="__codelineno-25-36" href="#__codelineno-25-36"></a><span class="w">    </span><span class="vi">@adj_list</span><span class="o">[</span><span class="n">vet2</span><span class="o">].</span><span class="n">delete</span><span class="p">(</span><span class="n">vet1</span><span class="p">)</span>
 <a id="__codelineno-25-37" name="__codelineno-25-37" href="#__codelineno-25-37"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-38" name="__codelineno-25-38" href="#__codelineno-25-38"></a>
-<a id="__codelineno-25-39" name="__codelineno-25-39" href="#__codelineno-25-39"></a><span class="w">  </span><span class="c1">### Add vertex ###</span>
+<a id="__codelineno-25-39" name="__codelineno-25-39" href="#__codelineno-25-39"></a><span class="w">  </span><span class="c1">### Add a vertex ###</span>
 <a id="__codelineno-25-40" name="__codelineno-25-40" href="#__codelineno-25-40"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">add_vertex</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span>
 <a id="__codelineno-25-41" name="__codelineno-25-41" href="#__codelineno-25-41"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="vi">@adj_list</span><span class="o">.</span><span class="n">include?</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span>
 <a id="__codelineno-25-42" name="__codelineno-25-42" href="#__codelineno-25-42"></a>
-<a id="__codelineno-25-43" name="__codelineno-25-43" href="#__codelineno-25-43"></a><span class="w">    </span><span class="c1"># Add a new linked list in the adjacency list</span>
+<a id="__codelineno-25-43" name="__codelineno-25-43" href="#__codelineno-25-43"></a><span class="w">    </span><span class="c1"># Add a new neighbor list to the adjacency list</span>
 <a id="__codelineno-25-44" name="__codelineno-25-44" href="#__codelineno-25-44"></a><span class="w">    </span><span class="vi">@adj_list</span><span class="o">[</span><span class="n">vet</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-25-45" name="__codelineno-25-45" href="#__codelineno-25-45"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-46" name="__codelineno-25-46" href="#__codelineno-25-46"></a>
-<a id="__codelineno-25-47" name="__codelineno-25-47" href="#__codelineno-25-47"></a><span class="w">  </span><span class="c1">### Delete vertex ###</span>
+<a id="__codelineno-25-47" name="__codelineno-25-47" href="#__codelineno-25-47"></a><span class="w">  </span><span class="c1">### Remove a vertex ###</span>
 <a id="__codelineno-25-48" name="__codelineno-25-48" href="#__codelineno-25-48"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">remove_vertex</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span>
 <a id="__codelineno-25-49" name="__codelineno-25-49" href="#__codelineno-25-49"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">ArgumentError</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="vi">@adj_list</span><span class="o">.</span><span class="n">include?</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span>
 <a id="__codelineno-25-50" name="__codelineno-25-50" href="#__codelineno-25-50"></a>
-<a id="__codelineno-25-51" name="__codelineno-25-51" href="#__codelineno-25-51"></a><span class="w">    </span><span class="c1"># Remove the linked list corresponding to vertex vet in the adjacency list</span>
+<a id="__codelineno-25-51" name="__codelineno-25-51" href="#__codelineno-25-51"></a><span class="w">    </span><span class="c1"># Remove the neighbor list for vertex vet from the adjacency list</span>
 <a id="__codelineno-25-52" name="__codelineno-25-52" href="#__codelineno-25-52"></a><span class="w">    </span><span class="vi">@adj_list</span><span class="o">.</span><span class="n">delete</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span>
-<a id="__codelineno-25-53" name="__codelineno-25-53" href="#__codelineno-25-53"></a><span class="w">    </span><span class="c1"># Traverse the linked lists of other vertices and remove all edges containing vet</span>
+<a id="__codelineno-25-53" name="__codelineno-25-53" href="#__codelineno-25-53"></a><span class="w">    </span><span class="c1"># Traverse the other neighbor lists and remove all edges incident to vet</span>
 <a id="__codelineno-25-54" name="__codelineno-25-54" href="#__codelineno-25-54"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">vertex</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="vi">@adj_list</span>
 <a id="__codelineno-25-55" name="__codelineno-25-55" href="#__codelineno-25-55"></a><span class="w">      </span><span class="vi">@adj_list</span><span class="o">[</span><span class="n">vertex</span><span class="o">.</span><span class="n">first</span><span class="o">].</span><span class="n">delete</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="vi">@adj_list</span><span class="o">[</span><span class="n">vertex</span><span class="o">.</span><span class="n">first</span><span class="o">].</span><span class="n">include?</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span>
 <a id="__codelineno-25-56" name="__codelineno-25-56" href="#__codelineno-25-56"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-25-57" name="__codelineno-25-57" href="#__codelineno-25-57"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-58" name="__codelineno-25-58" href="#__codelineno-25-58"></a>
-<a id="__codelineno-25-59" name="__codelineno-25-59" href="#__codelineno-25-59"></a><span class="w">  </span><span class="c1">### Print adjacency list ###</span>
+<a id="__codelineno-25-59" name="__codelineno-25-59" href="#__codelineno-25-59"></a><span class="w">  </span><span class="c1">### Print the adjacency list ###</span>
 <a id="__codelineno-25-60" name="__codelineno-25-60" href="#__codelineno-25-60"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="n">__print__</span>
 <a id="__codelineno-25-61" name="__codelineno-25-61" href="#__codelineno-25-61"></a><span class="w">    </span><span class="nb">puts</span><span class="w"> </span><span class="s1">'Adjacency list ='</span>
 <a id="__codelineno-25-62" name="__codelineno-25-62" href="#__codelineno-25-62"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">vertex</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="vi">@adj_list</span>
@@ -6994,9 +7071,14 @@
 </div>
 </div>
 </div>
-<h2 id="923-efficiency-comparison">9.2.3 &nbsp; Efficiency Comparison<a class="headerlink" href="#923-efficiency-comparison" title="Permanent link">&para;</a></h2>
-<p>Assuming the graph has <span class="arithmatex">\(n\)</span> vertices and <span class="arithmatex">\(m\)</span> edges, Table 9-2 compares the time efficiency and space efficiency of adjacency matrices and adjacency lists. Note that the adjacency list (linked list) corresponds to the implementation used in this section, while the adjacency list (hash table) refers specifically to the implementation where all linked lists are replaced with hash tables.</p>
-<p align="center"> Table 9-2 &nbsp; Comparison of adjacency matrix and adjacency list </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20Vertex%3A%0A%20%20%20%20%22%22%22Vertex%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%0Adef%20vals_to_vets%28vals%3A%20list%5Bint%5D%29%20-%3E%20list%5B%22Vertex%22%5D%3A%0A%20%20%20%20%22%22%22Given%20the%20list%20of%20values%20vals,%20return%20the%20list%20of%20vertices%20vets%22%22%22%0A%20%20%20%20return%20%5BVertex%28val%29%20for%20val%20in%20vals%5D%0A%0A%0Aclass%20GraphAdjList%3A%0A%20%20%20%20%22%22%22Undirected%20graph%20class%20implemented%20with%20an%20adjacency%20list%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20edges%3A%20list%5Blist%5BVertex%5D%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self.adj_list%20%3D%20dict%5BVertex,%20list%5BVertex%5D%5D%28%29%0A%20%20%20%20%20%20%20%20%23%20Add%20all%20vertices%20and%20edges%0A%20%20%20%20%20%20%20%20for%20edge%20in%20edges%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_vertex%28edge%5B0%5D%29%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_vertex%28edge%5B1%5D%29%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_edge%28edge%5B0%5D,%20edge%5B1%5D%29%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20number%20of%20vertices%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self.adj_list%29%0A%0A%20%20%20%20def%20add_edge%28self,%20vet1%3A%20Vertex,%20vet2%3A%20Vertex%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Add%20an%20edge%22%22%22%0A%20%20%20%20%20%20%20%20if%20vet1%20not%20in%20self.adj_list%20or%20vet2%20not%20in%20self.adj_list%20or%20vet1%20%3D%3D%20vet2%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20ValueError%28%29%0A%20%20%20%20%20%20%20%20%23%20Add%20edge%20vet1%20-%20vet2%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet1%5D.append%28vet2%29%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet2%5D.append%28vet1%29%0A%0A%20%20%20%20def%20remove_edge%28self,%20vet1%3A%20Vertex,%20vet2%3A%20Vertex%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Remove%20an%20edge%22%22%22%0A%20%20%20%20%20%20%20%20if%20vet1%20not%20in%20self.adj_list%20or%20vet2%20not%20in%20self.adj_list%20or%20vet1%20%3D%3D%20vet2%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20ValueError%28%29%0A%20%20%20%20%20%20%20%20%23%20Remove%20edge%20vet1%20-%20vet2%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet1%5D.remove%28vet2%29%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet2%5D.remove%28vet1%29%0A%0A%20%20%20%20def%20add_vertex%28self,%20vet%3A%20Vertex%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Add%20a%20vertex%22%22%22%0A%20%20%20%20%20%20%20%20if%20vet%20in%20self.adj_list%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%23%20Add%20a%20new%20list%20to%20the%20adjacency%20list%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet%5D%20%3D%20%5B%5D%0A%0A%20%20%20%20def%20remove_vertex%28self,%20vet%3A%20Vertex%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Remove%20a%20vertex%22%22%22%0A%20%20%20%20%20%20%20%20if%20vet%20not%20in%20self.adj_list%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20ValueError%28%29%0A%20%20%20%20%20%20%20%20%23%20Remove%20the%20list%20corresponding%20to%20vertex%20vet%20from%20the%20adjacency%20list%0A%20%20%20%20%20%20%20%20self.adj_list.pop%28vet%29%0A%20%20%20%20%20%20%20%20%23%20Traverse%20the%20lists%20of%20the%20other%20vertices%20and%20remove%20all%20edges%20incident%20to%20vet%0A%20%20%20%20%20%20%20%20for%20vertex%20in%20self.adj_list%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20vet%20in%20self.adj_list%5Bvertex%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20self.adj_list%5Bvertex%5D.remove%28vet%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20undirected%20graph%0A%20%20%20%20v%20%3D%20vals_to_vets%28%5B1,%203,%202,%205,%204%5D%29%0A%20%20%20%20edges%20%3D%20%5B%0A%20%20%20%20%20%20%20%20%5Bv%5B0%5D,%20v%5B1%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B0%5D,%20v%5B3%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B1%5D,%20v%5B2%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B2%5D,%20v%5B3%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B2%5D,%20v%5B4%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B3%5D,%20v%5B4%5D%5D,%0A%20%20%20%20%5D%0A%20%20%20%20graph%20%3D%20GraphAdjList%28edges%29%0A%20%20%20%20del%20edges%0A%0A%20%20%20%20%23%20Add%20an%20edge%0A%20%20%20%20graph.add_edge%28v%5B0%5D,%20v%5B2%5D%29%0A%0A%20%20%20%20%23%20Remove%20an%20edge%0A%20%20%20%20graph.remove_edge%28v%5B0%5D,%20v%5B1%5D%29%0A%0A%20%20%20%20%23%20Add%20a%20vertex%0A%20%20%20%20v5%20%3D%20Vertex%286%29%0A%20%20%20%20graph.add_vertex%28v5%29%0A%0A%20%20%20%20%23%20Remove%20a%20vertex%0A%20%20%20%20graph.remove_vertex%28v%5B1%5D%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=39&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20Vertex%3A%0A%20%20%20%20%22%22%22Vertex%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%0Adef%20vals_to_vets%28vals%3A%20list%5Bint%5D%29%20-%3E%20list%5B%22Vertex%22%5D%3A%0A%20%20%20%20%22%22%22Given%20the%20list%20of%20values%20vals,%20return%20the%20list%20of%20vertices%20vets%22%22%22%0A%20%20%20%20return%20%5BVertex%28val%29%20for%20val%20in%20vals%5D%0A%0A%0Aclass%20GraphAdjList%3A%0A%20%20%20%20%22%22%22Undirected%20graph%20class%20implemented%20with%20an%20adjacency%20list%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20edges%3A%20list%5Blist%5BVertex%5D%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self.adj_list%20%3D%20dict%5BVertex,%20list%5BVertex%5D%5D%28%29%0A%20%20%20%20%20%20%20%20%23%20Add%20all%20vertices%20and%20edges%0A%20%20%20%20%20%20%20%20for%20edge%20in%20edges%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_vertex%28edge%5B0%5D%29%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_vertex%28edge%5B1%5D%29%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_edge%28edge%5B0%5D,%20edge%5B1%5D%29%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20number%20of%20vertices%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self.adj_list%29%0A%0A%20%20%20%20def%20add_edge%28self,%20vet1%3A%20Vertex,%20vet2%3A%20Vertex%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Add%20an%20edge%22%22%22%0A%20%20%20%20%20%20%20%20if%20vet1%20not%20in%20self.adj_list%20or%20vet2%20not%20in%20self.adj_list%20or%20vet1%20%3D%3D%20vet2%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20ValueError%28%29%0A%20%20%20%20%20%20%20%20%23%20Add%20edge%20vet1%20-%20vet2%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet1%5D.append%28vet2%29%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet2%5D.append%28vet1%29%0A%0A%20%20%20%20def%20remove_edge%28self,%20vet1%3A%20Vertex,%20vet2%3A%20Vertex%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Remove%20an%20edge%22%22%22%0A%20%20%20%20%20%20%20%20if%20vet1%20not%20in%20self.adj_list%20or%20vet2%20not%20in%20self.adj_list%20or%20vet1%20%3D%3D%20vet2%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20ValueError%28%29%0A%20%20%20%20%20%20%20%20%23%20Remove%20edge%20vet1%20-%20vet2%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet1%5D.remove%28vet2%29%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet2%5D.remove%28vet1%29%0A%0A%20%20%20%20def%20add_vertex%28self,%20vet%3A%20Vertex%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Add%20a%20vertex%22%22%22%0A%20%20%20%20%20%20%20%20if%20vet%20in%20self.adj_list%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%23%20Add%20a%20new%20list%20to%20the%20adjacency%20list%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet%5D%20%3D%20%5B%5D%0A%0A%20%20%20%20def%20remove_vertex%28self,%20vet%3A%20Vertex%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Remove%20a%20vertex%22%22%22%0A%20%20%20%20%20%20%20%20if%20vet%20not%20in%20self.adj_list%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20ValueError%28%29%0A%20%20%20%20%20%20%20%20%23%20Remove%20the%20list%20corresponding%20to%20vertex%20vet%20from%20the%20adjacency%20list%0A%20%20%20%20%20%20%20%20self.adj_list.pop%28vet%29%0A%20%20%20%20%20%20%20%20%23%20Traverse%20the%20lists%20of%20the%20other%20vertices%20and%20remove%20all%20edges%20incident%20to%20vet%0A%20%20%20%20%20%20%20%20for%20vertex%20in%20self.adj_list%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20vet%20in%20self.adj_list%5Bvertex%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20self.adj_list%5Bvertex%5D.remove%28vet%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20undirected%20graph%0A%20%20%20%20v%20%3D%20vals_to_vets%28%5B1,%203,%202,%205,%204%5D%29%0A%20%20%20%20edges%20%3D%20%5B%0A%20%20%20%20%20%20%20%20%5Bv%5B0%5D,%20v%5B1%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B0%5D,%20v%5B3%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B1%5D,%20v%5B2%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B2%5D,%20v%5B3%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B2%5D,%20v%5B4%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B3%5D,%20v%5B4%5D%5D,%0A%20%20%20%20%5D%0A%20%20%20%20graph%20%3D%20GraphAdjList%28edges%29%0A%20%20%20%20del%20edges%0A%0A%20%20%20%20%23%20Add%20an%20edge%0A%20%20%20%20graph.add_edge%28v%5B0%5D,%20v%5B2%5D%29%0A%0A%20%20%20%20%23%20Remove%20an%20edge%0A%20%20%20%20graph.remove_edge%28v%5B0%5D,%20v%5B1%5D%29%0A%0A%20%20%20%20%23%20Add%20a%20vertex%0A%20%20%20%20v5%20%3D%20Vertex%286%29%0A%20%20%20%20graph.add_vertex%28v5%29%0A%0A%20%20%20%20%23%20Remove%20a%20vertex%0A%20%20%20%20graph.remove_vertex%28v%5B1%5D%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=39&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="923-comparing-efficiency">9.2.3 &nbsp; Comparing efficiency<a class="headerlink" href="#923-comparing-efficiency" title="Permanent link">&para;</a></h2>
+<p>For a graph with <span class="arithmatex">\(n\)</span> vertices and <span class="arithmatex">\(m\)</span> edges, Table 9-2 compares the time and space efficiency of adjacency matrices and adjacency lists. Here, "adjacency list (linked lists)" refers to the implementation in this section, while "adjacency list (hash tables)" specifically refers to an implementation in which every linked list is replaced by a hash table.</p>
+<p align="center"> Table 9-2 &nbsp; Comparing adjacency matrices and adjacency lists </p>
 
 <div class="center-table">
 <table>
@@ -7004,13 +7086,13 @@
 <tr>
 <th></th>
 <th>Adjacency matrix</th>
-<th>Adjacency list (linked list)</th>
-<th>Adjacency list (hash table)</th>
+<th>Adjacency list (linked lists)</th>
+<th>Adjacency list (hash tables)</th>
 </tr>
 </thead>
 <tbody>
 <tr>
-<td>Determine adjacency</td>
+<td>Check adjacency</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 <td><span class="arithmatex">\(O(n)\)</span></td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
@@ -7040,7 +7122,7 @@
 <td><span class="arithmatex">\(O(n)\)</span></td>
 </tr>
 <tr>
-<td>Memory space usage</td>
+<td>Memory usage</td>
 <td><span class="arithmatex">\(O(n^2)\)</span></td>
 <td><span class="arithmatex">\(O(n + m)\)</span></td>
 <td><span class="arithmatex">\(O(n + m)\)</span></td>
@@ -7048,7 +7130,7 @@
 </tbody>
 </table>
 </div>
-<p>Observing Table 9-2, it appears that the adjacency list (hash table) has the best time efficiency and space efficiency. However, in practice, operating on edges in the adjacency matrix is more efficient, requiring only a single array access or assignment operation. Overall, adjacency matrices embody the principle of "trading space for time", while adjacency lists embody "trading time for space".</p>
+<p>Table 9-2 suggests that an adjacency list using hash tables is the most efficient in both time and space. In practice, however, edge operations in an adjacency matrix are faster: each requires only a single array access or assignment. Overall, an adjacency matrix trades space for time, while an adjacency list trades time for space.</p>
 
 <!-- Source file information -->
 
@@ -7071,7 +7153,7 @@ aria-label="Footer"
   <a
     href="../graph/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 9.1 Graph"
+    aria-label="Previous: 9.1   Graphs"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -7083,7 +7165,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        9.1 Graph
+        9.1 &nbsp; Graphs
       </div>
     </div>
   </a>
@@ -7095,7 +7177,7 @@ aria-label="Footer"
   <a
     href="../graph_traversal/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 9.3 Graph Traversal"
+    aria-label="Next: 9.3   Graph traversal"
     rel="next"
   >
     <div class="md-footer__title">
@@ -7103,7 +7185,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        9.3 Graph Traversal
+        9.3 &nbsp; Graph traversal
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -7120,7 +7202,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_graph/graph_traversal.assets/graph_dfs_step11.png b/en/chapter_graph/graph_traversal.assets/graph_dfs_step11.png
index dbfc38675..c69d148d2 100644
Binary files a/en/chapter_graph/graph_traversal.assets/graph_dfs_step11.png and b/en/chapter_graph/graph_traversal.assets/graph_dfs_step11.png differ
diff --git a/en/chapter_graph/graph_traversal/index.html b/en/chapter_graph/graph_traversal/index.html
index fc3bdc2f1..89b639988 100644
--- a/en/chapter_graph/graph_traversal/index.html
+++ b/en/chapter_graph/graph_traversal/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>9.3 Graph Traversal - Hello Algo</title>
+        <title>9.3 Graph traversal - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -119,7 +119,7 @@
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     </svg>
-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              9.3 &nbsp; Graph Traversal
+              9.3 &nbsp; Graph traversal
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Dark mode"  type="radio" name="__palette" id="__palette_0">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -177,9 +177,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Light mode"  type="radio" name="__palette" id="__palette_1">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2380,7 +2380,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2401,7 +2401,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2422,7 +2422,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2449,7 +2449,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2485,7 +2485,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2502,7 +2502,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2528,22 +2528,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#931-breadth-first-search" class="md-nav__link">
+  <a href="#931-breadth-first-traversal" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.3.1 &nbsp; Breadth-First Search
+        9.3.1 &nbsp; Breadth-first traversal
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="9.3.1 Breadth-First Search">
+    <nav class="md-nav" aria-label="9.3.1 Breadth-first traversal">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-algorithm-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Algorithm Implementation
+        1. &nbsp; Algorithm implementation
       </span>
     </span>
   </a>
@@ -2554,7 +2554,7 @@
   <a href="#2-complexity-analysis" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Complexity Analysis
+        2. &nbsp; Complexity analysis
       </span>
     </span>
   </a>
@@ -2567,22 +2567,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#932-depth-first-search" class="md-nav__link">
+  <a href="#932-depth-first-traversal" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.3.2 &nbsp; Depth-First Search
+        9.3.2 &nbsp; Depth-first traversal
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="9.3.2 Depth-First Search">
+    <nav class="md-nav" aria-label="9.3.2 Depth-first traversal">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-algorithm-implementation_1" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Algorithm Implementation
+        1. &nbsp; Algorithm implementation
       </span>
     </span>
   </a>
@@ -2593,7 +2593,7 @@
   <a href="#2-complexity-analysis_1" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Complexity Analysis
+        2. &nbsp; Complexity analysis
       </span>
     </span>
   </a>
@@ -2628,7 +2628,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2655,7 +2655,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2723,7 +2723,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2744,7 +2744,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2765,7 +2765,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2792,7 +2792,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2819,7 +2819,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2846,7 +2846,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2873,7 +2873,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2900,7 +2900,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2927,7 +2927,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -3005,7 +3005,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3026,7 +3026,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3047,7 +3047,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3074,7 +3074,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3101,7 +3101,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3128,7 +3128,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3155,7 +3155,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3182,7 +3182,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3209,7 +3209,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3236,7 +3236,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3263,7 +3263,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3290,7 +3290,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3317,7 +3317,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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@@ -3695,7 +3695,7 @@
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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+  15.5 &nbsp; Summary
 
 
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+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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-  <a href="#931-breadth-first-search" class="md-nav__link">
+  <a href="#931-breadth-first-traversal" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.3.1 &nbsp; Breadth-First Search
+        9.3.1 &nbsp; Breadth-first traversal
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="9.3.1 Breadth-First Search">
+    <nav class="md-nav" aria-label="9.3.1 Breadth-first traversal">
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     <span class="md-ellipsis">
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-        1. &nbsp; Algorithm Implementation
+        1. &nbsp; Algorithm implementation
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-        2. &nbsp; Complexity Analysis
+        2. &nbsp; Complexity analysis
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 </li>
 
         <li class="md-nav__item">
-  <a href="#932-depth-first-search" class="md-nav__link">
+  <a href="#932-depth-first-traversal" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        9.3.2 &nbsp; Depth-First Search
+        9.3.2 &nbsp; Depth-first traversal
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="9.3.2 Depth-First Search">
+    <nav class="md-nav" aria-label="9.3.2 Depth-first traversal">
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+        1. &nbsp; Algorithm implementation
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     <span class="md-ellipsis">
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-        2. &nbsp; Complexity Analysis
+        2. &nbsp; Complexity analysis
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 <!-- Page content -->
-<h1 id="93-graph-traversal">9.3 &nbsp; Graph Traversal<a class="headerlink" href="#93-graph-traversal" title="Permanent link">&para;</a></h1>
-<p>Trees represent "one-to-many" relationships, while graphs have a higher degree of freedom and can represent any "many-to-many" relationships. Therefore, we can view trees as a special case of graphs. Clearly, <strong>tree traversal operations are also a special case of graph traversal operations</strong>.</p>
-<p>Both graphs and trees require the application of search algorithms to implement traversal operations. Graph traversal methods can also be divided into two types: <u>breadth-first traversal</u> and <u>depth-first traversal</u>.</p>
-<h2 id="931-breadth-first-search">9.3.1 &nbsp; Breadth-First Search<a class="headerlink" href="#931-breadth-first-search" title="Permanent link">&para;</a></h2>
-<p><strong>Breadth-first search proceeds from near to far: starting from a given node, it always visits the nearest vertices first and expands outward layer by layer</strong>. As shown in Figure 9-9, starting from the top-left vertex, first traverse all adjacent vertices of that vertex, then traverse all adjacent vertices of the next vertex, and so on, until all vertices have been visited.</p>
-<p><img alt="Breadth-first search of a graph" class="animation-figure" src="../graph_traversal.assets/graph_bfs.png" /></p>
-<p align="center"> Figure 9-9 &nbsp; Breadth-first search of a graph </p>
+<h1 id="93-graph-traversal">9.3 &nbsp; Graph traversal<a class="headerlink" href="#93-graph-traversal" title="Permanent link">&para;</a></h1>
+<p>A tree represents one-to-many relationships, whereas a graph allows more freedom and can represent arbitrary many-to-many relationships. We can therefore view a tree as a special case of a graph. Accordingly, <strong>tree traversal is also a special case of graph traversal</strong>.</p>
+<p>Both graphs and trees use search algorithms for traversal. Graph traversal can likewise be divided into two types: <u>breadth-first traversal</u> and <u>depth-first traversal</u>.</p>
+<h2 id="931-breadth-first-traversal">9.3.1 &nbsp; Breadth-first traversal<a class="headerlink" href="#931-breadth-first-traversal" title="Permanent link">&para;</a></h2>
+<p><strong>Breadth-first traversal visits vertices from near to far. Starting from a vertex, it always visits the nearest vertices first and expands outward layer by layer</strong>. In Figure 9-9, we start at the top-left vertex and visit all its adjacent vertices, then visit all the adjacent vertices of the next vertex, and so on until every vertex has been visited.</p>
+<p><img alt="Breadth-first traversal of a graph" class="animation-figure" src="../graph_traversal.assets/graph_bfs.png" /></p>
+<p align="center"> Figure 9-9 &nbsp; Breadth-first traversal of a graph </p>
 
-<h3 id="1-algorithm-implementation">1. &nbsp; Algorithm Implementation<a class="headerlink" href="#1-algorithm-implementation" title="Permanent link">&para;</a></h3>
-<p>BFS is typically implemented with the help of a queue, as shown in the code below. The queue has a "first in, first out" property, which aligns with the BFS idea of "near to far".</p>
+<h3 id="1-algorithm-implementation">1. &nbsp; Algorithm implementation<a class="headerlink" href="#1-algorithm-implementation" title="Permanent link">&para;</a></h3>
+<p>BFS is usually implemented with a queue, as shown in the code below. A queue's first-in, first-out behavior fits the near-to-far order of BFS.</p>
 <ol>
-<li>Add the starting vertex <code>startVet</code> to the queue and begin the loop.</li>
-<li>In each iteration of the loop, pop the vertex at the front of the queue and record it as visited, then add all adjacent vertices of that vertex to the back of the queue.</li>
-<li>Repeat step <code>2.</code> until all vertices have been visited.</li>
+<li>Enqueue the starting vertex <code>startVet</code> and begin the loop.</li>
+<li>In each iteration, dequeue the vertex at the front and record the visit, then enqueue all its adjacent vertices at the rear.</li>
+<li>Repeat step <code>2.</code> until every vertex has been visited.</li>
 </ol>
-<p>To prevent revisiting vertices, we use a hash set <code>visited</code> to record which nodes have been visited.</p>
+<p>To avoid visiting a vertex more than once, we use a hash set <code>visited</code> to track which vertices have already been visited.</p>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>A hash set can be viewed as a hash table that stores only <code>key</code> without storing <code>value</code>. It supports insertion, deletion, lookup, and update operations on <code>key</code> in <span class="arithmatex">\(O(1)\)</span> time. Based on the uniqueness of <code>key</code>, hash sets are typically used for data deduplication and similar scenarios.</p>
+<p>A hash set can be viewed as a hash table that stores only <code>key</code> entries, with no <code>value</code> entries. Adding, removing, looking up, and modifying a <code>key</code> take <span class="arithmatex">\(O(1)\)</span> time. Because every <code>key</code> is unique, hash sets are commonly used to remove duplicate data.</p>
 </div>
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 <div class="highlight"><span class="filename">graph_bfs.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">graph_bfs</span><span class="p">(</span><span class="n">graph</span><span class="p">:</span> <span class="n">GraphAdjList</span><span class="p">,</span> <span class="n">start_vet</span><span class="p">:</span> <span class="n">Vertex</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="n">Vertex</span><span class="p">]:</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Breadth-first traversal"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="c1"># Vertex traversal sequence</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Hash set recording visited vertices</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="n">visited</span> <span class="o">=</span> <span class="nb">set</span><span class="p">[</span><span class="n">Vertex</span><span class="p">]([</span><span class="n">start_vet</span><span class="p">])</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="c1"># Queue used to implement BFS</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="c1"># Use a queue to implement BFS</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="n">que</span> <span class="o">=</span> <span class="n">deque</span><span class="p">[</span><span class="n">Vertex</span><span class="p">]([</span><span class="n">start_vet</span><span class="p">])</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="c1"># Starting from vertex vet, loop until all vertices are visited</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="c1"># Starting from start_vet, loop until all reachable vertices have been visited</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="k">while</span> <span class="nb">len</span><span class="p">(</span><span class="n">que</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="n">vet</span> <span class="o">=</span> <span class="n">que</span><span class="o">.</span><span class="n">popleft</span><span class="p">()</span>  <span class="c1"># Dequeue the front vertex</span>
-<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span>  <span class="c1"># Record visited vertex</span>
-<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>        <span class="c1"># Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span>  <span class="c1"># Record the visited vertex</span>
+<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>        <span class="c1"># Traverse all neighbors of this vertex</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>        <span class="k">for</span> <span class="n">adj_vet</span> <span class="ow">in</span> <span class="n">graph</span><span class="o">.</span><span class="n">adj_list</span><span class="p">[</span><span class="n">vet</span><span class="p">]:</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>            <span class="k">if</span> <span class="n">adj_vet</span> <span class="ow">in</span> <span class="n">visited</span><span class="p">:</span>
-<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>                <span class="k">continue</span>  <span class="c1"># Skip vertices that have been visited</span>
-<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>            <span class="n">que</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">adj_vet</span><span class="p">)</span>  <span class="c1"># Only enqueue unvisited vertices</span>
+<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>                <span class="k">continue</span>  <span class="c1"># Skip vertices that have already been visited</span>
+<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>            <span class="n">que</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">adj_vet</span><span class="p">)</span>  <span class="c1"># Enqueue only unvisited vertices</span>
 <a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>            <span class="n">visited</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">adj_vet</span><span class="p">)</span>  <span class="c1"># Mark this vertex as visited</span>
-<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>    <span class="c1"># Return vertex traversal sequence</span>
+<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>    <span class="c1"># Return the vertex traversal sequence</span>
 <a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a>    <span class="k">return</span> <span class="n">res</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_bfs.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Breadth-first traversal */</span>
-<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="n">vector</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="n">graphBFS</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="o">&amp;</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">startVet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
-<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="n">unordered_set</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="n">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="n">startVet</span><span class="p">};</span>
-<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="c1">// Queue used to implement BFS</span>
+<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="c1">// Use a queue to implement BFS</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="n">queue</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="n">que</span><span class="p">;</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">    </span><span class="n">que</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">startVet</span><span class="p">);</span>
-<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">    </span><span class="c1">// Starting from vertex vet, loop until all vertices are visited</span>
+<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">    </span><span class="c1">// Starting from startVet, continue until all reachable vertices have been visited</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">que</span><span class="p">.</span><span class="n">empty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">que</span><span class="p">.</span><span class="n">front</span><span class="p">();</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">        </span><span class="n">que</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span><span class="w">          </span><span class="c1">// Dequeue the front vertex</span>
-<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record visited vertex</span>
-<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">        </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the visited vertex</span>
+<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">        </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="k">auto</span><span class="w"> </span><span class="n">adjVet</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="n">adjList</span><span class="p">[</span><span class="n">vet</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">visited</span><span class="p">.</span><span class="n">count</span><span class="p">(</span><span class="n">adjVet</span><span class="p">))</span>
-<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">                </span><span class="k">continue</span><span class="p">;</span><span class="w">            </span><span class="c1">// Skip vertices that have been visited</span>
-<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">            </span><span class="n">que</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">adjVet</span><span class="p">);</span><span class="w">        </span><span class="c1">// Only enqueue unvisited vertices</span>
+<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">                </span><span class="k">continue</span><span class="p">;</span><span class="w">            </span><span class="c1">// Skip vertices that have already been visited</span>
+<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">            </span><span class="n">que</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">adjVet</span><span class="p">);</span><span class="w">        </span><span class="c1">// Enqueue only unvisited vertices</span>
 <a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w">            </span><span class="n">visited</span><span class="p">.</span><span class="n">emplace</span><span class="p">(</span><span class="n">adjVet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
 <a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="w">    </span><span class="c1">// Return vertex traversal sequence</span>
+<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="w">    </span><span class="c1">// Return the vertex traversal sequence</span>
 <a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_bfs.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Breadth-first traversal */</span>
-<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="nf">graphBFS</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">startVet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
-<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="n">Set</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">HashSet</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="n">visited</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">startVet</span><span class="p">);</span>
-<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="c1">// Queue used to implement BFS</span>
+<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="c1">// Use a queue to implement BFS</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="n">Queue</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">que</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">LinkedList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="n">que</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">startVet</span><span class="p">);</span>
-<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="c1">// Starting from vertex vet, loop until all vertices are visited</span>
+<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="c1">// Starting from startVet, loop until all reachable vertices have been visited</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">que</span><span class="p">.</span><span class="na">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">        </span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">que</span><span class="p">.</span><span class="na">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// Dequeue the front vertex</span>
-<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w">            </span><span class="c1">// Record visited vertex</span>
-<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">        </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w">            </span><span class="c1">// Record the visited vertex</span>
+<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">        </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">adjVet</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="na">adjList</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">vet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">visited</span><span class="p">.</span><span class="na">contains</span><span class="p">(</span><span class="n">adjVet</span><span class="p">))</span>
-<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">                </span><span class="k">continue</span><span class="p">;</span><span class="w">        </span><span class="c1">// Skip vertices that have been visited</span>
-<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">            </span><span class="n">que</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">adjVet</span><span class="p">);</span><span class="w">   </span><span class="c1">// Only enqueue unvisited vertices</span>
+<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">                </span><span class="k">continue</span><span class="p">;</span><span class="w">        </span><span class="c1">// Skip vertices that have already been visited</span>
+<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">            </span><span class="n">que</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">adjVet</span><span class="p">);</span><span class="w">   </span><span class="c1">// Enqueue only unvisited vertices</span>
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">            </span><span class="n">visited</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">adjVet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
 <a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="w">    </span><span class="c1">// Return vertex traversal sequence</span>
+<a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="w">    </span><span class="c1">// Return the vertex traversal sequence</span>
 <a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_bfs.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Breadth-first traversal */</span>
-<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">GraphBFS</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">startVet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="n">HashSet</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="n">startVet</span><span class="p">];</span>
-<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="c1">// Queue used to implement BFS</span>
+<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="c1">// Use a queue to implement BFS</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="n">Queue</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">que</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">();</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="n">que</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="n">startVet</span><span class="p">);</span>
-<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">    </span><span class="c1">// Starting from vertex vet, loop until all vertices are visited</span>
+<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">    </span><span class="c1">// Starting from startVet, loop until all reachable vertices have been visited</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">que</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">que</span><span class="p">.</span><span class="n">Dequeue</span><span class="p">();</span><span class="w"> </span><span class="c1">// Dequeue the front vertex</span>
-<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w">               </span><span class="c1">// Record visited vertex</span>
+<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w">               </span><span class="c1">// Record the visited vertex</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">adjVet</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="n">adjList</span><span class="p">[</span><span class="n">vet</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">visited</span><span class="p">.</span><span class="n">Contains</span><span class="p">(</span><span class="n">adjVet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">                </span><span class="k">continue</span><span class="p">;</span><span class="w">          </span><span class="c1">// Skip vertices that have been visited</span>
+<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">                </span><span class="k">continue</span><span class="p">;</span><span class="w">          </span><span class="c1">// Skip vertices that have already been visited</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">            </span><span class="n">que</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="n">adjVet</span><span class="p">);</span><span class="w">   </span><span class="c1">// Only enqueue unvisited vertices</span>
+<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">            </span><span class="n">que</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="n">adjVet</span><span class="p">);</span><span class="w">   </span><span class="c1">// Enqueue only unvisited vertices</span>
 <a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">            </span><span class="n">visited</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">adjVet</span><span class="p">);</span><span class="w">   </span><span class="c1">// Mark this vertex as visited</span>
 <a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a>
-<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w">    </span><span class="c1">// Return vertex traversal sequence</span>
+<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w">    </span><span class="c1">// Return the vertex traversal sequence</span>
 <a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_bfs.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Breadth-first traversal */</span>
-<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="kd">func</span><span class="w"> </span><span class="nx">graphBFS</span><span class="p">(</span><span class="nx">g</span><span class="w"> </span><span class="o">*</span><span class="nx">graphAdjList</span><span class="p">,</span><span class="w"> </span><span class="nx">startVet</span><span class="w"> </span><span class="nx">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">[]</span><span class="nx">Vertex</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="nx">Vertex</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="nx">visited</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">(</span><span class="kt">map</span><span class="p">[</span><span class="nx">Vertex</span><span class="p">]</span><span class="kd">struct</span><span class="p">{})</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="nx">visited</span><span class="p">[</span><span class="nx">startVet</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kd">struct</span><span class="p">{}{}</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// Queue used to implement BFS, using slice to simulate queue</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// Use a queue for BFS, represented here by a slice</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="nx">queue</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="nx">Vertex</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">    </span><span class="nx">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">queue</span><span class="p">,</span><span class="w"> </span><span class="nx">startVet</span><span class="p">)</span>
-<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="c1">// Starting from vertex vet, loop until all vertices are visited</span>
+<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="c1">// Starting from startVet, loop until all reachable vertices have been visited</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">queue</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="c1">// Dequeue the front vertex</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="nx">vet</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">queue</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">        </span><span class="nx">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">queue</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
-<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">        </span><span class="c1">// Record visited vertex</span>
+<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">        </span><span class="c1">// Record the visited vertex</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">vet</span><span class="p">)</span>
-<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">        </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">        </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">adjVet</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">vet</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">            </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">isExist</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">visited</span><span class="p">[</span><span class="nx">adjVet</span><span class="p">]</span>
-<a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">            </span><span class="c1">// Only enqueue unvisited vertices</span>
+<a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">            </span><span class="c1">// Enqueue only unvisited vertices</span>
 <a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="nx">isExist</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="w">                </span><span class="nx">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">queue</span><span class="p">,</span><span class="w"> </span><span class="nx">adjVet</span><span class="p">)</span>
 <a id="__codelineno-4-25" name="__codelineno-4-25" href="#__codelineno-4-25"></a><span class="w">                </span><span class="nx">visited</span><span class="p">[</span><span class="nx">adjVet</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kd">struct</span><span class="p">{}{}</span>
 <a id="__codelineno-4-26" name="__codelineno-4-26" href="#__codelineno-4-26"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-4-27" name="__codelineno-4-27" href="#__codelineno-4-27"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-4-28" name="__codelineno-4-28" href="#__codelineno-4-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-29" name="__codelineno-4-29" href="#__codelineno-4-29"></a><span class="w">    </span><span class="c1">// Return vertex traversal sequence</span>
+<a id="__codelineno-4-29" name="__codelineno-4-29" href="#__codelineno-4-29"></a><span class="w">    </span><span class="c1">// Return the vertex traversal sequence</span>
 <a id="__codelineno-4-30" name="__codelineno-4-30" href="#__codelineno-4-30"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span>
 <a id="__codelineno-4-31" name="__codelineno-4-31" href="#__codelineno-4-31"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_bfs.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Breadth-first traversal */</span>
-<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="kd">func</span><span class="w"> </span><span class="nf">graphBFS</span><span class="p">(</span><span class="n">graph</span><span class="p">:</span><span class="w"> </span><span class="n">GraphAdjList</span><span class="p">,</span><span class="w"> </span><span class="n">startVet</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="n">Vertex</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="n">Vertex</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
-<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">visited</span><span class="p">:</span><span class="w"> </span><span class="n">Set</span><span class="p">&lt;</span><span class="n">Vertex</span><span class="p">&gt;</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[</span><span class="n">startVet</span><span class="p">]</span>
-<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="c1">// Queue used to implement BFS</span>
+<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="c1">// Use a queue to implement BFS</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">que</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="n">Vertex</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[</span><span class="n">startVet</span><span class="p">]</span>
-<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">    </span><span class="c1">// Starting from vertex vet, loop until all vertices are visited</span>
+<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">    </span><span class="c1">// Starting from startVet, loop until all reachable vertices have been visited</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="o">!</span><span class="n">que</span><span class="p">.</span><span class="bp">isEmpty</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">vet</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">que</span><span class="p">.</span><span class="n">removeFirst</span><span class="p">()</span><span class="w"> </span><span class="c1">// Dequeue the front vertex</span>
-<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="c1">// Record visited vertex</span>
-<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="c1">// Record the visited vertex</span>
+<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">adjVet</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="n">adjList</span><span class="p">[</span><span class="n">vet</span><span class="p">]</span><span class="w"> </span><span class="p">??</span><span class="w"> </span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">visited</span><span class="p">.</span><span class="bp">contains</span><span class="p">(</span><span class="n">adjVet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">                </span><span class="k">continue</span><span class="w"> </span><span class="c1">// Skip vertices that have been visited</span>
+<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">                </span><span class="k">continue</span><span class="w"> </span><span class="c1">// Skip vertices that have already been visited</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">            </span><span class="n">que</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">adjVet</span><span class="p">)</span><span class="w"> </span><span class="c1">// Only enqueue unvisited vertices</span>
+<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">            </span><span class="n">que</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">adjVet</span><span class="p">)</span><span class="w"> </span><span class="c1">// Enqueue only unvisited vertices</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w">            </span><span class="n">visited</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="n">adjVet</span><span class="p">)</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
 <a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="w">    </span><span class="c1">// Return vertex traversal sequence</span>
+<a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="w">    </span><span class="c1">// Return the vertex traversal sequence</span>
 <a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_bfs.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Breadth-first traversal */</span>
-<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="kd">function</span><span class="w"> </span><span class="nx">graphBFS</span><span class="p">(</span><span class="nx">graph</span><span class="p">,</span><span class="w"> </span><span class="nx">startVet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Set</span><span class="p">();</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="nx">visited</span><span class="p">.</span><span class="nx">add</span><span class="p">(</span><span class="nx">startVet</span><span class="p">);</span>
-<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="c1">// Queue used to implement BFS</span>
+<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="c1">// Use a queue to implement BFS</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">que</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="nx">startVet</span><span class="p">];</span>
-<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">    </span><span class="c1">// Starting from vertex vet, loop until all vertices are visited</span>
+<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">    </span><span class="c1">// Starting from startVet, loop until all reachable vertices have been visited</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">que</span><span class="p">.</span><span class="nx">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">vet</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">que</span><span class="p">.</span><span class="nx">shift</span><span class="p">();</span><span class="w"> </span><span class="c1">// Dequeue the front vertex</span>
-<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record visited vertex</span>
-<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">        </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the visited vertex</span>
+<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">        </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">adjVet</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">graph</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet</span><span class="p">)</span><span class="w"> </span><span class="o">??</span><span class="w"> </span><span class="p">[])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">visited</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">adjVet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">                </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have been visited</span>
+<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">                </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have already been visited</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">            </span><span class="nx">que</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">adjVet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Only enqueue unvisited vertices</span>
+<a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">            </span><span class="nx">que</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">adjVet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Enqueue only unvisited vertices</span>
 <a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">            </span><span class="nx">visited</span><span class="p">.</span><span class="nx">add</span><span class="p">(</span><span class="nx">adjVet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a><span class="w">    </span><span class="c1">// Return vertex traversal sequence</span>
+<a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a><span class="w">    </span><span class="c1">// Return the vertex traversal sequence</span>
 <a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-6-26" name="__codelineno-6-26" href="#__codelineno-6-26"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_bfs.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Breadth-first traversal */</span>
-<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="kd">function</span><span class="w"> </span><span class="nx">graphBFS</span><span class="p">(</span><span class="nx">graph</span><span class="o">:</span><span class="w"> </span><span class="kt">GraphAdjList</span><span class="p">,</span><span class="w"> </span><span class="nx">startVet</span><span class="o">:</span><span class="w"> </span><span class="kt">Vertex</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="nx">Vertex</span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">Vertex</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">visited</span><span class="o">:</span><span class="w"> </span><span class="kt">Set</span><span class="o">&lt;</span><span class="nx">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Set</span><span class="p">();</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="nx">visited</span><span class="p">.</span><span class="nx">add</span><span class="p">(</span><span class="nx">startVet</span><span class="p">);</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// Queue used to implement BFS</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// Use a queue to implement BFS</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">que</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="nx">startVet</span><span class="p">];</span>
-<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="c1">// Starting from vertex vet, loop until all vertices are visited</span>
+<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="c1">// Starting from startVet, loop until all reachable vertices have been visited</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">que</span><span class="p">.</span><span class="nx">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">vet</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">que</span><span class="p">.</span><span class="nx">shift</span><span class="p">();</span><span class="w"> </span><span class="c1">// Dequeue the front vertex</span>
-<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record visited vertex</span>
-<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">        </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the visited vertex</span>
+<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">        </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">adjVet</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">graph</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet</span><span class="p">)</span><span class="w"> </span><span class="o">??</span><span class="w"> </span><span class="p">[])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">visited</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">adjVet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">                </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have been visited</span>
+<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">                </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have already been visited</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">            </span><span class="nx">que</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">adjVet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Only enqueue unvisited</span>
+<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">            </span><span class="nx">que</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">adjVet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Enqueue only unvisited vertices</span>
 <a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">            </span><span class="nx">visited</span><span class="p">.</span><span class="nx">add</span><span class="p">(</span><span class="nx">adjVet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
 <a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">    </span><span class="c1">// Return vertex traversal sequence</span>
+<a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">    </span><span class="c1">// Return the vertex traversal sequence</span>
 <a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
 <a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="p">}</span>
 </code></pre></div>
@@ -5025,67 +5097,67 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_bfs.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Breadth-first traversal */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">graphBFS</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">startVet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="n">Set</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{};</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="n">visited</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">startVet</span><span class="p">);</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="c1">// Queue used to implement BFS</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="c1">// Use a queue to implement BFS</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">  </span><span class="n">Queue</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">que</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Queue</span><span class="p">();</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="n">que</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">startVet</span><span class="p">);</span>
-<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="c1">// Starting from vertex vet, loop until all vertices are visited</span>
+<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="c1">// Starting from startVet, loop until all reachable vertices have been visited</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">que</span><span class="p">.</span><span class="n">isNotEmpty</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">que</span><span class="p">.</span><span class="n">removeFirst</span><span class="p">();</span><span class="w"> </span><span class="c1">// Dequeue the front vertex</span>
-<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record visited vertex</span>
-<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">    </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the visited vertex</span>
+<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">    </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">adjVet</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="n">adjList</span><span class="p">[</span><span class="n">vet</span><span class="p">]</span><span class="o">!</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">visited</span><span class="p">.</span><span class="n">contains</span><span class="p">(</span><span class="n">adjVet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">        </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have been visited</span>
+<a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">        </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have already been visited</span>
 <a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">      </span><span class="p">}</span>
-<a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="w">      </span><span class="n">que</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">adjVet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Only enqueue unvisited vertices</span>
+<a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="w">      </span><span class="n">que</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">adjVet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Enqueue only unvisited vertices</span>
 <a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="w">      </span><span class="n">visited</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">adjVet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
 <a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="w">  </span><span class="c1">// Return vertex traversal sequence</span>
+<a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="w">  </span><span class="c1">// Return the vertex traversal sequence</span>
 <a id="__codelineno-8-26" name="__codelineno-8-26" href="#__codelineno-8-26"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_bfs.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Breadth-first traversal */</span>
-<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="k">fn</span><span class="w"> </span><span class="nf">graph_bfs</span><span class="p">(</span><span class="n">graph</span><span class="p">:</span><span class="w"> </span><span class="nc">GraphAdjList</span><span class="p">,</span><span class="w"> </span><span class="n">start_vet</span><span class="p">:</span><span class="w"> </span><span class="nc">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[];</span>
-<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashSet</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="n">visited</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="n">start_vet</span><span class="p">);</span>
-<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// Queue used to implement BFS</span>
+<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// Use a queue to implement BFS</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">que</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">VecDeque</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="n">que</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">start_vet</span><span class="p">);</span>
-<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="c1">// Starting from vertex vet, loop until all vertices are visited</span>
+<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="c1">// Start at start_vet and loop until all reachable vertices have been visited</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">que</span><span class="p">.</span><span class="n">pop_front</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record visited vertex</span>
+<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the visited vertex</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a>
-<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">        </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">        </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">adj_vets</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="n">adj_list</span><span class="p">.</span><span class="n">get</span><span class="p">(</span><span class="o">&amp;</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">            </span><span class="k">for</span><span class="w"> </span><span class="o">&amp;</span><span class="n">adj_vet</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">adj_vets</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">                </span><span class="k">if</span><span class="w"> </span><span class="n">visited</span><span class="p">.</span><span class="n">contains</span><span class="p">(</span><span class="o">&amp;</span><span class="n">adj_vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">                    </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have been visited</span>
+<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">                    </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have already been visited</span>
 <a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">                </span><span class="p">}</span>
-<a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="w">                </span><span class="n">que</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">adj_vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Only enqueue unvisited vertices</span>
+<a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="w">                </span><span class="n">que</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">adj_vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Enqueue only unvisited vertices</span>
 <a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="w">                </span><span class="n">visited</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="n">adj_vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
 <a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-26" name="__codelineno-9-26" href="#__codelineno-9-26"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="w">    </span><span class="c1">// Return vertex traversal sequence</span>
+<a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="w">    </span><span class="c1">// Return the vertex traversal sequence</span>
 <a id="__codelineno-9-28" name="__codelineno-9-28" href="#__codelineno-9-28"></a><span class="w">    </span><span class="n">res</span>
 <a id="__codelineno-9-29" name="__codelineno-9-29" href="#__codelineno-9-29"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_bfs.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Node queue structure */</span>
+<div class="highlight"><span class="filename">graph_bfs.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Vertex queue structure */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vertices</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">];</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="n">rear</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">;</span>
@@ -5098,7 +5170,7 @@
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">q</span><span class="p">;</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="p">}</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a>
-<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="kt">int</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">(</span><span class="n">Queue</span><span class="w"> </span><span class="o">*</span><span class="n">q</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">q</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="p">}</span>
@@ -5118,9 +5190,9 @@
 <a id="__codelineno-10-31" name="__codelineno-10-31" href="#__codelineno-10-31"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">vet</span><span class="p">;</span>
 <a id="__codelineno-10-32" name="__codelineno-10-32" href="#__codelineno-10-32"></a><span class="p">}</span>
 <a id="__codelineno-10-33" name="__codelineno-10-33" href="#__codelineno-10-33"></a>
-<a id="__codelineno-10-34" name="__codelineno-10-34" href="#__codelineno-10-34"></a><span class="cm">/* Check if vertex has been visited */</span>
+<a id="__codelineno-10-34" name="__codelineno-10-34" href="#__codelineno-10-34"></a><span class="cm">/* Check whether a vertex has been visited */</span>
 <a id="__codelineno-10-35" name="__codelineno-10-35" href="#__codelineno-10-35"></a><span class="kt">int</span><span class="w"> </span><span class="nf">isVisited</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="o">**</span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-36" name="__codelineno-10-36" href="#__codelineno-10-36"></a><span class="w">    </span><span class="c1">// Traverse to find node using O(n) time</span>
+<a id="__codelineno-10-36" name="__codelineno-10-36" href="#__codelineno-10-36"></a><span class="w">    </span><span class="c1">// Search for the vertex with a linear scan in O(n) time</span>
 <a id="__codelineno-10-37" name="__codelineno-10-37" href="#__codelineno-10-37"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-38" name="__codelineno-10-38" href="#__codelineno-10-38"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">visited</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span>
 <a id="__codelineno-10-39" name="__codelineno-10-39" href="#__codelineno-10-39"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -5129,22 +5201,22 @@
 <a id="__codelineno-10-42" name="__codelineno-10-42" href="#__codelineno-10-42"></a><span class="p">}</span>
 <a id="__codelineno-10-43" name="__codelineno-10-43" href="#__codelineno-10-43"></a>
 <a id="__codelineno-10-44" name="__codelineno-10-44" href="#__codelineno-10-44"></a><span class="cm">/* Breadth-first traversal */</span>
-<a id="__codelineno-10-45" name="__codelineno-10-45" href="#__codelineno-10-45"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-10-45" name="__codelineno-10-45" href="#__codelineno-10-45"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-10-46" name="__codelineno-10-46" href="#__codelineno-10-46"></a><span class="kt">void</span><span class="w"> </span><span class="nf">graphBFS</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="o">*</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">startVet</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">**</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">resSize</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">**</span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">visitedSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-47" name="__codelineno-10-47" href="#__codelineno-10-47"></a><span class="w">    </span><span class="c1">// Queue used to implement BFS</span>
+<a id="__codelineno-10-47" name="__codelineno-10-47" href="#__codelineno-10-47"></a><span class="w">    </span><span class="c1">// Use a queue to implement BFS</span>
 <a id="__codelineno-10-48" name="__codelineno-10-48" href="#__codelineno-10-48"></a><span class="w">    </span><span class="n">Queue</span><span class="w"> </span><span class="o">*</span><span class="n">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newQueue</span><span class="p">();</span>
 <a id="__codelineno-10-49" name="__codelineno-10-49" href="#__codelineno-10-49"></a><span class="w">    </span><span class="n">enqueue</span><span class="p">(</span><span class="n">queue</span><span class="p">,</span><span class="w"> </span><span class="n">startVet</span><span class="p">);</span>
 <a id="__codelineno-10-50" name="__codelineno-10-50" href="#__codelineno-10-50"></a><span class="w">    </span><span class="n">visited</span><span class="p">[(</span><span class="o">*</span><span class="n">visitedSize</span><span class="p">)</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">startVet</span><span class="p">;</span>
-<a id="__codelineno-10-51" name="__codelineno-10-51" href="#__codelineno-10-51"></a><span class="w">    </span><span class="c1">// Starting from vertex vet, loop until all vertices are visited</span>
+<a id="__codelineno-10-51" name="__codelineno-10-51" href="#__codelineno-10-51"></a><span class="w">    </span><span class="c1">// Starting from startVet, continue until all reachable vertices have been visited</span>
 <a id="__codelineno-10-52" name="__codelineno-10-52" href="#__codelineno-10-52"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">isEmpty</span><span class="p">(</span><span class="n">queue</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-53" name="__codelineno-10-53" href="#__codelineno-10-53"></a><span class="w">        </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dequeue</span><span class="p">(</span><span class="n">queue</span><span class="p">);</span><span class="w"> </span><span class="c1">// Dequeue the front vertex</span>
-<a id="__codelineno-10-54" name="__codelineno-10-54" href="#__codelineno-10-54"></a><span class="w">        </span><span class="n">res</span><span class="p">[(</span><span class="o">*</span><span class="n">resSize</span><span class="p">)</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">vet</span><span class="p">;</span><span class="w">      </span><span class="c1">// Record visited vertex</span>
-<a id="__codelineno-10-55" name="__codelineno-10-55" href="#__codelineno-10-55"></a><span class="w">        </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-10-54" name="__codelineno-10-54" href="#__codelineno-10-54"></a><span class="w">        </span><span class="n">res</span><span class="p">[(</span><span class="o">*</span><span class="n">resSize</span><span class="p">)</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">vet</span><span class="p">;</span><span class="w">      </span><span class="c1">// Record the visited vertex</span>
+<a id="__codelineno-10-55" name="__codelineno-10-55" href="#__codelineno-10-55"></a><span class="w">        </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-10-56" name="__codelineno-10-56" href="#__codelineno-10-56"></a><span class="w">        </span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findNode</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">vet</span><span class="p">);</span>
 <a id="__codelineno-10-57" name="__codelineno-10-57" href="#__codelineno-10-57"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-58" name="__codelineno-10-58" href="#__codelineno-10-58"></a><span class="w">            </span><span class="c1">// Skip vertices that have been visited</span>
+<a id="__codelineno-10-58" name="__codelineno-10-58" href="#__codelineno-10-58"></a><span class="w">            </span><span class="c1">// Skip vertices that have already been visited</span>
 <a id="__codelineno-10-59" name="__codelineno-10-59" href="#__codelineno-10-59"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">isVisited</span><span class="p">(</span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">visitedSize</span><span class="p">,</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">vertex</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-60" name="__codelineno-10-60" href="#__codelineno-10-60"></a><span class="w">                </span><span class="n">enqueue</span><span class="p">(</span><span class="n">queue</span><span class="p">,</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">vertex</span><span class="p">);</span><span class="w">             </span><span class="c1">// Only enqueue unvisited vertices</span>
+<a id="__codelineno-10-60" name="__codelineno-10-60" href="#__codelineno-10-60"></a><span class="w">                </span><span class="n">enqueue</span><span class="p">(</span><span class="n">queue</span><span class="p">,</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">vertex</span><span class="p">);</span><span class="w">             </span><span class="c1">// Enqueue only unvisited vertices</span>
 <a id="__codelineno-10-61" name="__codelineno-10-61" href="#__codelineno-10-61"></a><span class="w">                </span><span class="n">visited</span><span class="p">[(</span><span class="o">*</span><span class="n">visitedSize</span><span class="p">)</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">vertex</span><span class="p">;</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
 <a id="__codelineno-10-62" name="__codelineno-10-62" href="#__codelineno-10-62"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-10-63" name="__codelineno-10-63" href="#__codelineno-10-63"></a><span class="w">            </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
@@ -5157,29 +5229,29 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_bfs.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Breadth-first traversal */</span>
-<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">graphBFS</span><span class="p">(</span><span class="n">graph</span><span class="p">:</span><span class="w"> </span><span class="n">GraphAdjList</span><span class="p">,</span><span class="w"> </span><span class="n">startVet</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">):</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">Vertex?</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">Vertex?</span><span class="o">&gt;</span><span class="p">()</span>
-<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashSet</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="n">visited</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">startVet</span><span class="p">)</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// Queue used to implement BFS</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// Use a queue to implement BFS</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">que</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">LinkedList</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="n">que</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">startVet</span><span class="p">)</span>
-<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="c1">// Starting from vertex vet, loop until all vertices are visited</span>
+<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="c1">// Starting from startVet, loop until all reachable vertices have been visited</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">que</span><span class="p">.</span><span class="na">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">vet</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">que</span><span class="p">.</span><span class="na">poll</span><span class="p">()</span><span class="w"> </span><span class="c1">// Dequeue the front vertex</span>
-<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span><span class="w">         </span><span class="c1">// Record visited vertex</span>
-<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">        </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">        </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span><span class="w">         </span><span class="c1">// Record the visited vertex</span>
+<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">        </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">adjVet</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="na">adjList</span><span class="o">[</span><span class="n">vet</span><span class="o">]!!</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">visited</span><span class="p">.</span><span class="na">contains</span><span class="p">(</span><span class="n">adjVet</span><span class="p">))</span>
-<a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">                </span><span class="k">continue</span><span class="w">        </span><span class="c1">// Skip vertices that have been visited</span>
-<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">            </span><span class="n">que</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">adjVet</span><span class="p">)</span><span class="w">   </span><span class="c1">// Only enqueue unvisited vertices</span>
+<a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">                </span><span class="k">continue</span><span class="w">        </span><span class="c1">// Skip vertices that have already been visited</span>
+<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">            </span><span class="n">que</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">adjVet</span><span class="p">)</span><span class="w">   </span><span class="c1">// Enqueue only unvisited vertices</span>
 <a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">            </span><span class="n">visited</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">adjVet</span><span class="p">)</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
 <a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">    </span><span class="c1">// Return vertex traversal sequence</span>
+<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">    </span><span class="c1">// Return the vertex traversal sequence</span>
 <a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
 <a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="p">}</span>
 </code></pre></div>
@@ -5187,36 +5259,41 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_bfs.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Breadth-first traversal ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">graph_bfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">start_vet</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="c1"># Vertex traversal sequence</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
-<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="c1"># Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="c1"># Hash set recording visited vertices</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="n">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">Set</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="o">[</span><span class="n">start_vet</span><span class="o">]</span><span class="p">)</span>
-<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="c1"># Queue used to implement BFS</span>
+<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="c1"># Use a queue to implement BFS</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="n">que</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">start_vet</span><span class="o">]</span>
-<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="c1"># Starting from vertex vet, loop until all vertices are visited</span>
+<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="c1"># Start from start_vet and loop until all reachable vertices have been visited</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="n">que</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">    </span><span class="n">vet</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">que</span><span class="o">.</span><span class="n">shift</span><span class="w"> </span><span class="c1"># Dequeue the front vertex</span>
-<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">vet</span><span class="w"> </span><span class="c1"># Record visited vertex</span>
-<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="c1"># Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">vet</span><span class="w"> </span><span class="c1"># Record the visited vertex</span>
+<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="c1"># Traverse all neighbors of this vertex</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">adj_vet</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">graph</span><span class="o">.</span><span class="n">adj_list</span><span class="o">[</span><span class="n">vet</span><span class="o">]</span>
-<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">      </span><span class="k">next</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">visited</span><span class="o">.</span><span class="n">include?</span><span class="p">(</span><span class="n">adj_vet</span><span class="p">)</span><span class="w"> </span><span class="c1"># Skip vertices that have been visited</span>
-<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">      </span><span class="n">que</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">adj_vet</span><span class="w"> </span><span class="c1"># Only enqueue unvisited vertices</span>
+<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">      </span><span class="k">next</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">visited</span><span class="o">.</span><span class="n">include?</span><span class="p">(</span><span class="n">adj_vet</span><span class="p">)</span><span class="w"> </span><span class="c1"># Skip vertices that have already been visited</span>
+<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">      </span><span class="n">que</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">adj_vet</span><span class="w"> </span><span class="c1"># Enqueue only unvisited vertices</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">      </span><span class="n">visited</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">adj_vet</span><span class="p">)</span><span class="w"> </span><span class="c1"># Mark this vertex as visited</span>
 <a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="w">  </span><span class="k">end</span>
-<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">  </span><span class="c1"># Return vertex traversal sequence</span>
+<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">  </span><span class="c1"># Return the vertex traversal sequence</span>
 <a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w">  </span><span class="n">res</span>
 <a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<p>The code is relatively abstract; it is recommended to refer to Figure 9-10 to deepen understanding.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
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+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20collections%20import%20deque%0A%0Aclass%20Vertex%3A%0A%20%20%20%20%22%22%22Vertex%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%0Adef%20vals_to_vets%28vals%3A%20list%5Bint%5D%29%20-%3E%20list%5B%22Vertex%22%5D%3A%0A%20%20%20%20%22%22%22Given%20the%20list%20of%20values%20vals,%20return%20the%20list%20of%20vertices%20vets%22%22%22%0A%20%20%20%20return%20%5BVertex%28val%29%20for%20val%20in%20vals%5D%0A%0Aclass%20GraphAdjList%3A%0A%20%20%20%20%22%22%22Undirected%20graph%20class%20implemented%20with%20an%20adjacency%20list%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20edges%3A%20list%5Blist%5BVertex%5D%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self.adj_list%20%3D%20dict%5BVertex,%20list%5BVertex%5D%5D%28%29%0A%20%20%20%20%20%20%20%20for%20edge%20in%20edges%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_vertex%28edge%5B0%5D%29%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_vertex%28edge%5B1%5D%29%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_edge%28edge%5B0%5D,%20edge%5B1%5D%29%0A%0A%20%20%20%20def%20add_edge%28self,%20vet1%3A%20Vertex,%20vet2%3A%20Vertex%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Add%20an%20edge%22%22%22%0A%20%20%20%20%20%20%20%20if%20vet1%20not%20in%20self.adj_list%20or%20vet2%20not%20in%20self.adj_list%20or%20vet1%20%3D%3D%20vet2%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20ValueError%28%29%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet1%5D.append%28vet2%29%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet2%5D.append%28vet1%29%0A%0A%20%20%20%20def%20add_vertex%28self,%20vet%3A%20Vertex%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Add%20a%20vertex%22%22%22%0A%20%20%20%20%20%20%20%20if%20vet%20in%20self.adj_list%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet%5D%20%3D%20%5B%5D%0A%0A%0Adef%20graph_bfs%28graph%3A%20GraphAdjList,%20start_vet%3A%20Vertex%29%20-%3E%20list%5BVertex%5D%3A%0A%20%20%20%20%22%22%22Breadth-first%20traversal%22%22%22%0A%20%20%20%20%23%20Vertex%20traversal%20sequence%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20%23%20Hash%20set%20for%20recording%20visited%20vertices%0A%20%20%20%20visited%20%3D%20set%5BVertex%5D%28%5Bstart_vet%5D%29%0A%20%20%20%20%23%20Use%20a%20queue%20to%20implement%20BFS%0A%20%20%20%20que%20%3D%20deque%5BVertex%5D%28%5Bstart_vet%5D%29%0A%20%20%20%20%23%20Starting%20from%20start_vet,%20loop%20until%20all%20reachable%20vertices%20have%20been%20visited%0A%20%20%20%20while%20len%28que%29%20%3E%200%3A%0A%20%20%20%20%20%20%20%20vet%20%3D%20que.popleft%28%29%20%20%23%20Dequeue%20the%20front%20vertex%0A%20%20%20%20%20%20%20%20res.append%28vet%29%20%20%23%20Record%20the%20visited%20vertex%0A%20%20%20%20%20%20%20%20%23%20Traverse%20all%20neighbors%20of%20this%20vertex%0A%20%20%20%20%20%20%20%20for%20adj_vet%20in%20graph.adj_list%5Bvet%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20adj_vet%20in%20visited%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20continue%20%20%23%20Skip%20vertices%20that%20have%20already%20been%20visited%0A%20%20%20%20%20%20%20%20%20%20%20%20que.append%28adj_vet%29%20%20%23%20Enqueue%20only%20unvisited%20vertices%0A%20%20%20%20%20%20%20%20%20%20%20%20visited.add%28adj_vet%29%20%20%23%20Mark%20this%20vertex%20as%20visited%0A%20%20%20%20%23%20Return%20the%20vertex%20traversal%20sequence%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20undirected%20graph%0A%20%20%20%20v%20%3D%20vals_to_vets%28%5B0,%201,%202,%203,%204%5D%29%0A%20%20%20%20edges%20%3D%20%5B%0A%20%20%20%20%20%20%20%20%5Bv%5B0%5D,%20v%5B1%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B0%5D,%20v%5B3%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B1%5D,%20v%5B2%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B1%5D,%20v%5B4%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B3%5D,%20v%5B4%5D%5D,%0A%20%20%20%20%5D%0A%20%20%20%20graph%20%3D%20GraphAdjList%28edges%29%0A%20%20%20%20del%20edges%0A%0A%20%20%20%20%23%20Breadth-first%20traversal%0A%20%20%20%20res%20%3D%20graph_bfs%28graph,%20v%5B0%5D%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=131&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>The code is fairly abstract; compare it with Figure 9-10 to build your understanding.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:11"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">&lt;1&gt;</label><label for="__tabbed_2_2">&lt;2&gt;</label><label for="__tabbed_2_3">&lt;3&gt;</label><label for="__tabbed_2_4">&lt;4&gt;</label><label for="__tabbed_2_5">&lt;5&gt;</label><label for="__tabbed_2_6">&lt;6&gt;</label><label for="__tabbed_2_7">&lt;7&gt;</label><label for="__tabbed_2_8">&lt;8&gt;</label><label for="__tabbed_2_9">&lt;9&gt;</label><label for="__tabbed_2_10">&lt;10&gt;</label><label for="__tabbed_2_11">&lt;11&gt;</label></div>
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-<p><img alt="Steps of breadth-first search of a graph" class="animation-figure" src="../graph_traversal.assets/graph_bfs_step1.png" /></p>
+<p><img alt="Steps in breadth-first traversal of a graph" class="animation-figure" src="../graph_traversal.assets/graph_bfs_step1.png" /></p>
 </div>
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 <p><img alt="graph_bfs_step2" class="animation-figure" src="../graph_traversal.assets/graph_bfs_step2.png" /></p>
@@ -5250,42 +5327,42 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 9-10 &nbsp; Steps of breadth-first search of a graph </p>
+<p align="center"> Figure 9-10 &nbsp; Steps in breadth-first traversal of a graph </p>
 
 <div class="admonition question">
-<p class="admonition-title">Is the breadth-first traversal sequence unique?</p>
-<p>Not unique. Breadth-first search only requires traversing in a "near to far" order, <strong>and the traversal order of vertices at the same distance can be arbitrarily shuffled</strong>. Taking Figure 9-10 as an example, the visit order of vertices <span class="arithmatex">\(1\)</span> and <span class="arithmatex">\(3\)</span> can be swapped, as can the visit order of vertices <span class="arithmatex">\(2\)</span>, <span class="arithmatex">\(4\)</span>, and <span class="arithmatex">\(6\)</span>.</p>
+<p class="admonition-title">Is the breadth-first traversal order unique?</p>
+<p>No. Breadth-first traversal only requires vertices to be visited from near to far; <strong>vertices at the same distance may be visited in any order</strong>. In Figure 9-10, vertices <span class="arithmatex">\(1\)</span> and <span class="arithmatex">\(3\)</span> can be visited in either order, and vertices <span class="arithmatex">\(2\)</span>, <span class="arithmatex">\(4\)</span>, and <span class="arithmatex">\(6\)</span> can also be visited in any order.</p>
 </div>
-<h3 id="2-complexity-analysis">2. &nbsp; Complexity Analysis<a class="headerlink" href="#2-complexity-analysis" title="Permanent link">&para;</a></h3>
-<p><strong>Time complexity</strong>: All vertices will be enqueued and dequeued once, using <span class="arithmatex">\(O(|V|)\)</span> time; in the process of traversing adjacent vertices, since it is an undirected graph, all edges will be visited <span class="arithmatex">\(2\)</span> times, using <span class="arithmatex">\(O(2|E|)\)</span> time; overall using <span class="arithmatex">\(O(|V| + |E|)\)</span> time.</p>
-<p><strong>Space complexity</strong>: The list <code>res</code>, hash set <code>visited</code>, and queue <code>que</code> can contain at most <span class="arithmatex">\(|V|\)</span> vertices, using <span class="arithmatex">\(O(|V|)\)</span> space.</p>
-<h2 id="932-depth-first-search">9.3.2 &nbsp; Depth-First Search<a class="headerlink" href="#932-depth-first-search" title="Permanent link">&para;</a></h2>
-<p><strong>Depth-first search is a traversal method that prioritizes going as far as possible, then backtracks when no path remains</strong>. As shown in Figure 9-11, starting from the top-left vertex, visit an adjacent vertex of the current vertex, continuing until reaching a dead end, then return and continue going as far as possible before returning again, and so on, until all vertices have been traversed.</p>
-<p><img alt="Depth-first search of a graph" class="animation-figure" src="../graph_traversal.assets/graph_dfs.png" /></p>
-<p align="center"> Figure 9-11 &nbsp; Depth-first search of a graph </p>
+<h3 id="2-complexity-analysis">2. &nbsp; Complexity analysis<a class="headerlink" href="#2-complexity-analysis" title="Permanent link">&para;</a></h3>
+<p><strong>Time complexity</strong>: Every vertex is enqueued and dequeued once, taking <span class="arithmatex">\(O(|V|)\)</span> time. While traversing adjacent vertices, every edge is visited <span class="arithmatex">\(2\)</span> times because the graph is undirected, taking <span class="arithmatex">\(O(2|E|)\)</span> time. The total time is <span class="arithmatex">\(O(|V| + |E|)\)</span>.</p>
+<p><strong>Space complexity</strong>: The list <code>res</code>, hash set <code>visited</code>, and queue <code>que</code> each contain at most <span class="arithmatex">\(|V|\)</span> vertices, requiring <span class="arithmatex">\(O(|V|)\)</span> space.</p>
+<h2 id="932-depth-first-traversal">9.3.2 &nbsp; Depth-first traversal<a class="headerlink" href="#932-depth-first-traversal" title="Permanent link">&para;</a></h2>
+<p><strong>Depth-first traversal follows a path as far as possible and returns when it can go no farther</strong>. In Figure 9-11, we start at the top-left vertex and move to an adjacent vertex of the current vertex, continuing until we reach a dead end. We then return and explore another path as far as possible, repeating this process until every vertex has been visited.</p>
+<p><img alt="Depth-first traversal of a graph" class="animation-figure" src="../graph_traversal.assets/graph_dfs.png" /></p>
+<p align="center"> Figure 9-11 &nbsp; Depth-first traversal of a graph </p>
 
-<h3 id="1-algorithm-implementation_1">1. &nbsp; Algorithm Implementation<a class="headerlink" href="#1-algorithm-implementation_1" title="Permanent link">&para;</a></h3>
-<p>This "go as far as possible then return" algorithm paradigm is typically implemented using recursion. Similar to breadth-first search, in depth-first search we also need a hash set <code>visited</code> to record visited vertices and avoid revisiting.</p>
+<h3 id="1-algorithm-implementation_1">1. &nbsp; Algorithm implementation<a class="headerlink" href="#1-algorithm-implementation_1" title="Permanent link">&para;</a></h3>
+<p>This pattern of going as far as possible and then returning is usually implemented with recursion. As with breadth-first traversal, we use a hash set <code>visited</code> to track visited vertices and avoid visiting them again.</p>
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 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_dfs.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">:</span> <span class="n">GraphAdjList</span><span class="p">,</span> <span class="n">visited</span><span class="p">:</span> <span class="nb">set</span><span class="p">[</span><span class="n">Vertex</span><span class="p">],</span> <span class="n">res</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="n">Vertex</span><span class="p">],</span> <span class="n">vet</span><span class="p">:</span> <span class="n">Vertex</span><span class="p">):</span>
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Depth-first traversal helper function"""</span>
-<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span>  <span class="c1"># Record visited vertex</span>
+<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span>  <span class="c1"># Record the visited vertex</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="n">visited</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span>  <span class="c1"># Mark this vertex as visited</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Traverse all neighbors of this vertex</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="k">for</span> <span class="n">adjVet</span> <span class="ow">in</span> <span class="n">graph</span><span class="o">.</span><span class="n">adj_list</span><span class="p">[</span><span class="n">vet</span><span class="p">]:</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="k">if</span> <span class="n">adjVet</span> <span class="ow">in</span> <span class="n">visited</span><span class="p">:</span>
-<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>            <span class="k">continue</span>  <span class="c1"># Skip vertices that have been visited</span>
-<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>        <span class="c1"># Recursively visit adjacent vertices</span>
+<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>            <span class="k">continue</span>  <span class="c1"># Skip vertices that have already been visited</span>
+<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>        <span class="c1"># Recursively visit the neighboring vertex</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>        <span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span> <span class="n">visited</span><span class="p">,</span> <span class="n">res</span><span class="p">,</span> <span class="n">adjVet</span><span class="p">)</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a><span class="k">def</span><span class="w"> </span><span class="nf">graph_dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">:</span> <span class="n">GraphAdjList</span><span class="p">,</span> <span class="n">start_vet</span><span class="p">:</span> <span class="n">Vertex</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="n">Vertex</span><span class="p">]:</span>
 <a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a><span class="w">    </span><span class="sd">"""Depth-first traversal"""</span>
-<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>    <span class="c1"># Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>    <span class="c1"># Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>    <span class="c1"># Vertex traversal sequence</span>
 <a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>    <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
-<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>    <span class="c1"># Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>    <span class="c1"># Hash set recording visited vertices</span>
 <a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a>    <span class="n">visited</span> <span class="o">=</span> <span class="nb">set</span><span class="p">[</span><span class="n">Vertex</span><span class="p">]()</span>
 <a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a>    <span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span> <span class="n">visited</span><span class="p">,</span> <span class="n">res</span><span class="p">,</span> <span class="n">start_vet</span><span class="p">)</span>
 <a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a>    <span class="k">return</span> <span class="n">res</span>
@@ -5294,23 +5371,23 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_dfs.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Depth-first traversal helper function */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="o">&amp;</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">unordered_set</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w">   </span><span class="c1">// Record visited vertex</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w">   </span><span class="c1">// Record the visited vertex</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="n">visited</span><span class="p">.</span><span class="n">emplace</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
-<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">adjVet</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="n">adjList</span><span class="p">[</span><span class="n">vet</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">visited</span><span class="p">.</span><span class="n">count</span><span class="p">(</span><span class="n">adjVet</span><span class="p">))</span>
-<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have been visited</span>
-<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">        </span><span class="c1">// Recursively visit adjacent vertices</span>
+<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have already been visited</span>
+<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">        </span><span class="c1">// Recursively visit the neighboring vertex</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">        </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">adjVet</span><span class="p">);</span>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="p">}</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="cm">/* Depth-first traversal */</span>
-<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="n">vector</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="n">graphDFS</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="o">&amp;</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">startVet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
-<a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w">    </span><span class="n">unordered_set</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="n">visited</span><span class="p">;</span>
 <a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">startVet</span><span class="p">);</span>
 <a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
@@ -5320,23 +5397,23 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_dfs.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Depth-first traversal helper function */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Set</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w">     </span><span class="c1">// Record visited vertex</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w">     </span><span class="c1">// Record the visited vertex</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="n">visited</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
-<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">adjVet</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="na">adjList</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">vet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">visited</span><span class="p">.</span><span class="na">contains</span><span class="p">(</span><span class="n">adjVet</span><span class="p">))</span>
-<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have been visited</span>
-<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">        </span><span class="c1">// Recursively visit adjacent vertices</span>
+<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have already been visited</span>
+<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">        </span><span class="c1">// Recursively visit the neighboring vertex</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">        </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">adjVet</span><span class="p">);</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="p">}</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="cm">/* Depth-first traversal */</span>
-<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="nf">graphDFS</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">startVet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
-<a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w">    </span><span class="n">Set</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">HashSet</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">startVet</span><span class="p">);</span>
 <a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
@@ -5346,24 +5423,24 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_dfs.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Depth-first traversal helper function */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">DFS</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">HashSet</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w">     </span><span class="c1">// Record visited vertex</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w">     </span><span class="c1">// Record the visited vertex</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="n">visited</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
-<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">adjVet</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="n">adjList</span><span class="p">[</span><span class="n">vet</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">visited</span><span class="p">.</span><span class="n">Contains</span><span class="p">(</span><span class="n">adjVet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have been visited</span>
+<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have already been visited                             </span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">        </span><span class="c1">// Recursively visit adjacent vertices</span>
+<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">        </span><span class="c1">// Recursively visit the neighboring vertex</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">        </span><span class="n">DFS</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">adjVet</span><span class="p">);</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="p">}</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a>
 <a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="cm">/* Depth-first traversal */</span>
-<a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">GraphDFS</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">startVet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="w">    </span><span class="n">HashSet</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a><span class="w">    </span><span class="n">DFS</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">startVet</span><span class="p">);</span>
 <a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
@@ -5373,13 +5450,13 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_dfs.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Depth-first traversal helper function */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">g</span><span class="w"> </span><span class="o">*</span><span class="nx">graphAdjList</span><span class="p">,</span><span class="w"> </span><span class="nx">visited</span><span class="w"> </span><span class="kt">map</span><span class="p">[</span><span class="nx">Vertex</span><span class="p">]</span><span class="kd">struct</span><span class="p">{},</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="nx">Vertex</span><span class="p">,</span><span class="w"> </span><span class="nx">vet</span><span class="w"> </span><span class="nx">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// append operation returns a new reference, must reassign original reference to new slice's reference</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// append returns an updated slice, which must be assigned back through the original slice pointer</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="o">*</span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">vet</span><span class="p">)</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="nx">visited</span><span class="p">[</span><span class="nx">vet</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kd">struct</span><span class="p">{}{}</span>
-<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">adjVet</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">g</span><span class="p">.</span><span class="nx">adjList</span><span class="p">[</span><span class="nx">vet</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">        </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">isExist</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">visited</span><span class="p">[</span><span class="nx">adjVet</span><span class="p">]</span>
-<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">        </span><span class="c1">// Recursively visit adjacent vertices</span>
+<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">        </span><span class="c1">// Recursively visit the neighboring vertex</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="nx">isExist</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">            </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">g</span><span class="p">,</span><span class="w"> </span><span class="nx">visited</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">adjVet</span><span class="p">)</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">        </span><span class="p">}</span>
@@ -5387,14 +5464,14 @@
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="p">}</span>
 <a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="cm">/* Depth-first traversal */</span>
-<a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="kd">func</span><span class="w"> </span><span class="nx">graphDFS</span><span class="p">(</span><span class="nx">g</span><span class="w"> </span><span class="o">*</span><span class="nx">graphAdjList</span><span class="p">,</span><span class="w"> </span><span class="nx">startVet</span><span class="w"> </span><span class="nx">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">[]</span><span class="nx">Vertex</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="nx">Vertex</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-17-22" name="__codelineno-17-22" href="#__codelineno-17-22"></a><span class="w">    </span><span class="nx">visited</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">(</span><span class="kt">map</span><span class="p">[</span><span class="nx">Vertex</span><span class="p">]</span><span class="kd">struct</span><span class="p">{})</span>
 <a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a><span class="w">    </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">g</span><span class="p">,</span><span class="w"> </span><span class="nx">visited</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">startVet</span><span class="p">)</span>
-<a id="__codelineno-17-24" name="__codelineno-17-24" href="#__codelineno-17-24"></a><span class="w">    </span><span class="c1">// Return vertex traversal sequence</span>
+<a id="__codelineno-17-24" name="__codelineno-17-24" href="#__codelineno-17-24"></a><span class="w">    </span><span class="c1">// Return the vertex traversal sequence</span>
 <a id="__codelineno-17-25" name="__codelineno-17-25" href="#__codelineno-17-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span>
 <a id="__codelineno-17-26" name="__codelineno-17-26" href="#__codelineno-17-26"></a><span class="p">}</span>
 </code></pre></div>
@@ -5402,24 +5479,24 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_dfs.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Depth-first traversal helper function */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">:</span><span class="w"> </span><span class="n">GraphAdjList</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="n">Set</span><span class="p">&lt;</span><span class="n">Vertex</span><span class="p">&gt;,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="n">Vertex</span><span class="p">],</span><span class="w"> </span><span class="n">vet</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="c1">// Record visited vertex</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="c1">// Record the visited vertex</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="n">visited</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
-<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">adjVet</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="n">adjList</span><span class="p">[</span><span class="n">vet</span><span class="p">]</span><span class="w"> </span><span class="p">??</span><span class="w"> </span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">visited</span><span class="p">.</span><span class="bp">contains</span><span class="p">(</span><span class="n">adjVet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">            </span><span class="k">continue</span><span class="w"> </span><span class="c1">// Skip vertices that have been visited</span>
+<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">            </span><span class="k">continue</span><span class="w"> </span><span class="c1">// Skip vertices that have already been visited</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">        </span><span class="c1">// Recursively visit adjacent vertices</span>
+<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">        </span><span class="c1">// Recursively visit the neighboring vertex</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">        </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">:</span><span class="w"> </span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">vet</span><span class="p">:</span><span class="w"> </span><span class="n">adjVet</span><span class="p">)</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="p">}</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a>
 <a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="cm">/* Depth-first traversal */</span>
-<a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="kd">func</span><span class="w"> </span><span class="nf">graphDFS</span><span class="p">(</span><span class="n">graph</span><span class="p">:</span><span class="w"> </span><span class="n">GraphAdjList</span><span class="p">,</span><span class="w"> </span><span class="n">startVet</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="n">Vertex</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="n">Vertex</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
-<a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">visited</span><span class="p">:</span><span class="w"> </span><span class="n">Set</span><span class="p">&lt;</span><span class="n">Vertex</span><span class="p">&gt;</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
 <a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">:</span><span class="w"> </span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">vet</span><span class="p">:</span><span class="w"> </span><span class="n">startVet</span><span class="p">)</span>
 <a id="__codelineno-18-23" name="__codelineno-18-23" href="#__codelineno-18-23"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
@@ -5428,26 +5505,26 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_dfs.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Depth-first traversal */</span>
-<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="kd">function</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">graph</span><span class="p">,</span><span class="w"> </span><span class="nx">visited</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record visited vertex</span>
+<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the visited vertex</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="nx">visited</span><span class="p">.</span><span class="nx">add</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
-<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">adjVet</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">graph</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">visited</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">adjVet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have been visited</span>
+<a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have already been visited</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">        </span><span class="c1">// Recursively visit adjacent vertices</span>
+<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">        </span><span class="c1">// Recursively visit the neighboring vertex</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">        </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">graph</span><span class="p">,</span><span class="w"> </span><span class="nx">visited</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">adjVet</span><span class="p">);</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="p">}</span>
 <a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="cm">/* Depth-first traversal */</span>
-<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="kd">function</span><span class="w"> </span><span class="nx">graphDFS</span><span class="p">(</span><span class="nx">graph</span><span class="p">,</span><span class="w"> </span><span class="nx">startVet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Set</span><span class="p">();</span>
 <a id="__codelineno-19-23" name="__codelineno-19-23" href="#__codelineno-19-23"></a><span class="w">    </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">graph</span><span class="p">,</span><span class="w"> </span><span class="nx">visited</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">startVet</span><span class="p">);</span>
 <a id="__codelineno-19-24" name="__codelineno-19-24" href="#__codelineno-19-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
@@ -5462,24 +5539,24 @@
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">Vertex</span><span class="p">[],</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="nx">vet</span><span class="o">:</span><span class="w"> </span><span class="kt">Vertex</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record visited vertex</span>
+<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the visited vertex</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="nx">visited</span><span class="p">.</span><span class="nx">add</span><span class="p">(</span><span class="nx">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
-<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">adjVet</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">graph</span><span class="p">.</span><span class="nx">adjList</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">vet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">visited</span><span class="p">.</span><span class="nx">has</span><span class="p">(</span><span class="nx">adjVet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have been visited</span>
+<a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">            </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have already been visited</span>
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">        </span><span class="c1">// Recursively visit adjacent vertices</span>
+<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">        </span><span class="c1">// Recursively visit the neighboring vertex</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">        </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">graph</span><span class="p">,</span><span class="w"> </span><span class="nx">visited</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">adjVet</span><span class="p">);</span>
 <a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="p">}</span>
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="cm">/* Depth-first traversal */</span>
-<a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="kd">function</span><span class="w"> </span><span class="nx">graphDFS</span><span class="p">(</span><span class="nx">graph</span><span class="o">:</span><span class="w"> </span><span class="kt">GraphAdjList</span><span class="p">,</span><span class="w"> </span><span class="nx">startVet</span><span class="o">:</span><span class="w"> </span><span class="kt">Vertex</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="nx">Vertex</span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-23" name="__codelineno-20-23" href="#__codelineno-20-23"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">Vertex</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">visited</span><span class="o">:</span><span class="w"> </span><span class="kt">Set</span><span class="o">&lt;</span><span class="nx">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Set</span><span class="p">();</span>
 <a id="__codelineno-20-27" name="__codelineno-20-27" href="#__codelineno-20-27"></a><span class="w">    </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">graph</span><span class="p">,</span><span class="w"> </span><span class="nx">visited</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">startVet</span><span class="p">);</span>
 <a id="__codelineno-20-28" name="__codelineno-20-28" href="#__codelineno-20-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">res</span><span class="p">;</span>
@@ -5494,14 +5571,14 @@
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">,</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="n">Vertex</span><span class="w"> </span><span class="n">vet</span><span class="p">,</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record visited vertex</span>
+<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the visited vertex</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="n">visited</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
-<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="n">adjVet</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="n">adjList</span><span class="p">[</span><span class="n">vet</span><span class="p">]</span><span class="o">!</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">visited</span><span class="p">.</span><span class="n">contains</span><span class="p">(</span><span class="n">adjVet</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">      </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have been visited</span>
+<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">      </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have already been visited</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">    </span><span class="c1">// Recursively visit adjacent vertices</span>
+<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">    </span><span class="c1">// Recursively visit the neighboring vertex</span>
 <a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">adjVet</span><span class="p">);</span>
 <a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="p">}</span>
@@ -5510,7 +5587,7 @@
 <a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">graphDFS</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="n">startVet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">  </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">  </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">  </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-21-25" name="__codelineno-21-25" href="#__codelineno-21-25"></a><span class="w">  </span><span class="n">Set</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="n">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{};</span>
 <a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a><span class="w">  </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">startVet</span><span class="p">);</span>
 <a id="__codelineno-21-27" name="__codelineno-21-27" href="#__codelineno-21-27"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
@@ -5520,26 +5597,26 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_dfs.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Depth-first traversal helper function */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">GraphAdjList</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="n">HashSet</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">vet</span><span class="p">:</span><span class="w"> </span><span class="nc">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record visited vertex</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Record the visited vertex</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="n">visited</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="n">vet</span><span class="p">);</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
-<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">                         </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">                         </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">adj_vets</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="n">adj_list</span><span class="p">.</span><span class="n">get</span><span class="p">(</span><span class="o">&amp;</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="o">&amp;</span><span class="n">adj_vet</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">adj_vets</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">visited</span><span class="p">.</span><span class="n">contains</span><span class="p">(</span><span class="o">&amp;</span><span class="n">adj_vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">                </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have been visited</span>
+<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">                </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span><span class="c1">// Skip vertices that have already been visited</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">            </span><span class="c1">// Recursively visit adjacent vertices</span>
+<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">            </span><span class="c1">// Recursively visit the neighboring vertex</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">            </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">adj_vet</span><span class="p">);</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="p">}</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a>
 <a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="cm">/* Depth-first traversal */</span>
-<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="k">fn</span><span class="w"> </span><span class="nf">graph_dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">:</span><span class="w"> </span><span class="nc">GraphAdjList</span><span class="p">,</span><span class="w"> </span><span class="n">start_vet</span><span class="p">:</span><span class="w"> </span><span class="nc">Vertex</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Vertex</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[];</span>
-<a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashSet</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="o">&amp;</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">start_vet</span><span class="p">);</span>
 <a id="__codelineno-22-25" name="__codelineno-22-25" href="#__codelineno-22-25"></a>
@@ -5548,9 +5625,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">graph_dfs.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Check if vertex has been visited */</span>
+<div class="highlight"><span class="filename">graph_dfs.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Check whether a vertex has been visited */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">isVisited</span><span class="p">(</span><span class="n">Vertex</span><span class="w"> </span><span class="o">**</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Traverse to find node using O(n) time</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Search for the vertex with a linear scan in O(n) time</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
@@ -5561,14 +5638,14 @@
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="cm">/* Depth-first traversal helper function */</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="kt">void</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="o">*</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">**</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">resSize</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">    </span><span class="c1">// Record visited vertex</span>
+<a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">    </span><span class="c1">// Record the visited vertex</span>
 <a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">    </span><span class="n">res</span><span class="p">[(</span><span class="o">*</span><span class="n">resSize</span><span class="p">)</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">vet</span><span class="p">;</span>
-<a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w">    </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w">    </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w">    </span><span class="n">AdjListNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findNode</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">vet</span><span class="p">);</span>
 <a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w">        </span><span class="c1">// Skip vertices that have been visited</span>
+<a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w">        </span><span class="c1">// Skip vertices that have already been visited</span>
 <a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">isVisited</span><span class="p">(</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">resSize</span><span class="p">,</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">vertex</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="w">            </span><span class="c1">// Recursively visit adjacent vertices</span>
+<a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="w">            </span><span class="c1">// Recursively visit the neighboring vertex</span>
 <a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="w">            </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">resSize</span><span class="p">,</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">vertex</span><span class="p">);</span>
 <a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">        </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
@@ -5576,7 +5653,7 @@
 <a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="p">}</span>
 <a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a>
 <a id="__codelineno-23-28" name="__codelineno-23-28" href="#__codelineno-23-28"></a><span class="cm">/* Depth-first traversal */</span>
-<a id="__codelineno-23-29" name="__codelineno-23-29" href="#__codelineno-23-29"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-23-29" name="__codelineno-23-29" href="#__codelineno-23-29"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-23-30" name="__codelineno-23-30" href="#__codelineno-23-30"></a><span class="kt">void</span><span class="w"> </span><span class="nf">graphDFS</span><span class="p">(</span><span class="n">GraphAdjList</span><span class="w"> </span><span class="o">*</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">*</span><span class="n">startVet</span><span class="p">,</span><span class="w"> </span><span class="n">Vertex</span><span class="w"> </span><span class="o">**</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">resSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-31" name="__codelineno-23-31" href="#__codelineno-23-31"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">resSize</span><span class="p">,</span><span class="w"> </span><span class="n">startVet</span><span class="p">);</span>
 <a id="__codelineno-23-32" name="__codelineno-23-32" href="#__codelineno-23-32"></a><span class="p">}</span>
@@ -5590,23 +5667,23 @@
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">Vertex?</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="n">vet</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex?</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span><span class="w">     </span><span class="c1">// Record visited vertex</span>
+<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span><span class="w">     </span><span class="c1">// Record the visited vertex</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="n">visited</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="c1">// Mark this vertex as visited</span>
-<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="c1">// Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="c1">// Traverse all neighbors of this vertex</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">adjVet</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">graph</span><span class="p">.</span><span class="na">adjList</span><span class="o">[</span><span class="n">vet</span><span class="o">]!!</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">visited</span><span class="p">.</span><span class="na">contains</span><span class="p">(</span><span class="n">adjVet</span><span class="p">))</span>
-<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">            </span><span class="k">continue</span><span class="w">  </span><span class="c1">// Skip vertices that have been visited</span>
-<a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">        </span><span class="c1">// Recursively visit adjacent vertices</span>
+<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">            </span><span class="k">continue</span><span class="w">  </span><span class="c1">// Skip vertices that have already been visited</span>
+<a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">        </span><span class="c1">// Recursively visit the neighboring vertex</span>
 <a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">        </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">adjVet</span><span class="p">)</span>
 <a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="p">}</span>
 <a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a>
 <a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="cm">/* Depth-first traversal */</span>
-<a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="c1">// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="c1">// Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">graphDFS</span><span class="p">(</span><span class="n">graph</span><span class="p">:</span><span class="w"> </span><span class="n">GraphAdjList</span><span class="p">,</span><span class="w"> </span><span class="n">startVet</span><span class="p">:</span><span class="w"> </span><span class="n">Vertex?</span><span class="p">):</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">Vertex?</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">    </span><span class="c1">// Vertex traversal sequence</span>
 <a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">Vertex?</span><span class="o">&gt;</span><span class="p">()</span>
-<a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a><span class="w">    </span><span class="c1">// Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a><span class="w">    </span><span class="c1">// Hash set recording visited vertices</span>
 <a id="__codelineno-24-25" name="__codelineno-24-25" href="#__codelineno-24-25"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashSet</span><span class="o">&lt;</span><span class="n">Vertex?</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-24-26" name="__codelineno-24-26" href="#__codelineno-24-26"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">startVet</span><span class="p">)</span>
 <a id="__codelineno-24-27" name="__codelineno-24-27" href="#__codelineno-24-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span>
@@ -5616,22 +5693,22 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">graph_dfs.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Depth-first traversal helper function ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">vet</span><span class="p">)</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">vet</span><span class="w"> </span><span class="c1"># Record visited vertex</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">vet</span><span class="w"> </span><span class="c1"># Record the visited vertex</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="n">visited</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">vet</span><span class="p">)</span><span class="w"> </span><span class="c1"># Mark this vertex as visited</span>
-<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># Traverse all adjacent vertices of this vertex</span>
+<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># Traverse all neighbors of this vertex</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">adj_vet</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">graph</span><span class="o">.</span><span class="n">adj_list</span><span class="o">[</span><span class="n">vet</span><span class="o">]</span>
-<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">    </span><span class="k">next</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">visited</span><span class="o">.</span><span class="n">include?</span><span class="p">(</span><span class="n">adj_vet</span><span class="p">)</span><span class="w"> </span><span class="c1"># Skip vertices that have been visited</span>
-<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">    </span><span class="c1"># Recursively visit adjacent vertices</span>
+<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">    </span><span class="k">next</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">visited</span><span class="o">.</span><span class="n">include?</span><span class="p">(</span><span class="n">adj_vet</span><span class="p">)</span><span class="w"> </span><span class="c1"># Skip vertices that have already been visited</span>
+<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">    </span><span class="c1"># Recursively visit the neighboring vertex</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">adj_vet</span><span class="p">)</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="k">end</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="c1">### Depth-first traversal ###</span>
 <a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="k">def</span><span class="w"> </span><span class="nf">graph_dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">start_vet</span><span class="p">)</span>
-<a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">  </span><span class="c1"># Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex</span>
+<a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">  </span><span class="c1"># Represent the graph with an adjacency list to access all neighbors of a given vertex</span>
 <a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">  </span><span class="c1"># Vertex traversal sequence</span>
 <a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
-<a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w">  </span><span class="c1"># Hash set for recording vertices that have been visited</span>
+<a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w">  </span><span class="c1"># Hash set recording visited vertices</span>
 <a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="w">  </span><span class="n">visited</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">Set</span><span class="o">.</span><span class="n">new</span>
 <a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a><span class="w">  </span><span class="n">dfs</span><span class="p">(</span><span class="n">graph</span><span class="p">,</span><span class="w"> </span><span class="n">visited</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">start_vet</span><span class="p">)</span>
 <a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="w">  </span><span class="n">res</span>
@@ -5640,16 +5717,21 @@
 </div>
 </div>
 </div>
-<p>The algorithm flow of depth-first search is shown in Figure 9-12.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20Vertex%3A%0A%20%20%20%20%22%22%22Vertex%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%0Adef%20vals_to_vets%28vals%3A%20list%5Bint%5D%29%20-%3E%20list%5B%22Vertex%22%5D%3A%0A%20%20%20%20%22%22%22Given%20the%20list%20of%20values%20vals,%20return%20the%20list%20of%20vertices%20vets%22%22%22%0A%20%20%20%20return%20%5BVertex%28val%29%20for%20val%20in%20vals%5D%0A%0Aclass%20GraphAdjList%3A%0A%20%20%20%20%22%22%22Undirected%20graph%20class%20implemented%20with%20an%20adjacency%20list%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20edges%3A%20list%5Blist%5BVertex%5D%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self.adj_list%20%3D%20dict%5BVertex,%20list%5BVertex%5D%5D%28%29%0A%20%20%20%20%20%20%20%20for%20edge%20in%20edges%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_vertex%28edge%5B0%5D%29%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_vertex%28edge%5B1%5D%29%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_edge%28edge%5B0%5D,%20edge%5B1%5D%29%0A%0A%20%20%20%20def%20add_edge%28self,%20vet1%3A%20Vertex,%20vet2%3A%20Vertex%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Add%20an%20edge%22%22%22%0A%20%20%20%20%20%20%20%20if%20vet1%20not%20in%20self.adj_list%20or%20vet2%20not%20in%20self.adj_list%20or%20vet1%20%3D%3D%20vet2%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20ValueError%28%29%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet1%5D.append%28vet2%29%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet2%5D.append%28vet1%29%0A%0A%20%20%20%20def%20add_vertex%28self,%20vet%3A%20Vertex%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Add%20a%20vertex%22%22%22%0A%20%20%20%20%20%20%20%20if%20vet%20in%20self.adj_list%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet%5D%20%3D%20%5B%5D%0A%0A%0Adef%20dfs%28graph%3A%20GraphAdjList,%20visited%3A%20set%5BVertex%5D,%20res%3A%20list%5BVertex%5D,%20vet%3A%20Vertex%29%3A%0A%20%20%20%20%22%22%22Depth-first%20traversal%20helper%20function%22%22%22%0A%20%20%20%20res.append%28vet%29%20%20%23%20Record%20the%20visited%20vertex%0A%20%20%20%20visited.add%28vet%29%20%20%23%20Mark%20this%20vertex%20as%20visited%0A%20%20%20%20%23%20Traverse%20all%20neighbors%20of%20this%20vertex%0A%20%20%20%20for%20adjVet%20in%20graph.adj_list%5Bvet%5D%3A%0A%20%20%20%20%20%20%20%20if%20adjVet%20in%20visited%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20continue%20%20%23%20Skip%20vertices%20that%20have%20already%20been%20visited%0A%20%20%20%20%20%20%20%20%23%20Recursively%20visit%20the%20neighboring%20vertex%0A%20%20%20%20%20%20%20%20dfs%28graph,%20visited,%20res,%20adjVet%29%0A%0A%0Adef%20graph_dfs%28graph%3A%20GraphAdjList,%20start_vet%3A%20Vertex%29%20-%3E%20list%5BVertex%5D%3A%0A%20%20%20%20%22%22%22Depth-first%20traversal%22%22%22%0A%20%20%20%20%23%20Represent%20the%20graph%20with%20an%20adjacency%20list%20to%20access%20all%20neighbors%20of%20a%20given%20vertex%0A%20%20%20%20%23%20Vertex%20traversal%20sequence%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20%23%20Hash%20set%20for%20recording%20visited%20vertices%0A%20%20%20%20visited%20%3D%20set%5BVertex%5D%28%29%0A%20%20%20%20dfs%28graph,%20visited,%20res,%20start_vet%29%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20undirected%20graph%0A%20%20%20%20v%20%3D%20vals_to_vets%28%5B0,%201,%202,%203,%204%5D%29%0A%20%20%20%20edges%20%3D%20%5B%0A%20%20%20%20%20%20%20%20%5Bv%5B0%5D,%20v%5B1%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B0%5D,%20v%5B3%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B1%5D,%20v%5B2%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B1%5D,%20v%5B4%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B3%5D,%20v%5B4%5D%5D,%0A%20%20%20%20%5D%0A%20%20%20%20graph%20%3D%20GraphAdjList%28edges%29%0A%0A%20%20%20%20%23%20Depth-first%20traversal%0A%20%20%20%20res%20%3D%20graph_dfs%28graph,%20v%5B0%5D%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=130&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20Vertex%3A%0A%20%20%20%20%22%22%22Vertex%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%0Adef%20vals_to_vets%28vals%3A%20list%5Bint%5D%29%20-%3E%20list%5B%22Vertex%22%5D%3A%0A%20%20%20%20%22%22%22Given%20the%20list%20of%20values%20vals,%20return%20the%20list%20of%20vertices%20vets%22%22%22%0A%20%20%20%20return%20%5BVertex%28val%29%20for%20val%20in%20vals%5D%0A%0Aclass%20GraphAdjList%3A%0A%20%20%20%20%22%22%22Undirected%20graph%20class%20implemented%20with%20an%20adjacency%20list%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20edges%3A%20list%5Blist%5BVertex%5D%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self.adj_list%20%3D%20dict%5BVertex,%20list%5BVertex%5D%5D%28%29%0A%20%20%20%20%20%20%20%20for%20edge%20in%20edges%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_vertex%28edge%5B0%5D%29%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_vertex%28edge%5B1%5D%29%0A%20%20%20%20%20%20%20%20%20%20%20%20self.add_edge%28edge%5B0%5D,%20edge%5B1%5D%29%0A%0A%20%20%20%20def%20add_edge%28self,%20vet1%3A%20Vertex,%20vet2%3A%20Vertex%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Add%20an%20edge%22%22%22%0A%20%20%20%20%20%20%20%20if%20vet1%20not%20in%20self.adj_list%20or%20vet2%20not%20in%20self.adj_list%20or%20vet1%20%3D%3D%20vet2%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20ValueError%28%29%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet1%5D.append%28vet2%29%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet2%5D.append%28vet1%29%0A%0A%20%20%20%20def%20add_vertex%28self,%20vet%3A%20Vertex%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Add%20a%20vertex%22%22%22%0A%20%20%20%20%20%20%20%20if%20vet%20in%20self.adj_list%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20self.adj_list%5Bvet%5D%20%3D%20%5B%5D%0A%0A%0Adef%20dfs%28graph%3A%20GraphAdjList,%20visited%3A%20set%5BVertex%5D,%20res%3A%20list%5BVertex%5D,%20vet%3A%20Vertex%29%3A%0A%20%20%20%20%22%22%22Depth-first%20traversal%20helper%20function%22%22%22%0A%20%20%20%20res.append%28vet%29%20%20%23%20Record%20the%20visited%20vertex%0A%20%20%20%20visited.add%28vet%29%20%20%23%20Mark%20this%20vertex%20as%20visited%0A%20%20%20%20%23%20Traverse%20all%20neighbors%20of%20this%20vertex%0A%20%20%20%20for%20adjVet%20in%20graph.adj_list%5Bvet%5D%3A%0A%20%20%20%20%20%20%20%20if%20adjVet%20in%20visited%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20continue%20%20%23%20Skip%20vertices%20that%20have%20already%20been%20visited%0A%20%20%20%20%20%20%20%20%23%20Recursively%20visit%20the%20neighboring%20vertex%0A%20%20%20%20%20%20%20%20dfs%28graph,%20visited,%20res,%20adjVet%29%0A%0A%0Adef%20graph_dfs%28graph%3A%20GraphAdjList,%20start_vet%3A%20Vertex%29%20-%3E%20list%5BVertex%5D%3A%0A%20%20%20%20%22%22%22Depth-first%20traversal%22%22%22%0A%20%20%20%20%23%20Represent%20the%20graph%20with%20an%20adjacency%20list%20to%20access%20all%20neighbors%20of%20a%20given%20vertex%0A%20%20%20%20%23%20Vertex%20traversal%20sequence%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20%23%20Hash%20set%20for%20recording%20visited%20vertices%0A%20%20%20%20visited%20%3D%20set%5BVertex%5D%28%29%0A%20%20%20%20dfs%28graph,%20visited,%20res,%20start_vet%29%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20undirected%20graph%0A%20%20%20%20v%20%3D%20vals_to_vets%28%5B0,%201,%202,%203,%204%5D%29%0A%20%20%20%20edges%20%3D%20%5B%0A%20%20%20%20%20%20%20%20%5Bv%5B0%5D,%20v%5B1%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B0%5D,%20v%5B3%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B1%5D,%20v%5B2%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B1%5D,%20v%5B4%5D%5D,%0A%20%20%20%20%20%20%20%20%5Bv%5B3%5D,%20v%5B4%5D%5D,%0A%20%20%20%20%5D%0A%20%20%20%20graph%20%3D%20GraphAdjList%28edges%29%0A%0A%20%20%20%20%23%20Depth-first%20traversal%0A%20%20%20%20res%20%3D%20graph_dfs%28graph,%20v%5B0%5D%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=130&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Figure 9-12 shows the steps of depth-first traversal.</p>
 <ul>
-<li><strong>Straight dashed lines represent downward recursion</strong>, indicating that a new recursive method has been initiated to visit a new vertex.</li>
-<li><strong>Curved dashed lines represent upward backtracking</strong>, indicating that this recursive call has returned to the point where it was made.</li>
+<li><strong>Straight dashed lines show recursive descent</strong>, where a new recursive call begins to visit another vertex.</li>
+<li><strong>Curved dashed lines show backtracking</strong>, where a recursive call has returned to the point from which it was made.</li>
 </ul>
-<p>To deepen understanding, it is recommended to combine Figure 9-12 with the code to mentally simulate (or draw out) the entire DFS process, including when each recursive call begins and when it returns.</p>
+<p>To deepen your understanding, follow Figure 9-12 alongside the code and mentally simulate (or draw on paper) the entire DFS process, including when each recursive call begins and returns.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:11"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">&lt;1&gt;</label><label for="__tabbed_4_2">&lt;2&gt;</label><label for="__tabbed_4_3">&lt;3&gt;</label><label for="__tabbed_4_4">&lt;4&gt;</label><label for="__tabbed_4_5">&lt;5&gt;</label><label for="__tabbed_4_6">&lt;6&gt;</label><label for="__tabbed_4_7">&lt;7&gt;</label><label for="__tabbed_4_8">&lt;8&gt;</label><label for="__tabbed_4_9">&lt;9&gt;</label><label for="__tabbed_4_10">&lt;10&gt;</label><label for="__tabbed_4_11">&lt;11&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Steps of depth-first search of a graph" class="animation-figure" src="../graph_traversal.assets/graph_dfs_step1.png" /></p>
+<p><img alt="Steps in depth-first traversal of a graph" class="animation-figure" src="../graph_traversal.assets/graph_dfs_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="graph_dfs_step2" class="animation-figure" src="../graph_traversal.assets/graph_dfs_step2.png" /></p>
@@ -5683,16 +5765,16 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 9-12 &nbsp; Steps of depth-first search of a graph </p>
+<p align="center"> Figure 9-12 &nbsp; Steps in depth-first traversal of a graph </p>
 
 <div class="admonition question">
-<p class="admonition-title">Is the depth-first traversal sequence unique?</p>
-<p>Similar to breadth-first search, depth-first traversal sequences are also not unique. Given a vertex, any exploration direction may be chosen first; that is, the order of adjacent vertices can be arbitrarily rearranged and still constitute depth-first search.</p>
-<p>Taking tree traversal as an example, "root <span class="arithmatex">\(\rightarrow\)</span> left <span class="arithmatex">\(\rightarrow\)</span> right", "left <span class="arithmatex">\(\rightarrow\)</span> root <span class="arithmatex">\(\rightarrow\)</span> right", and "left <span class="arithmatex">\(\rightarrow\)</span> right <span class="arithmatex">\(\rightarrow\)</span> root" correspond to pre-order, in-order, and post-order traversals, respectively. They represent three different traversal priorities, yet all three belong to depth-first search.</p>
+<p class="admonition-title">Is the depth-first traversal order unique?</p>
+<p>Like breadth-first traversal, depth-first traversal does not have a unique order. From any vertex, we can explore in any direction first: rearranging the order of adjacent vertices still produces a depth-first traversal.</p>
+<p>For example, in tree traversal, "root <span class="arithmatex">\(\rightarrow\)</span> left <span class="arithmatex">\(\rightarrow\)</span> right", "left <span class="arithmatex">\(\rightarrow\)</span> root <span class="arithmatex">\(\rightarrow\)</span> right", and "left <span class="arithmatex">\(\rightarrow\)</span> right <span class="arithmatex">\(\rightarrow\)</span> root" correspond to preorder, inorder, and postorder traversal. They use three different visiting priorities, yet all are depth-first traversals.</p>
 </div>
-<h3 id="2-complexity-analysis_1">2. &nbsp; Complexity Analysis<a class="headerlink" href="#2-complexity-analysis_1" title="Permanent link">&para;</a></h3>
-<p><strong>Time complexity</strong>: All vertices will be visited <span class="arithmatex">\(1\)</span> time, using <span class="arithmatex">\(O(|V|)\)</span> time; all edges will be visited <span class="arithmatex">\(2\)</span> times, using <span class="arithmatex">\(O(2|E|)\)</span> time; overall using <span class="arithmatex">\(O(|V| + |E|)\)</span> time.</p>
-<p><strong>Space complexity</strong>: The list <code>res</code> and hash set <code>visited</code> can contain at most <span class="arithmatex">\(|V|\)</span> vertices, and the maximum recursion depth is <span class="arithmatex">\(|V|\)</span>, therefore using <span class="arithmatex">\(O(|V|)\)</span> space.</p>
+<h3 id="2-complexity-analysis_1">2. &nbsp; Complexity analysis<a class="headerlink" href="#2-complexity-analysis_1" title="Permanent link">&para;</a></h3>
+<p><strong>Time complexity</strong>: Every vertex is visited <span class="arithmatex">\(1\)</span> time, taking <span class="arithmatex">\(O(|V|)\)</span> time, and every edge is visited <span class="arithmatex">\(2\)</span> times, taking <span class="arithmatex">\(O(2|E|)\)</span> time. The total time is <span class="arithmatex">\(O(|V| + |E|)\)</span>.</p>
+<p><strong>Space complexity</strong>: The list <code>res</code> and hash set <code>visited</code> each contain at most <span class="arithmatex">\(|V|\)</span> vertices, and the maximum recursion depth is <span class="arithmatex">\(|V|\)</span>, so the space complexity is <span class="arithmatex">\(O(|V|)\)</span>.</p>
 
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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-  10.5 Searching Algorithms Revisited
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-  10.6 Summary
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-  10.7 Exercises
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-  Chapter 11. Sorting
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-  11.4 Insertion Sort
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-  11.5 Quick Sort
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-  11.6 Merge Sort
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-  11.7 Heap Sort
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
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-  13.2 Permutations Problem
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-  14.1 Introduction to Dynamic Programming
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-  14.2 Characteristics of Dynamic Programming Problems
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-  14.3 Dynamic Programming Problem-Solving Approach
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-  14.4 0-1 Knapsack Problem
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-  14.5 Unbounded Knapsack Problem
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-  14.6 Edit Distance Problem
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-  14.7 Summary
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-  14.8 Exercises
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+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
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-  15.4 Maximum Product Cutting Problem
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-  16.2 Contributing Together
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-  16.3 Glossary
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@@ -4577,18 +4649,18 @@
 
 
 <!-- Page content -->
-<h1 id="chapter-9-graph">Chapter 9. &nbsp; Graph<a class="headerlink" href="#chapter-9-graph" title="Permanent link">&para;</a></h1>
-<p><img alt="Graph" class="cover-image" src="../assets/covers/chapter_graph.jpg" /></p>
+<h1 id="chapter-9-graphs">Chapter 9. &nbsp; Graphs<a class="headerlink" href="#chapter-9-graphs" title="Permanent link">&para;</a></h1>
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 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
-<p>In the journey of life, we are like nodes, connected by countless invisible edges.</p>
-<p>Each encounter and parting leaves a unique mark on this vast network graph.</p>
+<p>On the journey of life, we are like nodes joined by countless invisible edges.</p>
+<p>Every meeting and parting leaves a unique mark on this vast network.</p>
 </div>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
-<li><a href="graph/">9.1 &nbsp; Graph</a></li>
-<li><a href="graph_operations/">9.2 &nbsp; Basic Operations on Graphs</a></li>
-<li><a href="graph_traversal/">9.3 &nbsp; Graph Traversal</a></li>
+<li><a href="graph/">9.1 &nbsp; Graphs</a></li>
+<li><a href="graph_operations/">9.2 &nbsp; Basic graph operations</a></li>
+<li><a href="graph_traversal/">9.3 &nbsp; Graph traversal</a></li>
 <li><a href="summary/">9.4 &nbsp; Summary</a></li>
 <li><a href="exercises/">9.5 &nbsp; Exercises</a></li>
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     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 8.5 Exercises"
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     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 9.1 Graph"
+    aria-label="Next: 9.1   Graphs"
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         Next
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+        9.1 &nbsp; Graphs
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     <div class="md-footer__button md-icon">
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2380,7 +2380,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
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@@ -2422,7 +2422,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
       </span>
     </span>
   </a>
@@ -2599,7 +2599,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2709,7 +2709,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2736,7 +2736,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2763,7 +2763,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2790,7 +2790,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2817,7 +2817,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2844,7 +2844,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2871,7 +2871,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2970,7 +2970,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2991,7 +2991,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3018,7 +3018,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3045,7 +3045,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3072,7 +3072,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3099,7 +3099,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3126,7 +3126,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3153,7 +3153,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3180,7 +3180,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3207,7 +3207,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
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   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3720,7 +3720,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3886,7 +3886,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4447,7 +4447,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4474,7 +4474,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4558,6 +4558,78 @@
 
 
 
+
+
+
+
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+
+
+
+
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+
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+
+
+
+
+    <li class="md-nav__item md-nav__item--nested">
+
+
+
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_20" >
+
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+          <div class="md-nav__link md-nav__container">
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@@ -4599,10 +4671,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
       </span>
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@@ -4658,28 +4730,28 @@
 
 <!-- Page content -->
 <h1 id="94-summary">9.4 &nbsp; Summary<a class="headerlink" href="#94-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
 <ul>
-<li>Graphs consist of vertices and edges and can be represented as a set of vertices and a set of edges.</li>
-<li>Compared with the linear relationships modeled by linked lists and the divide-and-conquer relationships modeled by trees, the network relationships modeled by graphs offer much greater flexibility and are therefore more complex.</li>
-<li>In directed graphs, edges have direction; in connected graphs, every vertex is reachable from any other vertex; and in weighted graphs, each edge carries a weight.</li>
-<li>Adjacency matrices use matrices to represent graphs, where each row (column) represents a vertex, and matrix elements represent edges, using <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(0\)</span> to indicate whether two vertices have an edge or not. Adjacency matrices are highly efficient for addition, deletion, lookup, and modification operations, but consume significant space.</li>
-<li>Adjacency lists use multiple linked lists to represent a graph: the <span class="arithmatex">\(i\)</span>-th linked list corresponds to vertex <span class="arithmatex">\(i\)</span> and stores all vertices adjacent to it. Compared with adjacency matrices, adjacency lists use less space, but edge lookups are less efficient because the linked list must be traversed.</li>
-<li>When linked lists in adjacency lists become too long, they can be converted to red-black trees or hash tables, thereby improving lookup efficiency.</li>
-<li>From an algorithmic perspective, adjacency matrices embody "trading space for time", while adjacency lists embody "trading time for space".</li>
-<li>Graphs can be used to model various real-world systems, such as social networks and subway lines.</li>
-<li>Trees are a special case of graphs, and tree traversal is a special case of graph traversal.</li>
-<li>Breadth-first search in graphs explores from near to far, expanding layer by layer, and is typically implemented with a queue.</li>
-<li>Depth-first search in graphs follows a path as deep as possible and backtracks when it can go no farther, and is commonly implemented with recursion.</li>
+<li>A graph consists of vertices and edges and can be represented using a vertex set and an edge set.</li>
+<li>Network relationships (graphs) allow more freedom than linear relationships (linked lists) or divide-and-conquer relationships (trees), making them more complex.</li>
+<li>Edges in a directed graph have a direction; all vertices in a connected graph are reachable; and each edge in a weighted graph has a weight.</li>
+<li>An adjacency matrix represents a graph with a matrix. Each row (or column) represents a vertex, and entries represent edges, with <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(0\)</span> indicating whether an edge exists between two vertices. Adding, removing, looking up, and modifying edges are efficient, but the matrix uses substantial space.</li>
+<li>An adjacency list represents a graph with multiple linked lists. The <span class="arithmatex">\(i\)</span>-th linked list corresponds to vertex <span class="arithmatex">\(i\)</span> and stores all of its adjacent vertices. It uses less space than an adjacency matrix, but finding an edge requires traversing a linked list, making lookups slower.</li>
+<li>A long linked list in an adjacency list can be replaced with a red-black tree or a hash table to speed up lookups.</li>
+<li>In terms of algorithm design, an adjacency matrix trades space for time, while an adjacency list trades time for space.</li>
+<li>Graphs can model many real-world systems, such as social networks and subway networks.</li>
+<li>A tree is a special case of a graph, and tree traversal is a special case of graph traversal.</li>
+<li>Breadth-first graph traversal searches from near to far, expanding layer by layer. It is usually implemented with a queue.</li>
+<li>Depth-first graph traversal follows a path as far as possible and backtracks when it can go no farther. It is often implemented with recursion.</li>
 </ul>
 <h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
 <p><strong>Q</strong>: Is a path defined as a sequence of vertices or a sequence of edges?</p>
-<p>The definitions in different language versions of Wikipedia are inconsistent: the English version states "a path is a sequence of edges", while the Chinese version states "a path is a sequence of vertices". The following is the original English text: In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.</p>
-<p>In this text, a path is viewed as a sequence of edges, not a sequence of vertices. This is because there may be multiple edges connecting two vertices, in which case each edge corresponds to a path.</p>
-<p><strong>Q</strong>: In a disconnected graph, will there be unreachable vertices?</p>
-<p>In a disconnected graph, if you start from one vertex, at least one other vertex will be unreachable. To traverse a disconnected graph, you need multiple starting points so that all connected components are covered.</p>
-<p><strong>Q</strong>: In an adjacency list, is there any required ordering for the vertices adjacent to a given vertex?</p>
-<p>They can appear in any order. In practice, however, they may need to be sorted according to specific rules, such as the order in which vertices were added or the order of vertex values, which helps when quickly finding a vertex with some extreme value.</p>
+<p>Wikipedia's language editions give different definitions: the English edition defines a path as a sequence of edges, while the Chinese edition defines it as a sequence of vertices. The English wording is: In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.</p>
+<p>In this book, a path is treated as a sequence of edges rather than vertices. This is because multiple edges can connect the same two vertices, in which case each edge corresponds to a path.</p>
+<p><strong>Q</strong>: Can some vertices in a disconnected graph remain unvisited?</p>
+<p>In a disconnected graph, at least one vertex cannot be reached from a given starting vertex. Traversing the entire graph requires multiple starting points so that all connected components are visited.</p>
+<p><strong>Q</strong>: Must the adjacent vertices in an adjacency list appear in a particular order?</p>
+<p>They may appear in any order. In practice, however, it can be useful to sort them by a specific rule, such as the order in which vertices were added or their values. This helps us quickly find a vertex with a particular extreme value.</p>
 
 <!-- Source file information -->
 
@@ -4702,7 +4774,7 @@ aria-label="Footer"
   <a
     href="../graph_traversal/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 9.3 Graph Traversal"
+    aria-label="Previous: 9.3   Graph traversal"
     rel="prev"
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@@ -4714,7 +4786,7 @@ aria-label="Footer"
         Previous
       </span>
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-        9.3 Graph Traversal
+        9.3 &nbsp; Graph traversal
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   <a
     href="../exercises/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 9.5 Exercises"
+    aria-label="Next: 9.5   Exercises"
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@@ -4751,7 +4823,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-  Before Starting
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+  Foreword
 
           </label>
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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-  0.3 Summary
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-  Chapter 1. Encounter with Algorithms
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-  3.5 Summary
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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           <ul class="md-nav__list" data-md-scrollfix>
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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+  15.5 &nbsp; Summary
 
 
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+  15.6 &nbsp; Exercises
 
 
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         <li class="md-nav__item">
-  <a href="#1561-concept-review" class="md-nav__link">
+  <a href="#1561-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        15.6.1 &nbsp; Concept Review
+        15.6.1 &nbsp; Check your understanding
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     </span>
   </a>
 
-    <nav class="md-nav" aria-label="15.6.1 Concept Review">
+    <nav class="md-nav" aria-label="15.6.1 Check your understanding">
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     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Is Choosing the Largest Coin Always Best?
+        1. &nbsp; Is choosing the largest coin always best?
       </span>
     </span>
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     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Which Item Should Go into the Knapsack First?
+        2. &nbsp; Which item should go into the knapsack first?
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     <span class="md-ellipsis">
       <span class="md-typeset">
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+        3. &nbsp; Which pointer should move next?
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   <a href="#1562-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        15.6.2 &nbsp; Programming Exercises
+        15.6.2 &nbsp; Programming exercises
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     </span>
   </a>
 
-    <nav class="md-nav" aria-label="15.6.2 Programming Exercises">
+    <nav class="md-nav" aria-label="15.6.2 Programming exercises">
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+        1. &nbsp; Fractional knapsack
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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@@ -4614,6 +4614,78 @@
 
 
 
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+  <a href="#1561-check-your-understanding" class="md-nav__link">
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       <span class="md-typeset">
-        15.6.1 &nbsp; Concept Review
+        15.6.1 &nbsp; Check your understanding
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     </span>
   </a>
 
-    <nav class="md-nav" aria-label="15.6.1 Concept Review">
+    <nav class="md-nav" aria-label="15.6.1 Check your understanding">
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   <a href="#1-is-choosing-the-largest-coin-always-best" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Is Choosing the Largest Coin Always Best?
+        1. &nbsp; Is choosing the largest coin always best?
       </span>
     </span>
   </a>
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     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Which Item Should Go into the Knapsack First?
+        2. &nbsp; Which item should go into the knapsack first?
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       <span class="md-typeset">
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+        3. &nbsp; Which pointer should move next?
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   <a href="#1562-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        15.6.2 &nbsp; Programming Exercises
+        15.6.2 &nbsp; Programming exercises
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-    <nav class="md-nav" aria-label="15.6.2 Programming Exercises">
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+        1. &nbsp; Fractional knapsack
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 <!-- Page content -->
 <h1 id="156-exercises">15.6 &nbsp; Exercises<a class="headerlink" href="#156-exercises" title="Permanent link">&para;</a></h1>
-<h2 id="1561-concept-review">15.6.1 &nbsp; Concept Review<a class="headerlink" href="#1561-concept-review" title="Permanent link">&para;</a></h2>
-<h3 id="1-is-choosing-the-largest-coin-always-best">1. &nbsp; Is Choosing the Largest Coin Always Best?<a class="headerlink" href="#1-is-choosing-the-largest-coin-always-best" title="Permanent link">&para;</a></h3>
+<h2 id="1561-check-your-understanding">15.6.1 &nbsp; Check your understanding<a class="headerlink" href="#1561-check-your-understanding" title="Permanent link">&para;</a></h2>
+<h3 id="1-is-choosing-the-largest-coin-always-best">1. &nbsp; Is choosing the largest coin always best?<a class="headerlink" href="#1-is-choosing-the-largest-coin-always-best" title="Permanent link">&para;</a></h3>
 <p>The coin denominations are <code>[1, 7, 10]</code>, and the target amount is 14.</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Follow the rule "always choose the largest denomination that does not exceed the remaining amount," and write the coins selected.</li>
-<li>Is there a solution using fewer coins? If so, give one; otherwise, explain why not.</li>
-<li>Does this example show that the greedy strategy is correct for every set of coin denominations?</li>
+<li>Follow the rule “always take the largest denomination that does not exceed the remaining amount,” and list the coins chosen.</li>
+<li>Is there a way to use fewer coins? If so, give one; if not, explain why.</li>
+<li>Can you conclude that this greedy strategy is correct for any set of coin denominations?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>The greedy strategy selects <code>10 + 1 + 1 + 1 + 1</code>, using 5 coins in total.</p>
+<p>The greedy strategy chooses <code>10 + 1 + 1 + 1 + 1</code>, using 5 coins in total.</p>
 </li>
 <li>
-<p>There is a solution using fewer coins: <code>7 + 7</code>, which uses only 2 coins.</p>
+<p>There is a solution with fewer coins: <code>7 + 7</code>, which uses only 2.</p>
 </li>
 <li>
-<p>No. This counterexample shows that, for arbitrary coin denominations, repeatedly choosing the largest currently available denomination does not necessarily minimize the number of coins.
-    The largest immediate choice may prevent a better combination later.</p>
+<p>No. This counterexample shows that choosing the largest available denomination at each step does not always minimize the number of coins for arbitrary denominations.
+    The largest immediate choice can rule out a better combination later.</p>
 </li>
 </ol>
 </details>
-<h3 id="2-which-item-should-go-into-the-knapsack-first">2. &nbsp; Which Item Should Go into the Knapsack First?<a class="headerlink" href="#2-which-item-should-go-into-the-knapsack-first" title="Permanent link">&para;</a></h3>
-<p>A knapsack with a capacity of 4 kilograms can hold the following items. A fraction of an item may be taken,
-and the value obtained is proportional to its weight:</p>
+<h3 id="2-which-item-should-go-into-the-knapsack-first">2. &nbsp; Which item should go into the knapsack first?<a class="headerlink" href="#2-which-item-should-go-into-the-knapsack-first" title="Permanent link">&para;</a></h3>
+<p>A knapsack with a capacity of 4 kg can hold the following items. You may take part of an item,
+and the value obtained is proportional to the weight taken.</p>
 <ul>
-<li>Item A: weight 4 kilograms, value 20.</li>
-<li>Item B: weight 3 kilograms, value 18.</li>
+<li>Item A weighs 4 kg and has a value of 20.</li>
+<li>Item B weighs 3 kg and has a value of 18.</li>
 </ul>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>What is the value per kilogram of each item? Which item should be placed in the knapsack first?</li>
-<li>Fill the knapsack using the greedy strategy for the fractional knapsack problem. What is the final value?</li>
-<li>When items can be divided and the knapsack limits total weight, should items be compared by total value or by value per kilogram? Why?</li>
+<li>What is each item's value per kilogram? Which item should you take first?</li>
+<li>If you fill the knapsack using the greedy strategy for fractional knapsack, what is the final value?</li>
+<li>When items can be divided and the knapsack limits total weight, should you compare total value or value per kilogram? Why?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>A has value <code>20 ÷ 4 = 5</code> per kilogram, while B has value <code>18 ÷ 3 = 6</code> per kilogram,
-    so B, with the higher value per unit weight, should be placed in the knapsack first.</p>
+<p>A's value per kilogram is <code>20 ÷ 4 = 5</code>, and B's is <code>18 ÷ 3 = 6</code>.
+    Take B first because it has the higher value per unit of weight.</p>
 </li>
 <li>
-<p>First take all of B, using 3 kilograms of capacity and gaining a value of 18. With 1 kilogram of capacity remaining,
-    take 1 kilogram of A, gaining a value of 5. The final value is <code>18 + 5 = 23</code>.</p>
+<p>Take all of B first, using 3 kg of capacity and gaining a value of 18. With 1 kg of capacity left,
+    take 1 kg of A for a value of 5. The final value is <code>18 + 5 = 23</code>.</p>
 </li>
 <li>
-<p>The knapsack limits total weight, and items can be divided, so they should be compared by value per unit weight.
-    Although A has a higher total value, its value per kilogram is lower than B's. Filling the knapsack with A first would yield only a value of 20.</p>
+<p>The knapsack limits total weight, and items can be divided, so compare value per unit of weight.
+    Although A has a higher total value, its value per kilogram is lower than B's. Taking all of A first would give a value of only 20.</p>
 </li>
 </ol>
 </details>
-<h3 id="3-which-pointer-should-move-next">3. &nbsp; Which Pointer Should Move Next?<a class="headerlink" href="#3-which-pointer-should-move-next" title="Permanent link">&para;</a></h3>
-<p>The partition heights are <code>[1, 8, 6, 2, 5]</code>. Use two pointers, one at each end, to find the maximum capacity.
-The capacity equals "the height of the shorter partition × the distance between the partitions' indices."</p>
+<h3 id="3-which-pointer-should-move-next">3. &nbsp; Which pointer should move next?<a class="headerlink" href="#3-which-pointer-should-move-next" title="Permanent link">&para;</a></h3>
+<p>The wall heights are <code>[1, 8, 6, 2, 5]</code>. Use two pointers, initially at opposite ends, to find the maximum container capacity.
+Capacity equals the height of the shorter of the two walls multiplied by the difference between their indices.</p>
 <!-- numbered-subquestions -->
 
 <ol>
 <li>Initially, the left pointer is at index 0 and the right pointer is at index 4. What is the current capacity? Which pointer should move next?</li>
-<li>After making the move chosen in Question 1, at which indices are the two pointers? What is the capacity now? Which pointer should move next?</li>
-<li>For the current pair of partitions, you could move either the pointer at the shorter partition or the pointer at the taller partition. Which move could still produce a greater capacity, and why?</li>
+<li>After making the move you chose in question 1, what are the two pointer indices? What is the capacity now, and which pointer should move next?</li>
+<li>For the current pair of walls, you could move the pointer at the shorter wall or the pointer at the taller wall. Which move still has a chance of producing a larger capacity? Why?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>The current capacity is <code>min(1, 5) × (4 - 0) = 4</code>. The left partition is shorter, so move the left pointer.</p>
+<p>The current capacity is <code>min(1, 5) × (4 - 0) = 4</code>. The left wall is shorter, so move the left pointer.</p>
 </li>
 <li>
-<p>After the left pointer moves, the pointers are at indices 1 and 4. The current capacity is
-    <code>min(8, 5) × (4 - 1) = 15</code>. The right partition is shorter, so move the right pointer next.</p>
+<p>After moving the left pointer, the pointers are at indices 1 and 4. The current capacity is
+    <code>min(8, 5) × (4 - 1) = 15</code>. The right wall is shorter, so move the right pointer next.</p>
 </li>
 <li>
-<p>Moving the pointer at the shorter partition is the only move that could still produce a greater capacity. If the taller partition's pointer moves, the distance certainly decreases while the height remains limited by the unmoved shorter partition,
-    so the capacity can only stay the same or decrease. Only by moving the shorter partition can a taller partition possibly be found.</p>
+<p>Only moving the pointer at the shorter wall can still produce a larger capacity. Moving the pointer at the taller wall reduces the distance between the walls, while the container's height remains limited by the shorter wall that has not moved.
+    That height can only stay the same or decrease, so the capacity cannot exceed its previous value. Moving the pointer at the shorter wall is the only way to have a chance of finding a taller wall.</p>
 </li>
 </ol>
 </details>
-<h2 id="1562-programming-exercises">15.6.2 &nbsp; Programming Exercises<a class="headerlink" href="#1562-programming-exercises" title="Permanent link">&para;</a></h2>
-<h3 id="1-fractional-knapsack">1. &nbsp; Fractional Knapsack<a class="headerlink" href="#1-fractional-knapsack" title="Permanent link">&para;</a></h3>
-<p>You are given equal-length arrays <code>wgt</code> and <code>val</code>, where <code>wgt[i] &gt; 0</code> and <code>val[i] &gt;= 0</code>. The knapsack has capacity <code>cap &gt;= 0</code>.
-There is only one of each item, but any fraction of an item may be placed in the knapsack.
-The value obtained is proportional to the fraction of the item's total weight that is included. Use a greedy algorithm
-and return the maximum total value the knapsack can hold as a real number.</p>
+<h2 id="1562-programming-exercises">15.6.2 &nbsp; Programming exercises<a class="headerlink" href="#1562-programming-exercises" title="Permanent link">&para;</a></h2>
+<h3 id="1-fractional-knapsack">1. &nbsp; Fractional knapsack<a class="headerlink" href="#1-fractional-knapsack" title="Permanent link">&para;</a></h3>
+<p>You are given arrays <code>wgt</code> and <code>val</code> of equal length, where <code>wgt[i] &gt; 0</code> and <code>val[i] &gt;= 0</code>, and a knapsack capacity <code>cap &gt;= 0</code>.
+There is one of each item, but you may take just part of it.
+The value obtained is proportional to the fraction of the item's total weight that you take. Use a greedy algorithm
+to return the maximum total value the knapsack can hold, as a real number.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>First calculate each item's value per unit weight as val[i] / wgt[i], keeping the fractional part of the division</li>
-<li>Place items with higher value per unit weight into the knapsack first</li>
-<li>If the remaining capacity is less than the current item's weight, take exactly the fraction that fills the knapsack and stop</li>
+<li>First calculate each item's value per unit of weight, val[i] / wgt[i]. Preserve the fractional part of the division result.</li>
+<li>Take items with a higher value per unit of weight first.</li>
+<li>If the remaining capacity is less than the current item's weight, take just enough of the item to fill the knapsack, then stop.</li>
 </ol>
 </details>
 
@@ -4888,7 +4960,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 15.5 Summary"
+    aria-label="Previous: 15.5   Summary"
     rel="prev"
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     <div class="md-footer__button md-icon">
@@ -4900,7 +4972,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        15.5 Summary
+        15.5 &nbsp; Summary
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@@ -4912,7 +4984,7 @@ aria-label="Footer"
   <a
     href="../../chapter_appendix/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: Chapter 16. &amp;nbsp; Appendix"
+    aria-label="Next: Chapter 16.   Appendix"
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@@ -4937,7 +5009,7 @@ aria-label="Footer"
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+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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@@ -41,7 +41,7 @@
 
 
 
-        <title>15.2 Fractional Knapsack Problem - Hello Algo</title>
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+              15.2 &nbsp; Fractional knapsack problem
 
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-  Before Starting
+  Foreword
 
 
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-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
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   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
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   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
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-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1265,7 +1265,7 @@
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-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
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@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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@@ -2378,7 +2378,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3283,7 +3283,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3304,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3352,7 +3352,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3379,7 +3379,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3406,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3433,7 +3433,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3460,7 +3460,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3526,7 +3526,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3568,7 +3568,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3595,7 +3595,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3622,7 +3622,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3649,7 +3649,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3676,7 +3676,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3773,7 +3773,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3815,7 +3815,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3869,7 +3869,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4072,7 +4072,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4114,7 +4114,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4193,10 +4193,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-greedy-strategy-determination" class="md-nav__link">
+  <a href="#1-choosing-a-greedy-strategy" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Greedy Strategy Determination
+        1. &nbsp; Choosing a greedy strategy
       </span>
     </span>
   </a>
@@ -4207,7 +4207,7 @@
   <a href="#2-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Code Implementation
+        2. &nbsp; Code implementation
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     </span>
   </a>
@@ -4215,10 +4215,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#3-correctness-proof" class="md-nav__link">
+  <a href="#3-proof-of-correctness" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Correctness Proof
+        3. &nbsp; Proof of correctness
       </span>
     </span>
   </a>
@@ -4248,7 +4248,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4275,7 +4275,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4389,7 +4389,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4431,7 +4431,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4458,7 +4458,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4485,7 +4485,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4569,6 +4569,78 @@
 
 
 
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+  <a href="#1-choosing-a-greedy-strategy" class="md-nav__link">
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-        1. &nbsp; Greedy Strategy Determination
+        1. &nbsp; Choosing a greedy strategy
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+  <a href="#3-proof-of-correctness" class="md-nav__link">
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-        3. &nbsp; Correctness Proof
+        3. &nbsp; Proof of correctness
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@@ -4679,35 +4751,35 @@
 
 
 <!-- Page content -->
-<h1 id="152-fractional-knapsack-problem">15.2 &nbsp; Fractional Knapsack Problem<a class="headerlink" href="#152-fractional-knapsack-problem" title="Permanent link">&para;</a></h1>
+<h1 id="152-fractional-knapsack-problem">15.2 &nbsp; Fractional knapsack problem<a class="headerlink" href="#152-fractional-knapsack-problem" title="Permanent link">&para;</a></h1>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given <span class="arithmatex">\(n\)</span> items, where the weight of the <span class="arithmatex">\(i\)</span>-th item is <span class="arithmatex">\(wgt[i-1]\)</span> and its value is <span class="arithmatex">\(val[i-1]\)</span>, and a knapsack with capacity <span class="arithmatex">\(cap\)</span>. Each item can be selected only once, <strong>but a fraction of an item may be selected, with its value proportional to the selected weight</strong>. What is the maximum total value that can be placed in the knapsack under the capacity constraint? An example is shown in Figure 15-3.</p>
+<p>Given <span class="arithmatex">\(n\)</span> items, where item <span class="arithmatex">\(i\)</span> has weight <span class="arithmatex">\(wgt[i-1]\)</span> and value <span class="arithmatex">\(val[i-1]\)</span>, and a knapsack with capacity <span class="arithmatex">\(cap\)</span>, each item may be selected only once. <strong>However, you may take a fraction of an item, with its value proportional to the weight taken</strong>. What is the maximum total value that fits within the knapsack's capacity? An example is shown in Figure 15-3.</p>
 </div>
-<p><img alt="Example data for the fractional knapsack problem" class="animation-figure" src="../fractional_knapsack_problem.assets/fractional_knapsack_example.png" /></p>
-<p align="center"> Figure 15-3 &nbsp; Example data for the fractional knapsack problem </p>
+<p><img alt="Example input for the fractional knapsack problem" class="animation-figure" src="../fractional_knapsack_problem.assets/fractional_knapsack_example.png" /></p>
+<p align="center"> Figure 15-3 &nbsp; Example input for the fractional knapsack problem </p>
 
-<p>The fractional knapsack problem is very similar overall to the 0-1 knapsack problem, with states including the current item <span class="arithmatex">\(i\)</span> and capacity <span class="arithmatex">\(c\)</span>, and the goal being to maximize value under the limited knapsack capacity.</p>
-<p>The difference is that this problem allows selecting only a fraction of an item. As shown in Figure 15-4, <strong>we can split an item arbitrarily and compute its value in proportion to the selected weight</strong>.</p>
+<p>The fractional knapsack problem is very similar to the 0-1 knapsack problem. The state consists of the current item <span class="arithmatex">\(i\)</span> and capacity <span class="arithmatex">\(c\)</span>, and the goal is to maximize value within the knapsack's capacity.</p>
+<p>The difference is that we may take only part of an item. As shown in Figure 15-4, <strong>we can divide items into arbitrary fractions and calculate their values in proportion to their weights</strong>.</p>
 <ol>
-<li>For item <span class="arithmatex">\(i\)</span>, its value per unit weight is <span class="arithmatex">\(val[i-1] / wgt[i-1]\)</span>, referred to as unit value.</li>
-<li>Suppose we put a portion of item <span class="arithmatex">\(i\)</span> with weight <span class="arithmatex">\(w\)</span> into the knapsack, then the value added to the knapsack is <span class="arithmatex">\(w \times val[i-1] / wgt[i-1]\)</span>.</li>
+<li>For item <span class="arithmatex">\(i\)</span>, the value per unit weight is <span class="arithmatex">\(val[i-1] / wgt[i-1]\)</span>, which we call its unit value.</li>
+<li>If we take a portion of item <span class="arithmatex">\(i\)</span> weighing <span class="arithmatex">\(w\)</span>, the value added to the knapsack is <span class="arithmatex">\(w \times val[i-1] / wgt[i-1]\)</span>.</li>
 </ol>
-<p><img alt="Value of items per unit weight" class="animation-figure" src="../fractional_knapsack_problem.assets/fractional_knapsack_unit_value.png" /></p>
-<p align="center"> Figure 15-4 &nbsp; Value of items per unit weight </p>
+<p><img alt="Value per unit weight of each item" class="animation-figure" src="../fractional_knapsack_problem.assets/fractional_knapsack_unit_value.png" /></p>
+<p align="center"> Figure 15-4 &nbsp; Value per unit weight of each item </p>
 
-<h3 id="1-greedy-strategy-determination">1. &nbsp; Greedy Strategy Determination<a class="headerlink" href="#1-greedy-strategy-determination" title="Permanent link">&para;</a></h3>
-<p>Maximizing the total value in the knapsack <strong>essentially means prioritizing items with higher value per unit weight</strong>. From this observation, we can derive the greedy strategy shown in Figure 15-5.</p>
+<h3 id="1-choosing-a-greedy-strategy">1. &nbsp; Choosing a greedy strategy<a class="headerlink" href="#1-choosing-a-greedy-strategy" title="Permanent link">&para;</a></h3>
+<p>Maximizing the total value in the knapsack <strong>essentially means maximizing the value obtained per unit weight</strong>. This leads to the greedy strategy shown in Figure 15-5.</p>
 <ol>
-<li>Sort items by unit value from high to low.</li>
-<li>Iterate through all items, <strong>greedily selecting the item with the highest unit value in each round</strong>.</li>
-<li>If the remaining knapsack capacity is insufficient, use a portion of the current item to fill the knapsack.</li>
+<li>Sort the items by unit value in descending order.</li>
+<li>Iterate through the items, <strong>greedily choosing the item with the highest unit value at each step</strong>.</li>
+<li>If the remaining capacity is insufficient, take just enough of the current item to fill the knapsack.</li>
 </ol>
 <p><img alt="Greedy strategy for the fractional knapsack problem" class="animation-figure" src="../fractional_knapsack_problem.assets/fractional_knapsack_greedy_strategy.png" /></p>
 <p align="center"> Figure 15-5 &nbsp; Greedy strategy for the fractional knapsack problem </p>
 
-<h3 id="2-code-implementation">2. &nbsp; Code Implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">&para;</a></h3>
-<p>We define an <code>Item</code> class so that items can be sorted by unit value. We then iterate through the sorted items greedily, stopping once the knapsack is full and returning the result:</p>
+<h3 id="2-code-implementation">2. &nbsp; Code implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">&para;</a></h3>
+<p>We define an <code>Item</code> class to make it easier to sort items by unit value. We repeatedly make greedy choices, stopping and returning the result when the knapsack is full:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
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 <div class="tabbed-block">
@@ -4719,22 +4791,22 @@
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">v</span> <span class="o">=</span> <span class="n">v</span>  <span class="c1"># Item value</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="k">def</span><span class="w"> </span><span class="nf">fractional_knapsack</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">val</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">cap</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="w">    </span><span class="sd">"""Fractional knapsack: Greedy algorithm"""</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="c1"># Create item list with two attributes: weight, value</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="w">    </span><span class="sd">"""Fractional knapsack: greedy algorithm"""</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="c1"># Create a list of items, each with two attributes: weight and value</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="n">items</span> <span class="o">=</span> <span class="p">[</span><span class="n">Item</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">w</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span> <span class="n">val</span><span class="p">)]</span>
-<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="c1"># Sort by unit value item.v / item.w from high to low</span>
+<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="c1"># Sort by value per unit weight, item.v / item.w, in descending order</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>    <span class="n">items</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">key</span><span class="o">=</span><span class="k">lambda</span> <span class="n">item</span><span class="p">:</span> <span class="n">item</span><span class="o">.</span><span class="n">v</span> <span class="o">/</span> <span class="n">item</span><span class="o">.</span><span class="n">w</span><span class="p">,</span> <span class="n">reverse</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
-<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>    <span class="c1"># Loop for greedy selection</span>
+<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>    <span class="c1"># Repeatedly make greedy choices</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="n">res</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>    <span class="k">for</span> <span class="n">item</span> <span class="ow">in</span> <span class="n">items</span><span class="p">:</span>
 <a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>        <span class="k">if</span> <span class="n">item</span><span class="o">.</span><span class="n">w</span> <span class="o">&lt;=</span> <span class="n">cap</span><span class="p">:</span>
-<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>            <span class="c1"># If remaining capacity is sufficient, put the entire current item into the knapsack</span>
+<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>            <span class="c1"># If enough capacity remains, put the entire current item in the knapsack</span>
 <a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>            <span class="n">res</span> <span class="o">+=</span> <span class="n">item</span><span class="o">.</span><span class="n">v</span>
 <a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>            <span class="n">cap</span> <span class="o">-=</span> <span class="n">item</span><span class="o">.</span><span class="n">w</span>
 <a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a>        <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>            <span class="c1"># If remaining capacity is insufficient, put part of the current item into the knapsack</span>
+<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>            <span class="c1"># If the remaining capacity is insufficient, put part of the current item in the knapsack</span>
 <a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a>            <span class="n">res</span> <span class="o">+=</span> <span class="p">(</span><span class="n">item</span><span class="o">.</span><span class="n">v</span> <span class="o">/</span> <span class="n">item</span><span class="o">.</span><span class="n">w</span><span class="p">)</span> <span class="o">*</span> <span class="n">cap</span>
-<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a>            <span class="c1"># No remaining capacity, so break out of the loop</span>
+<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a>            <span class="c1"># No capacity remains, so exit the loop</span>
 <a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a>            <span class="k">break</span>
 <a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a>    <span class="k">return</span> <span class="n">res</span>
 </code></pre></div>
@@ -4750,26 +4822,26 @@
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="p">};</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a>
-<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="cm">/* Fractional knapsack: Greedy algorithm */</span>
+<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="cm">/* Fractional knapsack: greedy algorithm */</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="kt">double</span><span class="w"> </span><span class="nf">fractionalKnapsack</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">    </span><span class="c1">// Create item list with two attributes: weight, value</span>
+<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">    </span><span class="c1">// Create a list of items, each with two attributes: weight and value</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">Item</span><span class="o">&gt;</span><span class="w"> </span><span class="n">items</span><span class="p">;</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">        </span><span class="n">items</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">Item</span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="p">]));</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">    </span><span class="c1">// Sort by unit value item.v / item.w from high to low</span>
+<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">    </span><span class="c1">// Sort by value per unit weight, item.v / item.w, in descending order</span>
 <a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">    </span><span class="n">sort</span><span class="p">(</span><span class="n">items</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span><span class="w"> </span><span class="n">items</span><span class="p">.</span><span class="n">end</span><span class="p">(),</span><span class="w"> </span><span class="p">[](</span><span class="n">Item</span><span class="w"> </span><span class="o">&amp;</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">Item</span><span class="w"> </span><span class="o">&amp;</span><span class="n">b</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">a</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">a</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">b</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">b</span><span class="p">.</span><span class="n">w</span><span class="p">;</span><span class="w"> </span><span class="p">});</span>
-<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">    </span><span class="c1">// Loop for greedy selection</span>
+<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices</span>
 <a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w">    </span><span class="kt">double</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="k">auto</span><span class="w"> </span><span class="o">&amp;</span><span class="n">item</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">items</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="w">            </span><span class="c1">// If remaining capacity is sufficient, put the entire current item into the knapsack</span>
+<a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="w">            </span><span class="c1">// If enough capacity remains, put the entire current item in the knapsack</span>
 <a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w">            </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="p">;</span>
 <a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="w">            </span><span class="n">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="p">;</span>
 <a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="w">            </span><span class="c1">// If remaining capacity is insufficient, put part of the current item into the knapsack</span>
+<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="w">            </span><span class="c1">// If the remaining capacity is insufficient, put part of the current item in the knapsack</span>
 <a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="w">            </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span>
-<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="w">            </span><span class="c1">// No remaining capacity, so break out of the loop</span>
+<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="w">            </span><span class="c1">// No capacity remains, so exit the loop</span>
 <a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="w">    </span><span class="p">}</span>
@@ -4789,26 +4861,26 @@
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="p">}</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a>
-<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="cm">/* Fractional knapsack: Greedy algorithm */</span>
+<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="cm">/* Fractional knapsack: greedy algorithm */</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="kt">double</span><span class="w"> </span><span class="nf">fractionalKnapsack</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">    </span><span class="c1">// Create item list with two attributes: weight, value</span>
+<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">    </span><span class="c1">// Create a list of items, each with two attributes: weight and value</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="n">Item</span><span class="o">[]</span><span class="w"> </span><span class="n">items</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Item</span><span class="o">[</span><span class="n">wgt</span><span class="p">.</span><span class="na">length</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">        </span><span class="n">items</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Item</span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">);</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">    </span><span class="c1">// Sort by unit value item.v / item.w from high to low</span>
+<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">    </span><span class="c1">// Sort by value per unit weight, item.v / item.w, in descending order</span>
 <a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">    </span><span class="n">Arrays</span><span class="p">.</span><span class="na">sort</span><span class="p">(</span><span class="n">items</span><span class="p">,</span><span class="w"> </span><span class="n">Comparator</span><span class="p">.</span><span class="na">comparingDouble</span><span class="p">(</span><span class="n">item</span><span class="w"> </span><span class="o">-&gt;</span><span class="w"> </span><span class="o">-</span><span class="p">((</span><span class="kt">double</span><span class="p">)</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">w</span><span class="p">)));</span>
-<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">    </span><span class="c1">// Loop for greedy selection</span>
+<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices</span>
 <a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="w">    </span><span class="kt">double</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Item</span><span class="w"> </span><span class="n">item</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">items</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">item</span><span class="p">.</span><span class="na">w</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="w">            </span><span class="c1">// If remaining capacity is sufficient, put the entire current item into the knapsack</span>
+<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="w">            </span><span class="c1">// If enough capacity remains, put the entire current item in the knapsack</span>
 <a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="w">            </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">v</span><span class="p">;</span>
 <a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a><span class="w">            </span><span class="n">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">w</span><span class="p">;</span>
 <a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="w">            </span><span class="c1">// If remaining capacity is insufficient, put part of the current item into the knapsack</span>
+<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="w">            </span><span class="c1">// If the remaining capacity is insufficient, put part of the current item in the knapsack</span>
 <a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a><span class="w">            </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">w</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span>
-<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="w">            </span><span class="c1">// No remaining capacity, so break out of the loop</span>
+<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="w">            </span><span class="c1">// No capacity remains, so exit the loop</span>
 <a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a><span class="w">    </span><span class="p">}</span>
@@ -4823,26 +4895,26 @@
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">v</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">v</span><span class="p">;</span><span class="w"> </span><span class="c1">// Item value</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="p">}</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a>
-<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="cm">/* Fractional knapsack: Greedy algorithm */</span>
+<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="cm">/* Fractional knapsack: greedy algorithm */</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="kt">double</span><span class="w"> </span><span class="nf">FractionalKnapsack</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="c1">// Create item list with two attributes: weight, value</span>
+<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="c1">// Create a list of items, each with two attributes: weight and value</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="n">Item</span><span class="p">[]</span><span class="w"> </span><span class="n">items</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Item</span><span class="p">[</span><span class="n">wgt</span><span class="p">.</span><span class="n">Length</span><span class="p">];</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="n">items</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Item</span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="c1">// Sort by unit value item.v / item.w from high to low</span>
+<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="c1">// Sort by value per unit weight, item.v / item.w, in descending order</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">    </span><span class="n">Array</span><span class="p">.</span><span class="n">Sort</span><span class="p">(</span><span class="n">items</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">y</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">y</span><span class="p">.</span><span class="n">w</span><span class="p">).</span><span class="n">CompareTo</span><span class="p">(</span><span class="n">x</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">x</span><span class="p">.</span><span class="n">w</span><span class="p">));</span>
-<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">    </span><span class="c1">// Loop for greedy selection</span>
+<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">    </span><span class="kt">double</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">Item</span><span class="w"> </span><span class="n">item</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">items</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">            </span><span class="c1">// If remaining capacity is sufficient, put the entire current item into the knapsack</span>
+<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">            </span><span class="c1">// If enough capacity remains, put the entire current item in the knapsack</span>
 <a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w">            </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="p">;</span>
 <a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w">            </span><span class="n">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="p">;</span>
 <a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w">            </span><span class="c1">// If remaining capacity is insufficient, put part of the current item into the knapsack</span>
+<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w">            </span><span class="c1">// If the remaining capacity is insufficient, put part of the current item in the knapsack</span>
 <a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="w">            </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span>
-<a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="w">            </span><span class="c1">// No remaining capacity, so break out of the loop</span>
+<a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="w">            </span><span class="c1">// No capacity remains, so exit the loop</span>
 <a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="w">    </span><span class="p">}</span>
@@ -4857,28 +4929,28 @@
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">v</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="c1">// Item value</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="p">}</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="cm">/* Fractional knapsack: Greedy algorithm */</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="cm">/* Fractional knapsack: greedy algorithm */</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="kd">func</span><span class="w"> </span><span class="nx">fractionalKnapsack</span><span class="p">(</span><span class="nx">wgt</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">float64</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// Create item list with two attributes: weight, value</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// Create a list of items, each with two attributes: weight and value</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="nx">items</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="nx">Item</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">wgt</span><span class="p">))</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">wgt</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="nx">items</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">Item</span><span class="p">{</span><span class="nx">wgt</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="p">]}</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="c1">// Sort by unit value item.v / item.w from high to low</span>
+<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="c1">// Sort by value per unit weight, item.v / item.w, in descending order</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="nx">sort</span><span class="p">.</span><span class="nx">Slice</span><span class="p">(</span><span class="nx">items</span><span class="p">,</span><span class="w"> </span><span class="kd">func</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">items</span><span class="p">[</span><span class="nx">i</span><span class="p">].</span><span class="nx">v</span><span class="p">)</span><span class="o">/</span><span class="kt">float64</span><span class="p">(</span><span class="nx">items</span><span class="p">[</span><span class="nx">i</span><span class="p">].</span><span class="nx">w</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">items</span><span class="p">[</span><span class="nx">j</span><span class="p">].</span><span class="nx">v</span><span class="p">)</span><span class="o">/</span><span class="kt">float64</span><span class="p">(</span><span class="nx">items</span><span class="p">[</span><span class="nx">j</span><span class="p">].</span><span class="nx">w</span><span class="p">)</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="p">})</span>
-<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">    </span><span class="c1">// Loop for greedy selection</span>
+<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mf">0.0</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">item</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">items</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">item</span><span class="p">.</span><span class="nx">w</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">cap</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">            </span><span class="c1">// If remaining capacity is sufficient, put the entire current item into the knapsack</span>
+<a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">            </span><span class="c1">// If enough capacity remains, put the entire current item in the knapsack</span>
 <a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w">            </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">item</span><span class="p">.</span><span class="nx">v</span><span class="p">)</span>
 <a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="w">            </span><span class="nx">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="nx">item</span><span class="p">.</span><span class="nx">w</span>
 <a id="__codelineno-4-25" name="__codelineno-4-25" href="#__codelineno-4-25"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-26" name="__codelineno-4-26" href="#__codelineno-4-26"></a><span class="w">            </span><span class="c1">// If remaining capacity is insufficient, put part of the current item into the knapsack</span>
+<a id="__codelineno-4-26" name="__codelineno-4-26" href="#__codelineno-4-26"></a><span class="w">            </span><span class="c1">// If the remaining capacity is insufficient, put part of the current item in the knapsack</span>
 <a id="__codelineno-4-27" name="__codelineno-4-27" href="#__codelineno-4-27"></a><span class="w">            </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">item</span><span class="p">.</span><span class="nx">v</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">item</span><span class="p">.</span><span class="nx">w</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">cap</span><span class="p">)</span>
-<a id="__codelineno-4-28" name="__codelineno-4-28" href="#__codelineno-4-28"></a><span class="w">            </span><span class="c1">// No remaining capacity, so break out of the loop</span>
+<a id="__codelineno-4-28" name="__codelineno-4-28" href="#__codelineno-4-28"></a><span class="w">            </span><span class="c1">// No capacity remains, so exit the loop</span>
 <a id="__codelineno-4-29" name="__codelineno-4-29" href="#__codelineno-4-29"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-4-30" name="__codelineno-4-30" href="#__codelineno-4-30"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-4-31" name="__codelineno-4-31" href="#__codelineno-4-31"></a><span class="w">    </span><span class="p">}</span>
@@ -4898,24 +4970,24 @@
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="p">}</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a>
-<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="cm">/* Fractional knapsack: Greedy algorithm */</span>
+<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="cm">/* Fractional knapsack: greedy algorithm */</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="kd">func</span><span class="w"> </span><span class="nf">fractionalKnapsack</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">cap</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Double</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">    </span><span class="c1">// Create item list with two attributes: weight, value</span>
+<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">    </span><span class="c1">// Create a list of items, each with two attributes: weight and value</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">items</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">zip</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">).</span><span class="bp">map</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">Item</span><span class="p">(</span><span class="n">w</span><span class="p">:</span><span class="w"> </span><span class="nv">$0</span><span class="p">,</span><span class="w"> </span><span class="n">v</span><span class="p">:</span><span class="w"> </span><span class="nv">$1</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">    </span><span class="c1">// Sort by unit value item.v / item.w from high to low</span>
+<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">    </span><span class="c1">// Sort by value per unit weight, item.v / item.w, in descending order</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">    </span><span class="n">items</span><span class="p">.</span><span class="bp">sort</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">-</span><span class="p">(</span><span class="nb">Double</span><span class="p">(</span><span class="nv">$0</span><span class="p">.</span><span class="n">v</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nb">Double</span><span class="p">(</span><span class="nv">$0</span><span class="p">.</span><span class="n">w</span><span class="p">))</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="o">-</span><span class="p">(</span><span class="nb">Double</span><span class="p">(</span><span class="nv">$1</span><span class="p">.</span><span class="n">v</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nb">Double</span><span class="p">(</span><span class="nv">$1</span><span class="p">.</span><span class="n">w</span><span class="p">))</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">    </span><span class="c1">// Loop for greedy selection</span>
+<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices</span>
 <a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mf">0.0</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">cap</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">cap</span>
 <a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">item</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">items</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="w">            </span><span class="c1">// If remaining capacity is sufficient, put the entire current item into the knapsack</span>
+<a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="w">            </span><span class="c1">// If enough capacity remains, put the entire current item in the knapsack</span>
 <a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="w">            </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nb">Double</span><span class="p">(</span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="p">)</span>
 <a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a><span class="w">            </span><span class="n">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span>
 <a id="__codelineno-5-26" name="__codelineno-5-26" href="#__codelineno-5-26"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-27" name="__codelineno-5-27" href="#__codelineno-5-27"></a><span class="w">            </span><span class="c1">// If remaining capacity is insufficient, put part of the current item into the knapsack</span>
+<a id="__codelineno-5-27" name="__codelineno-5-27" href="#__codelineno-5-27"></a><span class="w">            </span><span class="c1">// If the remaining capacity is insufficient, put part of the current item in the knapsack</span>
 <a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="w">            </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nb">Double</span><span class="p">(</span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nb">Double</span><span class="p">(</span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nb">Double</span><span class="p">(</span><span class="n">cap</span><span class="p">)</span>
-<a id="__codelineno-5-29" name="__codelineno-5-29" href="#__codelineno-5-29"></a><span class="w">            </span><span class="c1">// No remaining capacity, so break out of the loop</span>
+<a id="__codelineno-5-29" name="__codelineno-5-29" href="#__codelineno-5-29"></a><span class="w">            </span><span class="c1">// No capacity remains, so exit the loop</span>
 <a id="__codelineno-5-30" name="__codelineno-5-30" href="#__codelineno-5-30"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-5-31" name="__codelineno-5-31" href="#__codelineno-5-31"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-32" name="__codelineno-5-32" href="#__codelineno-5-32"></a><span class="w">    </span><span class="p">}</span>
@@ -4932,23 +5004,23 @@
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="p">}</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a>
-<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="cm">/* Fractional knapsack: Greedy algorithm */</span>
+<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="cm">/* Fractional knapsack: greedy algorithm */</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="kd">function</span><span class="w"> </span><span class="nx">fractionalKnapsack</span><span class="p">(</span><span class="nx">wgt</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">    </span><span class="c1">// Create item list with two attributes: weight, value</span>
+<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">    </span><span class="c1">// Create a list of items, each with two attributes: weight and value</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">items</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">wgt</span><span class="p">.</span><span class="nx">map</span><span class="p">((</span><span class="nx">w</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Item</span><span class="p">(</span><span class="nx">w</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="p">]));</span>
-<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="c1">// Sort by unit value item.v / item.w from high to low</span>
+<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="c1">// Sort by value per unit weight, item.v / item.w, in descending order</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">    </span><span class="nx">items</span><span class="p">.</span><span class="nx">sort</span><span class="p">((</span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">b</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">b</span><span class="p">.</span><span class="nx">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nx">b</span><span class="p">.</span><span class="nx">w</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">a</span><span class="p">.</span><span class="nx">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nx">a</span><span class="p">.</span><span class="nx">w</span><span class="p">);</span>
-<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">    </span><span class="c1">// Loop for greedy selection</span>
+<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">item</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">items</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">item</span><span class="p">.</span><span class="nx">w</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">            </span><span class="c1">// If remaining capacity is sufficient, put the entire current item into the knapsack</span>
+<a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">            </span><span class="c1">// If enough capacity remains, put the entire current item in the knapsack</span>
 <a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">            </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">item</span><span class="p">.</span><span class="nx">v</span><span class="p">;</span>
 <a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">            </span><span class="nx">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="nx">item</span><span class="p">.</span><span class="nx">w</span><span class="p">;</span>
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="w">            </span><span class="c1">// If remaining capacity is insufficient, put part of the current item into the knapsack</span>
+<a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="w">            </span><span class="c1">// If the remaining capacity is insufficient, put part of the current item in the knapsack</span>
 <a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a><span class="w">            </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">(</span><span class="nx">item</span><span class="p">.</span><span class="nx">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nx">item</span><span class="p">.</span><span class="nx">w</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span>
-<a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="w">            </span><span class="c1">// No remaining capacity, so break out of the loop</span>
+<a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="w">            </span><span class="c1">// No capacity remains, so exit the loop</span>
 <a id="__codelineno-6-26" name="__codelineno-6-26" href="#__codelineno-6-26"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-6-27" name="__codelineno-6-27" href="#__codelineno-6-27"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-6-28" name="__codelineno-6-28" href="#__codelineno-6-28"></a><span class="w">    </span><span class="p">}</span>
@@ -4968,23 +5040,23 @@
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="p">}</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a>
-<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="cm">/* Fractional knapsack: Greedy algorithm */</span>
+<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="cm">/* Fractional knapsack: greedy algorithm */</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="kd">function</span><span class="w"> </span><span class="nx">fractionalKnapsack</span><span class="p">(</span><span class="nx">wgt</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">cap</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">    </span><span class="c1">// Create item list with two attributes: weight, value</span>
+<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">    </span><span class="c1">// Create a list of items, each with two attributes: weight and value</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">items</span><span class="o">:</span><span class="w"> </span><span class="kt">Item</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">wgt</span><span class="p">.</span><span class="nx">map</span><span class="p">((</span><span class="nx">w</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Item</span><span class="p">(</span><span class="nx">w</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">[</span><span class="nx">i</span><span class="p">]));</span>
-<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="c1">// Sort by unit value item.v / item.w from high to low</span>
+<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="c1">// Sort by value per unit weight, item.v / item.w, in descending order</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">    </span><span class="nx">items</span><span class="p">.</span><span class="nx">sort</span><span class="p">((</span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">b</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">b</span><span class="p">.</span><span class="nx">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nx">b</span><span class="p">.</span><span class="nx">w</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">a</span><span class="p">.</span><span class="nx">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nx">a</span><span class="p">.</span><span class="nx">w</span><span class="p">);</span>
-<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">    </span><span class="c1">// Loop for greedy selection</span>
+<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">item</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">items</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">item</span><span class="p">.</span><span class="nx">w</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">            </span><span class="c1">// If remaining capacity is sufficient, put the entire current item into the knapsack</span>
+<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">            </span><span class="c1">// If enough capacity remains, put the entire current item in the knapsack</span>
 <a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">            </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">item</span><span class="p">.</span><span class="nx">v</span><span class="p">;</span>
 <a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">            </span><span class="nx">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="nx">item</span><span class="p">.</span><span class="nx">w</span><span class="p">;</span>
 <a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="w">            </span><span class="c1">// If remaining capacity is insufficient, put part of the current item into the knapsack</span>
+<a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="w">            </span><span class="c1">// If the remaining capacity is insufficient, put part of the current item in the knapsack</span>
 <a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="w">            </span><span class="nx">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">(</span><span class="nx">item</span><span class="p">.</span><span class="nx">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nx">item</span><span class="p">.</span><span class="nx">w</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">cap</span><span class="p">;</span>
-<a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w">            </span><span class="c1">// No remaining capacity, so break out of the loop</span>
+<a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w">            </span><span class="c1">// No capacity remains, so exit the loop</span>
 <a id="__codelineno-7-29" name="__codelineno-7-29" href="#__codelineno-7-29"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-7-30" name="__codelineno-7-30" href="#__codelineno-7-30"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-7-31" name="__codelineno-7-31" href="#__codelineno-7-31"></a><span class="w">    </span><span class="p">}</span>
@@ -5001,23 +5073,23 @@
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="n">Item</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">v</span><span class="p">);</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="p">}</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="cm">/* Fractional knapsack: Greedy algorithm */</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="cm">/* Fractional knapsack: greedy algorithm */</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="kt">double</span><span class="w"> </span><span class="n">fractionalKnapsack</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="c1">// Create item list with two attributes: weight, value</span>
+<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="c1">// Create a list of items, each with two attributes: weight and value</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Item</span><span class="o">&gt;</span><span class="w"> </span><span class="n">items</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">wgt</span><span class="p">.</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">Item</span><span class="p">(</span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="p">]));</span>
-<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="c1">// Sort by unit value item.v / item.w from high to low</span>
+<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="c1">// Sort by value per unit weight, item.v / item.w, in descending order</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">  </span><span class="n">items</span><span class="p">.</span><span class="n">sort</span><span class="p">((</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">b</span><span class="p">.</span><span class="n">w</span><span class="p">).</span><span class="n">compareTo</span><span class="p">(</span><span class="n">a</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">a</span><span class="p">.</span><span class="n">w</span><span class="p">));</span>
-<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="c1">// Loop for greedy selection</span>
+<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="c1">// Repeatedly make greedy choices</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">  </span><span class="kt">double</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Item</span><span class="w"> </span><span class="n">item</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">items</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">      </span><span class="c1">// If remaining capacity is sufficient, put the entire current item into the knapsack</span>
+<a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">      </span><span class="c1">// If enough capacity remains, put the entire current item in the knapsack</span>
 <a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">      </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="p">;</span>
 <a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="w">      </span><span class="n">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="p">;</span>
 <a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="w">      </span><span class="c1">// If remaining capacity is insufficient, put part of the current item into the knapsack</span>
+<a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="w">      </span><span class="c1">// If the remaining capacity is insufficient, put part of the current item in the knapsack</span>
 <a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="w">      </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">cap</span><span class="p">;</span>
-<a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="w">      </span><span class="c1">// No remaining capacity, so break out of the loop</span>
+<a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="w">      </span><span class="c1">// No capacity remains, so exit the loop</span>
 <a id="__codelineno-8-26" name="__codelineno-8-26" href="#__codelineno-8-26"></a><span class="w">      </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-8-28" name="__codelineno-8-28" href="#__codelineno-8-28"></a><span class="w">  </span><span class="p">}</span>
@@ -5038,31 +5110,31 @@
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="p">}</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a>
-<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="cm">/* Fractional knapsack: Greedy algorithm */</span>
+<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="cm">/* Fractional knapsack: greedy algorithm */</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="k">fn</span><span class="w"> </span><span class="nf">fractional_knapsack</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">cap</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">f64</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">    </span><span class="c1">// Create item list with two attributes: weight, value</span>
+<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">    </span><span class="c1">// Create a list of items, each with two attributes: weight and value</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">items</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">        </span><span class="p">.</span><span class="n">iter</span><span class="p">()</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">        </span><span class="p">.</span><span class="n">zip</span><span class="p">(</span><span class="n">val</span><span class="p">.</span><span class="n">iter</span><span class="p">())</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">        </span><span class="p">.</span><span class="n">map</span><span class="p">(</span><span class="o">|</span><span class="p">(</span><span class="o">&amp;</span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="n">v</span><span class="p">)</span><span class="o">|</span><span class="w"> </span><span class="n">Item</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="n">v</span><span class="p">))</span>
 <a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">        </span><span class="p">.</span><span class="n">collect</span><span class="p">::</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">Item</span><span class="o">&gt;&gt;</span><span class="p">();</span>
-<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">    </span><span class="c1">// Sort by unit value item.v / item.w from high to low</span>
+<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">    </span><span class="c1">// Sort by value per unit weight, item.v / item.w, in descending order</span>
 <a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="w">    </span><span class="n">items</span><span class="p">.</span><span class="n">sort_by</span><span class="p">(</span><span class="o">|</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="o">|</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="w">        </span><span class="p">(</span><span class="n">b</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">b</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="p">)</span>
 <a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a><span class="w">            </span><span class="p">.</span><span class="n">partial_cmp</span><span class="p">(</span><span class="o">&amp;</span><span class="p">(</span><span class="n">a</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">a</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="p">))</span>
 <a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a><span class="w">            </span><span class="p">.</span><span class="n">unwrap</span><span class="p">()</span>
 <a id="__codelineno-9-26" name="__codelineno-9-26" href="#__codelineno-9-26"></a><span class="w">    </span><span class="p">});</span>
-<a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="w">    </span><span class="c1">// Loop for greedy selection</span>
+<a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices</span>
 <a id="__codelineno-9-28" name="__codelineno-9-28" href="#__codelineno-9-28"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0.0</span><span class="p">;</span>
 <a id="__codelineno-9-29" name="__codelineno-9-29" href="#__codelineno-9-29"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">item</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="o">&amp;</span><span class="n">items</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-30" name="__codelineno-9-30" href="#__codelineno-9-30"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-31" name="__codelineno-9-31" href="#__codelineno-9-31"></a><span class="w">            </span><span class="c1">// If remaining capacity is sufficient, put the entire current item into the knapsack</span>
+<a id="__codelineno-9-31" name="__codelineno-9-31" href="#__codelineno-9-31"></a><span class="w">            </span><span class="c1">// If enough capacity remains, put the entire current item in the knapsack</span>
 <a id="__codelineno-9-32" name="__codelineno-9-32" href="#__codelineno-9-32"></a><span class="w">            </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="p">;</span>
 <a id="__codelineno-9-33" name="__codelineno-9-33" href="#__codelineno-9-33"></a><span class="w">            </span><span class="n">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="p">;</span>
 <a id="__codelineno-9-34" name="__codelineno-9-34" href="#__codelineno-9-34"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-35" name="__codelineno-9-35" href="#__codelineno-9-35"></a><span class="w">            </span><span class="c1">// If remaining capacity is insufficient, put part of the current item into the knapsack</span>
+<a id="__codelineno-9-35" name="__codelineno-9-35" href="#__codelineno-9-35"></a><span class="w">            </span><span class="c1">// If the remaining capacity is insufficient, put part of the current item in the knapsack</span>
 <a id="__codelineno-9-36" name="__codelineno-9-36" href="#__codelineno-9-36"></a><span class="w">            </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="n">w</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="p">;</span>
-<a id="__codelineno-9-37" name="__codelineno-9-37" href="#__codelineno-9-37"></a><span class="w">            </span><span class="c1">// No remaining capacity, so break out of the loop</span>
+<a id="__codelineno-9-37" name="__codelineno-9-37" href="#__codelineno-9-37"></a><span class="w">            </span><span class="c1">// No capacity remains, so exit the loop</span>
 <a id="__codelineno-9-38" name="__codelineno-9-38" href="#__codelineno-9-38"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-9-39" name="__codelineno-9-39" href="#__codelineno-9-39"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-40" name="__codelineno-9-40" href="#__codelineno-9-40"></a><span class="w">    </span><span class="p">}</span>
@@ -5077,24 +5149,24 @@
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">v</span><span class="p">;</span><span class="w"> </span><span class="c1">// Item value</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="p">}</span><span class="w"> </span><span class="n">Item</span><span class="p">;</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a>
-<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="cm">/* Fractional knapsack: Greedy algorithm */</span>
+<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="cm">/* Fractional knapsack: greedy algorithm */</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="kt">float</span><span class="w"> </span><span class="nf">fractionalKnapsack</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">wgt</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">itemCount</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="c1">// Create item list with two attributes: weight, value</span>
+<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="c1">// Create a list of items, each with two attributes: weight and value</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="n">Item</span><span class="w"> </span><span class="o">*</span><span class="n">items</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">Item</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">itemCount</span><span class="p">);</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">itemCount</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="n">items</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">Item</span><span class="p">){.</span><span class="n">w</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="p">.</span><span class="n">v</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">[</span><span class="n">i</span><span class="p">]};</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="c1">// Sort by unit value item.v / item.w from high to low</span>
+<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="c1">// Sort by value per unit weight, item.v / item.w, in descending order</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="n">qsort</span><span class="p">(</span><span class="n">items</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="kt">size_t</span><span class="p">)</span><span class="n">itemCount</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="n">Item</span><span class="p">),</span><span class="w"> </span><span class="n">sortByValueDensity</span><span class="p">);</span>
-<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">    </span><span class="c1">// Loop for greedy selection</span>
+<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices</span>
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">    </span><span class="kt">float</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0.0</span><span class="p">;</span>
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">itemCount</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">items</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">w</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">            </span><span class="c1">// If remaining capacity is sufficient, put the entire current item into the knapsack</span>
+<a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">            </span><span class="c1">// If enough capacity remains, put the entire current item in the knapsack</span>
 <a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">            </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">items</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">v</span><span class="p">;</span>
 <a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">            </span><span class="n">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">items</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">w</span><span class="p">;</span>
 <a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">            </span><span class="c1">// If remaining capacity is insufficient, put part of the current item into the knapsack</span>
+<a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">            </span><span class="c1">// If the remaining capacity is insufficient, put part of the current item in the knapsack</span>
 <a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="w">            </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">(</span><span class="kt">float</span><span class="p">)</span><span class="n">cap</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">items</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">w</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">items</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">v</span><span class="p">;</span>
 <a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="w">            </span><span class="n">cap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-10-27" name="__codelineno-10-27" href="#__codelineno-10-27"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
@@ -5108,31 +5180,31 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">fractional_knapsack.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Item */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">Item</span><span class="p">(</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">w</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="c1">// Item</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">w</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="c1">// Item weight</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">v</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w">  </span><span class="c1">// Item value</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="p">)</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a>
-<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="cm">/* Fractional knapsack: Greedy algorithm */</span>
+<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="cm">/* Fractional knapsack: greedy algorithm */</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">fractionalKnapsack</span><span class="p">(</span><span class="n">wgt</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Double</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// Create item list with two attributes: weight, value</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// Create a list of items, each with two attributes: weight and value</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">cap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">c</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">items</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arrayOfNulls</span><span class="o">&lt;</span><span class="n">Item</span><span class="o">&gt;</span><span class="p">(</span><span class="n">wgt</span><span class="p">.</span><span class="na">size</span><span class="p">)</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">wgt</span><span class="p">.</span><span class="na">indices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="n">items</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Item</span><span class="p">(</span><span class="n">wgt</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="c1">// Sort by unit value item.v / item.w from high to low</span>
+<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="c1">// Sort by value per unit weight, item.v / item.w, in descending order</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="n">items</span><span class="p">.</span><span class="na">sortBy</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">item</span><span class="p">:</span><span class="w"> </span><span class="n">Item?</span><span class="w"> </span><span class="o">-&gt;</span><span class="w"> </span><span class="o">-</span><span class="p">(</span><span class="n">item</span><span class="o">!!</span><span class="p">.</span><span class="na">v</span><span class="p">.</span><span class="na">toDouble</span><span class="p">()</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">w</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="c1">// Loop for greedy selection</span>
+<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0.0</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">item</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">items</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">item</span><span class="o">!!</span><span class="p">.</span><span class="na">w</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">            </span><span class="c1">// If remaining capacity is sufficient, put the entire current item into the knapsack</span>
+<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">            </span><span class="c1">// If enough capacity remains, put the entire current item in the knapsack</span>
 <a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">            </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">v</span>
 <a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w">            </span><span class="n">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">w</span>
 <a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w">            </span><span class="c1">// If remaining capacity is insufficient, put part of the current item into the knapsack</span>
+<a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w">            </span><span class="c1">// If the remaining capacity is insufficient, put part of the current item in the knapsack</span>
 <a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w">            </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">v</span><span class="p">.</span><span class="na">toDouble</span><span class="p">()</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">item</span><span class="p">.</span><span class="na">w</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">cap</span>
-<a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w">            </span><span class="c1">// No remaining capacity, so break out of the loop</span>
+<a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w">            </span><span class="c1">// No capacity remains, so exit the loop</span>
 <a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a><span class="w">    </span><span class="p">}</span>
@@ -5152,23 +5224,23 @@
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="k">end</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a>
-<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="c1">### Fractional knapsack: greedy ###</span>
+<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="c1">### Fractional knapsack: greedy algorithm ###</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="k">def</span><span class="w"> </span><span class="nf">fractional_knapsack</span><span class="p">(</span><span class="n">wgt</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span>
-<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">  </span><span class="c1"># Create item list with two attributes: weight, value</span>
+<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">  </span><span class="c1"># Create a list of items, each with two attributes: weight and value</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">  </span><span class="n">items</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">wgt</span><span class="o">.</span><span class="n">each_with_index</span><span class="o">.</span><span class="n">map</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="no">Item</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">  </span><span class="c1"># Sort by unit value item.v / item.w from high to low</span>
+<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">  </span><span class="c1"># Sort by value per unit weight, item.v / item.w, in descending order</span>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">  </span><span class="n">items</span><span class="o">.</span><span class="n">sort!</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="o">|</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="o">.</span><span class="n">v</span><span class="o">.</span><span class="n">to_f</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">b</span><span class="o">.</span><span class="n">w</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;=&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">v</span><span class="o">.</span><span class="n">to_f</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">a</span><span class="o">.</span><span class="n">w</span><span class="p">)</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">  </span><span class="c1"># Loop for greedy selection</span>
+<a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">  </span><span class="c1"># Repeatedly make greedy choices</span>
 <a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">item</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">items</span>
 <a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">item</span><span class="o">.</span><span class="n">w</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">cap</span>
-<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w">      </span><span class="c1"># If remaining capacity is sufficient, put the entire current item into the knapsack</span>
+<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w">      </span><span class="c1"># If enough capacity remains, put the entire current item in the knapsack</span>
 <a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="w">      </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">item</span><span class="o">.</span><span class="n">v</span>
 <a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a><span class="w">      </span><span class="n">cap</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">item</span><span class="o">.</span><span class="n">w</span>
 <a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="w">    </span><span class="k">else</span>
-<a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></a><span class="w">      </span><span class="c1"># If remaining capacity is insufficient, put part of the current item into the knapsack</span>
+<a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></a><span class="w">      </span><span class="c1"># If the remaining capacity is insufficient, put part of the current item in the knapsack</span>
 <a id="__codelineno-12-27" name="__codelineno-12-27" href="#__codelineno-12-27"></a><span class="w">      </span><span class="n">res</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">(</span><span class="n">item</span><span class="o">.</span><span class="n">v</span><span class="o">.</span><span class="n">to_f</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">item</span><span class="o">.</span><span class="n">w</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">cap</span>
-<a id="__codelineno-12-28" name="__codelineno-12-28" href="#__codelineno-12-28"></a><span class="w">      </span><span class="c1"># No remaining capacity, so break out of the loop</span>
+<a id="__codelineno-12-28" name="__codelineno-12-28" href="#__codelineno-12-28"></a><span class="w">      </span><span class="c1"># No capacity remains, so exit the loop</span>
 <a id="__codelineno-12-29" name="__codelineno-12-29" href="#__codelineno-12-29"></a><span class="w">      </span><span class="k">break</span>
 <a id="__codelineno-12-30" name="__codelineno-12-30" href="#__codelineno-12-30"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-12-31" name="__codelineno-12-31" href="#__codelineno-12-31"></a><span class="w">  </span><span class="k">end</span>
@@ -5178,14 +5250,19 @@
 </div>
 </div>
 </div>
-<p>Built-in sorting algorithms usually take <span class="arithmatex">\(O(n \log n)\)</span> time, and their space complexity is usually <span class="arithmatex">\(O(\log n)\)</span> or <span class="arithmatex">\(O(n)\)</span>, depending on the specific implementation of the programming language.</p>
-<p>Apart from sorting, in the worst case the entire item list needs to be traversed, <strong>therefore the time complexity is <span class="arithmatex">\(O(n)\)</span></strong>, where <span class="arithmatex">\(n\)</span> is the number of items.</p>
-<p>Since an <code>Item</code> object list is initialized, <strong>the space complexity is <span class="arithmatex">\(O(n)\)</span></strong>.</p>
-<h3 id="3-correctness-proof">3. &nbsp; Correctness Proof<a class="headerlink" href="#3-correctness-proof" title="Permanent link">&para;</a></h3>
-<p>We use proof by contradiction. Suppose item <span class="arithmatex">\(x\)</span> has the highest unit value, and some algorithm produces an optimal value <code>res</code>, but the resulting solution does not include item <span class="arithmatex">\(x\)</span>.</p>
-<p>Now remove one unit of weight from any item in the knapsack and replace it with one unit of weight from item <span class="arithmatex">\(x\)</span>. Since item <span class="arithmatex">\(x\)</span> has the highest unit value, the total value after the replacement must be greater than <code>res</code>. <strong>This contradicts the assumption that <code>res</code> is optimal, proving that any optimal solution must include item <span class="arithmatex">\(x\)</span></strong>.</p>
-<p>We can construct the same contradiction for the other items in the solution as well. In summary, <strong>items with higher unit value are always the better choice</strong>, which proves that the greedy strategy is effective.</p>
-<p>As shown in Figure 15-6, if we treat item weight and unit value as the horizontal and vertical axes of a two-dimensional chart, then the fractional knapsack problem can be viewed as "finding the maximum area enclosed within a bounded interval on the horizontal axis." This analogy helps explain the effectiveness of the greedy strategy from a geometric perspective.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20Item%3A%0A%20%20%20%20%22%22%22Item%22%22%22%0A%20%20%20%20def%20__init__%28self,%20w%3A%20int,%20v%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.w%20%3D%20w%20%20%23%20Item%20weight%0A%20%20%20%20%20%20%20%20self.v%20%3D%20v%20%20%23%20Item%20value%0A%0Adef%20fractional_knapsack%28wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20cap%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Fractional%20knapsack%3A%20greedy%20algorithm%22%22%22%0A%20%20%20%20%23%20Create%20a%20list%20of%20items,%20each%20with%20two%20attributes%3A%20weight%20and%20value%0A%20%20%20%20items%20%3D%20%5BItem%28w,%20v%29%20for%20w,%20v%20in%20zip%28wgt,%20val%29%5D%0A%20%20%20%20%23%20Sort%20by%20value%20per%20unit%20weight,%20item.v%20/%20item.w,%20in%20descending%20order%0A%20%20%20%20items.sort%28key%3Dlambda%20item%3A%20item.v%20/%20item.w,%20reverse%3DTrue%29%0A%20%20%20%20%23%20Repeatedly%20make%20greedy%20choices%0A%20%20%20%20res%20%3D%200%0A%20%20%20%20for%20item%20in%20items%3A%0A%20%20%20%20%20%20%20%20if%20item.w%20%3C%3D%20cap%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20enough%20capacity%20remains,%20put%20the%20entire%20current%20item%20in%20the%20knapsack%0A%20%20%20%20%20%20%20%20%20%20%20%20res%20%2B%3D%20item.v%0A%20%20%20%20%20%20%20%20%20%20%20%20cap%20-%3D%20item.w%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20the%20remaining%20capacity%20is%20insufficient,%20put%20part%20of%20the%20current%20item%20in%20the%20knapsack%0A%20%20%20%20%20%20%20%20%20%20%20%20res%20%2B%3D%20%28item.v%20/%20item.w%29%20%2A%20cap%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20No%20capacity%20remains,%20so%20exit%20the%20loop%0A%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10,%2020,%2030,%2040,%2050%5D%0A%20%20%20%20val%20%3D%20%5B50,%20120,%20150,%20210,%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20Greedy%20algorithm%0A%20%20%20%20res%20%3D%20fractional_knapsack%28wgt,%20val,%20cap%29%0A%20%20%20%20print%28f%22Maximum%20total%20value%20within%20the%20knapsack%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=8&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20Item%3A%0A%20%20%20%20%22%22%22Item%22%22%22%0A%20%20%20%20def%20__init__%28self,%20w%3A%20int,%20v%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.w%20%3D%20w%20%20%23%20Item%20weight%0A%20%20%20%20%20%20%20%20self.v%20%3D%20v%20%20%23%20Item%20value%0A%0Adef%20fractional_knapsack%28wgt%3A%20list%5Bint%5D,%20val%3A%20list%5Bint%5D,%20cap%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Fractional%20knapsack%3A%20greedy%20algorithm%22%22%22%0A%20%20%20%20%23%20Create%20a%20list%20of%20items,%20each%20with%20two%20attributes%3A%20weight%20and%20value%0A%20%20%20%20items%20%3D%20%5BItem%28w,%20v%29%20for%20w,%20v%20in%20zip%28wgt,%20val%29%5D%0A%20%20%20%20%23%20Sort%20by%20value%20per%20unit%20weight,%20item.v%20/%20item.w,%20in%20descending%20order%0A%20%20%20%20items.sort%28key%3Dlambda%20item%3A%20item.v%20/%20item.w,%20reverse%3DTrue%29%0A%20%20%20%20%23%20Repeatedly%20make%20greedy%20choices%0A%20%20%20%20res%20%3D%200%0A%20%20%20%20for%20item%20in%20items%3A%0A%20%20%20%20%20%20%20%20if%20item.w%20%3C%3D%20cap%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20enough%20capacity%20remains,%20put%20the%20entire%20current%20item%20in%20the%20knapsack%0A%20%20%20%20%20%20%20%20%20%20%20%20res%20%2B%3D%20item.v%0A%20%20%20%20%20%20%20%20%20%20%20%20cap%20-%3D%20item.w%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20the%20remaining%20capacity%20is%20insufficient,%20put%20part%20of%20the%20current%20item%20in%20the%20knapsack%0A%20%20%20%20%20%20%20%20%20%20%20%20res%20%2B%3D%20%28item.v%20/%20item.w%29%20%2A%20cap%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20No%20capacity%20remains,%20so%20exit%20the%20loop%0A%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20wgt%20%3D%20%5B10,%2020,%2030,%2040,%2050%5D%0A%20%20%20%20val%20%3D%20%5B50,%20120,%20150,%20210,%20240%5D%0A%20%20%20%20cap%20%3D%2050%0A%20%20%20%20n%20%3D%20len%28wgt%29%0A%0A%20%20%20%20%23%20Greedy%20algorithm%0A%20%20%20%20res%20%3D%20fractional_knapsack%28wgt,%20val,%20cap%29%0A%20%20%20%20print%28f%22Maximum%20total%20value%20within%20the%20knapsack%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=8&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Built-in sorting algorithms typically take <span class="arithmatex">\(O(n \log n)\)</span> time and <span class="arithmatex">\(O(\log n)\)</span> or <span class="arithmatex">\(O(n)\)</span> space, depending on the programming language's implementation.</p>
+<p>Apart from sorting, the worst case requires traversing the entire item list, <strong>so the time complexity is <span class="arithmatex">\(O(n)\)</span></strong>, where <span class="arithmatex">\(n\)</span> is the number of items.</p>
+<p>Initializing a list of <code>Item</code> objects requires <strong><span class="arithmatex">\(O(n)\)</span> space</strong>.</p>
+<h3 id="3-proof-of-correctness">3. &nbsp; Proof of correctness<a class="headerlink" href="#3-proof-of-correctness" title="Permanent link">&para;</a></h3>
+<p>We use proof by contradiction. Suppose item <span class="arithmatex">\(x\)</span> has the highest unit value, and an algorithm produces a maximum value of <code>res</code>, but its solution does not include item <span class="arithmatex">\(x\)</span>.</p>
+<p>Now remove one unit of weight of any item from the knapsack and replace it with one unit of weight of item <span class="arithmatex">\(x\)</span>. Since <span class="arithmatex">\(x\)</span> has the highest unit value, the total value after replacement must exceed <code>res</code>. <strong>This contradicts the assumption that <code>res</code> is optimal, showing that an optimal solution must include item <span class="arithmatex">\(x\)</span></strong>.</p>
+<p>We can construct the same contradiction for other items in the solution. Thus, <strong>an item with a higher unit value is always a better choice</strong>, which establishes the greedy strategy.</p>
+<p>As shown in Figure 15-6, if we plot item weight on the horizontal axis and unit value on the vertical axis, the fractional knapsack problem becomes a problem of maximizing the area within a limited horizontal interval. This analogy provides a geometric view of why the greedy strategy works.</p>
 <p><img alt="Geometric representation of the fractional knapsack problem" class="animation-figure" src="../fractional_knapsack_problem.assets/fractional_knapsack_area_chart.png" /></p>
 <p align="center"> Figure 15-6 &nbsp; Geometric representation of the fractional knapsack problem </p>
 
@@ -5210,7 +5287,7 @@ aria-label="Footer"
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-    aria-label="Previous: 15.1 Greedy Algorithm"
+    aria-label="Previous: 15.1   Greedy algorithms"
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@@ -5222,7 +5299,7 @@ aria-label="Footer"
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+        15.1 &nbsp; Greedy algorithms
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@@ -5234,7 +5311,7 @@ aria-label="Footer"
   <a
     href="../max_capacity_problem/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 15.3 Maximum Capacity Problem"
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@@ -5242,7 +5319,7 @@ aria-label="Footer"
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-        15.3 Maximum Capacity Problem
+        15.3 &nbsp; Maximum capacity problem
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@@ -5259,7 +5336,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
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diff --git a/en/chapter_greedy/greedy_algorithm/index.html b/en/chapter_greedy/greedy_algorithm/index.html
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@@ -6,7 +6,7 @@
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+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
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@@ -41,7 +41,7 @@
 
 
 
-        <title>15.1 Greedy Algorithm - Hello Algo</title>
+        <title>15.1 Greedy algorithms - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
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-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
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@@ -101,7 +101,7 @@
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@@ -119,7 +119,7 @@
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-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              15.1 &nbsp; Greedy Algorithm
+              15.1 &nbsp; Greedy algorithms
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Dark mode"  type="radio" name="__palette" id="__palette_0">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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       </label>
 
@@ -177,9 +177,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Light mode"  type="radio" name="__palette" id="__palette_1">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3283,7 +3283,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3304,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3352,7 +3352,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3379,7 +3379,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3406,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3433,7 +3433,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3460,7 +3460,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3526,7 +3526,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3547,7 +3547,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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+  15.1 &nbsp; Greedy algorithms
 
 
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-  <a href="#1511-advantages-and-limitations-of-greedy-algorithms" class="md-nav__link">
+  <a href="#1511-strengths-and-limitations-of-greedy-algorithms" class="md-nav__link">
     <span class="md-ellipsis">
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-        15.1.1 &nbsp; Advantages and Limitations of Greedy Algorithms
+        15.1.1 &nbsp; Strengths and limitations of greedy algorithms
       </span>
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 </li>
 
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+  <a href="#1512-properties-required-by-greedy-algorithms" class="md-nav__link">
     <span class="md-ellipsis">
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-        15.1.2 &nbsp; Characteristics of Greedy Algorithms
+        15.1.2 &nbsp; Properties required by greedy algorithms
       </span>
     </span>
   </a>
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 </li>
 
         <li class="md-nav__item">
-  <a href="#1513-steps-for-solving-problems-with-greedy-algorithms" class="md-nav__link">
+  <a href="#1513-steps-for-solving-a-problem-greedily" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        15.1.3 &nbsp; Steps for Solving Problems with Greedy Algorithms
+        15.1.3 &nbsp; Steps for solving a problem greedily
       </span>
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   </a>
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 </li>
 
         <li class="md-nav__item">
-  <a href="#1514-typical-problems-solved-by-greedy-algorithms" class="md-nav__link">
+  <a href="#1514-typical-greedy-problems" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        15.1.4 &nbsp; Typical Problems Solved by Greedy Algorithms
+        15.1.4 &nbsp; Typical greedy problems
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-  Chapter 16. Appendix
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+  16.2 &nbsp; Contribute to the book
 
 
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+  16.3 &nbsp; Glossary
 
 
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+  <a href="#1511-strengths-and-limitations-of-greedy-algorithms" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        15.1.1 &nbsp; Advantages and Limitations of Greedy Algorithms
+        15.1.1 &nbsp; Strengths and limitations of greedy algorithms
       </span>
     </span>
   </a>
@@ -4632,10 +4704,10 @@
 </li>
 
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-  <a href="#1512-characteristics-of-greedy-algorithms" class="md-nav__link">
+  <a href="#1512-properties-required-by-greedy-algorithms" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        15.1.2 &nbsp; Characteristics of Greedy Algorithms
+        15.1.2 &nbsp; Properties required by greedy algorithms
       </span>
     </span>
   </a>
@@ -4643,10 +4715,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1513-steps-for-solving-problems-with-greedy-algorithms" class="md-nav__link">
+  <a href="#1513-steps-for-solving-a-problem-greedily" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        15.1.3 &nbsp; Steps for Solving Problems with Greedy Algorithms
+        15.1.3 &nbsp; Steps for solving a problem greedily
       </span>
     </span>
   </a>
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 </li>
 
         <li class="md-nav__item">
-  <a href="#1514-typical-problems-solved-by-greedy-algorithms" class="md-nav__link">
+  <a href="#1514-typical-greedy-problems" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        15.1.4 &nbsp; Typical Problems Solved by Greedy Algorithms
+        15.1.4 &nbsp; Typical greedy problems
       </span>
     </span>
   </a>
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 <!-- Page content -->
-<h1 id="151-greedy-algorithm">15.1 &nbsp; Greedy Algorithm<a class="headerlink" href="#151-greedy-algorithm" title="Permanent link">&para;</a></h1>
-<p><u>Greedy algorithm</u> is a common approach to solving optimization problems. Its basic idea is to choose the option that appears best at each decision stage, that is, to greedily make locally optimal decisions in the hope of obtaining a globally optimal solution. Greedy algorithms are simple and efficient, and are widely used in many practical problems.</p>
-<p>Greedy algorithms and dynamic programming are both commonly used to solve optimization problems. They share some similarities, such as both relying on the optimal substructure property, but they work differently.</p>
+<h1 id="151-greedy-algorithms">15.1 &nbsp; Greedy algorithms<a class="headerlink" href="#151-greedy-algorithms" title="Permanent link">&para;</a></h1>
+<p>A <u>greedy algorithm</u> is a common approach to optimization problems. At each decision point, it chooses the option that looks best at that moment: it makes a locally optimal choice in the hope of reaching a globally optimal solution. Greedy algorithms are simple and efficient, with many practical applications.</p>
+<p>Both greedy algorithms and dynamic programming are commonly used for optimization. They share some features, such as relying on optimal substructure, but work differently.</p>
 <ul>
-<li>Dynamic programming considers all previous decisions when making the current decision, and uses solutions to past subproblems to construct the solution to the current subproblem.</li>
-<li>Greedy algorithms do not consider past decisions, but instead make greedy choices moving forward, continually reducing the problem size until the problem is solved.</li>
+<li>Dynamic programming considers the current decision in light of all decisions from earlier stages, using solutions to earlier subproblems to build the solution to the current subproblem.</li>
+<li>A greedy algorithm does not reconsider past decisions. It keeps moving forward, making greedy choices and shrinking the problem until it is solved.</li>
 </ul>
-<p>We will first understand how greedy algorithms work through the example problem "coin change." This problem was already introduced in the "Complete Knapsack Problem" chapter, so it should already be familiar to you.</p>
+<p>Let's explore how greedy algorithms work through the coin change problem, which you have already encountered in the section on the unbounded knapsack problem.</p>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given <span class="arithmatex">\(n\)</span> types of coins, where the denomination of the <span class="arithmatex">\(i\)</span>-th type is <span class="arithmatex">\(coins[i - 1]\)</span>, a target amount <span class="arithmatex">\(amt\)</span>, and an unlimited number of coins of each type, what is the minimum number of coins needed to make up the target amount? If the target amount cannot be made up, return <span class="arithmatex">\(-1\)</span>.</p>
+<p>Given <span class="arithmatex">\(n\)</span> coin denominations, where denomination <span class="arithmatex">\(i\)</span> has value <span class="arithmatex">\(coins[i - 1]\)</span>, and a target amount <span class="arithmatex">\(amt\)</span>, you may use each denomination any number of times. What is the minimum number of coins needed to make the target amount? Return <span class="arithmatex">\(-1\)</span> if it cannot be made.</p>
 </div>
-<p>The greedy strategy for this problem is shown in Figure 15-1. Given a target amount, <strong>we greedily choose the coin that does not exceed it and is closest to it</strong>, repeating this step until the target amount is made up.</p>
+<p>The greedy strategy is shown in Figure 15-1. Given the remaining target amount, <strong>we greedily choose the largest coin that does not exceed it</strong>, repeating this step until we have made the target amount.</p>
 <p><img alt="Greedy strategy for coin change" class="animation-figure" src="../greedy_algorithm.assets/coin_change_greedy_strategy.png" /></p>
 <p align="center"> Figure 15-1 &nbsp; Greedy strategy for coin change </p>
 
-<p>The implementation code is as follows:</p>
+<p>The implementation is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">coin_change_greedy.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">coin_change_greedy</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">amt</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Coin change: Greedy algorithm"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Assume coins list is sorted</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Coin change: greedy algorithm"""</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Assume the coins list is sorted</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="n">i</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">coins</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Loop to make greedy choices until no remaining amount</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Repeatedly make greedy choices until no amount remains</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="k">while</span> <span class="n">amt</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="c1"># Find the coin that is less than and closest to the remaining amount</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="c1"># Find the largest coin no greater than the remaining amount</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="k">while</span> <span class="n">i</span> <span class="o">&gt;</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">amt</span><span class="p">:</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>            <span class="n">i</span> <span class="o">-=</span> <span class="mi">1</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="c1"># Choose coins[i]</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="n">amt</span> <span class="o">-=</span> <span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="n">count</span> <span class="o">+=</span> <span class="mi">1</span>
-<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>    <span class="c1"># If no feasible solution is found, return -1</span>
+<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>    <span class="c1"># Return -1 if no feasible solution was found</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="k">return</span> <span class="n">count</span> <span class="k">if</span> <span class="n">amt</span> <span class="o">==</span> <span class="mi">0</span> <span class="k">else</span> <span class="o">-</span><span class="mi">1</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_greedy.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Coin change: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">coin_change_greedy.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Coin change: greedy algorithm */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeGreedy</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Assume coins list is sorted</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Assume the coins list is sorted</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Loop to make greedy choices until no remaining amount</span>
+<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until no amount remains</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="c1">// Find the coin that is less than and closest to the remaining amount</span>
+<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="c1">// Find the largest coin that does not exceed the remaining amount</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">            </span><span class="n">i</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="p">}</span>
@@ -4754,20 +4826,20 @@
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="n">amt</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">        </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">    </span><span class="c1">// If no feasible solution is found, return -1</span>
+<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">    </span><span class="c1">// Return -1 if no feasible solution was found</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
 <a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_greedy.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Coin change: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">coin_change_greedy.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Coin change: greedy algorithm */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeGreedy</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Assume coins list is sorted</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Assume the coins list is sorted</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// Loop to make greedy choices until no remaining amount</span>
+<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until no amount remains</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">        </span><span class="c1">// Find the coin that is less than and closest to the remaining amount</span>
+<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">        </span><span class="c1">// Find the largest coin no greater than the remaining amount</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">            </span><span class="n">i</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="p">}</span>
@@ -4775,20 +4847,20 @@
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="n">amt</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">        </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">    </span><span class="c1">// If no feasible solution is found, return -1</span>
+<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">    </span><span class="c1">// Return -1 if no feasible solution was found</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_greedy.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Coin change: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">coin_change_greedy.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Coin change: greedy algorithm */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">CoinChangeGreedy</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Assume coins list is sorted</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Assume the coins list is sorted</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Loop to make greedy choices until no remaining amount</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until no amount remains</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">        </span><span class="c1">// Find the coin that is less than and closest to the remaining amount</span>
+<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">        </span><span class="c1">// Find the largest coin no greater than the remaining amount</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">            </span><span class="n">i</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="p">}</span>
@@ -4796,20 +4868,20 @@
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="n">amt</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">        </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">    </span><span class="c1">// If no feasible solution is found, return -1</span>
+<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">    </span><span class="c1">// Return -1 if no feasible solution was found</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_greedy.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Coin change: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">coin_change_greedy.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Coin change: greedy algorithm */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">coinChangeGreedy</span><span class="p">(</span><span class="nx">coins</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Assume coins list is sorted</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Assume the coins list is sorted in ascending order</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">coins</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="nx">count</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="c1">// Loop to make greedy choices until no remaining amount</span>
+<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until no amount remains</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">        </span><span class="c1">// Find the coin that is less than and closest to the remaining amount</span>
+<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">        </span><span class="c1">// Find the largest coin no greater than the remaining amount</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">            </span><span class="nx">i</span><span class="o">--</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="p">}</span>
@@ -4817,7 +4889,7 @@
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">        </span><span class="nx">amt</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="nx">count</span><span class="o">++</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="c1">// If no feasible solution is found, return -1</span>
+<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="c1">// Return -1 if no feasible solution was found</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">    </span><span class="p">}</span>
@@ -4826,15 +4898,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_greedy.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Coin change: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">coin_change_greedy.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Coin change: greedy algorithm */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">coinChangeGreedy</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">amt</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Assume coins list is sorted</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Assume the coins list is sorted in ascending order</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="bp">count</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">amt</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">amt</span>
-<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// Loop to make greedy choices until no remaining amount</span>
+<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until no amount remains</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="c1">// Find the coin that is less than and closest to the remaining amount</span>
+<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="c1">// Find the largest coin no greater than the remaining amount</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">        </span><span class="p">}</span>
@@ -4842,20 +4914,20 @@
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="n">amt</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="bp">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">    </span><span class="c1">// If no feasible solution is found, return -1</span>
+<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">    </span><span class="c1">// Return -1 if no feasible solution was found</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">?</span><span class="w"> </span><span class="bp">count</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_greedy.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Coin change: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">coin_change_greedy.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Coin change: greedy algorithm */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">coinChangeGreedy</span><span class="p">(</span><span class="nx">coins</span><span class="p">,</span><span class="w"> </span><span class="nx">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Assume coins array is sorted</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Assume coins is sorted in ascending order</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">coins</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// Loop to make greedy choices until no remaining amount</span>
+<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until no amount remains</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">amt</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="c1">// Find the coin that is less than and closest to the remaining amount</span>
+<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="c1">// Find the largest coin no greater than the remaining amount</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">            </span><span class="nx">i</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="p">}</span>
@@ -4863,20 +4935,20 @@
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="nx">amt</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="nx">count</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">    </span><span class="c1">// If no feasible solution is found, return -1</span>
+<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">    </span><span class="c1">// Return -1 if no feasible solution was found</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_greedy.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Coin change: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">coin_change_greedy.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Coin change: greedy algorithm */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">coinChangeGreedy</span><span class="p">(</span><span class="nx">coins</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">amt</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Assume coins array is sorted</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Assume coins is sorted in ascending order</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">coins</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="c1">// Loop to make greedy choices until no remaining amount</span>
+<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until no amount remains</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">amt</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">        </span><span class="c1">// Find the coin that is less than and closest to the remaining amount</span>
+<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">        </span><span class="c1">// Find the largest coin no greater than the remaining amount</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">            </span><span class="nx">i</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="p">}</span>
@@ -4884,20 +4956,20 @@
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="nx">amt</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="nx">coins</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="nx">count</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="c1">// If no feasible solution is found, return -1</span>
+<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="c1">// Return -1 if no feasible solution was found</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">amt</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">count</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_greedy.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Coin change: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">coin_change_greedy.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Coin change: greedy algorithm */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">coinChangeGreedy</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Assume coins list is sorted</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Assume the coins list is sorted in ascending order</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
-<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="c1">// Loop to make greedy choices until no remaining amount</span>
+<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="c1">// Repeatedly make greedy choices until no amount remains</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">    </span><span class="c1">// Find the coin that is less than and closest to the remaining amount</span>
+<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">    </span><span class="c1">// Find the largest coin no greater than the remaining amount</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">      </span><span class="n">i</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="p">}</span>
@@ -4905,20 +4977,20 @@
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">    </span><span class="n">amt</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">  </span><span class="c1">// If no feasible solution is found, return -1</span>
+<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">  </span><span class="c1">// Return -1 if no feasible solution was found</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_greedy.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Coin change: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">coin_change_greedy.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Coin change: greedy algorithm */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">coin_change_greedy</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">amt</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Assume coins list is sorted</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Assume the coins list is sorted</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// Loop to make greedy choices until no remaining amount</span>
+<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until no amount remains</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">        </span><span class="c1">// Find the coin that is less than and closest to the remaining amount</span>
+<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">        </span><span class="c1">// Find the largest coin that does not exceed the remaining amount</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="p">}</span>
@@ -4926,7 +4998,7 @@
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">        </span><span class="n">amt</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">    </span><span class="c1">// If no feasible solution is found, return -1</span>
+<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">    </span><span class="c1">// Return -1 if no feasible solution was found</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">        </span><span class="n">count</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -4936,14 +5008,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_greedy.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Coin change: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">coin_change_greedy.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Coin change: greedy algorithm */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">coinChangeGreedy</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Assume coins list is sorted</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Assume the coins list is sorted</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="c1">// Loop to make greedy choices until no remaining amount</span>
+<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until no amount remains</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">amt</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">        </span><span class="c1">// Find the coin that is less than and closest to the remaining amount</span>
+<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">        </span><span class="c1">// Find the largest coin no greater than the remaining amount</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">            </span><span class="n">i</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">        </span><span class="p">}</span>
@@ -4951,21 +5023,21 @@
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="n">amt</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">coins</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">        </span><span class="n">count</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">    </span><span class="c1">// If no feasible solution is found, return -1</span>
+<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">    </span><span class="c1">// Return -1 if no feasible solution was found</span>
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_greedy.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Coin change: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">coin_change_greedy.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Coin change: greedy algorithm */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">coinChangeGreedy</span><span class="p">(</span><span class="n">coins</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">amt</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Assume coins list is sorted</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Assume coins is sorted in ascending order</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">am</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">amt</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="c1">// Loop to make greedy choices until no remaining amount</span>
+<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until no amount remains</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">am</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">        </span><span class="c1">// Find the coin that is less than and closest to the remaining amount</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">        </span><span class="c1">// Find the largest coin no greater than the remaining amount</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">am</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">            </span><span class="n">i</span><span class="o">--</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="p">}</span>
@@ -4973,20 +5045,20 @@
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">        </span><span class="n">am</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">        </span><span class="n">count</span><span class="o">++</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="c1">// If no feasible solution is found, return -1</span>
+<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="c1">// Return -1 if no feasible solution was found</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">am</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="o">-</span><span class="m">1</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">coin_change_greedy.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Coin change: greedy ###</span>
+<div class="highlight"><span class="filename">coin_change_greedy.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Coin change: greedy algorithm ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">coin_change_greedy</span><span class="p">(</span><span class="n">coins</span><span class="p">,</span><span class="w"> </span><span class="n">amt</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Assume coins list is sorted</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Assume the coins list is sorted in ascending order</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coins</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="n">count</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="c1"># Loop to make greedy choices until no remaining amount</span>
+<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="c1"># Repeatedly make greedy choices until no amount remains</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="n">amt</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="c1"># Find the coin that is less than and closest to the remaining amount</span>
+<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="c1"># Find the largest coin no greater than the remaining amount</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">amt</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="k">end</span>
@@ -4994,69 +5066,74 @@
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="n">amt</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">coins</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="n">count</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">  </span><span class="k">end</span>
-<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">  </span><span class="c1"># Return -1 if no solution found</span>
+<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">  </span><span class="c1"># Return -1 if no feasible solution was found</span>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">  </span><span class="n">amt</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<p>You may find yourself exclaiming, "So clean!" The greedy algorithm solves the coin change problem in only about ten lines of code.</p>
-<h2 id="1511-advantages-and-limitations-of-greedy-algorithms">15.1.1 &nbsp; Advantages and Limitations of Greedy Algorithms<a class="headerlink" href="#1511-advantages-and-limitations-of-greedy-algorithms" title="Permanent link">&para;</a></h2>
-<p><strong>Greedy algorithms are not only straightforward to apply and easy to implement, but are also usually very efficient</strong>. In the code above, if the smallest coin denomination is <span class="arithmatex">\(\min(coins)\)</span>, the greedy selection loop runs at most <span class="arithmatex">\(amt / \min(coins)\)</span> times, giving a time complexity of <span class="arithmatex">\(O(amt / \min(coins))\)</span>. This is an order of magnitude lower than the time complexity of the dynamic programming solution, <span class="arithmatex">\(O(n \times amt)\)</span>.</p>
-<p>However, <strong>for some coin denomination sets, greedy algorithms cannot find the optimal solution</strong>. Figure 15-2 shows two examples.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20coin_change_greedy%28coins%3A%20list%5Bint%5D,%20amt%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Coin%20change%3A%20greedy%20algorithm%22%22%22%0A%20%20%20%20%23%20Assume%20the%20coins%20list%20is%20sorted%0A%20%20%20%20i%20%3D%20len%28coins%29%20-%201%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20Repeatedly%20make%20greedy%20choices%20until%20no%20amount%20remains%0A%20%20%20%20while%20amt%20%3E%200%3A%0A%20%20%20%20%20%20%20%20%23%20Find%20the%20largest%20coin%20no%20greater%20than%20the%20remaining%20amount%0A%20%20%20%20%20%20%20%20while%20i%20%3E%200%20and%20coins%5Bi%5D%20%3E%20amt%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20-%3D%201%0A%20%20%20%20%20%20%20%20%23%20Choose%20coins%5Bi%5D%0A%20%20%20%20%20%20%20%20amt%20-%3D%20coins%5Bi%5D%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20%23%20Return%20-1%20if%20no%20feasible%20solution%20was%20found%0A%20%20%20%20return%20count%20if%20amt%20%3D%3D%200%20else%20-1%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Greedy%20algorithm%3A%20guaranteed%20to%20find%20a%20globally%20optimal%20solution%0A%20%20%20%20coins%20%3D%20%5B1,%205,%2010,%2020,%2050,%20100%5D%0A%20%20%20%20amt%20%3D%20186%0A%20%20%20%20res%20%3D%20coin_change_greedy%28coins,%20amt%29%0A%20%20%20%20print%28f%22%5Cncoins%20%3D%20%7Bcoins%7D,%20amt%20%3D%20%7Bamt%7D%22%29%0A%20%20%20%20print%28f%22Number%20of%20coins%20chosen%20by%20the%20greedy%20algorithm%20for%20amount%20%7Bamt%7D%3A%20%7Bres%7D%22%29%0A%0A%20%20%20%20%23%20Greedy%20algorithm%3A%20not%20guaranteed%20to%20find%20a%20globally%20optimal%20solution%0A%20%20%20%20coins%20%3D%20%5B1,%2020,%2050%5D%0A%20%20%20%20amt%20%3D%2060%0A%20%20%20%20res%20%3D%20coin_change_greedy%28coins,%20amt%29%0A%20%20%20%20print%28f%22%5Cncoins%20%3D%20%7Bcoins%7D,%20amt%20%3D%20%7Bamt%7D%22%29%0A%20%20%20%20print%28f%22Number%20of%20coins%20chosen%20by%20the%20greedy%20algorithm%20for%20amount%20%7Bamt%7D%3A%20%7Bres%7D%22%29%0A%20%20%20%20print%28f%22The%20actual%20minimum%20is%203%20coins%3A%2020%20%2B%2020%20%2B%2020%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20coin_change_greedy%28coins%3A%20list%5Bint%5D,%20amt%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Coin%20change%3A%20greedy%20algorithm%22%22%22%0A%20%20%20%20%23%20Assume%20the%20coins%20list%20is%20sorted%0A%20%20%20%20i%20%3D%20len%28coins%29%20-%201%0A%20%20%20%20count%20%3D%200%0A%20%20%20%20%23%20Repeatedly%20make%20greedy%20choices%20until%20no%20amount%20remains%0A%20%20%20%20while%20amt%20%3E%200%3A%0A%20%20%20%20%20%20%20%20%23%20Find%20the%20largest%20coin%20no%20greater%20than%20the%20remaining%20amount%0A%20%20%20%20%20%20%20%20while%20i%20%3E%200%20and%20coins%5Bi%5D%20%3E%20amt%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20-%3D%201%0A%20%20%20%20%20%20%20%20%23%20Choose%20coins%5Bi%5D%0A%20%20%20%20%20%20%20%20amt%20-%3D%20coins%5Bi%5D%0A%20%20%20%20%20%20%20%20count%20%2B%3D%201%0A%20%20%20%20%23%20Return%20-1%20if%20no%20feasible%20solution%20was%20found%0A%20%20%20%20return%20count%20if%20amt%20%3D%3D%200%20else%20-1%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Greedy%20algorithm%3A%20guaranteed%20to%20find%20a%20globally%20optimal%20solution%0A%20%20%20%20coins%20%3D%20%5B1,%205,%2010,%2020,%2050,%20100%5D%0A%20%20%20%20amt%20%3D%20186%0A%20%20%20%20res%20%3D%20coin_change_greedy%28coins,%20amt%29%0A%20%20%20%20print%28f%22%5Cncoins%20%3D%20%7Bcoins%7D,%20amt%20%3D%20%7Bamt%7D%22%29%0A%20%20%20%20print%28f%22Number%20of%20coins%20chosen%20by%20the%20greedy%20algorithm%20for%20amount%20%7Bamt%7D%3A%20%7Bres%7D%22%29%0A%0A%20%20%20%20%23%20Greedy%20algorithm%3A%20not%20guaranteed%20to%20find%20a%20globally%20optimal%20solution%0A%20%20%20%20coins%20%3D%20%5B1,%2020,%2050%5D%0A%20%20%20%20amt%20%3D%2060%0A%20%20%20%20res%20%3D%20coin_change_greedy%28coins,%20amt%29%0A%20%20%20%20print%28f%22%5Cncoins%20%3D%20%7Bcoins%7D,%20amt%20%3D%20%7Bamt%7D%22%29%0A%20%20%20%20print%28f%22Number%20of%20coins%20chosen%20by%20the%20greedy%20algorithm%20for%20amount%20%7Bamt%7D%3A%20%7Bres%7D%22%29%0A%20%20%20%20print%28f%22The%20actual%20minimum%20is%203%20coins%3A%2020%20%2B%2020%20%2B%2020%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>You might find yourself thinking, "So clean!" The greedy algorithm solves coin change in only about ten lines of code.</p>
+<h2 id="1511-strengths-and-limitations-of-greedy-algorithms">15.1.1 &nbsp; Strengths and limitations of greedy algorithms<a class="headerlink" href="#1511-strengths-and-limitations-of-greedy-algorithms" title="Permanent link">&para;</a></h2>
+<p><strong>Greedy algorithms are straightforward, easy to implement, and usually efficient</strong>. In the code above, let <span class="arithmatex">\(\min(coins)\)</span> be the smallest coin denomination. At most <span class="arithmatex">\(amt / \min(coins)\)</span> greedy choices are made, giving a time complexity of <span class="arithmatex">\(O(amt / \min(coins))\)</span>. This is an order of magnitude lower than the dynamic programming solution's <span class="arithmatex">\(O(n \times amt)\)</span> time complexity.</p>
+<p>However, <strong>for some sets of coin denominations, a greedy algorithm cannot find the optimal solution</strong>. Figure 15-2 illustrates two examples.</p>
 <ul>
-<li><strong>Positive example <span class="arithmatex">\(coins = [1, 5, 10, 20, 50, 100]\)</span></strong>: With this coin set, the greedy algorithm can find the optimal solution for any <span class="arithmatex">\(amt\)</span>.</li>
-<li><strong>Counterexample <span class="arithmatex">\(coins = [1, 20, 50]\)</span></strong>: Suppose <span class="arithmatex">\(amt = 60\)</span>. The greedy algorithm can only find the combination <span class="arithmatex">\(50 + 1 \times 10\)</span>, using <span class="arithmatex">\(11\)</span> coins in total, whereas dynamic programming can find the optimal solution <span class="arithmatex">\(20 + 20 + 20\)</span> using only <span class="arithmatex">\(3\)</span> coins.</li>
-<li><strong>Counterexample <span class="arithmatex">\(coins = [1, 49, 50]\)</span></strong>: Suppose <span class="arithmatex">\(amt = 98\)</span>. The greedy algorithm can only find the combination <span class="arithmatex">\(50 + 1 \times 48\)</span>, using <span class="arithmatex">\(49\)</span> coins in total, whereas dynamic programming can find the optimal solution <span class="arithmatex">\(49 + 49\)</span> using only <span class="arithmatex">\(2\)</span> coins.</li>
+<li><strong>Successful case: <span class="arithmatex">\(coins = [1, 5, 10, 20, 50, 100]\)</span></strong>. For these denominations, the greedy algorithm finds an optimal solution for any <span class="arithmatex">\(amt\)</span>.</li>
+<li><strong>Counterexample: <span class="arithmatex">\(coins = [1, 20, 50]\)</span></strong>. For <span class="arithmatex">\(amt = 60\)</span>, the greedy algorithm produces <span class="arithmatex">\(50 + 1 \times 10\)</span>, using <span class="arithmatex">\(11\)</span> coins. Dynamic programming finds the optimal solution <span class="arithmatex">\(20 + 20 + 20\)</span>, using only <span class="arithmatex">\(3\)</span> coins.</li>
+<li><strong>Counterexample: <span class="arithmatex">\(coins = [1, 49, 50]\)</span></strong>. For <span class="arithmatex">\(amt = 98\)</span>, the greedy algorithm produces <span class="arithmatex">\(50 + 1 \times 48\)</span>, using <span class="arithmatex">\(49\)</span> coins. Dynamic programming finds the optimal solution <span class="arithmatex">\(49 + 49\)</span>, using only <span class="arithmatex">\(2\)</span> coins.</li>
 </ul>
-<p><img alt="Examples where greedy algorithms cannot find the optimal solution" class="animation-figure" src="../greedy_algorithm.assets/coin_change_greedy_vs_dp.png" /></p>
-<p align="center"> Figure 15-2 &nbsp; Examples where greedy algorithms cannot find the optimal solution </p>
+<p><img alt="Examples where the greedy algorithm fails to find the optimal solution" class="animation-figure" src="../greedy_algorithm.assets/coin_change_greedy_vs_dp.png" /></p>
+<p align="center"> Figure 15-2 &nbsp; Examples where the greedy algorithm fails to find the optimal solution </p>
 
-<p>In other words, for the coin change problem, greedy algorithms cannot guarantee a globally optimal solution and may even produce very poor results. This problem is better solved with dynamic programming.</p>
-<p>In general, greedy algorithms are applicable in the following two situations.</p>
+<p>In other words, a greedy algorithm cannot guarantee a globally optimal solution to coin change and may produce a very poor solution. Dynamic programming is better suited to this problem.</p>
+<p>In general, greedy algorithms are useful in two situations.</p>
 <ol>
-<li><strong>The optimal solution can be guaranteed</strong>: In this case, greedy algorithms are often the best choice because they tend to be more efficient than backtracking and dynamic programming.</li>
-<li><strong>An approximately optimal solution can be found</strong>: Greedy algorithms are also useful in this case. For many complex problems, finding the global optimal solution is very difficult, so efficiently finding a suboptimal solution is already a very good outcome.</li>
+<li><strong>An optimal solution is guaranteed</strong>: A greedy algorithm is often the best choice because it is usually more efficient than backtracking or dynamic programming.</li>
+<li><strong>A near-optimal solution can be found</strong>: A greedy algorithm can still be useful. For many complex problems, finding a globally optimal solution is very difficult, so finding a good suboptimal solution efficiently is valuable.</li>
 </ol>
-<h2 id="1512-characteristics-of-greedy-algorithms">15.1.2 &nbsp; Characteristics of Greedy Algorithms<a class="headerlink" href="#1512-characteristics-of-greedy-algorithms" title="Permanent link">&para;</a></h2>
-<p>So the question arises: what kind of problems are suitable for solving with greedy algorithms? Or in other words, under what conditions can greedy algorithms guarantee finding the optimal solution?</p>
-<p>Compared to dynamic programming, the conditions for using greedy algorithms are stricter, mainly focusing on two properties of the problem.</p>
+<h2 id="1512-properties-required-by-greedy-algorithms">15.1.2 &nbsp; Properties required by greedy algorithms<a class="headerlink" href="#1512-properties-required-by-greedy-algorithms" title="Permanent link">&para;</a></h2>
+<p>What kinds of problems are suitable for greedy algorithms? More specifically, when can a greedy algorithm guarantee an optimal solution?</p>
+<p>Greedy algorithms have stricter requirements than dynamic programming. Two properties are particularly important.</p>
 <ul>
-<li><strong>Greedy choice property</strong>: Only when locally optimal choices can always lead to a globally optimal solution can greedy algorithms guarantee obtaining the optimal solution.</li>
-<li><strong>Optimal substructure</strong>: The optimal solution to the original problem contains the optimal solutions to subproblems.</li>
+<li><strong>Greedy-choice property</strong>: A greedy algorithm can guarantee an optimal solution only when locally optimal choices always lead to a globally optimal solution.</li>
+<li><strong>Optimal substructure</strong>: An optimal solution to the original problem contains optimal solutions to its subproblems.</li>
 </ul>
-<p>Optimal substructure has already been introduced in the "Dynamic Programming" chapter, so we won't elaborate on it here. It's worth noting that the optimal substructure of some problems is not obvious, but they can still be solved using greedy algorithms.</p>
-<p>We mainly explore methods for determining the greedy choice property. Although its description seems relatively simple, <strong>in practice, for many problems, proving the greedy choice property is not easy</strong>.</p>
-<p>For example, in the coin change problem, although we can easily provide counterexamples to disprove the greedy choice property, proving that it holds is much harder. If asked, <strong>under what conditions can a coin set be solved using a greedy algorithm</strong>? We often can only rely on intuition or examples to give a vague answer, and it is difficult to provide a rigorous mathematical proof.</p>
+<p>Optimal substructure was covered in the dynamic programming chapter, so we will not repeat that discussion here. Some problems have optimal substructure that is not immediately apparent, yet can still be solved greedily.</p>
+<p>We will focus on how to determine whether the greedy-choice property holds. Its description may sound simple, <strong>but proving it is difficult for many problems</strong>.</p>
+<p>For coin change, for example, it is easy to give counterexamples that disprove the greedy-choice property, but proving it is harder. If asked <strong>which conditions on the coin denominations allow a greedy solution</strong>, we may be able to give only a vague answer based on intuition or examples, rather than a rigorous mathematical proof.</p>
 <div class="admonition quote">
 <p class="admonition-title">Quote</p>
-<p>There is a paper that presents an <span class="arithmatex">\(O(n^3)\)</span> algorithm for determining whether a coin set can be solved optimally by a greedy algorithm for any amount.</p>
+<p>One paper gives an <span class="arithmatex">\(O(n^3)\)</span>-time algorithm for determining whether a set of coin denominations allows a greedy algorithm to find an optimal solution for every amount.</p>
 <p>Pearson, D. A polynomial-time algorithm for the change-making problem[J]. Operations Research Letters, 2005, 33(3): 231-234.</p>
 </div>
-<h2 id="1513-steps-for-solving-problems-with-greedy-algorithms">15.1.3 &nbsp; Steps for Solving Problems with Greedy Algorithms<a class="headerlink" href="#1513-steps-for-solving-problems-with-greedy-algorithms" title="Permanent link">&para;</a></h2>
-<p>The general process for solving greedy problems can be divided into the following three steps.</p>
+<h2 id="1513-steps-for-solving-a-problem-greedily">15.1.3 &nbsp; Steps for solving a problem greedily<a class="headerlink" href="#1513-steps-for-solving-a-problem-greedily" title="Permanent link">&para;</a></h2>
+<p>Solving a greedy problem generally involves three steps.</p>
 <ol>
-<li><strong>Problem analysis</strong>: Sort out and understand the characteristics of the problem, including state definitions, optimization objectives, and constraints. This step also appears in backtracking and dynamic programming.</li>
-<li><strong>Determine the greedy strategy</strong>: Decide how to make a greedy choice at each step. This strategy should reduce the problem size step by step and ultimately solve the entire problem.</li>
-<li><strong>Correctness proof</strong>: It is usually necessary to prove that the problem has both greedy choice property and optimal substructure. This step may require mathematical tools such as induction or proof by contradiction.</li>
+<li><strong>Analyze the problem</strong>: Understand its properties, including the state definition, optimization objective, and constraints. This step also appears in backtracking and dynamic programming.</li>
+<li><strong>Choose a greedy strategy</strong>: Decide how to make a greedy choice at each step. Each choice should reduce the problem's size, eventually solving the whole problem.</li>
+<li><strong>Prove correctness</strong>: Usually, we must establish the greedy-choice property and optimal substructure. This may require a mathematical proof, such as induction or proof by contradiction.</li>
 </ol>
-<p>Determining the greedy strategy is the core step in solving such problems, but it may not be easy in practice, mainly for the following reasons.</p>
+<p>Choosing the greedy strategy is the central step, but it can be difficult for the following reasons.</p>
 <ul>
-<li><strong>Greedy strategies vary greatly from problem to problem</strong>. For many problems, the greedy strategy is fairly intuitive and can be derived through rough reasoning and experimentation. For some complex problems, however, the greedy strategy may be deeply hidden, which strongly tests one's problem-solving experience and algorithmic ability.</li>
-<li><strong>Some greedy strategies are highly deceptive</strong>. We may confidently design a greedy strategy, write the solution code, and submit it, only to find that some test cases fail. This is because the designed greedy strategy is only "partially correct," as exemplified by the coin change problem discussed above.</li>
+<li><strong>Greedy strategies vary greatly from problem to problem</strong>. For many problems, the strategy is straightforward and can be found with a little thought and experimentation. For more complex problems, it may be well hidden, testing our problem-solving experience and algorithmic skills.</li>
+<li><strong>Some greedy strategies are misleading</strong>. We may confidently devise a strategy, implement it, and submit the code, only to find that it fails some test cases. The strategy is only partially correct; the coin change problem above is a typical example.</li>
 </ul>
-<p>To ensure correctness, we should give a rigorous mathematical proof of the greedy strategy, <strong>usually using proof by contradiction or mathematical induction</strong>.</p>
-<p>However, correctness proofs can also be difficult. If we have no clear direction, we usually resort to debugging against test cases, revising and validating the greedy strategy step by step.</p>
-<h2 id="1514-typical-problems-solved-by-greedy-algorithms">15.1.4 &nbsp; Typical Problems Solved by Greedy Algorithms<a class="headerlink" href="#1514-typical-problems-solved-by-greedy-algorithms" title="Permanent link">&para;</a></h2>
-<p>Greedy algorithms are often applied to optimization problems that satisfy greedy choice property and optimal substructure. Below are some typical greedy algorithm problems.</p>
+<p>To ensure correctness, we should give the greedy strategy a rigorous mathematical proof, <strong>usually using contradiction or mathematical induction</strong>.</p>
+<p>However, such a proof can itself be difficult. When we do not know where to start, we often debug against test cases, gradually refining and checking the greedy strategy.</p>
+<h2 id="1514-typical-greedy-problems">15.1.4 &nbsp; Typical greedy problems<a class="headerlink" href="#1514-typical-greedy-problems" title="Permanent link">&para;</a></h2>
+<p>Greedy algorithms often solve optimization problems with the greedy-choice property and optimal substructure. Some typical examples follow.</p>
 <ul>
-<li><strong>Coin change problem</strong>: With certain coin combinations, greedy algorithms can always obtain the optimal solution.</li>
-<li><strong>Interval scheduling problem</strong>: Suppose you have some tasks, each taking place during a period of time, and your goal is to complete as many tasks as possible. If you always choose the task that ends earliest, then the greedy algorithm can obtain the optimal solution.</li>
-<li><strong>Fractional knapsack problem</strong>: Given a set of items and a carrying capacity, your goal is to select a set of items such that the total weight does not exceed the carrying capacity and the total value is maximized. If you always choose the item with the highest value-to-weight ratio (value / weight), then the greedy algorithm can obtain the optimal solution in some cases.</li>
-<li><strong>Stock trading problem</strong>: Given a set of historical stock prices, you can make multiple trades, but if you already hold stocks, you cannot buy again before selling, and the goal is to obtain the maximum profit.</li>
-<li><strong>Huffman coding</strong>: Huffman coding is a greedy algorithm used for lossless data compression. By constructing a Huffman tree and always merging the two nodes with the lowest frequency, the resulting Huffman tree has the minimum weighted path length (encoding length).</li>
-<li><strong>Dijkstra's algorithm</strong>: For graphs with non-negative edge weights, it is a greedy algorithm for solving the shortest path problem from a given source vertex to all other vertices.</li>
+<li><strong>Coin change</strong>: For certain sets of coin denominations, a greedy algorithm always finds an optimal solution.</li>
+<li><strong>Interval scheduling</strong>: Given tasks that each occupy a time interval, the goal is to complete as many tasks as possible. Always choosing the available task that finishes earliest yields an optimal solution.</li>
+<li><strong>Fractional knapsack</strong>: Given items and a weight limit, choose items to maximize total value without exceeding the limit. Choosing the item with the highest value-to-weight ratio at each step can yield an optimal solution in some settings.</li>
+<li><strong>Stock trading</strong>: Given historical stock prices, you may buy and sell multiple times, but cannot buy again while holding stock until you sell it. The goal is to maximize profit.</li>
+<li><strong>Huffman coding</strong>: This greedy algorithm is used for lossless data compression. It builds a Huffman tree by repeatedly merging the two nodes with the lowest frequencies. The resulting tree has the minimum weighted path length (encoding length).</li>
+<li><strong>Dijkstra's algorithm</strong>: This greedy algorithm finds shortest paths from a given source vertex to every other vertex in a graph whose edge weights are all nonnegative.</li>
 </ul>
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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@@ -2420,7 +2420,7 @@
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-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
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@@ -2638,7 +2638,7 @@
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-  10.1 Binary Search
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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-  12.1 Divide and Conquer Algorithms
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4378,7 +4378,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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+
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 </nav>
                   </div>
@@ -4577,19 +4649,19 @@
 
 
 <!-- Page content -->
-<h1 id="chapter-15-greedy">Chapter 15. &nbsp; Greedy<a class="headerlink" href="#chapter-15-greedy" title="Permanent link">&para;</a></h1>
-<p><img alt="Greedy" class="cover-image" src="../assets/covers/chapter_greedy.jpg" /></p>
+<h1 id="chapter-15-greedy-algorithms">Chapter 15. &nbsp; Greedy algorithms<a class="headerlink" href="#chapter-15-greedy-algorithms" title="Permanent link">&para;</a></h1>
+<p><img alt="Greedy algorithms" class="cover-image" src="../assets/covers/chapter_greedy.jpg" /></p>
 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
-<p>Sunflowers turn toward the sun, always seeking the fullest growth possible.</p>
-<p>Through successive simple choices, greedy strategies gradually lead to the optimal solution.</p>
+<p>A sunflower turns toward the sun, always seeking the greatest opportunity to grow.</p>
+<p>Through a series of simple choices, a greedy strategy leads us step by step toward the best answer.</p>
 </div>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
-<li><a href="greedy_algorithm/">15.1 &nbsp; Greedy Algorithm</a></li>
-<li><a href="fractional_knapsack_problem/">15.2 &nbsp; Fractional Knapsack Problem</a></li>
-<li><a href="max_capacity_problem/">15.3 &nbsp; Maximum Capacity Problem</a></li>
-<li><a href="max_product_cutting_problem/">15.4 &nbsp; Maximum Product Cutting Problem</a></li>
+<li><a href="greedy_algorithm/">15.1 &nbsp; Greedy algorithms</a></li>
+<li><a href="fractional_knapsack_problem/">15.2 &nbsp; Fractional knapsack problem</a></li>
+<li><a href="max_capacity_problem/">15.3 &nbsp; Maximum capacity problem</a></li>
+<li><a href="max_product_cutting_problem/">15.4 &nbsp; Maximum product cutting problem</a></li>
 <li><a href="summary/">15.5 &nbsp; Summary</a></li>
 <li><a href="exercises/">15.6 &nbsp; Exercises</a></li>
 </ul>
@@ -4615,7 +4687,7 @@ aria-label="Footer"
   <a
     href="../chapter_dynamic_programming/exercises/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 14.8 Exercises"
+    aria-label="Previous: 14.8   Exercises"
     rel="prev"
   >
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@@ -4627,7 +4699,7 @@ aria-label="Footer"
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       </span>
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+        14.8 &nbsp; Exercises
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@@ -4639,7 +4711,7 @@ aria-label="Footer"
   <a
     href="./greedy_algorithm/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 15.1 Greedy Algorithm"
+    aria-label="Next: 15.1   Greedy algorithms"
     rel="next"
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@@ -4647,7 +4719,7 @@ aria-label="Footer"
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+        15.1 &nbsp; Greedy algorithms
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
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+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
     <div data-md-component="skip">
 
 
-        <a href="#153-max-capacity-problem" class="md-skip">
+        <a href="#153-maximum-capacity-problem" class="md-skip">
           Skip to content
         </a>
 
@@ -119,7 +119,7 @@
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-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              15.3 &nbsp; Max Capacity Problem
+              15.3 &nbsp; Maximum capacity problem
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3283,7 +3283,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3304,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3352,7 +3352,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
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+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
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+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
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-  13.5 Summary
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-  13.6 Exercises
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+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
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-  14.1 Introduction to Dynamic Programming
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-  14.2 Characteristics of Dynamic Programming Problems
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-  14.3 Dynamic Programming Problem-Solving Approach
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-  14.5 Unbounded Knapsack Problem
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+  14.6 &nbsp; Edit distance problem
 
 
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+  <a href="#1-choosing-a-greedy-strategy" class="md-nav__link">
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-        1. &nbsp; Greedy Strategy Determination
+        1. &nbsp; Choosing a greedy strategy
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-        1. &nbsp; Greedy Strategy Determination
+        1. &nbsp; Choosing a greedy strategy
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+        3. &nbsp; Proof of correctness
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 <!-- Page content -->
-<h1 id="153-max-capacity-problem">15.3 &nbsp; Max Capacity Problem<a class="headerlink" href="#153-max-capacity-problem" title="Permanent link">&para;</a></h1>
+<h1 id="153-maximum-capacity-problem">15.3 &nbsp; Maximum capacity problem<a class="headerlink" href="#153-maximum-capacity-problem" title="Permanent link">&para;</a></h1>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given an array <span class="arithmatex">\(ht\)</span>, where each element represents the height of a vertical partition. Any two partitions in the array, together with the space between them, can form a container.</p>
-<p>The capacity of the container equals the product of its height and width (that is, its area), where the height is determined by the shorter partition and the width is the difference between the array indices of the two partitions.</p>
-<p>Select two partitions in the array such that the capacity of the resulting container is maximized, and return that maximum capacity. An example is shown in Figure 15-7.</p>
+<p>Given an array <span class="arithmatex">\(ht\)</span>, each element represents the height of a vertical wall. Any two walls, together with the space between them, form a container.</p>
+<p>The container's capacity is its height multiplied by its width (an area). Its height is determined by the shorter wall, and its width is the difference between the two walls' array indices.</p>
+<p>Choose two walls from the array to maximize the container's capacity, and return that maximum capacity. An example is shown in Figure 15-7.</p>
 </div>
-<p><img alt="Example data for the max capacity problem" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_example.png" /></p>
-<p align="center"> Figure 15-7 &nbsp; Example data for the max capacity problem </p>
+<p><img alt="Example input for the maximum capacity problem" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_example.png" /></p>
+<p align="center"> Figure 15-7 &nbsp; Example input for the maximum capacity problem </p>
 
-<p>The container is formed by any two partitions, <strong>so the state of this problem is the indices of the two partitions, denoted by <span class="arithmatex">\([i, j]\)</span></strong>.</p>
-<p>According to the problem statement, capacity equals height multiplied by width, where the height is determined by the shorter partition and the width is the difference between the array indices of the two partitions. Let the capacity be <span class="arithmatex">\(cap[i, j]\)</span>; then we obtain the following formula:</p>
+<p>A container is formed by two walls, <strong>so the state of this problem is the pair of wall indices, denoted by <span class="arithmatex">\([i, j]\)</span></strong>.</p>
+<p>Capacity is height times width, with the height determined by the shorter wall and the width equal to the difference between the wall indices. Writing the capacity as <span class="arithmatex">\(cap[i, j]\)</span>, we have:</p>
 <div class="arithmatex">\[
 cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
 \]</div>
-<p>Let the array length be <span class="arithmatex">\(n\)</span>. Then the number of ways to choose two partitions (that is, the total number of states) is <span class="arithmatex">\(C_n^2 = \frac{n(n - 1)}{2}\)</span>. The most straightforward approach is to <strong>exhaustively enumerate all states</strong> to find the maximum capacity, which has a time complexity of <span class="arithmatex">\(O(n^2)\)</span>.</p>
-<h3 id="1-greedy-strategy-determination">1. &nbsp; Greedy Strategy Determination<a class="headerlink" href="#1-greedy-strategy-determination" title="Permanent link">&para;</a></h3>
-<p>This problem has a more efficient solution. As shown in Figure 15-8, consider a state <span class="arithmatex">\([i, j]\)</span> where <span class="arithmatex">\(i &lt; j\)</span> and <span class="arithmatex">\(ht[i] &lt; ht[j]\)</span>. In this case, <span class="arithmatex">\(i\)</span> is the shorter partition and <span class="arithmatex">\(j\)</span> is the taller partition.</p>
+<p>For an array of length <span class="arithmatex">\(n\)</span>, there are <span class="arithmatex">\(C_n^2 = \frac{n(n - 1)}{2}\)</span> pairs of walls (possible states). The most direct approach is to <strong>enumerate all states</strong> and find the maximum capacity, taking <span class="arithmatex">\(O(n^2)\)</span> time.</p>
+<h3 id="1-choosing-a-greedy-strategy">1. &nbsp; Choosing a greedy strategy<a class="headerlink" href="#1-choosing-a-greedy-strategy" title="Permanent link">&para;</a></h3>
+<p>A more efficient solution is possible. Consider the state <span class="arithmatex">\([i, j]\)</span> in Figure 15-8, where <span class="arithmatex">\(i &lt; j\)</span> and <span class="arithmatex">\(ht[i] &lt; ht[j]\)</span>: wall <span class="arithmatex">\(i\)</span> is shorter and wall <span class="arithmatex">\(j\)</span> is taller.</p>
 <p><img alt="Initial state" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_initial_state.png" /></p>
 <p align="center"> Figure 15-8 &nbsp; Initial state </p>
 
-<p>As shown in Figure 15-9, <strong>if we now move the taller partition <span class="arithmatex">\(j\)</span> inward toward the shorter partition <span class="arithmatex">\(i\)</span>, the capacity will definitely decrease</strong>.</p>
-<p>This is because after moving the taller partition <span class="arithmatex">\(j\)</span>, the width <span class="arithmatex">\(j-i\)</span> definitely decreases. Since the height is determined by the shorter partition, the height can only stay the same (<span class="arithmatex">\(i\)</span> remains the shorter partition) or decrease (<span class="arithmatex">\(j\)</span> becomes the shorter partition after being moved).</p>
-<p><img alt="State after moving the long partition inward" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_moving_long_board.png" /></p>
-<p align="center"> Figure 15-9 &nbsp; State after moving the long partition inward </p>
+<p>As shown in Figure 15-9, <strong>moving the taller wall <span class="arithmatex">\(j\)</span> inward toward the shorter wall <span class="arithmatex">\(i\)</span> must reduce the capacity</strong>.</p>
+<p>Moving wall <span class="arithmatex">\(j\)</span> inward always reduces the width <span class="arithmatex">\(j-i\)</span>. Since the shorter wall determines the height, the height can only stay the same (if <span class="arithmatex">\(i\)</span> remains shorter) or decrease (if the new wall <span class="arithmatex">\(j\)</span> is shorter).</p>
+<p><img alt="State after moving the taller wall inward" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_moving_long_board.png" /></p>
+<p align="center"> Figure 15-9 &nbsp; State after moving the taller wall inward </p>
 
-<p>Conversely, <strong>only by moving the shorter partition <span class="arithmatex">\(i\)</span> inward can the capacity possibly increase</strong>. Although the width will definitely decrease, <strong>the height may increase</strong> (the moved partition at <span class="arithmatex">\(i\)</span> may be taller). For example, in Figure 15-10, the area increases after moving the shorter partition.</p>
-<p><img alt="State after moving the short partition inward" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_moving_short_board.png" /></p>
-<p align="center"> Figure 15-10 &nbsp; State after moving the short partition inward </p>
+<p>Conversely, <strong>only moving the shorter wall <span class="arithmatex">\(i\)</span> inward can potentially increase the capacity</strong>. Although the width must decrease, <strong>the height may increase</strong> if the new wall <span class="arithmatex">\(i\)</span> is taller. Figure 15-10 shows an example where moving the shorter wall increases the area.</p>
+<p><img alt="State after moving the shorter wall inward" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_moving_short_board.png" /></p>
+<p align="center"> Figure 15-10 &nbsp; State after moving the shorter wall inward </p>
 
-<p>From this, we can derive the greedy strategy for this problem: initialize two pointers at the two ends, and in each round move the pointer corresponding to the shorter partition inward until the two pointers meet.</p>
-<p>Figure 15-11 shows the execution process of the greedy strategy.</p>
+<p>This gives us a greedy strategy: initialize two pointers at opposite ends of the container. At each step, move the pointer at the shorter wall inward until the pointers meet.</p>
+<p>Figure 15-11 illustrates the greedy process.</p>
 <ol>
-<li>In the initial state, pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> are at both ends of the array.</li>
-<li>Calculate the capacity of the current state <span class="arithmatex">\(cap[i, j]\)</span>, and update the maximum capacity.</li>
-<li>Compare the heights of partitions <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span>, and move the pointer corresponding to the shorter partition inward by one position.</li>
+<li>Initially, pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> are at opposite ends of the array.</li>
+<li>Calculate the current capacity <span class="arithmatex">\(cap[i, j]\)</span> and update the maximum capacity.</li>
+<li>Compare the heights of walls <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span>, and move the shorter wall inward by one position.</li>
 <li>Repeat steps <code>2.</code> and <code>3.</code> until <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> meet.</li>
 </ol>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:9"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label><label for="__tabbed_1_6">&lt;6&gt;</label><label for="__tabbed_1_7">&lt;7&gt;</label><label for="__tabbed_1_8">&lt;8&gt;</label><label for="__tabbed_1_9">&lt;9&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Greedy process for the max capacity problem" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_greedy_step1.png" /></p>
+<p><img alt="Greedy steps for the maximum capacity problem" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_greedy_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="max_capacity_greedy_step2" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_greedy_step2.png" /></p>
@@ -4748,26 +4820,26 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
 </div>
 </div>
 </div>
-<p align="center"> Figure 15-11 &nbsp; Greedy process for the max capacity problem </p>
+<p align="center"> Figure 15-11 &nbsp; Greedy steps for the maximum capacity problem </p>
 
-<h3 id="2-code-implementation">2. &nbsp; Code Implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">&para;</a></h3>
-<p>The code runs for at most <span class="arithmatex">\(n\)</span> rounds, <strong>so the time complexity is <span class="arithmatex">\(O(n)\)</span></strong>.</p>
-<p>Variables <span class="arithmatex">\(i\)</span>, <span class="arithmatex">\(j\)</span>, and <span class="arithmatex">\(res\)</span> use only a constant amount of extra space, <strong>so the space complexity is <span class="arithmatex">\(O(1)\)</span></strong>.</p>
+<h3 id="2-code-implementation">2. &nbsp; Code implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">&para;</a></h3>
+<p>The loop runs at most <span class="arithmatex">\(n\)</span> times, <strong>so the time complexity is <span class="arithmatex">\(O(n)\)</span></strong>.</p>
+<p>The variables <span class="arithmatex">\(i\)</span>, <span class="arithmatex">\(j\)</span>, and <span class="arithmatex">\(res\)</span> require constant extra space, <strong>so the space complexity is <span class="arithmatex">\(O(1)\)</span></strong>.</p>
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 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">max_capacity.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">max_capacity</span><span class="p">(</span><span class="n">ht</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">])</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Max capacity: Greedy algorithm"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Initialize i, j to be at both ends of the array</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Maximum container capacity: greedy algorithm"""</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Initialize i and j at opposite ends of the array</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">ht</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># Initial max capacity is 0</span>
+<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># The initial maximum capacity is 0</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="n">res</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="c1"># Loop for greedy selection until the two boards meet</span>
+<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="c1"># Repeatedly make greedy choices until the two sides meet</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="c1"># Update max capacity</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="c1"># Update the maximum capacity</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="n">cap</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">])</span> <span class="o">*</span> <span class="p">(</span><span class="n">j</span> <span class="o">-</span> <span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="n">res</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">cap</span><span class="p">)</span>
-<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="c1"># Move the shorter board inward</span>
+<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="c1"># Move the shorter side inward</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="k">if</span> <span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>        <span class="k">else</span><span class="p">:</span>
@@ -4776,18 +4848,18 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_capacity.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Max capacity: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_capacity.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Maximum container capacity: greedy algorithm */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">maxCapacity</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">ht</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Initialize i, j to be at both ends of the array</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Initialize i and j at opposite ends of the array</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ht</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="c1">// Initial max capacity is 0</span>
+<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="c1">// The initial maximum capacity is 0</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="c1">// Loop for greedy selection until the two boards meet</span>
+<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until the two sides meet</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">        </span><span class="c1">// Update max capacity</span>
+<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">        </span><span class="c1">// Update the maximum capacity</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">);</span>
-<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="c1">// Move the shorter board inward</span>
+<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="c1">// Move the shorter side inward</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -4799,18 +4871,18 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_capacity.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Max capacity: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_capacity.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Maximum container capacity: greedy algorithm */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">maxCapacity</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">ht</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Initialize i, j to be at both ends of the array</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Initialize i and j at opposite ends of the array</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ht</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="c1">// Initial max capacity is 0</span>
+<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="c1">// The initial maximum capacity is 0</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="c1">// Loop for greedy selection until the two boards meet</span>
+<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until the two sides meet</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">        </span><span class="c1">// Update max capacity</span>
+<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">        </span><span class="c1">// Update the maximum capacity</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">min</span><span class="p">(</span><span class="n">ht</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">ht</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">max</span><span class="p">(</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">);</span>
-<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="c1">// Move the shorter board inward</span>
+<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="c1">// Move the shorter side inward</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ht</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">ht</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -4822,18 +4894,18 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_capacity.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Max capacity: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_capacity.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Maximum container capacity: greedy algorithm */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">MaxCapacity</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">ht</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Initialize i, j to be at both ends of the array</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Initialize i and j at opposite ends of the array</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ht</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="c1">// Initial max capacity is 0</span>
+<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="c1">// The initial maximum capacity is 0</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="c1">// Loop for greedy selection until the two boards meet</span>
+<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until the two sides meet</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">        </span><span class="c1">// Update max capacity</span>
+<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">        </span><span class="c1">// Update the maximum capacity</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Min</span><span class="p">(</span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Max</span><span class="p">(</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">);</span>
-<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="c1">// Move the shorter board inward</span>
+<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="c1">// Move the shorter side inward</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -4845,18 +4917,18 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_capacity.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Max capacity: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_capacity.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Maximum container capacity: greedy algorithm */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">maxCapacity</span><span class="p">(</span><span class="nx">ht</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Initialize i, j to be at both ends of the array</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Initialize i and j at opposite ends of the array</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">ht</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
-<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="c1">// Initial max capacity is 0</span>
+<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="c1">// The initial maximum capacity is 0</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// Loop for greedy selection until the two boards meet</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until the two sides meet</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">        </span><span class="c1">// Update max capacity</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">        </span><span class="c1">// Update the maximum capacity</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">        </span><span class="nx">capacity</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Min</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">ht</span><span class="p">[</span><span class="nx">i</span><span class="p">]),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">ht</span><span class="p">[</span><span class="nx">j</span><span class="p">])))</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Max</span><span class="p">(</span><span class="kt">float64</span><span class="p">(</span><span class="nx">res</span><span class="p">),</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">capacity</span><span class="p">)))</span>
-<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="c1">// Move the shorter board inward</span>
+<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="c1">// Move the shorter side inward</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">ht</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">ht</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">            </span><span class="nx">i</span><span class="o">++</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -4868,18 +4940,18 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_capacity.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Max capacity: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_capacity.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Maximum container capacity: greedy algorithm */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">maxCapacity</span><span class="p">(</span><span class="n">ht</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Initialize i, j to be at both ends of the array</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Initialize i and j at opposite ends of the array</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ht</span><span class="p">.</span><span class="n">startIndex</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ht</span><span class="p">.</span><span class="n">endIndex</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="c1">// Initial max capacity is 0</span>
+<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="c1">// The initial maximum capacity is 0</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// Loop for greedy selection until the two boards meet</span>
+<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until the two sides meet</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="c1">// Update max capacity</span>
+<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="c1">// Update the maximum capacity</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">cap</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">min</span><span class="p">(</span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">max</span><span class="p">(</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span>
-<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">        </span><span class="c1">// Move the shorter board inward</span>
+<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">        </span><span class="c1">// Move the shorter side inward</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -4891,19 +4963,19 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_capacity.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Max capacity: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_capacity.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Maximum container capacity: greedy algorithm */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">maxCapacity</span><span class="p">(</span><span class="nx">ht</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Initialize i, j to be at both ends of the array</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Initialize i and j at opposite ends of the array</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">ht</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// Initial max capacity is 0</span>
+<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// The initial maximum capacity is 0</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="c1">// Loop for greedy selection until the two boards meet</span>
+<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until the two sides meet</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="c1">// Update max capacity</span>
+<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="c1">// Update the maximum capacity</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">cap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">ht</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">ht</span><span class="p">[</span><span class="nx">j</span><span class="p">])</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">);</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="p">);</span>
-<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="c1">// Move the shorter board inward</span>
+<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="c1">// Move the shorter side inward</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">ht</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">ht</span><span class="p">[</span><span class="nx">j</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -4915,19 +4987,19 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_capacity.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Max capacity: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_capacity.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Maximum container capacity: greedy algorithm */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">maxCapacity</span><span class="p">(</span><span class="nx">ht</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Initialize i, j to be at both ends of the array</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Initialize i and j at opposite ends of the array</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">ht</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="c1">// Initial max capacity is 0</span>
+<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="c1">// The initial maximum capacity is 0</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// Loop for greedy selection until the two boards meet</span>
+<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until the two sides meet</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="c1">// Update max capacity</span>
+<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="c1">// Update the maximum capacity</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">cap</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">min</span><span class="p">(</span><span class="nx">ht</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">ht</span><span class="p">[</span><span class="nx">j</span><span class="p">])</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">);</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">        </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">cap</span><span class="p">);</span>
-<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="c1">// Move the shorter board inward</span>
+<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="c1">// Move the shorter side inward</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">ht</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">ht</span><span class="p">[</span><span class="nx">j</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -4939,18 +5011,18 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_capacity.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Max capacity: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_capacity.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Maximum container capacity: greedy algorithm */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">maxCapacity</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">ht</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Initialize i, j to be at both ends of the array</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Initialize i and j at opposite ends of the array</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ht</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
-<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// Initial max capacity is 0</span>
+<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// The initial maximum capacity is 0</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
-<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// Loop for greedy selection until the two boards meet</span>
+<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// Repeatedly make greedy choices until the two sides meet</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="c1">// Update max capacity</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="c1">// Update the maximum capacity</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">);</span>
-<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="c1">// Move the shorter board inward</span>
+<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="c1">// Move the shorter side inward</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">      </span><span class="n">i</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -4962,19 +5034,19 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_capacity.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Max capacity: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_capacity.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Maximum container capacity: greedy algorithm */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">max_capacity</span><span class="p">(</span><span class="n">ht</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Initialize i, j to be at both ends of the array</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Initialize i and j at opposite ends of the array</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ht</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// Initial max capacity is 0</span>
+<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// The initial maximum capacity is 0</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="c1">// Loop for greedy selection until the two boards meet</span>
+<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until the two sides meet</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="c1">// Update max capacity</span>
+<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="c1">// Update the maximum capacity</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">min</span><span class="p">(</span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">max</span><span class="p">(</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">);</span>
-<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">        </span><span class="c1">// Move the shorter board inward</span>
+<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">        </span><span class="c1">// Move the shorter side inward</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -4986,19 +5058,19 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_capacity.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Max capacity: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_capacity.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Maximum container capacity: greedy algorithm */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">maxCapacity</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">ht</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">htLength</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Initialize i, j to be at both ends of the array</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Initialize i and j at opposite ends of the array</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">htLength</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="c1">// Initial max capacity is 0</span>
+<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="c1">// The initial maximum capacity is 0</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="c1">// Loop for greedy selection until the two boards meet</span>
+<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until the two sides meet</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">        </span><span class="c1">// Update max capacity</span>
+<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">        </span><span class="c1">// Update the maximum capacity</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMin</span><span class="p">(</span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">myMax</span><span class="p">(</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">capacity</span><span class="p">);</span>
-<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="c1">// Move the shorter board inward</span>
+<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="c1">// Move the shorter side inward</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ht</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">ht</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -5010,19 +5082,19 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_capacity.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Max capacity: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_capacity.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Maximum container capacity: greedy algorithm */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">maxCapacity</span><span class="p">(</span><span class="n">ht</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Initialize i, j to be at both ends of the array</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Initialize i and j at opposite ends of the array</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ht</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// Initial max capacity is 0</span>
+<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// The initial maximum capacity is 0</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="c1">// Loop for greedy selection until the two boards meet</span>
+<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="c1">// Repeatedly make greedy choices until the two sides meet</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">        </span><span class="c1">// Update max capacity</span>
+<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">        </span><span class="c1">// Update the maximum capacity</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">cap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">min</span><span class="p">(</span><span class="n">ht</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">ht</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="p">)</span>
-<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="c1">// Move the shorter board inward</span>
+<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="c1">// Move the shorter side inward</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ht</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">ht</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -5034,19 +5106,19 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_capacity.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Maximum capacity: greedy ###</span>
+<div class="highlight"><span class="filename">max_capacity.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Maximum container capacity: greedy algorithm ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">max_capacity</span><span class="p">(</span><span class="n">ht</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Initialize i, j to be at both ends of the array</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Initialize i and j at opposite ends of the array</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">ht</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Initial max capacity is 0</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># The initial maximum capacity is 0</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a>
-<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="c1"># Loop for greedy selection until the two boards meet</span>
+<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="c1"># Repeatedly make greedy choices until the two sides meet</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span>
-<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="c1"># Update max capacity</span>
+<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="c1"># Update the maximum capacity</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="n">cap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">ht</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">ht</span><span class="o">[</span><span class="n">j</span><span class="o">]].</span><span class="n">min</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">cap</span><span class="o">].</span><span class="n">max</span>
-<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="c1"># Move the shorter board inward</span>
+<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="c1"># Move the shorter side inward</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">ht</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">ht</span><span class="o">[</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">    </span><span class="k">else</span>
@@ -5060,17 +5132,22 @@ cap[i, j] = \min(ht[i], ht[j]) \times (j - i)
 </div>
 </div>
 </div>
-<h3 id="3-correctness-proof">3. &nbsp; Correctness Proof<a class="headerlink" href="#3-correctness-proof" title="Permanent link">&para;</a></h3>
-<p>The reason greedy is faster than exhaustive enumeration is that each round of greedy selection "skips" some states.</p>
-<p>For example, in state <span class="arithmatex">\(cap[i, j]\)</span>, suppose <span class="arithmatex">\(i\)</span> is the shorter partition and <span class="arithmatex">\(j\)</span> is the taller partition. If we greedily move the shorter partition <span class="arithmatex">\(i\)</span> inward by one position, the states shown in Figure 15-12 will be "skipped." <strong>This means that their capacities can no longer be checked later</strong>.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20max_capacity%28ht%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Maximum%20container%20capacity%3A%20greedy%20algorithm%22%22%22%0A%20%20%20%20%23%20Initialize%20i%20and%20j%20at%20opposite%20ends%20of%20the%20array%0A%20%20%20%20i,%20j%20%3D%200,%20len%28ht%29%20-%201%0A%20%20%20%20%23%20The%20initial%20maximum%20capacity%20is%200%0A%20%20%20%20res%20%3D%200%0A%20%20%20%20%23%20Repeatedly%20make%20greedy%20choices%20until%20the%20two%20sides%20meet%0A%20%20%20%20while%20i%20%3C%20j%3A%0A%20%20%20%20%20%20%20%20%23%20Update%20the%20maximum%20capacity%0A%20%20%20%20%20%20%20%20cap%20%3D%20min%28ht%5Bi%5D,%20ht%5Bj%5D%29%20%2A%20%28j%20-%20i%29%0A%20%20%20%20%20%20%20%20res%20%3D%20max%28res,%20cap%29%0A%20%20%20%20%20%20%20%20%23%20Move%20the%20shorter%20side%20inward%0A%20%20%20%20%20%20%20%20if%20ht%5Bi%5D%20%3C%20ht%5Bj%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20-%3D%201%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20ht%20%3D%20%5B3,%208,%205,%202,%207,%207,%203,%204%5D%0A%0A%20%20%20%20%23%20Greedy%20algorithm%0A%20%20%20%20res%20%3D%20max_capacity%28ht%29%0A%20%20%20%20print%28f%22Maximum%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20max_capacity%28ht%3A%20list%5Bint%5D%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Maximum%20container%20capacity%3A%20greedy%20algorithm%22%22%22%0A%20%20%20%20%23%20Initialize%20i%20and%20j%20at%20opposite%20ends%20of%20the%20array%0A%20%20%20%20i,%20j%20%3D%200,%20len%28ht%29%20-%201%0A%20%20%20%20%23%20The%20initial%20maximum%20capacity%20is%200%0A%20%20%20%20res%20%3D%200%0A%20%20%20%20%23%20Repeatedly%20make%20greedy%20choices%20until%20the%20two%20sides%20meet%0A%20%20%20%20while%20i%20%3C%20j%3A%0A%20%20%20%20%20%20%20%20%23%20Update%20the%20maximum%20capacity%0A%20%20%20%20%20%20%20%20cap%20%3D%20min%28ht%5Bi%5D,%20ht%5Bj%5D%29%20%2A%20%28j%20-%20i%29%0A%20%20%20%20%20%20%20%20res%20%3D%20max%28res,%20cap%29%0A%20%20%20%20%20%20%20%20%23%20Move%20the%20shorter%20side%20inward%0A%20%20%20%20%20%20%20%20if%20ht%5Bi%5D%20%3C%20ht%5Bj%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20-%3D%201%0A%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20ht%20%3D%20%5B3,%208,%205,%202,%207,%207,%203,%204%5D%0A%0A%20%20%20%20%23%20Greedy%20algorithm%0A%20%20%20%20res%20%3D%20max_capacity%28ht%29%0A%20%20%20%20print%28f%22Maximum%20capacity%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="3-proof-of-correctness">3. &nbsp; Proof of correctness<a class="headerlink" href="#3-proof-of-correctness" title="Permanent link">&para;</a></h3>
+<p>The greedy algorithm is faster than exhaustive search because each greedy choice skips some states.</p>
+<p>For example, in state <span class="arithmatex">\(cap[i, j]\)</span>, suppose <span class="arithmatex">\(i\)</span> is the shorter wall and <span class="arithmatex">\(j\)</span> the taller one. Moving the shorter wall <span class="arithmatex">\(i\)</span> inward by one position skips the states shown in Figure 15-12. <strong>Their capacities will never be checked later</strong>.</p>
 <div class="arithmatex">\[
 cap[i, i+1], cap[i, i+2], \dots, cap[i, j-2], cap[i, j-1]
 \]</div>
-<p><img alt="States skipped by moving the short partition" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_skipped_states.png" /></p>
-<p align="center"> Figure 15-12 &nbsp; States skipped by moving the short partition </p>
+<p><img alt="States skipped by moving the shorter wall" class="animation-figure" src="../max_capacity_problem.assets/max_capacity_skipped_states.png" /></p>
+<p align="center"> Figure 15-12 &nbsp; States skipped by moving the shorter wall </p>
 
-<p>A closer look shows that <strong>these skipped states are exactly the states obtained by moving the taller partition <span class="arithmatex">\(j\)</span> inward</strong>. We have already proven that moving the taller partition inward will definitely decrease the capacity. Therefore, none of the skipped states can be the optimal solution, <strong>so skipping them does not cause us to miss the optimum</strong>.</p>
-<p>The above analysis shows that moving the shorter partition is a "safe" operation, and that the greedy strategy is effective.</p>
+<p>Observe that <strong>the skipped states are precisely those obtained by moving the taller wall <span class="arithmatex">\(j\)</span> inward</strong>. We have already shown that moving the taller wall inward must reduce capacity. None of these skipped states can therefore be optimal, and <strong>skipping them cannot cause us to miss the optimal solution</strong>.</p>
+<p>This shows that moving the shorter wall is safe, and that the greedy strategy is valid.</p>
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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@@ -1992,7 +1992,7 @@
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2899,7 +2899,7 @@
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2920,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2947,7 +2947,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2974,7 +2974,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3001,7 +3001,7 @@
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3028,7 +3028,7 @@
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3055,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3082,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3109,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3136,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3163,7 +3163,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3190,7 +3190,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3217,7 +3217,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3283,7 +3283,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3304,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3325,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3352,7 +3352,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3379,7 +3379,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3406,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3433,7 +3433,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3460,7 +3460,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3526,7 +3526,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3568,7 +3568,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3595,7 +3595,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3622,7 +3622,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3649,7 +3649,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3676,7 +3676,7 @@
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3773,7 +3773,7 @@
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3794,7 +3794,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3815,7 +3815,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3869,7 +3869,7 @@
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3896,7 +3896,7 @@
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -3950,7 +3950,7 @@
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4072,7 +4072,7 @@
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4247,10 +4247,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-determining-the-greedy-strategy" class="md-nav__link">
+  <a href="#1-choosing-a-greedy-strategy" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Determining the Greedy Strategy
+        1. &nbsp; Choosing a greedy strategy
       </span>
     </span>
   </a>
@@ -4261,7 +4261,7 @@
   <a href="#2-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Code Implementation
+        2. &nbsp; Code implementation
       </span>
     </span>
   </a>
@@ -4269,10 +4269,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#3-correctness-proof" class="md-nav__link">
+  <a href="#3-proof-of-correctness" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Correctness Proof
+        3. &nbsp; Proof of correctness
       </span>
     </span>
   </a>
@@ -4302,7 +4302,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4389,7 +4389,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4431,7 +4431,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4458,7 +4458,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4485,7 +4485,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4569,6 +4569,78 @@
 
 
 
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-  <a href="#1-determining-the-greedy-strategy" class="md-nav__link">
+  <a href="#1-choosing-a-greedy-strategy" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Determining the Greedy Strategy
+        1. &nbsp; Choosing a greedy strategy
       </span>
     </span>
   </a>
@@ -4624,7 +4696,7 @@
   <a href="#2-code-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Code Implementation
+        2. &nbsp; Code implementation
       </span>
     </span>
   </a>
@@ -4632,10 +4704,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#3-correctness-proof" class="md-nav__link">
+  <a href="#3-proof-of-correctness" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Correctness Proof
+        3. &nbsp; Proof of correctness
       </span>
     </span>
   </a>
@@ -4679,25 +4751,25 @@
 
 
 <!-- Page content -->
-<h1 id="154-maximum-product-cutting-problem">15.4 &nbsp; Maximum Product Cutting Problem<a class="headerlink" href="#154-maximum-product-cutting-problem" title="Permanent link">&para;</a></h1>
+<h1 id="154-maximum-product-cutting-problem">15.4 &nbsp; Maximum product cutting problem<a class="headerlink" href="#154-maximum-product-cutting-problem" title="Permanent link">&para;</a></h1>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given a positive integer <span class="arithmatex">\(n\)</span>, split it into the sum of at least two positive integers and find the maximum product of the resulting integers, as shown in Figure 15-13.</p>
+<p>Given a positive integer <span class="arithmatex">\(n\)</span>, split it into a sum of at least two positive integers. What is the largest possible product of those integers? The problem is illustrated in Figure 15-13.</p>
 </div>
-<p><img alt="Problem definition of max product cutting" class="animation-figure" src="../max_product_cutting_problem.assets/max_product_cutting_definition.png" /></p>
-<p align="center"> Figure 15-13 &nbsp; Problem definition of max product cutting </p>
+<p><img alt="Definition of the maximum product cutting problem" class="animation-figure" src="../max_product_cutting_problem.assets/max_product_cutting_definition.png" /></p>
+<p align="center"> Figure 15-13 &nbsp; Definition of the maximum product cutting problem </p>
 
-<p>Suppose we split <span class="arithmatex">\(n\)</span> into <span class="arithmatex">\(m\)</span> integer factors, where the <span class="arithmatex">\(i\)</span>-th factor is denoted as <span class="arithmatex">\(n_i\)</span>, that is</p>
+<p>Suppose we split <span class="arithmatex">\(n\)</span> into <span class="arithmatex">\(m\)</span> integers that will be factors in the product. Denote the <span class="arithmatex">\(i\)</span>-th factor by <span class="arithmatex">\(n_i\)</span>, so that</p>
 <div class="arithmatex">\[
 n = \sum_{i=1}^{m}n_i
 \]</div>
-<p>The goal of this problem is to find the maximum product of all integer factors, namely</p>
+<p>Our goal is to maximize the product of all these integer factors:</p>
 <div class="arithmatex">\[
 \max(\prod_{i=1}^{m}n_i)
 \]</div>
-<p>We need to determine how many parts <span class="arithmatex">\(m\)</span> there should be and what each <span class="arithmatex">\(n_i\)</span> should be.</p>
-<h3 id="1-determining-the-greedy-strategy">1. &nbsp; Determining the Greedy Strategy<a class="headerlink" href="#1-determining-the-greedy-strategy" title="Permanent link">&para;</a></h3>
-<p>As a rule of thumb, the product of two integers is often greater than their sum. Suppose we split off a factor of <span class="arithmatex">\(2\)</span> from <span class="arithmatex">\(n\)</span>; the resulting product is <span class="arithmatex">\(2(n-2)\)</span>. We compare this product with <span class="arithmatex">\(n\)</span>:</p>
+<p>We need to determine how many parts <span class="arithmatex">\(m\)</span> to create and what each <span class="arithmatex">\(n_i\)</span> should be.</p>
+<h3 id="1-choosing-a-greedy-strategy">1. &nbsp; Choosing a greedy strategy<a class="headerlink" href="#1-choosing-a-greedy-strategy" title="Permanent link">&para;</a></h3>
+<p>The product of two integers is often greater than their sum. Suppose we split off a factor of <span class="arithmatex">\(2\)</span> from <span class="arithmatex">\(n\)</span>, giving the product <span class="arithmatex">\(2(n-2)\)</span>. Comparing this product with <span class="arithmatex">\(n\)</span> gives:</p>
 <div class="arithmatex">\[
 \begin{aligned}
 2(n-2) &amp; \geq n \newline
@@ -4705,336 +4777,341 @@ n = \sum_{i=1}^{m}n_i
 n &amp; \geq 4
 \end{aligned}
 \]</div>
-<p>As shown in Figure 15-14, when <span class="arithmatex">\(n \geq 4\)</span>, splitting out a <span class="arithmatex">\(2\)</span> will increase the product, <strong>which indicates that integers greater than or equal to <span class="arithmatex">\(4\)</span> should all be split</strong>.</p>
-<p><strong>Greedy strategy one</strong>: If the splitting scheme contains a factor <span class="arithmatex">\(\geq 4\)</span>, it should be split further. The final splitting scheme should contain only the factors <span class="arithmatex">\(1\)</span>, <span class="arithmatex">\(2\)</span>, and <span class="arithmatex">\(3\)</span>.</p>
-<p><img alt="Splitting causes product to increase" class="animation-figure" src="../max_product_cutting_problem.assets/max_product_cutting_greedy_infer1.png" /></p>
-<p align="center"> Figure 15-14 &nbsp; Splitting causes product to increase </p>
+<p>As shown in Figure 15-14, when <span class="arithmatex">\(n \geq 4\)</span>, splitting off a <span class="arithmatex">\(2\)</span> increases the product. <strong>This suggests that every integer greater than or equal to <span class="arithmatex">\(4\)</span> should be split further</strong>.</p>
+<p><strong>Greedy strategy 1</strong>: If a factor <span class="arithmatex">\(\geq 4\)</span> appears in a split, divide it further. The final split should contain only the factors <span class="arithmatex">\(1\)</span>, <span class="arithmatex">\(2\)</span>, and <span class="arithmatex">\(3\)</span>.</p>
+<p><img alt="Splitting increases the product" class="animation-figure" src="../max_product_cutting_problem.assets/max_product_cutting_greedy_infer1.png" /></p>
+<p align="center"> Figure 15-14 &nbsp; Splitting increases the product </p>
 
-<p>Next, consider which factor is optimal. Among the three factors <span class="arithmatex">\(1\)</span>, <span class="arithmatex">\(2\)</span>, and <span class="arithmatex">\(3\)</span>, clearly <span class="arithmatex">\(1\)</span> is the worst, because <span class="arithmatex">\(1 \times (n-1) &lt; n\)</span> always holds, meaning splitting out <span class="arithmatex">\(1\)</span> will actually decrease the product.</p>
-<p>As shown in Figure 15-15, when <span class="arithmatex">\(n = 6\)</span>, we have <span class="arithmatex">\(3 \times 3 &gt; 2 \times 2 \times 2\)</span>. <strong>This means that splitting out <span class="arithmatex">\(3\)</span> is better than splitting out <span class="arithmatex">\(2\)</span></strong>.</p>
-<p><strong>Greedy strategy two</strong>: In the splitting scheme, there should be at most two <span class="arithmatex">\(2\)</span>s, because three <span class="arithmatex">\(2\)</span>s can always be replaced by two <span class="arithmatex">\(3\)</span>s to obtain a larger product.</p>
-<p><img alt="Optimal splitting factor" class="animation-figure" src="../max_product_cutting_problem.assets/max_product_cutting_greedy_infer2.png" /></p>
-<p align="center"> Figure 15-15 &nbsp; Optimal splitting factor </p>
+<p>Next, consider which factor is best. Of <span class="arithmatex">\(1\)</span>, <span class="arithmatex">\(2\)</span>, and <span class="arithmatex">\(3\)</span>, clearly <span class="arithmatex">\(1\)</span> is the worst: <span class="arithmatex">\(1 \times (n-1) &lt; n\)</span> always holds, so splitting off a <span class="arithmatex">\(1\)</span> reduces the product.</p>
+<p>As shown in Figure 15-15, when <span class="arithmatex">\(n = 6\)</span>, we have <span class="arithmatex">\(3 \times 3 &gt; 2 \times 2 \times 2\)</span>. <strong>This means that splitting into factors of <span class="arithmatex">\(3\)</span> is better than splitting into factors of <span class="arithmatex">\(2\)</span></strong>.</p>
+<p><strong>Greedy strategy 2</strong>: A split should contain at most two factors of <span class="arithmatex">\(2\)</span>, because three <span class="arithmatex">\(2\)</span>s can always be replaced with two <span class="arithmatex">\(3\)</span>s to obtain a larger product.</p>
+<p><img alt="Optimal factors" class="animation-figure" src="../max_product_cutting_problem.assets/max_product_cutting_greedy_infer2.png" /></p>
+<p align="center"> Figure 15-15 &nbsp; Optimal factors </p>
 
-<p>In summary, the following greedy strategies can be derived.</p>
+<p>These observations lead to the following greedy strategy.</p>
 <ol>
-<li>Input integer <span class="arithmatex">\(n\)</span>, continuously split out factor <span class="arithmatex">\(3\)</span> until the remainder is <span class="arithmatex">\(0\)</span>, <span class="arithmatex">\(1\)</span>, or <span class="arithmatex">\(2\)</span>.</li>
-<li>When the remainder is <span class="arithmatex">\(0\)</span>, it means <span class="arithmatex">\(n\)</span> is a multiple of <span class="arithmatex">\(3\)</span>, so no further action is needed.</li>
-<li>When the remainder is <span class="arithmatex">\(2\)</span>, do not split it further; keep it as is.</li>
-<li>When the remainder is <span class="arithmatex">\(1\)</span>, since <span class="arithmatex">\(2 \times 2 &gt; 1 \times 3\)</span>, replace the final <span class="arithmatex">\(3\)</span> and the remaining <span class="arithmatex">\(1\)</span> with two <span class="arithmatex">\(2\)</span>s.</li>
+<li>Starting with <span class="arithmatex">\(n\)</span>, keep splitting off factors of <span class="arithmatex">\(3\)</span> until the remainder is <span class="arithmatex">\(0\)</span>, <span class="arithmatex">\(1\)</span>, or <span class="arithmatex">\(2\)</span>.</li>
+<li>If the remainder is <span class="arithmatex">\(0\)</span>, then <span class="arithmatex">\(n\)</span> is a multiple of <span class="arithmatex">\(3\)</span>, and no further action is needed.</li>
+<li>If the remainder is <span class="arithmatex">\(2\)</span>, keep it without splitting it further.</li>
+<li>If the remainder is <span class="arithmatex">\(1\)</span>, replace the last <span class="arithmatex">\(3\)</span> and the remaining <span class="arithmatex">\(1\)</span> with two <span class="arithmatex">\(2\)</span>s, since <span class="arithmatex">\(2 \times 2 &gt; 1 \times 3\)</span>.</li>
 </ol>
-<h3 id="2-code-implementation">2. &nbsp; Code Implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 15-16, we do not need loops to split the integer. Instead, we use integer division to obtain the number of <span class="arithmatex">\(3\)</span>s, denoted by <span class="arithmatex">\(a\)</span>, and the modulo operation to obtain the remainder <span class="arithmatex">\(b\)</span>, giving:</p>
+<h3 id="2-code-implementation">2. &nbsp; Code implementation<a class="headerlink" href="#2-code-implementation" title="Permanent link">&para;</a></h3>
+<p>As shown in Figure 15-16, we do not need a loop to split the integer. Floor division gives the number of <span class="arithmatex">\(3\)</span>s, denoted by <span class="arithmatex">\(a\)</span>, and the modulo operation gives the remainder <span class="arithmatex">\(b\)</span>. Thus:</p>
 <div class="arithmatex">\[
 n = 3 a + b
 \]</div>
-<p>Please note that for the edge case of <span class="arithmatex">\(n \leq 3\)</span>, a <span class="arithmatex">\(1\)</span> must be split out, with product <span class="arithmatex">\(1 \times (n - 1)\)</span>.</p>
+<p>For the boundary cases <span class="arithmatex">\(n \leq 3\)</span>, we must split off a <span class="arithmatex">\(1\)</span>, giving a product of <span class="arithmatex">\(1 \times (n - 1)\)</span>.</p>
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 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">max_product_cutting.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">max_product_cutting</span><span class="p">(</span><span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Max product cutting: Greedy algorithm"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># When n &lt;= 3, must cut out a 1</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Maximum product after splitting: greedy algorithm"""</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># When n &lt;= 3, one part must be 1</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="k">if</span> <span class="n">n</span> <span class="o">&lt;=</span> <span class="mi">3</span><span class="p">:</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="k">return</span> <span class="mi">1</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Greedily cut out 3, a is the number of 3s, b is the remainder</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Greedily split off parts of size 3; a is the number of 3s and b is the remainder</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="n">a</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">n</span> <span class="o">//</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">3</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="k">if</span> <span class="n">b</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="c1"># When the remainder is 1, convert a pair of 1 * 3 to 2 * 2</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="c1"># When the remainder is 1, replace one pair 1 * 3 with 2 * 2</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">a</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span> <span class="o">*</span> <span class="mi">2</span> <span class="o">*</span> <span class="mi">2</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="k">if</span> <span class="n">b</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
-<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="c1"># When the remainder is 2, do nothing</span>
+<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="c1"># When the remainder is 2, no adjustment is needed</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">a</span><span class="p">))</span> <span class="o">*</span> <span class="mi">2</span>
-<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>    <span class="c1"># When the remainder is 0, do nothing</span>
+<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>    <span class="c1"># When the remainder is 0, no adjustment is needed</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">a</span><span class="p">))</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_product_cutting.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Max product cutting: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_product_cutting.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Maximum product after splitting: greedy algorithm */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">maxProductCutting</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, must cut out a 1</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, one part must be 1</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="c1">// Greedily cut out 3, a is the number of 3s, b is the remainder</span>
+<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="c1">// Greedily split off parts of size 3; a is the number of 3s and b is the remainder</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, convert a pair of 1 * 3 to 2 * 2</span>
+<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, replace one pair 1 * 3 with 2 * 2</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, do nothing</span>
+<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, no adjustment is needed</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, do nothing</span>
+<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, no adjustment is needed</span>
 <a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">);</span>
 <a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_product_cutting.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Max product cutting: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_product_cutting.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Maximum product after splitting: greedy algorithm */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">maxProductCutting</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, must cut out a 1</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, one part must be 1</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="c1">// Greedily cut out 3, a is the number of 3s, b is the remainder</span>
+<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="c1">// Greedily split off parts of size 3; a is the number of 3s and b is the remainder</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, convert a pair of 1 * 3 to 2 * 2</span>
+<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, replace one pair 1 * 3 with 2 * 2</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, do nothing</span>
+<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, no adjustment is needed</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, do nothing</span>
+<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, no adjustment is needed</span>
 <a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">);</span>
 <a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_product_cutting.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Max product cutting: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_product_cutting.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Maximum product after splitting: greedy algorithm */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">MaxProductCutting</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, must cut out a 1</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, one part must be 1</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="c1">// Greedily cut out 3, a is the number of 3s, b is the remainder</span>
+<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="c1">// Greedily split off parts of size 3; a is the number of 3s and b is the remainder</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, convert a pair of 1 * 3 to 2 * 2</span>
+<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, replace one pair 1 * 3 with 2 * 2</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="n">Math</span><span class="p">.</span><span class="n">Pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, do nothing</span>
+<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, no adjustment is needed</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="n">Math</span><span class="p">.</span><span class="n">Pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, do nothing</span>
+<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, no adjustment is needed</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="n">Math</span><span class="p">.</span><span class="n">Pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">);</span>
 <a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_product_cutting.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Max product cutting: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_product_cutting.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Maximum product after splitting: greedy algorithm */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">maxProductCutting</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, must cut out a 1</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, one part must be 1</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// Greedily cut out 3, a is the number of 3s, b is the remainder</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// Greedily split off parts of size 3; a is the number of 3s and b is the remainder</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="nx">a</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">3</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="nx">b</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">3</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, convert a pair of 1 * 3 to 2 * 2</span>
+<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, replace one pair 1 * 3 with 2 * 2</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">a</span><span class="o">-</span><span class="mi">1</span><span class="p">)))</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, do nothing</span>
+<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, no adjustment is needed</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">a</span><span class="p">)))</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, do nothing</span>
+<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, no adjustment is needed</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">math</span><span class="p">.</span><span class="nx">Pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">a</span><span class="p">)))</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_product_cutting.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Max product cutting: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_product_cutting.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Maximum product after splitting: greedy algorithm */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">maxProductCutting</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, must cut out a 1</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, one part must be 1</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// Greedily cut out 3, a is the number of 3s, b is the remainder</span>
+<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// Greedily split off parts of size 3; a is the number of 3s and b is the remainder</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">3</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">b</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">3</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, convert a pair of 1 * 3 to 2 * 2</span>
+<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, replace one pair 1 * 3 with 2 * 2</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, do nothing</span>
+<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, no adjustment is needed</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, do nothing</span>
+<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, no adjustment is needed</span>
 <a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">)</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_product_cutting.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Max product cutting: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_product_cutting.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Maximum product after splitting: greedy algorithm */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">maxProductCutting</span><span class="p">(</span><span class="nx">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, must cut out a 1</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, one part must be 1</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mf">3</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="c1">// Greedily cut out 3, a is the number of 3s, b is the remainder</span>
+<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="c1">// Greedily split off parts of size 3; a is the number of 3s and b is the remainder</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">3</span><span class="p">);</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mf">3</span><span class="p">;</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">b</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, convert a pair of 1 * 3 to 2 * 2</span>
+<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, replace one pair 1 * 3 with 2 * 2</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">pow</span><span class="p">(</span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">b</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, do nothing</span>
+<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, no adjustment is needed</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">pow</span><span class="p">(</span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="nx">a</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, do nothing</span>
+<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, no adjustment is needed</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">pow</span><span class="p">(</span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="nx">a</span><span class="p">);</span>
 <a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_product_cutting.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Max product cutting: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_product_cutting.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Maximum product after splitting: greedy algorithm */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">maxProductCutting</span><span class="p">(</span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, must cut out a 1</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, one part must be 1</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mf">3</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="c1">// Greedily cut out 3, a is the number of 3s, b is the remainder</span>
+<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="c1">// Greedily split off parts of size 3; a is the number of 3s and b is the remainder</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">a</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">3</span><span class="p">);</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">b</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mf">3</span><span class="p">;</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">b</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, convert a pair of 1 * 3 to 2 * 2</span>
+<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, replace one pair 1 * 3 with 2 * 2</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">pow</span><span class="p">(</span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">b</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, do nothing</span>
+<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, no adjustment is needed</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">pow</span><span class="p">(</span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="nx">a</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, do nothing</span>
+<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, no adjustment is needed</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">pow</span><span class="p">(</span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="nx">a</span><span class="p">);</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_product_cutting.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Max product cutting: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_product_cutting.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Maximum product after splitting: greedy algorithm */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">maxProductCutting</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// When n &lt;= 3, must cut out a 1</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// When n &lt;= 3, one part must be 1</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="m">3</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// Greedily cut out 3, a is the number of 3s, b is the remainder</span>
+<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// Greedily split off parts of size 3; a is the number of 3s and b is the remainder</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">3</span><span class="p">;</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="m">3</span><span class="p">;</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="c1">// When the remainder is 1, convert a pair of 1 * 3 to 2 * 2</span>
+<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="c1">// When the remainder is 1, replace one pair 1 * 3 with 2 * 2</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">pow</span><span class="p">(</span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="m">2</span><span class="p">).</span><span class="n">toInt</span><span class="p">();</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">    </span><span class="c1">// When the remainder is 2, do nothing</span>
+<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">    </span><span class="c1">// When the remainder is 2, no adjustment is needed</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">pow</span><span class="p">(</span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="m">2</span><span class="p">).</span><span class="n">toInt</span><span class="p">();</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">  </span><span class="c1">// When the remainder is 0, do nothing</span>
+<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">  </span><span class="c1">// When the remainder is 0, no adjustment is needed</span>
 <a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">pow</span><span class="p">(</span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">).</span><span class="n">toInt</span><span class="p">();</span>
 <a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_product_cutting.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Max product cutting: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_product_cutting.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Maximum product after splitting: greedy algorithm */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">max_product_cutting</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, must cut out a 1</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, one part must be 1</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">3</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// Greedily cut out 3, a is the number of 3s, b is the remainder</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// Greedily split off parts of size 3; a is the number of 3s and b is the remainder</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, convert a pair of 1 * 3 to 2 * 2</span>
+<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, replace one pair 1 * 3 with 2 * 2</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">        </span><span class="mi">3_</span><span class="k">i32</span><span class="p">.</span><span class="n">pow</span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">u32</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="c1">// When the remainder is 2, do nothing</span>
+<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="c1">// When the remainder is 2, no adjustment is needed</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">        </span><span class="mi">3_</span><span class="k">i32</span><span class="p">.</span><span class="n">pow</span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">u32</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">        </span><span class="c1">// When the remainder is 0, do nothing</span>
+<a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">        </span><span class="c1">// When the remainder is 0, no adjustment is needed</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">        </span><span class="mi">3_</span><span class="k">i32</span><span class="p">.</span><span class="n">pow</span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">u32</span><span class="p">)</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_product_cutting.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Max product cutting: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_product_cutting.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Maximum product after splitting: greedy algorithm */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">maxProductCutting</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, must cut out a 1</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, one part must be 1</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="c1">// Greedily cut out 3, a is the number of 3s, b is the remainder</span>
+<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="c1">// Greedily split off parts of size 3; a is the number of 3s and b is the remainder</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, convert a pair of 1 * 3 to 2 * 2</span>
+<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, replace one pair 1 * 3 with 2 * 2</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, do nothing</span>
+<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, no adjustment is needed</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, do nothing</span>
+<a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, no adjustment is needed</span>
 <a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">pow</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">);</span>
 <a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_product_cutting.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Max product cutting: Greedy algorithm */</span>
+<div class="highlight"><span class="filename">max_product_cutting.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Maximum product after splitting: greedy algorithm */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">maxProductCutting</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, must cut out a 1</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// When n &lt;= 3, one part must be 1</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="m">3</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="c1">// Greedily cut out 3, a is the number of 3s, b is the remainder</span>
+<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="c1">// Greedily split off parts of size 3; a is the number of 3s and b is the remainder</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">3</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="m">3</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, convert a pair of 1 * 3 to 2 * 2</span>
+<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="c1">// When the remainder is 1, replace one pair 1 * 3 with 2 * 2</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">3.0</span><span class="p">.</span><span class="na">pow</span><span class="p">((</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)).</span><span class="na">toInt</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="m">2</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, do nothing</span>
+<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">        </span><span class="c1">// When the remainder is 2, no adjustment is needed</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">3.0</span><span class="p">.</span><span class="na">pow</span><span class="p">(</span><span class="n">a</span><span class="p">).</span><span class="na">toInt</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="m">2</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, do nothing</span>
+<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">    </span><span class="c1">// When the remainder is 0, no adjustment is needed</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="m">3.0</span><span class="p">.</span><span class="na">pow</span><span class="p">(</span><span class="n">a</span><span class="p">).</span><span class="na">toInt</span><span class="p">()</span>
 <a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">max_product_cutting.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Maximum cutting product: greedy ###</span>
+<div class="highlight"><span class="filename">max_product_cutting.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Maximum product after splitting: greedy algorithm ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">max_product_cutting</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># When n &lt;= 3, must cut out a 1</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># When n &lt;= 3, one part must be 1</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">3</span>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Greedily cut out 3, a is the number of 3s, b is the remainder</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Greedily split off parts of size 3; a is the number of 3s and b is the remainder</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">3</span>
-<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="c1"># When the remainder is 1, convert a pair of 1 * 3 to 2 * 2</span>
+<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="c1"># When the remainder is 1, replace one pair 1 * 3 with 2 * 2</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="mi">3</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">to_i</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="c1"># When the remainder is 2, do nothing</span>
+<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="c1"># When the remainder is 2, no adjustment is needed</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="mi">3</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">to_i</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">2</span>
-<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">  </span><span class="c1"># When the remainder is 0, do nothing</span>
+<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">  </span><span class="c1"># When the remainder is 0, no adjustment is needed</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">  </span><span class="mi">3</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="n">a</span><span class="p">)</span><span class="o">.</span><span class="n">to_i</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<p><img alt="Calculation method for max product cutting" class="animation-figure" src="../max_product_cutting_problem.assets/max_product_cutting_greedy_calculation.png" /></p>
-<p align="center"> Figure 15-16 &nbsp; Calculation method for max product cutting </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=import%20math%0A%0Adef%20max_product_cutting%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Maximum%20product%20after%20splitting%3A%20greedy%20algorithm%22%22%22%0A%20%20%20%20%23%20When%20n%20%3C%3D%203,%20one%20part%20must%20be%201%0A%20%20%20%20if%20n%20%3C%3D%203%3A%0A%20%20%20%20%20%20%20%20return%201%20%2A%20%28n%20-%201%29%0A%20%20%20%20%23%20Greedily%20split%20off%20parts%20of%20size%203%3B%20a%20is%20the%20number%20of%203s%20and%20b%20is%20the%20remainder%0A%20%20%20%20a,%20b%20%3D%20n%20//%203,%20n%20%25%203%0A%20%20%20%20if%20b%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20%23%20When%20the%20remainder%20is%201,%20replace%20one%20pair%201%20%2A%203%20with%202%20%2A%202%0A%20%20%20%20%20%20%20%20return%20int%28math.pow%283,%20a%20-%201%29%29%20%2A%202%20%2A%202%0A%20%20%20%20if%20b%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20%23%20When%20the%20remainder%20is%202,%20no%20adjustment%20is%20needed%0A%20%20%20%20%20%20%20%20return%20int%28math.pow%283,%20a%29%29%20%2A%202%0A%20%20%20%20%23%20When%20the%20remainder%20is%200,%20no%20adjustment%20is%20needed%0A%20%20%20%20return%20int%28math.pow%283,%20a%29%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%2058%0A%0A%20%20%20%20%23%20Greedy%20algorithm%0A%20%20%20%20res%20%3D%20max_product_cutting%28n%29%0A%20%20%20%20print%28f%22Maximum%20product%20after%20splitting%3A%20%7Bres%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=import%20math%0A%0Adef%20max_product_cutting%28n%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Maximum%20product%20after%20splitting%3A%20greedy%20algorithm%22%22%22%0A%20%20%20%20%23%20When%20n%20%3C%3D%203,%20one%20part%20must%20be%201%0A%20%20%20%20if%20n%20%3C%3D%203%3A%0A%20%20%20%20%20%20%20%20return%201%20%2A%20%28n%20-%201%29%0A%20%20%20%20%23%20Greedily%20split%20off%20parts%20of%20size%203%3B%20a%20is%20the%20number%20of%203s%20and%20b%20is%20the%20remainder%0A%20%20%20%20a,%20b%20%3D%20n%20//%203,%20n%20%25%203%0A%20%20%20%20if%20b%20%3D%3D%201%3A%0A%20%20%20%20%20%20%20%20%23%20When%20the%20remainder%20is%201,%20replace%20one%20pair%201%20%2A%203%20with%202%20%2A%202%0A%20%20%20%20%20%20%20%20return%20int%28math.pow%283,%20a%20-%201%29%29%20%2A%202%20%2A%202%0A%20%20%20%20if%20b%20%3D%3D%202%3A%0A%20%20%20%20%20%20%20%20%23%20When%20the%20remainder%20is%202,%20no%20adjustment%20is%20needed%0A%20%20%20%20%20%20%20%20return%20int%28math.pow%283,%20a%29%29%20%2A%202%0A%20%20%20%20%23%20When%20the%20remainder%20is%200,%20no%20adjustment%20is%20needed%0A%20%20%20%20return%20int%28math.pow%283,%20a%29%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20n%20%3D%2058%0A%0A%20%20%20%20%23%20Greedy%20algorithm%0A%20%20%20%20res%20%3D%20max_product_cutting%28n%29%0A%20%20%20%20print%28f%22Maximum%20product%20after%20splitting%3A%20%7Bres%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p><img alt="Calculating the maximum product" class="animation-figure" src="../max_product_cutting_problem.assets/max_product_cutting_greedy_calculation.png" /></p>
+<p align="center"> Figure 15-16 &nbsp; Calculating the maximum product </p>
 
-<p><strong>The time complexity depends on how exponentiation is implemented in the programming language</strong>. Taking Python as an example, there are three commonly used ways to compute powers.</p>
+<p><strong>The time complexity depends on how the programming language implements exponentiation</strong>. Python, for example, offers three common ways to compute powers.</p>
 <ul>
-<li>Both the operator <code>**</code> and the function <code>pow()</code> have time complexity <span class="arithmatex">\(O(\log⁡ a)\)</span>.</li>
-<li>The function <code>math.pow()</code> internally calls the C library's <code>pow()</code> function, which performs floating-point exponentiation, with time complexity <span class="arithmatex">\(O(1)\)</span>.</li>
+<li>The <code>**</code> operator and the <code>pow()</code> function both have a time complexity of <span class="arithmatex">\(O(\log⁡ a)\)</span>.</li>
+<li>The <code>math.pow()</code> function calls the C library's <code>pow()</code> function internally, performing floating-point exponentiation in <span class="arithmatex">\(O(1)\)</span> time.</li>
 </ul>
-<p>Variables <span class="arithmatex">\(a\)</span> and <span class="arithmatex">\(b\)</span> use a constant amount of extra space, <strong>therefore the space complexity is <span class="arithmatex">\(O(1)\)</span></strong>.</p>
-<h3 id="3-correctness-proof">3. &nbsp; Correctness Proof<a class="headerlink" href="#3-correctness-proof" title="Permanent link">&para;</a></h3>
-<p>We use proof by contradiction and consider only the case where <span class="arithmatex">\(n \geq 4\)</span>.</p>
+<p>The variables <span class="arithmatex">\(a\)</span> and <span class="arithmatex">\(b\)</span> use constant extra space, <strong>so the space complexity is <span class="arithmatex">\(O(1)\)</span></strong>.</p>
+<h3 id="3-proof-of-correctness">3. &nbsp; Proof of correctness<a class="headerlink" href="#3-proof-of-correctness" title="Permanent link">&para;</a></h3>
+<p>We use proof by contradiction, considering only <span class="arithmatex">\(n \geq 4\)</span>.</p>
 <ol>
-<li><strong>All factors <span class="arithmatex">\(\leq 3\)</span></strong>: Suppose the optimal splitting scheme includes a factor <span class="arithmatex">\(x \geq 4\)</span>. Then it can be further split into <span class="arithmatex">\(2(x-2)\)</span> to obtain a larger (or equal) product. This contradicts the assumption.</li>
-<li><strong>The splitting scheme does not contain <span class="arithmatex">\(1\)</span></strong>: Suppose the optimal splitting scheme includes a factor of <span class="arithmatex">\(1\)</span>. Then it can be merged into another factor to obtain a larger product. This contradicts the assumption.</li>
-<li><strong>The splitting scheme contains at most two <span class="arithmatex">\(2\)</span>s</strong>: Suppose the optimal splitting scheme includes three <span class="arithmatex">\(2\)</span>s. Then they can be replaced by two <span class="arithmatex">\(3\)</span>s, yielding a larger product. This contradicts the assumption.</li>
+<li><strong>All factors are <span class="arithmatex">\(\leq 3\)</span></strong>: Suppose an optimal split contains a factor <span class="arithmatex">\(x\)</span> that is <span class="arithmatex">\(\geq 4\)</span>. We can split it further to obtain <span class="arithmatex">\(2(x-2)\)</span>, giving a greater (or equal) product. This contradicts the assumption.</li>
+<li><strong>The split contains no <span class="arithmatex">\(1\)</span></strong>: Suppose an optimal split contains a factor of <span class="arithmatex">\(1\)</span>. We can merge it into another factor to obtain a larger product, contradicting the assumption.</li>
+<li><strong>The split contains at most two <span class="arithmatex">\(2\)</span>s</strong>: Suppose an optimal split contains three <span class="arithmatex">\(2\)</span>s. Replacing them with two <span class="arithmatex">\(3\)</span>s gives a larger product, contradicting the assumption.</li>
 </ol>
 
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@@ -5058,7 +5135,7 @@ aria-label="Footer"
   <a
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-    aria-label="Previous: 15.3 Maximum Capacity Problem"
+    aria-label="Previous: 15.3   Maximum capacity problem"
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@@ -5082,7 +5159,7 @@ aria-label="Footer"
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@@ -5107,7 +5184,7 @@ aria-label="Footer"
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+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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-  Before Starting
+  Foreword
 
 
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-  Before Starting
+  Foreword
 
           </label>
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
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-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
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   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
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-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
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-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
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-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
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-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
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+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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+  5.1 &nbsp; Stacks
 
 
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+  5.2 &nbsp; Queues
 
 
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+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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+  8.3 &nbsp; The top-k problem
 
 
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+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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+  Chapter 9. &nbsp; Graphs
 
 
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+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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+  9.3 &nbsp; Graph traversal
 
 
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+  9.4 &nbsp; Summary
 
 
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+  9.5 &nbsp; Exercises
 
 
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+  Chapter 10. &nbsp; Searching
 
 
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+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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+  Chapter 11. &nbsp; Sorting
 
 
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+  Chapter 11. &nbsp; Sorting
 
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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+  13.4 &nbsp; N-queens problem
 
 
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+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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         <li class="md-nav__item">
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+  <a href="#1-key-takeaways" class="md-nav__link">
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-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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@@ -4547,6 +4547,78 @@
 
 
 
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 <!-- Page content -->
 <h1 id="155-summary">15.5 &nbsp; Summary<a class="headerlink" href="#155-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
 <ul>
-<li>Greedy algorithms are typically used to solve optimization problems. The principle is to make locally optimal decisions at each decision stage in hopes of obtaining a globally optimal solution.</li>
-<li>Greedy algorithms iteratively make one greedy choice after another, transforming the problem into a smaller subproblem in each round, until the problem is solved.</li>
-<li>Greedy algorithms are not only simple to implement, but also have high problem-solving efficiency. Compared to dynamic programming, greedy algorithms typically have lower time complexity.</li>
-<li>In the coin change problem, for certain coin combinations, greedy algorithms can guarantee finding the optimal solution; for other coin combinations, however, greedy algorithms may find very poor solutions.</li>
-<li>Problems suitable for solving with greedy algorithms have two major properties: greedy choice property and optimal substructure. The greedy choice property represents the effectiveness of the greedy strategy.</li>
-<li>For some complex problems, proving the greedy choice property is not simple. Relatively speaking, disproving it is easier, such as in the coin change problem.</li>
-<li>Solving greedy problems mainly consists of three steps: problem analysis, determining the greedy strategy, and correctness proof. Among these, determining the greedy strategy is the core step, and correctness proof is often the main difficulty.</li>
-<li>The fractional knapsack problem, based on the 0-1 knapsack problem, allows selecting fractions of items, and therefore can be solved using greedy algorithms. The correctness of the greedy strategy can be proven using proof by contradiction.</li>
-<li>The max capacity problem can be solved using exhaustive enumeration with time complexity <span class="arithmatex">\(O(n^2)\)</span>. By designing a greedy strategy to move the shorter side inward in each round, the time complexity can be optimized to <span class="arithmatex">\(O(n)\)</span>.</li>
-<li>In the max product cutting problem, we successively derive two greedy strategies: integers <span class="arithmatex">\(\geq 4\)</span> should all continue to be split, and the optimal splitting factor is <span class="arithmatex">\(3\)</span>. The code includes exponentiation operations, and the time complexity depends on the implementation method of exponentiation, typically being <span class="arithmatex">\(O(1)\)</span> or <span class="arithmatex">\(O(\log n)\)</span>.</li>
+<li>Greedy algorithms are commonly used for optimization. At each decision point, they make a locally optimal choice in the hope of reaching a globally optimal solution.</li>
+<li>A greedy algorithm repeatedly makes greedy choices, turning the problem into a smaller subproblem at each step until it is solved.</li>
+<li>Greedy algorithms are easy to implement and efficient. Their time complexity is usually lower than that of dynamic programming.</li>
+<li>For some coin denominations, a greedy algorithm is guaranteed to find an optimal coin change solution. For others, it may produce a very poor solution.</li>
+<li>Problems suited to greedy algorithms have two properties: the greedy-choice property and optimal substructure. The greedy-choice property establishes that the strategy works.</li>
+<li>Proving the greedy-choice property can be difficult for complex problems. Disproving it is often easier, as coin change illustrates.</li>
+<li>Solving a greedy problem involves three main steps: analyze the problem, choose a greedy strategy, and prove correctness. Choosing the strategy is the central step, while proving correctness is often the hardest.</li>
+<li>The fractional knapsack problem extends 0-1 knapsack by allowing fractions of items, making a greedy solution possible. The strategy can be proved correct by contradiction.</li>
+<li>The maximum capacity problem can be solved by exhaustive search in <span class="arithmatex">\(O(n^2)\)</span> time. A greedy strategy that moves the shorter wall inward at each step reduces this to <span class="arithmatex">\(O(n)\)</span>.</li>
+<li>For the maximum product cutting problem, we derived two greedy strategies: continue splitting integers <span class="arithmatex">\(\geq 4\)</span>, and favor factors of <span class="arithmatex">\(3\)</span>. The implementation uses exponentiation, so its time complexity depends on the exponentiation method, usually <span class="arithmatex">\(O(1)\)</span> or <span class="arithmatex">\(O(\log n)\)</span>.</li>
 </ul>
 
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   <a
     href="../max_product_cutting_problem/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 15.4 Maximum Product Cutting Problem"
+    aria-label="Previous: 15.4   Maximum product cutting problem"
     rel="prev"
   >
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@@ -4683,7 +4755,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        15.4 Maximum Product Cutting Problem
+        15.4 &nbsp; Maximum product cutting problem
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   <a
     href="../exercises/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 15.6 Exercises"
+    aria-label="Next: 15.6   Exercises"
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+        15.6 &nbsp; Exercises
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@@ -4720,7 +4792,7 @@ aria-label="Footer"
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-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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+  Foreword
 
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-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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-  0.2 How to Use This Book
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-  0.3 Summary
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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-  Chapter 2. Complexity Analysis
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
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-  3.1 Classification of Data Structures
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-  3.2 Basic Data Types
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-  3.3 Number Encoding *
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  4.2 Linked List
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-  4.3 List
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-  4.4 Random-Access Memory and Cache *
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+  Chapter 6. &nbsp; Hash tables
 
 
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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         <li class="md-nav__item">
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+  <a href="#651-check-your-understanding" class="md-nav__link">
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-  <a href="#2-where-do-elements-go-after-a-hash-table-expands" class="md-nav__link">
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       <span class="md-typeset">
-        2. &nbsp; Where Do Elements Go After a Hash Table Expands?
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           <li class="md-nav__item">
-  <a href="#3-can-11-still-be-found-after-deleting-6" class="md-nav__link">
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       <span class="md-typeset">
-        3. &nbsp; Can 11 Still Be Found After Deleting 6?
+        3. &nbsp; Can we still find 11 after deleting 6?
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+    <nav class="md-nav" aria-label="6.5.2 Programming exercises">
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-  <a href="#1-compare-the-character-counts-of-two-strings" class="md-nav__link">
+  <a href="#1-comparing-the-characters-in-two-strings" class="md-nav__link">
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+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
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-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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+  7.3 &nbsp; Array representation of a binary tree
 
 
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+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
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+  8.2 &nbsp; Building a heap
 
 
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+  8.3 &nbsp; The top-k problem
 
 
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+  8.5 &nbsp; Exercises
 
 
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+  Chapter 9. &nbsp; Graphs
 
 
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@@ -2547,7 +2547,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2574,7 +2574,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2601,7 +2601,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2628,7 +2628,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2655,7 +2655,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2723,7 +2723,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2744,7 +2744,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2765,7 +2765,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2792,7 +2792,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2819,7 +2819,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2846,7 +2846,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2873,7 +2873,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2900,7 +2900,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2927,7 +2927,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -3005,7 +3005,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3026,7 +3026,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3047,7 +3047,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3074,7 +3074,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3101,7 +3101,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3128,7 +3128,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3155,7 +3155,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3182,7 +3182,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3209,7 +3209,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3236,7 +3236,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3263,7 +3263,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3290,7 +3290,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3317,7 +3317,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3344,7 +3344,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3410,7 +3410,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3431,7 +3431,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3452,7 +3452,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3479,7 +3479,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3506,7 +3506,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3533,7 +3533,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3560,7 +3560,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3587,7 +3587,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3653,7 +3653,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3674,7 +3674,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3695,7 +3695,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3722,7 +3722,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3749,7 +3749,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3776,7 +3776,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3803,7 +3803,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3830,7 +3830,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3900,7 +3900,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3921,7 +3921,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3942,7 +3942,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3969,7 +3969,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3996,7 +3996,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4023,7 +4023,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4050,7 +4050,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4077,7 +4077,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4104,7 +4104,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4131,7 +4131,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4197,7 +4197,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4218,7 +4218,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4239,7 +4239,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4266,7 +4266,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4293,7 +4293,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4320,7 +4320,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4347,7 +4347,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4374,7 +4374,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4434,7 +4434,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4455,7 +4455,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4476,7 +4476,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4503,7 +4503,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4530,7 +4530,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4614,6 +4614,78 @@
 
 
 
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@@ -4655,22 +4727,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#651-concept-review" class="md-nav__link">
+  <a href="#651-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.5.1 &nbsp; Concept Review
+        6.5.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="6.5.1 Concept Review">
+    <nav class="md-nav" aria-label="6.5.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-searching-after-a-hash-collision" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Searching After a Hash Collision
+        1. &nbsp; Searching after a hash collision
       </span>
     </span>
   </a>
@@ -4678,10 +4750,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-where-do-elements-go-after-a-hash-table-expands" class="md-nav__link">
+  <a href="#2-where-do-elements-go-when-a-hash-table-grows" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Where Do Elements Go After a Hash Table Expands?
+        2. &nbsp; Where do elements go when a hash table grows?
       </span>
     </span>
   </a>
@@ -4689,10 +4761,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-can-11-still-be-found-after-deleting-6" class="md-nav__link">
+  <a href="#3-can-we-still-find-11-after-deleting-6" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Can 11 Still Be Found After Deleting 6?
+        3. &nbsp; Can we still find 11 after deleting 6?
       </span>
     </span>
   </a>
@@ -4708,19 +4780,19 @@
   <a href="#652-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.5.2 &nbsp; Programming Exercises
+        6.5.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="6.5.2 Programming Exercises">
+    <nav class="md-nav" aria-label="6.5.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-compare-the-character-counts-of-two-strings" class="md-nav__link">
+  <a href="#1-comparing-the-characters-in-two-strings" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Compare the Character Counts of Two Strings
+        1. &nbsp; Comparing the characters in two strings
       </span>
     </span>
   </a>
@@ -4770,22 +4842,22 @@
 
 <!-- Page content -->
 <h1 id="65-exercises">6.5 &nbsp; Exercises<a class="headerlink" href="#65-exercises" title="Permanent link">&para;</a></h1>
-<h2 id="651-concept-review">6.5.1 &nbsp; Concept Review<a class="headerlink" href="#651-concept-review" title="Permanent link">&para;</a></h2>
-<h3 id="1-searching-after-a-hash-collision">1. &nbsp; Searching After a Hash Collision<a class="headerlink" href="#1-searching-after-a-hash-collision" title="Permanent link">&para;</a></h3>
-<p>A hash table has 5 buckets and uses the hash function <span class="arithmatex">\(h(x)=x \bmod 5\)</span>. When a collision occurs, elements are placed one after another in a list within that bucket.
-Insert <code>[1, 6, 11, 7]</code> in order:</p>
+<h2 id="651-check-your-understanding">6.5.1 &nbsp; Check your understanding<a class="headerlink" href="#651-check-your-understanding" title="Permanent link">&para;</a></h2>
+<h3 id="1-searching-after-a-hash-collision">1. &nbsp; Searching after a hash collision<a class="headerlink" href="#1-searching-after-a-hash-collision" title="Permanent link">&para;</a></h3>
+<p>A hash table has five buckets and uses the hash function <span class="arithmatex">\(h(x)=x \bmod 5\)</span>. When elements collide, they are added in order to the list in that bucket.
+Insert <code>[1, 6, 11, 7]</code> in sequence:</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Write the contents of buckets 0–4.</li>
-<li>When searching for 6, which bucket is checked first, and which elements are examined in order?</li>
-<li>Based on the bucket contents from Question 1, do later insertions overwrite earlier ones? Explain using this collision-resolution method.</li>
+<li>Write out the contents of buckets 0 through 4.</li>
+<li>When searching for 6, which bucket is visited first, and which elements are checked in order?</li>
+<li>Based on the bucket contents from question 1, did later elements overwrite earlier ones? Explain using this collision-handling method.</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>Because <span class="arithmatex">\(1\bmod5=6\bmod5=11\bmod5=1\)</span>, while <span class="arithmatex">\(7\bmod5=2\)</span>, the buckets are:</p>
+<p>Since <span class="arithmatex">\(1\bmod5=6\bmod5=11\bmod5=1\)</span> and <span class="arithmatex">\(7\bmod5=2\)</span>, the buckets contain:</p>
 <div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a>0: []
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>1: [1, 6, 11]
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>2: [7]
@@ -4794,27 +4866,27 @@ Insert <code>[1, 6, 11, 7]</code> in order:</p>
 </code></pre></div>
 </li>
 <li>
-<p>A search for 6 first goes to bucket 1, then compares 1 and 6 in order. The target is found on the second comparison.</p>
+<p>Searching for 6 first visits bucket 1 and compares against 1 and then 6, finding the target on the second comparison.</p>
 </li>
 <li>
-<p>Equal hash values mean only that the elements go into the same bucket, not that the elements are equal. Separate chaining keeps all colliding elements in the bucket
-    and compares them one at a time during a search, so 1, 6, and 11 do not overwrite one another.</p>
+<p>Equal hash values mean only that the elements land in the same bucket, not that the elements themselves are equal. Separate chaining keeps all colliding elements in the bucket
+    and compares them one by one during a search, so 1, 6, and 11 do not overwrite one another.</p>
 </li>
 </ol>
 </details>
-<h3 id="2-where-do-elements-go-after-a-hash-table-expands">2. &nbsp; Where Do Elements Go After a Hash Table Expands?<a class="headerlink" href="#2-where-do-elements-go-after-a-hash-table-expands" title="Permanent link">&para;</a></h3>
-<p>A hash table using separate chaining originally has 5 buckets and the hash function <span class="arithmatex">\(h(x)=x\bmod5\)</span>.
+<h3 id="2-where-do-elements-go-when-a-hash-table-grows">2. &nbsp; Where do elements go when a hash table grows?<a class="headerlink" href="#2-where-do-elements-go-when-a-hash-table-grows" title="Permanent link">&para;</a></h3>
+<p>A hash table using separate chaining initially has five buckets and the hash function <span class="arithmatex">\(h(x)=x\bmod5\)</span>.
 The keys <code>[1, 6, 11]</code> are all in bucket 1.</p>
-<p>The table is now expanded to 7 buckets, and the hash function becomes <span class="arithmatex">\(h(x)=x\bmod7\)</span>:</p>
+<p>The table is now expanded to seven buckets, and the hash function becomes <span class="arithmatex">\(h(x)=x\bmod7\)</span>:</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Calculate the new bucket number for 1, 6, and 11.</li>
+<li>Calculate the new bucket number for each of 1, 6, and 11.</li>
 <li>Which buckets contain elements after the expansion?</li>
-<li>During the expansion, can the list from the old bucket 1 simply be copied into the new bucket 1? Explain using the results from Questions 1 and 2.</li>
+<li>Can the list in the old bucket 1 simply be copied unchanged into the new bucket 1? Explain using your results from questions 1 and 2.</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
 <p>The new bucket numbers are:</p>
@@ -4825,56 +4897,56 @@ The keys <code>[1, 6, 11]</code> are all in bucket 1.</p>
 </ul>
 </li>
 <li>
-<p>Bucket 1 stores 1, bucket 4 stores 11, and bucket 6 stores 6. The three keys are no longer crowded into the same bucket.</p>
+<p>Bucket 1 holds 1, bucket 4 holds 11, and bucket 6 holds 6. The three keys no longer share one bucket.</p>
 </li>
 <li>
-<p>The list cannot be copied as is. A bucket number is calculated by taking the key modulo the number of buckets. When the bucket count changes from 5 to 7, a key's bucket number may change,
-    so the position of every key must be recalculated. If the old bucket 1 were copied unchanged, later searches using the new formula would go to bucket 6 for 6 and bucket 4 for 11
-    and would fail to find them.</p>
+<p>The list cannot simply be copied unchanged. A bucket number is the remainder of the key divided by the number of buckets. When that number changes from 5 to 7, a key's bucket may change,
+    so every key's position must be recalculated. If we copied the old bucket 1 unchanged, subsequent searches for 6 and 11 using the new formula
+    would visit buckets 6 and 4, respectively, and fail to find them.</p>
 </li>
 </ol>
 </details>
-<h3 id="3-can-11-still-be-found-after-deleting-6">3. &nbsp; Can 11 Still Be Found After Deleting 6?<a class="headerlink" href="#3-can-11-still-be-found-after-deleting-6" title="Permanent link">&para;</a></h3>
-<p>A hash table has 5 positions with indices <code>0–4</code> and uses the hash function <span class="arithmatex">\(h(x)=x\bmod5\)</span>.
-When a collision occurs, it searches to the right from the index produced by the hash function for the first empty position.</p>
+<h3 id="3-can-we-still-find-11-after-deleting-6">3. &nbsp; Can we still find 11 after deleting 6?<a class="headerlink" href="#3-can-we-still-find-11-after-deleting-6" title="Permanent link">&para;</a></h3>
+<p>A hash table has five positions with indices <code>0-4</code> and uses the hash function <span class="arithmatex">\(h(x)=x\bmod5\)</span>.
+When a collision occurs, it starts at the computed index and searches to the right for the first empty position.</p>
 <p>Insert <code>[1, 6, 11]</code> in order:</p>
 <!-- numbered-subquestions -->
 
 <ol>
 <li>At which index is each number ultimately stored?</li>
-<li>When searching for 11, which indices are examined in order?</li>
-<li>Suppose deleting 6 changes its position directly to an "unused empty position," and a search stops whenever it reaches an empty position.
-    What happens when searching for 11 afterward? Is this search result correct? If there is a problem, how can it be avoided?</li>
+<li>Which indices are checked, in order, when searching for 11?</li>
+<li>Suppose deleting 6 marks its position as an empty slot that has never been used, and a search stops on reaching an empty slot.
+    What happens when we search for 11 again? Is the result correct? If not, how can we avoid the problem?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
 <p>The number 1 is stored at index 1. The number 6 also maps to index 1, so after the collision it is stored at index 2.
-    The number 11 also starts at index 1, skips the occupied indices 1 and 2, and is ultimately stored at index 3.</p>
+    The number 11 likewise starts at index 1, skips occupied indices 1 and 2, and is finally stored at index 3.</p>
 </li>
 <li>
-<p>A search for 11 examines indices <code>1, 2, 3</code> in order and finds it at index 3.</p>
+<p>Searching for 11 checks indices <code>1, 2, 3</code> in order and finds it at index 3.</p>
 </li>
 <li>
-<p>If index 2 is changed to mean "never used," a search for 11 checks index 1 and then stops at index 2,
-    incorrectly concluding that 11 is absent. Deletion should leave a "deleted" marker.
-    A search that reaches this marker continues to the next index (wrapping from index 4 to index 0), while a later insertion may still reuse the position.</p>
+<p>If index 2 is marked as an empty slot that has never been used, a search for 11 checks index 1 and then stops at index 2,
+    incorrectly concluding that 11 is absent. Deletion should leave a deleted marker instead:
+    a search continues to the next index when it encounters this marker, wrapping from index 4 to index 0. Later insertions can still reuse the position.</p>
 </li>
 </ol>
 </details>
-<h2 id="652-programming-exercises">6.5.2 &nbsp; Programming Exercises<a class="headerlink" href="#652-programming-exercises" title="Permanent link">&para;</a></h2>
-<h3 id="1-compare-the-character-counts-of-two-strings">1. &nbsp; Compare the Character Counts of Two Strings<a class="headerlink" href="#1-compare-the-character-counts-of-two-strings" title="Permanent link">&para;</a></h3>
+<h2 id="652-programming-exercises">6.5.2 &nbsp; Programming exercises<a class="headerlink" href="#652-programming-exercises" title="Permanent link">&para;</a></h2>
+<h3 id="1-comparing-the-characters-in-two-strings">1. &nbsp; Comparing the characters in two strings<a class="headerlink" href="#1-comparing-the-characters-in-two-strings" title="Permanent link">&para;</a></h3>
 <p>Given two strings <code>s</code> and <code>t</code> containing only lowercase English letters,
-you may rearrange the characters in <code>s</code> in any order, but you may not add, remove, or replace characters.</p>
-<p>Determine whether the rearranged string can form <code>t</code>. Return <code>true</code> if it can and <code>false</code> otherwise.
-Use a hash table to record how many times each letter occurs. Do not sort the characters in the strings.</p>
+you may rearrange the characters in <code>s</code> freely, but may not add, delete, or replace any characters.</p>
+<p>Determine whether the rearrangement can produce <code>t</code>. Return <code>true</code> if it can, or <code>false</code> otherwise.
+Use a hash table to record the frequency of each letter; do not sort the characters in either string.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>If the two strings have different lengths, they cannot contain each character the same number of times</li>
-<li>Use a hash table to record the count of each letter; increment the corresponding count while scanning s</li>
-<li>Decrement the corresponding count while scanning t; the strings contain each character the same number of times only if every count is 0 at the end</li>
+<li>Strings of different lengths cannot contain the same characters with the same frequencies.</li>
+<li>Use a hash table to record each letter's count; increment the appropriate count by 1 while scanning s.</li>
+<li>Decrement the appropriate count by 1 while scanning t. The character frequencies match only if all counts end at 0.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/valid-anagram/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
@@ -4900,7 +4972,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 6.4 Summary"
+    aria-label="Previous: 6.4   Summary"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4912,7 +4984,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        6.4 Summary
+        6.4 &nbsp; Summary
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@@ -4924,7 +4996,7 @@ aria-label="Footer"
   <a
     href="../../chapter_tree/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: Chapter 7. &amp;nbsp; Tree"
+    aria-label="Next: Chapter 7.   Trees"
     rel="next"
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@@ -4932,7 +5004,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        Chapter 7. &nbsp; Tree
+        Chapter 7. &nbsp; Trees
       </div>
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     <div class="md-footer__button md-icon">
@@ -4949,7 +5021,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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diff --git a/en/chapter_hashing/hash_algorithm/index.html b/en/chapter_hashing/hash_algorithm/index.html
index 9999913f9..717b6acdb 100644
--- a/en/chapter_hashing/hash_algorithm/index.html
+++ b/en/chapter_hashing/hash_algorithm/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>6.3 Hash Algorithm - Hello Algo</title>
+        <title>6.3 Hash algorithms - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
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+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
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-        <a href="#63-hash-algorithm" class="md-skip">
+        <a href="#63-hash-algorithms" class="md-skip">
           Skip to content
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@@ -119,7 +119,7 @@
         <path
             d="M480 32c0-12.9-7.8-24.6-19.8-29.6s-25.7-2.2-34.9 6.9L381.7 53c-48 48-113.1 75-181 75H192 160 64c-35.3 0-64 28.7-64 64v96c0 35.3 28.7 64 64 64l0 128c0 17.7 14.3 32 32 32h64c17.7 0 32-14.3 32-32V352l8.7 0c67.9 0 133 27 181 75l43.6 43.6c9.2 9.2 22.9 11.9 34.9 6.9s19.8-16.6 19.8-29.6V300.4c18.6-8.8 32-32.5 32-60.4s-13.4-51.6-32-60.4V32zm-64 76.7V240 371.3C357.2 317.8 280.5 288 200.7 288H192V192h8.7c79.8 0 156.5-29.8 215.3-83.3z" />
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
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@@ -154,7 +154,7 @@
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           <span class="md-ellipsis">
 
-              6.3 &nbsp; Hash Algorithm
+              6.3 &nbsp; Hash algorithms
 
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1680,7 +1680,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1701,7 +1701,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1722,7 +1722,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1749,7 +1749,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1785,7 +1785,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1802,7 +1802,7 @@
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1828,10 +1828,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#631-goals-of-hash-algorithms" class="md-nav__link">
+  <a href="#631-goals-of-a-hash-algorithm" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.3.1 &nbsp; Goals of Hash Algorithms
+        6.3.1 &nbsp; Goals of a hash algorithm
       </span>
     </span>
   </a>
@@ -1839,10 +1839,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#632-design-of-hash-algorithms" class="md-nav__link">
+  <a href="#632-designing-a-hash-algorithm" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.3.2 &nbsp; Design of Hash Algorithms
+        6.3.2 &nbsp; Designing a hash algorithm
       </span>
     </span>
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@@ -1853,7 +1853,7 @@
   <a href="#633-common-hash-algorithms" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.3.3 &nbsp; Common Hash Algorithms
+        6.3.3 &nbsp; Common hash algorithms
       </span>
     </span>
   </a>
@@ -1861,10 +1861,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#634-hash-values-in-data-structures" class="md-nav__link">
+  <a href="#634-hash-values-of-data-structures" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.3.4 &nbsp; Hash Values in Data Structures
+        6.3.4 &nbsp; Hash values of data structures
       </span>
     </span>
   </a>
@@ -1894,7 +1894,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
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@@ -2031,7 +2031,7 @@
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-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2139,7 +2139,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
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-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4580,6 +4580,78 @@
 
 
 
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@@ -4621,10 +4693,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#631-goals-of-hash-algorithms" class="md-nav__link">
+  <a href="#631-goals-of-a-hash-algorithm" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.3.1 &nbsp; Goals of Hash Algorithms
+        6.3.1 &nbsp; Goals of a hash algorithm
       </span>
     </span>
   </a>
@@ -4632,10 +4704,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#632-design-of-hash-algorithms" class="md-nav__link">
+  <a href="#632-designing-a-hash-algorithm" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.3.2 &nbsp; Design of Hash Algorithms
+        6.3.2 &nbsp; Designing a hash algorithm
       </span>
     </span>
   </a>
@@ -4646,7 +4718,7 @@
   <a href="#633-common-hash-algorithms" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.3.3 &nbsp; Common Hash Algorithms
+        6.3.3 &nbsp; Common hash algorithms
       </span>
     </span>
   </a>
@@ -4654,10 +4726,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#634-hash-values-in-data-structures" class="md-nav__link">
+  <a href="#634-hash-values-of-data-structures" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.3.4 &nbsp; Hash Values in Data Structures
+        6.3.4 &nbsp; Hash values of data structures
       </span>
     </span>
   </a>
@@ -4701,43 +4773,43 @@
 
 
 <!-- Page content -->
-<h1 id="63-hash-algorithm">6.3 &nbsp; Hash Algorithm<a class="headerlink" href="#63-hash-algorithm" title="Permanent link">&para;</a></h1>
-<p>The previous two sections introduced the working principle of hash tables and the methods to handle hash collisions. However, both open addressing and separate chaining <strong>can only ensure that the hash table functions normally when hash collisions occur, but cannot reduce the frequency of hash collisions</strong>.</p>
-<p>If hash collisions occur too frequently, the performance of the hash table will deteriorate drastically. As shown in Figure 6-8, for a separate chaining hash table, in the ideal case, the key-value pairs are evenly distributed across the buckets, achieving optimal query efficiency; in the worst case, all key-value pairs are stored in the same bucket, degrading the time complexity to <span class="arithmatex">\(O(n)\)</span>.</p>
-<p><img alt="Ideal and worst cases of hash collisions" class="animation-figure" src="../hash_algorithm.assets/hash_collision_best_worst_condition.png" /></p>
-<p align="center"> Figure 6-8 &nbsp; Ideal and worst cases of hash collisions </p>
+<h1 id="63-hash-algorithms">6.3 &nbsp; Hash algorithms<a class="headerlink" href="#63-hash-algorithms" title="Permanent link">&para;</a></h1>
+<p>The previous two sections explained how hash tables work and how collisions are resolved. Whether we use open addressing or separate chaining, <strong>these techniques keep the table working when collisions occur, but do not reduce how often collisions happen</strong>.</p>
+<p>Frequent collisions can severely degrade performance. In the separate-chaining table shown below, the ideal case distributes pairs evenly among buckets for the best lookup efficiency. In the worst case, all pairs occupy the same bucket, degrading the time complexity to <span class="arithmatex">\(O(n)\)</span>.</p>
+<p><img alt="Best and worst cases for hash collisions" class="animation-figure" src="../hash_algorithm.assets/hash_collision_best_worst_condition.png" /></p>
+<p align="center"> Figure 6-8 &nbsp; Best and worst cases for hash collisions </p>
 
-<p><strong>The distribution of key-value pairs is determined by the hash function</strong>. Recall the steps of the hash function: first compute the hash value, then take it modulo the array length:</p>
+<p><strong>The hash function determines how key-value pairs are distributed</strong>. Recall its two steps: calculate a hash value, then take it modulo the array length:</p>
 <div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span>hash<span class="o">(</span>key<span class="o">)</span><span class="w"> </span>%<span class="w"> </span>capacity
 </code></pre></div>
-<p>Observing the above formula, when the hash table capacity <code>capacity</code> is fixed, <strong>the hash algorithm <code>hash()</code> determines the output value</strong>, thereby determining the distribution of key-value pairs in the hash table.</p>
-<p>This means that, to reduce the probability of hash collisions, we should focus on the design of the hash algorithm <code>hash()</code>.</p>
-<h2 id="631-goals-of-hash-algorithms">6.3.1 &nbsp; Goals of Hash Algorithms<a class="headerlink" href="#631-goals-of-hash-algorithms" title="Permanent link">&para;</a></h2>
-<p>To build a hash table that is both fast and robust, a hash algorithm should have the following properties:</p>
+<p>For a fixed <code>capacity</code>, <strong>the hash algorithm <code>hash()</code> determines the output</strong>, and therefore the distribution of pairs in the table.</p>
+<p>To lower the likelihood of collisions, we should therefore focus on designing <code>hash()</code>.</p>
+<h2 id="631-goals-of-a-hash-algorithm">6.3.1 &nbsp; Goals of a hash algorithm<a class="headerlink" href="#631-goals-of-a-hash-algorithm" title="Permanent link">&para;</a></h2>
+<p>To build a hash table that is both fast and reliable, its hash algorithm should have these properties.</p>
 <ul>
-<li><strong>Determinism</strong>: For the same input, the hash algorithm should always produce the same output. Only then can the hash table be reliable.</li>
-<li><strong>High efficiency</strong>: The process of computing the hash value should be fast enough. The smaller the computational overhead, the more practical the hash table.</li>
-<li><strong>Uniform distribution</strong>: The hash algorithm should ensure that key-value pairs are evenly distributed in the hash table. The more uniform the distribution, the lower the probability of hash collisions.</li>
+<li><strong>Determinism</strong>: The same input must always produce the same output, ensuring reliable lookup.</li>
+<li><strong>Efficiency</strong>: Calculating a hash should be fast. Lower computational overhead makes the table more practical.</li>
+<li><strong>Uniform distribution</strong>: The algorithm should distribute pairs evenly across the table. More uniform distribution means fewer collisions.</li>
 </ul>
-<p>In fact, hash algorithms are not only used to implement hash tables but are also widely applied in other fields.</p>
+<p>Hash algorithms also have many applications beyond hash tables.</p>
 <ul>
-<li><strong>Password storage</strong>: To protect the security of user passwords, systems usually do not store the plaintext passwords but rather the hash values of the passwords. When a user enters a password, the system calculates the hash value of the input and compares it with the stored hash value. If they match, the password is considered correct.</li>
-<li><strong>Data integrity check</strong>: The data sender can calculate the hash value of the data and send it along; the receiver can recalculate the hash value of the received data and compare it with the received hash value. If they match, the data is considered intact.</li>
+<li><strong>Password storage</strong>: To protect passwords, systems usually store password hashes rather than plaintext passwords. When a user enters a password, the system hashes it and compares the result with the stored hash. A match means the password is accepted.</li>
+<li><strong>Data-integrity checking</strong>: A sender can calculate a hash of the data and send it alongside the data. The receiver recomputes the hash and compares it with the received hash. A match indicates that the data is intact.</li>
 </ul>
-<p>For cryptographic applications, hash algorithms need stronger security properties to prevent reverse engineering, such as inferring the original password from a hash value.</p>
+<p>Cryptographic applications require stronger security properties to prevent reverse engineering, such as recovering an original password from its hash.</p>
 <ul>
-<li><strong>Unidirectionality</strong>: It should be impossible to deduce any information about the input data from the hash value.</li>
-<li><strong>Collision resistance</strong>: It should be extremely difficult to find two different inputs that produce the same hash value.</li>
-<li><strong>Avalanche effect</strong>: Minor changes in the input should lead to significant and unpredictable changes in the output.</li>
+<li><strong>One-wayness</strong>: The hash should not reveal any information about the input data.</li>
+<li><strong>Collision resistance</strong>: Finding two different inputs with the same hash should be extremely difficult.</li>
+<li><strong>Avalanche effect</strong>: A small input change should produce a large, unpredictable output change.</li>
 </ul>
-<p>Note that <strong>"uniform distribution" and "collision resistance" are two independent concepts</strong>. Satisfying uniform distribution does not necessarily mean collision resistance. For example, under random input <code>key</code>, the hash function <code>key % 100</code> can produce a uniformly distributed output. However, this hash algorithm is too simple, and all <code>key</code> with the same last two digits will have the same output, making it easy to deduce a usable <code>key</code> from the hash value, thereby cracking the password.</p>
-<h2 id="632-design-of-hash-algorithms">6.3.2 &nbsp; Design of Hash Algorithms<a class="headerlink" href="#632-design-of-hash-algorithms" title="Permanent link">&para;</a></h2>
-<p>The design of hash algorithms is a complex issue that requires consideration of many factors. However, for some less demanding scenarios, we can also design some simple hash algorithms.</p>
+<p><strong>Uniform distribution and collision resistance are independent concepts</strong>. Uniform output does not guarantee collision resistance. For example, with random keys, <code>key % 100</code> can produce uniform outputs. But this algorithm is too simple: all keys with the same last two digits give the same output. It is therefore easy to infer a usable key from a hash and crack a password.</p>
+<h2 id="632-designing-a-hash-algorithm">6.3.2 &nbsp; Designing a hash algorithm<a class="headerlink" href="#632-designing-a-hash-algorithm" title="Permanent link">&para;</a></h2>
+<p>Designing a hash algorithm is complex and involves many factors. For less demanding uses, however, we can devise simple algorithms.</p>
 <ul>
-<li><strong>Additive hash</strong>: Add up the ASCII codes of each character in the input and use the total sum as the hash value.</li>
-<li><strong>Multiplicative hash</strong>: Leverage the low correlation introduced by multiplication: multiply by a constant at each step and accumulate the ASCII codes of the characters into the hash value.</li>
-<li><strong>XOR hash</strong>: Accumulate the hash value by XORing each element of the input data.</li>
-<li><strong>Rotating hash</strong>: Accumulate the ASCII code of each character into a hash value, performing a rotation operation on the hash value before each accumulation.</li>
+<li><strong>Additive hashing</strong>: Add the ASCII codes of all input characters and use the sum as the hash.</li>
+<li><strong>Multiplicative hashing</strong>: Use multiplication to mix values, multiplying by a constant each round and accumulating character ASCII codes into the hash.</li>
+<li><strong>XOR hashing</strong>: Accumulate each input element into a hash using exclusive OR.</li>
+<li><strong>Rotational hashing</strong>: Accumulate each character's ASCII code into a hash, rotating the hash before each accumulation.</li>
 </ul>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
@@ -4767,7 +4839,7 @@
 <a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a>    <span class="k">return</span> <span class="nb">hash</span> <span class="o">%</span> <span class="n">modulus</span>
 <a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a>
 <a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="k">def</span><span class="w"> </span><span class="nf">rot_hash</span><span class="p">(</span><span class="n">key</span><span class="p">:</span> <span class="nb">str</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="w">    </span><span class="sd">"""Rotational hash"""</span>
+<a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="w">    </span><span class="sd">"""Rotating hash"""</span>
 <a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a>    <span class="nb">hash</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a>    <span class="n">modulus</span> <span class="o">=</span> <span class="mi">1000000007</span>
 <a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a>    <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">key</span><span class="p">:</span>
@@ -4806,7 +4878,7 @@
 <a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">&amp;</span><span class="w"> </span><span class="n">MODULUS</span><span class="p">;</span>
 <a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="p">}</span>
 <a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a>
-<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="cm">/* Rotational hash */</span>
+<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="cm">/* Rotating hash */</span>
 <a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="kt">int</span><span class="w"> </span><span class="nf">rotHash</span><span class="p">(</span><span class="n">string</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a><span class="w">    </span><span class="kt">long</span><span class="w"> </span><span class="kt">long</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a><span class="w">    </span><span class="k">const</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1000000007</span><span class="p">;</span>
@@ -4848,7 +4920,7 @@
 <a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">&amp;</span><span class="w"> </span><span class="n">MODULUS</span><span class="p">;</span>
 <a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="p">}</span>
 <a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a>
-<a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a><span class="cm">/* Rotational hash */</span>
+<a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a><span class="cm">/* Rotating hash */</span>
 <a id="__codelineno-3-32" name="__codelineno-3-32" href="#__codelineno-3-32"></a><span class="kt">int</span><span class="w"> </span><span class="nf">rotHash</span><span class="p">(</span><span class="n">String</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-33" name="__codelineno-3-33" href="#__codelineno-3-33"></a><span class="w">    </span><span class="kt">long</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-3-34" name="__codelineno-3-34" href="#__codelineno-3-34"></a><span class="w">    </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1000000007</span><span class="p">;</span>
@@ -4890,7 +4962,7 @@
 <a id="__codelineno-4-28" name="__codelineno-4-28" href="#__codelineno-4-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">&amp;</span><span class="w"> </span><span class="n">MODULUS</span><span class="p">;</span>
 <a id="__codelineno-4-29" name="__codelineno-4-29" href="#__codelineno-4-29"></a><span class="p">}</span>
 <a id="__codelineno-4-30" name="__codelineno-4-30" href="#__codelineno-4-30"></a>
-<a id="__codelineno-4-31" name="__codelineno-4-31" href="#__codelineno-4-31"></a><span class="cm">/* Rotational hash */</span>
+<a id="__codelineno-4-31" name="__codelineno-4-31" href="#__codelineno-4-31"></a><span class="cm">/* Rotating hash */</span>
 <a id="__codelineno-4-32" name="__codelineno-4-32" href="#__codelineno-4-32"></a><span class="kt">int</span><span class="w"> </span><span class="nf">RotHash</span><span class="p">(</span><span class="kt">string</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-33" name="__codelineno-4-33" href="#__codelineno-4-33"></a><span class="w">    </span><span class="kt">long</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-4-34" name="__codelineno-4-34" href="#__codelineno-4-34"></a><span class="w">    </span><span class="k">const</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1000000007</span><span class="p">;</span>
@@ -4938,7 +5010,7 @@
 <a id="__codelineno-5-34" name="__codelineno-5-34" href="#__codelineno-5-34"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">hash</span><span class="w"> </span><span class="o">&amp;</span><span class="w"> </span><span class="nx">modulus</span>
 <a id="__codelineno-5-35" name="__codelineno-5-35" href="#__codelineno-5-35"></a><span class="p">}</span>
 <a id="__codelineno-5-36" name="__codelineno-5-36" href="#__codelineno-5-36"></a>
-<a id="__codelineno-5-37" name="__codelineno-5-37" href="#__codelineno-5-37"></a><span class="cm">/* Rotational hash */</span>
+<a id="__codelineno-5-37" name="__codelineno-5-37" href="#__codelineno-5-37"></a><span class="cm">/* Rotating hash */</span>
 <a id="__codelineno-5-38" name="__codelineno-5-38" href="#__codelineno-5-38"></a><span class="kd">func</span><span class="w"> </span><span class="nx">rotHash</span><span class="p">(</span><span class="nx">key</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-39" name="__codelineno-5-39" href="#__codelineno-5-39"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nx">hash</span><span class="w"> </span><span class="kt">int64</span>
 <a id="__codelineno-5-40" name="__codelineno-5-40" href="#__codelineno-5-40"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nx">modulus</span><span class="w"> </span><span class="kt">int64</span>
@@ -4988,7 +5060,7 @@
 <a id="__codelineno-6-34" name="__codelineno-6-34" href="#__codelineno-6-34"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">&amp;</span><span class="w"> </span><span class="n">MODULUS</span>
 <a id="__codelineno-6-35" name="__codelineno-6-35" href="#__codelineno-6-35"></a><span class="p">}</span>
 <a id="__codelineno-6-36" name="__codelineno-6-36" href="#__codelineno-6-36"></a>
-<a id="__codelineno-6-37" name="__codelineno-6-37" href="#__codelineno-6-37"></a><span class="cm">/* Rotational hash */</span>
+<a id="__codelineno-6-37" name="__codelineno-6-37" href="#__codelineno-6-37"></a><span class="cm">/* Rotating hash */</span>
 <a id="__codelineno-6-38" name="__codelineno-6-38" href="#__codelineno-6-38"></a><span class="kd">func</span><span class="w"> </span><span class="nf">rotHash</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="nb">String</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-39" name="__codelineno-6-39" href="#__codelineno-6-39"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">hash</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-6-40" name="__codelineno-6-40" href="#__codelineno-6-40"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">MODULUS</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">1_000_000_007</span>
@@ -5032,7 +5104,7 @@
 <a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">hash</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="nx">MODULUS</span><span class="p">;</span>
 <a id="__codelineno-7-29" name="__codelineno-7-29" href="#__codelineno-7-29"></a><span class="p">}</span>
 <a id="__codelineno-7-30" name="__codelineno-7-30" href="#__codelineno-7-30"></a>
-<a id="__codelineno-7-31" name="__codelineno-7-31" href="#__codelineno-7-31"></a><span class="cm">/* Rotational hash */</span>
+<a id="__codelineno-7-31" name="__codelineno-7-31" href="#__codelineno-7-31"></a><span class="cm">/* Rotating hash */</span>
 <a id="__codelineno-7-32" name="__codelineno-7-32" href="#__codelineno-7-32"></a><span class="kd">function</span><span class="w"> </span><span class="nx">rotHash</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-33" name="__codelineno-7-33" href="#__codelineno-7-33"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-7-34" name="__codelineno-7-34" href="#__codelineno-7-34"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1000000007</span><span class="p">;</span>
@@ -5074,7 +5146,7 @@
 <a id="__codelineno-8-28" name="__codelineno-8-28" href="#__codelineno-8-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">hash</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="nx">MODULUS</span><span class="p">;</span>
 <a id="__codelineno-8-29" name="__codelineno-8-29" href="#__codelineno-8-29"></a><span class="p">}</span>
 <a id="__codelineno-8-30" name="__codelineno-8-30" href="#__codelineno-8-30"></a>
-<a id="__codelineno-8-31" name="__codelineno-8-31" href="#__codelineno-8-31"></a><span class="cm">/* Rotational hash */</span>
+<a id="__codelineno-8-31" name="__codelineno-8-31" href="#__codelineno-8-31"></a><span class="cm">/* Rotating hash */</span>
 <a id="__codelineno-8-32" name="__codelineno-8-32" href="#__codelineno-8-32"></a><span class="kd">function</span><span class="w"> </span><span class="nx">rotHash</span><span class="p">(</span><span class="nx">key</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-33" name="__codelineno-8-33" href="#__codelineno-8-33"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-8-34" name="__codelineno-8-34" href="#__codelineno-8-34"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1000000007</span><span class="p">;</span>
@@ -5116,7 +5188,7 @@
 <a id="__codelineno-9-28" name="__codelineno-9-28" href="#__codelineno-9-28"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">&amp;</span><span class="w"> </span><span class="n">MODULUS</span><span class="p">;</span>
 <a id="__codelineno-9-29" name="__codelineno-9-29" href="#__codelineno-9-29"></a><span class="p">}</span>
 <a id="__codelineno-9-30" name="__codelineno-9-30" href="#__codelineno-9-30"></a>
-<a id="__codelineno-9-31" name="__codelineno-9-31" href="#__codelineno-9-31"></a><span class="cm">/* Rotational hash */</span>
+<a id="__codelineno-9-31" name="__codelineno-9-31" href="#__codelineno-9-31"></a><span class="cm">/* Rotating hash */</span>
 <a id="__codelineno-9-32" name="__codelineno-9-32" href="#__codelineno-9-32"></a><span class="kt">int</span><span class="w"> </span><span class="n">rotHash</span><span class="p">(</span><span class="kt">String</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-33" name="__codelineno-9-33" href="#__codelineno-9-33"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-9-34" name="__codelineno-9-34" href="#__codelineno-9-34"></a><span class="w">  </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1000000007</span><span class="p">;</span>
@@ -5164,7 +5236,7 @@
 <a id="__codelineno-10-34" name="__codelineno-10-34" href="#__codelineno-10-34"></a><span class="w">    </span><span class="p">(</span><span class="n">hash</span><span class="w"> </span><span class="o">&amp;</span><span class="w"> </span><span class="n">MODULUS</span><span class="p">)</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span>
 <a id="__codelineno-10-35" name="__codelineno-10-35" href="#__codelineno-10-35"></a><span class="p">}</span>
 <a id="__codelineno-10-36" name="__codelineno-10-36" href="#__codelineno-10-36"></a>
-<a id="__codelineno-10-37" name="__codelineno-10-37" href="#__codelineno-10-37"></a><span class="cm">/* Rotational hash */</span>
+<a id="__codelineno-10-37" name="__codelineno-10-37" href="#__codelineno-10-37"></a><span class="cm">/* Rotating hash */</span>
 <a id="__codelineno-10-38" name="__codelineno-10-38" href="#__codelineno-10-38"></a><span class="k">fn</span><span class="w"> </span><span class="nf">rot_hash</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="kt">str</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-39" name="__codelineno-10-39" href="#__codelineno-10-39"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0_</span><span class="k">i64</span><span class="p">;</span>
 <a id="__codelineno-10-40" name="__codelineno-10-40" href="#__codelineno-10-40"></a><span class="w">    </span><span class="k">const</span><span class="w"> </span><span class="n">MODULUS</span><span class="p">:</span><span class="w"> </span><span class="kt">i64</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1000000007</span><span class="p">;</span>
@@ -5209,7 +5281,7 @@
 <a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">&amp;</span><span class="w"> </span><span class="n">MODULUS</span><span class="p">;</span>
 <a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a><span class="p">}</span>
 <a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a>
-<a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="cm">/* Rotational hash */</span>
+<a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="cm">/* Rotating hash */</span>
 <a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></a><span class="kt">int</span><span class="w"> </span><span class="nf">rotHash</span><span class="p">(</span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-34" name="__codelineno-11-34" href="#__codelineno-11-34"></a><span class="w">    </span><span class="kt">long</span><span class="w"> </span><span class="kt">long</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-11-35" name="__codelineno-11-35" href="#__codelineno-11-35"></a><span class="w">    </span><span class="k">const</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1000000007</span><span class="p">;</span>
@@ -5252,7 +5324,7 @@
 <a id="__codelineno-12-28" name="__codelineno-12-28" href="#__codelineno-12-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="n">and</span><span class="w"> </span><span class="n">MODULUS</span>
 <a id="__codelineno-12-29" name="__codelineno-12-29" href="#__codelineno-12-29"></a><span class="p">}</span>
 <a id="__codelineno-12-30" name="__codelineno-12-30" href="#__codelineno-12-30"></a>
-<a id="__codelineno-12-31" name="__codelineno-12-31" href="#__codelineno-12-31"></a><span class="cm">/* Rotational hash */</span>
+<a id="__codelineno-12-31" name="__codelineno-12-31" href="#__codelineno-12-31"></a><span class="cm">/* Rotating hash */</span>
 <a id="__codelineno-12-32" name="__codelineno-12-32" href="#__codelineno-12-32"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">rotHash</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">String</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-12-33" name="__codelineno-12-33" href="#__codelineno-12-33"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0L</span>
 <a id="__codelineno-12-34" name="__codelineno-12-34" href="#__codelineno-12-34"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">MODULUS</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1000000007</span>
@@ -5294,7 +5366,7 @@
 <a id="__codelineno-13-28" name="__codelineno-13-28" href="#__codelineno-13-28"></a><span class="w">  </span><span class="nb">hash</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">modulus</span>
 <a id="__codelineno-13-29" name="__codelineno-13-29" href="#__codelineno-13-29"></a><span class="k">end</span>
 <a id="__codelineno-13-30" name="__codelineno-13-30" href="#__codelineno-13-30"></a>
-<a id="__codelineno-13-31" name="__codelineno-13-31" href="#__codelineno-13-31"></a><span class="c1">### Rotational hash ###</span>
+<a id="__codelineno-13-31" name="__codelineno-13-31" href="#__codelineno-13-31"></a><span class="c1">### Rotating hash ###</span>
 <a id="__codelineno-13-32" name="__codelineno-13-32" href="#__codelineno-13-32"></a><span class="k">def</span><span class="w"> </span><span class="nf">rot_hash</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-13-33" name="__codelineno-13-33" href="#__codelineno-13-33"></a><span class="w">  </span><span class="nb">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-13-34" name="__codelineno-13-34" href="#__codelineno-13-34"></a><span class="w">  </span><span class="n">modulus</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1_000_000_007</span>
@@ -5307,9 +5379,14 @@
 </div>
 </div>
 </div>
-<p>We can observe that the final step of each hash algorithm is to take the result modulo the large prime <span class="arithmatex">\(1000000007\)</span>, ensuring that the hash value stays within a suitable range. This naturally raises a question: why emphasize using a prime modulus, and what are the drawbacks of using a composite modulus?</p>
-<p>In short: <strong>using a large prime as the modulus helps maximize the uniformity of hash values</strong>. Because a prime shares no common factors with other numbers, it can reduce periodic patterns introduced by the modulo operation and thus mitigate hash collisions.</p>
-<p>For example, suppose we choose the composite number <span class="arithmatex">\(9\)</span> as the modulus, which can be divided by <span class="arithmatex">\(3\)</span>, then all <code>key</code> divisible by <span class="arithmatex">\(3\)</span> will be mapped to hash values <span class="arithmatex">\(0\)</span>, <span class="arithmatex">\(3\)</span>, <span class="arithmatex">\(6\)</span>.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20add_hash%28key%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Additive%20hash%22%22%22%0A%20%20%20%20hash%20%3D%200%0A%20%20%20%20modulus%20%3D%201000000007%0A%20%20%20%20for%20c%20in%20key%3A%0A%20%20%20%20%20%20%20%20hash%20%2B%3D%20ord%28c%29%0A%20%20%20%20return%20hash%20%25%20modulus%0A%0A%0Adef%20mul_hash%28key%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Multiplicative%20hash%22%22%22%0A%20%20%20%20hash%20%3D%200%0A%20%20%20%20modulus%20%3D%201000000007%0A%20%20%20%20for%20c%20in%20key%3A%0A%20%20%20%20%20%20%20%20hash%20%3D%2031%20%2A%20hash%20%2B%20ord%28c%29%0A%20%20%20%20return%20hash%20%25%20modulus%0A%0A%0Adef%20xor_hash%28key%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22XOR%20hash%22%22%22%0A%20%20%20%20hash%20%3D%200%0A%20%20%20%20modulus%20%3D%201000000007%0A%20%20%20%20for%20c%20in%20key%3A%0A%20%20%20%20%20%20%20%20hash%20%5E%3D%20ord%28c%29%0A%20%20%20%20return%20hash%20%25%20modulus%0A%0A%0Adef%20rot_hash%28key%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Rotating%20hash%22%22%22%0A%20%20%20%20hash%20%3D%200%0A%20%20%20%20modulus%20%3D%201000000007%0A%20%20%20%20for%20c%20in%20key%3A%0A%20%20%20%20%20%20%20%20hash%20%3D%20%28hash%20%3C%3C%204%29%20%5E%20%28hash%20%3E%3E%2028%29%20%5E%20ord%28c%29%0A%20%20%20%20return%20hash%20%25%20modulus%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20key%20%3D%20%22Hello%20%E7%AE%97%E6%B3%95%22%0A%0A%20%20%20%20hash%20%3D%20add_hash%28key%29%0A%20%20%20%20print%28f%22Additive%20hash%20value%3A%20%7Bhash%7D%22%29%0A%0A%20%20%20%20hash%20%3D%20mul_hash%28key%29%0A%20%20%20%20print%28f%22Multiplicative%20hash%20value%3A%20%7Bhash%7D%22%29%0A%0A%20%20%20%20hash%20%3D%20xor_hash%28key%29%0A%20%20%20%20print%28f%22XOR%20hash%20value%3A%20%7Bhash%7D%22%29%0A%0A%20%20%20%20hash%20%3D%20rot_hash%28key%29%0A%20%20%20%20print%28f%22Rotating%20hash%20value%3A%20%7Bhash%7D%22%29%0A&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20add_hash%28key%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Additive%20hash%22%22%22%0A%20%20%20%20hash%20%3D%200%0A%20%20%20%20modulus%20%3D%201000000007%0A%20%20%20%20for%20c%20in%20key%3A%0A%20%20%20%20%20%20%20%20hash%20%2B%3D%20ord%28c%29%0A%20%20%20%20return%20hash%20%25%20modulus%0A%0A%0Adef%20mul_hash%28key%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Multiplicative%20hash%22%22%22%0A%20%20%20%20hash%20%3D%200%0A%20%20%20%20modulus%20%3D%201000000007%0A%20%20%20%20for%20c%20in%20key%3A%0A%20%20%20%20%20%20%20%20hash%20%3D%2031%20%2A%20hash%20%2B%20ord%28c%29%0A%20%20%20%20return%20hash%20%25%20modulus%0A%0A%0Adef%20xor_hash%28key%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22XOR%20hash%22%22%22%0A%20%20%20%20hash%20%3D%200%0A%20%20%20%20modulus%20%3D%201000000007%0A%20%20%20%20for%20c%20in%20key%3A%0A%20%20%20%20%20%20%20%20hash%20%5E%3D%20ord%28c%29%0A%20%20%20%20return%20hash%20%25%20modulus%0A%0A%0Adef%20rot_hash%28key%3A%20str%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Rotating%20hash%22%22%22%0A%20%20%20%20hash%20%3D%200%0A%20%20%20%20modulus%20%3D%201000000007%0A%20%20%20%20for%20c%20in%20key%3A%0A%20%20%20%20%20%20%20%20hash%20%3D%20%28hash%20%3C%3C%204%29%20%5E%20%28hash%20%3E%3E%2028%29%20%5E%20ord%28c%29%0A%20%20%20%20return%20hash%20%25%20modulus%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20key%20%3D%20%22Hello%20%E7%AE%97%E6%B3%95%22%0A%0A%20%20%20%20hash%20%3D%20add_hash%28key%29%0A%20%20%20%20print%28f%22Additive%20hash%20value%3A%20%7Bhash%7D%22%29%0A%0A%20%20%20%20hash%20%3D%20mul_hash%28key%29%0A%20%20%20%20print%28f%22Multiplicative%20hash%20value%3A%20%7Bhash%7D%22%29%0A%0A%20%20%20%20hash%20%3D%20xor_hash%28key%29%0A%20%20%20%20print%28f%22XOR%20hash%20value%3A%20%7Bhash%7D%22%29%0A%0A%20%20%20%20hash%20%3D%20rot_hash%28key%29%0A%20%20%20%20print%28f%22Rotating%20hash%20value%3A%20%7Bhash%7D%22%29%0A&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Each algorithm ends by taking the result modulo the large prime <span class="arithmatex">\(1000000007\)</span> to keep it within a suitable range. Why emphasize a prime modulus? What goes wrong with a composite modulus? This is an interesting question.</p>
+<p>The conclusion is that <strong>using a large prime modulus helps make hash values as uniformly distributed as possible</strong>. Because a prime shares no nontrivial factors with other numbers, it reduces periodic patterns introduced by the modulo operation and thus avoids collisions.</p>
+<p>For example, choose the composite modulus <span class="arithmatex">\(9\)</span>, which is divisible by <span class="arithmatex">\(3\)</span>. Every key divisible by <span class="arithmatex">\(3\)</span> then maps to one of just three hash values: <span class="arithmatex">\(0\)</span>, <span class="arithmatex">\(3\)</span>, or <span class="arithmatex">\(6\)</span>.</p>
 <div class="arithmatex">\[
 \begin{aligned}
 \text{modulus} &amp; = 9 \newline
@@ -5317,7 +5394,7 @@
 \text{hash} &amp; = \{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\dots \}
 \end{aligned}
 \]</div>
-<p>If the input <code>key</code> values happen to follow this kind of arithmetic progression, the hash values will cluster, worsening hash collisions. Now suppose we replace <code>modulus</code> with the prime number <span class="arithmatex">\(13\)</span>. Because <code>key</code> and <code>modulus</code> share no common factors, the output hash values become much more evenly distributed.</p>
+<p>If the input keys follow this arithmetic progression, the hash values cluster, increasing collisions. Now replace the modulus with the prime <span class="arithmatex">\(13\)</span>. With no common factor between the key pattern and the modulus, the output becomes noticeably more uniform.</p>
 <div class="arithmatex">\[
 \begin{aligned}
 \text{modulus} &amp; = 13 \newline
@@ -5325,16 +5402,16 @@
 \text{hash} &amp; = \{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \dots \}
 \end{aligned}
 \]</div>
-<p>It is worth noting that if the <code>key</code> is guaranteed to be randomly and uniformly distributed, then choosing a prime number or a composite number as the modulus can both produce uniformly distributed hash values. However, when the distribution of <code>key</code> has some periodicity, modulo a composite number is more likely to result in clustering.</p>
-<p>In summary, we usually choose a prime number as the modulus, and this prime number should be large enough to eliminate periodic patterns as much as possible, enhancing the robustness of the hash algorithm.</p>
-<h2 id="633-common-hash-algorithms">6.3.3 &nbsp; Common Hash Algorithms<a class="headerlink" href="#633-common-hash-algorithms" title="Permanent link">&para;</a></h2>
-<p>It is easy to see that the simple hash algorithms introduced above are fairly "fragile" and fall far short of the design goals of hash algorithms. For example, because addition and XOR are commutative, additive hash and XOR hash cannot distinguish strings with the same characters in a different order, which may worsen hash collisions and introduce security risks.</p>
-<p>In practice, we usually use some standard hash algorithms, such as MD5, SHA-1, SHA-2, and SHA-3. They can map input data of any length to a fixed-length hash value.</p>
-<p>Over the past century, hash algorithms have been in a continuous process of upgrading and optimization. Some researchers strive to improve the performance of hash algorithms, while others, including hackers, are dedicated to finding security issues in hash algorithms. Table 6-2 shows hash algorithms commonly used in practical applications.</p>
+<p>If the keys are already uniformly random, either a prime or a composite modulus can produce uniformly distributed hashes. When the keys have a periodic pattern, however, a composite modulus is more likely to cause clustering.</p>
+<p>We therefore usually choose a prime modulus, preferably a large one, to suppress periodic patterns as much as possible and make the algorithm more robust.</p>
+<h2 id="633-common-hash-algorithms">6.3.3 &nbsp; Common hash algorithms<a class="headerlink" href="#633-common-hash-algorithms" title="Permanent link">&para;</a></h2>
+<p>The simple algorithms above are fragile and fall well short of our design goals. Addition and XOR are commutative, for example, so additive and XOR hashing cannot distinguish strings containing the same characters in different orders. This can increase collisions and create security problems.</p>
+<p>In practice, standard algorithms such as MD5, SHA-1, SHA-2, and SHA-3 are commonly used. They map inputs of arbitrary length to fixed-length hash values.</p>
+<p>Over nearly a century, hash algorithms have continually evolved. Some researchers work to improve their performance, while others, along with hackers, search for security weaknesses. Table 6-2 lists commonly used algorithms.</p>
 <ul>
-<li>MD5 and SHA-1 have been successfully attacked multiple times and are thus abandoned in various security applications.</li>
-<li>SHA-2 series, especially SHA-256, is one of the most secure hash algorithms to date, with no successful attacks reported, hence commonly used in various security applications and protocols.</li>
-<li>SHA-3 has lower implementation costs and higher computational efficiency compared to SHA-2, but its current usage coverage is not as extensive as the SHA-2 series.</li>
+<li>MD5 and SHA-1 have been successfully attacked repeatedly and have been retired from security applications.</li>
+<li>SHA-256, part of the SHA-2 family, is among the most secure hash algorithms. No successful attack has been reported, so it is widely used in security applications and protocols.</li>
+<li>SHA-3 has lower implementation overhead and higher computational efficiency than SHA-2, but is not yet as widely used.</li>
 </ul>
 <p align="center"> Table 6-2 &nbsp; Common hash algorithms </p>
 
@@ -5351,218 +5428,218 @@
 </thead>
 <tbody>
 <tr>
-<td>Release Year</td>
+<td>Introduced</td>
 <td>1992</td>
 <td>1995</td>
 <td>2002</td>
 <td>2008</td>
 </tr>
 <tr>
-<td>Output Length</td>
+<td>Output length</td>
 <td>128 bit</td>
 <td>160 bit</td>
 <td>256/512 bit</td>
 <td>224/256/384/512 bit</td>
 </tr>
 <tr>
-<td>Hash Collisions</td>
-<td>Frequent</td>
-<td>Frequent</td>
+<td>Hash collisions</td>
+<td>More frequent</td>
+<td>More frequent</td>
 <td>Rare</td>
 <td>Rare</td>
 </tr>
 <tr>
-<td>Security Level</td>
-<td>Low, has been successfully attacked</td>
-<td>Low, has been successfully attacked</td>
+<td>Security</td>
+<td>Low; successfully attacked</td>
+<td>Low; successfully attacked</td>
 <td>High</td>
 <td>High</td>
 </tr>
 <tr>
 <td>Applications</td>
-<td>Abandoned, still used for data integrity checks</td>
-<td>Abandoned</td>
-<td>Cryptocurrency transaction verification, digital signatures, etc.</td>
-<td>Can be used to replace SHA-2</td>
+<td>Deprecated; still used for integrity checks</td>
+<td>Deprecated</td>
+<td>Cryptocurrency transaction verification, digital signatures, and more</td>
+<td>Can replace SHA-2</td>
 </tr>
 </tbody>
 </table>
 </div>
-<h2 id="634-hash-values-in-data-structures">6.3.4 &nbsp; Hash Values in Data Structures<a class="headerlink" href="#634-hash-values-in-data-structures" title="Permanent link">&para;</a></h2>
-<p>We know that hash table keys can be integers, floating-point numbers, strings, and other data types. Programming languages usually provide built-in hash algorithms for these types to compute bucket indices in a hash table. Taking Python as an example, we can call the <code>hash()</code> function to compute hash values for various data types.</p>
+<h2 id="634-hash-values-of-data-structures">6.3.4 &nbsp; Hash values of data structures<a class="headerlink" href="#634-hash-values-of-data-structures" title="Permanent link">&para;</a></h2>
+<p>Hash-table keys can be integers, floating-point numbers, strings, and other types. Programming languages usually provide built-in hashing for these types to calculate bucket indices. In Python, for example, <code>hash()</code> calculates hash values for various types.</p>
 <ul>
-<li>The hash values of integers and booleans are their own values.</li>
-<li>The calculation of hash values for floating-point numbers and strings is more complex, and interested readers are encouraged to study this on their own.</li>
-<li>The hash value of a tuple is obtained by hashing each of its elements and combining those results into a single hash value.</li>
-<li>An object's hash value is typically generated from its memory address. By overriding the object's hash method, it can instead be generated from the object's contents.</li>
+<li>Integers and Booleans have hash values equal to their own values.</li>
+<li>Hashing floating-point numbers and strings is more complex; interested readers can explore the details.</li>
+<li>A tuple's hash combines the hashes of its individual elements into one value.</li>
+<li>An object's hash is based on its memory address. Overriding its hash method allows hashing by content instead.</li>
 </ul>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>Be aware that the definition and methods of the built-in hash value calculation functions in different programming languages vary.</p>
+<p>Built-in hash functions differ in their definitions and implementations across programming languages.</p>
 </div>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">built_in_hash.py</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="n">num</span> <span class="o">=</span> <span class="mi">3</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="n">hash_num</span> <span class="o">=</span> <span class="nb">hash</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="c1"># Hash value of integer 3 is 3</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="c1"># The hash of the integer 3 is 3</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="n">bol</span> <span class="o">=</span> <span class="kc">True</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="n">hash_bol</span> <span class="o">=</span> <span class="nb">hash</span><span class="p">(</span><span class="n">bol</span><span class="p">)</span>
-<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="c1"># Hash value of boolean True is 1</span>
+<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="c1"># The hash of the Boolean True is 1</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="n">dec</span> <span class="o">=</span> <span class="mf">3.14159</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="n">hash_dec</span> <span class="o">=</span> <span class="nb">hash</span><span class="p">(</span><span class="n">dec</span><span class="p">)</span>
-<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="c1"># Hash value of decimal 3.14159 is 326484311674566659</span>
+<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="c1"># The hash of the floating-point number 3.14159 is 326484311674566659</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="nb">str</span> <span class="o">=</span> <span class="s2">"Hello 算法"</span>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="n">hash_str</span> <span class="o">=</span> <span class="nb">hash</span><span class="p">(</span><span class="nb">str</span><span class="p">)</span>
-<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="c1"># Hash value of string "Hello 算法" is 4617003410720528961</span>
+<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="c1"># The hash of the string “Hello 算法” is 4617003410720528961</span>
 <a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="n">tup</span> <span class="o">=</span> <span class="p">(</span><span class="mi">12836</span><span class="p">,</span> <span class="s2">"小哈"</span><span class="p">)</span>
 <a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="n">hash_tup</span> <span class="o">=</span> <span class="nb">hash</span><span class="p">(</span><span class="n">tup</span><span class="p">)</span>
-<a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="c1"># Hash value of tuple (12836, '小哈') is 1029005403108185979</span>
+<a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="c1"># The hash of the tuple (12836, '小哈') is 1029005403108185979</span>
 <a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a>
 <a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="n">obj</span> <span class="o">=</span> <span class="n">ListNode</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a><span class="n">hash_obj</span> <span class="o">=</span> <span class="nb">hash</span><span class="p">(</span><span class="n">obj</span><span class="p">)</span>
-<a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a><span class="c1"># Hash value of ListNode object at 0x1058fd810 is 274267521</span>
+<a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a><span class="c1"># The hash of the node object &lt;ListNode object at 0x1058fd810&gt; is 274267521</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">built_in_hash.cpp</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">size_t</span><span class="w"> </span><span class="n">hashNum</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hash</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">()(</span><span class="n">num</span><span class="p">);</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="c1">// Hash value of integer 3 is 3</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="c1">// The hash of the integer 3 is 3</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="kt">bool</span><span class="w"> </span><span class="n">bol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="kt">size_t</span><span class="w"> </span><span class="n">hashBol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hash</span><span class="o">&lt;</span><span class="kt">bool</span><span class="o">&gt;</span><span class="p">()(</span><span class="n">bol</span><span class="p">);</span>
-<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="c1">// Hash value of boolean 1 is 1</span>
+<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="c1">// The hash of the Boolean 1 is 1</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="kt">double</span><span class="w"> </span><span class="n">dec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">3.14159</span><span class="p">;</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="kt">size_t</span><span class="w"> </span><span class="n">hashDec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hash</span><span class="o">&lt;</span><span class="kt">double</span><span class="o">&gt;</span><span class="p">()(</span><span class="n">dec</span><span class="p">);</span>
-<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="c1">// Hash value of decimal 3.14159 is 4614256650576692846</span>
+<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="c1">// The hash of the floating-point number 3.14159 is 4614256650576692846</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="n">string</span><span class="w"> </span><span class="n">str</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"Hello 算法"</span><span class="p">;</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="kt">size_t</span><span class="w"> </span><span class="n">hashStr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hash</span><span class="o">&lt;</span><span class="n">string</span><span class="o">&gt;</span><span class="p">()(</span><span class="n">str</span><span class="p">);</span>
-<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="c1">// Hash value of string "Hello 算法" is 15466937326284535026</span>
+<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="c1">// The hash of the string “Hello 算法” is 15466937326284535026</span>
 <a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a>
-<a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="c1">// In C++, built-in std::hash() only provides hash values for basic data types</span>
-<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="c1">// Hash values for arrays and objects need to be implemented separately</span>
+<a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="c1">// In C++, built-in std::hash() provides hashing only for basic data types</span>
+<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="c1">// Hashing for arrays and objects must be implemented separately</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">built_in_hash.java</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashNum</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Integer</span><span class="p">.</span><span class="na">hashCode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="c1">// Hash value of integer 3 is 3</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="c1">// The hash of the integer 3 is 3</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="kt">boolean</span><span class="w"> </span><span class="n">bol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashBol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Boolean</span><span class="p">.</span><span class="na">hashCode</span><span class="p">(</span><span class="n">bol</span><span class="p">);</span>
-<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="c1">// Hash value of boolean true is 1231</span>
+<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="c1">// The hash of the Boolean true is 1231</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="kt">double</span><span class="w"> </span><span class="n">dec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">3.14159</span><span class="p">;</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashDec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Double</span><span class="p">.</span><span class="na">hashCode</span><span class="p">(</span><span class="n">dec</span><span class="p">);</span>
-<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="c1">// Hash value of decimal 3.14159 is -1340954729</span>
+<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="c1">// The hash of the floating-point number 3.14159 is -1340954729</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="n">String</span><span class="w"> </span><span class="n">str</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"Hello 算法"</span><span class="p">;</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashStr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">str</span><span class="p">.</span><span class="na">hashCode</span><span class="p">();</span>
-<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="c1">// Hash value of string "Hello 算法" is -727081396</span>
+<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="c1">// The hash of the string “Hello 算法” is -727081396</span>
 <a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a>
 <a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="n">Object</span><span class="o">[]</span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">12836</span><span class="p">,</span><span class="w"> </span><span class="s">"小哈"</span><span class="w"> </span><span class="p">};</span>
 <a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashTup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Arrays</span><span class="p">.</span><span class="na">hashCode</span><span class="p">(</span><span class="n">arr</span><span class="p">);</span>
-<a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="c1">// Hash value of array [12836, 小哈] is 1151158</span>
+<a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="c1">// The hash of the array [12836, 小哈] is 1151158</span>
 <a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a>
 <a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">obj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashObj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">obj</span><span class="p">.</span><span class="na">hashCode</span><span class="p">();</span>
-<a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="c1">// Hash value of ListNode object utils.ListNode@7dc5e7b4 is 2110121908</span>
+<a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="c1">// The hash of the node object utils.ListNode@7dc5e7b4 is 2110121908</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">built_in_hash.cs</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="p">;</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashNum</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">.</span><span class="n">GetHashCode</span><span class="p">();</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="c1">// Hash value of integer 3 is 3;</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="c1">// The hash of the integer 3 is 3;</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="kt">bool</span><span class="w"> </span><span class="n">bol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">true</span><span class="p">;</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashBol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">bol</span><span class="p">.</span><span class="n">GetHashCode</span><span class="p">();</span>
-<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="c1">// Hash value of boolean true is 1;</span>
+<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="c1">// The hash of the Boolean true is 1;</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="kt">double</span><span class="w"> </span><span class="n">dec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">3.14159</span><span class="p">;</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashDec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dec</span><span class="p">.</span><span class="n">GetHashCode</span><span class="p">();</span>
-<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="c1">// Hash value of decimal 3.14159 is -1340954729;</span>
+<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="c1">// The hash of the floating-point number 3.14159 is -1340954729;</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="kt">string</span><span class="w"> </span><span class="n">str</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"Hello 算法"</span><span class="p">;</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashStr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">str</span><span class="p">.</span><span class="n">GetHashCode</span><span class="p">();</span>
-<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="c1">// Hash value of string "Hello 算法" is -586107568;</span>
+<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="c1">// The hash of the string “Hello 算法” is -586107568;</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a>
 <a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="kt">object</span><span class="p">[]</span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">12836</span><span class="p">,</span><span class="w"> </span><span class="s">"小哈"</span><span class="p">];</span>
 <a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashTup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">.</span><span class="n">GetHashCode</span><span class="p">();</span>
-<a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="c1">// Hash value of array [12836, 小哈] is 42931033;</span>
+<a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="c1">// The hash of the array [12836, 小哈] is 42931033;</span>
 <a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a>
 <a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">obj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-17-22" name="__codelineno-17-22" href="#__codelineno-17-22"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashObj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">obj</span><span class="p">.</span><span class="n">GetHashCode</span><span class="p">();</span>
-<a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a><span class="c1">// Hash value of ListNode object 0 is 39053774;</span>
+<a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a><span class="c1">// The hash of the node object 0 is 39053774;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">built_in_hash.go</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="c1">// Go does not provide built-in hash code functions</span>
+<div class="highlight"><span class="filename">built_in_hash.go</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="c1">// Go provides no built-in hash-code function</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">built_in_hash.swift</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="kd">let</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">3</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">let</span><span class="w"> </span><span class="nv">hashNum</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">num</span><span class="p">.</span><span class="n">hashValue</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="c1">// Hash value of integer 3 is 9047044699613009734</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="c1">// The hash of the integer 3 is 9047044699613009734</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="kd">let</span><span class="w"> </span><span class="nv">bol</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">true</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="kd">let</span><span class="w"> </span><span class="nv">hashBol</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">bol</span><span class="p">.</span><span class="n">hashValue</span>
-<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="c1">// Hash value of boolean true is -4431640247352757451</span>
+<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="c1">// The hash of the Boolean true is -4431640247352757451</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="kd">let</span><span class="w"> </span><span class="nv">dec</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mf">3.14159</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="kd">let</span><span class="w"> </span><span class="nv">hashDec</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dec</span><span class="p">.</span><span class="n">hashValue</span>
-<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="c1">// Hash value of decimal 3.14159 is -2465384235396674631</span>
+<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="c1">// The hash of the floating-point number 3.14159 is -2465384235396674631</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="kd">let</span><span class="w"> </span><span class="nv">str</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"Hello 算法"</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="kd">let</span><span class="w"> </span><span class="nv">hashStr</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">str</span><span class="p">.</span><span class="n">hashValue</span>
-<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="c1">// Hash value of string "Hello 算法" is -7850626797806988787</span>
+<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="c1">// The hash of the string “Hello 算法” is -7850626797806988787</span>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a>
 <a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="kd">let</span><span class="w"> </span><span class="nv">arr</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[</span><span class="n">AnyHashable</span><span class="p">(</span><span class="mi">12836</span><span class="p">),</span><span class="w"> </span><span class="n">AnyHashable</span><span class="p">(</span><span class="s">"小哈"</span><span class="p">)]</span>
 <a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="kd">let</span><span class="w"> </span><span class="nv">hashTup</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">arr</span><span class="p">.</span><span class="n">hashValue</span>
-<a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="c1">// Hash value of array [AnyHashable(12836), AnyHashable("小哈")] is -2308633508154532996</span>
+<a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="c1">// The hash of the array [AnyHashable(12836), AnyHashable("小哈")] is -2308633508154532996</span>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a>
 <a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="kd">let</span><span class="w"> </span><span class="nv">obj</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="kd">let</span><span class="w"> </span><span class="nv">hashObj</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">obj</span><span class="p">.</span><span class="n">hashValue</span>
-<a id="__codelineno-19-23" name="__codelineno-19-23" href="#__codelineno-19-23"></a><span class="c1">// Hash value of ListNode object utils.ListNode is -2434780518035996159</span>
+<a id="__codelineno-19-23" name="__codelineno-19-23" href="#__codelineno-19-23"></a><span class="c1">// The hash of the node object utils.ListNode is -2434780518035996159</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">built_in_hash.js</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="c1">// JavaScript does not provide built-in hash code functions</span>
+<div class="highlight"><span class="filename">built_in_hash.js</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="c1">// JavaScript provides no built-in hash-code function</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">built_in_hash.ts</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="c1">// TypeScript does not provide built-in hash code functions</span>
+<div class="highlight"><span class="filename">built_in_hash.ts</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="c1">// TypeScript provides no built-in hash-code function</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">built_in_hash.dart</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="kt">int</span><span class="w"> </span><span class="kt">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">3</span><span class="p">;</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashNum</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">num</span><span class="p">.</span><span class="n">hashCode</span><span class="p">;</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="c1">// Hash value of integer 3 is 34803</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="c1">// The hash of the integer 3 is 34803</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="kt">bool</span><span class="w"> </span><span class="n">bol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashBol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">bol</span><span class="p">.</span><span class="n">hashCode</span><span class="p">;</span>
-<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="c1">// Hash value of boolean true is 1231</span>
+<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="c1">// The hash of the Boolean true is 1231</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="kt">double</span><span class="w"> </span><span class="n">dec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">3.14159</span><span class="p">;</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashDec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dec</span><span class="p">.</span><span class="n">hashCode</span><span class="p">;</span>
-<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="c1">// Hash value of decimal 3.14159 is 2570631074981783</span>
+<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="c1">// The hash of the floating-point number 3.14159 is 2570631074981783</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="kt">String</span><span class="w"> </span><span class="n">str</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"Hello 算法"</span><span class="p">;</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashStr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">str</span><span class="p">.</span><span class="n">hashCode</span><span class="p">;</span>
-<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="c1">// Hash value of string "Hello 算法" is 468167534</span>
+<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="c1">// The hash of the string “Hello 算法” is 468167534</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a>
 <a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="n">List</span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="m">12836</span><span class="p">,</span><span class="w"> </span><span class="s2">"小哈"</span><span class="p">];</span>
 <a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashArr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">.</span><span class="n">hashCode</span><span class="p">;</span>
-<a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="c1">// Hash value of array [12836, 小哈] is 976512528</span>
+<a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="c1">// The hash of the array [12836, 小哈] is 976512528</span>
 <a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a>
 <a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="n">ListNode</span><span class="w"> </span><span class="n">obj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="kt">int</span><span class="w"> </span><span class="n">hashObj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">obj</span><span class="p">.</span><span class="n">hashCode</span><span class="p">;</span>
-<a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="c1">// Hash value of ListNode object Instance of 'ListNode' is 1033450432</span>
+<a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="c1">// The hash of the node object Instance of 'ListNode' is 1033450432</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
@@ -5573,105 +5650,105 @@
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">num_hasher</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">DefaultHasher</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="n">num</span><span class="p">.</span><span class="n">hash</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">num_hasher</span><span class="p">);</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="kd">let</span><span class="w"> </span><span class="n">hash_num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num_hasher</span><span class="p">.</span><span class="n">finish</span><span class="p">();</span>
-<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="c1">// Hash value of integer 3 is 568126464209439262</span>
+<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="c1">// The hash of the integer 3 is 568126464209439262</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="kd">let</span><span class="w"> </span><span class="n">bol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span>
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">bol_hasher</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">DefaultHasher</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="n">bol</span><span class="p">.</span><span class="n">hash</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">bol_hasher</span><span class="p">);</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="kd">let</span><span class="w"> </span><span class="n">hash_bol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">bol_hasher</span><span class="p">.</span><span class="n">finish</span><span class="p">();</span>
-<a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="c1">// Hash value of boolean true is 4952851536318644461</span>
+<a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="c1">// The hash of the Boolean true is 4952851536318644461</span>
 <a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a>
 <a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="kd">let</span><span class="w"> </span><span class="n">dec</span><span class="p">:</span><span class="w"> </span><span class="kt">f32</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">3.14159</span><span class="p">;</span>
 <a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dec_hasher</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">DefaultHasher</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="n">dec</span><span class="p">.</span><span class="n">to_bits</span><span class="p">().</span><span class="n">hash</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">dec_hasher</span><span class="p">);</span>
 <a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="kd">let</span><span class="w"> </span><span class="n">hash_dec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dec_hasher</span><span class="p">.</span><span class="n">finish</span><span class="p">();</span>
-<a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="c1">// Hash value of decimal 3.14159 is 2566941990314602357</span>
+<a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="c1">// The hash of the floating-point number 3.14159 is 2566941990314602357</span>
 <a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a>
 <a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="kd">let</span><span class="w"> </span><span class="kt">str</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"Hello 算法"</span><span class="p">;</span>
 <a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">str_hasher</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">DefaultHasher</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="kt">str</span><span class="p">.</span><span class="n">hash</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">str_hasher</span><span class="p">);</span>
 <a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="kd">let</span><span class="w"> </span><span class="n">hash_str</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">str_hasher</span><span class="p">.</span><span class="n">finish</span><span class="p">();</span>
-<a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="c1">// Hash value of string "Hello 算法" is 16092673739211250988</span>
+<a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="c1">// The hash of the string “Hello 算法” is 16092673739211250988</span>
 <a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a>
 <a id="__codelineno-23-28" name="__codelineno-23-28" href="#__codelineno-23-28"></a><span class="kd">let</span><span class="w"> </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="o">&amp;</span><span class="mi">12836</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="s">"小哈"</span><span class="p">);</span>
 <a id="__codelineno-23-29" name="__codelineno-23-29" href="#__codelineno-23-29"></a><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">tup_hasher</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">DefaultHasher</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-23-30" name="__codelineno-23-30" href="#__codelineno-23-30"></a><span class="n">arr</span><span class="p">.</span><span class="n">hash</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">tup_hasher</span><span class="p">);</span>
 <a id="__codelineno-23-31" name="__codelineno-23-31" href="#__codelineno-23-31"></a><span class="kd">let</span><span class="w"> </span><span class="n">hash_tup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tup_hasher</span><span class="p">.</span><span class="n">finish</span><span class="p">();</span>
-<a id="__codelineno-23-32" name="__codelineno-23-32" href="#__codelineno-23-32"></a><span class="c1">// Hash value of tuple (12836, "小哈") is 1885128010422702749</span>
+<a id="__codelineno-23-32" name="__codelineno-23-32" href="#__codelineno-23-32"></a><span class="c1">// The hash of the tuple (12836, "小哈") is 1885128010422702749</span>
 <a id="__codelineno-23-33" name="__codelineno-23-33" href="#__codelineno-23-33"></a>
 <a id="__codelineno-23-34" name="__codelineno-23-34" href="#__codelineno-23-34"></a><span class="kd">let</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="mi">42</span><span class="p">);</span>
 <a id="__codelineno-23-35" name="__codelineno-23-35" href="#__codelineno-23-35"></a><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">hasher</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">DefaultHasher</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-23-36" name="__codelineno-23-36" href="#__codelineno-23-36"></a><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="p">.</span><span class="n">hash</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">hasher</span><span class="p">);</span>
 <a id="__codelineno-23-37" name="__codelineno-23-37" href="#__codelineno-23-37"></a><span class="kd">let</span><span class="w"> </span><span class="n">hash</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hasher</span><span class="p">.</span><span class="n">finish</span><span class="p">();</span>
-<a id="__codelineno-23-38" name="__codelineno-23-38" href="#__codelineno-23-38"></a><span class="c1">// Hash value of ListNode object RefCell { value: ListNode { val: 42, next: None } } is 15387811073369036852</span>
+<a id="__codelineno-23-38" name="__codelineno-23-38" href="#__codelineno-23-38"></a><span class="c1">// The hash of the node object RefCell { value: ListNode { val: 42, next: None } } is 15387811073369036852</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">built_in_hash.c</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="c1">// C does not provide built-in hash code functions</span>
+<div class="highlight"><span class="filename">built_in_hash.c</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="c1">// C provides no built-in hash-code function</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">built_in_hash.kt</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="kd">val</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">3</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="kd">val</span><span class="w"> </span><span class="nv">hashNum</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">.</span><span class="na">hashCode</span><span class="p">()</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="c1">// Hash value of integer 3 is 3</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="c1">// The hash of the integer 3 is 3</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="kd">val</span><span class="w"> </span><span class="nv">bol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="kd">val</span><span class="w"> </span><span class="nv">hashBol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">bol</span><span class="p">.</span><span class="na">hashCode</span><span class="p">()</span>
-<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="c1">// Hash value of boolean true is 1231</span>
+<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="c1">// The hash of the Boolean true is 1231</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="kd">val</span><span class="w"> </span><span class="nv">dec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">3.14159</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="kd">val</span><span class="w"> </span><span class="nv">hashDec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dec</span><span class="p">.</span><span class="na">hashCode</span><span class="p">()</span>
-<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="c1">// Hash value of decimal 3.14159 is -1340954729</span>
+<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="c1">// The hash of the floating-point number 3.14159 is -1340954729</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="kd">val</span><span class="w"> </span><span class="nv">str</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"Hello 算法"</span>
 <a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="kd">val</span><span class="w"> </span><span class="nv">hashStr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">str</span><span class="p">.</span><span class="na">hashCode</span><span class="p">()</span>
-<a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="c1">// Hash value of string "Hello 算法" is -727081396</span>
+<a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="c1">// The hash of the string “Hello 算法” is -727081396</span>
 <a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a>
 <a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="kd">val</span><span class="w"> </span><span class="nv">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arrayOf</span><span class="o">&lt;</span><span class="kt">Any</span><span class="o">&gt;</span><span class="p">(</span><span class="m">12836</span><span class="p">,</span><span class="w"> </span><span class="s">"小哈"</span><span class="p">)</span>
 <a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="kd">val</span><span class="w"> </span><span class="nv">hashTup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">.</span><span class="na">hashCode</span><span class="p">()</span>
-<a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="c1">// Hash value of array [12836, 小哈] is 189568618</span>
+<a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="c1">// The hash of the array [12836, 小哈] is 189568618</span>
 <a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a>
 <a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="kd">val</span><span class="w"> </span><span class="nv">obj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="m">0</span><span class="p">)</span>
 <a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="kd">val</span><span class="w"> </span><span class="nv">hashObj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">obj</span><span class="p">.</span><span class="na">hashCode</span><span class="p">()</span>
-<a id="__codelineno-25-23" name="__codelineno-25-23" href="#__codelineno-25-23"></a><span class="c1">// Hash value of ListNode object utils.ListNode@1d81eb93 is 495053715</span>
+<a id="__codelineno-25-23" name="__codelineno-25-23" href="#__codelineno-25-23"></a><span class="c1">// The hash of the node object utils.ListNode@1d81eb93 is 495053715</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">built_in_hash.rb</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span>
 <a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="n">hash_num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="o">.</span><span class="n">hash</span>
-<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a><span class="c1"># Hash value of integer 3 is -4385856518450339636</span>
+<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a><span class="c1"># The hash of the integer 3 is -4385856518450339636</span>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>
 <a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a><span class="n">bol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">true</span>
 <a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a><span class="n">hash_bol</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">bol</span><span class="o">.</span><span class="n">hash</span>
-<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a><span class="c1"># Hash value of boolean true is -1617938112149317027</span>
+<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a><span class="c1"># The hash of the Boolean true is -1617938112149317027</span>
 <a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>
 <a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a><span class="n">dec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">3</span><span class="o">.</span><span class="mi">14159</span>
 <a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a><span class="n">hash_dec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dec</span><span class="o">.</span><span class="n">hash</span>
-<a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a><span class="c1"># Hash value of decimal 3.14159 is -1479186995943067893</span>
+<a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a><span class="c1"># The hash of the floating-point number 3.14159 is -1479186995943067893</span>
 <a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a>
 <a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a><span class="n">str</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"Hello 算法"</span>
 <a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a><span class="n">hash_str</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">str</span><span class="o">.</span><span class="n">hash</span>
-<a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a><span class="c1"># Hash value of string "Hello 算法" is -4075943250025831763</span>
+<a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a><span class="c1"># The hash of the string “Hello 算法” is -4075943250025831763</span>
 <a id="__codelineno-26-16" name="__codelineno-26-16" href="#__codelineno-26-16"></a>
 <a id="__codelineno-26-17" name="__codelineno-26-17" href="#__codelineno-26-17"></a><span class="n">tup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="mi">12836</span><span class="p">,</span><span class="w"> </span><span class="s1">'小哈'</span><span class="o">]</span>
 <a id="__codelineno-26-18" name="__codelineno-26-18" href="#__codelineno-26-18"></a><span class="n">hash_tup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tup</span><span class="o">.</span><span class="n">hash</span>
-<a id="__codelineno-26-19" name="__codelineno-26-19" href="#__codelineno-26-19"></a><span class="c1"># Hash value of tuple (12836, '小哈') is 1999544809202288822</span>
+<a id="__codelineno-26-19" name="__codelineno-26-19" href="#__codelineno-26-19"></a><span class="c1"># The hash of the tuple (12836, '小哈') is 1999544809202288822</span>
 <a id="__codelineno-26-20" name="__codelineno-26-20" href="#__codelineno-26-20"></a>
 <a id="__codelineno-26-21" name="__codelineno-26-21" href="#__codelineno-26-21"></a><span class="n">obj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">ListNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-26-22" name="__codelineno-26-22" href="#__codelineno-26-22"></a><span class="n">hash_obj</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">obj</span><span class="o">.</span><span class="n">hash</span>
-<a id="__codelineno-26-23" name="__codelineno-26-23" href="#__codelineno-26-23"></a><span class="c1"># Hash value of ListNode object #&lt;ListNode:0x000078133140ab70&gt; is 4302940560806366381</span>
+<a id="__codelineno-26-23" name="__codelineno-26-23" href="#__codelineno-26-23"></a><span class="c1"># The hash of the node object #&lt;ListNode:0x000078133140ab70&gt; is 4302940560806366381</span>
 </code></pre></div>
 </div>
 </div>
 </div>
 <details class="pythontutor">
-<summary>Visualized Execution</summary>
-<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20num%20%3D%203%0A%20%20%20%20hash_num%20%3D%20hash%28num%29%0A%20%20%20%20%23%20%E6%95%B4%E6%95%B0%203%20%E7%9A%84%E5%93%88%E5%B8%8C%E5%80%BC%E4%B8%BA%203%0A%0A%20%20%20%20bol%20%3D%20True%0A%20%20%20%20hash_bol%20%3D%20hash%28bol%29%0A%20%20%20%20%23%20%E5%B8%83%E5%B0%94%E9%87%8F%20True%20%E7%9A%84%E5%93%88%E5%B8%8C%E5%80%BC%E4%B8%BA%201%0A%0A%20%20%20%20dec%20%3D%203.14159%0A%20%20%20%20hash_dec%20%3D%20hash%28dec%29%0A%20%20%20%20%23%20%E5%B0%8F%E6%95%B0%203.14159%20%E7%9A%84%E5%93%88%E5%B8%8C%E5%80%BC%E4%B8%BA%20326484311674566659%0A%0A%20%20%20%20str%20%3D%20%22Hello%20%E7%AE%97%E6%B3%95%22%0A%20%20%20%20hash_str%20%3D%20hash%28str%29%0A%20%20%20%20%23%20%E5%AD%97%E7%AC%A6%E4%B8%B2%E2%80%9CHello%20%E7%AE%97%E6%B3%95%E2%80%9D%E7%9A%84%E5%93%88%E5%B8%8C%E5%80%BC%E4%B8%BA%204617003410720528961%0A%0A%20%20%20%20tup%20%3D%20%2812836,%20%22%E5%B0%8F%E5%93%88%22%29%0A%20%20%20%20hash_tup%20%3D%20hash%28tup%29%0A%20%20%20%20%23%20%E5%85%83%E7%BB%84%20%2812836,%20'%E5%B0%8F%E5%93%88'%29%20%E7%9A%84%E5%93%88%E5%B8%8C%E5%80%BC%E4%B8%BA%201029005403108185979%0A%0A%20%20%20%20obj%20%3D%20ListNode%280%29%0A%20%20%20%20hash_obj%20%3D%20hash%28obj%29%0A%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%AF%B9%E8%B1%A1%20%3CListNode%20object%20at%200x1058fd810%3E%20%E7%9A%84%E5%93%88%E5%B8%8C%E5%80%BC%E4%B8%BA%20274267521&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=19&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20num%20%3D%203%0A%20%20%20%20hash_num%20%3D%20hash%28num%29%0A%20%20%20%20%23%20%E6%95%B4%E6%95%B0%203%20%E7%9A%84%E5%93%88%E5%B8%8C%E5%80%BC%E4%B8%BA%203%0A%0A%20%20%20%20bol%20%3D%20True%0A%20%20%20%20hash_bol%20%3D%20hash%28bol%29%0A%20%20%20%20%23%20%E5%B8%83%E5%B0%94%E9%87%8F%20True%20%E7%9A%84%E5%93%88%E5%B8%8C%E5%80%BC%E4%B8%BA%201%0A%0A%20%20%20%20dec%20%3D%203.14159%0A%20%20%20%20hash_dec%20%3D%20hash%28dec%29%0A%20%20%20%20%23%20%E5%B0%8F%E6%95%B0%203.14159%20%E7%9A%84%E5%93%88%E5%B8%8C%E5%80%BC%E4%B8%BA%20326484311674566659%0A%0A%20%20%20%20str%20%3D%20%22Hello%20%E7%AE%97%E6%B3%95%22%0A%20%20%20%20hash_str%20%3D%20hash%28str%29%0A%20%20%20%20%23%20%E5%AD%97%E7%AC%A6%E4%B8%B2%E2%80%9CHello%20%E7%AE%97%E6%B3%95%E2%80%9D%E7%9A%84%E5%93%88%E5%B8%8C%E5%80%BC%E4%B8%BA%204617003410720528961%0A%0A%20%20%20%20tup%20%3D%20%2812836,%20%22%E5%B0%8F%E5%93%88%22%29%0A%20%20%20%20hash_tup%20%3D%20hash%28tup%29%0A%20%20%20%20%23%20%E5%85%83%E7%BB%84%20%2812836,%20'%E5%B0%8F%E5%93%88'%29%20%E7%9A%84%E5%93%88%E5%B8%8C%E5%80%BC%E4%B8%BA%201029005403108185979%0A%0A%20%20%20%20obj%20%3D%20ListNode%280%29%0A%20%20%20%20hash_obj%20%3D%20hash%28obj%29%0A%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%AF%B9%E8%B1%A1%20%3CListNode%20object%20at%200x1058fd810%3E%20%E7%9A%84%E5%93%88%E5%B8%8C%E5%80%BC%E4%B8%BA%20274267521&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=19&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked-list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20num%20%3D%203%0A%20%20%20%20hash_num%20%3D%20hash%28num%29%0A%20%20%20%20%23%20The%20hash%20of%20the%20integer%203%20is%203%0A%0A%20%20%20%20bol%20%3D%20True%0A%20%20%20%20hash_bol%20%3D%20hash%28bol%29%0A%20%20%20%20%23%20The%20hash%20of%20the%20Boolean%20True%20is%201%0A%0A%20%20%20%20dec%20%3D%203.14159%0A%20%20%20%20hash_dec%20%3D%20hash%28dec%29%0A%20%20%20%20%23%20The%20hash%20of%20the%20floating-point%20number%203.14159%20is%20326484311674566659%0A%0A%20%20%20%20str%20%3D%20%22Hello%20%E7%AE%97%E6%B3%95%22%0A%20%20%20%20hash_str%20%3D%20hash%28str%29%0A%20%20%20%20%23%20The%20hash%20of%20the%20string%20%E2%80%9CHello%20%E7%AE%97%E6%B3%95%E2%80%9D%20is%204617003410720528961%0A%0A%20%20%20%20tup%20%3D%20%2812836%2C%20%22%E5%B0%8F%E5%93%88%22%29%0A%20%20%20%20hash_tup%20%3D%20hash%28tup%29%0A%20%20%20%20%23%20The%20hash%20of%20the%20tuple%20%2812836%2C%20%27%E5%B0%8F%E5%93%88%27%29%20is%201029005403108185979%0A%0A%20%20%20%20obj%20%3D%20ListNode%280%29%0A%20%20%20%20hash_obj%20%3D%20hash%28obj%29%0A%20%20%20%20%23%20The%20hash%20of%20the%20node%20object%20%3CListNode%20object%20at%200x1058fd810%3E%20is%20274267521&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=19&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked-list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20num%20%3D%203%0A%20%20%20%20hash_num%20%3D%20hash%28num%29%0A%20%20%20%20%23%20The%20hash%20of%20the%20integer%203%20is%203%0A%0A%20%20%20%20bol%20%3D%20True%0A%20%20%20%20hash_bol%20%3D%20hash%28bol%29%0A%20%20%20%20%23%20The%20hash%20of%20the%20Boolean%20True%20is%201%0A%0A%20%20%20%20dec%20%3D%203.14159%0A%20%20%20%20hash_dec%20%3D%20hash%28dec%29%0A%20%20%20%20%23%20The%20hash%20of%20the%20floating-point%20number%203.14159%20is%20326484311674566659%0A%0A%20%20%20%20str%20%3D%20%22Hello%20%E7%AE%97%E6%B3%95%22%0A%20%20%20%20hash_str%20%3D%20hash%28str%29%0A%20%20%20%20%23%20The%20hash%20of%20the%20string%20%E2%80%9CHello%20%E7%AE%97%E6%B3%95%E2%80%9D%20is%204617003410720528961%0A%0A%20%20%20%20tup%20%3D%20%2812836%2C%20%22%E5%B0%8F%E5%93%88%22%29%0A%20%20%20%20hash_tup%20%3D%20hash%28tup%29%0A%20%20%20%20%23%20The%20hash%20of%20the%20tuple%20%2812836%2C%20%27%E5%B0%8F%E5%93%88%27%29%20is%201029005403108185979%0A%0A%20%20%20%20obj%20%3D%20ListNode%280%29%0A%20%20%20%20hash_obj%20%3D%20hash%28obj%29%0A%20%20%20%20%23%20The%20hash%20of%20the%20node%20object%20%3CListNode%20object%20at%200x1058fd810%3E%20is%20274267521&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=19&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<p>In many programming languages, <strong>only immutable objects can serve as the <code>key</code> in a hash table</strong>. If we use a list (dynamic array) as a <code>key</code>, when the contents of the list change, its hash value also changes, and we would no longer be able to find the original <code>value</code> in the hash table.</p>
-<p>Although the member variables of a custom object (such as a linked list node) are mutable, it is hashable. <strong>This is because the hash value of an object is usually generated based on its memory address</strong>, and even if the contents of the object change, the memory address remains the same, so the hash value remains unchanged.</p>
-<p>You might have noticed that the hash values output in different consoles are different. <strong>This is because the Python interpreter adds a random salt to the string hash function each time it starts up</strong>. This approach effectively prevents HashDoS attacks and enhances the security of the hash algorithm.</p>
+<p>In many programming languages, <strong>only immutable objects can serve as hash-table keys</strong>. If a list, a dynamic array, were used as a key, changing its contents would change its hash, making the original value impossible to retrieve from the table.</p>
+<p>Custom objects, such as linked-list nodes, can be hashable even though their member variables are mutable. <strong>Their hash is usually based on their memory address</strong>, which stays the same even when their contents change.</p>
+<p>You may notice different hash values when running the program in different consoles. <strong>Each time the Python interpreter starts, it adds a random salt to its string hash function</strong>. This helps prevent HashDoS attacks and improves security.</p>
 
 <!-- Source file information -->
 
@@ -5694,7 +5771,7 @@ aria-label="Footer"
   <a
     href="../hash_collision/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 6.2 Hash Collision"
+    aria-label="Previous: 6.2   Hash collisions"
     rel="prev"
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         Previous
       </span>
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-        6.2 Hash Collision
+        6.2 &nbsp; Hash collisions
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   <a
     href="../summary/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 6.4 Summary"
+    aria-label="Next: 6.4   Summary"
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         Next
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-        6.4 Summary
+        6.4 &nbsp; Summary
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@@ -5743,7 +5820,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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diff --git a/en/chapter_hashing/hash_collision/index.html b/en/chapter_hashing/hash_collision/index.html
index a788748a9..196ba7750 100644
--- a/en/chapter_hashing/hash_collision/index.html
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       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>6.2 Hash Collision - Hello Algo</title>
+        <title>6.2 Hash collisions - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
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-        <a href="#62-hash-collision" class="md-skip">
+        <a href="#62-hash-collisions" class="md-skip">
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
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-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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-  Chapter 1. Encounter with Algorithms
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-  Chapter 1. Encounter with Algorithms
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-  1.2 What Is an Algorithm
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
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-  Chapter 3. Data Structures
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   <span class="md-ellipsis">
 
 
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-  3.2 Basic Data Types
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-  3.4 Character Encoding *
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-  3.5 Summary
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
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-  4.5 Summary
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
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+  5.1 &nbsp; Stacks
 
 
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   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1680,7 +1680,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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 </li>
 
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-  <a href="#623-choice-of-programming-languages" class="md-nav__link">
+  <a href="#623-choices-made-by-programming-languages" class="md-nav__link">
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-  6.4 Summary
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-  6.5 Exercises
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-  Chapter 7. Tree
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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@@ -2327,7 +2327,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2408,7 +2408,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
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-  Chapter 9. Graph
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@@ -2541,7 +2541,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2894,7 +2894,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2921,7 +2921,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2999,7 +2999,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3020,7 +3020,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3041,7 +3041,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3068,7 +3068,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3095,7 +3095,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3122,7 +3122,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3149,7 +3149,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3176,7 +3176,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3203,7 +3203,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3230,7 +3230,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3257,7 +3257,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3284,7 +3284,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3311,7 +3311,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3338,7 +3338,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3404,7 +3404,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3425,7 +3425,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3446,7 +3446,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3473,7 +3473,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3500,7 +3500,7 @@
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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@@ -3527,7 +3527,7 @@
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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@@ -3554,7 +3554,7 @@
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3647,7 +3647,7 @@
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3689,7 +3689,7 @@
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3743,7 +3743,7 @@
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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@@ -3770,7 +3770,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3797,7 +3797,7 @@
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3894,7 +3894,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3915,7 +3915,7 @@
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3936,7 +3936,7 @@
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4191,7 +4191,7 @@
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4428,7 +4428,7 @@
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4470,7 +4470,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4524,7 +4524,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4608,6 +4608,78 @@
 
 
 
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@@ -4652,7 +4724,7 @@
   <a href="#621-separate-chaining" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.2.1 &nbsp; Separate Chaining
+        6.2.1 &nbsp; Separate chaining
       </span>
     </span>
   </a>
@@ -4663,19 +4735,19 @@
   <a href="#622-open-addressing" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.2.2 &nbsp; Open Addressing
+        6.2.2 &nbsp; Open addressing
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="6.2.2 Open Addressing">
+    <nav class="md-nav" aria-label="6.2.2 Open addressing">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-linear-probing" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Linear Probing
+        1. &nbsp; Linear probing
       </span>
     </span>
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@@ -4686,7 +4758,7 @@
   <a href="#2-quadratic-probing" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Quadratic Probing
+        2. &nbsp; Quadratic probing
       </span>
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@@ -4697,7 +4769,7 @@
   <a href="#3-multiple-hashing" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Multiple Hashing
+        3. &nbsp; Multiple hashing
       </span>
     </span>
   </a>
@@ -4710,10 +4782,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#623-choice-of-programming-languages" class="md-nav__link">
+  <a href="#623-choices-made-by-programming-languages" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.2.3 &nbsp; Choice of Programming Languages
+        6.2.3 &nbsp; Choices made by programming languages
       </span>
     </span>
   </a>
@@ -4757,34 +4829,34 @@
 
 
 <!-- Page content -->
-<h1 id="62-hash-collision">6.2 &nbsp; Hash Collision<a class="headerlink" href="#62-hash-collision" title="Permanent link">&para;</a></h1>
-<p>The previous section mentioned that, <strong>in most cases, the input space of a hash function is much larger than the output space</strong>, so theoretically, hash collisions are inevitable. For example, if the input space is all integers and the output space is the array capacity size, then multiple integers will inevitably be mapped to the same bucket index.</p>
-<p>Hash collisions can lead to incorrect query results, severely impacting the usability of the hash table. To address this issue, whenever a hash collision occurs, we can perform hash table expansion until the collision disappears. This approach is simple, straightforward, and effective, but it is very inefficient because hash table expansion involves a large amount of data migration and hash value recalculation. To improve efficiency, we can adopt the following strategies:</p>
+<h1 id="62-hash-collisions">6.2 &nbsp; Hash collisions<a class="headerlink" href="#62-hash-collisions" title="Permanent link">&para;</a></h1>
+<p>As discussed in the previous section, <strong>a hash function's input space is usually much larger than its output space</strong>, so collisions are theoretically unavoidable. For example, if the input space is all integers and the output space is limited to the array's capacity, multiple integers must map to the same bucket index.</p>
+<p>Hash collisions can produce incorrect lookup results and make a hash table unusable. One solution is to expand the table whenever a collision occurs until the collision disappears. This simple, brute-force approach works, but is inefficient because expansion requires substantial data movement and hash computation. We can improve efficiency with the following strategies.</p>
 <ol>
-<li>Improve the hash table data structure so that <strong>the hash table can function normally when hash collisions occur</strong>.</li>
-<li>Only expand when necessary, that is, only when hash collisions are severe.</li>
+<li>Improve the data structure <strong>so that it works correctly even when collisions occur</strong>.</li>
+<li>Expand only when necessary, when collisions become sufficiently frequent.</li>
 </ol>
-<p>The main approaches to improving a hash table's structure are separate chaining and open addressing.</p>
-<h2 id="621-separate-chaining">6.2.1 &nbsp; Separate Chaining<a class="headerlink" href="#621-separate-chaining" title="Permanent link">&para;</a></h2>
-<p>In the original hash table, each bucket can store only one key-value pair. <u>Separate chaining</u> replaces the single element in each bucket with a linked list, treating each key-value pair as a node and storing all colliding key-value pairs in the same list. Figure 6-5 shows an example of a separate chaining hash table.</p>
-<p><img alt="Separate chaining hash table" class="animation-figure" src="../hash_collision.assets/hash_table_chaining.png" /></p>
-<p align="center"> Figure 6-5 &nbsp; Separate chaining hash table </p>
+<p>The main structural approaches are “separate chaining” and “open addressing.”</p>
+<h2 id="621-separate-chaining">6.2.1 &nbsp; Separate chaining<a class="headerlink" href="#621-separate-chaining" title="Permanent link">&para;</a></h2>
+<p>In the original hash table, each bucket holds only one key-value pair. <u>Separate chaining</u> replaces this single element with a linked list. Each pair becomes a node, and all colliding pairs are stored in the same list. An example is shown below.</p>
+<p><img alt="A hash table using separate chaining" class="animation-figure" src="../hash_collision.assets/hash_table_chaining.png" /></p>
+<p align="center"> Figure 6-5 &nbsp; A hash table using separate chaining </p>
 
-<p>In a hash table implemented with separate chaining, the basic operations work as follows:</p>
+<p>The operations change as follows.</p>
 <ul>
-<li><strong>Querying elements</strong>: Input <code>key</code>, compute the bucket index using the hash function, access the head of the corresponding linked list, and traverse the list while comparing keys until the target key-value pair is found.</li>
-<li><strong>Adding elements</strong>: First use the hash function to locate the corresponding linked list, then insert the node (key-value pair) into the list.</li>
-<li><strong>Deleting elements</strong>: Use the hash function to locate the corresponding linked list, then traverse it to find and delete the target node.</li>
+<li><strong>Lookup</strong>: Hash the input <code>key</code> to obtain a bucket index, access the list's head, and traverse the list comparing keys to find the target pair.</li>
+<li><strong>Insertion</strong>: Use the hash function to access the list's head, then add the node containing the key-value pair to the list.</li>
+<li><strong>Deletion</strong>: Use the hash function to locate the list, traverse it to find the target node, and remove that node.</li>
 </ul>
-<p>Separate chaining has the following limitations:</p>
+<p>Separate chaining has the following limitations.</p>
 <ul>
-<li><strong>Increased Space Usage</strong>: The linked list contains node pointers, which consume more memory space than arrays.</li>
-<li><strong>Reduced Query Efficiency</strong>: This is because linear traversal of the linked list is required to find the corresponding element.</li>
+<li><strong>More space</strong>: Linked lists store node pointers and consume more memory than arrays.</li>
+<li><strong>Slower lookup</strong>: Finding an element requires a linear scan of the list.</li>
 </ul>
-<p>The code below provides a simple implementation of a separate chaining hash table, with two things to note:</p>
+<p>The code below gives a simple implementation. Keep two points in mind.</p>
 <ul>
-<li>Lists (dynamic arrays) are used instead of linked lists to simplify the code. In this setup, the hash table (array) contains multiple buckets, each of which is a list.</li>
-<li>This implementation includes a hash table expansion method. When the load factor exceeds <span class="arithmatex">\(\frac{2}{3}\)</span>, we expand the hash table to <span class="arithmatex">\(2\)</span> times its original size.</li>
+<li>Lists, implemented as dynamic arrays, replace linked lists to simplify the code. The hash table is an array of buckets, each of which is a list.</li>
+<li>The implementation includes expansion: when the load factor exceeds <span class="arithmatex">\(\frac{2}{3}\)</span>, the table's capacity is doubled to <span class="arithmatex">\(2\)</span> times its original size.</li>
 </ul>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
@@ -4796,8 +4868,8 @@
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="w">        </span><span class="sd">"""Constructor"""</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Number of key-value pairs</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span> <span class="o">=</span> <span class="mi">4</span>  <span class="c1"># Hash table capacity</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">load_thres</span> <span class="o">=</span> <span class="mf">2.0</span> <span class="o">/</span> <span class="mf">3.0</span>  <span class="c1"># Load factor threshold for triggering expansion</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">extend_ratio</span> <span class="o">=</span> <span class="mi">2</span>  <span class="c1"># Expansion multiplier</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">load_thres</span> <span class="o">=</span> <span class="mf">2.0</span> <span class="o">/</span> <span class="mf">3.0</span>  <span class="c1"># Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">extend_ratio</span> <span class="o">=</span> <span class="mi">2</span>  <span class="c1"># Capacity growth factor</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">)]</span>  <span class="c1"># Bucket array</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">hash_func</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
@@ -4809,29 +4881,29 @@
 <a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span>
 <a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>
 <a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">str</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="w">        </span><span class="sd">"""Query operation"""</span>
+<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="w">        </span><span class="sd">"""Lookup operation"""</span>
 <a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>        <span class="n">index</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a>        <span class="n">bucket</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
-<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a>        <span class="c1"># Traverse bucket, if key is found, return corresponding val</span>
+<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a>        <span class="c1"># Traverse the bucket; if key is found, return its corresponding val</span>
 <a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a>        <span class="k">for</span> <span class="n">pair</span> <span class="ow">in</span> <span class="n">bucket</span><span class="p">:</span>
 <a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a>            <span class="k">if</span> <span class="n">pair</span><span class="o">.</span><span class="n">key</span> <span class="o">==</span> <span class="n">key</span><span class="p">:</span>
 <a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a>                <span class="k">return</span> <span class="n">pair</span><span class="o">.</span><span class="n">val</span>
-<a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a>        <span class="c1"># If key is not found, return None</span>
+<a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a>        <span class="c1"># Return None if key is not found</span>
 <a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a>        <span class="k">return</span> <span class="kc">None</span>
 <a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a>
 <a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">str</span><span class="p">):</span>
-<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a><span class="w">        </span><span class="sd">"""Add operation"""</span>
-<a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a>        <span class="c1"># When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a><span class="w">        </span><span class="sd">"""Insert operation"""</span>
+<a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a>        <span class="c1"># Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">load_factor</span><span class="p">()</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">load_thres</span><span class="p">:</span>
 <a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">extend</span><span class="p">()</span>
 <a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a>        <span class="n">index</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-0-37" name="__codelineno-0-37" href="#__codelineno-0-37"></a>        <span class="n">bucket</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
-<a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></a>        <span class="c1"># Traverse bucket, if specified key is encountered, update corresponding val and return</span>
+<a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></a>        <span class="c1"># Traverse the bucket; if the specified key is found, update its val and return</span>
 <a id="__codelineno-0-39" name="__codelineno-0-39" href="#__codelineno-0-39"></a>        <span class="k">for</span> <span class="n">pair</span> <span class="ow">in</span> <span class="n">bucket</span><span class="p">:</span>
 <a id="__codelineno-0-40" name="__codelineno-0-40" href="#__codelineno-0-40"></a>            <span class="k">if</span> <span class="n">pair</span><span class="o">.</span><span class="n">key</span> <span class="o">==</span> <span class="n">key</span><span class="p">:</span>
 <a id="__codelineno-0-41" name="__codelineno-0-41" href="#__codelineno-0-41"></a>                <span class="n">pair</span><span class="o">.</span><span class="n">val</span> <span class="o">=</span> <span class="n">val</span>
 <a id="__codelineno-0-42" name="__codelineno-0-42" href="#__codelineno-0-42"></a>                <span class="k">return</span>
-<a id="__codelineno-0-43" name="__codelineno-0-43" href="#__codelineno-0-43"></a>        <span class="c1"># If key does not exist, append key-value pair to the end</span>
+<a id="__codelineno-0-43" name="__codelineno-0-43" href="#__codelineno-0-43"></a>        <span class="c1"># If key is absent, append the key-value pair at the end</span>
 <a id="__codelineno-0-44" name="__codelineno-0-44" href="#__codelineno-0-44"></a>        <span class="n">pair</span> <span class="o">=</span> <span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-0-45" name="__codelineno-0-45" href="#__codelineno-0-45"></a>        <span class="n">bucket</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">pair</span><span class="p">)</span>
 <a id="__codelineno-0-46" name="__codelineno-0-46" href="#__codelineno-0-46"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">+=</span> <span class="mi">1</span>
@@ -4840,7 +4912,7 @@
 <a id="__codelineno-0-49" name="__codelineno-0-49" href="#__codelineno-0-49"></a><span class="w">        </span><span class="sd">"""Remove operation"""</span>
 <a id="__codelineno-0-50" name="__codelineno-0-50" href="#__codelineno-0-50"></a>        <span class="n">index</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-0-51" name="__codelineno-0-51" href="#__codelineno-0-51"></a>        <span class="n">bucket</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
-<a id="__codelineno-0-52" name="__codelineno-0-52" href="#__codelineno-0-52"></a>        <span class="c1"># Traverse bucket and remove key-value pair from it</span>
+<a id="__codelineno-0-52" name="__codelineno-0-52" href="#__codelineno-0-52"></a>        <span class="c1"># Traverse the bucket and remove the key-value pair</span>
 <a id="__codelineno-0-53" name="__codelineno-0-53" href="#__codelineno-0-53"></a>        <span class="k">for</span> <span class="n">pair</span> <span class="ow">in</span> <span class="n">bucket</span><span class="p">:</span>
 <a id="__codelineno-0-54" name="__codelineno-0-54" href="#__codelineno-0-54"></a>            <span class="k">if</span> <span class="n">pair</span><span class="o">.</span><span class="n">key</span> <span class="o">==</span> <span class="n">key</span><span class="p">:</span>
 <a id="__codelineno-0-55" name="__codelineno-0-55" href="#__codelineno-0-55"></a>                <span class="n">bucket</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">pair</span><span class="p">)</span>
@@ -4848,20 +4920,20 @@
 <a id="__codelineno-0-57" name="__codelineno-0-57" href="#__codelineno-0-57"></a>                <span class="k">break</span>
 <a id="__codelineno-0-58" name="__codelineno-0-58" href="#__codelineno-0-58"></a>
 <a id="__codelineno-0-59" name="__codelineno-0-59" href="#__codelineno-0-59"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">extend</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-<a id="__codelineno-0-60" name="__codelineno-0-60" href="#__codelineno-0-60"></a><span class="w">        </span><span class="sd">"""Expand hash table"""</span>
-<a id="__codelineno-0-61" name="__codelineno-0-61" href="#__codelineno-0-61"></a>        <span class="c1"># Temporarily store the original hash table</span>
+<a id="__codelineno-0-60" name="__codelineno-0-60" href="#__codelineno-0-60"></a><span class="w">        </span><span class="sd">"""Grow the hash table capacity"""</span>
+<a id="__codelineno-0-61" name="__codelineno-0-61" href="#__codelineno-0-61"></a>        <span class="c1"># Keep a reference to the original hash table</span>
 <a id="__codelineno-0-62" name="__codelineno-0-62" href="#__codelineno-0-62"></a>        <span class="n">buckets</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span>
-<a id="__codelineno-0-63" name="__codelineno-0-63" href="#__codelineno-0-63"></a>        <span class="c1"># Initialize expanded new hash table</span>
+<a id="__codelineno-0-63" name="__codelineno-0-63" href="#__codelineno-0-63"></a>        <span class="c1"># Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-0-64" name="__codelineno-0-64" href="#__codelineno-0-64"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span> <span class="o">*=</span> <span class="bp">self</span><span class="o">.</span><span class="n">extend_ratio</span>
 <a id="__codelineno-0-65" name="__codelineno-0-65" href="#__codelineno-0-65"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">)]</span>
 <a id="__codelineno-0-66" name="__codelineno-0-66" href="#__codelineno-0-66"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-0-67" name="__codelineno-0-67" href="#__codelineno-0-67"></a>        <span class="c1"># Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-0-67" name="__codelineno-0-67" href="#__codelineno-0-67"></a>        <span class="c1"># Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-0-68" name="__codelineno-0-68" href="#__codelineno-0-68"></a>        <span class="k">for</span> <span class="n">bucket</span> <span class="ow">in</span> <span class="n">buckets</span><span class="p">:</span>
 <a id="__codelineno-0-69" name="__codelineno-0-69" href="#__codelineno-0-69"></a>            <span class="k">for</span> <span class="n">pair</span> <span class="ow">in</span> <span class="n">bucket</span><span class="p">:</span>
 <a id="__codelineno-0-70" name="__codelineno-0-70" href="#__codelineno-0-70"></a>                <span class="bp">self</span><span class="o">.</span><span class="n">put</span><span class="p">(</span><span class="n">pair</span><span class="o">.</span><span class="n">key</span><span class="p">,</span> <span class="n">pair</span><span class="o">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-0-71" name="__codelineno-0-71" href="#__codelineno-0-71"></a>
 <a id="__codelineno-0-72" name="__codelineno-0-72" href="#__codelineno-0-72"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">print</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-<a id="__codelineno-0-73" name="__codelineno-0-73" href="#__codelineno-0-73"></a><span class="w">        </span><span class="sd">"""Print hash table"""</span>
+<a id="__codelineno-0-73" name="__codelineno-0-73" href="#__codelineno-0-73"></a><span class="w">        </span><span class="sd">"""Print the hash table"""</span>
 <a id="__codelineno-0-74" name="__codelineno-0-74" href="#__codelineno-0-74"></a>        <span class="k">for</span> <span class="n">bucket</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">:</span>
 <a id="__codelineno-0-75" name="__codelineno-0-75" href="#__codelineno-0-75"></a>            <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
 <a id="__codelineno-0-76" name="__codelineno-0-76" href="#__codelineno-0-76"></a>            <span class="k">for</span> <span class="n">pair</span> <span class="ow">in</span> <span class="n">bucket</span><span class="p">:</span>
@@ -4875,8 +4947,8 @@
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">  </span><span class="k">private</span><span class="o">:</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w">                       </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span><span class="w">                   </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="p">;</span><span class="w">               </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span><span class="w">                </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="p">;</span><span class="w">               </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span><span class="w">                </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">Pair</span><span class="w"> </span><span class="o">*&gt;&gt;</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span><span class="w"> </span><span class="c1">// Bucket array</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">  </span><span class="k">public</span><span class="o">:</span>
@@ -4905,34 +4977,34 @@
 <a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-35" name="__codelineno-1-35" href="#__codelineno-1-35"></a>
-<a id="__codelineno-1-36" name="__codelineno-1-36" href="#__codelineno-1-36"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-1-36" name="__codelineno-1-36" href="#__codelineno-1-36"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-1-37" name="__codelineno-1-37" href="#__codelineno-1-37"></a><span class="w">    </span><span class="n">string</span><span class="w"> </span><span class="n">get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-38" name="__codelineno-1-38" href="#__codelineno-1-38"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-1-39" name="__codelineno-1-39" href="#__codelineno-1-39"></a><span class="w">        </span><span class="c1">// Traverse bucket, if key is found, return corresponding val</span>
+<a id="__codelineno-1-39" name="__codelineno-1-39" href="#__codelineno-1-39"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if key is found, return its corresponding val</span>
 <a id="__codelineno-1-40" name="__codelineno-1-40" href="#__codelineno-1-40"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">pair</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-41" name="__codelineno-1-41" href="#__codelineno-1-41"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-42" name="__codelineno-1-42" href="#__codelineno-1-42"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-1-43" name="__codelineno-1-43" href="#__codelineno-1-43"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-1-44" name="__codelineno-1-44" href="#__codelineno-1-44"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-1-45" name="__codelineno-1-45" href="#__codelineno-1-45"></a><span class="w">        </span><span class="c1">// Return empty string if key not found</span>
+<a id="__codelineno-1-45" name="__codelineno-1-45" href="#__codelineno-1-45"></a><span class="w">        </span><span class="c1">// Return an empty string if key is not found</span>
 <a id="__codelineno-1-46" name="__codelineno-1-46" href="#__codelineno-1-46"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="s">""</span><span class="p">;</span>
 <a id="__codelineno-1-47" name="__codelineno-1-47" href="#__codelineno-1-47"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-48" name="__codelineno-1-48" href="#__codelineno-1-48"></a>
-<a id="__codelineno-1-49" name="__codelineno-1-49" href="#__codelineno-1-49"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-1-49" name="__codelineno-1-49" href="#__codelineno-1-49"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-1-50" name="__codelineno-1-50" href="#__codelineno-1-50"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">put</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-51" name="__codelineno-1-51" href="#__codelineno-1-51"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-1-51" name="__codelineno-1-51" href="#__codelineno-1-51"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-1-52" name="__codelineno-1-52" href="#__codelineno-1-52"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-53" name="__codelineno-1-53" href="#__codelineno-1-53"></a><span class="w">            </span><span class="n">extend</span><span class="p">();</span>
 <a id="__codelineno-1-54" name="__codelineno-1-54" href="#__codelineno-1-54"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-1-55" name="__codelineno-1-55" href="#__codelineno-1-55"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-1-56" name="__codelineno-1-56" href="#__codelineno-1-56"></a><span class="w">        </span><span class="c1">// Traverse bucket, if specified key is encountered, update corresponding val and return</span>
+<a id="__codelineno-1-56" name="__codelineno-1-56" href="#__codelineno-1-56"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if the specified key is found, update its val and return</span>
 <a id="__codelineno-1-57" name="__codelineno-1-57" href="#__codelineno-1-57"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">pair</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-58" name="__codelineno-1-58" href="#__codelineno-1-58"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-59" name="__codelineno-1-59" href="#__codelineno-1-59"></a><span class="w">                </span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-1-60" name="__codelineno-1-60" href="#__codelineno-1-60"></a><span class="w">                </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-1-61" name="__codelineno-1-61" href="#__codelineno-1-61"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-1-62" name="__codelineno-1-62" href="#__codelineno-1-62"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-1-63" name="__codelineno-1-63" href="#__codelineno-1-63"></a><span class="w">        </span><span class="c1">// If key does not exist, append key-value pair to the end</span>
+<a id="__codelineno-1-63" name="__codelineno-1-63" href="#__codelineno-1-63"></a><span class="w">        </span><span class="c1">// If key is absent, append the key-value pair at the end</span>
 <a id="__codelineno-1-64" name="__codelineno-1-64" href="#__codelineno-1-64"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">push_back</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">));</span>
 <a id="__codelineno-1-65" name="__codelineno-1-65" href="#__codelineno-1-65"></a><span class="w">        </span><span class="n">size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-1-66" name="__codelineno-1-66" href="#__codelineno-1-66"></a><span class="w">    </span><span class="p">}</span>
@@ -4941,11 +5013,11 @@
 <a id="__codelineno-1-69" name="__codelineno-1-69" href="#__codelineno-1-69"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-70" name="__codelineno-1-70" href="#__codelineno-1-70"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-1-71" name="__codelineno-1-71" href="#__codelineno-1-71"></a><span class="w">        </span><span class="k">auto</span><span class="w"> </span><span class="o">&amp;</span><span class="n">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
-<a id="__codelineno-1-72" name="__codelineno-1-72" href="#__codelineno-1-72"></a><span class="w">        </span><span class="c1">// Traverse bucket and remove key-value pair from it</span>
+<a id="__codelineno-1-72" name="__codelineno-1-72" href="#__codelineno-1-72"></a><span class="w">        </span><span class="c1">// Traverse the bucket and remove the key-value pair</span>
 <a id="__codelineno-1-73" name="__codelineno-1-73" href="#__codelineno-1-73"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">bucket</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-74" name="__codelineno-1-74" href="#__codelineno-1-74"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">bucket</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">-&gt;</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-75" name="__codelineno-1-75" href="#__codelineno-1-75"></a><span class="w">                </span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">bucket</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
-<a id="__codelineno-1-76" name="__codelineno-1-76" href="#__codelineno-1-76"></a><span class="w">                </span><span class="n">bucket</span><span class="p">.</span><span class="n">erase</span><span class="p">(</span><span class="n">bucket</span><span class="p">.</span><span class="n">begin</span><span class="p">()</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">i</span><span class="p">);</span><span class="w"> </span><span class="c1">// Remove key-value pair from it</span>
+<a id="__codelineno-1-76" name="__codelineno-1-76" href="#__codelineno-1-76"></a><span class="w">                </span><span class="n">bucket</span><span class="p">.</span><span class="n">erase</span><span class="p">(</span><span class="n">bucket</span><span class="p">.</span><span class="n">begin</span><span class="p">()</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">i</span><span class="p">);</span><span class="w"> </span><span class="c1">// Remove the key-value pair from the list</span>
 <a id="__codelineno-1-77" name="__codelineno-1-77" href="#__codelineno-1-77"></a><span class="w">                </span><span class="k">delete</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span><span class="w">                       </span><span class="c1">// Free memory</span>
 <a id="__codelineno-1-78" name="__codelineno-1-78" href="#__codelineno-1-78"></a><span class="w">                </span><span class="n">size</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-1-79" name="__codelineno-1-79" href="#__codelineno-1-79"></a><span class="w">                </span><span class="k">return</span><span class="p">;</span>
@@ -4953,16 +5025,16 @@
 <a id="__codelineno-1-81" name="__codelineno-1-81" href="#__codelineno-1-81"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-1-82" name="__codelineno-1-82" href="#__codelineno-1-82"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-83" name="__codelineno-1-83" href="#__codelineno-1-83"></a>
-<a id="__codelineno-1-84" name="__codelineno-1-84" href="#__codelineno-1-84"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-1-84" name="__codelineno-1-84" href="#__codelineno-1-84"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-1-85" name="__codelineno-1-85" href="#__codelineno-1-85"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-86" name="__codelineno-1-86" href="#__codelineno-1-86"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-1-86" name="__codelineno-1-86" href="#__codelineno-1-86"></a><span class="w">        </span><span class="c1">// Keep a copy of the original bucket array</span>
 <a id="__codelineno-1-87" name="__codelineno-1-87" href="#__codelineno-1-87"></a><span class="w">        </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="n">Pair</span><span class="w"> </span><span class="o">*&gt;&gt;</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span>
-<a id="__codelineno-1-88" name="__codelineno-1-88" href="#__codelineno-1-88"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-1-88" name="__codelineno-1-88" href="#__codelineno-1-88"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-1-89" name="__codelineno-1-89" href="#__codelineno-1-89"></a><span class="w">        </span><span class="n">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span>
 <a id="__codelineno-1-90" name="__codelineno-1-90" href="#__codelineno-1-90"></a><span class="w">        </span><span class="n">buckets</span><span class="p">.</span><span class="n">clear</span><span class="p">();</span>
 <a id="__codelineno-1-91" name="__codelineno-1-91" href="#__codelineno-1-91"></a><span class="w">        </span><span class="n">buckets</span><span class="p">.</span><span class="n">resize</span><span class="p">(</span><span class="n">capacity</span><span class="p">);</span>
 <a id="__codelineno-1-92" name="__codelineno-1-92" href="#__codelineno-1-92"></a><span class="w">        </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-1-93" name="__codelineno-1-93" href="#__codelineno-1-93"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-1-93" name="__codelineno-1-93" href="#__codelineno-1-93"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-1-94" name="__codelineno-1-94" href="#__codelineno-1-94"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="k">auto</span><span class="w"> </span><span class="o">&amp;</span><span class="n">bucket</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-95" name="__codelineno-1-95" href="#__codelineno-1-95"></a><span class="w">            </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">pair</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-96" name="__codelineno-1-96" href="#__codelineno-1-96"></a><span class="w">                </span><span class="n">put</span><span class="p">(</span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">);</span>
@@ -4972,7 +5044,7 @@
 <a id="__codelineno-1-100" name="__codelineno-1-100" href="#__codelineno-1-100"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-1-101" name="__codelineno-1-101" href="#__codelineno-1-101"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-102" name="__codelineno-1-102" href="#__codelineno-1-102"></a>
-<a id="__codelineno-1-103" name="__codelineno-1-103" href="#__codelineno-1-103"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-1-103" name="__codelineno-1-103" href="#__codelineno-1-103"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-1-104" name="__codelineno-1-104" href="#__codelineno-1-104"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-105" name="__codelineno-1-105" href="#__codelineno-1-105"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="k">auto</span><span class="w"> </span><span class="o">&amp;</span><span class="n">bucket</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-106" name="__codelineno-1-106" href="#__codelineno-1-106"></a><span class="w">            </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">"["</span><span class="p">;</span>
@@ -4990,8 +5062,8 @@
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kd">class</span> <span class="nc">HashMapChaining</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span><span class="w"> </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span><span class="w"> </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span><span class="w"> </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span><span class="w"> </span><span class="c1">// Bucket array</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -5016,36 +5088,36 @@
 <a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a>
-<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="w">    </span><span class="n">String</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a><span class="w">        </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
-<a id="__codelineno-2-35" name="__codelineno-2-35" href="#__codelineno-2-35"></a><span class="w">        </span><span class="c1">// Traverse bucket, if key is found, return corresponding val</span>
+<a id="__codelineno-2-35" name="__codelineno-2-35" href="#__codelineno-2-35"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if key is found, return its corresponding val</span>
 <a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-37" name="__codelineno-2-37" href="#__codelineno-2-37"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="na">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-38" name="__codelineno-2-38" href="#__codelineno-2-38"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="na">val</span><span class="p">;</span>
 <a id="__codelineno-2-39" name="__codelineno-2-39" href="#__codelineno-2-39"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-2-40" name="__codelineno-2-40" href="#__codelineno-2-40"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-2-41" name="__codelineno-2-41" href="#__codelineno-2-41"></a><span class="w">        </span><span class="c1">// If key is not found, return null</span>
+<a id="__codelineno-2-41" name="__codelineno-2-41" href="#__codelineno-2-41"></a><span class="w">        </span><span class="c1">// Return null if key is not found</span>
 <a id="__codelineno-2-42" name="__codelineno-2-42" href="#__codelineno-2-42"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-2-43" name="__codelineno-2-43" href="#__codelineno-2-43"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-44" name="__codelineno-2-44" href="#__codelineno-2-44"></a>
-<a id="__codelineno-2-45" name="__codelineno-2-45" href="#__codelineno-2-45"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-2-45" name="__codelineno-2-45" href="#__codelineno-2-45"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-2-46" name="__codelineno-2-46" href="#__codelineno-2-46"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-47" name="__codelineno-2-47" href="#__codelineno-2-47"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-2-47" name="__codelineno-2-47" href="#__codelineno-2-47"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-2-48" name="__codelineno-2-48" href="#__codelineno-2-48"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-49" name="__codelineno-2-49" href="#__codelineno-2-49"></a><span class="w">            </span><span class="n">extend</span><span class="p">();</span>
 <a id="__codelineno-2-50" name="__codelineno-2-50" href="#__codelineno-2-50"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-2-51" name="__codelineno-2-51" href="#__codelineno-2-51"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-2-52" name="__codelineno-2-52" href="#__codelineno-2-52"></a><span class="w">        </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
-<a id="__codelineno-2-53" name="__codelineno-2-53" href="#__codelineno-2-53"></a><span class="w">        </span><span class="c1">// Traverse bucket, if specified key is encountered, update corresponding val and return</span>
+<a id="__codelineno-2-53" name="__codelineno-2-53" href="#__codelineno-2-53"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if the specified key is found, update its val and return</span>
 <a id="__codelineno-2-54" name="__codelineno-2-54" href="#__codelineno-2-54"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-55" name="__codelineno-2-55" href="#__codelineno-2-55"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="na">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-56" name="__codelineno-2-56" href="#__codelineno-2-56"></a><span class="w">                </span><span class="n">pair</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-2-57" name="__codelineno-2-57" href="#__codelineno-2-57"></a><span class="w">                </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-2-58" name="__codelineno-2-58" href="#__codelineno-2-58"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-2-59" name="__codelineno-2-59" href="#__codelineno-2-59"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-2-60" name="__codelineno-2-60" href="#__codelineno-2-60"></a><span class="w">        </span><span class="c1">// If key does not exist, append key-value pair to the end</span>
+<a id="__codelineno-2-60" name="__codelineno-2-60" href="#__codelineno-2-60"></a><span class="w">        </span><span class="c1">// If key is absent, append the key-value pair at the end</span>
 <a id="__codelineno-2-61" name="__codelineno-2-61" href="#__codelineno-2-61"></a><span class="w">        </span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-2-62" name="__codelineno-2-62" href="#__codelineno-2-62"></a><span class="w">        </span><span class="n">bucket</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">pair</span><span class="p">);</span>
 <a id="__codelineno-2-63" name="__codelineno-2-63" href="#__codelineno-2-63"></a><span class="w">        </span><span class="n">size</span><span class="o">++</span><span class="p">;</span>
@@ -5055,7 +5127,7 @@
 <a id="__codelineno-2-67" name="__codelineno-2-67" href="#__codelineno-2-67"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-68" name="__codelineno-2-68" href="#__codelineno-2-68"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-2-69" name="__codelineno-2-69" href="#__codelineno-2-69"></a><span class="w">        </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
-<a id="__codelineno-2-70" name="__codelineno-2-70" href="#__codelineno-2-70"></a><span class="w">        </span><span class="c1">// Traverse bucket and remove key-value pair from it</span>
+<a id="__codelineno-2-70" name="__codelineno-2-70" href="#__codelineno-2-70"></a><span class="w">        </span><span class="c1">// Traverse the bucket and remove the key-value pair</span>
 <a id="__codelineno-2-71" name="__codelineno-2-71" href="#__codelineno-2-71"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-72" name="__codelineno-2-72" href="#__codelineno-2-72"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="na">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-73" name="__codelineno-2-73" href="#__codelineno-2-73"></a><span class="w">                </span><span class="n">bucket</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">pair</span><span class="p">);</span>
@@ -5065,18 +5137,18 @@
 <a id="__codelineno-2-77" name="__codelineno-2-77" href="#__codelineno-2-77"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-2-78" name="__codelineno-2-78" href="#__codelineno-2-78"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-79" name="__codelineno-2-79" href="#__codelineno-2-79"></a>
-<a id="__codelineno-2-80" name="__codelineno-2-80" href="#__codelineno-2-80"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-2-80" name="__codelineno-2-80" href="#__codelineno-2-80"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-2-81" name="__codelineno-2-81" href="#__codelineno-2-81"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="nf">extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-82" name="__codelineno-2-82" href="#__codelineno-2-82"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-2-82" name="__codelineno-2-82" href="#__codelineno-2-82"></a><span class="w">        </span><span class="c1">// Keep a reference to the original hash table</span>
 <a id="__codelineno-2-83" name="__codelineno-2-83" href="#__codelineno-2-83"></a><span class="w">        </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span>
-<a id="__codelineno-2-84" name="__codelineno-2-84" href="#__codelineno-2-84"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-2-84" name="__codelineno-2-84" href="#__codelineno-2-84"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-2-85" name="__codelineno-2-85" href="#__codelineno-2-85"></a><span class="w">        </span><span class="n">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span>
 <a id="__codelineno-2-86" name="__codelineno-2-86" href="#__codelineno-2-86"></a><span class="w">        </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">(</span><span class="n">capacity</span><span class="p">);</span>
 <a id="__codelineno-2-87" name="__codelineno-2-87" href="#__codelineno-2-87"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-88" name="__codelineno-2-88" href="#__codelineno-2-88"></a><span class="w">            </span><span class="n">buckets</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">());</span>
 <a id="__codelineno-2-89" name="__codelineno-2-89" href="#__codelineno-2-89"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-2-90" name="__codelineno-2-90" href="#__codelineno-2-90"></a><span class="w">        </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-2-91" name="__codelineno-2-91" href="#__codelineno-2-91"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-2-91" name="__codelineno-2-91" href="#__codelineno-2-91"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-2-92" name="__codelineno-2-92" href="#__codelineno-2-92"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-93" name="__codelineno-2-93" href="#__codelineno-2-93"></a><span class="w">            </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-94" name="__codelineno-2-94" href="#__codelineno-2-94"></a><span class="w">                </span><span class="n">put</span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="na">key</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="na">val</span><span class="p">);</span>
@@ -5084,7 +5156,7 @@
 <a id="__codelineno-2-96" name="__codelineno-2-96" href="#__codelineno-2-96"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-2-97" name="__codelineno-2-97" href="#__codelineno-2-97"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-98" name="__codelineno-2-98" href="#__codelineno-2-98"></a>
-<a id="__codelineno-2-99" name="__codelineno-2-99" href="#__codelineno-2-99"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-2-99" name="__codelineno-2-99" href="#__codelineno-2-99"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-2-100" name="__codelineno-2-100" href="#__codelineno-2-100"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-101" name="__codelineno-2-101" href="#__codelineno-2-101"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-102" name="__codelineno-2-102" href="#__codelineno-2-102"></a><span class="w">            </span><span class="n">List</span><span class="o">&lt;</span><span class="n">String</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
@@ -5102,8 +5174,8 @@
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">HashMapChaining</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span><span class="w"> </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span><span class="w"> </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span><span class="w"> </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span><span class="w"> </span><span class="c1">// Bucket array</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -5128,34 +5200,34 @@
 <a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a>
-<a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-3-32" name="__codelineno-3-32" href="#__codelineno-3-32"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">string?</span><span class="w"> </span><span class="n">Get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-33" name="__codelineno-3-33" href="#__codelineno-3-33"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-3-34" name="__codelineno-3-34" href="#__codelineno-3-34"></a><span class="w">        </span><span class="c1">// Traverse bucket, if key is found, return corresponding val</span>
+<a id="__codelineno-3-34" name="__codelineno-3-34" href="#__codelineno-3-34"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if key is found, return its corresponding val</span>
 <a id="__codelineno-3-35" name="__codelineno-3-35" href="#__codelineno-3-35"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-36" name="__codelineno-3-36" href="#__codelineno-3-36"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-37" name="__codelineno-3-37" href="#__codelineno-3-37"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-3-38" name="__codelineno-3-38" href="#__codelineno-3-38"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-3-39" name="__codelineno-3-39" href="#__codelineno-3-39"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-3-40" name="__codelineno-3-40" href="#__codelineno-3-40"></a><span class="w">        </span><span class="c1">// If key is not found, return null</span>
+<a id="__codelineno-3-40" name="__codelineno-3-40" href="#__codelineno-3-40"></a><span class="w">        </span><span class="c1">// Return null if key is not found</span>
 <a id="__codelineno-3-41" name="__codelineno-3-41" href="#__codelineno-3-41"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
 <a id="__codelineno-3-42" name="__codelineno-3-42" href="#__codelineno-3-42"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-43" name="__codelineno-3-43" href="#__codelineno-3-43"></a>
-<a id="__codelineno-3-44" name="__codelineno-3-44" href="#__codelineno-3-44"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-3-44" name="__codelineno-3-44" href="#__codelineno-3-44"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-3-45" name="__codelineno-3-45" href="#__codelineno-3-45"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">Put</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-46" name="__codelineno-3-46" href="#__codelineno-3-46"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-3-46" name="__codelineno-3-46" href="#__codelineno-3-46"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-3-47" name="__codelineno-3-47" href="#__codelineno-3-47"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">LoadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-48" name="__codelineno-3-48" href="#__codelineno-3-48"></a><span class="w">            </span><span class="n">Extend</span><span class="p">();</span>
 <a id="__codelineno-3-49" name="__codelineno-3-49" href="#__codelineno-3-49"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-3-50" name="__codelineno-3-50" href="#__codelineno-3-50"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-3-51" name="__codelineno-3-51" href="#__codelineno-3-51"></a><span class="w">        </span><span class="c1">// Traverse bucket, if specified key is encountered, update corresponding val and return</span>
+<a id="__codelineno-3-51" name="__codelineno-3-51" href="#__codelineno-3-51"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if the specified key is found, update its val and return</span>
 <a id="__codelineno-3-52" name="__codelineno-3-52" href="#__codelineno-3-52"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-53" name="__codelineno-3-53" href="#__codelineno-3-53"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-54" name="__codelineno-3-54" href="#__codelineno-3-54"></a><span class="w">                </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-3-55" name="__codelineno-3-55" href="#__codelineno-3-55"></a><span class="w">                </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-3-56" name="__codelineno-3-56" href="#__codelineno-3-56"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-3-57" name="__codelineno-3-57" href="#__codelineno-3-57"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-3-58" name="__codelineno-3-58" href="#__codelineno-3-58"></a><span class="w">        </span><span class="c1">// If key does not exist, append key-value pair to the end</span>
+<a id="__codelineno-3-58" name="__codelineno-3-58" href="#__codelineno-3-58"></a><span class="w">        </span><span class="c1">// If key is absent, append the key-value pair at the end</span>
 <a id="__codelineno-3-59" name="__codelineno-3-59" href="#__codelineno-3-59"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">Add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">));</span>
 <a id="__codelineno-3-60" name="__codelineno-3-60" href="#__codelineno-3-60"></a><span class="w">        </span><span class="n">size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-3-61" name="__codelineno-3-61" href="#__codelineno-3-61"></a><span class="w">    </span><span class="p">}</span>
@@ -5163,7 +5235,7 @@
 <a id="__codelineno-3-63" name="__codelineno-3-63" href="#__codelineno-3-63"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-3-64" name="__codelineno-3-64" href="#__codelineno-3-64"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">Remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-65" name="__codelineno-3-65" href="#__codelineno-3-65"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-3-66" name="__codelineno-3-66" href="#__codelineno-3-66"></a><span class="w">        </span><span class="c1">// Traverse bucket and remove key-value pair from it</span>
+<a id="__codelineno-3-66" name="__codelineno-3-66" href="#__codelineno-3-66"></a><span class="w">        </span><span class="c1">// Traverse the bucket and remove the key-value pair</span>
 <a id="__codelineno-3-67" name="__codelineno-3-67" href="#__codelineno-3-67"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">ToList</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-68" name="__codelineno-3-68" href="#__codelineno-3-68"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-69" name="__codelineno-3-69" href="#__codelineno-3-69"></a><span class="w">                </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">Remove</span><span class="p">(</span><span class="n">pair</span><span class="p">);</span>
@@ -5173,18 +5245,18 @@
 <a id="__codelineno-3-73" name="__codelineno-3-73" href="#__codelineno-3-73"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-3-74" name="__codelineno-3-74" href="#__codelineno-3-74"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-75" name="__codelineno-3-75" href="#__codelineno-3-75"></a>
-<a id="__codelineno-3-76" name="__codelineno-3-76" href="#__codelineno-3-76"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-3-76" name="__codelineno-3-76" href="#__codelineno-3-76"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-3-77" name="__codelineno-3-77" href="#__codelineno-3-77"></a><span class="w">    </span><span class="k">void</span><span class="w"> </span><span class="nf">Extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-78" name="__codelineno-3-78" href="#__codelineno-3-78"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-3-78" name="__codelineno-3-78" href="#__codelineno-3-78"></a><span class="w">        </span><span class="c1">// Keep a reference to the original hash table</span>
 <a id="__codelineno-3-79" name="__codelineno-3-79" href="#__codelineno-3-79"></a><span class="w">        </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span>
-<a id="__codelineno-3-80" name="__codelineno-3-80" href="#__codelineno-3-80"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-3-80" name="__codelineno-3-80" href="#__codelineno-3-80"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-3-81" name="__codelineno-3-81" href="#__codelineno-3-81"></a><span class="w">        </span><span class="n">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span>
 <a id="__codelineno-3-82" name="__codelineno-3-82" href="#__codelineno-3-82"></a><span class="w">        </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;&gt;</span><span class="p">(</span><span class="n">capacity</span><span class="p">);</span>
 <a id="__codelineno-3-83" name="__codelineno-3-83" href="#__codelineno-3-83"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-84" name="__codelineno-3-84" href="#__codelineno-3-84"></a><span class="w">            </span><span class="n">buckets</span><span class="p">.</span><span class="n">Add</span><span class="p">([]);</span>
 <a id="__codelineno-3-85" name="__codelineno-3-85" href="#__codelineno-3-85"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-3-86" name="__codelineno-3-86" href="#__codelineno-3-86"></a><span class="w">        </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-3-87" name="__codelineno-3-87" href="#__codelineno-3-87"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-3-87" name="__codelineno-3-87" href="#__codelineno-3-87"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-3-88" name="__codelineno-3-88" href="#__codelineno-3-88"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-89" name="__codelineno-3-89" href="#__codelineno-3-89"></a><span class="w">            </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-90" name="__codelineno-3-90" href="#__codelineno-3-90"></a><span class="w">                </span><span class="n">Put</span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
@@ -5192,7 +5264,7 @@
 <a id="__codelineno-3-92" name="__codelineno-3-92" href="#__codelineno-3-92"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-3-93" name="__codelineno-3-93" href="#__codelineno-3-93"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-94" name="__codelineno-3-94" href="#__codelineno-3-94"></a>
-<a id="__codelineno-3-95" name="__codelineno-3-95" href="#__codelineno-3-95"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-3-95" name="__codelineno-3-95" href="#__codelineno-3-95"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-3-96" name="__codelineno-3-96" href="#__codelineno-3-96"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">Print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-97" name="__codelineno-3-97" href="#__codelineno-3-97"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-98" name="__codelineno-3-98" href="#__codelineno-3-98"></a><span class="w">            </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">string</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
@@ -5212,8 +5284,8 @@
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">hashMapChaining</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="nx">size</span><span class="w">        </span><span class="kt">int</span><span class="w">      </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">capacity</span><span class="w">    </span><span class="kt">int</span><span class="w">      </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="nx">loadThres</span><span class="w">   </span><span class="kt">float64</span><span class="w">  </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="nx">extendRatio</span><span class="w"> </span><span class="kt">int</span><span class="w">      </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="nx">loadThres</span><span class="w">   </span><span class="kt">float64</span><span class="w">  </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="nx">extendRatio</span><span class="w"> </span><span class="kt">int</span><span class="w">      </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="nx">buckets</span><span class="w">     </span><span class="p">[][]</span><span class="nx">pair</span><span class="w"> </span><span class="c1">// Bucket array</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="p">}</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a>
@@ -5242,35 +5314,35 @@
 <a id="__codelineno-4-32" name="__codelineno-4-32" href="#__codelineno-4-32"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">m</span><span class="p">.</span><span class="nx">size</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">m</span><span class="p">.</span><span class="nx">capacity</span><span class="p">)</span>
 <a id="__codelineno-4-33" name="__codelineno-4-33" href="#__codelineno-4-33"></a><span class="p">}</span>
 <a id="__codelineno-4-34" name="__codelineno-4-34" href="#__codelineno-4-34"></a>
-<a id="__codelineno-4-35" name="__codelineno-4-35" href="#__codelineno-4-35"></a><span class="cm">/* Query operation */</span>
+<a id="__codelineno-4-35" name="__codelineno-4-35" href="#__codelineno-4-35"></a><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-4-36" name="__codelineno-4-36" href="#__codelineno-4-36"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">*</span><span class="nx">hashMapChaining</span><span class="p">)</span><span class="w"> </span><span class="nx">get</span><span class="p">(</span><span class="nx">key</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-37" name="__codelineno-4-37" href="#__codelineno-4-37"></a><span class="w">    </span><span class="nx">idx</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span>
 <a id="__codelineno-4-38" name="__codelineno-4-38" href="#__codelineno-4-38"></a><span class="w">    </span><span class="nx">bucket</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">idx</span><span class="p">]</span>
-<a id="__codelineno-4-39" name="__codelineno-4-39" href="#__codelineno-4-39"></a><span class="w">    </span><span class="c1">// Traverse bucket, if key is found, return corresponding val</span>
+<a id="__codelineno-4-39" name="__codelineno-4-39" href="#__codelineno-4-39"></a><span class="w">    </span><span class="c1">// Traverse the bucket; if key is found, return its corresponding val</span>
 <a id="__codelineno-4-40" name="__codelineno-4-40" href="#__codelineno-4-40"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">p</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-41" name="__codelineno-4-41" href="#__codelineno-4-41"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">p</span><span class="p">.</span><span class="nx">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">key</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-42" name="__codelineno-4-42" href="#__codelineno-4-42"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">p</span><span class="p">.</span><span class="nx">val</span>
 <a id="__codelineno-4-43" name="__codelineno-4-43" href="#__codelineno-4-43"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-4-44" name="__codelineno-4-44" href="#__codelineno-4-44"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-45" name="__codelineno-4-45" href="#__codelineno-4-45"></a><span class="w">    </span><span class="c1">// Return empty string if key not found</span>
+<a id="__codelineno-4-45" name="__codelineno-4-45" href="#__codelineno-4-45"></a><span class="w">    </span><span class="c1">// If key is not found, return an empty string</span>
 <a id="__codelineno-4-46" name="__codelineno-4-46" href="#__codelineno-4-46"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="s">""</span>
 <a id="__codelineno-4-47" name="__codelineno-4-47" href="#__codelineno-4-47"></a><span class="p">}</span>
 <a id="__codelineno-4-48" name="__codelineno-4-48" href="#__codelineno-4-48"></a>
-<a id="__codelineno-4-49" name="__codelineno-4-49" href="#__codelineno-4-49"></a><span class="cm">/* Add operation */</span>
+<a id="__codelineno-4-49" name="__codelineno-4-49" href="#__codelineno-4-49"></a><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-4-50" name="__codelineno-4-50" href="#__codelineno-4-50"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">*</span><span class="nx">hashMapChaining</span><span class="p">)</span><span class="w"> </span><span class="nx">put</span><span class="p">(</span><span class="nx">key</span><span class="w"> </span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-51" name="__codelineno-4-51" href="#__codelineno-4-51"></a><span class="w">    </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-4-51" name="__codelineno-4-51" href="#__codelineno-4-51"></a><span class="w">    </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-4-52" name="__codelineno-4-52" href="#__codelineno-4-52"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">loadThres</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-53" name="__codelineno-4-53" href="#__codelineno-4-53"></a><span class="w">        </span><span class="nx">m</span><span class="p">.</span><span class="nx">extend</span><span class="p">()</span>
 <a id="__codelineno-4-54" name="__codelineno-4-54" href="#__codelineno-4-54"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-55" name="__codelineno-4-55" href="#__codelineno-4-55"></a><span class="w">    </span><span class="nx">idx</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span>
-<a id="__codelineno-4-56" name="__codelineno-4-56" href="#__codelineno-4-56"></a><span class="w">    </span><span class="c1">// Traverse bucket, if specified key is encountered, update corresponding val and return</span>
+<a id="__codelineno-4-56" name="__codelineno-4-56" href="#__codelineno-4-56"></a><span class="w">    </span><span class="c1">// Traverse the bucket; if the specified key is found, update its val and return</span>
 <a id="__codelineno-4-57" name="__codelineno-4-57" href="#__codelineno-4-57"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">idx</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-58" name="__codelineno-4-58" href="#__codelineno-4-58"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">idx</span><span class="p">][</span><span class="nx">i</span><span class="p">].</span><span class="nx">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">key</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-59" name="__codelineno-4-59" href="#__codelineno-4-59"></a><span class="w">            </span><span class="nx">m</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">idx</span><span class="p">][</span><span class="nx">i</span><span class="p">].</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span>
 <a id="__codelineno-4-60" name="__codelineno-4-60" href="#__codelineno-4-60"></a><span class="w">            </span><span class="k">return</span>
 <a id="__codelineno-4-61" name="__codelineno-4-61" href="#__codelineno-4-61"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-4-62" name="__codelineno-4-62" href="#__codelineno-4-62"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-63" name="__codelineno-4-63" href="#__codelineno-4-63"></a><span class="w">    </span><span class="c1">// If key does not exist, append key-value pair to the end</span>
+<a id="__codelineno-4-63" name="__codelineno-4-63" href="#__codelineno-4-63"></a><span class="w">    </span><span class="c1">// If key is absent, append the key-value pair at the end</span>
 <a id="__codelineno-4-64" name="__codelineno-4-64" href="#__codelineno-4-64"></a><span class="w">    </span><span class="nx">p</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">pair</span><span class="p">{</span>
 <a id="__codelineno-4-65" name="__codelineno-4-65" href="#__codelineno-4-65"></a><span class="w">        </span><span class="nx">key</span><span class="p">:</span><span class="w"> </span><span class="nx">key</span><span class="p">,</span>
 <a id="__codelineno-4-66" name="__codelineno-4-66" href="#__codelineno-4-66"></a><span class="w">        </span><span class="nx">val</span><span class="p">:</span><span class="w"> </span><span class="nx">val</span><span class="p">,</span>
@@ -5282,10 +5354,10 @@
 <a id="__codelineno-4-72" name="__codelineno-4-72" href="#__codelineno-4-72"></a><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-4-73" name="__codelineno-4-73" href="#__codelineno-4-73"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">*</span><span class="nx">hashMapChaining</span><span class="p">)</span><span class="w"> </span><span class="nx">remove</span><span class="p">(</span><span class="nx">key</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-74" name="__codelineno-4-74" href="#__codelineno-4-74"></a><span class="w">    </span><span class="nx">idx</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span>
-<a id="__codelineno-4-75" name="__codelineno-4-75" href="#__codelineno-4-75"></a><span class="w">    </span><span class="c1">// Traverse bucket and remove key-value pair from it</span>
+<a id="__codelineno-4-75" name="__codelineno-4-75" href="#__codelineno-4-75"></a><span class="w">    </span><span class="c1">// Traverse the bucket and remove the key-value pair</span>
 <a id="__codelineno-4-76" name="__codelineno-4-76" href="#__codelineno-4-76"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">p</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">idx</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-77" name="__codelineno-4-77" href="#__codelineno-4-77"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">p</span><span class="p">.</span><span class="nx">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">key</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-78" name="__codelineno-4-78" href="#__codelineno-4-78"></a><span class="w">            </span><span class="c1">// Slice deletion</span>
+<a id="__codelineno-4-78" name="__codelineno-4-78" href="#__codelineno-4-78"></a><span class="w">            </span><span class="c1">// Remove the element from the slice</span>
 <a id="__codelineno-4-79" name="__codelineno-4-79" href="#__codelineno-4-79"></a><span class="w">            </span><span class="nx">m</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">idx</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">m</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">idx</span><span class="p">][:</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">idx</span><span class="p">][</span><span class="nx">i</span><span class="o">+</span><span class="mi">1</span><span class="p">:]...)</span>
 <a id="__codelineno-4-80" name="__codelineno-4-80" href="#__codelineno-4-80"></a><span class="w">            </span><span class="nx">m</span><span class="p">.</span><span class="nx">size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-4-81" name="__codelineno-4-81" href="#__codelineno-4-81"></a><span class="w">            </span><span class="k">break</span>
@@ -5293,22 +5365,22 @@
 <a id="__codelineno-4-83" name="__codelineno-4-83" href="#__codelineno-4-83"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-84" name="__codelineno-4-84" href="#__codelineno-4-84"></a><span class="p">}</span>
 <a id="__codelineno-4-85" name="__codelineno-4-85" href="#__codelineno-4-85"></a>
-<a id="__codelineno-4-86" name="__codelineno-4-86" href="#__codelineno-4-86"></a><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-4-86" name="__codelineno-4-86" href="#__codelineno-4-86"></a><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-4-87" name="__codelineno-4-87" href="#__codelineno-4-87"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">*</span><span class="nx">hashMapChaining</span><span class="p">)</span><span class="w"> </span><span class="nx">extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-88" name="__codelineno-4-88" href="#__codelineno-4-88"></a><span class="w">    </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-4-88" name="__codelineno-4-88" href="#__codelineno-4-88"></a><span class="w">    </span><span class="c1">// Save a copy of the original hash table</span>
 <a id="__codelineno-4-89" name="__codelineno-4-89" href="#__codelineno-4-89"></a><span class="w">    </span><span class="nx">tmpBuckets</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="nx">pair</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">m</span><span class="p">.</span><span class="nx">buckets</span><span class="p">))</span>
 <a id="__codelineno-4-90" name="__codelineno-4-90" href="#__codelineno-4-90"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">m</span><span class="p">.</span><span class="nx">buckets</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-91" name="__codelineno-4-91" href="#__codelineno-4-91"></a><span class="w">        </span><span class="nx">tmpBuckets</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="nx">pair</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">m</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">]))</span>
 <a id="__codelineno-4-92" name="__codelineno-4-92" href="#__codelineno-4-92"></a><span class="w">        </span><span class="nb">copy</span><span class="p">(</span><span class="nx">tmpBuckets</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">])</span>
 <a id="__codelineno-4-93" name="__codelineno-4-93" href="#__codelineno-4-93"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-94" name="__codelineno-4-94" href="#__codelineno-4-94"></a><span class="w">    </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-4-94" name="__codelineno-4-94" href="#__codelineno-4-94"></a><span class="w">    </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-4-95" name="__codelineno-4-95" href="#__codelineno-4-95"></a><span class="w">    </span><span class="nx">m</span><span class="p">.</span><span class="nx">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">extendRatio</span>
 <a id="__codelineno-4-96" name="__codelineno-4-96" href="#__codelineno-4-96"></a><span class="w">    </span><span class="nx">m</span><span class="p">.</span><span class="nx">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="nx">pair</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">capacity</span><span class="p">)</span>
 <a id="__codelineno-4-97" name="__codelineno-4-97" href="#__codelineno-4-97"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">capacity</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-98" name="__codelineno-4-98" href="#__codelineno-4-98"></a><span class="w">        </span><span class="nx">m</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="nx">pair</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-4-99" name="__codelineno-4-99" href="#__codelineno-4-99"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-100" name="__codelineno-4-100" href="#__codelineno-4-100"></a><span class="w">    </span><span class="nx">m</span><span class="p">.</span><span class="nx">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-4-101" name="__codelineno-4-101" href="#__codelineno-4-101"></a><span class="w">    </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-4-101" name="__codelineno-4-101" href="#__codelineno-4-101"></a><span class="w">    </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-4-102" name="__codelineno-4-102" href="#__codelineno-4-102"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">tmpBuckets</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-103" name="__codelineno-4-103" href="#__codelineno-4-103"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">p</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-104" name="__codelineno-4-104" href="#__codelineno-4-104"></a><span class="w">            </span><span class="nx">m</span><span class="p">.</span><span class="nx">put</span><span class="p">(</span><span class="nx">p</span><span class="p">.</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">p</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span>
@@ -5316,7 +5388,7 @@
 <a id="__codelineno-4-106" name="__codelineno-4-106" href="#__codelineno-4-106"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-107" name="__codelineno-4-107" href="#__codelineno-4-107"></a><span class="p">}</span>
 <a id="__codelineno-4-108" name="__codelineno-4-108" href="#__codelineno-4-108"></a>
-<a id="__codelineno-4-109" name="__codelineno-4-109" href="#__codelineno-4-109"></a><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-4-109" name="__codelineno-4-109" href="#__codelineno-4-109"></a><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-4-110" name="__codelineno-4-110" href="#__codelineno-4-110"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">*</span><span class="nx">hashMapChaining</span><span class="p">)</span><span class="w"> </span><span class="nb">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-111" name="__codelineno-4-111" href="#__codelineno-4-111"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nx">builder</span><span class="w"> </span><span class="nx">strings</span><span class="p">.</span><span class="nx">Builder</span>
 <a id="__codelineno-4-112" name="__codelineno-4-112" href="#__codelineno-4-112"></a>
@@ -5337,8 +5409,8 @@
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">HashMapChaining</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">size</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">capacity</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">loadThres</span><span class="p">:</span><span class="w"> </span><span class="nb">Double</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">extendRatio</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">loadThres</span><span class="p">:</span><span class="w"> </span><span class="nb">Double</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">extendRatio</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">buckets</span><span class="p">:</span><span class="w"> </span><span class="p">[[</span><span class="n">Pair</span><span class="p">]]</span><span class="w"> </span><span class="c1">// Bucket array</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -5360,36 +5432,36 @@
 <a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a><span class="w">        </span><span class="nb">Double</span><span class="p">(</span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nb">Double</span><span class="p">(</span><span class="n">capacity</span><span class="p">)</span>
 <a id="__codelineno-5-26" name="__codelineno-5-26" href="#__codelineno-5-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-27" name="__codelineno-5-27" href="#__codelineno-5-27"></a>
-<a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-5-29" name="__codelineno-5-29" href="#__codelineno-5-29"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">String</span><span class="p">?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-30" name="__codelineno-5-30" href="#__codelineno-5-30"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-5-31" name="__codelineno-5-31" href="#__codelineno-5-31"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">bucket</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
-<a id="__codelineno-5-32" name="__codelineno-5-32" href="#__codelineno-5-32"></a><span class="w">        </span><span class="c1">// Traverse bucket, if key is found, return corresponding val</span>
+<a id="__codelineno-5-32" name="__codelineno-5-32" href="#__codelineno-5-32"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if key is found, return its corresponding val</span>
 <a id="__codelineno-5-33" name="__codelineno-5-33" href="#__codelineno-5-33"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-34" name="__codelineno-5-34" href="#__codelineno-5-34"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">key</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-35" name="__codelineno-5-35" href="#__codelineno-5-35"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span>
 <a id="__codelineno-5-36" name="__codelineno-5-36" href="#__codelineno-5-36"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-5-37" name="__codelineno-5-37" href="#__codelineno-5-37"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-5-38" name="__codelineno-5-38" href="#__codelineno-5-38"></a><span class="w">        </span><span class="c1">// Return nil if key not found</span>
+<a id="__codelineno-5-38" name="__codelineno-5-38" href="#__codelineno-5-38"></a><span class="w">        </span><span class="c1">// Return nil if key is not found</span>
 <a id="__codelineno-5-39" name="__codelineno-5-39" href="#__codelineno-5-39"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-5-40" name="__codelineno-5-40" href="#__codelineno-5-40"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-41" name="__codelineno-5-41" href="#__codelineno-5-41"></a>
-<a id="__codelineno-5-42" name="__codelineno-5-42" href="#__codelineno-5-42"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-5-42" name="__codelineno-5-42" href="#__codelineno-5-42"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-5-43" name="__codelineno-5-43" href="#__codelineno-5-43"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="nb">String</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-44" name="__codelineno-5-44" href="#__codelineno-5-44"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-5-44" name="__codelineno-5-44" href="#__codelineno-5-44"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-5-45" name="__codelineno-5-45" href="#__codelineno-5-45"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">loadThres</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-46" name="__codelineno-5-46" href="#__codelineno-5-46"></a><span class="w">            </span><span class="bp">extend</span><span class="p">()</span>
 <a id="__codelineno-5-47" name="__codelineno-5-47" href="#__codelineno-5-47"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-48" name="__codelineno-5-48" href="#__codelineno-5-48"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-5-49" name="__codelineno-5-49" href="#__codelineno-5-49"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">bucket</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
-<a id="__codelineno-5-50" name="__codelineno-5-50" href="#__codelineno-5-50"></a><span class="w">        </span><span class="c1">// Traverse bucket, if specified key is encountered, update corresponding val and return</span>
+<a id="__codelineno-5-50" name="__codelineno-5-50" href="#__codelineno-5-50"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if the specified key is found, update its val and return</span>
 <a id="__codelineno-5-51" name="__codelineno-5-51" href="#__codelineno-5-51"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-52" name="__codelineno-5-52" href="#__codelineno-5-52"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">key</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-53" name="__codelineno-5-53" href="#__codelineno-5-53"></a><span class="w">                </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">val</span>
 <a id="__codelineno-5-54" name="__codelineno-5-54" href="#__codelineno-5-54"></a><span class="w">                </span><span class="k">return</span>
 <a id="__codelineno-5-55" name="__codelineno-5-55" href="#__codelineno-5-55"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-5-56" name="__codelineno-5-56" href="#__codelineno-5-56"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-5-57" name="__codelineno-5-57" href="#__codelineno-5-57"></a><span class="w">        </span><span class="c1">// If key does not exist, append key-value pair to the end</span>
+<a id="__codelineno-5-57" name="__codelineno-5-57" href="#__codelineno-5-57"></a><span class="w">        </span><span class="c1">// If key is absent, append the key-value pair at the end</span>
 <a id="__codelineno-5-58" name="__codelineno-5-58" href="#__codelineno-5-58"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">pair</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-5-59" name="__codelineno-5-59" href="#__codelineno-5-59"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">append</span><span class="p">(</span><span class="n">pair</span><span class="p">)</span>
 <a id="__codelineno-5-60" name="__codelineno-5-60" href="#__codelineno-5-60"></a><span class="w">        </span><span class="n">size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
@@ -5399,7 +5471,7 @@
 <a id="__codelineno-5-64" name="__codelineno-5-64" href="#__codelineno-5-64"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-65" name="__codelineno-5-65" href="#__codelineno-5-65"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-5-66" name="__codelineno-5-66" href="#__codelineno-5-66"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">bucket</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
-<a id="__codelineno-5-67" name="__codelineno-5-67" href="#__codelineno-5-67"></a><span class="w">        </span><span class="c1">// Traverse bucket and remove key-value pair from it</span>
+<a id="__codelineno-5-67" name="__codelineno-5-67" href="#__codelineno-5-67"></a><span class="w">        </span><span class="c1">// Traverse the bucket and remove the key-value pair</span>
 <a id="__codelineno-5-68" name="__codelineno-5-68" href="#__codelineno-5-68"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">pairIndex</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="p">)</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="p">.</span><span class="n">enumerated</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-69" name="__codelineno-5-69" href="#__codelineno-5-69"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">key</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-70" name="__codelineno-5-70" href="#__codelineno-5-70"></a><span class="w">                </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">remove</span><span class="p">(</span><span class="n">at</span><span class="p">:</span><span class="w"> </span><span class="n">pairIndex</span><span class="p">)</span>
@@ -5409,15 +5481,15 @@
 <a id="__codelineno-5-74" name="__codelineno-5-74" href="#__codelineno-5-74"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-75" name="__codelineno-5-75" href="#__codelineno-5-75"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-76" name="__codelineno-5-76" href="#__codelineno-5-76"></a>
-<a id="__codelineno-5-77" name="__codelineno-5-77" href="#__codelineno-5-77"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-5-77" name="__codelineno-5-77" href="#__codelineno-5-77"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-5-78" name="__codelineno-5-78" href="#__codelineno-5-78"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-79" name="__codelineno-5-79" href="#__codelineno-5-79"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-5-79" name="__codelineno-5-79" href="#__codelineno-5-79"></a><span class="w">        </span><span class="c1">// Save the original bucket array</span>
 <a id="__codelineno-5-80" name="__codelineno-5-80" href="#__codelineno-5-80"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">bucketsTmp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">buckets</span>
-<a id="__codelineno-5-81" name="__codelineno-5-81" href="#__codelineno-5-81"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-5-81" name="__codelineno-5-81" href="#__codelineno-5-81"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-5-82" name="__codelineno-5-82" href="#__codelineno-5-82"></a><span class="w">        </span><span class="n">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="n">extendRatio</span>
 <a id="__codelineno-5-83" name="__codelineno-5-83" href="#__codelineno-5-83"></a><span class="w">        </span><span class="n">buckets</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="p">[],</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">capacity</span><span class="p">)</span>
 <a id="__codelineno-5-84" name="__codelineno-5-84" href="#__codelineno-5-84"></a><span class="w">        </span><span class="n">size</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-5-85" name="__codelineno-5-85" href="#__codelineno-5-85"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-5-85" name="__codelineno-5-85" href="#__codelineno-5-85"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-5-86" name="__codelineno-5-86" href="#__codelineno-5-86"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-87" name="__codelineno-5-87" href="#__codelineno-5-87"></a><span class="w">            </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-88" name="__codelineno-5-88" href="#__codelineno-5-88"></a><span class="w">                </span><span class="n">put</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
@@ -5425,7 +5497,7 @@
 <a id="__codelineno-5-90" name="__codelineno-5-90" href="#__codelineno-5-90"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-91" name="__codelineno-5-91" href="#__codelineno-5-91"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-92" name="__codelineno-5-92" href="#__codelineno-5-92"></a>
-<a id="__codelineno-5-93" name="__codelineno-5-93" href="#__codelineno-5-93"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-5-93" name="__codelineno-5-93" href="#__codelineno-5-93"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-5-94" name="__codelineno-5-94" href="#__codelineno-5-94"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-95" name="__codelineno-5-95" href="#__codelineno-5-95"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-96" name="__codelineno-5-96" href="#__codelineno-5-96"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">bucket</span><span class="p">.</span><span class="bp">map</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="s">"</span><span class="si">\(</span><span class="nv">$0</span><span class="p">.</span><span class="n">key</span><span class="si">)</span><span class="s"> -&gt; </span><span class="si">\(</span><span class="nv">$0</span><span class="p">.</span><span class="n">val</span><span class="si">)</span><span class="s">"</span><span class="w"> </span><span class="p">}</span>
@@ -5440,8 +5512,8 @@
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">HashMapChaining</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="n">#size</span><span class="p">;</span><span class="w"> </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="n">#capacity</span><span class="p">;</span><span class="w"> </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="n">#loadThres</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="n">#extendRatio</span><span class="p">;</span><span class="w"> </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="n">#loadThres</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="n">#extendRatio</span><span class="p">;</span><span class="w"> </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="n">#buckets</span><span class="p">;</span><span class="w"> </span><span class="c1">// Bucket array</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -5463,36 +5535,36 @@
 <a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="p">;</span>
 <a id="__codelineno-6-26" name="__codelineno-6-26" href="#__codelineno-6-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-27" name="__codelineno-6-27" href="#__codelineno-6-27"></a>
-<a id="__codelineno-6-28" name="__codelineno-6-28" href="#__codelineno-6-28"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-6-28" name="__codelineno-6-28" href="#__codelineno-6-28"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-6-29" name="__codelineno-6-29" href="#__codelineno-6-29"></a><span class="w">    </span><span class="nx">get</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-30" name="__codelineno-6-30" href="#__codelineno-6-30"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
 <a id="__codelineno-6-31" name="__codelineno-6-31" href="#__codelineno-6-31"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">];</span>
-<a id="__codelineno-6-32" name="__codelineno-6-32" href="#__codelineno-6-32"></a><span class="w">        </span><span class="c1">// Traverse bucket, if key is found, return corresponding val</span>
+<a id="__codelineno-6-32" name="__codelineno-6-32" href="#__codelineno-6-32"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if key is found, return its corresponding val</span>
 <a id="__codelineno-6-33" name="__codelineno-6-33" href="#__codelineno-6-33"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-34" name="__codelineno-6-34" href="#__codelineno-6-34"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pair</span><span class="p">.</span><span class="nx">key</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-35" name="__codelineno-6-35" href="#__codelineno-6-35"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="nx">pair</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-6-36" name="__codelineno-6-36" href="#__codelineno-6-36"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-6-37" name="__codelineno-6-37" href="#__codelineno-6-37"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-6-38" name="__codelineno-6-38" href="#__codelineno-6-38"></a><span class="w">        </span><span class="c1">// If key is not found, return null</span>
+<a id="__codelineno-6-38" name="__codelineno-6-38" href="#__codelineno-6-38"></a><span class="w">        </span><span class="c1">// Return null if key was not found</span>
 <a id="__codelineno-6-39" name="__codelineno-6-39" href="#__codelineno-6-39"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-6-40" name="__codelineno-6-40" href="#__codelineno-6-40"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-41" name="__codelineno-6-41" href="#__codelineno-6-41"></a>
-<a id="__codelineno-6-42" name="__codelineno-6-42" href="#__codelineno-6-42"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-6-42" name="__codelineno-6-42" href="#__codelineno-6-42"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-6-43" name="__codelineno-6-43" href="#__codelineno-6-43"></a><span class="w">    </span><span class="nx">put</span><span class="p">(</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-44" name="__codelineno-6-44" href="#__codelineno-6-44"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-6-44" name="__codelineno-6-44" href="#__codelineno-6-44"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-6-45" name="__codelineno-6-45" href="#__codelineno-6-45"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-46" name="__codelineno-6-46" href="#__codelineno-6-46"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#extend</span><span class="p">();</span>
 <a id="__codelineno-6-47" name="__codelineno-6-47" href="#__codelineno-6-47"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-6-48" name="__codelineno-6-48" href="#__codelineno-6-48"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
 <a id="__codelineno-6-49" name="__codelineno-6-49" href="#__codelineno-6-49"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">];</span>
-<a id="__codelineno-6-50" name="__codelineno-6-50" href="#__codelineno-6-50"></a><span class="w">        </span><span class="c1">// Traverse bucket, if specified key is encountered, update corresponding val and return</span>
+<a id="__codelineno-6-50" name="__codelineno-6-50" href="#__codelineno-6-50"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if the specified key is found, update its val and return</span>
 <a id="__codelineno-6-51" name="__codelineno-6-51" href="#__codelineno-6-51"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-52" name="__codelineno-6-52" href="#__codelineno-6-52"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pair</span><span class="p">.</span><span class="nx">key</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-53" name="__codelineno-6-53" href="#__codelineno-6-53"></a><span class="w">                </span><span class="nx">pair</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-6-54" name="__codelineno-6-54" href="#__codelineno-6-54"></a><span class="w">                </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-6-55" name="__codelineno-6-55" href="#__codelineno-6-55"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-6-56" name="__codelineno-6-56" href="#__codelineno-6-56"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-6-57" name="__codelineno-6-57" href="#__codelineno-6-57"></a><span class="w">        </span><span class="c1">// If key does not exist, append key-value pair to the end</span>
+<a id="__codelineno-6-57" name="__codelineno-6-57" href="#__codelineno-6-57"></a><span class="w">        </span><span class="c1">// If key is absent, append the key-value pair at the end</span>
 <a id="__codelineno-6-58" name="__codelineno-6-58" href="#__codelineno-6-58"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Pair</span><span class="p">(</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-6-59" name="__codelineno-6-59" href="#__codelineno-6-59"></a><span class="w">        </span><span class="nx">bucket</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">pair</span><span class="p">);</span>
 <a id="__codelineno-6-60" name="__codelineno-6-60" href="#__codelineno-6-60"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="o">++</span><span class="p">;</span>
@@ -5502,7 +5574,7 @@
 <a id="__codelineno-6-64" name="__codelineno-6-64" href="#__codelineno-6-64"></a><span class="w">    </span><span class="nx">remove</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-65" name="__codelineno-6-65" href="#__codelineno-6-65"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
 <a id="__codelineno-6-66" name="__codelineno-6-66" href="#__codelineno-6-66"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">];</span>
-<a id="__codelineno-6-67" name="__codelineno-6-67" href="#__codelineno-6-67"></a><span class="w">        </span><span class="c1">// Traverse bucket and remove key-value pair from it</span>
+<a id="__codelineno-6-67" name="__codelineno-6-67" href="#__codelineno-6-67"></a><span class="w">        </span><span class="c1">// Traverse the bucket and remove the key-value pair</span>
 <a id="__codelineno-6-68" name="__codelineno-6-68" href="#__codelineno-6-68"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">bucket</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-69" name="__codelineno-6-69" href="#__codelineno-6-69"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">bucket</span><span class="p">[</span><span class="nx">i</span><span class="p">].</span><span class="nx">key</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-70" name="__codelineno-6-70" href="#__codelineno-6-70"></a><span class="w">                </span><span class="nx">bucket</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
@@ -5512,15 +5584,15 @@
 <a id="__codelineno-6-74" name="__codelineno-6-74" href="#__codelineno-6-74"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-6-75" name="__codelineno-6-75" href="#__codelineno-6-75"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-76" name="__codelineno-6-76" href="#__codelineno-6-76"></a>
-<a id="__codelineno-6-77" name="__codelineno-6-77" href="#__codelineno-6-77"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-6-77" name="__codelineno-6-77" href="#__codelineno-6-77"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-6-78" name="__codelineno-6-78" href="#__codelineno-6-78"></a><span class="w">    </span><span class="n">#extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-79" name="__codelineno-6-79" href="#__codelineno-6-79"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-6-79" name="__codelineno-6-79" href="#__codelineno-6-79"></a><span class="w">        </span><span class="c1">// Keep a reference to the original hash table</span>
 <a id="__codelineno-6-80" name="__codelineno-6-80" href="#__codelineno-6-80"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">bucketsTmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">;</span>
-<a id="__codelineno-6-81" name="__codelineno-6-81" href="#__codelineno-6-81"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-6-81" name="__codelineno-6-81" href="#__codelineno-6-81"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-6-82" name="__codelineno-6-82" href="#__codelineno-6-82"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#extendRatio</span><span class="p">;</span>
 <a id="__codelineno-6-83" name="__codelineno-6-83" href="#__codelineno-6-83"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">null</span><span class="p">).</span><span class="nx">map</span><span class="p">((</span><span class="nx">x</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="p">[]);</span>
 <a id="__codelineno-6-84" name="__codelineno-6-84" href="#__codelineno-6-84"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-6-85" name="__codelineno-6-85" href="#__codelineno-6-85"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-6-85" name="__codelineno-6-85" href="#__codelineno-6-85"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-6-86" name="__codelineno-6-86" href="#__codelineno-6-86"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">bucketsTmp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-87" name="__codelineno-6-87" href="#__codelineno-6-87"></a><span class="w">            </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-88" name="__codelineno-6-88" href="#__codelineno-6-88"></a><span class="w">                </span><span class="k">this</span><span class="p">.</span><span class="nx">put</span><span class="p">(</span><span class="nx">pair</span><span class="p">.</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">pair</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span>
@@ -5528,7 +5600,7 @@
 <a id="__codelineno-6-90" name="__codelineno-6-90" href="#__codelineno-6-90"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-6-91" name="__codelineno-6-91" href="#__codelineno-6-91"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-92" name="__codelineno-6-92" href="#__codelineno-6-92"></a>
-<a id="__codelineno-6-93" name="__codelineno-6-93" href="#__codelineno-6-93"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-6-93" name="__codelineno-6-93" href="#__codelineno-6-93"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-6-94" name="__codelineno-6-94" href="#__codelineno-6-94"></a><span class="w">    </span><span class="nx">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-95" name="__codelineno-6-95" href="#__codelineno-6-95"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-96" name="__codelineno-6-96" href="#__codelineno-6-96"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
@@ -5546,8 +5618,8 @@
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">HashMapChaining</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="n">#size</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="n">#capacity</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="n">#loadThres</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="n">#extendRatio</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="n">#loadThres</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="n">#extendRatio</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="n">#buckets</span><span class="o">:</span><span class="w"> </span><span class="nx">Pair</span><span class="p">[][];</span><span class="w"> </span><span class="c1">// Bucket array</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -5569,36 +5641,36 @@
 <a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="p">;</span>
 <a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a>
-<a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-7-29" name="__codelineno-7-29" href="#__codelineno-7-29"></a><span class="w">    </span><span class="nx">get</span><span class="p">(</span><span class="nx">key</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-30" name="__codelineno-7-30" href="#__codelineno-7-30"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
 <a id="__codelineno-7-31" name="__codelineno-7-31" href="#__codelineno-7-31"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">];</span>
-<a id="__codelineno-7-32" name="__codelineno-7-32" href="#__codelineno-7-32"></a><span class="w">        </span><span class="c1">// Traverse bucket, if key is found, return corresponding val</span>
+<a id="__codelineno-7-32" name="__codelineno-7-32" href="#__codelineno-7-32"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if key is found, return its corresponding val</span>
 <a id="__codelineno-7-33" name="__codelineno-7-33" href="#__codelineno-7-33"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-34" name="__codelineno-7-34" href="#__codelineno-7-34"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pair</span><span class="p">.</span><span class="nx">key</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-35" name="__codelineno-7-35" href="#__codelineno-7-35"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="nx">pair</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-7-36" name="__codelineno-7-36" href="#__codelineno-7-36"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-7-37" name="__codelineno-7-37" href="#__codelineno-7-37"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-7-38" name="__codelineno-7-38" href="#__codelineno-7-38"></a><span class="w">        </span><span class="c1">// If key is not found, return null</span>
+<a id="__codelineno-7-38" name="__codelineno-7-38" href="#__codelineno-7-38"></a><span class="w">        </span><span class="c1">// Return null if key was not found</span>
 <a id="__codelineno-7-39" name="__codelineno-7-39" href="#__codelineno-7-39"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-7-40" name="__codelineno-7-40" href="#__codelineno-7-40"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-41" name="__codelineno-7-41" href="#__codelineno-7-41"></a>
-<a id="__codelineno-7-42" name="__codelineno-7-42" href="#__codelineno-7-42"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-7-42" name="__codelineno-7-42" href="#__codelineno-7-42"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-7-43" name="__codelineno-7-43" href="#__codelineno-7-43"></a><span class="w">    </span><span class="nx">put</span><span class="p">(</span><span class="nx">key</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-44" name="__codelineno-7-44" href="#__codelineno-7-44"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-7-44" name="__codelineno-7-44" href="#__codelineno-7-44"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-7-45" name="__codelineno-7-45" href="#__codelineno-7-45"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-46" name="__codelineno-7-46" href="#__codelineno-7-46"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#extend</span><span class="p">();</span>
 <a id="__codelineno-7-47" name="__codelineno-7-47" href="#__codelineno-7-47"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-7-48" name="__codelineno-7-48" href="#__codelineno-7-48"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
 <a id="__codelineno-7-49" name="__codelineno-7-49" href="#__codelineno-7-49"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">];</span>
-<a id="__codelineno-7-50" name="__codelineno-7-50" href="#__codelineno-7-50"></a><span class="w">        </span><span class="c1">// Traverse bucket, if specified key is encountered, update corresponding val and return</span>
+<a id="__codelineno-7-50" name="__codelineno-7-50" href="#__codelineno-7-50"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if the specified key is found, update its val and return</span>
 <a id="__codelineno-7-51" name="__codelineno-7-51" href="#__codelineno-7-51"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-52" name="__codelineno-7-52" href="#__codelineno-7-52"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pair</span><span class="p">.</span><span class="nx">key</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-53" name="__codelineno-7-53" href="#__codelineno-7-53"></a><span class="w">                </span><span class="nx">pair</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-7-54" name="__codelineno-7-54" href="#__codelineno-7-54"></a><span class="w">                </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-7-55" name="__codelineno-7-55" href="#__codelineno-7-55"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-7-56" name="__codelineno-7-56" href="#__codelineno-7-56"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-7-57" name="__codelineno-7-57" href="#__codelineno-7-57"></a><span class="w">        </span><span class="c1">// If key does not exist, append key-value pair to the end</span>
+<a id="__codelineno-7-57" name="__codelineno-7-57" href="#__codelineno-7-57"></a><span class="w">        </span><span class="c1">// If key is absent, append the key-value pair at the end</span>
 <a id="__codelineno-7-58" name="__codelineno-7-58" href="#__codelineno-7-58"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Pair</span><span class="p">(</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-7-59" name="__codelineno-7-59" href="#__codelineno-7-59"></a><span class="w">        </span><span class="nx">bucket</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">pair</span><span class="p">);</span>
 <a id="__codelineno-7-60" name="__codelineno-7-60" href="#__codelineno-7-60"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="o">++</span><span class="p">;</span>
@@ -5608,7 +5680,7 @@
 <a id="__codelineno-7-64" name="__codelineno-7-64" href="#__codelineno-7-64"></a><span class="w">    </span><span class="nx">remove</span><span class="p">(</span><span class="nx">key</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-65" name="__codelineno-7-65" href="#__codelineno-7-65"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
 <a id="__codelineno-7-66" name="__codelineno-7-66" href="#__codelineno-7-66"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">];</span>
-<a id="__codelineno-7-67" name="__codelineno-7-67" href="#__codelineno-7-67"></a><span class="w">        </span><span class="c1">// Traverse bucket and remove key-value pair from it</span>
+<a id="__codelineno-7-67" name="__codelineno-7-67" href="#__codelineno-7-67"></a><span class="w">        </span><span class="c1">// Traverse the bucket and remove the key-value pair</span>
 <a id="__codelineno-7-68" name="__codelineno-7-68" href="#__codelineno-7-68"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">bucket</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-69" name="__codelineno-7-69" href="#__codelineno-7-69"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">bucket</span><span class="p">[</span><span class="nx">i</span><span class="p">].</span><span class="nx">key</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-70" name="__codelineno-7-70" href="#__codelineno-7-70"></a><span class="w">                </span><span class="nx">bucket</span><span class="p">.</span><span class="nx">splice</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
@@ -5618,15 +5690,15 @@
 <a id="__codelineno-7-74" name="__codelineno-7-74" href="#__codelineno-7-74"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-7-75" name="__codelineno-7-75" href="#__codelineno-7-75"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-76" name="__codelineno-7-76" href="#__codelineno-7-76"></a>
-<a id="__codelineno-7-77" name="__codelineno-7-77" href="#__codelineno-7-77"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-7-77" name="__codelineno-7-77" href="#__codelineno-7-77"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-7-78" name="__codelineno-7-78" href="#__codelineno-7-78"></a><span class="w">    </span><span class="n">#extend</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-79" name="__codelineno-7-79" href="#__codelineno-7-79"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-7-79" name="__codelineno-7-79" href="#__codelineno-7-79"></a><span class="w">        </span><span class="c1">// Keep a reference to the original hash table</span>
 <a id="__codelineno-7-80" name="__codelineno-7-80" href="#__codelineno-7-80"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">bucketsTmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">;</span>
-<a id="__codelineno-7-81" name="__codelineno-7-81" href="#__codelineno-7-81"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-7-81" name="__codelineno-7-81" href="#__codelineno-7-81"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-7-82" name="__codelineno-7-82" href="#__codelineno-7-82"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#extendRatio</span><span class="p">;</span>
 <a id="__codelineno-7-83" name="__codelineno-7-83" href="#__codelineno-7-83"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">null</span><span class="p">).</span><span class="nx">map</span><span class="p">((</span><span class="nx">x</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="p">[]);</span>
 <a id="__codelineno-7-84" name="__codelineno-7-84" href="#__codelineno-7-84"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-7-85" name="__codelineno-7-85" href="#__codelineno-7-85"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-7-85" name="__codelineno-7-85" href="#__codelineno-7-85"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-7-86" name="__codelineno-7-86" href="#__codelineno-7-86"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">bucketsTmp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-87" name="__codelineno-7-87" href="#__codelineno-7-87"></a><span class="w">            </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-88" name="__codelineno-7-88" href="#__codelineno-7-88"></a><span class="w">                </span><span class="k">this</span><span class="p">.</span><span class="nx">put</span><span class="p">(</span><span class="nx">pair</span><span class="p">.</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">pair</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span>
@@ -5634,7 +5706,7 @@
 <a id="__codelineno-7-90" name="__codelineno-7-90" href="#__codelineno-7-90"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-7-91" name="__codelineno-7-91" href="#__codelineno-7-91"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-92" name="__codelineno-7-92" href="#__codelineno-7-92"></a>
-<a id="__codelineno-7-93" name="__codelineno-7-93" href="#__codelineno-7-93"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-7-93" name="__codelineno-7-93" href="#__codelineno-7-93"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-7-94" name="__codelineno-7-94" href="#__codelineno-7-94"></a><span class="w">    </span><span class="nx">print</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-95" name="__codelineno-7-95" href="#__codelineno-7-95"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-96" name="__codelineno-7-96" href="#__codelineno-7-96"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
@@ -5652,8 +5724,8 @@
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">HashMapChaining</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span><span class="w"> </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span><span class="w"> </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span><span class="w"> </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span><span class="w"> </span><span class="c1">// Bucket array</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="cm">/* Constructor */</span>
@@ -5675,36 +5747,36 @@
 <a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-8-26" name="__codelineno-8-26" href="#__codelineno-8-26"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a>
-<a id="__codelineno-8-28" name="__codelineno-8-28" href="#__codelineno-8-28"></a><span class="w">  </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-8-28" name="__codelineno-8-28" href="#__codelineno-8-28"></a><span class="w">  </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-8-29" name="__codelineno-8-29" href="#__codelineno-8-29"></a><span class="w">  </span><span class="kt">String</span><span class="o">?</span><span class="w"> </span><span class="kd">get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-30" name="__codelineno-8-30" href="#__codelineno-8-30"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-8-31" name="__codelineno-8-31" href="#__codelineno-8-31"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
-<a id="__codelineno-8-32" name="__codelineno-8-32" href="#__codelineno-8-32"></a><span class="w">    </span><span class="c1">// Traverse bucket, if key is found, return corresponding val</span>
+<a id="__codelineno-8-32" name="__codelineno-8-32" href="#__codelineno-8-32"></a><span class="w">    </span><span class="c1">// Traverse the bucket; if key is found, return its corresponding val</span>
 <a id="__codelineno-8-33" name="__codelineno-8-33" href="#__codelineno-8-33"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-34" name="__codelineno-8-34" href="#__codelineno-8-34"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-35" name="__codelineno-8-35" href="#__codelineno-8-35"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-8-36" name="__codelineno-8-36" href="#__codelineno-8-36"></a><span class="w">      </span><span class="p">}</span>
 <a id="__codelineno-8-37" name="__codelineno-8-37" href="#__codelineno-8-37"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-8-38" name="__codelineno-8-38" href="#__codelineno-8-38"></a><span class="w">    </span><span class="c1">// If key is not found, return null</span>
+<a id="__codelineno-8-38" name="__codelineno-8-38" href="#__codelineno-8-38"></a><span class="w">    </span><span class="c1">// Return null if key was not found</span>
 <a id="__codelineno-8-39" name="__codelineno-8-39" href="#__codelineno-8-39"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-8-40" name="__codelineno-8-40" href="#__codelineno-8-40"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-41" name="__codelineno-8-41" href="#__codelineno-8-41"></a>
-<a id="__codelineno-8-42" name="__codelineno-8-42" href="#__codelineno-8-42"></a><span class="w">  </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-8-42" name="__codelineno-8-42" href="#__codelineno-8-42"></a><span class="w">  </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-8-43" name="__codelineno-8-43" href="#__codelineno-8-43"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">put</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="kt">String</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-44" name="__codelineno-8-44" href="#__codelineno-8-44"></a><span class="w">    </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-8-44" name="__codelineno-8-44" href="#__codelineno-8-44"></a><span class="w">    </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-8-45" name="__codelineno-8-45" href="#__codelineno-8-45"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-46" name="__codelineno-8-46" href="#__codelineno-8-46"></a><span class="w">      </span><span class="n">extend</span><span class="p">();</span>
 <a id="__codelineno-8-47" name="__codelineno-8-47" href="#__codelineno-8-47"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-8-48" name="__codelineno-8-48" href="#__codelineno-8-48"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-8-49" name="__codelineno-8-49" href="#__codelineno-8-49"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
-<a id="__codelineno-8-50" name="__codelineno-8-50" href="#__codelineno-8-50"></a><span class="w">    </span><span class="c1">// Traverse bucket, if specified key is encountered, update corresponding val and return</span>
+<a id="__codelineno-8-50" name="__codelineno-8-50" href="#__codelineno-8-50"></a><span class="w">    </span><span class="c1">// Traverse the bucket; if the specified key is found, update its val and return</span>
 <a id="__codelineno-8-51" name="__codelineno-8-51" href="#__codelineno-8-51"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-52" name="__codelineno-8-52" href="#__codelineno-8-52"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-53" name="__codelineno-8-53" href="#__codelineno-8-53"></a><span class="w">        </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-8-54" name="__codelineno-8-54" href="#__codelineno-8-54"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-8-55" name="__codelineno-8-55" href="#__codelineno-8-55"></a><span class="w">      </span><span class="p">}</span>
 <a id="__codelineno-8-56" name="__codelineno-8-56" href="#__codelineno-8-56"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-8-57" name="__codelineno-8-57" href="#__codelineno-8-57"></a><span class="w">    </span><span class="c1">// If key does not exist, append key-value pair to the end</span>
+<a id="__codelineno-8-57" name="__codelineno-8-57" href="#__codelineno-8-57"></a><span class="w">    </span><span class="c1">// If key is absent, append the key-value pair at the end</span>
 <a id="__codelineno-8-58" name="__codelineno-8-58" href="#__codelineno-8-58"></a><span class="w">    </span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-8-59" name="__codelineno-8-59" href="#__codelineno-8-59"></a><span class="w">    </span><span class="n">bucket</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">pair</span><span class="p">);</span>
 <a id="__codelineno-8-60" name="__codelineno-8-60" href="#__codelineno-8-60"></a><span class="w">    </span><span class="n">size</span><span class="o">++</span><span class="p">;</span>
@@ -5714,7 +5786,7 @@
 <a id="__codelineno-8-64" name="__codelineno-8-64" href="#__codelineno-8-64"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-65" name="__codelineno-8-65" href="#__codelineno-8-65"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-8-66" name="__codelineno-8-66" href="#__codelineno-8-66"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
-<a id="__codelineno-8-67" name="__codelineno-8-67" href="#__codelineno-8-67"></a><span class="w">    </span><span class="c1">// Traverse bucket and remove key-value pair from it</span>
+<a id="__codelineno-8-67" name="__codelineno-8-67" href="#__codelineno-8-67"></a><span class="w">    </span><span class="c1">// Traverse the bucket and remove the key-value pair</span>
 <a id="__codelineno-8-68" name="__codelineno-8-68" href="#__codelineno-8-68"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-69" name="__codelineno-8-69" href="#__codelineno-8-69"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-70" name="__codelineno-8-70" href="#__codelineno-8-70"></a><span class="w">        </span><span class="n">bucket</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">pair</span><span class="p">);</span>
@@ -5724,15 +5796,15 @@
 <a id="__codelineno-8-74" name="__codelineno-8-74" href="#__codelineno-8-74"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-8-75" name="__codelineno-8-75" href="#__codelineno-8-75"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-76" name="__codelineno-8-76" href="#__codelineno-8-76"></a>
-<a id="__codelineno-8-77" name="__codelineno-8-77" href="#__codelineno-8-77"></a><span class="w">  </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-8-77" name="__codelineno-8-77" href="#__codelineno-8-77"></a><span class="w">  </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-8-78" name="__codelineno-8-78" href="#__codelineno-8-78"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-79" name="__codelineno-8-79" href="#__codelineno-8-79"></a><span class="w">    </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-8-79" name="__codelineno-8-79" href="#__codelineno-8-79"></a><span class="w">    </span><span class="c1">// Keep a reference to the original hash table</span>
 <a id="__codelineno-8-80" name="__codelineno-8-80" href="#__codelineno-8-80"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span>
-<a id="__codelineno-8-81" name="__codelineno-8-81" href="#__codelineno-8-81"></a><span class="w">    </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-8-81" name="__codelineno-8-81" href="#__codelineno-8-81"></a><span class="w">    </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-8-82" name="__codelineno-8-82" href="#__codelineno-8-82"></a><span class="w">    </span><span class="n">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span>
 <a id="__codelineno-8-83" name="__codelineno-8-83" href="#__codelineno-8-83"></a><span class="w">    </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">capacity</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">_</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">[]);</span>
 <a id="__codelineno-8-84" name="__codelineno-8-84" href="#__codelineno-8-84"></a><span class="w">    </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
-<a id="__codelineno-8-85" name="__codelineno-8-85" href="#__codelineno-8-85"></a><span class="w">    </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-8-85" name="__codelineno-8-85" href="#__codelineno-8-85"></a><span class="w">    </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-8-86" name="__codelineno-8-86" href="#__codelineno-8-86"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-87" name="__codelineno-8-87" href="#__codelineno-8-87"></a><span class="w">      </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-88" name="__codelineno-8-88" href="#__codelineno-8-88"></a><span class="w">        </span><span class="n">put</span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
@@ -5740,7 +5812,7 @@
 <a id="__codelineno-8-90" name="__codelineno-8-90" href="#__codelineno-8-90"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-8-91" name="__codelineno-8-91" href="#__codelineno-8-91"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-92" name="__codelineno-8-92" href="#__codelineno-8-92"></a>
-<a id="__codelineno-8-93" name="__codelineno-8-93" href="#__codelineno-8-93"></a><span class="w">  </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-8-93" name="__codelineno-8-93" href="#__codelineno-8-93"></a><span class="w">  </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-8-94" name="__codelineno-8-94" href="#__codelineno-8-94"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">printHashMap</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-95" name="__codelineno-8-95" href="#__codelineno-8-95"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-96" name="__codelineno-8-96" href="#__codelineno-8-96"></a><span class="w">      </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">String</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
@@ -5789,7 +5861,7 @@
 <a id="__codelineno-9-33" name="__codelineno-9-33" href="#__codelineno-9-33"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="nb">String</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-34" name="__codelineno-9-34" href="#__codelineno-9-34"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-9-35" name="__codelineno-9-35" href="#__codelineno-9-35"></a>
-<a id="__codelineno-9-36" name="__codelineno-9-36" href="#__codelineno-9-36"></a><span class="w">        </span><span class="c1">// Traverse bucket and remove key-value pair from it</span>
+<a id="__codelineno-9-36" name="__codelineno-9-36" href="#__codelineno-9-36"></a><span class="w">        </span><span class="c1">// Traverse the bucket and remove the key-value pair</span>
 <a id="__codelineno-9-37" name="__codelineno-9-37" href="#__codelineno-9-37"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="p">)</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">iter_mut</span><span class="p">().</span><span class="n">enumerate</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-38" name="__codelineno-9-38" href="#__codelineno-9-38"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">p</span><span class="p">.</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-39" name="__codelineno-9-39" href="#__codelineno-9-39"></a><span class="w">                </span><span class="kd">let</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">remove</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
@@ -5798,21 +5870,21 @@
 <a id="__codelineno-9-42" name="__codelineno-9-42" href="#__codelineno-9-42"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-9-43" name="__codelineno-9-43" href="#__codelineno-9-43"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-44" name="__codelineno-9-44" href="#__codelineno-9-44"></a>
-<a id="__codelineno-9-45" name="__codelineno-9-45" href="#__codelineno-9-45"></a><span class="w">        </span><span class="c1">// If key is not found, return None</span>
+<a id="__codelineno-9-45" name="__codelineno-9-45" href="#__codelineno-9-45"></a><span class="w">        </span><span class="c1">// Return None if key is not found</span>
 <a id="__codelineno-9-46" name="__codelineno-9-46" href="#__codelineno-9-46"></a><span class="w">        </span><span class="nb">None</span>
 <a id="__codelineno-9-47" name="__codelineno-9-47" href="#__codelineno-9-47"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-48" name="__codelineno-9-48" href="#__codelineno-9-48"></a>
-<a id="__codelineno-9-49" name="__codelineno-9-49" href="#__codelineno-9-49"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-9-49" name="__codelineno-9-49" href="#__codelineno-9-49"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-9-50" name="__codelineno-9-50" href="#__codelineno-9-50"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">extend</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-51" name="__codelineno-9-51" href="#__codelineno-9-51"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-9-51" name="__codelineno-9-51" href="#__codelineno-9-51"></a><span class="w">        </span><span class="c1">// Temporarily take ownership of the original buckets</span>
 <a id="__codelineno-9-52" name="__codelineno-9-52" href="#__codelineno-9-52"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">buckets_tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">mem</span><span class="p">::</span><span class="n">take</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">);</span>
 <a id="__codelineno-9-53" name="__codelineno-9-53" href="#__codelineno-9-53"></a>
-<a id="__codelineno-9-54" name="__codelineno-9-54" href="#__codelineno-9-54"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-9-54" name="__codelineno-9-54" href="#__codelineno-9-54"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-9-55" name="__codelineno-9-55" href="#__codelineno-9-55"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">extend_ratio</span><span class="p">;</span>
 <a id="__codelineno-9-56" name="__codelineno-9-56" href="#__codelineno-9-56"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">capacity</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">];</span>
 <a id="__codelineno-9-57" name="__codelineno-9-57" href="#__codelineno-9-57"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-9-58" name="__codelineno-9-58" href="#__codelineno-9-58"></a>
-<a id="__codelineno-9-59" name="__codelineno-9-59" href="#__codelineno-9-59"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-9-59" name="__codelineno-9-59" href="#__codelineno-9-59"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-9-60" name="__codelineno-9-60" href="#__codelineno-9-60"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets_tmp</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-61" name="__codelineno-9-61" href="#__codelineno-9-61"></a><span class="w">            </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-62" name="__codelineno-9-62" href="#__codelineno-9-62"></a><span class="w">                </span><span class="bp">self</span><span class="p">.</span><span class="n">put</span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
@@ -5820,7 +5892,7 @@
 <a id="__codelineno-9-64" name="__codelineno-9-64" href="#__codelineno-9-64"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-65" name="__codelineno-9-65" href="#__codelineno-9-65"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-66" name="__codelineno-9-66" href="#__codelineno-9-66"></a>
-<a id="__codelineno-9-67" name="__codelineno-9-67" href="#__codelineno-9-67"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-9-67" name="__codelineno-9-67" href="#__codelineno-9-67"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-9-68" name="__codelineno-9-68" href="#__codelineno-9-68"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">print</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-69" name="__codelineno-9-69" href="#__codelineno-9-69"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="o">&amp;</span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-70" name="__codelineno-9-70" href="#__codelineno-9-70"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
@@ -5831,16 +5903,16 @@
 <a id="__codelineno-9-75" name="__codelineno-9-75" href="#__codelineno-9-75"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-76" name="__codelineno-9-76" href="#__codelineno-9-76"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-77" name="__codelineno-9-77" href="#__codelineno-9-77"></a>
-<a id="__codelineno-9-78" name="__codelineno-9-78" href="#__codelineno-9-78"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-9-78" name="__codelineno-9-78" href="#__codelineno-9-78"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-9-79" name="__codelineno-9-79" href="#__codelineno-9-79"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="nb">String</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-80" name="__codelineno-9-80" href="#__codelineno-9-80"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-9-80" name="__codelineno-9-80" href="#__codelineno-9-80"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-9-81" name="__codelineno-9-81" href="#__codelineno-9-81"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">load_factor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">load_thres</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-82" name="__codelineno-9-82" href="#__codelineno-9-82"></a><span class="w">            </span><span class="bp">self</span><span class="p">.</span><span class="n">extend</span><span class="p">();</span>
 <a id="__codelineno-9-83" name="__codelineno-9-83" href="#__codelineno-9-83"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-84" name="__codelineno-9-84" href="#__codelineno-9-84"></a>
 <a id="__codelineno-9-85" name="__codelineno-9-85" href="#__codelineno-9-85"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-9-86" name="__codelineno-9-86" href="#__codelineno-9-86"></a>
-<a id="__codelineno-9-87" name="__codelineno-9-87" href="#__codelineno-9-87"></a><span class="w">        </span><span class="c1">// Traverse bucket, if specified key is encountered, update corresponding val and return</span>
+<a id="__codelineno-9-87" name="__codelineno-9-87" href="#__codelineno-9-87"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if the specified key is found, update its val and return</span>
 <a id="__codelineno-9-88" name="__codelineno-9-88" href="#__codelineno-9-88"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">iter_mut</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-89" name="__codelineno-9-89" href="#__codelineno-9-89"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-90" name="__codelineno-9-90" href="#__codelineno-9-90"></a><span class="w">                </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
@@ -5848,24 +5920,24 @@
 <a id="__codelineno-9-92" name="__codelineno-9-92" href="#__codelineno-9-92"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-9-93" name="__codelineno-9-93" href="#__codelineno-9-93"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-94" name="__codelineno-9-94" href="#__codelineno-9-94"></a>
-<a id="__codelineno-9-95" name="__codelineno-9-95" href="#__codelineno-9-95"></a><span class="w">        </span><span class="c1">// If key does not exist, append key-value pair to the end</span>
+<a id="__codelineno-9-95" name="__codelineno-9-95" href="#__codelineno-9-95"></a><span class="w">        </span><span class="c1">// If key is absent, append the key-value pair at the end</span>
 <a id="__codelineno-9-96" name="__codelineno-9-96" href="#__codelineno-9-96"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Pair</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="p">};</span>
 <a id="__codelineno-9-97" name="__codelineno-9-97" href="#__codelineno-9-97"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">push</span><span class="p">(</span><span class="n">pair</span><span class="p">);</span>
 <a id="__codelineno-9-98" name="__codelineno-9-98" href="#__codelineno-9-98"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-99" name="__codelineno-9-99" href="#__codelineno-9-99"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-100" name="__codelineno-9-100" href="#__codelineno-9-100"></a>
-<a id="__codelineno-9-101" name="__codelineno-9-101" href="#__codelineno-9-101"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-9-101" name="__codelineno-9-101" href="#__codelineno-9-101"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-9-102" name="__codelineno-9-102" href="#__codelineno-9-102"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="kt">str</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-103" name="__codelineno-9-103" href="#__codelineno-9-103"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-9-104" name="__codelineno-9-104" href="#__codelineno-9-104"></a>
-<a id="__codelineno-9-105" name="__codelineno-9-105" href="#__codelineno-9-105"></a><span class="w">        </span><span class="c1">// Traverse bucket, if key is found, return corresponding val</span>
+<a id="__codelineno-9-105" name="__codelineno-9-105" href="#__codelineno-9-105"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if key is found, return its corresponding val</span>
 <a id="__codelineno-9-106" name="__codelineno-9-106" href="#__codelineno-9-106"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">iter</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-107" name="__codelineno-9-107" href="#__codelineno-9-107"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-108" name="__codelineno-9-108" href="#__codelineno-9-108"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="o">&amp;</span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-9-109" name="__codelineno-9-109" href="#__codelineno-9-109"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-9-110" name="__codelineno-9-110" href="#__codelineno-9-110"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-111" name="__codelineno-9-111" href="#__codelineno-9-111"></a>
-<a id="__codelineno-9-112" name="__codelineno-9-112" href="#__codelineno-9-112"></a><span class="w">        </span><span class="c1">// If key is not found, return None</span>
+<a id="__codelineno-9-112" name="__codelineno-9-112" href="#__codelineno-9-112"></a><span class="w">        </span><span class="c1">// Return None if key is not found</span>
 <a id="__codelineno-9-113" name="__codelineno-9-113" href="#__codelineno-9-113"></a><span class="w">        </span><span class="nb">None</span>
 <a id="__codelineno-9-114" name="__codelineno-9-114" href="#__codelineno-9-114"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-115" name="__codelineno-9-115" href="#__codelineno-9-115"></a><span class="p">}</span>
@@ -5882,8 +5954,8 @@
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w">         </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span><span class="w">     </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span><span class="w">  </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span><span class="w">  </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">    </span><span class="n">Node</span><span class="w"> </span><span class="o">**</span><span class="n">buckets</span><span class="p">;</span><span class="w">   </span><span class="c1">// Bucket array</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="p">}</span><span class="w"> </span><span class="n">HashMapChaining</span><span class="p">;</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a>
@@ -5926,10 +5998,10 @@
 <a id="__codelineno-10-52" name="__codelineno-10-52" href="#__codelineno-10-52"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-10-53" name="__codelineno-10-53" href="#__codelineno-10-53"></a><span class="p">}</span>
 <a id="__codelineno-10-54" name="__codelineno-10-54" href="#__codelineno-10-54"></a>
-<a id="__codelineno-10-55" name="__codelineno-10-55" href="#__codelineno-10-55"></a><span class="cm">/* Query operation */</span>
+<a id="__codelineno-10-55" name="__codelineno-10-55" href="#__codelineno-10-55"></a><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-10-56" name="__codelineno-10-56" href="#__codelineno-10-56"></a><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="nf">get</span><span class="p">(</span><span class="n">HashMapChaining</span><span class="w"> </span><span class="o">*</span><span class="n">hashMap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-57" name="__codelineno-10-57" href="#__codelineno-10-57"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">hashMap</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-10-58" name="__codelineno-10-58" href="#__codelineno-10-58"></a><span class="w">    </span><span class="c1">// Traverse bucket, if key is found, return corresponding val</span>
+<a id="__codelineno-10-58" name="__codelineno-10-58" href="#__codelineno-10-58"></a><span class="w">    </span><span class="c1">// Traverse the bucket; if key is found, return its corresponding val</span>
 <a id="__codelineno-10-59" name="__codelineno-10-59" href="#__codelineno-10-59"></a><span class="w">    </span><span class="n">Node</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
 <a id="__codelineno-10-60" name="__codelineno-10-60" href="#__codelineno-10-60"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-61" name="__codelineno-10-61" href="#__codelineno-10-61"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5937,26 +6009,26 @@
 <a id="__codelineno-10-63" name="__codelineno-10-63" href="#__codelineno-10-63"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-10-64" name="__codelineno-10-64" href="#__codelineno-10-64"></a><span class="w">        </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-10-65" name="__codelineno-10-65" href="#__codelineno-10-65"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-66" name="__codelineno-10-66" href="#__codelineno-10-66"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="s">""</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return empty string if key not found</span>
+<a id="__codelineno-10-66" name="__codelineno-10-66" href="#__codelineno-10-66"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="s">""</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return an empty string if key is not found</span>
 <a id="__codelineno-10-67" name="__codelineno-10-67" href="#__codelineno-10-67"></a><span class="p">}</span>
 <a id="__codelineno-10-68" name="__codelineno-10-68" href="#__codelineno-10-68"></a>
-<a id="__codelineno-10-69" name="__codelineno-10-69" href="#__codelineno-10-69"></a><span class="cm">/* Add operation */</span>
+<a id="__codelineno-10-69" name="__codelineno-10-69" href="#__codelineno-10-69"></a><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-10-70" name="__codelineno-10-70" href="#__codelineno-10-70"></a><span class="kt">void</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="n">HashMapChaining</span><span class="w"> </span><span class="o">*</span><span class="n">hashMap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-71" name="__codelineno-10-71" href="#__codelineno-10-71"></a><span class="w">    </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-10-71" name="__codelineno-10-71" href="#__codelineno-10-71"></a><span class="w">    </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-10-72" name="__codelineno-10-72" href="#__codelineno-10-72"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">loadFactor</span><span class="p">(</span><span class="n">hashMap</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-73" name="__codelineno-10-73" href="#__codelineno-10-73"></a><span class="w">        </span><span class="n">extend</span><span class="p">(</span><span class="n">hashMap</span><span class="p">);</span>
 <a id="__codelineno-10-74" name="__codelineno-10-74" href="#__codelineno-10-74"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-75" name="__codelineno-10-75" href="#__codelineno-10-75"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">hashMap</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-10-76" name="__codelineno-10-76" href="#__codelineno-10-76"></a><span class="w">    </span><span class="c1">// Traverse bucket, if specified key is encountered, update corresponding val and return</span>
+<a id="__codelineno-10-76" name="__codelineno-10-76" href="#__codelineno-10-76"></a><span class="w">    </span><span class="c1">// Traverse the bucket; if the specified key is found, update its val and return</span>
 <a id="__codelineno-10-77" name="__codelineno-10-77" href="#__codelineno-10-77"></a><span class="w">    </span><span class="n">Node</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
 <a id="__codelineno-10-78" name="__codelineno-10-78" href="#__codelineno-10-78"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-79" name="__codelineno-10-79" href="#__codelineno-10-79"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-80" name="__codelineno-10-80" href="#__codelineno-10-80"></a><span class="w">            </span><span class="n">strcpy</span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span><span class="w"> </span><span class="c1">// If specified key is found, update corresponding val and return</span>
+<a id="__codelineno-10-80" name="__codelineno-10-80" href="#__codelineno-10-80"></a><span class="w">            </span><span class="n">strcpy</span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span><span class="w"> </span><span class="c1">// If the specified key is found, update its val and return</span>
 <a id="__codelineno-10-81" name="__codelineno-10-81" href="#__codelineno-10-81"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-10-82" name="__codelineno-10-82" href="#__codelineno-10-82"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-10-83" name="__codelineno-10-83" href="#__codelineno-10-83"></a><span class="w">        </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-10-84" name="__codelineno-10-84" href="#__codelineno-10-84"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-85" name="__codelineno-10-85" href="#__codelineno-10-85"></a><span class="w">    </span><span class="c1">// If key not found, add key-value pair to list head</span>
+<a id="__codelineno-10-85" name="__codelineno-10-85" href="#__codelineno-10-85"></a><span class="w">    </span><span class="c1">// If key is absent, add the key-value pair at the head of the linked list</span>
 <a id="__codelineno-10-86" name="__codelineno-10-86" href="#__codelineno-10-86"></a><span class="w">    </span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">newPair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">Pair</span><span class="p">));</span>
 <a id="__codelineno-10-87" name="__codelineno-10-87" href="#__codelineno-10-87"></a><span class="w">    </span><span class="n">newPair</span><span class="o">-&gt;</span><span class="n">key</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">key</span><span class="p">;</span>
 <a id="__codelineno-10-88" name="__codelineno-10-88" href="#__codelineno-10-88"></a><span class="w">    </span><span class="n">strcpy</span><span class="p">(</span><span class="n">newPair</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
@@ -5967,19 +6039,19 @@
 <a id="__codelineno-10-93" name="__codelineno-10-93" href="#__codelineno-10-93"></a><span class="w">    </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-10-94" name="__codelineno-10-94" href="#__codelineno-10-94"></a><span class="p">}</span>
 <a id="__codelineno-10-95" name="__codelineno-10-95" href="#__codelineno-10-95"></a>
-<a id="__codelineno-10-96" name="__codelineno-10-96" href="#__codelineno-10-96"></a><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-10-96" name="__codelineno-10-96" href="#__codelineno-10-96"></a><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-10-97" name="__codelineno-10-97" href="#__codelineno-10-97"></a><span class="kt">void</span><span class="w"> </span><span class="nf">extend</span><span class="p">(</span><span class="n">HashMapChaining</span><span class="w"> </span><span class="o">*</span><span class="n">hashMap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-98" name="__codelineno-10-98" href="#__codelineno-10-98"></a><span class="w">    </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-10-98" name="__codelineno-10-98" href="#__codelineno-10-98"></a><span class="w">    </span><span class="c1">// Keep a reference to the original hash table</span>
 <a id="__codelineno-10-99" name="__codelineno-10-99" href="#__codelineno-10-99"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">oldCapacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-10-100" name="__codelineno-10-100" href="#__codelineno-10-100"></a><span class="w">    </span><span class="n">Node</span><span class="w"> </span><span class="o">**</span><span class="n">oldBuckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">;</span>
-<a id="__codelineno-10-101" name="__codelineno-10-101" href="#__codelineno-10-101"></a><span class="w">    </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-10-101" name="__codelineno-10-101" href="#__codelineno-10-101"></a><span class="w">    </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-10-102" name="__codelineno-10-102" href="#__codelineno-10-102"></a><span class="w">    </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">extendRatio</span><span class="p">;</span>
 <a id="__codelineno-10-103" name="__codelineno-10-103" href="#__codelineno-10-103"></a><span class="w">    </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">Node</span><span class="w"> </span><span class="o">**</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">capacity</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="n">Node</span><span class="w"> </span><span class="o">*</span><span class="p">));</span>
 <a id="__codelineno-10-104" name="__codelineno-10-104" href="#__codelineno-10-104"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">capacity</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-105" name="__codelineno-10-105" href="#__codelineno-10-105"></a><span class="w">        </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-10-106" name="__codelineno-10-106" href="#__codelineno-10-106"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-107" name="__codelineno-10-107" href="#__codelineno-10-107"></a><span class="w">    </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-10-108" name="__codelineno-10-108" href="#__codelineno-10-108"></a><span class="w">    </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-10-108" name="__codelineno-10-108" href="#__codelineno-10-108"></a><span class="w">    </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-10-109" name="__codelineno-10-109" href="#__codelineno-10-109"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">oldCapacity</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-110" name="__codelineno-10-110" href="#__codelineno-10-110"></a><span class="w">        </span><span class="n">Node</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">oldBuckets</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-10-111" name="__codelineno-10-111" href="#__codelineno-10-111"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6002,7 +6074,7 @@
 <a id="__codelineno-10-128" name="__codelineno-10-128" href="#__codelineno-10-128"></a><span class="w">    </span><span class="n">Node</span><span class="w"> </span><span class="o">*</span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-10-129" name="__codelineno-10-129" href="#__codelineno-10-129"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-130" name="__codelineno-10-130" href="#__codelineno-10-130"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-131" name="__codelineno-10-131" href="#__codelineno-10-131"></a><span class="w">            </span><span class="c1">// Remove key-value pair from it</span>
+<a id="__codelineno-10-131" name="__codelineno-10-131" href="#__codelineno-10-131"></a><span class="w">            </span><span class="c1">// Remove the key-value pair from the list</span>
 <a id="__codelineno-10-132" name="__codelineno-10-132" href="#__codelineno-10-132"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-133" name="__codelineno-10-133" href="#__codelineno-10-133"></a><span class="w">                </span><span class="n">pre</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-10-134" name="__codelineno-10-134" href="#__codelineno-10-134"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -6019,7 +6091,7 @@
 <a id="__codelineno-10-145" name="__codelineno-10-145" href="#__codelineno-10-145"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-146" name="__codelineno-10-146" href="#__codelineno-10-146"></a><span class="p">}</span>
 <a id="__codelineno-10-147" name="__codelineno-10-147" href="#__codelineno-10-147"></a>
-<a id="__codelineno-10-148" name="__codelineno-10-148" href="#__codelineno-10-148"></a><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-10-148" name="__codelineno-10-148" href="#__codelineno-10-148"></a><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-10-149" name="__codelineno-10-149" href="#__codelineno-10-149"></a><span class="kt">void</span><span class="w"> </span><span class="nf">print</span><span class="p">(</span><span class="n">HashMapChaining</span><span class="w"> </span><span class="o">*</span><span class="n">hashMap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-150" name="__codelineno-10-150" href="#__codelineno-10-150"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">capacity</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-151" name="__codelineno-10-151" href="#__codelineno-10-151"></a><span class="w">        </span><span class="n">Node</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
@@ -6038,8 +6110,8 @@
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">HashMapChaining</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">size</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">capacity</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">loadThres</span><span class="p">:</span><span class="w"> </span><span class="kt">Double</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">extendRatio</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">loadThres</span><span class="p">:</span><span class="w"> </span><span class="kt">Double</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">extendRatio</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">buckets</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="c1">// Bucket array</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -6064,34 +6136,34 @@
 <a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">capacity</span><span class="p">).</span><span class="na">toDouble</span><span class="p">()</span>
 <a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a>
-<a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">String?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-11-34" name="__codelineno-11-34" href="#__codelineno-11-34"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span>
-<a id="__codelineno-11-35" name="__codelineno-11-35" href="#__codelineno-11-35"></a><span class="w">        </span><span class="c1">// Traverse bucket, if key is found, return corresponding val</span>
+<a id="__codelineno-11-35" name="__codelineno-11-35" href="#__codelineno-11-35"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if key is found, return its corresponding val</span>
 <a id="__codelineno-11-36" name="__codelineno-11-36" href="#__codelineno-11-36"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-37" name="__codelineno-11-37" href="#__codelineno-11-37"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="na">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="na">_val</span>
 <a id="__codelineno-11-38" name="__codelineno-11-38" href="#__codelineno-11-38"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-11-39" name="__codelineno-11-39" href="#__codelineno-11-39"></a><span class="w">        </span><span class="c1">// If key is not found, return null</span>
+<a id="__codelineno-11-39" name="__codelineno-11-39" href="#__codelineno-11-39"></a><span class="w">        </span><span class="c1">// Return null if key is not found</span>
 <a id="__codelineno-11-40" name="__codelineno-11-40" href="#__codelineno-11-40"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span>
 <a id="__codelineno-11-41" name="__codelineno-11-41" href="#__codelineno-11-41"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-42" name="__codelineno-11-42" href="#__codelineno-11-42"></a>
-<a id="__codelineno-11-43" name="__codelineno-11-43" href="#__codelineno-11-43"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-11-43" name="__codelineno-11-43" href="#__codelineno-11-43"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-11-44" name="__codelineno-11-44" href="#__codelineno-11-44"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">String</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-45" name="__codelineno-11-45" href="#__codelineno-11-45"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-11-45" name="__codelineno-11-45" href="#__codelineno-11-45"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-11-46" name="__codelineno-11-46" href="#__codelineno-11-46"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-47" name="__codelineno-11-47" href="#__codelineno-11-47"></a><span class="w">            </span><span class="n">extend</span><span class="p">()</span>
 <a id="__codelineno-11-48" name="__codelineno-11-48" href="#__codelineno-11-48"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-11-49" name="__codelineno-11-49" href="#__codelineno-11-49"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-11-50" name="__codelineno-11-50" href="#__codelineno-11-50"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span>
-<a id="__codelineno-11-51" name="__codelineno-11-51" href="#__codelineno-11-51"></a><span class="w">        </span><span class="c1">// Traverse bucket, if specified key is encountered, update corresponding val and return</span>
+<a id="__codelineno-11-51" name="__codelineno-11-51" href="#__codelineno-11-51"></a><span class="w">        </span><span class="c1">// Traverse the bucket; if the specified key is found, update its val and return</span>
 <a id="__codelineno-11-52" name="__codelineno-11-52" href="#__codelineno-11-52"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-53" name="__codelineno-11-53" href="#__codelineno-11-53"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="na">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-54" name="__codelineno-11-54" href="#__codelineno-11-54"></a><span class="w">                </span><span class="n">pair</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_val</span>
 <a id="__codelineno-11-55" name="__codelineno-11-55" href="#__codelineno-11-55"></a><span class="w">                </span><span class="k">return</span>
 <a id="__codelineno-11-56" name="__codelineno-11-56" href="#__codelineno-11-56"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-11-57" name="__codelineno-11-57" href="#__codelineno-11-57"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-11-58" name="__codelineno-11-58" href="#__codelineno-11-58"></a><span class="w">        </span><span class="c1">// If key does not exist, append key-value pair to the end</span>
+<a id="__codelineno-11-58" name="__codelineno-11-58" href="#__codelineno-11-58"></a><span class="w">        </span><span class="c1">// If key is absent, append the key-value pair at the end</span>
 <a id="__codelineno-11-59" name="__codelineno-11-59" href="#__codelineno-11-59"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">)</span>
 <a id="__codelineno-11-60" name="__codelineno-11-60" href="#__codelineno-11-60"></a><span class="w">        </span><span class="n">bucket</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">pair</span><span class="p">)</span>
 <a id="__codelineno-11-61" name="__codelineno-11-61" href="#__codelineno-11-61"></a><span class="w">        </span><span class="n">size</span><span class="o">++</span>
@@ -6101,7 +6173,7 @@
 <a id="__codelineno-11-65" name="__codelineno-11-65" href="#__codelineno-11-65"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-66" name="__codelineno-11-66" href="#__codelineno-11-66"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-11-67" name="__codelineno-11-67" href="#__codelineno-11-67"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span>
-<a id="__codelineno-11-68" name="__codelineno-11-68" href="#__codelineno-11-68"></a><span class="w">        </span><span class="c1">// Traverse bucket and remove key-value pair from it</span>
+<a id="__codelineno-11-68" name="__codelineno-11-68" href="#__codelineno-11-68"></a><span class="w">        </span><span class="c1">// Traverse the bucket and remove the key-value pair</span>
 <a id="__codelineno-11-69" name="__codelineno-11-69" href="#__codelineno-11-69"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-70" name="__codelineno-11-70" href="#__codelineno-11-70"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="na">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-71" name="__codelineno-11-71" href="#__codelineno-11-71"></a><span class="w">                </span><span class="n">bucket</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">pair</span><span class="p">)</span>
@@ -6111,19 +6183,19 @@
 <a id="__codelineno-11-75" name="__codelineno-11-75" href="#__codelineno-11-75"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-11-76" name="__codelineno-11-76" href="#__codelineno-11-76"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-77" name="__codelineno-11-77" href="#__codelineno-11-77"></a>
-<a id="__codelineno-11-78" name="__codelineno-11-78" href="#__codelineno-11-78"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-11-78" name="__codelineno-11-78" href="#__codelineno-11-78"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-11-79" name="__codelineno-11-79" href="#__codelineno-11-79"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-80" name="__codelineno-11-80" href="#__codelineno-11-80"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-11-80" name="__codelineno-11-80" href="#__codelineno-11-80"></a><span class="w">        </span><span class="c1">// Keep a reference to the original hash table</span>
 <a id="__codelineno-11-81" name="__codelineno-11-81" href="#__codelineno-11-81"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">bucketsTmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span>
-<a id="__codelineno-11-82" name="__codelineno-11-82" href="#__codelineno-11-82"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-11-82" name="__codelineno-11-82" href="#__codelineno-11-82"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-11-83" name="__codelineno-11-83" href="#__codelineno-11-83"></a><span class="w">        </span><span class="n">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="n">extendRatio</span>
-<a id="__codelineno-11-84" name="__codelineno-11-84" href="#__codelineno-11-84"></a><span class="w">        </span><span class="c1">// mutablelist has no fixed size</span>
+<a id="__codelineno-11-84" name="__codelineno-11-84" href="#__codelineno-11-84"></a><span class="w">        </span><span class="c1">// A mutable list has no fixed size</span>
 <a id="__codelineno-11-85" name="__codelineno-11-85" href="#__codelineno-11-85"></a><span class="w">        </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="p">()</span>
 <a id="__codelineno-11-86" name="__codelineno-11-86" href="#__codelineno-11-86"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-87" name="__codelineno-11-87" href="#__codelineno-11-87"></a><span class="w">            </span><span class="n">buckets</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">mutableListOf</span><span class="p">())</span>
 <a id="__codelineno-11-88" name="__codelineno-11-88" href="#__codelineno-11-88"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-11-89" name="__codelineno-11-89" href="#__codelineno-11-89"></a><span class="w">        </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-11-90" name="__codelineno-11-90" href="#__codelineno-11-90"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-11-90" name="__codelineno-11-90" href="#__codelineno-11-90"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-11-91" name="__codelineno-11-91" href="#__codelineno-11-91"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-92" name="__codelineno-11-92" href="#__codelineno-11-92"></a><span class="w">            </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-93" name="__codelineno-11-93" href="#__codelineno-11-93"></a><span class="w">                </span><span class="n">put</span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="na">key</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="na">_val</span><span class="p">)</span>
@@ -6131,7 +6203,7 @@
 <a id="__codelineno-11-95" name="__codelineno-11-95" href="#__codelineno-11-95"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-11-96" name="__codelineno-11-96" href="#__codelineno-11-96"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-97" name="__codelineno-11-97" href="#__codelineno-11-97"></a>
-<a id="__codelineno-11-98" name="__codelineno-11-98" href="#__codelineno-11-98"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-11-98" name="__codelineno-11-98" href="#__codelineno-11-98"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-11-99" name="__codelineno-11-99" href="#__codelineno-11-99"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-100" name="__codelineno-11-100" href="#__codelineno-11-100"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-101" name="__codelineno-11-101" href="#__codelineno-11-101"></a><span class="w">            </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="kt">String</span><span class="o">&gt;</span><span class="p">()</span>
@@ -6147,14 +6219,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map_chaining.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Hash map with chaining ###</span>
+<div class="highlight"><span class="filename">hash_map_chaining.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Hash table using chaining ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">HashMapChaining</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1">### Constructor ###</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">initialize</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Number of key-value pairs</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">    </span><span class="vi">@capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">4</span><span class="w"> </span><span class="c1"># Hash table capacity</span>
-<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">    </span><span class="vi">@load_thres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="o">.</span><span class="mi">0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">3</span><span class="o">.</span><span class="mi">0</span><span class="w"> </span><span class="c1"># Load factor threshold for triggering expansion</span>
-<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="vi">@extend_ratio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1"># Expansion multiplier</span>
+<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">    </span><span class="vi">@load_thres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="o">.</span><span class="mi">0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">3</span><span class="o">.</span><span class="mi">0</span><span class="w"> </span><span class="c1"># Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="vi">@extend_ratio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1"># Capacity growth factor</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">    </span><span class="vi">@buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="vi">@capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">[]</span><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="c1"># Bucket array</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a>
@@ -6168,42 +6240,42 @@
 <a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="vi">@capacity</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a>
-<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w">  </span><span class="c1">### Query operation ###</span>
+<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w">  </span><span class="c1">### Lookup operation ###</span>
 <a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a><span class="w">    </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="w">    </span><span class="n">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span>
-<a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></a><span class="w">    </span><span class="c1"># Traverse bucket, if key is found, return corresponding val</span>
+<a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></a><span class="w">    </span><span class="c1"># Traverse the bucket; if key is found, return its corresponding val</span>
 <a id="__codelineno-12-27" name="__codelineno-12-27" href="#__codelineno-12-27"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span>
 <a id="__codelineno-12-28" name="__codelineno-12-28" href="#__codelineno-12-28"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">pair</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">pair</span><span class="o">.</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span>
 <a id="__codelineno-12-29" name="__codelineno-12-29" href="#__codelineno-12-29"></a><span class="w">    </span><span class="k">end</span>
-<a id="__codelineno-12-30" name="__codelineno-12-30" href="#__codelineno-12-30"></a><span class="w">    </span><span class="c1"># Return nil if key not found</span>
+<a id="__codelineno-12-30" name="__codelineno-12-30" href="#__codelineno-12-30"></a><span class="w">    </span><span class="c1"># Return nil if key is not found</span>
 <a id="__codelineno-12-31" name="__codelineno-12-31" href="#__codelineno-12-31"></a><span class="w">    </span><span class="kp">nil</span>
 <a id="__codelineno-12-32" name="__codelineno-12-32" href="#__codelineno-12-32"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-33" name="__codelineno-12-33" href="#__codelineno-12-33"></a>
-<a id="__codelineno-12-34" name="__codelineno-12-34" href="#__codelineno-12-34"></a><span class="w">  </span><span class="c1">### Add operation ###</span>
+<a id="__codelineno-12-34" name="__codelineno-12-34" href="#__codelineno-12-34"></a><span class="w">  </span><span class="c1">### Insert operation ###</span>
 <a id="__codelineno-12-35" name="__codelineno-12-35" href="#__codelineno-12-35"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
-<a id="__codelineno-12-36" name="__codelineno-12-36" href="#__codelineno-12-36"></a><span class="w">    </span><span class="c1"># When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-12-36" name="__codelineno-12-36" href="#__codelineno-12-36"></a><span class="w">    </span><span class="c1"># Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-12-37" name="__codelineno-12-37" href="#__codelineno-12-37"></a><span class="w">    </span><span class="kp">extend</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">load_factor</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="vi">@load_thres</span>
 <a id="__codelineno-12-38" name="__codelineno-12-38" href="#__codelineno-12-38"></a><span class="w">    </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-12-39" name="__codelineno-12-39" href="#__codelineno-12-39"></a><span class="w">    </span><span class="n">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span>
-<a id="__codelineno-12-40" name="__codelineno-12-40" href="#__codelineno-12-40"></a><span class="w">    </span><span class="c1"># Traverse bucket, if specified key is encountered, update corresponding val and return</span>
+<a id="__codelineno-12-40" name="__codelineno-12-40" href="#__codelineno-12-40"></a><span class="w">    </span><span class="c1"># Traverse the bucket; if the specified key is found, update its val and return</span>
 <a id="__codelineno-12-41" name="__codelineno-12-41" href="#__codelineno-12-41"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span>
 <a id="__codelineno-12-42" name="__codelineno-12-42" href="#__codelineno-12-42"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">pair</span><span class="o">.</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span>
 <a id="__codelineno-12-43" name="__codelineno-12-43" href="#__codelineno-12-43"></a><span class="w">        </span><span class="n">pair</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span>
 <a id="__codelineno-12-44" name="__codelineno-12-44" href="#__codelineno-12-44"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-12-45" name="__codelineno-12-45" href="#__codelineno-12-45"></a><span class="w">      </span><span class="k">end</span>
 <a id="__codelineno-12-46" name="__codelineno-12-46" href="#__codelineno-12-46"></a><span class="w">    </span><span class="k">end</span>
-<a id="__codelineno-12-47" name="__codelineno-12-47" href="#__codelineno-12-47"></a><span class="w">    </span><span class="c1"># If key does not exist, append key-value pair to the end</span>
+<a id="__codelineno-12-47" name="__codelineno-12-47" href="#__codelineno-12-47"></a><span class="w">    </span><span class="c1"># If key is absent, append the key-value pair at the end</span>
 <a id="__codelineno-12-48" name="__codelineno-12-48" href="#__codelineno-12-48"></a><span class="w">    </span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">Pair</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-12-49" name="__codelineno-12-49" href="#__codelineno-12-49"></a><span class="w">    </span><span class="n">bucket</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">pair</span>
 <a id="__codelineno-12-50" name="__codelineno-12-50" href="#__codelineno-12-50"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-51" name="__codelineno-12-51" href="#__codelineno-12-51"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-52" name="__codelineno-12-52" href="#__codelineno-12-52"></a>
-<a id="__codelineno-12-53" name="__codelineno-12-53" href="#__codelineno-12-53"></a><span class="w">  </span><span class="c1">### Delete operation ###</span>
+<a id="__codelineno-12-53" name="__codelineno-12-53" href="#__codelineno-12-53"></a><span class="w">  </span><span class="c1">### Remove operation ###</span>
 <a id="__codelineno-12-54" name="__codelineno-12-54" href="#__codelineno-12-54"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-12-55" name="__codelineno-12-55" href="#__codelineno-12-55"></a><span class="w">    </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-12-56" name="__codelineno-12-56" href="#__codelineno-12-56"></a><span class="w">    </span><span class="n">bucket</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span>
-<a id="__codelineno-12-57" name="__codelineno-12-57" href="#__codelineno-12-57"></a><span class="w">    </span><span class="c1"># Traverse bucket and remove key-value pair from it</span>
+<a id="__codelineno-12-57" name="__codelineno-12-57" href="#__codelineno-12-57"></a><span class="w">    </span><span class="c1"># Traverse the bucket and remove the key-value pair</span>
 <a id="__codelineno-12-58" name="__codelineno-12-58" href="#__codelineno-12-58"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span>
 <a id="__codelineno-12-59" name="__codelineno-12-59" href="#__codelineno-12-59"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">pair</span><span class="o">.</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span>
 <a id="__codelineno-12-60" name="__codelineno-12-60" href="#__codelineno-12-60"></a><span class="w">        </span><span class="n">bucket</span><span class="o">.</span><span class="n">delete</span><span class="p">(</span><span class="n">pair</span><span class="p">)</span>
@@ -6213,15 +6285,15 @@
 <a id="__codelineno-12-64" name="__codelineno-12-64" href="#__codelineno-12-64"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-12-65" name="__codelineno-12-65" href="#__codelineno-12-65"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-66" name="__codelineno-12-66" href="#__codelineno-12-66"></a>
-<a id="__codelineno-12-67" name="__codelineno-12-67" href="#__codelineno-12-67"></a><span class="w">  </span><span class="c1">### Expand hash table ###</span>
+<a id="__codelineno-12-67" name="__codelineno-12-67" href="#__codelineno-12-67"></a><span class="w">  </span><span class="c1">### Grow the hash table capacity ###</span>
 <a id="__codelineno-12-68" name="__codelineno-12-68" href="#__codelineno-12-68"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">extend</span>
-<a id="__codelineno-12-69" name="__codelineno-12-69" href="#__codelineno-12-69"></a><span class="w">    </span><span class="c1"># Temporarily store original hash table</span>
+<a id="__codelineno-12-69" name="__codelineno-12-69" href="#__codelineno-12-69"></a><span class="w">    </span><span class="c1"># Temporarily store the original hash table</span>
 <a id="__codelineno-12-70" name="__codelineno-12-70" href="#__codelineno-12-70"></a><span class="w">    </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@buckets</span>
-<a id="__codelineno-12-71" name="__codelineno-12-71" href="#__codelineno-12-71"></a><span class="w">    </span><span class="c1"># Initialize expanded new hash table</span>
+<a id="__codelineno-12-71" name="__codelineno-12-71" href="#__codelineno-12-71"></a><span class="w">    </span><span class="c1"># Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-12-72" name="__codelineno-12-72" href="#__codelineno-12-72"></a><span class="w">    </span><span class="vi">@capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="vi">@extend_ratio</span>
 <a id="__codelineno-12-73" name="__codelineno-12-73" href="#__codelineno-12-73"></a><span class="w">    </span><span class="vi">@buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="vi">@capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">[]</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-12-74" name="__codelineno-12-74" href="#__codelineno-12-74"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-12-75" name="__codelineno-12-75" href="#__codelineno-12-75"></a><span class="w">    </span><span class="c1"># Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-12-75" name="__codelineno-12-75" href="#__codelineno-12-75"></a><span class="w">    </span><span class="c1"># Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-12-76" name="__codelineno-12-76" href="#__codelineno-12-76"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span>
 <a id="__codelineno-12-77" name="__codelineno-12-77" href="#__codelineno-12-77"></a><span class="w">      </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span>
 <a id="__codelineno-12-78" name="__codelineno-12-78" href="#__codelineno-12-78"></a><span class="w">        </span><span class="n">put</span><span class="p">(</span><span class="n">pair</span><span class="o">.</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="o">.</span><span class="n">val</span><span class="p">)</span>
@@ -6229,7 +6301,7 @@
 <a id="__codelineno-12-80" name="__codelineno-12-80" href="#__codelineno-12-80"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-12-81" name="__codelineno-12-81" href="#__codelineno-12-81"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-82" name="__codelineno-12-82" href="#__codelineno-12-82"></a>
-<a id="__codelineno-12-83" name="__codelineno-12-83" href="#__codelineno-12-83"></a><span class="w">  </span><span class="c1">### Print hash table ###</span>
+<a id="__codelineno-12-83" name="__codelineno-12-83" href="#__codelineno-12-83"></a><span class="w">  </span><span class="c1">### Print the hash table ###</span>
 <a id="__codelineno-12-84" name="__codelineno-12-84" href="#__codelineno-12-84"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">print</span>
 <a id="__codelineno-12-85" name="__codelineno-12-85" href="#__codelineno-12-85"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="vi">@buckets</span>
 <a id="__codelineno-12-86" name="__codelineno-12-86" href="#__codelineno-12-86"></a><span class="w">      </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
@@ -6244,29 +6316,34 @@
 </div>
 </div>
 </div>
-<p>It's worth noting that when the linked list becomes very long, the query time <span class="arithmatex">\(O(n)\)</span> is poor. <strong>In this case, the linked list can be converted into an AVL tree or a red-black tree</strong>, reducing the time complexity of lookups to <span class="arithmatex">\(O(\log n)\)</span>.</p>
-<h2 id="622-open-addressing">6.2.2 &nbsp; Open Addressing<a class="headerlink" href="#622-open-addressing" title="Permanent link">&para;</a></h2>
-<p><u>Open addressing</u> does not introduce additional data structures. Instead, it handles hash collisions through repeated probing. Common probing strategies include linear probing, quadratic probing, and multiple hashing.</p>
-<p>Let's use linear probing as an example to introduce the mechanism of open addressing hash tables.</p>
-<h3 id="1-linear-probing">1. &nbsp; Linear Probing<a class="headerlink" href="#1-linear-probing" title="Permanent link">&para;</a></h3>
-<p>Linear probing uses a fixed step size to probe sequentially, so its operations differ somewhat from those of an ordinary hash table.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20Pair%3A%0A%20%20%20%20%22%22%22Key-value%20pair%22%22%22%0A%20%20%20%20def%20__init__%28self,%20key%3A%20int,%20val%3A%20str%29%3A%0A%20%20%20%20%20%20%20%20self.key%20%3D%20key%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%0Aclass%20HashMapChaining%3A%0A%20%20%20%20%22%22%22Hash%20table%20with%20separate%20chaining%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self.size%20%3D%200%0A%20%20%20%20%20%20%20%20self.capacity%20%3D%204%0A%20%20%20%20%20%20%20%20self.load_thres%20%3D%202.0%20/%203.0%0A%20%20%20%20%20%20%20%20self.extend_ratio%20%3D%202%0A%20%20%20%20%20%20%20%20self.buckets%20%3D%20%5B%5B%5D%20for%20_%20in%20range%28self.capacity%29%5D%0A%0A%20%20%20%20def%20hash_func%28self,%20key%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Hash%20function%22%22%22%0A%20%20%20%20%20%20%20%20return%20key%20%25%20self.capacity%0A%0A%20%20%20%20def%20load_factor%28self%29%20-%3E%20float%3A%0A%20%20%20%20%20%20%20%20%22%22%22Load%20factor%22%22%22%0A%20%20%20%20%20%20%20%20return%20self.size%20/%20self.capacity%0A%0A%20%20%20%20def%20get%28self,%20key%3A%20int%29%20-%3E%20str%20%7C%20None%3A%0A%20%20%20%20%20%20%20%20%22%22%22Lookup%20operation%22%22%22%0A%20%20%20%20%20%20%20%20index%20%3D%20self.hash_func%28key%29%0A%20%20%20%20%20%20%20%20bucket%20%3D%20self.buckets%5Bindex%5D%0A%20%20%20%20%20%20%20%20for%20pair%20in%20bucket%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20pair.key%20%3D%3D%20key%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20return%20pair.val%0A%20%20%20%20%20%20%20%20return%20None%0A%0A%20%20%20%20def%20put%28self,%20key%3A%20int,%20val%3A%20str%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Insert%20operation%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.load_factor%28%29%20%3E%20self.load_thres%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.extend%28%29%0A%20%20%20%20%20%20%20%20index%20%3D%20self.hash_func%28key%29%0A%20%20%20%20%20%20%20%20bucket%20%3D%20self.buckets%5Bindex%5D%0A%20%20%20%20%20%20%20%20for%20pair%20in%20bucket%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20pair.key%20%3D%3D%20key%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20pair.val%20%3D%20val%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20pair%20%3D%20Pair%28key,%20val%29%0A%20%20%20%20%20%20%20%20bucket.append%28pair%29%0A%20%20%20%20%20%20%20%20self.size%20%2B%3D%201%0A%0A%20%20%20%20def%20remove%28self,%20key%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Remove%20operation%22%22%22%0A%20%20%20%20%20%20%20%20index%20%3D%20self.hash_func%28key%29%0A%20%20%20%20%20%20%20%20bucket%20%3D%20self.buckets%5Bindex%5D%0A%20%20%20%20%20%20%20%20for%20pair%20in%20bucket%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20pair.key%20%3D%3D%20key%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20bucket.remove%28pair%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20self.size%20-%3D%201%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20break%0A%0A%20%20%20%20def%20extend%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Grow%20the%20hash%20table%20capacity%22%22%22%0A%20%20%20%20%20%20%20%20buckets%20%3D%20self.buckets%0A%20%20%20%20%20%20%20%20self.capacity%20%2A%3D%20self.extend_ratio%0A%20%20%20%20%20%20%20%20self.buckets%20%3D%20%5B%5B%5D%20for%20_%20in%20range%28self.capacity%29%5D%0A%20%20%20%20%20%20%20%20self.size%20%3D%200%0A%20%20%20%20%20%20%20%20for%20bucket%20in%20buckets%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20for%20pair%20in%20bucket%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20self.put%28pair.key,%20pair.val%29%0A%0A%20%20%20%20def%20print%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Print%20the%20hash%20table%22%22%22%0A%20%20%20%20%20%20%20%20for%20bucket%20in%20self.buckets%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20for%20pair%20in%20bucket%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20res.append%28str%28pair.key%29%20%2B%20%22%20-%3E%20%22%20%2B%20pair.val%29%0A%20%20%20%20%20%20%20%20%20%20%20%20print%28res%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20hash%20table%0A%20%20%20%20hashmap%20%3D%20HashMapChaining%28%29%0A%0A%20%20%20%20%23%20Insert%20operation%0A%20%20%20%20hashmap.put%2812836,%20%22%E5%B0%8F%E5%93%88%22%29%0A%20%20%20%20hashmap.put%2815937,%20%22%E5%B0%8F%E5%95%B0%22%29%0A%20%20%20%20hashmap.put%2816750,%20%22%E5%B0%8F%E7%AE%97%22%29%0A%20%20%20%20hashmap.put%2813276,%20%22%E5%B0%8F%E6%B3%95%22%29%0A%20%20%20%20hashmap.put%2810583,%20%22%E5%B0%8F%E9%B8%AD%22%29%0A%0A%20%20%20%20%23%20Lookup%20operation%0A%20%20%20%20name%20%3D%20hashmap.get%2813276%29%0A%0A%20%20%20%20%23%20Remove%20operation%0A%20%20%20%20hashmap.remove%2812836%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-fronte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</iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20Pair%3A%0A%20%20%20%20%22%22%22Key-value%20pair%22%22%22%0A%20%20%20%20def%20__init__%28self,%20key%3A%20int,%20val%3A%20str%29%3A%0A%20%20%20%20%20%20%20%20self.key%20%3D%20key%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%0Aclass%20HashMapChaining%3A%0A%20%20%20%20%22%22%22Hash%20table%20with%20separate%20chaining%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self.size%20%3D%200%0A%20%20%20%20%20%20%20%20self.capacity%20%3D%204%0A%20%20%20%20%20%20%20%20self.load_thres%20%3D%202.0%20/%203.0%0A%20%20%20%20%20%20%20%20self.extend_ratio%20%3D%202%0A%20%20%20%20%20%20%20%20self.buckets%20%3D%20%5B%5B%5D%20for%20_%20in%20range%28self.capacity%29%5D%0A%0A%20%20%20%20def%20hash_func%28self,%20key%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Hash%20function%22%22%22%0A%20%20%20%20%20%20%20%20return%20key%20%25%20self.capacity%0A%0A%20%20%20%20def%20load_factor%28self%29%20-%3E%20float%3A%0A%20%20%20%20%20%20%20%20%22%22%22Load%20factor%22%22%22%0A%20%20%20%20%20%20%20%20return%20self.size%20/%20self.capacity%0A%0A%20%20%20%20def%20get%28self,%20key%3A%20int%29%20-%3E%20str%20%7C%20None%3A%0A%20%20%20%20%20%20%20%20%22%22%22Lookup%20operation%22%22%22%0A%20%20%20%20%20%20%20%20index%20%3D%20self.hash_func%28key%29%0A%20%20%20%20%20%20%20%20bucket%20%3D%20self.buckets%5Bindex%5D%0A%20%20%20%20%20%20%20%20for%20pair%20in%20bucket%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20pair.key%20%3D%3D%20key%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20return%20pair.val%0A%20%20%20%20%20%20%20%20return%20None%0A%0A%20%20%20%20def%20put%28self,%20key%3A%20int,%20val%3A%20str%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Insert%20operation%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.load_factor%28%29%20%3E%20self.load_thres%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.extend%28%29%0A%20%20%20%20%20%20%20%20index%20%3D%20self.hash_func%28key%29%0A%20%20%20%20%20%20%20%20bucket%20%3D%20self.buckets%5Bindex%5D%0A%20%20%20%20%20%20%20%20for%20pair%20in%20bucket%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20pair.key%20%3D%3D%20key%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20pair.val%20%3D%20val%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20pair%20%3D%20Pair%28key,%20val%29%0A%20%20%20%20%20%20%20%20bucket.append%28pair%29%0A%20%20%20%20%20%20%20%20self.size%20%2B%3D%201%0A%0A%20%20%20%20def%20remove%28self,%20key%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Remove%20operation%22%22%22%0A%20%20%20%20%20%20%20%20index%20%3D%20self.hash_func%28key%29%0A%20%20%20%20%20%20%20%20bucket%20%3D%20self.buckets%5Bindex%5D%0A%20%20%20%20%20%20%20%20for%20pair%20in%20bucket%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20pair.key%20%3D%3D%20key%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20bucket.remove%28pair%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20self.size%20-%3D%201%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20break%0A%0A%20%20%20%20def%20extend%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Grow%20the%20hash%20table%20capacity%22%22%22%0A%20%20%20%20%20%20%20%20buckets%20%3D%20self.buckets%0A%20%20%20%20%20%20%20%20self.capacity%20%2A%3D%20self.extend_ratio%0A%20%20%20%20%20%20%20%20self.buckets%20%3D%20%5B%5B%5D%20for%20_%20in%20range%28self.capacity%29%5D%0A%20%20%20%20%20%20%20%20self.size%20%3D%200%0A%20%20%20%20%20%20%20%20for%20bucket%20in%20buckets%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20for%20pair%20in%20bucket%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20self.put%28pair.key,%20pair.val%29%0A%0A%20%20%20%20def%20print%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Print%20the%20hash%20table%22%22%22%0A%20%20%20%20%20%20%20%20for%20bucket%20in%20self.buckets%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20for%20pair%20in%20bucket%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20res.append%28str%28pair.key%29%20%2B%20%22%20-%3E%20%22%20%2B%20pair.val%29%0A%20%20%20%20%20%20%20%20%20%20%20%20print%28res%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20hash%20table%0A%20%20%20%20hashmap%20%3D%20HashMapChaining%28%29%0A%0A%20%20%20%20%23%20Insert%20operation%0A%20%20%20%20hashmap.put%2812836,%20%22%E5%B0%8F%E5%93%88%22%29%0A%20%20%20%20hashmap.put%2815937,%20%22%E5%B0%8F%E5%95%B0%22%29%0A%20%20%20%20hashmap.put%2816750,%20%22%E5%B0%8F%E7%AE%97%22%29%0A%20%20%20%20hashmap.put%2813276,%20%22%E5%B0%8F%E6%B3%95%22%29%0A%20%20%20%20hashmap.put%2810583,%20%22%E5%B0%8F%E9%B8%AD%22%29%0A%0A%20%20%20%20%23%20Lookup%20operation%0A%20%20%20%20name%20%3D%20hashmap.get%2813276%29%0A%0A%20%20%20%20%23%20Remove%20operation%0A%20%20%20%20hashmap.remove%2812836%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-front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target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>When a list becomes long, its <span class="arithmatex">\(O(n)\)</span> lookup time is poor. <strong>Replacing it with an AVL tree or red-black tree</strong> improves lookup time to <span class="arithmatex">\(O(\log n)\)</span>.</p>
+<h2 id="622-open-addressing">6.2.2 &nbsp; Open addressing<a class="headerlink" href="#622-open-addressing" title="Permanent link">&para;</a></h2>
+<p><u>Open addressing</u> handles collisions by probing multiple locations without introducing another data structure. Common strategies include linear probing, quadratic probing, and multiple hashing.</p>
+<p>We use linear probing to explain how an open-addressing hash table works.</p>
+<h3 id="1-linear-probing">1. &nbsp; Linear probing<a class="headerlink" href="#1-linear-probing" title="Permanent link">&para;</a></h3>
+<p>Linear probing searches with a fixed step size. Its operations differ from those of a basic hash table.</p>
 <ul>
-<li><strong>Inserting elements</strong>: Compute the bucket index using the hash function. If the bucket is already occupied, continue probing forward from the collision position with a fixed step size (usually <span class="arithmatex">\(1\)</span>) until an empty bucket is found, then insert the element there.</li>
-<li><strong>Searching for elements</strong>: If a collision occurs, continue probing forward with the same step size until the corresponding element is found and return its <code>value</code>; if an empty bucket is encountered, the target element is not in the hash table, so return <code>None</code>.</li>
+<li><strong>Insertion</strong>: Calculate the bucket index using the hash function. If the bucket is occupied, scan forward from that position with a fixed step size, usually <span class="arithmatex">\(1\)</span>, until an empty bucket is found, then insert the element.</li>
+<li><strong>Lookup</strong>: If a collision is encountered, scan forward with the same step size until the target is found, then return its <code>value</code>. An empty bucket means the target is absent, so return <code>None</code>.</li>
 </ul>
-<p>Figure 6-6 shows the distribution of key-value pairs in an open-addressing hash table that uses linear probing. Under this hash function, keys with the same last two digits are mapped to the same bucket. Linear probing then places them in that bucket and the subsequent buckets.</p>
-<p><img alt="Distribution of key-value pairs in open addressing (linear probing) hash table" class="animation-figure" src="../hash_collision.assets/hash_table_linear_probing.png" /></p>
-<p align="center"> Figure 6-6 &nbsp; Distribution of key-value pairs in open addressing (linear probing) hash table </p>
+<p>Figure 6-6 shows key-value pairs in an open-addressing table using linear probing. With this hash function, keys sharing the same last two digits map to the same bucket. Linear probing stores them successively in that bucket and the following buckets.</p>
+<p><img alt="Distribution of key-value pairs with linear probing" class="animation-figure" src="../hash_collision.assets/hash_table_linear_probing.png" /></p>
+<p align="center"> Figure 6-6 &nbsp; Distribution of key-value pairs with linear probing </p>
 
-<p>However, <strong>linear probing is prone to clustering</strong>. Specifically, the longer a contiguous occupied region in the array becomes, the more likely new collisions are to occur within that region. This in turn makes the cluster grow even further, creating a vicious cycle that gradually degrades the efficiency of insertion, deletion, lookup, and update operations.</p>
-<p>It's important to note that <strong>we cannot directly delete elements from an open-addressing hash table</strong>. Deleting an element creates an empty bucket <code>None</code> in the array. During lookup, once linear probing reaches that empty bucket, it stops, which means any elements stored farther along the probe sequence become unreachable. As a result, the program may incorrectly conclude that those elements do not exist, as shown in Figure 6-7.</p>
-<p><img alt="Query issues caused by deletion in open addressing" class="animation-figure" src="../hash_collision.assets/hash_table_open_addressing_deletion.png" /></p>
-<p align="center"> Figure 6-7 &nbsp; Query issues caused by deletion in open addressing </p>
+<p>However, <strong>linear probing is prone to clustering</strong>. The longer a run of occupied positions becomes, the more likely a collision is to occur within it, causing the cluster to grow further. This feedback loop eventually degrades insertion, deletion, lookup, and update performance.</p>
+<p><strong>We cannot simply delete an element from an open-addressing table</strong>. Deletion would leave an empty bucket, <code>None</code>. A lookup stops when linear probing reaches such a bucket, making later elements in the probe sequence inaccessible and potentially reporting them as absent, as shown below.</p>
+<p><img alt="Lookup problems caused by deletion in open addressing" class="animation-figure" src="../hash_collision.assets/hash_table_open_addressing_deletion.png" /></p>
+<p align="center"> Figure 6-7 &nbsp; Lookup problems caused by deletion in open addressing </p>
 
-<p>To solve this problem, we can adopt <u>lazy deletion</u>: instead of directly removing an element from the hash table, <strong>use a constant <code>TOMBSTONE</code> to mark the bucket</strong>. Under this mechanism, both <code>None</code> and <code>TOMBSTONE</code> denote buckets that can accept key-value pairs. The difference is that when linear probing encounters <code>TOMBSTONE</code>, it must continue probing, because key-value pairs may still exist farther along the sequence.</p>
-<p>However, <strong>lazy deletion may accelerate hash-table performance degradation</strong>. Each deletion leaves behind a marker, and as the number of <code>TOMBSTONE</code> entries grows, search time increases as well, because linear probing may need to skip over multiple tombstones before finding the target element.</p>
-<p>To address this, we can record the index of the first <code>TOMBSTONE</code> encountered during linear probing and swap the found target element into that position. The benefit is that each query or insertion can move elements closer to their ideal positions, that is, closer to where probing begins, which improves lookup efficiency.</p>
-<p>The code below implements an open addressing (linear probing) hash table with lazy deletion. To make better use of the hash table space, we treat the hash table as a "circular array". When going beyond the end of the array, we return to the beginning and continue traversing.</p>
+<p>To solve this, use <u>lazy deletion</u>: rather than simply removing the entry, <strong>mark its bucket with the constant <code>TOMBSTONE</code></strong>. Both <code>None</code> and <code>TOMBSTONE</code> represent available buckets that can hold a new pair. Unlike <code>None</code>, however, a <code>TOMBSTONE</code> does not stop a search, because further pairs may lie beyond it.</p>
+<p>However, <strong>lazy deletion can accelerate performance degradation</strong>. Each deletion leaves a marker. As <code>TOMBSTONE</code> entries accumulate, searches may need to skip many of them before reaching the target, increasing search time.</p>
+<p>To address this, record the first <code>TOMBSTONE</code> encountered during probing and swap the target entry with it when found. On each lookup or insertion, this moves the entry closer to its ideal location—the start of the probe sequence—and improves lookup efficiency.</p>
+<p>The code below implements linear probing with lazy deletion. To use the table's space more fully, treat the array as circular: after reaching the end, wrap around to the beginning.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -6277,10 +6354,10 @@
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="w">        </span><span class="sd">"""Constructor"""</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Number of key-value pairs</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span> <span class="o">=</span> <span class="mi">4</span>  <span class="c1"># Hash table capacity</span>
-<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">load_thres</span> <span class="o">=</span> <span class="mf">2.0</span> <span class="o">/</span> <span class="mf">3.0</span>  <span class="c1"># Load factor threshold for triggering expansion</span>
-<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">extend_ratio</span> <span class="o">=</span> <span class="mi">2</span>  <span class="c1"># Expansion multiplier</span>
+<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">load_thres</span> <span class="o">=</span> <span class="mf">2.0</span> <span class="o">/</span> <span class="mf">3.0</span>  <span class="c1"># Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">extend_ratio</span> <span class="o">=</span> <span class="mi">2</span>  <span class="c1"># Capacity growth factor</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="n">Pair</span> <span class="o">|</span> <span class="kc">None</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span>  <span class="c1"># Bucket array</span>
-<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">TOMBSTONE</span> <span class="o">=</span> <span class="n">Pair</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="s2">"-1"</span><span class="p">)</span>  <span class="c1"># Removal marker</span>
+<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">TOMBSTONE</span> <span class="o">=</span> <span class="n">Pair</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="s2">"-1"</span><span class="p">)</span>  <span class="c1"># Tombstone marker</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>
 <a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">hash_func</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a><span class="w">        </span><span class="sd">"""Hash function"""</span>
@@ -6291,76 +6368,76 @@
 <a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">/</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span>
 <a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a>
 <a id="__codelineno-13-21" name="__codelineno-13-21" href="#__codelineno-13-21"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">find_bucket</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a><span class="w">        </span><span class="sd">"""Search for bucket index corresponding to key"""</span>
+<a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a><span class="w">        </span><span class="sd">"""Find the bucket index for key"""</span>
 <a id="__codelineno-13-23" name="__codelineno-13-23" href="#__codelineno-13-23"></a>        <span class="n">index</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-13-24" name="__codelineno-13-24" href="#__codelineno-13-24"></a>        <span class="n">first_tombstone</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
-<a id="__codelineno-13-25" name="__codelineno-13-25" href="#__codelineno-13-25"></a>        <span class="c1"># Linear probing, break when encountering an empty bucket</span>
+<a id="__codelineno-13-25" name="__codelineno-13-25" href="#__codelineno-13-25"></a>        <span class="c1"># Probe linearly until an empty bucket is reached</span>
 <a id="__codelineno-13-26" name="__codelineno-13-26" href="#__codelineno-13-26"></a>        <span class="k">while</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-13-27" name="__codelineno-13-27" href="#__codelineno-13-27"></a>            <span class="c1"># If key is encountered, return the corresponding bucket index</span>
+<a id="__codelineno-13-27" name="__codelineno-13-27" href="#__codelineno-13-27"></a>            <span class="c1"># If key is found, return its bucket index</span>
 <a id="__codelineno-13-28" name="__codelineno-13-28" href="#__codelineno-13-28"></a>            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">.</span><span class="n">key</span> <span class="o">==</span> <span class="n">key</span><span class="p">:</span>
-<a id="__codelineno-13-29" name="__codelineno-13-29" href="#__codelineno-13-29"></a>                <span class="c1"># If a removal marker was encountered before, move the key-value pair to that index</span>
+<a id="__codelineno-13-29" name="__codelineno-13-29" href="#__codelineno-13-29"></a>                <span class="c1"># If a tombstone was encountered earlier, move the key-value pair to that index</span>
 <a id="__codelineno-13-30" name="__codelineno-13-30" href="#__codelineno-13-30"></a>                <span class="k">if</span> <span class="n">first_tombstone</span> <span class="o">!=</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-13-31" name="__codelineno-13-31" href="#__codelineno-13-31"></a>                    <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">first_tombstone</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
 <a id="__codelineno-13-32" name="__codelineno-13-32" href="#__codelineno-13-32"></a>                    <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">TOMBSTONE</span>
-<a id="__codelineno-13-33" name="__codelineno-13-33" href="#__codelineno-13-33"></a>                    <span class="k">return</span> <span class="n">first_tombstone</span>  <span class="c1"># Return the moved bucket index</span>
-<a id="__codelineno-13-34" name="__codelineno-13-34" href="#__codelineno-13-34"></a>                <span class="k">return</span> <span class="n">index</span>  <span class="c1"># Return bucket index</span>
-<a id="__codelineno-13-35" name="__codelineno-13-35" href="#__codelineno-13-35"></a>            <span class="c1"># Record the first removal marker encountered</span>
+<a id="__codelineno-13-33" name="__codelineno-13-33" href="#__codelineno-13-33"></a>                    <span class="k">return</span> <span class="n">first_tombstone</span>  <span class="c1"># Return the bucket index after the move</span>
+<a id="__codelineno-13-34" name="__codelineno-13-34" href="#__codelineno-13-34"></a>                <span class="k">return</span> <span class="n">index</span>  <span class="c1"># Return the bucket index</span>
+<a id="__codelineno-13-35" name="__codelineno-13-35" href="#__codelineno-13-35"></a>            <span class="c1"># Record the first tombstone encountered</span>
 <a id="__codelineno-13-36" name="__codelineno-13-36" href="#__codelineno-13-36"></a>            <span class="k">if</span> <span class="n">first_tombstone</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="ow">is</span> <span class="bp">self</span><span class="o">.</span><span class="n">TOMBSTONE</span><span class="p">:</span>
 <a id="__codelineno-13-37" name="__codelineno-13-37" href="#__codelineno-13-37"></a>                <span class="n">first_tombstone</span> <span class="o">=</span> <span class="n">index</span>
-<a id="__codelineno-13-38" name="__codelineno-13-38" href="#__codelineno-13-38"></a>            <span class="c1"># Calculate bucket index, wrap around to the head if past the tail</span>
+<a id="__codelineno-13-38" name="__codelineno-13-38" href="#__codelineno-13-38"></a>            <span class="c1"># Calculate the bucket index, wrapping to the beginning after the end</span>
 <a id="__codelineno-13-39" name="__codelineno-13-39" href="#__codelineno-13-39"></a>            <span class="n">index</span> <span class="o">=</span> <span class="p">(</span><span class="n">index</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span>
-<a id="__codelineno-13-40" name="__codelineno-13-40" href="#__codelineno-13-40"></a>        <span class="c1"># If key does not exist, return the index for insertion</span>
+<a id="__codelineno-13-40" name="__codelineno-13-40" href="#__codelineno-13-40"></a>        <span class="c1"># If key is absent, return the index for insertion</span>
 <a id="__codelineno-13-41" name="__codelineno-13-41" href="#__codelineno-13-41"></a>        <span class="k">return</span> <span class="n">index</span> <span class="k">if</span> <span class="n">first_tombstone</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span> <span class="k">else</span> <span class="n">first_tombstone</span>
 <a id="__codelineno-13-42" name="__codelineno-13-42" href="#__codelineno-13-42"></a>
 <a id="__codelineno-13-43" name="__codelineno-13-43" href="#__codelineno-13-43"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">str</span><span class="p">:</span>
-<a id="__codelineno-13-44" name="__codelineno-13-44" href="#__codelineno-13-44"></a><span class="w">        </span><span class="sd">"""Query operation"""</span>
-<a id="__codelineno-13-45" name="__codelineno-13-45" href="#__codelineno-13-45"></a>        <span class="c1"># Search for bucket index corresponding to key</span>
+<a id="__codelineno-13-44" name="__codelineno-13-44" href="#__codelineno-13-44"></a><span class="w">        </span><span class="sd">"""Lookup operation"""</span>
+<a id="__codelineno-13-45" name="__codelineno-13-45" href="#__codelineno-13-45"></a>        <span class="c1"># Find the bucket index for key</span>
 <a id="__codelineno-13-46" name="__codelineno-13-46" href="#__codelineno-13-46"></a>        <span class="n">index</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">find_bucket</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-13-47" name="__codelineno-13-47" href="#__codelineno-13-47"></a>        <span class="c1"># If key-value pair is found, return corresponding val</span>
+<a id="__codelineno-13-47" name="__codelineno-13-47" href="#__codelineno-13-47"></a>        <span class="c1"># If the key-value pair is found, return its val</span>
 <a id="__codelineno-13-48" name="__codelineno-13-48" href="#__codelineno-13-48"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">TOMBSTONE</span><span class="p">]:</span>
 <a id="__codelineno-13-49" name="__codelineno-13-49" href="#__codelineno-13-49"></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">.</span><span class="n">val</span>
-<a id="__codelineno-13-50" name="__codelineno-13-50" href="#__codelineno-13-50"></a>        <span class="c1"># If key-value pair does not exist, return None</span>
+<a id="__codelineno-13-50" name="__codelineno-13-50" href="#__codelineno-13-50"></a>        <span class="c1"># Return None if the key-value pair is absent</span>
 <a id="__codelineno-13-51" name="__codelineno-13-51" href="#__codelineno-13-51"></a>        <span class="k">return</span> <span class="kc">None</span>
 <a id="__codelineno-13-52" name="__codelineno-13-52" href="#__codelineno-13-52"></a>
 <a id="__codelineno-13-53" name="__codelineno-13-53" href="#__codelineno-13-53"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">str</span><span class="p">):</span>
-<a id="__codelineno-13-54" name="__codelineno-13-54" href="#__codelineno-13-54"></a><span class="w">        </span><span class="sd">"""Add operation"""</span>
-<a id="__codelineno-13-55" name="__codelineno-13-55" href="#__codelineno-13-55"></a>        <span class="c1"># When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-13-54" name="__codelineno-13-54" href="#__codelineno-13-54"></a><span class="w">        </span><span class="sd">"""Insert operation"""</span>
+<a id="__codelineno-13-55" name="__codelineno-13-55" href="#__codelineno-13-55"></a>        <span class="c1"># Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-13-56" name="__codelineno-13-56" href="#__codelineno-13-56"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">load_factor</span><span class="p">()</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">load_thres</span><span class="p">:</span>
 <a id="__codelineno-13-57" name="__codelineno-13-57" href="#__codelineno-13-57"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">extend</span><span class="p">()</span>
-<a id="__codelineno-13-58" name="__codelineno-13-58" href="#__codelineno-13-58"></a>        <span class="c1"># Search for bucket index corresponding to key</span>
+<a id="__codelineno-13-58" name="__codelineno-13-58" href="#__codelineno-13-58"></a>        <span class="c1"># Find the bucket index for key</span>
 <a id="__codelineno-13-59" name="__codelineno-13-59" href="#__codelineno-13-59"></a>        <span class="n">index</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">find_bucket</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-13-60" name="__codelineno-13-60" href="#__codelineno-13-60"></a>        <span class="c1"># If key-value pair is found, overwrite val and return</span>
+<a id="__codelineno-13-60" name="__codelineno-13-60" href="#__codelineno-13-60"></a>        <span class="c1"># If the key-value pair is found, overwrite val and return</span>
 <a id="__codelineno-13-61" name="__codelineno-13-61" href="#__codelineno-13-61"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">TOMBSTONE</span><span class="p">]:</span>
 <a id="__codelineno-13-62" name="__codelineno-13-62" href="#__codelineno-13-62"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">.</span><span class="n">val</span> <span class="o">=</span> <span class="n">val</span>
 <a id="__codelineno-13-63" name="__codelineno-13-63" href="#__codelineno-13-63"></a>            <span class="k">return</span>
-<a id="__codelineno-13-64" name="__codelineno-13-64" href="#__codelineno-13-64"></a>        <span class="c1"># If key-value pair does not exist, add the key-value pair</span>
+<a id="__codelineno-13-64" name="__codelineno-13-64" href="#__codelineno-13-64"></a>        <span class="c1"># If the key-value pair is absent, insert it</span>
 <a id="__codelineno-13-65" name="__codelineno-13-65" href="#__codelineno-13-65"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-13-66" name="__codelineno-13-66" href="#__codelineno-13-66"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">+=</span> <span class="mi">1</span>
 <a id="__codelineno-13-67" name="__codelineno-13-67" href="#__codelineno-13-67"></a>
 <a id="__codelineno-13-68" name="__codelineno-13-68" href="#__codelineno-13-68"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-13-69" name="__codelineno-13-69" href="#__codelineno-13-69"></a><span class="w">        </span><span class="sd">"""Remove operation"""</span>
-<a id="__codelineno-13-70" name="__codelineno-13-70" href="#__codelineno-13-70"></a>        <span class="c1"># Search for bucket index corresponding to key</span>
+<a id="__codelineno-13-70" name="__codelineno-13-70" href="#__codelineno-13-70"></a>        <span class="c1"># Find the bucket index for key</span>
 <a id="__codelineno-13-71" name="__codelineno-13-71" href="#__codelineno-13-71"></a>        <span class="n">index</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">find_bucket</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-13-72" name="__codelineno-13-72" href="#__codelineno-13-72"></a>        <span class="c1"># If key-value pair is found, overwrite it with removal marker</span>
+<a id="__codelineno-13-72" name="__codelineno-13-72" href="#__codelineno-13-72"></a>        <span class="c1"># If the key-value pair is found, replace it with a tombstone</span>
 <a id="__codelineno-13-73" name="__codelineno-13-73" href="#__codelineno-13-73"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">TOMBSTONE</span><span class="p">]:</span>
 <a id="__codelineno-13-74" name="__codelineno-13-74" href="#__codelineno-13-74"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">TOMBSTONE</span>
 <a id="__codelineno-13-75" name="__codelineno-13-75" href="#__codelineno-13-75"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">-=</span> <span class="mi">1</span>
 <a id="__codelineno-13-76" name="__codelineno-13-76" href="#__codelineno-13-76"></a>
 <a id="__codelineno-13-77" name="__codelineno-13-77" href="#__codelineno-13-77"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">extend</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-<a id="__codelineno-13-78" name="__codelineno-13-78" href="#__codelineno-13-78"></a><span class="w">        </span><span class="sd">"""Expand hash table"""</span>
-<a id="__codelineno-13-79" name="__codelineno-13-79" href="#__codelineno-13-79"></a>        <span class="c1"># Temporarily store the original hash table</span>
+<a id="__codelineno-13-78" name="__codelineno-13-78" href="#__codelineno-13-78"></a><span class="w">        </span><span class="sd">"""Grow the hash table capacity"""</span>
+<a id="__codelineno-13-79" name="__codelineno-13-79" href="#__codelineno-13-79"></a>        <span class="c1"># Keep a reference to the original hash table</span>
 <a id="__codelineno-13-80" name="__codelineno-13-80" href="#__codelineno-13-80"></a>        <span class="n">buckets_tmp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span>
-<a id="__codelineno-13-81" name="__codelineno-13-81" href="#__codelineno-13-81"></a>        <span class="c1"># Initialize expanded new hash table</span>
+<a id="__codelineno-13-81" name="__codelineno-13-81" href="#__codelineno-13-81"></a>        <span class="c1"># Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-13-82" name="__codelineno-13-82" href="#__codelineno-13-82"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span> <span class="o">*=</span> <span class="bp">self</span><span class="o">.</span><span class="n">extend_ratio</span>
 <a id="__codelineno-13-83" name="__codelineno-13-83" href="#__codelineno-13-83"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span>
 <a id="__codelineno-13-84" name="__codelineno-13-84" href="#__codelineno-13-84"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">size</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-13-85" name="__codelineno-13-85" href="#__codelineno-13-85"></a>        <span class="c1"># Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-13-85" name="__codelineno-13-85" href="#__codelineno-13-85"></a>        <span class="c1"># Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-13-86" name="__codelineno-13-86" href="#__codelineno-13-86"></a>        <span class="k">for</span> <span class="n">pair</span> <span class="ow">in</span> <span class="n">buckets_tmp</span><span class="p">:</span>
 <a id="__codelineno-13-87" name="__codelineno-13-87" href="#__codelineno-13-87"></a>            <span class="k">if</span> <span class="n">pair</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">[</span><span class="kc">None</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">TOMBSTONE</span><span class="p">]:</span>
 <a id="__codelineno-13-88" name="__codelineno-13-88" href="#__codelineno-13-88"></a>                <span class="bp">self</span><span class="o">.</span><span class="n">put</span><span class="p">(</span><span class="n">pair</span><span class="o">.</span><span class="n">key</span><span class="p">,</span> <span class="n">pair</span><span class="o">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-13-89" name="__codelineno-13-89" href="#__codelineno-13-89"></a>
 <a id="__codelineno-13-90" name="__codelineno-13-90" href="#__codelineno-13-90"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">print</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-<a id="__codelineno-13-91" name="__codelineno-13-91" href="#__codelineno-13-91"></a><span class="w">        </span><span class="sd">"""Print hash table"""</span>
+<a id="__codelineno-13-91" name="__codelineno-13-91" href="#__codelineno-13-91"></a><span class="w">        </span><span class="sd">"""Print the hash table"""</span>
 <a id="__codelineno-13-92" name="__codelineno-13-92" href="#__codelineno-13-92"></a>        <span class="k">for</span> <span class="n">pair</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">:</span>
 <a id="__codelineno-13-93" name="__codelineno-13-93" href="#__codelineno-13-93"></a>            <span class="k">if</span> <span class="n">pair</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-13-94" name="__codelineno-13-94" href="#__codelineno-13-94"></a>                <span class="nb">print</span><span class="p">(</span><span class="s2">"None"</span><span class="p">)</span>
@@ -6376,10 +6453,10 @@
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">  </span><span class="k">private</span><span class="o">:</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w">                             </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">4</span><span class="p">;</span><span class="w">                     </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="k">const</span><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">3.0</span><span class="p">;</span><span class="w">     </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="k">const</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">            </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="k">const</span><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">3.0</span><span class="p">;</span><span class="w">     </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="k">const</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">            </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">Pair</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span><span class="w">               </span><span class="c1">// Bucket array</span>
-<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="mi">-1</span><span class="p">,</span><span class="w"> </span><span class="s">"-1"</span><span class="p">);</span><span class="w"> </span><span class="c1">// Removal marker</span>
+<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="mi">-1</span><span class="p">,</span><span class="w"> </span><span class="s">"-1"</span><span class="p">);</span><span class="w"> </span><span class="c1">// Tombstone marker</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">  </span><span class="k">public</span><span class="o">:</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -6406,68 +6483,68 @@
 <a id="__codelineno-14-33" name="__codelineno-14-33" href="#__codelineno-14-33"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-14-34" name="__codelineno-14-34" href="#__codelineno-14-34"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-35" name="__codelineno-14-35" href="#__codelineno-14-35"></a>
-<a id="__codelineno-14-36" name="__codelineno-14-36" href="#__codelineno-14-36"></a><span class="w">    </span><span class="cm">/* Search for bucket index corresponding to key */</span>
+<a id="__codelineno-14-36" name="__codelineno-14-36" href="#__codelineno-14-36"></a><span class="w">    </span><span class="cm">/* Find the bucket index for key */</span>
 <a id="__codelineno-14-37" name="__codelineno-14-37" href="#__codelineno-14-37"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-38" name="__codelineno-14-38" href="#__codelineno-14-38"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-14-39" name="__codelineno-14-39" href="#__codelineno-14-39"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
-<a id="__codelineno-14-40" name="__codelineno-14-40" href="#__codelineno-14-40"></a><span class="w">        </span><span class="c1">// Linear probing, break when encountering an empty bucket</span>
+<a id="__codelineno-14-40" name="__codelineno-14-40" href="#__codelineno-14-40"></a><span class="w">        </span><span class="c1">// Probe linearly until an empty bucket is reached</span>
 <a id="__codelineno-14-41" name="__codelineno-14-41" href="#__codelineno-14-41"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-42" name="__codelineno-14-42" href="#__codelineno-14-42"></a><span class="w">            </span><span class="c1">// If key is encountered, return the corresponding bucket index</span>
+<a id="__codelineno-14-42" name="__codelineno-14-42" href="#__codelineno-14-42"></a><span class="w">            </span><span class="c1">// If key is found, return its bucket index</span>
 <a id="__codelineno-14-43" name="__codelineno-14-43" href="#__codelineno-14-43"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">-&gt;</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-44" name="__codelineno-14-44" href="#__codelineno-14-44"></a><span class="w">                </span><span class="c1">// If a removal marker was encountered before, move the key-value pair to that index</span>
+<a id="__codelineno-14-44" name="__codelineno-14-44" href="#__codelineno-14-44"></a><span class="w">                </span><span class="c1">// If a tombstone was encountered earlier, move the key-value pair to that index</span>
 <a id="__codelineno-14-45" name="__codelineno-14-45" href="#__codelineno-14-45"></a><span class="w">                </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">-1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-46" name="__codelineno-14-46" href="#__codelineno-14-46"></a><span class="w">                    </span><span class="n">buckets</span><span class="p">[</span><span class="n">firstTombstone</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
 <a id="__codelineno-14-47" name="__codelineno-14-47" href="#__codelineno-14-47"></a><span class="w">                    </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">;</span>
-<a id="__codelineno-14-48" name="__codelineno-14-48" href="#__codelineno-14-48"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the moved bucket index</span>
+<a id="__codelineno-14-48" name="__codelineno-14-48" href="#__codelineno-14-48"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index after the move</span>
 <a id="__codelineno-14-49" name="__codelineno-14-49" href="#__codelineno-14-49"></a><span class="w">                </span><span class="p">}</span>
-<a id="__codelineno-14-50" name="__codelineno-14-50" href="#__codelineno-14-50"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return bucket index</span>
+<a id="__codelineno-14-50" name="__codelineno-14-50" href="#__codelineno-14-50"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index</span>
 <a id="__codelineno-14-51" name="__codelineno-14-51" href="#__codelineno-14-51"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-14-52" name="__codelineno-14-52" href="#__codelineno-14-52"></a><span class="w">            </span><span class="c1">// Record the first removal marker encountered</span>
+<a id="__codelineno-14-52" name="__codelineno-14-52" href="#__codelineno-14-52"></a><span class="w">            </span><span class="c1">// Record the first tombstone encountered</span>
 <a id="__codelineno-14-53" name="__codelineno-14-53" href="#__codelineno-14-53"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">-1</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-54" name="__codelineno-14-54" href="#__codelineno-14-54"></a><span class="w">                </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span>
 <a id="__codelineno-14-55" name="__codelineno-14-55" href="#__codelineno-14-55"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-14-56" name="__codelineno-14-56" href="#__codelineno-14-56"></a><span class="w">            </span><span class="c1">// Calculate bucket index, wrap around to the head if past the tail</span>
+<a id="__codelineno-14-56" name="__codelineno-14-56" href="#__codelineno-14-56"></a><span class="w">            </span><span class="c1">// Calculate the bucket index, wrapping to the beginning after the end</span>
 <a id="__codelineno-14-57" name="__codelineno-14-57" href="#__codelineno-14-57"></a><span class="w">            </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-14-58" name="__codelineno-14-58" href="#__codelineno-14-58"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-14-59" name="__codelineno-14-59" href="#__codelineno-14-59"></a><span class="w">        </span><span class="c1">// If key does not exist, return the index for insertion</span>
+<a id="__codelineno-14-59" name="__codelineno-14-59" href="#__codelineno-14-59"></a><span class="w">        </span><span class="c1">// If key is absent, return the index for insertion</span>
 <a id="__codelineno-14-60" name="__codelineno-14-60" href="#__codelineno-14-60"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">-1</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">firstTombstone</span><span class="p">;</span>
 <a id="__codelineno-14-61" name="__codelineno-14-61" href="#__codelineno-14-61"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-62" name="__codelineno-14-62" href="#__codelineno-14-62"></a>
-<a id="__codelineno-14-63" name="__codelineno-14-63" href="#__codelineno-14-63"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-14-63" name="__codelineno-14-63" href="#__codelineno-14-63"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-14-64" name="__codelineno-14-64" href="#__codelineno-14-64"></a><span class="w">    </span><span class="n">string</span><span class="w"> </span><span class="n">get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-65" name="__codelineno-14-65" href="#__codelineno-14-65"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-14-65" name="__codelineno-14-65" href="#__codelineno-14-65"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-14-66" name="__codelineno-14-66" href="#__codelineno-14-66"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-14-67" name="__codelineno-14-67" href="#__codelineno-14-67"></a><span class="w">        </span><span class="c1">// If key-value pair is found, return corresponding val</span>
+<a id="__codelineno-14-67" name="__codelineno-14-67" href="#__codelineno-14-67"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, return its val</span>
 <a id="__codelineno-14-68" name="__codelineno-14-68" href="#__codelineno-14-68"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-69" name="__codelineno-14-69" href="#__codelineno-14-69"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-14-70" name="__codelineno-14-70" href="#__codelineno-14-70"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-14-71" name="__codelineno-14-71" href="#__codelineno-14-71"></a><span class="w">        </span><span class="c1">// Return empty string if key-value pair does not exist</span>
+<a id="__codelineno-14-71" name="__codelineno-14-71" href="#__codelineno-14-71"></a><span class="w">        </span><span class="c1">// Return an empty string if the key-value pair does not exist</span>
 <a id="__codelineno-14-72" name="__codelineno-14-72" href="#__codelineno-14-72"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="s">""</span><span class="p">;</span>
 <a id="__codelineno-14-73" name="__codelineno-14-73" href="#__codelineno-14-73"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-74" name="__codelineno-14-74" href="#__codelineno-14-74"></a>
-<a id="__codelineno-14-75" name="__codelineno-14-75" href="#__codelineno-14-75"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-14-75" name="__codelineno-14-75" href="#__codelineno-14-75"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-14-76" name="__codelineno-14-76" href="#__codelineno-14-76"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">put</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-77" name="__codelineno-14-77" href="#__codelineno-14-77"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-14-77" name="__codelineno-14-77" href="#__codelineno-14-77"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-14-78" name="__codelineno-14-78" href="#__codelineno-14-78"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-79" name="__codelineno-14-79" href="#__codelineno-14-79"></a><span class="w">            </span><span class="n">extend</span><span class="p">();</span>
 <a id="__codelineno-14-80" name="__codelineno-14-80" href="#__codelineno-14-80"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-14-81" name="__codelineno-14-81" href="#__codelineno-14-81"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-14-81" name="__codelineno-14-81" href="#__codelineno-14-81"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-14-82" name="__codelineno-14-82" href="#__codelineno-14-82"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-14-83" name="__codelineno-14-83" href="#__codelineno-14-83"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite val and return</span>
+<a id="__codelineno-14-83" name="__codelineno-14-83" href="#__codelineno-14-83"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, overwrite val and return</span>
 <a id="__codelineno-14-84" name="__codelineno-14-84" href="#__codelineno-14-84"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-85" name="__codelineno-14-85" href="#__codelineno-14-85"></a><span class="w">            </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-14-86" name="__codelineno-14-86" href="#__codelineno-14-86"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-14-87" name="__codelineno-14-87" href="#__codelineno-14-87"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-14-88" name="__codelineno-14-88" href="#__codelineno-14-88"></a><span class="w">        </span><span class="c1">// If key-value pair does not exist, add the key-value pair</span>
+<a id="__codelineno-14-88" name="__codelineno-14-88" href="#__codelineno-14-88"></a><span class="w">        </span><span class="c1">// If the key-value pair is absent, insert it</span>
 <a id="__codelineno-14-89" name="__codelineno-14-89" href="#__codelineno-14-89"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-14-90" name="__codelineno-14-90" href="#__codelineno-14-90"></a><span class="w">        </span><span class="n">size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-14-91" name="__codelineno-14-91" href="#__codelineno-14-91"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-92" name="__codelineno-14-92" href="#__codelineno-14-92"></a>
 <a id="__codelineno-14-93" name="__codelineno-14-93" href="#__codelineno-14-93"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-14-94" name="__codelineno-14-94" href="#__codelineno-14-94"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-95" name="__codelineno-14-95" href="#__codelineno-14-95"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-14-95" name="__codelineno-14-95" href="#__codelineno-14-95"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-14-96" name="__codelineno-14-96" href="#__codelineno-14-96"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-14-97" name="__codelineno-14-97" href="#__codelineno-14-97"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite it with removal marker</span>
+<a id="__codelineno-14-97" name="__codelineno-14-97" href="#__codelineno-14-97"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, replace it with a tombstone</span>
 <a id="__codelineno-14-98" name="__codelineno-14-98" href="#__codelineno-14-98"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-99" name="__codelineno-14-99" href="#__codelineno-14-99"></a><span class="w">            </span><span class="k">delete</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
 <a id="__codelineno-14-100" name="__codelineno-14-100" href="#__codelineno-14-100"></a><span class="w">            </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">;</span>
@@ -6475,15 +6552,15 @@
 <a id="__codelineno-14-102" name="__codelineno-14-102" href="#__codelineno-14-102"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-14-103" name="__codelineno-14-103" href="#__codelineno-14-103"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-104" name="__codelineno-14-104" href="#__codelineno-14-104"></a>
-<a id="__codelineno-14-105" name="__codelineno-14-105" href="#__codelineno-14-105"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-14-105" name="__codelineno-14-105" href="#__codelineno-14-105"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-14-106" name="__codelineno-14-106" href="#__codelineno-14-106"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-107" name="__codelineno-14-107" href="#__codelineno-14-107"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-14-107" name="__codelineno-14-107" href="#__codelineno-14-107"></a><span class="w">        </span><span class="c1">// Keep a copy of the original bucket array</span>
 <a id="__codelineno-14-108" name="__codelineno-14-108" href="#__codelineno-14-108"></a><span class="w">        </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">Pair</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span>
-<a id="__codelineno-14-109" name="__codelineno-14-109" href="#__codelineno-14-109"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-14-109" name="__codelineno-14-109" href="#__codelineno-14-109"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-14-110" name="__codelineno-14-110" href="#__codelineno-14-110"></a><span class="w">        </span><span class="n">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span>
 <a id="__codelineno-14-111" name="__codelineno-14-111" href="#__codelineno-14-111"></a><span class="w">        </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">Pair</span><span class="w"> </span><span class="o">*&gt;</span><span class="p">(</span><span class="n">capacity</span><span class="p">,</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">);</span>
 <a id="__codelineno-14-112" name="__codelineno-14-112" href="#__codelineno-14-112"></a><span class="w">        </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-14-113" name="__codelineno-14-113" href="#__codelineno-14-113"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-14-113" name="__codelineno-14-113" href="#__codelineno-14-113"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-14-114" name="__codelineno-14-114" href="#__codelineno-14-114"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">pair</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-115" name="__codelineno-14-115" href="#__codelineno-14-115"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-116" name="__codelineno-14-116" href="#__codelineno-14-116"></a><span class="w">                </span><span class="n">put</span><span class="p">(</span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">);</span>
@@ -6492,7 +6569,7 @@
 <a id="__codelineno-14-119" name="__codelineno-14-119" href="#__codelineno-14-119"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-14-120" name="__codelineno-14-120" href="#__codelineno-14-120"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-121" name="__codelineno-14-121" href="#__codelineno-14-121"></a>
-<a id="__codelineno-14-122" name="__codelineno-14-122" href="#__codelineno-14-122"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-14-122" name="__codelineno-14-122" href="#__codelineno-14-122"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-14-123" name="__codelineno-14-123" href="#__codelineno-14-123"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-124" name="__codelineno-14-124" href="#__codelineno-14-124"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">pair</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-125" name="__codelineno-14-125" href="#__codelineno-14-125"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6512,10 +6589,10 @@
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kd">class</span> <span class="nc">HashMapOpenAddressing</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">4</span><span class="p">;</span><span class="w"> </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">final</span><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">3.0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">final</span><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">3.0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="n">Pair</span><span class="o">[]</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span><span class="w"> </span><span class="c1">// Bucket array</span>
-<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">final</span><span class="w"> </span><span class="n">Pair</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="s">"-1"</span><span class="p">);</span><span class="w"> </span><span class="c1">// Removal marker</span>
+<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">final</span><span class="w"> </span><span class="n">Pair</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="s">"-1"</span><span class="p">);</span><span class="w"> </span><span class="c1">// Tombstone marker</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="nf">HashMapOpenAddressing</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
@@ -6533,83 +6610,83 @@
 <a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-15-24" name="__codelineno-15-24" href="#__codelineno-15-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-25" name="__codelineno-15-25" href="#__codelineno-15-25"></a>
-<a id="__codelineno-15-26" name="__codelineno-15-26" href="#__codelineno-15-26"></a><span class="w">    </span><span class="cm">/* Search for bucket index corresponding to key */</span>
+<a id="__codelineno-15-26" name="__codelineno-15-26" href="#__codelineno-15-26"></a><span class="w">    </span><span class="cm">/* Find the bucket index for key */</span>
 <a id="__codelineno-15-27" name="__codelineno-15-27" href="#__codelineno-15-27"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">findBucket</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-28" name="__codelineno-15-28" href="#__codelineno-15-28"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-15-29" name="__codelineno-15-29" href="#__codelineno-15-29"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-15-30" name="__codelineno-15-30" href="#__codelineno-15-30"></a><span class="w">        </span><span class="c1">// Linear probing, break when encountering an empty bucket</span>
+<a id="__codelineno-15-30" name="__codelineno-15-30" href="#__codelineno-15-30"></a><span class="w">        </span><span class="c1">// Probe linearly until an empty bucket is reached</span>
 <a id="__codelineno-15-31" name="__codelineno-15-31" href="#__codelineno-15-31"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-32" name="__codelineno-15-32" href="#__codelineno-15-32"></a><span class="w">            </span><span class="c1">// If key is encountered, return the corresponding bucket index</span>
+<a id="__codelineno-15-32" name="__codelineno-15-32" href="#__codelineno-15-32"></a><span class="w">            </span><span class="c1">// If key is found, return its bucket index</span>
 <a id="__codelineno-15-33" name="__codelineno-15-33" href="#__codelineno-15-33"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="p">.</span><span class="na">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-34" name="__codelineno-15-34" href="#__codelineno-15-34"></a><span class="w">                </span><span class="c1">// If a removal marker was encountered before, move the key-value pair to that index</span>
+<a id="__codelineno-15-34" name="__codelineno-15-34" href="#__codelineno-15-34"></a><span class="w">                </span><span class="c1">// If a tombstone was encountered earlier, move the key-value pair to that index</span>
 <a id="__codelineno-15-35" name="__codelineno-15-35" href="#__codelineno-15-35"></a><span class="w">                </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-36" name="__codelineno-15-36" href="#__codelineno-15-36"></a><span class="w">                    </span><span class="n">buckets</span><span class="o">[</span><span class="n">firstTombstone</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-37" name="__codelineno-15-37" href="#__codelineno-15-37"></a><span class="w">                    </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">;</span>
-<a id="__codelineno-15-38" name="__codelineno-15-38" href="#__codelineno-15-38"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the moved bucket index</span>
+<a id="__codelineno-15-38" name="__codelineno-15-38" href="#__codelineno-15-38"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index after the move</span>
 <a id="__codelineno-15-39" name="__codelineno-15-39" href="#__codelineno-15-39"></a><span class="w">                </span><span class="p">}</span>
-<a id="__codelineno-15-40" name="__codelineno-15-40" href="#__codelineno-15-40"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return bucket index</span>
+<a id="__codelineno-15-40" name="__codelineno-15-40" href="#__codelineno-15-40"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index</span>
 <a id="__codelineno-15-41" name="__codelineno-15-41" href="#__codelineno-15-41"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-15-42" name="__codelineno-15-42" href="#__codelineno-15-42"></a><span class="w">            </span><span class="c1">// Record the first removal marker encountered</span>
+<a id="__codelineno-15-42" name="__codelineno-15-42" href="#__codelineno-15-42"></a><span class="w">            </span><span class="c1">// Record the first tombstone encountered</span>
 <a id="__codelineno-15-43" name="__codelineno-15-43" href="#__codelineno-15-43"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-44" name="__codelineno-15-44" href="#__codelineno-15-44"></a><span class="w">                </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span>
 <a id="__codelineno-15-45" name="__codelineno-15-45" href="#__codelineno-15-45"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-15-46" name="__codelineno-15-46" href="#__codelineno-15-46"></a><span class="w">            </span><span class="c1">// Calculate bucket index, wrap around to the head if past the tail</span>
+<a id="__codelineno-15-46" name="__codelineno-15-46" href="#__codelineno-15-46"></a><span class="w">            </span><span class="c1">// Calculate the bucket index, wrapping to the beginning after the end</span>
 <a id="__codelineno-15-47" name="__codelineno-15-47" href="#__codelineno-15-47"></a><span class="w">            </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-15-48" name="__codelineno-15-48" href="#__codelineno-15-48"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-15-49" name="__codelineno-15-49" href="#__codelineno-15-49"></a><span class="w">        </span><span class="c1">// If key does not exist, return the index for insertion</span>
+<a id="__codelineno-15-49" name="__codelineno-15-49" href="#__codelineno-15-49"></a><span class="w">        </span><span class="c1">// If key is absent, return the index for insertion</span>
 <a id="__codelineno-15-50" name="__codelineno-15-50" href="#__codelineno-15-50"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">firstTombstone</span><span class="p">;</span>
 <a id="__codelineno-15-51" name="__codelineno-15-51" href="#__codelineno-15-51"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-52" name="__codelineno-15-52" href="#__codelineno-15-52"></a>
-<a id="__codelineno-15-53" name="__codelineno-15-53" href="#__codelineno-15-53"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-15-53" name="__codelineno-15-53" href="#__codelineno-15-53"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-15-54" name="__codelineno-15-54" href="#__codelineno-15-54"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-55" name="__codelineno-15-55" href="#__codelineno-15-55"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-15-55" name="__codelineno-15-55" href="#__codelineno-15-55"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-15-56" name="__codelineno-15-56" href="#__codelineno-15-56"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-15-57" name="__codelineno-15-57" href="#__codelineno-15-57"></a><span class="w">        </span><span class="c1">// If key-value pair is found, return corresponding val</span>
+<a id="__codelineno-15-57" name="__codelineno-15-57" href="#__codelineno-15-57"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, return its val</span>
 <a id="__codelineno-15-58" name="__codelineno-15-58" href="#__codelineno-15-58"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-59" name="__codelineno-15-59" href="#__codelineno-15-59"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="p">.</span><span class="na">val</span><span class="p">;</span>
 <a id="__codelineno-15-60" name="__codelineno-15-60" href="#__codelineno-15-60"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-15-61" name="__codelineno-15-61" href="#__codelineno-15-61"></a><span class="w">        </span><span class="c1">// If key-value pair does not exist, return null</span>
+<a id="__codelineno-15-61" name="__codelineno-15-61" href="#__codelineno-15-61"></a><span class="w">        </span><span class="c1">// Return null if the key-value pair is absent</span>
 <a id="__codelineno-15-62" name="__codelineno-15-62" href="#__codelineno-15-62"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-15-63" name="__codelineno-15-63" href="#__codelineno-15-63"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-64" name="__codelineno-15-64" href="#__codelineno-15-64"></a>
-<a id="__codelineno-15-65" name="__codelineno-15-65" href="#__codelineno-15-65"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-15-65" name="__codelineno-15-65" href="#__codelineno-15-65"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-15-66" name="__codelineno-15-66" href="#__codelineno-15-66"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-67" name="__codelineno-15-67" href="#__codelineno-15-67"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-15-67" name="__codelineno-15-67" href="#__codelineno-15-67"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-15-68" name="__codelineno-15-68" href="#__codelineno-15-68"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-69" name="__codelineno-15-69" href="#__codelineno-15-69"></a><span class="w">            </span><span class="n">extend</span><span class="p">();</span>
 <a id="__codelineno-15-70" name="__codelineno-15-70" href="#__codelineno-15-70"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-15-71" name="__codelineno-15-71" href="#__codelineno-15-71"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-15-71" name="__codelineno-15-71" href="#__codelineno-15-71"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-15-72" name="__codelineno-15-72" href="#__codelineno-15-72"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-15-73" name="__codelineno-15-73" href="#__codelineno-15-73"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite val and return</span>
+<a id="__codelineno-15-73" name="__codelineno-15-73" href="#__codelineno-15-73"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, overwrite val and return</span>
 <a id="__codelineno-15-74" name="__codelineno-15-74" href="#__codelineno-15-74"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-75" name="__codelineno-15-75" href="#__codelineno-15-75"></a><span class="w">            </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-15-76" name="__codelineno-15-76" href="#__codelineno-15-76"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-15-77" name="__codelineno-15-77" href="#__codelineno-15-77"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-15-78" name="__codelineno-15-78" href="#__codelineno-15-78"></a><span class="w">        </span><span class="c1">// If key-value pair does not exist, add the key-value pair</span>
+<a id="__codelineno-15-78" name="__codelineno-15-78" href="#__codelineno-15-78"></a><span class="w">        </span><span class="c1">// If the key-value pair is absent, insert it</span>
 <a id="__codelineno-15-79" name="__codelineno-15-79" href="#__codelineno-15-79"></a><span class="w">        </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-15-80" name="__codelineno-15-80" href="#__codelineno-15-80"></a><span class="w">        </span><span class="n">size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-15-81" name="__codelineno-15-81" href="#__codelineno-15-81"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-82" name="__codelineno-15-82" href="#__codelineno-15-82"></a>
 <a id="__codelineno-15-83" name="__codelineno-15-83" href="#__codelineno-15-83"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-15-84" name="__codelineno-15-84" href="#__codelineno-15-84"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-85" name="__codelineno-15-85" href="#__codelineno-15-85"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-15-85" name="__codelineno-15-85" href="#__codelineno-15-85"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-15-86" name="__codelineno-15-86" href="#__codelineno-15-86"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-15-87" name="__codelineno-15-87" href="#__codelineno-15-87"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite it with removal marker</span>
+<a id="__codelineno-15-87" name="__codelineno-15-87" href="#__codelineno-15-87"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, replace it with a tombstone</span>
 <a id="__codelineno-15-88" name="__codelineno-15-88" href="#__codelineno-15-88"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-89" name="__codelineno-15-89" href="#__codelineno-15-89"></a><span class="w">            </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">;</span>
 <a id="__codelineno-15-90" name="__codelineno-15-90" href="#__codelineno-15-90"></a><span class="w">            </span><span class="n">size</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-15-91" name="__codelineno-15-91" href="#__codelineno-15-91"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-15-92" name="__codelineno-15-92" href="#__codelineno-15-92"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-93" name="__codelineno-15-93" href="#__codelineno-15-93"></a>
-<a id="__codelineno-15-94" name="__codelineno-15-94" href="#__codelineno-15-94"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-15-94" name="__codelineno-15-94" href="#__codelineno-15-94"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-15-95" name="__codelineno-15-95" href="#__codelineno-15-95"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-96" name="__codelineno-15-96" href="#__codelineno-15-96"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-15-96" name="__codelineno-15-96" href="#__codelineno-15-96"></a><span class="w">        </span><span class="c1">// Keep a reference to the original hash table</span>
 <a id="__codelineno-15-97" name="__codelineno-15-97" href="#__codelineno-15-97"></a><span class="w">        </span><span class="n">Pair</span><span class="o">[]</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span>
-<a id="__codelineno-15-98" name="__codelineno-15-98" href="#__codelineno-15-98"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-15-98" name="__codelineno-15-98" href="#__codelineno-15-98"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-15-99" name="__codelineno-15-99" href="#__codelineno-15-99"></a><span class="w">        </span><span class="n">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span>
 <a id="__codelineno-15-100" name="__codelineno-15-100" href="#__codelineno-15-100"></a><span class="w">        </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Pair</span><span class="o">[</span><span class="n">capacity</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-101" name="__codelineno-15-101" href="#__codelineno-15-101"></a><span class="w">        </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-15-102" name="__codelineno-15-102" href="#__codelineno-15-102"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-15-102" name="__codelineno-15-102" href="#__codelineno-15-102"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-15-103" name="__codelineno-15-103" href="#__codelineno-15-103"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-104" name="__codelineno-15-104" href="#__codelineno-15-104"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-105" name="__codelineno-15-105" href="#__codelineno-15-105"></a><span class="w">                </span><span class="n">put</span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="na">key</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="na">val</span><span class="p">);</span>
@@ -6617,7 +6694,7 @@
 <a id="__codelineno-15-107" name="__codelineno-15-107" href="#__codelineno-15-107"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-15-108" name="__codelineno-15-108" href="#__codelineno-15-108"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-109" name="__codelineno-15-109" href="#__codelineno-15-109"></a>
-<a id="__codelineno-15-110" name="__codelineno-15-110" href="#__codelineno-15-110"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-15-110" name="__codelineno-15-110" href="#__codelineno-15-110"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-15-111" name="__codelineno-15-111" href="#__codelineno-15-111"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-112" name="__codelineno-15-112" href="#__codelineno-15-112"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-113" name="__codelineno-15-113" href="#__codelineno-15-113"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6637,10 +6714,10 @@
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">HashMapOpenAddressing</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">4</span><span class="p">;</span><span class="w"> </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">3.0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">3.0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="n">Pair</span><span class="p">[]</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span><span class="w"> </span><span class="c1">// Bucket array</span>
-<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="n">Pair</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="s">"-1"</span><span class="p">);</span><span class="w"> </span><span class="c1">// Removal marker</span>
+<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="n">Pair</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="s">"-1"</span><span class="p">);</span><span class="w"> </span><span class="c1">// Tombstone marker</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nf">HashMapOpenAddressing</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
@@ -6658,83 +6735,83 @@
 <a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-16-24" name="__codelineno-16-24" href="#__codelineno-16-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-25" name="__codelineno-16-25" href="#__codelineno-16-25"></a>
-<a id="__codelineno-16-26" name="__codelineno-16-26" href="#__codelineno-16-26"></a><span class="w">    </span><span class="cm">/* Search for bucket index corresponding to key */</span>
+<a id="__codelineno-16-26" name="__codelineno-16-26" href="#__codelineno-16-26"></a><span class="w">    </span><span class="cm">/* Find the bucket index for key */</span>
 <a id="__codelineno-16-27" name="__codelineno-16-27" href="#__codelineno-16-27"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="nf">FindBucket</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-28" name="__codelineno-16-28" href="#__codelineno-16-28"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-16-29" name="__codelineno-16-29" href="#__codelineno-16-29"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-16-30" name="__codelineno-16-30" href="#__codelineno-16-30"></a><span class="w">        </span><span class="c1">// Linear probing, break when encountering an empty bucket</span>
+<a id="__codelineno-16-30" name="__codelineno-16-30" href="#__codelineno-16-30"></a><span class="w">        </span><span class="c1">// Probe linearly until an empty bucket is reached</span>
 <a id="__codelineno-16-31" name="__codelineno-16-31" href="#__codelineno-16-31"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-32" name="__codelineno-16-32" href="#__codelineno-16-32"></a><span class="w">            </span><span class="c1">// If key is encountered, return the corresponding bucket index</span>
+<a id="__codelineno-16-32" name="__codelineno-16-32" href="#__codelineno-16-32"></a><span class="w">            </span><span class="c1">// If key is found, return its bucket index</span>
 <a id="__codelineno-16-33" name="__codelineno-16-33" href="#__codelineno-16-33"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-34" name="__codelineno-16-34" href="#__codelineno-16-34"></a><span class="w">                </span><span class="c1">// If a removal marker was encountered before, move the key-value pair to that index</span>
+<a id="__codelineno-16-34" name="__codelineno-16-34" href="#__codelineno-16-34"></a><span class="w">                </span><span class="c1">// If a tombstone was encountered earlier, move the key-value pair to that index</span>
 <a id="__codelineno-16-35" name="__codelineno-16-35" href="#__codelineno-16-35"></a><span class="w">                </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-36" name="__codelineno-16-36" href="#__codelineno-16-36"></a><span class="w">                    </span><span class="n">buckets</span><span class="p">[</span><span class="n">firstTombstone</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
 <a id="__codelineno-16-37" name="__codelineno-16-37" href="#__codelineno-16-37"></a><span class="w">                    </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">;</span>
-<a id="__codelineno-16-38" name="__codelineno-16-38" href="#__codelineno-16-38"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the moved bucket index</span>
+<a id="__codelineno-16-38" name="__codelineno-16-38" href="#__codelineno-16-38"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index after the move</span>
 <a id="__codelineno-16-39" name="__codelineno-16-39" href="#__codelineno-16-39"></a><span class="w">                </span><span class="p">}</span>
-<a id="__codelineno-16-40" name="__codelineno-16-40" href="#__codelineno-16-40"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return bucket index</span>
+<a id="__codelineno-16-40" name="__codelineno-16-40" href="#__codelineno-16-40"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index</span>
 <a id="__codelineno-16-41" name="__codelineno-16-41" href="#__codelineno-16-41"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-16-42" name="__codelineno-16-42" href="#__codelineno-16-42"></a><span class="w">            </span><span class="c1">// Record the first removal marker encountered</span>
+<a id="__codelineno-16-42" name="__codelineno-16-42" href="#__codelineno-16-42"></a><span class="w">            </span><span class="c1">// Record the first tombstone encountered</span>
 <a id="__codelineno-16-43" name="__codelineno-16-43" href="#__codelineno-16-43"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-44" name="__codelineno-16-44" href="#__codelineno-16-44"></a><span class="w">                </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span>
 <a id="__codelineno-16-45" name="__codelineno-16-45" href="#__codelineno-16-45"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-16-46" name="__codelineno-16-46" href="#__codelineno-16-46"></a><span class="w">            </span><span class="c1">// Calculate bucket index, wrap around to the head if past the tail</span>
+<a id="__codelineno-16-46" name="__codelineno-16-46" href="#__codelineno-16-46"></a><span class="w">            </span><span class="c1">// Calculate the bucket index, wrapping to the beginning after the end</span>
 <a id="__codelineno-16-47" name="__codelineno-16-47" href="#__codelineno-16-47"></a><span class="w">            </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-16-48" name="__codelineno-16-48" href="#__codelineno-16-48"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-16-49" name="__codelineno-16-49" href="#__codelineno-16-49"></a><span class="w">        </span><span class="c1">// If key does not exist, return the index for insertion</span>
+<a id="__codelineno-16-49" name="__codelineno-16-49" href="#__codelineno-16-49"></a><span class="w">        </span><span class="c1">// If key is absent, return the index for insertion</span>
 <a id="__codelineno-16-50" name="__codelineno-16-50" href="#__codelineno-16-50"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">firstTombstone</span><span class="p">;</span>
 <a id="__codelineno-16-51" name="__codelineno-16-51" href="#__codelineno-16-51"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-52" name="__codelineno-16-52" href="#__codelineno-16-52"></a>
-<a id="__codelineno-16-53" name="__codelineno-16-53" href="#__codelineno-16-53"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-16-53" name="__codelineno-16-53" href="#__codelineno-16-53"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-16-54" name="__codelineno-16-54" href="#__codelineno-16-54"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">string?</span><span class="w"> </span><span class="n">Get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-55" name="__codelineno-16-55" href="#__codelineno-16-55"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-16-55" name="__codelineno-16-55" href="#__codelineno-16-55"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-16-56" name="__codelineno-16-56" href="#__codelineno-16-56"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">FindBucket</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-16-57" name="__codelineno-16-57" href="#__codelineno-16-57"></a><span class="w">        </span><span class="c1">// If key-value pair is found, return corresponding val</span>
+<a id="__codelineno-16-57" name="__codelineno-16-57" href="#__codelineno-16-57"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, return its val</span>
 <a id="__codelineno-16-58" name="__codelineno-16-58" href="#__codelineno-16-58"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-59" name="__codelineno-16-59" href="#__codelineno-16-59"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-16-60" name="__codelineno-16-60" href="#__codelineno-16-60"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-16-61" name="__codelineno-16-61" href="#__codelineno-16-61"></a><span class="w">        </span><span class="c1">// If key-value pair does not exist, return null</span>
+<a id="__codelineno-16-61" name="__codelineno-16-61" href="#__codelineno-16-61"></a><span class="w">        </span><span class="c1">// Return null if the key-value pair is absent</span>
 <a id="__codelineno-16-62" name="__codelineno-16-62" href="#__codelineno-16-62"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
 <a id="__codelineno-16-63" name="__codelineno-16-63" href="#__codelineno-16-63"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-64" name="__codelineno-16-64" href="#__codelineno-16-64"></a>
-<a id="__codelineno-16-65" name="__codelineno-16-65" href="#__codelineno-16-65"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-16-65" name="__codelineno-16-65" href="#__codelineno-16-65"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-16-66" name="__codelineno-16-66" href="#__codelineno-16-66"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">Put</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-67" name="__codelineno-16-67" href="#__codelineno-16-67"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-16-67" name="__codelineno-16-67" href="#__codelineno-16-67"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-16-68" name="__codelineno-16-68" href="#__codelineno-16-68"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">LoadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-69" name="__codelineno-16-69" href="#__codelineno-16-69"></a><span class="w">            </span><span class="n">Extend</span><span class="p">();</span>
 <a id="__codelineno-16-70" name="__codelineno-16-70" href="#__codelineno-16-70"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-16-71" name="__codelineno-16-71" href="#__codelineno-16-71"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-16-71" name="__codelineno-16-71" href="#__codelineno-16-71"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-16-72" name="__codelineno-16-72" href="#__codelineno-16-72"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">FindBucket</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-16-73" name="__codelineno-16-73" href="#__codelineno-16-73"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite val and return</span>
+<a id="__codelineno-16-73" name="__codelineno-16-73" href="#__codelineno-16-73"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, overwrite val and return</span>
 <a id="__codelineno-16-74" name="__codelineno-16-74" href="#__codelineno-16-74"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-75" name="__codelineno-16-75" href="#__codelineno-16-75"></a><span class="w">            </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-16-76" name="__codelineno-16-76" href="#__codelineno-16-76"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-16-77" name="__codelineno-16-77" href="#__codelineno-16-77"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-16-78" name="__codelineno-16-78" href="#__codelineno-16-78"></a><span class="w">        </span><span class="c1">// If key-value pair does not exist, add the key-value pair</span>
+<a id="__codelineno-16-78" name="__codelineno-16-78" href="#__codelineno-16-78"></a><span class="w">        </span><span class="c1">// If the key-value pair is absent, insert it</span>
 <a id="__codelineno-16-79" name="__codelineno-16-79" href="#__codelineno-16-79"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-16-80" name="__codelineno-16-80" href="#__codelineno-16-80"></a><span class="w">        </span><span class="n">size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-16-81" name="__codelineno-16-81" href="#__codelineno-16-81"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-82" name="__codelineno-16-82" href="#__codelineno-16-82"></a>
 <a id="__codelineno-16-83" name="__codelineno-16-83" href="#__codelineno-16-83"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-16-84" name="__codelineno-16-84" href="#__codelineno-16-84"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">Remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-85" name="__codelineno-16-85" href="#__codelineno-16-85"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-16-85" name="__codelineno-16-85" href="#__codelineno-16-85"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-16-86" name="__codelineno-16-86" href="#__codelineno-16-86"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">FindBucket</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-16-87" name="__codelineno-16-87" href="#__codelineno-16-87"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite it with removal marker</span>
+<a id="__codelineno-16-87" name="__codelineno-16-87" href="#__codelineno-16-87"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, replace it with a tombstone</span>
 <a id="__codelineno-16-88" name="__codelineno-16-88" href="#__codelineno-16-88"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-89" name="__codelineno-16-89" href="#__codelineno-16-89"></a><span class="w">            </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">;</span>
 <a id="__codelineno-16-90" name="__codelineno-16-90" href="#__codelineno-16-90"></a><span class="w">            </span><span class="n">size</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-16-91" name="__codelineno-16-91" href="#__codelineno-16-91"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-16-92" name="__codelineno-16-92" href="#__codelineno-16-92"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-93" name="__codelineno-16-93" href="#__codelineno-16-93"></a>
-<a id="__codelineno-16-94" name="__codelineno-16-94" href="#__codelineno-16-94"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-16-94" name="__codelineno-16-94" href="#__codelineno-16-94"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-16-95" name="__codelineno-16-95" href="#__codelineno-16-95"></a><span class="w">    </span><span class="k">void</span><span class="w"> </span><span class="nf">Extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-96" name="__codelineno-16-96" href="#__codelineno-16-96"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-16-96" name="__codelineno-16-96" href="#__codelineno-16-96"></a><span class="w">        </span><span class="c1">// Keep a reference to the original hash table</span>
 <a id="__codelineno-16-97" name="__codelineno-16-97" href="#__codelineno-16-97"></a><span class="w">        </span><span class="n">Pair</span><span class="p">[]</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span>
-<a id="__codelineno-16-98" name="__codelineno-16-98" href="#__codelineno-16-98"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-16-98" name="__codelineno-16-98" href="#__codelineno-16-98"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-16-99" name="__codelineno-16-99" href="#__codelineno-16-99"></a><span class="w">        </span><span class="n">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span>
 <a id="__codelineno-16-100" name="__codelineno-16-100" href="#__codelineno-16-100"></a><span class="w">        </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Pair</span><span class="p">[</span><span class="n">capacity</span><span class="p">];</span>
 <a id="__codelineno-16-101" name="__codelineno-16-101" href="#__codelineno-16-101"></a><span class="w">        </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-16-102" name="__codelineno-16-102" href="#__codelineno-16-102"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-16-102" name="__codelineno-16-102" href="#__codelineno-16-102"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-16-103" name="__codelineno-16-103" href="#__codelineno-16-103"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-104" name="__codelineno-16-104" href="#__codelineno-16-104"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-105" name="__codelineno-16-105" href="#__codelineno-16-105"></a><span class="w">                </span><span class="n">Put</span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
@@ -6742,7 +6819,7 @@
 <a id="__codelineno-16-107" name="__codelineno-16-107" href="#__codelineno-16-107"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-16-108" name="__codelineno-16-108" href="#__codelineno-16-108"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-109" name="__codelineno-16-109" href="#__codelineno-16-109"></a>
-<a id="__codelineno-16-110" name="__codelineno-16-110" href="#__codelineno-16-110"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-16-110" name="__codelineno-16-110" href="#__codelineno-16-110"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-16-111" name="__codelineno-16-111" href="#__codelineno-16-111"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">Print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-112" name="__codelineno-16-112" href="#__codelineno-16-112"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-113" name="__codelineno-16-113" href="#__codelineno-16-113"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6762,10 +6839,10 @@
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">hashMapOpenAddressing</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="nx">size</span><span class="w">        </span><span class="kt">int</span><span class="w">     </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="nx">capacity</span><span class="w">    </span><span class="kt">int</span><span class="w">     </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="nx">loadThres</span><span class="w">   </span><span class="kt">float64</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="nx">extendRatio</span><span class="w"> </span><span class="kt">int</span><span class="w">     </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="nx">loadThres</span><span class="w">   </span><span class="kt">float64</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="nx">extendRatio</span><span class="w"> </span><span class="kt">int</span><span class="w">     </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="nx">buckets</span><span class="w">     </span><span class="p">[]</span><span class="o">*</span><span class="nx">pair</span><span class="w"> </span><span class="c1">// Bucket array</span>
-<a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">    </span><span class="nx">TOMBSTONE</span><span class="w">   </span><span class="o">*</span><span class="nx">pair</span><span class="w">   </span><span class="c1">// Removal marker</span>
+<a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">    </span><span class="nx">TOMBSTONE</span><span class="w">   </span><span class="o">*</span><span class="nx">pair</span><span class="w">   </span><span class="c1">// Tombstone marker</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="p">}</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="cm">/* Constructor */</span>
@@ -6782,79 +6859,79 @@
 <a id="__codelineno-17-22" name="__codelineno-17-22" href="#__codelineno-17-22"></a>
 <a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a><span class="cm">/* Hash function */</span>
 <a id="__codelineno-17-24" name="__codelineno-17-24" href="#__codelineno-17-24"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">hashMapOpenAddressing</span><span class="p">)</span><span class="w"> </span><span class="nx">hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-25" name="__codelineno-17-25" href="#__codelineno-17-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">key</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">capacity</span><span class="w"> </span><span class="c1">// Calculate hash value based on key</span>
+<a id="__codelineno-17-25" name="__codelineno-17-25" href="#__codelineno-17-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">key</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">capacity</span><span class="w"> </span><span class="c1">// Calculate the hash value from the key</span>
 <a id="__codelineno-17-26" name="__codelineno-17-26" href="#__codelineno-17-26"></a><span class="p">}</span>
 <a id="__codelineno-17-27" name="__codelineno-17-27" href="#__codelineno-17-27"></a>
 <a id="__codelineno-17-28" name="__codelineno-17-28" href="#__codelineno-17-28"></a><span class="cm">/* Load factor */</span>
 <a id="__codelineno-17-29" name="__codelineno-17-29" href="#__codelineno-17-29"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">hashMapOpenAddressing</span><span class="p">)</span><span class="w"> </span><span class="nx">loadFactor</span><span class="p">()</span><span class="w"> </span><span class="kt">float64</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-30" name="__codelineno-17-30" href="#__codelineno-17-30"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">h</span><span class="p">.</span><span class="nx">size</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">h</span><span class="p">.</span><span class="nx">capacity</span><span class="p">)</span><span class="w"> </span><span class="c1">// Calculate current load factor</span>
+<a id="__codelineno-17-30" name="__codelineno-17-30" href="#__codelineno-17-30"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">h</span><span class="p">.</span><span class="nx">size</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">h</span><span class="p">.</span><span class="nx">capacity</span><span class="p">)</span><span class="w"> </span><span class="c1">// Calculate the current load factor</span>
 <a id="__codelineno-17-31" name="__codelineno-17-31" href="#__codelineno-17-31"></a><span class="p">}</span>
 <a id="__codelineno-17-32" name="__codelineno-17-32" href="#__codelineno-17-32"></a>
-<a id="__codelineno-17-33" name="__codelineno-17-33" href="#__codelineno-17-33"></a><span class="cm">/* Search for bucket index corresponding to key */</span>
+<a id="__codelineno-17-33" name="__codelineno-17-33" href="#__codelineno-17-33"></a><span class="cm">/* Find the bucket index for key */</span>
 <a id="__codelineno-17-34" name="__codelineno-17-34" href="#__codelineno-17-34"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">hashMapOpenAddressing</span><span class="p">)</span><span class="w"> </span><span class="nx">findBucket</span><span class="p">(</span><span class="nx">key</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-35" name="__codelineno-17-35" href="#__codelineno-17-35"></a><span class="w">    </span><span class="nx">index</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="c1">// Get initial index</span>
-<a id="__codelineno-17-36" name="__codelineno-17-36" href="#__codelineno-17-36"></a><span class="w">    </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w">     </span><span class="c1">// Record position of first TOMBSTONE encountered</span>
+<a id="__codelineno-17-35" name="__codelineno-17-35" href="#__codelineno-17-35"></a><span class="w">    </span><span class="nx">index</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="c1">// Get the initial index</span>
+<a id="__codelineno-17-36" name="__codelineno-17-36" href="#__codelineno-17-36"></a><span class="w">    </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w">     </span><span class="c1">// Record the position of the first TOMBSTONE encountered</span>
 <a id="__codelineno-17-37" name="__codelineno-17-37" href="#__codelineno-17-37"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-38" name="__codelineno-17-38" href="#__codelineno-17-38"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">].</span><span class="nx">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">key</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-39" name="__codelineno-17-39" href="#__codelineno-17-39"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-40" name="__codelineno-17-40" href="#__codelineno-17-40"></a><span class="w">                </span><span class="c1">// If a removal marker was encountered before, move the key-value pair to that index</span>
+<a id="__codelineno-17-40" name="__codelineno-17-40" href="#__codelineno-17-40"></a><span class="w">                </span><span class="c1">// If a tombstone was encountered earlier, move the key-value pair to that index</span>
 <a id="__codelineno-17-41" name="__codelineno-17-41" href="#__codelineno-17-41"></a><span class="w">                </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">firstTombstone</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span>
 <a id="__codelineno-17-42" name="__codelineno-17-42" href="#__codelineno-17-42"></a><span class="w">                </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">TOMBSTONE</span>
-<a id="__codelineno-17-43" name="__codelineno-17-43" href="#__codelineno-17-43"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="c1">// Return the moved bucket index</span>
+<a id="__codelineno-17-43" name="__codelineno-17-43" href="#__codelineno-17-43"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="c1">// Return the bucket index after the move</span>
 <a id="__codelineno-17-44" name="__codelineno-17-44" href="#__codelineno-17-44"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-17-45" name="__codelineno-17-45" href="#__codelineno-17-45"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="c1">// Return found index</span>
+<a id="__codelineno-17-45" name="__codelineno-17-45" href="#__codelineno-17-45"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="c1">// Return the index found</span>
 <a id="__codelineno-17-46" name="__codelineno-17-46" href="#__codelineno-17-46"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-17-47" name="__codelineno-17-47" href="#__codelineno-17-47"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">TOMBSTONE</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-48" name="__codelineno-17-48" href="#__codelineno-17-48"></a><span class="w">            </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="c1">// Record position of first deletion marker encountered</span>
+<a id="__codelineno-17-48" name="__codelineno-17-48" href="#__codelineno-17-48"></a><span class="w">            </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="c1">// Record the position of the first tombstone encountered</span>
 <a id="__codelineno-17-49" name="__codelineno-17-49" href="#__codelineno-17-49"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-17-50" name="__codelineno-17-50" href="#__codelineno-17-50"></a><span class="w">        </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">capacity</span><span class="w"> </span><span class="c1">// Linear probing, wrap around to head if past tail</span>
+<a id="__codelineno-17-50" name="__codelineno-17-50" href="#__codelineno-17-50"></a><span class="w">        </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">capacity</span><span class="w"> </span><span class="c1">// Probe linearly, wrapping to the beginning after the end</span>
 <a id="__codelineno-17-51" name="__codelineno-17-51" href="#__codelineno-17-51"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-52" name="__codelineno-17-52" href="#__codelineno-17-52"></a><span class="w">    </span><span class="c1">// If key does not exist, return the index for insertion</span>
+<a id="__codelineno-17-52" name="__codelineno-17-52" href="#__codelineno-17-52"></a><span class="w">    </span><span class="c1">// If key is absent, return the index for insertion</span>
 <a id="__codelineno-17-53" name="__codelineno-17-53" href="#__codelineno-17-53"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-54" name="__codelineno-17-54" href="#__codelineno-17-54"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">firstTombstone</span>
 <a id="__codelineno-17-55" name="__codelineno-17-55" href="#__codelineno-17-55"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-56" name="__codelineno-17-56" href="#__codelineno-17-56"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">index</span>
 <a id="__codelineno-17-57" name="__codelineno-17-57" href="#__codelineno-17-57"></a><span class="p">}</span>
 <a id="__codelineno-17-58" name="__codelineno-17-58" href="#__codelineno-17-58"></a>
-<a id="__codelineno-17-59" name="__codelineno-17-59" href="#__codelineno-17-59"></a><span class="cm">/* Query operation */</span>
+<a id="__codelineno-17-59" name="__codelineno-17-59" href="#__codelineno-17-59"></a><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-17-60" name="__codelineno-17-60" href="#__codelineno-17-60"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">hashMapOpenAddressing</span><span class="p">)</span><span class="w"> </span><span class="nx">get</span><span class="p">(</span><span class="nx">key</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-61" name="__codelineno-17-61" href="#__codelineno-17-61"></a><span class="w">    </span><span class="nx">index</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">findBucket</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-17-61" name="__codelineno-17-61" href="#__codelineno-17-61"></a><span class="w">    </span><span class="nx">index</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">findBucket</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-17-62" name="__codelineno-17-62" href="#__codelineno-17-62"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">TOMBSTONE</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-63" name="__codelineno-17-63" href="#__codelineno-17-63"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">].</span><span class="nx">val</span><span class="w"> </span><span class="c1">// If key-value pair is found, return corresponding val</span>
+<a id="__codelineno-17-63" name="__codelineno-17-63" href="#__codelineno-17-63"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">].</span><span class="nx">val</span><span class="w"> </span><span class="c1">// If the key-value pair is found, return its val</span>
 <a id="__codelineno-17-64" name="__codelineno-17-64" href="#__codelineno-17-64"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-65" name="__codelineno-17-65" href="#__codelineno-17-65"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="s">""</span><span class="w"> </span><span class="c1">// Return "" if key-value pair does not exist</span>
+<a id="__codelineno-17-65" name="__codelineno-17-65" href="#__codelineno-17-65"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="s">""</span><span class="w"> </span><span class="c1">// If the key-value pair is absent, return ""</span>
 <a id="__codelineno-17-66" name="__codelineno-17-66" href="#__codelineno-17-66"></a><span class="p">}</span>
 <a id="__codelineno-17-67" name="__codelineno-17-67" href="#__codelineno-17-67"></a>
-<a id="__codelineno-17-68" name="__codelineno-17-68" href="#__codelineno-17-68"></a><span class="cm">/* Add operation */</span>
+<a id="__codelineno-17-68" name="__codelineno-17-68" href="#__codelineno-17-68"></a><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-17-69" name="__codelineno-17-69" href="#__codelineno-17-69"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">hashMapOpenAddressing</span><span class="p">)</span><span class="w"> </span><span class="nx">put</span><span class="p">(</span><span class="nx">key</span><span class="w"> </span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-70" name="__codelineno-17-70" href="#__codelineno-17-70"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">loadThres</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-71" name="__codelineno-17-71" href="#__codelineno-17-71"></a><span class="w">        </span><span class="nx">h</span><span class="p">.</span><span class="nx">extend</span><span class="p">()</span><span class="w"> </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-17-71" name="__codelineno-17-71" href="#__codelineno-17-71"></a><span class="w">        </span><span class="nx">h</span><span class="p">.</span><span class="nx">extend</span><span class="p">()</span><span class="w"> </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-17-72" name="__codelineno-17-72" href="#__codelineno-17-72"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-73" name="__codelineno-17-73" href="#__codelineno-17-73"></a><span class="w">    </span><span class="nx">index</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">findBucket</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-17-73" name="__codelineno-17-73" href="#__codelineno-17-73"></a><span class="w">    </span><span class="nx">index</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">findBucket</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-17-74" name="__codelineno-17-74" href="#__codelineno-17-74"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">TOMBSTONE</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-75" name="__codelineno-17-75" href="#__codelineno-17-75"></a><span class="w">        </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">pair</span><span class="p">{</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">}</span><span class="w"> </span><span class="c1">// If key-value pair does not exist, add the key-value pair</span>
+<a id="__codelineno-17-75" name="__codelineno-17-75" href="#__codelineno-17-75"></a><span class="w">        </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">pair</span><span class="p">{</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">}</span><span class="w"> </span><span class="c1">// If the key-value pair is absent, insert it</span>
 <a id="__codelineno-17-76" name="__codelineno-17-76" href="#__codelineno-17-76"></a><span class="w">        </span><span class="nx">h</span><span class="p">.</span><span class="nx">size</span><span class="o">++</span>
 <a id="__codelineno-17-77" name="__codelineno-17-77" href="#__codelineno-17-77"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-78" name="__codelineno-17-78" href="#__codelineno-17-78"></a><span class="w">        </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">].</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="c1">// If key-value pair found, overwrite val</span>
+<a id="__codelineno-17-78" name="__codelineno-17-78" href="#__codelineno-17-78"></a><span class="w">        </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">].</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="c1">// If the key-value pair is found, overwrite val</span>
 <a id="__codelineno-17-79" name="__codelineno-17-79" href="#__codelineno-17-79"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-80" name="__codelineno-17-80" href="#__codelineno-17-80"></a><span class="p">}</span>
 <a id="__codelineno-17-81" name="__codelineno-17-81" href="#__codelineno-17-81"></a>
 <a id="__codelineno-17-82" name="__codelineno-17-82" href="#__codelineno-17-82"></a><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-17-83" name="__codelineno-17-83" href="#__codelineno-17-83"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">hashMapOpenAddressing</span><span class="p">)</span><span class="w"> </span><span class="nx">remove</span><span class="p">(</span><span class="nx">key</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-84" name="__codelineno-17-84" href="#__codelineno-17-84"></a><span class="w">    </span><span class="nx">index</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">findBucket</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-17-84" name="__codelineno-17-84" href="#__codelineno-17-84"></a><span class="w">    </span><span class="nx">index</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">findBucket</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-17-85" name="__codelineno-17-85" href="#__codelineno-17-85"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">TOMBSTONE</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-86" name="__codelineno-17-86" href="#__codelineno-17-86"></a><span class="w">        </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">TOMBSTONE</span><span class="w"> </span><span class="c1">// If key-value pair is found, overwrite it with removal marker</span>
+<a id="__codelineno-17-86" name="__codelineno-17-86" href="#__codelineno-17-86"></a><span class="w">        </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">TOMBSTONE</span><span class="w"> </span><span class="c1">// If the key-value pair is found, replace it with a tombstone</span>
 <a id="__codelineno-17-87" name="__codelineno-17-87" href="#__codelineno-17-87"></a><span class="w">        </span><span class="nx">h</span><span class="p">.</span><span class="nx">size</span><span class="o">--</span>
 <a id="__codelineno-17-88" name="__codelineno-17-88" href="#__codelineno-17-88"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-89" name="__codelineno-17-89" href="#__codelineno-17-89"></a><span class="p">}</span>
 <a id="__codelineno-17-90" name="__codelineno-17-90" href="#__codelineno-17-90"></a>
-<a id="__codelineno-17-91" name="__codelineno-17-91" href="#__codelineno-17-91"></a><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-17-91" name="__codelineno-17-91" href="#__codelineno-17-91"></a><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-17-92" name="__codelineno-17-92" href="#__codelineno-17-92"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">hashMapOpenAddressing</span><span class="p">)</span><span class="w"> </span><span class="nx">extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-93" name="__codelineno-17-93" href="#__codelineno-17-93"></a><span class="w">    </span><span class="nx">oldBuckets</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="w">               </span><span class="c1">// Temporarily store the original hash table</span>
-<a id="__codelineno-17-94" name="__codelineno-17-94" href="#__codelineno-17-94"></a><span class="w">    </span><span class="nx">h</span><span class="p">.</span><span class="nx">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">extendRatio</span><span class="w">           </span><span class="c1">// Update capacity</span>
-<a id="__codelineno-17-95" name="__codelineno-17-95" href="#__codelineno-17-95"></a><span class="w">    </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="o">*</span><span class="nx">pair</span><span class="p">,</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">capacity</span><span class="p">)</span><span class="w"> </span><span class="c1">// Initialize expanded new hash table</span>
-<a id="__codelineno-17-96" name="__codelineno-17-96" href="#__codelineno-17-96"></a><span class="w">    </span><span class="nx">h</span><span class="p">.</span><span class="nx">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w">                            </span><span class="c1">// Reset size</span>
-<a id="__codelineno-17-97" name="__codelineno-17-97" href="#__codelineno-17-97"></a><span class="w">    </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-17-93" name="__codelineno-17-93" href="#__codelineno-17-93"></a><span class="w">    </span><span class="nx">oldBuckets</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="w">               </span><span class="c1">// Keep a reference to the original hash table</span>
+<a id="__codelineno-17-94" name="__codelineno-17-94" href="#__codelineno-17-94"></a><span class="w">    </span><span class="nx">h</span><span class="p">.</span><span class="nx">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">extendRatio</span><span class="w">           </span><span class="c1">// Update the capacity</span>
+<a id="__codelineno-17-95" name="__codelineno-17-95" href="#__codelineno-17-95"></a><span class="w">    </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="o">*</span><span class="nx">pair</span><span class="p">,</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">capacity</span><span class="p">)</span><span class="w"> </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
+<a id="__codelineno-17-96" name="__codelineno-17-96" href="#__codelineno-17-96"></a><span class="w">    </span><span class="nx">h</span><span class="p">.</span><span class="nx">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w">                            </span><span class="c1">// Reset the size</span>
+<a id="__codelineno-17-97" name="__codelineno-17-97" href="#__codelineno-17-97"></a><span class="w">    </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-17-98" name="__codelineno-17-98" href="#__codelineno-17-98"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">oldBuckets</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-99" name="__codelineno-17-99" href="#__codelineno-17-99"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">TOMBSTONE</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-100" name="__codelineno-17-100" href="#__codelineno-17-100"></a><span class="w">            </span><span class="nx">h</span><span class="p">.</span><span class="nx">put</span><span class="p">(</span><span class="nx">pair</span><span class="p">.</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">pair</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span>
@@ -6862,7 +6939,7 @@
 <a id="__codelineno-17-102" name="__codelineno-17-102" href="#__codelineno-17-102"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-103" name="__codelineno-17-103" href="#__codelineno-17-103"></a><span class="p">}</span>
 <a id="__codelineno-17-104" name="__codelineno-17-104" href="#__codelineno-17-104"></a>
-<a id="__codelineno-17-105" name="__codelineno-17-105" href="#__codelineno-17-105"></a><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-17-105" name="__codelineno-17-105" href="#__codelineno-17-105"></a><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-17-106" name="__codelineno-17-106" href="#__codelineno-17-106"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">hashMapOpenAddressing</span><span class="p">)</span><span class="w"> </span><span class="nb">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-107" name="__codelineno-17-107" href="#__codelineno-17-107"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">buckets</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-108" name="__codelineno-17-108" href="#__codelineno-17-108"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
@@ -6881,10 +6958,10 @@
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">HashMapOpenAddressing</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">size</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">capacity</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">loadThres</span><span class="p">:</span><span class="w"> </span><span class="nb">Double</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">extendRatio</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">loadThres</span><span class="p">:</span><span class="w"> </span><span class="nb">Double</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">extendRatio</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">buckets</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="n">Pair</span><span class="p">?]</span><span class="w"> </span><span class="c1">// Bucket array</span>
-<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">TOMBSTONE</span><span class="p">:</span><span class="w"> </span><span class="n">Pair</span><span class="w"> </span><span class="c1">// Removal marker</span>
+<a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">TOMBSTONE</span><span class="p">:</span><span class="w"> </span><span class="n">Pair</span><span class="w"> </span><span class="c1">// Tombstone marker</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="kd">init</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
@@ -6906,83 +6983,83 @@
 <a id="__codelineno-18-27" name="__codelineno-18-27" href="#__codelineno-18-27"></a><span class="w">        </span><span class="nb">Double</span><span class="p">(</span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nb">Double</span><span class="p">(</span><span class="n">capacity</span><span class="p">)</span>
 <a id="__codelineno-18-28" name="__codelineno-18-28" href="#__codelineno-18-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-29" name="__codelineno-18-29" href="#__codelineno-18-29"></a>
-<a id="__codelineno-18-30" name="__codelineno-18-30" href="#__codelineno-18-30"></a><span class="w">    </span><span class="cm">/* Search for bucket index corresponding to key */</span>
+<a id="__codelineno-18-30" name="__codelineno-18-30" href="#__codelineno-18-30"></a><span class="w">    </span><span class="cm">/* Find the bucket index for key */</span>
 <a id="__codelineno-18-31" name="__codelineno-18-31" href="#__codelineno-18-31"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-32" name="__codelineno-18-32" href="#__codelineno-18-32"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-18-33" name="__codelineno-18-33" href="#__codelineno-18-33"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">firstTombstone</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
-<a id="__codelineno-18-34" name="__codelineno-18-34" href="#__codelineno-18-34"></a><span class="w">        </span><span class="c1">// Linear probing, break when encountering an empty bucket</span>
+<a id="__codelineno-18-34" name="__codelineno-18-34" href="#__codelineno-18-34"></a><span class="w">        </span><span class="c1">// Probe linearly until an empty bucket is reached</span>
 <a id="__codelineno-18-35" name="__codelineno-18-35" href="#__codelineno-18-35"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-36" name="__codelineno-18-36" href="#__codelineno-18-36"></a><span class="w">            </span><span class="c1">// If key is encountered, return the corresponding bucket index</span>
+<a id="__codelineno-18-36" name="__codelineno-18-36" href="#__codelineno-18-36"></a><span class="w">            </span><span class="c1">// If key is found, return its bucket index</span>
 <a id="__codelineno-18-37" name="__codelineno-18-37" href="#__codelineno-18-37"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">!</span><span class="p">.</span><span class="n">key</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">key</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-38" name="__codelineno-18-38" href="#__codelineno-18-38"></a><span class="w">                </span><span class="c1">// If a removal marker was encountered before, move the key-value pair to that index</span>
+<a id="__codelineno-18-38" name="__codelineno-18-38" href="#__codelineno-18-38"></a><span class="w">                </span><span class="c1">// If a tombstone was encountered earlier, move the key-value pair to that index</span>
 <a id="__codelineno-18-39" name="__codelineno-18-39" href="#__codelineno-18-39"></a><span class="w">                </span><span class="k">if</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-40" name="__codelineno-18-40" href="#__codelineno-18-40"></a><span class="w">                    </span><span class="n">buckets</span><span class="p">[</span><span class="n">firstTombstone</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
 <a id="__codelineno-18-41" name="__codelineno-18-41" href="#__codelineno-18-41"></a><span class="w">                    </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">TOMBSTONE</span>
-<a id="__codelineno-18-42" name="__codelineno-18-42" href="#__codelineno-18-42"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="c1">// Return the moved bucket index</span>
+<a id="__codelineno-18-42" name="__codelineno-18-42" href="#__codelineno-18-42"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="c1">// Return the bucket index after the move</span>
 <a id="__codelineno-18-43" name="__codelineno-18-43" href="#__codelineno-18-43"></a><span class="w">                </span><span class="p">}</span>
-<a id="__codelineno-18-44" name="__codelineno-18-44" href="#__codelineno-18-44"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="c1">// Return bucket index</span>
+<a id="__codelineno-18-44" name="__codelineno-18-44" href="#__codelineno-18-44"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="c1">// Return the bucket index</span>
 <a id="__codelineno-18-45" name="__codelineno-18-45" href="#__codelineno-18-45"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-18-46" name="__codelineno-18-46" href="#__codelineno-18-46"></a><span class="w">            </span><span class="c1">// Record the first removal marker encountered</span>
+<a id="__codelineno-18-46" name="__codelineno-18-46" href="#__codelineno-18-46"></a><span class="w">            </span><span class="c1">// Record the first tombstone encountered</span>
 <a id="__codelineno-18-47" name="__codelineno-18-47" href="#__codelineno-18-47"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-48" name="__codelineno-18-48" href="#__codelineno-18-48"></a><span class="w">                </span><span class="n">firstTombstone</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">index</span>
 <a id="__codelineno-18-49" name="__codelineno-18-49" href="#__codelineno-18-49"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-18-50" name="__codelineno-18-50" href="#__codelineno-18-50"></a><span class="w">            </span><span class="c1">// Calculate bucket index, wrap around to the head if past the tail</span>
+<a id="__codelineno-18-50" name="__codelineno-18-50" href="#__codelineno-18-50"></a><span class="w">            </span><span class="c1">// Calculate the bucket index, wrapping to the beginning after the end</span>
 <a id="__codelineno-18-51" name="__codelineno-18-51" href="#__codelineno-18-51"></a><span class="w">            </span><span class="n">index</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span>
 <a id="__codelineno-18-52" name="__codelineno-18-52" href="#__codelineno-18-52"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-53" name="__codelineno-18-53" href="#__codelineno-18-53"></a><span class="w">        </span><span class="c1">// If key does not exist, return the index for insertion</span>
+<a id="__codelineno-18-53" name="__codelineno-18-53" href="#__codelineno-18-53"></a><span class="w">        </span><span class="c1">// If key is absent, return the index for insertion</span>
 <a id="__codelineno-18-54" name="__codelineno-18-54" href="#__codelineno-18-54"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">?</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">firstTombstone</span>
 <a id="__codelineno-18-55" name="__codelineno-18-55" href="#__codelineno-18-55"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-56" name="__codelineno-18-56" href="#__codelineno-18-56"></a>
-<a id="__codelineno-18-57" name="__codelineno-18-57" href="#__codelineno-18-57"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-18-57" name="__codelineno-18-57" href="#__codelineno-18-57"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-18-58" name="__codelineno-18-58" href="#__codelineno-18-58"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">String</span><span class="p">?</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-59" name="__codelineno-18-59" href="#__codelineno-18-59"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-18-59" name="__codelineno-18-59" href="#__codelineno-18-59"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-18-60" name="__codelineno-18-60" href="#__codelineno-18-60"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-18-61" name="__codelineno-18-61" href="#__codelineno-18-61"></a><span class="w">        </span><span class="c1">// If key-value pair is found, return corresponding val</span>
+<a id="__codelineno-18-61" name="__codelineno-18-61" href="#__codelineno-18-61"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, return its val</span>
 <a id="__codelineno-18-62" name="__codelineno-18-62" href="#__codelineno-18-62"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-63" name="__codelineno-18-63" href="#__codelineno-18-63"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">!</span><span class="p">.</span><span class="n">val</span>
 <a id="__codelineno-18-64" name="__codelineno-18-64" href="#__codelineno-18-64"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-65" name="__codelineno-18-65" href="#__codelineno-18-65"></a><span class="w">        </span><span class="c1">// If key-value pair does not exist, return null</span>
+<a id="__codelineno-18-65" name="__codelineno-18-65" href="#__codelineno-18-65"></a><span class="w">        </span><span class="c1">// Return nil if the key-value pair is absent</span>
 <a id="__codelineno-18-66" name="__codelineno-18-66" href="#__codelineno-18-66"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-18-67" name="__codelineno-18-67" href="#__codelineno-18-67"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-68" name="__codelineno-18-68" href="#__codelineno-18-68"></a>
-<a id="__codelineno-18-69" name="__codelineno-18-69" href="#__codelineno-18-69"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-18-69" name="__codelineno-18-69" href="#__codelineno-18-69"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-18-70" name="__codelineno-18-70" href="#__codelineno-18-70"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="nb">String</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-71" name="__codelineno-18-71" href="#__codelineno-18-71"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-18-71" name="__codelineno-18-71" href="#__codelineno-18-71"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-18-72" name="__codelineno-18-72" href="#__codelineno-18-72"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">loadThres</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-73" name="__codelineno-18-73" href="#__codelineno-18-73"></a><span class="w">            </span><span class="bp">extend</span><span class="p">()</span>
 <a id="__codelineno-18-74" name="__codelineno-18-74" href="#__codelineno-18-74"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-75" name="__codelineno-18-75" href="#__codelineno-18-75"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-18-75" name="__codelineno-18-75" href="#__codelineno-18-75"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-18-76" name="__codelineno-18-76" href="#__codelineno-18-76"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-18-77" name="__codelineno-18-77" href="#__codelineno-18-77"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite val and return</span>
+<a id="__codelineno-18-77" name="__codelineno-18-77" href="#__codelineno-18-77"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, overwrite val and return</span>
 <a id="__codelineno-18-78" name="__codelineno-18-78" href="#__codelineno-18-78"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-79" name="__codelineno-18-79" href="#__codelineno-18-79"></a><span class="w">            </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">val</span>
 <a id="__codelineno-18-80" name="__codelineno-18-80" href="#__codelineno-18-80"></a><span class="w">            </span><span class="k">return</span>
 <a id="__codelineno-18-81" name="__codelineno-18-81" href="#__codelineno-18-81"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-82" name="__codelineno-18-82" href="#__codelineno-18-82"></a><span class="w">        </span><span class="c1">// If key-value pair does not exist, add the key-value pair</span>
+<a id="__codelineno-18-82" name="__codelineno-18-82" href="#__codelineno-18-82"></a><span class="w">        </span><span class="c1">// If the key-value pair is absent, insert it</span>
 <a id="__codelineno-18-83" name="__codelineno-18-83" href="#__codelineno-18-83"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-18-84" name="__codelineno-18-84" href="#__codelineno-18-84"></a><span class="w">        </span><span class="n">size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-18-85" name="__codelineno-18-85" href="#__codelineno-18-85"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-86" name="__codelineno-18-86" href="#__codelineno-18-86"></a>
 <a id="__codelineno-18-87" name="__codelineno-18-87" href="#__codelineno-18-87"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-18-88" name="__codelineno-18-88" href="#__codelineno-18-88"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-89" name="__codelineno-18-89" href="#__codelineno-18-89"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-18-89" name="__codelineno-18-89" href="#__codelineno-18-89"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-18-90" name="__codelineno-18-90" href="#__codelineno-18-90"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-18-91" name="__codelineno-18-91" href="#__codelineno-18-91"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite it with removal marker</span>
+<a id="__codelineno-18-91" name="__codelineno-18-91" href="#__codelineno-18-91"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, replace it with a tombstone</span>
 <a id="__codelineno-18-92" name="__codelineno-18-92" href="#__codelineno-18-92"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-93" name="__codelineno-18-93" href="#__codelineno-18-93"></a><span class="w">            </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">TOMBSTONE</span>
 <a id="__codelineno-18-94" name="__codelineno-18-94" href="#__codelineno-18-94"></a><span class="w">            </span><span class="n">size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-18-95" name="__codelineno-18-95" href="#__codelineno-18-95"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-18-96" name="__codelineno-18-96" href="#__codelineno-18-96"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-97" name="__codelineno-18-97" href="#__codelineno-18-97"></a>
-<a id="__codelineno-18-98" name="__codelineno-18-98" href="#__codelineno-18-98"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-18-98" name="__codelineno-18-98" href="#__codelineno-18-98"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-18-99" name="__codelineno-18-99" href="#__codelineno-18-99"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-100" name="__codelineno-18-100" href="#__codelineno-18-100"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-18-100" name="__codelineno-18-100" href="#__codelineno-18-100"></a><span class="w">        </span><span class="c1">// Save the original bucket array</span>
 <a id="__codelineno-18-101" name="__codelineno-18-101" href="#__codelineno-18-101"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">bucketsTmp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">buckets</span>
-<a id="__codelineno-18-102" name="__codelineno-18-102" href="#__codelineno-18-102"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-18-102" name="__codelineno-18-102" href="#__codelineno-18-102"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-18-103" name="__codelineno-18-103" href="#__codelineno-18-103"></a><span class="w">        </span><span class="n">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="n">extendRatio</span>
 <a id="__codelineno-18-104" name="__codelineno-18-104" href="#__codelineno-18-104"></a><span class="w">        </span><span class="n">buckets</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">capacity</span><span class="p">)</span>
 <a id="__codelineno-18-105" name="__codelineno-18-105" href="#__codelineno-18-105"></a><span class="w">        </span><span class="n">size</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-18-106" name="__codelineno-18-106" href="#__codelineno-18-106"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-18-106" name="__codelineno-18-106" href="#__codelineno-18-106"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-18-107" name="__codelineno-18-107" href="#__codelineno-18-107"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-108" name="__codelineno-18-108" href="#__codelineno-18-108"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nv">pair</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-109" name="__codelineno-18-109" href="#__codelineno-18-109"></a><span class="w">                </span><span class="n">put</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
@@ -6990,7 +7067,7 @@
 <a id="__codelineno-18-111" name="__codelineno-18-111" href="#__codelineno-18-111"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-18-112" name="__codelineno-18-112" href="#__codelineno-18-112"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-113" name="__codelineno-18-113" href="#__codelineno-18-113"></a>
-<a id="__codelineno-18-114" name="__codelineno-18-114" href="#__codelineno-18-114"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-18-114" name="__codelineno-18-114" href="#__codelineno-18-114"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-18-115" name="__codelineno-18-115" href="#__codelineno-18-115"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-116" name="__codelineno-18-116" href="#__codelineno-18-116"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-117" name="__codelineno-18-117" href="#__codelineno-18-117"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
@@ -7010,19 +7087,19 @@
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">HashMapOpenAddressing</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="n">#size</span><span class="p">;</span><span class="w"> </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="n">#capacity</span><span class="p">;</span><span class="w"> </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="n">#loadThres</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="n">#extendRatio</span><span class="p">;</span><span class="w"> </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="n">#loadThres</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="n">#extendRatio</span><span class="p">;</span><span class="w"> </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="n">#buckets</span><span class="p">;</span><span class="w"> </span><span class="c1">// Bucket array</span>
-<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="n">#TOMBSTONE</span><span class="p">;</span><span class="w"> </span><span class="c1">// Removal marker</span>
+<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="n">#TOMBSTONE</span><span class="p">;</span><span class="w"> </span><span class="c1">// Tombstone marker</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">4</span><span class="p">;</span><span class="w"> </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#loadThres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">3.0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#loadThres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">3.0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">null</span><span class="p">);</span><span class="w"> </span><span class="c1">// Bucket array</span>
-<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#TOMBSTONE</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Pair</span><span class="p">(</span><span class="o">-</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="s1">'-1'</span><span class="p">);</span><span class="w"> </span><span class="c1">// Removal marker</span>
+<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#TOMBSTONE</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Pair</span><span class="p">(</span><span class="o">-</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="s1">'-1'</span><span class="p">);</span><span class="w"> </span><span class="c1">// Tombstone marker</span>
 <a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">    </span><span class="cm">/* Hash function */</span>
@@ -7035,60 +7112,60 @@
 <a id="__codelineno-19-27" name="__codelineno-19-27" href="#__codelineno-19-27"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="p">;</span>
 <a id="__codelineno-19-28" name="__codelineno-19-28" href="#__codelineno-19-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-29" name="__codelineno-19-29" href="#__codelineno-19-29"></a>
-<a id="__codelineno-19-30" name="__codelineno-19-30" href="#__codelineno-19-30"></a><span class="w">    </span><span class="cm">/* Search for bucket index corresponding to key */</span>
+<a id="__codelineno-19-30" name="__codelineno-19-30" href="#__codelineno-19-30"></a><span class="w">    </span><span class="cm">/* Find the bucket index for key */</span>
 <a id="__codelineno-19-31" name="__codelineno-19-31" href="#__codelineno-19-31"></a><span class="w">    </span><span class="n">#findBucket</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-32" name="__codelineno-19-32" href="#__codelineno-19-32"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
 <a id="__codelineno-19-33" name="__codelineno-19-33" href="#__codelineno-19-33"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-19-34" name="__codelineno-19-34" href="#__codelineno-19-34"></a><span class="w">        </span><span class="c1">// Linear probing, break when encountering an empty bucket</span>
+<a id="__codelineno-19-34" name="__codelineno-19-34" href="#__codelineno-19-34"></a><span class="w">        </span><span class="c1">// Probe linearly until an empty bucket is reached</span>
 <a id="__codelineno-19-35" name="__codelineno-19-35" href="#__codelineno-19-35"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-36" name="__codelineno-19-36" href="#__codelineno-19-36"></a><span class="w">            </span><span class="c1">// If key is encountered, return the corresponding bucket index</span>
+<a id="__codelineno-19-36" name="__codelineno-19-36" href="#__codelineno-19-36"></a><span class="w">            </span><span class="c1">// If key is found, return its bucket index</span>
 <a id="__codelineno-19-37" name="__codelineno-19-37" href="#__codelineno-19-37"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">].</span><span class="nx">key</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-38" name="__codelineno-19-38" href="#__codelineno-19-38"></a><span class="w">                </span><span class="c1">// If a removal marker was encountered before, move the key-value pair to that index</span>
+<a id="__codelineno-19-38" name="__codelineno-19-38" href="#__codelineno-19-38"></a><span class="w">                </span><span class="c1">// If a tombstone was encountered earlier, move the key-value pair to that index</span>
 <a id="__codelineno-19-39" name="__codelineno-19-39" href="#__codelineno-19-39"></a><span class="w">                </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-40" name="__codelineno-19-40" href="#__codelineno-19-40"></a><span class="w">                    </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">firstTombstone</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">];</span>
 <a id="__codelineno-19-41" name="__codelineno-19-41" href="#__codelineno-19-41"></a><span class="w">                    </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#TOMBSTONE</span><span class="p">;</span>
-<a id="__codelineno-19-42" name="__codelineno-19-42" href="#__codelineno-19-42"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="nx">firstTombstone</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the moved bucket index</span>
+<a id="__codelineno-19-42" name="__codelineno-19-42" href="#__codelineno-19-42"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="nx">firstTombstone</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index after the move</span>
 <a id="__codelineno-19-43" name="__codelineno-19-43" href="#__codelineno-19-43"></a><span class="w">                </span><span class="p">}</span>
-<a id="__codelineno-19-44" name="__codelineno-19-44" href="#__codelineno-19-44"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return bucket index</span>
+<a id="__codelineno-19-44" name="__codelineno-19-44" href="#__codelineno-19-44"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index</span>
 <a id="__codelineno-19-45" name="__codelineno-19-45" href="#__codelineno-19-45"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-19-46" name="__codelineno-19-46" href="#__codelineno-19-46"></a><span class="w">            </span><span class="c1">// Record the first removal marker encountered</span>
+<a id="__codelineno-19-46" name="__codelineno-19-46" href="#__codelineno-19-46"></a><span class="w">            </span><span class="c1">// Record the first tombstone encountered</span>
 <a id="__codelineno-19-47" name="__codelineno-19-47" href="#__codelineno-19-47"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span>
 <a id="__codelineno-19-48" name="__codelineno-19-48" href="#__codelineno-19-48"></a><span class="w">                </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="w"> </span><span class="o">&amp;&amp;</span>
 <a id="__codelineno-19-49" name="__codelineno-19-49" href="#__codelineno-19-49"></a><span class="w">                </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#TOMBSTONE</span>
 <a id="__codelineno-19-50" name="__codelineno-19-50" href="#__codelineno-19-50"></a><span class="w">            </span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-51" name="__codelineno-19-51" href="#__codelineno-19-51"></a><span class="w">                </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span>
 <a id="__codelineno-19-52" name="__codelineno-19-52" href="#__codelineno-19-52"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-19-53" name="__codelineno-19-53" href="#__codelineno-19-53"></a><span class="w">            </span><span class="c1">// Calculate bucket index, wrap around to the head if past the tail</span>
+<a id="__codelineno-19-53" name="__codelineno-19-53" href="#__codelineno-19-53"></a><span class="w">            </span><span class="c1">// Calculate the bucket index, wrapping to the beginning after the end</span>
 <a id="__codelineno-19-54" name="__codelineno-19-54" href="#__codelineno-19-54"></a><span class="w">            </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="p">;</span>
 <a id="__codelineno-19-55" name="__codelineno-19-55" href="#__codelineno-19-55"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-19-56" name="__codelineno-19-56" href="#__codelineno-19-56"></a><span class="w">        </span><span class="c1">// If key does not exist, return the index for insertion</span>
+<a id="__codelineno-19-56" name="__codelineno-19-56" href="#__codelineno-19-56"></a><span class="w">        </span><span class="c1">// If key is absent, return the index for insertion</span>
 <a id="__codelineno-19-57" name="__codelineno-19-57" href="#__codelineno-19-57"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">firstTombstone</span><span class="p">;</span>
 <a id="__codelineno-19-58" name="__codelineno-19-58" href="#__codelineno-19-58"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-59" name="__codelineno-19-59" href="#__codelineno-19-59"></a>
-<a id="__codelineno-19-60" name="__codelineno-19-60" href="#__codelineno-19-60"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-19-60" name="__codelineno-19-60" href="#__codelineno-19-60"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-19-61" name="__codelineno-19-61" href="#__codelineno-19-61"></a><span class="w">    </span><span class="nx">get</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-62" name="__codelineno-19-62" href="#__codelineno-19-62"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-19-62" name="__codelineno-19-62" href="#__codelineno-19-62"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-19-63" name="__codelineno-19-63" href="#__codelineno-19-63"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#findBucket</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
-<a id="__codelineno-19-64" name="__codelineno-19-64" href="#__codelineno-19-64"></a><span class="w">        </span><span class="c1">// If key-value pair is found, return corresponding val</span>
+<a id="__codelineno-19-64" name="__codelineno-19-64" href="#__codelineno-19-64"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, return its val</span>
 <a id="__codelineno-19-65" name="__codelineno-19-65" href="#__codelineno-19-65"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span>
 <a id="__codelineno-19-66" name="__codelineno-19-66" href="#__codelineno-19-66"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span>
 <a id="__codelineno-19-67" name="__codelineno-19-67" href="#__codelineno-19-67"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#TOMBSTONE</span>
 <a id="__codelineno-19-68" name="__codelineno-19-68" href="#__codelineno-19-68"></a><span class="w">        </span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-69" name="__codelineno-19-69" href="#__codelineno-19-69"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">].</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-19-70" name="__codelineno-19-70" href="#__codelineno-19-70"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-19-71" name="__codelineno-19-71" href="#__codelineno-19-71"></a><span class="w">        </span><span class="c1">// If key-value pair does not exist, return null</span>
+<a id="__codelineno-19-71" name="__codelineno-19-71" href="#__codelineno-19-71"></a><span class="w">        </span><span class="c1">// Return null if the key-value pair does not exist</span>
 <a id="__codelineno-19-72" name="__codelineno-19-72" href="#__codelineno-19-72"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-19-73" name="__codelineno-19-73" href="#__codelineno-19-73"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-74" name="__codelineno-19-74" href="#__codelineno-19-74"></a>
-<a id="__codelineno-19-75" name="__codelineno-19-75" href="#__codelineno-19-75"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-19-75" name="__codelineno-19-75" href="#__codelineno-19-75"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-19-76" name="__codelineno-19-76" href="#__codelineno-19-76"></a><span class="w">    </span><span class="nx">put</span><span class="p">(</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-77" name="__codelineno-19-77" href="#__codelineno-19-77"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-19-77" name="__codelineno-19-77" href="#__codelineno-19-77"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-19-78" name="__codelineno-19-78" href="#__codelineno-19-78"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-79" name="__codelineno-19-79" href="#__codelineno-19-79"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#extend</span><span class="p">();</span>
 <a id="__codelineno-19-80" name="__codelineno-19-80" href="#__codelineno-19-80"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-19-81" name="__codelineno-19-81" href="#__codelineno-19-81"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-19-81" name="__codelineno-19-81" href="#__codelineno-19-81"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-19-82" name="__codelineno-19-82" href="#__codelineno-19-82"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#findBucket</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
-<a id="__codelineno-19-83" name="__codelineno-19-83" href="#__codelineno-19-83"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite val and return</span>
+<a id="__codelineno-19-83" name="__codelineno-19-83" href="#__codelineno-19-83"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, overwrite val and return</span>
 <a id="__codelineno-19-84" name="__codelineno-19-84" href="#__codelineno-19-84"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span>
 <a id="__codelineno-19-85" name="__codelineno-19-85" href="#__codelineno-19-85"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span>
 <a id="__codelineno-19-86" name="__codelineno-19-86" href="#__codelineno-19-86"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#TOMBSTONE</span>
@@ -7096,16 +7173,16 @@
 <a id="__codelineno-19-88" name="__codelineno-19-88" href="#__codelineno-19-88"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">].</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-19-89" name="__codelineno-19-89" href="#__codelineno-19-89"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-19-90" name="__codelineno-19-90" href="#__codelineno-19-90"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-19-91" name="__codelineno-19-91" href="#__codelineno-19-91"></a><span class="w">        </span><span class="c1">// If key-value pair does not exist, add the key-value pair</span>
+<a id="__codelineno-19-91" name="__codelineno-19-91" href="#__codelineno-19-91"></a><span class="w">        </span><span class="c1">// If the key-value pair is absent, insert it</span>
 <a id="__codelineno-19-92" name="__codelineno-19-92" href="#__codelineno-19-92"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Pair</span><span class="p">(</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-19-93" name="__codelineno-19-93" href="#__codelineno-19-93"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-19-94" name="__codelineno-19-94" href="#__codelineno-19-94"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-95" name="__codelineno-19-95" href="#__codelineno-19-95"></a>
 <a id="__codelineno-19-96" name="__codelineno-19-96" href="#__codelineno-19-96"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-19-97" name="__codelineno-19-97" href="#__codelineno-19-97"></a><span class="w">    </span><span class="nx">remove</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-98" name="__codelineno-19-98" href="#__codelineno-19-98"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-19-98" name="__codelineno-19-98" href="#__codelineno-19-98"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-19-99" name="__codelineno-19-99" href="#__codelineno-19-99"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#findBucket</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
-<a id="__codelineno-19-100" name="__codelineno-19-100" href="#__codelineno-19-100"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite it with removal marker</span>
+<a id="__codelineno-19-100" name="__codelineno-19-100" href="#__codelineno-19-100"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, replace it with a tombstone</span>
 <a id="__codelineno-19-101" name="__codelineno-19-101" href="#__codelineno-19-101"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span>
 <a id="__codelineno-19-102" name="__codelineno-19-102" href="#__codelineno-19-102"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span>
 <a id="__codelineno-19-103" name="__codelineno-19-103" href="#__codelineno-19-103"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#TOMBSTONE</span>
@@ -7115,15 +7192,15 @@
 <a id="__codelineno-19-107" name="__codelineno-19-107" href="#__codelineno-19-107"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-19-108" name="__codelineno-19-108" href="#__codelineno-19-108"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-109" name="__codelineno-19-109" href="#__codelineno-19-109"></a>
-<a id="__codelineno-19-110" name="__codelineno-19-110" href="#__codelineno-19-110"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-19-110" name="__codelineno-19-110" href="#__codelineno-19-110"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-19-111" name="__codelineno-19-111" href="#__codelineno-19-111"></a><span class="w">    </span><span class="n">#extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-112" name="__codelineno-19-112" href="#__codelineno-19-112"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-19-112" name="__codelineno-19-112" href="#__codelineno-19-112"></a><span class="w">        </span><span class="c1">// Keep a reference to the original hash table</span>
 <a id="__codelineno-19-113" name="__codelineno-19-113" href="#__codelineno-19-113"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">bucketsTmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">;</span>
-<a id="__codelineno-19-114" name="__codelineno-19-114" href="#__codelineno-19-114"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-19-114" name="__codelineno-19-114" href="#__codelineno-19-114"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-19-115" name="__codelineno-19-115" href="#__codelineno-19-115"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#extendRatio</span><span class="p">;</span>
 <a id="__codelineno-19-116" name="__codelineno-19-116" href="#__codelineno-19-116"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#capacity</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">null</span><span class="p">);</span>
 <a id="__codelineno-19-117" name="__codelineno-19-117" href="#__codelineno-19-117"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-19-118" name="__codelineno-19-118" href="#__codelineno-19-118"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-19-118" name="__codelineno-19-118" href="#__codelineno-19-118"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-19-119" name="__codelineno-19-119" href="#__codelineno-19-119"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">bucketsTmp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-120" name="__codelineno-19-120" href="#__codelineno-19-120"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pair</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-121" name="__codelineno-19-121" href="#__codelineno-19-121"></a><span class="w">                </span><span class="k">this</span><span class="p">.</span><span class="nx">put</span><span class="p">(</span><span class="nx">pair</span><span class="p">.</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">pair</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span>
@@ -7131,7 +7208,7 @@
 <a id="__codelineno-19-123" name="__codelineno-19-123" href="#__codelineno-19-123"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-19-124" name="__codelineno-19-124" href="#__codelineno-19-124"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-125" name="__codelineno-19-125" href="#__codelineno-19-125"></a>
-<a id="__codelineno-19-126" name="__codelineno-19-126" href="#__codelineno-19-126"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-19-126" name="__codelineno-19-126" href="#__codelineno-19-126"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-19-127" name="__codelineno-19-127" href="#__codelineno-19-127"></a><span class="w">    </span><span class="nx">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-128" name="__codelineno-19-128" href="#__codelineno-19-128"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-129" name="__codelineno-19-129" href="#__codelineno-19-129"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pair</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7151,19 +7228,19 @@
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">HashMapOpenAddressing</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">size</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">capacity</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">loadThres</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">extendRatio</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">loadThres</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">extendRatio</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">buckets</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="nx">Pair</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="o">&gt;</span><span class="p">;</span><span class="w"> </span><span class="c1">// Bucket array</span>
-<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">TOMBSTONE</span><span class="o">:</span><span class="w"> </span><span class="kt">Pair</span><span class="p">;</span><span class="w"> </span><span class="c1">// Removal marker</span>
+<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">TOMBSTONE</span><span class="o">:</span><span class="w"> </span><span class="kt">Pair</span><span class="p">;</span><span class="w"> </span><span class="c1">// Tombstone marker</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">4</span><span class="p">;</span><span class="w"> </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">loadThres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">3.0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">loadThres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">3.0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">null</span><span class="p">);</span><span class="w"> </span><span class="c1">// Bucket array</span>
-<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">TOMBSTONE</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Pair</span><span class="p">(</span><span class="o">-</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="s1">'-1'</span><span class="p">);</span><span class="w"> </span><span class="c1">// Removal marker</span>
+<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">TOMBSTONE</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Pair</span><span class="p">(</span><span class="o">-</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="s1">'-1'</span><span class="p">);</span><span class="w"> </span><span class="c1">// Tombstone marker</span>
 <a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">    </span><span class="cm">/* Hash function */</span>
@@ -7176,60 +7253,60 @@
 <a id="__codelineno-20-27" name="__codelineno-20-27" href="#__codelineno-20-27"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">;</span>
 <a id="__codelineno-20-28" name="__codelineno-20-28" href="#__codelineno-20-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-29" name="__codelineno-20-29" href="#__codelineno-20-29"></a>
-<a id="__codelineno-20-30" name="__codelineno-20-30" href="#__codelineno-20-30"></a><span class="w">    </span><span class="cm">/* Search for bucket index corresponding to key */</span>
+<a id="__codelineno-20-30" name="__codelineno-20-30" href="#__codelineno-20-30"></a><span class="w">    </span><span class="cm">/* Find the bucket index for key */</span>
 <a id="__codelineno-20-31" name="__codelineno-20-31" href="#__codelineno-20-31"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">findBucket</span><span class="p">(</span><span class="nx">key</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-32" name="__codelineno-20-32" href="#__codelineno-20-32"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
 <a id="__codelineno-20-33" name="__codelineno-20-33" href="#__codelineno-20-33"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-20-34" name="__codelineno-20-34" href="#__codelineno-20-34"></a><span class="w">        </span><span class="c1">// Linear probing, break when encountering an empty bucket</span>
+<a id="__codelineno-20-34" name="__codelineno-20-34" href="#__codelineno-20-34"></a><span class="w">        </span><span class="c1">// Probe linearly until an empty bucket is reached</span>
 <a id="__codelineno-20-35" name="__codelineno-20-35" href="#__codelineno-20-35"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-36" name="__codelineno-20-36" href="#__codelineno-20-36"></a><span class="w">            </span><span class="c1">// If key is encountered, return the corresponding bucket index</span>
+<a id="__codelineno-20-36" name="__codelineno-20-36" href="#__codelineno-20-36"></a><span class="w">            </span><span class="c1">// If key is found, return its bucket index</span>
 <a id="__codelineno-20-37" name="__codelineno-20-37" href="#__codelineno-20-37"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="o">!</span><span class="p">.</span><span class="nx">key</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-38" name="__codelineno-20-38" href="#__codelineno-20-38"></a><span class="w">                </span><span class="c1">// If a removal marker was encountered before, move the key-value pair to that index</span>
+<a id="__codelineno-20-38" name="__codelineno-20-38" href="#__codelineno-20-38"></a><span class="w">                </span><span class="c1">// If a tombstone was encountered earlier, move the key-value pair to that index</span>
 <a id="__codelineno-20-39" name="__codelineno-20-39" href="#__codelineno-20-39"></a><span class="w">                </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-40" name="__codelineno-20-40" href="#__codelineno-20-40"></a><span class="w">                    </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">firstTombstone</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">];</span>
 <a id="__codelineno-20-41" name="__codelineno-20-41" href="#__codelineno-20-41"></a><span class="w">                    </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">TOMBSTONE</span><span class="p">;</span>
-<a id="__codelineno-20-42" name="__codelineno-20-42" href="#__codelineno-20-42"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="nx">firstTombstone</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the moved bucket index</span>
+<a id="__codelineno-20-42" name="__codelineno-20-42" href="#__codelineno-20-42"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="nx">firstTombstone</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index after the move</span>
 <a id="__codelineno-20-43" name="__codelineno-20-43" href="#__codelineno-20-43"></a><span class="w">                </span><span class="p">}</span>
-<a id="__codelineno-20-44" name="__codelineno-20-44" href="#__codelineno-20-44"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return bucket index</span>
+<a id="__codelineno-20-44" name="__codelineno-20-44" href="#__codelineno-20-44"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index</span>
 <a id="__codelineno-20-45" name="__codelineno-20-45" href="#__codelineno-20-45"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-20-46" name="__codelineno-20-46" href="#__codelineno-20-46"></a><span class="w">            </span><span class="c1">// Record the first removal marker encountered</span>
+<a id="__codelineno-20-46" name="__codelineno-20-46" href="#__codelineno-20-46"></a><span class="w">            </span><span class="c1">// Record the first tombstone encountered</span>
 <a id="__codelineno-20-47" name="__codelineno-20-47" href="#__codelineno-20-47"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span>
 <a id="__codelineno-20-48" name="__codelineno-20-48" href="#__codelineno-20-48"></a><span class="w">                </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="w"> </span><span class="o">&amp;&amp;</span>
 <a id="__codelineno-20-49" name="__codelineno-20-49" href="#__codelineno-20-49"></a><span class="w">                </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">TOMBSTONE</span>
 <a id="__codelineno-20-50" name="__codelineno-20-50" href="#__codelineno-20-50"></a><span class="w">            </span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-51" name="__codelineno-20-51" href="#__codelineno-20-51"></a><span class="w">                </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">index</span><span class="p">;</span>
 <a id="__codelineno-20-52" name="__codelineno-20-52" href="#__codelineno-20-52"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-20-53" name="__codelineno-20-53" href="#__codelineno-20-53"></a><span class="w">            </span><span class="c1">// Calculate bucket index, wrap around to the head if past the tail</span>
+<a id="__codelineno-20-53" name="__codelineno-20-53" href="#__codelineno-20-53"></a><span class="w">            </span><span class="c1">// Calculate the bucket index, wrapping to the beginning after the end</span>
 <a id="__codelineno-20-54" name="__codelineno-20-54" href="#__codelineno-20-54"></a><span class="w">            </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">;</span>
 <a id="__codelineno-20-55" name="__codelineno-20-55" href="#__codelineno-20-55"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-20-56" name="__codelineno-20-56" href="#__codelineno-20-56"></a><span class="w">        </span><span class="c1">// If key does not exist, return the index for insertion</span>
+<a id="__codelineno-20-56" name="__codelineno-20-56" href="#__codelineno-20-56"></a><span class="w">        </span><span class="c1">// If key is absent, return the index for insertion</span>
 <a id="__codelineno-20-57" name="__codelineno-20-57" href="#__codelineno-20-57"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">firstTombstone</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">firstTombstone</span><span class="p">;</span>
 <a id="__codelineno-20-58" name="__codelineno-20-58" href="#__codelineno-20-58"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-59" name="__codelineno-20-59" href="#__codelineno-20-59"></a>
-<a id="__codelineno-20-60" name="__codelineno-20-60" href="#__codelineno-20-60"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-20-60" name="__codelineno-20-60" href="#__codelineno-20-60"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-20-61" name="__codelineno-20-61" href="#__codelineno-20-61"></a><span class="w">    </span><span class="nx">get</span><span class="p">(</span><span class="nx">key</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-62" name="__codelineno-20-62" href="#__codelineno-20-62"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-20-62" name="__codelineno-20-62" href="#__codelineno-20-62"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-20-63" name="__codelineno-20-63" href="#__codelineno-20-63"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">findBucket</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
-<a id="__codelineno-20-64" name="__codelineno-20-64" href="#__codelineno-20-64"></a><span class="w">        </span><span class="c1">// If key-value pair is found, return corresponding val</span>
+<a id="__codelineno-20-64" name="__codelineno-20-64" href="#__codelineno-20-64"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, return its val</span>
 <a id="__codelineno-20-65" name="__codelineno-20-65" href="#__codelineno-20-65"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span>
 <a id="__codelineno-20-66" name="__codelineno-20-66" href="#__codelineno-20-66"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span>
 <a id="__codelineno-20-67" name="__codelineno-20-67" href="#__codelineno-20-67"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">TOMBSTONE</span>
 <a id="__codelineno-20-68" name="__codelineno-20-68" href="#__codelineno-20-68"></a><span class="w">        </span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-69" name="__codelineno-20-69" href="#__codelineno-20-69"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="o">!</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-20-70" name="__codelineno-20-70" href="#__codelineno-20-70"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-20-71" name="__codelineno-20-71" href="#__codelineno-20-71"></a><span class="w">        </span><span class="c1">// If key-value pair does not exist, return null</span>
+<a id="__codelineno-20-71" name="__codelineno-20-71" href="#__codelineno-20-71"></a><span class="w">        </span><span class="c1">// Return null if the key-value pair does not exist</span>
 <a id="__codelineno-20-72" name="__codelineno-20-72" href="#__codelineno-20-72"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-20-73" name="__codelineno-20-73" href="#__codelineno-20-73"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-74" name="__codelineno-20-74" href="#__codelineno-20-74"></a>
-<a id="__codelineno-20-75" name="__codelineno-20-75" href="#__codelineno-20-75"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-20-75" name="__codelineno-20-75" href="#__codelineno-20-75"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-20-76" name="__codelineno-20-76" href="#__codelineno-20-76"></a><span class="w">    </span><span class="nx">put</span><span class="p">(</span><span class="nx">key</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-77" name="__codelineno-20-77" href="#__codelineno-20-77"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-20-77" name="__codelineno-20-77" href="#__codelineno-20-77"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-20-78" name="__codelineno-20-78" href="#__codelineno-20-78"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-79" name="__codelineno-20-79" href="#__codelineno-20-79"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">extend</span><span class="p">();</span>
 <a id="__codelineno-20-80" name="__codelineno-20-80" href="#__codelineno-20-80"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-20-81" name="__codelineno-20-81" href="#__codelineno-20-81"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-20-81" name="__codelineno-20-81" href="#__codelineno-20-81"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-20-82" name="__codelineno-20-82" href="#__codelineno-20-82"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">findBucket</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
-<a id="__codelineno-20-83" name="__codelineno-20-83" href="#__codelineno-20-83"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite val and return</span>
+<a id="__codelineno-20-83" name="__codelineno-20-83" href="#__codelineno-20-83"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, overwrite val and return</span>
 <a id="__codelineno-20-84" name="__codelineno-20-84" href="#__codelineno-20-84"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span>
 <a id="__codelineno-20-85" name="__codelineno-20-85" href="#__codelineno-20-85"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span>
 <a id="__codelineno-20-86" name="__codelineno-20-86" href="#__codelineno-20-86"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">TOMBSTONE</span>
@@ -7237,16 +7314,16 @@
 <a id="__codelineno-20-88" name="__codelineno-20-88" href="#__codelineno-20-88"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="o">!</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-20-89" name="__codelineno-20-89" href="#__codelineno-20-89"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-20-90" name="__codelineno-20-90" href="#__codelineno-20-90"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-20-91" name="__codelineno-20-91" href="#__codelineno-20-91"></a><span class="w">        </span><span class="c1">// If key-value pair does not exist, add the key-value pair</span>
+<a id="__codelineno-20-91" name="__codelineno-20-91" href="#__codelineno-20-91"></a><span class="w">        </span><span class="c1">// If the key-value pair is absent, insert it</span>
 <a id="__codelineno-20-92" name="__codelineno-20-92" href="#__codelineno-20-92"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Pair</span><span class="p">(</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-20-93" name="__codelineno-20-93" href="#__codelineno-20-93"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-20-94" name="__codelineno-20-94" href="#__codelineno-20-94"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-95" name="__codelineno-20-95" href="#__codelineno-20-95"></a>
 <a id="__codelineno-20-96" name="__codelineno-20-96" href="#__codelineno-20-96"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-20-97" name="__codelineno-20-97" href="#__codelineno-20-97"></a><span class="w">    </span><span class="nx">remove</span><span class="p">(</span><span class="nx">key</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-98" name="__codelineno-20-98" href="#__codelineno-20-98"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-20-98" name="__codelineno-20-98" href="#__codelineno-20-98"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-20-99" name="__codelineno-20-99" href="#__codelineno-20-99"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">findBucket</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
-<a id="__codelineno-20-100" name="__codelineno-20-100" href="#__codelineno-20-100"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite it with removal marker</span>
+<a id="__codelineno-20-100" name="__codelineno-20-100" href="#__codelineno-20-100"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, replace it with a tombstone</span>
 <a id="__codelineno-20-101" name="__codelineno-20-101" href="#__codelineno-20-101"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span>
 <a id="__codelineno-20-102" name="__codelineno-20-102" href="#__codelineno-20-102"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span>
 <a id="__codelineno-20-103" name="__codelineno-20-103" href="#__codelineno-20-103"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">TOMBSTONE</span>
@@ -7256,15 +7333,15 @@
 <a id="__codelineno-20-107" name="__codelineno-20-107" href="#__codelineno-20-107"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-20-108" name="__codelineno-20-108" href="#__codelineno-20-108"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-109" name="__codelineno-20-109" href="#__codelineno-20-109"></a>
-<a id="__codelineno-20-110" name="__codelineno-20-110" href="#__codelineno-20-110"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-20-110" name="__codelineno-20-110" href="#__codelineno-20-110"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-20-111" name="__codelineno-20-111" href="#__codelineno-20-111"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">extend</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-112" name="__codelineno-20-112" href="#__codelineno-20-112"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-20-112" name="__codelineno-20-112" href="#__codelineno-20-112"></a><span class="w">        </span><span class="c1">// Keep a reference to the original hash table</span>
 <a id="__codelineno-20-113" name="__codelineno-20-113" href="#__codelineno-20-113"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">bucketsTmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">;</span>
-<a id="__codelineno-20-114" name="__codelineno-20-114" href="#__codelineno-20-114"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-20-114" name="__codelineno-20-114" href="#__codelineno-20-114"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-20-115" name="__codelineno-20-115" href="#__codelineno-20-115"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">extendRatio</span><span class="p">;</span>
 <a id="__codelineno-20-116" name="__codelineno-20-116" href="#__codelineno-20-116"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">null</span><span class="p">);</span>
 <a id="__codelineno-20-117" name="__codelineno-20-117" href="#__codelineno-20-117"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-20-118" name="__codelineno-20-118" href="#__codelineno-20-118"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-20-118" name="__codelineno-20-118" href="#__codelineno-20-118"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-20-119" name="__codelineno-20-119" href="#__codelineno-20-119"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">bucketsTmp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-120" name="__codelineno-20-120" href="#__codelineno-20-120"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pair</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-121" name="__codelineno-20-121" href="#__codelineno-20-121"></a><span class="w">                </span><span class="k">this</span><span class="p">.</span><span class="nx">put</span><span class="p">(</span><span class="nx">pair</span><span class="p">.</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">pair</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span>
@@ -7272,7 +7349,7 @@
 <a id="__codelineno-20-123" name="__codelineno-20-123" href="#__codelineno-20-123"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-20-124" name="__codelineno-20-124" href="#__codelineno-20-124"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-125" name="__codelineno-20-125" href="#__codelineno-20-125"></a>
-<a id="__codelineno-20-126" name="__codelineno-20-126" href="#__codelineno-20-126"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-20-126" name="__codelineno-20-126" href="#__codelineno-20-126"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-20-127" name="__codelineno-20-127" href="#__codelineno-20-127"></a><span class="w">    </span><span class="nx">print</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-128" name="__codelineno-20-128" href="#__codelineno-20-128"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-129" name="__codelineno-20-129" href="#__codelineno-20-129"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pair</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7292,10 +7369,10 @@
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">HashMapOpenAddressing</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">_size</span><span class="p">;</span><span class="w"> </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">_capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">4</span><span class="p">;</span><span class="w"> </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="kt">double</span><span class="w"> </span><span class="n">_loadThres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2.0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">3.0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">_extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="kt">double</span><span class="w"> </span><span class="n">_loadThres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2.0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">3.0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">_extendRatio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">?&gt;</span><span class="w"> </span><span class="n">_buckets</span><span class="p">;</span><span class="w"> </span><span class="c1">// Bucket array</span>
-<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="n">Pair</span><span class="w"> </span><span class="n">_TOMBSTONE</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="o">-</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="s2">"-1"</span><span class="p">);</span><span class="w"> </span><span class="c1">// Removal marker</span>
+<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="n">Pair</span><span class="w"> </span><span class="n">_TOMBSTONE</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="o">-</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="s2">"-1"</span><span class="p">);</span><span class="w"> </span><span class="c1">// Tombstone marker</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">  </span><span class="n">HashMapOpenAddressing</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
@@ -7313,83 +7390,83 @@
 <a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">_capacity</span><span class="p">;</span>
 <a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-25" name="__codelineno-21-25" href="#__codelineno-21-25"></a>
-<a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a><span class="w">  </span><span class="cm">/* Search for bucket index corresponding to key */</span>
+<a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a><span class="w">  </span><span class="cm">/* Find the bucket index for key */</span>
 <a id="__codelineno-21-27" name="__codelineno-21-27" href="#__codelineno-21-27"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-28" name="__codelineno-21-28" href="#__codelineno-21-28"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-21-29" name="__codelineno-21-29" href="#__codelineno-21-29"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">;</span>
-<a id="__codelineno-21-30" name="__codelineno-21-30" href="#__codelineno-21-30"></a><span class="w">    </span><span class="c1">// Linear probing, break when encountering an empty bucket</span>
+<a id="__codelineno-21-30" name="__codelineno-21-30" href="#__codelineno-21-30"></a><span class="w">    </span><span class="c1">// Probe linearly until an empty bucket is reached</span>
 <a id="__codelineno-21-31" name="__codelineno-21-31" href="#__codelineno-21-31"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-32" name="__codelineno-21-32" href="#__codelineno-21-32"></a><span class="w">      </span><span class="c1">// If key is encountered, return the corresponding bucket index</span>
+<a id="__codelineno-21-32" name="__codelineno-21-32" href="#__codelineno-21-32"></a><span class="w">      </span><span class="c1">// If key is found, return its bucket index</span>
 <a id="__codelineno-21-33" name="__codelineno-21-33" href="#__codelineno-21-33"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">!</span><span class="p">.</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-34" name="__codelineno-21-34" href="#__codelineno-21-34"></a><span class="w">        </span><span class="c1">// If a removal marker was encountered before, move the key-value pair to that index</span>
+<a id="__codelineno-21-34" name="__codelineno-21-34" href="#__codelineno-21-34"></a><span class="w">        </span><span class="c1">// If a tombstone was encountered earlier, move the key-value pair to that index</span>
 <a id="__codelineno-21-35" name="__codelineno-21-35" href="#__codelineno-21-35"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-36" name="__codelineno-21-36" href="#__codelineno-21-36"></a><span class="w">          </span><span class="n">_buckets</span><span class="p">[</span><span class="n">firstTombstone</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
 <a id="__codelineno-21-37" name="__codelineno-21-37" href="#__codelineno-21-37"></a><span class="w">          </span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_TOMBSTONE</span><span class="p">;</span>
-<a id="__codelineno-21-38" name="__codelineno-21-38" href="#__codelineno-21-38"></a><span class="w">          </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the moved bucket index</span>
+<a id="__codelineno-21-38" name="__codelineno-21-38" href="#__codelineno-21-38"></a><span class="w">          </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index after the move</span>
 <a id="__codelineno-21-39" name="__codelineno-21-39" href="#__codelineno-21-39"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-21-40" name="__codelineno-21-40" href="#__codelineno-21-40"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return bucket index</span>
+<a id="__codelineno-21-40" name="__codelineno-21-40" href="#__codelineno-21-40"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index</span>
 <a id="__codelineno-21-41" name="__codelineno-21-41" href="#__codelineno-21-41"></a><span class="w">      </span><span class="p">}</span>
-<a id="__codelineno-21-42" name="__codelineno-21-42" href="#__codelineno-21-42"></a><span class="w">      </span><span class="c1">// Record the first removal marker encountered</span>
+<a id="__codelineno-21-42" name="__codelineno-21-42" href="#__codelineno-21-42"></a><span class="w">      </span><span class="c1">// Record the first tombstone encountered</span>
 <a id="__codelineno-21-43" name="__codelineno-21-43" href="#__codelineno-21-43"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">_TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-44" name="__codelineno-21-44" href="#__codelineno-21-44"></a><span class="w">        </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span>
 <a id="__codelineno-21-45" name="__codelineno-21-45" href="#__codelineno-21-45"></a><span class="w">      </span><span class="p">}</span>
-<a id="__codelineno-21-46" name="__codelineno-21-46" href="#__codelineno-21-46"></a><span class="w">      </span><span class="c1">// Calculate bucket index, wrap around to the head if past the tail</span>
+<a id="__codelineno-21-46" name="__codelineno-21-46" href="#__codelineno-21-46"></a><span class="w">      </span><span class="c1">// Calculate the bucket index, wrapping to the beginning after the end</span>
 <a id="__codelineno-21-47" name="__codelineno-21-47" href="#__codelineno-21-47"></a><span class="w">      </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">_capacity</span><span class="p">;</span>
 <a id="__codelineno-21-48" name="__codelineno-21-48" href="#__codelineno-21-48"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-21-49" name="__codelineno-21-49" href="#__codelineno-21-49"></a><span class="w">    </span><span class="c1">// If key does not exist, return the index for insertion</span>
+<a id="__codelineno-21-49" name="__codelineno-21-49" href="#__codelineno-21-49"></a><span class="w">    </span><span class="c1">// If key is absent, return the index for insertion</span>
 <a id="__codelineno-21-50" name="__codelineno-21-50" href="#__codelineno-21-50"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">firstTombstone</span><span class="p">;</span>
 <a id="__codelineno-21-51" name="__codelineno-21-51" href="#__codelineno-21-51"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-52" name="__codelineno-21-52" href="#__codelineno-21-52"></a>
-<a id="__codelineno-21-53" name="__codelineno-21-53" href="#__codelineno-21-53"></a><span class="w">  </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-21-53" name="__codelineno-21-53" href="#__codelineno-21-53"></a><span class="w">  </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-21-54" name="__codelineno-21-54" href="#__codelineno-21-54"></a><span class="w">  </span><span class="kt">String</span><span class="o">?</span><span class="w"> </span><span class="kd">get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-55" name="__codelineno-21-55" href="#__codelineno-21-55"></a><span class="w">    </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-21-55" name="__codelineno-21-55" href="#__codelineno-21-55"></a><span class="w">    </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-21-56" name="__codelineno-21-56" href="#__codelineno-21-56"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-21-57" name="__codelineno-21-57" href="#__codelineno-21-57"></a><span class="w">    </span><span class="c1">// If key-value pair is found, return corresponding val</span>
+<a id="__codelineno-21-57" name="__codelineno-21-57" href="#__codelineno-21-57"></a><span class="w">    </span><span class="c1">// If the key-value pair is found, return its val</span>
 <a id="__codelineno-21-58" name="__codelineno-21-58" href="#__codelineno-21-58"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">_TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-59" name="__codelineno-21-59" href="#__codelineno-21-59"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-21-60" name="__codelineno-21-60" href="#__codelineno-21-60"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-21-61" name="__codelineno-21-61" href="#__codelineno-21-61"></a><span class="w">    </span><span class="c1">// If key-value pair does not exist, return null</span>
+<a id="__codelineno-21-61" name="__codelineno-21-61" href="#__codelineno-21-61"></a><span class="w">    </span><span class="c1">// Return null if the key-value pair does not exist</span>
 <a id="__codelineno-21-62" name="__codelineno-21-62" href="#__codelineno-21-62"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-21-63" name="__codelineno-21-63" href="#__codelineno-21-63"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-64" name="__codelineno-21-64" href="#__codelineno-21-64"></a>
-<a id="__codelineno-21-65" name="__codelineno-21-65" href="#__codelineno-21-65"></a><span class="w">  </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-21-65" name="__codelineno-21-65" href="#__codelineno-21-65"></a><span class="w">  </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-21-66" name="__codelineno-21-66" href="#__codelineno-21-66"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">put</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="kt">String</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-67" name="__codelineno-21-67" href="#__codelineno-21-67"></a><span class="w">    </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-21-67" name="__codelineno-21-67" href="#__codelineno-21-67"></a><span class="w">    </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-21-68" name="__codelineno-21-68" href="#__codelineno-21-68"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">_loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-69" name="__codelineno-21-69" href="#__codelineno-21-69"></a><span class="w">      </span><span class="n">extend</span><span class="p">();</span>
 <a id="__codelineno-21-70" name="__codelineno-21-70" href="#__codelineno-21-70"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-21-71" name="__codelineno-21-71" href="#__codelineno-21-71"></a><span class="w">    </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-21-71" name="__codelineno-21-71" href="#__codelineno-21-71"></a><span class="w">    </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-21-72" name="__codelineno-21-72" href="#__codelineno-21-72"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-21-73" name="__codelineno-21-73" href="#__codelineno-21-73"></a><span class="w">    </span><span class="c1">// If key-value pair is found, overwrite val and return</span>
+<a id="__codelineno-21-73" name="__codelineno-21-73" href="#__codelineno-21-73"></a><span class="w">    </span><span class="c1">// If the key-value pair is found, overwrite val and return</span>
 <a id="__codelineno-21-74" name="__codelineno-21-74" href="#__codelineno-21-74"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">_TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-75" name="__codelineno-21-75" href="#__codelineno-21-75"></a><span class="w">      </span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-21-76" name="__codelineno-21-76" href="#__codelineno-21-76"></a><span class="w">      </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-21-77" name="__codelineno-21-77" href="#__codelineno-21-77"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-21-78" name="__codelineno-21-78" href="#__codelineno-21-78"></a><span class="w">    </span><span class="c1">// If key-value pair does not exist, add the key-value pair</span>
+<a id="__codelineno-21-78" name="__codelineno-21-78" href="#__codelineno-21-78"></a><span class="w">    </span><span class="c1">// If the key-value pair is absent, insert it</span>
 <a id="__codelineno-21-79" name="__codelineno-21-79" href="#__codelineno-21-79"></a><span class="w">    </span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-21-80" name="__codelineno-21-80" href="#__codelineno-21-80"></a><span class="w">    </span><span class="n">_size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-21-81" name="__codelineno-21-81" href="#__codelineno-21-81"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-82" name="__codelineno-21-82" href="#__codelineno-21-82"></a>
 <a id="__codelineno-21-83" name="__codelineno-21-83" href="#__codelineno-21-83"></a><span class="w">  </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-21-84" name="__codelineno-21-84" href="#__codelineno-21-84"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-85" name="__codelineno-21-85" href="#__codelineno-21-85"></a><span class="w">    </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-21-85" name="__codelineno-21-85" href="#__codelineno-21-85"></a><span class="w">    </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-21-86" name="__codelineno-21-86" href="#__codelineno-21-86"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-21-87" name="__codelineno-21-87" href="#__codelineno-21-87"></a><span class="w">    </span><span class="c1">// If key-value pair is found, overwrite it with removal marker</span>
+<a id="__codelineno-21-87" name="__codelineno-21-87" href="#__codelineno-21-87"></a><span class="w">    </span><span class="c1">// If the key-value pair is found, replace it with a tombstone</span>
 <a id="__codelineno-21-88" name="__codelineno-21-88" href="#__codelineno-21-88"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">_TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-89" name="__codelineno-21-89" href="#__codelineno-21-89"></a><span class="w">      </span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_TOMBSTONE</span><span class="p">;</span>
 <a id="__codelineno-21-90" name="__codelineno-21-90" href="#__codelineno-21-90"></a><span class="w">      </span><span class="n">_size</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-21-91" name="__codelineno-21-91" href="#__codelineno-21-91"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-21-92" name="__codelineno-21-92" href="#__codelineno-21-92"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-93" name="__codelineno-21-93" href="#__codelineno-21-93"></a>
-<a id="__codelineno-21-94" name="__codelineno-21-94" href="#__codelineno-21-94"></a><span class="w">  </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-21-94" name="__codelineno-21-94" href="#__codelineno-21-94"></a><span class="w">  </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-21-95" name="__codelineno-21-95" href="#__codelineno-21-95"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-96" name="__codelineno-21-96" href="#__codelineno-21-96"></a><span class="w">    </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-21-96" name="__codelineno-21-96" href="#__codelineno-21-96"></a><span class="w">    </span><span class="c1">// Keep a reference to the original hash table</span>
 <a id="__codelineno-21-97" name="__codelineno-21-97" href="#__codelineno-21-97"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">?&gt;</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_buckets</span><span class="p">;</span>
-<a id="__codelineno-21-98" name="__codelineno-21-98" href="#__codelineno-21-98"></a><span class="w">    </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-21-98" name="__codelineno-21-98" href="#__codelineno-21-98"></a><span class="w">    </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-21-99" name="__codelineno-21-99" href="#__codelineno-21-99"></a><span class="w">    </span><span class="n">_capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="n">_extendRatio</span><span class="p">;</span>
 <a id="__codelineno-21-100" name="__codelineno-21-100" href="#__codelineno-21-100"></a><span class="w">    </span><span class="n">_buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">_capacity</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="kc">null</span><span class="p">);</span>
 <a id="__codelineno-21-101" name="__codelineno-21-101" href="#__codelineno-21-101"></a><span class="w">    </span><span class="n">_size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
-<a id="__codelineno-21-102" name="__codelineno-21-102" href="#__codelineno-21-102"></a><span class="w">    </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-21-102" name="__codelineno-21-102" href="#__codelineno-21-102"></a><span class="w">    </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-21-103" name="__codelineno-21-103" href="#__codelineno-21-103"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="o">?</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-104" name="__codelineno-21-104" href="#__codelineno-21-104"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">_TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-105" name="__codelineno-21-105" href="#__codelineno-21-105"></a><span class="w">        </span><span class="n">put</span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
@@ -7397,7 +7474,7 @@
 <a id="__codelineno-21-107" name="__codelineno-21-107" href="#__codelineno-21-107"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-21-108" name="__codelineno-21-108" href="#__codelineno-21-108"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-109" name="__codelineno-21-109" href="#__codelineno-21-109"></a>
-<a id="__codelineno-21-110" name="__codelineno-21-110" href="#__codelineno-21-110"></a><span class="w">  </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-21-110" name="__codelineno-21-110" href="#__codelineno-21-110"></a><span class="w">  </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-21-111" name="__codelineno-21-111" href="#__codelineno-21-111"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">printHashMap</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-112" name="__codelineno-21-112" href="#__codelineno-21-112"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="o">?</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">_buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-113" name="__codelineno-21-113" href="#__codelineno-21-113"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7417,10 +7494,10 @@
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">struct</span><span class="w"> </span><span class="nc">HashMapOpenAddressing</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="n">size</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w">                </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="n">capacity</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w">            </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="n">load_thres</span><span class="p">:</span><span class="w"> </span><span class="kt">f64</span><span class="p">,</span><span class="w">            </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="n">extend_ratio</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w">        </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="n">load_thres</span><span class="p">:</span><span class="w"> </span><span class="kt">f64</span><span class="p">,</span><span class="w">            </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="n">extend_ratio</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w">        </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="n">buckets</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="c1">// Bucket array</span>
-<a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">    </span><span class="n">TOMBSTONE</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="p">,</span><span class="w">    </span><span class="c1">// Removal marker</span>
+<a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">    </span><span class="n">TOMBSTONE</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="p">,</span><span class="w">    </span><span class="c1">// Tombstone marker</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="p">}</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a>
 <a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="k">impl</span><span class="w"> </span><span class="n">HashMapOpenAddressing</span><span class="w"> </span><span class="p">{</span>
@@ -7449,30 +7526,30 @@
 <a id="__codelineno-22-34" name="__codelineno-22-34" href="#__codelineno-22-34"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">capacity</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span>
 <a id="__codelineno-22-35" name="__codelineno-22-35" href="#__codelineno-22-35"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-36" name="__codelineno-22-36" href="#__codelineno-22-36"></a>
-<a id="__codelineno-22-37" name="__codelineno-22-37" href="#__codelineno-22-37"></a><span class="w">    </span><span class="cm">/* Search for bucket index corresponding to key */</span>
+<a id="__codelineno-22-37" name="__codelineno-22-37" href="#__codelineno-22-37"></a><span class="w">    </span><span class="cm">/* Find the bucket index for key */</span>
 <a id="__codelineno-22-38" name="__codelineno-22-38" href="#__codelineno-22-38"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">find_bucket</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-39" name="__codelineno-22-39" href="#__codelineno-22-39"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-22-40" name="__codelineno-22-40" href="#__codelineno-22-40"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">first_tombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-22-41" name="__codelineno-22-41" href="#__codelineno-22-41"></a><span class="w">        </span><span class="c1">// Linear probing, break when encountering an empty bucket</span>
+<a id="__codelineno-22-41" name="__codelineno-22-41" href="#__codelineno-22-41"></a><span class="w">        </span><span class="c1">// Probe linearly until an empty bucket is reached</span>
 <a id="__codelineno-22-42" name="__codelineno-22-42" href="#__codelineno-22-42"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">is_some</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-43" name="__codelineno-22-43" href="#__codelineno-22-43"></a><span class="w">            </span><span class="c1">// If key is found, return corresponding bucket index</span>
+<a id="__codelineno-22-43" name="__codelineno-22-43" href="#__codelineno-22-43"></a><span class="w">            </span><span class="c1">// If key is found, return the corresponding bucket index</span>
 <a id="__codelineno-22-44" name="__codelineno-22-44" href="#__codelineno-22-44"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">as_ref</span><span class="p">().</span><span class="n">unwrap</span><span class="p">().</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-45" name="__codelineno-22-45" href="#__codelineno-22-45"></a><span class="w">                </span><span class="c1">// If deletion marker was encountered before, move key-value pair to that index</span>
+<a id="__codelineno-22-45" name="__codelineno-22-45" href="#__codelineno-22-45"></a><span class="w">                </span><span class="c1">// If a tombstone was encountered earlier, move the key-value pair to that index</span>
 <a id="__codelineno-22-46" name="__codelineno-22-46" href="#__codelineno-22-46"></a><span class="w">                </span><span class="k">if</span><span class="w"> </span><span class="n">first_tombstone</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-47" name="__codelineno-22-47" href="#__codelineno-22-47"></a><span class="w">                    </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">first_tombstone</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">take</span><span class="p">();</span>
 <a id="__codelineno-22-48" name="__codelineno-22-48" href="#__codelineno-22-48"></a><span class="w">                    </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">TOMBSTONE</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
-<a id="__codelineno-22-49" name="__codelineno-22-49" href="#__codelineno-22-49"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="n">first_tombstone</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the moved bucket index</span>
+<a id="__codelineno-22-49" name="__codelineno-22-49" href="#__codelineno-22-49"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="n">first_tombstone</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index after the move</span>
 <a id="__codelineno-22-50" name="__codelineno-22-50" href="#__codelineno-22-50"></a><span class="w">                </span><span class="p">}</span>
-<a id="__codelineno-22-51" name="__codelineno-22-51" href="#__codelineno-22-51"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return bucket index</span>
+<a id="__codelineno-22-51" name="__codelineno-22-51" href="#__codelineno-22-51"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index</span>
 <a id="__codelineno-22-52" name="__codelineno-22-52" href="#__codelineno-22-52"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-22-53" name="__codelineno-22-53" href="#__codelineno-22-53"></a><span class="w">            </span><span class="c1">// Record the first removal marker encountered</span>
+<a id="__codelineno-22-53" name="__codelineno-22-53" href="#__codelineno-22-53"></a><span class="w">            </span><span class="c1">// Record the first tombstone encountered</span>
 <a id="__codelineno-22-54" name="__codelineno-22-54" href="#__codelineno-22-54"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">first_tombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-55" name="__codelineno-22-55" href="#__codelineno-22-55"></a><span class="w">                </span><span class="n">first_tombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
 <a id="__codelineno-22-56" name="__codelineno-22-56" href="#__codelineno-22-56"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-22-57" name="__codelineno-22-57" href="#__codelineno-22-57"></a><span class="w">            </span><span class="c1">// Calculate bucket index, wrap around to the head if past the tail</span>
+<a id="__codelineno-22-57" name="__codelineno-22-57" href="#__codelineno-22-57"></a><span class="w">            </span><span class="c1">// Calculate the bucket index, wrapping to the beginning after the end</span>
 <a id="__codelineno-22-58" name="__codelineno-22-58" href="#__codelineno-22-58"></a><span class="w">            </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-22-59" name="__codelineno-22-59" href="#__codelineno-22-59"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-22-60" name="__codelineno-22-60" href="#__codelineno-22-60"></a><span class="w">        </span><span class="c1">// If key does not exist, return the index for insertion</span>
+<a id="__codelineno-22-60" name="__codelineno-22-60" href="#__codelineno-22-60"></a><span class="w">        </span><span class="c1">// If key is absent, return the index for insertion</span>
 <a id="__codelineno-22-61" name="__codelineno-22-61" href="#__codelineno-22-61"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">first_tombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-62" name="__codelineno-22-62" href="#__codelineno-22-62"></a><span class="w">            </span><span class="n">index</span>
 <a id="__codelineno-22-63" name="__codelineno-22-63" href="#__codelineno-22-63"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -7480,57 +7557,57 @@
 <a id="__codelineno-22-65" name="__codelineno-22-65" href="#__codelineno-22-65"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-66" name="__codelineno-22-66" href="#__codelineno-22-66"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-67" name="__codelineno-22-67" href="#__codelineno-22-67"></a>
-<a id="__codelineno-22-68" name="__codelineno-22-68" href="#__codelineno-22-68"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-22-68" name="__codelineno-22-68" href="#__codelineno-22-68"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-22-69" name="__codelineno-22-69" href="#__codelineno-22-69"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="kt">str</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-70" name="__codelineno-22-70" href="#__codelineno-22-70"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-22-70" name="__codelineno-22-70" href="#__codelineno-22-70"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-22-71" name="__codelineno-22-71" href="#__codelineno-22-71"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">find_bucket</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-22-72" name="__codelineno-22-72" href="#__codelineno-22-72"></a><span class="w">        </span><span class="c1">// If key-value pair is found, return corresponding val</span>
+<a id="__codelineno-22-72" name="__codelineno-22-72" href="#__codelineno-22-72"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, return its val</span>
 <a id="__codelineno-22-73" name="__codelineno-22-73" href="#__codelineno-22-73"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">is_some</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-74" name="__codelineno-22-74" href="#__codelineno-22-74"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">as_ref</span><span class="p">().</span><span class="n">map</span><span class="p">(</span><span class="o">|</span><span class="n">pair</span><span class="o">|</span><span class="w"> </span><span class="o">&amp;</span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="o">&amp;</span><span class="kt">str</span><span class="p">);</span>
 <a id="__codelineno-22-75" name="__codelineno-22-75" href="#__codelineno-22-75"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-22-76" name="__codelineno-22-76" href="#__codelineno-22-76"></a><span class="w">        </span><span class="c1">// If key-value pair does not exist, return null</span>
+<a id="__codelineno-22-76" name="__codelineno-22-76" href="#__codelineno-22-76"></a><span class="w">        </span><span class="c1">// If the key-value pair does not exist, return None</span>
 <a id="__codelineno-22-77" name="__codelineno-22-77" href="#__codelineno-22-77"></a><span class="w">        </span><span class="nb">None</span>
 <a id="__codelineno-22-78" name="__codelineno-22-78" href="#__codelineno-22-78"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-79" name="__codelineno-22-79" href="#__codelineno-22-79"></a>
-<a id="__codelineno-22-80" name="__codelineno-22-80" href="#__codelineno-22-80"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-22-80" name="__codelineno-22-80" href="#__codelineno-22-80"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-22-81" name="__codelineno-22-81" href="#__codelineno-22-81"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="nb">String</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-82" name="__codelineno-22-82" href="#__codelineno-22-82"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-22-82" name="__codelineno-22-82" href="#__codelineno-22-82"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-22-83" name="__codelineno-22-83" href="#__codelineno-22-83"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">load_factor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">load_thres</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-84" name="__codelineno-22-84" href="#__codelineno-22-84"></a><span class="w">            </span><span class="bp">self</span><span class="p">.</span><span class="n">extend</span><span class="p">();</span>
 <a id="__codelineno-22-85" name="__codelineno-22-85" href="#__codelineno-22-85"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-22-86" name="__codelineno-22-86" href="#__codelineno-22-86"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-22-86" name="__codelineno-22-86" href="#__codelineno-22-86"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-22-87" name="__codelineno-22-87" href="#__codelineno-22-87"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">find_bucket</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-22-88" name="__codelineno-22-88" href="#__codelineno-22-88"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite val and return</span>
+<a id="__codelineno-22-88" name="__codelineno-22-88" href="#__codelineno-22-88"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, overwrite val and return</span>
 <a id="__codelineno-22-89" name="__codelineno-22-89" href="#__codelineno-22-89"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">is_some</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-90" name="__codelineno-22-90" href="#__codelineno-22-90"></a><span class="w">            </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">as_mut</span><span class="p">().</span><span class="n">unwrap</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-22-91" name="__codelineno-22-91" href="#__codelineno-22-91"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-22-92" name="__codelineno-22-92" href="#__codelineno-22-92"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-22-93" name="__codelineno-22-93" href="#__codelineno-22-93"></a><span class="w">        </span><span class="c1">// If key-value pair does not exist, add the key-value pair</span>
+<a id="__codelineno-22-93" name="__codelineno-22-93" href="#__codelineno-22-93"></a><span class="w">        </span><span class="c1">// If the key-value pair is absent, insert it</span>
 <a id="__codelineno-22-94" name="__codelineno-22-94" href="#__codelineno-22-94"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="p">});</span>
 <a id="__codelineno-22-95" name="__codelineno-22-95" href="#__codelineno-22-95"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-22-96" name="__codelineno-22-96" href="#__codelineno-22-96"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-97" name="__codelineno-22-97" href="#__codelineno-22-97"></a>
 <a id="__codelineno-22-98" name="__codelineno-22-98" href="#__codelineno-22-98"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-22-99" name="__codelineno-22-99" href="#__codelineno-22-99"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-100" name="__codelineno-22-100" href="#__codelineno-22-100"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-22-100" name="__codelineno-22-100" href="#__codelineno-22-100"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-22-101" name="__codelineno-22-101" href="#__codelineno-22-101"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">find_bucket</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-22-102" name="__codelineno-22-102" href="#__codelineno-22-102"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite it with removal marker</span>
+<a id="__codelineno-22-102" name="__codelineno-22-102" href="#__codelineno-22-102"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, replace it with a tombstone</span>
 <a id="__codelineno-22-103" name="__codelineno-22-103" href="#__codelineno-22-103"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">is_some</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-104" name="__codelineno-22-104" href="#__codelineno-22-104"></a><span class="w">            </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">TOMBSTONE</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-22-105" name="__codelineno-22-105" href="#__codelineno-22-105"></a><span class="w">            </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-22-106" name="__codelineno-22-106" href="#__codelineno-22-106"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-107" name="__codelineno-22-107" href="#__codelineno-22-107"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-108" name="__codelineno-22-108" href="#__codelineno-22-108"></a>
-<a id="__codelineno-22-109" name="__codelineno-22-109" href="#__codelineno-22-109"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-22-109" name="__codelineno-22-109" href="#__codelineno-22-109"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-22-110" name="__codelineno-22-110" href="#__codelineno-22-110"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">extend</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-111" name="__codelineno-22-111" href="#__codelineno-22-111"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-22-111" name="__codelineno-22-111" href="#__codelineno-22-111"></a><span class="w">        </span><span class="c1">// Temporarily save a copy of the original buckets</span>
 <a id="__codelineno-22-112" name="__codelineno-22-112" href="#__codelineno-22-112"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">buckets_tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
-<a id="__codelineno-22-113" name="__codelineno-22-113" href="#__codelineno-22-113"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-22-113" name="__codelineno-22-113" href="#__codelineno-22-113"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-22-114" name="__codelineno-22-114" href="#__codelineno-22-114"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">extend_ratio</span><span class="p">;</span>
 <a id="__codelineno-22-115" name="__codelineno-22-115" href="#__codelineno-22-115"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="nb">None</span><span class="p">;</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">capacity</span><span class="p">];</span>
 <a id="__codelineno-22-116" name="__codelineno-22-116" href="#__codelineno-22-116"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-22-117" name="__codelineno-22-117" href="#__codelineno-22-117"></a>
-<a id="__codelineno-22-118" name="__codelineno-22-118" href="#__codelineno-22-118"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-22-118" name="__codelineno-22-118" href="#__codelineno-22-118"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-22-119" name="__codelineno-22-119" href="#__codelineno-22-119"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets_tmp</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-120" name="__codelineno-22-120" href="#__codelineno-22-120"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">is_none</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">TOMBSTONE</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-121" name="__codelineno-22-121" href="#__codelineno-22-121"></a><span class="w">                </span><span class="k">continue</span><span class="p">;</span>
@@ -7540,7 +7617,7 @@
 <a id="__codelineno-22-125" name="__codelineno-22-125" href="#__codelineno-22-125"></a><span class="w">            </span><span class="bp">self</span><span class="p">.</span><span class="n">put</span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-22-126" name="__codelineno-22-126" href="#__codelineno-22-126"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-127" name="__codelineno-22-127" href="#__codelineno-22-127"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-128" name="__codelineno-22-128" href="#__codelineno-22-128"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-22-128" name="__codelineno-22-128" href="#__codelineno-22-128"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-22-129" name="__codelineno-22-129" href="#__codelineno-22-129"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">print</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-130" name="__codelineno-22-130" href="#__codelineno-22-130"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="o">&amp;</span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-131" name="__codelineno-22-131" href="#__codelineno-22-131"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">is_none</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
@@ -7561,10 +7638,10 @@
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w">         </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span><span class="w">     </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span><span class="w">  </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="kt">double</span><span class="w"> </span><span class="n">loadThres</span><span class="p">;</span><span class="w"> </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">extendRatio</span><span class="p">;</span><span class="w">  </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="n">Pair</span><span class="w"> </span><span class="o">**</span><span class="n">buckets</span><span class="p">;</span><span class="w">   </span><span class="c1">// Bucket array</span>
-<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">TOMBSTONE</span><span class="p">;</span><span class="w">  </span><span class="c1">// Removal marker</span>
+<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">TOMBSTONE</span><span class="p">;</span><span class="w">  </span><span class="c1">// Tombstone marker</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="p">}</span><span class="w"> </span><span class="n">HashMapOpenAddressing</span><span class="p">;</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a>
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="cm">/* Constructor */</span>
@@ -7606,54 +7683,54 @@
 <a id="__codelineno-23-47" name="__codelineno-23-47" href="#__codelineno-23-47"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="p">)</span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-23-48" name="__codelineno-23-48" href="#__codelineno-23-48"></a><span class="p">}</span>
 <a id="__codelineno-23-49" name="__codelineno-23-49" href="#__codelineno-23-49"></a>
-<a id="__codelineno-23-50" name="__codelineno-23-50" href="#__codelineno-23-50"></a><span class="cm">/* Search for bucket index corresponding to key */</span>
+<a id="__codelineno-23-50" name="__codelineno-23-50" href="#__codelineno-23-50"></a><span class="cm">/* Find the bucket index for key */</span>
 <a id="__codelineno-23-51" name="__codelineno-23-51" href="#__codelineno-23-51"></a><span class="kt">int</span><span class="w"> </span><span class="nf">findBucket</span><span class="p">(</span><span class="n">HashMapOpenAddressing</span><span class="w"> </span><span class="o">*</span><span class="n">hashMap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-52" name="__codelineno-23-52" href="#__codelineno-23-52"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">hashMap</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-23-53" name="__codelineno-23-53" href="#__codelineno-23-53"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
-<a id="__codelineno-23-54" name="__codelineno-23-54" href="#__codelineno-23-54"></a><span class="w">    </span><span class="c1">// Linear probing, break when encountering an empty bucket</span>
+<a id="__codelineno-23-54" name="__codelineno-23-54" href="#__codelineno-23-54"></a><span class="w">    </span><span class="c1">// Probe linearly until an empty bucket is reached</span>
 <a id="__codelineno-23-55" name="__codelineno-23-55" href="#__codelineno-23-55"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-56" name="__codelineno-23-56" href="#__codelineno-23-56"></a><span class="w">        </span><span class="c1">// If key is encountered, return the corresponding bucket index</span>
+<a id="__codelineno-23-56" name="__codelineno-23-56" href="#__codelineno-23-56"></a><span class="w">        </span><span class="c1">// If key is found, return its bucket index</span>
 <a id="__codelineno-23-57" name="__codelineno-23-57" href="#__codelineno-23-57"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">-&gt;</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-58" name="__codelineno-23-58" href="#__codelineno-23-58"></a><span class="w">            </span><span class="c1">// If a removal marker was encountered before, move the key-value pair to that index</span>
+<a id="__codelineno-23-58" name="__codelineno-23-58" href="#__codelineno-23-58"></a><span class="w">            </span><span class="c1">// If a tombstone was encountered earlier, move the key-value pair to that index</span>
 <a id="__codelineno-23-59" name="__codelineno-23-59" href="#__codelineno-23-59"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">-1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-60" name="__codelineno-23-60" href="#__codelineno-23-60"></a><span class="w">                </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">firstTombstone</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
 <a id="__codelineno-23-61" name="__codelineno-23-61" href="#__codelineno-23-61"></a><span class="w">                </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">TOMBSTONE</span><span class="p">;</span>
-<a id="__codelineno-23-62" name="__codelineno-23-62" href="#__codelineno-23-62"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the moved bucket index</span>
+<a id="__codelineno-23-62" name="__codelineno-23-62" href="#__codelineno-23-62"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index after the move</span>
 <a id="__codelineno-23-63" name="__codelineno-23-63" href="#__codelineno-23-63"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-23-64" name="__codelineno-23-64" href="#__codelineno-23-64"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return bucket index</span>
+<a id="__codelineno-23-64" name="__codelineno-23-64" href="#__codelineno-23-64"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the bucket index</span>
 <a id="__codelineno-23-65" name="__codelineno-23-65" href="#__codelineno-23-65"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-23-66" name="__codelineno-23-66" href="#__codelineno-23-66"></a><span class="w">        </span><span class="c1">// Record the first removal marker encountered</span>
+<a id="__codelineno-23-66" name="__codelineno-23-66" href="#__codelineno-23-66"></a><span class="w">        </span><span class="c1">// Record the first tombstone encountered</span>
 <a id="__codelineno-23-67" name="__codelineno-23-67" href="#__codelineno-23-67"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">-1</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-68" name="__codelineno-23-68" href="#__codelineno-23-68"></a><span class="w">            </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span>
 <a id="__codelineno-23-69" name="__codelineno-23-69" href="#__codelineno-23-69"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-23-70" name="__codelineno-23-70" href="#__codelineno-23-70"></a><span class="w">        </span><span class="c1">// Calculate bucket index, wrap around to the head if past the tail</span>
+<a id="__codelineno-23-70" name="__codelineno-23-70" href="#__codelineno-23-70"></a><span class="w">        </span><span class="c1">// Calculate the bucket index, wrapping to the beginning after the end</span>
 <a id="__codelineno-23-71" name="__codelineno-23-71" href="#__codelineno-23-71"></a><span class="w">        </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-23-72" name="__codelineno-23-72" href="#__codelineno-23-72"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-73" name="__codelineno-23-73" href="#__codelineno-23-73"></a><span class="w">    </span><span class="c1">// If key does not exist, return the index for insertion</span>
+<a id="__codelineno-23-73" name="__codelineno-23-73" href="#__codelineno-23-73"></a><span class="w">    </span><span class="c1">// If key is absent, return the index for insertion</span>
 <a id="__codelineno-23-74" name="__codelineno-23-74" href="#__codelineno-23-74"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">-1</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">firstTombstone</span><span class="p">;</span>
 <a id="__codelineno-23-75" name="__codelineno-23-75" href="#__codelineno-23-75"></a><span class="p">}</span>
 <a id="__codelineno-23-76" name="__codelineno-23-76" href="#__codelineno-23-76"></a>
-<a id="__codelineno-23-77" name="__codelineno-23-77" href="#__codelineno-23-77"></a><span class="cm">/* Query operation */</span>
+<a id="__codelineno-23-77" name="__codelineno-23-77" href="#__codelineno-23-77"></a><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-23-78" name="__codelineno-23-78" href="#__codelineno-23-78"></a><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="nf">get</span><span class="p">(</span><span class="n">HashMapOpenAddressing</span><span class="w"> </span><span class="o">*</span><span class="n">hashMap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-79" name="__codelineno-23-79" href="#__codelineno-23-79"></a><span class="w">    </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-23-79" name="__codelineno-23-79" href="#__codelineno-23-79"></a><span class="w">    </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-23-80" name="__codelineno-23-80" href="#__codelineno-23-80"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">hashMap</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-23-81" name="__codelineno-23-81" href="#__codelineno-23-81"></a><span class="w">    </span><span class="c1">// If key-value pair is found, return corresponding val</span>
+<a id="__codelineno-23-81" name="__codelineno-23-81" href="#__codelineno-23-81"></a><span class="w">    </span><span class="c1">// If the key-value pair is found, return its val</span>
 <a id="__codelineno-23-82" name="__codelineno-23-82" href="#__codelineno-23-82"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-83" name="__codelineno-23-83" href="#__codelineno-23-83"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-23-84" name="__codelineno-23-84" href="#__codelineno-23-84"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-85" name="__codelineno-23-85" href="#__codelineno-23-85"></a><span class="w">    </span><span class="c1">// Return empty string if key-value pair does not exist</span>
+<a id="__codelineno-23-85" name="__codelineno-23-85" href="#__codelineno-23-85"></a><span class="w">    </span><span class="c1">// Return an empty string if the key-value pair does not exist</span>
 <a id="__codelineno-23-86" name="__codelineno-23-86" href="#__codelineno-23-86"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="s">""</span><span class="p">;</span>
 <a id="__codelineno-23-87" name="__codelineno-23-87" href="#__codelineno-23-87"></a><span class="p">}</span>
 <a id="__codelineno-23-88" name="__codelineno-23-88" href="#__codelineno-23-88"></a>
-<a id="__codelineno-23-89" name="__codelineno-23-89" href="#__codelineno-23-89"></a><span class="cm">/* Add operation */</span>
+<a id="__codelineno-23-89" name="__codelineno-23-89" href="#__codelineno-23-89"></a><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-23-90" name="__codelineno-23-90" href="#__codelineno-23-90"></a><span class="kt">void</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="n">HashMapOpenAddressing</span><span class="w"> </span><span class="o">*</span><span class="n">hashMap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-91" name="__codelineno-23-91" href="#__codelineno-23-91"></a><span class="w">    </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-23-91" name="__codelineno-23-91" href="#__codelineno-23-91"></a><span class="w">    </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-23-92" name="__codelineno-23-92" href="#__codelineno-23-92"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">loadFactor</span><span class="p">(</span><span class="n">hashMap</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-93" name="__codelineno-23-93" href="#__codelineno-23-93"></a><span class="w">        </span><span class="n">extend</span><span class="p">(</span><span class="n">hashMap</span><span class="p">);</span>
 <a id="__codelineno-23-94" name="__codelineno-23-94" href="#__codelineno-23-94"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-95" name="__codelineno-23-95" href="#__codelineno-23-95"></a><span class="w">    </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-23-95" name="__codelineno-23-95" href="#__codelineno-23-95"></a><span class="w">    </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-23-96" name="__codelineno-23-96" href="#__codelineno-23-96"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">hashMap</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-23-97" name="__codelineno-23-97" href="#__codelineno-23-97"></a><span class="w">    </span><span class="c1">// If key-value pair is found, overwrite val and return</span>
+<a id="__codelineno-23-97" name="__codelineno-23-97" href="#__codelineno-23-97"></a><span class="w">    </span><span class="c1">// If the key-value pair is found, overwrite val and return</span>
 <a id="__codelineno-23-98" name="__codelineno-23-98" href="#__codelineno-23-98"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-99" name="__codelineno-23-99" href="#__codelineno-23-99"></a><span class="w">        </span><span class="n">free</span><span class="p">(</span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-23-100" name="__codelineno-23-100" href="#__codelineno-23-100"></a><span class="w">        </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">strlen</span><span class="p">(</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">));</span>
@@ -7661,7 +7738,7 @@
 <a id="__codelineno-23-102" name="__codelineno-23-102" href="#__codelineno-23-102"></a><span class="w">        </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">[</span><span class="n">strlen</span><span class="p">(</span><span class="n">val</span><span class="p">)]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="sc">'\0'</span><span class="p">;</span>
 <a id="__codelineno-23-103" name="__codelineno-23-103" href="#__codelineno-23-103"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-23-104" name="__codelineno-23-104" href="#__codelineno-23-104"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-105" name="__codelineno-23-105" href="#__codelineno-23-105"></a><span class="w">    </span><span class="c1">// If key-value pair does not exist, add the key-value pair</span>
+<a id="__codelineno-23-105" name="__codelineno-23-105" href="#__codelineno-23-105"></a><span class="w">    </span><span class="c1">// If the key-value pair is absent, insert it</span>
 <a id="__codelineno-23-106" name="__codelineno-23-106" href="#__codelineno-23-106"></a><span class="w">    </span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">Pair</span><span class="p">));</span>
 <a id="__codelineno-23-107" name="__codelineno-23-107" href="#__codelineno-23-107"></a><span class="w">    </span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">key</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">key</span><span class="p">;</span>
 <a id="__codelineno-23-108" name="__codelineno-23-108" href="#__codelineno-23-108"></a><span class="w">    </span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">strlen</span><span class="p">(</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">));</span>
@@ -7674,9 +7751,9 @@
 <a id="__codelineno-23-115" name="__codelineno-23-115" href="#__codelineno-23-115"></a>
 <a id="__codelineno-23-116" name="__codelineno-23-116" href="#__codelineno-23-116"></a><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-23-117" name="__codelineno-23-117" href="#__codelineno-23-117"></a><span class="kt">void</span><span class="w"> </span><span class="nf">removeItem</span><span class="p">(</span><span class="n">HashMapOpenAddressing</span><span class="w"> </span><span class="o">*</span><span class="n">hashMap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-118" name="__codelineno-23-118" href="#__codelineno-23-118"></a><span class="w">    </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-23-118" name="__codelineno-23-118" href="#__codelineno-23-118"></a><span class="w">    </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-23-119" name="__codelineno-23-119" href="#__codelineno-23-119"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">hashMap</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-23-120" name="__codelineno-23-120" href="#__codelineno-23-120"></a><span class="w">    </span><span class="c1">// If key-value pair is found, overwrite it with removal marker</span>
+<a id="__codelineno-23-120" name="__codelineno-23-120" href="#__codelineno-23-120"></a><span class="w">    </span><span class="c1">// If the key-value pair is found, replace it with a tombstone</span>
 <a id="__codelineno-23-121" name="__codelineno-23-121" href="#__codelineno-23-121"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-122" name="__codelineno-23-122" href="#__codelineno-23-122"></a><span class="w">        </span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
 <a id="__codelineno-23-123" name="__codelineno-23-123" href="#__codelineno-23-123"></a><span class="w">        </span><span class="n">free</span><span class="p">(</span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">);</span>
@@ -7686,16 +7763,16 @@
 <a id="__codelineno-23-127" name="__codelineno-23-127" href="#__codelineno-23-127"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-23-128" name="__codelineno-23-128" href="#__codelineno-23-128"></a><span class="p">}</span>
 <a id="__codelineno-23-129" name="__codelineno-23-129" href="#__codelineno-23-129"></a>
-<a id="__codelineno-23-130" name="__codelineno-23-130" href="#__codelineno-23-130"></a><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-23-130" name="__codelineno-23-130" href="#__codelineno-23-130"></a><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-23-131" name="__codelineno-23-131" href="#__codelineno-23-131"></a><span class="kt">void</span><span class="w"> </span><span class="nf">extend</span><span class="p">(</span><span class="n">HashMapOpenAddressing</span><span class="w"> </span><span class="o">*</span><span class="n">hashMap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-132" name="__codelineno-23-132" href="#__codelineno-23-132"></a><span class="w">    </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-23-132" name="__codelineno-23-132" href="#__codelineno-23-132"></a><span class="w">    </span><span class="c1">// Keep a reference to the original hash table</span>
 <a id="__codelineno-23-133" name="__codelineno-23-133" href="#__codelineno-23-133"></a><span class="w">    </span><span class="n">Pair</span><span class="w"> </span><span class="o">**</span><span class="n">bucketsTmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">;</span>
 <a id="__codelineno-23-134" name="__codelineno-23-134" href="#__codelineno-23-134"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">oldCapacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">capacity</span><span class="p">;</span>
-<a id="__codelineno-23-135" name="__codelineno-23-135" href="#__codelineno-23-135"></a><span class="w">    </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-23-135" name="__codelineno-23-135" href="#__codelineno-23-135"></a><span class="w">    </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-23-136" name="__codelineno-23-136" href="#__codelineno-23-136"></a><span class="w">    </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">extendRatio</span><span class="p">;</span>
 <a id="__codelineno-23-137" name="__codelineno-23-137" href="#__codelineno-23-137"></a><span class="w">    </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="o">**</span><span class="p">)</span><span class="n">calloc</span><span class="p">(</span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">capacity</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="p">));</span>
 <a id="__codelineno-23-138" name="__codelineno-23-138" href="#__codelineno-23-138"></a><span class="w">    </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-23-139" name="__codelineno-23-139" href="#__codelineno-23-139"></a><span class="w">    </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-23-139" name="__codelineno-23-139" href="#__codelineno-23-139"></a><span class="w">    </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-23-140" name="__codelineno-23-140" href="#__codelineno-23-140"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">oldCapacity</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-141" name="__codelineno-23-141" href="#__codelineno-23-141"></a><span class="w">        </span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-23-142" name="__codelineno-23-142" href="#__codelineno-23-142"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7707,7 +7784,7 @@
 <a id="__codelineno-23-148" name="__codelineno-23-148" href="#__codelineno-23-148"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">bucketsTmp</span><span class="p">);</span>
 <a id="__codelineno-23-149" name="__codelineno-23-149" href="#__codelineno-23-149"></a><span class="p">}</span>
 <a id="__codelineno-23-150" name="__codelineno-23-150" href="#__codelineno-23-150"></a>
-<a id="__codelineno-23-151" name="__codelineno-23-151" href="#__codelineno-23-151"></a><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-23-151" name="__codelineno-23-151" href="#__codelineno-23-151"></a><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-23-152" name="__codelineno-23-152" href="#__codelineno-23-152"></a><span class="kt">void</span><span class="w"> </span><span class="nf">print</span><span class="p">(</span><span class="n">HashMapOpenAddressing</span><span class="w"> </span><span class="o">*</span><span class="n">hashMap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-153" name="__codelineno-23-153" href="#__codelineno-23-153"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">capacity</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-154" name="__codelineno-23-154" href="#__codelineno-23-154"></a><span class="w">        </span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashMap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
@@ -7727,10 +7804,10 @@
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">HashMapOpenAddressing</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">size</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w">               </span><span class="c1">// Number of key-value pairs</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">capacity</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w">           </span><span class="c1">// Hash table capacity</span>
-<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">val</span><span class="w"> </span><span class="nv">loadThres</span><span class="p">:</span><span class="w"> </span><span class="kt">Double</span><span class="w">       </span><span class="c1">// Load factor threshold for triggering expansion</span>
-<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">val</span><span class="w"> </span><span class="nv">extendRatio</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w">        </span><span class="c1">// Expansion multiplier</span>
+<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">val</span><span class="w"> </span><span class="nv">loadThres</span><span class="p">:</span><span class="w"> </span><span class="kt">Double</span><span class="w">       </span><span class="c1">// Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">val</span><span class="w"> </span><span class="nv">extendRatio</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w">        </span><span class="c1">// Capacity growth factor</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">buckets</span><span class="p">:</span><span class="w"> </span><span class="n">Array</span><span class="o">&lt;</span><span class="n">Pair?</span><span class="o">&gt;</span><span class="w">   </span><span class="c1">// Bucket array</span>
-<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">val</span><span class="w"> </span><span class="nv">TOMBSTONE</span><span class="p">:</span><span class="w"> </span><span class="n">Pair</span><span class="w">         </span><span class="c1">// Removal marker</span>
+<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">val</span><span class="w"> </span><span class="nv">TOMBSTONE</span><span class="p">:</span><span class="w"> </span><span class="n">Pair</span><span class="w">         </span><span class="c1">// Tombstone marker</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">    </span><span class="k">init</span><span class="w"> </span><span class="p">{</span>
@@ -7752,83 +7829,83 @@
 <a id="__codelineno-24-27" name="__codelineno-24-27" href="#__codelineno-24-27"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">capacity</span><span class="p">).</span><span class="na">toDouble</span><span class="p">()</span>
 <a id="__codelineno-24-28" name="__codelineno-24-28" href="#__codelineno-24-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-29" name="__codelineno-24-29" href="#__codelineno-24-29"></a>
-<a id="__codelineno-24-30" name="__codelineno-24-30" href="#__codelineno-24-30"></a><span class="w">    </span><span class="cm">/* Search for bucket index corresponding to key */</span>
+<a id="__codelineno-24-30" name="__codelineno-24-30" href="#__codelineno-24-30"></a><span class="w">    </span><span class="cm">/* Find the bucket index for key */</span>
 <a id="__codelineno-24-31" name="__codelineno-24-31" href="#__codelineno-24-31"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-32" name="__codelineno-24-32" href="#__codelineno-24-32"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-24-33" name="__codelineno-24-33" href="#__codelineno-24-33"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">-</span><span class="m">1</span>
-<a id="__codelineno-24-34" name="__codelineno-24-34" href="#__codelineno-24-34"></a><span class="w">        </span><span class="c1">// Linear probing, break when encountering an empty bucket</span>
+<a id="__codelineno-24-34" name="__codelineno-24-34" href="#__codelineno-24-34"></a><span class="w">        </span><span class="c1">// Probe linearly until an empty bucket is reached</span>
 <a id="__codelineno-24-35" name="__codelineno-24-35" href="#__codelineno-24-35"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-36" name="__codelineno-24-36" href="#__codelineno-24-36"></a><span class="w">            </span><span class="c1">// If key is encountered, return the corresponding bucket index</span>
+<a id="__codelineno-24-36" name="__codelineno-24-36" href="#__codelineno-24-36"></a><span class="w">            </span><span class="c1">// If key is found, return its bucket index</span>
 <a id="__codelineno-24-37" name="__codelineno-24-37" href="#__codelineno-24-37"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]?.</span><span class="na">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-38" name="__codelineno-24-38" href="#__codelineno-24-38"></a><span class="w">                </span><span class="c1">// If a removal marker was encountered before, move the key-value pair to that index</span>
+<a id="__codelineno-24-38" name="__codelineno-24-38" href="#__codelineno-24-38"></a><span class="w">                </span><span class="c1">// If a tombstone was encountered earlier, move the key-value pair to that index</span>
 <a id="__codelineno-24-39" name="__codelineno-24-39" href="#__codelineno-24-39"></a><span class="w">                </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-40" name="__codelineno-24-40" href="#__codelineno-24-40"></a><span class="w">                    </span><span class="n">buckets</span><span class="o">[</span><span class="n">firstTombstone</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span>
 <a id="__codelineno-24-41" name="__codelineno-24-41" href="#__codelineno-24-41"></a><span class="w">                    </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TOMBSTONE</span>
-<a id="__codelineno-24-42" name="__codelineno-24-42" href="#__codelineno-24-42"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="c1">// Return the moved bucket index</span>
+<a id="__codelineno-24-42" name="__codelineno-24-42" href="#__codelineno-24-42"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="n">firstTombstone</span><span class="w"> </span><span class="c1">// Return the bucket index after the move</span>
 <a id="__codelineno-24-43" name="__codelineno-24-43" href="#__codelineno-24-43"></a><span class="w">                </span><span class="p">}</span>
-<a id="__codelineno-24-44" name="__codelineno-24-44" href="#__codelineno-24-44"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="c1">// Return bucket index</span>
+<a id="__codelineno-24-44" name="__codelineno-24-44" href="#__codelineno-24-44"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="c1">// Return the bucket index</span>
 <a id="__codelineno-24-45" name="__codelineno-24-45" href="#__codelineno-24-45"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-24-46" name="__codelineno-24-46" href="#__codelineno-24-46"></a><span class="w">            </span><span class="c1">// Record the first removal marker encountered</span>
+<a id="__codelineno-24-46" name="__codelineno-24-46" href="#__codelineno-24-46"></a><span class="w">            </span><span class="c1">// Record the first tombstone encountered</span>
 <a id="__codelineno-24-47" name="__codelineno-24-47" href="#__codelineno-24-47"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-48" name="__codelineno-24-48" href="#__codelineno-24-48"></a><span class="w">                </span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span>
 <a id="__codelineno-24-49" name="__codelineno-24-49" href="#__codelineno-24-49"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-24-50" name="__codelineno-24-50" href="#__codelineno-24-50"></a><span class="w">            </span><span class="c1">// Calculate bucket index, wrap around to the head if past the tail</span>
+<a id="__codelineno-24-50" name="__codelineno-24-50" href="#__codelineno-24-50"></a><span class="w">            </span><span class="c1">// Calculate the bucket index, wrapping to the beginning after the end</span>
 <a id="__codelineno-24-51" name="__codelineno-24-51" href="#__codelineno-24-51"></a><span class="w">            </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span>
 <a id="__codelineno-24-52" name="__codelineno-24-52" href="#__codelineno-24-52"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-24-53" name="__codelineno-24-53" href="#__codelineno-24-53"></a><span class="w">        </span><span class="c1">// If key does not exist, return the index for insertion</span>
+<a id="__codelineno-24-53" name="__codelineno-24-53" href="#__codelineno-24-53"></a><span class="w">        </span><span class="c1">// If key is absent, return the index for insertion</span>
 <a id="__codelineno-24-54" name="__codelineno-24-54" href="#__codelineno-24-54"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">firstTombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="n">firstTombstone</span>
 <a id="__codelineno-24-55" name="__codelineno-24-55" href="#__codelineno-24-55"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-56" name="__codelineno-24-56" href="#__codelineno-24-56"></a>
-<a id="__codelineno-24-57" name="__codelineno-24-57" href="#__codelineno-24-57"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-24-57" name="__codelineno-24-57" href="#__codelineno-24-57"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-24-58" name="__codelineno-24-58" href="#__codelineno-24-58"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">String?</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-59" name="__codelineno-24-59" href="#__codelineno-24-59"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-24-59" name="__codelineno-24-59" href="#__codelineno-24-59"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-24-60" name="__codelineno-24-60" href="#__codelineno-24-60"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-24-61" name="__codelineno-24-61" href="#__codelineno-24-61"></a><span class="w">        </span><span class="c1">// If key-value pair is found, return corresponding val</span>
+<a id="__codelineno-24-61" name="__codelineno-24-61" href="#__codelineno-24-61"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, return its val</span>
 <a id="__codelineno-24-62" name="__codelineno-24-62" href="#__codelineno-24-62"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-63" name="__codelineno-24-63" href="#__codelineno-24-63"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]?.</span><span class="na">_val</span>
 <a id="__codelineno-24-64" name="__codelineno-24-64" href="#__codelineno-24-64"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-24-65" name="__codelineno-24-65" href="#__codelineno-24-65"></a><span class="w">        </span><span class="c1">// If key-value pair does not exist, return null</span>
+<a id="__codelineno-24-65" name="__codelineno-24-65" href="#__codelineno-24-65"></a><span class="w">        </span><span class="c1">// Return null if the key-value pair is absent</span>
 <a id="__codelineno-24-66" name="__codelineno-24-66" href="#__codelineno-24-66"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span>
 <a id="__codelineno-24-67" name="__codelineno-24-67" href="#__codelineno-24-67"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-68" name="__codelineno-24-68" href="#__codelineno-24-68"></a>
-<a id="__codelineno-24-69" name="__codelineno-24-69" href="#__codelineno-24-69"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-24-69" name="__codelineno-24-69" href="#__codelineno-24-69"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-24-70" name="__codelineno-24-70" href="#__codelineno-24-70"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">String</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-71" name="__codelineno-24-71" href="#__codelineno-24-71"></a><span class="w">        </span><span class="c1">// When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-24-71" name="__codelineno-24-71" href="#__codelineno-24-71"></a><span class="w">        </span><span class="c1">// Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-24-72" name="__codelineno-24-72" href="#__codelineno-24-72"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">loadFactor</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">loadThres</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-73" name="__codelineno-24-73" href="#__codelineno-24-73"></a><span class="w">            </span><span class="n">extend</span><span class="p">()</span>
 <a id="__codelineno-24-74" name="__codelineno-24-74" href="#__codelineno-24-74"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-24-75" name="__codelineno-24-75" href="#__codelineno-24-75"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-24-75" name="__codelineno-24-75" href="#__codelineno-24-75"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-24-76" name="__codelineno-24-76" href="#__codelineno-24-76"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-24-77" name="__codelineno-24-77" href="#__codelineno-24-77"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite val and return</span>
+<a id="__codelineno-24-77" name="__codelineno-24-77" href="#__codelineno-24-77"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, overwrite val and return</span>
 <a id="__codelineno-24-78" name="__codelineno-24-78" href="#__codelineno-24-78"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-79" name="__codelineno-24-79" href="#__codelineno-24-79"></a><span class="w">            </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]!!</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_val</span>
 <a id="__codelineno-24-80" name="__codelineno-24-80" href="#__codelineno-24-80"></a><span class="w">            </span><span class="k">return</span>
 <a id="__codelineno-24-81" name="__codelineno-24-81" href="#__codelineno-24-81"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-24-82" name="__codelineno-24-82" href="#__codelineno-24-82"></a><span class="w">        </span><span class="c1">// If key-value pair does not exist, add the key-value pair</span>
+<a id="__codelineno-24-82" name="__codelineno-24-82" href="#__codelineno-24-82"></a><span class="w">        </span><span class="c1">// If the key-value pair is absent, insert it</span>
 <a id="__codelineno-24-83" name="__codelineno-24-83" href="#__codelineno-24-83"></a><span class="w">        </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">)</span>
 <a id="__codelineno-24-84" name="__codelineno-24-84" href="#__codelineno-24-84"></a><span class="w">        </span><span class="n">size</span><span class="o">++</span>
 <a id="__codelineno-24-85" name="__codelineno-24-85" href="#__codelineno-24-85"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-86" name="__codelineno-24-86" href="#__codelineno-24-86"></a>
 <a id="__codelineno-24-87" name="__codelineno-24-87" href="#__codelineno-24-87"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-24-88" name="__codelineno-24-88" href="#__codelineno-24-88"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-89" name="__codelineno-24-89" href="#__codelineno-24-89"></a><span class="w">        </span><span class="c1">// Search for bucket index corresponding to key</span>
+<a id="__codelineno-24-89" name="__codelineno-24-89" href="#__codelineno-24-89"></a><span class="w">        </span><span class="c1">// Find the bucket index for key</span>
 <a id="__codelineno-24-90" name="__codelineno-24-90" href="#__codelineno-24-90"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">findBucket</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-24-91" name="__codelineno-24-91" href="#__codelineno-24-91"></a><span class="w">        </span><span class="c1">// If key-value pair is found, overwrite it with removal marker</span>
+<a id="__codelineno-24-91" name="__codelineno-24-91" href="#__codelineno-24-91"></a><span class="w">        </span><span class="c1">// If the key-value pair is found, replace it with a tombstone</span>
 <a id="__codelineno-24-92" name="__codelineno-24-92" href="#__codelineno-24-92"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-93" name="__codelineno-24-93" href="#__codelineno-24-93"></a><span class="w">            </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TOMBSTONE</span>
 <a id="__codelineno-24-94" name="__codelineno-24-94" href="#__codelineno-24-94"></a><span class="w">            </span><span class="n">size</span><span class="o">--</span>
 <a id="__codelineno-24-95" name="__codelineno-24-95" href="#__codelineno-24-95"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-24-96" name="__codelineno-24-96" href="#__codelineno-24-96"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-97" name="__codelineno-24-97" href="#__codelineno-24-97"></a>
-<a id="__codelineno-24-98" name="__codelineno-24-98" href="#__codelineno-24-98"></a><span class="w">    </span><span class="cm">/* Expand hash table */</span>
+<a id="__codelineno-24-98" name="__codelineno-24-98" href="#__codelineno-24-98"></a><span class="w">    </span><span class="cm">/* Grow the hash table capacity */</span>
 <a id="__codelineno-24-99" name="__codelineno-24-99" href="#__codelineno-24-99"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">extend</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-100" name="__codelineno-24-100" href="#__codelineno-24-100"></a><span class="w">        </span><span class="c1">// Temporarily store the original hash table</span>
+<a id="__codelineno-24-100" name="__codelineno-24-100" href="#__codelineno-24-100"></a><span class="w">        </span><span class="c1">// Keep a reference to the original hash table</span>
 <a id="__codelineno-24-101" name="__codelineno-24-101" href="#__codelineno-24-101"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">bucketsTmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span>
-<a id="__codelineno-24-102" name="__codelineno-24-102" href="#__codelineno-24-102"></a><span class="w">        </span><span class="c1">// Initialize expanded new hash table</span>
+<a id="__codelineno-24-102" name="__codelineno-24-102" href="#__codelineno-24-102"></a><span class="w">        </span><span class="c1">// Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-24-103" name="__codelineno-24-103" href="#__codelineno-24-103"></a><span class="w">        </span><span class="n">capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="n">extendRatio</span>
 <a id="__codelineno-24-104" name="__codelineno-24-104" href="#__codelineno-24-104"></a><span class="w">        </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arrayOfNulls</span><span class="p">(</span><span class="n">capacity</span><span class="p">)</span>
 <a id="__codelineno-24-105" name="__codelineno-24-105" href="#__codelineno-24-105"></a><span class="w">        </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-24-106" name="__codelineno-24-106" href="#__codelineno-24-106"></a><span class="w">        </span><span class="c1">// Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-24-106" name="__codelineno-24-106" href="#__codelineno-24-106"></a><span class="w">        </span><span class="c1">// Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-24-107" name="__codelineno-24-107" href="#__codelineno-24-107"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucketsTmp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-108" name="__codelineno-24-108" href="#__codelineno-24-108"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">TOMBSTONE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-109" name="__codelineno-24-109" href="#__codelineno-24-109"></a><span class="w">                </span><span class="n">put</span><span class="p">(</span><span class="n">pair</span><span class="p">.</span><span class="na">key</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="na">_val</span><span class="p">)</span>
@@ -7836,7 +7913,7 @@
 <a id="__codelineno-24-111" name="__codelineno-24-111" href="#__codelineno-24-111"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-24-112" name="__codelineno-24-112" href="#__codelineno-24-112"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-113" name="__codelineno-24-113" href="#__codelineno-24-113"></a>
-<a id="__codelineno-24-114" name="__codelineno-24-114" href="#__codelineno-24-114"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-24-114" name="__codelineno-24-114" href="#__codelineno-24-114"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-24-115" name="__codelineno-24-115" href="#__codelineno-24-115"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-116" name="__codelineno-24-116" href="#__codelineno-24-116"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-117" name="__codelineno-24-117" href="#__codelineno-24-117"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pair</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7852,16 +7929,16 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map_open_addressing.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Hash map with open addressing ###</span>
+<div class="highlight"><span class="filename">hash_map_open_addressing.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Hash table with open addressing ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">HashMapOpenAddressing</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="no">TOMBSTONE</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">Pair</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="s1">'-1'</span><span class="p">)</span><span class="w"> </span><span class="c1"># Removal marker</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="no">TOMBSTONE</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">Pair</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="s1">'-1'</span><span class="p">)</span><span class="w"> </span><span class="c1"># Tombstone marker</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1">### Constructor ###</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">initialize</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Number of key-value pairs</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">    </span><span class="vi">@capacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">4</span><span class="w"> </span><span class="c1"># Hash table capacity</span>
-<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">    </span><span class="vi">@load_thres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="o">.</span><span class="mi">0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">3</span><span class="o">.</span><span class="mi">0</span><span class="w"> </span><span class="c1"># Load factor threshold for triggering expansion</span>
-<a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">    </span><span class="vi">@extend_ratio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1"># Expansion multiplier</span>
+<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">    </span><span class="vi">@load_thres</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="o">.</span><span class="mi">0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">3</span><span class="o">.</span><span class="mi">0</span><span class="w"> </span><span class="c1"># Load factor threshold that triggers capacity growth</span>
+<a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">    </span><span class="vi">@extend_ratio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1"># Capacity growth factor</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">    </span><span class="vi">@buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="vi">@capacity</span><span class="p">)</span><span class="w"> </span><span class="c1"># Bucket array</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a>
@@ -7875,83 +7952,83 @@
 <a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="vi">@capacity</span>
 <a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-23" name="__codelineno-25-23" href="#__codelineno-25-23"></a>
-<a id="__codelineno-25-24" name="__codelineno-25-24" href="#__codelineno-25-24"></a><span class="w">  </span><span class="c1">### Search bucket index for key ###</span>
+<a id="__codelineno-25-24" name="__codelineno-25-24" href="#__codelineno-25-24"></a><span class="w">  </span><span class="c1">### Find the bucket index for key ###</span>
 <a id="__codelineno-25-25" name="__codelineno-25-25" href="#__codelineno-25-25"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">find_bucket</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-25-26" name="__codelineno-25-26" href="#__codelineno-25-26"></a><span class="w">    </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-25-27" name="__codelineno-25-27" href="#__codelineno-25-27"></a><span class="w">    </span><span class="n">first_tombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
-<a id="__codelineno-25-28" name="__codelineno-25-28" href="#__codelineno-25-28"></a><span class="w">    </span><span class="c1"># Linear probing, break when encountering an empty bucket</span>
+<a id="__codelineno-25-28" name="__codelineno-25-28" href="#__codelineno-25-28"></a><span class="w">    </span><span class="c1"># Probe linearly until an empty bucket is reached</span>
 <a id="__codelineno-25-29" name="__codelineno-25-29" href="#__codelineno-25-29"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="o">!</span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">].</span><span class="n">nil?</span>
-<a id="__codelineno-25-30" name="__codelineno-25-30" href="#__codelineno-25-30"></a><span class="w">      </span><span class="c1"># If key is encountered, return the corresponding bucket index</span>
+<a id="__codelineno-25-30" name="__codelineno-25-30" href="#__codelineno-25-30"></a><span class="w">      </span><span class="c1"># If key is found, return its bucket index</span>
 <a id="__codelineno-25-31" name="__codelineno-25-31" href="#__codelineno-25-31"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">].</span><span class="n">key</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">key</span>
-<a id="__codelineno-25-32" name="__codelineno-25-32" href="#__codelineno-25-32"></a><span class="w">        </span><span class="c1"># If a removal marker was encountered before, move the key-value pair to that index</span>
+<a id="__codelineno-25-32" name="__codelineno-25-32" href="#__codelineno-25-32"></a><span class="w">        </span><span class="c1"># If a tombstone was encountered earlier, move the key-value pair to that index</span>
 <a id="__codelineno-25-33" name="__codelineno-25-33" href="#__codelineno-25-33"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">first_tombstone</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-25-34" name="__codelineno-25-34" href="#__codelineno-25-34"></a><span class="w">          </span><span class="vi">@buckets</span><span class="o">[</span><span class="n">first_tombstone</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span>
 <a id="__codelineno-25-35" name="__codelineno-25-35" href="#__codelineno-25-35"></a><span class="w">          </span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">TOMBSTONE</span>
-<a id="__codelineno-25-36" name="__codelineno-25-36" href="#__codelineno-25-36"></a><span class="w">          </span><span class="k">return</span><span class="w"> </span><span class="n">first_tombstone</span><span class="w"> </span><span class="c1"># Return the moved bucket index</span>
+<a id="__codelineno-25-36" name="__codelineno-25-36" href="#__codelineno-25-36"></a><span class="w">          </span><span class="k">return</span><span class="w"> </span><span class="n">first_tombstone</span><span class="w"> </span><span class="c1"># Return the bucket index after the move</span>
 <a id="__codelineno-25-37" name="__codelineno-25-37" href="#__codelineno-25-37"></a><span class="w">        </span><span class="k">end</span>
-<a id="__codelineno-25-38" name="__codelineno-25-38" href="#__codelineno-25-38"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="c1"># Return bucket index</span>
+<a id="__codelineno-25-38" name="__codelineno-25-38" href="#__codelineno-25-38"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="c1"># Return the bucket index</span>
 <a id="__codelineno-25-39" name="__codelineno-25-39" href="#__codelineno-25-39"></a><span class="w">      </span><span class="k">end</span>
-<a id="__codelineno-25-40" name="__codelineno-25-40" href="#__codelineno-25-40"></a><span class="w">      </span><span class="c1"># Record the first removal marker encountered</span>
+<a id="__codelineno-25-40" name="__codelineno-25-40" href="#__codelineno-25-40"></a><span class="w">      </span><span class="c1"># Record the first tombstone encountered</span>
 <a id="__codelineno-25-41" name="__codelineno-25-41" href="#__codelineno-25-41"></a><span class="w">      </span><span class="n">first_tombstone</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">first_tombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="no">TOMBSTONE</span>
-<a id="__codelineno-25-42" name="__codelineno-25-42" href="#__codelineno-25-42"></a><span class="w">      </span><span class="c1"># Calculate bucket index, wrap around to the head if past the tail</span>
+<a id="__codelineno-25-42" name="__codelineno-25-42" href="#__codelineno-25-42"></a><span class="w">      </span><span class="c1"># Calculate the bucket index, wrapping to the beginning after the end</span>
 <a id="__codelineno-25-43" name="__codelineno-25-43" href="#__codelineno-25-43"></a><span class="w">      </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="vi">@capacity</span>
 <a id="__codelineno-25-44" name="__codelineno-25-44" href="#__codelineno-25-44"></a><span class="w">    </span><span class="k">end</span>
-<a id="__codelineno-25-45" name="__codelineno-25-45" href="#__codelineno-25-45"></a><span class="w">    </span><span class="c1"># If key does not exist, return the index for insertion</span>
+<a id="__codelineno-25-45" name="__codelineno-25-45" href="#__codelineno-25-45"></a><span class="w">    </span><span class="c1"># If key is absent, return the index for insertion</span>
 <a id="__codelineno-25-46" name="__codelineno-25-46" href="#__codelineno-25-46"></a><span class="w">    </span><span class="n">first_tombstone</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">first_tombstone</span>
 <a id="__codelineno-25-47" name="__codelineno-25-47" href="#__codelineno-25-47"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-48" name="__codelineno-25-48" href="#__codelineno-25-48"></a>
-<a id="__codelineno-25-49" name="__codelineno-25-49" href="#__codelineno-25-49"></a><span class="w">  </span><span class="c1">### Query operation ###</span>
+<a id="__codelineno-25-49" name="__codelineno-25-49" href="#__codelineno-25-49"></a><span class="w">  </span><span class="c1">### Lookup operation ###</span>
 <a id="__codelineno-25-50" name="__codelineno-25-50" href="#__codelineno-25-50"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-25-51" name="__codelineno-25-51" href="#__codelineno-25-51"></a><span class="w">    </span><span class="c1"># Search for bucket index corresponding to key</span>
+<a id="__codelineno-25-51" name="__codelineno-25-51" href="#__codelineno-25-51"></a><span class="w">    </span><span class="c1"># Find the bucket index for key</span>
 <a id="__codelineno-25-52" name="__codelineno-25-52" href="#__codelineno-25-52"></a><span class="w">    </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">find_bucket</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-25-53" name="__codelineno-25-53" href="#__codelineno-25-53"></a><span class="w">    </span><span class="c1"># If key-value pair is found, return corresponding val</span>
+<a id="__codelineno-25-53" name="__codelineno-25-53" href="#__codelineno-25-53"></a><span class="w">    </span><span class="c1"># If the key-value pair is found, return its val</span>
 <a id="__codelineno-25-54" name="__codelineno-25-54" href="#__codelineno-25-54"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">].</span><span class="n">val</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="o">[</span><span class="kp">nil</span><span class="p">,</span><span class="w"> </span><span class="no">TOMBSTONE</span><span class="o">].</span><span class="n">include?</span><span class="p">(</span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="p">)</span>
-<a id="__codelineno-25-55" name="__codelineno-25-55" href="#__codelineno-25-55"></a><span class="w">    </span><span class="c1"># Return nil if key-value pair does not exist</span>
+<a id="__codelineno-25-55" name="__codelineno-25-55" href="#__codelineno-25-55"></a><span class="w">    </span><span class="c1"># Return nil if the key-value pair does not exist</span>
 <a id="__codelineno-25-56" name="__codelineno-25-56" href="#__codelineno-25-56"></a><span class="w">    </span><span class="kp">nil</span>
 <a id="__codelineno-25-57" name="__codelineno-25-57" href="#__codelineno-25-57"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-58" name="__codelineno-25-58" href="#__codelineno-25-58"></a>
-<a id="__codelineno-25-59" name="__codelineno-25-59" href="#__codelineno-25-59"></a><span class="w">  </span><span class="c1">### Add operation ###</span>
+<a id="__codelineno-25-59" name="__codelineno-25-59" href="#__codelineno-25-59"></a><span class="w">  </span><span class="c1">### Insert operation ###</span>
 <a id="__codelineno-25-60" name="__codelineno-25-60" href="#__codelineno-25-60"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
-<a id="__codelineno-25-61" name="__codelineno-25-61" href="#__codelineno-25-61"></a><span class="w">    </span><span class="c1"># When load factor exceeds threshold, perform expansion</span>
+<a id="__codelineno-25-61" name="__codelineno-25-61" href="#__codelineno-25-61"></a><span class="w">    </span><span class="c1"># Grow the capacity when the load factor exceeds the threshold</span>
 <a id="__codelineno-25-62" name="__codelineno-25-62" href="#__codelineno-25-62"></a><span class="w">    </span><span class="kp">extend</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">load_factor</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="vi">@load_thres</span>
-<a id="__codelineno-25-63" name="__codelineno-25-63" href="#__codelineno-25-63"></a><span class="w">    </span><span class="c1"># Search for bucket index corresponding to key</span>
+<a id="__codelineno-25-63" name="__codelineno-25-63" href="#__codelineno-25-63"></a><span class="w">    </span><span class="c1"># Find the bucket index for key</span>
 <a id="__codelineno-25-64" name="__codelineno-25-64" href="#__codelineno-25-64"></a><span class="w">    </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">find_bucket</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-25-65" name="__codelineno-25-65" href="#__codelineno-25-65"></a><span class="w">    </span><span class="c1"># If key-value pair found, overwrite val and return</span>
+<a id="__codelineno-25-65" name="__codelineno-25-65" href="#__codelineno-25-65"></a><span class="w">    </span><span class="c1"># If the key-value pair is found, overwrite val and return</span>
 <a id="__codelineno-25-66" name="__codelineno-25-66" href="#__codelineno-25-66"></a><span class="w">    </span><span class="k">unless</span><span class="w"> </span><span class="o">[</span><span class="kp">nil</span><span class="p">,</span><span class="w"> </span><span class="no">TOMBSTONE</span><span class="o">].</span><span class="n">include?</span><span class="p">(</span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-25-67" name="__codelineno-25-67" href="#__codelineno-25-67"></a><span class="w">      </span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">].</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span>
 <a id="__codelineno-25-68" name="__codelineno-25-68" href="#__codelineno-25-68"></a><span class="w">      </span><span class="k">return</span>
 <a id="__codelineno-25-69" name="__codelineno-25-69" href="#__codelineno-25-69"></a><span class="w">    </span><span class="k">end</span>
-<a id="__codelineno-25-70" name="__codelineno-25-70" href="#__codelineno-25-70"></a><span class="w">    </span><span class="c1"># If key-value pair does not exist, add the key-value pair</span>
+<a id="__codelineno-25-70" name="__codelineno-25-70" href="#__codelineno-25-70"></a><span class="w">    </span><span class="c1"># If the key-value pair is absent, insert it</span>
 <a id="__codelineno-25-71" name="__codelineno-25-71" href="#__codelineno-25-71"></a><span class="w">    </span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">Pair</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-25-72" name="__codelineno-25-72" href="#__codelineno-25-72"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-25-73" name="__codelineno-25-73" href="#__codelineno-25-73"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-74" name="__codelineno-25-74" href="#__codelineno-25-74"></a>
-<a id="__codelineno-25-75" name="__codelineno-25-75" href="#__codelineno-25-75"></a><span class="w">  </span><span class="c1">### Delete operation ###</span>
+<a id="__codelineno-25-75" name="__codelineno-25-75" href="#__codelineno-25-75"></a><span class="w">  </span><span class="c1">### Remove operation ###</span>
 <a id="__codelineno-25-76" name="__codelineno-25-76" href="#__codelineno-25-76"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-25-77" name="__codelineno-25-77" href="#__codelineno-25-77"></a><span class="w">    </span><span class="c1"># Search for bucket index corresponding to key</span>
+<a id="__codelineno-25-77" name="__codelineno-25-77" href="#__codelineno-25-77"></a><span class="w">    </span><span class="c1"># Find the bucket index for key</span>
 <a id="__codelineno-25-78" name="__codelineno-25-78" href="#__codelineno-25-78"></a><span class="w">    </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">find_bucket</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-25-79" name="__codelineno-25-79" href="#__codelineno-25-79"></a><span class="w">    </span><span class="c1"># If key-value pair is found, overwrite it with removal marker</span>
+<a id="__codelineno-25-79" name="__codelineno-25-79" href="#__codelineno-25-79"></a><span class="w">    </span><span class="c1"># If the key-value pair is found, replace it with a tombstone</span>
 <a id="__codelineno-25-80" name="__codelineno-25-80" href="#__codelineno-25-80"></a><span class="w">    </span><span class="k">unless</span><span class="w"> </span><span class="o">[</span><span class="kp">nil</span><span class="p">,</span><span class="w"> </span><span class="no">TOMBSTONE</span><span class="o">].</span><span class="n">include?</span><span class="p">(</span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-25-81" name="__codelineno-25-81" href="#__codelineno-25-81"></a><span class="w">      </span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">TOMBSTONE</span>
 <a id="__codelineno-25-82" name="__codelineno-25-82" href="#__codelineno-25-82"></a><span class="w">      </span><span class="vi">@size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-25-83" name="__codelineno-25-83" href="#__codelineno-25-83"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-25-84" name="__codelineno-25-84" href="#__codelineno-25-84"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-85" name="__codelineno-25-85" href="#__codelineno-25-85"></a>
-<a id="__codelineno-25-86" name="__codelineno-25-86" href="#__codelineno-25-86"></a><span class="w">  </span><span class="c1">### Expand hash table ###</span>
+<a id="__codelineno-25-86" name="__codelineno-25-86" href="#__codelineno-25-86"></a><span class="w">  </span><span class="c1">### Grow the hash table capacity ###</span>
 <a id="__codelineno-25-87" name="__codelineno-25-87" href="#__codelineno-25-87"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">extend</span>
-<a id="__codelineno-25-88" name="__codelineno-25-88" href="#__codelineno-25-88"></a><span class="w">    </span><span class="c1"># Temporarily store the original hash table</span>
+<a id="__codelineno-25-88" name="__codelineno-25-88" href="#__codelineno-25-88"></a><span class="w">    </span><span class="c1"># Keep a reference to the original hash table</span>
 <a id="__codelineno-25-89" name="__codelineno-25-89" href="#__codelineno-25-89"></a><span class="w">    </span><span class="n">buckets_tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@buckets</span>
-<a id="__codelineno-25-90" name="__codelineno-25-90" href="#__codelineno-25-90"></a><span class="w">    </span><span class="c1"># Initialize expanded new hash table</span>
+<a id="__codelineno-25-90" name="__codelineno-25-90" href="#__codelineno-25-90"></a><span class="w">    </span><span class="c1"># Initialize a new hash table with the expanded capacity</span>
 <a id="__codelineno-25-91" name="__codelineno-25-91" href="#__codelineno-25-91"></a><span class="w">    </span><span class="vi">@capacity</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="vi">@extend_ratio</span>
 <a id="__codelineno-25-92" name="__codelineno-25-92" href="#__codelineno-25-92"></a><span class="w">    </span><span class="vi">@buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="vi">@capacity</span><span class="p">)</span>
 <a id="__codelineno-25-93" name="__codelineno-25-93" href="#__codelineno-25-93"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-25-94" name="__codelineno-25-94" href="#__codelineno-25-94"></a><span class="w">    </span><span class="c1"># Move key-value pairs from original hash table to new hash table</span>
+<a id="__codelineno-25-94" name="__codelineno-25-94" href="#__codelineno-25-94"></a><span class="w">    </span><span class="c1"># Move the key-value pairs from the original hash table to the new one</span>
 <a id="__codelineno-25-95" name="__codelineno-25-95" href="#__codelineno-25-95"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets_tmp</span>
 <a id="__codelineno-25-96" name="__codelineno-25-96" href="#__codelineno-25-96"></a><span class="w">      </span><span class="n">put</span><span class="p">(</span><span class="n">pair</span><span class="o">.</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="o">.</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="o">[</span><span class="kp">nil</span><span class="p">,</span><span class="w"> </span><span class="no">TOMBSTONE</span><span class="o">].</span><span class="n">include?</span><span class="p">(</span><span class="n">pair</span><span class="p">)</span>
 <a id="__codelineno-25-97" name="__codelineno-25-97" href="#__codelineno-25-97"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-25-98" name="__codelineno-25-98" href="#__codelineno-25-98"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-99" name="__codelineno-25-99" href="#__codelineno-25-99"></a>
-<a id="__codelineno-25-100" name="__codelineno-25-100" href="#__codelineno-25-100"></a><span class="w">  </span><span class="c1">### Print hash table ###</span>
+<a id="__codelineno-25-100" name="__codelineno-25-100" href="#__codelineno-25-100"></a><span class="w">  </span><span class="c1">### Print the hash table ###</span>
 <a id="__codelineno-25-101" name="__codelineno-25-101" href="#__codelineno-25-101"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">print</span>
 <a id="__codelineno-25-102" name="__codelineno-25-102" href="#__codelineno-25-102"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="vi">@buckets</span>
 <a id="__codelineno-25-103" name="__codelineno-25-103" href="#__codelineno-25-103"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">pair</span><span class="o">.</span><span class="n">nil?</span>
@@ -7968,35 +8045,35 @@
 </div>
 </div>
 </div>
-<h3 id="2-quadratic-probing">2. &nbsp; Quadratic Probing<a class="headerlink" href="#2-quadratic-probing" title="Permanent link">&para;</a></h3>
-<p>Quadratic probing is similar to linear probing and is one of the common strategies for open addressing. When a collision occurs, quadratic probing does not simply skip a fixed number of steps but skips a number of steps equal to the "square of the number of probes", i.e., <span class="arithmatex">\(1, 4, 9, \dots\)</span> steps.</p>
-<p>Quadratic probing has the following advantages:</p>
+<h3 id="2-quadratic-probing">2. &nbsp; Quadratic probing<a class="headerlink" href="#2-quadratic-probing" title="Permanent link">&para;</a></h3>
+<p>Like linear probing, quadratic probing is a common open-addressing strategy. On a collision, it probes at distances determined by the square of the probe number—<span class="arithmatex">\(1, 4, 9, \dots\)</span>—rather than a fixed step size.</p>
+<p>Its main advantages are as follows.</p>
 <ul>
-<li>Quadratic probing attempts to alleviate the clustering effect of linear probing by skipping distances equal to the square of the probe count.</li>
-<li>Quadratic probing skips larger distances to find empty positions, which helps to distribute data more evenly.</li>
+<li>Squared probe distances aim to reduce the clustering caused by linear probing.</li>
+<li>Larger jumps help distribute entries more evenly while searching for empty positions.</li>
 </ul>
-<p>However, quadratic probing is not perfect:</p>
+<p>However, quadratic probing is not perfect.</p>
 <ul>
-<li>Clustering still exists, i.e., some positions are more likely to be occupied than others.</li>
-<li>Due to the growth of squares, quadratic probing may not probe the entire hash table, meaning that even if there are empty buckets in the hash table, quadratic probing may not be able to access them.</li>
+<li>Clustering still occurs: some positions are more likely to be occupied than others.</li>
+<li>The growing squared distances may prevent probing the entire table, so an empty bucket may remain unreachable.</li>
 </ul>
-<h3 id="3-multiple-hashing">3. &nbsp; Multiple Hashing<a class="headerlink" href="#3-multiple-hashing" title="Permanent link">&para;</a></h3>
-<p>As the name suggests, multiple hashing uses multiple hash functions <span class="arithmatex">\(f_1(x)\)</span>, <span class="arithmatex">\(f_2(x)\)</span>, <span class="arithmatex">\(f_3(x)\)</span>, <span class="arithmatex">\(\dots\)</span> for probing.</p>
+<h3 id="3-multiple-hashing">3. &nbsp; Multiple hashing<a class="headerlink" href="#3-multiple-hashing" title="Permanent link">&para;</a></h3>
+<p>As its name suggests, multiple hashing probes using several hash functions, <span class="arithmatex">\(f_1(x)\)</span>, <span class="arithmatex">\(f_2(x)\)</span>, <span class="arithmatex">\(f_3(x)\)</span>, <span class="arithmatex">\(\dots\)</span>.</p>
 <ul>
-<li><strong>Inserting elements</strong>: If hash function <span class="arithmatex">\(f_1(x)\)</span> encounters a conflict, try <span class="arithmatex">\(f_2(x)\)</span>, and so on, until an empty position is found and the element is inserted.</li>
-<li><strong>Searching for elements</strong>: Search in the same order of hash functions until the target element is found and return it; if an empty position is encountered or all hash functions have been tried, it indicates the element is not in the hash table, then return <code>None</code>.</li>
+<li><strong>Insertion</strong>: If <span class="arithmatex">\(f_1(x)\)</span> produces a collision, try <span class="arithmatex">\(f_2(x)\)</span>, and so on, inserting the element when an empty position is found.</li>
+<li><strong>Lookup</strong>: Try the hash functions in the same order and return when the target is found. If an empty position is reached or all functions have been tried, the target is absent; return <code>None</code>.</li>
 </ul>
-<p>Compared with linear probing, multiple hashing is less prone to clustering, but using multiple hash functions introduces additional computational overhead.</p>
+<p>Multiple hashing is less prone to clustering than linear probing, but computing several hash functions adds overhead.</p>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>Please note that hash tables based on open addressing, including linear probing, quadratic probing, and multiple hashing, all have the problem that elements cannot be deleted directly.</p>
+<p>All open-addressing strategies—linear probing, quadratic probing, and multiple hashing—share the restriction that elements cannot simply be deleted.</p>
 </div>
-<h2 id="623-choice-of-programming-languages">6.2.3 &nbsp; Choice of Programming Languages<a class="headerlink" href="#623-choice-of-programming-languages" title="Permanent link">&para;</a></h2>
-<p>Different programming languages adopt different hash table implementation strategies. Here are a few examples:</p>
+<h2 id="623-choices-made-by-programming-languages">6.2.3 &nbsp; Choices made by programming languages<a class="headerlink" href="#623-choices-made-by-programming-languages" title="Permanent link">&para;</a></h2>
+<p>Programming languages use different hash-table implementations. Here are a few examples.</p>
 <ul>
-<li>Python uses open addressing. The <code>dict</code> dictionary uses pseudo-random numbers for probing.</li>
-<li>Java uses separate chaining. Since JDK 1.8, when the array length in <code>HashMap</code> reaches 64 and the length of a linked list reaches 8, the linked list is converted to a red-black tree to improve search performance.</li>
-<li>Starting with Go 1.24, the built-in <code>map</code> uses an open-addressed implementation based on Swiss Tables by default. Go 1.23 and earlier use separate chaining, with up to 8 key-value pairs per bucket and overflow buckets linked when that capacity is exceeded.</li>
+<li>Python uses open addressing. Its <code>dict</code> probes using pseudorandom values.</li>
+<li>Java uses separate chaining. Since JDK 1.8, when a <code>HashMap</code>'s underlying array has length at least 64 and a chain reaches length 8, the chain is converted to a red-black tree to improve lookup performance.</li>
+<li>Since Go 1.24, the built-in <code>map</code> uses open addressing based on Swiss Tables by default. Go 1.23 and earlier use separate chaining, with up to 8 key-value pairs per bucket and linked overflow buckets when that capacity is exceeded.</li>
 </ul>
 
 <!-- Source file information -->
@@ -8020,7 +8097,7 @@ aria-label="Footer"
   <a
     href="../hash_map/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 6.1 Hash Table"
+    aria-label="Previous: 6.1   Hash tables"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -8032,7 +8109,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        6.1 Hash Table
+        6.1 &nbsp; Hash tables
       </div>
     </div>
   </a>
@@ -8044,7 +8121,7 @@ aria-label="Footer"
   <a
     href="../hash_algorithm/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 6.3 Hash Algorithm"
+    aria-label="Next: 6.3   Hash algorithms"
     rel="next"
   >
     <div class="md-footer__title">
@@ -8052,7 +8129,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        6.3 Hash Algorithm
+        6.3 &nbsp; Hash algorithms
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -8069,7 +8146,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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diff --git a/en/chapter_hashing/hash_map/index.html b/en/chapter_hashing/hash_map/index.html
index c0593523d..78867b30d 100644
--- a/en/chapter_hashing/hash_map/index.html
+++ b/en/chapter_hashing/hash_map/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>6.1 Hash Table - Hello Algo</title>
+        <title>6.1 Hash tables - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
     <div data-md-component="skip">
 
 
-        <a href="#61-hash-table" class="md-skip">
+        <a href="#61-hash-tables" class="md-skip">
           Skip to content
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@@ -119,7 +119,7 @@
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             d="M480 32c0-12.9-7.8-24.6-19.8-29.6s-25.7-2.2-34.9 6.9L381.7 53c-48 48-113.1 75-181 75H192 160 64c-35.3 0-64 28.7-64 64v96c0 35.3 28.7 64 64 64l0 128c0 17.7 14.3 32 32 32h64c17.7 0 32-14.3 32-32V352l8.7 0c67.9 0 133 27 181 75l43.6 43.6c9.2 9.2 22.9 11.9 34.9 6.9s19.8-16.6 19.8-29.6V300.4c18.6-8.8 32-32.5 32-60.4s-13.4-51.6-32-60.4V32zm-64 76.7V240 371.3C357.2 317.8 280.5 288 200.7 288H192V192h8.7c79.8 0 156.5-29.8 215.3-83.3z" />
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-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              6.1 &nbsp; Hash Table
+              6.1 &nbsp; Hash tables
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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@@ -177,9 +177,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
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@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
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@@ -737,7 +737,7 @@
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
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   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1680,7 +1680,7 @@
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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@@ -1731,7 +1731,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1748,7 +1748,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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@@ -1777,7 +1777,7 @@
   <a href="#611-common-hash-table-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.1.1 &nbsp; Common Hash Table Operations
+        6.1.1 &nbsp; Common hash-table operations
       </span>
     </span>
   </a>
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 </li>
 
         <li class="md-nav__item">
-  <a href="#612-simple-hash-table-implementation" class="md-nav__link">
+  <a href="#612-a-simple-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.1.2 &nbsp; Simple Hash Table Implementation
+        6.1.2 &nbsp; A simple implementation
       </span>
     </span>
   </a>
@@ -1796,10 +1796,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#613-hash-collision-and-resizing" class="md-nav__link">
+  <a href="#613-collisions-and-expansion" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.1.3 &nbsp; Hash Collision and Resizing
+        6.1.3 &nbsp; Collisions and expansion
       </span>
     </span>
   </a>
@@ -1829,7 +1829,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1978,7 +1978,7 @@
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
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@@ -2020,7 +2020,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2074,7 +2074,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2101,7 +2101,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2128,7 +2128,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2155,7 +2155,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2182,7 +2182,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2246,7 +2246,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2267,7 +2267,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2288,7 +2288,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2315,7 +2315,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2342,7 +2342,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2369,7 +2369,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2396,7 +2396,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2460,7 +2460,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2481,7 +2481,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2502,7 +2502,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2529,7 +2529,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2556,7 +2556,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2583,7 +2583,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2610,7 +2610,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2678,7 +2678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2699,7 +2699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2720,7 +2720,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2747,7 +2747,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2774,7 +2774,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2801,7 +2801,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2828,7 +2828,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2855,7 +2855,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2882,7 +2882,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2960,7 +2960,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3002,7 +3002,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3029,7 +3029,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3056,7 +3056,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3083,7 +3083,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3110,7 +3110,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3137,7 +3137,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3164,7 +3164,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3191,7 +3191,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3218,7 +3218,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3245,7 +3245,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3272,7 +3272,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3299,7 +3299,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3365,7 +3365,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3386,7 +3386,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3407,7 +3407,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3434,7 +3434,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3461,7 +3461,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3488,7 +3488,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3515,7 +3515,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3542,7 +3542,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3608,7 +3608,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3629,7 +3629,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3650,7 +3650,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3677,7 +3677,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3704,7 +3704,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3731,7 +3731,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3758,7 +3758,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3785,7 +3785,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3855,7 +3855,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3876,7 +3876,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3897,7 +3897,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3924,7 +3924,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3951,7 +3951,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3978,7 +3978,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4005,7 +4005,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4032,7 +4032,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4059,7 +4059,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4086,7 +4086,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4152,7 +4152,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4173,7 +4173,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4194,7 +4194,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4221,7 +4221,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4248,7 +4248,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4275,7 +4275,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4302,7 +4302,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4329,7 +4329,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4389,7 +4389,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4410,7 +4410,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4431,7 +4431,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4458,7 +4458,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4485,7 +4485,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4569,6 +4569,78 @@
 
 
 
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@@ -4613,7 +4685,7 @@
   <a href="#611-common-hash-table-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.1.1 &nbsp; Common Hash Table Operations
+        6.1.1 &nbsp; Common hash-table operations
       </span>
     </span>
   </a>
@@ -4621,10 +4693,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#612-simple-hash-table-implementation" class="md-nav__link">
+  <a href="#612-a-simple-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.1.2 &nbsp; Simple Hash Table Implementation
+        6.1.2 &nbsp; A simple implementation
       </span>
     </span>
   </a>
@@ -4632,10 +4704,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#613-hash-collision-and-resizing" class="md-nav__link">
+  <a href="#613-collisions-and-expansion" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        6.1.3 &nbsp; Hash Collision and Resizing
+        6.1.3 &nbsp; Collisions and expansion
       </span>
     </span>
   </a>
@@ -4679,19 +4751,19 @@
 
 
 <!-- Page content -->
-<h1 id="61-hash-table">6.1 &nbsp; Hash Table<a class="headerlink" href="#61-hash-table" title="Permanent link">&para;</a></h1>
-<p>A <u>hash table</u>, also known as a <u>hash map</u>, stores mappings from keys <code>key</code> to values <code>value</code>, enabling efficient lookups. Specifically, given a key <code>key</code>, we can retrieve the corresponding value <code>value</code> from a hash table in <span class="arithmatex">\(O(1)\)</span> time.</p>
-<p>As shown below, suppose we have <span class="arithmatex">\(n\)</span> students, each with two pieces of information: a name and a student ID. If we want to support the query "given a student ID, return the corresponding name," we can use the hash table shown below.</p>
-<p><img alt="Abstract representation of a hash table" class="animation-figure" src="../hash_map.assets/hash_table_lookup.png" /></p>
-<p align="center"> Figure 6-1 &nbsp; Abstract representation of a hash table </p>
+<h1 id="61-hash-tables">6.1 &nbsp; Hash tables<a class="headerlink" href="#61-hash-tables" title="Permanent link">&para;</a></h1>
+<p>A <u>hash table</u>, also called a <u>hash map</u>, supports efficient lookup by mapping a <code>key</code> to a <code>value</code>. Given a key, it can retrieve the corresponding value in <span class="arithmatex">\(O(1)\)</span> time.</p>
+<p>Suppose we have <span class="arithmatex">\(n\)</span> students, each with a name and a student ID. To support the query “given a student ID, return the corresponding name,” use the hash table shown below.</p>
+<p><img alt="An abstract view of hash-table lookup" class="animation-figure" src="../hash_map.assets/hash_table_lookup.png" /></p>
+<p align="center"> Figure 6-1 &nbsp; An abstract view of hash-table lookup </p>
 
-<p>In addition to hash tables, arrays and linked lists can also implement query functionality. Their efficiency comparison is shown in the following table.</p>
+<p>Arrays and linked lists also support lookup. Table 6-1 compares their efficiency.</p>
 <ul>
-<li><strong>Adding elements</strong>: Simply add elements to the end of the array (linked list), using <span class="arithmatex">\(O(1)\)</span> time.</li>
-<li><strong>Querying elements</strong>: Since the array (linked list) is unordered, all elements need to be traversed, using <span class="arithmatex">\(O(n)\)</span> time.</li>
-<li><strong>Deleting elements</strong>: The element must first be located, then deleted from the array (linked list), using <span class="arithmatex">\(O(n)\)</span> time.</li>
+<li><strong>Insertion</strong>: append an element to the end of the array or linked list in <span class="arithmatex">\(O(1)\)</span> time.</li>
+<li><strong>Lookup</strong>: because the array or list is unsorted, searching may require visiting every element, taking <span class="arithmatex">\(O(n)\)</span> time.</li>
+<li><strong>Deletion</strong>: first find the element, then remove it from the array or list, taking <span class="arithmatex">\(O(n)\)</span> time.</li>
 </ul>
-<p align="center"> Table 6-1 &nbsp; Comparison of element query efficiency </p>
+<p align="center"> Table 6-1 &nbsp; Comparison of element-lookup efficiency </p>
 
 <div class="center-table">
 <table>
@@ -4699,25 +4771,25 @@
 <tr>
 <th></th>
 <th>Array</th>
-<th>Linked List</th>
-<th>Hash Table</th>
+<th>Linked list</th>
+<th>Hash table</th>
 </tr>
 </thead>
 <tbody>
 <tr>
-<td>Find element</td>
+<td>Find an element</td>
 <td><span class="arithmatex">\(O(n)\)</span></td>
 <td><span class="arithmatex">\(O(n)\)</span></td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
-<td>Add element</td>
+<td>Insert an element</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
-<td>Delete element</td>
+<td>Delete an element</td>
 <td><span class="arithmatex">\(O(n)\)</span></td>
 <td><span class="arithmatex">\(O(n)\)</span></td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
@@ -4725,494 +4797,494 @@
 </tbody>
 </table>
 </div>
-<p>As we can see, <strong>insertion, deletion, lookup, and update operations in a hash table all have time complexity <span class="arithmatex">\(O(1)\)</span></strong>, making hash tables highly efficient.</p>
-<h2 id="611-common-hash-table-operations">6.1.1 &nbsp; Common Hash Table Operations<a class="headerlink" href="#611-common-hash-table-operations" title="Permanent link">&para;</a></h2>
-<p>Common operations on hash tables include: initialization, query operations, adding key-value pairs, and deleting key-value pairs. Example code is as follows:</p>
+<p><strong>Hash-table insertion, deletion, lookup, and updating all take <span class="arithmatex">\(O(1)\)</span> time</strong>, making them highly efficient.</p>
+<h2 id="611-common-hash-table-operations">6.1.1 &nbsp; Common hash-table operations<a class="headerlink" href="#611-common-hash-table-operations" title="Permanent link">&para;</a></h2>
+<p>Common operations include initialization, lookup, adding key-value pairs, and deleting pairs. Examples follow:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Initialize hash table</span>
+<div class="highlight"><span class="filename">hash_map.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Initialize a hash table</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="n">hmap</span><span class="p">:</span> <span class="nb">dict</span> <span class="o">=</span> <span class="p">{}</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>
-<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="c1"># Add operation</span>
-<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="c1"># Add key-value pair (key, value) to hash table</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="n">hmap</span><span class="p">[</span><span class="mi">12836</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"XiaoHa"</span>
-<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="n">hmap</span><span class="p">[</span><span class="mi">15937</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"XiaoLuo"</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="n">hmap</span><span class="p">[</span><span class="mi">16750</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"XiaoSuan"</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="n">hmap</span><span class="p">[</span><span class="mi">13276</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"XiaoFa"</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="n">hmap</span><span class="p">[</span><span class="mi">10583</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"XiaoYa"</span>
+<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="c1"># Insertion</span>
+<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="c1"># Add key-value pairs (key, value) to the hash table</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="n">hmap</span><span class="p">[</span><span class="mi">12836</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"小哈"</span>
+<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="n">hmap</span><span class="p">[</span><span class="mi">15937</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"小啰"</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="n">hmap</span><span class="p">[</span><span class="mi">16750</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"小算"</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="n">hmap</span><span class="p">[</span><span class="mi">13276</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"小法"</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="n">hmap</span><span class="p">[</span><span class="mi">10583</span><span class="p">]</span> <span class="o">=</span> <span class="s2">"小鸭"</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>
-<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="c1"># Query operation</span>
-<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="c1"># Input key into hash table to get value</span>
+<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="c1"># Lookup</span>
+<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="c1"># Look up key in the hash table to obtain value</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="n">name</span><span class="p">:</span> <span class="nb">str</span> <span class="o">=</span> <span class="n">hmap</span><span class="p">[</span><span class="mi">15937</span><span class="p">]</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>
-<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="c1"># Delete operation</span>
-<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="c1"># Delete key-value pair (key, value) from hash table</span>
+<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="c1"># Deletion</span>
+<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="c1"># Remove a key-value pair (key, value) from the hash table</span>
 <a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="n">hmap</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="mi">10583</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Initialize hash table */</span>
+<div class="highlight"><span class="filename">hash_map.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Initialize a hash table */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="n">unordered_map</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="n">string</span><span class="o">&gt;</span><span class="w"> </span><span class="n">map</span><span class="p">;</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a>
-<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="cm">/* Add operation */</span>
-<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="c1">// Add key-value pair (key, value) to hash table</span>
-<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="n">map</span><span class="p">[</span><span class="mi">12836</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"XiaoHa"</span><span class="p">;</span>
-<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="n">map</span><span class="p">[</span><span class="mi">15937</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"XiaoLuo"</span><span class="p">;</span>
-<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">map</span><span class="p">[</span><span class="mi">16750</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"XiaoSuan"</span><span class="p">;</span>
-<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="n">map</span><span class="p">[</span><span class="mi">13276</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"XiaoFa"</span><span class="p">;</span>
-<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="n">map</span><span class="p">[</span><span class="mi">10583</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"XiaoYa"</span><span class="p">;</span>
+<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="cm">/* Insertion */</span>
+<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="c1">// Add key-value pairs (key, value) to the hash table</span>
+<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="n">map</span><span class="p">[</span><span class="mi">12836</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"小哈"</span><span class="p">;</span>
+<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="n">map</span><span class="p">[</span><span class="mi">15937</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"小啰"</span><span class="p">;</span>
+<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">map</span><span class="p">[</span><span class="mi">16750</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"小算"</span><span class="p">;</span>
+<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="n">map</span><span class="p">[</span><span class="mi">13276</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"小法"</span><span class="p">;</span>
+<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="n">map</span><span class="p">[</span><span class="mi">10583</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"小鸭"</span><span class="p">;</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a>
-<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="cm">/* Query operation */</span>
-<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="c1">// Input key into hash table to get value</span>
+<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="cm">/* Lookup */</span>
+<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="c1">// Look up key in the hash table to obtain value</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="n">string</span><span class="w"> </span><span class="n">name</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">map</span><span class="p">[</span><span class="mi">15937</span><span class="p">];</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a>
-<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="cm">/* Delete operation */</span>
-<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="c1">// Delete key-value pair (key, value) from hash table</span>
+<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="cm">/* Deletion */</span>
+<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="c1">// Remove a key-value pair (key, value) from the hash table</span>
 <a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="n">map</span><span class="p">.</span><span class="n">erase</span><span class="p">(</span><span class="mi">10583</span><span class="p">);</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Initialize hash table */</span>
+<div class="highlight"><span class="filename">hash_map.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Initialize a hash table */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="n">Map</span><span class="o">&lt;</span><span class="n">Integer</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="o">&gt;</span><span class="w"> </span><span class="n">map</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">HashMap</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a>
-<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="cm">/* Add operation */</span>
-<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="c1">// Add key-value pair (key, value) to hash table</span>
-<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="n">map</span><span class="p">.</span><span class="na">put</span><span class="p">(</span><span class="mi">12836</span><span class="p">,</span><span class="w"> </span><span class="s">"XiaoHa"</span><span class="p">);</span>
-<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="n">map</span><span class="p">.</span><span class="na">put</span><span class="p">(</span><span class="mi">15937</span><span class="p">,</span><span class="w"> </span><span class="s">"XiaoLuo"</span><span class="p">);</span>
-<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="n">map</span><span class="p">.</span><span class="na">put</span><span class="p">(</span><span class="mi">16750</span><span class="p">,</span><span class="w"> </span><span class="s">"XiaoSuan"</span><span class="p">);</span>
-<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="n">map</span><span class="p">.</span><span class="na">put</span><span class="p">(</span><span class="mi">13276</span><span class="p">,</span><span class="w"> </span><span class="s">"XiaoFa"</span><span class="p">);</span>
-<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="n">map</span><span class="p">.</span><span class="na">put</span><span class="p">(</span><span class="mi">10583</span><span class="p">,</span><span class="w"> </span><span class="s">"XiaoYa"</span><span class="p">);</span>
+<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="cm">/* Insertion */</span>
+<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="c1">// Add key-value pairs (key, value) to the hash table</span>
+<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="n">map</span><span class="p">.</span><span class="na">put</span><span class="p">(</span><span class="mi">12836</span><span class="p">,</span><span class="w"> </span><span class="s">"小哈"</span><span class="p">);</span>
+<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="n">map</span><span class="p">.</span><span class="na">put</span><span class="p">(</span><span class="mi">15937</span><span class="p">,</span><span class="w"> </span><span class="s">"小啰"</span><span class="p">);</span>
+<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="n">map</span><span class="p">.</span><span class="na">put</span><span class="p">(</span><span class="mi">16750</span><span class="p">,</span><span class="w"> </span><span class="s">"小算"</span><span class="p">);</span>
+<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="n">map</span><span class="p">.</span><span class="na">put</span><span class="p">(</span><span class="mi">13276</span><span class="p">,</span><span class="w"> </span><span class="s">"小法"</span><span class="p">);</span>
+<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="n">map</span><span class="p">.</span><span class="na">put</span><span class="p">(</span><span class="mi">10583</span><span class="p">,</span><span class="w"> </span><span class="s">"小鸭"</span><span class="p">);</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a>
-<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="cm">/* Query operation */</span>
-<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="c1">// Input key into hash table to get value</span>
+<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="cm">/* Lookup */</span>
+<a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="c1">// Look up key in the hash table to obtain value</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="n">String</span><span class="w"> </span><span class="n">name</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">map</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="mi">15937</span><span class="p">);</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a>
-<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="cm">/* Delete operation */</span>
-<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="c1">// Delete key-value pair (key, value) from hash table</span>
+<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="cm">/* Deletion */</span>
+<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="c1">// Remove a key-value pair (key, value) from the hash table</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="n">map</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="mi">10583</span><span class="p">);</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Initialize hash table */</span>
+<div class="highlight"><span class="filename">hash_map.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Initialize a hash table */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="n">Dictionary</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">string</span><span class="o">&gt;</span><span class="w"> </span><span class="n">map</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
-<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Add key-value pair (key, value) to hash table</span>
-<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="p">{</span><span class="w"> </span><span class="mi">12836</span><span class="p">,</span><span class="w"> </span><span class="s">"XiaoHa"</span><span class="w"> </span><span class="p">},</span>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="p">{</span><span class="w"> </span><span class="mi">15937</span><span class="p">,</span><span class="w"> </span><span class="s">"XiaoLuo"</span><span class="w"> </span><span class="p">},</span>
-<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="p">{</span><span class="w"> </span><span class="mi">16750</span><span class="p">,</span><span class="w"> </span><span class="s">"XiaoSuan"</span><span class="w"> </span><span class="p">},</span>
-<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="p">{</span><span class="w"> </span><span class="mi">13276</span><span class="p">,</span><span class="w"> </span><span class="s">"XiaoFa"</span><span class="w"> </span><span class="p">},</span>
-<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="p">{</span><span class="w"> </span><span class="mi">10583</span><span class="p">,</span><span class="w"> </span><span class="s">"XiaoYa"</span><span class="w"> </span><span class="p">}</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="cm">/* Insertion */</span>
+<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Add key-value pairs (key, value) to the hash table</span>
+<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="p">{</span><span class="w"> </span><span class="mi">12836</span><span class="p">,</span><span class="w"> </span><span class="s">"小哈"</span><span class="w"> </span><span class="p">},</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="p">{</span><span class="w"> </span><span class="mi">15937</span><span class="p">,</span><span class="w"> </span><span class="s">"小啰"</span><span class="w"> </span><span class="p">},</span>
+<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="p">{</span><span class="w"> </span><span class="mi">16750</span><span class="p">,</span><span class="w"> </span><span class="s">"小算"</span><span class="w"> </span><span class="p">},</span>
+<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="p">{</span><span class="w"> </span><span class="mi">13276</span><span class="p">,</span><span class="w"> </span><span class="s">"小法"</span><span class="w"> </span><span class="p">},</span>
+<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="p">{</span><span class="w"> </span><span class="mi">10583</span><span class="p">,</span><span class="w"> </span><span class="s">"小鸭"</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="p">};</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a>
-<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="cm">/* Query operation */</span>
-<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="c1">// Input key into hash table to get value</span>
+<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="cm">/* Lookup */</span>
+<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="c1">// Look up key in the hash table to obtain value</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="kt">string</span><span class="w"> </span><span class="n">name</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">map</span><span class="p">[</span><span class="mi">15937</span><span class="p">];</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a>
-<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="cm">/* Delete operation */</span>
-<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="c1">// Delete key-value pair (key, value) from hash table</span>
+<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="cm">/* Deletion */</span>
+<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="c1">// Remove a key-value pair (key, value) from the hash table</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="n">map</span><span class="p">.</span><span class="n">Remove</span><span class="p">(</span><span class="mi">10583</span><span class="p">);</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map_test.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Initialize hash table */</span>
+<div class="highlight"><span class="filename">hash_map_test.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Initialize a hash table */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="nx">hmap</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">(</span><span class="kt">map</span><span class="p">[</span><span class="kt">int</span><span class="p">]</span><span class="kt">string</span><span class="p">)</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a>
-<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="cm">/* Add operation */</span>
-<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="c1">// Add key-value pair (key, value) to hash table</span>
-<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="nx">hmap</span><span class="p">[</span><span class="mi">12836</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"XiaoHa"</span>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="nx">hmap</span><span class="p">[</span><span class="mi">15937</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"XiaoLuo"</span>
-<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="nx">hmap</span><span class="p">[</span><span class="mi">16750</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"XiaoSuan"</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="nx">hmap</span><span class="p">[</span><span class="mi">13276</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"XiaoFa"</span>
-<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="nx">hmap</span><span class="p">[</span><span class="mi">10583</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"XiaoYa"</span>
+<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="cm">/* Insertion */</span>
+<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="c1">// Add key-value pairs (key, value) to the hash table</span>
+<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="nx">hmap</span><span class="p">[</span><span class="mi">12836</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"小哈"</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="nx">hmap</span><span class="p">[</span><span class="mi">15937</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"小啰"</span>
+<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="nx">hmap</span><span class="p">[</span><span class="mi">16750</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"小算"</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="nx">hmap</span><span class="p">[</span><span class="mi">13276</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"小法"</span>
+<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="nx">hmap</span><span class="p">[</span><span class="mi">10583</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"小鸭"</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a>
-<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="cm">/* Query operation */</span>
-<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="c1">// Input key into hash table to get value</span>
+<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="cm">/* Lookup */</span>
+<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="c1">// Look up key in the hash table to obtain value</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="nx">name</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">hmap</span><span class="p">[</span><span class="mi">15937</span><span class="p">]</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a>
-<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="cm">/* Delete operation */</span>
-<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="c1">// Delete key-value pair (key, value) from hash table</span>
+<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="cm">/* Deletion */</span>
+<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="c1">// Remove a key-value pair (key, value) from the hash table</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="nb">delete</span><span class="p">(</span><span class="nx">hmap</span><span class="p">,</span><span class="w"> </span><span class="mi">10583</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Initialize hash table */</span>
+<div class="highlight"><span class="filename">hash_map.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Initialize a hash table */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">var</span><span class="w"> </span><span class="nv">map</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">:</span><span class="w"> </span><span class="nb">String</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[:]</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a>
-<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="cm">/* Add operation */</span>
-<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="c1">// Add key-value pair (key, value) to hash table</span>
-<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="bp">map</span><span class="p">[</span><span class="mi">12836</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"XiaoHa"</span>
-<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="bp">map</span><span class="p">[</span><span class="mi">15937</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"XiaoLuo"</span>
-<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="bp">map</span><span class="p">[</span><span class="mi">16750</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"XiaoSuan"</span>
-<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="bp">map</span><span class="p">[</span><span class="mi">13276</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"XiaoFa"</span>
-<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="bp">map</span><span class="p">[</span><span class="mi">10583</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"XiaoYa"</span>
+<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="cm">/* Insertion */</span>
+<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="c1">// Add key-value pairs (key, value) to the hash table</span>
+<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="bp">map</span><span class="p">[</span><span class="mi">12836</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"小哈"</span>
+<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="bp">map</span><span class="p">[</span><span class="mi">15937</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"小啰"</span>
+<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="bp">map</span><span class="p">[</span><span class="mi">16750</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"小算"</span>
+<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="bp">map</span><span class="p">[</span><span class="mi">13276</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"小法"</span>
+<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="bp">map</span><span class="p">[</span><span class="mi">10583</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="s">"小鸭"</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a>
-<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="cm">/* Query operation */</span>
-<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="c1">// Input key into hash table to get value</span>
+<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="cm">/* Lookup */</span>
+<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="c1">// Look up key in the hash table to obtain value</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="kd">let</span><span class="w"> </span><span class="nv">name</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">map</span><span class="p">[</span><span class="mi">15937</span><span class="p">]</span><span class="o">!</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a>
-<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="cm">/* Delete operation */</span>
-<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="c1">// Delete key-value pair (key, value) from hash table</span>
+<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="cm">/* Deletion */</span>
+<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="c1">// Remove a key-value pair (key, value) from the hash table</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="bp">map</span><span class="p">.</span><span class="n">removeValue</span><span class="p">(</span><span class="n">forKey</span><span class="p">:</span><span class="w"> </span><span class="mi">10583</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Initialize hash table */</span>
+<div class="highlight"><span class="filename">hash_map.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Initialize a hash table */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">const</span><span class="w"> </span><span class="nx">map</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Map</span><span class="p">();</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="cm">/* Add operation */</span>
-<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="c1">// Add key-value pair (key, value) to hash table</span>
-<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">12836</span><span class="p">,</span><span class="w"> </span><span class="s1">'XiaoHa'</span><span class="p">);</span>
-<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">15937</span><span class="p">,</span><span class="w"> </span><span class="s1">'XiaoLuo'</span><span class="p">);</span>
-<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">16750</span><span class="p">,</span><span class="w"> </span><span class="s1">'XiaoSuan'</span><span class="p">);</span>
-<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">13276</span><span class="p">,</span><span class="w"> </span><span class="s1">'XiaoFa'</span><span class="p">);</span>
-<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">10583</span><span class="p">,</span><span class="w"> </span><span class="s1">'XiaoYa'</span><span class="p">);</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="cm">/* Insertion */</span>
+<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="c1">// Add key-value pairs (key, value) to the hash table</span>
+<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">12836</span><span class="p">,</span><span class="w"> </span><span class="s1">'小哈'</span><span class="p">);</span>
+<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">15937</span><span class="p">,</span><span class="w"> </span><span class="s1">'小啰'</span><span class="p">);</span>
+<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">16750</span><span class="p">,</span><span class="w"> </span><span class="s1">'小算'</span><span class="p">);</span>
+<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">13276</span><span class="p">,</span><span class="w"> </span><span class="s1">'小法'</span><span class="p">);</span>
+<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">10583</span><span class="p">,</span><span class="w"> </span><span class="s1">'小鸭'</span><span class="p">);</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a>
-<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="cm">/* Query operation */</span>
-<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="c1">// Input key into hash table to get value</span>
+<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="cm">/* Lookup */</span>
+<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="c1">// Look up key in the hash table to obtain value</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="kd">let</span><span class="w"> </span><span class="nx">name</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">map</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="mf">15937</span><span class="p">);</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a>
-<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="cm">/* Delete operation */</span>
-<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="c1">// Delete key-value pair (key, value) from hash table</span>
+<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="cm">/* Deletion */</span>
+<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="c1">// Remove a key-value pair (key, value) from the hash table</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="nx">map</span><span class="p">.</span><span class="ow">delete</span><span class="p">(</span><span class="mf">10583</span><span class="p">);</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Initialize hash table */</span>
+<div class="highlight"><span class="filename">hash_map.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Initialize a hash table */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">const</span><span class="w"> </span><span class="nx">map</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Map</span><span class="o">&lt;</span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="kt">string</span><span class="o">&gt;</span><span class="p">();</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="cm">/* Add operation */</span>
-<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="c1">// Add key-value pair (key, value) to hash table</span>
-<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">12836</span><span class="p">,</span><span class="w"> </span><span class="s1">'XiaoHa'</span><span class="p">);</span>
-<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">15937</span><span class="p">,</span><span class="w"> </span><span class="s1">'XiaoLuo'</span><span class="p">);</span>
-<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">16750</span><span class="p">,</span><span class="w"> </span><span class="s1">'XiaoSuan'</span><span class="p">);</span>
-<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">13276</span><span class="p">,</span><span class="w"> </span><span class="s1">'XiaoFa'</span><span class="p">);</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">10583</span><span class="p">,</span><span class="w"> </span><span class="s1">'XiaoYa'</span><span class="p">);</span>
-<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nAfter adding, hash table is\nKey -&gt; Value'</span><span class="p">);</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="cm">/* Insertion */</span>
+<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="c1">// Add key-value pairs (key, value) to the hash table</span>
+<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">12836</span><span class="p">,</span><span class="w"> </span><span class="s1">'小哈'</span><span class="p">);</span>
+<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">15937</span><span class="p">,</span><span class="w"> </span><span class="s1">'小啰'</span><span class="p">);</span>
+<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">16750</span><span class="p">,</span><span class="w"> </span><span class="s1">'小算'</span><span class="p">);</span>
+<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">13276</span><span class="p">,</span><span class="w"> </span><span class="s1">'小法'</span><span class="p">);</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="nx">map</span><span class="p">.</span><span class="nx">set</span><span class="p">(</span><span class="mf">10583</span><span class="p">,</span><span class="w"> </span><span class="s1">'小鸭'</span><span class="p">);</span>
+<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nAfter insertion, the hash table is\nKey -&gt; Value'</span><span class="p">);</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="nx">map</span><span class="p">);</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a>
-<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="cm">/* Query operation */</span>
-<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="c1">// Input key into hash table to get value</span>
+<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="cm">/* Lookup */</span>
+<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="c1">// Look up key in the hash table to obtain value</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="kd">let</span><span class="w"> </span><span class="nx">name</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">map</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="mf">15937</span><span class="p">);</span>
-<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nInput student ID 15937, queried name '</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">name</span><span class="p">);</span>
+<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nLooking up student ID 15937 gives the name '</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">name</span><span class="p">);</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a>
-<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="cm">/* Delete operation */</span>
-<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="c1">// Delete key-value pair (key, value) from hash table</span>
+<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="cm">/* Deletion */</span>
+<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="c1">// Remove a key-value pair (key, value) from the hash table</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="nx">map</span><span class="p">.</span><span class="ow">delete</span><span class="p">(</span><span class="mf">10583</span><span class="p">);</span>
-<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nAfter deleting 10583, hash table is\nKey -&gt; Value'</span><span class="p">);</span>
+<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nAfter deleting 10583, the hash table is\nKey -&gt; Value'</span><span class="p">);</span>
 <a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="nx">map</span><span class="p">);</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Initialize hash table */</span>
+<div class="highlight"><span class="filename">hash_map.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Initialize a hash table */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="n">Map</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">String</span><span class="o">&gt;</span><span class="w"> </span><span class="n">map</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{};</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a>
-<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="cm">/* Add operation */</span>
-<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="c1">// Add key-value pair (key, value) to hash table</span>
-<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="n">map</span><span class="p">[</span><span class="m">12836</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"XiaoHa"</span><span class="p">;</span>
-<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="n">map</span><span class="p">[</span><span class="m">15937</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"XiaoLuo"</span><span class="p">;</span>
-<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="n">map</span><span class="p">[</span><span class="m">16750</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"XiaoSuan"</span><span class="p">;</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="n">map</span><span class="p">[</span><span class="m">13276</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"XiaoFa"</span><span class="p">;</span>
-<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="n">map</span><span class="p">[</span><span class="m">10583</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"XiaoYa"</span><span class="p">;</span>
+<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="cm">/* Insertion */</span>
+<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="c1">// Add key-value pairs (key, value) to the hash table</span>
+<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="n">map</span><span class="p">[</span><span class="m">12836</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"小哈"</span><span class="p">;</span>
+<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="n">map</span><span class="p">[</span><span class="m">15937</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"小啰"</span><span class="p">;</span>
+<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="n">map</span><span class="p">[</span><span class="m">16750</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"小算"</span><span class="p">;</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="n">map</span><span class="p">[</span><span class="m">13276</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"小法"</span><span class="p">;</span>
+<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="n">map</span><span class="p">[</span><span class="m">10583</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"小鸭"</span><span class="p">;</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a>
-<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="cm">/* Query operation */</span>
-<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="c1">// Input key into hash table to get value</span>
+<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="cm">/* Lookup */</span>
+<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="c1">// Look up key in the hash table to obtain value</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="kt">String</span><span class="w"> </span><span class="n">name</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">map</span><span class="p">[</span><span class="m">15937</span><span class="p">];</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a>
-<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="cm">/* Delete operation */</span>
-<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="c1">// Delete key-value pair (key, value) from hash table</span>
+<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="cm">/* Deletion */</span>
+<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="c1">// Remove a key-value pair (key, value) from the hash table</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="n">map</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="m">10583</span><span class="p">);</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">hash_map.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="k">use</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">collections</span><span class="p">::</span><span class="n">HashMap</span><span class="p">;</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="cm">/* Initialize hash table */</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="cm">/* Initialize a hash table */</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">map</span><span class="p">:</span><span class="w"> </span><span class="nc">HashMap</span><span class="o">&lt;</span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="nb">String</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashMap</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a>
-<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="cm">/* Add operation */</span>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="c1">// Add key-value pair (key, value) to hash table</span>
-<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="n">map</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">12836</span><span class="p">,</span><span class="w"> </span><span class="s">"XiaoHa"</span><span class="p">.</span><span class="n">to_string</span><span class="p">());</span>
-<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="n">map</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">15937</span><span class="p">,</span><span class="w"> </span><span class="s">"XiaoLuo"</span><span class="p">.</span><span class="n">to_string</span><span class="p">());</span>
-<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="n">map</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">16750</span><span class="p">,</span><span class="w"> </span><span class="s">"XiaoSuan"</span><span class="p">.</span><span class="n">to_string</span><span class="p">());</span>
-<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="n">map</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">13276</span><span class="p">,</span><span class="w"> </span><span class="s">"XiaoFa"</span><span class="p">.</span><span class="n">to_string</span><span class="p">());</span>
-<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="n">map</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">10583</span><span class="p">,</span><span class="w"> </span><span class="s">"XiaoYa"</span><span class="p">.</span><span class="n">to_string</span><span class="p">());</span>
+<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="cm">/* Insertion */</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="c1">// Add key-value pairs (key, value) to the hash table</span>
+<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="n">map</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">12836</span><span class="p">,</span><span class="w"> </span><span class="s">"小哈"</span><span class="p">.</span><span class="n">to_string</span><span class="p">());</span>
+<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="n">map</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">15937</span><span class="p">,</span><span class="w"> </span><span class="s">"小啰"</span><span class="p">.</span><span class="n">to_string</span><span class="p">());</span>
+<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="n">map</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">16750</span><span class="p">,</span><span class="w"> </span><span class="s">"小算"</span><span class="p">.</span><span class="n">to_string</span><span class="p">());</span>
+<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="n">map</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">13279</span><span class="p">,</span><span class="w"> </span><span class="s">"小法"</span><span class="p">.</span><span class="n">to_string</span><span class="p">());</span>
+<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="n">map</span><span class="p">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">10583</span><span class="p">,</span><span class="w"> </span><span class="s">"小鸭"</span><span class="p">.</span><span class="n">to_string</span><span class="p">());</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a>
-<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="cm">/* Query operation */</span>
-<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="c1">// Input key into hash table to get value</span>
+<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="cm">/* Lookup */</span>
+<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="c1">// Look up key in the hash table to obtain value</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="kd">let</span><span class="w"> </span><span class="n">_name</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="nb">String</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">map</span><span class="p">.</span><span class="n">get</span><span class="p">(</span><span class="o">&amp;</span><span class="mi">15937</span><span class="p">);</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a>
-<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="cm">/* Delete operation */</span>
-<a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="c1">// Delete key-value pair (key, value) from hash table</span>
+<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="cm">/* Deletion */</span>
+<a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="c1">// Remove a key-value pair (key, value) from the hash table</span>
 <a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="kd">let</span><span class="w"> </span><span class="n">_removed_value</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="nb">String</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">map</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="o">&amp;</span><span class="mi">10583</span><span class="p">);</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// C does not provide a built-in hash table</span>
+<div class="highlight"><span class="filename">hash_map.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// C provides no built-in hash table</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Initialize hash table */</span>
+<div class="highlight"><span class="filename">hash_map.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Initialize a hash table */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">val</span><span class="w"> </span><span class="nv">map</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashMap</span><span class="o">&lt;</span><span class="kt">Int</span><span class="p">,</span><span class="kt">String</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a>
-<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="cm">/* Add operation */</span>
-<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="c1">// Add key-value pair (key, value) to hash table</span>
-<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="n">map</span><span class="o">[</span><span class="m">12836</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"XiaoHa"</span>
-<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="n">map</span><span class="o">[</span><span class="m">15937</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"XiaoLuo"</span>
-<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="n">map</span><span class="o">[</span><span class="m">16750</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"XiaoSuan"</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="n">map</span><span class="o">[</span><span class="m">13276</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"XiaoFa"</span>
-<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="n">map</span><span class="o">[</span><span class="m">10583</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"XiaoYa"</span>
+<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="cm">/* Insertion */</span>
+<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="c1">// Add key-value pairs (key, value) to the hash table</span>
+<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="n">map</span><span class="o">[</span><span class="m">12836</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"小哈"</span>
+<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="n">map</span><span class="o">[</span><span class="m">15937</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"小啰"</span>
+<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="n">map</span><span class="o">[</span><span class="m">16750</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"小算"</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="n">map</span><span class="o">[</span><span class="m">13276</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"小法"</span>
+<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="n">map</span><span class="o">[</span><span class="m">10583</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">"小鸭"</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a>
-<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="cm">/* Query operation */</span>
-<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="c1">// Input key into hash table to get value</span>
+<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="cm">/* Lookup */</span>
+<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="c1">// Look up key in the hash table to obtain value</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="kd">val</span><span class="w"> </span><span class="nv">name</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">map</span><span class="o">[</span><span class="m">15937</span><span class="o">]</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a>
-<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="cm">/* Delete operation */</span>
-<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="c1">// Delete key-value pair (key, value) from hash table</span>
+<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="cm">/* Deletion */</span>
+<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="c1">// Remove a key-value pair (key, value) from the hash table</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="n">map</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="m">10583</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Initialize hash table</span>
+<div class="highlight"><span class="filename">hash_map.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Initialize a hash table</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="n">hmap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{}</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a>
-<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="c1"># Add operation</span>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="c1"># Add key-value pair (key, value) to hash table</span>
-<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="n">hmap</span><span class="o">[</span><span class="mi">12836</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"XiaoHa"</span>
-<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="n">hmap</span><span class="o">[</span><span class="mi">15937</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"XiaoLuo"</span>
-<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="n">hmap</span><span class="o">[</span><span class="mi">16750</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"XiaoSuan"</span>
-<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="n">hmap</span><span class="o">[</span><span class="mi">13276</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"XiaoFa"</span>
-<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="n">hmap</span><span class="o">[</span><span class="mi">10583</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"XiaoYa"</span>
+<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="c1"># Insertion</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="c1"># Add key-value pairs (key, value) to the hash table</span>
+<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="n">hmap</span><span class="o">[</span><span class="mi">12836</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"小哈"</span>
+<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="n">hmap</span><span class="o">[</span><span class="mi">15937</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"小啰"</span>
+<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="n">hmap</span><span class="o">[</span><span class="mi">16750</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"小算"</span>
+<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="n">hmap</span><span class="o">[</span><span class="mi">13276</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"小法"</span>
+<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="n">hmap</span><span class="o">[</span><span class="mi">10583</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s2">"小鸭"</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a>
-<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="c1"># Query operation</span>
-<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="c1"># Input key into hash table to get value</span>
+<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="c1"># Lookup</span>
+<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="c1"># Look up key in the hash table to obtain value</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="nb">name</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hmap</span><span class="o">[</span><span class="mi">15937</span><span class="o">]</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a>
-<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="c1"># Delete operation</span>
-<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="c1"># Delete key-value pair (key, value) from hash table</span>
+<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="c1"># Deletion</span>
+<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="c1"># Remove a key-value pair (key, value) from the hash table</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="n">hmap</span><span class="o">.</span><span class="n">delete</span><span class="p">(</span><span class="mi">10583</span><span class="p">)</span>
 </code></pre></div>
 </div>
 </div>
 </div>
 <details class="pythontutor">
-<summary>Visualized Execution</summary>
-<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%93%88%E5%B8%8C%E8%A1%A8%0A%20%20%20%20hmap%20%3D%20%7B%7D%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%B7%BB%E5%8A%A0%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%9C%A8%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E6%B7%BB%E5%8A%A0%E9%94%AE%E5%80%BC%E5%AF%B9%20%28key,%20value%29%0A%20%20%20%20hmap%5B12836%5D%20%3D%20%22%E5%B0%8F%E5%93%88%22%0A%20%20%20%20hmap%5B15937%5D%20%3D%20%22%E5%B0%8F%E5%95%B0%22%0A%20%20%20%20hmap%5B16750%5D%20%3D%20%22%E5%B0%8F%E7%AE%97%22%0A%20%20%20%20hmap%5B13276%5D%20%3D%20%22%E5%B0%8F%E6%B3%95%22%0A%20%20%20%20hmap%5B10583%5D%20%3D%20%22%E5%B0%8F%E9%B8%AD%22%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%9F%A5%E8%AF%A2%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%90%91%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E8%BE%93%E5%85%A5%E9%94%AE%20key%20%EF%BC%8C%E5%BE%97%E5%88%B0%E5%80%BC%20value%0A%20%20%20%20name%20%3D%20hmap%5B15937%5D%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%9C%A8%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E5%88%A0%E9%99%A4%E9%94%AE%E5%80%BC%E5%AF%B9%20%28key,%20value%29%0A%20%20%20%20hmap.pop%2810583%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=2&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%93%88%E5%B8%8C%E8%A1%A8%0A%20%20%20%20hmap%20%3D%20%7B%7D%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%B7%BB%E5%8A%A0%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%9C%A8%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E6%B7%BB%E5%8A%A0%E9%94%AE%E5%80%BC%E5%AF%B9%20%28key,%20value%29%0A%20%20%20%20hmap%5B12836%5D%20%3D%20%22%E5%B0%8F%E5%93%88%22%0A%20%20%20%20hmap%5B15937%5D%20%3D%20%22%E5%B0%8F%E5%95%B0%22%0A%20%20%20%20hmap%5B16750%5D%20%3D%20%22%E5%B0%8F%E7%AE%97%22%0A%20%20%20%20hmap%5B13276%5D%20%3D%20%22%E5%B0%8F%E6%B3%95%22%0A%20%20%20%20hmap%5B10583%5D%20%3D%20%22%E5%B0%8F%E9%B8%AD%22%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%9F%A5%E8%AF%A2%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%90%91%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E8%BE%93%E5%85%A5%E9%94%AE%20key%20%EF%BC%8C%E5%BE%97%E5%88%B0%E5%80%BC%20value%0A%20%20%20%20name%20%3D%20hmap%5B15937%5D%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%9C%A8%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E5%88%A0%E9%99%A4%E9%94%AE%E5%80%BC%E5%AF%B9%20%28key,%20value%29%0A%20%20%20%20hmap.pop%2810583%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=2&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20hash%20table%0A%20%20%20%20hmap%20%3D%20%7B%7D%0A%20%20%20%20%0A%20%20%20%20%23%20Insertion%0A%20%20%20%20%23%20Add%20key-value%20pairs%20%28key%2C%20value%29%20to%20the%20hash%20table%0A%20%20%20%20hmap%5B12836%5D%20%3D%20%22%E5%B0%8F%E5%93%88%22%0A%20%20%20%20hmap%5B15937%5D%20%3D%20%22%E5%B0%8F%E5%95%B0%22%0A%20%20%20%20hmap%5B16750%5D%20%3D%20%22%E5%B0%8F%E7%AE%97%22%0A%20%20%20%20hmap%5B13276%5D%20%3D%20%22%E5%B0%8F%E6%B3%95%22%0A%20%20%20%20hmap%5B10583%5D%20%3D%20%22%E5%B0%8F%E9%B8%AD%22%0A%20%20%20%20%0A%20%20%20%20%23%20Lookup%0A%20%20%20%20%23%20Look%20up%20key%20in%20the%20hash%20table%20to%20obtain%20value%0A%20%20%20%20name%20%3D%20hmap%5B15937%5D%0A%20%20%20%20%0A%20%20%20%20%23%20Deletion%0A%20%20%20%20%23%20Remove%20a%20key-value%20pair%20%28key%2C%20value%29%20from%20the%20hash%20table%0A%20%20%20%20hmap.pop%2810583%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=2&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20hash%20table%0A%20%20%20%20hmap%20%3D%20%7B%7D%0A%20%20%20%20%0A%20%20%20%20%23%20Insertion%0A%20%20%20%20%23%20Add%20key-value%20pairs%20%28key%2C%20value%29%20to%20the%20hash%20table%0A%20%20%20%20hmap%5B12836%5D%20%3D%20%22%E5%B0%8F%E5%93%88%22%0A%20%20%20%20hmap%5B15937%5D%20%3D%20%22%E5%B0%8F%E5%95%B0%22%0A%20%20%20%20hmap%5B16750%5D%20%3D%20%22%E5%B0%8F%E7%AE%97%22%0A%20%20%20%20hmap%5B13276%5D%20%3D%20%22%E5%B0%8F%E6%B3%95%22%0A%20%20%20%20hmap%5B10583%5D%20%3D%20%22%E5%B0%8F%E9%B8%AD%22%0A%20%20%20%20%0A%20%20%20%20%23%20Lookup%0A%20%20%20%20%23%20Look%20up%20key%20in%20the%20hash%20table%20to%20obtain%20value%0A%20%20%20%20name%20%3D%20hmap%5B15937%5D%0A%20%20%20%20%0A%20%20%20%20%23%20Deletion%0A%20%20%20%20%23%20Remove%20a%20key-value%20pair%20%28key%2C%20value%29%20from%20the%20hash%20table%0A%20%20%20%20hmap.pop%2810583%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=2&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<p>There are three common ways to traverse a hash table: traversing key-value pairs, traversing keys, and traversing values. Example code is as follows:</p>
+<p>There are three common ways to traverse a hash table: visit key-value pairs, keys only, or values only. Examples follow:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="c1"># Traverse hash table</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="c1"># Traverse key-value pairs key-&gt;value</span>
+<div class="highlight"><span class="filename">hash_map.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="c1"># Traverse the hash table</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="c1"># Traverse key-value pairs: key-&gt;value</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="n">hmap</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="nb">print</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="s2">"-&gt;"</span><span class="p">,</span> <span class="n">value</span><span class="p">)</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="c1"># Traverse keys only</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="c1"># Traverse keys only: key</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a><span class="k">for</span> <span class="n">key</span> <span class="ow">in</span> <span class="n">hmap</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>    <span class="nb">print</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a><span class="c1"># Traverse values only</span>
+<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a><span class="c1"># Traverse values only: value</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a><span class="k">for</span> <span class="n">value</span> <span class="ow">in</span> <span class="n">hmap</span><span class="o">.</span><span class="n">values</span><span class="p">():</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>    <span class="nb">print</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Traverse hash table */</span>
-<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="c1">// Traverse key-value pairs key-&gt;value</span>
+<div class="highlight"><span class="filename">hash_map.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Traverse the hash table */</span>
+<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="c1">// Traverse key-value pairs: key-&gt;value</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="k">auto</span><span class="w"> </span><span class="n">kv</span><span class="o">:</span><span class="w"> </span><span class="n">map</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">kv</span><span class="p">.</span><span class="n">first</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">" -&gt; "</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">kv</span><span class="p">.</span><span class="n">second</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="p">}</span>
-<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="c1">// Traverse using iterator key-&gt;value</span>
+<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="c1">// Use an iterator to traverse key-&gt;value pairs</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="k">auto</span><span class="w"> </span><span class="n">iter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">map</span><span class="p">.</span><span class="n">begin</span><span class="p">();</span><span class="w"> </span><span class="n">iter</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">map</span><span class="p">.</span><span class="n">end</span><span class="p">();</span><span class="w"> </span><span class="n">iter</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">iter</span><span class="o">-&gt;</span><span class="n">first</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">"-&gt;"</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">iter</span><span class="o">-&gt;</span><span class="n">second</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Traverse hash table */</span>
-<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="c1">// Traverse key-value pairs key-&gt;value</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Map</span><span class="p">.</span><span class="na">Entry</span><span class="o">&lt;</span><span class="n">Integer</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="o">&gt;</span><span class="w"> </span><span class="n">kv</span><span class="p">:</span><span class="w"> </span><span class="n">map</span><span class="p">.</span><span class="na">entrySet</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
+<div class="highlight"><span class="filename">hash_map.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Traverse the hash table */</span>
+<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="c1">// Traverse key-value pairs: key-&gt;value</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Map</span><span class="p">.</span><span class="na">Entry</span><span class="w"> </span><span class="o">&lt;</span><span class="n">Integer</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="o">&gt;</span><span class="w"> </span><span class="n">kv</span><span class="p">:</span><span class="w"> </span><span class="n">map</span><span class="p">.</span><span class="na">entrySet</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="n">kv</span><span class="p">.</span><span class="na">getKey</span><span class="p">()</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="s">" -&gt; "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">kv</span><span class="p">.</span><span class="na">getValue</span><span class="p">());</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="p">}</span>
-<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="c1">// Traverse keys only</span>
+<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="c1">// Traverse keys only: key</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">map</span><span class="p">.</span><span class="na">keySet</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="p">}</span>
-<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="c1">// Traverse values only</span>
+<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="c1">// Traverse values only: value</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">String</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">map</span><span class="p">.</span><span class="na">values</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">    </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Traverse hash table */</span>
-<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="c1">// Traverse key-value pairs Key-&gt;Value</span>
+<div class="highlight"><span class="filename">hash_map.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Traverse the hash table */</span>
+<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="c1">// Traverse key-value pairs: Key-&gt;Value</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">var</span><span class="w"> </span><span class="n">kv</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">map</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="n">kv</span><span class="p">.</span><span class="n">Key</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="s">" -&gt; "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">kv</span><span class="p">.</span><span class="n">Value</span><span class="p">);</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="p">}</span>
-<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="c1">// Traverse keys only</span>
+<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="c1">// Traverse keys only: key</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">map</span><span class="p">.</span><span class="n">Keys</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="p">}</span>
-<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="c1">// Traverse values only</span>
+<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="c1">// Traverse values only: value</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">string</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">map</span><span class="p">.</span><span class="n">Values</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">    </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map_test.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Traverse hash table */</span>
-<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="c1">// Traverse key-value pairs key-&gt;value</span>
+<div class="highlight"><span class="filename">hash_map_test.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Traverse the hash table */</span>
+<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="c1">// Traverse key-value pairs: key-&gt;value</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="k">for</span><span class="w"> </span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">value</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">hmap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="s">"-&gt;"</span><span class="p">,</span><span class="w"> </span><span class="nx">value</span><span class="p">)</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="p">}</span>
-<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="c1">// Traverse keys only</span>
+<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="c1">// Traverse keys only: key</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="k">for</span><span class="w"> </span><span class="nx">key</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">hmap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="p">}</span>
-<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="c1">// Traverse values only</span>
+<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="c1">// Traverse values only: value</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">value</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">hmap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="nx">value</span><span class="p">)</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Traverse hash table */</span>
-<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="c1">// Traverse key-value pairs Key-&gt;Value</span>
+<div class="highlight"><span class="filename">hash_map.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Traverse the hash table */</span>
+<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="c1">// Traverse key-value pairs: Key-&gt;Value</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">value</span><span class="p">)</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="bp">map</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="bp">print</span><span class="p">(</span><span class="s">"</span><span class="si">\(</span><span class="n">key</span><span class="si">)</span><span class="s"> -&gt; </span><span class="si">\(</span><span class="n">value</span><span class="si">)</span><span class="s">"</span><span class="p">)</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="p">}</span>
-<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="c1">// Traverse keys only</span>
+<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="c1">// Traverse keys only: Key</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="k">for</span><span class="w"> </span><span class="n">key</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="bp">map</span><span class="p">.</span><span class="n">keys</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="bp">print</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="p">}</span>
-<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="c1">// Traverse values only</span>
+<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="c1">// Traverse values only: Value</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="k">for</span><span class="w"> </span><span class="n">value</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="bp">map</span><span class="p">.</span><span class="n">values</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">    </span><span class="bp">print</span><span class="p">(</span><span class="n">value</span><span class="p">)</span>
 <a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Traverse hash table */</span>
-<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nTraverse key-value pairs Key-&gt;Value'</span><span class="p">);</span>
+<div class="highlight"><span class="filename">hash_map.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Traverse the hash table */</span>
+<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nTraverse key-value pairs: Key-&gt;Value'</span><span class="p">);</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="p">[</span><span class="nx">k</span><span class="p">,</span><span class="w"> </span><span class="nx">v</span><span class="p">]</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">map</span><span class="p">.</span><span class="nx">entries</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="nx">k</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="s1">' -&gt; '</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">v</span><span class="p">);</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="p">}</span>
-<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nTraverse keys only Key'</span><span class="p">);</span>
+<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nTraverse keys only: Key'</span><span class="p">);</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">k</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">map</span><span class="p">.</span><span class="nx">keys</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="nx">k</span><span class="p">);</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="p">}</span>
-<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nTraverse values only Value'</span><span class="p">);</span>
+<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nTraverse values only: Value'</span><span class="p">);</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">v</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">map</span><span class="p">.</span><span class="nx">values</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="nx">v</span><span class="p">);</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Traverse hash table */</span>
-<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nTraverse key-value pairs Key-&gt;Value'</span><span class="p">);</span>
+<div class="highlight"><span class="filename">hash_map.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Traverse the hash table */</span>
+<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nTraverse key-value pairs: Key-&gt;Value'</span><span class="p">);</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="p">[</span><span class="nx">k</span><span class="p">,</span><span class="w"> </span><span class="nx">v</span><span class="p">]</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">map</span><span class="p">.</span><span class="nx">entries</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="nx">k</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="s1">' -&gt; '</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">v</span><span class="p">);</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="p">}</span>
-<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nTraverse keys only Key'</span><span class="p">);</span>
+<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nTraverse keys only: Key'</span><span class="p">);</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">k</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">map</span><span class="p">.</span><span class="nx">keys</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="nx">k</span><span class="p">);</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="p">}</span>
-<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nTraverse values only Value'</span><span class="p">);</span>
+<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="s1">'\nTraverse values only: Value'</span><span class="p">);</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">v</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">map</span><span class="p">.</span><span class="nx">values</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">    </span><span class="nx">console</span><span class="p">.</span><span class="nx">info</span><span class="p">(</span><span class="nx">v</span><span class="p">);</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Traverse hash table */</span>
-<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="c1">// Traverse key-value pairs Key-&gt;Value</span>
+<div class="highlight"><span class="filename">hash_map.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Traverse the hash table */</span>
+<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="c1">// Traverse key-value pairs: Key-&gt;Value</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="n">map</span><span class="p">.</span><span class="n">forEach</span><span class="p">((</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">value</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="n">print</span><span class="p">(</span><span class="s1">'</span><span class="si">$</span><span class="n">key</span><span class="s1"> -&gt; </span><span class="si">$</span><span class="n">value</span><span class="s1">'</span><span class="p">);</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="p">});</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a>
-<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="c1">// Traverse keys only</span>
+<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="c1">// Traverse keys only: Key</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="n">map</span><span class="p">.</span><span class="n">keys</span><span class="p">.</span><span class="n">forEach</span><span class="p">((</span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="n">print</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="p">});</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a>
-<a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="c1">// Traverse values only</span>
+<a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="c1">// Traverse values only: Value</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="n">map</span><span class="p">.</span><span class="n">values</span><span class="p">.</span><span class="n">forEach</span><span class="p">((</span><span class="n">value</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">  </span><span class="n">print</span><span class="p">(</span><span class="n">value</span><span class="p">);</span>
 <a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="p">});</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Traverse hash table */</span>
-<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="c1">// Traverse key-value pairs Key-&gt;Value</span>
+<div class="highlight"><span class="filename">hash_map.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Traverse the hash table */</span>
+<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="c1">// Traverse key-value pairs: Key-&gt;Value</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">value</span><span class="p">)</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="o">&amp;</span><span class="n">map</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{key} -&gt; {value}"</span><span class="p">);</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="p">}</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a>
-<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="c1">// Traverse keys only</span>
+<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="c1">// Traverse keys only: Key</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="k">for</span><span class="w"> </span><span class="n">key</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">map</span><span class="p">.</span><span class="n">keys</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">    </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{key}"</span><span class="p">);</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="p">}</span>
 <a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a>
-<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="c1">// Traverse values only</span>
+<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="c1">// Traverse values only: Value</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="k">for</span><span class="w"> </span><span class="n">value</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">map</span><span class="p">.</span><span class="n">values</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">    </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{value}"</span><span class="p">);</span>
 <a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="c1">// C does not provide a built-in hash table</span>
+<div class="highlight"><span class="filename">hash_map.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="c1">// C provides no built-in hash table</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Traverse hash table */</span>
-<a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="c1">// Traverse key-value pairs key-&gt;value</span>
+<div class="highlight"><span class="filename">hash_map.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Traverse the hash table */</span>
+<a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="c1">// Traverse key-value pairs: key-&gt;value</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="k">for</span><span class="w"> </span><span class="p">((</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">value</span><span class="p">)</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">map</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="n">println</span><span class="p">(</span><span class="s">"</span><span class="si">$</span><span class="n">key</span><span class="s"> -&gt; </span><span class="si">$</span><span class="n">value</span><span class="s">"</span><span class="p">)</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="p">}</span>
-<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="c1">// Traverse keys only</span>
+<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="c1">// Traverse keys only: key</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">key</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">map</span><span class="p">.</span><span class="na">keys</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="n">println</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="p">}</span>
-<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="c1">// Traverse values only</span>
+<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="c1">// Traverse values only: value</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">_val</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">map</span><span class="p">.</span><span class="na">values</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="n">println</span><span class="p">(</span><span class="n">_val</span><span class="p">)</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">hash_map.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1"># Traverse hash table</span>
-<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="c1"># Traverse key-value pairs key-&gt;value</span>
+<div class="highlight"><span class="filename">hash_map.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1"># Traverse the hash table</span>
+<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="c1"># Traverse key-value pairs: key-&gt;value</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="n">hmap</span><span class="o">.</span><span class="n">entries</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">value</span><span class="o">|</span><span class="w"> </span><span class="nb">puts</span><span class="w"> </span><span class="s2">"</span><span class="si">#{</span><span class="n">key</span><span class="si">}</span><span class="s2"> -&gt; </span><span class="si">#{</span><span class="n">value</span><span class="si">}</span><span class="s2">"</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a>
-<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="c1"># Traverse keys only</span>
+<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="c1"># Traverse keys only: key</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="n">hmap</span><span class="o">.</span><span class="n">keys</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">key</span><span class="o">|</span><span class="w"> </span><span class="nb">puts</span><span class="w"> </span><span class="n">key</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a>
-<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="c1"># Traverse values only</span>
+<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="c1"># Traverse values only: value</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="n">hmap</span><span class="o">.</span><span class="n">values</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">val</span><span class="o">|</span><span class="w"> </span><span class="nb">puts</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="p">}</span>
 </code></pre></div>
 </div>
 </div>
 </div>
 <details class="pythontutor">
-<summary>Visualized Execution</summary>
-<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%93%88%E5%B8%8C%E8%A1%A8%0A%20%20%20%20hmap%20%3D%20%7B%7D%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%B7%BB%E5%8A%A0%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%9C%A8%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E6%B7%BB%E5%8A%A0%E9%94%AE%E5%80%BC%E5%AF%B9%20%28key,%20value%29%0A%20%20%20%20hmap%5B12836%5D%20%3D%20%22%E5%B0%8F%E5%93%88%22%0A%20%20%20%20hmap%5B15937%5D%20%3D%20%22%E5%B0%8F%E5%95%B0%22%0A%20%20%20%20hmap%5B16750%5D%20%3D%20%22%E5%B0%8F%E7%AE%97%22%0A%20%20%20%20hmap%5B13276%5D%20%3D%20%22%E5%B0%8F%E6%B3%95%22%0A%20%20%20%20hmap%5B10583%5D%20%3D%20%22%E5%B0%8F%E9%B8%AD%22%0A%20%20%20%20%0A%20%20%20%20%23%20%E9%81%8D%E5%8E%86%E5%93%88%E5%B8%8C%E8%A1%A8%0A%20%20%20%20%23%20%E9%81%8D%E5%8E%86%E9%94%AE%E5%80%BC%E5%AF%B9%20key-%3Evalue%0A%20%20%20%20for%20key,%20value%20in%20hmap.items%28%29%3A%0A%20%20%20%20%20%20%20%20print%28key,%20%22-%3E%22,%20value%29%0A%20%20%20%20%23%20%E5%8D%95%E7%8B%AC%E9%81%8D%E5%8E%86%E9%94%AE%20key%0A%20%20%20%20for%20key%20in%20hmap.keys%28%29%3A%0A%20%20%20%20%20%20%20%20print%28key%29%0A%20%20%20%20%23%20%E5%8D%95%E7%8B%AC%E9%81%8D%E5%8E%86%E5%80%BC%20value%0A%20%20%20%20for%20value%20in%20hmap.values%28%29%3A%0A%20%20%20%20%20%20%20%20print%28value%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=8&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%93%88%E5%B8%8C%E8%A1%A8%0A%20%20%20%20hmap%20%3D%20%7B%7D%0A%20%20%20%20%0A%20%20%20%20%23%20%E6%B7%BB%E5%8A%A0%E6%93%8D%E4%BD%9C%0A%20%20%20%20%23%20%E5%9C%A8%E5%93%88%E5%B8%8C%E8%A1%A8%E4%B8%AD%E6%B7%BB%E5%8A%A0%E9%94%AE%E5%80%BC%E5%AF%B9%20%28key,%20value%29%0A%20%20%20%20hmap%5B12836%5D%20%3D%20%22%E5%B0%8F%E5%93%88%22%0A%20%20%20%20hmap%5B15937%5D%20%3D%20%22%E5%B0%8F%E5%95%B0%22%0A%20%20%20%20hmap%5B16750%5D%20%3D%20%22%E5%B0%8F%E7%AE%97%22%0A%20%20%20%20hmap%5B13276%5D%20%3D%20%22%E5%B0%8F%E6%B3%95%22%0A%20%20%20%20hmap%5B10583%5D%20%3D%20%22%E5%B0%8F%E9%B8%AD%22%0A%20%20%20%20%0A%20%20%20%20%23%20%E9%81%8D%E5%8E%86%E5%93%88%E5%B8%8C%E8%A1%A8%0A%20%20%20%20%23%20%E9%81%8D%E5%8E%86%E9%94%AE%E5%80%BC%E5%AF%B9%20key-%3Evalue%0A%20%20%20%20for%20key,%20value%20in%20hmap.items%28%29%3A%0A%20%20%20%20%20%20%20%20print%28key,%20%22-%3E%22,%20value%29%0A%20%20%20%20%23%20%E5%8D%95%E7%8B%AC%E9%81%8D%E5%8E%86%E9%94%AE%20key%0A%20%20%20%20for%20key%20in%20hmap.keys%28%29%3A%0A%20%20%20%20%20%20%20%20print%28key%29%0A%20%20%20%20%23%20%E5%8D%95%E7%8B%AC%E9%81%8D%E5%8E%86%E5%80%BC%20value%0A%20%20%20%20for%20value%20in%20hmap.values%28%29%3A%0A%20%20%20%20%20%20%20%20print%28value%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=8&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20hash%20table%0A%20%20%20%20hmap%20%3D%20%7B%7D%0A%20%20%20%20%0A%20%20%20%20%23%20Insertion%0A%20%20%20%20%23%20Add%20key-value%20pairs%20%28key%2C%20value%29%20to%20the%20hash%20table%0A%20%20%20%20hmap%5B12836%5D%20%3D%20%22%E5%B0%8F%E5%93%88%22%0A%20%20%20%20hmap%5B15937%5D%20%3D%20%22%E5%B0%8F%E5%95%B0%22%0A%20%20%20%20hmap%5B16750%5D%20%3D%20%22%E5%B0%8F%E7%AE%97%22%0A%20%20%20%20hmap%5B13276%5D%20%3D%20%22%E5%B0%8F%E6%B3%95%22%0A%20%20%20%20hmap%5B10583%5D%20%3D%20%22%E5%B0%8F%E9%B8%AD%22%0A%20%20%20%20%0A%20%20%20%20%23%20Traverse%20the%20hash%20table%0A%20%20%20%20%23%20Traverse%20key-value%20pairs%3A%20key-%3Evalue%0A%20%20%20%20for%20key%2C%20value%20in%20hmap.items%28%29%3A%0A%20%20%20%20%20%20%20%20print%28key%2C%20%22-%3E%22%2C%20value%29%0A%20%20%20%20%23%20Traverse%20keys%20only%3A%20key%0A%20%20%20%20for%20key%20in%20hmap.keys%28%29%3A%0A%20%20%20%20%20%20%20%20print%28key%29%0A%20%20%20%20%23%20Traverse%20values%20only%3A%20value%0A%20%20%20%20for%20value%20in%20hmap.values%28%29%3A%0A%20%20%20%20%20%20%20%20print%28value%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=8&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20hash%20table%0A%20%20%20%20hmap%20%3D%20%7B%7D%0A%20%20%20%20%0A%20%20%20%20%23%20Insertion%0A%20%20%20%20%23%20Add%20key-value%20pairs%20%28key%2C%20value%29%20to%20the%20hash%20table%0A%20%20%20%20hmap%5B12836%5D%20%3D%20%22%E5%B0%8F%E5%93%88%22%0A%20%20%20%20hmap%5B15937%5D%20%3D%20%22%E5%B0%8F%E5%95%B0%22%0A%20%20%20%20hmap%5B16750%5D%20%3D%20%22%E5%B0%8F%E7%AE%97%22%0A%20%20%20%20hmap%5B13276%5D%20%3D%20%22%E5%B0%8F%E6%B3%95%22%0A%20%20%20%20hmap%5B10583%5D%20%3D%20%22%E5%B0%8F%E9%B8%AD%22%0A%20%20%20%20%0A%20%20%20%20%23%20Traverse%20the%20hash%20table%0A%20%20%20%20%23%20Traverse%20key-value%20pairs%3A%20key-%3Evalue%0A%20%20%20%20for%20key%2C%20value%20in%20hmap.items%28%29%3A%0A%20%20%20%20%20%20%20%20print%28key%2C%20%22-%3E%22%2C%20value%29%0A%20%20%20%20%23%20Traverse%20keys%20only%3A%20key%0A%20%20%20%20for%20key%20in%20hmap.keys%28%29%3A%0A%20%20%20%20%20%20%20%20print%28key%29%0A%20%20%20%20%23%20Traverse%20values%20only%3A%20value%0A%20%20%20%20for%20value%20in%20hmap.values%28%29%3A%0A%20%20%20%20%20%20%20%20print%28value%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=8&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<h2 id="612-simple-hash-table-implementation">6.1.2 &nbsp; Simple Hash Table Implementation<a class="headerlink" href="#612-simple-hash-table-implementation" title="Permanent link">&para;</a></h2>
-<p>Let's start with the simplest case: <strong>implementing a hash table with just an array</strong>. In a hash table, each empty slot in the array is called a <u>bucket</u>, and each bucket can store one key-value pair. A lookup therefore consists of finding the bucket for <code>key</code> and reading the <code>value</code> stored there.</p>
-<p>So how do we find the right bucket for a given <code>key</code>? We do this with a <u>hash function</u>. A hash function maps a larger input space to a smaller output space. In a hash table, the input space is the set of all <code>key</code>s, and the output space is the set of all buckets (array indices). In other words, given a <code>key</code>, <strong>the hash function tells us where the corresponding key-value pair should be stored in the array</strong>.</p>
-<p>Given a <code>key</code>, computing the bucket index involves the following two steps:</p>
+<h2 id="612-a-simple-implementation">6.1.2 &nbsp; A simple implementation<a class="headerlink" href="#612-a-simple-implementation" title="Permanent link">&para;</a></h2>
+<p>Start with the simplest case: <strong>implement a hash table using just one array</strong>. Each slot is called a <u>bucket</u> and holds one key-value pair. A lookup finds the bucket corresponding to <code>key</code> and retrieves its <code>value</code>.</p>
+<p>How do we locate the bucket for a key? A <u>hash function</u> maps a larger input space into a smaller output space. Here, the inputs are all possible keys, and the outputs are bucket indices. <strong>Given a key, the hash function tells us where its key-value pair is stored in the array</strong>.</p>
+<p>For an input <code>key</code>, the calculation has two steps.</p>
 <ol>
-<li>Use a hash algorithm <code>hash()</code> to compute a hash value.</li>
-<li>Take that hash value modulo the number of buckets (array length), <code>capacity</code>, to obtain the bucket (array index) <code>index</code> corresponding to the <code>key</code>.</li>
+<li>Use a hash algorithm, <code>hash()</code>, to calculate a hash value.</li>
+<li>Take that value modulo the number of buckets, the array length <code>capacity</code>, to obtain the bucket index <code>index</code>.</li>
 </ol>
 <div class="highlight"><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span>hash<span class="o">(</span>key<span class="o">)</span><span class="w"> </span>%<span class="w"> </span>capacity
 </code></pre></div>
-<p>We can then use <code>index</code> to access the corresponding bucket in the hash table and retrieve the <code>value</code>.</p>
-<p>Suppose the array length is <code>capacity = 100</code> and the hash algorithm is <code>hash(key) = key</code>. Then the hash function is <code>key % 100</code>. Figure 6-2 illustrates how this hash function works, using student ID as <code>key</code> and name as <code>value</code>.</p>
-<p><img alt="Working principle of hash function" class="animation-figure" src="../hash_map.assets/hash_function.png" /></p>
-<p align="center"> Figure 6-2 &nbsp; Working principle of hash function </p>
+<p>Use <code>index</code> to access the bucket and retrieve <code>value</code>.</p>
+<p>If <code>capacity = 100</code> and <code>hash(key) = key</code>, the resulting hash function is <code>key % 100</code>. Figure 6-2 illustrates it with student IDs as keys and names as values.</p>
+<p><img alt="How a hash function works" class="animation-figure" src="../hash_map.assets/hash_function.png" /></p>
+<p align="center"> Figure 6-2 &nbsp; How a hash function works </p>
 
-<p>The following code implements a simple hash table. Here, we encapsulate <code>key</code> and <code>value</code> into a class <code>Pair</code> to represent a key-value pair.</p>
+<p>The code below implements a simple hash table. The <code>Pair</code> class groups a <code>key</code> and <code>value</code> into a key-value pair.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -5224,11 +5296,11 @@
 <a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">val</span> <span class="o">=</span> <span class="n">val</span>
 <a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a>
 <a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayHashMap</span><span class="p">:</span>
-<a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">    </span><span class="sd">"""Hash table based on array implementation"""</span>
+<a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">    </span><span class="sd">"""Hash table implemented with an array"""</span>
 <a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a>
 <a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a>    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
 <a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="w">        </span><span class="sd">"""Constructor"""</span>
-<a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a>        <span class="c1"># Initialize array with 100 buckets</span>
+<a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a>        <span class="c1"># Initialize an array containing 100 buckets</span>
 <a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="n">Pair</span> <span class="o">|</span> <span class="kc">None</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="kc">None</span><span class="p">]</span> <span class="o">*</span> <span class="mi">100</span>
 <a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a>
 <a id="__codelineno-27-16" name="__codelineno-27-16" href="#__codelineno-27-16"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">hash_func</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
@@ -5237,7 +5309,7 @@
 <a id="__codelineno-27-19" name="__codelineno-27-19" href="#__codelineno-27-19"></a>        <span class="k">return</span> <span class="n">index</span>
 <a id="__codelineno-27-20" name="__codelineno-27-20" href="#__codelineno-27-20"></a>
 <a id="__codelineno-27-21" name="__codelineno-27-21" href="#__codelineno-27-21"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">str</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-27-22" name="__codelineno-27-22" href="#__codelineno-27-22"></a><span class="w">        </span><span class="sd">"""Query operation"""</span>
+<a id="__codelineno-27-22" name="__codelineno-27-22" href="#__codelineno-27-22"></a><span class="w">        </span><span class="sd">"""Lookup operation"""</span>
 <a id="__codelineno-27-23" name="__codelineno-27-23" href="#__codelineno-27-23"></a>        <span class="n">index</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-27-24" name="__codelineno-27-24" href="#__codelineno-27-24"></a>        <span class="n">pair</span><span class="p">:</span> <span class="n">Pair</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
 <a id="__codelineno-27-25" name="__codelineno-27-25" href="#__codelineno-27-25"></a>        <span class="k">if</span> <span class="n">pair</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
@@ -5245,7 +5317,7 @@
 <a id="__codelineno-27-27" name="__codelineno-27-27" href="#__codelineno-27-27"></a>        <span class="k">return</span> <span class="n">pair</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-27-28" name="__codelineno-27-28" href="#__codelineno-27-28"></a>
 <a id="__codelineno-27-29" name="__codelineno-27-29" href="#__codelineno-27-29"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">str</span><span class="p">):</span>
-<a id="__codelineno-27-30" name="__codelineno-27-30" href="#__codelineno-27-30"></a><span class="w">        </span><span class="sd">"""Add and update operation"""</span>
+<a id="__codelineno-27-30" name="__codelineno-27-30" href="#__codelineno-27-30"></a><span class="w">        </span><span class="sd">"""Insert and update operations"""</span>
 <a id="__codelineno-27-31" name="__codelineno-27-31" href="#__codelineno-27-31"></a>        <span class="n">pair</span> <span class="o">=</span> <span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-27-32" name="__codelineno-27-32" href="#__codelineno-27-32"></a>        <span class="n">index</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-27-33" name="__codelineno-27-33" href="#__codelineno-27-33"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="n">pair</span>
@@ -5253,7 +5325,7 @@
 <a id="__codelineno-27-35" name="__codelineno-27-35" href="#__codelineno-27-35"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-27-36" name="__codelineno-27-36" href="#__codelineno-27-36"></a><span class="w">        </span><span class="sd">"""Remove operation"""</span>
 <a id="__codelineno-27-37" name="__codelineno-27-37" href="#__codelineno-27-37"></a>        <span class="n">index</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-27-38" name="__codelineno-27-38" href="#__codelineno-27-38"></a>        <span class="c1"># Set to None to represent removal</span>
+<a id="__codelineno-27-38" name="__codelineno-27-38" href="#__codelineno-27-38"></a>        <span class="c1"># Set to None to mark removal</span>
 <a id="__codelineno-27-39" name="__codelineno-27-39" href="#__codelineno-27-39"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span>
 <a id="__codelineno-27-40" name="__codelineno-27-40" href="#__codelineno-27-40"></a>
 <a id="__codelineno-27-41" name="__codelineno-27-41" href="#__codelineno-27-41"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">entry_set</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="n">Pair</span><span class="p">]:</span>
@@ -5281,7 +5353,7 @@
 <a id="__codelineno-27-63" name="__codelineno-27-63" href="#__codelineno-27-63"></a>        <span class="k">return</span> <span class="n">result</span>
 <a id="__codelineno-27-64" name="__codelineno-27-64" href="#__codelineno-27-64"></a>
 <a id="__codelineno-27-65" name="__codelineno-27-65" href="#__codelineno-27-65"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">print</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-<a id="__codelineno-27-66" name="__codelineno-27-66" href="#__codelineno-27-66"></a><span class="w">        </span><span class="sd">"""Print hash table"""</span>
+<a id="__codelineno-27-66" name="__codelineno-27-66" href="#__codelineno-27-66"></a><span class="w">        </span><span class="sd">"""Print the hash table"""</span>
 <a id="__codelineno-27-67" name="__codelineno-27-67" href="#__codelineno-27-67"></a>        <span class="k">for</span> <span class="n">pair</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">:</span>
 <a id="__codelineno-27-68" name="__codelineno-27-68" href="#__codelineno-27-68"></a>            <span class="k">if</span> <span class="n">pair</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-27-69" name="__codelineno-27-69" href="#__codelineno-27-69"></a>                <span class="nb">print</span><span class="p">(</span><span class="n">pair</span><span class="o">.</span><span class="n">key</span><span class="p">,</span> <span class="s2">"-&gt;"</span><span class="p">,</span> <span class="n">pair</span><span class="o">.</span><span class="n">val</span><span class="p">)</span>
@@ -5299,14 +5371,14 @@
 <a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="p">};</span>
 <a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a>
-<a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="cm">/* Hash table based on array implementation */</span>
+<a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="cm">/* Hash table implemented with an array */</span>
 <a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayHashMap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a><span class="w">  </span><span class="k">private</span><span class="o">:</span>
 <a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">Pair</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span>
 <a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a>
 <a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a><span class="w">  </span><span class="k">public</span><span class="o">:</span>
 <a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a><span class="w">    </span><span class="n">ArrayHashMap</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-19" name="__codelineno-28-19" href="#__codelineno-28-19"></a><span class="w">        </span><span class="c1">// Initialize array with 100 buckets</span>
+<a id="__codelineno-28-19" name="__codelineno-28-19" href="#__codelineno-28-19"></a><span class="w">        </span><span class="c1">// Initialize an array containing 100 buckets</span>
 <a id="__codelineno-28-20" name="__codelineno-28-20" href="#__codelineno-28-20"></a><span class="w">        </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">Pair</span><span class="w"> </span><span class="o">*&gt;</span><span class="p">(</span><span class="mi">100</span><span class="p">);</span>
 <a id="__codelineno-28-21" name="__codelineno-28-21" href="#__codelineno-28-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-22" name="__codelineno-28-22" href="#__codelineno-28-22"></a>
@@ -5324,7 +5396,7 @@
 <a id="__codelineno-28-34" name="__codelineno-28-34" href="#__codelineno-28-34"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span>
 <a id="__codelineno-28-35" name="__codelineno-28-35" href="#__codelineno-28-35"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-36" name="__codelineno-28-36" href="#__codelineno-28-36"></a>
-<a id="__codelineno-28-37" name="__codelineno-28-37" href="#__codelineno-28-37"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-28-37" name="__codelineno-28-37" href="#__codelineno-28-37"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-28-38" name="__codelineno-28-38" href="#__codelineno-28-38"></a><span class="w">    </span><span class="n">string</span><span class="w"> </span><span class="n">get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-39" name="__codelineno-28-39" href="#__codelineno-28-39"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-28-40" name="__codelineno-28-40" href="#__codelineno-28-40"></a><span class="w">        </span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
@@ -5333,7 +5405,7 @@
 <a id="__codelineno-28-43" name="__codelineno-28-43" href="#__codelineno-28-43"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pair</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-28-44" name="__codelineno-28-44" href="#__codelineno-28-44"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-45" name="__codelineno-28-45" href="#__codelineno-28-45"></a>
-<a id="__codelineno-28-46" name="__codelineno-28-46" href="#__codelineno-28-46"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-28-46" name="__codelineno-28-46" href="#__codelineno-28-46"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-28-47" name="__codelineno-28-47" href="#__codelineno-28-47"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">put</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-48" name="__codelineno-28-48" href="#__codelineno-28-48"></a><span class="w">        </span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-28-49" name="__codelineno-28-49" href="#__codelineno-28-49"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
@@ -5343,7 +5415,7 @@
 <a id="__codelineno-28-53" name="__codelineno-28-53" href="#__codelineno-28-53"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-28-54" name="__codelineno-28-54" href="#__codelineno-28-54"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-55" name="__codelineno-28-55" href="#__codelineno-28-55"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-28-56" name="__codelineno-28-56" href="#__codelineno-28-56"></a><span class="w">        </span><span class="c1">// Free memory and set to nullptr</span>
+<a id="__codelineno-28-56" name="__codelineno-28-56" href="#__codelineno-28-56"></a><span class="w">        </span><span class="c1">// Free the memory and set the pointer to nullptr</span>
 <a id="__codelineno-28-57" name="__codelineno-28-57" href="#__codelineno-28-57"></a><span class="w">        </span><span class="k">delete</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
 <a id="__codelineno-28-58" name="__codelineno-28-58" href="#__codelineno-28-58"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">;</span>
 <a id="__codelineno-28-59" name="__codelineno-28-59" href="#__codelineno-28-59"></a><span class="w">    </span><span class="p">}</span>
@@ -5381,7 +5453,7 @@
 <a id="__codelineno-28-91" name="__codelineno-28-91" href="#__codelineno-28-91"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">valueSet</span><span class="p">;</span>
 <a id="__codelineno-28-92" name="__codelineno-28-92" href="#__codelineno-28-92"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-93" name="__codelineno-28-93" href="#__codelineno-28-93"></a>
-<a id="__codelineno-28-94" name="__codelineno-28-94" href="#__codelineno-28-94"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-28-94" name="__codelineno-28-94" href="#__codelineno-28-94"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-28-95" name="__codelineno-28-95" href="#__codelineno-28-95"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-96" name="__codelineno-28-96" href="#__codelineno-28-96"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">kv</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">pairSet</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-97" name="__codelineno-28-97" href="#__codelineno-28-97"></a><span class="w">            </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">kv</span><span class="o">-&gt;</span><span class="n">key</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">" -&gt; "</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">kv</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
@@ -5402,12 +5474,12 @@
 <a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="p">}</span>
 <a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a>
-<a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="cm">/* Hash table based on array implementation */</span>
+<a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="cm">/* Hash table implemented with an array */</span>
 <a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="kd">class</span> <span class="nc">ArrayHashMap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span>
 <a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a>
 <a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="nf">ArrayHashMap</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></a><span class="w">        </span><span class="c1">// Initialize array with 100 buckets</span>
+<a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></a><span class="w">        </span><span class="c1">// Initialize an array containing 100 buckets</span>
 <a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a><span class="w">        </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-29-19" name="__codelineno-29-19" href="#__codelineno-29-19"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">100</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-20" name="__codelineno-29-20" href="#__codelineno-29-20"></a><span class="w">            </span><span class="n">buckets</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="kc">null</span><span class="p">);</span>
@@ -5420,7 +5492,7 @@
 <a id="__codelineno-29-27" name="__codelineno-29-27" href="#__codelineno-29-27"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span>
 <a id="__codelineno-29-28" name="__codelineno-29-28" href="#__codelineno-29-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-29" name="__codelineno-29-29" href="#__codelineno-29-29"></a>
-<a id="__codelineno-29-30" name="__codelineno-29-30" href="#__codelineno-29-30"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-29-30" name="__codelineno-29-30" href="#__codelineno-29-30"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-29-31" name="__codelineno-29-31" href="#__codelineno-29-31"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-32" name="__codelineno-29-32" href="#__codelineno-29-32"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-29-33" name="__codelineno-29-33" href="#__codelineno-29-33"></a><span class="w">        </span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">index</span><span class="p">);</span>
@@ -5429,7 +5501,7 @@
 <a id="__codelineno-29-36" name="__codelineno-29-36" href="#__codelineno-29-36"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="na">val</span><span class="p">;</span>
 <a id="__codelineno-29-37" name="__codelineno-29-37" href="#__codelineno-29-37"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-38" name="__codelineno-29-38" href="#__codelineno-29-38"></a>
-<a id="__codelineno-29-39" name="__codelineno-29-39" href="#__codelineno-29-39"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-29-39" name="__codelineno-29-39" href="#__codelineno-29-39"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-29-40" name="__codelineno-29-40" href="#__codelineno-29-40"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-41" name="__codelineno-29-41" href="#__codelineno-29-41"></a><span class="w">        </span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-29-42" name="__codelineno-29-42" href="#__codelineno-29-42"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
@@ -5439,7 +5511,7 @@
 <a id="__codelineno-29-46" name="__codelineno-29-46" href="#__codelineno-29-46"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-29-47" name="__codelineno-29-47" href="#__codelineno-29-47"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-48" name="__codelineno-29-48" href="#__codelineno-29-48"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-29-49" name="__codelineno-29-49" href="#__codelineno-29-49"></a><span class="w">        </span><span class="c1">// Set to null to represent deletion</span>
+<a id="__codelineno-29-49" name="__codelineno-29-49" href="#__codelineno-29-49"></a><span class="w">        </span><span class="c1">// Set the bucket to null to mark it as removed</span>
 <a id="__codelineno-29-50" name="__codelineno-29-50" href="#__codelineno-29-50"></a><span class="w">        </span><span class="n">buckets</span><span class="p">.</span><span class="na">set</span><span class="p">(</span><span class="n">index</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">);</span>
 <a id="__codelineno-29-51" name="__codelineno-29-51" href="#__codelineno-29-51"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-52" name="__codelineno-29-52" href="#__codelineno-29-52"></a>
@@ -5473,7 +5545,7 @@
 <a id="__codelineno-29-80" name="__codelineno-29-80" href="#__codelineno-29-80"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">valueSet</span><span class="p">;</span>
 <a id="__codelineno-29-81" name="__codelineno-29-81" href="#__codelineno-29-81"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-82" name="__codelineno-29-82" href="#__codelineno-29-82"></a>
-<a id="__codelineno-29-83" name="__codelineno-29-83" href="#__codelineno-29-83"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-29-83" name="__codelineno-29-83" href="#__codelineno-29-83"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-29-84" name="__codelineno-29-84" href="#__codelineno-29-84"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-85" name="__codelineno-29-85" href="#__codelineno-29-85"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">kv</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">pairSet</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-86" name="__codelineno-29-86" href="#__codelineno-29-86"></a><span class="w">            </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="n">kv</span><span class="p">.</span><span class="na">key</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="s">" -&gt; "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">kv</span><span class="p">.</span><span class="na">val</span><span class="p">);</span>
@@ -5483,17 +5555,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_hash_map.cs</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Key-value pair int-&gt;string */</span>
+<div class="highlight"><span class="filename">array_hash_map.cs</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Key-value pair: int-&gt;string */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="k">class</span><span class="w"> </span><span class="nf">Pair</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">key</span><span class="p">;</span>
 <a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="p">}</span>
 <a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a>
-<a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="cm">/* Hash table based on array implementation */</span>
+<a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="cm">/* Hash table implemented with an array */</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayHashMap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">?&gt;</span><span class="w"> </span><span class="n">buckets</span><span class="p">;</span>
 <a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nf">ArrayHashMap</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">        </span><span class="c1">// Initialize array with 100 buckets</span>
+<a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">        </span><span class="c1">// Initialize an array containing 100 buckets</span>
 <a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="w">        </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">100</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="w">            </span><span class="n">buckets</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="k">null</span><span class="p">);</span>
@@ -5506,7 +5578,7 @@
 <a id="__codelineno-30-21" name="__codelineno-30-21" href="#__codelineno-30-21"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span>
 <a id="__codelineno-30-22" name="__codelineno-30-22" href="#__codelineno-30-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-23" name="__codelineno-30-23" href="#__codelineno-30-23"></a>
-<a id="__codelineno-30-24" name="__codelineno-30-24" href="#__codelineno-30-24"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-30-24" name="__codelineno-30-24" href="#__codelineno-30-24"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-30-25" name="__codelineno-30-25" href="#__codelineno-30-25"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">string?</span><span class="w"> </span><span class="n">Get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-26" name="__codelineno-30-26" href="#__codelineno-30-26"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-30-27" name="__codelineno-30-27" href="#__codelineno-30-27"></a><span class="w">        </span><span class="n">Pair</span><span class="o">?</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
@@ -5514,7 +5586,7 @@
 <a id="__codelineno-30-29" name="__codelineno-30-29" href="#__codelineno-30-29"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-30-30" name="__codelineno-30-30" href="#__codelineno-30-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-31" name="__codelineno-30-31" href="#__codelineno-30-31"></a>
-<a id="__codelineno-30-32" name="__codelineno-30-32" href="#__codelineno-30-32"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-30-32" name="__codelineno-30-32" href="#__codelineno-30-32"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-30-33" name="__codelineno-30-33" href="#__codelineno-30-33"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">Put</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-34" name="__codelineno-30-34" href="#__codelineno-30-34"></a><span class="w">        </span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-30-35" name="__codelineno-30-35" href="#__codelineno-30-35"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
@@ -5524,7 +5596,7 @@
 <a id="__codelineno-30-39" name="__codelineno-30-39" href="#__codelineno-30-39"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-30-40" name="__codelineno-30-40" href="#__codelineno-30-40"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">Remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-41" name="__codelineno-30-41" href="#__codelineno-30-41"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-30-42" name="__codelineno-30-42" href="#__codelineno-30-42"></a><span class="w">        </span><span class="c1">// Set to null to represent deletion</span>
+<a id="__codelineno-30-42" name="__codelineno-30-42" href="#__codelineno-30-42"></a><span class="w">        </span><span class="c1">// Set the bucket to null to mark it as removed</span>
 <a id="__codelineno-30-43" name="__codelineno-30-43" href="#__codelineno-30-43"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
 <a id="__codelineno-30-44" name="__codelineno-30-44" href="#__codelineno-30-44"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-45" name="__codelineno-30-45" href="#__codelineno-30-45"></a>
@@ -5558,7 +5630,7 @@
 <a id="__codelineno-30-73" name="__codelineno-30-73" href="#__codelineno-30-73"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">valueSet</span><span class="p">;</span>
 <a id="__codelineno-30-74" name="__codelineno-30-74" href="#__codelineno-30-74"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-75" name="__codelineno-30-75" href="#__codelineno-30-75"></a>
-<a id="__codelineno-30-76" name="__codelineno-30-76" href="#__codelineno-30-76"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-30-76" name="__codelineno-30-76" href="#__codelineno-30-76"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-30-77" name="__codelineno-30-77" href="#__codelineno-30-77"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">Print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-78" name="__codelineno-30-78" href="#__codelineno-30-78"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="n">kv</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">PairSet</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-79" name="__codelineno-30-79" href="#__codelineno-30-79"></a><span class="w">            </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="n">kv</span><span class="p">.</span><span class="n">key</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="s">" -&gt; "</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">kv</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
@@ -5574,14 +5646,14 @@
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="nx">val</span><span class="w"> </span><span class="kt">string</span>
 <a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="p">}</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a>
-<a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="cm">/* Hash table based on array implementation */</span>
+<a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="cm">/* Hash table implemented with an array */</span>
 <a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="kd">type</span><span class="w"> </span><span class="nx">arrayHashMap</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">    </span><span class="nx">buckets</span><span class="w"> </span><span class="p">[]</span><span class="o">*</span><span class="nx">pair</span>
 <a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="p">}</span>
 <a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a>
-<a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="cm">/* Initialize hash table */</span>
+<a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="cm">/* Initialize the hash table */</span>
 <a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a><span class="kd">func</span><span class="w"> </span><span class="nx">newArrayHashMap</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayHashMap</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="w">    </span><span class="c1">// Initialize array with 100 buckets</span>
+<a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="w">    </span><span class="c1">// Initialize an array containing 100 buckets</span>
 <a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="w">    </span><span class="nx">buckets</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="o">*</span><span class="nx">pair</span><span class="p">,</span><span class="w"> </span><span class="mi">100</span><span class="p">)</span>
 <a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">arrayHashMap</span><span class="p">{</span><span class="nx">buckets</span><span class="p">:</span><span class="w"> </span><span class="nx">buckets</span><span class="p">}</span>
 <a id="__codelineno-31-17" name="__codelineno-31-17" href="#__codelineno-31-17"></a><span class="p">}</span>
@@ -5592,7 +5664,7 @@
 <a id="__codelineno-31-22" name="__codelineno-31-22" href="#__codelineno-31-22"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">index</span>
 <a id="__codelineno-31-23" name="__codelineno-31-23" href="#__codelineno-31-23"></a><span class="p">}</span>
 <a id="__codelineno-31-24" name="__codelineno-31-24" href="#__codelineno-31-24"></a>
-<a id="__codelineno-31-25" name="__codelineno-31-25" href="#__codelineno-31-25"></a><span class="cm">/* Query operation */</span>
+<a id="__codelineno-31-25" name="__codelineno-31-25" href="#__codelineno-31-25"></a><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-31-26" name="__codelineno-31-26" href="#__codelineno-31-26"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">a</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayHashMap</span><span class="p">)</span><span class="w"> </span><span class="nx">get</span><span class="p">(</span><span class="nx">key</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-27" name="__codelineno-31-27" href="#__codelineno-31-27"></a><span class="w">    </span><span class="nx">index</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">a</span><span class="p">.</span><span class="nx">hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span>
 <a id="__codelineno-31-28" name="__codelineno-31-28" href="#__codelineno-31-28"></a><span class="w">    </span><span class="nx">pair</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">a</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span>
@@ -5602,7 +5674,7 @@
 <a id="__codelineno-31-32" name="__codelineno-31-32" href="#__codelineno-31-32"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">pair</span><span class="p">.</span><span class="nx">val</span>
 <a id="__codelineno-31-33" name="__codelineno-31-33" href="#__codelineno-31-33"></a><span class="p">}</span>
 <a id="__codelineno-31-34" name="__codelineno-31-34" href="#__codelineno-31-34"></a>
-<a id="__codelineno-31-35" name="__codelineno-31-35" href="#__codelineno-31-35"></a><span class="cm">/* Add operation */</span>
+<a id="__codelineno-31-35" name="__codelineno-31-35" href="#__codelineno-31-35"></a><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-31-36" name="__codelineno-31-36" href="#__codelineno-31-36"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">a</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayHashMap</span><span class="p">)</span><span class="w"> </span><span class="nx">put</span><span class="p">(</span><span class="nx">key</span><span class="w"> </span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-37" name="__codelineno-31-37" href="#__codelineno-31-37"></a><span class="w">    </span><span class="nx">pair</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">pair</span><span class="p">{</span><span class="nx">key</span><span class="p">:</span><span class="w"> </span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">:</span><span class="w"> </span><span class="nx">val</span><span class="p">}</span>
 <a id="__codelineno-31-38" name="__codelineno-31-38" href="#__codelineno-31-38"></a><span class="w">    </span><span class="nx">index</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">a</span><span class="p">.</span><span class="nx">hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span>
@@ -5612,11 +5684,11 @@
 <a id="__codelineno-31-42" name="__codelineno-31-42" href="#__codelineno-31-42"></a><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-31-43" name="__codelineno-31-43" href="#__codelineno-31-43"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">a</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayHashMap</span><span class="p">)</span><span class="w"> </span><span class="nx">remove</span><span class="p">(</span><span class="nx">key</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-44" name="__codelineno-31-44" href="#__codelineno-31-44"></a><span class="w">    </span><span class="nx">index</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">a</span><span class="p">.</span><span class="nx">hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span>
-<a id="__codelineno-31-45" name="__codelineno-31-45" href="#__codelineno-31-45"></a><span class="w">    </span><span class="c1">// Set to nil to delete</span>
+<a id="__codelineno-31-45" name="__codelineno-31-45" href="#__codelineno-31-45"></a><span class="w">    </span><span class="c1">// Set to nil to mark the entry as removed</span>
 <a id="__codelineno-31-46" name="__codelineno-31-46" href="#__codelineno-31-46"></a><span class="w">    </span><span class="nx">a</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-31-47" name="__codelineno-31-47" href="#__codelineno-31-47"></a><span class="p">}</span>
 <a id="__codelineno-31-48" name="__codelineno-31-48" href="#__codelineno-31-48"></a>
-<a id="__codelineno-31-49" name="__codelineno-31-49" href="#__codelineno-31-49"></a><span class="cm">/* Get all key pairs */</span>
+<a id="__codelineno-31-49" name="__codelineno-31-49" href="#__codelineno-31-49"></a><span class="cm">/* Get all key-value pairs */</span>
 <a id="__codelineno-31-50" name="__codelineno-31-50" href="#__codelineno-31-50"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">a</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayHashMap</span><span class="p">)</span><span class="w"> </span><span class="nx">pairSet</span><span class="p">()</span><span class="w"> </span><span class="p">[]</span><span class="o">*</span><span class="nx">pair</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-51" name="__codelineno-31-51" href="#__codelineno-31-51"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nx">pairs</span><span class="w"> </span><span class="p">[]</span><span class="o">*</span><span class="nx">pair</span>
 <a id="__codelineno-31-52" name="__codelineno-31-52" href="#__codelineno-31-52"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">a</span><span class="p">.</span><span class="nx">buckets</span><span class="w"> </span><span class="p">{</span>
@@ -5649,7 +5721,7 @@
 <a id="__codelineno-31-79" name="__codelineno-31-79" href="#__codelineno-31-79"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">values</span>
 <a id="__codelineno-31-80" name="__codelineno-31-80" href="#__codelineno-31-80"></a><span class="p">}</span>
 <a id="__codelineno-31-81" name="__codelineno-31-81" href="#__codelineno-31-81"></a>
-<a id="__codelineno-31-82" name="__codelineno-31-82" href="#__codelineno-31-82"></a><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-31-82" name="__codelineno-31-82" href="#__codelineno-31-82"></a><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-31-83" name="__codelineno-31-83" href="#__codelineno-31-83"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">a</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayHashMap</span><span class="p">)</span><span class="w"> </span><span class="nb">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-84" name="__codelineno-31-84" href="#__codelineno-31-84"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">a</span><span class="p">.</span><span class="nx">buckets</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-85" name="__codelineno-31-85" href="#__codelineno-31-85"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
@@ -5675,12 +5747,12 @@
 <a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="p">}</span>
 <a id="__codelineno-32-15" name="__codelineno-32-15" href="#__codelineno-32-15"></a>
-<a id="__codelineno-32-16" name="__codelineno-32-16" href="#__codelineno-32-16"></a><span class="cm">/* Hash table based on array implementation */</span>
+<a id="__codelineno-32-16" name="__codelineno-32-16" href="#__codelineno-32-16"></a><span class="cm">/* Hash table implemented with an array */</span>
 <a id="__codelineno-32-17" name="__codelineno-32-17" href="#__codelineno-32-17"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayHashMap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-18" name="__codelineno-32-18" href="#__codelineno-32-18"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">buckets</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="n">Pair</span><span class="p">?]</span>
 <a id="__codelineno-32-19" name="__codelineno-32-19" href="#__codelineno-32-19"></a>
 <a id="__codelineno-32-20" name="__codelineno-32-20" href="#__codelineno-32-20"></a><span class="w">    </span><span class="kd">init</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-21" name="__codelineno-32-21" href="#__codelineno-32-21"></a><span class="w">        </span><span class="c1">// Initialize array with 100 buckets</span>
+<a id="__codelineno-32-21" name="__codelineno-32-21" href="#__codelineno-32-21"></a><span class="w">        </span><span class="c1">// Initialize an array containing 100 buckets</span>
 <a id="__codelineno-32-22" name="__codelineno-32-22" href="#__codelineno-32-22"></a><span class="w">        </span><span class="n">buckets</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="mi">100</span><span class="p">)</span>
 <a id="__codelineno-32-23" name="__codelineno-32-23" href="#__codelineno-32-23"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-24" name="__codelineno-32-24" href="#__codelineno-32-24"></a>
@@ -5690,14 +5762,14 @@
 <a id="__codelineno-32-28" name="__codelineno-32-28" href="#__codelineno-32-28"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">index</span>
 <a id="__codelineno-32-29" name="__codelineno-32-29" href="#__codelineno-32-29"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-30" name="__codelineno-32-30" href="#__codelineno-32-30"></a>
-<a id="__codelineno-32-31" name="__codelineno-32-31" href="#__codelineno-32-31"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-32-31" name="__codelineno-32-31" href="#__codelineno-32-31"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-32-32" name="__codelineno-32-32" href="#__codelineno-32-32"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">String</span><span class="p">?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-33" name="__codelineno-32-33" href="#__codelineno-32-33"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-32-34" name="__codelineno-32-34" href="#__codelineno-32-34"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">pair</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
 <a id="__codelineno-32-35" name="__codelineno-32-35" href="#__codelineno-32-35"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pair</span><span class="p">?.</span><span class="n">val</span>
 <a id="__codelineno-32-36" name="__codelineno-32-36" href="#__codelineno-32-36"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-37" name="__codelineno-32-37" href="#__codelineno-32-37"></a>
-<a id="__codelineno-32-38" name="__codelineno-32-38" href="#__codelineno-32-38"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-32-38" name="__codelineno-32-38" href="#__codelineno-32-38"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-32-39" name="__codelineno-32-39" href="#__codelineno-32-39"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="nb">String</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-40" name="__codelineno-32-40" href="#__codelineno-32-40"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">pair</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-32-41" name="__codelineno-32-41" href="#__codelineno-32-41"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">key</span><span class="p">)</span>
@@ -5707,7 +5779,7 @@
 <a id="__codelineno-32-45" name="__codelineno-32-45" href="#__codelineno-32-45"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-32-46" name="__codelineno-32-46" href="#__codelineno-32-46"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-47" name="__codelineno-32-47" href="#__codelineno-32-47"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-32-48" name="__codelineno-32-48" href="#__codelineno-32-48"></a><span class="w">        </span><span class="c1">// Set to nil to delete</span>
+<a id="__codelineno-32-48" name="__codelineno-32-48" href="#__codelineno-32-48"></a><span class="w">        </span><span class="c1">// Set to nil to mark the entry as removed</span>
 <a id="__codelineno-32-49" name="__codelineno-32-49" href="#__codelineno-32-49"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-32-50" name="__codelineno-32-50" href="#__codelineno-32-50"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-51" name="__codelineno-32-51" href="#__codelineno-32-51"></a>
@@ -5726,7 +5798,7 @@
 <a id="__codelineno-32-64" name="__codelineno-32-64" href="#__codelineno-32-64"></a><span class="w">        </span><span class="n">buckets</span><span class="p">.</span><span class="n">compactMap</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="nv">$0</span><span class="p">?.</span><span class="n">val</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-32-65" name="__codelineno-32-65" href="#__codelineno-32-65"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-66" name="__codelineno-32-66" href="#__codelineno-32-66"></a>
-<a id="__codelineno-32-67" name="__codelineno-32-67" href="#__codelineno-32-67"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-32-67" name="__codelineno-32-67" href="#__codelineno-32-67"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-32-68" name="__codelineno-32-68" href="#__codelineno-32-68"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-69" name="__codelineno-32-69" href="#__codelineno-32-69"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">pairSet</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-70" name="__codelineno-32-70" href="#__codelineno-32-70"></a><span class="w">            </span><span class="n">Swift</span><span class="p">.</span><span class="bp">print</span><span class="p">(</span><span class="s">"</span><span class="si">\(</span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="si">)</span><span class="s"> -&gt; </span><span class="si">\(</span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="si">)</span><span class="s">"</span><span class="p">)</span>
@@ -5736,7 +5808,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_hash_map.js</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Key-value pair Number -&gt; String */</span>
+<div class="highlight"><span class="filename">array_hash_map.js</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Key-value pair: Number -&gt; String */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">Pair</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">key</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">key</span><span class="p">;</span>
@@ -5744,11 +5816,11 @@
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="p">}</span>
 <a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a>
-<a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="cm">/* Hash table based on array implementation */</span>
+<a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="cm">/* Hash table implemented with an array */</span>
 <a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="kd">class</span><span class="w"> </span><span class="nx">ArrayHashMap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">    </span><span class="n">#buckets</span><span class="p">;</span>
 <a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">        </span><span class="c1">// Initialize array with 100 buckets</span>
+<a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">        </span><span class="c1">// Initialize an array containing 100 buckets</span>
 <a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">100</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">null</span><span class="p">);</span>
 <a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a>
@@ -5757,7 +5829,7 @@
 <a id="__codelineno-33-19" name="__codelineno-33-19" href="#__codelineno-33-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">key</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mf">100</span><span class="p">;</span>
 <a id="__codelineno-33-20" name="__codelineno-33-20" href="#__codelineno-33-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-21" name="__codelineno-33-21" href="#__codelineno-33-21"></a>
-<a id="__codelineno-33-22" name="__codelineno-33-22" href="#__codelineno-33-22"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-33-22" name="__codelineno-33-22" href="#__codelineno-33-22"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-33-23" name="__codelineno-33-23" href="#__codelineno-33-23"></a><span class="w">    </span><span class="nx">get</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-24" name="__codelineno-33-24" href="#__codelineno-33-24"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
 <a id="__codelineno-33-25" name="__codelineno-33-25" href="#__codelineno-33-25"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">];</span>
@@ -5765,7 +5837,7 @@
 <a id="__codelineno-33-27" name="__codelineno-33-27" href="#__codelineno-33-27"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">pair</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-33-28" name="__codelineno-33-28" href="#__codelineno-33-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-29" name="__codelineno-33-29" href="#__codelineno-33-29"></a>
-<a id="__codelineno-33-30" name="__codelineno-33-30" href="#__codelineno-33-30"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-33-30" name="__codelineno-33-30" href="#__codelineno-33-30"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-33-31" name="__codelineno-33-31" href="#__codelineno-33-31"></a><span class="w">    </span><span class="nx">set</span><span class="p">(</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-32" name="__codelineno-33-32" href="#__codelineno-33-32"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
 <a id="__codelineno-33-33" name="__codelineno-33-33" href="#__codelineno-33-33"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Pair</span><span class="p">(</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
@@ -5774,7 +5846,7 @@
 <a id="__codelineno-33-36" name="__codelineno-33-36" href="#__codelineno-33-36"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-33-37" name="__codelineno-33-37" href="#__codelineno-33-37"></a><span class="w">    </span><span class="ow">delete</span><span class="p">(</span><span class="nx">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-38" name="__codelineno-33-38" href="#__codelineno-33-38"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
-<a id="__codelineno-33-39" name="__codelineno-33-39" href="#__codelineno-33-39"></a><span class="w">        </span><span class="c1">// Set to null to represent deletion</span>
+<a id="__codelineno-33-39" name="__codelineno-33-39" href="#__codelineno-33-39"></a><span class="w">        </span><span class="c1">// Set the bucket to null to mark it as deleted</span>
 <a id="__codelineno-33-40" name="__codelineno-33-40" href="#__codelineno-33-40"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-33-41" name="__codelineno-33-41" href="#__codelineno-33-41"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-42" name="__codelineno-33-42" href="#__codelineno-33-42"></a>
@@ -5811,7 +5883,7 @@
 <a id="__codelineno-33-73" name="__codelineno-33-73" href="#__codelineno-33-73"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">arr</span><span class="p">;</span>
 <a id="__codelineno-33-74" name="__codelineno-33-74" href="#__codelineno-33-74"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-75" name="__codelineno-33-75" href="#__codelineno-33-75"></a>
-<a id="__codelineno-33-76" name="__codelineno-33-76" href="#__codelineno-33-76"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-33-76" name="__codelineno-33-76" href="#__codelineno-33-76"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-33-77" name="__codelineno-33-77" href="#__codelineno-33-77"></a><span class="w">    </span><span class="nx">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-78" name="__codelineno-33-78" href="#__codelineno-33-78"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">pairSet</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">entries</span><span class="p">();</span>
 <a id="__codelineno-33-79" name="__codelineno-33-79" href="#__codelineno-33-79"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">pairSet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5822,7 +5894,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_hash_map.ts</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Key-value pair Number -&gt; String */</span>
+<div class="highlight"><span class="filename">array_hash_map.ts</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Key-value pair: Number -&gt; String */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">Pair</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nx">key</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span>
 <a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">;</span>
@@ -5833,12 +5905,12 @@
 <a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="p">}</span>
 <a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a>
-<a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="cm">/* Hash table based on array implementation */</span>
+<a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="cm">/* Hash table implemented with an array */</span>
 <a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="kd">class</span><span class="w"> </span><span class="nx">ArrayHashMap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="k">readonly</span><span class="w"> </span><span class="nx">buckets</span><span class="o">:</span><span class="w"> </span><span class="p">(</span><span class="nx">Pair</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">)[];</span>
 <a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a>
 <a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-17" name="__codelineno-34-17" href="#__codelineno-34-17"></a><span class="w">        </span><span class="c1">// Initialize array with 100 buckets</span>
+<a id="__codelineno-34-17" name="__codelineno-34-17" href="#__codelineno-34-17"></a><span class="w">        </span><span class="c1">// Initialize an array containing 100 buckets</span>
 <a id="__codelineno-34-18" name="__codelineno-34-18" href="#__codelineno-34-18"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">100</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="kc">null</span><span class="p">);</span>
 <a id="__codelineno-34-19" name="__codelineno-34-19" href="#__codelineno-34-19"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-34-20" name="__codelineno-34-20" href="#__codelineno-34-20"></a>
@@ -5847,7 +5919,7 @@
 <a id="__codelineno-34-23" name="__codelineno-34-23" href="#__codelineno-34-23"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">key</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mf">100</span><span class="p">;</span>
 <a id="__codelineno-34-24" name="__codelineno-34-24" href="#__codelineno-34-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-34-25" name="__codelineno-34-25" href="#__codelineno-34-25"></a>
-<a id="__codelineno-34-26" name="__codelineno-34-26" href="#__codelineno-34-26"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-34-26" name="__codelineno-34-26" href="#__codelineno-34-26"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-34-27" name="__codelineno-34-27" href="#__codelineno-34-27"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nx">get</span><span class="p">(</span><span class="nx">key</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-28" name="__codelineno-34-28" href="#__codelineno-34-28"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
 <a id="__codelineno-34-29" name="__codelineno-34-29" href="#__codelineno-34-29"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">];</span>
@@ -5855,7 +5927,7 @@
 <a id="__codelineno-34-31" name="__codelineno-34-31" href="#__codelineno-34-31"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">pair</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-34-32" name="__codelineno-34-32" href="#__codelineno-34-32"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-34-33" name="__codelineno-34-33" href="#__codelineno-34-33"></a>
-<a id="__codelineno-34-34" name="__codelineno-34-34" href="#__codelineno-34-34"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-34-34" name="__codelineno-34-34" href="#__codelineno-34-34"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-34-35" name="__codelineno-34-35" href="#__codelineno-34-35"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nx">set</span><span class="p">(</span><span class="nx">key</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">string</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-36" name="__codelineno-34-36" href="#__codelineno-34-36"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
 <a id="__codelineno-34-37" name="__codelineno-34-37" href="#__codelineno-34-37"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">Pair</span><span class="p">(</span><span class="nx">key</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
@@ -5864,7 +5936,7 @@
 <a id="__codelineno-34-40" name="__codelineno-34-40" href="#__codelineno-34-40"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-34-41" name="__codelineno-34-41" href="#__codelineno-34-41"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="ow">delete</span><span class="p">(</span><span class="nx">key</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-42" name="__codelineno-34-42" href="#__codelineno-34-42"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">hashFunc</span><span class="p">(</span><span class="nx">key</span><span class="p">);</span>
-<a id="__codelineno-34-43" name="__codelineno-34-43" href="#__codelineno-34-43"></a><span class="w">        </span><span class="c1">// Set to null to represent deletion</span>
+<a id="__codelineno-34-43" name="__codelineno-34-43" href="#__codelineno-34-43"></a><span class="w">        </span><span class="c1">// Set the bucket to null to mark it as deleted</span>
 <a id="__codelineno-34-44" name="__codelineno-34-44" href="#__codelineno-34-44"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-34-45" name="__codelineno-34-45" href="#__codelineno-34-45"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-34-46" name="__codelineno-34-46" href="#__codelineno-34-46"></a>
@@ -5901,7 +5973,7 @@
 <a id="__codelineno-34-77" name="__codelineno-34-77" href="#__codelineno-34-77"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">arr</span><span class="p">;</span>
 <a id="__codelineno-34-78" name="__codelineno-34-78" href="#__codelineno-34-78"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-34-79" name="__codelineno-34-79" href="#__codelineno-34-79"></a>
-<a id="__codelineno-34-80" name="__codelineno-34-80" href="#__codelineno-34-80"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-34-80" name="__codelineno-34-80" href="#__codelineno-34-80"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-34-81" name="__codelineno-34-81" href="#__codelineno-34-81"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nx">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-82" name="__codelineno-34-82" href="#__codelineno-34-82"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">pairSet</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">entries</span><span class="p">();</span>
 <a id="__codelineno-34-83" name="__codelineno-34-83" href="#__codelineno-34-83"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">pair</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">pairSet</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5919,12 +5991,12 @@
 <a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">  </span><span class="n">Pair</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="p">}</span>
 <a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a>
-<a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="cm">/* Hash table based on array implementation */</span>
+<a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="cm">/* Hash table implemented with an array */</span>
 <a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayHashMap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">?&gt;</span><span class="w"> </span><span class="n">_buckets</span><span class="p">;</span>
 <a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a>
 <a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="w">  </span><span class="n">ArrayHashMap</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="w">    </span><span class="c1">// Initialize array with 100 buckets</span>
+<a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="w">    </span><span class="c1">// Initialize an array containing 100 buckets</span>
 <a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="w">    </span><span class="n">_buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">100</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">);</span>
 <a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-35-16" name="__codelineno-35-16" href="#__codelineno-35-16"></a>
@@ -5934,7 +6006,7 @@
 <a id="__codelineno-35-20" name="__codelineno-35-20" href="#__codelineno-35-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">index</span><span class="p">;</span>
 <a id="__codelineno-35-21" name="__codelineno-35-21" href="#__codelineno-35-21"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-35-22" name="__codelineno-35-22" href="#__codelineno-35-22"></a>
-<a id="__codelineno-35-23" name="__codelineno-35-23" href="#__codelineno-35-23"></a><span class="w">  </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-35-23" name="__codelineno-35-23" href="#__codelineno-35-23"></a><span class="w">  </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-35-24" name="__codelineno-35-24" href="#__codelineno-35-24"></a><span class="w">  </span><span class="kt">String</span><span class="o">?</span><span class="w"> </span><span class="kd">get</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-25" name="__codelineno-35-25" href="#__codelineno-35-25"></a><span class="w">    </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-35-26" name="__codelineno-35-26" href="#__codelineno-35-26"></a><span class="w">    </span><span class="kd">final</span><span class="w"> </span><span class="n">Pair</span><span class="o">?</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_buckets</span><span class="p">[</span><span class="n">index</span><span class="p">];</span>
@@ -5944,7 +6016,7 @@
 <a id="__codelineno-35-30" name="__codelineno-35-30" href="#__codelineno-35-30"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-35-31" name="__codelineno-35-31" href="#__codelineno-35-31"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-35-32" name="__codelineno-35-32" href="#__codelineno-35-32"></a>
-<a id="__codelineno-35-33" name="__codelineno-35-33" href="#__codelineno-35-33"></a><span class="w">  </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-35-33" name="__codelineno-35-33" href="#__codelineno-35-33"></a><span class="w">  </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-35-34" name="__codelineno-35-34" href="#__codelineno-35-34"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">put</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="kt">String</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-35" name="__codelineno-35-35" href="#__codelineno-35-35"></a><span class="w">    </span><span class="kd">final</span><span class="w"> </span><span class="n">Pair</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-35-36" name="__codelineno-35-36" href="#__codelineno-35-36"></a><span class="w">    </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
@@ -5990,7 +6062,7 @@
 <a id="__codelineno-35-76" name="__codelineno-35-76" href="#__codelineno-35-76"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">valueSet</span><span class="p">;</span>
 <a id="__codelineno-35-77" name="__codelineno-35-77" href="#__codelineno-35-77"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-35-78" name="__codelineno-35-78" href="#__codelineno-35-78"></a>
-<a id="__codelineno-35-79" name="__codelineno-35-79" href="#__codelineno-35-79"></a><span class="w">  </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-35-79" name="__codelineno-35-79" href="#__codelineno-35-79"></a><span class="w">  </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-35-80" name="__codelineno-35-80" href="#__codelineno-35-80"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">printHashMap</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-81" name="__codelineno-35-81" href="#__codelineno-35-81"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">final</span><span class="w"> </span><span class="n">Pair</span><span class="w"> </span><span class="n">kv</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">pairSet</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-82" name="__codelineno-35-82" href="#__codelineno-35-82"></a><span class="w">      </span><span class="n">print</span><span class="p">(</span><span class="s2">"</span><span class="si">${</span><span class="n">kv</span><span class="p">.</span><span class="n">key</span><span class="si">}</span><span class="s2"> -&gt; </span><span class="si">${</span><span class="n">kv</span><span class="p">.</span><span class="n">val</span><span class="si">}</span><span class="s2">"</span><span class="p">);</span>
@@ -6007,14 +6079,14 @@
 <a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="nb">String</span><span class="p">,</span>
 <a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="p">}</span>
 <a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a>
-<a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="cm">/* Hash table based on array implementation */</span>
+<a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="cm">/* Hash table implemented with an array */</span>
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="k">pub</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">ArrayHashMap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="w">    </span><span class="n">buckets</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="p">}</span>
 <a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a>
 <a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="k">impl</span><span class="w"> </span><span class="n">ArrayHashMap</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-14" name="__codelineno-36-14" href="#__codelineno-36-14"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">new</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">ArrayHashMap</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a><span class="w">        </span><span class="c1">// Initialize array with 100 buckets</span>
+<a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a><span class="w">        </span><span class="c1">// Initialize an array containing 100 buckets</span>
 <a id="__codelineno-36-16" name="__codelineno-36-16" href="#__codelineno-36-16"></a><span class="w">        </span><span class="bp">Self</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-17" name="__codelineno-36-17" href="#__codelineno-36-17"></a><span class="w">            </span><span class="n">buckets</span><span class="p">:</span><span class="w"> </span><span class="nc">vec</span><span class="o">!</span><span class="p">[</span><span class="nb">None</span><span class="p">;</span><span class="w"> </span><span class="mi">100</span><span class="p">],</span>
 <a id="__codelineno-36-18" name="__codelineno-36-18" href="#__codelineno-36-18"></a><span class="w">        </span><span class="p">}</span>
@@ -6025,13 +6097,13 @@
 <a id="__codelineno-36-23" name="__codelineno-36-23" href="#__codelineno-36-23"></a><span class="w">        </span><span class="n">key</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">100</span>
 <a id="__codelineno-36-24" name="__codelineno-36-24" href="#__codelineno-36-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-36-25" name="__codelineno-36-25" href="#__codelineno-36-25"></a>
-<a id="__codelineno-36-26" name="__codelineno-36-26" href="#__codelineno-36-26"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-36-26" name="__codelineno-36-26" href="#__codelineno-36-26"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-36-27" name="__codelineno-36-27" href="#__codelineno-36-27"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="nb">String</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-28" name="__codelineno-36-28" href="#__codelineno-36-28"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-36-29" name="__codelineno-36-29" href="#__codelineno-36-29"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">].</span><span class="n">as_ref</span><span class="p">().</span><span class="n">map</span><span class="p">(</span><span class="o">|</span><span class="n">pair</span><span class="o">|</span><span class="w"> </span><span class="o">&amp;</span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-36-30" name="__codelineno-36-30" href="#__codelineno-36-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-36-31" name="__codelineno-36-31" href="#__codelineno-36-31"></a>
-<a id="__codelineno-36-32" name="__codelineno-36-32" href="#__codelineno-36-32"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-36-32" name="__codelineno-36-32" href="#__codelineno-36-32"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-36-33" name="__codelineno-36-33" href="#__codelineno-36-33"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="kt">str</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-34" name="__codelineno-36-34" href="#__codelineno-36-34"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-36-35" name="__codelineno-36-35" href="#__codelineno-36-35"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">Pair</span><span class="w"> </span><span class="p">{</span>
@@ -6043,7 +6115,7 @@
 <a id="__codelineno-36-41" name="__codelineno-36-41" href="#__codelineno-36-41"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-36-42" name="__codelineno-36-42" href="#__codelineno-36-42"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-43" name="__codelineno-36-43" href="#__codelineno-36-43"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">);</span>
-<a id="__codelineno-36-44" name="__codelineno-36-44" href="#__codelineno-36-44"></a><span class="w">        </span><span class="c1">// Set to None to represent removal</span>
+<a id="__codelineno-36-44" name="__codelineno-36-44" href="#__codelineno-36-44"></a><span class="w">        </span><span class="c1">// Set to None to mark removal</span>
 <a id="__codelineno-36-45" name="__codelineno-36-45" href="#__codelineno-36-45"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">None</span><span class="p">;</span>
 <a id="__codelineno-36-46" name="__codelineno-36-46" href="#__codelineno-36-46"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-36-47" name="__codelineno-36-47" href="#__codelineno-36-47"></a>
@@ -6071,7 +6143,7 @@
 <a id="__codelineno-36-69" name="__codelineno-36-69" href="#__codelineno-36-69"></a><span class="w">            </span><span class="p">.</span><span class="n">collect</span><span class="p">()</span>
 <a id="__codelineno-36-70" name="__codelineno-36-70" href="#__codelineno-36-70"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-36-71" name="__codelineno-36-71" href="#__codelineno-36-71"></a>
-<a id="__codelineno-36-72" name="__codelineno-36-72" href="#__codelineno-36-72"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-36-72" name="__codelineno-36-72" href="#__codelineno-36-72"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-36-73" name="__codelineno-36-73" href="#__codelineno-36-73"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">print</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-74" name="__codelineno-36-74" href="#__codelineno-36-74"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">pair</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">entry_set</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-75" name="__codelineno-36-75" href="#__codelineno-36-75"></a><span class="w">            </span><span class="fm">println!</span><span class="p">(</span><span class="s">"{} -&gt; {}"</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
@@ -6081,13 +6153,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_hash_map.c</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Key-value pair int-&gt;string */</span>
+<div class="highlight"><span class="filename">array_hash_map.c</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Key-value pair: int -&gt; string */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">;</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="p">}</span><span class="w"> </span><span class="n">Pair</span><span class="p">;</span>
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a>
-<a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="cm">/* Hash table based on array implementation */</span>
+<a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="cm">/* Hash table implemented with an array */</span>
 <a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">    </span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">buckets</span><span class="p">[</span><span class="n">MAX_SIZE</span><span class="p">];</span>
 <a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="p">}</span><span class="w"> </span><span class="n">ArrayHashMap</span><span class="p">;</span>
@@ -6112,7 +6184,7 @@
 <a id="__codelineno-37-29" name="__codelineno-37-29" href="#__codelineno-37-29"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">hmap</span><span class="p">);</span>
 <a id="__codelineno-37-30" name="__codelineno-37-30" href="#__codelineno-37-30"></a><span class="p">}</span>
 <a id="__codelineno-37-31" name="__codelineno-37-31" href="#__codelineno-37-31"></a>
-<a id="__codelineno-37-32" name="__codelineno-37-32" href="#__codelineno-37-32"></a><span class="cm">/* Add operation */</span>
+<a id="__codelineno-37-32" name="__codelineno-37-32" href="#__codelineno-37-32"></a><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-37-33" name="__codelineno-37-33" href="#__codelineno-37-33"></a><span class="kt">void</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="n">ArrayHashMap</span><span class="w"> </span><span class="o">*</span><span class="n">hmap</span><span class="p">,</span><span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-34" name="__codelineno-37-34" href="#__codelineno-37-34"></a><span class="w">    </span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">Pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">Pair</span><span class="p">));</span>
 <a id="__codelineno-37-35" name="__codelineno-37-35" href="#__codelineno-37-35"></a><span class="w">    </span><span class="n">Pair</span><span class="o">-&gt;</span><span class="n">key</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">key</span><span class="p">;</span>
@@ -6136,7 +6208,7 @@
 <a id="__codelineno-37-53" name="__codelineno-37-53" href="#__codelineno-37-53"></a><span class="w">    </span><span class="n">Pair</span><span class="w"> </span><span class="o">*</span><span class="n">entries</span><span class="p">;</span>
 <a id="__codelineno-37-54" name="__codelineno-37-54" href="#__codelineno-37-54"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-37-55" name="__codelineno-37-55" href="#__codelineno-37-55"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-37-56" name="__codelineno-37-56" href="#__codelineno-37-56"></a><span class="w">    </span><span class="cm">/* Count valid key-value pairs */</span>
+<a id="__codelineno-37-56" name="__codelineno-37-56" href="#__codelineno-37-56"></a><span class="w">    </span><span class="cm">/* Count stored key-value pairs */</span>
 <a id="__codelineno-37-57" name="__codelineno-37-57" href="#__codelineno-37-57"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-58" name="__codelineno-37-58" href="#__codelineno-37-58"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">hmap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-59" name="__codelineno-37-59" href="#__codelineno-37-59"></a><span class="w">            </span><span class="n">total</span><span class="o">++</span><span class="p">;</span>
@@ -6160,7 +6232,7 @@
 <a id="__codelineno-37-77" name="__codelineno-37-77" href="#__codelineno-37-77"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">keys</span><span class="p">;</span>
 <a id="__codelineno-37-78" name="__codelineno-37-78" href="#__codelineno-37-78"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-37-79" name="__codelineno-37-79" href="#__codelineno-37-79"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-37-80" name="__codelineno-37-80" href="#__codelineno-37-80"></a><span class="w">    </span><span class="cm">/* Count valid key-value pairs */</span>
+<a id="__codelineno-37-80" name="__codelineno-37-80" href="#__codelineno-37-80"></a><span class="w">    </span><span class="cm">/* Count stored key-value pairs */</span>
 <a id="__codelineno-37-81" name="__codelineno-37-81" href="#__codelineno-37-81"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-82" name="__codelineno-37-82" href="#__codelineno-37-82"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">hmap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-83" name="__codelineno-37-83" href="#__codelineno-37-83"></a><span class="w">            </span><span class="n">total</span><span class="o">++</span><span class="p">;</span>
@@ -6182,7 +6254,7 @@
 <a id="__codelineno-37-99" name="__codelineno-37-99" href="#__codelineno-37-99"></a><span class="w">    </span><span class="kt">char</span><span class="w"> </span><span class="o">**</span><span class="n">vals</span><span class="p">;</span>
 <a id="__codelineno-37-100" name="__codelineno-37-100" href="#__codelineno-37-100"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-37-101" name="__codelineno-37-101" href="#__codelineno-37-101"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">total</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-37-102" name="__codelineno-37-102" href="#__codelineno-37-102"></a><span class="w">    </span><span class="cm">/* Count valid key-value pairs */</span>
+<a id="__codelineno-37-102" name="__codelineno-37-102" href="#__codelineno-37-102"></a><span class="w">    </span><span class="cm">/* Count stored key-value pairs */</span>
 <a id="__codelineno-37-103" name="__codelineno-37-103" href="#__codelineno-37-103"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-104" name="__codelineno-37-104" href="#__codelineno-37-104"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">hmap</span><span class="o">-&gt;</span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-105" name="__codelineno-37-105" href="#__codelineno-37-105"></a><span class="w">            </span><span class="n">total</span><span class="o">++</span><span class="p">;</span>
@@ -6199,7 +6271,7 @@
 <a id="__codelineno-37-116" name="__codelineno-37-116" href="#__codelineno-37-116"></a><span class="w">    </span><span class="n">set</span><span class="o">-&gt;</span><span class="n">len</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">total</span><span class="p">;</span>
 <a id="__codelineno-37-117" name="__codelineno-37-117" href="#__codelineno-37-117"></a><span class="p">}</span>
 <a id="__codelineno-37-118" name="__codelineno-37-118" href="#__codelineno-37-118"></a>
-<a id="__codelineno-37-119" name="__codelineno-37-119" href="#__codelineno-37-119"></a><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-37-119" name="__codelineno-37-119" href="#__codelineno-37-119"></a><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-37-120" name="__codelineno-37-120" href="#__codelineno-37-120"></a><span class="kt">void</span><span class="w"> </span><span class="nf">print</span><span class="p">(</span><span class="n">ArrayHashMap</span><span class="w"> </span><span class="o">*</span><span class="n">hmap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-121" name="__codelineno-37-121" href="#__codelineno-37-121"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-37-122" name="__codelineno-37-122" href="#__codelineno-37-122"></a><span class="w">    </span><span class="n">MapSet</span><span class="w"> </span><span class="n">set</span><span class="p">;</span>
@@ -6219,9 +6291,9 @@
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">String</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="p">)</span>
 <a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a>
-<a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="cm">/* Hash table based on array implementation */</span>
+<a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="cm">/* Hash table implemented with an array */</span>
 <a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayHashMap</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">    </span><span class="c1">// Initialize array with 100 buckets</span>
+<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">    </span><span class="c1">// Initialize an array containing 100 buckets</span>
 <a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">val</span><span class="w"> </span><span class="nv">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arrayOfNulls</span><span class="o">&lt;</span><span class="n">Pair</span><span class="o">&gt;</span><span class="p">(</span><span class="m">100</span><span class="p">)</span>
 <a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a>
 <a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="w">    </span><span class="cm">/* Hash function */</span>
@@ -6230,14 +6302,14 @@
 <a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">index</span>
 <a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-38-17" name="__codelineno-38-17" href="#__codelineno-38-17"></a>
-<a id="__codelineno-38-18" name="__codelineno-38-18" href="#__codelineno-38-18"></a><span class="w">    </span><span class="cm">/* Query operation */</span>
+<a id="__codelineno-38-18" name="__codelineno-38-18" href="#__codelineno-38-18"></a><span class="w">    </span><span class="cm">/* Lookup operation */</span>
 <a id="__codelineno-38-19" name="__codelineno-38-19" href="#__codelineno-38-19"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">String?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-38-20" name="__codelineno-38-20" href="#__codelineno-38-20"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-38-21" name="__codelineno-38-21" href="#__codelineno-38-21"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">?:</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span>
 <a id="__codelineno-38-22" name="__codelineno-38-22" href="#__codelineno-38-22"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pair</span><span class="p">.</span><span class="na">_val</span>
 <a id="__codelineno-38-23" name="__codelineno-38-23" href="#__codelineno-38-23"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-38-24" name="__codelineno-38-24" href="#__codelineno-38-24"></a>
-<a id="__codelineno-38-25" name="__codelineno-38-25" href="#__codelineno-38-25"></a><span class="w">    </span><span class="cm">/* Add operation */</span>
+<a id="__codelineno-38-25" name="__codelineno-38-25" href="#__codelineno-38-25"></a><span class="w">    </span><span class="cm">/* Insert operation */</span>
 <a id="__codelineno-38-26" name="__codelineno-38-26" href="#__codelineno-38-26"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">String</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-38-27" name="__codelineno-38-27" href="#__codelineno-38-27"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">)</span>
 <a id="__codelineno-38-28" name="__codelineno-38-28" href="#__codelineno-38-28"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
@@ -6247,7 +6319,7 @@
 <a id="__codelineno-38-32" name="__codelineno-38-32" href="#__codelineno-38-32"></a><span class="w">    </span><span class="cm">/* Remove operation */</span>
 <a id="__codelineno-38-33" name="__codelineno-38-33" href="#__codelineno-38-33"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">key</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-38-34" name="__codelineno-38-34" href="#__codelineno-38-34"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hashFunc</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-38-35" name="__codelineno-38-35" href="#__codelineno-38-35"></a><span class="w">        </span><span class="c1">// Set to null to represent deletion</span>
+<a id="__codelineno-38-35" name="__codelineno-38-35" href="#__codelineno-38-35"></a><span class="w">        </span><span class="c1">// Set the bucket to null to mark it as removed</span>
 <a id="__codelineno-38-36" name="__codelineno-38-36" href="#__codelineno-38-36"></a><span class="w">        </span><span class="n">buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span>
 <a id="__codelineno-38-37" name="__codelineno-38-37" href="#__codelineno-38-37"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-38-38" name="__codelineno-38-38" href="#__codelineno-38-38"></a>
@@ -6281,7 +6353,7 @@
 <a id="__codelineno-38-66" name="__codelineno-38-66" href="#__codelineno-38-66"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">valueSet</span>
 <a id="__codelineno-38-67" name="__codelineno-38-67" href="#__codelineno-38-67"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-38-68" name="__codelineno-38-68" href="#__codelineno-38-68"></a>
-<a id="__codelineno-38-69" name="__codelineno-38-69" href="#__codelineno-38-69"></a><span class="w">    </span><span class="cm">/* Print hash table */</span>
+<a id="__codelineno-38-69" name="__codelineno-38-69" href="#__codelineno-38-69"></a><span class="w">    </span><span class="cm">/* Print the hash table */</span>
 <a id="__codelineno-38-70" name="__codelineno-38-70" href="#__codelineno-38-70"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-38-71" name="__codelineno-38-71" href="#__codelineno-38-71"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">kv</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">pairSet</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-38-72" name="__codelineno-38-72" href="#__codelineno-38-72"></a><span class="w">            </span><span class="kd">val</span><span class="w"> </span><span class="nv">key</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">kv</span><span class="p">.</span><span class="na">key</span>
@@ -6303,11 +6375,11 @@
 <a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a><span class="k">end</span>
 <a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a>
-<a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a><span class="c1">### Hash map based on array ###</span>
+<a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a><span class="c1">### Hash table implemented with an array ###</span>
 <a id="__codelineno-39-12" name="__codelineno-39-12" href="#__codelineno-39-12"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayHashMap</span>
 <a id="__codelineno-39-13" name="__codelineno-39-13" href="#__codelineno-39-13"></a><span class="w">  </span><span class="c1">### Constructor ###</span>
 <a id="__codelineno-39-14" name="__codelineno-39-14" href="#__codelineno-39-14"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">initialize</span>
-<a id="__codelineno-39-15" name="__codelineno-39-15" href="#__codelineno-39-15"></a><span class="w">    </span><span class="c1"># Initialize array with 100 buckets</span>
+<a id="__codelineno-39-15" name="__codelineno-39-15" href="#__codelineno-39-15"></a><span class="w">    </span><span class="c1"># Initialize an array containing 100 buckets</span>
 <a id="__codelineno-39-16" name="__codelineno-39-16" href="#__codelineno-39-16"></a><span class="w">    </span><span class="vi">@buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
 <a id="__codelineno-39-17" name="__codelineno-39-17" href="#__codelineno-39-17"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-39-18" name="__codelineno-39-18" href="#__codelineno-39-18"></a>
@@ -6316,7 +6388,7 @@
 <a id="__codelineno-39-21" name="__codelineno-39-21" href="#__codelineno-39-21"></a><span class="w">    </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">key</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">100</span>
 <a id="__codelineno-39-22" name="__codelineno-39-22" href="#__codelineno-39-22"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-39-23" name="__codelineno-39-23" href="#__codelineno-39-23"></a>
-<a id="__codelineno-39-24" name="__codelineno-39-24" href="#__codelineno-39-24"></a><span class="w">  </span><span class="c1">### Query operation ###</span>
+<a id="__codelineno-39-24" name="__codelineno-39-24" href="#__codelineno-39-24"></a><span class="w">  </span><span class="c1">### Lookup operation ###</span>
 <a id="__codelineno-39-25" name="__codelineno-39-25" href="#__codelineno-39-25"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-39-26" name="__codelineno-39-26" href="#__codelineno-39-26"></a><span class="w">    </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-39-27" name="__codelineno-39-27" href="#__codelineno-39-27"></a><span class="w">    </span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span>
@@ -6325,17 +6397,17 @@
 <a id="__codelineno-39-30" name="__codelineno-39-30" href="#__codelineno-39-30"></a><span class="w">    </span><span class="n">pair</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-39-31" name="__codelineno-39-31" href="#__codelineno-39-31"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-39-32" name="__codelineno-39-32" href="#__codelineno-39-32"></a>
-<a id="__codelineno-39-33" name="__codelineno-39-33" href="#__codelineno-39-33"></a><span class="w">  </span><span class="c1">### Add operation ###</span>
+<a id="__codelineno-39-33" name="__codelineno-39-33" href="#__codelineno-39-33"></a><span class="w">  </span><span class="c1">### Insert operation ###</span>
 <a id="__codelineno-39-34" name="__codelineno-39-34" href="#__codelineno-39-34"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-39-35" name="__codelineno-39-35" href="#__codelineno-39-35"></a><span class="w">    </span><span class="n">pair</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">Pair</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-39-36" name="__codelineno-39-36" href="#__codelineno-39-36"></a><span class="w">    </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-39-37" name="__codelineno-39-37" href="#__codelineno-39-37"></a><span class="w">    </span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pair</span>
 <a id="__codelineno-39-38" name="__codelineno-39-38" href="#__codelineno-39-38"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-39-39" name="__codelineno-39-39" href="#__codelineno-39-39"></a>
-<a id="__codelineno-39-40" name="__codelineno-39-40" href="#__codelineno-39-40"></a><span class="w">  </span><span class="c1">### Delete operation ###</span>
+<a id="__codelineno-39-40" name="__codelineno-39-40" href="#__codelineno-39-40"></a><span class="w">  </span><span class="c1">### Remove operation ###</span>
 <a id="__codelineno-39-41" name="__codelineno-39-41" href="#__codelineno-39-41"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
 <a id="__codelineno-39-42" name="__codelineno-39-42" href="#__codelineno-39-42"></a><span class="w">    </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
-<a id="__codelineno-39-43" name="__codelineno-39-43" href="#__codelineno-39-43"></a><span class="w">    </span><span class="c1"># Set to nil to delete</span>
+<a id="__codelineno-39-43" name="__codelineno-39-43" href="#__codelineno-39-43"></a><span class="w">    </span><span class="c1"># Set the bucket to nil to indicate removal</span>
 <a id="__codelineno-39-44" name="__codelineno-39-44" href="#__codelineno-39-44"></a><span class="w">    </span><span class="vi">@buckets</span><span class="o">[</span><span class="n">index</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">nil</span>
 <a id="__codelineno-39-45" name="__codelineno-39-45" href="#__codelineno-39-45"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-39-46" name="__codelineno-39-46" href="#__codelineno-39-46"></a>
@@ -6360,7 +6432,7 @@
 <a id="__codelineno-39-65" name="__codelineno-39-65" href="#__codelineno-39-65"></a><span class="w">    </span><span class="n">result</span>
 <a id="__codelineno-39-66" name="__codelineno-39-66" href="#__codelineno-39-66"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-39-67" name="__codelineno-39-67" href="#__codelineno-39-67"></a>
-<a id="__codelineno-39-68" name="__codelineno-39-68" href="#__codelineno-39-68"></a><span class="w">  </span><span class="c1">### Print hash table ###</span>
+<a id="__codelineno-39-68" name="__codelineno-39-68" href="#__codelineno-39-68"></a><span class="w">  </span><span class="c1">### Print the hash table ###</span>
 <a id="__codelineno-39-69" name="__codelineno-39-69" href="#__codelineno-39-69"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">print</span>
 <a id="__codelineno-39-70" name="__codelineno-39-70" href="#__codelineno-39-70"></a><span class="w">    </span><span class="vi">@buckets</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">pair</span><span class="o">|</span><span class="w"> </span><span class="nb">puts</span><span class="w"> </span><span class="s2">"</span><span class="si">#{</span><span class="n">pair</span><span class="o">.</span><span class="n">key</span><span class="si">}</span><span class="s2"> -&gt; </span><span class="si">#{</span><span class="n">pair</span><span class="o">.</span><span class="n">val</span><span class="si">}</span><span class="s2">"</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="n">pair</span><span class="o">.</span><span class="n">nil?</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-39-71" name="__codelineno-39-71" href="#__codelineno-39-71"></a><span class="w">  </span><span class="k">end</span>
@@ -6369,23 +6441,28 @@
 </div>
 </div>
 </div>
-<h2 id="613-hash-collision-and-resizing">6.1.3 &nbsp; Hash Collision and Resizing<a class="headerlink" href="#613-hash-collision-and-resizing" title="Permanent link">&para;</a></h2>
-<p>Fundamentally, a hash function maps the input space consisting of all <code>key</code>s to the output space consisting of all array indices, and the input space is often much larger than the output space. Therefore, <strong>in theory, different inputs must sometimes map to the same output</strong>.</p>
-<p>For the hash function in the above example, when the input <code>key</code>s have the same last two digits, the hash function produces the same output. For example, when querying two students with IDs 12836 and 20336, we get:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20Pair%3A%0A%20%20%20%20%22%22%22Key-value%20pair%22%22%22%0A%20%20%20%20def%20__init__%28self,%20key%3A%20int,%20val%3A%20str%29%3A%0A%20%20%20%20%20%20%20%20self.key%20%3D%20key%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%0A%0Aclass%20ArrayHashMap%3A%0A%20%20%20%20%22%22%22Hash%20table%20implemented%20with%20an%20array%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Initialize%20an%20array%20containing%2020%20buckets%0A%20%20%20%20%20%20%20%20self.buckets%3A%20list%5BPair%20%7C%20None%5D%20%3D%20%5BNone%5D%20%2A%2020%0A%0A%20%20%20%20def%20hash_func%28self,%20key%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Hash%20function%22%22%22%0A%20%20%20%20%20%20%20%20index%20%3D%20key%20%25%2020%0A%20%20%20%20%20%20%20%20return%20index%0A%0A%20%20%20%20def%20get%28self,%20key%3A%20int%29%20-%3E%20str%3A%0A%20%20%20%20%20%20%20%20%22%22%22Lookup%20operation%22%22%22%0A%20%20%20%20%20%20%20%20index%3A%20int%20%3D%20self.hash_func%28key%29%0A%20%20%20%20%20%20%20%20pair%3A%20Pair%20%3D%20self.buckets%5Bindex%5D%0A%20%20%20%20%20%20%20%20if%20pair%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%20%20%20%20return%20pair.val%0A%0A%20%20%20%20def%20put%28self,%20key%3A%20int,%20val%3A%20str%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Insert%20operation%22%22%22%0A%20%20%20%20%20%20%20%20pair%20%3D%20Pair%28key,%20val%29%0A%20%20%20%20%20%20%20%20index%3A%20int%20%3D%20self.hash_func%28key%29%0A%20%20%20%20%20%20%20%20self.buckets%5Bindex%5D%20%3D%20pair%0A%0A%20%20%20%20def%20remove%28self,%20key%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Remove%20operation%22%22%22%0A%20%20%20%20%20%20%20%20index%3A%20int%20%3D%20self.hash_func%28key%29%0A%20%20%20%20%20%20%20%20%23%20Set%20to%20None%20to%20mark%20removal%0A%20%20%20%20%20%20%20%20self.buckets%5Bindex%5D%20%3D%20None%0A%0A%20%20%20%20def%20entry_set%28self%29%20-%3E%20list%5BPair%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20all%20key-value%20pairs%22%22%22%0A%20%20%20%20%20%20%20%20result%3A%20list%5BPair%5D%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20for%20pair%20in%20self.buckets%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20pair%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20result.append%28pair%29%0A%20%20%20%20%20%20%20%20return%20result%0A%0A%20%20%20%20def%20key_set%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20all%20keys%22%22%22%0A%20%20%20%20%20%20%20%20result%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20for%20pair%20in%20self.buckets%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20pair%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20result.append%28pair.key%29%0A%20%20%20%20%20%20%20%20return%20result%0A%0A%20%20%20%20def%20value_set%28self%29%20-%3E%20list%5Bstr%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20all%20values%22%22%22%0A%20%20%20%20%20%20%20%20result%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20for%20pair%20in%20self.buckets%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20pair%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20result.append%28pair.val%29%0A%20%20%20%20%20%20%20%20return%20result%0A%0A%20%20%20%20def%20print%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Print%20the%20hash%20table%22%22%22%0A%20%20%20%20%20%20%20%20for%20pair%20in%20self.buckets%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20pair%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20print%28pair.key,%20%22-%3E%22,%20pair.val%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20hash%20table%0A%20%20%20%20hmap%20%3D%20ArrayHashMap%28%29%0A%0A%20%20%20%20%23%20Insert%20operation%0A%20%20%20%20hmap.put%2812836,%20%22%E5%B0%8F%E5%93%88%22%29%0A%20%20%20%20hmap.put%2815937,%20%22%E5%B0%8F%E5%95%B0%22%29%0A%20%20%20%20hmap.put%2816750,%20%22%E5%B0%8F%E7%AE%97%22%29%0A%20%20%20%20hmap.put%2813276,%20%22%E5%B0%8F%E6%B3%95%22%29%0A%20%20%20%20hmap.put%2810583,%20%22%E5%B0%8F%E9%B8%AD%22%29%0A%0A%20%20%20%20%23%20Lookup%20operation%0A%20%20%20%20name%20%3D%20hmap.get%2815937%29%0A%0A%20%20%20%20%23%20Remove%20operation%0A%20%20%20%20hmap.remove%2810583%29%0A%0A%20%20%20%20%23%20Traverse%20the%20hash%20table%0A%20%20%20%20print%28%22%5CnTraverse%20key-value%20pairs%3A%20Key-%3EValue%22%29%0A%20%20%20%20for%20pair%20in%20hmap.entry_set%28%29%3A%0A%20%20%20%20%20%20%20%20print%28pair.key,%20%22-%3E%22,%20pair.val%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20Pair%3A%0A%20%20%20%20%22%22%22Key-value%20pair%22%22%22%0A%20%20%20%20def%20__init__%28self,%20key%3A%20int,%20val%3A%20str%29%3A%0A%20%20%20%20%20%20%20%20self.key%20%3D%20key%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%0A%0Aclass%20ArrayHashMap%3A%0A%20%20%20%20%22%22%22Hash%20table%20implemented%20with%20an%20array%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Initialize%20an%20array%20containing%2020%20buckets%0A%20%20%20%20%20%20%20%20self.buckets%3A%20list%5BPair%20%7C%20None%5D%20%3D%20%5BNone%5D%20%2A%2020%0A%0A%20%20%20%20def%20hash_func%28self,%20key%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Hash%20function%22%22%22%0A%20%20%20%20%20%20%20%20index%20%3D%20key%20%25%2020%0A%20%20%20%20%20%20%20%20return%20index%0A%0A%20%20%20%20def%20get%28self,%20key%3A%20int%29%20-%3E%20str%3A%0A%20%20%20%20%20%20%20%20%22%22%22Lookup%20operation%22%22%22%0A%20%20%20%20%20%20%20%20index%3A%20int%20%3D%20self.hash_func%28key%29%0A%20%20%20%20%20%20%20%20pair%3A%20Pair%20%3D%20self.buckets%5Bindex%5D%0A%20%20%20%20%20%20%20%20if%20pair%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%20%20%20%20return%20pair.val%0A%0A%20%20%20%20def%20put%28self,%20key%3A%20int,%20val%3A%20str%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Insert%20operation%22%22%22%0A%20%20%20%20%20%20%20%20pair%20%3D%20Pair%28key,%20val%29%0A%20%20%20%20%20%20%20%20index%3A%20int%20%3D%20self.hash_func%28key%29%0A%20%20%20%20%20%20%20%20self.buckets%5Bindex%5D%20%3D%20pair%0A%0A%20%20%20%20def%20remove%28self,%20key%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Remove%20operation%22%22%22%0A%20%20%20%20%20%20%20%20index%3A%20int%20%3D%20self.hash_func%28key%29%0A%20%20%20%20%20%20%20%20%23%20Set%20to%20None%20to%20mark%20removal%0A%20%20%20%20%20%20%20%20self.buckets%5Bindex%5D%20%3D%20None%0A%0A%20%20%20%20def%20entry_set%28self%29%20-%3E%20list%5BPair%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20all%20key-value%20pairs%22%22%22%0A%20%20%20%20%20%20%20%20result%3A%20list%5BPair%5D%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20for%20pair%20in%20self.buckets%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20pair%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20result.append%28pair%29%0A%20%20%20%20%20%20%20%20return%20result%0A%0A%20%20%20%20def%20key_set%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20all%20keys%22%22%22%0A%20%20%20%20%20%20%20%20result%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20for%20pair%20in%20self.buckets%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20pair%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20result.append%28pair.key%29%0A%20%20%20%20%20%20%20%20return%20result%0A%0A%20%20%20%20def%20value_set%28self%29%20-%3E%20list%5Bstr%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20all%20values%22%22%22%0A%20%20%20%20%20%20%20%20result%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20for%20pair%20in%20self.buckets%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20pair%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20result.append%28pair.val%29%0A%20%20%20%20%20%20%20%20return%20result%0A%0A%20%20%20%20def%20print%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Print%20the%20hash%20table%22%22%22%0A%20%20%20%20%20%20%20%20for%20pair%20in%20self.buckets%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20pair%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20print%28pair.key,%20%22-%3E%22,%20pair.val%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20hash%20table%0A%20%20%20%20hmap%20%3D%20ArrayHashMap%28%29%0A%0A%20%20%20%20%23%20Insert%20operation%0A%20%20%20%20hmap.put%2812836,%20%22%E5%B0%8F%E5%93%88%22%29%0A%20%20%20%20hmap.put%2815937,%20%22%E5%B0%8F%E5%95%B0%22%29%0A%20%20%20%20hmap.put%2816750,%20%22%E5%B0%8F%E7%AE%97%22%29%0A%20%20%20%20hmap.put%2813276,%20%22%E5%B0%8F%E6%B3%95%22%29%0A%20%20%20%20hmap.put%2810583,%20%22%E5%B0%8F%E9%B8%AD%22%29%0A%0A%20%20%20%20%23%20Lookup%20operation%0A%20%20%20%20name%20%3D%20hmap.get%2815937%29%0A%0A%20%20%20%20%23%20Remove%20operation%0A%20%20%20%20hmap.remove%2810583%29%0A%0A%20%20%20%20%23%20Traverse%20the%20hash%20table%0A%20%20%20%20print%28%22%5CnTraverse%20key-value%20pairs%3A%20Key-%3EValue%22%29%0A%20%20%20%20for%20pair%20in%20hmap.entry_set%28%29%3A%0A%20%20%20%20%20%20%20%20print%28pair.key,%20%22-%3E%22,%20pair.val%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="613-collisions-and-expansion">6.1.3 &nbsp; Collisions and expansion<a class="headerlink" href="#613-collisions-and-expansion" title="Permanent link">&para;</a></h2>
+<p>A hash function maps the input space of all possible keys into the output space of array indices. The input space is usually far larger, <strong>so multiple inputs must theoretically map to the same output</strong>.</p>
+<p>For the function above, keys with the same last two digits produce the same output. Looking up student IDs 12836 and 20336 gives:</p>
 <div class="highlight"><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="m">12836</span><span class="w"> </span>%<span class="w"> </span><span class="nv">100</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">36</span>
 <a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="m">20336</span><span class="w"> </span>%<span class="w"> </span><span class="nv">100</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">36</span>
 </code></pre></div>
-<p>As shown below, two student IDs now point to the same name, which is clearly incorrect. We call this situation, where multiple inputs map to the same output, a <u>hash collision</u>.</p>
-<p><img alt="Hash collision example" class="animation-figure" src="../hash_map.assets/hash_collision.png" /></p>
-<p align="center"> Figure 6-3 &nbsp; Hash collision example </p>
+<p>As shown below, both IDs point to the same name, which is clearly incorrect. This situation, where different inputs produce the same output, is called a <u>hash collision</u>.</p>
+<p><img alt="An example of a hash collision" class="animation-figure" src="../hash_map.assets/hash_collision.png" /></p>
+<p align="center"> Figure 6-3 &nbsp; An example of a hash collision </p>
 
-<p>It's easy to see that the larger the hash table capacity <span class="arithmatex">\(n\)</span>, the lower the probability that multiple <code>key</code>s will be assigned to the same bucket, and the fewer collisions. Therefore, <strong>we can reduce hash collisions by expanding the hash table</strong>.</p>
-<p>As shown in Figure 6-4, before expansion, the key-value pairs <code>(136, A)</code> and <code>(236, D)</code> collided, but after expansion, the collision disappears.</p>
-<p><img alt="Hash table resizing" class="animation-figure" src="../hash_map.assets/hash_table_reshash.png" /></p>
-<p align="center"> Figure 6-4 &nbsp; Hash table resizing </p>
+<p>A larger table capacity <span class="arithmatex">\(n\)</span> makes it less likely that several keys share a bucket, reducing collisions. <strong>Expanding the hash table can therefore reduce hash collisions</strong>.</p>
+<p>In the example below, <code>(136, A)</code> and <code>(236, D)</code> collide before expansion but no longer collide afterward.</p>
+<p><img alt="Expanding a hash table" class="animation-figure" src="../hash_map.assets/hash_table_reshash.png" /></p>
+<p align="center"> Figure 6-4 &nbsp; Expanding a hash table </p>
 
-<p>Like resizing an array, resizing a hash table requires migrating all key-value pairs from the original table to the new table, which is expensive. In addition, because the hash table capacity <code>capacity</code> changes, we must recompute the storage location of every key-value pair using the hash function, which further increases the cost of resizing. For this reason, programming languages typically reserve a sufficiently large hash table capacity to avoid frequent resizing.</p>
-<p>The <u>load factor</u> is an important concept in hash tables. It is defined as the number of elements in the hash table divided by the number of buckets and is used to measure the severity of hash collisions. <strong>It is also commonly used as a threshold for triggering hash table resizing</strong>. For example, in Java, when the load factor exceeds <span class="arithmatex">\(0.75\)</span>, the system expands the hash table to twice its original size.</p>
+<p>As with array expansion, expanding a hash table requires moving every pair into a new table, which is time-consuming. Because <code>capacity</code> changes, the hash function must also recalculate every pair's position, adding further computation. Programming languages therefore usually reserve enough capacity to avoid frequent expansion.</p>
+<p>The <u>load factor</u> is the number of elements divided by the number of buckets. It measures how crowded the table is and <strong>often serves as the trigger for expansion</strong>. For example, in Java, a load factor above <span class="arithmatex">\(0.75\)</span> causes the capacity to grow to <span class="arithmatex">\(2\)</span> times its previous size.</p>
 
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@@ -6408,7 +6485,7 @@ aria-label="Footer"
   <a
     href="../"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: Chapter 6. &amp;nbsp; Hashing"
+    aria-label="Previous: Chapter 6.   Hash tables"
     rel="prev"
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         Previous
       </span>
       <div class="md-ellipsis">
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+        Chapter 6. &nbsp; Hash tables
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   <a
     href="../hash_collision/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 6.2 Hash Collision"
+    aria-label="Next: 6.2   Hash collisions"
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+        6.2 &nbsp; Hash collisions
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@@ -6457,7 +6534,7 @@ aria-label="Footer"
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-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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@@ -41,7 +41,7 @@
 
 
 
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+        <title>Chapter 6. Hash tables - Hello Algo</title>
 
 
 
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-  Before Starting
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-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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-  0.3 Summary
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
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-  1.3 Summary
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-  2.1 Algorithm Efficiency Evaluation
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-  2.6 Exercises
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-  Chapter 3. Data Structures
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-  3.1 Classification of Data Structures
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.6 Exercises
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  8.2 Heap Construction Operation
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-  9.2 Basic Operations on Graphs
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-  Chapter 10. Searching
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.5 Searching Algorithms Revisited
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-  10.7 Exercises
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-  11.6 Merge Sort
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-  11.7 Heap Sort
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-  11.8 Bucket Sort
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-  11.9 Counting Sort
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-  14.5 Unbounded Knapsack Problem
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-  14.8 Exercises
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-  16.3 Glossary
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+
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+  Print edition (Chinese)
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 </nav>
                   </div>
@@ -4577,18 +4649,18 @@
 
 
 <!-- Page content -->
-<h1 id="chapter-6-hashing">Chapter 6. &nbsp; Hashing<a class="headerlink" href="#chapter-6-hashing" title="Permanent link">&para;</a></h1>
-<p><img alt="Hashing" class="cover-image" src="../assets/covers/chapter_hashing.jpg" /></p>
+<h1 id="chapter-6-hash-tables">Chapter 6. &nbsp; Hash tables<a class="headerlink" href="#chapter-6-hash-tables" title="Permanent link">&para;</a></h1>
+<p><img alt="Hash tables" class="cover-image" src="../assets/covers/chapter_hashing.jpg" /></p>
 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
-<p>In the world of computing, a hash table is like a clever librarian.</p>
-<p>It knows how to compute call numbers, allowing it to quickly locate the desired book.</p>
+<p>In the world of computers, a hash table is like a knowledgeable librarian.</p>
+<p>Knowing how to calculate a call number lets the librarian find the desired book quickly.</p>
 </div>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
-<li><a href="hash_map/">6.1 &nbsp; Hash Table</a></li>
-<li><a href="hash_collision/">6.2 &nbsp; Hash Collision</a></li>
-<li><a href="hash_algorithm/">6.3 &nbsp; Hash Algorithm</a></li>
+<li><a href="hash_map/">6.1 &nbsp; Hash tables</a></li>
+<li><a href="hash_collision/">6.2 &nbsp; Hash collisions</a></li>
+<li><a href="hash_algorithm/">6.3 &nbsp; Hash algorithms</a></li>
 <li><a href="summary/">6.4 &nbsp; Summary</a></li>
 <li><a href="exercises/">6.5 &nbsp; Exercises</a></li>
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     href="../chapter_stack_and_queue/exercises/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 5.5 Exercises"
+    aria-label="Previous: 5.5   Exercises"
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@@ -4626,7 +4698,7 @@ aria-label="Footer"
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+        5.5 &nbsp; Exercises
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   <a
     href="./hash_map/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 6.1 Hash Table"
+    aria-label="Next: 6.1   Hash tables"
     rel="next"
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     <div class="md-footer__title">
@@ -4646,7 +4718,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        6.1 Hash Table
+        6.1 &nbsp; Hash tables
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -4663,7 +4735,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
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@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
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@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1265,7 +1265,7 @@
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-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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@@ -1722,7 +1722,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
       </span>
     </span>
   </a>
@@ -1899,7 +1899,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1967,7 +1967,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2009,7 +2009,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -2036,7 +2036,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2063,7 +2063,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2090,7 +2090,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2117,7 +2117,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2144,7 +2144,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2171,7 +2171,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2235,7 +2235,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2256,7 +2256,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2277,7 +2277,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2304,7 +2304,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2331,7 +2331,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2358,7 +2358,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2449,7 +2449,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2491,7 +2491,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2518,7 +2518,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2545,7 +2545,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2572,7 +2572,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2599,7 +2599,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2709,7 +2709,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2736,7 +2736,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2763,7 +2763,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2790,7 +2790,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2817,7 +2817,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2844,7 +2844,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2871,7 +2871,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2970,7 +2970,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2991,7 +2991,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3018,7 +3018,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3045,7 +3045,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3072,7 +3072,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3099,7 +3099,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3126,7 +3126,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3153,7 +3153,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3180,7 +3180,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3207,7 +3207,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3234,7 +3234,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3375,7 +3375,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3396,7 +3396,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3639,7 +3639,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3666,7 +3666,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3720,7 +3720,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3886,7 +3886,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3967,7 +3967,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4021,7 +4021,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4141,7 +4141,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4183,7 +4183,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4264,7 +4264,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4447,7 +4447,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4558,6 +4558,78 @@
 
 
 
+
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         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
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-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
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 <!-- Page content -->
 <h1 id="64-summary">6.4 &nbsp; Summary<a class="headerlink" href="#64-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
 <ul>
-<li>Given an input <code>key</code>, a hash table can retrieve the corresponding <code>value</code> in <span class="arithmatex">\(O(1)\)</span> time, which is highly efficient.</li>
-<li>Common hash table operations include querying, adding key-value pairs, deleting key-value pairs, and traversing the hash table.</li>
-<li>The hash function maps a <code>key</code> to an array index, allowing access to the corresponding bucket and retrieval of the <code>value</code>.</li>
-<li>Two different keys may end up with the same array index after hashing, leading to erroneous query results. This phenomenon is known as hash collision.</li>
-<li>The larger the capacity of the hash table, the lower the probability of hash collisions. Therefore, hash table expansion can mitigate hash collisions. Similar to array expansion, hash table expansion is costly.</li>
-<li>The load factor, defined as the number of elements divided by the number of buckets, reflects the severity of hash collisions and is often used as a condition to trigger hash table expansion.</li>
-<li>Separate chaining addresses hash collisions by storing all colliding elements in the same linked list. However, excessively long linked lists can reduce query efficiency, which can be improved by further converting the linked lists into red-black trees.</li>
-<li>Open addressing handles hash collisions through multiple probing. Linear probing uses a fixed step size but cannot delete elements and is prone to clustering. Double hashing uses multiple hash functions for probing, which reduces clustering compared to linear probing but increases computational overhead.</li>
-<li>Different programming languages adopt various hash table implementations. For example, Java's <code>HashMap</code> uses separate chaining, while Python's <code>dict</code> employs open addressing.</li>
-<li>In hash tables, we desire hash algorithms with determinism, high efficiency, and uniform distribution. In cryptography, hash algorithms should also possess collision resistance and the avalanche effect.</li>
-<li>Hash algorithms typically use large prime numbers as moduli to maximize the uniform distribution of hash values and reduce hash collisions.</li>
-<li>Common hash algorithms include MD5, SHA-1, SHA-2, and SHA-3. MD5 is often used for file integrity checks, while SHA-2 is commonly used in secure applications and protocols.</li>
-<li>Programming languages usually provide built-in hash algorithms for data types to calculate bucket indices in hash tables. Generally, only immutable objects are hashable.</li>
+<li>Given a <code>key</code>, a hash table can retrieve its <code>value</code> in <span class="arithmatex">\(O(1)\)</span> time, making lookup highly efficient.</li>
+<li>Common operations include lookup, adding and deleting key-value pairs, and traversing the table.</li>
+<li>A hash function maps a <code>key</code> to an array index, allowing access to the corresponding bucket and its <code>value</code>.</li>
+<li>Two different keys may map to the same array index and cause incorrect lookup results. This is called a hash collision.</li>
+<li>A larger capacity lowers the likelihood of collisions, so expanding a hash table can alleviate them. Like array expansion, hash-table expansion is expensive.</li>
+<li>The load factor is the number of elements divided by the number of buckets. It indicates how crowded the table is and is commonly used to trigger expansion.</li>
+<li>Separate chaining replaces a single entry with a linked list and stores all colliding entries in that list. Long lists slow lookup; converting them to red-black trees can improve efficiency.</li>
+<li>Open addressing resolves collisions through repeated probing. Linear probing uses a fixed step size, is prone to clustering, and does not allow straightforward deletion. Multiple hashing uses several hash functions, reducing clustering but adding computational cost.</li>
+<li>Programming languages use different implementations: Java's <code>HashMap</code> uses separate chaining, while Python's <code>dict</code> uses open addressing.</li>
+<li>For hash tables, we want deterministic, efficient hashing with uniformly distributed outputs. Cryptographic hashing also requires collision resistance and an avalanche effect.</li>
+<li>Hash algorithms often use a large prime modulus to promote uniform hash values and reduce collisions.</li>
+<li>Common hash algorithms include MD5, SHA-1, SHA-2, and SHA-3. MD5 is often used for file-integrity checks, while SHA-2 is widely used in security applications and protocols.</li>
+<li>Programming languages usually provide built-in hashing for data types to calculate bucket indices. In general, only immutable objects are hashable.</li>
 </ul>
 <h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
-<p><strong>Q</strong>: When does the time complexity of a hash table degrade to <span class="arithmatex">\(O(n)\)</span>?</p>
-<p>The time complexity of a hash table can degrade to <span class="arithmatex">\(O(n)\)</span> when hash collisions are severe. When the hash function is well-designed, the capacity is set appropriately, and collisions are evenly distributed, the time complexity is <span class="arithmatex">\(O(1)\)</span>. We usually consider the time complexity to be <span class="arithmatex">\(O(1)\)</span> when using built-in hash tables in programming languages.</p>
-<p><strong>Q</strong>: Why not use the hash function <span class="arithmatex">\(f(x) = x\)</span>? This would eliminate collisions.</p>
-<p>Under the hash function <span class="arithmatex">\(f(x) = x\)</span>, each element corresponds to a unique bucket index, which is equivalent to an array. However, the input space is usually much larger than the output space (array length), so the last step of a hash function is often to take the modulo of the array length. In other words, the goal of a hash table is to map a larger state space to a smaller one while providing <span class="arithmatex">\(O(1)\)</span> query efficiency.</p>
-<p><strong>Q</strong>: Why can hash tables be more efficient than arrays, linked lists, or binary trees, even though hash tables are implemented using these structures?</p>
-<p>Firstly, hash tables have higher time efficiency but lower space efficiency. A significant portion of memory in hash tables remains unused.</p>
-<p>Secondly, hash tables are only more time-efficient in specific use cases. If a feature can be implemented with the same time complexity using an array or a linked list, it's usually faster than using a hash table. This is because the computation of the hash function incurs overhead, making the constant factor in the time complexity larger.</p>
-<p>Lastly, the time complexity of hash tables can degrade. For example, in separate chaining, we perform search operations in a linked list or red-black tree, which still risks degrading to <span class="arithmatex">\(O(n)\)</span> time.</p>
-<p><strong>Q</strong>: Does double hashing also have the flaw of not being able to delete elements directly? Can space marked as deleted be reused?</p>
-<p>Double hashing is a form of open addressing, and all open addressing methods have the drawback of not being able to delete elements directly; they require marking elements as deleted. Marked spaces can be reused. When inserting new elements into the hash table, and the hash function points to a position marked as deleted, that position can be used by the new element. This maintains the probing sequence of the hash table while ensuring efficient use of space.</p>
-<p><strong>Q</strong>: Why do hash collisions occur during the search process in linear probing?</p>
-<p>During the search process, the hash function points to the corresponding bucket and key-value pair. If the <code>key</code> doesn't match, it indicates a hash collision. Therefore, linear probing will search downward at a predetermined step size until the correct key-value pair is found or the search fails.</p>
-<p><strong>Q</strong>: Why can expanding a hash table alleviate hash collisions?</p>
-<p>The last step of a hash function often involves taking the modulo of the array length <span class="arithmatex">\(n\)</span>, to keep the output within the array index range. When expanding, the array length <span class="arithmatex">\(n\)</span> changes, and the indices corresponding to the keys may also change. Keys that were previously mapped to the same bucket might be distributed across multiple buckets after expansion, thereby mitigating hash collisions.</p>
-<p><strong>Q</strong>: If the goal is efficient access, why not just use an array directly?</p>
-<p>When the <code>key</code> values are continuous integers within a small range, an array is indeed a simple and efficient choice. But when the <code>key</code> is of another type, such as a string, we need a hash function to map the <code>key</code> to an array index and then store the element in a bucket array. That structure is precisely what a hash table is.</p>
+<p><strong>Q</strong>: When does a hash table have <span class="arithmatex">\(O(n)\)</span> time complexity?</p>
+<p>When collisions are severe, its time complexity degrades to <span class="arithmatex">\(O(n)\)</span>. With a well-designed hash function, suitable capacity, and evenly distributed collisions, it is <span class="arithmatex">\(O(1)\)</span>. We usually assume <span class="arithmatex">\(O(1)\)</span> time when using a language's built-in hash table.</p>
+<p><strong>Q</strong>: Why not use the hash function <span class="arithmatex">\(f(x) = x\)</span>? Wouldn't that avoid collisions?</p>
+<p>With <span class="arithmatex">\(f(x) = x\)</span>, each element has a unique bucket index, making the structure equivalent to an array. But the input space is usually far larger than the output space, the array's length, so the final hashing step usually takes a remainder modulo that length. A hash table maps a larger state space into a smaller one while providing <span class="arithmatex">\(O(1)\)</span> lookup.</p>
+<p><strong>Q</strong>: Hash tables are implemented with arrays, linked lists, and binary trees. How can they be faster than those structures?</p>
+<p>First, greater time efficiency comes at the cost of space efficiency: a significant portion of a hash table's memory is unused.</p>
+<p>Second, the improvement applies only to particular operations. If an array or linked list can perform an operation with the same time complexity, it is usually faster because computing a hash adds overhead and increases the constant factor.</p>
+<p>Finally, hash-table performance can degrade. For example, separate chaining searches within linked lists or red-black trees, and there is still a risk of degeneration to <span class="arithmatex">\(O(n)\)</span> time.</p>
+<p><strong>Q</strong>: Does multiple hashing also prevent direct deletion? Can a deleted slot be reused?</p>
+<p>Multiple hashing is a form of open addressing, and all open-addressing methods require deletion markers rather than simply emptying a slot. A marked slot can be reused. When inserting a new element, if the hash functions locate such a slot, the new element can occupy it. This preserves the probe sequence while making effective use of space.</p>
+<p><strong>Q</strong>: Why can a collision occur during a lookup with linear probing?</p>
+<p>The hash function leads to a bucket and a key-value pair, but the stored <code>key</code> may not match the requested one. That indicates a collision. Linear probing then continues with its predefined step size until it finds the correct pair or determines that none exists.</p>
+<p><strong>Q</strong>: Why does expanding a hash table reduce collisions?</p>
+<p>The final hashing step often takes the remainder modulo the array length <span class="arithmatex">\(n\)</span> to produce a valid index. After expansion, <span class="arithmatex">\(n\)</span> changes, so the index of a given <code>key</code> may change too. Keys that previously shared a bucket may be redistributed among several buckets, reducing collisions.</p>
+<p><strong>Q</strong>: If fast access is the goal, why not just use an array?</p>
+<p>When keys are consecutive integers in a small range, an array is simple and efficient. But for other key types, such as strings, we need a hash function to map keys to array indices and a bucket array to store entries. That structure is a hash table.</p>
 
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-  0.1 About This Book
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-  0.2 How to Use This Book
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-  Chapter 1. Encounter with Algorithms
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2166,7 +2166,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2187,7 +2187,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2208,7 +2208,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2244,7 +2244,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2261,7 +2261,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2287,10 +2287,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#821-implementing-with-element-insertion" class="md-nav__link">
+  <a href="#821-building-through-insertion" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.2.1 &nbsp; Implementing with Element Insertion
+        8.2.1 &nbsp; Building through insertion
       </span>
     </span>
   </a>
@@ -2298,10 +2298,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#822-implementing-through-heapify-traversal" class="md-nav__link">
+  <a href="#822-building-by-sifting-nodes-down" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.2.2 &nbsp; Implementing Through Heapify Traversal
+        8.2.2 &nbsp; Building by sifting nodes down
       </span>
     </span>
   </a>
@@ -2312,7 +2312,7 @@
   <a href="#823-complexity-analysis" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.2.3 &nbsp; Complexity Analysis
+        8.2.3 &nbsp; Complexity analysis
       </span>
     </span>
   </a>
@@ -2342,7 +2342,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2369,7 +2369,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2396,7 +2396,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2460,7 +2460,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2481,7 +2481,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2502,7 +2502,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2529,7 +2529,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2556,7 +2556,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2583,7 +2583,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2610,7 +2610,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2678,7 +2678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2699,7 +2699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2720,7 +2720,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2747,7 +2747,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2774,7 +2774,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2801,7 +2801,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2828,7 +2828,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2855,7 +2855,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2882,7 +2882,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2960,7 +2960,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2981,7 +2981,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3002,7 +3002,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3029,7 +3029,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3056,7 +3056,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3083,7 +3083,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3110,7 +3110,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3137,7 +3137,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3164,7 +3164,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3191,7 +3191,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3218,7 +3218,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3245,7 +3245,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3272,7 +3272,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3299,7 +3299,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3365,7 +3365,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3386,7 +3386,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3407,7 +3407,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3434,7 +3434,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3461,7 +3461,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3488,7 +3488,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3515,7 +3515,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3542,7 +3542,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3608,7 +3608,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3629,7 +3629,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3650,7 +3650,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3677,7 +3677,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3704,7 +3704,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3731,7 +3731,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3758,7 +3758,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3785,7 +3785,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3855,7 +3855,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3876,7 +3876,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3897,7 +3897,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3924,7 +3924,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3951,7 +3951,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3978,7 +3978,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4005,7 +4005,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4032,7 +4032,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4059,7 +4059,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4086,7 +4086,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4152,7 +4152,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4173,7 +4173,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4194,7 +4194,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4221,7 +4221,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4248,7 +4248,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4275,7 +4275,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4302,7 +4302,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4329,7 +4329,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4389,7 +4389,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4410,7 +4410,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4431,7 +4431,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4458,7 +4458,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4485,7 +4485,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4569,6 +4569,78 @@
 
 
 
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+  Print edition (Chinese)
+
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+
+
+
+          </ul>
+        </nav>
+
+    </li>
+
+
+
   </ul>
 </nav>
                   </div>
@@ -4610,10 +4682,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#821-implementing-with-element-insertion" class="md-nav__link">
+  <a href="#821-building-through-insertion" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.2.1 &nbsp; Implementing with Element Insertion
+        8.2.1 &nbsp; Building through insertion
       </span>
     </span>
   </a>
@@ -4621,10 +4693,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#822-implementing-through-heapify-traversal" class="md-nav__link">
+  <a href="#822-building-by-sifting-nodes-down" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.2.2 &nbsp; Implementing Through Heapify Traversal
+        8.2.2 &nbsp; Building by sifting nodes down
       </span>
     </span>
   </a>
@@ -4635,7 +4707,7 @@
   <a href="#823-complexity-analysis" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.2.3 &nbsp; Complexity Analysis
+        8.2.3 &nbsp; Complexity analysis
       </span>
     </span>
   </a>
@@ -4679,39 +4751,39 @@
 
 
 <!-- Page content -->
-<h1 id="82-heap-construction-operation">8.2 &nbsp; Heap Construction Operation<a class="headerlink" href="#82-heap-construction-operation" title="Permanent link">&para;</a></h1>
-<p>In some cases, we want to build a heap using all elements of a list, and this process is called "heap construction operation."</p>
-<h2 id="821-implementing-with-element-insertion">8.2.1 &nbsp; Implementing with Element Insertion<a class="headerlink" href="#821-implementing-with-element-insertion" title="Permanent link">&para;</a></h2>
-<p>We first create an empty heap, then iterate through the list, performing the "element insertion operation" on each element in sequence. This means appending the element to the end of the heap and then performing "bottom-to-top" heapify on that element.</p>
-<p>Each time an element is inserted into the heap, the heap's length increases by one. Since nodes are added to the binary tree sequentially from top to bottom, the heap is constructed "from top to bottom."</p>
-<p>Given <span class="arithmatex">\(n\)</span> elements, each element's insertion operation takes <span class="arithmatex">\(O(\log{n})\)</span> time, so the time complexity of this heap construction method is <span class="arithmatex">\(O(n \log n)\)</span>.</p>
-<h2 id="822-implementing-through-heapify-traversal">8.2.2 &nbsp; Implementing Through Heapify Traversal<a class="headerlink" href="#822-implementing-through-heapify-traversal" title="Permanent link">&para;</a></h2>
-<p>In fact, we can implement a more efficient heap construction method in two steps.</p>
+<h1 id="82-building-a-heap">8.2 &nbsp; Building a heap<a class="headerlink" href="#82-building-a-heap" title="Permanent link">&para;</a></h1>
+<p>Sometimes we want to construct a heap from all the elements of a list. This process is called “building a heap.”</p>
+<h2 id="821-building-through-insertion">8.2.1 &nbsp; Building through insertion<a class="headerlink" href="#821-building-through-insertion" title="Permanent link">&para;</a></h2>
+<p>First create an empty heap, then traverse the list and insert each element: append it to the end of the heap, then sift it upward.</p>
+<p>Each insertion increases the heap's length by one. Since nodes are added to the binary tree from the top down, this constructs the heap “top down.”</p>
+<p>For <span class="arithmatex">\(n\)</span> elements, each insertion takes <span class="arithmatex">\(O(\log{n})\)</span> time, so this method takes <span class="arithmatex">\(O(n \log n)\)</span> overall.</p>
+<h2 id="822-building-by-sifting-nodes-down">8.2.2 &nbsp; Building by sifting nodes down<a class="headerlink" href="#822-building-by-sifting-nodes-down" title="Permanent link">&para;</a></h2>
+<p>We can build a heap more efficiently in two steps.</p>
 <ol>
-<li>Add all elements of the list as-is to the heap, at which point the heap property is not yet satisfied.</li>
-<li>Traverse the heap in reverse order (reverse of level-order traversal), performing "top-to-bottom heapify" on each non-leaf node in sequence.</li>
+<li>Copy all list elements into the heap without changing their order. The heap property may not yet hold.</li>
+<li>Traverse the heap in reverse level order, sifting down each non-leaf node.</li>
 </ol>
-<p><strong>After heapifying a node, the subtree rooted at that node becomes a valid sub-heap</strong>. Since we traverse in reverse order, the heap is constructed "from bottom to top."</p>
-<p>The reason for choosing reverse-order traversal is that it ensures the subtrees beneath the current node are already valid sub-heaps, so heapifying the current node is effective.</p>
-<p>It's worth noting that <strong>since leaf nodes have no children, they are naturally valid sub-heaps and do not require heapification</strong>. As shown in the code below, the last non-leaf node is the parent of the last node; we start from that node and heapify while traversing in reverse order:</p>
+<p><strong>After a node is sifted down, the subtree rooted at that node is a valid heap</strong>. Since the traversal is reversed, the heap is built “bottom up.”</p>
+<p>Reverse traversal ensures that the subtrees below the current node are already valid heaps, which is necessary for sifting down the current node to work correctly.</p>
+<p><strong>Leaf nodes have no children and are already valid heaps, so they need no sifting</strong>. As the code below shows, the last non-leaf node is the parent of the last node. Start there and traverse backward, sifting each node down:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">my_heap.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]):</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Constructor, build heap based on input list"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Add list elements to heap as is</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Constructor: build a heap from the input list"""</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Place the list elements in the heap without changing their order</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">max_heap</span> <span class="o">=</span> <span class="n">nums</span>
-<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># Heapify all nodes except leaf nodes</span>
+<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># Sift down from every non-leaf node</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">parent</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">),</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">sift_down</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Constructor, build heap based on input list */</span>
+<div class="highlight"><span class="filename">my_heap.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Constructor: build a heap from the input list */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="n">MaxHeap</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Add list elements to heap as is</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Place the list elements in the heap without changing their order</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="n">maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span>
-<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="c1">// Heapify all nodes except leaf nodes</span>
+<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="c1">// Sift down from every non-leaf node</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">        </span><span class="n">siftDown</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="p">}</span>
@@ -4719,11 +4791,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Constructor, build heap based on input list */</span>
+<div class="highlight"><span class="filename">my_heap.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Constructor: build a heap from the input list */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="n">MaxHeap</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Add list elements to heap as is</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Place the list elements in the heap without changing their order</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="n">maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span>
-<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="c1">// Heapify all nodes except leaf nodes</span>
+<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="c1">// Sift down from every non-leaf node</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">        </span><span class="n">siftDown</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="p">}</span>
@@ -4731,11 +4803,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Constructor, build heap from input list */</span>
+<div class="highlight"><span class="filename">my_heap.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Constructor: build a heap from an input list */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="n">MaxHeap</span><span class="p">(</span><span class="n">IEnumerable</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Add list elements to heap as is</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Place the list elements in the heap without changing their order</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="n">maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">(</span><span class="n">nums</span><span class="p">);</span>
-<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="c1">// Heapify all nodes except leaf nodes</span>
+<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="c1">// Sift down from every non-leaf node</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="kt">var</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Parent</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">Size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">        </span><span class="n">SiftDown</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
@@ -4744,12 +4816,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Constructor, build heap from slice */</span>
+<div class="highlight"><span class="filename">my_heap.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Constructor: build a heap from a slice */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">newMaxHeap</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">any</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">maxHeap</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Add list elements to heap as is</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Place the list elements in the heap without changing their order</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">h</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">maxHeap</span><span class="p">{</span><span class="nx">data</span><span class="p">:</span><span class="w"> </span><span class="nx">nums</span><span class="p">}</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">parent</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="c1">// Heapify all nodes except leaf nodes</span>
+<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="c1">// Sift down from every non-leaf node</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="nx">h</span><span class="p">.</span><span class="nx">siftDown</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">h</span>
@@ -4757,11 +4829,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Constructor, build heap based on input list */</span>
+<div class="highlight"><span class="filename">my_heap.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Constructor: build a heap from the input list */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">init</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Add list elements to heap as is</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Place the list elements in the heap without changing their order</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="n">maxHeap</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span>
-<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="c1">// Heapify all nodes except leaf nodes</span>
+<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="c1">// Sift down from every non-leaf node</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)).</span><span class="n">reversed</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">        </span><span class="n">siftDown</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="p">}</span>
@@ -4769,11 +4841,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Constructor, build empty heap or build heap from input list */</span>
+<div class="highlight"><span class="filename">my_heap.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Constructor: create an empty heap or build a heap from the input list */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kr">constructor</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Add list elements to heap as is</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Place the list elements in the heap without changing their order</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">#maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="p">[]</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="p">[...</span><span class="nx">nums</span><span class="p">];</span>
-<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="c1">// Heapify all nodes except leaf nodes</span>
+<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="c1">// Sift down from every non-leaf node</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#parent</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#siftDown</span><span class="p">(</span><span class="nx">i</span><span class="p">);</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="p">}</span>
@@ -4781,11 +4853,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Constructor, build empty heap or build heap from input list */</span>
+<div class="highlight"><span class="filename">my_heap.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Constructor: create an empty heap or build a heap from the input list */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kr">constructor</span><span class="p">(</span><span class="nx">nums?</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Add list elements to heap as is</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Place the list elements in the heap without changing their order</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="p">[]</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="p">[...</span><span class="nx">nums</span><span class="p">];</span>
-<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="c1">// Heapify all nodes except leaf nodes</span>
+<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="c1">// Sift down from every non-leaf node</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">parent</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">siftDown</span><span class="p">(</span><span class="nx">i</span><span class="p">);</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="p">}</span>
@@ -4793,11 +4865,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Constructor, build heap based on input list */</span>
+<div class="highlight"><span class="filename">my_heap.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Constructor: build a heap from the input list */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="n">MaxHeap</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Add list elements to heap as is</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Place the list elements in the heap without changing their order</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="n">_maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span>
-<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// Heapify all nodes except leaf nodes</span>
+<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// Sift down from every non-leaf node</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_parent</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">    </span><span class="n">siftDown</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="p">}</span>
@@ -4805,11 +4877,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Constructor, build heap based on input list */</span>
+<div class="highlight"><span class="filename">my_heap.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Constructor: build a heap from the input list */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">new</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">Self</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Add list elements to heap as is</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Place the list elements in the heap without changing their order</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">heap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MaxHeap</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">max_heap</span><span class="p">:</span><span class="w"> </span><span class="nc">nums</span><span class="w"> </span><span class="p">};</span>
-<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="c1">// Heapify all nodes except leaf nodes</span>
+<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="c1">// Sift down from every non-leaf node</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="o">..=</span><span class="bp">Self</span><span class="p">::</span><span class="n">parent</span><span class="p">(</span><span class="n">heap</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)).</span><span class="n">rev</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">        </span><span class="n">heap</span><span class="p">.</span><span class="n">sift_down</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="p">}</span>
@@ -4818,14 +4890,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Constructor, build heap from slice */</span>
+<div class="highlight"><span class="filename">my_heap.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Constructor: build a heap from the input array */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="nf">newMaxHeap</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Push all elements to heap</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Place all elements in the heap</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="n">maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">MaxHeap</span><span class="p">));</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="p">;</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="n">memcpy</span><span class="p">(</span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">data</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">        </span><span class="c1">// Heapify all nodes except leaf nodes</span>
+<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">        </span><span class="c1">// Sift down from every non-leaf node</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">        </span><span class="n">siftDown</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">;</span>
@@ -4835,32 +4907,32 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">my_heap.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Max heap */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">MaxHeap</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;?</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Use list instead of array, no need to consider capacity expansion</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Use a resizable list instead of an array to avoid managing capacity growth</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">val</span><span class="w"> </span><span class="nv">maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a>
-<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="cm">/* Constructor, build heap based on input list */</span>
+<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="cm">/* Constructor: build a heap from the input list */</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="k">init</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">        </span><span class="c1">// Add list elements to heap as is</span>
+<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">        </span><span class="c1">// Place the list elements in the heap without changing their order</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">        </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">addAll</span><span class="p">(</span><span class="n">nums</span><span class="o">!!</span><span class="p">)</span>
-<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">        </span><span class="c1">// Heapify all nodes except leaf nodes</span>
+<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">        </span><span class="c1">// Sift down from every non-leaf node</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="n">downTo</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">            </span><span class="n">siftDown</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a>
-<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="cm">/* Get index of left child node */</span>
+<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="cm">/* Get the left child index */</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">fun</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a>
-<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">    </span><span class="cm">/* Get index of right child node */</span>
+<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">    </span><span class="cm">/* Get the right child index */</span>
 <a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">fun</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">2</span>
 <a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a>
-<a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w">    </span><span class="cm">/* Get index of parent node */</span>
+<a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w">    </span><span class="cm">/* Get the parent index */</span>
 <a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">fun</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="c1">// Floor division</span>
+<a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="c1">// Integer division (truncates toward zero in Kotlin)</span>
 <a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a>
 <a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a><span class="w">    </span><span class="cm">/* Swap elements */</span>
@@ -4870,81 +4942,81 @@
 <a id="__codelineno-11-35" name="__codelineno-11-35" href="#__codelineno-11-35"></a><span class="w">        </span><span class="n">maxHeap</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span>
 <a id="__codelineno-11-36" name="__codelineno-11-36" href="#__codelineno-11-36"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-37" name="__codelineno-11-37" href="#__codelineno-11-37"></a>
-<a id="__codelineno-11-38" name="__codelineno-11-38" href="#__codelineno-11-38"></a><span class="w">    </span><span class="cm">/* Get heap size */</span>
+<a id="__codelineno-11-38" name="__codelineno-11-38" href="#__codelineno-11-38"></a><span class="w">    </span><span class="cm">/* Get the heap size */</span>
 <a id="__codelineno-11-39" name="__codelineno-11-39" href="#__codelineno-11-39"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">size</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-40" name="__codelineno-11-40" href="#__codelineno-11-40"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-11-41" name="__codelineno-11-41" href="#__codelineno-11-41"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-42" name="__codelineno-11-42" href="#__codelineno-11-42"></a>
-<a id="__codelineno-11-43" name="__codelineno-11-43" href="#__codelineno-11-43"></a><span class="w">    </span><span class="cm">/* Check if heap is empty */</span>
+<a id="__codelineno-11-43" name="__codelineno-11-43" href="#__codelineno-11-43"></a><span class="w">    </span><span class="cm">/* Check whether the heap is empty */</span>
 <a id="__codelineno-11-44" name="__codelineno-11-44" href="#__codelineno-11-44"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">():</span><span class="w"> </span><span class="kt">Boolean</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-45" name="__codelineno-11-45" href="#__codelineno-11-45"></a><span class="w">        </span><span class="cm">/* Check if heap is empty */</span>
+<a id="__codelineno-11-45" name="__codelineno-11-45" href="#__codelineno-11-45"></a><span class="w">        </span><span class="cm">/* Check whether the heap is empty */</span>
 <a id="__codelineno-11-46" name="__codelineno-11-46" href="#__codelineno-11-46"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-11-47" name="__codelineno-11-47" href="#__codelineno-11-47"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-48" name="__codelineno-11-48" href="#__codelineno-11-48"></a>
-<a id="__codelineno-11-49" name="__codelineno-11-49" href="#__codelineno-11-49"></a><span class="w">    </span><span class="cm">/* Access top element */</span>
+<a id="__codelineno-11-49" name="__codelineno-11-49" href="#__codelineno-11-49"></a><span class="w">    </span><span class="cm">/* Access the top element of the heap */</span>
 <a id="__codelineno-11-50" name="__codelineno-11-50" href="#__codelineno-11-50"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">peek</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-51" name="__codelineno-11-51" href="#__codelineno-11-51"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">[</span><span class="m">0</span><span class="o">]</span>
 <a id="__codelineno-11-52" name="__codelineno-11-52" href="#__codelineno-11-52"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-53" name="__codelineno-11-53" href="#__codelineno-11-53"></a>
-<a id="__codelineno-11-54" name="__codelineno-11-54" href="#__codelineno-11-54"></a><span class="w">    </span><span class="cm">/* Element enters heap */</span>
+<a id="__codelineno-11-54" name="__codelineno-11-54" href="#__codelineno-11-54"></a><span class="w">    </span><span class="cm">/* Push an element onto the heap */</span>
 <a id="__codelineno-11-55" name="__codelineno-11-55" href="#__codelineno-11-55"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-56" name="__codelineno-11-56" href="#__codelineno-11-56"></a><span class="w">        </span><span class="c1">// Add node</span>
+<a id="__codelineno-11-56" name="__codelineno-11-56" href="#__codelineno-11-56"></a><span class="w">        </span><span class="c1">// Add a node</span>
 <a id="__codelineno-11-57" name="__codelineno-11-57" href="#__codelineno-11-57"></a><span class="w">        </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">_val</span><span class="p">)</span>
-<a id="__codelineno-11-58" name="__codelineno-11-58" href="#__codelineno-11-58"></a><span class="w">        </span><span class="c1">// Heapify from bottom to top</span>
+<a id="__codelineno-11-58" name="__codelineno-11-58" href="#__codelineno-11-58"></a><span class="w">        </span><span class="c1">// Sift up</span>
 <a id="__codelineno-11-59" name="__codelineno-11-59" href="#__codelineno-11-59"></a><span class="w">        </span><span class="n">siftUp</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-11-60" name="__codelineno-11-60" href="#__codelineno-11-60"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-61" name="__codelineno-11-61" href="#__codelineno-11-61"></a>
-<a id="__codelineno-11-62" name="__codelineno-11-62" href="#__codelineno-11-62"></a><span class="w">    </span><span class="cm">/* Starting from node i, heapify from bottom to top */</span>
+<a id="__codelineno-11-62" name="__codelineno-11-62" href="#__codelineno-11-62"></a><span class="w">    </span><span class="cm">/* Sift up starting from node i */</span>
 <a id="__codelineno-11-63" name="__codelineno-11-63" href="#__codelineno-11-63"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">fun</span><span class="w"> </span><span class="nf">siftUp</span><span class="p">(</span><span class="nb">it</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-64" name="__codelineno-11-64" href="#__codelineno-11-64"></a><span class="w">        </span><span class="c1">// Kotlin function parameters are immutable, so create temporary variable</span>
+<a id="__codelineno-11-64" name="__codelineno-11-64" href="#__codelineno-11-64"></a><span class="w">        </span><span class="c1">// Kotlin function parameters are immutable, so create a temporary variable</span>
 <a id="__codelineno-11-65" name="__codelineno-11-65" href="#__codelineno-11-65"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">it</span>
 <a id="__codelineno-11-66" name="__codelineno-11-66" href="#__codelineno-11-66"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-67" name="__codelineno-11-67" href="#__codelineno-11-67"></a><span class="w">            </span><span class="c1">// Get parent node of node i</span>
+<a id="__codelineno-11-67" name="__codelineno-11-67" href="#__codelineno-11-67"></a><span class="w">            </span><span class="c1">// Get the parent of node i</span>
 <a id="__codelineno-11-68" name="__codelineno-11-68" href="#__codelineno-11-68"></a><span class="w">            </span><span class="kd">val</span><span class="w"> </span><span class="nv">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-<a id="__codelineno-11-69" name="__codelineno-11-69" href="#__codelineno-11-69"></a><span class="w">            </span><span class="c1">// When "crossing root node" or "node needs no repair", end heapify</span>
+<a id="__codelineno-11-69" name="__codelineno-11-69" href="#__codelineno-11-69"></a><span class="w">            </span><span class="c1">// Stop sifting when the root has been passed or the node needs no repair</span>
 <a id="__codelineno-11-70" name="__codelineno-11-70" href="#__codelineno-11-70"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">[</span><span class="n">p</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="k">break</span>
-<a id="__codelineno-11-71" name="__codelineno-11-71" href="#__codelineno-11-71"></a><span class="w">            </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-11-71" name="__codelineno-11-71" href="#__codelineno-11-71"></a><span class="w">            </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-11-72" name="__codelineno-11-72" href="#__codelineno-11-72"></a><span class="w">            </span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="p">)</span>
-<a id="__codelineno-11-73" name="__codelineno-11-73" href="#__codelineno-11-73"></a><span class="w">            </span><span class="c1">// Loop upward heapify</span>
+<a id="__codelineno-11-73" name="__codelineno-11-73" href="#__codelineno-11-73"></a><span class="w">            </span><span class="c1">// Continue sifting up</span>
 <a id="__codelineno-11-74" name="__codelineno-11-74" href="#__codelineno-11-74"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">p</span>
 <a id="__codelineno-11-75" name="__codelineno-11-75" href="#__codelineno-11-75"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-11-76" name="__codelineno-11-76" href="#__codelineno-11-76"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-77" name="__codelineno-11-77" href="#__codelineno-11-77"></a>
-<a id="__codelineno-11-78" name="__codelineno-11-78" href="#__codelineno-11-78"></a><span class="w">    </span><span class="cm">/* Element exits heap */</span>
+<a id="__codelineno-11-78" name="__codelineno-11-78" href="#__codelineno-11-78"></a><span class="w">    </span><span class="cm">/* Pop an element from the heap */</span>
 <a id="__codelineno-11-79" name="__codelineno-11-79" href="#__codelineno-11-79"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">pop</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-80" name="__codelineno-11-80" href="#__codelineno-11-80"></a><span class="w">        </span><span class="c1">// Handle empty case</span>
+<a id="__codelineno-11-80" name="__codelineno-11-80" href="#__codelineno-11-80"></a><span class="w">        </span><span class="c1">// Handle an empty heap</span>
 <a id="__codelineno-11-81" name="__codelineno-11-81" href="#__codelineno-11-81"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">()</span>
-<a id="__codelineno-11-82" name="__codelineno-11-82" href="#__codelineno-11-82"></a><span class="w">        </span><span class="c1">// Delete node</span>
+<a id="__codelineno-11-82" name="__codelineno-11-82" href="#__codelineno-11-82"></a><span class="w">        </span><span class="c1">// Swap the root with the rightmost leaf on the last level (swap the first and last elements)</span>
 <a id="__codelineno-11-83" name="__codelineno-11-83" href="#__codelineno-11-83"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
-<a id="__codelineno-11-84" name="__codelineno-11-84" href="#__codelineno-11-84"></a><span class="w">        </span><span class="c1">// Remove node</span>
+<a id="__codelineno-11-84" name="__codelineno-11-84" href="#__codelineno-11-84"></a><span class="w">        </span><span class="c1">// Remove a node</span>
 <a id="__codelineno-11-85" name="__codelineno-11-85" href="#__codelineno-11-85"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">_val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">removeAt</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
-<a id="__codelineno-11-86" name="__codelineno-11-86" href="#__codelineno-11-86"></a><span class="w">        </span><span class="c1">// Return top element</span>
+<a id="__codelineno-11-86" name="__codelineno-11-86" href="#__codelineno-11-86"></a><span class="w">        </span><span class="c1">// Sift down</span>
 <a id="__codelineno-11-87" name="__codelineno-11-87" href="#__codelineno-11-87"></a><span class="w">        </span><span class="n">siftDown</span><span class="p">(</span><span class="m">0</span><span class="p">)</span>
-<a id="__codelineno-11-88" name="__codelineno-11-88" href="#__codelineno-11-88"></a><span class="w">        </span><span class="c1">// Return heap top element</span>
+<a id="__codelineno-11-88" name="__codelineno-11-88" href="#__codelineno-11-88"></a><span class="w">        </span><span class="c1">// Return the top element of the heap</span>
 <a id="__codelineno-11-89" name="__codelineno-11-89" href="#__codelineno-11-89"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">_val</span>
 <a id="__codelineno-11-90" name="__codelineno-11-90" href="#__codelineno-11-90"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-91" name="__codelineno-11-91" href="#__codelineno-11-91"></a>
-<a id="__codelineno-11-92" name="__codelineno-11-92" href="#__codelineno-11-92"></a><span class="w">    </span><span class="cm">/* Starting from node i, heapify from top to bottom */</span>
+<a id="__codelineno-11-92" name="__codelineno-11-92" href="#__codelineno-11-92"></a><span class="w">    </span><span class="cm">/* Sift down starting from node i */</span>
 <a id="__codelineno-11-93" name="__codelineno-11-93" href="#__codelineno-11-93"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">fun</span><span class="w"> </span><span class="nf">siftDown</span><span class="p">(</span><span class="nb">it</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-94" name="__codelineno-11-94" href="#__codelineno-11-94"></a><span class="w">        </span><span class="c1">// Kotlin function parameters are immutable, so create temporary variable</span>
+<a id="__codelineno-11-94" name="__codelineno-11-94" href="#__codelineno-11-94"></a><span class="w">        </span><span class="c1">// Kotlin function parameters are immutable, so create a temporary variable</span>
 <a id="__codelineno-11-95" name="__codelineno-11-95" href="#__codelineno-11-95"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">it</span>
 <a id="__codelineno-11-96" name="__codelineno-11-96" href="#__codelineno-11-96"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-97" name="__codelineno-11-97" href="#__codelineno-11-97"></a><span class="w">            </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-11-97" name="__codelineno-11-97" href="#__codelineno-11-97"></a><span class="w">            </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-11-98" name="__codelineno-11-98" href="#__codelineno-11-98"></a><span class="w">            </span><span class="kd">val</span><span class="w"> </span><span class="nv">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-11-99" name="__codelineno-11-99" href="#__codelineno-11-99"></a><span class="w">            </span><span class="kd">val</span><span class="w"> </span><span class="nv">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-11-100" name="__codelineno-11-100" href="#__codelineno-11-100"></a><span class="w">            </span><span class="kd">var</span><span class="w"> </span><span class="nv">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-11-101" name="__codelineno-11-101" href="#__codelineno-11-101"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">[</span><span class="n">l</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span>
 <a id="__codelineno-11-102" name="__codelineno-11-102" href="#__codelineno-11-102"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">[</span><span class="n">r</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span>
-<a id="__codelineno-11-103" name="__codelineno-11-103" href="#__codelineno-11-103"></a><span class="w">            </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-11-103" name="__codelineno-11-103" href="#__codelineno-11-103"></a><span class="w">            </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-11-104" name="__codelineno-11-104" href="#__codelineno-11-104"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="k">break</span>
-<a id="__codelineno-11-105" name="__codelineno-11-105" href="#__codelineno-11-105"></a><span class="w">            </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-11-105" name="__codelineno-11-105" href="#__codelineno-11-105"></a><span class="w">            </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-11-106" name="__codelineno-11-106" href="#__codelineno-11-106"></a><span class="w">            </span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">ma</span><span class="p">)</span>
-<a id="__codelineno-11-107" name="__codelineno-11-107" href="#__codelineno-11-107"></a><span class="w">            </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-11-107" name="__codelineno-11-107" href="#__codelineno-11-107"></a><span class="w">            </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-11-108" name="__codelineno-11-108" href="#__codelineno-11-108"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span>
 <a id="__codelineno-11-109" name="__codelineno-11-109" href="#__codelineno-11-109"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-11-110" name="__codelineno-11-110" href="#__codelineno-11-110"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-111" name="__codelineno-11-111" href="#__codelineno-11-111"></a>
-<a id="__codelineno-11-112" name="__codelineno-11-112" href="#__codelineno-11-112"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-11-112" name="__codelineno-11-112" href="#__codelineno-11-112"></a><span class="w">    </span><span class="cm">/* Print the heap (binary tree) */</span>
 <a id="__codelineno-11-113" name="__codelineno-11-113" href="#__codelineno-11-113"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-114" name="__codelineno-11-114" href="#__codelineno-11-114"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">PriorityQueue</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">-&gt;</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-11-115" name="__codelineno-11-115" href="#__codelineno-11-115"></a><span class="w">        </span><span class="n">queue</span><span class="p">.</span><span class="na">addAll</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">)</span>
@@ -4954,11 +5026,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Constructor, build heap from input list ###</span>
+<div class="highlight"><span class="filename">my_heap.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Constructor: build a heap from the input list ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">initialize</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Add list elements to heap as is</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Place the list elements in the heap without changing their order</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="vi">@max_heap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Heapify all nodes except leaf nodes</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Sift down from every non-leaf node</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="n">parent</span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">downto</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">    </span><span class="n">sift_down</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="k">end</span>
@@ -4967,33 +5039,38 @@
 </div>
 </div>
 </div>
-<h2 id="823-complexity-analysis">8.2.3 &nbsp; Complexity Analysis<a class="headerlink" href="#823-complexity-analysis" title="Permanent link">&para;</a></h2>
-<p>Next, let's attempt to derive the time complexity of this second heap construction method.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20MaxHeap%3A%0A%20%20%20%20%22%22%22Max%20heap%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%3A%20build%20a%20heap%20from%20the%20input%20list%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Place%20the%20list%20elements%20in%20the%20heap%20without%20changing%20their%20order%0A%20%20%20%20%20%20%20%20self.max_heap%20%3D%20nums%0A%20%20%20%20%20%20%20%20%23%20Sift%20down%20from%20every%20non-leaf%20node%0A%20%20%20%20%20%20%20%20for%20i%20in%20range%28self.parent%28self.size%28%29%20-%201%29,%20-1,%20-1%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.sift_down%28i%29%0A%0A%20%20%20%20def%20left%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20left%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%201%0A%0A%20%20%20%20def%20right%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20right%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%202%0A%0A%20%20%20%20def%20parent%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20parent%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%20%28i%20-%201%29%20//%202%20%20%23%20Floor%20division%0A%0A%20%20%20%20def%20swap%28self,%20i%3A%20int,%20j%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Swap%20elements%22%22%22%0A%20%20%20%20%20%20%20%20self.max_heap%5Bi%5D,%20self.max_heap%5Bj%5D%20%3D%20self.max_heap%5Bj%5D,%20self.max_heap%5Bi%5D%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20heap%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self.max_heap%29%0A%0A%20%20%20%20def%20sift_down%28self,%20i%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Sift%20down%20starting%20from%20node%20i%22%22%22%0A%20%20%20%20%20%20%20%20while%20True%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Find%20the%20node%20with%20the%20largest%20value%20among%20i,%20l,%20and%20r,%20and%20store%20its%20index%20in%20ma%0A%20%20%20%20%20%20%20%20%20%20%20%20l,%20r,%20ma%20%3D%20self.left%28i%29,%20self.right%28i%29,%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20l%20%3C%20self.size%28%29%20and%20self.max_heap%5Bl%5D%20%3E%20self.max_heap%5Bma%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20ma%20%3D%20l%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20r%20%3C%20self.size%28%29%20and%20self.max_heap%5Br%5D%20%3E%20self.max_heap%5Bma%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20ma%20%3D%20r%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Stop%20if%20node%20i%20is%20largest%20or%20both%20child%20indices%20l%20and%20r%20are%20out%20of%20bounds%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20ma%20%3D%3D%20i%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Swap%20the%20two%20nodes%0A%20%20%20%20%20%20%20%20%20%20%20%20self.swap%28i,%20ma%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Continue%20sifting%20down%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20ma%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20max%20heap%0A%20%20%20%20max_heap%20%3D%20MaxHeap%28%5B1,%202,%203,%204,%205%5D%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20MaxHeap%3A%0A%20%20%20%20%22%22%22Max%20heap%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%3A%20build%20a%20heap%20from%20the%20input%20list%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Place%20the%20list%20elements%20in%20the%20heap%20without%20changing%20their%20order%0A%20%20%20%20%20%20%20%20self.max_heap%20%3D%20nums%0A%20%20%20%20%20%20%20%20%23%20Sift%20down%20from%20every%20non-leaf%20node%0A%20%20%20%20%20%20%20%20for%20i%20in%20range%28self.parent%28self.size%28%29%20-%201%29,%20-1,%20-1%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.sift_down%28i%29%0A%0A%20%20%20%20def%20left%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20left%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%201%0A%0A%20%20%20%20def%20right%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20right%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%202%0A%0A%20%20%20%20def%20parent%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20parent%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%20%28i%20-%201%29%20//%202%20%20%23%20Floor%20division%0A%0A%20%20%20%20def%20swap%28self,%20i%3A%20int,%20j%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Swap%20elements%22%22%22%0A%20%20%20%20%20%20%20%20self.max_heap%5Bi%5D,%20self.max_heap%5Bj%5D%20%3D%20self.max_heap%5Bj%5D,%20self.max_heap%5Bi%5D%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20heap%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self.max_heap%29%0A%0A%20%20%20%20def%20sift_down%28self,%20i%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Sift%20down%20starting%20from%20node%20i%22%22%22%0A%20%20%20%20%20%20%20%20while%20True%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Find%20the%20node%20with%20the%20largest%20value%20among%20i,%20l,%20and%20r,%20and%20store%20its%20index%20in%20ma%0A%20%20%20%20%20%20%20%20%20%20%20%20l,%20r,%20ma%20%3D%20self.left%28i%29,%20self.right%28i%29,%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20l%20%3C%20self.size%28%29%20and%20self.max_heap%5Bl%5D%20%3E%20self.max_heap%5Bma%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20ma%20%3D%20l%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20r%20%3C%20self.size%28%29%20and%20self.max_heap%5Br%5D%20%3E%20self.max_heap%5Bma%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20ma%20%3D%20r%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Stop%20if%20node%20i%20is%20largest%20or%20both%20child%20indices%20l%20and%20r%20are%20out%20of%20bounds%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20ma%20%3D%3D%20i%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Swap%20the%20two%20nodes%0A%20%20%20%20%20%20%20%20%20%20%20%20self.swap%28i,%20ma%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Continue%20sifting%20down%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20ma%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20max%20heap%0A%20%20%20%20max_heap%20%3D%20MaxHeap%28%5B1,%202,%203,%204,%205%5D%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="823-complexity-analysis">8.2.3 &nbsp; Complexity analysis<a class="headerlink" href="#823-complexity-analysis" title="Permanent link">&para;</a></h2>
+<p>Let us derive the time complexity of the second method.</p>
 <ul>
-<li>Assuming the complete binary tree has <span class="arithmatex">\(n\)</span> nodes, then the number of leaf nodes is <span class="arithmatex">\((n + 1) / 2\)</span>, where <span class="arithmatex">\(/\)</span> is floor division. Therefore, the number of nodes that need heapification is <span class="arithmatex">\(n / 2\)</span>.</li>
-<li>In the top-to-bottom heapify process, each node can sink at most to a leaf node, so the maximum number of iterations is the height of the binary tree, <span class="arithmatex">\(\log n\)</span>.</li>
+<li>A complete binary tree with <span class="arithmatex">\(n\)</span> nodes has <span class="arithmatex">\((n + 1) / 2\)</span> leaves, where <span class="arithmatex">\(/\)</span> denotes integer division rounded down. Therefore, <span class="arithmatex">\(n / 2\)</span> nodes need sifting.</li>
+<li>A node can move at most as far as a leaf during sifting, so the maximum number of iterations is the tree height, <span class="arithmatex">\(\log n\)</span>.</li>
 </ul>
-<p>Multiplying these two together, we get a time complexity of <span class="arithmatex">\(O(n \log n)\)</span> for the heap construction process. <strong>However, this estimate is not accurate because it doesn't account for the property that binary trees have far more nodes at lower levels than at upper levels</strong>.</p>
-<p>Let's perform a more accurate calculation. To simplify the analysis, assume a "perfect binary tree" with <span class="arithmatex">\(n\)</span> nodes and height <span class="arithmatex">\(h\)</span>; this assumption does not affect the correctness of the result.</p>
-<p><img alt="Node count at each level of a perfect binary tree" class="animation-figure" src="../build_heap.assets/heapify_operations_count.png" /></p>
-<p align="center"> Figure 8-5 &nbsp; Node count at each level of a perfect binary tree </p>
+<p>Multiplying these gives a time complexity of <span class="arithmatex">\(O(n \log n)\)</span>. <strong>However, this estimate is not tight, because it ignores the fact that there are far more nodes near the bottom of the tree than near the top</strong>.</p>
+<p>For a more accurate calculation, assume a perfect binary tree with <span class="arithmatex">\(n\)</span> nodes and height <span class="arithmatex">\(h\)</span>. This simplifies the calculation without affecting the validity of the result.</p>
+<p><img alt="Number of nodes at each level of a perfect binary tree" class="animation-figure" src="../build_heap.assets/heapify_operations_count.png" /></p>
+<p align="center"> Figure 8-5 &nbsp; Number of nodes at each level of a perfect binary tree </p>
 
-<p>As shown in Figure 8-5, the maximum number of iterations for a node's "top-to-bottom heapify" equals the distance from that node to a leaf node, which is precisely the node's height. Therefore, we can sum the "number of nodes <span class="arithmatex">\(\times\)</span> node height" at each level to <strong>obtain the total number of heapify iterations for all nodes</strong>.</p>
+<p>As shown above, the maximum number of sift-down iterations for a node equals its distance to a leaf, which is its height. Summing “number of nodes <span class="arithmatex">\(\times\)</span> node height” over all levels therefore gives <strong>the total number of sifting iterations across all nodes</strong>.</p>
 <div class="arithmatex">\[
 T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \dots + 2^{(h-1)}\times1
 \]</div>
-<p>Simplifying the expression above requires some high-school sequence algebra. First, multiply <span class="arithmatex">\(T(h)\)</span> by <span class="arithmatex">\(2\)</span> to get:</p>
+<p>To simplify this expression, use a standard series technique. First multiply <span class="arithmatex">\(T(h)\)</span> by <span class="arithmatex">\(2\)</span>:</p>
 <div class="arithmatex">\[
 \begin{aligned}
 T(h) &amp; = 2^0h + 2^1(h-1) + 2^2(h-2) + \dots + 2^{h-1}\times1 \newline
 2 T(h) &amp; = 2^1h + 2^2(h-1) + 2^3(h-2) + \dots + 2^{h}\times1 \newline
 \end{aligned}
 \]</div>
-<p>Using subtraction of shifted sums, subtract the first equation <span class="arithmatex">\(T(h)\)</span> from the second equation <span class="arithmatex">\(2 T(h)\)</span> to get:</p>
+<p>Subtract the first expression, <span class="arithmatex">\(T(h)\)</span>, from the shifted expression, <span class="arithmatex">\(2 T(h)\)</span>:</p>
 <div class="arithmatex">\[
 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \dots + 2^{h-1} + 2^h
 \]</div>
-<p>Observing the above expression, we find that <span class="arithmatex">\(T(h)\)</span> is a geometric series, which can be calculated directly using the sum formula, yielding a time complexity of:</p>
+<p>The powers of two in <span class="arithmatex">\(T(h)\)</span> form a geometric series. Applying the sum formula gives:</p>
 <div class="arithmatex">\[
 \begin{aligned}
 T(h) &amp; = 2 \frac{1 - 2^h}{1 - 2} - h \newline
@@ -5001,7 +5078,7 @@ T(h) &amp; = 2 \frac{1 - 2^h}{1 - 2} - h \newline
 &amp; = O(2^h)
 \end{aligned}
 \]</div>
-<p>Furthermore, a perfect binary tree with height <span class="arithmatex">\(h\)</span> has <span class="arithmatex">\(n = 2^{h+1} - 1\)</span> nodes, so the complexity is <span class="arithmatex">\(O(2^h) = O(n)\)</span>. This derivation shows that <strong>the time complexity of building a heap from an input list is <span class="arithmatex">\(O(n)\)</span>, which is highly efficient</strong>.</p>
+<p>A perfect binary tree of height <span class="arithmatex">\(h\)</span> has <span class="arithmatex">\(n = 2^{h+1} - 1\)</span> nodes, so <span class="arithmatex">\(O(2^h) = O(n)\)</span>. This shows that <strong>building a heap from an input list takes <span class="arithmatex">\(O(n)\)</span> time and is highly efficient</strong>.</p>
 
 <!-- Source file information -->
 
@@ -5024,7 +5101,7 @@ aria-label="Footer"
   <a
     href="../heap/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 8.1 Heap"
+    aria-label="Previous: 8.1   Heaps"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -5036,7 +5113,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        8.1 Heap
+        8.1 &nbsp; Heaps
       </div>
     </div>
   </a>
@@ -5048,7 +5125,7 @@ aria-label="Footer"
   <a
     href="../top_k/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 8.3 Top-k Problem"
+    aria-label="Next: 8.3   The top-k problem"
     rel="next"
   >
     <div class="md-footer__title">
@@ -5056,7 +5133,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        8.3 Top-k Problem
+        8.3 &nbsp; The top-k problem
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -5073,7 +5150,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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@@ -6,7 +6,7 @@
       <meta charset="utf-8">
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+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
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-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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@@ -177,9 +177,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Light mode"  type="radio" name="__palette" id="__palette_1">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2166,7 +2166,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2187,7 +2187,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2208,7 +2208,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2235,7 +2235,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2262,7 +2262,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2289,7 +2289,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2325,7 +2325,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2342,7 +2342,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2368,22 +2368,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#851-concept-review" class="md-nav__link">
+  <a href="#851-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.5.1 &nbsp; Concept Review
+        8.5.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="8.5.1 Concept Review">
+    <nav class="md-nav" aria-label="8.5.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-how-does-the-heap-change-after-inserting-10" class="md-nav__link">
+  <a href="#1-restoring-the-heap-after-inserting-10" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; How Does the Heap Change After Inserting 10?
+        1. &nbsp; Restoring the heap after inserting 10
       </span>
     </span>
   </a>
@@ -2391,10 +2391,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-check-parentchild-relationships-in-a-min-heap" class="md-nav__link">
+  <a href="#2-checking-parent-child-relationships-in-a-min-heap" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Check Parent–Child Relationships in a Min Heap
+        2. &nbsp; Checking parent-child relationships in a min heap
       </span>
     </span>
   </a>
@@ -2402,10 +2402,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-keep-the-three-largest-numbers-with-a-min-heap" class="md-nav__link">
+  <a href="#3-keeping-the-three-largest-numbers-with-a-min-heap" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Keep the Three Largest Numbers with a Min Heap
+        3. &nbsp; Keeping the three largest numbers with a min heap
       </span>
     </span>
   </a>
@@ -2421,19 +2421,19 @@
   <a href="#852-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.5.2 &nbsp; Programming Exercises
+        8.5.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="8.5.2 Programming Exercises">
+    <nav class="md-nav" aria-label="8.5.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-find-the-kth-largest-element-in-an-array" class="md-nav__link">
+  <a href="#1-finding-the-kth-largest-element-in-an-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Find the Kth Largest Element in an Array
+        1. &nbsp; Finding the kth largest element in an array
       </span>
     </span>
   </a>
@@ -2505,7 +2505,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2526,7 +2526,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2547,7 +2547,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2574,7 +2574,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2601,7 +2601,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2628,7 +2628,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2655,7 +2655,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2723,7 +2723,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2744,7 +2744,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2765,7 +2765,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2792,7 +2792,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2819,7 +2819,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2846,7 +2846,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2873,7 +2873,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2900,7 +2900,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2927,7 +2927,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -3005,7 +3005,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3026,7 +3026,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3047,7 +3047,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3074,7 +3074,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3101,7 +3101,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3128,7 +3128,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3155,7 +3155,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3182,7 +3182,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3209,7 +3209,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3236,7 +3236,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3263,7 +3263,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3290,7 +3290,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3317,7 +3317,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3344,7 +3344,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3410,7 +3410,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3431,7 +3431,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3452,7 +3452,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3479,7 +3479,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3506,7 +3506,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3533,7 +3533,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3560,7 +3560,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3587,7 +3587,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3653,7 +3653,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3674,7 +3674,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3695,7 +3695,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3722,7 +3722,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3749,7 +3749,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3776,7 +3776,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3803,7 +3803,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3830,7 +3830,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3900,7 +3900,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3921,7 +3921,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3942,7 +3942,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3969,7 +3969,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3996,7 +3996,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4023,7 +4023,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4050,7 +4050,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4077,7 +4077,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4104,7 +4104,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4131,7 +4131,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4197,7 +4197,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4218,7 +4218,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4239,7 +4239,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4266,7 +4266,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4293,7 +4293,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4320,7 +4320,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4347,7 +4347,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4374,7 +4374,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4434,7 +4434,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4455,7 +4455,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4476,7 +4476,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4503,7 +4503,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4530,7 +4530,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4614,6 +4614,78 @@
 
 
 
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@@ -4655,22 +4727,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#851-concept-review" class="md-nav__link">
+  <a href="#851-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.5.1 &nbsp; Concept Review
+        8.5.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="8.5.1 Concept Review">
+    <nav class="md-nav" aria-label="8.5.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-how-does-the-heap-change-after-inserting-10" class="md-nav__link">
+  <a href="#1-restoring-the-heap-after-inserting-10" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; How Does the Heap Change After Inserting 10?
+        1. &nbsp; Restoring the heap after inserting 10
       </span>
     </span>
   </a>
@@ -4678,10 +4750,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-check-parentchild-relationships-in-a-min-heap" class="md-nav__link">
+  <a href="#2-checking-parent-child-relationships-in-a-min-heap" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Check Parent–Child Relationships in a Min Heap
+        2. &nbsp; Checking parent-child relationships in a min heap
       </span>
     </span>
   </a>
@@ -4689,10 +4761,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-keep-the-three-largest-numbers-with-a-min-heap" class="md-nav__link">
+  <a href="#3-keeping-the-three-largest-numbers-with-a-min-heap" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Keep the Three Largest Numbers with a Min Heap
+        3. &nbsp; Keeping the three largest numbers with a min heap
       </span>
     </span>
   </a>
@@ -4708,19 +4780,19 @@
   <a href="#852-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.5.2 &nbsp; Programming Exercises
+        8.5.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="8.5.2 Programming Exercises">
+    <nav class="md-nav" aria-label="8.5.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-find-the-kth-largest-element-in-an-array" class="md-nav__link">
+  <a href="#1-finding-the-kth-largest-element-in-an-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Find the Kth Largest Element in an Array
+        1. &nbsp; Finding the kth largest element in an array
       </span>
     </span>
   </a>
@@ -4770,74 +4842,74 @@
 
 <!-- Page content -->
 <h1 id="85-exercises">8.5 &nbsp; Exercises<a class="headerlink" href="#85-exercises" title="Permanent link">&para;</a></h1>
-<h2 id="851-concept-review">8.5.1 &nbsp; Concept Review<a class="headerlink" href="#851-concept-review" title="Permanent link">&para;</a></h2>
-<h3 id="1-how-does-the-heap-change-after-inserting-10">1. &nbsp; How Does the Heap Change After Inserting 10?<a class="headerlink" href="#1-how-does-the-heap-change-after-inserting-10" title="Permanent link">&para;</a></h3>
-<p>The array <code>[9, 7, 8, 3, 5]</code> represents a max heap. Now insert the number 10.</p>
+<h2 id="851-check-your-understanding">8.5.1 &nbsp; Check your understanding<a class="headerlink" href="#851-check-your-understanding" title="Permanent link">&para;</a></h2>
+<h3 id="1-restoring-the-heap-after-inserting-10">1. &nbsp; Restoring the heap after inserting 10<a class="headerlink" href="#1-restoring-the-heap-after-inserting-10" title="Permanent link">&para;</a></h3>
+<p>The array <code>[9, 7, 8, 3, 5]</code> represents a max heap. Insert the number 10.</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>First append 10 to the end of the array. What is the value of its parent node?</li>
-<li>Starting from the new node, perform bottom-to-top heapify and write the array after each swap.</li>
-<li>What is the final top element of the heap? How many swaps occur in total?</li>
+<li>After appending 10 to the array, what is its parent's value?</li>
+<li>Sift up from the new node. Write the array after each swap.</li>
+<li>What is the final root element? How many swaps were made?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>After 10 is appended, its index is 5, so its parent's index is
-    <span class="arithmatex">\(\lfloor(5-1)/2\rfloor=2\)</span>. The parent node's value is 8.</p>
+<p>The appended 10 has index 5, so its parent's index is
+    <span class="arithmatex">\(\lfloor(5-1)/2\rfloor=2\)</span>, and its parent's value is 8.</p>
 </li>
 <li>
-<p>Since 10 is greater than 8, the array after the first swap is <code>[9, 7, 10, 3, 5, 8]</code>.
-    Since 10 is also greater than its parent 9, the array after the second swap is <code>[10, 7, 9, 3, 5, 8]</code>.
-    The value 10 has now reached the root node, so heapification is complete.</p>
+<p>Since 10 is greater than 8, the first swap produces <code>[9, 7, 10, 3, 5, 8]</code>.
+    Since 10 is also greater than its new parent, 9, the second swap produces <code>[10, 7, 9, 3, 5, 8]</code>.
+    The value 10 has now reached the root, so sifting stops.</p>
 </li>
 <li>
-<p>The final top element is 10, and 2 swaps occur in total.</p>
+<p>The final root is 10, and there were two swaps.</p>
 </li>
 </ol>
 </details>
-<h3 id="2-check-parentchild-relationships-in-a-min-heap">2. &nbsp; Check Parent–Child Relationships in a Min Heap<a class="headerlink" href="#2-check-parentchild-relationships-in-a-min-heap" title="Permanent link">&para;</a></h3>
-<p>The array <code>[1, 4, 3, 7, 6, 2]</code> represents a complete binary tree. In a min heap, every parent node must be no greater than its children.
+<h3 id="2-checking-parent-child-relationships-in-a-min-heap">2. &nbsp; Checking parent-child relationships in a min heap<a class="headerlink" href="#2-checking-parent-child-relationships-in-a-min-heap" title="Permanent link">&para;</a></h3>
+<p>The array <code>[1, 4, 3, 7, 6, 2]</code> represents a complete binary tree. A min heap requires each parent to be no greater than its children.
 For index <span class="arithmatex">\(i\)</span>, the left and right child indices are <span class="arithmatex">\(2i+1\)</span> and <span class="arithmatex">\(2i+2\)</span>, respectively.</p>
 <!-- numbered-subquestions -->
 
 <ol>
 <li>What are the indices and values of the children of index 2?</li>
-<li>The node at index 2 has value 3. Does it violate the min-heap rule with its child? If so, which two elements should be swapped?</li>
-<li>Based on your answer to Question 2, write the array after the swap if the rule is violated; otherwise, explain why no swap is needed. Finally, check the remaining parent–child relationships.</li>
+<li>The node at index 2 has value 3. Does its relationship with its children violate the min-heap property? If so, which two elements should be swapped?</li>
+<li>Based on your answer to question 2, write the array after the swap if one is needed, or explain why no swap is necessary. Finally, check the other parent-child relationships.</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>The left child of index 2 is at index 5 and has value 2. The right child's index is 6, but the array has length 6, so the right child does not exist.</p>
+<p>The left child of index 2 is at index 5 and has value 2. Its right child would be at index 6, but the array has length 6, so that child does not exist.</p>
 </li>
 <li>
-<p>The parent value 3 is greater than the child value 2, violating the min-heap rule. The elements at indices 2 and 5 should be swapped.</p>
+<p>The parent value 3 exceeds the child value 2, violating the min-heap property. Swap the elements at indices 2 and 5.</p>
 </li>
 <li>
-<p>After the swap, the array is <code>[1, 4, 2, 7, 6, 3]</code>. Check each relationship:
-    <code>1 ≤ 4</code>, <code>1 ≤ 2</code>; <code>4 ≤ 7</code>, <code>4 ≤ 6</code>; and <code>2 ≤ 3</code>.
-    Every parent is now no greater than its children, so the min-heap rule is satisfied.</p>
+<p>The resulting array is <code>[1, 4, 2, 7, 6, 3]</code>. Checking each relationship gives:
+    <code>1 ≤ 4</code>, <code>1 ≤ 2</code>; <code>4 ≤ 7</code>, <code>4 ≤ 6</code>; <code>2 ≤ 3</code>.
+    Every parent is no greater than its children, so the min-heap property now holds.</p>
 </li>
 </ol>
 </details>
-<h3 id="3-keep-the-three-largest-numbers-with-a-min-heap">3. &nbsp; Keep the Three Largest Numbers with a Min Heap<a class="headerlink" href="#3-keep-the-three-largest-numbers-with-a-min-heap" title="Permanent link">&para;</a></h3>
-<p>To keep the 3 largest numbers from the data stream <code>[4, 1, 7, 3, 8]</code>, maintain a min heap containing no more than 3 elements.</p>
-<p>First insert the first 3 numbers into the min heap in order. Once the heap is full, for each new number:
-if it is greater than the top element, remove the top and insert the new number; otherwise, leave the heap unchanged.</p>
-<p>After each number is read, write the numbers kept in the heap and the top element.
-Write the kept numbers as a set; you do not need to give their order in the heap array.</p>
+<h3 id="3-keeping-the-three-largest-numbers-with-a-min-heap">3. &nbsp; Keeping the three largest numbers with a min heap<a class="headerlink" href="#3-keeping-the-three-largest-numbers-with-a-min-heap" title="Permanent link">&para;</a></h3>
+<p>To keep the three largest numbers from the stream <code>[4, 1, 7, 3, 8]</code>, maintain a min heap of at most three elements.</p>
+<p>Insert the first three numbers in order. Once the heap is full, for each new number:
+if it is greater than the root, remove the root and insert the new number; otherwise, leave the heap unchanged.</p>
+<p>After reading each number, write the numbers retained in the heap and the root value.
+You need only give the retained numbers as a set, not their arrangement in the heap array.</p>
 <details class="success">
-<summary>Answer</summary>
-<p>The result after each number is read is:</p>
+<summary>Reference answer</summary>
+<p>The results after each number are:</p>
 <table>
 <thead>
 <tr>
 <th>Number read</th>
-<th>Numbers kept</th>
-<th>Top</th>
+<th>Retained numbers</th>
+<th>Root</th>
 </tr>
 </thead>
 <tbody>
@@ -4868,19 +4940,19 @@ Write the kept numbers as a set; you do not need to give their order in the heap
 </tr>
 </tbody>
 </table>
-<p>Once the heap is full, its top is the smallest of the numbers currently kept. A new number replaces the top only when it is greater than the top.
-The final set <code>{4, 7, 8}</code> contains exactly the 3 largest numbers.</p>
+<p>Once the heap is full, its root is the smallest retained number. Replace the root with the new number
+only when the new number is larger. The final set <code>{4, 7, 8}</code> contains exactly the three largest numbers.</p>
 </details>
-<h2 id="852-programming-exercises">8.5.2 &nbsp; Programming Exercises<a class="headerlink" href="#852-programming-exercises" title="Permanent link">&para;</a></h2>
-<h3 id="1-find-the-kth-largest-element-in-an-array">1. &nbsp; Find the Kth Largest Element in an Array<a class="headerlink" href="#1-find-the-kth-largest-element-in-an-array" title="Permanent link">&para;</a></h3>
-<p>Given an integer array <code>nums</code> and an integer <span class="arithmatex">\(k\)</span>, where <span class="arithmatex">\(1 \le k \le n\)</span> and <span class="arithmatex">\(n\)</span> is the array's length, return the element that would appear at position <span class="arithmatex">\(k\)</span> if the array were arranged from largest to smallest.</p>
-<p>Count duplicate elements separately. For example, the second-largest element of <code>[5, 5, 2]</code> is still 5. Use a min heap containing no more than <span class="arithmatex">\(k\)</span> elements.</p>
+<h2 id="852-programming-exercises">8.5.2 &nbsp; Programming exercises<a class="headerlink" href="#852-programming-exercises" title="Permanent link">&para;</a></h2>
+<h3 id="1-finding-the-kth-largest-element-in-an-array">1. &nbsp; Finding the kth largest element in an array<a class="headerlink" href="#1-finding-the-kth-largest-element-in-an-array" title="Permanent link">&para;</a></h3>
+<p>Given an integer array <code>nums</code> and an integer <span class="arithmatex">\(k\)</span>, where <span class="arithmatex">\(1 \le k \le n\)</span> and <span class="arithmatex">\(n\)</span> is the array length, return the element in position <span class="arithmatex">\(k\)</span> when the array is ordered from largest to smallest.</p>
+<p>Count duplicate elements separately. For example, the second largest element of <code>[5, 5, 2]</code> is still 5. Use a min heap of at most <span class="arithmatex">\(k\)</span> elements.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>The kth largest element is the smallest among the k largest numbers</li>
-<li>Insert each number into the min heap, and remove the smallest value whenever the heap's size exceeds k</li>
-<li>After the traversal, the heap contains the k largest numbers, and its top is the answer</li>
+<li>The kth largest element is the smallest of the k largest numbers.</li>
+<li>Insert each number into the min heap; if its size exceeds k, remove the minimum.</li>
+<li>When the traversal ends, the heap retains the k largest numbers, and its root is the answer.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/kth-largest-element-in-an-array/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
@@ -4906,7 +4978,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 8.4 Summary"
+    aria-label="Previous: 8.4   Summary"
     rel="prev"
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@@ -4918,7 +4990,7 @@ aria-label="Footer"
         Previous
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-        8.4 Summary
+        8.4 &nbsp; Summary
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@@ -4930,7 +5002,7 @@ aria-label="Footer"
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     class="md-footer__link md-footer__link--next"
-    aria-label="Next: Chapter 9. &amp;nbsp; Graph"
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@@ -4938,7 +5010,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        Chapter 9. &nbsp; Graph
+        Chapter 9. &nbsp; Graphs
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -4955,7 +5027,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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diff --git a/en/chapter_heap/heap/index.html b/en/chapter_heap/heap/index.html
index 458301ccb..19619a23f 100644
--- a/en/chapter_heap/heap/index.html
+++ b/en/chapter_heap/heap/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>8.1 Heap - Hello Algo</title>
+        <title>8.1 Heaps - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
     <div data-md-component="skip">
 
 
-        <a href="#81-heap" class="md-skip">
+        <a href="#81-heaps" class="md-skip">
           Skip to content
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@@ -119,7 +119,7 @@
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             d="M480 32c0-12.9-7.8-24.6-19.8-29.6s-25.7-2.2-34.9 6.9L381.7 53c-48 48-113.1 75-181 75H192 160 64c-35.3 0-64 28.7-64 64v96c0 35.3 28.7 64 64 64l0 128c0 17.7 14.3 32 32 32h64c17.7 0 32-14.3 32-32V352l8.7 0c67.9 0 133 27 181 75l43.6 43.6c9.2 9.2 22.9 11.9 34.9 6.9s19.8-16.6 19.8-29.6V300.4c18.6-8.8 32-32.5 32-60.4s-13.4-51.6-32-60.4V32zm-64 76.7V240 371.3C357.2 317.8 280.5 288 200.7 288H192V192h8.7c79.8 0 156.5-29.8 215.3-83.3z" />
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-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              8.1 &nbsp; Heap
+              8.1 &nbsp; Heaps
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2166,7 +2166,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2187,7 +2187,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2217,7 +2217,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2234,7 +2234,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2263,7 +2263,7 @@
   <a href="#811-common-heap-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
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+        8.1.1 &nbsp; Common heap operations
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     </span>
   </a>
@@ -2271,22 +2271,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#812-implementation-of-the-heap" class="md-nav__link">
+  <a href="#812-implementing-a-heap" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.1.2 &nbsp; Implementation of the Heap
+        8.1.2 &nbsp; Implementing a heap
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     </span>
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-    <nav class="md-nav" aria-label="8.1.2 Implementation of the Heap">
+    <nav class="md-nav" aria-label="8.1.2 Implementing a heap">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-heap-storage-and-representation" class="md-nav__link">
+  <a href="#1-storage-and-representation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Heap Storage and Representation
+        1. &nbsp; Storage and representation
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     </span>
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@@ -2294,10 +2294,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-accessing-the-heap-top-element" class="md-nav__link">
+  <a href="#2-accessing-the-root" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Accessing the Heap Top Element
+        2. &nbsp; Accessing the root
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@@ -2305,10 +2305,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-inserting-an-element-into-the-heap" class="md-nav__link">
+  <a href="#3-inserting-an-element" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Inserting an Element Into the Heap
+        3. &nbsp; Inserting an element
       </span>
     </span>
   </a>
@@ -2316,10 +2316,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#4-removing-the-heap-top-element" class="md-nav__link">
+  <a href="#4-removing-the-root" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4. &nbsp; Removing the Heap Top Element
+        4. &nbsp; Removing the root
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     </span>
   </a>
@@ -2332,10 +2332,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#813-common-applications-of-heaps" class="md-nav__link">
+  <a href="#813-common-applications" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.1.3 &nbsp; Common Applications of Heaps
+        8.1.3 &nbsp; Common applications
       </span>
     </span>
   </a>
@@ -2365,7 +2365,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2392,7 +2392,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2419,7 +2419,7 @@
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2510,7 +2510,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2552,7 +2552,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2606,7 +2606,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2728,7 +2728,7 @@
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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@@ -3010,7 +3010,7 @@
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
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-  11.6 Merge Sort
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
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+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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+  Chapter 13. &nbsp; Backtracking
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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+  13.3 &nbsp; Subset sum problem
 
 
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+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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+  Chapter 14. &nbsp; Dynamic programming
 
 
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+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
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+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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+  15.5 &nbsp; Summary
 
 
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+  15.6 &nbsp; Exercises
 
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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@@ -4619,6 +4619,78 @@
 
 
 
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@@ -4663,7 +4735,7 @@
   <a href="#811-common-heap-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.1.1 &nbsp; Common Heap Operations
+        8.1.1 &nbsp; Common heap operations
       </span>
     </span>
   </a>
@@ -4671,22 +4743,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#812-implementation-of-the-heap" class="md-nav__link">
+  <a href="#812-implementing-a-heap" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.1.2 &nbsp; Implementation of the Heap
+        8.1.2 &nbsp; Implementing a heap
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="8.1.2 Implementation of the Heap">
+    <nav class="md-nav" aria-label="8.1.2 Implementing a heap">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-heap-storage-and-representation" class="md-nav__link">
+  <a href="#1-storage-and-representation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Heap Storage and Representation
+        1. &nbsp; Storage and representation
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 </li>
 
           <li class="md-nav__item">
-  <a href="#2-accessing-the-heap-top-element" class="md-nav__link">
+  <a href="#2-accessing-the-root" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Accessing the Heap Top Element
+        2. &nbsp; Accessing the root
       </span>
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   </a>
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 </li>
 
           <li class="md-nav__item">
-  <a href="#3-inserting-an-element-into-the-heap" class="md-nav__link">
+  <a href="#3-inserting-an-element" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Inserting an Element Into the Heap
+        3. &nbsp; Inserting an element
       </span>
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@@ -4716,10 +4788,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#4-removing-the-heap-top-element" class="md-nav__link">
+  <a href="#4-removing-the-root" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4. &nbsp; Removing the Heap Top Element
+        4. &nbsp; Removing the root
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-  <a href="#813-common-applications-of-heaps" class="md-nav__link">
+  <a href="#813-common-applications" class="md-nav__link">
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       <span class="md-typeset">
-        8.1.3 &nbsp; Common Applications of Heaps
+        8.1.3 &nbsp; Common applications
       </span>
     </span>
   </a>
@@ -4779,32 +4851,32 @@
 
 
 <!-- Page content -->
-<h1 id="81-heap">8.1 &nbsp; Heap<a class="headerlink" href="#81-heap" title="Permanent link">&para;</a></h1>
-<p>A <u>heap</u> is a complete binary tree that satisfies specific conditions and can be mainly categorized into two types, as shown in Figure 8-1.</p>
+<h1 id="81-heaps">8.1 &nbsp; Heaps<a class="headerlink" href="#81-heaps" title="Permanent link">&para;</a></h1>
+<p>A <u>heap</u> is a complete binary tree that satisfies a particular ordering condition. There are two main types, shown below.</p>
 <ul>
-<li><u>min heap</u>: The value of any node <span class="arithmatex">\(\leq\)</span> the values of its child nodes.</li>
-<li><u>max heap</u>: The value of any node <span class="arithmatex">\(\geq\)</span> the values of its child nodes.</li>
+<li><u>Min heap</u>: every node's value is <span class="arithmatex">\(\leq\)</span> the values of its children.</li>
+<li><u>Max heap</u>: every node's value is <span class="arithmatex">\(\geq\)</span> the values of its children.</li>
 </ul>
-<p><img alt="Min heap and max heap" class="animation-figure" src="../heap.assets/min_heap_and_max_heap.png" /></p>
-<p align="center"> Figure 8-1 &nbsp; Min heap and max heap </p>
+<p><img alt="A min heap and a max heap" class="animation-figure" src="../heap.assets/min_heap_and_max_heap.png" /></p>
+<p align="center"> Figure 8-1 &nbsp; A min heap and a max heap </p>
 
-<p>As a special case of a complete binary tree, heaps have the following characteristics.</p>
+<p>As a special kind of complete binary tree, a heap has the following properties.</p>
 <ul>
-<li>The bottom layer nodes are filled from left to right, and nodes in other layers are fully filled.</li>
-<li>We call the root node of the binary tree the "heap top" and the bottom-rightmost node the "heap bottom."</li>
-<li>For max heaps (min heaps), the value of the heap top element (root node) is the largest (smallest).</li>
+<li>The lowest level is filled from left to right, and all other levels are full.</li>
+<li>The root is the “top” of the heap, while the rightmost node on the lowest level is its “last” node.</li>
+<li>In a max heap or min heap, the root holds the largest or smallest value, respectively.</li>
 </ul>
-<h2 id="811-common-heap-operations">8.1.1 &nbsp; Common Heap Operations<a class="headerlink" href="#811-common-heap-operations" title="Permanent link">&para;</a></h2>
-<p>It should be noted that many programming languages provide a <u>priority queue</u>, an abstract data structure defined as a queue whose elements are ordered by priority.</p>
-<p>In fact, <strong>heaps are typically used to implement priority queues, with max heaps corresponding to priority queues where elements are dequeued in descending order</strong>. From a usage perspective, we can regard "priority queue" and "heap" as equivalent data structures. Therefore, this book does not make a special distinction between the two and uniformly refers to them as "heap."</p>
-<p>Common heap operations are shown in Table 8-1, and method names need to be determined based on the programming language.</p>
-<p align="center"> Table 8-1 &nbsp; Efficiency of Heap Operations </p>
+<h2 id="811-common-heap-operations">8.1.1 &nbsp; Common heap operations<a class="headerlink" href="#811-common-heap-operations" title="Permanent link">&para;</a></h2>
+<p>Many programming languages provide a <u>priority queue</u>, an abstract data structure whose elements are removed according to priority.</p>
+<p><strong>Heaps are commonly used to implement priority queues. A max heap corresponds to a priority queue that removes elements from largest to smallest</strong>. From the user's perspective, the two can be treated as equivalent. This book therefore does not distinguish them strictly and uses the term “heap” for both.</p>
+<p>Table 8-1 lists common operations. Method names depend on the programming language.</p>
+<p align="center"> Table 8-1 &nbsp; Efficiency of heap operations </p>
 
 <div class="center-table">
 <table>
 <thead>
 <tr>
-<th>Method name</th>
+<th>Method</th>
 <th>Description</th>
 <th>Time complexity</th>
 </tr>
@@ -4812,34 +4884,34 @@
 <tbody>
 <tr>
 <td><code>push()</code></td>
-<td>Insert an element into the heap</td>
+<td>Insert an element</td>
 <td><span class="arithmatex">\(O(\log n)\)</span></td>
 </tr>
 <tr>
 <td><code>pop()</code></td>
-<td>Remove the heap top element</td>
+<td>Remove the root element</td>
 <td><span class="arithmatex">\(O(\log n)\)</span></td>
 </tr>
 <tr>
 <td><code>peek()</code></td>
-<td>Access the heap top element (max/min value for max/min heap)</td>
+<td>Access the root: the maximum in a max heap or minimum in a min heap</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
 <td><code>size()</code></td>
-<td>Get the number of elements in the heap</td>
+<td>Get the number of elements</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
 <td><code>isEmpty()</code></td>
-<td>Check if the heap is empty</td>
+<td>Check whether the heap is empty</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 </tbody>
 </table>
 </div>
-<p>In practical applications, we can directly use the heap class (or priority queue class) provided by programming languages.</p>
-<p>Similar to "ascending order" and "descending order" in sorting algorithms, we can implement conversion between "min heap" and "max heap" by setting a <code>flag</code> or modifying the <code>Comparator</code>. The code is as follows:</p>
+<p>In practice, use the heap or priority-queue class provided by the language.</p>
+<p>Just as a sorting algorithm can order values in ascending or descending order, we can switch between min heaps and max heaps by setting a <code>flag</code> or changing the <code>Comparator</code>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
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 <div class="tabbed-block">
@@ -4849,21 +4921,21 @@
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="n">max_heap</span><span class="p">,</span> <span class="n">flag</span> <span class="o">=</span> <span class="p">[],</span> <span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="c1"># Python's heapq module implements a min heap by default</span>
-<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="c1"># Consider negating elements before pushing them to the heap, which inverts the size relationship and thus implements a max heap</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="c1"># In this example, flag = 1 corresponds to a min heap, flag = -1 corresponds to a max heap</span>
+<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="c1"># Negate elements before insertion to reverse their order and implement a max heap</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="c1"># Here, flag = 1 gives a min heap and flag = -1 gives a max heap</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="c1"># Push elements into the heap</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="c1"># Insert elements into the heap</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="n">heapq</span><span class="o">.</span><span class="n">heappush</span><span class="p">(</span><span class="n">max_heap</span><span class="p">,</span> <span class="n">flag</span> <span class="o">*</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="n">heapq</span><span class="o">.</span><span class="n">heappush</span><span class="p">(</span><span class="n">max_heap</span><span class="p">,</span> <span class="n">flag</span> <span class="o">*</span> <span class="mi">3</span><span class="p">)</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="n">heapq</span><span class="o">.</span><span class="n">heappush</span><span class="p">(</span><span class="n">max_heap</span><span class="p">,</span> <span class="n">flag</span> <span class="o">*</span> <span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="n">heapq</span><span class="o">.</span><span class="n">heappush</span><span class="p">(</span><span class="n">max_heap</span><span class="p">,</span> <span class="n">flag</span> <span class="o">*</span> <span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="n">heapq</span><span class="o">.</span><span class="n">heappush</span><span class="p">(</span><span class="n">max_heap</span><span class="p">,</span> <span class="n">flag</span> <span class="o">*</span> <span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>
-<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="c1"># Get the heap top element</span>
+<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="c1"># Access the root element</span>
 <a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="n">peek</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">flag</span> <span class="o">*</span> <span class="n">max_heap</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="c1"># 5</span>
 <a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>
-<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="c1"># Remove the heap top element</span>
-<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="c1"># The removed elements will form a descending sequence</span>
+<a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="c1"># Remove root elements</span>
+<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="c1"># Removed elements form a descending sequence</span>
 <a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="n">val</span> <span class="o">=</span> <span class="n">flag</span> <span class="o">*</span> <span class="n">heapq</span><span class="o">.</span><span class="n">heappop</span><span class="p">(</span><span class="n">max_heap</span><span class="p">)</span> <span class="c1"># 5</span>
 <a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a><span class="n">val</span> <span class="o">=</span> <span class="n">flag</span> <span class="o">*</span> <span class="n">heapq</span><span class="o">.</span><span class="n">heappop</span><span class="p">(</span><span class="n">max_heap</span><span class="p">)</span> <span class="c1"># 4</span>
 <a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a><span class="n">val</span> <span class="o">=</span> <span class="n">flag</span> <span class="o">*</span> <span class="n">heapq</span><span class="o">.</span><span class="n">heappop</span><span class="p">(</span><span class="n">max_heap</span><span class="p">)</span> <span class="c1"># 3</span>
@@ -4873,10 +4945,10 @@
 <a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a><span class="c1"># Get the heap size</span>
 <a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a><span class="n">size</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">max_heap</span><span class="p">)</span>
 <a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a>
-<a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a><span class="c1"># Check if the heap is empty</span>
+<a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a><span class="c1"># Check whether the heap is empty</span>
 <a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a><span class="n">is_empty</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="ow">not</span> <span class="n">max_heap</span>
 <a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a>
-<a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a><span class="c1"># Build a heap from an input list</span>
+<a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a><span class="c1"># Build a heap from a list</span>
 <a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a><span class="n">min_heap</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">]</span>
 <a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a><span class="n">heapq</span><span class="o">.</span><span class="n">heapify</span><span class="p">(</span><span class="n">min_heap</span><span class="p">)</span>
 </code></pre></div>
@@ -4888,18 +4960,18 @@
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="c1">// Initialize a max heap</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="n">priority_queue</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">less</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">;</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a>
-<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="cm">/* Push elements into the heap */</span>
+<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="cm">/* Insert elements into the heap */</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">maxHeap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="n">maxHeap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="n">maxHeap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="n">maxHeap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="n">maxHeap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a>
-<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="cm">/* Get the heap top element */</span>
+<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="cm">/* Access the root element */</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">top</span><span class="p">();</span><span class="w"> </span><span class="c1">// 5</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a>
-<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="cm">/* Remove the heap top element */</span>
-<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="c1">// The removed elements will form a descending sequence</span>
+<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="cm">/* Remove root elements */</span>
+<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="c1">// Removed elements form a descending sequence</span>
 <a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="n">maxHeap</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span><span class="w"> </span><span class="c1">// 5</span>
 <a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="n">maxHeap</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span><span class="w"> </span><span class="c1">// 4</span>
 <a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="n">maxHeap</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span><span class="w"> </span><span class="c1">// 3</span>
@@ -4909,10 +4981,10 @@
 <a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="cm">/* Get the heap size */</span>
 <a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
 <a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a>
-<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="cm">/* Check if the heap is empty */</span>
+<a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="cm">/* Check whether the heap is empty */</span>
 <a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">empty</span><span class="p">();</span>
 <a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a>
-<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="cm">/* Build a heap from an input list */</span>
+<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="cm">/* Build a heap from a list */</span>
 <a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">input</span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">};</span>
 <a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="n">priority_queue</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">greater</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">minHeap</span><span class="p">(</span><span class="n">input</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span><span class="w"> </span><span class="n">input</span><span class="p">.</span><span class="n">end</span><span class="p">());</span>
 </code></pre></div>
@@ -4921,21 +4993,21 @@
 <div class="highlight"><span class="filename">heap.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Initialize a heap */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="c1">// Initialize a min heap</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="n">Queue</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">minHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">PriorityQueue</span><span class="o">&lt;&gt;</span><span class="p">();</span>
-<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="c1">// Initialize a max heap (use lambda expression to modify Comparator)</span>
+<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="c1">// Initialize a max heap by changing the Comparator with a lambda expression</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="n">Queue</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">PriorityQueue</span><span class="o">&lt;&gt;</span><span class="p">((</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="p">)</span><span class="w"> </span><span class="o">-&gt;</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">a</span><span class="p">);</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a>
-<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="cm">/* Push elements into the heap */</span>
+<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="cm">/* Insert elements into the heap */</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="n">maxHeap</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="n">maxHeap</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="n">maxHeap</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="n">maxHeap</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="n">maxHeap</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a>
-<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="cm">/* Get the heap top element */</span>
+<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="cm">/* Access the root element */</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">peek</span><span class="p">();</span><span class="w"> </span><span class="c1">// 5</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a>
-<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="cm">/* Remove the heap top element */</span>
-<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="c1">// The removed elements will form a descending sequence</span>
+<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="cm">/* Remove root elements */</span>
+<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="c1">// Removed elements form a descending sequence</span>
 <a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// 5</span>
 <a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// 4</span>
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// 3</span>
@@ -4945,10 +5017,10 @@
 <a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="cm">/* Get the heap size */</span>
 <a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">size</span><span class="p">();</span>
 <a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a>
-<a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a><span class="cm">/* Check if the heap is empty */</span>
+<a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a><span class="cm">/* Check whether the heap is empty */</span>
 <a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="kt">boolean</span><span class="w"> </span><span class="n">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">isEmpty</span><span class="p">();</span>
 <a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a>
-<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="cm">/* Build a heap from an input list */</span>
+<a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="cm">/* Build a heap from a list */</span>
 <a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="n">minHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">PriorityQueue</span><span class="o">&lt;&gt;</span><span class="p">(</span><span class="n">Arrays</span><span class="p">.</span><span class="na">asList</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">));</span>
 </code></pre></div>
 </div>
@@ -4956,21 +5028,21 @@
 <div class="highlight"><span class="filename">heap.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Initialize a heap */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="c1">// Initialize a min heap</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="n">PriorityQueue</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">minHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">();</span>
-<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="c1">// Initialize a max heap (use lambda expression to modify Comparer)</span>
+<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="c1">// Initialize a max heap by changing the Comparer with a lambda expression</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="n">PriorityQueue</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="n">Comparer</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">.</span><span class="n">Create</span><span class="p">((</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">y</span><span class="p">.</span><span class="n">CompareTo</span><span class="p">(</span><span class="n">x</span><span class="p">)));</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a>
-<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="cm">/* Push elements into the heap */</span>
+<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="cm">/* Insert elements into the heap */</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="n">maxHeap</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="n">maxHeap</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="n">maxHeap</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">);</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="n">maxHeap</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="n">maxHeap</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a>
-<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="cm">/* Get the heap top element */</span>
+<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="cm">/* Access the root element */</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">Peek</span><span class="p">();</span><span class="c1">//5</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a>
-<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="cm">/* Remove the heap top element */</span>
-<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="c1">// The removed elements will form a descending sequence</span>
+<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="cm">/* Remove root elements */</span>
+<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="c1">// Removed elements form a descending sequence</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">Dequeue</span><span class="p">();</span><span class="w">  </span><span class="c1">// 5</span>
 <a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">Dequeue</span><span class="p">();</span><span class="w">  </span><span class="c1">// 4</span>
 <a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">Dequeue</span><span class="p">();</span><span class="w">  </span><span class="c1">// 3</span>
@@ -4980,50 +5052,50 @@
 <a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="cm">/* Get the heap size */</span>
 <a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">Count</span><span class="p">;</span>
 <a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a>
-<a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="cm">/* Check if the heap is empty */</span>
+<a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="cm">/* Check whether the heap is empty */</span>
 <a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a>
-<a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a><span class="cm">/* Build a heap from an input list */</span>
+<a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a><span class="cm">/* Build a heap from a list */</span>
 <a id="__codelineno-3-32" name="__codelineno-3-32" href="#__codelineno-3-32"></a><span class="n">minHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">PriorityQueue</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="p">([(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">),</span><span class="w"> </span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">),</span><span class="w"> </span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">),</span><span class="w"> </span><span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">),</span><span class="w"> </span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">)]);</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="c1">// In Go, we can construct a max heap of integers by implementing heap.Interface</span>
+<div class="highlight"><span class="filename">heap.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="c1">// In Go, implement heap.Interface to build an integer max heap</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="c1">// Implementing heap.Interface also requires implementing sort.Interface</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="kd">type</span><span class="w"> </span><span class="nx">intHeap</span><span class="w"> </span><span class="p">[]</span><span class="kt">any</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a>
-<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="c1">// Push implements the heap.Interface method for pushing an element into the heap</span>
+<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="c1">// Push implements the heap.Interface method for inserting an element</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">intHeap</span><span class="p">)</span><span class="w"> </span><span class="nx">Push</span><span class="p">(</span><span class="nx">x</span><span class="w"> </span><span class="kt">any</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// Push and Pop use pointer receiver as parameters</span>
-<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="c1">// because they not only adjust the slice contents but also modify the slice length</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="c1">// Push and Pop use pointer receivers</span>
+<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="c1">// because they change the slice length as well as its contents.</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="o">*</span><span class="nx">h</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="o">*</span><span class="nx">h</span><span class="p">,</span><span class="w"> </span><span class="nx">x</span><span class="p">.(</span><span class="kt">int</span><span class="p">))</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="p">}</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a>
-<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="c1">// Pop implements the heap.Interface method for popping the heap top element</span>
+<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="c1">// Pop implements the heap.Interface method used to remove the root element</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">intHeap</span><span class="p">)</span><span class="w"> </span><span class="nx">Pop</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="c1">// The element to be removed is stored at the end</span>
+<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="c1">// The element to remove has been placed at the end</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="nx">last</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">h</span><span class="p">)[</span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">h</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="o">*</span><span class="nx">h</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">h</span><span class="p">)[:</span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">h</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">last</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="p">}</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a>
-<a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="c1">// Len is a sort.Interface method</span>
+<a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="c1">// Len implements a sort.Interface method</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">intHeap</span><span class="p">)</span><span class="w"> </span><span class="nx">Len</span><span class="p">()</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">h</span><span class="p">)</span>
 <a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="p">}</span>
 <a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a>
-<a id="__codelineno-4-25" name="__codelineno-4-25" href="#__codelineno-4-25"></a><span class="c1">// Less is a sort.Interface method</span>
+<a id="__codelineno-4-25" name="__codelineno-4-25" href="#__codelineno-4-25"></a><span class="c1">// Less implements a sort.Interface method</span>
 <a id="__codelineno-4-26" name="__codelineno-4-26" href="#__codelineno-4-26"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">intHeap</span><span class="p">)</span><span class="w"> </span><span class="nx">Less</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-27" name="__codelineno-4-27" href="#__codelineno-4-27"></a><span class="w">    </span><span class="c1">// To implement a min heap, change this to a less-than sign</span>
+<a id="__codelineno-4-27" name="__codelineno-4-27" href="#__codelineno-4-27"></a><span class="w">    </span><span class="c1">// For a min heap, change this to a less-than comparison</span>
 <a id="__codelineno-4-28" name="__codelineno-4-28" href="#__codelineno-4-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">h</span><span class="p">)[</span><span class="nx">i</span><span class="p">].(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">h</span><span class="p">)[</span><span class="nx">j</span><span class="p">].(</span><span class="kt">int</span><span class="p">)</span>
 <a id="__codelineno-4-29" name="__codelineno-4-29" href="#__codelineno-4-29"></a><span class="p">}</span>
 <a id="__codelineno-4-30" name="__codelineno-4-30" href="#__codelineno-4-30"></a>
-<a id="__codelineno-4-31" name="__codelineno-4-31" href="#__codelineno-4-31"></a><span class="c1">// Swap is a sort.Interface method</span>
+<a id="__codelineno-4-31" name="__codelineno-4-31" href="#__codelineno-4-31"></a><span class="c1">// Swap implements a sort.Interface method</span>
 <a id="__codelineno-4-32" name="__codelineno-4-32" href="#__codelineno-4-32"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">intHeap</span><span class="p">)</span><span class="w"> </span><span class="nx">Swap</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-33" name="__codelineno-4-33" href="#__codelineno-4-33"></a><span class="w">    </span><span class="p">(</span><span class="o">*</span><span class="nx">h</span><span class="p">)[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">h</span><span class="p">)[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">h</span><span class="p">)[</span><span class="nx">j</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">h</span><span class="p">)[</span><span class="nx">i</span><span class="p">]</span>
 <a id="__codelineno-4-34" name="__codelineno-4-34" href="#__codelineno-4-34"></a><span class="p">}</span>
 <a id="__codelineno-4-35" name="__codelineno-4-35" href="#__codelineno-4-35"></a>
-<a id="__codelineno-4-36" name="__codelineno-4-36" href="#__codelineno-4-36"></a><span class="c1">// Top gets the heap top element</span>
+<a id="__codelineno-4-36" name="__codelineno-4-36" href="#__codelineno-4-36"></a><span class="c1">// Top returns the root element</span>
 <a id="__codelineno-4-37" name="__codelineno-4-37" href="#__codelineno-4-37"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">intHeap</span><span class="p">)</span><span class="w"> </span><span class="nx">Top</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-38" name="__codelineno-4-38" href="#__codelineno-4-38"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">h</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-4-39" name="__codelineno-4-39" href="#__codelineno-4-39"></a><span class="p">}</span>
@@ -5034,20 +5106,20 @@
 <a id="__codelineno-4-44" name="__codelineno-4-44" href="#__codelineno-4-44"></a><span class="w">    </span><span class="c1">// Initialize a max heap</span>
 <a id="__codelineno-4-45" name="__codelineno-4-45" href="#__codelineno-4-45"></a><span class="w">    </span><span class="nx">maxHeap</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">intHeap</span><span class="p">{}</span>
 <a id="__codelineno-4-46" name="__codelineno-4-46" href="#__codelineno-4-46"></a><span class="w">    </span><span class="nx">heap</span><span class="p">.</span><span class="nx">Init</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">)</span>
-<a id="__codelineno-4-47" name="__codelineno-4-47" href="#__codelineno-4-47"></a><span class="w">    </span><span class="cm">/* Push elements into the heap */</span>
-<a id="__codelineno-4-48" name="__codelineno-4-48" href="#__codelineno-4-48"></a><span class="w">    </span><span class="c1">// Call heap.Interface methods to add elements</span>
+<a id="__codelineno-4-47" name="__codelineno-4-47" href="#__codelineno-4-47"></a><span class="w">    </span><span class="cm">/* Insert elements into the heap */</span>
+<a id="__codelineno-4-48" name="__codelineno-4-48" href="#__codelineno-4-48"></a><span class="w">    </span><span class="c1">// Call the heap.Interface methods to add elements</span>
 <a id="__codelineno-4-49" name="__codelineno-4-49" href="#__codelineno-4-49"></a><span class="w">    </span><span class="nx">heap</span><span class="p">.</span><span class="nx">Push</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-4-50" name="__codelineno-4-50" href="#__codelineno-4-50"></a><span class="w">    </span><span class="nx">heap</span><span class="p">.</span><span class="nx">Push</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span>
 <a id="__codelineno-4-51" name="__codelineno-4-51" href="#__codelineno-4-51"></a><span class="w">    </span><span class="nx">heap</span><span class="p">.</span><span class="nx">Push</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-4-52" name="__codelineno-4-52" href="#__codelineno-4-52"></a><span class="w">    </span><span class="nx">heap</span><span class="p">.</span><span class="nx">Push</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-4-53" name="__codelineno-4-53" href="#__codelineno-4-53"></a><span class="w">    </span><span class="nx">heap</span><span class="p">.</span><span class="nx">Push</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-4-54" name="__codelineno-4-54" href="#__codelineno-4-54"></a>
-<a id="__codelineno-4-55" name="__codelineno-4-55" href="#__codelineno-4-55"></a><span class="w">    </span><span class="cm">/* Get the heap top element */</span>
+<a id="__codelineno-4-55" name="__codelineno-4-55" href="#__codelineno-4-55"></a><span class="w">    </span><span class="cm">/* Access the root element */</span>
 <a id="__codelineno-4-56" name="__codelineno-4-56" href="#__codelineno-4-56"></a><span class="w">    </span><span class="nx">top</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">maxHeap</span><span class="p">.</span><span class="nx">Top</span><span class="p">()</span>
-<a id="__codelineno-4-57" name="__codelineno-4-57" href="#__codelineno-4-57"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Printf</span><span class="p">(</span><span class="s">"Heap top element is %d\n"</span><span class="p">,</span><span class="w"> </span><span class="nx">top</span><span class="p">)</span>
+<a id="__codelineno-4-57" name="__codelineno-4-57" href="#__codelineno-4-57"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Printf</span><span class="p">(</span><span class="s">"The root element is %d\n"</span><span class="p">,</span><span class="w"> </span><span class="nx">top</span><span class="p">)</span>
 <a id="__codelineno-4-58" name="__codelineno-4-58" href="#__codelineno-4-58"></a>
-<a id="__codelineno-4-59" name="__codelineno-4-59" href="#__codelineno-4-59"></a><span class="w">    </span><span class="cm">/* Remove the heap top element */</span>
-<a id="__codelineno-4-60" name="__codelineno-4-60" href="#__codelineno-4-60"></a><span class="w">    </span><span class="c1">// Call heap.Interface methods to remove elements</span>
+<a id="__codelineno-4-59" name="__codelineno-4-59" href="#__codelineno-4-59"></a><span class="w">    </span><span class="cm">/* Remove root elements */</span>
+<a id="__codelineno-4-60" name="__codelineno-4-60" href="#__codelineno-4-60"></a><span class="w">    </span><span class="c1">// Call the heap.Interface methods to remove elements</span>
 <a id="__codelineno-4-61" name="__codelineno-4-61" href="#__codelineno-4-61"></a><span class="w">    </span><span class="nx">heap</span><span class="p">.</span><span class="nx">Pop</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="c1">// 5</span>
 <a id="__codelineno-4-62" name="__codelineno-4-62" href="#__codelineno-4-62"></a><span class="w">    </span><span class="nx">heap</span><span class="p">.</span><span class="nx">Pop</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="c1">// 4</span>
 <a id="__codelineno-4-63" name="__codelineno-4-63" href="#__codelineno-4-63"></a><span class="w">    </span><span class="nx">heap</span><span class="p">.</span><span class="nx">Pop</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="c1">// 3</span>
@@ -5056,30 +5128,30 @@
 <a id="__codelineno-4-66" name="__codelineno-4-66" href="#__codelineno-4-66"></a>
 <a id="__codelineno-4-67" name="__codelineno-4-67" href="#__codelineno-4-67"></a><span class="w">    </span><span class="cm">/* Get the heap size */</span>
 <a id="__codelineno-4-68" name="__codelineno-4-68" href="#__codelineno-4-68"></a><span class="w">    </span><span class="nx">size</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">maxHeap</span><span class="p">)</span>
-<a id="__codelineno-4-69" name="__codelineno-4-69" href="#__codelineno-4-69"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Printf</span><span class="p">(</span><span class="s">"Number of heap elements is %d\n"</span><span class="p">,</span><span class="w"> </span><span class="nx">size</span><span class="p">)</span>
+<a id="__codelineno-4-69" name="__codelineno-4-69" href="#__codelineno-4-69"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Printf</span><span class="p">(</span><span class="s">"The heap contains %d elements\n"</span><span class="p">,</span><span class="w"> </span><span class="nx">size</span><span class="p">)</span>
 <a id="__codelineno-4-70" name="__codelineno-4-70" href="#__codelineno-4-70"></a>
-<a id="__codelineno-4-71" name="__codelineno-4-71" href="#__codelineno-4-71"></a><span class="w">    </span><span class="cm">/* Check if the heap is empty */</span>
+<a id="__codelineno-4-71" name="__codelineno-4-71" href="#__codelineno-4-71"></a><span class="w">    </span><span class="cm">/* Check whether the heap is empty */</span>
 <a id="__codelineno-4-72" name="__codelineno-4-72" href="#__codelineno-4-72"></a><span class="w">    </span><span class="nx">isEmpty</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-4-73" name="__codelineno-4-73" href="#__codelineno-4-73"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Printf</span><span class="p">(</span><span class="s">"Is the heap empty? %t\n"</span><span class="p">,</span><span class="w"> </span><span class="nx">isEmpty</span><span class="p">)</span>
+<a id="__codelineno-4-73" name="__codelineno-4-73" href="#__codelineno-4-73"></a><span class="w">    </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Printf</span><span class="p">(</span><span class="s">"Is the heap empty: %t\n"</span><span class="p">,</span><span class="w"> </span><span class="nx">isEmpty</span><span class="p">)</span>
 <a id="__codelineno-4-74" name="__codelineno-4-74" href="#__codelineno-4-74"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">heap.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Initialize a heap */</span>
-<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Swift's Heap type supports both max heaps and min heaps, and requires importing swift-collections</span>
+<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Swift's Heap type supports both minimum and maximum operations and requires swift-collections</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="kd">var</span><span class="w"> </span><span class="nv">heap</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Heap</span><span class="p">&lt;</span><span class="nb">Int</span><span class="p">&gt;()</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a>
-<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="cm">/* Push elements into the heap */</span>
+<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="cm">/* Insert elements into the heap */</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="n">heap</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="n">heap</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="n">heap</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="n">heap</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="n">heap</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a>
-<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="cm">/* Get the heap top element */</span>
+<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="cm">/* Access the root element */</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="kd">var</span><span class="w"> </span><span class="nv">peek</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="bp">max</span><span class="p">()</span><span class="o">!</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a>
-<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="cm">/* Remove the heap top element */</span>
+<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="cm">/* Remove root elements */</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="n">peek</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="n">removeMax</span><span class="p">()</span><span class="w"> </span><span class="c1">// 5</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="n">peek</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="n">removeMax</span><span class="p">()</span><span class="w"> </span><span class="c1">// 4</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="n">peek</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="n">removeMax</span><span class="p">()</span><span class="w"> </span><span class="c1">// 3</span>
@@ -5089,23 +5161,23 @@
 <a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="cm">/* Get the heap size */</span>
 <a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="kd">let</span><span class="w"> </span><span class="nv">size</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a>
-<a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a><span class="cm">/* Check if the heap is empty */</span>
+<a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a><span class="cm">/* Check whether the heap is empty */</span>
 <a id="__codelineno-5-26" name="__codelineno-5-26" href="#__codelineno-5-26"></a><span class="kd">let</span><span class="w"> </span><span class="nv">isEmpty</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="bp">isEmpty</span>
 <a id="__codelineno-5-27" name="__codelineno-5-27" href="#__codelineno-5-27"></a>
-<a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="cm">/* Build a heap from an input list */</span>
+<a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="cm">/* Build a heap from a list */</span>
 <a id="__codelineno-5-29" name="__codelineno-5-29" href="#__codelineno-5-29"></a><span class="kd">let</span><span class="w"> </span><span class="nv">heap2</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Heap</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">])</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="c1">// JavaScript does not provide a built-in Heap class</span>
+<div class="highlight"><span class="filename">heap.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="c1">// JavaScript provides no built-in Heap class</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="c1">// TypeScript does not provide a built-in Heap class</span>
+<div class="highlight"><span class="filename">heap.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="c1">// TypeScript provides no built-in Heap class</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="c1">// Dart does not provide a built-in Heap class</span>
+<div class="highlight"><span class="filename">heap.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="c1">// Dart provides no built-in Heap class</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
@@ -5118,18 +5190,18 @@
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="c1">// Initialize a max heap</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">max_heap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">BinaryHeap</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a>
-<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="cm">/* Push elements into the heap */</span>
+<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="cm">/* Insert elements into the heap */</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="n">max_heap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="n">max_heap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="n">max_heap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="n">max_heap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="n">max_heap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a>
-<a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="cm">/* Get the heap top element */</span>
+<a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="cm">/* Access the root element */</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="kd">let</span><span class="w"> </span><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max_heap</span><span class="p">.</span><span class="n">peek</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span><span class="w">  </span><span class="c1">// 5</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a>
-<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="cm">/* Remove the heap top element */</span>
-<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="c1">// The removed elements will form a descending sequence</span>
+<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="cm">/* Remove root elements */</span>
+<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="c1">// Removed elements form a descending sequence</span>
 <a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="kd">let</span><span class="w"> </span><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max_heap</span><span class="p">.</span><span class="n">pop</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span><span class="w">   </span><span class="c1">// 5</span>
 <a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="kd">let</span><span class="w"> </span><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max_heap</span><span class="p">.</span><span class="n">pop</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span><span class="w">   </span><span class="c1">// 4</span>
 <a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a><span class="kd">let</span><span class="w"> </span><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max_heap</span><span class="p">.</span><span class="n">pop</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span><span class="w">   </span><span class="c1">// 3</span>
@@ -5139,36 +5211,36 @@
 <a id="__codelineno-9-28" name="__codelineno-9-28" href="#__codelineno-9-28"></a><span class="cm">/* Get the heap size */</span>
 <a id="__codelineno-9-29" name="__codelineno-9-29" href="#__codelineno-9-29"></a><span class="kd">let</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max_heap</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
 <a id="__codelineno-9-30" name="__codelineno-9-30" href="#__codelineno-9-30"></a>
-<a id="__codelineno-9-31" name="__codelineno-9-31" href="#__codelineno-9-31"></a><span class="cm">/* Check if the heap is empty */</span>
+<a id="__codelineno-9-31" name="__codelineno-9-31" href="#__codelineno-9-31"></a><span class="cm">/* Check whether the heap is empty */</span>
 <a id="__codelineno-9-32" name="__codelineno-9-32" href="#__codelineno-9-32"></a><span class="kd">let</span><span class="w"> </span><span class="n">is_empty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max_heap</span><span class="p">.</span><span class="n">is_empty</span><span class="p">();</span>
 <a id="__codelineno-9-33" name="__codelineno-9-33" href="#__codelineno-9-33"></a>
-<a id="__codelineno-9-34" name="__codelineno-9-34" href="#__codelineno-9-34"></a><span class="cm">/* Build a heap from an input list */</span>
+<a id="__codelineno-9-34" name="__codelineno-9-34" href="#__codelineno-9-34"></a><span class="cm">/* Build a heap from a list */</span>
 <a id="__codelineno-9-35" name="__codelineno-9-35" href="#__codelineno-9-35"></a><span class="kd">let</span><span class="w"> </span><span class="n">min_heap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">BinaryHeap</span><span class="p">::</span><span class="n">from</span><span class="p">(</span><span class="fm">vec!</span><span class="p">[</span><span class="n">Reverse</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span><span class="w"> </span><span class="n">Reverse</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span><span class="w"> </span><span class="n">Reverse</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span><span class="w"> </span><span class="n">Reverse</span><span class="p">(</span><span class="mi">5</span><span class="p">),</span><span class="w"> </span><span class="n">Reverse</span><span class="p">(</span><span class="mi">4</span><span class="p">)]);</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// C does not provide a built-in Heap class</span>
+<div class="highlight"><span class="filename">heap.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// C provides no built-in Heap class</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">heap.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Initialize a heap */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="c1">// Initialize a min heap</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="kd">var</span><span class="w"> </span><span class="nv">minHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">PriorityQueue</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span>
-<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="c1">// Initialize a max heap (use lambda expression to modify Comparator)</span>
+<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="c1">// Initialize a max heap by changing the Comparator with a lambda expression</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="kd">val</span><span class="w"> </span><span class="nv">maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">PriorityQueue</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">a</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">-&gt;</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a>
-<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="cm">/* Push elements into the heap */</span>
+<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="cm">/* Insert elements into the heap */</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="n">maxHeap</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="n">maxHeap</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="m">3</span><span class="p">)</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="n">maxHeap</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="m">2</span><span class="p">)</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="n">maxHeap</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="m">5</span><span class="p">)</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="n">maxHeap</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="m">4</span><span class="p">)</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a>
-<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="cm">/* Get the heap top element */</span>
+<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="cm">/* Access the root element */</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="kd">var</span><span class="w"> </span><span class="nv">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">peek</span><span class="p">()</span><span class="w"> </span><span class="c1">// 5</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a>
-<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="cm">/* Remove the heap top element */</span>
-<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="c1">// The removed elements will form a descending sequence</span>
+<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="cm">/* Remove root elements */</span>
+<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="c1">// Removed elements form a descending sequence</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">poll</span><span class="p">()</span><span class="w"> </span><span class="c1">// 5</span>
 <a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">poll</span><span class="p">()</span><span class="w"> </span><span class="c1">// 4</span>
 <a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">poll</span><span class="p">()</span><span class="w"> </span><span class="c1">// 3</span>
@@ -5178,250 +5250,250 @@
 <a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="cm">/* Get the heap size */</span>
 <a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="kd">val</span><span class="w"> </span><span class="nv">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a>
-<a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="cm">/* Check if the heap is empty */</span>
+<a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="cm">/* Check whether the heap is empty */</span>
 <a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="kd">val</span><span class="w"> </span><span class="nv">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">isEmpty</span><span class="p">()</span>
 <a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a>
-<a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a><span class="cm">/* Build a heap from an input list */</span>
+<a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a><span class="cm">/* Build a heap from a list */</span>
 <a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="n">minHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">PriorityQueue</span><span class="p">(</span><span class="n">mutableListOf</span><span class="p">(</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="m">5</span><span class="p">,</span><span class="w"> </span><span class="m">4</span><span class="p">))</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Ruby does not provide a built-in Heap class</span>
+<div class="highlight"><span class="filename">heap.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Ruby provides no built-in Heap class</span>
 </code></pre></div>
 </div>
 </div>
 </div>
 <details class="pythontutor">
-<summary>Code Visualization</summary>
-<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=import%20heapq%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%B0%8F%E9%A1%B6%E5%A0%86%0A%20%20%20%20min_heap,%20flag%20%3D%20%5B%5D,%201%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%A4%A7%E9%A1%B6%E5%A0%86%0A%20%20%20%20max_heap,%20flag%20%3D%20%5B%5D,%20-1%0A%20%20%20%20%0A%20%20%20%20%23%20Python%20%E7%9A%84%20heapq%20%E6%A8%A1%E5%9D%97%E9%BB%98%E8%AE%A4%E5%AE%9E%E7%8E%B0%E5%B0%8F%E9%A1%B6%E5%A0%86%0A%20%20%20%20%23%20%E8%80%83%E8%99%91%E5%B0%86%E2%80%9C%E5%85%83%E7%B4%A0%E5%8F%96%E8%B4%9F%E2%80%9D%E5%90%8E%E5%86%8D%E5%85%A5%E5%A0%86%EF%BC%8C%E8%BF%99%E6%A0%B7%E5%B0%B1%E5%8F%AF%E4%BB%A5%E5%B0%86%E5%A4%A7%E5%B0%8F%E5%85%B3%E7%B3%BB%E9%A2%A0%E5%80%92%EF%BC%8C%E4%BB%8E%E8%80%8C%E5%AE%9E%E7%8E%B0%E5%A4%A7%E9%A1%B6%E5%A0%86%0A%20%20%20%20%23%20%E5%9C%A8%E6%9C%AC%E7%A4%BA%E4%BE%8B%E4%B8%AD%EF%BC%8Cflag%20%3D%201%20%E6%97%B6%E5%AF%B9%E5%BA%94%E5%B0%8F%E9%A1%B6%E5%A0%86%EF%BC%8Cflag%20%3D%20-1%20%E6%97%B6%E5%AF%B9%E5%BA%94%E5%A4%A7%E9%A1%B6%E5%A0%86%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%85%A5%E5%A0%86%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%201%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%203%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%202%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%205%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%204%29%0A%20%20%20%20%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%A0%86%E9%A1%B6%E5%85%83%E7%B4%A0%0A%20%20%20%20peek%20%3D%20flag%20*%20max_heap%5B0%5D%20%23%205%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%A0%86%E9%A1%B6%E5%85%83%E7%B4%A0%E5%87%BA%E5%A0%86%0A%20%20%20%20%23%20%E5%87%BA%E5%A0%86%E5%85%83%E7%B4%A0%E4%BC%9A%E5%BD%A2%E6%88%90%E4%B8%80%E4%B8%AA%E4%BB%8E%E5%A4%A7%E5%88%B0%E5%B0%8F%E7%9A%84%E5%BA%8F%E5%88%97%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%205%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%204%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%203%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%202%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%201%0A%20%20%20%20%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%A0%86%E5%A4%A7%E5%B0%8F%0A%20%20%20%20size%20%3D%20len%28max_heap%29%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%88%A4%E6%96%AD%E5%A0%86%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%0A%20%20%20%20is_empty%20%3D%20not%20max_heap%0A%20%20%20%20%0A%20%20%20%20%23%20%E8%BE%93%E5%85%A5%E5%88%97%E8%A1%A8%E5%B9%B6%E5%BB%BA%E5%A0%86%0A%20%20%20%20min_heap%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20heapq.heapify%28min_heap%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=import%20heapq%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%B0%8F%E9%A1%B6%E5%A0%86%0A%20%20%20%20min_heap,%20flag%20%3D%20%5B%5D,%201%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%A4%A7%E9%A1%B6%E5%A0%86%0A%20%20%20%20max_heap,%20flag%20%3D%20%5B%5D,%20-1%0A%20%20%20%20%0A%20%20%20%20%23%20Python%20%E7%9A%84%20heapq%20%E6%A8%A1%E5%9D%97%E9%BB%98%E8%AE%A4%E5%AE%9E%E7%8E%B0%E5%B0%8F%E9%A1%B6%E5%A0%86%0A%20%20%20%20%23%20%E8%80%83%E8%99%91%E5%B0%86%E2%80%9C%E5%85%83%E7%B4%A0%E5%8F%96%E8%B4%9F%E2%80%9D%E5%90%8E%E5%86%8D%E5%85%A5%E5%A0%86%EF%BC%8C%E8%BF%99%E6%A0%B7%E5%B0%B1%E5%8F%AF%E4%BB%A5%E5%B0%86%E5%A4%A7%E5%B0%8F%E5%85%B3%E7%B3%BB%E9%A2%A0%E5%80%92%EF%BC%8C%E4%BB%8E%E8%80%8C%E5%AE%9E%E7%8E%B0%E5%A4%A7%E9%A1%B6%E5%A0%86%0A%20%20%20%20%23%20%E5%9C%A8%E6%9C%AC%E7%A4%BA%E4%BE%8B%E4%B8%AD%EF%BC%8Cflag%20%3D%201%20%E6%97%B6%E5%AF%B9%E5%BA%94%E5%B0%8F%E9%A1%B6%E5%A0%86%EF%BC%8Cflag%20%3D%20-1%20%E6%97%B6%E5%AF%B9%E5%BA%94%E5%A4%A7%E9%A1%B6%E5%A0%86%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%85%A5%E5%A0%86%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%201%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%203%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%202%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%205%29%0A%20%20%20%20heapq.heappush%28max_heap,%20flag%20*%204%29%0A%20%20%20%20%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%A0%86%E9%A1%B6%E5%85%83%E7%B4%A0%0A%20%20%20%20peek%20%3D%20flag%20*%20max_heap%5B0%5D%20%23%205%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%A0%86%E9%A1%B6%E5%85%83%E7%B4%A0%E5%87%BA%E5%A0%86%0A%20%20%20%20%23%20%E5%87%BA%E5%A0%86%E5%85%83%E7%B4%A0%E4%BC%9A%E5%BD%A2%E6%88%90%E4%B8%80%E4%B8%AA%E4%BB%8E%E5%A4%A7%E5%88%B0%E5%B0%8F%E7%9A%84%E5%BA%8F%E5%88%97%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%205%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%204%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%203%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%202%0A%20%20%20%20val%20%3D%20flag%20*%20heapq.heappop%28max_heap%29%20%23%201%0A%20%20%20%20%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%A0%86%E5%A4%A7%E5%B0%8F%0A%20%20%20%20size%20%3D%20len%28max_heap%29%0A%20%20%20%20%0A%20%20%20%20%23%20%E5%88%A4%E6%96%AD%E5%A0%86%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%0A%20%20%20%20is_empty%20%3D%20not%20max_heap%0A%20%20%20%20%0A%20%20%20%20%23%20%E8%BE%93%E5%85%A5%E5%88%97%E8%A1%A8%E5%B9%B6%E5%BB%BA%E5%A0%86%0A%20%20%20%20min_heap%20%3D%20%5B1,%203,%202,%205,%204%5D%0A%20%20%20%20heapq.heapify%28min_heap%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=import%20heapq%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20min%20heap%0A%20%20%20%20min_heap%2C%20flag%20%3D%20%5B%5D%2C%201%0A%20%20%20%20%23%20Initialize%20a%20max%20heap%0A%20%20%20%20max_heap%2C%20flag%20%3D%20%5B%5D%2C%20-1%0A%20%20%20%20%0A%20%20%20%20%23%20Python%27s%20heapq%20module%20implements%20a%20min%20heap%20by%20default%0A%20%20%20%20%23%20Negate%20elements%20before%20insertion%20to%20reverse%20their%20order%20and%20implement%20a%20max%20heap%0A%20%20%20%20%23%20Here%2C%20flag%20%3D%201%20gives%20a%20min%20heap%20and%20flag%20%3D%20-1%20gives%20a%20max%20heap%0A%20%20%20%20%0A%20%20%20%20%23%20Insert%20elements%20into%20the%20heap%0A%20%20%20%20heapq.heappush%28max_heap%2C%20flag%20%2A%201%29%0A%20%20%20%20heapq.heappush%28max_heap%2C%20flag%20%2A%203%29%0A%20%20%20%20heapq.heappush%28max_heap%2C%20flag%20%2A%202%29%0A%20%20%20%20heapq.heappush%28max_heap%2C%20flag%20%2A%205%29%0A%20%20%20%20heapq.heappush%28max_heap%2C%20flag%20%2A%204%29%0A%20%20%20%20%0A%20%20%20%20%23%20Access%20the%20root%20element%0A%20%20%20%20peek%20%3D%20flag%20%2A%20max_heap%5B0%5D%20%23%205%0A%20%20%20%20%0A%20%20%20%20%23%20Remove%20root%20elements%0A%20%20%20%20%23%20Removed%20elements%20form%20a%20descending%20sequence%0A%20%20%20%20val%20%3D%20flag%20%2A%20heapq.heappop%28max_heap%29%20%23%205%0A%20%20%20%20val%20%3D%20flag%20%2A%20heapq.heappop%28max_heap%29%20%23%204%0A%20%20%20%20val%20%3D%20flag%20%2A%20heapq.heappop%28max_heap%29%20%23%203%0A%20%20%20%20val%20%3D%20flag%20%2A%20heapq.heappop%28max_heap%29%20%23%202%0A%20%20%20%20val%20%3D%20flag%20%2A%20heapq.heappop%28max_heap%29%20%23%201%0A%20%20%20%20%0A%20%20%20%20%23%20Get%20the%20heap%20size%0A%20%20%20%20size%20%3D%20len%28max_heap%29%0A%20%20%20%20%0A%20%20%20%20%23%20Check%20whether%20the%20heap%20is%20empty%0A%20%20%20%20is_empty%20%3D%20not%20max_heap%0A%20%20%20%20%0A%20%20%20%20%23%20Build%20a%20heap%20from%20a%20list%0A%20%20%20%20min_heap%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D%0A%20%20%20%20heapq.heapify%28min_heap%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=import%20heapq%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20min%20heap%0A%20%20%20%20min_heap%2C%20flag%20%3D%20%5B%5D%2C%201%0A%20%20%20%20%23%20Initialize%20a%20max%20heap%0A%20%20%20%20max_heap%2C%20flag%20%3D%20%5B%5D%2C%20-1%0A%20%20%20%20%0A%20%20%20%20%23%20Python%27s%20heapq%20module%20implements%20a%20min%20heap%20by%20default%0A%20%20%20%20%23%20Negate%20elements%20before%20insertion%20to%20reverse%20their%20order%20and%20implement%20a%20max%20heap%0A%20%20%20%20%23%20Here%2C%20flag%20%3D%201%20gives%20a%20min%20heap%20and%20flag%20%3D%20-1%20gives%20a%20max%20heap%0A%20%20%20%20%0A%20%20%20%20%23%20Insert%20elements%20into%20the%20heap%0A%20%20%20%20heapq.heappush%28max_heap%2C%20flag%20%2A%201%29%0A%20%20%20%20heapq.heappush%28max_heap%2C%20flag%20%2A%203%29%0A%20%20%20%20heapq.heappush%28max_heap%2C%20flag%20%2A%202%29%0A%20%20%20%20heapq.heappush%28max_heap%2C%20flag%20%2A%205%29%0A%20%20%20%20heapq.heappush%28max_heap%2C%20flag%20%2A%204%29%0A%20%20%20%20%0A%20%20%20%20%23%20Access%20the%20root%20element%0A%20%20%20%20peek%20%3D%20flag%20%2A%20max_heap%5B0%5D%20%23%205%0A%20%20%20%20%0A%20%20%20%20%23%20Remove%20root%20elements%0A%20%20%20%20%23%20Removed%20elements%20form%20a%20descending%20sequence%0A%20%20%20%20val%20%3D%20flag%20%2A%20heapq.heappop%28max_heap%29%20%23%205%0A%20%20%20%20val%20%3D%20flag%20%2A%20heapq.heappop%28max_heap%29%20%23%204%0A%20%20%20%20val%20%3D%20flag%20%2A%20heapq.heappop%28max_heap%29%20%23%203%0A%20%20%20%20val%20%3D%20flag%20%2A%20heapq.heappop%28max_heap%29%20%23%202%0A%20%20%20%20val%20%3D%20flag%20%2A%20heapq.heappop%28max_heap%29%20%23%201%0A%20%20%20%20%0A%20%20%20%20%23%20Get%20the%20heap%20size%0A%20%20%20%20size%20%3D%20len%28max_heap%29%0A%20%20%20%20%0A%20%20%20%20%23%20Check%20whether%20the%20heap%20is%20empty%0A%20%20%20%20is_empty%20%3D%20not%20max_heap%0A%20%20%20%20%0A%20%20%20%20%23%20Build%20a%20heap%20from%20a%20list%0A%20%20%20%20min_heap%20%3D%20%5B1%2C%203%2C%202%2C%205%2C%204%5D%0A%20%20%20%20heapq.heapify%28min_heap%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<h2 id="812-implementation-of-the-heap">8.1.2 &nbsp; Implementation of the Heap<a class="headerlink" href="#812-implementation-of-the-heap" title="Permanent link">&para;</a></h2>
-<p>The following implementation is for a max heap. To convert it to a min heap, simply reverse all comparison logic related to ordering (for example, replace <span class="arithmatex">\(\geq\)</span> with <span class="arithmatex">\(\leq\)</span>). Interested readers are encouraged to implement this on their own.</p>
-<h3 id="1-heap-storage-and-representation">1. &nbsp; Heap Storage and Representation<a class="headerlink" href="#1-heap-storage-and-representation" title="Permanent link">&para;</a></h3>
-<p>As mentioned in the "Binary Tree" chapter, complete binary trees are well-suited for array representation. Since heaps are a type of complete binary tree, <strong>we will use arrays to store heaps</strong>.</p>
-<p>When representing a binary tree with an array, elements represent node values, and indexes represent node positions in the binary tree. <strong>Parent-child relationships are represented through index-mapping formulas</strong>.</p>
-<p>As shown in Figure 8-2, given an index <span class="arithmatex">\(i\)</span>, the index of its left child is <span class="arithmatex">\(2i + 1\)</span>, the index of its right child is <span class="arithmatex">\(2i + 2\)</span>, and the index of its parent is <span class="arithmatex">\((i - 1) / 2\)</span> (floor division). When an index is out of bounds, it indicates a null node or that the node does not exist.</p>
-<p><img alt="Representation and storage of heaps" class="animation-figure" src="../heap.assets/representation_of_heap.png" /></p>
-<p align="center"> Figure 8-2 &nbsp; Representation and storage of heaps </p>
+<h2 id="812-implementing-a-heap">8.1.2 &nbsp; Implementing a heap<a class="headerlink" href="#812-implementing-a-heap" title="Permanent link">&para;</a></h2>
+<p>The following implementation is a max heap. To turn it into a min heap, reverse every ordering comparison, for example replacing <span class="arithmatex">\(\geq\)</span> with <span class="arithmatex">\(\leq\)</span>. Interested readers can try this themselves.</p>
+<h3 id="1-storage-and-representation">1. &nbsp; Storage and representation<a class="headerlink" href="#1-storage-and-representation" title="Permanent link">&para;</a></h3>
+<p>The binary-tree chapter showed that complete binary trees are well suited to array representation. Since a heap is a complete binary tree, <strong>we will store it in an array</strong>.</p>
+<p>In an array representation, elements store node values and indices identify positions in the tree. <strong>Index-mapping formulas take the place of node pointers</strong>.</p>
+<p>As shown below, a node at index <span class="arithmatex">\(i\)</span> has its left child at <span class="arithmatex">\(2i + 1\)</span>, its right child at <span class="arithmatex">\(2i + 2\)</span>, and its parent at <span class="arithmatex">\((i - 1) / 2\)</span>, rounded down. An out-of-bounds index represents a missing node.</p>
+<p><img alt="Representing and storing a heap" class="animation-figure" src="../heap.assets/representation_of_heap.png" /></p>
+<p align="center"> Figure 8-2 &nbsp; Representing and storing a heap </p>
 
-<p>We can encapsulate the index mapping formula into functions for convenient subsequent use:</p>
+<p>Wrap the index formulas in functions for convenient use:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">my_heap.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Get index of left child node"""</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Get the left child index"""</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="k">def</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a><span class="w">    </span><span class="sd">"""Get index of right child node"""</span>
+<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a><span class="w">    </span><span class="sd">"""Get the right child index"""</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>    <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">2</span>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a><span class="k">def</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a><span class="w">    </span><span class="sd">"""Get index of parent node"""</span>
+<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a><span class="w">    </span><span class="sd">"""Get the parent index"""</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>    <span class="k">return</span> <span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>  <span class="c1"># Floor division</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Get index of left child node */</span>
+<div class="highlight"><span class="filename">my_heap.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Get the left child index */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="p">}</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a>
-<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="cm">/* Get index of right child node */</span>
+<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="cm">/* Get the right child index */</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="kt">int</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="p">}</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a>
-<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="cm">/* Get index of parent node */</span>
+<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="cm">/* Get the parent index */</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="kt">int</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Floor division</span>
+<a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Integer division truncates toward zero</span>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Get index of left child node */</span>
+<div class="highlight"><span class="filename">my_heap.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Get the left child index */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="p">}</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a>
-<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="cm">/* Get index of right child node */</span>
+<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="cm">/* Get the right child index */</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="kt">int</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="p">}</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a>
-<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="cm">/* Get index of parent node */</span>
+<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="cm">/* Get the parent index */</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="kt">int</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Floor division</span>
+<a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Integer division (truncates toward zero in Java)</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Get index of left child node */</span>
+<div class="highlight"><span class="filename">my_heap.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Get the left child index */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Left</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="p">}</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a>
-<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="cm">/* Get index of right child node */</span>
+<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="cm">/* Get the right child index */</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Right</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="p">}</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a>
-<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="cm">/* Get index of parent node */</span>
+<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="cm">/* Get the parent index */</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Parent</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Floor division</span>
+<a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Integer division (truncates toward zero in C#)</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Get index of left child node */</span>
+<div class="highlight"><span class="filename">my_heap.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Get the left child index */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="nx">left</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="o">*</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="p">}</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a>
-<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="cm">/* Get index of right child node */</span>
+<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="cm">/* Get the right child index */</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="nx">right</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="o">*</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="p">}</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a>
-<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="cm">/* Get index of parent node */</span>
+<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="cm">/* Get the parent index */</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="nx">parent</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">    </span><span class="c1">// Floor division</span>
+<a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">    </span><span class="c1">// Integer division (truncates toward zero in Go)</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Get index of left child node */</span>
+<div class="highlight"><span class="filename">my_heap.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Get the left child index */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="p">}</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a>
-<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="cm">/* Get index of right child node */</span>
+<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="cm">/* Get the right child index */</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="kd">func</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="p">}</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a>
-<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="cm">/* Get index of parent node */</span>
+<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="cm">/* Get the parent index */</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="kd">func</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">    </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// Floor division</span>
+<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">    </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// Integer division truncates toward zero</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Get index of left child node */</span>
+<div class="highlight"><span class="filename">my_heap.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Get the left child index */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="n">#left</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="p">}</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a>
-<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="cm">/* Get index of right child node */</span>
+<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="cm">/* Get the right child index */</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="n">#right</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="p">}</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a>
-<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="cm">/* Get index of parent node */</span>
+<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="cm">/* Get the parent index */</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="n">#parent</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">((</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Floor division</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Get index of left child node */</span>
+<div class="highlight"><span class="filename">my_heap.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Get the left child index */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="nx">left</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="p">}</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a>
-<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="cm">/* Get index of right child node */</span>
+<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="cm">/* Get the right child index */</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="nx">right</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="p">}</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a>
-<a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="cm">/* Get index of parent node */</span>
+<a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="cm">/* Get the parent index */</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="nx">parent</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">((</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Floor division</span>
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Get index of left child node */</span>
+<div class="highlight"><span class="filename">my_heap.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Get the left child index */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">_left</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="p">}</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a>
-<a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="cm">/* Get index of right child node */</span>
+<a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="cm">/* Get the right child index */</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="kt">int</span><span class="w"> </span><span class="n">_right</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">2</span><span class="p">;</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="p">}</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a>
-<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="cm">/* Get index of parent node */</span>
+<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="cm">/* Get the parent index */</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="kt">int</span><span class="w"> </span><span class="n">_parent</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Floor division</span>
+<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Integer division</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Get index of left child node */</span>
+<div class="highlight"><span class="filename">my_heap.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Get the left child index */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="p">}</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a>
-<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="cm">/* Get index of right child node */</span>
+<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="cm">/* Get the right child index */</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="k">fn</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">    </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="p">}</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a>
-<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="cm">/* Get index of parent node */</span>
+<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="cm">/* Get the parent index */</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="k">fn</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">    </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// Floor division</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Get index of left child node */</span>
+<div class="highlight"><span class="filename">my_heap.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Get the left child index */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="p">}</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a>
-<a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="cm">/* Get index of right child node */</span>
+<a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="cm">/* Get the right child index */</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="kt">int</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="p">}</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a>
-<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="cm">/* Get index of parent node */</span>
+<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="cm">/* Get the parent index */</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="kt">int</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Round down</span>
+<a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Integer division truncates toward zero</span>
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Get index of left child node */</span>
+<div class="highlight"><span class="filename">my_heap.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Get the left child index */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="p">}</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a>
-<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="cm">/* Get index of right child node */</span>
+<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="cm">/* Get the right child index */</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">2</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="p">}</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a>
-<a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="cm">/* Get index of parent node */</span>
+<a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="cm">/* Get the parent index */</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="c1">// Floor division</span>
+<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="c1">// Integer division (truncates toward zero in Kotlin)</span>
 <a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Get left child index ###</span>
+<div class="highlight"><span class="filename">my_heap.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Get the left child index ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="k">end</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a>
-<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="c1">### Get right child index ###</span>
+<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="c1">### Get the right child index ###</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="k">def</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">  </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="k">end</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a>
-<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="c1">### Get parent node index ###</span>
+<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="c1">### Get the parent index ###</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="k">def</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">  </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w">     </span><span class="c1"># Floor division</span>
 <a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="k">end</span>
@@ -5429,95 +5501,95 @@
 </div>
 </div>
 </div>
-<h3 id="2-accessing-the-heap-top-element">2. &nbsp; Accessing the Heap Top Element<a class="headerlink" href="#2-accessing-the-heap-top-element" title="Permanent link">&para;</a></h3>
-<p>The heap top element is the root node of the binary tree, which is also the first element of the list:</p>
+<h3 id="2-accessing-the-root">2. &nbsp; Accessing the root<a class="headerlink" href="#2-accessing-the-root" title="Permanent link">&para;</a></h3>
+<p>The heap's root is the first element of the list:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">my_heap.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">peek</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Access top element"""</span>
+<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Access the top element of the heap"""</span>
 <a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_heap</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Access top element */</span>
+<div class="highlight"><span class="filename">my_heap.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Access the top element of the heap */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Access top element */</span>
+<div class="highlight"><span class="filename">my_heap.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Access the top element of the heap */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Access top element */</span>
+<div class="highlight"><span class="filename">my_heap.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Access the top element of the heap */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Access top element */</span>
+<div class="highlight"><span class="filename">my_heap.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Access the top element of the heap */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="nx">peek</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Access top element */</span>
+<div class="highlight"><span class="filename">my_heap.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Access the top element of the heap */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">peek</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="n">maxHeap</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Access top element */</span>
+<div class="highlight"><span class="filename">my_heap.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Access the top element of the heap */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="nx">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#maxHeap</span><span class="p">[</span><span class="mf">0</span><span class="p">];</span>
 <a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Access top element */</span>
+<div class="highlight"><span class="filename">my_heap.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Access the top element of the heap */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="nx">peek</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">maxHeap</span><span class="p">[</span><span class="mf">0</span><span class="p">];</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Access top element */</span>
+<div class="highlight"><span class="filename">my_heap.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Access the top element of the heap */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">[</span><span class="m">0</span><span class="p">];</span>
 <a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Access top element */</span>
+<div class="highlight"><span class="filename">my_heap.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Access the top element of the heap */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">peek</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="bp">self</span><span class="p">.</span><span class="n">max_heap</span><span class="p">.</span><span class="n">first</span><span class="p">().</span><span class="n">copied</span><span class="p">()</span>
 <a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Access top element */</span>
+<div class="highlight"><span class="filename">my_heap.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Access the top element of the heap */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">peek</span><span class="p">(</span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="n">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">data</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Access top element */</span>
+<div class="highlight"><span class="filename">my_heap.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Access the top element of the heap */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">peek</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">[</span><span class="m">0</span><span class="o">]</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Access heap top element ###</span>
+<div class="highlight"><span class="filename">my_heap.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Access the top element of the heap ###</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">peek</span>
 <a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="vi">@max_heap</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="k">end</span>
@@ -5525,13 +5597,18 @@
 </div>
 </div>
 </div>
-<h3 id="3-inserting-an-element-into-the-heap">3. &nbsp; Inserting an Element Into the Heap<a class="headerlink" href="#3-inserting-an-element-into-the-heap" title="Permanent link">&para;</a></h3>
-<p>Given an element <code>val</code>, we first add it to the bottom of the heap. After insertion, because <code>val</code> may be larger than other elements in the heap, the heap property may be violated. <strong>Therefore, we need to restore the heap property along the path from the inserted node to the root</strong>. This operation is called <u>heapify</u>.</p>
-<p>Starting from the inserted node, <strong>perform heapify from bottom to top</strong>. As shown in Figure 8-3, we compare the inserted node with its parent, and if the inserted node is larger, we swap them. We continue this process from bottom to top until we move past the root or reach a node that no longer needs to be swapped.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20MaxHeap%3A%0A%20%20%20%20%22%22%22Max%20heap%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Place%20the%20list%20elements%20in%20the%20heap%20without%20changing%20their%20order%0A%20%20%20%20%20%20%20%20self.max_heap%20%3D%20nums%0A%0A%20%20%20%20def%20left%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20left%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%201%0A%0A%20%20%20%20def%20right%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20right%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%202%0A%0A%20%20%20%20def%20parent%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20parent%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%20%28i%20-%201%29%20//%202%20%20%23%20Floor%20division%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20heap%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self.max_heap%29%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22Check%20whether%20the%20heap%20is%20empty%22%22%22%0A%20%20%20%20%20%20%20%20return%20self.size%28%29%20%3D%3D%200%0A%0A%20%20%20%20def%20peek%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Access%20the%20top%20element%20of%20the%20heap%22%22%22%0A%20%20%20%20%20%20%20%20return%20self.max_heap%5B0%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20max%20heap%0A%20%20%20%20%23%20Note%20that%20the%20input%20array%20is%20already%20a%20valid%20heap%20%0A%20%20%20%20max_heap%20%3D%20MaxHeap%28%5B9,%208,%206,%206,%207,%205,%202,%201,%204,%203,%206,%202%5D%29%0A%0A%20%20%20%20%23%20Get%20the%20top%20element%20of%20the%20heap%0A%20%20%20%20peek%20%3D%20max_heap.peek%28%29%0A%20%20%20%20print%28f%22%5CnTop%20element%20of%20the%20heap%3A%20%7Bpeek%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20MaxHeap%3A%0A%20%20%20%20%22%22%22Max%20heap%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Place%20the%20list%20elements%20in%20the%20heap%20without%20changing%20their%20order%0A%20%20%20%20%20%20%20%20self.max_heap%20%3D%20nums%0A%0A%20%20%20%20def%20left%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20left%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%201%0A%0A%20%20%20%20def%20right%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20right%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%202%0A%0A%20%20%20%20def%20parent%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20parent%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%20%28i%20-%201%29%20//%202%20%20%23%20Floor%20division%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20heap%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self.max_heap%29%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22Check%20whether%20the%20heap%20is%20empty%22%22%22%0A%20%20%20%20%20%20%20%20return%20self.size%28%29%20%3D%3D%200%0A%0A%20%20%20%20def%20peek%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Access%20the%20top%20element%20of%20the%20heap%22%22%22%0A%20%20%20%20%20%20%20%20return%20self.max_heap%5B0%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20max%20heap%0A%20%20%20%20%23%20Note%20that%20the%20input%20array%20is%20already%20a%20valid%20heap%20%0A%20%20%20%20max_heap%20%3D%20MaxHeap%28%5B9,%208,%206,%206,%207,%205,%202,%201,%204,%203,%206,%202%5D%29%0A%0A%20%20%20%20%23%20Get%20the%20top%20element%20of%20the%20heap%0A%20%20%20%20peek%20%3D%20max_heap.peek%28%29%0A%20%20%20%20print%28f%22%5CnTop%20element%20of%20the%20heap%3A%20%7Bpeek%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=7&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="3-inserting-an-element">3. &nbsp; Inserting an element<a class="headerlink" href="#3-inserting-an-element" title="Permanent link">&para;</a></h3>
+<p>To insert <code>val</code>, first append it at the end of the heap. Since it may be larger than other elements, this can violate the heap property. <strong>We must restore the property along the path from the inserted node to the root</strong>, an operation called <u>heapifying</u>.</p>
+<p>Start at the inserted node and <strong>sift upward</strong>. As shown below, compare it with its parent and swap them if the inserted value is larger. Repeat upward until the root is reached or no swap is needed.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:9"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">&lt;1&gt;</label><label for="__tabbed_4_2">&lt;2&gt;</label><label for="__tabbed_4_3">&lt;3&gt;</label><label for="__tabbed_4_4">&lt;4&gt;</label><label for="__tabbed_4_5">&lt;5&gt;</label><label for="__tabbed_4_6">&lt;6&gt;</label><label for="__tabbed_4_7">&lt;7&gt;</label><label for="__tabbed_4_8">&lt;8&gt;</label><label for="__tabbed_4_9">&lt;9&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Steps of inserting an element into the heap" class="animation-figure" src="../heap.assets/heap_push_step1.png" /></p>
+<p><img alt="Steps for inserting into a heap" class="animation-figure" src="../heap.assets/heap_push_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="heap_push_step2" class="animation-figure" src="../heap.assets/heap_push_step2.png" /></p>
@@ -5559,343 +5636,343 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 8-3 &nbsp; Steps of inserting an element into the heap </p>
+<p align="center"> Figure 8-3 &nbsp; Steps for inserting into a heap </p>
 
-<p>Given a total of <span class="arithmatex">\(n\)</span> nodes, the tree height is <span class="arithmatex">\(O(\log n)\)</span>. Thus, the number of loop iterations in the heapify operation is at most <span class="arithmatex">\(O(\log n)\)</span>, <strong>making the time complexity of the element insertion operation <span class="arithmatex">\(O(\log n)\)</span></strong>. The code is as follows:</p>
+<p>With <span class="arithmatex">\(n\)</span> nodes, the tree height is <span class="arithmatex">\(O(\log n)\)</span>, so sifting takes at most <span class="arithmatex">\(O(\log n)\)</span> iterations. <strong>Insertion therefore takes <span class="arithmatex">\(O(\log n)\)</span> time</strong>. The code is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">my_heap.py</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Element enters heap"""</span>
-<a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="c1"># Add node</span>
+<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Push an element onto the heap"""</span>
+<a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="c1"># Add a node</span>
 <a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">max_heap</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
-<a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>    <span class="c1"># Heapify from bottom to top</span>
+<a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>    <span class="c1"># Sift up</span>
 <a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">sift_up</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a>
 <a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a><span class="k">def</span><span class="w"> </span><span class="nf">sift_up</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a><span class="w">    </span><span class="sd">"""Starting from node i, heapify from bottom to top"""</span>
+<a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a><span class="w">    </span><span class="sd">"""Sift up starting from node i"""</span>
 <a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a>    <span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
-<a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a>        <span class="c1"># Get parent node of node i</span>
+<a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a>        <span class="c1"># Get the parent of node i</span>
 <a id="__codelineno-39-12" name="__codelineno-39-12" href="#__codelineno-39-12"></a>        <span class="n">p</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">parent</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-<a id="__codelineno-39-13" name="__codelineno-39-13" href="#__codelineno-39-13"></a>        <span class="c1"># When "crossing root node" or "node needs no repair", end heapify</span>
+<a id="__codelineno-39-13" name="__codelineno-39-13" href="#__codelineno-39-13"></a>        <span class="c1"># Stop sifting when the root has been passed or the node needs no repair</span>
 <a id="__codelineno-39-14" name="__codelineno-39-14" href="#__codelineno-39-14"></a>        <span class="k">if</span> <span class="n">p</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_heap</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_heap</span><span class="p">[</span><span class="n">p</span><span class="p">]:</span>
 <a id="__codelineno-39-15" name="__codelineno-39-15" href="#__codelineno-39-15"></a>            <span class="k">break</span>
-<a id="__codelineno-39-16" name="__codelineno-39-16" href="#__codelineno-39-16"></a>        <span class="c1"># Swap two nodes</span>
+<a id="__codelineno-39-16" name="__codelineno-39-16" href="#__codelineno-39-16"></a>        <span class="c1"># Swap the two nodes</span>
 <a id="__codelineno-39-17" name="__codelineno-39-17" href="#__codelineno-39-17"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
-<a id="__codelineno-39-18" name="__codelineno-39-18" href="#__codelineno-39-18"></a>        <span class="c1"># Loop upward heapify</span>
+<a id="__codelineno-39-18" name="__codelineno-39-18" href="#__codelineno-39-18"></a>        <span class="c1"># Continue sifting up</span>
 <a id="__codelineno-39-19" name="__codelineno-39-19" href="#__codelineno-39-19"></a>        <span class="n">i</span> <span class="o">=</span> <span class="n">p</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Element enters heap */</span>
+<div class="highlight"><span class="filename">my_heap.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Push an element onto the heap */</span>
 <a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="c1">// Add node</span>
+<a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="c1">// Add a node</span>
 <a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">    </span><span class="c1">// Heapify from bottom to top</span>
+<a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">    </span><span class="c1">// Sift up</span>
 <a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w">    </span><span class="n">siftUp</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="p">}</span>
 <a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a>
-<a id="__codelineno-40-9" name="__codelineno-40-9" href="#__codelineno-40-9"></a><span class="cm">/* Starting from node i, heapify from bottom to top */</span>
+<a id="__codelineno-40-9" name="__codelineno-40-9" href="#__codelineno-40-9"></a><span class="cm">/* Sift up starting from node i */</span>
 <a id="__codelineno-40-10" name="__codelineno-40-10" href="#__codelineno-40-10"></a><span class="kt">void</span><span class="w"> </span><span class="nf">siftUp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-11" name="__codelineno-40-11" href="#__codelineno-40-11"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nb">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-12" name="__codelineno-40-12" href="#__codelineno-40-12"></a><span class="w">        </span><span class="c1">// Get parent node of node i</span>
+<a id="__codelineno-40-12" name="__codelineno-40-12" href="#__codelineno-40-12"></a><span class="w">        </span><span class="c1">// Get the parent of node i</span>
 <a id="__codelineno-40-13" name="__codelineno-40-13" href="#__codelineno-40-13"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
-<a id="__codelineno-40-14" name="__codelineno-40-14" href="#__codelineno-40-14"></a><span class="w">        </span><span class="c1">// When "crossing root node" or "node needs no repair", end heapify</span>
+<a id="__codelineno-40-14" name="__codelineno-40-14" href="#__codelineno-40-14"></a><span class="w">        </span><span class="c1">// Stop sifting when the root has been passed or the node needs no repair</span>
 <a id="__codelineno-40-15" name="__codelineno-40-15" href="#__codelineno-40-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">p</span><span class="p">])</span>
 <a id="__codelineno-40-16" name="__codelineno-40-16" href="#__codelineno-40-16"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
-<a id="__codelineno-40-17" name="__codelineno-40-17" href="#__codelineno-40-17"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-40-17" name="__codelineno-40-17" href="#__codelineno-40-17"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-40-18" name="__codelineno-40-18" href="#__codelineno-40-18"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">p</span><span class="p">]);</span>
-<a id="__codelineno-40-19" name="__codelineno-40-19" href="#__codelineno-40-19"></a><span class="w">        </span><span class="c1">// Loop upward heapify</span>
+<a id="__codelineno-40-19" name="__codelineno-40-19" href="#__codelineno-40-19"></a><span class="w">        </span><span class="c1">// Continue sifting up</span>
 <a id="__codelineno-40-20" name="__codelineno-40-20" href="#__codelineno-40-20"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">p</span><span class="p">;</span>
 <a id="__codelineno-40-21" name="__codelineno-40-21" href="#__codelineno-40-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-40-22" name="__codelineno-40-22" href="#__codelineno-40-22"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Element enters heap */</span>
+<div class="highlight"><span class="filename">my_heap.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Push an element onto the heap */</span>
 <a id="__codelineno-41-2" name="__codelineno-41-2" href="#__codelineno-41-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="c1">// Add node</span>
+<a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="c1">// Add a node</span>
 <a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">    </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">    </span><span class="c1">// Heapify from bottom to top</span>
+<a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">    </span><span class="c1">// Sift up</span>
 <a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a><span class="w">    </span><span class="n">siftUp</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="p">}</span>
 <a id="__codelineno-41-8" name="__codelineno-41-8" href="#__codelineno-41-8"></a>
-<a id="__codelineno-41-9" name="__codelineno-41-9" href="#__codelineno-41-9"></a><span class="cm">/* Starting from node i, heapify from bottom to top */</span>
+<a id="__codelineno-41-9" name="__codelineno-41-9" href="#__codelineno-41-9"></a><span class="cm">/* Sift up starting from node i */</span>
 <a id="__codelineno-41-10" name="__codelineno-41-10" href="#__codelineno-41-10"></a><span class="kt">void</span><span class="w"> </span><span class="nf">siftUp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-11" name="__codelineno-41-11" href="#__codelineno-41-11"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-12" name="__codelineno-41-12" href="#__codelineno-41-12"></a><span class="w">        </span><span class="c1">// Get parent node of node i</span>
+<a id="__codelineno-41-12" name="__codelineno-41-12" href="#__codelineno-41-12"></a><span class="w">        </span><span class="c1">// Get the parent of node i</span>
 <a id="__codelineno-41-13" name="__codelineno-41-13" href="#__codelineno-41-13"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
-<a id="__codelineno-41-14" name="__codelineno-41-14" href="#__codelineno-41-14"></a><span class="w">        </span><span class="c1">// When "crossing root node" or "node needs no repair", end heapify</span>
+<a id="__codelineno-41-14" name="__codelineno-41-14" href="#__codelineno-41-14"></a><span class="w">        </span><span class="c1">// Stop sifting when the root has been passed or the node needs no repair</span>
 <a id="__codelineno-41-15" name="__codelineno-41-15" href="#__codelineno-41-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">p</span><span class="p">))</span>
 <a id="__codelineno-41-16" name="__codelineno-41-16" href="#__codelineno-41-16"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
-<a id="__codelineno-41-17" name="__codelineno-41-17" href="#__codelineno-41-17"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-41-17" name="__codelineno-41-17" href="#__codelineno-41-17"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-41-18" name="__codelineno-41-18" href="#__codelineno-41-18"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="p">);</span>
-<a id="__codelineno-41-19" name="__codelineno-41-19" href="#__codelineno-41-19"></a><span class="w">        </span><span class="c1">// Loop upward heapify</span>
+<a id="__codelineno-41-19" name="__codelineno-41-19" href="#__codelineno-41-19"></a><span class="w">        </span><span class="c1">// Continue sifting up</span>
 <a id="__codelineno-41-20" name="__codelineno-41-20" href="#__codelineno-41-20"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">p</span><span class="p">;</span>
 <a id="__codelineno-41-21" name="__codelineno-41-21" href="#__codelineno-41-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-41-22" name="__codelineno-41-22" href="#__codelineno-41-22"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Element enters heap */</span>
+<div class="highlight"><span class="filename">my_heap.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Push an element onto the heap */</span>
 <a id="__codelineno-42-2" name="__codelineno-42-2" href="#__codelineno-42-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Push</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="c1">// Add node</span>
+<a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="c1">// Add a node</span>
 <a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">    </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">    </span><span class="c1">// Heapify from bottom to top</span>
+<a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">    </span><span class="c1">// Sift up</span>
 <a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a><span class="w">    </span><span class="n">SiftUp</span><span class="p">(</span><span class="n">Size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="p">}</span>
 <a id="__codelineno-42-8" name="__codelineno-42-8" href="#__codelineno-42-8"></a>
-<a id="__codelineno-42-9" name="__codelineno-42-9" href="#__codelineno-42-9"></a><span class="cm">/* Starting from node i, heapify from bottom to top */</span>
+<a id="__codelineno-42-9" name="__codelineno-42-9" href="#__codelineno-42-9"></a><span class="cm">/* Sift up starting from node i */</span>
 <a id="__codelineno-42-10" name="__codelineno-42-10" href="#__codelineno-42-10"></a><span class="k">void</span><span class="w"> </span><span class="nf">SiftUp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-11" name="__codelineno-42-11" href="#__codelineno-42-11"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="k">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-12" name="__codelineno-42-12" href="#__codelineno-42-12"></a><span class="w">        </span><span class="c1">// Get parent node of node i</span>
+<a id="__codelineno-42-12" name="__codelineno-42-12" href="#__codelineno-42-12"></a><span class="w">        </span><span class="c1">// Get the parent of node i</span>
 <a id="__codelineno-42-13" name="__codelineno-42-13" href="#__codelineno-42-13"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Parent</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
-<a id="__codelineno-42-14" name="__codelineno-42-14" href="#__codelineno-42-14"></a><span class="w">        </span><span class="c1">// If 'past root node' or 'node needs no repair', end heapify</span>
+<a id="__codelineno-42-14" name="__codelineno-42-14" href="#__codelineno-42-14"></a><span class="w">        </span><span class="c1">// Stop sifting if the root has been passed or the node needs no adjustment</span>
 <a id="__codelineno-42-15" name="__codelineno-42-15" href="#__codelineno-42-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">p</span><span class="p">])</span>
 <a id="__codelineno-42-16" name="__codelineno-42-16" href="#__codelineno-42-16"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
-<a id="__codelineno-42-17" name="__codelineno-42-17" href="#__codelineno-42-17"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-42-17" name="__codelineno-42-17" href="#__codelineno-42-17"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-42-18" name="__codelineno-42-18" href="#__codelineno-42-18"></a><span class="w">        </span><span class="n">Swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="p">);</span>
-<a id="__codelineno-42-19" name="__codelineno-42-19" href="#__codelineno-42-19"></a><span class="w">        </span><span class="c1">// Loop upward heapify</span>
+<a id="__codelineno-42-19" name="__codelineno-42-19" href="#__codelineno-42-19"></a><span class="w">        </span><span class="c1">// Continue sifting up</span>
 <a id="__codelineno-42-20" name="__codelineno-42-20" href="#__codelineno-42-20"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">p</span><span class="p">;</span>
 <a id="__codelineno-42-21" name="__codelineno-42-21" href="#__codelineno-42-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-42-22" name="__codelineno-42-22" href="#__codelineno-42-22"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Element enters heap */</span>
+<div class="highlight"><span class="filename">my_heap.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Push an element onto the heap */</span>
 <a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="nx">push</span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="kt">any</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="c1">// Add node</span>
+<a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="c1">// Add a node</span>
 <a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span>
-<a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w">    </span><span class="c1">// Heapify from bottom to top</span>
+<a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w">    </span><span class="c1">// Sift up</span>
 <a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a><span class="w">    </span><span class="nx">h</span><span class="p">.</span><span class="nx">siftUp</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="p">}</span>
 <a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a>
-<a id="__codelineno-43-9" name="__codelineno-43-9" href="#__codelineno-43-9"></a><span class="cm">/* Starting from node i, heapify from bottom to top */</span>
+<a id="__codelineno-43-9" name="__codelineno-43-9" href="#__codelineno-43-9"></a><span class="cm">/* Sift up starting from node i */</span>
 <a id="__codelineno-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="nx">siftUp</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">true</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-12" name="__codelineno-43-12" href="#__codelineno-43-12"></a><span class="w">        </span><span class="c1">// Get parent node of node i</span>
+<a id="__codelineno-43-12" name="__codelineno-43-12" href="#__codelineno-43-12"></a><span class="w">        </span><span class="c1">// Get the parent of node i</span>
 <a id="__codelineno-43-13" name="__codelineno-43-13" href="#__codelineno-43-13"></a><span class="w">        </span><span class="nx">p</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">parent</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span>
-<a id="__codelineno-43-14" name="__codelineno-43-14" href="#__codelineno-43-14"></a><span class="w">        </span><span class="c1">// When "crossing root node" or "node needs no repair", end heapify</span>
+<a id="__codelineno-43-14" name="__codelineno-43-14" href="#__codelineno-43-14"></a><span class="w">        </span><span class="c1">// Stop sifting when the root has been passed or the node needs no repair</span>
 <a id="__codelineno-43-15" name="__codelineno-43-15" href="#__codelineno-43-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">p</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">[</span><span class="nx">i</span><span class="p">].(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">[</span><span class="nx">p</span><span class="p">].(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-16" name="__codelineno-43-16" href="#__codelineno-43-16"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-43-17" name="__codelineno-43-17" href="#__codelineno-43-17"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-43-18" name="__codelineno-43-18" href="#__codelineno-43-18"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-43-18" name="__codelineno-43-18" href="#__codelineno-43-18"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-43-19" name="__codelineno-43-19" href="#__codelineno-43-19"></a><span class="w">        </span><span class="nx">h</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">p</span><span class="p">)</span>
-<a id="__codelineno-43-20" name="__codelineno-43-20" href="#__codelineno-43-20"></a><span class="w">        </span><span class="c1">// Loop upward heapify</span>
+<a id="__codelineno-43-20" name="__codelineno-43-20" href="#__codelineno-43-20"></a><span class="w">        </span><span class="c1">// Continue sifting up</span>
 <a id="__codelineno-43-21" name="__codelineno-43-21" href="#__codelineno-43-21"></a><span class="w">        </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">p</span>
 <a id="__codelineno-43-22" name="__codelineno-43-22" href="#__codelineno-43-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-43-23" name="__codelineno-43-23" href="#__codelineno-43-23"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Element enters heap */</span>
+<div class="highlight"><span class="filename">my_heap.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Push an element onto the heap */</span>
 <a id="__codelineno-44-2" name="__codelineno-44-2" href="#__codelineno-44-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a><span class="w">    </span><span class="c1">// Add node</span>
+<a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a><span class="w">    </span><span class="c1">// Add a node</span>
 <a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a><span class="w">    </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
-<a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">    </span><span class="c1">// Heapify from bottom to top</span>
+<a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">    </span><span class="c1">// Sift up</span>
 <a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></a><span class="w">    </span><span class="n">siftUp</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-44-7" name="__codelineno-44-7" href="#__codelineno-44-7"></a><span class="p">}</span>
 <a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a>
-<a id="__codelineno-44-9" name="__codelineno-44-9" href="#__codelineno-44-9"></a><span class="cm">/* Starting from node i, heapify from bottom to top */</span>
+<a id="__codelineno-44-9" name="__codelineno-44-9" href="#__codelineno-44-9"></a><span class="cm">/* Sift up starting from node i */</span>
 <a id="__codelineno-44-10" name="__codelineno-44-10" href="#__codelineno-44-10"></a><span class="kd">func</span><span class="w"> </span><span class="nf">siftUp</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-11" name="__codelineno-44-11" href="#__codelineno-44-11"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-44-12" name="__codelineno-44-12" href="#__codelineno-44-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="kc">true</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-44-13" name="__codelineno-44-13" href="#__codelineno-44-13"></a><span class="w">        </span><span class="c1">// Get parent node of node i</span>
+<a id="__codelineno-44-13" name="__codelineno-44-13" href="#__codelineno-44-13"></a><span class="w">        </span><span class="c1">// Get the parent of node i</span>
 <a id="__codelineno-44-14" name="__codelineno-44-14" href="#__codelineno-44-14"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">p</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
-<a id="__codelineno-44-15" name="__codelineno-44-15" href="#__codelineno-44-15"></a><span class="w">        </span><span class="c1">// When "crossing root node" or "node needs no repair", end heapify</span>
+<a id="__codelineno-44-15" name="__codelineno-44-15" href="#__codelineno-44-15"></a><span class="w">        </span><span class="c1">// Stop when the parent index is invalid or the node already satisfies the heap order</span>
 <a id="__codelineno-44-16" name="__codelineno-44-16" href="#__codelineno-44-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">p</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-17" name="__codelineno-44-17" href="#__codelineno-44-17"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-44-18" name="__codelineno-44-18" href="#__codelineno-44-18"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-44-19" name="__codelineno-44-19" href="#__codelineno-44-19"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-44-19" name="__codelineno-44-19" href="#__codelineno-44-19"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-44-20" name="__codelineno-44-20" href="#__codelineno-44-20"></a><span class="w">        </span><span class="bp">swap</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="n">p</span><span class="p">)</span>
-<a id="__codelineno-44-21" name="__codelineno-44-21" href="#__codelineno-44-21"></a><span class="w">        </span><span class="c1">// Loop upward heapify</span>
+<a id="__codelineno-44-21" name="__codelineno-44-21" href="#__codelineno-44-21"></a><span class="w">        </span><span class="c1">// Continue sifting up</span>
 <a id="__codelineno-44-22" name="__codelineno-44-22" href="#__codelineno-44-22"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">p</span>
 <a id="__codelineno-44-23" name="__codelineno-44-23" href="#__codelineno-44-23"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-44-24" name="__codelineno-44-24" href="#__codelineno-44-24"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Element enters heap */</span>
+<div class="highlight"><span class="filename">my_heap.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Push an element onto the heap */</span>
 <a id="__codelineno-45-2" name="__codelineno-45-2" href="#__codelineno-45-2"></a><span class="nx">push</span><span class="p">(</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="c1">// Add node</span>
+<a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="c1">// Add a node</span>
 <a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">#maxHeap</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">val</span><span class="p">);</span>
-<a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">    </span><span class="c1">// Heapify from bottom to top</span>
+<a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">    </span><span class="c1">// Sift up</span>
 <a id="__codelineno-45-6" name="__codelineno-45-6" href="#__codelineno-45-6"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">#siftUp</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-45-7" name="__codelineno-45-7" href="#__codelineno-45-7"></a><span class="p">}</span>
 <a id="__codelineno-45-8" name="__codelineno-45-8" href="#__codelineno-45-8"></a>
-<a id="__codelineno-45-9" name="__codelineno-45-9" href="#__codelineno-45-9"></a><span class="cm">/* Starting from node i, heapify from bottom to top */</span>
+<a id="__codelineno-45-9" name="__codelineno-45-9" href="#__codelineno-45-9"></a><span class="cm">/* Sift up starting from node i */</span>
 <a id="__codelineno-45-10" name="__codelineno-45-10" href="#__codelineno-45-10"></a><span class="n">#siftUp</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-11" name="__codelineno-45-11" href="#__codelineno-45-11"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-12" name="__codelineno-45-12" href="#__codelineno-45-12"></a><span class="w">        </span><span class="c1">// Get parent node of node i</span>
+<a id="__codelineno-45-12" name="__codelineno-45-12" href="#__codelineno-45-12"></a><span class="w">        </span><span class="c1">// Get the parent of node i</span>
 <a id="__codelineno-45-13" name="__codelineno-45-13" href="#__codelineno-45-13"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#parent</span><span class="p">(</span><span class="nx">i</span><span class="p">);</span>
-<a id="__codelineno-45-14" name="__codelineno-45-14" href="#__codelineno-45-14"></a><span class="w">        </span><span class="c1">// When "crossing root node" or "node needs no repair", end heapify</span>
+<a id="__codelineno-45-14" name="__codelineno-45-14" href="#__codelineno-45-14"></a><span class="w">        </span><span class="c1">// Stop sifting when the root has been passed or the node needs no repair</span>
 <a id="__codelineno-45-15" name="__codelineno-45-15" href="#__codelineno-45-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">p</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#maxHeap</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#maxHeap</span><span class="p">[</span><span class="nx">p</span><span class="p">])</span><span class="w"> </span><span class="k">break</span><span class="p">;</span>
-<a id="__codelineno-45-16" name="__codelineno-45-16" href="#__codelineno-45-16"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-45-16" name="__codelineno-45-16" href="#__codelineno-45-16"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-45-17" name="__codelineno-45-17" href="#__codelineno-45-17"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#swap</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">p</span><span class="p">);</span>
-<a id="__codelineno-45-18" name="__codelineno-45-18" href="#__codelineno-45-18"></a><span class="w">        </span><span class="c1">// Loop upward heapify</span>
+<a id="__codelineno-45-18" name="__codelineno-45-18" href="#__codelineno-45-18"></a><span class="w">        </span><span class="c1">// Continue sifting up</span>
 <a id="__codelineno-45-19" name="__codelineno-45-19" href="#__codelineno-45-19"></a><span class="w">        </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">p</span><span class="p">;</span>
 <a id="__codelineno-45-20" name="__codelineno-45-20" href="#__codelineno-45-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-45-21" name="__codelineno-45-21" href="#__codelineno-45-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Element enters heap */</span>
+<div class="highlight"><span class="filename">my_heap.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Push an element onto the heap */</span>
 <a id="__codelineno-46-2" name="__codelineno-46-2" href="#__codelineno-46-2"></a><span class="nx">push</span><span class="p">(</span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="c1">// Add node</span>
+<a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="c1">// Add a node</span>
 <a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">maxHeap</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">val</span><span class="p">);</span>
-<a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">    </span><span class="c1">// Heapify from bottom to top</span>
+<a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">    </span><span class="c1">// Sift up</span>
 <a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">siftUp</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="p">}</span>
 <a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a>
-<a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="cm">/* Starting from node i, heapify from bottom to top */</span>
+<a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="cm">/* Sift up starting from node i */</span>
 <a id="__codelineno-46-10" name="__codelineno-46-10" href="#__codelineno-46-10"></a><span class="nx">siftUp</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-11" name="__codelineno-46-11" href="#__codelineno-46-11"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w">        </span><span class="c1">// Get parent node of node i</span>
+<a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w">        </span><span class="c1">// Get the parent of node i</span>
 <a id="__codelineno-46-13" name="__codelineno-46-13" href="#__codelineno-46-13"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">parent</span><span class="p">(</span><span class="nx">i</span><span class="p">);</span>
-<a id="__codelineno-46-14" name="__codelineno-46-14" href="#__codelineno-46-14"></a><span class="w">        </span><span class="c1">// When "crossing root node" or "node needs no repair", end heapify</span>
+<a id="__codelineno-46-14" name="__codelineno-46-14" href="#__codelineno-46-14"></a><span class="w">        </span><span class="c1">// Stop sifting when the root has been passed or the node needs no repair</span>
 <a id="__codelineno-46-15" name="__codelineno-46-15" href="#__codelineno-46-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">p</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">maxHeap</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">maxHeap</span><span class="p">[</span><span class="nx">p</span><span class="p">])</span><span class="w"> </span><span class="k">break</span><span class="p">;</span>
-<a id="__codelineno-46-16" name="__codelineno-46-16" href="#__codelineno-46-16"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-46-16" name="__codelineno-46-16" href="#__codelineno-46-16"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-46-17" name="__codelineno-46-17" href="#__codelineno-46-17"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">p</span><span class="p">);</span>
-<a id="__codelineno-46-18" name="__codelineno-46-18" href="#__codelineno-46-18"></a><span class="w">        </span><span class="c1">// Loop upward heapify</span>
+<a id="__codelineno-46-18" name="__codelineno-46-18" href="#__codelineno-46-18"></a><span class="w">        </span><span class="c1">// Continue sifting up</span>
 <a id="__codelineno-46-19" name="__codelineno-46-19" href="#__codelineno-46-19"></a><span class="w">        </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">p</span><span class="p">;</span>
 <a id="__codelineno-46-20" name="__codelineno-46-20" href="#__codelineno-46-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-46-21" name="__codelineno-46-21" href="#__codelineno-46-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Element enters heap */</span>
+<div class="highlight"><span class="filename">my_heap.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Push an element onto the heap */</span>
 <a id="__codelineno-47-2" name="__codelineno-47-2" href="#__codelineno-47-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">push</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="c1">// Add node</span>
+<a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="c1">// Add a node</span>
 <a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="n">_maxHeap</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w">  </span><span class="c1">// Heapify from bottom to top</span>
+<a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w">  </span><span class="c1">// Sift up</span>
 <a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a><span class="w">  </span><span class="n">siftUp</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-47-7" name="__codelineno-47-7" href="#__codelineno-47-7"></a><span class="p">}</span>
 <a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a>
-<a id="__codelineno-47-9" name="__codelineno-47-9" href="#__codelineno-47-9"></a><span class="cm">/* Starting from node i, heapify from bottom to top */</span>
+<a id="__codelineno-47-9" name="__codelineno-47-9" href="#__codelineno-47-9"></a><span class="cm">/* Sift up starting from node i */</span>
 <a id="__codelineno-47-10" name="__codelineno-47-10" href="#__codelineno-47-10"></a><span class="kt">void</span><span class="w"> </span><span class="n">siftUp</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-11" name="__codelineno-47-11" href="#__codelineno-47-11"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-12" name="__codelineno-47-12" href="#__codelineno-47-12"></a><span class="w">    </span><span class="c1">// Get parent node of node i</span>
+<a id="__codelineno-47-12" name="__codelineno-47-12" href="#__codelineno-47-12"></a><span class="w">    </span><span class="c1">// Get the parent of node i</span>
 <a id="__codelineno-47-13" name="__codelineno-47-13" href="#__codelineno-47-13"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_parent</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
-<a id="__codelineno-47-14" name="__codelineno-47-14" href="#__codelineno-47-14"></a><span class="w">    </span><span class="c1">// When "crossing root node" or "node needs no repair", end heapify</span>
+<a id="__codelineno-47-14" name="__codelineno-47-14" href="#__codelineno-47-14"></a><span class="w">    </span><span class="c1">// Stop sifting when the root has been passed or the node needs no repair</span>
 <a id="__codelineno-47-15" name="__codelineno-47-15" href="#__codelineno-47-15"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">[</span><span class="n">p</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-16" name="__codelineno-47-16" href="#__codelineno-47-16"></a><span class="w">      </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-47-17" name="__codelineno-47-17" href="#__codelineno-47-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-47-18" name="__codelineno-47-18" href="#__codelineno-47-18"></a><span class="w">    </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-47-18" name="__codelineno-47-18" href="#__codelineno-47-18"></a><span class="w">    </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-47-19" name="__codelineno-47-19" href="#__codelineno-47-19"></a><span class="w">    </span><span class="n">_swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="p">);</span>
-<a id="__codelineno-47-20" name="__codelineno-47-20" href="#__codelineno-47-20"></a><span class="w">    </span><span class="c1">// Loop upward heapify</span>
+<a id="__codelineno-47-20" name="__codelineno-47-20" href="#__codelineno-47-20"></a><span class="w">    </span><span class="c1">// Continue sifting up</span>
 <a id="__codelineno-47-21" name="__codelineno-47-21" href="#__codelineno-47-21"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">p</span><span class="p">;</span>
 <a id="__codelineno-47-22" name="__codelineno-47-22" href="#__codelineno-47-22"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-47-23" name="__codelineno-47-23" href="#__codelineno-47-23"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Element enters heap */</span>
+<div class="highlight"><span class="filename">my_heap.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Push an element onto the heap */</span>
 <a id="__codelineno-48-2" name="__codelineno-48-2" href="#__codelineno-48-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="c1">// Add node</span>
+<a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="c1">// Add a node</span>
 <a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w">    </span><span class="bp">self</span><span class="p">.</span><span class="n">max_heap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w">    </span><span class="c1">// Heapify from bottom to top</span>
+<a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w">    </span><span class="c1">// Sift up</span>
 <a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a><span class="w">    </span><span class="bp">self</span><span class="p">.</span><span class="n">sift_up</span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="p">}</span>
 <a id="__codelineno-48-8" name="__codelineno-48-8" href="#__codelineno-48-8"></a>
-<a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a><span class="cm">/* Starting from node i, heapify from bottom to top */</span>
+<a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a><span class="cm">/* Sift up starting from node i */</span>
 <a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="k">fn</span><span class="w"> </span><span class="nf">sift_up</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-11" name="__codelineno-48-11" href="#__codelineno-48-11"></a><span class="w">    </span><span class="k">loop</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-12" name="__codelineno-48-12" href="#__codelineno-48-12"></a><span class="w">        </span><span class="c1">// Node i is already the heap root, end heapification</span>
+<a id="__codelineno-48-12" name="__codelineno-48-12" href="#__codelineno-48-12"></a><span class="w">        </span><span class="c1">// Stop sifting up if node i is already the root</span>
 <a id="__codelineno-48-13" name="__codelineno-48-13" href="#__codelineno-48-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-14" name="__codelineno-48-14" href="#__codelineno-48-14"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-48-15" name="__codelineno-48-15" href="#__codelineno-48-15"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-48-16" name="__codelineno-48-16" href="#__codelineno-48-16"></a><span class="w">        </span><span class="c1">// Get parent node of node i</span>
+<a id="__codelineno-48-16" name="__codelineno-48-16" href="#__codelineno-48-16"></a><span class="w">        </span><span class="c1">// Get the parent of node i</span>
 <a id="__codelineno-48-17" name="__codelineno-48-17" href="#__codelineno-48-17"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">parent</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
-<a id="__codelineno-48-18" name="__codelineno-48-18" href="#__codelineno-48-18"></a><span class="w">        </span><span class="c1">// When "node needs no repair", end heapification</span>
+<a id="__codelineno-48-18" name="__codelineno-48-18" href="#__codelineno-48-18"></a><span class="w">        </span><span class="c1">// Stop sifting up when the node already satisfies the heap property</span>
 <a id="__codelineno-48-19" name="__codelineno-48-19" href="#__codelineno-48-19"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">max_heap</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">max_heap</span><span class="p">[</span><span class="n">p</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-20" name="__codelineno-48-20" href="#__codelineno-48-20"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-48-21" name="__codelineno-48-21" href="#__codelineno-48-21"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-48-22" name="__codelineno-48-22" href="#__codelineno-48-22"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-48-22" name="__codelineno-48-22" href="#__codelineno-48-22"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-48-23" name="__codelineno-48-23" href="#__codelineno-48-23"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="p">);</span>
-<a id="__codelineno-48-24" name="__codelineno-48-24" href="#__codelineno-48-24"></a><span class="w">        </span><span class="c1">// Loop upward heapify</span>
+<a id="__codelineno-48-24" name="__codelineno-48-24" href="#__codelineno-48-24"></a><span class="w">        </span><span class="c1">// Continue sifting up</span>
 <a id="__codelineno-48-25" name="__codelineno-48-25" href="#__codelineno-48-25"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">p</span><span class="p">;</span>
 <a id="__codelineno-48-26" name="__codelineno-48-26" href="#__codelineno-48-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-48-27" name="__codelineno-48-27" href="#__codelineno-48-27"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Element enters heap */</span>
+<div class="highlight"><span class="filename">my_heap.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Push an element onto the heap */</span>
 <a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="c1">// By default, should not add this many nodes</span>
+<a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="c1">// Normally, the number of nodes should remain below this limit</span>
 <a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">        </span><span class="n">printf</span><span class="p">(</span><span class="s">"heap is full!"</span><span class="p">);</span>
+<a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">        </span><span class="n">printf</span><span class="p">(</span><span class="s">"Heap is full!"</span><span class="p">);</span>
 <a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w">    </span><span class="c1">// Add node</span>
+<a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w">    </span><span class="c1">// Add a node</span>
 <a id="__codelineno-49-9" name="__codelineno-49-9" href="#__codelineno-49-9"></a><span class="w">    </span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">data</span><span class="p">[</span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-49-10" name="__codelineno-49-10" href="#__codelineno-49-10"></a><span class="w">    </span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-49-11" name="__codelineno-49-11" href="#__codelineno-49-11"></a>
-<a id="__codelineno-49-12" name="__codelineno-49-12" href="#__codelineno-49-12"></a><span class="w">    </span><span class="c1">// Heapify from bottom to top</span>
+<a id="__codelineno-49-12" name="__codelineno-49-12" href="#__codelineno-49-12"></a><span class="w">    </span><span class="c1">// Sift up</span>
 <a id="__codelineno-49-13" name="__codelineno-49-13" href="#__codelineno-49-13"></a><span class="w">    </span><span class="n">siftUp</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-49-14" name="__codelineno-49-14" href="#__codelineno-49-14"></a><span class="p">}</span>
 <a id="__codelineno-49-15" name="__codelineno-49-15" href="#__codelineno-49-15"></a>
-<a id="__codelineno-49-16" name="__codelineno-49-16" href="#__codelineno-49-16"></a><span class="cm">/* Starting from node i, heapify from bottom to top */</span>
+<a id="__codelineno-49-16" name="__codelineno-49-16" href="#__codelineno-49-16"></a><span class="cm">/* Sift up starting from node i */</span>
 <a id="__codelineno-49-17" name="__codelineno-49-17" href="#__codelineno-49-17"></a><span class="kt">void</span><span class="w"> </span><span class="nf">siftUp</span><span class="p">(</span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-18" name="__codelineno-49-18" href="#__codelineno-49-18"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nb">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-19" name="__codelineno-49-19" href="#__codelineno-49-19"></a><span class="w">        </span><span class="c1">// Get parent node of node i</span>
+<a id="__codelineno-49-19" name="__codelineno-49-19" href="#__codelineno-49-19"></a><span class="w">        </span><span class="c1">// Get the parent of node i</span>
 <a id="__codelineno-49-20" name="__codelineno-49-20" href="#__codelineno-49-20"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
-<a id="__codelineno-49-21" name="__codelineno-49-21" href="#__codelineno-49-21"></a><span class="w">        </span><span class="c1">// When "crossing root node" or "node needs no repair", end heapify</span>
+<a id="__codelineno-49-21" name="__codelineno-49-21" href="#__codelineno-49-21"></a><span class="w">        </span><span class="c1">// Stop sifting when the root has been passed or the node needs no repair</span>
 <a id="__codelineno-49-22" name="__codelineno-49-22" href="#__codelineno-49-22"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">data</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">data</span><span class="p">[</span><span class="n">p</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-23" name="__codelineno-49-23" href="#__codelineno-49-23"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-49-24" name="__codelineno-49-24" href="#__codelineno-49-24"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-49-25" name="__codelineno-49-25" href="#__codelineno-49-25"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-49-25" name="__codelineno-49-25" href="#__codelineno-49-25"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-49-26" name="__codelineno-49-26" href="#__codelineno-49-26"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="p">);</span>
-<a id="__codelineno-49-27" name="__codelineno-49-27" href="#__codelineno-49-27"></a><span class="w">        </span><span class="c1">// Loop upward heapify</span>
+<a id="__codelineno-49-27" name="__codelineno-49-27" href="#__codelineno-49-27"></a><span class="w">        </span><span class="c1">// Continue sifting up</span>
 <a id="__codelineno-49-28" name="__codelineno-49-28" href="#__codelineno-49-28"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">p</span><span class="p">;</span>
 <a id="__codelineno-49-29" name="__codelineno-49-29" href="#__codelineno-49-29"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-49-30" name="__codelineno-49-30" href="#__codelineno-49-30"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Element enters heap */</span>
+<div class="highlight"><span class="filename">my_heap.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Push an element onto the heap */</span>
 <a id="__codelineno-50-2" name="__codelineno-50-2" href="#__codelineno-50-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="c1">// Add node</span>
+<a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="c1">// Add a node</span>
 <a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">_val</span><span class="p">)</span>
-<a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">    </span><span class="c1">// Heapify from bottom to top</span>
+<a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">    </span><span class="c1">// Sift up</span>
 <a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">    </span><span class="n">siftUp</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="p">}</span>
 <a id="__codelineno-50-8" name="__codelineno-50-8" href="#__codelineno-50-8"></a>
-<a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="cm">/* Starting from node i, heapify from bottom to top */</span>
+<a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="cm">/* Sift up starting from node i */</span>
 <a id="__codelineno-50-10" name="__codelineno-50-10" href="#__codelineno-50-10"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">siftUp</span><span class="p">(</span><span class="nb">it</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-11" name="__codelineno-50-11" href="#__codelineno-50-11"></a><span class="w">    </span><span class="c1">// Kotlin function parameters are immutable, so create temporary variable</span>
+<a id="__codelineno-50-11" name="__codelineno-50-11" href="#__codelineno-50-11"></a><span class="w">    </span><span class="c1">// Kotlin function parameters are immutable, so create a temporary variable</span>
 <a id="__codelineno-50-12" name="__codelineno-50-12" href="#__codelineno-50-12"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">it</span>
 <a id="__codelineno-50-13" name="__codelineno-50-13" href="#__codelineno-50-13"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-14" name="__codelineno-50-14" href="#__codelineno-50-14"></a><span class="w">        </span><span class="c1">// Get parent node of node i</span>
+<a id="__codelineno-50-14" name="__codelineno-50-14" href="#__codelineno-50-14"></a><span class="w">        </span><span class="c1">// Get the parent of node i</span>
 <a id="__codelineno-50-15" name="__codelineno-50-15" href="#__codelineno-50-15"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-<a id="__codelineno-50-16" name="__codelineno-50-16" href="#__codelineno-50-16"></a><span class="w">        </span><span class="c1">// When "crossing root node" or "node needs no repair", end heapify</span>
+<a id="__codelineno-50-16" name="__codelineno-50-16" href="#__codelineno-50-16"></a><span class="w">        </span><span class="c1">// Stop sifting when the root has been passed or the node needs no repair</span>
 <a id="__codelineno-50-17" name="__codelineno-50-17" href="#__codelineno-50-17"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">[</span><span class="n">p</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="k">break</span>
-<a id="__codelineno-50-18" name="__codelineno-50-18" href="#__codelineno-50-18"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-50-18" name="__codelineno-50-18" href="#__codelineno-50-18"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-50-19" name="__codelineno-50-19" href="#__codelineno-50-19"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">p</span><span class="p">)</span>
-<a id="__codelineno-50-20" name="__codelineno-50-20" href="#__codelineno-50-20"></a><span class="w">        </span><span class="c1">// Loop upward heapify</span>
+<a id="__codelineno-50-20" name="__codelineno-50-20" href="#__codelineno-50-20"></a><span class="w">        </span><span class="c1">// Continue sifting up</span>
 <a id="__codelineno-50-21" name="__codelineno-50-21" href="#__codelineno-50-21"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">p</span>
 <a id="__codelineno-50-22" name="__codelineno-50-22" href="#__codelineno-50-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-50-23" name="__codelineno-50-23" href="#__codelineno-50-23"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Push element to heap ###</span>
+<div class="highlight"><span class="filename">my_heap.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Push an element onto the heap ###</span>
 <a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
-<a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">  </span><span class="c1"># Add node</span>
+<a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">  </span><span class="c1"># Add a node</span>
 <a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="w">  </span><span class="vi">@max_heap</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">val</span>
-<a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">  </span><span class="c1"># Heapify from bottom to top</span>
+<a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">  </span><span class="c1"># Sift up</span>
 <a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">  </span><span class="n">sift_up</span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="k">end</span>
 <a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a>
-<a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="c1">### Heapify from node i, bottom to top ###</span>
+<a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="c1">### Sift up starting from node i ###</span>
 <a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a><span class="k">def</span><span class="w"> </span><span class="nf">sift_up</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-51-11" name="__codelineno-51-11" href="#__codelineno-51-11"></a><span class="w">  </span><span class="kp">loop</span><span class="w"> </span><span class="k">do</span>
-<a id="__codelineno-51-12" name="__codelineno-51-12" href="#__codelineno-51-12"></a><span class="w">    </span><span class="c1"># Get parent node of node i</span>
+<a id="__codelineno-51-12" name="__codelineno-51-12" href="#__codelineno-51-12"></a><span class="w">    </span><span class="c1"># Get the parent of node i</span>
 <a id="__codelineno-51-13" name="__codelineno-51-13" href="#__codelineno-51-13"></a><span class="w">    </span><span class="nb">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-<a id="__codelineno-51-14" name="__codelineno-51-14" href="#__codelineno-51-14"></a><span class="w">    </span><span class="c1"># When "crossing root node" or "node needs no repair", end heapify</span>
+<a id="__codelineno-51-14" name="__codelineno-51-14" href="#__codelineno-51-14"></a><span class="w">    </span><span class="c1"># Stop sifting when the root has been passed or the node needs no repair</span>
 <a id="__codelineno-51-15" name="__codelineno-51-15" href="#__codelineno-51-15"></a><span class="w">    </span><span class="k">break</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nb">p</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="vi">@max_heap</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="vi">@max_heap</span><span class="o">[</span><span class="nb">p</span><span class="o">]</span>
-<a id="__codelineno-51-16" name="__codelineno-51-16" href="#__codelineno-51-16"></a><span class="w">    </span><span class="c1"># Swap two nodes</span>
+<a id="__codelineno-51-16" name="__codelineno-51-16" href="#__codelineno-51-16"></a><span class="w">    </span><span class="c1"># Swap the two nodes</span>
 <a id="__codelineno-51-17" name="__codelineno-51-17" href="#__codelineno-51-17"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="nb">p</span><span class="p">)</span>
-<a id="__codelineno-51-18" name="__codelineno-51-18" href="#__codelineno-51-18"></a><span class="w">    </span><span class="c1"># Loop upward heapify</span>
+<a id="__codelineno-51-18" name="__codelineno-51-18" href="#__codelineno-51-18"></a><span class="w">    </span><span class="c1"># Continue sifting up</span>
 <a id="__codelineno-51-19" name="__codelineno-51-19" href="#__codelineno-51-19"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">p</span>
 <a id="__codelineno-51-20" name="__codelineno-51-20" href="#__codelineno-51-20"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-51-21" name="__codelineno-51-21" href="#__codelineno-51-21"></a><span class="k">end</span>
@@ -5903,18 +5980,23 @@
 </div>
 </div>
 </div>
-<h3 id="4-removing-the-heap-top-element">4. &nbsp; Removing the Heap Top Element<a class="headerlink" href="#4-removing-the-heap-top-element" title="Permanent link">&para;</a></h3>
-<p>The heap top element is the root node of the binary tree, which is the first element of the list. If we directly remove the first element from the list, all node indexes in the binary tree would change, making subsequent repair with heapify difficult. To minimize changes in element indexes, we use the following steps.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20MaxHeap%3A%0A%20%20%20%20%22%22%22Max%20heap%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Place%20the%20list%20elements%20in%20the%20heap%20without%20changing%20their%20order%0A%20%20%20%20%20%20%20%20self.max_heap%20%3D%20nums%0A%0A%20%20%20%20def%20left%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20left%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%201%0A%0A%20%20%20%20def%20right%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20right%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%202%0A%0A%20%20%20%20def%20parent%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20parent%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%20%28i%20-%201%29%20//%202%20%20%23%20Floor%20division%0A%0A%20%20%20%20def%20swap%28self,%20i%3A%20int,%20j%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Swap%20elements%22%22%22%0A%20%20%20%20%20%20%20%20self.max_heap%5Bi%5D,%20self.max_heap%5Bj%5D%20%3D%20self.max_heap%5Bj%5D,%20self.max_heap%5Bi%5D%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20heap%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self.max_heap%29%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22Check%20whether%20the%20heap%20is%20empty%22%22%22%0A%20%20%20%20%20%20%20%20return%20self.size%28%29%20%3D%3D%200%0A%0A%20%20%20%20def%20push%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Push%20an%20element%20onto%20the%20heap%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Add%20a%20node%0A%20%20%20%20%20%20%20%20self.max_heap.append%28val%29%0A%20%20%20%20%20%20%20%20%23%20Sift%20up%0A%20%20%20%20%20%20%20%20self.sift_up%28self.size%28%29%20-%201%29%0A%0A%20%20%20%20def%20sift_up%28self,%20i%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Sift%20up%20starting%20from%20node%20i%22%22%22%0A%20%20%20%20%20%20%20%20while%20True%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Get%20the%20parent%20of%20node%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20p%20%3D%20self.parent%28i%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Stop%20sifting%20when%20the%20root%20has%20been%20passed%20or%20the%20node%20needs%20no%20repair%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20p%20%3C%200%20or%20self.max_heap%5Bi%5D%20%3C%3D%20self.max_heap%5Bp%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Swap%20the%20two%20nodes%0A%20%20%20%20%20%20%20%20%20%20%20%20self.swap%28i,%20p%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Continue%20sifting%20up%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20p%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20max%20heap%0A%20%20%20%20%23%20Note%20that%20the%20input%20array%20is%20already%20a%20valid%20heap%0A%20%20%20%20max_heap%20%3D%20MaxHeap%28%5B9,%208,%206,%206,%207,%205,%202,%201,%204,%203,%206,%202%5D%29%0A%0A%20%20%20%20%23%20Push%20an%20element%20onto%20the%20heap%0A%20%20%20%20val%20%3D%207%0A%20%20%20%20max_heap.push%28val%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=8&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20MaxHeap%3A%0A%20%20%20%20%22%22%22Max%20heap%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Place%20the%20list%20elements%20in%20the%20heap%20without%20changing%20their%20order%0A%20%20%20%20%20%20%20%20self.max_heap%20%3D%20nums%0A%0A%20%20%20%20def%20left%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20left%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%201%0A%0A%20%20%20%20def%20right%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20right%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%202%0A%0A%20%20%20%20def%20parent%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20parent%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%20%28i%20-%201%29%20//%202%20%20%23%20Floor%20division%0A%0A%20%20%20%20def%20swap%28self,%20i%3A%20int,%20j%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Swap%20elements%22%22%22%0A%20%20%20%20%20%20%20%20self.max_heap%5Bi%5D,%20self.max_heap%5Bj%5D%20%3D%20self.max_heap%5Bj%5D,%20self.max_heap%5Bi%5D%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20heap%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self.max_heap%29%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22Check%20whether%20the%20heap%20is%20empty%22%22%22%0A%20%20%20%20%20%20%20%20return%20self.size%28%29%20%3D%3D%200%0A%0A%20%20%20%20def%20push%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Push%20an%20element%20onto%20the%20heap%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Add%20a%20node%0A%20%20%20%20%20%20%20%20self.max_heap.append%28val%29%0A%20%20%20%20%20%20%20%20%23%20Sift%20up%0A%20%20%20%20%20%20%20%20self.sift_up%28self.size%28%29%20-%201%29%0A%0A%20%20%20%20def%20sift_up%28self,%20i%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Sift%20up%20starting%20from%20node%20i%22%22%22%0A%20%20%20%20%20%20%20%20while%20True%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Get%20the%20parent%20of%20node%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20p%20%3D%20self.parent%28i%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Stop%20sifting%20when%20the%20root%20has%20been%20passed%20or%20the%20node%20needs%20no%20repair%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20p%20%3C%200%20or%20self.max_heap%5Bi%5D%20%3C%3D%20self.max_heap%5Bp%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Swap%20the%20two%20nodes%0A%20%20%20%20%20%20%20%20%20%20%20%20self.swap%28i,%20p%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Continue%20sifting%20up%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20p%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20max%20heap%0A%20%20%20%20%23%20Note%20that%20the%20input%20array%20is%20already%20a%20valid%20heap%0A%20%20%20%20max_heap%20%3D%20MaxHeap%28%5B9,%208,%206,%206,%207,%205,%202,%201,%204,%203,%206,%202%5D%29%0A%0A%20%20%20%20%23%20Push%20an%20element%20onto%20the%20heap%0A%20%20%20%20val%20%3D%207%0A%20%20%20%20max_heap.push%28val%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=8&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="4-removing-the-root">4. &nbsp; Removing the root<a class="headerlink" href="#4-removing-the-root" title="Permanent link">&para;</a></h3>
+<p>The root is the first element of the list. Removing it directly would change every remaining index, making it difficult to restore the heap property. To minimize index changes, use the following steps.</p>
 <ol>
-<li>Swap the heap top element with the heap bottom element (swap the root node with the rightmost leaf node).</li>
-<li>After swapping, remove the heap bottom from the list (note that since we've swapped, we're actually removing the original heap top element).</li>
-<li>Starting from the root node, <strong>perform heapify from top to bottom</strong>.</li>
+<li>Swap the root with the last element, the rightmost leaf.</li>
+<li>Remove the last element from the list. Because of the swap, this is the original root.</li>
+<li>Starting at the root, <strong>sift downward</strong>.</li>
 </ol>
-<p>As shown in Figure 8-4, <strong>the direction of "top-to-bottom heapify" is opposite to "bottom-to-top heapify"</strong>. We compare the root node's value with its two children and swap it with the largest child. Then loop this operation until we pass a leaf node or encounter a node that doesn't need swapping.</p>
+<p>As shown below, <strong>sifting down proceeds in the opposite direction from sifting up</strong>. Compare the root with its two children and swap it with the larger child when needed. Continue downward until a leaf is reached or no swap is necessary.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="6:10"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">&lt;1&gt;</label><label for="__tabbed_6_2">&lt;2&gt;</label><label for="__tabbed_6_3">&lt;3&gt;</label><label for="__tabbed_6_4">&lt;4&gt;</label><label for="__tabbed_6_5">&lt;5&gt;</label><label for="__tabbed_6_6">&lt;6&gt;</label><label for="__tabbed_6_7">&lt;7&gt;</label><label for="__tabbed_6_8">&lt;8&gt;</label><label for="__tabbed_6_9">&lt;9&gt;</label><label for="__tabbed_6_10">&lt;10&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Steps of removing the heap top element" class="animation-figure" src="../heap.assets/heap_pop_step1.png" /></p>
+<p><img alt="Steps for removing the root of a heap" class="animation-figure" src="../heap.assets/heap_pop_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="heap_pop_step2" class="animation-figure" src="../heap.assets/heap_pop_step2.png" /></p>
@@ -5945,174 +6027,174 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 8-4 &nbsp; Steps of removing the heap top element </p>
+<p align="center"> Figure 8-4 &nbsp; Steps for removing the root of a heap </p>
 
-<p>Similar to the element insertion operation, the time complexity of the heap top element removal operation is also <span class="arithmatex">\(O(\log n)\)</span>. The code is as follows:</p>
+<p>Like insertion, removing the root takes <span class="arithmatex">\(O(\log n)\)</span> time. The code is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="7:13"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><input id="__tabbed_7_13" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Python</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Java</label><label for="__tabbed_7_4">C#</label><label for="__tabbed_7_5">Go</label><label for="__tabbed_7_6">Swift</label><label for="__tabbed_7_7">JS</label><label for="__tabbed_7_8">TS</label><label for="__tabbed_7_9">Dart</label><label for="__tabbed_7_10">Rust</label><label for="__tabbed_7_11">C</label><label for="__tabbed_7_12">Kotlin</label><label for="__tabbed_7_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">my_heap.py</span><pre><span></span><code><a id="__codelineno-52-1" name="__codelineno-52-1" href="#__codelineno-52-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">pop</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Element exits heap"""</span>
-<a id="__codelineno-52-3" name="__codelineno-52-3" href="#__codelineno-52-3"></a>    <span class="c1"># Handle empty case</span>
+<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Pop an element from the heap"""</span>
+<a id="__codelineno-52-3" name="__codelineno-52-3" href="#__codelineno-52-3"></a>    <span class="c1"># Handle an empty heap</span>
 <a id="__codelineno-52-4" name="__codelineno-52-4" href="#__codelineno-52-4"></a>    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_empty</span><span class="p">():</span>
 <a id="__codelineno-52-5" name="__codelineno-52-5" href="#__codelineno-52-5"></a>        <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Heap is empty"</span><span class="p">)</span>
-<a id="__codelineno-52-6" name="__codelineno-52-6" href="#__codelineno-52-6"></a>    <span class="c1"># Swap root node with rightmost leaf node (swap first element with last element)</span>
+<a id="__codelineno-52-6" name="__codelineno-52-6" href="#__codelineno-52-6"></a>    <span class="c1"># Swap the root with the rightmost leaf on the last level (swap the first and last elements)</span>
 <a id="__codelineno-52-7" name="__codelineno-52-7" href="#__codelineno-52-7"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">swap</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-52-8" name="__codelineno-52-8" href="#__codelineno-52-8"></a>    <span class="c1"># Delete node</span>
+<a id="__codelineno-52-8" name="__codelineno-52-8" href="#__codelineno-52-8"></a>    <span class="c1"># Remove a node</span>
 <a id="__codelineno-52-9" name="__codelineno-52-9" href="#__codelineno-52-9"></a>    <span class="n">val</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_heap</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
-<a id="__codelineno-52-10" name="__codelineno-52-10" href="#__codelineno-52-10"></a>    <span class="c1"># Heapify from top to bottom</span>
+<a id="__codelineno-52-10" name="__codelineno-52-10" href="#__codelineno-52-10"></a>    <span class="c1"># Sift down</span>
 <a id="__codelineno-52-11" name="__codelineno-52-11" href="#__codelineno-52-11"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">sift_down</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-52-12" name="__codelineno-52-12" href="#__codelineno-52-12"></a>    <span class="c1"># Return top element</span>
+<a id="__codelineno-52-12" name="__codelineno-52-12" href="#__codelineno-52-12"></a>    <span class="c1"># Return the top element of the heap</span>
 <a id="__codelineno-52-13" name="__codelineno-52-13" href="#__codelineno-52-13"></a>    <span class="k">return</span> <span class="n">val</span>
 <a id="__codelineno-52-14" name="__codelineno-52-14" href="#__codelineno-52-14"></a>
 <a id="__codelineno-52-15" name="__codelineno-52-15" href="#__codelineno-52-15"></a><span class="k">def</span><span class="w"> </span><span class="nf">sift_down</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-52-16" name="__codelineno-52-16" href="#__codelineno-52-16"></a><span class="w">    </span><span class="sd">"""Starting from node i, heapify from top to bottom"""</span>
+<a id="__codelineno-52-16" name="__codelineno-52-16" href="#__codelineno-52-16"></a><span class="w">    </span><span class="sd">"""Sift down starting from node i"""</span>
 <a id="__codelineno-52-17" name="__codelineno-52-17" href="#__codelineno-52-17"></a>    <span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
-<a id="__codelineno-52-18" name="__codelineno-52-18" href="#__codelineno-52-18"></a>        <span class="c1"># Find node with largest value among i, l, r, denoted as ma</span>
+<a id="__codelineno-52-18" name="__codelineno-52-18" href="#__codelineno-52-18"></a>        <span class="c1"># Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-52-19" name="__codelineno-52-19" href="#__codelineno-52-19"></a>        <span class="n">l</span><span class="p">,</span> <span class="n">r</span><span class="p">,</span> <span class="n">ma</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">left</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="n">i</span>
 <a id="__codelineno-52-20" name="__codelineno-52-20" href="#__codelineno-52-20"></a>        <span class="k">if</span> <span class="n">l</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_heap</span><span class="p">[</span><span class="n">l</span><span class="p">]</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_heap</span><span class="p">[</span><span class="n">ma</span><span class="p">]:</span>
 <a id="__codelineno-52-21" name="__codelineno-52-21" href="#__codelineno-52-21"></a>            <span class="n">ma</span> <span class="o">=</span> <span class="n">l</span>
 <a id="__codelineno-52-22" name="__codelineno-52-22" href="#__codelineno-52-22"></a>        <span class="k">if</span> <span class="n">r</span> <span class="o">&lt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_heap</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">&gt;</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_heap</span><span class="p">[</span><span class="n">ma</span><span class="p">]:</span>
 <a id="__codelineno-52-23" name="__codelineno-52-23" href="#__codelineno-52-23"></a>            <span class="n">ma</span> <span class="o">=</span> <span class="n">r</span>
-<a id="__codelineno-52-24" name="__codelineno-52-24" href="#__codelineno-52-24"></a>        <span class="c1"># If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-52-24" name="__codelineno-52-24" href="#__codelineno-52-24"></a>        <span class="c1"># Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-52-25" name="__codelineno-52-25" href="#__codelineno-52-25"></a>        <span class="k">if</span> <span class="n">ma</span> <span class="o">==</span> <span class="n">i</span><span class="p">:</span>
 <a id="__codelineno-52-26" name="__codelineno-52-26" href="#__codelineno-52-26"></a>            <span class="k">break</span>
-<a id="__codelineno-52-27" name="__codelineno-52-27" href="#__codelineno-52-27"></a>        <span class="c1"># Swap two nodes</span>
+<a id="__codelineno-52-27" name="__codelineno-52-27" href="#__codelineno-52-27"></a>        <span class="c1"># Swap the two nodes</span>
 <a id="__codelineno-52-28" name="__codelineno-52-28" href="#__codelineno-52-28"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">ma</span><span class="p">)</span>
-<a id="__codelineno-52-29" name="__codelineno-52-29" href="#__codelineno-52-29"></a>        <span class="c1"># Loop downward heapify</span>
+<a id="__codelineno-52-29" name="__codelineno-52-29" href="#__codelineno-52-29"></a>        <span class="c1"># Continue sifting down</span>
 <a id="__codelineno-52-30" name="__codelineno-52-30" href="#__codelineno-52-30"></a>        <span class="n">i</span> <span class="o">=</span> <span class="n">ma</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Element exits heap */</span>
+<div class="highlight"><span class="filename">my_heap.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Pop an element from the heap */</span>
 <a id="__codelineno-53-2" name="__codelineno-53-2" href="#__codelineno-53-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="w">    </span><span class="c1">// Handle empty case</span>
+<a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="w">    </span><span class="c1">// Handle an empty heap</span>
 <a id="__codelineno-53-4" name="__codelineno-53-4" href="#__codelineno-53-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-5" name="__codelineno-53-5" href="#__codelineno-53-5"></a><span class="w">        </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Heap is empty"</span><span class="p">);</span>
 <a id="__codelineno-53-6" name="__codelineno-53-6" href="#__codelineno-53-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-53-7" name="__codelineno-53-7" href="#__codelineno-53-7"></a><span class="w">    </span><span class="c1">// Delete node</span>
+<a id="__codelineno-53-7" name="__codelineno-53-7" href="#__codelineno-53-7"></a><span class="w">    </span><span class="c1">// Swap the root with the rightmost leaf on the last level (swap the first and last elements)</span>
 <a id="__codelineno-53-8" name="__codelineno-53-8" href="#__codelineno-53-8"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
-<a id="__codelineno-53-9" name="__codelineno-53-9" href="#__codelineno-53-9"></a><span class="w">    </span><span class="c1">// Remove node</span>
+<a id="__codelineno-53-9" name="__codelineno-53-9" href="#__codelineno-53-9"></a><span class="w">    </span><span class="c1">// Remove a node</span>
 <a id="__codelineno-53-10" name="__codelineno-53-10" href="#__codelineno-53-10"></a><span class="w">    </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span>
-<a id="__codelineno-53-11" name="__codelineno-53-11" href="#__codelineno-53-11"></a><span class="w">    </span><span class="c1">// Return top element</span>
+<a id="__codelineno-53-11" name="__codelineno-53-11" href="#__codelineno-53-11"></a><span class="w">    </span><span class="c1">// Sift down</span>
 <a id="__codelineno-53-12" name="__codelineno-53-12" href="#__codelineno-53-12"></a><span class="w">    </span><span class="n">siftDown</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-53-13" name="__codelineno-53-13" href="#__codelineno-53-13"></a><span class="p">}</span>
 <a id="__codelineno-53-14" name="__codelineno-53-14" href="#__codelineno-53-14"></a>
-<a id="__codelineno-53-15" name="__codelineno-53-15" href="#__codelineno-53-15"></a><span class="cm">/* Starting from node i, heapify from top to bottom */</span>
+<a id="__codelineno-53-15" name="__codelineno-53-15" href="#__codelineno-53-15"></a><span class="cm">/* Sift down starting from node i */</span>
 <a id="__codelineno-53-16" name="__codelineno-53-16" href="#__codelineno-53-16"></a><span class="kt">void</span><span class="w"> </span><span class="nf">siftDown</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-17" name="__codelineno-53-17" href="#__codelineno-53-17"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nb">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-53-18" name="__codelineno-53-18" href="#__codelineno-53-18"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-53-18" name="__codelineno-53-18" href="#__codelineno-53-18"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-53-19" name="__codelineno-53-19" href="#__codelineno-53-19"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-53-20" name="__codelineno-53-20" href="#__codelineno-53-20"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">l</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span>
 <a id="__codelineno-53-21" name="__codelineno-53-21" href="#__codelineno-53-21"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span><span class="p">;</span>
 <a id="__codelineno-53-22" name="__codelineno-53-22" href="#__codelineno-53-22"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span>
 <a id="__codelineno-53-23" name="__codelineno-53-23" href="#__codelineno-53-23"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="p">;</span>
-<a id="__codelineno-53-24" name="__codelineno-53-24" href="#__codelineno-53-24"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-53-24" name="__codelineno-53-24" href="#__codelineno-53-24"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-53-25" name="__codelineno-53-25" href="#__codelineno-53-25"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-53-26" name="__codelineno-53-26" href="#__codelineno-53-26"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-53-27" name="__codelineno-53-27" href="#__codelineno-53-27"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">]);</span>
-<a id="__codelineno-53-28" name="__codelineno-53-28" href="#__codelineno-53-28"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-53-28" name="__codelineno-53-28" href="#__codelineno-53-28"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-53-29" name="__codelineno-53-29" href="#__codelineno-53-29"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span><span class="p">;</span>
 <a id="__codelineno-53-30" name="__codelineno-53-30" href="#__codelineno-53-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-53-31" name="__codelineno-53-31" href="#__codelineno-53-31"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Element exits heap */</span>
+<div class="highlight"><span class="filename">my_heap.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Pop an element from the heap */</span>
 <a id="__codelineno-54-2" name="__codelineno-54-2" href="#__codelineno-54-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="w">    </span><span class="c1">// Handle empty case</span>
+<a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="w">    </span><span class="c1">// Handle an empty heap</span>
 <a id="__codelineno-54-4" name="__codelineno-54-4" href="#__codelineno-54-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-54-5" name="__codelineno-54-5" href="#__codelineno-54-5"></a><span class="w">        </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">();</span>
-<a id="__codelineno-54-6" name="__codelineno-54-6" href="#__codelineno-54-6"></a><span class="w">    </span><span class="c1">// Delete node</span>
+<a id="__codelineno-54-6" name="__codelineno-54-6" href="#__codelineno-54-6"></a><span class="w">    </span><span class="c1">// Swap the root with the rightmost leaf on the last level (swap the first and last elements)</span>
 <a id="__codelineno-54-7" name="__codelineno-54-7" href="#__codelineno-54-7"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-54-8" name="__codelineno-54-8" href="#__codelineno-54-8"></a><span class="w">    </span><span class="c1">// Remove node</span>
+<a id="__codelineno-54-8" name="__codelineno-54-8" href="#__codelineno-54-8"></a><span class="w">    </span><span class="c1">// Remove a node</span>
 <a id="__codelineno-54-9" name="__codelineno-54-9" href="#__codelineno-54-9"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-54-10" name="__codelineno-54-10" href="#__codelineno-54-10"></a><span class="w">    </span><span class="c1">// Return top element</span>
+<a id="__codelineno-54-10" name="__codelineno-54-10" href="#__codelineno-54-10"></a><span class="w">    </span><span class="c1">// Sift down</span>
 <a id="__codelineno-54-11" name="__codelineno-54-11" href="#__codelineno-54-11"></a><span class="w">    </span><span class="n">siftDown</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
-<a id="__codelineno-54-12" name="__codelineno-54-12" href="#__codelineno-54-12"></a><span class="w">    </span><span class="c1">// Return heap top element</span>
+<a id="__codelineno-54-12" name="__codelineno-54-12" href="#__codelineno-54-12"></a><span class="w">    </span><span class="c1">// Return the top element of the heap</span>
 <a id="__codelineno-54-13" name="__codelineno-54-13" href="#__codelineno-54-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-54-14" name="__codelineno-54-14" href="#__codelineno-54-14"></a><span class="p">}</span>
 <a id="__codelineno-54-15" name="__codelineno-54-15" href="#__codelineno-54-15"></a>
-<a id="__codelineno-54-16" name="__codelineno-54-16" href="#__codelineno-54-16"></a><span class="cm">/* Starting from node i, heapify from top to bottom */</span>
+<a id="__codelineno-54-16" name="__codelineno-54-16" href="#__codelineno-54-16"></a><span class="cm">/* Sift down starting from node i */</span>
 <a id="__codelineno-54-17" name="__codelineno-54-17" href="#__codelineno-54-17"></a><span class="kt">void</span><span class="w"> </span><span class="nf">siftDown</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-18" name="__codelineno-54-18" href="#__codelineno-54-18"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-54-19" name="__codelineno-54-19" href="#__codelineno-54-19"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-54-19" name="__codelineno-54-19" href="#__codelineno-54-19"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-54-20" name="__codelineno-54-20" href="#__codelineno-54-20"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-54-21" name="__codelineno-54-21" href="#__codelineno-54-21"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">l</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">ma</span><span class="p">))</span>
 <a id="__codelineno-54-22" name="__codelineno-54-22" href="#__codelineno-54-22"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span><span class="p">;</span>
 <a id="__codelineno-54-23" name="__codelineno-54-23" href="#__codelineno-54-23"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">ma</span><span class="p">))</span>
 <a id="__codelineno-54-24" name="__codelineno-54-24" href="#__codelineno-54-24"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="p">;</span>
-<a id="__codelineno-54-25" name="__codelineno-54-25" href="#__codelineno-54-25"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-54-25" name="__codelineno-54-25" href="#__codelineno-54-25"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-54-26" name="__codelineno-54-26" href="#__codelineno-54-26"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-54-27" name="__codelineno-54-27" href="#__codelineno-54-27"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
-<a id="__codelineno-54-28" name="__codelineno-54-28" href="#__codelineno-54-28"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-54-28" name="__codelineno-54-28" href="#__codelineno-54-28"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-54-29" name="__codelineno-54-29" href="#__codelineno-54-29"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">ma</span><span class="p">);</span>
-<a id="__codelineno-54-30" name="__codelineno-54-30" href="#__codelineno-54-30"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-54-30" name="__codelineno-54-30" href="#__codelineno-54-30"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-54-31" name="__codelineno-54-31" href="#__codelineno-54-31"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span><span class="p">;</span>
 <a id="__codelineno-54-32" name="__codelineno-54-32" href="#__codelineno-54-32"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-54-33" name="__codelineno-54-33" href="#__codelineno-54-33"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Element exits heap */</span>
+<div class="highlight"><span class="filename">my_heap.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Pop an element from the heap */</span>
 <a id="__codelineno-55-2" name="__codelineno-55-2" href="#__codelineno-55-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="w">    </span><span class="c1">// Handle empty case</span>
+<a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="w">    </span><span class="c1">// Handle an empty heap</span>
 <a id="__codelineno-55-4" name="__codelineno-55-4" href="#__codelineno-55-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">IsEmpty</span><span class="p">())</span>
 <a id="__codelineno-55-5" name="__codelineno-55-5" href="#__codelineno-55-5"></a><span class="w">        </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">IndexOutOfRangeException</span><span class="p">();</span>
-<a id="__codelineno-55-6" name="__codelineno-55-6" href="#__codelineno-55-6"></a><span class="w">    </span><span class="c1">// Delete node</span>
+<a id="__codelineno-55-6" name="__codelineno-55-6" href="#__codelineno-55-6"></a><span class="w">    </span><span class="c1">// Swap the root with the rightmost leaf on the last level (swap the first and last elements)</span>
 <a id="__codelineno-55-7" name="__codelineno-55-7" href="#__codelineno-55-7"></a><span class="w">    </span><span class="n">Swap</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">Size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-55-8" name="__codelineno-55-8" href="#__codelineno-55-8"></a><span class="w">    </span><span class="c1">// Remove node</span>
+<a id="__codelineno-55-8" name="__codelineno-55-8" href="#__codelineno-55-8"></a><span class="w">    </span><span class="c1">// Remove a node</span>
 <a id="__codelineno-55-9" name="__codelineno-55-9" href="#__codelineno-55-9"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">Last</span><span class="p">();</span>
 <a id="__codelineno-55-10" name="__codelineno-55-10" href="#__codelineno-55-10"></a><span class="w">    </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">RemoveAt</span><span class="p">(</span><span class="n">Size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-55-11" name="__codelineno-55-11" href="#__codelineno-55-11"></a><span class="w">    </span><span class="c1">// Return top element</span>
+<a id="__codelineno-55-11" name="__codelineno-55-11" href="#__codelineno-55-11"></a><span class="w">    </span><span class="c1">// Sift down</span>
 <a id="__codelineno-55-12" name="__codelineno-55-12" href="#__codelineno-55-12"></a><span class="w">    </span><span class="n">SiftDown</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
-<a id="__codelineno-55-13" name="__codelineno-55-13" href="#__codelineno-55-13"></a><span class="w">    </span><span class="c1">// Return heap top element</span>
+<a id="__codelineno-55-13" name="__codelineno-55-13" href="#__codelineno-55-13"></a><span class="w">    </span><span class="c1">// Return the top element of the heap</span>
 <a id="__codelineno-55-14" name="__codelineno-55-14" href="#__codelineno-55-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-55-15" name="__codelineno-55-15" href="#__codelineno-55-15"></a><span class="p">}</span>
 <a id="__codelineno-55-16" name="__codelineno-55-16" href="#__codelineno-55-16"></a>
-<a id="__codelineno-55-17" name="__codelineno-55-17" href="#__codelineno-55-17"></a><span class="cm">/* Starting from node i, heapify from top to bottom */</span>
+<a id="__codelineno-55-17" name="__codelineno-55-17" href="#__codelineno-55-17"></a><span class="cm">/* Sift down starting from node i */</span>
 <a id="__codelineno-55-18" name="__codelineno-55-18" href="#__codelineno-55-18"></a><span class="k">void</span><span class="w"> </span><span class="nf">SiftDown</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-19" name="__codelineno-55-19" href="#__codelineno-55-19"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="k">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-55-20" name="__codelineno-55-20" href="#__codelineno-55-20"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-55-20" name="__codelineno-55-20" href="#__codelineno-55-20"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-55-21" name="__codelineno-55-21" href="#__codelineno-55-21"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Left</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Right</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-55-22" name="__codelineno-55-22" href="#__codelineno-55-22"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">Size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">l</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span>
 <a id="__codelineno-55-23" name="__codelineno-55-23" href="#__codelineno-55-23"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span><span class="p">;</span>
 <a id="__codelineno-55-24" name="__codelineno-55-24" href="#__codelineno-55-24"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">Size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span>
 <a id="__codelineno-55-25" name="__codelineno-55-25" href="#__codelineno-55-25"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="p">;</span>
-<a id="__codelineno-55-26" name="__codelineno-55-26" href="#__codelineno-55-26"></a><span class="w">        </span><span class="c1">// If 'node i is largest' or 'past leaf node', end heapify</span>
+<a id="__codelineno-55-26" name="__codelineno-55-26" href="#__codelineno-55-26"></a><span class="w">        </span><span class="c1">// Stop sifting if node i has the largest value or has no children</span>
 <a id="__codelineno-55-27" name="__codelineno-55-27" href="#__codelineno-55-27"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span>
-<a id="__codelineno-55-28" name="__codelineno-55-28" href="#__codelineno-55-28"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-55-28" name="__codelineno-55-28" href="#__codelineno-55-28"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-55-29" name="__codelineno-55-29" href="#__codelineno-55-29"></a><span class="w">        </span><span class="n">Swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">ma</span><span class="p">);</span>
-<a id="__codelineno-55-30" name="__codelineno-55-30" href="#__codelineno-55-30"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-55-30" name="__codelineno-55-30" href="#__codelineno-55-30"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-55-31" name="__codelineno-55-31" href="#__codelineno-55-31"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span><span class="p">;</span>
 <a id="__codelineno-55-32" name="__codelineno-55-32" href="#__codelineno-55-32"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-55-33" name="__codelineno-55-33" href="#__codelineno-55-33"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Element exits heap */</span>
+<div class="highlight"><span class="filename">my_heap.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Pop an element from the heap */</span>
 <a id="__codelineno-56-2" name="__codelineno-56-2" href="#__codelineno-56-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="nx">pop</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w">    </span><span class="c1">// Handle empty case</span>
+<a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w">    </span><span class="c1">// Handle an empty heap</span>
 <a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-5" name="__codelineno-56-5" href="#__codelineno-56-5"></a><span class="w">        </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="s">"error"</span><span class="p">)</span>
 <a id="__codelineno-56-6" name="__codelineno-56-6" href="#__codelineno-56-6"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-56-7" name="__codelineno-56-7" href="#__codelineno-56-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-56-8" name="__codelineno-56-8" href="#__codelineno-56-8"></a><span class="w">    </span><span class="c1">// Delete node</span>
+<a id="__codelineno-56-8" name="__codelineno-56-8" href="#__codelineno-56-8"></a><span class="w">    </span><span class="c1">// Swap the root with the rightmost leaf on the last level (swap the first and last elements)</span>
 <a id="__codelineno-56-9" name="__codelineno-56-9" href="#__codelineno-56-9"></a><span class="w">    </span><span class="nx">h</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-56-10" name="__codelineno-56-10" href="#__codelineno-56-10"></a><span class="w">    </span><span class="c1">// Remove node</span>
+<a id="__codelineno-56-10" name="__codelineno-56-10" href="#__codelineno-56-10"></a><span class="w">    </span><span class="c1">// Remove a node</span>
 <a id="__codelineno-56-11" name="__codelineno-56-11" href="#__codelineno-56-11"></a><span class="w">    </span><span class="nx">val</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-56-12" name="__codelineno-56-12" href="#__codelineno-56-12"></a><span class="w">    </span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">[:</span><span class="nb">len</span><span class="p">(</span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
-<a id="__codelineno-56-13" name="__codelineno-56-13" href="#__codelineno-56-13"></a><span class="w">    </span><span class="c1">// Return top element</span>
+<a id="__codelineno-56-13" name="__codelineno-56-13" href="#__codelineno-56-13"></a><span class="w">    </span><span class="c1">// Sift down</span>
 <a id="__codelineno-56-14" name="__codelineno-56-14" href="#__codelineno-56-14"></a><span class="w">    </span><span class="nx">h</span><span class="p">.</span><span class="nx">siftDown</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-56-15" name="__codelineno-56-15" href="#__codelineno-56-15"></a>
-<a id="__codelineno-56-16" name="__codelineno-56-16" href="#__codelineno-56-16"></a><span class="w">    </span><span class="c1">// Return heap top element</span>
+<a id="__codelineno-56-16" name="__codelineno-56-16" href="#__codelineno-56-16"></a><span class="w">    </span><span class="c1">// Return the top element of the heap</span>
 <a id="__codelineno-56-17" name="__codelineno-56-17" href="#__codelineno-56-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">val</span>
 <a id="__codelineno-56-18" name="__codelineno-56-18" href="#__codelineno-56-18"></a><span class="p">}</span>
 <a id="__codelineno-56-19" name="__codelineno-56-19" href="#__codelineno-56-19"></a>
-<a id="__codelineno-56-20" name="__codelineno-56-20" href="#__codelineno-56-20"></a><span class="cm">/* Starting from node i, heapify from top to bottom */</span>
+<a id="__codelineno-56-20" name="__codelineno-56-20" href="#__codelineno-56-20"></a><span class="cm">/* Sift down starting from node i */</span>
 <a id="__codelineno-56-21" name="__codelineno-56-21" href="#__codelineno-56-21"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">h</span><span class="w"> </span><span class="o">*</span><span class="nx">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="nx">siftDown</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-22" name="__codelineno-56-22" href="#__codelineno-56-22"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">true</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-56-23" name="__codelineno-56-23" href="#__codelineno-56-23"></a><span class="w">        </span><span class="c1">// Find node with maximum value among nodes i, l, r, denoted as max</span>
+<a id="__codelineno-56-23" name="__codelineno-56-23" href="#__codelineno-56-23"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in max</span>
 <a id="__codelineno-56-24" name="__codelineno-56-24" href="#__codelineno-56-24"></a><span class="w">        </span><span class="nx">l</span><span class="p">,</span><span class="w"> </span><span class="nx">r</span><span class="p">,</span><span class="w"> </span><span class="nx">max</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">left</span><span class="p">(</span><span class="nx">i</span><span class="p">),</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">right</span><span class="p">(</span><span class="nx">i</span><span class="p">),</span><span class="w"> </span><span class="nx">i</span>
 <a id="__codelineno-56-25" name="__codelineno-56-25" href="#__codelineno-56-25"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">[</span><span class="nx">l</span><span class="p">].(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">[</span><span class="nx">max</span><span class="p">].(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-26" name="__codelineno-56-26" href="#__codelineno-56-26"></a><span class="w">            </span><span class="nx">max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">l</span>
@@ -6120,40 +6202,40 @@
 <a id="__codelineno-56-28" name="__codelineno-56-28" href="#__codelineno-56-28"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">[</span><span class="nx">r</span><span class="p">].(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">data</span><span class="p">[</span><span class="nx">max</span><span class="p">].(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-29" name="__codelineno-56-29" href="#__codelineno-56-29"></a><span class="w">            </span><span class="nx">max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">r</span>
 <a id="__codelineno-56-30" name="__codelineno-56-30" href="#__codelineno-56-30"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-56-31" name="__codelineno-56-31" href="#__codelineno-56-31"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-56-31" name="__codelineno-56-31" href="#__codelineno-56-31"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-56-32" name="__codelineno-56-32" href="#__codelineno-56-32"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">max</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-33" name="__codelineno-56-33" href="#__codelineno-56-33"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-56-34" name="__codelineno-56-34" href="#__codelineno-56-34"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-56-35" name="__codelineno-56-35" href="#__codelineno-56-35"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-56-35" name="__codelineno-56-35" href="#__codelineno-56-35"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-56-36" name="__codelineno-56-36" href="#__codelineno-56-36"></a><span class="w">        </span><span class="nx">h</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">max</span><span class="p">)</span>
-<a id="__codelineno-56-37" name="__codelineno-56-37" href="#__codelineno-56-37"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-56-37" name="__codelineno-56-37" href="#__codelineno-56-37"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-56-38" name="__codelineno-56-38" href="#__codelineno-56-38"></a><span class="w">        </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">max</span>
 <a id="__codelineno-56-39" name="__codelineno-56-39" href="#__codelineno-56-39"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-56-40" name="__codelineno-56-40" href="#__codelineno-56-40"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Element exits heap */</span>
+<div class="highlight"><span class="filename">my_heap.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Pop an element from the heap */</span>
 <a id="__codelineno-57-2" name="__codelineno-57-2" href="#__codelineno-57-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">pop</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="w">    </span><span class="c1">// Handle empty case</span>
+<a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="w">    </span><span class="c1">// Handle an empty heap</span>
 <a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="bp">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-5" name="__codelineno-57-5" href="#__codelineno-57-5"></a><span class="w">        </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Heap is empty"</span><span class="p">)</span>
 <a id="__codelineno-57-6" name="__codelineno-57-6" href="#__codelineno-57-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-57-7" name="__codelineno-57-7" href="#__codelineno-57-7"></a><span class="w">    </span><span class="c1">// Delete node</span>
+<a id="__codelineno-57-7" name="__codelineno-57-7" href="#__codelineno-57-7"></a><span class="w">    </span><span class="c1">// Swap the root with the rightmost leaf on the last level (swap the first and last elements)</span>
 <a id="__codelineno-57-8" name="__codelineno-57-8" href="#__codelineno-57-8"></a><span class="w">    </span><span class="bp">swap</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-57-9" name="__codelineno-57-9" href="#__codelineno-57-9"></a><span class="w">    </span><span class="c1">// Remove node</span>
+<a id="__codelineno-57-9" name="__codelineno-57-9" href="#__codelineno-57-9"></a><span class="w">    </span><span class="c1">// Remove a node</span>
 <a id="__codelineno-57-10" name="__codelineno-57-10" href="#__codelineno-57-10"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">at</span><span class="p">:</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-57-11" name="__codelineno-57-11" href="#__codelineno-57-11"></a><span class="w">    </span><span class="c1">// Return top element</span>
+<a id="__codelineno-57-11" name="__codelineno-57-11" href="#__codelineno-57-11"></a><span class="w">    </span><span class="c1">// Sift down</span>
 <a id="__codelineno-57-12" name="__codelineno-57-12" href="#__codelineno-57-12"></a><span class="w">    </span><span class="n">siftDown</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-57-13" name="__codelineno-57-13" href="#__codelineno-57-13"></a><span class="w">    </span><span class="c1">// Return heap top element</span>
+<a id="__codelineno-57-13" name="__codelineno-57-13" href="#__codelineno-57-13"></a><span class="w">    </span><span class="c1">// Return the top element of the heap</span>
 <a id="__codelineno-57-14" name="__codelineno-57-14" href="#__codelineno-57-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">val</span>
 <a id="__codelineno-57-15" name="__codelineno-57-15" href="#__codelineno-57-15"></a><span class="p">}</span>
 <a id="__codelineno-57-16" name="__codelineno-57-16" href="#__codelineno-57-16"></a>
-<a id="__codelineno-57-17" name="__codelineno-57-17" href="#__codelineno-57-17"></a><span class="cm">/* Starting from node i, heapify from top to bottom */</span>
+<a id="__codelineno-57-17" name="__codelineno-57-17" href="#__codelineno-57-17"></a><span class="cm">/* Sift down starting from node i */</span>
 <a id="__codelineno-57-18" name="__codelineno-57-18" href="#__codelineno-57-18"></a><span class="kd">func</span><span class="w"> </span><span class="nf">siftDown</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-19" name="__codelineno-57-19" href="#__codelineno-57-19"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-57-20" name="__codelineno-57-20" href="#__codelineno-57-20"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="kc">true</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-57-21" name="__codelineno-57-21" href="#__codelineno-57-21"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-57-21" name="__codelineno-57-21" href="#__codelineno-57-21"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-57-22" name="__codelineno-57-22" href="#__codelineno-57-22"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">l</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kr">left</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-57-23" name="__codelineno-57-23" href="#__codelineno-57-23"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">r</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kr">right</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-57-24" name="__codelineno-57-24" href="#__codelineno-57-24"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">ma</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i</span>
@@ -6163,141 +6245,141 @@
 <a id="__codelineno-57-28" name="__codelineno-57-28" href="#__codelineno-57-28"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">(),</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-29" name="__codelineno-57-29" href="#__codelineno-57-29"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">r</span>
 <a id="__codelineno-57-30" name="__codelineno-57-30" href="#__codelineno-57-30"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-57-31" name="__codelineno-57-31" href="#__codelineno-57-31"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-57-31" name="__codelineno-57-31" href="#__codelineno-57-31"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-57-32" name="__codelineno-57-32" href="#__codelineno-57-32"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-33" name="__codelineno-57-33" href="#__codelineno-57-33"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-57-34" name="__codelineno-57-34" href="#__codelineno-57-34"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-57-35" name="__codelineno-57-35" href="#__codelineno-57-35"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-57-35" name="__codelineno-57-35" href="#__codelineno-57-35"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-57-36" name="__codelineno-57-36" href="#__codelineno-57-36"></a><span class="w">        </span><span class="bp">swap</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="n">ma</span><span class="p">)</span>
-<a id="__codelineno-57-37" name="__codelineno-57-37" href="#__codelineno-57-37"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-57-37" name="__codelineno-57-37" href="#__codelineno-57-37"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-57-38" name="__codelineno-57-38" href="#__codelineno-57-38"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ma</span>
 <a id="__codelineno-57-39" name="__codelineno-57-39" href="#__codelineno-57-39"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-57-40" name="__codelineno-57-40" href="#__codelineno-57-40"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Element exits heap */</span>
+<div class="highlight"><span class="filename">my_heap.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Pop an element from the heap */</span>
 <a id="__codelineno-58-2" name="__codelineno-58-2" href="#__codelineno-58-2"></a><span class="nx">pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a><span class="w">    </span><span class="c1">// Handle empty case</span>
+<a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a><span class="w">    </span><span class="c1">// Handle an empty heap</span>
 <a id="__codelineno-58-4" name="__codelineno-58-4" href="#__codelineno-58-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Heap is empty'</span><span class="p">);</span>
-<a id="__codelineno-58-5" name="__codelineno-58-5" href="#__codelineno-58-5"></a><span class="w">    </span><span class="c1">// Delete node</span>
+<a id="__codelineno-58-5" name="__codelineno-58-5" href="#__codelineno-58-5"></a><span class="w">    </span><span class="c1">// Swap the root with the rightmost leaf on the last level (swap the first and last elements)</span>
 <a id="__codelineno-58-6" name="__codelineno-58-6" href="#__codelineno-58-6"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">#swap</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-58-7" name="__codelineno-58-7" href="#__codelineno-58-7"></a><span class="w">    </span><span class="c1">// Remove node</span>
+<a id="__codelineno-58-7" name="__codelineno-58-7" href="#__codelineno-58-7"></a><span class="w">    </span><span class="c1">// Remove a node</span>
 <a id="__codelineno-58-8" name="__codelineno-58-8" href="#__codelineno-58-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#maxHeap</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
-<a id="__codelineno-58-9" name="__codelineno-58-9" href="#__codelineno-58-9"></a><span class="w">    </span><span class="c1">// Return top element</span>
+<a id="__codelineno-58-9" name="__codelineno-58-9" href="#__codelineno-58-9"></a><span class="w">    </span><span class="c1">// Sift down</span>
 <a id="__codelineno-58-10" name="__codelineno-58-10" href="#__codelineno-58-10"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">#siftDown</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
-<a id="__codelineno-58-11" name="__codelineno-58-11" href="#__codelineno-58-11"></a><span class="w">    </span><span class="c1">// Return heap top element</span>
+<a id="__codelineno-58-11" name="__codelineno-58-11" href="#__codelineno-58-11"></a><span class="w">    </span><span class="c1">// Return the top element of the heap</span>
 <a id="__codelineno-58-12" name="__codelineno-58-12" href="#__codelineno-58-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-58-13" name="__codelineno-58-13" href="#__codelineno-58-13"></a><span class="p">}</span>
 <a id="__codelineno-58-14" name="__codelineno-58-14" href="#__codelineno-58-14"></a>
-<a id="__codelineno-58-15" name="__codelineno-58-15" href="#__codelineno-58-15"></a><span class="cm">/* Starting from node i, heapify from top to bottom */</span>
+<a id="__codelineno-58-15" name="__codelineno-58-15" href="#__codelineno-58-15"></a><span class="cm">/* Sift down starting from node i */</span>
 <a id="__codelineno-58-16" name="__codelineno-58-16" href="#__codelineno-58-16"></a><span class="n">#siftDown</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-17" name="__codelineno-58-17" href="#__codelineno-58-17"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-58-18" name="__codelineno-58-18" href="#__codelineno-58-18"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-58-18" name="__codelineno-58-18" href="#__codelineno-58-18"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-58-19" name="__codelineno-58-19" href="#__codelineno-58-19"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#left</span><span class="p">(</span><span class="nx">i</span><span class="p">),</span>
 <a id="__codelineno-58-20" name="__codelineno-58-20" href="#__codelineno-58-20"></a><span class="w">            </span><span class="nx">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#right</span><span class="p">(</span><span class="nx">i</span><span class="p">);</span>
 <a id="__codelineno-58-21" name="__codelineno-58-21" href="#__codelineno-58-21"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-58-22" name="__codelineno-58-22" href="#__codelineno-58-22"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#maxHeap</span><span class="p">[</span><span class="nx">l</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#maxHeap</span><span class="p">[</span><span class="nx">ma</span><span class="p">])</span><span class="w"> </span><span class="nx">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">l</span><span class="p">;</span>
 <a id="__codelineno-58-23" name="__codelineno-58-23" href="#__codelineno-58-23"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#maxHeap</span><span class="p">[</span><span class="nx">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#maxHeap</span><span class="p">[</span><span class="nx">ma</span><span class="p">])</span><span class="w"> </span><span class="nx">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">r</span><span class="p">;</span>
-<a id="__codelineno-58-24" name="__codelineno-58-24" href="#__codelineno-58-24"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-58-24" name="__codelineno-58-24" href="#__codelineno-58-24"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-58-25" name="__codelineno-58-25" href="#__codelineno-58-25"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">ma</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span>
-<a id="__codelineno-58-26" name="__codelineno-58-26" href="#__codelineno-58-26"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-58-26" name="__codelineno-58-26" href="#__codelineno-58-26"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-58-27" name="__codelineno-58-27" href="#__codelineno-58-27"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#swap</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">ma</span><span class="p">);</span>
-<a id="__codelineno-58-28" name="__codelineno-58-28" href="#__codelineno-58-28"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-58-28" name="__codelineno-58-28" href="#__codelineno-58-28"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-58-29" name="__codelineno-58-29" href="#__codelineno-58-29"></a><span class="w">        </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">ma</span><span class="p">;</span>
 <a id="__codelineno-58-30" name="__codelineno-58-30" href="#__codelineno-58-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-58-31" name="__codelineno-58-31" href="#__codelineno-58-31"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Element exits heap */</span>
+<div class="highlight"><span class="filename">my_heap.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Pop an element from the heap */</span>
 <a id="__codelineno-59-2" name="__codelineno-59-2" href="#__codelineno-59-2"></a><span class="nx">pop</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="w">    </span><span class="c1">// Handle empty case</span>
-<a id="__codelineno-59-4" name="__codelineno-59-4" href="#__codelineno-59-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">RangeError</span><span class="p">(</span><span class="s1">'Heap is empty.'</span><span class="p">);</span>
-<a id="__codelineno-59-5" name="__codelineno-59-5" href="#__codelineno-59-5"></a><span class="w">    </span><span class="c1">// Delete node</span>
+<a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="w">    </span><span class="c1">// Handle an empty heap</span>
+<a id="__codelineno-59-4" name="__codelineno-59-4" href="#__codelineno-59-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">RangeError</span><span class="p">(</span><span class="s1">'Heap is empty'</span><span class="p">);</span>
+<a id="__codelineno-59-5" name="__codelineno-59-5" href="#__codelineno-59-5"></a><span class="w">    </span><span class="c1">// Swap the root with the rightmost leaf on the last level (swap the first and last elements)</span>
 <a id="__codelineno-59-6" name="__codelineno-59-6" href="#__codelineno-59-6"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-59-7" name="__codelineno-59-7" href="#__codelineno-59-7"></a><span class="w">    </span><span class="c1">// Remove node</span>
+<a id="__codelineno-59-7" name="__codelineno-59-7" href="#__codelineno-59-7"></a><span class="w">    </span><span class="c1">// Remove a node</span>
 <a id="__codelineno-59-8" name="__codelineno-59-8" href="#__codelineno-59-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">maxHeap</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
-<a id="__codelineno-59-9" name="__codelineno-59-9" href="#__codelineno-59-9"></a><span class="w">    </span><span class="c1">// Return top element</span>
+<a id="__codelineno-59-9" name="__codelineno-59-9" href="#__codelineno-59-9"></a><span class="w">    </span><span class="c1">// Sift down</span>
 <a id="__codelineno-59-10" name="__codelineno-59-10" href="#__codelineno-59-10"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">siftDown</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
-<a id="__codelineno-59-11" name="__codelineno-59-11" href="#__codelineno-59-11"></a><span class="w">    </span><span class="c1">// Return heap top element</span>
+<a id="__codelineno-59-11" name="__codelineno-59-11" href="#__codelineno-59-11"></a><span class="w">    </span><span class="c1">// Return the top element of the heap</span>
 <a id="__codelineno-59-12" name="__codelineno-59-12" href="#__codelineno-59-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-59-13" name="__codelineno-59-13" href="#__codelineno-59-13"></a><span class="p">}</span>
 <a id="__codelineno-59-14" name="__codelineno-59-14" href="#__codelineno-59-14"></a>
-<a id="__codelineno-59-15" name="__codelineno-59-15" href="#__codelineno-59-15"></a><span class="cm">/* Starting from node i, heapify from top to bottom */</span>
+<a id="__codelineno-59-15" name="__codelineno-59-15" href="#__codelineno-59-15"></a><span class="cm">/* Sift down starting from node i */</span>
 <a id="__codelineno-59-16" name="__codelineno-59-16" href="#__codelineno-59-16"></a><span class="nx">siftDown</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-17" name="__codelineno-59-17" href="#__codelineno-59-17"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-59-18" name="__codelineno-59-18" href="#__codelineno-59-18"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-59-18" name="__codelineno-59-18" href="#__codelineno-59-18"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-59-19" name="__codelineno-59-19" href="#__codelineno-59-19"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">left</span><span class="p">(</span><span class="nx">i</span><span class="p">),</span>
 <a id="__codelineno-59-20" name="__codelineno-59-20" href="#__codelineno-59-20"></a><span class="w">            </span><span class="nx">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">right</span><span class="p">(</span><span class="nx">i</span><span class="p">);</span>
 <a id="__codelineno-59-21" name="__codelineno-59-21" href="#__codelineno-59-21"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-59-22" name="__codelineno-59-22" href="#__codelineno-59-22"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">maxHeap</span><span class="p">[</span><span class="nx">l</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">maxHeap</span><span class="p">[</span><span class="nx">ma</span><span class="p">])</span><span class="w"> </span><span class="nx">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">l</span><span class="p">;</span>
 <a id="__codelineno-59-23" name="__codelineno-59-23" href="#__codelineno-59-23"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">maxHeap</span><span class="p">[</span><span class="nx">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">maxHeap</span><span class="p">[</span><span class="nx">ma</span><span class="p">])</span><span class="w"> </span><span class="nx">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">r</span><span class="p">;</span>
-<a id="__codelineno-59-24" name="__codelineno-59-24" href="#__codelineno-59-24"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-59-24" name="__codelineno-59-24" href="#__codelineno-59-24"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-59-25" name="__codelineno-59-25" href="#__codelineno-59-25"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">ma</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span>
-<a id="__codelineno-59-26" name="__codelineno-59-26" href="#__codelineno-59-26"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-59-26" name="__codelineno-59-26" href="#__codelineno-59-26"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-59-27" name="__codelineno-59-27" href="#__codelineno-59-27"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">ma</span><span class="p">);</span>
-<a id="__codelineno-59-28" name="__codelineno-59-28" href="#__codelineno-59-28"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-59-28" name="__codelineno-59-28" href="#__codelineno-59-28"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-59-29" name="__codelineno-59-29" href="#__codelineno-59-29"></a><span class="w">        </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">ma</span><span class="p">;</span>
 <a id="__codelineno-59-30" name="__codelineno-59-30" href="#__codelineno-59-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-59-31" name="__codelineno-59-31" href="#__codelineno-59-31"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Element exits heap */</span>
+<div class="highlight"><span class="filename">my_heap.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Pop an element from the heap */</span>
 <a id="__codelineno-60-2" name="__codelineno-60-2" href="#__codelineno-60-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-60-3" name="__codelineno-60-3" href="#__codelineno-60-3"></a><span class="w">  </span><span class="c1">// Handle empty case</span>
+<a id="__codelineno-60-3" name="__codelineno-60-3" href="#__codelineno-60-3"></a><span class="w">  </span><span class="c1">// Handle an empty heap</span>
 <a id="__codelineno-60-4" name="__codelineno-60-4" href="#__codelineno-60-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">Exception</span><span class="p">(</span><span class="s1">'Heap is empty'</span><span class="p">);</span>
-<a id="__codelineno-60-5" name="__codelineno-60-5" href="#__codelineno-60-5"></a><span class="w">  </span><span class="c1">// Delete node</span>
+<a id="__codelineno-60-5" name="__codelineno-60-5" href="#__codelineno-60-5"></a><span class="w">  </span><span class="c1">// Swap the root with the rightmost leaf on the last level (swap the first and last elements)</span>
 <a id="__codelineno-60-6" name="__codelineno-60-6" href="#__codelineno-60-6"></a><span class="w">  </span><span class="n">_swap</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
-<a id="__codelineno-60-7" name="__codelineno-60-7" href="#__codelineno-60-7"></a><span class="w">  </span><span class="c1">// Remove node</span>
+<a id="__codelineno-60-7" name="__codelineno-60-7" href="#__codelineno-60-7"></a><span class="w">  </span><span class="c1">// Remove a node</span>
 <a id="__codelineno-60-8" name="__codelineno-60-8" href="#__codelineno-60-8"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">.</span><span class="n">removeLast</span><span class="p">();</span>
-<a id="__codelineno-60-9" name="__codelineno-60-9" href="#__codelineno-60-9"></a><span class="w">  </span><span class="c1">// Return top element</span>
+<a id="__codelineno-60-9" name="__codelineno-60-9" href="#__codelineno-60-9"></a><span class="w">  </span><span class="c1">// Sift down</span>
 <a id="__codelineno-60-10" name="__codelineno-60-10" href="#__codelineno-60-10"></a><span class="w">  </span><span class="n">siftDown</span><span class="p">(</span><span class="m">0</span><span class="p">);</span>
-<a id="__codelineno-60-11" name="__codelineno-60-11" href="#__codelineno-60-11"></a><span class="w">  </span><span class="c1">// Return heap top element</span>
+<a id="__codelineno-60-11" name="__codelineno-60-11" href="#__codelineno-60-11"></a><span class="w">  </span><span class="c1">// Return the top element of the heap</span>
 <a id="__codelineno-60-12" name="__codelineno-60-12" href="#__codelineno-60-12"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-60-13" name="__codelineno-60-13" href="#__codelineno-60-13"></a><span class="p">}</span>
 <a id="__codelineno-60-14" name="__codelineno-60-14" href="#__codelineno-60-14"></a>
-<a id="__codelineno-60-15" name="__codelineno-60-15" href="#__codelineno-60-15"></a><span class="cm">/* Starting from node i, heapify from top to bottom */</span>
+<a id="__codelineno-60-15" name="__codelineno-60-15" href="#__codelineno-60-15"></a><span class="cm">/* Sift down starting from node i */</span>
 <a id="__codelineno-60-16" name="__codelineno-60-16" href="#__codelineno-60-16"></a><span class="kt">void</span><span class="w"> </span><span class="n">siftDown</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-17" name="__codelineno-60-17" href="#__codelineno-60-17"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-60-18" name="__codelineno-60-18" href="#__codelineno-60-18"></a><span class="w">    </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-60-18" name="__codelineno-60-18" href="#__codelineno-60-18"></a><span class="w">    </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-60-19" name="__codelineno-60-19" href="#__codelineno-60-19"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_left</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-60-20" name="__codelineno-60-20" href="#__codelineno-60-20"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_right</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-60-21" name="__codelineno-60-21" href="#__codelineno-60-21"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-60-22" name="__codelineno-60-22" href="#__codelineno-60-22"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">[</span><span class="n">l</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span><span class="p">;</span>
 <a id="__codelineno-60-23" name="__codelineno-60-23" href="#__codelineno-60-23"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">_maxHeap</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="p">;</span>
-<a id="__codelineno-60-24" name="__codelineno-60-24" href="#__codelineno-60-24"></a><span class="w">    </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-60-24" name="__codelineno-60-24" href="#__codelineno-60-24"></a><span class="w">    </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-60-25" name="__codelineno-60-25" href="#__codelineno-60-25"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span>
-<a id="__codelineno-60-26" name="__codelineno-60-26" href="#__codelineno-60-26"></a><span class="w">    </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-60-26" name="__codelineno-60-26" href="#__codelineno-60-26"></a><span class="w">    </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-60-27" name="__codelineno-60-27" href="#__codelineno-60-27"></a><span class="w">    </span><span class="n">_swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">ma</span><span class="p">);</span>
-<a id="__codelineno-60-28" name="__codelineno-60-28" href="#__codelineno-60-28"></a><span class="w">    </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-60-28" name="__codelineno-60-28" href="#__codelineno-60-28"></a><span class="w">    </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-60-29" name="__codelineno-60-29" href="#__codelineno-60-29"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span><span class="p">;</span>
 <a id="__codelineno-60-30" name="__codelineno-60-30" href="#__codelineno-60-30"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-60-31" name="__codelineno-60-31" href="#__codelineno-60-31"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Element exits heap */</span>
+<div class="highlight"><span class="filename">my_heap.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Pop an element from the heap */</span>
 <a id="__codelineno-61-2" name="__codelineno-61-2" href="#__codelineno-61-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">pop</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-61-3" name="__codelineno-61-3" href="#__codelineno-61-3"></a><span class="w">    </span><span class="c1">// Handle empty case</span>
+<a id="__codelineno-61-3" name="__codelineno-61-3" href="#__codelineno-61-3"></a><span class="w">    </span><span class="c1">// Handle an empty heap</span>
 <a id="__codelineno-61-4" name="__codelineno-61-4" href="#__codelineno-61-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">is_empty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-61-5" name="__codelineno-61-5" href="#__codelineno-61-5"></a><span class="w">        </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"index out of bounds"</span><span class="p">);</span>
+<a id="__codelineno-61-5" name="__codelineno-61-5" href="#__codelineno-61-5"></a><span class="w">        </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">);</span>
 <a id="__codelineno-61-6" name="__codelineno-61-6" href="#__codelineno-61-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-61-7" name="__codelineno-61-7" href="#__codelineno-61-7"></a><span class="w">    </span><span class="c1">// Delete node</span>
+<a id="__codelineno-61-7" name="__codelineno-61-7" href="#__codelineno-61-7"></a><span class="w">    </span><span class="c1">// Swap the root with the rightmost leaf on the last level (swap the first and last elements)</span>
 <a id="__codelineno-61-8" name="__codelineno-61-8" href="#__codelineno-61-8"></a><span class="w">    </span><span class="bp">self</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-61-9" name="__codelineno-61-9" href="#__codelineno-61-9"></a><span class="w">    </span><span class="c1">// Remove node</span>
+<a id="__codelineno-61-9" name="__codelineno-61-9" href="#__codelineno-61-9"></a><span class="w">    </span><span class="c1">// Remove a node</span>
 <a id="__codelineno-61-10" name="__codelineno-61-10" href="#__codelineno-61-10"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">max_heap</span><span class="p">.</span><span class="n">pop</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span>
-<a id="__codelineno-61-11" name="__codelineno-61-11" href="#__codelineno-61-11"></a><span class="w">    </span><span class="c1">// Return top element</span>
+<a id="__codelineno-61-11" name="__codelineno-61-11" href="#__codelineno-61-11"></a><span class="w">    </span><span class="c1">// Sift down</span>
 <a id="__codelineno-61-12" name="__codelineno-61-12" href="#__codelineno-61-12"></a><span class="w">    </span><span class="bp">self</span><span class="p">.</span><span class="n">sift_down</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
-<a id="__codelineno-61-13" name="__codelineno-61-13" href="#__codelineno-61-13"></a><span class="w">    </span><span class="c1">// Return heap top element</span>
+<a id="__codelineno-61-13" name="__codelineno-61-13" href="#__codelineno-61-13"></a><span class="w">    </span><span class="c1">// Return the top element of the heap</span>
 <a id="__codelineno-61-14" name="__codelineno-61-14" href="#__codelineno-61-14"></a><span class="w">    </span><span class="n">val</span>
 <a id="__codelineno-61-15" name="__codelineno-61-15" href="#__codelineno-61-15"></a><span class="p">}</span>
 <a id="__codelineno-61-16" name="__codelineno-61-16" href="#__codelineno-61-16"></a>
-<a id="__codelineno-61-17" name="__codelineno-61-17" href="#__codelineno-61-17"></a><span class="cm">/* Starting from node i, heapify from top to bottom */</span>
+<a id="__codelineno-61-17" name="__codelineno-61-17" href="#__codelineno-61-17"></a><span class="cm">/* Sift down starting from node i */</span>
 <a id="__codelineno-61-18" name="__codelineno-61-18" href="#__codelineno-61-18"></a><span class="k">fn</span><span class="w"> </span><span class="nf">sift_down</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-19" name="__codelineno-61-19" href="#__codelineno-61-19"></a><span class="w">    </span><span class="k">loop</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-61-20" name="__codelineno-61-20" href="#__codelineno-61-20"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-61-20" name="__codelineno-61-20" href="#__codelineno-61-20"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-61-21" name="__codelineno-61-21" href="#__codelineno-61-21"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">r</span><span class="p">,</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">ma</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="bp">Self</span><span class="p">::</span><span class="n">left</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">right</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-61-22" name="__codelineno-61-22" href="#__codelineno-61-22"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">max_heap</span><span class="p">[</span><span class="n">l</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">max_heap</span><span class="p">[</span><span class="n">ma</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-23" name="__codelineno-61-23" href="#__codelineno-61-23"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span><span class="p">;</span>
@@ -6305,42 +6387,42 @@
 <a id="__codelineno-61-25" name="__codelineno-61-25" href="#__codelineno-61-25"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">max_heap</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">max_heap</span><span class="p">[</span><span class="n">ma</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-26" name="__codelineno-61-26" href="#__codelineno-61-26"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="p">;</span>
 <a id="__codelineno-61-27" name="__codelineno-61-27" href="#__codelineno-61-27"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-61-28" name="__codelineno-61-28" href="#__codelineno-61-28"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-61-28" name="__codelineno-61-28" href="#__codelineno-61-28"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-61-29" name="__codelineno-61-29" href="#__codelineno-61-29"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-30" name="__codelineno-61-30" href="#__codelineno-61-30"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-61-31" name="__codelineno-61-31" href="#__codelineno-61-31"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-61-32" name="__codelineno-61-32" href="#__codelineno-61-32"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-61-32" name="__codelineno-61-32" href="#__codelineno-61-32"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-61-33" name="__codelineno-61-33" href="#__codelineno-61-33"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">ma</span><span class="p">);</span>
-<a id="__codelineno-61-34" name="__codelineno-61-34" href="#__codelineno-61-34"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-61-34" name="__codelineno-61-34" href="#__codelineno-61-34"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-61-35" name="__codelineno-61-35" href="#__codelineno-61-35"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span><span class="p">;</span>
 <a id="__codelineno-61-36" name="__codelineno-61-36" href="#__codelineno-61-36"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-61-37" name="__codelineno-61-37" href="#__codelineno-61-37"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Element exits heap */</span>
+<div class="highlight"><span class="filename">my_heap.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Pop an element from the heap */</span>
 <a id="__codelineno-62-2" name="__codelineno-62-2" href="#__codelineno-62-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">pop</span><span class="p">(</span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="n">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-62-3" name="__codelineno-62-3" href="#__codelineno-62-3"></a><span class="w">    </span><span class="c1">// Handle empty case</span>
+<a id="__codelineno-62-3" name="__codelineno-62-3" href="#__codelineno-62-3"></a><span class="w">    </span><span class="c1">// Handle an empty heap</span>
 <a id="__codelineno-62-4" name="__codelineno-62-4" href="#__codelineno-62-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-62-5" name="__codelineno-62-5" href="#__codelineno-62-5"></a><span class="w">        </span><span class="n">printf</span><span class="p">(</span><span class="s">"heap is empty!"</span><span class="p">);</span>
+<a id="__codelineno-62-5" name="__codelineno-62-5" href="#__codelineno-62-5"></a><span class="w">        </span><span class="n">printf</span><span class="p">(</span><span class="s">"Heap is empty!"</span><span class="p">);</span>
 <a id="__codelineno-62-6" name="__codelineno-62-6" href="#__codelineno-62-6"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
 <a id="__codelineno-62-7" name="__codelineno-62-7" href="#__codelineno-62-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-62-8" name="__codelineno-62-8" href="#__codelineno-62-8"></a><span class="w">    </span><span class="c1">// Delete node</span>
+<a id="__codelineno-62-8" name="__codelineno-62-8" href="#__codelineno-62-8"></a><span class="w">    </span><span class="c1">// Swap the root with the rightmost leaf on the last level (swap the first and last elements)</span>
 <a id="__codelineno-62-9" name="__codelineno-62-9" href="#__codelineno-62-9"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-62-10" name="__codelineno-62-10" href="#__codelineno-62-10"></a><span class="w">    </span><span class="c1">// Remove node</span>
+<a id="__codelineno-62-10" name="__codelineno-62-10" href="#__codelineno-62-10"></a><span class="w">    </span><span class="c1">// Remove a node</span>
 <a id="__codelineno-62-11" name="__codelineno-62-11" href="#__codelineno-62-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">data</span><span class="p">[</span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-62-12" name="__codelineno-62-12" href="#__codelineno-62-12"></a><span class="w">    </span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">size</span><span class="o">--</span><span class="p">;</span>
-<a id="__codelineno-62-13" name="__codelineno-62-13" href="#__codelineno-62-13"></a><span class="w">    </span><span class="c1">// Return top element</span>
+<a id="__codelineno-62-13" name="__codelineno-62-13" href="#__codelineno-62-13"></a><span class="w">    </span><span class="c1">// Sift down</span>
 <a id="__codelineno-62-14" name="__codelineno-62-14" href="#__codelineno-62-14"></a><span class="w">    </span><span class="n">siftDown</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-62-15" name="__codelineno-62-15" href="#__codelineno-62-15"></a>
-<a id="__codelineno-62-16" name="__codelineno-62-16" href="#__codelineno-62-16"></a><span class="w">    </span><span class="c1">// Return heap top element</span>
+<a id="__codelineno-62-16" name="__codelineno-62-16" href="#__codelineno-62-16"></a><span class="w">    </span><span class="c1">// Return the top element of the heap</span>
 <a id="__codelineno-62-17" name="__codelineno-62-17" href="#__codelineno-62-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-62-18" name="__codelineno-62-18" href="#__codelineno-62-18"></a><span class="p">}</span>
 <a id="__codelineno-62-19" name="__codelineno-62-19" href="#__codelineno-62-19"></a>
-<a id="__codelineno-62-20" name="__codelineno-62-20" href="#__codelineno-62-20"></a><span class="cm">/* Starting from node i, heapify from top to bottom */</span>
+<a id="__codelineno-62-20" name="__codelineno-62-20" href="#__codelineno-62-20"></a><span class="cm">/* Sift down starting from node i */</span>
 <a id="__codelineno-62-21" name="__codelineno-62-21" href="#__codelineno-62-21"></a><span class="kt">void</span><span class="w"> </span><span class="nf">siftDown</span><span class="p">(</span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-22" name="__codelineno-62-22" href="#__codelineno-62-22"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nb">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-62-23" name="__codelineno-62-23" href="#__codelineno-62-23"></a><span class="w">        </span><span class="c1">// Find node with maximum value among nodes i, l, r, denoted as max</span>
+<a id="__codelineno-62-23" name="__codelineno-62-23" href="#__codelineno-62-23"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in max</span>
 <a id="__codelineno-62-24" name="__codelineno-62-24" href="#__codelineno-62-24"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-62-25" name="__codelineno-62-25" href="#__codelineno-62-25"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-62-26" name="__codelineno-62-26" href="#__codelineno-62-26"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
@@ -6350,83 +6432,83 @@
 <a id="__codelineno-62-30" name="__codelineno-62-30" href="#__codelineno-62-30"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">data</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">data</span><span class="p">[</span><span class="n">max</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-31" name="__codelineno-62-31" href="#__codelineno-62-31"></a><span class="w">            </span><span class="n">max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="p">;</span>
 <a id="__codelineno-62-32" name="__codelineno-62-32" href="#__codelineno-62-32"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-62-33" name="__codelineno-62-33" href="#__codelineno-62-33"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-62-33" name="__codelineno-62-33" href="#__codelineno-62-33"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-62-34" name="__codelineno-62-34" href="#__codelineno-62-34"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">max</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-35" name="__codelineno-62-35" href="#__codelineno-62-35"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-62-36" name="__codelineno-62-36" href="#__codelineno-62-36"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-62-37" name="__codelineno-62-37" href="#__codelineno-62-37"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-62-37" name="__codelineno-62-37" href="#__codelineno-62-37"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-62-38" name="__codelineno-62-38" href="#__codelineno-62-38"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">max</span><span class="p">);</span>
-<a id="__codelineno-62-39" name="__codelineno-62-39" href="#__codelineno-62-39"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-62-39" name="__codelineno-62-39" href="#__codelineno-62-39"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-62-40" name="__codelineno-62-40" href="#__codelineno-62-40"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">;</span>
 <a id="__codelineno-62-41" name="__codelineno-62-41" href="#__codelineno-62-41"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-62-42" name="__codelineno-62-42" href="#__codelineno-62-42"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Element exits heap */</span>
+<div class="highlight"><span class="filename">my_heap.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Pop an element from the heap */</span>
 <a id="__codelineno-63-2" name="__codelineno-63-2" href="#__codelineno-63-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">pop</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-63-3" name="__codelineno-63-3" href="#__codelineno-63-3"></a><span class="w">    </span><span class="c1">// Handle empty case</span>
+<a id="__codelineno-63-3" name="__codelineno-63-3" href="#__codelineno-63-3"></a><span class="w">    </span><span class="c1">// Handle an empty heap</span>
 <a id="__codelineno-63-4" name="__codelineno-63-4" href="#__codelineno-63-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">()</span>
-<a id="__codelineno-63-5" name="__codelineno-63-5" href="#__codelineno-63-5"></a><span class="w">    </span><span class="c1">// Delete node</span>
+<a id="__codelineno-63-5" name="__codelineno-63-5" href="#__codelineno-63-5"></a><span class="w">    </span><span class="c1">// Swap the root with the rightmost leaf on the last level (swap the first and last elements)</span>
 <a id="__codelineno-63-6" name="__codelineno-63-6" href="#__codelineno-63-6"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
-<a id="__codelineno-63-7" name="__codelineno-63-7" href="#__codelineno-63-7"></a><span class="w">    </span><span class="c1">// Remove node</span>
+<a id="__codelineno-63-7" name="__codelineno-63-7" href="#__codelineno-63-7"></a><span class="w">    </span><span class="c1">// Remove a node</span>
 <a id="__codelineno-63-8" name="__codelineno-63-8" href="#__codelineno-63-8"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">_val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">maxHeap</span><span class="p">.</span><span class="na">removeAt</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
-<a id="__codelineno-63-9" name="__codelineno-63-9" href="#__codelineno-63-9"></a><span class="w">    </span><span class="c1">// Return top element</span>
+<a id="__codelineno-63-9" name="__codelineno-63-9" href="#__codelineno-63-9"></a><span class="w">    </span><span class="c1">// Sift down</span>
 <a id="__codelineno-63-10" name="__codelineno-63-10" href="#__codelineno-63-10"></a><span class="w">    </span><span class="n">siftDown</span><span class="p">(</span><span class="m">0</span><span class="p">)</span>
-<a id="__codelineno-63-11" name="__codelineno-63-11" href="#__codelineno-63-11"></a><span class="w">    </span><span class="c1">// Return heap top element</span>
+<a id="__codelineno-63-11" name="__codelineno-63-11" href="#__codelineno-63-11"></a><span class="w">    </span><span class="c1">// Return the top element of the heap</span>
 <a id="__codelineno-63-12" name="__codelineno-63-12" href="#__codelineno-63-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_val</span>
 <a id="__codelineno-63-13" name="__codelineno-63-13" href="#__codelineno-63-13"></a><span class="p">}</span>
 <a id="__codelineno-63-14" name="__codelineno-63-14" href="#__codelineno-63-14"></a>
-<a id="__codelineno-63-15" name="__codelineno-63-15" href="#__codelineno-63-15"></a><span class="cm">/* Starting from node i, heapify from top to bottom */</span>
+<a id="__codelineno-63-15" name="__codelineno-63-15" href="#__codelineno-63-15"></a><span class="cm">/* Sift down starting from node i */</span>
 <a id="__codelineno-63-16" name="__codelineno-63-16" href="#__codelineno-63-16"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">siftDown</span><span class="p">(</span><span class="nb">it</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-63-17" name="__codelineno-63-17" href="#__codelineno-63-17"></a><span class="w">    </span><span class="c1">// Kotlin function parameters are immutable, so create temporary variable</span>
+<a id="__codelineno-63-17" name="__codelineno-63-17" href="#__codelineno-63-17"></a><span class="w">    </span><span class="c1">// Kotlin function parameters are immutable, so create a temporary variable</span>
 <a id="__codelineno-63-18" name="__codelineno-63-18" href="#__codelineno-63-18"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">it</span>
 <a id="__codelineno-63-19" name="__codelineno-63-19" href="#__codelineno-63-19"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-63-20" name="__codelineno-63-20" href="#__codelineno-63-20"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-63-20" name="__codelineno-63-20" href="#__codelineno-63-20"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-63-21" name="__codelineno-63-21" href="#__codelineno-63-21"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-63-22" name="__codelineno-63-22" href="#__codelineno-63-22"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-63-23" name="__codelineno-63-23" href="#__codelineno-63-23"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-63-24" name="__codelineno-63-24" href="#__codelineno-63-24"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">[</span><span class="n">l</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span>
 <a id="__codelineno-63-25" name="__codelineno-63-25" href="#__codelineno-63-25"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">[</span><span class="n">r</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span>
-<a id="__codelineno-63-26" name="__codelineno-63-26" href="#__codelineno-63-26"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-63-26" name="__codelineno-63-26" href="#__codelineno-63-26"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-63-27" name="__codelineno-63-27" href="#__codelineno-63-27"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="k">break</span>
-<a id="__codelineno-63-28" name="__codelineno-63-28" href="#__codelineno-63-28"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-63-28" name="__codelineno-63-28" href="#__codelineno-63-28"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-63-29" name="__codelineno-63-29" href="#__codelineno-63-29"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">ma</span><span class="p">)</span>
-<a id="__codelineno-63-30" name="__codelineno-63-30" href="#__codelineno-63-30"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-63-30" name="__codelineno-63-30" href="#__codelineno-63-30"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-63-31" name="__codelineno-63-31" href="#__codelineno-63-31"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span>
 <a id="__codelineno-63-32" name="__codelineno-63-32" href="#__codelineno-63-32"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-63-33" name="__codelineno-63-33" href="#__codelineno-63-33"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">my_heap.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Pop element from heap ###</span>
+<div class="highlight"><span class="filename">my_heap.rb</span><pre><span></span><code><a id="__codelineno-64-1" name="__codelineno-64-1" href="#__codelineno-64-1"></a><span class="c1">### Pop an element from the heap ###</span>
 <a id="__codelineno-64-2" name="__codelineno-64-2" href="#__codelineno-64-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">pop</span>
-<a id="__codelineno-64-3" name="__codelineno-64-3" href="#__codelineno-64-3"></a><span class="w">  </span><span class="c1"># Handle empty case</span>
+<a id="__codelineno-64-3" name="__codelineno-64-3" href="#__codelineno-64-3"></a><span class="w">  </span><span class="c1"># Handle an empty heap</span>
 <a id="__codelineno-64-4" name="__codelineno-64-4" href="#__codelineno-64-4"></a><span class="w">  </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="p">,</span><span class="w"> </span><span class="s2">"Heap is empty"</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">is_empty?</span>
-<a id="__codelineno-64-5" name="__codelineno-64-5" href="#__codelineno-64-5"></a><span class="w">  </span><span class="c1"># Delete node</span>
+<a id="__codelineno-64-5" name="__codelineno-64-5" href="#__codelineno-64-5"></a><span class="w">  </span><span class="c1"># Swap the root with the rightmost leaf on the last level (swap the first and last elements)</span>
 <a id="__codelineno-64-6" name="__codelineno-64-6" href="#__codelineno-64-6"></a><span class="w">  </span><span class="n">swap</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-64-7" name="__codelineno-64-7" href="#__codelineno-64-7"></a><span class="w">  </span><span class="c1"># Remove node</span>
+<a id="__codelineno-64-7" name="__codelineno-64-7" href="#__codelineno-64-7"></a><span class="w">  </span><span class="c1"># Remove a node</span>
 <a id="__codelineno-64-8" name="__codelineno-64-8" href="#__codelineno-64-8"></a><span class="w">  </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@max_heap</span><span class="o">.</span><span class="n">pop</span>
-<a id="__codelineno-64-9" name="__codelineno-64-9" href="#__codelineno-64-9"></a><span class="w">  </span><span class="c1"># Return top element</span>
+<a id="__codelineno-64-9" name="__codelineno-64-9" href="#__codelineno-64-9"></a><span class="w">  </span><span class="c1"># Sift down</span>
 <a id="__codelineno-64-10" name="__codelineno-64-10" href="#__codelineno-64-10"></a><span class="w">  </span><span class="n">sift_down</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-64-11" name="__codelineno-64-11" href="#__codelineno-64-11"></a><span class="w">  </span><span class="c1"># Return heap top element</span>
+<a id="__codelineno-64-11" name="__codelineno-64-11" href="#__codelineno-64-11"></a><span class="w">  </span><span class="c1"># Return the top element of the heap</span>
 <a id="__codelineno-64-12" name="__codelineno-64-12" href="#__codelineno-64-12"></a><span class="w">  </span><span class="n">val</span>
 <a id="__codelineno-64-13" name="__codelineno-64-13" href="#__codelineno-64-13"></a><span class="k">end</span>
 <a id="__codelineno-64-14" name="__codelineno-64-14" href="#__codelineno-64-14"></a>
-<a id="__codelineno-64-15" name="__codelineno-64-15" href="#__codelineno-64-15"></a><span class="c1">### Heapify from node i, top to bottom ###</span>
+<a id="__codelineno-64-15" name="__codelineno-64-15" href="#__codelineno-64-15"></a><span class="c1">### Sift down starting from node i ###</span>
 <a id="__codelineno-64-16" name="__codelineno-64-16" href="#__codelineno-64-16"></a><span class="k">def</span><span class="w"> </span><span class="nf">sift_down</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-64-17" name="__codelineno-64-17" href="#__codelineno-64-17"></a><span class="w">  </span><span class="kp">loop</span><span class="w"> </span><span class="k">do</span>
-<a id="__codelineno-64-18" name="__codelineno-64-18" href="#__codelineno-64-18"></a><span class="w">    </span><span class="c1"># If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-64-18" name="__codelineno-64-18" href="#__codelineno-64-18"></a><span class="w">    </span><span class="c1"># Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-64-19" name="__codelineno-64-19" href="#__codelineno-64-19"></a><span class="w">    </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">r</span><span class="p">,</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="w"> </span><span class="n">right</span><span class="p">(</span><span class="n">i</span><span class="p">),</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-64-20" name="__codelineno-64-20" href="#__codelineno-64-20"></a><span class="w">    </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="vi">@max_heap</span><span class="o">[</span><span class="n">l</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="vi">@max_heap</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span>
 <a id="__codelineno-64-21" name="__codelineno-64-21" href="#__codelineno-64-21"></a><span class="w">    </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="vi">@max_heap</span><span class="o">[</span><span class="n">r</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="vi">@max_heap</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span>
 <a id="__codelineno-64-22" name="__codelineno-64-22" href="#__codelineno-64-22"></a>
-<a id="__codelineno-64-23" name="__codelineno-64-23" href="#__codelineno-64-23"></a><span class="w">    </span><span class="c1"># Swap two nodes</span>
+<a id="__codelineno-64-23" name="__codelineno-64-23" href="#__codelineno-64-23"></a><span class="w">    </span><span class="c1"># Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-64-24" name="__codelineno-64-24" href="#__codelineno-64-24"></a><span class="w">    </span><span class="k">break</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-64-25" name="__codelineno-64-25" href="#__codelineno-64-25"></a>
-<a id="__codelineno-64-26" name="__codelineno-64-26" href="#__codelineno-64-26"></a><span class="w">    </span><span class="c1"># Swap two nodes</span>
+<a id="__codelineno-64-26" name="__codelineno-64-26" href="#__codelineno-64-26"></a><span class="w">    </span><span class="c1"># Swap the two nodes</span>
 <a id="__codelineno-64-27" name="__codelineno-64-27" href="#__codelineno-64-27"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">ma</span><span class="p">)</span>
-<a id="__codelineno-64-28" name="__codelineno-64-28" href="#__codelineno-64-28"></a><span class="w">    </span><span class="c1"># Loop downwards heapification</span>
+<a id="__codelineno-64-28" name="__codelineno-64-28" href="#__codelineno-64-28"></a><span class="w">    </span><span class="c1"># Continue sifting down</span>
 <a id="__codelineno-64-29" name="__codelineno-64-29" href="#__codelineno-64-29"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span>
 <a id="__codelineno-64-30" name="__codelineno-64-30" href="#__codelineno-64-30"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-64-31" name="__codelineno-64-31" href="#__codelineno-64-31"></a><span class="k">end</span>
@@ -6434,11 +6516,16 @@
 </div>
 </div>
 </div>
-<h2 id="813-common-applications-of-heaps">8.1.3 &nbsp; Common Applications of Heaps<a class="headerlink" href="#813-common-applications-of-heaps" title="Permanent link">&para;</a></h2>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20MaxHeap%3A%0A%20%20%20%20%22%22%22Max%20heap%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Place%20the%20list%20elements%20in%20the%20heap%20without%20changing%20their%20order%0A%20%20%20%20%20%20%20%20self.max_heap%20%3D%20nums%0A%0A%20%20%20%20def%20left%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20left%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%201%0A%0A%20%20%20%20def%20right%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20right%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%202%0A%0A%20%20%20%20def%20parent%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20parent%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%20%28i%20-%201%29%20//%202%20%20%23%20Floor%20division%0A%0A%20%20%20%20def%20swap%28self,%20i%3A%20int,%20j%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Swap%20elements%22%22%22%0A%20%20%20%20%20%20%20%20self.max_heap%5Bi%5D,%20self.max_heap%5Bj%5D%20%3D%20self.max_heap%5Bj%5D,%20self.max_heap%5Bi%5D%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20heap%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self.max_heap%29%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22Check%20whether%20the%20heap%20is%20empty%22%22%22%0A%20%20%20%20%20%20%20%20return%20self.size%28%29%20%3D%3D%200%0A%0A%20%20%20%20def%20pop%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Pop%20an%20element%20from%20the%20heap%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Handle%20an%20empty%20heap%0A%20%20%20%20%20%20%20%20if%20self.is_empty%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22Heap%20is%20empty%22%29%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20root%20with%20the%20rightmost%20leaf%20on%20the%20last%20level%20%28swap%20the%20first%20and%20last%20elements%29%0A%20%20%20%20%20%20%20%20self.swap%280,%20self.size%28%29%20-%201%29%0A%20%20%20%20%20%20%20%20%23%20Remove%20a%20node%0A%20%20%20%20%20%20%20%20val%20%3D%20self.max_heap.pop%28%29%0A%20%20%20%20%20%20%20%20%23%20Sift%20down%0A%20%20%20%20%20%20%20%20self.sift_down%280%29%0A%20%20%20%20%20%20%20%20%23%20Return%20the%20top%20element%20of%20the%20heap%0A%20%20%20%20%20%20%20%20return%20val%0A%0A%20%20%20%20def%20sift_down%28self,%20i%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Sift%20down%20starting%20from%20node%20i%22%22%22%0A%20%20%20%20%20%20%20%20while%20True%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Find%20the%20node%20with%20the%20largest%20value%20among%20i,%20l,%20and%20r,%20and%20store%20its%20index%20in%20ma%0A%20%20%20%20%20%20%20%20%20%20%20%20l,%20r,%20ma%20%3D%20self.left%28i%29,%20self.right%28i%29,%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20l%20%3C%20self.size%28%29%20and%20self.max_heap%5Bl%5D%20%3E%20self.max_heap%5Bma%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20ma%20%3D%20l%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20r%20%3C%20self.size%28%29%20and%20self.max_heap%5Br%5D%20%3E%20self.max_heap%5Bma%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20ma%20%3D%20r%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Stop%20if%20node%20i%20is%20largest%20or%20both%20child%20indices%20l%20and%20r%20are%20out%20of%20bounds%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20ma%20%3D%3D%20i%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Swap%20the%20two%20nodes%0A%20%20%20%20%20%20%20%20%20%20%20%20self.swap%28i,%20ma%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Continue%20sifting%20down%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20ma%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20max%20heap%0A%20%20%20%20%23%20Note%20that%20the%20input%20array%20is%20already%20a%20valid%20heap%0A%20%20%20%20max_heap%20%3D%20MaxHeap%28%5B9,%208,%207,%206,%207,%206,%202,%201,%204,%203,%206,%202,%205%5D%29%0A%0A%20%20%20%20%23%20Pop%20the%20top%20element%20from%20the%20heap%0A%20%20%20%20peek%20%3D%20max_heap.pop%28%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=8&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20MaxHeap%3A%0A%20%20%20%20%22%22%22Max%20heap%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Place%20the%20list%20elements%20in%20the%20heap%20without%20changing%20their%20order%0A%20%20%20%20%20%20%20%20self.max_heap%20%3D%20nums%0A%0A%20%20%20%20def%20left%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20left%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%201%0A%0A%20%20%20%20def%20right%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20right%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%202%0A%0A%20%20%20%20def%20parent%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20parent%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%20%28i%20-%201%29%20//%202%20%20%23%20Floor%20division%0A%0A%20%20%20%20def%20swap%28self,%20i%3A%20int,%20j%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Swap%20elements%22%22%22%0A%20%20%20%20%20%20%20%20self.max_heap%5Bi%5D,%20self.max_heap%5Bj%5D%20%3D%20self.max_heap%5Bj%5D,%20self.max_heap%5Bi%5D%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20heap%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self.max_heap%29%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22Check%20whether%20the%20heap%20is%20empty%22%22%22%0A%20%20%20%20%20%20%20%20return%20self.size%28%29%20%3D%3D%200%0A%0A%20%20%20%20def%20pop%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Pop%20an%20element%20from%20the%20heap%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Handle%20an%20empty%20heap%0A%20%20%20%20%20%20%20%20if%20self.is_empty%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22Heap%20is%20empty%22%29%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20root%20with%20the%20rightmost%20leaf%20on%20the%20last%20level%20%28swap%20the%20first%20and%20last%20elements%29%0A%20%20%20%20%20%20%20%20self.swap%280,%20self.size%28%29%20-%201%29%0A%20%20%20%20%20%20%20%20%23%20Remove%20a%20node%0A%20%20%20%20%20%20%20%20val%20%3D%20self.max_heap.pop%28%29%0A%20%20%20%20%20%20%20%20%23%20Sift%20down%0A%20%20%20%20%20%20%20%20self.sift_down%280%29%0A%20%20%20%20%20%20%20%20%23%20Return%20the%20top%20element%20of%20the%20heap%0A%20%20%20%20%20%20%20%20return%20val%0A%0A%20%20%20%20def%20sift_down%28self,%20i%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Sift%20down%20starting%20from%20node%20i%22%22%22%0A%20%20%20%20%20%20%20%20while%20True%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Find%20the%20node%20with%20the%20largest%20value%20among%20i,%20l,%20and%20r,%20and%20store%20its%20index%20in%20ma%0A%20%20%20%20%20%20%20%20%20%20%20%20l,%20r,%20ma%20%3D%20self.left%28i%29,%20self.right%28i%29,%20i%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20l%20%3C%20self.size%28%29%20and%20self.max_heap%5Bl%5D%20%3E%20self.max_heap%5Bma%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20ma%20%3D%20l%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20r%20%3C%20self.size%28%29%20and%20self.max_heap%5Br%5D%20%3E%20self.max_heap%5Bma%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20ma%20%3D%20r%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Stop%20if%20node%20i%20is%20largest%20or%20both%20child%20indices%20l%20and%20r%20are%20out%20of%20bounds%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20ma%20%3D%3D%20i%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Swap%20the%20two%20nodes%0A%20%20%20%20%20%20%20%20%20%20%20%20self.swap%28i,%20ma%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Continue%20sifting%20down%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20ma%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20max%20heap%0A%20%20%20%20%23%20Note%20that%20the%20input%20array%20is%20already%20a%20valid%20heap%0A%20%20%20%20max_heap%20%3D%20MaxHeap%28%5B9,%208,%207,%206,%207,%206,%202,%201,%204,%203,%206,%202,%205%5D%29%0A%0A%20%20%20%20%23%20Pop%20the%20top%20element%20from%20the%20heap%0A%20%20%20%20peek%20%3D%20max_heap.pop%28%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=8&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="813-common-applications">8.1.3 &nbsp; Common applications<a class="headerlink" href="#813-common-applications" title="Permanent link">&para;</a></h2>
 <ul>
-<li><strong>Priority queue</strong>: Heaps are typically the preferred data structure for implementing priority queues. The time complexity of both enqueue and dequeue operations is <span class="arithmatex">\(O(\log n)\)</span>, and heap construction has a time complexity of <span class="arithmatex">\(O(n)\)</span>, making these operations highly efficient.</li>
-<li><strong>Heap sort</strong>: Given a set of data, we can build a heap with them and then continuously perform element removal operations to obtain sorted data. However, we usually use a more elegant approach to implement heap sort, as detailed in the "Heap Sort" chapter.</li>
-<li><strong>Getting the largest <span class="arithmatex">\(k\)</span> elements</strong>: This is a classic algorithm problem and also a typical application, such as selecting the top 10 trending news items for Weibo Hot Search or the top 10 best-selling products.</li>
+<li><strong>Priority queues</strong>: Heaps are usually the first choice for implementing priority queues. Enqueueing and dequeueing both take <span class="arithmatex">\(O(\log n)\)</span> time, and building a heap takes <span class="arithmatex">\(O(n)\)</span>.</li>
+<li><strong>Heap sort</strong>: Build a heap from a dataset, then repeatedly remove its root to obtain sorted data. In practice, a more elegant implementation is usually used; see the heap-sort section.</li>
+<li><strong>Finding the largest <span class="arithmatex">\(k\)</span> elements</strong>: This is both a classic algorithm problem and a common application, such as selecting the 10 hottest news stories for Weibo's trending list or the 10 best-selling products.</li>
 </ul>
 
 <!-- Source file information -->
@@ -6462,7 +6549,7 @@ aria-label="Footer"
   <a
     href="../"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: Chapter 8. &amp;nbsp; Heap"
+    aria-label="Previous: Chapter 8.   Heaps"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -6474,7 +6561,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        Chapter 8. &nbsp; Heap
+        Chapter 8. &nbsp; Heaps
       </div>
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   </a>
@@ -6486,7 +6573,7 @@ aria-label="Footer"
   <a
     href="../build_heap/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 8.2 Heap Construction Operation"
+    aria-label="Next: 8.2   Building a heap"
     rel="next"
   >
     <div class="md-footer__title">
@@ -6494,7 +6581,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        8.2 Heap Construction Operation
+        8.2 &nbsp; Building a heap
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -6511,7 +6598,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_heap/index.html b/en/chapter_heap/index.html
index 825604035..23b86a179 100644
--- a/en/chapter_heap/index.html
+++ b/en/chapter_heap/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>Chapter 8. Heap - Hello Algo</title>
+        <title>Chapter 8. Heaps - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
     <div data-md-component="skip">
 
 
-        <a href="#chapter-8-heap" class="md-skip">
+        <a href="#chapter-8-heaps" class="md-skip">
           Skip to content
         </a>
 
@@ -119,7 +119,7 @@
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             d="M480 32c0-12.9-7.8-24.6-19.8-29.6s-25.7-2.2-34.9 6.9L381.7 53c-48 48-113.1 75-181 75H192 160 64c-35.3 0-64 28.7-64 64v96c0 35.3 28.7 64 64 64l0 128c0 17.7 14.3 32 32 32h64c17.7 0 32-14.3 32-32V352l8.7 0c67.9 0 133 27 181 75l43.6 43.6c9.2 9.2 22.9 11.9 34.9 6.9s19.8-16.6 19.8-29.6V300.4c18.6-8.8 32-32.5 32-60.4s-13.4-51.6-32-60.4V32zm-64 76.7V240 371.3C357.2 317.8 280.5 288 200.7 288H192V192h8.7c79.8 0 156.5-29.8 215.3-83.3z" />
     </svg>
-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              Chapter 8. &nbsp; Heap
+              Chapter 8. &nbsp; Heaps
 
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         </div>
@@ -167,9 +167,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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@@ -177,9 +177,9 @@
 
 
 
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@@ -349,7 +349,7 @@
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-  Before Starting
+  Foreword
 
 
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-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
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   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
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   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
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   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
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   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
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   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2289,7 +2289,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2380,7 +2380,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
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@@ -2422,7 +2422,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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@@ -2449,7 +2449,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2476,7 +2476,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2503,7 +2503,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2530,7 +2530,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2598,7 +2598,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2640,7 +2640,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2694,7 +2694,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2721,7 +2721,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2748,7 +2748,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2775,7 +2775,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2802,7 +2802,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2880,7 +2880,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2922,7 +2922,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2976,7 +2976,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3003,7 +3003,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3030,7 +3030,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3057,7 +3057,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3084,7 +3084,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3111,7 +3111,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3138,7 +3138,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3165,7 +3165,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3192,7 +3192,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3219,7 +3219,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3285,7 +3285,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3306,7 +3306,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3327,7 +3327,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3381,7 +3381,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3408,7 +3408,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3435,7 +3435,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3462,7 +3462,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3528,7 +3528,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3570,7 +3570,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3597,7 +3597,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3624,7 +3624,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3651,7 +3651,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3678,7 +3678,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3705,7 +3705,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3775,7 +3775,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3796,7 +3796,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3817,7 +3817,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3844,7 +3844,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3871,7 +3871,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3898,7 +3898,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -3925,7 +3925,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -3952,7 +3952,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -3979,7 +3979,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4006,7 +4006,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4072,7 +4072,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4093,7 +4093,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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 <!-- Page content -->
-<h1 id="chapter-8-heap">Chapter 8. &nbsp; Heap<a class="headerlink" href="#chapter-8-heap" title="Permanent link">&para;</a></h1>
-<p><img alt="Heap" class="cover-image" src="../assets/covers/chapter_heap.jpg" /></p>
+<h1 id="chapter-8-heaps">Chapter 8. &nbsp; Heaps<a class="headerlink" href="#chapter-8-heaps" title="Permanent link">&para;</a></h1>
+<p><img alt="Heaps" class="cover-image" src="../assets/covers/chapter_heap.jpg" /></p>
 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
-<p>Heaps are like mountain peaks, rising layer upon layer, each with a distinct shape.</p>
-<p>The peaks rise and fall at varying heights, yet the tallest peak always catches the eye first.</p>
+<p>Heaps resemble mountain ranges, rising in layers of varied shapes.</p>
+<p>The peaks stand at different heights, yet the highest always catches the eye first.</p>
 </div>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
-<li><a href="heap/">8.1 &nbsp; Heap</a></li>
-<li><a href="build_heap/">8.2 &nbsp; Heap Construction Operation</a></li>
-<li><a href="top_k/">8.3 &nbsp; Top-k Problem</a></li>
+<li><a href="heap/">8.1 &nbsp; Heaps</a></li>
+<li><a href="build_heap/">8.2 &nbsp; Building a heap</a></li>
+<li><a href="top_k/">8.3 &nbsp; The top-k problem</a></li>
 <li><a href="summary/">8.4 &nbsp; Summary</a></li>
 <li><a href="exercises/">8.5 &nbsp; Exercises</a></li>
 </ul>
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   <a
     href="../chapter_tree/exercises/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 7.7 Exercises"
+    aria-label="Previous: 7.7   Exercises"
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@@ -4626,7 +4698,7 @@ aria-label="Footer"
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-        7.7 Exercises
+        7.7 &nbsp; Exercises
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   <a
     href="./heap/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 8.1 Heap"
+    aria-label="Next: 8.1   Heaps"
     rel="next"
   >
     <div class="md-footer__title">
@@ -4646,7 +4718,7 @@ aria-label="Footer"
         Next
       </span>
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-        8.1 Heap
+        8.1 &nbsp; Heaps
       </div>
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     <div class="md-footer__button md-icon">
@@ -4663,7 +4735,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-  2.6 Exercises
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-  7.5 AVL Tree *
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+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
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+  8.2 &nbsp; Building a heap
 
 
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+  8.4 &nbsp; Summary
 
 
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+  <a href="#1-key-takeaways" class="md-nav__link">
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+        1. &nbsp; Key takeaways
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-  8.5 Exercises
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-  9.2 Basic Operations on Graphs
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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+  Chapter 11. &nbsp; Sorting
 
 
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+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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+  11.10 &nbsp; Radix sort
 
 
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-  11.12 Exercises
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+  12.3 &nbsp; Building a binary tree
 
 
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+  13.2 &nbsp; Permutations problem
 
 
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+  13.4 &nbsp; N-queens problem
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
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+  15.6 &nbsp; Exercises
 
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
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-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
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   </a>
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 <!-- Page content -->
 <h1 id="84-summary">8.4 &nbsp; Summary<a class="headerlink" href="#84-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
 <ul>
-<li>A heap is a complete binary tree. Depending on the property it satisfies, it can be classified as either a max heap or a min heap. The top element of a max heap (min heap) is the largest (smallest) element.</li>
-<li>A priority queue is a queue in which elements are dequeued according to priority, and it is typically implemented using a heap.</li>
-<li>Common heap operations and their corresponding time complexities include inserting an element <span class="arithmatex">\(O(\log n)\)</span>, removing the top element <span class="arithmatex">\(O(\log n)\)</span>, and accessing the top element <span class="arithmatex">\(O(1)\)</span>.</li>
-<li>Complete binary trees are well-suited for array representation, so we typically use arrays to store heaps.</li>
-<li>Heapify operations are used to maintain the heap property and are employed in both element insertion and removal operations.</li>
-<li>Building a heap from <span class="arithmatex">\(n\)</span> input elements can be optimized to <span class="arithmatex">\(O(n)\)</span>, which is highly efficient.</li>
-<li>Top-k is a classic algorithmic problem that can be solved efficiently using a heap, with a time complexity of <span class="arithmatex">\(O(n \log k)\)</span>.</li>
+<li>A heap is a complete binary tree. Depending on its ordering condition, it is either a max heap or a min heap. The root holds the largest or smallest element, respectively.</li>
+<li>A priority queue removes elements according to their priority and is usually implemented with a heap.</li>
+<li>Common heap operations and their time complexities include insertion in <span class="arithmatex">\(O(\log n)\)</span>, removing the root in <span class="arithmatex">\(O(\log n)\)</span>, and accessing the root in <span class="arithmatex">\(O(1)\)</span>.</li>
+<li>Complete binary trees are well suited to array representation, so heaps are usually stored in arrays.</li>
+<li>Sifting restores the heap property and is used during both insertion and removal.</li>
+<li>Building a heap from <span class="arithmatex">\(n\)</span> input elements can be optimized to <span class="arithmatex">\(O(n)\)</span> time.</li>
+<li>Top-k is a classic algorithm problem that a heap can solve efficiently in <span class="arithmatex">\(O(n \log k)\)</span> time.</li>
 </ul>
 <h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
-<p><strong>Q</strong>: Does the term "heap" in data structures mean the same thing as "heap" in memory management?</p>
-<p>They are not the same concept; they simply share the same name. In computer systems, the heap is part of dynamic memory allocation, and programs can use it to store data at runtime. A program can request a certain amount of heap memory to store complex structures such as objects and arrays. When the data is no longer needed, the program must release that memory to prevent memory leaks. Compared with stack memory, heap memory requires more careful management and use; improper handling can lead to problems such as memory leaks and dangling pointers.</p>
+<p><strong>Q</strong>: Is the heap data structure the same concept as the heap in memory management?</p>
+<p>No; they simply share a name. In computer memory, the heap is used for dynamic memory allocation, allowing a running program to store data. A program can request heap memory for structures such as objects and arrays. When that data is no longer needed, the memory must be released to prevent leaks. Compared with stack memory, heap memory requires more careful management; misuse can lead to memory leaks, dangling pointers, and other problems.</p>
 
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@@ -4693,7 +4765,7 @@ aria-label="Footer"
   <a
     href="../top_k/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 8.3 Top-k Problem"
+    aria-label="Previous: 8.3   The top-k problem"
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@@ -4705,7 +4777,7 @@ aria-label="Footer"
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+        8.3 &nbsp; The top-k problem
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   <a
     href="../exercises/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 8.5 Exercises"
+    aria-label="Next: 8.5   Exercises"
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-        8.5 Exercises
+        8.5 &nbsp; Exercises
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@@ -4742,7 +4814,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>8.3 Top-k Problem - Hello Algo</title>
+        <title>8.3 The top-k problem - Hello Algo</title>
 
 
 
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+        <a href="#83-the-top-k-problem" class="md-skip">
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         </a>
 
@@ -119,7 +119,7 @@
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-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              8.3 &nbsp; Top-k Problem
+              8.3 &nbsp; The top-k problem
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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       </label>
 
@@ -177,9 +177,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Light mode"  type="radio" name="__palette" id="__palette_1">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2166,7 +2166,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2187,7 +2187,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2208,7 +2208,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2235,7 +2235,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2271,7 +2271,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2288,7 +2288,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2314,10 +2314,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#831-method-1-iterative-selection" class="md-nav__link">
+  <a href="#831-method-1-repeated-selection" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.3.1 &nbsp; Method 1: Iterative Selection
+        8.3.1 &nbsp; Method 1: Repeated selection
       </span>
     </span>
   </a>
@@ -2336,10 +2336,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#833-method-3-heap" class="md-nav__link">
+  <a href="#833-method-3-a-heap" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.3.3 &nbsp; Method 3: Heap
+        8.3.3 &nbsp; Method 3: A heap
       </span>
     </span>
   </a>
@@ -2369,7 +2369,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2396,7 +2396,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2460,7 +2460,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2481,7 +2481,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2502,7 +2502,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2529,7 +2529,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2556,7 +2556,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2583,7 +2583,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2610,7 +2610,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2678,7 +2678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2699,7 +2699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2720,7 +2720,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2747,7 +2747,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2774,7 +2774,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2801,7 +2801,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2828,7 +2828,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2855,7 +2855,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2882,7 +2882,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2960,7 +2960,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2981,7 +2981,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3002,7 +3002,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3029,7 +3029,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3056,7 +3056,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3083,7 +3083,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3110,7 +3110,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3137,7 +3137,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3164,7 +3164,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3191,7 +3191,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3218,7 +3218,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3245,7 +3245,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3272,7 +3272,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3299,7 +3299,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3365,7 +3365,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3386,7 +3386,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3407,7 +3407,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3434,7 +3434,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3461,7 +3461,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3488,7 +3488,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3515,7 +3515,7 @@
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3542,7 +3542,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3608,7 +3608,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3629,7 +3629,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3650,7 +3650,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3677,7 +3677,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3704,7 +3704,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3731,7 +3731,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3758,7 +3758,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3855,7 +3855,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3876,7 +3876,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3897,7 +3897,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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@@ -3951,7 +3951,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3978,7 +3978,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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@@ -4005,7 +4005,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4032,7 +4032,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4152,7 +4152,7 @@
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4173,7 +4173,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
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@@ -4610,10 +4682,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#831-method-1-iterative-selection" class="md-nav__link">
+  <a href="#831-method-1-repeated-selection" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.3.1 &nbsp; Method 1: Iterative Selection
+        8.3.1 &nbsp; Method 1: Repeated selection
       </span>
     </span>
   </a>
@@ -4632,10 +4704,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#833-method-3-heap" class="md-nav__link">
+  <a href="#833-method-3-a-heap" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        8.3.3 &nbsp; Method 3: Heap
+        8.3.3 &nbsp; Method 3: A heap
       </span>
     </span>
   </a>
@@ -4679,40 +4751,40 @@
 
 
 <!-- Page content -->
-<h1 id="83-top-k-problem">8.3 &nbsp; Top-k Problem<a class="headerlink" href="#83-top-k-problem" title="Permanent link">&para;</a></h1>
+<h1 id="83-the-top-k-problem">8.3 &nbsp; The top-k problem<a class="headerlink" href="#83-the-top-k-problem" title="Permanent link">&para;</a></h1>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given an unordered array <code>nums</code> of length <span class="arithmatex">\(n\)</span>, return the largest <span class="arithmatex">\(k\)</span> elements in the array.</p>
+<p>Given an unsorted array <code>nums</code> of length <span class="arithmatex">\(n\)</span>, return its largest <span class="arithmatex">\(k\)</span> elements.</p>
 </div>
-<p>For this problem, we will first introduce two relatively straightforward solutions, followed by a more efficient heap-based solution.</p>
-<h2 id="831-method-1-iterative-selection">8.3.1 &nbsp; Method 1: Iterative Selection<a class="headerlink" href="#831-method-1-iterative-selection" title="Permanent link">&para;</a></h2>
-<p>We can perform <span class="arithmatex">\(k\)</span> rounds of traversal as shown in Figure 8-6, extracting the <span class="arithmatex">\(1^{st}\)</span>, <span class="arithmatex">\(2^{nd}\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(k^{th}\)</span> largest elements in each round, with a time complexity of <span class="arithmatex">\(O(nk)\)</span>.</p>
-<p>This method is only suitable when <span class="arithmatex">\(k \ll n\)</span>, because when <span class="arithmatex">\(k\)</span> is close to <span class="arithmatex">\(n\)</span>, the time complexity approaches <span class="arithmatex">\(O(n^2)\)</span>, making it very inefficient.</p>
-<p><img alt="Traversing to find the largest k elements" class="animation-figure" src="../top_k.assets/top_k_traversal.png" /></p>
-<p align="center"> Figure 8-6 &nbsp; Traversing to find the largest k elements </p>
+<p>We first introduce two straightforward solutions, then a more efficient solution using a heap.</p>
+<h2 id="831-method-1-repeated-selection">8.3.1 &nbsp; Method 1: Repeated selection<a class="headerlink" href="#831-method-1-repeated-selection" title="Permanent link">&para;</a></h2>
+<p>As shown below, scan the array <span class="arithmatex">\(k\)</span> times, selecting the <span class="arithmatex">\(1\)</span>st, <span class="arithmatex">\(2\)</span>nd, <span class="arithmatex">\(\dots\)</span>, and <span class="arithmatex">\(k\)</span>th largest elements in successive rounds. This takes <span class="arithmatex">\(O(nk)\)</span> time.</p>
+<p>This method is suitable only when <span class="arithmatex">\(k \ll n\)</span>. As <span class="arithmatex">\(k\)</span> approaches <span class="arithmatex">\(n\)</span>, its time complexity approaches <span class="arithmatex">\(O(n^2)\)</span>, making it very slow.</p>
+<p><img alt="Finding the largest k elements by repeated scans" class="animation-figure" src="../top_k.assets/top_k_traversal.png" /></p>
+<p align="center"> Figure 8-6 &nbsp; Finding the largest k elements by repeated scans </p>
 
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>When <span class="arithmatex">\(k = n\)</span>, we can obtain a complete sorted sequence, which is equivalent to the "selection sort" algorithm.</p>
+<p>When <span class="arithmatex">\(k = n\)</span>, this produces a fully sorted sequence and is equivalent to selection sort.</p>
 </div>
 <h2 id="832-method-2-sorting">8.3.2 &nbsp; Method 2: Sorting<a class="headerlink" href="#832-method-2-sorting" title="Permanent link">&para;</a></h2>
-<p>As shown in Figure 8-7, we can first sort the array <code>nums</code>, then return the rightmost <span class="arithmatex">\(k\)</span> elements, with a time complexity of <span class="arithmatex">\(O(n \log n)\)</span>.</p>
-<p>Clearly, this method does more work than necessary, because we only need to find the largest <span class="arithmatex">\(k\)</span> elements rather than sort the other elements.</p>
-<p><img alt="Sorting to find the largest k elements" class="animation-figure" src="../top_k.assets/top_k_sorting.png" /></p>
-<p align="center"> Figure 8-7 &nbsp; Sorting to find the largest k elements </p>
+<p>As shown below, sort <code>nums</code> and return its rightmost <span class="arithmatex">\(k\)</span> elements. This takes <span class="arithmatex">\(O(n \log n)\)</span> time.</p>
+<p>This method does more work than necessary: we need only the largest <span class="arithmatex">\(k\)</span> elements, not a sorted order for all the others.</p>
+<p><img alt="Finding the largest k elements by sorting" class="animation-figure" src="../top_k.assets/top_k_sorting.png" /></p>
+<p align="center"> Figure 8-7 &nbsp; Finding the largest k elements by sorting </p>
 
-<h2 id="833-method-3-heap">8.3.3 &nbsp; Method 3: Heap<a class="headerlink" href="#833-method-3-heap" title="Permanent link">&para;</a></h2>
-<p>We can solve the Top-k problem more efficiently with a heap, as shown in Figure 8-8.</p>
+<h2 id="833-method-3-a-heap">8.3.3 &nbsp; Method 3: A heap<a class="headerlink" href="#833-method-3-a-heap" title="Permanent link">&para;</a></h2>
+<p>A heap solves the top-k problem more efficiently, following the steps below.</p>
 <ol>
-<li>Initialize a min heap, where the heap top element is the smallest.</li>
-<li>First, insert the first <span class="arithmatex">\(k\)</span> elements of the array into the heap in sequence.</li>
-<li>Starting from the <span class="arithmatex">\((k + 1)^{th}\)</span> element, if the current element is greater than the heap top element, remove the heap top element and insert the current element into the heap.</li>
-<li>After traversal is complete, the heap contains the largest <span class="arithmatex">\(k\)</span> elements.</li>
+<li>Initialize a min heap, whose root holds its smallest element.</li>
+<li>Insert the first <span class="arithmatex">\(k\)</span> array elements into the heap one by one.</li>
+<li>Starting with element <span class="arithmatex">\(k + 1\)</span>, if the current element is larger than the root, remove the root and insert the current element.</li>
+<li>After the traversal, the heap contains the largest <span class="arithmatex">\(k\)</span> elements.</li>
 </ol>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:9"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label><label for="__tabbed_1_6">&lt;6&gt;</label><label for="__tabbed_1_7">&lt;7&gt;</label><label for="__tabbed_1_8">&lt;8&gt;</label><label for="__tabbed_1_9">&lt;9&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Finding the largest k elements using a heap" class="animation-figure" src="../top_k.assets/top_k_heap_step1.png" /></p>
+<p><img alt="Finding the largest k elements with a heap" class="animation-figure" src="../top_k.assets/top_k_heap_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="top_k_heap_step2" class="animation-figure" src="../top_k.assets/top_k_heap_step2.png" /></p>
@@ -4740,22 +4812,22 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 8-8 &nbsp; Finding the largest k elements using a heap </p>
+<p align="center"> Figure 8-8 &nbsp; Finding the largest k elements with a heap </p>
 
-<p>Example code is as follows:</p>
+<p>An implementation follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">top_k.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">top_k_heap</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">k</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Find the largest k elements in array based on heap"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Initialize min heap</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Use a heap to find the k largest elements in an array"""</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Initialize a min heap</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="n">heap</span> <span class="o">=</span> <span class="p">[]</span>
-<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># Enter the first k elements of array into heap</span>
+<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># Push the first k array elements onto the heap</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="n">heapq</span><span class="o">.</span><span class="n">heappush</span><span class="p">(</span><span class="n">heap</span><span class="p">,</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="c1"># Starting from the (k+1)th element, maintain heap length as k</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="c1"># Starting with element k+1, keep the heap size at k</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)):</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="c1"># If current element is greater than top element, top element exits heap, current element enters heap</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="c1"># If the current element is larger than the heap top, pop the top element and push the current element</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">heap</span><span class="p">[</span><span class="mi">0</span><span class="p">]:</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>            <span class="n">heapq</span><span class="o">.</span><span class="n">heappop</span><span class="p">(</span><span class="n">heap</span><span class="p">)</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>            <span class="n">heapq</span><span class="o">.</span><span class="n">heappush</span><span class="p">(</span><span class="n">heap</span><span class="p">,</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
@@ -4763,17 +4835,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">top_k.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Find the largest k elements in array based on heap */</span>
+<div class="highlight"><span class="filename">top_k.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Use a heap to find the k largest elements in an array */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="n">priority_queue</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">greater</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">topKHeap</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Python's heapq module implements min heap by default</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Initialize a min heap</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="n">priority_queue</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">greater</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">heap</span><span class="p">;</span>
-<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="c1">// Enter the first k elements of array into heap</span>
+<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="c1">// Push the first k array elements onto the heap</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">        </span><span class="n">heap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="c1">// Starting from the (k+1)th element, maintain heap length as k</span>
+<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="c1">// Starting with element k+1, keep the heap size at k</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="c1">// If current element is greater than top element, top element exits heap, current element enters heap</span>
+<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="c1">// If the current element is larger than the heap top, pop the top element and push the current element</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="n">top</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">            </span><span class="n">heap</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">            </span><span class="n">heap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
@@ -4784,17 +4856,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">top_k.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Find the largest k elements in array based on heap */</span>
+<div class="highlight"><span class="filename">top_k.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Use a heap to find the k largest elements in an array */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="n">Queue</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="nf">topKHeap</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Python's heapq module implements min heap by default</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Initialize a min heap</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="n">Queue</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">heap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">PriorityQueue</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="p">();</span>
-<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="c1">// Enter the first k elements of array into heap</span>
+<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="c1">// Push the first k array elements onto the heap</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">        </span><span class="n">heap</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">);</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="c1">// Starting from the (k+1)th element, maintain heap length as k</span>
+<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="c1">// Starting with element k+1, keep the heap size at k</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="c1">// If current element is greater than top element, top element exits heap, current element enters heap</span>
+<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="c1">// If the current element is larger than the heap top, pop the top element and push the current element</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="na">peek</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">            </span><span class="n">heap</span><span class="p">.</span><span class="na">poll</span><span class="p">();</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">            </span><span class="n">heap</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">);</span>
@@ -4805,17 +4877,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">top_k.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Find the largest k elements in array based on heap */</span>
+<div class="highlight"><span class="filename">top_k.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Use a heap to find the k largest elements in an array */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="n">PriorityQueue</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">TopKHeap</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Python's heapq module implements min heap by default</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Initialize a min heap</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="n">PriorityQueue</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">heap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">();</span>
-<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="c1">// Enter the first k elements of array into heap</span>
+<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="c1">// Push the first k array elements onto the heap</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">        </span><span class="n">heap</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="c1">// Starting from the (k+1)th element, maintain heap length as k</span>
+<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="c1">// Starting with element k+1, keep the heap size at k</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="c1">// If current element is greater than top element, top element exits heap, current element enters heap</span>
+<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="c1">// If the current element is larger than the heap top, pop the top element and push the current element</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="n">Peek</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">            </span><span class="n">heap</span><span class="p">.</span><span class="n">Dequeue</span><span class="p">();</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">            </span><span class="n">heap</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
@@ -4826,18 +4898,18 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">top_k.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Find the largest k elements in array based on heap */</span>
+<div class="highlight"><span class="filename">top_k.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Use a heap to find the k largest elements in an array */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">topKHeap</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">k</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">minHeap</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Python's heapq module implements min heap by default</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Initialize a min heap</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">h</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">minHeap</span><span class="p">{}</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="nx">heap</span><span class="p">.</span><span class="nx">Init</span><span class="p">(</span><span class="nx">h</span><span class="p">)</span>
-<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="c1">// Enter the first k elements of array into heap</span>
+<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="c1">// Push the first k array elements onto the heap</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">k</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">        </span><span class="nx">heap</span><span class="p">.</span><span class="nx">Push</span><span class="p">(</span><span class="nx">h</span><span class="p">,</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">])</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="c1">// Starting from the (k+1)th element, maintain heap length as k</span>
+<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="c1">// Starting with element k+1, keep the heap size at k</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">k</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="c1">// If current element is greater than top element, top element exits heap, current element enters heap</span>
+<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="c1">// If the current element is larger than the heap top, pop the top element and push the current element</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">h</span><span class="p">.</span><span class="nx">Top</span><span class="p">().(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">            </span><span class="nx">heap</span><span class="p">.</span><span class="nx">Pop</span><span class="p">(</span><span class="nx">h</span><span class="p">)</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">            </span><span class="nx">heap</span><span class="p">.</span><span class="nx">Push</span><span class="p">(</span><span class="nx">h</span><span class="p">,</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">])</span>
@@ -4848,13 +4920,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">top_k.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Find the largest k elements in array based on heap */</span>
+<div class="highlight"><span class="filename">top_k.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Use a heap to find the k largest elements in an array */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">topKHeap</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">k</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Initialize min heap and build heap with first k elements</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Build a heap from the first k elements and use it as a min heap</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">heap</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Heap</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="kr">prefix</span><span class="p">(</span><span class="n">k</span><span class="p">))</span>
-<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="c1">// Starting from the (k+1)th element, maintain heap length as k</span>
+<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="c1">// Starting with element k+1, keep the heap size at k</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="bp">dropFirst</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">        </span><span class="c1">// If current element is greater than top element, top element exits heap, current element enters heap</span>
+<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">        </span><span class="c1">// If the current element exceeds the heap minimum, remove the minimum and insert the current element</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="bp">min</span><span class="p">()</span><span class="o">!</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">            </span><span class="kc">_</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="n">removeMin</span><span class="p">()</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">            </span><span class="n">heap</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
@@ -4865,107 +4937,107 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">top_k.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Element enters heap */</span>
+<div class="highlight"><span class="filename">top_k.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Push an element onto the heap */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">pushMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Negate element</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Negate the value</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="nx">maxHeap</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="o">-</span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="p">}</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a>
-<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="cm">/* Element exits heap */</span>
+<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="cm">/* Pop an element from the heap */</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="kd">function</span><span class="w"> </span><span class="nx">popMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="c1">// Negate element</span>
+<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="c1">// Negate the value</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="nx">maxHeap</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="p">}</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a>
-<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="cm">/* Access top element */</span>
+<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="cm">/* Access the top element of the heap */</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="kd">function</span><span class="w"> </span><span class="nx">peekMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">    </span><span class="c1">// Negate element</span>
+<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">    </span><span class="c1">// Negate the value</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="nx">maxHeap</span><span class="p">.</span><span class="nx">peek</span><span class="p">();</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="p">}</span>
 <a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a>
-<a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="cm">/* Extract elements from heap */</span>
+<a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="cm">/* Get the elements in the heap */</span>
 <a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="kd">function</span><span class="w"> </span><span class="nx">getMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">    </span><span class="c1">// Negate element</span>
+<a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">    </span><span class="c1">// Negate the value</span>
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">maxHeap</span><span class="p">.</span><span class="nx">getMaxHeap</span><span class="p">().</span><span class="nx">map</span><span class="p">((</span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="o">-</span><span class="nx">num</span><span class="p">);</span>
 <a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="p">}</span>
 <a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a>
-<a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="cm">/* Find the largest k elements in array based on heap */</span>
+<a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="cm">/* Use a heap to find the k largest elements in an array */</span>
 <a id="__codelineno-6-26" name="__codelineno-6-26" href="#__codelineno-6-26"></a><span class="kd">function</span><span class="w"> </span><span class="nx">topKHeap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-27" name="__codelineno-6-27" href="#__codelineno-6-27"></a><span class="w">    </span><span class="c1">// Python's heapq module implements min heap by default</span>
-<a id="__codelineno-6-28" name="__codelineno-6-28" href="#__codelineno-6-28"></a><span class="w">    </span><span class="c1">// Note: We negate all heap elements to simulate min heap using max heap</span>
+<a id="__codelineno-6-27" name="__codelineno-6-27" href="#__codelineno-6-27"></a><span class="w">    </span><span class="c1">// Initialize a min heap</span>
+<a id="__codelineno-6-28" name="__codelineno-6-28" href="#__codelineno-6-28"></a><span class="w">    </span><span class="c1">// Note: negate all values in the heap to simulate a min heap with a max heap</span>
 <a id="__codelineno-6-29" name="__codelineno-6-29" href="#__codelineno-6-29"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">MaxHeap</span><span class="p">([]);</span>
-<a id="__codelineno-6-30" name="__codelineno-6-30" href="#__codelineno-6-30"></a><span class="w">    </span><span class="c1">// Enter the first k elements of array into heap</span>
+<a id="__codelineno-6-30" name="__codelineno-6-30" href="#__codelineno-6-30"></a><span class="w">    </span><span class="c1">// Push the first k array elements onto the heap</span>
 <a id="__codelineno-6-31" name="__codelineno-6-31" href="#__codelineno-6-31"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">k</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-32" name="__codelineno-6-32" href="#__codelineno-6-32"></a><span class="w">        </span><span class="nx">pushMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]);</span>
 <a id="__codelineno-6-33" name="__codelineno-6-33" href="#__codelineno-6-33"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-34" name="__codelineno-6-34" href="#__codelineno-6-34"></a><span class="w">    </span><span class="c1">// Starting from the (k+1)th element, maintain heap length as k</span>
+<a id="__codelineno-6-34" name="__codelineno-6-34" href="#__codelineno-6-34"></a><span class="w">    </span><span class="c1">// Starting with element k+1, keep the heap size at k</span>
 <a id="__codelineno-6-35" name="__codelineno-6-35" href="#__codelineno-6-35"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">k</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-36" name="__codelineno-6-36" href="#__codelineno-6-36"></a><span class="w">        </span><span class="c1">// If current element is greater than top element, top element exits heap, current element enters heap</span>
+<a id="__codelineno-6-36" name="__codelineno-6-36" href="#__codelineno-6-36"></a><span class="w">        </span><span class="c1">// If the current element is larger than the heap top, pop the top element and push the current element</span>
 <a id="__codelineno-6-37" name="__codelineno-6-37" href="#__codelineno-6-37"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">peekMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-38" name="__codelineno-6-38" href="#__codelineno-6-38"></a><span class="w">            </span><span class="nx">popMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">);</span>
 <a id="__codelineno-6-39" name="__codelineno-6-39" href="#__codelineno-6-39"></a><span class="w">            </span><span class="nx">pushMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]);</span>
 <a id="__codelineno-6-40" name="__codelineno-6-40" href="#__codelineno-6-40"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-6-41" name="__codelineno-6-41" href="#__codelineno-6-41"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-42" name="__codelineno-6-42" href="#__codelineno-6-42"></a><span class="w">    </span><span class="c1">// Return elements in heap</span>
+<a id="__codelineno-6-42" name="__codelineno-6-42" href="#__codelineno-6-42"></a><span class="w">    </span><span class="c1">// Return the elements in the heap</span>
 <a id="__codelineno-6-43" name="__codelineno-6-43" href="#__codelineno-6-43"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">getMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">);</span>
 <a id="__codelineno-6-44" name="__codelineno-6-44" href="#__codelineno-6-44"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">top_k.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Element enters heap */</span>
+<div class="highlight"><span class="filename">top_k.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Push an element onto the heap */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">pushMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="o">:</span><span class="w"> </span><span class="kt">MaxHeap</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Negate element</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Negate the value</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="nx">maxHeap</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="o">-</span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="p">}</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a>
-<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="cm">/* Element exits heap */</span>
+<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="cm">/* Pop an element from the heap */</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="kd">function</span><span class="w"> </span><span class="nx">popMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="o">:</span><span class="w"> </span><span class="kt">MaxHeap</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// Negate element</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// Negate the value</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="nx">maxHeap</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="p">}</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a>
-<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="cm">/* Access top element */</span>
+<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="cm">/* Access the top element of the heap */</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="kd">function</span><span class="w"> </span><span class="nx">peekMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="o">:</span><span class="w"> </span><span class="kt">MaxHeap</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="c1">// Negate element</span>
+<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="c1">// Negate the value</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="nx">maxHeap</span><span class="p">.</span><span class="nx">peek</span><span class="p">();</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="p">}</span>
 <a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a>
-<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="cm">/* Extract elements from heap */</span>
+<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="cm">/* Get the elements in the heap */</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="kd">function</span><span class="w"> </span><span class="nx">getMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="o">:</span><span class="w"> </span><span class="kt">MaxHeap</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">    </span><span class="c1">// Negate element</span>
+<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">    </span><span class="c1">// Negate the value</span>
 <a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">maxHeap</span><span class="p">.</span><span class="nx">getMaxHeap</span><span class="p">().</span><span class="nx">map</span><span class="p">((</span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="o">-</span><span class="nx">num</span><span class="p">);</span>
 <a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="p">}</span>
 <a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a>
-<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="cm">/* Find the largest k elements in array based on heap */</span>
+<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="cm">/* Use a heap to find the k largest elements in an array */</span>
 <a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="kd">function</span><span class="w"> </span><span class="nx">topKHeap</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">k</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="w">    </span><span class="c1">// Python's heapq module implements min heap by default</span>
-<a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w">    </span><span class="c1">// Note: We negate all heap elements to simulate min heap using max heap</span>
+<a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="w">    </span><span class="c1">// Initialize a min heap</span>
+<a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w">    </span><span class="c1">// Note: negate all values in the heap to simulate a min heap with a max heap</span>
 <a id="__codelineno-7-29" name="__codelineno-7-29" href="#__codelineno-7-29"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">MaxHeap</span><span class="p">([]);</span>
-<a id="__codelineno-7-30" name="__codelineno-7-30" href="#__codelineno-7-30"></a><span class="w">    </span><span class="c1">// Enter the first k elements of array into heap</span>
+<a id="__codelineno-7-30" name="__codelineno-7-30" href="#__codelineno-7-30"></a><span class="w">    </span><span class="c1">// Push the first k array elements onto the heap</span>
 <a id="__codelineno-7-31" name="__codelineno-7-31" href="#__codelineno-7-31"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">k</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-32" name="__codelineno-7-32" href="#__codelineno-7-32"></a><span class="w">        </span><span class="nx">pushMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]);</span>
 <a id="__codelineno-7-33" name="__codelineno-7-33" href="#__codelineno-7-33"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-34" name="__codelineno-7-34" href="#__codelineno-7-34"></a><span class="w">    </span><span class="c1">// Starting from the (k+1)th element, maintain heap length as k</span>
+<a id="__codelineno-7-34" name="__codelineno-7-34" href="#__codelineno-7-34"></a><span class="w">    </span><span class="c1">// Starting with element k+1, keep the heap size at k</span>
 <a id="__codelineno-7-35" name="__codelineno-7-35" href="#__codelineno-7-35"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">k</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-36" name="__codelineno-7-36" href="#__codelineno-7-36"></a><span class="w">        </span><span class="c1">// If current element is greater than top element, top element exits heap, current element enters heap</span>
+<a id="__codelineno-7-36" name="__codelineno-7-36" href="#__codelineno-7-36"></a><span class="w">        </span><span class="c1">// If the current element is larger than the heap top, pop the top element and push the current element</span>
 <a id="__codelineno-7-37" name="__codelineno-7-37" href="#__codelineno-7-37"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">peekMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-38" name="__codelineno-7-38" href="#__codelineno-7-38"></a><span class="w">            </span><span class="nx">popMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">);</span>
 <a id="__codelineno-7-39" name="__codelineno-7-39" href="#__codelineno-7-39"></a><span class="w">            </span><span class="nx">pushMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]);</span>
 <a id="__codelineno-7-40" name="__codelineno-7-40" href="#__codelineno-7-40"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-7-41" name="__codelineno-7-41" href="#__codelineno-7-41"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-42" name="__codelineno-7-42" href="#__codelineno-7-42"></a><span class="w">    </span><span class="c1">// Return elements in heap</span>
+<a id="__codelineno-7-42" name="__codelineno-7-42" href="#__codelineno-7-42"></a><span class="w">    </span><span class="c1">// Return the elements in the heap</span>
 <a id="__codelineno-7-43" name="__codelineno-7-43" href="#__codelineno-7-43"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">getMinHeap</span><span class="p">(</span><span class="nx">maxHeap</span><span class="p">);</span>
 <a id="__codelineno-7-44" name="__codelineno-7-44" href="#__codelineno-7-44"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">top_k.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Find the largest k elements in array based on heap */</span>
+<div class="highlight"><span class="filename">top_k.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Use a heap to find the k largest elements in an array */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="n">MinHeap</span><span class="w"> </span><span class="n">topKHeap</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Initialize min heap, push first k elements of array to heap</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Initialize a min heap with the first k elements of the array</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="n">MinHeap</span><span class="w"> </span><span class="n">heap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MinHeap</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">sublist</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">));</span>
-<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// Starting from the (k+1)th element, maintain heap length as k</span>
+<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// Starting with element k+1, keep the heap size at k</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">    </span><span class="c1">// If current element is greater than top element, top element exits heap, current element enters heap</span>
+<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">    </span><span class="c1">// If the current element is larger than the heap top, pop the top element and push the current element</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="n">peek</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">      </span><span class="n">heap</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">      </span><span class="n">heap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
@@ -4976,17 +5048,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">top_k.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Find the largest k elements in array based on heap */</span>
+<div class="highlight"><span class="filename">top_k.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Use a heap to find the k largest elements in an array */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">top_k_heap</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">BinaryHeap</span><span class="o">&lt;</span><span class="n">Reverse</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// BinaryHeap is a max heap, use Reverse to negate elements to implement min heap</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// BinaryHeap is a max heap; Reverse reverses the comparison order to implement a min heap</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">heap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">BinaryHeap</span><span class="p">::</span><span class="o">&lt;</span><span class="n">Reverse</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
-<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="c1">// Enter the first k elements of array into heap</span>
+<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="c1">// Push the first k array elements onto the heap</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="o">&amp;</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">iter</span><span class="p">().</span><span class="n">take</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">        </span><span class="n">heap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">Reverse</span><span class="p">(</span><span class="n">num</span><span class="p">));</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// Starting from the (k+1)th element, maintain heap length as k</span>
+<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// Starting with element k+1, keep the heap size at k</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="o">&amp;</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">iter</span><span class="p">().</span><span class="n">skip</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="c1">// If current element is greater than top element, top element exits heap, current element enters heap</span>
+<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="c1">// If the current element is larger than the heap top, pop the top element and push the current element</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="n">peek</span><span class="p">().</span><span class="n">unwrap</span><span class="p">().</span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">            </span><span class="n">heap</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">            </span><span class="n">heap</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">Reverse</span><span class="p">(</span><span class="n">num</span><span class="p">));</span>
@@ -4997,27 +5069,27 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">top_k.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Element enters heap */</span>
+<div class="highlight"><span class="filename">top_k.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Push an element onto the heap */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">pushMinHeap</span><span class="p">(</span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Negate element</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Negate the element</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="n">push</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="o">-</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="p">}</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a>
-<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="cm">/* Element exits heap */</span>
+<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="cm">/* Pop an element from the heap */</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="kt">int</span><span class="w"> </span><span class="nf">popMinHeap</span><span class="p">(</span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="n">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="c1">// Negate element</span>
+<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="c1">// Negate the element</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="n">pop</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">);</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="p">}</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a>
-<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="cm">/* Access top element */</span>
+<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="cm">/* Access the top element of the heap */</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="kt">int</span><span class="w"> </span><span class="nf">peekMinHeap</span><span class="p">(</span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="n">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="c1">// Negate element</span>
+<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="c1">// Negate the element</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="n">peek</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">);</span>
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="p">}</span>
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a>
-<a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="cm">/* Extract elements from heap */</span>
+<a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="cm">/* Get the elements in the heap */</span>
 <a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="nf">getMinHeap</span><span class="p">(</span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="n">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">    </span><span class="c1">// Negate all heap elements and store in res array</span>
+<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">    </span><span class="c1">// Negate every element in the heap and store the results in array res</span>
 <a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">-</span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">data</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
@@ -5025,9 +5097,9 @@
 <a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-10-27" name="__codelineno-10-27" href="#__codelineno-10-27"></a><span class="p">}</span>
 <a id="__codelineno-10-28" name="__codelineno-10-28" href="#__codelineno-10-28"></a>
-<a id="__codelineno-10-29" name="__codelineno-10-29" href="#__codelineno-10-29"></a><span class="cm">/* Extract elements from heap */</span>
+<a id="__codelineno-10-29" name="__codelineno-10-29" href="#__codelineno-10-29"></a><span class="cm">/* Get the elements in the heap */</span>
 <a id="__codelineno-10-30" name="__codelineno-10-30" href="#__codelineno-10-30"></a><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="nf">getMinHeap</span><span class="p">(</span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="n">maxHeap</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-31" name="__codelineno-10-31" href="#__codelineno-10-31"></a><span class="w">    </span><span class="c1">// Negate all heap elements and store in res array</span>
+<a id="__codelineno-10-31" name="__codelineno-10-31" href="#__codelineno-10-31"></a><span class="w">    </span><span class="c1">// Negate every element in the heap and store the results in array res</span>
 <a id="__codelineno-10-32" name="__codelineno-10-32" href="#__codelineno-10-32"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-10-33" name="__codelineno-10-33" href="#__codelineno-10-33"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-34" name="__codelineno-10-34" href="#__codelineno-10-34"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">-</span><span class="n">maxHeap</span><span class="o">-&gt;</span><span class="n">data</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
@@ -5035,19 +5107,19 @@
 <a id="__codelineno-10-36" name="__codelineno-10-36" href="#__codelineno-10-36"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-10-37" name="__codelineno-10-37" href="#__codelineno-10-37"></a><span class="p">}</span>
 <a id="__codelineno-10-38" name="__codelineno-10-38" href="#__codelineno-10-38"></a>
-<a id="__codelineno-10-39" name="__codelineno-10-39" href="#__codelineno-10-39"></a><span class="c1">// Function to find k largest elements in array using heap</span>
+<a id="__codelineno-10-39" name="__codelineno-10-39" href="#__codelineno-10-39"></a><span class="c1">// Use a heap to find the k largest elements in an array</span>
 <a id="__codelineno-10-40" name="__codelineno-10-40" href="#__codelineno-10-40"></a><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="nf">topKHeap</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">sizeNums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-41" name="__codelineno-10-41" href="#__codelineno-10-41"></a><span class="w">    </span><span class="c1">// Python's heapq module implements min heap by default</span>
-<a id="__codelineno-10-42" name="__codelineno-10-42" href="#__codelineno-10-42"></a><span class="w">    </span><span class="c1">// Note: We negate all heap elements to simulate min heap using max heap</span>
+<a id="__codelineno-10-41" name="__codelineno-10-41" href="#__codelineno-10-41"></a><span class="w">    </span><span class="c1">// Initialize a min heap</span>
+<a id="__codelineno-10-42" name="__codelineno-10-42" href="#__codelineno-10-42"></a><span class="w">    </span><span class="c1">// Note: negate every element to simulate a min heap with a max heap</span>
 <a id="__codelineno-10-43" name="__codelineno-10-43" href="#__codelineno-10-43"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">empty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-10-44" name="__codelineno-10-44" href="#__codelineno-10-44"></a><span class="w">    </span><span class="n">MaxHeap</span><span class="w"> </span><span class="o">*</span><span class="n">maxHeap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newMaxHeap</span><span class="p">(</span><span class="n">empty</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
-<a id="__codelineno-10-45" name="__codelineno-10-45" href="#__codelineno-10-45"></a><span class="w">    </span><span class="c1">// Enter the first k elements of array into heap</span>
+<a id="__codelineno-10-45" name="__codelineno-10-45" href="#__codelineno-10-45"></a><span class="w">    </span><span class="c1">// Push the first k array elements onto the heap</span>
 <a id="__codelineno-10-46" name="__codelineno-10-46" href="#__codelineno-10-46"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-47" name="__codelineno-10-47" href="#__codelineno-10-47"></a><span class="w">        </span><span class="n">pushMinHeap</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
 <a id="__codelineno-10-48" name="__codelineno-10-48" href="#__codelineno-10-48"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-49" name="__codelineno-10-49" href="#__codelineno-10-49"></a><span class="w">    </span><span class="c1">// Starting from the (k+1)th element, maintain heap length as k</span>
+<a id="__codelineno-10-49" name="__codelineno-10-49" href="#__codelineno-10-49"></a><span class="w">    </span><span class="c1">// Starting with element k+1, keep the heap size at k</span>
 <a id="__codelineno-10-50" name="__codelineno-10-50" href="#__codelineno-10-50"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">sizeNums</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-51" name="__codelineno-10-51" href="#__codelineno-10-51"></a><span class="w">        </span><span class="c1">// If current element is greater than top element, top element exits heap, current element enters heap</span>
+<a id="__codelineno-10-51" name="__codelineno-10-51" href="#__codelineno-10-51"></a><span class="w">        </span><span class="c1">// If the current element is larger than the heap top, pop the top element and push the current element</span>
 <a id="__codelineno-10-52" name="__codelineno-10-52" href="#__codelineno-10-52"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">peekMinHeap</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-53" name="__codelineno-10-53" href="#__codelineno-10-53"></a><span class="w">            </span><span class="n">popMinHeap</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">);</span>
 <a id="__codelineno-10-54" name="__codelineno-10-54" href="#__codelineno-10-54"></a><span class="w">            </span><span class="n">pushMinHeap</span><span class="p">(</span><span class="n">maxHeap</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
@@ -5061,17 +5133,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">top_k.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Find the largest k elements in array based on heap */</span>
+<div class="highlight"><span class="filename">top_k.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Use a heap to find the k largest elements in an array */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">topKHeap</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="n">Queue</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Python's heapq module implements min heap by default</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Initialize a min heap</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">heap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">PriorityQueue</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span>
-<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="c1">// Enter the first k elements of array into heap</span>
+<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="c1">// Push the first k array elements onto the heap</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">        </span><span class="n">heap</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// Starting from the (k+1)th element, maintain heap length as k</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// Starting with element k+1, keep the heap size at k</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">k</span><span class="p">..</span><span class="o">&lt;</span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="c1">// If current element is greater than top element, top element exits heap, current element enters heap</span>
+<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="c1">// If the current element is larger than the heap top, pop the top element and push the current element</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">heap</span><span class="p">.</span><span class="na">peek</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">            </span><span class="n">heap</span><span class="p">.</span><span class="na">poll</span><span class="p">()</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">            </span><span class="n">heap</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">)</span>
@@ -5082,20 +5154,20 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">top_k.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Find largest k elements in array using heap ###</span>
+<div class="highlight"><span class="filename">top_k.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Use a heap to find the k largest elements in an array ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">top_k_heap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Python's heapq module implements min heap by default</span>
-<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="c1"># Note: We negate all heap elements to simulate min heap using max heap</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Initialize a min heap</span>
+<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="c1"># Note: negate all values in the heap to simulate a min heap with a max heap</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="n">max_heap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">MaxHeap</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="o">[]</span><span class="p">)</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a>
-<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="c1"># Enter the first k elements of array into heap</span>
+<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="c1"># Push the first k array elements onto the heap</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">k</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">    </span><span class="n">push_min_heap</span><span class="p">(</span><span class="n">max_heap</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a>
-<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">  </span><span class="c1"># Starting from the (k+1)th element, maintain heap length as k</span>
+<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">  </span><span class="c1"># Starting with element k+1, keep the heap size at k</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">k</span><span class="o">...</span><span class="n">nums</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="c1"># If current element is greater than top element, top element exits heap, current element enters heap</span>
+<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="c1"># If the current element is larger than the heap top, pop the top element and push the current element</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">peek_min_heap</span><span class="p">(</span><span class="n">max_heap</span><span class="p">)</span>
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">      </span><span class="n">pop_min_heap</span><span class="p">(</span><span class="n">max_heap</span><span class="p">)</span>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">      </span><span class="n">push_min_heap</span><span class="p">(</span><span class="n">max_heap</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">)</span>
@@ -5108,8 +5180,13 @@
 </div>
 </div>
 </div>
-<p>A total of <span class="arithmatex">\(n\)</span> rounds of heap insertions and removals are performed, with the heap's maximum length being <span class="arithmatex">\(k\)</span>, so the time complexity is <span class="arithmatex">\(O(n \log k)\)</span>. This method is very efficient; when <span class="arithmatex">\(k\)</span> is small, the time complexity approaches <span class="arithmatex">\(O(n)\)</span>; when <span class="arithmatex">\(k\)</span> is large, the time complexity does not exceed <span class="arithmatex">\(O(n \log n)\)</span>.</p>
-<p>Additionally, this method is well suited to dynamic data streams. As new data arrives, we can continuously maintain the elements in the heap, enabling dynamic updates to the largest <span class="arithmatex">\(k\)</span> elements.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=import%20heapq%0A%0Adef%20top_k_heap%28nums%3A%20list%5Bint%5D,%20k%3A%20int%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%22%22%22Use%20a%20heap%20to%20find%20the%20k%20largest%20elements%20in%20an%20array%22%22%22%0A%20%20%20%20%23%20Initialize%20a%20min%20heap%0A%20%20%20%20heap%20%3D%20%5B%5D%0A%20%20%20%20%23%20Push%20the%20first%20k%20array%20elements%20onto%20the%20heap%0A%20%20%20%20for%20i%20in%20range%28k%29%3A%0A%20%20%20%20%20%20%20%20heapq.heappush%28heap,%20nums%5Bi%5D%29%0A%20%20%20%20%23%20Starting%20with%20element%20k%2B1,%20keep%20the%20heap%20size%20at%20k%0A%20%20%20%20for%20i%20in%20range%28k,%20len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20%23%20If%20the%20current%20element%20is%20larger%20than%20the%20heap%20top,%20pop%20the%20top%20element%20and%20push%20the%20current%20element%0A%20%20%20%20%20%20%20%20if%20nums%5Bi%5D%20%3E%20heap%5B0%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20heapq.heappop%28heap%29%0A%20%20%20%20%20%20%20%20%20%20%20%20heapq.heappush%28heap,%20nums%5Bi%5D%29%0A%20%20%20%20return%20heap%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B1,%207,%206,%203,%202%5D%0A%20%20%20%20k%20%3D%203%0A%0A%20%20%20%20res%20%3D%20top_k_heap%28nums,%20k%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=import%20heapq%0A%0Adef%20top_k_heap%28nums%3A%20list%5Bint%5D,%20k%3A%20int%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%22%22%22Use%20a%20heap%20to%20find%20the%20k%20largest%20elements%20in%20an%20array%22%22%22%0A%20%20%20%20%23%20Initialize%20a%20min%20heap%0A%20%20%20%20heap%20%3D%20%5B%5D%0A%20%20%20%20%23%20Push%20the%20first%20k%20array%20elements%20onto%20the%20heap%0A%20%20%20%20for%20i%20in%20range%28k%29%3A%0A%20%20%20%20%20%20%20%20heapq.heappush%28heap,%20nums%5Bi%5D%29%0A%20%20%20%20%23%20Starting%20with%20element%20k%2B1,%20keep%20the%20heap%20size%20at%20k%0A%20%20%20%20for%20i%20in%20range%28k,%20len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20%23%20If%20the%20current%20element%20is%20larger%20than%20the%20heap%20top,%20pop%20the%20top%20element%20and%20push%20the%20current%20element%0A%20%20%20%20%20%20%20%20if%20nums%5Bi%5D%20%3E%20heap%5B0%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20heapq.heappop%28heap%29%0A%20%20%20%20%20%20%20%20%20%20%20%20heapq.heappush%28heap,%20nums%5Bi%5D%29%0A%20%20%20%20return%20heap%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B1,%207,%206,%203,%202%5D%0A%20%20%20%20k%20%3D%203%0A%0A%20%20%20%20res%20%3D%20top_k_heap%28nums,%20k%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>There are <span class="arithmatex">\(n\)</span> rounds of heap insertion and removal to consider, and the heap has at most <span class="arithmatex">\(k\)</span> elements, so the time complexity is <span class="arithmatex">\(O(n \log k)\)</span>. This is highly efficient: when <span class="arithmatex">\(k\)</span> is small, the complexity approaches <span class="arithmatex">\(O(n)\)</span>; when <span class="arithmatex">\(k\)</span> is large, it does not exceed <span class="arithmatex">\(O(n \log n)\)</span>.</p>
+<p>This approach also suits dynamic data streams. As new data arrives, we can maintain the heap to update the largest <span class="arithmatex">\(k\)</span> elements continuously.</p>
 
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   <a
     href="../build_heap/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 8.2 Heap Construction Operation"
+    aria-label="Previous: 8.2   Building a heap"
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+        8.2 &nbsp; Building a heap
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     href="../summary/"
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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@@ -39,7 +39,7 @@
 
 
 
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+        <title>Foreword - Hello Algo</title>
 
 
 
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+        <a href="#foreword" class="md-skip">
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+  7.4 &nbsp; Binary search trees
 
 
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@@ -4551,21 +4623,21 @@
 
 
 <!-- Page content -->
-<h1 id="preface">Preface<a class="headerlink" href="#preface" title="Permanent link">&para;</a></h1>
-<p>A few years ago, I shared the "Sword for Offer" problem solutions on LeetCode, receiving encouragement and support from many readers. During interactions with readers, the most frequently asked question I encountered was "how to get started with algorithms." Gradually, I developed a keen interest in this question.</p>
-<p>Diving straight into problem-solving seems to be the most popular approach—it's simple, direct, and effective. However, problem-solving is like playing Minesweeper: those with strong self-learning abilities can successfully defuse the mines one by one, while those with insufficient foundations may end up bruised and battered, retreating step by step in frustration. Reading through textbooks is also a common practice, but for job seekers, graduation theses, resume submissions, and preparations for written tests and interviews have already consumed most of their energy, making working through thick books an arduous challenge.</p>
-<p>If you're facing similar struggles, then it's fortunate that this book has "found" you. This book is my answer to this question—even if it may not be the optimal solution, it is at least a positive attempt. While this book alone won't directly land you a job offer, it will guide you through the "landscape" of data structures and algorithms, help you understand the shapes, sizes, and distributions of different "mines," and enable you to master various "mine-clearing methods." With these skills, I believe you can tackle problems and read technical literature more confidently, gradually building a complete knowledge system.</p>
-<p>I deeply agree with Professor Feynman's words: "Knowledge isn't free. You have to pay attention." In this sense, this book is not entirely "free." In order to live up to the precious "attention" you invest in this book, I will do my utmost and devote my greatest "attention" to completing this work.</p>
-<p>I'm keenly aware of the limits of my knowledge and experience. Although the content of this book has been refined over a period of time, there are certainly still many errors, and I sincerely welcome critiques and corrections from teachers and fellow students.</p>
-<p><img alt="Hello Algorithms" class="cover-image" src="../assets/covers/chapter_hello_algo.jpg" /></p>
+<h1 id="foreword">Foreword<a class="headerlink" href="#foreword" title="Permanent link">&para;</a></h1>
+<p>A few years ago, I shared a series of solutions to the “Coding Interviews” problems on LeetCode and received encouragement and support from many readers. In our conversations, one question came up more than any other: “How do I get started with algorithms?” Over time, I became deeply interested in this question.</p>
+<p>Jumping into practice problems with little preparation seems to be the most popular approach: simple, direct, and effective. Yet solving problems is rather like playing Minesweeper. Strong independent learners can clear the mines one by one, while those with a weak foundation may run into one setback after another and gradually retreat in frustration. Reading a textbook from cover to cover is another common approach. For job seekers, however, theses, applications, written tests, and interview preparation already consume most of their energy, making a thick textbook a formidable challenge.</p>
+<p>If you face similar difficulties, I am glad this book has found you. It is my answer to that question—perhaps not the optimal solution, but at least a constructive attempt. Although this book alone will not secure a job offer, it will guide you through the “knowledge map” of data structures and algorithms. It will show you the shapes, sizes, and locations of different “mines” and teach you ways to clear them. With these skills, I believe you can approach practice problems and technical reading more confidently and gradually build a complete body of knowledge.</p>
+<p>I wholeheartedly agree with Professor Feynman's words: “Knowledge isn't free. You have to pay attention.” In this sense, the book is not entirely “free.” To honor the precious attention you invest in it, I will devote as much of my own attention as I can to writing it.</p>
+<p>I am aware of the limits of my knowledge. Although the content has undergone considerable refinement, many errors surely remain. I sincerely welcome corrections and suggestions from teachers and fellow learners.</p>
+<p><img alt="Hello Algo" class="cover-image" src="../assets/covers/chapter_hello_algo.jpg" /></p>
 <div style="text-align: center;">
-    <h2 style="margin-top: 0.8em; margin-bottom: 0.8em;">Hello, Algorithms!</h2>
+    <h2 style="margin-top: 0.8em; margin-bottom: 0.8em;">Hello, algorithms!</h2>
 </div>
 
-<p>The advent of computers has brought tremendous changes to the world. With their high-speed computing capabilities and excellent programmability, they have become the ideal medium for executing algorithms and processing data. Whether it's the realistic graphics in video games, the intelligent decision-making in autonomous driving, AlphaGo's brilliant Go matches, or ChatGPT's natural interactions, these applications are all striking demonstrations of algorithms at work on computers.</p>
-<p>In fact, before the advent of computers, algorithms and data structures already existed in every corner of the world. Early algorithms were relatively simple, such as ancient counting methods and tool-making procedures. As civilization progressed, algorithms gradually became more refined and complex. From the ingenious craftsmanship of master artisans, to industrial products that liberate productive forces, to the scientific laws governing the operation of the universe, behind almost every ordinary or astonishing thing lies ingenious algorithmic thinking.</p>
-<p>Similarly, data structures are everywhere: from large-scale social networks to small subway systems, many systems can be modeled as "graphs"; from a nation to a family, the primary organizational forms of society exhibit characteristics of "trees"; winter clothing is like a "stack," where the first item put on is the last to be taken off; a badminton tube is like a "queue," with items inserted at one end and retrieved from the other; a dictionary is like a "hash table," enabling quick lookup of target entries.</p>
-<p>This book aims to help readers understand the core concepts of algorithms and data structures through clear and accessible animated illustrations and runnable code examples, and to implement them in code. Building on this foundation, the book endeavors to reveal the vivid manifestations of algorithms in the complex world and showcase the beauty of algorithms. I hope this book can be of help to you!</p>
+<p>Computers have transformed the world. Their speed and remarkable programmability make them an ideal medium for executing algorithms and processing data. From the lifelike graphics of video games and the intelligent decisions of self-driving cars to AlphaGo's brilliant games and ChatGPT's natural interactions, these applications are all compelling expressions of algorithms running on computers.</p>
+<p>In fact, algorithms and data structures existed throughout the world long before computers. Early algorithms were relatively simple, such as ancient counting methods and steps for making tools. As civilization advanced, algorithms became more refined and complex. From exquisite craftsmanship and industrial products that boosted productivity to the scientific laws governing the universe, ingenious algorithmic ideas lie behind almost everything, whether ordinary or astonishing.</p>
+<p>Data structures are equally widespread. Systems ranging from social networks to subway routes can be modeled as “graphs.” Social organizations, from nations to families, often have the characteristics of “trees.” Winter clothing behaves like a “stack”: what goes on first comes off last. A tube of shuttlecocks is like a “queue,” with items inserted at one end and removed at the other. A dictionary resembles a “hash table,” allowing quick lookup of a desired entry.</p>
+<p>Through clear animated illustrations and runnable code examples, this book aims to help readers understand the core concepts of algorithms and data structures and implement them through programming. Building on that foundation, it seeks to reveal how algorithms take shape in our complex world and to show their beauty. I hope you find it helpful!</p>
 
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-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
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+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              1.1 &nbsp; Algorithms Are Everywhere
+              1.1 &nbsp; Algorithms are everywhere
 
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@@ -167,9 +167,9 @@
 
 
 
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-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
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-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -577,7 +577,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -598,7 +598,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -628,7 +628,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -656,7 +656,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -683,7 +683,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -749,7 +749,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -770,7 +770,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -791,7 +791,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -818,7 +818,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -845,7 +845,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -872,7 +872,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -899,7 +899,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -926,7 +926,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -992,7 +992,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1013,7 +1013,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1034,7 +1034,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1061,7 +1061,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1088,7 +1088,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1115,7 +1115,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1142,7 +1142,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1169,7 +1169,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1235,7 +1235,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1256,7 +1256,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1277,7 +1277,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1304,7 +1304,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1331,7 +1331,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1358,7 +1358,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1385,7 +1385,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1412,7 +1412,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1476,7 +1476,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1497,7 +1497,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1518,7 +1518,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1545,7 +1545,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1572,7 +1572,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1599,7 +1599,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1626,7 +1626,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1690,7 +1690,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1711,7 +1711,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1732,7 +1732,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1759,7 +1759,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1786,7 +1786,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1813,7 +1813,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1840,7 +1840,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1908,7 +1908,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1929,7 +1929,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1950,7 +1950,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1977,7 +1977,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2004,7 +2004,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2031,7 +2031,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2058,7 +2058,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2085,7 +2085,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2112,7 +2112,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2176,7 +2176,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2197,7 +2197,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2218,7 +2218,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2245,7 +2245,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2272,7 +2272,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2299,7 +2299,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2326,7 +2326,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2390,7 +2390,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2411,7 +2411,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2432,7 +2432,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2459,7 +2459,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2486,7 +2486,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2513,7 +2513,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2540,7 +2540,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2608,7 +2608,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2629,7 +2629,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2650,7 +2650,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2677,7 +2677,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2704,7 +2704,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2731,7 +2731,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2758,7 +2758,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2785,7 +2785,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2812,7 +2812,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2890,7 +2890,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2911,7 +2911,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2932,7 +2932,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2959,7 +2959,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2986,7 +2986,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3013,7 +3013,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3040,7 +3040,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3067,7 +3067,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3094,7 +3094,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3121,7 +3121,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3148,7 +3148,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3175,7 +3175,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3202,7 +3202,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3229,7 +3229,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3295,7 +3295,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3316,7 +3316,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3337,7 +3337,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3364,7 +3364,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3391,7 +3391,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3418,7 +3418,7 @@
   <span class="md-ellipsis">
 
 
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 <!-- Page content -->
-<h1 id="11-algorithms-are-everywhere">1.1 &nbsp; Algorithms Are Everywhere<a class="headerlink" href="#11-algorithms-are-everywhere" title="Permanent link">&para;</a></h1>
-<p>When we hear the term "algorithm," we naturally think of mathematics. However, many algorithms do not involve complex mathematics but rely more on basic logic, which can be seen everywhere in our daily lives.</p>
-<p>Before we formally explore algorithms, here's an interesting fact worth sharing: <strong>you have already learned many algorithms without realizing it, and you are used to applying them in daily life</strong>. Let me give a few specific examples to illustrate this point.</p>
-<p><strong>Example 1: Looking Up a Dictionary</strong>. In an English dictionary, words are listed alphabetically. Assuming we're searching for a word that starts with the letter <span class="arithmatex">\(r\)</span>, this is typically done in the following way:</p>
+<h1 id="11-algorithms-are-everywhere">1.1 &nbsp; Algorithms are everywhere<a class="headerlink" href="#11-algorithms-are-everywhere" title="Permanent link">&para;</a></h1>
+<p>When we hear the word “algorithm,” we naturally think of mathematics. In fact, many algorithms do not involve complex mathematics. Instead, they rely on basic logic that we encounter throughout everyday life.</p>
+<p>Before we formally explore algorithms, here is an interesting fact: <strong>you have already learned many algorithms without realizing it, and you routinely use them in everyday life</strong>. The following examples illustrate this point.</p>
+<p><strong>Example 1: Looking up a word in a dictionary</strong>. In a Chinese dictionary, each character has a corresponding pinyin pronunciation, and entries are arranged alphabetically by pinyin. Suppose we want to find a character whose pinyin begins with <span class="arithmatex">\(r\)</span>. We would usually follow the process shown in Figure 1-1.</p>
 <ol>
-<li>Open the dictionary to about halfway and check the first word on that page; suppose it starts with the letter <span class="arithmatex">\(m\)</span>.</li>
-<li>Since <span class="arithmatex">\(r\)</span> comes after <span class="arithmatex">\(m\)</span> in the alphabet, the first half can be ignored and the search space is narrowed down to the second half.</li>
-<li>Repeat steps <code>1.</code> and <code>2.</code> until you find the page where the word starts with <span class="arithmatex">\(r\)</span>.</li>
+<li>Open the dictionary roughly halfway through and check the initial letter on that page. Suppose it is <span class="arithmatex">\(m\)</span>.</li>
+<li>Since <span class="arithmatex">\(r\)</span> comes after <span class="arithmatex">\(m\)</span> in the pinyin alphabet, rule out the first half of the dictionary and narrow the search to the second half.</li>
+<li>Repeat steps <code>1.</code> and <code>2.</code> until you find the pages whose pinyin entries begin with <span class="arithmatex">\(r\)</span>.</li>
 </ol>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:5"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Process of looking up a dictionary" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step1.png" /></p>
+<p><img alt="Steps for looking up a word in a dictionary" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step1.png" /></p>
 </div>
 <div class="tabbed-block">
-<p><img alt="Binary search in dictionary step 2" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step2.png" /></p>
+<p><img alt="binary_search_dictionary_step2" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step2.png" /></p>
 </div>
 <div class="tabbed-block">
-<p><img alt="Binary search in dictionary step 3" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step3.png" /></p>
+<p><img alt="binary_search_dictionary_step3" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step3.png" /></p>
 </div>
 <div class="tabbed-block">
-<p><img alt="Binary search in dictionary step 4" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step4.png" /></p>
+<p><img alt="binary_search_dictionary_step4" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step4.png" /></p>
 </div>
 <div class="tabbed-block">
-<p><img alt="Binary search in dictionary step 5" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step5.png" /></p>
+<p><img alt="binary_search_dictionary_step5" class="animation-figure" src="../algorithms_are_everywhere.assets/binary_search_dictionary_step5.png" /></p>
 </div>
 </div>
 </div>
-<p align="center"> Figure 1-1 &nbsp; Process of looking up a dictionary </p>
+<p align="center"> Figure 1-1 &nbsp; Steps for looking up a word in a dictionary </p>
 
-<p>Looking up a dictionary, an essential skill for elementary school students is actually the famous "Binary Search" algorithm. From a data structure perspective, we can consider the dictionary as a sorted "array"; from an algorithmic perspective, the series of actions taken to look up a word in the dictionary can be viewed as the algorithm "Binary Search."</p>
-<p><strong>Example 2: Organizing Playing Cards</strong>. When playing cards, we need to arrange the cards in our hands in ascending order, as shown in the following process.</p>
+<p>This dictionary skill, learned in elementary school, is actually the well-known “binary search” algorithm. From a data structure perspective, we can view the dictionary as a sorted “array”; from an algorithm perspective, the sequence of lookup operations above is “binary search.”</p>
+<p><strong>Example 2: Arranging playing cards</strong>. When playing cards, we need to arrange our hand from smallest to largest each round. The process is shown in Figure 1-2.</p>
 <ol>
-<li>Divide the playing cards into "ordered" and "unordered" sections, assuming initially the leftmost card is already in order.</li>
-<li>Take out a card from the unordered section and insert it into the correct position in the ordered section; after this, the leftmost two cards are in order.</li>
-<li>Repeat step <code>2</code> until all cards are in order.</li>
+<li>Divide the cards into “sorted” and “unsorted” sections, initially treating the single leftmost card as sorted.</li>
+<li>Take a card from the unsorted section and insert it into the correct position in the sorted section. The two leftmost cards are now sorted.</li>
+<li>Repeat step <code>2.</code>, inserting one card from the unsorted section into the sorted section each round, until all the cards are sorted.</li>
 </ol>
-<p><img alt="Process of sorting a deck of cards" class="animation-figure" src="../algorithms_are_everywhere.assets/playing_cards_sorting.png" /></p>
-<p align="center"> Figure 1-2 &nbsp; Process of sorting a deck of cards </p>
+<p><img alt="Steps for sorting playing cards" class="animation-figure" src="../algorithms_are_everywhere.assets/playing_cards_sorting.png" /></p>
+<p align="center"> Figure 1-2 &nbsp; Steps for sorting playing cards </p>
 
-<p>The above method of organizing playing cards is essentially the "Insertion Sort" algorithm, which is very efficient for small datasets. Many programming languages' built-in sorting implementations use insertion sort internally.</p>
-<p><strong>Example 3: Making Change</strong>. Assume making a purchase of <span class="arithmatex">\(69\)</span> at a supermarket. If you give the cashier <span class="arithmatex">\(100\)</span>, they will need to provide you with <span class="arithmatex">\(31\)</span> in change. This process can be clearly understood as illustrated in Figure 1-3.</p>
+<p>This method of arranging playing cards is essentially “insertion sort,” an algorithm that is very efficient for small datasets. Insertion sort appears in the library sorting functions of many programming languages.</p>
+<p><strong>Example 3: Making change</strong>. Suppose we buy goods costing <span class="arithmatex">\(69\)</span> yuan at a supermarket and pay the cashier <span class="arithmatex">\(100\)</span> yuan. The cashier needs to give us <span class="arithmatex">\(31\)</span> yuan in change and naturally reasons through the process shown in Figure 1-3.</p>
 <ol>
-<li>The available denominations smaller than <span class="arithmatex">\(31\)</span> are <span class="arithmatex">\(1\)</span>, <span class="arithmatex">\(5\)</span>, <span class="arithmatex">\(10\)</span>, and <span class="arithmatex">\(20\)</span>.</li>
-<li>Take out the largest <span class="arithmatex">\(20\)</span> from the options, leaving <span class="arithmatex">\(31 - 20 = 11\)</span>.</li>
-<li>Take out the largest <span class="arithmatex">\(10\)</span> from the remaining options, leaving <span class="arithmatex">\(11 - 10 = 1\)</span>.</li>
-<li>Take out the largest <span class="arithmatex">\(1\)</span> from the remaining options, leaving <span class="arithmatex">\(1 - 1 = 0\)</span>.</li>
-<li>Complete change-making, the solution is <span class="arithmatex">\(20 + 10 + 1 = 31\)</span>.</li>
+<li>The available denominations smaller than <span class="arithmatex">\(31\)</span> yuan are <span class="arithmatex">\(1\)</span>, <span class="arithmatex">\(5\)</span>, <span class="arithmatex">\(10\)</span>, and <span class="arithmatex">\(20\)</span> yuan.</li>
+<li>Take the largest available denomination, <span class="arithmatex">\(20\)</span> yuan, leaving <span class="arithmatex">\(31 - 20 = 11\)</span> yuan.</li>
+<li>Take the largest denomination that fits the remaining amount, <span class="arithmatex">\(10\)</span> yuan, leaving <span class="arithmatex">\(11 - 10 = 1\)</span> yuan.</li>
+<li>Take the largest denomination that fits the remaining amount, <span class="arithmatex">\(1\)</span> yuan, leaving <span class="arithmatex">\(1 - 1 = 0\)</span> yuan.</li>
+<li>The change is complete: <span class="arithmatex">\(20 + 10 + 1 = 31\)</span> yuan.</li>
 </ol>
-<p><img alt="Process of making change" class="animation-figure" src="../algorithms_are_everywhere.assets/greedy_change.png" /></p>
-<p align="center"> Figure 1-3 &nbsp; Process of making change </p>
+<p><img alt="The process of making change" class="animation-figure" src="../algorithms_are_everywhere.assets/greedy_change.png" /></p>
+<p align="center"> Figure 1-3 &nbsp; The process of making change </p>
 
-<p>In the steps above, we choose what seems to be the best option at each stage by using the largest denomination available, which leads to an effective way to make change. From a data structures and algorithms perspective, this approach is known as a "Greedy" algorithm.</p>
-<p>From cooking a meal to interstellar travel, almost all problem-solving involves algorithms. The advent of computers allows us to store data structures in memory and write code to call the CPU and GPU to execute algorithms. In this way, we can transfer real-life problems to computers and solve various complex issues in a more efficient way.</p>
+<p>At each step above, we make the choice that looks best at the moment—using the largest possible denomination—and eventually obtain a valid way to make change. In terms of data structures and algorithms, this is a “greedy” algorithm.</p>
+<p>From cooking a meal to traveling between stars, solving almost any problem involves algorithms. Computers allow us to store data structures in memory through programming and write code that calls on CPUs and GPUs to execute algorithms. We can therefore bring real-world problems to computers and solve complex problems more efficiently.</p>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>If concepts such as data structures, algorithms, arrays, and binary search still feel only half-familiar, keep reading. This book will guide you into the world of data structures and algorithms.</p>
+<p>If concepts such as data structures, algorithms, arrays, and binary search still seem unclear, keep reading. This book will guide you into the world of data structures and algorithms.</p>
 </div>
 
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+        <a href="#chapter-1-getting-to-know-algorithms" class="md-skip">
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           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              Chapter 1. &nbsp; Encounter with Algorithms
+              Chapter 1. &nbsp; Getting to know algorithms
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Dark mode"  type="radio" name="__palette" id="__palette_0">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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       </label>
 
@@ -177,9 +177,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Light mode"  type="radio" name="__palette" id="__palette_1">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -577,7 +577,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -598,7 +598,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -619,7 +619,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -646,7 +646,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -673,7 +673,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -739,7 +739,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -760,7 +760,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -781,7 +781,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -808,7 +808,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -835,7 +835,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -862,7 +862,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -889,7 +889,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -916,7 +916,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -982,7 +982,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1003,7 +1003,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1024,7 +1024,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1051,7 +1051,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1078,7 +1078,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1105,7 +1105,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1132,7 +1132,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1159,7 +1159,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1225,7 +1225,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1246,7 +1246,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1267,7 +1267,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1294,7 +1294,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1321,7 +1321,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1348,7 +1348,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1375,7 +1375,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1402,7 +1402,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1466,7 +1466,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1487,7 +1487,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1508,7 +1508,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1535,7 +1535,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1562,7 +1562,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1589,7 +1589,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1616,7 +1616,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1680,7 +1680,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1701,7 +1701,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1722,7 +1722,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1749,7 +1749,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1776,7 +1776,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1803,7 +1803,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1830,7 +1830,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1898,7 +1898,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1919,7 +1919,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1940,7 +1940,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1967,7 +1967,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1994,7 +1994,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2021,7 +2021,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2048,7 +2048,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2075,7 +2075,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2102,7 +2102,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2166,7 +2166,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2187,7 +2187,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2208,7 +2208,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2235,7 +2235,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2262,7 +2262,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2289,7 +2289,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2316,7 +2316,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2380,7 +2380,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2401,7 +2401,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2422,7 +2422,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2449,7 +2449,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2476,7 +2476,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2503,7 +2503,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2530,7 +2530,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2598,7 +2598,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2619,7 +2619,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2640,7 +2640,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2694,7 +2694,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2721,7 +2721,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2748,7 +2748,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2775,7 +2775,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2802,7 +2802,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2880,7 +2880,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2901,7 +2901,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2922,7 +2922,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2976,7 +2976,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3003,7 +3003,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3030,7 +3030,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3057,7 +3057,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3084,7 +3084,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3111,7 +3111,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3138,7 +3138,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3165,7 +3165,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3192,7 +3192,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3219,7 +3219,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3285,7 +3285,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3306,7 +3306,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3327,7 +3327,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3381,7 +3381,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3408,7 +3408,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3435,7 +3435,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3462,7 +3462,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3528,7 +3528,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3549,7 +3549,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3570,7 +3570,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3597,7 +3597,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3624,7 +3624,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3651,7 +3651,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3678,7 +3678,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3705,7 +3705,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3775,7 +3775,7 @@
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 <!-- Page content -->
-<h1 id="chapter-1-encounter-with-algorithms">Chapter 1. &nbsp; Encounter with Algorithms<a class="headerlink" href="#chapter-1-encounter-with-algorithms" title="Permanent link">&para;</a></h1>
-<p><img alt="Encounter with Algorithms" class="cover-image" src="../assets/covers/chapter_introduction.jpg" /></p>
+<h1 id="chapter-1-getting-to-know-algorithms">Chapter 1. &nbsp; Getting to know algorithms<a class="headerlink" href="#chapter-1-getting-to-know-algorithms" title="Permanent link">&para;</a></h1>
+<p><img alt="Getting to know algorithms" class="cover-image" src="../assets/covers/chapter_introduction.jpg" /></p>
 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
-<p>A young girl dances gracefully, intertwined with data, her skirt flowing with the melody of algorithms.</p>
-<p>She invites you to dance with her. Follow her steps closely and enter the world of algorithms, full of logic and beauty.</p>
+<p>A young woman dances gracefully among the data, the melody of algorithms rippling through her skirt.</p>
+<p>She invites you to join her. Follow her steps into a world of algorithms filled with logic and beauty.</p>
 </div>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
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-<li><a href="what_is_dsa/">1.2 &nbsp; What Is an Algorithm</a></li>
+<li><a href="algorithms_are_everywhere/">1.1 &nbsp; Algorithms are everywhere</a></li>
+<li><a href="what_is_dsa/">1.2 &nbsp; What is an algorithm?</a></li>
 <li><a href="summary/">1.3 &nbsp; Summary</a></li>
 </ul>
 
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     class="md-footer__link md-footer__link--prev"
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   <a
     href="./algorithms_are_everywhere/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 1.1 Algorithms Are Everywhere"
+    aria-label="Next: 1.1   Algorithms are everywhere"
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     <div class="md-footer__title">
@@ -4644,7 +4716,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        1.1 Algorithms Are Everywhere
+        1.1 &nbsp; Algorithms are everywhere
       </div>
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
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+  5.3 &nbsp; Deques
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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+  6.3 &nbsp; Hash algorithms
 
 
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+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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+  Chapter 7. &nbsp; Trees
 
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+  7.1 &nbsp; Binary trees
 
 
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+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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+  11.4 &nbsp; Insertion sort
 
 
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+  11.9 &nbsp; Counting sort
 
 
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+  11.10 &nbsp; Radix sort
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
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+  14.6 &nbsp; Edit distance problem
 
 
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+  Chapter 15. &nbsp; Greedy algorithms
 
 
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+  15.1 &nbsp; Greedy algorithms
 
 
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+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
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 <!-- Page content -->
 <h1 id="13-summary">1.3 &nbsp; Summary<a class="headerlink" href="#13-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
 <ul>
-<li>Algorithms are ubiquitous in daily life and are not some distant, esoteric body of knowledge. In fact, we have already learned many algorithms unconsciously and use them to solve problems big and small in life.</li>
-<li>The principle of looking up a dictionary is consistent with the binary search algorithm. Binary search embodies the important algorithmic idea of divide and conquer.</li>
-<li>The process of organizing playing cards is very similar to the insertion sort algorithm. Insertion sort is suitable for sorting small datasets.</li>
-<li>The steps of making change are essentially a greedy algorithm, where the best choice is made at each step based on the current situation.</li>
-<li>An algorithm is a set of instructions or operational steps that solves a specific problem within a finite amount of time, while a data structure is a way of organizing and storing data in a computer.</li>
-<li>Data structures and algorithms are closely connected. Data structures are the foundation of algorithms, and algorithms breathe life into data structures.</li>
-<li>We can compare data structures and algorithms to assembling building blocks. The blocks represent data, the way they are shaped and connected represents the data structure, and the steps used to assemble them correspond to the algorithm.</li>
+<li>Algorithms are everywhere in daily life; they are not remote, inaccessible knowledge. In fact, we have already learned many algorithms without realizing it and use them to solve problems large and small.</li>
+<li>Looking up a word in a dictionary follows the same principle as binary search. Binary search illustrates the important algorithmic idea of divide and conquer.</li>
+<li>Arranging playing cards closely resembles insertion sort. Insertion sort is well suited to small datasets.</li>
+<li>Making change is essentially a greedy algorithm: at each step, make the choice that looks best at the moment.</li>
+<li>An algorithm is a set of instructions or steps that solves a specific problem within a finite time. A data structure is a way of organizing and storing data in a computer.</li>
+<li>Data structures and algorithms are closely connected. Data structures provide the foundation for algorithms, while algorithms bring data structures to life.</li>
+<li>We can compare data structures and algorithms to assembling building blocks: the blocks represent data; their shapes and connections represent data structures; and the assembly steps represent algorithms.</li>
 </ul>
 <h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
-<p><strong>Q</strong>: As a programmer, I have never used algorithms to solve problems in my daily work. Common algorithms are already encapsulated by programming languages and can be used directly. Does this mean that the problems in our work have not yet reached the level where algorithms are needed?</p>
-<p>If we compare specific work skills to "techniques" in martial arts, then fundamental subjects should be more like "internal skills".</p>
-<p>I believe the significance of learning algorithms (and other fundamental subjects) is not that you will need to implement them from scratch at work, but that the knowledge you gain enables you to make sound professional judgments when solving problems, thereby improving the overall quality of your work. Here is a simple example. Every programming language has a built-in sorting function:</p>
+<p><strong>Q</strong>: As a programmer, I have never used algorithms to solve problems in my daily work. Common algorithms are already built into programming languages, so I can simply use them. Does this mean the problems we encounter at work are not difficult enough to require algorithms?</p>
+<p>If specific job skills are like the techniques of a martial art, foundational subjects are more like the underlying strength that makes those techniques effective.</p>
+<p>I believe the purpose of learning algorithms—and other foundational subjects—is not to implement them from scratch at work. It is to use what we have learned to respond and make judgments professionally when solving problems, improving the overall quality of our work. For a simple example, every programming language has a built-in sorting function:</p>
 <ul>
-<li>If we have not studied data structures and algorithms, we might simply feed any given data to this sorting function. It runs smoothly with good performance, and there doesn't seem to be any problem.</li>
-<li>But if we have studied algorithms, we would know that the time complexity of the built-in sorting function is <span class="arithmatex">\(O(n \log n)\)</span>. However, if the given data consists of integers with a fixed number of digits (such as student IDs), we can use the more efficient "radix sort", reducing the time complexity to <span class="arithmatex">\(O(nk)\)</span>, where <span class="arithmatex">\(k\)</span> is the number of digits. When the data volume is very large, the saved running time can create significant value (reduced costs, improved experience, etc.).</li>
+<li>Without studying data structures and algorithms, we might pass any dataset to this sorting function. It runs smoothly and performs reasonably well, so nothing seems wrong.</li>
+<li>After studying algorithms, however, we know that the built-in sorting function has a time complexity of <span class="arithmatex">\(O(n \log n)\)</span>. If the data consists of integers with a fixed number of digits, such as student IDs, we can use the more efficient “radix sort” to reduce the time complexity to <span class="arithmatex">\(O(nk)\)</span>, where <span class="arithmatex">\(k\)</span> is the number of digits. For large datasets, the time saved can create considerable value through lower costs, a better user experience, and so on.</li>
 </ul>
-<p>In engineering, many problems are difficult to solve optimally, and many others are only solved "well enough." The difficulty of a problem depends, on the one hand, on the nature of the problem itself and, on the other hand, on the knowledge of the person examining it. The more complete a person's knowledge and the more experience they have, the deeper their analysis will be, and the more elegantly the problem can be solved.</p>
-<p><strong>Q</strong>: Now that we let AI agents write almost all our code, do we still need to learn algorithms?</p>
-<p>To quote Mu Li: “An agent is an amplifier of ability—it multiplies your ability by 10, rather than adding 10 to it.”</p>
-<p>Since AI amplifies our abilities, the abilities being multiplied matter too. I think two abilities are especially worth developing. The first is communication, particularly the ability to work with AI: making goals clear, providing the necessary context, and correcting course in time. Explaining a problem to AI also helps us think it through ourselves. The second is foundational knowledge. AI can fill in the details, but we need to understand the basic concepts ourselves to ask the right questions, judge whether answers are correct, and turn its output into useful results.</p>
-<p>One distinctive thing about AI is that, unlike earlier tools, it does not have clearly defined functional boundaries. It is a little like water, taking different shapes in different people's hands. The greater a person's abilities and the better their taste, the more pronounced AI's amplifying effect becomes.</p>
+<p>In engineering, optimal solutions are difficult to reach for many problems, and many others are solved only “well enough.” How difficult a problem appears depends both on its nature and on the knowledge of the person examining it. The more knowledge and experience we have, the more deeply we can analyze a problem and the more elegantly we can solve it.</p>
+<p><strong>Q</strong>: Now that AI agents write almost all our code, do we still need to learn algorithms?</p>
+<p>To quote Mu Li: “An agent is an ability multiplier: it multiplies your ability by 10. It is not an adder that adds 10 to your ability.”</p>
+<p>If AI multiplies our ability, then that starting ability matters too. I think two kinds of ability are especially worth developing. The first is communication, particularly collaboration with AI: state the goal clearly, supply the necessary context, and correct the course promptly. Explaining a problem to AI also helps us clarify it for ourselves. The second is foundational knowledge. AI can fill in the details, but we need to understand the basic concepts ourselves to ask relevant questions, judge what is correct, and turn its output into useful results.</p>
+<p>One distinctive feature of AI is that, unlike earlier tools, it has no clear boundaries to its functionality. It is a little like water, taking different shapes in different people's hands. The stronger a person's abilities and the better their taste, the more pronounced AI's multiplying effect becomes.</p>
 
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@@ -41,7 +41,7 @@
 
 
 
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-  2.1 Algorithm Efficiency Evaluation
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-  2.2 Iteration and Recursion
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+  8.5 &nbsp; Exercises
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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+  9.5 &nbsp; Exercises
 
 
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+  Chapter 10. &nbsp; Searching
 
 
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+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
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-  10.4 Hash Optimization Strategy
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-  10.5 Searching Algorithms Revisited
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-  10.6 Summary
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-  10.7 Exercises
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-  15.4 Maximum Product Cutting Problem
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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@@ -4610,10 +4682,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#121-algorithm-definition" class="md-nav__link">
+  <a href="#121-definition-of-an-algorithm" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1.2.1 &nbsp; Algorithm Definition
+        1.2.1 &nbsp; Definition of an algorithm
       </span>
     </span>
   </a>
@@ -4621,10 +4693,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#122-data-structure-definition" class="md-nav__link">
+  <a href="#122-definition-of-a-data-structure" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1.2.2 &nbsp; Data Structure Definition
+        1.2.2 &nbsp; Definition of a data structure
       </span>
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       <span class="md-typeset">
-        1.2.3 &nbsp; The Relationship Between Data Structures and Algorithms
+        1.2.3 &nbsp; The relationship between data structures and algorithms
       </span>
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 <!-- Page content -->
-<h1 id="12-what-is-an-algorithm">1.2 &nbsp; What Is an Algorithm<a class="headerlink" href="#12-what-is-an-algorithm" title="Permanent link">&para;</a></h1>
-<h2 id="121-algorithm-definition">1.2.1 &nbsp; Algorithm Definition<a class="headerlink" href="#121-algorithm-definition" title="Permanent link">&para;</a></h2>
-<p>An <u>algorithm</u> is a set of instructions or operational steps that solves a specific problem within a finite amount of time. It has the following characteristics.</p>
+<h1 id="12-what-is-an-algorithm">1.2 &nbsp; What is an algorithm?<a class="headerlink" href="#12-what-is-an-algorithm" title="Permanent link">&para;</a></h1>
+<h2 id="121-definition-of-an-algorithm">1.2.1 &nbsp; Definition of an algorithm<a class="headerlink" href="#121-definition-of-an-algorithm" title="Permanent link">&para;</a></h2>
+<p>An <u>algorithm</u> is a set of instructions or steps that solves a specific problem within a finite time. It has the following properties.</p>
 <ul>
-<li>The problem is well-defined, with clear input and output definitions.</li>
-<li>It is feasible and can be completed with finite steps, time, and memory.</li>
-<li>Each step has a definite meaning, and under the same input and operating conditions, the output is always the same.</li>
+<li>The problem is well defined, with clear definitions of its inputs and outputs.</li>
+<li>The algorithm is feasible: it can finish within a finite number of steps, using finite time and memory.</li>
+<li>Every step has a definite meaning. Given the same inputs and execution conditions, the output is always the same.</li>
 </ul>
-<h2 id="122-data-structure-definition">1.2.2 &nbsp; Data Structure Definition<a class="headerlink" href="#122-data-structure-definition" title="Permanent link">&para;</a></h2>
-<p>A <u>data structure</u> is a way of organizing and storing data, including the data itself, the relationships between data elements, and the methods used to operate on them. It has the following design objectives.</p>
+<h2 id="122-definition-of-a-data-structure">1.2.2 &nbsp; Definition of a data structure<a class="headerlink" href="#122-definition-of-a-data-structure" title="Permanent link">&para;</a></h2>
+<p>A <u>data structure</u> is a way of organizing and storing data. It encompasses the data itself, the relationships between data items, and the methods used to operate on them. Its design aims to achieve the following goals.</p>
 <ul>
-<li>Occupy as little space as possible to save computer memory.</li>
-<li>Data operations should be as fast as possible, covering data access, addition, deletion, update, etc.</li>
-<li>Provide a concise data representation and logical information so that algorithms can run efficiently.</li>
+<li>Use as little space as possible to conserve computer memory.</li>
+<li>Make data operations as fast as possible, including access, insertion, deletion, and updating.</li>
+<li>Provide a concise representation of data and its logical relationships so that algorithms can run efficiently.</li>
 </ul>
-<p><strong>Data structure design is a process full of trade-offs</strong>. If we want to achieve improvements in one aspect, we often need to make compromises in another aspect. Here are two examples.</p>
+<p><strong>Designing a data structure involves tradeoffs</strong>. Improving one aspect often requires compromising another. Here are two examples.</p>
 <ul>
-<li>Compared to arrays, linked lists are more convenient for data addition and deletion operations but sacrifice data access speed.</li>
-<li>Compared to linked lists, graphs provide richer logical information but require larger memory space.</li>
+<li>Compared with arrays, linked lists make insertion and deletion easier, at the expense of data access speed.</li>
+<li>Compared with linked lists, graphs provide richer information about logical relationships but require more memory.</li>
 </ul>
-<h2 id="123-the-relationship-between-data-structures-and-algorithms">1.2.3 &nbsp; The Relationship Between Data Structures and Algorithms<a class="headerlink" href="#123-the-relationship-between-data-structures-and-algorithms" title="Permanent link">&para;</a></h2>
-<p>As shown in Figure 1-4, data structures and algorithms are highly related and tightly coupled, specifically manifested in the following three aspects.</p>
+<h2 id="123-the-relationship-between-data-structures-and-algorithms">1.2.3 &nbsp; The relationship between data structures and algorithms<a class="headerlink" href="#123-the-relationship-between-data-structures-and-algorithms" title="Permanent link">&para;</a></h2>
+<p>As Figure 1-4 shows, data structures and algorithms are closely intertwined in three ways.</p>
 <ul>
-<li>Data structures are the foundation of algorithms. Data structures provide algorithms with structured storage of data and methods for operating on data.</li>
-<li>Algorithms breathe life into data structures. Data structures themselves only store data information; combined with algorithms, they can solve specific problems.</li>
-<li>Algorithms can usually be implemented based on different data structures, but execution efficiency may vary greatly. Choosing the appropriate data structure is key.</li>
+<li>Data structures are the foundation of algorithms. They provide algorithms with data in a structured form and methods for operating on that data.</li>
+<li>Algorithms bring data structures to life. A data structure by itself only stores information; it must be combined with an algorithm to solve a specific problem.</li>
+<li>An algorithm can often be implemented using different data structures, but its efficiency may vary greatly. Choosing the right data structure is crucial.</li>
 </ul>
 <p><img alt="The relationship between data structures and algorithms" class="animation-figure" src="../what_is_dsa.assets/relationship_between_data_structure_and_algorithm.png" /></p>
 <p align="center"> Figure 1-4 &nbsp; The relationship between data structures and algorithms </p>
 
-<p>Data structures and algorithms are like assembling building blocks as shown in Figure 1-5. A set of building blocks, in addition to containing many parts, also comes with detailed assembly instructions. By following the instructions step by step, we can assemble an exquisite building block model.</p>
-<p><img alt="Assembling blocks" class="animation-figure" src="../what_is_dsa.assets/assembling_blocks.png" /></p>
-<p align="center"> Figure 1-5 &nbsp; Assembling blocks </p>
+<p>Data structures and algorithms are like assembling the building blocks shown in Figure 1-5. A set of building blocks includes not only many pieces but also detailed assembly instructions. By following those instructions step by step, we can build an intricate model.</p>
+<p><img alt="Assembling building blocks" class="animation-figure" src="../what_is_dsa.assets/assembling_blocks.png" /></p>
+<p align="center"> Figure 1-5 &nbsp; Assembling building blocks </p>
 
-<p>The detailed correspondence between the two is shown in Table 1-1.</p>
-<p align="center"> Table 1-1 &nbsp; Comparing data structures and algorithms to assembling building blocks </p>
+<p>Table 1-1 describes this analogy in detail.</p>
+<p align="center"> Table 1-1 &nbsp; Comparing data structures and algorithms to building blocks </p>
 
 <div class="center-table">
 <table>
 <thead>
 <tr>
 <th>Data structures and algorithms</th>
-<th>Assembling building blocks</th>
+<th>Building blocks</th>
 </tr>
 </thead>
 <tbody>
 <tr>
 <td>Input data</td>
-<td>Unassembled building blocks</td>
+<td>Unassembled blocks</td>
 </tr>
 <tr>
 <td>Data structure</td>
-<td>Organization form of building blocks, including shape, size, connection method, etc.</td>
+<td>How the blocks are organized, including their shapes, sizes, and connections</td>
 </tr>
 <tr>
 <td>Algorithm</td>
-<td>A series of operational steps to assemble the blocks into the target form</td>
+<td>A sequence of steps for assembling the blocks into the desired form</td>
 </tr>
 <tr>
 <td>Output data</td>
-<td>Building block model</td>
+<td>The finished model</td>
 </tr>
 </tbody>
 </table>
 </div>
-<p>It is worth noting that data structures and algorithms are independent of programming languages. That is why this book can provide implementations in multiple programming languages.</p>
+<p>Data structures and algorithms are independent of programming languages. This is why the book can provide implementations in multiple languages.</p>
 <div class="admonition tip">
-<p class="admonition-title">Conventional abbreviation</p>
-<p>In actual discussions, we usually abbreviate "data structures and algorithms" as "algorithms". For example, the well-known LeetCode algorithm problems actually examine knowledge of both data structures and algorithms.</p>
+<p class="admonition-title">A common shorthand</p>
+<p>In everyday discussions, we often shorten “data structures and algorithms” to “algorithms.” For example, the familiar algorithm problems on LeetCode actually test knowledge of both data structures and algorithms.</p>
 </div>
 
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       <div class="md-ellipsis">
-        1.1 Algorithms Are Everywhere
+        1.1 &nbsp; Algorithms are everywhere
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m13.13 22.19-1.63-3.83c1.57-.58 3.04-1.36 4.4-2.27zM5.64 12.5l-3.83-1.63 6.1-2.77C7 9.46 6.22 10.93 5.64 12.5M19.22 4c.28 0 .53 0 .74.05.17 1.39-.02 4.25-3.3 7.53-1.7 1.71-3.73 3.02-6.01 3.89l-2.15-2.1c.92-2.31 2.23-4.34 3.92-6.03C15.18 4.58 17.64 4 19.22 4m0-2c-1.98 0-4.98.69-8.22 3.93-2.19 2.19-3.5 4.6-4.35 6.71-.28.75-.09 1.57.46 2.13l2.13 2.12c.38.38.89.61 1.42.61.23 0 .47-.06.7-.15A19.1 19.1 0 0 0 18.07 13c5.66-5.66 3.54-10.61 3.54-10.61S20.7 2 19.22 2m-4.68 7.46c-.78-.78-.78-2.05 0-2.83s2.05-.78 2.83 0c.77.78.78 2.05 0 2.83s-2.05.78-2.83 0m-5.66 7.07-1.41-1.41zM6.24 22l3.64-3.64c-.34-.09-.67-.24-.97-.45L4.83 22zM2 22h1.41l4.77-4.76-1.42-1.41L2 20.59zm0-2.83 4.09-4.08c-.21-.3-.36-.62-.45-.97L2 17.76z"/></svg>
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+  <span class="md-ellipsis">
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+  Chapter 0. &nbsp; Preface
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+  Chapter 0. &nbsp; Preface
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+  <span class="md-ellipsis">
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+  0.1 &nbsp; About this book
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+
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+    </li>
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+      <a href="../chapter_preface/suggestions/" class="md-nav__link">
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+
+  <span class="md-ellipsis">
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+  Chapter 1. &nbsp; Getting to know algorithms
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+  </span>
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+  <span class="md-ellipsis">
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+  1.1 &nbsp; Algorithms are everywhere
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+  1.2 &nbsp; What is an algorithm?
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+  </span>
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+      <a href="../chapter_introduction/summary/" class="md-nav__link">
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+
+
+  <span class="md-ellipsis">
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+  1.3 &nbsp; Summary
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+
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+
+
+      </a>
+    </li>
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+          </ul>
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+
+  <span class="md-ellipsis">
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+  Chapter 2. &nbsp; Complexity analysis
+
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+  </span>
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+            </a>
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+          <label class="md-nav__title" for="__nav_4">
+            <span class="md-nav__icon md-icon"></span>
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+  Chapter 2. &nbsp; Complexity analysis
+
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+      <a href="../chapter_computational_complexity/performance_evaluation/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
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+  2.1 &nbsp; Evaluating algorithm efficiency
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+
+  </span>
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+
+      </a>
+    </li>
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+
+
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+
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+
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+      <a href="../chapter_computational_complexity/iteration_and_recursion/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
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+
+  2.2 &nbsp; Iteration and recursion
+
+
+  </span>
+
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+
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+    </li>
+
+
+
+
+
+
+
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+
+    <li class="md-nav__item">
+      <a href="../chapter_computational_complexity/time_complexity/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  2.3 &nbsp; Time complexity
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+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_computational_complexity/space_complexity/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  2.4 &nbsp; Space complexity
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_computational_complexity/summary/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  2.5 &nbsp; Summary
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_computational_complexity/exercises/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
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+
+  2.6 &nbsp; Exercises
+
+
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+
+
+      </a>
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+          </ul>
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M11 13.5v8H3v-8zm-2 2H5v4h4zM12 2l5.5 9h-11zm0 3.86L10.08 9h3.84zM17.5 13c2.5 0 4.5 2 4.5 4.5S20 22 17.5 22 13 20 13 17.5s2-4.5 4.5-4.5m0 2a2.5 2.5 0 0 0-2.5 2.5 2.5 2.5 0 0 0 2.5 2.5 2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-2.5-2.5"/></svg>
+
+  <span class="md-ellipsis">
+
+
+  Chapter 3. &nbsp; Data structures
+
+
+  </span>
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+
+            </a>
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+          <label class="md-nav__title" for="__nav_5">
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+
+  Chapter 3. &nbsp; Data structures
+
+          </label>
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+
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+
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+      <a href="../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  3.1 &nbsp; Classifying data structures
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+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
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+
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+
+
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+
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+      <a href="../chapter_data_structure/basic_data_types/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
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+  3.2 &nbsp; Basic data types
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+
+  </span>
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+
+
+      </a>
+    </li>
+
+
+
+
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+
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+
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+
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+      <a href="../chapter_data_structure/number_encoding/" class="md-nav__link">
+
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+  <span class="md-ellipsis">
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+  3.3 &nbsp; Number encoding *
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+
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+
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+
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+      <a href="../chapter_data_structure/character_encoding/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  3.4 &nbsp; Character encoding *
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+
+  </span>
+
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+
+      </a>
+    </li>
+
+
+
+
+
+
+
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+    <li class="md-nav__item">
+      <a href="../chapter_data_structure/summary/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
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+
+  3.5 &nbsp; Summary
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+
+  </span>
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+
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+
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+
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+
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+      <a href="../chapter_data_structure/exercises/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
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+
+  3.6 &nbsp; Exercises
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+
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+
+  <span class="md-ellipsis">
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+  Chapter 4. &nbsp; Arrays and linked lists
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+  Chapter 4. &nbsp; Arrays and linked lists
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+
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+  <span class="md-ellipsis">
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+  4.1 &nbsp; Arrays
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+
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+
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+
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+
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+      <a href="../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
+
+
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+  <span class="md-ellipsis">
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+  4.2 &nbsp; Linked lists
+
+
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+
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+    </li>
+
+
+
+
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+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_array_and_linkedlist/list/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  4.3 &nbsp; Lists
+
+
+  </span>
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+
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+    </li>
+
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+
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+      <a href="../chapter_array_and_linkedlist/ram_and_cache/" class="md-nav__link">
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+
+  <span class="md-ellipsis">
+
+
+  4.4 &nbsp; Memory and cache *
+
+
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+
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+    </li>
+
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+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_array_and_linkedlist/summary/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
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+
+  4.5 &nbsp; Summary
+
+
+  </span>
+
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+
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+    </li>
+
+
+
+
+
+
+
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+      <a href="../chapter_array_and_linkedlist/exercises/" class="md-nav__link">
+
+
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+  <span class="md-ellipsis">
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+  4.6 &nbsp; Exercises
+
+
+  </span>
+
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+            <a href="../chapter_stack_and_queue/" class="md-nav__link ">
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+
+
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17.36 20.2v-5.38h1.79V22H3v-7.18h1.8v5.38zM6.77 14.32l.37-1.76 8.79 1.85-.37 1.76zm1.16-4.21.76-1.61 8.14 3.78-.76 1.62zm2.26-3.99 1.15-1.38 6.9 5.76-1.15 1.37zm4.45-4.25L20 9.08l-1.44 1.07-5.36-7.21zM6.59 18.41v-1.8h8.98v1.8z"/></svg>
+
+  <span class="md-ellipsis">
+
+
+  Chapter 5. &nbsp; Stacks and queues
+
+
+  </span>
+
+
+
+            </a>
+
+
+              <label class="md-nav__link " for="__nav_7" id="__nav_7_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+
+          </div>
+
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_7">
+            <span class="md-nav__icon md-icon"></span>
+
+
+  Chapter 5. &nbsp; Stacks and queues
+
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_stack_and_queue/stack/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  5.1 &nbsp; Stacks
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_stack_and_queue/queue/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  5.2 &nbsp; Queues
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_stack_and_queue/deque/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  5.3 &nbsp; Deques
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_stack_and_queue/summary/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  5.4 &nbsp; Summary
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_stack_and_queue/exercises/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  5.5 &nbsp; Exercises
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+          </ul>
+        </nav>
+
+    </li>
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item md-nav__item--nested">
+
+
+
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
+
+
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_hashing/" class="md-nav__link ">
+
+
+
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.3 17.89c1.32-2.1.7-4.89-1.41-6.21a4.52 4.52 0 0 0-6.21 1.41C10.36 15.2 11 18 13.09 19.3c1.47.92 3.33.92 4.8 0L21 22.39 22.39 21zm-2-.62c-.98.98-2.56.97-3.54 0-.97-.98-.97-2.56.01-3.54.97-.97 2.55-.97 3.53 0 .96.99.95 2.57-.03 3.54zM19 4H5a2 2 0 0 0-2 2v12a2 2 0 0 0 2 2h5.81a6.3 6.3 0 0 1-1.31-2H5v-4h4.18c.16-.71.43-1.39.82-2H5V8h6v2.81a6.3 6.3 0 0 1 2-1.31V8h6v2a6.499 6.499 0 0 1 2 2V6a2 2 0 0 0-2-2"/></svg>
+
+  <span class="md-ellipsis">
+
+
+  Chapter 6. &nbsp; Hash tables
+
+
+  </span>
+
+
+
+            </a>
+
+
+              <label class="md-nav__link " for="__nav_8" id="__nav_8_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+
+          </div>
+
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_8">
+            <span class="md-nav__icon md-icon"></span>
+
+
+  Chapter 6. &nbsp; Hash tables
+
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_hashing/hash_map/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  6.1 &nbsp; Hash tables
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_hashing/hash_collision/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  6.2 &nbsp; Hash collisions
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_hashing/hash_algorithm/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  6.3 &nbsp; Hash algorithms
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_hashing/summary/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  6.4 &nbsp; Summary
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_hashing/exercises/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  6.5 &nbsp; Exercises
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+          </ul>
+        </nav>
+
+    </li>
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item md-nav__item--nested">
+
+
+
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
+
+
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_tree/" class="md-nav__link ">
+
+
+
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19.5 17c-.14 0-.26 0-.39.04L17.5 13.8c.45-.45.75-1.09.75-1.8a2.5 2.5 0 0 0-2.5-2.5c-.14 0-.25 0-.4.04L13.74 6.3c.47-.46.76-1.09.76-1.8a2.5 2.5 0 0 0-5 0c0 .7.29 1.34.76 1.79L8.65 9.54c-.15-.04-.26-.04-.4-.04a2.5 2.5 0 0 0-2.5 2.5c0 .71.29 1.34.75 1.79l-1.61 3.25C4.76 17 4.64 17 4.5 17a2.5 2.5 0 0 0 0 5A2.5 2.5 0 0 0 7 19.5c0-.7-.29-1.34-.76-1.79l1.62-3.25c.14.04.26.04.39.04s.25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0A2.5 2.5 0 0 0 12 17c-.13 0-.26 0-.39.04L10 13.8c.45-.45.75-1.09.75-1.8 0-.7-.29-1.33-.75-1.79l1.61-3.25c.13.04.26.04.39.04s.26 0 .39-.04L14 10.21a2.5 2.5 0 0 0 1.75 4.29c.13 0 .25 0 .38-.04l1.63 3.25c-.47.45-.76 1.09-.76 1.79a2.5 2.5 0 0 0 5 0 2.5 2.5 0 0 0-2.5-2.5m-15 3.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1m8.5-1c0 .55-.45 1-1 1s-1-.45-1-1 .45-1 1-1 1 .45 1 1M7.25 12c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1M11 4.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m3.75 7.5c0-.55.45-1 1-1s1 .45 1 1-.45 1-1 1-1-.45-1-1m4.75 8.5c-.55 0-1-.45-1-1s.45-1 1-1 1 .45 1 1-.45 1-1 1"/></svg>
+
+  <span class="md-ellipsis">
+
+
+  Chapter 7. &nbsp; Trees
+
+
+  </span>
+
+
+
+            </a>
+
+
+              <label class="md-nav__link " for="__nav_9" id="__nav_9_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+
+          </div>
+
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_9">
+            <span class="md-nav__icon md-icon"></span>
+
+
+  Chapter 7. &nbsp; Trees
+
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_tree/binary_tree/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  7.1 &nbsp; Binary trees
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_tree/binary_tree_traversal/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  7.2 &nbsp; Traversing a binary tree
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_tree/array_representation_of_tree/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  7.3 &nbsp; Array representation of a binary tree
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_tree/binary_search_tree/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  7.4 &nbsp; Binary search trees
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_tree/avl_tree/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  7.5 &nbsp; AVL trees *
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_tree/summary/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  7.6 &nbsp; Summary
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_tree/exercises/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  7.7 &nbsp; Exercises
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+          </ul>
+        </nav>
+
+    </li>
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item md-nav__item--nested">
+
+
+
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
+
+
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_heap/" class="md-nav__link ">
+
+
+
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 1a2.5 2.5 0 0 0-2.5 2.5A2.5 2.5 0 0 0 11 5.79V7H7a2 2 0 0 0-2 2v.71A2.5 2.5 0 0 0 3.5 12 2.5 2.5 0 0 0 5 14.29V15H4a2 2 0 0 0-2 2v1.21A2.5 2.5 0 0 0 .5 20.5 2.5 2.5 0 0 0 3 23a2.5 2.5 0 0 0 2.5-2.5A2.5 2.5 0 0 0 4 18.21V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 9 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2H7v-.71A2.5 2.5 0 0 0 8.5 12 2.5 2.5 0 0 0 7 9.71V9h10v.71A2.5 2.5 0 0 0 15.5 12a2.5 2.5 0 0 0 1.5 2.29V15h-1a2 2 0 0 0-2 2v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 15 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17h4v1.21a2.5 2.5 0 0 0-1.5 2.29A2.5 2.5 0 0 0 21 23a2.5 2.5 0 0 0 2.5-2.5 2.5 2.5 0 0 0-1.5-2.29V17a2 2 0 0 0-2-2h-1v-.71A2.5 2.5 0 0 0 20.5 12 2.5 2.5 0 0 0 19 9.71V9a2 2 0 0 0-2-2h-4V5.79a2.5 2.5 0 0 0 1.5-2.29A2.5 2.5 0 0 0 12 1m0 1.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M6 11a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m12 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1M3 19.5a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1m6 0a1 1 0 0 1 1 1 1 1 0 0 1-1 1 1 1 0 0 1-1-1 1 1 0 0 1 1-1"/></svg>
+
+  <span class="md-ellipsis">
+
+
+  Chapter 8. &nbsp; Heaps
+
+
+  </span>
+
+
+
+            </a>
+
+
+              <label class="md-nav__link " for="__nav_10" id="__nav_10_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+
+          </div>
+
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_10">
+            <span class="md-nav__icon md-icon"></span>
+
+
+  Chapter 8. &nbsp; Heaps
+
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_heap/heap/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  8.1 &nbsp; Heaps
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_heap/build_heap/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  8.2 &nbsp; Building a heap
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_heap/top_k/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  8.3 &nbsp; The top-k problem
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_heap/summary/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  8.4 &nbsp; Summary
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_heap/exercises/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  8.5 &nbsp; Exercises
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+          </ul>
+        </nav>
+
+    </li>
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item md-nav__item--nested">
+
+
+
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
+
+
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_graph/" class="md-nav__link ">
+
+
+
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m12 5.37-.44-.06L6 14.9c.24.21.4.48.47.78h11.06c.07-.3.23-.57.47-.78l-5.56-9.59zM6.6 16.53l4.28 2.53c.29-.27.69-.43 1.12-.43s.83.16 1.12.43l4.28-2.53zM12 22a1.68 1.68 0 0 1-1.68-1.68l.09-.56-4.3-2.55c-.31.36-.76.58-1.27.58a1.68 1.68 0 0 1-1.68-1.68c0-.79.53-1.45 1.26-1.64V9.36c-.83-.11-1.47-.82-1.47-1.68A1.68 1.68 0 0 1 4.63 6c.55 0 1.03.26 1.34.66l4.41-2.53-.06-.45c0-.93.75-1.68 1.68-1.68s1.68.75 1.68 1.68l-.06.45 4.41 2.53c.31-.4.79-.66 1.34-.66a1.68 1.68 0 0 1 1.68 1.68c0 .86-.64 1.57-1.47 1.68v5.11c.73.19 1.26.85 1.26 1.64a1.68 1.68 0 0 1-1.68 1.68c-.51 0-.96-.22-1.27-.58l-4.3 2.55.09.56A1.68 1.68 0 0 1 12 22M10.8 4.86 6.3 7.44l.02.24c0 .71-.44 1.32-1.06 1.57l.03 5.25zm2.4 0 5.51 9.64.03-5.25c-.62-.25-1.06-.86-1.06-1.57l.02-.24z"/></svg>
+
+  <span class="md-ellipsis">
+
+
+  Chapter 9. &nbsp; Graphs
+
+
+  </span>
+
+
+
+            </a>
+
+
+              <label class="md-nav__link " for="__nav_11" id="__nav_11_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+
+          </div>
+
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_11">
+            <span class="md-nav__icon md-icon"></span>
+
+
+  Chapter 9. &nbsp; Graphs
+
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_graph/graph/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  9.1 &nbsp; Graphs
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_graph/graph_operations/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  9.2 &nbsp; Basic graph operations
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_graph/graph_traversal/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  9.3 &nbsp; Graph traversal
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_graph/summary/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  9.4 &nbsp; Summary
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_graph/exercises/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  9.5 &nbsp; Exercises
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+          </ul>
+        </nav>
+
+    </li>
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item md-nav__item--nested">
+
+
+
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" >
+
+
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_searching/" class="md-nav__link ">
+
+
+
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19.31 18.9 3.08 3.1L21 23.39l-3.12-3.07c-.69.43-1.51.68-2.38.68-2.5 0-4.5-2-4.5-4.5s2-4.5 4.5-4.5 4.5 2 4.5 4.5c0 .88-.25 1.71-.69 2.4m-3.81.1a2.5 2.5 0 0 0 0-5 2.5 2.5 0 0 0 0 5M21 4v2H3V4zM3 16v-2h6v2zm0-5V9h18v2h-2.03c-1.01-.63-2.2-1-3.47-1s-2.46.37-3.47 1z"/></svg>
+
+  <span class="md-ellipsis">
+
+
+  Chapter 10. &nbsp; Searching
+
+
+  </span>
+
+
+
+            </a>
+
+
+              <label class="md-nav__link " for="__nav_12" id="__nav_12_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+
+          </div>
+
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_12">
+            <span class="md-nav__icon md-icon"></span>
+
+
+  Chapter 10. &nbsp; Searching
+
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_searching/binary_search/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  10.1 &nbsp; Binary search
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_searching/binary_search_insertion/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  10.2 &nbsp; Finding an insertion point with binary search
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_searching/binary_search_edge/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  10.3 &nbsp; Finding boundaries with binary search
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  10.4 &nbsp; Optimizing with hashing
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  10.5 &nbsp; Search algorithms revisited
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_searching/summary/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  10.6 &nbsp; Summary
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_searching/exercises/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  10.7 &nbsp; Exercises
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+          </ul>
+        </nav>
+
+    </li>
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item md-nav__item--nested">
+
+
+
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
+
+
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_sorting/" class="md-nav__link ">
+
+
+
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 17h3l-4 4-4-4h3V3h2M2 17h10v2H2M6 5v2H2V5m0 6h7v2H2z"/></svg>
+
+  <span class="md-ellipsis">
+
+
+  Chapter 11. &nbsp; Sorting
+
+
+  </span>
+
+
+
+            </a>
+
+
+              <label class="md-nav__link " for="__nav_13" id="__nav_13_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+
+          </div>
+
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_13">
+            <span class="md-nav__icon md-icon"></span>
+
+
+  Chapter 11. &nbsp; Sorting
+
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/sorting_algorithm/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  11.1 &nbsp; Sorting algorithms
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/selection_sort/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  11.2 &nbsp; Selection sort
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/bubble_sort/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  11.3 &nbsp; Bubble sort
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/insertion_sort/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  11.4 &nbsp; Insertion sort
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/quick_sort/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  11.5 &nbsp; Quick sort
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/merge_sort/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  11.6 &nbsp; Merge sort
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/heap_sort/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  11.7 &nbsp; Heap sort
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/bucket_sort/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  11.8 &nbsp; Bucket sort
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/counting_sort/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  11.9 &nbsp; Counting sort
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/radix_sort/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  11.10 &nbsp; Radix sort
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/summary/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  11.11 &nbsp; Summary
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_sorting/exercises/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  11.12 &nbsp; Exercises
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+          </ul>
+        </nav>
+
+    </li>
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item md-nav__item--nested">
+
+
+
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
+
+
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_divide_and_conquer/" class="md-nav__link ">
+
+
+
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17 7v2h5V7zM2 9v6h5V9zm10 0v2H9v2h3v2l3-3zm5 2v2h5v-2zm0 4v2h5v-2z"/></svg>
+
+  <span class="md-ellipsis">
+
+
+  Chapter 12. &nbsp; Divide and conquer
+
+
+  </span>
+
+
+
+            </a>
+
+
+              <label class="md-nav__link " for="__nav_14" id="__nav_14_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+
+          </div>
+
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_14">
+            <span class="md-nav__icon md-icon"></span>
+
+
+  Chapter 12. &nbsp; Divide and conquer
+
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_divide_and_conquer/divide_and_conquer/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  12.1 &nbsp; Divide and conquer
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_divide_and_conquer/binary_search_recur/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  12.2 &nbsp; Divide-and-conquer search
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_divide_and_conquer/build_binary_tree_problem/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  12.3 &nbsp; Building a binary tree
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_divide_and_conquer/hanota_problem/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  12.4 &nbsp; Tower of Hanoi
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_divide_and_conquer/summary/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  12.5 &nbsp; Summary
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_divide_and_conquer/exercises/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  12.6 &nbsp; Exercises
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+          </ul>
+        </nav>
+
+    </li>
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item md-nav__item--nested">
+
+
+
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
+
+
+          <div class="md-nav__link md-nav__container">
+            <a href="../chapter_backtracking/" class="md-nav__link ">
+
+
+
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M18 15a3 3 0 0 1 3 3 3 3 0 0 1-3 3 2.99 2.99 0 0 1-2.83-2H14v-2h1.17c.41-1.17 1.52-2 2.83-2m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m0-9a1.43 1.43 0 0 0 1.43-1.43 1.43 1.43 0 1 0-2.86 0A1.43 1.43 0 0 0 18 8m0-5.43a4 4 0 0 1 4 4C22 9.56 18 14 18 14s-4-4.44-4-7.43a4 4 0 0 1 4-4M8.83 17H10v2H8.83A2.99 2.99 0 0 1 6 21a3 3 0 0 1-3-3c0-1.31.83-2.42 2-2.83V14h2v1.17c.85.3 1.53.98 1.83 1.83M6 17a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1M6 3a3 3 0 0 1 3 3c0 1.31-.83 2.42-2 2.83V10H5V8.83A2.99 2.99 0 0 1 3 6a3 3 0 0 1 3-3m0 2a1 1 0 0 0-1 1 1 1 0 0 0 1 1 1 1 0 0 0 1-1 1 1 0 0 0-1-1m5 14v-2h2v2zm-4-6H5v-2h2z"/></svg>
+
+  <span class="md-ellipsis">
+
+
+  Chapter 13. &nbsp; Backtracking
+
+
+  </span>
+
+
+
+            </a>
+
+
+              <label class="md-nav__link " for="__nav_15" id="__nav_15_label" tabindex="0">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+
+          </div>
+
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_15_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_15">
+            <span class="md-nav__icon md-icon"></span>
+
+
+  Chapter 13. &nbsp; Backtracking
+
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  13.1 &nbsp; Backtracking algorithms
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_backtracking/permutations_problem/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  13.2 &nbsp; Permutations problem
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  13.3 &nbsp; Subset sum problem
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_backtracking/n_queens_problem/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  13.4 &nbsp; N-queens problem
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_backtracking/summary/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  13.5 &nbsp; Summary
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item">
+      <a href="../chapter_backtracking/exercises/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
+
+
+  13.6 &nbsp; Exercises
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+          </ul>
+        </nav>
+
+    </li>
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+    <li class="md-nav__item md-nav__item--nested">
+
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+
+  <span class="md-ellipsis">
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+  Chapter 14. &nbsp; Dynamic programming
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+  14.1 &nbsp; Introduction to dynamic programming
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+
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+
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+    </li>
+
+
+
+
+
+
+
+
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+
+    <li class="md-nav__item">
+      <a href="../chapter_dynamic_programming/dp_problem_features/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
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+
+  14.2 &nbsp; Properties of dynamic programming problems
+
+
+  </span>
+
+
+
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+    </li>
+
+
+
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+
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+      <a href="../chapter_dynamic_programming/dp_solution_pipeline/" class="md-nav__link">
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+
+
+  <span class="md-ellipsis">
+
+
+  14.3 &nbsp; Solving dynamic programming problems
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
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+
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+      <a href="../chapter_dynamic_programming/knapsack_problem/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
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+
+  14.4 &nbsp; 0-1 knapsack problem
+
+
+  </span>
+
+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
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+
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+      <a href="../chapter_dynamic_programming/unbounded_knapsack_problem/" class="md-nav__link">
+
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+
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+
+  14.5 &nbsp; Unbounded knapsack problem
+
+
+  </span>
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+
+
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+    </li>
+
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+      <a href="../chapter_dynamic_programming/edit_distance_problem/" class="md-nav__link">
+
+
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+  <span class="md-ellipsis">
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+
+  14.6 &nbsp; Edit distance problem
+
+
+  </span>
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+
+
+      </a>
+    </li>
+
+
+
+
+
+
+
+
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+
+    <li class="md-nav__item">
+      <a href="../chapter_dynamic_programming/summary/" class="md-nav__link">
+
+
+
+  <span class="md-ellipsis">
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+
+  14.7 &nbsp; Summary
+
+
+  </span>
+
+
+
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+    </li>
+
+
+
+
+
+
+
+
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+
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+      <a href="../chapter_dynamic_programming/exercises/" class="md-nav__link">
+
+
+
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+
+  14.8 &nbsp; Exercises
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+
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+
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 3c3.88 0 7 3.14 7 7 0 2.8-1.63 5.19-4 6.31V21H9v-3H8c-1.11 0-2-.89-2-2v-3H4.5c-.42 0-.66-.5-.42-.81L6 9.66A7.003 7.003 0 0 1 13 3m0-2C8.41 1 4.61 4.42 4.06 8.9L2.5 11h-.03l-.02.03c-.55.76-.62 1.76-.19 2.59.36.69 1 1.17 1.74 1.32V16c0 1.85 1.28 3.42 3 3.87V23h11v-5.5c2.5-1.67 4-4.44 4-7.5 0-4.97-4.04-9-9-9m4 7.83c0 1.54-1.36 2.77-3.42 4.64L13 14l-.58-.53C10.36 11.6 9 10.37 9 8.83c0-1.2.96-2.19 2.16-2.2h.04c.69 0 1.35.31 1.8.83.45-.52 1.11-.83 1.8-.83 1.2-.01 2.2.96 2.2 2.16z"/></svg>
+
+  <span class="md-ellipsis">
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+  Chapter 15. &nbsp; Greedy algorithms
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+  15.1 &nbsp; Greedy algorithms
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+  15.2 &nbsp; Fractional knapsack problem
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+  15.3 &nbsp; Maximum capacity problem
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+
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+
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+      <a href="../chapter_greedy/max_product_cutting_problem/" class="md-nav__link">
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+  15.4 &nbsp; Maximum product cutting problem
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+
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+      <a href="../chapter_greedy/summary/" class="md-nav__link">
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+  15.5 &nbsp; Summary
+
+
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+
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+
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+
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+      <a href="../chapter_greedy/exercises/" class="md-nav__link">
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+
+  15.6 &nbsp; Exercises
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+
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+  16.3 &nbsp; Glossary
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+<h1 id="print-edition-chinese">Print edition (Chinese)<a class="headerlink" href="#print-edition-chinese" title="Permanent link">&para;</a></h1>
+<p>After a long period of refinement, the Simplified Chinese print edition of <em>Hello Algo</em> is finally here! A line of poetry captures how I feel:</p>
+<p align="center">Chase the wind and moon without pause; beyond the open plains lie the mountains of spring.</p>
+
+<p><img alt="" class="animation-figure" src="index.assets/paper_book_overview.jpg" /></p>
+<p>The following video introduces the print edition and shares some of my thoughts on:</p>
+<ul>
+<li>The importance of learning data structures and algorithms.</li>
+<li>Why I chose Python for the print edition.</li>
+<li>What sharing knowledge means to me.</li>
+</ul>
+<blockquote>
+<p>I am new to making videos. Please show your support with a like, a coin, and a favorite—thank you!</p>
+</blockquote>
+<div class="video-container">
+    <iframe src="//player.bilibili.com/player.html?aid=1051597767&bvid=BV1QH4y157uC&cid=1462564112&p=1&autoplay=0" scrolling="no" border="0" frameborder="no" framespacing="0" allowfullscreen="true"> </iframe>
+</div>
+
+<p>Here are some snapshots of the print edition:</p>
+<p><img alt="" class="animation-figure" src="index.assets/paper_book_chapter_heap.jpg" /></p>
+<p><img alt="" class="animation-figure" src="index.assets/paper_book_avl_tree.jpg" /></p>
+<h2 id="strengths-and-limitations">Strengths and limitations<a class="headerlink" href="#strengths-and-limitations" title="Permanent link">&para;</a></h2>
+<p>Here are a few things about the print edition that may pleasantly surprise you:</p>
+<ul>
+<li>Full-color printing makes the most of the book's animation-style illustrations.</li>
+<li>Carefully selected paper reproduces colors faithfully while preserving the distinctive feel of a printed book.</li>
+<li>Formatting is more standardized than in the web edition; for example, formulas in figures are italicized.</li>
+<li>A fold-out mind map and bookmarks are included at no extra cost.</li>
+<li>The print, web, and PDF editions have synchronized content, so you can switch freely between them.</li>
+</ul>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>Synchronizing the print and web editions is difficult, so there may be minor differences. Thank you for your understanding!</p>
+</div>
+<p>Of course, there are also a few points to consider before buying:</p>
+<ul>
+<li>The book uses Python, which may not be your main language. You can treat Python as pseudocode and focus on understanding the ideas.</li>
+<li>Full-color printing greatly improves the reading experience for illustrations and code, but costs somewhat more than black-and-white printing.</li>
+</ul>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>“Print quality” and “price” are like “time efficiency” and “space efficiency” in algorithms: it is difficult to optimize both. In my view, print quality corresponds to time efficiency and deserves greater priority.</p>
+</div>
+<h2 id="where-to-buy">Where to buy<a class="headerlink" href="#where-to-buy" title="Permanent link">&para;</a></h2>
+<p>If you are interested in the Simplified Chinese print edition, consider getting a copy. We have arranged a 50% discount on the new book. Follow <a href="https://3.cn/1X-qmTD3">this link</a> or scan the QR code below:</p>
+<p><img alt="" class="animation-figure" src="index.assets/book_jd_link.jpg" /></p>
+<h2 id="afterword">Afterword<a class="headerlink" href="#afterword" title="Permanent link">&para;</a></h2>
+<p>At first, I underestimated the work involved in publishing a print edition. I assumed that as long as the open-source project was well maintained, some form of automation could generate the printed book. Experience showed that producing a printed book differs greatly from updating an open-source project, and moving between the two requires considerable extra work.</p>
+<p>There is still a long road from a first draft to a manuscript ready for publication. It takes close collaboration and sustained refinement by the author and the publisher's planning, editorial, design, and marketing teams. My thanks go to Turing's commissioning editor Junhua Wang and everyone at Posts &amp; Telecom Press and Turing Community who took part in publishing this book!</p>
+<p>I hope this book will help you!</p>
+
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\ No newline at end of file
diff --git a/en/chapter_preface/about_the_book/index.html b/en/chapter_preface/about_the_book/index.html
index 36bd6a0d8..cce56bfaa 100644
--- a/en/chapter_preface/about_the_book/index.html
+++ b/en/chapter_preface/about_the_book/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>0.1 About This Book - Hello Algo</title>
+        <title>0.1 About this book - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -119,7 +119,7 @@
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-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              0.1 &nbsp; About This Book
+              0.1 &nbsp; About this book
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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@@ -177,9 +177,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -421,7 +421,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -442,7 +442,7 @@
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -472,7 +472,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -489,7 +489,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -515,10 +515,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#011-target-audience" class="md-nav__link">
+  <a href="#011-who-this-book-is-for" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        0.1.1 &nbsp; Target Audience
+        0.1.1 &nbsp; Who this book is for
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     </span>
   </a>
@@ -526,10 +526,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#012-content-structure" class="md-nav__link">
+  <a href="#012-organization" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        0.1.2 &nbsp; Content Structure
+        0.1.2 &nbsp; Organization
       </span>
     </span>
   </a>
@@ -537,10 +537,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#013-acknowledgements" class="md-nav__link">
+  <a href="#013-acknowledgments" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        0.1.3 &nbsp; Acknowledgements
+        0.1.3 &nbsp; Acknowledgments
       </span>
     </span>
   </a>
@@ -570,7 +570,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -597,7 +597,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -657,7 +657,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -678,7 +678,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -699,7 +699,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -726,7 +726,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -753,7 +753,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -819,7 +819,7 @@
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
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   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
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-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1347,7 +1347,7 @@
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-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1588,7 +1588,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1802,7 +1802,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1883,7 +1883,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1978,7 +1978,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2020,7 +2020,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -2047,7 +2047,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2074,7 +2074,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2101,7 +2101,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2128,7 +2128,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2155,7 +2155,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2182,7 +2182,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2246,7 +2246,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2267,7 +2267,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2288,7 +2288,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2315,7 +2315,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2342,7 +2342,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2369,7 +2369,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2396,7 +2396,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2460,7 +2460,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2481,7 +2481,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2502,7 +2502,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2529,7 +2529,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2556,7 +2556,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2583,7 +2583,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2610,7 +2610,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2678,7 +2678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2699,7 +2699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2720,7 +2720,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2747,7 +2747,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2774,7 +2774,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2801,7 +2801,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2828,7 +2828,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2855,7 +2855,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2882,7 +2882,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2960,7 +2960,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2981,7 +2981,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3002,7 +3002,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3029,7 +3029,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3056,7 +3056,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3083,7 +3083,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3110,7 +3110,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3137,7 +3137,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3164,7 +3164,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3191,7 +3191,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3218,7 +3218,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3245,7 +3245,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3272,7 +3272,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3299,7 +3299,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3365,7 +3365,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3386,7 +3386,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3407,7 +3407,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3434,7 +3434,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3461,7 +3461,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3488,7 +3488,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3515,7 +3515,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3608,7 +3608,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3650,7 +3650,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3704,7 +3704,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3731,7 +3731,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3758,7 +3758,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3855,7 +3855,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3876,7 +3876,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3897,7 +3897,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3924,7 +3924,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3951,7 +3951,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3978,7 +3978,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4005,7 +4005,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4032,7 +4032,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4059,7 +4059,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4086,7 +4086,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4152,7 +4152,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4173,7 +4173,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4194,7 +4194,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4221,7 +4221,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4248,7 +4248,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4275,7 +4275,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4302,7 +4302,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4329,7 +4329,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4389,7 +4389,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4431,7 +4431,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4458,7 +4458,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4485,7 +4485,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4569,6 +4569,78 @@
 
 
 
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+  Print edition (Chinese)
+
+          </label>
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 <!-- Page content -->
-<h1 id="01-about-this-book">0.1 &nbsp; About This Book<a class="headerlink" href="#01-about-this-book" title="Permanent link">&para;</a></h1>
-<p>This project aims to create an open-source, free, beginner-friendly introductory tutorial on data structures and algorithms.</p>
+<h1 id="01-about-this-book">0.1 &nbsp; About this book<a class="headerlink" href="#01-about-this-book" title="Permanent link">&para;</a></h1>
+<p>This project aims to create an open-source, free, beginner-friendly introduction to data structures and algorithms.</p>
 <ul>
-<li>The entire book uses animated illustrations, with clear and easy-to-understand content and a smooth learning curve, guiding beginners through the landscape of data structures and algorithms.</li>
-<li>The source code can be run with one click, helping readers improve their programming skills through practice and understand how algorithms work and the underlying implementation of data structures.</li>
-<li>We encourage readers to learn from each other, and everyone is welcome to ask questions and share insights in the comments section, making progress together through discussion and exchange.</li>
+<li>Animated illustrations throughout the book make concepts clear and accessible, with a gentle learning curve that guides beginners through the landscape of data structures and algorithms.</li>
+<li>Source code runs with one click, helping readers improve their programming skills through practice and understand how algorithms work and how data structures are implemented.</li>
+<li>We encourage readers to learn together. Ask questions and share insights in the comments, and make progress through discussion.</li>
 </ul>
-<h2 id="011-target-audience">0.1.1 &nbsp; Target Audience<a class="headerlink" href="#011-target-audience" title="Permanent link">&para;</a></h2>
-<p>If you are an algorithm beginner who has never studied algorithms, or if you already have some problem-solving experience but only a hazy understanding of data structures and algorithms, then this book is tailor-made for you!</p>
-<p>If you have already accumulated a certain amount of problem-solving experience and are familiar with most question types, this book can help you review and organize your algorithm knowledge system, and the repository's source code can be used as a "problem-solving toolkit" or "algorithm dictionary."</p>
-<p>If you are an algorithm "expert," we look forward to receiving your valuable suggestions, or <a href="https://www.hello-algo.com/chapter_appendix/contribution/">joining us as a contributor</a>.</p>
+<h2 id="011-who-this-book-is-for">0.1.1 &nbsp; Who this book is for<a class="headerlink" href="#011-who-this-book-is-for" title="Permanent link">&para;</a></h2>
+<p>If you are new to algorithms, have never studied them, or have some experience with practice problems but only a hazy understanding of data structures and algorithms—sometimes feeling you understand, sometimes not—this book is written for you!</p>
+<p>If you have already solved a substantial number of problems and know most common problem types, this book can help you review and organize your knowledge. The repository's source code can serve as a “problem-solving toolkit” or an “algorithm dictionary.”</p>
+<p>If you are an expert in algorithms, we welcome your valuable suggestions and invite you to <a href="../../chapter_appendix/contribution/">contribute to the book</a>.</p>
 <div class="admonition success">
 <p class="admonition-title">Prerequisites</p>
-<p>You need basic programming knowledge in at least one language and the ability to read and write simple code.</p>
+<p>You should have basic programming skills in at least one language and be able to read and write simple code.</p>
 </div>
-<h2 id="012-content-structure">0.1.2 &nbsp; Content Structure<a class="headerlink" href="#012-content-structure" title="Permanent link">&para;</a></h2>
-<p>The main content of this book is shown in Figure 0-1.</p>
+<h2 id="012-organization">0.1.2 &nbsp; Organization<a class="headerlink" href="#012-organization" title="Permanent link">&para;</a></h2>
+<p>Figure 0-1 shows the book's main topics.</p>
 <ul>
-<li><strong>Complexity analysis</strong>: Evaluation dimensions and methods for data structures and algorithms. Methods for calculating time complexity and space complexity, common types, examples, etc.</li>
-<li><strong>Data structures</strong>: Classification methods for basic data types and data structures. Definitions, advantages and disadvantages, common operations, common types, typical applications, implementation methods, and more for data structures such as arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graphs.</li>
-<li><strong>Algorithms</strong>: The definition, advantages and disadvantages, efficiency, application scenarios, problem-solving steps, and example problems of algorithms such as searching, sorting, divide and conquer, backtracking, dynamic programming, and greedy algorithms.</li>
+<li><strong>Complexity analysis</strong>: Criteria and methods for evaluating data structures and algorithms, including how to derive time and space complexity, common complexity classes, and examples.</li>
+<li><strong>Data structures</strong>: Basic data types and ways to classify data structures; definitions, strengths, weaknesses, common operations, variants, typical applications, and implementations of arrays, linked lists, stacks, queues, hash tables, trees, heaps, graphs, and more.</li>
+<li><strong>Algorithms</strong>: Definitions, strengths, weaknesses, efficiency, applications, solution steps, and example problems for searching, sorting, divide and conquer, backtracking, dynamic programming, greedy algorithms, and more.</li>
 </ul>
-<p><img alt="Main content of this book" class="animation-figure" src="../about_the_book.assets/hello_algo_mindmap.png" /></p>
-<p align="center"> Figure 0-1 &nbsp; Main content of this book </p>
+<p><img alt="The book's main topics" class="animation-figure" src="../about_the_book.assets/hello_algo_mindmap.png" /></p>
+<p align="center"> Figure 0-1 &nbsp; The book's main topics </p>
 
-<h2 id="013-acknowledgements">0.1.3 &nbsp; Acknowledgements<a class="headerlink" href="#013-acknowledgements" title="Permanent link">&para;</a></h2>
-<p>This book has been continuously improved through the joint efforts of many contributors in the open-source community. Thanks to every contributor who invested time and effort, they are (in the order automatically generated by GitHub): krahets, coderonion, Gonglja, nuomi1, Reanon, justin-tse, hpstory, danielsss, curtishd, night-cruise, S-N-O-R-L-A-X, rongyi, msk397, gvenusleo, khoaxuantu, rivertwilight, K3v123, gyt95, zhuoqinyue, yuelinxin, Zuoxun, mingXta, Phoenix0415, FangYuan33, GN-Yu, longsizhuo, pengchzn, QiLOL, Cathay-Chen, guowei-gong, xBLACKICEx, IsChristina, JoseHung, qualifier1024, hello-ikun, magentaqin, Guanngxu, thomasq0, sunshinesDL, L-Super, Transmigration-zhou, WSL0809, Slone123c, lhxsm, yuan0221, what-is-me, theNefelibatas, Shyam-Chen, sangxiaai, longranger2, codeberg-user, xiongsp, JeffersonHuang, prinpal, seven1240, Wonderdch, malone6, xiaomiusa87, gaofer, bluebean-cloud, a16su, SamJin98, hongyun-robot, nanlei, XiaChuerwu, yd-j, iron-irax, mgisr, steventimes, junminhong, heshuyue, danny900714, Nigh, Dr-XYZ, MolDuM, XC-Zero, reeswell, PXG-XPG, NI-SW, Horbin-Magician, Enlightenus, YangXuanyi, xjr7670, beatrix-chan, DullSword, qq909244296, iStig, boloboloda, hts0000, gledfish, fbigm, echo1937, jiaxianhua, wenjianmin, keshida, kilikilikid, lclc6, lwbaptx, linyejoe2, liuxjerry, szu17dmy, dshlstarr, Yucao-cy, coderlef, czruby, bongbongbakudan, beintentional, ZongYangL, ZhongYuuu, ZhongGuanbin, hezhizhen, linzeyan, ZJKung, JTCPOWI, KawaiiAsh, luluxia, xb534, ztkuaikuai, yw-1021, ElaBosak233, baagod, zhouLion, yishangzhang, yi427, yanedie, yabo083, weibk, wangwang105, th1nk3r-ing, tao363, 4yDX3906, syd168, sslmj2020, smilelsb, siqyka, selear, sdshaoda, Xi-Row, popozhu, nuquist19, noobcodemaker, XiaoK29, chadyi, lyl625760, lucaswangdev, llql1211, 0130w, shanghai-Jerry, EJackYang, Javesun99, eltociear, lipusheng, KNChiu, BlindTerran, ShiMaRing, lovelock, FreddieLi, FloranceYeh, fanchenggang, gltianwen, goerll, nedchu, curly210102, CuB3y0nd, KraHsu, CarrotDLaw, youshaoXG, bubble9um, Asashishi, Asa0oo0o0o, fanenr, eagleanurag, akshiterate, 52coder, foursevenlove, KorsChen, hopkings2008, yang-le, realwujing, Evilrabbit520, Umer-Jahangir, Turing-1024-Lee, Suremotoo, paoxiaomooo, Chieko-Seren, Senrian, Allen-Scai, 19santosh99, ymmmas, Risuntsy, Richard-Zhang1019, RafaelCaso, qingpeng9802, primexiao, Urbaner3, codetypess, nidhoggfgg, MwumLi, CreatorMetaSky, martinx, ZnYang2018, hugtyftg, logan-qiu, psychelzh, Kunchen-Luo, Keynman, and KeiichiKasai.</p>
-<p>The code review work for this book was completed by coderonion, curtishd, Gonglja, gvenusleo, hpstory, justin-tse, khoaxuantu, krahets, night-cruise, nuomi1, Reanon and rongyi (in alphabetical order). Thanks to them for the time and effort they put in; they helped keep the code consistent and standardized across the different language versions.</p>
-<p>The English version of this book was reviewed by yuelinxin, K3v123, magentaqin, QiLOL, Phoenix0415, SamJin98, yanedie, RafaelCaso, pengchzn and thomasq0; the Japanese version was reviewed by eltociear; the Russian version was reviewed by И. А. Шевкун and Yuyan Huang; and the Traditional Chinese version was reviewed by Shyam-Chen and Dr-XYZ. Thanks to their contributions, this book is able to serve a broader readership, and we are deeply grateful to them.</p>
-<p>The ePub ebook generation tool for this book was developed by zhongfq. We thank him for his contribution, which provides readers with a more flexible way to read.</p>
-<p>During the creation of this book, I received help from many people.</p>
+<h2 id="013-acknowledgments">0.1.3 &nbsp; Acknowledgments<a class="headerlink" href="#013-acknowledgments" title="Permanent link">&para;</a></h2>
+<p>This book continues to improve through the collective efforts of many contributors in the open-source community. Thank you to everyone who has invested time and energy in writing it. In the order automatically generated by GitHub, they are: krahets, coderonion, Gonglja, nuomi1, Reanon, justin-tse, hpstory, danielsss, curtishd, night-cruise, S-N-O-R-L-A-X, rongyi, msk397, gvenusleo, khoaxuantu, rivertwilight, K3v123, gyt95, zhuoqinyue, yuelinxin, Zuoxun, mingXta, Phoenix0415, FangYuan33, GN-Yu, longsizhuo, pengchzn, QiLOL, Cathay-Chen, guowei-gong, xBLACKICEx, IsChristina, JoseHung, qualifier1024, hello-ikun, magentaqin, Guanngxu, thomasq0, sunshinesDL, L-Super, Transmigration-zhou, WSL0809, Slone123c, lhxsm, yuan0221, what-is-me, theNefelibatas, Shyam-Chen, sangxiaai, longranger2, codeberg-user, xiongsp, JeffersonHuang, prinpal, seven1240, Wonderdch, malone6, xiaomiusa87, gaofer, bluebean-cloud, a16su, SamJin98, hongyun-robot, nanlei, XiaChuerwu, yd-j, iron-irax, mgisr, steventimes, junminhong, heshuyue, danny900714, Nigh, Dr-XYZ, MolDuM, XC-Zero, reeswell, PXG-XPG, NI-SW, Horbin-Magician, Enlightenus, YangXuanyi, xjr7670, beatrix-chan, DullSword, qq909244296, iStig, boloboloda, hts0000, gledfish, fbigm, echo1937, jiaxianhua, wenjianmin, keshida, kilikilikid, lclc6, lwbaptx, linyejoe2, liuxjerry, szu17dmy, dshlstarr, Yucao-cy, coderlef, czruby, bongbongbakudan, beintentional, ZongYangL, ZhongYuuu, ZhongGuanbin, hezhizhen, linzeyan, ZJKung, JTCPOWI, KawaiiAsh, luluxia, xb534, ztkuaikuai, yw-1021, ElaBosak233, baagod, zhouLion, yishangzhang, yi427, yanedie, yabo083, weibk, wangwang105, th1nk3r-ing, tao363, 4yDX3906, syd168, sslmj2020, smilelsb, siqyka, selear, sdshaoda, Xi-Row, popozhu, nuquist19, noobcodemaker, XiaoK29, chadyi, lyl625760, lucaswangdev, llql1211, 0130w, shanghai-Jerry, EJackYang, Javesun99, eltociear, lipusheng, KNChiu, BlindTerran, ShiMaRing, lovelock, FreddieLi, FloranceYeh, fanchenggang, gltianwen, goerll, nedchu, curly210102, CuB3y0nd, KraHsu, CarrotDLaw, youshaoXG, bubble9um, Asashishi, Asa0oo0o0o, fanenr, eagleanurag, akshiterate, 52coder, foursevenlove, KorsChen, hopkings2008, yang-le, realwujing, Evilrabbit520, Umer-Jahangir, Turing-1024-Lee, Suremotoo, paoxiaomooo, Chieko-Seren, Senrian, Allen-Scai, 19santosh99, ymmmas, Risuntsy, Richard-Zhang1019, RafaelCaso, qingpeng9802, primexiao, Urbaner3, codetypess, nidhoggfgg, MwumLi, CreatorMetaSky, martinx, ZnYang2018, hugtyftg, logan-qiu, psychelzh, Kunchen-Luo, Keynman and KeiichiKasai.</p>
+<p>The code was reviewed by coderonion, curtishd, Gonglja, gvenusleo, hpstory, justin-tse, khoaxuantu, krahets, night-cruise, nuomi1, Reanon, and rongyi, listed alphabetically. Thank you for the time and effort you devoted to ensuring consistent, well-written code across languages.</p>
+<p>The English edition was reviewed by yuelinxin, K3v123, magentaqin, QiLOL, Phoenix0415, SamJin98, yanedie, RafaelCaso, pengchzn, and thomasq0; the Japanese edition by eltociear; the Russian edition by И. А. Шевкун and Yuyan Huang; and the Traditional Chinese edition by Shyam-Chen and Dr-XYZ. Their contributions have made the book accessible to a wider readership. Thank you all.</p>
+<p>The tool for generating the book's ePub edition was developed by zhongfq. Thank you for giving readers a more flexible way to read.</p>
+<p>Many people helped me while I was writing this book.</p>
 <ul>
-<li>Thanks to my mentor at the company, Dr. Li Xi, who encouraged me to "take action quickly" during a conversation, strengthening my determination to write this book;</li>
-<li>Thanks to my girlfriend Bubble, the first reader of this book, who provided many valuable suggestions from the perspective of an algorithm beginner, making this book more approachable for beginners;</li>
-<li>Thanks to Tengbao, Qibao, and Feibao for coming up with a creative name for this book, evoking everyone's fond memories of writing their first line of code "Hello World!";</li>
-<li>Thanks to Xiaoquan for providing professional help in intellectual property rights, which played an important role in the improvement of this open-source book;</li>
-<li>Thanks to Sutong for designing the beautiful cover and logo for this book, and for patiently revising them many times at my perfectionist insistence;</li>
-<li>Thanks to @squidfunk for the typesetting suggestions, as well as for developing the open-source documentation theme <a href="https://github.com/squidfunk/mkdocs-material">Material-for-MkDocs</a>.</li>
+<li>Thank you to my mentor at work, Dr. Xi Li. During a long conversation, your encouragement to “get started” strengthened my resolve to write this book;</li>
+<li>Thank you to my girlfriend Paopao, the book's first reader. Your many suggestions from a beginner's perspective have made it more accessible to newcomers;</li>
+<li>Thank you to Tengbao, Qibao, and Feibao for giving the book a creative name that brings back fond memories of writing our first line of code, “Hello World!”;</li>
+<li>Thank you to Xiaoquan for your professional help with intellectual property, which played an important role in improving this open-source book;</li>
+<li>Thank you to Sutong for designing the beautiful cover and logo, and for patiently revising them many times to satisfy my perfectionism;</li>
+<li>Thank you to @squidfunk for your typography advice and for developing the open-source documentation theme <a href="https://github.com/squidfunk/mkdocs-material">Material-for-MkDocs</a>.</li>
 </ul>
-<p>During the writing process, I read many textbooks and articles on data structures and algorithms. These works served as excellent models for this book and helped ensure the accuracy and quality of its content. I would like to thank all the teachers and predecessors for their outstanding contributions!</p>
-<p>This book advocates a hands-on approach to learning, and in this respect I was deeply inspired by <a href="https://github.com/d2l-ai/d2l-zh">Dive into Deep Learning</a>. I highly recommend this excellent work to all readers.</p>
-<p><strong>Heartfelt thanks to my parents. It is your support and encouragement that gave me the opportunity to pursue this enjoyable project</strong>.</p>
+<p>While writing, I read many textbooks and articles on data structures and algorithms. These works provided excellent models and helped ensure the accuracy and quality of this book. My thanks go to all the teachers and predecessors whose outstanding contributions made this possible!</p>
+<p>The book advocates learning through both thought and hands-on practice, an approach deeply inspired by <a href="https://github.com/d2l-ai/d2l-zh"><em>Dive into Deep Learning</em></a>. I warmly recommend this excellent book to every reader.</p>
+<p><strong>My heartfelt thanks to my parents. Your constant support and encouragement have given me the opportunity to do this wonderfully engaging work</strong>.</p>
 
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-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
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-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -421,7 +421,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
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@@ -463,7 +463,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -517,7 +517,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -577,7 +577,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
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   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
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-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -739,7 +739,7 @@
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
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   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1225,7 +1225,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1267,7 +1267,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
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@@ -1508,7 +1508,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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@@ -1722,7 +1722,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1898,7 +1898,7 @@
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1940,7 +1940,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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@@ -1967,7 +1967,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1994,7 +1994,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2075,7 +2075,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2166,7 +2166,7 @@
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
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@@ -2208,7 +2208,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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@@ -2262,7 +2262,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2289,7 +2289,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2316,7 +2316,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2380,7 +2380,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
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@@ -2422,7 +2422,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2476,7 +2476,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2503,7 +2503,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2530,7 +2530,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2598,7 +2598,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2640,7 +2640,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2694,7 +2694,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2721,7 +2721,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2748,7 +2748,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2775,7 +2775,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2802,7 +2802,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2880,7 +2880,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2901,7 +2901,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2922,7 +2922,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2976,7 +2976,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3003,7 +3003,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3030,7 +3030,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3057,7 +3057,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3084,7 +3084,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3111,7 +3111,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3138,7 +3138,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3165,7 +3165,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3192,7 +3192,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3219,7 +3219,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3285,7 +3285,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3306,7 +3306,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3327,7 +3327,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3381,7 +3381,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3408,7 +3408,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3435,7 +3435,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3462,7 +3462,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3528,7 +3528,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3570,7 +3570,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3597,7 +3597,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3624,7 +3624,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3651,7 +3651,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3678,7 +3678,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3705,7 +3705,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3775,7 +3775,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3796,7 +3796,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3817,7 +3817,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3844,7 +3844,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3871,7 +3871,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3898,7 +3898,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -3925,7 +3925,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -3952,7 +3952,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -3979,7 +3979,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4006,7 +4006,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4072,7 +4072,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4093,7 +4093,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4114,7 +4114,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4141,7 +4141,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4168,7 +4168,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  16.2 Contributing Together
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 <p><img alt="Preface" class="cover-image" src="../assets/covers/chapter_preface.jpg" /></p>
 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
-<p>Algorithms are like a beautiful symphony, each line of code flows like a melody.</p>
-<p>May this book gently resonate in your mind, leaving a unique and profound melody.</p>
+<p>Algorithms are like a beautiful symphony, with each line of code flowing to its rhythm.</p>
+<p>May this book play softly in your mind, leaving a melody that is both distinctive and lasting.</p>
 </div>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
-<li><a href="about_the_book/">0.1 &nbsp; About This Book</a></li>
-<li><a href="suggestions/">0.2 &nbsp; How to Use This Book</a></li>
+<li><a href="about_the_book/">0.1 &nbsp; About this book</a></li>
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 <li><a href="summary/">0.3 &nbsp; Summary</a></li>
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@@ -4624,7 +4696,7 @@ aria-label="Footer"
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     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 0.1 About This Book"
+    aria-label="Next: 0.1   About this book"
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@@ -4644,7 +4716,7 @@ aria-label="Footer"
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-        0.1 About This Book
+        0.1 &nbsp; About this book
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
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         <meta name="author" content="krahets">
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-        <title>0.2 How to Use This Book - Hello Algo</title>
+        <title>0.2 How to use this book - Hello Algo</title>
 
 
 
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1782,7 +1782,7 @@
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1803,7 +1803,7 @@
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1824,7 +1824,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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@@ -1878,7 +1878,7 @@
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2042,7 +2042,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2096,7 +2096,7 @@
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2150,7 +2150,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2177,7 +2177,7 @@
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2268,7 +2268,7 @@
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2310,7 +2310,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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@@ -2482,7 +2482,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2524,7 +2524,7 @@
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-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2742,7 +2742,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2982,7 +2982,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3003,7 +3003,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
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@@ -3024,7 +3024,7 @@
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3078,7 +3078,7 @@
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3105,7 +3105,7 @@
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3132,7 +3132,7 @@
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3159,7 +3159,7 @@
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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@@ -3186,7 +3186,7 @@
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3213,7 +3213,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3240,7 +3240,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3267,7 +3267,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3294,7 +3294,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3321,7 +3321,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3387,7 +3387,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3408,7 +3408,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3429,7 +3429,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3456,7 +3456,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3483,7 +3483,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3510,7 +3510,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3537,7 +3537,7 @@
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3564,7 +3564,7 @@
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3630,7 +3630,7 @@
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3672,7 +3672,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3699,7 +3699,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3726,7 +3726,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3753,7 +3753,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3780,7 +3780,7 @@
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3877,7 +3877,7 @@
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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@@ -3898,7 +3898,7 @@
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3919,7 +3919,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3973,7 +3973,7 @@
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4000,7 +4000,7 @@
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4027,7 +4027,7 @@
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4054,7 +4054,7 @@
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4081,7 +4081,7 @@
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4108,7 +4108,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4174,7 +4174,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4195,7 +4195,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4216,7 +4216,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4243,7 +4243,7 @@
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4297,7 +4297,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4324,7 +4324,7 @@
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4351,7 +4351,7 @@
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4411,7 +4411,7 @@
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4453,7 +4453,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4480,7 +4480,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4507,7 +4507,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4591,6 +4591,78 @@
 
 
 
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-  <a href="#021-writing-style-conventions" class="md-nav__link">
+  <a href="#021-writing-conventions" class="md-nav__link">
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       <span class="md-typeset">
-        0.2.1 &nbsp; Writing Style Conventions
+        0.2.1 &nbsp; Writing conventions
       </span>
     </span>
   </a>
@@ -4643,10 +4715,10 @@
 </li>
 
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-  <a href="#022-learning-efficiently-with-animated-illustrations" class="md-nav__link">
+  <a href="#022-learn-efficiently-with-animated-illustrations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        0.2.2 &nbsp; Learning Efficiently with Animated Illustrations
+        0.2.2 &nbsp; Learn efficiently with animated illustrations
       </span>
     </span>
   </a>
@@ -4654,10 +4726,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#023-deepening-understanding-through-code-practice" class="md-nav__link">
+  <a href="#023-deepen-your-understanding-through-coding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        0.2.3 &nbsp; Deepening Understanding Through Code Practice
+        0.2.3 &nbsp; Deepen your understanding through coding
       </span>
     </span>
   </a>
@@ -4665,10 +4737,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#024-growing-together-through-questions-and-discussions" class="md-nav__link">
+  <a href="#024-learn-together-through-questions-and-discussion" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        0.2.4 &nbsp; Growing Together Through Questions and Discussions
+        0.2.4 &nbsp; Learn together through questions and discussion
       </span>
     </span>
   </a>
@@ -4676,10 +4748,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#025-algorithm-learning-roadmap" class="md-nav__link">
+  <a href="#025-a-learning-path-for-algorithms" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        0.2.5 &nbsp; Algorithm Learning Roadmap
+        0.2.5 &nbsp; A learning path for algorithms
       </span>
     </span>
   </a>
@@ -4723,209 +4795,209 @@
 
 
 <!-- Page content -->
-<h1 id="02-how-to-use-this-book">0.2 &nbsp; How to Use This Book<a class="headerlink" href="#02-how-to-use-this-book" title="Permanent link">&para;</a></h1>
+<h1 id="02-how-to-use-this-book">0.2 &nbsp; How to use this book<a class="headerlink" href="#02-how-to-use-this-book" title="Permanent link">&para;</a></h1>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>For the best reading experience, it is recommended that you read through this section.</p>
+<p>For the best reading experience, we recommend reading this section in full.</p>
 </div>
-<h2 id="021-writing-style-conventions">0.2.1 &nbsp; Writing Style Conventions<a class="headerlink" href="#021-writing-style-conventions" title="Permanent link">&para;</a></h2>
+<h2 id="021-writing-conventions">0.2.1 &nbsp; Writing conventions<a class="headerlink" href="#021-writing-conventions" title="Permanent link">&para;</a></h2>
 <ul>
-<li>Sections marked with <code>*</code> after the title are optional and somewhat more challenging. If you're short on time, you can skip them on your first pass.</li>
-<li>Technical terms are shown in bold (in the print and PDF editions) or underlined (in the web edition), such as <u>array</u>. They are worth remembering, as they will help when reading technical literature.</li>
-<li>Key content and summary statements will be <strong>bolded</strong>, and such text deserves special attention.</li>
-<li>Words and phrases with specific meanings will be marked with "quotation marks" to avoid ambiguity.</li>
-<li>When terminology differs across programming languages, this book follows Python conventions; for example, it uses <code>None</code> to represent "null".</li>
-<li>This book partially relaxes conventional programming-language comment styles in favor of a more compact layout. Comments are mainly divided into three types: title comments, content comments, and multi-line comments.</li>
+<li>Sections with <code>*</code> after the title are optional and cover more difficult material. If your time is limited, you can skip them at first.</li>
+<li>Technical terms appear in bold in the print and PDF editions or are underlined in the web edition, such as <u>array</u>. Remembering these terms will help you read technical literature.</li>
+<li>Key points and summary statements appear in <strong>bold</strong> and deserve particular attention.</li>
+<li>Words and phrases used in a specific sense appear in “quotation marks” to avoid ambiguity.</li>
+<li>When terminology differs between programming languages, this book follows Python; for example, it uses <code>None</code> to represent a null value.</li>
+<li>For a more compact layout, this book sometimes departs from a language's usual commenting conventions. Comments fall into three main categories: heading comments, explanatory comments, and multiline comments.</li>
 </ul>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="sd">"""Title comment, used to label functions, classes, test cases, etc."""</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="sd">"""Heading comment identifying a function, class, test case, and so on"""</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="c1"># Content comment, used to explain code in detail</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="c1"># Explanatory comment describing the code in detail</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="sd">"""</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="sd">Multi-line</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="sd">Multiline</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="sd">comment</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="sd">"""</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Title comment, used to label functions, classes, test cases, etc. */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Heading comment identifying a function, class, test case, and so on */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="c1">// Content comment, used to explain code in detail</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="c1">// Explanatory comment describing the code in detail</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="cm">/**</span>
-<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="cm"> * Multi-line</span>
+<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="cm"> * Multiline</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="cm"> * comment</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="cm"> */</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Title comment, used to label functions, classes, test cases, etc. */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Heading comment identifying a function, class, test case, and so on */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="c1">// Content comment, used to explain code in detail</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="c1">// Explanatory comment describing the code in detail</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="cm">/**</span>
-<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="cm"> * Multi-line</span>
+<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="cm"> * Multiline</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="cm"> * comment</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="cm"> */</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Title comment, used to label functions, classes, test cases, etc. */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Heading comment identifying a function, class, test case, and so on */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="c1">// Content comment, used to explain code in detail</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="c1">// Explanatory comment describing the code in detail</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="cm">/**</span>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="cm"> * Multi-line</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="cm"> * Multiline</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="cm"> * comment</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="cm"> */</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Title comment, used to label functions, classes, test cases, etc. */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Heading comment identifying a function, class, test case, and so on */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="c1">// Content comment, used to explain code in detail</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="c1">// Explanatory comment describing the code in detail</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="cm">/**</span>
-<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="cm"> * Multi-line</span>
+<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="cm"> * Multiline</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="cm"> * comment</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="cm"> */</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Title comment, used to label functions, classes, test cases, etc. */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Heading comment identifying a function, class, test case, and so on */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="c1">// Content comment, used to explain code in detail</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="c1">// Explanatory comment describing the code in detail</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="cm">/**</span>
-<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="cm"> * Multi-line</span>
+<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="cm"> * Multiline</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="cm"> * comment</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="cm"> */</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Title comment, used to label functions, classes, test cases, etc. */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Heading comment identifying a function, class, test case, and so on */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="c1">// Content comment, used to explain code in detail</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="c1">// Explanatory comment describing the code in detail</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="cm">/**</span>
-<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="cm"> * Multi-line</span>
+<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="cm"> * Multiline</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="cm"> * comment</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="cm"> */</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Title comment, used to label functions, classes, test cases, etc. */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Heading comment identifying a function, class, test case, and so on */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="c1">// Content comment, used to explain code in detail</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="c1">// Explanatory comment describing the code in detail</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="cm">/**</span>
-<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="cm"> * Multi-line</span>
+<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="cm"> * Multiline</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="cm"> * comment</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="cm"> */</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Title comment, used to label functions, classes, test cases, etc. */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Heading comment identifying a function, class, test case, and so on */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="c1">// Content comment, used to explain code in detail</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="c1">// Explanatory comment describing the code in detail</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="cm">/**</span>
-<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="cm"> * Multi-line</span>
+<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="cm"> * Multiline</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="cm"> * comment</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="cm"> */</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Title comment, used to label functions, classes, test cases, etc. */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Heading comment identifying a function, class, test case, and so on */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="c1">// Content comment, used to explain code in detail</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="c1">// Explanatory comment describing the code in detail</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a>
-<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="c1">// Multi-line</span>
+<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="c1">// Multiline</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="c1">// comment</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Title comment, used to label functions, classes, test cases, etc. */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Heading comment identifying a function, class, test case, and so on */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="c1">// Content comment, used to explain code in detail</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="c1">// Explanatory comment describing the code in detail</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="cm">/**</span>
-<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="cm"> * Multi-line</span>
+<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="cm"> * Multiline</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="cm"> * comment</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="cm"> */</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Title comment, used to label functions, classes, test cases, etc. */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Heading comment identifying a function, class, test case, and so on */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="c1">// Content comment, used to explain code in detail</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="c1">// Explanatory comment describing the code in detail</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="cm">/**</span>
-<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="cm"> * Multi-line</span>
+<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="cm"> * Multiline</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="cm"> * comment</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="cm"> */</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Title comment, used to label functions, classes, test cases, etc. ###</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Heading comment identifying a function, class, test case, and so on ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="c1"># Content comment, used to explain code in detail</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="c1"># Explanatory comment describing the code in detail</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="c1"># Multi-line</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="c1"># Multiline</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="c1"># comment</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h2 id="022-learning-efficiently-with-animated-illustrations">0.2.2 &nbsp; Learning Efficiently with Animated Illustrations<a class="headerlink" href="#022-learning-efficiently-with-animated-illustrations" title="Permanent link">&para;</a></h2>
-<p>Compared with plain text, videos and images have higher information density and a clearer structure, making them easier to understand. In this book, <strong>key concepts and challenging topics are presented mainly through animated illustrations</strong>, with text serving as explanation and supplement.</p>
-<p>If, while reading this book, you encounter an animated illustration like the one shown below, <strong>treat the illustration as primary and the text as supplementary</strong>, and use both together to understand the content.</p>
-<p><img alt="Example of animated illustrations" class="animation-figure" src="../../index.assets/animation.gif" /></p>
-<p align="center"> Figure 0-2 &nbsp; Example of animated illustrations </p>
+<h2 id="022-learn-efficiently-with-animated-illustrations">0.2.2 &nbsp; Learn efficiently with animated illustrations<a class="headerlink" href="#022-learn-efficiently-with-animated-illustrations" title="Permanent link">&para;</a></h2>
+<p>Compared with text, videos and images convey information in a denser, more structured form that is easier to understand. In this book, <strong>key and difficult concepts are presented mainly through animated illustrations</strong>, with text providing explanations and additional detail.</p>
+<p>When a passage includes an animated illustration like the one below, <strong>focus on the illustration and use the text to support it</strong>, drawing on both to understand the material.</p>
+<p><img alt="An example of an animated illustration" class="animation-figure" src="../../index.assets/animation.gif" /></p>
+<p align="center"> Figure 0-2 &nbsp; An example of an animated illustration </p>
 
-<h2 id="023-deepening-understanding-through-code-practice">0.2.3 &nbsp; Deepening Understanding Through Code Practice<a class="headerlink" href="#023-deepening-understanding-through-code-practice" title="Permanent link">&para;</a></h2>
-<p>The accompanying code for this book is hosted in the <a href="https://github.com/krahets/hello-algo">GitHub repository</a>. As shown in Figure 0-3, <strong>the source code comes with test cases and can be run with one click</strong>.</p>
-<p>If time permits, <strong>it is recommended that you type out the code yourself</strong>. If you have limited study time, please at least read through and run all the code.</p>
-<p>Compared with simply reading code, writing it yourself often brings greater rewards. <strong>Hands-on practice is where real learning happens</strong>.</p>
-<p><img alt="Example of running code" class="animation-figure" src="../../index.assets/running_code.gif" /></p>
-<p align="center"> Figure 0-3 &nbsp; Example of running code </p>
+<h2 id="023-deepen-your-understanding-through-coding">0.2.3 &nbsp; Deepen your understanding through coding<a class="headerlink" href="#023-deepen-your-understanding-through-coding" title="Permanent link">&para;</a></h2>
+<p>The accompanying code is hosted in the <a href="https://github.com/krahets/hello-algo">GitHub repository</a>. As Figure 0-3 shows, <strong>the source code includes test cases and can be run with one click</strong>.</p>
+<p>If time permits, <strong>we recommend typing out the code yourself while referring to the examples</strong>. If your study time is limited, at least read and run all the code.</p>
+<p>Writing code often teaches us more than reading it. <strong>Hands-on practice is where real learning happens</strong>.</p>
+<p><img alt="An example of running code" class="animation-figure" src="../../index.assets/running_code.gif" /></p>
+<p align="center"> Figure 0-3 &nbsp; An example of running code </p>
 
-<p>Getting the code running mainly involves three preliminary steps.</p>
-<p><strong>Step 1: Install the local programming environment</strong>. Please follow the <a href="https://www.hello-algo.com/chapter_appendix/installation/">tutorial</a> in the appendix. If it is already installed, you can skip this step.</p>
-<p><strong>Step 2: Clone or download the code repository</strong>. Visit the <a href="https://github.com/krahets/hello-algo">GitHub repository</a>. If you have already installed <a href="https://git-scm.com/downloads">Git</a>, you can clone this repository with the following command:</p>
+<p>Getting ready to run the code involves three main steps.</p>
+<p><strong>Step 1: Set up a local programming environment</strong>. Follow the <a href="../../chapter_appendix/installation/">tutorial</a> in the appendix. If your environment is already set up, skip this step.</p>
+<p><strong>Step 2: Clone or download the repository</strong>. Visit the <a href="https://github.com/krahets/hello-algo">GitHub repository</a>. If you have <a href="https://git-scm.com/downloads">Git</a> installed, clone the repository with the following command:</p>
 <div class="highlight"><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a>git<span class="w"> </span>clone<span class="w"> </span>https://github.com/krahets/hello-algo.git
 </code></pre></div>
-<p>Alternatively, you can click the "Download ZIP" button shown below to download a ZIP archive of the repository directly and then extract it locally.</p>
-<p><img alt="Clone repository and download code" class="animation-figure" src="../suggestions.assets/download_code.png" /></p>
-<p align="center"> Figure 0-4 &nbsp; Clone repository and download code </p>
+<p>Alternatively, click “Download ZIP” in the location shown below to download the code as an archive, then extract it on your computer.</p>
+<p><img alt="Cloning the repository and downloading the code" class="animation-figure" src="../suggestions.assets/download_code.png" /></p>
+<p align="center"> Figure 0-4 &nbsp; Cloning the repository and downloading the code </p>
 
-<p><strong>Step 3: Run the source code</strong>. As shown in Figure 0-5, for code blocks with file names at the top, we can find the corresponding source code files in the <code>codes</code> folder of the repository. The source code files can be run with one click, which will help you save unnecessary debugging time and allow you to focus on learning content.</p>
-<p><img alt="Code blocks and corresponding source code files" class="animation-figure" src="../suggestions.assets/code_md_to_repo.png" /></p>
-<p align="center"> Figure 0-5 &nbsp; Code blocks and corresponding source code files </p>
+<p><strong>Step 3: Run the source code</strong>. As shown below, when a code block has a filename at the top, you can find the corresponding source file in the repository's <code>en/codes</code> folder. These files can be run with one click, saving unnecessary debugging time so you can focus on learning.</p>
+<p><img alt="A code block labeled with its source filename" class="animation-figure" src="../suggestions.assets/code_md_to_repo.png" /></p>
+<p align="center"> Figure 0-5 &nbsp; A code block labeled with its source filename </p>
 
-<p>In addition to running code locally, <strong>the web version also supports visual execution of Python code</strong> (implemented based on <a href="https://pythontutor.com/">pythontutor</a>). As shown in Figure 0-6, you can click "Visual Run" below the code block to expand the view and observe the execution process of the algorithm code; you can also click "Full Screen View" for a better viewing experience.</p>
-<p><img alt="Visual running of Python code" class="animation-figure" src="../suggestions.assets/pythontutor_example.png" /></p>
-<p align="center"> Figure 0-6 &nbsp; Visual running of Python code </p>
+<p>In addition to running code locally, <strong>the web edition supports visual execution of Python code</strong> through <a href="https://pythontutor.com/">pythontutor</a>. Click “Visualize execution” beneath a code block to expand the view and watch the algorithm run. You can also click “View full screen” for a larger display, as shown in Figure 0-6.</p>
+<p><img alt="Python code visualized in PythonTutor" class="animation-figure" src="../suggestions.assets/pythontutor_example.png" /></p>
+<p align="center"> Figure 0-6 &nbsp; Python code visualized in PythonTutor </p>
 
-<h2 id="024-growing-together-through-questions-and-discussions">0.2.4 &nbsp; Growing Together Through Questions and Discussions<a class="headerlink" href="#024-growing-together-through-questions-and-discussions" title="Permanent link">&para;</a></h2>
-<p>When reading this book, please do not skip over points that you still do not fully understand. <strong>Feel free to ask your questions in the comments section</strong>, and my friends and I will do our best to answer them, usually within two days.</p>
-<p>As shown in Figure 0-7, the web version has a comments section at the bottom of each chapter. I encourage you to pay close attention to the discussions there. On the one hand, you can learn about the problems that others encounter, thereby filling gaps in your own understanding and prompting deeper thought. On the other hand, I hope you will generously answer other readers' questions, share your insights, and help others improve.</p>
-<p><img alt="Example of comments section" class="animation-figure" src="../../index.assets/comment.gif" /></p>
-<p align="center"> Figure 0-7 &nbsp; Example of comments section </p>
+<h2 id="024-learn-together-through-questions-and-discussion">0.2.4 &nbsp; Learn together through questions and discussion<a class="headerlink" href="#024-learn-together-through-questions-and-discussion" title="Permanent link">&para;</a></h2>
+<p>As you read, do not simply skip concepts you have not understood. <strong>You are welcome to ask questions in the comments</strong>. My fellow contributors and I will do our best to answer, usually within two days.</p>
+<p>As shown below, every section in the web edition has a comments area at the bottom. I hope you will read it regularly. Other readers' questions can reveal gaps in your own understanding and encourage deeper thought. I also hope you will generously answer their questions, share your insights, and help them make progress.</p>
+<p><img alt="An example of the comments area" class="animation-figure" src="../../index.assets/comment.gif" /></p>
+<p align="center"> Figure 0-7 &nbsp; An example of the comments area </p>
 
-<h2 id="025-algorithm-learning-roadmap">0.2.5 &nbsp; Algorithm Learning Roadmap<a class="headerlink" href="#025-algorithm-learning-roadmap" title="Permanent link">&para;</a></h2>
-<p>Overall, we can divide the process of learning data structures and algorithms into three stages.</p>
+<h2 id="025-a-learning-path-for-algorithms">0.2.5 &nbsp; A learning path for algorithms<a class="headerlink" href="#025-a-learning-path-for-algorithms" title="Permanent link">&para;</a></h2>
+<p>Broadly speaking, learning data structures and algorithms can be divided into three stages.</p>
 <ol>
-<li><strong>Stage 1: Algorithm introduction</strong>. We need to familiarize ourselves with the characteristics and usage of various data structures, and learn the principles, processes, uses, and efficiency of different algorithms.</li>
-<li><strong>Stage 2: Practice algorithm problems</strong>. It is recommended to start with popular problems and solve at least 100 of them first, so that you become familiar with mainstream algorithm questions. When you first begin practicing problems, "knowledge forgetting" may feel like a challenge, but rest assured, this is very normal. We can review problems according to the "Ebbinghaus forgetting curve", and after 3-5 rounds of repetition, they usually stick firmly in memory. For recommended problem lists and practice plans, please see this <a href="https://github.com/krahets/LeetCode-Book">GitHub repository</a>.</li>
-<li><strong>Stage 3: Building a knowledge system</strong>. In terms of learning, we can read algorithm column articles, problem-solving frameworks, and algorithm textbooks to continuously enrich our knowledge system. In terms of practicing problems, we can try advanced problem-solving strategies, such as categorization by topic, one problem multiple solutions, one solution multiple problems, etc. Related problem-solving insights can be found in various communities.</li>
+<li><strong>Stage 1: Learn the fundamentals</strong>. Become familiar with the properties and uses of data structures, and study how different algorithms work, their steps, applications, and efficiency.</li>
+<li><strong>Stage 2: Solve practice problems</strong>. We recommend starting with popular problems and solving at least 100 to become familiar with common algorithmic problems. When you first begin, forgetting what you have learned may be a challenge, but this is normal. Review problems according to the “Ebbinghaus forgetting curve”; after 3–5 rounds of repetition, they will usually be firmly in memory. See this <a href="https://github.com/krahets/LeetCode-Book">GitHub repository</a> for recommended problem lists and practice plans.</li>
+<li><strong>Stage 3: Build a systematic understanding</strong>. Read algorithm articles, problem-solving frameworks, and textbooks to keep expanding your knowledge. In your practice, try more advanced strategies: group problems by topic, solve one problem in several ways, or apply one solution to several problems. You can find advice on these strategies in various communities.</li>
 </ol>
-<p>As shown in Figure 0-8, the content of this book mainly covers "Stage 1", aiming to help you more efficiently carry out Stage 2 and Stage 3 learning.</p>
-<p><img alt="Algorithm learning roadmap" class="animation-figure" src="../suggestions.assets/learning_route.png" /></p>
-<p align="center"> Figure 0-8 &nbsp; Algorithm learning roadmap </p>
+<p>As Figure 0-8 shows, this book mainly covers “Stage 1,” helping you approach Stages 2 and 3 more efficiently.</p>
+<p><img alt="A learning path for algorithms" class="animation-figure" src="../suggestions.assets/learning_route.png" /></p>
+<p align="center"> Figure 0-8 &nbsp; A learning path for algorithms </p>
 
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-    aria-label="Previous: 0.1 About This Book"
+    aria-label="Previous: 0.1   About this book"
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
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   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
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-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
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-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
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-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
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-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
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@@ -1998,7 +1998,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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@@ -2052,7 +2052,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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@@ -2079,7 +2079,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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@@ -2106,7 +2106,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2133,7 +2133,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2224,7 +2224,7 @@
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2266,7 +2266,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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@@ -2438,7 +2438,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2480,7 +2480,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2507,7 +2507,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2534,7 +2534,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2561,7 +2561,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2677,7 +2677,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2698,7 +2698,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2725,7 +2725,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2752,7 +2752,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2779,7 +2779,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2806,7 +2806,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2833,7 +2833,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2938,7 +2938,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2980,7 +2980,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3034,7 +3034,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3061,7 +3061,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3088,7 +3088,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3115,7 +3115,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3142,7 +3142,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3169,7 +3169,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3196,7 +3196,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3343,7 +3343,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3709,7 +3709,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4130,7 +4130,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4253,7 +4253,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
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@@ -4409,7 +4409,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4436,7 +4436,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4547,6 +4547,78 @@
 
 
 
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+  Print edition (Chinese)
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
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-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
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@@ -4636,14 +4708,14 @@
 
 <!-- Page content -->
 <h1 id="03-summary">0.3 &nbsp; Summary<a class="headerlink" href="#03-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
 <ul>
-<li>The main audience of this book is algorithm beginners. If you already have some background, this book can help you systematically review algorithm knowledge, and the source code in the book can also be used as a "problem-solving toolkit."</li>
-<li>The content of the book mainly includes three parts: complexity analysis, data structures, and algorithms, covering most topics in this field.</li>
-<li>For algorithm novices, reading an introductory book during the initial learning stage is crucial, as it can help you avoid many detours.</li>
-<li>The animated illustrations in the book are usually used to introduce key concepts and challenging topics. When reading this book, you should pay more attention to these topics.</li>
-<li>Practice is the best way to learn programming. It is strongly recommended to run the source code and type the code yourself.</li>
-<li>The web version of this book has a comments section for each chapter, where you are welcome to share your questions and insights at any time.</li>
+<li>This book is primarily for beginners in algorithms. If you already have some background, it can help you review your knowledge systematically, and its source code can serve as a toolkit for solving practice problems.</li>
+<li>The book has three main parts—complexity analysis, data structures, and algorithms—covering most topics in the field.</li>
+<li>For beginners, reading an introductory book early on is essential and can help avoid many detours.</li>
+<li>The animated illustrations usually explain key or challenging concepts. Pay particular attention to them as you read.</li>
+<li>Practice is the best way to learn programming. We strongly recommend running the source code and writing it out yourself.</li>
+<li>Every section in the web edition has a comments area. You are welcome to share your questions and insights at any time.</li>
 </ul>
 
 <!-- Source file information -->
@@ -4667,7 +4739,7 @@ aria-label="Footer"
   <a
     href="../suggestions/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 0.2 How to Use This Book"
+    aria-label="Previous: 0.2   How to use this book"
     rel="prev"
   >
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@@ -4679,7 +4751,7 @@ aria-label="Footer"
         Previous
       </span>
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-        0.2 How to Use This Book
+        0.2 &nbsp; How to use this book
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@@ -4691,7 +4763,7 @@ aria-label="Footer"
   <a
     href="../../chapter_introduction/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: Chapter 1. &amp;nbsp; Encounter with Algorithms"
+    aria-label="Next: Chapter 1.   Getting to know algorithms"
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@@ -4699,7 +4771,7 @@ aria-label="Footer"
         Next
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       <div class="md-ellipsis">
-        Chapter 1. &nbsp; Encounter with Algorithms
+        Chapter 1. &nbsp; Getting to know algorithms
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@@ -4716,7 +4788,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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diff --git a/en/chapter_reference/index.html b/en/chapter_reference/index.html
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -18,6 +18,8 @@
         <link rel="prev" href="../chapter_appendix/terminology/">
 
 
+        <link rel="next" href="../chapter_paperbook/">
+
 
 
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-  6.5 Exercises
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -2418,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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@@ -2472,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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@@ -2499,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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@@ -2594,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2615,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2636,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2663,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2690,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2717,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2744,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2771,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2798,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2876,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2897,7 +2899,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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@@ -2918,7 +2920,7 @@
   <span class="md-ellipsis">
 
 
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   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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@@ -3026,7 +3028,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3053,7 +3055,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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@@ -3080,7 +3082,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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@@ -3107,7 +3109,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3134,7 +3136,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3302,7 +3304,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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@@ -3323,7 +3325,7 @@
   <span class="md-ellipsis">
 
 
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-  12.2 Divide and Conquer Search Strategy
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   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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@@ -3404,7 +3406,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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@@ -3431,7 +3433,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3458,7 +3460,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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@@ -3524,7 +3526,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3545,7 +3547,7 @@
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -3566,7 +3568,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3593,7 +3595,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3620,7 +3622,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3647,7 +3649,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3674,7 +3676,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3701,7 +3703,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3771,7 +3773,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3792,7 +3794,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3813,7 +3815,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3840,7 +3842,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3867,7 +3869,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3894,7 +3896,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -3921,7 +3923,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -3948,7 +3950,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -3975,7 +3977,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4002,7 +4004,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4068,7 +4070,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4089,7 +4091,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4110,7 +4112,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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@@ -4137,7 +4139,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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@@ -4164,7 +4166,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4191,7 +4193,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4218,7 +4220,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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@@ -4245,7 +4247,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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@@ -4305,7 +4307,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4326,7 +4328,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4347,7 +4349,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4374,7 +4376,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4401,7 +4403,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4487,6 +4489,78 @@
 
 
 
+
+
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+
+  <span class="md-ellipsis">
+
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+  Print edition (Chinese)
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 </nav>
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@@ -4555,11 +4629,11 @@
 <p>[1] Thomas H. Cormen, et al. Introduction to Algorithms (3<sup>rd</sup> Edition).</p>
 <p>[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1<sup>st</sup> Edition).</p>
 <p>[3] Robert Sedgewick, et al. Algorithms (4<sup>th</sup> Edition).</p>
-<p>[4] Yan Weimin. Data Structures (C Language Version).</p>
-<p>[5] Deng Junhui. Data Structures (C++ Language Version, Third Edition).</p>
-<p>[6] Mark Allen Weiss, translated by Chen Yue. Data Structures and Algorithm Analysis in Java (Third Edition).</p>
-<p>[7] Cheng Jie. Data Structures in Plain Language.</p>
-<p>[8] Wang Zheng. The Beauty of Data Structures and Algorithms.</p>
+<p>[4] Weimin Yan. Data Structures (C Edition).</p>
+<p>[5] Junhui Deng. Data Structures (C++ Edition, 3<sup>rd</sup> Edition).</p>
+<p>[6] Mark Allen Weiss; translated by Yue Chen. Data Structures and Algorithm Analysis in Java (3<sup>rd</sup> Edition).</p>
+<p>[7] Jie Cheng. A Friendly Guide to Data Structures.</p>
+<p>[8] Zheng Wang. The Beauty of Data Structures and Algorithms.</p>
 <p>[9] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6<sup>th</sup> Edition).</p>
 <p>[10] Aston Zhang, et al. Dive into Deep Learning.</p>
 
@@ -4584,7 +4658,7 @@ aria-label="Footer"
   <a
     href="../chapter_appendix/terminology/"
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-    aria-label="Previous: 16.3 Glossary"
+    aria-label="Previous: 16.3   Glossary"
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@@ -4596,7 +4670,7 @@ aria-label="Footer"
         Previous
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-        16.3 Glossary
+        16.3 &nbsp; Glossary
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     </div>
   </a>
@@ -4604,6 +4678,27 @@ aria-label="Footer"
 
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 <!-- Comment system -->
diff --git a/en/chapter_searching/binary_search/index.html b/en/chapter_searching/binary_search/index.html
index f07c7e82f..7c3dbe52a 100644
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>10.1 Binary Search - Hello Algo</title>
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@@ -154,7 +154,7 @@
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         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2598,7 +2598,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2619,7 +2619,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2649,7 +2649,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2666,7 +2666,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2692,10 +2692,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1011-interval-representation-methods" class="md-nav__link">
+  <a href="#1011-ways-to-represent-the-interval" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.1.1 &nbsp; Interval Representation Methods
+        10.1.1 &nbsp; Ways to represent the interval
       </span>
     </span>
   </a>
@@ -2703,10 +2703,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1012-advantages-and-limitations" class="md-nav__link">
+  <a href="#1012-strengths-and-limitations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.1.2 &nbsp; Advantages and Limitations
+        10.1.2 &nbsp; Strengths and limitations
       </span>
     </span>
   </a>
@@ -2736,7 +2736,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2763,7 +2763,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2790,7 +2790,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2817,7 +2817,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2844,7 +2844,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2871,7 +2871,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2970,7 +2970,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2991,7 +2991,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3018,7 +3018,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3045,7 +3045,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3072,7 +3072,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3099,7 +3099,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3126,7 +3126,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3153,7 +3153,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3180,7 +3180,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3207,7 +3207,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3234,7 +3234,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3261,7 +3261,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3288,7 +3288,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3375,7 +3375,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3396,7 +3396,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3423,7 +3423,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3450,7 +3450,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3477,7 +3477,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3504,7 +3504,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3531,7 +3531,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3597,7 +3597,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3618,7 +3618,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3639,7 +3639,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3666,7 +3666,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3693,7 +3693,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3720,7 +3720,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3747,7 +3747,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3774,7 +3774,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3844,7 +3844,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3865,7 +3865,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3886,7 +3886,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3913,7 +3913,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3940,7 +3940,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3967,7 +3967,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -3994,7 +3994,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4021,7 +4021,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4048,7 +4048,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4075,7 +4075,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4141,7 +4141,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4162,7 +4162,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4183,7 +4183,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4210,7 +4210,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4237,7 +4237,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4264,7 +4264,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4291,7 +4291,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4318,7 +4318,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4378,7 +4378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4399,7 +4399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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@@ -4420,7 +4420,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4447,7 +4447,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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@@ -4558,6 +4558,78 @@
 
 
 
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         <li class="md-nav__item">
-  <a href="#1011-interval-representation-methods" class="md-nav__link">
+  <a href="#1011-ways-to-represent-the-interval" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.1.1 &nbsp; Interval Representation Methods
+        10.1.1 &nbsp; Ways to represent the interval
       </span>
     </span>
   </a>
@@ -4610,10 +4682,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1012-advantages-and-limitations" class="md-nav__link">
+  <a href="#1012-strengths-and-limitations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.1.2 &nbsp; Advantages and Limitations
+        10.1.2 &nbsp; Strengths and limitations
       </span>
     </span>
   </a>
@@ -4657,31 +4729,31 @@
 
 
 <!-- Page content -->
-<h1 id="101-binary-search">10.1 &nbsp; Binary Search<a class="headerlink" href="#101-binary-search" title="Permanent link">&para;</a></h1>
-<p><u>Binary search</u> is an efficient search algorithm based on the divide-and-conquer strategy. It leverages the sorted order of the data to reduce the search range by half in each round until the target element is found or the search interval becomes empty.</p>
+<h1 id="101-binary-search">10.1 &nbsp; Binary search<a class="headerlink" href="#101-binary-search" title="Permanent link">&para;</a></h1>
+<p><u>Binary search</u> is an efficient search algorithm based on divide and conquer. It uses the sorted order of the data to halve the search range each round, continuing until it finds the target element or the search interval becomes empty.</p>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given an array <code>nums</code> of length <span class="arithmatex">\(n\)</span> with elements arranged in ascending order and no duplicates, search for and return the index of element <code>target</code> in the array. If the array does not contain the element, return <span class="arithmatex">\(-1\)</span>. An example is shown in Figure 10-1.</p>
+<p>Given an array <code>nums</code> of length <span class="arithmatex">\(n\)</span>, with distinct elements arranged in ascending order, find and return the index of <code>target</code>. If the array does not contain this element, return <span class="arithmatex">\(-1\)</span>. An example is shown below.</p>
 </div>
-<p><img alt="Binary search example data" class="animation-figure" src="../binary_search.assets/binary_search_example.png" /></p>
-<p align="center"> Figure 10-1 &nbsp; Binary search example data </p>
+<p><img alt="Example data for binary search" class="animation-figure" src="../binary_search.assets/binary_search_example.png" /></p>
+<p align="center"> Figure 10-1 &nbsp; Example data for binary search </p>
 
-<p>As shown in Figure 10-2, we first initialize pointers <span class="arithmatex">\(i = 0\)</span> and <span class="arithmatex">\(j = n - 1\)</span>, pointing to the first and last elements of the array respectively, representing the search interval <span class="arithmatex">\([0, n - 1]\)</span>. Note that square brackets denote a closed interval, which includes the boundary values themselves.</p>
-<p>Next, perform the following two steps in a loop:</p>
+<p>As shown below, initialize pointers <span class="arithmatex">\(i = 0\)</span> and <span class="arithmatex">\(j = n - 1\)</span> to the first and last elements, respectively. They represent the search interval <span class="arithmatex">\([0, n - 1]\)</span>. Square brackets denote a closed interval, which includes its endpoints.</p>
+<p>Next, repeat the following two steps.</p>
 <ol>
-<li>Calculate the midpoint index <span class="arithmatex">\(m = \lfloor {(i + j) / 2} \rfloor\)</span>, where <span class="arithmatex">\(\lfloor \: \rfloor\)</span> denotes the floor operation.</li>
-<li>Compare <code>nums[m]</code> and <code>target</code>, which results in three cases:<ol>
-<li>When <code>nums[m] &lt; target</code>, it indicates that <code>target</code> is in the interval <span class="arithmatex">\([m + 1, j]\)</span>, so execute <span class="arithmatex">\(i = m + 1\)</span>.</li>
-<li>When <code>nums[m] &gt; target</code>, it indicates that <code>target</code> is in the interval <span class="arithmatex">\([i, m - 1]\)</span>, so execute <span class="arithmatex">\(j = m - 1\)</span>.</li>
-<li>When <code>nums[m] = target</code>, it indicates that <code>target</code> has been found, so return index <span class="arithmatex">\(m\)</span>.</li>
+<li>Calculate the midpoint index <span class="arithmatex">\(m = \lfloor {(i + j) / 2} \rfloor\)</span>, where <span class="arithmatex">\(\lfloor \: \rfloor\)</span> denotes rounding down.</li>
+<li>Compare <code>nums[m]</code> with <code>target</code>. There are three cases.<ol>
+<li>If <code>nums[m] &lt; target</code>, then <code>target</code> lies in <span class="arithmatex">\([m + 1, j]\)</span>, so set <span class="arithmatex">\(i = m + 1\)</span>.</li>
+<li>If <code>nums[m] &gt; target</code>, then <code>target</code> lies in <span class="arithmatex">\([i, m - 1]\)</span>, so set <span class="arithmatex">\(j = m - 1\)</span>.</li>
+<li>If <code>nums[m] = target</code>, we have found <code>target</code>, so return index <span class="arithmatex">\(m\)</span>.</li>
 </ol>
 </li>
 </ol>
-<p>If the array does not contain the target element, the search interval will eventually become empty. In this case, return <span class="arithmatex">\(-1\)</span>.</p>
+<p>If the array does not contain the target, the search interval eventually becomes empty. Return <span class="arithmatex">\(-1\)</span> in this case.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:7"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label><label for="__tabbed_1_6">&lt;6&gt;</label><label for="__tabbed_1_7">&lt;7&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Binary search process" class="animation-figure" src="../binary_search.assets/binary_search_step1.png" /></p>
+<p><img alt="The binary search process" class="animation-figure" src="../binary_search.assets/binary_search_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="binary_search_step2" class="animation-figure" src="../binary_search.assets/binary_search_step2.png" /></p>
@@ -4703,267 +4775,267 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 10-2 &nbsp; Binary search process </p>
+<p align="center"> Figure 10-2 &nbsp; The binary search process </p>
 
-<p>It's worth noting that since both <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> are of <code>int</code> type, <strong><span class="arithmatex">\(i + j\)</span> may exceed the range of the <code>int</code> type</strong>. To avoid integer overflow, we typically use the formula <span class="arithmatex">\(m = \lfloor {i + (j - i) / 2} \rfloor\)</span> to calculate the midpoint.</p>
-<p>The code is shown below:</p>
+<p>Because <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> are both of type <code>int</code>, <strong><span class="arithmatex">\(i + j\)</span> may exceed the range of <code>int</code></strong>. To avoid integer overflow, we usually calculate the midpoint using <span class="arithmatex">\(m = \lfloor {i + (j - i) / 2} \rfloor\)</span>.</p>
+<p>The implementation is shown below:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">binary_search</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Binary search (closed interval)"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
-<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># Loop, exit when the search interval is empty (empty when i &gt; j)</span>
+<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># Loop until the search interval is empty (when i &gt; j)</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="n">j</span><span class="p">:</span>
-<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="c1"># In theory, Python numbers can be infinitely large (depending on memory size), no need to consider large number overflow</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>  <span class="c1"># Calculate midpoint index m</span>
+<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="c1"># Python integers can grow arbitrarily large, subject to available memory, so integer overflow is not a concern</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>  <span class="c1"># Calculate the midpoint index m</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">target</span><span class="p">:</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>            <span class="n">i</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c1"># This means target is in the interval [m+1, j]</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>            <span class="n">i</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c1"># In this case, target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="k">elif</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">target</span><span class="p">:</span>
-<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>            <span class="n">j</span> <span class="o">=</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># This means target is in the interval [i, m-1]</span>
+<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>            <span class="n">j</span> <span class="o">=</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># In this case, target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>            <span class="k">return</span> <span class="n">m</span>  <span class="c1"># Found the target element, return its index</span>
-<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>  <span class="c1"># Target element not found, return -1</span>
+<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>            <span class="k">return</span> <span class="n">m</span>  <span class="c1"># The target element was found; return its index</span>
+<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>  <span class="c1"># The target element was not found; return -1</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Binary search (closed interval on both sides) */</span>
+<div class="highlight"><span class="filename">binary_search.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Binary search (closed interval) */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearch</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i &gt; j)</span>
+<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i &gt; j)</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
-<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w">    </span><span class="c1">// This means target is in the interval [m+1, j]</span>
+<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w">    </span><span class="c1">// In this case, target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// This means target is in the interval [i, m-1]</span>
+<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// In this case, target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Binary search (closed interval on both sides) */</span>
+<div class="highlight"><span class="filename">binary_search.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Binary search (closed interval) */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearch</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i &gt; j)</span>
+<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i &gt; j)</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
-<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// This means target is in the interval [m+1, j]</span>
+<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// In this case, target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// This means target is in the interval [i, m-1]</span>
+<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// In this case, target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Binary search (closed interval on both sides) */</span>
+<div class="highlight"><span class="filename">binary_search.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Binary search (closed interval) */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">BinarySearch</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i &gt; j)</span>
+<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i &gt; j)</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">   </span><span class="c1">// Calculate the midpoint index m</span>
-<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w">      </span><span class="c1">// This means target is in the interval [m+1, j]</span>
+<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w">      </span><span class="c1">// In this case, target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="nf">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// This means target is in the interval [i, m-1]</span>
+<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="nf">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// In this case, target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="k">else</span><span class="w">                       </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="k">else</span><span class="w">                       </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Binary search (closed interval on both sides) */</span>
+<div class="highlight"><span class="filename">binary_search.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Binary search (closed interval) */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">binarySearch</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
-<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i &gt; j)</span>
+<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i &gt; j)</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="o">-</span><span class="nx">i</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="w">      </span><span class="c1">// Calculate the midpoint index m</span>
-<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// This means target is in the interval [m+1, j]</span>
+<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// In this case, target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// This means target is in the interval [i, m-1]</span>
+<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// In this case, target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Binary search (closed interval on both sides) */</span>
+<div class="highlight"><span class="filename">binary_search.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Binary search (closed interval) */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">binarySearch</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">startIndex</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">endIndex</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i &gt; j)</span>
+<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i &gt; j)</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
-<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// This means target is in the interval [m+1, j]</span>
+<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// In this case, target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// This means target is in the interval [i, m-1]</span>
+<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// In this case, target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Binary search (closed interval on both sides) */</span>
+<div class="highlight"><span class="filename">binary_search.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Binary search (closed interval) */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearch</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i &gt; j)</span>
+<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i &gt; j)</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="c1">// Calculate midpoint index m, use parseInt() to round down</span>
+<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="c1">// Calculate midpoint index m, using parseInt() to round down</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">parseInt</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span>
-<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">            </span><span class="c1">// This means target is in the interval [m+1, j]</span>
+<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">            </span><span class="c1">// In this case, target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span>
-<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">            </span><span class="c1">// This means target is in the interval [i, m-1]</span>
+<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">            </span><span class="c1">// In this case, target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Binary search (closed interval on both sides) */</span>
+<div class="highlight"><span class="filename">binary_search.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Binary search (closed interval) */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearch</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i &gt; j)</span>
+<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i &gt; j)</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">        </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">            </span><span class="c1">// This means target is in the interval [m+1, j]</span>
+<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">            </span><span class="c1">// In this case, target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">            </span><span class="c1">// This means target is in the interval [i, m-1]</span>
+<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">            </span><span class="c1">// In this case, target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">            </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">            </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Binary search (closed interval on both sides) */</span>
+<div class="highlight"><span class="filename">binary_search.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Binary search (closed interval) */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">binarySearch</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
-<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// Loop, exit when the search interval is empty (empty when i &gt; j)</span>
+<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// Loop until the search interval is empty (when i &gt; j)</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">      </span><span class="c1">// This means target is in the interval [m+1, j]</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">      </span><span class="c1">// In this case, target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">      </span><span class="c1">// This means target is in the interval [i, m-1]</span>
+<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">      </span><span class="c1">// In this case, target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">      </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">      </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">  </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">  </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Binary search (closed interval on both sides) */</span>
+<div class="highlight"><span class="filename">binary_search.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Binary search (closed interval) */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">binary_search</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i &gt; j)</span>
+<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i &gt; j)</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">            </span><span class="c1">// This means target is in the interval [m+1, j]</span>
+<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">            </span><span class="c1">// In this case, target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">            </span><span class="c1">// This means target is in the interval [i, m-1]</span>
+<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">            </span><span class="c1">// In this case, target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">            </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">            </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Binary search (closed interval on both sides) */</span>
+<div class="highlight"><span class="filename">binary_search.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Binary search (closed interval) */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearch</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">len</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">len</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i &gt; j)</span>
+<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i &gt; j)</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
-<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w">    </span><span class="c1">// This means target is in the interval [m+1, j]</span>
+<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w">    </span><span class="c1">// In this case, target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// This means target is in the interval [i, m-1]</span>
+<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// In this case, target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Binary search (closed interval on both sides) */</span>
+<div class="highlight"><span class="filename">binary_search.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Binary search (closed interval) */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">binarySearch</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i &gt; j)</span>
+<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i &gt; j)</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// This means target is in the interval [m+1, j]</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// In this case, target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// This means target is in the interval [i, m-1]</span>
+<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// In this case, target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="k">else</span><span class="w">  </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="k">else</span><span class="w">  </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="m">1</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="p">}</span>
 </code></pre></div>
@@ -4971,337 +5043,347 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Binary search (closed interval) ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">binary_search</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a>
-<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="c1"># Loop, exit when the search interval is empty (empty when i &gt; j)</span>
+<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="c1"># Loop until the search interval is empty (when i &gt; j)</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span>
-<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="c1"># In theory, Ruby numbers can be infinitely large (limited by memory), no need to consider overflow</span>
+<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="c1"># Ruby integers can grow arbitrarily large, limited by available memory, so integer overflow is not a concern here</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">    </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w">   </span><span class="c1"># Calculate the midpoint index m</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span>
-<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># This means target is in the interval [m+1, j]</span>
+<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># In this case, target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="k">elsif</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span>
-<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># This means target is in the interval [i, m-1]</span>
+<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># In this case, target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">    </span><span class="k">else</span>
-<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="w">  </span><span class="c1"># Found the target element, return its index</span>
+<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="w">  </span><span class="c1"># The target element was found; return its index</span>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a>
-<a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="w">  </span><span class="o">-</span><span class="mi">1</span><span class="w">  </span><span class="c1"># Target element not found, return -1</span>
+<a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="w">  </span><span class="o">-</span><span class="mi">1</span><span class="w">  </span><span class="c1"># The target element was not found; return -1</span>
 <a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<p><strong>Time complexity is <span class="arithmatex">\(O(\log n)\)</span></strong>: In the binary search loop, the interval is reduced by half each round, so the number of iterations is <span class="arithmatex">\(\log_2 n\)</span>.</p>
-<p><strong>Space complexity is <span class="arithmatex">\(O(1)\)</span></strong>: Pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> use constant-size space.</p>
-<h2 id="1011-interval-representation-methods">10.1.1 &nbsp; Interval Representation Methods<a class="headerlink" href="#1011-interval-representation-methods" title="Permanent link">&para;</a></h2>
-<p>In addition to the closed interval mentioned above, another common interval representation is the "left-closed right-open" interval, defined as <span class="arithmatex">\([0, n)\)</span>, meaning that the left boundary is inclusive while the right boundary is exclusive. Under this representation, the interval <span class="arithmatex">\([i, j)\)</span> is empty when <span class="arithmatex">\(i = j\)</span>.</p>
-<p>We can implement a binary search algorithm with the same functionality based on this representation:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20binary_search%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Binary%20search%20%28closed%20interval%29%22%22%22%0A%20%20%20%20%23%20Initialize%20the%20closed%20interval%20%5B0,%20n-1%5D,%20with%20i%20and%20j%20pointing%20to%20the%20first%20and%20last%20elements%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%20-%201%0A%20%20%20%20%23%20Loop%20until%20the%20search%20interval%20is%20empty%20%28when%20i%20%3E%20j%29%0A%20%20%20%20while%20i%20%3C%3D%20j%3A%0A%20%20%20%20%20%20%20%20%23%20Python%20integers%20can%20grow%20arbitrarily%20large,%20subject%20to%20available%20memory,%20so%20integer%20overflow%20is%20not%20a%20concern%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20Calculate%20the%20midpoint%20index%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20In%20this%20case,%20target%20must%20be%20in%20%5Bm%2B1,%20j%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20In%20this%20case,%20target%20must%20be%20in%20%5Bi,%20m-1%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20m%20%20%23%20The%20target%20element%20was%20found%3B%20return%20its%20index%0A%20%20%20%20return%20-1%20%20%23%20The%20target%20element%20was%20not%20found%3B%20return%20-1%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%208,%2012,%2015,%2023,%2026,%2031,%2035%5D%0A%0A%20%20%20%20%23%20Binary%20search%20%28closed%20interval%29%0A%20%20%20%20index%20%3D%20binary_search%28nums,%20target%29%0A%20%20%20%20print%28%22Index%20of%20target%20element%206%20%3D%20%22,%20index%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20binary_search%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Binary%20search%20%28closed%20interval%29%22%22%22%0A%20%20%20%20%23%20Initialize%20the%20closed%20interval%20%5B0,%20n-1%5D,%20with%20i%20and%20j%20pointing%20to%20the%20first%20and%20last%20elements%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%20-%201%0A%20%20%20%20%23%20Loop%20until%20the%20search%20interval%20is%20empty%20%28when%20i%20%3E%20j%29%0A%20%20%20%20while%20i%20%3C%3D%20j%3A%0A%20%20%20%20%20%20%20%20%23%20Python%20integers%20can%20grow%20arbitrarily%20large,%20subject%20to%20available%20memory,%20so%20integer%20overflow%20is%20not%20a%20concern%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20Calculate%20the%20midpoint%20index%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20In%20this%20case,%20target%20must%20be%20in%20%5Bm%2B1,%20j%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20In%20this%20case,%20target%20must%20be%20in%20%5Bi,%20m-1%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20m%20%20%23%20The%20target%20element%20was%20found%3B%20return%20its%20index%0A%20%20%20%20return%20-1%20%20%23%20The%20target%20element%20was%20not%20found%3B%20return%20-1%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%208,%2012,%2015,%2023,%2026,%2031,%2035%5D%0A%0A%20%20%20%20%23%20Binary%20search%20%28closed%20interval%29%0A%20%20%20%20index%20%3D%20binary_search%28nums,%20target%29%0A%20%20%20%20print%28%22Index%20of%20target%20element%206%20%3D%20%22,%20index%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p><strong>Time complexity is <span class="arithmatex">\(O(\log n)\)</span></strong>: the search interval is halved each round, so the loop runs <span class="arithmatex">\(\log_2 n\)</span> times.</p>
+<p><strong>Space complexity is <span class="arithmatex">\(O(1)\)</span></strong>: pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> use constant space.</p>
+<h2 id="1011-ways-to-represent-the-interval">10.1.1 &nbsp; Ways to represent the interval<a class="headerlink" href="#1011-ways-to-represent-the-interval" title="Permanent link">&para;</a></h2>
+<p>Besides the closed interval above, another common representation is a “left-closed, right-open” interval, written <span class="arithmatex">\([0, n)\)</span>. It includes the left endpoint but excludes the right endpoint. With this representation, <span class="arithmatex">\([i, j)\)</span> is empty when <span class="arithmatex">\(i = j\)</span>.</p>
+<p>We can implement an equivalent binary search algorithm using this representation:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">binary_search_lcro</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Binary search (left-closed right-open interval)"""</span>
-<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Binary search (left-closed, right-open interval)"""</span>
+<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Loop, exit when the search interval is empty (empty when i = j)</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Loop until the search interval is empty (when i = j)</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
-<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>  <span class="c1"># Calculate midpoint index m</span>
+<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>  <span class="c1"># Calculate the midpoint index m</span>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">target</span><span class="p">:</span>
-<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>            <span class="n">i</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c1"># This means target is in the interval [m+1, j)</span>
+<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>            <span class="n">i</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c1"># In this case, target must be in [m+1, j) if it exists</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>        <span class="k">elif</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">target</span><span class="p">:</span>
-<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>            <span class="n">j</span> <span class="o">=</span> <span class="n">m</span>  <span class="c1"># This means target is in the interval [i, m)</span>
+<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>            <span class="n">j</span> <span class="o">=</span> <span class="n">m</span>  <span class="c1"># In this case, target must be in [i, m) if it exists</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>        <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>            <span class="k">return</span> <span class="n">m</span>  <span class="c1"># Found the target element, return its index</span>
-<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>    <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>  <span class="c1"># Target element not found, return -1</span>
+<a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>            <span class="k">return</span> <span class="n">m</span>  <span class="c1"># The target element was found; return its index</span>
+<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>    <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>  <span class="c1"># The target element was not found; return -1</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Binary search (left-closed right-open interval) */</span>
+<div class="highlight"><span class="filename">binary_search.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Binary search (left-closed, right-open interval) */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchLCRO</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i = j)</span>
+<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i = j)</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
-<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w">    </span><span class="c1">// This means target is in the interval [m+1, j)</span>
+<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w">    </span><span class="c1">// In this case, target must be in [m+1, j) if it exists</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// This means target is in the interval [i, m)</span>
+<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// In this case, target must be in [i, m) if it exists</span>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
-<a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Binary search (left-closed right-open interval) */</span>
+<div class="highlight"><span class="filename">binary_search.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Binary search (left-closed, right-open interval) */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchLCRO</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
-<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i = j)</span>
+<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i = j)</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
-<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// This means target is in the interval [m+1, j)</span>
+<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// In this case, target must be in [m+1, j) if it exists</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// This means target is in the interval [i, m)</span>
+<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// In this case, target must be in [i, m) if it exists</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
-<a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Binary search (left-closed right-open interval) */</span>
+<div class="highlight"><span class="filename">binary_search.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Binary search (left-closed, right-open interval) */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">BinarySearchLCRO</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
-<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i = j)</span>
+<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i = j)</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">   </span><span class="c1">// Calculate the midpoint index m</span>
-<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w">      </span><span class="c1">// This means target is in the interval [m+1, j)</span>
+<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w">      </span><span class="c1">// In this case, target must be in [m+1, j) if it exists</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="nf">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// This means target is in the interval [i, m)</span>
+<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="nf">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// In this case, target must be in [i, m) if it exists</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
-<a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">        </span><span class="k">else</span><span class="w">                       </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">        </span><span class="k">else</span><span class="w">                       </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Binary search (left-closed right-open interval) */</span>
+<div class="highlight"><span class="filename">binary_search.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Binary search (left-closed, right-open interval) */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">binarySearchLCRO</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span>
-<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i = j)</span>
+<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i = j)</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="o">-</span><span class="nx">i</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="w">      </span><span class="c1">// Calculate the midpoint index m</span>
-<a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// This means target is in the interval [m+1, j)</span>
+<a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// In this case, target must be in [m+1, j) if it exists</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// This means target is in the interval [i, m)</span>
+<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// In this case, target must be in [i, m) if it exists</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span>
-<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Binary search (left-closed right-open interval) */</span>
+<div class="highlight"><span class="filename">binary_search.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Binary search (left-closed, right-open interval) */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">binarySearchLCRO</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">startIndex</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">endIndex</span>
-<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i = j)</span>
+<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i = j)</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
-<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// This means target is in the interval [m+1, j)</span>
+<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// In this case, target must be in [m+1, j) if it exists</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// This means target is in the interval [i, m)</span>
+<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// In this case, target must be in [i, m) if it exists</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">m</span>
-<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span>
 <a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Binary search (left-closed right-open interval) */</span>
+<div class="highlight"><span class="filename">binary_search.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Binary search (left-closed, right-open interval) */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearchLCRO</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i = j)</span>
+<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i = j)</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">        </span><span class="c1">// Calculate midpoint index m, use parseInt() to round down</span>
+<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">        </span><span class="c1">// Calculate midpoint index m, using parseInt() to round down</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">parseInt</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span>
-<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">            </span><span class="c1">// This means target is in the interval [m+1, j)</span>
+<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">            </span><span class="c1">// In this case, target must be in [m+1, j) if it exists</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span>
-<a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">            </span><span class="c1">// This means target is in the interval [i, m)</span>
+<a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">            </span><span class="c1">// In this case, target must be in [i, m) if it exists</span>
 <a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span>
-<a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">        </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">        </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span>
 <a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Binary search (left-closed right-open interval) */</span>
+<div class="highlight"><span class="filename">binary_search.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Binary search (left-closed, right-open interval) */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearchLCRO</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1</span>
+<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i = j)</span>
+<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i = j)</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">        </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">            </span><span class="c1">// This means target is in the interval [m+1, j)</span>
+<a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">            </span><span class="c1">// In this case, target must be in [m+1, j) if it exists</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">            </span><span class="c1">// This means target is in the interval [i, m)</span>
+<a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">            </span><span class="c1">// In this case, target must be in [i, m) if it exists</span>
 <a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">            </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">            </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span>
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Binary search (left-closed right-open interval) */</span>
+<div class="highlight"><span class="filename">binary_search.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Binary search (left-closed, right-open interval) */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">binarySearchLCRO</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1</span>
+<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
-<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="c1">// Loop, exit when the search interval is empty (empty when i = j)</span>
+<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="c1">// Loop until the search interval is empty (when i = j)</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">      </span><span class="c1">// This means target is in the interval [m+1, j)</span>
+<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">      </span><span class="c1">// In this case, target must be in [m+1, j) if it exists</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">      </span><span class="c1">// This means target is in the interval [i, m)</span>
+<a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">      </span><span class="c1">// In this case, target must be in [i, m) if it exists</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">      </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">      </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="w">  </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="w">  </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Binary search (left-closed right-open interval) */</span>
+<div class="highlight"><span class="filename">binary_search.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Binary search (left-closed, right-open interval) */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">binary_search_lcro</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
-<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i = j)</span>
+<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i = j)</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">            </span><span class="c1">// This means target is in the interval [m+1, j)</span>
+<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">            </span><span class="c1">// In this case, target must be in [m+1, j) if it exists</span>
 <a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">            </span><span class="c1">// This means target is in the interval [i, m)</span>
+<a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">            </span><span class="c1">// In this case, target must be in [i, m) if it exists</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">            </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">            </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Binary search (left-closed right-open interval) */</span>
+<div class="highlight"><span class="filename">binary_search.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Binary search (left-closed, right-open interval) */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchLCRO</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">len</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">len</span><span class="p">;</span>
-<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i = j)</span>
+<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i = j)</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
-<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w">    </span><span class="c1">// This means target is in the interval [m+1, j)</span>
+<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w">    </span><span class="c1">// In this case, target must be in [m+1, j) if it exists</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// This means target is in the interval [i, m)</span>
+<a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// In this case, target must be in [i, m) if it exists</span>
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
-<a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
 <a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Binary search (left-closed right-open interval) */</span>
+<div class="highlight"><span class="filename">binary_search.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Binary search (left-closed, right-open interval) */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">binarySearchLCRO</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1</span>
+<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span>
-<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="c1">// Loop, exit when the search interval is empty (empty when i = j)</span>
+<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="c1">// Loop until the search interval is empty (when i = j)</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
-<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// This means target is in the interval [m+1, j)</span>
+<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// In this case, target must be in [m+1, j) if it exists</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// This means target is in the interval [i, m)</span>
+<a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="c1">// In this case, target must be in [i, m) if it exists</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span>
-<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">        </span><span class="k">else</span><span class="w">  </span><span class="c1">// Found the target element, return its index</span>
+<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">        </span><span class="k">else</span><span class="w">  </span><span class="c1">// The target element was found; return its index</span>
 <a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span>
 <a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">    </span><span class="c1">// Target element not found, return -1</span>
+<a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">    </span><span class="c1">// The target element was not found; return -1</span>
 <a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="m">1</span>
 <a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Binary search (left-closed right-open interval) ###</span>
+<div class="highlight"><span class="filename">binary_search.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Binary search (left-closed, right-open interval) ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">binary_search_lcro</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a>
-<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="c1"># Loop, exit when the search interval is empty (empty when i = j)</span>
+<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="c1"># Loop until the search interval is empty (when i = j)</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">    </span><span class="c1"># Calculate the midpoint index m</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">    </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span>
-<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># This means target is in the interval [m+1, j)</span>
+<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># In this case, target must be in [m+1, j) if it exists</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">    </span><span class="k">elsif</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span>
-<a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># This means target is in the interval [i, m)</span>
+<a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># In this case, target must be in [i, m) if it exists</span>
 <a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">    </span><span class="k">else</span>
-<a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="w">  </span><span class="c1"># Found the target element, return its index</span>
+<a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="w">  </span><span class="c1"># The target element was found; return its index</span>
 <a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a>
-<a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a><span class="w">  </span><span class="o">-</span><span class="mi">1</span><span class="w">  </span><span class="c1"># Target element not found, return -1</span>
+<a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a><span class="w">  </span><span class="o">-</span><span class="mi">1</span><span class="w">  </span><span class="c1"># The target element was not found; return -1</span>
 <a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<p>As shown in Figure 10-3, under the two interval representations, the initialization, loop condition, and interval narrowing operations of the binary search algorithm are all different.</p>
-<p>Since both the left and right boundaries in the "closed interval" representation are defined as closed, the operations to narrow the interval through pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> are also symmetric. This makes it less error-prone, <strong>so the "closed interval" approach is generally recommended</strong>.</p>
-<p><img alt="Two interval definitions" class="animation-figure" src="../binary_search.assets/binary_search_ranges.png" /></p>
-<p align="center"> Figure 10-3 &nbsp; Two interval definitions </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20binary_search_lcro%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Binary%20search%20%28left-closed,%20right-open%20interval%29%22%22%22%0A%20%20%20%20%23%20Initialize%20the%20left-closed,%20right-open%20interval%20%5B0,%20n%29,%20with%20i%20at%20the%20first%20element%20and%20j%20one%20past%20the%20last%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%0A%20%20%20%20%23%20Loop%20until%20the%20search%20interval%20is%20empty%20%28when%20i%20%3D%20j%29%0A%20%20%20%20while%20i%20%3C%20j%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20Calculate%20the%20midpoint%20index%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20In%20this%20case,%20target%20must%20be%20in%20%5Bm%2B1,%20j%29%20if%20it%20exists%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20%20%23%20In%20this%20case,%20target%20must%20be%20in%20%5Bi,%20m%29%20if%20it%20exists%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20m%20%20%23%20The%20target%20element%20was%20found%3B%20return%20its%20index%0A%20%20%20%20return%20-1%20%20%23%20The%20target%20element%20was%20not%20found%3B%20return%20-1%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%208,%2012,%2015,%2023,%2026,%2031,%2035%5D%0A%0A%20%20%20%20%23%20Binary%20search%20%28left-closed,%20right-open%20interval%29%0A%20%20%20%20index%20%3D%20binary_search_lcro%28nums,%20target%29%0A%20%20%20%20print%28%22Index%20of%20target%20element%206%20%3D%20%22,%20index%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20binary_search_lcro%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Binary%20search%20%28left-closed,%20right-open%20interval%29%22%22%22%0A%20%20%20%20%23%20Initialize%20the%20left-closed,%20right-open%20interval%20%5B0,%20n%29,%20with%20i%20at%20the%20first%20element%20and%20j%20one%20past%20the%20last%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%0A%20%20%20%20%23%20Loop%20until%20the%20search%20interval%20is%20empty%20%28when%20i%20%3D%20j%29%0A%20%20%20%20while%20i%20%3C%20j%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20Calculate%20the%20midpoint%20index%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20In%20this%20case,%20target%20must%20be%20in%20%5Bm%2B1,%20j%29%20if%20it%20exists%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20%20%23%20In%20this%20case,%20target%20must%20be%20in%20%5Bi,%20m%29%20if%20it%20exists%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20m%20%20%23%20The%20target%20element%20was%20found%3B%20return%20its%20index%0A%20%20%20%20return%20-1%20%20%23%20The%20target%20element%20was%20not%20found%3B%20return%20-1%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%208,%2012,%2015,%2023,%2026,%2031,%2035%5D%0A%0A%20%20%20%20%23%20Binary%20search%20%28left-closed,%20right-open%20interval%29%0A%20%20%20%20index%20%3D%20binary_search_lcro%28nums,%20target%29%0A%20%20%20%20print%28%22Index%20of%20target%20element%206%20%3D%20%22,%20index%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>As shown below, the two interval representations require different initialization, loop conditions, and operations for narrowing the interval.</p>
+<p>Since a closed interval includes both endpoints, the operations that narrow the interval through pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> are symmetric. This makes mistakes less likely, <strong>so we generally recommend the closed-interval version</strong>.</p>
+<p><img alt="Two ways to define the interval" class="animation-figure" src="../binary_search.assets/binary_search_ranges.png" /></p>
+<p align="center"> Figure 10-3 &nbsp; Two ways to define the interval </p>
 
-<h2 id="1012-advantages-and-limitations">10.1.2 &nbsp; Advantages and Limitations<a class="headerlink" href="#1012-advantages-and-limitations" title="Permanent link">&para;</a></h2>
-<p>Binary search offers good performance in both time and space.</p>
+<h2 id="1012-strengths-and-limitations">10.1.2 &nbsp; Strengths and limitations<a class="headerlink" href="#1012-strengths-and-limitations" title="Permanent link">&para;</a></h2>
+<p>Binary search performs well in both time and space.</p>
 <ul>
-<li>Binary search has high time efficiency. With large data volumes, the logarithmic time complexity has significant advantages. For example, when the data size <span class="arithmatex">\(n = 2^{20}\)</span>, linear search requires <span class="arithmatex">\(2^{20} = 1048576\)</span> iterations, while binary search only needs <span class="arithmatex">\(\log_2 2^{20} = 20\)</span> iterations.</li>
-<li>Binary search requires no extra space. Compared to searching algorithms that require additional space (such as hash-based search), binary search is more space-efficient.</li>
+<li>Binary search is time-efficient. Its logarithmic time complexity offers a significant advantage for large datasets. For example, when <span class="arithmatex">\(n = 2^{20}\)</span>, linear search requires <span class="arithmatex">\(2^{20} = 1048576\)</span> iterations, whereas binary search needs only <span class="arithmatex">\(\log_2 2^{20} = 20\)</span>.</li>
+<li>Binary search requires no extra storage. It uses less space than search algorithms that rely on additional storage, such as hash lookup.</li>
 </ul>
-<p>However, binary search is not suitable for all situations, mainly for the following reasons:</p>
+<p>However, binary search is not suitable in every situation, mainly for the following reasons.</p>
 <ul>
-<li>Binary search is only applicable to sorted data. If the input data is unsorted, sorting specifically to use binary search would be counterproductive, as sorting algorithms typically have a time complexity of <span class="arithmatex">\(O(n \log n)\)</span>, which is higher than both linear search and binary search. For scenarios with frequent element insertions, keeping the array sorted requires inserting elements at specific positions with a time complexity of <span class="arithmatex">\(O(n)\)</span>, which is also very expensive.</li>
-<li>Binary search is only applicable to arrays. Binary search requires non-contiguous, jump-style access to elements, and this kind of access is inefficient in linked lists, making it unsuitable for linked lists or linked-list-based data structures.</li>
-<li>For small data volumes, linear search performs better. In linear search, each round requires only 1 comparison operation; while in binary search, it requires 1 addition, 1 division, 1-3 comparison operations, and 1 addition (subtraction), totaling 4-6 unit operations. Therefore, when the data volume <span class="arithmatex">\(n\)</span> is small, linear search is actually faster than binary search.</li>
+<li>Binary search requires sorted data. Sorting unsorted input solely to perform a binary search is not worthwhile, because sorting usually takes <span class="arithmatex">\(O(n \log n)\)</span> time, more than either linear or binary search. When insertions are frequent, preserving the array's order requires inserting each element at a specific position, an expensive <span class="arithmatex">\(O(n)\)</span> operation.</li>
+<li>Binary search is suitable only for arrays. It needs to jump between nonadjacent elements, which is inefficient in a linked list. It is therefore unsuitable for linked lists and data structures based on them.</li>
+<li>Linear search performs better on small datasets. Each iteration of linear search needs only one comparison. Binary search needs one addition, one division, 1–3 comparisons, and one more addition or subtraction, totaling 4–6 elementary operations. When <span class="arithmatex">\(n\)</span> is small, linear search can therefore be faster.</li>
 </ul>
 
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@@ -5325,7 +5407,7 @@ aria-label="Footer"
   <a
     href="../"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: Chapter 10. &amp;nbsp; Searching"
+    aria-label="Previous: Chapter 10.   Searching"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -5349,7 +5431,7 @@ aria-label="Footer"
   <a
     href="../binary_search_insertion/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 10.2 Binary Search Insertion Point"
+    aria-label="Next: 10.2   Finding an insertion point with binary search"
     rel="next"
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     <div class="md-footer__title">
@@ -5357,7 +5439,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        10.2 Binary Search Insertion Point
+        10.2 &nbsp; Finding an insertion point with binary search
       </div>
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@@ -5374,7 +5456,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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diff --git a/en/chapter_searching/binary_search_edge/index.html b/en/chapter_searching/binary_search_edge/index.html
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       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>10.3 Binary Search Boundaries - Hello Algo</title>
+        <title>10.3 Finding boundaries with binary search - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
     <div data-md-component="skip">
 
 
-        <a href="#103-binary-search-boundaries" class="md-skip">
+        <a href="#103-finding-boundaries-with-binary-search" class="md-skip">
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@@ -119,7 +119,7 @@
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
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-              10.3 &nbsp; Binary Search Boundaries
+              10.3 &nbsp; Finding boundaries with binary search
 
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-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
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   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
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   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
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   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
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-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
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   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
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-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
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   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2598,7 +2598,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2619,7 +2619,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2640,7 +2640,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2703,7 +2703,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2720,7 +2720,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2749,7 +2749,7 @@
   <a href="#1031-finding-the-left-boundary" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.3.1 &nbsp; Finding the Left Boundary
+        10.3.1 &nbsp; Finding the left boundary
       </span>
     </span>
   </a>
@@ -2760,19 +2760,19 @@
   <a href="#1032-finding-the-right-boundary" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.3.2 &nbsp; Finding the Right Boundary
+        10.3.2 &nbsp; Finding the right boundary
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="10.3.2 Finding the Right Boundary">
+    <nav class="md-nav" aria-label="10.3.2 Finding the right boundary">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-reusing-left-boundary-search" class="md-nav__link">
+  <a href="#1-reusing-the-left-boundary-search" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Reusing Left Boundary Search
+        1. &nbsp; Reusing the left-boundary search
       </span>
     </span>
   </a>
@@ -2780,10 +2780,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-converting-to-element-search" class="md-nav__link">
+  <a href="#2-turning-a-boundary-search-into-an-element-search" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Converting to Element Search
+        2. &nbsp; Turning a boundary search into an element search
       </span>
     </span>
   </a>
@@ -2818,7 +2818,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2845,7 +2845,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2872,7 +2872,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2899,7 +2899,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2977,7 +2977,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2998,7 +2998,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3019,7 +3019,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3046,7 +3046,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3073,7 +3073,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3100,7 +3100,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3127,7 +3127,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3154,7 +3154,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3181,7 +3181,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3208,7 +3208,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3235,7 +3235,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3262,7 +3262,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3289,7 +3289,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3316,7 +3316,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3382,7 +3382,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3403,7 +3403,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3424,7 +3424,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3451,7 +3451,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3478,7 +3478,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3505,7 +3505,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3532,7 +3532,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3559,7 +3559,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3625,7 +3625,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3646,7 +3646,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3667,7 +3667,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3694,7 +3694,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3721,7 +3721,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3748,7 +3748,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3775,7 +3775,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3802,7 +3802,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3872,7 +3872,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3893,7 +3893,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3914,7 +3914,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3941,7 +3941,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3968,7 +3968,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3995,7 +3995,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4022,7 +4022,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4049,7 +4049,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4076,7 +4076,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4103,7 +4103,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4169,7 +4169,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4190,7 +4190,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4211,7 +4211,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4238,7 +4238,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4265,7 +4265,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4292,7 +4292,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4319,7 +4319,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4346,7 +4346,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4406,7 +4406,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4427,7 +4427,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4448,7 +4448,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4475,7 +4475,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4502,7 +4502,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4586,6 +4586,78 @@
 
 
 
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   <a href="#1031-finding-the-left-boundary" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.3.1 &nbsp; Finding the Left Boundary
+        10.3.1 &nbsp; Finding the left boundary
       </span>
     </span>
   </a>
@@ -4641,19 +4713,19 @@
   <a href="#1032-finding-the-right-boundary" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.3.2 &nbsp; Finding the Right Boundary
+        10.3.2 &nbsp; Finding the right boundary
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="10.3.2 Finding the Right Boundary">
+    <nav class="md-nav" aria-label="10.3.2 Finding the right boundary">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-reusing-left-boundary-search" class="md-nav__link">
+  <a href="#1-reusing-the-left-boundary-search" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Reusing Left Boundary Search
+        1. &nbsp; Reusing the left-boundary search
       </span>
     </span>
   </a>
@@ -4661,10 +4733,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-converting-to-element-search" class="md-nav__link">
+  <a href="#2-turning-a-boundary-search-into-an-element-search" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Converting to Element Search
+        2. &nbsp; Turning a boundary search into an element search
       </span>
     </span>
   </a>
@@ -4713,436 +4785,446 @@
 
 
 <!-- Page content -->
-<h1 id="103-binary-search-boundaries">10.3 &nbsp; Binary Search Boundaries<a class="headerlink" href="#103-binary-search-boundaries" title="Permanent link">&para;</a></h1>
-<h2 id="1031-finding-the-left-boundary">10.3.1 &nbsp; Finding the Left Boundary<a class="headerlink" href="#1031-finding-the-left-boundary" title="Permanent link">&para;</a></h2>
+<h1 id="103-finding-boundaries-with-binary-search">10.3 &nbsp; Finding boundaries with binary search<a class="headerlink" href="#103-finding-boundaries-with-binary-search" title="Permanent link">&para;</a></h1>
+<h2 id="1031-finding-the-left-boundary">10.3.1 &nbsp; Finding the left boundary<a class="headerlink" href="#1031-finding-the-left-boundary" title="Permanent link">&para;</a></h2>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given a sorted array <code>nums</code> of length <span class="arithmatex">\(n\)</span> that may contain duplicate elements, return the index of the leftmost occurrence of <code>target</code>. If the array does not contain <code>target</code>, return <span class="arithmatex">\(-1\)</span>.</p>
+<p>Given a sorted array <code>nums</code> of length <span class="arithmatex">\(n\)</span> that may contain duplicate elements, return the index of the leftmost occurrence of <code>target</code>. If the array does not contain this element, return <span class="arithmatex">\(-1\)</span>.</p>
 </div>
-<p>Recall the method for finding the insertion point with binary search. After the search completes, <span class="arithmatex">\(i\)</span> points to the leftmost <code>target</code>, <strong>so finding the insertion point is essentially finding the index of the leftmost <code>target</code></strong>.</p>
-<p>Consider implementing the left boundary search using the insertion point finding function. Note that the array may not contain <code>target</code>, which could result in the following two cases:</p>
+<p>Recall the method for finding an insertion point with binary search: when the search finishes, <span class="arithmatex">\(i\)</span> points to the leftmost <code>target</code>. <strong>Finding the insertion point therefore amounts to finding the index of the leftmost <code>target</code></strong>.</p>
+<p>We can use the insertion-point function to find the left boundary. However, <code>target</code> may be absent from the array, leading to either of the following outcomes.</p>
 <ul>
-<li>The insertion point index <span class="arithmatex">\(i\)</span> is out of bounds.</li>
-<li>The element <code>nums[i]</code> is not equal to <code>target</code>.</li>
+<li>The insertion index <span class="arithmatex">\(i\)</span> is out of bounds.</li>
+<li><code>nums[i]</code> is not equal to <code>target</code>.</li>
 </ul>
-<p>When either of these situations occurs, simply return <span class="arithmatex">\(-1\)</span>. The code is shown below:</p>
+<p>In either case, simply return <span class="arithmatex">\(-1\)</span>. The code is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_edge.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">binary_search_left_edge</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Binary search for the leftmost target"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Equivalent to finding the insertion point of target</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Find the leftmost occurrence of target using binary search"""</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># This is equivalent to finding the insertion point of target</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="n">i</span> <span class="o">=</span> <span class="n">binary_search_insertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">target</span><span class="p">)</span>
-<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># Target not found, return -1</span>
+<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="c1"># If target was not found, return -1</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="ow">or</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">!=</span> <span class="n">target</span><span class="p">:</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="c1"># Found target, return index i</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="c1"># Target was found; return index i</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="k">return</span> <span class="n">i</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Binary search for the leftmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Find the leftmost occurrence of target using binary search */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchLeftEdge</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Equivalent to finding the insertion point of target</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// This is equivalent to finding the insertion point of target</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">);</span>
-<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="c1">// Found target, return index i</span>
+<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="c1">// Target was found; return index i</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Binary search for the leftmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Find the leftmost occurrence of target using binary search */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchLeftEdge</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Equivalent to finding the insertion point of target</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// This is equivalent to finding the insertion point of target</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binary_search_insertion</span><span class="p">.</span><span class="na">binarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">);</span>
-<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="c1">// Found target, return index i</span>
+<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="c1">// Target was found; return index i</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Binary search for the leftmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Find the leftmost occurrence of target using binary search */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">BinarySearchLeftEdge</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Equivalent to finding the insertion point of target</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// This is equivalent to finding the insertion point of target</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binary_search_insertion</span><span class="p">.</span><span class="n">BinarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">);</span>
-<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="c1">// Found target, return index i</span>
+<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="c1">// Target was found; return index i</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Binary search for the leftmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Find the leftmost occurrence of target using binary search */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">binarySearchLeftEdge</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Equivalent to finding the insertion point of target</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// This is equivalent to finding the insertion point of target</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">binarySearchInsertion</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span>
-<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// Found target, return index i</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// Target was found; return index i</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Binary search for the leftmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Find the leftmost occurrence of target using binary search */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">binarySearchLeftEdge</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Equivalent to finding the insertion point of target</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// This is equivalent to finding the insertion point of target</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">binarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
-<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">endIndex</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="c1">// Found target, return index i</span>
+<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="c1">// Target was found; return index i</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Binary search for the leftmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Find the leftmost occurrence of target using binary search */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearchLeftEdge</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Equivalent to finding the insertion point of target</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// This is equivalent to finding the insertion point of target</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">binarySearchInsertion</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">);</span>
-<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="c1">// Found target, return index i</span>
+<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="c1">// Target was found; return index i</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Binary search for the leftmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Find the leftmost occurrence of target using binary search */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearchLeftEdge</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Equivalent to finding the insertion point of target</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// This is equivalent to finding the insertion point of target</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">binarySearchInsertion</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">);</span>
-<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// Found target, return index i</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// Target was found; return index i</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Binary search for the leftmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Find the leftmost occurrence of target using binary search */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">binarySearchLeftEdge</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Equivalent to finding the insertion point of target</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// This is equivalent to finding the insertion point of target</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">);</span>
-<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="c1">// Found target, return index i</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="c1">// Target was found; return index i</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Binary search for the leftmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Find the leftmost occurrence of target using binary search */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">binary_search_left_edge</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Equivalent to finding the insertion point of target</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// This is equivalent to finding the insertion point of target</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binary_search_insertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">);</span>
-<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// Found target, return index i</span>
+<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// Target was found; return index i</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">    </span><span class="n">i</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Binary search for the leftmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Find the leftmost occurrence of target using binary search */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchLeftEdge</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">numSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Equivalent to finding the insertion point of target</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// This is equivalent to finding the insertion point of target</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">numSize</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">);</span>
-<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">numSize</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="c1">// Found target, return index i</span>
+<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="c1">// Target was found; return index i</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Binary search for the leftmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Find the leftmost occurrence of target using binary search */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">binarySearchLeftEdge</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Equivalent to finding the insertion point of target</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// This is equivalent to finding the insertion point of target</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
-<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="m">1</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// Found target, return index i</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// Target was found; return index i</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Binary search leftmost target ###</span>
+<div class="highlight"><span class="filename">binary_search_edge.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Find the leftmost occurrence of target using binary search ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">binary_search_left_edge</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Equivalent to finding the insertion point of target</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># This is equivalent to finding the insertion point of target</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binary_search_insertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a>
-<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="c1"># Target not found, return -1</span>
+<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="c1"># If target was not found, return -1</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a>
-<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="c1"># Found target, return index i</span>
+<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="c1"># Target was found; return index i</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h2 id="1032-finding-the-right-boundary">10.3.2 &nbsp; Finding the Right Boundary<a class="headerlink" href="#1032-finding-the-right-boundary" title="Permanent link">&para;</a></h2>
-<p>So how do we find the rightmost <code>target</code>? The most direct approach is to modify the code and replace the pointer shrinking operation in the <code>nums[m] == target</code> case. The code is omitted here; interested readers can implement it themselves.</p>
-<p>Below we introduce two more clever methods.</p>
-<h3 id="1-reusing-left-boundary-search">1. &nbsp; Reusing Left Boundary Search<a class="headerlink" href="#1-reusing-left-boundary-search" title="Permanent link">&para;</a></h3>
-<p>In fact, we can use the function for finding the leftmost <code>target</code> to find the rightmost <code>target</code>. The specific method is: <strong>convert finding the rightmost <code>target</code> into finding the leftmost <code>target + 1</code></strong>.</p>
-<p>As shown in Figure 10-7, after the search completes, the pointer <span class="arithmatex">\(i\)</span> points to the leftmost <code>target + 1</code> (if it exists), while <span class="arithmatex">\(j\)</span> points to the rightmost <code>target</code>, <strong>so we can return <span class="arithmatex">\(j\)</span></strong>.</p>
-<p><img alt="Converting right boundary search to left boundary search" class="animation-figure" src="../binary_search_edge.assets/binary_search_right_edge_by_left_edge.png" /></p>
-<p align="center"> Figure 10-7 &nbsp; Converting right boundary search to left boundary search </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20binary_search_insertion%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20an%20insertion%20point%20using%20binary%20search%20%28with%20duplicates%29%22%22%22%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%20-%201%20%20%23%20Initialize%20the%20closed%20interval%20%5B0,%20n-1%5D%0A%20%20%20%20while%20i%20%3C%3D%20j%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20Calculate%20the%20midpoint%20index%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20Target%20must%20be%20in%20%5Bm%2B1,%20j%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20Target%20must%20be%20in%20%5Bi,%20m-1%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20Search%20%5Bi,%20m-1%5D%20for%20the%20insertion%20point%20before%20equal%20elements%0A%20%20%20%20%23%20Return%20insertion%20point%20i%0A%20%20%20%20return%20i%0A%0Adef%20binary_search_left_edge%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20the%20leftmost%20occurrence%20of%20target%20using%20binary%20search%22%22%22%0A%20%20%20%20%23%20This%20is%20equivalent%20to%20finding%20the%20insertion%20point%20of%20target%0A%20%20%20%20i%20%3D%20binary_search_insertion%28nums,%20target%29%0A%20%20%20%20%23%20If%20target%20was%20not%20found,%20return%20-1%0A%20%20%20%20if%20i%20%3D%3D%20len%28nums%29%20or%20nums%5Bi%5D%20%21%3D%20target%3A%0A%20%20%20%20%20%20%20%20return%20-1%0A%20%20%20%20%23%20Target%20was%20found%3B%20return%20index%20i%0A%20%20%20%20return%20i%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Array%20containing%20duplicate%20elements%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%206,%206,%206,%206,%2010,%2012,%2015%5D%0A%20%20%20%20%23%20Find%20a%20boundary%20using%20binary%20search%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20index%20%3D%20binary_search_left_edge%28nums,%20target%29%0A%20%20%20%20print%28f%22Index%20of%20the%20leftmost%20occurrence%20of%20%7Btarget%7D%3A%20%7Bindex%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20binary_search_insertion%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20an%20insertion%20point%20using%20binary%20search%20%28with%20duplicates%29%22%22%22%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%20-%201%20%20%23%20Initialize%20the%20closed%20interval%20%5B0,%20n-1%5D%0A%20%20%20%20while%20i%20%3C%3D%20j%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20Calculate%20the%20midpoint%20index%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20Target%20must%20be%20in%20%5Bm%2B1,%20j%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20Target%20must%20be%20in%20%5Bi,%20m-1%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20Search%20%5Bi,%20m-1%5D%20for%20the%20insertion%20point%20before%20equal%20elements%0A%20%20%20%20%23%20Return%20insertion%20point%20i%0A%20%20%20%20return%20i%0A%0Adef%20binary_search_left_edge%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20the%20leftmost%20occurrence%20of%20target%20using%20binary%20search%22%22%22%0A%20%20%20%20%23%20This%20is%20equivalent%20to%20finding%20the%20insertion%20point%20of%20target%0A%20%20%20%20i%20%3D%20binary_search_insertion%28nums,%20target%29%0A%20%20%20%20%23%20If%20target%20was%20not%20found,%20return%20-1%0A%20%20%20%20if%20i%20%3D%3D%20len%28nums%29%20or%20nums%5Bi%5D%20%21%3D%20target%3A%0A%20%20%20%20%20%20%20%20return%20-1%0A%20%20%20%20%23%20Target%20was%20found%3B%20return%20index%20i%0A%20%20%20%20return%20i%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Array%20containing%20duplicate%20elements%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%206,%206,%206,%206,%2010,%2012,%2015%5D%0A%20%20%20%20%23%20Find%20a%20boundary%20using%20binary%20search%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20index%20%3D%20binary_search_left_edge%28nums,%20target%29%0A%20%20%20%20print%28f%22Index%20of%20the%20leftmost%20occurrence%20of%20%7Btarget%7D%3A%20%7Bindex%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1032-finding-the-right-boundary">10.3.2 &nbsp; Finding the right boundary<a class="headerlink" href="#1032-finding-the-right-boundary" title="Permanent link">&para;</a></h2>
+<p>How can we find the rightmost <code>target</code>? The most direct approach is to modify the pointer update for <code>nums[m] == target</code>. We omit the code here; interested readers can implement it themselves.</p>
+<p>Here are two shortcuts.</p>
+<h3 id="1-reusing-the-left-boundary-search">1. &nbsp; Reusing the left-boundary search<a class="headerlink" href="#1-reusing-the-left-boundary-search" title="Permanent link">&para;</a></h3>
+<p>We can use the function for finding the leftmost element to find the rightmost one: <strong>turn the search for the rightmost <code>target</code> into a search for the leftmost <code>target + 1</code></strong>.</p>
+<p>As shown below, when the search finishes, pointer <span class="arithmatex">\(i\)</span> points to the leftmost <code>target + 1</code>, if it exists, while <span class="arithmatex">\(j\)</span> points to the rightmost <code>target</code>. <strong>We therefore return <span class="arithmatex">\(j\)</span></strong>.</p>
+<p><img alt="Turning a right-boundary search into a left-boundary search" class="animation-figure" src="../binary_search_edge.assets/binary_search_right_edge_by_left_edge.png" /></p>
+<p align="center"> Figure 10-7 &nbsp; Turning a right-boundary search into a left-boundary search </p>
 
-<p>Note that the returned insertion point is <span class="arithmatex">\(i\)</span>, so we need to subtract <span class="arithmatex">\(1\)</span> from it to obtain <span class="arithmatex">\(j\)</span>:</p>
+<p>The returned insertion point is <span class="arithmatex">\(i\)</span>, so subtract <span class="arithmatex">\(1\)</span> to obtain <span class="arithmatex">\(j\)</span>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_edge.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">binary_search_right_edge</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Binary search for the rightmost target"""</span>
-<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># Convert to finding the leftmost target + 1</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Find the rightmost occurrence of target using binary search"""</span>
+<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># Reduce the problem to finding the leftmost insertion point of target + 1</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="n">i</span> <span class="o">=</span> <span class="n">binary_search_insertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">target</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># j points to the rightmost target, i points to the first element greater than target</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># j is the candidate rightmost target index; i is the index of the first element greater than target, or len(nums)</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="n">j</span> <span class="o">=</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span>
-<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>    <span class="c1"># Target not found, return -1</span>
+<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>    <span class="c1"># If target was not found, return -1</span>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>    <span class="k">if</span> <span class="n">j</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">or</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">!=</span> <span class="n">target</span><span class="p">:</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>        <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
-<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>    <span class="c1"># Found target, return index j</span>
+<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>    <span class="c1"># Target was found; return index j</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>    <span class="k">return</span> <span class="n">j</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Binary search for the rightmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Find the rightmost occurrence of target using binary search */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchRightEdge</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// Convert to finding the leftmost target + 1</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// Reduce the problem to finding the leftmost insertion point of target + 1</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="c1">// j points to the rightmost target, i points to the first element greater than target</span>
+<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="c1">// j points to the rightmost target, while i points to the first element greater than target</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">-1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">    </span><span class="c1">// Found target, return index j</span>
+<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">    </span><span class="c1">// Target was found; return index j</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Binary search for the rightmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Find the rightmost occurrence of target using binary search */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchRightEdge</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Convert to finding the leftmost target + 1</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Reduce the problem to finding the leftmost insertion point of target + 1</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binary_search_insertion</span><span class="p">.</span><span class="na">binarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="c1">// j points to the rightmost target, i points to the first element greater than target</span>
+<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="c1">// j points to the rightmost target, while i points to the first element greater than target</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="c1">// Found target, return index j</span>
+<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="c1">// Target was found; return index j</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Binary search for the rightmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Find the rightmost occurrence of target using binary search */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">BinarySearchRightEdge</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// Convert to finding the leftmost target + 1</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// Reduce the problem to finding the leftmost insertion point of target + 1</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binary_search_insertion</span><span class="p">.</span><span class="n">BinarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="c1">// j points to the rightmost target, i points to the first element greater than target</span>
+<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="c1">// j is the candidate index of the rightmost target; i is the insertion point after all elements &lt;= target</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="c1">// Found target, return index j</span>
+<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="c1">// Target was found; return index j</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Binary search for the rightmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Find the rightmost occurrence of target using binary search */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">binarySearchRightEdge</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Convert to finding the leftmost target + 1</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// For integer values, find the leftmost insertion point of target + 1 (assuming no overflow)</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">binarySearchInsertion</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="c1">// j points to the rightmost target, i points to the first element greater than target</span>
+<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="c1">// i is the index of the first element greater than target (or len(nums)); j is the candidate index of the rightmost target</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="c1">// Found target, return index j</span>
+<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="c1">// Target was found; return index j</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">j</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Binary search for the rightmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Find the rightmost occurrence of target using binary search */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">binarySearchRightEdge</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Convert to finding the leftmost target + 1</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Reduce the problem to finding the leftmost insertion point of target + 1</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">binarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="c1">// j points to the rightmost target, i points to the first element greater than target</span>
+<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="c1">// i is the first index with a value greater than target (or nums.endIndex); j is the candidate index of the rightmost target</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="c1">// Found target, return index j</span>
+<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="c1">// Target was found; return index j</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">j</span>
 <a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Binary search for the rightmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Find the rightmost occurrence of target using binary search */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearchRightEdge</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Convert to finding the leftmost target + 1</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Reduce the problem to finding the leftmost insertion point of target + 1</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">binarySearchInsertion</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="c1">// j points to the rightmost target, i points to the first element greater than target</span>
+<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="c1">// j points to the rightmost target, while i points to the first element greater than target</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">    </span><span class="c1">// Found target, return index j</span>
+<a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">    </span><span class="c1">// Target was found; return index j</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">j</span><span class="p">;</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Binary search for the rightmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Find the rightmost occurrence of target using binary search */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearchRightEdge</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// Convert to finding the leftmost target + 1</span>
+<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// Reduce the problem to finding the leftmost insertion point of target + 1</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">binarySearchInsertion</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="c1">// j points to the rightmost target, i points to the first element greater than target</span>
+<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="c1">// For integer elements, i is the first index greater than target (or the array length); j is the candidate rightmost target index</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="c1">// Found target, return index j</span>
+<a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="c1">// Target was found; return index j</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">j</span><span class="p">;</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Binary search for the rightmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Find the rightmost occurrence of target using binary search */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">binarySearchRightEdge</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// Convert to finding the leftmost target + 1</span>
+<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// Reduce the problem to finding the leftmost insertion point of target + 1</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
-<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="c1">// j points to the rightmost target, i points to the first element greater than target</span>
+<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="c1">// i is the first index greater than target (or the array length); j is the candidate rightmost target index</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
-<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">  </span><span class="c1">// Found target, return index j</span>
+<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">  </span><span class="c1">// Target was found; return index j</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Binary search for the rightmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Find the rightmost occurrence of target using binary search */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">binary_search_right_edge</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Convert to finding the leftmost target + 1</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Reduce the problem to finding the leftmost insertion point of target + 1</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binary_search_insertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="c1">// j points to the rightmost target, i points to the first element greater than target</span>
+<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="c1">// j points to the rightmost target, while i points to the first element greater than target</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">    </span><span class="c1">// Found target, return index j</span>
+<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">    </span><span class="c1">// Target was found; return index j</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">    </span><span class="n">j</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Binary search for the rightmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Find the rightmost occurrence of target using binary search */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchRightEdge</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">numSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Convert to finding the leftmost target + 1</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Reduce the problem to finding the leftmost insertion point of target + 1</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">numSize</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="c1">// j points to the rightmost target, i points to the first element greater than target</span>
+<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="c1">// j points to the rightmost target, while i points to the first element greater than target</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">-1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="c1">// Found target, return index j</span>
+<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="c1">// Target was found; return index j</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">j</span><span class="p">;</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Binary search for the rightmost target */</span>
+<div class="highlight"><span class="filename">binary_search_edge.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Find the rightmost occurrence of target using binary search */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">binarySearchRightEdge</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Convert to finding the leftmost target + 1</span>
+<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// For integer values, find the leftmost insertion point of target + 1 (assuming no overflow)</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
-<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="c1">// j points to the rightmost target, i points to the first element greater than target</span>
+<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="c1">// i is the index of the first element greater than target (or the array length); j is the candidate rightmost target index</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="c1">// Target not found, return -1</span>
+<a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="c1">// If target was not found, return -1</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="m">1</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">    </span><span class="c1">// Found target, return index j</span>
+<a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">    </span><span class="c1">// Target was found; return index j</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">j</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_edge.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Binary search rightmost target ###</span>
+<div class="highlight"><span class="filename">binary_search_edge.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Find the rightmost occurrence of target using binary search ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">binary_search_right_edge</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Convert to finding the leftmost target + 1</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># For integer elements, find the leftmost insertion point of target + 1</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">binary_search_insertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a>
-<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="c1"># j points to the rightmost target, i points to the first element greater than target</span>
+<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="c1"># i is the index of the first element greater than target (or the array length); j is the candidate rightmost target index</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a>
-<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="c1"># Target not found, return -1</span>
+<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="c1"># If target was not found, return -1</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">target</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a>
-<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">  </span><span class="n">j</span><span class="w"> </span><span class="c1"># Found target, return index j</span>
+<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">  </span><span class="n">j</span><span class="w"> </span><span class="c1"># Target was found; return index j</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h3 id="2-converting-to-element-search">2. &nbsp; Converting to Element Search<a class="headerlink" href="#2-converting-to-element-search" title="Permanent link">&para;</a></h3>
-<p>We know that when the array does not contain <code>target</code>, <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> will eventually point to the first element greater than <code>target</code> and the rightmost element less than <code>target</code>, respectively.</p>
-<p>Therefore, as shown in Figure 10-8, we can construct an element that does not exist in the array to find the left and right boundaries.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20binary_search_insertion%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20an%20insertion%20point%20using%20binary%20search%20%28with%20duplicates%29%22%22%22%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%20-%201%20%20%23%20Initialize%20the%20closed%20interval%20%5B0,%20n-1%5D%0A%20%20%20%20while%20i%20%3C%3D%20j%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20Calculate%20the%20midpoint%20index%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20Target%20must%20be%20in%20%5Bm%2B1,%20j%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20Target%20must%20be%20in%20%5Bi,%20m-1%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20Search%20%5Bi,%20m-1%5D%20for%20the%20insertion%20point%20before%20equal%20elements%0A%20%20%20%20%23%20Return%20insertion%20point%20i%0A%20%20%20%20return%20i%0A%0Adef%20binary_search_right_edge%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20the%20rightmost%20occurrence%20of%20target%20using%20binary%20search%22%22%22%0A%20%20%20%20%23%20Reduce%20the%20problem%20to%20finding%20the%20leftmost%20insertion%20point%20of%20target%20%2B%201%0A%20%20%20%20i%20%3D%20binary_search_insertion%28nums,%20target%20%2B%201%29%0A%20%20%20%20%23%20j%20is%20the%20candidate%20index%20of%20the%20rightmost%20target%3B%20i%20is%20the%20index%20of%20the%20first%20element%20greater%20than%20target,%20or%20len%28nums%29%0A%20%20%20%20j%20%3D%20i%20-%201%0A%20%20%20%20%23%20If%20target%20was%20not%20found,%20return%20-1%0A%20%20%20%20if%20j%20%3D%3D%20-1%20or%20nums%5Bj%5D%20%21%3D%20target%3A%0A%20%20%20%20%20%20%20%20return%20-1%0A%20%20%20%20%23%20Target%20was%20found%3B%20return%20index%20j%0A%20%20%20%20return%20j%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Array%20containing%20duplicate%20elements%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%206,%206,%206,%206,%2010,%2012,%2015%5D%0A%20%20%20%20%23%20Find%20a%20boundary%20using%20binary%20search%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20index%20%3D%20binary_search_right_edge%28nums,%20target%29%0A%20%20%20%20print%28f%22Index%20of%20the%20rightmost%20occurrence%20of%20%7Btarget%7D%3A%20%7Bindex%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20binary_search_insertion%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20an%20insertion%20point%20using%20binary%20search%20%28with%20duplicates%29%22%22%22%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%20-%201%20%20%23%20Initialize%20the%20closed%20interval%20%5B0,%20n-1%5D%0A%20%20%20%20while%20i%20%3C%3D%20j%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20Calculate%20the%20midpoint%20index%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20Target%20must%20be%20in%20%5Bm%2B1,%20j%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20Target%20must%20be%20in%20%5Bi,%20m-1%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20Search%20%5Bi,%20m-1%5D%20for%20the%20insertion%20point%20before%20equal%20elements%0A%20%20%20%20%23%20Return%20insertion%20point%20i%0A%20%20%20%20return%20i%0A%0Adef%20binary_search_right_edge%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20the%20rightmost%20occurrence%20of%20target%20using%20binary%20search%22%22%22%0A%20%20%20%20%23%20Reduce%20the%20problem%20to%20finding%20the%20leftmost%20insertion%20point%20of%20target%20%2B%201%0A%20%20%20%20i%20%3D%20binary_search_insertion%28nums,%20target%20%2B%201%29%0A%20%20%20%20%23%20j%20is%20the%20candidate%20index%20of%20the%20rightmost%20target%3B%20i%20is%20the%20index%20of%20the%20first%20element%20greater%20than%20target,%20or%20len%28nums%29%0A%20%20%20%20j%20%3D%20i%20-%201%0A%20%20%20%20%23%20If%20target%20was%20not%20found,%20return%20-1%0A%20%20%20%20if%20j%20%3D%3D%20-1%20or%20nums%5Bj%5D%20%21%3D%20target%3A%0A%20%20%20%20%20%20%20%20return%20-1%0A%20%20%20%20%23%20Target%20was%20found%3B%20return%20index%20j%0A%20%20%20%20return%20j%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Array%20containing%20duplicate%20elements%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%206,%206,%206,%206,%2010,%2012,%2015%5D%0A%20%20%20%20%23%20Find%20a%20boundary%20using%20binary%20search%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20index%20%3D%20binary_search_right_edge%28nums,%20target%29%0A%20%20%20%20print%28f%22Index%20of%20the%20rightmost%20occurrence%20of%20%7Btarget%7D%3A%20%7Bindex%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="2-turning-a-boundary-search-into-an-element-search">2. &nbsp; Turning a boundary search into an element search<a class="headerlink" href="#2-turning-a-boundary-search-into-an-element-search" title="Permanent link">&para;</a></h3>
+<p>When the array does not contain <code>target</code>, <span class="arithmatex">\(i\)</span> ultimately points to the first element greater than <code>target</code>, while <span class="arithmatex">\(j\)</span> points to the rightmost element smaller than <code>target</code>.</p>
+<p>As shown below, we can therefore construct a value absent from the array and use it to find the left and right boundaries.</p>
 <ul>
-<li>Finding the leftmost <code>target</code>: This can be converted to finding <code>target - 0.5</code> and returning the pointer <span class="arithmatex">\(i\)</span>.</li>
-<li>Finding the rightmost <code>target</code>: This can be converted to finding <code>target + 0.5</code> and returning the pointer <span class="arithmatex">\(j\)</span>.</li>
+<li>To find the leftmost <code>target</code>, search for <code>target - 0.5</code> and return pointer <span class="arithmatex">\(i\)</span>.</li>
+<li>To find the rightmost <code>target</code>, search for <code>target + 0.5</code> and return pointer <span class="arithmatex">\(j\)</span>.</li>
 </ul>
-<p><img alt="Converting boundary search to element search" class="animation-figure" src="../binary_search_edge.assets/binary_search_edge_by_element.png" /></p>
-<p align="center"> Figure 10-8 &nbsp; Converting boundary search to element search </p>
+<p><img alt="Turning a boundary search into an element search" class="animation-figure" src="../binary_search_edge.assets/binary_search_edge_by_element.png" /></p>
+<p align="center"> Figure 10-8 &nbsp; Turning a boundary search into an element search </p>
 
-<p>The code is omitted here, but the following two points are worth noting:</p>
+<p>The code is omitted here, but keep two points in mind.</p>
 <ul>
-<li>Since the given array does not contain decimal values, we do not need to worry about how to handle equality.</li>
-<li>Because this method introduces decimals, the variable <code>target</code> in the function needs to be changed to a floating-point type (Python does not require this change).</li>
+<li>The given array contains no fractional values, so we do not need to handle equality.</li>
+<li>Since this method introduces fractional values, change the function's <code>target</code> variable to a floating-point type. No change is needed in Python.</li>
 </ul>
 
 <!-- Source file information -->
@@ -5166,7 +5248,7 @@ aria-label="Footer"
   <a
     href="../binary_search_insertion/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 10.2 Binary Search Insertion Point"
+    aria-label="Previous: 10.2   Finding an insertion point with binary search"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -5178,7 +5260,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        10.2 Binary Search Insertion Point
+        10.2 &nbsp; Finding an insertion point with binary search
       </div>
     </div>
   </a>
@@ -5190,7 +5272,7 @@ aria-label="Footer"
   <a
     href="../replace_linear_by_hashing/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 10.4 Hash Optimization Strategy"
+    aria-label="Next: 10.4   Optimizing with hashing"
     rel="next"
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@@ -5198,7 +5280,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        10.4 Hash Optimization Strategy
+        10.4 &nbsp; Optimizing with hashing
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -5215,7 +5297,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>10.2 Binary Search Insertion Point - Hello Algo</title>
+        <title>10.2 Finding an insertion point with binary search - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
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+        <a href="#102-finding-an-insertion-point-with-binary-search" class="md-skip">
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@@ -154,7 +154,7 @@
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
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-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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-  Chapter 1. Encounter with Algorithms
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-  Chapter 1. Encounter with Algorithms
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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-  1.3 Summary
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-  Chapter 2. Complexity Analysis
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-  2.2 Iteration and Recursion
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
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-  3.2 Basic Data Types
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-  3.4 Character Encoding *
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+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
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+  5.2 &nbsp; Queues
 
 
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+  5.3 &nbsp; Deques
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
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+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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@@ -2420,7 +2420,7 @@
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+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
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         <li class="md-nav__item">
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+  <a href="#1021-when-there-are-no-duplicates" class="md-nav__link">
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+        10.2.1 &nbsp; When there are no duplicates
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     </span>
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 </li>
 
         <li class="md-nav__item">
-  <a href="#1022-case-with-duplicate-elements" class="md-nav__link">
+  <a href="#1022-when-duplicates-are-present" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.2.2 &nbsp; Case with Duplicate Elements
+        10.2.2 &nbsp; When duplicates are present
       </span>
     </span>
   </a>
@@ -2763,7 +2763,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2970,7 +2970,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2991,7 +2991,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3018,7 +3018,7 @@
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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@@ -3207,7 +3207,7 @@
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3261,7 +3261,7 @@
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3288,7 +3288,7 @@
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
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@@ -3396,7 +3396,7 @@
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-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4558,6 +4558,78 @@
 
 
 
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@@ -4599,10 +4671,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1021-case-without-duplicate-elements" class="md-nav__link">
+  <a href="#1021-when-there-are-no-duplicates" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.2.1 &nbsp; Case Without Duplicate Elements
+        10.2.1 &nbsp; When there are no duplicates
       </span>
     </span>
   </a>
@@ -4610,10 +4682,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1022-case-with-duplicate-elements" class="md-nav__link">
+  <a href="#1022-when-duplicates-are-present" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.2.2 &nbsp; Case with Duplicate Elements
+        10.2.2 &nbsp; When duplicates are present
       </span>
     </span>
   </a>
@@ -4657,266 +4729,266 @@
 
 
 <!-- Page content -->
-<h1 id="102-binary-search-insertion-point">10.2 &nbsp; Binary Search Insertion Point<a class="headerlink" href="#102-binary-search-insertion-point" title="Permanent link">&para;</a></h1>
-<p>Binary search can be used not only to search for target elements, but also to solve many variant problems, such as finding the insertion position of a target element.</p>
-<h2 id="1021-case-without-duplicate-elements">10.2.1 &nbsp; Case Without Duplicate Elements<a class="headerlink" href="#1021-case-without-duplicate-elements" title="Permanent link">&para;</a></h2>
+<h1 id="102-finding-an-insertion-point-with-binary-search">10.2 &nbsp; Finding an insertion point with binary search<a class="headerlink" href="#102-finding-an-insertion-point-with-binary-search" title="Permanent link">&para;</a></h1>
+<p>Binary search can do more than find a target element. It also solves many variations of the problem, such as finding where to insert a target.</p>
+<h2 id="1021-when-there-are-no-duplicates">10.2.1 &nbsp; When there are no duplicates<a class="headerlink" href="#1021-when-there-are-no-duplicates" title="Permanent link">&para;</a></h2>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given a sorted array <code>nums</code> of length <span class="arithmatex">\(n\)</span> and an element <code>target</code>, where the array contains no duplicate elements, insert <code>target</code> into <code>nums</code> while maintaining its sorted order. If <code>target</code> already exists in the array, insert it to its left. Return the index of <code>target</code> after insertion. An example is shown below.</p>
+<p>Given a sorted array <code>nums</code> of length <span class="arithmatex">\(n\)</span> with no duplicate elements and a value <code>target</code>, insert <code>target</code> into <code>nums</code> while preserving its order. If <code>target</code> is already present, insert the new value to its left. Return the index of the newly inserted <code>target</code>, as illustrated below.</p>
 </div>
-<p><img alt="Binary search insertion point example data" class="animation-figure" src="../binary_search_insertion.assets/binary_search_insertion_example.png" /></p>
-<p align="center"> Figure 10-4 &nbsp; Binary search insertion point example data </p>
+<p><img alt="Example data for finding an insertion point" class="animation-figure" src="../binary_search_insertion.assets/binary_search_insertion_example.png" /></p>
+<p align="center"> Figure 10-4 &nbsp; Example data for finding an insertion point </p>
 
-<p>If we want to reuse the binary search code from the previous section, we need to answer the following two questions.</p>
-<p><strong>Question 1</strong>: When the array contains <code>target</code>, is the insertion point index the same as that element's index?</p>
-<p>The problem requires inserting <code>target</code> to the left of equal elements, which means the newly inserted <code>target</code> replaces the position of the original <code>target</code>. In other words, <strong>when the array contains <code>target</code>, the insertion point index is the index of that <code>target</code></strong>.</p>
-<p><strong>Question 2</strong>: When the array does not contain <code>target</code>, what is the insertion point index?</p>
-<p>To analyze this further, consider the binary search process: when <code>nums[m] &lt; target</code>, <span class="arithmatex">\(i\)</span> moves, meaning that pointer <span class="arithmatex">\(i\)</span> is approaching elements greater than or equal to <code>target</code>. Similarly, pointer <span class="arithmatex">\(j\)</span> is always approaching elements less than or equal to <code>target</code>.</p>
-<p>Therefore, when the binary search ends, <span class="arithmatex">\(i\)</span> must point to the first element greater than <code>target</code>, and <span class="arithmatex">\(j\)</span> must point to the rightmost element less than <code>target</code>. <strong>It follows that when the array does not contain <code>target</code>, the insertion index is <span class="arithmatex">\(i\)</span></strong>. The code is shown below:</p>
+<p>To reuse the binary search code from the previous section, we need to answer two questions.</p>
+<p><strong>Question 1</strong>: If the array contains <code>target</code>, is the insertion index the index of that element?</p>
+<p>The problem asks us to insert <code>target</code> to the left of the equal element, so the new <code>target</code> takes the original one's position. Thus, <strong>if the array contains <code>target</code>, the insertion index is the index of that <code>target</code></strong>.</p>
+<p><strong>Question 2</strong>: If the array does not contain <code>target</code>, which index is the insertion point?</p>
+<p>Consider the search process more closely. Pointer <span class="arithmatex">\(i\)</span> moves when <code>nums[m] &lt; target</code>, so <span class="arithmatex">\(i\)</span> approaches elements greater than or equal to <code>target</code>. Similarly, pointer <span class="arithmatex">\(j\)</span> keeps approaching elements less than or equal to <code>target</code>.</p>
+<p>When the search ends, <span class="arithmatex">\(i\)</span> points to the first element greater than <code>target</code>, and <span class="arithmatex">\(j\)</span> points to the rightmost element smaller than <code>target</code>. <strong>Thus, when <code>target</code> is absent, the insertion index is <span class="arithmatex">\(i\)</span></strong>. The code is as follows:</p>
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 <div class="highlight"><span class="filename">binary_search_insertion.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">binary_search_insertion_simple</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Binary search for insertion point (no duplicate elements)"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Find an insertion point using binary search (no duplicates)"""</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="n">j</span><span class="p">:</span>
-<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>  <span class="c1"># Calculate midpoint index m</span>
+<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>  <span class="c1"># Calculate the midpoint index m</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">target</span><span class="p">:</span>
-<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>            <span class="n">i</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c1"># target is in the interval [m+1, j]</span>
+<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>            <span class="n">i</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c1"># Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="k">elif</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">target</span><span class="p">:</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>            <span class="n">j</span> <span class="o">=</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># target is in the interval [i, m-1]</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>            <span class="n">j</span> <span class="o">=</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>            <span class="k">return</span> <span class="n">m</span>  <span class="c1"># Found target, return insertion point m</span>
-<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="c1"># Target not found, return insertion point i</span>
+<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>            <span class="k">return</span> <span class="n">m</span>  <span class="c1"># Target was found; return insertion point m</span>
+<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="c1"># Target was not found; return insertion point i</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>    <span class="k">return</span> <span class="n">i</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Binary search for insertion point (no duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Find an insertion point using binary search (no duplicates) */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchInsertionSimple</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// Found target, return insertion point m</span>
+<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target was found; return insertion point m</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">    </span><span class="c1">// Target not found, return insertion point i</span>
+<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">    </span><span class="c1">// Target was not found; return insertion point i</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Binary search for insertion point (no duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Find an insertion point using binary search (no duplicates) */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchInsertionSimple</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// Found target, return insertion point m</span>
+<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target was found; return insertion point m</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">    </span><span class="c1">// Target not found, return insertion point i</span>
+<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">    </span><span class="c1">// Target was not found; return insertion point i</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Binary search for insertion point (no duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Find an insertion point using binary search (no duplicates) */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">BinarySearchInsertionSimple</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// Found target, return insertion point m</span>
+<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target was found; return insertion point m</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="c1">// Target not found, return insertion point i</span>
+<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="c1">// Target was not found; return insertion point i</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Binary search for insertion point (no duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Find an insertion point using binary search (no duplicates) */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">binarySearchInsertionSimple</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="o">-</span><span class="nx">i</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">            </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">            </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">            </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">            </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">            </span><span class="c1">// Found target, return insertion point m</span>
+<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">            </span><span class="c1">// Target was found; return insertion point m</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">    </span><span class="c1">// Target not found, return insertion point i</span>
+<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">    </span><span class="c1">// Target was not found; return insertion point i</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Binary search for insertion point (no duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Find an insertion point using binary search (no duplicates) */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">binarySearchInsertionSimple</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">startIndex</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">endIndex</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="c1">// Found target, return insertion point m</span>
+<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="c1">// Target was found; return insertion point m</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">    </span><span class="c1">// Target not found, return insertion point i</span>
+<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">    </span><span class="c1">// Target was not found; return insertion point i</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Binary search for insertion point (no duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Find an insertion point using binary search (no duplicates) */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearchInsertionSimple</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
-<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Calculate midpoint index m, use Math.floor() to round down</span>
+<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Calculate midpoint index m, using Math.floor() to round down</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// Found target, return insertion point m</span>
+<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target was found; return insertion point m</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">    </span><span class="c1">// Target not found, return insertion point i</span>
+<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">    </span><span class="c1">// Target was not found; return insertion point i</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Binary search for insertion point (no duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Find an insertion point using binary search (no duplicates) */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearchInsertionSimple</span><span class="p">(</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
-<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Calculate midpoint index m, use Math.floor() to round down</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Calculate midpoint index m, using Math.floor() to round down</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// Found target, return insertion point m</span>
+<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target was found; return insertion point m</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">    </span><span class="c1">// Target not found, return insertion point i</span>
+<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">    </span><span class="c1">// Target was not found; return insertion point i</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Binary search for insertion point (no duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Find an insertion point using binary search (no duplicates) */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">binarySearchInsertionSimple</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// Found target, return insertion point m</span>
+<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target was found; return insertion point m</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">  </span><span class="c1">// Target not found, return insertion point i</span>
+<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">  </span><span class="c1">// Target was not found; return insertion point i</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Binary search for insertion point (no duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Find an insertion point using binary search (no duplicates) */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">binary_search_insertion_simple</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">    </span><span class="c1">// Target not found, return insertion point i</span>
+<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">    </span><span class="c1">// Target was not found; return insertion point i</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">    </span><span class="n">i</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Binary search for insertion point (no duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Find an insertion point using binary search (no duplicates) */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchInsertionSimple</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">numSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">numSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">numSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// Found target, return insertion point m</span>
+<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target was found; return insertion point m</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="c1">// Target not found, return insertion point i</span>
+<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="c1">// Target was not found; return insertion point i</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Binary search for insertion point (no duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Find an insertion point using binary search (no duplicates) */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">binarySearchInsertionSimple</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="c1">// Found target, return insertion point m</span>
+<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="c1">// Target was found; return insertion point m</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="c1">// Target not found, return insertion point i</span>
+<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="c1">// Target was not found; return insertion point i</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Binary search insertion point (no duplicates) ###</span>
+<div class="highlight"><span class="filename">binary_search_insertion.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Find an insertion point using binary search (no duplicates) ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">binary_search_insertion_simple</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span>
@@ -4924,45 +4996,50 @@
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span>
-<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># target is in the interval [m+1, j]</span>
+<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">    </span><span class="k">elsif</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span>
-<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># target is in the interval [i, m-1]</span>
+<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="k">else</span>
-<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="w">  </span><span class="c1"># Found target, return insertion point m</span>
+<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">m</span><span class="w">  </span><span class="c1"># Target was found; return insertion point m</span>
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a>
-<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="c1"># Target not found, return insertion point i</span>
+<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="c1"># Target was not found; return insertion point i</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h2 id="1022-case-with-duplicate-elements">10.2.2 &nbsp; Case with Duplicate Elements<a class="headerlink" href="#1022-case-with-duplicate-elements" title="Permanent link">&para;</a></h2>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20binary_search_insertion_simple%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20an%20insertion%20point%20using%20binary%20search%20%28no%20duplicates%29%22%22%22%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%20-%201%20%20%23%20Initialize%20the%20closed%20interval%20%5B0,%20n-1%5D%0A%20%20%20%20while%20i%20%3C%3D%20j%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20Calculate%20the%20midpoint%20index%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20Target%20must%20be%20in%20%5Bm%2B1,%20j%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20Target%20must%20be%20in%20%5Bi,%20m-1%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20m%20%20%23%20Target%20was%20found%3B%20return%20insertion%20point%20m%0A%20%20%20%20%23%20Target%20was%20not%20found%3B%20return%20insertion%20point%20i%0A%20%20%20%20return%20i%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Array%20with%20no%20duplicate%20elements%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%208,%2012,%2015,%2023,%2026,%2031,%2035%5D%0A%20%20%20%20%23%20Find%20an%20insertion%20point%20using%20binary%20search%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20index%20%3D%20binary_search_insertion_simple%28nums,%20target%29%0A%20%20%20%20print%28f%22Insertion%20index%20for%20element%20%7Btarget%7D%3A%20%7Bindex%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20binary_search_insertion_simple%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20an%20insertion%20point%20using%20binary%20search%20%28no%20duplicates%29%22%22%22%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%20-%201%20%20%23%20Initialize%20the%20closed%20interval%20%5B0,%20n-1%5D%0A%20%20%20%20while%20i%20%3C%3D%20j%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20Calculate%20the%20midpoint%20index%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20Target%20must%20be%20in%20%5Bm%2B1,%20j%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20Target%20must%20be%20in%20%5Bi,%20m-1%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20m%20%20%23%20Target%20was%20found%3B%20return%20insertion%20point%20m%0A%20%20%20%20%23%20Target%20was%20not%20found%3B%20return%20insertion%20point%20i%0A%20%20%20%20return%20i%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Array%20with%20no%20duplicate%20elements%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%208,%2012,%2015,%2023,%2026,%2031,%2035%5D%0A%20%20%20%20%23%20Find%20an%20insertion%20point%20using%20binary%20search%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20index%20%3D%20binary_search_insertion_simple%28nums,%20target%29%0A%20%20%20%20print%28f%22Insertion%20index%20for%20element%20%7Btarget%7D%3A%20%7Bindex%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1022-when-duplicates-are-present">10.2.2 &nbsp; When duplicates are present<a class="headerlink" href="#1022-when-duplicates-are-present" title="Permanent link">&para;</a></h2>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Based on the previous problem, assume the array may contain duplicate elements, with everything else remaining the same.</p>
+<p>Modify the previous problem to allow duplicate elements in the array. All other requirements remain the same.</p>
 </div>
-<p>Suppose there are multiple <code>target</code> elements in the array. Ordinary binary search can only return the index of one <code>target</code>, <strong>and cannot determine how many <code>target</code> elements are to the left and right of that element</strong>.</p>
-<p>The problem requires inserting the target element at the leftmost position, <strong>so we need to find the index of the leftmost <code>target</code> in the array</strong>. A straightforward initial approach is to follow the steps shown in Figure 10-5:</p>
+<p>If the array contains multiple occurrences of <code>target</code>, ordinary binary search can return the index of one occurrence, <strong>but cannot determine how many more occurrences lie to its left or right</strong>.</p>
+<p>The target must be inserted as far left as possible, <strong>so we need the index of the leftmost <code>target</code></strong>. One initial approach follows the steps below.</p>
 <ol>
-<li>Perform binary search to obtain the index of any <code>target</code>, denoted as <span class="arithmatex">\(k\)</span>.</li>
-<li>Starting from index <span class="arithmatex">\(k\)</span>, perform linear traversal to the left, and return when the leftmost <code>target</code> is found.</li>
+<li>Run binary search to find the index of any <code>target</code>, denoted by <span class="arithmatex">\(k\)</span>.</li>
+<li>Starting at index <span class="arithmatex">\(k\)</span>, scan leftward and return when the leftmost <code>target</code> is reached.</li>
 </ol>
-<p><img alt="Linear search for insertion point of duplicate elements" class="animation-figure" src="../binary_search_insertion.assets/binary_search_insertion_naive.png" /></p>
-<p align="center"> Figure 10-5 &nbsp; Linear search for insertion point of duplicate elements </p>
+<p><img alt="Finding the insertion point among duplicates with a linear scan" class="animation-figure" src="../binary_search_insertion.assets/binary_search_insertion_naive.png" /></p>
+<p align="center"> Figure 10-5 &nbsp; Finding the insertion point among duplicates with a linear scan </p>
 
-<p>Although this method works, it includes linear search, resulting in a time complexity of <span class="arithmatex">\(O(n)\)</span>. When the array contains many duplicate <code>target</code> elements, this method is very inefficient.</p>
-<p>Now consider extending the binary search code. As shown in Figure 10-6, the overall process remains unchanged: in each iteration, we first compute the midpoint index <span class="arithmatex">\(m\)</span>, then compare <code>target</code> with <code>nums[m]</code>, leading to the following cases:</p>
+<p>Although this works, it includes a linear search and therefore takes <span class="arithmatex">\(O(n)\)</span> time. It is inefficient when the array contains many occurrences of <code>target</code>.</p>
+<p>Now consider extending binary search itself. As shown below, the overall process remains the same: each round calculates midpoint index <span class="arithmatex">\(m\)</span> and compares <code>target</code> with <code>nums[m]</code>. The cases are as follows.</p>
 <ul>
-<li>When <code>nums[m] &lt; target</code> or <code>nums[m] &gt; target</code>, it means <code>target</code> has not been found yet, so use the standard interval-shrinking operation of binary search to <strong>move pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> closer to <code>target</code></strong>.</li>
-<li>When <code>nums[m] == target</code>, it means elements less than <code>target</code> are in the interval <span class="arithmatex">\([i, m - 1]\)</span>, so use <span class="arithmatex">\(j = m - 1\)</span> to shrink the interval, thereby <strong>moving pointer <span class="arithmatex">\(j\)</span> closer to elements less than <code>target</code></strong>.</li>
+<li>If <code>nums[m] &lt; target</code> or <code>nums[m] &gt; target</code>, we have not found <code>target</code>. Narrow the interval as in ordinary binary search, <strong>bringing pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> closer to <code>target</code></strong>.</li>
+<li>If <code>nums[m] == target</code>, elements smaller than <code>target</code> lie in <span class="arithmatex">\([i, m - 1]\)</span>. Narrow the interval by setting <span class="arithmatex">\(j = m - 1\)</span>, <strong>bringing pointer <span class="arithmatex">\(j\)</span> closer to elements smaller than <code>target</code></strong>.</li>
 </ul>
-<p>After the loop completes, <span class="arithmatex">\(i\)</span> points to the leftmost <code>target</code>, and <span class="arithmatex">\(j\)</span> points to the rightmost element less than <code>target</code>, <strong>so index <span class="arithmatex">\(i\)</span> is the insertion point</strong>.</p>
+<p>When the loop finishes, <span class="arithmatex">\(i\)</span> points to the leftmost <code>target</code>, and <span class="arithmatex">\(j\)</span> points to the rightmost element smaller than <code>target</code>. <strong>Index <span class="arithmatex">\(i\)</span> is therefore the insertion point</strong>.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:8"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">&lt;1&gt;</label><label for="__tabbed_2_2">&lt;2&gt;</label><label for="__tabbed_2_3">&lt;3&gt;</label><label for="__tabbed_2_4">&lt;4&gt;</label><label for="__tabbed_2_5">&lt;5&gt;</label><label for="__tabbed_2_6">&lt;6&gt;</label><label for="__tabbed_2_7">&lt;7&gt;</label><label for="__tabbed_2_8">&lt;8&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Steps for binary search insertion point of duplicate elements" class="animation-figure" src="../binary_search_insertion.assets/binary_search_insertion_step1.png" /></p>
+<p><img alt="Steps for finding an insertion point among duplicates" class="animation-figure" src="../binary_search_insertion.assets/binary_search_insertion_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="binary_search_insertion_step2" class="animation-figure" src="../binary_search_insertion.assets/binary_search_insertion_step2.png" /></p>
@@ -4987,40 +5064,40 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 10-6 &nbsp; Steps for binary search insertion point of duplicate elements </p>
+<p align="center"> Figure 10-6 &nbsp; Steps for finding an insertion point among duplicates </p>
 
-<p>Observe the following code: the branches <code>nums[m] &gt; target</code> and <code>nums[m] == target</code> perform the same operation, so they can be merged.</p>
-<p>Even so, we can still keep the conditional branches expanded, as the logic is clearer and more readable.</p>
+<p>In the code below, the branches for <code>nums[m] &gt; target</code> and <code>nums[m] == target</code> perform the same operation, so they can be combined.</p>
+<p>Even so, we can leave them separate to make the logic clearer and easier to read.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_insertion.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">binary_search_insertion</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Binary search for insertion point (with duplicate elements)"""</span>
-<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Find an insertion point using binary search (with duplicates)"""</span>
+<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="n">j</span><span class="p">:</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>        <span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>  <span class="c1"># Calculate midpoint index m</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>        <span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="n">j</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>  <span class="c1"># Calculate the midpoint index m</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>        <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">target</span><span class="p">:</span>
-<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>            <span class="n">i</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c1"># target is in the interval [m+1, j]</span>
+<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>            <span class="n">i</span> <span class="o">=</span> <span class="n">m</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c1"># Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="k">elif</span> <span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">target</span><span class="p">:</span>
-<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>            <span class="n">j</span> <span class="o">=</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># target is in the interval [i, m-1]</span>
+<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>            <span class="n">j</span> <span class="o">=</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>        <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>            <span class="n">j</span> <span class="o">=</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># The rightmost element less than target is in the interval [i, m-1]</span>
+<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>            <span class="n">j</span> <span class="o">=</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># Continue in [i, m-1] to find the rightmost element smaller than target</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>    <span class="c1"># Return insertion point i</span>
 <a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>    <span class="k">return</span> <span class="n">i</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Binary search for insertion point (with duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Find an insertion point using binary search (with duplicates) */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchInsertion</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The rightmost element less than target is in the interval [i, m-1]</span>
+<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Continue searching [i, m-1] for the insertion point before any equal elements</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">    </span><span class="c1">// Return insertion point i</span>
@@ -5029,17 +5106,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Binary search for insertion point (with duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Find an insertion point using binary search (with duplicates) */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchInsertion</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The rightmost element less than target is in the interval [i, m-1]</span>
+<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Continue in [i, m-1] to find the rightmost element smaller than target</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">    </span><span class="c1">// Return insertion point i</span>
@@ -5048,17 +5125,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Binary search for insertion point (with duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Find an insertion point using binary search (with duplicates) */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">BinarySearchInsertion</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The rightmost element less than target is in the interval [i, m-1]</span>
+<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Continue in [i, m-1] to find the rightmost element smaller than target</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">    </span><span class="c1">// Return insertion point i</span>
@@ -5067,21 +5144,21 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Binary search for insertion point (with duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Find an insertion point using binary search (with duplicates) */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">binarySearchInsertion</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">        </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="o">-</span><span class="nx">i</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">            </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">            </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">            </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">            </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">            </span><span class="c1">// The rightmost element less than target is in the interval [i, m-1]</span>
+<a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">            </span><span class="c1">// Continue in [i, m-1] to find the rightmost element smaller than target</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="w">    </span><span class="p">}</span>
@@ -5091,19 +5168,19 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Binary search for insertion point (with duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Find an insertion point using binary search (with duplicates) */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">binarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">startIndex</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">endIndex</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// The rightmost element less than target is in the interval [i, m-1]</span>
+<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Continue in [i, m-1] to find the rightmost element smaller than target</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">    </span><span class="c1">// Return insertion point i</span>
@@ -5112,18 +5189,18 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Binary search for insertion point (with duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Find an insertion point using binary search (with duplicates) */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearchInsertion</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
-<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Calculate midpoint index m, use Math.floor() to round down</span>
+<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Calculate midpoint index m, using Math.floor() to round down</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The rightmost element less than target is in the interval [i, m-1]</span>
+<a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Continue in [i, m-1] to find the rightmost element smaller than target</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">    </span><span class="c1">// Return insertion point i</span>
@@ -5132,18 +5209,18 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Binary search for insertion point (with duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Find an insertion point using binary search (with duplicates) */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">binarySearchInsertion</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span>
-<a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Calculate midpoint index m, use Math.floor() to round down</span>
+<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Calculate midpoint index m, using Math.floor() to round down</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The rightmost element less than target is in the interval [i, m-1]</span>
+<a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Continue in [i, m-1] to find the rightmost element smaller than target</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">    </span><span class="c1">// Return insertion point i</span>
@@ -5152,17 +5229,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Binary search for insertion point (with duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Find an insertion point using binary search (with duplicates) */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">binarySearchInsertion</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The rightmost element less than target is in the interval [i, m-1]</span>
+<a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Continue in [i, m-1] to find the rightmost element smaller than target</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">  </span><span class="c1">// Return insertion point i</span>
@@ -5171,17 +5248,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Binary search for insertion point (with duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Find an insertion point using binary search (with duplicates) */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">binary_search_insertion</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The rightmost element less than target is in the interval [i, m-1]</span>
+<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Continue searching [i, m-1] for the insertion point before any equal elements</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">    </span><span class="c1">// Return insertion point i</span>
@@ -5190,17 +5267,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Binary search for insertion point (with duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Find an insertion point using binary search (with duplicates) */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">binarySearchInsertion</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">numSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">numSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">numSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">m</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The rightmost element less than target is in the interval [i, m-1]</span>
+<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Continue in [i, m-1] to find the rightmost element smaller than target</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">    </span><span class="c1">// Return insertion point i</span>
@@ -5209,18 +5286,18 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Binary search for insertion point (with duplicate elements) */</span>
+<div class="highlight"><span class="filename">binary_search_insertion.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Find an insertion point using binary search (with duplicates) */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">binarySearchInsertion</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="c1">// Calculate the midpoint index m</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// target is in the interval [m+1, j]</span>
+<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// target is in the interval [i, m-1]</span>
+<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// The rightmost element less than target is in the interval [i, m-1]</span>
+<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Continue in [i, m-1] to find the rightmost element smaller than target</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">    </span><span class="c1">// Return insertion point i</span>
@@ -5229,9 +5306,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_insertion.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Binary search insertion point (with duplicates) ###</span>
+<div class="highlight"><span class="filename">binary_search_insertion.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Find an insertion point using binary search (with duplicates) ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">binary_search_insertion</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Initialize closed interval [0, n-1]</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Initialize the closed interval [0, n-1]</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">j</span>
@@ -5239,11 +5316,11 @@
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">    </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">target</span>
-<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># target is in the interval [m+1, j]</span>
+<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Target must be in [m+1, j] if it exists</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">    </span><span class="k">elsif</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">m</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">target</span>
-<a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># target is in the interval [i, m-1]</span>
+<a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Target must be in [i, m-1] if it exists</span>
 <a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">    </span><span class="k">else</span>
-<a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># The rightmost element less than target is in the interval [i, m-1]</span>
+<a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Continue in [i, m-1] to find the rightmost element smaller than target</span>
 <a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a>
@@ -5253,12 +5330,17 @@
 </div>
 </div>
 </div>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20binary_search_insertion%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20an%20insertion%20point%20using%20binary%20search%20%28with%20duplicates%29%22%22%22%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%20-%201%20%20%23%20Initialize%20the%20closed%20interval%20%5B0,%20n-1%5D%0A%20%20%20%20while%20i%20%3C%3D%20j%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20Calculate%20the%20midpoint%20index%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20Target%20must%20be%20in%20%5Bm%2B1,%20j%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20Target%20must%20be%20in%20%5Bi,%20m-1%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20Search%20%5Bi,%20m-1%5D%20for%20the%20insertion%20point%20before%20equal%20elements%0A%20%20%20%20%23%20Return%20insertion%20point%20i%0A%20%20%20%20return%20i%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Array%20containing%20duplicate%20elements%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%206,%206,%206,%206,%2010,%2012,%2015%5D%0A%20%20%20%20%23%20Find%20an%20insertion%20point%20using%20binary%20search%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20index%20%3D%20binary_search_insertion%28nums,%20target%29%0A%20%20%20%20print%28f%22Insertion%20index%20for%20element%20%7Btarget%7D%3A%20%7Bindex%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20binary_search_insertion%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Find%20an%20insertion%20point%20using%20binary%20search%20%28with%20duplicates%29%22%22%22%0A%20%20%20%20i,%20j%20%3D%200,%20len%28nums%29%20-%201%20%20%23%20Initialize%20the%20closed%20interval%20%5B0,%20n-1%5D%0A%20%20%20%20while%20i%20%3C%3D%20j%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20%28i%20%2B%20j%29%20//%202%20%20%23%20Calculate%20the%20midpoint%20index%20m%0A%20%20%20%20%20%20%20%20if%20nums%5Bm%5D%20%3C%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%3D%20m%20%2B%201%20%20%23%20Target%20must%20be%20in%20%5Bm%2B1,%20j%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20elif%20nums%5Bm%5D%20%3E%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20Target%20must%20be%20in%20%5Bi,%20m-1%5D%20if%20it%20exists%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%3D%20m%20-%201%20%20%23%20Search%20%5Bi,%20m-1%5D%20for%20the%20insertion%20point%20before%20equal%20elements%0A%20%20%20%20%23%20Return%20insertion%20point%20i%0A%20%20%20%20return%20i%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Array%20containing%20duplicate%20elements%0A%20%20%20%20nums%20%3D%20%5B1,%203,%206,%206,%206,%206,%206,%2010,%2012,%2015%5D%0A%20%20%20%20%23%20Find%20an%20insertion%20point%20using%20binary%20search%0A%20%20%20%20target%20%3D%206%0A%20%20%20%20index%20%3D%20binary_search_insertion%28nums,%20target%29%0A%20%20%20%20print%28f%22Insertion%20index%20for%20element%20%7Btarget%7D%3A%20%7Bindex%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>The code in this section uses the "closed interval" approach throughout. Interested readers can implement the "left-closed, right-open" approach themselves.</p>
+<p>All the code in this section uses closed intervals. Interested readers can implement versions using left-closed, right-open intervals.</p>
 </div>
-<p>Overall, binary search is simply a matter of setting separate search targets for pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span>. The target may be a specific element (such as <code>target</code>) or a range of elements (such as elements less than <code>target</code>).</p>
-<p>With each iteration of binary search, pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> gradually approach their preset targets. Ultimately, they either find the answer or stop after crossing the boundary.</p>
+<p>Viewed more broadly, binary search simply assigns a search goal to each of pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span>. The goal may be a particular element, such as <code>target</code>, or a range of elements, such as those smaller than <code>target</code>.</p>
+<p>As the interval is repeatedly halved, pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> approach their assigned goals. Eventually they either find the answer or stop after crossing a boundary.</p>
 
 <!-- Source file information -->
 
@@ -5281,7 +5363,7 @@ aria-label="Footer"
   <a
     href="../binary_search/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 10.1 Binary Search"
+    aria-label="Previous: 10.1   Binary search"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -5293,7 +5375,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        10.1 Binary Search
+        10.1 &nbsp; Binary search
       </div>
     </div>
   </a>
@@ -5305,7 +5387,7 @@ aria-label="Footer"
   <a
     href="../binary_search_edge/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 10.3 Binary Search Boundaries"
+    aria-label="Next: 10.3   Finding boundaries with binary search"
     rel="next"
   >
     <div class="md-footer__title">
@@ -5313,7 +5395,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        10.3 Binary Search Boundaries
+        10.3 &nbsp; Finding boundaries with binary search
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -5330,7 +5412,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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index ff9c6228b..6386b5d3b 100644
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -67,8 +67,8 @@
 
 
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-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-  Before Starting
+  Foreword
 
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           <ul class="md-nav__list" data-md-scrollfix>
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-  Chapter 0. Preface
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
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@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
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+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
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   <span class="md-ellipsis">
 
 
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-  3.2 Basic Data Types
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-  3.3 Number Encoding *
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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   <span class="md-ellipsis">
 
 
-  8.4 Summary
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-  8.5 Exercises
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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@@ -2598,7 +2598,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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@@ -2619,7 +2619,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -2640,7 +2640,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2694,7 +2694,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2721,7 +2721,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2748,7 +2748,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2775,7 +2775,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2811,7 +2811,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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+  <a href="#1071-check-your-understanding" class="md-nav__link">
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     </span>
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   <a href="#1-how-binary-search-narrows-the-search-interval" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
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 </li>
 
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+  <a href="#2-left-and-right-boundaries-of-repeated-elements" class="md-nav__link">
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       <span class="md-typeset">
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       <span class="md-typeset">
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-        10.7.2 &nbsp; Programming Exercises
+        10.7.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="10.7.2 Programming Exercises">
+    <nav class="md-nav" aria-label="10.7.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-binary-search-in-a-sorted-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Binary Search in a Sorted Array
+        1. &nbsp; Binary search in a sorted array
       </span>
     </span>
   </a>
@@ -2927,10 +2927,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-insertion-point-in-a-sorted-array" class="md-nav__link">
+  <a href="#2-finding-an-insertion-position-in-a-sorted-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Insertion Point in a Sorted Array
+        2. &nbsp; Finding an insertion position in a sorted array
       </span>
     </span>
   </a>
@@ -3016,7 +3016,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3037,7 +3037,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3058,7 +3058,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3085,7 +3085,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3112,7 +3112,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3139,7 +3139,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3166,7 +3166,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3193,7 +3193,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3220,7 +3220,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3247,7 +3247,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3274,7 +3274,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3301,7 +3301,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3328,7 +3328,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3355,7 +3355,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3421,7 +3421,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3442,7 +3442,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3463,7 +3463,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3490,7 +3490,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3517,7 +3517,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3544,7 +3544,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3571,7 +3571,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3598,7 +3598,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3664,7 +3664,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3685,7 +3685,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3706,7 +3706,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3733,7 +3733,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3760,7 +3760,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3787,7 +3787,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3814,7 +3814,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3841,7 +3841,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3911,7 +3911,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3932,7 +3932,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3953,7 +3953,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3980,7 +3980,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -4007,7 +4007,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4034,7 +4034,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4061,7 +4061,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4088,7 +4088,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4115,7 +4115,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4142,7 +4142,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4208,7 +4208,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4229,7 +4229,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4250,7 +4250,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4277,7 +4277,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4304,7 +4304,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4331,7 +4331,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4358,7 +4358,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4385,7 +4385,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4445,7 +4445,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4466,7 +4466,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4487,7 +4487,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4514,7 +4514,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4541,7 +4541,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4625,6 +4625,78 @@
 
 
 
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@@ -4666,22 +4738,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1071-concept-review" class="md-nav__link">
+  <a href="#1071-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.7.1 &nbsp; Concept Review
+        10.7.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="10.7.1 Concept Review">
+    <nav class="md-nav" aria-label="10.7.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-how-binary-search-narrows-the-search-interval" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; How Binary Search Narrows the Search Interval
+        1. &nbsp; How binary search narrows the search interval
       </span>
     </span>
   </a>
@@ -4689,10 +4761,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-left-and-right-boundaries-of-duplicate-elements" class="md-nav__link">
+  <a href="#2-left-and-right-boundaries-of-repeated-elements" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Left and Right Boundaries of Duplicate Elements
+        2. &nbsp; Left and right boundaries of repeated elements
       </span>
     </span>
   </a>
@@ -4703,7 +4775,7 @@
   <a href="#3-choosing-a-search-method-for-different-data" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Choosing a Search Method for Different Data
+        3. &nbsp; Choosing a search method for different data
       </span>
     </span>
   </a>
@@ -4719,19 +4791,19 @@
   <a href="#1072-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.7.2 &nbsp; Programming Exercises
+        10.7.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="10.7.2 Programming Exercises">
+    <nav class="md-nav" aria-label="10.7.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-binary-search-in-a-sorted-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Binary Search in a Sorted Array
+        1. &nbsp; Binary search in a sorted array
       </span>
     </span>
   </a>
@@ -4739,10 +4811,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-insertion-point-in-a-sorted-array" class="md-nav__link">
+  <a href="#2-finding-an-insertion-position-in-a-sorted-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Insertion Point in a Sorted Array
+        2. &nbsp; Finding an insertion position in a sorted array
       </span>
     </span>
   </a>
@@ -4792,22 +4864,22 @@
 
 <!-- Page content -->
 <h1 id="107-exercises">10.7 &nbsp; Exercises<a class="headerlink" href="#107-exercises" title="Permanent link">&para;</a></h1>
-<h2 id="1071-concept-review">10.7.1 &nbsp; Concept Review<a class="headerlink" href="#1071-concept-review" title="Permanent link">&para;</a></h2>
-<h3 id="1-how-binary-search-narrows-the-search-interval">1. &nbsp; How Binary Search Narrows the Search Interval<a class="headerlink" href="#1-how-binary-search-narrows-the-search-interval" title="Permanent link">&para;</a></h3>
+<h2 id="1071-check-your-understanding">10.7.1 &nbsp; Check your understanding<a class="headerlink" href="#1071-check-your-understanding" title="Permanent link">&para;</a></h2>
+<h3 id="1-how-binary-search-narrows-the-search-interval">1. &nbsp; How binary search narrows the search interval<a class="headerlink" href="#1-how-binary-search-narrows-the-search-interval" title="Permanent link">&para;</a></h3>
 <p>Search for 16 in the sorted array <code>[2, 5, 8, 12, 16, 23, 38]</code>.
-Use the closed interval <code>[i, j]</code> and calculate the midpoint as
+Use a closed interval <code>[i, j]</code> and the midpoint
 <span class="arithmatex">\(m=i+(j-i)/2\)</span>, rounded down.</p>
-<p>For each round, write <code>(i, j, m)</code>, the middle element, and how the interval is narrowed next,
+<p>For each iteration, write <code>(i, j, m)</code>, the midpoint element, and how the interval is narrowed next,
 until the target is found.</p>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <p>The search proceeds as follows:</p>
 <table>
 <thead>
 <tr>
-<th>Round</th>
+<th>Iteration</th>
 <th><code>(i, j, m)</code></th>
-<th>Middle element</th>
+<th>Midpoint element</th>
 <th>Next step</th>
 </tr>
 </thead>
@@ -4816,13 +4888,13 @@ until the target is found.</p>
 <td>1</td>
 <td><code>(0, 6, 3)</code></td>
 <td>12</td>
-<td><code>12 &lt; 16</code>, set <code>i = 4</code></td>
+<td><code>12 &lt; 16</code>, so set <code>i = 4</code></td>
 </tr>
 <tr>
 <td>2</td>
 <td><code>(4, 6, 5)</code></td>
 <td>23</td>
-<td><code>23 &gt; 16</code>, set <code>j = 4</code></td>
+<td><code>23 &gt; 16</code>, so set <code>j = 4</code></td>
 </tr>
 <tr>
 <td>3</td>
@@ -4832,80 +4904,79 @@ until the target is found.</p>
 </tr>
 </tbody>
 </table>
-<p>Because the array is sorted, when the middle value is smaller than the target, the middle and everything to its left can be excluded.
-When the middle value is greater than the target, the middle and everything to its right can be excluded.</p>
+<p>Since the array is sorted, when the midpoint value is below the target, we can exclude the midpoint and everything to its left.
+When it is above the target, we can exclude the midpoint and everything to its right.</p>
 </details>
-<h3 id="2-left-and-right-boundaries-of-duplicate-elements">2. &nbsp; Left and Right Boundaries of Duplicate Elements<a class="headerlink" href="#2-left-and-right-boundaries-of-duplicate-elements" title="Permanent link">&para;</a></h3>
-<p>Search for the number 2 in the array <code>[1, 2, 2, 2, 4, 6]</code>.
-A student uses binary search, returns immediately after finding the target at index 2, and says,
-"Index 2 is the left boundary of the number 2."</p>
+<h3 id="2-left-and-right-boundaries-of-repeated-elements">2. &nbsp; Left and right boundaries of repeated elements<a class="headerlink" href="#2-left-and-right-boundaries-of-repeated-elements" title="Permanent link">&para;</a></h3>
+<p>Search for 2 in the array <code>[1, 2, 2, 2, 4, 6]</code>.
+A student uses binary search, finds the target at index 2, and immediately returns, saying:
+“Index 2 is the left boundary of the number 2.”</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Is the student's statement correct? What are the left and right boundaries of 2? Explain why.</li>
-<li>When searching for the left boundary, if the middle element equals the target, which side should be searched next?</li>
-<li>When searching for the right boundary, which side should be searched next? State only the direction; you do not need to write the complete search process.</li>
+<li>Is the student correct? What are the left and right boundaries of 2? Explain.</li>
+<li>When finding the left boundary, which side should we continue searching if the midpoint element equals the target?</li>
+<li>Which side should we continue searching when finding the right boundary? State only the direction; you do not need to give the full search process.</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>The student's statement is incorrect. Returning as soon as a 2 is found guarantees only that some occurrence of 2 was found, not the leftmost or rightmost one.
+<p>The student is incorrect. Returning immediately upon finding a 2 guarantees only that some occurrence has been found, not the leftmost or rightmost one.
     Here, the left boundary is index 1 and the right boundary is index 3.</p>
 </li>
 <li>
-<p>When searching for the left boundary, continue searching on the left even if the middle element equals 2.
+<p>When finding the left boundary, continue searching to the left even if the midpoint element equals 2.
     For example, with a closed interval, set <code>j = m - 1</code>.</p>
 </li>
 <li>
-<p>When searching for the right boundary, continue searching on the right after the middle element equals 2.
+<p>When finding the right boundary, continue searching to the right after the midpoint element equals 2.
     For example, set <code>i = m + 1</code>.</p>
 </li>
 </ol>
 </details>
-<h3 id="3-choosing-a-search-method-for-different-data">3. &nbsp; Choosing a Search Method for Different Data<a class="headerlink" href="#3-choosing-a-search-method-for-different-data" title="Permanent link">&para;</a></h3>
-<p>For each situation below, choose an appropriate method from "linear search," "binary search," and "hash table," and explain why:</p>
+<h3 id="3-choosing-a-search-method-for-different-data">3. &nbsp; Choosing a search method for different data<a class="headerlink" href="#3-choosing-a-search-method-for-different-data" title="Permanent link">&para;</a></h3>
+<p>Choose a suitable method from linear search, binary search, and a hash table for each situation, and explain:</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Repeatedly search among <span class="arithmatex">\(10^7\)</span> integers that are already sorted and never change, without building any additional data structure.</li>
-<li>Repeatedly test whether a key exists in a collection with frequent insertions and deletions. The collection need not remain sorted, and no range searches are needed.</li>
-<li>Search an unsorted array for a value only once.</li>
+<li>Repeatedly searching <span class="arithmatex">\(10^7\)</span> integers that are already sorted and will not change, without building an additional data structure.</li>
+<li>Repeatedly checking whether a key exists in a data set with frequent insertions and deletions, with no need to preserve order or perform range queries.</li>
+<li>Searching for a value only once in an unsorted array.</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
-<li>Binary search: the data is sorted and static, so searching takes <span class="arithmatex">\(O(\log n)\)</span> time without extra space.</li>
-<li>Hash table: when the hash function distributes keys fairly evenly among the buckets, insertion, deletion, and lookup by key all take <span class="arithmatex">\(O(1)\)</span> time on average.</li>
+<li>Binary search: the data is sorted and static, giving <span class="arithmatex">\(O(\log n)\)</span> time without an additional data structure.</li>
+<li>A hash table: when the hash function distributes keys fairly evenly among buckets, insertion, deletion, and key lookup all have average time complexity <span class="arithmatex">\(O(1)\)</span>.</li>
 <li>
-<p>Scan directly from beginning to end. When searching only once, sorting the array or building a hash table still requires processing the entire array first,
-    so neither reduces the total work for this single task.</p>
-<p>The choice depends on whether the data is sorted, whether an additional structure is allowed, how many searches are needed, and which operations must be supported.</p>
+<p>Scan directly from beginning to end. For a single search, sorting or building a hash table would first require processing the entire array,
+    without reducing the total work for this task.</p>
+<p>The choice depends on whether the data is sorted, whether additional structures are allowed, how many queries there are, and which operations are needed.</p>
 </li>
 </ol>
 </details>
-<h2 id="1072-programming-exercises">10.7.2 &nbsp; Programming Exercises<a class="headerlink" href="#1072-programming-exercises" title="Permanent link">&para;</a></h2>
-<h3 id="1-binary-search-in-a-sorted-array">1. &nbsp; Binary Search in a Sorted Array<a class="headerlink" href="#1-binary-search-in-a-sorted-array" title="Permanent link">&para;</a></h3>
-<p>Given an integer array <code>nums</code> in strictly increasing order and a target value <code>target</code>, use binary search to find <code>target</code>. If it exists, return its array index; otherwise, return -1.</p>
+<h2 id="1072-programming-exercises">10.7.2 &nbsp; Programming exercises<a class="headerlink" href="#1072-programming-exercises" title="Permanent link">&para;</a></h2>
+<h3 id="1-binary-search-in-a-sorted-array">1. &nbsp; Binary search in a sorted array<a class="headerlink" href="#1-binary-search-in-a-sorted-array" title="Permanent link">&para;</a></h3>
+<p>Given an integer array <code>nums</code> in strictly increasing order and a target value <code>target</code>, find <code>target</code> using binary search. Return its array index if it exists, or -1 otherwise.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>The initial interval is left = 0 and right = n - 1; it is nonempty while left &lt;= right</li>
-<li>Calculate the midpoint with mid = left + (right - left) // 2</li>
-<li>If <code>nums[mid]</code> is less than <code>target</code>, move the left boundary to <code>mid + 1</code>; if <code>nums[mid]</code> is greater than <code>target</code>, move the right boundary to <code>mid - 1</code>; if they are equal, return immediately</li>
+<li>The initial interval has left = 0 and right = n - 1; it is nonempty while left &lt;= right.</li>
+<li>Calculate the midpoint with mid = left + (right - left) // 2.</li>
+<li>If <code>nums[mid]</code> is less than <code>target</code>, move the left boundary to <code>mid + 1</code>; if <code>nums[mid]</code> is greater than <code>target</code>, move the right boundary to <code>mid - 1</code>; return immediately if they are equal.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/binary-search/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
-<h3 id="2-insertion-point-in-a-sorted-array">2. &nbsp; Insertion Point in a Sorted Array<a class="headerlink" href="#2-insertion-point-in-a-sorted-array" title="Permanent link">&para;</a></h3>
+<h3 id="2-finding-an-insertion-position-in-a-sorted-array">2. &nbsp; Finding an insertion position in a sorted array<a class="headerlink" href="#2-finding-an-insertion-position-in-a-sorted-array" title="Permanent link">&para;</a></h3>
 <p>You are given an integer array <code>nums</code> in strictly increasing order and a target value <code>target</code>.</p>
-<p>Return the index of <code>target</code> if it is already in the array. Otherwise, return the insertion point at which <code>target</code> can be inserted while keeping the array in strictly increasing order.
-The insertion point may be 0 or may equal the array's length. Use binary search.</p>
+<p>If <code>target</code> is already in the array, return its index. Otherwise, return the position at which <code>target</code> can be inserted while preserving strictly increasing order. The answer may be 0 or the array length. Use binary search.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>The answer may be 0 or the array length n</li>
-<li>With a closed interval, if <code>nums[mid]</code> is greater than or equal to <code>target</code>, set <code>right = mid - 1</code> and continue checking farther left; otherwise, set <code>left = mid + 1</code></li>
-<li>When the loop ends, left is the insertion point</li>
+<li>The answer may be 0 or the array length n.</li>
+<li>With a closed interval, if <code>nums[mid]</code> is greater than or equal to <code>target</code>, set <code>right = mid - 1</code> to keep checking farther left; otherwise, set <code>left = mid + 1</code>.</li>
+<li>When the loop ends, left is the insertion position.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/search-insert-position/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
@@ -4931,7 +5002,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 10.6 Summary"
+    aria-label="Previous: 10.6   Summary"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4943,7 +5014,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        10.6 Summary
+        10.6 &nbsp; Summary
       </div>
     </div>
   </a>
@@ -4955,7 +5026,7 @@ aria-label="Footer"
   <a
     href="../../chapter_sorting/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: Chapter 11. &amp;nbsp; Sorting"
+    aria-label="Next: Chapter 11.   Sorting"
     rel="next"
   >
     <div class="md-footer__title">
@@ -4980,7 +5051,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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diff --git a/en/chapter_searching/index.html b/en/chapter_searching/index.html
index ade1ad2a2..a0a15bafe 100644
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       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
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@@ -67,8 +67,8 @@
 
 
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-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
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@@ -1506,7 +1506,7 @@
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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@@ -2378,7 +2378,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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@@ -2399,7 +2399,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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@@ -2598,7 +2598,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2619,7 +2619,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2640,7 +2640,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2694,7 +2694,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2721,7 +2721,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2748,7 +2748,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2775,7 +2775,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2802,7 +2802,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2880,7 +2880,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2901,7 +2901,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -2922,7 +2922,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2976,7 +2976,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3003,7 +3003,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3030,7 +3030,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3057,7 +3057,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3084,7 +3084,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3111,7 +3111,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3138,7 +3138,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3165,7 +3165,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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@@ -3192,7 +3192,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3219,7 +3219,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3285,7 +3285,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3306,7 +3306,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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@@ -3327,7 +3327,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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@@ -3408,7 +3408,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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@@ -3435,7 +3435,7 @@
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3462,7 +3462,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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@@ -3528,7 +3528,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3549,7 +3549,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -3570,7 +3570,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3597,7 +3597,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3624,7 +3624,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3651,7 +3651,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3678,7 +3678,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3705,7 +3705,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3775,7 +3775,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3796,7 +3796,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3817,7 +3817,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3844,7 +3844,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3871,7 +3871,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3898,7 +3898,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -3925,7 +3925,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -3952,7 +3952,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -3979,7 +3979,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4006,7 +4006,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4072,7 +4072,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4093,7 +4093,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4114,7 +4114,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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@@ -4141,7 +4141,7 @@
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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@@ -4168,7 +4168,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4195,7 +4195,7 @@
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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@@ -4222,7 +4222,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4249,7 +4249,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4309,7 +4309,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4330,7 +4330,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4351,7 +4351,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4378,7 +4378,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4405,7 +4405,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4489,6 +4489,78 @@
 
 
 
+
+
+
+
+
+
+
+
+
+
+
+
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+    <li class="md-nav__item md-nav__item--nested">
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+
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+            <a href="../chapter_paperbook/" class="md-nav__link ">
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19 2-5 4.5v11l5-4.5zM6.5 5C4.55 5 2.45 5.4 1 6.5v14.66c0 .25.25.5.5.5.1 0 .15-.07.25-.07 1.35-.65 3.3-1.09 4.75-1.09 1.95 0 4.05.4 5.5 1.5 1.35-.85 3.8-1.5 5.5-1.5 1.65 0 3.35.31 4.75 1.06.1.05.15.03.25.03.25 0 .5-.25.5-.5V6.5c-.6-.45-1.25-.75-2-1V19c-1.1-.35-2.3-.5-3.5-.5-1.7 0-4.15.65-5.5 1.5V6.5C10.55 5.4 8.45 5 6.5 5"/></svg>
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+
+
+  Print edition (Chinese)
+
+
+  </span>
+
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+    <span class="md-status md-status--new" title="Recently added">
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+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
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+
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 </nav>
                   </div>
@@ -4581,16 +4653,16 @@
 <p><img alt="Searching" class="cover-image" src="../assets/covers/chapter_searching.jpg" /></p>
 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
-<p>Searching is an adventure into the unknown, where we may need to traverse every corner of the mysterious space, or we may be able to quickly lock onto the target.</p>
-<p>In this journey of discovery, each exploration may yield an unexpected answer.</p>
+<p>Searching is an adventure into the unknown. We may need to explore every corner of a mysterious space, or we may be able to pinpoint our target quickly.</p>
+<p>On this journey, every exploration may reveal an unexpected answer.</p>
 </div>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
-<li><a href="binary_search/">10.1 &nbsp; Binary Search</a></li>
-<li><a href="binary_search_insertion/">10.2 &nbsp; Binary Search Insertion Point</a></li>
-<li><a href="binary_search_edge/">10.3 &nbsp; Binary Search Boundaries</a></li>
-<li><a href="replace_linear_by_hashing/">10.4 &nbsp; Hash Optimization Strategy</a></li>
-<li><a href="searching_algorithm_revisited/">10.5 &nbsp; Searching Algorithms Revisited</a></li>
+<li><a href="binary_search/">10.1 &nbsp; Binary search</a></li>
+<li><a href="binary_search_insertion/">10.2 &nbsp; Finding an insertion point with binary search</a></li>
+<li><a href="binary_search_edge/">10.3 &nbsp; Finding boundaries with binary search</a></li>
+<li><a href="replace_linear_by_hashing/">10.4 &nbsp; Optimizing with hashing</a></li>
+<li><a href="searching_algorithm_revisited/">10.5 &nbsp; Search algorithms revisited</a></li>
 <li><a href="summary/">10.6 &nbsp; Summary</a></li>
 <li><a href="exercises/">10.7 &nbsp; Exercises</a></li>
 </ul>
@@ -4616,7 +4688,7 @@ aria-label="Footer"
   <a
     href="../chapter_graph/exercises/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 9.5 Exercises"
+    aria-label="Previous: 9.5   Exercises"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4628,7 +4700,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        9.5 Exercises
+        9.5 &nbsp; Exercises
       </div>
     </div>
   </a>
@@ -4640,7 +4712,7 @@ aria-label="Footer"
   <a
     href="./binary_search/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 10.1 Binary Search"
+    aria-label="Next: 10.1   Binary search"
     rel="next"
   >
     <div class="md-footer__title">
@@ -4648,7 +4720,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        10.1 Binary Search
+        10.1 &nbsp; Binary search
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -4665,7 +4737,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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--- a/en/chapter_searching/replace_linear_by_hashing/index.html
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       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>10.4 Hash Optimization Strategy - Hello Algo</title>
+        <title>10.4 Optimizing with hashing - Hello Algo</title>
 
 
 
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-        <a href="#104-hash-optimization-strategy" class="md-skip">
+        <a href="#104-optimizing-with-hashing" class="md-skip">
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
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-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2598,7 +2598,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2619,7 +2619,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2640,7 +2640,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2694,7 +2694,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2730,7 +2730,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2747,7 +2747,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2776,7 +2776,7 @@
   <a href="#1041-linear-search-trading-time-for-space" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.4.1 &nbsp; Linear Search: Trading Time for Space
+        10.4.1 &nbsp; Linear search: trading time for space
       </span>
     </span>
   </a>
@@ -2784,10 +2784,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1042-hash-based-search-trading-space-for-time" class="md-nav__link">
+  <a href="#1042-hash-lookup-trading-space-for-time" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.4.2 &nbsp; Hash-Based Search: Trading Space for Time
+        10.4.2 &nbsp; Hash lookup: trading space for time
       </span>
     </span>
   </a>
@@ -2817,7 +2817,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2844,7 +2844,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2871,7 +2871,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2970,7 +2970,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2991,7 +2991,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3018,7 +3018,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3045,7 +3045,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3072,7 +3072,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3099,7 +3099,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3126,7 +3126,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3153,7 +3153,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3180,7 +3180,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3207,7 +3207,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3234,7 +3234,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3261,7 +3261,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3288,7 +3288,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3375,7 +3375,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3396,7 +3396,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3423,7 +3423,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3450,7 +3450,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3477,7 +3477,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3504,7 +3504,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3531,7 +3531,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3597,7 +3597,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3618,7 +3618,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3639,7 +3639,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3666,7 +3666,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3693,7 +3693,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3720,7 +3720,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3747,7 +3747,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3774,7 +3774,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3844,7 +3844,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3865,7 +3865,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3886,7 +3886,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3913,7 +3913,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3940,7 +3940,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3967,7 +3967,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -3994,7 +3994,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4021,7 +4021,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4048,7 +4048,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4075,7 +4075,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4141,7 +4141,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4162,7 +4162,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4183,7 +4183,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4210,7 +4210,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4237,7 +4237,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4264,7 +4264,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4291,7 +4291,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4318,7 +4318,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4378,7 +4378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4399,7 +4399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4420,7 +4420,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4447,7 +4447,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4474,7 +4474,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4558,6 +4558,78 @@
 
 
 
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@@ -4602,7 +4674,7 @@
   <a href="#1041-linear-search-trading-time-for-space" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.4.1 &nbsp; Linear Search: Trading Time for Space
+        10.4.1 &nbsp; Linear search: trading time for space
       </span>
     </span>
   </a>
@@ -4610,10 +4682,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1042-hash-based-search-trading-space-for-time" class="md-nav__link">
+  <a href="#1042-hash-lookup-trading-space-for-time" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.4.2 &nbsp; Hash-Based Search: Trading Space for Time
+        10.4.2 &nbsp; Hash lookup: trading space for time
       </span>
     </span>
   </a>
@@ -4657,24 +4729,24 @@
 
 
 <!-- Page content -->
-<h1 id="104-hash-optimization-strategy">10.4 &nbsp; Hash Optimization Strategy<a class="headerlink" href="#104-hash-optimization-strategy" title="Permanent link">&para;</a></h1>
-<p>In algorithm problems, <strong>we often reduce the time complexity of algorithms by replacing linear search with hash-based search</strong>. Let's use an algorithm problem to deepen our understanding.</p>
+<h1 id="104-optimizing-with-hashing">10.4 &nbsp; Optimizing with hashing<a class="headerlink" href="#104-optimizing-with-hashing" title="Permanent link">&para;</a></h1>
+<p>In algorithm problems, <strong>we often reduce time complexity by replacing linear search with hash lookup</strong>. Let us explore this strategy through a problem.</p>
 <div class="admonition question">
 <p class="admonition-title">Question</p>
-<p>Given an integer array <code>nums</code> and a target value <code>target</code>, find two elements in the array whose sum is <code>target</code>, and return their indices. Any solution will do.</p>
+<p>Given an integer array <code>nums</code> and a target value <code>target</code>, find two elements whose sum equals <code>target</code> and return their array indices. Any valid solution is acceptable.</p>
 </div>
-<h2 id="1041-linear-search-trading-time-for-space">10.4.1 &nbsp; Linear Search: Trading Time for Space<a class="headerlink" href="#1041-linear-search-trading-time-for-space" title="Permanent link">&para;</a></h2>
-<p>Consider directly traversing all possible combinations. As shown in Figure 10-9, we use nested loops and check in each iteration whether the sum of two integers is <code>target</code>. If so, return their indices.</p>
-<p><img alt="Linear search solution for two sum" class="animation-figure" src="../replace_linear_by_hashing.assets/two_sum_brute_force.png" /></p>
-<p align="center"> Figure 10-9 &nbsp; Linear search solution for two sum </p>
+<h2 id="1041-linear-search-trading-time-for-space">10.4.1 &nbsp; Linear search: trading time for space<a class="headerlink" href="#1041-linear-search-trading-time-for-space" title="Permanent link">&para;</a></h2>
+<p>Consider examining every possible pair directly. As shown below, use two nested loops and check whether the two integers sum to <code>target</code> each time. If they do, return their indices.</p>
+<p><img alt="Solving two sum with linear search" class="animation-figure" src="../replace_linear_by_hashing.assets/two_sum_brute_force.png" /></p>
+<p align="center"> Figure 10-9 &nbsp; Solving two sum with linear search </p>
 
-<p>The code is shown below:</p>
+<p>The code is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">two_sum.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">two_sum_brute_force</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Method 1: Brute force enumeration"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Two nested loops, time complexity is O(n^2)</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Method 1: brute-force enumeration"""</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Two nested loops give a time complexity of O(n^2)</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)):</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>            <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
@@ -4683,10 +4755,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Method 1: Brute force enumeration */</span>
+<div class="highlight"><span class="filename">two_sum.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Method 1: brute-force enumeration */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">twoSumBruteForce</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// Two nested loops, time complexity is O(n^2)</span>
+<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// Two nested loops give a time complexity of O(n^2)</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
@@ -4698,10 +4770,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Method 1: Brute force enumeration */</span>
+<div class="highlight"><span class="filename">two_sum.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Method 1: brute-force enumeration */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="nf">twoSumBruteForce</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
-<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// Two nested loops, time complexity is O(n^2)</span>
+<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// Two nested loops give a time complexity of O(n^2)</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
@@ -4713,10 +4785,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Method 1: Brute force enumeration */</span>
+<div class="highlight"><span class="filename">two_sum.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Method 1: brute-force enumeration */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="nf">TwoSumBruteForce</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
-<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Two nested loops, time complexity is O(n^2)</span>
+<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Two nested loops give a time complexity of O(n^2)</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
@@ -4728,10 +4800,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Method 1: Brute force enumeration */</span>
+<div class="highlight"><span class="filename">two_sum.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Method 1: brute-force enumeration */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">twoSumBruteForce</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="nx">size</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span>
-<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// Two nested loops, time complexity is O(n^2)</span>
+<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// Two nested loops give a time complexity of O(n^2)</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">size</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">size</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="o">+</span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="p">{</span>
@@ -4744,9 +4816,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Method 1: Brute force enumeration */</span>
+<div class="highlight"><span class="filename">two_sum.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Method 1: brute-force enumeration */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">twoSumBruteForce</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Two nested loops, time complexity is O(n^2)</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Two nested loops give a time complexity of O(n^2)</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="bp">dropLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="bp">dropFirst</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
@@ -4759,10 +4831,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Method 1: Brute force enumeration */</span>
+<div class="highlight"><span class="filename">two_sum.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Method 1: brute-force enumeration */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">twoSumBruteForce</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// Two nested loops, time complexity is O(n^2)</span>
+<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// Two nested loops give a time complexity of O(n^2)</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4775,10 +4847,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Method 1: Brute force enumeration */</span>
+<div class="highlight"><span class="filename">two_sum.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Method 1: brute-force enumeration */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">twoSumBruteForce</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// Two nested loops, time complexity is O(n^2)</span>
+<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// Two nested loops give a time complexity of O(n^2)</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4791,10 +4863,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Method 1: Brute force enumeration */</span>
+<div class="highlight"><span class="filename">two_sum.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Method 1: brute-force enumeration */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">twoSumBruteForce</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
-<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// Two nested loops, time complexity is O(n^2)</span>
+<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// Two nested loops give a time complexity of O(n^2)</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">];</span>
@@ -4805,10 +4877,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Method 1: Brute force enumeration */</span>
+<div class="highlight"><span class="filename">two_sum.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Method 1: brute-force enumeration */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">two_sum_brute_force</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
-<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="c1">// Two nested loops, time complexity is O(n^2)</span>
+<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="c1">// Two nested loops give a time complexity of O(n^2)</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">size</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="w"> </span><span class="p">{</span>
@@ -4821,7 +4893,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Method 1: Brute force enumeration */</span>
+<div class="highlight"><span class="filename">two_sum.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Method 1: brute-force enumeration */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="nf">twoSumBruteForce</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">numsSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">returnSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">numsSize</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">numsSize</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4839,10 +4911,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Method 1: Brute force enumeration */</span>
+<div class="highlight"><span class="filename">two_sum.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Method 1: brute-force enumeration */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">twoSumBruteForce</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="n">IntArray</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span>
-<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="c1">// Two nested loops, time complexity is O(n^2)</span>
+<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="c1">// Two nested loops give a time complexity of O(n^2)</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">intArrayOf</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">)</span>
@@ -4853,9 +4925,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Method 1: Brute force enumeration ###</span>
+<div class="highlight"><span class="filename">two_sum.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Method 1: brute-force enumeration ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">two_sum_brute_force</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Two nested loops, time complexity is O(n^2)</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Two nested loops give a time complexity of O(n^2)</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="p">(</span><span class="n">nums</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="o">...</span><span class="n">nums</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="o">[</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">target</span>
@@ -4868,17 +4940,22 @@
 </div>
 </div>
 </div>
-<p>This method has a time complexity of <span class="arithmatex">\(O(n^2)\)</span> and a space complexity of <span class="arithmatex">\(O(1)\)</span>, making it very time-consuming on large inputs.</p>
-<h2 id="1042-hash-based-search-trading-space-for-time">10.4.2 &nbsp; Hash-Based Search: Trading Space for Time<a class="headerlink" href="#1042-hash-based-search-trading-space-for-time" title="Permanent link">&para;</a></h2>
-<p>Consider using a hash table whose keys are array elements and whose values are their indices. Traverse the array and perform the steps shown in Figure 10-10 in each iteration:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 441px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20two_sum_brute_force%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%22%22%22Method%201%3A%20brute-force%20enumeration%22%22%22%0A%20%20%20%20%23%20Two%20nested%20loops%20give%20a%20time%20complexity%20of%20O%28n%5E2%29%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%20-%201%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28i%20%2B%201,%20len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20nums%5Bi%5D%20%2B%20nums%5Bj%5D%20%3D%3D%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20return%20%5Bi,%20j%5D%0A%20%20%20%20return%20%5B%5D%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B2,%207,%2011,%2015%5D%0A%20%20%20%20target%20%3D%2013%0A%20%20%20%20res%20%3D%20two_sum_brute_force%28nums,%20target%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20two_sum_brute_force%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%22%22%22Method%201%3A%20brute-force%20enumeration%22%22%22%0A%20%20%20%20%23%20Two%20nested%20loops%20give%20a%20time%20complexity%20of%20O%28n%5E2%29%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%20-%201%29%3A%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28i%20%2B%201,%20len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20nums%5Bi%5D%20%2B%20nums%5Bj%5D%20%3D%3D%20target%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20return%20%5Bi,%20j%5D%0A%20%20%20%20return%20%5B%5D%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B2,%207,%2011,%2015%5D%0A%20%20%20%20target%20%3D%2013%0A%20%20%20%20res%20%3D%20two_sum_brute_force%28nums,%20target%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>This approach takes <span class="arithmatex">\(O(n^2)\)</span> time and <span class="arithmatex">\(O(1)\)</span> space, making it very slow for large datasets.</p>
+<h2 id="1042-hash-lookup-trading-space-for-time">10.4.2 &nbsp; Hash lookup: trading space for time<a class="headerlink" href="#1042-hash-lookup-trading-space-for-time" title="Permanent link">&para;</a></h2>
+<p>Now use a hash table whose keys are array elements and whose values are their indices. Iterate through the array, performing the steps below each round.</p>
 <ol>
-<li>Check if the number <code>target - nums[i]</code> is in the hash table. If so, directly return the indices of these two elements.</li>
-<li>Add the key-value pair <code>nums[i]</code> and index <code>i</code> to the hash table.</li>
+<li>Check whether <code>target - nums[i]</code> is in the hash table. If it is, return the indices of the two elements.</li>
+<li>Add the key-value pair consisting of <code>nums[i]</code> and index <code>i</code> to the hash table.</li>
 </ol>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:3"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">&lt;1&gt;</label><label for="__tabbed_2_2">&lt;2&gt;</label><label for="__tabbed_2_3">&lt;3&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Hash table solution for two sum" class="animation-figure" src="../replace_linear_by_hashing.assets/two_sum_hashtable_step1.png" /></p>
+<p><img alt="Solving two sum with an auxiliary hash table" class="animation-figure" src="../replace_linear_by_hashing.assets/two_sum_hashtable_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="two_sum_hashtable_step2" class="animation-figure" src="../replace_linear_by_hashing.assets/two_sum_hashtable_step2.png" /></p>
@@ -4888,17 +4965,17 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 10-10 &nbsp; Hash table solution for two sum </p>
+<p align="center"> Figure 10-10 &nbsp; Solving two sum with an auxiliary hash table </p>
 
-<p>The implementation is shown below and requires only a single loop:</p>
+<p>The implementation below requires only one loop:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">two_sum.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">two_sum_hash_table</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">target</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Method 2: Auxiliary hash table"""</span>
-<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># Auxiliary hash table, space complexity is O(n)</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Method 2: auxiliary hash table"""</span>
+<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># The auxiliary hash table uses O(n) space</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="n">dic</span> <span class="o">=</span> <span class="p">{}</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Single loop, time complexity is O(n)</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># A single loop with average O(1) hash operations gives an average time complexity of O(n)</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)):</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="k">if</span> <span class="n">target</span> <span class="o">-</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="ow">in</span> <span class="n">dic</span><span class="p">:</span>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>            <span class="k">return</span> <span class="p">[</span><span class="n">dic</span><span class="p">[</span><span class="n">target</span> <span class="o">-</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]],</span> <span class="n">i</span><span class="p">]</span>
@@ -4907,12 +4984,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Method 2: Auxiliary hash table */</span>
+<div class="highlight"><span class="filename">two_sum.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Method 2: auxiliary hash table */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">twoSumHashTable</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="c1">// Auxiliary hash table, space complexity is O(n)</span>
+<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="c1">// The auxiliary hash table uses O(n) space</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="n">unordered_map</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dic</span><span class="p">;</span>
-<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="c1">// Single loop, time complexity is O(n)</span>
+<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="c1">// A single loop gives a time complexity of O(n)</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">dic</span><span class="p">.</span><span class="n">find</span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">dic</span><span class="p">.</span><span class="n">end</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="p">{</span><span class="n">dic</span><span class="p">[</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]],</span><span class="w"> </span><span class="n">i</span><span class="p">};</span>
@@ -4924,12 +5001,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Method 2: Auxiliary hash table */</span>
+<div class="highlight"><span class="filename">two_sum.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Method 2: auxiliary hash table */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="nf">twoSumHashTable</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
-<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="c1">// Auxiliary hash table, space complexity is O(n)</span>
+<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="c1">// The auxiliary hash table uses O(n) space</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="n">Map</span><span class="o">&lt;</span><span class="n">Integer</span><span class="p">,</span><span class="w"> </span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dic</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">HashMap</span><span class="o">&lt;&gt;</span><span class="p">();</span>
-<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="c1">// Single loop, time complexity is O(n)</span>
+<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="c1">// A single loop gives a time complexity of O(n)</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">dic</span><span class="p">.</span><span class="na">containsKey</span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">dic</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">),</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">};</span>
@@ -4941,12 +5018,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Method 2: Auxiliary hash table */</span>
+<div class="highlight"><span class="filename">two_sum.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Method 2: auxiliary hash table */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="nf">TwoSumHashTable</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
-<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="c1">// Auxiliary hash table, space complexity is O(n)</span>
+<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="c1">// The auxiliary hash table uses O(n) space</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="n">Dictionary</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dic</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="c1">// Single loop, time complexity is O(n)</span>
+<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="c1">// A single loop gives a time complexity of O(n)</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">dic</span><span class="p">.</span><span class="n">ContainsKey</span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="n">dic</span><span class="p">[</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]],</span><span class="w"> </span><span class="n">i</span><span class="p">];</span>
@@ -4958,11 +5035,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Method 2: Auxiliary hash table */</span>
+<div class="highlight"><span class="filename">two_sum.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Method 2: auxiliary hash table */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">twoSumHashTable</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Auxiliary hash table, space complexity is O(n)</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// The auxiliary hash table uses O(n) space</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="nx">hashTable</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="kt">map</span><span class="p">[</span><span class="kt">int</span><span class="p">]</span><span class="kt">int</span><span class="p">{}</span>
-<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="c1">// Single loop, time complexity is O(n)</span>
+<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="c1">// A single loop gives a time complexity of O(n)</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">idx</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">preIdx</span><span class="p">,</span><span class="w"> </span><span class="nx">ok</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">hashTable</span><span class="p">[</span><span class="nx">target</span><span class="o">-</span><span class="nx">val</span><span class="p">];</span><span class="w"> </span><span class="nx">ok</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">{</span><span class="nx">preIdx</span><span class="p">,</span><span class="w"> </span><span class="nx">idx</span><span class="p">}</span>
@@ -4974,11 +5051,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Method 2: Auxiliary hash table */</span>
+<div class="highlight"><span class="filename">two_sum.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Method 2: auxiliary hash table */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">twoSumHashTable</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Auxiliary hash table, space complexity is O(n)</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// The auxiliary hash table uses O(n) space</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">dic</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[:]</span>
-<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="c1">// Single loop, time complexity is O(n)</span>
+<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="c1">// A single loop gives a time complexity of O(n)</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">dic</span><span class="p">[</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">]</span>
@@ -4990,11 +5067,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Method 2: Auxiliary hash table */</span>
+<div class="highlight"><span class="filename">two_sum.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Method 2: auxiliary hash table */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">twoSumHashTable</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Auxiliary hash table, space complexity is O(n)</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// The auxiliary hash table uses O(n) space</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{};</span>
-<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="c1">// Single loop, time complexity is O(n)</span>
+<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="c1">// A single loop gives a time complexity of O(n)</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">m</span><span class="p">[</span><span class="nx">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]]</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">undefined</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="nx">m</span><span class="p">[</span><span class="nx">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]],</span><span class="w"> </span><span class="nx">i</span><span class="p">];</span>
@@ -5007,11 +5084,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Method 2: Auxiliary hash table */</span>
+<div class="highlight"><span class="filename">two_sum.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Method 2: auxiliary hash table */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">twoSumHashTable</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">target</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// Auxiliary hash table, space complexity is O(n)</span>
+<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// The auxiliary hash table uses O(n) space</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">m</span><span class="o">:</span><span class="w"> </span><span class="kt">Map</span><span class="o">&lt;</span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="kt">number</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Map</span><span class="p">();</span>
-<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="c1">// Single loop, time complexity is O(n)</span>
+<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="c1">// A single loop gives a time complexity of O(n)</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">m</span><span class="p">.</span><span class="nx">get</span><span class="p">(</span><span class="nx">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]);</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">index</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">undefined</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5025,12 +5102,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Method 2: Auxiliary hash table */</span>
+<div class="highlight"><span class="filename">two_sum.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Method 2: auxiliary hash table */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">twoSumHashTable</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
-<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="c1">// Auxiliary hash table, space complexity is O(n)</span>
+<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="c1">// The auxiliary hash table uses O(n) space</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="n">Map</span><span class="o">&lt;</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">dic</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashMap</span><span class="p">();</span>
-<a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="c1">// Single loop, time complexity is O(n)</span>
+<a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="c1">// A single loop gives a time complexity of O(n)</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">dic</span><span class="p">.</span><span class="n">containsKey</span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="p">[</span><span class="n">dic</span><span class="p">[</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="o">!</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">];</span>
@@ -5042,11 +5119,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Method 2: Auxiliary hash table */</span>
+<div class="highlight"><span class="filename">two_sum.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Method 2: auxiliary hash table */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">two_sum_hash_table</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Auxiliary hash table, space complexity is O(n)</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// The auxiliary hash table uses O(n) space</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">dic</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashMap</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
-<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="c1">// Single loop, time complexity is O(n)</span>
+<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="c1">// A single loop gives a time complexity of O(n)</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">iter</span><span class="p">().</span><span class="n">enumerate</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">        </span><span class="k">match</span><span class="w"> </span><span class="n">dic</span><span class="p">.</span><span class="n">get</span><span class="p">(</span><span class="o">&amp;</span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">num</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">            </span><span class="nb">Some</span><span class="p">(</span><span class="n">v</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="fm">vec!</span><span class="p">[</span><span class="o">*</span><span class="n">v</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">]),</span>
@@ -5065,14 +5142,14 @@
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="n">UT_hash_handle</span><span class="w"> </span><span class="n">hh</span><span class="p">;</span><span class="w"> </span><span class="c1">// Implemented using uthash.h</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="p">}</span><span class="w"> </span><span class="n">HashTable</span><span class="p">;</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a>
-<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="cm">/* Hash table lookup */</span>
+<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="cm">/* Look up a key in the hash table */</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="n">HashTable</span><span class="w"> </span><span class="o">*</span><span class="nf">find</span><span class="p">(</span><span class="n">HashTable</span><span class="w"> </span><span class="o">*</span><span class="n">h</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">    </span><span class="n">HashTable</span><span class="w"> </span><span class="o">*</span><span class="n">tmp</span><span class="p">;</span>
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="n">HASH_FIND_INT</span><span class="p">(</span><span class="n">h</span><span class="p">,</span><span class="w"> </span><span class="o">&amp;</span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="n">tmp</span><span class="p">);</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="p">}</span>
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a>
-<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="cm">/* Hash table element insertion */</span>
+<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="cm">/* Insert an entry into the hash table */</span>
 <a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">HashTable</span><span class="w"> </span><span class="o">**</span><span class="n">h</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">key</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w">    </span><span class="n">HashTable</span><span class="w"> </span><span class="o">*</span><span class="n">t</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">find</span><span class="p">(</span><span class="o">*</span><span class="n">h</span><span class="p">,</span><span class="w"> </span><span class="n">key</span><span class="p">);</span>
 <a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">t</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5084,7 +5161,7 @@
 <a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="p">}</span>
 <a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a>
-<a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a><span class="cm">/* Method 2: Auxiliary hash table */</span>
+<a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a><span class="cm">/* Method 2: auxiliary hash table */</span>
 <a id="__codelineno-23-28" name="__codelineno-23-28" href="#__codelineno-23-28"></a><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="nf">twoSumHashTable</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">numsSize</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">target</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">returnSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-29" name="__codelineno-23-29" href="#__codelineno-23-29"></a><span class="w">    </span><span class="n">HashTable</span><span class="w"> </span><span class="o">*</span><span class="n">hashtable</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-23-30" name="__codelineno-23-30" href="#__codelineno-23-30"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">numsSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5103,12 +5180,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Method 2: Auxiliary hash table */</span>
+<div class="highlight"><span class="filename">two_sum.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Method 2: auxiliary hash table */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">twoSumHashTable</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="n">IntArray</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span>
-<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="c1">// Auxiliary hash table, space complexity is O(n)</span>
+<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="c1">// The auxiliary hash table uses O(n) space</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">dic</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">HashMap</span><span class="o">&lt;</span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span>
-<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="c1">// Single loop, time complexity is O(n)</span>
+<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="c1">// A single loop gives a time complexity of O(n)</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">dic</span><span class="p">.</span><span class="na">containsKey</span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">intArrayOf</span><span class="p">(</span><span class="n">dic</span><span class="o">[</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]]!!</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
@@ -5120,11 +5197,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">two_sum.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Method 2: Auxiliary hash table ###</span>
+<div class="highlight"><span class="filename">two_sum.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Method 2: auxiliary hash table ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">two_sum_hash_table</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">target</span><span class="p">)</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Auxiliary hash table, space complexity is O(n)</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># The auxiliary hash table uses O(n) space</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="n">dic</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{}</span>
-<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># Single loop, time complexity is O(n)</span>
+<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># A single loop gives a time complexity of O(n)</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">nums</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">[</span><span class="n">dic</span><span class="o">[</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]]</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">dic</span><span class="o">.</span><span class="n">has_key?</span><span class="p">(</span><span class="n">target</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a>
@@ -5137,8 +5214,13 @@
 </div>
 </div>
 </div>
-<p>This method reduces the time complexity from <span class="arithmatex">\(O(n^2)\)</span> to <span class="arithmatex">\(O(n)\)</span> through hash-based search, greatly improving runtime efficiency.</p>
-<p>Since an additional hash table needs to be maintained, the space complexity is <span class="arithmatex">\(O(n)\)</span>. <strong>Nevertheless, this method offers a more balanced overall time-space trade-off, making it the optimal solution to this problem</strong>.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 477px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20two_sum_hash_table%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%22%22%22Method%202%3A%20auxiliary%20hash%20table%22%22%22%0A%20%20%20%20%23%20The%20auxiliary%20hash%20table%20uses%20O%28n%29%20space%0A%20%20%20%20dic%20%3D%20%7B%7D%0A%20%20%20%20%23%20A%20single%20loop%20gives%20average%20O%28n%29%20time%20with%20average%20O%281%29%20hash-table%20operations%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20if%20target%20-%20nums%5Bi%5D%20in%20dic%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20%5Bdic%5Btarget%20-%20nums%5Bi%5D%5D,%20i%5D%0A%20%20%20%20%20%20%20%20dic%5Bnums%5Bi%5D%5D%20%3D%20i%0A%20%20%20%20return%20%5B%5D%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B2,%207,%2011,%2015%5D%0A%20%20%20%20target%20%3D%2013%0A%20%20%20%20res%20%3D%20two_sum_hash_table%28nums,%20target%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20two_sum_hash_table%28nums%3A%20list%5Bint%5D,%20target%3A%20int%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%22%22%22Method%202%3A%20auxiliary%20hash%20table%22%22%22%0A%20%20%20%20%23%20The%20auxiliary%20hash%20table%20uses%20O%28n%29%20space%0A%20%20%20%20dic%20%3D%20%7B%7D%0A%20%20%20%20%23%20A%20single%20loop%20gives%20average%20O%28n%29%20time%20with%20average%20O%281%29%20hash-table%20operations%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20if%20target%20-%20nums%5Bi%5D%20in%20dic%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20%5Bdic%5Btarget%20-%20nums%5Bi%5D%5D,%20i%5D%0A%20%20%20%20%20%20%20%20dic%5Bnums%5Bi%5D%5D%20%3D%20i%0A%20%20%20%20return%20%5B%5D%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B2,%207,%2011,%2015%5D%0A%20%20%20%20target%20%3D%2013%0A%20%20%20%20res%20%3D%20two_sum_hash_table%28nums,%20target%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Hash lookup reduces the time complexity from <span class="arithmatex">\(O(n^2)\)</span> to <span class="arithmatex">\(O(n)\)</span>, greatly improving efficiency.</p>
+<p>Maintaining an additional hash table requires <span class="arithmatex">\(O(n)\)</span> space. <strong>Even so, this approach offers a better balance of time and space efficiency, making it the optimal solution to this problem</strong>.</p>
 
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@@ -5161,7 +5243,7 @@ aria-label="Footer"
   <a
     href="../binary_search_edge/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 10.3 Binary Search Boundaries"
+    aria-label="Previous: 10.3   Finding boundaries with binary search"
     rel="prev"
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     <div class="md-footer__button md-icon">
@@ -5173,7 +5255,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        10.3 Binary Search Boundaries
+        10.3 &nbsp; Finding boundaries with binary search
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   </a>
@@ -5185,7 +5267,7 @@ aria-label="Footer"
   <a
     href="../searching_algorithm_revisited/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 10.5 Searching Algorithms Revisited"
+    aria-label="Next: 10.5   Search algorithms revisited"
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     <div class="md-footer__title">
@@ -5193,7 +5275,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        10.5 Searching Algorithms Revisited
+        10.5 &nbsp; Search algorithms revisited
       </div>
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     <div class="md-footer__button md-icon">
@@ -5210,7 +5292,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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diff --git a/en/chapter_searching/searching_algorithm_revisited/index.html b/en/chapter_searching/searching_algorithm_revisited/index.html
index 727d77af0..cd1eef89a 100644
--- a/en/chapter_searching/searching_algorithm_revisited/index.html
+++ b/en/chapter_searching/searching_algorithm_revisited/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>10.5 Searching Algorithms Revisited - Hello Algo</title>
+        <title>10.5 Search algorithms revisited - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
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-        <a href="#105-searching-algorithms-revisited" class="md-skip">
+        <a href="#105-search-algorithms-revisited" class="md-skip">
           Skip to content
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@@ -119,7 +119,7 @@
         <path
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-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              10.5 &nbsp; Searching Algorithms Revisited
+              10.5 &nbsp; Search algorithms revisited
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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@@ -177,9 +177,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2598,7 +2598,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2619,7 +2619,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2640,7 +2640,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2694,7 +2694,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2721,7 +2721,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2757,7 +2757,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2774,7 +2774,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2803,7 +2803,7 @@
   <a href="#1051-brute-force-search" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.5.1 &nbsp; Brute-Force Search
+        10.5.1 &nbsp; Brute-force search
       </span>
     </span>
   </a>
@@ -2814,7 +2814,7 @@
   <a href="#1052-adaptive-search" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.5.2 &nbsp; Adaptive Search
+        10.5.2 &nbsp; Adaptive search
       </span>
     </span>
   </a>
@@ -2822,10 +2822,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1053-search-method-selection" class="md-nav__link">
+  <a href="#1053-choosing-a-search-method" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.5.3 &nbsp; Search Method Selection
+        10.5.3 &nbsp; Choosing a search method
       </span>
     </span>
   </a>
@@ -2855,7 +2855,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2882,7 +2882,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2960,7 +2960,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2981,7 +2981,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3002,7 +3002,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3029,7 +3029,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3056,7 +3056,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3083,7 +3083,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3110,7 +3110,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3137,7 +3137,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3164,7 +3164,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3191,7 +3191,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3218,7 +3218,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3245,7 +3245,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3272,7 +3272,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3299,7 +3299,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3365,7 +3365,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3386,7 +3386,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3407,7 +3407,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3434,7 +3434,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3461,7 +3461,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3488,7 +3488,7 @@
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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@@ -3515,7 +3515,7 @@
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3542,7 +3542,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3608,7 +3608,7 @@
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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@@ -3650,7 +3650,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3731,7 +3731,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3758,7 +3758,7 @@
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3855,7 +3855,7 @@
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4569,6 +4569,78 @@
 
 
 
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@@ -4613,7 +4685,7 @@
   <a href="#1051-brute-force-search" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.5.1 &nbsp; Brute-Force Search
+        10.5.1 &nbsp; Brute-force search
       </span>
     </span>
   </a>
@@ -4624,7 +4696,7 @@
   <a href="#1052-adaptive-search" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.5.2 &nbsp; Adaptive Search
+        10.5.2 &nbsp; Adaptive search
       </span>
     </span>
   </a>
@@ -4632,10 +4704,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1053-search-method-selection" class="md-nav__link">
+  <a href="#1053-choosing-a-search-method" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        10.5.3 &nbsp; Search Method Selection
+        10.5.3 &nbsp; Choosing a search method
       </span>
     </span>
   </a>
@@ -4679,42 +4751,42 @@
 
 
 <!-- Page content -->
-<h1 id="105-searching-algorithms-revisited">10.5 &nbsp; Searching Algorithms Revisited<a class="headerlink" href="#105-searching-algorithms-revisited" title="Permanent link">&para;</a></h1>
-<p><u>Searching algorithms</u> are used to search for one or a group of elements that meet specific conditions in data structures (such as arrays, linked lists, trees, or graphs).</p>
-<p>Searching algorithms can be divided into the following two categories based on their implementation approach:</p>
+<h1 id="105-search-algorithms-revisited">10.5 &nbsp; Search algorithms revisited<a class="headerlink" href="#105-search-algorithms-revisited" title="Permanent link">&para;</a></h1>
+<p>A <u>search algorithm</u> finds one or more elements that satisfy specified conditions in a data structure, such as an array, linked list, tree, or graph.</p>
+<p>Search algorithms fall into two categories according to their approach.</p>
 <ul>
-<li><strong>Locating target elements by traversing the data structure</strong>, such as traversing arrays, linked lists, trees, and graphs.</li>
-<li><strong>Achieving efficient element lookup by leveraging the way data is organized or prior information about the data</strong>, such as binary search, hash-based search, and binary search tree search.</li>
+<li><strong>Locate the target by traversing the data structure</strong>, as in traversals of arrays, linked lists, trees, and graphs.</li>
+<li><strong>Find elements efficiently by using the organization of the data or prior information about it</strong>, as in binary search, hash lookup, and binary search tree lookup.</li>
 </ul>
-<p>As these topics have already been introduced in earlier chapters, searching algorithms should already be familiar to us. In this section, we revisit them from a more systematic perspective.</p>
-<h2 id="1051-brute-force-search">10.5.1 &nbsp; Brute-Force Search<a class="headerlink" href="#1051-brute-force-search" title="Permanent link">&para;</a></h2>
-<p>Brute-force search locates target elements by traversing each element of the data structure.</p>
+<p>These ideas have all appeared in earlier chapters, so search algorithms are already familiar to us. In this section, we revisit them from a more systematic perspective.</p>
+<h2 id="1051-brute-force-search">10.5.1 &nbsp; Brute-force search<a class="headerlink" href="#1051-brute-force-search" title="Permanent link">&para;</a></h2>
+<p>Brute-force search locates a target by visiting the elements of a data structure one by one.</p>
 <ul>
-<li>"Linear search" is applicable to linear data structures such as arrays and linked lists. It starts from one end of the data structure and accesses elements one by one until the target element is found or the other end is reached without finding the target element.</li>
-<li>"Breadth-first search" and "depth-first search" are two traversal strategies for graphs and trees. Breadth-first search starts from the initial node and searches layer by layer, visiting nodes from near to far. Depth-first search starts from the initial node, follows a path to the end, then backtracks and tries other paths until the entire data structure is traversed.</li>
+<li>“Linear search” applies to linear data structures such as arrays and linked lists. Starting at one end, it visits elements in sequence until it finds the target or reaches the other end without finding it.</li>
+<li>“Breadth-first search” and “depth-first search” are two strategies for traversing graphs and trees. Breadth-first search starts at an initial node and searches level by level, visiting nearby nodes before more distant ones. Depth-first search follows a path as far as possible, then backtracks and tries other paths until the whole structure has been traversed.</li>
 </ul>
-<p>The advantage of brute-force search is that it is simple and has good generality, <strong>requiring no data preprocessing or additional data structures</strong>.</p>
-<p>However, <strong>the time complexity of such algorithms is <span class="arithmatex">\(O(n)\)</span></strong>, where <span class="arithmatex">\(n\)</span> is the number of elements, so performance is poor when dealing with large amounts of data.</p>
-<h2 id="1052-adaptive-search">10.5.2 &nbsp; Adaptive Search<a class="headerlink" href="#1052-adaptive-search" title="Permanent link">&para;</a></h2>
-<p>Adaptive search leverages properties of the data itself (such as sorted order) to optimize the search process and locate target elements more efficiently.</p>
+<p>Brute-force search is simple and broadly applicable. <strong>It requires neither preprocessing the data nor building an additional data structure</strong>.</p>
+<p>However, <strong>these algorithms take <span class="arithmatex">\(O(n)\)</span> time</strong>, where <span class="arithmatex">\(n\)</span> is the number of elements, so they perform poorly on large datasets.</p>
+<h2 id="1052-adaptive-search">10.5.2 &nbsp; Adaptive search<a class="headerlink" href="#1052-adaptive-search" title="Permanent link">&para;</a></h2>
+<p>Adaptive search uses properties of the data, such as its ordering, to optimize the search process and locate the target more efficiently.</p>
 <ul>
-<li>"Binary search" uses the orderliness of data to achieve efficient searching, applicable only to arrays.</li>
-<li>"Hash-based search" uses hash tables to store searchable data as key-value pairs, thereby enabling efficient queries.</li>
-<li>"Tree search" operates on specific tree structures (such as binary search trees), quickly ruling out nodes by comparing node values to locate the target element.</li>
+<li>“Binary search” uses sorted data to search efficiently and applies only to arrays.</li>
+<li>“Hash lookup” uses a hash table to map lookup keys to target data as key-value pairs.</li>
+<li>“Tree search” uses comparisons of node values in a particular tree structure, such as a binary search tree, to rule out nodes quickly and locate the target.</li>
 </ul>
-<p>The advantage of such algorithms is high efficiency, <strong>with time complexity reaching <span class="arithmatex">\(O(\log n)\)</span> or even <span class="arithmatex">\(O(1)\)</span></strong>.</p>
-<p>However, <strong>using these algorithms often requires data preprocessing</strong>. For example, binary search requires pre-sorting the array, while hash-based search and tree search both require additional data structures, and maintaining these data structures also requires extra time and space overhead.</p>
+<p>These algorithms are efficient, <strong>achieving time complexities of <span class="arithmatex">\(O(\log n)\)</span> or even <span class="arithmatex">\(O(1)\)</span></strong>.</p>
+<p>However, <strong>they often require preprocessing the data</strong>. Binary search requires sorting the array in advance; hash lookup and tree search require additional data structures, whose maintenance also costs time and space.</p>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>Adaptive search algorithms are often called lookup algorithms, <strong>mainly used to quickly retrieve target elements in specific data structures</strong>.</p>
+<p>Adaptive search algorithms are often called lookup algorithms. <strong>Their main purpose is to retrieve target elements quickly from specific data structures</strong>.</p>
 </div>
-<h2 id="1053-search-method-selection">10.5.3 &nbsp; Search Method Selection<a class="headerlink" href="#1053-search-method-selection" title="Permanent link">&para;</a></h2>
-<p>Given a dataset of size <span class="arithmatex">\(n\)</span>, we can use linear search, binary search, tree search, hash-based search, and other methods to search for the target element. The working principles of each method are shown in Figure 10-11.</p>
-<p><img alt="Multiple search strategies" class="animation-figure" src="../searching_algorithm_revisited.assets/searching_algorithms.png" /></p>
-<p align="center"> Figure 10-11 &nbsp; Multiple search strategies </p>
+<h2 id="1053-choosing-a-search-method">10.5.3 &nbsp; Choosing a search method<a class="headerlink" href="#1053-choosing-a-search-method" title="Permanent link">&para;</a></h2>
+<p>Given a dataset of size <span class="arithmatex">\(n\)</span>, we can find a target using linear search, binary search, tree search, hash lookup, and other methods. Figure 10-11 illustrates how these methods work.</p>
+<p><img alt="Different search strategies" class="animation-figure" src="../searching_algorithm_revisited.assets/searching_algorithms.png" /></p>
+<p align="center"> Figure 10-11 &nbsp; Different search strategies </p>
 
-<p>The efficiency and characteristics of these methods are summarized in Table 10-1.</p>
-<p align="center"> Table 10-1 &nbsp; Comparison of search algorithm efficiency </p>
+<p>Table 10-1 compares their efficiency and properties.</p>
+<p align="center"> Table 10-1 &nbsp; Efficiency of lookup algorithms </p>
 
 <div class="center-table">
 <table>
@@ -4724,26 +4796,26 @@
 <th>Linear search</th>
 <th>Binary search</th>
 <th>Tree search</th>
-<th>Hash-based search</th>
+<th>Hash lookup</th>
 </tr>
 </thead>
 <tbody>
 <tr>
-<td>Search element</td>
+<td>Find an element</td>
 <td><span class="arithmatex">\(O(n)\)</span></td>
 <td><span class="arithmatex">\(O(\log n)\)</span></td>
 <td><span class="arithmatex">\(O(\log n)\)</span></td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
-<td>Insert element</td>
+<td>Insert an element</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 <td><span class="arithmatex">\(O(n)\)</span></td>
 <td><span class="arithmatex">\(O(\log n)\)</span></td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
-<td>Delete element</td>
+<td>Delete an element</td>
 <td><span class="arithmatex">\(O(n)\)</span></td>
 <td><span class="arithmatex">\(O(n)\)</span></td>
 <td><span class="arithmatex">\(O(\log n)\)</span></td>
@@ -4757,48 +4829,48 @@
 <td><span class="arithmatex">\(O(n)\)</span></td>
 </tr>
 <tr>
-<td>Data preprocessing</td>
+<td>Preprocessing</td>
 <td>/</td>
-<td>Sorting <span class="arithmatex">\(O(n \log n)\)</span></td>
-<td>Tree building <span class="arithmatex">\(O(n \log n)\)</span></td>
-<td>Hash table building <span class="arithmatex">\(O(n)\)</span></td>
+<td>Sort <span class="arithmatex">\(O(n \log n)\)</span></td>
+<td>Build a tree <span class="arithmatex">\(O(n \log n)\)</span></td>
+<td>Build a hash table <span class="arithmatex">\(O(n)\)</span></td>
 </tr>
 <tr>
-<td>Data ordered</td>
-<td>Unordered</td>
-<td>Ordered</td>
-<td>Ordered</td>
-<td>Unordered</td>
+<td>Data ordering</td>
+<td>Unsorted</td>
+<td>Sorted</td>
+<td>Sorted</td>
+<td>Unsorted</td>
 </tr>
 </tbody>
 </table>
 </div>
-<p>The choice of search algorithm also depends on data volume, search performance requirements, data query and update frequency, etc.</p>
+<p>The choice also depends on dataset size, required search performance, and how frequently the data is queried and updated.</p>
 <p><strong>Linear search</strong></p>
 <ul>
-<li>Good generality, requiring no data preprocessing operations. If we need to query the data only once, the preprocessing required by the other three methods can take longer than the linear search itself.</li>
-<li>Suitable for small data volumes, where time complexity has less impact on efficiency.</li>
-<li>Suitable for scenarios with high data update frequency, as this method does not require any additional data maintenance.</li>
+<li>It is broadly applicable and requires no preprocessing. If we need only one lookup, preprocessing for the other three methods takes longer than a linear search itself.</li>
+<li>It suits small datasets, where time complexity has less influence on actual performance.</li>
+<li>It suits frequently updated data because no additional data structure needs to be maintained.</li>
 </ul>
 <p><strong>Binary search</strong></p>
 <ul>
-<li>Suitable for large datasets, with stable performance and a worst-case time complexity of <span class="arithmatex">\(O(\log n)\)</span>.</li>
-<li>Data volume cannot be too large, as storing arrays requires contiguous memory space.</li>
-<li>Not suitable for scenarios with frequent data insertion and deletion, as maintaining a sorted array has high overhead.</li>
+<li>It suits large datasets, with predictable performance and a worst-case time complexity of <span class="arithmatex">\(O(\log n)\)</span>.</li>
+<li>The dataset cannot be excessively large, because an array requires contiguous memory.</li>
+<li>It is unsuitable for frequent insertions and deletions, since maintaining a sorted array is expensive.</li>
 </ul>
-<p><strong>Hash-based search</strong></p>
+<p><strong>Hash lookup</strong></p>
 <ul>
-<li>Suitable for scenarios with high query performance requirements, with an average time complexity of <span class="arithmatex">\(O(1)\)</span>.</li>
-<li>Not suitable for scenarios requiring ordered data or range searches, as hash tables cannot maintain the data in sorted order.</li>
-<li>High dependence on hash functions and hash collision handling strategies, with significant risk of performance degradation.</li>
-<li>Not suitable for excessively large data volumes, as hash tables require extra space to minimize collisions and thus provide good query performance.</li>
+<li>It suits situations demanding very fast queries, with an average time complexity of <span class="arithmatex">\(O(1)\)</span>.</li>
+<li>It is unsuitable when sorted data or range queries are needed, because a hash table does not preserve order.</li>
+<li>It depends heavily on the hash function and collision-resolution strategy, creating a greater risk of performance degradation.</li>
+<li>It is unsuitable for excessively large datasets, because a hash table needs extra space to minimize collisions and maintain good query performance.</li>
 </ul>
 <p><strong>Tree search</strong></p>
 <ul>
-<li>Suitable for massive datasets, as tree nodes are stored non-contiguously in memory.</li>
-<li>Suitable for scenarios that require maintaining ordered data or performing range searches.</li>
-<li>During continuous node insertion and deletion, binary search trees may become skewed, degrading time complexity to <span class="arithmatex">\(O(n)\)</span>.</li>
-<li>If AVL trees or red-black trees are used, all operations can consistently run in <span class="arithmatex">\(O(\log n)\)</span> time, though maintaining tree balance adds extra overhead.</li>
+<li>It suits very large datasets, because tree nodes are stored at separate locations in memory.</li>
+<li>It suits situations requiring sorted data or range queries.</li>
+<li>Repeated insertions and deletions can make a binary search tree skewed, degrading its time complexity to <span class="arithmatex">\(O(n)\)</span>.</li>
+<li>An AVL tree or red-black tree keeps operations at <span class="arithmatex">\(O(\log n)\)</span>, but maintaining balance adds overhead.</li>
 </ul>
 
 <!-- Source file information -->
@@ -4822,7 +4894,7 @@ aria-label="Footer"
   <a
     href="../replace_linear_by_hashing/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 10.4 Hash Optimization Strategy"
+    aria-label="Previous: 10.4   Optimizing with hashing"
     rel="prev"
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@@ -4834,7 +4906,7 @@ aria-label="Footer"
         Previous
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-        10.4 Hash Optimization Strategy
+        10.4 &nbsp; Optimizing with hashing
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@@ -4846,7 +4918,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 10.6 Summary"
+    aria-label="Next: 10.6   Summary"
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@@ -4854,7 +4926,7 @@ aria-label="Footer"
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       <div class="md-ellipsis">
-        10.6 Summary
+        10.6 &nbsp; Summary
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@@ -4871,7 +4943,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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index be787e95c..9e55742b8 100644
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       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -67,8 +67,8 @@
 
 
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-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
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           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
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   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2598,7 +2598,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2619,7 +2619,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2640,7 +2640,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2694,7 +2694,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2721,7 +2721,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2748,7 +2748,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2784,7 +2784,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2801,7 +2801,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2827,10 +2827,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
       </span>
     </span>
   </a>
@@ -2860,7 +2860,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2938,7 +2938,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2959,7 +2959,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2980,7 +2980,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3007,7 +3007,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3034,7 +3034,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3061,7 +3061,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3088,7 +3088,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3115,7 +3115,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3142,7 +3142,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3169,7 +3169,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3196,7 +3196,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3223,7 +3223,7 @@
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
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-  12.2 Divide and Conquer Search Strategy
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.6 Exercises
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  Chapter 14. Dynamic Programming
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-  14.1 Introduction to Dynamic Programming
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4436,7 +4436,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4463,7 +4463,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4547,6 +4547,78 @@
 
 
 
+
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@@ -4588,10 +4660,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
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-        1. &nbsp; Key Review
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@@ -4636,14 +4708,14 @@
 
 <!-- Page content -->
 <h1 id="106-summary">10.6 &nbsp; Summary<a class="headerlink" href="#106-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
 <ul>
-<li>Binary search relies on ordered data and searches by repeatedly halving the search interval. It requires the input data to be sorted and applies only to arrays or array-based data structures.</li>
-<li>Brute-force search locates data by traversing the data structure. Linear search applies to arrays and linked lists, while breadth-first search and depth-first search apply to graphs and trees. These algorithms are broadly applicable and require no data preprocessing, but their relatively high time complexity is <span class="arithmatex">\(O(n)\)</span>.</li>
-<li>Hash-based search, tree search, and binary search are efficient search methods that can quickly locate target elements in specific data structures. Such algorithms are highly efficient with time complexity reaching <span class="arithmatex">\(O(\log n)\)</span> or even <span class="arithmatex">\(O(1)\)</span>, but typically require additional data structures.</li>
-<li>In practice, we need to analyze factors such as data scale, search performance requirements, and data query and update frequency to choose the appropriate search method.</li>
-<li>Linear search is suitable for small datasets or data that is updated frequently; binary search is suitable for large sorted datasets; hash-based search is suitable when high query efficiency is required and range queries are unnecessary; tree search is suitable for large dynamic datasets that must maintain order and support range queries.</li>
-<li>Replacing linear search with hash-based search is a commonly used strategy to optimize runtime, reducing time complexity from <span class="arithmatex">\(O(n)\)</span> to <span class="arithmatex">\(O(1)\)</span>.</li>
+<li>Binary search relies on sorted data and repeatedly halves the search interval. It requires ordered input and applies only to arrays or data structures based on arrays.</li>
+<li>Brute-force search locates data by traversing a data structure. Linear search suits arrays and linked lists; breadth-first and depth-first search suit graphs and trees. These algorithms are broadly applicable and require no preprocessing, but their <span class="arithmatex">\(O(n)\)</span> time complexity is relatively high.</li>
+<li>Hash lookup, tree search, and binary search efficiently locate targets in particular data structures. Their time complexities can be <span class="arithmatex">\(O(\log n)\)</span> or even <span class="arithmatex">\(O(1)\)</span>, but they usually rely on additional data structures.</li>
+<li>In practice, choose a search method by considering dataset size, required search performance, query frequency, update frequency, and related factors.</li>
+<li>Linear search suits small or frequently updated datasets; binary search suits large, sorted datasets; hash lookup suits fast queries without range-query requirements; tree search suits large, dynamic datasets that need ordering and range queries.</li>
+<li>Replacing linear search with hash lookup is a common optimization that reduces lookup time from <span class="arithmatex">\(O(n)\)</span> to <span class="arithmatex">\(O(1)\)</span>.</li>
 </ul>
 
 <!-- Source file information -->
@@ -4667,7 +4739,7 @@ aria-label="Footer"
   <a
     href="../searching_algorithm_revisited/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 10.5 Searching Algorithms Revisited"
+    aria-label="Previous: 10.5   Search algorithms revisited"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4679,7 +4751,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        10.5 Searching Algorithms Revisited
+        10.5 &nbsp; Search algorithms revisited
       </div>
     </div>
   </a>
@@ -4691,7 +4763,7 @@ aria-label="Footer"
   <a
     href="../exercises/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 10.7 Exercises"
+    aria-label="Next: 10.7   Exercises"
     rel="next"
   >
     <div class="md-footer__title">
@@ -4699,7 +4771,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        10.7 Exercises
+        10.7 &nbsp; Exercises
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -4716,7 +4788,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>11.3 Bubble Sort - Hello Algo</title>
+        <title>11.3 Bubble sort - Hello Algo</title>
 
 
 
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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-  Chapter 1. Encounter with Algorithms
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
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-  1.3 Summary
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-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1265,7 +1265,7 @@
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-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
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@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2880,7 +2880,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2901,7 +2901,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2922,7 +2922,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2985,7 +2985,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3002,7 +3002,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3028,10 +3028,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1131-algorithm-flow" class="md-nav__link">
+  <a href="#1131-algorithm-steps" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.3.1 &nbsp; Algorithm Flow
+        11.3.1 &nbsp; Algorithm steps
       </span>
     </span>
   </a>
@@ -3039,10 +3039,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1132-efficiency-optimization" class="md-nav__link">
+  <a href="#1132-improving-efficiency" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.3.2 &nbsp; Efficiency Optimization
+        11.3.2 &nbsp; Improving efficiency
       </span>
     </span>
   </a>
@@ -3050,10 +3050,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1133-algorithm-characteristics" class="md-nav__link">
+  <a href="#1133-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.3.3 &nbsp; Algorithm Characteristics
+        11.3.3 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -3083,7 +3083,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3110,7 +3110,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3137,7 +3137,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3164,7 +3164,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3191,7 +3191,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3218,7 +3218,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3245,7 +3245,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3272,7 +3272,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3299,7 +3299,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3365,7 +3365,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3386,7 +3386,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3407,7 +3407,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3434,7 +3434,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
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   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3731,7 +3731,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3758,7 +3758,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3855,7 +3855,7 @@
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3897,7 +3897,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3924,7 +3924,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3951,7 +3951,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4152,7 +4152,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4275,7 +4275,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
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   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4458,7 +4458,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4485,7 +4485,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4569,6 +4569,78 @@
 
 
 
+
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19 2-5 4.5v11l5-4.5zM6.5 5C4.55 5 2.45 5.4 1 6.5v14.66c0 .25.25.5.5.5.1 0 .15-.07.25-.07 1.35-.65 3.3-1.09 4.75-1.09 1.95 0 4.05.4 5.5 1.5 1.35-.85 3.8-1.5 5.5-1.5 1.65 0 3.35.31 4.75 1.06.1.05.15.03.25.03.25 0 .5-.25.5-.5V6.5c-.6-.45-1.25-.75-2-1V19c-1.1-.35-2.3-.5-3.5-.5-1.7 0-4.15.65-5.5 1.5V6.5C10.55 5.4 8.45 5 6.5 5"/></svg>
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+  Print edition (Chinese)
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@@ -4610,10 +4682,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1131-algorithm-flow" class="md-nav__link">
+  <a href="#1131-algorithm-steps" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.3.1 &nbsp; Algorithm Flow
+        11.3.1 &nbsp; Algorithm steps
       </span>
     </span>
   </a>
@@ -4621,10 +4693,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1132-efficiency-optimization" class="md-nav__link">
+  <a href="#1132-improving-efficiency" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.3.2 &nbsp; Efficiency Optimization
+        11.3.2 &nbsp; Improving efficiency
       </span>
     </span>
   </a>
@@ -4632,10 +4704,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1133-algorithm-characteristics" class="md-nav__link">
+  <a href="#1133-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.3.3 &nbsp; Algorithm Characteristics
+        11.3.3 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -4679,13 +4751,13 @@
 
 
 <!-- Page content -->
-<h1 id="113-bubble-sort">11.3 &nbsp; Bubble Sort<a class="headerlink" href="#113-bubble-sort" title="Permanent link">&para;</a></h1>
-<p><u>Bubble sort</u> sorts an array by continuously comparing and swapping adjacent elements. This process resembles bubbles rising from the bottom to the top, hence the name bubble sort.</p>
-<p>As shown in Figure 11-4, the bubbling process can be simulated using element swaps: starting from the leftmost end of the array and traversing to the right, compare each pair of adjacent elements, and if "left element &gt; right element", swap them. After the traversal is complete, the largest element is moved to the rightmost end of the array.</p>
+<h1 id="113-bubble-sort">11.3 &nbsp; Bubble sort<a class="headerlink" href="#113-bubble-sort" title="Permanent link">&para;</a></h1>
+<p><u>Bubble sort</u> sorts by repeatedly comparing and swapping adjacent elements. The process resembles bubbles rising from the bottom to the top, which gives the algorithm its name.</p>
+<p>As shown below, we can simulate bubbling through swaps. Scan from the left end of the array to the right, comparing each pair of adjacent elements. If the left element is greater than the right, swap them. After the scan, the largest element has moved to the far right.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:7"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label><label for="__tabbed_1_6">&lt;6&gt;</label><label for="__tabbed_1_7">&lt;7&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Simulating bubble sort using element swaps" class="animation-figure" src="../bubble_sort.assets/bubble_operation_step1.png" /></p>
+<p><img alt="Simulating bubbling through element swaps" class="animation-figure" src="../bubble_sort.assets/bubble_operation_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="bubble_operation_step2" class="animation-figure" src="../bubble_sort.assets/bubble_operation_step2.png" /></p>
@@ -4707,29 +4779,29 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 11-4 &nbsp; Simulating bubble sort using element swaps </p>
+<p align="center"> Figure 11-4 &nbsp; Simulating bubbling through element swaps </p>
 
-<h2 id="1131-algorithm-flow">11.3.1 &nbsp; Algorithm Flow<a class="headerlink" href="#1131-algorithm-flow" title="Permanent link">&para;</a></h2>
-<p>Assume the array has length <span class="arithmatex">\(n\)</span>. The steps of bubble sort are shown in Figure 11-5.</p>
+<h2 id="1131-algorithm-steps">11.3.1 &nbsp; Algorithm steps<a class="headerlink" href="#1131-algorithm-steps" title="Permanent link">&para;</a></h2>
+<p>For an array of length <span class="arithmatex">\(n\)</span>, bubble sort proceeds as shown below.</p>
 <ol>
-<li>First, perform "bubbling" on <span class="arithmatex">\(n\)</span> elements, <strong>swapping the largest element of the array to its correct position</strong>.</li>
-<li>Next, perform "bubbling" on the remaining <span class="arithmatex">\(n - 1\)</span> elements, <strong>swapping the second largest element to its correct position</strong>.</li>
-<li>And so on. After <span class="arithmatex">\(n - 1\)</span> rounds of "bubbling", <strong>the largest <span class="arithmatex">\(n - 1\)</span> elements have all been swapped to their correct positions</strong>.</li>
-<li>The only remaining element must be the smallest element, requiring no sorting, so the array sorting is complete.</li>
+<li>First, perform a bubbling pass over all <span class="arithmatex">\(n\)</span> elements, <strong>moving the largest element to its correct position</strong>.</li>
+<li>Next, perform a pass over the remaining <span class="arithmatex">\(n - 1\)</span> elements, <strong>moving the second-largest element to its correct position</strong>.</li>
+<li>Continue in this way. After <span class="arithmatex">\(n - 1\)</span> passes, <strong>the largest <span class="arithmatex">\(n - 1\)</span> elements are all in their correct positions</strong>.</li>
+<li>The one remaining element must be the smallest and needs no further sorting. The array is now sorted.</li>
 </ol>
-<p><img alt="Bubble sort flow" class="animation-figure" src="../bubble_sort.assets/bubble_sort_overview.png" /></p>
-<p align="center"> Figure 11-5 &nbsp; Bubble sort flow </p>
+<p><img alt="The bubble sort process" class="animation-figure" src="../bubble_sort.assets/bubble_sort_overview.png" /></p>
+<p align="center"> Figure 11-5 &nbsp; The bubble sort process </p>
 
-<p>Example code is as follows:</p>
+<p>An implementation follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">bubble_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]):</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Bubble sort"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
-<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="c1"># Outer loop: unsorted interval is [0, i]</span>
+<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="c1"># Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="c1"># Inner loop: swap the largest element in the unsorted interval [0, i] to the rightmost end of the interval</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="c1"># Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>            <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]:</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>                <span class="c1"># Swap nums[j] and nums[j + 1]</span>
@@ -4739,13 +4811,13 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Bubble sort */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">bubbleSort</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
-<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">                </span><span class="c1">// Using std::swap() function here</span>
+<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">                </span><span class="c1">// Use the std::swap() function here</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">                </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="p">}</span>
@@ -4756,9 +4828,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Bubble sort */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">bubbleSort</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
@@ -4774,9 +4846,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Bubble sort */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">BubbleSort</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
@@ -4790,9 +4862,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Bubble sort */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">bubbleSort</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
@@ -4806,9 +4878,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Bubble sort */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">bubbleSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="bp">dropFirst</span><span class="p">().</span><span class="n">reversed</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
@@ -4822,9 +4894,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Bubble sort */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">bubbleSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
@@ -4840,9 +4912,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Bubble sort */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">bubbleSort</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
@@ -4858,9 +4930,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Bubble sort */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">bubbleSort</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">    </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">    </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">        </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
@@ -4876,9 +4948,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Bubble sort */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">bubble_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="o">..</span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()).</span><span class="n">rev</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">i</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
@@ -4892,9 +4964,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Bubble sort */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">bubbleSort</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">                </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
@@ -4909,9 +4981,9 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Bubble sort */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">bubbleSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="n">downTo</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
@@ -4928,9 +5000,9 @@
 <div class="highlight"><span class="filename">bubble_sort.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Bubble sort ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">bubble_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="c1"># Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="c1"># Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">downto</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">    </span><span class="c1"># Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">    </span><span class="c1"># Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">i</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">        </span><span class="c1"># Swap nums[j] and nums[j + 1]</span>
@@ -4943,45 +5015,50 @@
 </div>
 </div>
 </div>
-<h2 id="1132-efficiency-optimization">11.3.2 &nbsp; Efficiency Optimization<a class="headerlink" href="#1132-efficiency-optimization" title="Permanent link">&para;</a></h2>
-<p>We can observe that if no swaps occur during a round of "bubbling", the array is already sorted and the algorithm can return immediately. Therefore, we can add a flag <code>flag</code> to detect this situation and terminate as soon as it occurs.</p>
-<p>After this optimization, the worst-case and average-case time complexities of bubble sort remain <span class="arithmatex">\(O(n^2)\)</span>; however, when the input array is already sorted, the best-case time complexity becomes <span class="arithmatex">\(O(n)\)</span>.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 477px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20bubble_sort%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Bubble%20sort%22%22%22%0A%20%20%20%20n%20%3D%20len%28nums%29%0A%20%20%20%20%23%20Outer%20loop%3A%20the%20unsorted%20range%20is%20%5B0,%20i%5D%0A%20%20%20%20for%20i%20in%20range%28n%20-%201,%200,%20-1%29%3A%0A%20%20%20%20%20%20%20%20%23%20Inner%20loop%3A%20move%20the%20largest%20element%20in%20the%20unsorted%20range%20%5B0,%20i%5D%20to%20its%20right%20end%20through%20swaps%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28i%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20nums%5Bj%5D%20%3E%20nums%5Bj%20%2B%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Swap%20nums%5Bj%5D%20and%20nums%5Bj%20%2B%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bj%5D,%20nums%5Bj%20%2B%201%5D%20%3D%20nums%5Bj%20%2B%201%5D,%20nums%5Bj%5D%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B4,%201,%203,%201,%205,%202%5D%0A%20%20%20%20bubble_sort%28nums%29%0A%20%20%20%20print%28%22After%20bubble%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20bubble_sort%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Bubble%20sort%22%22%22%0A%20%20%20%20n%20%3D%20len%28nums%29%0A%20%20%20%20%23%20Outer%20loop%3A%20the%20unsorted%20range%20is%20%5B0,%20i%5D%0A%20%20%20%20for%20i%20in%20range%28n%20-%201,%200,%20-1%29%3A%0A%20%20%20%20%20%20%20%20%23%20Inner%20loop%3A%20move%20the%20largest%20element%20in%20the%20unsorted%20range%20%5B0,%20i%5D%20to%20its%20right%20end%20through%20swaps%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28i%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20nums%5Bj%5D%20%3E%20nums%5Bj%20%2B%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Swap%20nums%5Bj%5D%20and%20nums%5Bj%20%2B%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bj%5D,%20nums%5Bj%20%2B%201%5D%20%3D%20nums%5Bj%20%2B%201%5D,%20nums%5Bj%5D%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B4,%201,%203,%201,%205,%202%5D%0A%20%20%20%20bubble_sort%28nums%29%0A%20%20%20%20print%28%22After%20bubble%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1132-improving-efficiency">11.3.2 &nbsp; Improving efficiency<a class="headerlink" href="#1132-improving-efficiency" title="Permanent link">&para;</a></h2>
+<p>If a bubbling pass performs no swaps, the array is already sorted and we can return immediately. We can therefore add a variable <code>flag</code> to detect this situation and stop as soon as it occurs.</p>
+<p>With this optimization, the worst-case and average time complexities remain <span class="arithmatex">\(O(n^2)\)</span>. However, a fully sorted input achieves the best-case time complexity of <span class="arithmatex">\(O(n)\)</span>.</p>
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 <div class="highlight"><span class="filename">bubble_sort.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">bubble_sort_with_flag</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]):</span>
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Bubble sort (flag optimization)"""</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
-<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="c1"># Outer loop: unsorted interval is [0, i]</span>
+<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="c1"># Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>        <span class="n">flag</span> <span class="o">=</span> <span class="kc">False</span>  <span class="c1"># Initialize flag</span>
-<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="c1"># Inner loop: swap the largest element in the unsorted interval [0, i] to the rightmost end of the interval</span>
+<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>        <span class="n">flag</span> <span class="o">=</span> <span class="kc">False</span>  <span class="c1"># Initialize the flag</span>
+<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="c1"># Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">):</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>            <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]:</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>                <span class="c1"># Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>                <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
-<a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>                <span class="n">flag</span> <span class="o">=</span> <span class="kc">True</span>  <span class="c1"># Record element swap</span>
+<a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>                <span class="n">flag</span> <span class="o">=</span> <span class="kc">True</span>  <span class="c1"># Record that a swap occurred</span>
 <a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>        <span class="k">if</span> <span class="ow">not</span> <span class="n">flag</span><span class="p">:</span>
-<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>            <span class="k">break</span>  <span class="c1"># No elements were swapped in this round of "bubbling", exit directly</span>
+<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>            <span class="k">break</span>  <span class="c1"># Exit immediately if this pass made no swaps</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Bubble sort (flag optimization)*/</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">bubbleSortWithFlag</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">        </span><span class="kt">bool</span><span class="w"> </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">false</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize flag</span>
-<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">        </span><span class="kt">bool</span><span class="w"> </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">false</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the flag</span>
+<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
-<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">                </span><span class="c1">// Using std::swap() function here</span>
+<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">                </span><span class="c1">// Use the std::swap() function here</span>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">                </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
-<a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">                </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record element swap</span>
+<a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">                </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">true</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record that a swap occurred</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">flag</span><span class="p">)</span>
-<a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span><span class="w"> </span><span class="c1">// No elements were swapped in this round of "bubbling", exit directly</span>
+<a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span><span class="w"> </span><span class="c1">// Exit immediately if this pass made no swaps</span>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="p">}</span>
 </code></pre></div>
@@ -4989,59 +5066,59 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Bubble sort (flag optimization) */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">bubbleSortWithFlag</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">        </span><span class="kt">boolean</span><span class="w"> </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize flag</span>
-<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">        </span><span class="kt">boolean</span><span class="w"> </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the flag</span>
+<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">                </span><span class="kt">int</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">                </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">                </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
-<a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">                </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record element swap</span>
+<a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">                </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record that a swap occurred</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">flag</span><span class="p">)</span>
-<a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span><span class="w"> </span><span class="c1">// No elements were swapped in this round of "bubbling", exit directly</span>
+<a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span><span class="w"> </span><span class="c1">// Exit immediately if this pass made no swaps</span>
 <a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">bubble_sort.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Bubble sort (flag optimization) */</span>
+<div class="highlight"><span class="filename">bubble_sort.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Bubble sort (flag optimization)*/</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">BubbleSortWithFlag</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">        </span><span class="kt">bool</span><span class="w"> </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">false</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize flag</span>
-<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">        </span><span class="kt">bool</span><span class="w"> </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">false</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the flag</span>
+<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">                </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]);</span>
-<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">                </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">true</span><span class="p">;</span><span class="w">  </span><span class="c1">// Record element swap</span>
+<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">                </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">true</span><span class="p">;</span><span class="w">  </span><span class="c1">// Record that a swap occurred</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">flag</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span><span class="w">     </span><span class="c1">// No elements were swapped in this round of "bubbling", exit directly</span>
+<a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">flag</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span><span class="w">     </span><span class="c1">// Exit immediately if this pass made no swaps</span>
 <a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">bubble_sort.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Bubble sort (flag optimization) */</span>
+<div class="highlight"><span class="filename">bubble_sort.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Bubble sort (flag optimization)*/</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">bubbleSortWithFlag</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">        </span><span class="nx">flag</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="kc">false</span><span class="w"> </span><span class="c1">// Initialize flag</span>
-<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">        </span><span class="nx">flag</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="kc">false</span><span class="w"> </span><span class="c1">// Initialize the flag</span>
+<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">                </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="o">+</span><span class="mi">1</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span>
-<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">                </span><span class="nx">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="w"> </span><span class="c1">// Record element swap</span>
+<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">                </span><span class="nx">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="w"> </span><span class="c1">// Record that a swap occurred</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">flag</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">false</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// No elements were swapped in this round of "bubbling", exit directly</span>
+<a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">flag</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">false</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// Exit immediately if this pass made no swaps</span>
 <a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">    </span><span class="p">}</span>
@@ -5049,19 +5126,19 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">bubble_sort.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Bubble sort (flag optimization) */</span>
+<div class="highlight"><span class="filename">bubble_sort.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Bubble sort (flag optimization)*/</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">bubbleSortWithFlag</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="bp">dropFirst</span><span class="p">().</span><span class="n">reversed</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">flag</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">false</span><span class="w"> </span><span class="c1">// Initialize flag</span>
+<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">flag</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">false</span><span class="w"> </span><span class="c1">// Initialize the flag</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">                </span><span class="n">nums</span><span class="p">.</span><span class="n">swapAt</span><span class="p">(</span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">                </span><span class="n">flag</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">true</span><span class="w"> </span><span class="c1">// Record element swap</span>
+<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">                </span><span class="n">flag</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">true</span><span class="w"> </span><span class="c1">// Record that a swap occurred</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">flag</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// No elements were swapped in this round of "bubbling", exit directly</span>
+<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">flag</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// Exit immediately if this pass made no swaps</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">    </span><span class="p">}</span>
@@ -5069,64 +5146,64 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">bubble_sort.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Bubble sort (flag optimization) */</span>
+<div class="highlight"><span class="filename">bubble_sort.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Bubble sort (flag optimization)*/</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">bubbleSortWithFlag</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize flag</span>
-<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the flag</span>
+<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">                </span><span class="kd">let</span><span class="w"> </span><span class="nx">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">                </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">                </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span><span class="p">;</span>
-<a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">                </span><span class="nx">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record element swap</span>
+<a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">                </span><span class="nx">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record that a swap occurred</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="nx">flag</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span><span class="w"> </span><span class="c1">// No elements were swapped in this round of "bubbling", exit directly</span>
+<a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="nx">flag</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span><span class="w"> </span><span class="c1">// Exit immediately if this pass made no swaps</span>
 <a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">bubble_sort.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Bubble sort (flag optimization) */</span>
+<div class="highlight"><span class="filename">bubble_sort.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Bubble sort (flag optimization)*/</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">bubbleSortWithFlag</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize flag</span>
-<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the flag</span>
+<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">                </span><span class="kd">let</span><span class="w"> </span><span class="nx">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">                </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">                </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span><span class="p">;</span>
-<a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">                </span><span class="nx">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record element swap</span>
+<a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">                </span><span class="nx">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record that a swap occurred</span>
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="nx">flag</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span><span class="w"> </span><span class="c1">// No elements were swapped in this round of "bubbling", exit directly</span>
+<a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="nx">flag</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span><span class="w"> </span><span class="c1">// Exit immediately if this pass made no swaps</span>
 <a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">bubble_sort.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Bubble sort (flag optimization) */</span>
+<div class="highlight"><span class="filename">bubble_sort.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Bubble sort (flag optimization)*/</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">bubbleSortWithFlag</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">    </span><span class="kt">bool</span><span class="w"> </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize flag</span>
-<a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">    </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">    </span><span class="kt">bool</span><span class="w"> </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the flag</span>
+<a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">    </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">        </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
-<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">        </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record element swap</span>
+<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">        </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record that a swap occurred</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">      </span><span class="p">}</span>
 <a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">flag</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span><span class="w"> </span><span class="c1">// No elements were swapped in this round of "bubbling", exit directly</span>
+<a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">flag</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span><span class="w"> </span><span class="c1">// Exit immediately if this pass made no swaps</span>
 <a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="p">}</span>
 </code></pre></div>
@@ -5134,31 +5211,31 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Bubble sort (flag optimization) */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">bubble_sort_with_flag</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="o">..</span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()).</span><span class="n">rev</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize flag</span>
-<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="p">;</span><span class="w"> </span><span class="c1">// Initialize the flag</span>
+<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">i</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">                </span><span class="n">nums</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">                </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record element swap</span>
+<a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">                </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record that a swap occurred</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">flag</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span><span class="w"> </span><span class="c1">// No elements were swapped in this round of "bubbling", exit directly</span>
+<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span><span class="w"> </span><span class="c1">// Exit immediately if this pass made no swaps</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">        </span><span class="p">};</span>
 <a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">bubble_sort.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Bubble sort (flag optimization) */</span>
+<div class="highlight"><span class="filename">bubble_sort.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Bubble sort (flag optimization)*/</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">bubbleSortWithFlag</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">        </span><span class="kt">bool</span><span class="w"> </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">false</span><span class="p">;</span>
-<a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">                </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
@@ -5176,53 +5253,58 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bubble_sort.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Bubble sort (flag optimization) */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">bubbleSortWithFlag</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="n">downTo</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="w"> </span><span class="c1">// Initialize flag</span>
-<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">        </span><span class="c1">// Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">false</span><span class="w"> </span><span class="c1">// Initialize the flag</span>
+<a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">        </span><span class="c1">// Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">                </span><span class="c1">// Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">                </span><span class="kd">val</span><span class="w"> </span><span class="nv">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">                </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="o">]</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">                </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span>
-<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">                </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="w"> </span><span class="c1">// Record element swap</span>
+<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">                </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">true</span><span class="w"> </span><span class="c1">// Record that a swap occurred</span>
 <a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">flag</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="w"> </span><span class="c1">// No elements were swapped in this round of "bubbling", exit directly</span>
+<a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">flag</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="w"> </span><span class="c1">// Exit immediately if this pass made no swaps</span>
 <a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">bubble_sort.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Bubble sort (flag optimization) ###</span>
+<div class="highlight"><span class="filename">bubble_sort.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Bubble sort (flag optimization)###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">bubble_sort_with_flag</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="c1"># Outer loop: unsorted range is [0, i]</span>
+<a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="c1"># Outer loop: the unsorted range is [0, i]</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">downto</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">    </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">false</span><span class="w"> </span><span class="c1"># Initialize flag</span>
+<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">    </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">false</span><span class="w"> </span><span class="c1"># Initialize the flag</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a>
-<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">    </span><span class="c1"># Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range</span>
+<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">    </span><span class="c1"># Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">i</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">        </span><span class="c1"># Swap nums[j] and nums[j + 1]</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span>
-<a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">        </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">true</span><span class="w"> </span><span class="c1"># Record element swap</span>
+<a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">        </span><span class="n">flag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">true</span><span class="w"> </span><span class="c1"># Record that a swap occurred</span>
 <a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">      </span><span class="k">end</span>
 <a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a>
-<a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w">    </span><span class="k">break</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="n">flag</span><span class="w"> </span><span class="c1"># No elements were swapped in this round of "bubbling", exit directly</span>
+<a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w">    </span><span class="k">break</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="n">flag</span><span class="w"> </span><span class="c1"># Exit immediately if this pass made no swaps</span>
 <a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h2 id="1133-algorithm-characteristics">11.3.3 &nbsp; Algorithm Characteristics<a class="headerlink" href="#1133-algorithm-characteristics" title="Permanent link">&para;</a></h2>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20bubble_sort_with_flag%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Bubble%20sort%20%28flag%20optimization%29%22%22%22%0A%20%20%20%20n%20%3D%20len%28nums%29%0A%20%20%20%20%23%20Outer%20loop%3A%20the%20unsorted%20range%20is%20%5B0,%20i%5D%0A%20%20%20%20for%20i%20in%20range%28n%20-%201,%200,%20-1%29%3A%0A%20%20%20%20%20%20%20%20flag%20%3D%20False%20%20%23%20Initialize%20the%20flag%0A%20%20%20%20%20%20%20%20%23%20Inner%20loop%3A%20move%20the%20largest%20element%20in%20the%20unsorted%20range%20%5B0,%20i%5D%20to%20its%20right%20end%20through%20swaps%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28i%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20nums%5Bj%5D%20%3E%20nums%5Bj%20%2B%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Swap%20nums%5Bj%5D%20and%20nums%5Bj%20%2B%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bj%5D,%20nums%5Bj%20%2B%201%5D%20%3D%20nums%5Bj%20%2B%201%5D,%20nums%5Bj%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20flag%20%3D%20True%20%20%23%20Record%20that%20a%20swap%20occurred%0A%20%20%20%20%20%20%20%20if%20not%20flag%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20break%20%20%23%20Exit%20immediately%20if%20this%20pass%20made%20no%20swaps%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B4,%201,%203,%201,%205,%202%5D%0A%20%20%20%20bubble_sort_with_flag%28nums%29%0A%20%20%20%20print%28%22After%20bubble%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20bubble_sort_with_flag%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Bubble%20sort%20%28flag%20optimization%29%22%22%22%0A%20%20%20%20n%20%3D%20len%28nums%29%0A%20%20%20%20%23%20Outer%20loop%3A%20the%20unsorted%20range%20is%20%5B0,%20i%5D%0A%20%20%20%20for%20i%20in%20range%28n%20-%201,%200,%20-1%29%3A%0A%20%20%20%20%20%20%20%20flag%20%3D%20False%20%20%23%20Initialize%20the%20flag%0A%20%20%20%20%20%20%20%20%23%20Inner%20loop%3A%20move%20the%20largest%20element%20in%20the%20unsorted%20range%20%5B0,%20i%5D%20to%20its%20right%20end%20through%20swaps%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28i%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20nums%5Bj%5D%20%3E%20nums%5Bj%20%2B%201%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Swap%20nums%5Bj%5D%20and%20nums%5Bj%20%2B%201%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bj%5D,%20nums%5Bj%20%2B%201%5D%20%3D%20nums%5Bj%20%2B%201%5D,%20nums%5Bj%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20flag%20%3D%20True%20%20%23%20Record%20that%20a%20swap%20occurred%0A%20%20%20%20%20%20%20%20if%20not%20flag%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20break%20%20%23%20Exit%20immediately%20if%20this%20pass%20made%20no%20swaps%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B4,%201,%203,%201,%205,%202%5D%0A%20%20%20%20bubble_sort_with_flag%28nums%29%0A%20%20%20%20print%28%22After%20bubble%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1133-algorithm-properties">11.3.3 &nbsp; Algorithm properties<a class="headerlink" href="#1133-algorithm-properties" title="Permanent link">&para;</a></h2>
 <ul>
-<li><strong>Time complexity is <span class="arithmatex">\(O(n^2)\)</span>; adaptive</strong>: In successive rounds of "bubbling", the traversed portion of the array has lengths <span class="arithmatex">\(n - 1\)</span>, <span class="arithmatex">\(n - 2\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, <span class="arithmatex">\(1\)</span>, for a total of <span class="arithmatex">\((n - 1) n / 2\)</span>. After introducing the <code>flag</code> optimization, the best-case time complexity can reach <span class="arithmatex">\(O(n)\)</span>.</li>
-<li><strong>Space complexity of <span class="arithmatex">\(O(1)\)</span>, in-place sorting</strong>: Pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> use a constant amount of extra space.</li>
-<li><strong>Stable sorting</strong>: Equal elements are not swapped during "bubbling".</li>
+<li><strong>Time complexity: <span class="arithmatex">\(O(n^2)\)</span>; adaptive</strong>. Successive passes traverse <span class="arithmatex">\(n - 1\)</span>, <span class="arithmatex">\(n - 2\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, and <span class="arithmatex">\(1\)</span> positions, totaling <span class="arithmatex">\((n - 1) n / 2\)</span>. With the <code>flag</code> optimization, the best-case time complexity is <span class="arithmatex">\(O(n)\)</span>.</li>
+<li><strong>Space complexity: <span class="arithmatex">\(O(1)\)</span>; in-place</strong>. Pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> use constant extra space.</li>
+<li><strong>Stable</strong>. Equal elements are not swapped during a bubbling pass.</li>
 </ul>
 
 <!-- Source file information -->
@@ -5246,7 +5328,7 @@ aria-label="Footer"
   <a
     href="../selection_sort/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 11.2 Selection Sort"
+    aria-label="Previous: 11.2   Selection sort"
     rel="prev"
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     <div class="md-footer__button md-icon">
@@ -5258,7 +5340,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        11.2 Selection Sort
+        11.2 &nbsp; Selection sort
       </div>
     </div>
   </a>
@@ -5270,7 +5352,7 @@ aria-label="Footer"
   <a
     href="../insertion_sort/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 11.4 Insertion Sort"
+    aria-label="Next: 11.4   Insertion sort"
     rel="next"
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     <div class="md-footer__title">
@@ -5278,7 +5360,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        11.4 Insertion Sort
+        11.4 &nbsp; Insertion sort
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -5295,7 +5377,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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diff --git a/en/chapter_sorting/bucket_sort/index.html b/en/chapter_sorting/bucket_sort/index.html
index 470b8e83d..b511f05ce 100644
--- a/en/chapter_sorting/bucket_sort/index.html
+++ b/en/chapter_sorting/bucket_sort/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>11.8 Bucket Sort - Hello Algo</title>
+        <title>11.8 Bucket sort - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
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-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              11.8 &nbsp; Bucket Sort
+              11.8 &nbsp; Bucket sort
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Dark mode"  type="radio" name="__palette" id="__palette_0">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -177,9 +177,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Light mode"  type="radio" name="__palette" id="__palette_1">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1917,7 +1917,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2880,7 +2880,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2901,7 +2901,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2922,7 +2922,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2976,7 +2976,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3003,7 +3003,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3030,7 +3030,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3057,7 +3057,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3084,7 +3084,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3120,7 +3120,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3137,7 +3137,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3163,10 +3163,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1181-algorithm-flow" class="md-nav__link">
+  <a href="#1181-algorithm-steps" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.8.1 &nbsp; Algorithm Flow
+        11.8.1 &nbsp; Algorithm steps
       </span>
     </span>
   </a>
@@ -3174,10 +3174,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1182-algorithm-characteristics" class="md-nav__link">
+  <a href="#1182-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.8.2 &nbsp; Algorithm Characteristics
+        11.8.2 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -3185,10 +3185,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1183-how-to-achieve-even-distribution" class="md-nav__link">
+  <a href="#1183-distributing-elements-evenly" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.8.3 &nbsp; How to Achieve Even Distribution
+        11.8.3 &nbsp; Distributing elements evenly
       </span>
     </span>
   </a>
@@ -3218,7 +3218,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3245,7 +3245,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3272,7 +3272,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3299,7 +3299,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3365,7 +3365,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3386,7 +3386,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3407,7 +3407,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3434,7 +3434,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3461,7 +3461,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3488,7 +3488,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3515,7 +3515,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3542,7 +3542,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3608,7 +3608,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3629,7 +3629,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3650,7 +3650,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3677,7 +3677,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3704,7 +3704,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3731,7 +3731,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3758,7 +3758,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3785,7 +3785,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3855,7 +3855,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3876,7 +3876,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3897,7 +3897,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3924,7 +3924,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3951,7 +3951,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3978,7 +3978,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4005,7 +4005,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4032,7 +4032,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4059,7 +4059,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4086,7 +4086,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4152,7 +4152,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4173,7 +4173,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4194,7 +4194,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4221,7 +4221,7 @@
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4248,7 +4248,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4302,7 +4302,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4431,7 +4431,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4458,7 +4458,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4569,6 +4569,78 @@
 
 
 
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@@ -4610,10 +4682,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1181-algorithm-flow" class="md-nav__link">
+  <a href="#1181-algorithm-steps" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.8.1 &nbsp; Algorithm Flow
+        11.8.1 &nbsp; Algorithm steps
       </span>
     </span>
   </a>
@@ -4621,10 +4693,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1182-algorithm-characteristics" class="md-nav__link">
+  <a href="#1182-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.8.2 &nbsp; Algorithm Characteristics
+        11.8.2 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -4632,10 +4704,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1183-how-to-achieve-even-distribution" class="md-nav__link">
+  <a href="#1183-distributing-elements-evenly" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.8.3 &nbsp; How to Achieve Even Distribution
+        11.8.3 &nbsp; Distributing elements evenly
       </span>
     </span>
   </a>
@@ -4679,18 +4751,18 @@
 
 
 <!-- Page content -->
-<h1 id="118-bucket-sort">11.8 &nbsp; Bucket Sort<a class="headerlink" href="#118-bucket-sort" title="Permanent link">&para;</a></h1>
-<p>The sorting algorithms discussed earlier are all comparison-based sorting algorithms, which sort by comparing the relative order of elements. The worst-case time complexity of such algorithms has a lower bound of <span class="arithmatex">\(\Omega(n \log n)\)</span>. Next, we will explore several non-comparison sorting algorithms, whose time complexity can be linear.</p>
-<p><u>Bucket sort</u> is a typical application of the divide-and-conquer strategy. It works by creating a sequence of ordered buckets, each corresponding to a data range, and distributing the data evenly among them. The elements within each bucket are then sorted separately. Finally, all buckets are merged in order.</p>
-<h2 id="1181-algorithm-flow">11.8.1 &nbsp; Algorithm Flow<a class="headerlink" href="#1181-algorithm-flow" title="Permanent link">&para;</a></h2>
-<p>Consider an array of length <span class="arithmatex">\(n\)</span>, whose elements are floating-point numbers in the range <span class="arithmatex">\([0, 1)\)</span>. The flow of bucket sort is shown in Figure 11-13.</p>
+<h1 id="118-bucket-sort">11.8 &nbsp; Bucket sort<a class="headerlink" href="#118-bucket-sort" title="Permanent link">&para;</a></h1>
+<p>The sorting algorithms discussed so far are “comparison-based”: they sort by comparing elements. Their worst-case time complexity has a lower bound of <span class="arithmatex">\(\Omega(n \log n)\)</span>. We now turn to several “non-comparison” sorting algorithms, which can achieve linear time complexity.</p>
+<p><u>Bucket sort</u> is a classic application of divide and conquer. It creates ordered buckets, each corresponding to a range of values, and distributes the data evenly among them. It then sorts each bucket separately and combines the results in bucket order.</p>
+<h2 id="1181-algorithm-steps">11.8.1 &nbsp; Algorithm steps<a class="headerlink" href="#1181-algorithm-steps" title="Permanent link">&para;</a></h2>
+<p>Consider an array of length <span class="arithmatex">\(n\)</span> containing floating-point values in <span class="arithmatex">\([0, 1)\)</span>. Bucket sort proceeds as shown below.</p>
 <ol>
-<li>Initialize <span class="arithmatex">\(k\)</span> buckets and distribute the <span class="arithmatex">\(n\)</span> elements into the <span class="arithmatex">\(k\)</span> buckets.</li>
-<li>Sort each bucket separately (here we use the built-in sorting function of the programming language).</li>
-<li>Merge the results in order from smallest to largest bucket.</li>
+<li>Initialize <span class="arithmatex">\(k\)</span> buckets and distribute the <span class="arithmatex">\(n\)</span> elements among the <span class="arithmatex">\(k\)</span> buckets.</li>
+<li>Sort each bucket separately, using the programming language's built-in sorting function here.</li>
+<li>Combine the results in ascending bucket order.</li>
 </ol>
-<p><img alt="Bucket sort algorithm flow" class="animation-figure" src="../bucket_sort.assets/bucket_sort_overview.png" /></p>
-<p align="center"> Figure 11-13 &nbsp; Bucket sort algorithm flow </p>
+<p><img alt="The bucket sort process" class="animation-figure" src="../bucket_sort.assets/bucket_sort_overview.png" /></p>
+<p align="center"> Figure 11-13 &nbsp; The bucket sort process </p>
 
 <p>The code is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
@@ -4698,20 +4770,20 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bucket_sort.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">bucket_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">float</span><span class="p">]):</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Bucket sort"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Initialize k = n/2 buckets, expected to allocate 2 elements per bucket</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Initialize k = n/2 buckets, expecting 2 elements per bucket</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="n">k</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="n">buckets</span> <span class="o">=</span> <span class="p">[[]</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">k</span><span class="p">)]</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># 1. Distribute array elements into various buckets</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># 1. Distribute the array elements among the buckets</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="k">for</span> <span class="n">num</span> <span class="ow">in</span> <span class="n">nums</span><span class="p">:</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="c1"># Input data range is [0, 1), use num * k to map to index range [0, k-1]</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="c1"># The input range is [0, 1); use num * k to map each value to an index in [0, k-1]</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="n">i</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">num</span> <span class="o">*</span> <span class="n">k</span><span class="p">)</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="c1"># Add num to bucket i</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="c1"># 2. Sort each bucket</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>    <span class="k">for</span> <span class="n">bucket</span> <span class="ow">in</span> <span class="n">buckets</span><span class="p">:</span>
-<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>        <span class="c1"># Use built-in sorting function, can also replace with other sorting algorithms</span>
+<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>        <span class="c1"># Use the built-in sorting function, or replace it with another sorting algorithm</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>        <span class="n">bucket</span><span class="o">.</span><span class="n">sort</span><span class="p">()</span>
-<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>    <span class="c1"># 3. Traverse buckets to merge results</span>
+<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>    <span class="c1"># 3. Traverse the buckets to combine the results</span>
 <a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>    <span class="k">for</span> <span class="n">bucket</span> <span class="ow">in</span> <span class="n">buckets</span><span class="p">:</span>
 <a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>        <span class="k">for</span> <span class="n">num</span> <span class="ow">in</span> <span class="n">bucket</span><span class="p">:</span>
@@ -4722,22 +4794,22 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bucket_sort.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Bucket sort */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">bucketSort</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">float</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Initialize k = n/2 buckets, expected to allocate 2 elements per bucket</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Initialize k = n/2 buckets, expecting 2 elements per bucket</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">float</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">buckets</span><span class="p">(</span><span class="n">k</span><span class="p">);</span>
-<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// 1. Distribute array elements into various buckets</span>
+<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// 1. Distribute the array elements among the buckets</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="c1">// Input data range is [0, 1), use num * k to map to index range [0, k-1]</span>
+<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="c1">// The input range is [0, 1); use num * k to map each value to an index in [0, k-1]</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">k</span><span class="p">;</span>
-<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="c1">// Add num to bucket bucket_idx</span>
+<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="c1">// Add num to bucket i</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">push_back</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">    </span><span class="c1">// 2. Sort each bucket</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">float</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">bucket</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">        </span><span class="c1">// Use built-in sorting function, can also replace with other sorting algorithms</span>
+<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">        </span><span class="c1">// Use the built-in sorting function, or replace it with another sorting algorithm</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">        </span><span class="n">sort</span><span class="p">(</span><span class="n">bucket</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span><span class="w"> </span><span class="n">bucket</span><span class="p">.</span><span class="n">end</span><span class="p">());</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">    </span><span class="c1">// 3. Traverse buckets to merge results</span>
+<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">    </span><span class="c1">// 3. Traverse the buckets to combine the results</span>
 <a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">float</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">bucket</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4750,25 +4822,25 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bucket_sort.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Bucket sort */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">bucketSort</span><span class="p">(</span><span class="kt">float</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Initialize k = n/2 buckets, expected to allocate 2 elements per bucket</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Initialize k = n/2 buckets, expecting 2 elements per bucket</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Float</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">        </span><span class="n">buckets</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">());</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="c1">// 1. Distribute array elements into various buckets</span>
+<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="c1">// 1. Distribute the array elements among the buckets</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="c1">// Input data range is [0, 1), use num * k to map to index range [0, k-1]</span>
+<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="c1">// The input range is [0, 1); use num * k to map each value to an index in [0, k-1]</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">k</span><span class="p">);</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="c1">// Add num to bucket i</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">        </span><span class="n">buckets</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">i</span><span class="p">).</span><span class="na">add</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">    </span><span class="c1">// 2. Sort each bucket</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Float</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">        </span><span class="c1">// Use built-in sorting function, can also replace with other sorting algorithms</span>
+<a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">        </span><span class="c1">// Use the built-in sorting function, or replace it with another sorting algorithm</span>
 <a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">        </span><span class="n">Collections</span><span class="p">.</span><span class="na">sort</span><span class="p">(</span><span class="n">bucket</span><span class="p">);</span>
 <a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">    </span><span class="c1">// 3. Traverse buckets to merge results</span>
+<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">    </span><span class="c1">// 3. Traverse the buckets to combine the results</span>
 <a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="n">Float</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4781,25 +4853,25 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bucket_sort.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Bucket sort */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">BucketSort</span><span class="p">(</span><span class="kt">float</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Initialize k = n/2 buckets, expected to allocate 2 elements per bucket</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Initialize k = n/2 buckets, expecting 2 elements per bucket</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">float</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">        </span><span class="n">buckets</span><span class="p">.</span><span class="n">Add</span><span class="p">([]);</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="c1">// 1. Distribute array elements into various buckets</span>
+<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="c1">// 1. Distribute the array elements among the buckets</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="c1">// Input data range is [0, 1), use num * k to map to index range [0, k-1]</span>
+<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="c1">// The input range is [0, 1); use num * k to map each value to an index in [0, k-1]</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="p">)(</span><span class="n">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">k</span><span class="p">);</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="c1">// Add num to bucket i</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">Add</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">    </span><span class="c1">// 2. Sort each bucket</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">float</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">        </span><span class="c1">// Use built-in sorting function, can also replace with other sorting algorithms</span>
+<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">        </span><span class="c1">// Use the built-in sorting function, or replace it with another sorting algorithm</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">        </span><span class="n">bucket</span><span class="p">.</span><span class="n">Sort</span><span class="p">();</span>
 <a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w">    </span><span class="c1">// 3. Traverse buckets to merge results</span>
+<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w">    </span><span class="c1">// 3. Traverse the buckets to combine the results</span>
 <a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">float</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w">        </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4812,25 +4884,25 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bucket_sort.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Bucket sort */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">bucketSort</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">float64</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Initialize k = n/2 buckets, expected to allocate 2 elements per bucket</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Initialize k = floor(n/2) buckets, expecting about 2 elements per bucket for uniformly distributed input</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">k</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="nx">buckets</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([][]</span><span class="kt">float64</span><span class="p">,</span><span class="w"> </span><span class="nx">k</span><span class="p">)</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">k</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">float64</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// 1. Distribute array elements into various buckets</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// 1. Distribute the array elements among the buckets</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="c1">// Input data range is [0, 1), use num * k to map to index range [0, k-1]</span>
+<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="c1">// The input range is [0, 1); use num * k to map each value to an index in [0, k-1]</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="kt">int</span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="kt">float64</span><span class="p">(</span><span class="nx">k</span><span class="p">))</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">        </span><span class="c1">// Add num to bucket i</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="c1">// 2. Sort each bucket</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">k</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">        </span><span class="c1">// Use built-in slice sorting function, can also be replaced with other sorting algorithms</span>
+<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">        </span><span class="c1">// Use the standard slice-sorting function, or replace it with another sorting algorithm</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">        </span><span class="nx">sort</span><span class="p">.</span><span class="nx">Float64s</span><span class="p">(</span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">])</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">    </span><span class="c1">// 3. Traverse buckets to merge results</span>
+<a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">    </span><span class="c1">// 3. Traverse the buckets to combine the results</span>
 <a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">    </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">buckets</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="p">{</span>
@@ -4844,22 +4916,22 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bucket_sort.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Bucket sort */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">bucketSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Double</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Initialize k = n/2 buckets, expected to allocate 2 elements per bucket</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Initialize k = n/2 buckets, expecting 2 elements per bucket</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">k</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">count</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">buckets</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">k</span><span class="p">).</span><span class="bp">map</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">[</span><span class="nb">Double</span><span class="p">]()</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// 1. Distribute array elements into various buckets</span>
+<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// 1. Distribute the array elements among the buckets</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">        </span><span class="c1">// Input data range is [0, 1), use num * k to map to index range [0, k-1]</span>
+<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">        </span><span class="c1">// The input range is [0, 1); use num * k to map each value to an index in [0, k-1]</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Int</span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nb">Double</span><span class="p">(</span><span class="n">k</span><span class="p">))</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="c1">// Add num to bucket i</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">append</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">    </span><span class="c1">// 2. Sort each bucket</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">.</span><span class="bp">indices</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="c1">// Use built-in sorting function, can also replace with other sorting algorithms</span>
+<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="c1">// Use the built-in sorting function, or replace it with another sorting algorithm</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="bp">sort</span><span class="p">()</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">    </span><span class="c1">// 3. Traverse buckets to merge results</span>
+<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">    </span><span class="c1">// 3. Traverse the buckets to combine the results</span>
 <a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">startIndex</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="p">{</span>
@@ -4873,25 +4945,25 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bucket_sort.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Bucket sort */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">bucketSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Initialize k = n/2 buckets, expected to allocate 2 elements per bucket</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Initialize ceil(n/2) buckets, aiming for about 2 elements per bucket</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">k</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">        </span><span class="nx">buckets</span><span class="p">.</span><span class="nx">push</span><span class="p">([]);</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="c1">// 1. Distribute array elements into various buckets</span>
+<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="c1">// 1. Distribute the array elements among the buckets</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="c1">// Input data range is [0, 1), use num * k to map to index range [0, k-1]</span>
+<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="c1">// The input range is [0, 1); floor(num * k) maps values to indices from 0 to ceil(k)-1</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">k</span><span class="p">);</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="c1">// Add num to bucket i</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">].</span><span class="nx">push</span><span class="p">(</span><span class="nx">num</span><span class="p">);</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">    </span><span class="c1">// 2. Sort each bucket</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">        </span><span class="c1">// Use built-in sorting function, can also replace with other sorting algorithms</span>
+<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">        </span><span class="c1">// Use the built-in sorting function, or replace it with another sorting algorithm</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">        </span><span class="nx">bucket</span><span class="p">.</span><span class="nx">sort</span><span class="p">((</span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">b</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">b</span><span class="p">);</span>
 <a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">    </span><span class="c1">// 3. Traverse buckets to merge results</span>
+<a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">    </span><span class="c1">// 3. Traverse the buckets to combine the results</span>
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4904,25 +4976,25 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bucket_sort.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Bucket sort */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">bucketSort</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Initialize k = n/2 buckets, expected to allocate 2 elements per bucket</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Set k = n/2 and create ceil(k) buckets, aiming for about 2 elements per bucket</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">buckets</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[][]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">k</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">        </span><span class="nx">buckets</span><span class="p">.</span><span class="nx">push</span><span class="p">([]);</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// 1. Distribute array elements into various buckets</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// 1. Distribute the array elements among the buckets</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="c1">// Input data range is [0, 1), use num * k to map to index range [0, k-1]</span>
+<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="c1">// The input range is [0, 1); floor(num * k) maps each value to an index in [0, ceil(k)-1]</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">k</span><span class="p">);</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="c1">// Add num to bucket i</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="nx">buckets</span><span class="p">[</span><span class="nx">i</span><span class="p">].</span><span class="nx">push</span><span class="p">(</span><span class="nx">num</span><span class="p">);</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="c1">// 2. Sort each bucket</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">        </span><span class="c1">// Use built-in sorting function, can also replace with other sorting algorithms</span>
+<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">        </span><span class="c1">// Use the built-in sorting function, or replace it with another sorting algorithm</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">        </span><span class="nx">bucket</span><span class="p">.</span><span class="nx">sort</span><span class="p">((</span><span class="nx">a</span><span class="p">,</span><span class="w"> </span><span class="nx">b</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="nx">a</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">b</span><span class="p">);</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">    </span><span class="c1">// 3. Traverse buckets to merge results</span>
+<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">    </span><span class="c1">// 3. Traverse the buckets to combine the results</span>
 <a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">bucket</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4935,22 +5007,22 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bucket_sort.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Bucket sort */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">bucketSort</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">double</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Initialize k = n/2 buckets, expected to allocate 2 elements per bucket</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Initialize k = floor(n/2) buckets, aiming for about 2 elements per bucket</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">;</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">double</span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">generate</span><span class="p">(</span><span class="n">k</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">[]);</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a>
-<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// 1. Distribute array elements into various buckets</span>
+<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="c1">// 1. Distribute the array elements among the buckets</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="c1">// Input data range is [0, 1), use _num * k to map to index range [0, k-1]</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="c1">// The input range is [0, 1); use _num * k to map each value to an index in [0, k-1]</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">_num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">k</span><span class="p">).</span><span class="n">toInt</span><span class="p">();</span>
-<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="c1">// Add _num to bucket bucket_idx</span>
+<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="c1">// Add _num to bucket i</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">add</span><span class="p">(</span><span class="n">_num</span><span class="p">);</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">  </span><span class="c1">// 2. Sort each bucket</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">double</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">    </span><span class="n">bucket</span><span class="p">.</span><span class="n">sort</span><span class="p">();</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">  </span><span class="c1">// 3. Traverse buckets to merge results</span>
+<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">  </span><span class="c1">// 3. Traverse the buckets to combine the results</span>
 <a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">double</span><span class="o">&gt;</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">double</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4963,22 +5035,22 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bucket_sort.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Bucket sort */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">bucket_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">f64</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Initialize k = n/2 buckets, expected to allocate 2 elements per bucket</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Initialize k = n/2 buckets, expecting 2 elements per bucket</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="fm">vec!</span><span class="p">[];</span><span class="w"> </span><span class="n">k</span><span class="p">];</span>
-<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// 1. Distribute array elements into various buckets</span>
+<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// 1. Distribute the array elements among the buckets</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="o">&amp;</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">iter</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">        </span><span class="c1">// Input data range is [0, 1), use num * k to map to index range [0, k-1]</span>
+<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">        </span><span class="c1">// The input range is [0, 1); use num * k to map each value to an index in [0, k-1]</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">f64</span><span class="p">)</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">;</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="c1">// Add num to bucket i</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">].</span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">    </span><span class="c1">// 2. Sort each bucket</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="n">buckets</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">        </span><span class="c1">// Use built-in sorting function, can also replace with other sorting algorithms</span>
+<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">        </span><span class="c1">// Use the built-in sorting function, or replace it with another sorting algorithm</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">        </span><span class="n">bucket</span><span class="p">.</span><span class="n">sort_by</span><span class="p">(</span><span class="o">|</span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="o">|</span><span class="w"> </span><span class="n">a</span><span class="p">.</span><span class="n">partial_cmp</span><span class="p">(</span><span class="n">b</span><span class="p">).</span><span class="n">unwrap</span><span class="p">());</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">    </span><span class="c1">// 3. Traverse buckets to merge results</span>
+<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">    </span><span class="c1">// 3. Traverse the buckets to combine the results</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">.</span><span class="n">iter</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="o">&amp;</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="p">.</span><span class="n">iter</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
@@ -4993,14 +5065,14 @@
 <div class="highlight"><span class="filename">bucket_sort.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Bucket sort */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">bucketSort</span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">                                 </span><span class="c1">// Initialize k = n/2 buckets</span>
-<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">sizes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="n">k</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span><span class="w">          </span><span class="c1">// Record each bucket's size</span>
+<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">sizes</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="n">k</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span><span class="w">          </span><span class="c1">// Record the size of each bucket</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="kt">float</span><span class="w"> </span><span class="o">**</span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="n">k</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="o">*</span><span class="p">));</span><span class="w"> </span><span class="c1">// Array of dynamic arrays (buckets)</span>
-<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="c1">// Pre-allocate sufficient space for each bucket</span>
+<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="c1">// Preallocate enough space for each bucket</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">float</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">float</span><span class="p">));</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">        </span><span class="n">sizes</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// 1. Distribute array elements into various buckets</span>
+<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// 1. Distribute the array elements among the buckets</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">idx</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="p">)(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">k</span><span class="p">);</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">        </span><span class="n">buckets</span><span class="p">[</span><span class="n">idx</span><span class="p">][</span><span class="n">sizes</span><span class="p">[</span><span class="n">idx</span><span class="p">]</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
@@ -5009,7 +5081,7 @@
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">        </span><span class="n">qsort</span><span class="p">(</span><span class="n">buckets</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">sizes</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">float</span><span class="p">),</span><span class="w"> </span><span class="n">compare</span><span class="p">);</span>
 <a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">    </span><span class="c1">// 3. Merge sorted buckets</span>
+<a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">    </span><span class="c1">// 3. Merge the sorted buckets</span>
 <a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">idx</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">k</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">sizes</span><span class="p">[</span><span class="n">i</span><span class="p">];</span><span class="w"> </span><span class="o">++</span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5024,25 +5096,25 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bucket_sort.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Bucket sort */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">bucketSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">FloatArray</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Initialize k = n/2 buckets, expected to allocate 2 elements per bucket</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Initialize k = floor(n/2) buckets, expecting about 2 elements per bucket</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">2</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Float</span><span class="o">&gt;&gt;</span><span class="p">()</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">        </span><span class="n">buckets</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">mutableListOf</span><span class="p">())</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// 1. Distribute array elements into various buckets</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// 1. Distribute the array elements among the buckets</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="c1">// Input data range is [0, 1), use num * k to map to index range [0, k-1]</span>
+<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="c1">// The input range is [0, 1); use num * k to map each value to an index in [0, k-1]</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">k</span><span class="p">).</span><span class="na">toInt</span><span class="p">()</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="c1">// Add num to bucket i</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">        </span><span class="n">buckets</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="c1">// 2. Sort each bucket</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">        </span><span class="c1">// Use built-in sorting function, can also replace with other sorting algorithms</span>
+<a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">        </span><span class="c1">// Use the built-in sorting function, or replace it with another sorting algorithm</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">        </span><span class="n">bucket</span><span class="p">.</span><span class="na">sort</span><span class="p">()</span>
 <a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">    </span><span class="c1">// 3. Traverse buckets to merge results</span>
+<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">    </span><span class="c1">// 3. Traverse the buckets to combine the results</span>
 <a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">bucket</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">buckets</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">bucket</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5055,13 +5127,13 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">bucket_sort.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Bucket sort ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">bucket_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Initialize k = n/2 buckets, expected to allocate 2 elements per bucket</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Initialize k = floor(n/2) buckets, aiming for about 2 elements per bucket</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="n">buckets</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">[]</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a>
-<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="c1"># 1. Distribute array elements into various buckets</span>
+<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="c1"># 1. Distribute the array elements among the buckets</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="n">nums</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="o">|</span><span class="n">num</span><span class="o">|</span>
-<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">    </span><span class="c1"># Input data range is [0, 1), use num * k to map to index range [0, k-1]</span>
+<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">    </span><span class="c1"># The input range is [0, 1); use num * k to map each value to an index in [0, k-1]</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="o">.</span><span class="n">to_i</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="c1"># Add num to bucket i</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">    </span><span class="n">buckets</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">num</span>
@@ -5069,11 +5141,11 @@
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">  </span><span class="c1"># 2. Sort each bucket</span>
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">  </span><span class="n">buckets</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="o">|</span><span class="n">bucket</span><span class="o">|</span>
-<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">    </span><span class="c1"># Use built-in sorting function, can also replace with other sorting algorithms</span>
+<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">    </span><span class="c1"># Use the built-in sorting function, or replace it with another sorting algorithm</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">    </span><span class="n">bucket</span><span class="o">.</span><span class="n">sort!</span>
 <a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a>
-<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">  </span><span class="c1"># 3. Traverse buckets to merge results</span>
+<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">  </span><span class="c1"># 3. Traverse the buckets to combine the results</span>
 <a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="w">  </span><span class="n">buckets</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="o">|</span><span class="n">bucket</span><span class="o">|</span>
 <a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a><span class="w">    </span><span class="n">bucket</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="o">|</span><span class="n">num</span><span class="o">|</span>
@@ -5086,24 +5158,29 @@
 </div>
 </div>
 </div>
-<h2 id="1182-algorithm-characteristics">11.8.2 &nbsp; Algorithm Characteristics<a class="headerlink" href="#1182-algorithm-characteristics" title="Permanent link">&para;</a></h2>
-<p>Bucket sort is suitable for processing very large datasets. For example, suppose the input contains 1 million elements, and limited memory prevents the system from loading all of them at once. In that case, the data can be divided into 1000 buckets, each bucket can be sorted separately, and the results can then be merged.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20bucket_sort%28nums%3A%20list%5Bfloat%5D%29%3A%0A%20%20%20%20%22%22%22Bucket%20sort%22%22%22%0A%20%20%20%20%23%20Initialize%20k%20%3D%20n/2%20buckets,%20expecting%202%20elements%20per%20bucket%0A%20%20%20%20k%20%3D%20len%28nums%29%20//%202%0A%20%20%20%20buckets%20%3D%20%5B%5B%5D%20for%20_%20in%20range%28k%29%5D%0A%20%20%20%20%23%201.%20Distribute%20the%20array%20elements%20among%20the%20buckets%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20%23%20The%20input%20range%20is%20%5B0,%201%29%3B%20use%20num%20%2A%20k%20to%20map%20each%20value%20to%20an%20index%20in%20%5B0,%20k-1%5D%0A%20%20%20%20%20%20%20%20i%20%3D%20int%28num%20%2A%20k%29%0A%20%20%20%20%20%20%20%20%23%20Add%20num%20to%20bucket%20i%0A%20%20%20%20%20%20%20%20buckets%5Bi%5D.append%28num%29%0A%20%20%20%20%23%202.%20Sort%20each%20bucket%0A%20%20%20%20for%20bucket%20in%20buckets%3A%0A%20%20%20%20%20%20%20%20%23%20Use%20the%20built-in%20sorting%20function,%20or%20replace%20it%20with%20another%20sorting%20algorithm%0A%20%20%20%20%20%20%20%20bucket.sort%28%29%0A%20%20%20%20%23%203.%20Traverse%20the%20buckets%20to%20combine%20the%20results%0A%20%20%20%20i%20%3D%200%0A%20%20%20%20for%20bucket%20in%20buckets%3A%0A%20%20%20%20%20%20%20%20for%20num%20in%20bucket%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bi%5D%20%3D%20num%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%0A%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Assume%20the%20input%20consists%20of%20floating-point%20numbers%20in%20%5B0,%201%29%0A%20%20%20%20nums%20%3D%20%5B0.49,%200.96,%200.82,%200.09,%200.57,%200.43,%200.91,%200.75,%200.15,%200.37%5D%0A%20%20%20%20bucket_sort%28nums%29%0A%20%20%20%20print%28%22After%20bucket%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20bucket_sort%28nums%3A%20list%5Bfloat%5D%29%3A%0A%20%20%20%20%22%22%22Bucket%20sort%22%22%22%0A%20%20%20%20%23%20Initialize%20k%20%3D%20n/2%20buckets,%20expecting%202%20elements%20per%20bucket%0A%20%20%20%20k%20%3D%20len%28nums%29%20//%202%0A%20%20%20%20buckets%20%3D%20%5B%5B%5D%20for%20_%20in%20range%28k%29%5D%0A%20%20%20%20%23%201.%20Distribute%20the%20array%20elements%20among%20the%20buckets%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20%23%20The%20input%20range%20is%20%5B0,%201%29%3B%20use%20num%20%2A%20k%20to%20map%20each%20value%20to%20an%20index%20in%20%5B0,%20k-1%5D%0A%20%20%20%20%20%20%20%20i%20%3D%20int%28num%20%2A%20k%29%0A%20%20%20%20%20%20%20%20%23%20Add%20num%20to%20bucket%20i%0A%20%20%20%20%20%20%20%20buckets%5Bi%5D.append%28num%29%0A%20%20%20%20%23%202.%20Sort%20each%20bucket%0A%20%20%20%20for%20bucket%20in%20buckets%3A%0A%20%20%20%20%20%20%20%20%23%20Use%20the%20built-in%20sorting%20function,%20or%20replace%20it%20with%20another%20sorting%20algorithm%0A%20%20%20%20%20%20%20%20bucket.sort%28%29%0A%20%20%20%20%23%203.%20Traverse%20the%20buckets%20to%20combine%20the%20results%0A%20%20%20%20i%20%3D%200%0A%20%20%20%20for%20bucket%20in%20buckets%3A%0A%20%20%20%20%20%20%20%20for%20num%20in%20bucket%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bi%5D%20%3D%20num%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%0A%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Assume%20the%20input%20consists%20of%20floating-point%20numbers%20in%20%5B0,%201%29%0A%20%20%20%20nums%20%3D%20%5B0.49,%200.96,%200.82,%200.09,%200.57,%200.43,%200.91,%200.75,%200.15,%200.37%5D%0A%20%20%20%20bucket_sort%28nums%29%0A%20%20%20%20print%28%22After%20bucket%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1182-algorithm-properties">11.8.2 &nbsp; Algorithm properties<a class="headerlink" href="#1182-algorithm-properties" title="Permanent link">&para;</a></h2>
+<p>Bucket sort is useful for very large datasets. For example, suppose the input contains one million elements and memory limitations prevent loading all of them at once. We can divide them into 1,000 buckets, sort the buckets separately, and combine the results.</p>
 <ul>
-<li><strong>Time complexity is <span class="arithmatex">\(O(n + k)\)</span></strong>: Assuming the elements are evenly distributed across the buckets, each bucket contains <span class="arithmatex">\(\frac{n}{k}\)</span> elements. If sorting a single bucket takes <span class="arithmatex">\(O(\frac{n}{k} \log\frac{n}{k})\)</span> time, then sorting all buckets takes <span class="arithmatex">\(O(n \log\frac{n}{k})\)</span> time. <strong>When the number of buckets <span class="arithmatex">\(k\)</span> is relatively large, the time complexity approaches <span class="arithmatex">\(O(n)\)</span></strong>. Merging the results requires traversing all buckets and elements, which takes <span class="arithmatex">\(O(n + k)\)</span> time. In the worst case, all data is placed into a single bucket, and sorting that bucket takes <span class="arithmatex">\(O(n^2)\)</span> time.</li>
-<li><strong>Space complexity is <span class="arithmatex">\(O(n + k)\)</span>, and bucket sort is not in-place</strong>: It requires extra space for <span class="arithmatex">\(k\)</span> buckets and a total of <span class="arithmatex">\(n\)</span> elements.</li>
-<li>Whether bucket sort is stable depends on whether the algorithm for sorting elements within buckets is stable.</li>
+<li><strong>Time complexity: <span class="arithmatex">\(O(n + k)\)</span></strong>. If the elements are evenly distributed, each bucket contains <span class="arithmatex">\(\frac{n}{k}\)</span> elements. If sorting one bucket takes <span class="arithmatex">\(O(\frac{n}{k} \log\frac{n}{k})\)</span> time, sorting all buckets takes <span class="arithmatex">\(O(n \log\frac{n}{k})\)</span>. <strong>When the number of buckets <span class="arithmatex">\(k\)</span> is relatively large, this approaches <span class="arithmatex">\(O(n)\)</span></strong>. Combining the results visits all buckets and elements, taking <span class="arithmatex">\(O(n + k)\)</span> time. In the worst case, all data falls into one bucket, and sorting it takes <span class="arithmatex">\(O(n^2)\)</span> time.</li>
+<li><strong>Space complexity: <span class="arithmatex">\(O(n + k)\)</span>; not in-place</strong>. Extra space is needed for <span class="arithmatex">\(k\)</span> buckets and a total of <span class="arithmatex">\(n\)</span> elements.</li>
+<li>Whether bucket sort is stable depends on whether the algorithm used to sort within each bucket is stable.</li>
 </ul>
-<h2 id="1183-how-to-achieve-even-distribution">11.8.3 &nbsp; How to Achieve Even Distribution<a class="headerlink" href="#1183-how-to-achieve-even-distribution" title="Permanent link">&para;</a></h2>
-<p>In theory, bucket sort can achieve <span class="arithmatex">\(O(n)\)</span> time complexity. <strong>The key is to distribute the elements evenly across the buckets</strong>, because real-world data is often not uniformly distributed. For example, suppose we want to divide all products on Taobao evenly into 10 buckets by price range, but the price distribution is uneven: there are many products priced below 100 yuan and very few priced above 1000 yuan. If the price range is divided evenly into 10 intervals, the numbers of products in the buckets will differ greatly.</p>
-<p>To achieve a more even distribution, we can first choose a rough boundary and partition the data into 3 buckets. <strong>After that, buckets containing more products can be further divided into 3 buckets until the numbers of elements in all buckets are roughly equal</strong>.</p>
-<p>As shown in Figure 11-14, this method essentially builds a recursion tree whose goal is to make the leaf nodes as balanced as possible. Of course, the data does not have to be split into 3 buckets in every round; the specific partitioning strategy can be chosen flexibly based on the characteristics of the data.</p>
-<p><img alt="Recursively dividing buckets" class="animation-figure" src="../bucket_sort.assets/scatter_in_buckets_recursively.png" /></p>
-<p align="center"> Figure 11-14 &nbsp; Recursively dividing buckets </p>
+<h2 id="1183-distributing-elements-evenly">11.8.3 &nbsp; Distributing elements evenly<a class="headerlink" href="#1183-distributing-elements-evenly" title="Permanent link">&para;</a></h2>
+<p>Bucket sort can theoretically achieve <span class="arithmatex">\(O(n)\)</span> time. <strong>The key is distributing elements evenly among the buckets</strong>, since real data is often unevenly distributed. Suppose we want to divide all products on Taobao evenly into 10 buckets by price. There are many products below 100 yuan but few above 1,000 yuan. Splitting the price range into 10 equal intervals would leave very different numbers of products in the buckets.</p>
+<p>To obtain an even distribution, we can first choose rough boundaries and divide the data into 3 buckets. <strong>Then split the more crowded buckets into 3 smaller buckets, repeating until all buckets contain roughly equal numbers of elements</strong>.</p>
+<p>As shown below, this effectively creates a recursion tree whose leaf values should be as balanced as possible. We need not use 3 buckets at every step; the subdivision can be chosen to suit the data.</p>
+<p><img alt="Recursively subdividing buckets" class="animation-figure" src="../bucket_sort.assets/scatter_in_buckets_recursively.png" /></p>
+<p align="center"> Figure 11-14 &nbsp; Recursively subdividing buckets </p>
 
-<p>If we know the probability distribution of product prices in advance, <strong>we can set the price boundaries for each bucket according to that distribution</strong>. Notably, the data distribution does not need to be measured exactly; it can also be approximated with a probability model chosen to fit the characteristics of the data.</p>
-<p>As shown in Figure 11-15, we assume that product prices follow a normal distribution, which allows us to reasonably set price intervals to evenly distribute products to each bucket.</p>
-<p><img alt="Dividing buckets based on probability distribution" class="animation-figure" src="../bucket_sort.assets/scatter_in_buckets_distribution.png" /></p>
-<p align="center"> Figure 11-15 &nbsp; Dividing buckets based on probability distribution </p>
+<p>If we know the probability distribution of product prices in advance, <strong>we can choose bucket boundaries according to that distribution</strong>. We do not necessarily need to measure the distribution explicitly; we can approximate it with a probability model suited to the data.</p>
+<p>In Figure 11-15, we assume prices follow a normal distribution. This lets us choose suitable price intervals and distribute products evenly among the buckets.</p>
+<p><img alt="Defining buckets from a probability distribution" class="animation-figure" src="../bucket_sort.assets/scatter_in_buckets_distribution.png" /></p>
+<p align="center"> Figure 11-15 &nbsp; Defining buckets from a probability distribution </p>
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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@@ -3111,7 +3111,7 @@
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3190,10 +3190,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1191-simple-implementation" class="md-nav__link">
+  <a href="#1191-a-simple-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.9.1 &nbsp; Simple Implementation
+        11.9.1 &nbsp; A simple implementation
       </span>
     </span>
   </a>
@@ -3201,10 +3201,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1192-complete-implementation" class="md-nav__link">
+  <a href="#1192-a-complete-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.9.2 &nbsp; Complete Implementation
+        11.9.2 &nbsp; A complete implementation
       </span>
     </span>
   </a>
@@ -3212,10 +3212,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1193-algorithm-characteristics" class="md-nav__link">
+  <a href="#1193-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.9.3 &nbsp; Algorithm Characteristics
+        11.9.3 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -3256,7 +3256,7 @@
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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@@ -3283,7 +3283,7 @@
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4580,6 +4580,78 @@
 
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1191-simple-implementation" class="md-nav__link">
+  <a href="#1191-a-simple-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.9.1 &nbsp; Simple Implementation
+        11.9.1 &nbsp; A simple implementation
       </span>
     </span>
   </a>
@@ -4632,10 +4704,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1192-complete-implementation" class="md-nav__link">
+  <a href="#1192-a-complete-implementation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.9.2 &nbsp; Complete Implementation
+        11.9.2 &nbsp; A complete implementation
       </span>
     </span>
   </a>
@@ -4643,10 +4715,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1193-algorithm-characteristics" class="md-nav__link">
+  <a href="#1193-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.9.3 &nbsp; Algorithm Characteristics
+        11.9.3 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -4701,17 +4773,17 @@
 
 
 <!-- Page content -->
-<h1 id="119-counting-sort">11.9 &nbsp; Counting Sort<a class="headerlink" href="#119-counting-sort" title="Permanent link">&para;</a></h1>
-<p><u>Counting sort</u> sorts by counting the occurrences of elements and is typically applied to integer arrays.</p>
-<h2 id="1191-simple-implementation">11.9.1 &nbsp; Simple Implementation<a class="headerlink" href="#1191-simple-implementation" title="Permanent link">&para;</a></h2>
-<p>Let's start with a simple example. Given an array <code>nums</code> of length <span class="arithmatex">\(n\)</span>, where the elements are all "non-negative integers", the overall flow of counting sort is shown in Figure 11-16.</p>
+<h1 id="119-counting-sort">11.9 &nbsp; Counting sort<a class="headerlink" href="#119-counting-sort" title="Permanent link">&para;</a></h1>
+<p><u>Counting sort</u> sorts by counting how often each element occurs. It is typically used for integer arrays.</p>
+<h2 id="1191-a-simple-implementation">11.9.1 &nbsp; A simple implementation<a class="headerlink" href="#1191-a-simple-implementation" title="Permanent link">&para;</a></h2>
+<p>Start with a simple example: an array <code>nums</code> of length <span class="arithmatex">\(n\)</span> containing nonnegative integers. The overall process is shown below.</p>
 <ol>
-<li>Traverse the array to find the largest number, denoted as <span class="arithmatex">\(m\)</span>, and then create an auxiliary array <code>counter</code> of length <span class="arithmatex">\(m + 1\)</span>.</li>
-<li><strong>Use <code>counter</code> to count how many times each number appears in <code>nums</code></strong>, where <code>counter[num]</code> stores the number of occurrences of <code>num</code>. This is simple: traverse <code>nums</code> (denote the current number by <code>num</code>) and increment <code>counter[num]</code> by <span class="arithmatex">\(1\)</span> each time.</li>
-<li><strong>Because the indices of <code>counter</code> are naturally ordered, the numbers are effectively already sorted</strong>. Next, traverse <code>counter</code> and write the numbers back into <code>nums</code> in ascending order according to their occurrence counts.</li>
+<li>Traverse the array to find its maximum value, denoted by <span class="arithmatex">\(m\)</span>, then create an auxiliary array <code>counter</code> of length <span class="arithmatex">\(m + 1\)</span>.</li>
+<li><strong>Use <code>counter</code> to count occurrences of each number in <code>nums</code></strong>. Here, <code>counter[num]</code> is the number of occurrences of <code>num</code>. Simply traverse <code>nums</code> and increment <code>counter[num]</code> by <span class="arithmatex">\(1\)</span> for each current value <code>num</code>.</li>
+<li><strong>Because the indices of <code>counter</code> are inherently ordered, the numbers are effectively already sorted</strong>. Traverse <code>counter</code> and write each number into <code>nums</code> in ascending order, repeating it according to its count.</li>
 </ol>
-<p><img alt="Counting sort flow" class="animation-figure" src="../counting_sort.assets/counting_sort_overview.png" /></p>
-<p align="center"> Figure 11-16 &nbsp; Counting sort flow </p>
+<p><img alt="The counting sort process" class="animation-figure" src="../counting_sort.assets/counting_sort_overview.png" /></p>
+<p align="center"> Figure 11-16 &nbsp; The counting sort process </p>
 
 <p>The code is as follows:</p>
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@@ -4719,40 +4791,38 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">counting_sort_naive</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]):</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Counting sort"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Simple implementation, cannot be used for sorting objects</span>
-<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="c1"># 1. Count the maximum element m in the array</span>
-<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="n">m</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="k">for</span> <span class="n">num</span> <span class="ow">in</span> <span class="n">nums</span><span class="p">:</span>
-<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="n">m</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">num</span><span class="p">)</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="c1"># 2. Count the occurrence of each number</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="c1"># counter[num] represents the occurrence of num</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="n">counter</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="k">for</span> <span class="n">num</span> <span class="ow">in</span> <span class="n">nums</span><span class="p">:</span>
-<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>    <span class="c1"># 3. Traverse counter, filling each element back into the original array nums</span>
-<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
-<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="k">for</span> <span class="n">num</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
-<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>        <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]):</span>
-<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>            <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">num</span>
-<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Simple implementation; cannot be used to sort objects</span>
+<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="c1"># 1. Find the maximum array element m</span>
+<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="n">m</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># 2. Count the occurrences of each number</span>
+<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="c1"># counter[num] is the number of occurrences of num</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="n">counter</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="k">for</span> <span class="n">num</span> <span class="ow">in</span> <span class="n">nums</span><span class="p">:</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
+<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="c1"># 3. Traverse counter and fill the original array nums with the elements</span>
+<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
+<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>    <span class="k">for</span> <span class="n">num</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
+<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>        <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]):</span>
+<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>            <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">num</span>
+<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="c1">// Simple implementation, cannot be used for sorting objects</span>
+<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="c1">// Simple implementation; cannot be used to sort objects</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="kt">void</span><span class="w"> </span><span class="nf">countingSortNaive</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">        </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">counter</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">    </span><span class="c1">// 3. Traverse counter, filling each element back into the original array nums</span>
+<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">    </span><span class="c1">// 3. Traverse counter and fill the original array nums with the elements</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">num</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">];</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4764,20 +4834,20 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="c1">// Simple implementation, cannot be used for sorting objects</span>
+<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="c1">// Simple implementation; cannot be used to sort objects</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="kt">void</span><span class="w"> </span><span class="nf">countingSortNaive</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">        </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">max</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]++</span><span class="p">;</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="c1">// 3. Traverse counter, filling each element back into the original array nums</span>
+<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="c1">// 3. Traverse counter and fill the original array nums with the elements</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">num</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4789,20 +4859,20 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="c1">// Simple implementation, cannot be used for sorting objects</span>
+<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="c1">// Simple implementation; cannot be used to sort objects</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="k">void</span><span class="w"> </span><span class="nf">CountingSortNaive</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">        </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Max</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">    </span><span class="c1">// 3. Traverse counter, filling each element back into the original array nums</span>
+<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">    </span><span class="c1">// 3. Traverse counter and fill the original array nums with the elements</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">num</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">];</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4814,22 +4884,22 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="c1">// Simple implementation, cannot be used for sorting objects</span>
+<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="c1">// Simple implementation; cannot be used to sort objects</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="kd">func</span><span class="w"> </span><span class="nx">countingSortNaive</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">            </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="nx">counter</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">]</span><span class="o">++</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="c1">// 3. Traverse counter, filling each element back into the original array nums</span>
+<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="c1">// 3. Traverse counter and fill the original array nums with the elements</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="o">+</span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">num</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">];</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">            </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span>
@@ -4841,17 +4911,17 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Simple implementation, cannot be used for sorting objects</span>
+<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Simple implementation; cannot be used to sort objects</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="kd">func</span><span class="w"> </span><span class="nf">countingSortNaive</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">max</span><span class="p">()</span><span class="o">!</span>
-<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">counter</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">    </span><span class="c1">// 3. Traverse counter, filling each element back into the original array nums</span>
+<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">    </span><span class="c1">// 3. Traverse counter and fill the original array nums with the elements</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="kc">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
@@ -4864,17 +4934,17 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="c1">// Simple implementation, cannot be used for sorting objects</span>
+<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="c1">// Simple implementation; cannot be used to sort objects</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="kd">function</span><span class="w"> </span><span class="nx">countingSortNaive</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(...</span><span class="nx">nums</span><span class="p">);</span>
-<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">]</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="c1">// 3. Traverse counter, filling each element back into the original array nums</span>
+<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="c1">// 3. Traverse counter and fill the original array nums with the elements</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">num</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">];</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4886,17 +4956,17 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="c1">// Simple implementation, cannot be used for sorting objects</span>
+<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="c1">// Simple implementation; cannot be used to sort objects</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="kd">function</span><span class="w"> </span><span class="nx">countingSortNaive</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">m</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(...</span><span class="nx">nums</span><span class="p">);</span>
-<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">counter</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">]</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="c1">// 3. Traverse counter, filling each element back into the original array nums</span>
+<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="c1">// 3. Traverse counter and fill the original array nums with the elements</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">num</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">];</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4908,20 +4978,20 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="c1">// Simple implementation, cannot be used for sorting objects</span>
+<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="c1">// Simple implementation; cannot be used to sort objects</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="kt">void</span><span class="w"> </span><span class="n">countingSortNaive</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">    </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">_num</span><span class="p">);</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">  </span><span class="c1">// counter[_num] represents occurrence count of _num</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">  </span><span class="c1">// counter[_num] is the number of occurrences of _num</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">    </span><span class="n">counter</span><span class="p">[</span><span class="n">_num</span><span class="p">]</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="c1">// 3. Traverse counter, filling each element back into the original array nums</span>
+<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="c1">// 3. Traverse counter and fill the original array nums with the elements</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">_num</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">_num</span><span class="p">];</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4933,17 +5003,17 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="c1">// Simple implementation, cannot be used for sorting objects</span>
+<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="c1">// Simple implementation; cannot be used to sort objects</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="k">fn</span><span class="w"> </span><span class="nf">counting_sort_naive</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">.</span><span class="n">iter</span><span class="p">().</span><span class="n">max</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span>
-<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="o">&amp;</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">iter</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="c1">// 3. Traverse counter, filling each element back into the original array nums</span>
+<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="c1">// 3. Traverse counter and fill the original array nums with the elements</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">_</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
@@ -4956,22 +5026,22 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="c1">// Simple implementation, cannot be used for sorting objects</span>
+<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="c1">// Simple implementation; cannot be used to sort objects</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="kt">void</span><span class="w"> </span><span class="nf">countingSortNaive</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">            </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">    </span><span class="c1">// 3. Traverse counter, filling each element back into the original array nums</span>
+<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">    </span><span class="c1">// 3. Traverse counter and fill the original array nums with the elements</span>
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">num</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">];</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -4985,20 +5055,20 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="c1">// Simple implementation, cannot be used for sorting objects</span>
+<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="c1">// Simple implementation; cannot be used to sort objects</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">countingSortNaive</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">        </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]++</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="c1">// 3. Traverse counter, filling each element back into the original array nums</span>
+<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="c1">// 3. Traverse counter and fill the original array nums with the elements</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
@@ -5014,15 +5084,15 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Counting sort ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">counting_sort_naive</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Simple implementation, cannot be used for sorting objects</span>
-<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="c1"># 1. Count the maximum element m in the array</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Simple implementation; cannot be used to sort objects</span>
+<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="c1"># 1. Find the maximum array element m</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="n">nums</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">num</span><span class="o">|</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="o">].</span><span class="n">max</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="c1"># 2. Count the occurrence of each number</span>
-<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="c1"># counter[num] represents the occurrence of num</span>
+<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="c1"># 2. Count the occurrences of each number</span>
+<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="c1"># counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="n">nums</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">num</span><span class="o">|</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">  </span><span class="c1"># 3. Traverse counter, filling each element back into the original array nums</span>
+<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">  </span><span class="c1"># 3. Traverse counter and fill the original array nums with the elements</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="k">do</span>
@@ -5035,26 +5105,31 @@
 </div>
 </div>
 </div>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20counting_sort_naive%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Counting%20sort%22%22%22%0A%20%20%20%20%23%20Simple%20implementation%3B%20cannot%20be%20used%20to%20sort%20objects%0A%20%20%20%20%23%201.%20Find%20the%20maximum%20array%20element%20m%0A%20%20%20%20m%20%3D%200%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20max%28m,%20num%29%0A%20%20%20%20%23%202.%20Count%20the%20occurrences%20of%20each%20number%0A%20%20%20%20%23%20counter%5Bnum%5D%20is%20the%20number%20of%20occurrences%20of%20num%0A%20%20%20%20counter%20%3D%20%5B0%5D%20%2A%20%28m%20%2B%201%29%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20counter%5Bnum%5D%20%2B%3D%201%0A%20%20%20%20%23%203.%20Traverse%20counter%20and%20fill%20the%20original%20array%20nums%20with%20the%20elements%0A%20%20%20%20i%20%3D%200%0A%20%20%20%20for%20num%20in%20range%28m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20_%20in%20range%28counter%5Bnum%5D%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bi%5D%20%3D%20num%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B1,%200,%201,%202,%200,%204,%200,%202,%202,%204%5D%0A%20%20%20%20counting_sort_naive%28nums%29%0A%20%20%20%20print%28f%22After%20counting%20sort%20%28without%20support%20for%20objects%29,%20nums%20%3D%20%7Bnums%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20counting_sort_naive%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Counting%20sort%22%22%22%0A%20%20%20%20%23%20Simple%20implementation%3B%20cannot%20be%20used%20to%20sort%20objects%0A%20%20%20%20%23%201.%20Find%20the%20maximum%20array%20element%20m%0A%20%20%20%20m%20%3D%200%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20m%20%3D%20max%28m,%20num%29%0A%20%20%20%20%23%202.%20Count%20the%20occurrences%20of%20each%20number%0A%20%20%20%20%23%20counter%5Bnum%5D%20is%20the%20number%20of%20occurrences%20of%20num%0A%20%20%20%20counter%20%3D%20%5B0%5D%20%2A%20%28m%20%2B%201%29%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20counter%5Bnum%5D%20%2B%3D%201%0A%20%20%20%20%23%203.%20Traverse%20counter%20and%20fill%20the%20original%20array%20nums%20with%20the%20elements%0A%20%20%20%20i%20%3D%200%0A%20%20%20%20for%20num%20in%20range%28m%20%2B%201%29%3A%0A%20%20%20%20%20%20%20%20for%20_%20in%20range%28counter%5Bnum%5D%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bi%5D%20%3D%20num%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B1,%200,%201,%202,%200,%204,%200,%202,%202,%204%5D%0A%20%20%20%20counting_sort_naive%28nums%29%0A%20%20%20%20print%28f%22After%20counting%20sort%20%28without%20support%20for%20objects%29,%20nums%20%3D%20%7Bnums%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
 <div class="admonition note">
-<p class="admonition-title">Connection between counting sort and bucket sort</p>
-<p>From the perspective of bucket sort, each index of the counting array <code>counter</code> can be viewed as a bucket, and the counting process can be seen as distributing elements into their corresponding buckets. Essentially, counting sort is a special case of bucket sort for integer data.</p>
+<p class="admonition-title">How counting sort relates to bucket sort</p>
+<p>From a bucket-sort perspective, each index of <code>counter</code> is a bucket, and counting occurrences is equivalent to distributing elements into their corresponding buckets. Counting sort is essentially a special case of bucket sort for integer data.</p>
 </div>
-<h2 id="1192-complete-implementation">11.9.2 &nbsp; Complete Implementation<a class="headerlink" href="#1192-complete-implementation" title="Permanent link">&para;</a></h2>
-<p>Observant readers may have noticed that <strong>if the input consists of objects, step <code>3.</code> above no longer works</strong>. Suppose the input consists of product objects and we want to sort them by price (a member variable of the class); the above algorithm can only produce the sorted order of the prices themselves.</p>
-<p>So how can we obtain the sorted order of the original data? We first compute the prefix sums of <code>counter</code>. As the name suggests, the prefix sum at index <code>i</code>, <code>prefix[i]</code>, equals the sum of the elements from index <code>0</code> through <code>i</code>:</p>
+<h2 id="1192-a-complete-implementation">11.9.2 &nbsp; A complete implementation<a class="headerlink" href="#1192-a-complete-implementation" title="Permanent link">&para;</a></h2>
+<p>You may have noticed that <strong>step <code>3.</code> above fails when the input consists of objects</strong>. Suppose the inputs are product objects and we want to sort them by price, a member variable of the class. The algorithm above can produce only the sorted prices.</p>
+<p>How can we obtain the sorted original data? First, calculate the “prefix sums” of <code>counter</code>. The prefix sum <code>prefix[i]</code> is the sum of the array entries up to index <code>i</code>:</p>
 <div class="arithmatex">\[
 \text{prefix}[i] = \sum_{j=0}^i \text{counter[j]}
 \]</div>
-<p><strong>The prefix sum has a clear interpretation: <code>prefix[num] - 1</code> gives the index of the last occurrence of element <code>num</code> in the result array <code>res</code></strong>. This information is crucial because it tells us where each element should be placed in the result array. Next, we traverse the original array <code>nums</code> in reverse, and for each element <code>num</code>, perform the following two steps.</p>
+<p><strong>The prefix sum has a precise meaning: <code>prefix[num] - 1</code> is the index of the last occurrence of <code>num</code> in the result array <code>res</code></strong>. This tells us where each element belongs. Next, traverse the original array <code>nums</code> in reverse and perform two steps for each element <code>num</code>.</p>
 <ol>
-<li>Place <code>num</code> at index <code>prefix[num] - 1</code> of the array <code>res</code>.</li>
-<li>Decrease the prefix sum <code>prefix[num]</code> by <span class="arithmatex">\(1\)</span> to get the index for the next placement of <code>num</code>.</li>
+<li>Place <code>num</code> at index <code>prefix[num] - 1</code> in <code>res</code>.</li>
+<li>Decrease <code>prefix[num]</code> by <span class="arithmatex">\(1\)</span> to obtain the index for placing the next occurrence of <code>num</code>.</li>
 </ol>
-<p>After the traversal is complete, the array <code>res</code> contains the sorted result, and finally <code>res</code> is used to overwrite the original array <code>nums</code>. The complete counting sort flow is shown in Figure 11-17.</p>
+<p>After the traversal, <code>res</code> contains the sorted result. Finally, copy <code>res</code> over the original array <code>nums</code>. Figure 11-17 shows the complete process.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:8"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">&lt;1&gt;</label><label for="__tabbed_2_2">&lt;2&gt;</label><label for="__tabbed_2_3">&lt;3&gt;</label><label for="__tabbed_2_4">&lt;4&gt;</label><label for="__tabbed_2_5">&lt;5&gt;</label><label for="__tabbed_2_6">&lt;6&gt;</label><label for="__tabbed_2_7">&lt;7&gt;</label><label for="__tabbed_2_8">&lt;8&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Counting sort steps" class="animation-figure" src="../counting_sort.assets/counting_sort_step1.png" /></p>
+<p><img alt="Steps of counting sort" class="animation-figure" src="../counting_sort.assets/counting_sort_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="counting_sort_step2" class="animation-figure" src="../counting_sort.assets/counting_sort_step2.png" /></p>
@@ -5079,103 +5154,103 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 11-17 &nbsp; Counting sort steps </p>
+<p align="center"> Figure 11-17 &nbsp; Steps of counting sort </p>
 
-<p>The counting sort implementation is shown below:</p>
+<p>The implementation is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">counting_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]):</span>
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Counting sort"""</span>
-<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># Complete implementation, can sort objects and is a stable sort</span>
-<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="c1"># 1. Count the maximum element m in the array</span>
+<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># Full implementation; stable and adaptable to sorting objects</span>
+<a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="c1"># 1. Find the maximum array element m</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="n">m</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
-<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="c1"># 2. Count the occurrence of each number</span>
-<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>    <span class="c1"># counter[num] represents the occurrence of num</span>
+<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="c1"># 2. Count the occurrences of each number</span>
+<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>    <span class="c1"># counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>    <span class="n">counter</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">m</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>    <span class="k">for</span> <span class="n">num</span> <span class="ow">in</span> <span class="n">nums</span><span class="p">:</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>        <span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
-<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>    <span class="c1"># 3. Calculate the prefix sum of counter, converting "occurrence count" to "tail index"</span>
-<a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>    <span class="c1"># counter[num]-1 is the last index where num appears in res</span>
+<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>    <span class="c1"># 3. Compute prefix sums of counter to convert occurrence counts into end positions</span>
+<a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>    <span class="c1"># That is, counter[num]-1 is the index of the last occurrence of num in res</span>
 <a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
 <a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>        <span class="n">counter</span><span class="p">[</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-<a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>    <span class="c1"># 4. Traverse nums in reverse order, placing each element into the result array res</span>
-<a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>    <span class="c1"># Initialize the array res to record results</span>
+<a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>    <span class="c1"># 4. Traverse nums in reverse and place each element in the result array res</span>
+<a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>    <span class="c1"># Initialize array res to store the result</span>
 <a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
 <a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a>    <span class="n">res</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span>
 <a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a>        <span class="n">num</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-13-21" name="__codelineno-13-21" href="#__codelineno-13-21"></a>        <span class="n">res</span><span class="p">[</span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">num</span>  <span class="c1"># Place num at the corresponding index</span>
-<a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a>        <span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>  <span class="c1"># Decrement the prefix sum by 1, getting the next index to place num</span>
-<a id="__codelineno-13-23" name="__codelineno-13-23" href="#__codelineno-13-23"></a>    <span class="c1"># Use result array res to overwrite the original array nums</span>
+<a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a>        <span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>  <span class="c1"># Decrement the prefix sum by 1 to determine where to place the next occurrence of num</span>
+<a id="__codelineno-13-23" name="__codelineno-13-23" href="#__codelineno-13-23"></a>    <span class="c1"># Overwrite the original array nums with the result array res</span>
 <a id="__codelineno-13-24" name="__codelineno-13-24" href="#__codelineno-13-24"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-13-25" name="__codelineno-13-25" href="#__codelineno-13-25"></a>        <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="c1">// Complete implementation, can sort objects and is a stable sort</span>
+<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="c1">// Stable implementation that can be adapted to sort objects</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="kt">void</span><span class="w"> </span><span class="nf">countingSort</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">        </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">counter</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">    </span><span class="c1">// 3. Calculate the prefix sum of counter, converting "occurrence count" to "tail index"</span>
-<a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">    </span><span class="c1">// counter[num]-1 is the last index where num appears in res</span>
+<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">    </span><span class="c1">// 3. Compute prefix sums of counter to convert occurrence counts into end positions</span>
+<a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">    </span><span class="c1">// That is, counter[num]-1 is the index of the last occurrence of num in res</span>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse order, placing each element into the result array res</span>
-<a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="w">    </span><span class="c1">// Initialize the array res to record results</span>
+<a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse and place each element in the result array res</span>
+<a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="w">    </span><span class="c1">// Initialize array res to store the result</span>
 <a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
 <a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">(</span><span class="n">n</span><span class="p">);</span>
 <a id="__codelineno-14-24" name="__codelineno-14-24" href="#__codelineno-14-24"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-25" name="__codelineno-14-25" href="#__codelineno-14-25"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-14-26" name="__codelineno-14-26" href="#__codelineno-14-26"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span><span class="w"> </span><span class="c1">// Place num at the corresponding index</span>
-<a id="__codelineno-14-27" name="__codelineno-14-27" href="#__codelineno-14-27"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w">              </span><span class="c1">// Decrement the prefix sum by 1, getting the next index to place num</span>
+<a id="__codelineno-14-27" name="__codelineno-14-27" href="#__codelineno-14-27"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w">              </span><span class="c1">// Decrement the prefix sum by 1 to determine where to place the next occurrence of num</span>
 <a id="__codelineno-14-28" name="__codelineno-14-28" href="#__codelineno-14-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-29" name="__codelineno-14-29" href="#__codelineno-14-29"></a><span class="w">    </span><span class="c1">// Use result array res to overwrite the original array nums</span>
+<a id="__codelineno-14-29" name="__codelineno-14-29" href="#__codelineno-14-29"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result array res</span>
 <a id="__codelineno-14-30" name="__codelineno-14-30" href="#__codelineno-14-30"></a><span class="w">    </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-14-31" name="__codelineno-14-31" href="#__codelineno-14-31"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="c1">// Complete implementation, can sort objects and is a stable sort</span>
+<a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="c1">// Full implementation; stable and adaptable to sorting objects</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="kt">void</span><span class="w"> </span><span class="nf">countingSort</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">        </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">max</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]++</span><span class="p">;</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="c1">// 3. Calculate the prefix sum of counter, converting "occurrence count" to "tail index"</span>
-<a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">    </span><span class="c1">// counter[num]-1 is the last index where num appears in res</span>
+<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="c1">// 3. Compute prefix sums of counter to convert occurrence counts into end positions</span>
+<a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">    </span><span class="c1">// That is, counter[num]-1 is the index of the last occurrence of num in res</span>
 <a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse order, placing each element into the result array res</span>
-<a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="w">    </span><span class="c1">// Initialize the array res to record results</span>
+<a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse and place each element in the result array res</span>
+<a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="w">    </span><span class="c1">// Initialize array res to store the result</span>
 <a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
 <a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-24" name="__codelineno-15-24" href="#__codelineno-15-24"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-25" name="__codelineno-15-25" href="#__codelineno-15-25"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-26" name="__codelineno-15-26" href="#__codelineno-15-26"></a><span class="w">        </span><span class="n">res</span><span class="o">[</span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span><span class="w"> </span><span class="c1">// Place num at the corresponding index</span>
-<a id="__codelineno-15-27" name="__codelineno-15-27" href="#__codelineno-15-27"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrement the prefix sum by 1, getting the next index to place num</span>
+<a id="__codelineno-15-27" name="__codelineno-15-27" href="#__codelineno-15-27"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrement the prefix sum by 1 to determine where to place the next occurrence of num</span>
 <a id="__codelineno-15-28" name="__codelineno-15-28" href="#__codelineno-15-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-29" name="__codelineno-15-29" href="#__codelineno-15-29"></a><span class="w">    </span><span class="c1">// Use result array res to overwrite the original array nums</span>
+<a id="__codelineno-15-29" name="__codelineno-15-29" href="#__codelineno-15-29"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result array res</span>
 <a id="__codelineno-15-30" name="__codelineno-15-30" href="#__codelineno-15-30"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-31" name="__codelineno-15-31" href="#__codelineno-15-31"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">res</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-15-32" name="__codelineno-15-32" href="#__codelineno-15-32"></a><span class="w">    </span><span class="p">}</span>
@@ -5184,34 +5259,34 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="c1">// Complete implementation, can sort objects and is a stable sort</span>
+<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="c1">// Full implementation; stable and adaptable to sorting objects</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="k">void</span><span class="w"> </span><span class="nf">CountingSort</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">        </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Max</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">    </span><span class="c1">// 3. Calculate the prefix sum of counter, converting "occurrence count" to "tail index"</span>
-<a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">    </span><span class="c1">// counter[num]-1 is the last index where num appears in res</span>
+<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">    </span><span class="c1">// 3. Compute prefix sums of counter to convert occurrence counts into end positions</span>
+<a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">    </span><span class="c1">// That is, counter[num]-1 is the index of the last occurrence of num in res</span>
 <a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse order, placing each element into the result array res</span>
-<a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="w">    </span><span class="c1">// Initialize the array res to record results</span>
+<a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse and place each element in the result array res</span>
+<a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="w">    </span><span class="c1">// Initialize array res to store the result</span>
 <a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
 <a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="p">];</span>
 <a id="__codelineno-16-24" name="__codelineno-16-24" href="#__codelineno-16-24"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-25" name="__codelineno-16-25" href="#__codelineno-16-25"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-16-26" name="__codelineno-16-26" href="#__codelineno-16-26"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span><span class="w"> </span><span class="c1">// Place num at the corresponding index</span>
-<a id="__codelineno-16-27" name="__codelineno-16-27" href="#__codelineno-16-27"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrement the prefix sum by 1, getting the next index to place num</span>
+<a id="__codelineno-16-27" name="__codelineno-16-27" href="#__codelineno-16-27"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrement the prefix sum by 1 to determine where to place the next occurrence of num</span>
 <a id="__codelineno-16-28" name="__codelineno-16-28" href="#__codelineno-16-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-16-29" name="__codelineno-16-29" href="#__codelineno-16-29"></a><span class="w">    </span><span class="c1">// Use result array res to overwrite the original array nums</span>
+<a id="__codelineno-16-29" name="__codelineno-16-29" href="#__codelineno-16-29"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result array res</span>
 <a id="__codelineno-16-30" name="__codelineno-16-30" href="#__codelineno-16-30"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-31" name="__codelineno-16-31" href="#__codelineno-16-31"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-16-32" name="__codelineno-16-32" href="#__codelineno-16-32"></a><span class="w">    </span><span class="p">}</span>
@@ -5220,68 +5295,68 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="c1">// Complete implementation, can sort objects and is a stable sort</span>
+<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="c1">// Full implementation; stable and adaptable to sorting objects</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="kd">func</span><span class="w"> </span><span class="nx">countingSort</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="nx">m</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">            </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">    </span><span class="nx">counter</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">]</span><span class="o">++</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">    </span><span class="c1">// 3. Calculate the prefix sum of counter, converting "occurrence count" to "tail index"</span>
-<a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="w">    </span><span class="c1">// counter[num]-1 is the last index where num appears in res</span>
+<a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">    </span><span class="c1">// 3. Compute prefix sums of counter to convert occurrence counts into end positions</span>
+<a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="w">    </span><span class="c1">// That is, counter[num]-1 is the index of the last occurrence of num in res</span>
 <a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">counter</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
 <a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-22" name="__codelineno-17-22" href="#__codelineno-17-22"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse order, placing each element into the result array res</span>
-<a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a><span class="w">    </span><span class="c1">// Initialize the array res to record results</span>
+<a id="__codelineno-17-22" name="__codelineno-17-22" href="#__codelineno-17-22"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse and place each element in the result array res</span>
+<a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a><span class="w">    </span><span class="c1">// Initialize array res to store the result</span>
 <a id="__codelineno-17-24" name="__codelineno-17-24" href="#__codelineno-17-24"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span>
 <a id="__codelineno-17-25" name="__codelineno-17-25" href="#__codelineno-17-25"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
 <a id="__codelineno-17-26" name="__codelineno-17-26" href="#__codelineno-17-26"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-27" name="__codelineno-17-27" href="#__codelineno-17-27"></a><span class="w">        </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
 <a id="__codelineno-17-28" name="__codelineno-17-28" href="#__codelineno-17-28"></a><span class="w">        </span><span class="c1">// Place num at the corresponding index</span>
 <a id="__codelineno-17-29" name="__codelineno-17-29" href="#__codelineno-17-29"></a><span class="w">        </span><span class="nx">res</span><span class="p">[</span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">]</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span>
-<a id="__codelineno-17-30" name="__codelineno-17-30" href="#__codelineno-17-30"></a><span class="w">        </span><span class="c1">// Decrement the prefix sum by 1, getting the next index to place num</span>
+<a id="__codelineno-17-30" name="__codelineno-17-30" href="#__codelineno-17-30"></a><span class="w">        </span><span class="c1">// Decrement the prefix sum by 1 to determine where to place the next occurrence of num</span>
 <a id="__codelineno-17-31" name="__codelineno-17-31" href="#__codelineno-17-31"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">]</span><span class="o">--</span>
 <a id="__codelineno-17-32" name="__codelineno-17-32" href="#__codelineno-17-32"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-33" name="__codelineno-17-33" href="#__codelineno-17-33"></a><span class="w">    </span><span class="c1">// Use result array res to overwrite the original array nums</span>
+<a id="__codelineno-17-33" name="__codelineno-17-33" href="#__codelineno-17-33"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result array res</span>
 <a id="__codelineno-17-34" name="__codelineno-17-34" href="#__codelineno-17-34"></a><span class="w">    </span><span class="nb">copy</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span>
 <a id="__codelineno-17-35" name="__codelineno-17-35" href="#__codelineno-17-35"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="c1">// Complete implementation, can sort objects and is a stable sort</span>
+<a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="c1">// Full implementation; stable and adaptable to sorting objects</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="kd">func</span><span class="w"> </span><span class="nf">countingSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">max</span><span class="p">()</span><span class="o">!</span>
-<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">counter</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">    </span><span class="c1">// 3. Calculate the prefix sum of counter, converting "occurrence count" to "tail index"</span>
-<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">    </span><span class="c1">// counter[num]-1 is the last index where num appears in res</span>
+<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">    </span><span class="c1">// 3. Compute prefix sums of counter to convert occurrence counts into end positions</span>
+<a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">    </span><span class="c1">// That is, counter[num]-1 is the index of the last occurrence of num in res</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse order, placing each element into the result array res</span>
-<a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">    </span><span class="c1">// Initialize the array res to record results</span>
+<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse and place each element in the result array res</span>
+<a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">    </span><span class="c1">// Initialize array res to store the result</span>
 <a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">count</span><span class="p">)</span>
 <a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="n">reversed</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="c1">// Place num at the corresponding index</span>
-<a id="__codelineno-18-23" name="__codelineno-18-23" href="#__codelineno-18-23"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Decrement the prefix sum by 1, getting the next index to place num</span>
+<a id="__codelineno-18-23" name="__codelineno-18-23" href="#__codelineno-18-23"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Decrement the prefix sum by 1 to determine where to place the next occurrence of num</span>
 <a id="__codelineno-18-24" name="__codelineno-18-24" href="#__codelineno-18-24"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-25" name="__codelineno-18-25" href="#__codelineno-18-25"></a><span class="w">    </span><span class="c1">// Use result array res to overwrite the original array nums</span>
+<a id="__codelineno-18-25" name="__codelineno-18-25" href="#__codelineno-18-25"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result array res</span>
 <a id="__codelineno-18-26" name="__codelineno-18-26" href="#__codelineno-18-26"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-27" name="__codelineno-18-27" href="#__codelineno-18-27"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-18-28" name="__codelineno-18-28" href="#__codelineno-18-28"></a><span class="w">    </span><span class="p">}</span>
@@ -5290,31 +5365,31 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="c1">// Complete implementation, can sort objects and is a stable sort</span>
+<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="c1">// Full implementation; stable and adaptable to sorting objects</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="kd">function</span><span class="w"> </span><span class="nx">countingSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(...</span><span class="nx">nums</span><span class="p">);</span>
-<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">]</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">    </span><span class="c1">// 3. Calculate the prefix sum of counter, converting "occurrence count" to "tail index"</span>
-<a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">    </span><span class="c1">// counter[num]-1 is the last index where num appears in res</span>
+<a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">    </span><span class="c1">// 3. Compute prefix sums of counter to convert occurrence counts into end positions</span>
+<a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">    </span><span class="c1">// That is, counter[num]-1 is the index of the last occurrence of num in res</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">counter</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse order, placing each element into the result array res</span>
-<a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">    </span><span class="c1">// Initialize the array res to record results</span>
+<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse and place each element in the result array res</span>
+<a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">    </span><span class="c1">// Initialize array res to store the result</span>
 <a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">);</span>
 <a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-19-23" name="__codelineno-19-23" href="#__codelineno-19-23"></a><span class="w">        </span><span class="nx">res</span><span class="p">[</span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span><span class="w"> </span><span class="c1">// Place num at the corresponding index</span>
-<a id="__codelineno-19-24" name="__codelineno-19-24" href="#__codelineno-19-24"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrement the prefix sum by 1, getting the next index to place num</span>
+<a id="__codelineno-19-24" name="__codelineno-19-24" href="#__codelineno-19-24"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrement the prefix sum by 1 to determine where to place the next occurrence of num</span>
 <a id="__codelineno-19-25" name="__codelineno-19-25" href="#__codelineno-19-25"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-26" name="__codelineno-19-26" href="#__codelineno-19-26"></a><span class="w">    </span><span class="c1">// Use result array res to overwrite the original array nums</span>
+<a id="__codelineno-19-26" name="__codelineno-19-26" href="#__codelineno-19-26"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result array res</span>
 <a id="__codelineno-19-27" name="__codelineno-19-27" href="#__codelineno-19-27"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-28" name="__codelineno-19-28" href="#__codelineno-19-28"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">res</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-19-29" name="__codelineno-19-29" href="#__codelineno-19-29"></a><span class="w">    </span><span class="p">}</span>
@@ -5323,31 +5398,31 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="c1">// Complete implementation, can sort objects and is a stable sort</span>
+<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="c1">// Full stable implementation; this approach can be adapted to sort objects</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="kd">function</span><span class="w"> </span><span class="nx">countingSort</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">m</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(...</span><span class="nx">nums</span><span class="p">);</span>
-<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">counter</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">]</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">    </span><span class="c1">// 3. Calculate the prefix sum of counter, converting "occurrence count" to "tail index"</span>
-<a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">    </span><span class="c1">// counter[num]-1 is the last index where num appears in res</span>
+<a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">    </span><span class="c1">// 3. Compute prefix sums of counter to convert occurrence counts into end positions</span>
+<a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">    </span><span class="c1">// That is, counter[num]-1 is the index of the last occurrence of num in res</span>
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">counter</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse order, placing each element into the result array res</span>
-<a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">    </span><span class="c1">// Initialize the array res to record results</span>
+<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse and place each element in the result array res</span>
+<a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">    </span><span class="c1">// Initialize array res to store the result</span>
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">(</span><span class="nx">n</span><span class="p">);</span>
 <a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-20-23" name="__codelineno-20-23" href="#__codelineno-20-23"></a><span class="w">        </span><span class="nx">res</span><span class="p">[</span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span><span class="w"> </span><span class="c1">// Place num at the corresponding index</span>
-<a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrement the prefix sum by 1, getting the next index to place num</span>
+<a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">num</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrement the prefix sum by 1 to determine where to place the next occurrence of num</span>
 <a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="w">    </span><span class="c1">// Use result array res to overwrite the original array nums</span>
+<a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result array res</span>
 <a id="__codelineno-20-27" name="__codelineno-20-27" href="#__codelineno-20-27"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-28" name="__codelineno-20-28" href="#__codelineno-20-28"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">res</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-20-29" name="__codelineno-20-29" href="#__codelineno-20-29"></a><span class="w">    </span><span class="p">}</span>
@@ -5356,100 +5431,100 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="c1">// Complete implementation, can sort objects and is a stable sort</span>
+<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="c1">// Full stable implementation; this approach can be adapted to sort objects</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="kt">void</span><span class="w"> </span><span class="n">countingSort</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">    </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">_num</span><span class="p">);</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="c1">// counter[_num] represents occurrence count of _num</span>
+<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="c1">// counter[_num] is the number of occurrences of _num</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">    </span><span class="n">counter</span><span class="p">[</span><span class="n">_num</span><span class="p">]</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">  </span><span class="c1">// 3. Calculate the prefix sum of counter, converting "occurrence count" to "tail index"</span>
-<a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">  </span><span class="c1">// That is, counter[_num]-1 is the last occurrence index of _num in res</span>
+<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">  </span><span class="c1">// 3. Compute prefix sums of counter to convert occurrence counts into end positions</span>
+<a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">  </span><span class="c1">// That is, counter[_num]-1 is the index of the last occurrence of _num in res</span>
 <a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">    </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="w">  </span><span class="c1">// 4. Traverse nums in reverse order, placing each element into the result array res</span>
-<a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="w">  </span><span class="c1">// Initialize the array res to record results</span>
+<a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="w">  </span><span class="c1">// 4. Traverse nums in reverse and place each element in the result array res</span>
+<a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="w">  </span><span class="c1">// Initialize array res to store the result</span>
 <a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-25" name="__codelineno-21-25" href="#__codelineno-21-25"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
-<a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a><span class="w">    </span><span class="n">res</span><span class="p">[</span><span class="n">counter</span><span class="p">[</span><span class="n">_num</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span><span class="w"> </span><span class="c1">// Place _num at corresponding index</span>
-<a id="__codelineno-21-27" name="__codelineno-21-27" href="#__codelineno-21-27"></a><span class="w">    </span><span class="n">counter</span><span class="p">[</span><span class="n">_num</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrement prefix sum by 1 to get next placement index for _num</span>
+<a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a><span class="w">    </span><span class="n">res</span><span class="p">[</span><span class="n">counter</span><span class="p">[</span><span class="n">_num</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span><span class="w"> </span><span class="c1">// Place _num at the corresponding index</span>
+<a id="__codelineno-21-27" name="__codelineno-21-27" href="#__codelineno-21-27"></a><span class="w">    </span><span class="n">counter</span><span class="p">[</span><span class="n">_num</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrement the prefix sum by 1 to determine where to place the next occurrence of _num</span>
 <a id="__codelineno-21-28" name="__codelineno-21-28" href="#__codelineno-21-28"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-29" name="__codelineno-21-29" href="#__codelineno-21-29"></a><span class="w">  </span><span class="c1">// Use result array res to overwrite the original array nums</span>
+<a id="__codelineno-21-29" name="__codelineno-21-29" href="#__codelineno-21-29"></a><span class="w">  </span><span class="c1">// Overwrite the original array nums with the result array res</span>
 <a id="__codelineno-21-30" name="__codelineno-21-30" href="#__codelineno-21-30"></a><span class="w">  </span><span class="n">nums</span><span class="p">.</span><span class="n">setAll</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-21-31" name="__codelineno-21-31" href="#__codelineno-21-31"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="c1">// Complete implementation, can sort objects and is a stable sort</span>
+<a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="c1">// Stable implementation that can be adapted to sort objects</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="k">fn</span><span class="w"> </span><span class="nf">counting_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">.</span><span class="n">iter</span><span class="p">().</span><span class="n">max</span><span class="p">().</span><span class="n">unwrap</span><span class="p">()</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">;</span>
-<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="o">&amp;</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">iter</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">    </span><span class="c1">// 3. Calculate the prefix sum of counter, converting "occurrence count" to "tail index"</span>
-<a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">    </span><span class="c1">// counter[num]-1 is the last index where num appears in res</span>
+<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">    </span><span class="c1">// 3. Compute prefix sums of counter to convert occurrence counts into end positions</span>
+<a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">    </span><span class="c1">// That is, counter[num]-1 is the index of the last occurrence of num in res</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse order, placing each element into the result array res</span>
-<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">    </span><span class="c1">// Initialize the array res to record results</span>
+<a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse and place each element in the result array res</span>
+<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">    </span><span class="c1">// Initialize array res to store the result</span>
 <a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
 <a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="p">];</span>
 <a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="p">).</span><span class="n">rev</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span><span class="w"> </span><span class="c1">// Place num at the corresponding index</span>
-<a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrement the prefix sum by 1, getting the next index to place num</span>
+<a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrement the prefix sum by 1 to determine where to place the next occurrence of num</span>
 <a id="__codelineno-22-25" name="__codelineno-22-25" href="#__codelineno-22-25"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-26" name="__codelineno-22-26" href="#__codelineno-22-26"></a><span class="w">    </span><span class="c1">// Use result array res to overwrite the original array nums</span>
+<a id="__codelineno-22-26" name="__codelineno-22-26" href="#__codelineno-22-26"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result array res</span>
 <a id="__codelineno-22-27" name="__codelineno-22-27" href="#__codelineno-22-27"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="n">copy_from_slice</span><span class="p">(</span><span class="o">&amp;</span><span class="n">res</span><span class="p">)</span>
 <a id="__codelineno-22-28" name="__codelineno-22-28" href="#__codelineno-22-28"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="c1">// Complete implementation, can sort objects and is a stable sort</span>
+<a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="c1">// Full, stable implementation; can be adapted to sort objects</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="kt">void</span><span class="w"> </span><span class="nf">countingSort</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">            </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">calloc</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]]</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w">    </span><span class="c1">// 3. Calculate the prefix sum of counter, converting "occurrence count" to "tail index"</span>
-<a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">    </span><span class="c1">// counter[num]-1 is the last index where num appears in res</span>
+<a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w">    </span><span class="c1">// 3. Compute prefix sums of counter to convert occurrence counts into end positions</span>
+<a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">    </span><span class="c1">// That is, counter[num]-1 is the index of the last occurrence of num in res</span>
 <a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse order, placing each element into the result array res</span>
-<a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="w">    </span><span class="c1">// Initialize the array res to record results</span>
+<a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse and place each element in the result array res</span>
+<a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="w">    </span><span class="c1">// Initialize array res to store the result</span>
 <a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">size</span><span class="p">);</span>
 <a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span><span class="w"> </span><span class="c1">// Place num at the corresponding index</span>
-<a id="__codelineno-23-28" name="__codelineno-23-28" href="#__codelineno-23-28"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w">              </span><span class="c1">// Decrement the prefix sum by 1, getting the next index to place num</span>
+<a id="__codelineno-23-28" name="__codelineno-23-28" href="#__codelineno-23-28"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">num</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w">              </span><span class="c1">// Decrement the prefix sum by 1 to determine where to place the next occurrence of num</span>
 <a id="__codelineno-23-29" name="__codelineno-23-29" href="#__codelineno-23-29"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-30" name="__codelineno-23-30" href="#__codelineno-23-30"></a><span class="w">    </span><span class="c1">// Use result array res to overwrite the original array nums</span>
+<a id="__codelineno-23-30" name="__codelineno-23-30" href="#__codelineno-23-30"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result array res</span>
 <a id="__codelineno-23-31" name="__codelineno-23-31" href="#__codelineno-23-31"></a><span class="w">    </span><span class="n">memcpy</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
 <a id="__codelineno-23-32" name="__codelineno-23-32" href="#__codelineno-23-32"></a><span class="w">    </span><span class="c1">// 5. Free memory</span>
 <a id="__codelineno-23-33" name="__codelineno-23-33" href="#__codelineno-23-33"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">res</span><span class="p">);</span>
@@ -5459,34 +5534,34 @@
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Counting sort */</span>
-<a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="c1">// Complete implementation, can sort objects and is a stable sort</span>
+<a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="c1">// Full implementation; stable and adaptable to sorting objects</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">countingSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="c1">// 1. Count the maximum element m in the array</span>
+<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="c1">// 1. Find the maximum array element m</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">        </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrence of each number</span>
-<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="c1">// counter[num] represents the occurrence of num</span>
+<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="c1">// 2. Count the occurrences of each number</span>
+<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="c1">// counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]++</span>
 <a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">    </span><span class="c1">// 3. Calculate the prefix sum of counter, converting "occurrence count" to "tail index"</span>
-<a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">    </span><span class="c1">// counter[num]-1 is the last index where num appears in res</span>
+<a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">    </span><span class="c1">// 3. Compute prefix sums of counter to convert occurrence counts into end positions</span>
+<a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">    </span><span class="c1">// That is, counter[num]-1 is the index of the last occurrence of num in res</span>
 <a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse order, placing each element into the result array res</span>
-<a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="w">    </span><span class="c1">// Initialize the array res to record results</span>
+<a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w">    </span><span class="c1">// 4. Traverse nums in reverse and place each element in the result array res</span>
+<a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="w">    </span><span class="c1">// Initialize array res to store the result</span>
 <a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="n">downTo</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-25" name="__codelineno-24-25" href="#__codelineno-24-25"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-24-26" name="__codelineno-24-26" href="#__codelineno-24-26"></a><span class="w">        </span><span class="n">res</span><span class="o">[</span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="c1">// Place num at the corresponding index</span>
-<a id="__codelineno-24-27" name="__codelineno-24-27" href="#__codelineno-24-27"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]--</span><span class="w"> </span><span class="c1">// Decrement the prefix sum by 1, getting the next index to place num</span>
+<a id="__codelineno-24-27" name="__codelineno-24-27" href="#__codelineno-24-27"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]--</span><span class="w"> </span><span class="c1">// Decrement the prefix sum by 1 to determine where to place the next occurrence of num</span>
 <a id="__codelineno-24-28" name="__codelineno-24-28" href="#__codelineno-24-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-29" name="__codelineno-24-29" href="#__codelineno-24-29"></a><span class="w">    </span><span class="c1">// Use result array res to overwrite the original array nums</span>
+<a id="__codelineno-24-29" name="__codelineno-24-29" href="#__codelineno-24-29"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result array res</span>
 <a id="__codelineno-24-30" name="__codelineno-24-30" href="#__codelineno-24-30"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-31" name="__codelineno-24-31" href="#__codelineno-24-31"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">res</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-24-32" name="__codelineno-24-32" href="#__codelineno-24-32"></a><span class="w">    </span><span class="p">}</span>
@@ -5496,42 +5571,47 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">counting_sort.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Counting sort ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">counting_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Complete implementation, can sort objects and is a stable sort</span>
-<a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="c1"># 1. Count the maximum element m in the array</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Full, stable implementation; the approach can be adapted to sort objects</span>
+<a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="c1"># 1. Find the maximum array element m</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">max</span>
-<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="c1"># 2. Count the occurrence of each number</span>
-<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="c1"># counter[num] represents the occurrence of num</span>
+<a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="c1"># 2. Count the occurrences of each number</span>
+<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="c1"># counter[num] is the number of occurrences of num</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">  </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="n">nums</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">num</span><span class="o">|</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="c1"># 3. Calculate the prefix sum of counter, converting "occurrence count" to "tail index"</span>
-<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">  </span><span class="c1"># counter[num]-1 is the last index where num appears in res</span>
+<a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="c1"># 3. Compute prefix sums of counter to convert occurrence counts into end positions</span>
+<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">  </span><span class="c1"># That is, counter[num]-1 is the index of the last occurrence of num in res</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">  </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">m</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">  </span><span class="c1"># 4. Traverse nums in reverse, fill elements into result array res</span>
-<a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">  </span><span class="c1"># Initialize the array res to record results</span>
+<a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">  </span><span class="c1"># 4. Traverse nums in reverse and place each element in the result array res</span>
+<a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">  </span><span class="c1"># Initialize array res to store the result</span>
 <a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w">  </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">downto</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span>
 <a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w">    </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="w">    </span><span class="n">res</span><span class="o">[</span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="c1"># Place num at the corresponding index</span>
-<a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a><span class="w">    </span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Decrement the prefix sum by 1, getting the next index to place num</span>
+<a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a><span class="w">    </span><span class="n">counter</span><span class="o">[</span><span class="n">num</span><span class="o">]</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Decrement the prefix sum by 1 to determine where to place the next occurrence of num</span>
 <a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="w">  </span><span class="k">end</span>
-<a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="w">  </span><span class="c1"># Use result array res to overwrite the original array nums</span>
+<a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="w">  </span><span class="c1"># Overwrite the original array nums with the result array res</span>
 <a id="__codelineno-25-23" name="__codelineno-25-23" href="#__codelineno-25-23"></a><span class="w">  </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">res</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-25-24" name="__codelineno-25-24" href="#__codelineno-25-24"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h2 id="1193-algorithm-characteristics">11.9.3 &nbsp; Algorithm Characteristics<a class="headerlink" href="#1193-algorithm-characteristics" title="Permanent link">&para;</a></h2>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20counting_sort%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Counting%20sort%22%22%22%0A%20%20%20%20%23%20Full%20implementation%3B%20stable%20and%20adaptable%20to%20sorting%20objects%0A%20%20%20%20%23%201.%20Find%20the%20maximum%20array%20element%20m%0A%20%20%20%20m%20%3D%20max%28nums%29%0A%20%20%20%20%23%202.%20Count%20the%20occurrences%20of%20each%20number%0A%20%20%20%20%23%20counter%5Bnum%5D%20is%20the%20number%20of%20occurrences%20of%20num%0A%20%20%20%20counter%20%3D%20%5B0%5D%20%2A%20%28m%20%2B%201%29%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20counter%5Bnum%5D%20%2B%3D%201%0A%20%20%20%20%23%203.%20Compute%20prefix%20sums%20of%20counter%20to%20convert%20occurrence%20counts%20into%20end%20positions%0A%20%20%20%20%23%20That%20is,%20counter%5Bnum%5D-1%20is%20the%20index%20of%20the%20last%20occurrence%20of%20num%20in%20res%0A%20%20%20%20for%20i%20in%20range%28m%29%3A%0A%20%20%20%20%20%20%20%20counter%5Bi%20%2B%201%5D%20%2B%3D%20counter%5Bi%5D%0A%20%20%20%20%23%204.%20Traverse%20nums%20in%20reverse%20and%20place%20each%20element%20in%20the%20result%20array%20res%0A%20%20%20%20%23%20Initialize%20array%20res%20to%20store%20the%20result%0A%20%20%20%20n%20%3D%20len%28nums%29%0A%20%20%20%20res%20%3D%20%5B0%5D%20%2A%20n%0A%20%20%20%20for%20i%20in%20range%28n%20-%201,%20-1,%20-1%29%3A%0A%20%20%20%20%20%20%20%20num%20%3D%20nums%5Bi%5D%0A%20%20%20%20%20%20%20%20res%5Bcounter%5Bnum%5D%20-%201%5D%20%3D%20num%20%20%23%20Place%20num%20at%20the%20corresponding%20index%0A%20%20%20%20%20%20%20%20counter%5Bnum%5D%20-%3D%201%20%20%23%20Decrement%20the%20prefix%20sum%20by%201%20to%20determine%20where%20to%20place%20the%20next%20occurrence%20of%20num%0A%20%20%20%20%23%20Overwrite%20the%20original%20array%20nums%20with%20the%20result%20array%20res%0A%20%20%20%20for%20i%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20nums%5Bi%5D%20%3D%20res%5Bi%5D%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B1,%200,%201,%202,%200,%204,%200,%202,%202,%204%5D%0A%20%20%20%20counting_sort%28nums%29%0A%20%20%20%20print%28f%22After%20counting%20sort,%20nums%20%3D%20%7Bnums%7D%22%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20counting_sort%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Counting%20sort%22%22%22%0A%20%20%20%20%23%20Full%20implementation%3B%20stable%20and%20adaptable%20to%20sorting%20objects%0A%20%20%20%20%23%201.%20Find%20the%20maximum%20array%20element%20m%0A%20%20%20%20m%20%3D%20max%28nums%29%0A%20%20%20%20%23%202.%20Count%20the%20occurrences%20of%20each%20number%0A%20%20%20%20%23%20counter%5Bnum%5D%20is%20the%20number%20of%20occurrences%20of%20num%0A%20%20%20%20counter%20%3D%20%5B0%5D%20%2A%20%28m%20%2B%201%29%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20counter%5Bnum%5D%20%2B%3D%201%0A%20%20%20%20%23%203.%20Compute%20prefix%20sums%20of%20counter%20to%20convert%20occurrence%20counts%20into%20end%20positions%0A%20%20%20%20%23%20That%20is,%20counter%5Bnum%5D-1%20is%20the%20index%20of%20the%20last%20occurrence%20of%20num%20in%20res%0A%20%20%20%20for%20i%20in%20range%28m%29%3A%0A%20%20%20%20%20%20%20%20counter%5Bi%20%2B%201%5D%20%2B%3D%20counter%5Bi%5D%0A%20%20%20%20%23%204.%20Traverse%20nums%20in%20reverse%20and%20place%20each%20element%20in%20the%20result%20array%20res%0A%20%20%20%20%23%20Initialize%20array%20res%20to%20store%20the%20result%0A%20%20%20%20n%20%3D%20len%28nums%29%0A%20%20%20%20res%20%3D%20%5B0%5D%20%2A%20n%0A%20%20%20%20for%20i%20in%20range%28n%20-%201,%20-1,%20-1%29%3A%0A%20%20%20%20%20%20%20%20num%20%3D%20nums%5Bi%5D%0A%20%20%20%20%20%20%20%20res%5Bcounter%5Bnum%5D%20-%201%5D%20%3D%20num%20%20%23%20Place%20num%20at%20the%20corresponding%20index%0A%20%20%20%20%20%20%20%20counter%5Bnum%5D%20-%3D%201%20%20%23%20Decrement%20the%20prefix%20sum%20by%201%20to%20determine%20where%20to%20place%20the%20next%20occurrence%20of%20num%0A%20%20%20%20%23%20Overwrite%20the%20original%20array%20nums%20with%20the%20result%20array%20res%0A%20%20%20%20for%20i%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20nums%5Bi%5D%20%3D%20res%5Bi%5D%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B1,%200,%201,%202,%200,%204,%200,%202,%202,%204%5D%0A%20%20%20%20counting_sort%28nums%29%0A%20%20%20%20print%28f%22After%20counting%20sort,%20nums%20%3D%20%7Bnums%7D%22%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1193-algorithm-properties">11.9.3 &nbsp; Algorithm properties<a class="headerlink" href="#1193-algorithm-properties" title="Permanent link">&para;</a></h2>
 <ul>
-<li><strong>Time complexity is <span class="arithmatex">\(O(n + m)\)</span>, and counting sort is non-adaptive</strong>: Traversing <code>nums</code> and <code>counter</code> both takes linear time. In general, when <span class="arithmatex">\(n \gg m\)</span>, the time complexity approaches <span class="arithmatex">\(O(n)\)</span>.</li>
-<li><strong>Space complexity of <span class="arithmatex">\(O(n + m)\)</span>, non-in-place sorting</strong>: Uses arrays <code>res</code> and <code>counter</code> of lengths <span class="arithmatex">\(n\)</span> and <span class="arithmatex">\(m\)</span> respectively.</li>
-<li><strong>Stable sorting</strong>: Since elements are filled into <code>res</code> in a "right-to-left" order, traversing <code>nums</code> in reverse can avoid changing the relative positions of equal elements, thereby achieving stable sorting. In fact, traversing <code>nums</code> in forward order can also yield correct sorting results, but the result would be unstable.</li>
+<li><strong>Time complexity: <span class="arithmatex">\(O(n + m)\)</span>; not adaptive</strong>. Traversing <code>nums</code> and <code>counter</code> each takes linear time. Usually <span class="arithmatex">\(n \gg m\)</span>, so the complexity approaches <span class="arithmatex">\(O(n)\)</span>.</li>
+<li><strong>Space complexity: <span class="arithmatex">\(O(n + m)\)</span>; not in-place</strong>. The algorithm uses arrays <code>res</code> and <code>counter</code> of lengths <span class="arithmatex">\(n\)</span> and <span class="arithmatex">\(m\)</span>, respectively.</li>
+<li><strong>Stable</strong>. Since elements are placed into <code>res</code> from right to left, traversing <code>nums</code> in reverse preserves the relative order of equal elements. A forward traversal also produces a correctly sorted result, but it is not stable.</li>
 </ul>
 <h2 id="1194-limitations">11.9.4 &nbsp; Limitations<a class="headerlink" href="#1194-limitations" title="Permanent link">&para;</a></h2>
-<p>At this point, you might think counting sort is quite ingenious because it achieves efficient sorting simply by counting occurrences. However, the prerequisites for using counting sort are fairly restrictive.</p>
-<p><strong>Counting sort is only applicable to non-negative integers</strong>. To apply it to other types of data, you must ensure that they can be converted to non-negative integers without changing the relative ordering of the elements. For example, for an integer array containing negative numbers, you can first add a constant to every number to shift them into the non-negative range, and then shift them back after sorting.</p>
-<p><strong>Counting sort is well suited to cases with many elements but a small value range</strong>. For example, in the above scenario, <span class="arithmatex">\(m\)</span> cannot be too large; otherwise, it consumes too much space. And when <span class="arithmatex">\(n \ll m\)</span>, counting sort takes <span class="arithmatex">\(O(m)\)</span> time, which may be slower than sorting algorithms with <span class="arithmatex">\(O(n \log n)\)</span> time complexity.</p>
+<p>Counting sort may seem ingenious: it sorts efficiently simply by counting occurrences. However, its prerequisites are relatively restrictive.</p>
+<p><strong>Counting sort applies only to nonnegative integers</strong>. For other data types, the values must be convertible to nonnegative integers without changing their relative order. For example, if an integer array contains negative numbers, add a constant to every element to make them all positive, sort, and then reverse the transformation.</p>
+<p><strong>Counting sort suits large datasets with a relatively small value range</strong>. In the example above, <span class="arithmatex">\(m\)</span> must not be too large, or the algorithm will consume excessive space. When <span class="arithmatex">\(n \ll m\)</span>, counting sort takes <span class="arithmatex">\(O(m)\)</span> time and may be slower than an <span class="arithmatex">\(O(n \log n)\)</span> sorting algorithm.</p>
 
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-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
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       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
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@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
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@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
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@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2880,7 +2880,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2901,7 +2901,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2922,7 +2922,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2976,7 +2976,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3003,7 +3003,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3030,7 +3030,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3057,7 +3057,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3084,7 +3084,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3111,7 +3111,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3138,7 +3138,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3165,7 +3165,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3192,7 +3192,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3228,7 +3228,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3245,7 +3245,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3271,22 +3271,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#11121-concept-review" class="md-nav__link">
+  <a href="#11121-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.12.1 &nbsp; Concept Review
+        11.12.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="11.12.1 Concept Review">
+    <nav class="md-nav" aria-label="11.12.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-the-first-few-rounds-of-selection-sort-and-bubble-sort" class="md-nav__link">
+  <a href="#1-the-first-passes-of-selection-sort-and-bubble-sort" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; The First Few Rounds of Selection Sort and Bubble Sort
+        1. &nbsp; The first passes of selection sort and bubble sort
       </span>
     </span>
   </a>
@@ -3294,10 +3294,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-can-equal-elements-change-their-relative-order" class="md-nav__link">
+  <a href="#2-does-the-order-of-equal-elements-change" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Can Equal Elements Change Their Relative Order?
+        2. &nbsp; Does the order of equal elements change?
       </span>
     </span>
   </a>
@@ -3305,10 +3305,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-compare-counting-sort-and-radix-sort" class="md-nav__link">
+  <a href="#3-comparing-counting-sort-and-radix-sort" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Compare Counting Sort and Radix Sort
+        3. &nbsp; Comparing counting sort and radix sort
       </span>
     </span>
   </a>
@@ -3324,19 +3324,19 @@
   <a href="#11122-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.12.2 &nbsp; Programming Exercises
+        11.12.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="11.12.2 Programming Exercises">
+    <nav class="md-nav" aria-label="11.12.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-sort-an-array-with-merge-sort" class="md-nav__link">
+  <a href="#1-sorting-an-array-with-merge-sort" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Sort an Array with Merge Sort
+        1. &nbsp; Sorting an array with merge sort
       </span>
     </span>
   </a>
@@ -3344,10 +3344,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-sort-an-integer-array-with-counting-sort" class="md-nav__link">
+  <a href="#2-sorting-an-integer-array-with-counting-sort" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Sort an Integer Array with Counting Sort
+        2. &nbsp; Sorting an integer array with counting sort
       </span>
     </span>
   </a>
@@ -3421,7 +3421,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3442,7 +3442,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3463,7 +3463,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3490,7 +3490,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3517,7 +3517,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3544,7 +3544,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3571,7 +3571,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3598,7 +3598,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3664,7 +3664,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3685,7 +3685,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3706,7 +3706,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3733,7 +3733,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3760,7 +3760,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3787,7 +3787,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3814,7 +3814,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3841,7 +3841,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3911,7 +3911,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3932,7 +3932,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3953,7 +3953,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3980,7 +3980,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -4007,7 +4007,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4034,7 +4034,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4061,7 +4061,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4088,7 +4088,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4115,7 +4115,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4142,7 +4142,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4208,7 +4208,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4229,7 +4229,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4250,7 +4250,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4277,7 +4277,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4304,7 +4304,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4331,7 +4331,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4358,7 +4358,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4445,7 +4445,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4466,7 +4466,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4487,7 +4487,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4514,7 +4514,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4541,7 +4541,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4625,6 +4625,78 @@
 
 
 
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@@ -4666,22 +4738,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#11121-concept-review" class="md-nav__link">
+  <a href="#11121-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.12.1 &nbsp; Concept Review
+        11.12.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="11.12.1 Concept Review">
+    <nav class="md-nav" aria-label="11.12.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-the-first-few-rounds-of-selection-sort-and-bubble-sort" class="md-nav__link">
+  <a href="#1-the-first-passes-of-selection-sort-and-bubble-sort" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; The First Few Rounds of Selection Sort and Bubble Sort
+        1. &nbsp; The first passes of selection sort and bubble sort
       </span>
     </span>
   </a>
@@ -4689,10 +4761,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-can-equal-elements-change-their-relative-order" class="md-nav__link">
+  <a href="#2-does-the-order-of-equal-elements-change" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Can Equal Elements Change Their Relative Order?
+        2. &nbsp; Does the order of equal elements change?
       </span>
     </span>
   </a>
@@ -4700,10 +4772,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-compare-counting-sort-and-radix-sort" class="md-nav__link">
+  <a href="#3-comparing-counting-sort-and-radix-sort" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Compare Counting Sort and Radix Sort
+        3. &nbsp; Comparing counting sort and radix sort
       </span>
     </span>
   </a>
@@ -4719,19 +4791,19 @@
   <a href="#11122-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.12.2 &nbsp; Programming Exercises
+        11.12.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="11.12.2 Programming Exercises">
+    <nav class="md-nav" aria-label="11.12.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-sort-an-array-with-merge-sort" class="md-nav__link">
+  <a href="#1-sorting-an-array-with-merge-sort" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Sort an Array with Merge Sort
+        1. &nbsp; Sorting an array with merge sort
       </span>
     </span>
   </a>
@@ -4739,10 +4811,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-sort-an-integer-array-with-counting-sort" class="md-nav__link">
+  <a href="#2-sorting-an-integer-array-with-counting-sort" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Sort an Integer Array with Counting Sort
+        2. &nbsp; Sorting an integer array with counting sort
       </span>
     </span>
   </a>
@@ -4792,24 +4864,24 @@
 
 <!-- Page content -->
 <h1 id="1112-exercises">11.12 &nbsp; Exercises<a class="headerlink" href="#1112-exercises" title="Permanent link">&para;</a></h1>
-<h2 id="11121-concept-review">11.12.1 &nbsp; Concept Review<a class="headerlink" href="#11121-concept-review" title="Permanent link">&para;</a></h2>
-<h3 id="1-the-first-few-rounds-of-selection-sort-and-bubble-sort">1. &nbsp; The First Few Rounds of Selection Sort and Bubble Sort<a class="headerlink" href="#1-the-first-few-rounds-of-selection-sort-and-bubble-sort" title="Permanent link">&para;</a></h3>
-<p>Given the array <code>[4, 2, 5, 1, 3]</code>, sort it in ascending order in both parts below.</p>
+<h2 id="11121-check-your-understanding">11.12.1 &nbsp; Check your understanding<a class="headerlink" href="#11121-check-your-understanding" title="Permanent link">&para;</a></h2>
+<h3 id="1-the-first-passes-of-selection-sort-and-bubble-sort">1. &nbsp; The first passes of selection sort and bubble sort<a class="headerlink" href="#1-the-first-passes-of-selection-sort-and-bubble-sort" title="Permanent link">&para;</a></h3>
+<p>Given the array <code>[4, 2, 5, 1, 3]</code>, sort in ascending order throughout this exercise.</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Simulate the first two rounds of selection sort. Write the array after each round and identify which positions are now fixed.</li>
-<li>Simulate the first round of bubble sort. Write the resulting array and the number of swaps, and identify which position is now fixed.</li>
+<li>Simulate the first two passes of selection sort. Write the array after each pass and identify the positions whose final values are now fixed.</li>
+<li>Simulate the first pass of bubble sort. Write the array and the number of swaps, and identify the position whose final value is now fixed.</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>The first two rounds are:</p>
+<p>The first two passes are:</p>
 <table>
 <thead>
 <tr>
-<th>Round</th>
+<th>Pass</th>
 <th>Array</th>
 <th>Explanation</th>
 </tr>
@@ -4818,7 +4890,7 @@
 <tr>
 <td>1</td>
 <td><code>[1, 2, 5, 4, 3]</code></td>
-<td>The smallest element, 1, is swapped with the first element</td>
+<td>Swap the minimum element, 1, with the first element</td>
 </tr>
 <tr>
 <td>2</td>
@@ -4827,87 +4899,87 @@
 </tr>
 </tbody>
 </table>
-<p>The first two positions are now fixed. Later rounds need to find the smallest element only within <code>[5, 4, 3]</code>.</p>
+<p>The first two positions are now fixed. Subsequent passes need only continue selecting the minimum from <code>[5, 4, 3]</code>.</p>
 </li>
 <li>
-<p>Compare adjacent elements in order: swap 4 and 2; do not swap 4 and 5; swap 5 and 1; then swap 5 and 3.
-    The result is <code>[2, 4, 1, 3, 5]</code>, after 3 swaps. The largest element, 5, has moved to the end of the array, so the last position is now fixed.</p>
+<p>Compare adjacent elements in order: swap 4 and 2, leave 4 and 5 unchanged, swap 5 and 1, then swap 5 and 3.
+    The result is <code>[2, 4, 1, 3, 5]</code>, with three swaps. The maximum element, 5, has moved to the end, fixing the last position.</p>
 </li>
 </ol>
 </details>
-<h3 id="2-can-equal-elements-change-their-relative-order">2. &nbsp; Can Equal Elements Change Their Relative Order?<a class="headerlink" href="#2-can-equal-elements-change-their-relative-order" title="Permanent link">&para;</a></h3>
-<p>In the array <span class="arithmatex">\([2_a, 2_b, 1]\)</span>, <span class="arithmatex">\(2_a\)</span> and <span class="arithmatex">\(2_b\)</span> have equal values, but their subscripts mark their original order.</p>
+<h3 id="2-does-the-order-of-equal-elements-change">2. &nbsp; Does the order of equal elements change?<a class="headerlink" href="#2-does-the-order-of-equal-elements-change" title="Permanent link">&para;</a></h3>
+<p>In the array <span class="arithmatex">\([2_a, 2_b, 1]\)</span>, <span class="arithmatex">\(2_a\)</span> and <span class="arithmatex">\(2_b\)</span> have equal values; the subscripts mark their original order.</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Write the array after the first round of selection sort. Has the relative order of <span class="arithmatex">\(2_a\)</span> and <span class="arithmatex">\(2_b\)</span> changed?</li>
-<li>Write the array after the first round of bubble sort. Has the relative order of <span class="arithmatex">\(2_a\)</span> and <span class="arithmatex">\(2_b\)</span> changed?</li>
-<li>Based on the first two questions, explain how the two sorting algorithms differ in preserving the original order of equal elements.</li>
+<li>Write the array after the first pass of selection sort. Has the relative order of <span class="arithmatex">\(2_a\)</span> and <span class="arithmatex">\(2_b\)</span> changed?</li>
+<li>Write the array after the first pass of bubble sort. Has the relative order of <span class="arithmatex">\(2_a\)</span> and <span class="arithmatex">\(2_b\)</span> changed?</li>
+<li>Based on your answers, explain how the two algorithms differ in preserving the original order of equal elements.</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>In its first round, selection sort selects the smallest element, 1, and swaps it with the first element, <span class="arithmatex">\(2_a\)</span>, producing
-    <span class="arithmatex">\([1, 2_b, 2_a]\)</span>. The relative order has changed because <span class="arithmatex">\(2_a\)</span> has moved behind <span class="arithmatex">\(2_b\)</span>.</p>
+<p>The first pass of selection sort selects the minimum element, 1, and swaps it with the first element, <span class="arithmatex">\(2_a\)</span>, producing
+    <span class="arithmatex">\([1, 2_b, 2_a]\)</span>. The swap moves <span class="arithmatex">\(2_a\)</span> after <span class="arithmatex">\(2_b\)</span>, changing their relative order.</p>
 </li>
 <li>
-<p>Bubble sort first compares <span class="arithmatex">\(2_a\)</span> and <span class="arithmatex">\(2_b\)</span>. Because they are equal, it does not swap them. It then compares <span class="arithmatex">\(2_b\)</span> and 1 and swaps them,
-    producing <span class="arithmatex">\([2_a, 1, 2_b]\)</span> after the first round. <span class="arithmatex">\(2_a\)</span> is still before <span class="arithmatex">\(2_b\)</span>, so their relative order has not changed.</p>
+<p>Bubble sort first compares <span class="arithmatex">\(2_a\)</span> with <span class="arithmatex">\(2_b\)</span> and does not swap them because they are equal. It then compares and swaps <span class="arithmatex">\(2_b\)</span> and 1.
+    After the first pass, the array is <span class="arithmatex">\([2_a, 1, 2_b]\)</span>. Since <span class="arithmatex">\(2_a\)</span> still precedes <span class="arithmatex">\(2_b\)</span>, their relative order is unchanged.</p>
 </li>
 <li>
-<p>In this example, selection sort changes the original order of equal elements. Bubble sort swaps adjacent elements only when the left element is greater than the right one.
-    Equal elements are not swapped, so their original relative order is preserved.</p>
+<p>In this example, selection sort changes the original order of equal elements. Bubble sort swaps adjacent elements only when the left element
+    is greater than the right one. Equal elements are never swapped with each other, so their original order is preserved.</p>
 </li>
 </ol>
 </details>
-<h3 id="3-compare-counting-sort-and-radix-sort">3. &nbsp; Compare Counting Sort and Radix Sort<a class="headerlink" href="#3-compare-counting-sort-and-radix-sort" title="Permanent link">&para;</a></h3>
-<p>A school needs to sort many student ID numbers, each exactly 8 digits long. Answer the following questions:</p>
+<h3 id="3-comparing-counting-sort-and-radix-sort">3. &nbsp; Comparing counting sort and radix sort<a class="headerlink" href="#3-comparing-counting-sort-and-radix-sort" title="Permanent link">&para;</a></h3>
+<p>A school needs to sort many student IDs, each exactly eight digits long. Answer the following:</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>How many rounds does radix sort need when it starts with the least significant digit?</li>
-<li>If the student IDs are treated directly as integers for counting sort, why would the count array need many entries that are never used?</li>
-<li>Based on the first two questions, which would you choose for sorting many fixed-length 8-digit student IDs: counting sort or radix sort? Why?</li>
+<li>How many passes does radix sort need when starting with the least significant digit?</li>
+<li>If counting sort treats each student ID directly as an integer, why would many counting entries go unused?</li>
+<li>Based on your answers, which would you choose for sorting many eight-digit student IDs: counting sort or radix sort? Why?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>A student ID has 8 digits, so 8 rounds are needed from the least significant to the most significant digit. Each round groups values only by 0–9.</p>
+<p>Each ID has eight digits, so there are eight passes, from the least to the most significant digit. Each pass groups digits only by values 0 through 9.</p>
 </li>
 <li>
-<p>Direct counting would require an entry for every possible 8-digit value, but only a small fraction of those values are actually assigned to students.
-    Most entries in the count array would remain 0.</p>
+<p>Direct counting reserves an entry for every possible eight-digit value, but actual students use only a small fraction of them.
+    Most counts would remain 0.</p>
 </li>
 <li>
-<p>Radix sort is the better choice. It uses the facts that the length is fixed and each digit has only 10 possible values, requiring only 8 rounds of stable grouping.
-    Counting sort, if it used the entire 8-digit ID as an integer index, would require count-array entries for many values that never occur.</p>
+<p>Choose radix sort. It uses the fixed number of digits and the ten possible values per digit, requiring only eight stable grouping passes.
+    Counting sort using each entire eight-digit ID as an integer index would reserve entries for many values that never occur.</p>
 </li>
 </ol>
 </details>
-<h2 id="11122-programming-exercises">11.12.2 &nbsp; Programming Exercises<a class="headerlink" href="#11122-programming-exercises" title="Permanent link">&para;</a></h2>
-<h3 id="1-sort-an-array-with-merge-sort">1. &nbsp; Sort an Array with Merge Sort<a class="headerlink" href="#1-sort-an-array-with-merge-sort" title="Permanent link">&para;</a></h3>
-<p>Given an integer array <code>nums</code>, implement merge sort yourself, arrange its elements in nondecreasing order, and return the result. Do not call the language's built-in sorting function.</p>
+<h2 id="11122-programming-exercises">11.12.2 &nbsp; Programming exercises<a class="headerlink" href="#11122-programming-exercises" title="Permanent link">&para;</a></h2>
+<h3 id="1-sorting-an-array-with-merge-sort">1. &nbsp; Sorting an array with merge sort<a class="headerlink" href="#1-sorting-an-array-with-merge-sort" title="Permanent link">&para;</a></h3>
+<p>Given an integer array <code>nums</code>, implement merge sort yourself to arrange its elements in nondecreasing order and return the result. Do not call the language's built-in sorting function.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>An interval of length at most 1 is already sorted</li>
-<li>Divide the interval in half at its midpoint and recursively sort both halves</li>
-<li>Use two pointers to merge the two sorted halves, then write the result back into the original array</li>
+<li>An interval of length at most 1 is already sorted.</li>
+<li>Split the interval in half at the midpoint and recursively sort each half.</li>
+<li>Use two pointers to merge the sorted halves, then write the result back to the original array.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/sort-an-array/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
-<h3 id="2-sort-an-integer-array-with-counting-sort">2. &nbsp; Sort an Integer Array with Counting Sort<a class="headerlink" href="#2-sort-an-integer-array-with-counting-sort" title="Permanent link">&para;</a></h3>
-<p>You are given an integer array <code>nums</code> and a non-negative integer <span class="arithmatex">\(K\)</span>. Every element of the array is between <span class="arithmatex">\(0\)</span> and <span class="arithmatex">\(K\)</span>.</p>
-<p>Implement counting sort, write the result back into <code>nums</code> in nondecreasing order, and return <code>nums</code>.
-Do not determine the order by comparing elements, and do not call the language's built-in sorting function.</p>
+<h3 id="2-sorting-an-integer-array-with-counting-sort">2. &nbsp; Sorting an integer array with counting sort<a class="headerlink" href="#2-sorting-an-integer-array-with-counting-sort" title="Permanent link">&para;</a></h3>
+<p>You are given an integer array <code>nums</code> and a nonnegative integer <span class="arithmatex">\(K\)</span>. Every element lies between <span class="arithmatex">\(0\)</span> and <span class="arithmatex">\(K\)</span>.</p>
+<p>Implement counting sort, write the elements back into <code>nums</code> in nondecreasing order, and return <code>nums</code>.
+Do not determine the order by comparing elements with one another, and do not use the language's built-in sorting function.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>Because every element is between 0 and K, use each element's value directly as an index in the counting array</li>
-<li>Scan nums once and increment the count at the corresponding position</li>
-<li>Then scan the counting array from 0 to K; if the value x occurs a certain number of times, write x into nums that many consecutive times</li>
+<li>Since every element is between 0 and K, its value can directly index the counting array.</li>
+<li>First scan nums, incrementing the count at the corresponding position by 1.</li>
+<li>Then scan the counting array from 0 through K. If x occurs a given number of times, write that many consecutive copies of x into nums.</li>
 </ol>
 </details>
 
@@ -4932,7 +5004,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 11.11 Summary"
+    aria-label="Previous: 11.11   Summary"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4944,7 +5016,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        11.11 Summary
+        11.11 &nbsp; Summary
       </div>
     </div>
   </a>
@@ -4956,7 +5028,7 @@ aria-label="Footer"
   <a
     href="../../chapter_divide_and_conquer/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: Chapter 12. &amp;nbsp; Divide and Conquer"
+    aria-label="Next: Chapter 12.   Divide and conquer"
     rel="next"
   >
     <div class="md-footer__title">
@@ -4964,7 +5036,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        Chapter 12. &nbsp; Divide and Conquer
+        Chapter 12. &nbsp; Divide and conquer
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -4981,7 +5053,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_sorting/heap_sort/index.html b/en/chapter_sorting/heap_sort/index.html
index b8ba9a0e4..4ef1533bf 100644
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       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>11.7 Heap Sort - Hello Algo</title>
+        <title>11.7 Heap sort - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
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-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-              11.7 &nbsp; Heap Sort
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-  Before Starting
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-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
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-  Chapter 0. Preface
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
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@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
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-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
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@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
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   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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@@ -2378,7 +2378,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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@@ -2596,7 +2596,7 @@
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2880,7 +2880,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2901,7 +2901,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2922,7 +2922,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2976,7 +2976,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3003,7 +3003,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3030,7 +3030,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3057,7 +3057,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3093,7 +3093,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3110,7 +3110,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3136,10 +3136,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1171-algorithm-flow" class="md-nav__link">
+  <a href="#1171-algorithm-steps" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.7.1 &nbsp; Algorithm Flow
+        11.7.1 &nbsp; Algorithm steps
       </span>
     </span>
   </a>
@@ -3147,10 +3147,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1172-algorithm-characteristics" class="md-nav__link">
+  <a href="#1172-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.7.2 &nbsp; Algorithm Characteristics
+        11.7.2 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -3180,7 +3180,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3207,7 +3207,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3234,7 +3234,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3261,7 +3261,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3288,7 +3288,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3375,7 +3375,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3396,7 +3396,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3423,7 +3423,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3450,7 +3450,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3477,7 +3477,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3504,7 +3504,7 @@
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3531,7 +3531,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3597,7 +3597,7 @@
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3618,7 +3618,7 @@
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3639,7 +3639,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3666,7 +3666,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3693,7 +3693,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3720,7 +3720,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3747,7 +3747,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3774,7 +3774,7 @@
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3844,7 +3844,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3865,7 +3865,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3886,7 +3886,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3913,7 +3913,7 @@
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3940,7 +3940,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3967,7 +3967,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -3994,7 +3994,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4021,7 +4021,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4048,7 +4048,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4075,7 +4075,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4141,7 +4141,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4162,7 +4162,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4183,7 +4183,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4210,7 +4210,7 @@
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4378,7 +4378,7 @@
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4420,7 +4420,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4447,7 +4447,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4474,7 +4474,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4558,6 +4558,78 @@
 
 
 
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@@ -4599,10 +4671,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1171-algorithm-flow" class="md-nav__link">
+  <a href="#1171-algorithm-steps" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.7.1 &nbsp; Algorithm Flow
+        11.7.1 &nbsp; Algorithm steps
       </span>
     </span>
   </a>
@@ -4610,10 +4682,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1172-algorithm-characteristics" class="md-nav__link">
+  <a href="#1172-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.7.2 &nbsp; Algorithm Characteristics
+        11.7.2 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -4657,33 +4729,33 @@
 
 
 <!-- Page content -->
-<h1 id="117-heap-sort">11.7 &nbsp; Heap Sort<a class="headerlink" href="#117-heap-sort" title="Permanent link">&para;</a></h1>
+<h1 id="117-heap-sort">11.7 &nbsp; Heap sort<a class="headerlink" href="#117-heap-sort" title="Permanent link">&para;</a></h1>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>Before reading this section, please ensure you have completed the "Heap" chapter.</p>
+<p>Before reading this section, make sure you have completed the chapter on heaps.</p>
 </div>
-<p><u>Heap sort</u> is an efficient sorting algorithm based on the heap data structure. We can implement heap sort using the heap construction and element removal operations introduced earlier.</p>
+<p><u>Heap sort</u> is an efficient sorting algorithm built on the heap data structure. We can implement it with the heap-building and removal operations we have already learned.</p>
 <ol>
-<li>Input the array and build a min-heap, at which point the smallest element is at the heap top.</li>
-<li>Continuously perform element removal operations and record the removed elements in order to obtain a sequence sorted in ascending order.</li>
+<li>Build a min heap from the input array. The smallest element is now at the root.</li>
+<li>Repeatedly remove the root and record the removed elements to obtain an ascending sequence.</li>
 </ol>
-<p>Although the above method is feasible, it requires an additional array to save the popped elements, which is quite wasteful of space. In practice, we usually use a more elegant implementation method.</p>
-<h2 id="1171-algorithm-flow">11.7.1 &nbsp; Algorithm Flow<a class="headerlink" href="#1171-algorithm-flow" title="Permanent link">&para;</a></h2>
-<p>Assume the array length is <span class="arithmatex">\(n\)</span>. The flow of heap sort is shown in Figure 11-12.</p>
+<p>Although this works, it requires an extra array to store the removed elements, wasting space. In practice, we usually use a more elegant implementation.</p>
+<h2 id="1171-algorithm-steps">11.7.1 &nbsp; Algorithm steps<a class="headerlink" href="#1171-algorithm-steps" title="Permanent link">&para;</a></h2>
+<p>For an array of length <span class="arithmatex">\(n\)</span>, heap sort proceeds as shown below.</p>
 <ol>
-<li>Input the array and build a max-heap. After completion, the largest element is at the heap top.</li>
-<li>Swap the heap top element (first element) with the heap bottom element (last element). After the swap is complete, reduce the heap length by <span class="arithmatex">\(1\)</span> and increase the count of sorted elements by <span class="arithmatex">\(1\)</span>.</li>
-<li>Starting from the heap top element, perform a top-to-bottom heapify operation (sift down). After heapify is complete, the heap property is restored.</li>
-<li>Repeat steps <code>2.</code> and <code>3.</code> After <span class="arithmatex">\(n - 1\)</span> rounds, the array is sorted.</li>
+<li>Build a max heap from the input array. The largest element is now at the root.</li>
+<li>Swap the root, the first element, with the last element of the heap. Reduce the heap's length by <span class="arithmatex">\(1\)</span> and increase the number of sorted elements by <span class="arithmatex">\(1\)</span>.</li>
+<li>Starting at the root, sift down to restore the heap property.</li>
+<li>Repeat steps <code>2.</code> and <code>3.</code>. After <span class="arithmatex">\(n - 1\)</span> rounds, the array is sorted.</li>
 </ol>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>In fact, the element removal operation also includes steps <code>2.</code> and <code>3.</code>, with the additional step of removing the element.</p>
+<p>In fact, removing the root of a heap also includes steps <code>2.</code> and <code>3.</code>, with the additional step of removing an element from the array.</p>
 </div>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label><label for="__tabbed_1_6">&lt;6&gt;</label><label for="__tabbed_1_7">&lt;7&gt;</label><label for="__tabbed_1_8">&lt;8&gt;</label><label for="__tabbed_1_9">&lt;9&gt;</label><label for="__tabbed_1_10">&lt;10&gt;</label><label for="__tabbed_1_11">&lt;11&gt;</label><label for="__tabbed_1_12">&lt;12&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Heap sort steps" class="animation-figure" src="../heap_sort.assets/heap_sort_step1.png" /></p>
+<p><img alt="Steps of heap sort" class="animation-figure" src="../heap_sort.assets/heap_sort_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="heap_sort_step2" class="animation-figure" src="../heap_sort.assets/heap_sort_step2.png" /></p>
@@ -4720,16 +4792,16 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 11-12 &nbsp; Heap sort steps </p>
+<p align="center"> Figure 11-12 &nbsp; Steps of heap sort </p>
 
-<p>In the code below, we use the same <code>sift_down()</code> function for top-to-bottom heapify as in the "Heap" chapter. It is worth noting that since the heap length decreases as the largest element is extracted, we need to add a length parameter <span class="arithmatex">\(n\)</span> to <code>sift_down()</code> to specify the current effective length of the heap. The code is as follows:</p>
+<p>The implementation uses the same <code>sift_down()</code> function introduced in the heap chapter. Because the heap shrinks as maximum elements are extracted, add a length parameter <span class="arithmatex">\(n\)</span> to specify its current valid length. The code is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">heap_sort.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">sift_down</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">n</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Heap length is n, start heapifying node i, from top to bottom"""</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Sift down from node i in a heap of size n"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
-<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>        <span class="c1"># Determine the largest node among i, l, r, noted as ma</span>
+<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>        <span class="c1"># Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="n">l</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="n">r</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">2</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="n">ma</span> <span class="o">=</span> <span class="n">i</span>
@@ -4737,32 +4809,32 @@
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>            <span class="n">ma</span> <span class="o">=</span> <span class="n">l</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="k">if</span> <span class="n">r</span> <span class="o">&lt;</span> <span class="n">n</span> <span class="ow">and</span> <span class="n">nums</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">]:</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>            <span class="n">ma</span> <span class="o">=</span> <span class="n">r</span>
-<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="c1"># If node i is the largest or indices l, r are out of bounds, no further heapification needed, break</span>
+<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="c1"># Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="k">if</span> <span class="n">ma</span> <span class="o">==</span> <span class="n">i</span><span class="p">:</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>            <span class="k">break</span>
-<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>        <span class="c1"># Swap two nodes</span>
+<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>        <span class="c1"># Swap the two nodes</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>        <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>        <span class="c1"># Loop downwards heapification</span>
+<a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>        <span class="c1"># Continue sifting down</span>
 <a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>        <span class="n">i</span> <span class="o">=</span> <span class="n">ma</span>
 <a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>
 <a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a><span class="k">def</span><span class="w"> </span><span class="nf">heap_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]):</span>
 <a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="w">    </span><span class="sd">"""Heap sort"""</span>
-<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>    <span class="c1"># Build heap operation: heapify all nodes except leaves</span>
+<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>    <span class="c1"># Build the heap by sifting down from every non-leaf node</span>
 <a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a>        <span class="n">sift_down</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">),</span> <span class="n">i</span><span class="p">)</span>
-<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a>    <span class="c1"># Extract the largest element from the heap and repeat for n-1 rounds</span>
+<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a>    <span class="c1"># Extract the largest element from the heap, repeating for n-1 rounds</span>
 <a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
-<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a>        <span class="c1"># Swap the root node with the rightmost leaf node (swap the first element with the last element)</span>
+<a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a>        <span class="c1"># Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)</span>
 <a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a>        <span class="n">nums</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
-<a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a>        <span class="c1"># Start heapifying the root node, from top to bottom</span>
+<a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a>        <span class="c1"># Sift down starting at the root</span>
 <a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a>        <span class="n">sift_down</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap_sort.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Heap length is n, start heapifying node i, from top to bottom */</span>
+<div class="highlight"><span class="filename">heap_sort.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Sift down from node i in a heap of size n */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">siftDown</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nb">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
@@ -4770,38 +4842,38 @@
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span><span class="p">;</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="p">;</span>
-<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">]);</span>
-<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span><span class="p">;</span>
 <a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="p">}</span>
 <a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a>
 <a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="cm">/* Heap sort */</span>
 <a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="kt">void</span><span class="w"> </span><span class="nf">heapSort</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w">    </span><span class="c1">// Build heap operation: heapify all nodes except leaves</span>
+<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w">    </span><span class="c1">// Build the heap by sifting down from every non-leaf node</span>
 <a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="o">--</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a><span class="w">        </span><span class="n">siftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">(),</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap and repeat for n-1 rounds</span>
+<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap, repeating for n-1 rounds</span>
 <a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="o">--</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="w">        </span><span class="c1">// Delete node</span>
+<a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="w">        </span><span class="c1">// Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)</span>
 <a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="w">        </span><span class="c1">// Start heapifying the root node, from top to bottom</span>
+<a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="w">        </span><span class="c1">// Sift down starting at the root</span>
 <a id="__codelineno-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a><span class="w">        </span><span class="n">siftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-1-35" name="__codelineno-1-35" href="#__codelineno-1-35"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-36" name="__codelineno-1-36" href="#__codelineno-1-36"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap_sort.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Heap length is n, start heapifying node i, from top to bottom */</span>
+<div class="highlight"><span class="filename">heap_sort.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Sift down from node i in a heap of size n */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">siftDown</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
@@ -4809,41 +4881,41 @@
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span><span class="p">;</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">r</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="p">;</span>
-<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
-<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span>
-<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span><span class="p">;</span>
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="p">}</span>
 <a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a>
 <a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="cm">/* Heap sort */</span>
 <a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="kt">void</span><span class="w"> </span><span class="nf">heapSort</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="w">    </span><span class="c1">// Build heap operation: heapify all nodes except leaves</span>
+<a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="w">    </span><span class="c1">// Build the heap by sifting down from every non-leaf node</span>
 <a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a><span class="w">        </span><span class="n">siftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap and repeat for n-1 rounds</span>
+<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap, repeating for n-1 rounds</span>
 <a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="w">        </span><span class="c1">// Delete node</span>
+<a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="w">        </span><span class="c1">// Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)</span>
 <a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-35" name="__codelineno-2-35" href="#__codelineno-2-35"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
-<a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a><span class="w">        </span><span class="c1">// Start heapifying the root node, from top to bottom</span>
+<a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a><span class="w">        </span><span class="c1">// Sift down starting at the root</span>
 <a id="__codelineno-2-37" name="__codelineno-2-37" href="#__codelineno-2-37"></a><span class="w">        </span><span class="n">siftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-2-38" name="__codelineno-2-38" href="#__codelineno-2-38"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-39" name="__codelineno-2-39" href="#__codelineno-2-39"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap_sort.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Heap length is n, start heapifying node i, from top to bottom */</span>
+<div class="highlight"><span class="filename">heap_sort.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Sift down from node i in a heap of size n */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">SiftDown</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="k">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
@@ -4851,37 +4923,37 @@
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span><span class="p">;</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="p">;</span>
-<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
-<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">        </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">]);</span>
-<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span><span class="p">;</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="p">}</span>
 <a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a>
 <a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="cm">/* Heap sort */</span>
 <a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="k">void</span><span class="w"> </span><span class="nf">HeapSort</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w">    </span><span class="c1">// Build heap operation: heapify all nodes except leaves</span>
+<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w">    </span><span class="c1">// Build the heap by sifting down from every non-leaf node</span>
 <a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="w">        </span><span class="n">SiftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap and repeat for n-1 rounds</span>
+<a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap, repeating for n-1 rounds</span>
 <a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a><span class="w">        </span><span class="c1">// Delete node</span>
+<a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a><span class="w">        </span><span class="c1">// Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)</span>
 <a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a><span class="w">        </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]);</span>
-<a id="__codelineno-3-32" name="__codelineno-3-32" href="#__codelineno-3-32"></a><span class="w">        </span><span class="c1">// Start heapifying the root node, from top to bottom</span>
+<a id="__codelineno-3-32" name="__codelineno-3-32" href="#__codelineno-3-32"></a><span class="w">        </span><span class="c1">// Sift down starting at the root</span>
 <a id="__codelineno-3-33" name="__codelineno-3-33" href="#__codelineno-3-33"></a><span class="w">        </span><span class="n">SiftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-3-34" name="__codelineno-3-34" href="#__codelineno-3-34"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-35" name="__codelineno-3-35" href="#__codelineno-3-35"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap_sort.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Heap length is n, start heapifying node i, from top to bottom */</span>
+<div class="highlight"><span class="filename">heap_sort.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Sift down from node i in a heap of size n */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">siftDown</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="kc">true</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">        </span><span class="nx">l</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">2</span><span class="o">*</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="nx">r</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">2</span><span class="o">*</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="nx">ma</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">i</span>
@@ -4891,39 +4963,39 @@
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">nums</span><span class="p">)[</span><span class="nx">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">nums</span><span class="p">)[</span><span class="nx">ma</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">            </span><span class="nx">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">r</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">        </span><span class="p">(</span><span class="o">*</span><span class="nx">nums</span><span class="p">)[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">nums</span><span class="p">)[</span><span class="nx">ma</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">nums</span><span class="p">)[</span><span class="nx">ma</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">nums</span><span class="p">)[</span><span class="nx">i</span><span class="p">]</span>
-<a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">        </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">ma</span>
 <a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="p">}</span>
 <a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a>
 <a id="__codelineno-4-25" name="__codelineno-4-25" href="#__codelineno-4-25"></a><span class="cm">/* Heap sort */</span>
 <a id="__codelineno-4-26" name="__codelineno-4-26" href="#__codelineno-4-26"></a><span class="kd">func</span><span class="w"> </span><span class="nx">heapSort</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-27" name="__codelineno-4-27" href="#__codelineno-4-27"></a><span class="w">    </span><span class="c1">// Build heap operation: heapify all nodes except leaves</span>
+<a id="__codelineno-4-27" name="__codelineno-4-27" href="#__codelineno-4-27"></a><span class="w">    </span><span class="c1">// Build the heap by sifting down from every non-leaf node</span>
 <a id="__codelineno-4-28" name="__codelineno-4-28" href="#__codelineno-4-28"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">nums</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-29" name="__codelineno-4-29" href="#__codelineno-4-29"></a><span class="w">        </span><span class="nx">siftDown</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">nums</span><span class="p">),</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span>
 <a id="__codelineno-4-30" name="__codelineno-4-30" href="#__codelineno-4-30"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-31" name="__codelineno-4-31" href="#__codelineno-4-31"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap and repeat for n-1 rounds</span>
+<a id="__codelineno-4-31" name="__codelineno-4-31" href="#__codelineno-4-31"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap, repeating for n-1 rounds</span>
 <a id="__codelineno-4-32" name="__codelineno-4-32" href="#__codelineno-4-32"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="o">*</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-33" name="__codelineno-4-33" href="#__codelineno-4-33"></a><span class="w">        </span><span class="c1">// Delete node</span>
+<a id="__codelineno-4-33" name="__codelineno-4-33" href="#__codelineno-4-33"></a><span class="w">        </span><span class="c1">// Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)</span>
 <a id="__codelineno-4-34" name="__codelineno-4-34" href="#__codelineno-4-34"></a><span class="w">        </span><span class="p">(</span><span class="o">*</span><span class="nx">nums</span><span class="p">)[</span><span class="mi">0</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">nums</span><span class="p">)[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">nums</span><span class="p">)[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="nx">nums</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
-<a id="__codelineno-4-35" name="__codelineno-4-35" href="#__codelineno-4-35"></a><span class="w">        </span><span class="c1">// Start heapifying the root node, from top to bottom</span>
+<a id="__codelineno-4-35" name="__codelineno-4-35" href="#__codelineno-4-35"></a><span class="w">        </span><span class="c1">// Sift down starting at the root</span>
 <a id="__codelineno-4-36" name="__codelineno-4-36" href="#__codelineno-4-36"></a><span class="w">        </span><span class="nx">siftDown</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-4-37" name="__codelineno-4-37" href="#__codelineno-4-37"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-38" name="__codelineno-4-38" href="#__codelineno-4-38"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap_sort.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Heap length is n, start heapifying node i, from top to bottom */</span>
+<div class="highlight"><span class="filename">heap_sort.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Sift down from node i in a heap of size n */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">siftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="kc">true</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">l</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">r</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">ma</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i</span>
@@ -4933,38 +5005,38 @@
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">r</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w">        </span><span class="n">nums</span><span class="p">.</span><span class="n">swapAt</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">ma</span><span class="p">)</span>
-<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ma</span>
 <a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="p">}</span>
 <a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a>
 <a id="__codelineno-5-26" name="__codelineno-5-26" href="#__codelineno-5-26"></a><span class="cm">/* Heap sort */</span>
 <a id="__codelineno-5-27" name="__codelineno-5-27" href="#__codelineno-5-27"></a><span class="kd">func</span><span class="w"> </span><span class="nf">heapSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="w">    </span><span class="c1">// Build heap operation: heapify all nodes except leaves</span>
+<a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="w">    </span><span class="c1">// Build the heap by sifting down from every non-leaf node</span>
 <a id="__codelineno-5-29" name="__codelineno-5-29" href="#__codelineno-5-29"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="bp">stride</span><span class="p">(</span><span class="n">from</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">count</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">through</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">by</span><span class="p">:</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-30" name="__codelineno-5-30" href="#__codelineno-5-30"></a><span class="w">        </span><span class="n">siftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">count</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-5-31" name="__codelineno-5-31" href="#__codelineno-5-31"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-32" name="__codelineno-5-32" href="#__codelineno-5-32"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap and repeat for n-1 rounds</span>
+<a id="__codelineno-5-32" name="__codelineno-5-32" href="#__codelineno-5-32"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap, repeating for n-1 rounds</span>
 <a id="__codelineno-5-33" name="__codelineno-5-33" href="#__codelineno-5-33"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="bp">dropFirst</span><span class="p">().</span><span class="n">reversed</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-34" name="__codelineno-5-34" href="#__codelineno-5-34"></a><span class="w">        </span><span class="c1">// Delete node</span>
+<a id="__codelineno-5-34" name="__codelineno-5-34" href="#__codelineno-5-34"></a><span class="w">        </span><span class="c1">// Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)</span>
 <a id="__codelineno-5-35" name="__codelineno-5-35" href="#__codelineno-5-35"></a><span class="w">        </span><span class="n">nums</span><span class="p">.</span><span class="n">swapAt</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
-<a id="__codelineno-5-36" name="__codelineno-5-36" href="#__codelineno-5-36"></a><span class="w">        </span><span class="c1">// Start heapifying the root node, from top to bottom</span>
+<a id="__codelineno-5-36" name="__codelineno-5-36" href="#__codelineno-5-36"></a><span class="w">        </span><span class="c1">// Sift down starting at the root</span>
 <a id="__codelineno-5-37" name="__codelineno-5-37" href="#__codelineno-5-37"></a><span class="w">        </span><span class="n">siftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-5-38" name="__codelineno-5-38" href="#__codelineno-5-38"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-39" name="__codelineno-5-39" href="#__codelineno-5-39"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap_sort.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Heap length is n, start heapifying node i, from top to bottom */</span>
+<div class="highlight"><span class="filename">heap_sort.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Sift down from node i in a heap of size n */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">siftDown</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
@@ -4974,38 +5046,38 @@
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">ma</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">            </span><span class="nx">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">r</span><span class="p">;</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">ma</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">        </span><span class="p">[</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">ma</span><span class="p">]]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="nx">nums</span><span class="p">[</span><span class="nx">ma</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]];</span>
-<a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">        </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">ma</span><span class="p">;</span>
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="p">}</span>
 <a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a>
 <a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="cm">/* Heap sort */</span>
 <a id="__codelineno-6-26" name="__codelineno-6-26" href="#__codelineno-6-26"></a><span class="kd">function</span><span class="w"> </span><span class="nx">heapSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-27" name="__codelineno-6-27" href="#__codelineno-6-27"></a><span class="w">    </span><span class="c1">// Build heap operation: heapify all nodes except leaves</span>
+<a id="__codelineno-6-27" name="__codelineno-6-27" href="#__codelineno-6-27"></a><span class="w">    </span><span class="c1">// Build the heap by sifting down from every non-leaf node</span>
 <a id="__codelineno-6-28" name="__codelineno-6-28" href="#__codelineno-6-28"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-29" name="__codelineno-6-29" href="#__codelineno-6-29"></a><span class="w">        </span><span class="nx">siftDown</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">);</span>
 <a id="__codelineno-6-30" name="__codelineno-6-30" href="#__codelineno-6-30"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-31" name="__codelineno-6-31" href="#__codelineno-6-31"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap and repeat for n-1 rounds</span>
+<a id="__codelineno-6-31" name="__codelineno-6-31" href="#__codelineno-6-31"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap, repeating for n-1 rounds</span>
 <a id="__codelineno-6-32" name="__codelineno-6-32" href="#__codelineno-6-32"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-33" name="__codelineno-6-33" href="#__codelineno-6-33"></a><span class="w">        </span><span class="c1">// Delete node</span>
+<a id="__codelineno-6-33" name="__codelineno-6-33" href="#__codelineno-6-33"></a><span class="w">        </span><span class="c1">// Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)</span>
 <a id="__codelineno-6-34" name="__codelineno-6-34" href="#__codelineno-6-34"></a><span class="w">        </span><span class="p">[</span><span class="nx">nums</span><span class="p">[</span><span class="mf">0</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="mf">0</span><span class="p">]];</span>
-<a id="__codelineno-6-35" name="__codelineno-6-35" href="#__codelineno-6-35"></a><span class="w">        </span><span class="c1">// Start heapifying the root node, from top to bottom</span>
+<a id="__codelineno-6-35" name="__codelineno-6-35" href="#__codelineno-6-35"></a><span class="w">        </span><span class="c1">// Sift down starting at the root</span>
 <a id="__codelineno-6-36" name="__codelineno-6-36" href="#__codelineno-6-36"></a><span class="w">        </span><span class="nx">siftDown</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-6-37" name="__codelineno-6-37" href="#__codelineno-6-37"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-38" name="__codelineno-6-38" href="#__codelineno-6-38"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap_sort.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Heap length is n, start heapifying node i, from top to bottom */</span>
+<div class="highlight"><span class="filename">heap_sort.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Sift down from node i in a heap of size n */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">siftDown</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">n</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
@@ -5015,77 +5087,77 @@
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">ma</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">            </span><span class="nx">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">r</span><span class="p">;</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">ma</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">        </span><span class="p">[</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">ma</span><span class="p">]]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="nx">nums</span><span class="p">[</span><span class="nx">ma</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]];</span>
-<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">        </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">ma</span><span class="p">;</span>
 <a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="p">}</span>
 <a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a>
 <a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="cm">/* Heap sort */</span>
 <a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="kd">function</span><span class="w"> </span><span class="nx">heapSort</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="w">    </span><span class="c1">// Build heap operation: heapify all nodes except leaves</span>
+<a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="w">    </span><span class="c1">// Build the heap by sifting down from every non-leaf node</span>
 <a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-29" name="__codelineno-7-29" href="#__codelineno-7-29"></a><span class="w">        </span><span class="nx">siftDown</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">);</span>
 <a id="__codelineno-7-30" name="__codelineno-7-30" href="#__codelineno-7-30"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-31" name="__codelineno-7-31" href="#__codelineno-7-31"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap and repeat for n-1 rounds</span>
+<a id="__codelineno-7-31" name="__codelineno-7-31" href="#__codelineno-7-31"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap, repeating for n-1 rounds</span>
 <a id="__codelineno-7-32" name="__codelineno-7-32" href="#__codelineno-7-32"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-33" name="__codelineno-7-33" href="#__codelineno-7-33"></a><span class="w">        </span><span class="c1">// Delete node</span>
+<a id="__codelineno-7-33" name="__codelineno-7-33" href="#__codelineno-7-33"></a><span class="w">        </span><span class="c1">// Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)</span>
 <a id="__codelineno-7-34" name="__codelineno-7-34" href="#__codelineno-7-34"></a><span class="w">        </span><span class="p">[</span><span class="nx">nums</span><span class="p">[</span><span class="mf">0</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="mf">0</span><span class="p">]];</span>
-<a id="__codelineno-7-35" name="__codelineno-7-35" href="#__codelineno-7-35"></a><span class="w">        </span><span class="c1">// Start heapifying the root node, from top to bottom</span>
+<a id="__codelineno-7-35" name="__codelineno-7-35" href="#__codelineno-7-35"></a><span class="w">        </span><span class="c1">// Sift down starting at the root</span>
 <a id="__codelineno-7-36" name="__codelineno-7-36" href="#__codelineno-7-36"></a><span class="w">        </span><span class="nx">siftDown</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-7-37" name="__codelineno-7-37" href="#__codelineno-7-37"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-38" name="__codelineno-7-38" href="#__codelineno-7-38"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap_sort.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Heap length is n, start heapifying node i, from top to bottom */</span>
+<div class="highlight"><span class="filename">heap_sort.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Sift down from node i in a heap of size n */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">siftDown</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">    </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">    </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">2</span><span class="p">;</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">l</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span><span class="p">;</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="p">;</span>
-<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span>
-<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">];</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span>
-<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">    </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">    </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span><span class="p">;</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="p">}</span>
 <a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a>
 <a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="cm">/* Heap sort */</span>
 <a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="kt">void</span><span class="w"> </span><span class="n">heapSort</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="w">  </span><span class="c1">// Build heap operation: heapify all nodes except leaves</span>
+<a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="w">  </span><span class="c1">// Build the heap by sifting down from every non-leaf node</span>
 <a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="w">    </span><span class="n">siftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-8-26" name="__codelineno-8-26" href="#__codelineno-8-26"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a><span class="w">  </span><span class="c1">// Extract the largest element from the heap and repeat for n-1 rounds</span>
+<a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a><span class="w">  </span><span class="c1">// Extract the largest element from the heap, repeating for n-1 rounds</span>
 <a id="__codelineno-8-28" name="__codelineno-8-28" href="#__codelineno-8-28"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-29" name="__codelineno-8-29" href="#__codelineno-8-29"></a><span class="w">    </span><span class="c1">// Delete node</span>
+<a id="__codelineno-8-29" name="__codelineno-8-29" href="#__codelineno-8-29"></a><span class="w">    </span><span class="c1">// Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)</span>
 <a id="__codelineno-8-30" name="__codelineno-8-30" href="#__codelineno-8-30"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="m">0</span><span class="p">];</span>
 <a id="__codelineno-8-31" name="__codelineno-8-31" href="#__codelineno-8-31"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="m">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-8-32" name="__codelineno-8-32" href="#__codelineno-8-32"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
-<a id="__codelineno-8-33" name="__codelineno-8-33" href="#__codelineno-8-33"></a><span class="w">    </span><span class="c1">// Start heapifying the root node, from top to bottom</span>
+<a id="__codelineno-8-33" name="__codelineno-8-33" href="#__codelineno-8-33"></a><span class="w">    </span><span class="c1">// Sift down starting at the root</span>
 <a id="__codelineno-8-34" name="__codelineno-8-34" href="#__codelineno-8-34"></a><span class="w">    </span><span class="n">siftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-8-35" name="__codelineno-8-35" href="#__codelineno-8-35"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-36" name="__codelineno-8-36" href="#__codelineno-8-36"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap_sort.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Heap length is n, start heapifying node i, from top to bottom */</span>
+<div class="highlight"><span class="filename">heap_sort.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Sift down from node i in a heap of size n */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">sift_down</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="k">loop</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
@@ -5095,38 +5167,38 @@
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="p">;</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">        </span><span class="n">nums</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">ma</span><span class="p">);</span>
-<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span><span class="p">;</span>
 <a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="p">}</span>
 <a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a>
 <a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a><span class="cm">/* Heap sort */</span>
 <a id="__codelineno-9-26" name="__codelineno-9-26" href="#__codelineno-9-26"></a><span class="k">fn</span><span class="w"> </span><span class="nf">heap_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="w">    </span><span class="c1">// Build heap operation: heapify all nodes except leaves</span>
+<a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="w">    </span><span class="c1">// Build the heap by sifting down from every non-leaf node</span>
 <a id="__codelineno-9-28" name="__codelineno-9-28" href="#__codelineno-9-28"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="o">..</span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">).</span><span class="n">rev</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-29" name="__codelineno-9-29" href="#__codelineno-9-29"></a><span class="w">        </span><span class="n">sift_down</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">(),</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-9-30" name="__codelineno-9-30" href="#__codelineno-9-30"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-31" name="__codelineno-9-31" href="#__codelineno-9-31"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap and repeat for n-1 rounds</span>
+<a id="__codelineno-9-31" name="__codelineno-9-31" href="#__codelineno-9-31"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap, repeating for n-1 rounds</span>
 <a id="__codelineno-9-32" name="__codelineno-9-32" href="#__codelineno-9-32"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="o">..</span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()).</span><span class="n">rev</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-33" name="__codelineno-9-33" href="#__codelineno-9-33"></a><span class="w">        </span><span class="c1">// Delete node</span>
+<a id="__codelineno-9-33" name="__codelineno-9-33" href="#__codelineno-9-33"></a><span class="w">        </span><span class="c1">// Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)</span>
 <a id="__codelineno-9-34" name="__codelineno-9-34" href="#__codelineno-9-34"></a><span class="w">        </span><span class="n">nums</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
-<a id="__codelineno-9-35" name="__codelineno-9-35" href="#__codelineno-9-35"></a><span class="w">        </span><span class="c1">// Start heapifying the root node, from top to bottom</span>
+<a id="__codelineno-9-35" name="__codelineno-9-35" href="#__codelineno-9-35"></a><span class="w">        </span><span class="c1">// Sift down starting at the root</span>
 <a id="__codelineno-9-36" name="__codelineno-9-36" href="#__codelineno-9-36"></a><span class="w">        </span><span class="n">sift_down</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-9-37" name="__codelineno-9-37" href="#__codelineno-9-37"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-38" name="__codelineno-9-38" href="#__codelineno-9-38"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap_sort.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Heap length is n, start heapifying node i, from top to bottom */</span>
+<div class="highlight"><span class="filename">heap_sort.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Sift down from node i in a heap of size n */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">siftDown</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
@@ -5134,43 +5206,43 @@
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span><span class="p">;</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">r</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">])</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="p">;</span>
-<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">];</span>
 <a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">ma</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span>
-<a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span><span class="p">;</span>
 <a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="p">}</span>
 <a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a>
 <a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="cm">/* Heap sort */</span>
 <a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="kt">void</span><span class="w"> </span><span class="nf">heapSort</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-27" name="__codelineno-10-27" href="#__codelineno-10-27"></a><span class="w">    </span><span class="c1">// Build heap operation: heapify all nodes except leaves</span>
+<a id="__codelineno-10-27" name="__codelineno-10-27" href="#__codelineno-10-27"></a><span class="w">    </span><span class="c1">// Build the heap by sifting down from every non-leaf node</span>
 <a id="__codelineno-10-28" name="__codelineno-10-28" href="#__codelineno-10-28"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="o">--</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-29" name="__codelineno-10-29" href="#__codelineno-10-29"></a><span class="w">        </span><span class="n">siftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-10-30" name="__codelineno-10-30" href="#__codelineno-10-30"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-31" name="__codelineno-10-31" href="#__codelineno-10-31"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap and repeat for n-1 rounds</span>
+<a id="__codelineno-10-31" name="__codelineno-10-31" href="#__codelineno-10-31"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap, repeating for n-1 rounds</span>
 <a id="__codelineno-10-32" name="__codelineno-10-32" href="#__codelineno-10-32"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="o">--</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-33" name="__codelineno-10-33" href="#__codelineno-10-33"></a><span class="w">        </span><span class="c1">// Delete node</span>
+<a id="__codelineno-10-33" name="__codelineno-10-33" href="#__codelineno-10-33"></a><span class="w">        </span><span class="c1">// Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)</span>
 <a id="__codelineno-10-34" name="__codelineno-10-34" href="#__codelineno-10-34"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span>
 <a id="__codelineno-10-35" name="__codelineno-10-35" href="#__codelineno-10-35"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-10-36" name="__codelineno-10-36" href="#__codelineno-10-36"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
-<a id="__codelineno-10-37" name="__codelineno-10-37" href="#__codelineno-10-37"></a><span class="w">        </span><span class="c1">// Start heapifying the root node, from top to bottom</span>
+<a id="__codelineno-10-37" name="__codelineno-10-37" href="#__codelineno-10-37"></a><span class="w">        </span><span class="c1">// Sift down starting at the root</span>
 <a id="__codelineno-10-38" name="__codelineno-10-38" href="#__codelineno-10-38"></a><span class="w">        </span><span class="n">siftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-10-39" name="__codelineno-10-39" href="#__codelineno-10-39"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-40" name="__codelineno-10-40" href="#__codelineno-10-40"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap_sort.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Heap length is n, start heapifying node i, from top to bottom */</span>
+<div class="highlight"><span class="filename">heap_sort.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Sift down from node i in a heap of size n */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">siftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">li</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">li</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="kc">true</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">        </span><span class="c1">// If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">        </span><span class="c1">// Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">2</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span>
@@ -5178,66 +5250,66 @@
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">r</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span><span class="p">)</span><span class="w"> </span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">            </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span>
-<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="c1">// Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">            </span><span class="k">break</span>
-<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">        </span><span class="c1">// Swap two nodes</span>
+<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">        </span><span class="c1">// Swap the two nodes</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span>
-<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">        </span><span class="c1">// Loop downwards heapification</span>
+<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">        </span><span class="c1">// Continue sifting down</span>
 <a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">        </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span>
 <a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="p">}</span>
 <a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a>
 <a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="cm">/* Heap sort */</span>
 <a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">heapSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w">    </span><span class="c1">// Build heap operation: heapify all nodes except leaves</span>
+<a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w">    </span><span class="c1">// Build the heap by sifting down from every non-leaf node</span>
 <a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="n">downTo</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="w">        </span><span class="n">siftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap and repeat for n-1 rounds</span>
+<a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a><span class="w">    </span><span class="c1">// Extract the largest element from the heap, repeating for n-1 rounds</span>
 <a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="n">downTo</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></a><span class="w">        </span><span class="c1">// Delete node</span>
+<a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></a><span class="w">        </span><span class="c1">// Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)</span>
 <a id="__codelineno-11-34" name="__codelineno-11-34" href="#__codelineno-11-34"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="m">0</span><span class="o">]</span>
 <a id="__codelineno-11-35" name="__codelineno-11-35" href="#__codelineno-11-35"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="m">0</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-11-36" name="__codelineno-11-36" href="#__codelineno-11-36"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span>
-<a id="__codelineno-11-37" name="__codelineno-11-37" href="#__codelineno-11-37"></a><span class="w">        </span><span class="c1">// Start heapifying the root node, from top to bottom</span>
+<a id="__codelineno-11-37" name="__codelineno-11-37" href="#__codelineno-11-37"></a><span class="w">        </span><span class="c1">// Sift down starting at the root</span>
 <a id="__codelineno-11-38" name="__codelineno-11-38" href="#__codelineno-11-38"></a><span class="w">        </span><span class="n">siftDown</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">)</span>
 <a id="__codelineno-11-39" name="__codelineno-11-39" href="#__codelineno-11-39"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-40" name="__codelineno-11-40" href="#__codelineno-11-40"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">heap_sort.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Heap length is n, heapify from node i, top to bottom ###</span>
+<div class="highlight"><span class="filename">heap_sort.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Sift down from node i in a heap of size n ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">sift_down</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="kp">true</span>
-<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">    </span><span class="c1"># If node i is largest or indices l, r are out of bounds, no need to continue heapify, break</span>
+<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">    </span><span class="c1"># Find the node with the largest value among i, l, and r, and store its index in ma</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">    </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">    </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">    </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">l</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">    </span><span class="n">ma</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">r</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span>
-<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="c1"># Swap two nodes</span>
+<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="c1"># Stop if node i is largest or both child indices l and r are out of bounds</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="k">break</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">ma</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">i</span>
-<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">    </span><span class="c1"># Swap two nodes</span>
+<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">    </span><span class="c1"># Swap the two nodes</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">ma</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
-<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="c1"># Loop downwards heapification</span>
+<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="c1"># Continue sifting down</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ma</span>
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="k">end</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a>
 <a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="c1">### Heap sort ###</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="k">def</span><span class="w"> </span><span class="nf">heap_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
-<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">  </span><span class="c1"># Build heap operation: heapify all nodes except leaves</span>
+<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">  </span><span class="c1"># Build the heap by sifting down from every non-leaf node</span>
 <a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w">  </span><span class="p">(</span><span class="n">nums</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">downto</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span>
 <a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="w">    </span><span class="n">sift_down</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">length</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a><span class="w">  </span><span class="k">end</span>
-<a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="w">  </span><span class="c1"># Extract the largest element from the heap and repeat for n-1 rounds</span>
+<a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="w">  </span><span class="c1"># Extract the largest element from the heap, repeating for n-1 rounds</span>
 <a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></a><span class="w">  </span><span class="p">(</span><span class="n">nums</span><span class="o">.</span><span class="n">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">downto</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span>
-<a id="__codelineno-12-27" name="__codelineno-12-27" href="#__codelineno-12-27"></a><span class="w">    </span><span class="c1"># Delete node</span>
+<a id="__codelineno-12-27" name="__codelineno-12-27" href="#__codelineno-12-27"></a><span class="w">    </span><span class="c1"># Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)</span>
 <a id="__codelineno-12-28" name="__codelineno-12-28" href="#__codelineno-12-28"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="mi">0</span><span class="o">]</span>
-<a id="__codelineno-12-29" name="__codelineno-12-29" href="#__codelineno-12-29"></a><span class="w">    </span><span class="c1"># Start heapifying the root node, from top to bottom</span>
+<a id="__codelineno-12-29" name="__codelineno-12-29" href="#__codelineno-12-29"></a><span class="w">    </span><span class="c1"># Sift down starting at the root</span>
 <a id="__codelineno-12-30" name="__codelineno-12-30" href="#__codelineno-12-30"></a><span class="w">    </span><span class="n">sift_down</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-12-31" name="__codelineno-12-31" href="#__codelineno-12-31"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-32" name="__codelineno-12-32" href="#__codelineno-12-32"></a><span class="k">end</span>
@@ -5245,11 +5317,16 @@
 </div>
 </div>
 </div>
-<h2 id="1172-algorithm-characteristics">11.7.2 &nbsp; Algorithm Characteristics<a class="headerlink" href="#1172-algorithm-characteristics" title="Permanent link">&para;</a></h2>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20sift_down%28nums%3A%20list%5Bint%5D,%20n%3A%20int,%20i%3A%20int%29%3A%0A%20%20%20%20%22%22%22Sift%20down%20from%20node%20i%20in%20a%20heap%20of%20size%20n%22%22%22%0A%20%20%20%20while%20True%3A%0A%20%20%20%20%20%20%20%20%23%20Find%20the%20node%20with%20the%20largest%20value%20among%20i,%20l,%20and%20r,%20and%20store%20its%20index%20in%20ma%0A%20%20%20%20%20%20%20%20l%20%3D%202%20%2A%20i%20%2B%201%0A%20%20%20%20%20%20%20%20r%20%3D%202%20%2A%20i%20%2B%202%0A%20%20%20%20%20%20%20%20ma%20%3D%20i%0A%20%20%20%20%20%20%20%20if%20l%20%3C%20n%20and%20nums%5Bl%5D%20%3E%20nums%5Bma%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20ma%20%3D%20l%0A%20%20%20%20%20%20%20%20if%20r%20%3C%20n%20and%20nums%5Br%5D%20%3E%20nums%5Bma%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20ma%20%3D%20r%0A%20%20%20%20%20%20%20%20%23%20Stop%20if%20node%20i%20is%20largest%20or%20both%20child%20indices%20l%20and%20r%20are%20out%20of%20bounds%0A%20%20%20%20%20%20%20%20if%20ma%20%3D%3D%20i%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20two%20nodes%0A%20%20%20%20%20%20%20%20nums%5Bi%5D,%20nums%5Bma%5D%20%3D%20nums%5Bma%5D,%20nums%5Bi%5D%0A%20%20%20%20%20%20%20%20%23%20Continue%20sifting%20down%0A%20%20%20%20%20%20%20%20i%20%3D%20ma%0A%0Adef%20heap_sort%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Heap%20sort%22%22%22%0A%20%20%20%20%23%20Build%20the%20heap%20by%20sifting%20down%20from%20every%20non-leaf%20node%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%20//%202%20-%201,%20-1,%20-1%29%3A%0A%20%20%20%20%20%20%20%20sift_down%28nums,%20len%28nums%29,%20i%29%0A%20%20%20%20%23%20Extract%20the%20largest%20element%20from%20the%20heap,%20repeating%20for%20n-1%20rounds%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%20-%201,%200,%20-1%29%3A%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20root%20with%20the%20rightmost%20leaf%20on%20the%20last%20level%20%28the%20first%20and%20last%20elements%20of%20the%20current%20heap%29%0A%20%20%20%20%20%20%20%20nums%5B0%5D,%20nums%5Bi%5D%20%3D%20nums%5Bi%5D,%20nums%5B0%5D%0A%20%20%20%20%20%20%20%20%23%20Sift%20down%20starting%20at%20the%20root%0A%20%20%20%20%20%20%20%20sift_down%28nums,%20i,%200%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B4,%201,%203,%201,%205,%202%5D%0A%20%20%20%20heap_sort%28nums%29%0A%20%20%20%20print%28%22After%20heap%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20sift_down%28nums%3A%20list%5Bint%5D,%20n%3A%20int,%20i%3A%20int%29%3A%0A%20%20%20%20%22%22%22Sift%20down%20from%20node%20i%20in%20a%20heap%20of%20size%20n%22%22%22%0A%20%20%20%20while%20True%3A%0A%20%20%20%20%20%20%20%20%23%20Find%20the%20node%20with%20the%20largest%20value%20among%20i,%20l,%20and%20r,%20and%20store%20its%20index%20in%20ma%0A%20%20%20%20%20%20%20%20l%20%3D%202%20%2A%20i%20%2B%201%0A%20%20%20%20%20%20%20%20r%20%3D%202%20%2A%20i%20%2B%202%0A%20%20%20%20%20%20%20%20ma%20%3D%20i%0A%20%20%20%20%20%20%20%20if%20l%20%3C%20n%20and%20nums%5Bl%5D%20%3E%20nums%5Bma%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20ma%20%3D%20l%0A%20%20%20%20%20%20%20%20if%20r%20%3C%20n%20and%20nums%5Br%5D%20%3E%20nums%5Bma%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20ma%20%3D%20r%0A%20%20%20%20%20%20%20%20%23%20Stop%20if%20node%20i%20is%20largest%20or%20both%20child%20indices%20l%20and%20r%20are%20out%20of%20bounds%0A%20%20%20%20%20%20%20%20if%20ma%20%3D%3D%20i%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20two%20nodes%0A%20%20%20%20%20%20%20%20nums%5Bi%5D,%20nums%5Bma%5D%20%3D%20nums%5Bma%5D,%20nums%5Bi%5D%0A%20%20%20%20%20%20%20%20%23%20Continue%20sifting%20down%0A%20%20%20%20%20%20%20%20i%20%3D%20ma%0A%0Adef%20heap_sort%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Heap%20sort%22%22%22%0A%20%20%20%20%23%20Build%20the%20heap%20by%20sifting%20down%20from%20every%20non-leaf%20node%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%20//%202%20-%201,%20-1,%20-1%29%3A%0A%20%20%20%20%20%20%20%20sift_down%28nums,%20len%28nums%29,%20i%29%0A%20%20%20%20%23%20Extract%20the%20largest%20element%20from%20the%20heap,%20repeating%20for%20n-1%20rounds%0A%20%20%20%20for%20i%20in%20range%28len%28nums%29%20-%201,%200,%20-1%29%3A%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20root%20with%20the%20rightmost%20leaf%20on%20the%20last%20level%20%28the%20first%20and%20last%20elements%20of%20the%20current%20heap%29%0A%20%20%20%20%20%20%20%20nums%5B0%5D,%20nums%5Bi%5D%20%3D%20nums%5Bi%5D,%20nums%5B0%5D%0A%20%20%20%20%20%20%20%20%23%20Sift%20down%20starting%20at%20the%20root%0A%20%20%20%20%20%20%20%20sift_down%28nums,%20i,%200%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B4,%201,%203,%201,%205,%202%5D%0A%20%20%20%20heap_sort%28nums%29%0A%20%20%20%20print%28%22After%20heap%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1172-algorithm-properties">11.7.2 &nbsp; Algorithm properties<a class="headerlink" href="#1172-algorithm-properties" title="Permanent link">&para;</a></h2>
 <ul>
-<li><strong>Time complexity is <span class="arithmatex">\(O(n \log n)\)</span>; heap sort is non-adaptive</strong>: Heap construction takes <span class="arithmatex">\(O(n)\)</span> time. Extracting the largest element from the heap takes <span class="arithmatex">\(O(\log n)\)</span> time, and this is repeated for a total of <span class="arithmatex">\(n - 1\)</span> rounds.</li>
-<li><strong>Space complexity is <span class="arithmatex">\(O(1)\)</span>; heap sort is in-place</strong>: A few pointer variables use <span class="arithmatex">\(O(1)\)</span> space. Element swapping and heapify are both performed on the original array.</li>
-<li><strong>Unstable sorting</strong>: When swapping the heap top element and heap bottom element, the relative positions of equal elements may change.</li>
+<li><strong>Time complexity: <span class="arithmatex">\(O(n \log n)\)</span>; not adaptive</strong>. Building the heap takes <span class="arithmatex">\(O(n)\)</span> time. Extracting its maximum takes <span class="arithmatex">\(O(\log n)\)</span> time and is repeated <span class="arithmatex">\(n - 1\)</span> times.</li>
+<li><strong>Space complexity: <span class="arithmatex">\(O(1)\)</span>; in-place</strong>. A few pointer variables use <span class="arithmatex">\(O(1)\)</span> space. Swapping and sifting take place within the original array.</li>
+<li><strong>Not stable</strong>. Swapping the root with the last element of the heap may change the relative order of equal elements.</li>
 </ul>
 
 <!-- Source file information -->
@@ -5273,7 +5350,7 @@ aria-label="Footer"
   <a
     href="../merge_sort/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 11.6 Merge Sort"
+    aria-label="Previous: 11.6   Merge sort"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -5285,7 +5362,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        11.6 Merge Sort
+        11.6 &nbsp; Merge sort
       </div>
     </div>
   </a>
@@ -5297,7 +5374,7 @@ aria-label="Footer"
   <a
     href="../bucket_sort/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 11.8 Bucket Sort"
+    aria-label="Next: 11.8   Bucket sort"
     rel="next"
   >
     <div class="md-footer__title">
@@ -5305,7 +5382,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        11.8 Bucket Sort
+        11.8 &nbsp; Bucket sort
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -5322,7 +5399,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_sorting/index.html b/en/chapter_sorting/index.html
index 149e00d17..e084d41bc 100644
--- a/en/chapter_sorting/index.html
+++ b/en/chapter_sorting/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
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@@ -167,9 +167,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
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@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
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@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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           <ul class="md-nav__list" data-md-scrollfix>
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   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
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@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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@@ -2287,7 +2287,7 @@
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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@@ -2378,7 +2378,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2880,7 +2880,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2901,7 +2901,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2922,7 +2922,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3003,7 +3003,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3030,7 +3030,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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@@ -3057,7 +3057,7 @@
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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@@ -3084,7 +3084,7 @@
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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@@ -3111,7 +3111,7 @@
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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@@ -3138,7 +3138,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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@@ -3192,7 +3192,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
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@@ -3327,7 +3327,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
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   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4072,7 +4072,7 @@
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4114,7 +4114,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4309,7 +4309,7 @@
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4351,7 +4351,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4378,7 +4378,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4405,7 +4405,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4489,6 +4489,78 @@
 
 
 
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@@ -4581,21 +4653,21 @@
 <p><img alt="Sorting" class="cover-image" src="../assets/covers/chapter_sorting.jpg" /></p>
 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
-<p>Sorting is like a magic key that transforms chaos into order, enabling us to understand and process data more efficiently.</p>
-<p>From simple ascending order to more complex classification schemes, sorting reveals the harmonious beauty of data.</p>
+<p>Sorting is like a magic key that turns chaos into order, helping us understand and process data more efficiently.</p>
+<p>From a simple ascending sequence to a complex arrangement by category, sorting reveals the harmonious beauty of data.</p>
 </div>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
-<li><a href="sorting_algorithm/">11.1 &nbsp; Sorting Algorithm</a></li>
-<li><a href="selection_sort/">11.2 &nbsp; Selection Sort</a></li>
-<li><a href="bubble_sort/">11.3 &nbsp; Bubble Sort</a></li>
-<li><a href="insertion_sort/">11.4 &nbsp; Insertion Sort</a></li>
-<li><a href="quick_sort/">11.5 &nbsp; Quick Sort</a></li>
-<li><a href="merge_sort/">11.6 &nbsp; Merge Sort</a></li>
-<li><a href="heap_sort/">11.7 &nbsp; Heap Sort</a></li>
-<li><a href="bucket_sort/">11.8 &nbsp; Bucket Sort</a></li>
-<li><a href="counting_sort/">11.9 &nbsp; Counting Sort</a></li>
-<li><a href="radix_sort/">11.10 &nbsp; Radix Sort</a></li>
+<li><a href="sorting_algorithm/">11.1 &nbsp; Sorting algorithms</a></li>
+<li><a href="selection_sort/">11.2 &nbsp; Selection sort</a></li>
+<li><a href="bubble_sort/">11.3 &nbsp; Bubble sort</a></li>
+<li><a href="insertion_sort/">11.4 &nbsp; Insertion sort</a></li>
+<li><a href="quick_sort/">11.5 &nbsp; Quick sort</a></li>
+<li><a href="merge_sort/">11.6 &nbsp; Merge sort</a></li>
+<li><a href="heap_sort/">11.7 &nbsp; Heap sort</a></li>
+<li><a href="bucket_sort/">11.8 &nbsp; Bucket sort</a></li>
+<li><a href="counting_sort/">11.9 &nbsp; Counting sort</a></li>
+<li><a href="radix_sort/">11.10 &nbsp; Radix sort</a></li>
 <li><a href="summary/">11.11 &nbsp; Summary</a></li>
 <li><a href="exercises/">11.12 &nbsp; Exercises</a></li>
 </ul>
@@ -4621,7 +4693,7 @@ aria-label="Footer"
   <a
     href="../chapter_searching/exercises/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 10.7 Exercises"
+    aria-label="Previous: 10.7   Exercises"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4633,7 +4705,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        10.7 Exercises
+        10.7 &nbsp; Exercises
       </div>
     </div>
   </a>
@@ -4645,7 +4717,7 @@ aria-label="Footer"
   <a
     href="./sorting_algorithm/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 11.1 Sorting Algorithm"
+    aria-label="Next: 11.1   Sorting algorithms"
     rel="next"
   >
     <div class="md-footer__title">
@@ -4653,7 +4725,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        11.1 Sorting Algorithm
+        11.1 &nbsp; Sorting algorithms
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -4670,7 +4742,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>11.4 Insertion Sort - Hello Algo</title>
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
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-  3.6 Exercises
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-  4.6 Exercises
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-  5.5 Exercises
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.5 Exercises
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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@@ -2596,7 +2596,7 @@
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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-  10.1 Binary Search
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   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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         <li class="md-nav__item">
-  <a href="#1141-algorithm-flow" class="md-nav__link">
+  <a href="#1141-algorithm-steps" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.4.1 &nbsp; Algorithm Flow
+        11.4.1 &nbsp; Algorithm steps
       </span>
     </span>
   </a>
@@ -3066,10 +3066,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1142-algorithm-characteristics" class="md-nav__link">
+  <a href="#1142-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.4.2 &nbsp; Algorithm Characteristics
+        11.4.2 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -3077,10 +3077,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1143-advantages-of-insertion-sort" class="md-nav__link">
+  <a href="#1143-why-use-insertion-sort" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.4.3 &nbsp; Advantages of Insertion Sort
+        11.4.3 &nbsp; Why use insertion sort?
       </span>
     </span>
   </a>
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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+  13.3 &nbsp; Subset sum problem
 
 
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+  13.4 &nbsp; N-queens problem
 
 
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+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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+  15.1 &nbsp; Greedy algorithms
 
 
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+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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+  15.5 &nbsp; Summary
 
 
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+  15.6 &nbsp; Exercises
 
 
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+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
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-  <a href="#1141-algorithm-flow" class="md-nav__link">
+  <a href="#1141-algorithm-steps" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.4.1 &nbsp; Algorithm Flow
+        11.4.1 &nbsp; Algorithm steps
       </span>
     </span>
   </a>
@@ -4621,10 +4693,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1142-algorithm-characteristics" class="md-nav__link">
+  <a href="#1142-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.4.2 &nbsp; Algorithm Characteristics
+        11.4.2 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -4632,10 +4704,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1143-advantages-of-insertion-sort" class="md-nav__link">
+  <a href="#1143-why-use-insertion-sort" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.4.3 &nbsp; Advantages of Insertion Sort
+        11.4.3 &nbsp; Why use insertion sort?
       </span>
     </span>
   </a>
@@ -4679,53 +4751,53 @@
 
 
 <!-- Page content -->
-<h1 id="114-insertion-sort">11.4 &nbsp; Insertion Sort<a class="headerlink" href="#114-insertion-sort" title="Permanent link">&para;</a></h1>
-<p><u>Insertion sort</u> is a simple sorting algorithm that works very similarly to the process of manually sorting a deck of cards.</p>
-<p>Specifically, we select a base element from the unsorted portion, compare it one by one with the elements in the sorted portion to its left, and insert it into the correct position.</p>
-<p>Figure 11-6 illustrates how an element is inserted into an array. Let the base element be <code>base</code>. We need to shift all elements between the target index and <code>base</code> one position to the right, and then assign <code>base</code> to the target index.</p>
-<p><img alt="Single insertion operation" class="animation-figure" src="../insertion_sort.assets/insertion_operation.png" /></p>
-<p align="center"> Figure 11-6 &nbsp; Single insertion operation </p>
+<h1 id="114-insertion-sort">11.4 &nbsp; Insertion sort<a class="headerlink" href="#114-insertion-sort" title="Permanent link">&para;</a></h1>
+<p><u>Insertion sort</u> is a simple sorting algorithm that works much like arranging a hand of playing cards.</p>
+<p>Choose a base element from the unsorted section, compare it with the elements in the sorted section to its left, and insert it into the correct position.</p>
+<p>Figure 11-6 shows how to insert an element into an array. Let the base element be <code>base</code>. Shift all elements between the target index and <code>base</code> one position to the right, then assign <code>base</code> to the target index.</p>
+<p><img alt="A single insertion operation" class="animation-figure" src="../insertion_sort.assets/insertion_operation.png" /></p>
+<p align="center"> Figure 11-6 &nbsp; A single insertion operation </p>
 
-<h2 id="1141-algorithm-flow">11.4.1 &nbsp; Algorithm Flow<a class="headerlink" href="#1141-algorithm-flow" title="Permanent link">&para;</a></h2>
-<p>The overall flow of insertion sort is shown in Figure 11-7.</p>
+<h2 id="1141-algorithm-steps">11.4.1 &nbsp; Algorithm steps<a class="headerlink" href="#1141-algorithm-steps" title="Permanent link">&para;</a></h2>
+<p>The overall insertion sort process is shown below.</p>
 <ol>
-<li>Initially, the first element of the array is already sorted.</li>
-<li>Select the second element of the array as <code>base</code>, and after inserting it into the correct position, <strong>the first 2 elements of the array are sorted</strong>.</li>
-<li>Select the third element as <code>base</code>, and after inserting it into the correct position, <strong>the first 3 elements of the array are sorted</strong>.</li>
-<li>And so on. In the last round, select the last element as <code>base</code>, and after inserting it into the correct position, <strong>all elements are sorted</strong>.</li>
+<li>Initially, the first element of the array is treated as sorted.</li>
+<li>Select the second element as <code>base</code> and insert it into the correct position. <strong>The first two elements are now sorted</strong>.</li>
+<li>Select the third element as <code>base</code> and insert it into the correct position. <strong>The first three elements are now sorted</strong>.</li>
+<li>Continue in this way. In the final round, select the last element as <code>base</code> and insert it into the correct position. <strong>All elements are now sorted</strong>.</li>
 </ol>
-<p><img alt="Insertion sort flow" class="animation-figure" src="../insertion_sort.assets/insertion_sort_overview.png" /></p>
-<p align="center"> Figure 11-7 &nbsp; Insertion sort flow </p>
+<p><img alt="The insertion sort process" class="animation-figure" src="../insertion_sort.assets/insertion_sort_overview.png" /></p>
+<p align="center"> Figure 11-7 &nbsp; The insertion sort process </p>
 
-<p>Example code is as follows:</p>
+<p>An implementation follows:</p>
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 <div class="tabbed-block">
 <div class="highlight"><span class="filename">insertion_sort.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">insertion_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]):</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Insertion sort"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Outer loop: sorted interval is [0, i-1]</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Outer loop: the sorted range is [0, i-1]</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)):</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="n">base</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="n">j</span> <span class="o">=</span> <span class="n">i</span> <span class="o">-</span> <span class="mi">1</span>
-<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="c1"># Inner loop: insert base into the correct position within the sorted interval [0, i-1]</span>
+<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="c1"># Inner loop: insert base at the correct position in the sorted range [0, i-1]</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="k">while</span> <span class="n">j</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">&gt;</span> <span class="n">base</span><span class="p">:</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>            <span class="n">nums</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>  <span class="c1"># Move nums[j] to the right by one position</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>            <span class="n">nums</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>  <span class="c1"># Shift nums[j] one position to the right</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>            <span class="n">j</span> <span class="o">-=</span> <span class="mi">1</span>
-<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="n">nums</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">base</span>  <span class="c1"># Assign base to the correct position</span>
+<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="n">nums</span><span class="p">[</span><span class="n">j</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">base</span>  <span class="c1"># Place base at the correct position</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">insertion_sort.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Insertion sort */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insertionSort</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Outer loop: sorted interval is [0, i-1]</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Outer loop: the sorted range is [0, i-1]</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">        </span><span class="c1">// Inner loop: insert base into the correct position within the sorted interval [0, i-1]</span>
+<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">        </span><span class="c1">// Inner loop: insert base at the correct position in the sorted range [0, i-1]</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">base</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">            </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span><span class="w"> </span><span class="c1">// Move nums[j] to the right by one position</span>
+<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">            </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span><span class="w"> </span><span class="c1">// Shift nums[j] one position to the right</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w"> </span><span class="c1">// Assign base to the correct position</span>
+<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w"> </span><span class="c1">// Place base at the correct position</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="p">}</span>
 </code></pre></div>
@@ -4733,15 +4805,15 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">insertion_sort.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Insertion sort */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insertionSort</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Outer loop: sorted interval is [0, i-1]</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Outer loop: the sorted range is [0, i-1]</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">        </span><span class="c1">// Inner loop: insert base into the correct position within the sorted interval [0, i-1]</span>
+<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">        </span><span class="c1">// Inner loop: insert base at the correct position in the sorted range [0, i-1]</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">base</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">            </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// Move nums[j] to the right by one position</span>
+<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">            </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">;</span><span class="w"> </span><span class="c1">// Shift nums[j] one position to the right</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w">        </span><span class="c1">// Assign base to the correct position</span>
+<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w">        </span><span class="c1">// Place base at the correct position</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="p">}</span>
 </code></pre></div>
@@ -4749,15 +4821,15 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">insertion_sort.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Insertion sort */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">InsertionSort</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Outer loop: sorted interval is [0, i-1]</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Outer loop: the sorted range is [0, i-1]</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">bas</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">        </span><span class="c1">// Inner loop: insert base into the correct position within the sorted interval [0, i-1]</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">        </span><span class="c1">// Inner loop: insert bas at the correct position in the sorted range [0, i-1]</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">bas</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">            </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span><span class="w"> </span><span class="c1">// Move nums[j] to the right by one position</span>
+<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">            </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span><span class="w"> </span><span class="c1">// Shift nums[j] one position to the right</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">bas</span><span class="p">;</span><span class="w">         </span><span class="c1">// Assign base to the correct position</span>
+<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">bas</span><span class="p">;</span><span class="w">         </span><span class="c1">// Place bas at the correct position</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="p">}</span>
 </code></pre></div>
@@ -4765,16 +4837,16 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">insertion_sort.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Insertion sort */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">insertionSort</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Outer loop: sorted interval is [0, i-1]</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Outer loop: the sorted range is [0, i-1]</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">);</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">        </span><span class="nx">base</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="c1">// Inner loop: insert base into the correct position within the sorted interval [0, i-1]</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="c1">// Inner loop: insert base at the correct position in the sorted range [0, i-1]</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">base</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">            </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="c1">// Move nums[j] to the right by one position</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">            </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="c1">// Shift nums[j] one position to the right</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">            </span><span class="nx">j</span><span class="o">--</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">base</span><span class="w"> </span><span class="c1">// Assign base to the correct position</span>
+<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">base</span><span class="w"> </span><span class="c1">// Place base at the correct position</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="p">}</span>
 </code></pre></div>
@@ -4782,16 +4854,16 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">insertion_sort.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Insertion sort */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">insertionSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Outer loop: sorted interval is [0, i-1]</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Outer loop: the sorted range is [0, i-1]</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="bp">dropFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">base</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">        </span><span class="c1">// Inner loop: insert base into the correct position within the sorted interval [0, i-1]</span>
+<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">        </span><span class="c1">// Inner loop: insert base at the correct position in the sorted range [0, i-1]</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">            </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="c1">// Move nums[j] to the right by one position</span>
+<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">            </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="c1">// Shift nums[j] one position to the right</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="c1">// Assign base to the correct position</span>
+<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="c1">// Place base at the correct position</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="p">}</span>
 </code></pre></div>
@@ -4799,16 +4871,16 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">insertion_sort.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Insertion sort */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">insertionSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Outer loop: sorted interval is [0, i-1]</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Outer loop: the sorted range is [0, i-1]</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">        </span><span class="c1">// Inner loop: insert base into the correct position within the sorted interval [0, i-1]</span>
+<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">        </span><span class="c1">// Inner loop: insert base at the correct position in the sorted range [0, i-1]</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">base</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">            </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span><span class="w"> </span><span class="c1">// Move nums[j] to the right by one position</span>
+<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">            </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span><span class="w"> </span><span class="c1">// Shift nums[j] one position to the right</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">            </span><span class="nx">j</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">base</span><span class="p">;</span><span class="w"> </span><span class="c1">// Assign base to the correct position</span>
+<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">base</span><span class="p">;</span><span class="w"> </span><span class="c1">// Place base at the correct position</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="p">}</span>
 </code></pre></div>
@@ -4816,16 +4888,16 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">insertion_sort.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Insertion sort */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">insertionSort</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Outer loop: sorted interval is [0, i-1]</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Outer loop: the sorted range is [0, i-1]</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
-<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">        </span><span class="c1">// Inner loop: insert base into the correct position within the sorted interval [0, i-1]</span>
+<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">        </span><span class="c1">// Inner loop: insert base at the correct position in the sorted range [0, i-1]</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">base</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">            </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span><span class="w"> </span><span class="c1">// Move nums[j] to the right by one position</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">            </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span><span class="w"> </span><span class="c1">// Shift nums[j] one position to the right</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">            </span><span class="nx">j</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">base</span><span class="p">;</span><span class="w"> </span><span class="c1">// Assign base to the correct position</span>
+<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">base</span><span class="p">;</span><span class="w"> </span><span class="c1">// Place base at the correct position</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="p">}</span>
 </code></pre></div>
@@ -4833,15 +4905,15 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">insertion_sort.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Insertion sort */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">insertionSort</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Outer loop: sorted interval is [0, i-1]</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Outer loop: the sorted range is [0, i-1]</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
-<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">    </span><span class="c1">// Inner loop: insert base into the correct position within the sorted interval [0, i-1]</span>
+<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">    </span><span class="c1">// Inner loop: insert base at the correct position in the sorted range [0, i-1]</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">base</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">      </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span><span class="w"> </span><span class="c1">// Move nums[j] to the right by one position</span>
+<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">      </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span><span class="w"> </span><span class="c1">// Shift nums[j] one position to the right</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">      </span><span class="n">j</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w"> </span><span class="c1">// Assign base to the correct position</span>
+<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w"> </span><span class="c1">// Place base at the correct position</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="p">}</span>
 </code></pre></div>
@@ -4849,15 +4921,15 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">insertion_sort.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Insertion sort */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">insertion_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Outer loop: sorted interval is [0, i-1]</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Outer loop: the sorted range is [0, i-1]</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="n">base</span><span class="p">,</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">);</span>
-<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">        </span><span class="c1">// Inner loop: insert base into the correct position within the sorted interval [0, i-1]</span>
+<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">        </span><span class="c1">// Inner loop: insert base at the correct position in the sorted range [0, i-1]</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">            </span><span class="n">nums</span><span class="p">[(</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">];</span><span class="w"> </span><span class="c1">// Move nums[j] to the right by one position</span>
+<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">            </span><span class="n">nums</span><span class="p">[(</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">];</span><span class="w"> </span><span class="c1">// Shift nums[j] one position to the right</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="n">nums</span><span class="p">[(</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w"> </span><span class="c1">// Assign base to the correct position</span>
+<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="n">nums</span><span class="p">[(</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">base</span><span class="p">;</span><span class="w"> </span><span class="c1">// Place base at the correct position</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="p">}</span>
 </code></pre></div>
@@ -4865,16 +4937,16 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">insertion_sort.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Insertion sort */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insertionSort</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Outer loop: sorted interval is [0, i-1]</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Outer loop: the sorted range is [0, i-1]</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
-<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">        </span><span class="c1">// Inner loop: insert base into the correct position within the sorted interval [0, i-1]</span>
+<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">        </span><span class="c1">// Inner loop: insert base at the correct position in the sorted range [0, i-1]</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">base</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">            </span><span class="c1">// Move nums[j] to the right by one position</span>
+<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">            </span><span class="c1">// Shift nums[j] one position to the right</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">            </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="c1">// Assign base to the correct position</span>
+<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="c1">// Place base at the correct position</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">base</span><span class="p">;</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="p">}</span>
@@ -4883,16 +4955,16 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">insertion_sort.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Insertion sort */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">insertionSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Outer loop: sorted elements are 1, 2, ..., n</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">//Outer loop: the sorted prefix grows to 1, 2, ..., n elements</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">indices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">        </span><span class="c1">// Inner loop: insert base into the correct position within the sorted interval [0, i-1]</span>
+<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">        </span><span class="c1">// Inner loop: insert base at the correct position in the sorted range [0, i-1]</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">base</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">            </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="c1">// Move nums[j] to the right by one position</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">            </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="c1">// Shift nums[j] one position to the right</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">base</span><span class="w">        </span><span class="c1">// Assign base to the correct position</span>
+<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">base</span><span class="w">        </span><span class="c1">// Place base at the correct position</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="p">}</span>
 </code></pre></div>
@@ -4901,37 +4973,42 @@
 <div class="highlight"><span class="filename">insertion_sort.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Insertion sort ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">insertion_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="c1"># Outer loop: sorted interval is [0, i-1]</span>
+<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="c1"># Outer loop: the sorted range is [0, i-1]</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">...</span><span class="n">n</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">    </span><span class="n">base</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">    </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span>
-<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="c1"># Inner loop: insert base into the correct position within the sorted interval [0, i-1]</span>
+<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="c1"># Inner loop: insert base at the correct position in the sorted range [0, i-1]</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">base</span>
-<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">      </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="c1"># Move nums[j] to the right by one position</span>
+<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">      </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="c1"># Shift nums[j] one position to the right</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">    </span><span class="k">end</span>
-<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="c1"># Assign base to the correct position</span>
+<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">base</span><span class="w"> </span><span class="c1"># Place base at the correct position</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h2 id="1142-algorithm-characteristics">11.4.2 &nbsp; Algorithm Characteristics<a class="headerlink" href="#1142-algorithm-characteristics" title="Permanent link">&para;</a></h2>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 513px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20insertion_sort%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Insertion%20sort%22%22%22%0A%20%20%20%20%23%20Outer%20loop%3A%20the%20sorted%20range%20is%20%5B0,%20i-1%5D%0A%20%20%20%20for%20i%20in%20range%281,%20len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20base%20%3D%20nums%5Bi%5D%0A%20%20%20%20%20%20%20%20j%20%3D%20i%20-%201%0A%20%20%20%20%20%20%20%20%23%20Inner%20loop%3A%20insert%20base%20at%20the%20correct%20position%20in%20the%20sorted%20range%20%5B0,%20i-1%5D%0A%20%20%20%20%20%20%20%20while%20j%20%3E%3D%200%20and%20nums%5Bj%5D%20%3E%20base%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bj%20%2B%201%5D%20%3D%20nums%5Bj%5D%20%20%23%20Shift%20nums%5Bj%5D%20one%20position%20to%20the%20right%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20-%3D%201%0A%20%20%20%20%20%20%20%20nums%5Bj%20%2B%201%5D%20%3D%20base%20%20%23%20Place%20base%20at%20the%20correct%20position%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B4,%201,%203,%201,%205,%202%5D%0A%20%20%20%20insertion_sort%28nums%29%0A%20%20%20%20print%28%22After%20insertion%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20insertion_sort%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Insertion%20sort%22%22%22%0A%20%20%20%20%23%20Outer%20loop%3A%20the%20sorted%20range%20is%20%5B0,%20i-1%5D%0A%20%20%20%20for%20i%20in%20range%281,%20len%28nums%29%29%3A%0A%20%20%20%20%20%20%20%20base%20%3D%20nums%5Bi%5D%0A%20%20%20%20%20%20%20%20j%20%3D%20i%20-%201%0A%20%20%20%20%20%20%20%20%23%20Inner%20loop%3A%20insert%20base%20at%20the%20correct%20position%20in%20the%20sorted%20range%20%5B0,%20i-1%5D%0A%20%20%20%20%20%20%20%20while%20j%20%3E%3D%200%20and%20nums%5Bj%5D%20%3E%20base%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20nums%5Bj%20%2B%201%5D%20%3D%20nums%5Bj%5D%20%20%23%20Shift%20nums%5Bj%5D%20one%20position%20to%20the%20right%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20-%3D%201%0A%20%20%20%20%20%20%20%20nums%5Bj%20%2B%201%5D%20%3D%20base%20%20%23%20Place%20base%20at%20the%20correct%20position%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B4,%201,%203,%201,%205,%202%5D%0A%20%20%20%20insertion_sort%28nums%29%0A%20%20%20%20print%28%22After%20insertion%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1142-algorithm-properties">11.4.2 &nbsp; Algorithm properties<a class="headerlink" href="#1142-algorithm-properties" title="Permanent link">&para;</a></h2>
 <ul>
-<li><strong>Time complexity of <span class="arithmatex">\(O(n^2)\)</span>, adaptive sorting</strong>: In the worst case, the insertion operations require <span class="arithmatex">\(n - 1\)</span>, <span class="arithmatex">\(n-2\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, and <span class="arithmatex">\(1\)</span> iterations, respectively, summing to <span class="arithmatex">\((n - 1) n / 2\)</span>, so the time complexity is <span class="arithmatex">\(O(n^2)\)</span>. When the data is already sorted, each insertion operation terminates early. When the input array is completely sorted, insertion sort achieves its best-case time complexity of <span class="arithmatex">\(O(n)\)</span>.</li>
-<li><strong>Space complexity of <span class="arithmatex">\(O(1)\)</span>, in-place sorting</strong>: Pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> use a constant amount of extra space.</li>
-<li><strong>Stable sorting</strong>: During insertion, we place elements to the right of equal elements, so their relative order is unchanged.</li>
+<li><strong>Time complexity: <span class="arithmatex">\(O(n^2)\)</span>; adaptive</strong>. In the worst case, the insertion operations require <span class="arithmatex">\(n - 1\)</span>, <span class="arithmatex">\(n-2\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, and <span class="arithmatex">\(1\)</span> iterations, totaling <span class="arithmatex">\((n - 1) n / 2\)</span>, so the time complexity is <span class="arithmatex">\(O(n^2)\)</span>. For ordered data, an insertion can stop early. A fully sorted input achieves the best-case time complexity of <span class="arithmatex">\(O(n)\)</span>.</li>
+<li><strong>Space complexity: <span class="arithmatex">\(O(1)\)</span>; in-place</strong>. Pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> use constant extra space.</li>
+<li><strong>Stable</strong>. Each element is inserted to the right of equal elements, preserving their order.</li>
 </ul>
-<h2 id="1143-advantages-of-insertion-sort">11.4.3 &nbsp; Advantages of Insertion Sort<a class="headerlink" href="#1143-advantages-of-insertion-sort" title="Permanent link">&para;</a></h2>
-<p>The time complexity of insertion sort is <span class="arithmatex">\(O(n^2)\)</span>, while the time complexity of quick sort, which we will learn about next, is <span class="arithmatex">\(O(n \log n)\)</span>. Although insertion sort has a higher time complexity, <strong>it is usually faster on small datasets</strong>.</p>
-<p>This conclusion is similar to the one about when linear search and binary search are applicable. Algorithms such as quick sort, with <span class="arithmatex">\(O(n \log n)\)</span> complexity, are divide-and-conquer sorting algorithms and often involve more primitive operations. When the dataset is small, the values of <span class="arithmatex">\(n^2\)</span> and <span class="arithmatex">\(n \log n\)</span> are relatively close, so asymptotic complexity does not dominate; instead, the number of primitive operations per round becomes the deciding factor.</p>
-<p>In fact, the built-in sorting functions of many programming languages (such as Java) use insertion sort. The general idea is: for large arrays, use divide-and-conquer sorting algorithms such as quick sort; for short arrays, use insertion sort directly.</p>
-<p>Although bubble sort, selection sort, and insertion sort all have a time complexity of <span class="arithmatex">\(O(n^2)\)</span>, in actual situations, <strong>insertion sort is used significantly more frequently than bubble sort and selection sort</strong>, mainly for the following reasons.</p>
+<h2 id="1143-why-use-insertion-sort">11.4.3 &nbsp; Why use insertion sort?<a class="headerlink" href="#1143-why-use-insertion-sort" title="Permanent link">&para;</a></h2>
+<p>Insertion sort takes <span class="arithmatex">\(O(n^2)\)</span> time, while quick sort, which we will study shortly, takes <span class="arithmatex">\(O(n \log n)\)</span>. Despite its higher time complexity, <strong>insertion sort is usually faster for small datasets</strong>.</p>
+<p>This resembles the comparison between linear and binary search. Algorithms such as quick sort, with <span class="arithmatex">\(O(n \log n)\)</span> time complexity, use divide and conquer and tend to involve more elementary operations. When the dataset is small, <span class="arithmatex">\(n^2\)</span> and <span class="arithmatex">\(n \log n\)</span> are relatively close, so asymptotic complexity is not the dominant factor; the number of elementary operations per round matters more.</p>
+<p>In fact, the built-in sorting functions of many programming languages, such as Java, use insertion sort. The general approach is to use a divide-and-conquer algorithm such as quick sort for long arrays and insertion sort directly for short ones.</p>
+<p>Although bubble sort, selection sort, and insertion sort all take <span class="arithmatex">\(O(n^2)\)</span> time, <strong>insertion sort is used much more often in practice</strong>, mainly for the following reasons.</p>
 <ul>
-<li>Bubble sort is implemented through element swaps, which require a temporary variable and involve 3 primitive operations; insertion sort is implemented through element assignment and requires only 1 primitive operation. Therefore, <strong>bubble sort usually has higher computational overhead than insertion sort</strong>.</li>
-<li>Selection sort has a time complexity of <span class="arithmatex">\(O(n^2)\)</span> in any case. <strong>If given a set of partially ordered data, insertion sort is usually more efficient than selection sort</strong>.</li>
-<li>Selection sort is unstable and cannot be applied to multi-level sorting.</li>
+<li>Bubble sort swaps elements, using a temporary variable and three elementary operations. Insertion sort shifts elements by assignment, requiring just one operation. <strong>Bubble sort therefore usually has higher computational overhead</strong>.</li>
+<li>Selection sort takes <span class="arithmatex">\(O(n^2)\)</span> time in every case. <strong>For partially sorted data, insertion sort is usually more efficient than selection sort</strong>.</li>
+<li>Selection sort is not stable and therefore cannot be used for multilevel sorting.</li>
 </ul>
 
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   <a
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-    aria-label="Previous: 11.3 Bubble Sort"
+    aria-label="Previous: 11.3   Bubble sort"
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     href="../quick_sort/"
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-    aria-label="Next: 11.5 Quick Sort"
+    aria-label="Next: 11.5   Quick sort"
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
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@@ -41,7 +41,7 @@
 
 
 
-        <title>11.6 Merge Sort - Hello Algo</title>
+        <title>11.6 Merge sort - Hello Algo</title>
 
 
 
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-  0.2 How to Use This Book
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-  0.3 Summary
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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-  2.6 Exercises
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-  Chapter 3. Data Structures
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-  3.1 Classification of Data Structures
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-  3.2 Basic Data Types
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-  3.3 Number Encoding *
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-  3.4 Character Encoding *
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+  6.2 &nbsp; Hash collisions
 
 
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+  6.3 &nbsp; Hash algorithms
 
 
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+  7.2 &nbsp; Traversing a binary tree
 
 
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+  7.4 &nbsp; Binary search trees
 
 
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+  7.5 &nbsp; AVL trees *
 
 
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+  Chapter 8. &nbsp; Heaps
 
 
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+  8.2 &nbsp; Building a heap
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
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@@ -2638,7 +2638,7 @@
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-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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@@ -2922,7 +2922,7 @@
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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@@ -3109,10 +3109,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1161-algorithm-flow" class="md-nav__link">
+  <a href="#1161-algorithm-steps" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.6.1 &nbsp; Algorithm Flow
+        11.6.1 &nbsp; Algorithm steps
       </span>
     </span>
   </a>
@@ -3120,10 +3120,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1162-algorithm-characteristics" class="md-nav__link">
+  <a href="#1162-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.6.2 &nbsp; Algorithm Characteristics
+        11.6.2 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -3131,10 +3131,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1163-linked-list-sorting" class="md-nav__link">
+  <a href="#1163-sorting-linked-lists" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.6.3 &nbsp; Linked List Sorting
+        11.6.3 &nbsp; Sorting linked lists
       </span>
     </span>
   </a>
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
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-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4569,6 +4569,78 @@
 
 
 
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         <li class="md-nav__item">
-  <a href="#1161-algorithm-flow" class="md-nav__link">
+  <a href="#1161-algorithm-steps" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.6.1 &nbsp; Algorithm Flow
+        11.6.1 &nbsp; Algorithm steps
       </span>
     </span>
   </a>
@@ -4621,10 +4693,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1162-algorithm-characteristics" class="md-nav__link">
+  <a href="#1162-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.6.2 &nbsp; Algorithm Characteristics
+        11.6.2 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -4632,10 +4704,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1163-linked-list-sorting" class="md-nav__link">
+  <a href="#1163-sorting-linked-lists" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.6.3 &nbsp; Linked List Sorting
+        11.6.3 &nbsp; Sorting linked lists
       </span>
     </span>
   </a>
@@ -4679,26 +4751,26 @@
 
 
 <!-- Page content -->
-<h1 id="116-merge-sort">11.6 &nbsp; Merge Sort<a class="headerlink" href="#116-merge-sort" title="Permanent link">&para;</a></h1>
-<p><u>Merge sort</u> is a sorting algorithm based on a divide-and-conquer strategy, consisting of the "divide" and "merge" phases shown in Figure 11-10.</p>
+<h1 id="116-merge-sort">11.6 &nbsp; Merge sort<a class="headerlink" href="#116-merge-sort" title="Permanent link">&para;</a></h1>
+<p><u>Merge sort</u> is a divide-and-conquer sorting algorithm with the “division” and “merging” phases shown below.</p>
 <ol>
-<li><strong>Divide phase</strong>: Recursively split the array at the midpoint, reducing the problem of sorting a long array to the problem of sorting shorter arrays.</li>
-<li><strong>Merge phase</strong>: When a sub-array has length 1, stop dividing and start merging, continuously combining the shorter sorted sub-arrays on the left and right into a longer sorted array until the process is complete.</li>
+<li><strong>Division</strong>: recursively split the array at its midpoint, turning the problem of sorting a long array into problems of sorting shorter arrays.</li>
+<li><strong>Merging</strong>: stop dividing when a subarray has length 1, then repeatedly merge two shorter sorted arrays into one longer sorted array until the process is complete.</li>
 </ol>
-<p><img alt="Divide and merge phases of merge sort" class="animation-figure" src="../merge_sort.assets/merge_sort_overview.png" /></p>
-<p align="center"> Figure 11-10 &nbsp; Divide and merge phases of merge sort </p>
+<p><img alt="The division and merging phases of merge sort" class="animation-figure" src="../merge_sort.assets/merge_sort_overview.png" /></p>
+<p align="center"> Figure 11-10 &nbsp; The division and merging phases of merge sort </p>
 
-<h2 id="1161-algorithm-flow">11.6.1 &nbsp; Algorithm Flow<a class="headerlink" href="#1161-algorithm-flow" title="Permanent link">&para;</a></h2>
-<p>As shown in Figure 11-11, the "divide phase" recursively splits the array from the midpoint into two sub-arrays from top to bottom.</p>
+<h2 id="1161-algorithm-steps">11.6.1 &nbsp; Algorithm steps<a class="headerlink" href="#1161-algorithm-steps" title="Permanent link">&para;</a></h2>
+<p>As shown below, the division phase works from the top down, recursively splitting each array into two subarrays at its midpoint.</p>
 <ol>
-<li>Calculate the array midpoint <code>mid</code>, recursively divide the left sub-array (interval <code>[left, mid]</code>) and right sub-array (interval <code>[mid + 1, right]</code>).</li>
-<li>Repeat step <code>1.</code> recursively until a sub-array has length 1.</li>
+<li>Calculate the midpoint <code>mid</code>, then recursively divide the left subarray, <code>[left, mid]</code>, and the right subarray, <code>[mid + 1, right]</code>.</li>
+<li>Repeat step <code>1.</code> recursively until each subarray has length 1.</li>
 </ol>
-<p>The "merge phase" merges the left and right sub-arrays into a sorted array from bottom to top. Note that merging starts from sub-arrays of length 1, so every sub-array involved in this phase is already sorted.</p>
+<p>The merging phase works from the bottom up, combining the left and right subarrays into a sorted array. Merging begins with subarrays of length 1, so every subarray involved in this phase is already sorted.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:10"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label><label for="__tabbed_1_6">&lt;6&gt;</label><label for="__tabbed_1_7">&lt;7&gt;</label><label for="__tabbed_1_8">&lt;8&gt;</label><label for="__tabbed_1_9">&lt;9&gt;</label><label for="__tabbed_1_10">&lt;10&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Merge sort steps" class="animation-figure" src="../merge_sort.assets/merge_sort_step1.png" /></p>
+<p><img alt="Steps of merge sort" class="animation-figure" src="../merge_sort.assets/merge_sort_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="merge_sort_step2" class="animation-figure" src="../merge_sort.assets/merge_sort_step2.png" /></p>
@@ -4729,25 +4801,25 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 11-11 &nbsp; Merge sort steps </p>
+<p align="center"> Figure 11-11 &nbsp; Steps of merge sort </p>
 
-<p>The recursive order of merge sort is consistent with the post-order traversal of a binary tree.</p>
+<p>Merge sort follows the same recursive order as a postorder traversal of a binary tree.</p>
 <ul>
-<li><strong>Post-order traversal</strong>: First recursively traverse the left subtree, then recursively traverse the right subtree, and finally process the root node.</li>
-<li><strong>Merge sort</strong>: First recursively process the left sub-array, then recursively process the right sub-array, and finally perform the merge.</li>
+<li><strong>Postorder traversal</strong>: recurse into the left subtree, then the right subtree, and finally process the root.</li>
+<li><strong>Merge sort</strong>: recurse into the left subarray, then the right subarray, and finally merge them.</li>
 </ul>
-<p>The implementation of merge sort is shown in the code below. Note that the interval to be merged in <code>nums</code> is <code>[left, right]</code>, while the corresponding interval in <code>tmp</code> is <code>[0, right - left]</code>.</p>
+<p>The implementation follows. The interval to merge in <code>nums</code> is <code>[left, right]</code>, while its corresponding interval in <code>tmp</code> is <code>[0, right - left]</code>.</p>
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 <div class="highlight"><span class="filename">merge_sort.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">merge</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">left</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">mid</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">right</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Merge left subarray and right subarray"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Left subarray interval is [left, mid], right subarray interval is [mid+1, right]</span>
-<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="c1"># Create a temporary array tmp to store the merged results</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Merge the left and right subarrays"""</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># The left subarray spans [left, mid], and the right subarray spans [mid+1, right]</span>
+<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="c1"># Create a temporary array tmp to store the merged result</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="n">tmp</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">right</span> <span class="o">-</span> <span class="n">left</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Initialize the start indices of the left and right subarrays</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Initialize the starting indices of the left and right subarrays</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">left</span><span class="p">,</span> <span class="n">mid</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="c1"># While both subarrays still have elements, compare and copy the smaller element into the temporary array</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="c1"># While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="n">mid</span> <span class="ow">and</span> <span class="n">j</span> <span class="o">&lt;=</span> <span class="n">right</span><span class="p">:</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;=</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]:</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>            <span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
@@ -4765,32 +4837,32 @@
 <a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a>        <span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
 <a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a>        <span class="n">j</span> <span class="o">+=</span> <span class="mi">1</span>
 <a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a>        <span class="n">k</span> <span class="o">+=</span> <span class="mi">1</span>
-<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a>    <span class="c1"># Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval</span>
+<a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a>    <span class="c1"># Copy the elements of temporary array tmp back into the corresponding range of the original array nums</span>
 <a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a>    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">tmp</span><span class="p">)):</span>
 <a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a>        <span class="n">nums</span><span class="p">[</span><span class="n">left</span> <span class="o">+</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
 <a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a>
 <a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a><span class="k">def</span><span class="w"> </span><span class="nf">merge_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">left</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">right</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a><span class="w">    </span><span class="sd">"""Merge sort"""</span>
-<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a>    <span class="c1"># Termination condition</span>
+<a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a>    <span class="c1"># Base case</span>
 <a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a>    <span class="k">if</span> <span class="n">left</span> <span class="o">&gt;=</span> <span class="n">right</span><span class="p">:</span>
-<a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a>        <span class="k">return</span>  <span class="c1"># Terminate recursion when subarray length is 1</span>
-<a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a>    <span class="c1"># Divide and conquer stage</span>
-<a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a>    <span class="n">mid</span> <span class="o">=</span> <span class="p">(</span><span class="n">left</span> <span class="o">+</span> <span class="n">right</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>  <span class="c1"># Calculate midpoint</span>
-<a id="__codelineno-0-37" name="__codelineno-0-37" href="#__codelineno-0-37"></a>    <span class="n">merge_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">left</span><span class="p">,</span> <span class="n">mid</span><span class="p">)</span>  <span class="c1"># Recursively process the left subarray</span>
-<a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></a>    <span class="n">merge_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">mid</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>  <span class="c1"># Recursively process the right subarray</span>
-<a id="__codelineno-0-39" name="__codelineno-0-39" href="#__codelineno-0-39"></a>    <span class="c1"># Merge stage</span>
+<a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a>        <span class="k">return</span>  <span class="c1"># Stop recursing when the subarray has at most one element</span>
+<a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a>    <span class="c1"># Divide phase</span>
+<a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a>    <span class="n">mid</span> <span class="o">=</span> <span class="p">(</span><span class="n">left</span> <span class="o">+</span> <span class="n">right</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span> <span class="c1"># Calculate the midpoint</span>
+<a id="__codelineno-0-37" name="__codelineno-0-37" href="#__codelineno-0-37"></a>    <span class="n">merge_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">left</span><span class="p">,</span> <span class="n">mid</span><span class="p">)</span>  <span class="c1"># Recurse on the left subarray</span>
+<a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></a>    <span class="n">merge_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">mid</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>  <span class="c1"># Recurse on the right subarray</span>
+<a id="__codelineno-0-39" name="__codelineno-0-39" href="#__codelineno-0-39"></a>    <span class="c1"># Merge phase</span>
 <a id="__codelineno-0-40" name="__codelineno-0-40" href="#__codelineno-0-40"></a>    <span class="n">merge</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">left</span><span class="p">,</span> <span class="n">mid</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">merge_sort.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Merge left subarray and right subarray */</span>
+<div class="highlight"><span class="filename">merge_sort.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Merge the left and right subarrays */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">merge</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Left subarray interval is [left, mid], right subarray interval is [mid+1, right]</span>
-<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged results</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// The left subarray spans [left, mid], and the right subarray spans [mid+1, right]</span>
+<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged result</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">tmp</span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Initialize the start indices of the left and right subarrays</span>
+<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Initialize the starting indices of the left and right subarrays</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="c1">// While both subarrays still have elements, compare and copy the smaller element into the temporary array</span>
+<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="c1">// While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">            </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="o">++</span><span class="p">];</span>
@@ -4804,7 +4876,7 @@
 <a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">        </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="o">++</span><span class="p">];</span>
 <a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w">    </span><span class="c1">// Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval</span>
+<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w">    </span><span class="c1">// Copy the elements of temporary array tmp back into the corresponding range of the original array nums</span>
 <a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">k</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="p">];</span>
 <a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w">    </span><span class="p">}</span>
@@ -4812,27 +4884,27 @@
 <a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a>
 <a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="cm">/* Merge sort */</span>
 <a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="kt">void</span><span class="w"> </span><span class="nf">mergeSort</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span><span class="w"> </span><span class="c1">// Terminate recursion when subarray length is 1</span>
-<a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="w">    </span><span class="c1">// Divide and conquer stage</span>
-<a id="__codelineno-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">    </span><span class="c1">// Calculate midpoint</span>
-<a id="__codelineno-1-35" name="__codelineno-1-35" href="#__codelineno-1-35"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">);</span><span class="w">      </span><span class="c1">// Recursively process the left subarray</span>
-<a id="__codelineno-1-36" name="__codelineno-1-36" href="#__codelineno-1-36"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively process the right subarray</span>
-<a id="__codelineno-1-37" name="__codelineno-1-37" href="#__codelineno-1-37"></a><span class="w">    </span><span class="c1">// Merge stage</span>
+<a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span><span class="w"> </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
+<a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a><span class="w">    </span><span class="c1">// Divide phase</span>
+<a id="__codelineno-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">    </span><span class="c1">// Calculate the midpoint</span>
+<a id="__codelineno-1-35" name="__codelineno-1-35" href="#__codelineno-1-35"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">);</span><span class="w">      </span><span class="c1">// Recurse on the left subarray</span>
+<a id="__codelineno-1-36" name="__codelineno-1-36" href="#__codelineno-1-36"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recurse on the right subarray</span>
+<a id="__codelineno-1-37" name="__codelineno-1-37" href="#__codelineno-1-37"></a><span class="w">    </span><span class="c1">// Merge phase</span>
 <a id="__codelineno-1-38" name="__codelineno-1-38" href="#__codelineno-1-38"></a><span class="w">    </span><span class="n">merge</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-1-39" name="__codelineno-1-39" href="#__codelineno-1-39"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">merge_sort.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Merge left subarray and right subarray */</span>
+<div class="highlight"><span class="filename">merge_sort.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Merge the left and right subarrays */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">merge</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Left subarray interval is [left, mid], right subarray interval is [mid+1, right]</span>
-<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged results</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// The left subarray spans [left, mid], and the right subarray spans [mid+1, right]</span>
+<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged result</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
-<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// Initialize the start indices of the left and right subarrays</span>
+<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// Initialize the starting indices of the left and right subarrays</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="c1">// While both subarrays still have elements, compare and copy the smaller element into the temporary array</span>
+<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="c1">// While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">            </span><span class="n">tmp</span><span class="o">[</span><span class="n">k</span><span class="o">++]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">++]</span><span class="p">;</span>
@@ -4846,7 +4918,7 @@
 <a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">        </span><span class="n">tmp</span><span class="o">[</span><span class="n">k</span><span class="o">++]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">++]</span><span class="p">;</span>
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="w">    </span><span class="c1">// Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval</span>
+<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="w">    </span><span class="c1">// Copy the elements of temporary array tmp back into the corresponding range of the original array nums</span>
 <a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="na">length</span><span class="p">;</span><span class="w"> </span><span class="n">k</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">k</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="o">[</span><span class="n">k</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="w">    </span><span class="p">}</span>
@@ -4854,27 +4926,27 @@
 <a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a>
 <a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a><span class="cm">/* Merge sort */</span>
 <a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="kt">void</span><span class="w"> </span><span class="nf">mergeSort</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span><span class="w"> </span><span class="c1">// Terminate recursion when subarray length is 1</span>
-<a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a><span class="w">    </span><span class="c1">// Divide and conquer stage</span>
-<a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate midpoint</span>
-<a id="__codelineno-2-35" name="__codelineno-2-35" href="#__codelineno-2-35"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively process the left subarray</span>
-<a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively process the right subarray</span>
-<a id="__codelineno-2-37" name="__codelineno-2-37" href="#__codelineno-2-37"></a><span class="w">    </span><span class="c1">// Merge stage</span>
+<a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span><span class="w"> </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
+<a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a><span class="w">    </span><span class="c1">// Divide phase</span>
+<a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint</span>
+<a id="__codelineno-2-35" name="__codelineno-2-35" href="#__codelineno-2-35"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recurse on the left subarray</span>
+<a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recurse on the right subarray</span>
+<a id="__codelineno-2-37" name="__codelineno-2-37" href="#__codelineno-2-37"></a><span class="w">    </span><span class="c1">// Merge phase</span>
 <a id="__codelineno-2-38" name="__codelineno-2-38" href="#__codelineno-2-38"></a><span class="w">    </span><span class="n">merge</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-2-39" name="__codelineno-2-39" href="#__codelineno-2-39"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">merge_sort.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Merge left subarray and right subarray */</span>
+<div class="highlight"><span class="filename">merge_sort.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Merge the left and right subarrays */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Merge</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Left subarray interval is [left, mid], right subarray interval is [mid+1, right]</span>
-<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged results</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// The left subarray spans [left, mid], and the right subarray spans [mid+1, right]</span>
+<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged result</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Initialize the start indices of the left and right subarrays</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Initialize the starting indices of the left and right subarrays</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="c1">// While both subarrays still have elements, compare and copy the smaller element into the temporary array</span>
+<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="c1">// While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">            </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="o">++</span><span class="p">];</span>
@@ -4888,7 +4960,7 @@
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">        </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="o">++</span><span class="p">];</span>
 <a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w">    </span><span class="c1">// Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval</span>
+<a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w">    </span><span class="c1">// Copy the elements of temporary array tmp back into the corresponding range of the original array nums</span>
 <a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="p">];</span>
 <a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="w">    </span><span class="p">}</span>
@@ -4896,26 +4968,26 @@
 <a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a>
 <a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="cm">/* Merge sort */</span>
 <a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="k">void</span><span class="w"> </span><span class="nf">MergeSort</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a><span class="w">    </span><span class="c1">// Termination condition</span>
-<a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span><span class="w">       </span><span class="c1">// Terminate recursion when subarray length is 1</span>
-<a id="__codelineno-3-32" name="__codelineno-3-32" href="#__codelineno-3-32"></a><span class="w">    </span><span class="c1">// Divide and conquer stage</span>
-<a id="__codelineno-3-33" name="__codelineno-3-33" href="#__codelineno-3-33"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">    </span><span class="c1">// Calculate midpoint</span>
-<a id="__codelineno-3-34" name="__codelineno-3-34" href="#__codelineno-3-34"></a><span class="w">    </span><span class="n">MergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">);</span><span class="w">      </span><span class="c1">// Recursively process the left subarray</span>
-<a id="__codelineno-3-35" name="__codelineno-3-35" href="#__codelineno-3-35"></a><span class="w">    </span><span class="n">MergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively process the right subarray</span>
-<a id="__codelineno-3-36" name="__codelineno-3-36" href="#__codelineno-3-36"></a><span class="w">    </span><span class="c1">// Merge stage</span>
+<a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a><span class="w">    </span><span class="c1">// Base case</span>
+<a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span><span class="w">       </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
+<a id="__codelineno-3-32" name="__codelineno-3-32" href="#__codelineno-3-32"></a><span class="w">    </span><span class="c1">// Divide phase</span>
+<a id="__codelineno-3-33" name="__codelineno-3-33" href="#__codelineno-3-33"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">    </span><span class="c1">// Calculate the midpoint</span>
+<a id="__codelineno-3-34" name="__codelineno-3-34" href="#__codelineno-3-34"></a><span class="w">    </span><span class="n">MergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">);</span><span class="w">      </span><span class="c1">// Recurse on the left subarray</span>
+<a id="__codelineno-3-35" name="__codelineno-3-35" href="#__codelineno-3-35"></a><span class="w">    </span><span class="n">MergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recurse on the right subarray</span>
+<a id="__codelineno-3-36" name="__codelineno-3-36" href="#__codelineno-3-36"></a><span class="w">    </span><span class="c1">// Merge phase</span>
 <a id="__codelineno-3-37" name="__codelineno-3-37" href="#__codelineno-3-37"></a><span class="w">    </span><span class="n">Merge</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-3-38" name="__codelineno-3-38" href="#__codelineno-3-38"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">merge_sort.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Merge left subarray and right subarray */</span>
+<div class="highlight"><span class="filename">merge_sort.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Merge the left and right subarrays */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">merge</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Left subarray interval is [left, mid], right subarray interval is [mid+1, right]</span>
-<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged results</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// The left subarray spans [left, mid], and the right subarray spans [mid+1, right]</span>
+<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged result</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="nx">tmp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="o">-</span><span class="nx">left</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="c1">// Initialize the start indices of the left and right subarrays</span>
+<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="c1">// Initialize the starting indices of the left and right subarrays</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">,</span><span class="w"> </span><span class="nx">k</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="c1">// While both subarrays still have elements, compare and copy the smaller element into the temporary array</span>
+<a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="c1">// While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">mid</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">            </span><span class="nx">tmp</span><span class="p">[</span><span class="nx">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
@@ -4937,7 +5009,7 @@
 <a id="__codelineno-4-27" name="__codelineno-4-27" href="#__codelineno-4-27"></a><span class="w">        </span><span class="nx">j</span><span class="o">++</span>
 <a id="__codelineno-4-28" name="__codelineno-4-28" href="#__codelineno-4-28"></a><span class="w">        </span><span class="nx">k</span><span class="o">++</span>
 <a id="__codelineno-4-29" name="__codelineno-4-29" href="#__codelineno-4-29"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-30" name="__codelineno-4-30" href="#__codelineno-4-30"></a><span class="w">    </span><span class="c1">// Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval</span>
+<a id="__codelineno-4-30" name="__codelineno-4-30" href="#__codelineno-4-30"></a><span class="w">    </span><span class="c1">// Copy the elements of temporary array tmp back into the corresponding range of the original array nums</span>
 <a id="__codelineno-4-31" name="__codelineno-4-31" href="#__codelineno-4-31"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">k</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">k</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">tmp</span><span class="p">);</span><span class="w"> </span><span class="nx">k</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-32" name="__codelineno-4-32" href="#__codelineno-4-32"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="o">+</span><span class="nx">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span><span class="p">[</span><span class="nx">k</span><span class="p">]</span>
 <a id="__codelineno-4-33" name="__codelineno-4-33" href="#__codelineno-4-33"></a><span class="w">    </span><span class="p">}</span>
@@ -4945,28 +5017,28 @@
 <a id="__codelineno-4-35" name="__codelineno-4-35" href="#__codelineno-4-35"></a>
 <a id="__codelineno-4-36" name="__codelineno-4-36" href="#__codelineno-4-36"></a><span class="cm">/* Merge sort */</span>
 <a id="__codelineno-4-37" name="__codelineno-4-37" href="#__codelineno-4-37"></a><span class="kd">func</span><span class="w"> </span><span class="nx">mergeSort</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-38" name="__codelineno-4-38" href="#__codelineno-4-38"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-4-38" name="__codelineno-4-38" href="#__codelineno-4-38"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-4-39" name="__codelineno-4-39" href="#__codelineno-4-39"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-40" name="__codelineno-4-40" href="#__codelineno-4-40"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-4-41" name="__codelineno-4-41" href="#__codelineno-4-41"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-42" name="__codelineno-4-42" href="#__codelineno-4-42"></a><span class="w">    </span><span class="c1">// Divide and conquer stage</span>
+<a id="__codelineno-4-42" name="__codelineno-4-42" href="#__codelineno-4-42"></a><span class="w">    </span><span class="c1">// Divide phase</span>
 <a id="__codelineno-4-43" name="__codelineno-4-43" href="#__codelineno-4-43"></a><span class="w">    </span><span class="nx">mid</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-4-44" name="__codelineno-4-44" href="#__codelineno-4-44"></a><span class="w">    </span><span class="nx">mergeSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="p">)</span>
 <a id="__codelineno-4-45" name="__codelineno-4-45" href="#__codelineno-4-45"></a><span class="w">    </span><span class="nx">mergeSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span>
-<a id="__codelineno-4-46" name="__codelineno-4-46" href="#__codelineno-4-46"></a><span class="w">    </span><span class="c1">// Merge stage</span>
+<a id="__codelineno-4-46" name="__codelineno-4-46" href="#__codelineno-4-46"></a><span class="w">    </span><span class="c1">// Merge phase</span>
 <a id="__codelineno-4-47" name="__codelineno-4-47" href="#__codelineno-4-47"></a><span class="w">    </span><span class="nx">merge</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span>
 <a id="__codelineno-4-48" name="__codelineno-4-48" href="#__codelineno-4-48"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">merge_sort.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Merge left subarray and right subarray */</span>
+<div class="highlight"><span class="filename">merge_sort.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Merge the left and right subarrays */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">merge</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Left subarray interval is [left, mid], right subarray interval is [mid+1, right]</span>
-<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged results</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// The left subarray spans [left, mid], and the right subarray spans [mid+1, right]</span>
+<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged result</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">tmp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="kr">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="kr">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// Initialize the start indices of the left and right subarrays</span>
+<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="c1">// Initialize the starting indices of the left and right subarrays</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="c1">// While both subarrays still have elements, compare and copy the smaller element into the temporary array</span>
+<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">    </span><span class="c1">// While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="kr">right</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">            </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
@@ -4988,7 +5060,7 @@
 <a id="__codelineno-5-27" name="__codelineno-5-27" href="#__codelineno-5-27"></a><span class="w">        </span><span class="n">j</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="w">        </span><span class="n">k</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-5-29" name="__codelineno-5-29" href="#__codelineno-5-29"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-30" name="__codelineno-5-30" href="#__codelineno-5-30"></a><span class="w">    </span><span class="c1">// Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval</span>
+<a id="__codelineno-5-30" name="__codelineno-5-30" href="#__codelineno-5-30"></a><span class="w">    </span><span class="c1">// Copy the elements of temporary array tmp back into the corresponding range of the original array nums</span>
 <a id="__codelineno-5-31" name="__codelineno-5-31" href="#__codelineno-5-31"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="bp">indices</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-32" name="__codelineno-5-32" href="#__codelineno-5-32"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="kr">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
 <a id="__codelineno-5-33" name="__codelineno-5-33" href="#__codelineno-5-33"></a><span class="w">    </span><span class="p">}</span>
@@ -4996,30 +5068,30 @@
 <a id="__codelineno-5-35" name="__codelineno-5-35" href="#__codelineno-5-35"></a>
 <a id="__codelineno-5-36" name="__codelineno-5-36" href="#__codelineno-5-36"></a><span class="cm">/* Merge sort */</span>
 <a id="__codelineno-5-37" name="__codelineno-5-37" href="#__codelineno-5-37"></a><span class="kd">func</span><span class="w"> </span><span class="nf">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-38" name="__codelineno-5-38" href="#__codelineno-5-38"></a><span class="w">    </span><span class="c1">// Termination condition</span>
-<a id="__codelineno-5-39" name="__codelineno-5-39" href="#__codelineno-5-39"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="kr">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="kr">right</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// Terminate recursion when subarray length is 1</span>
+<a id="__codelineno-5-38" name="__codelineno-5-38" href="#__codelineno-5-38"></a><span class="w">    </span><span class="c1">// Base case</span>
+<a id="__codelineno-5-39" name="__codelineno-5-39" href="#__codelineno-5-39"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="kr">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="kr">right</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
 <a id="__codelineno-5-40" name="__codelineno-5-40" href="#__codelineno-5-40"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-5-41" name="__codelineno-5-41" href="#__codelineno-5-41"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-42" name="__codelineno-5-42" href="#__codelineno-5-42"></a><span class="w">    </span><span class="c1">// Divide and conquer stage</span>
-<a id="__codelineno-5-43" name="__codelineno-5-43" href="#__codelineno-5-43"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">mid</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kr">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="kr">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="kr">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// Calculate midpoint</span>
-<a id="__codelineno-5-44" name="__codelineno-5-44" href="#__codelineno-5-44"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="n">mid</span><span class="p">)</span><span class="w"> </span><span class="c1">// Recursively process the left subarray</span>
-<a id="__codelineno-5-45" name="__codelineno-5-45" href="#__codelineno-5-45"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="kr">right</span><span class="p">)</span><span class="w"> </span><span class="c1">// Recursively process the right subarray</span>
-<a id="__codelineno-5-46" name="__codelineno-5-46" href="#__codelineno-5-46"></a><span class="w">    </span><span class="c1">// Merge stage</span>
+<a id="__codelineno-5-42" name="__codelineno-5-42" href="#__codelineno-5-42"></a><span class="w">    </span><span class="c1">// Divide phase</span>
+<a id="__codelineno-5-43" name="__codelineno-5-43" href="#__codelineno-5-43"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">mid</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kr">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="kr">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="kr">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1">// Calculate the midpoint</span>
+<a id="__codelineno-5-44" name="__codelineno-5-44" href="#__codelineno-5-44"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="n">mid</span><span class="p">)</span><span class="w"> </span><span class="c1">// Recurse on the left subarray</span>
+<a id="__codelineno-5-45" name="__codelineno-5-45" href="#__codelineno-5-45"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="kr">right</span><span class="p">)</span><span class="w"> </span><span class="c1">// Recurse on the right subarray</span>
+<a id="__codelineno-5-46" name="__codelineno-5-46" href="#__codelineno-5-46"></a><span class="w">    </span><span class="c1">// Merge phase</span>
 <a id="__codelineno-5-47" name="__codelineno-5-47" href="#__codelineno-5-47"></a><span class="w">    </span><span class="n">merge</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">:</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="kr">right</span><span class="p">)</span>
 <a id="__codelineno-5-48" name="__codelineno-5-48" href="#__codelineno-5-48"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">merge_sort.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Merge left subarray and right subarray */</span>
+<div class="highlight"><span class="filename">merge_sort.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Merge the left and right subarrays */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">merge</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Left subarray interval is [left, mid], right subarray interval is [mid+1, right]</span>
-<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged results</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// The left subarray spans [left, mid], and the right subarray spans [mid+1, right]</span>
+<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged result</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// Initialize the start indices of the left and right subarrays</span>
+<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="c1">// Initialize the starting indices of the left and right subarrays</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">        </span><span class="nx">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="c1">// While both subarrays still have elements, compare and copy the smaller element into the temporary array</span>
+<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="c1">// While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">mid</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">            </span><span class="nx">tmp</span><span class="p">[</span><span class="nx">k</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="o">++</span><span class="p">];</span>
@@ -5034,7 +5106,7 @@
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="w">        </span><span class="nx">tmp</span><span class="p">[</span><span class="nx">k</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="o">++</span><span class="p">];</span>
 <a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="w">    </span><span class="c1">// Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval</span>
+<a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="w">    </span><span class="c1">// Copy the elements of temporary array tmp back into the corresponding range of the original array nums</span>
 <a id="__codelineno-6-26" name="__codelineno-6-26" href="#__codelineno-6-26"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="nx">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">k</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">tmp</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">k</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-27" name="__codelineno-6-27" href="#__codelineno-6-27"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span><span class="p">[</span><span class="nx">k</span><span class="p">];</span>
 <a id="__codelineno-6-28" name="__codelineno-6-28" href="#__codelineno-6-28"></a><span class="w">    </span><span class="p">}</span>
@@ -5042,28 +5114,28 @@
 <a id="__codelineno-6-30" name="__codelineno-6-30" href="#__codelineno-6-30"></a>
 <a id="__codelineno-6-31" name="__codelineno-6-31" href="#__codelineno-6-31"></a><span class="cm">/* Merge sort */</span>
 <a id="__codelineno-6-32" name="__codelineno-6-32" href="#__codelineno-6-32"></a><span class="kd">function</span><span class="w"> </span><span class="nx">mergeSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-33" name="__codelineno-6-33" href="#__codelineno-6-33"></a><span class="w">    </span><span class="c1">// Termination condition</span>
-<a id="__codelineno-6-34" name="__codelineno-6-34" href="#__codelineno-6-34"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span><span class="w"> </span><span class="c1">// Terminate recursion when subarray length is 1</span>
-<a id="__codelineno-6-35" name="__codelineno-6-35" href="#__codelineno-6-35"></a><span class="w">    </span><span class="c1">// Divide and conquer stage</span>
-<a id="__codelineno-6-36" name="__codelineno-6-36" href="#__codelineno-6-36"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Calculate midpoint</span>
-<a id="__codelineno-6-37" name="__codelineno-6-37" href="#__codelineno-6-37"></a><span class="w">    </span><span class="nx">mergeSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively process the left subarray</span>
-<a id="__codelineno-6-38" name="__codelineno-6-38" href="#__codelineno-6-38"></a><span class="w">    </span><span class="nx">mergeSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively process the right subarray</span>
-<a id="__codelineno-6-39" name="__codelineno-6-39" href="#__codelineno-6-39"></a><span class="w">    </span><span class="c1">// Merge stage</span>
+<a id="__codelineno-6-33" name="__codelineno-6-33" href="#__codelineno-6-33"></a><span class="w">    </span><span class="c1">// Base case</span>
+<a id="__codelineno-6-34" name="__codelineno-6-34" href="#__codelineno-6-34"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span><span class="w"> </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
+<a id="__codelineno-6-35" name="__codelineno-6-35" href="#__codelineno-6-35"></a><span class="w">    </span><span class="c1">// Divide phase</span>
+<a id="__codelineno-6-36" name="__codelineno-6-36" href="#__codelineno-6-36"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Calculate the midpoint</span>
+<a id="__codelineno-6-37" name="__codelineno-6-37" href="#__codelineno-6-37"></a><span class="w">    </span><span class="nx">mergeSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recurse on the left subarray</span>
+<a id="__codelineno-6-38" name="__codelineno-6-38" href="#__codelineno-6-38"></a><span class="w">    </span><span class="nx">mergeSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recurse on the right subarray</span>
+<a id="__codelineno-6-39" name="__codelineno-6-39" href="#__codelineno-6-39"></a><span class="w">    </span><span class="c1">// Merge phase</span>
 <a id="__codelineno-6-40" name="__codelineno-6-40" href="#__codelineno-6-40"></a><span class="w">    </span><span class="nx">merge</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span>
 <a id="__codelineno-6-41" name="__codelineno-6-41" href="#__codelineno-6-41"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">merge_sort.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Merge left subarray and right subarray */</span>
+<div class="highlight"><span class="filename">merge_sort.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Merge the left and right subarrays */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">merge</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">left</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Left subarray interval is [left, mid], right subarray interval is [mid+1, right]</span>
-<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged results</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// The left subarray spans [left, mid], and the right subarray spans [mid+1, right]</span>
+<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged result</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="c1">// Initialize the start indices of the left and right subarrays</span>
+<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="c1">// Initialize the starting indices of the left and right subarrays</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">        </span><span class="nx">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
-<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="c1">// While both subarrays still have elements, compare and copy the smaller element into the temporary array</span>
+<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="c1">// While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">mid</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">            </span><span class="nx">tmp</span><span class="p">[</span><span class="nx">k</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="o">++</span><span class="p">];</span>
@@ -5078,7 +5150,7 @@
 <a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">        </span><span class="nx">tmp</span><span class="p">[</span><span class="nx">k</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="o">++</span><span class="p">];</span>
 <a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">    </span><span class="c1">// Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval</span>
+<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">    </span><span class="c1">// Copy the elements of temporary array tmp back into the corresponding range of the original array nums</span>
 <a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="nx">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">k</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">tmp</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span><span class="w"> </span><span class="nx">k</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span><span class="p">[</span><span class="nx">k</span><span class="p">];</span>
 <a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w">    </span><span class="p">}</span>
@@ -5086,26 +5158,26 @@
 <a id="__codelineno-7-30" name="__codelineno-7-30" href="#__codelineno-7-30"></a>
 <a id="__codelineno-7-31" name="__codelineno-7-31" href="#__codelineno-7-31"></a><span class="cm">/* Merge sort */</span>
 <a id="__codelineno-7-32" name="__codelineno-7-32" href="#__codelineno-7-32"></a><span class="kd">function</span><span class="w"> </span><span class="nx">mergeSort</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">left</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-33" name="__codelineno-7-33" href="#__codelineno-7-33"></a><span class="w">    </span><span class="c1">// Termination condition</span>
-<a id="__codelineno-7-34" name="__codelineno-7-34" href="#__codelineno-7-34"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span><span class="w"> </span><span class="c1">// Terminate recursion when subarray length is 1</span>
-<a id="__codelineno-7-35" name="__codelineno-7-35" href="#__codelineno-7-35"></a><span class="w">    </span><span class="c1">// Divide and conquer stage</span>
-<a id="__codelineno-7-36" name="__codelineno-7-36" href="#__codelineno-7-36"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Calculate midpoint</span>
-<a id="__codelineno-7-37" name="__codelineno-7-37" href="#__codelineno-7-37"></a><span class="w">    </span><span class="nx">mergeSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively process the left subarray</span>
-<a id="__codelineno-7-38" name="__codelineno-7-38" href="#__codelineno-7-38"></a><span class="w">    </span><span class="nx">mergeSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively process the right subarray</span>
-<a id="__codelineno-7-39" name="__codelineno-7-39" href="#__codelineno-7-39"></a><span class="w">    </span><span class="c1">// Merge stage</span>
+<a id="__codelineno-7-33" name="__codelineno-7-33" href="#__codelineno-7-33"></a><span class="w">    </span><span class="c1">// Base case</span>
+<a id="__codelineno-7-34" name="__codelineno-7-34" href="#__codelineno-7-34"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span><span class="w"> </span><span class="c1">// Stop recursing when the subarray has zero or one element</span>
+<a id="__codelineno-7-35" name="__codelineno-7-35" href="#__codelineno-7-35"></a><span class="w">    </span><span class="c1">// Divide phase</span>
+<a id="__codelineno-7-36" name="__codelineno-7-36" href="#__codelineno-7-36"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nx">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Calculate the midpoint</span>
+<a id="__codelineno-7-37" name="__codelineno-7-37" href="#__codelineno-7-37"></a><span class="w">    </span><span class="nx">mergeSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recurse on the left subarray</span>
+<a id="__codelineno-7-38" name="__codelineno-7-38" href="#__codelineno-7-38"></a><span class="w">    </span><span class="nx">mergeSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recurse on the right subarray</span>
+<a id="__codelineno-7-39" name="__codelineno-7-39" href="#__codelineno-7-39"></a><span class="w">    </span><span class="c1">// Merge phase</span>
 <a id="__codelineno-7-40" name="__codelineno-7-40" href="#__codelineno-7-40"></a><span class="w">    </span><span class="nx">merge</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span>
 <a id="__codelineno-7-41" name="__codelineno-7-41" href="#__codelineno-7-41"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">merge_sort.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Merge left subarray and right subarray */</span>
+<div class="highlight"><span class="filename">merge_sort.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Merge the left and right subarrays */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">merge</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Left subarray interval is [left, mid], right subarray interval is [mid+1, right]</span>
-<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// Create a temporary array tmp to store the merged results</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// The left subarray spans [left, mid], and the right subarray spans [mid+1, right]</span>
+<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// Create a temporary array tmp to store the merged result</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
-<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="c1">// Initialize the start indices of the left and right subarrays</span>
+<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="c1">// Initialize the starting indices of the left and right subarrays</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
-<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="c1">// While both subarrays still have elements, compare and copy the smaller element into the temporary array</span>
+<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="c1">// While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">      </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="o">++</span><span class="p">];</span>
@@ -5119,7 +5191,7 @@
 <a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">    </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="o">++</span><span class="p">];</span>
 <a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="w">  </span><span class="c1">// Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval</span>
+<a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="w">  </span><span class="c1">// Copy the elements of temporary array tmp back into the corresponding range of the original array nums</span>
 <a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span><span class="n">k</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="p">];</span>
 <a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="w">  </span><span class="p">}</span>
@@ -5127,27 +5199,27 @@
 <a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a>
 <a id="__codelineno-8-28" name="__codelineno-8-28" href="#__codelineno-8-28"></a><span class="cm">/* Merge sort */</span>
 <a id="__codelineno-8-29" name="__codelineno-8-29" href="#__codelineno-8-29"></a><span class="kt">void</span><span class="w"> </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-30" name="__codelineno-8-30" href="#__codelineno-8-30"></a><span class="w">  </span><span class="c1">// Termination condition</span>
-<a id="__codelineno-8-31" name="__codelineno-8-31" href="#__codelineno-8-31"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span><span class="w"> </span><span class="c1">// Terminate recursion when subarray length is 1</span>
-<a id="__codelineno-8-32" name="__codelineno-8-32" href="#__codelineno-8-32"></a><span class="w">  </span><span class="c1">// Divide and conquer stage</span>
-<a id="__codelineno-8-33" name="__codelineno-8-33" href="#__codelineno-8-33"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate midpoint</span>
-<a id="__codelineno-8-34" name="__codelineno-8-34" href="#__codelineno-8-34"></a><span class="w">  </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively process the left subarray</span>
-<a id="__codelineno-8-35" name="__codelineno-8-35" href="#__codelineno-8-35"></a><span class="w">  </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively process the right subarray</span>
-<a id="__codelineno-8-36" name="__codelineno-8-36" href="#__codelineno-8-36"></a><span class="w">  </span><span class="c1">// Merge stage</span>
+<a id="__codelineno-8-30" name="__codelineno-8-30" href="#__codelineno-8-30"></a><span class="w">  </span><span class="c1">// Base case</span>
+<a id="__codelineno-8-31" name="__codelineno-8-31" href="#__codelineno-8-31"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span><span class="w"> </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
+<a id="__codelineno-8-32" name="__codelineno-8-32" href="#__codelineno-8-32"></a><span class="w">  </span><span class="c1">// Divide phase</span>
+<a id="__codelineno-8-33" name="__codelineno-8-33" href="#__codelineno-8-33"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint</span>
+<a id="__codelineno-8-34" name="__codelineno-8-34" href="#__codelineno-8-34"></a><span class="w">  </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recurse on the left subarray</span>
+<a id="__codelineno-8-35" name="__codelineno-8-35" href="#__codelineno-8-35"></a><span class="w">  </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recurse on the right subarray</span>
+<a id="__codelineno-8-36" name="__codelineno-8-36" href="#__codelineno-8-36"></a><span class="w">  </span><span class="c1">// Merge phase</span>
 <a id="__codelineno-8-37" name="__codelineno-8-37" href="#__codelineno-8-37"></a><span class="w">  </span><span class="n">merge</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-8-38" name="__codelineno-8-38" href="#__codelineno-8-38"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">merge_sort.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Merge left subarray and right subarray */</span>
+<div class="highlight"><span class="filename">merge_sort.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Merge the left and right subarrays */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">merge</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">left</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Left subarray interval is [left, mid], right subarray interval is [mid+1, right]</span>
-<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged results</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// The left subarray spans [left, mid], and the right subarray spans [mid+1, right]</span>
+<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged result</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">tmp_size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">tmp_size</span><span class="p">];</span>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// Initialize the start indices of the left and right subarrays</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// Initialize the starting indices of the left and right subarrays</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
-<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// While both subarrays still have elements, compare and copy the smaller element into the temporary array</span>
+<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">            </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
@@ -5169,7 +5241,7 @@
 <a id="__codelineno-9-28" name="__codelineno-9-28" href="#__codelineno-9-28"></a><span class="w">        </span><span class="n">k</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-29" name="__codelineno-9-29" href="#__codelineno-9-29"></a><span class="w">        </span><span class="n">j</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-30" name="__codelineno-9-30" href="#__codelineno-9-30"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-31" name="__codelineno-9-31" href="#__codelineno-9-31"></a><span class="w">    </span><span class="c1">// Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval</span>
+<a id="__codelineno-9-31" name="__codelineno-9-31" href="#__codelineno-9-31"></a><span class="w">    </span><span class="c1">// Copy the elements of temporary array tmp back into the corresponding range of the original array nums</span>
 <a id="__codelineno-9-32" name="__codelineno-9-32" href="#__codelineno-9-32"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">tmp_size</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-33" name="__codelineno-9-33" href="#__codelineno-9-33"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="p">];</span>
 <a id="__codelineno-9-34" name="__codelineno-9-34" href="#__codelineno-9-34"></a><span class="w">    </span><span class="p">}</span>
@@ -5177,31 +5249,31 @@
 <a id="__codelineno-9-36" name="__codelineno-9-36" href="#__codelineno-9-36"></a>
 <a id="__codelineno-9-37" name="__codelineno-9-37" href="#__codelineno-9-37"></a><span class="cm">/* Merge sort */</span>
 <a id="__codelineno-9-38" name="__codelineno-9-38" href="#__codelineno-9-38"></a><span class="k">fn</span><span class="w"> </span><span class="nf">merge_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">left</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-39" name="__codelineno-9-39" href="#__codelineno-9-39"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-9-39" name="__codelineno-9-39" href="#__codelineno-9-39"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-9-40" name="__codelineno-9-40" href="#__codelineno-9-40"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-41" name="__codelineno-9-41" href="#__codelineno-9-41"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span><span class="w"> </span><span class="c1">// Terminate recursion when subarray length is 1</span>
+<a id="__codelineno-9-41" name="__codelineno-9-41" href="#__codelineno-9-41"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span><span class="w"> </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
 <a id="__codelineno-9-42" name="__codelineno-9-42" href="#__codelineno-9-42"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-43" name="__codelineno-9-43" href="#__codelineno-9-43"></a>
-<a id="__codelineno-9-44" name="__codelineno-9-44" href="#__codelineno-9-44"></a><span class="w">    </span><span class="c1">// Divide and conquer stage</span>
-<a id="__codelineno-9-45" name="__codelineno-9-45" href="#__codelineno-9-45"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate midpoint</span>
-<a id="__codelineno-9-46" name="__codelineno-9-46" href="#__codelineno-9-46"></a><span class="w">    </span><span class="n">merge_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively process the left subarray</span>
-<a id="__codelineno-9-47" name="__codelineno-9-47" href="#__codelineno-9-47"></a><span class="w">    </span><span class="n">merge_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively process the right subarray</span>
+<a id="__codelineno-9-44" name="__codelineno-9-44" href="#__codelineno-9-44"></a><span class="w">    </span><span class="c1">// Divide phase</span>
+<a id="__codelineno-9-45" name="__codelineno-9-45" href="#__codelineno-9-45"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w"> </span><span class="c1">// Calculate the midpoint</span>
+<a id="__codelineno-9-46" name="__codelineno-9-46" href="#__codelineno-9-46"></a><span class="w">    </span><span class="n">merge_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recurse on the left subarray</span>
+<a id="__codelineno-9-47" name="__codelineno-9-47" href="#__codelineno-9-47"></a><span class="w">    </span><span class="n">merge_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recurse on the right subarray</span>
 <a id="__codelineno-9-48" name="__codelineno-9-48" href="#__codelineno-9-48"></a>
-<a id="__codelineno-9-49" name="__codelineno-9-49" href="#__codelineno-9-49"></a><span class="w">    </span><span class="c1">// Merge stage</span>
+<a id="__codelineno-9-49" name="__codelineno-9-49" href="#__codelineno-9-49"></a><span class="w">    </span><span class="c1">// Merge phase</span>
 <a id="__codelineno-9-50" name="__codelineno-9-50" href="#__codelineno-9-50"></a><span class="w">    </span><span class="n">merge</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-9-51" name="__codelineno-9-51" href="#__codelineno-9-51"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">merge_sort.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Merge left subarray and right subarray */</span>
+<div class="highlight"><span class="filename">merge_sort.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Merge the left and right subarrays */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">merge</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Left subarray interval is [left, mid], right subarray interval is [mid+1, right]</span>
-<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged results</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// The left subarray spans [left, mid], and the right subarray spans [mid+1, right]</span>
+<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged result</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">tmpSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="n">tmpSize</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
-<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="c1">// Initialize the start indices of the left and right subarrays</span>
+<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">    </span><span class="c1">// Initialize the starting indices of the left and right subarrays</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="c1">// While both subarrays still have elements, compare and copy the smaller element into the temporary array</span>
+<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="c1">// While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">            </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="o">++</span><span class="p">];</span>
@@ -5216,7 +5288,7 @@
 <a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">        </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="o">++</span><span class="p">];</span>
 <a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">    </span><span class="c1">// Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval</span>
+<a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">    </span><span class="c1">// Copy the elements of temporary array tmp back into the corresponding range of the original array nums</span>
 <a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">tmpSize</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">k</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">[</span><span class="n">k</span><span class="p">];</span>
 <a id="__codelineno-10-27" name="__codelineno-10-27" href="#__codelineno-10-27"></a><span class="w">    </span><span class="p">}</span>
@@ -5226,29 +5298,29 @@
 <a id="__codelineno-10-31" name="__codelineno-10-31" href="#__codelineno-10-31"></a>
 <a id="__codelineno-10-32" name="__codelineno-10-32" href="#__codelineno-10-32"></a><span class="cm">/* Merge sort */</span>
 <a id="__codelineno-10-33" name="__codelineno-10-33" href="#__codelineno-10-33"></a><span class="kt">void</span><span class="w"> </span><span class="nf">mergeSort</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-34" name="__codelineno-10-34" href="#__codelineno-10-34"></a><span class="w">    </span><span class="c1">// Termination condition</span>
+<a id="__codelineno-10-34" name="__codelineno-10-34" href="#__codelineno-10-34"></a><span class="w">    </span><span class="c1">// Base case</span>
 <a id="__codelineno-10-35" name="__codelineno-10-35" href="#__codelineno-10-35"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-10-36" name="__codelineno-10-36" href="#__codelineno-10-36"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span><span class="w"> </span><span class="c1">// Terminate recursion when subarray length is 1</span>
-<a id="__codelineno-10-37" name="__codelineno-10-37" href="#__codelineno-10-37"></a><span class="w">    </span><span class="c1">// Divide and conquer stage</span>
-<a id="__codelineno-10-38" name="__codelineno-10-38" href="#__codelineno-10-38"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">    </span><span class="c1">// Calculate midpoint</span>
-<a id="__codelineno-10-39" name="__codelineno-10-39" href="#__codelineno-10-39"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">);</span><span class="w">      </span><span class="c1">// Recursively process the left subarray</span>
-<a id="__codelineno-10-40" name="__codelineno-10-40" href="#__codelineno-10-40"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively process the right subarray</span>
-<a id="__codelineno-10-41" name="__codelineno-10-41" href="#__codelineno-10-41"></a><span class="w">    </span><span class="c1">// Merge stage</span>
+<a id="__codelineno-10-36" name="__codelineno-10-36" href="#__codelineno-10-36"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span><span class="w"> </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
+<a id="__codelineno-10-37" name="__codelineno-10-37" href="#__codelineno-10-37"></a><span class="w">    </span><span class="c1">// Divide phase</span>
+<a id="__codelineno-10-38" name="__codelineno-10-38" href="#__codelineno-10-38"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span><span class="w">    </span><span class="c1">// Calculate the midpoint</span>
+<a id="__codelineno-10-39" name="__codelineno-10-39" href="#__codelineno-10-39"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">);</span><span class="w">      </span><span class="c1">// Recurse on the left subarray</span>
+<a id="__codelineno-10-40" name="__codelineno-10-40" href="#__codelineno-10-40"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recurse on the right subarray</span>
+<a id="__codelineno-10-41" name="__codelineno-10-41" href="#__codelineno-10-41"></a><span class="w">    </span><span class="c1">// Merge phase</span>
 <a id="__codelineno-10-42" name="__codelineno-10-42" href="#__codelineno-10-42"></a><span class="w">    </span><span class="n">merge</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-10-43" name="__codelineno-10-43" href="#__codelineno-10-43"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">merge_sort.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Merge left subarray and right subarray */</span>
+<div class="highlight"><span class="filename">merge_sort.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Merge the left and right subarrays */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">merge</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Left subarray interval is [left, mid], right subarray interval is [mid+1, right]</span>
-<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged results</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// The left subarray spans [left, mid], and the right subarray spans [mid+1, right]</span>
+<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="c1">// Create a temporary array tmp to store the merged result</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
-<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// Initialize the start indices of the left and right subarrays</span>
+<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// Initialize the starting indices of the left and right subarrays</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
-<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="c1">// While both subarrays still have elements, compare and copy the smaller element into the temporary array</span>
+<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="c1">// While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">            </span><span class="n">tmp</span><span class="o">[</span><span class="n">k</span><span class="o">++]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">++]</span>
@@ -5262,7 +5334,7 @@
 <a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">        </span><span class="n">tmp</span><span class="o">[</span><span class="n">k</span><span class="o">++]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">++]</span>
 <a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">    </span><span class="c1">// Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval</span>
+<a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">    </span><span class="c1">// Copy the elements of temporary array tmp back into the corresponding range of the original array nums</span>
 <a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="na">indices</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">l</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="o">[</span><span class="n">l</span><span class="o">]</span>
 <a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w">    </span><span class="p">}</span>
@@ -5270,26 +5342,26 @@
 <a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a>
 <a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a><span class="cm">/* Merge sort */</span>
 <a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="w">    </span><span class="c1">// Termination condition</span>
-<a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w">  </span><span class="c1">// Terminate recursion when subarray length is 1</span>
-<a id="__codelineno-11-34" name="__codelineno-11-34" href="#__codelineno-11-34"></a><span class="w">    </span><span class="c1">// Divide and conquer stage</span>
-<a id="__codelineno-11-35" name="__codelineno-11-35" href="#__codelineno-11-35"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="c1">// Calculate midpoint</span>
-<a id="__codelineno-11-36" name="__codelineno-11-36" href="#__codelineno-11-36"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">)</span><span class="w"> </span><span class="c1">// Recursively process the left subarray</span>
-<a id="__codelineno-11-37" name="__codelineno-11-37" href="#__codelineno-11-37"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="c1">// Recursively process the right subarray</span>
-<a id="__codelineno-11-38" name="__codelineno-11-38" href="#__codelineno-11-38"></a><span class="w">    </span><span class="c1">// Merge stage</span>
+<a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="w">    </span><span class="c1">// Base case</span>
+<a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w">  </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
+<a id="__codelineno-11-34" name="__codelineno-11-34" href="#__codelineno-11-34"></a><span class="w">    </span><span class="c1">// Divide phase</span>
+<a id="__codelineno-11-35" name="__codelineno-11-35" href="#__codelineno-11-35"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="c1">// Calculate the midpoint</span>
+<a id="__codelineno-11-36" name="__codelineno-11-36" href="#__codelineno-11-36"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">)</span><span class="w"> </span><span class="c1">// Recurse on the left subarray</span>
+<a id="__codelineno-11-37" name="__codelineno-11-37" href="#__codelineno-11-37"></a><span class="w">    </span><span class="n">mergeSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="c1">// Recurse on the right subarray</span>
+<a id="__codelineno-11-38" name="__codelineno-11-38" href="#__codelineno-11-38"></a><span class="w">    </span><span class="c1">// Merge phase</span>
 <a id="__codelineno-11-39" name="__codelineno-11-39" href="#__codelineno-11-39"></a><span class="w">    </span><span class="n">merge</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
 <a id="__codelineno-11-40" name="__codelineno-11-40" href="#__codelineno-11-40"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">merge_sort.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Merge left and right subarrays ###</span>
+<div class="highlight"><span class="filename">merge_sort.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Merge the left and right subarrays ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">merge</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Left subarray interval is [left, mid], right subarray interval is [mid+1, right]</span>
-<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="c1"># Create temporary array tmp to store merged result</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># The left subarray spans [left, mid], and the right subarray spans [mid+1, right]</span>
+<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="c1"># Create a temporary array tmp to store the merged result</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="c1"># Initialize the start indices of the left and right subarrays</span>
+<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="c1"># Initialize the starting indices of the left and right subarrays</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="c1"># While both subarrays still have elements, compare and copy the smaller element into the temporary array</span>
+<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">  </span><span class="c1"># While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">right</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">      </span><span class="n">tmp</span><span class="o">[</span><span class="n">k</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
@@ -5311,7 +5383,7 @@
 <a id="__codelineno-12-27" name="__codelineno-12-27" href="#__codelineno-12-27"></a><span class="w">    </span><span class="n">j</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-28" name="__codelineno-12-28" href="#__codelineno-12-28"></a><span class="w">    </span><span class="n">k</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-29" name="__codelineno-12-29" href="#__codelineno-12-29"></a><span class="w">  </span><span class="k">end</span>
-<a id="__codelineno-12-30" name="__codelineno-12-30" href="#__codelineno-12-30"></a><span class="w">  </span><span class="c1"># Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval</span>
+<a id="__codelineno-12-30" name="__codelineno-12-30" href="#__codelineno-12-30"></a><span class="w">  </span><span class="c1"># Copy the elements of temporary array tmp back into the corresponding range of the original array nums</span>
 <a id="__codelineno-12-31" name="__codelineno-12-31" href="#__codelineno-12-31"></a><span class="w">  </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">tmp</span><span class="o">.</span><span class="n">length</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="o">|</span><span class="n">k</span><span class="o">|</span>
 <a id="__codelineno-12-32" name="__codelineno-12-32" href="#__codelineno-12-32"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">k</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="o">[</span><span class="n">k</span><span class="o">]</span>
 <a id="__codelineno-12-33" name="__codelineno-12-33" href="#__codelineno-12-33"></a><span class="w">  </span><span class="k">end</span>
@@ -5319,33 +5391,38 @@
 <a id="__codelineno-12-35" name="__codelineno-12-35" href="#__codelineno-12-35"></a>
 <a id="__codelineno-12-36" name="__codelineno-12-36" href="#__codelineno-12-36"></a><span class="c1">### Merge sort ###</span>
 <a id="__codelineno-12-37" name="__codelineno-12-37" href="#__codelineno-12-37"></a><span class="k">def</span><span class="w"> </span><span class="nf">merge_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-12-38" name="__codelineno-12-38" href="#__codelineno-12-38"></a><span class="w">  </span><span class="c1"># Termination condition</span>
-<a id="__codelineno-12-39" name="__codelineno-12-39" href="#__codelineno-12-39"></a><span class="w">  </span><span class="c1"># Terminate recursion when subarray length is 1</span>
+<a id="__codelineno-12-38" name="__codelineno-12-38" href="#__codelineno-12-38"></a><span class="w">  </span><span class="c1"># Base case</span>
+<a id="__codelineno-12-39" name="__codelineno-12-39" href="#__codelineno-12-39"></a><span class="w">  </span><span class="c1"># Stop recursing when the subarray contains at most one element</span>
 <a id="__codelineno-12-40" name="__codelineno-12-40" href="#__codelineno-12-40"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span>
-<a id="__codelineno-12-41" name="__codelineno-12-41" href="#__codelineno-12-41"></a><span class="w">  </span><span class="c1"># Divide and conquer stage</span>
-<a id="__codelineno-12-42" name="__codelineno-12-42" href="#__codelineno-12-42"></a><span class="w">  </span><span class="n">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1"># Calculate midpoint</span>
-<a id="__codelineno-12-43" name="__codelineno-12-43" href="#__codelineno-12-43"></a><span class="w">  </span><span class="n">merge_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">)</span><span class="w"> </span><span class="c1"># Recursively process the left subarray</span>
-<a id="__codelineno-12-44" name="__codelineno-12-44" href="#__codelineno-12-44"></a><span class="w">  </span><span class="n">merge_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="c1"># Recursively process the right subarray</span>
-<a id="__codelineno-12-45" name="__codelineno-12-45" href="#__codelineno-12-45"></a><span class="w">  </span><span class="c1"># Merge stage</span>
+<a id="__codelineno-12-41" name="__codelineno-12-41" href="#__codelineno-12-41"></a><span class="w">  </span><span class="c1"># Divide phase</span>
+<a id="__codelineno-12-42" name="__codelineno-12-42" href="#__codelineno-12-42"></a><span class="w">  </span><span class="n">mid</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="c1"># Calculate the midpoint</span>
+<a id="__codelineno-12-43" name="__codelineno-12-43" href="#__codelineno-12-43"></a><span class="w">  </span><span class="n">merge_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">)</span><span class="w"> </span><span class="c1"># Recurse on the left subarray</span>
+<a id="__codelineno-12-44" name="__codelineno-12-44" href="#__codelineno-12-44"></a><span class="w">  </span><span class="n">merge_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="c1"># Recurse on the right subarray</span>
+<a id="__codelineno-12-45" name="__codelineno-12-45" href="#__codelineno-12-45"></a><span class="w">  </span><span class="c1"># Merge phase</span>
 <a id="__codelineno-12-46" name="__codelineno-12-46" href="#__codelineno-12-46"></a><span class="w">  </span><span class="n">merge</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
 <a id="__codelineno-12-47" name="__codelineno-12-47" href="#__codelineno-12-47"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h2 id="1162-algorithm-characteristics">11.6.2 &nbsp; Algorithm Characteristics<a class="headerlink" href="#1162-algorithm-characteristics" title="Permanent link">&para;</a></h2>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20merge%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20mid%3A%20int,%20right%3A%20int%29%3A%0A%20%20%20%20%22%22%22Merge%20the%20left%20and%20right%20subarrays%22%22%22%0A%20%20%20%20%23%20The%20left%20subarray%20spans%20%5Bleft,%20mid%5D,%20and%20the%20right%20subarray%20spans%20%5Bmid%2B1,%20right%5D%0A%20%20%20%20%23%20Create%20a%20temporary%20array%20tmp%20to%20store%20the%20merged%20result%0A%20%20%20%20tmp%20%3D%20%5B0%5D%20%2A%20%28right%20-%20left%20%2B%201%29%0A%20%20%20%20%23%20Initialize%20the%20starting%20indices%20of%20the%20left%20and%20right%20subarrays%0A%20%20%20%20i,%20j,%20k%20%3D%20left,%20mid%20%2B%201,%200%0A%20%20%20%20%23%20While%20both%20subarrays%20have%20elements%20remaining,%20compare%20them%20and%20copy%20the%20smaller%20element%20into%20the%20temporary%20array%0A%20%20%20%20while%20i%20%3C%3D%20mid%20and%20j%20%3C%3D%20right%3A%0A%20%20%20%20%20%20%20%20if%20nums%5Bi%5D%20%3C%3D%20nums%5Bj%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20tmp%5Bk%5D%20%3D%20nums%5Bi%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20tmp%5Bk%5D%20%3D%20nums%5Bj%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%2B%3D%201%0A%20%20%20%20%20%20%20%20k%20%2B%3D%201%0A%20%20%20%20%23%20Copy%20the%20remaining%20elements%20of%20the%20left%20and%20right%20subarrays%20into%20the%20temporary%20array%0A%20%20%20%20while%20i%20%3C%3D%20mid%3A%0A%20%20%20%20%20%20%20%20tmp%5Bk%5D%20%3D%20nums%5Bi%5D%0A%20%20%20%20%20%20%20%20i%20%2B%3D%201%0A%20%20%20%20%20%20%20%20k%20%2B%3D%201%0A%20%20%20%20while%20j%20%3C%3D%20right%3A%0A%20%20%20%20%20%20%20%20tmp%5Bk%5D%20%3D%20nums%5Bj%5D%0A%20%20%20%20%20%20%20%20j%20%2B%3D%201%0A%20%20%20%20%20%20%20%20k%20%2B%3D%201%0A%20%20%20%20%23%20Copy%20the%20elements%20of%20temporary%20array%20tmp%20back%20into%20the%20corresponding%20range%20of%20the%20original%20array%20nums%0A%20%20%20%20for%20k%20in%20range%280,%20len%28tmp%29%29%3A%0A%20%20%20%20%20%20%20%20nums%5Bleft%20%2B%20k%5D%20%3D%20tmp%5Bk%5D%0A%0A%0Adef%20merge_sort%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20right%3A%20int%29%3A%0A%20%20%20%20%22%22%22Merge%20sort%22%22%22%0A%20%20%20%20%23%20Base%20case%0A%20%20%20%20if%20left%20%3E%3D%20right%3A%0A%20%20%20%20%20%20%20%20return%20%20%23%20Stop%20recursing%20when%20the%20subarray%20has%20at%20most%20one%20element%0A%20%20%20%20%23%20Divide%20phase%0A%20%20%20%20mid%20%3D%20%28left%20%2B%20right%29%20//%202%20%20%23%20Calculate%20the%20midpoint%0A%20%20%20%20merge_sort%28nums,%20left,%20mid%29%20%20%23%20Recurse%20on%20the%20left%20subarray%0A%20%20%20%20merge_sort%28nums,%20mid%20%2B%201,%20right%29%20%20%23%20Recurse%20on%20the%20right%20subarray%0A%20%20%20%20%23%20Merge%20phase%0A%20%20%20%20merge%28nums,%20left,%20mid,%20right%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B7,%203,%202,%206,%200,%201,%205,%204%5D%0A%20%20%20%20merge_sort%28nums,%200,%20len%28nums%29%20-%201%29%0A%20%20%20%20print%28%22After%20merge%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20merge%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20mid%3A%20int,%20right%3A%20int%29%3A%0A%20%20%20%20%22%22%22Merge%20the%20left%20and%20right%20subarrays%22%22%22%0A%20%20%20%20%23%20The%20left%20subarray%20spans%20%5Bleft,%20mid%5D,%20and%20the%20right%20subarray%20spans%20%5Bmid%2B1,%20right%5D%0A%20%20%20%20%23%20Create%20a%20temporary%20array%20tmp%20to%20store%20the%20merged%20result%0A%20%20%20%20tmp%20%3D%20%5B0%5D%20%2A%20%28right%20-%20left%20%2B%201%29%0A%20%20%20%20%23%20Initialize%20the%20starting%20indices%20of%20the%20left%20and%20right%20subarrays%0A%20%20%20%20i,%20j,%20k%20%3D%20left,%20mid%20%2B%201,%200%0A%20%20%20%20%23%20While%20both%20subarrays%20have%20elements%20remaining,%20compare%20them%20and%20copy%20the%20smaller%20element%20into%20the%20temporary%20array%0A%20%20%20%20while%20i%20%3C%3D%20mid%20and%20j%20%3C%3D%20right%3A%0A%20%20%20%20%20%20%20%20if%20nums%5Bi%5D%20%3C%3D%20nums%5Bj%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20tmp%5Bk%5D%20%3D%20nums%5Bi%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20tmp%5Bk%5D%20%3D%20nums%5Bj%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%2B%3D%201%0A%20%20%20%20%20%20%20%20k%20%2B%3D%201%0A%20%20%20%20%23%20Copy%20the%20remaining%20elements%20of%20the%20left%20and%20right%20subarrays%20into%20the%20temporary%20array%0A%20%20%20%20while%20i%20%3C%3D%20mid%3A%0A%20%20%20%20%20%20%20%20tmp%5Bk%5D%20%3D%20nums%5Bi%5D%0A%20%20%20%20%20%20%20%20i%20%2B%3D%201%0A%20%20%20%20%20%20%20%20k%20%2B%3D%201%0A%20%20%20%20while%20j%20%3C%3D%20right%3A%0A%20%20%20%20%20%20%20%20tmp%5Bk%5D%20%3D%20nums%5Bj%5D%0A%20%20%20%20%20%20%20%20j%20%2B%3D%201%0A%20%20%20%20%20%20%20%20k%20%2B%3D%201%0A%20%20%20%20%23%20Copy%20the%20elements%20of%20temporary%20array%20tmp%20back%20into%20the%20corresponding%20range%20of%20the%20original%20array%20nums%0A%20%20%20%20for%20k%20in%20range%280,%20len%28tmp%29%29%3A%0A%20%20%20%20%20%20%20%20nums%5Bleft%20%2B%20k%5D%20%3D%20tmp%5Bk%5D%0A%0A%0Adef%20merge_sort%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20right%3A%20int%29%3A%0A%20%20%20%20%22%22%22Merge%20sort%22%22%22%0A%20%20%20%20%23%20Base%20case%0A%20%20%20%20if%20left%20%3E%3D%20right%3A%0A%20%20%20%20%20%20%20%20return%20%20%23%20Stop%20recursing%20when%20the%20subarray%20has%20at%20most%20one%20element%0A%20%20%20%20%23%20Divide%20phase%0A%20%20%20%20mid%20%3D%20%28left%20%2B%20right%29%20//%202%20%20%23%20Calculate%20the%20midpoint%0A%20%20%20%20merge_sort%28nums,%20left,%20mid%29%20%20%23%20Recurse%20on%20the%20left%20subarray%0A%20%20%20%20merge_sort%28nums,%20mid%20%2B%201,%20right%29%20%20%23%20Recurse%20on%20the%20right%20subarray%0A%20%20%20%20%23%20Merge%20phase%0A%20%20%20%20merge%28nums,%20left,%20mid,%20right%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B7,%203,%202,%206,%200,%201,%205,%204%5D%0A%20%20%20%20merge_sort%28nums,%200,%20len%28nums%29%20-%201%29%0A%20%20%20%20print%28%22After%20merge%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1162-algorithm-properties">11.6.2 &nbsp; Algorithm properties<a class="headerlink" href="#1162-algorithm-properties" title="Permanent link">&para;</a></h2>
 <ul>
-<li><strong>Time complexity is <span class="arithmatex">\(O(n \log n)\)</span>; merge sort is non-adaptive</strong>: The divide phase produces a recursion tree of height <span class="arithmatex">\(\log n\)</span>, and the total number of operations performed during merging at each level is <span class="arithmatex">\(n\)</span>, so the overall time complexity is <span class="arithmatex">\(O(n \log n)\)</span>.</li>
-<li><strong>Space complexity is <span class="arithmatex">\(O(n)\)</span>; merge sort is not in-place</strong>: The recursion depth is <span class="arithmatex">\(\log n\)</span>, which uses <span class="arithmatex">\(O(\log n)\)</span> stack-frame space. The merge operation requires an auxiliary array, which uses <span class="arithmatex">\(O(n)\)</span> additional space.</li>
-<li><strong>Stable sort</strong>: During merging, the relative order of equal elements remains unchanged.</li>
+<li><strong>Time complexity: <span class="arithmatex">\(O(n \log n)\)</span>; not adaptive</strong>. Division produces a recursion tree of height <span class="arithmatex">\(\log n\)</span>. Merging performs a total of <span class="arithmatex">\(n\)</span> operations at each level, giving <span class="arithmatex">\(O(n \log n)\)</span> overall.</li>
+<li><strong>Space complexity: <span class="arithmatex">\(O(n)\)</span>; not in-place</strong>. The recursion depth is <span class="arithmatex">\(\log n\)</span>, requiring <span class="arithmatex">\(O(\log n)\)</span> space for stack frames. Merging uses an auxiliary array requiring <span class="arithmatex">\(O(n)\)</span> extra space.</li>
+<li><strong>Stable</strong>. Merging preserves the order of equal elements.</li>
 </ul>
-<h2 id="1163-linked-list-sorting">11.6.3 &nbsp; Linked List Sorting<a class="headerlink" href="#1163-linked-list-sorting" title="Permanent link">&para;</a></h2>
-<p>For linked lists, merge sort has significant advantages over other sorting algorithms, <strong>and it can reduce the space complexity of the sorting task to <span class="arithmatex">\(O(1)\)</span></strong>.</p>
+<h2 id="1163-sorting-linked-lists">11.6.3 &nbsp; Sorting linked lists<a class="headerlink" href="#1163-sorting-linked-lists" title="Permanent link">&para;</a></h2>
+<p>Merge sort has a significant advantage over other sorting algorithms for linked lists: <strong>it can reduce the space complexity of sorting a linked list to <span class="arithmatex">\(O(1)\)</span></strong>.</p>
 <ul>
-<li><strong>Divide phase</strong>: Iteration can be used instead of recursion to split the linked list, thereby eliminating the stack-frame space used by recursion.</li>
-<li><strong>Merge phase</strong>: In linked lists, node insertion and deletion require only pointer updates, so the merge phase (merging two short sorted linked lists into one longer sorted linked list) does not require creating an additional linked list.</li>
+<li><strong>Division</strong>: use iteration instead of recursion to divide the list, eliminating the space needed for recursive stack frames.</li>
+<li><strong>Merging</strong>: inserting and deleting linked-list nodes requires only changing references or pointers. Merging two short sorted lists into a longer one therefore needs no additional list.</li>
 </ul>
-<p>The specific implementation details are quite complex, and interested readers can consult related materials for learning.</p>
+<p>The implementation is fairly involved. Interested readers can consult further resources for details.</p>
 
 <!-- Source file information -->
 
@@ -5368,7 +5445,7 @@ aria-label="Footer"
   <a
     href="../quick_sort/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 11.5 Quick Sort"
+    aria-label="Previous: 11.5   Quick sort"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -5380,7 +5457,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        11.5 Quick Sort
+        11.5 &nbsp; Quick sort
       </div>
     </div>
   </a>
@@ -5392,7 +5469,7 @@ aria-label="Footer"
   <a
     href="../heap_sort/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 11.7 Heap Sort"
+    aria-label="Next: 11.7   Heap sort"
     rel="next"
   >
     <div class="md-footer__title">
@@ -5400,7 +5477,7 @@ aria-label="Footer"
         Next
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-        11.7 Heap Sort
+        11.7 &nbsp; Heap sort
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     <div class="md-footer__button md-icon">
@@ -5417,7 +5494,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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diff --git a/en/chapter_sorting/quick_sort/index.html b/en/chapter_sorting/quick_sort/index.html
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>11.5 Quick Sort - Hello Algo</title>
+        <title>11.5 Quick sort - Hello Algo</title>
 
 
 
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-  Before Starting
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-  Before Starting
+  Foreword
 
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   <span class="md-ellipsis">
 
 
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
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   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
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+  Chapter 2. &nbsp; Complexity analysis
 
 
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   <span class="md-ellipsis">
 
 
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-  2.2 Iteration and Recursion
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
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-  4.5 Summary
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2880,7 +2880,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2901,7 +2901,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2922,7 +2922,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2976,7 +2976,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3003,7 +3003,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3039,7 +3039,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3056,7 +3056,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3082,10 +3082,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1151-algorithm-flow" class="md-nav__link">
+  <a href="#1151-algorithm-steps" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.5.1 &nbsp; Algorithm Flow
+        11.5.1 &nbsp; Algorithm steps
       </span>
     </span>
   </a>
@@ -3093,10 +3093,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1152-algorithm-characteristics" class="md-nav__link">
+  <a href="#1152-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.5.2 &nbsp; Algorithm Characteristics
+        11.5.2 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -3104,10 +3104,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1153-why-is-quick-sort-fast" class="md-nav__link">
+  <a href="#1153-why-is-quick-sort-quick" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.5.3 &nbsp; Why Is Quick Sort Fast
+        11.5.3 &nbsp; Why is quick sort quick?
       </span>
     </span>
   </a>
@@ -3115,10 +3115,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1154-pivot-optimization" class="md-nav__link">
+  <a href="#1154-improving-pivot-selection" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.5.4 &nbsp; Pivot Optimization
+        11.5.4 &nbsp; Improving pivot selection
       </span>
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   </a>
@@ -3126,10 +3126,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1155-recursive-depth-optimization" class="md-nav__link">
+  <a href="#1155-reducing-recursion-depth" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.5.5 &nbsp; Recursive Depth Optimization
+        11.5.5 &nbsp; Reducing recursion depth
       </span>
     </span>
   </a>
@@ -3159,7 +3159,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3240,7 +3240,7 @@
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3429,7 +3429,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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+  15.2 &nbsp; Fractional knapsack problem
 
 
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+  15.3 &nbsp; Maximum capacity problem
 
 
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+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  Chapter 16. Appendix
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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+  <a href="#1151-algorithm-steps" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.5.1 &nbsp; Algorithm Flow
+        11.5.1 &nbsp; Algorithm steps
       </span>
     </span>
   </a>
@@ -4643,10 +4715,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1152-algorithm-characteristics" class="md-nav__link">
+  <a href="#1152-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.5.2 &nbsp; Algorithm Characteristics
+        11.5.2 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -4654,10 +4726,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1153-why-is-quick-sort-fast" class="md-nav__link">
+  <a href="#1153-why-is-quick-sort-quick" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.5.3 &nbsp; Why Is Quick Sort Fast
+        11.5.3 &nbsp; Why is quick sort quick?
       </span>
     </span>
   </a>
@@ -4665,10 +4737,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1154-pivot-optimization" class="md-nav__link">
+  <a href="#1154-improving-pivot-selection" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.5.4 &nbsp; Pivot Optimization
+        11.5.4 &nbsp; Improving pivot selection
       </span>
     </span>
   </a>
@@ -4676,10 +4748,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1155-recursive-depth-optimization" class="md-nav__link">
+  <a href="#1155-reducing-recursion-depth" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.5.5 &nbsp; Recursive Depth Optimization
+        11.5.5 &nbsp; Reducing recursion depth
       </span>
     </span>
   </a>
@@ -4723,18 +4795,18 @@
 
 
 <!-- Page content -->
-<h1 id="115-quick-sort">11.5 &nbsp; Quick Sort<a class="headerlink" href="#115-quick-sort" title="Permanent link">&para;</a></h1>
-<p><u>Quick sort</u> is an efficient and widely used sorting algorithm based on the divide-and-conquer strategy.</p>
-<p>The core operation of quick sort is "sentinel partitioning", whose goal is to select an element as the "pivot", move all elements smaller than the pivot to its left, and move all elements larger than the pivot to its right. Specifically, the process is shown in Figure 11-8.</p>
+<h1 id="115-quick-sort">11.5 &nbsp; Quick sort<a class="headerlink" href="#115-quick-sort" title="Permanent link">&para;</a></h1>
+<p><u>Quick sort</u> is an efficient, widely used sorting algorithm based on divide and conquer.</p>
+<p>Its core operation is “partitioning”: choose an array element as the “pivot,” move all smaller elements to its left, and move all larger elements to its right. The partitioning process is shown below.</p>
 <ol>
-<li>Select the leftmost element as the pivot, and initialize two pointers <code>i</code> and <code>j</code> at the two ends of the array.</li>
-<li>Enter a loop. In each round, use <code>i</code> (<code>j</code>) to find the first element larger (smaller) than the pivot, and then swap the two elements.</li>
-<li>Repeat step <code>2.</code> until <code>i</code> and <code>j</code> meet, then swap the pivot into the boundary position between the two sub-arrays.</li>
+<li>Choose the leftmost element as the pivot and initialize pointers <code>i</code> and <code>j</code> at the two ends of the array.</li>
+<li>In a loop, use <code>i</code> to find the first element greater than the pivot and <code>j</code> to find the first element smaller than it, then swap those elements.</li>
+<li>Repeat step <code>2.</code> until <code>i</code> and <code>j</code> meet. Finally, swap the pivot into the boundary between the two subarrays.</li>
 </ol>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:9"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label><label for="__tabbed_1_6">&lt;6&gt;</label><label for="__tabbed_1_7">&lt;7&gt;</label><label for="__tabbed_1_8">&lt;8&gt;</label><label for="__tabbed_1_9">&lt;9&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Sentinel partitioning steps" class="animation-figure" src="../quick_sort.assets/pivot_division_step1.png" /></p>
+<p><img alt="Steps of partitioning" class="animation-figure" src="../quick_sort.assets/pivot_division_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="pivot_division_step2" class="animation-figure" src="../quick_sort.assets/pivot_division_step2.png" /></p>
@@ -4762,331 +4834,336 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 11-8 &nbsp; Sentinel partitioning steps </p>
+<p align="center"> Figure 11-8 &nbsp; Steps of partitioning </p>
 
-<p>After sentinel partitioning, the original array is divided into three parts: the left sub-array, the pivot, and the right sub-array, such that "any element in the left sub-array <span class="arithmatex">\(\leq\)</span> the pivot <span class="arithmatex">\(\leq\)</span> any element in the right sub-array". Therefore, we only need to sort the two sub-arrays next.</p>
+<p>After partitioning, the original array has three parts: the left subarray, the pivot, and the right subarray. Every element in the left subarray is <span class="arithmatex">\(\leq\)</span> the pivot, which is <span class="arithmatex">\(\leq\)</span> every element in the right subarray. We now need only sort the two subarrays.</p>
 <div class="admonition note">
-<p class="admonition-title">Divide-and-conquer strategy of quick sort</p>
-<p>The essence of sentinel partitioning is to simplify the sorting problem of a longer array into the sorting problems of two shorter arrays.</p>
+<p class="admonition-title">Quick sort's divide-and-conquer strategy</p>
+<p>Partitioning reduces the problem of sorting one long array to sorting two shorter arrays.</p>
 </div>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">partition</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">left</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">right</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Sentinel partition"""</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Two-pointer partitioning"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Use nums[left] as the pivot</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">left</span><span class="p">,</span> <span class="n">right</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">j</span> <span class="ow">and</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">]:</span>
-<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>            <span class="n">j</span> <span class="o">-=</span> <span class="mi">1</span>  <span class="c1"># Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>            <span class="n">j</span> <span class="o">-=</span> <span class="mi">1</span>  <span class="c1"># Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">j</span> <span class="ow">and</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;=</span> <span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">]:</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>  <span class="c1"># Search from left to right for the first element greater than the pivot</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="c1"># Swap elements</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>  <span class="c1"># Scan from left to right for the first element larger than the pivot</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="c1"># Swap the elements</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="c1"># Swap the pivot to the boundary between the two subarrays</span>
+<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="c1"># Swap the pivot into the boundary position between the two subarrays</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>    <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>    <span class="k">return</span> <span class="n">i</span>  <span class="c1"># Return the index of the pivot</span>
+<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>    <span class="k">return</span> <span class="n">i</span>  <span class="c1"># Return the pivot index</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Sentinel partition */</span>
+<div class="highlight"><span class="filename">quick_sort.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Two-pointer partitioning */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">partition</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span>
-<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w">                </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w">                </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span>
-<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w">                </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w">                </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]);</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">]);</span><span class="w">  </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w">                   </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">]);</span><span class="w">  </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w">                   </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Swap elements */</span>
+<div class="highlight"><span class="filename">quick_sort.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Swap the elements */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">swap</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="p">}</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a>
-<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="cm">/* Sentinel partition */</span>
+<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="cm">/* Two-pointer partitioning */</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="kt">int</span><span class="w"> </span><span class="nf">partition</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span><span class="p">)</span>
-<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w">          </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w">          </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span><span class="p">)</span>
-<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w">          </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w">          </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w">  </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w">             </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w">  </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w">             </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Swap elements */</span>
+<div class="highlight"><span class="filename">quick_sort.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Swap the elements */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Swap</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]);</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="p">}</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="cm">/* Sentinel partition */</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="cm">/* Two-pointer partitioning */</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Partition</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span>
-<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w">          </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w">          </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span>
-<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w">          </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w">          </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">        </span><span class="n">Swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">    </span><span class="n">Swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w">  </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w">             </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">    </span><span class="n">Swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w">  </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w">             </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Sentinel partition */</span>
+<div class="highlight"><span class="filename">quick_sort.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Two-pointer partitioning */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">quickSort</span><span class="p">)</span><span class="w"> </span><span class="nx">partition</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">            </span><span class="nx">j</span><span class="o">--</span><span class="w"> </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">            </span><span class="nx">j</span><span class="o">--</span><span class="w"> </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">            </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">            </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="c1">// Swap elements</span>
+<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="c1">// Swap the elements</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
+<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">    </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
-<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Sentinel partition */</span>
+<div class="highlight"><span class="filename">quick_sort.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Two-pointer partitioning */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kr">left</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kr">right</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="kr">left</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="kr">left</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="n">nums</span><span class="p">.</span><span class="n">swapAt</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="n">swapAt</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kr">left</span><span class="p">)</span><span class="w"> </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="n">swapAt</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kr">left</span><span class="p">)</span><span class="w"> </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Swap elements */</span>
+<div class="highlight"><span class="filename">quick_sort.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Swap the elements */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span><span class="p">;</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="p">}</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a>
-<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="cm">/* Sentinel partition */</span>
+<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="cm">/* Two-pointer partitioning */</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="nx">partition</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">right</span><span class="p">;</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">        </span><span class="c1">// Swap elements</span>
+<a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">        </span><span class="c1">// Swap the elements</span>
 <a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Swap elements */</span>
+<div class="highlight"><span class="filename">quick_sort.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Swap the elements */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">];</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span><span class="p">;</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="p">}</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a>
-<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="cm">/* Sentinel partition */</span>
+<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="cm">/* Two-pointer partitioning */</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="nx">partition</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">left</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">right</span><span class="p">;</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">            </span><span class="nx">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">            </span><span class="nx">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">        </span><span class="c1">// Swap elements</span>
+<a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">        </span><span class="c1">// Swap the elements</span>
 <a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Swap elements */</span>
+<div class="highlight"><span class="filename">quick_sort.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Swap the elements */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">_swap</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="p">}</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a>
-<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="cm">/* Sentinel partition */</span>
+<a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="cm">/* Two-pointer partitioning */</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="kt">int</span><span class="w"> </span><span class="n">_partition</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">  </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span><span class="w"> </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
-<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span><span class="w"> </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">    </span><span class="n">_swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">  </span><span class="n">_swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">  </span><span class="n">_swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Sentinel partition */</span>
+<div class="highlight"><span class="filename">quick_sort.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Two-pointer partitioning */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">left</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">        </span><span class="n">nums</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Swap elements */</span>
+<div class="highlight"><span class="filename">quick_sort.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Swap the elements */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">swap</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">];</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">;</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="p">}</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a>
-<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="cm">/* Sentinel partition */</span>
+<a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="cm">/* Two-pointer partitioning */</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="kt">int</span><span class="w"> </span><span class="nf">partition</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">        </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span>
 <a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">    </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
+<a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">    </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
 <a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span>
-<a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">    </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">    </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Swap elements */</span>
+<div class="highlight"><span class="filename">quick_sort.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Swap the elements */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="p">}</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a>
-<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="cm">/* Sentinel partition */</span>
+<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="cm">/* Two-pointer partitioning */</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span><span class="p">)</span>
-<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="w">           </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="w">           </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span><span class="p">)</span>
-<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="w">           </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="w">           </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w">  </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w">   </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w">              </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w">   </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w">              </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Sentinel partition ###</span>
+<div class="highlight"><span class="filename">quick_sort.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Two-pointer partitioning ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Use nums[left] as the pivot</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span>
-<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span>
-<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="k">end</span>
-<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">    </span><span class="c1"># Swap elements</span>
+<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">    </span><span class="c1"># Swap the elements</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">  </span><span class="k">end</span>
-<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">  </span><span class="c1"># Swap the pivot to the boundary between the two subarrays</span>
+<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">  </span><span class="c1"># Swap the pivot into the boundary position between the two subarrays</span>
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">  </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
-<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="c1"># Return the index of the pivot</span>
+<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="c1"># Return the pivot index</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h2 id="1151-algorithm-flow">11.5.1 &nbsp; Algorithm Flow<a class="headerlink" href="#1151-algorithm-flow" title="Permanent link">&para;</a></h2>
-<p>The overall flow of quick sort is shown in Figure 11-9.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20partition%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20right%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Two-pointer%20partitioning%22%22%22%0A%20%20%20%20%23%20Partition%20using%20the%20pivot%20stored%20at%20nums%5Bleft%5D%0A%20%20%20%20i,%20j%20%3D%20left,%20right%0A%20%20%20%20while%20i%20%3C%20j%3A%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bj%5D%20%3E%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20-%3D%201%20%20%23%20Scan%20from%20right%20to%20left%20for%20the%20first%20element%20smaller%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bi%5D%20%3C%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%20%20%23%20Scan%20from%20left%20to%20right%20for%20the%20first%20element%20larger%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20elements%0A%20%20%20%20%20%20%20%20nums%5Bi%5D,%20nums%5Bj%5D%20%3D%20nums%5Bj%5D,%20nums%5Bi%5D%0A%20%20%20%20%23%20Swap%20the%20pivot%20into%20the%20boundary%20position%20between%20the%20two%20subarrays%0A%20%20%20%20nums%5Bi%5D,%20nums%5Bleft%5D%20%3D%20nums%5Bleft%5D,%20nums%5Bi%5D%0A%20%20%20%20return%20i%20%20%23%20Return%20the%20pivot%20index%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B2,%204,%201,%200,%203,%205%5D%0A%20%20%20%20partition%28nums,%200,%20len%28nums%29%20-%201%29%0A%20%20%20%20print%28%22After%20two-pointer%20partitioning,%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20partition%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20right%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Two-pointer%20partitioning%22%22%22%0A%20%20%20%20%23%20Partition%20using%20the%20pivot%20stored%20at%20nums%5Bleft%5D%0A%20%20%20%20i,%20j%20%3D%20left,%20right%0A%20%20%20%20while%20i%20%3C%20j%3A%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bj%5D%20%3E%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20-%3D%201%20%20%23%20Scan%20from%20right%20to%20left%20for%20the%20first%20element%20smaller%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bi%5D%20%3C%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%20%20%23%20Scan%20from%20left%20to%20right%20for%20the%20first%20element%20larger%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20elements%0A%20%20%20%20%20%20%20%20nums%5Bi%5D,%20nums%5Bj%5D%20%3D%20nums%5Bj%5D,%20nums%5Bi%5D%0A%20%20%20%20%23%20Swap%20the%20pivot%20into%20the%20boundary%20position%20between%20the%20two%20subarrays%0A%20%20%20%20nums%5Bi%5D,%20nums%5Bleft%5D%20%3D%20nums%5Bleft%5D,%20nums%5Bi%5D%0A%20%20%20%20return%20i%20%20%23%20Return%20the%20pivot%20index%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B2,%204,%201,%200,%203,%205%5D%0A%20%20%20%20partition%28nums,%200,%20len%28nums%29%20-%201%29%0A%20%20%20%20print%28%22After%20two-pointer%20partitioning,%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1151-algorithm-steps">11.5.1 &nbsp; Algorithm steps<a class="headerlink" href="#1151-algorithm-steps" title="Permanent link">&para;</a></h2>
+<p>The overall quick sort process is shown below.</p>
 <ol>
-<li>First, perform one "sentinel partitioning" on the original array to obtain the unsorted left sub-array and right sub-array.</li>
-<li>Then, recursively perform "sentinel partitioning" on the left sub-array and right sub-array respectively.</li>
-<li>Continue recursively until the sub-array length is 1, at which point sorting of the entire array is complete.</li>
+<li>Partition the original array to obtain unsorted left and right subarrays.</li>
+<li>Recursively partition the left and right subarrays.</li>
+<li>Continue until each subarray has length 1, completing the sort.</li>
 </ol>
-<p><img alt="Quick sort flow" class="animation-figure" src="../quick_sort.assets/quick_sort_overview.png" /></p>
-<p align="center"> Figure 11-9 &nbsp; Quick sort flow </p>
+<p><img alt="The quick sort process" class="animation-figure" src="../quick_sort.assets/quick_sort_overview.png" /></p>
+<p align="center"> Figure 11-9 &nbsp; The quick sort process </p>
 
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 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">quick_sort</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">left</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">right</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Quick sort"""</span>
-<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># Terminate recursion when subarray length is 1</span>
+<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># Stop recursing when the subarray has at most one element</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="k">if</span> <span class="n">left</span> <span class="o">&gt;=</span> <span class="n">right</span><span class="p">:</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>        <span class="k">return</span>
-<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="c1"># Sentinel partition</span>
+<a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="c1"># Two-pointer partitioning</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>    <span class="n">pivot</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">left</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>    <span class="c1"># Recursively process the left subarray and right subarray</span>
+<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>    <span class="c1"># Recurse on the left and right subarrays</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">quick_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">left</span><span class="p">,</span> <span class="n">pivot</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">quick_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">pivot</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>
 </code></pre></div>
@@ -5094,12 +5171,12 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Quick sort */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">quickSort</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// Terminate recursion when subarray length is 1</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="c1">// Sentinel partition</span>
+<a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="c1">// Two-pointer partitioning</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="c1">// Recursively process the left subarray and right subarray</span>
+<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="c1">// Recurse on the left and right subarrays</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">    </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="p">}</span>
@@ -5108,12 +5185,12 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Quick sort */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">quickSort</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Terminate recursion when subarray length is 1</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="c1">// Sentinel partition</span>
+<a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="c1">// Two-pointer partitioning</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="c1">// Recursively process the left subarray and right subarray</span>
+<a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="c1">// Recurse on the left and right subarrays</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="p">}</span>
@@ -5122,12 +5199,12 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Quick sort */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">QuickSort</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// Terminate recursion when subarray length is 1</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="c1">// Sentinel partition</span>
+<a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="c1">// Two-pointer partitioning</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="c1">// Recursively process the left subarray and right subarray</span>
+<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="c1">// Recurse on the left and right subarrays</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="n">QuickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="n">QuickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="p">}</span>
@@ -5136,13 +5213,13 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Quick sort */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">quickSort</span><span class="p">)</span><span class="w"> </span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Terminate recursion when subarray length is 1</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="c1">// Sentinel partition</span>
+<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="c1">// Two-pointer partitioning</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">    </span><span class="nx">pivot</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">partition</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span>
-<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">    </span><span class="c1">// Recursively process the left subarray and right subarray</span>
+<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">    </span><span class="c1">// Recurse on the left and right subarrays</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">    </span><span class="nx">q</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="nx">q</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="p">}</span>
@@ -5151,13 +5228,13 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Quick sort */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Terminate recursion when subarray length is 1</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="kr">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="kr">right</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="c1">// Sentinel partition</span>
+<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="c1">// Two-pointer partitioning</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">pivot</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="kr">right</span><span class="p">)</span>
-<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="c1">// Recursively process the left subarray and right subarray</span>
+<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="c1">// Recurse on the left and right subarrays</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="kr">right</span><span class="p">)</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="p">}</span>
@@ -5166,11 +5243,11 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Quick sort */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Terminate recursion when subarray length is 1</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="c1">// Sentinel partition</span>
+<a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="c1">// Two-pointer partitioning</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">partition</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span>
-<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="c1">// Recursively process the left subarray and right subarray</span>
+<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="c1">// Recurse on the left and right subarrays</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="p">}</span>
@@ -5179,13 +5256,13 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Quick sort */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">left</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// Terminate recursion when subarray length is 1</span>
+<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// Stop recursing when the subarray has zero or one element</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="c1">// Sentinel partition</span>
+<a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="c1">// Two-pointer partitioning</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">partition</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span>
-<a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="c1">// Recursively process the left subarray and right subarray</span>
+<a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="c1">// Recurse on the left and right subarrays</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="p">}</span>
@@ -5194,11 +5271,11 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Quick sort */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">quickSort</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// Terminate recursion when subarray length is 1</span>
+<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="c1">// Sentinel partition</span>
+<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="c1">// Two-pointer partitioning</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="c1">// Recursively process the left subarray and right subarray</span>
+<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="c1">// Recurse on the left and right subarrays</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="p">}</span>
@@ -5207,13 +5284,13 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Quick sort */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">quick_sort</span><span class="p">(</span><span class="n">left</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Terminate recursion when subarray length is 1</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="c1">// Sentinel partition</span>
+<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="c1">// Two-pointer partitioning</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
-<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">    </span><span class="c1">// Recursively process the left subarray and right subarray</span>
+<a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">    </span><span class="c1">// Recurse on the left and right subarrays</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">    </span><span class="bp">Self</span><span class="p">::</span><span class="n">quick_sort</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">);</span>
 <a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">    </span><span class="bp">Self</span><span class="p">::</span><span class="n">quick_sort</span><span class="p">(</span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">);</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="p">}</span>
@@ -5222,13 +5299,13 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Quick sort */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">quickSort</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Terminate recursion when subarray length is 1</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="c1">// Sentinel partition</span>
+<a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="c1">// Two-pointer partitioning</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">    </span><span class="c1">// Recursively process the left subarray and right subarray</span>
+<a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">    </span><span class="c1">// Recurse on the left and right subarrays</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">    </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">    </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="p">}</span>
@@ -5237,24 +5314,24 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Quick sort */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Terminate recursion when subarray length is 1</span>
+<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Stop recursing when the subarray has at most one element</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="k">return</span>
-<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="c1">// Sentinel partition</span>
+<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="c1">// Two-pointer partitioning</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="c1">// Recursively process the left subarray and right subarray</span>
+<a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="c1">// Recurse on the left and right subarrays</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Quick sort class ###</span>
+<div class="highlight"><span class="filename">quick_sort.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Quick sort ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">quick_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Recurse when subarray length is not 1</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Continue while the subarray contains more than one element</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span>
-<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">    </span><span class="c1"># Sentinel partition</span>
+<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">    </span><span class="c1"># Two-pointer partitioning</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">    </span><span class="n">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">    </span><span class="c1"># Recursively process the left subarray and right subarray</span>
+<a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">    </span><span class="c1"># Recurse on the left and right subarrays</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">    </span><span class="n">quick_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">    </span><span class="n">quick_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="k">end</span>
@@ -5264,25 +5341,30 @@
 </div>
 </div>
 </div>
-<h2 id="1152-algorithm-characteristics">11.5.2 &nbsp; Algorithm Characteristics<a class="headerlink" href="#1152-algorithm-characteristics" title="Permanent link">&para;</a></h2>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20partition%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20right%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Two-pointer%20partitioning%22%22%22%0A%20%20%20%20%23%20Partition%20using%20the%20pivot%20stored%20at%20nums%5Bleft%5D%0A%20%20%20%20i,%20j%20%3D%20left,%20right%0A%20%20%20%20while%20i%20%3C%20j%3A%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bj%5D%20%3E%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20-%3D%201%20%20%23%20Scan%20from%20right%20to%20left%20for%20the%20first%20element%20smaller%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bi%5D%20%3C%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%20%20%23%20Scan%20from%20left%20to%20right%20for%20the%20first%20element%20larger%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20elements%0A%20%20%20%20%20%20%20%20nums%5Bi%5D,%20nums%5Bj%5D%20%3D%20nums%5Bj%5D,%20nums%5Bi%5D%0A%20%20%20%20%23%20Swap%20the%20pivot%20into%20the%20boundary%20position%20between%20the%20two%20subarrays%0A%20%20%20%20nums%5Bi%5D,%20nums%5Bleft%5D%20%3D%20nums%5Bleft%5D,%20nums%5Bi%5D%0A%20%20%20%20return%20i%20%20%23%20Return%20the%20pivot%20index%0A%0Adef%20quick_sort%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20right%3A%20int%29%3A%0A%20%20%20%20%22%22%22Quick%20sort%22%22%22%0A%20%20%20%20%23%20Stop%20recursing%20when%20the%20subarray%20has%20at%20most%20one%20element%0A%20%20%20%20if%20left%20%3E%3D%20right%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Two-pointer%20partitioning%0A%20%20%20%20pivot%20%3D%20partition%28nums,%20left,%20right%29%0A%20%20%20%20%23%20Recurse%20on%20the%20left%20and%20right%20subarrays%0A%20%20%20%20quick_sort%28nums,%20left,%20pivot%20-%201%29%0A%20%20%20%20quick_sort%28nums,%20pivot%20%2B%201,%20right%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Quick%20sort%0A%20%20%20%20nums%20%3D%20%5B2,%204,%201,%200,%203,%205%5D%0A%20%20%20%20quick_sort%28nums,%200,%20len%28nums%29%20-%201%29%0A%20%20%20%20print%28%22After%20quick%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20partition%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20right%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Two-pointer%20partitioning%22%22%22%0A%20%20%20%20%23%20Partition%20using%20the%20pivot%20stored%20at%20nums%5Bleft%5D%0A%20%20%20%20i,%20j%20%3D%20left,%20right%0A%20%20%20%20while%20i%20%3C%20j%3A%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bj%5D%20%3E%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20-%3D%201%20%20%23%20Scan%20from%20right%20to%20left%20for%20the%20first%20element%20smaller%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bi%5D%20%3C%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%20%20%23%20Scan%20from%20left%20to%20right%20for%20the%20first%20element%20larger%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20elements%0A%20%20%20%20%20%20%20%20nums%5Bi%5D,%20nums%5Bj%5D%20%3D%20nums%5Bj%5D,%20nums%5Bi%5D%0A%20%20%20%20%23%20Swap%20the%20pivot%20into%20the%20boundary%20position%20between%20the%20two%20subarrays%0A%20%20%20%20nums%5Bi%5D,%20nums%5Bleft%5D%20%3D%20nums%5Bleft%5D,%20nums%5Bi%5D%0A%20%20%20%20return%20i%20%20%23%20Return%20the%20pivot%20index%0A%0Adef%20quick_sort%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20right%3A%20int%29%3A%0A%20%20%20%20%22%22%22Quick%20sort%22%22%22%0A%20%20%20%20%23%20Stop%20recursing%20when%20the%20subarray%20has%20at%20most%20one%20element%0A%20%20%20%20if%20left%20%3E%3D%20right%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Two-pointer%20partitioning%0A%20%20%20%20pivot%20%3D%20partition%28nums,%20left,%20right%29%0A%20%20%20%20%23%20Recurse%20on%20the%20left%20and%20right%20subarrays%0A%20%20%20%20quick_sort%28nums,%20left,%20pivot%20-%201%29%0A%20%20%20%20quick_sort%28nums,%20pivot%20%2B%201,%20right%29%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Quick%20sort%0A%20%20%20%20nums%20%3D%20%5B2,%204,%201,%200,%203,%205%5D%0A%20%20%20%20quick_sort%28nums,%200,%20len%28nums%29%20-%201%29%0A%20%20%20%20print%28%22After%20quick%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1152-algorithm-properties">11.5.2 &nbsp; Algorithm properties<a class="headerlink" href="#1152-algorithm-properties" title="Permanent link">&para;</a></h2>
 <ul>
-<li><strong>Time complexity of <span class="arithmatex">\(O(n \log n)\)</span>, non-adaptive sorting</strong>: On average, sentinel partitioning produces <span class="arithmatex">\(\log n\)</span> recursive levels, and the total number of loop iterations across each level is <span class="arithmatex">\(n\)</span>, so the overall time complexity is <span class="arithmatex">\(O(n \log n)\)</span>. In the worst case, each round of sentinel partitioning splits an array of length <span class="arithmatex">\(n\)</span> into sub-arrays of lengths <span class="arithmatex">\(0\)</span> and <span class="arithmatex">\(n - 1\)</span>. The recursion depth then reaches <span class="arithmatex">\(n\)</span>, with <span class="arithmatex">\(n\)</span> loop iterations at each level, yielding an overall time complexity of <span class="arithmatex">\(O(n^2)\)</span>.</li>
-<li><strong>Space complexity of <span class="arithmatex">\(O(n)\)</span>, in-place sorting</strong>: In the case where the input array is completely reversed, the worst recursive depth reaches <span class="arithmatex">\(n\)</span>, using <span class="arithmatex">\(O(n)\)</span> stack frame space. The sorting operation is performed on the original array without the aid of an additional array.</li>
-<li><strong>Unstable sorting</strong>: In the last step of sentinel partitioning, the pivot may be swapped to the right of an equal element.</li>
+<li><strong>Time complexity: <span class="arithmatex">\(O(n \log n)\)</span>; not adaptive</strong>. On average, partitioning produces <span class="arithmatex">\(\log n\)</span> recursive levels, with <span class="arithmatex">\(n\)</span> loop iterations in total at each level, giving <span class="arithmatex">\(O(n \log n)\)</span> time. In the worst case, each partition splits an array of length <span class="arithmatex">\(n\)</span> into subarrays of lengths <span class="arithmatex">\(0\)</span> and <span class="arithmatex">\(n - 1\)</span>. The recursion reaches <span class="arithmatex">\(n\)</span> levels, each involving up to <span class="arithmatex">\(n\)</span> iterations, for <span class="arithmatex">\(O(n^2)\)</span> time overall.</li>
+<li><strong>Space complexity: <span class="arithmatex">\(O(n)\)</span>; in-place</strong>. A fully reverse-sorted input reaches the worst-case recursion depth of <span class="arithmatex">\(n\)</span>, requiring <span class="arithmatex">\(O(n)\)</span> space for stack frames. Sorting takes place in the original array without an auxiliary array.</li>
+<li><strong>Not stable</strong>. In the final partitioning step, the pivot may be swapped to the right of an equal element.</li>
 </ul>
-<h2 id="1153-why-is-quick-sort-fast">11.5.3 &nbsp; Why Is Quick Sort Fast<a class="headerlink" href="#1153-why-is-quick-sort-fast" title="Permanent link">&para;</a></h2>
-<p>As the name suggests, quick sort has clear efficiency advantages. Although its average time complexity is the same as that of "merge sort" and "heap sort", quick sort is usually faster in practice for the following reasons.</p>
+<h2 id="1153-why-is-quick-sort-quick">11.5.3 &nbsp; Why is quick sort quick?<a class="headerlink" href="#1153-why-is-quick-sort-quick" title="Permanent link">&para;</a></h2>
+<p>As its name suggests, quick sort has an efficiency advantage. Although its average time complexity matches that of merge sort and heap sort, it is usually faster, mainly for the following reasons.</p>
 <ul>
-<li><strong>The worst case is unlikely to occur</strong>: Although the worst-case time complexity of quick sort is <span class="arithmatex">\(O(n^2)\)</span> and its performance is less predictable than that of merge sort, quick sort runs in <span class="arithmatex">\(O(n \log n)\)</span> time in the vast majority of cases.</li>
-<li><strong>High cache efficiency</strong>: During sentinel partitioning, the system can load the entire sub-array into cache, so accessing elements is relatively efficient. By contrast, algorithms such as "heap sort" require non-contiguous access to elements and therefore do not enjoy this advantage.</li>
-<li><strong>Small constant factors</strong>: Among the three algorithms above, quick sort performs the fewest comparisons, assignments, and swaps in total. This is similar to why "insertion sort" is faster than "bubble sort".</li>
+<li><strong>The worst case is unlikely</strong>. Quick sort's worst-case time complexity is <span class="arithmatex">\(O(n^2)\)</span>, making its performance less predictable than merge sort's, but it runs in <span class="arithmatex">\(O(n \log n)\)</span> time in the vast majority of cases.</li>
+<li><strong>Efficient cache use</strong>. During partitioning, the system can load the whole subarray into cache, making element access efficient. Algorithms such as heap sort jump between elements and do not share this advantage.</li>
+<li><strong>Small constant factors</strong>. Of these three algorithms, quick sort performs the fewest comparisons, assignments, swaps, and similar operations overall. This resembles the reason insertion sort is faster than bubble sort.</li>
 </ul>
-<h2 id="1154-pivot-optimization">11.5.4 &nbsp; Pivot Optimization<a class="headerlink" href="#1154-pivot-optimization" title="Permanent link">&para;</a></h2>
-<p><strong>Quick sort can become less time-efficient for certain inputs</strong>. Consider an extreme example in which the input array is in completely descending order. Because we choose the leftmost element as the pivot, once sentinel partitioning is complete, the pivot is swapped to the far right of the array, leaving a left sub-array of length <span class="arithmatex">\(n - 1\)</span> and a right sub-array of length <span class="arithmatex">\(0\)</span>. If this continues recursively, each round of sentinel partitioning produces one sub-array of length <span class="arithmatex">\(0\)</span>, the divide-and-conquer strategy breaks down, and quick sort degenerates into an approximation of "bubble sort".</p>
-<p>To reduce the chance of this happening, <strong>we can optimize the pivot selection strategy used in sentinel partitioning</strong>. For example, we can choose a pivot at random. However, if we are unlucky and repeatedly pick poor pivots, performance can still be unsatisfactory.</p>
-<p>It should be noted that programming languages usually generate "pseudo-random numbers". If we construct a specific test case against a pseudo-random sequence, quick sort can still suffer degraded performance.</p>
-<p>To improve further, we can choose three candidate elements from the array, usually the first, last, and middle elements, <strong>and use the median of the three as the pivot</strong>. This greatly increases the chance that the pivot is "neither too small nor too large". We can also choose more candidate elements to further improve the robustness of the algorithm. With this method, the probability that the time complexity degrades to <span class="arithmatex">\(O(n^2)\)</span> is significantly reduced.</p>
-<p>Example code is as follows:</p>
+<h2 id="1154-improving-pivot-selection">11.5.4 &nbsp; Improving pivot selection<a class="headerlink" href="#1154-improving-pivot-selection" title="Permanent link">&para;</a></h2>
+<p><strong>Quick sort can become slower on certain inputs</strong>. Consider a fully reverse-sorted array. Because the leftmost element is chosen as the pivot, partitioning moves it to the far right, leaving subarrays of lengths <span class="arithmatex">\(n - 1\)</span> and <span class="arithmatex">\(0\)</span>. Continuing recursively produces a subarray of length <span class="arithmatex">\(0\)</span> after every partition. Divide and conquer becomes ineffective, and quick sort degenerates into a process resembling bubble sort.</p>
+<p>To reduce this risk, <strong>we can improve how the pivot is selected</strong>. For example, choose a random element as the pivot. With bad luck, however, every pivot may still be poor, leading to disappointing performance.</p>
+<p>Programming languages generally produce pseudorandom numbers. A test case constructed specifically for a pseudorandom sequence can still degrade quick sort's performance.</p>
+<p>For further improvement, choose three candidate elements, usually the first, last, and middle elements, <strong>and use their median as the pivot</strong>. This greatly increases the chance that the pivot is neither too small nor too large. We can use more candidates to improve robustness further. This approach greatly reduces the likelihood of degradation to <span class="arithmatex">\(O(n^2)\)</span>.</p>
+<p>An implementation follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:13"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><input id="__tabbed_4_13" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Python</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Java</label><label for="__tabbed_4_4">C#</label><label for="__tabbed_4_5">Go</label><label for="__tabbed_4_6">Swift</label><label for="__tabbed_4_7">JS</label><label for="__tabbed_4_8">TS</label><label for="__tabbed_4_9">Dart</label><label for="__tabbed_4_10">Rust</label><label for="__tabbed_4_11">C</label><label for="__tabbed_4_12">Kotlin</label><label for="__tabbed_4_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -5290,29 +5372,29 @@
 <a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Select the median of three candidate elements"""</span>
 <a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="n">l</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">r</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">mid</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">right</span><span class="p">]</span>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="k">if</span> <span class="p">(</span><span class="n">l</span> <span class="o">&lt;=</span> <span class="n">m</span> <span class="o">&lt;=</span> <span class="n">r</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">r</span> <span class="o">&lt;=</span> <span class="n">m</span> <span class="o">&lt;=</span> <span class="n">l</span><span class="p">):</span>
-<a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>        <span class="k">return</span> <span class="n">mid</span>  <span class="c1"># m is between l and r</span>
+<a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>        <span class="k">return</span> <span class="n">mid</span>  <span class="c1"># m lies between l and r</span>
 <a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="k">if</span> <span class="p">(</span><span class="n">m</span> <span class="o">&lt;=</span> <span class="n">l</span> <span class="o">&lt;=</span> <span class="n">r</span><span class="p">)</span> <span class="ow">or</span> <span class="p">(</span><span class="n">r</span> <span class="o">&lt;=</span> <span class="n">l</span> <span class="o">&lt;=</span> <span class="n">m</span><span class="p">):</span>
-<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>        <span class="k">return</span> <span class="n">left</span>  <span class="c1"># l is between m and r</span>
+<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>        <span class="k">return</span> <span class="n">left</span>  <span class="c1"># l lies between m and r</span>
 <a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>    <span class="k">return</span> <span class="n">right</span>
 <a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>
 <a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a><span class="k">def</span><span class="w"> </span><span class="nf">partition</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">left</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">right</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a><span class="w">    </span><span class="sd">"""Sentinel partition (median of three)"""</span>
-<a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a>    <span class="c1"># Use nums[left] as the pivot</span>
+<a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a><span class="w">    </span><span class="sd">"""Two-pointer partitioning (median of three)"""</span>
+<a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a>    <span class="c1"># Select the median of the leftmost, middle, and rightmost elements as the pivot</span>
 <a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>    <span class="n">med</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">median_three</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">left</span><span class="p">,</span> <span class="p">(</span><span class="n">left</span> <span class="o">+</span> <span class="n">right</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a>    <span class="c1"># Swap the median to the array's leftmost position</span>
+<a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a>    <span class="c1"># Swap the median into the leftmost position of the current subarray</span>
 <a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a>    <span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">med</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">med</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">]</span>
 <a id="__codelineno-26-16" name="__codelineno-26-16" href="#__codelineno-26-16"></a>    <span class="c1"># Use nums[left] as the pivot</span>
 <a id="__codelineno-26-17" name="__codelineno-26-17" href="#__codelineno-26-17"></a>    <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">left</span><span class="p">,</span> <span class="n">right</span>
 <a id="__codelineno-26-18" name="__codelineno-26-18" href="#__codelineno-26-18"></a>    <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">j</span><span class="p">:</span>
 <a id="__codelineno-26-19" name="__codelineno-26-19" href="#__codelineno-26-19"></a>        <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">j</span> <span class="ow">and</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">]:</span>
-<a id="__codelineno-26-20" name="__codelineno-26-20" href="#__codelineno-26-20"></a>            <span class="n">j</span> <span class="o">-=</span> <span class="mi">1</span>  <span class="c1"># Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-26-20" name="__codelineno-26-20" href="#__codelineno-26-20"></a>            <span class="n">j</span> <span class="o">-=</span> <span class="mi">1</span>  <span class="c1"># Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-26-21" name="__codelineno-26-21" href="#__codelineno-26-21"></a>        <span class="k">while</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="n">j</span> <span class="ow">and</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">&lt;=</span> <span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">]:</span>
-<a id="__codelineno-26-22" name="__codelineno-26-22" href="#__codelineno-26-22"></a>            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>  <span class="c1"># Search from left to right for the first element greater than the pivot</span>
-<a id="__codelineno-26-23" name="__codelineno-26-23" href="#__codelineno-26-23"></a>        <span class="c1"># Swap elements</span>
+<a id="__codelineno-26-22" name="__codelineno-26-22" href="#__codelineno-26-22"></a>            <span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>  <span class="c1"># Scan from left to right for the first element larger than the pivot</span>
+<a id="__codelineno-26-23" name="__codelineno-26-23" href="#__codelineno-26-23"></a>        <span class="c1"># Swap the elements</span>
 <a id="__codelineno-26-24" name="__codelineno-26-24" href="#__codelineno-26-24"></a>        <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-<a id="__codelineno-26-25" name="__codelineno-26-25" href="#__codelineno-26-25"></a>    <span class="c1"># Swap the pivot to the boundary between the two subarrays</span>
+<a id="__codelineno-26-25" name="__codelineno-26-25" href="#__codelineno-26-25"></a>    <span class="c1"># Swap the pivot into the boundary position between the two subarrays</span>
 <a id="__codelineno-26-26" name="__codelineno-26-26" href="#__codelineno-26-26"></a>    <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
-<a id="__codelineno-26-27" name="__codelineno-26-27" href="#__codelineno-26-27"></a>    <span class="k">return</span> <span class="n">i</span>  <span class="c1"># Return the index of the pivot</span>
+<a id="__codelineno-26-27" name="__codelineno-26-27" href="#__codelineno-26-27"></a>    <span class="k">return</span> <span class="n">i</span>  <span class="c1"># Return the pivot index</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
@@ -5320,29 +5402,29 @@
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">medianThree</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">],</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">mid</span><span class="p">],</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">right</span><span class="p">];</span>
 <a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="p">))</span>
-<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="p">;</span><span class="w"> </span><span class="c1">// m is between l and r</span>
+<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="p">;</span><span class="w"> </span><span class="c1">// m lies between l and r</span>
 <a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">))</span>
-<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// l is between m and r</span>
+<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// l lies between m and r</span>
 <a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="p">}</span>
 <a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a>
-<a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="cm">/* Sentinel partition (median of three) */</span>
+<a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="cm">/* Two-pointer partitioning (median of three) */</span>
 <a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="kt">int</span><span class="w"> </span><span class="nf">partition</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">    </span><span class="c1">// Select the median of three candidate elements</span>
 <a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">med</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">medianThree</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="w">    </span><span class="c1">// Swap the median to the array's leftmost position</span>
+<a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="w">    </span><span class="c1">// Swap the median into the leftmost position of the current subarray</span>
 <a id="__codelineno-27-16" name="__codelineno-27-16" href="#__codelineno-27-16"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">med</span><span class="p">]);</span>
 <a id="__codelineno-27-17" name="__codelineno-27-17" href="#__codelineno-27-17"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-27-18" name="__codelineno-27-18" href="#__codelineno-27-18"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-27-19" name="__codelineno-27-19" href="#__codelineno-27-19"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-20" name="__codelineno-27-20" href="#__codelineno-27-20"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span>
-<a id="__codelineno-27-21" name="__codelineno-27-21" href="#__codelineno-27-21"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w">                </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-27-21" name="__codelineno-27-21" href="#__codelineno-27-21"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w">                </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-27-22" name="__codelineno-27-22" href="#__codelineno-27-22"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span>
-<a id="__codelineno-27-23" name="__codelineno-27-23" href="#__codelineno-27-23"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w">                </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-27-23" name="__codelineno-27-23" href="#__codelineno-27-23"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w">                </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-27-24" name="__codelineno-27-24" href="#__codelineno-27-24"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]);</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-27-25" name="__codelineno-27-25" href="#__codelineno-27-25"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-27-26" name="__codelineno-27-26" href="#__codelineno-27-26"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">]);</span><span class="w">  </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-27-27" name="__codelineno-27-27" href="#__codelineno-27-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w">                   </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-27-26" name="__codelineno-27-26" href="#__codelineno-27-26"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">]);</span><span class="w">  </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-27-27" name="__codelineno-27-27" href="#__codelineno-27-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w">                   </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-27-28" name="__codelineno-27-28" href="#__codelineno-27-28"></a><span class="p">}</span>
 </code></pre></div>
 </div>
@@ -5351,29 +5433,29 @@
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">medianThree</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">mid</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">right</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="p">))</span>
-<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="p">;</span><span class="w"> </span><span class="c1">// m is between l and r</span>
+<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="p">;</span><span class="w"> </span><span class="c1">// m lies between l and r</span>
 <a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">))</span>
-<a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// l is between m and r</span>
+<a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// l lies between m and r</span>
 <a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="p">}</span>
 <a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a>
-<a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="cm">/* Sentinel partition (median of three) */</span>
+<a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="cm">/* Two-pointer partitioning (median of three) */</span>
 <a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="kt">int</span><span class="w"> </span><span class="nf">partition</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">    </span><span class="c1">// Select the median of three candidate elements</span>
 <a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">med</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">medianThree</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="w">    </span><span class="c1">// Swap the median to the array's leftmost position</span>
+<a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="w">    </span><span class="c1">// Swap the median into the leftmost position of the current subarray</span>
 <a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">med</span><span class="p">);</span>
 <a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-28-19" name="__codelineno-28-19" href="#__codelineno-28-19"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-20" name="__codelineno-28-20" href="#__codelineno-28-20"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span><span class="p">)</span>
-<a id="__codelineno-28-21" name="__codelineno-28-21" href="#__codelineno-28-21"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w">          </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-28-21" name="__codelineno-28-21" href="#__codelineno-28-21"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w">          </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-28-22" name="__codelineno-28-22" href="#__codelineno-28-22"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span><span class="p">)</span>
-<a id="__codelineno-28-23" name="__codelineno-28-23" href="#__codelineno-28-23"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w">          </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-28-23" name="__codelineno-28-23" href="#__codelineno-28-23"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w">          </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-28-24" name="__codelineno-28-24" href="#__codelineno-28-24"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-28-25" name="__codelineno-28-25" href="#__codelineno-28-25"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-28-26" name="__codelineno-28-26" href="#__codelineno-28-26"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w">  </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-28-27" name="__codelineno-28-27" href="#__codelineno-28-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w">             </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-28-26" name="__codelineno-28-26" href="#__codelineno-28-26"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w">  </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-28-27" name="__codelineno-28-27" href="#__codelineno-28-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w">             </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-28-28" name="__codelineno-28-28" href="#__codelineno-28-28"></a><span class="p">}</span>
 </code></pre></div>
 </div>
@@ -5382,29 +5464,29 @@
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">MedianThree</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">],</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">mid</span><span class="p">],</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">right</span><span class="p">];</span>
 <a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="p">))</span>
-<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="p">;</span><span class="w"> </span><span class="c1">// m is between l and r</span>
+<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="p">;</span><span class="w"> </span><span class="c1">// m lies between l and r</span>
 <a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">))</span>
-<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// l is between m and r</span>
+<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// l lies between m and r</span>
 <a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="p">}</span>
 <a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a>
-<a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="cm">/* Sentinel partition (median of three) */</span>
+<a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="cm">/* Two-pointer partitioning (median of three) */</span>
 <a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Partition</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">    </span><span class="c1">// Select the median of three candidate elements</span>
 <a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">med</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">MedianThree</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w">    </span><span class="c1">// Swap the median to the array's leftmost position</span>
+<a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w">    </span><span class="c1">// Swap the median into the leftmost position of the current subarray</span>
 <a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="w">    </span><span class="n">Swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">med</span><span class="p">);</span>
 <a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-29-19" name="__codelineno-29-19" href="#__codelineno-29-19"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-20" name="__codelineno-29-20" href="#__codelineno-29-20"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span>
-<a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w">          </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w">          </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-29-22" name="__codelineno-29-22" href="#__codelineno-29-22"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span>
-<a id="__codelineno-29-23" name="__codelineno-29-23" href="#__codelineno-29-23"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w">          </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-29-23" name="__codelineno-29-23" href="#__codelineno-29-23"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w">          </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-29-24" name="__codelineno-29-24" href="#__codelineno-29-24"></a><span class="w">        </span><span class="n">Swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-29-25" name="__codelineno-29-25" href="#__codelineno-29-25"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-29-26" name="__codelineno-29-26" href="#__codelineno-29-26"></a><span class="w">    </span><span class="n">Swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w">  </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-29-27" name="__codelineno-29-27" href="#__codelineno-29-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w">             </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-29-26" name="__codelineno-29-26" href="#__codelineno-29-26"></a><span class="w">    </span><span class="n">Swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w">  </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-29-27" name="__codelineno-29-27" href="#__codelineno-29-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w">             </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-29-28" name="__codelineno-29-28" href="#__codelineno-29-28"></a><span class="p">}</span>
 </code></pre></div>
 </div>
@@ -5413,35 +5495,35 @@
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">quickSortMedian</span><span class="p">)</span><span class="w"> </span><span class="nx">medianThree</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">mid</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="nx">l</span><span class="p">,</span><span class="w"> </span><span class="nx">m</span><span class="p">,</span><span class="w"> </span><span class="nx">r</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">mid</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">right</span><span class="p">]</span>
 <a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="nx">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">l</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">mid</span><span class="w"> </span><span class="c1">// m is between l and r</span>
+<a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">mid</span><span class="w"> </span><span class="c1">// m lies between l and r</span>
 <a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="nx">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="c1">// l is between m and r</span>
+<a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="c1">// l lies between m and r</span>
 <a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">right</span>
 <a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="p">}</span>
 <a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a>
-<a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="cm">/* Sentinel partition (median of three) */</span>
+<a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="cm">/* Two-pointer partitioning (median of three)*/</span>
 <a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">quickSortMedian</span><span class="p">)</span><span class="w"> </span><span class="nx">partition</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
+<a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="w">    </span><span class="c1">// Select the median of the left, middle, and right elements as the pivot</span>
 <a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="w">    </span><span class="nx">med</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">medianThree</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="nx">left</span><span class="o">+</span><span class="nx">right</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span>
-<a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></a><span class="w">    </span><span class="c1">// Swap the median to the array's leftmost position</span>
+<a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></a><span class="w">    </span><span class="c1">// Swap the median into the leftmost position of the current subarray</span>
 <a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a><span class="w">    </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">med</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">med</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">]</span>
 <a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-30-20" name="__codelineno-30-20" href="#__codelineno-30-20"></a><span class="w">    </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span>
 <a id="__codelineno-30-21" name="__codelineno-30-21" href="#__codelineno-30-21"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-22" name="__codelineno-30-22" href="#__codelineno-30-22"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-23" name="__codelineno-30-23" href="#__codelineno-30-23"></a><span class="w">            </span><span class="nx">j</span><span class="o">--</span><span class="w"> </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-30-23" name="__codelineno-30-23" href="#__codelineno-30-23"></a><span class="w">            </span><span class="nx">j</span><span class="o">--</span><span class="w"> </span><span class="c1">//Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-30-24" name="__codelineno-30-24" href="#__codelineno-30-24"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-30-25" name="__codelineno-30-25" href="#__codelineno-30-25"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-26" name="__codelineno-30-26" href="#__codelineno-30-26"></a><span class="w">            </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-30-26" name="__codelineno-30-26" href="#__codelineno-30-26"></a><span class="w">            </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="c1">//Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-30-27" name="__codelineno-30-27" href="#__codelineno-30-27"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-30-28" name="__codelineno-30-28" href="#__codelineno-30-28"></a><span class="w">        </span><span class="c1">// Swap elements</span>
+<a id="__codelineno-30-28" name="__codelineno-30-28" href="#__codelineno-30-28"></a><span class="w">        </span><span class="c1">//Swap the elements</span>
 <a id="__codelineno-30-29" name="__codelineno-30-29" href="#__codelineno-30-29"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
 <a id="__codelineno-30-30" name="__codelineno-30-30" href="#__codelineno-30-30"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-31" name="__codelineno-30-31" href="#__codelineno-30-31"></a><span class="w">    </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
+<a id="__codelineno-30-31" name="__codelineno-30-31" href="#__codelineno-30-31"></a><span class="w">    </span><span class="c1">//Swap the pivot into the boundary position between the two subarrays</span>
 <a id="__codelineno-30-32" name="__codelineno-30-32" href="#__codelineno-30-32"></a><span class="w">    </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
-<a id="__codelineno-30-33" name="__codelineno-30-33" href="#__codelineno-30-33"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-30-33" name="__codelineno-30-33" href="#__codelineno-30-33"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="c1">//Return the pivot index</span>
 <a id="__codelineno-30-34" name="__codelineno-30-34" href="#__codelineno-30-34"></a><span class="p">}</span>
 </code></pre></div>
 </div>
@@ -5452,19 +5534,19 @@
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">mid</span><span class="p">]</span>
 <a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">r</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="kr">right</span><span class="p">]</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="c1">// m is between l and r</span>
+<a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="c1">// m lies between l and r</span>
 <a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kr">left</span><span class="w"> </span><span class="c1">// l is between m and r</span>
+<a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kr">left</span><span class="w"> </span><span class="c1">// l lies between m and r</span>
 <a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="kr">right</span>
 <a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a><span class="p">}</span>
 <a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a>
-<a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="cm">/* Sentinel partition (median of three) */</span>
+<a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="cm">/* Two-pointer partitioning (median of three) */</span>
 <a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="kd">func</span><span class="w"> </span><span class="nf">partitionMedian</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-17" name="__codelineno-31-17" href="#__codelineno-31-17"></a><span class="w">    </span><span class="c1">// Select the median of three candidate elements</span>
 <a id="__codelineno-31-18" name="__codelineno-31-18" href="#__codelineno-31-18"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">med</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">medianThree</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">:</span><span class="w"> </span><span class="kr">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="kr">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="kr">left</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="kr">right</span><span class="p">)</span>
-<a id="__codelineno-31-19" name="__codelineno-31-19" href="#__codelineno-31-19"></a><span class="w">    </span><span class="c1">// Swap the median to the array's leftmost position</span>
+<a id="__codelineno-31-19" name="__codelineno-31-19" href="#__codelineno-31-19"></a><span class="w">    </span><span class="c1">// Swap the median into the leftmost position of the current subarray</span>
 <a id="__codelineno-31-20" name="__codelineno-31-20" href="#__codelineno-31-20"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="n">swapAt</span><span class="p">(</span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="n">med</span><span class="p">)</span>
 <a id="__codelineno-31-21" name="__codelineno-31-21" href="#__codelineno-31-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="bp">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="kr">right</span><span class="p">)</span>
 <a id="__codelineno-31-22" name="__codelineno-31-22" href="#__codelineno-31-22"></a><span class="p">}</span>
@@ -5476,14 +5558,14 @@
 <a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">],</span>
 <a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">mid</span><span class="p">],</span>
 <a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">        </span><span class="nx">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">right</span><span class="p">];</span>
-<a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">    </span><span class="c1">// m is between l and r</span>
+<a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">    </span><span class="c1">// m lies between l and r</span>
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="nx">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="nx">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">l</span><span class="p">))</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">mid</span><span class="p">;</span>
-<a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="c1">// l is between m and r</span>
+<a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="c1">// l lies between m and r</span>
 <a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="nx">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="nx">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">))</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">left</span><span class="p">;</span>
 <a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">right</span><span class="p">;</span>
 <a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="p">}</span>
 <a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a>
-<a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="cm">/* Sentinel partition (median of three) */</span>
+<a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="cm">/* Two-pointer partitioning (median of three) */</span>
 <a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="nx">partition</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-15" name="__codelineno-32-15" href="#__codelineno-32-15"></a><span class="w">    </span><span class="c1">// Select the median of three candidate elements</span>
 <a id="__codelineno-32-16" name="__codelineno-32-16" href="#__codelineno-32-16"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">med</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">medianThree</span><span class="p">(</span>
@@ -5492,18 +5574,18 @@
 <a id="__codelineno-32-19" name="__codelineno-32-19" href="#__codelineno-32-19"></a><span class="w">        </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">((</span><span class="nx">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">),</span>
 <a id="__codelineno-32-20" name="__codelineno-32-20" href="#__codelineno-32-20"></a><span class="w">        </span><span class="nx">right</span>
 <a id="__codelineno-32-21" name="__codelineno-32-21" href="#__codelineno-32-21"></a><span class="w">    </span><span class="p">);</span>
-<a id="__codelineno-32-22" name="__codelineno-32-22" href="#__codelineno-32-22"></a><span class="w">    </span><span class="c1">// Swap the median to the array's leftmost position</span>
+<a id="__codelineno-32-22" name="__codelineno-32-22" href="#__codelineno-32-22"></a><span class="w">    </span><span class="c1">// Swap the median into the leftmost position of the current subarray</span>
 <a id="__codelineno-32-23" name="__codelineno-32-23" href="#__codelineno-32-23"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">med</span><span class="p">);</span>
 <a id="__codelineno-32-24" name="__codelineno-32-24" href="#__codelineno-32-24"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-32-25" name="__codelineno-32-25" href="#__codelineno-32-25"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span>
 <a id="__codelineno-32-26" name="__codelineno-32-26" href="#__codelineno-32-26"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">right</span><span class="p">;</span>
 <a id="__codelineno-32-27" name="__codelineno-32-27" href="#__codelineno-32-27"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-28" name="__codelineno-32-28" href="#__codelineno-32-28"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">])</span><span class="w"> </span><span class="nx">j</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
-<a id="__codelineno-32-29" name="__codelineno-32-29" href="#__codelineno-32-29"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">])</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-32-28" name="__codelineno-32-28" href="#__codelineno-32-28"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">])</span><span class="w"> </span><span class="nx">j</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-32-29" name="__codelineno-32-29" href="#__codelineno-32-29"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">])</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-32-30" name="__codelineno-32-30" href="#__codelineno-32-30"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-32-31" name="__codelineno-32-31" href="#__codelineno-32-31"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-32-32" name="__codelineno-32-32" href="#__codelineno-32-32"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-32-33" name="__codelineno-32-33" href="#__codelineno-32-33"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-32-32" name="__codelineno-32-32" href="#__codelineno-32-32"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-32-33" name="__codelineno-32-33" href="#__codelineno-32-33"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-32-34" name="__codelineno-32-34" href="#__codelineno-32-34"></a><span class="p">}</span>
 </code></pre></div>
 </div>
@@ -5518,14 +5600,14 @@
 <a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">],</span>
 <a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">        </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">mid</span><span class="p">],</span>
 <a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="w">        </span><span class="nx">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">right</span><span class="p">];</span>
-<a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">    </span><span class="c1">// m is between l and r</span>
+<a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">    </span><span class="c1">// m lies between l and r</span>
 <a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="nx">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="nx">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">l</span><span class="p">))</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">mid</span><span class="p">;</span>
-<a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">    </span><span class="c1">// l is between m and r</span>
+<a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">    </span><span class="c1">// l lies between m and r</span>
 <a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="nx">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="nx">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">))</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">left</span><span class="p">;</span>
 <a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">right</span><span class="p">;</span>
 <a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a><span class="p">}</span>
 <a id="__codelineno-33-17" name="__codelineno-33-17" href="#__codelineno-33-17"></a>
-<a id="__codelineno-33-18" name="__codelineno-33-18" href="#__codelineno-33-18"></a><span class="cm">/* Sentinel partition (median of three) */</span>
+<a id="__codelineno-33-18" name="__codelineno-33-18" href="#__codelineno-33-18"></a><span class="cm">/* Two-pointer partitioning (median of three) */</span>
 <a id="__codelineno-33-19" name="__codelineno-33-19" href="#__codelineno-33-19"></a><span class="nx">partition</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">left</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-20" name="__codelineno-33-20" href="#__codelineno-33-20"></a><span class="w">    </span><span class="c1">// Select the median of three candidate elements</span>
 <a id="__codelineno-33-21" name="__codelineno-33-21" href="#__codelineno-33-21"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">med</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">medianThree</span><span class="p">(</span>
@@ -5534,22 +5616,22 @@
 <a id="__codelineno-33-24" name="__codelineno-33-24" href="#__codelineno-33-24"></a><span class="w">        </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">((</span><span class="nx">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">),</span>
 <a id="__codelineno-33-25" name="__codelineno-33-25" href="#__codelineno-33-25"></a><span class="w">        </span><span class="nx">right</span>
 <a id="__codelineno-33-26" name="__codelineno-33-26" href="#__codelineno-33-26"></a><span class="w">    </span><span class="p">);</span>
-<a id="__codelineno-33-27" name="__codelineno-33-27" href="#__codelineno-33-27"></a><span class="w">    </span><span class="c1">// Swap the median to the array's leftmost position</span>
+<a id="__codelineno-33-27" name="__codelineno-33-27" href="#__codelineno-33-27"></a><span class="w">    </span><span class="c1">// Swap the median into the leftmost position of the current subarray</span>
 <a id="__codelineno-33-28" name="__codelineno-33-28" href="#__codelineno-33-28"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">med</span><span class="p">);</span>
 <a id="__codelineno-33-29" name="__codelineno-33-29" href="#__codelineno-33-29"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-33-30" name="__codelineno-33-30" href="#__codelineno-33-30"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span>
 <a id="__codelineno-33-31" name="__codelineno-33-31" href="#__codelineno-33-31"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">right</span><span class="p">;</span>
 <a id="__codelineno-33-32" name="__codelineno-33-32" href="#__codelineno-33-32"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-33" name="__codelineno-33-33" href="#__codelineno-33-33"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-34" name="__codelineno-33-34" href="#__codelineno-33-34"></a><span class="w">            </span><span class="nx">j</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-33-34" name="__codelineno-33-34" href="#__codelineno-33-34"></a><span class="w">            </span><span class="nx">j</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-33-35" name="__codelineno-33-35" href="#__codelineno-33-35"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-33-36" name="__codelineno-33-36" href="#__codelineno-33-36"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">left</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-37" name="__codelineno-33-37" href="#__codelineno-33-37"></a><span class="w">            </span><span class="nx">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-33-37" name="__codelineno-33-37" href="#__codelineno-33-37"></a><span class="w">            </span><span class="nx">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-33-38" name="__codelineno-33-38" href="#__codelineno-33-38"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-33-39" name="__codelineno-33-39" href="#__codelineno-33-39"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-33-40" name="__codelineno-33-40" href="#__codelineno-33-40"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-33-41" name="__codelineno-33-41" href="#__codelineno-33-41"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-33-42" name="__codelineno-33-42" href="#__codelineno-33-42"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-33-41" name="__codelineno-33-41" href="#__codelineno-33-41"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">swap</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-33-42" name="__codelineno-33-42" href="#__codelineno-33-42"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-33-43" name="__codelineno-33-43" href="#__codelineno-33-43"></a><span class="p">}</span>
 </code></pre></div>
 </div>
@@ -5558,27 +5640,27 @@
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">_medianThree</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">],</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">mid</span><span class="p">],</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">right</span><span class="p">];</span>
 <a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="p">))</span>
-<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="p">;</span><span class="w"> </span><span class="c1">// m is between l and r</span>
+<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="p">;</span><span class="w"> </span><span class="c1">// m lies between l and r</span>
 <a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">))</span>
-<a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// l is between m and r</span>
+<a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// l lies between m and r</span>
 <a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="p">}</span>
 <a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a>
-<a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="cm">/* Sentinel partition (median of three) */</span>
+<a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="cm">/* Two-pointer partitioning (median of three) */</span>
 <a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="kt">int</span><span class="w"> </span><span class="n">_partition</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">  </span><span class="c1">// Select the median of three candidate elements</span>
 <a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">med</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_medianThree</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">  </span><span class="c1">// Swap the median to the array's leftmost position</span>
+<a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">  </span><span class="c1">// Swap the median into the leftmost position of the current subarray</span>
 <a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w">  </span><span class="n">_swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">med</span><span class="p">);</span>
 <a id="__codelineno-34-17" name="__codelineno-34-17" href="#__codelineno-34-17"></a><span class="w">  </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-34-18" name="__codelineno-34-18" href="#__codelineno-34-18"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-34-19" name="__codelineno-34-19" href="#__codelineno-34-19"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-20" name="__codelineno-34-20" href="#__codelineno-34-20"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span><span class="w"> </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
-<a id="__codelineno-34-21" name="__codelineno-34-21" href="#__codelineno-34-21"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-34-20" name="__codelineno-34-20" href="#__codelineno-34-20"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span><span class="w"> </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-34-21" name="__codelineno-34-21" href="#__codelineno-34-21"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-34-22" name="__codelineno-34-22" href="#__codelineno-34-22"></a><span class="w">    </span><span class="n">_swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-34-23" name="__codelineno-34-23" href="#__codelineno-34-23"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-34-24" name="__codelineno-34-24" href="#__codelineno-34-24"></a><span class="w">  </span><span class="n">_swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-34-25" name="__codelineno-34-25" href="#__codelineno-34-25"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-34-24" name="__codelineno-34-24" href="#__codelineno-34-24"></a><span class="w">  </span><span class="n">_swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-34-25" name="__codelineno-34-25" href="#__codelineno-34-25"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-34-26" name="__codelineno-34-26" href="#__codelineno-34-26"></a><span class="p">}</span>
 </code></pre></div>
 </div>
@@ -5587,33 +5669,33 @@
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">median_three</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">left</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">mid</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">right</span><span class="p">]);</span>
 <a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="p">;</span><span class="w"> </span><span class="c1">// m is between l and r</span>
+<a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="p">;</span><span class="w"> </span><span class="c1">// m lies between l and r</span>
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// l is between m and r</span>
+<a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// l lies between m and r</span>
 <a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w">    </span><span class="n">right</span>
 <a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="p">}</span>
 <a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a>
-<a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="cm">/* Sentinel partition (median of three) */</span>
+<a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="cm">/* Two-pointer partitioning (median of three) */</span>
 <a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="k">fn</span><span class="w"> </span><span class="nf">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">left</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">    </span><span class="c1">// Select the median of three candidate elements</span>
 <a id="__codelineno-35-16" name="__codelineno-35-16" href="#__codelineno-35-16"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">med</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">median_three</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-35-17" name="__codelineno-35-17" href="#__codelineno-35-17"></a><span class="w">    </span><span class="c1">// Swap the median to the array's leftmost position</span>
+<a id="__codelineno-35-17" name="__codelineno-35-17" href="#__codelineno-35-17"></a><span class="w">    </span><span class="c1">// Swap the median into the leftmost position of the current subarray</span>
 <a id="__codelineno-35-18" name="__codelineno-35-18" href="#__codelineno-35-18"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">med</span><span class="p">);</span>
 <a id="__codelineno-35-19" name="__codelineno-35-19" href="#__codelineno-35-19"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-35-20" name="__codelineno-35-20" href="#__codelineno-35-20"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-35-21" name="__codelineno-35-21" href="#__codelineno-35-21"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-22" name="__codelineno-35-22" href="#__codelineno-35-22"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-23" name="__codelineno-35-23" href="#__codelineno-35-23"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-35-23" name="__codelineno-35-23" href="#__codelineno-35-23"></a><span class="w">            </span><span class="n">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-35-24" name="__codelineno-35-24" href="#__codelineno-35-24"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-35-25" name="__codelineno-35-25" href="#__codelineno-35-25"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-26" name="__codelineno-35-26" href="#__codelineno-35-26"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-35-26" name="__codelineno-35-26" href="#__codelineno-35-26"></a><span class="w">            </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-35-27" name="__codelineno-35-27" href="#__codelineno-35-27"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-35-28" name="__codelineno-35-28" href="#__codelineno-35-28"></a><span class="w">        </span><span class="n">nums</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-35-29" name="__codelineno-35-29" href="#__codelineno-35-29"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-35-30" name="__codelineno-35-30" href="#__codelineno-35-30"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-35-31" name="__codelineno-35-31" href="#__codelineno-35-31"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-35-30" name="__codelineno-35-30" href="#__codelineno-35-30"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-35-31" name="__codelineno-35-31" href="#__codelineno-35-31"></a><span class="w">    </span><span class="n">i</span><span class="w"> </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-35-32" name="__codelineno-35-32" href="#__codelineno-35-32"></a><span class="p">}</span>
 </code></pre></div>
 </div>
@@ -5622,29 +5704,29 @@
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">medianThree</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">],</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">mid</span><span class="p">],</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">right</span><span class="p">];</span>
 <a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="p">))</span>
-<a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="p">;</span><span class="w"> </span><span class="c1">// m is between l and r</span>
+<a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="p">;</span><span class="w"> </span><span class="c1">// m lies between l and r</span>
 <a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="n">m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">))</span>
-<a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// l is between m and r</span>
+<a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// l lies between m and r</span>
 <a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="p">}</span>
 <a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a>
-<a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="cm">/* Sentinel partition (median of three) */</span>
+<a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="cm">/* Two-pointer partitioning (median of three) */</span>
 <a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="kt">int</span><span class="w"> </span><span class="nf">partitionMedian</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">    </span><span class="c1">// Select the median of three candidate elements</span>
 <a id="__codelineno-36-14" name="__codelineno-36-14" href="#__codelineno-36-14"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">med</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">medianThree</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a><span class="w">    </span><span class="c1">// Swap the median to the array's leftmost position</span>
+<a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a><span class="w">    </span><span class="c1">// Swap the median into the leftmost position of the array</span>
 <a id="__codelineno-36-16" name="__codelineno-36-16" href="#__codelineno-36-16"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">med</span><span class="p">);</span>
 <a id="__codelineno-36-17" name="__codelineno-36-17" href="#__codelineno-36-17"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-36-18" name="__codelineno-36-18" href="#__codelineno-36-18"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-36-19" name="__codelineno-36-19" href="#__codelineno-36-19"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-20" name="__codelineno-36-20" href="#__codelineno-36-20"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span>
-<a id="__codelineno-36-21" name="__codelineno-36-21" href="#__codelineno-36-21"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-36-21" name="__codelineno-36-21" href="#__codelineno-36-21"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-36-22" name="__codelineno-36-22" href="#__codelineno-36-22"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">left</span><span class="p">])</span>
-<a id="__codelineno-36-23" name="__codelineno-36-23" href="#__codelineno-36-23"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w">          </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-36-23" name="__codelineno-36-23" href="#__codelineno-36-23"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="p">;</span><span class="w">          </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-36-24" name="__codelineno-36-24" href="#__codelineno-36-24"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-36-25" name="__codelineno-36-25" href="#__codelineno-36-25"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-36-26" name="__codelineno-36-26" href="#__codelineno-36-26"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-36-27" name="__codelineno-36-27" href="#__codelineno-36-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w">            </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-36-26" name="__codelineno-36-26" href="#__codelineno-36-26"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-36-27" name="__codelineno-36-27" href="#__codelineno-36-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="p">;</span><span class="w">            </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-36-28" name="__codelineno-36-28" href="#__codelineno-36-28"></a><span class="p">}</span>
 </code></pre></div>
 </div>
@@ -5655,105 +5737,110 @@
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">mid</span><span class="o">]</span>
 <a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">right</span><span class="o">]</span>
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="n">m</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">l</span><span class="p">..</span><span class="na">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">m</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">r</span><span class="p">..</span><span class="na">l</span><span class="p">))</span>
-<a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="w">  </span><span class="c1">// m is between l and r</span>
+<a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="w">  </span><span class="c1">// m lies between l and r</span>
 <a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">((</span><span class="n">l</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">m</span><span class="p">..</span><span class="na">r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">r</span><span class="p">..</span><span class="na">m</span><span class="p">))</span>
-<a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="c1">// l is between m and r</span>
+<a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="c1">// l lies between m and r</span>
 <a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">right</span>
 <a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="p">}</span>
 <a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a>
-<a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="cm">/* Sentinel partition (median of three) */</span>
+<a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="cm">/* Two-pointer partitioning (median of three) */</span>
 <a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">partitionMedian</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-15" name="__codelineno-37-15" href="#__codelineno-37-15"></a><span class="w">    </span><span class="c1">// Select the median of three candidate elements</span>
 <a id="__codelineno-37-16" name="__codelineno-37-16" href="#__codelineno-37-16"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">med</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">medianThree</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-37-17" name="__codelineno-37-17" href="#__codelineno-37-17"></a><span class="w">    </span><span class="c1">// Swap the median to the array's leftmost position</span>
+<a id="__codelineno-37-17" name="__codelineno-37-17" href="#__codelineno-37-17"></a><span class="w">    </span><span class="c1">// Swap the median into the leftmost position of the current subarray</span>
 <a id="__codelineno-37-18" name="__codelineno-37-18" href="#__codelineno-37-18"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">med</span><span class="p">)</span>
 <a id="__codelineno-37-19" name="__codelineno-37-19" href="#__codelineno-37-19"></a><span class="w">    </span><span class="c1">// Use nums[left] as the pivot</span>
 <a id="__codelineno-37-20" name="__codelineno-37-20" href="#__codelineno-37-20"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span>
 <a id="__codelineno-37-21" name="__codelineno-37-21" href="#__codelineno-37-21"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span>
 <a id="__codelineno-37-22" name="__codelineno-37-22" href="#__codelineno-37-22"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-23" name="__codelineno-37-23" href="#__codelineno-37-23"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span><span class="p">)</span>
-<a id="__codelineno-37-24" name="__codelineno-37-24" href="#__codelineno-37-24"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="w">                      </span><span class="c1">// Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-37-24" name="__codelineno-37-24" href="#__codelineno-37-24"></a><span class="w">            </span><span class="n">j</span><span class="o">--</span><span class="w">                      </span><span class="c1">// Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-37-25" name="__codelineno-37-25" href="#__codelineno-37-25"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span><span class="p">)</span>
-<a id="__codelineno-37-26" name="__codelineno-37-26" href="#__codelineno-37-26"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="w">                      </span><span class="c1">// Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-37-26" name="__codelineno-37-26" href="#__codelineno-37-26"></a><span class="w">            </span><span class="n">i</span><span class="o">++</span><span class="w">                      </span><span class="c1">// Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-37-27" name="__codelineno-37-27" href="#__codelineno-37-27"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="p">)</span><span class="w">             </span><span class="c1">// Swap these two elements</span>
 <a id="__codelineno-37-28" name="__codelineno-37-28" href="#__codelineno-37-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-37-29" name="__codelineno-37-29" href="#__codelineno-37-29"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w">              </span><span class="c1">// Swap the pivot to the boundary between the two subarrays</span>
-<a id="__codelineno-37-30" name="__codelineno-37-30" href="#__codelineno-37-30"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w">                         </span><span class="c1">// Return the index of the pivot</span>
+<a id="__codelineno-37-29" name="__codelineno-37-29" href="#__codelineno-37-29"></a><span class="w">    </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">)</span><span class="w">              </span><span class="c1">// Swap the pivot into the boundary position between the two subarrays</span>
+<a id="__codelineno-37-30" name="__codelineno-37-30" href="#__codelineno-37-30"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">i</span><span class="w">                         </span><span class="c1">// Return the pivot index</span>
 <a id="__codelineno-37-31" name="__codelineno-37-31" href="#__codelineno-37-31"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Select median of three candidate elements ###</span>
+<div class="highlight"><span class="filename">quick_sort.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Select the median of three candidate elements ###</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">median_three</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">mid</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
 <a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1"># Select the median of three candidate elements</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="n">_l</span><span class="p">,</span><span class="w"> </span><span class="n">_m</span><span class="p">,</span><span class="w"> </span><span class="n">_r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">mid</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">right</span><span class="o">]</span>
-<a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">  </span><span class="c1"># m is between l and r</span>
+<a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">  </span><span class="c1"># _m lies between _l and _r</span>
 <a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">mid</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">_m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">_m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">_r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">_r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">_m</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">_m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">_l</span><span class="p">)</span>
-<a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="w">  </span><span class="c1"># l is between m and r</span>
+<a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="w">  </span><span class="c1"># _l lies between _m and _r</span>
 <a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_m</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">_l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">_l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">_r</span><span class="p">)</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="p">(</span><span class="n">_r</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">_l</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">_l</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">_m</span><span class="p">)</span>
 <a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">right</span>
 <a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="k">end</span>
 <a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a>
-<a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="c1">### Sentinel partition (median of three) ###</span>
+<a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="c1">### Two-pointer partitioning (median of three)###</span>
 <a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="k">def</span><span class="w"> </span><span class="nf">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="w">  </span><span class="c1">### Use nums[left] as pivot</span>
+<a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="w">  </span><span class="c1">### Use nums[left] as the pivot</span>
 <a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a><span class="w">  </span><span class="n">med</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">median_three</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w">  </span><span class="c1"># Swap median to leftmost position of array</span>
+<a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w">  </span><span class="c1"># Swap the median into the leftmost position of the current subarray</span>
 <a id="__codelineno-38-17" name="__codelineno-38-17" href="#__codelineno-38-17"></a><span class="w">  </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">med</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">med</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span>
 <a id="__codelineno-38-18" name="__codelineno-38-18" href="#__codelineno-38-18"></a><span class="w">  </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span>
 <a id="__codelineno-38-19" name="__codelineno-38-19" href="#__codelineno-38-19"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span>
 <a id="__codelineno-38-20" name="__codelineno-38-20" href="#__codelineno-38-20"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span>
-<a id="__codelineno-38-21" name="__codelineno-38-21" href="#__codelineno-38-21"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Search from right to left for the first element smaller than the pivot</span>
+<a id="__codelineno-38-21" name="__codelineno-38-21" href="#__codelineno-38-21"></a><span class="w">      </span><span class="n">j</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Scan from right to left for the first element smaller than the pivot</span>
 <a id="__codelineno-38-22" name="__codelineno-38-22" href="#__codelineno-38-22"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-38-23" name="__codelineno-38-23" href="#__codelineno-38-23"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span>
-<a id="__codelineno-38-24" name="__codelineno-38-24" href="#__codelineno-38-24"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Search from left to right for the first element greater than the pivot</span>
+<a id="__codelineno-38-24" name="__codelineno-38-24" href="#__codelineno-38-24"></a><span class="w">      </span><span class="n">i</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Scan from left to right for the first element larger than the pivot</span>
 <a id="__codelineno-38-25" name="__codelineno-38-25" href="#__codelineno-38-25"></a><span class="w">    </span><span class="k">end</span>
-<a id="__codelineno-38-26" name="__codelineno-38-26" href="#__codelineno-38-26"></a><span class="w">    </span><span class="c1"># Swap elements</span>
+<a id="__codelineno-38-26" name="__codelineno-38-26" href="#__codelineno-38-26"></a><span class="w">    </span><span class="c1"># Swap the elements</span>
 <a id="__codelineno-38-27" name="__codelineno-38-27" href="#__codelineno-38-27"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-38-28" name="__codelineno-38-28" href="#__codelineno-38-28"></a><span class="w">  </span><span class="k">end</span>
-<a id="__codelineno-38-29" name="__codelineno-38-29" href="#__codelineno-38-29"></a><span class="w">  </span><span class="c1"># Swap the pivot to the boundary between the two subarrays</span>
+<a id="__codelineno-38-29" name="__codelineno-38-29" href="#__codelineno-38-29"></a><span class="w">  </span><span class="c1"># Swap the pivot into the boundary position between the two subarrays</span>
 <a id="__codelineno-38-30" name="__codelineno-38-30" href="#__codelineno-38-30"></a><span class="w">  </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">left</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
-<a id="__codelineno-38-31" name="__codelineno-38-31" href="#__codelineno-38-31"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="c1"># Return the index of the pivot</span>
+<a id="__codelineno-38-31" name="__codelineno-38-31" href="#__codelineno-38-31"></a><span class="w">  </span><span class="n">i</span><span class="w"> </span><span class="c1"># Return the pivot index</span>
 <a id="__codelineno-38-32" name="__codelineno-38-32" href="#__codelineno-38-32"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h2 id="1155-recursive-depth-optimization">11.5.5 &nbsp; Recursive Depth Optimization<a class="headerlink" href="#1155-recursive-depth-optimization" title="Permanent link">&para;</a></h2>
-<p><strong>Quick sort may also use more space for certain inputs</strong>. Consider a fully sorted input array. Let the length of the current sub-array in the recursion be <span class="arithmatex">\(m\)</span>. Each round of sentinel partitioning produces a left sub-array of length <span class="arithmatex">\(0\)</span> and a right sub-array of length <span class="arithmatex">\(m - 1\)</span>, which means each recursive call reduces the problem size by only one element. The recursion tree can therefore reach a height of <span class="arithmatex">\(n - 1\)</span>, requiring <span class="arithmatex">\(O(n)\)</span> stack-frame space.</p>
-<p>To prevent stack frames from accumulating, we can compare the lengths of the two sub-arrays after each round of sentinel partitioning, <strong>and recurse only on the shorter one</strong>. Because the shorter sub-array has length at most <span class="arithmatex">\(n / 2\)</span>, this method ensures that the recursion depth does not exceed <span class="arithmatex">\(\log n\)</span>, reducing the worst-case space complexity to <span class="arithmatex">\(O(\log n)\)</span>. The code is shown below:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20median_three%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20mid%3A%20int,%20right%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Select%20the%20median%20of%20three%20candidate%20elements%22%22%22%0A%20%20%20%20l,%20m,%20r%20%3D%20nums%5Bleft%5D,%20nums%5Bmid%5D,%20nums%5Bright%5D%0A%20%20%20%20if%20%28l%20%3C%3D%20m%20%3C%3D%20r%29%20or%20%28r%20%3C%3D%20m%20%3C%3D%20l%29%3A%0A%20%20%20%20%20%20%20%20return%20mid%20%20%23%20m%20lies%20between%20l%20and%20r%0A%20%20%20%20if%20%28m%20%3C%3D%20l%20%3C%3D%20r%29%20or%20%28r%20%3C%3D%20l%20%3C%3D%20m%29%3A%0A%20%20%20%20%20%20%20%20return%20left%20%20%23%20l%20lies%20between%20m%20and%20r%0A%20%20%20%20return%20right%0A%0Adef%20partition%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20right%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Two-pointer%20partitioning%20%28median%20of%20three%29%22%22%22%0A%20%20%20%20%23%20Partition%20using%20the%20pivot%20stored%20at%20nums%5Bleft%5D%0A%20%20%20%20med%20%3D%20median_three%28nums,%20left,%20%28left%20%2B%20right%29%20//%202,%20right%29%0A%20%20%20%20%23%20Swap%20the%20median%20into%20the%20leftmost%20position%20of%20the%20current%20subarray%0A%20%20%20%20nums%5Bleft%5D,%20nums%5Bmed%5D%20%3D%20nums%5Bmed%5D,%20nums%5Bleft%5D%0A%20%20%20%20%23%20Partition%20using%20the%20pivot%20stored%20at%20nums%5Bleft%5D%0A%20%20%20%20i,%20j%20%3D%20left,%20right%0A%20%20%20%20while%20i%20%3C%20j%3A%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bj%5D%20%3E%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20-%3D%201%20%20%23%20Scan%20from%20right%20to%20left%20for%20the%20first%20element%20smaller%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bi%5D%20%3C%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%20%20%23%20Scan%20from%20left%20to%20right%20for%20the%20first%20element%20larger%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20elements%0A%20%20%20%20%20%20%20%20nums%5Bi%5D,%20nums%5Bj%5D%20%3D%20nums%5Bj%5D,%20nums%5Bi%5D%0A%20%20%20%20%23%20Swap%20the%20pivot%20into%20the%20boundary%20position%20between%20the%20two%20subarrays%0A%20%20%20%20nums%5Bi%5D,%20nums%5Bleft%5D%20%3D%20nums%5Bleft%5D,%20nums%5Bi%5D%0A%20%20%20%20return%20i%20%20%23%20Return%20the%20pivot%20index%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Median-of-three%20pivot%20optimization%0A%20%20%20%20nums%20%3D%20%5B2,%204,%201,%200,%203,%205%5D%0A%20%20%20%20partition%28nums,%200,%20len%28nums%29%20-%201%29%0A%20%20%20%20print%28%22After%20partitioning%20with%20a%20median-of-three%20pivot,%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20median_three%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20mid%3A%20int,%20right%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Select%20the%20median%20of%20three%20candidate%20elements%22%22%22%0A%20%20%20%20l,%20m,%20r%20%3D%20nums%5Bleft%5D,%20nums%5Bmid%5D,%20nums%5Bright%5D%0A%20%20%20%20if%20%28l%20%3C%3D%20m%20%3C%3D%20r%29%20or%20%28r%20%3C%3D%20m%20%3C%3D%20l%29%3A%0A%20%20%20%20%20%20%20%20return%20mid%20%20%23%20m%20lies%20between%20l%20and%20r%0A%20%20%20%20if%20%28m%20%3C%3D%20l%20%3C%3D%20r%29%20or%20%28r%20%3C%3D%20l%20%3C%3D%20m%29%3A%0A%20%20%20%20%20%20%20%20return%20left%20%20%23%20l%20lies%20between%20m%20and%20r%0A%20%20%20%20return%20right%0A%0Adef%20partition%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20right%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Two-pointer%20partitioning%20%28median%20of%20three%29%22%22%22%0A%20%20%20%20%23%20Partition%20using%20the%20pivot%20stored%20at%20nums%5Bleft%5D%0A%20%20%20%20med%20%3D%20median_three%28nums,%20left,%20%28left%20%2B%20right%29%20//%202,%20right%29%0A%20%20%20%20%23%20Swap%20the%20median%20into%20the%20leftmost%20position%20of%20the%20current%20subarray%0A%20%20%20%20nums%5Bleft%5D,%20nums%5Bmed%5D%20%3D%20nums%5Bmed%5D,%20nums%5Bleft%5D%0A%20%20%20%20%23%20Partition%20using%20the%20pivot%20stored%20at%20nums%5Bleft%5D%0A%20%20%20%20i,%20j%20%3D%20left,%20right%0A%20%20%20%20while%20i%20%3C%20j%3A%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bj%5D%20%3E%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20-%3D%201%20%20%23%20Scan%20from%20right%20to%20left%20for%20the%20first%20element%20smaller%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bi%5D%20%3C%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%20%20%23%20Scan%20from%20left%20to%20right%20for%20the%20first%20element%20larger%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20elements%0A%20%20%20%20%20%20%20%20nums%5Bi%5D,%20nums%5Bj%5D%20%3D%20nums%5Bj%5D,%20nums%5Bi%5D%0A%20%20%20%20%23%20Swap%20the%20pivot%20into%20the%20boundary%20position%20between%20the%20two%20subarrays%0A%20%20%20%20nums%5Bi%5D,%20nums%5Bleft%5D%20%3D%20nums%5Bleft%5D,%20nums%5Bi%5D%0A%20%20%20%20return%20i%20%20%23%20Return%20the%20pivot%20index%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Median-of-three%20pivot%20optimization%0A%20%20%20%20nums%20%3D%20%5B2,%204,%201,%200,%203,%205%5D%0A%20%20%20%20partition%28nums,%200,%20len%28nums%29%20-%201%29%0A%20%20%20%20print%28%22After%20partitioning%20with%20a%20median-of-three%20pivot,%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1155-reducing-recursion-depth">11.5.5 &nbsp; Reducing recursion depth<a class="headerlink" href="#1155-reducing-recursion-depth" title="Permanent link">&para;</a></h2>
+<p><strong>Quick sort may use considerable space on certain inputs</strong>. For a fully sorted input, let the current subarray have length <span class="arithmatex">\(m\)</span>. Each partition produces a left subarray of length <span class="arithmatex">\(0\)</span> and a right subarray of length <span class="arithmatex">\(m - 1\)</span>. Each recursive call therefore reduces the problem size by only one element, producing a recursion tree of height <span class="arithmatex">\(n - 1\)</span> and requiring <span class="arithmatex">\(O(n)\)</span> space for stack frames.</p>
+<p>To prevent stack frames from accumulating, compare the two subarray lengths after each partition and <strong>recurse only on the shorter one</strong>. Since that subarray has length at most <span class="arithmatex">\(n / 2\)</span>, the recursion depth is at most <span class="arithmatex">\(\log n\)</span>, reducing the worst-case space complexity to <span class="arithmatex">\(O(\log n)\)</span>. The code is as follows:</p>
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 <div class="highlight"><span class="filename">quick_sort.py</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">quick_sort</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">left</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">right</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Quick sort (recursion depth optimization)"""</span>
-<a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="c1"># Terminate when subarray length is 1</span>
+<a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="c1"># Stop when the subarray has at most one element</span>
 <a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="k">while</span> <span class="n">left</span> <span class="o">&lt;</span> <span class="n">right</span><span class="p">:</span>
-<a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>        <span class="c1"># Sentinel partition operation</span>
+<a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>        <span class="c1"># Perform two-pointer partitioning</span>
 <a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a>        <span class="n">pivot</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">left</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a>        <span class="c1"># Perform quick sort on the shorter of the two subarrays</span>
+<a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a>        <span class="c1"># Apply quick sort to the shorter of the two subarrays</span>
 <a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a>        <span class="k">if</span> <span class="n">pivot</span> <span class="o">-</span> <span class="n">left</span> <span class="o">&lt;</span> <span class="n">right</span> <span class="o">-</span> <span class="n">pivot</span><span class="p">:</span>
 <a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">quick_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">left</span><span class="p">,</span> <span class="n">pivot</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>  <span class="c1"># Recursively sort the left subarray</span>
-<a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a>            <span class="n">left</span> <span class="o">=</span> <span class="n">pivot</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c1"># Remaining unsorted interval is [pivot + 1, right]</span>
+<a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a>            <span class="n">left</span> <span class="o">=</span> <span class="n">pivot</span> <span class="o">+</span> <span class="mi">1</span>  <span class="c1"># The remaining unsorted range is [pivot + 1, right]</span>
 <a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a>        <span class="k">else</span><span class="p">:</span>
 <a id="__codelineno-39-12" name="__codelineno-39-12" href="#__codelineno-39-12"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">quick_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">pivot</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">right</span><span class="p">)</span>  <span class="c1"># Recursively sort the right subarray</span>
-<a id="__codelineno-39-13" name="__codelineno-39-13" href="#__codelineno-39-13"></a>            <span class="n">right</span> <span class="o">=</span> <span class="n">pivot</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># Remaining unsorted interval is [left, pivot - 1]</span>
+<a id="__codelineno-39-13" name="__codelineno-39-13" href="#__codelineno-39-13"></a>            <span class="n">right</span> <span class="o">=</span> <span class="n">pivot</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># The remaining unsorted range is [left, pivot - 1]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Quick sort (recursion depth optimization) */</span>
 <a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">quickSort</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="c1">// Terminate when subarray length is 1</span>
+<a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="c1">// Stop when the subarray has at most one element</span>
 <a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">        </span><span class="c1">// Sentinel partition operation</span>
+<a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">        </span><span class="c1">// Perform two-pointer partitioning</span>
 <a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="w">        </span><span class="c1">// Perform quick sort on the shorter of the two subarrays</span>
+<a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="w">        </span><span class="c1">// Apply quick sort to the shorter of the two subarrays</span>
 <a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-9" name="__codelineno-40-9" href="#__codelineno-40-9"></a><span class="w">            </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively sort the left subarray</span>
-<a id="__codelineno-40-10" name="__codelineno-40-10" href="#__codelineno-40-10"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                 </span><span class="c1">// Remaining unsorted interval is [pivot + 1, right]</span>
+<a id="__codelineno-40-10" name="__codelineno-40-10" href="#__codelineno-40-10"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                 </span><span class="c1">// The remaining unsorted range is [pivot + 1, right]</span>
 <a id="__codelineno-40-11" name="__codelineno-40-11" href="#__codelineno-40-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-12" name="__codelineno-40-12" href="#__codelineno-40-12"></a><span class="w">            </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively sort the right subarray</span>
-<a id="__codelineno-40-13" name="__codelineno-40-13" href="#__codelineno-40-13"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                 </span><span class="c1">// Remaining unsorted interval is [left, pivot - 1]</span>
+<a id="__codelineno-40-13" name="__codelineno-40-13" href="#__codelineno-40-13"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">                 </span><span class="c1">// The remaining unsorted range is [left, pivot - 1]</span>
 <a id="__codelineno-40-14" name="__codelineno-40-14" href="#__codelineno-40-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-40-15" name="__codelineno-40-15" href="#__codelineno-40-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-40-16" name="__codelineno-40-16" href="#__codelineno-40-16"></a><span class="p">}</span>
@@ -5762,17 +5849,17 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Quick sort (recursion depth optimization) */</span>
 <a id="__codelineno-41-2" name="__codelineno-41-2" href="#__codelineno-41-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">quickSort</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="c1">// Terminate when subarray length is 1</span>
+<a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="c1">// Stop when the subarray has at most one element</span>
 <a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">        </span><span class="c1">// Sentinel partition operation</span>
+<a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">        </span><span class="c1">// Perform two-pointer partitioning</span>
 <a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="w">        </span><span class="c1">// Perform quick sort on the shorter of the two subarrays</span>
+<a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="w">        </span><span class="c1">// Apply quick sort to the shorter of the two subarrays</span>
 <a id="__codelineno-41-8" name="__codelineno-41-8" href="#__codelineno-41-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-9" name="__codelineno-41-9" href="#__codelineno-41-9"></a><span class="w">            </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively sort the left subarray</span>
-<a id="__codelineno-41-10" name="__codelineno-41-10" href="#__codelineno-41-10"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Remaining unsorted interval is [pivot + 1, right]</span>
+<a id="__codelineno-41-10" name="__codelineno-41-10" href="#__codelineno-41-10"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The remaining unsorted range is [pivot + 1, right]</span>
 <a id="__codelineno-41-11" name="__codelineno-41-11" href="#__codelineno-41-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-12" name="__codelineno-41-12" href="#__codelineno-41-12"></a><span class="w">            </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively sort the right subarray</span>
-<a id="__codelineno-41-13" name="__codelineno-41-13" href="#__codelineno-41-13"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Remaining unsorted interval is [left, pivot - 1]</span>
+<a id="__codelineno-41-13" name="__codelineno-41-13" href="#__codelineno-41-13"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The remaining unsorted range is [left, pivot - 1]</span>
 <a id="__codelineno-41-14" name="__codelineno-41-14" href="#__codelineno-41-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-41-15" name="__codelineno-41-15" href="#__codelineno-41-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-41-16" name="__codelineno-41-16" href="#__codelineno-41-16"></a><span class="p">}</span>
@@ -5781,36 +5868,36 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Quick sort (recursion depth optimization) */</span>
 <a id="__codelineno-42-2" name="__codelineno-42-2" href="#__codelineno-42-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">QuickSort</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="c1">// Terminate when subarray length is 1</span>
+<a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="c1">// Stop when the subarray has at most one element</span>
 <a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">        </span><span class="c1">// Sentinel partition operation</span>
+<a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">        </span><span class="c1">// Perform two-pointer partitioning</span>
 <a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="w">        </span><span class="c1">// Perform quick sort on the shorter of the two subarrays</span>
+<a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="w">        </span><span class="c1">// Apply quick sort to the shorter of the two subarrays</span>
 <a id="__codelineno-42-8" name="__codelineno-42-8" href="#__codelineno-42-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-9" name="__codelineno-42-9" href="#__codelineno-42-9"></a><span class="w">            </span><span class="n">QuickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w">  </span><span class="c1">// Recursively sort the left subarray</span>
-<a id="__codelineno-42-10" name="__codelineno-42-10" href="#__codelineno-42-10"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// Remaining unsorted interval is [pivot + 1, right]</span>
+<a id="__codelineno-42-10" name="__codelineno-42-10" href="#__codelineno-42-10"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w">  </span><span class="c1">// The remaining unsorted range is [pivot + 1, right]</span>
 <a id="__codelineno-42-11" name="__codelineno-42-11" href="#__codelineno-42-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-12" name="__codelineno-42-12" href="#__codelineno-42-12"></a><span class="w">            </span><span class="n">QuickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively sort the right subarray</span>
-<a id="__codelineno-42-13" name="__codelineno-42-13" href="#__codelineno-42-13"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Remaining unsorted interval is [left, pivot - 1]</span>
+<a id="__codelineno-42-13" name="__codelineno-42-13" href="#__codelineno-42-13"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The remaining unsorted range is [left, pivot - 1]</span>
 <a id="__codelineno-42-14" name="__codelineno-42-14" href="#__codelineno-42-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-42-15" name="__codelineno-42-15" href="#__codelineno-42-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-42-16" name="__codelineno-42-16" href="#__codelineno-42-16"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Quick sort (recursion depth optimization) */</span>
+<div class="highlight"><span class="filename">quick_sort.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Quick sort (recursion depth optimization)*/</span>
 <a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">quickSortTailCall</span><span class="p">)</span><span class="w"> </span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="c1">// Terminate when subarray length is 1</span>
+<a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="c1">// Stop when the subarray has at most one element</span>
 <a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w">        </span><span class="c1">// Sentinel partition operation</span>
+<a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w">        </span><span class="c1">// Perform two-pointer partitioning</span>
 <a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a><span class="w">        </span><span class="nx">pivot</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">partition</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span>
-<a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="w">        </span><span class="c1">// Perform quick sort on the shorter of the two subarrays</span>
+<a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="w">        </span><span class="c1">// Apply quick sort to the shorter of the two subarrays</span>
 <a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">pivot</span><span class="o">-</span><span class="nx">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">right</span><span class="o">-</span><span class="nx">pivot</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-9" name="__codelineno-43-9" href="#__codelineno-43-9"></a><span class="w">            </span><span class="nx">q</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="c1">// Recursively sort the left subarray</span>
-<a id="__codelineno-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></a><span class="w">            </span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w">                 </span><span class="c1">// Remaining unsorted interval is [pivot + 1, right]</span>
+<a id="__codelineno-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></a><span class="w">            </span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w">                 </span><span class="c1">// The remaining unsorted range is [pivot + 1, right]</span>
 <a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-12" name="__codelineno-43-12" href="#__codelineno-43-12"></a><span class="w">            </span><span class="nx">q</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="c1">// Recursively sort the right subarray</span>
-<a id="__codelineno-43-13" name="__codelineno-43-13" href="#__codelineno-43-13"></a><span class="w">            </span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w">                 </span><span class="c1">// Remaining unsorted interval is [left, pivot - 1]</span>
+<a id="__codelineno-43-13" name="__codelineno-43-13" href="#__codelineno-43-13"></a><span class="w">            </span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w">                 </span><span class="c1">// The remaining unsorted range is [left, pivot - 1]</span>
 <a id="__codelineno-43-14" name="__codelineno-43-14" href="#__codelineno-43-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-43-15" name="__codelineno-43-15" href="#__codelineno-43-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-43-16" name="__codelineno-43-16" href="#__codelineno-43-16"></a><span class="p">}</span>
@@ -5821,17 +5908,17 @@
 <a id="__codelineno-44-2" name="__codelineno-44-2" href="#__codelineno-44-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">quickSortTailCall</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kr">left</span>
 <a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">right</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kr">right</span>
-<a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">    </span><span class="c1">// Terminate when subarray length is 1</span>
+<a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">    </span><span class="c1">// Stop when the subarray has at most one element</span>
 <a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="kr">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="kr">right</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-44-7" name="__codelineno-44-7" href="#__codelineno-44-7"></a><span class="w">        </span><span class="c1">// Sentinel partition operation</span>
+<a id="__codelineno-44-7" name="__codelineno-44-7" href="#__codelineno-44-7"></a><span class="w">        </span><span class="c1">// Perform two-pointer partitioning</span>
 <a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">pivot</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="kr">right</span><span class="p">)</span>
-<a id="__codelineno-44-9" name="__codelineno-44-9" href="#__codelineno-44-9"></a><span class="w">        </span><span class="c1">// Perform quick sort on the shorter of the two subarrays</span>
+<a id="__codelineno-44-9" name="__codelineno-44-9" href="#__codelineno-44-9"></a><span class="w">        </span><span class="c1">// Apply quick sort to the shorter of the two subarrays</span>
 <a id="__codelineno-44-10" name="__codelineno-44-10" href="#__codelineno-44-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="kr">left</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="p">(</span><span class="kr">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-11" name="__codelineno-44-11" href="#__codelineno-44-11"></a><span class="w">            </span><span class="n">quickSortTailCall</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="c1">// Recursively sort the left subarray</span>
-<a id="__codelineno-44-12" name="__codelineno-44-12" href="#__codelineno-44-12"></a><span class="w">            </span><span class="kr">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Remaining unsorted interval is [pivot + 1, right]</span>
+<a id="__codelineno-44-12" name="__codelineno-44-12" href="#__codelineno-44-12"></a><span class="w">            </span><span class="kr">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// The remaining unsorted range is [pivot + 1, right]</span>
 <a id="__codelineno-44-13" name="__codelineno-44-13" href="#__codelineno-44-13"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-14" name="__codelineno-44-14" href="#__codelineno-44-14"></a><span class="w">            </span><span class="n">quickSortTailCall</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kr">left</span><span class="p">:</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="kr">right</span><span class="p">:</span><span class="w"> </span><span class="kr">right</span><span class="p">)</span><span class="w"> </span><span class="c1">// Recursively sort the right subarray</span>
-<a id="__codelineno-44-15" name="__codelineno-44-15" href="#__codelineno-44-15"></a><span class="w">            </span><span class="kr">right</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Remaining unsorted interval is [left, pivot - 1]</span>
+<a id="__codelineno-44-15" name="__codelineno-44-15" href="#__codelineno-44-15"></a><span class="w">            </span><span class="kr">right</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// The remaining unsorted range is [left, pivot - 1]</span>
 <a id="__codelineno-44-16" name="__codelineno-44-16" href="#__codelineno-44-16"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-44-17" name="__codelineno-44-17" href="#__codelineno-44-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-44-18" name="__codelineno-44-18" href="#__codelineno-44-18"></a><span class="p">}</span>
@@ -5840,17 +5927,17 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Quick sort (recursion depth optimization) */</span>
 <a id="__codelineno-45-2" name="__codelineno-45-2" href="#__codelineno-45-2"></a><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="c1">// Terminate when subarray length is 1</span>
+<a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="c1">// Stop when the subarray has at most one element</span>
 <a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">        </span><span class="c1">// Sentinel partition operation</span>
+<a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">        </span><span class="c1">// Perform two-pointer partitioning</span>
 <a id="__codelineno-45-6" name="__codelineno-45-6" href="#__codelineno-45-6"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">partition</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span>
-<a id="__codelineno-45-7" name="__codelineno-45-7" href="#__codelineno-45-7"></a><span class="w">        </span><span class="c1">// Perform quick sort on the shorter of the two subarrays</span>
+<a id="__codelineno-45-7" name="__codelineno-45-7" href="#__codelineno-45-7"></a><span class="w">        </span><span class="c1">// Apply quick sort to the shorter of the two subarrays</span>
 <a id="__codelineno-45-8" name="__codelineno-45-8" href="#__codelineno-45-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-9" name="__codelineno-45-9" href="#__codelineno-45-9"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively sort the left subarray</span>
-<a id="__codelineno-45-10" name="__codelineno-45-10" href="#__codelineno-45-10"></a><span class="w">            </span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Remaining unsorted interval is [pivot + 1, right]</span>
+<a id="__codelineno-45-10" name="__codelineno-45-10" href="#__codelineno-45-10"></a><span class="w">            </span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The remaining unsorted range is [pivot + 1, right]</span>
 <a id="__codelineno-45-11" name="__codelineno-45-11" href="#__codelineno-45-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-12" name="__codelineno-45-12" href="#__codelineno-45-12"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively sort the right subarray</span>
-<a id="__codelineno-45-13" name="__codelineno-45-13" href="#__codelineno-45-13"></a><span class="w">            </span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Remaining unsorted interval is [left, pivot - 1]</span>
+<a id="__codelineno-45-13" name="__codelineno-45-13" href="#__codelineno-45-13"></a><span class="w">            </span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The remaining unsorted range is [left, pivot - 1]</span>
 <a id="__codelineno-45-14" name="__codelineno-45-14" href="#__codelineno-45-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-45-15" name="__codelineno-45-15" href="#__codelineno-45-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-45-16" name="__codelineno-45-16" href="#__codelineno-45-16"></a><span class="p">}</span>
@@ -5859,17 +5946,17 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Quick sort (recursion depth optimization) */</span>
 <a id="__codelineno-46-2" name="__codelineno-46-2" href="#__codelineno-46-2"></a><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">left</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="c1">// Terminate when subarray length is 1</span>
+<a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="c1">// Stop when the subarray has zero or one element</span>
 <a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">        </span><span class="c1">// Sentinel partition operation</span>
+<a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">        </span><span class="c1">// Perform two-pointer partitioning</span>
 <a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">partition</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span>
-<a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="w">        </span><span class="c1">// Perform quick sort on the shorter of the two subarrays</span>
+<a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="w">        </span><span class="c1">// Apply quick sort to the shorter of the two subarrays</span>
 <a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nx">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively sort the left subarray</span>
-<a id="__codelineno-46-10" name="__codelineno-46-10" href="#__codelineno-46-10"></a><span class="w">            </span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Remaining unsorted interval is [pivot + 1, right]</span>
+<a id="__codelineno-46-10" name="__codelineno-46-10" href="#__codelineno-46-10"></a><span class="w">            </span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The remaining unsorted range is [pivot + 1, right]</span>
 <a id="__codelineno-46-11" name="__codelineno-46-11" href="#__codelineno-46-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">quickSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively sort the right subarray</span>
-<a id="__codelineno-46-13" name="__codelineno-46-13" href="#__codelineno-46-13"></a><span class="w">            </span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Remaining unsorted interval is [left, pivot - 1]</span>
+<a id="__codelineno-46-13" name="__codelineno-46-13" href="#__codelineno-46-13"></a><span class="w">            </span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The remaining unsorted range is [left, pivot - 1]</span>
 <a id="__codelineno-46-14" name="__codelineno-46-14" href="#__codelineno-46-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-46-15" name="__codelineno-46-15" href="#__codelineno-46-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-46-16" name="__codelineno-46-16" href="#__codelineno-46-16"></a><span class="p">}</span>
@@ -5878,17 +5965,17 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Quick sort (recursion depth optimization) */</span>
 <a id="__codelineno-47-2" name="__codelineno-47-2" href="#__codelineno-47-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">quickSort</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="c1">// Terminate when subarray length is 1</span>
+<a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="c1">// Stop when the subarray has at most one element</span>
 <a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w">    </span><span class="c1">// Sentinel partition operation</span>
+<a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w">    </span><span class="c1">// Perform two-pointer partitioning</span>
 <a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-47-7" name="__codelineno-47-7" href="#__codelineno-47-7"></a><span class="w">    </span><span class="c1">// Perform quick sort on the shorter of the two subarrays</span>
+<a id="__codelineno-47-7" name="__codelineno-47-7" href="#__codelineno-47-7"></a><span class="w">    </span><span class="c1">// Apply quick sort to the shorter of the two subarrays</span>
 <a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-9" name="__codelineno-47-9" href="#__codelineno-47-9"></a><span class="w">      </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively sort the left subarray</span>
-<a id="__codelineno-47-10" name="__codelineno-47-10" href="#__codelineno-47-10"></a><span class="w">      </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Remaining unsorted interval is [pivot + 1, right]</span>
+<a id="__codelineno-47-10" name="__codelineno-47-10" href="#__codelineno-47-10"></a><span class="w">      </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The remaining unsorted range is [pivot + 1, right]</span>
 <a id="__codelineno-47-11" name="__codelineno-47-11" href="#__codelineno-47-11"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-12" name="__codelineno-47-12" href="#__codelineno-47-12"></a><span class="w">      </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively sort the right subarray</span>
-<a id="__codelineno-47-13" name="__codelineno-47-13" href="#__codelineno-47-13"></a><span class="w">      </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Remaining unsorted interval is [left, pivot - 1]</span>
+<a id="__codelineno-47-13" name="__codelineno-47-13" href="#__codelineno-47-13"></a><span class="w">      </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The remaining unsorted range is [left, pivot - 1]</span>
 <a id="__codelineno-47-14" name="__codelineno-47-14" href="#__codelineno-47-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-47-15" name="__codelineno-47-15" href="#__codelineno-47-15"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-47-16" name="__codelineno-47-16" href="#__codelineno-47-16"></a><span class="p">}</span>
@@ -5897,17 +5984,17 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Quick sort (recursion depth optimization) */</span>
 <a id="__codelineno-48-2" name="__codelineno-48-2" href="#__codelineno-48-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">quick_sort</span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">left</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">right</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="c1">// Terminate when subarray length is 1</span>
+<a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="c1">// Stop when the subarray has at most one element</span>
 <a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w">        </span><span class="c1">// Sentinel partition operation</span>
+<a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w">        </span><span class="c1">// Perform two-pointer partitioning</span>
 <a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">;</span>
-<a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="w">        </span><span class="c1">// Perform quick sort on the shorter of the two subarrays</span>
+<a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="w">        </span><span class="c1">// Apply quick sort to the shorter of the two subarrays</span>
 <a id="__codelineno-48-8" name="__codelineno-48-8" href="#__codelineno-48-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a><span class="w">            </span><span class="bp">Self</span><span class="p">::</span><span class="n">quick_sort</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively sort the left subarray</span>
-<a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Remaining unsorted interval is [pivot + 1, right]</span>
+<a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The remaining unsorted range is [pivot + 1, right]</span>
 <a id="__codelineno-48-11" name="__codelineno-48-11" href="#__codelineno-48-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-12" name="__codelineno-48-12" href="#__codelineno-48-12"></a><span class="w">            </span><span class="bp">Self</span><span class="p">::</span><span class="n">quick_sort</span><span class="p">(</span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="p">);</span><span class="w"> </span><span class="c1">// Recursively sort the right subarray</span>
-<a id="__codelineno-48-13" name="__codelineno-48-13" href="#__codelineno-48-13"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Remaining unsorted interval is [left, pivot - 1]</span>
+<a id="__codelineno-48-13" name="__codelineno-48-13" href="#__codelineno-48-13"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// The remaining unsorted range is [left, pivot - 1]</span>
 <a id="__codelineno-48-14" name="__codelineno-48-14" href="#__codelineno-48-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-48-15" name="__codelineno-48-15" href="#__codelineno-48-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-48-16" name="__codelineno-48-16" href="#__codelineno-48-16"></a><span class="p">}</span>
@@ -5916,20 +6003,20 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Quick sort (recursion depth optimization) */</span>
 <a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">quickSortTailCall</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="c1">// Terminate when subarray length is 1</span>
+<a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="c1">// Stop when the subarray has at most one element</span>
 <a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">        </span><span class="c1">// Sentinel partition operation</span>
+<a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">        </span><span class="c1">// Perform two-pointer partitioning</span>
 <a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">        </span><span class="c1">// Perform quick sort on the shorter of the two subarrays</span>
+<a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">        </span><span class="c1">// Apply quick sort to the shorter of the two subarrays</span>
 <a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-9" name="__codelineno-49-9" href="#__codelineno-49-9"></a><span class="w">            </span><span class="c1">// Recursively sort the left subarray</span>
 <a id="__codelineno-49-10" name="__codelineno-49-10" href="#__codelineno-49-10"></a><span class="w">            </span><span class="n">quickSortTailCall</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-49-11" name="__codelineno-49-11" href="#__codelineno-49-11"></a><span class="w">            </span><span class="c1">// Remaining unsorted interval is [pivot + 1, right]</span>
+<a id="__codelineno-49-11" name="__codelineno-49-11" href="#__codelineno-49-11"></a><span class="w">            </span><span class="c1">// The remaining unsorted range is [pivot + 1, right]</span>
 <a id="__codelineno-49-12" name="__codelineno-49-12" href="#__codelineno-49-12"></a><span class="w">            </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-49-13" name="__codelineno-49-13" href="#__codelineno-49-13"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-14" name="__codelineno-49-14" href="#__codelineno-49-14"></a><span class="w">            </span><span class="c1">// Recursively sort the right subarray</span>
 <a id="__codelineno-49-15" name="__codelineno-49-15" href="#__codelineno-49-15"></a><span class="w">            </span><span class="n">quickSortTailCall</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-49-16" name="__codelineno-49-16" href="#__codelineno-49-16"></a><span class="w">            </span><span class="c1">// Remaining unsorted interval is [left, pivot - 1]</span>
+<a id="__codelineno-49-16" name="__codelineno-49-16" href="#__codelineno-49-16"></a><span class="w">            </span><span class="c1">// The remaining unsorted range is [left, pivot - 1]</span>
 <a id="__codelineno-49-17" name="__codelineno-49-17" href="#__codelineno-49-17"></a><span class="w">            </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-49-18" name="__codelineno-49-18" href="#__codelineno-49-18"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-49-19" name="__codelineno-49-19" href="#__codelineno-49-19"></a><span class="w">    </span><span class="p">}</span>
@@ -5939,38 +6026,38 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">quick_sort.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Quick sort (recursion depth optimization) */</span>
 <a id="__codelineno-50-2" name="__codelineno-50-2" href="#__codelineno-50-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">quickSortTailCall</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="c1">// Terminate when subarray length is 1</span>
+<a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="c1">// Stop when the subarray has at most one element</span>
 <a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left</span>
 <a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right</span>
 <a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="w">        </span><span class="c1">// Sentinel partition operation</span>
+<a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="w">        </span><span class="c1">// Perform two-pointer partitioning</span>
 <a id="__codelineno-50-8" name="__codelineno-50-8" href="#__codelineno-50-8"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">r</span><span class="p">)</span>
-<a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="w">        </span><span class="c1">// Perform quick sort on the shorter of the two subarrays</span>
+<a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="w">        </span><span class="c1">// Apply quick sort to the shorter of the two subarrays</span>
 <a id="__codelineno-50-10" name="__codelineno-50-10" href="#__codelineno-50-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">l</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">pivot</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-11" name="__codelineno-50-11" href="#__codelineno-50-11"></a><span class="w">            </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="c1">// Recursively sort the left subarray</span>
-<a id="__codelineno-50-12" name="__codelineno-50-12" href="#__codelineno-50-12"></a><span class="w">            </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Remaining unsorted interval is [pivot + 1, right]</span>
+<a id="__codelineno-50-12" name="__codelineno-50-12" href="#__codelineno-50-12"></a><span class="w">            </span><span class="n">l</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// The remaining unsorted range is [pivot + 1, r]</span>
 <a id="__codelineno-50-13" name="__codelineno-50-13" href="#__codelineno-50-13"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-14" name="__codelineno-50-14" href="#__codelineno-50-14"></a><span class="w">            </span><span class="n">quickSort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">r</span><span class="p">)</span><span class="w"> </span><span class="c1">// Recursively sort the right subarray</span>
-<a id="__codelineno-50-15" name="__codelineno-50-15" href="#__codelineno-50-15"></a><span class="w">            </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Remaining unsorted interval is [left, pivot - 1]</span>
+<a id="__codelineno-50-15" name="__codelineno-50-15" href="#__codelineno-50-15"></a><span class="w">            </span><span class="n">r</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// The remaining unsorted range is [l, pivot - 1]</span>
 <a id="__codelineno-50-16" name="__codelineno-50-16" href="#__codelineno-50-16"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-50-17" name="__codelineno-50-17" href="#__codelineno-50-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-50-18" name="__codelineno-50-18" href="#__codelineno-50-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">quick_sort.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Quick sort (recursion depth optimization) ###</span>
+<div class="highlight"><span class="filename">quick_sort.rb</span><pre><span></span><code><a id="__codelineno-51-1" name="__codelineno-51-1" href="#__codelineno-51-1"></a><span class="c1">### Quick sort (recursion depth optimization)###</span>
 <a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">quick_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">  </span><span class="c1"># Recurse when subarray length is not 1</span>
+<a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">  </span><span class="c1"># Continue while the subarray contains more than one element</span>
 <a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span>
-<a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">    </span><span class="c1"># Sentinel partition</span>
+<a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">    </span><span class="c1"># Two-pointer partitioning</span>
 <a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">    </span><span class="n">pivot</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">partition</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="w">    </span><span class="c1"># Perform quick sort on the shorter of the two subarrays</span>
+<a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="w">    </span><span class="c1"># Apply quick sort to the shorter of the two subarrays</span>
 <a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">pivot</span>
 <a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w">      </span><span class="n">quick_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a><span class="w">      </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Remaining unsorted interval is [pivot + 1, right]</span>
+<a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a><span class="w">      </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># The remaining unsorted range is [pivot + 1, right]</span>
 <a id="__codelineno-51-11" name="__codelineno-51-11" href="#__codelineno-51-11"></a><span class="w">    </span><span class="k">else</span>
 <a id="__codelineno-51-12" name="__codelineno-51-12" href="#__codelineno-51-12"></a><span class="w">      </span><span class="n">quick_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">right</span><span class="p">)</span>
-<a id="__codelineno-51-13" name="__codelineno-51-13" href="#__codelineno-51-13"></a><span class="w">      </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Remaining unsorted interval is [left, pivot - 1]</span>
+<a id="__codelineno-51-13" name="__codelineno-51-13" href="#__codelineno-51-13"></a><span class="w">      </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pivot</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># The remaining unsorted range is [left, pivot - 1]</span>
 <a id="__codelineno-51-14" name="__codelineno-51-14" href="#__codelineno-51-14"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-51-15" name="__codelineno-51-15" href="#__codelineno-51-15"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-51-16" name="__codelineno-51-16" href="#__codelineno-51-16"></a><span class="k">end</span>
@@ -5978,6 +6065,11 @@
 </div>
 </div>
 </div>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20partition%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20right%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Two-pointer%20partitioning%22%22%22%0A%20%20%20%20%23%20Partition%20using%20the%20pivot%20stored%20at%20nums%5Bleft%5D%0A%20%20%20%20i,%20j%20%3D%20left,%20right%0A%20%20%20%20while%20i%20%3C%20j%3A%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bj%5D%20%3E%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20-%3D%201%20%20%23%20Scan%20from%20right%20to%20left%20for%20the%20first%20element%20smaller%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bi%5D%20%3C%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%20%20%23%20Scan%20from%20left%20to%20right%20for%20the%20first%20element%20larger%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20elements%0A%20%20%20%20%20%20%20%20nums%5Bi%5D,%20nums%5Bj%5D%20%3D%20nums%5Bj%5D,%20nums%5Bi%5D%0A%20%20%20%20%23%20Swap%20the%20pivot%20into%20the%20boundary%20position%20between%20the%20two%20subarrays%0A%20%20%20%20nums%5Bi%5D,%20nums%5Bleft%5D%20%3D%20nums%5Bleft%5D,%20nums%5Bi%5D%0A%20%20%20%20return%20i%20%20%23%20Return%20the%20pivot%20index%0A%0Adef%20quick_sort%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20right%3A%20int%29%3A%0A%20%20%20%20%22%22%22Quick%20sort%20%28recursion%20depth%20optimization%29%22%22%22%0A%20%20%20%20%23%20Stop%20when%20the%20subarray%20has%20at%20most%20one%20element%0A%20%20%20%20while%20left%20%3C%20right%3A%0A%20%20%20%20%20%20%20%20%23%20Perform%20two-pointer%20partitioning%0A%20%20%20%20%20%20%20%20pivot%20%3D%20partition%28nums,%20left,%20right%29%0A%20%20%20%20%20%20%20%20%23%20Apply%20quick%20sort%20to%20the%20shorter%20of%20the%20two%20subarrays%0A%20%20%20%20%20%20%20%20if%20pivot%20-%20left%20%3C%20right%20-%20pivot%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20quick_sort%28nums,%20left,%20pivot%20-%201%29%20%20%23%20Recursively%20sort%20the%20left%20subarray%0A%20%20%20%20%20%20%20%20%20%20%20%20left%20%3D%20pivot%20%2B%201%20%20%23%20The%20remaining%20unsorted%20range%20is%20%5Bpivot%20%2B%201,%20right%5D%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20quick_sort%28nums,%20pivot%20%2B%201,%20right%29%20%20%23%20Recursively%20sort%20the%20right%20subarray%0A%20%20%20%20%20%20%20%20%20%20%20%20right%20%3D%20pivot%20-%201%20%20%23%20The%20remaining%20unsorted%20range%20is%20%5Bleft,%20pivot%20-%201%5D%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Quick%20sort%20%28recursion%20depth%20optimization%29%0A%20%20%20%20nums%20%3D%20%5B2,%204,%201,%200,%203,%205%5D%0A%20%20%20%20quick_sort%28nums,%200,%20len%28nums%29%20-%201%29%0A%20%20%20%20print%28%22After%20quick%20sort%20with%20recursion%20depth%20optimization,%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20partition%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20right%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Two-pointer%20partitioning%22%22%22%0A%20%20%20%20%23%20Partition%20using%20the%20pivot%20stored%20at%20nums%5Bleft%5D%0A%20%20%20%20i,%20j%20%3D%20left,%20right%0A%20%20%20%20while%20i%20%3C%20j%3A%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bj%5D%20%3E%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20-%3D%201%20%20%23%20Scan%20from%20right%20to%20left%20for%20the%20first%20element%20smaller%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20while%20i%20%3C%20j%20and%20nums%5Bi%5D%20%3C%3D%20nums%5Bleft%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20i%20%2B%3D%201%20%20%23%20Scan%20from%20left%20to%20right%20for%20the%20first%20element%20larger%20than%20the%20pivot%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20elements%0A%20%20%20%20%20%20%20%20nums%5Bi%5D,%20nums%5Bj%5D%20%3D%20nums%5Bj%5D,%20nums%5Bi%5D%0A%20%20%20%20%23%20Swap%20the%20pivot%20into%20the%20boundary%20position%20between%20the%20two%20subarrays%0A%20%20%20%20nums%5Bi%5D,%20nums%5Bleft%5D%20%3D%20nums%5Bleft%5D,%20nums%5Bi%5D%0A%20%20%20%20return%20i%20%20%23%20Return%20the%20pivot%20index%0A%0Adef%20quick_sort%28nums%3A%20list%5Bint%5D,%20left%3A%20int,%20right%3A%20int%29%3A%0A%20%20%20%20%22%22%22Quick%20sort%20%28recursion%20depth%20optimization%29%22%22%22%0A%20%20%20%20%23%20Stop%20when%20the%20subarray%20has%20at%20most%20one%20element%0A%20%20%20%20while%20left%20%3C%20right%3A%0A%20%20%20%20%20%20%20%20%23%20Perform%20two-pointer%20partitioning%0A%20%20%20%20%20%20%20%20pivot%20%3D%20partition%28nums,%20left,%20right%29%0A%20%20%20%20%20%20%20%20%23%20Apply%20quick%20sort%20to%20the%20shorter%20of%20the%20two%20subarrays%0A%20%20%20%20%20%20%20%20if%20pivot%20-%20left%20%3C%20right%20-%20pivot%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20quick_sort%28nums,%20left,%20pivot%20-%201%29%20%20%23%20Recursively%20sort%20the%20left%20subarray%0A%20%20%20%20%20%20%20%20%20%20%20%20left%20%3D%20pivot%20%2B%201%20%20%23%20The%20remaining%20unsorted%20range%20is%20%5Bpivot%20%2B%201,%20right%5D%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20quick_sort%28nums,%20pivot%20%2B%201,%20right%29%20%20%23%20Recursively%20sort%20the%20right%20subarray%0A%20%20%20%20%20%20%20%20%20%20%20%20right%20%3D%20pivot%20-%201%20%20%23%20The%20remaining%20unsorted%20range%20is%20%5Bleft,%20pivot%20-%201%5D%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Quick%20sort%20%28recursion%20depth%20optimization%29%0A%20%20%20%20nums%20%3D%20%5B2,%204,%201,%200,%203,%205%5D%0A%20%20%20%20quick_sort%28nums,%200,%20len%28nums%29%20-%201%29%0A%20%20%20%20print%28%22After%20quick%20sort%20with%20recursion%20depth%20optimization,%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
 
 <!-- Source file information -->
 
@@ -6000,7 +6092,7 @@ aria-label="Footer"
   <a
     href="../insertion_sort/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 11.4 Insertion Sort"
+    aria-label="Previous: 11.4   Insertion sort"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -6012,7 +6104,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        11.4 Insertion Sort
+        11.4 &nbsp; Insertion sort
       </div>
     </div>
   </a>
@@ -6024,7 +6116,7 @@ aria-label="Footer"
   <a
     href="../merge_sort/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 11.6 Merge Sort"
+    aria-label="Next: 11.6   Merge sort"
     rel="next"
   >
     <div class="md-footer__title">
@@ -6032,7 +6124,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        11.6 Merge Sort
+        11.6 &nbsp; Merge sort
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -6049,7 +6141,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_sorting/radix_sort/index.html b/en/chapter_sorting/radix_sort/index.html
index 02b2ba335..86deae91e 100644
--- a/en/chapter_sorting/radix_sort/index.html
+++ b/en/chapter_sorting/radix_sort/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>11.10 Radix Sort - Hello Algo</title>
+        <title>11.10 Radix sort - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -119,7 +119,7 @@
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     </svg>
-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              11.10 &nbsp; Radix Sort
+              11.10 &nbsp; Radix sort
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Dark mode"  type="radio" name="__palette" id="__palette_0">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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@@ -177,9 +177,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
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@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
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@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2880,7 +2880,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2901,7 +2901,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2922,7 +2922,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2976,7 +2976,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3003,7 +3003,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3030,7 +3030,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3057,7 +3057,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3084,7 +3084,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3111,7 +3111,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3138,7 +3138,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3174,7 +3174,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3191,7 +3191,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3217,10 +3217,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#11101-algorithm-flow" class="md-nav__link">
+  <a href="#11101-algorithm-steps" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.10.1 &nbsp; Algorithm Flow
+        11.10.1 &nbsp; Algorithm steps
       </span>
     </span>
   </a>
@@ -3228,10 +3228,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#11102-algorithm-characteristics" class="md-nav__link">
+  <a href="#11102-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.10.2 &nbsp; Algorithm Characteristics
+        11.10.2 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -3261,7 +3261,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3288,7 +3288,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3375,7 +3375,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3396,7 +3396,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3423,7 +3423,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3450,7 +3450,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3477,7 +3477,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3504,7 +3504,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3531,7 +3531,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3597,7 +3597,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3618,7 +3618,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3639,7 +3639,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3666,7 +3666,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3693,7 +3693,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3720,7 +3720,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3747,7 +3747,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3774,7 +3774,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3844,7 +3844,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3865,7 +3865,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3886,7 +3886,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3913,7 +3913,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3940,7 +3940,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3967,7 +3967,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -3994,7 +3994,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4021,7 +4021,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4048,7 +4048,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4075,7 +4075,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4141,7 +4141,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4162,7 +4162,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4183,7 +4183,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4210,7 +4210,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4237,7 +4237,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4264,7 +4264,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4291,7 +4291,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4378,7 +4378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -4420,7 +4420,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4558,6 +4558,78 @@
 
 
 
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+  <a href="#11101-algorithm-steps" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.10.1 &nbsp; Algorithm Flow
+        11.10.1 &nbsp; Algorithm steps
       </span>
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   </a>
@@ -4610,10 +4682,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#11102-algorithm-characteristics" class="md-nav__link">
+  <a href="#11102-algorithm-properties" class="md-nav__link">
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       <span class="md-typeset">
-        11.10.2 &nbsp; Algorithm Characteristics
+        11.10.2 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -4657,201 +4729,201 @@
 
 
 <!-- Page content -->
-<h1 id="1110-radix-sort">11.10 &nbsp; Radix Sort<a class="headerlink" href="#1110-radix-sort" title="Permanent link">&para;</a></h1>
-<p>The previous section introduced counting sort, which is suitable when the number of items <span class="arithmatex">\(n\)</span> is large but the value range <span class="arithmatex">\(m\)</span> is small. Suppose we need to sort <span class="arithmatex">\(n = 10^6\)</span> student IDs, each of which is an 8-digit number. Then the value range <span class="arithmatex">\(m = 10^8\)</span> is very large. Using counting sort would require a large amount of memory, whereas radix sort avoids this problem.</p>
-<p><u>Radix sort</u> is based on the same core idea as counting sort: it also sorts by counting occurrences. Building on this, radix sort exploits the positional relationship among digits and sorts them one digit at a time to obtain the final result.</p>
-<h2 id="11101-algorithm-flow">11.10.1 &nbsp; Algorithm Flow<a class="headerlink" href="#11101-algorithm-flow" title="Permanent link">&para;</a></h2>
-<p>Taking student ID data as an example, assume the lowest digit is the <span class="arithmatex">\(1\)</span>st digit and the highest digit is the <span class="arithmatex">\(8\)</span>th digit. The flow of radix sort is shown in Figure 11-18.</p>
+<h1 id="1110-radix-sort">11.10 &nbsp; Radix sort<a class="headerlink" href="#1110-radix-sort" title="Permanent link">&para;</a></h1>
+<p>The previous section introduced counting sort, which suits a large number of elements <span class="arithmatex">\(n\)</span> with a small value range <span class="arithmatex">\(m\)</span>. Suppose we need to sort <span class="arithmatex">\(n = 10^6\)</span> student IDs, each an <span class="arithmatex">\(8\)</span>-digit number. The value range <span class="arithmatex">\(m = 10^8\)</span> is very large, so counting sort would require substantial memory. Radix sort avoids this problem.</p>
+<p><u>Radix sort</u> shares counting sort's core idea of sorting by counting occurrences. It also uses the relative significance of digit positions, sorting one digit at a time to obtain the final order.</p>
+<h2 id="11101-algorithm-steps">11.10.1 &nbsp; Algorithm steps<a class="headerlink" href="#11101-algorithm-steps" title="Permanent link">&para;</a></h2>
+<p>For the student IDs, number the least significant digit as <span class="arithmatex">\(1\)</span> and the most significant as <span class="arithmatex">\(8\)</span>. Radix sort proceeds as shown below.</p>
 <ol>
-<li>Initialize the digit <span class="arithmatex">\(k = 1\)</span>.</li>
-<li>Perform "counting sort" on the <span class="arithmatex">\(k\)</span>th digit of the student IDs. After completion, the data will be sorted from smallest to largest according to the <span class="arithmatex">\(k\)</span>th digit.</li>
-<li>Increase <span class="arithmatex">\(k\)</span> by <span class="arithmatex">\(1\)</span>, then return to step <code>2.</code> and continue iterating until all digits are sorted, at which point the process ends.</li>
+<li>Initialize the digit position <span class="arithmatex">\(k = 1\)</span>.</li>
+<li>Apply counting sort to the <span class="arithmatex">\(k\)</span>th digit of each student ID. The data is now sorted in ascending order by its <span class="arithmatex">\(k\)</span>th digit.</li>
+<li>Increase the digit position <span class="arithmatex">\(k\)</span> by <span class="arithmatex">\(1\)</span> and return to step <code>2.</code>. Continue until all digit positions have been processed.</li>
 </ol>
-<p><img alt="Radix sort algorithm flow" class="animation-figure" src="../radix_sort.assets/radix_sort_overview.png" /></p>
-<p align="center"> Figure 11-18 &nbsp; Radix sort algorithm flow </p>
+<p><img alt="The radix sort process" class="animation-figure" src="../radix_sort.assets/radix_sort_overview.png" /></p>
+<p align="center"> Figure 11-18 &nbsp; The radix sort process </p>
 
-<p>Next, let us look at the code. For a number <span class="arithmatex">\(x\)</span> in base <span class="arithmatex">\(d\)</span>, its <span class="arithmatex">\(k\)</span>th digit <span class="arithmatex">\(x_k\)</span> can be obtained with the following formula:</p>
+<p>Now consider the implementation. For a number <span class="arithmatex">\(x\)</span> in base <span class="arithmatex">\(d\)</span>, its <span class="arithmatex">\(k\)</span>th digit <span class="arithmatex">\(x_k\)</span> can be obtained with the following formula:</p>
 <div class="arithmatex">\[
 x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 \]</div>
-<p>Here, <span class="arithmatex">\(\lfloor a \rfloor\)</span> denotes rounding the floating-point number <span class="arithmatex">\(a\)</span> down, and <span class="arithmatex">\(\bmod \: d\)</span> denotes taking the remainder modulo <span class="arithmatex">\(d\)</span>. For student ID data, <span class="arithmatex">\(d = 10\)</span> and <span class="arithmatex">\(k \in [1, 8]\)</span>.</p>
-<p>Additionally, we need to slightly modify the counting sort code to make it sort based on the <span class="arithmatex">\(k\)</span>th digit of the number:</p>
+<p>Here, <span class="arithmatex">\(\lfloor a \rfloor\)</span> rounds the floating-point value <span class="arithmatex">\(a\)</span> down, and <span class="arithmatex">\(\bmod \: d\)</span> takes the remainder modulo <span class="arithmatex">\(d\)</span>. For the student IDs, <span class="arithmatex">\(d = 10\)</span> and <span class="arithmatex">\(k \in [1, 8]\)</span>.</p>
+<p>We also need to modify counting sort slightly so that it sorts by the <span class="arithmatex">\(k\)</span>th digit:</p>
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 <div class="highlight"><span class="filename">radix_sort.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">digit</span><span class="p">(</span><span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">exp</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Get the k-th digit of element num, where exp = 10^(k-1)"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Passing exp instead of k can avoid repeated expensive exponentiation here</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Get the k-th digit of num, where exp = 10^(k-1)"""</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Passing exp instead of k avoids repeatedly performing expensive exponentiation here</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="k">return</span> <span class="p">(</span><span class="n">num</span> <span class="o">//</span> <span class="n">exp</span><span class="p">)</span> <span class="o">%</span> <span class="mi">10</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="k">def</span><span class="w"> </span><span class="nf">counting_sort_digit</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">],</span> <span class="n">exp</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="w">    </span><span class="sd">"""Counting sort (based on nums k-th digit)"""</span>
-<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="c1"># Decimal digit range is 0~9, therefore need a bucket array of length 10</span>
+<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="w">    </span><span class="sd">"""Counting sort (sort nums by the k-th digit)"""</span>
+<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="c1"># Decimal digits range from 0 to 9, so the bucket array needs length 10</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>    <span class="n">counter</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="mi">10</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
-<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="c1"># Count the occurrence of digits 0~9</span>
+<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>    <span class="c1"># Count the occurrences of each digit from 0 to 9</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
-<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="n">d</span> <span class="o">=</span> <span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">exp</span><span class="p">)</span>  <span class="c1"># Get the k-th digit of nums[i], noted as d</span>
-<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>        <span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>  <span class="c1"># Count the occurrence of digit d</span>
-<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="c1"># Calculate prefix sum, converting "occurrence count" into "array index"</span>
+<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="n">d</span> <span class="o">=</span> <span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">exp</span><span class="p">)</span>  <span class="c1"># Get the k-th digit of nums[i] and store it in d</span>
+<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>        <span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>  <span class="c1"># Count the occurrences of digit d</span>
+<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="c1"># Compute prefix sums to convert occurrence counts into array positions</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">):</span>
 <a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>        <span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="n">counter</span><span class="p">[</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>
-<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>    <span class="c1"># Traverse in reverse, based on bucket statistics, place each element into res</span>
+<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a>    <span class="c1"># Traverse in reverse and place each element into res according to the bucket counts</span>
 <a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a>    <span class="n">res</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">n</span>
 <a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">):</span>
 <a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a>        <span class="n">d</span> <span class="o">=</span> <span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">exp</span><span class="p">)</span>
-<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>        <span class="n">j</span> <span class="o">=</span> <span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># Get the index j for d in the array</span>
+<a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a>        <span class="n">j</span> <span class="o">=</span> <span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">-</span> <span class="mi">1</span>  <span class="c1"># Get the destination index j for digit d in the array</span>
 <a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a>        <span class="n">res</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>  <span class="c1"># Place the current element at index j</span>
-<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a>        <span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>  <span class="c1"># Decrease the count of d by 1</span>
-<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a>    <span class="c1"># Use result to overwrite the original array nums</span>
+<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a>        <span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span> <span class="o">-=</span> <span class="mi">1</span>  <span class="c1"># Decrease the count for d by 1</span>
+<a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a>    <span class="c1"># Overwrite the original array nums with the result</span>
 <a id="__codelineno-0-26" name="__codelineno-0-26" href="#__codelineno-0-26"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-0-27" name="__codelineno-0-27" href="#__codelineno-0-27"></a>        <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-0-28" name="__codelineno-0-28" href="#__codelineno-0-28"></a>
 <a id="__codelineno-0-29" name="__codelineno-0-29" href="#__codelineno-0-29"></a><span class="k">def</span><span class="w"> </span><span class="nf">radix_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]):</span>
 <a id="__codelineno-0-30" name="__codelineno-0-30" href="#__codelineno-0-30"></a><span class="w">    </span><span class="sd">"""Radix sort"""</span>
-<a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a>    <span class="c1"># Get the maximum element of the array, used to determine the maximum number of digits</span>
+<a id="__codelineno-0-31" name="__codelineno-0-31" href="#__codelineno-0-31"></a>    <span class="c1"># Get the largest array element to determine the maximum number of digits</span>
 <a id="__codelineno-0-32" name="__codelineno-0-32" href="#__codelineno-0-32"></a>    <span class="n">m</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
-<a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a>    <span class="c1"># Traverse from the lowest to the highest digit</span>
+<a id="__codelineno-0-33" name="__codelineno-0-33" href="#__codelineno-0-33"></a>    <span class="c1"># Process digits from least significant to most significant</span>
 <a id="__codelineno-0-34" name="__codelineno-0-34" href="#__codelineno-0-34"></a>    <span class="n">exp</span> <span class="o">=</span> <span class="mi">1</span>
 <a id="__codelineno-0-35" name="__codelineno-0-35" href="#__codelineno-0-35"></a>    <span class="k">while</span> <span class="n">exp</span> <span class="o">&lt;=</span> <span class="n">m</span><span class="p">:</span>
-<a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a>        <span class="c1"># Perform counting sort on the k-th digit of array elements</span>
+<a id="__codelineno-0-36" name="__codelineno-0-36" href="#__codelineno-0-36"></a>        <span class="c1"># Apply counting sort to the k-th digit of the array elements</span>
 <a id="__codelineno-0-37" name="__codelineno-0-37" href="#__codelineno-0-37"></a>        <span class="c1"># k = 1 -&gt; exp = 1</span>
 <a id="__codelineno-0-38" name="__codelineno-0-38" href="#__codelineno-0-38"></a>        <span class="c1"># k = 2 -&gt; exp = 10</span>
-<a id="__codelineno-0-39" name="__codelineno-0-39" href="#__codelineno-0-39"></a>        <span class="c1"># i.e., exp = 10^(k-1)</span>
+<a id="__codelineno-0-39" name="__codelineno-0-39" href="#__codelineno-0-39"></a>        <span class="c1"># That is, exp = 10^(k-1)</span>
 <a id="__codelineno-0-40" name="__codelineno-0-40" href="#__codelineno-0-40"></a>        <span class="n">counting_sort_digit</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
 <a id="__codelineno-0-41" name="__codelineno-0-41" href="#__codelineno-0-41"></a>        <span class="n">exp</span> <span class="o">*=</span> <span class="mi">10</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">radix_sort.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Get the k-th digit of element num, where exp = 10^(k-1) */</span>
+<div class="highlight"><span class="filename">radix_sort.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Get the k-th digit of num, where exp = 10^(k-1) */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">digit</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k can avoid repeated expensive exponentiation here</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k avoids repeatedly performing expensive exponentiation here</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="p">}</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a>
-<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="cm">/* Counting sort (based on nums k-th digit) */</span>
+<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="cm">/* Counting sort (sort nums by the k-th digit) */</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="kt">void</span><span class="w"> </span><span class="nf">countingSortDigit</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="c1">// Decimal digit range is 0~9, therefore need a bucket array of length 10</span>
+<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">    </span><span class="c1">// Decimal digits range from 0 to 9, so the bucket array needs length 10</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">counter</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="c1">// Count the occurrence of digits 0~9</span>
+<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">    </span><span class="c1">// Count the occurrences of each digit from 0 to 9</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i], noted as d</span>
-<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">++</span><span class="p">;</span><span class="w">                </span><span class="c1">// Count the occurrence of digit d</span>
+<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i] and store it in d</span>
+<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">++</span><span class="p">;</span><span class="w">                </span><span class="c1">// Count the occurrences of digit d</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">    </span><span class="c1">// Calculate prefix sum, converting "occurrence count" into "array index"</span>
+<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">    </span><span class="c1">// Compute prefix sums to convert occurrence counts into array positions</span>
 <a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse, based on bucket statistics, place each element into res</span>
+<a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse and place each element into res according to the bucket counts</span>
 <a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-24" name="__codelineno-1-24" href="#__codelineno-1-24"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span>
-<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the index j for d in the array</span>
+<a id="__codelineno-1-25" name="__codelineno-1-25" href="#__codelineno-1-25"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the destination index j for digit d in the array</span>
 <a id="__codelineno-1-26" name="__codelineno-1-26" href="#__codelineno-1-26"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span><span class="w">       </span><span class="c1">// Place the current element at index j</span>
-<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w">           </span><span class="c1">// Decrease the count of d by 1</span>
+<a id="__codelineno-1-27" name="__codelineno-1-27" href="#__codelineno-1-27"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w">           </span><span class="c1">// Decrease the count for d by 1</span>
 <a id="__codelineno-1-28" name="__codelineno-1-28" href="#__codelineno-1-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="w">    </span><span class="c1">// Use result to overwrite the original array nums</span>
+<a id="__codelineno-1-29" name="__codelineno-1-29" href="#__codelineno-1-29"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result</span>
 <a id="__codelineno-1-30" name="__codelineno-1-30" href="#__codelineno-1-30"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span>
 <a id="__codelineno-1-31" name="__codelineno-1-31" href="#__codelineno-1-31"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-1-32" name="__codelineno-1-32" href="#__codelineno-1-32"></a><span class="p">}</span>
 <a id="__codelineno-1-33" name="__codelineno-1-33" href="#__codelineno-1-33"></a>
 <a id="__codelineno-1-34" name="__codelineno-1-34" href="#__codelineno-1-34"></a><span class="cm">/* Radix sort */</span>
 <a id="__codelineno-1-35" name="__codelineno-1-35" href="#__codelineno-1-35"></a><span class="kt">void</span><span class="w"> </span><span class="nf">radixSort</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-36" name="__codelineno-1-36" href="#__codelineno-1-36"></a><span class="w">    </span><span class="c1">// Get the maximum element of the array, used to determine the maximum number of digits</span>
+<a id="__codelineno-1-36" name="__codelineno-1-36" href="#__codelineno-1-36"></a><span class="w">    </span><span class="c1">// Get the largest array element to determine the maximum number of digits</span>
 <a id="__codelineno-1-37" name="__codelineno-1-37" href="#__codelineno-1-37"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">*</span><span class="n">max_element</span><span class="p">(</span><span class="n">nums</span><span class="p">.</span><span class="n">begin</span><span class="p">(),</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">end</span><span class="p">());</span>
-<a id="__codelineno-1-38" name="__codelineno-1-38" href="#__codelineno-1-38"></a><span class="w">    </span><span class="c1">// Traverse from the lowest to the highest digit</span>
+<a id="__codelineno-1-38" name="__codelineno-1-38" href="#__codelineno-1-38"></a><span class="w">    </span><span class="c1">// Process digits from least significant to most significant</span>
 <a id="__codelineno-1-39" name="__codelineno-1-39" href="#__codelineno-1-39"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">10</span><span class="p">)</span>
-<a id="__codelineno-1-40" name="__codelineno-1-40" href="#__codelineno-1-40"></a><span class="w">        </span><span class="c1">// Perform counting sort on the k-th digit of array elements</span>
+<a id="__codelineno-1-40" name="__codelineno-1-40" href="#__codelineno-1-40"></a><span class="w">        </span><span class="c1">// Apply counting sort to the k-th digit of the array elements</span>
 <a id="__codelineno-1-41" name="__codelineno-1-41" href="#__codelineno-1-41"></a><span class="w">        </span><span class="c1">// k = 1 -&gt; exp = 1</span>
 <a id="__codelineno-1-42" name="__codelineno-1-42" href="#__codelineno-1-42"></a><span class="w">        </span><span class="c1">// k = 2 -&gt; exp = 10</span>
-<a id="__codelineno-1-43" name="__codelineno-1-43" href="#__codelineno-1-43"></a><span class="w">        </span><span class="c1">// i.e., exp = 10^(k-1)</span>
+<a id="__codelineno-1-43" name="__codelineno-1-43" href="#__codelineno-1-43"></a><span class="w">        </span><span class="c1">// That is, exp = 10^(k-1)</span>
 <a id="__codelineno-1-44" name="__codelineno-1-44" href="#__codelineno-1-44"></a><span class="w">        </span><span class="n">countingSortDigit</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span>
 <a id="__codelineno-1-45" name="__codelineno-1-45" href="#__codelineno-1-45"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">radix_sort.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Get the k-th digit of element num, where exp = 10^(k-1) */</span>
+<div class="highlight"><span class="filename">radix_sort.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Get the k-th digit of num, where exp = 10^(k-1) */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">digit</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k can avoid repeated expensive exponentiation here</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k avoids repeatedly performing expensive exponentiation here</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="p">}</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a>
-<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="cm">/* Counting sort (based on nums k-th digit) */</span>
+<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="cm">/* Counting sort (sort nums by the k-th digit) */</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="kt">void</span><span class="w"> </span><span class="nf">countingSortDigit</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="c1">// Decimal digit range is 0~9, therefore need a bucket array of length 10</span>
+<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">    </span><span class="c1">// Decimal digits range from 0 to 9, so the bucket array needs length 10</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="mi">10</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
-<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="c1">// Count the occurrence of digits 0~9</span>
+<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">    </span><span class="c1">// Count the occurrences of each digit from 0 to 9</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i], noted as d</span>
-<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]++</span><span class="p">;</span><span class="w">                </span><span class="c1">// Count the occurrence of digit d</span>
+<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i] and store it in d</span>
+<a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]++</span><span class="p">;</span><span class="w">                </span><span class="c1">// Count the occurrences of digit d</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">    </span><span class="c1">// Calculate prefix sum, converting "occurrence count" into "array index"</span>
+<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">    </span><span class="c1">// Compute prefix sums to convert occurrence counts into array positions</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse, based on bucket statistics, place each element into res</span>
+<a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse and place each element into res according to the bucket counts</span>
 <a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="w">    </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">n</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-24" name="__codelineno-2-24" href="#__codelineno-2-24"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span>
-<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the index j for d in the array</span>
+<a id="__codelineno-2-25" name="__codelineno-2-25" href="#__codelineno-2-25"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the destination index j for digit d in the array</span>
 <a id="__codelineno-2-26" name="__codelineno-2-26" href="#__codelineno-2-26"></a><span class="w">        </span><span class="n">res</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span><span class="w">       </span><span class="c1">// Place the current element at index j</span>
-<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]--</span><span class="p">;</span><span class="w">           </span><span class="c1">// Decrease the count of d by 1</span>
+<a id="__codelineno-2-27" name="__codelineno-2-27" href="#__codelineno-2-27"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]--</span><span class="p">;</span><span class="w">           </span><span class="c1">// Decrease the count for d by 1</span>
 <a id="__codelineno-2-28" name="__codelineno-2-28" href="#__codelineno-2-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="w">    </span><span class="c1">// Use result to overwrite the original array nums</span>
+<a id="__codelineno-2-29" name="__codelineno-2-29" href="#__codelineno-2-29"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result</span>
 <a id="__codelineno-2-30" name="__codelineno-2-30" href="#__codelineno-2-30"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span>
 <a id="__codelineno-2-31" name="__codelineno-2-31" href="#__codelineno-2-31"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">res</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-32" name="__codelineno-2-32" href="#__codelineno-2-32"></a><span class="p">}</span>
 <a id="__codelineno-2-33" name="__codelineno-2-33" href="#__codelineno-2-33"></a>
 <a id="__codelineno-2-34" name="__codelineno-2-34" href="#__codelineno-2-34"></a><span class="cm">/* Radix sort */</span>
 <a id="__codelineno-2-35" name="__codelineno-2-35" href="#__codelineno-2-35"></a><span class="kt">void</span><span class="w"> </span><span class="nf">radixSort</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a><span class="w">    </span><span class="c1">// Get the maximum element of the array, used to determine the maximum number of digits</span>
+<a id="__codelineno-2-36" name="__codelineno-2-36" href="#__codelineno-2-36"></a><span class="w">    </span><span class="c1">// Get the largest array element to determine the maximum number of digits</span>
 <a id="__codelineno-2-37" name="__codelineno-2-37" href="#__codelineno-2-37"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Integer</span><span class="p">.</span><span class="na">MIN_VALUE</span><span class="p">;</span>
 <a id="__codelineno-2-38" name="__codelineno-2-38" href="#__codelineno-2-38"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span>
 <a id="__codelineno-2-39" name="__codelineno-2-39" href="#__codelineno-2-39"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">m</span><span class="p">)</span>
 <a id="__codelineno-2-40" name="__codelineno-2-40" href="#__codelineno-2-40"></a><span class="w">            </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
-<a id="__codelineno-2-41" name="__codelineno-2-41" href="#__codelineno-2-41"></a><span class="w">    </span><span class="c1">// Traverse from the lowest to the highest digit</span>
+<a id="__codelineno-2-41" name="__codelineno-2-41" href="#__codelineno-2-41"></a><span class="w">    </span><span class="c1">// Process digits from least significant to most significant</span>
 <a id="__codelineno-2-42" name="__codelineno-2-42" href="#__codelineno-2-42"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">10</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-43" name="__codelineno-2-43" href="#__codelineno-2-43"></a><span class="w">        </span><span class="c1">// Perform counting sort on the k-th digit of array elements</span>
+<a id="__codelineno-2-43" name="__codelineno-2-43" href="#__codelineno-2-43"></a><span class="w">        </span><span class="c1">// Apply counting sort to the k-th digit of the array elements</span>
 <a id="__codelineno-2-44" name="__codelineno-2-44" href="#__codelineno-2-44"></a><span class="w">        </span><span class="c1">// k = 1 -&gt; exp = 1</span>
 <a id="__codelineno-2-45" name="__codelineno-2-45" href="#__codelineno-2-45"></a><span class="w">        </span><span class="c1">// k = 2 -&gt; exp = 10</span>
-<a id="__codelineno-2-46" name="__codelineno-2-46" href="#__codelineno-2-46"></a><span class="w">        </span><span class="c1">// i.e., exp = 10^(k-1)</span>
+<a id="__codelineno-2-46" name="__codelineno-2-46" href="#__codelineno-2-46"></a><span class="w">        </span><span class="c1">// That is, exp = 10^(k-1)</span>
 <a id="__codelineno-2-47" name="__codelineno-2-47" href="#__codelineno-2-47"></a><span class="w">        </span><span class="n">countingSortDigit</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span>
 <a id="__codelineno-2-48" name="__codelineno-2-48" href="#__codelineno-2-48"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-49" name="__codelineno-2-49" href="#__codelineno-2-49"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">radix_sort.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Get the k-th digit of element num, where exp = 10^(k-1) */</span>
+<div class="highlight"><span class="filename">radix_sort.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Get the k-th digit of num, where exp = 10^(k-1) */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Digit</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k can avoid repeated expensive exponentiation here</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k avoids repeatedly performing expensive exponentiation here</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="p">}</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a>
-<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="cm">/* Counting sort (based on nums k-th digit) */</span>
+<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="cm">/* Counting sort (sort nums by the k-th digit) */</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="k">void</span><span class="w"> </span><span class="nf">CountingSortDigit</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="c1">// Decimal digit range is 0~9, therefore need a bucket array of length 10</span>
+<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">    </span><span class="c1">// Decimal digits range from 0 to 9, so the bucket array needs length 10</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="mi">10</span><span class="p">];</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
-<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="c1">// Count the occurrence of digits 0~9</span>
+<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">    </span><span class="c1">// Count the occurrences of each digit from 0 to 9</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i], noted as d</span>
-<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">++</span><span class="p">;</span><span class="w">                </span><span class="c1">// Count the occurrence of digit d</span>
+<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i] and store it in d</span>
+<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">++</span><span class="p">;</span><span class="w">                </span><span class="c1">// Count the occurrences of digit d</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">    </span><span class="c1">// Calculate prefix sum, converting "occurrence count" into "array index"</span>
+<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">    </span><span class="c1">// Compute prefix sums to convert occurrence counts into array positions</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse, based on bucket statistics, place each element into res</span>
+<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse and place each element into res according to the bucket counts</span>
 <a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">n</span><span class="p">];</span>
 <a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span>
-<a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the index j for d in the array</span>
+<a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the destination index j for digit d in the array</span>
 <a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span><span class="w">       </span><span class="c1">// Place the current element at index j</span>
-<a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w">           </span><span class="c1">// Decrease the count of d by 1</span>
+<a id="__codelineno-3-27" name="__codelineno-3-27" href="#__codelineno-3-27"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w">           </span><span class="c1">// Decrease the count for d by 1</span>
 <a id="__codelineno-3-28" name="__codelineno-3-28" href="#__codelineno-3-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="w">    </span><span class="c1">// Use result to overwrite the original array nums</span>
+<a id="__codelineno-3-29" name="__codelineno-3-29" href="#__codelineno-3-29"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result</span>
 <a id="__codelineno-3-30" name="__codelineno-3-30" href="#__codelineno-3-30"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-31" name="__codelineno-3-31" href="#__codelineno-3-31"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-3-32" name="__codelineno-3-32" href="#__codelineno-3-32"></a><span class="w">    </span><span class="p">}</span>
@@ -4859,52 +4931,52 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-3-34" name="__codelineno-3-34" href="#__codelineno-3-34"></a>
 <a id="__codelineno-3-35" name="__codelineno-3-35" href="#__codelineno-3-35"></a><span class="cm">/* Radix sort */</span>
 <a id="__codelineno-3-36" name="__codelineno-3-36" href="#__codelineno-3-36"></a><span class="k">void</span><span class="w"> </span><span class="nf">RadixSort</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-37" name="__codelineno-3-37" href="#__codelineno-3-37"></a><span class="w">    </span><span class="c1">// Get the maximum element of the array, used to determine the maximum number of digits</span>
+<a id="__codelineno-3-37" name="__codelineno-3-37" href="#__codelineno-3-37"></a><span class="w">    </span><span class="c1">// Get the largest array element to determine the maximum number of digits</span>
 <a id="__codelineno-3-38" name="__codelineno-3-38" href="#__codelineno-3-38"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">int</span><span class="p">.</span><span class="n">MinValue</span><span class="p">;</span>
 <a id="__codelineno-3-39" name="__codelineno-3-39" href="#__codelineno-3-39"></a><span class="w">    </span><span class="k">foreach</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-40" name="__codelineno-3-40" href="#__codelineno-3-40"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-3-41" name="__codelineno-3-41" href="#__codelineno-3-41"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-42" name="__codelineno-3-42" href="#__codelineno-3-42"></a><span class="w">    </span><span class="c1">// Traverse from the lowest to the highest digit</span>
+<a id="__codelineno-3-42" name="__codelineno-3-42" href="#__codelineno-3-42"></a><span class="w">    </span><span class="c1">// Process digits from least significant to most significant</span>
 <a id="__codelineno-3-43" name="__codelineno-3-43" href="#__codelineno-3-43"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">10</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-44" name="__codelineno-3-44" href="#__codelineno-3-44"></a><span class="w">        </span><span class="c1">// Perform counting sort on the k-th digit of array elements</span>
+<a id="__codelineno-3-44" name="__codelineno-3-44" href="#__codelineno-3-44"></a><span class="w">        </span><span class="c1">// Apply counting sort to the k-th digit of the array elements</span>
 <a id="__codelineno-3-45" name="__codelineno-3-45" href="#__codelineno-3-45"></a><span class="w">        </span><span class="c1">// k = 1 -&gt; exp = 1</span>
 <a id="__codelineno-3-46" name="__codelineno-3-46" href="#__codelineno-3-46"></a><span class="w">        </span><span class="c1">// k = 2 -&gt; exp = 10</span>
-<a id="__codelineno-3-47" name="__codelineno-3-47" href="#__codelineno-3-47"></a><span class="w">        </span><span class="c1">// i.e., exp = 10^(k-1)</span>
+<a id="__codelineno-3-47" name="__codelineno-3-47" href="#__codelineno-3-47"></a><span class="w">        </span><span class="c1">// That is, exp = 10^(k-1)</span>
 <a id="__codelineno-3-48" name="__codelineno-3-48" href="#__codelineno-3-48"></a><span class="w">        </span><span class="n">CountingSortDigit</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span>
 <a id="__codelineno-3-49" name="__codelineno-3-49" href="#__codelineno-3-49"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-50" name="__codelineno-3-50" href="#__codelineno-3-50"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">radix_sort.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Get the k-th digit of element num, where exp = 10^(k-1) */</span>
+<div class="highlight"><span class="filename">radix_sort.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Get the k-th digit of num, where exp = 10^(k-1) */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">digit</span><span class="p">(</span><span class="nx">num</span><span class="p">,</span><span class="w"> </span><span class="nx">exp</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k can avoid repeated expensive exponentiation here</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k avoids repeatedly performing expensive exponentiation here</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nx">exp</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">10</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="p">}</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="cm">/* Counting sort (based on nums k-th digit) */</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="cm">/* Counting sort (sort nums by the k-th digit) */</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="kd">func</span><span class="w"> </span><span class="nx">countingSortDigit</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">exp</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// Decimal digit range is 0~9, therefore need a bucket array of length 10</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="c1">// Decimal digits range from 0 to 9, so the bucket array needs length 10</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">    </span><span class="nx">counter</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">10</span><span class="p">)</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span>
-<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="c1">// Count the occurrence of digits 0~9</span>
+<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">    </span><span class="c1">// Count the occurrences of each digit from 0 to 9</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="nx">d</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">digit</span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">exp</span><span class="p">)</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i], noted as d</span>
-<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="o">++</span><span class="w">             </span><span class="c1">// Count the occurrence of digit d</span>
+<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="nx">d</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">digit</span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">exp</span><span class="p">)</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i] and store it in d</span>
+<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="o">++</span><span class="w">             </span><span class="c1">// Count the occurrences of digit d</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="c1">// Calculate prefix sum, converting "occurrence count" into "array index"</span>
+<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="c1">// Compute prefix sums to convert occurrence counts into array positions</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">counter</span><span class="p">[</span><span class="nx">i</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse, based on bucket statistics, place each element into res</span>
+<a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse and place each element into res according to the bucket counts</span>
 <a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">n</span><span class="p">)</span>
 <a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="w">        </span><span class="nx">d</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">digit</span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">exp</span><span class="p">)</span>
-<a id="__codelineno-4-25" name="__codelineno-4-25" href="#__codelineno-4-25"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Get the index j for d in the array</span>
+<a id="__codelineno-4-25" name="__codelineno-4-25" href="#__codelineno-4-25"></a><span class="w">        </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Get the destination index j for digit d in the array</span>
 <a id="__codelineno-4-26" name="__codelineno-4-26" href="#__codelineno-4-26"></a><span class="w">        </span><span class="nx">res</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w">    </span><span class="c1">// Place the current element at index j</span>
-<a id="__codelineno-4-27" name="__codelineno-4-27" href="#__codelineno-4-27"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="o">--</span><span class="w">        </span><span class="c1">// Decrease the count of d by 1</span>
+<a id="__codelineno-4-27" name="__codelineno-4-27" href="#__codelineno-4-27"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="o">--</span><span class="w">        </span><span class="c1">// Decrease the count for d by 1</span>
 <a id="__codelineno-4-28" name="__codelineno-4-28" href="#__codelineno-4-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-29" name="__codelineno-4-29" href="#__codelineno-4-29"></a><span class="w">    </span><span class="c1">// Use result to overwrite the original array nums</span>
+<a id="__codelineno-4-29" name="__codelineno-4-29" href="#__codelineno-4-29"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result</span>
 <a id="__codelineno-4-30" name="__codelineno-4-30" href="#__codelineno-4-30"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-31" name="__codelineno-4-31" href="#__codelineno-4-31"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">res</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
 <a id="__codelineno-4-32" name="__codelineno-4-32" href="#__codelineno-4-32"></a><span class="w">    </span><span class="p">}</span>
@@ -4912,53 +4984,53 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-4-34" name="__codelineno-4-34" href="#__codelineno-4-34"></a>
 <a id="__codelineno-4-35" name="__codelineno-4-35" href="#__codelineno-4-35"></a><span class="cm">/* Radix sort */</span>
 <a id="__codelineno-4-36" name="__codelineno-4-36" href="#__codelineno-4-36"></a><span class="kd">func</span><span class="w"> </span><span class="nx">radixSort</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-37" name="__codelineno-4-37" href="#__codelineno-4-37"></a><span class="w">    </span><span class="c1">// Get the maximum element of the array, used to determine the maximum number of digits</span>
+<a id="__codelineno-4-37" name="__codelineno-4-37" href="#__codelineno-4-37"></a><span class="w">    </span><span class="c1">// Get the largest array element to determine the maximum number of digits</span>
 <a id="__codelineno-4-38" name="__codelineno-4-38" href="#__codelineno-4-38"></a><span class="w">    </span><span class="nx">max</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">math</span><span class="p">.</span><span class="nx">MinInt</span>
 <a id="__codelineno-4-39" name="__codelineno-4-39" href="#__codelineno-4-39"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">_</span><span class="p">,</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="k">range</span><span class="w"> </span><span class="nx">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-40" name="__codelineno-4-40" href="#__codelineno-4-40"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">max</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-41" name="__codelineno-4-41" href="#__codelineno-4-41"></a><span class="w">            </span><span class="nx">max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span>
 <a id="__codelineno-4-42" name="__codelineno-4-42" href="#__codelineno-4-42"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-4-43" name="__codelineno-4-43" href="#__codelineno-4-43"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-44" name="__codelineno-4-44" href="#__codelineno-4-44"></a><span class="w">    </span><span class="c1">// Traverse from the lowest to the highest digit</span>
+<a id="__codelineno-4-44" name="__codelineno-4-44" href="#__codelineno-4-44"></a><span class="w">    </span><span class="c1">// Process digits from least significant to most significant</span>
 <a id="__codelineno-4-45" name="__codelineno-4-45" href="#__codelineno-4-45"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">exp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">max</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">exp</span><span class="p">;</span><span class="w"> </span><span class="nx">exp</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">10</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-46" name="__codelineno-4-46" href="#__codelineno-4-46"></a><span class="w">        </span><span class="c1">// Perform counting sort on the k-th digit of array elements</span>
+<a id="__codelineno-4-46" name="__codelineno-4-46" href="#__codelineno-4-46"></a><span class="w">        </span><span class="c1">// Apply counting sort to the k-th digit of the array elements</span>
 <a id="__codelineno-4-47" name="__codelineno-4-47" href="#__codelineno-4-47"></a><span class="w">        </span><span class="c1">// k = 1 -&gt; exp = 1</span>
 <a id="__codelineno-4-48" name="__codelineno-4-48" href="#__codelineno-4-48"></a><span class="w">        </span><span class="c1">// k = 2 -&gt; exp = 10</span>
-<a id="__codelineno-4-49" name="__codelineno-4-49" href="#__codelineno-4-49"></a><span class="w">        </span><span class="c1">// i.e., exp = 10^(k-1)</span>
+<a id="__codelineno-4-49" name="__codelineno-4-49" href="#__codelineno-4-49"></a><span class="w">        </span><span class="c1">// That is, exp = 10^(k-1)</span>
 <a id="__codelineno-4-50" name="__codelineno-4-50" href="#__codelineno-4-50"></a><span class="w">        </span><span class="nx">countingSortDigit</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">exp</span><span class="p">)</span>
 <a id="__codelineno-4-51" name="__codelineno-4-51" href="#__codelineno-4-51"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-52" name="__codelineno-4-52" href="#__codelineno-4-52"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">radix_sort.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Get the k-th digit of element num, where exp = 10^(k-1) */</span>
+<div class="highlight"><span class="filename">radix_sort.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Get the k-th digit of num, where exp = 10^(k-1) */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">digit</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k can avoid repeated expensive exponentiation here</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k avoids repeatedly performing expensive exponentiation here</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">10</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="p">}</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a>
-<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="cm">/* Counting sort (based on nums k-th digit) */</span>
+<a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="cm">/* Counting sort (sort nums by the k-th digit) */</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="kd">func</span><span class="w"> </span><span class="nf">countingSortDigit</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="c1">// Decimal digit range is 0~9, therefore need a bucket array of length 10</span>
+<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">    </span><span class="c1">// Decimal digits range from 0 to 9, so the bucket array needs length 10</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">counter</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="mi">10</span><span class="p">)</span>
-<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">    </span><span class="c1">// Count the occurrence of digits 0~9</span>
+<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">    </span><span class="c1">// Count the occurrences of each digit from 0 to 9</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">d</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">:</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i], noted as d</span>
-<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Count the occurrence of digit d</span>
+<a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">d</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">:</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i] and store it in d</span>
+<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Count the occurrences of digit d</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">    </span><span class="c1">// Calculate prefix sum, converting "occurrence count" into "array index"</span>
+<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">    </span><span class="c1">// Compute prefix sums to convert occurrence counts into array positions</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="mi">10</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w">    </span><span class="c1">// Traverse in reverse, based on bucket statistics, place each element into res</span>
+<a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w">    </span><span class="c1">// Traverse in reverse and place each element into res according to the bucket counts</span>
 <a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">count</span><span class="p">)</span>
 <a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="n">reversed</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">d</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">:</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span>
-<a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Get the index j for d in the array</span>
+<a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Get the destination index j for digit d in the array</span>
 <a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="c1">// Place the current element at index j</span>
-<a id="__codelineno-5-26" name="__codelineno-5-26" href="#__codelineno-5-26"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Decrease the count of d by 1</span>
+<a id="__codelineno-5-26" name="__codelineno-5-26" href="#__codelineno-5-26"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Decrease the count for d by 1</span>
 <a id="__codelineno-5-27" name="__codelineno-5-27" href="#__codelineno-5-27"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="w">    </span><span class="c1">// Use result to overwrite the original array nums</span>
+<a id="__codelineno-5-28" name="__codelineno-5-28" href="#__codelineno-5-28"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result</span>
 <a id="__codelineno-5-29" name="__codelineno-5-29" href="#__codelineno-5-29"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-30" name="__codelineno-5-30" href="#__codelineno-5-30"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-5-31" name="__codelineno-5-31" href="#__codelineno-5-31"></a><span class="w">    </span><span class="p">}</span>
@@ -4966,54 +5038,54 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-5-33" name="__codelineno-5-33" href="#__codelineno-5-33"></a>
 <a id="__codelineno-5-34" name="__codelineno-5-34" href="#__codelineno-5-34"></a><span class="cm">/* Radix sort */</span>
 <a id="__codelineno-5-35" name="__codelineno-5-35" href="#__codelineno-5-35"></a><span class="kd">func</span><span class="w"> </span><span class="nf">radixSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-36" name="__codelineno-5-36" href="#__codelineno-5-36"></a><span class="w">    </span><span class="c1">// Get the maximum element of the array, used to determine the maximum number of digits</span>
+<a id="__codelineno-5-36" name="__codelineno-5-36" href="#__codelineno-5-36"></a><span class="w">    </span><span class="c1">// Get the largest array element to determine the maximum number of digits</span>
 <a id="__codelineno-5-37" name="__codelineno-5-37" href="#__codelineno-5-37"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Int</span><span class="p">.</span><span class="bp">min</span>
 <a id="__codelineno-5-38" name="__codelineno-5-38" href="#__codelineno-5-38"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-39" name="__codelineno-5-39" href="#__codelineno-5-39"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-40" name="__codelineno-5-40" href="#__codelineno-5-40"></a><span class="w">            </span><span class="n">m</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-5-41" name="__codelineno-5-41" href="#__codelineno-5-41"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-42" name="__codelineno-5-42" href="#__codelineno-5-42"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-43" name="__codelineno-5-43" href="#__codelineno-5-43"></a><span class="w">    </span><span class="c1">// Traverse from the lowest to the highest digit</span>
+<a id="__codelineno-5-43" name="__codelineno-5-43" href="#__codelineno-5-43"></a><span class="w">    </span><span class="c1">// Process digits from least significant to most significant</span>
 <a id="__codelineno-5-44" name="__codelineno-5-44" href="#__codelineno-5-44"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">sequence</span><span class="p">(</span><span class="bp">first</span><span class="p">:</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">next</span><span class="p">:</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="p">(</span><span class="nv">$0</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">10</span><span class="p">)</span><span class="w"> </span><span class="p">?</span><span class="w"> </span><span class="nv">$0</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">10</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">})</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-45" name="__codelineno-5-45" href="#__codelineno-5-45"></a><span class="w">        </span><span class="c1">// Perform counting sort on the k-th digit of array elements</span>
+<a id="__codelineno-5-45" name="__codelineno-5-45" href="#__codelineno-5-45"></a><span class="w">        </span><span class="c1">// Apply counting sort to the k-th digit of the array elements</span>
 <a id="__codelineno-5-46" name="__codelineno-5-46" href="#__codelineno-5-46"></a><span class="w">        </span><span class="c1">// k = 1 -&gt; exp = 1</span>
 <a id="__codelineno-5-47" name="__codelineno-5-47" href="#__codelineno-5-47"></a><span class="w">        </span><span class="c1">// k = 2 -&gt; exp = 10</span>
-<a id="__codelineno-5-48" name="__codelineno-5-48" href="#__codelineno-5-48"></a><span class="w">        </span><span class="c1">// i.e., exp = 10^(k-1)</span>
+<a id="__codelineno-5-48" name="__codelineno-5-48" href="#__codelineno-5-48"></a><span class="w">        </span><span class="c1">// That is, exp = 10^(k-1)</span>
 <a id="__codelineno-5-49" name="__codelineno-5-49" href="#__codelineno-5-49"></a><span class="w">        </span><span class="n">countingSortDigit</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="p">&amp;</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">:</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span>
 <a id="__codelineno-5-50" name="__codelineno-5-50" href="#__codelineno-5-50"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-51" name="__codelineno-5-51" href="#__codelineno-5-51"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">radix_sort.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Get the k-th digit of element num, where exp = 10^(k-1) */</span>
+<div class="highlight"><span class="filename">radix_sort.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Get the k-th digit of num, where exp = 10^(k-1) */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">digit</span><span class="p">(</span><span class="nx">num</span><span class="p">,</span><span class="w"> </span><span class="nx">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k can avoid repeated expensive exponentiation here</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k avoids repeatedly performing expensive exponentiation here</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nx">exp</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mf">10</span><span class="p">;</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="p">}</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a>
-<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="cm">/* Counting sort (based on nums k-th digit) */</span>
+<a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="cm">/* Counting sort (sort nums by the k-th digit) */</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="kd">function</span><span class="w"> </span><span class="nx">countingSortDigit</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="c1">// Decimal digit range is 0~9, therefore need a bucket array of length 10</span>
+<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">    </span><span class="c1">// Decimal digits range from 0 to 9, so the bucket array needs length 10</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">10</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="c1">// Count the occurrence of digits 0~9</span>
+<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="c1">// Count the occurrences of each digit from 0 to 9</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">digit</span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">exp</span><span class="p">);</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i], noted as d</span>
-<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Count the occurrence of digit d</span>
+<a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">digit</span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">exp</span><span class="p">);</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i] and store it in d</span>
+<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Count the occurrences of digit d</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">    </span><span class="c1">// Calculate prefix sum, converting "occurrence count" into "array index"</span>
+<a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="w">    </span><span class="c1">// Compute prefix sums to convert occurrence counts into array positions</span>
 <a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">10</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">counter</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse, based on bucket statistics, place each element into res</span>
+<a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse and place each element into res according to the bucket counts</span>
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">digit</span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">exp</span><span class="p">);</span>
-<a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the index j for d in the array</span>
+<a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the destination index j for digit d in the array</span>
 <a id="__codelineno-6-26" name="__codelineno-6-26" href="#__codelineno-6-26"></a><span class="w">        </span><span class="nx">res</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span><span class="w"> </span><span class="c1">// Place the current element at index j</span>
-<a id="__codelineno-6-27" name="__codelineno-6-27" href="#__codelineno-6-27"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrease the count of d by 1</span>
+<a id="__codelineno-6-27" name="__codelineno-6-27" href="#__codelineno-6-27"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrease the count for d by 1</span>
 <a id="__codelineno-6-28" name="__codelineno-6-28" href="#__codelineno-6-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-29" name="__codelineno-6-29" href="#__codelineno-6-29"></a><span class="w">    </span><span class="c1">// Use result to overwrite the original array nums</span>
+<a id="__codelineno-6-29" name="__codelineno-6-29" href="#__codelineno-6-29"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result</span>
 <a id="__codelineno-6-30" name="__codelineno-6-30" href="#__codelineno-6-30"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-31" name="__codelineno-6-31" href="#__codelineno-6-31"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">res</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-6-32" name="__codelineno-6-32" href="#__codelineno-6-32"></a><span class="w">    </span><span class="p">}</span>
@@ -5021,49 +5093,49 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-6-34" name="__codelineno-6-34" href="#__codelineno-6-34"></a>
 <a id="__codelineno-6-35" name="__codelineno-6-35" href="#__codelineno-6-35"></a><span class="cm">/* Radix sort */</span>
 <a id="__codelineno-6-36" name="__codelineno-6-36" href="#__codelineno-6-36"></a><span class="kd">function</span><span class="w"> </span><span class="nx">radixSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-37" name="__codelineno-6-37" href="#__codelineno-6-37"></a><span class="w">    </span><span class="c1">// Get the maximum element of the array, used to determine the maximum number of digits</span>
+<a id="__codelineno-6-37" name="__codelineno-6-37" href="#__codelineno-6-37"></a><span class="w">    </span><span class="c1">// Get the largest array element to determine the maximum number of digits</span>
 <a id="__codelineno-6-38" name="__codelineno-6-38" href="#__codelineno-6-38"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(...</span><span class="w"> </span><span class="nx">nums</span><span class="p">);</span>
-<a id="__codelineno-6-39" name="__codelineno-6-39" href="#__codelineno-6-39"></a><span class="w">    </span><span class="c1">// Traverse from the lowest to the highest digit</span>
+<a id="__codelineno-6-39" name="__codelineno-6-39" href="#__codelineno-6-39"></a><span class="w">    </span><span class="c1">// Process digits from least significant to most significant</span>
 <a id="__codelineno-6-40" name="__codelineno-6-40" href="#__codelineno-6-40"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">exp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">exp</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">exp</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mf">10</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-41" name="__codelineno-6-41" href="#__codelineno-6-41"></a><span class="w">        </span><span class="c1">// Perform counting sort on the k-th digit of array elements</span>
+<a id="__codelineno-6-41" name="__codelineno-6-41" href="#__codelineno-6-41"></a><span class="w">        </span><span class="c1">// Apply counting sort to the k-th digit of the array elements</span>
 <a id="__codelineno-6-42" name="__codelineno-6-42" href="#__codelineno-6-42"></a><span class="w">        </span><span class="c1">// k = 1 -&gt; exp = 1</span>
 <a id="__codelineno-6-43" name="__codelineno-6-43" href="#__codelineno-6-43"></a><span class="w">        </span><span class="c1">// k = 2 -&gt; exp = 10</span>
-<a id="__codelineno-6-44" name="__codelineno-6-44" href="#__codelineno-6-44"></a><span class="w">        </span><span class="c1">// i.e., exp = 10^(k-1)</span>
+<a id="__codelineno-6-44" name="__codelineno-6-44" href="#__codelineno-6-44"></a><span class="w">        </span><span class="c1">// That is, exp = 10^(k-1)</span>
 <a id="__codelineno-6-45" name="__codelineno-6-45" href="#__codelineno-6-45"></a><span class="w">        </span><span class="nx">countingSortDigit</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">exp</span><span class="p">);</span>
 <a id="__codelineno-6-46" name="__codelineno-6-46" href="#__codelineno-6-46"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-47" name="__codelineno-6-47" href="#__codelineno-6-47"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">radix_sort.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Get the k-th digit of element num, where exp = 10^(k-1) */</span>
+<div class="highlight"><span class="filename">radix_sort.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Get the k-th digit of num, where exp = 10^(k-1) */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">digit</span><span class="p">(</span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">exp</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k can avoid repeated expensive exponentiation here</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k avoids repeatedly performing expensive exponentiation here</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nx">exp</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mf">10</span><span class="p">;</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="p">}</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a>
-<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="cm">/* Counting sort (based on nums k-th digit) */</span>
+<a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="cm">/* Counting sort (sort nums by the k-th digit) */</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="kd">function</span><span class="w"> </span><span class="nx">countingSortDigit</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[],</span><span class="w"> </span><span class="nx">exp</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// Decimal digit range is 0~9, therefore need a bucket array of length 10</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">    </span><span class="c1">// Decimal digits range from 0 to 9, so the bucket array needs length 10</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="mf">10</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="c1">// Count the occurrence of digits 0~9</span>
+<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="c1">// Count the occurrences of each digit from 0 to 9</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">digit</span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">exp</span><span class="p">);</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i], noted as d</span>
-<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Count the occurrence of digit d</span>
+<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">digit</span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">exp</span><span class="p">);</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i] and store it in d</span>
+<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Count the occurrences of digit d</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">    </span><span class="c1">// Calculate prefix sum, converting "occurrence count" into "array index"</span>
+<a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">    </span><span class="c1">// Compute prefix sums to convert occurrence counts into array positions</span>
 <a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">10</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="nx">counter</span><span class="p">[</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse, based on bucket statistics, place each element into res</span>
+<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse and place each element into res according to the bucket counts</span>
 <a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">n</span><span class="p">).</span><span class="nx">fill</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
 <a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">digit</span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">exp</span><span class="p">);</span>
-<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the index j for d in the array</span>
+<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the destination index j for digit d in the array</span>
 <a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="w">        </span><span class="nx">res</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span><span class="w"> </span><span class="c1">// Place the current element at index j</span>
-<a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrease the count of d by 1</span>
+<a id="__codelineno-7-27" name="__codelineno-7-27" href="#__codelineno-7-27"></a><span class="w">        </span><span class="nx">counter</span><span class="p">[</span><span class="nx">d</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrease the count for d by 1</span>
 <a id="__codelineno-7-28" name="__codelineno-7-28" href="#__codelineno-7-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-29" name="__codelineno-7-29" href="#__codelineno-7-29"></a><span class="w">    </span><span class="c1">// Use result to overwrite the original array nums</span>
+<a id="__codelineno-7-29" name="__codelineno-7-29" href="#__codelineno-7-29"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result</span>
 <a id="__codelineno-7-30" name="__codelineno-7-30" href="#__codelineno-7-30"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-31" name="__codelineno-7-31" href="#__codelineno-7-31"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">res</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-7-32" name="__codelineno-7-32" href="#__codelineno-7-32"></a><span class="w">    </span><span class="p">}</span>
@@ -5071,106 +5143,106 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-7-34" name="__codelineno-7-34" href="#__codelineno-7-34"></a>
 <a id="__codelineno-7-35" name="__codelineno-7-35" href="#__codelineno-7-35"></a><span class="cm">/* Radix sort */</span>
 <a id="__codelineno-7-36" name="__codelineno-7-36" href="#__codelineno-7-36"></a><span class="kd">function</span><span class="w"> </span><span class="nx">radixSort</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-37" name="__codelineno-7-37" href="#__codelineno-7-37"></a><span class="w">    </span><span class="c1">// Get the maximum element of the array, used to determine the maximum number of digits</span>
+<a id="__codelineno-7-37" name="__codelineno-7-37" href="#__codelineno-7-37"></a><span class="w">    </span><span class="c1">// Get the largest array element to determine the maximum number of digits</span>
 <a id="__codelineno-7-38" name="__codelineno-7-38" href="#__codelineno-7-38"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">m</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(...</span><span class="w"> </span><span class="nx">nums</span><span class="p">);</span>
-<a id="__codelineno-7-39" name="__codelineno-7-39" href="#__codelineno-7-39"></a><span class="w">    </span><span class="c1">// Traverse from the lowest to the highest digit</span>
+<a id="__codelineno-7-39" name="__codelineno-7-39" href="#__codelineno-7-39"></a><span class="w">    </span><span class="c1">// Process digits from least significant to most significant</span>
 <a id="__codelineno-7-40" name="__codelineno-7-40" href="#__codelineno-7-40"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">exp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">exp</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="nx">m</span><span class="p">;</span><span class="w"> </span><span class="nx">exp</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mf">10</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-41" name="__codelineno-7-41" href="#__codelineno-7-41"></a><span class="w">        </span><span class="c1">// Perform counting sort on the k-th digit of array elements</span>
+<a id="__codelineno-7-41" name="__codelineno-7-41" href="#__codelineno-7-41"></a><span class="w">        </span><span class="c1">// Apply counting sort to the k-th digit of the array elements</span>
 <a id="__codelineno-7-42" name="__codelineno-7-42" href="#__codelineno-7-42"></a><span class="w">        </span><span class="c1">// k = 1 -&gt; exp = 1</span>
 <a id="__codelineno-7-43" name="__codelineno-7-43" href="#__codelineno-7-43"></a><span class="w">        </span><span class="c1">// k = 2 -&gt; exp = 10</span>
-<a id="__codelineno-7-44" name="__codelineno-7-44" href="#__codelineno-7-44"></a><span class="w">        </span><span class="c1">// i.e., exp = 10^(k-1)</span>
+<a id="__codelineno-7-44" name="__codelineno-7-44" href="#__codelineno-7-44"></a><span class="w">        </span><span class="c1">// That is, exp = 10^(k-1)</span>
 <a id="__codelineno-7-45" name="__codelineno-7-45" href="#__codelineno-7-45"></a><span class="w">        </span><span class="nx">countingSortDigit</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">exp</span><span class="p">);</span>
 <a id="__codelineno-7-46" name="__codelineno-7-46" href="#__codelineno-7-46"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-47" name="__codelineno-7-47" href="#__codelineno-7-47"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">radix_sort.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Get k-th digit of element _num, where exp = 10^(k-1) */</span>
+<div class="highlight"><span class="filename">radix_sort.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Get the k-th digit of _num, where exp = 10^(k-1) */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Passing exp instead of k can avoid repeated expensive exponentiation here</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Passing exp instead of k avoids repeatedly performing expensive exponentiation here</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">_num</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="m">10</span><span class="p">;</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="p">}</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a>
-<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="cm">/* Counting sort (based on nums k-th digit) */</span>
+<a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="cm">/* Counting sort (sort nums by the k-th digit) */</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="kt">void</span><span class="w"> </span><span class="n">countingSortDigit</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="c1">// Decimal digit range is 0~9, therefore need a bucket array of length 10</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">  </span><span class="c1">// Decimal digits range from 0 to 9, so the bucket array needs length 10</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="m">10</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
-<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="c1">// Count the occurrence of digits 0~9</span>
+<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">  </span><span class="c1">// Count the occurrences of each digit from 0 to 9</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i], noted as d</span>
-<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">    </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Count the occurrence of digit d</span>
+<a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i] and store it in d</span>
+<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">    </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Count the occurrences of digit d</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">  </span><span class="c1">// Calculate prefix sum, converting "occurrence count" into "array index"</span>
+<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">  </span><span class="c1">// Compute prefix sums to convert occurrence counts into array positions</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">10</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="w">    </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">];</span>
 <a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="w">  </span><span class="c1">// Traverse in reverse, based on bucket statistics, place each element into res</span>
+<a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="w">  </span><span class="c1">// Traverse in reverse and place each element into res according to the bucket counts</span>
 <a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span>
-<a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the index j for d in the array</span>
+<a id="__codelineno-8-25" name="__codelineno-8-25" href="#__codelineno-8-25"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the destination index j for digit d in the array</span>
 <a id="__codelineno-8-26" name="__codelineno-8-26" href="#__codelineno-8-26"></a><span class="w">    </span><span class="n">res</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span><span class="w"> </span><span class="c1">// Place the current element at index j</span>
-<a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a><span class="w">    </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrease the count of d by 1</span>
+<a id="__codelineno-8-27" name="__codelineno-8-27" href="#__codelineno-8-27"></a><span class="w">    </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrease the count for d by 1</span>
 <a id="__codelineno-8-28" name="__codelineno-8-28" href="#__codelineno-8-28"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-29" name="__codelineno-8-29" href="#__codelineno-8-29"></a><span class="w">  </span><span class="c1">// Use result to overwrite the original array nums</span>
+<a id="__codelineno-8-29" name="__codelineno-8-29" href="#__codelineno-8-29"></a><span class="w">  </span><span class="c1">// Overwrite the original array nums with the result</span>
 <a id="__codelineno-8-30" name="__codelineno-8-30" href="#__codelineno-8-30"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-8-31" name="__codelineno-8-31" href="#__codelineno-8-31"></a><span class="p">}</span>
 <a id="__codelineno-8-32" name="__codelineno-8-32" href="#__codelineno-8-32"></a>
 <a id="__codelineno-8-33" name="__codelineno-8-33" href="#__codelineno-8-33"></a><span class="cm">/* Radix sort */</span>
 <a id="__codelineno-8-34" name="__codelineno-8-34" href="#__codelineno-8-34"></a><span class="kt">void</span><span class="w"> </span><span class="n">radixSort</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-35" name="__codelineno-8-35" href="#__codelineno-8-35"></a><span class="w">  </span><span class="c1">// Get the maximum element of the array, used to determine the maximum number of digits</span>
-<a id="__codelineno-8-36" name="__codelineno-8-36" href="#__codelineno-8-36"></a><span class="w">  </span><span class="c1">// In Dart, int length is 64 bits</span>
+<a id="__codelineno-8-35" name="__codelineno-8-35" href="#__codelineno-8-35"></a><span class="w">  </span><span class="c1">// Get the largest array element to determine the maximum number of digits</span>
+<a id="__codelineno-8-36" name="__codelineno-8-36" href="#__codelineno-8-36"></a><span class="w">  </span><span class="c1">// On native platforms, Dart int values are 64-bit integers</span>
 <a id="__codelineno-8-37" name="__codelineno-8-37" href="#__codelineno-8-37"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="m">63</span><span class="p">;</span>
 <a id="__codelineno-8-38" name="__codelineno-8-38" href="#__codelineno-8-38"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_num</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span>
-<a id="__codelineno-8-39" name="__codelineno-8-39" href="#__codelineno-8-39"></a><span class="w">  </span><span class="c1">// Traverse from the lowest to the highest digit</span>
+<a id="__codelineno-8-39" name="__codelineno-8-39" href="#__codelineno-8-39"></a><span class="w">  </span><span class="c1">// Process digits from least significant to most significant</span>
 <a id="__codelineno-8-40" name="__codelineno-8-40" href="#__codelineno-8-40"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="m">10</span><span class="p">)</span>
-<a id="__codelineno-8-41" name="__codelineno-8-41" href="#__codelineno-8-41"></a><span class="w">    </span><span class="c1">// Perform counting sort on the k-th digit of array elements</span>
+<a id="__codelineno-8-41" name="__codelineno-8-41" href="#__codelineno-8-41"></a><span class="w">    </span><span class="c1">// Apply counting sort to the k-th digit of the array elements</span>
 <a id="__codelineno-8-42" name="__codelineno-8-42" href="#__codelineno-8-42"></a><span class="w">    </span><span class="c1">// k = 1 -&gt; exp = 1</span>
 <a id="__codelineno-8-43" name="__codelineno-8-43" href="#__codelineno-8-43"></a><span class="w">    </span><span class="c1">// k = 2 -&gt; exp = 10</span>
-<a id="__codelineno-8-44" name="__codelineno-8-44" href="#__codelineno-8-44"></a><span class="w">    </span><span class="c1">// i.e., exp = 10^(k-1)</span>
+<a id="__codelineno-8-44" name="__codelineno-8-44" href="#__codelineno-8-44"></a><span class="w">    </span><span class="c1">// That is, exp = 10^(k-1)</span>
 <a id="__codelineno-8-45" name="__codelineno-8-45" href="#__codelineno-8-45"></a><span class="w">    </span><span class="n">countingSortDigit</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span>
 <a id="__codelineno-8-46" name="__codelineno-8-46" href="#__codelineno-8-46"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">radix_sort.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Get the k-th digit of element num, where exp = 10^(k-1) */</span>
+<div class="highlight"><span class="filename">radix_sort.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Get the k-th digit of num, where exp = 10^(k-1) */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">digit</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k can avoid repeated expensive exponentiation here</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k avoids repeatedly performing expensive exponentiation here</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">((</span><span class="n">num</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">10</span><span class="p">)</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">;</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="p">}</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="cm">/* Counting sort (based on nums k-th digit) */</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="cm">/* Counting sort (sort nums by the k-th digit) */</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="k">fn</span><span class="w"> </span><span class="nf">counting_sort_digit</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// Decimal digit range is 0~9, therefore need a bucket array of length 10</span>
+<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="c1">// Decimal digits range from 0 to 9, so the bucket array needs length 10</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="mi">10</span><span class="p">];</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
-<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="c1">// Count the occurrence of digits 0~9</span>
+<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">    </span><span class="c1">// Count the occurrences of each digit from 0 to 9</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i], noted as d</span>
-<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Count the occurrence of digit d</span>
+<a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i] and store it in d</span>
+<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Count the occurrences of digit d</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">    </span><span class="c1">// Calculate prefix sum, converting "occurrence count" into "array index"</span>
+<a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">    </span><span class="c1">// Compute prefix sums to convert occurrence counts into array positions</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="mi">10</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse, based on bucket statistics, place each element into res</span>
+<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse and place each element into res according to the bucket counts</span>
 <a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="p">];</span>
 <a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="p">).</span><span class="n">rev</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span>
-<a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the index j for d in the array</span>
+<a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the destination index j for digit d in the array</span>
 <a id="__codelineno-9-26" name="__codelineno-9-26" href="#__codelineno-9-26"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span><span class="w"> </span><span class="c1">// Place the current element at index j</span>
-<a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrease the count of d by 1</span>
+<a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Decrease the count for d by 1</span>
 <a id="__codelineno-9-28" name="__codelineno-9-28" href="#__codelineno-9-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-9-29" name="__codelineno-9-29" href="#__codelineno-9-29"></a><span class="w">    </span><span class="c1">// Use result to overwrite the original array nums</span>
+<a id="__codelineno-9-29" name="__codelineno-9-29" href="#__codelineno-9-29"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result</span>
 <a id="__codelineno-9-30" name="__codelineno-9-30" href="#__codelineno-9-30"></a><span class="w">    </span><span class="n">nums</span><span class="p">.</span><span class="n">copy_from_slice</span><span class="p">(</span><span class="o">&amp;</span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-9-31" name="__codelineno-9-31" href="#__codelineno-9-31"></a><span class="p">}</span>
 <a id="__codelineno-9-32" name="__codelineno-9-32" href="#__codelineno-9-32"></a>
 <a id="__codelineno-9-33" name="__codelineno-9-33" href="#__codelineno-9-33"></a><span class="cm">/* Radix sort */</span>
 <a id="__codelineno-9-34" name="__codelineno-9-34" href="#__codelineno-9-34"></a><span class="k">fn</span><span class="w"> </span><span class="nf">radix_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="p">[</span><span class="kt">i32</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-35" name="__codelineno-9-35" href="#__codelineno-9-35"></a><span class="w">    </span><span class="c1">// Get the maximum element of the array, used to determine the maximum number of digits</span>
+<a id="__codelineno-9-35" name="__codelineno-9-35" href="#__codelineno-9-35"></a><span class="w">    </span><span class="c1">// Get the largest array element to determine the maximum number of digits</span>
 <a id="__codelineno-9-36" name="__codelineno-9-36" href="#__codelineno-9-36"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">.</span><span class="n">into_iter</span><span class="p">().</span><span class="n">max</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span>
-<a id="__codelineno-9-37" name="__codelineno-9-37" href="#__codelineno-9-37"></a><span class="w">    </span><span class="c1">// Traverse from the lowest to the highest digit</span>
+<a id="__codelineno-9-37" name="__codelineno-9-37" href="#__codelineno-9-37"></a><span class="w">    </span><span class="c1">// Process digits from least significant to most significant</span>
 <a id="__codelineno-9-38" name="__codelineno-9-38" href="#__codelineno-9-38"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-9-39" name="__codelineno-9-39" href="#__codelineno-9-39"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-40" name="__codelineno-9-40" href="#__codelineno-9-40"></a><span class="w">        </span><span class="n">counting_sort_digit</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span>
@@ -5180,37 +5252,37 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">radix_sort.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Get the k-th digit of element num, where exp = 10^(k-1) */</span>
+<div class="highlight"><span class="filename">radix_sort.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Get the k-th digit of num, where exp = 10^(k-1) */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">digit</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k can avoid repeated expensive exponentiation here</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k avoids repeatedly performing expensive exponentiation here</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="p">}</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a>
-<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="cm">/* Counting sort (based on nums k-th digit) */</span>
+<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="cm">/* Counting sort (sort nums by the k-th digit) */</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="kt">void</span><span class="w"> </span><span class="nf">countingSortDigit</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="c1">// Decimal digit range is 0~9, therefore need a bucket array of length 10</span>
+<a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="c1">// Decimal digits range from 0 to 9, so the bucket array needs length 10</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">((</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">10</span><span class="p">));</span>
-<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="n">memset</span><span class="p">(</span><span class="n">counter</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">10</span><span class="p">);</span><span class="w"> </span><span class="c1">// Initialize to 0 to support subsequent memory release</span>
-<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="c1">// Count the occurrence of digits 0~9</span>
+<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="n">memset</span><span class="p">(</span><span class="n">counter</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="mi">10</span><span class="p">);</span><span class="w"> </span><span class="c1">// Initialize the counts to 0</span>
+<a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="c1">// Count the occurrences of each digit from 0 to 9</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">        </span><span class="c1">// Get the k-th digit of nums[i], noted as d</span>
+<a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">        </span><span class="c1">// Get the k-th digit of nums[i] and store it in d</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span>
-<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="c1">// Count the occurrence of digit d</span>
+<a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">        </span><span class="c1">// Count the occurrences of digit d</span>
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">    </span><span class="c1">// Calculate prefix sum, converting "occurrence count" into "array index"</span>
+<a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">    </span><span class="c1">// Compute prefix sums to convert occurrence counts into array positions</span>
 <a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">10</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">    </span><span class="c1">// Traverse in reverse, based on bucket statistics, place each element into res</span>
+<a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">    </span><span class="c1">// Traverse in reverse and place each element into res according to the bucket counts</span>
 <a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">size</span><span class="p">);</span>
 <a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">--</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span>
-<a id="__codelineno-10-27" name="__codelineno-10-27" href="#__codelineno-10-27"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the index j for d in the array</span>
+<a id="__codelineno-10-27" name="__codelineno-10-27" href="#__codelineno-10-27"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Get the destination index j for digit d in the array</span>
 <a id="__codelineno-10-28" name="__codelineno-10-28" href="#__codelineno-10-28"></a><span class="w">        </span><span class="n">res</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span><span class="w">       </span><span class="c1">// Place the current element at index j</span>
-<a id="__codelineno-10-29" name="__codelineno-10-29" href="#__codelineno-10-29"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w">           </span><span class="c1">// Decrease the count of d by 1</span>
+<a id="__codelineno-10-29" name="__codelineno-10-29" href="#__codelineno-10-29"></a><span class="w">        </span><span class="n">counter</span><span class="p">[</span><span class="n">d</span><span class="p">]</span><span class="o">--</span><span class="p">;</span><span class="w">           </span><span class="c1">// Decrease the count for d by 1</span>
 <a id="__codelineno-10-30" name="__codelineno-10-30" href="#__codelineno-10-30"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-31" name="__codelineno-10-31" href="#__codelineno-10-31"></a><span class="w">    </span><span class="c1">// Use result to overwrite the original array nums</span>
+<a id="__codelineno-10-31" name="__codelineno-10-31" href="#__codelineno-10-31"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result</span>
 <a id="__codelineno-10-32" name="__codelineno-10-32" href="#__codelineno-10-32"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-33" name="__codelineno-10-33" href="#__codelineno-10-33"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-10-34" name="__codelineno-10-34" href="#__codelineno-10-34"></a><span class="w">    </span><span class="p">}</span>
@@ -5221,69 +5293,69 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 <a id="__codelineno-10-39" name="__codelineno-10-39" href="#__codelineno-10-39"></a>
 <a id="__codelineno-10-40" name="__codelineno-10-40" href="#__codelineno-10-40"></a><span class="cm">/* Radix sort */</span>
 <a id="__codelineno-10-41" name="__codelineno-10-41" href="#__codelineno-10-41"></a><span class="kt">void</span><span class="w"> </span><span class="nf">radixSort</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-42" name="__codelineno-10-42" href="#__codelineno-10-42"></a><span class="w">    </span><span class="c1">// Get the maximum element of the array, used to determine the maximum number of digits</span>
+<a id="__codelineno-10-42" name="__codelineno-10-42" href="#__codelineno-10-42"></a><span class="w">    </span><span class="c1">// Get the largest array element to determine the maximum number of digits</span>
 <a id="__codelineno-10-43" name="__codelineno-10-43" href="#__codelineno-10-43"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">INT32_MIN</span><span class="p">;</span>
 <a id="__codelineno-10-44" name="__codelineno-10-44" href="#__codelineno-10-44"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-45" name="__codelineno-10-45" href="#__codelineno-10-45"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">max</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-46" name="__codelineno-10-46" href="#__codelineno-10-46"></a><span class="w">            </span><span class="n">max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-10-47" name="__codelineno-10-47" href="#__codelineno-10-47"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-10-48" name="__codelineno-10-48" href="#__codelineno-10-48"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-49" name="__codelineno-10-49" href="#__codelineno-10-49"></a><span class="w">    </span><span class="c1">// Traverse from the lowest to the highest digit</span>
+<a id="__codelineno-10-49" name="__codelineno-10-49" href="#__codelineno-10-49"></a><span class="w">    </span><span class="c1">// Process digits from least significant to most significant</span>
 <a id="__codelineno-10-50" name="__codelineno-10-50" href="#__codelineno-10-50"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">max</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">exp</span><span class="p">;</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">10</span><span class="p">)</span>
-<a id="__codelineno-10-51" name="__codelineno-10-51" href="#__codelineno-10-51"></a><span class="w">        </span><span class="c1">// Perform counting sort on the k-th digit of array elements</span>
+<a id="__codelineno-10-51" name="__codelineno-10-51" href="#__codelineno-10-51"></a><span class="w">        </span><span class="c1">// Apply counting sort to the k-th digit of the array elements</span>
 <a id="__codelineno-10-52" name="__codelineno-10-52" href="#__codelineno-10-52"></a><span class="w">        </span><span class="c1">// k = 1 -&gt; exp = 1</span>
 <a id="__codelineno-10-53" name="__codelineno-10-53" href="#__codelineno-10-53"></a><span class="w">        </span><span class="c1">// k = 2 -&gt; exp = 10</span>
-<a id="__codelineno-10-54" name="__codelineno-10-54" href="#__codelineno-10-54"></a><span class="w">        </span><span class="c1">// i.e., exp = 10^(k-1)</span>
+<a id="__codelineno-10-54" name="__codelineno-10-54" href="#__codelineno-10-54"></a><span class="w">        </span><span class="c1">// That is, exp = 10^(k-1)</span>
 <a id="__codelineno-10-55" name="__codelineno-10-55" href="#__codelineno-10-55"></a><span class="w">        </span><span class="n">countingSortDigit</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">);</span>
 <a id="__codelineno-10-56" name="__codelineno-10-56" href="#__codelineno-10-56"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">radix_sort.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Get the k-th digit of element num, where exp = 10^(k-1) */</span>
+<div class="highlight"><span class="filename">radix_sort.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Get the k-th digit of num, where exp = 10^(k-1) */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">digit</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k can avoid repeated expensive exponentiation here</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Passing exp instead of k avoids repeatedly performing expensive exponentiation here</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="m">10</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="p">}</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a>
-<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="cm">/* Counting sort (based on nums k-th digit) */</span>
+<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="cm">/* Counting sort (sort nums by the k-th digit) */</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">countingSortDigit</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// Decimal digit range is 0~9, therefore need a bucket array of length 10</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">    </span><span class="c1">// Decimal digits range from 0 to 9, so the bucket array needs length 10</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="m">10</span><span class="p">)</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span>
-<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="c1">// Count the occurrence of digits 0~9</span>
+<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">    </span><span class="c1">// Count the occurrences of each digit from 0 to 9</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i], noted as d</span>
-<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]++</span><span class="w">                </span><span class="c1">// Count the occurrence of digit d</span>
+<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="c1">// Get the k-th digit of nums[i] and store it in d</span>
+<a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]++</span><span class="w">                </span><span class="c1">// Count the occurrences of digit d</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="c1">// Calculate prefix sum, converting "occurrence count" into "array index"</span>
+<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="c1">// Compute prefix sums to convert occurrence counts into array positions</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="m">9</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span>
 <a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse, based on bucket statistics, place each element into res</span>
+<a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="w">    </span><span class="c1">// Traverse in reverse and place each element into res according to the bucket counts</span>
 <a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="n">downTo</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-24" name="__codelineno-11-24" href="#__codelineno-11-24"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span>
-<a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Get the index j for d in the array</span>
+<a id="__codelineno-11-25" name="__codelineno-11-25" href="#__codelineno-11-25"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="c1">// Get the destination index j for digit d in the array</span>
 <a id="__codelineno-11-26" name="__codelineno-11-26" href="#__codelineno-11-26"></a><span class="w">        </span><span class="n">res</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w">       </span><span class="c1">// Place the current element at index j</span>
-<a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]--</span><span class="w">           </span><span class="c1">// Decrease the count of d by 1</span>
+<a id="__codelineno-11-27" name="__codelineno-11-27" href="#__codelineno-11-27"></a><span class="w">        </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]--</span><span class="w">           </span><span class="c1">// Decrease the count for d by 1</span>
 <a id="__codelineno-11-28" name="__codelineno-11-28" href="#__codelineno-11-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="w">    </span><span class="c1">// Use result to overwrite the original array nums</span>
+<a id="__codelineno-11-29" name="__codelineno-11-29" href="#__codelineno-11-29"></a><span class="w">    </span><span class="c1">// Overwrite the original array nums with the result</span>
 <a id="__codelineno-11-30" name="__codelineno-11-30" href="#__codelineno-11-30"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span>
 <a id="__codelineno-11-31" name="__codelineno-11-31" href="#__codelineno-11-31"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">res</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-11-32" name="__codelineno-11-32" href="#__codelineno-11-32"></a><span class="p">}</span>
 <a id="__codelineno-11-33" name="__codelineno-11-33" href="#__codelineno-11-33"></a>
 <a id="__codelineno-11-34" name="__codelineno-11-34" href="#__codelineno-11-34"></a><span class="cm">/* Radix sort */</span>
 <a id="__codelineno-11-35" name="__codelineno-11-35" href="#__codelineno-11-35"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">radixSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-36" name="__codelineno-11-36" href="#__codelineno-11-36"></a><span class="w">    </span><span class="c1">// Get the maximum element of the array, used to determine the maximum number of digits</span>
+<a id="__codelineno-11-36" name="__codelineno-11-36" href="#__codelineno-11-36"></a><span class="w">    </span><span class="c1">// Get the largest array element to determine the maximum number of digits</span>
 <a id="__codelineno-11-37" name="__codelineno-11-37" href="#__codelineno-11-37"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kt">Int</span><span class="p">.</span><span class="na">MIN_VALUE</span>
 <a id="__codelineno-11-38" name="__codelineno-11-38" href="#__codelineno-11-38"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-11-39" name="__codelineno-11-39" href="#__codelineno-11-39"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">exp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span>
-<a id="__codelineno-11-40" name="__codelineno-11-40" href="#__codelineno-11-40"></a><span class="w">    </span><span class="c1">// Traverse from the lowest to the highest digit</span>
+<a id="__codelineno-11-40" name="__codelineno-11-40" href="#__codelineno-11-40"></a><span class="w">    </span><span class="c1">// Process digits from least significant to most significant</span>
 <a id="__codelineno-11-41" name="__codelineno-11-41" href="#__codelineno-11-41"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">exp</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-42" name="__codelineno-11-42" href="#__codelineno-11-42"></a><span class="w">        </span><span class="c1">// Perform counting sort on the k-th digit of array elements</span>
+<a id="__codelineno-11-42" name="__codelineno-11-42" href="#__codelineno-11-42"></a><span class="w">        </span><span class="c1">// Apply counting sort to the k-th digit of the array elements</span>
 <a id="__codelineno-11-43" name="__codelineno-11-43" href="#__codelineno-11-43"></a><span class="w">        </span><span class="c1">// k = 1 -&gt; exp = 1</span>
 <a id="__codelineno-11-44" name="__codelineno-11-44" href="#__codelineno-11-44"></a><span class="w">        </span><span class="c1">// k = 2 -&gt; exp = 10</span>
-<a id="__codelineno-11-45" name="__codelineno-11-45" href="#__codelineno-11-45"></a><span class="w">        </span><span class="c1">// i.e., exp = 10^(k-1)</span>
+<a id="__codelineno-11-45" name="__codelineno-11-45" href="#__codelineno-11-45"></a><span class="w">        </span><span class="c1">// That is, exp = 10^(k-1)</span>
 <a id="__codelineno-11-46" name="__codelineno-11-46" href="#__codelineno-11-46"></a><span class="w">        </span><span class="n">countingSortDigit</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span>
 <a id="__codelineno-11-47" name="__codelineno-11-47" href="#__codelineno-11-47"></a><span class="w">        </span><span class="n">exp</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="m">10</span>
 <a id="__codelineno-11-48" name="__codelineno-11-48" href="#__codelineno-11-48"></a><span class="w">    </span><span class="p">}</span>
@@ -5291,47 +5363,47 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">radix_sort.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Get k-th digit of element num, where exp = 10^(k-1) ###</span>
+<div class="highlight"><span class="filename">radix_sort.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Get the k-th digit of num, where exp = 10^(k-1) ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">digit</span><span class="p">(</span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Passing exp instead of k avoids expensive exponentiation calculations</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Passing exp instead of k avoids repeatedly performing expensive exponentiation here</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="p">(</span><span class="n">num</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="mi">10</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="k">end</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a>
-<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="c1">### Counting sort (sort by k-th digit of nums) ###</span>
+<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="c1">### Counting sort (sort nums by the k-th digit)###</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="k">def</span><span class="w"> </span><span class="nf">counting_sort_digit</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span>
-<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="c1"># Decimal digit range is 0~9, therefore need a bucket array of length 10</span>
+<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">  </span><span class="c1"># Decimal digits range from 0 to 9, so the bucket array needs length 10</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">  </span><span class="n">counter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">  </span><span class="c1"># Count the occurrence of digits 0~9</span>
+<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">  </span><span class="c1"># Count the occurrences of each digit from 0 to 9</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">n</span>
-<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="c1"># Get the k-th digit of nums[i], noted as d</span>
-<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">    </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Count the occurrence of digit d</span>
+<a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span><span class="w"> </span><span class="c1"># Get the k-th digit of nums[i] and store it in d</span>
+<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">    </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Count the occurrences of digit d</span>
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">  </span><span class="k">end</span>
-<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">  </span><span class="c1"># Calculate prefix sum, converting "occurrence count" into "array index"</span>
+<a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="w">  </span><span class="c1"># Compute prefix sums to convert occurrence counts into array positions</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="w">  </span><span class="p">(</span><span class="mi">1</span><span class="o">...</span><span class="mi">10</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="o">]</span><span class="w"> </span><span class="p">}</span>
-<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">  </span><span class="c1"># Traverse in reverse, based on bucket statistics, place each element into res</span>
+<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="w">  </span><span class="c1"># Traverse in reverse and place each element into res according to the bucket counts</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">downto</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="w">    </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">digit</span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span>
-<a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="w">    </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Get the index j for d in the array</span>
+<a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="w">    </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Get the destination index j for digit d in the array</span>
 <a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a><span class="w">    </span><span class="n">res</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="c1"># Place the current element at index j</span>
-<a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="w">    </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Decrease the count of d by 1</span>
+<a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="w">    </span><span class="n">counter</span><span class="o">[</span><span class="n">d</span><span class="o">]</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Decrease the count for d by 1</span>
 <a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></a><span class="w">  </span><span class="k">end</span>
-<a id="__codelineno-12-27" name="__codelineno-12-27" href="#__codelineno-12-27"></a><span class="w">  </span><span class="c1"># Use result to overwrite the original array nums</span>
+<a id="__codelineno-12-27" name="__codelineno-12-27" href="#__codelineno-12-27"></a><span class="w">  </span><span class="c1"># Overwrite the original array nums with the result</span>
 <a id="__codelineno-12-28" name="__codelineno-12-28" href="#__codelineno-12-28"></a><span class="w">  </span><span class="p">(</span><span class="mi">0</span><span class="o">...</span><span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">each</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">|</span><span class="n">i</span><span class="o">|</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">res</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-12-29" name="__codelineno-12-29" href="#__codelineno-12-29"></a><span class="k">end</span>
 <a id="__codelineno-12-30" name="__codelineno-12-30" href="#__codelineno-12-30"></a>
 <a id="__codelineno-12-31" name="__codelineno-12-31" href="#__codelineno-12-31"></a><span class="c1">### Radix sort ###</span>
 <a id="__codelineno-12-32" name="__codelineno-12-32" href="#__codelineno-12-32"></a><span class="k">def</span><span class="w"> </span><span class="nf">radix_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
-<a id="__codelineno-12-33" name="__codelineno-12-33" href="#__codelineno-12-33"></a><span class="w">  </span><span class="c1"># Get the maximum element of the array, used to determine the maximum number of digits</span>
+<a id="__codelineno-12-33" name="__codelineno-12-33" href="#__codelineno-12-33"></a><span class="w">  </span><span class="c1"># Get the largest array element to determine the maximum number of digits</span>
 <a id="__codelineno-12-34" name="__codelineno-12-34" href="#__codelineno-12-34"></a><span class="w">  </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">max</span>
-<a id="__codelineno-12-35" name="__codelineno-12-35" href="#__codelineno-12-35"></a><span class="w">  </span><span class="c1"># Traverse from the lowest to the highest digit</span>
+<a id="__codelineno-12-35" name="__codelineno-12-35" href="#__codelineno-12-35"></a><span class="w">  </span><span class="c1"># Process digits from least significant to most significant</span>
 <a id="__codelineno-12-36" name="__codelineno-12-36" href="#__codelineno-12-36"></a><span class="w">  </span><span class="n">exp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-12-37" name="__codelineno-12-37" href="#__codelineno-12-37"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="n">exp</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">m</span>
-<a id="__codelineno-12-38" name="__codelineno-12-38" href="#__codelineno-12-38"></a><span class="w">    </span><span class="c1"># Perform counting sort on the k-th digit of array elements</span>
+<a id="__codelineno-12-38" name="__codelineno-12-38" href="#__codelineno-12-38"></a><span class="w">    </span><span class="c1"># Apply counting sort to the k-th digit of the array elements</span>
 <a id="__codelineno-12-39" name="__codelineno-12-39" href="#__codelineno-12-39"></a><span class="w">    </span><span class="c1"># k = 1 -&gt; exp = 1</span>
 <a id="__codelineno-12-40" name="__codelineno-12-40" href="#__codelineno-12-40"></a><span class="w">    </span><span class="c1"># k = 2 -&gt; exp = 10</span>
-<a id="__codelineno-12-41" name="__codelineno-12-41" href="#__codelineno-12-41"></a><span class="w">    </span><span class="c1"># i.e., exp = 10^(k-1)</span>
+<a id="__codelineno-12-41" name="__codelineno-12-41" href="#__codelineno-12-41"></a><span class="w">    </span><span class="c1"># That is, exp = 10^(k-1)</span>
 <a id="__codelineno-12-42" name="__codelineno-12-42" href="#__codelineno-12-42"></a><span class="w">    </span><span class="n">counting_sort_digit</span><span class="p">(</span><span class="n">nums</span><span class="p">,</span><span class="w"> </span><span class="n">exp</span><span class="p">)</span>
 <a id="__codelineno-12-43" name="__codelineno-12-43" href="#__codelineno-12-43"></a><span class="w">    </span><span class="n">exp</span><span class="w"> </span><span class="o">*=</span><span class="w"> </span><span class="mi">10</span>
 <a id="__codelineno-12-44" name="__codelineno-12-44" href="#__codelineno-12-44"></a><span class="w">  </span><span class="k">end</span>
@@ -5340,16 +5412,21 @@ x_k = \lfloor\frac{x}{d^{k-1}}\rfloor \bmod d
 </div>
 </div>
 </div>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20digit%28num%3A%20int,%20exp%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Get%20the%20k-th%20digit%20of%20num,%20where%20exp%20%3D%2010%5E%28k-1%29%22%22%22%0A%20%20%20%20%23%20Passing%20exp%20instead%20of%20k%20avoids%20repeatedly%20performing%20expensive%20exponentiation%20here%0A%20%20%20%20return%20%28num%20//%20exp%29%20%25%2010%0A%0Adef%20counting_sort_digit%28nums%3A%20list%5Bint%5D,%20exp%3A%20int%29%3A%0A%20%20%20%20%22%22%22Counting%20sort%20%28sort%20nums%20by%20the%20k-th%20digit%29%22%22%22%0A%20%20%20%20%23%20Decimal%20digits%20range%20from%200%20to%209,%20so%20the%20bucket%20array%20needs%20length%2010%0A%20%20%20%20counter%20%3D%20%5B0%5D%20%2A%2010%0A%20%20%20%20n%20%3D%20len%28nums%29%0A%20%20%20%20%23%20Count%20the%20occurrences%20of%20each%20digit%20from%200%20to%209%0A%20%20%20%20for%20i%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20d%20%3D%20digit%28nums%5Bi%5D,%20exp%29%20%20%23%20Get%20the%20k-th%20digit%20of%20nums%5Bi%5D%20and%20store%20it%20in%20d%0A%20%20%20%20%20%20%20%20counter%5Bd%5D%20%2B%3D%201%20%20%23%20Count%20the%20occurrences%20of%20digit%20d%0A%20%20%20%20%23%20Compute%20prefix%20sums%20to%20convert%20occurrence%20counts%20into%20array%20positions%0A%20%20%20%20for%20i%20in%20range%281,%2010%29%3A%0A%20%20%20%20%20%20%20%20counter%5Bi%5D%20%2B%3D%20counter%5Bi%20-%201%5D%0A%20%20%20%20%23%20Traverse%20in%20reverse%20and%20place%20each%20element%20into%20res%20according%20to%20the%20bucket%20counts%0A%20%20%20%20res%20%3D%20%5B0%5D%20%2A%20n%0A%20%20%20%20for%20i%20in%20range%28n%20-%201,%20-1,%20-1%29%3A%0A%20%20%20%20%20%20%20%20d%20%3D%20digit%28nums%5Bi%5D,%20exp%29%0A%20%20%20%20%20%20%20%20j%20%3D%20counter%5Bd%5D%20-%201%20%20%23%20Get%20the%20destination%20index%20j%20for%20digit%20d%20in%20the%20array%0A%20%20%20%20%20%20%20%20res%5Bj%5D%20%3D%20nums%5Bi%5D%20%20%23%20Place%20the%20current%20element%20at%20index%20j%0A%20%20%20%20%20%20%20%20counter%5Bd%5D%20-%3D%201%20%20%23%20Decrease%20the%20count%20for%20d%20by%201%0A%20%20%20%20%23%20Overwrite%20the%20original%20array%20nums%20with%20the%20result%0A%20%20%20%20for%20i%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20nums%5Bi%5D%20%3D%20res%5Bi%5D%0A%0Adef%20radix_sort%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Radix%20sort%22%22%22%0A%20%20%20%20%23%20Get%20the%20largest%20array%20element%20to%20determine%20the%20maximum%20number%20of%20digits%0A%20%20%20%20m%20%3D%20max%28nums%29%0A%20%20%20%20%23%20Process%20digits%20from%20least%20significant%20to%20most%20significant%0A%20%20%20%20exp%20%3D%201%0A%20%20%20%20while%20exp%20%3C%3D%20m%3A%0A%20%20%20%20%20%20%20%20%23%20Apply%20counting%20sort%20to%20the%20k-th%20digit%20of%20the%20array%20elements%0A%20%20%20%20%20%20%20%20%23%20k%20%3D%201%20-%3E%20exp%20%3D%201%0A%20%20%20%20%20%20%20%20%23%20k%20%3D%202%20-%3E%20exp%20%3D%2010%0A%20%20%20%20%20%20%20%20%23%20That%20is,%20exp%20%3D%2010%5E%28k-1%29%0A%20%20%20%20%20%20%20%20counting_sort_digit%28nums,%20exp%29%0A%20%20%20%20%20%20%20%20exp%20%2A%3D%2010%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Radix%20sort%0A%20%20%20%20nums%20%3D%20%5B%0A%20%20%20%20%20%20%20%20105,%0A%20%20%20%20%20%20%20%20356,%0A%20%20%20%20%20%20%20%20428,%0A%20%20%20%20%20%20%20%20348,%0A%20%20%20%20%20%20%20%20818,%0A%20%20%20%20%5D%0A%20%20%20%20radix_sort%28nums%29%0A%20%20%20%20print%28%22After%20radix%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20digit%28num%3A%20int,%20exp%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%22%22%22Get%20the%20k-th%20digit%20of%20num,%20where%20exp%20%3D%2010%5E%28k-1%29%22%22%22%0A%20%20%20%20%23%20Passing%20exp%20instead%20of%20k%20avoids%20repeatedly%20performing%20expensive%20exponentiation%20here%0A%20%20%20%20return%20%28num%20//%20exp%29%20%25%2010%0A%0Adef%20counting_sort_digit%28nums%3A%20list%5Bint%5D,%20exp%3A%20int%29%3A%0A%20%20%20%20%22%22%22Counting%20sort%20%28sort%20nums%20by%20the%20k-th%20digit%29%22%22%22%0A%20%20%20%20%23%20Decimal%20digits%20range%20from%200%20to%209,%20so%20the%20bucket%20array%20needs%20length%2010%0A%20%20%20%20counter%20%3D%20%5B0%5D%20%2A%2010%0A%20%20%20%20n%20%3D%20len%28nums%29%0A%20%20%20%20%23%20Count%20the%20occurrences%20of%20each%20digit%20from%200%20to%209%0A%20%20%20%20for%20i%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20d%20%3D%20digit%28nums%5Bi%5D,%20exp%29%20%20%23%20Get%20the%20k-th%20digit%20of%20nums%5Bi%5D%20and%20store%20it%20in%20d%0A%20%20%20%20%20%20%20%20counter%5Bd%5D%20%2B%3D%201%20%20%23%20Count%20the%20occurrences%20of%20digit%20d%0A%20%20%20%20%23%20Compute%20prefix%20sums%20to%20convert%20occurrence%20counts%20into%20array%20positions%0A%20%20%20%20for%20i%20in%20range%281,%2010%29%3A%0A%20%20%20%20%20%20%20%20counter%5Bi%5D%20%2B%3D%20counter%5Bi%20-%201%5D%0A%20%20%20%20%23%20Traverse%20in%20reverse%20and%20place%20each%20element%20into%20res%20according%20to%20the%20bucket%20counts%0A%20%20%20%20res%20%3D%20%5B0%5D%20%2A%20n%0A%20%20%20%20for%20i%20in%20range%28n%20-%201,%20-1,%20-1%29%3A%0A%20%20%20%20%20%20%20%20d%20%3D%20digit%28nums%5Bi%5D,%20exp%29%0A%20%20%20%20%20%20%20%20j%20%3D%20counter%5Bd%5D%20-%201%20%20%23%20Get%20the%20destination%20index%20j%20for%20digit%20d%20in%20the%20array%0A%20%20%20%20%20%20%20%20res%5Bj%5D%20%3D%20nums%5Bi%5D%20%20%23%20Place%20the%20current%20element%20at%20index%20j%0A%20%20%20%20%20%20%20%20counter%5Bd%5D%20-%3D%201%20%20%23%20Decrease%20the%20count%20for%20d%20by%201%0A%20%20%20%20%23%20Overwrite%20the%20original%20array%20nums%20with%20the%20result%0A%20%20%20%20for%20i%20in%20range%28n%29%3A%0A%20%20%20%20%20%20%20%20nums%5Bi%5D%20%3D%20res%5Bi%5D%0A%0Adef%20radix_sort%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Radix%20sort%22%22%22%0A%20%20%20%20%23%20Get%20the%20largest%20array%20element%20to%20determine%20the%20maximum%20number%20of%20digits%0A%20%20%20%20m%20%3D%20max%28nums%29%0A%20%20%20%20%23%20Process%20digits%20from%20least%20significant%20to%20most%20significant%0A%20%20%20%20exp%20%3D%201%0A%20%20%20%20while%20exp%20%3C%3D%20m%3A%0A%20%20%20%20%20%20%20%20%23%20Apply%20counting%20sort%20to%20the%20k-th%20digit%20of%20the%20array%20elements%0A%20%20%20%20%20%20%20%20%23%20k%20%3D%201%20-%3E%20exp%20%3D%201%0A%20%20%20%20%20%20%20%20%23%20k%20%3D%202%20-%3E%20exp%20%3D%2010%0A%20%20%20%20%20%20%20%20%23%20That%20is,%20exp%20%3D%2010%5E%28k-1%29%0A%20%20%20%20%20%20%20%20counting_sort_digit%28nums,%20exp%29%0A%20%20%20%20%20%20%20%20exp%20%2A%3D%2010%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Radix%20sort%0A%20%20%20%20nums%20%3D%20%5B%0A%20%20%20%20%20%20%20%20105,%0A%20%20%20%20%20%20%20%20356,%0A%20%20%20%20%20%20%20%20428,%0A%20%20%20%20%20%20%20%20348,%0A%20%20%20%20%20%20%20%20818,%0A%20%20%20%20%5D%0A%20%20%20%20radix_sort%28nums%29%0A%20%20%20%20print%28%22After%20radix%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=6&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
 <div class="admonition question">
-<p class="admonition-title">Why start sorting from the lowest digit?</p>
-<p>In successive sorting passes, a later pass overrides the result of an earlier one. For example, if the first pass yields <span class="arithmatex">\(a &lt; b\)</span> but the second yields <span class="arithmatex">\(a &gt; b\)</span>, then the result of the second pass prevails. Because higher-order digits have higher priority than lower-order digits, we should sort the lower digits first and then the higher digits.</p>
+<p class="admonition-title">Why start with the least significant digit?</p>
+<p>In successive rounds, a later sort takes precedence over an earlier one. For example, if the first round orders two elements as <span class="arithmatex">\(a &lt; b\)</span> but the second orders them as <span class="arithmatex">\(a &gt; b\)</span>, the second result replaces the first. Higher digits have greater priority, so lower digits must be sorted first.</p>
 </div>
-<h2 id="11102-algorithm-characteristics">11.10.2 &nbsp; Algorithm Characteristics<a class="headerlink" href="#11102-algorithm-characteristics" title="Permanent link">&para;</a></h2>
-<p>Compared with counting sort, radix sort is suitable for larger value ranges, <strong>but only when the data can be represented with a fixed number of digits and that digit count is not too large</strong>. For example, floating-point numbers are not well suited to radix sort because the digit count <span class="arithmatex">\(k\)</span> can be too large, potentially leading to time complexity <span class="arithmatex">\(O(nk) \gg O(n^2)\)</span>.</p>
+<h2 id="11102-algorithm-properties">11.10.2 &nbsp; Algorithm properties<a class="headerlink" href="#11102-algorithm-properties" title="Permanent link">&para;</a></h2>
+<p>Compared with counting sort, radix sort handles a larger range of values, <strong>provided the data can be represented with a fixed number of digits and that number is not too large</strong>. For example, floating-point numbers are unsuitable because their digit count <span class="arithmatex">\(k\)</span> may be so large that <span class="arithmatex">\(O(nk) \gg O(n^2)\)</span>.</p>
 <ul>
-<li><strong>Time complexity of <span class="arithmatex">\(O(nk)\)</span>, non-adaptive sorting</strong>: Let the number of items be <span class="arithmatex">\(n\)</span>, let the values be represented in base <span class="arithmatex">\(d\)</span>, and let the maximum number of digits be <span class="arithmatex">\(k\)</span>. Counting sort on one digit takes <span class="arithmatex">\(O(n + d)\)</span> time, so sorting all <span class="arithmatex">\(k\)</span> digits takes <span class="arithmatex">\(O((n + d)k)\)</span> time. In practice, <span class="arithmatex">\(d\)</span> and <span class="arithmatex">\(k\)</span> are usually relatively small, so the overall time complexity approaches <span class="arithmatex">\(O(n)\)</span>.</li>
-<li><strong>Space complexity of <span class="arithmatex">\(O(n + d)\)</span>, non-in-place sorting</strong>: Same as counting sort, radix sort requires auxiliary arrays <code>res</code> and <code>counter</code> of lengths <span class="arithmatex">\(n\)</span> and <span class="arithmatex">\(d\)</span>.</li>
-<li><strong>Stable sort</strong>: When counting sort is stable, radix sort is also stable; when counting sort is unstable, radix sort cannot guarantee correct sorting results.</li>
+<li><strong>Time complexity: <span class="arithmatex">\(O(nk)\)</span>; not adaptive</strong>. For <span class="arithmatex">\(n\)</span> elements represented in base <span class="arithmatex">\(d\)</span> with at most <span class="arithmatex">\(k\)</span> digits, counting sort on one digit takes <span class="arithmatex">\(O(n + d)\)</span> time. Sorting all <span class="arithmatex">\(k\)</span> digits takes <span class="arithmatex">\(O((n + d)k)\)</span>. Usually <span class="arithmatex">\(d\)</span> and <span class="arithmatex">\(k\)</span> are relatively small, so the complexity approaches <span class="arithmatex">\(O(n)\)</span>.</li>
+<li><strong>Space complexity: <span class="arithmatex">\(O(n + d)\)</span>; not in-place</strong>. Like counting sort, radix sort requires arrays <code>res</code> and <code>counter</code> of lengths <span class="arithmatex">\(n\)</span> and <span class="arithmatex">\(d\)</span>.</li>
+<li><strong>Stable</strong>. Radix sort is stable when the counting sort used within it is stable. If counting sort is not stable, radix sort cannot guarantee a correctly sorted result.</li>
 </ul>
 
 <!-- Source file information -->
@@ -5373,7 +5450,7 @@ aria-label="Footer"
   <a
     href="../counting_sort/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 11.9 Counting Sort"
+    aria-label="Previous: 11.9   Counting sort"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -5385,7 +5462,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        11.9 Counting Sort
+        11.9 &nbsp; Counting sort
       </div>
     </div>
   </a>
@@ -5397,7 +5474,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 11.11 Summary"
+    aria-label="Next: 11.11   Summary"
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     <div class="md-footer__title">
@@ -5405,7 +5482,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        11.11 Summary
+        11.11 &nbsp; Summary
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -5422,7 +5499,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>11.2 Selection Sort - Hello Algo</title>
+        <title>11.2 Selection sort - Hello Algo</title>
 
 
 
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-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
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       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
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@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1896,7 +1896,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1965,7 +1965,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1992,7 +1992,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2019,7 +2019,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2046,7 +2046,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2073,7 +2073,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2100,7 +2100,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2164,7 +2164,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2185,7 +2185,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2206,7 +2206,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2233,7 +2233,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2260,7 +2260,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2287,7 +2287,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2314,7 +2314,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2378,7 +2378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2399,7 +2399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2420,7 +2420,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2447,7 +2447,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2474,7 +2474,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2501,7 +2501,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2528,7 +2528,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2617,7 +2617,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2665,7 +2665,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2692,7 +2692,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2719,7 +2719,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2746,7 +2746,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2773,7 +2773,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2800,7 +2800,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2880,7 +2880,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2901,7 +2901,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2922,7 +2922,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -2958,7 +2958,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -2975,7 +2975,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3001,10 +3001,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1121-algorithm-characteristics" class="md-nav__link">
+  <a href="#1121-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.2.1 &nbsp; Algorithm Characteristics
+        11.2.1 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -3034,7 +3034,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3061,7 +3061,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3088,7 +3088,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3115,7 +3115,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3142,7 +3142,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3169,7 +3169,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3196,7 +3196,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3223,7 +3223,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3250,7 +3250,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3277,7 +3277,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3343,7 +3343,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3364,7 +3364,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3385,7 +3385,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3412,7 +3412,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3439,7 +3439,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3466,7 +3466,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3493,7 +3493,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3520,7 +3520,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3586,7 +3586,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3607,7 +3607,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3628,7 +3628,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3655,7 +3655,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3682,7 +3682,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3709,7 +3709,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3736,7 +3736,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3763,7 +3763,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3833,7 +3833,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3854,7 +3854,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3875,7 +3875,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3902,7 +3902,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3929,7 +3929,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3956,7 +3956,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -3983,7 +3983,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4010,7 +4010,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4037,7 +4037,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4064,7 +4064,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4130,7 +4130,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4151,7 +4151,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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@@ -4172,7 +4172,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4199,7 +4199,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4226,7 +4226,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4253,7 +4253,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4280,7 +4280,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4307,7 +4307,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4367,7 +4367,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4388,7 +4388,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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@@ -4409,7 +4409,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4436,7 +4436,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4463,7 +4463,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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@@ -4547,6 +4547,78 @@
 
 
 
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         <li class="md-nav__item">
-  <a href="#1121-algorithm-characteristics" class="md-nav__link">
+  <a href="#1121-algorithm-properties" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.2.1 &nbsp; Algorithm Characteristics
+        11.2.1 &nbsp; Algorithm properties
       </span>
     </span>
   </a>
@@ -4635,20 +4707,20 @@
 
 
 <!-- Page content -->
-<h1 id="112-selection-sort">11.2 &nbsp; Selection Sort<a class="headerlink" href="#112-selection-sort" title="Permanent link">&para;</a></h1>
-<p><u>Selection sort</u> works very simply: in each round, it selects the smallest element from the unsorted interval and places it at the end of the sorted interval.</p>
-<p>Assume the array has length <span class="arithmatex">\(n\)</span>. The procedure of selection sort is shown in Figure 11-2.</p>
+<h1 id="112-selection-sort">11.2 &nbsp; Selection sort<a class="headerlink" href="#112-selection-sort" title="Permanent link">&para;</a></h1>
+<p><u>Selection sort</u> works very simply: in each round, select the smallest element from the unsorted section and place it at the end of the sorted section.</p>
+<p>For an array of length <span class="arithmatex">\(n\)</span>, the algorithm proceeds as shown below.</p>
 <ol>
-<li>Initially, all elements are unsorted, i.e., the unsorted (index) interval is <span class="arithmatex">\([0, n-1]\)</span>.</li>
-<li>Select the smallest element in the interval <span class="arithmatex">\([0, n-1]\)</span> and swap it with the element at index <span class="arithmatex">\(0\)</span>. After completion, the first element of the array is sorted.</li>
-<li>Select the smallest element in the interval <span class="arithmatex">\([1, n-1]\)</span> and swap it with the element at index <span class="arithmatex">\(1\)</span>. After completion, the first 2 elements of the array are sorted.</li>
-<li>And so on. After <span class="arithmatex">\(n - 1\)</span> rounds of selection and swapping, the first <span class="arithmatex">\(n - 1\)</span> elements of the array are sorted.</li>
-<li>The only remaining element must be the largest, so no further sorting is needed and the array is sorted.</li>
+<li>Initially, all elements are unsorted, so the unsorted index interval is <span class="arithmatex">\([0, n-1]\)</span>.</li>
+<li>Select the smallest element in <span class="arithmatex">\([0, n-1]\)</span> and swap it with the element at index <span class="arithmatex">\(0\)</span>. The first element is now sorted.</li>
+<li>Select the smallest element in <span class="arithmatex">\([1, n-1]\)</span> and swap it with the element at index <span class="arithmatex">\(1\)</span>. The first two elements are now sorted.</li>
+<li>Continue in this way. After <span class="arithmatex">\(n - 1\)</span> rounds of selection and swapping, the first <span class="arithmatex">\(n - 1\)</span> elements are sorted.</li>
+<li>The remaining element must be the largest and needs no sorting. The array is now sorted.</li>
 </ol>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:11"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label><label for="__tabbed_1_5">&lt;5&gt;</label><label for="__tabbed_1_6">&lt;6&gt;</label><label for="__tabbed_1_7">&lt;7&gt;</label><label for="__tabbed_1_8">&lt;8&gt;</label><label for="__tabbed_1_9">&lt;9&gt;</label><label for="__tabbed_1_10">&lt;10&gt;</label><label for="__tabbed_1_11">&lt;11&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Selection sort steps" class="animation-figure" src="../selection_sort.assets/selection_sort_step1.png" /></p>
+<p><img alt="Steps of selection sort" class="animation-figure" src="../selection_sort.assets/selection_sort_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="selection_sort_step2" class="animation-figure" src="../selection_sort.assets/selection_sort_step2.png" /></p>
@@ -4682,23 +4754,23 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 11-2 &nbsp; Selection sort steps </p>
+<p align="center"> Figure 11-2 &nbsp; Steps of selection sort </p>
 
-<p>In the code, we use <span class="arithmatex">\(k\)</span> to track the smallest element within the unsorted interval:</p>
+<p>In the code, <span class="arithmatex">\(k\)</span> tracks the smallest element in the unsorted section:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">selection_sort.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">selection_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]):</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Selection sort"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
-<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="c1"># Outer loop: unsorted interval is [i, n-1]</span>
+<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="c1"># Outer loop: the unsorted range is [i, n-1]</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="c1"># Inner loop: find the smallest element within the unsorted interval</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="c1"># Inner loop: find the smallest element in the unsorted range</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="n">k</span> <span class="o">=</span> <span class="n">i</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>            <span class="k">if</span> <span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">]:</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>                <span class="n">k</span> <span class="o">=</span> <span class="n">j</span>  <span class="c1"># Record the index of the smallest element</span>
-<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="c1"># Swap the smallest element with the first element of the unsorted interval</span>
+<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="c1"># Swap the smallest element with the first element of the unsorted range</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">],</span> <span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 </code></pre></div>
 </div>
@@ -4706,15 +4778,15 @@
 <div class="highlight"><span class="filename">selection_sort.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Selection sort */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">selectionSort</span><span class="p">(</span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
-<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted interval is [i, n-1]</span>
+<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [i, n-1]</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element within the unsorted interval</span>
+<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element in the unsorted range</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">])</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">                </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record the index of the smallest element</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted interval</span>
+<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted range</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="n">swap</span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">]);</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="p">}</span>
@@ -4724,15 +4796,15 @@
 <div class="highlight"><span class="filename">selection_sort.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Selection sort */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">selectionSort</span><span class="p">(</span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
-<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted interval is [i, n-1]</span>
+<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [i, n-1]</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element within the unsorted interval</span>
+<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element in the unsorted range</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">k</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">                </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record the index of the smallest element</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted interval</span>
+<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted range</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">k</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">k</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span>
@@ -4744,15 +4816,15 @@
 <div class="highlight"><span class="filename">selection_sort.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Selection sort */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">SelectionSort</span><span class="p">(</span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
-<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted interval is [i, n-1]</span>
+<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [i, n-1]</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element within the unsorted interval</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element in the unsorted range</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">])</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">                </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record the index of the smallest element</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted interval</span>
+<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted range</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">]);</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="p">}</span>
@@ -4762,9 +4834,9 @@
 <div class="highlight"><span class="filename">selection_sort.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Selection sort */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">selectionSort</span><span class="p">(</span><span class="nx">nums</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="nx">n</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span>
-<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted interval is [i, n-1]</span>
+<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [i, n-1]</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="o">-</span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element within the unsorted interval</span>
+<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element in the unsorted range</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">        </span><span class="nx">k</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">i</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">k</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
@@ -4772,7 +4844,7 @@
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">                </span><span class="nx">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">j</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted interval</span>
+<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted range</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">k</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="p">}</span>
@@ -4782,16 +4854,16 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">selection_sort.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Selection sort */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">selectionSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted interval is [i, n-1]</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [i, n-1]</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="bp">dropLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element within the unsorted interval</span>
+<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element in the unsorted range</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">k</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="bp">indices</span><span class="p">.</span><span class="bp">dropFirst</span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">                </span><span class="n">k</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="c1">// Record the index of the smallest element</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted interval</span>
+<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted range</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="n">nums</span><span class="p">.</span><span class="n">swapAt</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">)</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="p">}</span>
@@ -4801,16 +4873,16 @@
 <div class="highlight"><span class="filename">selection_sort.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Selection sort */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">selectionSort</span><span class="p">(</span><span class="nx">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted interval is [i, n-1]</span>
+<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [i, n-1]</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element within the unsorted interval</span>
+<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element in the unsorted range</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">k</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">                </span><span class="nx">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">j</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record the index of the smallest element</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted interval</span>
+<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted range</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">        </span><span class="p">[</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">k</span><span class="p">]]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="nx">nums</span><span class="p">[</span><span class="nx">k</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]];</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="p">}</span>
@@ -4820,16 +4892,16 @@
 <div class="highlight"><span class="filename">selection_sort.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Selection sort */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">selectionSort</span><span class="p">(</span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
-<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted interval is [i, n-1]</span>
+<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [i, n-1]</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element within the unsorted interval</span>
+<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element in the unsorted range</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="p">;</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">n</span><span class="p">;</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">k</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">                </span><span class="nx">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">j</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record the index of the smallest element</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted interval</span>
+<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted range</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">        </span><span class="p">[</span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">k</span><span class="p">]]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="nx">nums</span><span class="p">[</span><span class="nx">k</span><span class="p">],</span><span class="w"> </span><span class="nx">nums</span><span class="p">[</span><span class="nx">i</span><span class="p">]];</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="p">}</span>
@@ -4839,14 +4911,14 @@
 <div class="highlight"><span class="filename">selection_sort.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Selection sort */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">selectionSort</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
-<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// Outer loop: unsorted interval is [i, n-1]</span>
+<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// Outer loop: the unsorted range is [i, n-1]</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">    </span><span class="c1">// Inner loop: find the smallest element within the unsorted interval</span>
+<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">    </span><span class="c1">// Inner loop: find the smallest element in the unsorted range</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">])</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record the index of the smallest element</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="c1">// Swap the smallest element with the first element of the unsorted interval</span>
+<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="c1">// Swap the smallest element with the first element of the unsorted range</span>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">];</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">    </span><span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span>
@@ -4861,16 +4933,16 @@
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted interval is [i, n-1]</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [i, n-1]</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element within the unsorted interval</span>
+<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element in the unsorted range</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="o">..</span><span class="n">n</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">                </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record the index of the smallest element</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted interval</span>
+<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted range</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">        </span><span class="n">nums</span><span class="p">.</span><span class="n">swap</span><span class="p">(</span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">);</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="p">}</span>
@@ -4879,15 +4951,15 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">selection_sort.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Selection sort */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">selectionSort</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">nums</span><span class="p">[],</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted interval is [i, n-1]</span>
+<a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [i, n-1]</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element within the unsorted interval</span>
+<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element in the unsorted range</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">])</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">                </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="p">;</span><span class="w"> </span><span class="c1">// Record the index of the smallest element</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted interval</span>
+<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted range</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">];</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">;</span>
@@ -4899,15 +4971,15 @@
 <div class="highlight"><span class="filename">selection_sort.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Selection sort */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">selectionSort</span><span class="p">(</span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span>
-<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="c1">// Outer loop: unsorted interval is [i, n-1]</span>
+<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="c1">// Outer loop: the unsorted range is [i, n-1]</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span>
-<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element within the unsorted interval</span>
+<a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">        </span><span class="c1">// Inner loop: find the smallest element in the unsorted range</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">k</span><span class="o">]</span><span class="p">)</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">                </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="c1">// Record the index of the smallest element</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted interval</span>
+<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="c1">// Swap the smallest element with the first element of the unsorted range</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">k</span><span class="o">]</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">k</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span>
@@ -4919,16 +4991,16 @@
 <div class="highlight"><span class="filename">selection_sort.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Selection sort ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">selection_sort</span><span class="p">(</span><span class="n">nums</span><span class="p">)</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">.</span><span class="n">length</span>
-<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="c1"># Outer loop: unsorted interval is [i, n-1]</span>
+<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="c1"># Outer loop: the unsorted range is [i, n-1]</span>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">    </span><span class="c1"># Inner loop: find the smallest element within the unsorted interval</span>
+<a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">    </span><span class="c1"># Inner loop: find the smallest element in the unsorted range</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">    </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">i</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="o">...</span><span class="n">n</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">k</span><span class="o">]</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">        </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="c1"># Record the index of the smallest element</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">      </span><span class="k">end</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">    </span><span class="k">end</span>
-<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="c1"># Swap the smallest element with the first element of the unsorted interval</span>
+<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="c1"># Swap the smallest element with the first element of the unsorted range</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">k</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">k</span><span class="o">]</span><span class="p">,</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="k">end</span>
@@ -4936,14 +5008,19 @@
 </div>
 </div>
 </div>
-<h2 id="1121-algorithm-characteristics">11.2.1 &nbsp; Algorithm Characteristics<a class="headerlink" href="#1121-algorithm-characteristics" title="Permanent link">&para;</a></h2>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 531px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=def%20selection_sort%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Selection%20sort%22%22%22%0A%20%20%20%20n%20%3D%20len%28nums%29%0A%20%20%20%20%23%20Outer%20loop%3A%20the%20unsorted%20range%20is%20%5Bi,%20n-1%5D%0A%20%20%20%20for%20i%20in%20range%28n%20-%201%29%3A%0A%20%20%20%20%20%20%20%20%23%20Inner%20loop%3A%20find%20the%20smallest%20element%20in%20the%20unsorted%20range%0A%20%20%20%20%20%20%20%20k%20%3D%20i%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28i%20%2B%201,%20n%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20nums%5Bj%5D%20%3C%20nums%5Bk%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20k%20%3D%20j%20%20%23%20Record%20the%20index%20of%20the%20smallest%20element%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20smallest%20element%20with%20the%20first%20element%20of%20the%20unsorted%20range%0A%20%20%20%20%20%20%20%20nums%5Bi%5D,%20nums%5Bk%5D%20%3D%20nums%5Bk%5D,%20nums%5Bi%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B4,%201,%203,%201,%205,%202%5D%0A%20%20%20%20selection_sort%28nums%29%0A%20%20%20%20print%28%22After%20selection%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=def%20selection_sort%28nums%3A%20list%5Bint%5D%29%3A%0A%20%20%20%20%22%22%22Selection%20sort%22%22%22%0A%20%20%20%20n%20%3D%20len%28nums%29%0A%20%20%20%20%23%20Outer%20loop%3A%20the%20unsorted%20range%20is%20%5Bi,%20n-1%5D%0A%20%20%20%20for%20i%20in%20range%28n%20-%201%29%3A%0A%20%20%20%20%20%20%20%20%23%20Inner%20loop%3A%20find%20the%20smallest%20element%20in%20the%20unsorted%20range%0A%20%20%20%20%20%20%20%20k%20%3D%20i%0A%20%20%20%20%20%20%20%20for%20j%20in%20range%28i%20%2B%201,%20n%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20nums%5Bj%5D%20%3C%20nums%5Bk%5D%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20k%20%3D%20j%20%20%23%20Record%20the%20index%20of%20the%20smallest%20element%0A%20%20%20%20%20%20%20%20%23%20Swap%20the%20smallest%20element%20with%20the%20first%20element%20of%20the%20unsorted%20range%0A%20%20%20%20%20%20%20%20nums%5Bi%5D,%20nums%5Bk%5D%20%3D%20nums%5Bk%5D,%20nums%5Bi%5D%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20nums%20%3D%20%5B4,%201,%203,%201,%205,%202%5D%0A%20%20%20%20selection_sort%28nums%29%0A%20%20%20%20print%28%22After%20selection%20sort,%20nums%20%3D%22,%20nums%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="1121-algorithm-properties">11.2.1 &nbsp; Algorithm properties<a class="headerlink" href="#1121-algorithm-properties" title="Permanent link">&para;</a></h2>
 <ul>
-<li><strong>Time complexity <span class="arithmatex">\(O(n^2)\)</span>, non-adaptive sorting</strong>: The outer loop has <span class="arithmatex">\(n - 1\)</span> rounds in total. The inner loop runs <span class="arithmatex">\(n - 1\)</span> times in the first round and <span class="arithmatex">\(1\)</span> time in the last round. Thus, it runs <span class="arithmatex">\(n - 1\)</span>, <span class="arithmatex">\(n - 2\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, and <span class="arithmatex">\(1\)</span> times across the rounds, summing to <span class="arithmatex">\(\frac{n(n - 1)}{2}\)</span>.</li>
-<li><strong>Space complexity <span class="arithmatex">\(O(1)\)</span>, in-place sorting</strong>: Pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> use a constant amount of extra space.</li>
-<li><strong>Unstable sorting</strong>: As shown in Figure 11-3, element <code>nums[i]</code> may be swapped to the right of an element equal to it, causing a change in their relative order.</li>
+<li><strong>Time complexity: <span class="arithmatex">\(O(n^2)\)</span>; not adaptive</strong>. The outer loop runs <span class="arithmatex">\(n - 1\)</span> times. The inner loop runs <span class="arithmatex">\(n - 1\)</span> times in the first round and <span class="arithmatex">\(1\)</span> time in the last, giving successive counts of <span class="arithmatex">\(n - 1\)</span>, <span class="arithmatex">\(n - 2\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, and <span class="arithmatex">\(1\)</span>, which sum to <span class="arithmatex">\(\frac{n(n - 1)}{2}\)</span>.</li>
+<li><strong>Space complexity: <span class="arithmatex">\(O(1)\)</span>; in-place</strong>. Pointers <span class="arithmatex">\(i\)</span> and <span class="arithmatex">\(j\)</span> use constant extra space.</li>
+<li><strong>Not stable</strong>. As shown below, <code>nums[i]</code> may be swapped to the right of an equal element, changing their relative order.</li>
 </ul>
-<p><img alt="Selection sort non-stability example" class="animation-figure" src="../selection_sort.assets/selection_sort_instability.png" /></p>
-<p align="center"> Figure 11-3 &nbsp; Selection sort non-stability example </p>
+<p><img alt="An example showing that selection sort is not stable" class="animation-figure" src="../selection_sort.assets/selection_sort_instability.png" /></p>
+<p align="center"> Figure 11-3 &nbsp; An example showing that selection sort is not stable </p>
 
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+    aria-label="Previous: 11.1   Sorting algorithms"
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+    aria-label="Next: 11.3   Bubble sort"
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+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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+        <title>11.1 Sorting algorithms - Hello Algo</title>
 
 
 
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-  0.2 How to Use This Book
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-  Chapter 1. Encounter with Algorithms
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-  1.1 Algorithms Are Everywhere
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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           <ul class="md-nav__list" data-md-scrollfix>
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
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-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
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   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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@@ -2638,7 +2638,7 @@
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-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1111-evaluation-dimensions" class="md-nav__link">
+  <a href="#1111-evaluation-criteria" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.1.1 &nbsp; Evaluation Dimensions
+        11.1.1 &nbsp; Evaluation criteria
       </span>
     </span>
   </a>
@@ -2985,10 +2985,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1112-ideal-sorting-algorithm" class="md-nav__link">
+  <a href="#1112-the-ideal-sorting-algorithm" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.1.2 &nbsp; Ideal Sorting Algorithm
+        11.1.2 &nbsp; The ideal sorting algorithm
       </span>
     </span>
   </a>
@@ -3018,7 +3018,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3396,7 +3396,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4558,6 +4558,78 @@
 
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1111-evaluation-dimensions" class="md-nav__link">
+  <a href="#1111-evaluation-criteria" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.1.1 &nbsp; Evaluation Dimensions
+        11.1.1 &nbsp; Evaluation criteria
       </span>
     </span>
   </a>
@@ -4610,10 +4682,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#1112-ideal-sorting-algorithm" class="md-nav__link">
+  <a href="#1112-the-ideal-sorting-algorithm" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        11.1.2 &nbsp; Ideal Sorting Algorithm
+        11.1.2 &nbsp; The ideal sorting algorithm
       </span>
     </span>
   </a>
@@ -4657,18 +4729,18 @@
 
 
 <!-- Page content -->
-<h1 id="111-sorting-algorithm">11.1 &nbsp; Sorting Algorithm<a class="headerlink" href="#111-sorting-algorithm" title="Permanent link">&para;</a></h1>
-<p>A <u>sorting algorithm</u> arranges a set of data in a specific order. Sorting algorithms have extensive applications because ordered data can usually be searched, analyzed, and processed more efficiently.</p>
-<p>As shown in Figure 11-1, the data being sorted can be integers, floating-point numbers, characters, strings, and so on. The sorting rule can be defined as needed, such as numerical order, ASCII order, or a custom rule.</p>
-<p><img alt="Data type and criterion examples" class="animation-figure" src="../sorting_algorithm.assets/sorting_examples.png" /></p>
-<p align="center"> Figure 11-1 &nbsp; Data type and criterion examples </p>
+<h1 id="111-sorting-algorithms">11.1 &nbsp; Sorting algorithms<a class="headerlink" href="#111-sorting-algorithms" title="Permanent link">&para;</a></h1>
+<p>A <u>sorting algorithm</u> arranges a dataset in a specified order. Sorting is widely used because ordered data is often easier to search, analyze, and process efficiently.</p>
+<p>As shown below, the data may consist of integers, floating-point numbers, characters, strings, and other types. The ordering rule can be chosen as needed: numerical magnitude, ASCII character order, or a custom rule.</p>
+<p><img alt="Examples of data types and ordering rules" class="animation-figure" src="../sorting_algorithm.assets/sorting_examples.png" /></p>
+<p align="center"> Figure 11-1 &nbsp; Examples of data types and ordering rules </p>
 
-<h2 id="1111-evaluation-dimensions">11.1.1 &nbsp; Evaluation Dimensions<a class="headerlink" href="#1111-evaluation-dimensions" title="Permanent link">&para;</a></h2>
-<p><strong>Execution efficiency</strong>: We expect the time complexity of sorting algorithms to be as low as possible, with a smaller total number of operations (reducing the constant factor in time complexity). For large data volumes, execution efficiency is particularly important.</p>
-<p><strong>In-place property</strong>: As the name implies, <u>in-place sorting</u> achieves sorting by operating directly on the original array without requiring additional auxiliary arrays, thus saving memory. Typically, in-place sorting involves fewer data movement operations and runs faster.</p>
-<p><strong>Stability</strong>: <u>Stable sorting</u> ensures that the relative order of equal elements in the array does not change after sorting is completed.</p>
-<p>Stable sorting is a necessary condition for multi-level sorting scenarios. Suppose we have a table storing student information, where column 1 and column 2 are name and age, respectively. In this case, <u>unstable sorting</u> may cause the ordered nature of the input data to be lost:</p>
-<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># The input data is sorted by name</span>
+<h2 id="1111-evaluation-criteria">11.1.1 &nbsp; Evaluation criteria<a class="headerlink" href="#1111-evaluation-criteria" title="Permanent link">&para;</a></h2>
+<p><strong>Efficiency</strong>: We want low time complexity and few operations overall, corresponding to smaller constant factors in the complexity. Efficiency is especially important for large datasets.</p>
+<p><strong>In-place operation</strong>: An <u>in-place sort</u> works directly on the original array without an auxiliary array, saving memory. It usually moves data fewer times and runs faster as well.</p>
+<p><strong>Stability</strong>: A <u>stable sort</u> preserves the relative order of equal elements in the array.</p>
+<p>Stable sorting is necessary for multilevel sorting. Suppose a table stores student names in its first column and ages in its second. An <u>unstable sort</u> may destroy the input's existing order:</p>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># The input is already sorted by name</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="c1"># (name, age)</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="w">  </span><span class="o">(</span><span class="s1">'A'</span>,<span class="w"> </span><span class="m">19</span><span class="o">)</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="w">  </span><span class="o">(</span><span class="s1">'B'</span>,<span class="w"> </span><span class="m">18</span><span class="o">)</span>
@@ -4676,20 +4748,20 @@
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="w">  </span><span class="o">(</span><span class="s1">'D'</span>,<span class="w"> </span><span class="m">19</span><span class="o">)</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="w">  </span><span class="o">(</span><span class="s1">'E'</span>,<span class="w"> </span><span class="m">23</span><span class="o">)</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="c1"># Suppose we use an unstable sorting algorithm to sort the list by age.</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="c1"># In the result, the relative positions of ('D', 19) and ('A', 19) change,</span>
-<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="c1"># so the property that the input data is sorted by name is lost.</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="c1"># Suppose an unstable algorithm sorts the list by age.</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="c1"># The relative positions of ('D', 19) and ('A', 19) change,</span>
+<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="c1"># so the original ordering by name is lost</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="w">  </span><span class="o">(</span><span class="s1">'B'</span>,<span class="w"> </span><span class="m">18</span><span class="o">)</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="w">  </span><span class="o">(</span><span class="s1">'D'</span>,<span class="w"> </span><span class="m">19</span><span class="o">)</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="w">  </span><span class="o">(</span><span class="s1">'A'</span>,<span class="w"> </span><span class="m">19</span><span class="o">)</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="w">  </span><span class="o">(</span><span class="s1">'C'</span>,<span class="w"> </span><span class="m">21</span><span class="o">)</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="w">  </span><span class="o">(</span><span class="s1">'E'</span>,<span class="w"> </span><span class="m">23</span><span class="o">)</span>
 </code></pre></div>
-<p><strong>Adaptability</strong>: <u>Adaptive sorting</u> can utilize the existing order information in the input data to reduce the amount of computation, achieving better time efficiency. The best-case time complexity of adaptive sorting algorithms is typically better than the average time complexity.</p>
-<p><strong>Comparison-based or non-comparison</strong>: <u>Comparison-based sorting</u> relies on comparison operators (<span class="arithmatex">\(&lt;\)</span>, <span class="arithmatex">\(=\)</span>, <span class="arithmatex">\(&gt;\)</span>) to determine the relative order of elements, thereby sorting the entire array. Its worst-case time complexity has a lower bound of <span class="arithmatex">\(\Omega(n \log n)\)</span>. <u>Non-comparison sorting</u> does not use comparison operators and can achieve a time complexity of <span class="arithmatex">\(O(n)\)</span>, but its versatility is relatively limited.</p>
-<h2 id="1112-ideal-sorting-algorithm">11.1.2 &nbsp; Ideal Sorting Algorithm<a class="headerlink" href="#1112-ideal-sorting-algorithm" title="Permanent link">&para;</a></h2>
-<p><strong>Fast, in-place, stable, adaptive, and broadly applicable</strong>. Clearly, no sorting algorithm has been discovered to date that combines all of these characteristics. Therefore, when selecting a sorting algorithm, it is necessary to decide based on the specific characteristics of the data and the requirements of the problem.</p>
-<p>Next, we will examine various sorting algorithms and analyze their advantages and disadvantages based on the evaluation dimensions above.</p>
+<p><strong>Adaptiveness</strong>: An <u>adaptive sort</u> uses ordering already present in the input to reduce computation and improve efficiency. Its best-case time complexity is usually better than its average complexity.</p>
+<p><strong>Comparison-based or not</strong>: A <u>comparison-based sort</u> uses comparison operators (<span class="arithmatex">\(&lt;\)</span>, <span class="arithmatex">\(=\)</span>, <span class="arithmatex">\(&gt;\)</span>) to determine the relative order of elements and sort the array. Its worst-case time complexity has a lower bound of <span class="arithmatex">\(\Omega(n \log n)\)</span>. A <u>non-comparison sort</u> does not use comparison operators and can achieve <span class="arithmatex">\(O(n)\)</span> time, but is less broadly applicable.</p>
+<h2 id="1112-the-ideal-sorting-algorithm">11.1.2 &nbsp; The ideal sorting algorithm<a class="headerlink" href="#1112-the-ideal-sorting-algorithm" title="Permanent link">&para;</a></h2>
+<p><strong>Fast, in-place, stable, adaptive, and broadly applicable</strong>. No sorting algorithm with all these properties has yet been found. We must therefore choose according to the characteristics of the data and the requirements of the problem.</p>
+<p>Next, we will study different sorting algorithms and analyze their strengths and weaknesses using these criteria.</p>
 
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     class="md-footer__link md-footer__link--prev"
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+  0.1 &nbsp; About this book
 
 
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-  0.2 How to Use This Book
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-  0.3 Summary
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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-  2.6 Exercises
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-  3.1 Classification of Data Structures
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-  3.2 Basic Data Types
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-  3.3 Number Encoding *
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
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+  Chapter 6. &nbsp; Hash tables
 
 
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+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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+  Chapter 9. &nbsp; Graphs
 
 
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+  9.1 &nbsp; Graphs
 
 
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+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
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+  10.5 &nbsp; Search algorithms revisited
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
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+  <a href="#1-key-takeaways" class="md-nav__link">
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+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.8 Exercises
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+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
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+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
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+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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+  <a href="#1-key-takeaways" class="md-nav__link">
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-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
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 <!-- Page content -->
 <h1 id="1111-summary">11.11 &nbsp; Summary<a class="headerlink" href="#1111-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
 <ul>
-<li>Bubble sort achieves sorting by swapping adjacent elements. By adding a flag to enable early return, we can optimize the best-case time complexity of bubble sort to <span class="arithmatex">\(O(n)\)</span>.</li>
-<li>In each round, insertion sort inserts an element from the unsorted portion into its correct position in the sorted portion. Although insertion sort has a time complexity of <span class="arithmatex">\(O(n^2)\)</span>, it remains very popular for small sorting tasks because each operation is relatively lightweight.</li>
-<li>Quick sort relies on sentinel partitioning. In sentinel partitioning, repeatedly choosing the worst possible pivot can degrade the time complexity to <span class="arithmatex">\(O(n^2)\)</span>. Choosing a median-based pivot or a random pivot can reduce the probability of this degradation. By recursing on the shorter subarray first, we can effectively reduce the recursion depth and optimize the space complexity to <span class="arithmatex">\(O(\log n)\)</span>.</li>
-<li>Merge sort includes two phases: divide and merge, which typically embody the divide-and-conquer strategy. In merge sort, sorting an array requires creating auxiliary arrays, with a space complexity of <span class="arithmatex">\(O(n)\)</span>; however, the space complexity of sorting a linked list can be optimized to <span class="arithmatex">\(O(1)\)</span>.</li>
-<li>Bucket sort consists of three steps: distributing data into buckets, sorting within buckets, and merging results. It also embodies the divide-and-conquer strategy and is suitable for very large data volumes. The key to bucket sort is distributing data evenly.</li>
-<li>Counting sort is a special case of bucket sort, which achieves sorting by counting the number of occurrences of data. Counting sort is suitable for situations where the data volume is large but the data range is limited, and requires that data can be converted to positive integers.</li>
-<li>Radix sort achieves data sorting by sorting digit by digit, requiring that data can be represented as fixed-digit numbers.</li>
-<li>Overall, we hope to find a sorting algorithm that is efficient, stable, in-place, and adaptive. However, as with other data structures and algorithms, no sorting algorithm can satisfy all of these criteria at the same time. In practice, we need to choose the appropriate sorting algorithm based on the characteristics of the data.</li>
-<li>Figure 11-19 compares mainstream sorting algorithms in terms of efficiency, stability, in-place property, and adaptability.</li>
+<li>Bubble sort sorts by swapping adjacent elements. Adding a flag to stop early improves its best-case time complexity to <span class="arithmatex">\(O(n)\)</span>.</li>
+<li>Insertion sort inserts one element from the unsorted section into its correct position in the sorted section each round. Although its time complexity is <span class="arithmatex">\(O(n^2)\)</span>, its low operation count makes it popular for small datasets.</li>
+<li>Quick sort sorts through partitioning. Repeatedly choosing the worst pivot can degrade its time complexity to <span class="arithmatex">\(O(n^2)\)</span>. Median or random pivot selection reduces this risk. Recursing on the shorter subarray reduces recursion depth and improves space complexity to <span class="arithmatex">\(O(\log n)\)</span>.</li>
+<li>Merge sort has division and merging phases, exemplifying divide and conquer. Sorting an array requires an auxiliary array and <span class="arithmatex">\(O(n)\)</span> space, while sorting a linked list can be optimized to use <span class="arithmatex">\(O(1)\)</span> space.</li>
+<li>Bucket sort has three steps: distribute data into buckets, sort within each bucket, and combine the results. It also uses divide and conquer and suits very large datasets. The key is distributing data evenly.</li>
+<li>Counting sort is a special case of bucket sort that sorts by counting occurrences. It suits large datasets with a limited value range and requires data that can be converted to positive integers.</li>
+<li>Radix sort sorts one digit at a time and requires data that can be represented as numbers with a fixed number of digits.</li>
+<li>Ideally, a sorting algorithm would be efficient, stable, in-place, and adaptive. As with other data structures and algorithms, however, no single sorting algorithm meets all these requirements. In practice, choose one according to the data's properties.</li>
+<li>Figure 11-19 compares common sorting algorithms in efficiency, stability, in-place operation, and adaptiveness.</li>
 </ul>
-<p><img alt="Sorting algorithm comparison" class="animation-figure" src="../summary.assets/sorting_algorithms_comparison.png" /></p>
-<p align="center"> Figure 11-19 &nbsp; Sorting algorithm comparison </p>
+<p><img alt="Comparison of sorting algorithms" class="animation-figure" src="../summary.assets/sorting_algorithms_comparison.png" /></p>
+<p align="center"> Figure 11-19 &nbsp; Comparison of sorting algorithms </p>
 
 <h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
-<p><strong>Q</strong>: In what situations is the stability of sorting algorithms necessary?</p>
-<p>In reality, we may sort based on a certain attribute of objects. For example, students have two attributes: name and height. We want to implement multi-level sorting: first sort by name to get <code>(A, 180) (B, 185) (C, 170) (D, 170)</code>; then sort by height. Because the sorting algorithm is unstable, we may get <code>(D, 170) (C, 170) (A, 180) (B, 185)</code>.</p>
-<p>We can see that students D and C have swapped positions, destroying the ordering by name, which is not what we want.</p>
-<p><strong>Q</strong>: Can the order of "searching from right to left" and "searching from left to right" in sentinel partitioning be swapped?</p>
-<p>No. When we use the leftmost element as the pivot, we must first "search from right to left" and then "search from left to right". This conclusion is somewhat counterintuitive; let's analyze the reason.</p>
-<p>The last step of sentinel partitioning <code>partition()</code> is to swap <code>nums[left]</code> and <code>nums[i]</code>. After the swap is complete, the elements to the left of the pivot are all <code>&lt;=</code> the pivot, <strong>which requires that <code>nums[left] &gt;= nums[i]</code> must hold before the last swap</strong>. Suppose we first "search from left to right", then if we cannot find an element larger than the pivot, <strong>we will exit the loop when <code>i == j</code>, at which point it may be that <code>nums[j] == nums[i] &gt; nums[left]</code></strong>. In other words, the last swap operation will swap an element larger than the pivot to the leftmost end of the array, causing sentinel partitioning to fail.</p>
-<p>For example, given the array <code>[0, 0, 0, 0, 1]</code>, if we first "search from left to right", the array after sentinel partitioning is <code>[1, 0, 0, 0, 0]</code>, which is incorrect.</p>
-<p>By the same reasoning, if we select <code>nums[right]</code> as the pivot, the order is reversed: we must first "search from left to right".</p>
-<p><strong>Q</strong>: Regarding the optimization of recursion depth in quick sort, why can selecting the shorter array ensure that the recursion depth does not exceed <span class="arithmatex">\(\log n\)</span>?</p>
-<p>Recursion depth is the number of recursive calls that have not yet returned. Each round of sentinel partitioning divides the original array into two sub-arrays. After this optimization, the sub-array selected for further recursion is at most half the length of the original array. In the worst case, if it is always half as long, the final recursion depth is <span class="arithmatex">\(\log n\)</span>.</p>
-<p>Reviewing the original quick sort, we may continuously recurse on the longer array. In the worst case, it would be <span class="arithmatex">\(n\)</span>, <span class="arithmatex">\(n - 1\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, <span class="arithmatex">\(1\)</span>, with a recursion depth of <span class="arithmatex">\(n\)</span>. Recursion depth optimization can avoid this situation.</p>
-<p><strong>Q</strong>: When all elements in the array are equal, is the time complexity of quick sort <span class="arithmatex">\(O(n^2)\)</span>? How should this degenerate case be handled?</p>
-<p>Yes. In this case, the array can be partitioned into three parts through sentinel partitioning: less than, equal to, and greater than the pivot. We then recurse only on the less-than and greater-than parts. With this approach, an array whose elements are all equal can be sorted in just one round of sentinel partitioning.</p>
-<p><strong>Q</strong>: Why is the worst-case time complexity of bucket sort <span class="arithmatex">\(O(n^2)\)</span>?</p>
-<p>In the worst case, all elements are distributed into the same bucket. If we use an <span class="arithmatex">\(O(n^2)\)</span> algorithm to sort these elements, the time complexity will be <span class="arithmatex">\(O(n^2)\)</span>.</p>
+<p><strong>Q</strong>: When is stability necessary in a sorting algorithm?</p>
+<p>We often sort objects by one of their attributes. For example, students have names and heights, and we want a multilevel sort. First sort by name to obtain <code>(A, 180) (B, 185) (C, 170) (D, 170)</code>, then sort by height. An unstable algorithm may produce <code>(D, 170) (C, 170) (A, 180) (B, 185)</code>.</p>
+<p>Students D and C have switched positions, breaking the ordering by name. This is not what we want.</p>
+<p><strong>Q</strong>: Can the right-to-left and left-to-right scans in partitioning be reversed?</p>
+<p>No. When the leftmost element is the pivot, the right-to-left scan must come first. This may seem counterintuitive, so let us examine why.</p>
+<p>The final step of <code>partition()</code> swaps <code>nums[left]</code> and <code>nums[i]</code>. Afterward, every element to the left of the pivot must be <code>&lt;=</code> the pivot, <strong>so <code>nums[left] &gt;= nums[i]</code> must hold before that swap</strong>. If we scan left to right first and fail to find a larger element, <strong>the loop stops at <code>i == j</code>, where it is possible that <code>nums[j] == nums[i] &gt; nums[left]</code></strong>. The final swap would then move an element larger than the pivot to the far left, causing partitioning to fail.</p>
+<p>For example, given <code>[0, 0, 0, 0, 1]</code>, scanning left to right first produces <code>[1, 0, 0, 0, 0]</code> after partitioning, which is incorrect.</p>
+<p>Conversely, if we choose <code>nums[right]</code> as the pivot, we must scan left to right first.</p>
+<p><strong>Q</strong>: In the recursion-depth optimization for quick sort, why does choosing the shorter subarray keep the depth at most <span class="arithmatex">\(\log n\)</span>?</p>
+<p>Recursion depth is the number of recursive calls that have not yet returned. Each partition divides the array into two subarrays. After the optimization, the subarray chosen for recursion is at most half the original length. Even in the worst case, where it is always exactly half, the depth is <span class="arithmatex">\(\log n\)</span>.</p>
+<p>In the original quick sort, we may repeatedly recurse on the larger subarray, with lengths <span class="arithmatex">\(n\)</span>, <span class="arithmatex">\(n - 1\)</span>, <span class="arithmatex">\(\dots\)</span>, <span class="arithmatex">\(2\)</span>, and <span class="arithmatex">\(1\)</span> in the worst case, giving depth <span class="arithmatex">\(n\)</span>. The optimization prevents this.</p>
+<p><strong>Q</strong>: If all array elements are equal, does quick sort take <span class="arithmatex">\(O(n^2)\)</span> time? How can we handle this degeneration?</p>
+<p>Yes. In this case, consider partitioning into three sections: elements smaller than, equal to, and greater than the pivot. Recurse only on the smaller and greater sections. If all elements are equal, a single partition then completes the sort.</p>
+<p><strong>Q</strong>: Why is bucket sort's worst-case time complexity <span class="arithmatex">\(O(n^2)\)</span>?</p>
+<p>In the worst case, all elements fall into one bucket. If an <span class="arithmatex">\(O(n^2)\)</span> algorithm is used to sort that bucket, the overall time complexity is <span class="arithmatex">\(O(n^2)\)</span>.</p>
 
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-        <title>5.3 Deque - Hello Algo</title>
+        <title>5.3 Deques - Hello Algo</title>
 
 
 
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+        <a href="#53-deques" class="md-skip">
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-              5.3 &nbsp; Deque
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-  Before Starting
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-  3.1 Classification of Data Structures
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
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-  3.4 Character Encoding *
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-  5.5 Exercises
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+  6.1 &nbsp; Hash tables
 
 
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@@ -1857,7 +1857,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1884,7 +1884,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1911,7 +1911,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1938,7 +1938,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -2006,7 +2006,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -2027,7 +2027,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2048,7 +2048,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -2075,7 +2075,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2102,7 +2102,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2129,7 +2129,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2156,7 +2156,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2183,7 +2183,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2210,7 +2210,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2274,7 +2274,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2295,7 +2295,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2316,7 +2316,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2343,7 +2343,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2370,7 +2370,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2397,7 +2397,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2424,7 +2424,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2488,7 +2488,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2509,7 +2509,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2530,7 +2530,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2557,7 +2557,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2584,7 +2584,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2611,7 +2611,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2638,7 +2638,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2706,7 +2706,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2727,7 +2727,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2748,7 +2748,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2775,7 +2775,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2802,7 +2802,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2829,7 +2829,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2856,7 +2856,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2883,7 +2883,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2910,7 +2910,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2988,7 +2988,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3009,7 +3009,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3030,7 +3030,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3057,7 +3057,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3084,7 +3084,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3111,7 +3111,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3138,7 +3138,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3165,7 +3165,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3192,7 +3192,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3219,7 +3219,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3246,7 +3246,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3273,7 +3273,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3300,7 +3300,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3327,7 +3327,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3393,7 +3393,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3414,7 +3414,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3435,7 +3435,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3462,7 +3462,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3489,7 +3489,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3516,7 +3516,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3543,7 +3543,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3570,7 +3570,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3636,7 +3636,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3657,7 +3657,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3678,7 +3678,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3705,7 +3705,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3732,7 +3732,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3759,7 +3759,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3786,7 +3786,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3813,7 +3813,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3883,7 +3883,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3904,7 +3904,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3925,7 +3925,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3952,7 +3952,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3979,7 +3979,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4006,7 +4006,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4033,7 +4033,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4060,7 +4060,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4087,7 +4087,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4114,7 +4114,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4180,7 +4180,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4201,7 +4201,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4222,7 +4222,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4249,7 +4249,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4276,7 +4276,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4303,7 +4303,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4330,7 +4330,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4357,7 +4357,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4417,7 +4417,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4438,7 +4438,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4459,7 +4459,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4486,7 +4486,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4513,7 +4513,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4597,6 +4597,78 @@
 
 
 
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 </nav>
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@@ -4641,7 +4713,7 @@
   <a href="#531-common-deque-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.3.1 &nbsp; Common Deque Operations
+        5.3.1 &nbsp; Common deque operations
       </span>
     </span>
   </a>
@@ -4649,22 +4721,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#532-deque-implementation" class="md-nav__link">
+  <a href="#532-implementing-a-deque" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.3.2 &nbsp; Deque Implementation *
+        5.3.2 &nbsp; Implementing a deque *
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="5.3.2 Deque Implementation *">
+    <nav class="md-nav" aria-label="5.3.2 Implementing a deque *">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-doubly-linked-list-implementation" class="md-nav__link">
+  <a href="#1-implementation-with-a-doubly-linked-list" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Doubly Linked List Implementation
+        1. &nbsp; Implementation with a doubly linked list
       </span>
     </span>
   </a>
@@ -4672,10 +4744,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-array-implementation" class="md-nav__link">
+  <a href="#2-implementation-with-an-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Array Implementation
+        2. &nbsp; Implementation with an array
       </span>
     </span>
   </a>
@@ -4688,10 +4760,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#533-deque-applications" class="md-nav__link">
+  <a href="#533-applications" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.3.3 &nbsp; Deque Applications
+        5.3.3 &nbsp; Applications
       </span>
     </span>
   </a>
@@ -4735,14 +4807,14 @@
 
 
 <!-- Page content -->
-<h1 id="53-deque">5.3 &nbsp; Deque<a class="headerlink" href="#53-deque" title="Permanent link">&para;</a></h1>
-<p>In a queue, we can only remove elements from the front or add elements at the rear. As shown in Figure 5-7, a <u>double-ended queue (deque)</u> provides greater flexibility, allowing elements to be added or removed at both the front and the rear.</p>
-<p><img alt="Operations of deque" class="animation-figure" src="../deque.assets/deque_operations.png" /></p>
-<p align="center"> Figure 5-7 &nbsp; Operations of deque </p>
+<h1 id="53-deques">5.3 &nbsp; Deques<a class="headerlink" href="#53-deques" title="Permanent link">&para;</a></h1>
+<p>A queue allows removal only at the front and insertion only at the rear. As shown below, a <u>double-ended queue</u>, or deque, is more flexible: elements can be added or removed at either end.</p>
+<p><img alt="Deque operations" class="animation-figure" src="../deque.assets/deque_operations.png" /></p>
+<p align="center"> Figure 5-7 &nbsp; Deque operations </p>
 
-<h2 id="531-common-deque-operations">5.3.1 &nbsp; Common Deque Operations<a class="headerlink" href="#531-common-deque-operations" title="Permanent link">&para;</a></h2>
-<p>The common operations on a deque are shown in Table 5-3. The specific method names depend on the programming language used.</p>
-<p align="center"> Table 5-3 &nbsp; Efficiency of Deque Operations </p>
+<h2 id="531-common-deque-operations">5.3.1 &nbsp; Common deque operations<a class="headerlink" href="#531-common-deque-operations" title="Permanent link">&para;</a></h2>
+<p>Table 5-3 lists common operations. Actual method names depend on the programming language.</p>
+<p align="center"> Table 5-3 &nbsp; Efficiency of deque operations </p>
 
 <div class="center-table">
 <table>
@@ -4750,57 +4822,57 @@
 <tr>
 <th>Method</th>
 <th>Description</th>
-<th>Time Complexity</th>
+<th>Time complexity</th>
 </tr>
 </thead>
 <tbody>
 <tr>
 <td><code>push_first()</code></td>
-<td>Add element to front</td>
+<td>Add an element at the front</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
 <td><code>push_last()</code></td>
-<td>Add element to rear</td>
+<td>Add an element at the rear</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
 <td><code>pop_first()</code></td>
-<td>Remove front element</td>
+<td>Remove the front element</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
 <td><code>pop_last()</code></td>
-<td>Remove rear element</td>
+<td>Remove the rear element</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
 <td><code>peek_first()</code></td>
-<td>Access front element</td>
+<td>Access the front element</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
 <td><code>peek_last()</code></td>
-<td>Access rear element</td>
+<td>Access the rear element</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 </tbody>
 </table>
 </div>
-<p>Similarly, we can directly use the deque classes provided by the programming language:</p>
+<p>Again, we can use a deque class already provided by the programming language:</p>
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 <div class="highlight"><span class="filename">deque.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">from</span><span class="w"> </span><span class="nn">collections</span><span class="w"> </span><span class="kn">import</span> <span class="n">deque</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="c1"># Initialize deque</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="c1"># Initialize a deque</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="n">deq</span><span class="p">:</span> <span class="n">deque</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="n">deque</span><span class="p">()</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="c1"># Enqueue elements</span>
-<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="n">deq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>      <span class="c1"># Add to rear</span>
+<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="n">deq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>      <span class="c1"># Add at the rear</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="n">deq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="n">deq</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="n">deq</span><span class="o">.</span><span class="n">appendleft</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>  <span class="c1"># Add to front</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="n">deq</span><span class="o">.</span><span class="n">appendleft</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>  <span class="c1"># Add at the front</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="n">deq</span><span class="o">.</span><span class="n">appendleft</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="c1"># Access elements</span>
@@ -4808,25 +4880,25 @@
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="n">rear</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">deq</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>  <span class="c1"># Rear element</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a>
 <a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a><span class="c1"># Dequeue elements</span>
-<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="n">pop_front</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">deq</span><span class="o">.</span><span class="n">popleft</span><span class="p">()</span>  <span class="c1"># Front element dequeue</span>
-<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="n">pop_rear</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">deq</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>       <span class="c1"># Rear element dequeue</span>
+<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="n">pop_front</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">deq</span><span class="o">.</span><span class="n">popleft</span><span class="p">()</span>  <span class="c1"># Remove the front element</span>
+<a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="n">pop_rear</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">deq</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>       <span class="c1"># Remove the rear element</span>
 <a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>
-<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="c1"># Get deque length</span>
+<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="c1"># Get the deque size</span>
 <a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="n">size</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">deq</span><span class="p">)</span>
 <a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a>
-<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a><span class="c1"># Check if deque is empty</span>
+<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a><span class="c1"># Check whether the deque is empty</span>
 <a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a><span class="n">is_empty</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">deq</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">deque.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Initialize deque */</span>
+<div class="highlight"><span class="filename">deque.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Initialize a deque */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="n">deque</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">deque</span><span class="p">;</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="cm">/* Enqueue elements */</span>
-<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span><span class="w">   </span><span class="c1">// Add to rear</span>
+<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span><span class="w">   </span><span class="c1">// Add at the rear</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
-<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_front</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span><span class="w">  </span><span class="c1">// Add to front</span>
+<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_front</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span><span class="w">  </span><span class="c1">// Add at the front</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_front</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="cm">/* Access elements */</span>
@@ -4834,25 +4906,25 @@
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="kt">int</span><span class="w"> </span><span class="n">back</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">back</span><span class="p">();</span><span class="w">   </span><span class="c1">// Rear element</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="cm">/* Dequeue elements */</span>
-<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="n">deque</span><span class="p">.</span><span class="n">pop_front</span><span class="p">();</span><span class="w">  </span><span class="c1">// Front element dequeue</span>
-<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="n">deque</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span><span class="w">   </span><span class="c1">// Rear element dequeue</span>
+<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="n">deque</span><span class="p">.</span><span class="n">pop_front</span><span class="p">();</span><span class="w">  </span><span class="c1">// Remove the front element</span>
+<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="n">deque</span><span class="p">.</span><span class="n">pop_back</span><span class="p">();</span><span class="w">   </span><span class="c1">// Remove the rear element</span>
 <a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a>
-<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="cm">/* Get deque length */</span>
+<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
 <a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a>
-<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="cm">/* Check if deque is empty */</span>
+<a id="__codelineno-1-22" name="__codelineno-1-22" href="#__codelineno-1-22"></a><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-1-23" name="__codelineno-1-23" href="#__codelineno-1-23"></a><span class="kt">bool</span><span class="w"> </span><span class="n">empty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">empty</span><span class="p">();</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">deque.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Initialize deque */</span>
+<div class="highlight"><span class="filename">deque.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Initialize a deque */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="n">Deque</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">deque</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">LinkedList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="cm">/* Enqueue elements */</span>
-<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="n">deque</span><span class="p">.</span><span class="na">offerLast</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span><span class="w">   </span><span class="c1">// Add to rear</span>
+<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="n">deque</span><span class="p">.</span><span class="na">offerLast</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span><span class="w">   </span><span class="c1">// Add at the rear</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="n">deque</span><span class="p">.</span><span class="na">offerLast</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="n">deque</span><span class="p">.</span><span class="na">offerLast</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
-<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="n">deque</span><span class="p">.</span><span class="na">offerFirst</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span><span class="w">  </span><span class="c1">// Add to front</span>
+<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="n">deque</span><span class="p">.</span><span class="na">offerFirst</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span><span class="w">  </span><span class="c1">// Add at the front</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="n">deque</span><span class="p">.</span><span class="na">offerFirst</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="cm">/* Access elements */</span>
@@ -4860,26 +4932,26 @@
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="kt">int</span><span class="w"> </span><span class="n">peekLast</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="na">peekLast</span><span class="p">();</span><span class="w">    </span><span class="c1">// Rear element</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="cm">/* Dequeue elements */</span>
-<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="kt">int</span><span class="w"> </span><span class="n">popFirst</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="na">pollFirst</span><span class="p">();</span><span class="w">  </span><span class="c1">// Front element dequeue</span>
-<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="kt">int</span><span class="w"> </span><span class="n">popLast</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="na">pollLast</span><span class="p">();</span><span class="w">    </span><span class="c1">// Rear element dequeue</span>
+<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="kt">int</span><span class="w"> </span><span class="n">popFirst</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="na">pollFirst</span><span class="p">();</span><span class="w">  </span><span class="c1">// Remove the front element</span>
+<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="kt">int</span><span class="w"> </span><span class="n">popLast</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="na">pollLast</span><span class="p">();</span><span class="w">    </span><span class="c1">// Remove the rear element</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a>
-<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="cm">/* Get deque length */</span>
+<a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="na">size</span><span class="p">();</span>
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a>
-<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="cm">/* Check if deque is empty */</span>
+<a id="__codelineno-2-22" name="__codelineno-2-22" href="#__codelineno-2-22"></a><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-2-23" name="__codelineno-2-23" href="#__codelineno-2-23"></a><span class="kt">boolean</span><span class="w"> </span><span class="n">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="na">isEmpty</span><span class="p">();</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">deque.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Initialize deque */</span>
-<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="c1">// In C#, use LinkedList as a deque</span>
+<div class="highlight"><span class="filename">deque.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Initialize a deque */</span>
+<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="c1">// In C#, use a LinkedList as a deque</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="n">LinkedList</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">deque</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">();</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="cm">/* Enqueue elements */</span>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="n">deque</span><span class="p">.</span><span class="n">AddLast</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span><span class="w">   </span><span class="c1">// Add to rear</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="n">deque</span><span class="p">.</span><span class="n">AddLast</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span><span class="w">   </span><span class="c1">// Add at the rear</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="n">deque</span><span class="p">.</span><span class="n">AddLast</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="n">deque</span><span class="p">.</span><span class="n">AddLast</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
-<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="n">deque</span><span class="p">.</span><span class="n">AddFirst</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span><span class="w">  </span><span class="c1">// Add to front</span>
+<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="n">deque</span><span class="p">.</span><span class="n">AddFirst</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span><span class="w">  </span><span class="c1">// Add at the front</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="n">deque</span><span class="p">.</span><span class="n">AddFirst</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="cm">/* Access elements */</span>
@@ -4887,26 +4959,26 @@
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="kt">int</span><span class="w"> </span><span class="n">peekLast</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">Last</span><span class="p">.</span><span class="n">Value</span><span class="p">;</span><span class="w">    </span><span class="c1">// Rear element</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="cm">/* Dequeue elements */</span>
-<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="n">deque</span><span class="p">.</span><span class="n">RemoveFirst</span><span class="p">();</span><span class="w">  </span><span class="c1">// Front element dequeue</span>
-<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="n">deque</span><span class="p">.</span><span class="n">RemoveLast</span><span class="p">();</span><span class="w">   </span><span class="c1">// Rear element dequeue</span>
+<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="n">deque</span><span class="p">.</span><span class="n">RemoveFirst</span><span class="p">();</span><span class="w">  </span><span class="c1">// Remove the front element</span>
+<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="n">deque</span><span class="p">.</span><span class="n">RemoveLast</span><span class="p">();</span><span class="w">   </span><span class="c1">// Remove the rear element</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a>
-<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="cm">/* Get deque length */</span>
+<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">Count</span><span class="p">;</span>
 <a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a>
-<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="cm">/* Check if deque is empty */</span>
+<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">deque_test.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Initialize deque */</span>
-<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="c1">// In Go, use list as a deque</span>
+<div class="highlight"><span class="filename">deque_test.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Initialize a deque */</span>
+<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="c1">// In Go, use a list as a deque</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="nx">deque</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">list</span><span class="p">.</span><span class="nx">New</span><span class="p">()</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="cm">/* Enqueue elements */</span>
-<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">PushBack</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="w">      </span><span class="c1">// Add to rear</span>
+<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">PushBack</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="w">      </span><span class="c1">// Add at the rear</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">PushBack</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">PushBack</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">PushFront</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="w">     </span><span class="c1">// Add to front</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">PushFront</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="w">     </span><span class="c1">// Add at the front</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">PushFront</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="cm">/* Access elements */</span>
@@ -4914,26 +4986,26 @@
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="nx">rear</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">deque</span><span class="p">.</span><span class="nx">Back</span><span class="p">()</span><span class="w">   </span><span class="c1">// Rear element</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="cm">/* Dequeue elements */</span>
-<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">Remove</span><span class="p">(</span><span class="nx">front</span><span class="p">)</span><span class="w">    </span><span class="c1">// Front element dequeue</span>
-<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">Remove</span><span class="p">(</span><span class="nx">rear</span><span class="p">)</span><span class="w">     </span><span class="c1">// Rear element dequeue</span>
+<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">Remove</span><span class="p">(</span><span class="nx">front</span><span class="p">)</span><span class="w">    </span><span class="c1">// Remove the front element</span>
+<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">Remove</span><span class="p">(</span><span class="nx">rear</span><span class="p">)</span><span class="w">     </span><span class="c1">// Remove the rear element</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a>
-<a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="cm">/* Get deque length */</span>
+<a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="nx">size</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">deque</span><span class="p">.</span><span class="nx">Len</span><span class="p">()</span>
 <a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a>
-<a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="cm">/* Check if deque is empty */</span>
+<a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-4-24" name="__codelineno-4-24" href="#__codelineno-4-24"></a><span class="nx">isEmpty</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">deque</span><span class="p">.</span><span class="nx">Len</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">deque.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Initialize deque */</span>
-<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Swift does not have a built-in deque class, can use Array as a deque</span>
+<div class="highlight"><span class="filename">deque.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Initialize a deque */</span>
+<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Swift has no built-in deque class; use an Array as a deque</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="kd">var</span><span class="w"> </span><span class="nv">deque</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="cm">/* Enqueue elements */</span>
-<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="n">deque</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="c1">// Add to rear</span>
+<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="n">deque</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="c1">// Add at the rear</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="n">deque</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="n">deque</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
-<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="n">deque</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">at</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// Add to front</span>
+<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="n">deque</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="n">at</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1">// Add at the front</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="n">deque</span><span class="p">.</span><span class="bp">insert</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="n">at</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="cm">/* Access elements */</span>
@@ -4941,27 +5013,27 @@
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="kd">let</span><span class="w"> </span><span class="nv">peekLast</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="bp">last</span><span class="p">!</span><span class="w"> </span><span class="c1">// Rear element</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="cm">/* Dequeue elements */</span>
-<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="c1">// When using Array simulation, popFirst has O(n) complexity</span>
-<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="kd">let</span><span class="w"> </span><span class="nv">popFirst</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">removeFirst</span><span class="p">()</span><span class="w"> </span><span class="c1">// Front element dequeue</span>
-<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="kd">let</span><span class="w"> </span><span class="nv">popLast</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span><span class="w"> </span><span class="c1">// Rear element dequeue</span>
+<a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="c1">// When using an Array, popFirst takes O(n) time</span>
+<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="kd">let</span><span class="w"> </span><span class="nv">popFirst</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">removeFirst</span><span class="p">()</span><span class="w"> </span><span class="c1">// Remove the front element</span>
+<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="kd">let</span><span class="w"> </span><span class="nv">popLast</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span><span class="w"> </span><span class="c1">// Remove the rear element</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a>
-<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="cm">/* Get deque length */</span>
+<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="kd">let</span><span class="w"> </span><span class="nv">size</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a>
-<a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="cm">/* Check if deque is empty */</span>
+<a id="__codelineno-5-24" name="__codelineno-5-24" href="#__codelineno-5-24"></a><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-5-25" name="__codelineno-5-25" href="#__codelineno-5-25"></a><span class="kd">let</span><span class="w"> </span><span class="nv">isEmpty</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="bp">isEmpty</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">deque.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Initialize deque */</span>
-<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="c1">// JavaScript does not have a built-in deque, can only use Array as a deque</span>
+<div class="highlight"><span class="filename">deque.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Initialize a deque */</span>
+<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="c1">// JavaScript has no built-in deque; use an Array instead</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="kd">const</span><span class="w"> </span><span class="nx">deque</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="cm">/* Enqueue elements */</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">2</span><span class="p">);</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">5</span><span class="p">);</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">4</span><span class="p">);</span>
-<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="c1">// Please note that since it's an array, unshift() has O(n) time complexity</span>
+<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="c1">// Note that this is an array, so unshift() takes O(n) time</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">unshift</span><span class="p">(</span><span class="mf">3</span><span class="p">);</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">unshift</span><span class="p">(</span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a>
@@ -4970,27 +5042,27 @@
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="kd">const</span><span class="w"> </span><span class="nx">peekLast</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">deque</span><span class="p">[</span><span class="nx">deque</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="cm">/* Dequeue elements */</span>
-<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="c1">// Please note that since it's an array, shift() has O(n) time complexity</span>
+<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="c1">// Note that this is an array, so shift() takes O(n) time</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="kd">const</span><span class="w"> </span><span class="nx">popFront</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">deque</span><span class="p">.</span><span class="nx">shift</span><span class="p">();</span>
 <a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="kd">const</span><span class="w"> </span><span class="nx">popBack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">deque</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a>
-<a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="cm">/* Get deque length */</span>
+<a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="kd">const</span><span class="w"> </span><span class="nx">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">deque</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-6-24" name="__codelineno-6-24" href="#__codelineno-6-24"></a>
-<a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="cm">/* Check if deque is empty */</span>
+<a id="__codelineno-6-25" name="__codelineno-6-25" href="#__codelineno-6-25"></a><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-6-26" name="__codelineno-6-26" href="#__codelineno-6-26"></a><span class="kd">const</span><span class="w"> </span><span class="nx">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">size</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">deque.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Initialize deque */</span>
-<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="c1">// TypeScript does not have a built-in deque, can only use Array as a deque</span>
+<div class="highlight"><span class="filename">deque.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Initialize a deque */</span>
+<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="c1">// TypeScript has no built-in deque; use an Array instead</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="kd">const</span><span class="w"> </span><span class="nx">deque</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="cm">/* Enqueue elements */</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">2</span><span class="p">);</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">5</span><span class="p">);</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">4</span><span class="p">);</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="c1">// Please note that since it's an array, unshift() has O(n) time complexity</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="c1">// Note that this is an array, so unshift() takes O(n) time</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">unshift</span><span class="p">(</span><span class="mf">3</span><span class="p">);</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="nx">deque</span><span class="p">.</span><span class="nx">unshift</span><span class="p">(</span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a>
@@ -4999,27 +5071,27 @@
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="kd">const</span><span class="w"> </span><span class="nx">peekLast</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">deque</span><span class="p">[</span><span class="nx">deque</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="cm">/* Dequeue elements */</span>
-<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="c1">// Please note that since it's an array, shift() has O(n) time complexity</span>
+<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="c1">// Note that this is an array, so shift() takes O(n) time</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="kd">const</span><span class="w"> </span><span class="nx">popFront</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">deque</span><span class="p">.</span><span class="nx">shift</span><span class="p">()</span><span class="w"> </span><span class="kr">as</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="kd">const</span><span class="w"> </span><span class="nx">popBack</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">deque</span><span class="p">.</span><span class="nx">pop</span><span class="p">()</span><span class="w"> </span><span class="kr">as</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span>
 <a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a>
-<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="cm">/* Get deque length */</span>
+<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="kd">const</span><span class="w"> </span><span class="nx">size</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">deque</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-7-24" name="__codelineno-7-24" href="#__codelineno-7-24"></a>
-<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="cm">/* Check if deque is empty */</span>
+<a id="__codelineno-7-25" name="__codelineno-7-25" href="#__codelineno-7-25"></a><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-7-26" name="__codelineno-7-26" href="#__codelineno-7-26"></a><span class="kd">const</span><span class="w"> </span><span class="nx">isEmpty</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">size</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">deque.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Initialize deque */</span>
+<div class="highlight"><span class="filename">deque.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Initialize a deque */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="c1">// In Dart, Queue is defined as a deque</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="n">Queue</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">deque</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Queue</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="p">();</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="cm">/* Enqueue elements */</span>
-<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="n">deque</span><span class="p">.</span><span class="n">addLast</span><span class="p">(</span><span class="m">2</span><span class="p">);</span><span class="w">  </span><span class="c1">// Add to rear</span>
+<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="n">deque</span><span class="p">.</span><span class="n">addLast</span><span class="p">(</span><span class="m">2</span><span class="p">);</span><span class="w">  </span><span class="c1">// Add at the rear</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="n">deque</span><span class="p">.</span><span class="n">addLast</span><span class="p">(</span><span class="m">5</span><span class="p">);</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="n">deque</span><span class="p">.</span><span class="n">addLast</span><span class="p">(</span><span class="m">4</span><span class="p">);</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="n">deque</span><span class="p">.</span><span class="n">addFirst</span><span class="p">(</span><span class="m">3</span><span class="p">);</span><span class="w"> </span><span class="c1">// Add to front</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="n">deque</span><span class="p">.</span><span class="n">addFirst</span><span class="p">(</span><span class="m">3</span><span class="p">);</span><span class="w"> </span><span class="c1">// Add at the front</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="n">deque</span><span class="p">.</span><span class="n">addFirst</span><span class="p">(</span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a>
 <a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="cm">/* Access elements */</span>
@@ -5027,25 +5099,25 @@
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="kt">int</span><span class="w"> </span><span class="n">peekLast</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">last</span><span class="p">;</span><span class="w">   </span><span class="c1">// Rear element</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="cm">/* Dequeue elements */</span>
-<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="kt">int</span><span class="w"> </span><span class="n">popFirst</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">removeFirst</span><span class="p">();</span><span class="w"> </span><span class="c1">// Front element dequeue</span>
-<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="kt">int</span><span class="w"> </span><span class="n">popLast</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">removeLast</span><span class="p">();</span><span class="w">   </span><span class="c1">// Rear element dequeue</span>
+<a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="kt">int</span><span class="w"> </span><span class="n">popFirst</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">removeFirst</span><span class="p">();</span><span class="w"> </span><span class="c1">// Remove the front element</span>
+<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="kt">int</span><span class="w"> </span><span class="n">popLast</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">removeLast</span><span class="p">();</span><span class="w">   </span><span class="c1">// Remove the rear element</span>
 <a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a>
-<a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="cm">/* Get deque length */</span>
+<a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a>
-<a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="cm">/* Check if deque is empty */</span>
+<a id="__codelineno-8-23" name="__codelineno-8-23" href="#__codelineno-8-23"></a><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-8-24" name="__codelineno-8-24" href="#__codelineno-8-24"></a><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">isEmpty</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">deque.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Initialize deque */</span>
+<div class="highlight"><span class="filename">deque.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Initialize a deque */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">deque</span><span class="p">:</span><span class="w"> </span><span class="nc">VecDeque</span><span class="o">&lt;</span><span class="kt">u32</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">VecDeque</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="cm">/* Enqueue elements */</span>
-<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span><span class="w">  </span><span class="c1">// Add to rear</span>
+<a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span><span class="w">  </span><span class="c1">// Add at the rear</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
-<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_front</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span><span class="w"> </span><span class="c1">// Add to front</span>
+<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_front</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span><span class="w"> </span><span class="c1">// Add at the front</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_front</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="cm">/* Access elements */</span>
@@ -5055,31 +5127,31 @@
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="p">}</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="cm">/* Dequeue elements */</span>
-<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">pop_front</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">pop_front</span><span class="p">()</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// Front element dequeue</span>
+<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">pop_front</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">pop_front</span><span class="p">()</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="c1">// Remove the front element</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="p">}</span>
-<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">pop_rear</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">pop_back</span><span class="p">()</span><span class="w"> </span><span class="p">{</span><span class="w">   </span><span class="c1">// Rear element dequeue</span>
+<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">pop_rear</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">pop_back</span><span class="p">()</span><span class="w"> </span><span class="p">{</span><span class="w">   </span><span class="c1">// Remove the rear element</span>
 <a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="p">}</span>
 <a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a>
-<a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="cm">/* Get deque length */</span>
+<a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a><span class="kd">let</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
 <a id="__codelineno-9-25" name="__codelineno-9-25" href="#__codelineno-9-25"></a>
-<a id="__codelineno-9-26" name="__codelineno-9-26" href="#__codelineno-9-26"></a><span class="cm">/* Check if deque is empty */</span>
+<a id="__codelineno-9-26" name="__codelineno-9-26" href="#__codelineno-9-26"></a><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-9-27" name="__codelineno-9-27" href="#__codelineno-9-27"></a><span class="kd">let</span><span class="w"> </span><span class="n">is_empty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">is_empty</span><span class="p">();</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">deque.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// C does not provide a built-in deque</span>
+<div class="highlight"><span class="filename">deque.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// C provides no built-in deque</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">deque.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Initialize deque */</span>
+<div class="highlight"><span class="filename">deque.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Initialize a deque */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">val</span><span class="w"> </span><span class="nv">deque</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">LinkedList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="cm">/* Enqueue elements */</span>
-<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="n">deque</span><span class="p">.</span><span class="na">offerLast</span><span class="p">(</span><span class="m">2</span><span class="p">)</span><span class="w">  </span><span class="c1">// Add to rear</span>
+<a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="n">deque</span><span class="p">.</span><span class="na">offerLast</span><span class="p">(</span><span class="m">2</span><span class="p">)</span><span class="w">  </span><span class="c1">// Add at the rear</span>
 <a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="n">deque</span><span class="p">.</span><span class="na">offerLast</span><span class="p">(</span><span class="m">5</span><span class="p">)</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="n">deque</span><span class="p">.</span><span class="na">offerLast</span><span class="p">(</span><span class="m">4</span><span class="p">)</span>
-<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="n">deque</span><span class="p">.</span><span class="na">offerFirst</span><span class="p">(</span><span class="m">3</span><span class="p">)</span><span class="w"> </span><span class="c1">// Add to front</span>
+<a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="n">deque</span><span class="p">.</span><span class="na">offerFirst</span><span class="p">(</span><span class="m">3</span><span class="p">)</span><span class="w"> </span><span class="c1">// Add at the front</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="n">deque</span><span class="p">.</span><span class="na">offerFirst</span><span class="p">(</span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="cm">/* Access elements */</span>
@@ -5087,26 +5159,26 @@
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="kd">val</span><span class="w"> </span><span class="nv">peekLast</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="na">peekLast</span><span class="p">()</span><span class="w">   </span><span class="c1">// Rear element</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="cm">/* Dequeue elements */</span>
-<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="kd">val</span><span class="w"> </span><span class="nv">popFirst</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="na">pollFirst</span><span class="p">()</span><span class="w"> </span><span class="c1">// Front element dequeue</span>
-<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="kd">val</span><span class="w"> </span><span class="nv">popLast</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="na">pollLast</span><span class="p">()</span><span class="w">   </span><span class="c1">// Rear element dequeue</span>
+<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="kd">val</span><span class="w"> </span><span class="nv">popFirst</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="na">pollFirst</span><span class="p">()</span><span class="w"> </span><span class="c1">// Remove the front element</span>
+<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="kd">val</span><span class="w"> </span><span class="nv">popLast</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="na">pollLast</span><span class="p">()</span><span class="w">   </span><span class="c1">// Remove the rear element</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a>
-<a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="cm">/* Get deque length */</span>
+<a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="kd">val</span><span class="w"> </span><span class="nv">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a>
-<a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="cm">/* Check if deque is empty */</span>
+<a id="__codelineno-11-22" name="__codelineno-11-22" href="#__codelineno-11-22"></a><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-11-23" name="__codelineno-11-23" href="#__codelineno-11-23"></a><span class="kd">val</span><span class="w"> </span><span class="nv">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="na">isEmpty</span><span class="p">()</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">deque.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Initialize deque</span>
-<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="c1"># Ruby does not have a built-in deque, can only use Array as a deque</span>
+<div class="highlight"><span class="filename">deque.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Initialize a deque</span>
+<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="c1"># Ruby has no built-in deque; use an Array instead</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="n">deque</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="c1"># Enqueue elements</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="n">deque</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="n">deque</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">5</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="n">deque</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">4</span>
-<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="c1"># Please note that since it's an array, Array#unshift has O(n) time complexity</span>
+<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="c1"># Note that this is an array, so Array#unshift takes O(n) time</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="n">deque</span><span class="o">.</span><span class="n">unshift</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="n">deque</span><span class="o">.</span><span class="n">unshift</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a>
@@ -5115,34 +5187,34 @@
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="n">peek_last</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="o">.</span><span class="n">last</span>
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="c1"># Dequeue elements</span>
-<a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="c1"># Please note that since it's an array, Array#shift has O(n) time complexity</span>
+<a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="c1"># Note that this is an array, so Array#shift takes O(n) time</span>
 <a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="n">pop_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="o">.</span><span class="n">shift</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="n">pop_back</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="o">.</span><span class="n">pop</span>
 <a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a>
-<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="c1"># Get deque length</span>
+<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="c1"># Get the deque size</span>
 <a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-12-24" name="__codelineno-12-24" href="#__codelineno-12-24"></a>
-<a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="c1"># Check if deque is empty</span>
+<a id="__codelineno-12-25" name="__codelineno-12-25" href="#__codelineno-12-25"></a><span class="c1"># Check whether the deque is empty</span>
 <a id="__codelineno-12-26" name="__codelineno-12-26" href="#__codelineno-12-26"></a><span class="n">is_empty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">size</span><span class="o">.</span><span class="n">zero?</span>
 </code></pre></div>
 </div>
 </div>
 </div>
 <details class="pythontutor">
-<summary>Code Visualization</summary>
-<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=from%20collections%20import%20deque%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%0A%20%20%20%20deq%20%3D%20deque%28%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%85%A5%E9%98%9F%0A%20%20%20%20deq.append%282%29%20%20%23%20%E6%B7%BB%E5%8A%A0%E8%87%B3%E9%98%9F%E5%B0%BE%0A%20%20%20%20deq.append%285%29%0A%20%20%20%20deq.append%284%29%0A%20%20%20%20deq.appendleft%283%29%20%20%23%20%E6%B7%BB%E5%8A%A0%E8%87%B3%E9%98%9F%E9%A6%96%0A%20%20%20%20deq.appendleft%281%29%0A%20%20%20%20print%28%22%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%20deque%20%3D%22,%20deq%29%0A%0A%20%20%20%20%23%20%E8%AE%BF%E9%97%AE%E5%85%83%E7%B4%A0%0A%20%20%20%20front%20%3D%20deq%5B0%5D%20%20%23%20%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%20front%20%3D%22,%20front%29%0A%20%20%20%20rear%20%3D%20deq%5B-1%5D%20%20%23%20%E9%98%9F%E5%B0%BE%E5%85%83%E7%B4%A0%0A%20%20%20%20print%28%22%E9%98%9F%E5%B0%BE%E5%85%83%E7%B4%A0%20rear%20%3D%22,%20rear%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20pop_front%20%3D%20deq.popleft%28%29%20%20%23%20%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%87%BA%E9%98%9F%E5%85%83%E7%B4%A0%20%20pop_front%20%3D%22,%20pop_front%29%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%87%BA%E9%98%9F%E5%90%8E%20deque%20%3D%22,%20deq%29%0A%20%20%20%20pop_rear%20%3D%20deq.pop%28%29%20%20%23%20%E9%98%9F%E5%B0%BE%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20print%28%22%E9%98%9F%E5%B0%BE%E5%87%BA%E9%98%9F%E5%85%83%E7%B4%A0%20%20pop_rear%20%3D%22,%20pop_rear%29%0A%20%20%20%20print%28%22%E9%98%9F%E5%B0%BE%E5%87%BA%E9%98%9F%E5%90%8E%20deque%20%3D%22,%20deq%29%0A%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E7%9A%84%E9%95%BF%E5%BA%A6%0A%20%20%20%20size%20%3D%20len%28deq%29%0A%20%20%20%20print%28%22%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E9%95%BF%E5%BA%A6%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20%E5%88%A4%E6%96%AD%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%0A%20%20%20%20is_empty%20%3D%20len%28deq%29%20%3D%3D%200%0A%20%20%20%20print%28%22%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%20%3D%22,%20is_empty%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20collections%20import%20deque%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%0A%20%20%20%20deq%20%3D%20deque%28%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%85%A5%E9%98%9F%0A%20%20%20%20deq.append%282%29%20%20%23%20%E6%B7%BB%E5%8A%A0%E8%87%B3%E9%98%9F%E5%B0%BE%0A%20%20%20%20deq.append%285%29%0A%20%20%20%20deq.append%284%29%0A%20%20%20%20deq.appendleft%283%29%20%20%23%20%E6%B7%BB%E5%8A%A0%E8%87%B3%E9%98%9F%E9%A6%96%0A%20%20%20%20deq.appendleft%281%29%0A%20%20%20%20print%28%22%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%20deque%20%3D%22,%20deq%29%0A%0A%20%20%20%20%23%20%E8%AE%BF%E9%97%AE%E5%85%83%E7%B4%A0%0A%20%20%20%20front%20%3D%20deq%5B0%5D%20%20%23%20%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%20front%20%3D%22,%20front%29%0A%20%20%20%20rear%20%3D%20deq%5B-1%5D%20%20%23%20%E9%98%9F%E5%B0%BE%E5%85%83%E7%B4%A0%0A%20%20%20%20print%28%22%E9%98%9F%E5%B0%BE%E5%85%83%E7%B4%A0%20rear%20%3D%22,%20rear%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20pop_front%20%3D%20deq.popleft%28%29%20%20%23%20%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%87%BA%E9%98%9F%E5%85%83%E7%B4%A0%20%20pop_front%20%3D%22,%20pop_front%29%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%87%BA%E9%98%9F%E5%90%8E%20deque%20%3D%22,%20deq%29%0A%20%20%20%20pop_rear%20%3D%20deq.pop%28%29%20%20%23%20%E9%98%9F%E5%B0%BE%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20print%28%22%E9%98%9F%E5%B0%BE%E5%87%BA%E9%98%9F%E5%85%83%E7%B4%A0%20%20pop_rear%20%3D%22,%20pop_rear%29%0A%20%20%20%20print%28%22%E9%98%9F%E5%B0%BE%E5%87%BA%E9%98%9F%E5%90%8E%20deque%20%3D%22,%20deq%29%0A%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E7%9A%84%E9%95%BF%E5%BA%A6%0A%20%20%20%20size%20%3D%20len%28deq%29%0A%20%20%20%20print%28%22%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E9%95%BF%E5%BA%A6%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20%E5%88%A4%E6%96%AD%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%0A%20%20%20%20is_empty%20%3D%20len%28deq%29%20%3D%3D%200%0A%20%20%20%20print%28%22%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%20%3D%22,%20is_empty%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=from%20collections%20import%20deque%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20deque%0A%20%20%20%20deq%20%3D%20deque%28%29%0A%0A%20%20%20%20%23%20Enqueue%20elements%0A%20%20%20%20deq.append%282%29%20%20%23%20Add%20at%20the%20rear%0A%20%20%20%20deq.append%285%29%0A%20%20%20%20deq.append%284%29%0A%20%20%20%20deq.appendleft%283%29%20%20%23%20Add%20at%20the%20front%0A%20%20%20%20deq.appendleft%281%29%0A%20%20%20%20print%28%22Deque%3A%20deque%20%3D%22%2C%20deq%29%0A%0A%20%20%20%20%23%20Access%20elements%0A%20%20%20%20front%20%3D%20deq%5B0%5D%20%20%23%20Front%20element%0A%20%20%20%20print%28%22Front%20element%3A%20front%20%3D%22%2C%20front%29%0A%20%20%20%20rear%20%3D%20deq%5B-1%5D%20%20%23%20Rear%20element%0A%20%20%20%20print%28%22Rear%20element%3A%20rear%20%3D%22%2C%20rear%29%0A%0A%20%20%20%20%23%20Dequeue%20elements%0A%20%20%20%20pop_front%20%3D%20deq.popleft%28%29%20%20%23%20Remove%20the%20front%20element%0A%20%20%20%20print%28%22Element%20removed%20from%20the%20front%3A%20pop_front%20%3D%22%2C%20pop_front%29%0A%20%20%20%20print%28%22After%20removing%20from%20the%20front%3A%20deque%20%3D%22%2C%20deq%29%0A%20%20%20%20pop_rear%20%3D%20deq.pop%28%29%20%20%23%20Remove%20the%20rear%20element%0A%20%20%20%20print%28%22Element%20removed%20from%20the%20rear%3A%20pop_rear%20%3D%22%2C%20pop_rear%29%0A%20%20%20%20print%28%22After%20removing%20from%20the%20rear%3A%20deque%20%3D%22%2C%20deq%29%0A%0A%20%20%20%20%23%20Get%20the%20deque%20size%0A%20%20%20%20size%20%3D%20len%28deq%29%0A%20%20%20%20print%28%22Deque%20size%3A%20size%20%3D%22%2C%20size%29%0A%0A%20%20%20%20%23%20Check%20whether%20the%20deque%20is%20empty%0A%20%20%20%20is_empty%20%3D%20len%28deq%29%20%3D%3D%200%0A%20%20%20%20print%28%22Is%20the%20deque%20empty%20%3D%22%2C%20is_empty%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20collections%20import%20deque%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20deque%0A%20%20%20%20deq%20%3D%20deque%28%29%0A%0A%20%20%20%20%23%20Enqueue%20elements%0A%20%20%20%20deq.append%282%29%20%20%23%20Add%20at%20the%20rear%0A%20%20%20%20deq.append%285%29%0A%20%20%20%20deq.append%284%29%0A%20%20%20%20deq.appendleft%283%29%20%20%23%20Add%20at%20the%20front%0A%20%20%20%20deq.appendleft%281%29%0A%20%20%20%20print%28%22Deque%3A%20deque%20%3D%22%2C%20deq%29%0A%0A%20%20%20%20%23%20Access%20elements%0A%20%20%20%20front%20%3D%20deq%5B0%5D%20%20%23%20Front%20element%0A%20%20%20%20print%28%22Front%20element%3A%20front%20%3D%22%2C%20front%29%0A%20%20%20%20rear%20%3D%20deq%5B-1%5D%20%20%23%20Rear%20element%0A%20%20%20%20print%28%22Rear%20element%3A%20rear%20%3D%22%2C%20rear%29%0A%0A%20%20%20%20%23%20Dequeue%20elements%0A%20%20%20%20pop_front%20%3D%20deq.popleft%28%29%20%20%23%20Remove%20the%20front%20element%0A%20%20%20%20print%28%22Element%20removed%20from%20the%20front%3A%20pop_front%20%3D%22%2C%20pop_front%29%0A%20%20%20%20print%28%22After%20removing%20from%20the%20front%3A%20deque%20%3D%22%2C%20deq%29%0A%20%20%20%20pop_rear%20%3D%20deq.pop%28%29%20%20%23%20Remove%20the%20rear%20element%0A%20%20%20%20print%28%22Element%20removed%20from%20the%20rear%3A%20pop_rear%20%3D%22%2C%20pop_rear%29%0A%20%20%20%20print%28%22After%20removing%20from%20the%20rear%3A%20deque%20%3D%22%2C%20deq%29%0A%0A%20%20%20%20%23%20Get%20the%20deque%20size%0A%20%20%20%20size%20%3D%20len%28deq%29%0A%20%20%20%20print%28%22Deque%20size%3A%20size%20%3D%22%2C%20size%29%0A%0A%20%20%20%20%23%20Check%20whether%20the%20deque%20is%20empty%0A%20%20%20%20is_empty%20%3D%20len%28deq%29%20%3D%3D%200%0A%20%20%20%20print%28%22Is%20the%20deque%20empty%20%3D%22%2C%20is_empty%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<h2 id="532-deque-implementation">5.3.2 &nbsp; Deque Implementation *<a class="headerlink" href="#532-deque-implementation" title="Permanent link">&para;</a></h2>
-<p>The implementation of a deque is similar to that of a queue. You can choose either a linked list or an array as the underlying data structure.</p>
-<h3 id="1-doubly-linked-list-implementation">1. &nbsp; Doubly Linked List Implementation<a class="headerlink" href="#1-doubly-linked-list-implementation" title="Permanent link">&para;</a></h3>
-<p>Reviewing the previous section, we used a regular singly linked list to implement a queue because it conveniently allows deleting the head node (corresponding to dequeue) and adding new nodes after the tail node (corresponding to enqueue).</p>
-<p>For a deque, both the front and rear can perform enqueue and dequeue operations. In other words, a deque needs to implement operations in the opposite direction as well. For this reason, we use a "doubly linked list" as the underlying data structure for the deque.</p>
-<p>As shown in Figure 5-8, we treat the head and tail nodes of the doubly linked list as the front and rear of the deque, implementing functionality to add and remove nodes at both ends.</p>
+<h2 id="532-implementing-a-deque">5.3.2 &nbsp; Implementing a deque *<a class="headerlink" href="#532-implementing-a-deque" title="Permanent link">&para;</a></h2>
+<p>A deque is implemented much like a queue, using either a linked list or an array as the underlying structure.</p>
+<h3 id="1-implementation-with-a-doubly-linked-list">1. &nbsp; Implementation with a doubly linked list<a class="headerlink" href="#1-implementation-with-a-doubly-linked-list" title="Permanent link">&para;</a></h3>
+<p>In the previous section, we used a singly linked list for a queue because it easily supports removing the head for dequeueing and adding after the tail for enqueueing.</p>
+<p>A deque supports both insertion and removal at both ends. It therefore needs the symmetric operations in the other direction as well. For this reason, we use a doubly linked list.</p>
+<p>As shown below, treat the list's head and tail as the deque's front and rear, and support adding and removing nodes at both ends.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:5"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">&lt;1&gt;</label><label for="__tabbed_2_2">&lt;2&gt;</label><label for="__tabbed_2_3">&lt;3&gt;</label><label for="__tabbed_2_4">&lt;4&gt;</label><label for="__tabbed_2_5">&lt;5&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Enqueue and dequeue operations in linked list implementation of deque" class="animation-figure" src="../deque.assets/linkedlist_deque_step1.png" /></p>
+<p><img alt="Insertion and removal in a linked-list deque" class="animation-figure" src="../deque.assets/linkedlist_deque_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="linkedlist_deque_push_last" class="animation-figure" src="../deque.assets/linkedlist_deque_step2_push_last.png" /></p>
@@ -5158,9 +5230,9 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 5-8 &nbsp; Enqueue and dequeue operations in linked list implementation of deque </p>
+<p align="center"> Figure 5-8 &nbsp; Insertion and removal in a linked-list deque </p>
 
-<p>The implementation code is shown below:</p>
+<p>The implementation is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -5170,24 +5242,24 @@
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="w">        </span><span class="sd">"""Constructor"""</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">val</span>
-<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span>  <span class="c1"># Successor node reference</span>
-<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">prev</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span>  <span class="c1"># Predecessor node reference</span>
+<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">next</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span>  <span class="c1"># Reference to the next node</span>
+<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">prev</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span>  <span class="c1"># Reference to the previous node</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a><span class="k">class</span><span class="w"> </span><span class="nc">LinkedListDeque</span><span class="p">:</span>
-<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a><span class="w">    </span><span class="sd">"""Double-ended queue based on doubly linked list implementation"""</span>
+<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a><span class="w">    </span><span class="sd">"""Deque implemented with a doubly linked list"""</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>
 <a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
 <a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a><span class="w">        </span><span class="sd">"""Constructor"""</span>
 <a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_front</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span>  <span class="c1"># Head node front</span>
 <a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span>  <span class="c1"># Tail node rear</span>
-<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Length of the double-ended queue</span>
+<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Deque size</span>
 <a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a>
 <a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a><span class="w">        </span><span class="sd">"""Get the length of the double-ended queue"""</span>
+<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a><span class="w">        </span><span class="sd">"""Get the deque size"""</span>
 <a id="__codelineno-13-21" name="__codelineno-13-21" href="#__codelineno-13-21"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span>
 <a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a>
 <a id="__codelineno-13-23" name="__codelineno-13-23" href="#__codelineno-13-23"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">is_empty</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">bool</span><span class="p">:</span>
-<a id="__codelineno-13-24" name="__codelineno-13-24" href="#__codelineno-13-24"></a><span class="w">        </span><span class="sd">"""Check if the double-ended queue is empty"""</span>
+<a id="__codelineno-13-24" name="__codelineno-13-24" href="#__codelineno-13-24"></a><span class="w">        </span><span class="sd">"""Check whether the deque is empty"""</span>
 <a id="__codelineno-13-25" name="__codelineno-13-25" href="#__codelineno-13-25"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">==</span> <span class="mi">0</span>
 <a id="__codelineno-13-26" name="__codelineno-13-26" href="#__codelineno-13-26"></a>
 <a id="__codelineno-13-27" name="__codelineno-13-27" href="#__codelineno-13-27"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">is_front</span><span class="p">:</span> <span class="nb">bool</span><span class="p">):</span>
@@ -5196,78 +5268,78 @@
 <a id="__codelineno-13-30" name="__codelineno-13-30" href="#__codelineno-13-30"></a>        <span class="c1"># If the linked list is empty, make both front and rear point to node</span>
 <a id="__codelineno-13-31" name="__codelineno-13-31" href="#__codelineno-13-31"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_empty</span><span class="p">():</span>
 <a id="__codelineno-13-32" name="__codelineno-13-32" href="#__codelineno-13-32"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span> <span class="o">=</span> <span class="n">node</span>
-<a id="__codelineno-13-33" name="__codelineno-13-33" href="#__codelineno-13-33"></a>        <span class="c1"># Front of the queue enqueue operation</span>
+<a id="__codelineno-13-33" name="__codelineno-13-33" href="#__codelineno-13-33"></a>        <span class="c1"># Enqueue at the front</span>
 <a id="__codelineno-13-34" name="__codelineno-13-34" href="#__codelineno-13-34"></a>        <span class="k">elif</span> <span class="n">is_front</span><span class="p">:</span>
 <a id="__codelineno-13-35" name="__codelineno-13-35" href="#__codelineno-13-35"></a>            <span class="c1"># Add node to the head of the linked list</span>
 <a id="__codelineno-13-36" name="__codelineno-13-36" href="#__codelineno-13-36"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_front</span><span class="o">.</span><span class="n">prev</span> <span class="o">=</span> <span class="n">node</span>
 <a id="__codelineno-13-37" name="__codelineno-13-37" href="#__codelineno-13-37"></a>            <span class="n">node</span><span class="o">.</span><span class="n">next</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_front</span>
-<a id="__codelineno-13-38" name="__codelineno-13-38" href="#__codelineno-13-38"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">=</span> <span class="n">node</span>  <span class="c1"># Update head node</span>
-<a id="__codelineno-13-39" name="__codelineno-13-39" href="#__codelineno-13-39"></a>        <span class="c1"># Rear of the queue enqueue operation</span>
+<a id="__codelineno-13-38" name="__codelineno-13-38" href="#__codelineno-13-38"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">=</span> <span class="n">node</span>  <span class="c1"># Update the head node</span>
+<a id="__codelineno-13-39" name="__codelineno-13-39" href="#__codelineno-13-39"></a>        <span class="c1"># Enqueue at the rear</span>
 <a id="__codelineno-13-40" name="__codelineno-13-40" href="#__codelineno-13-40"></a>        <span class="k">else</span><span class="p">:</span>
 <a id="__codelineno-13-41" name="__codelineno-13-41" href="#__codelineno-13-41"></a>            <span class="c1"># Add node to the tail of the linked list</span>
 <a id="__codelineno-13-42" name="__codelineno-13-42" href="#__codelineno-13-42"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span><span class="o">.</span><span class="n">next</span> <span class="o">=</span> <span class="n">node</span>
 <a id="__codelineno-13-43" name="__codelineno-13-43" href="#__codelineno-13-43"></a>            <span class="n">node</span><span class="o">.</span><span class="n">prev</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span>
-<a id="__codelineno-13-44" name="__codelineno-13-44" href="#__codelineno-13-44"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span> <span class="o">=</span> <span class="n">node</span>  <span class="c1"># Update tail node</span>
-<a id="__codelineno-13-45" name="__codelineno-13-45" href="#__codelineno-13-45"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">+=</span> <span class="mi">1</span>  <span class="c1"># Update queue length</span>
+<a id="__codelineno-13-44" name="__codelineno-13-44" href="#__codelineno-13-44"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span> <span class="o">=</span> <span class="n">node</span>  <span class="c1"># Update the tail node</span>
+<a id="__codelineno-13-45" name="__codelineno-13-45" href="#__codelineno-13-45"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">+=</span> <span class="mi">1</span>  <span class="c1"># Update the deque size</span>
 <a id="__codelineno-13-46" name="__codelineno-13-46" href="#__codelineno-13-46"></a>
 <a id="__codelineno-13-47" name="__codelineno-13-47" href="#__codelineno-13-47"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">push_first</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-13-48" name="__codelineno-13-48" href="#__codelineno-13-48"></a><span class="w">        </span><span class="sd">"""Front of the queue enqueue"""</span>
+<a id="__codelineno-13-48" name="__codelineno-13-48" href="#__codelineno-13-48"></a><span class="w">        </span><span class="sd">"""Enqueue at the front"""</span>
 <a id="__codelineno-13-49" name="__codelineno-13-49" href="#__codelineno-13-49"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="kc">True</span><span class="p">)</span>
 <a id="__codelineno-13-50" name="__codelineno-13-50" href="#__codelineno-13-50"></a>
 <a id="__codelineno-13-51" name="__codelineno-13-51" href="#__codelineno-13-51"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">push_last</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-13-52" name="__codelineno-13-52" href="#__codelineno-13-52"></a><span class="w">        </span><span class="sd">"""Rear of the queue enqueue"""</span>
+<a id="__codelineno-13-52" name="__codelineno-13-52" href="#__codelineno-13-52"></a><span class="w">        </span><span class="sd">"""Enqueue at the rear"""</span>
 <a id="__codelineno-13-53" name="__codelineno-13-53" href="#__codelineno-13-53"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">,</span> <span class="kc">False</span><span class="p">)</span>
 <a id="__codelineno-13-54" name="__codelineno-13-54" href="#__codelineno-13-54"></a>
 <a id="__codelineno-13-55" name="__codelineno-13-55" href="#__codelineno-13-55"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">pop</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">is_front</span><span class="p">:</span> <span class="nb">bool</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-13-56" name="__codelineno-13-56" href="#__codelineno-13-56"></a><span class="w">        </span><span class="sd">"""Dequeue operation"""</span>
 <a id="__codelineno-13-57" name="__codelineno-13-57" href="#__codelineno-13-57"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_empty</span><span class="p">():</span>
-<a id="__codelineno-13-58" name="__codelineno-13-58" href="#__codelineno-13-58"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Double-ended queue is empty"</span><span class="p">)</span>
-<a id="__codelineno-13-59" name="__codelineno-13-59" href="#__codelineno-13-59"></a>        <span class="c1"># When the deque has one element, front and rear point to the same node</span>
+<a id="__codelineno-13-58" name="__codelineno-13-58" href="#__codelineno-13-58"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Deque is empty"</span><span class="p">)</span>
+<a id="__codelineno-13-59" name="__codelineno-13-59" href="#__codelineno-13-59"></a>        <span class="c1"># When the deque size is 1, the head and tail point to the same node</span>
 <a id="__codelineno-13-60" name="__codelineno-13-60" href="#__codelineno-13-60"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-13-61" name="__codelineno-13-61" href="#__codelineno-13-61"></a>            <span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_front</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-13-62" name="__codelineno-13-62" href="#__codelineno-13-62"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">=</span> <span class="kc">None</span>
 <a id="__codelineno-13-63" name="__codelineno-13-63" href="#__codelineno-13-63"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span> <span class="o">=</span> <span class="kc">None</span>
-<a id="__codelineno-13-64" name="__codelineno-13-64" href="#__codelineno-13-64"></a>        <span class="c1"># Front of the queue dequeue operation</span>
+<a id="__codelineno-13-64" name="__codelineno-13-64" href="#__codelineno-13-64"></a>        <span class="c1"># Dequeue from the front</span>
 <a id="__codelineno-13-65" name="__codelineno-13-65" href="#__codelineno-13-65"></a>        <span class="k">elif</span> <span class="n">is_front</span><span class="p">:</span>
-<a id="__codelineno-13-66" name="__codelineno-13-66" href="#__codelineno-13-66"></a>            <span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_front</span><span class="o">.</span><span class="n">val</span>  <span class="c1"># Temporarily store head node value</span>
-<a id="__codelineno-13-67" name="__codelineno-13-67" href="#__codelineno-13-67"></a>            <span class="c1"># Delete head node</span>
+<a id="__codelineno-13-66" name="__codelineno-13-66" href="#__codelineno-13-66"></a>            <span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_front</span><span class="o">.</span><span class="n">val</span>  <span class="c1"># Save the head node value</span>
+<a id="__codelineno-13-67" name="__codelineno-13-67" href="#__codelineno-13-67"></a>            <span class="c1"># Remove the head node</span>
 <a id="__codelineno-13-68" name="__codelineno-13-68" href="#__codelineno-13-68"></a>            <span class="n">fnext</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_front</span><span class="o">.</span><span class="n">next</span>
 <a id="__codelineno-13-69" name="__codelineno-13-69" href="#__codelineno-13-69"></a>            <span class="n">fnext</span><span class="o">.</span><span class="n">prev</span> <span class="o">=</span> <span class="kc">None</span>
 <a id="__codelineno-13-70" name="__codelineno-13-70" href="#__codelineno-13-70"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_front</span><span class="o">.</span><span class="n">next</span> <span class="o">=</span> <span class="kc">None</span>
-<a id="__codelineno-13-71" name="__codelineno-13-71" href="#__codelineno-13-71"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">=</span> <span class="n">fnext</span>  <span class="c1"># Update head node</span>
-<a id="__codelineno-13-72" name="__codelineno-13-72" href="#__codelineno-13-72"></a>        <span class="c1"># Rear of the queue dequeue operation</span>
+<a id="__codelineno-13-71" name="__codelineno-13-71" href="#__codelineno-13-71"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">=</span> <span class="n">fnext</span>  <span class="c1"># Update the head node</span>
+<a id="__codelineno-13-72" name="__codelineno-13-72" href="#__codelineno-13-72"></a>        <span class="c1"># Dequeue from the rear</span>
 <a id="__codelineno-13-73" name="__codelineno-13-73" href="#__codelineno-13-73"></a>        <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-13-74" name="__codelineno-13-74" href="#__codelineno-13-74"></a>            <span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span><span class="o">.</span><span class="n">val</span>  <span class="c1"># Temporarily store tail node value</span>
-<a id="__codelineno-13-75" name="__codelineno-13-75" href="#__codelineno-13-75"></a>            <span class="c1"># Delete tail node</span>
+<a id="__codelineno-13-74" name="__codelineno-13-74" href="#__codelineno-13-74"></a>            <span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span><span class="o">.</span><span class="n">val</span>  <span class="c1"># Save the tail node value</span>
+<a id="__codelineno-13-75" name="__codelineno-13-75" href="#__codelineno-13-75"></a>            <span class="c1"># Remove the tail node</span>
 <a id="__codelineno-13-76" name="__codelineno-13-76" href="#__codelineno-13-76"></a>            <span class="n">rprev</span><span class="p">:</span> <span class="n">ListNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span><span class="o">.</span><span class="n">prev</span>
 <a id="__codelineno-13-77" name="__codelineno-13-77" href="#__codelineno-13-77"></a>            <span class="n">rprev</span><span class="o">.</span><span class="n">next</span> <span class="o">=</span> <span class="kc">None</span>
 <a id="__codelineno-13-78" name="__codelineno-13-78" href="#__codelineno-13-78"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span><span class="o">.</span><span class="n">prev</span> <span class="o">=</span> <span class="kc">None</span>
-<a id="__codelineno-13-79" name="__codelineno-13-79" href="#__codelineno-13-79"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span> <span class="o">=</span> <span class="n">rprev</span>  <span class="c1"># Update tail node</span>
-<a id="__codelineno-13-80" name="__codelineno-13-80" href="#__codelineno-13-80"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">-=</span> <span class="mi">1</span>  <span class="c1"># Update queue length</span>
+<a id="__codelineno-13-79" name="__codelineno-13-79" href="#__codelineno-13-79"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span> <span class="o">=</span> <span class="n">rprev</span>  <span class="c1"># Update the tail node</span>
+<a id="__codelineno-13-80" name="__codelineno-13-80" href="#__codelineno-13-80"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">-=</span> <span class="mi">1</span>  <span class="c1"># Update the deque size</span>
 <a id="__codelineno-13-81" name="__codelineno-13-81" href="#__codelineno-13-81"></a>        <span class="k">return</span> <span class="n">val</span>
 <a id="__codelineno-13-82" name="__codelineno-13-82" href="#__codelineno-13-82"></a>
 <a id="__codelineno-13-83" name="__codelineno-13-83" href="#__codelineno-13-83"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">pop_first</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-84" name="__codelineno-13-84" href="#__codelineno-13-84"></a><span class="w">        </span><span class="sd">"""Front of the queue dequeue"""</span>
+<a id="__codelineno-13-84" name="__codelineno-13-84" href="#__codelineno-13-84"></a><span class="w">        </span><span class="sd">"""Dequeue from the front"""</span>
 <a id="__codelineno-13-85" name="__codelineno-13-85" href="#__codelineno-13-85"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
 <a id="__codelineno-13-86" name="__codelineno-13-86" href="#__codelineno-13-86"></a>
 <a id="__codelineno-13-87" name="__codelineno-13-87" href="#__codelineno-13-87"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">pop_last</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-88" name="__codelineno-13-88" href="#__codelineno-13-88"></a><span class="w">        </span><span class="sd">"""Rear of the queue dequeue"""</span>
+<a id="__codelineno-13-88" name="__codelineno-13-88" href="#__codelineno-13-88"></a><span class="w">        </span><span class="sd">"""Dequeue from the rear"""</span>
 <a id="__codelineno-13-89" name="__codelineno-13-89" href="#__codelineno-13-89"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">pop</span><span class="p">(</span><span class="kc">False</span><span class="p">)</span>
 <a id="__codelineno-13-90" name="__codelineno-13-90" href="#__codelineno-13-90"></a>
 <a id="__codelineno-13-91" name="__codelineno-13-91" href="#__codelineno-13-91"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">peek_first</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-92" name="__codelineno-13-92" href="#__codelineno-13-92"></a><span class="w">        </span><span class="sd">"""Access front of the queue element"""</span>
+<a id="__codelineno-13-92" name="__codelineno-13-92" href="#__codelineno-13-92"></a><span class="w">        </span><span class="sd">"""Access the front element"""</span>
 <a id="__codelineno-13-93" name="__codelineno-13-93" href="#__codelineno-13-93"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_empty</span><span class="p">():</span>
-<a id="__codelineno-13-94" name="__codelineno-13-94" href="#__codelineno-13-94"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Double-ended queue is empty"</span><span class="p">)</span>
+<a id="__codelineno-13-94" name="__codelineno-13-94" href="#__codelineno-13-94"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Deque is empty"</span><span class="p">)</span>
 <a id="__codelineno-13-95" name="__codelineno-13-95" href="#__codelineno-13-95"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_front</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-13-96" name="__codelineno-13-96" href="#__codelineno-13-96"></a>
 <a id="__codelineno-13-97" name="__codelineno-13-97" href="#__codelineno-13-97"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">peek_last</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-98" name="__codelineno-13-98" href="#__codelineno-13-98"></a><span class="w">        </span><span class="sd">"""Access rear of the queue element"""</span>
+<a id="__codelineno-13-98" name="__codelineno-13-98" href="#__codelineno-13-98"></a><span class="w">        </span><span class="sd">"""Access the rear element"""</span>
 <a id="__codelineno-13-99" name="__codelineno-13-99" href="#__codelineno-13-99"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_empty</span><span class="p">():</span>
-<a id="__codelineno-13-100" name="__codelineno-13-100" href="#__codelineno-13-100"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Double-ended queue is empty"</span><span class="p">)</span>
+<a id="__codelineno-13-100" name="__codelineno-13-100" href="#__codelineno-13-100"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Deque is empty"</span><span class="p">)</span>
 <a id="__codelineno-13-101" name="__codelineno-13-101" href="#__codelineno-13-101"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-13-102" name="__codelineno-13-102" href="#__codelineno-13-102"></a>
 <a id="__codelineno-13-103" name="__codelineno-13-103" href="#__codelineno-13-103"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">to_array</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
-<a id="__codelineno-13-104" name="__codelineno-13-104" href="#__codelineno-13-104"></a><span class="w">        </span><span class="sd">"""Return array for printing"""</span>
+<a id="__codelineno-13-104" name="__codelineno-13-104" href="#__codelineno-13-104"></a><span class="w">        </span><span class="sd">"""Return an array for printing"""</span>
 <a id="__codelineno-13-105" name="__codelineno-13-105" href="#__codelineno-13-105"></a>        <span class="n">node</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_front</span>
 <a id="__codelineno-13-106" name="__codelineno-13-106" href="#__codelineno-13-106"></a>        <span class="n">res</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span>
 <a id="__codelineno-13-107" name="__codelineno-13-107" href="#__codelineno-13-107"></a>        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()):</span>
@@ -5280,17 +5352,17 @@
 <div class="highlight"><span class="filename">linkedlist_deque.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Doubly linked list node */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="k">struct</span><span class="w"> </span><span class="nc">DoublyListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w">              </span><span class="c1">// Node value</span>
-<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Successor node pointer</span>
-<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Predecessor node pointer</span>
+<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Pointer to the next node</span>
+<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Pointer to the previous node</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="n">DoublyListNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">val</span><span class="p">(</span><span class="n">val</span><span class="p">),</span><span class="w"> </span><span class="n">prev</span><span class="p">(</span><span class="nb">nullptr</span><span class="p">),</span><span class="w"> </span><span class="n">next</span><span class="p">(</span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="p">};</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a>
-<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="cm">/* Double-ended queue based on doubly linked list implementation */</span>
+<a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="cm">/* Deque implemented with a doubly linked list */</span>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="k">class</span><span class="w"> </span><span class="nc">LinkedListDeque</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">  </span><span class="k">private</span><span class="o">:</span>
-<a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">    </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node front, tail node rear</span>
-<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">              </span><span class="c1">// Length of the double-ended queue</span>
+<a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">    </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node front and tail node rear</span>
+<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">              </span><span class="c1">// Deque size</span>
 <a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a>
 <a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">  </span><span class="k">public</span><span class="o">:</span>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -5299,7 +5371,7 @@
 <a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a>
 <a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="w">    </span><span class="cm">/* Destructor */</span>
 <a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a><span class="w">    </span><span class="o">~</span><span class="n">LinkedListDeque</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a><span class="w">        </span><span class="c1">// Traverse linked list to delete nodes and free memory</span>
+<a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a><span class="w">        </span><span class="c1">// Traverse the linked list and delete each node to free its memory</span>
 <a id="__codelineno-14-24" name="__codelineno-14-24" href="#__codelineno-14-24"></a><span class="w">        </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="k">pre</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span>
 <a id="__codelineno-14-25" name="__codelineno-14-25" href="#__codelineno-14-25"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-26" name="__codelineno-14-26" href="#__codelineno-14-26"></a><span class="w">            </span><span class="k">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
@@ -5308,12 +5380,12 @@
 <a id="__codelineno-14-29" name="__codelineno-14-29" href="#__codelineno-14-29"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-14-30" name="__codelineno-14-30" href="#__codelineno-14-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-31" name="__codelineno-14-31" href="#__codelineno-14-31"></a>
-<a id="__codelineno-14-32" name="__codelineno-14-32" href="#__codelineno-14-32"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-14-32" name="__codelineno-14-32" href="#__codelineno-14-32"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-14-33" name="__codelineno-14-33" href="#__codelineno-14-33"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-34" name="__codelineno-14-34" href="#__codelineno-14-34"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-14-35" name="__codelineno-14-35" href="#__codelineno-14-35"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-36" name="__codelineno-14-36" href="#__codelineno-14-36"></a>
-<a id="__codelineno-14-37" name="__codelineno-14-37" href="#__codelineno-14-37"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-14-37" name="__codelineno-14-37" href="#__codelineno-14-37"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-14-38" name="__codelineno-14-38" href="#__codelineno-14-38"></a><span class="w">    </span><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-39" name="__codelineno-14-39" href="#__codelineno-14-39"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-14-40" name="__codelineno-14-40" href="#__codelineno-14-40"></a><span class="w">    </span><span class="p">}</span>
@@ -5324,28 +5396,28 @@
 <a id="__codelineno-14-45" name="__codelineno-14-45" href="#__codelineno-14-45"></a><span class="w">        </span><span class="c1">// If the linked list is empty, make both front and rear point to node</span>
 <a id="__codelineno-14-46" name="__codelineno-14-46" href="#__codelineno-14-46"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-14-47" name="__codelineno-14-47" href="#__codelineno-14-47"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
-<a id="__codelineno-14-48" name="__codelineno-14-48" href="#__codelineno-14-48"></a><span class="w">        </span><span class="c1">// Front of the queue enqueue operation</span>
+<a id="__codelineno-14-48" name="__codelineno-14-48" href="#__codelineno-14-48"></a><span class="w">        </span><span class="c1">// Enqueue at the front</span>
 <a id="__codelineno-14-49" name="__codelineno-14-49" href="#__codelineno-14-49"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-50" name="__codelineno-14-50" href="#__codelineno-14-50"></a><span class="w">            </span><span class="c1">// Add node to the head of the linked list</span>
 <a id="__codelineno-14-51" name="__codelineno-14-51" href="#__codelineno-14-51"></a><span class="w">            </span><span class="n">front</span><span class="o">-&gt;</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-14-52" name="__codelineno-14-52" href="#__codelineno-14-52"></a><span class="w">            </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span>
-<a id="__codelineno-14-53" name="__codelineno-14-53" href="#__codelineno-14-53"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update head node</span>
-<a id="__codelineno-14-54" name="__codelineno-14-54" href="#__codelineno-14-54"></a><span class="w">        </span><span class="c1">// Rear of the queue enqueue operation</span>
+<a id="__codelineno-14-53" name="__codelineno-14-53" href="#__codelineno-14-53"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the head node</span>
+<a id="__codelineno-14-54" name="__codelineno-14-54" href="#__codelineno-14-54"></a><span class="w">        </span><span class="c1">// Enqueue at the rear</span>
 <a id="__codelineno-14-55" name="__codelineno-14-55" href="#__codelineno-14-55"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-56" name="__codelineno-14-56" href="#__codelineno-14-56"></a><span class="w">            </span><span class="c1">// Add node to the tail of the linked list</span>
 <a id="__codelineno-14-57" name="__codelineno-14-57" href="#__codelineno-14-57"></a><span class="w">            </span><span class="n">rear</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-14-58" name="__codelineno-14-58" href="#__codelineno-14-58"></a><span class="w">            </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="p">;</span>
-<a id="__codelineno-14-59" name="__codelineno-14-59" href="#__codelineno-14-59"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-14-59" name="__codelineno-14-59" href="#__codelineno-14-59"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-14-60" name="__codelineno-14-60" href="#__codelineno-14-60"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-14-61" name="__codelineno-14-61" href="#__codelineno-14-61"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-14-61" name="__codelineno-14-61" href="#__codelineno-14-61"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the deque size</span>
 <a id="__codelineno-14-62" name="__codelineno-14-62" href="#__codelineno-14-62"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-63" name="__codelineno-14-63" href="#__codelineno-14-63"></a>
-<a id="__codelineno-14-64" name="__codelineno-14-64" href="#__codelineno-14-64"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-14-64" name="__codelineno-14-64" href="#__codelineno-14-64"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-14-65" name="__codelineno-14-65" href="#__codelineno-14-65"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">pushFirst</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-66" name="__codelineno-14-66" href="#__codelineno-14-66"></a><span class="w">        </span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="nb">true</span><span class="p">);</span>
 <a id="__codelineno-14-67" name="__codelineno-14-67" href="#__codelineno-14-67"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-68" name="__codelineno-14-68" href="#__codelineno-14-68"></a>
-<a id="__codelineno-14-69" name="__codelineno-14-69" href="#__codelineno-14-69"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-14-69" name="__codelineno-14-69" href="#__codelineno-14-69"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-14-70" name="__codelineno-14-70" href="#__codelineno-14-70"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">pushLast</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-71" name="__codelineno-14-71" href="#__codelineno-14-71"></a><span class="w">        </span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span>
 <a id="__codelineno-14-72" name="__codelineno-14-72" href="#__codelineno-14-72"></a><span class="w">    </span><span class="p">}</span>
@@ -5353,63 +5425,63 @@
 <a id="__codelineno-14-74" name="__codelineno-14-74" href="#__codelineno-14-74"></a><span class="w">    </span><span class="cm">/* Dequeue operation */</span>
 <a id="__codelineno-14-75" name="__codelineno-14-75" href="#__codelineno-14-75"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">pop</span><span class="p">(</span><span class="kt">bool</span><span class="w"> </span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-76" name="__codelineno-14-76" href="#__codelineno-14-76"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
-<a id="__codelineno-14-77" name="__codelineno-14-77" href="#__codelineno-14-77"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Queue is empty"</span><span class="p">);</span>
+<a id="__codelineno-14-77" name="__codelineno-14-77" href="#__codelineno-14-77"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Deque is empty"</span><span class="p">);</span>
 <a id="__codelineno-14-78" name="__codelineno-14-78" href="#__codelineno-14-78"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
-<a id="__codelineno-14-79" name="__codelineno-14-79" href="#__codelineno-14-79"></a><span class="w">        </span><span class="c1">// When the deque has one element, front and rear point to the same node</span>
+<a id="__codelineno-14-79" name="__codelineno-14-79" href="#__codelineno-14-79"></a><span class="w">        </span><span class="c1">// When the deque size is 1, the head and tail point to the same node</span>
 <a id="__codelineno-14-80" name="__codelineno-14-80" href="#__codelineno-14-80"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-81" name="__codelineno-14-81" href="#__codelineno-14-81"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
-<a id="__codelineno-14-82" name="__codelineno-14-82" href="#__codelineno-14-82"></a><span class="w">            </span><span class="k">delete</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Free the shared node once</span>
+<a id="__codelineno-14-82" name="__codelineno-14-82" href="#__codelineno-14-82"></a><span class="w">            </span><span class="k">delete</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Both point to the same node, so free it only once</span>
 <a id="__codelineno-14-83" name="__codelineno-14-83" href="#__codelineno-14-83"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">;</span>
 <a id="__codelineno-14-84" name="__codelineno-14-84" href="#__codelineno-14-84"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">;</span>
 <a id="__codelineno-14-85" name="__codelineno-14-85" href="#__codelineno-14-85"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-14-86" name="__codelineno-14-86" href="#__codelineno-14-86"></a><span class="w">        </span><span class="c1">// Temporarily store head node value</span>
+<a id="__codelineno-14-86" name="__codelineno-14-86" href="#__codelineno-14-86"></a><span class="w">        </span><span class="c1">// Dequeue from the front</span>
 <a id="__codelineno-14-87" name="__codelineno-14-87" href="#__codelineno-14-87"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-88" name="__codelineno-14-88" href="#__codelineno-14-88"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Delete head node</span>
-<a id="__codelineno-14-89" name="__codelineno-14-89" href="#__codelineno-14-89"></a><span class="w">            </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-14-88" name="__codelineno-14-88" href="#__codelineno-14-88"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Save the head node value</span>
+<a id="__codelineno-14-89" name="__codelineno-14-89" href="#__codelineno-14-89"></a><span class="w">            </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-14-90" name="__codelineno-14-90" href="#__codelineno-14-90"></a><span class="w">            </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">fNext</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-14-91" name="__codelineno-14-91" href="#__codelineno-14-91"></a><span class="w">            </span><span class="n">fNext</span><span class="o">-&gt;</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">;</span>
 <a id="__codelineno-14-92" name="__codelineno-14-92" href="#__codelineno-14-92"></a><span class="w">            </span><span class="n">front</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">;</span>
 <a id="__codelineno-14-93" name="__codelineno-14-93" href="#__codelineno-14-93"></a><span class="w">            </span><span class="k">delete</span><span class="w"> </span><span class="n">front</span><span class="p">;</span>
-<a id="__codelineno-14-94" name="__codelineno-14-94" href="#__codelineno-14-94"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fNext</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update head node</span>
-<a id="__codelineno-14-95" name="__codelineno-14-95" href="#__codelineno-14-95"></a><span class="w">        </span><span class="c1">// Temporarily store tail node value</span>
+<a id="__codelineno-14-94" name="__codelineno-14-94" href="#__codelineno-14-94"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fNext</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the head node</span>
+<a id="__codelineno-14-95" name="__codelineno-14-95" href="#__codelineno-14-95"></a><span class="w">        </span><span class="c1">// Dequeue from the rear</span>
 <a id="__codelineno-14-96" name="__codelineno-14-96" href="#__codelineno-14-96"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-97" name="__codelineno-14-97" href="#__codelineno-14-97"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Delete tail node</span>
-<a id="__codelineno-14-98" name="__codelineno-14-98" href="#__codelineno-14-98"></a><span class="w">            </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-14-97" name="__codelineno-14-97" href="#__codelineno-14-97"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Save the tail node value</span>
+<a id="__codelineno-14-98" name="__codelineno-14-98" href="#__codelineno-14-98"></a><span class="w">            </span><span class="c1">// Remove the tail node</span>
 <a id="__codelineno-14-99" name="__codelineno-14-99" href="#__codelineno-14-99"></a><span class="w">            </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">rPrev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="o">-&gt;</span><span class="n">prev</span><span class="p">;</span>
 <a id="__codelineno-14-100" name="__codelineno-14-100" href="#__codelineno-14-100"></a><span class="w">            </span><span class="n">rPrev</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">;</span>
 <a id="__codelineno-14-101" name="__codelineno-14-101" href="#__codelineno-14-101"></a><span class="w">            </span><span class="n">rear</span><span class="o">-&gt;</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">;</span>
 <a id="__codelineno-14-102" name="__codelineno-14-102" href="#__codelineno-14-102"></a><span class="w">            </span><span class="k">delete</span><span class="w"> </span><span class="n">rear</span><span class="p">;</span>
-<a id="__codelineno-14-103" name="__codelineno-14-103" href="#__codelineno-14-103"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rPrev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-14-103" name="__codelineno-14-103" href="#__codelineno-14-103"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rPrev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-14-104" name="__codelineno-14-104" href="#__codelineno-14-104"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-14-105" name="__codelineno-14-105" href="#__codelineno-14-105"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-14-105" name="__codelineno-14-105" href="#__codelineno-14-105"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the deque size</span>
 <a id="__codelineno-14-106" name="__codelineno-14-106" href="#__codelineno-14-106"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-14-107" name="__codelineno-14-107" href="#__codelineno-14-107"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-108" name="__codelineno-14-108" href="#__codelineno-14-108"></a>
-<a id="__codelineno-14-109" name="__codelineno-14-109" href="#__codelineno-14-109"></a><span class="w">    </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-14-109" name="__codelineno-14-109" href="#__codelineno-14-109"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-14-110" name="__codelineno-14-110" href="#__codelineno-14-110"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">popFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-111" name="__codelineno-14-111" href="#__codelineno-14-111"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pop</span><span class="p">(</span><span class="nb">true</span><span class="p">);</span>
 <a id="__codelineno-14-112" name="__codelineno-14-112" href="#__codelineno-14-112"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-113" name="__codelineno-14-113" href="#__codelineno-14-113"></a>
-<a id="__codelineno-14-114" name="__codelineno-14-114" href="#__codelineno-14-114"></a><span class="w">    </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-14-114" name="__codelineno-14-114" href="#__codelineno-14-114"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-14-115" name="__codelineno-14-115" href="#__codelineno-14-115"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">popLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-116" name="__codelineno-14-116" href="#__codelineno-14-116"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pop</span><span class="p">(</span><span class="nb">false</span><span class="p">);</span>
 <a id="__codelineno-14-117" name="__codelineno-14-117" href="#__codelineno-14-117"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-118" name="__codelineno-14-118" href="#__codelineno-14-118"></a>
-<a id="__codelineno-14-119" name="__codelineno-14-119" href="#__codelineno-14-119"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-14-119" name="__codelineno-14-119" href="#__codelineno-14-119"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-14-120" name="__codelineno-14-120" href="#__codelineno-14-120"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">peekFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-121" name="__codelineno-14-121" href="#__codelineno-14-121"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-14-122" name="__codelineno-14-122" href="#__codelineno-14-122"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Deque is empty"</span><span class="p">);</span>
 <a id="__codelineno-14-123" name="__codelineno-14-123" href="#__codelineno-14-123"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">front</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-14-124" name="__codelineno-14-124" href="#__codelineno-14-124"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-125" name="__codelineno-14-125" href="#__codelineno-14-125"></a>
-<a id="__codelineno-14-126" name="__codelineno-14-126" href="#__codelineno-14-126"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-14-126" name="__codelineno-14-126" href="#__codelineno-14-126"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-14-127" name="__codelineno-14-127" href="#__codelineno-14-127"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">peekLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-128" name="__codelineno-14-128" href="#__codelineno-14-128"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-14-129" name="__codelineno-14-129" href="#__codelineno-14-129"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Deque is empty"</span><span class="p">);</span>
 <a id="__codelineno-14-130" name="__codelineno-14-130" href="#__codelineno-14-130"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">rear</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-14-131" name="__codelineno-14-131" href="#__codelineno-14-131"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-132" name="__codelineno-14-132" href="#__codelineno-14-132"></a>
-<a id="__codelineno-14-133" name="__codelineno-14-133" href="#__codelineno-14-133"></a><span class="w">    </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-14-133" name="__codelineno-14-133" href="#__codelineno-14-133"></a><span class="w">    </span><span class="cm">/* Return a vector for printing */</span>
 <a id="__codelineno-14-134" name="__codelineno-14-134" href="#__codelineno-14-134"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">toVector</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-135" name="__codelineno-14-135" href="#__codelineno-14-135"></a><span class="w">        </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span>
 <a id="__codelineno-14-136" name="__codelineno-14-136" href="#__codelineno-14-136"></a><span class="w">        </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">(</span><span class="n">size</span><span class="p">());</span>
@@ -5426,8 +5498,8 @@
 <div class="highlight"><span class="filename">linkedlist_deque.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Doubly linked list node */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kd">class</span> <span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Node value</span>
-<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Successor node reference</span>
-<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Predecessor node reference</span>
+<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference to the next node</span>
+<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference to the previous node</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="n">ListNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
@@ -5435,21 +5507,21 @@
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="p">}</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a>
-<a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="cm">/* Double-ended queue based on doubly linked list implementation */</span>
+<a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="cm">/* Deque implemented with a doubly linked list */</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="kd">class</span> <span class="nc">LinkedListDeque</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="n">rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node front, tail node rear</span>
-<a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Length of the double-ended queue</span>
+<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="n">rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node front and tail node rear</span>
+<a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Deque size</span>
 <a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a>
 <a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="nf">LinkedListDeque</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a>
-<a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-24" name="__codelineno-15-24" href="#__codelineno-15-24"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-15-25" name="__codelineno-15-25" href="#__codelineno-15-25"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-26" name="__codelineno-15-26" href="#__codelineno-15-26"></a>
-<a id="__codelineno-15-27" name="__codelineno-15-27" href="#__codelineno-15-27"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-15-27" name="__codelineno-15-27" href="#__codelineno-15-27"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-15-28" name="__codelineno-15-28" href="#__codelineno-15-28"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">boolean</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-29" name="__codelineno-15-29" href="#__codelineno-15-29"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-15-30" name="__codelineno-15-30" href="#__codelineno-15-30"></a><span class="w">    </span><span class="p">}</span>
@@ -5460,28 +5532,28 @@
 <a id="__codelineno-15-35" name="__codelineno-15-35" href="#__codelineno-15-35"></a><span class="w">        </span><span class="c1">// If the linked list is empty, make both front and rear point to node</span>
 <a id="__codelineno-15-36" name="__codelineno-15-36" href="#__codelineno-15-36"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-15-37" name="__codelineno-15-37" href="#__codelineno-15-37"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
-<a id="__codelineno-15-38" name="__codelineno-15-38" href="#__codelineno-15-38"></a><span class="w">        </span><span class="c1">// Front of the queue enqueue operation</span>
+<a id="__codelineno-15-38" name="__codelineno-15-38" href="#__codelineno-15-38"></a><span class="w">        </span><span class="c1">// Enqueue at the front</span>
 <a id="__codelineno-15-39" name="__codelineno-15-39" href="#__codelineno-15-39"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-40" name="__codelineno-15-40" href="#__codelineno-15-40"></a><span class="w">            </span><span class="c1">// Add node to the head of the linked list</span>
 <a id="__codelineno-15-41" name="__codelineno-15-41" href="#__codelineno-15-41"></a><span class="w">            </span><span class="n">front</span><span class="p">.</span><span class="na">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-15-42" name="__codelineno-15-42" href="#__codelineno-15-42"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span>
-<a id="__codelineno-15-43" name="__codelineno-15-43" href="#__codelineno-15-43"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update head node</span>
-<a id="__codelineno-15-44" name="__codelineno-15-44" href="#__codelineno-15-44"></a><span class="w">        </span><span class="c1">// Rear of the queue enqueue operation</span>
+<a id="__codelineno-15-43" name="__codelineno-15-43" href="#__codelineno-15-43"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the head node</span>
+<a id="__codelineno-15-44" name="__codelineno-15-44" href="#__codelineno-15-44"></a><span class="w">        </span><span class="c1">// Enqueue at the rear</span>
 <a id="__codelineno-15-45" name="__codelineno-15-45" href="#__codelineno-15-45"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-46" name="__codelineno-15-46" href="#__codelineno-15-46"></a><span class="w">            </span><span class="c1">// Add node to the tail of the linked list</span>
 <a id="__codelineno-15-47" name="__codelineno-15-47" href="#__codelineno-15-47"></a><span class="w">            </span><span class="n">rear</span><span class="p">.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-15-48" name="__codelineno-15-48" href="#__codelineno-15-48"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="na">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="p">;</span>
-<a id="__codelineno-15-49" name="__codelineno-15-49" href="#__codelineno-15-49"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-15-49" name="__codelineno-15-49" href="#__codelineno-15-49"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-15-50" name="__codelineno-15-50" href="#__codelineno-15-50"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-15-51" name="__codelineno-15-51" href="#__codelineno-15-51"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-15-51" name="__codelineno-15-51" href="#__codelineno-15-51"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the queue size</span>
 <a id="__codelineno-15-52" name="__codelineno-15-52" href="#__codelineno-15-52"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-53" name="__codelineno-15-53" href="#__codelineno-15-53"></a>
-<a id="__codelineno-15-54" name="__codelineno-15-54" href="#__codelineno-15-54"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-15-54" name="__codelineno-15-54" href="#__codelineno-15-54"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-15-55" name="__codelineno-15-55" href="#__codelineno-15-55"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">pushFirst</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-56" name="__codelineno-15-56" href="#__codelineno-15-56"></a><span class="w">        </span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kc">true</span><span class="p">);</span>
 <a id="__codelineno-15-57" name="__codelineno-15-57" href="#__codelineno-15-57"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-58" name="__codelineno-15-58" href="#__codelineno-15-58"></a>
-<a id="__codelineno-15-59" name="__codelineno-15-59" href="#__codelineno-15-59"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-15-59" name="__codelineno-15-59" href="#__codelineno-15-59"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-15-60" name="__codelineno-15-60" href="#__codelineno-15-60"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">pushLast</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-61" name="__codelineno-15-61" href="#__codelineno-15-61"></a><span class="w">        </span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">);</span>
 <a id="__codelineno-15-62" name="__codelineno-15-62" href="#__codelineno-15-62"></a><span class="w">    </span><span class="p">}</span>
@@ -5491,58 +5563,58 @@
 <a id="__codelineno-15-66" name="__codelineno-15-66" href="#__codelineno-15-66"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-15-67" name="__codelineno-15-67" href="#__codelineno-15-67"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">();</span>
 <a id="__codelineno-15-68" name="__codelineno-15-68" href="#__codelineno-15-68"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
-<a id="__codelineno-15-69" name="__codelineno-15-69" href="#__codelineno-15-69"></a><span class="w">        </span><span class="c1">// When the deque has one element, front and rear point to the same node</span>
+<a id="__codelineno-15-69" name="__codelineno-15-69" href="#__codelineno-15-69"></a><span class="w">        </span><span class="c1">// When the queue size is 1, the head and tail point to the same node</span>
 <a id="__codelineno-15-70" name="__codelineno-15-70" href="#__codelineno-15-70"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-71" name="__codelineno-15-71" href="#__codelineno-15-71"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">.</span><span class="na">val</span><span class="p">;</span>
 <a id="__codelineno-15-72" name="__codelineno-15-72" href="#__codelineno-15-72"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-15-73" name="__codelineno-15-73" href="#__codelineno-15-73"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-15-74" name="__codelineno-15-74" href="#__codelineno-15-74"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-15-75" name="__codelineno-15-75" href="#__codelineno-15-75"></a><span class="w">        </span><span class="c1">// Temporarily store head node value</span>
+<a id="__codelineno-15-75" name="__codelineno-15-75" href="#__codelineno-15-75"></a><span class="w">        </span><span class="c1">// Dequeue from the front</span>
 <a id="__codelineno-15-76" name="__codelineno-15-76" href="#__codelineno-15-76"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-77" name="__codelineno-15-77" href="#__codelineno-15-77"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">.</span><span class="na">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Delete head node</span>
-<a id="__codelineno-15-78" name="__codelineno-15-78" href="#__codelineno-15-78"></a><span class="w">            </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-15-77" name="__codelineno-15-77" href="#__codelineno-15-77"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">.</span><span class="na">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Save the head node value</span>
+<a id="__codelineno-15-78" name="__codelineno-15-78" href="#__codelineno-15-78"></a><span class="w">            </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-15-79" name="__codelineno-15-79" href="#__codelineno-15-79"></a><span class="w">            </span><span class="n">ListNode</span><span class="w"> </span><span class="n">fNext</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">.</span><span class="na">next</span><span class="p">;</span>
 <a id="__codelineno-15-80" name="__codelineno-15-80" href="#__codelineno-15-80"></a><span class="w">            </span><span class="n">fNext</span><span class="p">.</span><span class="na">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-15-81" name="__codelineno-15-81" href="#__codelineno-15-81"></a><span class="w">            </span><span class="n">front</span><span class="p">.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-15-82" name="__codelineno-15-82" href="#__codelineno-15-82"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fNext</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update head node</span>
-<a id="__codelineno-15-83" name="__codelineno-15-83" href="#__codelineno-15-83"></a><span class="w">        </span><span class="c1">// Temporarily store tail node value</span>
+<a id="__codelineno-15-82" name="__codelineno-15-82" href="#__codelineno-15-82"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fNext</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the head node</span>
+<a id="__codelineno-15-83" name="__codelineno-15-83" href="#__codelineno-15-83"></a><span class="w">        </span><span class="c1">// Dequeue from the rear</span>
 <a id="__codelineno-15-84" name="__codelineno-15-84" href="#__codelineno-15-84"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-85" name="__codelineno-15-85" href="#__codelineno-15-85"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="p">.</span><span class="na">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Delete tail node</span>
-<a id="__codelineno-15-86" name="__codelineno-15-86" href="#__codelineno-15-86"></a><span class="w">            </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-15-85" name="__codelineno-15-85" href="#__codelineno-15-85"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="p">.</span><span class="na">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Save the tail node value</span>
+<a id="__codelineno-15-86" name="__codelineno-15-86" href="#__codelineno-15-86"></a><span class="w">            </span><span class="c1">// Remove the tail node</span>
 <a id="__codelineno-15-87" name="__codelineno-15-87" href="#__codelineno-15-87"></a><span class="w">            </span><span class="n">ListNode</span><span class="w"> </span><span class="n">rPrev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="p">.</span><span class="na">prev</span><span class="p">;</span>
 <a id="__codelineno-15-88" name="__codelineno-15-88" href="#__codelineno-15-88"></a><span class="w">            </span><span class="n">rPrev</span><span class="p">.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-15-89" name="__codelineno-15-89" href="#__codelineno-15-89"></a><span class="w">            </span><span class="n">rear</span><span class="p">.</span><span class="na">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-15-90" name="__codelineno-15-90" href="#__codelineno-15-90"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rPrev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-15-90" name="__codelineno-15-90" href="#__codelineno-15-90"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rPrev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-15-91" name="__codelineno-15-91" href="#__codelineno-15-91"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-15-92" name="__codelineno-15-92" href="#__codelineno-15-92"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-15-92" name="__codelineno-15-92" href="#__codelineno-15-92"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the queue size</span>
 <a id="__codelineno-15-93" name="__codelineno-15-93" href="#__codelineno-15-93"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-15-94" name="__codelineno-15-94" href="#__codelineno-15-94"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-95" name="__codelineno-15-95" href="#__codelineno-15-95"></a>
-<a id="__codelineno-15-96" name="__codelineno-15-96" href="#__codelineno-15-96"></a><span class="w">    </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-15-96" name="__codelineno-15-96" href="#__codelineno-15-96"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-15-97" name="__codelineno-15-97" href="#__codelineno-15-97"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">popFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-98" name="__codelineno-15-98" href="#__codelineno-15-98"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pop</span><span class="p">(</span><span class="kc">true</span><span class="p">);</span>
 <a id="__codelineno-15-99" name="__codelineno-15-99" href="#__codelineno-15-99"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-100" name="__codelineno-15-100" href="#__codelineno-15-100"></a>
-<a id="__codelineno-15-101" name="__codelineno-15-101" href="#__codelineno-15-101"></a><span class="w">    </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-15-101" name="__codelineno-15-101" href="#__codelineno-15-101"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-15-102" name="__codelineno-15-102" href="#__codelineno-15-102"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">popLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-103" name="__codelineno-15-103" href="#__codelineno-15-103"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pop</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span>
 <a id="__codelineno-15-104" name="__codelineno-15-104" href="#__codelineno-15-104"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-105" name="__codelineno-15-105" href="#__codelineno-15-105"></a>
-<a id="__codelineno-15-106" name="__codelineno-15-106" href="#__codelineno-15-106"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-15-106" name="__codelineno-15-106" href="#__codelineno-15-106"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-15-107" name="__codelineno-15-107" href="#__codelineno-15-107"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">peekFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-108" name="__codelineno-15-108" href="#__codelineno-15-108"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-15-109" name="__codelineno-15-109" href="#__codelineno-15-109"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">();</span>
 <a id="__codelineno-15-110" name="__codelineno-15-110" href="#__codelineno-15-110"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">front</span><span class="p">.</span><span class="na">val</span><span class="p">;</span>
 <a id="__codelineno-15-111" name="__codelineno-15-111" href="#__codelineno-15-111"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-112" name="__codelineno-15-112" href="#__codelineno-15-112"></a>
-<a id="__codelineno-15-113" name="__codelineno-15-113" href="#__codelineno-15-113"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-15-113" name="__codelineno-15-113" href="#__codelineno-15-113"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-15-114" name="__codelineno-15-114" href="#__codelineno-15-114"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">peekLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-115" name="__codelineno-15-115" href="#__codelineno-15-115"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-15-116" name="__codelineno-15-116" href="#__codelineno-15-116"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">();</span>
 <a id="__codelineno-15-117" name="__codelineno-15-117" href="#__codelineno-15-117"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">rear</span><span class="p">.</span><span class="na">val</span><span class="p">;</span>
 <a id="__codelineno-15-118" name="__codelineno-15-118" href="#__codelineno-15-118"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-119" name="__codelineno-15-119" href="#__codelineno-15-119"></a>
-<a id="__codelineno-15-120" name="__codelineno-15-120" href="#__codelineno-15-120"></a><span class="w">    </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-15-120" name="__codelineno-15-120" href="#__codelineno-15-120"></a><span class="w">    </span><span class="cm">/* Return an array for printing */</span>
 <a id="__codelineno-15-121" name="__codelineno-15-121" href="#__codelineno-15-121"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="nf">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-122" name="__codelineno-15-122" href="#__codelineno-15-122"></a><span class="w">        </span><span class="n">ListNode</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span>
 <a id="__codelineno-15-123" name="__codelineno-15-123" href="#__codelineno-15-123"></a><span class="w">        </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">size</span><span class="p">()</span><span class="o">]</span><span class="p">;</span>
@@ -5559,26 +5631,26 @@
 <div class="highlight"><span class="filename">linkedlist_deque.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Doubly linked list node */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">class</span><span class="w"> </span><span class="nf">ListNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w">       </span><span class="c1">// Node value</span>
-<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span><span class="w"> </span><span class="c1">// Successor node reference</span>
-<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span><span class="w"> </span><span class="c1">// Predecessor node reference</span>
+<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference to the next node</span>
+<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference to the previous node</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="p">}</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a>
-<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="cm">/* Double-ended queue based on doubly linked list implementation */</span>
+<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="cm">/* Deque implemented with a doubly linked list */</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="k">class</span><span class="w"> </span><span class="nc">LinkedListDeque</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="n">rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node front, tail node rear</span>
-<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">      </span><span class="c1">// Length of the double-ended queue</span>
+<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="n">rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node front and tail node rear</span>
+<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">      </span><span class="c1">// Deque size</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nf">LinkedListDeque</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
 <a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">        </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
 <a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a>
-<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a>
-<a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-16-24" name="__codelineno-16-24" href="#__codelineno-16-24"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="nf">IsEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-25" name="__codelineno-16-25" href="#__codelineno-16-25"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nf">Size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-16-26" name="__codelineno-16-26" href="#__codelineno-16-26"></a><span class="w">    </span><span class="p">}</span>
@@ -5591,30 +5663,30 @@
 <a id="__codelineno-16-33" name="__codelineno-16-33" href="#__codelineno-16-33"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-16-34" name="__codelineno-16-34" href="#__codelineno-16-34"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-16-35" name="__codelineno-16-35" href="#__codelineno-16-35"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-16-36" name="__codelineno-16-36" href="#__codelineno-16-36"></a><span class="w">        </span><span class="c1">// Front of the queue enqueue operation</span>
+<a id="__codelineno-16-36" name="__codelineno-16-36" href="#__codelineno-16-36"></a><span class="w">        </span><span class="c1">// Enqueue at the front</span>
 <a id="__codelineno-16-37" name="__codelineno-16-37" href="#__codelineno-16-37"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="nf">if</span><span class="w"> </span><span class="p">(</span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-38" name="__codelineno-16-38" href="#__codelineno-16-38"></a><span class="w">            </span><span class="c1">// Add node to the head of the linked list</span>
 <a id="__codelineno-16-39" name="__codelineno-16-39" href="#__codelineno-16-39"></a><span class="w">            </span><span class="n">front</span><span class="o">!</span><span class="p">.</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-16-40" name="__codelineno-16-40" href="#__codelineno-16-40"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span>
-<a id="__codelineno-16-41" name="__codelineno-16-41" href="#__codelineno-16-41"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update head node</span>
+<a id="__codelineno-16-41" name="__codelineno-16-41" href="#__codelineno-16-41"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the head node                           </span>
 <a id="__codelineno-16-42" name="__codelineno-16-42" href="#__codelineno-16-42"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-16-43" name="__codelineno-16-43" href="#__codelineno-16-43"></a><span class="w">        </span><span class="c1">// Rear of the queue enqueue operation</span>
+<a id="__codelineno-16-43" name="__codelineno-16-43" href="#__codelineno-16-43"></a><span class="w">        </span><span class="c1">// Enqueue at the rear</span>
 <a id="__codelineno-16-44" name="__codelineno-16-44" href="#__codelineno-16-44"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-45" name="__codelineno-16-45" href="#__codelineno-16-45"></a><span class="w">            </span><span class="c1">// Add node to the tail of the linked list</span>
 <a id="__codelineno-16-46" name="__codelineno-16-46" href="#__codelineno-16-46"></a><span class="w">            </span><span class="n">rear</span><span class="o">!</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-16-47" name="__codelineno-16-47" href="#__codelineno-16-47"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="p">;</span>
-<a id="__codelineno-16-48" name="__codelineno-16-48" href="#__codelineno-16-48"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w">  </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-16-48" name="__codelineno-16-48" href="#__codelineno-16-48"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w">  </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-16-49" name="__codelineno-16-49" href="#__codelineno-16-49"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-16-50" name="__codelineno-16-50" href="#__codelineno-16-50"></a>
-<a id="__codelineno-16-51" name="__codelineno-16-51" href="#__codelineno-16-51"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-16-51" name="__codelineno-16-51" href="#__codelineno-16-51"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the queue size</span>
 <a id="__codelineno-16-52" name="__codelineno-16-52" href="#__codelineno-16-52"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-53" name="__codelineno-16-53" href="#__codelineno-16-53"></a>
-<a id="__codelineno-16-54" name="__codelineno-16-54" href="#__codelineno-16-54"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-16-54" name="__codelineno-16-54" href="#__codelineno-16-54"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-16-55" name="__codelineno-16-55" href="#__codelineno-16-55"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">PushFirst</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-56" name="__codelineno-16-56" href="#__codelineno-16-56"></a><span class="w">        </span><span class="n">Push</span><span class="p">(</span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="k">true</span><span class="p">);</span>
 <a id="__codelineno-16-57" name="__codelineno-16-57" href="#__codelineno-16-57"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-58" name="__codelineno-16-58" href="#__codelineno-16-58"></a>
-<a id="__codelineno-16-59" name="__codelineno-16-59" href="#__codelineno-16-59"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-16-59" name="__codelineno-16-59" href="#__codelineno-16-59"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-16-60" name="__codelineno-16-60" href="#__codelineno-16-60"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">PushLast</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-61" name="__codelineno-16-61" href="#__codelineno-16-61"></a><span class="w">        </span><span class="n">Push</span><span class="p">(</span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="k">false</span><span class="p">);</span>
 <a id="__codelineno-16-62" name="__codelineno-16-62" href="#__codelineno-16-62"></a><span class="w">    </span><span class="p">}</span>
@@ -5624,60 +5696,60 @@
 <a id="__codelineno-16-66" name="__codelineno-16-66" href="#__codelineno-16-66"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">IsEmpty</span><span class="p">())</span>
 <a id="__codelineno-16-67" name="__codelineno-16-67" href="#__codelineno-16-67"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">Exception</span><span class="p">();</span>
 <a id="__codelineno-16-68" name="__codelineno-16-68" href="#__codelineno-16-68"></a><span class="w">        </span><span class="kt">int?</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
-<a id="__codelineno-16-69" name="__codelineno-16-69" href="#__codelineno-16-69"></a><span class="w">        </span><span class="c1">// When the deque has one element, front and rear point to the same node</span>
+<a id="__codelineno-16-69" name="__codelineno-16-69" href="#__codelineno-16-69"></a><span class="w">        </span><span class="c1">// When the queue size is 1, the head and tail point to the same node</span>
 <a id="__codelineno-16-70" name="__codelineno-16-70" href="#__codelineno-16-70"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-71" name="__codelineno-16-71" href="#__codelineno-16-71"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-16-72" name="__codelineno-16-72" href="#__codelineno-16-72"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
 <a id="__codelineno-16-73" name="__codelineno-16-73" href="#__codelineno-16-73"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
 <a id="__codelineno-16-74" name="__codelineno-16-74" href="#__codelineno-16-74"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-16-75" name="__codelineno-16-75" href="#__codelineno-16-75"></a><span class="w">        </span><span class="c1">// Temporarily store head node value</span>
+<a id="__codelineno-16-75" name="__codelineno-16-75" href="#__codelineno-16-75"></a><span class="w">        </span><span class="c1">// Dequeue from the front</span>
 <a id="__codelineno-16-76" name="__codelineno-16-76" href="#__codelineno-16-76"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="nf">if</span><span class="w"> </span><span class="p">(</span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-77" name="__codelineno-16-77" href="#__codelineno-16-77"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Delete head node</span>
-<a id="__codelineno-16-78" name="__codelineno-16-78" href="#__codelineno-16-78"></a><span class="w">            </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-16-77" name="__codelineno-16-77" href="#__codelineno-16-77"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Save the head node value</span>
+<a id="__codelineno-16-78" name="__codelineno-16-78" href="#__codelineno-16-78"></a><span class="w">            </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-16-79" name="__codelineno-16-79" href="#__codelineno-16-79"></a><span class="w">            </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">fNext</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="o">?.</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-16-80" name="__codelineno-16-80" href="#__codelineno-16-80"></a><span class="w">            </span><span class="n">fNext</span><span class="o">!</span><span class="p">.</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
 <a id="__codelineno-16-81" name="__codelineno-16-81" href="#__codelineno-16-81"></a><span class="w">            </span><span class="n">front</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
-<a id="__codelineno-16-82" name="__codelineno-16-82" href="#__codelineno-16-82"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fNext</span><span class="p">;</span><span class="w">   </span><span class="c1">// Update head node</span>
+<a id="__codelineno-16-82" name="__codelineno-16-82" href="#__codelineno-16-82"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fNext</span><span class="p">;</span><span class="w">   </span><span class="c1">// Update the head node</span>
 <a id="__codelineno-16-83" name="__codelineno-16-83" href="#__codelineno-16-83"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-16-84" name="__codelineno-16-84" href="#__codelineno-16-84"></a><span class="w">        </span><span class="c1">// Temporarily store tail node value</span>
+<a id="__codelineno-16-84" name="__codelineno-16-84" href="#__codelineno-16-84"></a><span class="w">        </span><span class="c1">// Dequeue from the rear</span>
 <a id="__codelineno-16-85" name="__codelineno-16-85" href="#__codelineno-16-85"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-86" name="__codelineno-16-86" href="#__codelineno-16-86"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">;</span><span class="w">  </span><span class="c1">// Delete tail node</span>
-<a id="__codelineno-16-87" name="__codelineno-16-87" href="#__codelineno-16-87"></a><span class="w">            </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-16-86" name="__codelineno-16-86" href="#__codelineno-16-86"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">;</span><span class="w">  </span><span class="c1">// Save the tail node value</span>
+<a id="__codelineno-16-87" name="__codelineno-16-87" href="#__codelineno-16-87"></a><span class="w">            </span><span class="c1">// Remove the tail node</span>
 <a id="__codelineno-16-88" name="__codelineno-16-88" href="#__codelineno-16-88"></a><span class="w">            </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">rPrev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="o">?.</span><span class="n">prev</span><span class="p">;</span>
 <a id="__codelineno-16-89" name="__codelineno-16-89" href="#__codelineno-16-89"></a><span class="w">            </span><span class="n">rPrev</span><span class="o">!</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
 <a id="__codelineno-16-90" name="__codelineno-16-90" href="#__codelineno-16-90"></a><span class="w">            </span><span class="n">rear</span><span class="p">.</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
-<a id="__codelineno-16-91" name="__codelineno-16-91" href="#__codelineno-16-91"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rPrev</span><span class="p">;</span><span class="w">    </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-16-91" name="__codelineno-16-91" href="#__codelineno-16-91"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rPrev</span><span class="p">;</span><span class="w">    </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-16-92" name="__codelineno-16-92" href="#__codelineno-16-92"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-16-93" name="__codelineno-16-93" href="#__codelineno-16-93"></a>
-<a id="__codelineno-16-94" name="__codelineno-16-94" href="#__codelineno-16-94"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-16-94" name="__codelineno-16-94" href="#__codelineno-16-94"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the queue size</span>
 <a id="__codelineno-16-95" name="__codelineno-16-95" href="#__codelineno-16-95"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-16-96" name="__codelineno-16-96" href="#__codelineno-16-96"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-97" name="__codelineno-16-97" href="#__codelineno-16-97"></a>
-<a id="__codelineno-16-98" name="__codelineno-16-98" href="#__codelineno-16-98"></a><span class="w">    </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-16-98" name="__codelineno-16-98" href="#__codelineno-16-98"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-16-99" name="__codelineno-16-99" href="#__codelineno-16-99"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int?</span><span class="w"> </span><span class="n">PopFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-100" name="__codelineno-16-100" href="#__codelineno-16-100"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nf">Pop</span><span class="p">(</span><span class="k">true</span><span class="p">);</span>
 <a id="__codelineno-16-101" name="__codelineno-16-101" href="#__codelineno-16-101"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-102" name="__codelineno-16-102" href="#__codelineno-16-102"></a>
-<a id="__codelineno-16-103" name="__codelineno-16-103" href="#__codelineno-16-103"></a><span class="w">    </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-16-103" name="__codelineno-16-103" href="#__codelineno-16-103"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-16-104" name="__codelineno-16-104" href="#__codelineno-16-104"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int?</span><span class="w"> </span><span class="n">PopLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-105" name="__codelineno-16-105" href="#__codelineno-16-105"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nf">Pop</span><span class="p">(</span><span class="k">false</span><span class="p">);</span>
 <a id="__codelineno-16-106" name="__codelineno-16-106" href="#__codelineno-16-106"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-107" name="__codelineno-16-107" href="#__codelineno-16-107"></a>
-<a id="__codelineno-16-108" name="__codelineno-16-108" href="#__codelineno-16-108"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-16-108" name="__codelineno-16-108" href="#__codelineno-16-108"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-16-109" name="__codelineno-16-109" href="#__codelineno-16-109"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int?</span><span class="w"> </span><span class="n">PeekFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-110" name="__codelineno-16-110" href="#__codelineno-16-110"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">IsEmpty</span><span class="p">())</span>
 <a id="__codelineno-16-111" name="__codelineno-16-111" href="#__codelineno-16-111"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">Exception</span><span class="p">();</span>
 <a id="__codelineno-16-112" name="__codelineno-16-112" href="#__codelineno-16-112"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">front</span><span class="o">?.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-16-113" name="__codelineno-16-113" href="#__codelineno-16-113"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-114" name="__codelineno-16-114" href="#__codelineno-16-114"></a>
-<a id="__codelineno-16-115" name="__codelineno-16-115" href="#__codelineno-16-115"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-16-115" name="__codelineno-16-115" href="#__codelineno-16-115"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-16-116" name="__codelineno-16-116" href="#__codelineno-16-116"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int?</span><span class="w"> </span><span class="n">PeekLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-117" name="__codelineno-16-117" href="#__codelineno-16-117"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">IsEmpty</span><span class="p">())</span>
 <a id="__codelineno-16-118" name="__codelineno-16-118" href="#__codelineno-16-118"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">Exception</span><span class="p">();</span>
 <a id="__codelineno-16-119" name="__codelineno-16-119" href="#__codelineno-16-119"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">rear</span><span class="o">?.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-16-120" name="__codelineno-16-120" href="#__codelineno-16-120"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-121" name="__codelineno-16-121" href="#__codelineno-16-121"></a>
-<a id="__codelineno-16-122" name="__codelineno-16-122" href="#__codelineno-16-122"></a><span class="w">    </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-16-122" name="__codelineno-16-122" href="#__codelineno-16-122"></a><span class="w">    </span><span class="cm">/* Return an array for printing */</span>
 <a id="__codelineno-16-123" name="__codelineno-16-123" href="#__codelineno-16-123"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int?</span><span class="p">[]</span><span class="w"> </span><span class="n">ToArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-124" name="__codelineno-16-124" href="#__codelineno-16-124"></a><span class="w">        </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span>
 <a id="__codelineno-16-125" name="__codelineno-16-125" href="#__codelineno-16-125"></a><span class="w">        </span><span class="kt">int?</span><span class="p">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int?</span><span class="p">[</span><span class="n">Size</span><span class="p">()];</span>
@@ -5692,30 +5764,30 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_deque.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Double-ended queue based on doubly linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_deque.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Deque implemented with a doubly linked list */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">linkedListDeque</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Use built-in package list</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Use the standard library list package</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="nx">data</span><span class="w"> </span><span class="o">*</span><span class="nx">list</span><span class="p">.</span><span class="nx">List</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="p">}</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a>
-<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="cm">/* Initialize deque */</span>
+<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="cm">/* Initialize the deque */</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="kd">func</span><span class="w"> </span><span class="nx">newLinkedListDeque</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListDeque</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">linkedListDeque</span><span class="p">{</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">        </span><span class="nx">data</span><span class="p">:</span><span class="w"> </span><span class="nx">list</span><span class="p">.</span><span class="nx">New</span><span class="p">(),</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="p">}</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a>
-<a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="cm">/* Front element enqueue */</span>
+<a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="cm">/* Enqueue an element at the front */</span>
 <a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">pushFirst</span><span class="p">(</span><span class="nx">value</span><span class="w"> </span><span class="kt">any</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">    </span><span class="nx">s</span><span class="p">.</span><span class="nx">data</span><span class="p">.</span><span class="nx">PushFront</span><span class="p">(</span><span class="nx">value</span><span class="p">)</span>
 <a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="p">}</span>
 <a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a>
-<a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="cm">/* Rear element enqueue */</span>
+<a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="cm">/* Enqueue an element at the rear */</span>
 <a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">pushLast</span><span class="p">(</span><span class="nx">value</span><span class="w"> </span><span class="kt">any</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="w">    </span><span class="nx">s</span><span class="p">.</span><span class="nx">data</span><span class="p">.</span><span class="nx">PushBack</span><span class="p">(</span><span class="nx">value</span><span class="p">)</span>
 <a id="__codelineno-17-22" name="__codelineno-17-22" href="#__codelineno-17-22"></a><span class="p">}</span>
 <a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a>
-<a id="__codelineno-17-24" name="__codelineno-17-24" href="#__codelineno-17-24"></a><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-17-24" name="__codelineno-17-24" href="#__codelineno-17-24"></a><span class="cm">/* Dequeue the front element */</span>
 <a id="__codelineno-17-25" name="__codelineno-17-25" href="#__codelineno-17-25"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">popFirst</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-26" name="__codelineno-17-26" href="#__codelineno-17-26"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-27" name="__codelineno-17-27" href="#__codelineno-17-27"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
@@ -5725,7 +5797,7 @@
 <a id="__codelineno-17-31" name="__codelineno-17-31" href="#__codelineno-17-31"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">e</span><span class="p">.</span><span class="nx">Value</span>
 <a id="__codelineno-17-32" name="__codelineno-17-32" href="#__codelineno-17-32"></a><span class="p">}</span>
 <a id="__codelineno-17-33" name="__codelineno-17-33" href="#__codelineno-17-33"></a>
-<a id="__codelineno-17-34" name="__codelineno-17-34" href="#__codelineno-17-34"></a><span class="cm">/* Rear element dequeue */</span>
+<a id="__codelineno-17-34" name="__codelineno-17-34" href="#__codelineno-17-34"></a><span class="cm">/* Dequeue the rear element */</span>
 <a id="__codelineno-17-35" name="__codelineno-17-35" href="#__codelineno-17-35"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">popLast</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-36" name="__codelineno-17-36" href="#__codelineno-17-36"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-37" name="__codelineno-17-37" href="#__codelineno-17-37"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
@@ -5735,7 +5807,7 @@
 <a id="__codelineno-17-41" name="__codelineno-17-41" href="#__codelineno-17-41"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">e</span><span class="p">.</span><span class="nx">Value</span>
 <a id="__codelineno-17-42" name="__codelineno-17-42" href="#__codelineno-17-42"></a><span class="p">}</span>
 <a id="__codelineno-17-43" name="__codelineno-17-43" href="#__codelineno-17-43"></a>
-<a id="__codelineno-17-44" name="__codelineno-17-44" href="#__codelineno-17-44"></a><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-17-44" name="__codelineno-17-44" href="#__codelineno-17-44"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-17-45" name="__codelineno-17-45" href="#__codelineno-17-45"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">peekFirst</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-46" name="__codelineno-17-46" href="#__codelineno-17-46"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-47" name="__codelineno-17-47" href="#__codelineno-17-47"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
@@ -5744,7 +5816,7 @@
 <a id="__codelineno-17-50" name="__codelineno-17-50" href="#__codelineno-17-50"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">e</span><span class="p">.</span><span class="nx">Value</span>
 <a id="__codelineno-17-51" name="__codelineno-17-51" href="#__codelineno-17-51"></a><span class="p">}</span>
 <a id="__codelineno-17-52" name="__codelineno-17-52" href="#__codelineno-17-52"></a>
-<a id="__codelineno-17-53" name="__codelineno-17-53" href="#__codelineno-17-53"></a><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-17-53" name="__codelineno-17-53" href="#__codelineno-17-53"></a><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-17-54" name="__codelineno-17-54" href="#__codelineno-17-54"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">peekLast</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-55" name="__codelineno-17-55" href="#__codelineno-17-55"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-56" name="__codelineno-17-56" href="#__codelineno-17-56"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
@@ -5753,17 +5825,17 @@
 <a id="__codelineno-17-59" name="__codelineno-17-59" href="#__codelineno-17-59"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">e</span><span class="p">.</span><span class="nx">Value</span>
 <a id="__codelineno-17-60" name="__codelineno-17-60" href="#__codelineno-17-60"></a><span class="p">}</span>
 <a id="__codelineno-17-61" name="__codelineno-17-61" href="#__codelineno-17-61"></a>
-<a id="__codelineno-17-62" name="__codelineno-17-62" href="#__codelineno-17-62"></a><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-17-62" name="__codelineno-17-62" href="#__codelineno-17-62"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-17-63" name="__codelineno-17-63" href="#__codelineno-17-63"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-64" name="__codelineno-17-64" href="#__codelineno-17-64"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">data</span><span class="p">.</span><span class="nx">Len</span><span class="p">()</span>
 <a id="__codelineno-17-65" name="__codelineno-17-65" href="#__codelineno-17-65"></a><span class="p">}</span>
 <a id="__codelineno-17-66" name="__codelineno-17-66" href="#__codelineno-17-66"></a>
-<a id="__codelineno-17-67" name="__codelineno-17-67" href="#__codelineno-17-67"></a><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-17-67" name="__codelineno-17-67" href="#__codelineno-17-67"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-17-68" name="__codelineno-17-68" href="#__codelineno-17-68"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-69" name="__codelineno-17-69" href="#__codelineno-17-69"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">data</span><span class="p">.</span><span class="nx">Len</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-17-70" name="__codelineno-17-70" href="#__codelineno-17-70"></a><span class="p">}</span>
 <a id="__codelineno-17-71" name="__codelineno-17-71" href="#__codelineno-17-71"></a>
-<a id="__codelineno-17-72" name="__codelineno-17-72" href="#__codelineno-17-72"></a><span class="cm">/* Get List for printing */</span>
+<a id="__codelineno-17-72" name="__codelineno-17-72" href="#__codelineno-17-72"></a><span class="cm">/* Get the list for printing */</span>
 <a id="__codelineno-17-73" name="__codelineno-17-73" href="#__codelineno-17-73"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">toList</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="nx">list</span><span class="p">.</span><span class="nx">List</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-74" name="__codelineno-17-74" href="#__codelineno-17-74"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">data</span>
 <a id="__codelineno-17-75" name="__codelineno-17-75" href="#__codelineno-17-75"></a><span class="p">}</span>
@@ -5773,30 +5845,30 @@
 <div class="highlight"><span class="filename">linkedlist_deque.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Doubly linked list node */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">val</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Node value</span>
-<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">next</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Successor node reference</span>
-<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="kr">weak</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">prev</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Predecessor node reference</span>
+<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">next</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Reference to the next node</span>
+<a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="kr">weak</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">prev</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Reference to the previous node</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="kd">init</span><span class="p">(</span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">        </span><span class="kc">self</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">val</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="p">}</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a>
-<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="cm">/* Double-ended queue based on doubly linked list implementation */</span>
+<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="cm">/* Deque implemented with a doubly linked list */</span>
 <a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="kd">class</span><span class="w"> </span><span class="nc">LinkedListDeque</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">front</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Head node front</span>
 <a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">rear</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Tail node rear</span>
-<a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">_size</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Length of the double-ended queue</span>
+<a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">_size</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Deque size</span>
 <a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a>
 <a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">    </span><span class="kd">init</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a>
-<a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-18-23" name="__codelineno-18-23" href="#__codelineno-18-23"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-24" name="__codelineno-18-24" href="#__codelineno-18-24"></a><span class="w">        </span><span class="n">_size</span>
 <a id="__codelineno-18-25" name="__codelineno-18-25" href="#__codelineno-18-25"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-26" name="__codelineno-18-26" href="#__codelineno-18-26"></a>
-<a id="__codelineno-18-27" name="__codelineno-18-27" href="#__codelineno-18-27"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-18-27" name="__codelineno-18-27" href="#__codelineno-18-27"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-18-28" name="__codelineno-18-28" href="#__codelineno-18-28"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-29" name="__codelineno-18-29" href="#__codelineno-18-29"></a><span class="w">        </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-18-30" name="__codelineno-18-30" href="#__codelineno-18-30"></a><span class="w">    </span><span class="p">}</span>
@@ -5809,29 +5881,29 @@
 <a id="__codelineno-18-37" name="__codelineno-18-37" href="#__codelineno-18-37"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-18-38" name="__codelineno-18-38" href="#__codelineno-18-38"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-18-39" name="__codelineno-18-39" href="#__codelineno-18-39"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-40" name="__codelineno-18-40" href="#__codelineno-18-40"></a><span class="w">        </span><span class="c1">// Front of the queue enqueue operation</span>
+<a id="__codelineno-18-40" name="__codelineno-18-40" href="#__codelineno-18-40"></a><span class="w">        </span><span class="c1">// Enqueue at the front</span>
 <a id="__codelineno-18-41" name="__codelineno-18-41" href="#__codelineno-18-41"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">isFront</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-42" name="__codelineno-18-42" href="#__codelineno-18-42"></a><span class="w">            </span><span class="c1">// Add node to the head of the linked list</span>
 <a id="__codelineno-18-43" name="__codelineno-18-43" href="#__codelineno-18-43"></a><span class="w">            </span><span class="n">front</span><span class="p">?.</span><span class="n">prev</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-18-44" name="__codelineno-18-44" href="#__codelineno-18-44"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">front</span>
-<a id="__codelineno-18-45" name="__codelineno-18-45" href="#__codelineno-18-45"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1">// Update head node</span>
+<a id="__codelineno-18-45" name="__codelineno-18-45" href="#__codelineno-18-45"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1">// Update the head node</span>
 <a id="__codelineno-18-46" name="__codelineno-18-46" href="#__codelineno-18-46"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-47" name="__codelineno-18-47" href="#__codelineno-18-47"></a><span class="w">        </span><span class="c1">// Rear of the queue enqueue operation</span>
+<a id="__codelineno-18-47" name="__codelineno-18-47" href="#__codelineno-18-47"></a><span class="w">        </span><span class="c1">// Enqueue at the rear</span>
 <a id="__codelineno-18-48" name="__codelineno-18-48" href="#__codelineno-18-48"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-49" name="__codelineno-18-49" href="#__codelineno-18-49"></a><span class="w">            </span><span class="c1">// Add node to the tail of the linked list</span>
 <a id="__codelineno-18-50" name="__codelineno-18-50" href="#__codelineno-18-50"></a><span class="w">            </span><span class="n">rear</span><span class="p">?.</span><span class="n">next</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-18-51" name="__codelineno-18-51" href="#__codelineno-18-51"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="n">prev</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">rear</span>
-<a id="__codelineno-18-52" name="__codelineno-18-52" href="#__codelineno-18-52"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-18-52" name="__codelineno-18-52" href="#__codelineno-18-52"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-18-53" name="__codelineno-18-53" href="#__codelineno-18-53"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-54" name="__codelineno-18-54" href="#__codelineno-18-54"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-18-54" name="__codelineno-18-54" href="#__codelineno-18-54"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Update the deque size</span>
 <a id="__codelineno-18-55" name="__codelineno-18-55" href="#__codelineno-18-55"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-56" name="__codelineno-18-56" href="#__codelineno-18-56"></a>
-<a id="__codelineno-18-57" name="__codelineno-18-57" href="#__codelineno-18-57"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-18-57" name="__codelineno-18-57" href="#__codelineno-18-57"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-18-58" name="__codelineno-18-58" href="#__codelineno-18-58"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">pushFirst</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-59" name="__codelineno-18-59" href="#__codelineno-18-59"></a><span class="w">        </span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="n">isFront</span><span class="p">:</span><span class="w"> </span><span class="kc">true</span><span class="p">)</span>
 <a id="__codelineno-18-60" name="__codelineno-18-60" href="#__codelineno-18-60"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-61" name="__codelineno-18-61" href="#__codelineno-18-61"></a>
-<a id="__codelineno-18-62" name="__codelineno-18-62" href="#__codelineno-18-62"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-18-62" name="__codelineno-18-62" href="#__codelineno-18-62"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-18-63" name="__codelineno-18-63" href="#__codelineno-18-63"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">pushLast</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-64" name="__codelineno-18-64" href="#__codelineno-18-64"></a><span class="w">        </span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="n">isFront</span><span class="p">:</span><span class="w"> </span><span class="kc">false</span><span class="p">)</span>
 <a id="__codelineno-18-65" name="__codelineno-18-65" href="#__codelineno-18-65"></a><span class="w">    </span><span class="p">}</span>
@@ -5842,45 +5914,45 @@
 <a id="__codelineno-18-70" name="__codelineno-18-70" href="#__codelineno-18-70"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Deque is empty"</span><span class="p">)</span>
 <a id="__codelineno-18-71" name="__codelineno-18-71" href="#__codelineno-18-71"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-18-72" name="__codelineno-18-72" href="#__codelineno-18-72"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">val</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span>
-<a id="__codelineno-18-73" name="__codelineno-18-73" href="#__codelineno-18-73"></a><span class="w">        </span><span class="c1">// When the deque has one element, front and rear point to the same node</span>
+<a id="__codelineno-18-73" name="__codelineno-18-73" href="#__codelineno-18-73"></a><span class="w">        </span><span class="c1">// When the deque size is 1, the head and tail point to the same node</span>
 <a id="__codelineno-18-74" name="__codelineno-18-74" href="#__codelineno-18-74"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">_size</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-75" name="__codelineno-18-75" href="#__codelineno-18-75"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">front</span><span class="p">!.</span><span class="n">val</span>
 <a id="__codelineno-18-76" name="__codelineno-18-76" href="#__codelineno-18-76"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-18-77" name="__codelineno-18-77" href="#__codelineno-18-77"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-18-78" name="__codelineno-18-78" href="#__codelineno-18-78"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-79" name="__codelineno-18-79" href="#__codelineno-18-79"></a><span class="w">        </span><span class="c1">// Temporarily store head node value</span>
+<a id="__codelineno-18-79" name="__codelineno-18-79" href="#__codelineno-18-79"></a><span class="w">        </span><span class="c1">// Dequeue from the front</span>
 <a id="__codelineno-18-80" name="__codelineno-18-80" href="#__codelineno-18-80"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">isFront</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-81" name="__codelineno-18-81" href="#__codelineno-18-81"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">front</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="c1">// Delete head node</span>
-<a id="__codelineno-18-82" name="__codelineno-18-82" href="#__codelineno-18-82"></a><span class="w">            </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-18-81" name="__codelineno-18-81" href="#__codelineno-18-81"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">front</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="c1">// Save the head node value</span>
+<a id="__codelineno-18-82" name="__codelineno-18-82" href="#__codelineno-18-82"></a><span class="w">            </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-18-83" name="__codelineno-18-83" href="#__codelineno-18-83"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="nv">fNext</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">front</span><span class="p">?.</span><span class="n">next</span>
 <a id="__codelineno-18-84" name="__codelineno-18-84" href="#__codelineno-18-84"></a><span class="w">            </span><span class="n">fNext</span><span class="p">?.</span><span class="n">prev</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-18-85" name="__codelineno-18-85" href="#__codelineno-18-85"></a><span class="w">            </span><span class="n">front</span><span class="p">?.</span><span class="n">next</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">nil</span>
-<a id="__codelineno-18-86" name="__codelineno-18-86" href="#__codelineno-18-86"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">fNext</span><span class="w"> </span><span class="c1">// Update head node</span>
+<a id="__codelineno-18-86" name="__codelineno-18-86" href="#__codelineno-18-86"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">fNext</span><span class="w"> </span><span class="c1">// Update the head node</span>
 <a id="__codelineno-18-87" name="__codelineno-18-87" href="#__codelineno-18-87"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-88" name="__codelineno-18-88" href="#__codelineno-18-88"></a><span class="w">        </span><span class="c1">// Temporarily store tail node value</span>
+<a id="__codelineno-18-88" name="__codelineno-18-88" href="#__codelineno-18-88"></a><span class="w">        </span><span class="c1">// Dequeue from the rear</span>
 <a id="__codelineno-18-89" name="__codelineno-18-89" href="#__codelineno-18-89"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-90" name="__codelineno-18-90" href="#__codelineno-18-90"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">rear</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="c1">// Delete tail node</span>
-<a id="__codelineno-18-91" name="__codelineno-18-91" href="#__codelineno-18-91"></a><span class="w">            </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-18-90" name="__codelineno-18-90" href="#__codelineno-18-90"></a><span class="w">            </span><span class="n">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">rear</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="c1">// Save the tail node value</span>
+<a id="__codelineno-18-91" name="__codelineno-18-91" href="#__codelineno-18-91"></a><span class="w">            </span><span class="c1">// Remove the tail node</span>
 <a id="__codelineno-18-92" name="__codelineno-18-92" href="#__codelineno-18-92"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="nv">rPrev</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">rear</span><span class="p">?.</span><span class="n">prev</span>
 <a id="__codelineno-18-93" name="__codelineno-18-93" href="#__codelineno-18-93"></a><span class="w">            </span><span class="n">rPrev</span><span class="p">?.</span><span class="n">next</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-18-94" name="__codelineno-18-94" href="#__codelineno-18-94"></a><span class="w">            </span><span class="n">rear</span><span class="p">?.</span><span class="n">prev</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="kc">nil</span>
-<a id="__codelineno-18-95" name="__codelineno-18-95" href="#__codelineno-18-95"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">rPrev</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-18-95" name="__codelineno-18-95" href="#__codelineno-18-95"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">rPrev</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-18-96" name="__codelineno-18-96" href="#__codelineno-18-96"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-97" name="__codelineno-18-97" href="#__codelineno-18-97"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-18-97" name="__codelineno-18-97" href="#__codelineno-18-97"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1">// Update the deque size</span>
 <a id="__codelineno-18-98" name="__codelineno-18-98" href="#__codelineno-18-98"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">val</span>
 <a id="__codelineno-18-99" name="__codelineno-18-99" href="#__codelineno-18-99"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-100" name="__codelineno-18-100" href="#__codelineno-18-100"></a>
-<a id="__codelineno-18-101" name="__codelineno-18-101" href="#__codelineno-18-101"></a><span class="w">    </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-18-101" name="__codelineno-18-101" href="#__codelineno-18-101"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-18-102" name="__codelineno-18-102" href="#__codelineno-18-102"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">popFirst</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-103" name="__codelineno-18-103" href="#__codelineno-18-103"></a><span class="w">        </span><span class="n">pop</span><span class="p">(</span><span class="n">isFront</span><span class="p">:</span><span class="w"> </span><span class="kc">true</span><span class="p">)</span>
 <a id="__codelineno-18-104" name="__codelineno-18-104" href="#__codelineno-18-104"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-105" name="__codelineno-18-105" href="#__codelineno-18-105"></a>
-<a id="__codelineno-18-106" name="__codelineno-18-106" href="#__codelineno-18-106"></a><span class="w">    </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-18-106" name="__codelineno-18-106" href="#__codelineno-18-106"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-18-107" name="__codelineno-18-107" href="#__codelineno-18-107"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">popLast</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-108" name="__codelineno-18-108" href="#__codelineno-18-108"></a><span class="w">        </span><span class="n">pop</span><span class="p">(</span><span class="n">isFront</span><span class="p">:</span><span class="w"> </span><span class="kc">false</span><span class="p">)</span>
 <a id="__codelineno-18-109" name="__codelineno-18-109" href="#__codelineno-18-109"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-110" name="__codelineno-18-110" href="#__codelineno-18-110"></a>
-<a id="__codelineno-18-111" name="__codelineno-18-111" href="#__codelineno-18-111"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-18-111" name="__codelineno-18-111" href="#__codelineno-18-111"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-18-112" name="__codelineno-18-112" href="#__codelineno-18-112"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">peekFirst</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-113" name="__codelineno-18-113" href="#__codelineno-18-113"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-114" name="__codelineno-18-114" href="#__codelineno-18-114"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Deque is empty"</span><span class="p">)</span>
@@ -5888,7 +5960,7 @@
 <a id="__codelineno-18-116" name="__codelineno-18-116" href="#__codelineno-18-116"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">front</span><span class="p">!.</span><span class="n">val</span>
 <a id="__codelineno-18-117" name="__codelineno-18-117" href="#__codelineno-18-117"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-118" name="__codelineno-18-118" href="#__codelineno-18-118"></a>
-<a id="__codelineno-18-119" name="__codelineno-18-119" href="#__codelineno-18-119"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-18-119" name="__codelineno-18-119" href="#__codelineno-18-119"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-18-120" name="__codelineno-18-120" href="#__codelineno-18-120"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">peekLast</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-121" name="__codelineno-18-121" href="#__codelineno-18-121"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-122" name="__codelineno-18-122" href="#__codelineno-18-122"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Deque is empty"</span><span class="p">)</span>
@@ -5896,7 +5968,7 @@
 <a id="__codelineno-18-124" name="__codelineno-18-124" href="#__codelineno-18-124"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">rear</span><span class="p">!.</span><span class="n">val</span>
 <a id="__codelineno-18-125" name="__codelineno-18-125" href="#__codelineno-18-125"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-126" name="__codelineno-18-126" href="#__codelineno-18-126"></a>
-<a id="__codelineno-18-127" name="__codelineno-18-127" href="#__codelineno-18-127"></a><span class="w">    </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-18-127" name="__codelineno-18-127" href="#__codelineno-18-127"></a><span class="w">    </span><span class="cm">/* Return an array for printing */</span>
 <a id="__codelineno-18-128" name="__codelineno-18-128" href="#__codelineno-18-128"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-129" name="__codelineno-18-129" href="#__codelineno-18-129"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">front</span>
 <a id="__codelineno-18-130" name="__codelineno-18-130" href="#__codelineno-18-130"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">size</span><span class="p">())</span>
@@ -5912,8 +5984,8 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">linkedlist_deque.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Doubly linked list node */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">ListNode</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="nx">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Predecessor node reference (pointer)</span>
-<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="nx">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Successor node reference (pointer)</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="nx">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference (pointer) to the previous node</span>
+<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="nx">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference (pointer) to the next node</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="nx">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Node value</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5923,11 +5995,11 @@
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="p">}</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a>
-<a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="cm">/* Double-ended queue based on doubly linked list implementation */</span>
+<a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="cm">/* Deque implemented with a doubly linked list */</span>
 <a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="kd">class</span><span class="w"> </span><span class="nx">LinkedListDeque</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">    </span><span class="n">#front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node front</span>
 <a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">    </span><span class="n">#rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Tail node rear</span>
-<a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">    </span><span class="n">#queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Length of the double-ended queue</span>
+<a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">    </span><span class="n">#queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Deque size</span>
 <a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
@@ -5935,7 +6007,7 @@
 <a id="__codelineno-19-23" name="__codelineno-19-23" href="#__codelineno-19-23"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-19-24" name="__codelineno-19-24" href="#__codelineno-19-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-25" name="__codelineno-19-25" href="#__codelineno-19-25"></a>
-<a id="__codelineno-19-26" name="__codelineno-19-26" href="#__codelineno-19-26"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue operation */</span>
+<a id="__codelineno-19-26" name="__codelineno-19-26" href="#__codelineno-19-26"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-19-27" name="__codelineno-19-27" href="#__codelineno-19-27"></a><span class="w">    </span><span class="nx">pushLast</span><span class="p">(</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-28" name="__codelineno-19-28" href="#__codelineno-19-28"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-19-29" name="__codelineno-19-29" href="#__codelineno-19-29"></a><span class="w">        </span><span class="c1">// If the linked list is empty, make both front and rear point to node</span>
@@ -5946,12 +6018,12 @@
 <a id="__codelineno-19-34" name="__codelineno-19-34" href="#__codelineno-19-34"></a><span class="w">            </span><span class="c1">// Add node to the tail of the linked list</span>
 <a id="__codelineno-19-35" name="__codelineno-19-35" href="#__codelineno-19-35"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-19-36" name="__codelineno-19-36" href="#__codelineno-19-36"></a><span class="w">            </span><span class="nx">node</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="p">;</span>
-<a id="__codelineno-19-37" name="__codelineno-19-37" href="#__codelineno-19-37"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-19-37" name="__codelineno-19-37" href="#__codelineno-19-37"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-19-38" name="__codelineno-19-38" href="#__codelineno-19-38"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-19-39" name="__codelineno-19-39" href="#__codelineno-19-39"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-19-40" name="__codelineno-19-40" href="#__codelineno-19-40"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-41" name="__codelineno-19-41" href="#__codelineno-19-41"></a>
-<a id="__codelineno-19-42" name="__codelineno-19-42" href="#__codelineno-19-42"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue operation */</span>
+<a id="__codelineno-19-42" name="__codelineno-19-42" href="#__codelineno-19-42"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-19-43" name="__codelineno-19-43" href="#__codelineno-19-43"></a><span class="w">    </span><span class="nx">pushFirst</span><span class="p">(</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-44" name="__codelineno-19-44" href="#__codelineno-19-44"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-19-45" name="__codelineno-19-45" href="#__codelineno-19-45"></a><span class="w">        </span><span class="c1">// If the linked list is empty, make both front and rear point to node</span>
@@ -5962,76 +6034,76 @@
 <a id="__codelineno-19-50" name="__codelineno-19-50" href="#__codelineno-19-50"></a><span class="w">            </span><span class="c1">// Add node to the head of the linked list</span>
 <a id="__codelineno-19-51" name="__codelineno-19-51" href="#__codelineno-19-51"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-19-52" name="__codelineno-19-52" href="#__codelineno-19-52"></a><span class="w">            </span><span class="nx">node</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">;</span>
-<a id="__codelineno-19-53" name="__codelineno-19-53" href="#__codelineno-19-53"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update head node</span>
+<a id="__codelineno-19-53" name="__codelineno-19-53" href="#__codelineno-19-53"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the head node</span>
 <a id="__codelineno-19-54" name="__codelineno-19-54" href="#__codelineno-19-54"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-19-55" name="__codelineno-19-55" href="#__codelineno-19-55"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-19-56" name="__codelineno-19-56" href="#__codelineno-19-56"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-57" name="__codelineno-19-57" href="#__codelineno-19-57"></a>
-<a id="__codelineno-19-58" name="__codelineno-19-58" href="#__codelineno-19-58"></a><span class="w">    </span><span class="cm">/* Temporarily store tail node value */</span>
+<a id="__codelineno-19-58" name="__codelineno-19-58" href="#__codelineno-19-58"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-19-59" name="__codelineno-19-59" href="#__codelineno-19-59"></a><span class="w">    </span><span class="nx">popLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-60" name="__codelineno-19-60" href="#__codelineno-19-60"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-61" name="__codelineno-19-61" href="#__codelineno-19-61"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-19-62" name="__codelineno-19-62" href="#__codelineno-19-62"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-19-63" name="__codelineno-19-63" href="#__codelineno-19-63"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">value</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Store tail node value</span>
-<a id="__codelineno-19-64" name="__codelineno-19-64" href="#__codelineno-19-64"></a><span class="w">        </span><span class="c1">// When the deque has one element, front and rear point to the same node</span>
+<a id="__codelineno-19-63" name="__codelineno-19-63" href="#__codelineno-19-63"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">value</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Store the tail node value</span>
+<a id="__codelineno-19-64" name="__codelineno-19-64" href="#__codelineno-19-64"></a><span class="w">        </span><span class="c1">// When the queue size is 1, the head and tail point to the same node</span>
 <a id="__codelineno-19-65" name="__codelineno-19-65" href="#__codelineno-19-65"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-66" name="__codelineno-19-66" href="#__codelineno-19-66"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-19-67" name="__codelineno-19-67" href="#__codelineno-19-67"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-19-68" name="__codelineno-19-68" href="#__codelineno-19-68"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-19-69" name="__codelineno-19-69" href="#__codelineno-19-69"></a><span class="w">        </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-19-69" name="__codelineno-19-69" href="#__codelineno-19-69"></a><span class="w">        </span><span class="c1">// Remove the tail node</span>
 <a id="__codelineno-19-70" name="__codelineno-19-70" href="#__codelineno-19-70"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-71" name="__codelineno-19-71" href="#__codelineno-19-71"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="nx">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="p">.</span><span class="nx">prev</span><span class="p">;</span>
 <a id="__codelineno-19-72" name="__codelineno-19-72" href="#__codelineno-19-72"></a><span class="w">            </span><span class="nx">temp</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-19-73" name="__codelineno-19-73" href="#__codelineno-19-73"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-19-74" name="__codelineno-19-74" href="#__codelineno-19-74"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-19-74" name="__codelineno-19-74" href="#__codelineno-19-74"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-19-75" name="__codelineno-19-75" href="#__codelineno-19-75"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-19-76" name="__codelineno-19-76" href="#__codelineno-19-76"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-19-77" name="__codelineno-19-77" href="#__codelineno-19-77"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">value</span><span class="p">;</span>
 <a id="__codelineno-19-78" name="__codelineno-19-78" href="#__codelineno-19-78"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-79" name="__codelineno-19-79" href="#__codelineno-19-79"></a>
-<a id="__codelineno-19-80" name="__codelineno-19-80" href="#__codelineno-19-80"></a><span class="w">    </span><span class="cm">/* Temporarily store head node value */</span>
+<a id="__codelineno-19-80" name="__codelineno-19-80" href="#__codelineno-19-80"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-19-81" name="__codelineno-19-81" href="#__codelineno-19-81"></a><span class="w">    </span><span class="nx">popFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-82" name="__codelineno-19-82" href="#__codelineno-19-82"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-83" name="__codelineno-19-83" href="#__codelineno-19-83"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-19-84" name="__codelineno-19-84" href="#__codelineno-19-84"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-19-85" name="__codelineno-19-85" href="#__codelineno-19-85"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">value</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Store the head node value</span>
-<a id="__codelineno-19-86" name="__codelineno-19-86" href="#__codelineno-19-86"></a><span class="w">        </span><span class="c1">// When the deque has one element, front and rear point to the same node</span>
+<a id="__codelineno-19-86" name="__codelineno-19-86" href="#__codelineno-19-86"></a><span class="w">        </span><span class="c1">// When the queue size is 1, the head and tail point to the same node</span>
 <a id="__codelineno-19-87" name="__codelineno-19-87" href="#__codelineno-19-87"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-88" name="__codelineno-19-88" href="#__codelineno-19-88"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-19-89" name="__codelineno-19-89" href="#__codelineno-19-89"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-19-90" name="__codelineno-19-90" href="#__codelineno-19-90"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-19-91" name="__codelineno-19-91" href="#__codelineno-19-91"></a><span class="w">        </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-19-91" name="__codelineno-19-91" href="#__codelineno-19-91"></a><span class="w">        </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-19-92" name="__codelineno-19-92" href="#__codelineno-19-92"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-93" name="__codelineno-19-93" href="#__codelineno-19-93"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="nx">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">.</span><span class="nx">next</span><span class="p">;</span>
 <a id="__codelineno-19-94" name="__codelineno-19-94" href="#__codelineno-19-94"></a><span class="w">            </span><span class="nx">temp</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-19-95" name="__codelineno-19-95" href="#__codelineno-19-95"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-19-96" name="__codelineno-19-96" href="#__codelineno-19-96"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update head node</span>
+<a id="__codelineno-19-96" name="__codelineno-19-96" href="#__codelineno-19-96"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the head node</span>
 <a id="__codelineno-19-97" name="__codelineno-19-97" href="#__codelineno-19-97"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-19-98" name="__codelineno-19-98" href="#__codelineno-19-98"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-19-99" name="__codelineno-19-99" href="#__codelineno-19-99"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">value</span><span class="p">;</span>
 <a id="__codelineno-19-100" name="__codelineno-19-100" href="#__codelineno-19-100"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-101" name="__codelineno-19-101" href="#__codelineno-19-101"></a>
-<a id="__codelineno-19-102" name="__codelineno-19-102" href="#__codelineno-19-102"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-19-102" name="__codelineno-19-102" href="#__codelineno-19-102"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-19-103" name="__codelineno-19-103" href="#__codelineno-19-103"></a><span class="w">    </span><span class="nx">peekLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-104" name="__codelineno-19-104" href="#__codelineno-19-104"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-19-105" name="__codelineno-19-105" href="#__codelineno-19-105"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-106" name="__codelineno-19-106" href="#__codelineno-19-106"></a>
-<a id="__codelineno-19-107" name="__codelineno-19-107" href="#__codelineno-19-107"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-19-107" name="__codelineno-19-107" href="#__codelineno-19-107"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-19-108" name="__codelineno-19-108" href="#__codelineno-19-108"></a><span class="w">    </span><span class="nx">peekFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-109" name="__codelineno-19-109" href="#__codelineno-19-109"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-19-110" name="__codelineno-19-110" href="#__codelineno-19-110"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-111" name="__codelineno-19-111" href="#__codelineno-19-111"></a>
-<a id="__codelineno-19-112" name="__codelineno-19-112" href="#__codelineno-19-112"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-19-112" name="__codelineno-19-112" href="#__codelineno-19-112"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-19-113" name="__codelineno-19-113" href="#__codelineno-19-113"></a><span class="w">    </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-114" name="__codelineno-19-114" href="#__codelineno-19-114"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="p">;</span>
 <a id="__codelineno-19-115" name="__codelineno-19-115" href="#__codelineno-19-115"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-116" name="__codelineno-19-116" href="#__codelineno-19-116"></a>
-<a id="__codelineno-19-117" name="__codelineno-19-117" href="#__codelineno-19-117"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-19-117" name="__codelineno-19-117" href="#__codelineno-19-117"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-19-118" name="__codelineno-19-118" href="#__codelineno-19-118"></a><span class="w">    </span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-119" name="__codelineno-19-119" href="#__codelineno-19-119"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-19-120" name="__codelineno-19-120" href="#__codelineno-19-120"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-121" name="__codelineno-19-121" href="#__codelineno-19-121"></a>
-<a id="__codelineno-19-122" name="__codelineno-19-122" href="#__codelineno-19-122"></a><span class="w">    </span><span class="cm">/* Print deque */</span>
+<a id="__codelineno-19-122" name="__codelineno-19-122" href="#__codelineno-19-122"></a><span class="w">    </span><span class="cm">/* Print the deque */</span>
 <a id="__codelineno-19-123" name="__codelineno-19-123" href="#__codelineno-19-123"></a><span class="w">    </span><span class="nx">print</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-124" name="__codelineno-19-124" href="#__codelineno-19-124"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-19-125" name="__codelineno-19-125" href="#__codelineno-19-125"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">;</span>
@@ -6047,8 +6119,8 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">linkedlist_deque.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Doubly linked list node */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">ListNode</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="nx">prev</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="p">;</span><span class="w"> </span><span class="c1">// Predecessor node reference (pointer)</span>
-<a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="nx">next</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="p">;</span><span class="w"> </span><span class="c1">// Successor node reference (pointer)</span>
+<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="nx">prev</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference (pointer) to the previous node</span>
+<a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="nx">next</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference (pointer) to the next node</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Node value</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6058,11 +6130,11 @@
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="p">}</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a>
-<a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="cm">/* Double-ended queue based on doubly linked list implementation */</span>
+<a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="cm">/* Deque implemented with a doubly linked list */</span>
 <a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="kd">class</span><span class="w"> </span><span class="nx">LinkedListDeque</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">front</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node front</span>
 <a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">rear</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="p">;</span><span class="w"> </span><span class="c1">// Tail node rear</span>
-<a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">queSize</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Length of the double-ended queue</span>
+<a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">queSize</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Deque size</span>
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
@@ -6070,7 +6142,7 @@
 <a id="__codelineno-20-23" name="__codelineno-20-23" href="#__codelineno-20-23"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a>
-<a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue operation */</span>
+<a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-20-27" name="__codelineno-20-27" href="#__codelineno-20-27"></a><span class="w">    </span><span class="nx">pushLast</span><span class="p">(</span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-28" name="__codelineno-20-28" href="#__codelineno-20-28"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">node</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-20-29" name="__codelineno-20-29" href="#__codelineno-20-29"></a><span class="w">        </span><span class="c1">// If the linked list is empty, make both front and rear point to node</span>
@@ -6081,12 +6153,12 @@
 <a id="__codelineno-20-34" name="__codelineno-20-34" href="#__codelineno-20-34"></a><span class="w">            </span><span class="c1">// Add node to the tail of the linked list</span>
 <a id="__codelineno-20-35" name="__codelineno-20-35" href="#__codelineno-20-35"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-20-36" name="__codelineno-20-36" href="#__codelineno-20-36"></a><span class="w">            </span><span class="nx">node</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="p">;</span>
-<a id="__codelineno-20-37" name="__codelineno-20-37" href="#__codelineno-20-37"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-20-37" name="__codelineno-20-37" href="#__codelineno-20-37"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-20-38" name="__codelineno-20-38" href="#__codelineno-20-38"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-20-39" name="__codelineno-20-39" href="#__codelineno-20-39"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-20-40" name="__codelineno-20-40" href="#__codelineno-20-40"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-41" name="__codelineno-20-41" href="#__codelineno-20-41"></a>
-<a id="__codelineno-20-42" name="__codelineno-20-42" href="#__codelineno-20-42"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue operation */</span>
+<a id="__codelineno-20-42" name="__codelineno-20-42" href="#__codelineno-20-42"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-20-43" name="__codelineno-20-43" href="#__codelineno-20-43"></a><span class="w">    </span><span class="nx">pushFirst</span><span class="p">(</span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-44" name="__codelineno-20-44" href="#__codelineno-20-44"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">node</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-20-45" name="__codelineno-20-45" href="#__codelineno-20-45"></a><span class="w">        </span><span class="c1">// If the linked list is empty, make both front and rear point to node</span>
@@ -6097,76 +6169,76 @@
 <a id="__codelineno-20-50" name="__codelineno-20-50" href="#__codelineno-20-50"></a><span class="w">            </span><span class="c1">// Add node to the head of the linked list</span>
 <a id="__codelineno-20-51" name="__codelineno-20-51" href="#__codelineno-20-51"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-20-52" name="__codelineno-20-52" href="#__codelineno-20-52"></a><span class="w">            </span><span class="nx">node</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="p">;</span>
-<a id="__codelineno-20-53" name="__codelineno-20-53" href="#__codelineno-20-53"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update head node</span>
+<a id="__codelineno-20-53" name="__codelineno-20-53" href="#__codelineno-20-53"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the head node</span>
 <a id="__codelineno-20-54" name="__codelineno-20-54" href="#__codelineno-20-54"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-20-55" name="__codelineno-20-55" href="#__codelineno-20-55"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-20-56" name="__codelineno-20-56" href="#__codelineno-20-56"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-57" name="__codelineno-20-57" href="#__codelineno-20-57"></a>
-<a id="__codelineno-20-58" name="__codelineno-20-58" href="#__codelineno-20-58"></a><span class="w">    </span><span class="cm">/* Temporarily store tail node value */</span>
+<a id="__codelineno-20-58" name="__codelineno-20-58" href="#__codelineno-20-58"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-20-59" name="__codelineno-20-59" href="#__codelineno-20-59"></a><span class="w">    </span><span class="nx">popLast</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-60" name="__codelineno-20-60" href="#__codelineno-20-60"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-61" name="__codelineno-20-61" href="#__codelineno-20-61"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-20-62" name="__codelineno-20-62" href="#__codelineno-20-62"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-20-63" name="__codelineno-20-63" href="#__codelineno-20-63"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">value</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="o">!</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Store tail node value</span>
-<a id="__codelineno-20-64" name="__codelineno-20-64" href="#__codelineno-20-64"></a><span class="w">        </span><span class="c1">// When the deque has one element, front and rear point to the same node</span>
+<a id="__codelineno-20-63" name="__codelineno-20-63" href="#__codelineno-20-63"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">value</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="o">!</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Store the tail node value</span>
+<a id="__codelineno-20-64" name="__codelineno-20-64" href="#__codelineno-20-64"></a><span class="w">        </span><span class="c1">// When the queue size is 1, the head and tail point to the same node</span>
 <a id="__codelineno-20-65" name="__codelineno-20-65" href="#__codelineno-20-65"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-66" name="__codelineno-20-66" href="#__codelineno-20-66"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-20-67" name="__codelineno-20-67" href="#__codelineno-20-67"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-20-68" name="__codelineno-20-68" href="#__codelineno-20-68"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-20-69" name="__codelineno-20-69" href="#__codelineno-20-69"></a><span class="w">        </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-20-69" name="__codelineno-20-69" href="#__codelineno-20-69"></a><span class="w">        </span><span class="c1">// Remove the tail node</span>
 <a id="__codelineno-20-70" name="__codelineno-20-70" href="#__codelineno-20-70"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-71" name="__codelineno-20-71" href="#__codelineno-20-71"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="nx">temp</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="o">!</span><span class="p">.</span><span class="nx">prev</span><span class="p">;</span>
 <a id="__codelineno-20-72" name="__codelineno-20-72" href="#__codelineno-20-72"></a><span class="w">            </span><span class="nx">temp</span><span class="o">!</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-20-73" name="__codelineno-20-73" href="#__codelineno-20-73"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="o">!</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-20-74" name="__codelineno-20-74" href="#__codelineno-20-74"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-20-74" name="__codelineno-20-74" href="#__codelineno-20-74"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-20-75" name="__codelineno-20-75" href="#__codelineno-20-75"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-20-76" name="__codelineno-20-76" href="#__codelineno-20-76"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-20-77" name="__codelineno-20-77" href="#__codelineno-20-77"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">value</span><span class="p">;</span>
 <a id="__codelineno-20-78" name="__codelineno-20-78" href="#__codelineno-20-78"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-79" name="__codelineno-20-79" href="#__codelineno-20-79"></a>
-<a id="__codelineno-20-80" name="__codelineno-20-80" href="#__codelineno-20-80"></a><span class="w">    </span><span class="cm">/* Temporarily store head node value */</span>
+<a id="__codelineno-20-80" name="__codelineno-20-80" href="#__codelineno-20-80"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-20-81" name="__codelineno-20-81" href="#__codelineno-20-81"></a><span class="w">    </span><span class="nx">popFirst</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-82" name="__codelineno-20-82" href="#__codelineno-20-82"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-83" name="__codelineno-20-83" href="#__codelineno-20-83"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-20-84" name="__codelineno-20-84" href="#__codelineno-20-84"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-20-85" name="__codelineno-20-85" href="#__codelineno-20-85"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">value</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Store the head node value</span>
-<a id="__codelineno-20-86" name="__codelineno-20-86" href="#__codelineno-20-86"></a><span class="w">        </span><span class="c1">// When the deque has one element, front and rear point to the same node</span>
+<a id="__codelineno-20-86" name="__codelineno-20-86" href="#__codelineno-20-86"></a><span class="w">        </span><span class="c1">// When the queue size is 1, the head and tail point to the same node</span>
 <a id="__codelineno-20-87" name="__codelineno-20-87" href="#__codelineno-20-87"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-88" name="__codelineno-20-88" href="#__codelineno-20-88"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-20-89" name="__codelineno-20-89" href="#__codelineno-20-89"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-20-90" name="__codelineno-20-90" href="#__codelineno-20-90"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-20-91" name="__codelineno-20-91" href="#__codelineno-20-91"></a><span class="w">        </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-20-91" name="__codelineno-20-91" href="#__codelineno-20-91"></a><span class="w">        </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-20-92" name="__codelineno-20-92" href="#__codelineno-20-92"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-93" name="__codelineno-20-93" href="#__codelineno-20-93"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="nx">temp</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="o">!</span><span class="p">.</span><span class="nx">next</span><span class="p">;</span>
 <a id="__codelineno-20-94" name="__codelineno-20-94" href="#__codelineno-20-94"></a><span class="w">            </span><span class="nx">temp</span><span class="o">!</span><span class="p">.</span><span class="nx">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-20-95" name="__codelineno-20-95" href="#__codelineno-20-95"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="o">!</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-20-96" name="__codelineno-20-96" href="#__codelineno-20-96"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update head node</span>
+<a id="__codelineno-20-96" name="__codelineno-20-96" href="#__codelineno-20-96"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the head node</span>
 <a id="__codelineno-20-97" name="__codelineno-20-97" href="#__codelineno-20-97"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-20-98" name="__codelineno-20-98" href="#__codelineno-20-98"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-20-99" name="__codelineno-20-99" href="#__codelineno-20-99"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">value</span><span class="p">;</span>
 <a id="__codelineno-20-100" name="__codelineno-20-100" href="#__codelineno-20-100"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-101" name="__codelineno-20-101" href="#__codelineno-20-101"></a>
-<a id="__codelineno-20-102" name="__codelineno-20-102" href="#__codelineno-20-102"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-20-102" name="__codelineno-20-102" href="#__codelineno-20-102"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-20-103" name="__codelineno-20-103" href="#__codelineno-20-103"></a><span class="w">    </span><span class="nx">peekLast</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-104" name="__codelineno-20-104" href="#__codelineno-20-104"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">this.rear.val</span><span class="p">;</span>
 <a id="__codelineno-20-105" name="__codelineno-20-105" href="#__codelineno-20-105"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-106" name="__codelineno-20-106" href="#__codelineno-20-106"></a>
-<a id="__codelineno-20-107" name="__codelineno-20-107" href="#__codelineno-20-107"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-20-107" name="__codelineno-20-107" href="#__codelineno-20-107"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-20-108" name="__codelineno-20-108" href="#__codelineno-20-108"></a><span class="w">    </span><span class="nx">peekFirst</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-109" name="__codelineno-20-109" href="#__codelineno-20-109"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">this.front.val</span><span class="p">;</span>
 <a id="__codelineno-20-110" name="__codelineno-20-110" href="#__codelineno-20-110"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-111" name="__codelineno-20-111" href="#__codelineno-20-111"></a>
-<a id="__codelineno-20-112" name="__codelineno-20-112" href="#__codelineno-20-112"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-20-112" name="__codelineno-20-112" href="#__codelineno-20-112"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-20-113" name="__codelineno-20-113" href="#__codelineno-20-113"></a><span class="w">    </span><span class="nx">size</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-114" name="__codelineno-20-114" href="#__codelineno-20-114"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="p">;</span>
 <a id="__codelineno-20-115" name="__codelineno-20-115" href="#__codelineno-20-115"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-116" name="__codelineno-20-116" href="#__codelineno-20-116"></a>
-<a id="__codelineno-20-117" name="__codelineno-20-117" href="#__codelineno-20-117"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-20-117" name="__codelineno-20-117" href="#__codelineno-20-117"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-20-118" name="__codelineno-20-118" href="#__codelineno-20-118"></a><span class="w">    </span><span class="nx">isEmpty</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-119" name="__codelineno-20-119" href="#__codelineno-20-119"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-20-120" name="__codelineno-20-120" href="#__codelineno-20-120"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-121" name="__codelineno-20-121" href="#__codelineno-20-121"></a>
-<a id="__codelineno-20-122" name="__codelineno-20-122" href="#__codelineno-20-122"></a><span class="w">    </span><span class="cm">/* Print deque */</span>
+<a id="__codelineno-20-122" name="__codelineno-20-122" href="#__codelineno-20-122"></a><span class="w">    </span><span class="cm">/* Print the deque */</span>
 <a id="__codelineno-20-123" name="__codelineno-20-123" href="#__codelineno-20-123"></a><span class="w">    </span><span class="nx">print</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-124" name="__codelineno-20-124" href="#__codelineno-20-124"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">arr</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-20-125" name="__codelineno-20-125" href="#__codelineno-20-125"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">temp</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="p">;</span>
@@ -6183,29 +6255,29 @@
 <div class="highlight"><span class="filename">linkedlist_deque.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Doubly linked list node */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Node value</span>
-<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Successor node reference</span>
-<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Predecessor node reference</span>
+<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference to the next node</span>
+<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference to the previous node</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="n">ListNode</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="p">{</span><span class="k">this</span><span class="p">.</span><span class="n">next</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">prev</span><span class="p">});</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="p">}</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a>
-<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="cm">/* Deque implemented based on doubly linked list */</span>
+<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="cm">/* Deque implemented with a doubly linked list */</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="kd">class</span><span class="w"> </span><span class="nc">LinkedListDeque</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">_front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node _front</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">_rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Tail node _rear</span>
-<a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">_queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Length of the double-ended queue</span>
+<a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">_queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Deque size</span>
 <a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a>
 <a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">  </span><span class="n">LinkedListDeque</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">_rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a>
-<a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="w">  </span><span class="cm">/* Get deque length */</span>
+<a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="w">  </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">_queSize</span><span class="p">;</span>
 <a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-25" name="__codelineno-21-25" href="#__codelineno-21-25"></a>
-<a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a><span class="w">  </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a><span class="w">  </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-21-27" name="__codelineno-21-27" href="#__codelineno-21-27"></a><span class="w">  </span><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-28" name="__codelineno-21-28" href="#__codelineno-21-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-21-29" name="__codelineno-21-29" href="#__codelineno-21-29"></a><span class="w">  </span><span class="p">}</span>
@@ -6214,89 +6286,89 @@
 <a id="__codelineno-21-32" name="__codelineno-21-32" href="#__codelineno-21-32"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">push</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="p">,</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-33" name="__codelineno-21-33" href="#__codelineno-21-33"></a><span class="w">    </span><span class="kd">final</span><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="n">_num</span><span class="p">);</span>
 <a id="__codelineno-21-34" name="__codelineno-21-34" href="#__codelineno-21-34"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-35" name="__codelineno-21-35" href="#__codelineno-21-35"></a><span class="w">      </span><span class="c1">// If list is empty, let both _front and _rear point to node</span>
+<a id="__codelineno-21-35" name="__codelineno-21-35" href="#__codelineno-21-35"></a><span class="w">      </span><span class="c1">// If the linked list is empty, make both _front and _rear point to node</span>
 <a id="__codelineno-21-36" name="__codelineno-21-36" href="#__codelineno-21-36"></a><span class="w">      </span><span class="n">_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-21-37" name="__codelineno-21-37" href="#__codelineno-21-37"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-38" name="__codelineno-21-38" href="#__codelineno-21-38"></a><span class="w">      </span><span class="c1">// Front of the queue enqueue operation</span>
+<a id="__codelineno-21-38" name="__codelineno-21-38" href="#__codelineno-21-38"></a><span class="w">      </span><span class="c1">// Enqueue at the front</span>
 <a id="__codelineno-21-39" name="__codelineno-21-39" href="#__codelineno-21-39"></a><span class="w">      </span><span class="c1">// Add node to the head of the linked list</span>
 <a id="__codelineno-21-40" name="__codelineno-21-40" href="#__codelineno-21-40"></a><span class="w">      </span><span class="n">_front</span><span class="o">!</span><span class="p">.</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-21-41" name="__codelineno-21-41" href="#__codelineno-21-41"></a><span class="w">      </span><span class="n">node</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_front</span><span class="p">;</span>
-<a id="__codelineno-21-42" name="__codelineno-21-42" href="#__codelineno-21-42"></a><span class="w">      </span><span class="n">_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update head node</span>
+<a id="__codelineno-21-42" name="__codelineno-21-42" href="#__codelineno-21-42"></a><span class="w">      </span><span class="n">_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the head node</span>
 <a id="__codelineno-21-43" name="__codelineno-21-43" href="#__codelineno-21-43"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-44" name="__codelineno-21-44" href="#__codelineno-21-44"></a><span class="w">      </span><span class="c1">// Rear of the queue enqueue operation</span>
+<a id="__codelineno-21-44" name="__codelineno-21-44" href="#__codelineno-21-44"></a><span class="w">      </span><span class="c1">// Enqueue at the rear</span>
 <a id="__codelineno-21-45" name="__codelineno-21-45" href="#__codelineno-21-45"></a><span class="w">      </span><span class="c1">// Add node to the tail of the linked list</span>
 <a id="__codelineno-21-46" name="__codelineno-21-46" href="#__codelineno-21-46"></a><span class="w">      </span><span class="n">_rear</span><span class="o">!</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-21-47" name="__codelineno-21-47" href="#__codelineno-21-47"></a><span class="w">      </span><span class="n">node</span><span class="p">.</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_rear</span><span class="p">;</span>
-<a id="__codelineno-21-48" name="__codelineno-21-48" href="#__codelineno-21-48"></a><span class="w">      </span><span class="n">_rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-21-48" name="__codelineno-21-48" href="#__codelineno-21-48"></a><span class="w">      </span><span class="n">_rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-21-49" name="__codelineno-21-49" href="#__codelineno-21-49"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-21-50" name="__codelineno-21-50" href="#__codelineno-21-50"></a><span class="w">    </span><span class="n">_queSize</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-21-50" name="__codelineno-21-50" href="#__codelineno-21-50"></a><span class="w">    </span><span class="n">_queSize</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the queue size</span>
 <a id="__codelineno-21-51" name="__codelineno-21-51" href="#__codelineno-21-51"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-52" name="__codelineno-21-52" href="#__codelineno-21-52"></a>
-<a id="__codelineno-21-53" name="__codelineno-21-53" href="#__codelineno-21-53"></a><span class="w">  </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-21-53" name="__codelineno-21-53" href="#__codelineno-21-53"></a><span class="w">  </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-21-54" name="__codelineno-21-54" href="#__codelineno-21-54"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">pushFirst</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-55" name="__codelineno-21-55" href="#__codelineno-21-55"></a><span class="w">    </span><span class="n">push</span><span class="p">(</span><span class="n">_num</span><span class="p">,</span><span class="w"> </span><span class="kc">true</span><span class="p">);</span>
 <a id="__codelineno-21-56" name="__codelineno-21-56" href="#__codelineno-21-56"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-57" name="__codelineno-21-57" href="#__codelineno-21-57"></a>
-<a id="__codelineno-21-58" name="__codelineno-21-58" href="#__codelineno-21-58"></a><span class="w">  </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-21-58" name="__codelineno-21-58" href="#__codelineno-21-58"></a><span class="w">  </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-21-59" name="__codelineno-21-59" href="#__codelineno-21-59"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">pushLast</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-60" name="__codelineno-21-60" href="#__codelineno-21-60"></a><span class="w">    </span><span class="n">push</span><span class="p">(</span><span class="n">_num</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">);</span>
 <a id="__codelineno-21-61" name="__codelineno-21-61" href="#__codelineno-21-61"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-62" name="__codelineno-21-62" href="#__codelineno-21-62"></a>
 <a id="__codelineno-21-63" name="__codelineno-21-63" href="#__codelineno-21-63"></a><span class="w">  </span><span class="cm">/* Dequeue operation */</span>
 <a id="__codelineno-21-64" name="__codelineno-21-64" href="#__codelineno-21-64"></a><span class="w">  </span><span class="kt">int</span><span class="o">?</span><span class="w"> </span><span class="n">pop</span><span class="p">(</span><span class="kt">bool</span><span class="w"> </span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-65" name="__codelineno-21-65" href="#__codelineno-21-65"></a><span class="w">    </span><span class="c1">// If queue is empty, return null directly</span>
+<a id="__codelineno-21-65" name="__codelineno-21-65" href="#__codelineno-21-65"></a><span class="w">    </span><span class="c1">// If the queue is empty, return null immediately</span>
 <a id="__codelineno-21-66" name="__codelineno-21-66" href="#__codelineno-21-66"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-67" name="__codelineno-21-67" href="#__codelineno-21-67"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-21-68" name="__codelineno-21-68" href="#__codelineno-21-68"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-21-69" name="__codelineno-21-69" href="#__codelineno-21-69"></a><span class="w">    </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
-<a id="__codelineno-21-70" name="__codelineno-21-70" href="#__codelineno-21-70"></a><span class="w">    </span><span class="c1">// When the deque has one element, front and rear point to the same node</span>
+<a id="__codelineno-21-70" name="__codelineno-21-70" href="#__codelineno-21-70"></a><span class="w">    </span><span class="c1">// When the queue size is 1, the head and tail point to the same node</span>
 <a id="__codelineno-21-71" name="__codelineno-21-71" href="#__codelineno-21-71"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-72" name="__codelineno-21-72" href="#__codelineno-21-72"></a><span class="w">      </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_front</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-21-73" name="__codelineno-21-73" href="#__codelineno-21-73"></a><span class="w">      </span><span class="n">_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-21-74" name="__codelineno-21-74" href="#__codelineno-21-74"></a><span class="w">      </span><span class="n">_rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-21-75" name="__codelineno-21-75" href="#__codelineno-21-75"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-21-76" name="__codelineno-21-76" href="#__codelineno-21-76"></a><span class="w">    </span><span class="c1">// Temporarily store head node value</span>
+<a id="__codelineno-21-76" name="__codelineno-21-76" href="#__codelineno-21-76"></a><span class="w">    </span><span class="c1">// Dequeue from the front</span>
 <a id="__codelineno-21-77" name="__codelineno-21-77" href="#__codelineno-21-77"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-78" name="__codelineno-21-78" href="#__codelineno-21-78"></a><span class="w">      </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_front</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Delete head node</span>
-<a id="__codelineno-21-79" name="__codelineno-21-79" href="#__codelineno-21-79"></a><span class="w">      </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-21-78" name="__codelineno-21-78" href="#__codelineno-21-78"></a><span class="w">      </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_front</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Save the head node value</span>
+<a id="__codelineno-21-79" name="__codelineno-21-79" href="#__codelineno-21-79"></a><span class="w">      </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-21-80" name="__codelineno-21-80" href="#__codelineno-21-80"></a><span class="w">      </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">fNext</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_front</span><span class="o">!</span><span class="p">.</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-21-81" name="__codelineno-21-81" href="#__codelineno-21-81"></a><span class="w">      </span><span class="n">fNext</span><span class="o">!</span><span class="p">.</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-21-82" name="__codelineno-21-82" href="#__codelineno-21-82"></a><span class="w">      </span><span class="n">_front</span><span class="o">!</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-21-83" name="__codelineno-21-83" href="#__codelineno-21-83"></a><span class="w">      </span><span class="n">_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fNext</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update head node</span>
-<a id="__codelineno-21-84" name="__codelineno-21-84" href="#__codelineno-21-84"></a><span class="w">    </span><span class="c1">// Temporarily store tail node value</span>
+<a id="__codelineno-21-83" name="__codelineno-21-83" href="#__codelineno-21-83"></a><span class="w">      </span><span class="n">_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fNext</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the head node</span>
+<a id="__codelineno-21-84" name="__codelineno-21-84" href="#__codelineno-21-84"></a><span class="w">    </span><span class="c1">// Dequeue from the rear</span>
 <a id="__codelineno-21-85" name="__codelineno-21-85" href="#__codelineno-21-85"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-86" name="__codelineno-21-86" href="#__codelineno-21-86"></a><span class="w">      </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_rear</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Delete tail node</span>
-<a id="__codelineno-21-87" name="__codelineno-21-87" href="#__codelineno-21-87"></a><span class="w">      </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-21-86" name="__codelineno-21-86" href="#__codelineno-21-86"></a><span class="w">      </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_rear</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Save the tail node value</span>
+<a id="__codelineno-21-87" name="__codelineno-21-87" href="#__codelineno-21-87"></a><span class="w">      </span><span class="c1">// Remove the tail node</span>
 <a id="__codelineno-21-88" name="__codelineno-21-88" href="#__codelineno-21-88"></a><span class="w">      </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">rPrev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_rear</span><span class="o">!</span><span class="p">.</span><span class="n">prev</span><span class="p">;</span>
 <a id="__codelineno-21-89" name="__codelineno-21-89" href="#__codelineno-21-89"></a><span class="w">      </span><span class="n">rPrev</span><span class="o">!</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-21-90" name="__codelineno-21-90" href="#__codelineno-21-90"></a><span class="w">      </span><span class="n">_rear</span><span class="o">!</span><span class="p">.</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-21-91" name="__codelineno-21-91" href="#__codelineno-21-91"></a><span class="w">      </span><span class="n">_rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rPrev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-21-91" name="__codelineno-21-91" href="#__codelineno-21-91"></a><span class="w">      </span><span class="n">_rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rPrev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-21-92" name="__codelineno-21-92" href="#__codelineno-21-92"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-21-93" name="__codelineno-21-93" href="#__codelineno-21-93"></a><span class="w">    </span><span class="n">_queSize</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-21-93" name="__codelineno-21-93" href="#__codelineno-21-93"></a><span class="w">    </span><span class="n">_queSize</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the queue size</span>
 <a id="__codelineno-21-94" name="__codelineno-21-94" href="#__codelineno-21-94"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-21-95" name="__codelineno-21-95" href="#__codelineno-21-95"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-96" name="__codelineno-21-96" href="#__codelineno-21-96"></a>
-<a id="__codelineno-21-97" name="__codelineno-21-97" href="#__codelineno-21-97"></a><span class="w">  </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-21-97" name="__codelineno-21-97" href="#__codelineno-21-97"></a><span class="w">  </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-21-98" name="__codelineno-21-98" href="#__codelineno-21-98"></a><span class="w">  </span><span class="kt">int</span><span class="o">?</span><span class="w"> </span><span class="n">popFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-99" name="__codelineno-21-99" href="#__codelineno-21-99"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">pop</span><span class="p">(</span><span class="kc">true</span><span class="p">);</span>
 <a id="__codelineno-21-100" name="__codelineno-21-100" href="#__codelineno-21-100"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-101" name="__codelineno-21-101" href="#__codelineno-21-101"></a>
-<a id="__codelineno-21-102" name="__codelineno-21-102" href="#__codelineno-21-102"></a><span class="w">  </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-21-102" name="__codelineno-21-102" href="#__codelineno-21-102"></a><span class="w">  </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-21-103" name="__codelineno-21-103" href="#__codelineno-21-103"></a><span class="w">  </span><span class="kt">int</span><span class="o">?</span><span class="w"> </span><span class="n">popLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-104" name="__codelineno-21-104" href="#__codelineno-21-104"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">pop</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span>
 <a id="__codelineno-21-105" name="__codelineno-21-105" href="#__codelineno-21-105"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-106" name="__codelineno-21-106" href="#__codelineno-21-106"></a>
-<a id="__codelineno-21-107" name="__codelineno-21-107" href="#__codelineno-21-107"></a><span class="w">  </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-21-107" name="__codelineno-21-107" href="#__codelineno-21-107"></a><span class="w">  </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-21-108" name="__codelineno-21-108" href="#__codelineno-21-108"></a><span class="w">  </span><span class="kt">int</span><span class="o">?</span><span class="w"> </span><span class="n">peekFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-109" name="__codelineno-21-109" href="#__codelineno-21-109"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_front</span><span class="o">?</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-21-110" name="__codelineno-21-110" href="#__codelineno-21-110"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-111" name="__codelineno-21-111" href="#__codelineno-21-111"></a>
-<a id="__codelineno-21-112" name="__codelineno-21-112" href="#__codelineno-21-112"></a><span class="w">  </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-21-112" name="__codelineno-21-112" href="#__codelineno-21-112"></a><span class="w">  </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-21-113" name="__codelineno-21-113" href="#__codelineno-21-113"></a><span class="w">  </span><span class="kt">int</span><span class="o">?</span><span class="w"> </span><span class="n">peekLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-114" name="__codelineno-21-114" href="#__codelineno-21-114"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_rear</span><span class="o">?</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-21-115" name="__codelineno-21-115" href="#__codelineno-21-115"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-116" name="__codelineno-21-116" href="#__codelineno-21-116"></a>
-<a id="__codelineno-21-117" name="__codelineno-21-117" href="#__codelineno-21-117"></a><span class="w">  </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-21-117" name="__codelineno-21-117" href="#__codelineno-21-117"></a><span class="w">  </span><span class="cm">/* Return an array for printing */</span>
 <a id="__codelineno-21-118" name="__codelineno-21-118" href="#__codelineno-21-118"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-119" name="__codelineno-21-119" href="#__codelineno-21-119"></a><span class="w">    </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_front</span><span class="p">;</span>
 <a id="__codelineno-21-120" name="__codelineno-21-120" href="#__codelineno-21-120"></a><span class="w">    </span><span class="kd">final</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
@@ -6313,8 +6385,8 @@
 <div class="highlight"><span class="filename">linkedlist_deque.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Doubly linked list node */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="nc">T</span><span class="p">,</span><span class="w">                                 </span><span class="c1">// Node value</span>
-<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="n">next</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="c1">// Successor node pointer</span>
-<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="n">prev</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="c1">// Predecessor node pointer</span>
+<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="n">next</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="c1">// Pointer to the next node</span>
+<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="n">prev</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="c1">// Pointer to the previous node</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="p">}</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="k">impl</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
@@ -6327,12 +6399,12 @@
 <a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="p">}</span>
 <a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a>
-<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="cm">/* Double-ended queue based on doubly linked list implementation */</span>
+<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="cm">/* Deque implemented with a doubly linked list */</span>
 <a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="cp">#[allow(dead_code)]</span>
 <a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="k">pub</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">LinkedListDeque</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w">    </span><span class="n">front</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="c1">// Head node front</span>
 <a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w">    </span><span class="n">rear</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span><span class="w">  </span><span class="c1">// Tail node rear</span>
-<a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w">    </span><span class="n">que_size</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w">                         </span><span class="c1">// Length of the double-ended queue</span>
+<a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w">    </span><span class="n">que_size</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w">                         </span><span class="c1">// Deque size</span>
 <a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="p">}</span>
 <a id="__codelineno-22-25" name="__codelineno-22-25" href="#__codelineno-22-25"></a>
 <a id="__codelineno-22-26" name="__codelineno-22-26" href="#__codelineno-22-26"></a><span class="k">impl</span><span class="o">&lt;</span><span class="n">T</span><span class="p">:</span><span class="w"> </span><span class="nb">Copy</span><span class="o">&gt;</span><span class="w"> </span><span class="n">LinkedListDeque</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
@@ -6344,12 +6416,12 @@
 <a id="__codelineno-22-32" name="__codelineno-22-32" href="#__codelineno-22-32"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-33" name="__codelineno-22-33" href="#__codelineno-22-33"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-34" name="__codelineno-22-34" href="#__codelineno-22-34"></a>
-<a id="__codelineno-22-35" name="__codelineno-22-35" href="#__codelineno-22-35"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-22-35" name="__codelineno-22-35" href="#__codelineno-22-35"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-22-36" name="__codelineno-22-36" href="#__codelineno-22-36"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-37" name="__codelineno-22-37" href="#__codelineno-22-37"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="p">;</span>
 <a id="__codelineno-22-38" name="__codelineno-22-38" href="#__codelineno-22-38"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-39" name="__codelineno-22-39" href="#__codelineno-22-39"></a>
-<a id="__codelineno-22-40" name="__codelineno-22-40" href="#__codelineno-22-40"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-22-40" name="__codelineno-22-40" href="#__codelineno-22-40"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-22-41" name="__codelineno-22-41" href="#__codelineno-22-41"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">is_empty</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-42" name="__codelineno-22-42" href="#__codelineno-22-42"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-22-43" name="__codelineno-22-43" href="#__codelineno-22-43"></a><span class="w">    </span><span class="p">}</span>
@@ -6357,7 +6429,7 @@
 <a id="__codelineno-22-45" name="__codelineno-22-45" href="#__codelineno-22-45"></a><span class="w">    </span><span class="cm">/* Enqueue operation */</span>
 <a id="__codelineno-22-46" name="__codelineno-22-46" href="#__codelineno-22-46"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nc">T</span><span class="p">,</span><span class="w"> </span><span class="n">is_front</span><span class="p">:</span><span class="w"> </span><span class="kt">bool</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-47" name="__codelineno-22-47" href="#__codelineno-22-47"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
-<a id="__codelineno-22-48" name="__codelineno-22-48" href="#__codelineno-22-48"></a><span class="w">        </span><span class="c1">// Front of the queue enqueue operation</span>
+<a id="__codelineno-22-48" name="__codelineno-22-48" href="#__codelineno-22-48"></a><span class="w">        </span><span class="c1">// Enqueue at the front</span>
 <a id="__codelineno-22-49" name="__codelineno-22-49" href="#__codelineno-22-49"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">is_front</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-50" name="__codelineno-22-50" href="#__codelineno-22-50"></a><span class="w">            </span><span class="k">match</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="p">.</span><span class="n">take</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-51" name="__codelineno-22-51" href="#__codelineno-22-51"></a><span class="w">                </span><span class="c1">// If the linked list is empty, make both front and rear point to node</span>
@@ -6369,11 +6441,11 @@
 <a id="__codelineno-22-57" name="__codelineno-22-57" href="#__codelineno-22-57"></a><span class="w">                </span><span class="nb">Some</span><span class="p">(</span><span class="n">old_front</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-58" name="__codelineno-22-58" href="#__codelineno-22-58"></a><span class="w">                    </span><span class="n">old_front</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-22-59" name="__codelineno-22-59" href="#__codelineno-22-59"></a><span class="w">                    </span><span class="n">node</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">old_front</span><span class="p">);</span>
-<a id="__codelineno-22-60" name="__codelineno-22-60" href="#__codelineno-22-60"></a><span class="w">                    </span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update head node</span>
+<a id="__codelineno-22-60" name="__codelineno-22-60" href="#__codelineno-22-60"></a><span class="w">                    </span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update the head node</span>
 <a id="__codelineno-22-61" name="__codelineno-22-61" href="#__codelineno-22-61"></a><span class="w">                </span><span class="p">}</span>
 <a id="__codelineno-22-62" name="__codelineno-22-62" href="#__codelineno-22-62"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-22-63" name="__codelineno-22-63" href="#__codelineno-22-63"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-22-64" name="__codelineno-22-64" href="#__codelineno-22-64"></a><span class="w">        </span><span class="c1">// Rear of the queue enqueue operation</span>
+<a id="__codelineno-22-64" name="__codelineno-22-64" href="#__codelineno-22-64"></a><span class="w">        </span><span class="c1">// Enqueue at the rear</span>
 <a id="__codelineno-22-65" name="__codelineno-22-65" href="#__codelineno-22-65"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-66" name="__codelineno-22-66" href="#__codelineno-22-66"></a><span class="w">            </span><span class="k">match</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">rear</span><span class="p">.</span><span class="n">take</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-67" name="__codelineno-22-67" href="#__codelineno-22-67"></a><span class="w">                </span><span class="c1">// If the linked list is empty, make both front and rear point to node</span>
@@ -6385,84 +6457,84 @@
 <a id="__codelineno-22-73" name="__codelineno-22-73" href="#__codelineno-22-73"></a><span class="w">                </span><span class="nb">Some</span><span class="p">(</span><span class="n">old_rear</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-74" name="__codelineno-22-74" href="#__codelineno-22-74"></a><span class="w">                    </span><span class="n">old_rear</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-22-75" name="__codelineno-22-75" href="#__codelineno-22-75"></a><span class="w">                    </span><span class="n">node</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">old_rear</span><span class="p">);</span>
-<a id="__codelineno-22-76" name="__codelineno-22-76" href="#__codelineno-22-76"></a><span class="w">                    </span><span class="bp">self</span><span class="p">.</span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-22-76" name="__codelineno-22-76" href="#__codelineno-22-76"></a><span class="w">                    </span><span class="bp">self</span><span class="p">.</span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-22-77" name="__codelineno-22-77" href="#__codelineno-22-77"></a><span class="w">                </span><span class="p">}</span>
 <a id="__codelineno-22-78" name="__codelineno-22-78" href="#__codelineno-22-78"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-22-79" name="__codelineno-22-79" href="#__codelineno-22-79"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-22-80" name="__codelineno-22-80" href="#__codelineno-22-80"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-22-80" name="__codelineno-22-80" href="#__codelineno-22-80"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the deque size</span>
 <a id="__codelineno-22-81" name="__codelineno-22-81" href="#__codelineno-22-81"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-82" name="__codelineno-22-82" href="#__codelineno-22-82"></a>
-<a id="__codelineno-22-83" name="__codelineno-22-83" href="#__codelineno-22-83"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-22-83" name="__codelineno-22-83" href="#__codelineno-22-83"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-22-84" name="__codelineno-22-84" href="#__codelineno-22-84"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">push_first</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nc">T</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-85" name="__codelineno-22-85" href="#__codelineno-22-85"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kc">true</span><span class="p">);</span>
 <a id="__codelineno-22-86" name="__codelineno-22-86" href="#__codelineno-22-86"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-87" name="__codelineno-22-87" href="#__codelineno-22-87"></a>
-<a id="__codelineno-22-88" name="__codelineno-22-88" href="#__codelineno-22-88"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-22-88" name="__codelineno-22-88" href="#__codelineno-22-88"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-22-89" name="__codelineno-22-89" href="#__codelineno-22-89"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">push_last</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nc">T</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-90" name="__codelineno-22-90" href="#__codelineno-22-90"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">);</span>
 <a id="__codelineno-22-91" name="__codelineno-22-91" href="#__codelineno-22-91"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-92" name="__codelineno-22-92" href="#__codelineno-22-92"></a>
 <a id="__codelineno-22-93" name="__codelineno-22-93" href="#__codelineno-22-93"></a><span class="w">    </span><span class="cm">/* Dequeue operation */</span>
 <a id="__codelineno-22-94" name="__codelineno-22-94" href="#__codelineno-22-94"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">pop</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">is_front</span><span class="p">:</span><span class="w"> </span><span class="kt">bool</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-95" name="__codelineno-22-95" href="#__codelineno-22-95"></a><span class="w">        </span><span class="c1">// If queue is empty, return None directly</span>
+<a id="__codelineno-22-95" name="__codelineno-22-95" href="#__codelineno-22-95"></a><span class="w">        </span><span class="c1">// If the deque is empty, return None immediately</span>
 <a id="__codelineno-22-96" name="__codelineno-22-96" href="#__codelineno-22-96"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">is_empty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-97" name="__codelineno-22-97" href="#__codelineno-22-97"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nb">None</span><span class="p">;</span>
 <a id="__codelineno-22-98" name="__codelineno-22-98" href="#__codelineno-22-98"></a><span class="w">        </span><span class="p">};</span>
-<a id="__codelineno-22-99" name="__codelineno-22-99" href="#__codelineno-22-99"></a><span class="w">        </span><span class="c1">// Temporarily store head node value</span>
+<a id="__codelineno-22-99" name="__codelineno-22-99" href="#__codelineno-22-99"></a><span class="w">        </span><span class="c1">// Dequeue from the front</span>
 <a id="__codelineno-22-100" name="__codelineno-22-100" href="#__codelineno-22-100"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">is_front</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-101" name="__codelineno-22-101" href="#__codelineno-22-101"></a><span class="w">            </span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="p">.</span><span class="n">take</span><span class="p">().</span><span class="n">map</span><span class="p">(</span><span class="o">|</span><span class="n">old_front</span><span class="o">|</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-102" name="__codelineno-22-102" href="#__codelineno-22-102"></a><span class="w">                </span><span class="k">match</span><span class="w"> </span><span class="n">old_front</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">next</span><span class="p">.</span><span class="n">take</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-103" name="__codelineno-22-103" href="#__codelineno-22-103"></a><span class="w">                    </span><span class="nb">Some</span><span class="p">(</span><span class="n">new_front</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-104" name="__codelineno-22-104" href="#__codelineno-22-104"></a><span class="w">                        </span><span class="n">new_front</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">prev</span><span class="p">.</span><span class="n">take</span><span class="p">();</span>
-<a id="__codelineno-22-105" name="__codelineno-22-105" href="#__codelineno-22-105"></a><span class="w">                        </span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">new_front</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update head node</span>
+<a id="__codelineno-22-105" name="__codelineno-22-105" href="#__codelineno-22-105"></a><span class="w">                        </span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">new_front</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update the head node</span>
 <a id="__codelineno-22-106" name="__codelineno-22-106" href="#__codelineno-22-106"></a><span class="w">                    </span><span class="p">}</span>
 <a id="__codelineno-22-107" name="__codelineno-22-107" href="#__codelineno-22-107"></a><span class="w">                    </span><span class="nb">None</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-108" name="__codelineno-22-108" href="#__codelineno-22-108"></a><span class="w">                        </span><span class="bp">self</span><span class="p">.</span><span class="n">rear</span><span class="p">.</span><span class="n">take</span><span class="p">();</span>
 <a id="__codelineno-22-109" name="__codelineno-22-109" href="#__codelineno-22-109"></a><span class="w">                    </span><span class="p">}</span>
 <a id="__codelineno-22-110" name="__codelineno-22-110" href="#__codelineno-22-110"></a><span class="w">                </span><span class="p">}</span>
-<a id="__codelineno-22-111" name="__codelineno-22-111" href="#__codelineno-22-111"></a><span class="w">                </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-22-111" name="__codelineno-22-111" href="#__codelineno-22-111"></a><span class="w">                </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the deque size</span>
 <a id="__codelineno-22-112" name="__codelineno-22-112" href="#__codelineno-22-112"></a><span class="w">                </span><span class="n">old_front</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span>
 <a id="__codelineno-22-113" name="__codelineno-22-113" href="#__codelineno-22-113"></a><span class="w">            </span><span class="p">})</span>
 <a id="__codelineno-22-114" name="__codelineno-22-114" href="#__codelineno-22-114"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-22-115" name="__codelineno-22-115" href="#__codelineno-22-115"></a><span class="w">        </span><span class="c1">// Temporarily store tail node value</span>
+<a id="__codelineno-22-115" name="__codelineno-22-115" href="#__codelineno-22-115"></a><span class="w">        </span><span class="c1">// Dequeue from the rear</span>
 <a id="__codelineno-22-116" name="__codelineno-22-116" href="#__codelineno-22-116"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-117" name="__codelineno-22-117" href="#__codelineno-22-117"></a><span class="w">            </span><span class="bp">self</span><span class="p">.</span><span class="n">rear</span><span class="p">.</span><span class="n">take</span><span class="p">().</span><span class="n">map</span><span class="p">(</span><span class="o">|</span><span class="n">old_rear</span><span class="o">|</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-118" name="__codelineno-22-118" href="#__codelineno-22-118"></a><span class="w">                </span><span class="k">match</span><span class="w"> </span><span class="n">old_rear</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">prev</span><span class="p">.</span><span class="n">take</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-119" name="__codelineno-22-119" href="#__codelineno-22-119"></a><span class="w">                    </span><span class="nb">Some</span><span class="p">(</span><span class="n">new_rear</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-120" name="__codelineno-22-120" href="#__codelineno-22-120"></a><span class="w">                        </span><span class="n">new_rear</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">next</span><span class="p">.</span><span class="n">take</span><span class="p">();</span>
-<a id="__codelineno-22-121" name="__codelineno-22-121" href="#__codelineno-22-121"></a><span class="w">                        </span><span class="bp">self</span><span class="p">.</span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">new_rear</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-22-121" name="__codelineno-22-121" href="#__codelineno-22-121"></a><span class="w">                        </span><span class="bp">self</span><span class="p">.</span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">new_rear</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-22-122" name="__codelineno-22-122" href="#__codelineno-22-122"></a><span class="w">                    </span><span class="p">}</span>
 <a id="__codelineno-22-123" name="__codelineno-22-123" href="#__codelineno-22-123"></a><span class="w">                    </span><span class="nb">None</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-124" name="__codelineno-22-124" href="#__codelineno-22-124"></a><span class="w">                        </span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="p">.</span><span class="n">take</span><span class="p">();</span>
 <a id="__codelineno-22-125" name="__codelineno-22-125" href="#__codelineno-22-125"></a><span class="w">                    </span><span class="p">}</span>
 <a id="__codelineno-22-126" name="__codelineno-22-126" href="#__codelineno-22-126"></a><span class="w">                </span><span class="p">}</span>
-<a id="__codelineno-22-127" name="__codelineno-22-127" href="#__codelineno-22-127"></a><span class="w">                </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-22-127" name="__codelineno-22-127" href="#__codelineno-22-127"></a><span class="w">                </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the deque size</span>
 <a id="__codelineno-22-128" name="__codelineno-22-128" href="#__codelineno-22-128"></a><span class="w">                </span><span class="n">old_rear</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span>
 <a id="__codelineno-22-129" name="__codelineno-22-129" href="#__codelineno-22-129"></a><span class="w">            </span><span class="p">})</span>
 <a id="__codelineno-22-130" name="__codelineno-22-130" href="#__codelineno-22-130"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-131" name="__codelineno-22-131" href="#__codelineno-22-131"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-132" name="__codelineno-22-132" href="#__codelineno-22-132"></a>
-<a id="__codelineno-22-133" name="__codelineno-22-133" href="#__codelineno-22-133"></a><span class="w">    </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-22-133" name="__codelineno-22-133" href="#__codelineno-22-133"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-22-134" name="__codelineno-22-134" href="#__codelineno-22-134"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">pop_first</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-135" name="__codelineno-22-135" href="#__codelineno-22-135"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">pop</span><span class="p">(</span><span class="kc">true</span><span class="p">);</span>
 <a id="__codelineno-22-136" name="__codelineno-22-136" href="#__codelineno-22-136"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-137" name="__codelineno-22-137" href="#__codelineno-22-137"></a>
-<a id="__codelineno-22-138" name="__codelineno-22-138" href="#__codelineno-22-138"></a><span class="w">    </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-22-138" name="__codelineno-22-138" href="#__codelineno-22-138"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-22-139" name="__codelineno-22-139" href="#__codelineno-22-139"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">pop_last</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-140" name="__codelineno-22-140" href="#__codelineno-22-140"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">pop</span><span class="p">(</span><span class="kc">false</span><span class="p">);</span>
 <a id="__codelineno-22-141" name="__codelineno-22-141" href="#__codelineno-22-141"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-142" name="__codelineno-22-142" href="#__codelineno-22-142"></a>
-<a id="__codelineno-22-143" name="__codelineno-22-143" href="#__codelineno-22-143"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-22-143" name="__codelineno-22-143" href="#__codelineno-22-143"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-22-144" name="__codelineno-22-144" href="#__codelineno-22-144"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">peek_first</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-145" name="__codelineno-22-145" href="#__codelineno-22-145"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="p">.</span><span class="n">as_ref</span><span class="p">()</span>
 <a id="__codelineno-22-146" name="__codelineno-22-146" href="#__codelineno-22-146"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-147" name="__codelineno-22-147" href="#__codelineno-22-147"></a>
-<a id="__codelineno-22-148" name="__codelineno-22-148" href="#__codelineno-22-148"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-22-148" name="__codelineno-22-148" href="#__codelineno-22-148"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-22-149" name="__codelineno-22-149" href="#__codelineno-22-149"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">peek_last</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-150" name="__codelineno-22-150" href="#__codelineno-22-150"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">rear</span><span class="p">.</span><span class="n">as_ref</span><span class="p">()</span>
 <a id="__codelineno-22-151" name="__codelineno-22-151" href="#__codelineno-22-151"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-152" name="__codelineno-22-152" href="#__codelineno-22-152"></a>
-<a id="__codelineno-22-153" name="__codelineno-22-153" href="#__codelineno-22-153"></a><span class="w">    </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-22-153" name="__codelineno-22-153" href="#__codelineno-22-153"></a><span class="w">    </span><span class="cm">/* Return a vector for printing */</span>
 <a id="__codelineno-22-154" name="__codelineno-22-154" href="#__codelineno-22-154"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">to_array</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">head</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-155" name="__codelineno-22-155" href="#__codelineno-22-155"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-22-156" name="__codelineno-22-156" href="#__codelineno-22-156"></a><span class="w">        </span><span class="k">fn</span><span class="w"> </span><span class="nf">recur</span><span class="o">&lt;</span><span class="n">T</span><span class="p">:</span><span class="w"> </span><span class="nb">Copy</span><span class="o">&gt;</span><span class="p">(</span><span class="n">cur</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6482,8 +6554,8 @@
 <div class="highlight"><span class="filename">linkedlist_deque.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Doubly linked list node */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">DoublyListNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w">                     </span><span class="c1">// Node value</span>
-<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">struct</span><span class="w"> </span><span class="nc">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Successor node</span>
-<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="k">struct</span><span class="w"> </span><span class="nc">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Predecessor node</span>
+<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">struct</span><span class="w"> </span><span class="nc">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">next</span><span class="p">;</span><span class="w"> </span><span class="c1">// Next node</span>
+<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="k">struct</span><span class="w"> </span><span class="nc">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">prev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Previous node</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="p">}</span><span class="w"> </span><span class="n">DoublyListNode</span><span class="p">;</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="cm">/* Constructor */</span>
@@ -6500,10 +6572,10 @@
 <a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="p">}</span>
 <a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a>
-<a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="cm">/* Double-ended queue based on doubly linked list implementation */</span>
+<a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="cm">/* Deque implemented with a doubly linked list */</span>
 <a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">    </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node front, tail node rear</span>
-<a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w">                  </span><span class="c1">// Length of the double-ended queue</span>
+<a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">    </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node front and tail node rear</span>
+<a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w">                  </span><span class="c1">// Deque size</span>
 <a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="p">}</span><span class="w"> </span><span class="n">LinkedListDeque</span><span class="p">;</span>
 <a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a>
 <a id="__codelineno-23-28" name="__codelineno-23-28" href="#__codelineno-23-28"></a><span class="cm">/* Constructor */</span>
@@ -6523,16 +6595,16 @@
 <a id="__codelineno-23-42" name="__codelineno-23-42" href="#__codelineno-23-42"></a><span class="w">        </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-23-43" name="__codelineno-23-43" href="#__codelineno-23-43"></a><span class="w">        </span><span class="n">free</span><span class="p">(</span><span class="n">tmp</span><span class="p">);</span>
 <a id="__codelineno-23-44" name="__codelineno-23-44" href="#__codelineno-23-44"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-45" name="__codelineno-23-45" href="#__codelineno-23-45"></a><span class="w">    </span><span class="c1">// Free deque structure</span>
+<a id="__codelineno-23-45" name="__codelineno-23-45" href="#__codelineno-23-45"></a><span class="w">    </span><span class="c1">// Free the deque structure</span>
 <a id="__codelineno-23-46" name="__codelineno-23-46" href="#__codelineno-23-46"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">deque</span><span class="p">);</span>
 <a id="__codelineno-23-47" name="__codelineno-23-47" href="#__codelineno-23-47"></a><span class="p">}</span>
 <a id="__codelineno-23-48" name="__codelineno-23-48" href="#__codelineno-23-48"></a>
-<a id="__codelineno-23-49" name="__codelineno-23-49" href="#__codelineno-23-49"></a><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-23-49" name="__codelineno-23-49" href="#__codelineno-23-49"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-23-50" name="__codelineno-23-50" href="#__codelineno-23-50"></a><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="n">LinkedListDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-51" name="__codelineno-23-51" href="#__codelineno-23-51"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-23-52" name="__codelineno-23-52" href="#__codelineno-23-52"></a><span class="p">}</span>
 <a id="__codelineno-23-53" name="__codelineno-23-53" href="#__codelineno-23-53"></a>
-<a id="__codelineno-23-54" name="__codelineno-23-54" href="#__codelineno-23-54"></a><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-23-54" name="__codelineno-23-54" href="#__codelineno-23-54"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-23-55" name="__codelineno-23-55" href="#__codelineno-23-55"></a><span class="kt">bool</span><span class="w"> </span><span class="nf">empty</span><span class="p">(</span><span class="n">LinkedListDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-56" name="__codelineno-23-56" href="#__codelineno-23-56"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="p">(</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-23-57" name="__codelineno-23-57" href="#__codelineno-23-57"></a><span class="p">}</span>
@@ -6540,44 +6612,44 @@
 <a id="__codelineno-23-59" name="__codelineno-23-59" href="#__codelineno-23-59"></a><span class="cm">/* Enqueue */</span>
 <a id="__codelineno-23-60" name="__codelineno-23-60" href="#__codelineno-23-60"></a><span class="kt">void</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="n">LinkedListDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-61" name="__codelineno-23-61" href="#__codelineno-23-61"></a><span class="w">    </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newDoublyListNode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
-<a id="__codelineno-23-62" name="__codelineno-23-62" href="#__codelineno-23-62"></a><span class="w">    </span><span class="c1">// If list is empty, set both front and rear to node</span>
+<a id="__codelineno-23-62" name="__codelineno-23-62" href="#__codelineno-23-62"></a><span class="w">    </span><span class="c1">// If the list is empty, make both front and rear point to node</span>
 <a id="__codelineno-23-63" name="__codelineno-23-63" href="#__codelineno-23-63"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">empty</span><span class="p">(</span><span class="n">deque</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-64" name="__codelineno-23-64" href="#__codelineno-23-64"></a><span class="w">        </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-23-65" name="__codelineno-23-65" href="#__codelineno-23-65"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-66" name="__codelineno-23-66" href="#__codelineno-23-66"></a><span class="w">    </span><span class="c1">// Front of the queue enqueue operation</span>
+<a id="__codelineno-23-66" name="__codelineno-23-66" href="#__codelineno-23-66"></a><span class="w">    </span><span class="c1">// Enqueue at the front</span>
 <a id="__codelineno-23-67" name="__codelineno-23-67" href="#__codelineno-23-67"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-68" name="__codelineno-23-68" href="#__codelineno-23-68"></a><span class="w">        </span><span class="c1">// Add node to the head of the linked list</span>
 <a id="__codelineno-23-69" name="__codelineno-23-69" href="#__codelineno-23-69"></a><span class="w">        </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="o">-&gt;</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-23-70" name="__codelineno-23-70" href="#__codelineno-23-70"></a><span class="w">        </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="p">;</span>
-<a id="__codelineno-23-71" name="__codelineno-23-71" href="#__codelineno-23-71"></a><span class="w">        </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update head node</span>
+<a id="__codelineno-23-71" name="__codelineno-23-71" href="#__codelineno-23-71"></a><span class="w">        </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the head node</span>
 <a id="__codelineno-23-72" name="__codelineno-23-72" href="#__codelineno-23-72"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-73" name="__codelineno-23-73" href="#__codelineno-23-73"></a><span class="w">    </span><span class="c1">// Rear of the queue enqueue operation</span>
+<a id="__codelineno-23-73" name="__codelineno-23-73" href="#__codelineno-23-73"></a><span class="w">    </span><span class="c1">// Enqueue at the rear</span>
 <a id="__codelineno-23-74" name="__codelineno-23-74" href="#__codelineno-23-74"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-75" name="__codelineno-23-75" href="#__codelineno-23-75"></a><span class="w">        </span><span class="c1">// Add node to the tail of the linked list</span>
 <a id="__codelineno-23-76" name="__codelineno-23-76" href="#__codelineno-23-76"></a><span class="w">        </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">rear</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-23-77" name="__codelineno-23-77" href="#__codelineno-23-77"></a><span class="w">        </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">rear</span><span class="p">;</span>
 <a id="__codelineno-23-78" name="__codelineno-23-78" href="#__codelineno-23-78"></a><span class="w">        </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-23-79" name="__codelineno-23-79" href="#__codelineno-23-79"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-80" name="__codelineno-23-80" href="#__codelineno-23-80"></a><span class="w">    </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-23-80" name="__codelineno-23-80" href="#__codelineno-23-80"></a><span class="w">    </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the queue size</span>
 <a id="__codelineno-23-81" name="__codelineno-23-81" href="#__codelineno-23-81"></a><span class="p">}</span>
 <a id="__codelineno-23-82" name="__codelineno-23-82" href="#__codelineno-23-82"></a>
-<a id="__codelineno-23-83" name="__codelineno-23-83" href="#__codelineno-23-83"></a><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-23-83" name="__codelineno-23-83" href="#__codelineno-23-83"></a><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-23-84" name="__codelineno-23-84" href="#__codelineno-23-84"></a><span class="kt">void</span><span class="w"> </span><span class="nf">pushFirst</span><span class="p">(</span><span class="n">LinkedListDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-85" name="__codelineno-23-85" href="#__codelineno-23-85"></a><span class="w">    </span><span class="n">push</span><span class="p">(</span><span class="n">deque</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="nb">true</span><span class="p">);</span>
 <a id="__codelineno-23-86" name="__codelineno-23-86" href="#__codelineno-23-86"></a><span class="p">}</span>
 <a id="__codelineno-23-87" name="__codelineno-23-87" href="#__codelineno-23-87"></a>
-<a id="__codelineno-23-88" name="__codelineno-23-88" href="#__codelineno-23-88"></a><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-23-88" name="__codelineno-23-88" href="#__codelineno-23-88"></a><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-23-89" name="__codelineno-23-89" href="#__codelineno-23-89"></a><span class="kt">void</span><span class="w"> </span><span class="nf">pushLast</span><span class="p">(</span><span class="n">LinkedListDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-90" name="__codelineno-23-90" href="#__codelineno-23-90"></a><span class="w">    </span><span class="n">push</span><span class="p">(</span><span class="n">deque</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span>
 <a id="__codelineno-23-91" name="__codelineno-23-91" href="#__codelineno-23-91"></a><span class="p">}</span>
 <a id="__codelineno-23-92" name="__codelineno-23-92" href="#__codelineno-23-92"></a>
-<a id="__codelineno-23-93" name="__codelineno-23-93" href="#__codelineno-23-93"></a><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-23-93" name="__codelineno-23-93" href="#__codelineno-23-93"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-23-94" name="__codelineno-23-94" href="#__codelineno-23-94"></a><span class="kt">int</span><span class="w"> </span><span class="nf">peekFirst</span><span class="p">(</span><span class="n">LinkedListDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-95" name="__codelineno-23-95" href="#__codelineno-23-95"></a><span class="w">    </span><span class="n">assert</span><span class="p">(</span><span class="n">size</span><span class="p">(</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="p">);</span>
 <a id="__codelineno-23-96" name="__codelineno-23-96" href="#__codelineno-23-96"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-23-97" name="__codelineno-23-97" href="#__codelineno-23-97"></a><span class="p">}</span>
 <a id="__codelineno-23-98" name="__codelineno-23-98" href="#__codelineno-23-98"></a>
-<a id="__codelineno-23-99" name="__codelineno-23-99" href="#__codelineno-23-99"></a><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-23-99" name="__codelineno-23-99" href="#__codelineno-23-99"></a><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-23-100" name="__codelineno-23-100" href="#__codelineno-23-100"></a><span class="kt">int</span><span class="w"> </span><span class="nf">peekLast</span><span class="p">(</span><span class="n">LinkedListDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-101" name="__codelineno-23-101" href="#__codelineno-23-101"></a><span class="w">    </span><span class="n">assert</span><span class="p">(</span><span class="n">size</span><span class="p">(</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">rear</span><span class="p">);</span>
 <a id="__codelineno-23-102" name="__codelineno-23-102" href="#__codelineno-23-102"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">rear</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
@@ -6588,49 +6660,49 @@
 <a id="__codelineno-23-107" name="__codelineno-23-107" href="#__codelineno-23-107"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">empty</span><span class="p">(</span><span class="n">deque</span><span class="p">))</span>
 <a id="__codelineno-23-108" name="__codelineno-23-108" href="#__codelineno-23-108"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
 <a id="__codelineno-23-109" name="__codelineno-23-109" href="#__codelineno-23-109"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
-<a id="__codelineno-23-110" name="__codelineno-23-110" href="#__codelineno-23-110"></a><span class="w">    </span><span class="c1">// When the deque has one element, front and rear point to the same node</span>
+<a id="__codelineno-23-110" name="__codelineno-23-110" href="#__codelineno-23-110"></a><span class="w">    </span><span class="c1">// When the queue size is 1, the head and tail point to the same node</span>
 <a id="__codelineno-23-111" name="__codelineno-23-111" href="#__codelineno-23-111"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-112" name="__codelineno-23-112" href="#__codelineno-23-112"></a><span class="w">        </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
-<a id="__codelineno-23-113" name="__codelineno-23-113" href="#__codelineno-23-113"></a><span class="w">        </span><span class="n">delDoublyListNode</span><span class="p">(</span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="p">);</span><span class="w"> </span><span class="c1">// Free the shared node once</span>
+<a id="__codelineno-23-113" name="__codelineno-23-113" href="#__codelineno-23-113"></a><span class="w">        </span><span class="n">delDoublyListNode</span><span class="p">(</span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="p">);</span><span class="w"> </span><span class="c1">// Both point to the same node, so free it only once</span>
 <a id="__codelineno-23-114" name="__codelineno-23-114" href="#__codelineno-23-114"></a><span class="w">        </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-23-115" name="__codelineno-23-115" href="#__codelineno-23-115"></a><span class="w">        </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-23-116" name="__codelineno-23-116" href="#__codelineno-23-116"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-117" name="__codelineno-23-117" href="#__codelineno-23-117"></a><span class="w">    </span><span class="c1">// Temporarily store head node value</span>
+<a id="__codelineno-23-117" name="__codelineno-23-117" href="#__codelineno-23-117"></a><span class="w">    </span><span class="c1">// Dequeue from the front</span>
 <a id="__codelineno-23-118" name="__codelineno-23-118" href="#__codelineno-23-118"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-119" name="__codelineno-23-119" href="#__codelineno-23-119"></a><span class="w">        </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peekFirst</span><span class="p">(</span><span class="n">deque</span><span class="p">);</span><span class="w"> </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-23-119" name="__codelineno-23-119" href="#__codelineno-23-119"></a><span class="w">        </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peekFirst</span><span class="p">(</span><span class="n">deque</span><span class="p">);</span><span class="w"> </span><span class="c1">// Save the head node value</span>
 <a id="__codelineno-23-120" name="__codelineno-23-120" href="#__codelineno-23-120"></a><span class="w">        </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">fNext</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-23-121" name="__codelineno-23-121" href="#__codelineno-23-121"></a><span class="w">        </span><span class="n">fNext</span><span class="o">-&gt;</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-23-122" name="__codelineno-23-122" href="#__codelineno-23-122"></a><span class="w">        </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-23-123" name="__codelineno-23-123" href="#__codelineno-23-123"></a><span class="w">        </span><span class="n">delDoublyListNode</span><span class="p">(</span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="p">);</span>
-<a id="__codelineno-23-124" name="__codelineno-23-124" href="#__codelineno-23-124"></a><span class="w">        </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fNext</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update head node</span>
+<a id="__codelineno-23-124" name="__codelineno-23-124" href="#__codelineno-23-124"></a><span class="w">        </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fNext</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the head node</span>
 <a id="__codelineno-23-125" name="__codelineno-23-125" href="#__codelineno-23-125"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-126" name="__codelineno-23-126" href="#__codelineno-23-126"></a><span class="w">    </span><span class="c1">// Temporarily store tail node value</span>
+<a id="__codelineno-23-126" name="__codelineno-23-126" href="#__codelineno-23-126"></a><span class="w">    </span><span class="c1">// Dequeue from the rear</span>
 <a id="__codelineno-23-127" name="__codelineno-23-127" href="#__codelineno-23-127"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-128" name="__codelineno-23-128" href="#__codelineno-23-128"></a><span class="w">        </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peekLast</span><span class="p">(</span><span class="n">deque</span><span class="p">);</span><span class="w"> </span><span class="c1">// Delete tail node</span>
+<a id="__codelineno-23-128" name="__codelineno-23-128" href="#__codelineno-23-128"></a><span class="w">        </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peekLast</span><span class="p">(</span><span class="n">deque</span><span class="p">);</span><span class="w"> </span><span class="c1">// Save the tail node value</span>
 <a id="__codelineno-23-129" name="__codelineno-23-129" href="#__codelineno-23-129"></a><span class="w">        </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">rPrev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">rear</span><span class="o">-&gt;</span><span class="n">prev</span><span class="p">;</span>
 <a id="__codelineno-23-130" name="__codelineno-23-130" href="#__codelineno-23-130"></a><span class="w">        </span><span class="n">rPrev</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-23-131" name="__codelineno-23-131" href="#__codelineno-23-131"></a><span class="w">        </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">rear</span><span class="o">-&gt;</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-23-132" name="__codelineno-23-132" href="#__codelineno-23-132"></a><span class="w">        </span><span class="n">delDoublyListNode</span><span class="p">(</span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">rear</span><span class="p">);</span>
-<a id="__codelineno-23-133" name="__codelineno-23-133" href="#__codelineno-23-133"></a><span class="w">        </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rPrev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-23-133" name="__codelineno-23-133" href="#__codelineno-23-133"></a><span class="w">        </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rPrev</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-23-134" name="__codelineno-23-134" href="#__codelineno-23-134"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-135" name="__codelineno-23-135" href="#__codelineno-23-135"></a><span class="w">    </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-23-135" name="__codelineno-23-135" href="#__codelineno-23-135"></a><span class="w">    </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update the queue size</span>
 <a id="__codelineno-23-136" name="__codelineno-23-136" href="#__codelineno-23-136"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-23-137" name="__codelineno-23-137" href="#__codelineno-23-137"></a><span class="p">}</span>
 <a id="__codelineno-23-138" name="__codelineno-23-138" href="#__codelineno-23-138"></a>
-<a id="__codelineno-23-139" name="__codelineno-23-139" href="#__codelineno-23-139"></a><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-23-139" name="__codelineno-23-139" href="#__codelineno-23-139"></a><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-23-140" name="__codelineno-23-140" href="#__codelineno-23-140"></a><span class="kt">int</span><span class="w"> </span><span class="nf">popFirst</span><span class="p">(</span><span class="n">LinkedListDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-141" name="__codelineno-23-141" href="#__codelineno-23-141"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">pop</span><span class="p">(</span><span class="n">deque</span><span class="p">,</span><span class="w"> </span><span class="nb">true</span><span class="p">);</span>
 <a id="__codelineno-23-142" name="__codelineno-23-142" href="#__codelineno-23-142"></a><span class="p">}</span>
 <a id="__codelineno-23-143" name="__codelineno-23-143" href="#__codelineno-23-143"></a>
-<a id="__codelineno-23-144" name="__codelineno-23-144" href="#__codelineno-23-144"></a><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-23-144" name="__codelineno-23-144" href="#__codelineno-23-144"></a><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-23-145" name="__codelineno-23-145" href="#__codelineno-23-145"></a><span class="kt">int</span><span class="w"> </span><span class="nf">popLast</span><span class="p">(</span><span class="n">LinkedListDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-146" name="__codelineno-23-146" href="#__codelineno-23-146"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">pop</span><span class="p">(</span><span class="n">deque</span><span class="p">,</span><span class="w"> </span><span class="nb">false</span><span class="p">);</span>
 <a id="__codelineno-23-147" name="__codelineno-23-147" href="#__codelineno-23-147"></a><span class="p">}</span>
 <a id="__codelineno-23-148" name="__codelineno-23-148" href="#__codelineno-23-148"></a>
-<a id="__codelineno-23-149" name="__codelineno-23-149" href="#__codelineno-23-149"></a><span class="cm">/* Print queue */</span>
+<a id="__codelineno-23-149" name="__codelineno-23-149" href="#__codelineno-23-149"></a><span class="cm">/* Print the queue */</span>
 <a id="__codelineno-23-150" name="__codelineno-23-150" href="#__codelineno-23-150"></a><span class="kt">void</span><span class="w"> </span><span class="nf">printLinkedListDeque</span><span class="p">(</span><span class="n">LinkedListDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-151" name="__codelineno-23-151" href="#__codelineno-23-151"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="p">);</span>
-<a id="__codelineno-23-152" name="__codelineno-23-152" href="#__codelineno-23-152"></a><span class="w">    </span><span class="c1">// Copy data from list to array</span>
+<a id="__codelineno-23-152" name="__codelineno-23-152" href="#__codelineno-23-152"></a><span class="w">    </span><span class="c1">// Copy the linked list data into an array</span>
 <a id="__codelineno-23-153" name="__codelineno-23-153" href="#__codelineno-23-153"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-23-154" name="__codelineno-23-154" href="#__codelineno-23-154"></a><span class="w">    </span><span class="n">DoublyListNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-23-155" name="__codelineno-23-155" href="#__codelineno-23-155"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6645,23 +6717,23 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">linkedlist_deque.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Doubly linked list node */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ListNode</span><span class="p">(</span><span class="kd">var</span><span class="w"> </span><span class="nv">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Node value</span>
-<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">next</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="c1">// Successor node reference</span>
-<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">prev</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="c1">// Predecessor node reference</span>
+<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// The node value is stored in _val</span>
+<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">next</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="c1">// Reference to the next node</span>
+<a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">prev</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="c1">// Reference to the previous node</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="p">}</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a>
-<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="cm">/* Double-ended queue based on doubly linked list implementation */</span>
+<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="cm">/* Deque implemented with a doubly linked list */</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="kd">class</span><span class="w"> </span><span class="nc">LinkedListDeque</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">front</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="c1">// Head node front</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">rear</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="c1">// Tail node rear</span>
-<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">queSize</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Length of the double-ended queue</span>
+<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">queSize</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Deque size</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a>
-<a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">size</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span>
 <a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a>
-<a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">():</span><span class="w"> </span><span class="kt">Boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">    </span><span class="p">}</span>
@@ -6673,28 +6745,28 @@
 <a id="__codelineno-24-28" name="__codelineno-24-28" href="#__codelineno-24-28"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-29" name="__codelineno-24-29" href="#__codelineno-24-29"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-24-30" name="__codelineno-24-30" href="#__codelineno-24-30"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span>
-<a id="__codelineno-24-31" name="__codelineno-24-31" href="#__codelineno-24-31"></a><span class="w">            </span><span class="c1">// Front of the queue enqueue operation</span>
+<a id="__codelineno-24-31" name="__codelineno-24-31" href="#__codelineno-24-31"></a><span class="w">            </span><span class="c1">// Enqueue at the front</span>
 <a id="__codelineno-24-32" name="__codelineno-24-32" href="#__codelineno-24-32"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-33" name="__codelineno-24-33" href="#__codelineno-24-33"></a><span class="w">            </span><span class="c1">// Add node to the head of the linked list</span>
 <a id="__codelineno-24-34" name="__codelineno-24-34" href="#__codelineno-24-34"></a><span class="w">            </span><span class="n">front</span><span class="o">?.</span><span class="na">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-24-35" name="__codelineno-24-35" href="#__codelineno-24-35"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span>
-<a id="__codelineno-24-36" name="__codelineno-24-36" href="#__codelineno-24-36"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1">// Update head node</span>
-<a id="__codelineno-24-37" name="__codelineno-24-37" href="#__codelineno-24-37"></a><span class="w">            </span><span class="c1">// Rear of the queue enqueue operation</span>
+<a id="__codelineno-24-36" name="__codelineno-24-36" href="#__codelineno-24-36"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1">// Update the head node</span>
+<a id="__codelineno-24-37" name="__codelineno-24-37" href="#__codelineno-24-37"></a><span class="w">            </span><span class="c1">// Enqueue at the rear</span>
 <a id="__codelineno-24-38" name="__codelineno-24-38" href="#__codelineno-24-38"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-39" name="__codelineno-24-39" href="#__codelineno-24-39"></a><span class="w">            </span><span class="c1">// Add node to the tail of the linked list</span>
 <a id="__codelineno-24-40" name="__codelineno-24-40" href="#__codelineno-24-40"></a><span class="w">            </span><span class="n">rear</span><span class="o">?.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-24-41" name="__codelineno-24-41" href="#__codelineno-24-41"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="na">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span>
-<a id="__codelineno-24-42" name="__codelineno-24-42" href="#__codelineno-24-42"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-24-42" name="__codelineno-24-42" href="#__codelineno-24-42"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-24-43" name="__codelineno-24-43" href="#__codelineno-24-43"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-24-44" name="__codelineno-24-44" href="#__codelineno-24-44"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-24-44" name="__codelineno-24-44" href="#__codelineno-24-44"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="w"> </span><span class="c1">// Update the queue size</span>
 <a id="__codelineno-24-45" name="__codelineno-24-45" href="#__codelineno-24-45"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-46" name="__codelineno-24-46" href="#__codelineno-24-46"></a>
-<a id="__codelineno-24-47" name="__codelineno-24-47" href="#__codelineno-24-47"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-24-47" name="__codelineno-24-47" href="#__codelineno-24-47"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-24-48" name="__codelineno-24-48" href="#__codelineno-24-48"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">pushFirst</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-49" name="__codelineno-24-49" href="#__codelineno-24-49"></a><span class="w">        </span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kc">true</span><span class="p">)</span>
 <a id="__codelineno-24-50" name="__codelineno-24-50" href="#__codelineno-24-50"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-51" name="__codelineno-24-51" href="#__codelineno-24-51"></a>
-<a id="__codelineno-24-52" name="__codelineno-24-52" href="#__codelineno-24-52"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-24-52" name="__codelineno-24-52" href="#__codelineno-24-52"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-24-53" name="__codelineno-24-53" href="#__codelineno-24-53"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">pushLast</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-54" name="__codelineno-24-54" href="#__codelineno-24-54"></a><span class="w">        </span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kc">false</span><span class="p">)</span>
 <a id="__codelineno-24-55" name="__codelineno-24-55" href="#__codelineno-24-55"></a><span class="w">    </span><span class="p">}</span>
@@ -6704,56 +6776,56 @@
 <a id="__codelineno-24-59" name="__codelineno-24-59" href="#__codelineno-24-59"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span>
 <a id="__codelineno-24-60" name="__codelineno-24-60" href="#__codelineno-24-60"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">()</span>
 <a id="__codelineno-24-61" name="__codelineno-24-61" href="#__codelineno-24-61"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span>
-<a id="__codelineno-24-62" name="__codelineno-24-62" href="#__codelineno-24-62"></a><span class="w">        </span><span class="c1">// When the deque has one element, front and rear point to the same node</span>
+<a id="__codelineno-24-62" name="__codelineno-24-62" href="#__codelineno-24-62"></a><span class="w">        </span><span class="c1">// When the queue size is 1, the head and tail point to the same node</span>
 <a id="__codelineno-24-63" name="__codelineno-24-63" href="#__codelineno-24-63"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-64" name="__codelineno-24-64" href="#__codelineno-24-64"></a><span class="w">            </span><span class="n">_val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="o">!!</span><span class="p">.</span><span class="na">_val</span>
 <a id="__codelineno-24-65" name="__codelineno-24-65" href="#__codelineno-24-65"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span>
 <a id="__codelineno-24-66" name="__codelineno-24-66" href="#__codelineno-24-66"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span>
 <a id="__codelineno-24-67" name="__codelineno-24-67" href="#__codelineno-24-67"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-24-68" name="__codelineno-24-68" href="#__codelineno-24-68"></a><span class="w">        </span><span class="c1">// Temporarily store head node value</span>
+<a id="__codelineno-24-68" name="__codelineno-24-68" href="#__codelineno-24-68"></a><span class="w">        </span><span class="c1">// Dequeue from the front</span>
 <a id="__codelineno-24-69" name="__codelineno-24-69" href="#__codelineno-24-69"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isFront</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-70" name="__codelineno-24-70" href="#__codelineno-24-70"></a><span class="w">            </span><span class="n">_val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="o">!!</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="c1">// Delete head node</span>
-<a id="__codelineno-24-71" name="__codelineno-24-71" href="#__codelineno-24-71"></a><span class="w">            </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-24-70" name="__codelineno-24-70" href="#__codelineno-24-70"></a><span class="w">            </span><span class="n">_val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="o">!!</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="c1">// Save the head node value</span>
+<a id="__codelineno-24-71" name="__codelineno-24-71" href="#__codelineno-24-71"></a><span class="w">            </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-24-72" name="__codelineno-24-72" href="#__codelineno-24-72"></a><span class="w">            </span><span class="kd">val</span><span class="w"> </span><span class="nv">fNext</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="o">!!</span><span class="p">.</span><span class="na">next</span>
 <a id="__codelineno-24-73" name="__codelineno-24-73" href="#__codelineno-24-73"></a><span class="w">            </span><span class="n">fNext</span><span class="o">!!</span><span class="p">.</span><span class="na">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span>
 <a id="__codelineno-24-74" name="__codelineno-24-74" href="#__codelineno-24-74"></a><span class="w">            </span><span class="n">front</span><span class="o">!!</span><span class="p">.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span>
-<a id="__codelineno-24-75" name="__codelineno-24-75" href="#__codelineno-24-75"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fNext</span><span class="w"> </span><span class="c1">// Update head node</span>
-<a id="__codelineno-24-76" name="__codelineno-24-76" href="#__codelineno-24-76"></a><span class="w">            </span><span class="c1">// Temporarily store tail node value</span>
+<a id="__codelineno-24-75" name="__codelineno-24-75" href="#__codelineno-24-75"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fNext</span><span class="w"> </span><span class="c1">// Update the head node</span>
+<a id="__codelineno-24-76" name="__codelineno-24-76" href="#__codelineno-24-76"></a><span class="w">            </span><span class="c1">// Dequeue from the rear</span>
 <a id="__codelineno-24-77" name="__codelineno-24-77" href="#__codelineno-24-77"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-78" name="__codelineno-24-78" href="#__codelineno-24-78"></a><span class="w">            </span><span class="n">_val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="o">!!</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="c1">// Delete tail node</span>
-<a id="__codelineno-24-79" name="__codelineno-24-79" href="#__codelineno-24-79"></a><span class="w">            </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-24-78" name="__codelineno-24-78" href="#__codelineno-24-78"></a><span class="w">            </span><span class="n">_val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="o">!!</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="c1">// Save the tail node value</span>
+<a id="__codelineno-24-79" name="__codelineno-24-79" href="#__codelineno-24-79"></a><span class="w">            </span><span class="c1">// Remove the tail node</span>
 <a id="__codelineno-24-80" name="__codelineno-24-80" href="#__codelineno-24-80"></a><span class="w">            </span><span class="kd">val</span><span class="w"> </span><span class="nv">rPrev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rear</span><span class="o">!!</span><span class="p">.</span><span class="na">prev</span>
 <a id="__codelineno-24-81" name="__codelineno-24-81" href="#__codelineno-24-81"></a><span class="w">            </span><span class="n">rPrev</span><span class="o">!!</span><span class="p">.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span>
 <a id="__codelineno-24-82" name="__codelineno-24-82" href="#__codelineno-24-82"></a><span class="w">            </span><span class="n">rear</span><span class="o">!!</span><span class="p">.</span><span class="na">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span>
-<a id="__codelineno-24-83" name="__codelineno-24-83" href="#__codelineno-24-83"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rPrev</span><span class="w"> </span><span class="c1">// Update tail node</span>
+<a id="__codelineno-24-83" name="__codelineno-24-83" href="#__codelineno-24-83"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rPrev</span><span class="w"> </span><span class="c1">// Update the tail node</span>
 <a id="__codelineno-24-84" name="__codelineno-24-84" href="#__codelineno-24-84"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-24-85" name="__codelineno-24-85" href="#__codelineno-24-85"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="w"> </span><span class="c1">// Update queue length</span>
+<a id="__codelineno-24-85" name="__codelineno-24-85" href="#__codelineno-24-85"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="w"> </span><span class="c1">// Update the queue size</span>
 <a id="__codelineno-24-86" name="__codelineno-24-86" href="#__codelineno-24-86"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">_val</span>
 <a id="__codelineno-24-87" name="__codelineno-24-87" href="#__codelineno-24-87"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-88" name="__codelineno-24-88" href="#__codelineno-24-88"></a>
-<a id="__codelineno-24-89" name="__codelineno-24-89" href="#__codelineno-24-89"></a><span class="w">    </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-24-89" name="__codelineno-24-89" href="#__codelineno-24-89"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-24-90" name="__codelineno-24-90" href="#__codelineno-24-90"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">popFirst</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-91" name="__codelineno-24-91" href="#__codelineno-24-91"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pop</span><span class="p">(</span><span class="kc">true</span><span class="p">)</span>
 <a id="__codelineno-24-92" name="__codelineno-24-92" href="#__codelineno-24-92"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-93" name="__codelineno-24-93" href="#__codelineno-24-93"></a>
-<a id="__codelineno-24-94" name="__codelineno-24-94" href="#__codelineno-24-94"></a><span class="w">    </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-24-94" name="__codelineno-24-94" href="#__codelineno-24-94"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-24-95" name="__codelineno-24-95" href="#__codelineno-24-95"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">popLast</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-96" name="__codelineno-24-96" href="#__codelineno-24-96"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">pop</span><span class="p">(</span><span class="kc">false</span><span class="p">)</span>
 <a id="__codelineno-24-97" name="__codelineno-24-97" href="#__codelineno-24-97"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-98" name="__codelineno-24-98" href="#__codelineno-24-98"></a>
-<a id="__codelineno-24-99" name="__codelineno-24-99" href="#__codelineno-24-99"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-24-99" name="__codelineno-24-99" href="#__codelineno-24-99"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-24-100" name="__codelineno-24-100" href="#__codelineno-24-100"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">peekFirst</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-101" name="__codelineno-24-101" href="#__codelineno-24-101"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">()</span>
 <a id="__codelineno-24-102" name="__codelineno-24-102" href="#__codelineno-24-102"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">front</span><span class="o">!!</span><span class="p">.</span><span class="na">_val</span>
 <a id="__codelineno-24-103" name="__codelineno-24-103" href="#__codelineno-24-103"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-104" name="__codelineno-24-104" href="#__codelineno-24-104"></a>
-<a id="__codelineno-24-105" name="__codelineno-24-105" href="#__codelineno-24-105"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-24-105" name="__codelineno-24-105" href="#__codelineno-24-105"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-24-106" name="__codelineno-24-106" href="#__codelineno-24-106"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">peekLast</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-107" name="__codelineno-24-107" href="#__codelineno-24-107"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">()</span>
 <a id="__codelineno-24-108" name="__codelineno-24-108" href="#__codelineno-24-108"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">rear</span><span class="o">!!</span><span class="p">.</span><span class="na">_val</span>
 <a id="__codelineno-24-109" name="__codelineno-24-109" href="#__codelineno-24-109"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-110" name="__codelineno-24-110" href="#__codelineno-24-110"></a>
-<a id="__codelineno-24-111" name="__codelineno-24-111" href="#__codelineno-24-111"></a><span class="w">    </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-24-111" name="__codelineno-24-111" href="#__codelineno-24-111"></a><span class="w">    </span><span class="cm">/* Return an array for printing */</span>
 <a id="__codelineno-24-112" name="__codelineno-24-112" href="#__codelineno-24-112"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">toArray</span><span class="p">():</span><span class="w"> </span><span class="n">IntArray</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-113" name="__codelineno-24-113" href="#__codelineno-24-113"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span>
 <a id="__codelineno-24-114" name="__codelineno-24-114" href="#__codelineno-24-114"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">size</span><span class="p">())</span>
@@ -6776,8 +6848,8 @@
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="c1">### Doubly linked list node</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="k">class</span><span class="w"> </span><span class="nc">ListNode</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="kp">attr_accessor</span><span class="w"> </span><span class="ss">:val</span>
-<a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="kp">attr_accessor</span><span class="w"> </span><span class="ss">:next</span><span class="w"> </span><span class="c1"># Successor node reference</span>
-<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">  </span><span class="kp">attr_accessor</span><span class="w"> </span><span class="ss">:prev</span><span class="w"> </span><span class="c1"># Predecessor node reference</span>
+<a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="kp">attr_accessor</span><span class="w"> </span><span class="ss">:next</span><span class="w"> </span><span class="c1"># Reference to the next node</span>
+<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">  </span><span class="kp">attr_accessor</span><span class="w"> </span><span class="ss">:prev</span><span class="w"> </span><span class="c1"># Reference to the previous node</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">  </span><span class="c1">### Constructor ###</span>
 <a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">initialize</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
@@ -6785,19 +6857,19 @@
 <a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="k">end</span>
 <a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a>
-<a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="c1">### Deque based on doubly linked list ###</span>
+<a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="c1">### Deque implemented with a doubly linked list ###</span>
 <a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a><span class="k">class</span><span class="w"> </span><span class="nc">LinkedListDeque</span>
-<a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="w">  </span><span class="c1">### Get deque length ###</span>
+<a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="w">  </span><span class="c1">### Get the deque size ###</span>
 <a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="w">  </span><span class="kp">attr_reader</span><span class="w"> </span><span class="ss">:size</span>
 <a id="__codelineno-25-23" name="__codelineno-25-23" href="#__codelineno-25-23"></a>
 <a id="__codelineno-25-24" name="__codelineno-25-24" href="#__codelineno-25-24"></a><span class="w">  </span><span class="c1">### Constructor ###</span>
 <a id="__codelineno-25-25" name="__codelineno-25-25" href="#__codelineno-25-25"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">initialize</span>
 <a id="__codelineno-25-26" name="__codelineno-25-26" href="#__codelineno-25-26"></a><span class="w">    </span><span class="vi">@front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">nil</span><span class="w">  </span><span class="c1"># Head node front</span>
 <a id="__codelineno-25-27" name="__codelineno-25-27" href="#__codelineno-25-27"></a><span class="w">    </span><span class="vi">@rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">nil</span><span class="w">   </span><span class="c1"># Tail node rear</span>
-<a id="__codelineno-25-28" name="__codelineno-25-28" href="#__codelineno-25-28"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w">     </span><span class="c1"># Length of the double-ended queue</span>
+<a id="__codelineno-25-28" name="__codelineno-25-28" href="#__codelineno-25-28"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w">     </span><span class="c1"># Deque size</span>
 <a id="__codelineno-25-29" name="__codelineno-25-29" href="#__codelineno-25-29"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-30" name="__codelineno-25-30" href="#__codelineno-25-30"></a>
-<a id="__codelineno-25-31" name="__codelineno-25-31" href="#__codelineno-25-31"></a><span class="w">  </span><span class="c1">### Check if deque is empty ###</span>
+<a id="__codelineno-25-31" name="__codelineno-25-31" href="#__codelineno-25-31"></a><span class="w">  </span><span class="c1">### Check whether the deque is empty ###</span>
 <a id="__codelineno-25-32" name="__codelineno-25-32" href="#__codelineno-25-32"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">is_empty?</span>
 <a id="__codelineno-25-33" name="__codelineno-25-33" href="#__codelineno-25-33"></a><span class="w">    </span><span class="n">size</span><span class="o">.</span><span class="n">zero?</span>
 <a id="__codelineno-25-34" name="__codelineno-25-34" href="#__codelineno-25-34"></a><span class="w">  </span><span class="k">end</span>
@@ -6805,31 +6877,31 @@
 <a id="__codelineno-25-36" name="__codelineno-25-36" href="#__codelineno-25-36"></a><span class="w">  </span><span class="c1">### Enqueue operation ###</span>
 <a id="__codelineno-25-37" name="__codelineno-25-37" href="#__codelineno-25-37"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="n">is_front</span><span class="p">)</span>
 <a id="__codelineno-25-38" name="__codelineno-25-38" href="#__codelineno-25-38"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">ListNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
-<a id="__codelineno-25-39" name="__codelineno-25-39" href="#__codelineno-25-39"></a><span class="w">    </span><span class="c1"># If list is empty, set both front and rear to node</span>
+<a id="__codelineno-25-39" name="__codelineno-25-39" href="#__codelineno-25-39"></a><span class="w">    </span><span class="c1"># If the list is empty, point both front and rear to node</span>
 <a id="__codelineno-25-40" name="__codelineno-25-40" href="#__codelineno-25-40"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">is_empty?</span>
 <a id="__codelineno-25-41" name="__codelineno-25-41" href="#__codelineno-25-41"></a><span class="w">      </span><span class="vi">@front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
-<a id="__codelineno-25-42" name="__codelineno-25-42" href="#__codelineno-25-42"></a><span class="w">    </span><span class="c1"># Front of the queue enqueue operation</span>
+<a id="__codelineno-25-42" name="__codelineno-25-42" href="#__codelineno-25-42"></a><span class="w">    </span><span class="c1"># Enqueue at the front</span>
 <a id="__codelineno-25-43" name="__codelineno-25-43" href="#__codelineno-25-43"></a><span class="w">    </span><span class="k">elsif</span><span class="w"> </span><span class="n">is_front</span>
 <a id="__codelineno-25-44" name="__codelineno-25-44" href="#__codelineno-25-44"></a><span class="w">      </span><span class="c1"># Add node to the head of the linked list</span>
 <a id="__codelineno-25-45" name="__codelineno-25-45" href="#__codelineno-25-45"></a><span class="w">      </span><span class="vi">@front</span><span class="o">.</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-25-46" name="__codelineno-25-46" href="#__codelineno-25-46"></a><span class="w">      </span><span class="n">node</span><span class="o">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@front</span>
-<a id="__codelineno-25-47" name="__codelineno-25-47" href="#__codelineno-25-47"></a><span class="w">      </span><span class="vi">@front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1"># Update head node</span>
-<a id="__codelineno-25-48" name="__codelineno-25-48" href="#__codelineno-25-48"></a><span class="w">    </span><span class="c1"># Rear of the queue enqueue operation</span>
+<a id="__codelineno-25-47" name="__codelineno-25-47" href="#__codelineno-25-47"></a><span class="w">      </span><span class="vi">@front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1"># Update the head node</span>
+<a id="__codelineno-25-48" name="__codelineno-25-48" href="#__codelineno-25-48"></a><span class="w">    </span><span class="c1"># Enqueue at the rear</span>
 <a id="__codelineno-25-49" name="__codelineno-25-49" href="#__codelineno-25-49"></a><span class="w">    </span><span class="k">else</span>
 <a id="__codelineno-25-50" name="__codelineno-25-50" href="#__codelineno-25-50"></a><span class="w">      </span><span class="c1"># Add node to the tail of the linked list</span>
 <a id="__codelineno-25-51" name="__codelineno-25-51" href="#__codelineno-25-51"></a><span class="w">      </span><span class="vi">@rear</span><span class="o">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-25-52" name="__codelineno-25-52" href="#__codelineno-25-52"></a><span class="w">      </span><span class="n">node</span><span class="o">.</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@rear</span>
-<a id="__codelineno-25-53" name="__codelineno-25-53" href="#__codelineno-25-53"></a><span class="w">      </span><span class="vi">@rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1"># Update tail node</span>
+<a id="__codelineno-25-53" name="__codelineno-25-53" href="#__codelineno-25-53"></a><span class="w">      </span><span class="vi">@rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1"># Update the tail node</span>
 <a id="__codelineno-25-54" name="__codelineno-25-54" href="#__codelineno-25-54"></a><span class="w">    </span><span class="k">end</span>
-<a id="__codelineno-25-55" name="__codelineno-25-55" href="#__codelineno-25-55"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Update queue length</span>
+<a id="__codelineno-25-55" name="__codelineno-25-55" href="#__codelineno-25-55"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Update the queue size</span>
 <a id="__codelineno-25-56" name="__codelineno-25-56" href="#__codelineno-25-56"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-57" name="__codelineno-25-57" href="#__codelineno-25-57"></a>
-<a id="__codelineno-25-58" name="__codelineno-25-58" href="#__codelineno-25-58"></a><span class="w">  </span><span class="c1">### Enqueue at front ###</span>
+<a id="__codelineno-25-58" name="__codelineno-25-58" href="#__codelineno-25-58"></a><span class="w">  </span><span class="c1">### Enqueue at the front ###</span>
 <a id="__codelineno-25-59" name="__codelineno-25-59" href="#__codelineno-25-59"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">push_first</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-25-60" name="__codelineno-25-60" href="#__codelineno-25-60"></a><span class="w">    </span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kp">true</span><span class="p">)</span>
 <a id="__codelineno-25-61" name="__codelineno-25-61" href="#__codelineno-25-61"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-62" name="__codelineno-25-62" href="#__codelineno-25-62"></a>
-<a id="__codelineno-25-63" name="__codelineno-25-63" href="#__codelineno-25-63"></a><span class="w">  </span><span class="c1">### Enqueue at rear ###</span>
+<a id="__codelineno-25-63" name="__codelineno-25-63" href="#__codelineno-25-63"></a><span class="w">  </span><span class="c1">### Enqueue at the rear ###</span>
 <a id="__codelineno-25-64" name="__codelineno-25-64" href="#__codelineno-25-64"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">push_last</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-25-65" name="__codelineno-25-65" href="#__codelineno-25-65"></a><span class="w">    </span><span class="n">push</span><span class="p">(</span><span class="n">num</span><span class="p">,</span><span class="w"> </span><span class="kp">false</span><span class="p">)</span>
 <a id="__codelineno-25-66" name="__codelineno-25-66" href="#__codelineno-25-66"></a><span class="w">  </span><span class="k">end</span>
@@ -6838,58 +6910,58 @@
 <a id="__codelineno-25-69" name="__codelineno-25-69" href="#__codelineno-25-69"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">pop</span><span class="p">(</span><span class="n">is_front</span><span class="p">)</span>
 <a id="__codelineno-25-70" name="__codelineno-25-70" href="#__codelineno-25-70"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="p">,</span><span class="w"> </span><span class="s1">'Deque is empty'</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">is_empty?</span>
 <a id="__codelineno-25-71" name="__codelineno-25-71" href="#__codelineno-25-71"></a>
-<a id="__codelineno-25-72" name="__codelineno-25-72" href="#__codelineno-25-72"></a><span class="w">    </span><span class="c1"># When the deque has one element, front and rear point to the same node</span>
+<a id="__codelineno-25-72" name="__codelineno-25-72" href="#__codelineno-25-72"></a><span class="w">    </span><span class="c1"># When the queue size is 1, the head and tail point to the same node</span>
 <a id="__codelineno-25-73" name="__codelineno-25-73" href="#__codelineno-25-73"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="vi">@size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-25-74" name="__codelineno-25-74" href="#__codelineno-25-74"></a><span class="w">      </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@front</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-25-75" name="__codelineno-25-75" href="#__codelineno-25-75"></a><span class="w">      </span><span class="vi">@front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">nil</span>
 <a id="__codelineno-25-76" name="__codelineno-25-76" href="#__codelineno-25-76"></a><span class="w">      </span><span class="vi">@rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">nil</span>
-<a id="__codelineno-25-77" name="__codelineno-25-77" href="#__codelineno-25-77"></a><span class="w">    </span><span class="c1"># Temporarily store head node value</span>
+<a id="__codelineno-25-77" name="__codelineno-25-77" href="#__codelineno-25-77"></a><span class="w">    </span><span class="c1"># Dequeue from the front</span>
 <a id="__codelineno-25-78" name="__codelineno-25-78" href="#__codelineno-25-78"></a><span class="w">    </span><span class="k">elsif</span><span class="w"> </span><span class="n">is_front</span>
-<a id="__codelineno-25-79" name="__codelineno-25-79" href="#__codelineno-25-79"></a><span class="w">      </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@front</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="c1"># Delete head node</span>
-<a id="__codelineno-25-80" name="__codelineno-25-80" href="#__codelineno-25-80"></a><span class="w">      </span><span class="c1"># Delete head node</span>
+<a id="__codelineno-25-79" name="__codelineno-25-79" href="#__codelineno-25-79"></a><span class="w">      </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@front</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="c1"># Save the head node value</span>
+<a id="__codelineno-25-80" name="__codelineno-25-80" href="#__codelineno-25-80"></a><span class="w">      </span><span class="c1"># Remove the head node</span>
 <a id="__codelineno-25-81" name="__codelineno-25-81" href="#__codelineno-25-81"></a><span class="w">      </span><span class="n">fnext</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@front</span><span class="o">.</span><span class="n">next</span>
 <a id="__codelineno-25-82" name="__codelineno-25-82" href="#__codelineno-25-82"></a><span class="w">      </span><span class="n">fnext</span><span class="o">.</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">nil</span>
 <a id="__codelineno-25-83" name="__codelineno-25-83" href="#__codelineno-25-83"></a><span class="w">      </span><span class="vi">@front</span><span class="o">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">nil</span>
-<a id="__codelineno-25-84" name="__codelineno-25-84" href="#__codelineno-25-84"></a><span class="w">      </span><span class="vi">@front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fnext</span><span class="w"> </span><span class="c1"># Update head node</span>
-<a id="__codelineno-25-85" name="__codelineno-25-85" href="#__codelineno-25-85"></a><span class="w">    </span><span class="c1"># Temporarily store tail node value</span>
+<a id="__codelineno-25-84" name="__codelineno-25-84" href="#__codelineno-25-84"></a><span class="w">      </span><span class="vi">@front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">fnext</span><span class="w"> </span><span class="c1"># Update the head node</span>
+<a id="__codelineno-25-85" name="__codelineno-25-85" href="#__codelineno-25-85"></a><span class="w">    </span><span class="c1"># Dequeue from the rear</span>
 <a id="__codelineno-25-86" name="__codelineno-25-86" href="#__codelineno-25-86"></a><span class="w">    </span><span class="k">else</span>
-<a id="__codelineno-25-87" name="__codelineno-25-87" href="#__codelineno-25-87"></a><span class="w">      </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@rear</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="c1"># Delete tail node</span>
-<a id="__codelineno-25-88" name="__codelineno-25-88" href="#__codelineno-25-88"></a><span class="w">      </span><span class="c1"># Update tail node</span>
+<a id="__codelineno-25-87" name="__codelineno-25-87" href="#__codelineno-25-87"></a><span class="w">      </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@rear</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="c1"># Save the tail node value</span>
+<a id="__codelineno-25-88" name="__codelineno-25-88" href="#__codelineno-25-88"></a><span class="w">      </span><span class="c1"># Remove the tail node</span>
 <a id="__codelineno-25-89" name="__codelineno-25-89" href="#__codelineno-25-89"></a><span class="w">      </span><span class="n">rprev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@rear</span><span class="o">.</span><span class="n">prev</span>
 <a id="__codelineno-25-90" name="__codelineno-25-90" href="#__codelineno-25-90"></a><span class="w">      </span><span class="n">rprev</span><span class="o">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">nil</span>
 <a id="__codelineno-25-91" name="__codelineno-25-91" href="#__codelineno-25-91"></a><span class="w">      </span><span class="vi">@rear</span><span class="o">.</span><span class="n">prev</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kp">nil</span>
-<a id="__codelineno-25-92" name="__codelineno-25-92" href="#__codelineno-25-92"></a><span class="w">      </span><span class="vi">@rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rprev</span><span class="w"> </span><span class="c1"># Update tail node</span>
+<a id="__codelineno-25-92" name="__codelineno-25-92" href="#__codelineno-25-92"></a><span class="w">      </span><span class="vi">@rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rprev</span><span class="w"> </span><span class="c1"># Update the tail node</span>
 <a id="__codelineno-25-93" name="__codelineno-25-93" href="#__codelineno-25-93"></a><span class="w">    </span><span class="k">end</span>
-<a id="__codelineno-25-94" name="__codelineno-25-94" href="#__codelineno-25-94"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Update queue length</span>
+<a id="__codelineno-25-94" name="__codelineno-25-94" href="#__codelineno-25-94"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="c1"># Update the queue size</span>
 <a id="__codelineno-25-95" name="__codelineno-25-95" href="#__codelineno-25-95"></a>
 <a id="__codelineno-25-96" name="__codelineno-25-96" href="#__codelineno-25-96"></a><span class="w">    </span><span class="n">val</span>
 <a id="__codelineno-25-97" name="__codelineno-25-97" href="#__codelineno-25-97"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-98" name="__codelineno-25-98" href="#__codelineno-25-98"></a>
-<a id="__codelineno-25-99" name="__codelineno-25-99" href="#__codelineno-25-99"></a><span class="w">  </span><span class="c1">### Dequeue from front ###</span>
+<a id="__codelineno-25-99" name="__codelineno-25-99" href="#__codelineno-25-99"></a><span class="w">  </span><span class="c1">### Dequeue from the front ###</span>
 <a id="__codelineno-25-100" name="__codelineno-25-100" href="#__codelineno-25-100"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">pop_first</span>
 <a id="__codelineno-25-101" name="__codelineno-25-101" href="#__codelineno-25-101"></a><span class="w">    </span><span class="n">pop</span><span class="p">(</span><span class="kp">true</span><span class="p">)</span>
 <a id="__codelineno-25-102" name="__codelineno-25-102" href="#__codelineno-25-102"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-103" name="__codelineno-25-103" href="#__codelineno-25-103"></a>
-<a id="__codelineno-25-104" name="__codelineno-25-104" href="#__codelineno-25-104"></a><span class="w">  </span><span class="c1">### Dequeue from front ###</span>
+<a id="__codelineno-25-104" name="__codelineno-25-104" href="#__codelineno-25-104"></a><span class="w">  </span><span class="c1">### Dequeue from the rear ###</span>
 <a id="__codelineno-25-105" name="__codelineno-25-105" href="#__codelineno-25-105"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">pop_last</span>
 <a id="__codelineno-25-106" name="__codelineno-25-106" href="#__codelineno-25-106"></a><span class="w">    </span><span class="n">pop</span><span class="p">(</span><span class="kp">false</span><span class="p">)</span>
 <a id="__codelineno-25-107" name="__codelineno-25-107" href="#__codelineno-25-107"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-108" name="__codelineno-25-108" href="#__codelineno-25-108"></a>
-<a id="__codelineno-25-109" name="__codelineno-25-109" href="#__codelineno-25-109"></a><span class="w">  </span><span class="c1">### Access front element ###</span>
+<a id="__codelineno-25-109" name="__codelineno-25-109" href="#__codelineno-25-109"></a><span class="w">  </span><span class="c1">### Access the front element ###</span>
 <a id="__codelineno-25-110" name="__codelineno-25-110" href="#__codelineno-25-110"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">peek_first</span>
 <a id="__codelineno-25-111" name="__codelineno-25-111" href="#__codelineno-25-111"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="p">,</span><span class="w"> </span><span class="s1">'Deque is empty'</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">is_empty?</span>
 <a id="__codelineno-25-112" name="__codelineno-25-112" href="#__codelineno-25-112"></a>
 <a id="__codelineno-25-113" name="__codelineno-25-113" href="#__codelineno-25-113"></a><span class="w">    </span><span class="vi">@front</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-25-114" name="__codelineno-25-114" href="#__codelineno-25-114"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-115" name="__codelineno-25-115" href="#__codelineno-25-115"></a>
-<a id="__codelineno-25-116" name="__codelineno-25-116" href="#__codelineno-25-116"></a><span class="w">  </span><span class="c1">### Access rear element ###</span>
+<a id="__codelineno-25-116" name="__codelineno-25-116" href="#__codelineno-25-116"></a><span class="w">  </span><span class="c1">### Access the rear element ###</span>
 <a id="__codelineno-25-117" name="__codelineno-25-117" href="#__codelineno-25-117"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">peek_last</span>
 <a id="__codelineno-25-118" name="__codelineno-25-118" href="#__codelineno-25-118"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="p">,</span><span class="w"> </span><span class="s1">'Deque is empty'</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">is_empty?</span>
 <a id="__codelineno-25-119" name="__codelineno-25-119" href="#__codelineno-25-119"></a>
 <a id="__codelineno-25-120" name="__codelineno-25-120" href="#__codelineno-25-120"></a><span class="w">    </span><span class="vi">@rear</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-25-121" name="__codelineno-25-121" href="#__codelineno-25-121"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-122" name="__codelineno-25-122" href="#__codelineno-25-122"></a>
-<a id="__codelineno-25-123" name="__codelineno-25-123" href="#__codelineno-25-123"></a><span class="w">  </span><span class="c1">### Return array for printing ###</span>
+<a id="__codelineno-25-123" name="__codelineno-25-123" href="#__codelineno-25-123"></a><span class="w">  </span><span class="c1">### Return an array for printing ###</span>
 <a id="__codelineno-25-124" name="__codelineno-25-124" href="#__codelineno-25-124"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">to_array</span>
 <a id="__codelineno-25-125" name="__codelineno-25-125" href="#__codelineno-25-125"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@front</span>
 <a id="__codelineno-25-126" name="__codelineno-25-126" href="#__codelineno-25-126"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
@@ -6904,12 +6976,12 @@
 </div>
 </div>
 </div>
-<h3 id="2-array-implementation">2. &nbsp; Array Implementation<a class="headerlink" href="#2-array-implementation" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 5-9, similar to implementing a queue based on an array, we can also use a circular array to implement a deque.</p>
+<h3 id="2-implementation-with-an-array">2. &nbsp; Implementation with an array<a class="headerlink" href="#2-implementation-with-an-array" title="Permanent link">&para;</a></h3>
+<p>As with an array-based queue, we can use a circular array to implement a deque, as shown below.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:5"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">&lt;1&gt;</label><label for="__tabbed_4_2">&lt;2&gt;</label><label for="__tabbed_4_3">&lt;3&gt;</label><label for="__tabbed_4_4">&lt;4&gt;</label><label for="__tabbed_4_5">&lt;5&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Enqueue and dequeue operations in array implementation of deque" class="animation-figure" src="../deque.assets/array_deque_step1.png" /></p>
+<p><img alt="Insertion and removal in an array-based deque" class="animation-figure" src="../deque.assets/array_deque_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="array_deque_push_last" class="animation-figure" src="../deque.assets/array_deque_step2_push_last.png" /></p>
@@ -6925,14 +6997,14 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 5-9 &nbsp; Enqueue and dequeue operations in array implementation of deque </p>
+<p align="center"> Figure 5-9 &nbsp; Insertion and removal in an array-based deque </p>
 
-<p>Based on the queue implementation, we only need to add methods for "enqueue at front" and "dequeue from rear":</p>
+<p>Starting with the queue implementation, we need only add methods for insertion at the front and removal at the rear:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">array_deque.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayDeque</span><span class="p">:</span>
-<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Double-ended queue based on circular array implementation"""</span>
+<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Deque implemented with a circular array"""</span>
 <a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">capacity</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a><span class="w">        </span><span class="sd">"""Constructor"""</span>
@@ -6941,78 +7013,78 @@
 <a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>
 <a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">capacity</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a><span class="w">        </span><span class="sd">"""Get the capacity of the double-ended queue"""</span>
+<a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a><span class="w">        </span><span class="sd">"""Get the deque capacity"""</span>
 <a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a>        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_nums</span><span class="p">)</span>
 <a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>
 <a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a><span class="w">        </span><span class="sd">"""Get the length of the double-ended queue"""</span>
+<a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a><span class="w">        </span><span class="sd">"""Get the deque size"""</span>
 <a id="__codelineno-26-16" name="__codelineno-26-16" href="#__codelineno-26-16"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span>
 <a id="__codelineno-26-17" name="__codelineno-26-17" href="#__codelineno-26-17"></a>
 <a id="__codelineno-26-18" name="__codelineno-26-18" href="#__codelineno-26-18"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">is_empty</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">bool</span><span class="p">:</span>
-<a id="__codelineno-26-19" name="__codelineno-26-19" href="#__codelineno-26-19"></a><span class="w">        </span><span class="sd">"""Check if the double-ended queue is empty"""</span>
+<a id="__codelineno-26-19" name="__codelineno-26-19" href="#__codelineno-26-19"></a><span class="w">        </span><span class="sd">"""Check whether the deque is empty"""</span>
 <a id="__codelineno-26-20" name="__codelineno-26-20" href="#__codelineno-26-20"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">==</span> <span class="mi">0</span>
 <a id="__codelineno-26-21" name="__codelineno-26-21" href="#__codelineno-26-21"></a>
 <a id="__codelineno-26-22" name="__codelineno-26-22" href="#__codelineno-26-22"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">index</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-23" name="__codelineno-26-23" href="#__codelineno-26-23"></a><span class="w">        </span><span class="sd">"""Calculate circular array index"""</span>
-<a id="__codelineno-26-24" name="__codelineno-26-24" href="#__codelineno-26-24"></a>        <span class="c1"># Use modulo operation to wrap the array head and tail together</span>
-<a id="__codelineno-26-25" name="__codelineno-26-25" href="#__codelineno-26-25"></a>        <span class="c1"># When i passes the tail of the array, return to the head</span>
-<a id="__codelineno-26-26" name="__codelineno-26-26" href="#__codelineno-26-26"></a>        <span class="c1"># When i passes the head of the array, return to the tail</span>
+<a id="__codelineno-26-23" name="__codelineno-26-23" href="#__codelineno-26-23"></a><span class="w">        </span><span class="sd">"""Calculate an index in the circular array"""</span>
+<a id="__codelineno-26-24" name="__codelineno-26-24" href="#__codelineno-26-24"></a>        <span class="c1"># Use modulo to connect the end of the array to its beginning</span>
+<a id="__codelineno-26-25" name="__codelineno-26-25" href="#__codelineno-26-25"></a>        <span class="c1"># When i passes the end of the array, wrap around to the beginning</span>
+<a id="__codelineno-26-26" name="__codelineno-26-26" href="#__codelineno-26-26"></a>        <span class="c1"># When i passes the beginning of the array, wrap around to the end</span>
 <a id="__codelineno-26-27" name="__codelineno-26-27" href="#__codelineno-26-27"></a>        <span class="k">return</span> <span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">())</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">()</span>
 <a id="__codelineno-26-28" name="__codelineno-26-28" href="#__codelineno-26-28"></a>
 <a id="__codelineno-26-29" name="__codelineno-26-29" href="#__codelineno-26-29"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">push_first</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-26-30" name="__codelineno-26-30" href="#__codelineno-26-30"></a><span class="w">        </span><span class="sd">"""Front of the queue enqueue"""</span>
+<a id="__codelineno-26-30" name="__codelineno-26-30" href="#__codelineno-26-30"></a><span class="w">        </span><span class="sd">"""Enqueue at the front"""</span>
 <a id="__codelineno-26-31" name="__codelineno-26-31" href="#__codelineno-26-31"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">():</span>
-<a id="__codelineno-26-32" name="__codelineno-26-32" href="#__codelineno-26-32"></a>            <span class="nb">print</span><span class="p">(</span><span class="s2">"Double-ended queue is full"</span><span class="p">)</span>
+<a id="__codelineno-26-32" name="__codelineno-26-32" href="#__codelineno-26-32"></a>            <span class="nb">print</span><span class="p">(</span><span class="s2">"Deque is full"</span><span class="p">)</span>
 <a id="__codelineno-26-33" name="__codelineno-26-33" href="#__codelineno-26-33"></a>            <span class="k">return</span>
-<a id="__codelineno-26-34" name="__codelineno-26-34" href="#__codelineno-26-34"></a>        <span class="c1"># Front pointer moves one position to the left</span>
-<a id="__codelineno-26-35" name="__codelineno-26-35" href="#__codelineno-26-35"></a>        <span class="c1"># Use modulo operation to wrap front around to the tail after passing the head of the array</span>
+<a id="__codelineno-26-34" name="__codelineno-26-34" href="#__codelineno-26-34"></a>        <span class="c1"># Move the front pointer one position to the left</span>
+<a id="__codelineno-26-35" name="__codelineno-26-35" href="#__codelineno-26-35"></a>        <span class="c1"># Use modulo to wrap front to the end when it passes the beginning of the array</span>
 <a id="__codelineno-26-36" name="__codelineno-26-36" href="#__codelineno-26-36"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-26-37" name="__codelineno-26-37" href="#__codelineno-26-37"></a>        <span class="c1"># Add num to the front of the queue</span>
+<a id="__codelineno-26-37" name="__codelineno-26-37" href="#__codelineno-26-37"></a>        <span class="c1"># Add num to the front</span>
 <a id="__codelineno-26-38" name="__codelineno-26-38" href="#__codelineno-26-38"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_nums</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_front</span><span class="p">]</span> <span class="o">=</span> <span class="n">num</span>
 <a id="__codelineno-26-39" name="__codelineno-26-39" href="#__codelineno-26-39"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">+=</span> <span class="mi">1</span>
 <a id="__codelineno-26-40" name="__codelineno-26-40" href="#__codelineno-26-40"></a>
 <a id="__codelineno-26-41" name="__codelineno-26-41" href="#__codelineno-26-41"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">push_last</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-26-42" name="__codelineno-26-42" href="#__codelineno-26-42"></a><span class="w">        </span><span class="sd">"""Rear of the queue enqueue"""</span>
+<a id="__codelineno-26-42" name="__codelineno-26-42" href="#__codelineno-26-42"></a><span class="w">        </span><span class="sd">"""Enqueue at the rear"""</span>
 <a id="__codelineno-26-43" name="__codelineno-26-43" href="#__codelineno-26-43"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">():</span>
-<a id="__codelineno-26-44" name="__codelineno-26-44" href="#__codelineno-26-44"></a>            <span class="nb">print</span><span class="p">(</span><span class="s2">"Double-ended queue is full"</span><span class="p">)</span>
+<a id="__codelineno-26-44" name="__codelineno-26-44" href="#__codelineno-26-44"></a>            <span class="nb">print</span><span class="p">(</span><span class="s2">"Deque is full"</span><span class="p">)</span>
 <a id="__codelineno-26-45" name="__codelineno-26-45" href="#__codelineno-26-45"></a>            <span class="k">return</span>
-<a id="__codelineno-26-46" name="__codelineno-26-46" href="#__codelineno-26-46"></a>        <span class="c1"># Calculate rear pointer, points to rear index + 1</span>
+<a id="__codelineno-26-46" name="__codelineno-26-46" href="#__codelineno-26-46"></a>        <span class="c1"># Calculate the rear pointer, which points one position past the last element</span>
 <a id="__codelineno-26-47" name="__codelineno-26-47" href="#__codelineno-26-47"></a>        <span class="n">rear</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">)</span>
-<a id="__codelineno-26-48" name="__codelineno-26-48" href="#__codelineno-26-48"></a>        <span class="c1"># Add num to the rear of the queue</span>
+<a id="__codelineno-26-48" name="__codelineno-26-48" href="#__codelineno-26-48"></a>        <span class="c1"># Add num to the rear</span>
 <a id="__codelineno-26-49" name="__codelineno-26-49" href="#__codelineno-26-49"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_nums</span><span class="p">[</span><span class="n">rear</span><span class="p">]</span> <span class="o">=</span> <span class="n">num</span>
 <a id="__codelineno-26-50" name="__codelineno-26-50" href="#__codelineno-26-50"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">+=</span> <span class="mi">1</span>
 <a id="__codelineno-26-51" name="__codelineno-26-51" href="#__codelineno-26-51"></a>
 <a id="__codelineno-26-52" name="__codelineno-26-52" href="#__codelineno-26-52"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">pop_first</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-53" name="__codelineno-26-53" href="#__codelineno-26-53"></a><span class="w">        </span><span class="sd">"""Front of the queue dequeue"""</span>
+<a id="__codelineno-26-53" name="__codelineno-26-53" href="#__codelineno-26-53"></a><span class="w">        </span><span class="sd">"""Dequeue from the front"""</span>
 <a id="__codelineno-26-54" name="__codelineno-26-54" href="#__codelineno-26-54"></a>        <span class="n">num</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">peek_first</span><span class="p">()</span>
-<a id="__codelineno-26-55" name="__codelineno-26-55" href="#__codelineno-26-55"></a>        <span class="c1"># Front pointer moves one position backward</span>
+<a id="__codelineno-26-55" name="__codelineno-26-55" href="#__codelineno-26-55"></a>        <span class="c1"># Advance the front pointer by one position</span>
 <a id="__codelineno-26-56" name="__codelineno-26-56" href="#__codelineno-26-56"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-26-57" name="__codelineno-26-57" href="#__codelineno-26-57"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">-=</span> <span class="mi">1</span>
 <a id="__codelineno-26-58" name="__codelineno-26-58" href="#__codelineno-26-58"></a>        <span class="k">return</span> <span class="n">num</span>
 <a id="__codelineno-26-59" name="__codelineno-26-59" href="#__codelineno-26-59"></a>
 <a id="__codelineno-26-60" name="__codelineno-26-60" href="#__codelineno-26-60"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">pop_last</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-61" name="__codelineno-26-61" href="#__codelineno-26-61"></a><span class="w">        </span><span class="sd">"""Rear of the queue dequeue"""</span>
+<a id="__codelineno-26-61" name="__codelineno-26-61" href="#__codelineno-26-61"></a><span class="w">        </span><span class="sd">"""Dequeue from the rear"""</span>
 <a id="__codelineno-26-62" name="__codelineno-26-62" href="#__codelineno-26-62"></a>        <span class="n">num</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">peek_last</span><span class="p">()</span>
 <a id="__codelineno-26-63" name="__codelineno-26-63" href="#__codelineno-26-63"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">-=</span> <span class="mi">1</span>
 <a id="__codelineno-26-64" name="__codelineno-26-64" href="#__codelineno-26-64"></a>        <span class="k">return</span> <span class="n">num</span>
 <a id="__codelineno-26-65" name="__codelineno-26-65" href="#__codelineno-26-65"></a>
 <a id="__codelineno-26-66" name="__codelineno-26-66" href="#__codelineno-26-66"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">peek_first</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-67" name="__codelineno-26-67" href="#__codelineno-26-67"></a><span class="w">        </span><span class="sd">"""Access front of the queue element"""</span>
+<a id="__codelineno-26-67" name="__codelineno-26-67" href="#__codelineno-26-67"></a><span class="w">        </span><span class="sd">"""Access the front element"""</span>
 <a id="__codelineno-26-68" name="__codelineno-26-68" href="#__codelineno-26-68"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_empty</span><span class="p">():</span>
-<a id="__codelineno-26-69" name="__codelineno-26-69" href="#__codelineno-26-69"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Double-ended queue is empty"</span><span class="p">)</span>
+<a id="__codelineno-26-69" name="__codelineno-26-69" href="#__codelineno-26-69"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Deque is empty"</span><span class="p">)</span>
 <a id="__codelineno-26-70" name="__codelineno-26-70" href="#__codelineno-26-70"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_nums</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_front</span><span class="p">]</span>
 <a id="__codelineno-26-71" name="__codelineno-26-71" href="#__codelineno-26-71"></a>
 <a id="__codelineno-26-72" name="__codelineno-26-72" href="#__codelineno-26-72"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">peek_last</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-73" name="__codelineno-26-73" href="#__codelineno-26-73"></a><span class="w">        </span><span class="sd">"""Access rear of the queue element"""</span>
+<a id="__codelineno-26-73" name="__codelineno-26-73" href="#__codelineno-26-73"></a><span class="w">        </span><span class="sd">"""Access the rear element"""</span>
 <a id="__codelineno-26-74" name="__codelineno-26-74" href="#__codelineno-26-74"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_empty</span><span class="p">():</span>
-<a id="__codelineno-26-75" name="__codelineno-26-75" href="#__codelineno-26-75"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Double-ended queue is empty"</span><span class="p">)</span>
-<a id="__codelineno-26-76" name="__codelineno-26-76" href="#__codelineno-26-76"></a>        <span class="c1"># Calculate tail element index</span>
+<a id="__codelineno-26-75" name="__codelineno-26-75" href="#__codelineno-26-75"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Deque is empty"</span><span class="p">)</span>
+<a id="__codelineno-26-76" name="__codelineno-26-76" href="#__codelineno-26-76"></a>        <span class="c1"># Calculate the index of the last element</span>
 <a id="__codelineno-26-77" name="__codelineno-26-77" href="#__codelineno-26-77"></a>        <span class="n">last</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-26-78" name="__codelineno-26-78" href="#__codelineno-26-78"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_nums</span><span class="p">[</span><span class="n">last</span><span class="p">]</span>
 <a id="__codelineno-26-79" name="__codelineno-26-79" href="#__codelineno-26-79"></a>
 <a id="__codelineno-26-80" name="__codelineno-26-80" href="#__codelineno-26-80"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">to_array</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
-<a id="__codelineno-26-81" name="__codelineno-26-81" href="#__codelineno-26-81"></a><span class="w">        </span><span class="sd">"""Return array for printing"""</span>
-<a id="__codelineno-26-82" name="__codelineno-26-82" href="#__codelineno-26-82"></a>        <span class="c1"># Only convert list elements within the valid length range</span>
+<a id="__codelineno-26-81" name="__codelineno-26-81" href="#__codelineno-26-81"></a><span class="w">        </span><span class="sd">"""Return an array for printing"""</span>
+<a id="__codelineno-26-82" name="__codelineno-26-82" href="#__codelineno-26-82"></a>        <span class="c1"># Convert only the list elements within the occupied range</span>
 <a id="__codelineno-26-83" name="__codelineno-26-83" href="#__codelineno-26-83"></a>        <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
 <a id="__codelineno-26-84" name="__codelineno-26-84" href="#__codelineno-26-84"></a>        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">):</span>
 <a id="__codelineno-26-85" name="__codelineno-26-85" href="#__codelineno-26-85"></a>            <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_nums</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">index</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">+</span> <span class="n">i</span><span class="p">)])</span>
@@ -7020,12 +7092,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_deque.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Double-ended queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_deque.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Deque implemented with a circular array */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayDeque</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">  </span><span class="k">private</span><span class="o">:</span>
-<a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array for storing double-ended queue elements</span>
-<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w">        </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w">      </span><span class="c1">// Double-ended queue length</span>
+<a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array for storing deque elements</span>
+<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w">        </span><span class="c1">// Front index, identifying the first element</span>
+<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w">      </span><span class="c1">// Deque size</span>
 <a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a>
 <a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">  </span><span class="k">public</span><span class="o">:</span>
 <a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -7034,91 +7106,91 @@
 <a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a>
-<a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="w">    </span><span class="cm">/* Get the capacity of the double-ended queue */</span>
+<a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="w">    </span><span class="cm">/* Get the deque capacity */</span>
 <a id="__codelineno-27-16" name="__codelineno-27-16" href="#__codelineno-27-16"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-17" name="__codelineno-27-17" href="#__codelineno-27-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
 <a id="__codelineno-27-18" name="__codelineno-27-18" href="#__codelineno-27-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-19" name="__codelineno-27-19" href="#__codelineno-27-19"></a>
-<a id="__codelineno-27-20" name="__codelineno-27-20" href="#__codelineno-27-20"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-27-20" name="__codelineno-27-20" href="#__codelineno-27-20"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-27-21" name="__codelineno-27-21" href="#__codelineno-27-21"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-22" name="__codelineno-27-22" href="#__codelineno-27-22"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-27-23" name="__codelineno-27-23" href="#__codelineno-27-23"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-24" name="__codelineno-27-24" href="#__codelineno-27-24"></a>
-<a id="__codelineno-27-25" name="__codelineno-27-25" href="#__codelineno-27-25"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-27-25" name="__codelineno-27-25" href="#__codelineno-27-25"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-27-26" name="__codelineno-27-26" href="#__codelineno-27-26"></a><span class="w">    </span><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-27" name="__codelineno-27-27" href="#__codelineno-27-27"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-27-28" name="__codelineno-27-28" href="#__codelineno-27-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-29" name="__codelineno-27-29" href="#__codelineno-27-29"></a>
-<a id="__codelineno-27-30" name="__codelineno-27-30" href="#__codelineno-27-30"></a><span class="w">    </span><span class="cm">/* Calculate circular array index */</span>
+<a id="__codelineno-27-30" name="__codelineno-27-30" href="#__codelineno-27-30"></a><span class="w">    </span><span class="cm">/* Calculate an index in the circular array */</span>
 <a id="__codelineno-27-31" name="__codelineno-27-31" href="#__codelineno-27-31"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-32" name="__codelineno-27-32" href="#__codelineno-27-32"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap the array head and tail together</span>
-<a id="__codelineno-27-33" name="__codelineno-27-33" href="#__codelineno-27-33"></a><span class="w">        </span><span class="c1">// When i passes the tail of the array, return to the head</span>
-<a id="__codelineno-27-34" name="__codelineno-27-34" href="#__codelineno-27-34"></a><span class="w">        </span><span class="c1">// When i passes the head of the array, return to the tail</span>
+<a id="__codelineno-27-32" name="__codelineno-27-32" href="#__codelineno-27-32"></a><span class="w">        </span><span class="c1">// Use modulo to connect the end of the array to its beginning</span>
+<a id="__codelineno-27-33" name="__codelineno-27-33" href="#__codelineno-27-33"></a><span class="w">        </span><span class="c1">// When i passes the end of the array, wrap around to the beginning</span>
+<a id="__codelineno-27-34" name="__codelineno-27-34" href="#__codelineno-27-34"></a><span class="w">        </span><span class="c1">// When i passes the beginning of the array, wrap around to the end</span>
 <a id="__codelineno-27-35" name="__codelineno-27-35" href="#__codelineno-27-35"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">();</span>
 <a id="__codelineno-27-36" name="__codelineno-27-36" href="#__codelineno-27-36"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-37" name="__codelineno-27-37" href="#__codelineno-27-37"></a>
-<a id="__codelineno-27-38" name="__codelineno-27-38" href="#__codelineno-27-38"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-27-38" name="__codelineno-27-38" href="#__codelineno-27-38"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-27-39" name="__codelineno-27-39" href="#__codelineno-27-39"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">pushFirst</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-40" name="__codelineno-27-40" href="#__codelineno-27-40"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-41" name="__codelineno-27-41" href="#__codelineno-27-41"></a><span class="w">            </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">"Double-ended queue is full"</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
+<a id="__codelineno-27-41" name="__codelineno-27-41" href="#__codelineno-27-41"></a><span class="w">            </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">"Deque is full"</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
 <a id="__codelineno-27-42" name="__codelineno-27-42" href="#__codelineno-27-42"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-27-43" name="__codelineno-27-43" href="#__codelineno-27-43"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-27-44" name="__codelineno-27-44" href="#__codelineno-27-44"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap front around to the tail after passing the head of the array</span>
-<a id="__codelineno-27-45" name="__codelineno-27-45" href="#__codelineno-27-45"></a><span class="w">        </span><span class="c1">// Add num to the front of the queue</span>
+<a id="__codelineno-27-44" name="__codelineno-27-44" href="#__codelineno-27-44"></a><span class="w">        </span><span class="c1">// Move the front index one position to the left</span>
+<a id="__codelineno-27-45" name="__codelineno-27-45" href="#__codelineno-27-45"></a><span class="w">        </span><span class="c1">// Use modulo to wrap front to the end when it passes the beginning of the array</span>
 <a id="__codelineno-27-46" name="__codelineno-27-46" href="#__codelineno-27-46"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-27-47" name="__codelineno-27-47" href="#__codelineno-27-47"></a><span class="w">        </span><span class="c1">// Add num to front of queue</span>
+<a id="__codelineno-27-47" name="__codelineno-27-47" href="#__codelineno-27-47"></a><span class="w">        </span><span class="c1">// Add num to the front</span>
 <a id="__codelineno-27-48" name="__codelineno-27-48" href="#__codelineno-27-48"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">front</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-27-49" name="__codelineno-27-49" href="#__codelineno-27-49"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-27-50" name="__codelineno-27-50" href="#__codelineno-27-50"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-51" name="__codelineno-27-51" href="#__codelineno-27-51"></a>
-<a id="__codelineno-27-52" name="__codelineno-27-52" href="#__codelineno-27-52"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-27-52" name="__codelineno-27-52" href="#__codelineno-27-52"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-27-53" name="__codelineno-27-53" href="#__codelineno-27-53"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">pushLast</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-54" name="__codelineno-27-54" href="#__codelineno-27-54"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-55" name="__codelineno-27-55" href="#__codelineno-27-55"></a><span class="w">            </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">"Double-ended queue is full"</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
+<a id="__codelineno-27-55" name="__codelineno-27-55" href="#__codelineno-27-55"></a><span class="w">            </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">"Deque is full"</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
 <a id="__codelineno-27-56" name="__codelineno-27-56" href="#__codelineno-27-56"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-27-57" name="__codelineno-27-57" href="#__codelineno-27-57"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-27-58" name="__codelineno-27-58" href="#__codelineno-27-58"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
+<a id="__codelineno-27-58" name="__codelineno-27-58" href="#__codelineno-27-58"></a><span class="w">        </span><span class="c1">// Calculate the rear index, one position past the last element</span>
 <a id="__codelineno-27-59" name="__codelineno-27-59" href="#__codelineno-27-59"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">queSize</span><span class="p">);</span>
-<a id="__codelineno-27-60" name="__codelineno-27-60" href="#__codelineno-27-60"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-27-60" name="__codelineno-27-60" href="#__codelineno-27-60"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-27-61" name="__codelineno-27-61" href="#__codelineno-27-61"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">rear</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-27-62" name="__codelineno-27-62" href="#__codelineno-27-62"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-27-63" name="__codelineno-27-63" href="#__codelineno-27-63"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-64" name="__codelineno-27-64" href="#__codelineno-27-64"></a>
-<a id="__codelineno-27-65" name="__codelineno-27-65" href="#__codelineno-27-65"></a><span class="w">    </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-27-65" name="__codelineno-27-65" href="#__codelineno-27-65"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-27-66" name="__codelineno-27-66" href="#__codelineno-27-66"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">popFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-67" name="__codelineno-27-67" href="#__codelineno-27-67"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peekFirst</span><span class="p">();</span>
-<a id="__codelineno-27-68" name="__codelineno-27-68" href="#__codelineno-27-68"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position</span>
+<a id="__codelineno-27-68" name="__codelineno-27-68" href="#__codelineno-27-68"></a><span class="w">        </span><span class="c1">// Advance the front index by one position</span>
 <a id="__codelineno-27-69" name="__codelineno-27-69" href="#__codelineno-27-69"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-27-70" name="__codelineno-27-70" href="#__codelineno-27-70"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-27-71" name="__codelineno-27-71" href="#__codelineno-27-71"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-27-72" name="__codelineno-27-72" href="#__codelineno-27-72"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-73" name="__codelineno-27-73" href="#__codelineno-27-73"></a>
-<a id="__codelineno-27-74" name="__codelineno-27-74" href="#__codelineno-27-74"></a><span class="w">    </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-27-74" name="__codelineno-27-74" href="#__codelineno-27-74"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-27-75" name="__codelineno-27-75" href="#__codelineno-27-75"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">popLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-76" name="__codelineno-27-76" href="#__codelineno-27-76"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peekLast</span><span class="p">();</span>
 <a id="__codelineno-27-77" name="__codelineno-27-77" href="#__codelineno-27-77"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-27-78" name="__codelineno-27-78" href="#__codelineno-27-78"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-27-79" name="__codelineno-27-79" href="#__codelineno-27-79"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-80" name="__codelineno-27-80" href="#__codelineno-27-80"></a>
-<a id="__codelineno-27-81" name="__codelineno-27-81" href="#__codelineno-27-81"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-27-81" name="__codelineno-27-81" href="#__codelineno-27-81"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-27-82" name="__codelineno-27-82" href="#__codelineno-27-82"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">peekFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-83" name="__codelineno-27-83" href="#__codelineno-27-83"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-27-84" name="__codelineno-27-84" href="#__codelineno-27-84"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Deque is empty"</span><span class="p">);</span>
 <a id="__codelineno-27-85" name="__codelineno-27-85" href="#__codelineno-27-85"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">front</span><span class="p">];</span>
 <a id="__codelineno-27-86" name="__codelineno-27-86" href="#__codelineno-27-86"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-87" name="__codelineno-27-87" href="#__codelineno-27-87"></a>
-<a id="__codelineno-27-88" name="__codelineno-27-88" href="#__codelineno-27-88"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-27-88" name="__codelineno-27-88" href="#__codelineno-27-88"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-27-89" name="__codelineno-27-89" href="#__codelineno-27-89"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">peekLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-90" name="__codelineno-27-90" href="#__codelineno-27-90"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-27-91" name="__codelineno-27-91" href="#__codelineno-27-91"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Deque is empty"</span><span class="p">);</span>
-<a id="__codelineno-27-92" name="__codelineno-27-92" href="#__codelineno-27-92"></a><span class="w">        </span><span class="c1">// Initialize double-ended queue</span>
+<a id="__codelineno-27-92" name="__codelineno-27-92" href="#__codelineno-27-92"></a><span class="w">        </span><span class="c1">// Calculate the index of the last element</span>
 <a id="__codelineno-27-93" name="__codelineno-27-93" href="#__codelineno-27-93"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">last</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-27-94" name="__codelineno-27-94" href="#__codelineno-27-94"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">last</span><span class="p">];</span>
 <a id="__codelineno-27-95" name="__codelineno-27-95" href="#__codelineno-27-95"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-96" name="__codelineno-27-96" href="#__codelineno-27-96"></a>
-<a id="__codelineno-27-97" name="__codelineno-27-97" href="#__codelineno-27-97"></a><span class="w">    </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-27-97" name="__codelineno-27-97" href="#__codelineno-27-97"></a><span class="w">    </span><span class="cm">/* Return a vector for printing */</span>
 <a id="__codelineno-27-98" name="__codelineno-27-98" href="#__codelineno-27-98"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">toVector</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-99" name="__codelineno-27-99" href="#__codelineno-27-99"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-27-99" name="__codelineno-27-99" href="#__codelineno-27-99"></a><span class="w">        </span><span class="c1">// Copy only the elements in the occupied range</span>
 <a id="__codelineno-27-100" name="__codelineno-27-100" href="#__codelineno-27-100"></a><span class="w">        </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">(</span><span class="n">queSize</span><span class="p">);</span>
 <a id="__codelineno-27-101" name="__codelineno-27-101" href="#__codelineno-27-101"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-102" name="__codelineno-27-102" href="#__codelineno-27-102"></a><span class="w">            </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">index</span><span class="p">(</span><span class="n">j</span><span class="p">)];</span>
@@ -7129,11 +7201,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_deque.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Double-ended queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_deque.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Deque implemented with a circular array */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kd">class</span> <span class="nc">ArrayDeque</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array for storing double-ended queue elements</span>
-<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Double-ended queue length</span>
+<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array that stores the deque elements</span>
+<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, pointing to the first element</span>
+<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Deque size</span>
 <a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a>
 <a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="nf">ArrayDeque</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7141,91 +7213,91 @@
 <a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a>
-<a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">    </span><span class="cm">/* Get the capacity of the double-ended queue */</span>
+<a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">    </span><span class="cm">/* Get the deque capacity */</span>
 <a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
 <a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a>
-<a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-28-19" name="__codelineno-28-19" href="#__codelineno-28-19"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-20" name="__codelineno-28-20" href="#__codelineno-28-20"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-28-21" name="__codelineno-28-21" href="#__codelineno-28-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-22" name="__codelineno-28-22" href="#__codelineno-28-22"></a>
-<a id="__codelineno-28-23" name="__codelineno-28-23" href="#__codelineno-28-23"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-28-23" name="__codelineno-28-23" href="#__codelineno-28-23"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-28-24" name="__codelineno-28-24" href="#__codelineno-28-24"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">boolean</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-25" name="__codelineno-28-25" href="#__codelineno-28-25"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-28-26" name="__codelineno-28-26" href="#__codelineno-28-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-27" name="__codelineno-28-27" href="#__codelineno-28-27"></a>
-<a id="__codelineno-28-28" name="__codelineno-28-28" href="#__codelineno-28-28"></a><span class="w">    </span><span class="cm">/* Calculate circular array index */</span>
+<a id="__codelineno-28-28" name="__codelineno-28-28" href="#__codelineno-28-28"></a><span class="w">    </span><span class="cm">/* Calculate an index in the circular array */</span>
 <a id="__codelineno-28-29" name="__codelineno-28-29" href="#__codelineno-28-29"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">index</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-30" name="__codelineno-28-30" href="#__codelineno-28-30"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap the array head and tail together</span>
-<a id="__codelineno-28-31" name="__codelineno-28-31" href="#__codelineno-28-31"></a><span class="w">        </span><span class="c1">// When i passes the tail of the array, return to the head</span>
-<a id="__codelineno-28-32" name="__codelineno-28-32" href="#__codelineno-28-32"></a><span class="w">        </span><span class="c1">// When i passes the head of the array, return to the tail</span>
+<a id="__codelineno-28-30" name="__codelineno-28-30" href="#__codelineno-28-30"></a><span class="w">        </span><span class="c1">// Use modulo to connect the end of the array to its beginning</span>
+<a id="__codelineno-28-31" name="__codelineno-28-31" href="#__codelineno-28-31"></a><span class="w">        </span><span class="c1">// When i passes the end of the array, wrap around to the beginning</span>
+<a id="__codelineno-28-32" name="__codelineno-28-32" href="#__codelineno-28-32"></a><span class="w">        </span><span class="c1">// When i passes the beginning of the array, wrap around to the end</span>
 <a id="__codelineno-28-33" name="__codelineno-28-33" href="#__codelineno-28-33"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">();</span>
 <a id="__codelineno-28-34" name="__codelineno-28-34" href="#__codelineno-28-34"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-35" name="__codelineno-28-35" href="#__codelineno-28-35"></a>
-<a id="__codelineno-28-36" name="__codelineno-28-36" href="#__codelineno-28-36"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-28-36" name="__codelineno-28-36" href="#__codelineno-28-36"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-28-37" name="__codelineno-28-37" href="#__codelineno-28-37"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">pushFirst</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-38" name="__codelineno-28-38" href="#__codelineno-28-38"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-39" name="__codelineno-28-39" href="#__codelineno-28-39"></a><span class="w">            </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="s">"Double-ended queue is full"</span><span class="p">);</span>
+<a id="__codelineno-28-39" name="__codelineno-28-39" href="#__codelineno-28-39"></a><span class="w">            </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="s">"Deque is full"</span><span class="p">);</span>
 <a id="__codelineno-28-40" name="__codelineno-28-40" href="#__codelineno-28-40"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-28-41" name="__codelineno-28-41" href="#__codelineno-28-41"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-28-42" name="__codelineno-28-42" href="#__codelineno-28-42"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap front around to the tail after passing the head of the array</span>
-<a id="__codelineno-28-43" name="__codelineno-28-43" href="#__codelineno-28-43"></a><span class="w">        </span><span class="c1">// Add num to the front of the queue</span>
+<a id="__codelineno-28-42" name="__codelineno-28-42" href="#__codelineno-28-42"></a><span class="w">        </span><span class="c1">// Move the front pointer one position to the left</span>
+<a id="__codelineno-28-43" name="__codelineno-28-43" href="#__codelineno-28-43"></a><span class="w">        </span><span class="c1">// Use modulo to wrap front to the end when it passes the beginning of the array</span>
 <a id="__codelineno-28-44" name="__codelineno-28-44" href="#__codelineno-28-44"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-28-45" name="__codelineno-28-45" href="#__codelineno-28-45"></a><span class="w">        </span><span class="c1">// Add num to front of queue</span>
+<a id="__codelineno-28-45" name="__codelineno-28-45" href="#__codelineno-28-45"></a><span class="w">        </span><span class="c1">// Add num to the front</span>
 <a id="__codelineno-28-46" name="__codelineno-28-46" href="#__codelineno-28-46"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">front</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-28-47" name="__codelineno-28-47" href="#__codelineno-28-47"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-28-48" name="__codelineno-28-48" href="#__codelineno-28-48"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-49" name="__codelineno-28-49" href="#__codelineno-28-49"></a>
-<a id="__codelineno-28-50" name="__codelineno-28-50" href="#__codelineno-28-50"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-28-50" name="__codelineno-28-50" href="#__codelineno-28-50"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-28-51" name="__codelineno-28-51" href="#__codelineno-28-51"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">pushLast</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-52" name="__codelineno-28-52" href="#__codelineno-28-52"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-53" name="__codelineno-28-53" href="#__codelineno-28-53"></a><span class="w">            </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="s">"Double-ended queue is full"</span><span class="p">);</span>
+<a id="__codelineno-28-53" name="__codelineno-28-53" href="#__codelineno-28-53"></a><span class="w">            </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="s">"Deque is full"</span><span class="p">);</span>
 <a id="__codelineno-28-54" name="__codelineno-28-54" href="#__codelineno-28-54"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-28-55" name="__codelineno-28-55" href="#__codelineno-28-55"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-28-56" name="__codelineno-28-56" href="#__codelineno-28-56"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
+<a id="__codelineno-28-56" name="__codelineno-28-56" href="#__codelineno-28-56"></a><span class="w">        </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
 <a id="__codelineno-28-57" name="__codelineno-28-57" href="#__codelineno-28-57"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">queSize</span><span class="p">);</span>
-<a id="__codelineno-28-58" name="__codelineno-28-58" href="#__codelineno-28-58"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-28-58" name="__codelineno-28-58" href="#__codelineno-28-58"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-28-59" name="__codelineno-28-59" href="#__codelineno-28-59"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">rear</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-28-60" name="__codelineno-28-60" href="#__codelineno-28-60"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-28-61" name="__codelineno-28-61" href="#__codelineno-28-61"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-62" name="__codelineno-28-62" href="#__codelineno-28-62"></a>
-<a id="__codelineno-28-63" name="__codelineno-28-63" href="#__codelineno-28-63"></a><span class="w">    </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-28-63" name="__codelineno-28-63" href="#__codelineno-28-63"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-28-64" name="__codelineno-28-64" href="#__codelineno-28-64"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">popFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-65" name="__codelineno-28-65" href="#__codelineno-28-65"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peekFirst</span><span class="p">();</span>
-<a id="__codelineno-28-66" name="__codelineno-28-66" href="#__codelineno-28-66"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position</span>
+<a id="__codelineno-28-66" name="__codelineno-28-66" href="#__codelineno-28-66"></a><span class="w">        </span><span class="c1">// Advance the front pointer by one position</span>
 <a id="__codelineno-28-67" name="__codelineno-28-67" href="#__codelineno-28-67"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-28-68" name="__codelineno-28-68" href="#__codelineno-28-68"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-28-69" name="__codelineno-28-69" href="#__codelineno-28-69"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-28-70" name="__codelineno-28-70" href="#__codelineno-28-70"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-71" name="__codelineno-28-71" href="#__codelineno-28-71"></a>
-<a id="__codelineno-28-72" name="__codelineno-28-72" href="#__codelineno-28-72"></a><span class="w">    </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-28-72" name="__codelineno-28-72" href="#__codelineno-28-72"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-28-73" name="__codelineno-28-73" href="#__codelineno-28-73"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">popLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-74" name="__codelineno-28-74" href="#__codelineno-28-74"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peekLast</span><span class="p">();</span>
 <a id="__codelineno-28-75" name="__codelineno-28-75" href="#__codelineno-28-75"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-28-76" name="__codelineno-28-76" href="#__codelineno-28-76"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-28-77" name="__codelineno-28-77" href="#__codelineno-28-77"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-78" name="__codelineno-28-78" href="#__codelineno-28-78"></a>
-<a id="__codelineno-28-79" name="__codelineno-28-79" href="#__codelineno-28-79"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-28-79" name="__codelineno-28-79" href="#__codelineno-28-79"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-28-80" name="__codelineno-28-80" href="#__codelineno-28-80"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">peekFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-81" name="__codelineno-28-81" href="#__codelineno-28-81"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-28-82" name="__codelineno-28-82" href="#__codelineno-28-82"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">();</span>
 <a id="__codelineno-28-83" name="__codelineno-28-83" href="#__codelineno-28-83"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">front</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-84" name="__codelineno-28-84" href="#__codelineno-28-84"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-85" name="__codelineno-28-85" href="#__codelineno-28-85"></a>
-<a id="__codelineno-28-86" name="__codelineno-28-86" href="#__codelineno-28-86"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-28-86" name="__codelineno-28-86" href="#__codelineno-28-86"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-28-87" name="__codelineno-28-87" href="#__codelineno-28-87"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">peekLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-88" name="__codelineno-28-88" href="#__codelineno-28-88"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-28-89" name="__codelineno-28-89" href="#__codelineno-28-89"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">();</span>
-<a id="__codelineno-28-90" name="__codelineno-28-90" href="#__codelineno-28-90"></a><span class="w">        </span><span class="c1">// Initialize double-ended queue</span>
+<a id="__codelineno-28-90" name="__codelineno-28-90" href="#__codelineno-28-90"></a><span class="w">        </span><span class="c1">// Calculate the index of the last element</span>
 <a id="__codelineno-28-91" name="__codelineno-28-91" href="#__codelineno-28-91"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">last</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-28-92" name="__codelineno-28-92" href="#__codelineno-28-92"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">last</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-93" name="__codelineno-28-93" href="#__codelineno-28-93"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-94" name="__codelineno-28-94" href="#__codelineno-28-94"></a>
-<a id="__codelineno-28-95" name="__codelineno-28-95" href="#__codelineno-28-95"></a><span class="w">    </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-28-95" name="__codelineno-28-95" href="#__codelineno-28-95"></a><span class="w">    </span><span class="cm">/* Return an array for printing */</span>
 <a id="__codelineno-28-96" name="__codelineno-28-96" href="#__codelineno-28-96"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="nf">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-97" name="__codelineno-28-97" href="#__codelineno-28-97"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-28-97" name="__codelineno-28-97" href="#__codelineno-28-97"></a><span class="w">        </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-28-98" name="__codelineno-28-98" href="#__codelineno-28-98"></a><span class="w">        </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">queSize</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-99" name="__codelineno-28-99" href="#__codelineno-28-99"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-100" name="__codelineno-28-100" href="#__codelineno-28-100"></a><span class="w">            </span><span class="n">res</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">index</span><span class="p">(</span><span class="n">j</span><span class="p">)</span><span class="o">]</span><span class="p">;</span>
@@ -7236,11 +7308,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_deque.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Double-ended queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_deque.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Deque implemented with a circular array */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayDeque</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span><span class="w">  </span><span class="c1">// Array for storing double-ended queue elements</span>
-<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w">   </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Double-ended queue length</span>
+<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span><span class="w">  </span><span class="c1">// Array that stores the deque elements</span>
+<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w">   </span><span class="c1">// Front pointer, pointing to the first element</span>
+<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Deque size</span>
 <a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a>
 <a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nf">ArrayDeque</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7248,73 +7320,73 @@
 <a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a>
-<a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">    </span><span class="cm">/* Get the capacity of the double-ended queue */</span>
+<a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">    </span><span class="cm">/* Get the deque capacity */</span>
 <a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="nf">Capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
 <a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></a>
-<a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-29-19" name="__codelineno-29-19" href="#__codelineno-29-19"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-20" name="__codelineno-29-20" href="#__codelineno-29-20"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-22" name="__codelineno-29-22" href="#__codelineno-29-22"></a>
-<a id="__codelineno-29-23" name="__codelineno-29-23" href="#__codelineno-29-23"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-29-23" name="__codelineno-29-23" href="#__codelineno-29-23"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-29-24" name="__codelineno-29-24" href="#__codelineno-29-24"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="nf">IsEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-25" name="__codelineno-29-25" href="#__codelineno-29-25"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-29-26" name="__codelineno-29-26" href="#__codelineno-29-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-27" name="__codelineno-29-27" href="#__codelineno-29-27"></a>
-<a id="__codelineno-29-28" name="__codelineno-29-28" href="#__codelineno-29-28"></a><span class="w">    </span><span class="cm">/* Calculate circular array index */</span>
+<a id="__codelineno-29-28" name="__codelineno-29-28" href="#__codelineno-29-28"></a><span class="w">    </span><span class="cm">/* Calculate an index in the circular array */</span>
 <a id="__codelineno-29-29" name="__codelineno-29-29" href="#__codelineno-29-29"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="nf">Index</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-30" name="__codelineno-29-30" href="#__codelineno-29-30"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap the array head and tail together</span>
-<a id="__codelineno-29-31" name="__codelineno-29-31" href="#__codelineno-29-31"></a><span class="w">        </span><span class="c1">// When i passes the tail of the array, return to the head</span>
-<a id="__codelineno-29-32" name="__codelineno-29-32" href="#__codelineno-29-32"></a><span class="w">        </span><span class="c1">// When i passes the head of the array, return to the tail</span>
+<a id="__codelineno-29-30" name="__codelineno-29-30" href="#__codelineno-29-30"></a><span class="w">        </span><span class="c1">// Use modulo to connect the end of the array to its beginning</span>
+<a id="__codelineno-29-31" name="__codelineno-29-31" href="#__codelineno-29-31"></a><span class="w">        </span><span class="c1">// When i passes the end of the array, wrap around to the beginning</span>
+<a id="__codelineno-29-32" name="__codelineno-29-32" href="#__codelineno-29-32"></a><span class="w">        </span><span class="c1">// When i passes the beginning of the array, wrap around to the end</span>
 <a id="__codelineno-29-33" name="__codelineno-29-33" href="#__codelineno-29-33"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">Capacity</span><span class="p">())</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">Capacity</span><span class="p">();</span>
 <a id="__codelineno-29-34" name="__codelineno-29-34" href="#__codelineno-29-34"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-35" name="__codelineno-29-35" href="#__codelineno-29-35"></a>
-<a id="__codelineno-29-36" name="__codelineno-29-36" href="#__codelineno-29-36"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-29-36" name="__codelineno-29-36" href="#__codelineno-29-36"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-29-37" name="__codelineno-29-37" href="#__codelineno-29-37"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">PushFirst</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-38" name="__codelineno-29-38" href="#__codelineno-29-38"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">Capacity</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-39" name="__codelineno-29-39" href="#__codelineno-29-39"></a><span class="w">            </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="s">"Double-ended queue is full"</span><span class="p">);</span>
+<a id="__codelineno-29-39" name="__codelineno-29-39" href="#__codelineno-29-39"></a><span class="w">            </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="s">"Deque is full"</span><span class="p">);</span>
 <a id="__codelineno-29-40" name="__codelineno-29-40" href="#__codelineno-29-40"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-29-41" name="__codelineno-29-41" href="#__codelineno-29-41"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-29-42" name="__codelineno-29-42" href="#__codelineno-29-42"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap front around to the tail after passing the head of the array</span>
-<a id="__codelineno-29-43" name="__codelineno-29-43" href="#__codelineno-29-43"></a><span class="w">        </span><span class="c1">// Add num to the front of the queue</span>
+<a id="__codelineno-29-42" name="__codelineno-29-42" href="#__codelineno-29-42"></a><span class="w">        </span><span class="c1">// Move the front pointer one position to the left</span>
+<a id="__codelineno-29-43" name="__codelineno-29-43" href="#__codelineno-29-43"></a><span class="w">        </span><span class="c1">// Use modulo to wrap front to the end when it passes the beginning of the array</span>
 <a id="__codelineno-29-44" name="__codelineno-29-44" href="#__codelineno-29-44"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-29-45" name="__codelineno-29-45" href="#__codelineno-29-45"></a><span class="w">        </span><span class="c1">// Add num to front of queue</span>
+<a id="__codelineno-29-45" name="__codelineno-29-45" href="#__codelineno-29-45"></a><span class="w">        </span><span class="c1">// Add num to the front</span>
 <a id="__codelineno-29-46" name="__codelineno-29-46" href="#__codelineno-29-46"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">front</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-29-47" name="__codelineno-29-47" href="#__codelineno-29-47"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-29-48" name="__codelineno-29-48" href="#__codelineno-29-48"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-49" name="__codelineno-29-49" href="#__codelineno-29-49"></a>
-<a id="__codelineno-29-50" name="__codelineno-29-50" href="#__codelineno-29-50"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-29-50" name="__codelineno-29-50" href="#__codelineno-29-50"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-29-51" name="__codelineno-29-51" href="#__codelineno-29-51"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">PushLast</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-52" name="__codelineno-29-52" href="#__codelineno-29-52"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">Capacity</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-53" name="__codelineno-29-53" href="#__codelineno-29-53"></a><span class="w">            </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="s">"Double-ended queue is full"</span><span class="p">);</span>
+<a id="__codelineno-29-53" name="__codelineno-29-53" href="#__codelineno-29-53"></a><span class="w">            </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="s">"Deque is full"</span><span class="p">);</span>
 <a id="__codelineno-29-54" name="__codelineno-29-54" href="#__codelineno-29-54"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-29-55" name="__codelineno-29-55" href="#__codelineno-29-55"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-29-56" name="__codelineno-29-56" href="#__codelineno-29-56"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
+<a id="__codelineno-29-56" name="__codelineno-29-56" href="#__codelineno-29-56"></a><span class="w">        </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
 <a id="__codelineno-29-57" name="__codelineno-29-57" href="#__codelineno-29-57"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">queSize</span><span class="p">);</span>
-<a id="__codelineno-29-58" name="__codelineno-29-58" href="#__codelineno-29-58"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-29-58" name="__codelineno-29-58" href="#__codelineno-29-58"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-29-59" name="__codelineno-29-59" href="#__codelineno-29-59"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">rear</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-29-60" name="__codelineno-29-60" href="#__codelineno-29-60"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-29-61" name="__codelineno-29-61" href="#__codelineno-29-61"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-62" name="__codelineno-29-62" href="#__codelineno-29-62"></a>
-<a id="__codelineno-29-63" name="__codelineno-29-63" href="#__codelineno-29-63"></a><span class="w">    </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-29-63" name="__codelineno-29-63" href="#__codelineno-29-63"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-29-64" name="__codelineno-29-64" href="#__codelineno-29-64"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">PopFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-65" name="__codelineno-29-65" href="#__codelineno-29-65"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">PeekFirst</span><span class="p">();</span>
-<a id="__codelineno-29-66" name="__codelineno-29-66" href="#__codelineno-29-66"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position</span>
+<a id="__codelineno-29-66" name="__codelineno-29-66" href="#__codelineno-29-66"></a><span class="w">        </span><span class="c1">// Advance the front pointer by one position</span>
 <a id="__codelineno-29-67" name="__codelineno-29-67" href="#__codelineno-29-67"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-29-68" name="__codelineno-29-68" href="#__codelineno-29-68"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-29-69" name="__codelineno-29-69" href="#__codelineno-29-69"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-29-70" name="__codelineno-29-70" href="#__codelineno-29-70"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-71" name="__codelineno-29-71" href="#__codelineno-29-71"></a>
-<a id="__codelineno-29-72" name="__codelineno-29-72" href="#__codelineno-29-72"></a><span class="w">    </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-29-72" name="__codelineno-29-72" href="#__codelineno-29-72"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-29-73" name="__codelineno-29-73" href="#__codelineno-29-73"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">PopLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-74" name="__codelineno-29-74" href="#__codelineno-29-74"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">PeekLast</span><span class="p">();</span>
 <a id="__codelineno-29-75" name="__codelineno-29-75" href="#__codelineno-29-75"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-29-76" name="__codelineno-29-76" href="#__codelineno-29-76"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-29-77" name="__codelineno-29-77" href="#__codelineno-29-77"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-78" name="__codelineno-29-78" href="#__codelineno-29-78"></a>
-<a id="__codelineno-29-79" name="__codelineno-29-79" href="#__codelineno-29-79"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-29-79" name="__codelineno-29-79" href="#__codelineno-29-79"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-29-80" name="__codelineno-29-80" href="#__codelineno-29-80"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">PeekFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-81" name="__codelineno-29-81" href="#__codelineno-29-81"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">IsEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-82" name="__codelineno-29-82" href="#__codelineno-29-82"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">InvalidOperationException</span><span class="p">();</span>
@@ -7322,19 +7394,19 @@
 <a id="__codelineno-29-84" name="__codelineno-29-84" href="#__codelineno-29-84"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">front</span><span class="p">];</span>
 <a id="__codelineno-29-85" name="__codelineno-29-85" href="#__codelineno-29-85"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-86" name="__codelineno-29-86" href="#__codelineno-29-86"></a>
-<a id="__codelineno-29-87" name="__codelineno-29-87" href="#__codelineno-29-87"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-29-87" name="__codelineno-29-87" href="#__codelineno-29-87"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-29-88" name="__codelineno-29-88" href="#__codelineno-29-88"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">PeekLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-89" name="__codelineno-29-89" href="#__codelineno-29-89"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">IsEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-90" name="__codelineno-29-90" href="#__codelineno-29-90"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">InvalidOperationException</span><span class="p">();</span>
 <a id="__codelineno-29-91" name="__codelineno-29-91" href="#__codelineno-29-91"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-29-92" name="__codelineno-29-92" href="#__codelineno-29-92"></a><span class="w">        </span><span class="c1">// Initialize double-ended queue</span>
+<a id="__codelineno-29-92" name="__codelineno-29-92" href="#__codelineno-29-92"></a><span class="w">        </span><span class="c1">// Calculate the index of the last element</span>
 <a id="__codelineno-29-93" name="__codelineno-29-93" href="#__codelineno-29-93"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">last</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-29-94" name="__codelineno-29-94" href="#__codelineno-29-94"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">last</span><span class="p">];</span>
 <a id="__codelineno-29-95" name="__codelineno-29-95" href="#__codelineno-29-95"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-96" name="__codelineno-29-96" href="#__codelineno-29-96"></a>
-<a id="__codelineno-29-97" name="__codelineno-29-97" href="#__codelineno-29-97"></a><span class="w">    </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-29-97" name="__codelineno-29-97" href="#__codelineno-29-97"></a><span class="w">    </span><span class="cm">/* Return an array for printing */</span>
 <a id="__codelineno-29-98" name="__codelineno-29-98" href="#__codelineno-29-98"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="nf">ToArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-99" name="__codelineno-29-99" href="#__codelineno-29-99"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-29-99" name="__codelineno-29-99" href="#__codelineno-29-99"></a><span class="w">        </span><span class="c1">// Copy only the elements within the occupied range</span>
 <a id="__codelineno-29-100" name="__codelineno-29-100" href="#__codelineno-29-100"></a><span class="w">        </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">queSize</span><span class="p">];</span>
 <a id="__codelineno-29-101" name="__codelineno-29-101" href="#__codelineno-29-101"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-102" name="__codelineno-29-102" href="#__codelineno-29-102"></a><span class="w">            </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">Index</span><span class="p">(</span><span class="n">j</span><span class="p">)];</span>
@@ -7345,15 +7417,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_deque.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Double-ended queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_deque.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Deque implemented with a circular array */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">arrayDeque</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="nx">nums</span><span class="w">        </span><span class="p">[]</span><span class="kt">int</span><span class="w"> </span><span class="c1">// Array for storing double-ended queue elements</span>
-<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="nx">front</span><span class="w">       </span><span class="kt">int</span><span class="w">   </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">    </span><span class="nx">queSize</span><span class="w">     </span><span class="kt">int</span><span class="w">   </span><span class="c1">// Double-ended queue length</span>
-<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="nx">queCapacity</span><span class="w"> </span><span class="kt">int</span><span class="w">   </span><span class="c1">// Queue capacity (maximum number of elements)</span>
+<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="nx">nums</span><span class="w">        </span><span class="p">[]</span><span class="kt">int</span><span class="w"> </span><span class="c1">// Array that stores the deque elements</span>
+<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="nx">front</span><span class="w">       </span><span class="kt">int</span><span class="w">   </span><span class="c1">// Front pointer, pointing to the first element</span>
+<a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">    </span><span class="nx">queSize</span><span class="w">     </span><span class="kt">int</span><span class="w">   </span><span class="c1">// Deque size</span>
+<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="nx">queCapacity</span><span class="w"> </span><span class="kt">int</span><span class="w">   </span><span class="c1">// Queue capacity (the maximum number of elements)</span>
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="p">}</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a>
-<a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="cm">/* Access front of the queue element */</span>
+<a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="cm">/* Initialize the queue */</span>
 <a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="kd">func</span><span class="w"> </span><span class="nx">newArrayDeque</span><span class="p">(</span><span class="nx">queCapacity</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayDeque</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">arrayDeque</span><span class="p">{</span>
 <a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="w">        </span><span class="nx">nums</span><span class="p">:</span><span class="w">        </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">queCapacity</span><span class="p">),</span>
@@ -7363,64 +7435,64 @@
 <a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></a><span class="p">}</span>
 <a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a>
-<a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></a><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></a><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-30-20" name="__codelineno-30-20" href="#__codelineno-30-20"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-21" name="__codelineno-30-21" href="#__codelineno-30-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span>
 <a id="__codelineno-30-22" name="__codelineno-30-22" href="#__codelineno-30-22"></a><span class="p">}</span>
 <a id="__codelineno-30-23" name="__codelineno-30-23" href="#__codelineno-30-23"></a>
-<a id="__codelineno-30-24" name="__codelineno-30-24" href="#__codelineno-30-24"></a><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-30-24" name="__codelineno-30-24" href="#__codelineno-30-24"></a><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-30-25" name="__codelineno-30-25" href="#__codelineno-30-25"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-26" name="__codelineno-30-26" href="#__codelineno-30-26"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-30-27" name="__codelineno-30-27" href="#__codelineno-30-27"></a><span class="p">}</span>
 <a id="__codelineno-30-28" name="__codelineno-30-28" href="#__codelineno-30-28"></a>
-<a id="__codelineno-30-29" name="__codelineno-30-29" href="#__codelineno-30-29"></a><span class="cm">/* Calculate circular array index */</span>
+<a id="__codelineno-30-29" name="__codelineno-30-29" href="#__codelineno-30-29"></a><span class="cm">/* Calculate an index in the circular array */</span>
 <a id="__codelineno-30-30" name="__codelineno-30-30" href="#__codelineno-30-30"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">index</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-31" name="__codelineno-30-31" href="#__codelineno-30-31"></a><span class="w">    </span><span class="c1">// Use modulo operation to wrap the array head and tail together</span>
-<a id="__codelineno-30-32" name="__codelineno-30-32" href="#__codelineno-30-32"></a><span class="w">    </span><span class="c1">// When i passes the tail of the array, return to the head</span>
-<a id="__codelineno-30-33" name="__codelineno-30-33" href="#__codelineno-30-33"></a><span class="w">    </span><span class="c1">// When i passes the head of the array, return to the tail</span>
+<a id="__codelineno-30-31" name="__codelineno-30-31" href="#__codelineno-30-31"></a><span class="w">    </span><span class="c1">// Use modulo to connect the end of the array to its beginning</span>
+<a id="__codelineno-30-32" name="__codelineno-30-32" href="#__codelineno-30-32"></a><span class="w">    </span><span class="c1">// When i passes the end of the array, wrap around to the beginning</span>
+<a id="__codelineno-30-33" name="__codelineno-30-33" href="#__codelineno-30-33"></a><span class="w">    </span><span class="c1">// When i passes the beginning of the array, wrap around to the end</span>
 <a id="__codelineno-30-34" name="__codelineno-30-34" href="#__codelineno-30-34"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queCapacity</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queCapacity</span>
 <a id="__codelineno-30-35" name="__codelineno-30-35" href="#__codelineno-30-35"></a><span class="p">}</span>
 <a id="__codelineno-30-36" name="__codelineno-30-36" href="#__codelineno-30-36"></a>
-<a id="__codelineno-30-37" name="__codelineno-30-37" href="#__codelineno-30-37"></a><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-30-37" name="__codelineno-30-37" href="#__codelineno-30-37"></a><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-30-38" name="__codelineno-30-38" href="#__codelineno-30-38"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">pushFirst</span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-39" name="__codelineno-30-39" href="#__codelineno-30-39"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queCapacity</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-40" name="__codelineno-30-40" href="#__codelineno-30-40"></a><span class="w">        </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="s">"Double-ended queue is full"</span><span class="p">)</span>
+<a id="__codelineno-30-40" name="__codelineno-30-40" href="#__codelineno-30-40"></a><span class="w">        </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="s">"Deque is full"</span><span class="p">)</span>
 <a id="__codelineno-30-41" name="__codelineno-30-41" href="#__codelineno-30-41"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-30-42" name="__codelineno-30-42" href="#__codelineno-30-42"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-43" name="__codelineno-30-43" href="#__codelineno-30-43"></a><span class="w">    </span><span class="c1">// Use modulo operation to wrap front around to the tail after passing the head of the array</span>
-<a id="__codelineno-30-44" name="__codelineno-30-44" href="#__codelineno-30-44"></a><span class="w">    </span><span class="c1">// Add num to the front of the queue</span>
+<a id="__codelineno-30-43" name="__codelineno-30-43" href="#__codelineno-30-43"></a><span class="w">    </span><span class="c1">// Move the front pointer one position to the left</span>
+<a id="__codelineno-30-44" name="__codelineno-30-44" href="#__codelineno-30-44"></a><span class="w">    </span><span class="c1">// Use modulo to wrap front to the end when it passes the beginning of the array</span>
 <a id="__codelineno-30-45" name="__codelineno-30-45" href="#__codelineno-30-45"></a><span class="w">    </span><span class="nx">q</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">index</span><span class="p">(</span><span class="nx">q</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-30-46" name="__codelineno-30-46" href="#__codelineno-30-46"></a><span class="w">    </span><span class="c1">// Add num to front of queue</span>
+<a id="__codelineno-30-46" name="__codelineno-30-46" href="#__codelineno-30-46"></a><span class="w">    </span><span class="c1">// Add num to the front</span>
 <a id="__codelineno-30-47" name="__codelineno-30-47" href="#__codelineno-30-47"></a><span class="w">    </span><span class="nx">q</span><span class="p">.</span><span class="nx">nums</span><span class="p">[</span><span class="nx">q</span><span class="p">.</span><span class="nx">front</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span>
 <a id="__codelineno-30-48" name="__codelineno-30-48" href="#__codelineno-30-48"></a><span class="w">    </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="o">++</span>
 <a id="__codelineno-30-49" name="__codelineno-30-49" href="#__codelineno-30-49"></a><span class="p">}</span>
 <a id="__codelineno-30-50" name="__codelineno-30-50" href="#__codelineno-30-50"></a>
-<a id="__codelineno-30-51" name="__codelineno-30-51" href="#__codelineno-30-51"></a><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-30-51" name="__codelineno-30-51" href="#__codelineno-30-51"></a><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-30-52" name="__codelineno-30-52" href="#__codelineno-30-52"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">pushLast</span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-53" name="__codelineno-30-53" href="#__codelineno-30-53"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queCapacity</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-54" name="__codelineno-30-54" href="#__codelineno-30-54"></a><span class="w">        </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="s">"Double-ended queue is full"</span><span class="p">)</span>
+<a id="__codelineno-30-54" name="__codelineno-30-54" href="#__codelineno-30-54"></a><span class="w">        </span><span class="nx">fmt</span><span class="p">.</span><span class="nx">Println</span><span class="p">(</span><span class="s">"Deque is full"</span><span class="p">)</span>
 <a id="__codelineno-30-55" name="__codelineno-30-55" href="#__codelineno-30-55"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-30-56" name="__codelineno-30-56" href="#__codelineno-30-56"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-57" name="__codelineno-30-57" href="#__codelineno-30-57"></a><span class="w">    </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
+<a id="__codelineno-30-57" name="__codelineno-30-57" href="#__codelineno-30-57"></a><span class="w">    </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
 <a id="__codelineno-30-58" name="__codelineno-30-58" href="#__codelineno-30-58"></a><span class="w">    </span><span class="nx">rear</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">index</span><span class="p">(</span><span class="nx">q</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="p">)</span>
-<a id="__codelineno-30-59" name="__codelineno-30-59" href="#__codelineno-30-59"></a><span class="w">    </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-30-59" name="__codelineno-30-59" href="#__codelineno-30-59"></a><span class="w">    </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-30-60" name="__codelineno-30-60" href="#__codelineno-30-60"></a><span class="w">    </span><span class="nx">q</span><span class="p">.</span><span class="nx">nums</span><span class="p">[</span><span class="nx">rear</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span>
 <a id="__codelineno-30-61" name="__codelineno-30-61" href="#__codelineno-30-61"></a><span class="w">    </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="o">++</span>
 <a id="__codelineno-30-62" name="__codelineno-30-62" href="#__codelineno-30-62"></a><span class="p">}</span>
 <a id="__codelineno-30-63" name="__codelineno-30-63" href="#__codelineno-30-63"></a>
-<a id="__codelineno-30-64" name="__codelineno-30-64" href="#__codelineno-30-64"></a><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-30-64" name="__codelineno-30-64" href="#__codelineno-30-64"></a><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-30-65" name="__codelineno-30-65" href="#__codelineno-30-65"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">popFirst</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-66" name="__codelineno-30-66" href="#__codelineno-30-66"></a><span class="w">    </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">peekFirst</span><span class="p">()</span>
 <a id="__codelineno-30-67" name="__codelineno-30-67" href="#__codelineno-30-67"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-68" name="__codelineno-30-68" href="#__codelineno-30-68"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-30-69" name="__codelineno-30-69" href="#__codelineno-30-69"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-70" name="__codelineno-30-70" href="#__codelineno-30-70"></a><span class="w">    </span><span class="c1">// Move front pointer backward by one position</span>
+<a id="__codelineno-30-70" name="__codelineno-30-70" href="#__codelineno-30-70"></a><span class="w">    </span><span class="c1">// Advance the front pointer by one position</span>
 <a id="__codelineno-30-71" name="__codelineno-30-71" href="#__codelineno-30-71"></a><span class="w">    </span><span class="nx">q</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">index</span><span class="p">(</span><span class="nx">q</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-30-72" name="__codelineno-30-72" href="#__codelineno-30-72"></a><span class="w">    </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="o">--</span>
 <a id="__codelineno-30-73" name="__codelineno-30-73" href="#__codelineno-30-73"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span>
 <a id="__codelineno-30-74" name="__codelineno-30-74" href="#__codelineno-30-74"></a><span class="p">}</span>
 <a id="__codelineno-30-75" name="__codelineno-30-75" href="#__codelineno-30-75"></a>
-<a id="__codelineno-30-76" name="__codelineno-30-76" href="#__codelineno-30-76"></a><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-30-76" name="__codelineno-30-76" href="#__codelineno-30-76"></a><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-30-77" name="__codelineno-30-77" href="#__codelineno-30-77"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">popLast</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-78" name="__codelineno-30-78" href="#__codelineno-30-78"></a><span class="w">    </span><span class="nx">num</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">peekLast</span><span class="p">()</span>
 <a id="__codelineno-30-79" name="__codelineno-30-79" href="#__codelineno-30-79"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
@@ -7430,7 +7502,7 @@
 <a id="__codelineno-30-83" name="__codelineno-30-83" href="#__codelineno-30-83"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span>
 <a id="__codelineno-30-84" name="__codelineno-30-84" href="#__codelineno-30-84"></a><span class="p">}</span>
 <a id="__codelineno-30-85" name="__codelineno-30-85" href="#__codelineno-30-85"></a>
-<a id="__codelineno-30-86" name="__codelineno-30-86" href="#__codelineno-30-86"></a><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-30-86" name="__codelineno-30-86" href="#__codelineno-30-86"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-30-87" name="__codelineno-30-87" href="#__codelineno-30-87"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">peekFirst</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-88" name="__codelineno-30-88" href="#__codelineno-30-88"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-89" name="__codelineno-30-89" href="#__codelineno-30-89"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
@@ -7438,19 +7510,19 @@
 <a id="__codelineno-30-91" name="__codelineno-30-91" href="#__codelineno-30-91"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">nums</span><span class="p">[</span><span class="nx">q</span><span class="p">.</span><span class="nx">front</span><span class="p">]</span>
 <a id="__codelineno-30-92" name="__codelineno-30-92" href="#__codelineno-30-92"></a><span class="p">}</span>
 <a id="__codelineno-30-93" name="__codelineno-30-93" href="#__codelineno-30-93"></a>
-<a id="__codelineno-30-94" name="__codelineno-30-94" href="#__codelineno-30-94"></a><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-30-94" name="__codelineno-30-94" href="#__codelineno-30-94"></a><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-30-95" name="__codelineno-30-95" href="#__codelineno-30-95"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">peekLast</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-96" name="__codelineno-30-96" href="#__codelineno-30-96"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-97" name="__codelineno-30-97" href="#__codelineno-30-97"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-30-98" name="__codelineno-30-98" href="#__codelineno-30-98"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-99" name="__codelineno-30-99" href="#__codelineno-30-99"></a><span class="w">    </span><span class="c1">// Initialize double-ended queue</span>
+<a id="__codelineno-30-99" name="__codelineno-30-99" href="#__codelineno-30-99"></a><span class="w">    </span><span class="c1">// Calculate the index of the last element</span>
 <a id="__codelineno-30-100" name="__codelineno-30-100" href="#__codelineno-30-100"></a><span class="w">    </span><span class="nx">last</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">index</span><span class="p">(</span><span class="nx">q</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-30-101" name="__codelineno-30-101" href="#__codelineno-30-101"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">nums</span><span class="p">[</span><span class="nx">last</span><span class="p">]</span>
 <a id="__codelineno-30-102" name="__codelineno-30-102" href="#__codelineno-30-102"></a><span class="p">}</span>
 <a id="__codelineno-30-103" name="__codelineno-30-103" href="#__codelineno-30-103"></a>
-<a id="__codelineno-30-104" name="__codelineno-30-104" href="#__codelineno-30-104"></a><span class="cm">/* Get Slice for printing */</span>
+<a id="__codelineno-30-104" name="__codelineno-30-104" href="#__codelineno-30-104"></a><span class="cm">/* Get a slice for printing */</span>
 <a id="__codelineno-30-105" name="__codelineno-30-105" href="#__codelineno-30-105"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayDeque</span><span class="p">)</span><span class="w"> </span><span class="nx">toSlice</span><span class="p">()</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-106" name="__codelineno-30-106" href="#__codelineno-30-106"></a><span class="w">    </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-30-106" name="__codelineno-30-106" href="#__codelineno-30-106"></a><span class="w">    </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-30-107" name="__codelineno-30-107" href="#__codelineno-30-107"></a><span class="w">    </span><span class="nx">res</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="p">)</span>
 <a id="__codelineno-30-108" name="__codelineno-30-108" href="#__codelineno-30-108"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">front</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-109" name="__codelineno-30-109" href="#__codelineno-30-109"></a><span class="w">        </span><span class="nx">res</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">nums</span><span class="p">[</span><span class="nx">q</span><span class="p">.</span><span class="nx">index</span><span class="p">(</span><span class="nx">j</span><span class="p">)]</span>
@@ -7461,11 +7533,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_deque.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Double-ended queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_deque.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Deque implemented with a circular array */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayDeque</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="c1">// Array for storing double-ended queue elements</span>
-<a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">front</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">_size</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Double-ended queue length</span>
+<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="c1">// Array that stores the deque elements</span>
+<a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">front</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Front index, identifying the first element</span>
+<a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">_size</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Deque size</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a>
 <a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">    </span><span class="kd">init</span><span class="p">(</span><span class="n">capacity</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7474,73 +7546,73 @@
 <a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a>
-<a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="w">    </span><span class="cm">/* Get the capacity of the double-ended queue */</span>
+<a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="w">    </span><span class="cm">/* Get the deque capacity */</span>
 <a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="w">        </span><span class="n">nums</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-31-17" name="__codelineno-31-17" href="#__codelineno-31-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-18" name="__codelineno-31-18" href="#__codelineno-31-18"></a>
-<a id="__codelineno-31-19" name="__codelineno-31-19" href="#__codelineno-31-19"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-31-19" name="__codelineno-31-19" href="#__codelineno-31-19"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-31-20" name="__codelineno-31-20" href="#__codelineno-31-20"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-21" name="__codelineno-31-21" href="#__codelineno-31-21"></a><span class="w">        </span><span class="n">_size</span>
 <a id="__codelineno-31-22" name="__codelineno-31-22" href="#__codelineno-31-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-23" name="__codelineno-31-23" href="#__codelineno-31-23"></a>
-<a id="__codelineno-31-24" name="__codelineno-31-24" href="#__codelineno-31-24"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-31-24" name="__codelineno-31-24" href="#__codelineno-31-24"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-31-25" name="__codelineno-31-25" href="#__codelineno-31-25"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-26" name="__codelineno-31-26" href="#__codelineno-31-26"></a><span class="w">        </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-31-27" name="__codelineno-31-27" href="#__codelineno-31-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-28" name="__codelineno-31-28" href="#__codelineno-31-28"></a>
-<a id="__codelineno-31-29" name="__codelineno-31-29" href="#__codelineno-31-29"></a><span class="w">    </span><span class="cm">/* Calculate circular array index */</span>
+<a id="__codelineno-31-29" name="__codelineno-31-29" href="#__codelineno-31-29"></a><span class="w">    </span><span class="cm">/* Calculate an index in the circular array */</span>
 <a id="__codelineno-31-30" name="__codelineno-31-30" href="#__codelineno-31-30"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">func</span><span class="w"> </span><span class="nf">index</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-31" name="__codelineno-31-31" href="#__codelineno-31-31"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap the array head and tail together</span>
-<a id="__codelineno-31-32" name="__codelineno-31-32" href="#__codelineno-31-32"></a><span class="w">        </span><span class="c1">// When i passes the tail of the array, return to the head</span>
-<a id="__codelineno-31-33" name="__codelineno-31-33" href="#__codelineno-31-33"></a><span class="w">        </span><span class="c1">// When i passes the head of the array, return to the tail</span>
+<a id="__codelineno-31-31" name="__codelineno-31-31" href="#__codelineno-31-31"></a><span class="w">        </span><span class="c1">// Use modulo to connect the end of the array to its beginning</span>
+<a id="__codelineno-31-32" name="__codelineno-31-32" href="#__codelineno-31-32"></a><span class="w">        </span><span class="c1">// When i passes the end of the array, wrap around to the beginning</span>
+<a id="__codelineno-31-33" name="__codelineno-31-33" href="#__codelineno-31-33"></a><span class="w">        </span><span class="c1">// When i passes the beginning of the array, wrap around to the end</span>
 <a id="__codelineno-31-34" name="__codelineno-31-34" href="#__codelineno-31-34"></a><span class="w">        </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span>
 <a id="__codelineno-31-35" name="__codelineno-31-35" href="#__codelineno-31-35"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-36" name="__codelineno-31-36" href="#__codelineno-31-36"></a>
-<a id="__codelineno-31-37" name="__codelineno-31-37" href="#__codelineno-31-37"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-31-37" name="__codelineno-31-37" href="#__codelineno-31-37"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-31-38" name="__codelineno-31-38" href="#__codelineno-31-38"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">pushFirst</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-39" name="__codelineno-31-39" href="#__codelineno-31-39"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-40" name="__codelineno-31-40" href="#__codelineno-31-40"></a><span class="w">            </span><span class="bp">print</span><span class="p">(</span><span class="s">"Double-ended queue is full"</span><span class="p">)</span>
+<a id="__codelineno-31-40" name="__codelineno-31-40" href="#__codelineno-31-40"></a><span class="w">            </span><span class="bp">print</span><span class="p">(</span><span class="s">"Deque is full"</span><span class="p">)</span>
 <a id="__codelineno-31-41" name="__codelineno-31-41" href="#__codelineno-31-41"></a><span class="w">            </span><span class="k">return</span>
 <a id="__codelineno-31-42" name="__codelineno-31-42" href="#__codelineno-31-42"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-31-43" name="__codelineno-31-43" href="#__codelineno-31-43"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap front around to the tail after passing the head of the array</span>
-<a id="__codelineno-31-44" name="__codelineno-31-44" href="#__codelineno-31-44"></a><span class="w">        </span><span class="c1">// Add num to the front of the queue</span>
+<a id="__codelineno-31-43" name="__codelineno-31-43" href="#__codelineno-31-43"></a><span class="w">        </span><span class="c1">// Move the front index one position to the left</span>
+<a id="__codelineno-31-44" name="__codelineno-31-44" href="#__codelineno-31-44"></a><span class="w">        </span><span class="c1">// Use modulo to wrap front to the end when it passes the beginning of the array</span>
 <a id="__codelineno-31-45" name="__codelineno-31-45" href="#__codelineno-31-45"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">front</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-31-46" name="__codelineno-31-46" href="#__codelineno-31-46"></a><span class="w">        </span><span class="c1">// Add num to front of queue</span>
+<a id="__codelineno-31-46" name="__codelineno-31-46" href="#__codelineno-31-46"></a><span class="w">        </span><span class="c1">// Add num to the front</span>
 <a id="__codelineno-31-47" name="__codelineno-31-47" href="#__codelineno-31-47"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">front</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-31-48" name="__codelineno-31-48" href="#__codelineno-31-48"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-31-49" name="__codelineno-31-49" href="#__codelineno-31-49"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-50" name="__codelineno-31-50" href="#__codelineno-31-50"></a>
-<a id="__codelineno-31-51" name="__codelineno-31-51" href="#__codelineno-31-51"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-31-51" name="__codelineno-31-51" href="#__codelineno-31-51"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-31-52" name="__codelineno-31-52" href="#__codelineno-31-52"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">pushLast</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-53" name="__codelineno-31-53" href="#__codelineno-31-53"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-54" name="__codelineno-31-54" href="#__codelineno-31-54"></a><span class="w">            </span><span class="bp">print</span><span class="p">(</span><span class="s">"Double-ended queue is full"</span><span class="p">)</span>
+<a id="__codelineno-31-54" name="__codelineno-31-54" href="#__codelineno-31-54"></a><span class="w">            </span><span class="bp">print</span><span class="p">(</span><span class="s">"Deque is full"</span><span class="p">)</span>
 <a id="__codelineno-31-55" name="__codelineno-31-55" href="#__codelineno-31-55"></a><span class="w">            </span><span class="k">return</span>
 <a id="__codelineno-31-56" name="__codelineno-31-56" href="#__codelineno-31-56"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-31-57" name="__codelineno-31-57" href="#__codelineno-31-57"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
+<a id="__codelineno-31-57" name="__codelineno-31-57" href="#__codelineno-31-57"></a><span class="w">        </span><span class="c1">// Calculate the rear index, one position past the last element</span>
 <a id="__codelineno-31-58" name="__codelineno-31-58" href="#__codelineno-31-58"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">rear</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">size</span><span class="p">())</span>
-<a id="__codelineno-31-59" name="__codelineno-31-59" href="#__codelineno-31-59"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-31-59" name="__codelineno-31-59" href="#__codelineno-31-59"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-31-60" name="__codelineno-31-60" href="#__codelineno-31-60"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">rear</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-31-61" name="__codelineno-31-61" href="#__codelineno-31-61"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-31-62" name="__codelineno-31-62" href="#__codelineno-31-62"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-63" name="__codelineno-31-63" href="#__codelineno-31-63"></a>
-<a id="__codelineno-31-64" name="__codelineno-31-64" href="#__codelineno-31-64"></a><span class="w">    </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-31-64" name="__codelineno-31-64" href="#__codelineno-31-64"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-31-65" name="__codelineno-31-65" href="#__codelineno-31-65"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">popFirst</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-66" name="__codelineno-31-66" href="#__codelineno-31-66"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">peekFirst</span><span class="p">()</span>
-<a id="__codelineno-31-67" name="__codelineno-31-67" href="#__codelineno-31-67"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position</span>
+<a id="__codelineno-31-67" name="__codelineno-31-67" href="#__codelineno-31-67"></a><span class="w">        </span><span class="c1">// Advance the front index by one position</span>
 <a id="__codelineno-31-68" name="__codelineno-31-68" href="#__codelineno-31-68"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-31-69" name="__codelineno-31-69" href="#__codelineno-31-69"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-31-70" name="__codelineno-31-70" href="#__codelineno-31-70"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-31-71" name="__codelineno-31-71" href="#__codelineno-31-71"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-72" name="__codelineno-31-72" href="#__codelineno-31-72"></a>
-<a id="__codelineno-31-73" name="__codelineno-31-73" href="#__codelineno-31-73"></a><span class="w">    </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-31-73" name="__codelineno-31-73" href="#__codelineno-31-73"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-31-74" name="__codelineno-31-74" href="#__codelineno-31-74"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">popLast</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-75" name="__codelineno-31-75" href="#__codelineno-31-75"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">peekLast</span><span class="p">()</span>
 <a id="__codelineno-31-76" name="__codelineno-31-76" href="#__codelineno-31-76"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-31-77" name="__codelineno-31-77" href="#__codelineno-31-77"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-31-78" name="__codelineno-31-78" href="#__codelineno-31-78"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-79" name="__codelineno-31-79" href="#__codelineno-31-79"></a>
-<a id="__codelineno-31-80" name="__codelineno-31-80" href="#__codelineno-31-80"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-31-80" name="__codelineno-31-80" href="#__codelineno-31-80"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-31-81" name="__codelineno-31-81" href="#__codelineno-31-81"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">peekFirst</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-82" name="__codelineno-31-82" href="#__codelineno-31-82"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-83" name="__codelineno-31-83" href="#__codelineno-31-83"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Deque is empty"</span><span class="p">)</span>
@@ -7548,30 +7620,30 @@
 <a id="__codelineno-31-85" name="__codelineno-31-85" href="#__codelineno-31-85"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">front</span><span class="p">]</span>
 <a id="__codelineno-31-86" name="__codelineno-31-86" href="#__codelineno-31-86"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-87" name="__codelineno-31-87" href="#__codelineno-31-87"></a>
-<a id="__codelineno-31-88" name="__codelineno-31-88" href="#__codelineno-31-88"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-31-88" name="__codelineno-31-88" href="#__codelineno-31-88"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-31-89" name="__codelineno-31-89" href="#__codelineno-31-89"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">peekLast</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-90" name="__codelineno-31-90" href="#__codelineno-31-90"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-91" name="__codelineno-31-91" href="#__codelineno-31-91"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Deque is empty"</span><span class="p">)</span>
 <a id="__codelineno-31-92" name="__codelineno-31-92" href="#__codelineno-31-92"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-31-93" name="__codelineno-31-93" href="#__codelineno-31-93"></a><span class="w">        </span><span class="c1">// Initialize double-ended queue</span>
+<a id="__codelineno-31-93" name="__codelineno-31-93" href="#__codelineno-31-93"></a><span class="w">        </span><span class="c1">// Calculate the index of the last element</span>
 <a id="__codelineno-31-94" name="__codelineno-31-94" href="#__codelineno-31-94"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">last</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-31-95" name="__codelineno-31-95" href="#__codelineno-31-95"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="bp">last</span><span class="p">]</span>
 <a id="__codelineno-31-96" name="__codelineno-31-96" href="#__codelineno-31-96"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-97" name="__codelineno-31-97" href="#__codelineno-31-97"></a>
-<a id="__codelineno-31-98" name="__codelineno-31-98" href="#__codelineno-31-98"></a><span class="w">    </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-31-98" name="__codelineno-31-98" href="#__codelineno-31-98"></a><span class="w">    </span><span class="cm">/* Return an array for printing */</span>
 <a id="__codelineno-31-99" name="__codelineno-31-99" href="#__codelineno-31-99"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-100" name="__codelineno-31-100" href="#__codelineno-31-100"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-31-100" name="__codelineno-31-100" href="#__codelineno-31-100"></a><span class="w">        </span><span class="c1">// Include only the elements in the occupied range</span>
 <a id="__codelineno-31-101" name="__codelineno-31-101" href="#__codelineno-31-101"></a><span class="w">        </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">size</span><span class="p">()).</span><span class="bp">map</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">index</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nv">$0</span><span class="p">)]</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-31-102" name="__codelineno-31-102" href="#__codelineno-31-102"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-103" name="__codelineno-31-103" href="#__codelineno-31-103"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_deque.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Double-ended queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_deque.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Deque implemented with a circular array */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">ArrayDeque</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="n">#nums</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array for storing double-ended queue elements</span>
-<a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="n">#front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="n">#queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Double-ended queue length</span>
+<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="n">#nums</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array for storing deque elements</span>
+<a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="n">#front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, pointing to the first element</span>
+<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="n">#queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Deque size</span>
 <a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a>
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7580,89 +7652,89 @@
 <a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a>
-<a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="w">    </span><span class="cm">/* Get the capacity of the double-ended queue */</span>
+<a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="w">    </span><span class="cm">/* Get the deque capacity */</span>
 <a id="__codelineno-32-15" name="__codelineno-32-15" href="#__codelineno-32-15"></a><span class="w">    </span><span class="nx">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-16" name="__codelineno-32-16" href="#__codelineno-32-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-32-17" name="__codelineno-32-17" href="#__codelineno-32-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-18" name="__codelineno-32-18" href="#__codelineno-32-18"></a>
-<a id="__codelineno-32-19" name="__codelineno-32-19" href="#__codelineno-32-19"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-32-19" name="__codelineno-32-19" href="#__codelineno-32-19"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-32-20" name="__codelineno-32-20" href="#__codelineno-32-20"></a><span class="w">    </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-21" name="__codelineno-32-21" href="#__codelineno-32-21"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="p">;</span>
 <a id="__codelineno-32-22" name="__codelineno-32-22" href="#__codelineno-32-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-23" name="__codelineno-32-23" href="#__codelineno-32-23"></a>
-<a id="__codelineno-32-24" name="__codelineno-32-24" href="#__codelineno-32-24"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-32-24" name="__codelineno-32-24" href="#__codelineno-32-24"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-32-25" name="__codelineno-32-25" href="#__codelineno-32-25"></a><span class="w">    </span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-26" name="__codelineno-32-26" href="#__codelineno-32-26"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-32-27" name="__codelineno-32-27" href="#__codelineno-32-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-28" name="__codelineno-32-28" href="#__codelineno-32-28"></a>
-<a id="__codelineno-32-29" name="__codelineno-32-29" href="#__codelineno-32-29"></a><span class="w">    </span><span class="cm">/* Calculate circular array index */</span>
+<a id="__codelineno-32-29" name="__codelineno-32-29" href="#__codelineno-32-29"></a><span class="w">    </span><span class="cm">/* Calculate an index in the circular array */</span>
 <a id="__codelineno-32-30" name="__codelineno-32-30" href="#__codelineno-32-30"></a><span class="w">    </span><span class="nx">index</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-31" name="__codelineno-32-31" href="#__codelineno-32-31"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap the array head and tail together</span>
-<a id="__codelineno-32-32" name="__codelineno-32-32" href="#__codelineno-32-32"></a><span class="w">        </span><span class="c1">// When i passes the tail of the array, return to the head</span>
-<a id="__codelineno-32-33" name="__codelineno-32-33" href="#__codelineno-32-33"></a><span class="w">        </span><span class="c1">// When i passes the head of the array, return to the tail</span>
+<a id="__codelineno-32-31" name="__codelineno-32-31" href="#__codelineno-32-31"></a><span class="w">        </span><span class="c1">// Use modulo to connect the end of the array to its beginning</span>
+<a id="__codelineno-32-32" name="__codelineno-32-32" href="#__codelineno-32-32"></a><span class="w">        </span><span class="c1">// When i passes the end of the array, wrap around to the beginning</span>
+<a id="__codelineno-32-33" name="__codelineno-32-33" href="#__codelineno-32-33"></a><span class="w">        </span><span class="c1">// When i passes the beginning of the array, wrap around to the end</span>
 <a id="__codelineno-32-34" name="__codelineno-32-34" href="#__codelineno-32-34"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">())</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">();</span>
 <a id="__codelineno-32-35" name="__codelineno-32-35" href="#__codelineno-32-35"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-36" name="__codelineno-32-36" href="#__codelineno-32-36"></a>
-<a id="__codelineno-32-37" name="__codelineno-32-37" href="#__codelineno-32-37"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-32-37" name="__codelineno-32-37" href="#__codelineno-32-37"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-32-38" name="__codelineno-32-38" href="#__codelineno-32-38"></a><span class="w">    </span><span class="nx">pushFirst</span><span class="p">(</span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-39" name="__codelineno-32-39" href="#__codelineno-32-39"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-40" name="__codelineno-32-40" href="#__codelineno-32-40"></a><span class="w">            </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Double-ended queue is full'</span><span class="p">);</span>
+<a id="__codelineno-32-40" name="__codelineno-32-40" href="#__codelineno-32-40"></a><span class="w">            </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Deque is full'</span><span class="p">);</span>
 <a id="__codelineno-32-41" name="__codelineno-32-41" href="#__codelineno-32-41"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-32-42" name="__codelineno-32-42" href="#__codelineno-32-42"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-32-43" name="__codelineno-32-43" href="#__codelineno-32-43"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap front around to the tail after passing the head of the array</span>
-<a id="__codelineno-32-44" name="__codelineno-32-44" href="#__codelineno-32-44"></a><span class="w">        </span><span class="c1">// Add num to the front of the queue</span>
+<a id="__codelineno-32-43" name="__codelineno-32-43" href="#__codelineno-32-43"></a><span class="w">        </span><span class="c1">// Move the front pointer one position to the left</span>
+<a id="__codelineno-32-44" name="__codelineno-32-44" href="#__codelineno-32-44"></a><span class="w">        </span><span class="c1">// Use modulo to wrap front to the end when it passes the beginning of the array</span>
 <a id="__codelineno-32-45" name="__codelineno-32-45" href="#__codelineno-32-45"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">index</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-32-46" name="__codelineno-32-46" href="#__codelineno-32-46"></a><span class="w">        </span><span class="c1">// Add num to front of queue</span>
+<a id="__codelineno-32-46" name="__codelineno-32-46" href="#__codelineno-32-46"></a><span class="w">        </span><span class="c1">// Add num to the front</span>
 <a id="__codelineno-32-47" name="__codelineno-32-47" href="#__codelineno-32-47"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#nums</span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-32-48" name="__codelineno-32-48" href="#__codelineno-32-48"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-32-49" name="__codelineno-32-49" href="#__codelineno-32-49"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-50" name="__codelineno-32-50" href="#__codelineno-32-50"></a>
-<a id="__codelineno-32-51" name="__codelineno-32-51" href="#__codelineno-32-51"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-32-51" name="__codelineno-32-51" href="#__codelineno-32-51"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-32-52" name="__codelineno-32-52" href="#__codelineno-32-52"></a><span class="w">    </span><span class="nx">pushLast</span><span class="p">(</span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-53" name="__codelineno-32-53" href="#__codelineno-32-53"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-54" name="__codelineno-32-54" href="#__codelineno-32-54"></a><span class="w">            </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Double-ended queue is full'</span><span class="p">);</span>
+<a id="__codelineno-32-54" name="__codelineno-32-54" href="#__codelineno-32-54"></a><span class="w">            </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Deque is full'</span><span class="p">);</span>
 <a id="__codelineno-32-55" name="__codelineno-32-55" href="#__codelineno-32-55"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-32-56" name="__codelineno-32-56" href="#__codelineno-32-56"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-32-57" name="__codelineno-32-57" href="#__codelineno-32-57"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
+<a id="__codelineno-32-57" name="__codelineno-32-57" href="#__codelineno-32-57"></a><span class="w">        </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
 <a id="__codelineno-32-58" name="__codelineno-32-58" href="#__codelineno-32-58"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">index</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="p">);</span>
-<a id="__codelineno-32-59" name="__codelineno-32-59" href="#__codelineno-32-59"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-32-59" name="__codelineno-32-59" href="#__codelineno-32-59"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-32-60" name="__codelineno-32-60" href="#__codelineno-32-60"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#nums</span><span class="p">[</span><span class="nx">rear</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-32-61" name="__codelineno-32-61" href="#__codelineno-32-61"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-32-62" name="__codelineno-32-62" href="#__codelineno-32-62"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-63" name="__codelineno-32-63" href="#__codelineno-32-63"></a>
-<a id="__codelineno-32-64" name="__codelineno-32-64" href="#__codelineno-32-64"></a><span class="w">    </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-32-64" name="__codelineno-32-64" href="#__codelineno-32-64"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-32-65" name="__codelineno-32-65" href="#__codelineno-32-65"></a><span class="w">    </span><span class="nx">popFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-66" name="__codelineno-32-66" href="#__codelineno-32-66"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">peekFirst</span><span class="p">();</span>
-<a id="__codelineno-32-67" name="__codelineno-32-67" href="#__codelineno-32-67"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position</span>
+<a id="__codelineno-32-67" name="__codelineno-32-67" href="#__codelineno-32-67"></a><span class="w">        </span><span class="c1">// Advance the front pointer by one position</span>
 <a id="__codelineno-32-68" name="__codelineno-32-68" href="#__codelineno-32-68"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">index</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-32-69" name="__codelineno-32-69" href="#__codelineno-32-69"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-32-70" name="__codelineno-32-70" href="#__codelineno-32-70"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-32-71" name="__codelineno-32-71" href="#__codelineno-32-71"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-72" name="__codelineno-32-72" href="#__codelineno-32-72"></a>
-<a id="__codelineno-32-73" name="__codelineno-32-73" href="#__codelineno-32-73"></a><span class="w">    </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-32-73" name="__codelineno-32-73" href="#__codelineno-32-73"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-32-74" name="__codelineno-32-74" href="#__codelineno-32-74"></a><span class="w">    </span><span class="nx">popLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-75" name="__codelineno-32-75" href="#__codelineno-32-75"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">peekLast</span><span class="p">();</span>
 <a id="__codelineno-32-76" name="__codelineno-32-76" href="#__codelineno-32-76"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-32-77" name="__codelineno-32-77" href="#__codelineno-32-77"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-32-78" name="__codelineno-32-78" href="#__codelineno-32-78"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-79" name="__codelineno-32-79" href="#__codelineno-32-79"></a>
-<a id="__codelineno-32-80" name="__codelineno-32-80" href="#__codelineno-32-80"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-32-80" name="__codelineno-32-80" href="#__codelineno-32-80"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-32-81" name="__codelineno-32-81" href="#__codelineno-32-81"></a><span class="w">    </span><span class="nx">peekFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-82" name="__codelineno-32-82" href="#__codelineno-32-82"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'The Deque Is Empty.'</span><span class="p">);</span>
+<a id="__codelineno-32-82" name="__codelineno-32-82" href="#__codelineno-32-82"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Deque is empty'</span><span class="p">);</span>
 <a id="__codelineno-32-83" name="__codelineno-32-83" href="#__codelineno-32-83"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#nums</span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">];</span>
 <a id="__codelineno-32-84" name="__codelineno-32-84" href="#__codelineno-32-84"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-85" name="__codelineno-32-85" href="#__codelineno-32-85"></a>
-<a id="__codelineno-32-86" name="__codelineno-32-86" href="#__codelineno-32-86"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-32-86" name="__codelineno-32-86" href="#__codelineno-32-86"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-32-87" name="__codelineno-32-87" href="#__codelineno-32-87"></a><span class="w">    </span><span class="nx">peekLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-88" name="__codelineno-32-88" href="#__codelineno-32-88"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'The Deque Is Empty.'</span><span class="p">);</span>
-<a id="__codelineno-32-89" name="__codelineno-32-89" href="#__codelineno-32-89"></a><span class="w">        </span><span class="c1">// Initialize double-ended queue</span>
+<a id="__codelineno-32-88" name="__codelineno-32-88" href="#__codelineno-32-88"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Deque is empty'</span><span class="p">);</span>
+<a id="__codelineno-32-89" name="__codelineno-32-89" href="#__codelineno-32-89"></a><span class="w">        </span><span class="c1">// Calculate the index of the last element</span>
 <a id="__codelineno-32-90" name="__codelineno-32-90" href="#__codelineno-32-90"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">last</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">index</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-32-91" name="__codelineno-32-91" href="#__codelineno-32-91"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#nums</span><span class="p">[</span><span class="nx">last</span><span class="p">];</span>
 <a id="__codelineno-32-92" name="__codelineno-32-92" href="#__codelineno-32-92"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-93" name="__codelineno-32-93" href="#__codelineno-32-93"></a>
-<a id="__codelineno-32-94" name="__codelineno-32-94" href="#__codelineno-32-94"></a><span class="w">    </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-32-94" name="__codelineno-32-94" href="#__codelineno-32-94"></a><span class="w">    </span><span class="cm">/* Return an array for printing */</span>
 <a id="__codelineno-32-95" name="__codelineno-32-95" href="#__codelineno-32-95"></a><span class="w">    </span><span class="nx">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-96" name="__codelineno-32-96" href="#__codelineno-32-96"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-32-96" name="__codelineno-32-96" href="#__codelineno-32-96"></a><span class="w">        </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-32-97" name="__codelineno-32-97" href="#__codelineno-32-97"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-32-98" name="__codelineno-32-98" href="#__codelineno-32-98"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-99" name="__codelineno-32-99" href="#__codelineno-32-99"></a><span class="w">            </span><span class="nx">res</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#nums</span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="nx">index</span><span class="p">(</span><span class="nx">j</span><span class="p">)];</span>
@@ -7673,11 +7745,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_deque.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Double-ended queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_deque.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Deque implemented with a circular array */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">ArrayDeque</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[];</span><span class="w"> </span><span class="c1">// Array for storing double-ended queue elements</span>
-<a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">front</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">queSize</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Double-ended queue length</span>
+<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[];</span><span class="w"> </span><span class="c1">// Array for storing deque elements</span>
+<a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">front</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, pointing to the first element</span>
+<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">queSize</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Deque size</span>
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a>
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">capacity</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7686,89 +7758,89 @@
 <a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a>
-<a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">    </span><span class="cm">/* Get the capacity of the double-ended queue */</span>
+<a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">    </span><span class="cm">/* Get the deque capacity */</span>
 <a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="w">    </span><span class="nx">capacity</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-33-17" name="__codelineno-33-17" href="#__codelineno-33-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-18" name="__codelineno-33-18" href="#__codelineno-33-18"></a>
-<a id="__codelineno-33-19" name="__codelineno-33-19" href="#__codelineno-33-19"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-33-19" name="__codelineno-33-19" href="#__codelineno-33-19"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-33-20" name="__codelineno-33-20" href="#__codelineno-33-20"></a><span class="w">    </span><span class="nx">size</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-21" name="__codelineno-33-21" href="#__codelineno-33-21"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="p">;</span>
 <a id="__codelineno-33-22" name="__codelineno-33-22" href="#__codelineno-33-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-23" name="__codelineno-33-23" href="#__codelineno-33-23"></a>
-<a id="__codelineno-33-24" name="__codelineno-33-24" href="#__codelineno-33-24"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-33-24" name="__codelineno-33-24" href="#__codelineno-33-24"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-33-25" name="__codelineno-33-25" href="#__codelineno-33-25"></a><span class="w">    </span><span class="nx">isEmpty</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-26" name="__codelineno-33-26" href="#__codelineno-33-26"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-33-27" name="__codelineno-33-27" href="#__codelineno-33-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-28" name="__codelineno-33-28" href="#__codelineno-33-28"></a>
-<a id="__codelineno-33-29" name="__codelineno-33-29" href="#__codelineno-33-29"></a><span class="w">    </span><span class="cm">/* Calculate circular array index */</span>
+<a id="__codelineno-33-29" name="__codelineno-33-29" href="#__codelineno-33-29"></a><span class="w">    </span><span class="cm">/* Calculate an index in the circular array */</span>
 <a id="__codelineno-33-30" name="__codelineno-33-30" href="#__codelineno-33-30"></a><span class="w">    </span><span class="nx">index</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-31" name="__codelineno-33-31" href="#__codelineno-33-31"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap the array head and tail together</span>
-<a id="__codelineno-33-32" name="__codelineno-33-32" href="#__codelineno-33-32"></a><span class="w">        </span><span class="c1">// When i passes the tail of the array, return to the head</span>
-<a id="__codelineno-33-33" name="__codelineno-33-33" href="#__codelineno-33-33"></a><span class="w">        </span><span class="c1">// When i passes the head of the array, return to the tail</span>
+<a id="__codelineno-33-31" name="__codelineno-33-31" href="#__codelineno-33-31"></a><span class="w">        </span><span class="c1">// Use modulo to connect the end of the array to its beginning</span>
+<a id="__codelineno-33-32" name="__codelineno-33-32" href="#__codelineno-33-32"></a><span class="w">        </span><span class="c1">// When i passes the end of the array, wrap around to the beginning</span>
+<a id="__codelineno-33-33" name="__codelineno-33-33" href="#__codelineno-33-33"></a><span class="w">        </span><span class="c1">// When i passes the beginning of the array, wrap around to the end</span>
 <a id="__codelineno-33-34" name="__codelineno-33-34" href="#__codelineno-33-34"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">())</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">();</span>
 <a id="__codelineno-33-35" name="__codelineno-33-35" href="#__codelineno-33-35"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-36" name="__codelineno-33-36" href="#__codelineno-33-36"></a>
-<a id="__codelineno-33-37" name="__codelineno-33-37" href="#__codelineno-33-37"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-33-37" name="__codelineno-33-37" href="#__codelineno-33-37"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-33-38" name="__codelineno-33-38" href="#__codelineno-33-38"></a><span class="w">    </span><span class="nx">pushFirst</span><span class="p">(</span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-39" name="__codelineno-33-39" href="#__codelineno-33-39"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-40" name="__codelineno-33-40" href="#__codelineno-33-40"></a><span class="w">            </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Double-ended queue is full'</span><span class="p">);</span>
+<a id="__codelineno-33-40" name="__codelineno-33-40" href="#__codelineno-33-40"></a><span class="w">            </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Deque is full'</span><span class="p">);</span>
 <a id="__codelineno-33-41" name="__codelineno-33-41" href="#__codelineno-33-41"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-33-42" name="__codelineno-33-42" href="#__codelineno-33-42"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-33-43" name="__codelineno-33-43" href="#__codelineno-33-43"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap front around to the tail after passing the head of the array</span>
-<a id="__codelineno-33-44" name="__codelineno-33-44" href="#__codelineno-33-44"></a><span class="w">        </span><span class="c1">// Add num to the front of the queue</span>
+<a id="__codelineno-33-43" name="__codelineno-33-43" href="#__codelineno-33-43"></a><span class="w">        </span><span class="c1">// Move the front pointer one position to the left</span>
+<a id="__codelineno-33-44" name="__codelineno-33-44" href="#__codelineno-33-44"></a><span class="w">        </span><span class="c1">// Use modulo to wrap front to the end when it passes the beginning of the array</span>
 <a id="__codelineno-33-45" name="__codelineno-33-45" href="#__codelineno-33-45"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">index</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
-<a id="__codelineno-33-46" name="__codelineno-33-46" href="#__codelineno-33-46"></a><span class="w">        </span><span class="c1">// Add num to front of queue</span>
+<a id="__codelineno-33-46" name="__codelineno-33-46" href="#__codelineno-33-46"></a><span class="w">        </span><span class="c1">// Add num to the front</span>
 <a id="__codelineno-33-47" name="__codelineno-33-47" href="#__codelineno-33-47"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">nums</span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-33-48" name="__codelineno-33-48" href="#__codelineno-33-48"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-33-49" name="__codelineno-33-49" href="#__codelineno-33-49"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-50" name="__codelineno-33-50" href="#__codelineno-33-50"></a>
-<a id="__codelineno-33-51" name="__codelineno-33-51" href="#__codelineno-33-51"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-33-51" name="__codelineno-33-51" href="#__codelineno-33-51"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-33-52" name="__codelineno-33-52" href="#__codelineno-33-52"></a><span class="w">    </span><span class="nx">pushLast</span><span class="p">(</span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-53" name="__codelineno-33-53" href="#__codelineno-33-53"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-54" name="__codelineno-33-54" href="#__codelineno-33-54"></a><span class="w">            </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Double-ended queue is full'</span><span class="p">);</span>
+<a id="__codelineno-33-54" name="__codelineno-33-54" href="#__codelineno-33-54"></a><span class="w">            </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Deque is full'</span><span class="p">);</span>
 <a id="__codelineno-33-55" name="__codelineno-33-55" href="#__codelineno-33-55"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-33-56" name="__codelineno-33-56" href="#__codelineno-33-56"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-33-57" name="__codelineno-33-57" href="#__codelineno-33-57"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
+<a id="__codelineno-33-57" name="__codelineno-33-57" href="#__codelineno-33-57"></a><span class="w">        </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
 <a id="__codelineno-33-58" name="__codelineno-33-58" href="#__codelineno-33-58"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">rear</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">index</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="p">);</span>
-<a id="__codelineno-33-59" name="__codelineno-33-59" href="#__codelineno-33-59"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-33-59" name="__codelineno-33-59" href="#__codelineno-33-59"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-33-60" name="__codelineno-33-60" href="#__codelineno-33-60"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">nums</span><span class="p">[</span><span class="nx">rear</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-33-61" name="__codelineno-33-61" href="#__codelineno-33-61"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-33-62" name="__codelineno-33-62" href="#__codelineno-33-62"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-63" name="__codelineno-33-63" href="#__codelineno-33-63"></a>
-<a id="__codelineno-33-64" name="__codelineno-33-64" href="#__codelineno-33-64"></a><span class="w">    </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-33-64" name="__codelineno-33-64" href="#__codelineno-33-64"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-33-65" name="__codelineno-33-65" href="#__codelineno-33-65"></a><span class="w">    </span><span class="nx">popFirst</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-66" name="__codelineno-33-66" href="#__codelineno-33-66"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">peekFirst</span><span class="p">();</span>
-<a id="__codelineno-33-67" name="__codelineno-33-67" href="#__codelineno-33-67"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position</span>
+<a id="__codelineno-33-67" name="__codelineno-33-67" href="#__codelineno-33-67"></a><span class="w">        </span><span class="c1">// Advance the front pointer by one position</span>
 <a id="__codelineno-33-68" name="__codelineno-33-68" href="#__codelineno-33-68"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">index</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-33-69" name="__codelineno-33-69" href="#__codelineno-33-69"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-33-70" name="__codelineno-33-70" href="#__codelineno-33-70"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-33-71" name="__codelineno-33-71" href="#__codelineno-33-71"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-72" name="__codelineno-33-72" href="#__codelineno-33-72"></a>
-<a id="__codelineno-33-73" name="__codelineno-33-73" href="#__codelineno-33-73"></a><span class="w">    </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-33-73" name="__codelineno-33-73" href="#__codelineno-33-73"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-33-74" name="__codelineno-33-74" href="#__codelineno-33-74"></a><span class="w">    </span><span class="nx">popLast</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-75" name="__codelineno-33-75" href="#__codelineno-33-75"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">peekLast</span><span class="p">();</span>
 <a id="__codelineno-33-76" name="__codelineno-33-76" href="#__codelineno-33-76"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-33-77" name="__codelineno-33-77" href="#__codelineno-33-77"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-33-78" name="__codelineno-33-78" href="#__codelineno-33-78"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-79" name="__codelineno-33-79" href="#__codelineno-33-79"></a>
-<a id="__codelineno-33-80" name="__codelineno-33-80" href="#__codelineno-33-80"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-33-80" name="__codelineno-33-80" href="#__codelineno-33-80"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-33-81" name="__codelineno-33-81" href="#__codelineno-33-81"></a><span class="w">    </span><span class="nx">peekFirst</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-82" name="__codelineno-33-82" href="#__codelineno-33-82"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'The Deque Is Empty.'</span><span class="p">);</span>
+<a id="__codelineno-33-82" name="__codelineno-33-82" href="#__codelineno-33-82"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Deque is empty'</span><span class="p">);</span>
 <a id="__codelineno-33-83" name="__codelineno-33-83" href="#__codelineno-33-83"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">nums</span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="p">];</span>
 <a id="__codelineno-33-84" name="__codelineno-33-84" href="#__codelineno-33-84"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-85" name="__codelineno-33-85" href="#__codelineno-33-85"></a>
-<a id="__codelineno-33-86" name="__codelineno-33-86" href="#__codelineno-33-86"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-33-86" name="__codelineno-33-86" href="#__codelineno-33-86"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-33-87" name="__codelineno-33-87" href="#__codelineno-33-87"></a><span class="w">    </span><span class="nx">peekLast</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-88" name="__codelineno-33-88" href="#__codelineno-33-88"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'The Deque Is Empty.'</span><span class="p">);</span>
-<a id="__codelineno-33-89" name="__codelineno-33-89" href="#__codelineno-33-89"></a><span class="w">        </span><span class="c1">// Initialize double-ended queue</span>
+<a id="__codelineno-33-88" name="__codelineno-33-88" href="#__codelineno-33-88"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Deque is empty'</span><span class="p">);</span>
+<a id="__codelineno-33-89" name="__codelineno-33-89" href="#__codelineno-33-89"></a><span class="w">        </span><span class="c1">// Calculate the index of the last element</span>
 <a id="__codelineno-33-90" name="__codelineno-33-90" href="#__codelineno-33-90"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">last</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">index</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">);</span>
 <a id="__codelineno-33-91" name="__codelineno-33-91" href="#__codelineno-33-91"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">nums</span><span class="p">[</span><span class="nx">last</span><span class="p">];</span>
 <a id="__codelineno-33-92" name="__codelineno-33-92" href="#__codelineno-33-92"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-93" name="__codelineno-33-93" href="#__codelineno-33-93"></a>
-<a id="__codelineno-33-94" name="__codelineno-33-94" href="#__codelineno-33-94"></a><span class="w">    </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-33-94" name="__codelineno-33-94" href="#__codelineno-33-94"></a><span class="w">    </span><span class="cm">/* Return an array for printing */</span>
 <a id="__codelineno-33-95" name="__codelineno-33-95" href="#__codelineno-33-95"></a><span class="w">    </span><span class="nx">toArray</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-96" name="__codelineno-33-96" href="#__codelineno-33-96"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-33-96" name="__codelineno-33-96" href="#__codelineno-33-96"></a><span class="w">        </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-33-97" name="__codelineno-33-97" href="#__codelineno-33-97"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-33-98" name="__codelineno-33-98" href="#__codelineno-33-98"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-99" name="__codelineno-33-99" href="#__codelineno-33-99"></a><span class="w">            </span><span class="nx">res</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">nums</span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="nx">index</span><span class="p">(</span><span class="nx">j</span><span class="p">)];</span>
@@ -7779,11 +7851,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_deque.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Double-ended queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_deque.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Deque implemented with a circular array */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayDeque</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">_nums</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array for storing double-ended queue elements</span>
-<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">_front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">_queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Double-ended queue length</span>
+<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">_nums</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array for storing deque elements</span>
+<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">_front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, pointing to the first element</span>
+<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">_queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Deque size</span>
 <a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a>
 <a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">  </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">  </span><span class="n">ArrayDeque</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7791,71 +7863,71 @@
 <a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">_queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a>
-<a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">  </span><span class="cm">/* Get the capacity of the double-ended queue */</span>
+<a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">  </span><span class="cm">/* Get the deque capacity */</span>
 <a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-17" name="__codelineno-34-17" href="#__codelineno-34-17"></a>
-<a id="__codelineno-34-18" name="__codelineno-34-18" href="#__codelineno-34-18"></a><span class="w">  </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-34-18" name="__codelineno-34-18" href="#__codelineno-34-18"></a><span class="w">  </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-34-19" name="__codelineno-34-19" href="#__codelineno-34-19"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-20" name="__codelineno-34-20" href="#__codelineno-34-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_queSize</span><span class="p">;</span>
 <a id="__codelineno-34-21" name="__codelineno-34-21" href="#__codelineno-34-21"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-22" name="__codelineno-34-22" href="#__codelineno-34-22"></a>
-<a id="__codelineno-34-23" name="__codelineno-34-23" href="#__codelineno-34-23"></a><span class="w">  </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-34-23" name="__codelineno-34-23" href="#__codelineno-34-23"></a><span class="w">  </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-34-24" name="__codelineno-34-24" href="#__codelineno-34-24"></a><span class="w">  </span><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-25" name="__codelineno-34-25" href="#__codelineno-34-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-34-26" name="__codelineno-34-26" href="#__codelineno-34-26"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-27" name="__codelineno-34-27" href="#__codelineno-34-27"></a>
-<a id="__codelineno-34-28" name="__codelineno-34-28" href="#__codelineno-34-28"></a><span class="w">  </span><span class="cm">/* Calculate circular array index */</span>
+<a id="__codelineno-34-28" name="__codelineno-34-28" href="#__codelineno-34-28"></a><span class="w">  </span><span class="cm">/* Calculate an index in the circular array */</span>
 <a id="__codelineno-34-29" name="__codelineno-34-29" href="#__codelineno-34-29"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-30" name="__codelineno-34-30" href="#__codelineno-34-30"></a><span class="w">    </span><span class="c1">// Use modulo operation to wrap the array head and tail together</span>
-<a id="__codelineno-34-31" name="__codelineno-34-31" href="#__codelineno-34-31"></a><span class="w">    </span><span class="c1">// When i passes the tail of the array, return to the head</span>
-<a id="__codelineno-34-32" name="__codelineno-34-32" href="#__codelineno-34-32"></a><span class="w">    </span><span class="c1">// When i passes the head of the array, return to the tail</span>
+<a id="__codelineno-34-30" name="__codelineno-34-30" href="#__codelineno-34-30"></a><span class="w">    </span><span class="c1">// Use modulo to connect the end of the array to its beginning</span>
+<a id="__codelineno-34-31" name="__codelineno-34-31" href="#__codelineno-34-31"></a><span class="w">    </span><span class="c1">// When i passes the end of the array, wrap around to the beginning</span>
+<a id="__codelineno-34-32" name="__codelineno-34-32" href="#__codelineno-34-32"></a><span class="w">    </span><span class="c1">// When i passes the beginning of the array, wrap around to the end</span>
 <a id="__codelineno-34-33" name="__codelineno-34-33" href="#__codelineno-34-33"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">();</span>
 <a id="__codelineno-34-34" name="__codelineno-34-34" href="#__codelineno-34-34"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-35" name="__codelineno-34-35" href="#__codelineno-34-35"></a>
-<a id="__codelineno-34-36" name="__codelineno-34-36" href="#__codelineno-34-36"></a><span class="w">  </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-34-36" name="__codelineno-34-36" href="#__codelineno-34-36"></a><span class="w">  </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-34-37" name="__codelineno-34-37" href="#__codelineno-34-37"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">pushFirst</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-38" name="__codelineno-34-38" href="#__codelineno-34-38"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-39" name="__codelineno-34-39" href="#__codelineno-34-39"></a><span class="w">      </span><span class="k">throw</span><span class="w"> </span><span class="n">Exception</span><span class="p">(</span><span class="s2">"Double-ended queue is full"</span><span class="p">);</span>
+<a id="__codelineno-34-39" name="__codelineno-34-39" href="#__codelineno-34-39"></a><span class="w">      </span><span class="k">throw</span><span class="w"> </span><span class="n">Exception</span><span class="p">(</span><span class="s2">"Deque is full"</span><span class="p">);</span>
 <a id="__codelineno-34-40" name="__codelineno-34-40" href="#__codelineno-34-40"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-34-41" name="__codelineno-34-41" href="#__codelineno-34-41"></a><span class="w">    </span><span class="c1">// Use modulo operation to wrap front around to the tail after passing the head of the array</span>
-<a id="__codelineno-34-42" name="__codelineno-34-42" href="#__codelineno-34-42"></a><span class="w">    </span><span class="c1">// Use modulo operation to wrap _front from array head back to tail</span>
+<a id="__codelineno-34-41" name="__codelineno-34-41" href="#__codelineno-34-41"></a><span class="w">    </span><span class="c1">// Move the front pointer one position to the left</span>
+<a id="__codelineno-34-42" name="__codelineno-34-42" href="#__codelineno-34-42"></a><span class="w">    </span><span class="c1">// Use modulo to wrap _front to the end when it passes the beginning of the array</span>
 <a id="__codelineno-34-43" name="__codelineno-34-43" href="#__codelineno-34-43"></a><span class="w">    </span><span class="n">_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">_front</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
-<a id="__codelineno-34-44" name="__codelineno-34-44" href="#__codelineno-34-44"></a><span class="w">    </span><span class="c1">// Add _num to queue front</span>
+<a id="__codelineno-34-44" name="__codelineno-34-44" href="#__codelineno-34-44"></a><span class="w">    </span><span class="c1">// Add _num to the front</span>
 <a id="__codelineno-34-45" name="__codelineno-34-45" href="#__codelineno-34-45"></a><span class="w">    </span><span class="n">_nums</span><span class="p">[</span><span class="n">_front</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span>
 <a id="__codelineno-34-46" name="__codelineno-34-46" href="#__codelineno-34-46"></a><span class="w">    </span><span class="n">_queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-34-47" name="__codelineno-34-47" href="#__codelineno-34-47"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-48" name="__codelineno-34-48" href="#__codelineno-34-48"></a>
-<a id="__codelineno-34-49" name="__codelineno-34-49" href="#__codelineno-34-49"></a><span class="w">  </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-34-49" name="__codelineno-34-49" href="#__codelineno-34-49"></a><span class="w">  </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-34-50" name="__codelineno-34-50" href="#__codelineno-34-50"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">pushLast</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-51" name="__codelineno-34-51" href="#__codelineno-34-51"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-52" name="__codelineno-34-52" href="#__codelineno-34-52"></a><span class="w">      </span><span class="k">throw</span><span class="w"> </span><span class="n">Exception</span><span class="p">(</span><span class="s2">"Double-ended queue is full"</span><span class="p">);</span>
+<a id="__codelineno-34-52" name="__codelineno-34-52" href="#__codelineno-34-52"></a><span class="w">      </span><span class="k">throw</span><span class="w"> </span><span class="n">Exception</span><span class="p">(</span><span class="s2">"Deque is full"</span><span class="p">);</span>
 <a id="__codelineno-34-53" name="__codelineno-34-53" href="#__codelineno-34-53"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-34-54" name="__codelineno-34-54" href="#__codelineno-34-54"></a><span class="w">    </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
+<a id="__codelineno-34-54" name="__codelineno-34-54" href="#__codelineno-34-54"></a><span class="w">    </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
 <a id="__codelineno-34-55" name="__codelineno-34-55" href="#__codelineno-34-55"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">_front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">_queSize</span><span class="p">);</span>
-<a id="__codelineno-34-56" name="__codelineno-34-56" href="#__codelineno-34-56"></a><span class="w">    </span><span class="c1">// Add _num to queue rear</span>
+<a id="__codelineno-34-56" name="__codelineno-34-56" href="#__codelineno-34-56"></a><span class="w">    </span><span class="c1">// Add _num to the rear</span>
 <a id="__codelineno-34-57" name="__codelineno-34-57" href="#__codelineno-34-57"></a><span class="w">    </span><span class="n">_nums</span><span class="p">[</span><span class="n">rear</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span>
 <a id="__codelineno-34-58" name="__codelineno-34-58" href="#__codelineno-34-58"></a><span class="w">    </span><span class="n">_queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-34-59" name="__codelineno-34-59" href="#__codelineno-34-59"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-60" name="__codelineno-34-60" href="#__codelineno-34-60"></a>
-<a id="__codelineno-34-61" name="__codelineno-34-61" href="#__codelineno-34-61"></a><span class="w">  </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-34-61" name="__codelineno-34-61" href="#__codelineno-34-61"></a><span class="w">  </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-34-62" name="__codelineno-34-62" href="#__codelineno-34-62"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">popFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-63" name="__codelineno-34-63" href="#__codelineno-34-63"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peekFirst</span><span class="p">();</span>
-<a id="__codelineno-34-64" name="__codelineno-34-64" href="#__codelineno-34-64"></a><span class="w">    </span><span class="c1">// Move front pointer right by one</span>
+<a id="__codelineno-34-64" name="__codelineno-34-64" href="#__codelineno-34-64"></a><span class="w">    </span><span class="c1">// Move the front pointer one position to the right</span>
 <a id="__codelineno-34-65" name="__codelineno-34-65" href="#__codelineno-34-65"></a><span class="w">    </span><span class="n">_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">_front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-34-66" name="__codelineno-34-66" href="#__codelineno-34-66"></a><span class="w">    </span><span class="n">_queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-34-67" name="__codelineno-34-67" href="#__codelineno-34-67"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span>
 <a id="__codelineno-34-68" name="__codelineno-34-68" href="#__codelineno-34-68"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-69" name="__codelineno-34-69" href="#__codelineno-34-69"></a>
-<a id="__codelineno-34-70" name="__codelineno-34-70" href="#__codelineno-34-70"></a><span class="w">  </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-34-70" name="__codelineno-34-70" href="#__codelineno-34-70"></a><span class="w">  </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-34-71" name="__codelineno-34-71" href="#__codelineno-34-71"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">popLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-72" name="__codelineno-34-72" href="#__codelineno-34-72"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peekLast</span><span class="p">();</span>
 <a id="__codelineno-34-73" name="__codelineno-34-73" href="#__codelineno-34-73"></a><span class="w">    </span><span class="n">_queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-34-74" name="__codelineno-34-74" href="#__codelineno-34-74"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span>
 <a id="__codelineno-34-75" name="__codelineno-34-75" href="#__codelineno-34-75"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-76" name="__codelineno-34-76" href="#__codelineno-34-76"></a>
-<a id="__codelineno-34-77" name="__codelineno-34-77" href="#__codelineno-34-77"></a><span class="w">  </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-34-77" name="__codelineno-34-77" href="#__codelineno-34-77"></a><span class="w">  </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-34-78" name="__codelineno-34-78" href="#__codelineno-34-78"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">peekFirst</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-79" name="__codelineno-34-79" href="#__codelineno-34-79"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-80" name="__codelineno-34-80" href="#__codelineno-34-80"></a><span class="w">      </span><span class="k">throw</span><span class="w"> </span><span class="n">Exception</span><span class="p">(</span><span class="s2">"Deque is empty"</span><span class="p">);</span>
@@ -7863,19 +7935,19 @@
 <a id="__codelineno-34-82" name="__codelineno-34-82" href="#__codelineno-34-82"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_nums</span><span class="p">[</span><span class="n">_front</span><span class="p">];</span>
 <a id="__codelineno-34-83" name="__codelineno-34-83" href="#__codelineno-34-83"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-84" name="__codelineno-34-84" href="#__codelineno-34-84"></a>
-<a id="__codelineno-34-85" name="__codelineno-34-85" href="#__codelineno-34-85"></a><span class="w">  </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-34-85" name="__codelineno-34-85" href="#__codelineno-34-85"></a><span class="w">  </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-34-86" name="__codelineno-34-86" href="#__codelineno-34-86"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">peekLast</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-87" name="__codelineno-34-87" href="#__codelineno-34-87"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-88" name="__codelineno-34-88" href="#__codelineno-34-88"></a><span class="w">      </span><span class="k">throw</span><span class="w"> </span><span class="n">Exception</span><span class="p">(</span><span class="s2">"Deque is empty"</span><span class="p">);</span>
 <a id="__codelineno-34-89" name="__codelineno-34-89" href="#__codelineno-34-89"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-34-90" name="__codelineno-34-90" href="#__codelineno-34-90"></a><span class="w">    </span><span class="c1">// Initialize double-ended queue</span>
+<a id="__codelineno-34-90" name="__codelineno-34-90" href="#__codelineno-34-90"></a><span class="w">    </span><span class="c1">// Calculate the index of the last element</span>
 <a id="__codelineno-34-91" name="__codelineno-34-91" href="#__codelineno-34-91"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">last</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">_front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">_queSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-34-92" name="__codelineno-34-92" href="#__codelineno-34-92"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_nums</span><span class="p">[</span><span class="n">last</span><span class="p">];</span>
 <a id="__codelineno-34-93" name="__codelineno-34-93" href="#__codelineno-34-93"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-94" name="__codelineno-34-94" href="#__codelineno-34-94"></a>
-<a id="__codelineno-34-95" name="__codelineno-34-95" href="#__codelineno-34-95"></a><span class="w">  </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-34-95" name="__codelineno-34-95" href="#__codelineno-34-95"></a><span class="w">  </span><span class="cm">/* Return an array for printing */</span>
 <a id="__codelineno-34-96" name="__codelineno-34-96" href="#__codelineno-34-96"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-97" name="__codelineno-34-97" href="#__codelineno-34-97"></a><span class="w">    </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-34-97" name="__codelineno-34-97" href="#__codelineno-34-97"></a><span class="w">    </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-34-98" name="__codelineno-34-98" href="#__codelineno-34-98"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">_queSize</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-34-99" name="__codelineno-34-99" href="#__codelineno-34-99"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_front</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">_queSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-100" name="__codelineno-34-100" href="#__codelineno-34-100"></a><span class="w">      </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_nums</span><span class="p">[</span><span class="n">index</span><span class="p">(</span><span class="n">j</span><span class="p">)];</span>
@@ -7886,11 +7958,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_deque.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Double-ended queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_deque.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Deque implemented with a circular array */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">struct</span><span class="w"> </span><span class="nc">ArrayDeque</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="p">,</span><span class="w">    </span><span class="c1">// Array for storing double-ended queue elements</span>
-<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="n">front</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w">    </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">    </span><span class="n">que_size</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="c1">// Double-ended queue length</span>
+<a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="p">,</span><span class="w">    </span><span class="c1">// Array for storing deque elements</span>
+<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="n">front</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w">    </span><span class="c1">// Front index, identifying the first element</span>
+<a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">    </span><span class="n">que_size</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w"> </span><span class="c1">// Deque size</span>
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="p">}</span>
 <a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a>
 <a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="k">impl</span><span class="o">&lt;</span><span class="n">T</span><span class="p">:</span><span class="w"> </span><span class="nb">Copy</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nb">Default</span><span class="o">&gt;</span><span class="w"> </span><span class="n">ArrayDeque</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
@@ -7903,73 +7975,73 @@
 <a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-35-16" name="__codelineno-35-16" href="#__codelineno-35-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-17" name="__codelineno-35-17" href="#__codelineno-35-17"></a>
-<a id="__codelineno-35-18" name="__codelineno-35-18" href="#__codelineno-35-18"></a><span class="w">    </span><span class="cm">/* Get the capacity of the double-ended queue */</span>
+<a id="__codelineno-35-18" name="__codelineno-35-18" href="#__codelineno-35-18"></a><span class="w">    </span><span class="cm">/* Get the deque capacity */</span>
 <a id="__codelineno-35-19" name="__codelineno-35-19" href="#__codelineno-35-19"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">capacity</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-20" name="__codelineno-35-20" href="#__codelineno-35-20"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">nums</span><span class="p">.</span><span class="n">len</span><span class="p">()</span>
 <a id="__codelineno-35-21" name="__codelineno-35-21" href="#__codelineno-35-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-22" name="__codelineno-35-22" href="#__codelineno-35-22"></a>
-<a id="__codelineno-35-23" name="__codelineno-35-23" href="#__codelineno-35-23"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-35-23" name="__codelineno-35-23" href="#__codelineno-35-23"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-35-24" name="__codelineno-35-24" href="#__codelineno-35-24"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-25" name="__codelineno-35-25" href="#__codelineno-35-25"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span>
 <a id="__codelineno-35-26" name="__codelineno-35-26" href="#__codelineno-35-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-27" name="__codelineno-35-27" href="#__codelineno-35-27"></a>
-<a id="__codelineno-35-28" name="__codelineno-35-28" href="#__codelineno-35-28"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-35-28" name="__codelineno-35-28" href="#__codelineno-35-28"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-35-29" name="__codelineno-35-29" href="#__codelineno-35-29"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">is_empty</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-30" name="__codelineno-35-30" href="#__codelineno-35-30"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-35-31" name="__codelineno-35-31" href="#__codelineno-35-31"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-32" name="__codelineno-35-32" href="#__codelineno-35-32"></a>
-<a id="__codelineno-35-33" name="__codelineno-35-33" href="#__codelineno-35-33"></a><span class="w">    </span><span class="cm">/* Calculate circular array index */</span>
+<a id="__codelineno-35-33" name="__codelineno-35-33" href="#__codelineno-35-33"></a><span class="w">    </span><span class="cm">/* Calculate an index in the circular array */</span>
 <a id="__codelineno-35-34" name="__codelineno-35-34" href="#__codelineno-35-34"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">index</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-35" name="__codelineno-35-35" href="#__codelineno-35-35"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap the array head and tail together</span>
-<a id="__codelineno-35-36" name="__codelineno-35-36" href="#__codelineno-35-36"></a><span class="w">        </span><span class="c1">// When i passes the tail of the array, return to the head</span>
-<a id="__codelineno-35-37" name="__codelineno-35-37" href="#__codelineno-35-37"></a><span class="w">        </span><span class="c1">// When i passes the head of the array, return to the tail</span>
+<a id="__codelineno-35-35" name="__codelineno-35-35" href="#__codelineno-35-35"></a><span class="w">        </span><span class="c1">// Use modulo to connect the end of the array to its beginning</span>
+<a id="__codelineno-35-36" name="__codelineno-35-36" href="#__codelineno-35-36"></a><span class="w">        </span><span class="c1">// When i passes the end of the array, wrap around to the beginning</span>
+<a id="__codelineno-35-37" name="__codelineno-35-37" href="#__codelineno-35-37"></a><span class="w">        </span><span class="c1">// When i passes the beginning of the array, wrap around to the end</span>
 <a id="__codelineno-35-38" name="__codelineno-35-38" href="#__codelineno-35-38"></a><span class="w">        </span><span class="p">((</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span>
 <a id="__codelineno-35-39" name="__codelineno-35-39" href="#__codelineno-35-39"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-40" name="__codelineno-35-40" href="#__codelineno-35-40"></a>
-<a id="__codelineno-35-41" name="__codelineno-35-41" href="#__codelineno-35-41"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-35-41" name="__codelineno-35-41" href="#__codelineno-35-41"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-35-42" name="__codelineno-35-42" href="#__codelineno-35-42"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">push_first</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nc">T</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-43" name="__codelineno-35-43" href="#__codelineno-35-43"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-44" name="__codelineno-35-44" href="#__codelineno-35-44"></a><span class="w">            </span><span class="fm">println!</span><span class="p">(</span><span class="s">"Double-ended queue is full"</span><span class="p">);</span>
+<a id="__codelineno-35-44" name="__codelineno-35-44" href="#__codelineno-35-44"></a><span class="w">            </span><span class="fm">println!</span><span class="p">(</span><span class="s">"Deque is full"</span><span class="p">);</span>
 <a id="__codelineno-35-45" name="__codelineno-35-45" href="#__codelineno-35-45"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-35-46" name="__codelineno-35-46" href="#__codelineno-35-46"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-35-47" name="__codelineno-35-47" href="#__codelineno-35-47"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap front around to the tail after passing the head of the array</span>
-<a id="__codelineno-35-48" name="__codelineno-35-48" href="#__codelineno-35-48"></a><span class="w">        </span><span class="c1">// Add num to the front of the queue</span>
+<a id="__codelineno-35-47" name="__codelineno-35-47" href="#__codelineno-35-47"></a><span class="w">        </span><span class="c1">// Move the front index one position to the left</span>
+<a id="__codelineno-35-48" name="__codelineno-35-48" href="#__codelineno-35-48"></a><span class="w">        </span><span class="c1">// Use modulo to wrap front to the end when it passes the beginning of the array</span>
 <a id="__codelineno-35-49" name="__codelineno-35-49" href="#__codelineno-35-49"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">index</span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-35-50" name="__codelineno-35-50" href="#__codelineno-35-50"></a><span class="w">        </span><span class="c1">// Add num to front of queue</span>
+<a id="__codelineno-35-50" name="__codelineno-35-50" href="#__codelineno-35-50"></a><span class="w">        </span><span class="c1">// Add num to the front</span>
 <a id="__codelineno-35-51" name="__codelineno-35-51" href="#__codelineno-35-51"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">nums</span><span class="p">[</span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-35-52" name="__codelineno-35-52" href="#__codelineno-35-52"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-35-53" name="__codelineno-35-53" href="#__codelineno-35-53"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-54" name="__codelineno-35-54" href="#__codelineno-35-54"></a>
-<a id="__codelineno-35-55" name="__codelineno-35-55" href="#__codelineno-35-55"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-35-55" name="__codelineno-35-55" href="#__codelineno-35-55"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-35-56" name="__codelineno-35-56" href="#__codelineno-35-56"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">push_last</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nc">T</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-57" name="__codelineno-35-57" href="#__codelineno-35-57"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-58" name="__codelineno-35-58" href="#__codelineno-35-58"></a><span class="w">            </span><span class="fm">println!</span><span class="p">(</span><span class="s">"Double-ended queue is full"</span><span class="p">);</span>
+<a id="__codelineno-35-58" name="__codelineno-35-58" href="#__codelineno-35-58"></a><span class="w">            </span><span class="fm">println!</span><span class="p">(</span><span class="s">"Deque is full"</span><span class="p">);</span>
 <a id="__codelineno-35-59" name="__codelineno-35-59" href="#__codelineno-35-59"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-35-60" name="__codelineno-35-60" href="#__codelineno-35-60"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-35-61" name="__codelineno-35-61" href="#__codelineno-35-61"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
+<a id="__codelineno-35-61" name="__codelineno-35-61" href="#__codelineno-35-61"></a><span class="w">        </span><span class="c1">// Calculate the rear index, one position past the last element</span>
 <a id="__codelineno-35-62" name="__codelineno-35-62" href="#__codelineno-35-62"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">index</span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="p">);</span>
-<a id="__codelineno-35-63" name="__codelineno-35-63" href="#__codelineno-35-63"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-35-63" name="__codelineno-35-63" href="#__codelineno-35-63"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-35-64" name="__codelineno-35-64" href="#__codelineno-35-64"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">nums</span><span class="p">[</span><span class="n">rear</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-35-65" name="__codelineno-35-65" href="#__codelineno-35-65"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-35-66" name="__codelineno-35-66" href="#__codelineno-35-66"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-67" name="__codelineno-35-67" href="#__codelineno-35-67"></a>
-<a id="__codelineno-35-68" name="__codelineno-35-68" href="#__codelineno-35-68"></a><span class="w">    </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-35-68" name="__codelineno-35-68" href="#__codelineno-35-68"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-35-69" name="__codelineno-35-69" href="#__codelineno-35-69"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">pop_first</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">T</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-70" name="__codelineno-35-70" href="#__codelineno-35-70"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">peek_first</span><span class="p">();</span>
-<a id="__codelineno-35-71" name="__codelineno-35-71" href="#__codelineno-35-71"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position</span>
+<a id="__codelineno-35-71" name="__codelineno-35-71" href="#__codelineno-35-71"></a><span class="w">        </span><span class="c1">// Advance the front index by one position</span>
 <a id="__codelineno-35-72" name="__codelineno-35-72" href="#__codelineno-35-72"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">index</span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-35-73" name="__codelineno-35-73" href="#__codelineno-35-73"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-35-74" name="__codelineno-35-74" href="#__codelineno-35-74"></a><span class="w">        </span><span class="n">num</span>
 <a id="__codelineno-35-75" name="__codelineno-35-75" href="#__codelineno-35-75"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-76" name="__codelineno-35-76" href="#__codelineno-35-76"></a>
-<a id="__codelineno-35-77" name="__codelineno-35-77" href="#__codelineno-35-77"></a><span class="w">    </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-35-77" name="__codelineno-35-77" href="#__codelineno-35-77"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-35-78" name="__codelineno-35-78" href="#__codelineno-35-78"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">pop_last</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">T</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-79" name="__codelineno-35-79" href="#__codelineno-35-79"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">peek_last</span><span class="p">();</span>
 <a id="__codelineno-35-80" name="__codelineno-35-80" href="#__codelineno-35-80"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-35-81" name="__codelineno-35-81" href="#__codelineno-35-81"></a><span class="w">        </span><span class="n">num</span>
 <a id="__codelineno-35-82" name="__codelineno-35-82" href="#__codelineno-35-82"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-83" name="__codelineno-35-83" href="#__codelineno-35-83"></a>
-<a id="__codelineno-35-84" name="__codelineno-35-84" href="#__codelineno-35-84"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-35-84" name="__codelineno-35-84" href="#__codelineno-35-84"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-35-85" name="__codelineno-35-85" href="#__codelineno-35-85"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">peek_first</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">T</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-86" name="__codelineno-35-86" href="#__codelineno-35-86"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">is_empty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-87" name="__codelineno-35-87" href="#__codelineno-35-87"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"Deque is empty"</span><span class="p">)</span>
@@ -7977,19 +8049,19 @@
 <a id="__codelineno-35-89" name="__codelineno-35-89" href="#__codelineno-35-89"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">nums</span><span class="p">[</span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="p">]</span>
 <a id="__codelineno-35-90" name="__codelineno-35-90" href="#__codelineno-35-90"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-91" name="__codelineno-35-91" href="#__codelineno-35-91"></a>
-<a id="__codelineno-35-92" name="__codelineno-35-92" href="#__codelineno-35-92"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-35-92" name="__codelineno-35-92" href="#__codelineno-35-92"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-35-93" name="__codelineno-35-93" href="#__codelineno-35-93"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">peek_last</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">T</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-94" name="__codelineno-35-94" href="#__codelineno-35-94"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">is_empty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-95" name="__codelineno-35-95" href="#__codelineno-35-95"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"Deque is empty"</span><span class="p">)</span>
 <a id="__codelineno-35-96" name="__codelineno-35-96" href="#__codelineno-35-96"></a><span class="w">        </span><span class="p">};</span>
-<a id="__codelineno-35-97" name="__codelineno-35-97" href="#__codelineno-35-97"></a><span class="w">        </span><span class="c1">// Initialize double-ended queue</span>
+<a id="__codelineno-35-97" name="__codelineno-35-97" href="#__codelineno-35-97"></a><span class="w">        </span><span class="c1">// Calculate the index of the last element</span>
 <a id="__codelineno-35-98" name="__codelineno-35-98" href="#__codelineno-35-98"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">last</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">index</span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-35-99" name="__codelineno-35-99" href="#__codelineno-35-99"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">nums</span><span class="p">[</span><span class="n">last</span><span class="p">]</span>
 <a id="__codelineno-35-100" name="__codelineno-35-100" href="#__codelineno-35-100"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-101" name="__codelineno-35-101" href="#__codelineno-35-101"></a>
-<a id="__codelineno-35-102" name="__codelineno-35-102" href="#__codelineno-35-102"></a><span class="w">    </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-35-102" name="__codelineno-35-102" href="#__codelineno-35-102"></a><span class="w">    </span><span class="cm">/* Return a vector for printing */</span>
 <a id="__codelineno-35-103" name="__codelineno-35-103" href="#__codelineno-35-103"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">to_array</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-104" name="__codelineno-35-104" href="#__codelineno-35-104"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-35-104" name="__codelineno-35-104" href="#__codelineno-35-104"></a><span class="w">        </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-35-105" name="__codelineno-35-105" href="#__codelineno-35-105"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="fm">vec!</span><span class="p">[</span><span class="n">T</span><span class="p">::</span><span class="n">default</span><span class="p">();</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="p">];</span>
 <a id="__codelineno-35-106" name="__codelineno-35-106" href="#__codelineno-35-106"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="p">;</span>
 <a id="__codelineno-35-107" name="__codelineno-35-107" href="#__codelineno-35-107"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="p">{</span>
@@ -8002,18 +8074,18 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_deque.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Double-ended queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_deque.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Deque implemented with a circular array */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">;</span><span class="w">       </span><span class="c1">// Array for storing queue elements</span>
-<a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w">       </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w">     </span><span class="c1">// Rear pointer, points to rear + 1</span>
+<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">;</span><span class="w">       </span><span class="c1">// Array that stores the queue elements</span>
+<a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w">       </span><span class="c1">// Front pointer, pointing to the first element</span>
+<a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w">     </span><span class="c1">// Current number of elements in the deque</span>
 <a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queCapacity</span><span class="p">;</span><span class="w"> </span><span class="c1">// Queue capacity</span>
 <a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="p">}</span><span class="w"> </span><span class="n">ArrayDeque</span><span class="p">;</span>
 <a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a>
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="cm">/* Constructor */</span>
 <a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="n">ArrayDeque</span><span class="w"> </span><span class="o">*</span><span class="nf">newArrayDeque</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="w">    </span><span class="n">ArrayDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">ArrayDeque</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">ArrayDeque</span><span class="p">));</span>
-<a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="w">    </span><span class="c1">// Initialize array</span>
+<a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="w">    </span><span class="c1">// Initialize the array</span>
 <a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">    </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queCapacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-36-14" name="__codelineno-36-14" href="#__codelineno-36-14"></a><span class="w">    </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queCapacity</span><span class="p">);</span>
 <a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a><span class="w">    </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
@@ -8026,88 +8098,88 @@
 <a id="__codelineno-36-22" name="__codelineno-36-22" href="#__codelineno-36-22"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">deque</span><span class="p">);</span>
 <a id="__codelineno-36-23" name="__codelineno-36-23" href="#__codelineno-36-23"></a><span class="p">}</span>
 <a id="__codelineno-36-24" name="__codelineno-36-24" href="#__codelineno-36-24"></a>
-<a id="__codelineno-36-25" name="__codelineno-36-25" href="#__codelineno-36-25"></a><span class="cm">/* Get the capacity of the double-ended queue */</span>
+<a id="__codelineno-36-25" name="__codelineno-36-25" href="#__codelineno-36-25"></a><span class="cm">/* Get the deque capacity */</span>
 <a id="__codelineno-36-26" name="__codelineno-36-26" href="#__codelineno-36-26"></a><span class="kt">int</span><span class="w"> </span><span class="nf">capacity</span><span class="p">(</span><span class="n">ArrayDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-27" name="__codelineno-36-27" href="#__codelineno-36-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queCapacity</span><span class="p">;</span>
 <a id="__codelineno-36-28" name="__codelineno-36-28" href="#__codelineno-36-28"></a><span class="p">}</span>
 <a id="__codelineno-36-29" name="__codelineno-36-29" href="#__codelineno-36-29"></a>
-<a id="__codelineno-36-30" name="__codelineno-36-30" href="#__codelineno-36-30"></a><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-36-30" name="__codelineno-36-30" href="#__codelineno-36-30"></a><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-36-31" name="__codelineno-36-31" href="#__codelineno-36-31"></a><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="n">ArrayDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-32" name="__codelineno-36-32" href="#__codelineno-36-32"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-36-33" name="__codelineno-36-33" href="#__codelineno-36-33"></a><span class="p">}</span>
 <a id="__codelineno-36-34" name="__codelineno-36-34" href="#__codelineno-36-34"></a>
-<a id="__codelineno-36-35" name="__codelineno-36-35" href="#__codelineno-36-35"></a><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-36-35" name="__codelineno-36-35" href="#__codelineno-36-35"></a><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-36-36" name="__codelineno-36-36" href="#__codelineno-36-36"></a><span class="kt">bool</span><span class="w"> </span><span class="nf">empty</span><span class="p">(</span><span class="n">ArrayDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-37" name="__codelineno-36-37" href="#__codelineno-36-37"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-36-38" name="__codelineno-36-38" href="#__codelineno-36-38"></a><span class="p">}</span>
 <a id="__codelineno-36-39" name="__codelineno-36-39" href="#__codelineno-36-39"></a>
-<a id="__codelineno-36-40" name="__codelineno-36-40" href="#__codelineno-36-40"></a><span class="cm">/* Calculate circular array index */</span>
+<a id="__codelineno-36-40" name="__codelineno-36-40" href="#__codelineno-36-40"></a><span class="cm">/* Calculate an index in the circular array */</span>
 <a id="__codelineno-36-41" name="__codelineno-36-41" href="#__codelineno-36-41"></a><span class="kt">int</span><span class="w"> </span><span class="nf">dequeIndex</span><span class="p">(</span><span class="n">ArrayDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-42" name="__codelineno-36-42" href="#__codelineno-36-42"></a><span class="w">    </span><span class="c1">// Use modulo operation to wrap the array head and tail together</span>
-<a id="__codelineno-36-43" name="__codelineno-36-43" href="#__codelineno-36-43"></a><span class="w">    </span><span class="c1">// When i exceeds array end, wrap to head</span>
-<a id="__codelineno-36-44" name="__codelineno-36-44" href="#__codelineno-36-44"></a><span class="w">    </span><span class="c1">// When i passes the head of the array, return to the tail</span>
+<a id="__codelineno-36-42" name="__codelineno-36-42" href="#__codelineno-36-42"></a><span class="w">    </span><span class="c1">// Use modulo to connect the end of the array to its beginning</span>
+<a id="__codelineno-36-43" name="__codelineno-36-43" href="#__codelineno-36-43"></a><span class="w">    </span><span class="c1">// When i passes the end of the array, wrap to the beginning</span>
+<a id="__codelineno-36-44" name="__codelineno-36-44" href="#__codelineno-36-44"></a><span class="w">    </span><span class="c1">// When i passes the beginning of the array, wrap around to the end</span>
 <a id="__codelineno-36-45" name="__codelineno-36-45" href="#__codelineno-36-45"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">((</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">capacity</span><span class="p">(</span><span class="n">deque</span><span class="p">))</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">(</span><span class="n">deque</span><span class="p">));</span>
 <a id="__codelineno-36-46" name="__codelineno-36-46" href="#__codelineno-36-46"></a><span class="p">}</span>
 <a id="__codelineno-36-47" name="__codelineno-36-47" href="#__codelineno-36-47"></a>
-<a id="__codelineno-36-48" name="__codelineno-36-48" href="#__codelineno-36-48"></a><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-36-48" name="__codelineno-36-48" href="#__codelineno-36-48"></a><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-36-49" name="__codelineno-36-49" href="#__codelineno-36-49"></a><span class="kt">void</span><span class="w"> </span><span class="nf">pushFirst</span><span class="p">(</span><span class="n">ArrayDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-50" name="__codelineno-36-50" href="#__codelineno-36-50"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">(</span><span class="n">deque</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-51" name="__codelineno-36-51" href="#__codelineno-36-51"></a><span class="w">        </span><span class="n">printf</span><span class="p">(</span><span class="s">"Deque is full</span><span class="se">\r\n</span><span class="s">"</span><span class="p">);</span>
 <a id="__codelineno-36-52" name="__codelineno-36-52" href="#__codelineno-36-52"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-36-53" name="__codelineno-36-53" href="#__codelineno-36-53"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-36-54" name="__codelineno-36-54" href="#__codelineno-36-54"></a><span class="w">    </span><span class="c1">// Use modulo operation to wrap front around to the tail after passing the head of the array</span>
-<a id="__codelineno-36-55" name="__codelineno-36-55" href="#__codelineno-36-55"></a><span class="w">    </span><span class="c1">// Use modulo to wrap front from array head to rear</span>
+<a id="__codelineno-36-54" name="__codelineno-36-54" href="#__codelineno-36-54"></a><span class="w">    </span><span class="c1">// Move the front pointer one position to the left</span>
+<a id="__codelineno-36-55" name="__codelineno-36-55" href="#__codelineno-36-55"></a><span class="w">    </span><span class="c1">// Use modulo to wrap front to the end when it passes the beginning</span>
 <a id="__codelineno-36-56" name="__codelineno-36-56" href="#__codelineno-36-56"></a><span class="w">    </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dequeIndex</span><span class="p">(</span><span class="n">deque</span><span class="p">,</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
-<a id="__codelineno-36-57" name="__codelineno-36-57" href="#__codelineno-36-57"></a><span class="w">    </span><span class="c1">// Add num to queue front</span>
+<a id="__codelineno-36-57" name="__codelineno-36-57" href="#__codelineno-36-57"></a><span class="w">    </span><span class="c1">// Add num to the front</span>
 <a id="__codelineno-36-58" name="__codelineno-36-58" href="#__codelineno-36-58"></a><span class="w">    </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">nums</span><span class="p">[</span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-36-59" name="__codelineno-36-59" href="#__codelineno-36-59"></a><span class="w">    </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-36-60" name="__codelineno-36-60" href="#__codelineno-36-60"></a><span class="p">}</span>
 <a id="__codelineno-36-61" name="__codelineno-36-61" href="#__codelineno-36-61"></a>
-<a id="__codelineno-36-62" name="__codelineno-36-62" href="#__codelineno-36-62"></a><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-36-62" name="__codelineno-36-62" href="#__codelineno-36-62"></a><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-36-63" name="__codelineno-36-63" href="#__codelineno-36-63"></a><span class="kt">void</span><span class="w"> </span><span class="nf">pushLast</span><span class="p">(</span><span class="n">ArrayDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-64" name="__codelineno-36-64" href="#__codelineno-36-64"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">(</span><span class="n">deque</span><span class="p">))</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-65" name="__codelineno-36-65" href="#__codelineno-36-65"></a><span class="w">        </span><span class="n">printf</span><span class="p">(</span><span class="s">"Deque is full</span><span class="se">\r\n</span><span class="s">"</span><span class="p">);</span>
 <a id="__codelineno-36-66" name="__codelineno-36-66" href="#__codelineno-36-66"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-36-67" name="__codelineno-36-67" href="#__codelineno-36-67"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-36-68" name="__codelineno-36-68" href="#__codelineno-36-68"></a><span class="w">    </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
+<a id="__codelineno-36-68" name="__codelineno-36-68" href="#__codelineno-36-68"></a><span class="w">    </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
 <a id="__codelineno-36-69" name="__codelineno-36-69" href="#__codelineno-36-69"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dequeIndex</span><span class="p">(</span><span class="n">deque</span><span class="p">,</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="p">);</span>
-<a id="__codelineno-36-70" name="__codelineno-36-70" href="#__codelineno-36-70"></a><span class="w">    </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-36-70" name="__codelineno-36-70" href="#__codelineno-36-70"></a><span class="w">    </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-36-71" name="__codelineno-36-71" href="#__codelineno-36-71"></a><span class="w">    </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">nums</span><span class="p">[</span><span class="n">rear</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-36-72" name="__codelineno-36-72" href="#__codelineno-36-72"></a><span class="w">    </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-36-73" name="__codelineno-36-73" href="#__codelineno-36-73"></a><span class="p">}</span>
 <a id="__codelineno-36-74" name="__codelineno-36-74" href="#__codelineno-36-74"></a>
-<a id="__codelineno-36-75" name="__codelineno-36-75" href="#__codelineno-36-75"></a><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-36-75" name="__codelineno-36-75" href="#__codelineno-36-75"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-36-76" name="__codelineno-36-76" href="#__codelineno-36-76"></a><span class="kt">int</span><span class="w"> </span><span class="nf">peekFirst</span><span class="p">(</span><span class="n">ArrayDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-77" name="__codelineno-36-77" href="#__codelineno-36-77"></a><span class="w">    </span><span class="c1">// Access error: Deque is empty</span>
+<a id="__codelineno-36-77" name="__codelineno-36-77" href="#__codelineno-36-77"></a><span class="w">    </span><span class="c1">// Access error: the deque is empty</span>
 <a id="__codelineno-36-78" name="__codelineno-36-78" href="#__codelineno-36-78"></a><span class="w">    </span><span class="n">assert</span><span class="p">(</span><span class="n">empty</span><span class="p">(</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-36-79" name="__codelineno-36-79" href="#__codelineno-36-79"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">nums</span><span class="p">[</span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="p">];</span>
 <a id="__codelineno-36-80" name="__codelineno-36-80" href="#__codelineno-36-80"></a><span class="p">}</span>
 <a id="__codelineno-36-81" name="__codelineno-36-81" href="#__codelineno-36-81"></a>
-<a id="__codelineno-36-82" name="__codelineno-36-82" href="#__codelineno-36-82"></a><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-36-82" name="__codelineno-36-82" href="#__codelineno-36-82"></a><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-36-83" name="__codelineno-36-83" href="#__codelineno-36-83"></a><span class="kt">int</span><span class="w"> </span><span class="nf">peekLast</span><span class="p">(</span><span class="n">ArrayDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-84" name="__codelineno-36-84" href="#__codelineno-36-84"></a><span class="w">    </span><span class="c1">// Access error: Deque is empty</span>
+<a id="__codelineno-36-84" name="__codelineno-36-84" href="#__codelineno-36-84"></a><span class="w">    </span><span class="c1">// Access error: the deque is empty</span>
 <a id="__codelineno-36-85" name="__codelineno-36-85" href="#__codelineno-36-85"></a><span class="w">    </span><span class="n">assert</span><span class="p">(</span><span class="n">empty</span><span class="p">(</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-36-86" name="__codelineno-36-86" href="#__codelineno-36-86"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">last</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dequeIndex</span><span class="p">(</span><span class="n">deque</span><span class="p">,</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-36-87" name="__codelineno-36-87" href="#__codelineno-36-87"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">nums</span><span class="p">[</span><span class="n">last</span><span class="p">];</span>
 <a id="__codelineno-36-88" name="__codelineno-36-88" href="#__codelineno-36-88"></a><span class="p">}</span>
 <a id="__codelineno-36-89" name="__codelineno-36-89" href="#__codelineno-36-89"></a>
-<a id="__codelineno-36-90" name="__codelineno-36-90" href="#__codelineno-36-90"></a><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-36-90" name="__codelineno-36-90" href="#__codelineno-36-90"></a><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-36-91" name="__codelineno-36-91" href="#__codelineno-36-91"></a><span class="kt">int</span><span class="w"> </span><span class="nf">popFirst</span><span class="p">(</span><span class="n">ArrayDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-92" name="__codelineno-36-92" href="#__codelineno-36-92"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peekFirst</span><span class="p">(</span><span class="n">deque</span><span class="p">);</span>
-<a id="__codelineno-36-93" name="__codelineno-36-93" href="#__codelineno-36-93"></a><span class="w">    </span><span class="c1">// Move front pointer backward by one position</span>
+<a id="__codelineno-36-93" name="__codelineno-36-93" href="#__codelineno-36-93"></a><span class="w">    </span><span class="c1">// Advance the front pointer by one position</span>
 <a id="__codelineno-36-94" name="__codelineno-36-94" href="#__codelineno-36-94"></a><span class="w">    </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dequeIndex</span><span class="p">(</span><span class="n">deque</span><span class="p">,</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-36-95" name="__codelineno-36-95" href="#__codelineno-36-95"></a><span class="w">    </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-36-96" name="__codelineno-36-96" href="#__codelineno-36-96"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-36-97" name="__codelineno-36-97" href="#__codelineno-36-97"></a><span class="p">}</span>
 <a id="__codelineno-36-98" name="__codelineno-36-98" href="#__codelineno-36-98"></a>
-<a id="__codelineno-36-99" name="__codelineno-36-99" href="#__codelineno-36-99"></a><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-36-99" name="__codelineno-36-99" href="#__codelineno-36-99"></a><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-36-100" name="__codelineno-36-100" href="#__codelineno-36-100"></a><span class="kt">int</span><span class="w"> </span><span class="nf">popLast</span><span class="p">(</span><span class="n">ArrayDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-101" name="__codelineno-36-101" href="#__codelineno-36-101"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peekLast</span><span class="p">(</span><span class="n">deque</span><span class="p">);</span>
 <a id="__codelineno-36-102" name="__codelineno-36-102" href="#__codelineno-36-102"></a><span class="w">    </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-36-103" name="__codelineno-36-103" href="#__codelineno-36-103"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-36-104" name="__codelineno-36-104" href="#__codelineno-36-104"></a><span class="p">}</span>
 <a id="__codelineno-36-105" name="__codelineno-36-105" href="#__codelineno-36-105"></a>
-<a id="__codelineno-36-106" name="__codelineno-36-106" href="#__codelineno-36-106"></a><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-36-106" name="__codelineno-36-106" href="#__codelineno-36-106"></a><span class="cm">/* Return an array for printing */</span>
 <a id="__codelineno-36-107" name="__codelineno-36-107" href="#__codelineno-36-107"></a><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="nf">toArray</span><span class="p">(</span><span class="n">ArrayDeque</span><span class="w"> </span><span class="o">*</span><span class="n">deque</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">queSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-108" name="__codelineno-36-108" href="#__codelineno-36-108"></a><span class="w">    </span><span class="o">*</span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-36-109" name="__codelineno-36-109" href="#__codelineno-36-109"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">calloc</span><span class="p">(</span><span class="n">deque</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
@@ -8123,93 +8195,93 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">array_deque.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Constructor */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayDeque</span><span class="p">(</span><span class="n">capacity</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">capacity</span><span class="p">)</span><span class="w"> </span><span class="c1">// Array for storing double-ended queue elements</span>
-<a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">front</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">queSize</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Double-ended queue length</span>
+<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">capacity</span><span class="p">)</span><span class="w"> </span><span class="c1">// Array that stores the deque elements</span>
+<a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">front</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Front pointer, pointing to the first element</span>
+<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">queSize</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Deque size</span>
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a>
-<a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="cm">/* Get the capacity of the double-ended queue */</span>
+<a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="cm">/* Get the deque capacity */</span>
 <a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">capacity</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a>
-<a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="w">    </span><span class="cm">/* Get the length of the double-ended queue */</span>
+<a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="w">    </span><span class="cm">/* Get the deque size */</span>
 <a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">size</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span>
 <a id="__codelineno-37-15" name="__codelineno-37-15" href="#__codelineno-37-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-16" name="__codelineno-37-16" href="#__codelineno-37-16"></a>
-<a id="__codelineno-37-17" name="__codelineno-37-17" href="#__codelineno-37-17"></a><span class="w">    </span><span class="cm">/* Check if the double-ended queue is empty */</span>
+<a id="__codelineno-37-17" name="__codelineno-37-17" href="#__codelineno-37-17"></a><span class="w">    </span><span class="cm">/* Check whether the deque is empty */</span>
 <a id="__codelineno-37-18" name="__codelineno-37-18" href="#__codelineno-37-18"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">():</span><span class="w"> </span><span class="kt">Boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-19" name="__codelineno-37-19" href="#__codelineno-37-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-37-20" name="__codelineno-37-20" href="#__codelineno-37-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-21" name="__codelineno-37-21" href="#__codelineno-37-21"></a>
-<a id="__codelineno-37-22" name="__codelineno-37-22" href="#__codelineno-37-22"></a><span class="w">    </span><span class="cm">/* Calculate circular array index */</span>
+<a id="__codelineno-37-22" name="__codelineno-37-22" href="#__codelineno-37-22"></a><span class="w">    </span><span class="cm">/* Calculate an index in the circular array */</span>
 <a id="__codelineno-37-23" name="__codelineno-37-23" href="#__codelineno-37-23"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">fun</span><span class="w"> </span><span class="nf">index</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-24" name="__codelineno-37-24" href="#__codelineno-37-24"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap the array head and tail together</span>
-<a id="__codelineno-37-25" name="__codelineno-37-25" href="#__codelineno-37-25"></a><span class="w">        </span><span class="c1">// When i passes the tail of the array, return to the head</span>
-<a id="__codelineno-37-26" name="__codelineno-37-26" href="#__codelineno-37-26"></a><span class="w">        </span><span class="c1">// When i passes the head of the array, return to the tail</span>
+<a id="__codelineno-37-24" name="__codelineno-37-24" href="#__codelineno-37-24"></a><span class="w">        </span><span class="c1">// Use modulo to connect the end of the array to its beginning</span>
+<a id="__codelineno-37-25" name="__codelineno-37-25" href="#__codelineno-37-25"></a><span class="w">        </span><span class="c1">// When i passes the end of the array, wrap around to the beginning</span>
+<a id="__codelineno-37-26" name="__codelineno-37-26" href="#__codelineno-37-26"></a><span class="w">        </span><span class="c1">// When i passes the beginning of the array, wrap around to the end</span>
 <a id="__codelineno-37-27" name="__codelineno-37-27" href="#__codelineno-37-27"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span>
 <a id="__codelineno-37-28" name="__codelineno-37-28" href="#__codelineno-37-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-29" name="__codelineno-37-29" href="#__codelineno-37-29"></a>
-<a id="__codelineno-37-30" name="__codelineno-37-30" href="#__codelineno-37-30"></a><span class="w">    </span><span class="cm">/* Front of the queue enqueue */</span>
+<a id="__codelineno-37-30" name="__codelineno-37-30" href="#__codelineno-37-30"></a><span class="w">    </span><span class="cm">/* Enqueue at the front */</span>
 <a id="__codelineno-37-31" name="__codelineno-37-31" href="#__codelineno-37-31"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">pushFirst</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-32" name="__codelineno-37-32" href="#__codelineno-37-32"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-33" name="__codelineno-37-33" href="#__codelineno-37-33"></a><span class="w">            </span><span class="n">println</span><span class="p">(</span><span class="s">"Double-ended queue is full"</span><span class="p">)</span>
+<a id="__codelineno-37-33" name="__codelineno-37-33" href="#__codelineno-37-33"></a><span class="w">            </span><span class="n">println</span><span class="p">(</span><span class="s">"Deque is full"</span><span class="p">)</span>
 <a id="__codelineno-37-34" name="__codelineno-37-34" href="#__codelineno-37-34"></a><span class="w">            </span><span class="k">return</span>
 <a id="__codelineno-37-35" name="__codelineno-37-35" href="#__codelineno-37-35"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-37-36" name="__codelineno-37-36" href="#__codelineno-37-36"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap front around to the tail after passing the head of the array</span>
-<a id="__codelineno-37-37" name="__codelineno-37-37" href="#__codelineno-37-37"></a><span class="w">        </span><span class="c1">// Add num to the front of the queue</span>
+<a id="__codelineno-37-36" name="__codelineno-37-36" href="#__codelineno-37-36"></a><span class="w">        </span><span class="c1">// Move the front pointer one position to the left</span>
+<a id="__codelineno-37-37" name="__codelineno-37-37" href="#__codelineno-37-37"></a><span class="w">        </span><span class="c1">// Use modulo to wrap front to the end when it passes the beginning of the array</span>
 <a id="__codelineno-37-38" name="__codelineno-37-38" href="#__codelineno-37-38"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
-<a id="__codelineno-37-39" name="__codelineno-37-39" href="#__codelineno-37-39"></a><span class="w">        </span><span class="c1">// Add num to front of queue</span>
+<a id="__codelineno-37-39" name="__codelineno-37-39" href="#__codelineno-37-39"></a><span class="w">        </span><span class="c1">// Add num to the front</span>
 <a id="__codelineno-37-40" name="__codelineno-37-40" href="#__codelineno-37-40"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">front</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-37-41" name="__codelineno-37-41" href="#__codelineno-37-41"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span>
 <a id="__codelineno-37-42" name="__codelineno-37-42" href="#__codelineno-37-42"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-43" name="__codelineno-37-43" href="#__codelineno-37-43"></a>
-<a id="__codelineno-37-44" name="__codelineno-37-44" href="#__codelineno-37-44"></a><span class="w">    </span><span class="cm">/* Rear of the queue enqueue */</span>
+<a id="__codelineno-37-44" name="__codelineno-37-44" href="#__codelineno-37-44"></a><span class="w">    </span><span class="cm">/* Enqueue at the rear */</span>
 <a id="__codelineno-37-45" name="__codelineno-37-45" href="#__codelineno-37-45"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">pushLast</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-46" name="__codelineno-37-46" href="#__codelineno-37-46"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-47" name="__codelineno-37-47" href="#__codelineno-37-47"></a><span class="w">            </span><span class="n">println</span><span class="p">(</span><span class="s">"Double-ended queue is full"</span><span class="p">)</span>
+<a id="__codelineno-37-47" name="__codelineno-37-47" href="#__codelineno-37-47"></a><span class="w">            </span><span class="n">println</span><span class="p">(</span><span class="s">"Deque is full"</span><span class="p">)</span>
 <a id="__codelineno-37-48" name="__codelineno-37-48" href="#__codelineno-37-48"></a><span class="w">            </span><span class="k">return</span>
 <a id="__codelineno-37-49" name="__codelineno-37-49" href="#__codelineno-37-49"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-37-50" name="__codelineno-37-50" href="#__codelineno-37-50"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
+<a id="__codelineno-37-50" name="__codelineno-37-50" href="#__codelineno-37-50"></a><span class="w">        </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
 <a id="__codelineno-37-51" name="__codelineno-37-51" href="#__codelineno-37-51"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">queSize</span><span class="p">)</span>
-<a id="__codelineno-37-52" name="__codelineno-37-52" href="#__codelineno-37-52"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-37-52" name="__codelineno-37-52" href="#__codelineno-37-52"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-37-53" name="__codelineno-37-53" href="#__codelineno-37-53"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">rear</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-37-54" name="__codelineno-37-54" href="#__codelineno-37-54"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span>
 <a id="__codelineno-37-55" name="__codelineno-37-55" href="#__codelineno-37-55"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-56" name="__codelineno-37-56" href="#__codelineno-37-56"></a>
-<a id="__codelineno-37-57" name="__codelineno-37-57" href="#__codelineno-37-57"></a><span class="w">    </span><span class="cm">/* Rear of the queue dequeue */</span>
+<a id="__codelineno-37-57" name="__codelineno-37-57" href="#__codelineno-37-57"></a><span class="w">    </span><span class="cm">/* Dequeue from the front */</span>
 <a id="__codelineno-37-58" name="__codelineno-37-58" href="#__codelineno-37-58"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">popFirst</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-59" name="__codelineno-37-59" href="#__codelineno-37-59"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peekFirst</span><span class="p">()</span>
-<a id="__codelineno-37-60" name="__codelineno-37-60" href="#__codelineno-37-60"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position</span>
+<a id="__codelineno-37-60" name="__codelineno-37-60" href="#__codelineno-37-60"></a><span class="w">        </span><span class="c1">// Advance the front pointer by one position</span>
 <a id="__codelineno-37-61" name="__codelineno-37-61" href="#__codelineno-37-61"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-37-62" name="__codelineno-37-62" href="#__codelineno-37-62"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span>
 <a id="__codelineno-37-63" name="__codelineno-37-63" href="#__codelineno-37-63"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-37-64" name="__codelineno-37-64" href="#__codelineno-37-64"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-65" name="__codelineno-37-65" href="#__codelineno-37-65"></a>
-<a id="__codelineno-37-66" name="__codelineno-37-66" href="#__codelineno-37-66"></a><span class="w">    </span><span class="cm">/* Access rear of the queue element */</span>
+<a id="__codelineno-37-66" name="__codelineno-37-66" href="#__codelineno-37-66"></a><span class="w">    </span><span class="cm">/* Dequeue from the rear */</span>
 <a id="__codelineno-37-67" name="__codelineno-37-67" href="#__codelineno-37-67"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">popLast</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-68" name="__codelineno-37-68" href="#__codelineno-37-68"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peekLast</span><span class="p">()</span>
 <a id="__codelineno-37-69" name="__codelineno-37-69" href="#__codelineno-37-69"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span>
 <a id="__codelineno-37-70" name="__codelineno-37-70" href="#__codelineno-37-70"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-37-71" name="__codelineno-37-71" href="#__codelineno-37-71"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-72" name="__codelineno-37-72" href="#__codelineno-37-72"></a>
-<a id="__codelineno-37-73" name="__codelineno-37-73" href="#__codelineno-37-73"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-37-73" name="__codelineno-37-73" href="#__codelineno-37-73"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-37-74" name="__codelineno-37-74" href="#__codelineno-37-74"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">peekFirst</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-75" name="__codelineno-37-75" href="#__codelineno-37-75"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">()</span>
 <a id="__codelineno-37-76" name="__codelineno-37-76" href="#__codelineno-37-76"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">front</span><span class="o">]</span>
 <a id="__codelineno-37-77" name="__codelineno-37-77" href="#__codelineno-37-77"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-78" name="__codelineno-37-78" href="#__codelineno-37-78"></a>
-<a id="__codelineno-37-79" name="__codelineno-37-79" href="#__codelineno-37-79"></a><span class="w">    </span><span class="cm">/* Driver Code */</span>
+<a id="__codelineno-37-79" name="__codelineno-37-79" href="#__codelineno-37-79"></a><span class="w">    </span><span class="cm">/* Access the rear element */</span>
 <a id="__codelineno-37-80" name="__codelineno-37-80" href="#__codelineno-37-80"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">peekLast</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-81" name="__codelineno-37-81" href="#__codelineno-37-81"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">()</span>
-<a id="__codelineno-37-82" name="__codelineno-37-82" href="#__codelineno-37-82"></a><span class="w">        </span><span class="c1">// Initialize double-ended queue</span>
+<a id="__codelineno-37-82" name="__codelineno-37-82" href="#__codelineno-37-82"></a><span class="w">        </span><span class="c1">// Calculate the index of the last element</span>
 <a id="__codelineno-37-83" name="__codelineno-37-83" href="#__codelineno-37-83"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">last</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-37-84" name="__codelineno-37-84" href="#__codelineno-37-84"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">last</span><span class="o">]</span>
 <a id="__codelineno-37-85" name="__codelineno-37-85" href="#__codelineno-37-85"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-86" name="__codelineno-37-86" href="#__codelineno-37-86"></a>
-<a id="__codelineno-37-87" name="__codelineno-37-87" href="#__codelineno-37-87"></a><span class="w">    </span><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-37-87" name="__codelineno-37-87" href="#__codelineno-37-87"></a><span class="w">    </span><span class="cm">/* Return an array for printing */</span>
 <a id="__codelineno-37-88" name="__codelineno-37-88" href="#__codelineno-37-88"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">toArray</span><span class="p">():</span><span class="w"> </span><span class="n">IntArray</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-89" name="__codelineno-37-89" href="#__codelineno-37-89"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-37-89" name="__codelineno-37-89" href="#__codelineno-37-89"></a><span class="w">        </span><span class="c1">// Copy only the elements within the occupied range</span>
 <a id="__codelineno-37-90" name="__codelineno-37-90" href="#__codelineno-37-90"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">queSize</span><span class="p">)</span>
 <a id="__codelineno-37-91" name="__codelineno-37-91" href="#__codelineno-37-91"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-37-92" name="__codelineno-37-92" href="#__codelineno-37-92"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span>
@@ -8224,9 +8296,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_deque.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Deque based on circular array ###</span>
+<div class="highlight"><span class="filename">array_deque.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Deque implemented with a circular array ###</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayDeque</span>
-<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1">### Get deque length ###</span>
+<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1">### Get the deque size ###</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="kp">attr_reader</span><span class="w"> </span><span class="ss">:size</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a>
 <a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="c1">### Constructor ###</span>
@@ -8236,80 +8308,80 @@
 <a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a>
-<a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">  </span><span class="c1">### Get deque capacity ###</span>
+<a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">  </span><span class="c1">### Get the deque capacity ###</span>
 <a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">capacity</span>
 <a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a><span class="w">    </span><span class="vi">@nums</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-17" name="__codelineno-38-17" href="#__codelineno-38-17"></a>
-<a id="__codelineno-38-18" name="__codelineno-38-18" href="#__codelineno-38-18"></a><span class="w">  </span><span class="c1">### Check if deque is empty ###</span>
+<a id="__codelineno-38-18" name="__codelineno-38-18" href="#__codelineno-38-18"></a><span class="w">  </span><span class="c1">### Check whether the deque is empty ###</span>
 <a id="__codelineno-38-19" name="__codelineno-38-19" href="#__codelineno-38-19"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">is_empty?</span>
 <a id="__codelineno-38-20" name="__codelineno-38-20" href="#__codelineno-38-20"></a><span class="w">    </span><span class="n">size</span><span class="o">.</span><span class="n">zero?</span>
 <a id="__codelineno-38-21" name="__codelineno-38-21" href="#__codelineno-38-21"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-22" name="__codelineno-38-22" href="#__codelineno-38-22"></a>
-<a id="__codelineno-38-23" name="__codelineno-38-23" href="#__codelineno-38-23"></a><span class="w">  </span><span class="c1">### Enqueue at front ###</span>
+<a id="__codelineno-38-23" name="__codelineno-38-23" href="#__codelineno-38-23"></a><span class="w">  </span><span class="c1">### Enqueue at the front ###</span>
 <a id="__codelineno-38-24" name="__codelineno-38-24" href="#__codelineno-38-24"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">push_first</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-38-25" name="__codelineno-38-25" href="#__codelineno-38-25"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span>
-<a id="__codelineno-38-26" name="__codelineno-38-26" href="#__codelineno-38-26"></a><span class="w">      </span><span class="nb">puts</span><span class="w"> </span><span class="s1">'Double-ended queue is full'</span>
+<a id="__codelineno-38-26" name="__codelineno-38-26" href="#__codelineno-38-26"></a><span class="w">      </span><span class="nb">puts</span><span class="w"> </span><span class="s1">'Deque is full'</span>
 <a id="__codelineno-38-27" name="__codelineno-38-27" href="#__codelineno-38-27"></a><span class="w">      </span><span class="k">return</span>
 <a id="__codelineno-38-28" name="__codelineno-38-28" href="#__codelineno-38-28"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-38-29" name="__codelineno-38-29" href="#__codelineno-38-29"></a>
-<a id="__codelineno-38-30" name="__codelineno-38-30" href="#__codelineno-38-30"></a><span class="w">    </span><span class="c1"># Use modulo operation to wrap front around to the tail after passing the head of the array</span>
-<a id="__codelineno-38-31" name="__codelineno-38-31" href="#__codelineno-38-31"></a><span class="w">    </span><span class="c1"># Add num to the front of the queue</span>
+<a id="__codelineno-38-30" name="__codelineno-38-30" href="#__codelineno-38-30"></a><span class="w">    </span><span class="c1"># Move the front pointer one position to the left</span>
+<a id="__codelineno-38-31" name="__codelineno-38-31" href="#__codelineno-38-31"></a><span class="w">    </span><span class="c1"># Use modulo to wrap front to the end when it passes the beginning of the array</span>
 <a id="__codelineno-38-32" name="__codelineno-38-32" href="#__codelineno-38-32"></a><span class="w">    </span><span class="vi">@front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="vi">@front</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
-<a id="__codelineno-38-33" name="__codelineno-38-33" href="#__codelineno-38-33"></a><span class="w">    </span><span class="c1"># Add num to front of queue</span>
+<a id="__codelineno-38-33" name="__codelineno-38-33" href="#__codelineno-38-33"></a><span class="w">    </span><span class="c1"># Add num to the front</span>
 <a id="__codelineno-38-34" name="__codelineno-38-34" href="#__codelineno-38-34"></a><span class="w">    </span><span class="vi">@nums</span><span class="o">[</span><span class="vi">@front</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-38-35" name="__codelineno-38-35" href="#__codelineno-38-35"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-38-36" name="__codelineno-38-36" href="#__codelineno-38-36"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-37" name="__codelineno-38-37" href="#__codelineno-38-37"></a>
-<a id="__codelineno-38-38" name="__codelineno-38-38" href="#__codelineno-38-38"></a><span class="w">  </span><span class="c1">### Enqueue at rear ###</span>
+<a id="__codelineno-38-38" name="__codelineno-38-38" href="#__codelineno-38-38"></a><span class="w">  </span><span class="c1">### Enqueue at the rear ###</span>
 <a id="__codelineno-38-39" name="__codelineno-38-39" href="#__codelineno-38-39"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">push_last</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-38-40" name="__codelineno-38-40" href="#__codelineno-38-40"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span>
-<a id="__codelineno-38-41" name="__codelineno-38-41" href="#__codelineno-38-41"></a><span class="w">      </span><span class="nb">puts</span><span class="w"> </span><span class="s1">'Double-ended queue is full'</span>
+<a id="__codelineno-38-41" name="__codelineno-38-41" href="#__codelineno-38-41"></a><span class="w">      </span><span class="nb">puts</span><span class="w"> </span><span class="s1">'Deque is full'</span>
 <a id="__codelineno-38-42" name="__codelineno-38-42" href="#__codelineno-38-42"></a><span class="w">      </span><span class="k">return</span>
 <a id="__codelineno-38-43" name="__codelineno-38-43" href="#__codelineno-38-43"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-38-44" name="__codelineno-38-44" href="#__codelineno-38-44"></a>
-<a id="__codelineno-38-45" name="__codelineno-38-45" href="#__codelineno-38-45"></a><span class="w">    </span><span class="c1"># Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
+<a id="__codelineno-38-45" name="__codelineno-38-45" href="#__codelineno-38-45"></a><span class="w">    </span><span class="c1"># Calculate the rear pointer, which points one position past the last element</span>
 <a id="__codelineno-38-46" name="__codelineno-38-46" href="#__codelineno-38-46"></a><span class="w">    </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="vi">@front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">size</span><span class="p">)</span>
-<a id="__codelineno-38-47" name="__codelineno-38-47" href="#__codelineno-38-47"></a><span class="w">    </span><span class="c1"># Front pointer moves one position backward</span>
+<a id="__codelineno-38-47" name="__codelineno-38-47" href="#__codelineno-38-47"></a><span class="w">    </span><span class="c1"># Add num to the rear</span>
 <a id="__codelineno-38-48" name="__codelineno-38-48" href="#__codelineno-38-48"></a><span class="w">    </span><span class="vi">@nums</span><span class="o">[</span><span class="n">rear</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-38-49" name="__codelineno-38-49" href="#__codelineno-38-49"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-38-50" name="__codelineno-38-50" href="#__codelineno-38-50"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-51" name="__codelineno-38-51" href="#__codelineno-38-51"></a>
-<a id="__codelineno-38-52" name="__codelineno-38-52" href="#__codelineno-38-52"></a><span class="w">  </span><span class="c1">### Dequeue from front ###</span>
+<a id="__codelineno-38-52" name="__codelineno-38-52" href="#__codelineno-38-52"></a><span class="w">  </span><span class="c1">### Dequeue from the front ###</span>
 <a id="__codelineno-38-53" name="__codelineno-38-53" href="#__codelineno-38-53"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">pop_first</span>
 <a id="__codelineno-38-54" name="__codelineno-38-54" href="#__codelineno-38-54"></a><span class="w">    </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peek_first</span>
-<a id="__codelineno-38-55" name="__codelineno-38-55" href="#__codelineno-38-55"></a><span class="w">    </span><span class="c1"># Move front pointer backward by one position</span>
+<a id="__codelineno-38-55" name="__codelineno-38-55" href="#__codelineno-38-55"></a><span class="w">    </span><span class="c1"># Advance the front pointer by one position</span>
 <a id="__codelineno-38-56" name="__codelineno-38-56" href="#__codelineno-38-56"></a><span class="w">    </span><span class="vi">@front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="vi">@front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-38-57" name="__codelineno-38-57" href="#__codelineno-38-57"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-38-58" name="__codelineno-38-58" href="#__codelineno-38-58"></a><span class="w">    </span><span class="n">num</span>
 <a id="__codelineno-38-59" name="__codelineno-38-59" href="#__codelineno-38-59"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-60" name="__codelineno-38-60" href="#__codelineno-38-60"></a>
-<a id="__codelineno-38-61" name="__codelineno-38-61" href="#__codelineno-38-61"></a><span class="w">  </span><span class="c1">### Dequeue from rear ###</span>
+<a id="__codelineno-38-61" name="__codelineno-38-61" href="#__codelineno-38-61"></a><span class="w">  </span><span class="c1">### Dequeue from the rear ###</span>
 <a id="__codelineno-38-62" name="__codelineno-38-62" href="#__codelineno-38-62"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">pop_last</span>
 <a id="__codelineno-38-63" name="__codelineno-38-63" href="#__codelineno-38-63"></a><span class="w">    </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peek_last</span>
 <a id="__codelineno-38-64" name="__codelineno-38-64" href="#__codelineno-38-64"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-38-65" name="__codelineno-38-65" href="#__codelineno-38-65"></a><span class="w">    </span><span class="n">num</span>
 <a id="__codelineno-38-66" name="__codelineno-38-66" href="#__codelineno-38-66"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-67" name="__codelineno-38-67" href="#__codelineno-38-67"></a>
-<a id="__codelineno-38-68" name="__codelineno-38-68" href="#__codelineno-38-68"></a><span class="w">  </span><span class="c1">### Access front element ###</span>
+<a id="__codelineno-38-68" name="__codelineno-38-68" href="#__codelineno-38-68"></a><span class="w">  </span><span class="c1">### Access the front element ###</span>
 <a id="__codelineno-38-69" name="__codelineno-38-69" href="#__codelineno-38-69"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">peek_first</span>
 <a id="__codelineno-38-70" name="__codelineno-38-70" href="#__codelineno-38-70"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="p">,</span><span class="w"> </span><span class="s1">'Deque is empty'</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">is_empty?</span>
 <a id="__codelineno-38-71" name="__codelineno-38-71" href="#__codelineno-38-71"></a>
 <a id="__codelineno-38-72" name="__codelineno-38-72" href="#__codelineno-38-72"></a><span class="w">    </span><span class="vi">@nums</span><span class="o">[</span><span class="vi">@front</span><span class="o">]</span>
 <a id="__codelineno-38-73" name="__codelineno-38-73" href="#__codelineno-38-73"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-74" name="__codelineno-38-74" href="#__codelineno-38-74"></a>
-<a id="__codelineno-38-75" name="__codelineno-38-75" href="#__codelineno-38-75"></a><span class="w">  </span><span class="c1">### Access rear element ###</span>
+<a id="__codelineno-38-75" name="__codelineno-38-75" href="#__codelineno-38-75"></a><span class="w">  </span><span class="c1">### Access the rear element ###</span>
 <a id="__codelineno-38-76" name="__codelineno-38-76" href="#__codelineno-38-76"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">peek_last</span>
 <a id="__codelineno-38-77" name="__codelineno-38-77" href="#__codelineno-38-77"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="p">,</span><span class="w"> </span><span class="s1">'Deque is empty'</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">is_empty?</span>
 <a id="__codelineno-38-78" name="__codelineno-38-78" href="#__codelineno-38-78"></a>
-<a id="__codelineno-38-79" name="__codelineno-38-79" href="#__codelineno-38-79"></a><span class="w">    </span><span class="c1"># Initialize double-ended queue</span>
+<a id="__codelineno-38-79" name="__codelineno-38-79" href="#__codelineno-38-79"></a><span class="w">    </span><span class="c1"># Calculate the index of the last element</span>
 <a id="__codelineno-38-80" name="__codelineno-38-80" href="#__codelineno-38-80"></a><span class="w">    </span><span class="n">last</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">(</span><span class="vi">@front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-38-81" name="__codelineno-38-81" href="#__codelineno-38-81"></a><span class="w">    </span><span class="vi">@nums</span><span class="o">[</span><span class="n">last</span><span class="o">]</span>
 <a id="__codelineno-38-82" name="__codelineno-38-82" href="#__codelineno-38-82"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-83" name="__codelineno-38-83" href="#__codelineno-38-83"></a>
-<a id="__codelineno-38-84" name="__codelineno-38-84" href="#__codelineno-38-84"></a><span class="w">  </span><span class="c1">### Return array for printing ###</span>
+<a id="__codelineno-38-84" name="__codelineno-38-84" href="#__codelineno-38-84"></a><span class="w">  </span><span class="c1">### Return an array for printing ###</span>
 <a id="__codelineno-38-85" name="__codelineno-38-85" href="#__codelineno-38-85"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">to_array</span>
-<a id="__codelineno-38-86" name="__codelineno-38-86" href="#__codelineno-38-86"></a><span class="w">    </span><span class="c1"># Elements enqueue</span>
+<a id="__codelineno-38-86" name="__codelineno-38-86" href="#__codelineno-38-86"></a><span class="w">    </span><span class="c1"># Convert only the list elements within the occupied range</span>
 <a id="__codelineno-38-87" name="__codelineno-38-87" href="#__codelineno-38-87"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-38-88" name="__codelineno-38-88" href="#__codelineno-38-88"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">size</span>
 <a id="__codelineno-38-89" name="__codelineno-38-89" href="#__codelineno-38-89"></a><span class="w">      </span><span class="n">res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="vi">@nums</span><span class="o">[</span><span class="n">index</span><span class="p">(</span><span class="vi">@front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="o">]</span>
@@ -8319,11 +8391,11 @@
 <a id="__codelineno-38-93" name="__codelineno-38-93" href="#__codelineno-38-93"></a>
 <a id="__codelineno-38-94" name="__codelineno-38-94" href="#__codelineno-38-94"></a><span class="w">  </span><span class="kp">private</span>
 <a id="__codelineno-38-95" name="__codelineno-38-95" href="#__codelineno-38-95"></a>
-<a id="__codelineno-38-96" name="__codelineno-38-96" href="#__codelineno-38-96"></a><span class="w">  </span><span class="c1">### Calculate circular array index ###</span>
+<a id="__codelineno-38-96" name="__codelineno-38-96" href="#__codelineno-38-96"></a><span class="w">  </span><span class="c1">### Calculate an index in the circular array ###</span>
 <a id="__codelineno-38-97" name="__codelineno-38-97" href="#__codelineno-38-97"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">index</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-<a id="__codelineno-38-98" name="__codelineno-38-98" href="#__codelineno-38-98"></a><span class="w">    </span><span class="c1"># Use modulo operation to wrap the array head and tail together</span>
-<a id="__codelineno-38-99" name="__codelineno-38-99" href="#__codelineno-38-99"></a><span class="w">    </span><span class="c1"># When i passes the tail of the array, return to the head</span>
-<a id="__codelineno-38-100" name="__codelineno-38-100" href="#__codelineno-38-100"></a><span class="w">    </span><span class="c1"># When i passes the head of the array, return to the tail</span>
+<a id="__codelineno-38-98" name="__codelineno-38-98" href="#__codelineno-38-98"></a><span class="w">    </span><span class="c1"># Use modulo to connect the end of the array to its beginning</span>
+<a id="__codelineno-38-99" name="__codelineno-38-99" href="#__codelineno-38-99"></a><span class="w">    </span><span class="c1"># When i passes the end of the array, wrap around to the beginning</span>
+<a id="__codelineno-38-100" name="__codelineno-38-100" href="#__codelineno-38-100"></a><span class="w">    </span><span class="c1"># When i passes the beginning of the array, wrap around to the end</span>
 <a id="__codelineno-38-101" name="__codelineno-38-101" href="#__codelineno-38-101"></a><span class="w">    </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">capacity</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span>
 <a id="__codelineno-38-102" name="__codelineno-38-102" href="#__codelineno-38-102"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-103" name="__codelineno-38-103" href="#__codelineno-38-103"></a><span class="k">end</span>
@@ -8331,9 +8403,9 @@
 </div>
 </div>
 </div>
-<h2 id="533-deque-applications">5.3.3 &nbsp; Deque Applications<a class="headerlink" href="#533-deque-applications" title="Permanent link">&para;</a></h2>
-<p>A deque combines the logic of both stacks and queues. <strong>Therefore, it can implement all application scenarios of both, while providing greater flexibility</strong>.</p>
-<p>We know that the "undo" function in software is typically implemented using a stack: the system pushes each change operation onto the stack and then implements undo through pop. However, considering system resource limitations, software usually limits the number of undo steps (for example, only allowing 50 steps to be saved). When the stack length exceeds 50, the software needs to perform a deletion operation at the bottom of the stack (front of the queue). <strong>But a stack cannot implement this functionality, so a deque is needed to replace the stack</strong>. Note that the core logic of "undo" still follows the LIFO principle of a stack; it's just that the deque can more flexibly implement some additional logic.</p>
+<h2 id="533-applications">5.3.3 &nbsp; Applications<a class="headerlink" href="#533-applications" title="Permanent link">&para;</a></h2>
+<p>A deque combines stack and queue behavior, <strong>so it supports all their applications while offering greater flexibility</strong>.</p>
+<p>Software undo is usually implemented with a stack: the system uses <code>push</code> to record each change and <code>pop</code> to undo it. Because resources are limited, software often restricts the history to a fixed number of steps, such as <span class="arithmatex">\(50\)</span>. When the stack grows beyond <span class="arithmatex">\(50\)</span>, the oldest entry must be removed from the bottom, the front of the deque. <strong>A stack cannot provide this operation, so a deque is needed instead</strong>. The core undo behavior still follows last-in, first-out order; the deque simply provides flexibility for the additional logic.</p>
 
 <!-- Source file information -->
 
@@ -8356,7 +8428,7 @@ aria-label="Footer"
   <a
     href="../queue/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 5.2 Queue"
+    aria-label="Previous: 5.2   Queues"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -8368,7 +8440,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        5.2 Queue
+        5.2 &nbsp; Queues
       </div>
     </div>
   </a>
@@ -8380,7 +8452,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 5.4 Summary"
+    aria-label="Next: 5.4   Summary"
     rel="next"
   >
     <div class="md-footer__title">
@@ -8388,7 +8460,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        5.4 Summary
+        5.4 &nbsp; Summary
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -8405,7 +8477,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_stack_and_queue/exercises/index.html b/en/chapter_stack_and_queue/exercises/index.html
index 8c4a58a98..511a87a2b 100644
--- a/en/chapter_stack_and_queue/exercises/index.html
+++ b/en/chapter_stack_and_queue/exercises/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -119,7 +119,7 @@
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             d="M480 32c0-12.9-7.8-24.6-19.8-29.6s-25.7-2.2-34.9 6.9L381.7 53c-48 48-113.1 75-181 75H192 160 64c-35.3 0-64 28.7-64 64v96c0 35.3 28.7 64 64 64l0 128c0 17.7 14.3 32 32 32h64c17.7 0 32-14.3 32-32V352l8.7 0c67.9 0 133 27 181 75l43.6 43.6c9.2 9.2 22.9 11.9 34.9 6.9s19.8-16.6 19.8-29.6V300.4c18.6-8.8 32-32.5 32-60.4s-13.4-51.6-32-60.4V32zm-64 76.7V240 371.3C357.2 317.8 280.5 288 200.7 288H192V192h8.7c79.8 0 156.5-29.8 215.3-83.3z" />
     </svg>
-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -167,9 +167,9 @@
 
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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@@ -177,9 +177,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Light mode"  type="radio" name="__palette" id="__palette_1">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1466,7 +1466,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1487,7 +1487,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1508,7 +1508,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1535,7 +1535,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1562,7 +1562,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1589,7 +1589,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1625,7 +1625,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1642,7 +1642,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1668,22 +1668,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#551-concept-review" class="md-nav__link">
+  <a href="#551-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.5.1 &nbsp; Concept Review
+        5.5.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="5.5.1 Concept Review">
+    <nav class="md-nav" aria-label="5.5.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-which-element-leaves-a-stack-or-queue-first" class="md-nav__link">
+  <a href="#1-which-element-comes-out-first-in-a-stack-or-queue" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Which Element Leaves a Stack or Queue First?
+        1. &nbsp; Which element comes out first in a stack or queue?
       </span>
     </span>
   </a>
@@ -1694,7 +1694,7 @@
   <a href="#2-what-happens-when-the-rear-passes-the-end-of-the-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; What Happens When the Rear Passes the End of the Array?
+        2. &nbsp; What happens when the rear passes the end of the array?
       </span>
     </span>
   </a>
@@ -1705,7 +1705,7 @@
   <a href="#3-operations-at-both-ends-of-a-deque" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Operations at Both Ends of a Deque
+        3. &nbsp; Operations at both ends of a deque
       </span>
     </span>
   </a>
@@ -1721,19 +1721,19 @@
   <a href="#552-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.5.2 &nbsp; Programming Exercises
+        5.5.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="5.5.2 Programming Exercises">
+    <nav class="md-nav" aria-label="5.5.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-check-a-bracket-sequence" class="md-nav__link">
+  <a href="#1-checking-a-bracket-sequence" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Check a Bracket Sequence
+        1. &nbsp; Checking a bracket sequence
       </span>
     </span>
   </a>
@@ -1805,7 +1805,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1826,7 +1826,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1847,7 +1847,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1874,7 +1874,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1901,7 +1901,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1928,7 +1928,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1955,7 +1955,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -2023,7 +2023,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -2044,7 +2044,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2065,7 +2065,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -2092,7 +2092,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2119,7 +2119,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2146,7 +2146,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2173,7 +2173,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2200,7 +2200,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2227,7 +2227,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2291,7 +2291,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2312,7 +2312,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2333,7 +2333,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2360,7 +2360,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2387,7 +2387,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2414,7 +2414,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2441,7 +2441,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2505,7 +2505,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2526,7 +2526,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2547,7 +2547,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2574,7 +2574,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2601,7 +2601,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2628,7 +2628,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2655,7 +2655,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2723,7 +2723,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2744,7 +2744,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2765,7 +2765,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2792,7 +2792,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2819,7 +2819,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2846,7 +2846,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2873,7 +2873,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2900,7 +2900,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2927,7 +2927,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -3005,7 +3005,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3026,7 +3026,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3047,7 +3047,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3074,7 +3074,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3101,7 +3101,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3128,7 +3128,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3155,7 +3155,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3182,7 +3182,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3209,7 +3209,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3236,7 +3236,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3263,7 +3263,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3290,7 +3290,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3317,7 +3317,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3344,7 +3344,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3410,7 +3410,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3431,7 +3431,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3452,7 +3452,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3479,7 +3479,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3506,7 +3506,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3533,7 +3533,7 @@
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3560,7 +3560,7 @@
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3587,7 +3587,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3653,7 +3653,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3674,7 +3674,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3695,7 +3695,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3722,7 +3722,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3749,7 +3749,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3776,7 +3776,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3803,7 +3803,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3830,7 +3830,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3900,7 +3900,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3921,7 +3921,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3942,7 +3942,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3969,7 +3969,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3996,7 +3996,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4023,7 +4023,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4050,7 +4050,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4077,7 +4077,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4104,7 +4104,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4131,7 +4131,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4197,7 +4197,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4218,7 +4218,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4239,7 +4239,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4266,7 +4266,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4293,7 +4293,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4320,7 +4320,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4347,7 +4347,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4374,7 +4374,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4434,7 +4434,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4455,7 +4455,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4476,7 +4476,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4503,7 +4503,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4530,7 +4530,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4614,6 +4614,78 @@
 
 
 
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@@ -4655,22 +4727,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#551-concept-review" class="md-nav__link">
+  <a href="#551-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.5.1 &nbsp; Concept Review
+        5.5.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="5.5.1 Concept Review">
+    <nav class="md-nav" aria-label="5.5.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-which-element-leaves-a-stack-or-queue-first" class="md-nav__link">
+  <a href="#1-which-element-comes-out-first-in-a-stack-or-queue" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Which Element Leaves a Stack or Queue First?
+        1. &nbsp; Which element comes out first in a stack or queue?
       </span>
     </span>
   </a>
@@ -4681,7 +4753,7 @@
   <a href="#2-what-happens-when-the-rear-passes-the-end-of-the-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; What Happens When the Rear Passes the End of the Array?
+        2. &nbsp; What happens when the rear passes the end of the array?
       </span>
     </span>
   </a>
@@ -4692,7 +4764,7 @@
   <a href="#3-operations-at-both-ends-of-a-deque" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Operations at Both Ends of a Deque
+        3. &nbsp; Operations at both ends of a deque
       </span>
     </span>
   </a>
@@ -4708,19 +4780,19 @@
   <a href="#552-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.5.2 &nbsp; Programming Exercises
+        5.5.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="5.5.2 Programming Exercises">
+    <nav class="md-nav" aria-label="5.5.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-check-a-bracket-sequence" class="md-nav__link">
+  <a href="#1-checking-a-bracket-sequence" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Check a Bracket Sequence
+        1. &nbsp; Checking a bracket sequence
       </span>
     </span>
   </a>
@@ -4770,55 +4842,55 @@
 
 <!-- Page content -->
 <h1 id="55-exercises">5.5 &nbsp; Exercises<a class="headerlink" href="#55-exercises" title="Permanent link">&para;</a></h1>
-<h2 id="551-concept-review">5.5.1 &nbsp; Concept Review<a class="headerlink" href="#551-concept-review" title="Permanent link">&para;</a></h2>
-<h3 id="1-which-element-leaves-a-stack-or-queue-first">1. &nbsp; Which Element Leaves a Stack or Queue First?<a class="headerlink" href="#1-which-element-leaves-a-stack-or-queue-first" title="Permanent link">&para;</a></h3>
-<p>Prepare an empty stack <code>S</code> and an empty queue <code>Q</code>. Perform the same sequence of operations on each one:</p>
+<h2 id="551-check-your-understanding">5.5.1 &nbsp; Check your understanding<a class="headerlink" href="#551-check-your-understanding" title="Permanent link">&para;</a></h2>
+<h3 id="1-which-element-comes-out-first-in-a-stack-or-queue">1. &nbsp; Which element comes out first in a stack or queue?<a class="headerlink" href="#1-which-element-comes-out-first-in-a-stack-or-queue" title="Permanent link">&para;</a></h3>
+<p>Start with an empty stack <code>S</code> and an empty queue <code>Q</code>. Perform the same operations on each:</p>
 <p>Step 1: Add <code>A</code>.
 Step 2: Add <code>B</code>.
 Step 3: Remove and record one element.
 Step 4: Add <code>C</code>.
 Step 5: Keep removing and recording elements until the container is empty.</p>
-<p>Write the order in which elements are removed from <code>S</code> and from <code>Q</code>. Explain the difference using "last-in-first-out" or "first-in-first-out."</p>
+<p>Write the order in which elements are removed from <code>S</code> and from <code>Q</code>. Explain the difference using last in, first out or first in, first out.</p>
 <details class="success">
-<summary>Answer</summary>
-<p>The removal order for stack <code>S</code> is <code>B, C, A</code>. After the elements <code>A, B</code> are added, the most recently added element, <code>B</code>, is popped first. After <code>C</code> is added,
-<code>C, A</code> are popped in that order. This is "last-in-first-out."</p>
-<p>The removal order for queue <code>Q</code> is <code>A, B, C</code>. After the elements <code>A, B</code> are added, the earliest added element, <code>A</code>, is removed first.
-After <code>C</code> is added, <code>B, C</code> are removed in that order. This is "first-in-first-out."</p>
+<summary>Reference answer</summary>
+<p>The removal order for stack <code>S</code> is <code>B, C, A</code>. After adding <code>A, B</code>, we first pop <code>B</code>, the most recently added element, and then add <code>C</code>.
+We then pop <code>C, A</code> in order. This is last in, first out.</p>
+<p>The removal order for queue <code>Q</code> is <code>A, B, C</code>. After adding <code>A, B</code>, we first remove <code>A</code>, the earliest element added.
+We then add <code>C</code> and remove <code>B, C</code> in order. This is first in, first out.</p>
 </details>
-<h3 id="2-what-happens-when-the-rear-passes-the-end-of-the-array">2. &nbsp; What Happens When the Rear Passes the End of the Array?<a class="headerlink" href="#2-what-happens-when-the-rear-passes-the-end-of-the-array" title="Permanent link">&para;</a></h3>
-<p>A queue is implemented with a circular array of length 5, whose indices are <code>0–4</code>.
-Currently, <code>front = 3</code> and <code>size = 2</code>; <code>A, B</code> are stored at indices 3 and 4, respectively.</p>
+<h3 id="2-what-happens-when-the-rear-passes-the-end-of-the-array">2. &nbsp; What happens when the rear passes the end of the array?<a class="headerlink" href="#2-what-happens-when-the-rear-passes-the-end-of-the-array" title="Permanent link">&para;</a></h3>
+<p>A queue uses a circular array of length 5, with indices <code>0-4</code>.
+Currently, <code>front = 3</code> and <code>size = 2</code>, with queue elements <code>A, B</code> at indices 3 and 4, respectively.</p>
 <!-- numbered-subquestions -->
 
 <ol>
-<li>When <code>C</code> is enqueued, at which index should <code>C</code> be stored? What is <code>size</code> after the enqueue?</li>
-<li>Next, dequeue once. Which element is removed? What are the new values of <code>front</code> and <code>size</code>?</li>
-<li>What is the logical order from the front to the rear now? Does dequeuing require moving the other elements in the array? Why or why not?</li>
+<li>When enqueueing <code>C</code>, at which index should <code>C</code> be placed? What is <code>size</code> afterward?</li>
+<li>If we then dequeue once, which element is removed? What are the new values of <code>front</code> and <code>size</code>?</li>
+<li>What is the logical order from front to rear now? Does dequeuing require moving the other array elements? Why?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
 <p>The new element's position is
     <code>(front + size) % 5 = (3 + 2) % 5 = 0</code>,
-    so <code>C</code> is stored at index 0. After the enqueue, <code>size = 3</code>.</p>
+    so <code>C</code> goes at index 0. After enqueueing, <code>size = 3</code>.</p>
 </li>
 <li>
-<p>Dequeuing removes the current front element, <code>A</code>. The new front index is
+<p>Dequeueing removes the current front element, <code>A</code>. The new front index is
     <code>(3 + 1) % 5 = 4</code>, so <code>front = 4</code> and <code>size = 2</code>.</p>
 </li>
 <li>
-<p>The logical order of valid elements is <code>B, C</code>, with <code>B</code> at index 4 and <code>C</code> at index 0.
-    Dequeuing requires only changing <code>front</code> and <code>size</code>. The circular array uses the remainder operation to wrap the index back to the beginning,
-    so there is no need to shift all the other elements forward.</p>
+<p>The valid elements are logically ordered as <code>B, C</code>, with <code>B</code> at index 4 and <code>C</code> at index 0.
+    Dequeueing only advances <code>front</code> and updates <code>size</code>. The circular array uses the remainder operation to wrap indices back to the beginning,
+    so the other elements do not need to shift forward together.</p>
 </li>
 </ol>
 </details>
-<h3 id="3-operations-at-both-ends-of-a-deque">3. &nbsp; Operations at Both Ends of a Deque<a class="headerlink" href="#3-operations-at-both-ends-of-a-deque" title="Permanent link">&para;</a></h3>
-<p>Here, <code>push_first</code> adds an element at the front, <code>push_last</code> adds one at the rear,
-<code>pop_first</code> removes one from the front, and <code>pop_last</code> removes one from the rear.</p>
-<p>Perform the following operations on an empty deque <code>deq</code>:</p>
+<h3 id="3-operations-at-both-ends-of-a-deque">3. &nbsp; Operations at both ends of a deque<a class="headerlink" href="#3-operations-at-both-ends-of-a-deque" title="Permanent link">&para;</a></h3>
+<p>Here, <code>push_first</code> adds at the front, <code>push_last</code> adds at the rear,
+<code>pop_first</code> removes from the front, and <code>pop_last</code> removes from the rear.</p>
+<p>Starting with an empty deque <code>deq</code>, perform:</p>
 <ol>
 <li><code>push_last(A)</code></li>
 <li><code>push_last(B)</code></li>
@@ -4830,40 +4902,40 @@ Currently, <code>front = 3</code> and <code>size = 2</code>; <code>A, B</code> a
 <!-- numbered-subquestions -->
 
 <ol>
-<li>Which elements are returned by the two pop operations?</li>
-<li>After all operations are complete, which elements remain from front to rear?</li>
-<li>Examine the six operations. Can a queue that allows insertion only at the rear and removal only at the front perform all of them? If not, identify the operations it cannot perform. Then state whether a deque can perform them and explain why.</li>
+<li>Which elements are removed by the two pop operations?</li>
+<li>Which elements remain, from front to rear, after all the operations?</li>
+<li>Examine these six steps. Can a queue that allows additions only at the rear and removals only at the front perform them all? If not, identify the unsupported operations. Then explain whether a deque can perform them all, and why.</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
-<p>After the first three steps, the deque from front to rear is <code>[C, A, B]</code>.</p>
+<summary>Reference answer</summary>
+<p>After the first three steps, the deque contains <code>[C, A, B]</code> from front to rear.</p>
 <!-- numbered-subquestions -->
 <ol>
 <li>
-<p><code>pop_last()</code> removes <code>B</code>. After <code>D</code> is added, the deque is <code>[C, A, D]</code>,
-    and <code>pop_first()</code> then removes <code>C</code>.</p>
+<p><code>pop_last()</code> removes <code>B</code>. After adding <code>D</code>, the deque is <code>[C, A, D]</code>.
+    Then <code>pop_first()</code> removes <code>C</code>.</p>
 </li>
 <li>
-<p><code>[A, D]</code> remains.</p>
+<p>The remaining elements are <code>[A, D]</code>.</p>
 </li>
 <li>
-<p>A queue that permits insertion only at the rear and removal only at the front cannot perform all the operations.
-    Step 3, <code>push_first(C)</code>, requires insertion at the front, and Step 4, <code>pop_last()</code>, requires removal at the rear. Both are outside the operations supported by such a queue.
-    A deque permits insertion and removal at both ends, so it can perform all six operations.</p>
+<p>A queue that allows additions only at the rear and removals only at the front cannot perform every operation:
+    step 3, <code>push_first(C)</code>, adds at the front, and step 4, <code>pop_last()</code>, removes from the rear. Neither operation is supported by that queue.
+    A deque supports additions and removals at both ends, so it can perform all six steps.</p>
 </li>
 </ol>
 </details>
-<h2 id="552-programming-exercises">5.5.2 &nbsp; Programming Exercises<a class="headerlink" href="#552-programming-exercises" title="Permanent link">&para;</a></h2>
-<h3 id="1-check-a-bracket-sequence">1. &nbsp; Check a Bracket Sequence<a class="headerlink" href="#1-check-a-bracket-sequence" title="Permanent link">&para;</a></h3>
-<p>Given a string <code>s</code> containing only the three types of brackets <code>()</code>, <code>[]</code>, and <code>{}</code>, use a stack to determine whether the string is valid.</p>
-<p>A valid sequence must satisfy both conditions: every closing bracket matches the type of the most recent unmatched opening bracket,
-and no unmatched opening bracket remains after the traversal. Return a Boolean value for the result.</p>
+<h2 id="552-programming-exercises">5.5.2 &nbsp; Programming exercises<a class="headerlink" href="#552-programming-exercises" title="Permanent link">&para;</a></h2>
+<h3 id="1-checking-a-bracket-sequence">1. &nbsp; Checking a bracket sequence<a class="headerlink" href="#1-checking-a-bracket-sequence" title="Permanent link">&para;</a></h3>
+<p>Given a string <code>s</code> containing only the three types of brackets <code>()</code>, <code>[]</code>, and <code>{}</code>, use a stack to determine whether it is valid.</p>
+<p>A valid sequence must satisfy both conditions: each closing bracket matches the type of the most recent unmatched opening bracket,
+and no unmatched opening brackets remain at the end. Return a Boolean result.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>You can create a mapping from each closing bracket to its matching opening bracket</li>
-<li>When you encounter a closing bracket, first check whether the stack is empty, and then check whether the top matches</li>
-<li>The stack must also be empty after the traversal</li>
+<li>You can map each closing bracket to its matching opening bracket.</li>
+<li>When you encounter a closing bracket, first check whether the stack is empty, then check whether its top matches.</li>
+<li>The stack must also be empty when the scan finishes.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/valid-parentheses/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
@@ -4889,7 +4961,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 5.4 Summary"
+    aria-label="Previous: 5.4   Summary"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4901,7 +4973,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        5.4 Summary
+        5.4 &nbsp; Summary
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@@ -4913,7 +4985,7 @@ aria-label="Footer"
   <a
     href="../../chapter_hashing/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: Chapter 6. &amp;nbsp; Hashing"
+    aria-label="Next: Chapter 6.   Hash tables"
     rel="next"
   >
     <div class="md-footer__title">
@@ -4921,7 +4993,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        Chapter 6. &nbsp; Hashing
+        Chapter 6. &nbsp; Hash tables
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -4938,7 +5010,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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diff --git a/en/chapter_stack_and_queue/index.html b/en/chapter_stack_and_queue/index.html
index 20c06cdae..44e03cc0b 100644
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       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>Chapter 5. Stacks and Queues - Hello Algo</title>
+        <title>Chapter 5. Stacks and queues - Hello Algo</title>
 
 
 
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+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
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-  Before Starting
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-  Before Starting
+  Foreword
 
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-  Chapter 0. Preface
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-  0.1 About This Book
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
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-  1.3 Summary
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-  2.5 Summary
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
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   <span class="md-ellipsis">
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
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-  3.4 Character Encoding *
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-  3.5 Summary
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
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-  Chapter 4. Arrays and Linked Lists
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
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-  4.5 Summary
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
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+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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@@ -1680,7 +1680,7 @@
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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@@ -1722,7 +1722,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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@@ -1776,7 +1776,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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@@ -1803,7 +1803,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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-  7.1 Binary Tree
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   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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@@ -2075,7 +2075,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
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-  8.1 Heap
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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@@ -2289,7 +2289,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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@@ -2380,7 +2380,7 @@
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
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@@ -2422,7 +2422,7 @@
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-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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@@ -2476,7 +2476,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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@@ -2503,7 +2503,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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@@ -2530,7 +2530,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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@@ -2598,7 +2598,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2619,7 +2619,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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@@ -2640,7 +2640,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2694,7 +2694,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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@@ -2721,7 +2721,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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@@ -2748,7 +2748,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2775,7 +2775,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2802,7 +2802,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2880,7 +2880,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2901,7 +2901,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -2922,7 +2922,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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@@ -2976,7 +2976,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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@@ -3003,7 +3003,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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@@ -3030,7 +3030,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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@@ -3057,7 +3057,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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@@ -3084,7 +3084,7 @@
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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@@ -3111,7 +3111,7 @@
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
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+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
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-  12.5 Summary
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-  12.6 Exercises
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+  Chapter 13. &nbsp; Backtracking
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
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-  Chapter 14. Dynamic Programming
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-  14.1 Introduction to Dynamic Programming
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
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-  14.4 0-1 Knapsack Problem
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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@@ -4072,7 +4072,7 @@
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
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-  15.2 Fractional Knapsack Problem
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
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-  16.1 Programming Environment Installation
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
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+
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+  Print edition (Chinese)
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                   </div>
@@ -4577,18 +4649,18 @@
 
 
 <!-- Page content -->
-<h1 id="chapter-5-stacks-and-queues">Chapter 5. &nbsp; Stacks and Queues<a class="headerlink" href="#chapter-5-stacks-and-queues" title="Permanent link">&para;</a></h1>
-<p><img alt="Stacks and Queues" class="cover-image" src="../assets/covers/chapter_stack_and_queue.jpg" /></p>
+<h1 id="chapter-5-stacks-and-queues">Chapter 5. &nbsp; Stacks and queues<a class="headerlink" href="#chapter-5-stacks-and-queues" title="Permanent link">&para;</a></h1>
+<p><img alt="Stacks and queues" class="cover-image" src="../assets/covers/chapter_stack_and_queue.jpg" /></p>
 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
-<p>A stack is like cats piled on top of one another, while a queue is like cats lining up.</p>
-<p>They represent the logical relationships of LIFO (Last In, First Out) and FIFO (First In, First Out), respectively.</p>
+<p>A stack is like cats piled on top of one another; a queue is like cats standing in line.</p>
+<p>They represent last-in, first-out and first-in, first-out order, respectively.</p>
 </div>
 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
 <ul>
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-<li><a href="queue/">5.2 &nbsp; Queue</a></li>
-<li><a href="deque/">5.3 &nbsp; Deque</a></li>
+<li><a href="stack/">5.1 &nbsp; Stacks</a></li>
+<li><a href="queue/">5.2 &nbsp; Queues</a></li>
+<li><a href="deque/">5.3 &nbsp; Deques</a></li>
 <li><a href="summary/">5.4 &nbsp; Summary</a></li>
 <li><a href="exercises/">5.5 &nbsp; Exercises</a></li>
 </ul>
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   <a
     href="../chapter_array_and_linkedlist/exercises/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 4.6 Exercises"
+    aria-label="Previous: 4.6   Exercises"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4626,7 +4698,7 @@ aria-label="Footer"
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-        4.6 Exercises
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   <a
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     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 5.1 Stack"
+    aria-label="Next: 5.1   Stacks"
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         Next
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+        5.1 &nbsp; Stacks
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     <div class="md-footer__button md-icon">
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
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@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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@@ -1223,7 +1223,7 @@
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1265,7 +1265,7 @@
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-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.2.1 &nbsp; Common Queue Operations
+        5.2.1 &nbsp; Common queue operations
       </span>
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   </a>
@@ -1598,22 +1598,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#522-queue-implementation" class="md-nav__link">
+  <a href="#522-implementing-a-queue" class="md-nav__link">
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       <span class="md-typeset">
-        5.2.2 &nbsp; Queue Implementation
+        5.2.2 &nbsp; Implementing a queue
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="5.2.2 Queue Implementation">
+    <nav class="md-nav" aria-label="5.2.2 Implementing a queue">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-linked-list-implementation" class="md-nav__link">
+  <a href="#1-implementation-with-a-linked-list" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Linked List Implementation
+        1. &nbsp; Implementation with a linked list
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   </a>
@@ -1621,10 +1621,10 @@
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           <li class="md-nav__item">
-  <a href="#2-array-implementation" class="md-nav__link">
+  <a href="#2-implementation-with-an-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Array Implementation
+        2. &nbsp; Implementation with an array
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         <li class="md-nav__item">
-  <a href="#523-typical-applications-of-queue" class="md-nav__link">
+  <a href="#523-typical-applications" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.2.3 &nbsp; Typical Applications of Queue
+        5.2.3 &nbsp; Typical applications
       </span>
     </span>
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
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@@ -1830,7 +1830,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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@@ -2006,7 +2006,7 @@
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2048,7 +2048,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2102,7 +2102,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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@@ -2129,7 +2129,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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@@ -2156,7 +2156,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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@@ -2183,7 +2183,7 @@
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2316,7 +2316,7 @@
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-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2530,7 +2530,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2748,7 +2748,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2856,7 +2856,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2883,7 +2883,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2988,7 +2988,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
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@@ -3030,7 +3030,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
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   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
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+
+
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+
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+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="m19 2-5 4.5v11l5-4.5zM6.5 5C4.55 5 2.45 5.4 1 6.5v14.66c0 .25.25.5.5.5.1 0 .15-.07.25-.07 1.35-.65 3.3-1.09 4.75-1.09 1.95 0 4.05.4 5.5 1.5 1.35-.85 3.8-1.5 5.5-1.5 1.65 0 3.35.31 4.75 1.06.1.05.15.03.25.03.25 0 .5-.25.5-.5V6.5c-.6-.45-1.25-.75-2-1V19c-1.1-.35-2.3-.5-3.5-.5-1.7 0-4.15.65-5.5 1.5V6.5C10.55 5.4 8.45 5 6.5 5"/></svg>
+
+  <span class="md-ellipsis">
+
+
+  Print edition (Chinese)
+
+
+  </span>
+
+
+
+
+
+    <span class="md-status md-status--new" title="Recently added">
+    </span>
+
+
+
+
+            </a>
+
+          </div>
+
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
+            <span class="md-nav__icon md-icon"></span>
+
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+  Print edition (Chinese)
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   </ul>
 </nav>
                   </div>
@@ -4641,7 +4713,7 @@
   <a href="#521-common-queue-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.2.1 &nbsp; Common Queue Operations
+        5.2.1 &nbsp; Common queue operations
       </span>
     </span>
   </a>
@@ -4649,22 +4721,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#522-queue-implementation" class="md-nav__link">
+  <a href="#522-implementing-a-queue" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.2.2 &nbsp; Queue Implementation
+        5.2.2 &nbsp; Implementing a queue
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="5.2.2 Queue Implementation">
+    <nav class="md-nav" aria-label="5.2.2 Implementing a queue">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-linked-list-implementation" class="md-nav__link">
+  <a href="#1-implementation-with-a-linked-list" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Linked List Implementation
+        1. &nbsp; Implementation with a linked list
       </span>
     </span>
   </a>
@@ -4672,10 +4744,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-array-implementation" class="md-nav__link">
+  <a href="#2-implementation-with-an-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Array Implementation
+        2. &nbsp; Implementation with an array
       </span>
     </span>
   </a>
@@ -4688,10 +4760,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#523-typical-applications-of-queue" class="md-nav__link">
+  <a href="#523-typical-applications" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.2.3 &nbsp; Typical Applications of Queue
+        5.2.3 &nbsp; Typical applications
       </span>
     </span>
   </a>
@@ -4735,15 +4807,15 @@
 
 
 <!-- Page content -->
-<h1 id="52-queue">5.2 &nbsp; Queue<a class="headerlink" href="#52-queue" title="Permanent link">&para;</a></h1>
-<p>A <u>queue</u> is a linear data structure that follows the First In, First Out (FIFO) rule. As the name suggests, it models people lining up: newcomers continuously join the rear of the queue, while the people at the front leave one by one.</p>
-<p>As shown in Figure 5-4, we call the front of the queue the "front" and the end the "rear." The operation of adding an element to the rear is called "enqueue," and the operation of removing the front element is called "dequeue."</p>
-<p><img alt="FIFO rule of queue" class="animation-figure" src="../queue.assets/queue_operations.png" /></p>
-<p align="center"> Figure 5-4 &nbsp; FIFO rule of queue </p>
+<h1 id="52-queues">5.2 &nbsp; Queues<a class="headerlink" href="#52-queues" title="Permanent link">&para;</a></h1>
+<p>A <u>queue</u> is a linear data structure that follows a first-in, first-out rule. It models a line of people: newcomers join at the back, while those at the front leave one by one.</p>
+<p>As shown below, the two ends are called the “front” and the “rear.” Adding an element at the rear is called “enqueueing,” and removing an element from the front is called “dequeueing.”</p>
+<p><img alt="The first-in, first-out rule of a queue" class="animation-figure" src="../queue.assets/queue_operations.png" /></p>
+<p align="center"> Figure 5-4 &nbsp; The first-in, first-out rule of a queue </p>
 
-<h2 id="521-common-queue-operations">5.2.1 &nbsp; Common Queue Operations<a class="headerlink" href="#521-common-queue-operations" title="Permanent link">&para;</a></h2>
-<p>The common operations on a queue are shown in Table 5-2. Note that method names may vary across programming languages. Here, we use the same naming convention as for stacks.</p>
-<p align="center"> Table 5-2 &nbsp; Efficiency of Queue Operations </p>
+<h2 id="521-common-queue-operations">5.2.1 &nbsp; Common queue operations<a class="headerlink" href="#521-common-queue-operations" title="Permanent link">&para;</a></h2>
+<p>Table 5-2 lists common operations. Method names vary between programming languages; here we use the same names as for stacks.</p>
+<p align="center"> Table 5-2 &nbsp; Efficiency of queue operations </p>
 
 <div class="center-table">
 <table>
@@ -4751,37 +4823,37 @@
 <tr>
 <th>Method</th>
 <th>Description</th>
-<th>Time Complexity</th>
+<th>Time complexity</th>
 </tr>
 </thead>
 <tbody>
 <tr>
 <td><code>push()</code></td>
-<td>Enqueue element, add element to rear</td>
+<td>Enqueue an element at the rear</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
 <td><code>pop()</code></td>
-<td>Dequeue front element</td>
+<td>Dequeue the front element</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
 <td><code>peek()</code></td>
-<td>Access front element</td>
+<td>Access the front element</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 </tbody>
 </table>
 </div>
-<p>We can directly use the queue classes provided by the programming language:</p>
+<p>We can use a queue class already provided by the programming language:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">queue.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">from</span><span class="w"> </span><span class="nn">collections</span><span class="w"> </span><span class="kn">import</span> <span class="n">deque</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="c1"># Initialize queue</span>
-<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="c1"># In Python, we generally use the deque class as a queue</span>
-<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="c1"># Although queue.Queue() is a pure queue class, it is not very user-friendly, so it is not recommended</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="c1"># Initialize a queue</span>
+<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a><span class="c1"># In Python, we usually use the deque class as a queue</span>
+<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="c1"># Although queue.Queue() is a dedicated queue class, it is less convenient and is not recommended here</span>
 <a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="n">que</span><span class="p">:</span> <span class="n">deque</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="n">deque</span><span class="p">()</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="c1"># Enqueue elements</span>
@@ -4791,21 +4863,21 @@
 <a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="n">que</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="n">que</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>
-<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="c1"># Access front element</span>
+<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="c1"># Access the front element</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="n">front</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">que</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
 <a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>
-<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="c1"># Dequeue element</span>
+<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="c1"># Dequeue an element</span>
 <a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="n">pop</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">que</span><span class="o">.</span><span class="n">popleft</span><span class="p">()</span>
 <a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>
-<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="c1"># Get queue length</span>
+<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="c1"># Get the queue size</span>
 <a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="n">size</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">que</span><span class="p">)</span>
 <a id="__codelineno-0-23" name="__codelineno-0-23" href="#__codelineno-0-23"></a>
-<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a><span class="c1"># Check if queue is empty</span>
+<a id="__codelineno-0-24" name="__codelineno-0-24" href="#__codelineno-0-24"></a><span class="c1"># Check whether the queue is empty</span>
 <a id="__codelineno-0-25" name="__codelineno-0-25" href="#__codelineno-0-25"></a><span class="n">is_empty</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">que</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">queue.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Initialize queue */</span>
+<div class="highlight"><span class="filename">queue.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Initialize a queue */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="n">queue</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">queue</span><span class="p">;</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="cm">/* Enqueue elements */</span>
@@ -4815,21 +4887,21 @@
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">queue</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="n">queue</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a>
-<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="cm">/* Access front element */</span>
+<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">front</span><span class="p">();</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a>
-<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="cm">/* Dequeue element */</span>
+<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="cm">/* Dequeue an element */</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="n">queue</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a>
-<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="cm">/* Get queue length */</span>
+<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
 <a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a>
-<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="cm">/* Check if queue is empty */</span>
+<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="kt">bool</span><span class="w"> </span><span class="n">empty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">empty</span><span class="p">();</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">queue.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Initialize queue */</span>
+<div class="highlight"><span class="filename">queue.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Initialize a queue */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="n">Queue</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">LinkedList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="cm">/* Enqueue elements */</span>
@@ -4839,21 +4911,21 @@
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="n">queue</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="n">queue</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a>
-<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="cm">/* Access front element */</span>
+<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="na">peek</span><span class="p">();</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a>
-<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="cm">/* Dequeue element */</span>
+<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="cm">/* Dequeue an element */</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="kt">int</span><span class="w"> </span><span class="n">pop</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="na">poll</span><span class="p">();</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a>
-<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="cm">/* Get queue length */</span>
+<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="na">size</span><span class="p">();</span>
 <a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a>
-<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="cm">/* Check if queue is empty */</span>
+<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="kt">boolean</span><span class="w"> </span><span class="n">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="na">isEmpty</span><span class="p">();</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">queue.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Initialize queue */</span>
+<div class="highlight"><span class="filename">queue.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Initialize a queue */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="n">Queue</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">();</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="cm">/* Enqueue elements */</span>
@@ -4863,22 +4935,22 @@
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="n">queue</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="n">queue</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a>
-<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="cm">/* Access front element */</span>
+<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">Peek</span><span class="p">();</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a>
-<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="cm">/* Dequeue element */</span>
+<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="cm">/* Dequeue an element */</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="kt">int</span><span class="w"> </span><span class="n">pop</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">Dequeue</span><span class="p">();</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a>
-<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="cm">/* Get queue length */</span>
+<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">Count</span><span class="p">;</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a>
-<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="cm">/* Check if queue is empty */</span>
+<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">queue_test.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Initialize queue */</span>
-<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="c1">// In Go, use list as a queue</span>
+<div class="highlight"><span class="filename">queue_test.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Initialize a queue */</span>
+<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="c1">// In Go, use a list as a queue</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="nx">queue</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">list</span><span class="p">.</span><span class="nx">New</span><span class="p">()</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="cm">/* Enqueue elements */</span>
@@ -4888,23 +4960,23 @@
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="nx">queue</span><span class="p">.</span><span class="nx">PushBack</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="nx">queue</span><span class="p">.</span><span class="nx">PushBack</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a>
-<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="cm">/* Access front element */</span>
+<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="nx">peek</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">Front</span><span class="p">()</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a>
-<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="cm">/* Dequeue element */</span>
+<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="cm">/* Dequeue an element */</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="nx">pop</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">Front</span><span class="p">()</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="nx">queue</span><span class="p">.</span><span class="nx">Remove</span><span class="p">(</span><span class="nx">pop</span><span class="p">)</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a>
-<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="cm">/* Get queue length */</span>
+<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="nx">size</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">Len</span><span class="p">()</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a>
-<a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="cm">/* Check if queue is empty */</span>
+<a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="nx">isEmpty</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">Len</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">queue.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Initialize queue */</span>
-<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Swift does not have a built-in queue class, can use Array as a queue</span>
+<div class="highlight"><span class="filename">queue.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Initialize a queue */</span>
+<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Swift has no built-in queue class; use an Array as a queue</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="kd">var</span><span class="w"> </span><span class="nv">queue</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="cm">/* Enqueue elements */</span>
@@ -4914,23 +4986,23 @@
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="n">queue</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="n">queue</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a>
-<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="cm">/* Access front element */</span>
+<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="kd">let</span><span class="w"> </span><span class="nv">peek</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="bp">first</span><span class="p">!</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a>
-<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="cm">/* Dequeue element */</span>
-<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="c1">// Since it's an array, removeFirst has O(n) complexity</span>
+<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="cm">/* Dequeue an element */</span>
+<a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="c1">// Because this is an array, removeFirst takes O(n) time</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="kd">let</span><span class="w"> </span><span class="nv">pop</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">removeFirst</span><span class="p">()</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a>
-<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="cm">/* Get queue length */</span>
+<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="kd">let</span><span class="w"> </span><span class="nv">size</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a>
-<a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="cm">/* Check if queue is empty */</span>
+<a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-5-23" name="__codelineno-5-23" href="#__codelineno-5-23"></a><span class="kd">let</span><span class="w"> </span><span class="nv">isEmpty</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="bp">isEmpty</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">queue.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Initialize queue */</span>
-<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="c1">// JavaScript does not have a built-in queue, can use Array as a queue</span>
+<div class="highlight"><span class="filename">queue.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Initialize a queue */</span>
+<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="c1">// JavaScript has no built-in queue; use an Array as a queue</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="kd">const</span><span class="w"> </span><span class="nx">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="cm">/* Enqueue elements */</span>
@@ -4940,23 +5012,23 @@
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="nx">queue</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">5</span><span class="p">);</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="nx">queue</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">4</span><span class="p">);</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a>
-<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="cm">/* Access front element */</span>
+<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="kd">const</span><span class="w"> </span><span class="nx">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">queue</span><span class="p">[</span><span class="mf">0</span><span class="p">];</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a>
-<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="cm">/* Dequeue element */</span>
-<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="c1">// The underlying structure is an array, so shift() has O(n) time complexity</span>
+<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="cm">/* Dequeue an element */</span>
+<a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="c1">// The underlying structure is an array, so shift() takes O(n) time</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a><span class="kd">const</span><span class="w"> </span><span class="nx">pop</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">shift</span><span class="p">();</span>
 <a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a>
-<a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="cm">/* Get queue length */</span>
+<a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a><span class="kd">const</span><span class="w"> </span><span class="nx">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a>
-<a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="cm">/* Check if queue is empty */</span>
+<a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-6-23" name="__codelineno-6-23" href="#__codelineno-6-23"></a><span class="kd">const</span><span class="w"> </span><span class="nx">empty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">queue.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Initialize queue */</span>
-<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="c1">// TypeScript does not have a built-in queue, can use Array as a queue</span>
+<div class="highlight"><span class="filename">queue.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Initialize a queue */</span>
+<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="c1">// TypeScript has no built-in queue; use an Array as a queue</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="kd">const</span><span class="w"> </span><span class="nx">queue</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="cm">/* Enqueue elements */</span>
@@ -4966,23 +5038,23 @@
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="nx">queue</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">5</span><span class="p">);</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="nx">queue</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">4</span><span class="p">);</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a>
-<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="cm">/* Access front element */</span>
+<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="kd">const</span><span class="w"> </span><span class="nx">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">queue</span><span class="p">[</span><span class="mf">0</span><span class="p">];</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a>
-<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="cm">/* Dequeue element */</span>
-<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="c1">// The underlying structure is an array, so shift() has O(n) time complexity</span>
+<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="cm">/* Dequeue an element */</span>
+<a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="c1">// The underlying structure is an array, so shift() takes O(n) time</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="kd">const</span><span class="w"> </span><span class="nx">pop</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">shift</span><span class="p">();</span>
 <a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a>
-<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="cm">/* Get queue length */</span>
+<a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a><span class="kd">const</span><span class="w"> </span><span class="nx">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a>
-<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="cm">/* Check if queue is empty */</span>
+<a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-7-23" name="__codelineno-7-23" href="#__codelineno-7-23"></a><span class="kd">const</span><span class="w"> </span><span class="nx">empty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">queue.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Initialize queue */</span>
-<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="c1">// In Dart, the Queue class is a deque and can also be used as a queue</span>
+<div class="highlight"><span class="filename">queue.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Initialize a queue */</span>
+<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="c1">// In Dart, Queue is a deque and can also be used as a queue</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="n">Queue</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Queue</span><span class="p">();</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="cm">/* Enqueue elements */</span>
@@ -4992,22 +5064,22 @@
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="n">queue</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="m">5</span><span class="p">);</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="n">queue</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="m">4</span><span class="p">);</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a>
-<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="cm">/* Access front element */</span>
+<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">first</span><span class="p">;</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a>
-<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="cm">/* Dequeue element */</span>
+<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="cm">/* Dequeue an element */</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="kt">int</span><span class="w"> </span><span class="n">pop</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">removeFirst</span><span class="p">();</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a>
-<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="cm">/* Get queue length */</span>
+<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a>
-<a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="cm">/* Check if queue is empty */</span>
+<a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">isEmpty</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">queue.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Initialize deque */</span>
-<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="c1">// In Rust, use deque as a regular queue</span>
+<div class="highlight"><span class="filename">queue.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Initialize a deque */</span>
+<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="c1">// In Rust, use a deque as an ordinary queue</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">deque</span><span class="p">:</span><span class="w"> </span><span class="nc">VecDeque</span><span class="o">&lt;</span><span class="kt">u32</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">VecDeque</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="cm">/* Enqueue elements */</span>
@@ -5017,27 +5089,27 @@
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="n">deque</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a>
-<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="cm">/* Access front element */</span>
+<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">front</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">front</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="p">}</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a>
-<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="cm">/* Dequeue element */</span>
+<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="cm">/* Dequeue an element */</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">pop</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">pop_front</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="p">}</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a>
-<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="cm">/* Get queue length */</span>
+<a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="kd">let</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
 <a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a>
-<a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="cm">/* Check if queue is empty */</span>
+<a id="__codelineno-9-23" name="__codelineno-9-23" href="#__codelineno-9-23"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-9-24" name="__codelineno-9-24" href="#__codelineno-9-24"></a><span class="kd">let</span><span class="w"> </span><span class="n">is_empty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">deque</span><span class="p">.</span><span class="n">is_empty</span><span class="p">();</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">queue.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// C does not provide a built-in queue</span>
+<div class="highlight"><span class="filename">queue.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// C provides no built-in queue</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">queue.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Initialize queue */</span>
+<div class="highlight"><span class="filename">queue.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Initialize a queue */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">val</span><span class="w"> </span><span class="nv">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">LinkedList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="cm">/* Enqueue elements */</span>
@@ -5047,22 +5119,22 @@
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="n">queue</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="m">5</span><span class="p">)</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="n">queue</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="m">4</span><span class="p">)</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a>
-<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="cm">/* Access front element */</span>
+<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="kd">val</span><span class="w"> </span><span class="nv">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="na">peek</span><span class="p">()</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a>
-<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="cm">/* Dequeue element */</span>
+<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="cm">/* Dequeue an element */</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="kd">val</span><span class="w"> </span><span class="nv">pop</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="na">poll</span><span class="p">()</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a>
-<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="cm">/* Get queue length */</span>
+<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="kd">val</span><span class="w"> </span><span class="nv">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a>
-<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="cm">/* Check if queue is empty */</span>
+<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="kd">val</span><span class="w"> </span><span class="nv">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="na">isEmpty</span><span class="p">()</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">queue.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Initialize queue</span>
-<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="c1"># Ruby's built-in queue (Thread::Queue) does not have peek and traversal methods, can use Array as a queue</span>
+<div class="highlight"><span class="filename">queue.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Initialize a queue</span>
+<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="c1"># Ruby's built-in Thread::Queue has no peek or traversal methods; use an Array as a queue</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="n">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="c1"># Enqueue elements</span>
@@ -5072,35 +5144,35 @@
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="n">queue</span><span class="o">.</span><span class="n">push</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="n">queue</span><span class="o">.</span><span class="n">push</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a>
-<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="c1"># Access front element</span>
+<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="c1"># Access the front element of the queue</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="o">.</span><span class="n">first</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a>
-<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="c1"># Dequeue element</span>
-<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="c1"># Please note that since it's an array, Array#shift has O(n) time complexity</span>
+<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="c1"># Dequeue an element</span>
+<a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="c1"># Note that this is an array, so Array#shift takes O(n) time</span>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a><span class="n">pop</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="o">.</span><span class="n">shift</span>
 <a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a>
-<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="c1"># Get queue length</span>
+<a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="c1"># Get the queue size</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a>
-<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="c1"># Check if queue is empty</span>
+<a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="c1"># Check whether the queue is empty</span>
 <a id="__codelineno-12-23" name="__codelineno-12-23" href="#__codelineno-12-23"></a><span class="n">is_empty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="o">.</span><span class="n">empty?</span>
 </code></pre></div>
 </div>
 </div>
 </div>
 <details class="pythontutor">
-<summary>Code Visualization</summary>
-<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=from%20collections%20import%20deque%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E9%98%9F%E5%88%97%0A%20%20%20%20%23%20%E5%9C%A8%20Python%20%E4%B8%AD%EF%BC%8C%E6%88%91%E4%BB%AC%E4%B8%80%E8%88%AC%E5%B0%86%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E7%B1%BB%20deque%20%E7%9C%8B%E4%BD%9C%E9%98%9F%E5%88%97%E4%BD%BF%E7%94%A8%0A%20%20%20%20%23%20%E8%99%BD%E7%84%B6%20queue.Queue%28%29%20%E6%98%AF%E7%BA%AF%E6%AD%A3%E7%9A%84%E9%98%9F%E5%88%97%E7%B1%BB%EF%BC%8C%E4%BD%86%E4%B8%8D%E5%A4%AA%E5%A5%BD%E7%94%A8%0A%20%20%20%20que%20%3D%20deque%28%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%85%A5%E9%98%9F%0A%20%20%20%20que.append%281%29%0A%20%20%20%20que.append%283%29%0A%20%20%20%20que.append%282%29%0A%20%20%20%20que.append%285%29%0A%20%20%20%20que.append%284%29%0A%20%20%20%20print%28%22%E9%98%9F%E5%88%97%20que%20%3D%22,%20que%29%0A%0A%20%20%20%20%23%20%E8%AE%BF%E9%97%AE%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%0A%20%20%20%20front%20%3D%20que%5B0%5D%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%20front%20%3D%22,%20front%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20pop%20%3D%20que.popleft%28%29%0A%20%20%20%20print%28%22%E5%87%BA%E9%98%9F%E5%85%83%E7%B4%A0%20pop%20%3D%22,%20pop%29%0A%20%20%20%20print%28%22%E5%87%BA%E9%98%9F%E5%90%8E%20que%20%3D%22,%20que%29%0A%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E9%98%9F%E5%88%97%E7%9A%84%E9%95%BF%E5%BA%A6%0A%20%20%20%20size%20%3D%20len%28que%29%0A%20%20%20%20print%28%22%E9%98%9F%E5%88%97%E9%95%BF%E5%BA%A6%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20%E5%88%A4%E6%96%AD%E9%98%9F%E5%88%97%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%0A%20%20%20%20is_empty%20%3D%20len%28que%29%20%3D%3D%200%0A%20%20%20%20print%28%22%E9%98%9F%E5%88%97%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%20%3D%22,%20is_empty%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20collections%20import%20deque%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E9%98%9F%E5%88%97%0A%20%20%20%20%23%20%E5%9C%A8%20Python%20%E4%B8%AD%EF%BC%8C%E6%88%91%E4%BB%AC%E4%B8%80%E8%88%AC%E5%B0%86%E5%8F%8C%E5%90%91%E9%98%9F%E5%88%97%E7%B1%BB%20deque%20%E7%9C%8B%E4%BD%9C%E9%98%9F%E5%88%97%E4%BD%BF%E7%94%A8%0A%20%20%20%20%23%20%E8%99%BD%E7%84%B6%20queue.Queue%28%29%20%E6%98%AF%E7%BA%AF%E6%AD%A3%E7%9A%84%E9%98%9F%E5%88%97%E7%B1%BB%EF%BC%8C%E4%BD%86%E4%B8%8D%E5%A4%AA%E5%A5%BD%E7%94%A8%0A%20%20%20%20que%20%3D%20deque%28%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%85%A5%E9%98%9F%0A%20%20%20%20que.append%281%29%0A%20%20%20%20que.append%283%29%0A%20%20%20%20que.append%282%29%0A%20%20%20%20que.append%285%29%0A%20%20%20%20que.append%284%29%0A%20%20%20%20print%28%22%E9%98%9F%E5%88%97%20que%20%3D%22,%20que%29%0A%0A%20%20%20%20%23%20%E8%AE%BF%E9%97%AE%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%0A%20%20%20%20front%20%3D%20que%5B0%5D%0A%20%20%20%20print%28%22%E9%98%9F%E9%A6%96%E5%85%83%E7%B4%A0%20front%20%3D%22,%20front%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%87%BA%E9%98%9F%0A%20%20%20%20pop%20%3D%20que.popleft%28%29%0A%20%20%20%20print%28%22%E5%87%BA%E9%98%9F%E5%85%83%E7%B4%A0%20pop%20%3D%22,%20pop%29%0A%20%20%20%20print%28%22%E5%87%BA%E9%98%9F%E5%90%8E%20que%20%3D%22,%20que%29%0A%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E9%98%9F%E5%88%97%E7%9A%84%E9%95%BF%E5%BA%A6%0A%20%20%20%20size%20%3D%20len%28que%29%0A%20%20%20%20print%28%22%E9%98%9F%E5%88%97%E9%95%BF%E5%BA%A6%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20%E5%88%A4%E6%96%AD%E9%98%9F%E5%88%97%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%0A%20%20%20%20is_empty%20%3D%20len%28que%29%20%3D%3D%200%0A%20%20%20%20print%28%22%E9%98%9F%E5%88%97%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%20%3D%22,%20is_empty%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=from%20collections%20import%20deque%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20queue%0A%20%20%20%20%23%20In%20Python%2C%20we%20usually%20use%20the%20deque%20class%20as%20a%20queue%0A%20%20%20%20%23%20Although%20queue.Queue%28%29%20is%20a%20dedicated%20queue%20class%2C%20it%20is%20less%20convenient%0A%20%20%20%20que%20%3D%20deque%28%29%0A%0A%20%20%20%20%23%20Enqueue%20elements%0A%20%20%20%20que.append%281%29%0A%20%20%20%20que.append%283%29%0A%20%20%20%20que.append%282%29%0A%20%20%20%20que.append%285%29%0A%20%20%20%20que.append%284%29%0A%20%20%20%20print%28%22Queue%3A%20que%20%3D%22%2C%20que%29%0A%0A%20%20%20%20%23%20Access%20the%20front%20element%0A%20%20%20%20front%20%3D%20que%5B0%5D%0A%20%20%20%20print%28%22Front%20element%3A%20front%20%3D%22%2C%20front%29%0A%0A%20%20%20%20%23%20Dequeue%20an%20element%0A%20%20%20%20pop%20%3D%20que.popleft%28%29%0A%20%20%20%20print%28%22Dequeued%20element%3A%20pop%20%3D%22%2C%20pop%29%0A%20%20%20%20print%28%22After%20dequeuing%3A%20que%20%3D%22%2C%20que%29%0A%0A%20%20%20%20%23%20Get%20the%20queue%20size%0A%20%20%20%20size%20%3D%20len%28que%29%0A%20%20%20%20print%28%22Queue%20size%3A%20size%20%3D%22%2C%20size%29%0A%0A%20%20%20%20%23%20Check%20whether%20the%20queue%20is%20empty%0A%20%20%20%20is_empty%20%3D%20len%28que%29%20%3D%3D%200%0A%20%20%20%20print%28%22Is%20the%20queue%20empty%20%3D%22%2C%20is_empty%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20collections%20import%20deque%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20queue%0A%20%20%20%20%23%20In%20Python%2C%20we%20usually%20use%20the%20deque%20class%20as%20a%20queue%0A%20%20%20%20%23%20Although%20queue.Queue%28%29%20is%20a%20dedicated%20queue%20class%2C%20it%20is%20less%20convenient%0A%20%20%20%20que%20%3D%20deque%28%29%0A%0A%20%20%20%20%23%20Enqueue%20elements%0A%20%20%20%20que.append%281%29%0A%20%20%20%20que.append%283%29%0A%20%20%20%20que.append%282%29%0A%20%20%20%20que.append%285%29%0A%20%20%20%20que.append%284%29%0A%20%20%20%20print%28%22Queue%3A%20que%20%3D%22%2C%20que%29%0A%0A%20%20%20%20%23%20Access%20the%20front%20element%0A%20%20%20%20front%20%3D%20que%5B0%5D%0A%20%20%20%20print%28%22Front%20element%3A%20front%20%3D%22%2C%20front%29%0A%0A%20%20%20%20%23%20Dequeue%20an%20element%0A%20%20%20%20pop%20%3D%20que.popleft%28%29%0A%20%20%20%20print%28%22Dequeued%20element%3A%20pop%20%3D%22%2C%20pop%29%0A%20%20%20%20print%28%22After%20dequeuing%3A%20que%20%3D%22%2C%20que%29%0A%0A%20%20%20%20%23%20Get%20the%20queue%20size%0A%20%20%20%20size%20%3D%20len%28que%29%0A%20%20%20%20print%28%22Queue%20size%3A%20size%20%3D%22%2C%20size%29%0A%0A%20%20%20%20%23%20Check%20whether%20the%20queue%20is%20empty%0A%20%20%20%20is_empty%20%3D%20len%28que%29%20%3D%3D%200%0A%20%20%20%20print%28%22Is%20the%20queue%20empty%20%3D%22%2C%20is_empty%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<h2 id="522-queue-implementation">5.2.2 &nbsp; Queue Implementation<a class="headerlink" href="#522-queue-implementation" title="Permanent link">&para;</a></h2>
-<p>To implement a queue, we need a data structure that allows adding elements at one end and removing elements at the other end. Both linked lists and arrays meet this requirement.</p>
-<h3 id="1-linked-list-implementation">1. &nbsp; Linked List Implementation<a class="headerlink" href="#1-linked-list-implementation" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 5-5, we can treat the "head node" and "tail node" of a linked list as the "front" and "rear" of the queue, respectively, with the rule that nodes can only be added at the rear and removed from the front.</p>
+<h2 id="522-implementing-a-queue">5.2.2 &nbsp; Implementing a queue<a class="headerlink" href="#522-implementing-a-queue" title="Permanent link">&para;</a></h2>
+<p>A queue needs a data structure that allows insertion at one end and deletion at the other. Both linked lists and arrays can meet this requirement.</p>
+<h3 id="1-implementation-with-a-linked-list">1. &nbsp; Implementation with a linked list<a class="headerlink" href="#1-implementation-with-a-linked-list" title="Permanent link">&para;</a></h3>
+<p>As shown below, treat the linked list's head and tail as the queue's front and rear. Nodes may be added only at the rear and removed only from the front.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:3"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">&lt;1&gt;</label><label for="__tabbed_2_2">&lt;2&gt;</label><label for="__tabbed_2_3">&lt;3&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Enqueue and dequeue operations in linked list implementation of queue" class="animation-figure" src="../queue.assets/linkedlist_queue_step1.png" /></p>
+<p><img alt="Enqueue and dequeue in a linked-list queue" class="animation-figure" src="../queue.assets/linkedlist_queue_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="linkedlist_queue_push" class="animation-figure" src="../queue.assets/linkedlist_queue_step2_push.png" /></p>
@@ -5110,14 +5182,14 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 5-5 &nbsp; Enqueue and dequeue operations in linked list implementation of queue </p>
+<p align="center"> Figure 5-5 &nbsp; Enqueue and dequeue in a linked-list queue </p>
 
-<p>Below is the code for implementing a queue using a linked list:</p>
+<p>Here is an implementation using a linked list:</p>
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 <div class="highlight"><span class="filename">linkedlist_queue.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">LinkedListQueue</span><span class="p">:</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Queue based on linked list implementation"""</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Queue implemented with a linked list"""</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="w">        </span><span class="sd">"""Constructor"""</span>
@@ -5126,22 +5198,22 @@
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a><span class="w">        </span><span class="sd">"""Get the length of the queue"""</span>
+<a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a><span class="w">        </span><span class="sd">"""Get the queue size"""</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span>
 <a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>
 <a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">is_empty</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">bool</span><span class="p">:</span>
-<a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a><span class="w">        </span><span class="sd">"""Check if the queue is empty"""</span>
+<a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a><span class="w">        </span><span class="sd">"""Check whether the queue is empty"""</span>
 <a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">==</span> <span class="mi">0</span>
 <a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>
 <a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a><span class="w">        </span><span class="sd">"""Enqueue"""</span>
 <a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a>        <span class="c1"># Add num after the tail node</span>
 <a id="__codelineno-13-21" name="__codelineno-13-21" href="#__codelineno-13-21"></a>        <span class="n">node</span> <span class="o">=</span> <span class="n">ListNode</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
-<a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a>        <span class="c1"># If the queue is empty, make both front and rear point to the node</span>
+<a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a>        <span class="c1"># If the queue is empty, make both the head and tail point to this node</span>
 <a id="__codelineno-13-23" name="__codelineno-13-23" href="#__codelineno-13-23"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-13-24" name="__codelineno-13-24" href="#__codelineno-13-24"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">=</span> <span class="n">node</span>
 <a id="__codelineno-13-25" name="__codelineno-13-25" href="#__codelineno-13-25"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span> <span class="o">=</span> <span class="n">node</span>
-<a id="__codelineno-13-26" name="__codelineno-13-26" href="#__codelineno-13-26"></a>        <span class="c1"># If the queue is not empty, add the node after the tail node</span>
+<a id="__codelineno-13-26" name="__codelineno-13-26" href="#__codelineno-13-26"></a>        <span class="c1"># If the queue is not empty, add this node after the tail node</span>
 <a id="__codelineno-13-27" name="__codelineno-13-27" href="#__codelineno-13-27"></a>        <span class="k">else</span><span class="p">:</span>
 <a id="__codelineno-13-28" name="__codelineno-13-28" href="#__codelineno-13-28"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span><span class="o">.</span><span class="n">next</span> <span class="o">=</span> <span class="n">node</span>
 <a id="__codelineno-13-29" name="__codelineno-13-29" href="#__codelineno-13-29"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_rear</span> <span class="o">=</span> <span class="n">node</span>
@@ -5150,19 +5222,19 @@
 <a id="__codelineno-13-32" name="__codelineno-13-32" href="#__codelineno-13-32"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">pop</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-13-33" name="__codelineno-13-33" href="#__codelineno-13-33"></a><span class="w">        </span><span class="sd">"""Dequeue"""</span>
 <a id="__codelineno-13-34" name="__codelineno-13-34" href="#__codelineno-13-34"></a>        <span class="n">num</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">peek</span><span class="p">()</span>
-<a id="__codelineno-13-35" name="__codelineno-13-35" href="#__codelineno-13-35"></a>        <span class="c1"># Delete head node</span>
+<a id="__codelineno-13-35" name="__codelineno-13-35" href="#__codelineno-13-35"></a>        <span class="c1"># Remove the head node</span>
 <a id="__codelineno-13-36" name="__codelineno-13-36" href="#__codelineno-13-36"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_front</span><span class="o">.</span><span class="n">next</span>
 <a id="__codelineno-13-37" name="__codelineno-13-37" href="#__codelineno-13-37"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">-=</span> <span class="mi">1</span>
 <a id="__codelineno-13-38" name="__codelineno-13-38" href="#__codelineno-13-38"></a>        <span class="k">return</span> <span class="n">num</span>
 <a id="__codelineno-13-39" name="__codelineno-13-39" href="#__codelineno-13-39"></a>
 <a id="__codelineno-13-40" name="__codelineno-13-40" href="#__codelineno-13-40"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">peek</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-41" name="__codelineno-13-41" href="#__codelineno-13-41"></a><span class="w">        </span><span class="sd">"""Access front of the queue element"""</span>
+<a id="__codelineno-13-41" name="__codelineno-13-41" href="#__codelineno-13-41"></a><span class="w">        </span><span class="sd">"""Access the front element"""</span>
 <a id="__codelineno-13-42" name="__codelineno-13-42" href="#__codelineno-13-42"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_empty</span><span class="p">():</span>
 <a id="__codelineno-13-43" name="__codelineno-13-43" href="#__codelineno-13-43"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Queue is empty"</span><span class="p">)</span>
 <a id="__codelineno-13-44" name="__codelineno-13-44" href="#__codelineno-13-44"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_front</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-13-45" name="__codelineno-13-45" href="#__codelineno-13-45"></a>
 <a id="__codelineno-13-46" name="__codelineno-13-46" href="#__codelineno-13-46"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">to_list</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
-<a id="__codelineno-13-47" name="__codelineno-13-47" href="#__codelineno-13-47"></a><span class="w">        </span><span class="sd">"""Convert to list for printing"""</span>
+<a id="__codelineno-13-47" name="__codelineno-13-47" href="#__codelineno-13-47"></a><span class="w">        </span><span class="sd">"""Convert to a list for printing"""</span>
 <a id="__codelineno-13-48" name="__codelineno-13-48" href="#__codelineno-13-48"></a>        <span class="n">queue</span> <span class="o">=</span> <span class="p">[]</span>
 <a id="__codelineno-13-49" name="__codelineno-13-49" href="#__codelineno-13-49"></a>        <span class="n">temp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_front</span>
 <a id="__codelineno-13-50" name="__codelineno-13-50" href="#__codelineno-13-50"></a>        <span class="k">while</span> <span class="n">temp</span><span class="p">:</span>
@@ -5172,10 +5244,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_queue.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Queue based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_queue.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Queue implemented with a linked list */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">LinkedListQueue</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">  </span><span class="k">private</span><span class="o">:</span>
-<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node front, tail node rear</span>
+<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node front and tail node rear</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">  </span><span class="k">public</span><span class="o">:</span>
@@ -5186,16 +5258,16 @@
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">    </span><span class="o">~</span><span class="n">LinkedListQueue</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">        </span><span class="c1">// Traverse linked list to delete nodes and free memory</span>
+<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">        </span><span class="c1">// Traverse the linked list and delete each node to free its memory</span>
 <a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">        </span><span class="n">freeMemoryLinkedList</span><span class="p">(</span><span class="n">front</span><span class="p">);</span>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a>
-<a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a>
-<a id="__codelineno-14-24" name="__codelineno-14-24" href="#__codelineno-14-24"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-14-24" name="__codelineno-14-24" href="#__codelineno-14-24"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-14-25" name="__codelineno-14-25" href="#__codelineno-14-25"></a><span class="w">    </span><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-26" name="__codelineno-14-26" href="#__codelineno-14-26"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-14-27" name="__codelineno-14-27" href="#__codelineno-14-27"></a><span class="w">    </span><span class="p">}</span>
@@ -5204,12 +5276,12 @@
 <a id="__codelineno-14-30" name="__codelineno-14-30" href="#__codelineno-14-30"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="n">push</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-31" name="__codelineno-14-31" href="#__codelineno-14-31"></a><span class="w">        </span><span class="c1">// Add num after the tail node</span>
 <a id="__codelineno-14-32" name="__codelineno-14-32" href="#__codelineno-14-32"></a><span class="w">        </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
-<a id="__codelineno-14-33" name="__codelineno-14-33" href="#__codelineno-14-33"></a><span class="w">        </span><span class="c1">// If the queue is empty, make both front and rear point to the node</span>
+<a id="__codelineno-14-33" name="__codelineno-14-33" href="#__codelineno-14-33"></a><span class="w">        </span><span class="c1">// If the queue is empty, make both the head and tail point to this node</span>
 <a id="__codelineno-14-34" name="__codelineno-14-34" href="#__codelineno-14-34"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-35" name="__codelineno-14-35" href="#__codelineno-14-35"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-14-36" name="__codelineno-14-36" href="#__codelineno-14-36"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-14-37" name="__codelineno-14-37" href="#__codelineno-14-37"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-14-38" name="__codelineno-14-38" href="#__codelineno-14-38"></a><span class="w">        </span><span class="c1">// If the queue is not empty, add the node after the tail node</span>
+<a id="__codelineno-14-38" name="__codelineno-14-38" href="#__codelineno-14-38"></a><span class="w">        </span><span class="c1">// If the queue is not empty, add this node after the tail node</span>
 <a id="__codelineno-14-39" name="__codelineno-14-39" href="#__codelineno-14-39"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-40" name="__codelineno-14-40" href="#__codelineno-14-40"></a><span class="w">            </span><span class="n">rear</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-14-41" name="__codelineno-14-41" href="#__codelineno-14-41"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
@@ -5220,7 +5292,7 @@
 <a id="__codelineno-14-46" name="__codelineno-14-46" href="#__codelineno-14-46"></a><span class="w">    </span><span class="cm">/* Dequeue */</span>
 <a id="__codelineno-14-47" name="__codelineno-14-47" href="#__codelineno-14-47"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-48" name="__codelineno-14-48" href="#__codelineno-14-48"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peek</span><span class="p">();</span>
-<a id="__codelineno-14-49" name="__codelineno-14-49" href="#__codelineno-14-49"></a><span class="w">        </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-14-49" name="__codelineno-14-49" href="#__codelineno-14-49"></a><span class="w">        </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-14-50" name="__codelineno-14-50" href="#__codelineno-14-50"></a><span class="w">        </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span>
 <a id="__codelineno-14-51" name="__codelineno-14-51" href="#__codelineno-14-51"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-14-52" name="__codelineno-14-52" href="#__codelineno-14-52"></a><span class="w">        </span><span class="c1">// Free memory</span>
@@ -5229,14 +5301,14 @@
 <a id="__codelineno-14-55" name="__codelineno-14-55" href="#__codelineno-14-55"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-14-56" name="__codelineno-14-56" href="#__codelineno-14-56"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-57" name="__codelineno-14-57" href="#__codelineno-14-57"></a>
-<a id="__codelineno-14-58" name="__codelineno-14-58" href="#__codelineno-14-58"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-14-58" name="__codelineno-14-58" href="#__codelineno-14-58"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-14-59" name="__codelineno-14-59" href="#__codelineno-14-59"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-60" name="__codelineno-14-60" href="#__codelineno-14-60"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-14-61" name="__codelineno-14-61" href="#__codelineno-14-61"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Queue is empty"</span><span class="p">);</span>
 <a id="__codelineno-14-62" name="__codelineno-14-62" href="#__codelineno-14-62"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">front</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-14-63" name="__codelineno-14-63" href="#__codelineno-14-63"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-64" name="__codelineno-14-64" href="#__codelineno-14-64"></a>
-<a id="__codelineno-14-65" name="__codelineno-14-65" href="#__codelineno-14-65"></a><span class="w">    </span><span class="cm">/* Convert linked list to Vector and return */</span>
+<a id="__codelineno-14-65" name="__codelineno-14-65" href="#__codelineno-14-65"></a><span class="w">    </span><span class="cm">/* Convert the linked list to a vector and return it */</span>
 <a id="__codelineno-14-66" name="__codelineno-14-66" href="#__codelineno-14-66"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">toVector</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-67" name="__codelineno-14-67" href="#__codelineno-14-67"></a><span class="w">        </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span>
 <a id="__codelineno-14-68" name="__codelineno-14-68" href="#__codelineno-14-68"></a><span class="w">        </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">(</span><span class="n">size</span><span class="p">());</span>
@@ -5250,9 +5322,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_queue.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Queue based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_queue.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Queue implemented with a linked list */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kd">class</span> <span class="nc">LinkedListQueue</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="n">rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node front, tail node rear</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="n">rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node front and tail node rear</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="nf">LinkedListQueue</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
@@ -5260,12 +5332,12 @@
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">        </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a>
-<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a>
-<a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">boolean</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w">    </span><span class="p">}</span>
@@ -5274,11 +5346,11 @@
 <a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w">        </span><span class="c1">// Add num after the tail node</span>
 <a id="__codelineno-15-24" name="__codelineno-15-24" href="#__codelineno-15-24"></a><span class="w">        </span><span class="n">ListNode</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
-<a id="__codelineno-15-25" name="__codelineno-15-25" href="#__codelineno-15-25"></a><span class="w">        </span><span class="c1">// If the queue is empty, make both front and rear point to the node</span>
+<a id="__codelineno-15-25" name="__codelineno-15-25" href="#__codelineno-15-25"></a><span class="w">        </span><span class="c1">// If the queue is empty, make both the head and tail point to this node</span>
 <a id="__codelineno-15-26" name="__codelineno-15-26" href="#__codelineno-15-26"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-27" name="__codelineno-15-27" href="#__codelineno-15-27"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-15-28" name="__codelineno-15-28" href="#__codelineno-15-28"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
-<a id="__codelineno-15-29" name="__codelineno-15-29" href="#__codelineno-15-29"></a><span class="w">        </span><span class="c1">// If the queue is not empty, add the node after the tail node</span>
+<a id="__codelineno-15-29" name="__codelineno-15-29" href="#__codelineno-15-29"></a><span class="w">        </span><span class="c1">// If the queue is not empty, add this node after the tail node</span>
 <a id="__codelineno-15-30" name="__codelineno-15-30" href="#__codelineno-15-30"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-31" name="__codelineno-15-31" href="#__codelineno-15-31"></a><span class="w">            </span><span class="n">rear</span><span class="p">.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-15-32" name="__codelineno-15-32" href="#__codelineno-15-32"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
@@ -5289,20 +5361,20 @@
 <a id="__codelineno-15-37" name="__codelineno-15-37" href="#__codelineno-15-37"></a><span class="w">    </span><span class="cm">/* Dequeue */</span>
 <a id="__codelineno-15-38" name="__codelineno-15-38" href="#__codelineno-15-38"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-39" name="__codelineno-15-39" href="#__codelineno-15-39"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peek</span><span class="p">();</span>
-<a id="__codelineno-15-40" name="__codelineno-15-40" href="#__codelineno-15-40"></a><span class="w">        </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-15-40" name="__codelineno-15-40" href="#__codelineno-15-40"></a><span class="w">        </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-15-41" name="__codelineno-15-41" href="#__codelineno-15-41"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">.</span><span class="na">next</span><span class="p">;</span>
 <a id="__codelineno-15-42" name="__codelineno-15-42" href="#__codelineno-15-42"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-15-43" name="__codelineno-15-43" href="#__codelineno-15-43"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-15-44" name="__codelineno-15-44" href="#__codelineno-15-44"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-45" name="__codelineno-15-45" href="#__codelineno-15-45"></a>
-<a id="__codelineno-15-46" name="__codelineno-15-46" href="#__codelineno-15-46"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-15-46" name="__codelineno-15-46" href="#__codelineno-15-46"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-15-47" name="__codelineno-15-47" href="#__codelineno-15-47"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-48" name="__codelineno-15-48" href="#__codelineno-15-48"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-15-49" name="__codelineno-15-49" href="#__codelineno-15-49"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">();</span>
 <a id="__codelineno-15-50" name="__codelineno-15-50" href="#__codelineno-15-50"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">front</span><span class="p">.</span><span class="na">val</span><span class="p">;</span>
 <a id="__codelineno-15-51" name="__codelineno-15-51" href="#__codelineno-15-51"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-52" name="__codelineno-15-52" href="#__codelineno-15-52"></a>
-<a id="__codelineno-15-53" name="__codelineno-15-53" href="#__codelineno-15-53"></a><span class="w">    </span><span class="cm">/* Convert linked list to Array and return */</span>
+<a id="__codelineno-15-53" name="__codelineno-15-53" href="#__codelineno-15-53"></a><span class="w">    </span><span class="cm">/* Convert the linked list to an array and return it */</span>
 <a id="__codelineno-15-54" name="__codelineno-15-54" href="#__codelineno-15-54"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="nf">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-55" name="__codelineno-15-55" href="#__codelineno-15-55"></a><span class="w">        </span><span class="n">ListNode</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span>
 <a id="__codelineno-15-56" name="__codelineno-15-56" href="#__codelineno-15-56"></a><span class="w">        </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">size</span><span class="p">()</span><span class="o">]</span><span class="p">;</span>
@@ -5316,9 +5388,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_queue.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Queue based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_queue.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Queue implemented with a linked list */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">LinkedListQueue</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="n">rear</span><span class="p">;</span><span class="w">  </span><span class="c1">// Head node front, tail node rear</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="n">rear</span><span class="p">;</span><span class="w">  </span><span class="c1">// Head node front and tail node rear </span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nf">LinkedListQueue</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
@@ -5326,12 +5398,12 @@
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">        </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a>
-<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a>
-<a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="nf">IsEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nf">Size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w">    </span><span class="p">}</span>
@@ -5340,11 +5412,11 @@
 <a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="k">void</span><span class="w"> </span><span class="nf">Push</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">        </span><span class="c1">// Add num after the tail node</span>
 <a id="__codelineno-16-24" name="__codelineno-16-24" href="#__codelineno-16-24"></a><span class="w">        </span><span class="n">ListNode</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
-<a id="__codelineno-16-25" name="__codelineno-16-25" href="#__codelineno-16-25"></a><span class="w">        </span><span class="c1">// If the queue is empty, make both front and rear point to the node</span>
+<a id="__codelineno-16-25" name="__codelineno-16-25" href="#__codelineno-16-25"></a><span class="w">        </span><span class="c1">// If the queue is empty, make both the head and tail point to this node</span>
 <a id="__codelineno-16-26" name="__codelineno-16-26" href="#__codelineno-16-26"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-27" name="__codelineno-16-27" href="#__codelineno-16-27"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-16-28" name="__codelineno-16-28" href="#__codelineno-16-28"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
-<a id="__codelineno-16-29" name="__codelineno-16-29" href="#__codelineno-16-29"></a><span class="w">            </span><span class="c1">// If the queue is not empty, add the node after the tail node</span>
+<a id="__codelineno-16-29" name="__codelineno-16-29" href="#__codelineno-16-29"></a><span class="w">            </span><span class="c1">// If the queue is not empty, add this node after the tail node</span>
 <a id="__codelineno-16-30" name="__codelineno-16-30" href="#__codelineno-16-30"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">rear</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-31" name="__codelineno-16-31" href="#__codelineno-16-31"></a><span class="w">            </span><span class="n">rear</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-16-32" name="__codelineno-16-32" href="#__codelineno-16-32"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
@@ -5355,20 +5427,20 @@
 <a id="__codelineno-16-37" name="__codelineno-16-37" href="#__codelineno-16-37"></a><span class="w">    </span><span class="cm">/* Dequeue */</span>
 <a id="__codelineno-16-38" name="__codelineno-16-38" href="#__codelineno-16-38"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-39" name="__codelineno-16-39" href="#__codelineno-16-39"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Peek</span><span class="p">();</span>
-<a id="__codelineno-16-40" name="__codelineno-16-40" href="#__codelineno-16-40"></a><span class="w">        </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-16-40" name="__codelineno-16-40" href="#__codelineno-16-40"></a><span class="w">        </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-16-41" name="__codelineno-16-41" href="#__codelineno-16-41"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="o">?.</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-16-42" name="__codelineno-16-42" href="#__codelineno-16-42"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-16-43" name="__codelineno-16-43" href="#__codelineno-16-43"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-16-44" name="__codelineno-16-44" href="#__codelineno-16-44"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-45" name="__codelineno-16-45" href="#__codelineno-16-45"></a>
-<a id="__codelineno-16-46" name="__codelineno-16-46" href="#__codelineno-16-46"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-16-46" name="__codelineno-16-46" href="#__codelineno-16-46"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-16-47" name="__codelineno-16-47" href="#__codelineno-16-47"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-48" name="__codelineno-16-48" href="#__codelineno-16-48"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">IsEmpty</span><span class="p">())</span>
 <a id="__codelineno-16-49" name="__codelineno-16-49" href="#__codelineno-16-49"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">Exception</span><span class="p">();</span>
 <a id="__codelineno-16-50" name="__codelineno-16-50" href="#__codelineno-16-50"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">front</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-16-51" name="__codelineno-16-51" href="#__codelineno-16-51"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-52" name="__codelineno-16-52" href="#__codelineno-16-52"></a>
-<a id="__codelineno-16-53" name="__codelineno-16-53" href="#__codelineno-16-53"></a><span class="w">    </span><span class="cm">/* Convert linked list to Array and return */</span>
+<a id="__codelineno-16-53" name="__codelineno-16-53" href="#__codelineno-16-53"></a><span class="w">    </span><span class="cm">/* Convert the linked list to an array and return it */</span>
 <a id="__codelineno-16-54" name="__codelineno-16-54" href="#__codelineno-16-54"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="nf">ToArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-55" name="__codelineno-16-55" href="#__codelineno-16-55"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span>
 <a id="__codelineno-16-56" name="__codelineno-16-56" href="#__codelineno-16-56"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="p">[];</span>
@@ -5385,13 +5457,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_queue.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Queue based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_queue.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Queue implemented with a linked list */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">linkedListQueue</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Use built-in package list to implement queue</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Use the standard library list package to implement the queue</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="nx">data</span><span class="w"> </span><span class="o">*</span><span class="nx">list</span><span class="p">.</span><span class="nx">List</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="p">}</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a>
-<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="cm">/* Access front of the queue element */</span>
+<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="cm">/* Initialize the queue */</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="kd">func</span><span class="w"> </span><span class="nx">newLinkedListQueue</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListQueue</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">linkedListQueue</span><span class="p">{</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">        </span><span class="nx">data</span><span class="p">:</span><span class="w"> </span><span class="nx">list</span><span class="p">.</span><span class="nx">New</span><span class="p">(),</span>
@@ -5413,7 +5485,7 @@
 <a id="__codelineno-17-26" name="__codelineno-17-26" href="#__codelineno-17-26"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">e</span><span class="p">.</span><span class="nx">Value</span>
 <a id="__codelineno-17-27" name="__codelineno-17-27" href="#__codelineno-17-27"></a><span class="p">}</span>
 <a id="__codelineno-17-28" name="__codelineno-17-28" href="#__codelineno-17-28"></a>
-<a id="__codelineno-17-29" name="__codelineno-17-29" href="#__codelineno-17-29"></a><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-17-29" name="__codelineno-17-29" href="#__codelineno-17-29"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-17-30" name="__codelineno-17-30" href="#__codelineno-17-30"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListQueue</span><span class="p">)</span><span class="w"> </span><span class="nx">peek</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-31" name="__codelineno-17-31" href="#__codelineno-17-31"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-32" name="__codelineno-17-32" href="#__codelineno-17-32"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
@@ -5422,24 +5494,24 @@
 <a id="__codelineno-17-35" name="__codelineno-17-35" href="#__codelineno-17-35"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">e</span><span class="p">.</span><span class="nx">Value</span>
 <a id="__codelineno-17-36" name="__codelineno-17-36" href="#__codelineno-17-36"></a><span class="p">}</span>
 <a id="__codelineno-17-37" name="__codelineno-17-37" href="#__codelineno-17-37"></a>
-<a id="__codelineno-17-38" name="__codelineno-17-38" href="#__codelineno-17-38"></a><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-17-38" name="__codelineno-17-38" href="#__codelineno-17-38"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-17-39" name="__codelineno-17-39" href="#__codelineno-17-39"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListQueue</span><span class="p">)</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-40" name="__codelineno-17-40" href="#__codelineno-17-40"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">data</span><span class="p">.</span><span class="nx">Len</span><span class="p">()</span>
 <a id="__codelineno-17-41" name="__codelineno-17-41" href="#__codelineno-17-41"></a><span class="p">}</span>
 <a id="__codelineno-17-42" name="__codelineno-17-42" href="#__codelineno-17-42"></a>
-<a id="__codelineno-17-43" name="__codelineno-17-43" href="#__codelineno-17-43"></a><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-17-43" name="__codelineno-17-43" href="#__codelineno-17-43"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-17-44" name="__codelineno-17-44" href="#__codelineno-17-44"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListQueue</span><span class="p">)</span><span class="w"> </span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-45" name="__codelineno-17-45" href="#__codelineno-17-45"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">data</span><span class="p">.</span><span class="nx">Len</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-17-46" name="__codelineno-17-46" href="#__codelineno-17-46"></a><span class="p">}</span>
 <a id="__codelineno-17-47" name="__codelineno-17-47" href="#__codelineno-17-47"></a>
-<a id="__codelineno-17-48" name="__codelineno-17-48" href="#__codelineno-17-48"></a><span class="cm">/* Get List for printing */</span>
+<a id="__codelineno-17-48" name="__codelineno-17-48" href="#__codelineno-17-48"></a><span class="cm">/* Get the list for printing */</span>
 <a id="__codelineno-17-49" name="__codelineno-17-49" href="#__codelineno-17-49"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListQueue</span><span class="p">)</span><span class="w"> </span><span class="nx">toList</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="nx">list</span><span class="p">.</span><span class="nx">List</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-50" name="__codelineno-17-50" href="#__codelineno-17-50"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">data</span>
 <a id="__codelineno-17-51" name="__codelineno-17-51" href="#__codelineno-17-51"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_queue.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Queue based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_queue.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Queue implemented with a linked list */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">LinkedListQueue</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">front</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Head node</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">rear</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Tail node</span>
@@ -5449,12 +5521,12 @@
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a>
-<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">        </span><span class="n">_size</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a>
-<a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">        </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">    </span><span class="p">}</span>
@@ -5463,12 +5535,12 @@
 <a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-23" name="__codelineno-18-23" href="#__codelineno-18-23"></a><span class="w">        </span><span class="c1">// Add num after the tail node</span>
 <a id="__codelineno-18-24" name="__codelineno-18-24" href="#__codelineno-18-24"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
-<a id="__codelineno-18-25" name="__codelineno-18-25" href="#__codelineno-18-25"></a><span class="w">        </span><span class="c1">// If the queue is empty, make both front and rear point to the node</span>
+<a id="__codelineno-18-25" name="__codelineno-18-25" href="#__codelineno-18-25"></a><span class="w">        </span><span class="c1">// If the queue is empty, make both the head and tail point to this node</span>
 <a id="__codelineno-18-26" name="__codelineno-18-26" href="#__codelineno-18-26"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">front</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-27" name="__codelineno-18-27" href="#__codelineno-18-27"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-18-28" name="__codelineno-18-28" href="#__codelineno-18-28"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-18-29" name="__codelineno-18-29" href="#__codelineno-18-29"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-30" name="__codelineno-18-30" href="#__codelineno-18-30"></a><span class="w">        </span><span class="c1">// If the queue is not empty, add the node after the tail node</span>
+<a id="__codelineno-18-30" name="__codelineno-18-30" href="#__codelineno-18-30"></a><span class="w">        </span><span class="c1">// If the queue is not empty, add this node after the tail node</span>
 <a id="__codelineno-18-31" name="__codelineno-18-31" href="#__codelineno-18-31"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-32" name="__codelineno-18-32" href="#__codelineno-18-32"></a><span class="w">            </span><span class="n">rear</span><span class="p">?.</span><span class="n">next</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-18-33" name="__codelineno-18-33" href="#__codelineno-18-33"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span>
@@ -5480,13 +5552,13 @@
 <a id="__codelineno-18-39" name="__codelineno-18-39" href="#__codelineno-18-39"></a><span class="w">    </span><span class="p">@</span><span class="n">discardableResult</span>
 <a id="__codelineno-18-40" name="__codelineno-18-40" href="#__codelineno-18-40"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">pop</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-41" name="__codelineno-18-41" href="#__codelineno-18-41"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">peek</span><span class="p">()</span>
-<a id="__codelineno-18-42" name="__codelineno-18-42" href="#__codelineno-18-42"></a><span class="w">        </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-18-42" name="__codelineno-18-42" href="#__codelineno-18-42"></a><span class="w">        </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-18-43" name="__codelineno-18-43" href="#__codelineno-18-43"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">front</span><span class="p">?.</span><span class="n">next</span>
 <a id="__codelineno-18-44" name="__codelineno-18-44" href="#__codelineno-18-44"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-18-45" name="__codelineno-18-45" href="#__codelineno-18-45"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-18-46" name="__codelineno-18-46" href="#__codelineno-18-46"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-47" name="__codelineno-18-47" href="#__codelineno-18-47"></a>
-<a id="__codelineno-18-48" name="__codelineno-18-48" href="#__codelineno-18-48"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-18-48" name="__codelineno-18-48" href="#__codelineno-18-48"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-18-49" name="__codelineno-18-49" href="#__codelineno-18-49"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">peek</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-50" name="__codelineno-18-50" href="#__codelineno-18-50"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-51" name="__codelineno-18-51" href="#__codelineno-18-51"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Queue is empty"</span><span class="p">)</span>
@@ -5494,7 +5566,7 @@
 <a id="__codelineno-18-53" name="__codelineno-18-53" href="#__codelineno-18-53"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">front</span><span class="p">!.</span><span class="n">val</span>
 <a id="__codelineno-18-54" name="__codelineno-18-54" href="#__codelineno-18-54"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-55" name="__codelineno-18-55" href="#__codelineno-18-55"></a>
-<a id="__codelineno-18-56" name="__codelineno-18-56" href="#__codelineno-18-56"></a><span class="w">    </span><span class="cm">/* Convert linked list to Array and return */</span>
+<a id="__codelineno-18-56" name="__codelineno-18-56" href="#__codelineno-18-56"></a><span class="w">    </span><span class="cm">/* Convert the linked list to an array and return it */</span>
 <a id="__codelineno-18-57" name="__codelineno-18-57" href="#__codelineno-18-57"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-58" name="__codelineno-18-58" href="#__codelineno-18-58"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">front</span>
 <a id="__codelineno-18-59" name="__codelineno-18-59" href="#__codelineno-18-59"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">size</span><span class="p">())</span>
@@ -5508,10 +5580,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_queue.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Queue based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_queue.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Queue implemented with a linked list */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">LinkedListQueue</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="n">#front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front node #front</span>
-<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="n">#rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Rear node #rear</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="n">#front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node #front</span>
+<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="n">#rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Tail node #rear</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="n">#queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
@@ -5519,12 +5591,12 @@
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a>
-<a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">    </span><span class="nx">get</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="p">;</span>
 <a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a>
-<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">    </span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">    </span><span class="p">}</span>
@@ -5533,11 +5605,11 @@
 <a id="__codelineno-19-23" name="__codelineno-19-23" href="#__codelineno-19-23"></a><span class="w">    </span><span class="nx">push</span><span class="p">(</span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-24" name="__codelineno-19-24" href="#__codelineno-19-24"></a><span class="w">        </span><span class="c1">// Add num after the tail node</span>
 <a id="__codelineno-19-25" name="__codelineno-19-25" href="#__codelineno-19-25"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="nx">num</span><span class="p">);</span>
-<a id="__codelineno-19-26" name="__codelineno-19-26" href="#__codelineno-19-26"></a><span class="w">        </span><span class="c1">// If the queue is empty, make both front and rear point to the node</span>
+<a id="__codelineno-19-26" name="__codelineno-19-26" href="#__codelineno-19-26"></a><span class="w">        </span><span class="c1">// If the queue is empty, make both the head and tail point to this node</span>
 <a id="__codelineno-19-27" name="__codelineno-19-27" href="#__codelineno-19-27"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-28" name="__codelineno-19-28" href="#__codelineno-19-28"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-19-29" name="__codelineno-19-29" href="#__codelineno-19-29"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
-<a id="__codelineno-19-30" name="__codelineno-19-30" href="#__codelineno-19-30"></a><span class="w">            </span><span class="c1">// If the queue is not empty, add the node after the tail node</span>
+<a id="__codelineno-19-30" name="__codelineno-19-30" href="#__codelineno-19-30"></a><span class="w">            </span><span class="c1">// If the queue is not empty, add this node after the tail node</span>
 <a id="__codelineno-19-31" name="__codelineno-19-31" href="#__codelineno-19-31"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-32" name="__codelineno-19-32" href="#__codelineno-19-32"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-19-33" name="__codelineno-19-33" href="#__codelineno-19-33"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="n">#rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
@@ -5548,19 +5620,19 @@
 <a id="__codelineno-19-38" name="__codelineno-19-38" href="#__codelineno-19-38"></a><span class="w">    </span><span class="cm">/* Dequeue */</span>
 <a id="__codelineno-19-39" name="__codelineno-19-39" href="#__codelineno-19-39"></a><span class="w">    </span><span class="nx">pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-40" name="__codelineno-19-40" href="#__codelineno-19-40"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">peek</span><span class="p">();</span>
-<a id="__codelineno-19-41" name="__codelineno-19-41" href="#__codelineno-19-41"></a><span class="w">        </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-19-41" name="__codelineno-19-41" href="#__codelineno-19-41"></a><span class="w">        </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-19-42" name="__codelineno-19-42" href="#__codelineno-19-42"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">.</span><span class="nx">next</span><span class="p">;</span>
 <a id="__codelineno-19-43" name="__codelineno-19-43" href="#__codelineno-19-43"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-19-44" name="__codelineno-19-44" href="#__codelineno-19-44"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-19-45" name="__codelineno-19-45" href="#__codelineno-19-45"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-46" name="__codelineno-19-46" href="#__codelineno-19-46"></a>
-<a id="__codelineno-19-47" name="__codelineno-19-47" href="#__codelineno-19-47"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-19-47" name="__codelineno-19-47" href="#__codelineno-19-47"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-19-48" name="__codelineno-19-48" href="#__codelineno-19-48"></a><span class="w">    </span><span class="nx">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-49" name="__codelineno-19-49" href="#__codelineno-19-49"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Queue is empty'</span><span class="p">);</span>
 <a id="__codelineno-19-50" name="__codelineno-19-50" href="#__codelineno-19-50"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-19-51" name="__codelineno-19-51" href="#__codelineno-19-51"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-52" name="__codelineno-19-52" href="#__codelineno-19-52"></a>
-<a id="__codelineno-19-53" name="__codelineno-19-53" href="#__codelineno-19-53"></a><span class="w">    </span><span class="cm">/* Convert linked list to Array and return */</span>
+<a id="__codelineno-19-53" name="__codelineno-19-53" href="#__codelineno-19-53"></a><span class="w">    </span><span class="cm">/* Convert the linked list to an Array and return it */</span>
 <a id="__codelineno-19-54" name="__codelineno-19-54" href="#__codelineno-19-54"></a><span class="w">    </span><span class="nx">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-55" name="__codelineno-19-55" href="#__codelineno-19-55"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">;</span>
 <a id="__codelineno-19-56" name="__codelineno-19-56" href="#__codelineno-19-56"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">);</span>
@@ -5574,7 +5646,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_queue.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Queue based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_queue.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Queue implemented with a linked list */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">LinkedListQueue</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">front</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node front</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">rear</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span><span class="w"> </span><span class="c1">// Tail node rear</span>
@@ -5585,12 +5657,12 @@
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a>
-<a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">    </span><span class="nx">get</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="p">;</span>
 <a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a>
-<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">    </span><span class="nx">isEmpty</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">    </span><span class="p">}</span>
@@ -5599,11 +5671,11 @@
 <a id="__codelineno-20-23" name="__codelineno-20-23" href="#__codelineno-20-23"></a><span class="w">    </span><span class="nx">push</span><span class="p">(</span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">        </span><span class="c1">// Add num after the tail node</span>
 <a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">ListNode</span><span class="p">(</span><span class="nx">num</span><span class="p">);</span>
-<a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="w">        </span><span class="c1">// If the queue is empty, make both front and rear point to the node</span>
+<a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="w">        </span><span class="c1">// If the queue is empty, make both the head and tail point to this node</span>
 <a id="__codelineno-20-27" name="__codelineno-20-27" href="#__codelineno-20-27"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-28" name="__codelineno-20-28" href="#__codelineno-20-28"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-20-29" name="__codelineno-20-29" href="#__codelineno-20-29"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
-<a id="__codelineno-20-30" name="__codelineno-20-30" href="#__codelineno-20-30"></a><span class="w">            </span><span class="c1">// If the queue is not empty, add the node after the tail node</span>
+<a id="__codelineno-20-30" name="__codelineno-20-30" href="#__codelineno-20-30"></a><span class="w">            </span><span class="c1">// If the queue is not empty, add this node after the tail node</span>
 <a id="__codelineno-20-31" name="__codelineno-20-31" href="#__codelineno-20-31"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-32" name="__codelineno-20-32" href="#__codelineno-20-32"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="o">!</span><span class="p">.</span><span class="nx">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-20-33" name="__codelineno-20-33" href="#__codelineno-20-33"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
@@ -5615,19 +5687,19 @@
 <a id="__codelineno-20-39" name="__codelineno-20-39" href="#__codelineno-20-39"></a><span class="w">    </span><span class="nx">pop</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-40" name="__codelineno-20-40" href="#__codelineno-20-40"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">peek</span><span class="p">();</span>
 <a id="__codelineno-20-41" name="__codelineno-20-41" href="#__codelineno-20-41"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Queue is empty'</span><span class="p">);</span>
-<a id="__codelineno-20-42" name="__codelineno-20-42" href="#__codelineno-20-42"></a><span class="w">        </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-20-42" name="__codelineno-20-42" href="#__codelineno-20-42"></a><span class="w">        </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-20-43" name="__codelineno-20-43" href="#__codelineno-20-43"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="p">.</span><span class="nx">next</span><span class="p">;</span>
 <a id="__codelineno-20-44" name="__codelineno-20-44" href="#__codelineno-20-44"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-20-45" name="__codelineno-20-45" href="#__codelineno-20-45"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-20-46" name="__codelineno-20-46" href="#__codelineno-20-46"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-47" name="__codelineno-20-47" href="#__codelineno-20-47"></a>
-<a id="__codelineno-20-48" name="__codelineno-20-48" href="#__codelineno-20-48"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-20-48" name="__codelineno-20-48" href="#__codelineno-20-48"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-20-49" name="__codelineno-20-49" href="#__codelineno-20-49"></a><span class="w">    </span><span class="nx">peek</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-50" name="__codelineno-20-50" href="#__codelineno-20-50"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Queue is empty'</span><span class="p">);</span>
 <a id="__codelineno-20-51" name="__codelineno-20-51" href="#__codelineno-20-51"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="o">!</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-20-52" name="__codelineno-20-52" href="#__codelineno-20-52"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-53" name="__codelineno-20-53" href="#__codelineno-20-53"></a>
-<a id="__codelineno-20-54" name="__codelineno-20-54" href="#__codelineno-20-54"></a><span class="w">    </span><span class="cm">/* Convert linked list to Array and return */</span>
+<a id="__codelineno-20-54" name="__codelineno-20-54" href="#__codelineno-20-54"></a><span class="w">    </span><span class="cm">/* Convert the linked list to an Array and return it */</span>
 <a id="__codelineno-20-55" name="__codelineno-20-55" href="#__codelineno-20-55"></a><span class="w">    </span><span class="nx">toArray</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-56" name="__codelineno-20-56" href="#__codelineno-20-56"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="p">;</span>
 <a id="__codelineno-20-57" name="__codelineno-20-57" href="#__codelineno-20-57"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">);</span>
@@ -5641,37 +5713,37 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_queue.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Queue based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_queue.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Queue implemented with a linked list */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">LinkedListQueue</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">_front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Head node _front</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">_rear</span><span class="p">;</span><span class="w"> </span><span class="c1">// Tail node _rear</span>
-<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">_queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Queue length</span>
+<a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">_queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Queue size</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="n">LinkedListQueue</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">    </span><span class="n">_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">    </span><span class="n">_rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a>
-<a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">  </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">  </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_queSize</span><span class="p">;</span>
 <a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a>
-<a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">  </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">  </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">  </span><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a>
 <a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">  </span><span class="cm">/* Enqueue */</span>
 <a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">push</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">    </span><span class="c1">// Add _num after tail node</span>
+<a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">    </span><span class="c1">// Add _num after the tail node</span>
 <a id="__codelineno-21-25" name="__codelineno-21-25" href="#__codelineno-21-25"></a><span class="w">    </span><span class="kd">final</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="n">_num</span><span class="p">);</span>
-<a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a><span class="w">    </span><span class="c1">// If the queue is empty, make both front and rear point to the node</span>
+<a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a><span class="w">    </span><span class="c1">// If the queue is empty, make both the head and tail point to this node</span>
 <a id="__codelineno-21-27" name="__codelineno-21-27" href="#__codelineno-21-27"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_front</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-28" name="__codelineno-21-28" href="#__codelineno-21-28"></a><span class="w">      </span><span class="n">_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-21-29" name="__codelineno-21-29" href="#__codelineno-21-29"></a><span class="w">      </span><span class="n">_rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-21-30" name="__codelineno-21-30" href="#__codelineno-21-30"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-31" name="__codelineno-21-31" href="#__codelineno-21-31"></a><span class="w">      </span><span class="c1">// If the queue is not empty, add the node after the tail node</span>
+<a id="__codelineno-21-31" name="__codelineno-21-31" href="#__codelineno-21-31"></a><span class="w">      </span><span class="c1">// If the queue is not empty, add this node after the tail node</span>
 <a id="__codelineno-21-32" name="__codelineno-21-32" href="#__codelineno-21-32"></a><span class="w">      </span><span class="n">_rear</span><span class="o">!</span><span class="p">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-21-33" name="__codelineno-21-33" href="#__codelineno-21-33"></a><span class="w">      </span><span class="n">_rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-21-34" name="__codelineno-21-34" href="#__codelineno-21-34"></a><span class="w">    </span><span class="p">}</span>
@@ -5681,13 +5753,13 @@
 <a id="__codelineno-21-38" name="__codelineno-21-38" href="#__codelineno-21-38"></a><span class="w">  </span><span class="cm">/* Dequeue */</span>
 <a id="__codelineno-21-39" name="__codelineno-21-39" href="#__codelineno-21-39"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-40" name="__codelineno-21-40" href="#__codelineno-21-40"></a><span class="w">    </span><span class="kd">final</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peek</span><span class="p">();</span>
-<a id="__codelineno-21-41" name="__codelineno-21-41" href="#__codelineno-21-41"></a><span class="w">    </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-21-41" name="__codelineno-21-41" href="#__codelineno-21-41"></a><span class="w">    </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-21-42" name="__codelineno-21-42" href="#__codelineno-21-42"></a><span class="w">    </span><span class="n">_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_front</span><span class="o">!</span><span class="p">.</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-21-43" name="__codelineno-21-43" href="#__codelineno-21-43"></a><span class="w">    </span><span class="n">_queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-21-44" name="__codelineno-21-44" href="#__codelineno-21-44"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span>
 <a id="__codelineno-21-45" name="__codelineno-21-45" href="#__codelineno-21-45"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-46" name="__codelineno-21-46" href="#__codelineno-21-46"></a>
-<a id="__codelineno-21-47" name="__codelineno-21-47" href="#__codelineno-21-47"></a><span class="w">  </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-21-47" name="__codelineno-21-47" href="#__codelineno-21-47"></a><span class="w">  </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-21-48" name="__codelineno-21-48" href="#__codelineno-21-48"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-49" name="__codelineno-21-49" href="#__codelineno-21-49"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-50" name="__codelineno-21-50" href="#__codelineno-21-50"></a><span class="w">      </span><span class="k">throw</span><span class="w"> </span><span class="n">Exception</span><span class="p">(</span><span class="s1">'Queue is empty'</span><span class="p">);</span>
@@ -5695,7 +5767,7 @@
 <a id="__codelineno-21-52" name="__codelineno-21-52" href="#__codelineno-21-52"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_front</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-21-53" name="__codelineno-21-53" href="#__codelineno-21-53"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-54" name="__codelineno-21-54" href="#__codelineno-21-54"></a>
-<a id="__codelineno-21-55" name="__codelineno-21-55" href="#__codelineno-21-55"></a><span class="w">  </span><span class="cm">/* Convert linked list to Array and return */</span>
+<a id="__codelineno-21-55" name="__codelineno-21-55" href="#__codelineno-21-55"></a><span class="w">  </span><span class="cm">/* Convert the linked list to an array and return it */</span>
 <a id="__codelineno-21-56" name="__codelineno-21-56" href="#__codelineno-21-56"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-57" name="__codelineno-21-57" href="#__codelineno-21-57"></a><span class="w">    </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_front</span><span class="p">;</span>
 <a id="__codelineno-21-58" name="__codelineno-21-58" href="#__codelineno-21-58"></a><span class="w">    </span><span class="kd">final</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
@@ -5709,12 +5781,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_queue.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Queue based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_queue.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Queue implemented with a linked list */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="cp">#[allow(dead_code)]</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="k">pub</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">LinkedListQueue</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="n">front</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="c1">// Head node front</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="n">rear</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span><span class="w">  </span><span class="c1">// Tail node rear</span>
-<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="n">que_size</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w">                         </span><span class="c1">// Queue length</span>
+<a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">    </span><span class="n">que_size</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w">                         </span><span class="c1">// Queue size</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="p">}</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="k">impl</span><span class="o">&lt;</span><span class="n">T</span><span class="p">:</span><span class="w"> </span><span class="nb">Copy</span><span class="o">&gt;</span><span class="w"> </span><span class="n">LinkedListQueue</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
@@ -5726,12 +5798,12 @@
 <a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a>
-<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="p">;</span>
 <a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a>
-<a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">is_empty</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-25" name="__codelineno-22-25" href="#__codelineno-22-25"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-22-26" name="__codelineno-22-26" href="#__codelineno-22-26"></a><span class="w">    </span><span class="p">}</span>
@@ -5741,12 +5813,12 @@
 <a id="__codelineno-22-30" name="__codelineno-22-30" href="#__codelineno-22-30"></a><span class="w">        </span><span class="c1">// Add num after the tail node</span>
 <a id="__codelineno-22-31" name="__codelineno-22-31" href="#__codelineno-22-31"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">new_rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-22-32" name="__codelineno-22-32" href="#__codelineno-22-32"></a><span class="w">        </span><span class="k">match</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">rear</span><span class="p">.</span><span class="n">take</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-33" name="__codelineno-22-33" href="#__codelineno-22-33"></a><span class="w">            </span><span class="c1">// If the queue is not empty, add the node after the tail node</span>
+<a id="__codelineno-22-33" name="__codelineno-22-33" href="#__codelineno-22-33"></a><span class="w">            </span><span class="c1">// If the queue is not empty, add this node after the tail node</span>
 <a id="__codelineno-22-34" name="__codelineno-22-34" href="#__codelineno-22-34"></a><span class="w">            </span><span class="nb">Some</span><span class="p">(</span><span class="n">old_rear</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-35" name="__codelineno-22-35" href="#__codelineno-22-35"></a><span class="w">                </span><span class="n">old_rear</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">new_rear</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-22-36" name="__codelineno-22-36" href="#__codelineno-22-36"></a><span class="w">                </span><span class="bp">self</span><span class="p">.</span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">new_rear</span><span class="p">);</span>
 <a id="__codelineno-22-37" name="__codelineno-22-37" href="#__codelineno-22-37"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-22-38" name="__codelineno-22-38" href="#__codelineno-22-38"></a><span class="w">            </span><span class="c1">// If the queue is empty, make both front and rear point to the node</span>
+<a id="__codelineno-22-38" name="__codelineno-22-38" href="#__codelineno-22-38"></a><span class="w">            </span><span class="c1">// If the queue is empty, make both the head and tail point to this node</span>
 <a id="__codelineno-22-39" name="__codelineno-22-39" href="#__codelineno-22-39"></a><span class="w">            </span><span class="nb">None</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-40" name="__codelineno-22-40" href="#__codelineno-22-40"></a><span class="w">                </span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">new_rear</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-22-41" name="__codelineno-22-41" href="#__codelineno-22-41"></a><span class="w">                </span><span class="bp">self</span><span class="p">.</span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">new_rear</span><span class="p">);</span>
@@ -5771,12 +5843,12 @@
 <a id="__codelineno-22-60" name="__codelineno-22-60" href="#__codelineno-22-60"></a><span class="w">        </span><span class="p">})</span>
 <a id="__codelineno-22-61" name="__codelineno-22-61" href="#__codelineno-22-61"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-62" name="__codelineno-22-62" href="#__codelineno-22-62"></a>
-<a id="__codelineno-22-63" name="__codelineno-22-63" href="#__codelineno-22-63"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-22-63" name="__codelineno-22-63" href="#__codelineno-22-63"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-22-64" name="__codelineno-22-64" href="#__codelineno-22-64"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">peek</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-65" name="__codelineno-22-65" href="#__codelineno-22-65"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="p">.</span><span class="n">as_ref</span><span class="p">()</span>
 <a id="__codelineno-22-66" name="__codelineno-22-66" href="#__codelineno-22-66"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-67" name="__codelineno-22-67" href="#__codelineno-22-67"></a>
-<a id="__codelineno-22-68" name="__codelineno-22-68" href="#__codelineno-22-68"></a><span class="w">    </span><span class="cm">/* Convert linked list to Array and return */</span>
+<a id="__codelineno-22-68" name="__codelineno-22-68" href="#__codelineno-22-68"></a><span class="w">    </span><span class="cm">/* Convert the linked list to a vector and return it */</span>
 <a id="__codelineno-22-69" name="__codelineno-22-69" href="#__codelineno-22-69"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">to_array</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">head</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-70" name="__codelineno-22-70" href="#__codelineno-22-70"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-22-71" name="__codelineno-22-71" href="#__codelineno-22-71"></a>
@@ -5795,7 +5867,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_queue.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Queue based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_queue.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Queue implemented with a linked list */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">front</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">rear</span><span class="p">;</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span>
@@ -5818,30 +5890,30 @@
 <a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="w">        </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="o">-&gt;</span><span class="n">next</span><span class="p">;</span>
 <a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="w">        </span><span class="n">free</span><span class="p">(</span><span class="n">tmp</span><span class="p">);</span>
 <a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">    </span><span class="c1">// Free queue structure</span>
+<a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">    </span><span class="c1">// Free the queue structure</span>
 <a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">queue</span><span class="p">);</span>
 <a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="p">}</span>
 <a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a>
-<a id="__codelineno-23-28" name="__codelineno-23-28" href="#__codelineno-23-28"></a><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-23-28" name="__codelineno-23-28" href="#__codelineno-23-28"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-23-29" name="__codelineno-23-29" href="#__codelineno-23-29"></a><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="n">LinkedListQueue</span><span class="w"> </span><span class="o">*</span><span class="n">queue</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-30" name="__codelineno-23-30" href="#__codelineno-23-30"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-23-31" name="__codelineno-23-31" href="#__codelineno-23-31"></a><span class="p">}</span>
 <a id="__codelineno-23-32" name="__codelineno-23-32" href="#__codelineno-23-32"></a>
-<a id="__codelineno-23-33" name="__codelineno-23-33" href="#__codelineno-23-33"></a><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-23-33" name="__codelineno-23-33" href="#__codelineno-23-33"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-23-34" name="__codelineno-23-34" href="#__codelineno-23-34"></a><span class="kt">bool</span><span class="w"> </span><span class="nf">empty</span><span class="p">(</span><span class="n">LinkedListQueue</span><span class="w"> </span><span class="o">*</span><span class="n">queue</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-35" name="__codelineno-23-35" href="#__codelineno-23-35"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">size</span><span class="p">(</span><span class="n">queue</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-23-36" name="__codelineno-23-36" href="#__codelineno-23-36"></a><span class="p">}</span>
 <a id="__codelineno-23-37" name="__codelineno-23-37" href="#__codelineno-23-37"></a>
 <a id="__codelineno-23-38" name="__codelineno-23-38" href="#__codelineno-23-38"></a><span class="cm">/* Enqueue */</span>
 <a id="__codelineno-23-39" name="__codelineno-23-39" href="#__codelineno-23-39"></a><span class="kt">void</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="n">LinkedListQueue</span><span class="w"> </span><span class="o">*</span><span class="n">queue</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-40" name="__codelineno-23-40" href="#__codelineno-23-40"></a><span class="w">    </span><span class="c1">// Add node at tail</span>
+<a id="__codelineno-23-40" name="__codelineno-23-40" href="#__codelineno-23-40"></a><span class="w">    </span><span class="c1">// Add node at the tail</span>
 <a id="__codelineno-23-41" name="__codelineno-23-41" href="#__codelineno-23-41"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newListNode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
-<a id="__codelineno-23-42" name="__codelineno-23-42" href="#__codelineno-23-42"></a><span class="w">    </span><span class="c1">// If the queue is empty, make both front and rear point to the node</span>
+<a id="__codelineno-23-42" name="__codelineno-23-42" href="#__codelineno-23-42"></a><span class="w">    </span><span class="c1">// If the queue is empty, make both the head and tail point to this node</span>
 <a id="__codelineno-23-43" name="__codelineno-23-43" href="#__codelineno-23-43"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-44" name="__codelineno-23-44" href="#__codelineno-23-44"></a><span class="w">        </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-23-45" name="__codelineno-23-45" href="#__codelineno-23-45"></a><span class="w">        </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-23-46" name="__codelineno-23-46" href="#__codelineno-23-46"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-47" name="__codelineno-23-47" href="#__codelineno-23-47"></a><span class="w">    </span><span class="c1">// If the queue is not empty, add the node after the tail node</span>
+<a id="__codelineno-23-47" name="__codelineno-23-47" href="#__codelineno-23-47"></a><span class="w">    </span><span class="c1">// If the queue is not empty, add this node after the tail node</span>
 <a id="__codelineno-23-48" name="__codelineno-23-48" href="#__codelineno-23-48"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-49" name="__codelineno-23-49" href="#__codelineno-23-49"></a><span class="w">        </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">rear</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-23-50" name="__codelineno-23-50" href="#__codelineno-23-50"></a><span class="w">        </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
@@ -5849,7 +5921,7 @@
 <a id="__codelineno-23-52" name="__codelineno-23-52" href="#__codelineno-23-52"></a><span class="w">    </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-23-53" name="__codelineno-23-53" href="#__codelineno-23-53"></a><span class="p">}</span>
 <a id="__codelineno-23-54" name="__codelineno-23-54" href="#__codelineno-23-54"></a>
-<a id="__codelineno-23-55" name="__codelineno-23-55" href="#__codelineno-23-55"></a><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-23-55" name="__codelineno-23-55" href="#__codelineno-23-55"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-23-56" name="__codelineno-23-56" href="#__codelineno-23-56"></a><span class="kt">int</span><span class="w"> </span><span class="nf">peek</span><span class="p">(</span><span class="n">LinkedListQueue</span><span class="w"> </span><span class="o">*</span><span class="n">queue</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-57" name="__codelineno-23-57" href="#__codelineno-23-57"></a><span class="w">    </span><span class="n">assert</span><span class="p">(</span><span class="n">size</span><span class="p">(</span><span class="n">queue</span><span class="p">)</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="p">);</span>
 <a id="__codelineno-23-58" name="__codelineno-23-58" href="#__codelineno-23-58"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
@@ -5865,10 +5937,10 @@
 <a id="__codelineno-23-68" name="__codelineno-23-68" href="#__codelineno-23-68"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-23-69" name="__codelineno-23-69" href="#__codelineno-23-69"></a><span class="p">}</span>
 <a id="__codelineno-23-70" name="__codelineno-23-70" href="#__codelineno-23-70"></a>
-<a id="__codelineno-23-71" name="__codelineno-23-71" href="#__codelineno-23-71"></a><span class="cm">/* Print queue */</span>
+<a id="__codelineno-23-71" name="__codelineno-23-71" href="#__codelineno-23-71"></a><span class="cm">/* Print the queue */</span>
 <a id="__codelineno-23-72" name="__codelineno-23-72" href="#__codelineno-23-72"></a><span class="kt">void</span><span class="w"> </span><span class="nf">printLinkedListQueue</span><span class="p">(</span><span class="n">LinkedListQueue</span><span class="w"> </span><span class="o">*</span><span class="n">queue</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-73" name="__codelineno-23-73" href="#__codelineno-23-73"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="p">);</span>
-<a id="__codelineno-23-74" name="__codelineno-23-74" href="#__codelineno-23-74"></a><span class="w">    </span><span class="c1">// Copy data from list to array</span>
+<a id="__codelineno-23-74" name="__codelineno-23-74" href="#__codelineno-23-74"></a><span class="w">    </span><span class="c1">// Copy the linked list data into an array</span>
 <a id="__codelineno-23-75" name="__codelineno-23-75" href="#__codelineno-23-75"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">;</span>
 <a id="__codelineno-23-76" name="__codelineno-23-76" href="#__codelineno-23-76"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-23-77" name="__codelineno-23-77" href="#__codelineno-23-77"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5881,20 +5953,20 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_queue.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Queue based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_queue.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Queue implemented with a linked list */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">LinkedListQueue</span><span class="p">(</span>
-<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Head node front, tail node rear</span>
+<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Head node front and tail node rear</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">front</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">rear</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">queSize</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a>
-<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">size</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a>
-<a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">():</span><span class="w"> </span><span class="kt">Boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">    </span><span class="p">}</span>
@@ -5903,11 +5975,11 @@
 <a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="w">        </span><span class="c1">// Add num after the tail node</span>
 <a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">ListNode</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
-<a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">        </span><span class="c1">// If the queue is empty, make both front and rear point to the node</span>
+<a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">        </span><span class="c1">// If the queue is empty, make both the head and tail point to this node</span>
 <a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-25" name="__codelineno-24-25" href="#__codelineno-24-25"></a><span class="w">            </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-24-26" name="__codelineno-24-26" href="#__codelineno-24-26"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
-<a id="__codelineno-24-27" name="__codelineno-24-27" href="#__codelineno-24-27"></a><span class="w">            </span><span class="c1">// If the queue is not empty, add the node after the tail node</span>
+<a id="__codelineno-24-27" name="__codelineno-24-27" href="#__codelineno-24-27"></a><span class="w">            </span><span class="c1">// If the queue is not empty, add this node after the tail node</span>
 <a id="__codelineno-24-28" name="__codelineno-24-28" href="#__codelineno-24-28"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-29" name="__codelineno-24-29" href="#__codelineno-24-29"></a><span class="w">            </span><span class="n">rear</span><span class="o">?.</span><span class="na">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-24-30" name="__codelineno-24-30" href="#__codelineno-24-30"></a><span class="w">            </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
@@ -5918,19 +5990,19 @@
 <a id="__codelineno-24-35" name="__codelineno-24-35" href="#__codelineno-24-35"></a><span class="w">    </span><span class="cm">/* Dequeue */</span>
 <a id="__codelineno-24-36" name="__codelineno-24-36" href="#__codelineno-24-36"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">pop</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-37" name="__codelineno-24-37" href="#__codelineno-24-37"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peek</span><span class="p">()</span>
-<a id="__codelineno-24-38" name="__codelineno-24-38" href="#__codelineno-24-38"></a><span class="w">        </span><span class="c1">// Delete head node</span>
+<a id="__codelineno-24-38" name="__codelineno-24-38" href="#__codelineno-24-38"></a><span class="w">        </span><span class="c1">// Remove the head node</span>
 <a id="__codelineno-24-39" name="__codelineno-24-39" href="#__codelineno-24-39"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="o">?.</span><span class="na">next</span>
 <a id="__codelineno-24-40" name="__codelineno-24-40" href="#__codelineno-24-40"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span>
 <a id="__codelineno-24-41" name="__codelineno-24-41" href="#__codelineno-24-41"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-24-42" name="__codelineno-24-42" href="#__codelineno-24-42"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-43" name="__codelineno-24-43" href="#__codelineno-24-43"></a>
-<a id="__codelineno-24-44" name="__codelineno-24-44" href="#__codelineno-24-44"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-24-44" name="__codelineno-24-44" href="#__codelineno-24-44"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-24-45" name="__codelineno-24-45" href="#__codelineno-24-45"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">peek</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-46" name="__codelineno-24-46" href="#__codelineno-24-46"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">()</span>
 <a id="__codelineno-24-47" name="__codelineno-24-47" href="#__codelineno-24-47"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">front</span><span class="o">!!</span><span class="p">.</span><span class="na">_val</span>
 <a id="__codelineno-24-48" name="__codelineno-24-48" href="#__codelineno-24-48"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-49" name="__codelineno-24-49" href="#__codelineno-24-49"></a>
-<a id="__codelineno-24-50" name="__codelineno-24-50" href="#__codelineno-24-50"></a><span class="w">    </span><span class="cm">/* Convert linked list to Array and return */</span>
+<a id="__codelineno-24-50" name="__codelineno-24-50" href="#__codelineno-24-50"></a><span class="w">    </span><span class="cm">/* Convert the linked list to an array and return it */</span>
 <a id="__codelineno-24-51" name="__codelineno-24-51" href="#__codelineno-24-51"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">toArray</span><span class="p">():</span><span class="w"> </span><span class="n">IntArray</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-52" name="__codelineno-24-52" href="#__codelineno-24-52"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span>
 <a id="__codelineno-24-53" name="__codelineno-24-53" href="#__codelineno-24-53"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">size</span><span class="p">())</span>
@@ -5944,9 +6016,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_queue.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Queue based on linked list ###</span>
+<div class="highlight"><span class="filename">linkedlist_queue.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Queue implemented with a linked list ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">LinkedListQueue</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1">### Get queue length ###</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1">### Get the queue size ###</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="kp">attr_reader</span><span class="w"> </span><span class="ss">:size</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="c1">### Constructor ###</span>
@@ -5956,7 +6028,7 @@
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a>
-<a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">  </span><span class="c1">### Check if queue is empty ###</span>
+<a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">  </span><span class="c1">### Check whether the queue is empty ###</span>
 <a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">is_empty?</span>
 <a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">    </span><span class="vi">@front</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">  </span><span class="k">end</span>
@@ -5966,11 +6038,11 @@
 <a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a><span class="w">    </span><span class="c1"># Add num after the tail node</span>
 <a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">ListNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a>
-<a id="__codelineno-25-23" name="__codelineno-25-23" href="#__codelineno-25-23"></a><span class="w">    </span><span class="c1"># If queue is empty, set both front and rear to this node</span>
+<a id="__codelineno-25-23" name="__codelineno-25-23" href="#__codelineno-25-23"></a><span class="w">    </span><span class="c1"># If the queue is empty, point both the head and tail to this node</span>
 <a id="__codelineno-25-24" name="__codelineno-25-24" href="#__codelineno-25-24"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="vi">@front</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-25-25" name="__codelineno-25-25" href="#__codelineno-25-25"></a><span class="w">      </span><span class="vi">@front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-25-26" name="__codelineno-25-26" href="#__codelineno-25-26"></a><span class="w">      </span><span class="vi">@rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
-<a id="__codelineno-25-27" name="__codelineno-25-27" href="#__codelineno-25-27"></a><span class="w">    </span><span class="c1"># If queue is not empty, add this node after rear</span>
+<a id="__codelineno-25-27" name="__codelineno-25-27" href="#__codelineno-25-27"></a><span class="w">    </span><span class="c1"># If the queue is not empty, append this node after the tail</span>
 <a id="__codelineno-25-28" name="__codelineno-25-28" href="#__codelineno-25-28"></a><span class="w">    </span><span class="k">else</span>
 <a id="__codelineno-25-29" name="__codelineno-25-29" href="#__codelineno-25-29"></a><span class="w">      </span><span class="vi">@rear</span><span class="o">.</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-25-30" name="__codelineno-25-30" href="#__codelineno-25-30"></a><span class="w">      </span><span class="vi">@rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
@@ -5982,20 +6054,20 @@
 <a id="__codelineno-25-36" name="__codelineno-25-36" href="#__codelineno-25-36"></a><span class="w">  </span><span class="c1">### Dequeue ###</span>
 <a id="__codelineno-25-37" name="__codelineno-25-37" href="#__codelineno-25-37"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">pop</span>
 <a id="__codelineno-25-38" name="__codelineno-25-38" href="#__codelineno-25-38"></a><span class="w">    </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peek</span>
-<a id="__codelineno-25-39" name="__codelineno-25-39" href="#__codelineno-25-39"></a><span class="w">    </span><span class="c1"># Delete head node</span>
+<a id="__codelineno-25-39" name="__codelineno-25-39" href="#__codelineno-25-39"></a><span class="w">    </span><span class="c1"># Remove the head node</span>
 <a id="__codelineno-25-40" name="__codelineno-25-40" href="#__codelineno-25-40"></a><span class="w">    </span><span class="vi">@front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@front</span><span class="o">.</span><span class="n">next</span>
 <a id="__codelineno-25-41" name="__codelineno-25-41" href="#__codelineno-25-41"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-25-42" name="__codelineno-25-42" href="#__codelineno-25-42"></a><span class="w">    </span><span class="n">num</span>
 <a id="__codelineno-25-43" name="__codelineno-25-43" href="#__codelineno-25-43"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-44" name="__codelineno-25-44" href="#__codelineno-25-44"></a>
-<a id="__codelineno-25-45" name="__codelineno-25-45" href="#__codelineno-25-45"></a><span class="w">  </span><span class="c1">### Access front element ###</span>
+<a id="__codelineno-25-45" name="__codelineno-25-45" href="#__codelineno-25-45"></a><span class="w">  </span><span class="c1">### Access the front element ###</span>
 <a id="__codelineno-25-46" name="__codelineno-25-46" href="#__codelineno-25-46"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">peek</span>
 <a id="__codelineno-25-47" name="__codelineno-25-47" href="#__codelineno-25-47"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="p">,</span><span class="w"> </span><span class="s1">'Queue is empty'</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">is_empty?</span>
 <a id="__codelineno-25-48" name="__codelineno-25-48" href="#__codelineno-25-48"></a>
 <a id="__codelineno-25-49" name="__codelineno-25-49" href="#__codelineno-25-49"></a><span class="w">    </span><span class="vi">@front</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-25-50" name="__codelineno-25-50" href="#__codelineno-25-50"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-51" name="__codelineno-25-51" href="#__codelineno-25-51"></a>
-<a id="__codelineno-25-52" name="__codelineno-25-52" href="#__codelineno-25-52"></a><span class="w">  </span><span class="c1">### Convert linked list to Array and return ###</span>
+<a id="__codelineno-25-52" name="__codelineno-25-52" href="#__codelineno-25-52"></a><span class="w">  </span><span class="c1">### Convert the linked list to an Array and return it ###</span>
 <a id="__codelineno-25-53" name="__codelineno-25-53" href="#__codelineno-25-53"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">to_array</span>
 <a id="__codelineno-25-54" name="__codelineno-25-54" href="#__codelineno-25-54"></a><span class="w">    </span><span class="n">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-25-55" name="__codelineno-25-55" href="#__codelineno-25-55"></a><span class="w">    </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@front</span>
@@ -6010,19 +6082,24 @@
 </div>
 </div>
 </div>
-<h3 id="2-array-implementation">2. &nbsp; Array Implementation<a class="headerlink" href="#2-array-implementation" title="Permanent link">&para;</a></h3>
-<p>Deleting the first element in an array has a time complexity of <span class="arithmatex">\(O(n)\)</span>, which would make the dequeue operation inefficient. However, we can use the following clever method to avoid this problem.</p>
-<p>We can use a variable <code>front</code> to point to the index of the front element and maintain a variable <code>size</code> to record the queue length. We define <code>rear = front + size</code>, which calculates the position right after the rear element.</p>
-<p>Based on this design, <strong>the valid interval containing elements in the array is <code>[front, rear - 1]</code></strong>. The implementation methods for various operations are shown in Figure 5-6:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0A%0Aclass%20LinkedListQueue%3A%0A%20%20%20%20%22%22%22Queue%20implemented%20with%20a%20linked%20list%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self._front%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Head%20node%20front%0A%20%20%20%20%20%20%20%20self._rear%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Tail%20node%20rear%0A%20%20%20%20%20%20%20%20self._size%3A%20int%20%3D%200%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20queue%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20self._size%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22Check%20whether%20the%20queue%20is%20empty%22%22%22%0A%20%20%20%20%20%20%20%20return%20not%20self._front%0A%0A%20%20%20%20def%20push%28self,%20num%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Enqueue%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Add%20num%20after%20the%20tail%20node%0A%20%20%20%20%20%20%20%20node%20%3D%20ListNode%28num%29%0A%20%20%20%20%20%20%20%20%23%20If%20the%20queue%20is%20empty,%20make%20both%20the%20head%20and%20tail%20point%20to%20this%20node%0A%20%20%20%20%20%20%20%20if%20self._front%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self._front%20%3D%20node%0A%20%20%20%20%20%20%20%20%20%20%20%20self._rear%20%3D%20node%0A%20%20%20%20%20%20%20%20%23%20If%20the%20queue%20is%20not%20empty,%20add%20this%20node%20after%20the%20tail%20node%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self._rear.next%20%3D%20node%0A%20%20%20%20%20%20%20%20%20%20%20%20self._rear%20%3D%20node%0A%20%20%20%20%20%20%20%20self._size%20%2B%3D%201%0A%0A%20%20%20%20def%20pop%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Dequeue%22%22%22%0A%20%20%20%20%20%20%20%20num%20%3D%20self.peek%28%29%0A%20%20%20%20%20%20%20%20%23%20Remove%20the%20head%20node%0A%20%20%20%20%20%20%20%20self._front%20%3D%20self._front.next%0A%20%20%20%20%20%20%20%20self._size%20-%3D%201%0A%20%20%20%20%20%20%20%20return%20num%0A%0A%20%20%20%20def%20peek%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Access%20the%20front%20element%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.is_empty%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22Queue%20is%20empty%22%29%0A%20%20%20%20%20%20%20%20return%20self._front.val%0A%0A%20%20%20%20def%20to_list%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Convert%20to%20a%20list%20for%20printing%22%22%22%0A%20%20%20%20%20%20%20%20queue%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20temp%20%3D%20self._front%0A%20%20%20%20%20%20%20%20while%20temp%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20queue.append%28temp.val%29%0A%20%20%20%20%20%20%20%20%20%20%20%20temp%20%3D%20temp.next%0A%20%20%20%20%20%20%20%20return%20queue%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20queue%0A%20%20%20%20queue%20%3D%20LinkedListQueue%28%29%0A%0A%20%20%20%20%23%20Enqueue%20elements%0A%20%20%20%20queue.push%281%29%0A%20%20%20%20queue.push%283%29%0A%20%20%20%20queue.push%282%29%0A%20%20%20%20queue.push%285%29%0A%20%20%20%20queue.push%284%29%0A%20%20%20%20print%28%22Queue%20queue%20%3D%22,%20queue.to_list%28%29%29%0A%0A%20%20%20%20%23%20Access%20the%20front%20element%0A%20%20%20%20peek%20%3D%20queue.peek%28%29%0A%20%20%20%20print%28%22Front%20element%20peek%20%3D%22,%20peek%29%0A%0A%20%20%20%20%23%20Dequeue%20an%20element%0A%20%20%20%20pop_front%20%3D%20queue.pop%28%29%0A%20%20%20%20print%28%22Dequeued%20element%20pop_front%20%3D%22,%20pop_front%29%0A%20%20%20%20print%28%22After%20dequeuing,%20queue%20%3D%22,%20queue.to_list%28%29%29%0A%0A%20%20%20%20%23%20Get%20the%20queue%20size%0A%20%20%20%20size%20%3D%20queue.size%28%29%0A%20%20%20%20print%28%22Queue%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20Check%20whether%20the%20queue%20is%20empty%0A%20%20%20%20is_empty%20%3D%20queue.is_empty%28%29%0A%20%20%20%20print%28%22Is%20the%20queue%20empty%20%3D%22,%20is_empty%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0A%0Aclass%20LinkedListQueue%3A%0A%20%20%20%20%22%22%22Queue%20implemented%20with%20a%20linked%20list%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self._front%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Head%20node%20front%0A%20%20%20%20%20%20%20%20self._rear%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Tail%20node%20rear%0A%20%20%20%20%20%20%20%20self._size%3A%20int%20%3D%200%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20queue%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20self._size%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22Check%20whether%20the%20queue%20is%20empty%22%22%22%0A%20%20%20%20%20%20%20%20return%20not%20self._front%0A%0A%20%20%20%20def%20push%28self,%20num%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Enqueue%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Add%20num%20after%20the%20tail%20node%0A%20%20%20%20%20%20%20%20node%20%3D%20ListNode%28num%29%0A%20%20%20%20%20%20%20%20%23%20If%20the%20queue%20is%20empty,%20make%20both%20the%20head%20and%20tail%20point%20to%20this%20node%0A%20%20%20%20%20%20%20%20if%20self._front%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self._front%20%3D%20node%0A%20%20%20%20%20%20%20%20%20%20%20%20self._rear%20%3D%20node%0A%20%20%20%20%20%20%20%20%23%20If%20the%20queue%20is%20not%20empty,%20add%20this%20node%20after%20the%20tail%20node%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self._rear.next%20%3D%20node%0A%20%20%20%20%20%20%20%20%20%20%20%20self._rear%20%3D%20node%0A%20%20%20%20%20%20%20%20self._size%20%2B%3D%201%0A%0A%20%20%20%20def%20pop%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Dequeue%22%22%22%0A%20%20%20%20%20%20%20%20num%20%3D%20self.peek%28%29%0A%20%20%20%20%20%20%20%20%23%20Remove%20the%20head%20node%0A%20%20%20%20%20%20%20%20self._front%20%3D%20self._front.next%0A%20%20%20%20%20%20%20%20self._size%20-%3D%201%0A%20%20%20%20%20%20%20%20return%20num%0A%0A%20%20%20%20def%20peek%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Access%20the%20front%20element%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.is_empty%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22Queue%20is%20empty%22%29%0A%20%20%20%20%20%20%20%20return%20self._front.val%0A%0A%20%20%20%20def%20to_list%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Convert%20to%20a%20list%20for%20printing%22%22%22%0A%20%20%20%20%20%20%20%20queue%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20temp%20%3D%20self._front%0A%20%20%20%20%20%20%20%20while%20temp%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20queue.append%28temp.val%29%0A%20%20%20%20%20%20%20%20%20%20%20%20temp%20%3D%20temp.next%0A%20%20%20%20%20%20%20%20return%20queue%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20queue%0A%20%20%20%20queue%20%3D%20LinkedListQueue%28%29%0A%0A%20%20%20%20%23%20Enqueue%20elements%0A%20%20%20%20queue.push%281%29%0A%20%20%20%20queue.push%283%29%0A%20%20%20%20queue.push%282%29%0A%20%20%20%20queue.push%285%29%0A%20%20%20%20queue.push%284%29%0A%20%20%20%20print%28%22Queue%20queue%20%3D%22,%20queue.to_list%28%29%29%0A%0A%20%20%20%20%23%20Access%20the%20front%20element%0A%20%20%20%20peek%20%3D%20queue.peek%28%29%0A%20%20%20%20print%28%22Front%20element%20peek%20%3D%22,%20peek%29%0A%0A%20%20%20%20%23%20Dequeue%20an%20element%0A%20%20%20%20pop_front%20%3D%20queue.pop%28%29%0A%20%20%20%20print%28%22Dequeued%20element%20pop_front%20%3D%22,%20pop_front%29%0A%20%20%20%20print%28%22After%20dequeuing,%20queue%20%3D%22,%20queue.to_list%28%29%29%0A%0A%20%20%20%20%23%20Get%20the%20queue%20size%0A%20%20%20%20size%20%3D%20queue.size%28%29%0A%20%20%20%20print%28%22Queue%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20Check%20whether%20the%20queue%20is%20empty%0A%20%20%20%20is_empty%20%3D%20queue.is_empty%28%29%0A%20%20%20%20print%28%22Is%20the%20queue%20empty%20%3D%22,%20is_empty%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="2-implementation-with-an-array">2. &nbsp; Implementation with an array<a class="headerlink" href="#2-implementation-with-an-array" title="Permanent link">&para;</a></h3>
+<p>Removing the first element of an array takes <span class="arithmatex">\(O(n)\)</span> time, which would make dequeueing inefficient. A useful technique avoids this problem.</p>
+<p>Use a variable <code>front</code> for the index of the front element and a variable <code>size</code> for the queue's length. Define <code>rear = front + size</code>, so <code>rear</code> points to the position immediately after the last element.</p>
+<p>With this design, <strong>the occupied interval is <code>[front, rear - 1]</code></strong>. Operations work as shown below.</p>
 <ul>
-<li>Enqueue operation: Assign the input element to the <code>rear</code> index and increase <code>size</code> by 1.</li>
-<li>Dequeue operation: Simply increase <code>front</code> by 1 and decrease <code>size</code> by 1.</li>
+<li>Enqueue: assign the new element to index <code>rear</code> and increase <code>size</code> by 1.</li>
+<li>Dequeue: increase <code>front</code> by 1 and decrease <code>size</code> by 1.</li>
 </ul>
-<p>As you can see, both enqueue and dequeue operations require only one operation, with a time complexity of <span class="arithmatex">\(O(1)\)</span>.</p>
+<p>Both operations require only a constant amount of work and take <span class="arithmatex">\(O(1)\)</span> time.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:3"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">&lt;1&gt;</label><label for="__tabbed_4_2">&lt;2&gt;</label><label for="__tabbed_4_3">&lt;3&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Enqueue and dequeue operations in array implementation of queue" class="animation-figure" src="../queue.assets/array_queue_step1.png" /></p>
+<p><img alt="Enqueue and dequeue in an array-based queue" class="animation-figure" src="../queue.assets/array_queue_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="array_queue_push" class="animation-figure" src="../queue.assets/array_queue_step2_push.png" /></p>
@@ -6032,61 +6109,61 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 5-6 &nbsp; Enqueue and dequeue operations in array implementation of queue </p>
+<p align="center"> Figure 5-6 &nbsp; Enqueue and dequeue in an array-based queue </p>
 
-<p>You may notice a problem: as we continuously enqueue and dequeue, both <code>front</code> and <code>rear</code> move to the right. <strong>When they reach the end of the array, they cannot continue moving</strong>. To solve this problem, we can treat the array as a "circular array" with head and tail connected.</p>
-<p>For a circular array, we need to let <code>front</code> or <code>rear</code> wrap around to the beginning of the array when they cross the end. This periodic pattern can be implemented using the "modulo operation," as shown in the code below:</p>
+<p>There is a problem: repeated enqueueing and dequeueing move both <code>front</code> and <code>rear</code> to the right, <strong>and they cannot continue once they reach the end of the array</strong>. To solve this, treat the array as circular, with its end connected to its beginning.</p>
+<p>When <code>front</code> or <code>rear</code> passes the array's end, wrap it back to the beginning. The remainder operation implements this cyclic behavior:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label></div>
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 <div class="highlight"><span class="filename">array_queue.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayQueue</span><span class="p">:</span>
-<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Queue based on circular array implementation"""</span>
+<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Queue implemented with a circular array"""</span>
 <a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">size</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a><span class="w">        </span><span class="sd">"""Constructor"""</span>
-<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">size</span>  <span class="c1"># Array for storing queue elements</span>
-<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_front</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Queue length</span>
+<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_nums</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">size</span>  <span class="c1"># Array that stores the queue elements</span>
+<a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_front</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Front pointer, pointing to the first element</span>
+<a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">0</span>  <span class="c1"># Queue size</span>
 <a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>
 <a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">capacity</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a><span class="w">        </span><span class="sd">"""Get the capacity of the queue"""</span>
+<a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a><span class="w">        </span><span class="sd">"""Get the queue capacity"""</span>
 <a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a>        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_nums</span><span class="p">)</span>
 <a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>
 <a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a><span class="w">        </span><span class="sd">"""Get the length of the queue"""</span>
+<a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a><span class="w">        </span><span class="sd">"""Get the queue size"""</span>
 <a id="__codelineno-26-16" name="__codelineno-26-16" href="#__codelineno-26-16"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span>
 <a id="__codelineno-26-17" name="__codelineno-26-17" href="#__codelineno-26-17"></a>
 <a id="__codelineno-26-18" name="__codelineno-26-18" href="#__codelineno-26-18"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">is_empty</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">bool</span><span class="p">:</span>
-<a id="__codelineno-26-19" name="__codelineno-26-19" href="#__codelineno-26-19"></a><span class="w">        </span><span class="sd">"""Check if the queue is empty"""</span>
+<a id="__codelineno-26-19" name="__codelineno-26-19" href="#__codelineno-26-19"></a><span class="w">        </span><span class="sd">"""Check whether the queue is empty"""</span>
 <a id="__codelineno-26-20" name="__codelineno-26-20" href="#__codelineno-26-20"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">==</span> <span class="mi">0</span>
 <a id="__codelineno-26-21" name="__codelineno-26-21" href="#__codelineno-26-21"></a>
 <a id="__codelineno-26-22" name="__codelineno-26-22" href="#__codelineno-26-22"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-26-23" name="__codelineno-26-23" href="#__codelineno-26-23"></a><span class="w">        </span><span class="sd">"""Enqueue"""</span>
 <a id="__codelineno-26-24" name="__codelineno-26-24" href="#__codelineno-26-24"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">==</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">():</span>
 <a id="__codelineno-26-25" name="__codelineno-26-25" href="#__codelineno-26-25"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Queue is full"</span><span class="p">)</span>
-<a id="__codelineno-26-26" name="__codelineno-26-26" href="#__codelineno-26-26"></a>        <span class="c1"># Calculate rear pointer, points to rear index + 1</span>
-<a id="__codelineno-26-27" name="__codelineno-26-27" href="#__codelineno-26-27"></a>        <span class="c1"># Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
+<a id="__codelineno-26-26" name="__codelineno-26-26" href="#__codelineno-26-26"></a>        <span class="c1"># Calculate the rear pointer, which points one position past the last element</span>
+<a id="__codelineno-26-27" name="__codelineno-26-27" href="#__codelineno-26-27"></a>        <span class="c1"># Use modulo to wrap rear to the beginning when it passes the end of the array</span>
 <a id="__codelineno-26-28" name="__codelineno-26-28" href="#__codelineno-26-28"></a>        <span class="n">rear</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">)</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">()</span>
-<a id="__codelineno-26-29" name="__codelineno-26-29" href="#__codelineno-26-29"></a>        <span class="c1"># Add num to the rear of the queue</span>
+<a id="__codelineno-26-29" name="__codelineno-26-29" href="#__codelineno-26-29"></a>        <span class="c1"># Add num to the rear</span>
 <a id="__codelineno-26-30" name="__codelineno-26-30" href="#__codelineno-26-30"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_nums</span><span class="p">[</span><span class="n">rear</span><span class="p">]</span> <span class="o">=</span> <span class="n">num</span>
 <a id="__codelineno-26-31" name="__codelineno-26-31" href="#__codelineno-26-31"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">+=</span> <span class="mi">1</span>
 <a id="__codelineno-26-32" name="__codelineno-26-32" href="#__codelineno-26-32"></a>
 <a id="__codelineno-26-33" name="__codelineno-26-33" href="#__codelineno-26-33"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">pop</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
 <a id="__codelineno-26-34" name="__codelineno-26-34" href="#__codelineno-26-34"></a><span class="w">        </span><span class="sd">"""Dequeue"""</span>
 <a id="__codelineno-26-35" name="__codelineno-26-35" href="#__codelineno-26-35"></a>        <span class="n">num</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">peek</span><span class="p">()</span>
-<a id="__codelineno-26-36" name="__codelineno-26-36" href="#__codelineno-26-36"></a>        <span class="c1"># Front pointer moves one position backward, if it passes the tail, return to the head of the array</span>
+<a id="__codelineno-26-36" name="__codelineno-26-36" href="#__codelineno-26-36"></a>        <span class="c1"># Advance the front pointer by one position, wrapping to the beginning if it passes the end</span>
 <a id="__codelineno-26-37" name="__codelineno-26-37" href="#__codelineno-26-37"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">=</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_front</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="bp">self</span><span class="o">.</span><span class="n">capacity</span><span class="p">()</span>
 <a id="__codelineno-26-38" name="__codelineno-26-38" href="#__codelineno-26-38"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">-=</span> <span class="mi">1</span>
 <a id="__codelineno-26-39" name="__codelineno-26-39" href="#__codelineno-26-39"></a>        <span class="k">return</span> <span class="n">num</span>
 <a id="__codelineno-26-40" name="__codelineno-26-40" href="#__codelineno-26-40"></a>
 <a id="__codelineno-26-41" name="__codelineno-26-41" href="#__codelineno-26-41"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">peek</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-42" name="__codelineno-26-42" href="#__codelineno-26-42"></a><span class="w">        </span><span class="sd">"""Access front of the queue element"""</span>
+<a id="__codelineno-26-42" name="__codelineno-26-42" href="#__codelineno-26-42"></a><span class="w">        </span><span class="sd">"""Access the front element"""</span>
 <a id="__codelineno-26-43" name="__codelineno-26-43" href="#__codelineno-26-43"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_empty</span><span class="p">():</span>
 <a id="__codelineno-26-44" name="__codelineno-26-44" href="#__codelineno-26-44"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Queue is empty"</span><span class="p">)</span>
 <a id="__codelineno-26-45" name="__codelineno-26-45" href="#__codelineno-26-45"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_nums</span><span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">_front</span><span class="p">]</span>
 <a id="__codelineno-26-46" name="__codelineno-26-46" href="#__codelineno-26-46"></a>
 <a id="__codelineno-26-47" name="__codelineno-26-47" href="#__codelineno-26-47"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">to_list</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
-<a id="__codelineno-26-48" name="__codelineno-26-48" href="#__codelineno-26-48"></a><span class="w">        </span><span class="sd">"""Return list for printing"""</span>
+<a id="__codelineno-26-48" name="__codelineno-26-48" href="#__codelineno-26-48"></a><span class="w">        </span><span class="sd">"""Return a list for printing"""</span>
 <a id="__codelineno-26-49" name="__codelineno-26-49" href="#__codelineno-26-49"></a>        <span class="n">res</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span>
 <a id="__codelineno-26-50" name="__codelineno-26-50" href="#__codelineno-26-50"></a>        <span class="n">j</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_front</span>
 <a id="__codelineno-26-51" name="__codelineno-26-51" href="#__codelineno-26-51"></a>        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()):</span>
@@ -6096,17 +6173,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_queue.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_queue.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Queue implemented with a circular array */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayQueue</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">  </span><span class="k">private</span><span class="o">:</span>
-<a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">;</span><span class="w">       </span><span class="c1">// Array for storing queue elements</span>
-<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w">       </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w">     </span><span class="c1">// Queue length</span>
+<a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">;</span><span class="w">       </span><span class="c1">// Array that stores the queue elements</span>
+<a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w">       </span><span class="c1">// Front index, identifying the first element</span>
+<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w">     </span><span class="c1">// Queue size</span>
 <a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queCapacity</span><span class="p">;</span><span class="w"> </span><span class="c1">// Queue capacity</span>
 <a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a>
 <a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">  </span><span class="k">public</span><span class="o">:</span>
 <a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a><span class="w">    </span><span class="n">ArrayQueue</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="w">        </span><span class="c1">// Initialize array</span>
+<a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="w">        </span><span class="c1">// Initialize the array</span>
 <a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="w">        </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">capacity</span><span class="p">];</span>
 <a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">        </span><span class="n">queCapacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
@@ -6116,17 +6193,17 @@
 <a id="__codelineno-27-18" name="__codelineno-27-18" href="#__codelineno-27-18"></a><span class="w">        </span><span class="k">delete</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span>
 <a id="__codelineno-27-19" name="__codelineno-27-19" href="#__codelineno-27-19"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-20" name="__codelineno-27-20" href="#__codelineno-27-20"></a>
-<a id="__codelineno-27-21" name="__codelineno-27-21" href="#__codelineno-27-21"></a><span class="w">    </span><span class="cm">/* Get the capacity of the queue */</span>
+<a id="__codelineno-27-21" name="__codelineno-27-21" href="#__codelineno-27-21"></a><span class="w">    </span><span class="cm">/* Get the queue capacity */</span>
 <a id="__codelineno-27-22" name="__codelineno-27-22" href="#__codelineno-27-22"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-23" name="__codelineno-27-23" href="#__codelineno-27-23"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queCapacity</span><span class="p">;</span>
 <a id="__codelineno-27-24" name="__codelineno-27-24" href="#__codelineno-27-24"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-25" name="__codelineno-27-25" href="#__codelineno-27-25"></a>
-<a id="__codelineno-27-26" name="__codelineno-27-26" href="#__codelineno-27-26"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-27-26" name="__codelineno-27-26" href="#__codelineno-27-26"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-27-27" name="__codelineno-27-27" href="#__codelineno-27-27"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-28" name="__codelineno-27-28" href="#__codelineno-27-28"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-27-29" name="__codelineno-27-29" href="#__codelineno-27-29"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-30" name="__codelineno-27-30" href="#__codelineno-27-30"></a>
-<a id="__codelineno-27-31" name="__codelineno-27-31" href="#__codelineno-27-31"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-27-31" name="__codelineno-27-31" href="#__codelineno-27-31"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-27-32" name="__codelineno-27-32" href="#__codelineno-27-32"></a><span class="w">    </span><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-33" name="__codelineno-27-33" href="#__codelineno-27-33"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-27-34" name="__codelineno-27-34" href="#__codelineno-27-34"></a><span class="w">    </span><span class="p">}</span>
@@ -6137,10 +6214,10 @@
 <a id="__codelineno-27-39" name="__codelineno-27-39" href="#__codelineno-27-39"></a><span class="w">            </span><span class="n">cout</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="s">"Queue is full"</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">endl</span><span class="p">;</span>
 <a id="__codelineno-27-40" name="__codelineno-27-40" href="#__codelineno-27-40"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-27-41" name="__codelineno-27-41" href="#__codelineno-27-41"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-27-42" name="__codelineno-27-42" href="#__codelineno-27-42"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
-<a id="__codelineno-27-43" name="__codelineno-27-43" href="#__codelineno-27-43"></a><span class="w">        </span><span class="c1">// Add num to the rear of the queue</span>
+<a id="__codelineno-27-42" name="__codelineno-27-42" href="#__codelineno-27-42"></a><span class="w">        </span><span class="c1">// Calculate the rear index, one position past the last element</span>
+<a id="__codelineno-27-43" name="__codelineno-27-43" href="#__codelineno-27-43"></a><span class="w">        </span><span class="c1">// Use modulo to wrap rear to the beginning when it passes the end of the array</span>
 <a id="__codelineno-27-44" name="__codelineno-27-44" href="#__codelineno-27-44"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">queSize</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">queCapacity</span><span class="p">;</span>
-<a id="__codelineno-27-45" name="__codelineno-27-45" href="#__codelineno-27-45"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-27-45" name="__codelineno-27-45" href="#__codelineno-27-45"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-27-46" name="__codelineno-27-46" href="#__codelineno-27-46"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">rear</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-27-47" name="__codelineno-27-47" href="#__codelineno-27-47"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-27-48" name="__codelineno-27-48" href="#__codelineno-27-48"></a><span class="w">    </span><span class="p">}</span>
@@ -6148,22 +6225,22 @@
 <a id="__codelineno-27-50" name="__codelineno-27-50" href="#__codelineno-27-50"></a><span class="w">    </span><span class="cm">/* Dequeue */</span>
 <a id="__codelineno-27-51" name="__codelineno-27-51" href="#__codelineno-27-51"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-52" name="__codelineno-27-52" href="#__codelineno-27-52"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peek</span><span class="p">();</span>
-<a id="__codelineno-27-53" name="__codelineno-27-53" href="#__codelineno-27-53"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position, if it passes the tail, return to array head</span>
+<a id="__codelineno-27-53" name="__codelineno-27-53" href="#__codelineno-27-53"></a><span class="w">        </span><span class="c1">// Advance the front index by one position, wrapping to the beginning if it passes the end</span>
 <a id="__codelineno-27-54" name="__codelineno-27-54" href="#__codelineno-27-54"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">queCapacity</span><span class="p">;</span>
 <a id="__codelineno-27-55" name="__codelineno-27-55" href="#__codelineno-27-55"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-27-56" name="__codelineno-27-56" href="#__codelineno-27-56"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-27-57" name="__codelineno-27-57" href="#__codelineno-27-57"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-58" name="__codelineno-27-58" href="#__codelineno-27-58"></a>
-<a id="__codelineno-27-59" name="__codelineno-27-59" href="#__codelineno-27-59"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-27-59" name="__codelineno-27-59" href="#__codelineno-27-59"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-27-60" name="__codelineno-27-60" href="#__codelineno-27-60"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-61" name="__codelineno-27-61" href="#__codelineno-27-61"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-27-62" name="__codelineno-27-62" href="#__codelineno-27-62"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Queue is empty"</span><span class="p">);</span>
 <a id="__codelineno-27-63" name="__codelineno-27-63" href="#__codelineno-27-63"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">front</span><span class="p">];</span>
 <a id="__codelineno-27-64" name="__codelineno-27-64" href="#__codelineno-27-64"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-65" name="__codelineno-27-65" href="#__codelineno-27-65"></a>
-<a id="__codelineno-27-66" name="__codelineno-27-66" href="#__codelineno-27-66"></a><span class="w">    </span><span class="cm">/* Convert array to Vector and return */</span>
+<a id="__codelineno-27-66" name="__codelineno-27-66" href="#__codelineno-27-66"></a><span class="w">    </span><span class="cm">/* Convert the array to a vector and return it */</span>
 <a id="__codelineno-27-67" name="__codelineno-27-67" href="#__codelineno-27-67"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">toVector</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-68" name="__codelineno-27-68" href="#__codelineno-27-68"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-27-68" name="__codelineno-27-68" href="#__codelineno-27-68"></a><span class="w">        </span><span class="c1">// Copy only the elements in the occupied range</span>
 <a id="__codelineno-27-69" name="__codelineno-27-69" href="#__codelineno-27-69"></a><span class="w">        </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">arr</span><span class="p">(</span><span class="n">queSize</span><span class="p">);</span>
 <a id="__codelineno-27-70" name="__codelineno-27-70" href="#__codelineno-27-70"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-71" name="__codelineno-27-71" href="#__codelineno-27-71"></a><span class="w">            </span><span class="n">arr</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">queCapacity</span><span class="p">];</span>
@@ -6174,28 +6251,28 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_queue.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_queue.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Queue implemented with a circular array */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kd">class</span> <span class="nc">ArrayQueue</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array for storing queue elements</span>
-<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Queue length</span>
+<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array that stores the queue elements</span>
+<a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, pointing to the first element</span>
+<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Queue size</span>
 <a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a>
 <a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="nf">ArrayQueue</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">        </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">capacity</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a>
-<a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="w">    </span><span class="cm">/* Get the capacity of the queue */</span>
+<a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="w">    </span><span class="cm">/* Get the queue capacity */</span>
 <a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">length</span><span class="p">;</span>
 <a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a>
-<a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-19" name="__codelineno-28-19" href="#__codelineno-28-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-28-20" name="__codelineno-28-20" href="#__codelineno-28-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-21" name="__codelineno-28-21" href="#__codelineno-28-21"></a>
-<a id="__codelineno-28-22" name="__codelineno-28-22" href="#__codelineno-28-22"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-28-22" name="__codelineno-28-22" href="#__codelineno-28-22"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-28-23" name="__codelineno-28-23" href="#__codelineno-28-23"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">boolean</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-24" name="__codelineno-28-24" href="#__codelineno-28-24"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-28-25" name="__codelineno-28-25" href="#__codelineno-28-25"></a><span class="w">    </span><span class="p">}</span>
@@ -6206,10 +6283,10 @@
 <a id="__codelineno-28-30" name="__codelineno-28-30" href="#__codelineno-28-30"></a><span class="w">            </span><span class="n">System</span><span class="p">.</span><span class="na">out</span><span class="p">.</span><span class="na">println</span><span class="p">(</span><span class="s">"Queue is full"</span><span class="p">);</span>
 <a id="__codelineno-28-31" name="__codelineno-28-31" href="#__codelineno-28-31"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-28-32" name="__codelineno-28-32" href="#__codelineno-28-32"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-28-33" name="__codelineno-28-33" href="#__codelineno-28-33"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
-<a id="__codelineno-28-34" name="__codelineno-28-34" href="#__codelineno-28-34"></a><span class="w">        </span><span class="c1">// Add num to the rear of the queue</span>
+<a id="__codelineno-28-33" name="__codelineno-28-33" href="#__codelineno-28-33"></a><span class="w">        </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
+<a id="__codelineno-28-34" name="__codelineno-28-34" href="#__codelineno-28-34"></a><span class="w">        </span><span class="c1">// Use modulo to wrap rear to the beginning when it passes the end of the array</span>
 <a id="__codelineno-28-35" name="__codelineno-28-35" href="#__codelineno-28-35"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">queSize</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">();</span>
-<a id="__codelineno-28-36" name="__codelineno-28-36" href="#__codelineno-28-36"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-28-36" name="__codelineno-28-36" href="#__codelineno-28-36"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-28-37" name="__codelineno-28-37" href="#__codelineno-28-37"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">rear</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-28-38" name="__codelineno-28-38" href="#__codelineno-28-38"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-28-39" name="__codelineno-28-39" href="#__codelineno-28-39"></a><span class="w">    </span><span class="p">}</span>
@@ -6217,22 +6294,22 @@
 <a id="__codelineno-28-41" name="__codelineno-28-41" href="#__codelineno-28-41"></a><span class="w">    </span><span class="cm">/* Dequeue */</span>
 <a id="__codelineno-28-42" name="__codelineno-28-42" href="#__codelineno-28-42"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-43" name="__codelineno-28-43" href="#__codelineno-28-43"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peek</span><span class="p">();</span>
-<a id="__codelineno-28-44" name="__codelineno-28-44" href="#__codelineno-28-44"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position, if it passes the tail, return to array head</span>
+<a id="__codelineno-28-44" name="__codelineno-28-44" href="#__codelineno-28-44"></a><span class="w">        </span><span class="c1">// Advance the front pointer by one position, wrapping to the beginning if it passes the end</span>
 <a id="__codelineno-28-45" name="__codelineno-28-45" href="#__codelineno-28-45"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">();</span>
 <a id="__codelineno-28-46" name="__codelineno-28-46" href="#__codelineno-28-46"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-28-47" name="__codelineno-28-47" href="#__codelineno-28-47"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-28-48" name="__codelineno-28-48" href="#__codelineno-28-48"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-49" name="__codelineno-28-49" href="#__codelineno-28-49"></a>
-<a id="__codelineno-28-50" name="__codelineno-28-50" href="#__codelineno-28-50"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-28-50" name="__codelineno-28-50" href="#__codelineno-28-50"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-28-51" name="__codelineno-28-51" href="#__codelineno-28-51"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-52" name="__codelineno-28-52" href="#__codelineno-28-52"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-28-53" name="__codelineno-28-53" href="#__codelineno-28-53"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">();</span>
 <a id="__codelineno-28-54" name="__codelineno-28-54" href="#__codelineno-28-54"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">front</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-55" name="__codelineno-28-55" href="#__codelineno-28-55"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-56" name="__codelineno-28-56" href="#__codelineno-28-56"></a>
-<a id="__codelineno-28-57" name="__codelineno-28-57" href="#__codelineno-28-57"></a><span class="w">    </span><span class="cm">/* Return array */</span>
+<a id="__codelineno-28-57" name="__codelineno-28-57" href="#__codelineno-28-57"></a><span class="w">    </span><span class="cm">/* Return an array */</span>
 <a id="__codelineno-28-58" name="__codelineno-28-58" href="#__codelineno-28-58"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="nf">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-59" name="__codelineno-28-59" href="#__codelineno-28-59"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-28-59" name="__codelineno-28-59" href="#__codelineno-28-59"></a><span class="w">        </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-28-60" name="__codelineno-28-60" href="#__codelineno-28-60"></a><span class="w">        </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">queSize</span><span class="o">]</span><span class="p">;</span>
 <a id="__codelineno-28-61" name="__codelineno-28-61" href="#__codelineno-28-61"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-62" name="__codelineno-28-62" href="#__codelineno-28-62"></a><span class="w">            </span><span class="n">res</span><span class="o">[</span><span class="n">i</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">j</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span><span class="o">]</span><span class="p">;</span>
@@ -6243,28 +6320,28 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_queue.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_queue.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Queue implemented with a circular array */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayQueue</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span><span class="w">  </span><span class="c1">// Array for storing queue elements</span>
-<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w">   </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Queue length</span>
+<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">nums</span><span class="p">;</span><span class="w">  </span><span class="c1">// Array that stores the queue elements</span>
+<a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w">   </span><span class="c1">// Front pointer, pointing to the first element</span>
+<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Queue size</span>
 <a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a>
 <a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nf">ArrayQueue</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">        </span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">capacity</span><span class="p">];</span>
 <a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a>
-<a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="w">    </span><span class="cm">/* Get the capacity of the queue */</span>
+<a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="w">    </span><span class="cm">/* Get the queue capacity */</span>
 <a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="nf">Capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="n">Length</span><span class="p">;</span>
 <a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a>
-<a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-19" name="__codelineno-29-19" href="#__codelineno-29-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-29-20" name="__codelineno-29-20" href="#__codelineno-29-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a>
-<a id="__codelineno-29-22" name="__codelineno-29-22" href="#__codelineno-29-22"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-29-22" name="__codelineno-29-22" href="#__codelineno-29-22"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-29-23" name="__codelineno-29-23" href="#__codelineno-29-23"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="nf">IsEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-24" name="__codelineno-29-24" href="#__codelineno-29-24"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-29-25" name="__codelineno-29-25" href="#__codelineno-29-25"></a><span class="w">    </span><span class="p">}</span>
@@ -6275,10 +6352,10 @@
 <a id="__codelineno-29-30" name="__codelineno-29-30" href="#__codelineno-29-30"></a><span class="w">            </span><span class="n">Console</span><span class="p">.</span><span class="n">WriteLine</span><span class="p">(</span><span class="s">"Queue is full"</span><span class="p">);</span>
 <a id="__codelineno-29-31" name="__codelineno-29-31" href="#__codelineno-29-31"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-29-32" name="__codelineno-29-32" href="#__codelineno-29-32"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-29-33" name="__codelineno-29-33" href="#__codelineno-29-33"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
-<a id="__codelineno-29-34" name="__codelineno-29-34" href="#__codelineno-29-34"></a><span class="w">        </span><span class="c1">// Add num to the rear of the queue</span>
+<a id="__codelineno-29-33" name="__codelineno-29-33" href="#__codelineno-29-33"></a><span class="w">        </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
+<a id="__codelineno-29-34" name="__codelineno-29-34" href="#__codelineno-29-34"></a><span class="w">        </span><span class="c1">// Use modulo to wrap rear to the beginning when it passes the end of the array</span>
 <a id="__codelineno-29-35" name="__codelineno-29-35" href="#__codelineno-29-35"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">queSize</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">Capacity</span><span class="p">();</span>
-<a id="__codelineno-29-36" name="__codelineno-29-36" href="#__codelineno-29-36"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-29-36" name="__codelineno-29-36" href="#__codelineno-29-36"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-29-37" name="__codelineno-29-37" href="#__codelineno-29-37"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">rear</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-29-38" name="__codelineno-29-38" href="#__codelineno-29-38"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-29-39" name="__codelineno-29-39" href="#__codelineno-29-39"></a><span class="w">    </span><span class="p">}</span>
@@ -6286,22 +6363,22 @@
 <a id="__codelineno-29-41" name="__codelineno-29-41" href="#__codelineno-29-41"></a><span class="w">    </span><span class="cm">/* Dequeue */</span>
 <a id="__codelineno-29-42" name="__codelineno-29-42" href="#__codelineno-29-42"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-43" name="__codelineno-29-43" href="#__codelineno-29-43"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Peek</span><span class="p">();</span>
-<a id="__codelineno-29-44" name="__codelineno-29-44" href="#__codelineno-29-44"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position, if it passes the tail, return to array head</span>
+<a id="__codelineno-29-44" name="__codelineno-29-44" href="#__codelineno-29-44"></a><span class="w">        </span><span class="c1">// Advance the front pointer by one position, wrapping to the beginning if it passes the end</span>
 <a id="__codelineno-29-45" name="__codelineno-29-45" href="#__codelineno-29-45"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">Capacity</span><span class="p">();</span>
 <a id="__codelineno-29-46" name="__codelineno-29-46" href="#__codelineno-29-46"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-29-47" name="__codelineno-29-47" href="#__codelineno-29-47"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-29-48" name="__codelineno-29-48" href="#__codelineno-29-48"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-49" name="__codelineno-29-49" href="#__codelineno-29-49"></a>
-<a id="__codelineno-29-50" name="__codelineno-29-50" href="#__codelineno-29-50"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-29-50" name="__codelineno-29-50" href="#__codelineno-29-50"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-29-51" name="__codelineno-29-51" href="#__codelineno-29-51"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-52" name="__codelineno-29-52" href="#__codelineno-29-52"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">IsEmpty</span><span class="p">())</span>
 <a id="__codelineno-29-53" name="__codelineno-29-53" href="#__codelineno-29-53"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">Exception</span><span class="p">();</span>
 <a id="__codelineno-29-54" name="__codelineno-29-54" href="#__codelineno-29-54"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">front</span><span class="p">];</span>
 <a id="__codelineno-29-55" name="__codelineno-29-55" href="#__codelineno-29-55"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-56" name="__codelineno-29-56" href="#__codelineno-29-56"></a>
-<a id="__codelineno-29-57" name="__codelineno-29-57" href="#__codelineno-29-57"></a><span class="w">    </span><span class="cm">/* Return array */</span>
+<a id="__codelineno-29-57" name="__codelineno-29-57" href="#__codelineno-29-57"></a><span class="w">    </span><span class="cm">/* Return an array */</span>
 <a id="__codelineno-29-58" name="__codelineno-29-58" href="#__codelineno-29-58"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="nf">ToArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-59" name="__codelineno-29-59" href="#__codelineno-29-59"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-29-59" name="__codelineno-29-59" href="#__codelineno-29-59"></a><span class="w">        </span><span class="c1">// Copy only the elements within the occupied range</span>
 <a id="__codelineno-29-60" name="__codelineno-29-60" href="#__codelineno-29-60"></a><span class="w">        </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="p">[</span><span class="n">queSize</span><span class="p">];</span>
 <a id="__codelineno-29-61" name="__codelineno-29-61" href="#__codelineno-29-61"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-62" name="__codelineno-29-62" href="#__codelineno-29-62"></a><span class="w">            </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">Capacity</span><span class="p">()];</span>
@@ -6312,15 +6389,15 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_queue.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_queue.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Queue implemented with a circular array */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">arrayQueue</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="nx">nums</span><span class="w">        </span><span class="p">[]</span><span class="kt">int</span><span class="w"> </span><span class="c1">// Array for storing queue elements</span>
-<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="nx">front</span><span class="w">       </span><span class="kt">int</span><span class="w">   </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">    </span><span class="nx">queSize</span><span class="w">     </span><span class="kt">int</span><span class="w">   </span><span class="c1">// Queue length</span>
-<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="nx">queCapacity</span><span class="w"> </span><span class="kt">int</span><span class="w">   </span><span class="c1">// Queue capacity (maximum number of elements)</span>
+<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="nx">nums</span><span class="w">        </span><span class="p">[]</span><span class="kt">int</span><span class="w"> </span><span class="c1">// Array that stores the queue elements</span>
+<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="nx">front</span><span class="w">       </span><span class="kt">int</span><span class="w">   </span><span class="c1">// Front pointer, pointing to the first element</span>
+<a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">    </span><span class="nx">queSize</span><span class="w">     </span><span class="kt">int</span><span class="w">   </span><span class="c1">// Queue size</span>
+<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="nx">queCapacity</span><span class="w"> </span><span class="kt">int</span><span class="w">   </span><span class="c1">// Queue capacity (the maximum number of elements)</span>
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="p">}</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a>
-<a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="cm">/* Access front of the queue element */</span>
+<a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="cm">/* Initialize the queue */</span>
 <a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="kd">func</span><span class="w"> </span><span class="nx">newArrayQueue</span><span class="p">(</span><span class="nx">queCapacity</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayQueue</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">arrayQueue</span><span class="p">{</span>
 <a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="w">        </span><span class="nx">nums</span><span class="p">:</span><span class="w">        </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">queCapacity</span><span class="p">),</span>
@@ -6330,26 +6407,26 @@
 <a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></a><span class="p">}</span>
 <a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a>
-<a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></a><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-30-20" name="__codelineno-30-20" href="#__codelineno-30-20"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayQueue</span><span class="p">)</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-21" name="__codelineno-30-21" href="#__codelineno-30-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span>
 <a id="__codelineno-30-22" name="__codelineno-30-22" href="#__codelineno-30-22"></a><span class="p">}</span>
 <a id="__codelineno-30-23" name="__codelineno-30-23" href="#__codelineno-30-23"></a>
-<a id="__codelineno-30-24" name="__codelineno-30-24" href="#__codelineno-30-24"></a><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-30-24" name="__codelineno-30-24" href="#__codelineno-30-24"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-30-25" name="__codelineno-30-25" href="#__codelineno-30-25"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayQueue</span><span class="p">)</span><span class="w"> </span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-26" name="__codelineno-30-26" href="#__codelineno-30-26"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-30-27" name="__codelineno-30-27" href="#__codelineno-30-27"></a><span class="p">}</span>
 <a id="__codelineno-30-28" name="__codelineno-30-28" href="#__codelineno-30-28"></a>
 <a id="__codelineno-30-29" name="__codelineno-30-29" href="#__codelineno-30-29"></a><span class="cm">/* Enqueue */</span>
 <a id="__codelineno-30-30" name="__codelineno-30-30" href="#__codelineno-30-30"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayQueue</span><span class="p">)</span><span class="w"> </span><span class="nx">push</span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-31" name="__codelineno-30-31" href="#__codelineno-30-31"></a><span class="w">    </span><span class="c1">// When rear == queCapacity, queue is full</span>
+<a id="__codelineno-30-31" name="__codelineno-30-31" href="#__codelineno-30-31"></a><span class="w">    </span><span class="c1">// The queue is full when queSize == queCapacity</span>
 <a id="__codelineno-30-32" name="__codelineno-30-32" href="#__codelineno-30-32"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queCapacity</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-33" name="__codelineno-30-33" href="#__codelineno-30-33"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-30-34" name="__codelineno-30-34" href="#__codelineno-30-34"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-35" name="__codelineno-30-35" href="#__codelineno-30-35"></a><span class="w">    </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
-<a id="__codelineno-30-36" name="__codelineno-30-36" href="#__codelineno-30-36"></a><span class="w">    </span><span class="c1">// Add num to the rear of the queue</span>
+<a id="__codelineno-30-35" name="__codelineno-30-35" href="#__codelineno-30-35"></a><span class="w">    </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
+<a id="__codelineno-30-36" name="__codelineno-30-36" href="#__codelineno-30-36"></a><span class="w">    </span><span class="c1">// Use modulo to wrap rear to the beginning when it passes the end of the array</span>
 <a id="__codelineno-30-37" name="__codelineno-30-37" href="#__codelineno-30-37"></a><span class="w">    </span><span class="nx">rear</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queCapacity</span>
-<a id="__codelineno-30-38" name="__codelineno-30-38" href="#__codelineno-30-38"></a><span class="w">    </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-30-38" name="__codelineno-30-38" href="#__codelineno-30-38"></a><span class="w">    </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-30-39" name="__codelineno-30-39" href="#__codelineno-30-39"></a><span class="w">    </span><span class="nx">q</span><span class="p">.</span><span class="nx">nums</span><span class="p">[</span><span class="nx">rear</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span>
 <a id="__codelineno-30-40" name="__codelineno-30-40" href="#__codelineno-30-40"></a><span class="w">    </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="o">++</span>
 <a id="__codelineno-30-41" name="__codelineno-30-41" href="#__codelineno-30-41"></a><span class="p">}</span>
@@ -6361,13 +6438,13 @@
 <a id="__codelineno-30-47" name="__codelineno-30-47" href="#__codelineno-30-47"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-30-48" name="__codelineno-30-48" href="#__codelineno-30-48"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-49" name="__codelineno-30-49" href="#__codelineno-30-49"></a>
-<a id="__codelineno-30-50" name="__codelineno-30-50" href="#__codelineno-30-50"></a><span class="w">    </span><span class="c1">// Move front pointer backward by one position, if it passes the tail, return to array head</span>
+<a id="__codelineno-30-50" name="__codelineno-30-50" href="#__codelineno-30-50"></a><span class="w">    </span><span class="c1">// Advance the front pointer by one position, wrapping to the beginning if it passes the end</span>
 <a id="__codelineno-30-51" name="__codelineno-30-51" href="#__codelineno-30-51"></a><span class="w">    </span><span class="nx">q</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queCapacity</span>
 <a id="__codelineno-30-52" name="__codelineno-30-52" href="#__codelineno-30-52"></a><span class="w">    </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="o">--</span>
 <a id="__codelineno-30-53" name="__codelineno-30-53" href="#__codelineno-30-53"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span>
 <a id="__codelineno-30-54" name="__codelineno-30-54" href="#__codelineno-30-54"></a><span class="p">}</span>
 <a id="__codelineno-30-55" name="__codelineno-30-55" href="#__codelineno-30-55"></a>
-<a id="__codelineno-30-56" name="__codelineno-30-56" href="#__codelineno-30-56"></a><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-30-56" name="__codelineno-30-56" href="#__codelineno-30-56"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-30-57" name="__codelineno-30-57" href="#__codelineno-30-57"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayQueue</span><span class="p">)</span><span class="w"> </span><span class="nx">peek</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-58" name="__codelineno-30-58" href="#__codelineno-30-58"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-59" name="__codelineno-30-59" href="#__codelineno-30-59"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
@@ -6375,7 +6452,7 @@
 <a id="__codelineno-30-61" name="__codelineno-30-61" href="#__codelineno-30-61"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">nums</span><span class="p">[</span><span class="nx">q</span><span class="p">.</span><span class="nx">front</span><span class="p">]</span>
 <a id="__codelineno-30-62" name="__codelineno-30-62" href="#__codelineno-30-62"></a><span class="p">}</span>
 <a id="__codelineno-30-63" name="__codelineno-30-63" href="#__codelineno-30-63"></a>
-<a id="__codelineno-30-64" name="__codelineno-30-64" href="#__codelineno-30-64"></a><span class="cm">/* Get Slice for printing */</span>
+<a id="__codelineno-30-64" name="__codelineno-30-64" href="#__codelineno-30-64"></a><span class="cm">/* Get a slice for printing */</span>
 <a id="__codelineno-30-65" name="__codelineno-30-65" href="#__codelineno-30-65"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayQueue</span><span class="p">)</span><span class="w"> </span><span class="nx">toSlice</span><span class="p">()</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-66" name="__codelineno-30-66" href="#__codelineno-30-66"></a><span class="w">    </span><span class="nx">rear</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">(</span><span class="nx">q</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queSize</span><span class="p">)</span>
 <a id="__codelineno-30-67" name="__codelineno-30-67" href="#__codelineno-30-67"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">rear</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">q</span><span class="p">.</span><span class="nx">queCapacity</span><span class="w"> </span><span class="p">{</span>
@@ -6387,30 +6464,30 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_queue.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_queue.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Queue implemented with a circular array */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayQueue</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="c1">// Array for storing queue elements</span>
-<a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">front</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">_size</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Queue length</span>
+<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">nums</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="c1">// Array that stores the queue elements</span>
+<a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">front</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Front index, identifying the first element</span>
+<a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">_size</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Queue size</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a>
 <a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">    </span><span class="kd">init</span><span class="p">(</span><span class="n">capacity</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">        </span><span class="c1">// Initialize array</span>
+<a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">        </span><span class="c1">// Initialize the array</span>
 <a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">        </span><span class="n">nums</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">capacity</span><span class="p">)</span>
 <a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a>
-<a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="w">    </span><span class="cm">/* Get the capacity of the queue */</span>
+<a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="w">    </span><span class="cm">/* Get the queue capacity */</span>
 <a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="w">        </span><span class="n">nums</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-31-17" name="__codelineno-31-17" href="#__codelineno-31-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-18" name="__codelineno-31-18" href="#__codelineno-31-18"></a>
-<a id="__codelineno-31-19" name="__codelineno-31-19" href="#__codelineno-31-19"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-31-19" name="__codelineno-31-19" href="#__codelineno-31-19"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-31-20" name="__codelineno-31-20" href="#__codelineno-31-20"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-21" name="__codelineno-31-21" href="#__codelineno-31-21"></a><span class="w">        </span><span class="n">_size</span>
 <a id="__codelineno-31-22" name="__codelineno-31-22" href="#__codelineno-31-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-23" name="__codelineno-31-23" href="#__codelineno-31-23"></a>
-<a id="__codelineno-31-24" name="__codelineno-31-24" href="#__codelineno-31-24"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-31-24" name="__codelineno-31-24" href="#__codelineno-31-24"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-31-25" name="__codelineno-31-25" href="#__codelineno-31-25"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-26" name="__codelineno-31-26" href="#__codelineno-31-26"></a><span class="w">        </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-31-27" name="__codelineno-31-27" href="#__codelineno-31-27"></a><span class="w">    </span><span class="p">}</span>
@@ -6421,10 +6498,10 @@
 <a id="__codelineno-31-32" name="__codelineno-31-32" href="#__codelineno-31-32"></a><span class="w">            </span><span class="bp">print</span><span class="p">(</span><span class="s">"Queue is full"</span><span class="p">)</span>
 <a id="__codelineno-31-33" name="__codelineno-31-33" href="#__codelineno-31-33"></a><span class="w">            </span><span class="k">return</span>
 <a id="__codelineno-31-34" name="__codelineno-31-34" href="#__codelineno-31-34"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-31-35" name="__codelineno-31-35" href="#__codelineno-31-35"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
-<a id="__codelineno-31-36" name="__codelineno-31-36" href="#__codelineno-31-36"></a><span class="w">        </span><span class="c1">// Add num to the rear of the queue</span>
+<a id="__codelineno-31-35" name="__codelineno-31-35" href="#__codelineno-31-35"></a><span class="w">        </span><span class="c1">// Calculate the rear index, one position past the last element</span>
+<a id="__codelineno-31-36" name="__codelineno-31-36" href="#__codelineno-31-36"></a><span class="w">        </span><span class="c1">// Use modulo to wrap rear to the beginning when it passes the end of the array</span>
 <a id="__codelineno-31-37" name="__codelineno-31-37" href="#__codelineno-31-37"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">rear</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">size</span><span class="p">())</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span>
-<a id="__codelineno-31-38" name="__codelineno-31-38" href="#__codelineno-31-38"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-31-38" name="__codelineno-31-38" href="#__codelineno-31-38"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-31-39" name="__codelineno-31-39" href="#__codelineno-31-39"></a><span class="w">        </span><span class="n">nums</span><span class="p">[</span><span class="n">rear</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-31-40" name="__codelineno-31-40" href="#__codelineno-31-40"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-31-41" name="__codelineno-31-41" href="#__codelineno-31-41"></a><span class="w">    </span><span class="p">}</span>
@@ -6433,13 +6510,13 @@
 <a id="__codelineno-31-44" name="__codelineno-31-44" href="#__codelineno-31-44"></a><span class="w">    </span><span class="p">@</span><span class="n">discardableResult</span>
 <a id="__codelineno-31-45" name="__codelineno-31-45" href="#__codelineno-31-45"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">pop</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-46" name="__codelineno-31-46" href="#__codelineno-31-46"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">peek</span><span class="p">()</span>
-<a id="__codelineno-31-47" name="__codelineno-31-47" href="#__codelineno-31-47"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position, if it passes the tail, return to array head</span>
+<a id="__codelineno-31-47" name="__codelineno-31-47" href="#__codelineno-31-47"></a><span class="w">        </span><span class="c1">// Advance the front index by one position, wrapping to the beginning if it passes the end</span>
 <a id="__codelineno-31-48" name="__codelineno-31-48" href="#__codelineno-31-48"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span>
 <a id="__codelineno-31-49" name="__codelineno-31-49" href="#__codelineno-31-49"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-31-50" name="__codelineno-31-50" href="#__codelineno-31-50"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-31-51" name="__codelineno-31-51" href="#__codelineno-31-51"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-52" name="__codelineno-31-52" href="#__codelineno-31-52"></a>
-<a id="__codelineno-31-53" name="__codelineno-31-53" href="#__codelineno-31-53"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-31-53" name="__codelineno-31-53" href="#__codelineno-31-53"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-31-54" name="__codelineno-31-54" href="#__codelineno-31-54"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">peek</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-55" name="__codelineno-31-55" href="#__codelineno-31-55"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-56" name="__codelineno-31-56" href="#__codelineno-31-56"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Queue is empty"</span><span class="p">)</span>
@@ -6447,36 +6524,36 @@
 <a id="__codelineno-31-58" name="__codelineno-31-58" href="#__codelineno-31-58"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="n">front</span><span class="p">]</span>
 <a id="__codelineno-31-59" name="__codelineno-31-59" href="#__codelineno-31-59"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-60" name="__codelineno-31-60" href="#__codelineno-31-60"></a>
-<a id="__codelineno-31-61" name="__codelineno-31-61" href="#__codelineno-31-61"></a><span class="w">    </span><span class="cm">/* Return array */</span>
+<a id="__codelineno-31-61" name="__codelineno-31-61" href="#__codelineno-31-61"></a><span class="w">    </span><span class="cm">/* Return an array */</span>
 <a id="__codelineno-31-62" name="__codelineno-31-62" href="#__codelineno-31-62"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-63" name="__codelineno-31-63" href="#__codelineno-31-63"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-31-63" name="__codelineno-31-63" href="#__codelineno-31-63"></a><span class="w">        </span><span class="c1">// Include only the elements in the occupied range</span>
 <a id="__codelineno-31-64" name="__codelineno-31-64" href="#__codelineno-31-64"></a><span class="w">        </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">size</span><span class="p">()).</span><span class="bp">map</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">nums</span><span class="p">[</span><span class="nv">$0</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">()]</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-31-65" name="__codelineno-31-65" href="#__codelineno-31-65"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-66" name="__codelineno-31-66" href="#__codelineno-31-66"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_queue.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_queue.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Queue implemented with a circular array */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">ArrayQueue</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="n">#nums</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array for storing queue elements</span>
-<a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="n">#front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="n">#queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Queue length</span>
+<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="n">#nums</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array that stores the queue elements</span>
+<a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="n">#front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, pointing to the first element</span>
+<a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="n">#queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Queue size</span>
 <a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a>
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">capacity</span><span class="p">);</span>
 <a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a>
-<a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="w">    </span><span class="cm">/* Get the capacity of the queue */</span>
+<a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="w">    </span><span class="cm">/* Get the queue capacity */</span>
 <a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="w">    </span><span class="nx">get</span><span class="w"> </span><span class="nx">capacity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-15" name="__codelineno-32-15" href="#__codelineno-32-15"></a>
-<a id="__codelineno-32-16" name="__codelineno-32-16" href="#__codelineno-32-16"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-32-16" name="__codelineno-32-16" href="#__codelineno-32-16"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-32-17" name="__codelineno-32-17" href="#__codelineno-32-17"></a><span class="w">    </span><span class="nx">get</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-18" name="__codelineno-32-18" href="#__codelineno-32-18"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="p">;</span>
 <a id="__codelineno-32-19" name="__codelineno-32-19" href="#__codelineno-32-19"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-20" name="__codelineno-32-20" href="#__codelineno-32-20"></a>
-<a id="__codelineno-32-21" name="__codelineno-32-21" href="#__codelineno-32-21"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-32-21" name="__codelineno-32-21" href="#__codelineno-32-21"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-32-22" name="__codelineno-32-22" href="#__codelineno-32-22"></a><span class="w">    </span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-23" name="__codelineno-32-23" href="#__codelineno-32-23"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-32-24" name="__codelineno-32-24" href="#__codelineno-32-24"></a><span class="w">    </span><span class="p">}</span>
@@ -6487,10 +6564,10 @@
 <a id="__codelineno-32-29" name="__codelineno-32-29" href="#__codelineno-32-29"></a><span class="w">            </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Queue is full'</span><span class="p">);</span>
 <a id="__codelineno-32-30" name="__codelineno-32-30" href="#__codelineno-32-30"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-32-31" name="__codelineno-32-31" href="#__codelineno-32-31"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-32-32" name="__codelineno-32-32" href="#__codelineno-32-32"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
-<a id="__codelineno-32-33" name="__codelineno-32-33" href="#__codelineno-32-33"></a><span class="w">        </span><span class="c1">// Add num to the rear of the queue</span>
+<a id="__codelineno-32-32" name="__codelineno-32-32" href="#__codelineno-32-32"></a><span class="w">        </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
+<a id="__codelineno-32-33" name="__codelineno-32-33" href="#__codelineno-32-33"></a><span class="w">        </span><span class="c1">// Use modulo to wrap rear to the beginning when it passes the end of the array</span>
 <a id="__codelineno-32-34" name="__codelineno-32-34" href="#__codelineno-32-34"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">;</span>
-<a id="__codelineno-32-35" name="__codelineno-32-35" href="#__codelineno-32-35"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-32-35" name="__codelineno-32-35" href="#__codelineno-32-35"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-32-36" name="__codelineno-32-36" href="#__codelineno-32-36"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#nums</span><span class="p">[</span><span class="nx">rear</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-32-37" name="__codelineno-32-37" href="#__codelineno-32-37"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-32-38" name="__codelineno-32-38" href="#__codelineno-32-38"></a><span class="w">    </span><span class="p">}</span>
@@ -6498,21 +6575,21 @@
 <a id="__codelineno-32-40" name="__codelineno-32-40" href="#__codelineno-32-40"></a><span class="w">    </span><span class="cm">/* Dequeue */</span>
 <a id="__codelineno-32-41" name="__codelineno-32-41" href="#__codelineno-32-41"></a><span class="w">    </span><span class="nx">pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-42" name="__codelineno-32-42" href="#__codelineno-32-42"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">peek</span><span class="p">();</span>
-<a id="__codelineno-32-43" name="__codelineno-32-43" href="#__codelineno-32-43"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position, if it passes the tail, return to array head</span>
+<a id="__codelineno-32-43" name="__codelineno-32-43" href="#__codelineno-32-43"></a><span class="w">        </span><span class="c1">// Advance the front pointer by one position, wrapping to the beginning if it passes the end</span>
 <a id="__codelineno-32-44" name="__codelineno-32-44" href="#__codelineno-32-44"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">;</span>
 <a id="__codelineno-32-45" name="__codelineno-32-45" href="#__codelineno-32-45"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-32-46" name="__codelineno-32-46" href="#__codelineno-32-46"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-32-47" name="__codelineno-32-47" href="#__codelineno-32-47"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-48" name="__codelineno-32-48" href="#__codelineno-32-48"></a>
-<a id="__codelineno-32-49" name="__codelineno-32-49" href="#__codelineno-32-49"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-32-49" name="__codelineno-32-49" href="#__codelineno-32-49"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-32-50" name="__codelineno-32-50" href="#__codelineno-32-50"></a><span class="w">    </span><span class="nx">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-51" name="__codelineno-32-51" href="#__codelineno-32-51"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Queue is empty'</span><span class="p">);</span>
 <a id="__codelineno-32-52" name="__codelineno-32-52" href="#__codelineno-32-52"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#nums</span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">];</span>
 <a id="__codelineno-32-53" name="__codelineno-32-53" href="#__codelineno-32-53"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-54" name="__codelineno-32-54" href="#__codelineno-32-54"></a>
-<a id="__codelineno-32-55" name="__codelineno-32-55" href="#__codelineno-32-55"></a><span class="w">    </span><span class="cm">/* Return Array */</span>
+<a id="__codelineno-32-55" name="__codelineno-32-55" href="#__codelineno-32-55"></a><span class="w">    </span><span class="cm">/* Return an Array */</span>
 <a id="__codelineno-32-56" name="__codelineno-32-56" href="#__codelineno-32-56"></a><span class="w">    </span><span class="nx">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-57" name="__codelineno-32-57" href="#__codelineno-32-57"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-32-57" name="__codelineno-32-57" href="#__codelineno-32-57"></a><span class="w">        </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-32-58" name="__codelineno-32-58" href="#__codelineno-32-58"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">);</span>
 <a id="__codelineno-32-59" name="__codelineno-32-59" href="#__codelineno-32-59"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#front</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-60" name="__codelineno-32-60" href="#__codelineno-32-60"></a><span class="w">            </span><span class="nx">arr</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">];</span>
@@ -6523,28 +6600,28 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_queue.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_queue.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Queue implemented with a circular array */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">ArrayQueue</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[];</span><span class="w"> </span><span class="c1">// Array for storing queue elements</span>
-<a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">front</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">queSize</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Queue length</span>
+<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">nums</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[];</span><span class="w"> </span><span class="c1">// Array that stores the queue elements</span>
+<a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">front</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, pointing to the first element</span>
+<a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">queSize</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w"> </span><span class="c1">// Queue size</span>
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a>
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">capacity</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="nx">capacity</span><span class="p">);</span>
 <a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a>
-<a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="w">    </span><span class="cm">/* Get the capacity of the queue */</span>
+<a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="w">    </span><span class="cm">/* Get the queue capacity */</span>
 <a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">    </span><span class="nx">get</span><span class="w"> </span><span class="nx">capacity</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">nums</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a>
-<a id="__codelineno-33-17" name="__codelineno-33-17" href="#__codelineno-33-17"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-33-17" name="__codelineno-33-17" href="#__codelineno-33-17"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-33-18" name="__codelineno-33-18" href="#__codelineno-33-18"></a><span class="w">    </span><span class="nx">get</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-19" name="__codelineno-33-19" href="#__codelineno-33-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="p">;</span>
 <a id="__codelineno-33-20" name="__codelineno-33-20" href="#__codelineno-33-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-21" name="__codelineno-33-21" href="#__codelineno-33-21"></a>
-<a id="__codelineno-33-22" name="__codelineno-33-22" href="#__codelineno-33-22"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-33-22" name="__codelineno-33-22" href="#__codelineno-33-22"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-33-23" name="__codelineno-33-23" href="#__codelineno-33-23"></a><span class="w">    </span><span class="nx">isEmpty</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-24" name="__codelineno-33-24" href="#__codelineno-33-24"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-33-25" name="__codelineno-33-25" href="#__codelineno-33-25"></a><span class="w">    </span><span class="p">}</span>
@@ -6555,10 +6632,10 @@
 <a id="__codelineno-33-30" name="__codelineno-33-30" href="#__codelineno-33-30"></a><span class="w">            </span><span class="nx">console</span><span class="p">.</span><span class="nx">log</span><span class="p">(</span><span class="s1">'Queue is full'</span><span class="p">);</span>
 <a id="__codelineno-33-31" name="__codelineno-33-31" href="#__codelineno-33-31"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-33-32" name="__codelineno-33-32" href="#__codelineno-33-32"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-33-33" name="__codelineno-33-33" href="#__codelineno-33-33"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
-<a id="__codelineno-33-34" name="__codelineno-33-34" href="#__codelineno-33-34"></a><span class="w">        </span><span class="c1">// Add num to the rear of the queue</span>
+<a id="__codelineno-33-33" name="__codelineno-33-33" href="#__codelineno-33-33"></a><span class="w">        </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
+<a id="__codelineno-33-34" name="__codelineno-33-34" href="#__codelineno-33-34"></a><span class="w">        </span><span class="c1">// Use modulo to wrap rear to the beginning when it passes the end of the array</span>
 <a id="__codelineno-33-35" name="__codelineno-33-35" href="#__codelineno-33-35"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">;</span>
-<a id="__codelineno-33-36" name="__codelineno-33-36" href="#__codelineno-33-36"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-33-36" name="__codelineno-33-36" href="#__codelineno-33-36"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-33-37" name="__codelineno-33-37" href="#__codelineno-33-37"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">nums</span><span class="p">[</span><span class="nx">rear</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-33-38" name="__codelineno-33-38" href="#__codelineno-33-38"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-33-39" name="__codelineno-33-39" href="#__codelineno-33-39"></a><span class="w">    </span><span class="p">}</span>
@@ -6566,21 +6643,21 @@
 <a id="__codelineno-33-41" name="__codelineno-33-41" href="#__codelineno-33-41"></a><span class="w">    </span><span class="cm">/* Dequeue */</span>
 <a id="__codelineno-33-42" name="__codelineno-33-42" href="#__codelineno-33-42"></a><span class="w">    </span><span class="nx">pop</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-43" name="__codelineno-33-43" href="#__codelineno-33-43"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">peek</span><span class="p">();</span>
-<a id="__codelineno-33-44" name="__codelineno-33-44" href="#__codelineno-33-44"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position, if it passes the tail, return to array head</span>
+<a id="__codelineno-33-44" name="__codelineno-33-44" href="#__codelineno-33-44"></a><span class="w">        </span><span class="c1">// Advance the front pointer by one position, wrapping to the beginning if it passes the end</span>
 <a id="__codelineno-33-45" name="__codelineno-33-45" href="#__codelineno-33-45"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">;</span>
 <a id="__codelineno-33-46" name="__codelineno-33-46" href="#__codelineno-33-46"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-33-47" name="__codelineno-33-47" href="#__codelineno-33-47"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-33-48" name="__codelineno-33-48" href="#__codelineno-33-48"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-49" name="__codelineno-33-49" href="#__codelineno-33-49"></a>
-<a id="__codelineno-33-50" name="__codelineno-33-50" href="#__codelineno-33-50"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-33-50" name="__codelineno-33-50" href="#__codelineno-33-50"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-33-51" name="__codelineno-33-51" href="#__codelineno-33-51"></a><span class="w">    </span><span class="nx">peek</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-52" name="__codelineno-33-52" href="#__codelineno-33-52"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Queue is empty'</span><span class="p">);</span>
 <a id="__codelineno-33-53" name="__codelineno-33-53" href="#__codelineno-33-53"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">nums</span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="p">];</span>
 <a id="__codelineno-33-54" name="__codelineno-33-54" href="#__codelineno-33-54"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-55" name="__codelineno-33-55" href="#__codelineno-33-55"></a>
-<a id="__codelineno-33-56" name="__codelineno-33-56" href="#__codelineno-33-56"></a><span class="w">    </span><span class="cm">/* Return Array */</span>
+<a id="__codelineno-33-56" name="__codelineno-33-56" href="#__codelineno-33-56"></a><span class="w">    </span><span class="cm">/* Return an Array */</span>
 <a id="__codelineno-33-57" name="__codelineno-33-57" href="#__codelineno-33-57"></a><span class="w">    </span><span class="nx">toArray</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-58" name="__codelineno-33-58" href="#__codelineno-33-58"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-33-58" name="__codelineno-33-58" href="#__codelineno-33-58"></a><span class="w">        </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-33-59" name="__codelineno-33-59" href="#__codelineno-33-59"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">);</span>
 <a id="__codelineno-33-60" name="__codelineno-33-60" href="#__codelineno-33-60"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">front</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="nx">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-61" name="__codelineno-33-61" href="#__codelineno-33-61"></a><span class="w">            </span><span class="nx">arr</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">nums</span><span class="p">[</span><span class="nx">j</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">capacity</span><span class="p">];</span>
@@ -6591,28 +6668,28 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_queue.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_queue.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Queue implemented with a circular array */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayQueue</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">_nums</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array for storing queue elements</span>
-<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">_front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">_queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Queue length</span>
+<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">_nums</span><span class="p">;</span><span class="w"> </span><span class="c1">// Array that stores the queue elements</span>
+<a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">_front</span><span class="p">;</span><span class="w"> </span><span class="c1">// Front pointer, pointing to the first element</span>
+<a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">_queSize</span><span class="p">;</span><span class="w"> </span><span class="c1">// Queue size</span>
 <a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a>
 <a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">  </span><span class="n">ArrayQueue</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">    </span><span class="n">_nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">capacity</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w">    </span><span class="n">_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a>
-<a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="w">  </span><span class="cm">/* Get the capacity of the queue */</span>
+<a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="w">  </span><span class="cm">/* Get the queue capacity */</span>
 <a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">capaCity</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_nums</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a>
-<a id="__codelineno-34-17" name="__codelineno-34-17" href="#__codelineno-34-17"></a><span class="w">  </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-34-17" name="__codelineno-34-17" href="#__codelineno-34-17"></a><span class="w">  </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-34-18" name="__codelineno-34-18" href="#__codelineno-34-18"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-19" name="__codelineno-34-19" href="#__codelineno-34-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_queSize</span><span class="p">;</span>
 <a id="__codelineno-34-20" name="__codelineno-34-20" href="#__codelineno-34-20"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-21" name="__codelineno-34-21" href="#__codelineno-34-21"></a>
-<a id="__codelineno-34-22" name="__codelineno-34-22" href="#__codelineno-34-22"></a><span class="w">  </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-34-22" name="__codelineno-34-22" href="#__codelineno-34-22"></a><span class="w">  </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-34-23" name="__codelineno-34-23" href="#__codelineno-34-23"></a><span class="w">  </span><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-24" name="__codelineno-34-24" href="#__codelineno-34-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-34-25" name="__codelineno-34-25" href="#__codelineno-34-25"></a><span class="w">  </span><span class="p">}</span>
@@ -6622,10 +6699,10 @@
 <a id="__codelineno-34-29" name="__codelineno-34-29" href="#__codelineno-34-29"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capaCity</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-30" name="__codelineno-34-30" href="#__codelineno-34-30"></a><span class="w">      </span><span class="k">throw</span><span class="w"> </span><span class="n">Exception</span><span class="p">(</span><span class="s2">"Queue is full"</span><span class="p">);</span>
 <a id="__codelineno-34-31" name="__codelineno-34-31" href="#__codelineno-34-31"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-34-32" name="__codelineno-34-32" href="#__codelineno-34-32"></a><span class="w">    </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
-<a id="__codelineno-34-33" name="__codelineno-34-33" href="#__codelineno-34-33"></a><span class="w">    </span><span class="c1">// Add num to the rear of the queue</span>
+<a id="__codelineno-34-32" name="__codelineno-34-32" href="#__codelineno-34-32"></a><span class="w">    </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
+<a id="__codelineno-34-33" name="__codelineno-34-33" href="#__codelineno-34-33"></a><span class="w">    </span><span class="c1">// Use modulo to wrap rear to the beginning when it passes the end of the array</span>
 <a id="__codelineno-34-34" name="__codelineno-34-34" href="#__codelineno-34-34"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">_front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">_queSize</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capaCity</span><span class="p">();</span>
-<a id="__codelineno-34-35" name="__codelineno-34-35" href="#__codelineno-34-35"></a><span class="w">    </span><span class="c1">// Add _num to queue rear</span>
+<a id="__codelineno-34-35" name="__codelineno-34-35" href="#__codelineno-34-35"></a><span class="w">    </span><span class="c1">// Add _num to the rear</span>
 <a id="__codelineno-34-36" name="__codelineno-34-36" href="#__codelineno-34-36"></a><span class="w">    </span><span class="n">_nums</span><span class="p">[</span><span class="n">rear</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span>
 <a id="__codelineno-34-37" name="__codelineno-34-37" href="#__codelineno-34-37"></a><span class="w">    </span><span class="n">_queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-34-38" name="__codelineno-34-38" href="#__codelineno-34-38"></a><span class="w">  </span><span class="p">}</span>
@@ -6633,13 +6710,13 @@
 <a id="__codelineno-34-40" name="__codelineno-34-40" href="#__codelineno-34-40"></a><span class="w">  </span><span class="cm">/* Dequeue */</span>
 <a id="__codelineno-34-41" name="__codelineno-34-41" href="#__codelineno-34-41"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">pop</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-42" name="__codelineno-34-42" href="#__codelineno-34-42"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peek</span><span class="p">();</span>
-<a id="__codelineno-34-43" name="__codelineno-34-43" href="#__codelineno-34-43"></a><span class="w">    </span><span class="c1">// Move front pointer backward by one position, if it passes the tail, return to array head</span>
+<a id="__codelineno-34-43" name="__codelineno-34-43" href="#__codelineno-34-43"></a><span class="w">    </span><span class="c1">// Advance the front pointer by one position, wrapping to the beginning if it passes the end</span>
 <a id="__codelineno-34-44" name="__codelineno-34-44" href="#__codelineno-34-44"></a><span class="w">    </span><span class="n">_front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">_front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capaCity</span><span class="p">();</span>
 <a id="__codelineno-34-45" name="__codelineno-34-45" href="#__codelineno-34-45"></a><span class="w">    </span><span class="n">_queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-34-46" name="__codelineno-34-46" href="#__codelineno-34-46"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span>
 <a id="__codelineno-34-47" name="__codelineno-34-47" href="#__codelineno-34-47"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-48" name="__codelineno-34-48" href="#__codelineno-34-48"></a>
-<a id="__codelineno-34-49" name="__codelineno-34-49" href="#__codelineno-34-49"></a><span class="w">  </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-34-49" name="__codelineno-34-49" href="#__codelineno-34-49"></a><span class="w">  </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-34-50" name="__codelineno-34-50" href="#__codelineno-34-50"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-51" name="__codelineno-34-51" href="#__codelineno-34-51"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-52" name="__codelineno-34-52" href="#__codelineno-34-52"></a><span class="w">      </span><span class="k">throw</span><span class="w"> </span><span class="n">Exception</span><span class="p">(</span><span class="s2">"Queue is empty"</span><span class="p">);</span>
@@ -6647,9 +6724,9 @@
 <a id="__codelineno-34-54" name="__codelineno-34-54" href="#__codelineno-34-54"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_nums</span><span class="p">[</span><span class="n">_front</span><span class="p">];</span>
 <a id="__codelineno-34-55" name="__codelineno-34-55" href="#__codelineno-34-55"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-56" name="__codelineno-34-56" href="#__codelineno-34-56"></a>
-<a id="__codelineno-34-57" name="__codelineno-34-57" href="#__codelineno-34-57"></a><span class="w">  </span><span class="cm">/* Return Array */</span>
+<a id="__codelineno-34-57" name="__codelineno-34-57" href="#__codelineno-34-57"></a><span class="w">  </span><span class="cm">/* Return an array */</span>
 <a id="__codelineno-34-58" name="__codelineno-34-58" href="#__codelineno-34-58"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-59" name="__codelineno-34-59" href="#__codelineno-34-59"></a><span class="w">    </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-34-59" name="__codelineno-34-59" href="#__codelineno-34-59"></a><span class="w">    </span><span class="c1">// Convert only the list elements within the occupied range</span>
 <a id="__codelineno-34-60" name="__codelineno-34-60" href="#__codelineno-34-60"></a><span class="w">    </span><span class="kd">final</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">List</span><span class="p">.</span><span class="n">filled</span><span class="p">(</span><span class="n">_queSize</span><span class="p">,</span><span class="w"> </span><span class="m">0</span><span class="p">);</span>
 <a id="__codelineno-34-61" name="__codelineno-34-61" href="#__codelineno-34-61"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_front</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">_queSize</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">,</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-62" name="__codelineno-34-62" href="#__codelineno-34-62"></a><span class="w">      </span><span class="n">res</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_nums</span><span class="p">[</span><span class="n">j</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capaCity</span><span class="p">()];</span>
@@ -6660,11 +6737,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_queue.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_queue.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Queue implemented with a circular array */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">struct</span><span class="w"> </span><span class="nc">ArrayQueue</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="p">,</span><span class="w">      </span><span class="c1">// Array for storing queue elements</span>
-<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="n">front</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w">        </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">    </span><span class="n">que_size</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w">     </span><span class="c1">// Queue length</span>
+<a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="n">nums</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="p">,</span><span class="w">      </span><span class="c1">// Array that stores the queue elements</span>
+<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="n">front</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w">        </span><span class="c1">// Front index, identifying the first element</span>
+<a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">    </span><span class="n">que_size</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w">     </span><span class="c1">// Queue size</span>
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="n">que_capacity</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w"> </span><span class="c1">// Queue capacity</span>
 <a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="p">}</span>
 <a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a>
@@ -6679,17 +6756,17 @@
 <a id="__codelineno-35-17" name="__codelineno-35-17" href="#__codelineno-35-17"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-35-18" name="__codelineno-35-18" href="#__codelineno-35-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-19" name="__codelineno-35-19" href="#__codelineno-35-19"></a>
-<a id="__codelineno-35-20" name="__codelineno-35-20" href="#__codelineno-35-20"></a><span class="w">    </span><span class="cm">/* Get the capacity of the queue */</span>
+<a id="__codelineno-35-20" name="__codelineno-35-20" href="#__codelineno-35-20"></a><span class="w">    </span><span class="cm">/* Get the queue capacity */</span>
 <a id="__codelineno-35-21" name="__codelineno-35-21" href="#__codelineno-35-21"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">capacity</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-22" name="__codelineno-35-22" href="#__codelineno-35-22"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">que_capacity</span>
 <a id="__codelineno-35-23" name="__codelineno-35-23" href="#__codelineno-35-23"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-24" name="__codelineno-35-24" href="#__codelineno-35-24"></a>
-<a id="__codelineno-35-25" name="__codelineno-35-25" href="#__codelineno-35-25"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-35-25" name="__codelineno-35-25" href="#__codelineno-35-25"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-35-26" name="__codelineno-35-26" href="#__codelineno-35-26"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-27" name="__codelineno-35-27" href="#__codelineno-35-27"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span>
 <a id="__codelineno-35-28" name="__codelineno-35-28" href="#__codelineno-35-28"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-29" name="__codelineno-35-29" href="#__codelineno-35-29"></a>
-<a id="__codelineno-35-30" name="__codelineno-35-30" href="#__codelineno-35-30"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-35-30" name="__codelineno-35-30" href="#__codelineno-35-30"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-35-31" name="__codelineno-35-31" href="#__codelineno-35-31"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">is_empty</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-32" name="__codelineno-35-32" href="#__codelineno-35-32"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-35-33" name="__codelineno-35-33" href="#__codelineno-35-33"></a><span class="w">    </span><span class="p">}</span>
@@ -6700,10 +6777,10 @@
 <a id="__codelineno-35-38" name="__codelineno-35-38" href="#__codelineno-35-38"></a><span class="w">            </span><span class="fm">println!</span><span class="p">(</span><span class="s">"Queue is full"</span><span class="p">);</span>
 <a id="__codelineno-35-39" name="__codelineno-35-39" href="#__codelineno-35-39"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-35-40" name="__codelineno-35-40" href="#__codelineno-35-40"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-35-41" name="__codelineno-35-41" href="#__codelineno-35-41"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
-<a id="__codelineno-35-42" name="__codelineno-35-42" href="#__codelineno-35-42"></a><span class="w">        </span><span class="c1">// Add num to the rear of the queue</span>
+<a id="__codelineno-35-41" name="__codelineno-35-41" href="#__codelineno-35-41"></a><span class="w">        </span><span class="c1">// Calculate the rear index, one position past the last element</span>
+<a id="__codelineno-35-42" name="__codelineno-35-42" href="#__codelineno-35-42"></a><span class="w">        </span><span class="c1">// Use modulo to wrap rear to the beginning when it passes the end of the array</span>
 <a id="__codelineno-35-43" name="__codelineno-35-43" href="#__codelineno-35-43"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">que_capacity</span><span class="p">;</span>
-<a id="__codelineno-35-44" name="__codelineno-35-44" href="#__codelineno-35-44"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-35-44" name="__codelineno-35-44" href="#__codelineno-35-44"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-35-45" name="__codelineno-35-45" href="#__codelineno-35-45"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">nums</span><span class="p">[</span><span class="n">rear</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-35-46" name="__codelineno-35-46" href="#__codelineno-35-46"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-35-47" name="__codelineno-35-47" href="#__codelineno-35-47"></a><span class="w">    </span><span class="p">}</span>
@@ -6711,21 +6788,21 @@
 <a id="__codelineno-35-49" name="__codelineno-35-49" href="#__codelineno-35-49"></a><span class="w">    </span><span class="cm">/* Dequeue */</span>
 <a id="__codelineno-35-50" name="__codelineno-35-50" href="#__codelineno-35-50"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">pop</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">T</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-51" name="__codelineno-35-51" href="#__codelineno-35-51"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">peek</span><span class="p">();</span>
-<a id="__codelineno-35-52" name="__codelineno-35-52" href="#__codelineno-35-52"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position, if it passes the tail, return to array head</span>
+<a id="__codelineno-35-52" name="__codelineno-35-52" href="#__codelineno-35-52"></a><span class="w">        </span><span class="c1">// Advance the front index by one position, wrapping to the beginning if it passes the end</span>
 <a id="__codelineno-35-53" name="__codelineno-35-53" href="#__codelineno-35-53"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">que_capacity</span><span class="p">;</span>
 <a id="__codelineno-35-54" name="__codelineno-35-54" href="#__codelineno-35-54"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">que_size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-35-55" name="__codelineno-35-55" href="#__codelineno-35-55"></a><span class="w">        </span><span class="n">num</span>
 <a id="__codelineno-35-56" name="__codelineno-35-56" href="#__codelineno-35-56"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-57" name="__codelineno-35-57" href="#__codelineno-35-57"></a>
-<a id="__codelineno-35-58" name="__codelineno-35-58" href="#__codelineno-35-58"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-35-58" name="__codelineno-35-58" href="#__codelineno-35-58"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-35-59" name="__codelineno-35-59" href="#__codelineno-35-59"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">peek</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">T</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-60" name="__codelineno-35-60" href="#__codelineno-35-60"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">is_empty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-61" name="__codelineno-35-61" href="#__codelineno-35-61"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"index out of bounds"</span><span class="p">);</span>
+<a id="__codelineno-35-61" name="__codelineno-35-61" href="#__codelineno-35-61"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"Index out of bounds"</span><span class="p">);</span>
 <a id="__codelineno-35-62" name="__codelineno-35-62" href="#__codelineno-35-62"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-35-63" name="__codelineno-35-63" href="#__codelineno-35-63"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">nums</span><span class="p">[</span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">usize</span><span class="p">]</span>
 <a id="__codelineno-35-64" name="__codelineno-35-64" href="#__codelineno-35-64"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-65" name="__codelineno-35-65" href="#__codelineno-35-65"></a>
-<a id="__codelineno-35-66" name="__codelineno-35-66" href="#__codelineno-35-66"></a><span class="w">    </span><span class="cm">/* Return array */</span>
+<a id="__codelineno-35-66" name="__codelineno-35-66" href="#__codelineno-35-66"></a><span class="w">    </span><span class="cm">/* Return a vector */</span>
 <a id="__codelineno-35-67" name="__codelineno-35-67" href="#__codelineno-35-67"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">to_vector</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-68" name="__codelineno-35-68" href="#__codelineno-35-68"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">cap</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">que_capacity</span><span class="p">;</span>
 <a id="__codelineno-35-69" name="__codelineno-35-69" href="#__codelineno-35-69"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">front</span><span class="p">;</span>
@@ -6740,10 +6817,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_queue.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_queue.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Queue implemented with a circular array */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">;</span><span class="w">       </span><span class="c1">// Array for storing queue elements</span>
-<a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w">       </span><span class="c1">// Front pointer, points to the front of the queue element</span>
+<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">nums</span><span class="p">;</span><span class="w">       </span><span class="c1">// Array that stores the queue elements</span>
+<a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">front</span><span class="p">;</span><span class="w">       </span><span class="c1">// Front pointer, pointing to the first element</span>
 <a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queSize</span><span class="p">;</span><span class="w">     </span><span class="c1">// Current number of elements in the queue</span>
 <a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">queCapacity</span><span class="p">;</span><span class="w"> </span><span class="c1">// Queue capacity</span>
 <a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="p">}</span><span class="w"> </span><span class="n">ArrayQueue</span><span class="p">;</span>
@@ -6751,7 +6828,7 @@
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="cm">/* Constructor */</span>
 <a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="n">ArrayQueue</span><span class="w"> </span><span class="o">*</span><span class="nf">newArrayQueue</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">capacity</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="w">    </span><span class="n">ArrayQueue</span><span class="w"> </span><span class="o">*</span><span class="n">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">ArrayQueue</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">ArrayQueue</span><span class="p">));</span>
-<a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="w">    </span><span class="c1">// Initialize array</span>
+<a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="w">    </span><span class="c1">// Initialize the array</span>
 <a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">    </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queCapacity</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">capacity</span><span class="p">;</span>
 <a id="__codelineno-36-14" name="__codelineno-36-14" href="#__codelineno-36-14"></a><span class="w">    </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queCapacity</span><span class="p">);</span>
 <a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a><span class="w">    </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
@@ -6764,22 +6841,22 @@
 <a id="__codelineno-36-22" name="__codelineno-36-22" href="#__codelineno-36-22"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">queue</span><span class="p">);</span>
 <a id="__codelineno-36-23" name="__codelineno-36-23" href="#__codelineno-36-23"></a><span class="p">}</span>
 <a id="__codelineno-36-24" name="__codelineno-36-24" href="#__codelineno-36-24"></a>
-<a id="__codelineno-36-25" name="__codelineno-36-25" href="#__codelineno-36-25"></a><span class="cm">/* Get the capacity of the queue */</span>
+<a id="__codelineno-36-25" name="__codelineno-36-25" href="#__codelineno-36-25"></a><span class="cm">/* Get the queue capacity */</span>
 <a id="__codelineno-36-26" name="__codelineno-36-26" href="#__codelineno-36-26"></a><span class="kt">int</span><span class="w"> </span><span class="nf">capacity</span><span class="p">(</span><span class="n">ArrayQueue</span><span class="w"> </span><span class="o">*</span><span class="n">queue</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-27" name="__codelineno-36-27" href="#__codelineno-36-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queCapacity</span><span class="p">;</span>
 <a id="__codelineno-36-28" name="__codelineno-36-28" href="#__codelineno-36-28"></a><span class="p">}</span>
 <a id="__codelineno-36-29" name="__codelineno-36-29" href="#__codelineno-36-29"></a>
-<a id="__codelineno-36-30" name="__codelineno-36-30" href="#__codelineno-36-30"></a><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-36-30" name="__codelineno-36-30" href="#__codelineno-36-30"></a><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-36-31" name="__codelineno-36-31" href="#__codelineno-36-31"></a><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="n">ArrayQueue</span><span class="w"> </span><span class="o">*</span><span class="n">queue</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-32" name="__codelineno-36-32" href="#__codelineno-36-32"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-36-33" name="__codelineno-36-33" href="#__codelineno-36-33"></a><span class="p">}</span>
 <a id="__codelineno-36-34" name="__codelineno-36-34" href="#__codelineno-36-34"></a>
-<a id="__codelineno-36-35" name="__codelineno-36-35" href="#__codelineno-36-35"></a><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-36-35" name="__codelineno-36-35" href="#__codelineno-36-35"></a><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-36-36" name="__codelineno-36-36" href="#__codelineno-36-36"></a><span class="kt">bool</span><span class="w"> </span><span class="nf">empty</span><span class="p">(</span><span class="n">ArrayQueue</span><span class="w"> </span><span class="o">*</span><span class="n">queue</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-37" name="__codelineno-36-37" href="#__codelineno-36-37"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-36-38" name="__codelineno-36-38" href="#__codelineno-36-38"></a><span class="p">}</span>
 <a id="__codelineno-36-39" name="__codelineno-36-39" href="#__codelineno-36-39"></a>
-<a id="__codelineno-36-40" name="__codelineno-36-40" href="#__codelineno-36-40"></a><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-36-40" name="__codelineno-36-40" href="#__codelineno-36-40"></a><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-36-41" name="__codelineno-36-41" href="#__codelineno-36-41"></a><span class="kt">int</span><span class="w"> </span><span class="nf">peek</span><span class="p">(</span><span class="n">ArrayQueue</span><span class="w"> </span><span class="o">*</span><span class="n">queue</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-42" name="__codelineno-36-42" href="#__codelineno-36-42"></a><span class="w">    </span><span class="n">assert</span><span class="p">(</span><span class="n">size</span><span class="p">(</span><span class="n">queue</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">0</span><span class="p">);</span>
 <a id="__codelineno-36-43" name="__codelineno-36-43" href="#__codelineno-36-43"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">nums</span><span class="p">[</span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="p">];</span>
@@ -6791,10 +6868,10 @@
 <a id="__codelineno-36-49" name="__codelineno-36-49" href="#__codelineno-36-49"></a><span class="w">        </span><span class="n">printf</span><span class="p">(</span><span class="s">"Queue is full</span><span class="se">\r\n</span><span class="s">"</span><span class="p">);</span>
 <a id="__codelineno-36-50" name="__codelineno-36-50" href="#__codelineno-36-50"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-36-51" name="__codelineno-36-51" href="#__codelineno-36-51"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-36-52" name="__codelineno-36-52" href="#__codelineno-36-52"></a><span class="w">    </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
-<a id="__codelineno-36-53" name="__codelineno-36-53" href="#__codelineno-36-53"></a><span class="w">    </span><span class="c1">// Add num to the rear of the queue</span>
+<a id="__codelineno-36-52" name="__codelineno-36-52" href="#__codelineno-36-52"></a><span class="w">    </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
+<a id="__codelineno-36-53" name="__codelineno-36-53" href="#__codelineno-36-53"></a><span class="w">    </span><span class="c1">// Use modulo to wrap rear to the beginning when it passes the end of the array</span>
 <a id="__codelineno-36-54" name="__codelineno-36-54" href="#__codelineno-36-54"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queCapacity</span><span class="p">;</span>
-<a id="__codelineno-36-55" name="__codelineno-36-55" href="#__codelineno-36-55"></a><span class="w">    </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-36-55" name="__codelineno-36-55" href="#__codelineno-36-55"></a><span class="w">    </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-36-56" name="__codelineno-36-56" href="#__codelineno-36-56"></a><span class="w">    </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">nums</span><span class="p">[</span><span class="n">rear</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-36-57" name="__codelineno-36-57" href="#__codelineno-36-57"></a><span class="w">    </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-36-58" name="__codelineno-36-58" href="#__codelineno-36-58"></a><span class="p">}</span>
@@ -6802,13 +6879,13 @@
 <a id="__codelineno-36-60" name="__codelineno-36-60" href="#__codelineno-36-60"></a><span class="cm">/* Dequeue */</span>
 <a id="__codelineno-36-61" name="__codelineno-36-61" href="#__codelineno-36-61"></a><span class="kt">int</span><span class="w"> </span><span class="nf">pop</span><span class="p">(</span><span class="n">ArrayQueue</span><span class="w"> </span><span class="o">*</span><span class="n">queue</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-62" name="__codelineno-36-62" href="#__codelineno-36-62"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peek</span><span class="p">(</span><span class="n">queue</span><span class="p">);</span>
-<a id="__codelineno-36-63" name="__codelineno-36-63" href="#__codelineno-36-63"></a><span class="w">    </span><span class="c1">// Move front pointer backward by one position, if it passes the tail, return to array head</span>
+<a id="__codelineno-36-63" name="__codelineno-36-63" href="#__codelineno-36-63"></a><span class="w">    </span><span class="c1">// Advance the front pointer by one position, wrapping to the beginning if it passes the end</span>
 <a id="__codelineno-36-64" name="__codelineno-36-64" href="#__codelineno-36-64"></a><span class="w">    </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queCapacity</span><span class="p">;</span>
 <a id="__codelineno-36-65" name="__codelineno-36-65" href="#__codelineno-36-65"></a><span class="w">    </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="o">--</span><span class="p">;</span>
 <a id="__codelineno-36-66" name="__codelineno-36-66" href="#__codelineno-36-66"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-36-67" name="__codelineno-36-67" href="#__codelineno-36-67"></a><span class="p">}</span>
 <a id="__codelineno-36-68" name="__codelineno-36-68" href="#__codelineno-36-68"></a>
-<a id="__codelineno-36-69" name="__codelineno-36-69" href="#__codelineno-36-69"></a><span class="cm">/* Return array for printing */</span>
+<a id="__codelineno-36-69" name="__codelineno-36-69" href="#__codelineno-36-69"></a><span class="cm">/* Return an array for printing */</span>
 <a id="__codelineno-36-70" name="__codelineno-36-70" href="#__codelineno-36-70"></a><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="nf">toArray</span><span class="p">(</span><span class="n">ArrayQueue</span><span class="w"> </span><span class="o">*</span><span class="n">queue</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">queSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-71" name="__codelineno-36-71" href="#__codelineno-36-71"></a><span class="w">    </span><span class="o">*</span><span class="n">queSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="p">;</span>
 <a id="__codelineno-36-72" name="__codelineno-36-72" href="#__codelineno-36-72"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">calloc</span><span class="p">(</span><span class="n">queue</span><span class="o">-&gt;</span><span class="n">queSize</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">));</span>
@@ -6822,23 +6899,23 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_queue.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Queue based on circular array implementation */</span>
+<div class="highlight"><span class="filename">array_queue.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Queue implemented with a circular array */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayQueue</span><span class="p">(</span><span class="n">capacity</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">val</span><span class="w"> </span><span class="nv">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">capacity</span><span class="p">)</span><span class="w"> </span><span class="c1">// Array for storing queue elements</span>
-<a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">front</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">queSize</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Queue length</span>
+<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">val</span><span class="w"> </span><span class="nv">nums</span><span class="p">:</span><span class="w"> </span><span class="n">IntArray</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">capacity</span><span class="p">)</span><span class="w"> </span><span class="c1">// Array that stores the queue elements</span>
+<a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">front</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Front pointer, pointing to the first element</span>
+<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">queSize</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Queue size</span>
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a>
-<a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="cm">/* Get the capacity of the queue */</span>
+<a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="cm">/* Get the queue capacity */</span>
 <a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">capacity</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a>
-<a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="w">    </span><span class="cm">/* Get the length of the queue */</span>
+<a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="w">    </span><span class="cm">/* Get the queue size */</span>
 <a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">size</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span>
 <a id="__codelineno-37-15" name="__codelineno-37-15" href="#__codelineno-37-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-16" name="__codelineno-37-16" href="#__codelineno-37-16"></a>
-<a id="__codelineno-37-17" name="__codelineno-37-17" href="#__codelineno-37-17"></a><span class="w">    </span><span class="cm">/* Check if the queue is empty */</span>
+<a id="__codelineno-37-17" name="__codelineno-37-17" href="#__codelineno-37-17"></a><span class="w">    </span><span class="cm">/* Check whether the queue is empty */</span>
 <a id="__codelineno-37-18" name="__codelineno-37-18" href="#__codelineno-37-18"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">():</span><span class="w"> </span><span class="kt">Boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-19" name="__codelineno-37-19" href="#__codelineno-37-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">queSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-37-20" name="__codelineno-37-20" href="#__codelineno-37-20"></a><span class="w">    </span><span class="p">}</span>
@@ -6849,10 +6926,10 @@
 <a id="__codelineno-37-25" name="__codelineno-37-25" href="#__codelineno-37-25"></a><span class="w">            </span><span class="n">println</span><span class="p">(</span><span class="s">"Queue is full"</span><span class="p">)</span>
 <a id="__codelineno-37-26" name="__codelineno-37-26" href="#__codelineno-37-26"></a><span class="w">            </span><span class="k">return</span>
 <a id="__codelineno-37-27" name="__codelineno-37-27" href="#__codelineno-37-27"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-37-28" name="__codelineno-37-28" href="#__codelineno-37-28"></a><span class="w">        </span><span class="c1">// Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
-<a id="__codelineno-37-29" name="__codelineno-37-29" href="#__codelineno-37-29"></a><span class="w">        </span><span class="c1">// Add num to the rear of the queue</span>
+<a id="__codelineno-37-28" name="__codelineno-37-28" href="#__codelineno-37-28"></a><span class="w">        </span><span class="c1">// Calculate the rear pointer, which points one position past the last element</span>
+<a id="__codelineno-37-29" name="__codelineno-37-29" href="#__codelineno-37-29"></a><span class="w">        </span><span class="c1">// Use modulo to wrap rear to the beginning when it passes the end of the array</span>
 <a id="__codelineno-37-30" name="__codelineno-37-30" href="#__codelineno-37-30"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">queSize</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span>
-<a id="__codelineno-37-31" name="__codelineno-37-31" href="#__codelineno-37-31"></a><span class="w">        </span><span class="c1">// Front pointer moves one position backward</span>
+<a id="__codelineno-37-31" name="__codelineno-37-31" href="#__codelineno-37-31"></a><span class="w">        </span><span class="c1">// Add num to the rear</span>
 <a id="__codelineno-37-32" name="__codelineno-37-32" href="#__codelineno-37-32"></a><span class="w">        </span><span class="n">nums</span><span class="o">[</span><span class="n">rear</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-37-33" name="__codelineno-37-33" href="#__codelineno-37-33"></a><span class="w">        </span><span class="n">queSize</span><span class="o">++</span>
 <a id="__codelineno-37-34" name="__codelineno-37-34" href="#__codelineno-37-34"></a><span class="w">    </span><span class="p">}</span>
@@ -6860,21 +6937,21 @@
 <a id="__codelineno-37-36" name="__codelineno-37-36" href="#__codelineno-37-36"></a><span class="w">    </span><span class="cm">/* Dequeue */</span>
 <a id="__codelineno-37-37" name="__codelineno-37-37" href="#__codelineno-37-37"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">pop</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-38" name="__codelineno-37-38" href="#__codelineno-37-38"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peek</span><span class="p">()</span>
-<a id="__codelineno-37-39" name="__codelineno-37-39" href="#__codelineno-37-39"></a><span class="w">        </span><span class="c1">// Move front pointer backward by one position, if it passes the tail, return to array head</span>
+<a id="__codelineno-37-39" name="__codelineno-37-39" href="#__codelineno-37-39"></a><span class="w">        </span><span class="c1">// Advance the front pointer by one position, wrapping to the beginning if it passes the end</span>
 <a id="__codelineno-37-40" name="__codelineno-37-40" href="#__codelineno-37-40"></a><span class="w">        </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span><span class="p">()</span>
 <a id="__codelineno-37-41" name="__codelineno-37-41" href="#__codelineno-37-41"></a><span class="w">        </span><span class="n">queSize</span><span class="o">--</span>
 <a id="__codelineno-37-42" name="__codelineno-37-42" href="#__codelineno-37-42"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-37-43" name="__codelineno-37-43" href="#__codelineno-37-43"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-44" name="__codelineno-37-44" href="#__codelineno-37-44"></a>
-<a id="__codelineno-37-45" name="__codelineno-37-45" href="#__codelineno-37-45"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-37-45" name="__codelineno-37-45" href="#__codelineno-37-45"></a><span class="w">    </span><span class="cm">/* Access the front element */</span>
 <a id="__codelineno-37-46" name="__codelineno-37-46" href="#__codelineno-37-46"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">peek</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-47" name="__codelineno-37-47" href="#__codelineno-37-47"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">()</span>
 <a id="__codelineno-37-48" name="__codelineno-37-48" href="#__codelineno-37-48"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">nums</span><span class="o">[</span><span class="n">front</span><span class="o">]</span>
 <a id="__codelineno-37-49" name="__codelineno-37-49" href="#__codelineno-37-49"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-50" name="__codelineno-37-50" href="#__codelineno-37-50"></a>
-<a id="__codelineno-37-51" name="__codelineno-37-51" href="#__codelineno-37-51"></a><span class="w">    </span><span class="cm">/* Return array */</span>
+<a id="__codelineno-37-51" name="__codelineno-37-51" href="#__codelineno-37-51"></a><span class="w">    </span><span class="cm">/* Return an array */</span>
 <a id="__codelineno-37-52" name="__codelineno-37-52" href="#__codelineno-37-52"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">toArray</span><span class="p">():</span><span class="w"> </span><span class="n">IntArray</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-53" name="__codelineno-37-53" href="#__codelineno-37-53"></a><span class="w">        </span><span class="c1">// Elements enqueue</span>
+<a id="__codelineno-37-53" name="__codelineno-37-53" href="#__codelineno-37-53"></a><span class="w">        </span><span class="c1">// Copy only the elements within the occupied range</span>
 <a id="__codelineno-37-54" name="__codelineno-37-54" href="#__codelineno-37-54"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">queSize</span><span class="p">)</span>
 <a id="__codelineno-37-55" name="__codelineno-37-55" href="#__codelineno-37-55"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-37-56" name="__codelineno-37-56" href="#__codelineno-37-56"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">front</span>
@@ -6889,24 +6966,24 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_queue.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Queue based on circular array ###</span>
+<div class="highlight"><span class="filename">array_queue.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Queue implemented with a circular array ###</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayQueue</span>
-<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1">### Get queue length ###</span>
+<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1">### Get the queue size ###</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="kp">attr_reader</span><span class="w"> </span><span class="ss">:size</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a>
 <a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="c1">### Constructor ###</span>
 <a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">initialize</span><span class="p">(</span><span class="n">size</span><span class="p">)</span>
-<a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">    </span><span class="vi">@nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1"># Array for storing queue elements</span>
-<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">    </span><span class="vi">@front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Front pointer, points to the front of the queue element</span>
-<a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Queue length</span>
+<a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">    </span><span class="vi">@nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="c1"># Array that stores the queue elements</span>
+<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">    </span><span class="vi">@front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Front pointer, pointing to the first element</span>
+<a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="c1"># Queue size</span>
 <a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a>
-<a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">  </span><span class="c1">### Get queue capacity ###</span>
+<a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">  </span><span class="c1">### Get the queue capacity ###</span>
 <a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">capacity</span>
 <a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a><span class="w">    </span><span class="vi">@nums</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-17" name="__codelineno-38-17" href="#__codelineno-38-17"></a>
-<a id="__codelineno-38-18" name="__codelineno-38-18" href="#__codelineno-38-18"></a><span class="w">  </span><span class="c1">### Check if queue is empty ###</span>
+<a id="__codelineno-38-18" name="__codelineno-38-18" href="#__codelineno-38-18"></a><span class="w">  </span><span class="c1">### Check whether the queue is empty ###</span>
 <a id="__codelineno-38-19" name="__codelineno-38-19" href="#__codelineno-38-19"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">is_empty?</span>
 <a id="__codelineno-38-20" name="__codelineno-38-20" href="#__codelineno-38-20"></a><span class="w">    </span><span class="n">size</span><span class="o">.</span><span class="n">zero?</span>
 <a id="__codelineno-38-21" name="__codelineno-38-21" href="#__codelineno-38-21"></a><span class="w">  </span><span class="k">end</span>
@@ -6915,10 +6992,10 @@
 <a id="__codelineno-38-24" name="__codelineno-38-24" href="#__codelineno-38-24"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-38-25" name="__codelineno-38-25" href="#__codelineno-38-25"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="p">,</span><span class="w"> </span><span class="s1">'Queue is full'</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">capacity</span>
 <a id="__codelineno-38-26" name="__codelineno-38-26" href="#__codelineno-38-26"></a>
-<a id="__codelineno-38-27" name="__codelineno-38-27" href="#__codelineno-38-27"></a><span class="w">    </span><span class="c1"># Use modulo operation to wrap rear around to the head after passing the tail of the array</span>
-<a id="__codelineno-38-28" name="__codelineno-38-28" href="#__codelineno-38-28"></a><span class="w">    </span><span class="c1"># Add num to the rear of the queue</span>
+<a id="__codelineno-38-27" name="__codelineno-38-27" href="#__codelineno-38-27"></a><span class="w">    </span><span class="c1"># Calculate the rear pointer, which points one position past the last element</span>
+<a id="__codelineno-38-28" name="__codelineno-38-28" href="#__codelineno-38-28"></a><span class="w">    </span><span class="c1"># Use modulo to wrap rear to the beginning when it passes the end of the array</span>
 <a id="__codelineno-38-29" name="__codelineno-38-29" href="#__codelineno-38-29"></a><span class="w">    </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="vi">@front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span>
-<a id="__codelineno-38-30" name="__codelineno-38-30" href="#__codelineno-38-30"></a><span class="w">    </span><span class="c1"># Front pointer moves one position backward</span>
+<a id="__codelineno-38-30" name="__codelineno-38-30" href="#__codelineno-38-30"></a><span class="w">    </span><span class="c1"># Add num to the rear</span>
 <a id="__codelineno-38-31" name="__codelineno-38-31" href="#__codelineno-38-31"></a><span class="w">    </span><span class="vi">@nums</span><span class="o">[</span><span class="n">rear</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-38-32" name="__codelineno-38-32" href="#__codelineno-38-32"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-38-33" name="__codelineno-38-33" href="#__codelineno-38-33"></a><span class="w">  </span><span class="k">end</span>
@@ -6926,20 +7003,20 @@
 <a id="__codelineno-38-35" name="__codelineno-38-35" href="#__codelineno-38-35"></a><span class="w">  </span><span class="c1">### Dequeue ###</span>
 <a id="__codelineno-38-36" name="__codelineno-38-36" href="#__codelineno-38-36"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">pop</span>
 <a id="__codelineno-38-37" name="__codelineno-38-37" href="#__codelineno-38-37"></a><span class="w">    </span><span class="n">num</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">peek</span>
-<a id="__codelineno-38-38" name="__codelineno-38-38" href="#__codelineno-38-38"></a><span class="w">    </span><span class="c1"># Move front pointer backward by one position, if it passes the tail, return to array head</span>
+<a id="__codelineno-38-38" name="__codelineno-38-38" href="#__codelineno-38-38"></a><span class="w">    </span><span class="c1"># Advance the front pointer by one position, wrapping to the beginning if it passes the end</span>
 <a id="__codelineno-38-39" name="__codelineno-38-39" href="#__codelineno-38-39"></a><span class="w">    </span><span class="vi">@front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="vi">@front</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">%</span><span class="w"> </span><span class="n">capacity</span>
 <a id="__codelineno-38-40" name="__codelineno-38-40" href="#__codelineno-38-40"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-38-41" name="__codelineno-38-41" href="#__codelineno-38-41"></a><span class="w">    </span><span class="n">num</span>
 <a id="__codelineno-38-42" name="__codelineno-38-42" href="#__codelineno-38-42"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-43" name="__codelineno-38-43" href="#__codelineno-38-43"></a>
-<a id="__codelineno-38-44" name="__codelineno-38-44" href="#__codelineno-38-44"></a><span class="w">  </span><span class="c1">### Access front element ###</span>
+<a id="__codelineno-38-44" name="__codelineno-38-44" href="#__codelineno-38-44"></a><span class="w">  </span><span class="c1">### Access the front element ###</span>
 <a id="__codelineno-38-45" name="__codelineno-38-45" href="#__codelineno-38-45"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">peek</span>
 <a id="__codelineno-38-46" name="__codelineno-38-46" href="#__codelineno-38-46"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="p">,</span><span class="w"> </span><span class="s1">'Queue is empty'</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">is_empty?</span>
 <a id="__codelineno-38-47" name="__codelineno-38-47" href="#__codelineno-38-47"></a>
 <a id="__codelineno-38-48" name="__codelineno-38-48" href="#__codelineno-38-48"></a><span class="w">    </span><span class="vi">@nums</span><span class="o">[</span><span class="vi">@front</span><span class="o">]</span>
 <a id="__codelineno-38-49" name="__codelineno-38-49" href="#__codelineno-38-49"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-50" name="__codelineno-38-50" href="#__codelineno-38-50"></a>
-<a id="__codelineno-38-51" name="__codelineno-38-51" href="#__codelineno-38-51"></a><span class="w">  </span><span class="c1">### Return list for printing ###</span>
+<a id="__codelineno-38-51" name="__codelineno-38-51" href="#__codelineno-38-51"></a><span class="w">  </span><span class="c1">### Return a list for printing ###</span>
 <a id="__codelineno-38-52" name="__codelineno-38-52" href="#__codelineno-38-52"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">to_array</span>
 <a id="__codelineno-38-53" name="__codelineno-38-53" href="#__codelineno-38-53"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Array</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">size</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-38-54" name="__codelineno-38-54" href="#__codelineno-38-54"></a><span class="w">    </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@front</span>
@@ -6956,12 +7033,17 @@
 </div>
 </div>
 </div>
-<p>The queue implemented above still has limitations: its length is immutable. However, this problem is not difficult to solve. We can replace the array with a dynamic array to introduce an expansion mechanism. Interested readers can try to implement this themselves.</p>
-<p>The comparison conclusions for the two implementations are consistent with those for stacks and will not be repeated here.</p>
-<h2 id="523-typical-applications-of-queue">5.2.3 &nbsp; Typical Applications of Queue<a class="headerlink" href="#523-typical-applications-of-queue" title="Permanent link">&para;</a></h2>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20ArrayQueue%3A%0A%20%20%20%20%22%22%22Queue%20implemented%20with%20a%20circular%20array%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20size%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self._nums%3A%20list%5Bint%5D%20%3D%20%5B0%5D%20%2A%20size%20%20%23%20Array%20that%20stores%20the%20queue%20elements%0A%20%20%20%20%20%20%20%20self._front%3A%20int%20%3D%200%20%20%23%20Front%20pointer,%20pointing%20to%20the%20first%20element%0A%20%20%20%20%20%20%20%20self._size%3A%20int%20%3D%200%20%20%23%20Queue%20size%0A%0A%20%20%20%20def%20capacity%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20queue%20capacity%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self._nums%29%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20queue%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20self._size%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22Check%20whether%20the%20queue%20is%20empty%22%22%22%0A%20%20%20%20%20%20%20%20return%20self._size%20%3D%3D%200%0A%0A%20%20%20%20def%20push%28self,%20num%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Enqueue%22%22%22%0A%20%20%20%20%20%20%20%20if%20self._size%20%3D%3D%20self.capacity%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22Queue%20is%20full%22%29%0A%20%20%20%20%20%20%20%20%23%20Calculate%20the%20rear%20pointer,%20which%20points%20one%20position%20past%20the%20last%20element%0A%20%20%20%20%20%20%20%20%23%20Use%20modulo%20to%20wrap%20rear%20to%20the%20beginning%20when%20it%20passes%20the%20end%20of%20the%20array%0A%20%20%20%20%20%20%20%20rear%3A%20int%20%3D%20%28self._front%20%2B%20self._size%29%20%25%20self.capacity%28%29%0A%20%20%20%20%20%20%20%20%23%20Add%20num%20to%20the%20rear%0A%20%20%20%20%20%20%20%20self._nums%5Brear%5D%20%3D%20num%0A%20%20%20%20%20%20%20%20self._size%20%2B%3D%201%0A%0A%20%20%20%20def%20pop%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Dequeue%22%22%22%0A%20%20%20%20%20%20%20%20num%3A%20int%20%3D%20self.peek%28%29%0A%20%20%20%20%20%20%20%20%23%20Advance%20the%20front%20pointer%20by%20one%20position,%20wrapping%20to%20the%20beginning%20if%20it%20passes%20the%20end%0A%20%20%20%20%20%20%20%20self._front%20%3D%20%28self._front%20%2B%201%29%20%25%20self.capacity%28%29%0A%20%20%20%20%20%20%20%20self._size%20-%3D%201%0A%20%20%20%20%20%20%20%20return%20num%0A%0A%20%20%20%20def%20peek%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Access%20the%20front%20element%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.is_empty%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22Queue%20is%20empty%22%29%0A%20%20%20%20%20%20%20%20return%20self._nums%5Bself._front%5D%0A%0A%20%20%20%20def%20to_list%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Return%20a%20list%20for%20printing%22%22%22%0A%20%20%20%20%20%20%20%20res%20%3D%20%5B0%5D%20%2A%20self.size%28%29%0A%20%20%20%20%20%20%20%20j%3A%20int%20%3D%20self._front%0A%20%20%20%20%20%20%20%20for%20i%20in%20range%28self.size%28%29%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20res%5Bi%5D%20%3D%20self._nums%5B%28j%20%25%20self.capacity%28%29%29%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%2B%3D%201%0A%20%20%20%20%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20queue%0A%20%20%20%20queue%20%3D%20ArrayQueue%2810%29%0A%0A%20%20%20%20%23%20Enqueue%20elements%0A%20%20%20%20queue.push%281%29%0A%20%20%20%20queue.push%283%29%0A%20%20%20%20queue.push%282%29%0A%20%20%20%20queue.push%285%29%0A%20%20%20%20queue.push%284%29%0A%0A%20%20%20%20%23%20Access%20the%20front%20element%0A%20%20%20%20peek%20%3D%20queue.peek%28%29%0A%20%20%20%20print%28%22Front%20element%20peek%20%3D%22,%20peek%29%0A%0A%20%20%20%20%23%20Dequeue%20an%20element%0A%20%20%20%20pop%20%3D%20queue.pop%28%29%0A%20%20%20%20print%28%22Dequeued%20element%20pop%20%3D%22,%20pop%29%0A%0A%20%20%20%20%23%20Get%20the%20queue%20size%0A%20%20%20%20size%20%3D%20queue.size%28%29%0A%20%20%20%20print%28%22Queue%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20Check%20whether%20the%20queue%20is%20empty%0A%20%20%20%20is_empty%20%3D%20queue.is_empty%28%29%0A%20%20%20%20print%28%22Is%20the%20queue%20empty%20%3D%22,%20is_empty%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ArrayQueue%3A%0A%20%20%20%20%22%22%22Queue%20implemented%20with%20a%20circular%20array%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20size%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self._nums%3A%20list%5Bint%5D%20%3D%20%5B0%5D%20%2A%20size%20%20%23%20Array%20that%20stores%20the%20queue%20elements%0A%20%20%20%20%20%20%20%20self._front%3A%20int%20%3D%200%20%20%23%20Front%20pointer,%20pointing%20to%20the%20first%20element%0A%20%20%20%20%20%20%20%20self._size%3A%20int%20%3D%200%20%20%23%20Queue%20size%0A%0A%20%20%20%20def%20capacity%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20queue%20capacity%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self._nums%29%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20queue%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20self._size%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22Check%20whether%20the%20queue%20is%20empty%22%22%22%0A%20%20%20%20%20%20%20%20return%20self._size%20%3D%3D%200%0A%0A%20%20%20%20def%20push%28self,%20num%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Enqueue%22%22%22%0A%20%20%20%20%20%20%20%20if%20self._size%20%3D%3D%20self.capacity%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22Queue%20is%20full%22%29%0A%20%20%20%20%20%20%20%20%23%20Calculate%20the%20rear%20pointer,%20which%20points%20one%20position%20past%20the%20last%20element%0A%20%20%20%20%20%20%20%20%23%20Use%20modulo%20to%20wrap%20rear%20to%20the%20beginning%20when%20it%20passes%20the%20end%20of%20the%20array%0A%20%20%20%20%20%20%20%20rear%3A%20int%20%3D%20%28self._front%20%2B%20self._size%29%20%25%20self.capacity%28%29%0A%20%20%20%20%20%20%20%20%23%20Add%20num%20to%20the%20rear%0A%20%20%20%20%20%20%20%20self._nums%5Brear%5D%20%3D%20num%0A%20%20%20%20%20%20%20%20self._size%20%2B%3D%201%0A%0A%20%20%20%20def%20pop%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Dequeue%22%22%22%0A%20%20%20%20%20%20%20%20num%3A%20int%20%3D%20self.peek%28%29%0A%20%20%20%20%20%20%20%20%23%20Advance%20the%20front%20pointer%20by%20one%20position,%20wrapping%20to%20the%20beginning%20if%20it%20passes%20the%20end%0A%20%20%20%20%20%20%20%20self._front%20%3D%20%28self._front%20%2B%201%29%20%25%20self.capacity%28%29%0A%20%20%20%20%20%20%20%20self._size%20-%3D%201%0A%20%20%20%20%20%20%20%20return%20num%0A%0A%20%20%20%20def%20peek%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Access%20the%20front%20element%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.is_empty%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22Queue%20is%20empty%22%29%0A%20%20%20%20%20%20%20%20return%20self._nums%5Bself._front%5D%0A%0A%20%20%20%20def%20to_list%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Return%20a%20list%20for%20printing%22%22%22%0A%20%20%20%20%20%20%20%20res%20%3D%20%5B0%5D%20%2A%20self.size%28%29%0A%20%20%20%20%20%20%20%20j%3A%20int%20%3D%20self._front%0A%20%20%20%20%20%20%20%20for%20i%20in%20range%28self.size%28%29%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20res%5Bi%5D%20%3D%20self._nums%5B%28j%20%25%20self.capacity%28%29%29%5D%0A%20%20%20%20%20%20%20%20%20%20%20%20j%20%2B%3D%201%0A%20%20%20%20%20%20%20%20return%20res%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20queue%0A%20%20%20%20queue%20%3D%20ArrayQueue%2810%29%0A%0A%20%20%20%20%23%20Enqueue%20elements%0A%20%20%20%20queue.push%281%29%0A%20%20%20%20queue.push%283%29%0A%20%20%20%20queue.push%282%29%0A%20%20%20%20queue.push%285%29%0A%20%20%20%20queue.push%284%29%0A%0A%20%20%20%20%23%20Access%20the%20front%20element%0A%20%20%20%20peek%20%3D%20queue.peek%28%29%0A%20%20%20%20print%28%22Front%20element%20peek%20%3D%22,%20peek%29%0A%0A%20%20%20%20%23%20Dequeue%20an%20element%0A%20%20%20%20pop%20%3D%20queue.pop%28%29%0A%20%20%20%20print%28%22Dequeued%20element%20pop%20%3D%22,%20pop%29%0A%0A%20%20%20%20%23%20Get%20the%20queue%20size%0A%20%20%20%20size%20%3D%20queue.size%28%29%0A%20%20%20%20print%28%22Queue%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20Check%20whether%20the%20queue%20is%20empty%0A%20%20%20%20is_empty%20%3D%20queue.is_empty%28%29%0A%20%20%20%20print%28%22Is%20the%20queue%20empty%20%3D%22,%20is_empty%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>This queue still has a limitation: its capacity is fixed. We can address it by replacing the array with a dynamic array and adding resizing. Interested readers can try implementing this themselves.</p>
+<p>The comparison between these two implementations is the same as for stacks, so we will not repeat it here.</p>
+<h2 id="523-typical-applications">5.2.3 &nbsp; Typical applications<a class="headerlink" href="#523-typical-applications" title="Permanent link">&para;</a></h2>
 <ul>
-<li><strong>Taobao orders</strong>. After shoppers place orders, the orders are added to a queue, and the system subsequently processes the orders in the queue according to their sequence. During Double Eleven, massive orders are generated in a short time, and high concurrency becomes a key challenge that engineers need to tackle.</li>
-<li><strong>Various to-do tasks</strong>. Any scenario that needs to implement "first come, first served" functionality, such as a printer's task queue or a restaurant's order queue, can effectively maintain the processing order using queues.</li>
+<li><strong>Taobao orders</strong>. When a shopper places an order, it enters a queue, and the system processes orders in sequence. During the Singles' Day shopping festival, enormous numbers of orders arrive in a short time, making high concurrency a major engineering challenge.</li>
+<li><strong>Tasks awaiting processing</strong>. Queues preserve processing order wherever first-come, first-served behavior is needed, such as printer jobs or restaurant orders.</li>
 </ul>
 
 <!-- Source file information -->
@@ -6985,7 +7067,7 @@ aria-label="Footer"
   <a
     href="../stack/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 5.1 Stack"
+    aria-label="Previous: 5.1   Stacks"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -6997,7 +7079,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        5.1 Stack
+        5.1 &nbsp; Stacks
       </div>
     </div>
   </a>
@@ -7009,7 +7091,7 @@ aria-label="Footer"
   <a
     href="../deque/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 5.3 Deque"
+    aria-label="Next: 5.3   Deques"
     rel="next"
   >
     <div class="md-footer__title">
@@ -7017,7 +7099,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        5.3 Deque
+        5.3 &nbsp; Deques
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -7034,7 +7116,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_stack_and_queue/stack/index.html b/en/chapter_stack_and_queue/stack/index.html
index c978bdebd..cf8d1267c 100644
--- a/en/chapter_stack_and_queue/stack/index.html
+++ b/en/chapter_stack_and_queue/stack/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>5.1 Stack - Hello Algo</title>
+        <title>5.1 Stacks - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
     <div data-md-component="skip">
 
 
-        <a href="#51-stack" class="md-skip">
+        <a href="#51-stacks" class="md-skip">
           Skip to content
         </a>
 
@@ -119,7 +119,7 @@
         <path
             d="M480 32c0-12.9-7.8-24.6-19.8-29.6s-25.7-2.2-34.9 6.9L381.7 53c-48 48-113.1 75-181 75H192 160 64c-35.3 0-64 28.7-64 64v96c0 35.3 28.7 64 64 64l0 128c0 17.7 14.3 32 32 32h64c17.7 0 32-14.3 32-32V352l8.7 0c67.9 0 133 27 181 75l43.6 43.6c9.2 9.2 22.9 11.9 34.9 6.9s19.8-16.6 19.8-29.6V300.4c18.6-8.8 32-32.5 32-60.4s-13.4-51.6-32-60.4V32zm-64 76.7V240 371.3C357.2 317.8 280.5 288 200.7 288H192V192h8.7c79.8 0 156.5-29.8 215.3-83.3z" />
     </svg>
-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              5.1 &nbsp; Stack
+              5.1 &nbsp; Stacks
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1466,7 +1466,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1487,7 +1487,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1517,7 +1517,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1534,7 +1534,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1563,7 +1563,7 @@
   <a href="#511-common-stack-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.1.1 &nbsp; Common Stack Operations
+        5.1.1 &nbsp; Common stack operations
       </span>
     </span>
   </a>
@@ -1571,22 +1571,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#512-stack-implementation" class="md-nav__link">
+  <a href="#512-implementing-a-stack" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.1.2 &nbsp; Stack Implementation
+        5.1.2 &nbsp; Implementing a stack
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="5.1.2 Stack Implementation">
+    <nav class="md-nav" aria-label="5.1.2 Implementing a stack">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-linked-list-implementation" class="md-nav__link">
+  <a href="#1-implementation-with-a-linked-list" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Linked List Implementation
+        1. &nbsp; Implementation with a linked list
       </span>
     </span>
   </a>
@@ -1594,10 +1594,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-array-implementation" class="md-nav__link">
+  <a href="#2-implementation-with-an-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Array Implementation
+        2. &nbsp; Implementation with an array
       </span>
     </span>
   </a>
@@ -1610,10 +1610,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#513-comparison-of-the-two-implementations" class="md-nav__link">
+  <a href="#513-comparing-the-implementations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.1.3 &nbsp; Comparison of the Two Implementations
+        5.1.3 &nbsp; Comparing the implementations
       </span>
     </span>
   </a>
@@ -1621,10 +1621,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#514-typical-applications-of-stack" class="md-nav__link">
+  <a href="#514-typical-applications" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.1.4 &nbsp; Typical Applications of Stack
+        5.1.4 &nbsp; Typical applications
       </span>
     </span>
   </a>
@@ -1654,7 +1654,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1681,7 +1681,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1708,7 +1708,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1735,7 +1735,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1799,7 +1799,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1820,7 +1820,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1841,7 +1841,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1868,7 +1868,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1895,7 +1895,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1922,7 +1922,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1949,7 +1949,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -2017,7 +2017,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -2038,7 +2038,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2059,7 +2059,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -2086,7 +2086,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2113,7 +2113,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2140,7 +2140,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2167,7 +2167,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2194,7 +2194,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2221,7 +2221,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2285,7 +2285,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2306,7 +2306,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2327,7 +2327,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2354,7 +2354,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2381,7 +2381,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2408,7 +2408,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2435,7 +2435,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2499,7 +2499,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2520,7 +2520,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2541,7 +2541,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2568,7 +2568,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2595,7 +2595,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2622,7 +2622,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2649,7 +2649,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2717,7 +2717,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2738,7 +2738,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2759,7 +2759,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2786,7 +2786,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2813,7 +2813,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2840,7 +2840,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2867,7 +2867,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2894,7 +2894,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2921,7 +2921,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2999,7 +2999,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3020,7 +3020,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3041,7 +3041,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3068,7 +3068,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3095,7 +3095,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3122,7 +3122,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3149,7 +3149,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3176,7 +3176,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3203,7 +3203,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3230,7 +3230,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3257,7 +3257,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3284,7 +3284,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3311,7 +3311,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3338,7 +3338,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3404,7 +3404,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3425,7 +3425,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3446,7 +3446,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3473,7 +3473,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3500,7 +3500,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3527,7 +3527,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3554,7 +3554,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3581,7 +3581,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3647,7 +3647,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3668,7 +3668,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3689,7 +3689,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3716,7 +3716,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3743,7 +3743,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3770,7 +3770,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3797,7 +3797,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3824,7 +3824,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3894,7 +3894,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3915,7 +3915,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3936,7 +3936,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3963,7 +3963,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3990,7 +3990,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4017,7 +4017,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4044,7 +4044,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4071,7 +4071,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4098,7 +4098,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4125,7 +4125,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4191,7 +4191,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4212,7 +4212,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4233,7 +4233,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4260,7 +4260,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4287,7 +4287,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4314,7 +4314,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4341,7 +4341,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4368,7 +4368,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4428,7 +4428,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4449,7 +4449,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4470,7 +4470,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4497,7 +4497,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4524,7 +4524,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4608,6 +4608,78 @@
 
 
 
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@@ -4652,7 +4724,7 @@
   <a href="#511-common-stack-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.1.1 &nbsp; Common Stack Operations
+        5.1.1 &nbsp; Common stack operations
       </span>
     </span>
   </a>
@@ -4660,22 +4732,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#512-stack-implementation" class="md-nav__link">
+  <a href="#512-implementing-a-stack" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.1.2 &nbsp; Stack Implementation
+        5.1.2 &nbsp; Implementing a stack
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="5.1.2 Stack Implementation">
+    <nav class="md-nav" aria-label="5.1.2 Implementing a stack">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-linked-list-implementation" class="md-nav__link">
+  <a href="#1-implementation-with-a-linked-list" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Linked List Implementation
+        1. &nbsp; Implementation with a linked list
       </span>
     </span>
   </a>
@@ -4683,10 +4755,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-array-implementation" class="md-nav__link">
+  <a href="#2-implementation-with-an-array" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Array Implementation
+        2. &nbsp; Implementation with an array
       </span>
     </span>
   </a>
@@ -4699,10 +4771,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#513-comparison-of-the-two-implementations" class="md-nav__link">
+  <a href="#513-comparing-the-implementations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.1.3 &nbsp; Comparison of the Two Implementations
+        5.1.3 &nbsp; Comparing the implementations
       </span>
     </span>
   </a>
@@ -4710,10 +4782,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#514-typical-applications-of-stack" class="md-nav__link">
+  <a href="#514-typical-applications" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5.1.4 &nbsp; Typical Applications of Stack
+        5.1.4 &nbsp; Typical applications
       </span>
     </span>
   </a>
@@ -4757,16 +4829,16 @@
 
 
 <!-- Page content -->
-<h1 id="51-stack">5.1 &nbsp; Stack<a class="headerlink" href="#51-stack" title="Permanent link">&para;</a></h1>
-<p>A <u>stack</u> is a linear data structure that follows the Last In, First Out (LIFO) principle.</p>
-<p>We can compare a stack to a pile of plates on a table. If we specify that only one plate can be moved at a time, then to get the bottom plate, we must first remove the plates above it one by one. If we replace the plates with various types of elements (such as integers, characters, objects, etc.), we get the stack data structure.</p>
-<p>As shown in Figure 5-1, we call the top of the stacked elements the "top" and the bottom the "bottom." The operation of adding an element to the top is called "push," and the operation of removing the top element is called "pop."</p>
-<p><img alt="LIFO rule of stack" class="animation-figure" src="../stack.assets/stack_operations.png" /></p>
-<p align="center"> Figure 5-1 &nbsp; LIFO rule of stack </p>
+<h1 id="51-stacks">5.1 &nbsp; Stacks<a class="headerlink" href="#51-stacks" title="Permanent link">&para;</a></h1>
+<p>A <u>stack</u> is a linear data structure that follows a last-in, first-out rule.</p>
+<p>Think of a stack of plates on a table. If we can move only one plate at a time, reaching the bottom plate requires removing all the plates above it first. Replace the plates with elements such as integers, characters, or objects, and we have a stack.</p>
+<p>As shown below, the upper end is the “top” of the stack and the lower end is the “bottom.” Adding an element to the top is called “pushing,” and removing the top element is called “popping.”</p>
+<p><img alt="The last-in, first-out rule of a stack" class="animation-figure" src="../stack.assets/stack_operations.png" /></p>
+<p align="center"> Figure 5-1 &nbsp; The last-in, first-out rule of a stack </p>
 
-<h2 id="511-common-stack-operations">5.1.1 &nbsp; Common Stack Operations<a class="headerlink" href="#511-common-stack-operations" title="Permanent link">&para;</a></h2>
-<p>The common operations on a stack are shown in Table 5-1. The specific method names depend on the programming language used. Here, we use the common naming convention of <code>push()</code>, <code>pop()</code>, and <code>peek()</code>.</p>
-<p align="center"> Table 5-1 &nbsp; Efficiency of Stack Operations </p>
+<h2 id="511-common-stack-operations">5.1.1 &nbsp; Common stack operations<a class="headerlink" href="#511-common-stack-operations" title="Permanent link">&para;</a></h2>
+<p>Table 5-1 lists common operations. Actual method names depend on the programming language; here we use the familiar names <code>push()</code>, <code>pop()</code>, and <code>peek()</code>.</p>
+<p align="center"> Table 5-1 &nbsp; Efficiency of stack operations </p>
 
 <div class="center-table">
 <table>
@@ -4774,34 +4846,34 @@
 <tr>
 <th>Method</th>
 <th>Description</th>
-<th>Time Complexity</th>
+<th>Time complexity</th>
 </tr>
 </thead>
 <tbody>
 <tr>
 <td><code>push()</code></td>
-<td>Push element onto stack (add to top)</td>
+<td>Push an element onto the top</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
 <td><code>pop()</code></td>
-<td>Pop top element from stack</td>
+<td>Remove the top element</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 <tr>
 <td><code>peek()</code></td>
-<td>Access top element</td>
+<td>Access the top element</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 </tr>
 </tbody>
 </table>
 </div>
-<p>Typically, we can directly use the built-in stack class provided by the programming language. However, some languages may not provide a dedicated stack class. In such cases, we can use the language's "array" or "linked list" as a stack and simply avoid using operations unrelated to stack behavior.</p>
+<p>We can usually use a language's built-in stack class. If no dedicated class is provided, use an array or linked list as a stack and avoid operations unrelated to stack behavior.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">stack.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Initialize stack</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="c1"># Python does not have a built-in stack class, can use list as a stack</span>
+<div class="highlight"><span class="filename">stack.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Initialize a stack</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="c1"># Python has no built-in stack class; use a list as a stack</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="n">stack</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="p">[]</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="c1"># Push elements</span>
@@ -4811,21 +4883,21 @@
 <a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="n">stack</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="n">stack</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>
-<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="c1"># Access top element</span>
+<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="c1"># Access the top element</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="n">peek</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">stack</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>
-<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="c1"># Pop element</span>
+<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="c1"># Pop an element</span>
 <a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="n">pop</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">stack</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
 <a id="__codelineno-0-17" name="__codelineno-0-17" href="#__codelineno-0-17"></a>
-<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="c1"># Get stack length</span>
+<a id="__codelineno-0-18" name="__codelineno-0-18" href="#__codelineno-0-18"></a><span class="c1"># Get the stack size</span>
 <a id="__codelineno-0-19" name="__codelineno-0-19" href="#__codelineno-0-19"></a><span class="n">size</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">stack</span><span class="p">)</span>
 <a id="__codelineno-0-20" name="__codelineno-0-20" href="#__codelineno-0-20"></a>
-<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="c1"># Check if empty</span>
+<a id="__codelineno-0-21" name="__codelineno-0-21" href="#__codelineno-0-21"></a><span class="c1"># Check whether the stack is empty</span>
 <a id="__codelineno-0-22" name="__codelineno-0-22" href="#__codelineno-0-22"></a><span class="n">is_empty</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">stack</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">stack.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Initialize stack */</span>
+<div class="highlight"><span class="filename">stack.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Initialize a stack */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="n">stack</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">stack</span><span class="p">;</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="cm">/* Push elements */</span>
@@ -4835,21 +4907,21 @@
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="n">stack</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="n">stack</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a>
-<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="cm">/* Access top element */</span>
+<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="kt">int</span><span class="w"> </span><span class="n">top</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">top</span><span class="p">();</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a>
-<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="cm">/* Pop element */</span>
+<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="cm">/* Pop an element */</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="n">stack</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span><span class="w"> </span><span class="c1">// No return value</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a>
-<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="cm">/* Get stack length */</span>
+<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
 <a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a>
-<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="cm">/* Check if empty */</span>
+<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-1-21" name="__codelineno-1-21" href="#__codelineno-1-21"></a><span class="kt">bool</span><span class="w"> </span><span class="n">empty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">empty</span><span class="p">();</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">stack.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Initialize stack */</span>
+<div class="highlight"><span class="filename">stack.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Initialize a stack */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="n">Stack</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">Stack</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="cm">/* Push elements */</span>
@@ -4859,21 +4931,21 @@
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="n">stack</span><span class="p">.</span><span class="na">push</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="n">stack</span><span class="p">.</span><span class="na">push</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a>
-<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="cm">/* Access top element */</span>
+<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">peek</span><span class="p">();</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a>
-<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="cm">/* Pop element */</span>
+<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="cm">/* Pop an element */</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="kt">int</span><span class="w"> </span><span class="n">pop</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">pop</span><span class="p">();</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a>
-<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="cm">/* Get stack length */</span>
+<a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">size</span><span class="p">();</span>
 <a id="__codelineno-2-19" name="__codelineno-2-19" href="#__codelineno-2-19"></a>
-<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="cm">/* Check if empty */</span>
+<a id="__codelineno-2-20" name="__codelineno-2-20" href="#__codelineno-2-20"></a><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-2-21" name="__codelineno-2-21" href="#__codelineno-2-21"></a><span class="kt">boolean</span><span class="w"> </span><span class="n">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">isEmpty</span><span class="p">();</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">stack.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Initialize stack */</span>
+<div class="highlight"><span class="filename">stack.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Initialize a stack */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="n">Stack</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">();</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="cm">/* Push elements */</span>
@@ -4883,22 +4955,22 @@
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="n">stack</span><span class="p">.</span><span class="n">Push</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="n">stack</span><span class="p">.</span><span class="n">Push</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a>
-<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="cm">/* Access top element */</span>
+<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">Peek</span><span class="p">();</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a>
-<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="cm">/* Pop element */</span>
+<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="cm">/* Pop an element */</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="kt">int</span><span class="w"> </span><span class="n">pop</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">Pop</span><span class="p">();</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a>
-<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="cm">/* Get stack length */</span>
+<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">Count</span><span class="p">;</span>
 <a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a>
-<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="cm">/* Check if empty */</span>
+<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">stack_test.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Initialize stack */</span>
-<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="c1">// In Go, it is recommended to use Slice as a stack</span>
+<div class="highlight"><span class="filename">stack_test.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Initialize a stack */</span>
+<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="c1">// In Go, a slice is recommended for use as a stack</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="kd">var</span><span class="w"> </span><span class="nx">stack</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="cm">/* Push elements */</span>
@@ -4908,23 +4980,23 @@
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="nx">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">stack</span><span class="p">,</span><span class="w"> </span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="nx">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">stack</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a>
-<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="cm">/* Access top element */</span>
+<a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="nx">peek</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">stack</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="nx">stack</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a>
-<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="cm">/* Pop element */</span>
+<a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="cm">/* Pop an element */</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="nx">pop</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">stack</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="nx">stack</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="nx">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">stack</span><span class="p">[:</span><span class="nb">len</span><span class="p">(</span><span class="nx">stack</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a>
-<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="cm">/* Get stack length */</span>
+<a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="nx">size</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">stack</span><span class="p">)</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a>
-<a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="cm">/* Check if empty */</span>
+<a id="__codelineno-4-22" name="__codelineno-4-22" href="#__codelineno-4-22"></a><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-4-23" name="__codelineno-4-23" href="#__codelineno-4-23"></a><span class="nx">isEmpty</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">stack</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">stack.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Initialize stack */</span>
-<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Swift does not have a built-in stack class, can use Array as a stack</span>
+<div class="highlight"><span class="filename">stack.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Initialize a stack */</span>
+<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Swift has no built-in stack class; use an Array as a stack</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="kd">var</span><span class="w"> </span><span class="nv">stack</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="cm">/* Push elements */</span>
@@ -4934,22 +5006,22 @@
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="n">stack</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="n">stack</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a>
-<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="cm">/* Access top element */</span>
+<a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="kd">let</span><span class="w"> </span><span class="nv">peek</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="bp">last</span><span class="p">!</span>
 <a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a>
-<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="cm">/* Pop element */</span>
+<a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="cm">/* Pop an element */</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="kd">let</span><span class="w"> </span><span class="nv">pop</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a>
-<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="cm">/* Get stack length */</span>
+<a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="kd">let</span><span class="w"> </span><span class="nv">size</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a>
-<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="cm">/* Check if empty */</span>
+<a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-5-22" name="__codelineno-5-22" href="#__codelineno-5-22"></a><span class="kd">let</span><span class="w"> </span><span class="nv">isEmpty</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="bp">isEmpty</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">stack.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Initialize stack */</span>
-<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="c1">// JavaScript does not have a built-in stack class, can use Array as a stack</span>
+<div class="highlight"><span class="filename">stack.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Initialize a stack */</span>
+<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="c1">// JavaScript has no built-in stack class; use an Array as a stack</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="kd">const</span><span class="w"> </span><span class="nx">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="cm">/* Push elements */</span>
@@ -4959,22 +5031,22 @@
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="nx">stack</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">5</span><span class="p">);</span>
 <a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="nx">stack</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">4</span><span class="p">);</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a>
-<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="cm">/* Access top element */</span>
+<a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="kd">const</span><span class="w"> </span><span class="nx">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">stack</span><span class="p">[</span><span class="nx">stack</span><span class="p">.</span><span class="nx">length</span><span class="o">-</span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a>
-<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="cm">/* Pop element */</span>
+<a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="cm">/* Pop an element */</span>
 <a id="__codelineno-6-16" name="__codelineno-6-16" href="#__codelineno-6-16"></a><span class="kd">const</span><span class="w"> </span><span class="nx">pop</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">stack</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-6-17" name="__codelineno-6-17" href="#__codelineno-6-17"></a>
-<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="cm">/* Get stack length */</span>
+<a id="__codelineno-6-18" name="__codelineno-6-18" href="#__codelineno-6-18"></a><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-6-19" name="__codelineno-6-19" href="#__codelineno-6-19"></a><span class="kd">const</span><span class="w"> </span><span class="nx">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">stack</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-6-20" name="__codelineno-6-20" href="#__codelineno-6-20"></a>
-<a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="cm">/* Check if empty */</span>
+<a id="__codelineno-6-21" name="__codelineno-6-21" href="#__codelineno-6-21"></a><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-6-22" name="__codelineno-6-22" href="#__codelineno-6-22"></a><span class="kd">const</span><span class="w"> </span><span class="nx">is_empty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">stack</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">stack.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Initialize stack */</span>
-<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="c1">// TypeScript does not have a built-in stack class, can use Array as a stack</span>
+<div class="highlight"><span class="filename">stack.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Initialize a stack */</span>
+<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="c1">// TypeScript has no built-in stack class; use an Array as a stack</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="kd">const</span><span class="w"> </span><span class="nx">stack</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="cm">/* Push elements */</span>
@@ -4984,22 +5056,22 @@
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="nx">stack</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">5</span><span class="p">);</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="nx">stack</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="mf">4</span><span class="p">);</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a>
-<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="cm">/* Access top element */</span>
+<a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="kd">const</span><span class="w"> </span><span class="nx">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">stack</span><span class="p">[</span><span class="nx">stack</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a>
-<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="cm">/* Pop element */</span>
+<a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="cm">/* Pop an element */</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="kd">const</span><span class="w"> </span><span class="nx">pop</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">stack</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a>
-<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="cm">/* Get stack length */</span>
+<a id="__codelineno-7-18" name="__codelineno-7-18" href="#__codelineno-7-18"></a><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-7-19" name="__codelineno-7-19" href="#__codelineno-7-19"></a><span class="kd">const</span><span class="w"> </span><span class="nx">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">stack</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-7-20" name="__codelineno-7-20" href="#__codelineno-7-20"></a>
-<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="cm">/* Check if empty */</span>
+<a id="__codelineno-7-21" name="__codelineno-7-21" href="#__codelineno-7-21"></a><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-7-22" name="__codelineno-7-22" href="#__codelineno-7-22"></a><span class="kd">const</span><span class="w"> </span><span class="nx">is_empty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">stack</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">stack.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Initialize stack */</span>
-<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="c1">// Dart does not have a built-in stack class, can use List as a stack</span>
+<div class="highlight"><span class="filename">stack.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Initialize a stack */</span>
+<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="c1">// Dart has no built-in stack class; use a List as a stack</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="cm">/* Push elements */</span>
@@ -5009,22 +5081,22 @@
 <a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="n">stack</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="m">5</span><span class="p">);</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="n">stack</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="m">4</span><span class="p">);</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a>
-<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="cm">/* Access top element */</span>
+<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">last</span><span class="p">;</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a>
-<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="cm">/* Pop element */</span>
+<a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="cm">/* Pop an element */</span>
 <a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="kt">int</span><span class="w"> </span><span class="n">pop</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">removeLast</span><span class="p">();</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a>
-<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="cm">/* Get stack length */</span>
+<a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-8-19" name="__codelineno-8-19" href="#__codelineno-8-19"></a><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-8-20" name="__codelineno-8-20" href="#__codelineno-8-20"></a>
-<a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="cm">/* Check if empty */</span>
+<a id="__codelineno-8-21" name="__codelineno-8-21" href="#__codelineno-8-21"></a><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-8-22" name="__codelineno-8-22" href="#__codelineno-8-22"></a><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">isEmpty</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">stack.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Initialize stack */</span>
-<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="c1">// Use Vec as a stack</span>
+<div class="highlight"><span class="filename">stack.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Initialize a stack */</span>
+<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="c1">// Use a Vec as a stack</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">stack</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="cm">/* Push elements */</span>
@@ -5034,25 +5106,25 @@
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="n">stack</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="n">stack</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a>
-<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="cm">/* Access top element */</span>
+<a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="kd">let</span><span class="w"> </span><span class="n">top</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">last</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a>
-<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="cm">/* Pop element */</span>
+<a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="cm">/* Pop an element */</span>
 <a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="kd">let</span><span class="w"> </span><span class="n">pop</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">pop</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a>
-<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="cm">/* Get stack length */</span>
+<a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="kd">let</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">len</span><span class="p">();</span>
 <a id="__codelineno-9-20" name="__codelineno-9-20" href="#__codelineno-9-20"></a>
-<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="cm">/* Check if empty */</span>
+<a id="__codelineno-9-21" name="__codelineno-9-21" href="#__codelineno-9-21"></a><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-9-22" name="__codelineno-9-22" href="#__codelineno-9-22"></a><span class="kd">let</span><span class="w"> </span><span class="n">is_empty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">is_empty</span><span class="p">();</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">stack.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// C does not provide a built-in stack</span>
+<div class="highlight"><span class="filename">stack.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="c1">// C provides no built-in stack</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">stack.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Initialize stack */</span>
+<div class="highlight"><span class="filename">stack.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Initialize a stack */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">val</span><span class="w"> </span><span class="nv">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Stack</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="cm">/* Push elements */</span>
@@ -5062,22 +5134,22 @@
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="n">stack</span><span class="p">.</span><span class="na">push</span><span class="p">(</span><span class="m">5</span><span class="p">)</span>
 <a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="n">stack</span><span class="p">.</span><span class="na">push</span><span class="p">(</span><span class="m">4</span><span class="p">)</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a>
-<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="cm">/* Access top element */</span>
+<a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="kd">val</span><span class="w"> </span><span class="nv">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">peek</span><span class="p">()</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a>
-<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="cm">/* Pop element */</span>
+<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="cm">/* Pop an element */</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="kd">val</span><span class="w"> </span><span class="nv">pop</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">pop</span><span class="p">()</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a>
-<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="cm">/* Get stack length */</span>
+<a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="kd">val</span><span class="w"> </span><span class="nv">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-11-19" name="__codelineno-11-19" href="#__codelineno-11-19"></a>
-<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="cm">/* Check if empty */</span>
+<a id="__codelineno-11-20" name="__codelineno-11-20" href="#__codelineno-11-20"></a><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-11-21" name="__codelineno-11-21" href="#__codelineno-11-21"></a><span class="kd">val</span><span class="w"> </span><span class="nv">isEmpty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">isEmpty</span><span class="p">()</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">stack.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Initialize stack</span>
-<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="c1"># Ruby does not have a built-in stack class, can use Array as a stack</span>
+<div class="highlight"><span class="filename">stack.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1"># Initialize a stack</span>
+<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="c1"># Ruby has no built-in stack class; use an Array as a stack</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="n">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a>
 <a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="c1"># Push elements</span>
@@ -5087,36 +5159,36 @@
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="n">stack</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">5</span>
 <a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="n">stack</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">4</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a>
-<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="c1"># Access top element</span>
+<a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="c1"># Access the top element</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="n">peek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="o">.</span><span class="n">last</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a>
-<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="c1"># Pop element</span>
+<a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="c1"># Pop an element</span>
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="n">pop</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="o">.</span><span class="n">pop</span>
 <a id="__codelineno-12-17" name="__codelineno-12-17" href="#__codelineno-12-17"></a>
-<a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="c1"># Get stack length</span>
+<a id="__codelineno-12-18" name="__codelineno-12-18" href="#__codelineno-12-18"></a><span class="c1"># Get the stack size</span>
 <a id="__codelineno-12-19" name="__codelineno-12-19" href="#__codelineno-12-19"></a><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-12-20" name="__codelineno-12-20" href="#__codelineno-12-20"></a>
-<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="c1"># Check if empty</span>
+<a id="__codelineno-12-21" name="__codelineno-12-21" href="#__codelineno-12-21"></a><span class="c1"># Check whether the stack is empty</span>
 <a id="__codelineno-12-22" name="__codelineno-12-22" href="#__codelineno-12-22"></a><span class="n">is_empty</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stack</span><span class="o">.</span><span class="n">empty?</span>
 </code></pre></div>
 </div>
 </div>
 </div>
 <details class="pythontutor">
-<summary>Code Visualization</summary>
-<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E6%A0%88%0A%20%20%20%20%23%20Python%20%E6%B2%A1%E6%9C%89%E5%86%85%E7%BD%AE%E7%9A%84%E6%A0%88%E7%B1%BB%EF%BC%8C%E5%8F%AF%E4%BB%A5%E6%8A%8A%20list%20%E5%BD%93%E4%BD%9C%E6%A0%88%E6%9D%A5%E4%BD%BF%E7%94%A8%0A%20%20%20%20stack%20%3D%20%5B%5D%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%85%A5%E6%A0%88%0A%20%20%20%20stack.append%281%29%0A%20%20%20%20stack.append%283%29%0A%20%20%20%20stack.append%282%29%0A%20%20%20%20stack.append%285%29%0A%20%20%20%20stack.append%284%29%0A%20%20%20%20print%28%22%E6%A0%88%20stack%20%3D%22,%20stack%29%0A%0A%20%20%20%20%23%20%E8%AE%BF%E9%97%AE%E6%A0%88%E9%A1%B6%E5%85%83%E7%B4%A0%0A%20%20%20%20peek%20%3D%20stack%5B-1%5D%0A%20%20%20%20print%28%22%E6%A0%88%E9%A1%B6%E5%85%83%E7%B4%A0%20peek%20%3D%22,%20peek%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%87%BA%E6%A0%88%0A%20%20%20%20pop%20%3D%20stack.pop%28%29%0A%20%20%20%20print%28%22%E5%87%BA%E6%A0%88%E5%85%83%E7%B4%A0%20pop%20%3D%22,%20pop%29%0A%20%20%20%20print%28%22%E5%87%BA%E6%A0%88%E5%90%8E%20stack%20%3D%22,%20stack%29%0A%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E6%A0%88%E7%9A%84%E9%95%BF%E5%BA%A6%0A%20%20%20%20size%20%3D%20len%28stack%29%0A%20%20%20%20print%28%22%E6%A0%88%E7%9A%84%E9%95%BF%E5%BA%A6%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20%E5%88%A4%E6%96%AD%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%0A%20%20%20%20is_empty%20%3D%20len%28stack%29%20%3D%3D%200%0A%20%20%20%20print%28%22%E6%A0%88%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%20%3D%22,%20is_empty%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=2&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E6%A0%88%0A%20%20%20%20%23%20Python%20%E6%B2%A1%E6%9C%89%E5%86%85%E7%BD%AE%E7%9A%84%E6%A0%88%E7%B1%BB%EF%BC%8C%E5%8F%AF%E4%BB%A5%E6%8A%8A%20list%20%E5%BD%93%E4%BD%9C%E6%A0%88%E6%9D%A5%E4%BD%BF%E7%94%A8%0A%20%20%20%20stack%20%3D%20%5B%5D%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%85%A5%E6%A0%88%0A%20%20%20%20stack.append%281%29%0A%20%20%20%20stack.append%283%29%0A%20%20%20%20stack.append%282%29%0A%20%20%20%20stack.append%285%29%0A%20%20%20%20stack.append%284%29%0A%20%20%20%20print%28%22%E6%A0%88%20stack%20%3D%22,%20stack%29%0A%0A%20%20%20%20%23%20%E8%AE%BF%E9%97%AE%E6%A0%88%E9%A1%B6%E5%85%83%E7%B4%A0%0A%20%20%20%20peek%20%3D%20stack%5B-1%5D%0A%20%20%20%20print%28%22%E6%A0%88%E9%A1%B6%E5%85%83%E7%B4%A0%20peek%20%3D%22,%20peek%29%0A%0A%20%20%20%20%23%20%E5%85%83%E7%B4%A0%E5%87%BA%E6%A0%88%0A%20%20%20%20pop%20%3D%20stack.pop%28%29%0A%20%20%20%20print%28%22%E5%87%BA%E6%A0%88%E5%85%83%E7%B4%A0%20pop%20%3D%22,%20pop%29%0A%20%20%20%20print%28%22%E5%87%BA%E6%A0%88%E5%90%8E%20stack%20%3D%22,%20stack%29%0A%0A%20%20%20%20%23%20%E8%8E%B7%E5%8F%96%E6%A0%88%E7%9A%84%E9%95%BF%E5%BA%A6%0A%20%20%20%20size%20%3D%20len%28stack%29%0A%20%20%20%20print%28%22%E6%A0%88%E7%9A%84%E9%95%BF%E5%BA%A6%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20%E5%88%A4%E6%96%AD%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%0A%20%20%20%20is_empty%20%3D%20len%28stack%29%20%3D%3D%200%0A%20%20%20%20print%28%22%E6%A0%88%E6%98%AF%E5%90%A6%E4%B8%BA%E7%A9%BA%20%3D%22,%20is_empty%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=2&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20stack%0A%20%20%20%20%23%20Python%20has%20no%20built-in%20stack%20class%3B%20use%20a%20list%20as%20a%20stack%0A%20%20%20%20stack%20%3D%20%5B%5D%0A%0A%20%20%20%20%23%20Push%20elements%0A%20%20%20%20stack.append%281%29%0A%20%20%20%20stack.append%283%29%0A%20%20%20%20stack.append%282%29%0A%20%20%20%20stack.append%285%29%0A%20%20%20%20stack.append%284%29%0A%20%20%20%20print%28%22Stack%3A%20stack%20%3D%22%2C%20stack%29%0A%0A%20%20%20%20%23%20Access%20the%20top%20element%0A%20%20%20%20peek%20%3D%20stack%5B-1%5D%0A%20%20%20%20print%28%22Top%20element%3A%20peek%20%3D%22%2C%20peek%29%0A%0A%20%20%20%20%23%20Pop%20an%20element%0A%20%20%20%20pop%20%3D%20stack.pop%28%29%0A%20%20%20%20print%28%22Popped%20element%3A%20pop%20%3D%22%2C%20pop%29%0A%20%20%20%20print%28%22After%20popping%3A%20stack%20%3D%22%2C%20stack%29%0A%0A%20%20%20%20%23%20Get%20the%20stack%20size%0A%20%20%20%20size%20%3D%20len%28stack%29%0A%20%20%20%20print%28%22Stack%20size%3A%20size%20%3D%22%2C%20size%29%0A%0A%20%20%20%20%23%20Check%20whether%20the%20stack%20is%20empty%0A%20%20%20%20is_empty%20%3D%20len%28stack%29%20%3D%3D%200%0A%20%20%20%20print%28%22Is%20the%20stack%20empty%20%3D%22%2C%20is_empty%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=2&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20stack%0A%20%20%20%20%23%20Python%20has%20no%20built-in%20stack%20class%3B%20use%20a%20list%20as%20a%20stack%0A%20%20%20%20stack%20%3D%20%5B%5D%0A%0A%20%20%20%20%23%20Push%20elements%0A%20%20%20%20stack.append%281%29%0A%20%20%20%20stack.append%283%29%0A%20%20%20%20stack.append%282%29%0A%20%20%20%20stack.append%285%29%0A%20%20%20%20stack.append%284%29%0A%20%20%20%20print%28%22Stack%3A%20stack%20%3D%22%2C%20stack%29%0A%0A%20%20%20%20%23%20Access%20the%20top%20element%0A%20%20%20%20peek%20%3D%20stack%5B-1%5D%0A%20%20%20%20print%28%22Top%20element%3A%20peek%20%3D%22%2C%20peek%29%0A%0A%20%20%20%20%23%20Pop%20an%20element%0A%20%20%20%20pop%20%3D%20stack.pop%28%29%0A%20%20%20%20print%28%22Popped%20element%3A%20pop%20%3D%22%2C%20pop%29%0A%20%20%20%20print%28%22After%20popping%3A%20stack%20%3D%22%2C%20stack%29%0A%0A%20%20%20%20%23%20Get%20the%20stack%20size%0A%20%20%20%20size%20%3D%20len%28stack%29%0A%20%20%20%20print%28%22Stack%20size%3A%20size%20%3D%22%2C%20size%29%0A%0A%20%20%20%20%23%20Check%20whether%20the%20stack%20is%20empty%0A%20%20%20%20is_empty%20%3D%20len%28stack%29%20%3D%3D%200%0A%20%20%20%20print%28%22Is%20the%20stack%20empty%20%3D%22%2C%20is_empty%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=2&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<h2 id="512-stack-implementation">5.1.2 &nbsp; Stack Implementation<a class="headerlink" href="#512-stack-implementation" title="Permanent link">&para;</a></h2>
-<p>To gain a deeper understanding of how a stack operates, let's try implementing a stack class ourselves.</p>
-<p>A stack follows the LIFO principle, so we can only add or remove elements at the top. However, both arrays and linked lists allow adding and removing elements at any position. <strong>Therefore, a stack can be viewed as a restricted array or linked list</strong>. In other words, we can "shield" some irrelevant operations of arrays or linked lists so that their external logic conforms to the characteristics of a stack.</p>
-<h3 id="1-linked-list-implementation">1. &nbsp; Linked List Implementation<a class="headerlink" href="#1-linked-list-implementation" title="Permanent link">&para;</a></h3>
-<p>When implementing a stack using a linked list, we can treat the head node of the linked list as the top of the stack and the tail node as the base.</p>
-<p>As shown in Figure 5-2, for the push operation, we simply insert an element at the head of the linked list. This node insertion method is called the "head insertion method." For the pop operation, we just need to remove the head node from the linked list.</p>
+<h2 id="512-implementing-a-stack">5.1.2 &nbsp; Implementing a stack<a class="headerlink" href="#512-implementing-a-stack" title="Permanent link">&para;</a></h2>
+<p>To understand how a stack works more deeply, let us implement a stack class ourselves.</p>
+<p>A stack follows last-in, first-out order, so elements can be added or removed only at the top. Arrays and linked lists allow insertion and deletion at arbitrary positions, <strong>so a stack can be viewed as a restricted array or linked list</strong>. By hiding irrelevant operations, we make its public behavior conform to the stack's rules.</p>
+<h3 id="1-implementation-with-a-linked-list">1. &nbsp; Implementation with a linked list<a class="headerlink" href="#1-implementation-with-a-linked-list" title="Permanent link">&para;</a></h3>
+<p>When implementing a stack with a linked list, treat the head as the top and the tail as the bottom.</p>
+<p>As shown below, pushing simply inserts a node at the head, an operation called “head insertion.” Popping removes the head node.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:3"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">&lt;1&gt;</label><label for="__tabbed_2_2">&lt;2&gt;</label><label for="__tabbed_2_3">&lt;3&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Push and pop operations in linked list implementation of stack" class="animation-figure" src="../stack.assets/linkedlist_stack_step1.png" /></p>
+<p><img alt="Push and pop in a linked-list stack" class="animation-figure" src="../stack.assets/linkedlist_stack_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="linkedlist_stack_push" class="animation-figure" src="../stack.assets/linkedlist_stack_step2_push.png" /></p>
@@ -5126,14 +5198,14 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 5-2 &nbsp; Push and pop operations in linked list implementation of stack </p>
+<p align="center"> Figure 5-2 &nbsp; Push and pop in a linked-list stack </p>
 
-<p>Below is sample code for implementing a stack based on a linked list:</p>
+<p>Here is an implementation using a linked list:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">linkedlist_stack.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">LinkedListStack</span><span class="p">:</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Stack based on linked list implementation"""</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Stack implemented with a linked list"""</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="w">        </span><span class="sd">"""Constructor"""</span>
@@ -5141,11 +5213,11 @@
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_size</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">0</span>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a><span class="w">        </span><span class="sd">"""Get the length of the stack"""</span>
+<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a><span class="w">        </span><span class="sd">"""Get the stack size"""</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>
 <a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">is_empty</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">bool</span><span class="p">:</span>
-<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a><span class="w">        </span><span class="sd">"""Check if the stack is empty"""</span>
+<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a><span class="w">        </span><span class="sd">"""Check whether the stack is empty"""</span>
 <a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_size</span> <span class="o">==</span> <span class="mi">0</span>
 <a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>
 <a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
@@ -5163,13 +5235,13 @@
 <a id="__codelineno-13-29" name="__codelineno-13-29" href="#__codelineno-13-29"></a>        <span class="k">return</span> <span class="n">num</span>
 <a id="__codelineno-13-30" name="__codelineno-13-30" href="#__codelineno-13-30"></a>
 <a id="__codelineno-13-31" name="__codelineno-13-31" href="#__codelineno-13-31"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">peek</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-32" name="__codelineno-13-32" href="#__codelineno-13-32"></a><span class="w">        </span><span class="sd">"""Access top of the stack element"""</span>
+<a id="__codelineno-13-32" name="__codelineno-13-32" href="#__codelineno-13-32"></a><span class="w">        </span><span class="sd">"""Access the top element"""</span>
 <a id="__codelineno-13-33" name="__codelineno-13-33" href="#__codelineno-13-33"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_empty</span><span class="p">():</span>
 <a id="__codelineno-13-34" name="__codelineno-13-34" href="#__codelineno-13-34"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Stack is empty"</span><span class="p">)</span>
 <a id="__codelineno-13-35" name="__codelineno-13-35" href="#__codelineno-13-35"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_peek</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-13-36" name="__codelineno-13-36" href="#__codelineno-13-36"></a>
 <a id="__codelineno-13-37" name="__codelineno-13-37" href="#__codelineno-13-37"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">to_list</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
-<a id="__codelineno-13-38" name="__codelineno-13-38" href="#__codelineno-13-38"></a><span class="w">        </span><span class="sd">"""Convert to list for printing"""</span>
+<a id="__codelineno-13-38" name="__codelineno-13-38" href="#__codelineno-13-38"></a><span class="w">        </span><span class="sd">"""Convert to a list for printing"""</span>
 <a id="__codelineno-13-39" name="__codelineno-13-39" href="#__codelineno-13-39"></a>        <span class="n">arr</span> <span class="o">=</span> <span class="p">[]</span>
 <a id="__codelineno-13-40" name="__codelineno-13-40" href="#__codelineno-13-40"></a>        <span class="n">node</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_peek</span>
 <a id="__codelineno-13-41" name="__codelineno-13-41" href="#__codelineno-13-41"></a>        <span class="k">while</span> <span class="n">node</span><span class="p">:</span>
@@ -5180,11 +5252,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_stack.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Stack based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_stack.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Stack implemented with a linked list */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">LinkedListStack</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">  </span><span class="k">private</span><span class="o">:</span>
-<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">stackTop</span><span class="p">;</span><span class="w"> </span><span class="c1">// Use head node as stack top</span>
-<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">stkSize</span><span class="p">;</span><span class="w">        </span><span class="c1">// Stack length</span>
+<a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">stackTop</span><span class="p">;</span><span class="w"> </span><span class="c1">// Use the head node as the stack top</span>
+<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">stkSize</span><span class="p">;</span><span class="w">        </span><span class="c1">// Stack size</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">  </span><span class="k">public</span><span class="o">:</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="n">LinkedListStack</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
@@ -5193,16 +5265,16 @@
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">    </span><span class="o">~</span><span class="n">LinkedListStack</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">        </span><span class="c1">// Traverse linked list to delete nodes and free memory</span>
+<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">        </span><span class="c1">// Traverse the linked list and delete each node to free its memory</span>
 <a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">        </span><span class="n">freeMemoryLinkedList</span><span class="p">(</span><span class="n">stackTop</span><span class="p">);</span>
 <a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a>
-<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stkSize</span><span class="p">;</span>
 <a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a>
-<a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-14-24" name="__codelineno-14-24" href="#__codelineno-14-24"></a><span class="w">    </span><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-25" name="__codelineno-14-25" href="#__codelineno-14-25"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-14-26" name="__codelineno-14-26" href="#__codelineno-14-26"></a><span class="w">    </span><span class="p">}</span>
@@ -5226,14 +5298,14 @@
 <a id="__codelineno-14-44" name="__codelineno-14-44" href="#__codelineno-14-44"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-14-45" name="__codelineno-14-45" href="#__codelineno-14-45"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-46" name="__codelineno-14-46" href="#__codelineno-14-46"></a>
-<a id="__codelineno-14-47" name="__codelineno-14-47" href="#__codelineno-14-47"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-14-47" name="__codelineno-14-47" href="#__codelineno-14-47"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-14-48" name="__codelineno-14-48" href="#__codelineno-14-48"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">top</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-49" name="__codelineno-14-49" href="#__codelineno-14-49"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-14-50" name="__codelineno-14-50" href="#__codelineno-14-50"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Stack is empty"</span><span class="p">);</span>
 <a id="__codelineno-14-51" name="__codelineno-14-51" href="#__codelineno-14-51"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stackTop</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-14-52" name="__codelineno-14-52" href="#__codelineno-14-52"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-53" name="__codelineno-14-53" href="#__codelineno-14-53"></a>
-<a id="__codelineno-14-54" name="__codelineno-14-54" href="#__codelineno-14-54"></a><span class="w">    </span><span class="cm">/* Convert List to Array and return */</span>
+<a id="__codelineno-14-54" name="__codelineno-14-54" href="#__codelineno-14-54"></a><span class="w">    </span><span class="cm">/* Convert the linked list to a vector and return it */</span>
 <a id="__codelineno-14-55" name="__codelineno-14-55" href="#__codelineno-14-55"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">toVector</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-56" name="__codelineno-14-56" href="#__codelineno-14-56"></a><span class="w">        </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stackTop</span><span class="p">;</span>
 <a id="__codelineno-14-57" name="__codelineno-14-57" href="#__codelineno-14-57"></a><span class="w">        </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">(</span><span class="n">size</span><span class="p">());</span>
@@ -5247,21 +5319,21 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_stack.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Stack based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_stack.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Stack implemented with a linked list */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kd">class</span> <span class="nc">LinkedListStack</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">stackPeek</span><span class="p">;</span><span class="w"> </span><span class="c1">// Use head node as stack top</span>
-<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">stkSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Stack length</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="n">ListNode</span><span class="w"> </span><span class="n">stackPeek</span><span class="p">;</span><span class="w"> </span><span class="c1">// Use the head node as the stack top</span>
+<a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">stkSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Stack size</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="nf">LinkedListStack</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">        </span><span class="n">stackPeek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a>
-<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stkSize</span><span class="p">;</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a>
-<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">boolean</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">    </span><span class="p">}</span>
@@ -5282,14 +5354,14 @@
 <a id="__codelineno-15-33" name="__codelineno-15-33" href="#__codelineno-15-33"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-15-34" name="__codelineno-15-34" href="#__codelineno-15-34"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-35" name="__codelineno-15-35" href="#__codelineno-15-35"></a>
-<a id="__codelineno-15-36" name="__codelineno-15-36" href="#__codelineno-15-36"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-15-36" name="__codelineno-15-36" href="#__codelineno-15-36"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-15-37" name="__codelineno-15-37" href="#__codelineno-15-37"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-38" name="__codelineno-15-38" href="#__codelineno-15-38"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-15-39" name="__codelineno-15-39" href="#__codelineno-15-39"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">();</span>
 <a id="__codelineno-15-40" name="__codelineno-15-40" href="#__codelineno-15-40"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stackPeek</span><span class="p">.</span><span class="na">val</span><span class="p">;</span>
 <a id="__codelineno-15-41" name="__codelineno-15-41" href="#__codelineno-15-41"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-42" name="__codelineno-15-42" href="#__codelineno-15-42"></a>
-<a id="__codelineno-15-43" name="__codelineno-15-43" href="#__codelineno-15-43"></a><span class="w">    </span><span class="cm">/* Convert List to Array and return */</span>
+<a id="__codelineno-15-43" name="__codelineno-15-43" href="#__codelineno-15-43"></a><span class="w">    </span><span class="cm">/* Convert the list to an array and return it */</span>
 <a id="__codelineno-15-44" name="__codelineno-15-44" href="#__codelineno-15-44"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="nf">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-45" name="__codelineno-15-45" href="#__codelineno-15-45"></a><span class="w">        </span><span class="n">ListNode</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stackPeek</span><span class="p">;</span>
 <a id="__codelineno-15-46" name="__codelineno-15-46" href="#__codelineno-15-46"></a><span class="w">        </span><span class="kt">int</span><span class="o">[]</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="kt">int</span><span class="o">[</span><span class="n">size</span><span class="p">()</span><span class="o">]</span><span class="p">;</span>
@@ -5303,21 +5375,21 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_stack.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Stack based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_stack.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Stack implemented with a linked list */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">LinkedListStack</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">stackPeek</span><span class="p">;</span><span class="w">  </span><span class="c1">// Use head node as stack top</span>
-<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">stkSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">   </span><span class="c1">// Stack length</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">stackPeek</span><span class="p">;</span><span class="w">  </span><span class="c1">// Use the head node as the stack top</span>
+<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">stkSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">   </span><span class="c1">// Stack size</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nf">LinkedListStack</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">        </span><span class="n">stackPeek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a>
-<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stkSize</span><span class="p">;</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a>
-<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="nf">IsEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nf">Size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">    </span><span class="p">}</span>
@@ -5339,14 +5411,14 @@
 <a id="__codelineno-16-34" name="__codelineno-16-34" href="#__codelineno-16-34"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-16-35" name="__codelineno-16-35" href="#__codelineno-16-35"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-36" name="__codelineno-16-36" href="#__codelineno-16-36"></a>
-<a id="__codelineno-16-37" name="__codelineno-16-37" href="#__codelineno-16-37"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-16-37" name="__codelineno-16-37" href="#__codelineno-16-37"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-16-38" name="__codelineno-16-38" href="#__codelineno-16-38"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-39" name="__codelineno-16-39" href="#__codelineno-16-39"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">IsEmpty</span><span class="p">())</span>
 <a id="__codelineno-16-40" name="__codelineno-16-40" href="#__codelineno-16-40"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">Exception</span><span class="p">();</span>
 <a id="__codelineno-16-41" name="__codelineno-16-41" href="#__codelineno-16-41"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stackPeek</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-16-42" name="__codelineno-16-42" href="#__codelineno-16-42"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-43" name="__codelineno-16-43" href="#__codelineno-16-43"></a>
-<a id="__codelineno-16-44" name="__codelineno-16-44" href="#__codelineno-16-44"></a><span class="w">    </span><span class="cm">/* Convert List to Array and return */</span>
+<a id="__codelineno-16-44" name="__codelineno-16-44" href="#__codelineno-16-44"></a><span class="w">    </span><span class="cm">/* Convert the linked list to an array and return it */</span>
 <a id="__codelineno-16-45" name="__codelineno-16-45" href="#__codelineno-16-45"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="nf">ToArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-46" name="__codelineno-16-46" href="#__codelineno-16-46"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">stackPeek</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span>
 <a id="__codelineno-16-47" name="__codelineno-16-47" href="#__codelineno-16-47"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="p">[];</span>
@@ -5363,13 +5435,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_stack.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Stack based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_stack.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Stack implemented with a linked list */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">linkedListStack</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Use built-in package list to implement stack</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Use the standard library list package to implement the stack</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="nx">data</span><span class="w"> </span><span class="o">*</span><span class="nx">list</span><span class="p">.</span><span class="nx">List</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="p">}</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a>
-<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="cm">/* Access top of the stack element */</span>
+<a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="cm">/* Initialize the stack */</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="kd">func</span><span class="w"> </span><span class="nx">newLinkedListStack</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListStack</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">linkedListStack</span><span class="p">{</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">        </span><span class="nx">data</span><span class="p">:</span><span class="w"> </span><span class="nx">list</span><span class="p">.</span><span class="nx">New</span><span class="p">(),</span>
@@ -5391,7 +5463,7 @@
 <a id="__codelineno-17-26" name="__codelineno-17-26" href="#__codelineno-17-26"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">e</span><span class="p">.</span><span class="nx">Value</span>
 <a id="__codelineno-17-27" name="__codelineno-17-27" href="#__codelineno-17-27"></a><span class="p">}</span>
 <a id="__codelineno-17-28" name="__codelineno-17-28" href="#__codelineno-17-28"></a>
-<a id="__codelineno-17-29" name="__codelineno-17-29" href="#__codelineno-17-29"></a><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-17-29" name="__codelineno-17-29" href="#__codelineno-17-29"></a><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-17-30" name="__codelineno-17-30" href="#__codelineno-17-30"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListStack</span><span class="p">)</span><span class="w"> </span><span class="nx">peek</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-31" name="__codelineno-17-31" href="#__codelineno-17-31"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-32" name="__codelineno-17-32" href="#__codelineno-17-32"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
@@ -5400,38 +5472,38 @@
 <a id="__codelineno-17-35" name="__codelineno-17-35" href="#__codelineno-17-35"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">e</span><span class="p">.</span><span class="nx">Value</span>
 <a id="__codelineno-17-36" name="__codelineno-17-36" href="#__codelineno-17-36"></a><span class="p">}</span>
 <a id="__codelineno-17-37" name="__codelineno-17-37" href="#__codelineno-17-37"></a>
-<a id="__codelineno-17-38" name="__codelineno-17-38" href="#__codelineno-17-38"></a><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-17-38" name="__codelineno-17-38" href="#__codelineno-17-38"></a><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-17-39" name="__codelineno-17-39" href="#__codelineno-17-39"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListStack</span><span class="p">)</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-40" name="__codelineno-17-40" href="#__codelineno-17-40"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">data</span><span class="p">.</span><span class="nx">Len</span><span class="p">()</span>
 <a id="__codelineno-17-41" name="__codelineno-17-41" href="#__codelineno-17-41"></a><span class="p">}</span>
 <a id="__codelineno-17-42" name="__codelineno-17-42" href="#__codelineno-17-42"></a>
-<a id="__codelineno-17-43" name="__codelineno-17-43" href="#__codelineno-17-43"></a><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-17-43" name="__codelineno-17-43" href="#__codelineno-17-43"></a><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-17-44" name="__codelineno-17-44" href="#__codelineno-17-44"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListStack</span><span class="p">)</span><span class="w"> </span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-45" name="__codelineno-17-45" href="#__codelineno-17-45"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">data</span><span class="p">.</span><span class="nx">Len</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-17-46" name="__codelineno-17-46" href="#__codelineno-17-46"></a><span class="p">}</span>
 <a id="__codelineno-17-47" name="__codelineno-17-47" href="#__codelineno-17-47"></a>
-<a id="__codelineno-17-48" name="__codelineno-17-48" href="#__codelineno-17-48"></a><span class="cm">/* Get List for printing */</span>
+<a id="__codelineno-17-48" name="__codelineno-17-48" href="#__codelineno-17-48"></a><span class="cm">/* Get the list for printing */</span>
 <a id="__codelineno-17-49" name="__codelineno-17-49" href="#__codelineno-17-49"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">linkedListStack</span><span class="p">)</span><span class="w"> </span><span class="nx">toList</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="nx">list</span><span class="p">.</span><span class="nx">List</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-50" name="__codelineno-17-50" href="#__codelineno-17-50"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">data</span>
 <a id="__codelineno-17-51" name="__codelineno-17-51" href="#__codelineno-17-51"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_stack.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Stack based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_stack.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Stack implemented with a linked list */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">LinkedListStack</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">_peek</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Use head node as stack top</span>
-<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">_size</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Stack length</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">_peek</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Use the head node as the stack top</span>
+<a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">_size</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Stack size</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="kd">init</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">        </span><span class="n">_size</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a>
-<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">        </span><span class="n">_size</span>
 <a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a>
-<a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">        </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">    </span><span class="p">}</span>
@@ -5453,7 +5525,7 @@
 <a id="__codelineno-18-34" name="__codelineno-18-34" href="#__codelineno-18-34"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-18-35" name="__codelineno-18-35" href="#__codelineno-18-35"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-36" name="__codelineno-18-36" href="#__codelineno-18-36"></a>
-<a id="__codelineno-18-37" name="__codelineno-18-37" href="#__codelineno-18-37"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-18-37" name="__codelineno-18-37" href="#__codelineno-18-37"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-18-38" name="__codelineno-18-38" href="#__codelineno-18-38"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">peek</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-39" name="__codelineno-18-39" href="#__codelineno-18-39"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-40" name="__codelineno-18-40" href="#__codelineno-18-40"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Stack is empty"</span><span class="p">)</span>
@@ -5461,7 +5533,7 @@
 <a id="__codelineno-18-42" name="__codelineno-18-42" href="#__codelineno-18-42"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">_peek</span><span class="p">!.</span><span class="n">val</span>
 <a id="__codelineno-18-43" name="__codelineno-18-43" href="#__codelineno-18-43"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-44" name="__codelineno-18-44" href="#__codelineno-18-44"></a>
-<a id="__codelineno-18-45" name="__codelineno-18-45" href="#__codelineno-18-45"></a><span class="w">    </span><span class="cm">/* Convert List to Array and return */</span>
+<a id="__codelineno-18-45" name="__codelineno-18-45" href="#__codelineno-18-45"></a><span class="w">    </span><span class="cm">/* Convert the linked list to an array and return it */</span>
 <a id="__codelineno-18-46" name="__codelineno-18-46" href="#__codelineno-18-46"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-47" name="__codelineno-18-47" href="#__codelineno-18-47"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">_peek</span>
 <a id="__codelineno-18-48" name="__codelineno-18-48" href="#__codelineno-18-48"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="n">repeating</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="bp">count</span><span class="p">:</span><span class="w"> </span><span class="n">size</span><span class="p">())</span>
@@ -5475,21 +5547,21 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_stack.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Stack based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_stack.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Stack implemented with a linked list */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">LinkedListStack</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="n">#stackPeek</span><span class="p">;</span><span class="w"> </span><span class="c1">// Use head node as stack top</span>
-<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="n">#stkSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Stack length</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="n">#stackPeek</span><span class="p">;</span><span class="w"> </span><span class="c1">// Use the head node as the stack top</span>
+<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="n">#stkSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Stack size</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#stackPeek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a>
-<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">    </span><span class="nx">get</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#stkSize</span><span class="p">;</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a>
-<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">    </span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">    </span><span class="p">}</span>
@@ -5510,13 +5582,13 @@
 <a id="__codelineno-19-33" name="__codelineno-19-33" href="#__codelineno-19-33"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-19-34" name="__codelineno-19-34" href="#__codelineno-19-34"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-35" name="__codelineno-19-35" href="#__codelineno-19-35"></a>
-<a id="__codelineno-19-36" name="__codelineno-19-36" href="#__codelineno-19-36"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-19-36" name="__codelineno-19-36" href="#__codelineno-19-36"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-19-37" name="__codelineno-19-37" href="#__codelineno-19-37"></a><span class="w">    </span><span class="nx">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-38" name="__codelineno-19-38" href="#__codelineno-19-38"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="k">this</span><span class="p">.</span><span class="n">#stackPeek</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Stack is empty'</span><span class="p">);</span>
 <a id="__codelineno-19-39" name="__codelineno-19-39" href="#__codelineno-19-39"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#stackPeek</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-19-40" name="__codelineno-19-40" href="#__codelineno-19-40"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-41" name="__codelineno-19-41" href="#__codelineno-19-41"></a>
-<a id="__codelineno-19-42" name="__codelineno-19-42" href="#__codelineno-19-42"></a><span class="w">    </span><span class="cm">/* Convert linked list to Array and return */</span>
+<a id="__codelineno-19-42" name="__codelineno-19-42" href="#__codelineno-19-42"></a><span class="w">    </span><span class="cm">/* Convert the linked list to an Array and return it */</span>
 <a id="__codelineno-19-43" name="__codelineno-19-43" href="#__codelineno-19-43"></a><span class="w">    </span><span class="nx">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-44" name="__codelineno-19-44" href="#__codelineno-19-44"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#stackPeek</span><span class="p">;</span>
 <a id="__codelineno-19-45" name="__codelineno-19-45" href="#__codelineno-19-45"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">);</span>
@@ -5530,21 +5602,21 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_stack.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Stack based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_stack.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Stack implemented with a linked list */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">LinkedListStack</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">stackPeek</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span><span class="w"> </span><span class="c1">// Use head node as stack top</span>
-<a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">stkSize</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Stack length</span>
+<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">stackPeek</span><span class="o">:</span><span class="w"> </span><span class="kt">ListNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span><span class="w"> </span><span class="c1">// Use the head node as the stack top</span>
+<a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">stkSize</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Stack size</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">stackPeek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a>
-<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="nx">get</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">stkSize</span><span class="p">;</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a>
-<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">    </span><span class="nx">isEmpty</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">    </span><span class="p">}</span>
@@ -5566,13 +5638,13 @@
 <a id="__codelineno-20-34" name="__codelineno-20-34" href="#__codelineno-20-34"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">num</span><span class="p">;</span>
 <a id="__codelineno-20-35" name="__codelineno-20-35" href="#__codelineno-20-35"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-36" name="__codelineno-20-36" href="#__codelineno-20-36"></a>
-<a id="__codelineno-20-37" name="__codelineno-20-37" href="#__codelineno-20-37"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-20-37" name="__codelineno-20-37" href="#__codelineno-20-37"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-20-38" name="__codelineno-20-38" href="#__codelineno-20-38"></a><span class="w">    </span><span class="nx">peek</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-39" name="__codelineno-20-39" href="#__codelineno-20-39"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="k">this</span><span class="p">.</span><span class="nx">stackPeek</span><span class="p">)</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Stack is empty'</span><span class="p">);</span>
 <a id="__codelineno-20-40" name="__codelineno-20-40" href="#__codelineno-20-40"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">stackPeek</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-20-41" name="__codelineno-20-41" href="#__codelineno-20-41"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-42" name="__codelineno-20-42" href="#__codelineno-20-42"></a>
-<a id="__codelineno-20-43" name="__codelineno-20-43" href="#__codelineno-20-43"></a><span class="w">    </span><span class="cm">/* Convert linked list to Array and return */</span>
+<a id="__codelineno-20-43" name="__codelineno-20-43" href="#__codelineno-20-43"></a><span class="w">    </span><span class="cm">/* Convert the linked list to an Array and return it */</span>
 <a id="__codelineno-20-44" name="__codelineno-20-44" href="#__codelineno-20-44"></a><span class="w">    </span><span class="nx">toArray</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-45" name="__codelineno-20-45" href="#__codelineno-20-45"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">stackPeek</span><span class="p">;</span>
 <a id="__codelineno-20-46" name="__codelineno-20-46" href="#__codelineno-20-46"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nb">Array</span><span class="o">&lt;</span><span class="kt">number</span><span class="o">&gt;</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">);</span>
@@ -5586,21 +5658,21 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_stack.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Stack implemented based on linked list class */</span>
+<div class="highlight"><span class="filename">linkedlist_stack.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Stack implemented with a linked list */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">LinkedListStack</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">_stackPeek</span><span class="p">;</span><span class="w"> </span><span class="c1">// Use head node as stack top</span>
-<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">_stkSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Stack length</span>
+<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">_stackPeek</span><span class="p">;</span><span class="w"> </span><span class="c1">// Use the head node as the stack top</span>
+<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">_stkSize</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">;</span><span class="w"> </span><span class="c1">// Stack size</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="n">LinkedListStack</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">    </span><span class="n">_stackPeek</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a>
-<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_stkSize</span><span class="p">;</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a>
-<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">  </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">  </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">  </span><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_stkSize</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">  </span><span class="p">}</span>
@@ -5621,7 +5693,7 @@
 <a id="__codelineno-21-33" name="__codelineno-21-33" href="#__codelineno-21-33"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_num</span><span class="p">;</span>
 <a id="__codelineno-21-34" name="__codelineno-21-34" href="#__codelineno-21-34"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-35" name="__codelineno-21-35" href="#__codelineno-21-35"></a>
-<a id="__codelineno-21-36" name="__codelineno-21-36" href="#__codelineno-21-36"></a><span class="w">  </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-21-36" name="__codelineno-21-36" href="#__codelineno-21-36"></a><span class="w">  </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-21-37" name="__codelineno-21-37" href="#__codelineno-21-37"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-38" name="__codelineno-21-38" href="#__codelineno-21-38"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_stackPeek</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-39" name="__codelineno-21-39" href="#__codelineno-21-39"></a><span class="w">      </span><span class="k">throw</span><span class="w"> </span><span class="n">Exception</span><span class="p">(</span><span class="s2">"Stack is empty"</span><span class="p">);</span>
@@ -5629,7 +5701,7 @@
 <a id="__codelineno-21-41" name="__codelineno-21-41" href="#__codelineno-21-41"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_stackPeek</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-21-42" name="__codelineno-21-42" href="#__codelineno-21-42"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-43" name="__codelineno-21-43" href="#__codelineno-21-43"></a>
-<a id="__codelineno-21-44" name="__codelineno-21-44" href="#__codelineno-21-44"></a><span class="w">  </span><span class="cm">/* Convert linked list to List and return */</span>
+<a id="__codelineno-21-44" name="__codelineno-21-44" href="#__codelineno-21-44"></a><span class="w">  </span><span class="cm">/* Convert the linked list to a List and return it */</span>
 <a id="__codelineno-21-45" name="__codelineno-21-45" href="#__codelineno-21-45"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">toList</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-46" name="__codelineno-21-46" href="#__codelineno-21-46"></a><span class="w">    </span><span class="n">ListNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_stackPeek</span><span class="p">;</span>
 <a id="__codelineno-21-47" name="__codelineno-21-47" href="#__codelineno-21-47"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">list</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
@@ -5644,11 +5716,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_stack.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Stack based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_stack.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Stack implemented with a linked list */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="cp">#[allow(dead_code)]</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="k">pub</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">LinkedListStack</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="n">stack_peek</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="c1">// Use head node as stack top</span>
-<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="n">stk_size</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w">                              </span><span class="c1">// Stack length</span>
+<a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="n">stack_peek</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="c1">// Use the head node as the stack top</span>
+<a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="n">stk_size</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">,</span><span class="w">                              </span><span class="c1">// Stack size</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="p">}</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="k">impl</span><span class="o">&lt;</span><span class="n">T</span><span class="p">:</span><span class="w"> </span><span class="nb">Copy</span><span class="o">&gt;</span><span class="w"> </span><span class="n">LinkedListStack</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
@@ -5659,12 +5731,12 @@
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a>
-<a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">stk_size</span><span class="p">;</span>
 <a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a>
-<a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">is_empty</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w">    </span><span class="p">}</span>
@@ -5687,12 +5759,12 @@
 <a id="__codelineno-22-41" name="__codelineno-22-41" href="#__codelineno-22-41"></a><span class="w">        </span><span class="p">})</span>
 <a id="__codelineno-22-42" name="__codelineno-22-42" href="#__codelineno-22-42"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-43" name="__codelineno-22-43" href="#__codelineno-22-43"></a>
-<a id="__codelineno-22-44" name="__codelineno-22-44" href="#__codelineno-22-44"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-22-44" name="__codelineno-22-44" href="#__codelineno-22-44"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-22-45" name="__codelineno-22-45" href="#__codelineno-22-45"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">peek</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-46" name="__codelineno-22-46" href="#__codelineno-22-46"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">stack_peek</span><span class="p">.</span><span class="n">as_ref</span><span class="p">()</span>
 <a id="__codelineno-22-47" name="__codelineno-22-47" href="#__codelineno-22-47"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-48" name="__codelineno-22-48" href="#__codelineno-22-48"></a>
-<a id="__codelineno-22-49" name="__codelineno-22-49" href="#__codelineno-22-49"></a><span class="w">    </span><span class="cm">/* Convert List to Array and return */</span>
+<a id="__codelineno-22-49" name="__codelineno-22-49" href="#__codelineno-22-49"></a><span class="w">    </span><span class="cm">/* Convert the linked list to a vector and return it */</span>
 <a id="__codelineno-22-50" name="__codelineno-22-50" href="#__codelineno-22-50"></a><span class="w">    </span><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">to_array</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-51" name="__codelineno-22-51" href="#__codelineno-22-51"></a><span class="w">        </span><span class="k">fn</span><span class="w"> </span><span class="nf">_to_array</span><span class="o">&lt;</span><span class="n">T</span><span class="p">:</span><span class="w"> </span><span class="nb">Sized</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nb">Copy</span><span class="o">&gt;</span><span class="p">(</span><span class="n">head</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">ListNode</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;&gt;&gt;&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-52" name="__codelineno-22-52" href="#__codelineno-22-52"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">head</span><span class="w"> </span><span class="p">{</span>
@@ -5709,10 +5781,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_stack.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Stack based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_stack.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Stack implemented with a linked list */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">top</span><span class="p">;</span><span class="w"> </span><span class="c1">// Use head node as stack top</span>
-<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w">      </span><span class="c1">// Stack length</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">top</span><span class="p">;</span><span class="w"> </span><span class="c1">// Use the head node as the stack top</span>
+<a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span><span class="w">      </span><span class="c1">// Stack size</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="p">}</span><span class="w"> </span><span class="n">LinkedListStack</span><span class="p">;</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="cm">/* Constructor */</span>
@@ -5733,12 +5805,12 @@
 <a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">s</span><span class="p">);</span>
 <a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="p">}</span>
 <a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a>
-<a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="n">LinkedListStack</span><span class="w"> </span><span class="o">*</span><span class="n">s</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">s</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">;</span>
 <a id="__codelineno-23-28" name="__codelineno-23-28" href="#__codelineno-23-28"></a><span class="p">}</span>
 <a id="__codelineno-23-29" name="__codelineno-23-29" href="#__codelineno-23-29"></a>
-<a id="__codelineno-23-30" name="__codelineno-23-30" href="#__codelineno-23-30"></a><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-23-30" name="__codelineno-23-30" href="#__codelineno-23-30"></a><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-23-31" name="__codelineno-23-31" href="#__codelineno-23-31"></a><span class="kt">bool</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">(</span><span class="n">LinkedListStack</span><span class="w"> </span><span class="o">*</span><span class="n">s</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-32" name="__codelineno-23-32" href="#__codelineno-23-32"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">s</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-23-33" name="__codelineno-23-33" href="#__codelineno-23-33"></a><span class="p">}</span>
@@ -5746,13 +5818,13 @@
 <a id="__codelineno-23-35" name="__codelineno-23-35" href="#__codelineno-23-35"></a><span class="cm">/* Push */</span>
 <a id="__codelineno-23-36" name="__codelineno-23-36" href="#__codelineno-23-36"></a><span class="kt">void</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="n">LinkedListStack</span><span class="w"> </span><span class="o">*</span><span class="n">s</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-37" name="__codelineno-23-37" href="#__codelineno-23-37"></a><span class="w">    </span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">ListNode</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">ListNode</span><span class="p">));</span>
-<a id="__codelineno-23-38" name="__codelineno-23-38" href="#__codelineno-23-38"></a><span class="w">    </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="o">-&gt;</span><span class="n">top</span><span class="p">;</span><span class="w"> </span><span class="c1">// Update new node's pointer field</span>
-<a id="__codelineno-23-39" name="__codelineno-23-39" href="#__codelineno-23-39"></a><span class="w">    </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span><span class="w">     </span><span class="c1">// Update new node's data field</span>
-<a id="__codelineno-23-40" name="__codelineno-23-40" href="#__codelineno-23-40"></a><span class="w">    </span><span class="n">s</span><span class="o">-&gt;</span><span class="n">top</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w">       </span><span class="c1">// Update stack top</span>
-<a id="__codelineno-23-41" name="__codelineno-23-41" href="#__codelineno-23-41"></a><span class="w">    </span><span class="n">s</span><span class="o">-&gt;</span><span class="n">size</span><span class="o">++</span><span class="p">;</span><span class="w">           </span><span class="c1">// Update stack size</span>
+<a id="__codelineno-23-38" name="__codelineno-23-38" href="#__codelineno-23-38"></a><span class="w">    </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">next</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s</span><span class="o">-&gt;</span><span class="n">top</span><span class="p">;</span><span class="w"> </span><span class="c1">// Set the new node's pointer field</span>
+<a id="__codelineno-23-39" name="__codelineno-23-39" href="#__codelineno-23-39"></a><span class="w">    </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">num</span><span class="p">;</span><span class="w">     </span><span class="c1">// Set the new node's data field</span>
+<a id="__codelineno-23-40" name="__codelineno-23-40" href="#__codelineno-23-40"></a><span class="w">    </span><span class="n">s</span><span class="o">-&gt;</span><span class="n">top</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w">       </span><span class="c1">// Update the stack top</span>
+<a id="__codelineno-23-41" name="__codelineno-23-41" href="#__codelineno-23-41"></a><span class="w">    </span><span class="n">s</span><span class="o">-&gt;</span><span class="n">size</span><span class="o">++</span><span class="p">;</span><span class="w">           </span><span class="c1">// Update the stack size</span>
 <a id="__codelineno-23-42" name="__codelineno-23-42" href="#__codelineno-23-42"></a><span class="p">}</span>
 <a id="__codelineno-23-43" name="__codelineno-23-43" href="#__codelineno-23-43"></a>
-<a id="__codelineno-23-44" name="__codelineno-23-44" href="#__codelineno-23-44"></a><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-23-44" name="__codelineno-23-44" href="#__codelineno-23-44"></a><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-23-45" name="__codelineno-23-45" href="#__codelineno-23-45"></a><span class="kt">int</span><span class="w"> </span><span class="nf">peek</span><span class="p">(</span><span class="n">LinkedListStack</span><span class="w"> </span><span class="o">*</span><span class="n">s</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-46" name="__codelineno-23-46" href="#__codelineno-23-46"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-47" name="__codelineno-23-47" href="#__codelineno-23-47"></a><span class="w">        </span><span class="n">printf</span><span class="p">(</span><span class="s">"Stack is empty</span><span class="se">\n</span><span class="s">"</span><span class="p">);</span>
@@ -5774,18 +5846,18 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_stack.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Stack based on linked list implementation */</span>
+<div class="highlight"><span class="filename">linkedlist_stack.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Stack implemented with a linked list */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">LinkedListStack</span><span class="p">(</span>
-<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">stackPeek</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="c1">// Use head node as stack top</span>
-<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">stkSize</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Stack length</span>
+<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">stackPeek</span><span class="p">:</span><span class="w"> </span><span class="n">ListNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="c1">// Use the head node as the stack top</span>
+<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">stkSize</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="c1">// Stack size</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a>
-<a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">size</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stkSize</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a>
-<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">():</span><span class="w"> </span><span class="kt">Boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">    </span><span class="p">}</span>
@@ -5806,13 +5878,13 @@
 <a id="__codelineno-24-30" name="__codelineno-24-30" href="#__codelineno-24-30"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-24-31" name="__codelineno-24-31" href="#__codelineno-24-31"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-32" name="__codelineno-24-32" href="#__codelineno-24-32"></a>
-<a id="__codelineno-24-33" name="__codelineno-24-33" href="#__codelineno-24-33"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-24-33" name="__codelineno-24-33" href="#__codelineno-24-33"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-24-34" name="__codelineno-24-34" href="#__codelineno-24-34"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">peek</span><span class="p">():</span><span class="w"> </span><span class="kt">Int?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-35" name="__codelineno-24-35" href="#__codelineno-24-35"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">()</span>
 <a id="__codelineno-24-36" name="__codelineno-24-36" href="#__codelineno-24-36"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stackPeek</span><span class="o">?.</span><span class="na">_val</span>
 <a id="__codelineno-24-37" name="__codelineno-24-37" href="#__codelineno-24-37"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-38" name="__codelineno-24-38" href="#__codelineno-24-38"></a>
-<a id="__codelineno-24-39" name="__codelineno-24-39" href="#__codelineno-24-39"></a><span class="w">    </span><span class="cm">/* Convert List to Array and return */</span>
+<a id="__codelineno-24-39" name="__codelineno-24-39" href="#__codelineno-24-39"></a><span class="w">    </span><span class="cm">/* Convert the list to an array and return it */</span>
 <a id="__codelineno-24-40" name="__codelineno-24-40" href="#__codelineno-24-40"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">toArray</span><span class="p">():</span><span class="w"> </span><span class="n">IntArray</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-41" name="__codelineno-24-41" href="#__codelineno-24-41"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">stackPeek</span>
 <a id="__codelineno-24-42" name="__codelineno-24-42" href="#__codelineno-24-42"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">IntArray</span><span class="p">(</span><span class="n">size</span><span class="p">())</span>
@@ -5826,7 +5898,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">linkedlist_stack.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Stack based on linked list ###</span>
+<div class="highlight"><span class="filename">linkedlist_stack.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Stack implemented with a linked list ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">LinkedListStack</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="kp">attr_reader</span><span class="w"> </span><span class="ss">:size</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a>
@@ -5835,7 +5907,7 @@
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">    </span><span class="vi">@size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a>
-<a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="c1">### Check if stack is empty ###</span>
+<a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="c1">### Check whether the stack is empty ###</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">is_empty?</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">    </span><span class="vi">@peek</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">  </span><span class="k">end</span>
@@ -5856,14 +5928,14 @@
 <a id="__codelineno-25-28" name="__codelineno-25-28" href="#__codelineno-25-28"></a><span class="w">    </span><span class="n">num</span>
 <a id="__codelineno-25-29" name="__codelineno-25-29" href="#__codelineno-25-29"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-30" name="__codelineno-25-30" href="#__codelineno-25-30"></a>
-<a id="__codelineno-25-31" name="__codelineno-25-31" href="#__codelineno-25-31"></a><span class="w">  </span><span class="c1">### Access top element ###</span>
+<a id="__codelineno-25-31" name="__codelineno-25-31" href="#__codelineno-25-31"></a><span class="w">  </span><span class="c1">### Access the top element ###</span>
 <a id="__codelineno-25-32" name="__codelineno-25-32" href="#__codelineno-25-32"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">peek</span>
 <a id="__codelineno-25-33" name="__codelineno-25-33" href="#__codelineno-25-33"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="p">,</span><span class="w"> </span><span class="s1">'Stack is empty'</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">is_empty?</span>
 <a id="__codelineno-25-34" name="__codelineno-25-34" href="#__codelineno-25-34"></a>
 <a id="__codelineno-25-35" name="__codelineno-25-35" href="#__codelineno-25-35"></a><span class="w">    </span><span class="vi">@peek</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-25-36" name="__codelineno-25-36" href="#__codelineno-25-36"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-37" name="__codelineno-25-37" href="#__codelineno-25-37"></a>
-<a id="__codelineno-25-38" name="__codelineno-25-38" href="#__codelineno-25-38"></a><span class="w">  </span><span class="c1">### Convert linked list to Array and return ###</span>
+<a id="__codelineno-25-38" name="__codelineno-25-38" href="#__codelineno-25-38"></a><span class="w">  </span><span class="c1">### Convert the linked list to an Array and return it ###</span>
 <a id="__codelineno-25-39" name="__codelineno-25-39" href="#__codelineno-25-39"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">to_array</span>
 <a id="__codelineno-25-40" name="__codelineno-25-40" href="#__codelineno-25-40"></a><span class="w">    </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-25-41" name="__codelineno-25-41" href="#__codelineno-25-41"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@peek</span>
@@ -5878,12 +5950,17 @@
 </div>
 </div>
 </div>
-<h3 id="2-array-implementation">2. &nbsp; Array Implementation<a class="headerlink" href="#2-array-implementation" title="Permanent link">&para;</a></h3>
-<p>When implementing a stack using an array, we can treat the end of the array as the top of the stack. As shown in Figure 5-3, push and pop operations correspond to adding and removing elements at the end of the array, both with a time complexity of <span class="arithmatex">\(O(1)\)</span>.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0A%0Aclass%20LinkedListStack%3A%0A%20%20%20%20%22%22%22Stack%20implemented%20with%20a%20linked%20list%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self._peek%3A%20ListNode%20%7C%20None%20%3D%20None%0A%20%20%20%20%20%20%20%20self._size%3A%20int%20%3D%200%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20stack%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20self._size%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22Check%20whether%20the%20stack%20is%20empty%22%22%22%0A%20%20%20%20%20%20%20%20return%20not%20self._peek%0A%0A%20%20%20%20def%20push%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Push%22%22%22%0A%20%20%20%20%20%20%20%20node%20%3D%20ListNode%28val%29%0A%20%20%20%20%20%20%20%20node.next%20%3D%20self._peek%0A%20%20%20%20%20%20%20%20self._peek%20%3D%20node%0A%20%20%20%20%20%20%20%20self._size%20%2B%3D%201%0A%0A%20%20%20%20def%20pop%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Pop%22%22%22%0A%20%20%20%20%20%20%20%20num%20%3D%20self.peek%28%29%0A%20%20%20%20%20%20%20%20self._peek%20%3D%20self._peek.next%0A%20%20%20%20%20%20%20%20self._size%20-%3D%201%0A%20%20%20%20%20%20%20%20return%20num%0A%0A%20%20%20%20def%20peek%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Access%20the%20top%20element%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.is_empty%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22Stack%20is%20empty%22%29%0A%20%20%20%20%20%20%20%20return%20self._peek.val%0A%0A%20%20%20%20def%20to_list%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Convert%20to%20a%20list%20for%20printing%22%22%22%0A%20%20%20%20%20%20%20%20arr%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20node%20%3D%20self._peek%0A%20%20%20%20%20%20%20%20while%20node%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20arr.append%28node.val%29%0A%20%20%20%20%20%20%20%20%20%20%20%20node%20%3D%20node.next%0A%20%20%20%20%20%20%20%20arr.reverse%28%29%0A%20%20%20%20%20%20%20%20return%20arr%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20stack%0A%20%20%20%20stack%20%3D%20LinkedListStack%28%29%0A%0A%20%20%20%20%23%20Push%20elements%20onto%20the%20stack%0A%20%20%20%20stack.push%281%29%0A%20%20%20%20stack.push%283%29%0A%20%20%20%20stack.push%282%29%0A%20%20%20%20stack.push%285%29%0A%20%20%20%20stack.push%284%29%0A%20%20%20%20print%28%22Stack%20stack%20%3D%22,%20stack.to_list%28%29%29%0A%0A%20%20%20%20%23%20Access%20the%20top%20element%0A%20%20%20%20peek%20%3D%20stack.peek%28%29%0A%20%20%20%20print%28%22Top%20element%20peek%20%3D%22,%20peek%29%0A%0A%20%20%20%20%23%20Pop%20an%20element%20from%20the%20stack%0A%20%20%20%20pop%20%3D%20stack.pop%28%29%0A%20%20%20%20print%28%22Popped%20element%20pop%20%3D%22,%20pop%29%0A%20%20%20%20print%28%22After%20popping,%20stack%20%3D%22,%20stack.to_list%28%29%29%0A%0A%20%20%20%20%23%20Get%20the%20stack%20size%0A%20%20%20%20size%20%3D%20stack.size%28%29%0A%20%20%20%20print%28%22Stack%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20Check%20whether%20it%20is%20empty%0A%20%20%20%20is_empty%20%3D%20stack.is_empty%28%29%0A%20%20%20%20print%28%22Is%20the%20stack%20empty%20%3D%22,%20is_empty%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22Linked%20list%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20next%20node%0A%0A%0Aclass%20LinkedListStack%3A%0A%20%20%20%20%22%22%22Stack%20implemented%20with%20a%20linked%20list%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self._peek%3A%20ListNode%20%7C%20None%20%3D%20None%0A%20%20%20%20%20%20%20%20self._size%3A%20int%20%3D%200%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20stack%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20self._size%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22Check%20whether%20the%20stack%20is%20empty%22%22%22%0A%20%20%20%20%20%20%20%20return%20not%20self._peek%0A%0A%20%20%20%20def%20push%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Push%22%22%22%0A%20%20%20%20%20%20%20%20node%20%3D%20ListNode%28val%29%0A%20%20%20%20%20%20%20%20node.next%20%3D%20self._peek%0A%20%20%20%20%20%20%20%20self._peek%20%3D%20node%0A%20%20%20%20%20%20%20%20self._size%20%2B%3D%201%0A%0A%20%20%20%20def%20pop%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Pop%22%22%22%0A%20%20%20%20%20%20%20%20num%20%3D%20self.peek%28%29%0A%20%20%20%20%20%20%20%20self._peek%20%3D%20self._peek.next%0A%20%20%20%20%20%20%20%20self._size%20-%3D%201%0A%20%20%20%20%20%20%20%20return%20num%0A%0A%20%20%20%20def%20peek%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Access%20the%20top%20element%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.is_empty%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22Stack%20is%20empty%22%29%0A%20%20%20%20%20%20%20%20return%20self._peek.val%0A%0A%20%20%20%20def%20to_list%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Convert%20to%20a%20list%20for%20printing%22%22%22%0A%20%20%20%20%20%20%20%20arr%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20node%20%3D%20self._peek%0A%20%20%20%20%20%20%20%20while%20node%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20arr.append%28node.val%29%0A%20%20%20%20%20%20%20%20%20%20%20%20node%20%3D%20node.next%0A%20%20%20%20%20%20%20%20arr.reverse%28%29%0A%20%20%20%20%20%20%20%20return%20arr%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20stack%0A%20%20%20%20stack%20%3D%20LinkedListStack%28%29%0A%0A%20%20%20%20%23%20Push%20elements%20onto%20the%20stack%0A%20%20%20%20stack.push%281%29%0A%20%20%20%20stack.push%283%29%0A%20%20%20%20stack.push%282%29%0A%20%20%20%20stack.push%285%29%0A%20%20%20%20stack.push%284%29%0A%20%20%20%20print%28%22Stack%20stack%20%3D%22,%20stack.to_list%28%29%29%0A%0A%20%20%20%20%23%20Access%20the%20top%20element%0A%20%20%20%20peek%20%3D%20stack.peek%28%29%0A%20%20%20%20print%28%22Top%20element%20peek%20%3D%22,%20peek%29%0A%0A%20%20%20%20%23%20Pop%20an%20element%20from%20the%20stack%0A%20%20%20%20pop%20%3D%20stack.pop%28%29%0A%20%20%20%20print%28%22Popped%20element%20pop%20%3D%22,%20pop%29%0A%20%20%20%20print%28%22After%20popping,%20stack%20%3D%22,%20stack.to_list%28%29%29%0A%0A%20%20%20%20%23%20Get%20the%20stack%20size%0A%20%20%20%20size%20%3D%20stack.size%28%29%0A%20%20%20%20print%28%22Stack%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20Check%20whether%20it%20is%20empty%0A%20%20%20%20is_empty%20%3D%20stack.is_empty%28%29%0A%20%20%20%20print%28%22Is%20the%20stack%20empty%20%3D%22,%20is_empty%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=4&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="2-implementation-with-an-array">2. &nbsp; Implementation with an array<a class="headerlink" href="#2-implementation-with-an-array" title="Permanent link">&para;</a></h3>
+<p>For an array-based stack, use the end of the array as the top. As shown below, pushing and popping correspond to adding and removing the last element, both taking <span class="arithmatex">\(O(1)\)</span> time.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:3"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">&lt;1&gt;</label><label for="__tabbed_4_2">&lt;2&gt;</label><label for="__tabbed_4_3">&lt;3&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Push and pop operations in array implementation of stack" class="animation-figure" src="../stack.assets/array_stack_step1.png" /></p>
+<p><img alt="Push and pop in an array-based stack" class="animation-figure" src="../stack.assets/array_stack_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="array_stack_push" class="animation-figure" src="../stack.assets/array_stack_step2_push.png" /></p>
@@ -5893,25 +5970,25 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 5-3 &nbsp; Push and pop operations in array implementation of stack </p>
+<p align="center"> Figure 5-3 &nbsp; Push and pop in an array-based stack </p>
 
-<p>Since elements pushed onto the stack may increase continuously, we can use a dynamic array, which eliminates the need to handle array expansion ourselves. Here is the sample code:</p>
+<p>Since elements may keep arriving, use a dynamic array so that resizing is handled for us. An implementation follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">array_stack.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayStack</span><span class="p">:</span>
-<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Stack based on array implementation"""</span>
+<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Stack implemented with an array"""</span>
 <a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
 <a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a><span class="w">        </span><span class="sd">"""Constructor"""</span>
 <a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_stack</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="p">[]</span>
 <a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>
 <a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a><span class="w">        </span><span class="sd">"""Get the length of the stack"""</span>
+<a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a><span class="w">        </span><span class="sd">"""Get the stack size"""</span>
 <a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_stack</span><span class="p">)</span>
 <a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a>
 <a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">is_empty</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">bool</span><span class="p">:</span>
-<a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a><span class="w">        </span><span class="sd">"""Check if the stack is empty"""</span>
+<a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a><span class="w">        </span><span class="sd">"""Check whether the stack is empty"""</span>
 <a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()</span> <span class="o">==</span> <span class="mi">0</span>
 <a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a>
 <a id="__codelineno-26-16" name="__codelineno-26-16" href="#__codelineno-26-16"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">push</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">item</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
@@ -5925,29 +6002,29 @@
 <a id="__codelineno-26-24" name="__codelineno-26-24" href="#__codelineno-26-24"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_stack</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
 <a id="__codelineno-26-25" name="__codelineno-26-25" href="#__codelineno-26-25"></a>
 <a id="__codelineno-26-26" name="__codelineno-26-26" href="#__codelineno-26-26"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">peek</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-27" name="__codelineno-26-27" href="#__codelineno-26-27"></a><span class="w">        </span><span class="sd">"""Access top of the stack element"""</span>
+<a id="__codelineno-26-27" name="__codelineno-26-27" href="#__codelineno-26-27"></a><span class="w">        </span><span class="sd">"""Access the top element"""</span>
 <a id="__codelineno-26-28" name="__codelineno-26-28" href="#__codelineno-26-28"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">is_empty</span><span class="p">():</span>
 <a id="__codelineno-26-29" name="__codelineno-26-29" href="#__codelineno-26-29"></a>            <span class="k">raise</span> <span class="ne">IndexError</span><span class="p">(</span><span class="s2">"Stack is empty"</span><span class="p">)</span>
 <a id="__codelineno-26-30" name="__codelineno-26-30" href="#__codelineno-26-30"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_stack</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
 <a id="__codelineno-26-31" name="__codelineno-26-31" href="#__codelineno-26-31"></a>
 <a id="__codelineno-26-32" name="__codelineno-26-32" href="#__codelineno-26-32"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">to_list</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
-<a id="__codelineno-26-33" name="__codelineno-26-33" href="#__codelineno-26-33"></a><span class="w">        </span><span class="sd">"""Return list for printing"""</span>
+<a id="__codelineno-26-33" name="__codelineno-26-33" href="#__codelineno-26-33"></a><span class="w">        </span><span class="sd">"""Return a list for printing"""</span>
 <a id="__codelineno-26-34" name="__codelineno-26-34" href="#__codelineno-26-34"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_stack</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_stack.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Stack based on array implementation */</span>
+<div class="highlight"><span class="filename">array_stack.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Stack implemented with an array */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayStack</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">  </span><span class="k">private</span><span class="o">:</span>
 <a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">stack</span><span class="p">;</span>
 <a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a>
 <a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">  </span><span class="k">public</span><span class="o">:</span>
-<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
 <a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a>
-<a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">    </span><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="w">    </span><span class="p">}</span>
@@ -5964,14 +6041,14 @@
 <a id="__codelineno-27-26" name="__codelineno-27-26" href="#__codelineno-27-26"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">num</span><span class="p">;</span>
 <a id="__codelineno-27-27" name="__codelineno-27-27" href="#__codelineno-27-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-28" name="__codelineno-27-28" href="#__codelineno-27-28"></a>
-<a id="__codelineno-27-29" name="__codelineno-27-29" href="#__codelineno-27-29"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-27-29" name="__codelineno-27-29" href="#__codelineno-27-29"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-27-30" name="__codelineno-27-30" href="#__codelineno-27-30"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">top</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-31" name="__codelineno-27-31" href="#__codelineno-27-31"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-27-32" name="__codelineno-27-32" href="#__codelineno-27-32"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="n">out_of_range</span><span class="p">(</span><span class="s">"Stack is empty"</span><span class="p">);</span>
 <a id="__codelineno-27-33" name="__codelineno-27-33" href="#__codelineno-27-33"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">back</span><span class="p">();</span>
 <a id="__codelineno-27-34" name="__codelineno-27-34" href="#__codelineno-27-34"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-35" name="__codelineno-27-35" href="#__codelineno-27-35"></a>
-<a id="__codelineno-27-36" name="__codelineno-27-36" href="#__codelineno-27-36"></a><span class="w">    </span><span class="cm">/* Return Vector */</span>
+<a id="__codelineno-27-36" name="__codelineno-27-36" href="#__codelineno-27-36"></a><span class="w">    </span><span class="cm">/* Return a vector */</span>
 <a id="__codelineno-27-37" name="__codelineno-27-37" href="#__codelineno-27-37"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">toVector</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-38" name="__codelineno-27-38" href="#__codelineno-27-38"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">;</span>
 <a id="__codelineno-27-39" name="__codelineno-27-39" href="#__codelineno-27-39"></a><span class="w">    </span><span class="p">}</span>
@@ -5979,21 +6056,21 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_stack.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Stack based on array implementation */</span>
+<div class="highlight"><span class="filename">array_stack.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Stack implemented with an array */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kd">class</span> <span class="nc">ArrayStack</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">stack</span><span class="p">;</span>
 <a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a>
 <a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="nf">ArrayStack</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">        </span><span class="c1">// Initialize list (dynamic array)</span>
+<a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">        </span><span class="c1">// Initialize the list (dynamic array)</span>
 <a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">        </span><span class="n">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a>
-<a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">size</span><span class="p">();</span>
 <a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a>
-<a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">boolean</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a><span class="w">    </span><span class="p">}</span>
@@ -6010,14 +6087,14 @@
 <a id="__codelineno-28-29" name="__codelineno-28-29" href="#__codelineno-28-29"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">remove</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-28-30" name="__codelineno-28-30" href="#__codelineno-28-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-31" name="__codelineno-28-31" href="#__codelineno-28-31"></a>
-<a id="__codelineno-28-32" name="__codelineno-28-32" href="#__codelineno-28-32"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-28-32" name="__codelineno-28-32" href="#__codelineno-28-32"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-28-33" name="__codelineno-28-33" href="#__codelineno-28-33"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-34" name="__codelineno-28-34" href="#__codelineno-28-34"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span>
 <a id="__codelineno-28-35" name="__codelineno-28-35" href="#__codelineno-28-35"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">();</span>
 <a id="__codelineno-28-36" name="__codelineno-28-36" href="#__codelineno-28-36"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-28-37" name="__codelineno-28-37" href="#__codelineno-28-37"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-38" name="__codelineno-28-38" href="#__codelineno-28-38"></a>
-<a id="__codelineno-28-39" name="__codelineno-28-39" href="#__codelineno-28-39"></a><span class="w">    </span><span class="cm">/* Convert List to Array and return */</span>
+<a id="__codelineno-28-39" name="__codelineno-28-39" href="#__codelineno-28-39"></a><span class="w">    </span><span class="cm">/* Convert the list to an array and return it */</span>
 <a id="__codelineno-28-40" name="__codelineno-28-40" href="#__codelineno-28-40"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="n">Object</span><span class="o">[]</span><span class="w"> </span><span class="nf">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-41" name="__codelineno-28-41" href="#__codelineno-28-41"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">toArray</span><span class="p">();</span>
 <a id="__codelineno-28-42" name="__codelineno-28-42" href="#__codelineno-28-42"></a><span class="w">    </span><span class="p">}</span>
@@ -6025,20 +6102,20 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_stack.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Stack based on array implementation */</span>
+<div class="highlight"><span class="filename">array_stack.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Stack implemented with an array */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayStack</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">stack</span><span class="p">;</span>
 <a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="nf">ArrayStack</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">        </span><span class="c1">// Initialize list (dynamic array)</span>
+<a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">        </span><span class="c1">// Initialize the list (dynamic array)</span>
 <a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">        </span><span class="n">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a>
-<a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="n">Count</span><span class="p">;</span>
 <a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a>
-<a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="nf">IsEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nf">Size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></a><span class="w">    </span><span class="p">}</span>
@@ -6057,14 +6134,14 @@
 <a id="__codelineno-29-30" name="__codelineno-29-30" href="#__codelineno-29-30"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-29-31" name="__codelineno-29-31" href="#__codelineno-29-31"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-32" name="__codelineno-29-32" href="#__codelineno-29-32"></a>
-<a id="__codelineno-29-33" name="__codelineno-29-33" href="#__codelineno-29-33"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-29-33" name="__codelineno-29-33" href="#__codelineno-29-33"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-29-34" name="__codelineno-29-34" href="#__codelineno-29-34"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-35" name="__codelineno-29-35" href="#__codelineno-29-35"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">IsEmpty</span><span class="p">())</span>
 <a id="__codelineno-29-36" name="__codelineno-29-36" href="#__codelineno-29-36"></a><span class="w">            </span><span class="k">throw</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="nf">Exception</span><span class="p">();</span>
 <a id="__codelineno-29-37" name="__codelineno-29-37" href="#__codelineno-29-37"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">[</span><span class="n">Size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">];</span>
 <a id="__codelineno-29-38" name="__codelineno-29-38" href="#__codelineno-29-38"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-39" name="__codelineno-29-39" href="#__codelineno-29-39"></a>
-<a id="__codelineno-29-40" name="__codelineno-29-40" href="#__codelineno-29-40"></a><span class="w">    </span><span class="cm">/* Convert List to Array and return */</span>
+<a id="__codelineno-29-40" name="__codelineno-29-40" href="#__codelineno-29-40"></a><span class="w">    </span><span class="cm">/* Convert the list to an array and return it */</span>
 <a id="__codelineno-29-41" name="__codelineno-29-41" href="#__codelineno-29-41"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="p">[]</span><span class="w"> </span><span class="nf">ToArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-42" name="__codelineno-29-42" href="#__codelineno-29-42"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">[..</span><span class="w"> </span><span class="n">stack</span><span class="p">];</span>
 <a id="__codelineno-29-43" name="__codelineno-29-43" href="#__codelineno-29-43"></a><span class="w">    </span><span class="p">}</span>
@@ -6072,32 +6149,32 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_stack.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Stack based on array implementation */</span>
+<div class="highlight"><span class="filename">array_stack.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Stack implemented with an array */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">arrayStack</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="nx">data</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="w"> </span><span class="c1">// Data</span>
 <a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="p">}</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a>
-<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="cm">/* Access top of the stack element */</span>
+<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="cm">/* Initialize the stack */</span>
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="kd">func</span><span class="w"> </span><span class="nx">newArrayStack</span><span class="p">()</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayStack</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">arrayStack</span><span class="p">{</span>
-<a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">        </span><span class="c1">// Set stack length to 0, capacity to 16</span>
+<a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">        </span><span class="c1">// Set the stack length to 0 and its capacity to 16</span>
 <a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">        </span><span class="nx">data</span><span class="p">:</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">16</span><span class="p">),</span>
 <a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="p">}</span>
 <a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a>
-<a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="cm">/* Stack length */</span>
+<a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="cm">/* Stack size */</span>
 <a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayStack</span><span class="p">)</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">s</span><span class="p">.</span><span class="nx">data</span><span class="p">)</span>
 <a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></a><span class="p">}</span>
 <a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a>
-<a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></a><span class="cm">/* Is stack empty */</span>
+<a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></a><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-30-20" name="__codelineno-30-20" href="#__codelineno-30-20"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayStack</span><span class="p">)</span><span class="w"> </span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-21" name="__codelineno-30-21" href="#__codelineno-30-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-30-22" name="__codelineno-30-22" href="#__codelineno-30-22"></a><span class="p">}</span>
 <a id="__codelineno-30-23" name="__codelineno-30-23" href="#__codelineno-30-23"></a>
 <a id="__codelineno-30-24" name="__codelineno-30-24" href="#__codelineno-30-24"></a><span class="cm">/* Push */</span>
 <a id="__codelineno-30-25" name="__codelineno-30-25" href="#__codelineno-30-25"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayStack</span><span class="p">)</span><span class="w"> </span><span class="nx">push</span><span class="p">(</span><span class="nx">v</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-26" name="__codelineno-30-26" href="#__codelineno-30-26"></a><span class="w">    </span><span class="c1">// Slice will automatically expand</span>
+<a id="__codelineno-30-26" name="__codelineno-30-26" href="#__codelineno-30-26"></a><span class="w">    </span><span class="c1">// Appending grows the slice capacity automatically when needed</span>
 <a id="__codelineno-30-27" name="__codelineno-30-27" href="#__codelineno-30-27"></a><span class="w">    </span><span class="nx">s</span><span class="p">.</span><span class="nx">data</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">s</span><span class="p">.</span><span class="nx">data</span><span class="p">,</span><span class="w"> </span><span class="nx">v</span><span class="p">)</span>
 <a id="__codelineno-30-28" name="__codelineno-30-28" href="#__codelineno-30-28"></a><span class="p">}</span>
 <a id="__codelineno-30-29" name="__codelineno-30-29" href="#__codelineno-30-29"></a>
@@ -6108,7 +6185,7 @@
 <a id="__codelineno-30-34" name="__codelineno-30-34" href="#__codelineno-30-34"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">val</span>
 <a id="__codelineno-30-35" name="__codelineno-30-35" href="#__codelineno-30-35"></a><span class="p">}</span>
 <a id="__codelineno-30-36" name="__codelineno-30-36" href="#__codelineno-30-36"></a>
-<a id="__codelineno-30-37" name="__codelineno-30-37" href="#__codelineno-30-37"></a><span class="cm">/* Get stack top element */</span>
+<a id="__codelineno-30-37" name="__codelineno-30-37" href="#__codelineno-30-37"></a><span class="cm">/* Get the top element */</span>
 <a id="__codelineno-30-38" name="__codelineno-30-38" href="#__codelineno-30-38"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayStack</span><span class="p">)</span><span class="w"> </span><span class="nx">peek</span><span class="p">()</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-39" name="__codelineno-30-39" href="#__codelineno-30-39"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-40" name="__codelineno-30-40" href="#__codelineno-30-40"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
@@ -6117,28 +6194,28 @@
 <a id="__codelineno-30-43" name="__codelineno-30-43" href="#__codelineno-30-43"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">val</span>
 <a id="__codelineno-30-44" name="__codelineno-30-44" href="#__codelineno-30-44"></a><span class="p">}</span>
 <a id="__codelineno-30-45" name="__codelineno-30-45" href="#__codelineno-30-45"></a>
-<a id="__codelineno-30-46" name="__codelineno-30-46" href="#__codelineno-30-46"></a><span class="cm">/* Get Slice for printing */</span>
+<a id="__codelineno-30-46" name="__codelineno-30-46" href="#__codelineno-30-46"></a><span class="cm">/* Get a slice for printing */</span>
 <a id="__codelineno-30-47" name="__codelineno-30-47" href="#__codelineno-30-47"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">s</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayStack</span><span class="p">)</span><span class="w"> </span><span class="nx">toSlice</span><span class="p">()</span><span class="w"> </span><span class="p">[]</span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-48" name="__codelineno-30-48" href="#__codelineno-30-48"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">s</span><span class="p">.</span><span class="nx">data</span>
 <a id="__codelineno-30-49" name="__codelineno-30-49" href="#__codelineno-30-49"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_stack.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Stack based on array implementation */</span>
+<div class="highlight"><span class="filename">array_stack.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Stack implemented with an array */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayStack</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">stack</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span>
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a>
 <a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="kd">init</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">        </span><span class="c1">// Initialize list (dynamic array)</span>
+<a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">        </span><span class="c1">// Initialize the list (dynamic array)</span>
 <a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">        </span><span class="n">stack</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
 <a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a>
-<a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="w">        </span><span class="n">stack</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a>
-<a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-17" name="__codelineno-31-17" href="#__codelineno-31-17"></a><span class="w">        </span><span class="n">stack</span><span class="p">.</span><span class="bp">isEmpty</span>
 <a id="__codelineno-31-18" name="__codelineno-31-18" href="#__codelineno-31-18"></a><span class="w">    </span><span class="p">}</span>
@@ -6157,7 +6234,7 @@
 <a id="__codelineno-31-31" name="__codelineno-31-31" href="#__codelineno-31-31"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="bp">removeLast</span><span class="p">()</span>
 <a id="__codelineno-31-32" name="__codelineno-31-32" href="#__codelineno-31-32"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-33" name="__codelineno-31-33" href="#__codelineno-31-33"></a>
-<a id="__codelineno-31-34" name="__codelineno-31-34" href="#__codelineno-31-34"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-31-34" name="__codelineno-31-34" href="#__codelineno-31-34"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-31-35" name="__codelineno-31-35" href="#__codelineno-31-35"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">peek</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-36" name="__codelineno-31-36" href="#__codelineno-31-36"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-37" name="__codelineno-31-37" href="#__codelineno-31-37"></a><span class="w">            </span><span class="bp">fatalError</span><span class="p">(</span><span class="s">"Stack is empty"</span><span class="p">)</span>
@@ -6165,7 +6242,7 @@
 <a id="__codelineno-31-39" name="__codelineno-31-39" href="#__codelineno-31-39"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="bp">last</span><span class="p">!</span>
 <a id="__codelineno-31-40" name="__codelineno-31-40" href="#__codelineno-31-40"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-41" name="__codelineno-31-41" href="#__codelineno-31-41"></a>
-<a id="__codelineno-31-42" name="__codelineno-31-42" href="#__codelineno-31-42"></a><span class="w">    </span><span class="cm">/* Convert List to Array and return */</span>
+<a id="__codelineno-31-42" name="__codelineno-31-42" href="#__codelineno-31-42"></a><span class="w">    </span><span class="cm">/* Return the stack elements as an array */</span>
 <a id="__codelineno-31-43" name="__codelineno-31-43" href="#__codelineno-31-43"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-44" name="__codelineno-31-44" href="#__codelineno-31-44"></a><span class="w">        </span><span class="n">stack</span>
 <a id="__codelineno-31-45" name="__codelineno-31-45" href="#__codelineno-31-45"></a><span class="w">    </span><span class="p">}</span>
@@ -6173,19 +6250,19 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_stack.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Stack based on array implementation */</span>
+<div class="highlight"><span class="filename">array_stack.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Stack implemented with an array */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">ArrayStack</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="n">#stack</span><span class="p">;</span>
 <a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a>
-<a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w">    </span><span class="nx">get</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#stack</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a>
-<a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="w">    </span><span class="nx">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-15" name="__codelineno-32-15" href="#__codelineno-32-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#stack</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-32-16" name="__codelineno-32-16" href="#__codelineno-32-16"></a><span class="w">    </span><span class="p">}</span>
@@ -6201,13 +6278,13 @@
 <a id="__codelineno-32-26" name="__codelineno-32-26" href="#__codelineno-32-26"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#stack</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-32-27" name="__codelineno-32-27" href="#__codelineno-32-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-28" name="__codelineno-32-28" href="#__codelineno-32-28"></a>
-<a id="__codelineno-32-29" name="__codelineno-32-29" href="#__codelineno-32-29"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-32-29" name="__codelineno-32-29" href="#__codelineno-32-29"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-32-30" name="__codelineno-32-30" href="#__codelineno-32-30"></a><span class="w">    </span><span class="nx">top</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-31" name="__codelineno-32-31" href="#__codelineno-32-31"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Stack is empty'</span><span class="p">);</span>
 <a id="__codelineno-32-32" name="__codelineno-32-32" href="#__codelineno-32-32"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#stack</span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="n">#stack</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-32-33" name="__codelineno-32-33" href="#__codelineno-32-33"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-34" name="__codelineno-32-34" href="#__codelineno-32-34"></a>
-<a id="__codelineno-32-35" name="__codelineno-32-35" href="#__codelineno-32-35"></a><span class="w">    </span><span class="cm">/* Return Array */</span>
+<a id="__codelineno-32-35" name="__codelineno-32-35" href="#__codelineno-32-35"></a><span class="w">    </span><span class="cm">/* Return an Array */</span>
 <a id="__codelineno-32-36" name="__codelineno-32-36" href="#__codelineno-32-36"></a><span class="w">    </span><span class="nx">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-37" name="__codelineno-32-37" href="#__codelineno-32-37"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#stack</span><span class="p">;</span>
 <a id="__codelineno-32-38" name="__codelineno-32-38" href="#__codelineno-32-38"></a><span class="w">    </span><span class="p">}</span>
@@ -6215,19 +6292,19 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_stack.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Stack based on array implementation */</span>
+<div class="highlight"><span class="filename">array_stack.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Stack implemented with an array */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">ArrayStack</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="k">private</span><span class="w"> </span><span class="nx">stack</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[];</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a>
-<a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">    </span><span class="nx">get</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">stack</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a>
-<a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">    </span><span class="nx">isEmpty</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">stack</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a><span class="w">    </span><span class="p">}</span>
@@ -6243,13 +6320,13 @@
 <a id="__codelineno-33-26" name="__codelineno-33-26" href="#__codelineno-33-26"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">stack</span><span class="p">.</span><span class="nx">pop</span><span class="p">();</span>
 <a id="__codelineno-33-27" name="__codelineno-33-27" href="#__codelineno-33-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-28" name="__codelineno-33-28" href="#__codelineno-33-28"></a>
-<a id="__codelineno-33-29" name="__codelineno-33-29" href="#__codelineno-33-29"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-33-29" name="__codelineno-33-29" href="#__codelineno-33-29"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-33-30" name="__codelineno-33-30" href="#__codelineno-33-30"></a><span class="w">    </span><span class="nx">top</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-31" name="__codelineno-33-31" href="#__codelineno-33-31"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="ne">Error</span><span class="p">(</span><span class="s1">'Stack is empty'</span><span class="p">);</span>
 <a id="__codelineno-33-32" name="__codelineno-33-32" href="#__codelineno-33-32"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">stack</span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="nx">stack</span><span class="p">.</span><span class="nx">length</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">];</span>
 <a id="__codelineno-33-33" name="__codelineno-33-33" href="#__codelineno-33-33"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-34" name="__codelineno-33-34" href="#__codelineno-33-34"></a>
-<a id="__codelineno-33-35" name="__codelineno-33-35" href="#__codelineno-33-35"></a><span class="w">    </span><span class="cm">/* Return Array */</span>
+<a id="__codelineno-33-35" name="__codelineno-33-35" href="#__codelineno-33-35"></a><span class="w">    </span><span class="cm">/* Return an Array */</span>
 <a id="__codelineno-33-36" name="__codelineno-33-36" href="#__codelineno-33-36"></a><span class="w">    </span><span class="nx">toArray</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-37" name="__codelineno-33-37" href="#__codelineno-33-37"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">stack</span><span class="p">;</span>
 <a id="__codelineno-33-38" name="__codelineno-33-38" href="#__codelineno-33-38"></a><span class="w">    </span><span class="p">}</span>
@@ -6257,19 +6334,19 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_stack.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Stack based on array implementation */</span>
+<div class="highlight"><span class="filename">array_stack.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Stack implemented with an array */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayStack</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">_stack</span><span class="p">;</span>
 <a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="n">ArrayStack</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">    </span><span class="n">_stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a>
-<a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">  </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">  </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_stack</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a>
-<a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">  </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">  </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">  </span><span class="kt">bool</span><span class="w"> </span><span class="n">isEmpty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_stack</span><span class="p">.</span><span class="n">isEmpty</span><span class="p">;</span>
 <a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w">  </span><span class="p">}</span>
@@ -6287,7 +6364,7 @@
 <a id="__codelineno-34-28" name="__codelineno-34-28" href="#__codelineno-34-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_stack</span><span class="p">.</span><span class="n">removeLast</span><span class="p">();</span>
 <a id="__codelineno-34-29" name="__codelineno-34-29" href="#__codelineno-34-29"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-30" name="__codelineno-34-30" href="#__codelineno-34-30"></a>
-<a id="__codelineno-34-31" name="__codelineno-34-31" href="#__codelineno-34-31"></a><span class="w">  </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-34-31" name="__codelineno-34-31" href="#__codelineno-34-31"></a><span class="w">  </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-34-32" name="__codelineno-34-32" href="#__codelineno-34-32"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">peek</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-33" name="__codelineno-34-33" href="#__codelineno-34-33"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-34" name="__codelineno-34-34" href="#__codelineno-34-34"></a><span class="w">      </span><span class="k">throw</span><span class="w"> </span><span class="n">Exception</span><span class="p">(</span><span class="s2">"Stack is empty"</span><span class="p">);</span>
@@ -6295,31 +6372,31 @@
 <a id="__codelineno-34-36" name="__codelineno-34-36" href="#__codelineno-34-36"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_stack</span><span class="p">.</span><span class="n">last</span><span class="p">;</span>
 <a id="__codelineno-34-37" name="__codelineno-34-37" href="#__codelineno-34-37"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-38" name="__codelineno-34-38" href="#__codelineno-34-38"></a>
-<a id="__codelineno-34-39" name="__codelineno-34-39" href="#__codelineno-34-39"></a><span class="w">  </span><span class="cm">/* Convert stack to Array and return */</span>
+<a id="__codelineno-34-39" name="__codelineno-34-39" href="#__codelineno-34-39"></a><span class="w">  </span><span class="cm">/* Return the underlying stack array */</span>
 <a id="__codelineno-34-40" name="__codelineno-34-40" href="#__codelineno-34-40"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">toArray</span><span class="p">()</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">_stack</span><span class="p">;</span>
 <a id="__codelineno-34-41" name="__codelineno-34-41" href="#__codelineno-34-41"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_stack.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Stack based on array implementation */</span>
+<div class="highlight"><span class="filename">array_stack.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Stack implemented with an array */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">struct</span><span class="w"> </span><span class="nc">ArrayStack</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="n">stack</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="p">,</span>
 <a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="p">}</span>
 <a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a>
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="k">impl</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="n">ArrayStack</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="cm">/* Access top of the stack element */</span>
+<a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="cm">/* Initialize the stack */</span>
 <a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">new</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">ArrayStack</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">        </span><span class="n">ArrayStack</span><span class="p">::</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w">            </span><span class="n">stack</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="p">::</span><span class="n">new</span><span class="p">(),</span>
 <a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a>
-<a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">usize</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-16" name="__codelineno-35-16" href="#__codelineno-35-16"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">stack</span><span class="p">.</span><span class="n">len</span><span class="p">()</span>
 <a id="__codelineno-35-17" name="__codelineno-35-17" href="#__codelineno-35-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-18" name="__codelineno-35-18" href="#__codelineno-35-18"></a>
-<a id="__codelineno-35-19" name="__codelineno-35-19" href="#__codelineno-35-19"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-35-19" name="__codelineno-35-19" href="#__codelineno-35-19"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-35-20" name="__codelineno-35-20" href="#__codelineno-35-20"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">is_empty</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">bool</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-21" name="__codelineno-35-21" href="#__codelineno-35-21"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-35-22" name="__codelineno-35-22" href="#__codelineno-35-22"></a><span class="w">    </span><span class="p">}</span>
@@ -6334,7 +6411,7 @@
 <a id="__codelineno-35-31" name="__codelineno-35-31" href="#__codelineno-35-31"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">stack</span><span class="p">.</span><span class="n">pop</span><span class="p">()</span>
 <a id="__codelineno-35-32" name="__codelineno-35-32" href="#__codelineno-35-32"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-33" name="__codelineno-35-33" href="#__codelineno-35-33"></a>
-<a id="__codelineno-35-34" name="__codelineno-35-34" href="#__codelineno-35-34"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-35-34" name="__codelineno-35-34" href="#__codelineno-35-34"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-35-35" name="__codelineno-35-35" href="#__codelineno-35-35"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">peek</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-36" name="__codelineno-35-36" href="#__codelineno-35-36"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">is_empty</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-37" name="__codelineno-35-37" href="#__codelineno-35-37"></a><span class="w">            </span><span class="fm">panic!</span><span class="p">(</span><span class="s">"Stack is empty"</span><span class="p">)</span>
@@ -6342,7 +6419,7 @@
 <a id="__codelineno-35-39" name="__codelineno-35-39" href="#__codelineno-35-39"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">stack</span><span class="p">.</span><span class="n">last</span><span class="p">()</span>
 <a id="__codelineno-35-40" name="__codelineno-35-40" href="#__codelineno-35-40"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-41" name="__codelineno-35-41" href="#__codelineno-35-41"></a>
-<a id="__codelineno-35-42" name="__codelineno-35-42" href="#__codelineno-35-42"></a><span class="w">    </span><span class="cm">/* Return &amp;Vec */</span>
+<a id="__codelineno-35-42" name="__codelineno-35-42" href="#__codelineno-35-42"></a><span class="w">    </span><span class="cm">/* Return a reference to the vector */</span>
 <a id="__codelineno-35-43" name="__codelineno-35-43" href="#__codelineno-35-43"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">to_array</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kp">&amp;</span><span class="nb">Vec</span><span class="o">&lt;</span><span class="n">T</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-44" name="__codelineno-35-44" href="#__codelineno-35-44"></a><span class="w">        </span><span class="o">&amp;</span><span class="bp">self</span><span class="p">.</span><span class="n">stack</span>
 <a id="__codelineno-35-45" name="__codelineno-35-45" href="#__codelineno-35-45"></a><span class="w">    </span><span class="p">}</span>
@@ -6350,7 +6427,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_stack.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Stack based on array implementation */</span>
+<div class="highlight"><span class="filename">array_stack.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Stack implemented with an array */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">data</span><span class="p">;</span>
 <a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span>
@@ -6359,7 +6436,7 @@
 <a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="cm">/* Constructor */</span>
 <a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="n">ArrayStack</span><span class="w"> </span><span class="o">*</span><span class="nf">newArrayStack</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="w">    </span><span class="n">ArrayStack</span><span class="w"> </span><span class="o">*</span><span class="n">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">ArrayStack</span><span class="p">));</span>
-<a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="w">    </span><span class="c1">// Initialize with large capacity to avoid expansion</span>
+<a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="w">    </span><span class="c1">// Initialize with a large capacity to avoid resizing</span>
 <a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="w">    </span><span class="n">stack</span><span class="o">-&gt;</span><span class="n">data</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="p">);</span>
 <a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="w">    </span><span class="n">stack</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">;</span>
@@ -6371,12 +6448,12 @@
 <a id="__codelineno-36-19" name="__codelineno-36-19" href="#__codelineno-36-19"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">stack</span><span class="p">);</span>
 <a id="__codelineno-36-20" name="__codelineno-36-20" href="#__codelineno-36-20"></a><span class="p">}</span>
 <a id="__codelineno-36-21" name="__codelineno-36-21" href="#__codelineno-36-21"></a>
-<a id="__codelineno-36-22" name="__codelineno-36-22" href="#__codelineno-36-22"></a><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-36-22" name="__codelineno-36-22" href="#__codelineno-36-22"></a><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-36-23" name="__codelineno-36-23" href="#__codelineno-36-23"></a><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="n">ArrayStack</span><span class="w"> </span><span class="o">*</span><span class="n">stack</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-24" name="__codelineno-36-24" href="#__codelineno-36-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">;</span>
 <a id="__codelineno-36-25" name="__codelineno-36-25" href="#__codelineno-36-25"></a><span class="p">}</span>
 <a id="__codelineno-36-26" name="__codelineno-36-26" href="#__codelineno-36-26"></a>
-<a id="__codelineno-36-27" name="__codelineno-36-27" href="#__codelineno-36-27"></a><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-36-27" name="__codelineno-36-27" href="#__codelineno-36-27"></a><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-36-28" name="__codelineno-36-28" href="#__codelineno-36-28"></a><span class="kt">bool</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">(</span><span class="n">ArrayStack</span><span class="w"> </span><span class="o">*</span><span class="n">stack</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-29" name="__codelineno-36-29" href="#__codelineno-36-29"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-36-30" name="__codelineno-36-30" href="#__codelineno-36-30"></a><span class="p">}</span>
@@ -6391,7 +6468,7 @@
 <a id="__codelineno-36-39" name="__codelineno-36-39" href="#__codelineno-36-39"></a><span class="w">    </span><span class="n">stack</span><span class="o">-&gt;</span><span class="n">size</span><span class="o">++</span><span class="p">;</span>
 <a id="__codelineno-36-40" name="__codelineno-36-40" href="#__codelineno-36-40"></a><span class="p">}</span>
 <a id="__codelineno-36-41" name="__codelineno-36-41" href="#__codelineno-36-41"></a>
-<a id="__codelineno-36-42" name="__codelineno-36-42" href="#__codelineno-36-42"></a><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-36-42" name="__codelineno-36-42" href="#__codelineno-36-42"></a><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-36-43" name="__codelineno-36-43" href="#__codelineno-36-43"></a><span class="kt">int</span><span class="w"> </span><span class="nf">peek</span><span class="p">(</span><span class="n">ArrayStack</span><span class="w"> </span><span class="o">*</span><span class="n">stack</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-44" name="__codelineno-36-44" href="#__codelineno-36-44"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">stack</span><span class="o">-&gt;</span><span class="n">size</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-45" name="__codelineno-36-45" href="#__codelineno-36-45"></a><span class="w">        </span><span class="n">printf</span><span class="p">(</span><span class="s">"Stack is empty</span><span class="se">\n</span><span class="s">"</span><span class="p">);</span>
@@ -6409,17 +6486,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_stack.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Stack based on array implementation */</span>
+<div class="highlight"><span class="filename">array_stack.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Stack implemented with an array */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayStack</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="c1">// Initialize list (dynamic array)</span>
+<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="c1">// Initialize the list (dynamic array)</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">val</span><span class="w"> </span><span class="nv">stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a>
-<a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="cm">/* Get the length of the stack */</span>
+<a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="cm">/* Get the stack size */</span>
 <a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">size</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a>
-<a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="w">    </span><span class="cm">/* Check if the stack is empty */</span>
+<a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="w">    </span><span class="cm">/* Check whether the stack is empty */</span>
 <a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">isEmpty</span><span class="p">():</span><span class="w"> </span><span class="kt">Boolean</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="w">    </span><span class="p">}</span>
@@ -6435,13 +6512,13 @@
 <a id="__codelineno-37-24" name="__codelineno-37-24" href="#__codelineno-37-24"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">removeAt</span><span class="p">(</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-37-25" name="__codelineno-37-25" href="#__codelineno-37-25"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-26" name="__codelineno-37-26" href="#__codelineno-37-26"></a>
-<a id="__codelineno-37-27" name="__codelineno-37-27" href="#__codelineno-37-27"></a><span class="w">    </span><span class="cm">/* Return list for printing */</span>
+<a id="__codelineno-37-27" name="__codelineno-37-27" href="#__codelineno-37-27"></a><span class="w">    </span><span class="cm">/* Access the top element */</span>
 <a id="__codelineno-37-28" name="__codelineno-37-28" href="#__codelineno-37-28"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">peek</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-29" name="__codelineno-37-29" href="#__codelineno-37-29"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="k">throw</span><span class="w"> </span><span class="n">IndexOutOfBoundsException</span><span class="p">()</span>
 <a id="__codelineno-37-30" name="__codelineno-37-30" href="#__codelineno-37-30"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="o">[</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="o">]</span>
 <a id="__codelineno-37-31" name="__codelineno-37-31" href="#__codelineno-37-31"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-32" name="__codelineno-37-32" href="#__codelineno-37-32"></a>
-<a id="__codelineno-37-33" name="__codelineno-37-33" href="#__codelineno-37-33"></a><span class="w">    </span><span class="cm">/* Convert List to Array and return */</span>
+<a id="__codelineno-37-33" name="__codelineno-37-33" href="#__codelineno-37-33"></a><span class="w">    </span><span class="cm">/* Convert the list to an array and return it */</span>
 <a id="__codelineno-37-34" name="__codelineno-37-34" href="#__codelineno-37-34"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">toArray</span><span class="p">():</span><span class="w"> </span><span class="n">Array</span><span class="o">&lt;</span><span class="kt">Any</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-35" name="__codelineno-37-35" href="#__codelineno-37-35"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">stack</span><span class="p">.</span><span class="na">toTypedArray</span><span class="p">()</span>
 <a id="__codelineno-37-36" name="__codelineno-37-36" href="#__codelineno-37-36"></a><span class="w">    </span><span class="p">}</span>
@@ -6449,19 +6526,19 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_stack.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Stack based on array ###</span>
+<div class="highlight"><span class="filename">array_stack.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Stack implemented with an array ###</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayStack</span>
 <a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1">### Constructor ###</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">initialize</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="w">    </span><span class="vi">@stack</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a>
-<a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">  </span><span class="c1">### Get stack length ###</span>
+<a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">  </span><span class="c1">### Get the stack size ###</span>
 <a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">size</span>
 <a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">    </span><span class="vi">@stack</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a>
-<a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">  </span><span class="c1">### Check if stack is empty ###</span>
+<a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">  </span><span class="c1">### Check whether the stack is empty ###</span>
 <a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">is_empty?</span>
 <a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a><span class="w">    </span><span class="vi">@stack</span><span class="o">.</span><span class="n">empty?</span>
 <a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w">  </span><span class="k">end</span>
@@ -6478,14 +6555,14 @@
 <a id="__codelineno-38-27" name="__codelineno-38-27" href="#__codelineno-38-27"></a><span class="w">    </span><span class="vi">@stack</span><span class="o">.</span><span class="n">pop</span>
 <a id="__codelineno-38-28" name="__codelineno-38-28" href="#__codelineno-38-28"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-29" name="__codelineno-38-29" href="#__codelineno-38-29"></a>
-<a id="__codelineno-38-30" name="__codelineno-38-30" href="#__codelineno-38-30"></a><span class="w">  </span><span class="c1">### Access top element ###</span>
+<a id="__codelineno-38-30" name="__codelineno-38-30" href="#__codelineno-38-30"></a><span class="w">  </span><span class="c1">### Access the top element ###</span>
 <a id="__codelineno-38-31" name="__codelineno-38-31" href="#__codelineno-38-31"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">peek</span>
 <a id="__codelineno-38-32" name="__codelineno-38-32" href="#__codelineno-38-32"></a><span class="w">    </span><span class="k">raise</span><span class="w"> </span><span class="no">IndexError</span><span class="p">,</span><span class="w"> </span><span class="s1">'Stack is empty'</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">is_empty?</span>
 <a id="__codelineno-38-33" name="__codelineno-38-33" href="#__codelineno-38-33"></a>
 <a id="__codelineno-38-34" name="__codelineno-38-34" href="#__codelineno-38-34"></a><span class="w">    </span><span class="vi">@stack</span><span class="o">.</span><span class="n">last</span>
 <a id="__codelineno-38-35" name="__codelineno-38-35" href="#__codelineno-38-35"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-36" name="__codelineno-38-36" href="#__codelineno-38-36"></a>
-<a id="__codelineno-38-37" name="__codelineno-38-37" href="#__codelineno-38-37"></a><span class="w">  </span><span class="c1">### Return list for printing ###</span>
+<a id="__codelineno-38-37" name="__codelineno-38-37" href="#__codelineno-38-37"></a><span class="w">  </span><span class="c1">### Return a list for printing ###</span>
 <a id="__codelineno-38-38" name="__codelineno-38-38" href="#__codelineno-38-38"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">to_array</span>
 <a id="__codelineno-38-39" name="__codelineno-38-39" href="#__codelineno-38-39"></a><span class="w">    </span><span class="vi">@stack</span>
 <a id="__codelineno-38-40" name="__codelineno-38-40" href="#__codelineno-38-40"></a><span class="w">  </span><span class="k">end</span>
@@ -6494,25 +6571,30 @@
 </div>
 </div>
 </div>
-<h2 id="513-comparison-of-the-two-implementations">5.1.3 &nbsp; Comparison of the Two Implementations<a class="headerlink" href="#513-comparison-of-the-two-implementations" title="Permanent link">&para;</a></h2>
-<p><strong>Supported Operations</strong></p>
-<p>Both implementations support all operations defined by the stack. The array implementation additionally supports random access, but this goes beyond the stack definition and is generally not used.</p>
-<p><strong>Time Efficiency</strong></p>
-<p>In the array-based implementation, both push and pop operations occur in pre-allocated contiguous memory, which has good cache locality and is therefore more efficient. However, if pushing exceeds the array capacity, it triggers an expansion mechanism, causing the time complexity of that particular push operation to become <span class="arithmatex">\(O(n)\)</span>.</p>
-<p>In the linked list-based implementation, list expansion is very flexible, and there is no issue of reduced efficiency due to array expansion. However, the push operation requires initializing a node object and modifying pointers, so it is relatively less efficient. Nevertheless, if the pushed elements are already node objects, the initialization step can be omitted, thereby improving efficiency.</p>
-<p>In summary, when the elements pushed and popped are basic data types such as <code>int</code> or <code>double</code>, we can draw the following conclusions:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20ArrayStack%3A%0A%20%20%20%20%22%22%22Stack%20implemented%20with%20an%20array%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self._stack%3A%20list%5Bint%5D%20%3D%20%5B%5D%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20stack%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self._stack%29%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22Check%20whether%20the%20stack%20is%20empty%22%22%22%0A%20%20%20%20%20%20%20%20return%20self._stack%20%3D%3D%20%5B%5D%0A%0A%20%20%20%20def%20push%28self,%20item%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Push%22%22%22%0A%20%20%20%20%20%20%20%20self._stack.append%28item%29%0A%0A%20%20%20%20def%20pop%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Pop%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.is_empty%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22Stack%20is%20empty%22%29%0A%20%20%20%20%20%20%20%20return%20self._stack.pop%28%29%0A%0A%20%20%20%20def%20peek%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Access%20the%20top%20element%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.is_empty%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22Stack%20is%20empty%22%29%0A%20%20%20%20%20%20%20%20return%20self._stack%5B-1%5D%0A%0A%20%20%20%20def%20to_list%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Return%20a%20list%20for%20printing%22%22%22%0A%20%20%20%20%20%20%20%20return%20self._stack%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20stack%0A%20%20%20%20stack%20%3D%20ArrayStack%28%29%0A%0A%20%20%20%20%23%20Push%20elements%20onto%20the%20stack%0A%20%20%20%20stack.push%281%29%0A%20%20%20%20stack.push%283%29%0A%20%20%20%20stack.push%282%29%0A%20%20%20%20stack.push%285%29%0A%20%20%20%20stack.push%284%29%0A%20%20%20%20print%28%22Stack%20stack%20%3D%22,%20stack.to_list%28%29%29%0A%0A%20%20%20%20%23%20Access%20the%20top%20element%0A%20%20%20%20peek%20%3D%20stack.peek%28%29%0A%20%20%20%20print%28%22Top%20element%20peek%20%3D%22,%20peek%29%0A%0A%20%20%20%20%23%20Pop%20an%20element%20from%20the%20stack%0A%20%20%20%20pop%20%3D%20stack.pop%28%29%0A%20%20%20%20print%28%22Popped%20element%20pop%20%3D%22,%20pop%29%0A%20%20%20%20print%28%22After%20popping,%20stack%20%3D%22,%20stack.to_list%28%29%29%0A%0A%20%20%20%20%23%20Get%20the%20stack%20size%0A%20%20%20%20size%20%3D%20stack.size%28%29%0A%20%20%20%20print%28%22Stack%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20Check%20whether%20it%20is%20empty%0A%20%20%20%20is_empty%20%3D%20stack.is_empty%28%29%0A%20%20%20%20print%28%22Is%20the%20stack%20empty%20%3D%22,%20is_empty%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ArrayStack%3A%0A%20%20%20%20%22%22%22Stack%20implemented%20with%20an%20array%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self._stack%3A%20list%5Bint%5D%20%3D%20%5B%5D%0A%0A%20%20%20%20def%20size%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20stack%20size%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self._stack%29%0A%0A%20%20%20%20def%20is_empty%28self%29%20-%3E%20bool%3A%0A%20%20%20%20%20%20%20%20%22%22%22Check%20whether%20the%20stack%20is%20empty%22%22%22%0A%20%20%20%20%20%20%20%20return%20self._stack%20%3D%3D%20%5B%5D%0A%0A%20%20%20%20def%20push%28self,%20item%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Push%22%22%22%0A%20%20%20%20%20%20%20%20self._stack.append%28item%29%0A%0A%20%20%20%20def%20pop%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Pop%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.is_empty%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22Stack%20is%20empty%22%29%0A%20%20%20%20%20%20%20%20return%20self._stack.pop%28%29%0A%0A%20%20%20%20def%20peek%28self%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Access%20the%20top%20element%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.is_empty%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20raise%20IndexError%28%22Stack%20is%20empty%22%29%0A%20%20%20%20%20%20%20%20return%20self._stack%5B-1%5D%0A%0A%20%20%20%20def%20to_list%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Return%20a%20list%20for%20printing%22%22%22%0A%20%20%20%20%20%20%20%20return%20self._stack%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20stack%0A%20%20%20%20stack%20%3D%20ArrayStack%28%29%0A%0A%20%20%20%20%23%20Push%20elements%20onto%20the%20stack%0A%20%20%20%20stack.push%281%29%0A%20%20%20%20stack.push%283%29%0A%20%20%20%20stack.push%282%29%0A%20%20%20%20stack.push%285%29%0A%20%20%20%20stack.push%284%29%0A%20%20%20%20print%28%22Stack%20stack%20%3D%22,%20stack.to_list%28%29%29%0A%0A%20%20%20%20%23%20Access%20the%20top%20element%0A%20%20%20%20peek%20%3D%20stack.peek%28%29%0A%20%20%20%20print%28%22Top%20element%20peek%20%3D%22,%20peek%29%0A%0A%20%20%20%20%23%20Pop%20an%20element%20from%20the%20stack%0A%20%20%20%20pop%20%3D%20stack.pop%28%29%0A%20%20%20%20print%28%22Popped%20element%20pop%20%3D%22,%20pop%29%0A%20%20%20%20print%28%22After%20popping,%20stack%20%3D%22,%20stack.to_list%28%29%29%0A%0A%20%20%20%20%23%20Get%20the%20stack%20size%0A%20%20%20%20size%20%3D%20stack.size%28%29%0A%20%20%20%20print%28%22Stack%20size%20%3D%22,%20size%29%0A%0A%20%20%20%20%23%20Check%20whether%20it%20is%20empty%0A%20%20%20%20is_empty%20%3D%20stack.is_empty%28%29%0A%20%20%20%20print%28%22Is%20the%20stack%20empty%20%3D%22,%20is_empty%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="513-comparing-the-implementations">5.1.3 &nbsp; Comparing the implementations<a class="headerlink" href="#513-comparing-the-implementations" title="Permanent link">&para;</a></h2>
+<p><strong>Supported operations</strong></p>
+<p>Both implementations support all stack operations. An array also allows random access, but that is outside the stack abstraction and is generally not used.</p>
+<p><strong>Time efficiency</strong></p>
+<p>In an array-based stack, pushes and pops operate in preallocated contiguous memory, giving good cache locality and high efficiency. However, a push that exceeds the array's capacity triggers expansion, making that particular push an <span class="arithmatex">\(O(n)\)</span> operation.</p>
+<p>A linked list grows flexibly and avoids the slowdown associated with array expansion. However, pushing requires creating a node object and updating pointers, so it is relatively slower. If the element is already a node object, the creation step can be skipped to improve efficiency.</p>
+<p>For elements of primitive types such as <code>int</code> or <code>double</code>, we can therefore draw the following conclusions.</p>
 <ul>
-<li>The array-based stack implementation has reduced efficiency when expansion is triggered, but since expansion is an infrequent operation, the average efficiency is higher.</li>
-<li>The linked list-based stack implementation can provide more stable efficiency performance.</li>
+<li>An array-based stack slows down when expansion is triggered, but expansion is infrequent, so average efficiency is higher.</li>
+<li>A linked-list stack provides more consistent performance.</li>
 </ul>
-<p><strong>Space Efficiency</strong></p>
-<p>When initializing a list, the system allocates an "initial capacity" that may exceed the actual need. Additionally, the expansion mechanism typically expands at a specific ratio (e.g., 2x), and the capacity after expansion may also exceed actual needs. Therefore, <strong>the array-based stack implementation may cause some space wastage</strong>.</p>
-<p>However, since linked list nodes need to store additional pointers, <strong>the space occupied by linked list nodes is relatively large</strong>.</p>
-<p>In summary, we cannot simply determine which implementation is more memory-efficient and need to analyze the specific situation.</p>
-<h2 id="514-typical-applications-of-stack">5.1.4 &nbsp; Typical Applications of Stack<a class="headerlink" href="#514-typical-applications-of-stack" title="Permanent link">&para;</a></h2>
+<p><strong>Space efficiency</strong></p>
+<p>When a dynamic array is initialized, its initial capacity may exceed actual needs. Expansion typically multiplies the capacity by a fixed factor, such as 2, which may again leave excess space. <strong>An array-based stack can therefore waste some space</strong>.</p>
+<p>However, linked-list nodes also store pointers, <strong>so each node occupies relatively more space</strong>.</p>
+<p>We cannot say universally which implementation uses less memory; the answer depends on the situation.</p>
+<h2 id="514-typical-applications">5.1.4 &nbsp; Typical applications<a class="headerlink" href="#514-typical-applications" title="Permanent link">&para;</a></h2>
 <ul>
-<li><strong>Back and forward in browsers, undo and redo in software</strong>. Every time we open a new webpage, the browser pushes the previous page onto the stack, allowing us to return to the previous page via the back operation. The back operation is essentially performing a pop. To support both back and forward, two stacks are needed to work together.</li>
-<li><strong>Program memory management</strong>. Each time a function is called, the system adds a stack frame to the top of the stack to record the function's context information. During recursion, the downward recursive phase continuously performs push operations, while the upward backtracking phase continuously performs pop operations.</li>
+<li><strong>Browser back and forward navigation, and software undo and redo</strong>. When a new web page is opened, the browser pushes the previous page onto a stack. Going back pops that page. Supporting both back and forward requires two stacks working together.</li>
+<li><strong>Program memory management</strong>. Each function call adds a stack frame containing the function's context. During recursion, frames are pushed as calls descend and popped as they return.</li>
 </ul>
 
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   <a
     href="../"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: Chapter 5. &amp;nbsp; Stacks and Queues"
+    aria-label="Previous: Chapter 5.   Stacks and queues"
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-        Chapter 5. &nbsp; Stacks and Queues
+        Chapter 5. &nbsp; Stacks and queues
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   <a
     href="../queue/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 5.2 Queue"
+    aria-label="Next: 5.2   Queues"
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         Next
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+        5.2 &nbsp; Queues
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@@ -6585,7 +6667,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
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@@ -67,8 +67,8 @@
 
 
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-  Before Starting
+  Foreword
 
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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-  Chapter 0. Preface
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-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
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   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
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@@ -1265,7 +1265,7 @@
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-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.3 List
+  4.3 &nbsp; Lists
 
 
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-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
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@@ -1373,7 +1373,7 @@
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-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
       </span>
     </span>
   </a>
@@ -1685,7 +1685,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1791,7 +1791,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1872,7 +1872,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2009,7 +2009,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -2036,7 +2036,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2063,7 +2063,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2090,7 +2090,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2117,7 +2117,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2144,7 +2144,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2171,7 +2171,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2235,7 +2235,7 @@
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2256,7 +2256,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2277,7 +2277,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2304,7 +2304,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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@@ -2331,7 +2331,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2358,7 +2358,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2385,7 +2385,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2449,7 +2449,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2491,7 +2491,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2518,7 +2518,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2545,7 +2545,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2572,7 +2572,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2599,7 +2599,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2688,7 +2688,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2709,7 +2709,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2736,7 +2736,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2763,7 +2763,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2790,7 +2790,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2817,7 +2817,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2844,7 +2844,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2871,7 +2871,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2970,7 +2970,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2991,7 +2991,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3018,7 +3018,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3045,7 +3045,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3072,7 +3072,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3099,7 +3099,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3126,7 +3126,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3153,7 +3153,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3180,7 +3180,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3207,7 +3207,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3234,7 +3234,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3261,7 +3261,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3288,7 +3288,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3396,7 +3396,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3423,7 +3423,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3450,7 +3450,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3477,7 +3477,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3504,7 +3504,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3639,7 +3639,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3844,7 +3844,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3886,7 +3886,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3967,7 +3967,7 @@
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4021,7 +4021,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4141,7 +4141,7 @@
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4162,7 +4162,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4183,7 +4183,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4264,7 +4264,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4378,7 +4378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4420,7 +4420,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4447,7 +4447,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4474,7 +4474,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4558,6 +4558,78 @@
 
 
 
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+          </div>
+
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
+            <span class="md-nav__icon md-icon"></span>
+
+
+  Print edition (Chinese)
+
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+          <ul class="md-nav__list" data-md-scrollfix>
+
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+        </nav>
+
+    </li>
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@@ -4599,10 +4671,10 @@
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         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
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   </a>
@@ -4658,27 +4730,27 @@
 
 <!-- Page content -->
 <h1 id="54-summary">5.4 &nbsp; Summary<a class="headerlink" href="#54-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
 <ul>
-<li>A stack is a data structure that follows the LIFO principle and can be implemented using arrays or linked lists.</li>
-<li>In terms of time efficiency, the array implementation of a stack has higher average efficiency, but during expansion, the time complexity of a single push operation degrades to <span class="arithmatex">\(O(n)\)</span>. In contrast, the linked-list implementation of a stack offers more stable performance.</li>
-<li>In terms of space efficiency, the array implementation of a stack may lead to some degree of space wastage. However, it should be noted that the memory space occupied by linked list nodes is larger than that of array elements.</li>
-<li>A queue is a data structure that follows the FIFO principle and can also be implemented using arrays or linked lists. The conclusions regarding time efficiency and space efficiency comparisons for queues are similar to those for stacks mentioned above.</li>
-<li>A deque is a queue with greater flexibility that allows adding and removing elements at both ends.</li>
+<li>A stack follows last-in, first-out order and can be implemented with an array or linked list.</li>
+<li>An array-based stack offers higher average time efficiency, but a push that triggers expansion takes <span class="arithmatex">\(O(n)\)</span> time. A linked-list stack provides more consistent performance.</li>
+<li>An array-based stack may waste some space. However, linked-list nodes require more memory than individual array elements.</li>
+<li>A queue follows first-in, first-out order and can also be implemented with an array or linked list. Their time and space tradeoffs resemble those of stack implementations.</li>
+<li>A deque is a more flexible queue that allows insertion and removal at both ends.</li>
 </ul>
 <h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
-<p><strong>Q</strong>: Is the browser's forward and backward functionality implemented with a doubly linked list?</p>
-<p>The browser's forward and backward behavior is essentially an application of a "stack." When a user visits a new page, that page is added to the top of the stack; when the user clicks the back button, that page is popped from the top of the stack. A deque can conveniently support some additional operations, as mentioned in the "Deque" section.</p>
-<p><strong>Q</strong>: After popping from the stack, do we need to free the memory of the popped node?</p>
-<p>If the popped node will still be needed later, then memory does not need to be freed. If it won't be used afterward, languages like Java and Python have automatic garbage collection, so manual memory deallocation is not required; in C and C++, manual memory deallocation is necessary.</p>
-<p><strong>Q</strong>: A deque seems like two stacks joined together. What is its purpose?</p>
-<p>A deque is like a combination of a stack and a queue, or two stacks joined together. It combines the logic of both, so it can support all applications of stacks and queues while offering greater flexibility.</p>
-<p><strong>Q</strong>: How are undo and redo specifically implemented?</p>
-<p>Use two stacks: stack <code>A</code> for undo and stack <code>B</code> for redo.</p>
+<p><strong>Q</strong>: Are browser back and forward navigation implemented with a doubly linked list?</p>
+<p>Browser navigation fundamentally reflects stack behavior. Visiting a new page adds it to the top of a stack; clicking Back removes it. A deque conveniently supports additional operations, as discussed in the deque section.</p>
+<p><strong>Q</strong>: After popping a node, do we need to free its memory?</p>
+<p>If the node will still be used, do not free it. If it is no longer needed, languages such as <code>Java</code> and <code>Python</code> use automatic garbage collection, so manual deallocation is unnecessary. In <code>C</code> and <code>C++</code>, memory must be released manually.</p>
+<p><strong>Q</strong>: A deque looks like two stacks joined together. What is it useful for?</p>
+<p>A deque resembles a combination of a stack and a queue, or two stacks joined together. It combines stack and queue behavior, supporting all their applications with greater flexibility.</p>
+<p><strong>Q</strong>: How exactly do undo and redo work?</p>
+<p>Use two stacks: <code>A</code> for undo and <code>B</code> for redo.</p>
 <ol>
-<li>Whenever the user performs an operation, push this operation onto stack <code>A</code> and clear stack <code>B</code>.</li>
-<li>When the user performs "undo," pop the most recent operation from stack <code>A</code> and push it onto stack <code>B</code>.</li>
-<li>When the user performs "redo," pop the most recent operation from stack <code>B</code> and push it onto stack <code>A</code>.</li>
+<li>Whenever the user performs an action, push it onto <code>A</code> and clear <code>B</code>.</li>
+<li>To undo, pop the most recent action from <code>A</code> and push it onto <code>B</code>.</li>
+<li>To redo, pop the most recent action from <code>B</code> and push it onto <code>A</code>.</li>
 </ol>
 
 <!-- Source file information -->
@@ -4702,7 +4774,7 @@ aria-label="Footer"
   <a
     href="../deque/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 5.3 Deque"
+    aria-label="Previous: 5.3   Deques"
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@@ -4714,7 +4786,7 @@ aria-label="Footer"
         Previous
       </span>
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-        5.3 Deque
+        5.3 &nbsp; Deques
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@@ -4726,7 +4798,7 @@ aria-label="Footer"
   <a
     href="../exercises/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 5.5 Exercises"
+    aria-label="Next: 5.5   Exercises"
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@@ -4751,7 +4823,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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diff --git a/en/chapter_tree/array_representation_of_tree/index.html b/en/chapter_tree/array_representation_of_tree/index.html
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       <meta charset="utf-8">
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>7.3 Array Representation of Binary Trees - Hello Algo</title>
+        <title>7.3 Array representation of a binary tree - Hello Algo</title>
 
 
 
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-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
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+        <a href="#73-array-representation-of-a-binary-tree" class="md-skip">
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@@ -119,7 +119,7 @@
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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-              7.3 &nbsp; Array Representation of Binary Trees
+              7.3 &nbsp; Array representation of a binary tree
 
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   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
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-  Before Starting
+  Foreword
 
           </label>
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-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
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-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
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-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
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-  Chapter 1. Encounter with Algorithms
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-  1.1 Algorithms Are Everywhere
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-  1.2 What Is an Algorithm
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-  2.6 Exercises
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-  3.1 Classification of Data Structures
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-  3.2 Basic Data Types
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+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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+  <a href="#731-representing-a-perfect-binary-tree" class="md-nav__link">
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-        7.3.1 &nbsp; Representing Perfect Binary Trees
+        7.3.1 &nbsp; Representing a perfect binary tree
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-  <a href="#732-representing-any-binary-tree" class="md-nav__link">
+  <a href="#732-representing-an-arbitrary-binary-tree" class="md-nav__link">
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-        7.3.2 &nbsp; Representing Any Binary Tree
+        7.3.2 &nbsp; Representing an arbitrary binary tree
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-  <a href="#733-advantages-and-limitations" class="md-nav__link">
+  <a href="#733-strengths-and-limitations" class="md-nav__link">
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-        7.3.3 &nbsp; Advantages and Limitations
+        7.3.3 &nbsp; Strengths and limitations
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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+  16.3 &nbsp; Glossary
 
 
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         <li class="md-nav__item">
-  <a href="#731-representing-perfect-binary-trees" class="md-nav__link">
+  <a href="#731-representing-a-perfect-binary-tree" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.3.1 &nbsp; Representing Perfect Binary Trees
+        7.3.1 &nbsp; Representing a perfect binary tree
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-  <a href="#732-representing-any-binary-tree" class="md-nav__link">
+  <a href="#732-representing-an-arbitrary-binary-tree" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
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+        7.3.2 &nbsp; Representing an arbitrary binary tree
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   </a>
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-  <a href="#733-advantages-and-limitations" class="md-nav__link">
+  <a href="#733-strengths-and-limitations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.3.3 &nbsp; Advantages and Limitations
+        7.3.3 &nbsp; Strengths and limitations
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 <!-- Page content -->
-<h1 id="73-array-representation-of-binary-trees">7.3 &nbsp; Array Representation of Binary Trees<a class="headerlink" href="#73-array-representation-of-binary-trees" title="Permanent link">&para;</a></h1>
-<p>In the linked-list representation, the storage unit of a binary tree is a node <code>TreeNode</code>, and nodes are connected by pointers. The previous section introduced the basic operations of binary trees in this representation.</p>
-<p>So, can we use an array to represent a binary tree? The answer is yes.</p>
-<h2 id="731-representing-perfect-binary-trees">7.3.1 &nbsp; Representing Perfect Binary Trees<a class="headerlink" href="#731-representing-perfect-binary-trees" title="Permanent link">&para;</a></h2>
-<p>Let's analyze a simple case first. Given a perfect binary tree, we store all nodes in an array according to the order of level-order traversal, where each node corresponds to a unique array index.</p>
-<p>Based on the characteristics of level-order traversal, we can derive a "mapping formula" between parent node index and child node indices: <strong>If a node's index is <span class="arithmatex">\(i\)</span>, then its left child index is <span class="arithmatex">\(2i + 1\)</span> and its right child index is <span class="arithmatex">\(2i + 2\)</span></strong>. Figure 7-12 shows the mapping relationships between various node indices.</p>
+<h1 id="73-array-representation-of-a-binary-tree">7.3 &nbsp; Array representation of a binary tree<a class="headerlink" href="#73-array-representation-of-a-binary-tree" title="Permanent link">&para;</a></h1>
+<p>In a linked representation, a binary tree stores data in <code>TreeNode</code> nodes connected by pointers. The previous section introduced basic operations on binary trees in this form.</p>
+<p>Can we represent a binary tree with an array instead? Yes.</p>
+<h2 id="731-representing-a-perfect-binary-tree">7.3.1 &nbsp; Representing a perfect binary tree<a class="headerlink" href="#731-representing-a-perfect-binary-tree" title="Permanent link">&para;</a></h2>
+<p>Start with a simple case. Store every node of a perfect binary tree in an array in level order. Each node then corresponds to a unique array index.</p>
+<p>Level-order traversal gives us a mapping between parent and child indices: <strong>if a node has index <span class="arithmatex">\(i\)</span>, its left child has index <span class="arithmatex">\(2i + 1\)</span> and its right child has index <span class="arithmatex">\(2i + 2\)</span></strong>. Figure 7-12 illustrates this relationship.</p>
 <p><img alt="Array representation of a perfect binary tree" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_binary_tree.png" /></p>
 <p align="center"> Figure 7-12 &nbsp; Array representation of a perfect binary tree </p>
 
-<p><strong>The mapping formula plays a role similar to the node references (pointers) in linked lists</strong>. Given any node in the array, we can access its left (right) child node using the mapping formula.</p>
-<h2 id="732-representing-any-binary-tree">7.3.2 &nbsp; Representing Any Binary Tree<a class="headerlink" href="#732-representing-any-binary-tree" title="Permanent link">&para;</a></h2>
-<p>Perfect binary trees are a special case; in the middle levels of a binary tree, there are typically many <code>None</code> values. Since the level-order traversal sequence does not include these <code>None</code> values, we cannot infer the number and distribution of <code>None</code> values based on this sequence alone. <strong>This means multiple binary tree structures can correspond to the same level-order traversal sequence</strong>.</p>
-<p>As shown in Figure 7-13, given a non-perfect binary tree, the above method of array representation fails.</p>
-<p><img alt="Level-order traversal sequence corresponds to multiple binary tree possibilities" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_without_empty.png" /></p>
-<p align="center"> Figure 7-13 &nbsp; Level-order traversal sequence corresponds to multiple binary tree possibilities </p>
+<p><strong>The mapping formulas serve the same role as node references or pointers in a linked representation</strong>. Given any node in the array, we can use them to access its left or right child.</p>
+<h2 id="732-representing-an-arbitrary-binary-tree">7.3.2 &nbsp; Representing an arbitrary binary tree<a class="headerlink" href="#732-representing-an-arbitrary-binary-tree" title="Permanent link">&para;</a></h2>
+<p>A perfect binary tree is a special case. Other binary trees often contain many missing nodes, <code>None</code>, at intermediate levels. Ordinary level-order traversal omits these positions, so its sequence alone cannot tell us their number or locations. <strong>Multiple tree structures can therefore produce the same level-order sequence</strong>.</p>
+<p>As shown below, the array representation above no longer works for a non-perfect binary tree.</p>
+<p><img alt="Multiple binary trees can share a level-order sequence" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_without_empty.png" /></p>
+<p align="center"> Figure 7-13 &nbsp; Multiple binary trees can share a level-order sequence </p>
 
-<p>To solve this problem, <strong>we can explicitly write out all <code>None</code> values in the level-order traversal sequence</strong>. As shown in Figure 7-14, once we do this, the level-order traversal sequence can uniquely represent a binary tree. Example code is as follows:</p>
+<p>To solve this, <strong>include every <code>None</code> position explicitly in the level-order sequence</strong>. The sequence then represents the tree uniquely, as shown below. Example code follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Array representation of a binary tree</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="c1"># Using None to represent empty slots</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="c1"># Use None for empty positions</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="n">tree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="mi">15</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Array representation of a binary tree */</span>
-<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="c1">// Using the maximum integer value INT_MAX to mark empty slots</span>
+<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="c1">// Use INT_MAX, the maximum int value, for empty positions</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="mi">12</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="mi">15</span><span class="p">};</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Array representation of a binary tree */</span>
-<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="c1">// Using the Integer wrapper class allows for using null to mark empty slots</span>
+<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="c1">// Use the wrapper class Integer so that null can mark empty positions</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="n">Integer</span><span class="o">[]</span><span class="w"> </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mi">12</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mi">15</span><span class="w"> </span><span class="p">};</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Array representation of a binary tree */</span>
-<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="c1">// Using nullable int (int?) allows for using null to mark empty slots</span>
+<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="c1">// Use the nullable type int? so that null can mark empty positions</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="kt">int?</span><span class="p">[]</span><span class="w"> </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="mi">12</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="k">null</span><span class="p">,</span><span class="w"> </span><span class="mi">15</span><span class="p">];</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Array representation of a binary tree */</span>
-<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="c1">// Using an any type slice, allowing for nil to mark empty slots</span>
+<a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="c1">// Use a slice of any so that nil can mark empty positions</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="nx">tree</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="p">[]</span><span class="kt">any</span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="mi">12</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="mi">15</span><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Array representation of a binary tree */</span>
-<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Using optional Int (Int?) allows for using nil to mark empty slots</span>
+<a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="c1">// Use the optional type Int? so that nil can mark empty positions</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="kd">let</span><span class="w"> </span><span class="nv">tree</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">?]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="mi">12</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="mi">15</span><span class="p">]</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Array representation of a binary tree */</span>
-<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="c1">// Using null to represent empty slots</span>
+<a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="c1">// Use null for empty positions</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="kd">let</span><span class="w"> </span><span class="nx">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="mf">2</span><span class="p">,</span><span class="w"> </span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">4</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">6</span><span class="p">,</span><span class="w"> </span><span class="mf">7</span><span class="p">,</span><span class="w"> </span><span class="mf">8</span><span class="p">,</span><span class="w"> </span><span class="mf">9</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">12</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">15</span><span class="p">];</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Array representation of a binary tree */</span>
-<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="c1">// Using null to represent empty slots</span>
+<a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="c1">// Use null for empty positions</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="kd">let</span><span class="w"> </span><span class="nx">tree</span><span class="o">:</span><span class="w"> </span><span class="p">(</span><span class="kt">number</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">)[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="mf">1</span><span class="p">,</span><span class="w"> </span><span class="mf">2</span><span class="p">,</span><span class="w"> </span><span class="mf">3</span><span class="p">,</span><span class="w"> </span><span class="mf">4</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">6</span><span class="p">,</span><span class="w"> </span><span class="mf">7</span><span class="p">,</span><span class="w"> </span><span class="mf">8</span><span class="p">,</span><span class="w"> </span><span class="mf">9</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">12</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="mf">15</span><span class="p">];</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Array representation of a binary tree */</span>
-<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="c1">// Using nullable int (int?) allows for using null to mark empty slots</span>
+<a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="c1">// Use the nullable type int? so that null can mark empty positions</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">?&gt;</span><span class="w"> </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">4</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="m">6</span><span class="p">,</span><span class="w"> </span><span class="m">7</span><span class="p">,</span><span class="w"> </span><span class="m">8</span><span class="p">,</span><span class="w"> </span><span class="m">9</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="m">12</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="m">15</span><span class="p">];</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Array representation of a binary tree */</span>
-<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="c1">// Using None to mark empty slots</span>
+<a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="c1">// Use None for empty positions</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="kd">let</span><span class="w"> </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="nb">Some</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span><span class="w"> </span><span class="nb">None</span><span class="p">,</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">6</span><span class="p">),</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">7</span><span class="p">),</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">8</span><span class="p">),</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">9</span><span class="p">),</span><span class="w"> </span><span class="nb">None</span><span class="p">,</span><span class="w"> </span><span class="nb">None</span><span class="p">,</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">12</span><span class="p">),</span><span class="w"> </span><span class="nb">None</span><span class="p">,</span><span class="w"> </span><span class="nb">None</span><span class="p">,</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="mi">15</span><span class="p">)];</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Array representation of a binary tree */</span>
-<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="c1">// Using the maximum int value to mark empty slots, therefore, node values must not be INT_MAX</span>
+<a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="c1">// Use the maximum int value for empty positions, so node values must not equal INT_MAX</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="kt">int</span><span class="w"> </span><span class="n">tree</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="mi">12</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">,</span><span class="w"> </span><span class="mi">15</span><span class="p">};</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Array representation of a binary tree */</span>
-<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="c1">// Using null to represent empty slots</span>
+<a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="c1">// Use null for empty positions</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="kd">val</span><span class="w"> </span><span class="nv">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arrayOf</span><span class="p">(</span><span class="w"> </span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="m">4</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="m">6</span><span class="p">,</span><span class="w"> </span><span class="m">7</span><span class="p">,</span><span class="w"> </span><span class="m">8</span><span class="p">,</span><span class="w"> </span><span class="m">9</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="m">12</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="m">15</span><span class="w"> </span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Array representation of a binary tree ###</span>
-<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="c1"># Using nil to represent empty slots</span>
+<a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="c1"># Use nil for empty positions</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">3</span><span class="p">,</span><span class="w"> </span><span class="mi">4</span><span class="p">,</span><span class="w"> </span><span class="kp">nil</span><span class="p">,</span><span class="w"> </span><span class="mi">6</span><span class="p">,</span><span class="w"> </span><span class="mi">7</span><span class="p">,</span><span class="w"> </span><span class="mi">8</span><span class="p">,</span><span class="w"> </span><span class="mi">9</span><span class="p">,</span><span class="w"> </span><span class="kp">nil</span><span class="p">,</span><span class="w"> </span><span class="kp">nil</span><span class="p">,</span><span class="w"> </span><span class="mi">12</span><span class="p">,</span><span class="w"> </span><span class="kp">nil</span><span class="p">,</span><span class="w"> </span><span class="kp">nil</span><span class="p">,</span><span class="w"> </span><span class="mi">15</span><span class="o">]</span>
 </code></pre></div>
 </div>
@@ -4781,53 +4853,53 @@
 <p><img alt="Array representation of an arbitrary binary tree" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_with_empty.png" /></p>
 <p align="center"> Figure 7-14 &nbsp; Array representation of an arbitrary binary tree </p>
 
-<p>It's worth noting that <strong>complete binary trees are very well-suited for array representation</strong>. Recalling the definition of a complete binary tree, <code>None</code> only appears at the bottom level and towards the right, <strong>meaning all <code>None</code> values must appear at the end of the level-order traversal sequence</strong>.</p>
-<p>This means that when using an array to represent a complete binary tree, it's possible to omit storing all <code>None</code> values, which is very convenient. Figure 7-15 gives an example.</p>
+<p><strong>Complete binary trees are particularly well suited to array representation</strong>. By definition, missing positions occur only at the right end of the lowest level, <strong>so all <code>None</code> entries appear at the end of the level-order sequence</strong>.</p>
+<p>We can therefore omit all <code>None</code> entries when storing a complete binary tree in an array. An example is shown below.</p>
 <p><img alt="Array representation of a complete binary tree" class="animation-figure" src="../array_representation_of_tree.assets/array_representation_complete_binary_tree.png" /></p>
 <p align="center"> Figure 7-15 &nbsp; Array representation of a complete binary tree </p>
 
-<p>The following code implements a binary tree using an array representation, including the following operations:</p>
+<p>The following code implements an array-based binary tree with these operations.</p>
 <ul>
-<li>Given a node, obtain its value, left (right) child node, and parent node.</li>
-<li>Obtain the preorder, inorder, postorder, and level-order traversal sequences.</li>
+<li>Given a node, get its value, left or right child, and parent.</li>
+<li>Obtain preorder, inorder, postorder, and level-order traversal sequences.</li>
 </ul>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">array_binary_tree.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayBinaryTree</span><span class="p">:</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Binary tree class represented by array"""</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Binary tree class using an array representation"""</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">arr</span><span class="p">:</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span> <span class="o">|</span> <span class="kc">None</span><span class="p">]):</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="w">        </span><span class="sd">"""Constructor"""</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_tree</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">arr</span><span class="p">)</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
-<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a><span class="w">        </span><span class="sd">"""List capacity"""</span>
+<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a><span class="w">        </span><span class="sd">"""Length of the backing list"""</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>        <span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_tree</span><span class="p">)</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">val</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a><span class="w">        </span><span class="sd">"""Get value of node at index i"""</span>
-<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>        <span class="c1"># If index is out of bounds, return None, representing empty position</span>
+<a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a><span class="w">        </span><span class="sd">"""Get the value of the node at index i"""</span>
+<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>        <span class="c1"># If the index is out of bounds, return None to represent an empty position</span>
 <a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>        <span class="k">if</span> <span class="n">i</span> <span class="o">&lt;</span> <span class="mi">0</span> <span class="ow">or</span> <span class="n">i</span> <span class="o">&gt;=</span> <span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">():</span>
 <a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>            <span class="k">return</span> <span class="kc">None</span>
 <a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>        <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_tree</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a>
 <a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a><span class="w">        </span><span class="sd">"""Get index of left child node of node at index i"""</span>
+<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a><span class="w">        </span><span class="sd">"""Get the left child index of the node at index i"""</span>
 <a id="__codelineno-13-21" name="__codelineno-13-21" href="#__codelineno-13-21"></a>        <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
 <a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a>
 <a id="__codelineno-13-23" name="__codelineno-13-23" href="#__codelineno-13-23"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-13-24" name="__codelineno-13-24" href="#__codelineno-13-24"></a><span class="w">        </span><span class="sd">"""Get index of right child node of node at index i"""</span>
+<a id="__codelineno-13-24" name="__codelineno-13-24" href="#__codelineno-13-24"></a><span class="w">        </span><span class="sd">"""Get the right child index of the node at index i"""</span>
 <a id="__codelineno-13-25" name="__codelineno-13-25" href="#__codelineno-13-25"></a>        <span class="k">return</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">2</span>
 <a id="__codelineno-13-26" name="__codelineno-13-26" href="#__codelineno-13-26"></a>
 <a id="__codelineno-13-27" name="__codelineno-13-27" href="#__codelineno-13-27"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-13-28" name="__codelineno-13-28" href="#__codelineno-13-28"></a><span class="w">        </span><span class="sd">"""Get index of parent node of node at index i"""</span>
+<a id="__codelineno-13-28" name="__codelineno-13-28" href="#__codelineno-13-28"></a><span class="w">        </span><span class="sd">"""Get the parent index of the node at index i"""</span>
 <a id="__codelineno-13-29" name="__codelineno-13-29" href="#__codelineno-13-29"></a>        <span class="k">return</span> <span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>
 <a id="__codelineno-13-30" name="__codelineno-13-30" href="#__codelineno-13-30"></a>
 <a id="__codelineno-13-31" name="__codelineno-13-31" href="#__codelineno-13-31"></a>    <span class="k">def</span><span class="w"> </span><span class="nf">level_order</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
 <a id="__codelineno-13-32" name="__codelineno-13-32" href="#__codelineno-13-32"></a><span class="w">        </span><span class="sd">"""Level-order traversal"""</span>
 <a id="__codelineno-13-33" name="__codelineno-13-33" href="#__codelineno-13-33"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
-<a id="__codelineno-13-34" name="__codelineno-13-34" href="#__codelineno-13-34"></a>        <span class="c1"># Traverse array directly</span>
+<a id="__codelineno-13-34" name="__codelineno-13-34" href="#__codelineno-13-34"></a>        <span class="c1"># Traverse the array directly</span>
 <a id="__codelineno-13-35" name="__codelineno-13-35" href="#__codelineno-13-35"></a>        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">size</span><span class="p">()):</span>
 <a id="__codelineno-13-36" name="__codelineno-13-36" href="#__codelineno-13-36"></a>            <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-13-37" name="__codelineno-13-37" href="#__codelineno-13-37"></a>                <span class="bp">self</span><span class="o">.</span><span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
@@ -4869,7 +4941,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_binary_tree.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Binary tree class represented by array */</span>
+<div class="highlight"><span class="filename">array_binary_tree.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Binary tree class using an array representation */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayBinaryTree</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">  </span><span class="k">public</span><span class="o">:</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="cm">/* Constructor */</span>
@@ -4877,30 +4949,30 @@
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">        </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="p">;</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a>
-<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="cm">/* List capacity */</span>
+<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="cm">/* Number of positions in the array */</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">tree</span><span class="p">.</span><span class="n">size</span><span class="p">();</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a>
-<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">    </span><span class="cm">/* Get value of node at index i */</span>
+<a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">    </span><span class="cm">/* Get the value of the node at index i */</span>
 <a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">        </span><span class="c1">// Return INT_MAX if index out of bounds, representing empty position</span>
+<a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">        </span><span class="c1">// If the index is out of bounds, return INT_MAX to represent an empty position</span>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">())</span>
 <a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
 <a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">tree</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a>
-<a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a><span class="w">    </span><span class="cm">/* Get index of left child node of node at index i */</span>
+<a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a><span class="w">    </span><span class="cm">/* Get the left child index of the node at index i */</span>
 <a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">left</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-24" name="__codelineno-14-24" href="#__codelineno-14-24"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-14-25" name="__codelineno-14-25" href="#__codelineno-14-25"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-26" name="__codelineno-14-26" href="#__codelineno-14-26"></a>
-<a id="__codelineno-14-27" name="__codelineno-14-27" href="#__codelineno-14-27"></a><span class="w">    </span><span class="cm">/* Get index of right child node of node at index i */</span>
+<a id="__codelineno-14-27" name="__codelineno-14-27" href="#__codelineno-14-27"></a><span class="w">    </span><span class="cm">/* Get the right child index of the node at index i */</span>
 <a id="__codelineno-14-28" name="__codelineno-14-28" href="#__codelineno-14-28"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">right</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-29" name="__codelineno-14-29" href="#__codelineno-14-29"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-14-30" name="__codelineno-14-30" href="#__codelineno-14-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-31" name="__codelineno-14-31" href="#__codelineno-14-31"></a>
-<a id="__codelineno-14-32" name="__codelineno-14-32" href="#__codelineno-14-32"></a><span class="w">    </span><span class="cm">/* Get index of parent node of node at index i */</span>
+<a id="__codelineno-14-32" name="__codelineno-14-32" href="#__codelineno-14-32"></a><span class="w">    </span><span class="cm">/* Get the parent index of the node at index i */</span>
 <a id="__codelineno-14-33" name="__codelineno-14-33" href="#__codelineno-14-33"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-34" name="__codelineno-14-34" href="#__codelineno-14-34"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-14-35" name="__codelineno-14-35" href="#__codelineno-14-35"></a><span class="w">    </span><span class="p">}</span>
@@ -4908,7 +4980,7 @@
 <a id="__codelineno-14-37" name="__codelineno-14-37" href="#__codelineno-14-37"></a><span class="w">    </span><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-14-38" name="__codelineno-14-38" href="#__codelineno-14-38"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">levelOrder</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-39" name="__codelineno-14-39" href="#__codelineno-14-39"></a><span class="w">        </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
-<a id="__codelineno-14-40" name="__codelineno-14-40" href="#__codelineno-14-40"></a><span class="w">        </span><span class="c1">// Traverse array directly</span>
+<a id="__codelineno-14-40" name="__codelineno-14-40" href="#__codelineno-14-40"></a><span class="w">        </span><span class="c1">// Traverse the array directly</span>
 <a id="__codelineno-14-41" name="__codelineno-14-41" href="#__codelineno-14-41"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-42" name="__codelineno-14-42" href="#__codelineno-14-42"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">)</span>
 <a id="__codelineno-14-43" name="__codelineno-14-43" href="#__codelineno-14-43"></a><span class="w">                </span><span class="n">res</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">));</span>
@@ -4942,7 +5014,7 @@
 <a id="__codelineno-14-71" name="__codelineno-14-71" href="#__codelineno-14-71"></a>
 <a id="__codelineno-14-72" name="__codelineno-14-72" href="#__codelineno-14-72"></a><span class="w">    </span><span class="cm">/* Depth-first traversal */</span>
 <a id="__codelineno-14-73" name="__codelineno-14-73" href="#__codelineno-14-73"></a><span class="w">    </span><span class="kt">void</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">order</span><span class="p">,</span><span class="w"> </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="o">&amp;</span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-74" name="__codelineno-14-74" href="#__codelineno-14-74"></a><span class="w">        </span><span class="c1">// If empty position, return</span>
+<a id="__codelineno-14-74" name="__codelineno-14-74" href="#__codelineno-14-74"></a><span class="w">        </span><span class="c1">// Return if this position is empty</span>
 <a id="__codelineno-14-75" name="__codelineno-14-75" href="#__codelineno-14-75"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">)</span>
 <a id="__codelineno-14-76" name="__codelineno-14-76" href="#__codelineno-14-76"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-14-77" name="__codelineno-14-77" href="#__codelineno-14-77"></a><span class="w">        </span><span class="c1">// Preorder traversal</span>
@@ -4961,7 +5033,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_binary_tree.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Binary tree class represented by array */</span>
+<div class="highlight"><span class="filename">array_binary_tree.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Binary tree class using an array representation */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kd">class</span> <span class="nc">ArrayBinaryTree</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">tree</span><span class="p">;</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a>
@@ -4970,30 +5042,30 @@
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">        </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">(</span><span class="n">arr</span><span class="p">);</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a>
-<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="cm">/* List capacity */</span>
+<a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="cm">/* List size */</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">tree</span><span class="p">.</span><span class="na">size</span><span class="p">();</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a>
-<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="cm">/* Get value of node at index i */</span>
+<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="cm">/* Get the value of the node at index i */</span>
 <a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="n">Integer</span><span class="w"> </span><span class="nf">val</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">        </span><span class="c1">// If index out of bounds, return null to represent empty position</span>
+<a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">        </span><span class="c1">// If the index is out of bounds, return null to represent an empty position</span>
 <a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">())</span>
 <a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">tree</span><span class="p">.</span><span class="na">get</span><span class="p">(</span><span class="n">i</span><span class="p">);</span>
 <a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a>
-<a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w">    </span><span class="cm">/* Get index of left child node of node at index i */</span>
+<a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w">    </span><span class="cm">/* Get the left child index of the node at index i */</span>
 <a id="__codelineno-15-24" name="__codelineno-15-24" href="#__codelineno-15-24"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="n">Integer</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-25" name="__codelineno-15-25" href="#__codelineno-15-25"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-15-26" name="__codelineno-15-26" href="#__codelineno-15-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-27" name="__codelineno-15-27" href="#__codelineno-15-27"></a>
-<a id="__codelineno-15-28" name="__codelineno-15-28" href="#__codelineno-15-28"></a><span class="w">    </span><span class="cm">/* Get index of right child node of node at index i */</span>
+<a id="__codelineno-15-28" name="__codelineno-15-28" href="#__codelineno-15-28"></a><span class="w">    </span><span class="cm">/* Get the right child index of the node at index i */</span>
 <a id="__codelineno-15-29" name="__codelineno-15-29" href="#__codelineno-15-29"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="n">Integer</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-30" name="__codelineno-15-30" href="#__codelineno-15-30"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-15-31" name="__codelineno-15-31" href="#__codelineno-15-31"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-32" name="__codelineno-15-32" href="#__codelineno-15-32"></a>
-<a id="__codelineno-15-33" name="__codelineno-15-33" href="#__codelineno-15-33"></a><span class="w">    </span><span class="cm">/* Get index of parent node of node at index i */</span>
+<a id="__codelineno-15-33" name="__codelineno-15-33" href="#__codelineno-15-33"></a><span class="w">    </span><span class="cm">/* Get the parent index of the node at index i */</span>
 <a id="__codelineno-15-34" name="__codelineno-15-34" href="#__codelineno-15-34"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="n">Integer</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-35" name="__codelineno-15-35" href="#__codelineno-15-35"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-15-36" name="__codelineno-15-36" href="#__codelineno-15-36"></a><span class="w">    </span><span class="p">}</span>
@@ -5001,7 +5073,7 @@
 <a id="__codelineno-15-38" name="__codelineno-15-38" href="#__codelineno-15-38"></a><span class="w">    </span><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-15-39" name="__codelineno-15-39" href="#__codelineno-15-39"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="nf">levelOrder</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-40" name="__codelineno-15-40" href="#__codelineno-15-40"></a><span class="w">        </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
-<a id="__codelineno-15-41" name="__codelineno-15-41" href="#__codelineno-15-41"></a><span class="w">        </span><span class="c1">// Traverse array directly</span>
+<a id="__codelineno-15-41" name="__codelineno-15-41" href="#__codelineno-15-41"></a><span class="w">        </span><span class="c1">// Traverse the array directly</span>
 <a id="__codelineno-15-42" name="__codelineno-15-42" href="#__codelineno-15-42"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-43" name="__codelineno-15-43" href="#__codelineno-15-43"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-15-44" name="__codelineno-15-44" href="#__codelineno-15-44"></a><span class="w">                </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">));</span>
@@ -5011,7 +5083,7 @@
 <a id="__codelineno-15-48" name="__codelineno-15-48" href="#__codelineno-15-48"></a>
 <a id="__codelineno-15-49" name="__codelineno-15-49" href="#__codelineno-15-49"></a><span class="w">    </span><span class="cm">/* Depth-first traversal */</span>
 <a id="__codelineno-15-50" name="__codelineno-15-50" href="#__codelineno-15-50"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kt">void</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">Integer</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="n">String</span><span class="w"> </span><span class="n">order</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-51" name="__codelineno-15-51" href="#__codelineno-15-51"></a><span class="w">        </span><span class="c1">// If empty position, return</span>
+<a id="__codelineno-15-51" name="__codelineno-15-51" href="#__codelineno-15-51"></a><span class="w">        </span><span class="c1">// Return if this position is empty</span>
 <a id="__codelineno-15-52" name="__codelineno-15-52" href="#__codelineno-15-52"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-15-53" name="__codelineno-15-53" href="#__codelineno-15-53"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-15-54" name="__codelineno-15-54" href="#__codelineno-15-54"></a><span class="w">        </span><span class="c1">// Preorder traversal</span>
@@ -5051,34 +5123,34 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_binary_tree.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Binary tree class represented by array */</span>
+<div class="highlight"><span class="filename">array_binary_tree.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Binary tree class using an array representation */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">class</span><span class="w"> </span><span class="nf">ArrayBinaryTree</span><span class="p">(</span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int?</span><span class="o">&gt;</span><span class="w"> </span><span class="n">arr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int?</span><span class="o">&gt;</span><span class="w"> </span><span class="n">tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="n">arr</span><span class="p">);</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a>
-<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="cm">/* List capacity */</span>
+<a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="cm">/* List size */</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">tree</span><span class="p">.</span><span class="n">Count</span><span class="p">;</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a>
-<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="cm">/* Get value of node at index i */</span>
+<a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="cm">/* Get the value of the node at index i */</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int?</span><span class="w"> </span><span class="n">Val</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">        </span><span class="c1">// If index out of bounds, return null to represent empty position</span>
+<a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">        </span><span class="c1">// If the index is out of bounds, return null to represent an empty position</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">Size</span><span class="p">())</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
 <a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">tree</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a>
-<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">    </span><span class="cm">/* Get index of left child node of node at index i */</span>
+<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">    </span><span class="cm">/* Get the left child index of the node at index i */</span>
 <a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Left</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a>
-<a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">    </span><span class="cm">/* Get index of right child node of node at index i */</span>
+<a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">    </span><span class="cm">/* Get the right child index of the node at index i */</span>
 <a id="__codelineno-16-24" name="__codelineno-16-24" href="#__codelineno-16-24"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Right</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-25" name="__codelineno-16-25" href="#__codelineno-16-25"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-16-26" name="__codelineno-16-26" href="#__codelineno-16-26"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-27" name="__codelineno-16-27" href="#__codelineno-16-27"></a>
-<a id="__codelineno-16-28" name="__codelineno-16-28" href="#__codelineno-16-28"></a><span class="w">    </span><span class="cm">/* Get index of parent node of node at index i */</span>
+<a id="__codelineno-16-28" name="__codelineno-16-28" href="#__codelineno-16-28"></a><span class="w">    </span><span class="cm">/* Get the parent index of the node at index i */</span>
 <a id="__codelineno-16-29" name="__codelineno-16-29" href="#__codelineno-16-29"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="nf">Parent</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-30" name="__codelineno-16-30" href="#__codelineno-16-30"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span><span class="p">;</span>
 <a id="__codelineno-16-31" name="__codelineno-16-31" href="#__codelineno-16-31"></a><span class="w">    </span><span class="p">}</span>
@@ -5086,7 +5158,7 @@
 <a id="__codelineno-16-33" name="__codelineno-16-33" href="#__codelineno-16-33"></a><span class="w">    </span><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-16-34" name="__codelineno-16-34" href="#__codelineno-16-34"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">LevelOrder</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-35" name="__codelineno-16-35" href="#__codelineno-16-35"></a><span class="w">        </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-16-36" name="__codelineno-16-36" href="#__codelineno-16-36"></a><span class="w">        </span><span class="c1">// Traverse array directly</span>
+<a id="__codelineno-16-36" name="__codelineno-16-36" href="#__codelineno-16-36"></a><span class="w">        </span><span class="c1">// Traverse the array directly</span>
 <a id="__codelineno-16-37" name="__codelineno-16-37" href="#__codelineno-16-37"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">Size</span><span class="p">();</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-38" name="__codelineno-16-38" href="#__codelineno-16-38"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">Val</span><span class="p">(</span><span class="n">i</span><span class="p">).</span><span class="n">HasValue</span><span class="p">)</span>
 <a id="__codelineno-16-39" name="__codelineno-16-39" href="#__codelineno-16-39"></a><span class="w">                </span><span class="n">res</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">Val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">!</span><span class="p">.</span><span class="n">Value</span><span class="p">);</span>
@@ -5096,7 +5168,7 @@
 <a id="__codelineno-16-43" name="__codelineno-16-43" href="#__codelineno-16-43"></a>
 <a id="__codelineno-16-44" name="__codelineno-16-44" href="#__codelineno-16-44"></a><span class="w">    </span><span class="cm">/* Depth-first traversal */</span>
 <a id="__codelineno-16-45" name="__codelineno-16-45" href="#__codelineno-16-45"></a><span class="w">    </span><span class="k">void</span><span class="w"> </span><span class="nf">DFS</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">string</span><span class="w"> </span><span class="n">order</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-46" name="__codelineno-16-46" href="#__codelineno-16-46"></a><span class="w">        </span><span class="c1">// If empty position, return</span>
+<a id="__codelineno-16-46" name="__codelineno-16-46" href="#__codelineno-16-46"></a><span class="w">        </span><span class="c1">// Return if this position is empty</span>
 <a id="__codelineno-16-47" name="__codelineno-16-47" href="#__codelineno-16-47"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">Val</span><span class="p">(</span><span class="n">i</span><span class="p">).</span><span class="n">HasValue</span><span class="p">)</span>
 <a id="__codelineno-16-48" name="__codelineno-16-48" href="#__codelineno-16-48"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-16-49" name="__codelineno-16-49" href="#__codelineno-16-49"></a><span class="w">        </span><span class="c1">// Preorder traversal</span>
@@ -5136,7 +5208,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_binary_tree.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Binary tree class represented by array */</span>
+<div class="highlight"><span class="filename">array_binary_tree.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Binary tree type using an array representation */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">arrayBinaryTree</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="nx">tree</span><span class="w"> </span><span class="p">[]</span><span class="kt">any</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="p">}</span>
@@ -5148,31 +5220,31 @@
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="p">}</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a>
-<a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="cm">/* List capacity */</span>
+<a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="cm">/* Length of the backing slice */</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">abt</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayBinaryTree</span><span class="p">)</span><span class="w"> </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nb">len</span><span class="p">(</span><span class="nx">abt</span><span class="p">.</span><span class="nx">tree</span><span class="p">)</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="p">}</span>
 <a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a>
-<a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="cm">/* Get value of node at index i */</span>
+<a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="cm">/* Get the value of the node at index i */</span>
 <a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">abt</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayBinaryTree</span><span class="p">)</span><span class="w"> </span><span class="nx">val</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a><span class="w">    </span><span class="c1">// If index out of bounds, return null to represent empty position</span>
+<a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a><span class="w">    </span><span class="c1">// If the index is out of bounds, return nil to represent an empty position</span>
 <a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="nx">abt</span><span class="p">.</span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-22" name="__codelineno-17-22" href="#__codelineno-17-22"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-24" name="__codelineno-17-24" href="#__codelineno-17-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">abt</span><span class="p">.</span><span class="nx">tree</span><span class="p">[</span><span class="nx">i</span><span class="p">]</span>
 <a id="__codelineno-17-25" name="__codelineno-17-25" href="#__codelineno-17-25"></a><span class="p">}</span>
 <a id="__codelineno-17-26" name="__codelineno-17-26" href="#__codelineno-17-26"></a>
-<a id="__codelineno-17-27" name="__codelineno-17-27" href="#__codelineno-17-27"></a><span class="cm">/* Get index of left child node of node at index i */</span>
+<a id="__codelineno-17-27" name="__codelineno-17-27" href="#__codelineno-17-27"></a><span class="cm">/* Get the left child index of the node at index i */</span>
 <a id="__codelineno-17-28" name="__codelineno-17-28" href="#__codelineno-17-28"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">abt</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayBinaryTree</span><span class="p">)</span><span class="w"> </span><span class="nx">left</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-29" name="__codelineno-17-29" href="#__codelineno-17-29"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="o">*</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-17-30" name="__codelineno-17-30" href="#__codelineno-17-30"></a><span class="p">}</span>
 <a id="__codelineno-17-31" name="__codelineno-17-31" href="#__codelineno-17-31"></a>
-<a id="__codelineno-17-32" name="__codelineno-17-32" href="#__codelineno-17-32"></a><span class="cm">/* Get index of right child node of node at index i */</span>
+<a id="__codelineno-17-32" name="__codelineno-17-32" href="#__codelineno-17-32"></a><span class="cm">/* Get the right child index of the node at index i */</span>
 <a id="__codelineno-17-33" name="__codelineno-17-33" href="#__codelineno-17-33"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">abt</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayBinaryTree</span><span class="p">)</span><span class="w"> </span><span class="nx">right</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-34" name="__codelineno-17-34" href="#__codelineno-17-34"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">2</span><span class="o">*</span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-17-35" name="__codelineno-17-35" href="#__codelineno-17-35"></a><span class="p">}</span>
 <a id="__codelineno-17-36" name="__codelineno-17-36" href="#__codelineno-17-36"></a>
-<a id="__codelineno-17-37" name="__codelineno-17-37" href="#__codelineno-17-37"></a><span class="cm">/* Get index of parent node of node at index i */</span>
+<a id="__codelineno-17-37" name="__codelineno-17-37" href="#__codelineno-17-37"></a><span class="cm">/* Get the parent index of the node at index i */</span>
 <a id="__codelineno-17-38" name="__codelineno-17-38" href="#__codelineno-17-38"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">abt</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayBinaryTree</span><span class="p">)</span><span class="w"> </span><span class="nx">parent</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-39" name="__codelineno-17-39" href="#__codelineno-17-39"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-17-40" name="__codelineno-17-40" href="#__codelineno-17-40"></a><span class="p">}</span>
@@ -5180,7 +5252,7 @@
 <a id="__codelineno-17-42" name="__codelineno-17-42" href="#__codelineno-17-42"></a><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-17-43" name="__codelineno-17-43" href="#__codelineno-17-43"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">abt</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayBinaryTree</span><span class="p">)</span><span class="w"> </span><span class="nx">levelOrder</span><span class="p">()</span><span class="w"> </span><span class="p">[]</span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-44" name="__codelineno-17-44" href="#__codelineno-17-44"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="p">[]</span><span class="kt">any</span>
-<a id="__codelineno-17-45" name="__codelineno-17-45" href="#__codelineno-17-45"></a><span class="w">    </span><span class="c1">// Traverse array directly</span>
+<a id="__codelineno-17-45" name="__codelineno-17-45" href="#__codelineno-17-45"></a><span class="w">    </span><span class="c1">// Traverse the array directly</span>
 <a id="__codelineno-17-46" name="__codelineno-17-46" href="#__codelineno-17-46"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">abt</span><span class="p">.</span><span class="nx">size</span><span class="p">();</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-47" name="__codelineno-17-47" href="#__codelineno-17-47"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">abt</span><span class="p">.</span><span class="nx">val</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-48" name="__codelineno-17-48" href="#__codelineno-17-48"></a><span class="w">            </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">res</span><span class="p">,</span><span class="w"> </span><span class="nx">abt</span><span class="p">.</span><span class="nx">val</span><span class="p">(</span><span class="nx">i</span><span class="p">))</span>
@@ -5191,7 +5263,7 @@
 <a id="__codelineno-17-53" name="__codelineno-17-53" href="#__codelineno-17-53"></a>
 <a id="__codelineno-17-54" name="__codelineno-17-54" href="#__codelineno-17-54"></a><span class="cm">/* Depth-first traversal */</span>
 <a id="__codelineno-17-55" name="__codelineno-17-55" href="#__codelineno-17-55"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">abt</span><span class="w"> </span><span class="o">*</span><span class="nx">arrayBinaryTree</span><span class="p">)</span><span class="w"> </span><span class="nx">dfs</span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="kt">int</span><span class="p">,</span><span class="w"> </span><span class="nx">order</span><span class="w"> </span><span class="kt">string</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">*</span><span class="p">[]</span><span class="kt">any</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-56" name="__codelineno-17-56" href="#__codelineno-17-56"></a><span class="w">    </span><span class="c1">// If empty position, return</span>
+<a id="__codelineno-17-56" name="__codelineno-17-56" href="#__codelineno-17-56"></a><span class="w">    </span><span class="c1">// Return if this position is empty</span>
 <a id="__codelineno-17-57" name="__codelineno-17-57" href="#__codelineno-17-57"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">abt</span><span class="p">.</span><span class="nx">val</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-58" name="__codelineno-17-58" href="#__codelineno-17-58"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-17-59" name="__codelineno-17-59" href="#__codelineno-17-59"></a><span class="w">    </span><span class="p">}</span>
@@ -5234,7 +5306,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_binary_tree.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Binary tree class represented by array */</span>
+<div class="highlight"><span class="filename">array_binary_tree.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Binary tree class using an array representation */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayBinaryTree</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">var</span><span class="w"> </span><span class="nv">tree</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">?]</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a>
@@ -5243,31 +5315,31 @@
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">        </span><span class="n">tree</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">arr</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a>
-<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="cm">/* List capacity */</span>
+<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="cm">/* Length of the backing array */</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">size</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">        </span><span class="n">tree</span><span class="p">.</span><span class="bp">count</span>
 <a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a>
-<a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">    </span><span class="cm">/* Get value of node at index i */</span>
+<a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">    </span><span class="cm">/* Get the value of the node at index i */</span>
 <a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">val</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="p">?</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">        </span><span class="c1">// If index out of bounds, return null to represent empty position</span>
+<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">        </span><span class="c1">// If the index is out of bounds, return nil to represent an empty position</span>
 <a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">tree</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
 <a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-23" name="__codelineno-18-23" href="#__codelineno-18-23"></a>
-<a id="__codelineno-18-24" name="__codelineno-18-24" href="#__codelineno-18-24"></a><span class="w">    </span><span class="cm">/* Get index of left child node of node at index i */</span>
+<a id="__codelineno-18-24" name="__codelineno-18-24" href="#__codelineno-18-24"></a><span class="w">    </span><span class="cm">/* Get the left child index of the node at index i */</span>
 <a id="__codelineno-18-25" name="__codelineno-18-25" href="#__codelineno-18-25"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-26" name="__codelineno-18-26" href="#__codelineno-18-26"></a><span class="w">        </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-18-27" name="__codelineno-18-27" href="#__codelineno-18-27"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-28" name="__codelineno-18-28" href="#__codelineno-18-28"></a>
-<a id="__codelineno-18-29" name="__codelineno-18-29" href="#__codelineno-18-29"></a><span class="w">    </span><span class="cm">/* Get index of right child node of node at index i */</span>
+<a id="__codelineno-18-29" name="__codelineno-18-29" href="#__codelineno-18-29"></a><span class="w">    </span><span class="cm">/* Get the right child index of the node at index i */</span>
 <a id="__codelineno-18-30" name="__codelineno-18-30" href="#__codelineno-18-30"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-31" name="__codelineno-18-31" href="#__codelineno-18-31"></a><span class="w">        </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-18-32" name="__codelineno-18-32" href="#__codelineno-18-32"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-33" name="__codelineno-18-33" href="#__codelineno-18-33"></a>
-<a id="__codelineno-18-34" name="__codelineno-18-34" href="#__codelineno-18-34"></a><span class="w">    </span><span class="cm">/* Get index of parent node of node at index i */</span>
+<a id="__codelineno-18-34" name="__codelineno-18-34" href="#__codelineno-18-34"></a><span class="w">    </span><span class="cm">/* Get the parent index of the node at index i */</span>
 <a id="__codelineno-18-35" name="__codelineno-18-35" href="#__codelineno-18-35"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-36" name="__codelineno-18-36" href="#__codelineno-18-36"></a><span class="w">        </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-18-37" name="__codelineno-18-37" href="#__codelineno-18-37"></a><span class="w">    </span><span class="p">}</span>
@@ -5275,7 +5347,7 @@
 <a id="__codelineno-18-39" name="__codelineno-18-39" href="#__codelineno-18-39"></a><span class="w">    </span><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-18-40" name="__codelineno-18-40" href="#__codelineno-18-40"></a><span class="w">    </span><span class="kd">func</span><span class="w"> </span><span class="nf">levelOrder</span><span class="p">()</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-41" name="__codelineno-18-41" href="#__codelineno-18-41"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">res</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
-<a id="__codelineno-18-42" name="__codelineno-18-42" href="#__codelineno-18-42"></a><span class="w">        </span><span class="c1">// Traverse array directly</span>
+<a id="__codelineno-18-42" name="__codelineno-18-42" href="#__codelineno-18-42"></a><span class="w">        </span><span class="c1">// Traverse the array directly</span>
 <a id="__codelineno-18-43" name="__codelineno-18-43" href="#__codelineno-18-43"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">..</span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-44" name="__codelineno-18-44" href="#__codelineno-18-44"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nv">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-45" name="__codelineno-18-45" href="#__codelineno-18-45"></a><span class="w">                </span><span class="n">res</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
@@ -5286,7 +5358,7 @@
 <a id="__codelineno-18-50" name="__codelineno-18-50" href="#__codelineno-18-50"></a>
 <a id="__codelineno-18-51" name="__codelineno-18-51" href="#__codelineno-18-51"></a><span class="w">    </span><span class="cm">/* Depth-first traversal */</span>
 <a id="__codelineno-18-52" name="__codelineno-18-52" href="#__codelineno-18-52"></a><span class="w">    </span><span class="kd">private</span><span class="w"> </span><span class="kd">func</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">,</span><span class="w"> </span><span class="n">order</span><span class="p">:</span><span class="w"> </span><span class="nb">String</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kr">inout</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">])</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-53" name="__codelineno-18-53" href="#__codelineno-18-53"></a><span class="w">        </span><span class="c1">// If empty position, return</span>
+<a id="__codelineno-18-53" name="__codelineno-18-53" href="#__codelineno-18-53"></a><span class="w">        </span><span class="c1">// Return if this position is empty</span>
 <a id="__codelineno-18-54" name="__codelineno-18-54" href="#__codelineno-18-54"></a><span class="w">        </span><span class="k">guard</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nv">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-55" name="__codelineno-18-55" href="#__codelineno-18-55"></a><span class="w">            </span><span class="k">return</span>
 <a id="__codelineno-18-56" name="__codelineno-18-56" href="#__codelineno-18-56"></a><span class="w">        </span><span class="p">}</span>
@@ -5330,7 +5402,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_binary_tree.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Binary tree class represented by array */</span>
+<div class="highlight"><span class="filename">array_binary_tree.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Binary tree class using an array representation */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">ArrayBinaryTree</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="n">#tree</span><span class="p">;</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a>
@@ -5339,29 +5411,29 @@
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">arr</span><span class="p">;</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a>
-<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="cm">/* List capacity */</span>
+<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="cm">/* Array length */</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">    </span><span class="nx">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#tree</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a>
-<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">    </span><span class="cm">/* Get value of node at index i */</span>
+<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">    </span><span class="cm">/* Get the value of the node at index i */</span>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">    </span><span class="nx">val</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">        </span><span class="c1">// If index out of bounds, return null to represent empty position</span>
+<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">        </span><span class="c1">// If the index is out of bounds, return null to indicate an empty position</span>
 <a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">())</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#tree</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a>
-<a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="w">    </span><span class="cm">/* Get index of left child node of node at index i */</span>
+<a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="w">    </span><span class="cm">/* Get the left child index of the node at index i */</span>
 <a id="__codelineno-19-23" name="__codelineno-19-23" href="#__codelineno-19-23"></a><span class="w">    </span><span class="nx">left</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-24" name="__codelineno-19-24" href="#__codelineno-19-24"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-19-25" name="__codelineno-19-25" href="#__codelineno-19-25"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-26" name="__codelineno-19-26" href="#__codelineno-19-26"></a>
-<a id="__codelineno-19-27" name="__codelineno-19-27" href="#__codelineno-19-27"></a><span class="w">    </span><span class="cm">/* Get index of right child node of node at index i */</span>
+<a id="__codelineno-19-27" name="__codelineno-19-27" href="#__codelineno-19-27"></a><span class="w">    </span><span class="cm">/* Get the right child index of the node at index i */</span>
 <a id="__codelineno-19-28" name="__codelineno-19-28" href="#__codelineno-19-28"></a><span class="w">    </span><span class="nx">right</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-29" name="__codelineno-19-29" href="#__codelineno-19-29"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
 <a id="__codelineno-19-30" name="__codelineno-19-30" href="#__codelineno-19-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-31" name="__codelineno-19-31" href="#__codelineno-19-31"></a>
-<a id="__codelineno-19-32" name="__codelineno-19-32" href="#__codelineno-19-32"></a><span class="w">    </span><span class="cm">/* Get index of parent node of node at index i */</span>
+<a id="__codelineno-19-32" name="__codelineno-19-32" href="#__codelineno-19-32"></a><span class="w">    </span><span class="cm">/* Get the parent index of the node at index i */</span>
 <a id="__codelineno-19-33" name="__codelineno-19-33" href="#__codelineno-19-33"></a><span class="w">    </span><span class="nx">parent</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-34" name="__codelineno-19-34" href="#__codelineno-19-34"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">((</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Floor division</span>
 <a id="__codelineno-19-35" name="__codelineno-19-35" href="#__codelineno-19-35"></a><span class="w">    </span><span class="p">}</span>
@@ -5369,7 +5441,7 @@
 <a id="__codelineno-19-37" name="__codelineno-19-37" href="#__codelineno-19-37"></a><span class="w">    </span><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-19-38" name="__codelineno-19-38" href="#__codelineno-19-38"></a><span class="w">    </span><span class="nx">levelOrder</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-39" name="__codelineno-19-39" href="#__codelineno-19-39"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-19-40" name="__codelineno-19-40" href="#__codelineno-19-40"></a><span class="w">        </span><span class="c1">// Traverse array directly</span>
+<a id="__codelineno-19-40" name="__codelineno-19-40" href="#__codelineno-19-40"></a><span class="w">        </span><span class="c1">// Traverse the array directly</span>
 <a id="__codelineno-19-41" name="__codelineno-19-41" href="#__codelineno-19-41"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">();</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-42" name="__codelineno-19-42" href="#__codelineno-19-42"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="p">(</span><span class="nx">i</span><span class="p">));</span>
 <a id="__codelineno-19-43" name="__codelineno-19-43" href="#__codelineno-19-43"></a><span class="w">        </span><span class="p">}</span>
@@ -5378,7 +5450,7 @@
 <a id="__codelineno-19-46" name="__codelineno-19-46" href="#__codelineno-19-46"></a>
 <a id="__codelineno-19-47" name="__codelineno-19-47" href="#__codelineno-19-47"></a><span class="w">    </span><span class="cm">/* Depth-first traversal */</span>
 <a id="__codelineno-19-48" name="__codelineno-19-48" href="#__codelineno-19-48"></a><span class="w">    </span><span class="n">#dfs</span><span class="p">(</span><span class="nx">i</span><span class="p">,</span><span class="w"> </span><span class="nx">order</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-49" name="__codelineno-19-49" href="#__codelineno-19-49"></a><span class="w">        </span><span class="c1">// If empty position, return</span>
+<a id="__codelineno-19-49" name="__codelineno-19-49" href="#__codelineno-19-49"></a><span class="w">        </span><span class="c1">// Return if this position is empty</span>
 <a id="__codelineno-19-50" name="__codelineno-19-50" href="#__codelineno-19-50"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-19-51" name="__codelineno-19-51" href="#__codelineno-19-51"></a><span class="w">        </span><span class="c1">// Preorder traversal</span>
 <a id="__codelineno-19-52" name="__codelineno-19-52" href="#__codelineno-19-52"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">order</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="s1">'pre'</span><span class="p">)</span><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="p">(</span><span class="nx">i</span><span class="p">));</span>
@@ -5414,7 +5486,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_binary_tree.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Binary tree class represented by array */</span>
+<div class="highlight"><span class="filename">array_binary_tree.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Binary tree class using an array representation */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">ArrayBinaryTree</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="n">#tree</span><span class="o">:</span><span class="w"> </span><span class="p">(</span><span class="kt">number</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">)[];</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a>
@@ -5423,29 +5495,29 @@
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="n">#tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">arr</span><span class="p">;</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a>
-<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="cm">/* List capacity */</span>
+<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="cm">/* Array length, including empty positions */</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="nx">size</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#tree</span><span class="p">.</span><span class="nx">length</span><span class="p">;</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a>
-<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">    </span><span class="cm">/* Get value of node at index i */</span>
+<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">    </span><span class="cm">/* Get the value of the node at index i */</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">    </span><span class="nx">val</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">        </span><span class="c1">// If index out of bounds, return null to represent empty position</span>
+<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">        </span><span class="c1">// If the index is out of bounds, return null to indicate an empty position</span>
 <a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">())</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#tree</span><span class="p">[</span><span class="nx">i</span><span class="p">];</span>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a>
-<a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="w">    </span><span class="cm">/* Get index of left child node of node at index i */</span>
+<a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="w">    </span><span class="cm">/* Get the left child index of the node at index i */</span>
 <a id="__codelineno-20-23" name="__codelineno-20-23" href="#__codelineno-20-23"></a><span class="w">    </span><span class="nx">left</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-24" name="__codelineno-20-24" href="#__codelineno-20-24"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a>
-<a id="__codelineno-20-27" name="__codelineno-20-27" href="#__codelineno-20-27"></a><span class="w">    </span><span class="cm">/* Get index of right child node of node at index i */</span>
+<a id="__codelineno-20-27" name="__codelineno-20-27" href="#__codelineno-20-27"></a><span class="w">    </span><span class="cm">/* Get the right child index of the node at index i */</span>
 <a id="__codelineno-20-28" name="__codelineno-20-28" href="#__codelineno-20-28"></a><span class="w">    </span><span class="nx">right</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-29" name="__codelineno-20-29" href="#__codelineno-20-29"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mf">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">2</span><span class="p">;</span>
 <a id="__codelineno-20-30" name="__codelineno-20-30" href="#__codelineno-20-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-31" name="__codelineno-20-31" href="#__codelineno-20-31"></a>
-<a id="__codelineno-20-32" name="__codelineno-20-32" href="#__codelineno-20-32"></a><span class="w">    </span><span class="cm">/* Get index of parent node of node at index i */</span>
+<a id="__codelineno-20-32" name="__codelineno-20-32" href="#__codelineno-20-32"></a><span class="w">    </span><span class="cm">/* Get the parent index of the node at index i */</span>
 <a id="__codelineno-20-33" name="__codelineno-20-33" href="#__codelineno-20-33"></a><span class="w">    </span><span class="nx">parent</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-34" name="__codelineno-20-34" href="#__codelineno-20-34"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nb">Math</span><span class="p">.</span><span class="nx">floor</span><span class="p">((</span><span class="nx">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mf">2</span><span class="p">);</span><span class="w"> </span><span class="c1">// Floor division</span>
 <a id="__codelineno-20-35" name="__codelineno-20-35" href="#__codelineno-20-35"></a><span class="w">    </span><span class="p">}</span>
@@ -5453,7 +5525,7 @@
 <a id="__codelineno-20-37" name="__codelineno-20-37" href="#__codelineno-20-37"></a><span class="w">    </span><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-20-38" name="__codelineno-20-38" href="#__codelineno-20-38"></a><span class="w">    </span><span class="nx">levelOrder</span><span class="p">()</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-39" name="__codelineno-20-39" href="#__codelineno-20-39"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
-<a id="__codelineno-20-40" name="__codelineno-20-40" href="#__codelineno-20-40"></a><span class="w">        </span><span class="c1">// Traverse array directly</span>
+<a id="__codelineno-20-40" name="__codelineno-20-40" href="#__codelineno-20-40"></a><span class="w">        </span><span class="c1">// Traverse the array directly</span>
 <a id="__codelineno-20-41" name="__codelineno-20-41" href="#__codelineno-20-41"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kd">let</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span><span class="w"> </span><span class="nx">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">size</span><span class="p">();</span><span class="w"> </span><span class="nx">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-42" name="__codelineno-20-42" href="#__codelineno-20-42"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="p">(</span><span class="nx">i</span><span class="p">));</span>
 <a id="__codelineno-20-43" name="__codelineno-20-43" href="#__codelineno-20-43"></a><span class="w">        </span><span class="p">}</span>
@@ -5462,7 +5534,7 @@
 <a id="__codelineno-20-46" name="__codelineno-20-46" href="#__codelineno-20-46"></a>
 <a id="__codelineno-20-47" name="__codelineno-20-47" href="#__codelineno-20-47"></a><span class="w">    </span><span class="cm">/* Depth-first traversal */</span>
 <a id="__codelineno-20-48" name="__codelineno-20-48" href="#__codelineno-20-48"></a><span class="w">    </span><span class="n">#dfs</span><span class="p">(</span><span class="nx">i</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">order</span><span class="o">:</span><span class="w"> </span><span class="kt">Order</span><span class="p">,</span><span class="w"> </span><span class="nx">res</span><span class="o">:</span><span class="w"> </span><span class="p">(</span><span class="kt">number</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">)[])</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-49" name="__codelineno-20-49" href="#__codelineno-20-49"></a><span class="w">        </span><span class="c1">// If empty position, return</span>
+<a id="__codelineno-20-49" name="__codelineno-20-49" href="#__codelineno-20-49"></a><span class="w">        </span><span class="c1">// Return if this position is empty</span>
 <a id="__codelineno-20-50" name="__codelineno-20-50" href="#__codelineno-20-50"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="p">(</span><span class="nx">i</span><span class="p">)</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-20-51" name="__codelineno-20-51" href="#__codelineno-20-51"></a><span class="w">        </span><span class="c1">// Preorder traversal</span>
 <a id="__codelineno-20-52" name="__codelineno-20-52" href="#__codelineno-20-52"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">order</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="s1">'pre'</span><span class="p">)</span><span class="w"> </span><span class="nx">res</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="p">(</span><span class="nx">i</span><span class="p">));</span>
@@ -5498,38 +5570,38 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_binary_tree.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Binary tree class represented by array */</span>
+<div class="highlight"><span class="filename">array_binary_tree.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Binary tree class using an array representation */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayBinaryTree</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="kd">late</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">?&gt;</span><span class="w"> </span><span class="n">_tree</span><span class="p">;</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="cm">/* Constructor */</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="n">ArrayBinaryTree</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">_tree</span><span class="p">);</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a>
-<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="cm">/* List capacity */</span>
+<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="cm">/* Array length, including empty positions */</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_tree</span><span class="p">.</span><span class="n">length</span><span class="p">;</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a>
-<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="cm">/* Get value of node at index i */</span>
+<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="cm">/* Get the value of the node at index i */</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">  </span><span class="kt">int</span><span class="o">?</span><span class="w"> </span><span class="n">val</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">    </span><span class="c1">// If index out of bounds, return null to represent empty position</span>
+<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">    </span><span class="c1">// If the index is out of bounds, return null to indicate an empty position</span>
 <a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">_tree</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
 <a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a>
-<a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">  </span><span class="cm">/* Get index of left child node of node at index i */</span>
+<a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">  </span><span class="cm">/* Get the left child index of the node at index i */</span>
 <a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">  </span><span class="kt">int</span><span class="o">?</span><span class="w"> </span><span class="n">left</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-21-25" name="__codelineno-21-25" href="#__codelineno-21-25"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-26" name="__codelineno-21-26" href="#__codelineno-21-26"></a>
-<a id="__codelineno-21-27" name="__codelineno-21-27" href="#__codelineno-21-27"></a><span class="w">  </span><span class="cm">/* Get index of right child node of node at index i */</span>
+<a id="__codelineno-21-27" name="__codelineno-21-27" href="#__codelineno-21-27"></a><span class="w">  </span><span class="cm">/* Get the right child index of the node at index i */</span>
 <a id="__codelineno-21-28" name="__codelineno-21-28" href="#__codelineno-21-28"></a><span class="w">  </span><span class="kt">int</span><span class="o">?</span><span class="w"> </span><span class="n">right</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-29" name="__codelineno-21-29" href="#__codelineno-21-29"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">2</span><span class="p">;</span>
 <a id="__codelineno-21-30" name="__codelineno-21-30" href="#__codelineno-21-30"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-31" name="__codelineno-21-31" href="#__codelineno-21-31"></a>
-<a id="__codelineno-21-32" name="__codelineno-21-32" href="#__codelineno-21-32"></a><span class="w">  </span><span class="cm">/* Get index of parent node of node at index i */</span>
+<a id="__codelineno-21-32" name="__codelineno-21-32" href="#__codelineno-21-32"></a><span class="w">  </span><span class="cm">/* Get the parent index of the node at index i */</span>
 <a id="__codelineno-21-33" name="__codelineno-21-33" href="#__codelineno-21-33"></a><span class="w">  </span><span class="kt">int</span><span class="o">?</span><span class="w"> </span><span class="n">parent</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-34" name="__codelineno-21-34" href="#__codelineno-21-34"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">~/</span><span class="w"> </span><span class="m">2</span><span class="p">;</span>
 <a id="__codelineno-21-35" name="__codelineno-21-35" href="#__codelineno-21-35"></a><span class="w">  </span><span class="p">}</span>
@@ -5547,7 +5619,7 @@
 <a id="__codelineno-21-47" name="__codelineno-21-47" href="#__codelineno-21-47"></a>
 <a id="__codelineno-21-48" name="__codelineno-21-48" href="#__codelineno-21-48"></a><span class="w">  </span><span class="cm">/* Depth-first traversal */</span>
 <a id="__codelineno-21-49" name="__codelineno-21-49" href="#__codelineno-21-49"></a><span class="w">  </span><span class="kt">void</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">String</span><span class="w"> </span><span class="n">order</span><span class="p">,</span><span class="w"> </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">?&gt;</span><span class="w"> </span><span class="n">res</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-50" name="__codelineno-21-50" href="#__codelineno-21-50"></a><span class="w">    </span><span class="c1">// If empty position, return</span>
+<a id="__codelineno-21-50" name="__codelineno-21-50" href="#__codelineno-21-50"></a><span class="w">    </span><span class="c1">// Return if this position is empty</span>
 <a id="__codelineno-21-51" name="__codelineno-21-51" href="#__codelineno-21-51"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-52" name="__codelineno-21-52" href="#__codelineno-21-52"></a><span class="w">      </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-21-53" name="__codelineno-21-53" href="#__codelineno-21-53"></a><span class="w">    </span><span class="p">}</span>
@@ -5591,7 +5663,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_binary_tree.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Binary tree class represented by array */</span>
+<div class="highlight"><span class="filename">array_binary_tree.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Binary tree type using an array representation */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">struct</span><span class="w"> </span><span class="nc">ArrayBinaryTree</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="n">tree</span><span class="p">:</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="nb">Option</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;&gt;</span><span class="p">,</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="p">}</span>
@@ -5602,14 +5674,14 @@
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">        </span><span class="bp">Self</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">tree</span><span class="p">:</span><span class="w"> </span><span class="nc">arr</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a>
-<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">    </span><span class="cm">/* List capacity */</span>
+<a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">    </span><span class="cm">/* Number of positions in the array */</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">tree</span><span class="p">.</span><span class="n">len</span><span class="p">()</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">i32</span>
 <a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a>
-<a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">    </span><span class="cm">/* Get value of node at index i */</span>
+<a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">    </span><span class="cm">/* Get the value of the node at index i */</span>
 <a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">val</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w">        </span><span class="c1">// If index is out of bounds, return None, representing empty position</span>
+<a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w">        </span><span class="c1">// If the index is out of bounds, return None to represent an empty position</span>
 <a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">size</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w">            </span><span class="nb">None</span>
 <a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -5617,17 +5689,17 @@
 <a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-25" name="__codelineno-22-25" href="#__codelineno-22-25"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-26" name="__codelineno-22-26" href="#__codelineno-22-26"></a>
-<a id="__codelineno-22-27" name="__codelineno-22-27" href="#__codelineno-22-27"></a><span class="w">    </span><span class="cm">/* Get index of left child node of node at index i */</span>
+<a id="__codelineno-22-27" name="__codelineno-22-27" href="#__codelineno-22-27"></a><span class="w">    </span><span class="cm">/* Get the left child index of the node at index i */</span>
 <a id="__codelineno-22-28" name="__codelineno-22-28" href="#__codelineno-22-28"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-29" name="__codelineno-22-29" href="#__codelineno-22-29"></a><span class="w">        </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-22-30" name="__codelineno-22-30" href="#__codelineno-22-30"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-31" name="__codelineno-22-31" href="#__codelineno-22-31"></a>
-<a id="__codelineno-22-32" name="__codelineno-22-32" href="#__codelineno-22-32"></a><span class="w">    </span><span class="cm">/* Get index of right child node of node at index i */</span>
+<a id="__codelineno-22-32" name="__codelineno-22-32" href="#__codelineno-22-32"></a><span class="w">    </span><span class="cm">/* Get the right child index of the node at index i */</span>
 <a id="__codelineno-22-33" name="__codelineno-22-33" href="#__codelineno-22-33"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-34" name="__codelineno-22-34" href="#__codelineno-22-34"></a><span class="w">        </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-22-35" name="__codelineno-22-35" href="#__codelineno-22-35"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-36" name="__codelineno-22-36" href="#__codelineno-22-36"></a>
-<a id="__codelineno-22-37" name="__codelineno-22-37" href="#__codelineno-22-37"></a><span class="w">    </span><span class="cm">/* Get index of parent node of node at index i */</span>
+<a id="__codelineno-22-37" name="__codelineno-22-37" href="#__codelineno-22-37"></a><span class="w">    </span><span class="cm">/* Get the parent index of the node at index i */</span>
 <a id="__codelineno-22-38" name="__codelineno-22-38" href="#__codelineno-22-38"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-39" name="__codelineno-22-39" href="#__codelineno-22-39"></a><span class="w">        </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-22-40" name="__codelineno-22-40" href="#__codelineno-22-40"></a><span class="w">    </span><span class="p">}</span>
@@ -5683,7 +5755,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_binary_tree.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Binary tree structure in array representation */</span>
+<div class="highlight"><span class="filename">array_binary_tree.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Binary tree structure represented by an array */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">tree</span><span class="p">;</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">size</span><span class="p">;</span>
@@ -5704,14 +5776,14 @@
 <a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">abt</span><span class="p">);</span>
 <a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="p">}</span>
 <a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a>
-<a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="cm">/* List capacity */</span>
+<a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="cm">/* Array length */</span>
 <a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="kt">int</span><span class="w"> </span><span class="nf">size</span><span class="p">(</span><span class="n">ArrayBinaryTree</span><span class="w"> </span><span class="o">*</span><span class="n">abt</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">abt</span><span class="o">-&gt;</span><span class="n">size</span><span class="p">;</span>
 <a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="p">}</span>
 <a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a>
-<a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a><span class="cm">/* Get value of node at index i */</span>
+<a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a><span class="cm">/* Get the value of the node at index i */</span>
 <a id="__codelineno-23-28" name="__codelineno-23-28" href="#__codelineno-23-28"></a><span class="kt">int</span><span class="w"> </span><span class="nf">val</span><span class="p">(</span><span class="n">ArrayBinaryTree</span><span class="w"> </span><span class="o">*</span><span class="n">abt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-29" name="__codelineno-23-29" href="#__codelineno-23-29"></a><span class="w">    </span><span class="c1">// Return INT_MAX if index out of bounds, representing empty position</span>
+<a id="__codelineno-23-29" name="__codelineno-23-29" href="#__codelineno-23-29"></a><span class="w">    </span><span class="c1">// If the index is out of bounds, return INT_MAX to represent an empty position</span>
 <a id="__codelineno-23-30" name="__codelineno-23-30" href="#__codelineno-23-30"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">abt</span><span class="p">))</span>
 <a id="__codelineno-23-31" name="__codelineno-23-31" href="#__codelineno-23-31"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">;</span>
 <a id="__codelineno-23-32" name="__codelineno-23-32" href="#__codelineno-23-32"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">abt</span><span class="o">-&gt;</span><span class="n">tree</span><span class="p">[</span><span class="n">i</span><span class="p">];</span>
@@ -5721,7 +5793,7 @@
 <a id="__codelineno-23-36" name="__codelineno-23-36" href="#__codelineno-23-36"></a><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="nf">levelOrder</span><span class="p">(</span><span class="n">ArrayBinaryTree</span><span class="w"> </span><span class="o">*</span><span class="n">abt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">returnSize</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-37" name="__codelineno-23-37" href="#__codelineno-23-37"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">abt</span><span class="p">));</span>
 <a id="__codelineno-23-38" name="__codelineno-23-38" href="#__codelineno-23-38"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-23-39" name="__codelineno-23-39" href="#__codelineno-23-39"></a><span class="w">    </span><span class="c1">// Traverse array directly</span>
+<a id="__codelineno-23-39" name="__codelineno-23-39" href="#__codelineno-23-39"></a><span class="w">    </span><span class="c1">// Traverse the array directly</span>
 <a id="__codelineno-23-40" name="__codelineno-23-40" href="#__codelineno-23-40"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">size</span><span class="p">(</span><span class="n">abt</span><span class="p">);</span><span class="w"> </span><span class="n">i</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-41" name="__codelineno-23-41" href="#__codelineno-23-41"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">abt</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">)</span>
 <a id="__codelineno-23-42" name="__codelineno-23-42" href="#__codelineno-23-42"></a><span class="w">            </span><span class="n">res</span><span class="p">[</span><span class="n">index</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span><span class="p">(</span><span class="n">abt</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">);</span>
@@ -5732,7 +5804,7 @@
 <a id="__codelineno-23-47" name="__codelineno-23-47" href="#__codelineno-23-47"></a>
 <a id="__codelineno-23-48" name="__codelineno-23-48" href="#__codelineno-23-48"></a><span class="cm">/* Depth-first traversal */</span>
 <a id="__codelineno-23-49" name="__codelineno-23-49" href="#__codelineno-23-49"></a><span class="kt">void</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">ArrayBinaryTree</span><span class="w"> </span><span class="o">*</span><span class="n">abt</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">i</span><span class="p">,</span><span class="w"> </span><span class="kt">char</span><span class="w"> </span><span class="o">*</span><span class="n">order</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">res</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">index</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-50" name="__codelineno-23-50" href="#__codelineno-23-50"></a><span class="w">    </span><span class="c1">// If empty position, return</span>
+<a id="__codelineno-23-50" name="__codelineno-23-50" href="#__codelineno-23-50"></a><span class="w">    </span><span class="c1">// Return if this position is empty</span>
 <a id="__codelineno-23-51" name="__codelineno-23-51" href="#__codelineno-23-51"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="p">(</span><span class="n">abt</span><span class="p">,</span><span class="w"> </span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">INT_MAX</span><span class="p">)</span>
 <a id="__codelineno-23-52" name="__codelineno-23-52" href="#__codelineno-23-52"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-23-53" name="__codelineno-23-53" href="#__codelineno-23-53"></a><span class="w">    </span><span class="c1">// Preorder traversal</span>
@@ -5777,31 +5849,31 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_binary_tree.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Binary tree class represented by array */</span>
+<div class="highlight"><span class="filename">array_binary_tree.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Binary tree class using an array representation */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">ArrayBinaryTree</span><span class="p">(</span><span class="kd">val</span><span class="w"> </span><span class="nv">tree</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int?</span><span class="o">&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="cm">/* List capacity */</span>
+<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="cm">/* List size */</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">size</span><span class="p">():</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">tree</span><span class="p">.</span><span class="na">size</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a>
-<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="cm">/* Get value of node at index i */</span>
+<a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="cm">/* Get the value of the node at index i */</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">_val</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int?</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">        </span><span class="c1">// If index out of bounds, return null to represent empty position</span>
+<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">        </span><span class="c1">// If the index is out of bounds, return null to represent an empty position</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span><span class="p">())</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">tree</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a>
-<a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">    </span><span class="cm">/* Get index of left child node of node at index i */</span>
+<a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">    </span><span class="cm">/* Get the left child index of the node at index i */</span>
 <a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a>
-<a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w">    </span><span class="cm">/* Get index of right child node of node at index i */</span>
+<a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w">    </span><span class="cm">/* Get the right child index of the node at index i */</span>
 <a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">2</span>
 <a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a>
-<a id="__codelineno-24-25" name="__codelineno-24-25" href="#__codelineno-24-25"></a><span class="w">    </span><span class="cm">/* Get index of parent node of node at index i */</span>
+<a id="__codelineno-24-25" name="__codelineno-24-25" href="#__codelineno-24-25"></a><span class="w">    </span><span class="cm">/* Get the parent index of the node at index i */</span>
 <a id="__codelineno-24-26" name="__codelineno-24-26" href="#__codelineno-24-26"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-27" name="__codelineno-24-27" href="#__codelineno-24-27"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="m">2</span>
 <a id="__codelineno-24-28" name="__codelineno-24-28" href="#__codelineno-24-28"></a><span class="w">    </span><span class="p">}</span>
@@ -5809,7 +5881,7 @@
 <a id="__codelineno-24-30" name="__codelineno-24-30" href="#__codelineno-24-30"></a><span class="w">    </span><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-24-31" name="__codelineno-24-31" href="#__codelineno-24-31"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">levelOrder</span><span class="p">():</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int?</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-32" name="__codelineno-24-32" href="#__codelineno-24-32"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="kt">Int?</span><span class="o">&gt;</span><span class="p">()</span>
-<a id="__codelineno-24-33" name="__codelineno-24-33" href="#__codelineno-24-33"></a><span class="w">        </span><span class="c1">// Traverse array directly</span>
+<a id="__codelineno-24-33" name="__codelineno-24-33" href="#__codelineno-24-33"></a><span class="w">        </span><span class="c1">// Traverse the array directly</span>
 <a id="__codelineno-24-34" name="__codelineno-24-34" href="#__codelineno-24-34"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="m">0.</span><span class="p">.</span><span class="o">&lt;</span><span class="n">size</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-35" name="__codelineno-24-35" href="#__codelineno-24-35"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-24-36" name="__codelineno-24-36" href="#__codelineno-24-36"></a><span class="w">                </span><span class="n">res</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">_val</span><span class="p">(</span><span class="n">i</span><span class="p">))</span>
@@ -5819,7 +5891,7 @@
 <a id="__codelineno-24-40" name="__codelineno-24-40" href="#__codelineno-24-40"></a>
 <a id="__codelineno-24-41" name="__codelineno-24-41" href="#__codelineno-24-41"></a><span class="w">    </span><span class="cm">/* Depth-first traversal */</span>
 <a id="__codelineno-24-42" name="__codelineno-24-42" href="#__codelineno-24-42"></a><span class="w">    </span><span class="kd">fun</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">i</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">,</span><span class="w"> </span><span class="n">order</span><span class="p">:</span><span class="w"> </span><span class="kt">String</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int?</span><span class="o">&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-43" name="__codelineno-24-43" href="#__codelineno-24-43"></a><span class="w">        </span><span class="c1">// If empty position, return</span>
+<a id="__codelineno-24-43" name="__codelineno-24-43" href="#__codelineno-24-43"></a><span class="w">        </span><span class="c1">// Return if this position is empty</span>
 <a id="__codelineno-24-44" name="__codelineno-24-44" href="#__codelineno-24-44"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-24-45" name="__codelineno-24-45" href="#__codelineno-24-45"></a><span class="w">            </span><span class="k">return</span>
 <a id="__codelineno-24-46" name="__codelineno-24-46" href="#__codelineno-24-46"></a><span class="w">        </span><span class="c1">// Preorder traversal</span>
@@ -5859,37 +5931,37 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">array_binary_tree.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Array representation of binary tree class ###</span>
+<div class="highlight"><span class="filename">array_binary_tree.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Binary tree class using an array representation ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">ArrayBinaryTree</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1">### Constructor ###</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">initialize</span><span class="p">(</span><span class="n">arr</span><span class="p">)</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">    </span><span class="vi">@tree</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">arr</span><span class="o">.</span><span class="n">to_a</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a>
-<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">  </span><span class="c1">### List capacity ###</span>
+<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">  </span><span class="c1">### Length of the array representation ###</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">size</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">    </span><span class="vi">@tree</span><span class="o">.</span><span class="n">length</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a>
-<a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">  </span><span class="c1">### Get value of node at index i ###</span>
+<a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">  </span><span class="c1">### Get the value of the node at index i ###</span>
 <a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
-<a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">    </span><span class="c1"># Return nil if index out of bounds, representing empty position</span>
+<a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">    </span><span class="c1"># If the index is out of bounds, return nil to indicate an empty position</span>
 <a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="n">size</span>
 <a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a>
 <a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w">    </span><span class="vi">@tree</span><span class="o">[</span><span class="n">i</span><span class="o">]</span>
 <a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a>
-<a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="w">  </span><span class="c1">### Get left child index of node at index i ###</span>
+<a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="w">  </span><span class="c1">### Get the left child index of the node at index i ###</span>
 <a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">left</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-25-23" name="__codelineno-25-23" href="#__codelineno-25-23"></a><span class="w">    </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-25-24" name="__codelineno-25-24" href="#__codelineno-25-24"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-25" name="__codelineno-25-25" href="#__codelineno-25-25"></a>
-<a id="__codelineno-25-26" name="__codelineno-25-26" href="#__codelineno-25-26"></a><span class="w">  </span><span class="c1">### Get right child index of node at index i ###</span>
+<a id="__codelineno-25-26" name="__codelineno-25-26" href="#__codelineno-25-26"></a><span class="w">  </span><span class="c1">### Get the right child index of the node at index i ###</span>
 <a id="__codelineno-25-27" name="__codelineno-25-27" href="#__codelineno-25-27"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">right</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-25-28" name="__codelineno-25-28" href="#__codelineno-25-28"></a><span class="w">    </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-25-29" name="__codelineno-25-29" href="#__codelineno-25-29"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-30" name="__codelineno-25-30" href="#__codelineno-25-30"></a>
-<a id="__codelineno-25-31" name="__codelineno-25-31" href="#__codelineno-25-31"></a><span class="w">  </span><span class="c1">### Get parent node index of node at index i ###</span>
+<a id="__codelineno-25-31" name="__codelineno-25-31" href="#__codelineno-25-31"></a><span class="w">  </span><span class="c1">### Get the parent index of the node at index i ###</span>
 <a id="__codelineno-25-32" name="__codelineno-25-32" href="#__codelineno-25-32"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">parent</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
 <a id="__codelineno-25-33" name="__codelineno-25-33" href="#__codelineno-25-33"></a><span class="w">    </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">2</span>
 <a id="__codelineno-25-34" name="__codelineno-25-34" href="#__codelineno-25-34"></a><span class="w">  </span><span class="k">end</span>
@@ -5898,7 +5970,7 @@
 <a id="__codelineno-25-37" name="__codelineno-25-37" href="#__codelineno-25-37"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">level_order</span>
 <a id="__codelineno-25-38" name="__codelineno-25-38" href="#__codelineno-25-38"></a><span class="w">    </span><span class="vi">@res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-25-39" name="__codelineno-25-39" href="#__codelineno-25-39"></a>
-<a id="__codelineno-25-40" name="__codelineno-25-40" href="#__codelineno-25-40"></a><span class="w">    </span><span class="c1"># Traverse array directly</span>
+<a id="__codelineno-25-40" name="__codelineno-25-40" href="#__codelineno-25-40"></a><span class="w">    </span><span class="c1"># Traverse the array directly</span>
 <a id="__codelineno-25-41" name="__codelineno-25-41" href="#__codelineno-25-41"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">...</span><span class="n">size</span>
 <a id="__codelineno-25-42" name="__codelineno-25-42" href="#__codelineno-25-42"></a><span class="w">      </span><span class="vi">@res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-25-43" name="__codelineno-25-43" href="#__codelineno-25-43"></a><span class="w">    </span><span class="k">end</span>
@@ -5919,21 +5991,21 @@
 <a id="__codelineno-25-58" name="__codelineno-25-58" href="#__codelineno-25-58"></a><span class="w">    </span><span class="vi">@res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">val</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">order</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="ss">:post</span>
 <a id="__codelineno-25-59" name="__codelineno-25-59" href="#__codelineno-25-59"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-60" name="__codelineno-25-60" href="#__codelineno-25-60"></a>
-<a id="__codelineno-25-61" name="__codelineno-25-61" href="#__codelineno-25-61"></a><span class="w">  </span><span class="c1">### Pre-order traversal ###</span>
+<a id="__codelineno-25-61" name="__codelineno-25-61" href="#__codelineno-25-61"></a><span class="w">  </span><span class="c1">### Preorder traversal ###</span>
 <a id="__codelineno-25-62" name="__codelineno-25-62" href="#__codelineno-25-62"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">pre_order</span>
 <a id="__codelineno-25-63" name="__codelineno-25-63" href="#__codelineno-25-63"></a><span class="w">    </span><span class="vi">@res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-25-64" name="__codelineno-25-64" href="#__codelineno-25-64"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="ss">:pre</span><span class="p">)</span>
 <a id="__codelineno-25-65" name="__codelineno-25-65" href="#__codelineno-25-65"></a><span class="w">    </span><span class="vi">@res</span>
 <a id="__codelineno-25-66" name="__codelineno-25-66" href="#__codelineno-25-66"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-67" name="__codelineno-25-67" href="#__codelineno-25-67"></a>
-<a id="__codelineno-25-68" name="__codelineno-25-68" href="#__codelineno-25-68"></a><span class="w">  </span><span class="c1">### In-order traversal ###</span>
+<a id="__codelineno-25-68" name="__codelineno-25-68" href="#__codelineno-25-68"></a><span class="w">  </span><span class="c1">### Inorder traversal ###</span>
 <a id="__codelineno-25-69" name="__codelineno-25-69" href="#__codelineno-25-69"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">in_order</span>
 <a id="__codelineno-25-70" name="__codelineno-25-70" href="#__codelineno-25-70"></a><span class="w">    </span><span class="vi">@res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-25-71" name="__codelineno-25-71" href="#__codelineno-25-71"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="ss">:in</span><span class="p">)</span>
 <a id="__codelineno-25-72" name="__codelineno-25-72" href="#__codelineno-25-72"></a><span class="w">    </span><span class="vi">@res</span>
 <a id="__codelineno-25-73" name="__codelineno-25-73" href="#__codelineno-25-73"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-74" name="__codelineno-25-74" href="#__codelineno-25-74"></a>
-<a id="__codelineno-25-75" name="__codelineno-25-75" href="#__codelineno-25-75"></a><span class="w">  </span><span class="c1">### Post-order traversal ###</span>
+<a id="__codelineno-25-75" name="__codelineno-25-75" href="#__codelineno-25-75"></a><span class="w">  </span><span class="c1">### Postorder traversal ###</span>
 <a id="__codelineno-25-76" name="__codelineno-25-76" href="#__codelineno-25-76"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">post_order</span>
 <a id="__codelineno-25-77" name="__codelineno-25-77" href="#__codelineno-25-77"></a><span class="w">    </span><span class="vi">@res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-25-78" name="__codelineno-25-78" href="#__codelineno-25-78"></a><span class="w">    </span><span class="n">dfs</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="ss">:post</span><span class="p">)</span>
@@ -5944,18 +6016,23 @@
 </div>
 </div>
 </div>
-<h2 id="733-advantages-and-limitations">7.3.3 &nbsp; Advantages and Limitations<a class="headerlink" href="#733-advantages-and-limitations" title="Permanent link">&para;</a></h2>
-<p>The array representation of binary trees has the following advantages:</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%29%3A%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%20%20%20%20%20%20%20%20self.left%20%3D%20None%0A%20%20%20%20%20%20%20%20self.right%20%3D%20None%0A%0Aclass%20ArrayBinaryTree%3A%0A%20%20%20%20%22%22%22Binary%20tree%20class%20using%20an%20array%20representation%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20arr%3A%20list%5Bint%20%7C%20None%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self._tree%20%3D%20list%28arr%29%0A%0A%20%20%20%20def%20size%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Length%20of%20the%20backing%20list%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self._tree%29%0A%0A%20%20%20%20def%20val%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20value%20of%20the%20node%20at%20index%20i%22%22%22%0A%20%20%20%20%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20self.size%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%20%20%20%20return%20self._tree%5Bi%5D%0A%0A%20%20%20%20def%20left%28self,%20i%3A%20int%29%20-%3E%20int%20%7C%20None%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20left%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%201%0A%0A%20%20%20%20def%20right%28self,%20i%3A%20int%29%20-%3E%20int%20%7C%20None%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20right%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%202%0A%0A%20%20%20%20def%20parent%28self,%20i%3A%20int%29%20-%3E%20int%20%7C%20None%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20parent%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%20%28i%20-%201%29%20//%202%0A%0A%20%20%20%20def%20level_order%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Level-order%20traversal%22%22%22%0A%20%20%20%20%20%20%20%20self.res%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20%23%20Traverse%20the%20array%20directly%0A%20%20%20%20%20%20%20%20for%20i%20in%20range%28self.size%28%29%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20self.val%28i%29%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20self.res.append%28self.val%28i%29%29%0A%20%20%20%20%20%20%20%20return%20self.res%0A%0A%20%20%20%20def%20dfs%28self,%20i%3A%20int,%20order%3A%20str%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Depth-first%20traversal%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.val%28i%29%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%23%20Preorder%20traversal%0A%20%20%20%20%20%20%20%20if%20order%20%3D%3D%20%22pre%22%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.res.append%28self.val%28i%29%29%0A%20%20%20%20%20%20%20%20self.dfs%28self.left%28i%29,%20order%29%0A%20%20%20%20%20%20%20%20%23%20Inorder%20traversal%0A%20%20%20%20%20%20%20%20if%20order%20%3D%3D%20%22in%22%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.res.append%28self.val%28i%29%29%0A%20%20%20%20%20%20%20%20self.dfs%28self.right%28i%29,%20order%29%0A%20%20%20%20%20%20%20%20%23%20Postorder%20traversal%0A%20%20%20%20%20%20%20%20if%20order%20%3D%3D%20%22post%22%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.res.append%28self.val%28i%29%29%0A%0A%20%20%20%20def%20pre_order%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Preorder%20traversal%22%22%22%0A%20%20%20%20%20%20%20%20self.res%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20self.dfs%280,%20order%3D%22pre%22%29%0A%20%20%20%20%20%20%20%20return%20self.res%0A%0A%20%20%20%20def%20in_order%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Inorder%20traversal%22%22%22%0A%20%20%20%20%20%20%20%20self.res%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20self.dfs%280,%20order%3D%22in%22%29%0A%20%20%20%20%20%20%20%20return%20self.res%0A%0A%20%20%20%20def%20post_order%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Postorder%20traversal%22%22%22%0A%20%20%20%20%20%20%20%20self.res%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20self.dfs%280,%20order%3D%22post%22%29%0A%20%20%20%20%20%20%20%20return%20self.res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20binary%20tree%0A%20%20%20%20arr%20%3D%20%5B1,%202,%203,%204,%20None,%206,%20None%5D%0A%20%20%20%20abt%20%3D%20ArrayBinaryTree%28arr%29%0A%0A%20%20%20%20%23%20Access%20a%20node%0A%20%20%20%20i%20%3D%201%0A%20%20%20%20l,%20r,%20p%20%3D%20abt.left%28i%29,%20abt.right%28i%29,%20abt.parent%28i%29%0A%0A%20%20%20%20%23%20Traverse%20the%20tree%0A%20%20%20%20res%20%3D%20abt.level_order%28%29%0A%20%20%20%20res%20%3D%20abt.pre_order%28%29%0A%20%20%20%20res%20%3D%20abt.in_order%28%29%0A%20%20%20%20res%20%3D%20abt.post_order%28%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%29%3A%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%20%20%20%20%20%20%20%20self.left%20%3D%20None%0A%20%20%20%20%20%20%20%20self.right%20%3D%20None%0A%0Aclass%20ArrayBinaryTree%3A%0A%20%20%20%20%22%22%22Binary%20tree%20class%20using%20an%20array%20representation%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20arr%3A%20list%5Bint%20%7C%20None%5D%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self._tree%20%3D%20list%28arr%29%0A%0A%20%20%20%20def%20size%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Length%20of%20the%20backing%20list%22%22%22%0A%20%20%20%20%20%20%20%20return%20len%28self._tree%29%0A%0A%20%20%20%20def%20val%28self,%20i%3A%20int%29%20-%3E%20int%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20value%20of%20the%20node%20at%20index%20i%22%22%22%0A%20%20%20%20%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20self.size%28%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%20%20%20%20return%20self._tree%5Bi%5D%0A%0A%20%20%20%20def%20left%28self,%20i%3A%20int%29%20-%3E%20int%20%7C%20None%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20left%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%201%0A%0A%20%20%20%20def%20right%28self,%20i%3A%20int%29%20-%3E%20int%20%7C%20None%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20right%20child%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%202%20%2A%20i%20%2B%202%0A%0A%20%20%20%20def%20parent%28self,%20i%3A%20int%29%20-%3E%20int%20%7C%20None%3A%0A%20%20%20%20%20%20%20%20%22%22%22Get%20the%20parent%20index%22%22%22%0A%20%20%20%20%20%20%20%20return%20%28i%20-%201%29%20//%202%0A%0A%20%20%20%20def%20level_order%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Level-order%20traversal%22%22%22%0A%20%20%20%20%20%20%20%20self.res%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20%23%20Traverse%20the%20array%20directly%0A%20%20%20%20%20%20%20%20for%20i%20in%20range%28self.size%28%29%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20self.val%28i%29%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20self.res.append%28self.val%28i%29%29%0A%20%20%20%20%20%20%20%20return%20self.res%0A%0A%20%20%20%20def%20dfs%28self,%20i%3A%20int,%20order%3A%20str%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Depth-first%20traversal%22%22%22%0A%20%20%20%20%20%20%20%20if%20self.val%28i%29%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%23%20Preorder%20traversal%0A%20%20%20%20%20%20%20%20if%20order%20%3D%3D%20%22pre%22%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.res.append%28self.val%28i%29%29%0A%20%20%20%20%20%20%20%20self.dfs%28self.left%28i%29,%20order%29%0A%20%20%20%20%20%20%20%20%23%20Inorder%20traversal%0A%20%20%20%20%20%20%20%20if%20order%20%3D%3D%20%22in%22%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.res.append%28self.val%28i%29%29%0A%20%20%20%20%20%20%20%20self.dfs%28self.right%28i%29,%20order%29%0A%20%20%20%20%20%20%20%20%23%20Postorder%20traversal%0A%20%20%20%20%20%20%20%20if%20order%20%3D%3D%20%22post%22%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self.res.append%28self.val%28i%29%29%0A%0A%20%20%20%20def%20pre_order%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Preorder%20traversal%22%22%22%0A%20%20%20%20%20%20%20%20self.res%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20self.dfs%280,%20order%3D%22pre%22%29%0A%20%20%20%20%20%20%20%20return%20self.res%0A%0A%20%20%20%20def%20in_order%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Inorder%20traversal%22%22%22%0A%20%20%20%20%20%20%20%20self.res%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20self.dfs%280,%20order%3D%22in%22%29%0A%20%20%20%20%20%20%20%20return%20self.res%0A%0A%20%20%20%20def%20post_order%28self%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%20%20%20%20%22%22%22Postorder%20traversal%22%22%22%0A%20%20%20%20%20%20%20%20self.res%20%3D%20%5B%5D%0A%20%20%20%20%20%20%20%20self.dfs%280,%20order%3D%22post%22%29%0A%20%20%20%20%20%20%20%20return%20self.res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20binary%20tree%0A%20%20%20%20arr%20%3D%20%5B1,%202,%203,%204,%20None,%206,%20None%5D%0A%20%20%20%20abt%20%3D%20ArrayBinaryTree%28arr%29%0A%0A%20%20%20%20%23%20Access%20a%20node%0A%20%20%20%20i%20%3D%201%0A%20%20%20%20l,%20r,%20p%20%3D%20abt.left%28i%29,%20abt.right%28i%29,%20abt.parent%28i%29%0A%0A%20%20%20%20%23%20Traverse%20the%20tree%0A%20%20%20%20res%20%3D%20abt.level_order%28%29%0A%20%20%20%20res%20%3D%20abt.pre_order%28%29%0A%20%20%20%20res%20%3D%20abt.in_order%28%29%0A%20%20%20%20res%20%3D%20abt.post_order%28%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=5&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h2 id="733-strengths-and-limitations">7.3.3 &nbsp; Strengths and limitations<a class="headerlink" href="#733-strengths-and-limitations" title="Permanent link">&para;</a></h2>
+<p>Array representation offers the following advantages.</p>
 <ul>
-<li>Arrays are stored in contiguous memory space, which is cache-friendly, allowing faster access and traversal.</li>
-<li>It does not require storing pointers, which saves space.</li>
-<li>It allows random access to nodes.</li>
+<li>Contiguous memory is cache-friendly, making access and traversal fast.</li>
+<li>No pointers need to be stored, saving space.</li>
+<li>Nodes can be accessed randomly by index.</li>
 </ul>
-<p>However, the array representation also has some limitations:</p>
+<p>However, it also has limitations.</p>
 <ul>
-<li>Array storage requires contiguous memory space, so it is not suitable for storing trees with a large amount of data.</li>
-<li>Adding or removing nodes requires array insertion and deletion operations, which have lower efficiency.</li>
-<li>When there are many <code>None</code> values in the binary tree, the proportion of node data contained in the array is low, leading to lower space utilization.</li>
+<li>Arrays require contiguous memory, making them unsuitable for extremely large trees.</li>
+<li>Adding and removing nodes requires array insertion and deletion operations, which are inefficient.</li>
+<li>If the tree has many missing nodes, actual node data occupies only a small fraction of the array, wasting space.</li>
 </ul>
 
 <!-- Source file information -->
@@ -5979,7 +6056,7 @@ aria-label="Footer"
   <a
     href="../binary_tree_traversal/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 7.2 Binary Tree Traversal"
+    aria-label="Previous: 7.2   Traversing a binary tree"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -5991,7 +6068,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        7.2 Binary Tree Traversal
+        7.2 &nbsp; Traversing a binary tree
       </div>
     </div>
   </a>
@@ -6003,7 +6080,7 @@ aria-label="Footer"
   <a
     href="../binary_search_tree/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 7.4 Binary Search Tree"
+    aria-label="Next: 7.4   Binary search trees"
     rel="next"
   >
     <div class="md-footer__title">
@@ -6011,7 +6088,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        7.4 Binary Search Tree
+        7.4 &nbsp; Binary search trees
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -6028,7 +6105,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_tree/avl_tree/index.html b/en/chapter_tree/avl_tree/index.html
index 57e0ff088..11fb6ca74 100644
--- a/en/chapter_tree/avl_tree/index.html
+++ b/en/chapter_tree/avl_tree/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>7.5 AVL Tree * - Hello Algo</title>
+        <title>7.5 AVL trees * - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
     <div data-md-component="skip">
 
 
-        <a href="#75-avl-tree" class="md-skip">
+        <a href="#75-avl-trees" class="md-skip">
           Skip to content
         </a>
 
@@ -119,7 +119,7 @@
         <path
             d="M480 32c0-12.9-7.8-24.6-19.8-29.6s-25.7-2.2-34.9 6.9L381.7 53c-48 48-113.1 75-181 75H192 160 64c-35.3 0-64 28.7-64 64v96c0 35.3 28.7 64 64 64l0 128c0 17.7 14.3 32 32 32h64c17.7 0 32-14.3 32-32V352l8.7 0c67.9 0 133 27 181 75l43.6 43.6c9.2 9.2 22.9 11.9 34.9 6.9s19.8-16.6 19.8-29.6V300.4c18.6-8.8 32-32.5 32-60.4s-13.4-51.6-32-60.4V32zm-64 76.7V240 371.3C357.2 317.8 280.5 288 200.7 288H192V192h8.7c79.8 0 156.5-29.8 215.3-83.3z" />
     </svg>
-    <span>Welcome to contribute to Chinese-to-English translation! For more details, please refer to <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">CONTRIBUTING.md</a>.</span>
+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              7.5 &nbsp; AVL Tree *
+              7.5 &nbsp; AVL trees *
 
           </span>
         </div>
@@ -167,9 +167,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Dark mode"  type="radio" name="__palette" id="__palette_0">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -177,9 +177,9 @@
 
 
 
-    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Light mode"  type="radio" name="__palette" id="__palette_1">
+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1898,7 +1898,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1919,7 +1919,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1940,7 +1940,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1967,7 +1967,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1994,7 +1994,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2021,7 +2021,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2057,7 +2057,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2074,7 +2074,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2100,22 +2100,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#751-common-terminology-in-avl-trees" class="md-nav__link">
+  <a href="#751-avl-tree-terminology" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.5.1 &nbsp; Common Terminology in AVL Trees
+        7.5.1 &nbsp; AVL tree terminology
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.5.1 Common Terminology in AVL Trees">
+    <nav class="md-nav" aria-label="7.5.1 AVL tree terminology">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-node-height" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Node Height
+        1. &nbsp; Node height
       </span>
     </span>
   </a>
@@ -2123,10 +2123,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-node-balance-factor" class="md-nav__link">
+  <a href="#2-balance-factor" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Node Balance Factor
+        2. &nbsp; Balance factor
       </span>
     </span>
   </a>
@@ -2139,22 +2139,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#752-rotations-in-avl-trees" class="md-nav__link">
+  <a href="#752-avl-tree-rotations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.5.2 &nbsp; Rotations in AVL Trees
+        7.5.2 &nbsp; AVL tree rotations
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.5.2 Rotations in AVL Trees">
+    <nav class="md-nav" aria-label="7.5.2 AVL tree rotations">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-right-rotation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Right Rotation
+        1. &nbsp; Right rotation
       </span>
     </span>
   </a>
@@ -2165,7 +2165,7 @@
   <a href="#2-left-rotation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Left Rotation
+        2. &nbsp; Left rotation
       </span>
     </span>
   </a>
@@ -2173,10 +2173,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-left-rotation-then-right-rotation" class="md-nav__link">
+  <a href="#3-left-then-right-rotation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Left Rotation Then Right Rotation
+        3. &nbsp; Left-then-right rotation
       </span>
     </span>
   </a>
@@ -2184,10 +2184,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#4-right-rotation-then-left-rotation" class="md-nav__link">
+  <a href="#4-right-then-left-rotation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4. &nbsp; Right Rotation Then Left Rotation
+        4. &nbsp; Right-then-left rotation
       </span>
     </span>
   </a>
@@ -2195,10 +2195,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#5-choice-of-rotation" class="md-nav__link">
+  <a href="#5-choosing-a-rotation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5. &nbsp; Choice of Rotation
+        5. &nbsp; Choosing a rotation
       </span>
     </span>
   </a>
@@ -2211,22 +2211,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#753-common-operations-in-avl-trees" class="md-nav__link">
+  <a href="#753-common-avl-tree-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.5.3 &nbsp; Common Operations in AVL Trees
+        7.5.3 &nbsp; Common AVL tree operations
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.5.3 Common Operations in AVL Trees">
+    <nav class="md-nav" aria-label="7.5.3 Common AVL tree operations">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-node-insertion" class="md-nav__link">
+  <a href="#1-inserting-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Node Insertion
+        1. &nbsp; Inserting a node
       </span>
     </span>
   </a>
@@ -2234,10 +2234,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-node-removal" class="md-nav__link">
+  <a href="#2-deleting-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Node Removal
+        2. &nbsp; Deleting a node
       </span>
     </span>
   </a>
@@ -2245,10 +2245,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-node-search" class="md-nav__link">
+  <a href="#3-finding-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Node Search
+        3. &nbsp; Finding a node
       </span>
     </span>
   </a>
@@ -2261,10 +2261,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#754-typical-applications-of-avl-trees" class="md-nav__link">
+  <a href="#754-typical-applications" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.5.4 &nbsp; Typical Applications of AVL Trees
+        7.5.4 &nbsp; Typical applications
       </span>
     </span>
   </a>
@@ -2294,7 +2294,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2321,7 +2321,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2385,7 +2385,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2406,7 +2406,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2427,7 +2427,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2454,7 +2454,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2481,7 +2481,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2508,7 +2508,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2535,7 +2535,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2599,7 +2599,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2620,7 +2620,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2641,7 +2641,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2668,7 +2668,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2695,7 +2695,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2722,7 +2722,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2749,7 +2749,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2817,7 +2817,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2838,7 +2838,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2859,7 +2859,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2886,7 +2886,7 @@
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -3099,7 +3099,7 @@
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
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@@ -3141,7 +3141,7 @@
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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@@ -4624,7 +4624,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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@@ -4708,6 +4708,78 @@
 
 
 
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                   </div>
@@ -4749,22 +4821,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#751-common-terminology-in-avl-trees" class="md-nav__link">
+  <a href="#751-avl-tree-terminology" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.5.1 &nbsp; Common Terminology in AVL Trees
+        7.5.1 &nbsp; AVL tree terminology
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.5.1 Common Terminology in AVL Trees">
+    <nav class="md-nav" aria-label="7.5.1 AVL tree terminology">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-node-height" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Node Height
+        1. &nbsp; Node height
       </span>
     </span>
   </a>
@@ -4772,10 +4844,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-node-balance-factor" class="md-nav__link">
+  <a href="#2-balance-factor" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Node Balance Factor
+        2. &nbsp; Balance factor
       </span>
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   </a>
@@ -4788,22 +4860,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#752-rotations-in-avl-trees" class="md-nav__link">
+  <a href="#752-avl-tree-rotations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.5.2 &nbsp; Rotations in AVL Trees
+        7.5.2 &nbsp; AVL tree rotations
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.5.2 Rotations in AVL Trees">
+    <nav class="md-nav" aria-label="7.5.2 AVL tree rotations">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-right-rotation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Right Rotation
+        1. &nbsp; Right rotation
       </span>
     </span>
   </a>
@@ -4814,7 +4886,7 @@
   <a href="#2-left-rotation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Left Rotation
+        2. &nbsp; Left rotation
       </span>
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   </a>
@@ -4822,10 +4894,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-left-rotation-then-right-rotation" class="md-nav__link">
+  <a href="#3-left-then-right-rotation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Left Rotation Then Right Rotation
+        3. &nbsp; Left-then-right rotation
       </span>
     </span>
   </a>
@@ -4833,10 +4905,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#4-right-rotation-then-left-rotation" class="md-nav__link">
+  <a href="#4-right-then-left-rotation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4. &nbsp; Right Rotation Then Left Rotation
+        4. &nbsp; Right-then-left rotation
       </span>
     </span>
   </a>
@@ -4844,10 +4916,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#5-choice-of-rotation" class="md-nav__link">
+  <a href="#5-choosing-a-rotation" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        5. &nbsp; Choice of Rotation
+        5. &nbsp; Choosing a rotation
       </span>
     </span>
   </a>
@@ -4860,22 +4932,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#753-common-operations-in-avl-trees" class="md-nav__link">
+  <a href="#753-common-avl-tree-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.5.3 &nbsp; Common Operations in AVL Trees
+        7.5.3 &nbsp; Common AVL tree operations
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.5.3 Common Operations in AVL Trees">
+    <nav class="md-nav" aria-label="7.5.3 Common AVL tree operations">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-node-insertion" class="md-nav__link">
+  <a href="#1-inserting-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Node Insertion
+        1. &nbsp; Inserting a node
       </span>
     </span>
   </a>
@@ -4883,10 +4955,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-node-removal" class="md-nav__link">
+  <a href="#2-deleting-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Node Removal
+        2. &nbsp; Deleting a node
       </span>
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   </a>
@@ -4894,10 +4966,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-node-search" class="md-nav__link">
+  <a href="#3-finding-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Node Search
+        3. &nbsp; Finding a node
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@@ -4910,10 +4982,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#754-typical-applications-of-avl-trees" class="md-nav__link">
+  <a href="#754-typical-applications" class="md-nav__link">
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       <span class="md-typeset">
-        7.5.4 &nbsp; Typical Applications of AVL Trees
+        7.5.4 &nbsp; Typical applications
       </span>
     </span>
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@@ -4957,26 +5029,26 @@
 
 
 <!-- Page content -->
-<h1 id="75-avl-tree">7.5 &nbsp; AVL Tree *<a class="headerlink" href="#75-avl-tree" title="Permanent link">&para;</a></h1>
-<p>In the "Binary Search Tree" section, we mentioned that after multiple insertion and removal operations, a binary search tree may degenerate into a linked list. In this case, the time complexity of all operations degrades from <span class="arithmatex">\(O(\log n)\)</span> to <span class="arithmatex">\(O(n)\)</span>.</p>
-<p>As shown in Figure 7-24, after two node removal operations, this binary search tree will degrade into a linked list.</p>
-<p><img alt="Degradation of an AVL tree after removing nodes" class="animation-figure" src="../avl_tree.assets/avltree_degradation_from_removing_node.png" /></p>
-<p align="center"> Figure 7-24 &nbsp; Degradation of an AVL tree after removing nodes </p>
+<h1 id="75-avl-trees">7.5 &nbsp; AVL trees *<a class="headerlink" href="#75-avl-trees" title="Permanent link">&para;</a></h1>
+<p>The binary-search-tree section explained that repeated insertions and deletions can make a tree degenerate into a linked list. All operations then degrade from <span class="arithmatex">\(O(\log n)\)</span> to <span class="arithmatex">\(O(n)\)</span> time.</p>
+<p>In the example below, just two deletions turn the binary search tree into a linked list.</p>
+<p><img alt="An AVL-tree example degenerating after deletions" class="animation-figure" src="../avl_tree.assets/avltree_degradation_from_removing_node.png" /></p>
+<p align="center"> Figure 7-24 &nbsp; An AVL-tree example degenerating after deletions </p>
 
-<p>For example, in the perfect binary tree shown in Figure 7-25, after inserting two nodes, the tree will lean heavily to the left, and the time complexity of search operations will also degrade.</p>
-<p><img alt="Degradation of an AVL tree after inserting nodes" class="animation-figure" src="../avl_tree.assets/avltree_degradation_from_inserting_node.png" /></p>
-<p align="center"> Figure 7-25 &nbsp; Degradation of an AVL tree after inserting nodes </p>
+<p>Likewise, inserting two nodes into the perfect binary tree below makes it heavily skewed to the left, degrading lookup performance.</p>
+<p><img alt="An AVL-tree example degenerating after insertions" class="animation-figure" src="../avl_tree.assets/avltree_degradation_from_inserting_node.png" /></p>
+<p align="center"> Figure 7-25 &nbsp; An AVL-tree example degenerating after insertions </p>
 
-<p>In 1962, G. M. Adelson-Velsky and E. M. Landis proposed the <u>AVL tree</u> in their paper "An algorithm for the organization of information". The paper describes a series of operations that prevent an AVL tree from degenerating as nodes are inserted and removed, thereby keeping the time complexity of various operations at <span class="arithmatex">\(O(\log n)\)</span>. In other words, in scenarios that require frequent insertion, deletion, lookup, and update operations, AVL trees can maintain consistently efficient performance and therefore have strong practical value.</p>
-<h2 id="751-common-terminology-in-avl-trees">7.5.1 &nbsp; Common Terminology in AVL Trees<a class="headerlink" href="#751-common-terminology-in-avl-trees" title="Permanent link">&para;</a></h2>
-<p>An AVL tree is both a binary search tree and a balanced binary tree, simultaneously satisfying all the properties of these two types of binary trees, hence it is a <u>balanced binary search tree</u>.</p>
-<h3 id="1-node-height">1. &nbsp; Node Height<a class="headerlink" href="#1-node-height" title="Permanent link">&para;</a></h3>
-<p>Since the operations related to AVL trees require obtaining node heights, we need to add a <code>height</code> variable to the node class:</p>
+<p>In 1962, G. M. Adelson-Velsky and E. M. Landis introduced the <u>AVL tree</u> in their paper “An algorithm for the organization of information.” They described operations that prevent degeneration through repeated insertions and deletions, keeping the time complexity of operations at <span class="arithmatex">\(O(\log n)\)</span>. AVL trees therefore maintain efficient data operations when insertions, deletions, lookups, and updates are frequent, making them valuable in practice.</p>
+<h2 id="751-avl-tree-terminology">7.5.1 &nbsp; AVL tree terminology<a class="headerlink" href="#751-avl-tree-terminology" title="Permanent link">&para;</a></h2>
+<p>An AVL tree is both a binary search tree and a balanced binary tree, satisfying the properties of both. It is therefore a <u>balanced binary search tree</u>.</p>
+<h3 id="1-node-height">1. &nbsp; Node height<a class="headerlink" href="#1-node-height" title="Permanent link">&para;</a></h3>
+<p>AVL operations need node heights, so add a <code>height</code> variable to the node class:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">TreeNode</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""AVL tree node"""</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""AVL tree node class"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">val</span>                 <span class="c1"># Node value</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">height</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">0</span>                <span class="c1"># Node height</span>
@@ -4985,7 +5057,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* AVL tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* AVL tree node class */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="k">struct</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">{};</span><span class="w">          </span><span class="c1">// Node value</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w">     </span><span class="c1">// Node height</span>
@@ -4997,7 +5069,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* AVL tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* AVL tree node class */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kd">class</span> <span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w">        </span><span class="c1">// Node value</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="kd">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">height</span><span class="p">;</span><span class="w">     </span><span class="c1">// Node height</span>
@@ -5008,7 +5080,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* AVL tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* AVL tree node class */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">class</span><span class="w"> </span><span class="nf">TreeNode</span><span class="p">(</span><span class="kt">int?</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int?</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w">    </span><span class="c1">// Node value</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">height</span><span class="p">;</span><span class="w">      </span><span class="c1">// Node height</span>
@@ -5018,7 +5090,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* AVL tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* AVL tree node struct */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="nx">Val</span><span class="w">    </span><span class="kt">int</span><span class="w">       </span><span class="c1">// Node value</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">Height</span><span class="w"> </span><span class="kt">int</span><span class="w">       </span><span class="c1">// Node height</span>
@@ -5028,7 +5100,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* AVL tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* AVL tree node class */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">val</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Node value</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">height</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Node height</span>
@@ -5043,10 +5115,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* AVL tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* AVL tree node class */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="nx">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Node value</span>
-<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="nx">height</span><span class="p">;</span><span class="w"> </span><span class="c1">// Node height</span>
+<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="nx">height</span><span class="p">;</span><span class="w"> </span><span class="c1">//Node height</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="nx">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// Left child pointer</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="nx">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// Right child pointer</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">,</span><span class="w"> </span><span class="nx">height</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -5059,7 +5131,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* AVL tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* AVL tree node class */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w">            </span><span class="c1">// Node value</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="nx">height</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span><span class="w">         </span><span class="c1">// Node height</span>
@@ -5067,15 +5139,15 @@
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="nx">right</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span><span class="w"> </span><span class="c1">// Right child pointer</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">val?</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">height?</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">left?</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="nx">right?</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">0</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">val</span><span class="p">;</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">height</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">0</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">height</span><span class="p">;</span><span class="w"> </span>
-<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">left</span><span class="p">;</span><span class="w"> </span>
-<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">right</span><span class="p">;</span><span class="w"> </span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">height</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">0</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">height</span><span class="p">;</span>
+<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">left</span><span class="p">;</span>
+<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">right</span><span class="p">;</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* AVL tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* AVL tree node class */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w">         </span><span class="c1">// Node value</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">height</span><span class="p">;</span><span class="w">      </span><span class="c1">// Node height</span>
@@ -5089,7 +5161,7 @@
 <div class="highlight"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="k">use</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">rc</span><span class="p">::</span><span class="n">Rc</span><span class="p">;</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">use</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cell</span><span class="p">::</span><span class="n">RefCell</span><span class="p">;</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a>
-<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="cm">/* AVL tree node */</span>
+<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="cm">/* AVL tree node struct */</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="k">struct</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w">                               </span><span class="c1">// Node value</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="n">height</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w">                            </span><span class="c1">// Node height</span>
@@ -5111,7 +5183,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* AVL tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* AVL tree node struct */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">height</span><span class="p">;</span>
@@ -5133,7 +5205,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* AVL tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* AVL tree node class */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">TreeNode</span><span class="p">(</span><span class="kd">val</span><span class="w"> </span><span class="nv">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w">  </span><span class="c1">// Node value</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">height</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="w">          </span><span class="c1">// Node height</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">left</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="w">   </span><span class="c1">// Left child</span>
@@ -5158,80 +5230,80 @@
 </div>
 </div>
 </div>
-<p>The "node height" refers to the distance from that node to its farthest leaf node, i.e., the number of edges on the path. It is important to note that the height of a leaf node is <span class="arithmatex">\(0\)</span>, and the height of a null node is <span class="arithmatex">\(-1\)</span>. We will create two utility functions for getting and updating the height of a node:</p>
+<p>A node's height is the distance to its farthest leaf, measured in edges. A leaf has height <span class="arithmatex">\(0\)</span>, while a missing node has height <span class="arithmatex">\(-1\)</span>. We create two helper functions to retrieve and update a node's height:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">avl_tree.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">height</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">node</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Get node height"""</span>
-<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># Empty node height is -1, leaf node height is 0</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Get the node height"""</span>
+<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># An empty subtree has height -1; a leaf has height 0</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="k">if</span> <span class="n">node</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>        <span class="k">return</span> <span class="n">node</span><span class="o">.</span><span class="n">height</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="k">return</span> <span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a><span class="k">def</span><span class="w"> </span><span class="nf">update_height</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">node</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">):</span>
-<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a><span class="w">    </span><span class="sd">"""Update node height"""</span>
-<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>    <span class="c1"># Node height equals the height of the tallest subtree + 1</span>
+<a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a><span class="w">    </span><span class="sd">"""Update the node height"""</span>
+<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>    <span class="c1"># The node height equals the height of its taller subtree + 1</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>    <span class="n">node</span><span class="o">.</span><span class="n">height</span> <span class="o">=</span> <span class="nb">max</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">),</span> <span class="bp">self</span><span class="o">.</span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="p">)])</span> <span class="o">+</span> <span class="mi">1</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Get node height */</span>
+<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Get the node height */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">height</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// Empty node height is -1, leaf node height is 0</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// A null node has height -1; a leaf has height 0</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="mi">-1</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">height</span><span class="p">;</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="p">}</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a>
-<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="cm">/* Update node height */</span>
+<a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="cm">/* Update the node height */</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="kt">void</span><span class="w"> </span><span class="nf">updateHeight</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="c1">// Node height equals the height of the tallest subtree + 1</span>
+<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="c1">// The node height equals the height of its taller subtree + 1</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">    </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">),</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Get node height */</span>
+<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Get the node height */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">height</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// Empty node height is -1, leaf node height is 0</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// A null node has height -1; a leaf has height 0</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">height</span><span class="p">;</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="p">}</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a>
-<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="cm">/* Update node height */</span>
+<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="cm">/* Update the node height */</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="kt">void</span><span class="w"> </span><span class="nf">updateHeight</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="c1">// Node height equals the height of the tallest subtree + 1</span>
+<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="c1">// The node height equals the height of its taller subtree + 1</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="n">node</span><span class="p">.</span><span class="na">height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="na">max</span><span class="p">(</span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">),</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Get node height */</span>
+<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Get the node height */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">Height</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// Empty node height is -1, leaf node height is 0</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// A null node has height -1; a leaf has height 0</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">height</span><span class="p">;</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="p">}</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a>
-<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="cm">/* Update node height */</span>
+<a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="cm">/* Update the node height */</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="k">void</span><span class="w"> </span><span class="nf">UpdateHeight</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="c1">// Node height equals the height of the tallest subtree + 1</span>
+<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="c1">// The node height equals the height of its taller subtree + 1</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="n">node</span><span class="p">.</span><span class="n">height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Math</span><span class="p">.</span><span class="n">Max</span><span class="p">(</span><span class="n">Height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">),</span><span class="w"> </span><span class="n">Height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Get node height */</span>
+<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Get the node height */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">t</span><span class="w"> </span><span class="o">*</span><span class="nx">aVLTree</span><span class="p">)</span><span class="w"> </span><span class="nx">height</span><span class="p">(</span><span class="nx">node</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// Empty node height is -1, leaf node height is 0</span>
+<a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="c1">// A nil node has height -1; a leaf has height 0</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Height</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="p">}</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a>
-<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="cm">/* Update node height */</span>
+<a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="cm">/* Update the node height */</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">t</span><span class="w"> </span><span class="o">*</span><span class="nx">aVLTree</span><span class="p">)</span><span class="w"> </span><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">    </span><span class="nx">lh</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">height</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="p">)</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">    </span><span class="nx">rh</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">height</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span><span class="p">)</span>
-<a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">    </span><span class="c1">// Node height equals the height of the tallest subtree + 1</span>
+<a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">    </span><span class="c1">// The node height equals the height of its taller subtree + 1</span>
 <a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">lh</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">rh</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">        </span><span class="nx">node</span><span class="p">.</span><span class="nx">Height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">lh</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -5241,99 +5313,99 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Get node height */</span>
+<div class="highlight"><span class="filename">avl_tree.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Get the node height */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">height</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// Empty node height is -1, leaf node height is 0</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// A nil node has height -1; a leaf has height 0</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="n">node</span><span class="p">?.</span><span class="n">height</span><span class="w"> </span><span class="p">??</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="p">}</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a>
-<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="cm">/* Update node height */</span>
+<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="cm">/* Update the node height */</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="kd">func</span><span class="w"> </span><span class="nf">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="c1">// Node height equals the height of the tallest subtree + 1</span>
+<a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="c1">// The node height equals the height of its taller subtree + 1</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="n">node</span><span class="p">?.</span><span class="n">height</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="bp">max</span><span class="p">(</span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">left</span><span class="p">),</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">right</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Get node height */</span>
+<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Get the node height */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="nx">height</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// Empty node height is -1, leaf node height is 0</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// A null node has height -1; a leaf has height 0</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">height</span><span class="p">;</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="p">}</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a>
-<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="cm">/* Update node height */</span>
+<a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="cm">/* Update the node height */</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="n">#updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">    </span><span class="c1">// Node height equals the height of the tallest subtree + 1</span>
+<a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">    </span><span class="c1">// The node height equals the height of its taller subtree + 1</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="nx">node</span><span class="p">.</span><span class="nx">height</span><span class="w"> </span><span class="o">=</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">        </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">height</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">),</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">height</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Get node height */</span>
+<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Get the node height */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="nx">height</span><span class="p">(</span><span class="nx">node</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// Empty node height is -1, leaf node height is 0</span>
+<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// A null node has height -1; a leaf has height 0</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="o">-</span><span class="nx">1</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">node.height</span><span class="p">;</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="p">}</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a>
-<a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="cm">/* Update node height */</span>
+<a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="cm">/* Update the node height */</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="c1">// Node height equals the height of the tallest subtree + 1</span>
+<a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="c1">// The node height equals the height of its taller subtree + 1</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="nx">node</span><span class="p">.</span><span class="nx">height</span><span class="w"> </span><span class="o">=</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">        </span><span class="nb">Math</span><span class="p">.</span><span class="nx">max</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">height</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">),</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">height</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mf">1</span><span class="p">;</span>
 <a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Get node height */</span>
+<div class="highlight"><span class="filename">avl_tree.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Get the node height */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// Empty node height is -1, leaf node height is 0</span>
+<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// A null node has height -1; a leaf has height 0</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">height</span><span class="p">;</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="p">}</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a>
-<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="cm">/* Update node height */</span>
+<a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="cm">/* Update the node height */</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="kt">void</span><span class="w"> </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="c1">// Node height equals the height of the tallest subtree + 1</span>
+<a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="c1">// The node height equals the height of its taller subtree + 1</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="n">node</span><span class="o">!</span><span class="p">.</span><span class="n">height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">),</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">;</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Get node height */</span>
+<div class="highlight"><span class="filename">avl_tree.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Get the node height */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">height</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="nc">OptionTreeNodeRc</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// Empty node height is -1, leaf node height is 0</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// A null node has height -1; a leaf has height 0</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">match</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">        </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">height</span><span class="p">,</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">        </span><span class="nb">None</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">,</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="p">}</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a>
-<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="cm">/* Update node height */</span>
+<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="cm">/* Update the node height */</span>
 <a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="k">fn</span><span class="w"> </span><span class="nf">update_height</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="nc">OptionTreeNodeRc</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
-<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">        </span><span class="c1">// Node height equals the height of the tallest subtree + 1</span>
+<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">        </span><span class="c1">// The node height equals the height of its taller subtree + 1</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cmp</span><span class="p">::</span><span class="n">max</span><span class="p">(</span><span class="bp">Self</span><span class="p">::</span><span class="n">height</span><span class="p">(</span><span class="n">left</span><span class="p">),</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">height</span><span class="p">(</span><span class="n">right</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Get node height */</span>
+<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Get the node height */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">height</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// Empty node height is -1, leaf node height is 0</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// A null node has height -1; a leaf has height 0</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">height</span><span class="p">;</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="mi">-1</span><span class="p">;</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="p">}</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a>
-<a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="cm">/* Update node height */</span>
+<a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="cm">/* Update the node height */</span>
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="kt">void</span><span class="w"> </span><span class="nf">updateHeight</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">lh</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">);</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">rh</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">);</span>
-<a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">    </span><span class="c1">// Node height equals the height of the tallest subtree + 1</span>
+<a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">    </span><span class="c1">// The node height equals the height of its taller subtree + 1</span>
 <a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">lh</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">rh</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w">        </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">lh</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>
 <a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -5343,45 +5415,45 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Get node height */</span>
+<div class="highlight"><span class="filename">avl_tree.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Get the node height */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">height</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// Empty node height is -1, leaf node height is 0</span>
+<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// A null node has height -1; a leaf has height 0</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="o">?.</span><span class="na">height</span><span class="w"> </span><span class="o">?:</span><span class="w"> </span><span class="o">-</span><span class="m">1</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="p">}</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a>
-<a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="cm">/* Update node height */</span>
+<a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="cm">/* Update the node height */</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="c1">// Node height equals the height of the tallest subtree + 1</span>
+<a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="c1">// The node height equals the height of its taller subtree + 1</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="n">node</span><span class="o">?.</span><span class="na">height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">max</span><span class="p">(</span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="o">?.</span><span class="na">left</span><span class="p">),</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="o">?.</span><span class="na">right</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Get node height ###</span>
+<div class="highlight"><span class="filename">avl_tree.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Get the node height ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">height</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># Empty node height is -1, leaf node height is 0</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># A nil node has height -1; a leaf has height 0</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">height</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="k">end</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a>
-<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="c1">### Update node height ###</span>
+<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="c1">### Update the node height ###</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="k">def</span><span class="w"> </span><span class="nf">update_height</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">  </span><span class="c1"># Node height equals the height of the tallest subtree + 1</span>
+<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">  </span><span class="c1"># The node height equals the height of its taller subtree + 1</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">  </span><span class="n">node</span><span class="o">.</span><span class="n">height</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">),</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="p">)</span><span class="o">].</span><span class="n">max</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h3 id="2-node-balance-factor">2. &nbsp; Node Balance Factor<a class="headerlink" href="#2-node-balance-factor" title="Permanent link">&para;</a></h3>
-<p>The <u>balance factor</u> of a node is defined as the height of the node's left subtree minus the height of its right subtree, and the balance factor of a null node is defined as <span class="arithmatex">\(0\)</span>. We also encapsulate the function to obtain the node's balance factor for convenient subsequent use:</p>
+<h3 id="2-balance-factor">2. &nbsp; Balance factor<a class="headerlink" href="#2-balance-factor" title="Permanent link">&para;</a></h3>
+<p>A node's <u>balance factor</u> is its left subtree's height minus its right subtree's height. A missing node has balance factor <span class="arithmatex">\(0\)</span>. Wrap this calculation in a function for convenient use:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">avl_tree.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">balance_factor</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">node</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
-<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Get balance factor"""</span>
-<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="c1"># Empty node balance factor is 0</span>
+<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Get the balance factor"""</span>
+<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="c1"># An empty subtree has balance factor 0</span>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="k">if</span> <span class="n">node</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>        <span class="k">return</span> <span class="mi">0</span>
 <a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="c1"># Node balance factor = left subtree height - right subtree height</span>
@@ -5389,9 +5461,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Get balance factor */</span>
+<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Get the balance factor */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">balanceFactor</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="c1">// Empty node balance factor is 0</span>
+<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="c1">// A null node has balance factor 0</span>
 <a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span>
 <a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="c1">// Node balance factor = left subtree height - right subtree height</span>
@@ -5400,9 +5472,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Get balance factor */</span>
+<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Get the balance factor */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">balanceFactor</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="c1">// Empty node balance factor is 0</span>
+<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="c1">// A null node has balance factor 0</span>
 <a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="c1">// Node balance factor = left subtree height - right subtree height</span>
@@ -5411,9 +5483,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Get balance factor */</span>
+<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Get the balance factor */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">BalanceFactor</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="c1">// Empty node balance factor is 0</span>
+<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="c1">// A null node has balance factor 0</span>
 <a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">    </span><span class="c1">// Node balance factor = left subtree height - right subtree height</span>
 <a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nf">Height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">Height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">);</span>
@@ -5421,9 +5493,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Get balance factor */</span>
+<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Get the balance factor */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">t</span><span class="w"> </span><span class="o">*</span><span class="nx">aVLTree</span><span class="p">)</span><span class="w"> </span><span class="nx">balanceFactor</span><span class="p">(</span><span class="nx">node</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="c1">// Empty node balance factor is 0</span>
+<a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="c1">// A nil node has balance factor 0</span>
 <a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span>
 <a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">    </span><span class="p">}</span>
@@ -5433,9 +5505,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Get balance factor */</span>
+<div class="highlight"><span class="filename">avl_tree.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Get the balance factor */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="c1">// Empty node balance factor is 0</span>
+<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="c1">// A nil node has balance factor 0</span>
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="k">guard</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">    </span><span class="c1">// Node balance factor = left subtree height - right subtree height</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="kr">left</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="kr">right</span><span class="p">)</span>
@@ -5443,9 +5515,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Get balance factor */</span>
+<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Get the balance factor */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="nx">balanceFactor</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="c1">// Empty node balance factor is 0</span>
+<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="c1">// A null node has balance factor 0</span>
 <a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="c1">// Node balance factor = left subtree height - right subtree height</span>
 <a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">height</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">height</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">);</span>
@@ -5453,9 +5525,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Get balance factor */</span>
+<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Get the balance factor */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="nx">balanceFactor</span><span class="p">(</span><span class="nx">node</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="c1">// Empty node balance factor is 0</span>
+<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="c1">// A null node has balance factor 0</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="mf">0</span><span class="p">;</span>
 <a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="c1">// Node balance factor = left subtree height - right subtree height</span>
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">height</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">height</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">);</span>
@@ -5463,9 +5535,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Get balance factor */</span>
+<div class="highlight"><span class="filename">avl_tree.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Get the balance factor */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kt">int</span><span class="w"> </span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="c1">// Empty node balance factor is 0</span>
+<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="c1">// A null node has balance factor 0</span>
 <a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="m">0</span><span class="p">;</span>
 <a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="c1">// Node balance factor = left subtree height - right subtree height</span>
 <a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">);</span>
@@ -5473,10 +5545,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Get balance factor */</span>
+<div class="highlight"><span class="filename">avl_tree.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Get the balance factor */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">balance_factor</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="nc">OptionTreeNodeRc</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="kt">i32</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="k">match</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">        </span><span class="c1">// Empty node balance factor is 0</span>
+<a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">        </span><span class="c1">// A null node has balance factor 0</span>
 <a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">        </span><span class="nb">None</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span>
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">        </span><span class="c1">// Node balance factor = left subtree height - right subtree height</span>
 <a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">        </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
@@ -5487,9 +5559,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Get balance factor */</span>
+<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Get the balance factor */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="kt">int</span><span class="w"> </span><span class="nf">balanceFactor</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="c1">// Empty node balance factor is 0</span>
+<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="w">    </span><span class="c1">// A null node has balance factor 0</span>
 <a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">    </span><span class="p">}</span>
@@ -5499,9 +5571,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Get balance factor */</span>
+<div class="highlight"><span class="filename">avl_tree.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Get the balance factor */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">):</span><span class="w"> </span><span class="kt">Int</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="c1">// Empty node balance factor is 0</span>
+<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="c1">// A null node has balance factor 0</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="m">0</span>
 <a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">    </span><span class="c1">// Node balance factor = left subtree height - right subtree height</span>
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">height</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">)</span>
@@ -5509,9 +5581,9 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Get balance factor ###</span>
+<div class="highlight"><span class="filename">avl_tree.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Get the balance factor ###</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">balance_factor</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1"># Empty node balance factor is 0</span>
+<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1"># A nil node has balance factor 0</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a>
 <a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="c1"># Node balance factor = left subtree height - right subtree height</span>
@@ -5523,17 +5595,17 @@
 </div>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>Let the balance factor be <span class="arithmatex">\(f\)</span>, then the balance factor of any node in an AVL tree satisfies <span class="arithmatex">\(-1 \le f \le 1\)</span>.</p>
+<p>If a node's balance factor is <span class="arithmatex">\(f\)</span>, every node in an AVL tree satisfies <span class="arithmatex">\(-1 \le f \le 1\)</span>.</p>
 </div>
-<h2 id="752-rotations-in-avl-trees">7.5.2 &nbsp; Rotations in AVL Trees<a class="headerlink" href="#752-rotations-in-avl-trees" title="Permanent link">&para;</a></h2>
-<p>The characteristic of AVL trees lies in the "rotation" operation, which can restore balance to unbalanced nodes without affecting the inorder traversal sequence of the binary tree. In other words, <strong>rotation operations can both maintain the property of a "binary search tree" and make the tree return to a "balanced binary tree"</strong>.</p>
-<p>We call nodes with a balance factor absolute value <span class="arithmatex">\(&gt; 1\)</span> "unbalanced nodes". Depending on the imbalance situation, rotation operations are divided into four types: right rotation, left rotation, right rotation then left rotation, and left rotation then right rotation. Below we describe these rotation operations in detail.</p>
-<h3 id="1-right-rotation">1. &nbsp; Right Rotation<a class="headerlink" href="#1-right-rotation" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 7-26, the value below the node is the balance factor. From bottom to top, the first unbalanced node in the binary tree is "node 3". We focus on the subtree with this unbalanced node as the root, denoting the node as <code>node</code> and its left child as <code>child</code>, and perform a "right rotation" operation. After the right rotation is completed, the subtree regains balance and still maintains the properties of a binary search tree.</p>
+<h2 id="752-avl-tree-rotations">7.5.2 &nbsp; AVL tree rotations<a class="headerlink" href="#752-avl-tree-rotations" title="Permanent link">&para;</a></h2>
+<p>The defining operation of an AVL tree is rotation, which restores balance without changing the inorder traversal sequence. <strong>A rotation preserves the binary-search-tree property while restoring the balanced-tree property</strong>.</p>
+<p>A node whose balance factor has absolute value <span class="arithmatex">\(&gt; 1\)</span> is “unbalanced.” Depending on the imbalance, we use one of four rotations: right, left, right-then-left, or left-then-right. Let us examine them in detail.</p>
+<h3 id="1-right-rotation">1. &nbsp; Right rotation<a class="headerlink" href="#1-right-rotation" title="Permanent link">&para;</a></h3>
+<p>In Figure 7-26, balance factors appear beneath the nodes. Viewed from the bottom up, node 3 is the first unbalanced node. Focus on the subtree rooted there, naming its root <code>node</code> and its left child <code>child</code>. A right rotation restores balance while preserving the binary-search-tree property.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:4"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">&lt;1&gt;</label><label for="__tabbed_4_2">&lt;2&gt;</label><label for="__tabbed_4_3">&lt;3&gt;</label><label for="__tabbed_4_4">&lt;4&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Steps of right rotation" class="animation-figure" src="../avl_tree.assets/avltree_right_rotate_step1.png" /></p>
+<p><img alt="Steps of a right rotation" class="animation-figure" src="../avl_tree.assets/avltree_right_rotate_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="avltree_right_rotate_step2" class="animation-figure" src="../avl_tree.assets/avltree_right_rotate_step2.png" /></p>
@@ -5546,172 +5618,172 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 7-26 &nbsp; Steps of right rotation </p>
+<p align="center"> Figure 7-26 &nbsp; Steps of a right rotation </p>
 
-<p>As shown in Figure 7-27, when the <code>child</code> node has a right child (denoted as <code>grand_child</code>), a step needs to be added in the right rotation: set <code>grand_child</code> as the left child of <code>node</code>.</p>
-<p><img alt="Right rotation with grand_child" class="animation-figure" src="../avl_tree.assets/avltree_right_rotate_with_grandchild.png" /></p>
-<p align="center"> Figure 7-27 &nbsp; Right rotation with grand_child </p>
+<p>If <code>child</code> has a right child, denoted by <code>grand_child</code>, add one step to the rotation: make <code>grand_child</code> the left child of <code>node</code>, as shown below.</p>
+<p><img alt="A right rotation with grand_child" class="animation-figure" src="../avl_tree.assets/avltree_right_rotate_with_grandchild.png" /></p>
+<p align="center"> Figure 7-27 &nbsp; A right rotation with grand_child </p>
 
-<p>"Right rotation" is a figurative term; in practice, it is achieved by modifying node pointers, as shown in the following code:</p>
+<p>“Rotating right” is a visual description; the implementation updates node pointers:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">avl_tree.py</span><pre><span></span><code><a id="__codelineno-39-1" name="__codelineno-39-1" href="#__codelineno-39-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">right_rotate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">node</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Right rotation operation"""</span>
+<a id="__codelineno-39-2" name="__codelineno-39-2" href="#__codelineno-39-2"></a><span class="w">    </span><span class="sd">"""Right rotation"""</span>
 <a id="__codelineno-39-3" name="__codelineno-39-3" href="#__codelineno-39-3"></a>    <span class="n">child</span> <span class="o">=</span> <span class="n">node</span><span class="o">.</span><span class="n">left</span>
 <a id="__codelineno-39-4" name="__codelineno-39-4" href="#__codelineno-39-4"></a>    <span class="n">grand_child</span> <span class="o">=</span> <span class="n">child</span><span class="o">.</span><span class="n">right</span>
-<a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>    <span class="c1"># Using child as pivot, rotate node to the right</span>
+<a id="__codelineno-39-5" name="__codelineno-39-5" href="#__codelineno-39-5"></a>    <span class="c1"># Rotate node to the right around child</span>
 <a id="__codelineno-39-6" name="__codelineno-39-6" href="#__codelineno-39-6"></a>    <span class="n">child</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="n">node</span>
 <a id="__codelineno-39-7" name="__codelineno-39-7" href="#__codelineno-39-7"></a>    <span class="n">node</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">grand_child</span>
-<a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a>    <span class="c1"># Update node height</span>
+<a id="__codelineno-39-8" name="__codelineno-39-8" href="#__codelineno-39-8"></a>    <span class="c1"># Update the node height</span>
 <a id="__codelineno-39-9" name="__codelineno-39-9" href="#__codelineno-39-9"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">update_height</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-39-10" name="__codelineno-39-10" href="#__codelineno-39-10"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">update_height</span><span class="p">(</span><span class="n">child</span><span class="p">)</span>
-<a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a>    <span class="c1"># Return root node of subtree after rotation</span>
+<a id="__codelineno-39-11" name="__codelineno-39-11" href="#__codelineno-39-11"></a>    <span class="c1"># Return the root of the rotated subtree</span>
 <a id="__codelineno-39-12" name="__codelineno-39-12" href="#__codelineno-39-12"></a>    <span class="k">return</span> <span class="n">child</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Right rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-40-1" name="__codelineno-40-1" href="#__codelineno-40-1"></a><span class="cm">/* Right rotation */</span>
 <a id="__codelineno-40-2" name="__codelineno-40-2" href="#__codelineno-40-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">rightRotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-40-3" name="__codelineno-40-3" href="#__codelineno-40-3"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-40-4" name="__codelineno-40-4" href="#__codelineno-40-4"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the right</span>
+<a id="__codelineno-40-5" name="__codelineno-40-5" href="#__codelineno-40-5"></a><span class="w">    </span><span class="c1">// Rotate node to the right around child</span>
 <a id="__codelineno-40-6" name="__codelineno-40-6" href="#__codelineno-40-6"></a><span class="w">    </span><span class="n">child</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-40-7" name="__codelineno-40-7" href="#__codelineno-40-7"></a><span class="w">    </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grandChild</span><span class="p">;</span>
-<a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-40-8" name="__codelineno-40-8" href="#__codelineno-40-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-40-9" name="__codelineno-40-9" href="#__codelineno-40-9"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-40-10" name="__codelineno-40-10" href="#__codelineno-40-10"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">child</span><span class="p">);</span>
-<a id="__codelineno-40-11" name="__codelineno-40-11" href="#__codelineno-40-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-40-11" name="__codelineno-40-11" href="#__codelineno-40-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-40-12" name="__codelineno-40-12" href="#__codelineno-40-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-40-13" name="__codelineno-40-13" href="#__codelineno-40-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Right rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-41-1" name="__codelineno-41-1" href="#__codelineno-41-1"></a><span class="cm">/* Right rotation */</span>
 <a id="__codelineno-41-2" name="__codelineno-41-2" href="#__codelineno-41-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">rightRotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-41-3" name="__codelineno-41-3" href="#__codelineno-41-3"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">;</span>
 <a id="__codelineno-41-4" name="__codelineno-41-4" href="#__codelineno-41-4"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">.</span><span class="na">right</span><span class="p">;</span>
-<a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the right</span>
+<a id="__codelineno-41-5" name="__codelineno-41-5" href="#__codelineno-41-5"></a><span class="w">    </span><span class="c1">// Rotate node to the right around child</span>
 <a id="__codelineno-41-6" name="__codelineno-41-6" href="#__codelineno-41-6"></a><span class="w">    </span><span class="n">child</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-41-7" name="__codelineno-41-7" href="#__codelineno-41-7"></a><span class="w">    </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grandChild</span><span class="p">;</span>
-<a id="__codelineno-41-8" name="__codelineno-41-8" href="#__codelineno-41-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-41-8" name="__codelineno-41-8" href="#__codelineno-41-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-41-9" name="__codelineno-41-9" href="#__codelineno-41-9"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-41-10" name="__codelineno-41-10" href="#__codelineno-41-10"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">child</span><span class="p">);</span>
-<a id="__codelineno-41-11" name="__codelineno-41-11" href="#__codelineno-41-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-41-11" name="__codelineno-41-11" href="#__codelineno-41-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-41-12" name="__codelineno-41-12" href="#__codelineno-41-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-41-13" name="__codelineno-41-13" href="#__codelineno-41-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Right rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-42-1" name="__codelineno-42-1" href="#__codelineno-42-1"></a><span class="cm">/* Right rotation */</span>
 <a id="__codelineno-42-2" name="__codelineno-42-2" href="#__codelineno-42-2"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">RightRotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-42-3" name="__codelineno-42-3" href="#__codelineno-42-3"></a><span class="w">    </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">?.</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-42-4" name="__codelineno-42-4" href="#__codelineno-42-4"></a><span class="w">    </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="o">?.</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the right</span>
+<a id="__codelineno-42-5" name="__codelineno-42-5" href="#__codelineno-42-5"></a><span class="w">    </span><span class="c1">// Rotate node to the right around child</span>
 <a id="__codelineno-42-6" name="__codelineno-42-6" href="#__codelineno-42-6"></a><span class="w">    </span><span class="n">child</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-42-7" name="__codelineno-42-7" href="#__codelineno-42-7"></a><span class="w">    </span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grandChild</span><span class="p">;</span>
-<a id="__codelineno-42-8" name="__codelineno-42-8" href="#__codelineno-42-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-42-8" name="__codelineno-42-8" href="#__codelineno-42-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-42-9" name="__codelineno-42-9" href="#__codelineno-42-9"></a><span class="w">    </span><span class="n">UpdateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-42-10" name="__codelineno-42-10" href="#__codelineno-42-10"></a><span class="w">    </span><span class="n">UpdateHeight</span><span class="p">(</span><span class="n">child</span><span class="p">);</span>
-<a id="__codelineno-42-11" name="__codelineno-42-11" href="#__codelineno-42-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-42-11" name="__codelineno-42-11" href="#__codelineno-42-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-42-12" name="__codelineno-42-12" href="#__codelineno-42-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-42-13" name="__codelineno-42-13" href="#__codelineno-42-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Right rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-43-1" name="__codelineno-43-1" href="#__codelineno-43-1"></a><span class="cm">/* Right rotation */</span>
 <a id="__codelineno-43-2" name="__codelineno-43-2" href="#__codelineno-43-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">t</span><span class="w"> </span><span class="o">*</span><span class="nx">aVLTree</span><span class="p">)</span><span class="w"> </span><span class="nx">rightRotate</span><span class="p">(</span><span class="nx">node</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-43-3" name="__codelineno-43-3" href="#__codelineno-43-3"></a><span class="w">    </span><span class="nx">child</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span>
 <a id="__codelineno-43-4" name="__codelineno-43-4" href="#__codelineno-43-4"></a><span class="w">    </span><span class="nx">grandChild</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">child</span><span class="p">.</span><span class="nx">Right</span>
-<a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the right</span>
+<a id="__codelineno-43-5" name="__codelineno-43-5" href="#__codelineno-43-5"></a><span class="w">    </span><span class="c1">// Rotate node to the right around child</span>
 <a id="__codelineno-43-6" name="__codelineno-43-6" href="#__codelineno-43-6"></a><span class="w">    </span><span class="nx">child</span><span class="p">.</span><span class="nx">Right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span>
 <a id="__codelineno-43-7" name="__codelineno-43-7" href="#__codelineno-43-7"></a><span class="w">    </span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grandChild</span>
-<a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-43-8" name="__codelineno-43-8" href="#__codelineno-43-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-43-9" name="__codelineno-43-9" href="#__codelineno-43-9"></a><span class="w">    </span><span class="nx">t</span><span class="p">.</span><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span>
 <a id="__codelineno-43-10" name="__codelineno-43-10" href="#__codelineno-43-10"></a><span class="w">    </span><span class="nx">t</span><span class="p">.</span><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">child</span><span class="p">)</span>
-<a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-43-11" name="__codelineno-43-11" href="#__codelineno-43-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-43-12" name="__codelineno-43-12" href="#__codelineno-43-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">child</span>
 <a id="__codelineno-43-13" name="__codelineno-43-13" href="#__codelineno-43-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Right rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.swift</span><pre><span></span><code><a id="__codelineno-44-1" name="__codelineno-44-1" href="#__codelineno-44-1"></a><span class="cm">/* Right rotation */</span>
 <a id="__codelineno-44-2" name="__codelineno-44-2" href="#__codelineno-44-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-44-3" name="__codelineno-44-3" href="#__codelineno-44-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">child</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">left</span>
 <a id="__codelineno-44-4" name="__codelineno-44-4" href="#__codelineno-44-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">grandChild</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">child</span><span class="p">?.</span><span class="kr">right</span>
-<a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the right</span>
+<a id="__codelineno-44-5" name="__codelineno-44-5" href="#__codelineno-44-5"></a><span class="w">    </span><span class="c1">// Rotate node to the right around child</span>
 <a id="__codelineno-44-6" name="__codelineno-44-6" href="#__codelineno-44-6"></a><span class="w">    </span><span class="n">child</span><span class="p">?.</span><span class="kr">right</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-44-7" name="__codelineno-44-7" href="#__codelineno-44-7"></a><span class="w">    </span><span class="n">node</span><span class="p">?.</span><span class="kr">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">grandChild</span>
-<a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-44-8" name="__codelineno-44-8" href="#__codelineno-44-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-44-9" name="__codelineno-44-9" href="#__codelineno-44-9"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-44-10" name="__codelineno-44-10" href="#__codelineno-44-10"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">child</span><span class="p">)</span>
-<a id="__codelineno-44-11" name="__codelineno-44-11" href="#__codelineno-44-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-44-11" name="__codelineno-44-11" href="#__codelineno-44-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-44-12" name="__codelineno-44-12" href="#__codelineno-44-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">child</span>
 <a id="__codelineno-44-13" name="__codelineno-44-13" href="#__codelineno-44-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Right rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-45-1" name="__codelineno-45-1" href="#__codelineno-45-1"></a><span class="cm">/* Right rotation */</span>
 <a id="__codelineno-45-2" name="__codelineno-45-2" href="#__codelineno-45-2"></a><span class="n">#rightRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-45-3" name="__codelineno-45-3" href="#__codelineno-45-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">;</span>
 <a id="__codelineno-45-4" name="__codelineno-45-4" href="#__codelineno-45-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
-<a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the right</span>
+<a id="__codelineno-45-5" name="__codelineno-45-5" href="#__codelineno-45-5"></a><span class="w">    </span><span class="c1">// Rotate node to the right around child</span>
 <a id="__codelineno-45-6" name="__codelineno-45-6" href="#__codelineno-45-6"></a><span class="w">    </span><span class="nx">child</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-45-7" name="__codelineno-45-7" href="#__codelineno-45-7"></a><span class="w">    </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grandChild</span><span class="p">;</span>
-<a id="__codelineno-45-8" name="__codelineno-45-8" href="#__codelineno-45-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-45-8" name="__codelineno-45-8" href="#__codelineno-45-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-45-9" name="__codelineno-45-9" href="#__codelineno-45-9"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">#updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
 <a id="__codelineno-45-10" name="__codelineno-45-10" href="#__codelineno-45-10"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">#updateHeight</span><span class="p">(</span><span class="nx">child</span><span class="p">);</span>
-<a id="__codelineno-45-11" name="__codelineno-45-11" href="#__codelineno-45-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-45-11" name="__codelineno-45-11" href="#__codelineno-45-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-45-12" name="__codelineno-45-12" href="#__codelineno-45-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">child</span><span class="p">;</span>
 <a id="__codelineno-45-13" name="__codelineno-45-13" href="#__codelineno-45-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Right rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-46-1" name="__codelineno-46-1" href="#__codelineno-46-1"></a><span class="cm">/* Right rotation */</span>
 <a id="__codelineno-46-2" name="__codelineno-46-2" href="#__codelineno-46-2"></a><span class="nx">rightRotate</span><span class="p">(</span><span class="nx">node</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-46-3" name="__codelineno-46-3" href="#__codelineno-46-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">;</span>
 <a id="__codelineno-46-4" name="__codelineno-46-4" href="#__codelineno-46-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
-<a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the right</span>
+<a id="__codelineno-46-5" name="__codelineno-46-5" href="#__codelineno-46-5"></a><span class="w">    </span><span class="c1">// Rotate node to the right around child</span>
 <a id="__codelineno-46-6" name="__codelineno-46-6" href="#__codelineno-46-6"></a><span class="w">    </span><span class="nx">child</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-46-7" name="__codelineno-46-7" href="#__codelineno-46-7"></a><span class="w">    </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grandChild</span><span class="p">;</span>
-<a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-46-8" name="__codelineno-46-8" href="#__codelineno-46-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-46-9" name="__codelineno-46-9" href="#__codelineno-46-9"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
 <a id="__codelineno-46-10" name="__codelineno-46-10" href="#__codelineno-46-10"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">child</span><span class="p">);</span>
-<a id="__codelineno-46-11" name="__codelineno-46-11" href="#__codelineno-46-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-46-11" name="__codelineno-46-11" href="#__codelineno-46-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-46-12" name="__codelineno-46-12" href="#__codelineno-46-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">child</span><span class="p">;</span>
 <a id="__codelineno-46-13" name="__codelineno-46-13" href="#__codelineno-46-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Right rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.dart</span><pre><span></span><code><a id="__codelineno-47-1" name="__codelineno-47-1" href="#__codelineno-47-1"></a><span class="cm">/* Right rotation */</span>
 <a id="__codelineno-47-2" name="__codelineno-47-2" href="#__codelineno-47-2"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-47-3" name="__codelineno-47-3" href="#__codelineno-47-3"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">!</span><span class="p">.</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-47-4" name="__codelineno-47-4" href="#__codelineno-47-4"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="o">!</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w">  </span><span class="c1">// Using child as pivot, rotate node to the right</span>
+<a id="__codelineno-47-5" name="__codelineno-47-5" href="#__codelineno-47-5"></a><span class="w">  </span><span class="c1">// Rotate node to the right around child</span>
 <a id="__codelineno-47-6" name="__codelineno-47-6" href="#__codelineno-47-6"></a><span class="w">  </span><span class="n">child</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-47-7" name="__codelineno-47-7" href="#__codelineno-47-7"></a><span class="w">  </span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grandChild</span><span class="p">;</span>
-<a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a><span class="w">  </span><span class="c1">// Update node height</span>
+<a id="__codelineno-47-8" name="__codelineno-47-8" href="#__codelineno-47-8"></a><span class="w">  </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-47-9" name="__codelineno-47-9" href="#__codelineno-47-9"></a><span class="w">  </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-47-10" name="__codelineno-47-10" href="#__codelineno-47-10"></a><span class="w">  </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">child</span><span class="p">);</span>
-<a id="__codelineno-47-11" name="__codelineno-47-11" href="#__codelineno-47-11"></a><span class="w">  </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-47-11" name="__codelineno-47-11" href="#__codelineno-47-11"></a><span class="w">  </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-47-12" name="__codelineno-47-12" href="#__codelineno-47-12"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-47-13" name="__codelineno-47-13" href="#__codelineno-47-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Right rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.rs</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* Right rotation */</span>
 <a id="__codelineno-48-2" name="__codelineno-48-2" href="#__codelineno-48-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">right_rotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="nc">OptionTreeNodeRc</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">OptionTreeNodeRc</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-3" name="__codelineno-48-3" href="#__codelineno-48-3"></a><span class="w">    </span><span class="k">match</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-4" name="__codelineno-48-4" href="#__codelineno-48-4"></a><span class="w">        </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-48-5" name="__codelineno-48-5" href="#__codelineno-48-5"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span>
 <a id="__codelineno-48-6" name="__codelineno-48-6" href="#__codelineno-48-6"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="n">grand_child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
-<a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="w">            </span><span class="c1">// Using child as pivot, rotate node to the right</span>
+<a id="__codelineno-48-7" name="__codelineno-48-7" href="#__codelineno-48-7"></a><span class="w">            </span><span class="c1">// Rotate node to the right around child</span>
 <a id="__codelineno-48-8" name="__codelineno-48-8" href="#__codelineno-48-8"></a><span class="w">            </span><span class="n">child</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-48-9" name="__codelineno-48-9" href="#__codelineno-48-9"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grand_child</span><span class="p">;</span>
-<a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="w">            </span><span class="c1">// Update node height</span>
+<a id="__codelineno-48-10" name="__codelineno-48-10" href="#__codelineno-48-10"></a><span class="w">            </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-48-11" name="__codelineno-48-11" href="#__codelineno-48-11"></a><span class="w">            </span><span class="bp">Self</span><span class="p">::</span><span class="n">update_height</span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">));</span>
 <a id="__codelineno-48-12" name="__codelineno-48-12" href="#__codelineno-48-12"></a><span class="w">            </span><span class="bp">Self</span><span class="p">::</span><span class="n">update_height</span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">child</span><span class="p">.</span><span class="n">clone</span><span class="p">()));</span>
-<a id="__codelineno-48-13" name="__codelineno-48-13" href="#__codelineno-48-13"></a><span class="w">            </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-48-13" name="__codelineno-48-13" href="#__codelineno-48-13"></a><span class="w">            </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-48-14" name="__codelineno-48-14" href="#__codelineno-48-14"></a><span class="w">            </span><span class="nb">Some</span><span class="p">(</span><span class="n">child</span><span class="p">)</span>
 <a id="__codelineno-48-15" name="__codelineno-48-15" href="#__codelineno-48-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-48-16" name="__codelineno-48-16" href="#__codelineno-48-16"></a><span class="w">        </span><span class="nb">None</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="nb">None</span><span class="p">,</span>
@@ -5720,34 +5792,34 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Right rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-49-1" name="__codelineno-49-1" href="#__codelineno-49-1"></a><span class="cm">/* Right rotation */</span>
 <a id="__codelineno-49-2" name="__codelineno-49-2" href="#__codelineno-49-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">rightRotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-49-3" name="__codelineno-49-3" href="#__codelineno-49-3"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">child</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">grandChild</span><span class="p">;</span>
 <a id="__codelineno-49-4" name="__codelineno-49-4" href="#__codelineno-49-4"></a><span class="w">    </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-49-5" name="__codelineno-49-5" href="#__codelineno-49-5"></a><span class="w">    </span><span class="n">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the right</span>
+<a id="__codelineno-49-6" name="__codelineno-49-6" href="#__codelineno-49-6"></a><span class="w">    </span><span class="c1">// Rotate node to the right around child</span>
 <a id="__codelineno-49-7" name="__codelineno-49-7" href="#__codelineno-49-7"></a><span class="w">    </span><span class="n">child</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-49-8" name="__codelineno-49-8" href="#__codelineno-49-8"></a><span class="w">    </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grandChild</span><span class="p">;</span>
-<a id="__codelineno-49-9" name="__codelineno-49-9" href="#__codelineno-49-9"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-49-9" name="__codelineno-49-9" href="#__codelineno-49-9"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-49-10" name="__codelineno-49-10" href="#__codelineno-49-10"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-49-11" name="__codelineno-49-11" href="#__codelineno-49-11"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">child</span><span class="p">);</span>
-<a id="__codelineno-49-12" name="__codelineno-49-12" href="#__codelineno-49-12"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-49-12" name="__codelineno-49-12" href="#__codelineno-49-12"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-49-13" name="__codelineno-49-13" href="#__codelineno-49-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-49-14" name="__codelineno-49-14" href="#__codelineno-49-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Right rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.kt</span><pre><span></span><code><a id="__codelineno-50-1" name="__codelineno-50-1" href="#__codelineno-50-1"></a><span class="cm">/* Right rotation */</span>
 <a id="__codelineno-50-2" name="__codelineno-50-2" href="#__codelineno-50-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">):</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-50-3" name="__codelineno-50-3" href="#__codelineno-50-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">!!</span><span class="p">.</span><span class="na">left</span>
 <a id="__codelineno-50-4" name="__codelineno-50-4" href="#__codelineno-50-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="o">!!</span><span class="p">.</span><span class="na">right</span>
-<a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the right</span>
+<a id="__codelineno-50-5" name="__codelineno-50-5" href="#__codelineno-50-5"></a><span class="w">    </span><span class="c1">// Rotate node to the right around child</span>
 <a id="__codelineno-50-6" name="__codelineno-50-6" href="#__codelineno-50-6"></a><span class="w">    </span><span class="n">child</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-50-7" name="__codelineno-50-7" href="#__codelineno-50-7"></a><span class="w">    </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grandChild</span>
-<a id="__codelineno-50-8" name="__codelineno-50-8" href="#__codelineno-50-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-50-8" name="__codelineno-50-8" href="#__codelineno-50-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-50-9" name="__codelineno-50-9" href="#__codelineno-50-9"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-50-10" name="__codelineno-50-10" href="#__codelineno-50-10"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">child</span><span class="p">)</span>
-<a id="__codelineno-50-11" name="__codelineno-50-11" href="#__codelineno-50-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-50-11" name="__codelineno-50-11" href="#__codelineno-50-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-50-12" name="__codelineno-50-12" href="#__codelineno-50-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">child</span>
 <a id="__codelineno-50-13" name="__codelineno-50-13" href="#__codelineno-50-13"></a><span class="p">}</span>
 </code></pre></div>
@@ -5757,188 +5829,188 @@
 <a id="__codelineno-51-2" name="__codelineno-51-2" href="#__codelineno-51-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">right_rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-51-3" name="__codelineno-51-3" href="#__codelineno-51-3"></a><span class="w">  </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">left</span>
 <a id="__codelineno-51-4" name="__codelineno-51-4" href="#__codelineno-51-4"></a><span class="w">  </span><span class="n">grand_child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="o">.</span><span class="n">right</span>
-<a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">  </span><span class="c1"># Using child as pivot, rotate node to the right</span>
+<a id="__codelineno-51-5" name="__codelineno-51-5" href="#__codelineno-51-5"></a><span class="w">  </span><span class="c1"># Rotate node to the right around child</span>
 <a id="__codelineno-51-6" name="__codelineno-51-6" href="#__codelineno-51-6"></a><span class="w">  </span><span class="n">child</span><span class="o">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-51-7" name="__codelineno-51-7" href="#__codelineno-51-7"></a><span class="w">  </span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grand_child</span>
-<a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="w">  </span><span class="c1"># Update node height</span>
+<a id="__codelineno-51-8" name="__codelineno-51-8" href="#__codelineno-51-8"></a><span class="w">  </span><span class="c1"># Update the node height</span>
 <a id="__codelineno-51-9" name="__codelineno-51-9" href="#__codelineno-51-9"></a><span class="w">  </span><span class="n">update_height</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-51-10" name="__codelineno-51-10" href="#__codelineno-51-10"></a><span class="w">  </span><span class="n">update_height</span><span class="p">(</span><span class="n">child</span><span class="p">)</span>
-<a id="__codelineno-51-11" name="__codelineno-51-11" href="#__codelineno-51-11"></a><span class="w">  </span><span class="c1"># Return root node of subtree after rotation</span>
+<a id="__codelineno-51-11" name="__codelineno-51-11" href="#__codelineno-51-11"></a><span class="w">  </span><span class="c1"># Return the root of the rotated subtree</span>
 <a id="__codelineno-51-12" name="__codelineno-51-12" href="#__codelineno-51-12"></a><span class="w">  </span><span class="n">child</span>
 <a id="__codelineno-51-13" name="__codelineno-51-13" href="#__codelineno-51-13"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h3 id="2-left-rotation">2. &nbsp; Left Rotation<a class="headerlink" href="#2-left-rotation" title="Permanent link">&para;</a></h3>
-<p>Correspondingly, if considering the "mirror" of the above unbalanced binary tree, the "left rotation" operation shown in Figure 7-28 needs to be performed.</p>
-<p><img alt="Left rotation operation" class="animation-figure" src="../avl_tree.assets/avltree_left_rotate.png" /></p>
-<p align="center"> Figure 7-28 &nbsp; Left rotation operation </p>
+<h3 id="2-left-rotation">2. &nbsp; Left rotation<a class="headerlink" href="#2-left-rotation" title="Permanent link">&para;</a></h3>
+<p>For the mirror image of the unbalanced tree above, perform a left rotation, as shown below.</p>
+<p><img alt="A left rotation" class="animation-figure" src="../avl_tree.assets/avltree_left_rotate.png" /></p>
+<p align="center"> Figure 7-28 &nbsp; A left rotation </p>
 
-<p>Similarly, as shown in Figure 7-29, when the <code>child</code> node has a left child (denoted as <code>grand_child</code>), a step needs to be added in the left rotation: set <code>grand_child</code> as the right child of <code>node</code>.</p>
-<p><img alt="Left rotation with grand_child" class="animation-figure" src="../avl_tree.assets/avltree_left_rotate_with_grandchild.png" /></p>
-<p align="center"> Figure 7-29 &nbsp; Left rotation with grand_child </p>
+<p>Similarly, if <code>child</code> has a left child, denoted by <code>grand_child</code>, make it the right child of <code>node</code> during the rotation.</p>
+<p><img alt="A left rotation with grand_child" class="animation-figure" src="../avl_tree.assets/avltree_left_rotate_with_grandchild.png" /></p>
+<p align="center"> Figure 7-29 &nbsp; A left rotation with grand_child </p>
 
-<p>It can be observed that <strong>right rotation and left rotation operations are mirror symmetric in logic, and the two imbalance cases they solve are also symmetric</strong>. Based on symmetry, we only need to replace all <code>left</code> in the right rotation implementation code with <code>right</code>, and all <code>right</code> with <code>left</code>, to obtain the left rotation implementation code:</p>
+<p><strong>Right and left rotations are mirror images, as are the imbalances they resolve</strong>. To obtain left-rotation code, exchange every <code>left</code> and <code>right</code> in the right-rotation implementation:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="6:13"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><input id="__tabbed_6_13" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Python</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Java</label><label for="__tabbed_6_4">C#</label><label for="__tabbed_6_5">Go</label><label for="__tabbed_6_6">Swift</label><label for="__tabbed_6_7">JS</label><label for="__tabbed_6_8">TS</label><label for="__tabbed_6_9">Dart</label><label for="__tabbed_6_10">Rust</label><label for="__tabbed_6_11">C</label><label for="__tabbed_6_12">Kotlin</label><label for="__tabbed_6_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">avl_tree.py</span><pre><span></span><code><a id="__codelineno-52-1" name="__codelineno-52-1" href="#__codelineno-52-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">left_rotate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">node</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Left rotation operation"""</span>
+<a id="__codelineno-52-2" name="__codelineno-52-2" href="#__codelineno-52-2"></a><span class="w">    </span><span class="sd">"""Left rotation"""</span>
 <a id="__codelineno-52-3" name="__codelineno-52-3" href="#__codelineno-52-3"></a>    <span class="n">child</span> <span class="o">=</span> <span class="n">node</span><span class="o">.</span><span class="n">right</span>
 <a id="__codelineno-52-4" name="__codelineno-52-4" href="#__codelineno-52-4"></a>    <span class="n">grand_child</span> <span class="o">=</span> <span class="n">child</span><span class="o">.</span><span class="n">left</span>
-<a id="__codelineno-52-5" name="__codelineno-52-5" href="#__codelineno-52-5"></a>    <span class="c1"># Using child as pivot, rotate node to the left</span>
+<a id="__codelineno-52-5" name="__codelineno-52-5" href="#__codelineno-52-5"></a>    <span class="c1"># Rotate node to the left around child</span>
 <a id="__codelineno-52-6" name="__codelineno-52-6" href="#__codelineno-52-6"></a>    <span class="n">child</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">node</span>
 <a id="__codelineno-52-7" name="__codelineno-52-7" href="#__codelineno-52-7"></a>    <span class="n">node</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="n">grand_child</span>
-<a id="__codelineno-52-8" name="__codelineno-52-8" href="#__codelineno-52-8"></a>    <span class="c1"># Update node height</span>
+<a id="__codelineno-52-8" name="__codelineno-52-8" href="#__codelineno-52-8"></a>    <span class="c1"># Update the node height</span>
 <a id="__codelineno-52-9" name="__codelineno-52-9" href="#__codelineno-52-9"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">update_height</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-52-10" name="__codelineno-52-10" href="#__codelineno-52-10"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">update_height</span><span class="p">(</span><span class="n">child</span><span class="p">)</span>
-<a id="__codelineno-52-11" name="__codelineno-52-11" href="#__codelineno-52-11"></a>    <span class="c1"># Return root node of subtree after rotation</span>
+<a id="__codelineno-52-11" name="__codelineno-52-11" href="#__codelineno-52-11"></a>    <span class="c1"># Return the root of the rotated subtree</span>
 <a id="__codelineno-52-12" name="__codelineno-52-12" href="#__codelineno-52-12"></a>    <span class="k">return</span> <span class="n">child</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Left rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-53-1" name="__codelineno-53-1" href="#__codelineno-53-1"></a><span class="cm">/* Left rotation */</span>
 <a id="__codelineno-53-2" name="__codelineno-53-2" href="#__codelineno-53-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">leftRotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-53-3" name="__codelineno-53-3" href="#__codelineno-53-3"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-53-4" name="__codelineno-53-4" href="#__codelineno-53-4"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">;</span>
-<a id="__codelineno-53-5" name="__codelineno-53-5" href="#__codelineno-53-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the left</span>
+<a id="__codelineno-53-5" name="__codelineno-53-5" href="#__codelineno-53-5"></a><span class="w">    </span><span class="c1">// Rotate node to the left around child</span>
 <a id="__codelineno-53-6" name="__codelineno-53-6" href="#__codelineno-53-6"></a><span class="w">    </span><span class="n">child</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-53-7" name="__codelineno-53-7" href="#__codelineno-53-7"></a><span class="w">    </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grandChild</span><span class="p">;</span>
-<a id="__codelineno-53-8" name="__codelineno-53-8" href="#__codelineno-53-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-53-8" name="__codelineno-53-8" href="#__codelineno-53-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-53-9" name="__codelineno-53-9" href="#__codelineno-53-9"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-53-10" name="__codelineno-53-10" href="#__codelineno-53-10"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">child</span><span class="p">);</span>
-<a id="__codelineno-53-11" name="__codelineno-53-11" href="#__codelineno-53-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-53-11" name="__codelineno-53-11" href="#__codelineno-53-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-53-12" name="__codelineno-53-12" href="#__codelineno-53-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-53-13" name="__codelineno-53-13" href="#__codelineno-53-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Left rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-54-1" name="__codelineno-54-1" href="#__codelineno-54-1"></a><span class="cm">/* Left rotation */</span>
 <a id="__codelineno-54-2" name="__codelineno-54-2" href="#__codelineno-54-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">leftRotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-54-3" name="__codelineno-54-3" href="#__codelineno-54-3"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">;</span>
 <a id="__codelineno-54-4" name="__codelineno-54-4" href="#__codelineno-54-4"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">.</span><span class="na">left</span><span class="p">;</span>
-<a id="__codelineno-54-5" name="__codelineno-54-5" href="#__codelineno-54-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the left</span>
+<a id="__codelineno-54-5" name="__codelineno-54-5" href="#__codelineno-54-5"></a><span class="w">    </span><span class="c1">// Rotate node to the left around child</span>
 <a id="__codelineno-54-6" name="__codelineno-54-6" href="#__codelineno-54-6"></a><span class="w">    </span><span class="n">child</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-54-7" name="__codelineno-54-7" href="#__codelineno-54-7"></a><span class="w">    </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grandChild</span><span class="p">;</span>
-<a id="__codelineno-54-8" name="__codelineno-54-8" href="#__codelineno-54-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-54-8" name="__codelineno-54-8" href="#__codelineno-54-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-54-9" name="__codelineno-54-9" href="#__codelineno-54-9"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-54-10" name="__codelineno-54-10" href="#__codelineno-54-10"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">child</span><span class="p">);</span>
-<a id="__codelineno-54-11" name="__codelineno-54-11" href="#__codelineno-54-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-54-11" name="__codelineno-54-11" href="#__codelineno-54-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-54-12" name="__codelineno-54-12" href="#__codelineno-54-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-54-13" name="__codelineno-54-13" href="#__codelineno-54-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Left rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-55-1" name="__codelineno-55-1" href="#__codelineno-55-1"></a><span class="cm">/* Left rotation */</span>
 <a id="__codelineno-55-2" name="__codelineno-55-2" href="#__codelineno-55-2"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">LeftRotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-55-3" name="__codelineno-55-3" href="#__codelineno-55-3"></a><span class="w">    </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">?.</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-55-4" name="__codelineno-55-4" href="#__codelineno-55-4"></a><span class="w">    </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="o">?.</span><span class="n">left</span><span class="p">;</span>
-<a id="__codelineno-55-5" name="__codelineno-55-5" href="#__codelineno-55-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the left</span>
+<a id="__codelineno-55-5" name="__codelineno-55-5" href="#__codelineno-55-5"></a><span class="w">    </span><span class="c1">// Rotate node to the left around child</span>
 <a id="__codelineno-55-6" name="__codelineno-55-6" href="#__codelineno-55-6"></a><span class="w">    </span><span class="n">child</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-55-7" name="__codelineno-55-7" href="#__codelineno-55-7"></a><span class="w">    </span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grandChild</span><span class="p">;</span>
-<a id="__codelineno-55-8" name="__codelineno-55-8" href="#__codelineno-55-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-55-8" name="__codelineno-55-8" href="#__codelineno-55-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-55-9" name="__codelineno-55-9" href="#__codelineno-55-9"></a><span class="w">    </span><span class="n">UpdateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-55-10" name="__codelineno-55-10" href="#__codelineno-55-10"></a><span class="w">    </span><span class="n">UpdateHeight</span><span class="p">(</span><span class="n">child</span><span class="p">);</span>
-<a id="__codelineno-55-11" name="__codelineno-55-11" href="#__codelineno-55-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-55-11" name="__codelineno-55-11" href="#__codelineno-55-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-55-12" name="__codelineno-55-12" href="#__codelineno-55-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-55-13" name="__codelineno-55-13" href="#__codelineno-55-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Left rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-56-1" name="__codelineno-56-1" href="#__codelineno-56-1"></a><span class="cm">/* Left rotation */</span>
 <a id="__codelineno-56-2" name="__codelineno-56-2" href="#__codelineno-56-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">t</span><span class="w"> </span><span class="o">*</span><span class="nx">aVLTree</span><span class="p">)</span><span class="w"> </span><span class="nx">leftRotate</span><span class="p">(</span><span class="nx">node</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-56-3" name="__codelineno-56-3" href="#__codelineno-56-3"></a><span class="w">    </span><span class="nx">child</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span>
 <a id="__codelineno-56-4" name="__codelineno-56-4" href="#__codelineno-56-4"></a><span class="w">    </span><span class="nx">grandChild</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">child</span><span class="p">.</span><span class="nx">Left</span>
-<a id="__codelineno-56-5" name="__codelineno-56-5" href="#__codelineno-56-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the left</span>
+<a id="__codelineno-56-5" name="__codelineno-56-5" href="#__codelineno-56-5"></a><span class="w">    </span><span class="c1">// Rotate node to the left around child</span>
 <a id="__codelineno-56-6" name="__codelineno-56-6" href="#__codelineno-56-6"></a><span class="w">    </span><span class="nx">child</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span>
 <a id="__codelineno-56-7" name="__codelineno-56-7" href="#__codelineno-56-7"></a><span class="w">    </span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grandChild</span>
-<a id="__codelineno-56-8" name="__codelineno-56-8" href="#__codelineno-56-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-56-8" name="__codelineno-56-8" href="#__codelineno-56-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-56-9" name="__codelineno-56-9" href="#__codelineno-56-9"></a><span class="w">    </span><span class="nx">t</span><span class="p">.</span><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span>
 <a id="__codelineno-56-10" name="__codelineno-56-10" href="#__codelineno-56-10"></a><span class="w">    </span><span class="nx">t</span><span class="p">.</span><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">child</span><span class="p">)</span>
-<a id="__codelineno-56-11" name="__codelineno-56-11" href="#__codelineno-56-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-56-11" name="__codelineno-56-11" href="#__codelineno-56-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-56-12" name="__codelineno-56-12" href="#__codelineno-56-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">child</span>
 <a id="__codelineno-56-13" name="__codelineno-56-13" href="#__codelineno-56-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Left rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.swift</span><pre><span></span><code><a id="__codelineno-57-1" name="__codelineno-57-1" href="#__codelineno-57-1"></a><span class="cm">/* Left rotation */</span>
 <a id="__codelineno-57-2" name="__codelineno-57-2" href="#__codelineno-57-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-57-3" name="__codelineno-57-3" href="#__codelineno-57-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">child</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">right</span>
 <a id="__codelineno-57-4" name="__codelineno-57-4" href="#__codelineno-57-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">grandChild</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">child</span><span class="p">?.</span><span class="kr">left</span>
-<a id="__codelineno-57-5" name="__codelineno-57-5" href="#__codelineno-57-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the left</span>
+<a id="__codelineno-57-5" name="__codelineno-57-5" href="#__codelineno-57-5"></a><span class="w">    </span><span class="c1">// Rotate node to the left around child</span>
 <a id="__codelineno-57-6" name="__codelineno-57-6" href="#__codelineno-57-6"></a><span class="w">    </span><span class="n">child</span><span class="p">?.</span><span class="kr">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-57-7" name="__codelineno-57-7" href="#__codelineno-57-7"></a><span class="w">    </span><span class="n">node</span><span class="p">?.</span><span class="kr">right</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">grandChild</span>
-<a id="__codelineno-57-8" name="__codelineno-57-8" href="#__codelineno-57-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-57-8" name="__codelineno-57-8" href="#__codelineno-57-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-57-9" name="__codelineno-57-9" href="#__codelineno-57-9"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-57-10" name="__codelineno-57-10" href="#__codelineno-57-10"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">child</span><span class="p">)</span>
-<a id="__codelineno-57-11" name="__codelineno-57-11" href="#__codelineno-57-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-57-11" name="__codelineno-57-11" href="#__codelineno-57-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-57-12" name="__codelineno-57-12" href="#__codelineno-57-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">child</span>
 <a id="__codelineno-57-13" name="__codelineno-57-13" href="#__codelineno-57-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Left rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-58-1" name="__codelineno-58-1" href="#__codelineno-58-1"></a><span class="cm">/* Left rotation */</span>
 <a id="__codelineno-58-2" name="__codelineno-58-2" href="#__codelineno-58-2"></a><span class="n">#leftRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-58-3" name="__codelineno-58-3" href="#__codelineno-58-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
 <a id="__codelineno-58-4" name="__codelineno-58-4" href="#__codelineno-58-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span><span class="p">.</span><span class="nx">left</span><span class="p">;</span>
-<a id="__codelineno-58-5" name="__codelineno-58-5" href="#__codelineno-58-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the left</span>
+<a id="__codelineno-58-5" name="__codelineno-58-5" href="#__codelineno-58-5"></a><span class="w">    </span><span class="c1">// Rotate node to the left around child</span>
 <a id="__codelineno-58-6" name="__codelineno-58-6" href="#__codelineno-58-6"></a><span class="w">    </span><span class="nx">child</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-58-7" name="__codelineno-58-7" href="#__codelineno-58-7"></a><span class="w">    </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grandChild</span><span class="p">;</span>
-<a id="__codelineno-58-8" name="__codelineno-58-8" href="#__codelineno-58-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-58-8" name="__codelineno-58-8" href="#__codelineno-58-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-58-9" name="__codelineno-58-9" href="#__codelineno-58-9"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">#updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
 <a id="__codelineno-58-10" name="__codelineno-58-10" href="#__codelineno-58-10"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">#updateHeight</span><span class="p">(</span><span class="nx">child</span><span class="p">);</span>
-<a id="__codelineno-58-11" name="__codelineno-58-11" href="#__codelineno-58-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-58-11" name="__codelineno-58-11" href="#__codelineno-58-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-58-12" name="__codelineno-58-12" href="#__codelineno-58-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">child</span><span class="p">;</span>
 <a id="__codelineno-58-13" name="__codelineno-58-13" href="#__codelineno-58-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Left rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-59-1" name="__codelineno-59-1" href="#__codelineno-59-1"></a><span class="cm">/* Left rotation */</span>
 <a id="__codelineno-59-2" name="__codelineno-59-2" href="#__codelineno-59-2"></a><span class="nx">leftRotate</span><span class="p">(</span><span class="nx">node</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-59-3" name="__codelineno-59-3" href="#__codelineno-59-3"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
 <a id="__codelineno-59-4" name="__codelineno-59-4" href="#__codelineno-59-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span><span class="p">.</span><span class="nx">left</span><span class="p">;</span>
-<a id="__codelineno-59-5" name="__codelineno-59-5" href="#__codelineno-59-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the left</span>
+<a id="__codelineno-59-5" name="__codelineno-59-5" href="#__codelineno-59-5"></a><span class="w">    </span><span class="c1">// Rotate node to the left around child</span>
 <a id="__codelineno-59-6" name="__codelineno-59-6" href="#__codelineno-59-6"></a><span class="w">    </span><span class="nx">child</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-59-7" name="__codelineno-59-7" href="#__codelineno-59-7"></a><span class="w">    </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">grandChild</span><span class="p">;</span>
-<a id="__codelineno-59-8" name="__codelineno-59-8" href="#__codelineno-59-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-59-8" name="__codelineno-59-8" href="#__codelineno-59-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-59-9" name="__codelineno-59-9" href="#__codelineno-59-9"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
 <a id="__codelineno-59-10" name="__codelineno-59-10" href="#__codelineno-59-10"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">child</span><span class="p">);</span>
-<a id="__codelineno-59-11" name="__codelineno-59-11" href="#__codelineno-59-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-59-11" name="__codelineno-59-11" href="#__codelineno-59-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-59-12" name="__codelineno-59-12" href="#__codelineno-59-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">child</span><span class="p">;</span>
 <a id="__codelineno-59-13" name="__codelineno-59-13" href="#__codelineno-59-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Left rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.dart</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* Left rotation */</span>
 <a id="__codelineno-60-2" name="__codelineno-60-2" href="#__codelineno-60-2"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-60-3" name="__codelineno-60-3" href="#__codelineno-60-3"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">!</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-60-4" name="__codelineno-60-4" href="#__codelineno-60-4"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="o">!</span><span class="p">.</span><span class="n">left</span><span class="p">;</span>
-<a id="__codelineno-60-5" name="__codelineno-60-5" href="#__codelineno-60-5"></a><span class="w">  </span><span class="c1">// Using child as pivot, rotate node to the left</span>
+<a id="__codelineno-60-5" name="__codelineno-60-5" href="#__codelineno-60-5"></a><span class="w">  </span><span class="c1">// Rotate node to the left around child</span>
 <a id="__codelineno-60-6" name="__codelineno-60-6" href="#__codelineno-60-6"></a><span class="w">  </span><span class="n">child</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-60-7" name="__codelineno-60-7" href="#__codelineno-60-7"></a><span class="w">  </span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grandChild</span><span class="p">;</span>
-<a id="__codelineno-60-8" name="__codelineno-60-8" href="#__codelineno-60-8"></a><span class="w">  </span><span class="c1">// Update node height</span>
+<a id="__codelineno-60-8" name="__codelineno-60-8" href="#__codelineno-60-8"></a><span class="w">  </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-60-9" name="__codelineno-60-9" href="#__codelineno-60-9"></a><span class="w">  </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-60-10" name="__codelineno-60-10" href="#__codelineno-60-10"></a><span class="w">  </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">child</span><span class="p">);</span>
-<a id="__codelineno-60-11" name="__codelineno-60-11" href="#__codelineno-60-11"></a><span class="w">  </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-60-11" name="__codelineno-60-11" href="#__codelineno-60-11"></a><span class="w">  </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-60-12" name="__codelineno-60-12" href="#__codelineno-60-12"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-60-13" name="__codelineno-60-13" href="#__codelineno-60-13"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Left rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.rs</span><pre><span></span><code><a id="__codelineno-61-1" name="__codelineno-61-1" href="#__codelineno-61-1"></a><span class="cm">/* Left rotation */</span>
 <a id="__codelineno-61-2" name="__codelineno-61-2" href="#__codelineno-61-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">left_rotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="nc">OptionTreeNodeRc</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">OptionTreeNodeRc</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-3" name="__codelineno-61-3" href="#__codelineno-61-3"></a><span class="w">    </span><span class="k">match</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-4" name="__codelineno-61-4" href="#__codelineno-61-4"></a><span class="w">        </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-61-5" name="__codelineno-61-5" href="#__codelineno-61-5"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span>
 <a id="__codelineno-61-6" name="__codelineno-61-6" href="#__codelineno-61-6"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="n">grand_child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
-<a id="__codelineno-61-7" name="__codelineno-61-7" href="#__codelineno-61-7"></a><span class="w">            </span><span class="c1">// Using child as pivot, rotate node to the left</span>
+<a id="__codelineno-61-7" name="__codelineno-61-7" href="#__codelineno-61-7"></a><span class="w">            </span><span class="c1">// Rotate node to the left around child</span>
 <a id="__codelineno-61-8" name="__codelineno-61-8" href="#__codelineno-61-8"></a><span class="w">            </span><span class="n">child</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-61-9" name="__codelineno-61-9" href="#__codelineno-61-9"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grand_child</span><span class="p">;</span>
-<a id="__codelineno-61-10" name="__codelineno-61-10" href="#__codelineno-61-10"></a><span class="w">            </span><span class="c1">// Update node height</span>
+<a id="__codelineno-61-10" name="__codelineno-61-10" href="#__codelineno-61-10"></a><span class="w">            </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-61-11" name="__codelineno-61-11" href="#__codelineno-61-11"></a><span class="w">            </span><span class="bp">Self</span><span class="p">::</span><span class="n">update_height</span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">));</span>
 <a id="__codelineno-61-12" name="__codelineno-61-12" href="#__codelineno-61-12"></a><span class="w">            </span><span class="bp">Self</span><span class="p">::</span><span class="n">update_height</span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">child</span><span class="p">.</span><span class="n">clone</span><span class="p">()));</span>
-<a id="__codelineno-61-13" name="__codelineno-61-13" href="#__codelineno-61-13"></a><span class="w">            </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-61-13" name="__codelineno-61-13" href="#__codelineno-61-13"></a><span class="w">            </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-61-14" name="__codelineno-61-14" href="#__codelineno-61-14"></a><span class="w">            </span><span class="nb">Some</span><span class="p">(</span><span class="n">child</span><span class="p">)</span>
 <a id="__codelineno-61-15" name="__codelineno-61-15" href="#__codelineno-61-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-61-16" name="__codelineno-61-16" href="#__codelineno-61-16"></a><span class="w">        </span><span class="nb">None</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="nb">None</span><span class="p">,</span>
@@ -5947,34 +6019,34 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Left rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-62-1" name="__codelineno-62-1" href="#__codelineno-62-1"></a><span class="cm">/* Left rotation */</span>
 <a id="__codelineno-62-2" name="__codelineno-62-2" href="#__codelineno-62-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">leftRotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-62-3" name="__codelineno-62-3" href="#__codelineno-62-3"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">child</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">grandChild</span><span class="p">;</span>
 <a id="__codelineno-62-4" name="__codelineno-62-4" href="#__codelineno-62-4"></a><span class="w">    </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-62-5" name="__codelineno-62-5" href="#__codelineno-62-5"></a><span class="w">    </span><span class="n">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">;</span>
-<a id="__codelineno-62-6" name="__codelineno-62-6" href="#__codelineno-62-6"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the left</span>
+<a id="__codelineno-62-6" name="__codelineno-62-6" href="#__codelineno-62-6"></a><span class="w">    </span><span class="c1">// Rotate node to the left around child</span>
 <a id="__codelineno-62-7" name="__codelineno-62-7" href="#__codelineno-62-7"></a><span class="w">    </span><span class="n">child</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-62-8" name="__codelineno-62-8" href="#__codelineno-62-8"></a><span class="w">    </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grandChild</span><span class="p">;</span>
-<a id="__codelineno-62-9" name="__codelineno-62-9" href="#__codelineno-62-9"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-62-9" name="__codelineno-62-9" href="#__codelineno-62-9"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-62-10" name="__codelineno-62-10" href="#__codelineno-62-10"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-62-11" name="__codelineno-62-11" href="#__codelineno-62-11"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">child</span><span class="p">);</span>
-<a id="__codelineno-62-12" name="__codelineno-62-12" href="#__codelineno-62-12"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-62-12" name="__codelineno-62-12" href="#__codelineno-62-12"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-62-13" name="__codelineno-62-13" href="#__codelineno-62-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-62-14" name="__codelineno-62-14" href="#__codelineno-62-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Left rotation operation */</span>
+<div class="highlight"><span class="filename">avl_tree.kt</span><pre><span></span><code><a id="__codelineno-63-1" name="__codelineno-63-1" href="#__codelineno-63-1"></a><span class="cm">/* Left rotation */</span>
 <a id="__codelineno-63-2" name="__codelineno-63-2" href="#__codelineno-63-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">):</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-63-3" name="__codelineno-63-3" href="#__codelineno-63-3"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">!!</span><span class="p">.</span><span class="na">right</span>
 <a id="__codelineno-63-4" name="__codelineno-63-4" href="#__codelineno-63-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">grandChild</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="o">!!</span><span class="p">.</span><span class="na">left</span>
-<a id="__codelineno-63-5" name="__codelineno-63-5" href="#__codelineno-63-5"></a><span class="w">    </span><span class="c1">// Using child as pivot, rotate node to the left</span>
+<a id="__codelineno-63-5" name="__codelineno-63-5" href="#__codelineno-63-5"></a><span class="w">    </span><span class="c1">// Rotate node to the left around child</span>
 <a id="__codelineno-63-6" name="__codelineno-63-6" href="#__codelineno-63-6"></a><span class="w">    </span><span class="n">child</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-63-7" name="__codelineno-63-7" href="#__codelineno-63-7"></a><span class="w">    </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grandChild</span>
-<a id="__codelineno-63-8" name="__codelineno-63-8" href="#__codelineno-63-8"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-63-8" name="__codelineno-63-8" href="#__codelineno-63-8"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-63-9" name="__codelineno-63-9" href="#__codelineno-63-9"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-63-10" name="__codelineno-63-10" href="#__codelineno-63-10"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">child</span><span class="p">)</span>
-<a id="__codelineno-63-11" name="__codelineno-63-11" href="#__codelineno-63-11"></a><span class="w">    </span><span class="c1">// Return root node of subtree after rotation</span>
+<a id="__codelineno-63-11" name="__codelineno-63-11" href="#__codelineno-63-11"></a><span class="w">    </span><span class="c1">// Return the root of the rotated subtree</span>
 <a id="__codelineno-63-12" name="__codelineno-63-12" href="#__codelineno-63-12"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">child</span>
 <a id="__codelineno-63-13" name="__codelineno-63-13" href="#__codelineno-63-13"></a><span class="p">}</span>
 </code></pre></div>
@@ -5984,732 +6056,732 @@
 <a id="__codelineno-64-2" name="__codelineno-64-2" href="#__codelineno-64-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">left_rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-64-3" name="__codelineno-64-3" href="#__codelineno-64-3"></a><span class="w">  </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">right</span>
 <a id="__codelineno-64-4" name="__codelineno-64-4" href="#__codelineno-64-4"></a><span class="w">  </span><span class="n">grand_child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="o">.</span><span class="n">left</span>
-<a id="__codelineno-64-5" name="__codelineno-64-5" href="#__codelineno-64-5"></a><span class="w">  </span><span class="c1"># Using child as pivot, rotate node to the left</span>
+<a id="__codelineno-64-5" name="__codelineno-64-5" href="#__codelineno-64-5"></a><span class="w">  </span><span class="c1"># Rotate node to the left around child</span>
 <a id="__codelineno-64-6" name="__codelineno-64-6" href="#__codelineno-64-6"></a><span class="w">  </span><span class="n">child</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-64-7" name="__codelineno-64-7" href="#__codelineno-64-7"></a><span class="w">  </span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">grand_child</span>
-<a id="__codelineno-64-8" name="__codelineno-64-8" href="#__codelineno-64-8"></a><span class="w">  </span><span class="c1"># Update node height</span>
+<a id="__codelineno-64-8" name="__codelineno-64-8" href="#__codelineno-64-8"></a><span class="w">  </span><span class="c1"># Update the node height</span>
 <a id="__codelineno-64-9" name="__codelineno-64-9" href="#__codelineno-64-9"></a><span class="w">  </span><span class="n">update_height</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-64-10" name="__codelineno-64-10" href="#__codelineno-64-10"></a><span class="w">  </span><span class="n">update_height</span><span class="p">(</span><span class="n">child</span><span class="p">)</span>
-<a id="__codelineno-64-11" name="__codelineno-64-11" href="#__codelineno-64-11"></a><span class="w">  </span><span class="c1"># Return root node of subtree after rotation</span>
+<a id="__codelineno-64-11" name="__codelineno-64-11" href="#__codelineno-64-11"></a><span class="w">  </span><span class="c1"># Return the root of the rotated subtree</span>
 <a id="__codelineno-64-12" name="__codelineno-64-12" href="#__codelineno-64-12"></a><span class="w">  </span><span class="n">child</span>
 <a id="__codelineno-64-13" name="__codelineno-64-13" href="#__codelineno-64-13"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h3 id="3-left-rotation-then-right-rotation">3. &nbsp; Left Rotation Then Right Rotation<a class="headerlink" href="#3-left-rotation-then-right-rotation" title="Permanent link">&para;</a></h3>
-<p>For the unbalanced node 3 in Figure 7-30, using either left rotation or right rotation alone cannot restore the subtree to balance. In this case, a "left rotation" needs to be performed on <code>child</code> first, followed by a "right rotation" on <code>node</code>.</p>
-<p><img alt="Left-right rotation" class="animation-figure" src="../avl_tree.assets/avltree_left_right_rotate.png" /></p>
-<p align="center"> Figure 7-30 &nbsp; Left-right rotation </p>
+<h3 id="3-left-then-right-rotation">3. &nbsp; Left-then-right rotation<a class="headerlink" href="#3-left-then-right-rotation" title="Permanent link">&para;</a></h3>
+<p>For unbalanced node 3 below, neither a left nor a right rotation alone restores balance. First rotate <code>child</code> to the left, then rotate <code>node</code> to the right.</p>
+<p><img alt="A left-then-right rotation" class="animation-figure" src="../avl_tree.assets/avltree_left_right_rotate.png" /></p>
+<p align="center"> Figure 7-30 &nbsp; A left-then-right rotation </p>
 
-<h3 id="4-right-rotation-then-left-rotation">4. &nbsp; Right Rotation Then Left Rotation<a class="headerlink" href="#4-right-rotation-then-left-rotation" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 7-31, for the mirror case of the above unbalanced binary tree, a "right rotation" needs to be performed on <code>child</code> first, then a "left rotation" on <code>node</code>.</p>
-<p><img alt="Right-left rotation" class="animation-figure" src="../avl_tree.assets/avltree_right_left_rotate.png" /></p>
-<p align="center"> Figure 7-31 &nbsp; Right-left rotation </p>
+<h3 id="4-right-then-left-rotation">4. &nbsp; Right-then-left rotation<a class="headerlink" href="#4-right-then-left-rotation" title="Permanent link">&para;</a></h3>
+<p>For the mirror-image case below, first rotate <code>child</code> to the right, then rotate <code>node</code> to the left.</p>
+<p><img alt="A right-then-left rotation" class="animation-figure" src="../avl_tree.assets/avltree_right_left_rotate.png" /></p>
+<p align="center"> Figure 7-31 &nbsp; A right-then-left rotation </p>
 
-<h3 id="5-choice-of-rotation">5. &nbsp; Choice of Rotation<a class="headerlink" href="#5-choice-of-rotation" title="Permanent link">&para;</a></h3>
-<p>The four imbalances shown in Figure 7-32 correspond one-to-one with the above cases, requiring right rotation, left rotation then right rotation, right rotation then left rotation, and left rotation operations respectively.</p>
-<p><img alt="The four rotation cases of AVL tree" class="animation-figure" src="../avl_tree.assets/avltree_rotation_cases.png" /></p>
-<p align="center"> Figure 7-32 &nbsp; The four rotation cases of AVL tree </p>
+<h3 id="5-choosing-a-rotation">5. &nbsp; Choosing a rotation<a class="headerlink" href="#5-choosing-a-rotation" title="Permanent link">&para;</a></h3>
+<p>The four cases below correspond to the examples above and require, respectively, right, left-then-right, right-then-left, and left rotations.</p>
+<p><img alt="The four AVL rotation cases" class="animation-figure" src="../avl_tree.assets/avltree_rotation_cases.png" /></p>
+<p align="center"> Figure 7-32 &nbsp; The four AVL rotation cases </p>
 
-<p>As shown in Table 7-3, we determine which case the unbalanced node belongs to by judging the signs of the balance factor of the unbalanced node and the balance factor of its taller-side child node.</p>
-<p align="center"> Table 7-3 &nbsp; Conditions for Choosing Among the Four Rotation Cases </p>
+<p>As Table 7-3 shows, the signs of the unbalanced node's balance factor and that of the child on its taller side determine the case.</p>
+<p align="center"> Table 7-3 &nbsp; Conditions for choosing a rotation </p>
 
 <div class="center-table">
 <table>
 <thead>
 <tr>
-<th>Balance factor of the unbalanced node</th>
-<th>Balance factor of the child node</th>
-<th>Rotation method to apply</th>
+<th>Unbalanced node's balance factor</th>
+<th>Child's balance factor</th>
+<th>Rotation</th>
 </tr>
 </thead>
 <tbody>
 <tr>
-<td><span class="arithmatex">\(&gt; 1\)</span> (left-leaning tree)</td>
+<td><span class="arithmatex">\(&gt; 1\)</span> (left-heavy)</td>
 <td><span class="arithmatex">\(\geq 0\)</span></td>
-<td>Right rotation</td>
+<td>Right</td>
 </tr>
 <tr>
-<td><span class="arithmatex">\(&gt; 1\)</span> (left-leaning tree)</td>
+<td><span class="arithmatex">\(&gt; 1\)</span> (left-heavy)</td>
 <td><span class="arithmatex">\(&lt;0\)</span></td>
-<td>Left rotation then right rotation</td>
+<td>Left, then right</td>
 </tr>
 <tr>
-<td><span class="arithmatex">\(&lt; -1\)</span> (right-leaning tree)</td>
+<td><span class="arithmatex">\(&lt; -1\)</span> (right-heavy)</td>
 <td><span class="arithmatex">\(\leq 0\)</span></td>
-<td>Left rotation</td>
+<td>Left</td>
 </tr>
 <tr>
-<td><span class="arithmatex">\(&lt; -1\)</span> (right-leaning tree)</td>
+<td><span class="arithmatex">\(&lt; -1\)</span> (right-heavy)</td>
 <td><span class="arithmatex">\(&gt;0\)</span></td>
-<td>Right rotation then left rotation</td>
+<td>Right, then left</td>
 </tr>
 </tbody>
 </table>
 </div>
-<p>For ease of use, we encapsulate the rotation operations into a function. <strong>With this function, we can perform rotations for various imbalance situations, restoring balance to unbalanced nodes</strong>. The code is as follows:</p>
+<p>Wrap the rotation logic in one function. <strong>It can then handle every imbalance and restore the affected node's balance</strong>. The code is as follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="7:13"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><input id="__tabbed_7_13" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Python</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Java</label><label for="__tabbed_7_4">C#</label><label for="__tabbed_7_5">Go</label><label for="__tabbed_7_6">Swift</label><label for="__tabbed_7_7">JS</label><label for="__tabbed_7_8">TS</label><label for="__tabbed_7_9">Dart</label><label for="__tabbed_7_10">Rust</label><label for="__tabbed_7_11">C</label><label for="__tabbed_7_12">Kotlin</label><label for="__tabbed_7_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">avl_tree.py</span><pre><span></span><code><a id="__codelineno-65-1" name="__codelineno-65-1" href="#__codelineno-65-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">rotate</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">node</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-65-2" name="__codelineno-65-2" href="#__codelineno-65-2"></a><span class="w">    </span><span class="sd">"""Perform rotation operation to restore balance to this subtree"""</span>
-<a id="__codelineno-65-3" name="__codelineno-65-3" href="#__codelineno-65-3"></a>    <span class="c1"># Get balance factor of node</span>
+<a id="__codelineno-65-2" name="__codelineno-65-2" href="#__codelineno-65-2"></a><span class="w">    </span><span class="sd">"""Perform rotations to restore balance in this subtree"""</span>
+<a id="__codelineno-65-3" name="__codelineno-65-3" href="#__codelineno-65-3"></a>    <span class="c1"># Get the balance factor of node</span>
 <a id="__codelineno-65-4" name="__codelineno-65-4" href="#__codelineno-65-4"></a>    <span class="n">balance_factor</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">balance_factor</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-65-5" name="__codelineno-65-5" href="#__codelineno-65-5"></a>    <span class="c1"># Left-leaning tree</span>
+<a id="__codelineno-65-5" name="__codelineno-65-5" href="#__codelineno-65-5"></a>    <span class="c1"># Left-heavy tree</span>
 <a id="__codelineno-65-6" name="__codelineno-65-6" href="#__codelineno-65-6"></a>    <span class="k">if</span> <span class="n">balance_factor</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-65-7" name="__codelineno-65-7" href="#__codelineno-65-7"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">balance_factor</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">:</span>
-<a id="__codelineno-65-8" name="__codelineno-65-8" href="#__codelineno-65-8"></a>            <span class="c1"># Right rotation</span>
+<a id="__codelineno-65-8" name="__codelineno-65-8" href="#__codelineno-65-8"></a>            <span class="c1"># Rotate right</span>
 <a id="__codelineno-65-9" name="__codelineno-65-9" href="#__codelineno-65-9"></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">right_rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-65-10" name="__codelineno-65-10" href="#__codelineno-65-10"></a>        <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-65-11" name="__codelineno-65-11" href="#__codelineno-65-11"></a>            <span class="c1"># First left rotation then right rotation</span>
+<a id="__codelineno-65-11" name="__codelineno-65-11" href="#__codelineno-65-11"></a>            <span class="c1"># Rotate left, then right</span>
 <a id="__codelineno-65-12" name="__codelineno-65-12" href="#__codelineno-65-12"></a>            <span class="n">node</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">left_rotate</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
 <a id="__codelineno-65-13" name="__codelineno-65-13" href="#__codelineno-65-13"></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">right_rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-65-14" name="__codelineno-65-14" href="#__codelineno-65-14"></a>    <span class="c1"># Right-leaning tree</span>
+<a id="__codelineno-65-14" name="__codelineno-65-14" href="#__codelineno-65-14"></a>    <span class="c1"># Right-heavy tree</span>
 <a id="__codelineno-65-15" name="__codelineno-65-15" href="#__codelineno-65-15"></a>    <span class="k">elif</span> <span class="n">balance_factor</span> <span class="o">&lt;</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
 <a id="__codelineno-65-16" name="__codelineno-65-16" href="#__codelineno-65-16"></a>        <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">balance_factor</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mi">0</span><span class="p">:</span>
-<a id="__codelineno-65-17" name="__codelineno-65-17" href="#__codelineno-65-17"></a>            <span class="c1"># Left rotation</span>
+<a id="__codelineno-65-17" name="__codelineno-65-17" href="#__codelineno-65-17"></a>            <span class="c1"># Rotate left</span>
 <a id="__codelineno-65-18" name="__codelineno-65-18" href="#__codelineno-65-18"></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">left_rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-65-19" name="__codelineno-65-19" href="#__codelineno-65-19"></a>        <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-65-20" name="__codelineno-65-20" href="#__codelineno-65-20"></a>            <span class="c1"># First right rotation then left rotation</span>
+<a id="__codelineno-65-20" name="__codelineno-65-20" href="#__codelineno-65-20"></a>            <span class="c1"># Rotate right, then left</span>
 <a id="__codelineno-65-21" name="__codelineno-65-21" href="#__codelineno-65-21"></a>            <span class="n">node</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">right_rotate</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
 <a id="__codelineno-65-22" name="__codelineno-65-22" href="#__codelineno-65-22"></a>            <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">left_rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-65-23" name="__codelineno-65-23" href="#__codelineno-65-23"></a>    <span class="c1"># Balanced tree, no rotation needed, return directly</span>
+<a id="__codelineno-65-23" name="__codelineno-65-23" href="#__codelineno-65-23"></a>    <span class="c1"># The tree is balanced; return without rotating</span>
 <a id="__codelineno-65-24" name="__codelineno-65-24" href="#__codelineno-65-24"></a>    <span class="k">return</span> <span class="n">node</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="cm">/* Perform rotation operation to restore balance to this subtree */</span>
+<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-66-1" name="__codelineno-66-1" href="#__codelineno-66-1"></a><span class="cm">/* Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-66-2" name="__codelineno-66-2" href="#__codelineno-66-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">rotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-66-3" name="__codelineno-66-3" href="#__codelineno-66-3"></a><span class="w">    </span><span class="c1">// Get balance factor of node</span>
+<a id="__codelineno-66-3" name="__codelineno-66-3" href="#__codelineno-66-3"></a><span class="w">    </span><span class="c1">// Get the balance factor of node</span>
 <a id="__codelineno-66-4" name="__codelineno-66-4" href="#__codelineno-66-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">_balanceFactor</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-66-5" name="__codelineno-66-5" href="#__codelineno-66-5"></a><span class="w">    </span><span class="c1">// Left-leaning tree</span>
+<a id="__codelineno-66-5" name="__codelineno-66-5" href="#__codelineno-66-5"></a><span class="w">    </span><span class="c1">// Left-heavy tree</span>
 <a id="__codelineno-66-6" name="__codelineno-66-6" href="#__codelineno-66-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_balanceFactor</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-66-7" name="__codelineno-66-7" href="#__codelineno-66-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-66-8" name="__codelineno-66-8" href="#__codelineno-66-8"></a><span class="w">            </span><span class="c1">// Right rotation</span>
+<a id="__codelineno-66-8" name="__codelineno-66-8" href="#__codelineno-66-8"></a><span class="w">            </span><span class="c1">// Rotate right</span>
 <a id="__codelineno-66-9" name="__codelineno-66-9" href="#__codelineno-66-9"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-66-10" name="__codelineno-66-10" href="#__codelineno-66-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-66-11" name="__codelineno-66-11" href="#__codelineno-66-11"></a><span class="w">            </span><span class="c1">// First left rotation then right rotation</span>
+<a id="__codelineno-66-11" name="__codelineno-66-11" href="#__codelineno-66-11"></a><span class="w">            </span><span class="c1">// Rotate left, then right</span>
 <a id="__codelineno-66-12" name="__codelineno-66-12" href="#__codelineno-66-12"></a><span class="w">            </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">);</span>
 <a id="__codelineno-66-13" name="__codelineno-66-13" href="#__codelineno-66-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-66-14" name="__codelineno-66-14" href="#__codelineno-66-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-66-15" name="__codelineno-66-15" href="#__codelineno-66-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-66-16" name="__codelineno-66-16" href="#__codelineno-66-16"></a><span class="w">    </span><span class="c1">// Right-leaning tree</span>
+<a id="__codelineno-66-16" name="__codelineno-66-16" href="#__codelineno-66-16"></a><span class="w">    </span><span class="c1">// Right-heavy tree</span>
 <a id="__codelineno-66-17" name="__codelineno-66-17" href="#__codelineno-66-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_balanceFactor</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">-1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-66-18" name="__codelineno-66-18" href="#__codelineno-66-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-66-19" name="__codelineno-66-19" href="#__codelineno-66-19"></a><span class="w">            </span><span class="c1">// Left rotation</span>
+<a id="__codelineno-66-19" name="__codelineno-66-19" href="#__codelineno-66-19"></a><span class="w">            </span><span class="c1">// Rotate left</span>
 <a id="__codelineno-66-20" name="__codelineno-66-20" href="#__codelineno-66-20"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-66-21" name="__codelineno-66-21" href="#__codelineno-66-21"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-66-22" name="__codelineno-66-22" href="#__codelineno-66-22"></a><span class="w">            </span><span class="c1">// First right rotation then left rotation</span>
+<a id="__codelineno-66-22" name="__codelineno-66-22" href="#__codelineno-66-22"></a><span class="w">            </span><span class="c1">// Rotate right, then left</span>
 <a id="__codelineno-66-23" name="__codelineno-66-23" href="#__codelineno-66-23"></a><span class="w">            </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-66-24" name="__codelineno-66-24" href="#__codelineno-66-24"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-66-25" name="__codelineno-66-25" href="#__codelineno-66-25"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-66-26" name="__codelineno-66-26" href="#__codelineno-66-26"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-66-27" name="__codelineno-66-27" href="#__codelineno-66-27"></a><span class="w">    </span><span class="c1">// Balanced tree, no rotation needed, return directly</span>
+<a id="__codelineno-66-27" name="__codelineno-66-27" href="#__codelineno-66-27"></a><span class="w">    </span><span class="c1">// The tree is balanced; return without rotating</span>
 <a id="__codelineno-66-28" name="__codelineno-66-28" href="#__codelineno-66-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-66-29" name="__codelineno-66-29" href="#__codelineno-66-29"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="cm">/* Perform rotation operation to restore balance to this subtree */</span>
+<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-67-1" name="__codelineno-67-1" href="#__codelineno-67-1"></a><span class="cm">/* Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-67-2" name="__codelineno-67-2" href="#__codelineno-67-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">rotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-67-3" name="__codelineno-67-3" href="#__codelineno-67-3"></a><span class="w">    </span><span class="c1">// Get balance factor of node</span>
+<a id="__codelineno-67-3" name="__codelineno-67-3" href="#__codelineno-67-3"></a><span class="w">    </span><span class="c1">// Get the balance factor of node</span>
 <a id="__codelineno-67-4" name="__codelineno-67-4" href="#__codelineno-67-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">balanceFactor</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-67-5" name="__codelineno-67-5" href="#__codelineno-67-5"></a><span class="w">    </span><span class="c1">// Left-leaning tree</span>
+<a id="__codelineno-67-5" name="__codelineno-67-5" href="#__codelineno-67-5"></a><span class="w">    </span><span class="c1">// Left-heavy tree</span>
 <a id="__codelineno-67-6" name="__codelineno-67-6" href="#__codelineno-67-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-67-7" name="__codelineno-67-7" href="#__codelineno-67-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-67-8" name="__codelineno-67-8" href="#__codelineno-67-8"></a><span class="w">            </span><span class="c1">// Right rotation</span>
+<a id="__codelineno-67-8" name="__codelineno-67-8" href="#__codelineno-67-8"></a><span class="w">            </span><span class="c1">// Rotate right</span>
 <a id="__codelineno-67-9" name="__codelineno-67-9" href="#__codelineno-67-9"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-67-10" name="__codelineno-67-10" href="#__codelineno-67-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-67-11" name="__codelineno-67-11" href="#__codelineno-67-11"></a><span class="w">            </span><span class="c1">// First left rotation then right rotation</span>
+<a id="__codelineno-67-11" name="__codelineno-67-11" href="#__codelineno-67-11"></a><span class="w">            </span><span class="c1">// Rotate left, then right</span>
 <a id="__codelineno-67-12" name="__codelineno-67-12" href="#__codelineno-67-12"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">);</span>
 <a id="__codelineno-67-13" name="__codelineno-67-13" href="#__codelineno-67-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-67-14" name="__codelineno-67-14" href="#__codelineno-67-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-67-15" name="__codelineno-67-15" href="#__codelineno-67-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-67-16" name="__codelineno-67-16" href="#__codelineno-67-16"></a><span class="w">    </span><span class="c1">// Right-leaning tree</span>
+<a id="__codelineno-67-16" name="__codelineno-67-16" href="#__codelineno-67-16"></a><span class="w">    </span><span class="c1">// Right-heavy tree</span>
 <a id="__codelineno-67-17" name="__codelineno-67-17" href="#__codelineno-67-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-67-18" name="__codelineno-67-18" href="#__codelineno-67-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-67-19" name="__codelineno-67-19" href="#__codelineno-67-19"></a><span class="w">            </span><span class="c1">// Left rotation</span>
+<a id="__codelineno-67-19" name="__codelineno-67-19" href="#__codelineno-67-19"></a><span class="w">            </span><span class="c1">// Rotate left</span>
 <a id="__codelineno-67-20" name="__codelineno-67-20" href="#__codelineno-67-20"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-67-21" name="__codelineno-67-21" href="#__codelineno-67-21"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-67-22" name="__codelineno-67-22" href="#__codelineno-67-22"></a><span class="w">            </span><span class="c1">// First right rotation then left rotation</span>
+<a id="__codelineno-67-22" name="__codelineno-67-22" href="#__codelineno-67-22"></a><span class="w">            </span><span class="c1">// Rotate right, then left</span>
 <a id="__codelineno-67-23" name="__codelineno-67-23" href="#__codelineno-67-23"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">);</span>
 <a id="__codelineno-67-24" name="__codelineno-67-24" href="#__codelineno-67-24"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-67-25" name="__codelineno-67-25" href="#__codelineno-67-25"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-67-26" name="__codelineno-67-26" href="#__codelineno-67-26"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-67-27" name="__codelineno-67-27" href="#__codelineno-67-27"></a><span class="w">    </span><span class="c1">// Balanced tree, no rotation needed, return directly</span>
+<a id="__codelineno-67-27" name="__codelineno-67-27" href="#__codelineno-67-27"></a><span class="w">    </span><span class="c1">// The tree is balanced; return without rotating</span>
 <a id="__codelineno-67-28" name="__codelineno-67-28" href="#__codelineno-67-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-67-29" name="__codelineno-67-29" href="#__codelineno-67-29"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a><span class="cm">/* Perform rotation operation to restore balance to this subtree */</span>
+<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-68-1" name="__codelineno-68-1" href="#__codelineno-68-1"></a><span class="cm">/* Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-68-2" name="__codelineno-68-2" href="#__codelineno-68-2"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">Rotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-68-3" name="__codelineno-68-3" href="#__codelineno-68-3"></a><span class="w">    </span><span class="c1">// Get balance factor of node</span>
+<a id="__codelineno-68-3" name="__codelineno-68-3" href="#__codelineno-68-3"></a><span class="w">    </span><span class="c1">// Get the balance factor of node</span>
 <a id="__codelineno-68-4" name="__codelineno-68-4" href="#__codelineno-68-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">balanceFactorInt</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">BalanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-68-5" name="__codelineno-68-5" href="#__codelineno-68-5"></a><span class="w">    </span><span class="c1">// Left-leaning tree</span>
+<a id="__codelineno-68-5" name="__codelineno-68-5" href="#__codelineno-68-5"></a><span class="w">    </span><span class="c1">// Left-heavy tree</span>
 <a id="__codelineno-68-6" name="__codelineno-68-6" href="#__codelineno-68-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactorInt</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-68-7" name="__codelineno-68-7" href="#__codelineno-68-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">BalanceFactor</span><span class="p">(</span><span class="n">node</span><span class="o">?.</span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-68-8" name="__codelineno-68-8" href="#__codelineno-68-8"></a><span class="w">            </span><span class="c1">// Right rotation</span>
+<a id="__codelineno-68-8" name="__codelineno-68-8" href="#__codelineno-68-8"></a><span class="w">            </span><span class="c1">// Rotate right</span>
 <a id="__codelineno-68-9" name="__codelineno-68-9" href="#__codelineno-68-9"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nf">RightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-68-10" name="__codelineno-68-10" href="#__codelineno-68-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-68-11" name="__codelineno-68-11" href="#__codelineno-68-11"></a><span class="w">            </span><span class="c1">// First left rotation then right rotation</span>
+<a id="__codelineno-68-11" name="__codelineno-68-11" href="#__codelineno-68-11"></a><span class="w">            </span><span class="c1">// Rotate left, then right</span>
 <a id="__codelineno-68-12" name="__codelineno-68-12" href="#__codelineno-68-12"></a><span class="w">            </span><span class="n">node</span><span class="o">!</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">LeftRotate</span><span class="p">(</span><span class="n">node</span><span class="o">!</span><span class="p">.</span><span class="n">left</span><span class="p">);</span>
 <a id="__codelineno-68-13" name="__codelineno-68-13" href="#__codelineno-68-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nf">RightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-68-14" name="__codelineno-68-14" href="#__codelineno-68-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-68-15" name="__codelineno-68-15" href="#__codelineno-68-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-68-16" name="__codelineno-68-16" href="#__codelineno-68-16"></a><span class="w">    </span><span class="c1">// Right-leaning tree</span>
+<a id="__codelineno-68-16" name="__codelineno-68-16" href="#__codelineno-68-16"></a><span class="w">    </span><span class="c1">// Right-heavy tree</span>
 <a id="__codelineno-68-17" name="__codelineno-68-17" href="#__codelineno-68-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactorInt</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-68-18" name="__codelineno-68-18" href="#__codelineno-68-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">BalanceFactor</span><span class="p">(</span><span class="n">node</span><span class="o">?.</span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-68-19" name="__codelineno-68-19" href="#__codelineno-68-19"></a><span class="w">            </span><span class="c1">// Left rotation</span>
+<a id="__codelineno-68-19" name="__codelineno-68-19" href="#__codelineno-68-19"></a><span class="w">            </span><span class="c1">// Rotate left</span>
 <a id="__codelineno-68-20" name="__codelineno-68-20" href="#__codelineno-68-20"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nf">LeftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-68-21" name="__codelineno-68-21" href="#__codelineno-68-21"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-68-22" name="__codelineno-68-22" href="#__codelineno-68-22"></a><span class="w">            </span><span class="c1">// First right rotation then left rotation</span>
+<a id="__codelineno-68-22" name="__codelineno-68-22" href="#__codelineno-68-22"></a><span class="w">            </span><span class="c1">// Rotate right, then left</span>
 <a id="__codelineno-68-23" name="__codelineno-68-23" href="#__codelineno-68-23"></a><span class="w">            </span><span class="n">node</span><span class="o">!</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">RightRotate</span><span class="p">(</span><span class="n">node</span><span class="o">!</span><span class="p">.</span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-68-24" name="__codelineno-68-24" href="#__codelineno-68-24"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nf">LeftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-68-25" name="__codelineno-68-25" href="#__codelineno-68-25"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-68-26" name="__codelineno-68-26" href="#__codelineno-68-26"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-68-27" name="__codelineno-68-27" href="#__codelineno-68-27"></a><span class="w">    </span><span class="c1">// Balanced tree, no rotation needed, return directly</span>
+<a id="__codelineno-68-27" name="__codelineno-68-27" href="#__codelineno-68-27"></a><span class="w">    </span><span class="c1">// The tree is balanced; return without rotating</span>
 <a id="__codelineno-68-28" name="__codelineno-68-28" href="#__codelineno-68-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-68-29" name="__codelineno-68-29" href="#__codelineno-68-29"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a><span class="cm">/* Perform rotation operation to restore balance to this subtree */</span>
+<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-69-1" name="__codelineno-69-1" href="#__codelineno-69-1"></a><span class="cm">/* Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-69-2" name="__codelineno-69-2" href="#__codelineno-69-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">t</span><span class="w"> </span><span class="o">*</span><span class="nx">aVLTree</span><span class="p">)</span><span class="w"> </span><span class="nx">rotate</span><span class="p">(</span><span class="nx">node</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-69-3" name="__codelineno-69-3" href="#__codelineno-69-3"></a><span class="w">    </span><span class="c1">// Get balance factor of node</span>
-<a id="__codelineno-69-4" name="__codelineno-69-4" href="#__codelineno-69-4"></a><span class="w">    </span><span class="c1">// Go recommends short variables, here bf refers to t.balanceFactor</span>
+<a id="__codelineno-69-3" name="__codelineno-69-3" href="#__codelineno-69-3"></a><span class="w">    </span><span class="c1">// Get the balance factor of node</span>
+<a id="__codelineno-69-4" name="__codelineno-69-4" href="#__codelineno-69-4"></a><span class="w">    </span><span class="c1">// Go favors short local variable names; bf holds the result of t.balanceFactor(node)</span>
 <a id="__codelineno-69-5" name="__codelineno-69-5" href="#__codelineno-69-5"></a><span class="w">    </span><span class="nx">bf</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">balanceFactor</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span>
-<a id="__codelineno-69-6" name="__codelineno-69-6" href="#__codelineno-69-6"></a><span class="w">    </span><span class="c1">// Left-leaning tree</span>
+<a id="__codelineno-69-6" name="__codelineno-69-6" href="#__codelineno-69-6"></a><span class="w">    </span><span class="c1">// Left-heavy tree</span>
 <a id="__codelineno-69-7" name="__codelineno-69-7" href="#__codelineno-69-7"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">bf</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-69-8" name="__codelineno-69-8" href="#__codelineno-69-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">balanceFactor</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-69-9" name="__codelineno-69-9" href="#__codelineno-69-9"></a><span class="w">            </span><span class="c1">// Right rotation</span>
+<a id="__codelineno-69-9" name="__codelineno-69-9" href="#__codelineno-69-9"></a><span class="w">            </span><span class="c1">// Rotate right</span>
 <a id="__codelineno-69-10" name="__codelineno-69-10" href="#__codelineno-69-10"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">rightRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span>
 <a id="__codelineno-69-11" name="__codelineno-69-11" href="#__codelineno-69-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-69-12" name="__codelineno-69-12" href="#__codelineno-69-12"></a><span class="w">            </span><span class="c1">// First left rotation then right rotation</span>
+<a id="__codelineno-69-12" name="__codelineno-69-12" href="#__codelineno-69-12"></a><span class="w">            </span><span class="c1">// Rotate left, then right</span>
 <a id="__codelineno-69-13" name="__codelineno-69-13" href="#__codelineno-69-13"></a><span class="w">            </span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">leftRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="p">)</span>
 <a id="__codelineno-69-14" name="__codelineno-69-14" href="#__codelineno-69-14"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">rightRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span>
 <a id="__codelineno-69-15" name="__codelineno-69-15" href="#__codelineno-69-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-69-16" name="__codelineno-69-16" href="#__codelineno-69-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-69-17" name="__codelineno-69-17" href="#__codelineno-69-17"></a><span class="w">    </span><span class="c1">// Right-leaning tree</span>
+<a id="__codelineno-69-17" name="__codelineno-69-17" href="#__codelineno-69-17"></a><span class="w">    </span><span class="c1">// Right-heavy tree</span>
 <a id="__codelineno-69-18" name="__codelineno-69-18" href="#__codelineno-69-18"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">bf</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-69-19" name="__codelineno-69-19" href="#__codelineno-69-19"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">balanceFactor</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-69-20" name="__codelineno-69-20" href="#__codelineno-69-20"></a><span class="w">            </span><span class="c1">// Left rotation</span>
+<a id="__codelineno-69-20" name="__codelineno-69-20" href="#__codelineno-69-20"></a><span class="w">            </span><span class="c1">// Rotate left</span>
 <a id="__codelineno-69-21" name="__codelineno-69-21" href="#__codelineno-69-21"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">leftRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span>
 <a id="__codelineno-69-22" name="__codelineno-69-22" href="#__codelineno-69-22"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-69-23" name="__codelineno-69-23" href="#__codelineno-69-23"></a><span class="w">            </span><span class="c1">// First right rotation then left rotation</span>
+<a id="__codelineno-69-23" name="__codelineno-69-23" href="#__codelineno-69-23"></a><span class="w">            </span><span class="c1">// Rotate right, then left</span>
 <a id="__codelineno-69-24" name="__codelineno-69-24" href="#__codelineno-69-24"></a><span class="w">            </span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">rightRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span><span class="p">)</span>
 <a id="__codelineno-69-25" name="__codelineno-69-25" href="#__codelineno-69-25"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">leftRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span>
 <a id="__codelineno-69-26" name="__codelineno-69-26" href="#__codelineno-69-26"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-69-27" name="__codelineno-69-27" href="#__codelineno-69-27"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-69-28" name="__codelineno-69-28" href="#__codelineno-69-28"></a><span class="w">    </span><span class="c1">// Balanced tree, no rotation needed, return directly</span>
+<a id="__codelineno-69-28" name="__codelineno-69-28" href="#__codelineno-69-28"></a><span class="w">    </span><span class="c1">// The tree is balanced; return without rotating</span>
 <a id="__codelineno-69-29" name="__codelineno-69-29" href="#__codelineno-69-29"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span>
 <a id="__codelineno-69-30" name="__codelineno-69-30" href="#__codelineno-69-30"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.swift</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a><span class="cm">/* Perform rotation operation to restore balance to this subtree */</span>
+<div class="highlight"><span class="filename">avl_tree.swift</span><pre><span></span><code><a id="__codelineno-70-1" name="__codelineno-70-1" href="#__codelineno-70-1"></a><span class="cm">/* Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-70-2" name="__codelineno-70-2" href="#__codelineno-70-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-70-3" name="__codelineno-70-3" href="#__codelineno-70-3"></a><span class="w">    </span><span class="c1">// Get balance factor of node</span>
+<a id="__codelineno-70-3" name="__codelineno-70-3" href="#__codelineno-70-3"></a><span class="w">    </span><span class="c1">// Get the balance factor of node</span>
 <a id="__codelineno-70-4" name="__codelineno-70-4" href="#__codelineno-70-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">balanceFactor</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-70-5" name="__codelineno-70-5" href="#__codelineno-70-5"></a><span class="w">    </span><span class="c1">// Left-leaning tree</span>
+<a id="__codelineno-70-5" name="__codelineno-70-5" href="#__codelineno-70-5"></a><span class="w">    </span><span class="c1">// Left-heavy tree</span>
 <a id="__codelineno-70-6" name="__codelineno-70-6" href="#__codelineno-70-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">balanceFactor</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-70-7" name="__codelineno-70-7" href="#__codelineno-70-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="kc">self</span><span class="p">.</span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">left</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-70-8" name="__codelineno-70-8" href="#__codelineno-70-8"></a><span class="w">            </span><span class="c1">// Right rotation</span>
+<a id="__codelineno-70-8" name="__codelineno-70-8" href="#__codelineno-70-8"></a><span class="w">            </span><span class="c1">// Rotate right</span>
 <a id="__codelineno-70-9" name="__codelineno-70-9" href="#__codelineno-70-9"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-70-10" name="__codelineno-70-10" href="#__codelineno-70-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-70-11" name="__codelineno-70-11" href="#__codelineno-70-11"></a><span class="w">            </span><span class="c1">// First left rotation then right rotation</span>
+<a id="__codelineno-70-11" name="__codelineno-70-11" href="#__codelineno-70-11"></a><span class="w">            </span><span class="c1">// Rotate left, then right</span>
 <a id="__codelineno-70-12" name="__codelineno-70-12" href="#__codelineno-70-12"></a><span class="w">            </span><span class="n">node</span><span class="p">?.</span><span class="kr">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">left</span><span class="p">)</span>
 <a id="__codelineno-70-13" name="__codelineno-70-13" href="#__codelineno-70-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-70-14" name="__codelineno-70-14" href="#__codelineno-70-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-70-15" name="__codelineno-70-15" href="#__codelineno-70-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-70-16" name="__codelineno-70-16" href="#__codelineno-70-16"></a><span class="w">    </span><span class="c1">// Right-leaning tree</span>
+<a id="__codelineno-70-16" name="__codelineno-70-16" href="#__codelineno-70-16"></a><span class="w">    </span><span class="c1">// Right-heavy tree</span>
 <a id="__codelineno-70-17" name="__codelineno-70-17" href="#__codelineno-70-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">balanceFactor</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-70-18" name="__codelineno-70-18" href="#__codelineno-70-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="kc">self</span><span class="p">.</span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">right</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-70-19" name="__codelineno-70-19" href="#__codelineno-70-19"></a><span class="w">            </span><span class="c1">// Left rotation</span>
+<a id="__codelineno-70-19" name="__codelineno-70-19" href="#__codelineno-70-19"></a><span class="w">            </span><span class="c1">// Rotate left</span>
 <a id="__codelineno-70-20" name="__codelineno-70-20" href="#__codelineno-70-20"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-70-21" name="__codelineno-70-21" href="#__codelineno-70-21"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-70-22" name="__codelineno-70-22" href="#__codelineno-70-22"></a><span class="w">            </span><span class="c1">// First right rotation then left rotation</span>
+<a id="__codelineno-70-22" name="__codelineno-70-22" href="#__codelineno-70-22"></a><span class="w">            </span><span class="c1">// Rotate right, then left</span>
 <a id="__codelineno-70-23" name="__codelineno-70-23" href="#__codelineno-70-23"></a><span class="w">            </span><span class="n">node</span><span class="p">?.</span><span class="kr">right</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">right</span><span class="p">)</span>
 <a id="__codelineno-70-24" name="__codelineno-70-24" href="#__codelineno-70-24"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-70-25" name="__codelineno-70-25" href="#__codelineno-70-25"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-70-26" name="__codelineno-70-26" href="#__codelineno-70-26"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-70-27" name="__codelineno-70-27" href="#__codelineno-70-27"></a><span class="w">    </span><span class="c1">// Balanced tree, no rotation needed, return directly</span>
+<a id="__codelineno-70-27" name="__codelineno-70-27" href="#__codelineno-70-27"></a><span class="w">    </span><span class="c1">// The tree is balanced; return without rotating</span>
 <a id="__codelineno-70-28" name="__codelineno-70-28" href="#__codelineno-70-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-70-29" name="__codelineno-70-29" href="#__codelineno-70-29"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a><span class="cm">/* Perform rotation operation to restore balance to this subtree */</span>
+<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-71-1" name="__codelineno-71-1" href="#__codelineno-71-1"></a><span class="cm">/* Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-71-2" name="__codelineno-71-2" href="#__codelineno-71-2"></a><span class="n">#rotate</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-71-3" name="__codelineno-71-3" href="#__codelineno-71-3"></a><span class="w">    </span><span class="c1">// Get balance factor of node</span>
+<a id="__codelineno-71-3" name="__codelineno-71-3" href="#__codelineno-71-3"></a><span class="w">    </span><span class="c1">// Get the balance factor of node</span>
 <a id="__codelineno-71-4" name="__codelineno-71-4" href="#__codelineno-71-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">balanceFactor</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">balanceFactor</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
-<a id="__codelineno-71-5" name="__codelineno-71-5" href="#__codelineno-71-5"></a><span class="w">    </span><span class="c1">// Left-leaning tree</span>
+<a id="__codelineno-71-5" name="__codelineno-71-5" href="#__codelineno-71-5"></a><span class="w">    </span><span class="c1">// Left-heavy tree</span>
 <a id="__codelineno-71-6" name="__codelineno-71-6" href="#__codelineno-71-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">balanceFactor</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-71-7" name="__codelineno-71-7" href="#__codelineno-71-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">balanceFactor</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-71-8" name="__codelineno-71-8" href="#__codelineno-71-8"></a><span class="w">            </span><span class="c1">// Right rotation</span>
+<a id="__codelineno-71-8" name="__codelineno-71-8" href="#__codelineno-71-8"></a><span class="w">            </span><span class="c1">// Rotate right</span>
 <a id="__codelineno-71-9" name="__codelineno-71-9" href="#__codelineno-71-9"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#rightRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
 <a id="__codelineno-71-10" name="__codelineno-71-10" href="#__codelineno-71-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-71-11" name="__codelineno-71-11" href="#__codelineno-71-11"></a><span class="w">            </span><span class="c1">// First left rotation then right rotation</span>
+<a id="__codelineno-71-11" name="__codelineno-71-11" href="#__codelineno-71-11"></a><span class="w">            </span><span class="c1">// Rotate left, then right</span>
 <a id="__codelineno-71-12" name="__codelineno-71-12" href="#__codelineno-71-12"></a><span class="w">            </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#leftRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">);</span>
 <a id="__codelineno-71-13" name="__codelineno-71-13" href="#__codelineno-71-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#rightRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
 <a id="__codelineno-71-14" name="__codelineno-71-14" href="#__codelineno-71-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-71-15" name="__codelineno-71-15" href="#__codelineno-71-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-71-16" name="__codelineno-71-16" href="#__codelineno-71-16"></a><span class="w">    </span><span class="c1">// Right-leaning tree</span>
+<a id="__codelineno-71-16" name="__codelineno-71-16" href="#__codelineno-71-16"></a><span class="w">    </span><span class="c1">// Right-heavy tree</span>
 <a id="__codelineno-71-17" name="__codelineno-71-17" href="#__codelineno-71-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">balanceFactor</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-71-18" name="__codelineno-71-18" href="#__codelineno-71-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">balanceFactor</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-71-19" name="__codelineno-71-19" href="#__codelineno-71-19"></a><span class="w">            </span><span class="c1">// Left rotation</span>
+<a id="__codelineno-71-19" name="__codelineno-71-19" href="#__codelineno-71-19"></a><span class="w">            </span><span class="c1">// Rotate left</span>
 <a id="__codelineno-71-20" name="__codelineno-71-20" href="#__codelineno-71-20"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#leftRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
 <a id="__codelineno-71-21" name="__codelineno-71-21" href="#__codelineno-71-21"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-71-22" name="__codelineno-71-22" href="#__codelineno-71-22"></a><span class="w">            </span><span class="c1">// First right rotation then left rotation</span>
+<a id="__codelineno-71-22" name="__codelineno-71-22" href="#__codelineno-71-22"></a><span class="w">            </span><span class="c1">// Rotate right, then left</span>
 <a id="__codelineno-71-23" name="__codelineno-71-23" href="#__codelineno-71-23"></a><span class="w">            </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#rightRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">);</span>
 <a id="__codelineno-71-24" name="__codelineno-71-24" href="#__codelineno-71-24"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#leftRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
 <a id="__codelineno-71-25" name="__codelineno-71-25" href="#__codelineno-71-25"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-71-26" name="__codelineno-71-26" href="#__codelineno-71-26"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-71-27" name="__codelineno-71-27" href="#__codelineno-71-27"></a><span class="w">    </span><span class="c1">// Balanced tree, no rotation needed, return directly</span>
+<a id="__codelineno-71-27" name="__codelineno-71-27" href="#__codelineno-71-27"></a><span class="w">    </span><span class="c1">// The tree is balanced; return without rotating</span>
 <a id="__codelineno-71-28" name="__codelineno-71-28" href="#__codelineno-71-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-71-29" name="__codelineno-71-29" href="#__codelineno-71-29"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* Perform rotation operation to restore balance to this subtree */</span>
+<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-72-2" name="__codelineno-72-2" href="#__codelineno-72-2"></a><span class="nx">rotate</span><span class="p">(</span><span class="nx">node</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-72-3" name="__codelineno-72-3" href="#__codelineno-72-3"></a><span class="w">    </span><span class="c1">// Get balance factor of node</span>
+<a id="__codelineno-72-3" name="__codelineno-72-3" href="#__codelineno-72-3"></a><span class="w">    </span><span class="c1">// Get the balance factor of node</span>
 <a id="__codelineno-72-4" name="__codelineno-72-4" href="#__codelineno-72-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">balanceFactor</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">balanceFactor</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
-<a id="__codelineno-72-5" name="__codelineno-72-5" href="#__codelineno-72-5"></a><span class="w">    </span><span class="c1">// Left-leaning tree</span>
+<a id="__codelineno-72-5" name="__codelineno-72-5" href="#__codelineno-72-5"></a><span class="w">    </span><span class="c1">// Left-heavy tree</span>
 <a id="__codelineno-72-6" name="__codelineno-72-6" href="#__codelineno-72-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">balanceFactor</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-72-7" name="__codelineno-72-7" href="#__codelineno-72-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">balanceFactor</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-72-8" name="__codelineno-72-8" href="#__codelineno-72-8"></a><span class="w">            </span><span class="c1">// Right rotation</span>
+<a id="__codelineno-72-8" name="__codelineno-72-8" href="#__codelineno-72-8"></a><span class="w">            </span><span class="c1">// Rotate right</span>
 <a id="__codelineno-72-9" name="__codelineno-72-9" href="#__codelineno-72-9"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">rightRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
 <a id="__codelineno-72-10" name="__codelineno-72-10" href="#__codelineno-72-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-72-11" name="__codelineno-72-11" href="#__codelineno-72-11"></a><span class="w">            </span><span class="c1">// First left rotation then right rotation</span>
+<a id="__codelineno-72-11" name="__codelineno-72-11" href="#__codelineno-72-11"></a><span class="w">            </span><span class="c1">// Rotate left, then right</span>
 <a id="__codelineno-72-12" name="__codelineno-72-12" href="#__codelineno-72-12"></a><span class="w">            </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">leftRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">);</span>
 <a id="__codelineno-72-13" name="__codelineno-72-13" href="#__codelineno-72-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">rightRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
 <a id="__codelineno-72-14" name="__codelineno-72-14" href="#__codelineno-72-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-72-15" name="__codelineno-72-15" href="#__codelineno-72-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-72-16" name="__codelineno-72-16" href="#__codelineno-72-16"></a><span class="w">    </span><span class="c1">// Right-leaning tree</span>
+<a id="__codelineno-72-16" name="__codelineno-72-16" href="#__codelineno-72-16"></a><span class="w">    </span><span class="c1">// Right-heavy tree</span>
 <a id="__codelineno-72-17" name="__codelineno-72-17" href="#__codelineno-72-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">balanceFactor</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="o">-</span><span class="mf">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-72-18" name="__codelineno-72-18" href="#__codelineno-72-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">balanceFactor</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mf">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-72-19" name="__codelineno-72-19" href="#__codelineno-72-19"></a><span class="w">            </span><span class="c1">// Left rotation</span>
+<a id="__codelineno-72-19" name="__codelineno-72-19" href="#__codelineno-72-19"></a><span class="w">            </span><span class="c1">// Rotate left</span>
 <a id="__codelineno-72-20" name="__codelineno-72-20" href="#__codelineno-72-20"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">leftRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
 <a id="__codelineno-72-21" name="__codelineno-72-21" href="#__codelineno-72-21"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-72-22" name="__codelineno-72-22" href="#__codelineno-72-22"></a><span class="w">            </span><span class="c1">// First right rotation then left rotation</span>
+<a id="__codelineno-72-22" name="__codelineno-72-22" href="#__codelineno-72-22"></a><span class="w">            </span><span class="c1">// Rotate right, then left</span>
 <a id="__codelineno-72-23" name="__codelineno-72-23" href="#__codelineno-72-23"></a><span class="w">            </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">rightRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">);</span>
 <a id="__codelineno-72-24" name="__codelineno-72-24" href="#__codelineno-72-24"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">leftRotate</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
 <a id="__codelineno-72-25" name="__codelineno-72-25" href="#__codelineno-72-25"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-72-26" name="__codelineno-72-26" href="#__codelineno-72-26"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-72-27" name="__codelineno-72-27" href="#__codelineno-72-27"></a><span class="w">    </span><span class="c1">// Balanced tree, no rotation needed, return directly</span>
+<a id="__codelineno-72-27" name="__codelineno-72-27" href="#__codelineno-72-27"></a><span class="w">    </span><span class="c1">// The tree is balanced; return without rotating</span>
 <a id="__codelineno-72-28" name="__codelineno-72-28" href="#__codelineno-72-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-72-29" name="__codelineno-72-29" href="#__codelineno-72-29"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.dart</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="cm">/* Perform rotation operation to restore balance to this subtree */</span>
+<div class="highlight"><span class="filename">avl_tree.dart</span><pre><span></span><code><a id="__codelineno-73-1" name="__codelineno-73-1" href="#__codelineno-73-1"></a><span class="cm">/* Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-73-2" name="__codelineno-73-2" href="#__codelineno-73-2"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">rotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-73-3" name="__codelineno-73-3" href="#__codelineno-73-3"></a><span class="w">  </span><span class="c1">// Get balance factor of node</span>
+<a id="__codelineno-73-3" name="__codelineno-73-3" href="#__codelineno-73-3"></a><span class="w">  </span><span class="c1">// Get the balance factor of node</span>
 <a id="__codelineno-73-4" name="__codelineno-73-4" href="#__codelineno-73-4"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">factor</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-73-5" name="__codelineno-73-5" href="#__codelineno-73-5"></a><span class="w">  </span><span class="c1">// Left-leaning tree</span>
+<a id="__codelineno-73-5" name="__codelineno-73-5" href="#__codelineno-73-5"></a><span class="w">  </span><span class="c1">// Left-heavy tree</span>
 <a id="__codelineno-73-6" name="__codelineno-73-6" href="#__codelineno-73-6"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">factor</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-73-7" name="__codelineno-73-7" href="#__codelineno-73-7"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="o">!</span><span class="p">.</span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-73-8" name="__codelineno-73-8" href="#__codelineno-73-8"></a><span class="w">      </span><span class="c1">// Right rotation</span>
+<a id="__codelineno-73-8" name="__codelineno-73-8" href="#__codelineno-73-8"></a><span class="w">      </span><span class="c1">// Rotate right</span>
 <a id="__codelineno-73-9" name="__codelineno-73-9" href="#__codelineno-73-9"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-73-10" name="__codelineno-73-10" href="#__codelineno-73-10"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-73-11" name="__codelineno-73-11" href="#__codelineno-73-11"></a><span class="w">      </span><span class="c1">// First left rotation then right rotation</span>
+<a id="__codelineno-73-11" name="__codelineno-73-11" href="#__codelineno-73-11"></a><span class="w">      </span><span class="c1">// Rotate left, then right</span>
 <a id="__codelineno-73-12" name="__codelineno-73-12" href="#__codelineno-73-12"></a><span class="w">      </span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">);</span>
 <a id="__codelineno-73-13" name="__codelineno-73-13" href="#__codelineno-73-13"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-73-14" name="__codelineno-73-14" href="#__codelineno-73-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-73-15" name="__codelineno-73-15" href="#__codelineno-73-15"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-73-16" name="__codelineno-73-16" href="#__codelineno-73-16"></a><span class="w">  </span><span class="c1">// Right-leaning tree</span>
+<a id="__codelineno-73-16" name="__codelineno-73-16" href="#__codelineno-73-16"></a><span class="w">  </span><span class="c1">// Right-heavy tree</span>
 <a id="__codelineno-73-17" name="__codelineno-73-17" href="#__codelineno-73-17"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">factor</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-73-18" name="__codelineno-73-18" href="#__codelineno-73-18"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="o">!</span><span class="p">.</span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-73-19" name="__codelineno-73-19" href="#__codelineno-73-19"></a><span class="w">      </span><span class="c1">// Left rotation</span>
+<a id="__codelineno-73-19" name="__codelineno-73-19" href="#__codelineno-73-19"></a><span class="w">      </span><span class="c1">// Rotate left</span>
 <a id="__codelineno-73-20" name="__codelineno-73-20" href="#__codelineno-73-20"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-73-21" name="__codelineno-73-21" href="#__codelineno-73-21"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-73-22" name="__codelineno-73-22" href="#__codelineno-73-22"></a><span class="w">      </span><span class="c1">// First right rotation then left rotation</span>
+<a id="__codelineno-73-22" name="__codelineno-73-22" href="#__codelineno-73-22"></a><span class="w">      </span><span class="c1">// Rotate right, then left</span>
 <a id="__codelineno-73-23" name="__codelineno-73-23" href="#__codelineno-73-23"></a><span class="w">      </span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-73-24" name="__codelineno-73-24" href="#__codelineno-73-24"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-73-25" name="__codelineno-73-25" href="#__codelineno-73-25"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-73-26" name="__codelineno-73-26" href="#__codelineno-73-26"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-73-27" name="__codelineno-73-27" href="#__codelineno-73-27"></a><span class="w">  </span><span class="c1">// Balanced tree, no rotation needed, return directly</span>
+<a id="__codelineno-73-27" name="__codelineno-73-27" href="#__codelineno-73-27"></a><span class="w">  </span><span class="c1">// The tree is balanced; return without rotating</span>
 <a id="__codelineno-73-28" name="__codelineno-73-28" href="#__codelineno-73-28"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-73-29" name="__codelineno-73-29" href="#__codelineno-73-29"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.rs</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="cm">/* Perform rotation operation to restore balance to this subtree */</span>
+<div class="highlight"><span class="filename">avl_tree.rs</span><pre><span></span><code><a id="__codelineno-74-1" name="__codelineno-74-1" href="#__codelineno-74-1"></a><span class="cm">/* Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-74-2" name="__codelineno-74-2" href="#__codelineno-74-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="nc">OptionTreeNodeRc</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">OptionTreeNodeRc</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-74-3" name="__codelineno-74-3" href="#__codelineno-74-3"></a><span class="w">    </span><span class="c1">// Get balance factor of node</span>
+<a id="__codelineno-74-3" name="__codelineno-74-3" href="#__codelineno-74-3"></a><span class="w">    </span><span class="c1">// Get the balance factor of node</span>
 <a id="__codelineno-74-4" name="__codelineno-74-4" href="#__codelineno-74-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">balance_factor</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">balance_factor</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
-<a id="__codelineno-74-5" name="__codelineno-74-5" href="#__codelineno-74-5"></a><span class="w">    </span><span class="c1">// Left-leaning tree</span>
+<a id="__codelineno-74-5" name="__codelineno-74-5" href="#__codelineno-74-5"></a><span class="w">    </span><span class="c1">// Left-heavy tree</span>
 <a id="__codelineno-74-6" name="__codelineno-74-6" href="#__codelineno-74-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">balance_factor</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-74-7" name="__codelineno-74-7" href="#__codelineno-74-7"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">unwrap</span><span class="p">();</span>
 <a id="__codelineno-74-8" name="__codelineno-74-8" href="#__codelineno-74-8"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">balance_factor</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">())</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-74-9" name="__codelineno-74-9" href="#__codelineno-74-9"></a><span class="w">            </span><span class="c1">// Right rotation</span>
+<a id="__codelineno-74-9" name="__codelineno-74-9" href="#__codelineno-74-9"></a><span class="w">            </span><span class="c1">// Rotate right</span>
 <a id="__codelineno-74-10" name="__codelineno-74-10" href="#__codelineno-74-10"></a><span class="w">            </span><span class="bp">Self</span><span class="p">::</span><span class="n">right_rotate</span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">))</span>
 <a id="__codelineno-74-11" name="__codelineno-74-11" href="#__codelineno-74-11"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-74-12" name="__codelineno-74-12" href="#__codelineno-74-12"></a><span class="w">            </span><span class="c1">// First left rotation then right rotation</span>
+<a id="__codelineno-74-12" name="__codelineno-74-12" href="#__codelineno-74-12"></a><span class="w">            </span><span class="c1">// Rotate left, then right</span>
 <a id="__codelineno-74-13" name="__codelineno-74-13" href="#__codelineno-74-13"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-74-14" name="__codelineno-74-14" href="#__codelineno-74-14"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">left_rotate</span><span class="p">(</span><span class="n">left</span><span class="p">);</span>
 <a id="__codelineno-74-15" name="__codelineno-74-15" href="#__codelineno-74-15"></a><span class="w">            </span><span class="bp">Self</span><span class="p">::</span><span class="n">right_rotate</span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">))</span>
 <a id="__codelineno-74-16" name="__codelineno-74-16" href="#__codelineno-74-16"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-74-17" name="__codelineno-74-17" href="#__codelineno-74-17"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-74-18" name="__codelineno-74-18" href="#__codelineno-74-18"></a><span class="w">    </span><span class="c1">// Right-leaning tree</span>
+<a id="__codelineno-74-18" name="__codelineno-74-18" href="#__codelineno-74-18"></a><span class="w">    </span><span class="c1">// Right-heavy tree</span>
 <a id="__codelineno-74-19" name="__codelineno-74-19" href="#__codelineno-74-19"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">balance_factor</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-74-20" name="__codelineno-74-20" href="#__codelineno-74-20"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">unwrap</span><span class="p">();</span>
 <a id="__codelineno-74-21" name="__codelineno-74-21" href="#__codelineno-74-21"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">balance_factor</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">())</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-74-22" name="__codelineno-74-22" href="#__codelineno-74-22"></a><span class="w">            </span><span class="c1">// Left rotation</span>
+<a id="__codelineno-74-22" name="__codelineno-74-22" href="#__codelineno-74-22"></a><span class="w">            </span><span class="c1">// Rotate left</span>
 <a id="__codelineno-74-23" name="__codelineno-74-23" href="#__codelineno-74-23"></a><span class="w">            </span><span class="bp">Self</span><span class="p">::</span><span class="n">left_rotate</span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">))</span>
 <a id="__codelineno-74-24" name="__codelineno-74-24" href="#__codelineno-74-24"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-74-25" name="__codelineno-74-25" href="#__codelineno-74-25"></a><span class="w">            </span><span class="c1">// First right rotation then left rotation</span>
+<a id="__codelineno-74-25" name="__codelineno-74-25" href="#__codelineno-74-25"></a><span class="w">            </span><span class="c1">// Rotate right, then left</span>
 <a id="__codelineno-74-26" name="__codelineno-74-26" href="#__codelineno-74-26"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-74-27" name="__codelineno-74-27" href="#__codelineno-74-27"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">right_rotate</span><span class="p">(</span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-74-28" name="__codelineno-74-28" href="#__codelineno-74-28"></a><span class="w">            </span><span class="bp">Self</span><span class="p">::</span><span class="n">left_rotate</span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">))</span>
 <a id="__codelineno-74-29" name="__codelineno-74-29" href="#__codelineno-74-29"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-74-30" name="__codelineno-74-30" href="#__codelineno-74-30"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-74-31" name="__codelineno-74-31" href="#__codelineno-74-31"></a><span class="w">        </span><span class="c1">// Balanced tree, no rotation needed, return directly</span>
+<a id="__codelineno-74-31" name="__codelineno-74-31" href="#__codelineno-74-31"></a><span class="w">        </span><span class="c1">// The tree is balanced; return without rotating</span>
 <a id="__codelineno-74-32" name="__codelineno-74-32" href="#__codelineno-74-32"></a><span class="w">        </span><span class="n">node</span>
 <a id="__codelineno-74-33" name="__codelineno-74-33" href="#__codelineno-74-33"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-74-34" name="__codelineno-74-34" href="#__codelineno-74-34"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="cm">/* Perform rotation operation to restore balance to this subtree */</span>
+<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-75-1" name="__codelineno-75-1" href="#__codelineno-75-1"></a><span class="cm">/* Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-75-2" name="__codelineno-75-2" href="#__codelineno-75-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">rotate</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-75-3" name="__codelineno-75-3" href="#__codelineno-75-3"></a><span class="w">    </span><span class="c1">// Get balance factor of node</span>
+<a id="__codelineno-75-3" name="__codelineno-75-3" href="#__codelineno-75-3"></a><span class="w">    </span><span class="c1">// Get the balance factor of node</span>
 <a id="__codelineno-75-4" name="__codelineno-75-4" href="#__codelineno-75-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">bf</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-75-5" name="__codelineno-75-5" href="#__codelineno-75-5"></a><span class="w">    </span><span class="c1">// Left-leaning tree</span>
+<a id="__codelineno-75-5" name="__codelineno-75-5" href="#__codelineno-75-5"></a><span class="w">    </span><span class="c1">// Left-heavy tree</span>
 <a id="__codelineno-75-6" name="__codelineno-75-6" href="#__codelineno-75-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">bf</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-75-7" name="__codelineno-75-7" href="#__codelineno-75-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-75-8" name="__codelineno-75-8" href="#__codelineno-75-8"></a><span class="w">            </span><span class="c1">// Right rotation</span>
+<a id="__codelineno-75-8" name="__codelineno-75-8" href="#__codelineno-75-8"></a><span class="w">            </span><span class="c1">// Rotate right</span>
 <a id="__codelineno-75-9" name="__codelineno-75-9" href="#__codelineno-75-9"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-75-10" name="__codelineno-75-10" href="#__codelineno-75-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-75-11" name="__codelineno-75-11" href="#__codelineno-75-11"></a><span class="w">            </span><span class="c1">// First left rotation then right rotation</span>
+<a id="__codelineno-75-11" name="__codelineno-75-11" href="#__codelineno-75-11"></a><span class="w">            </span><span class="c1">// Rotate left, then right</span>
 <a id="__codelineno-75-12" name="__codelineno-75-12" href="#__codelineno-75-12"></a><span class="w">            </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">);</span>
 <a id="__codelineno-75-13" name="__codelineno-75-13" href="#__codelineno-75-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-75-14" name="__codelineno-75-14" href="#__codelineno-75-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-75-15" name="__codelineno-75-15" href="#__codelineno-75-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-75-16" name="__codelineno-75-16" href="#__codelineno-75-16"></a><span class="w">    </span><span class="c1">// Right-leaning tree</span>
+<a id="__codelineno-75-16" name="__codelineno-75-16" href="#__codelineno-75-16"></a><span class="w">    </span><span class="c1">// Right-heavy tree</span>
 <a id="__codelineno-75-17" name="__codelineno-75-17" href="#__codelineno-75-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">bf</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="mi">-1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-75-18" name="__codelineno-75-18" href="#__codelineno-75-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-75-19" name="__codelineno-75-19" href="#__codelineno-75-19"></a><span class="w">            </span><span class="c1">// Left rotation</span>
+<a id="__codelineno-75-19" name="__codelineno-75-19" href="#__codelineno-75-19"></a><span class="w">            </span><span class="c1">// Rotate left</span>
 <a id="__codelineno-75-20" name="__codelineno-75-20" href="#__codelineno-75-20"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-75-21" name="__codelineno-75-21" href="#__codelineno-75-21"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-75-22" name="__codelineno-75-22" href="#__codelineno-75-22"></a><span class="w">            </span><span class="c1">// First right rotation then left rotation</span>
+<a id="__codelineno-75-22" name="__codelineno-75-22" href="#__codelineno-75-22"></a><span class="w">            </span><span class="c1">// Rotate right, then left</span>
 <a id="__codelineno-75-23" name="__codelineno-75-23" href="#__codelineno-75-23"></a><span class="w">            </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-75-24" name="__codelineno-75-24" href="#__codelineno-75-24"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
 <a id="__codelineno-75-25" name="__codelineno-75-25" href="#__codelineno-75-25"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-75-26" name="__codelineno-75-26" href="#__codelineno-75-26"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-75-27" name="__codelineno-75-27" href="#__codelineno-75-27"></a><span class="w">    </span><span class="c1">// Balanced tree, no rotation needed, return directly</span>
+<a id="__codelineno-75-27" name="__codelineno-75-27" href="#__codelineno-75-27"></a><span class="w">    </span><span class="c1">// The tree is balanced; return without rotating</span>
 <a id="__codelineno-75-28" name="__codelineno-75-28" href="#__codelineno-75-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-75-29" name="__codelineno-75-29" href="#__codelineno-75-29"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.kt</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a><span class="cm">/* Perform rotation operation to restore balance to this subtree */</span>
+<div class="highlight"><span class="filename">avl_tree.kt</span><pre><span></span><code><a id="__codelineno-76-1" name="__codelineno-76-1" href="#__codelineno-76-1"></a><span class="cm">/* Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-76-2" name="__codelineno-76-2" href="#__codelineno-76-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">):</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-76-3" name="__codelineno-76-3" href="#__codelineno-76-3"></a><span class="w">    </span><span class="c1">// Get balance factor of node</span>
+<a id="__codelineno-76-3" name="__codelineno-76-3" href="#__codelineno-76-3"></a><span class="w">    </span><span class="c1">// Get the balance factor of node</span>
 <a id="__codelineno-76-4" name="__codelineno-76-4" href="#__codelineno-76-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">balanceFactor</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-76-5" name="__codelineno-76-5" href="#__codelineno-76-5"></a><span class="w">    </span><span class="c1">// Left-leaning tree</span>
+<a id="__codelineno-76-5" name="__codelineno-76-5" href="#__codelineno-76-5"></a><span class="w">    </span><span class="c1">// Left-heavy tree</span>
 <a id="__codelineno-76-6" name="__codelineno-76-6" href="#__codelineno-76-6"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-76-7" name="__codelineno-76-7" href="#__codelineno-76-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-76-8" name="__codelineno-76-8" href="#__codelineno-76-8"></a><span class="w">            </span><span class="c1">// Right rotation</span>
+<a id="__codelineno-76-8" name="__codelineno-76-8" href="#__codelineno-76-8"></a><span class="w">            </span><span class="c1">// Rotate right</span>
 <a id="__codelineno-76-9" name="__codelineno-76-9" href="#__codelineno-76-9"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-76-10" name="__codelineno-76-10" href="#__codelineno-76-10"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-76-11" name="__codelineno-76-11" href="#__codelineno-76-11"></a><span class="w">            </span><span class="c1">// First left rotation then right rotation</span>
+<a id="__codelineno-76-11" name="__codelineno-76-11" href="#__codelineno-76-11"></a><span class="w">            </span><span class="c1">// Rotate left, then right</span>
 <a id="__codelineno-76-12" name="__codelineno-76-12" href="#__codelineno-76-12"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">)</span>
 <a id="__codelineno-76-13" name="__codelineno-76-13" href="#__codelineno-76-13"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-76-14" name="__codelineno-76-14" href="#__codelineno-76-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-76-15" name="__codelineno-76-15" href="#__codelineno-76-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-76-16" name="__codelineno-76-16" href="#__codelineno-76-16"></a><span class="w">    </span><span class="c1">// Right-leaning tree</span>
+<a id="__codelineno-76-16" name="__codelineno-76-16" href="#__codelineno-76-16"></a><span class="w">    </span><span class="c1">// Right-heavy tree</span>
 <a id="__codelineno-76-17" name="__codelineno-76-17" href="#__codelineno-76-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="o">-</span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-76-18" name="__codelineno-76-18" href="#__codelineno-76-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">balanceFactor</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="m">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-76-19" name="__codelineno-76-19" href="#__codelineno-76-19"></a><span class="w">            </span><span class="c1">// Left rotation</span>
+<a id="__codelineno-76-19" name="__codelineno-76-19" href="#__codelineno-76-19"></a><span class="w">            </span><span class="c1">// Rotate left</span>
 <a id="__codelineno-76-20" name="__codelineno-76-20" href="#__codelineno-76-20"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-76-21" name="__codelineno-76-21" href="#__codelineno-76-21"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-76-22" name="__codelineno-76-22" href="#__codelineno-76-22"></a><span class="w">            </span><span class="c1">// First right rotation then left rotation</span>
+<a id="__codelineno-76-22" name="__codelineno-76-22" href="#__codelineno-76-22"></a><span class="w">            </span><span class="c1">// Rotate right, then left</span>
 <a id="__codelineno-76-23" name="__codelineno-76-23" href="#__codelineno-76-23"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rightRotate</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">)</span>
 <a id="__codelineno-76-24" name="__codelineno-76-24" href="#__codelineno-76-24"></a><span class="w">            </span><span class="k">return</span><span class="w"> </span><span class="n">leftRotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-76-25" name="__codelineno-76-25" href="#__codelineno-76-25"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-76-26" name="__codelineno-76-26" href="#__codelineno-76-26"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-76-27" name="__codelineno-76-27" href="#__codelineno-76-27"></a><span class="w">    </span><span class="c1">// Balanced tree, no rotation needed, return directly</span>
+<a id="__codelineno-76-27" name="__codelineno-76-27" href="#__codelineno-76-27"></a><span class="w">    </span><span class="c1">// The tree is balanced; return without rotating</span>
 <a id="__codelineno-76-28" name="__codelineno-76-28" href="#__codelineno-76-28"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-76-29" name="__codelineno-76-29" href="#__codelineno-76-29"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.rb</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="c1">### Perform rotation to rebalance subtree ###</span>
+<div class="highlight"><span class="filename">avl_tree.rb</span><pre><span></span><code><a id="__codelineno-77-1" name="__codelineno-77-1" href="#__codelineno-77-1"></a><span class="c1">### Perform rotations to restore balance in this subtree ###</span>
 <a id="__codelineno-77-2" name="__codelineno-77-2" href="#__codelineno-77-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-77-3" name="__codelineno-77-3" href="#__codelineno-77-3"></a><span class="w">  </span><span class="c1"># Get balance factor of node</span>
+<a id="__codelineno-77-3" name="__codelineno-77-3" href="#__codelineno-77-3"></a><span class="w">  </span><span class="c1"># Get the balance factor of node</span>
 <a id="__codelineno-77-4" name="__codelineno-77-4" href="#__codelineno-77-4"></a><span class="w">  </span><span class="n">balance_factor</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">balance_factor</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-77-5" name="__codelineno-77-5" href="#__codelineno-77-5"></a><span class="w">  </span><span class="c1"># Left-heavy tree</span>
 <a id="__codelineno-77-6" name="__codelineno-77-6" href="#__codelineno-77-6"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="n">balance_factor</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">1</span>
 <a id="__codelineno-77-7" name="__codelineno-77-7" href="#__codelineno-77-7"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">balance_factor</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-77-8" name="__codelineno-77-8" href="#__codelineno-77-8"></a><span class="w">      </span><span class="c1"># Right rotation</span>
+<a id="__codelineno-77-8" name="__codelineno-77-8" href="#__codelineno-77-8"></a><span class="w">      </span><span class="c1"># Rotate right</span>
 <a id="__codelineno-77-9" name="__codelineno-77-9" href="#__codelineno-77-9"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">right_rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-77-10" name="__codelineno-77-10" href="#__codelineno-77-10"></a><span class="w">    </span><span class="k">else</span>
-<a id="__codelineno-77-11" name="__codelineno-77-11" href="#__codelineno-77-11"></a><span class="w">      </span><span class="c1"># First left rotation then right rotation</span>
+<a id="__codelineno-77-11" name="__codelineno-77-11" href="#__codelineno-77-11"></a><span class="w">      </span><span class="c1"># Rotate left, then right</span>
 <a id="__codelineno-77-12" name="__codelineno-77-12" href="#__codelineno-77-12"></a><span class="w">      </span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left_rotate</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
 <a id="__codelineno-77-13" name="__codelineno-77-13" href="#__codelineno-77-13"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">right_rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-77-14" name="__codelineno-77-14" href="#__codelineno-77-14"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-77-15" name="__codelineno-77-15" href="#__codelineno-77-15"></a><span class="w">  </span><span class="c1"># Right-heavy tree</span>
 <a id="__codelineno-77-16" name="__codelineno-77-16" href="#__codelineno-77-16"></a><span class="w">  </span><span class="k">elsif</span><span class="w"> </span><span class="n">balance_factor</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="o">-</span><span class="mi">1</span>
 <a id="__codelineno-77-17" name="__codelineno-77-17" href="#__codelineno-77-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">balance_factor</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="mi">0</span>
-<a id="__codelineno-77-18" name="__codelineno-77-18" href="#__codelineno-77-18"></a><span class="w">      </span><span class="c1"># Left rotation</span>
+<a id="__codelineno-77-18" name="__codelineno-77-18" href="#__codelineno-77-18"></a><span class="w">      </span><span class="c1"># Rotate left</span>
 <a id="__codelineno-77-19" name="__codelineno-77-19" href="#__codelineno-77-19"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">left_rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-77-20" name="__codelineno-77-20" href="#__codelineno-77-20"></a><span class="w">    </span><span class="k">else</span>
-<a id="__codelineno-77-21" name="__codelineno-77-21" href="#__codelineno-77-21"></a><span class="w">      </span><span class="c1"># First right rotation then left rotation</span>
+<a id="__codelineno-77-21" name="__codelineno-77-21" href="#__codelineno-77-21"></a><span class="w">      </span><span class="c1"># Rotate right, then left</span>
 <a id="__codelineno-77-22" name="__codelineno-77-22" href="#__codelineno-77-22"></a><span class="w">      </span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">right_rotate</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
 <a id="__codelineno-77-23" name="__codelineno-77-23" href="#__codelineno-77-23"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="n">left_rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-77-24" name="__codelineno-77-24" href="#__codelineno-77-24"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-77-25" name="__codelineno-77-25" href="#__codelineno-77-25"></a><span class="w">  </span><span class="k">end</span>
-<a id="__codelineno-77-26" name="__codelineno-77-26" href="#__codelineno-77-26"></a><span class="w">  </span><span class="c1"># Balanced tree, no rotation needed, return directly</span>
+<a id="__codelineno-77-26" name="__codelineno-77-26" href="#__codelineno-77-26"></a><span class="w">  </span><span class="c1"># The tree is balanced; return without rotating</span>
 <a id="__codelineno-77-27" name="__codelineno-77-27" href="#__codelineno-77-27"></a><span class="w">  </span><span class="n">node</span>
 <a id="__codelineno-77-28" name="__codelineno-77-28" href="#__codelineno-77-28"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h2 id="753-common-operations-in-avl-trees">7.5.3 &nbsp; Common Operations in AVL Trees<a class="headerlink" href="#753-common-operations-in-avl-trees" title="Permanent link">&para;</a></h2>
-<h3 id="1-node-insertion">1. &nbsp; Node Insertion<a class="headerlink" href="#1-node-insertion" title="Permanent link">&para;</a></h3>
-<p>The node insertion operation in AVL trees is similar in principle to that in binary search trees. The only difference is that after inserting a node in an AVL tree, a series of unbalanced nodes may appear on the path from that node to the root. Therefore, <strong>we need to start from this node and perform rotation operations from bottom to top, restoring balance to all unbalanced nodes</strong>. The code is as follows:</p>
+<h2 id="753-common-avl-tree-operations">7.5.3 &nbsp; Common AVL tree operations<a class="headerlink" href="#753-common-avl-tree-operations" title="Permanent link">&para;</a></h2>
+<h3 id="1-inserting-a-node">1. &nbsp; Inserting a node<a class="headerlink" href="#1-inserting-a-node" title="Permanent link">&para;</a></h3>
+<p>Insertion largely follows the binary-search-tree procedure. The difference is that inserting a node may unbalance nodes along the path to the root. <strong>Starting at the inserted node, work upward and perform rotations to restore balance at every affected node</strong>:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="8:13"><input checked="checked" id="__tabbed_8_1" name="__tabbed_8" type="radio" /><input id="__tabbed_8_2" name="__tabbed_8" type="radio" /><input id="__tabbed_8_3" name="__tabbed_8" type="radio" /><input id="__tabbed_8_4" name="__tabbed_8" type="radio" /><input id="__tabbed_8_5" name="__tabbed_8" type="radio" /><input id="__tabbed_8_6" name="__tabbed_8" type="radio" /><input id="__tabbed_8_7" name="__tabbed_8" type="radio" /><input id="__tabbed_8_8" name="__tabbed_8" type="radio" /><input id="__tabbed_8_9" name="__tabbed_8" type="radio" /><input id="__tabbed_8_10" name="__tabbed_8" type="radio" /><input id="__tabbed_8_11" name="__tabbed_8" type="radio" /><input id="__tabbed_8_12" name="__tabbed_8" type="radio" /><input id="__tabbed_8_13" name="__tabbed_8" type="radio" /><div class="tabbed-labels"><label for="__tabbed_8_1">Python</label><label for="__tabbed_8_2">C++</label><label for="__tabbed_8_3">Java</label><label for="__tabbed_8_4">C#</label><label for="__tabbed_8_5">Go</label><label for="__tabbed_8_6">Swift</label><label for="__tabbed_8_7">JS</label><label for="__tabbed_8_8">TS</label><label for="__tabbed_8_9">Dart</label><label for="__tabbed_8_10">Rust</label><label for="__tabbed_8_11">C</label><label for="__tabbed_8_12">Kotlin</label><label for="__tabbed_8_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">avl_tree.py</span><pre><span></span><code><a id="__codelineno-78-1" name="__codelineno-78-1" href="#__codelineno-78-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">):</span>
-<a id="__codelineno-78-2" name="__codelineno-78-2" href="#__codelineno-78-2"></a><span class="w">    </span><span class="sd">"""Insert node"""</span>
+<a id="__codelineno-78-2" name="__codelineno-78-2" href="#__codelineno-78-2"></a><span class="w">    </span><span class="sd">"""Insert a node"""</span>
 <a id="__codelineno-78-3" name="__codelineno-78-3" href="#__codelineno-78-3"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">_root</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">insert_helper</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_root</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-78-4" name="__codelineno-78-4" href="#__codelineno-78-4"></a>
 <a id="__codelineno-78-5" name="__codelineno-78-5" href="#__codelineno-78-5"></a><span class="k">def</span><span class="w"> </span><span class="nf">insert_helper</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">node</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">TreeNode</span><span class="p">:</span>
-<a id="__codelineno-78-6" name="__codelineno-78-6" href="#__codelineno-78-6"></a><span class="w">    </span><span class="sd">"""Recursively insert node (helper method)"""</span>
+<a id="__codelineno-78-6" name="__codelineno-78-6" href="#__codelineno-78-6"></a><span class="w">    </span><span class="sd">"""Insert a node recursively (helper method)"""</span>
 <a id="__codelineno-78-7" name="__codelineno-78-7" href="#__codelineno-78-7"></a>    <span class="k">if</span> <span class="n">node</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-78-8" name="__codelineno-78-8" href="#__codelineno-78-8"></a>        <span class="k">return</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
-<a id="__codelineno-78-9" name="__codelineno-78-9" href="#__codelineno-78-9"></a>    <span class="c1"># 1. Find insertion position and insert node</span>
+<a id="__codelineno-78-9" name="__codelineno-78-9" href="#__codelineno-78-9"></a>    <span class="c1"># 1. Find the insertion position and insert the node</span>
 <a id="__codelineno-78-10" name="__codelineno-78-10" href="#__codelineno-78-10"></a>    <span class="k">if</span> <span class="n">val</span> <span class="o">&lt;</span> <span class="n">node</span><span class="o">.</span><span class="n">val</span><span class="p">:</span>
 <a id="__codelineno-78-11" name="__codelineno-78-11" href="#__codelineno-78-11"></a>        <span class="n">node</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">insert_helper</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-78-12" name="__codelineno-78-12" href="#__codelineno-78-12"></a>    <span class="k">elif</span> <span class="n">val</span> <span class="o">&gt;</span> <span class="n">node</span><span class="o">.</span><span class="n">val</span><span class="p">:</span>
 <a id="__codelineno-78-13" name="__codelineno-78-13" href="#__codelineno-78-13"></a>        <span class="n">node</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">insert_helper</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-78-14" name="__codelineno-78-14" href="#__codelineno-78-14"></a>    <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-78-15" name="__codelineno-78-15" href="#__codelineno-78-15"></a>        <span class="c1"># Duplicate node not inserted, return directly</span>
+<a id="__codelineno-78-15" name="__codelineno-78-15" href="#__codelineno-78-15"></a>        <span class="c1"># Return without inserting a duplicate node</span>
 <a id="__codelineno-78-16" name="__codelineno-78-16" href="#__codelineno-78-16"></a>        <span class="k">return</span> <span class="n">node</span>
-<a id="__codelineno-78-17" name="__codelineno-78-17" href="#__codelineno-78-17"></a>    <span class="c1"># Update node height</span>
+<a id="__codelineno-78-17" name="__codelineno-78-17" href="#__codelineno-78-17"></a>    <span class="c1"># Update the node height</span>
 <a id="__codelineno-78-18" name="__codelineno-78-18" href="#__codelineno-78-18"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">update_height</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-78-19" name="__codelineno-78-19" href="#__codelineno-78-19"></a>    <span class="c1"># 2. Perform rotation operation to restore balance to this subtree</span>
+<a id="__codelineno-78-19" name="__codelineno-78-19" href="#__codelineno-78-19"></a>    <span class="c1"># 2. Perform rotations to restore balance in this subtree</span>
 <a id="__codelineno-78-20" name="__codelineno-78-20" href="#__codelineno-78-20"></a>    <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-79-1" name="__codelineno-79-1" href="#__codelineno-79-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-79-1" name="__codelineno-79-1" href="#__codelineno-79-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-79-2" name="__codelineno-79-2" href="#__codelineno-79-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-3" name="__codelineno-79-3" href="#__codelineno-79-3"></a><span class="w">    </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-79-4" name="__codelineno-79-4" href="#__codelineno-79-4"></a><span class="p">}</span>
 <a id="__codelineno-79-5" name="__codelineno-79-5" href="#__codelineno-79-5"></a>
-<a id="__codelineno-79-6" name="__codelineno-79-6" href="#__codelineno-79-6"></a><span class="cm">/* Recursively insert node (helper method) */</span>
+<a id="__codelineno-79-6" name="__codelineno-79-6" href="#__codelineno-79-6"></a><span class="cm">/* Insert a node recursively (helper method) */</span>
 <a id="__codelineno-79-7" name="__codelineno-79-7" href="#__codelineno-79-7"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">insertHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-79-8" name="__codelineno-79-8" href="#__codelineno-79-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span>
 <a id="__codelineno-79-9" name="__codelineno-79-9" href="#__codelineno-79-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-79-10" name="__codelineno-79-10" href="#__codelineno-79-10"></a><span class="w">    </span><span class="cm">/* 1. Find insertion position and insert node */</span>
+<a id="__codelineno-79-10" name="__codelineno-79-10" href="#__codelineno-79-10"></a><span class="w">    </span><span class="cm">/* 1. Find the insertion position and insert the node */</span>
 <a id="__codelineno-79-11" name="__codelineno-79-11" href="#__codelineno-79-11"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-79-12" name="__codelineno-79-12" href="#__codelineno-79-12"></a><span class="w">        </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-79-13" name="__codelineno-79-13" href="#__codelineno-79-13"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-79-14" name="__codelineno-79-14" href="#__codelineno-79-14"></a><span class="w">        </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-79-15" name="__codelineno-79-15" href="#__codelineno-79-15"></a><span class="w">    </span><span class="k">else</span>
-<a id="__codelineno-79-16" name="__codelineno-79-16" href="#__codelineno-79-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w">    </span><span class="c1">// Duplicate node not inserted, return directly</span>
-<a id="__codelineno-79-17" name="__codelineno-79-17" href="#__codelineno-79-17"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update node height</span>
-<a id="__codelineno-79-18" name="__codelineno-79-18" href="#__codelineno-79-18"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-79-16" name="__codelineno-79-16" href="#__codelineno-79-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w">    </span><span class="c1">// Return without inserting a duplicate node</span>
+<a id="__codelineno-79-17" name="__codelineno-79-17" href="#__codelineno-79-17"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-79-18" name="__codelineno-79-18" href="#__codelineno-79-18"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-79-19" name="__codelineno-79-19" href="#__codelineno-79-19"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-79-20" name="__codelineno-79-20" href="#__codelineno-79-20"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-79-20" name="__codelineno-79-20" href="#__codelineno-79-20"></a><span class="w">    </span><span class="c1">// Return the root of the subtree</span>
 <a id="__codelineno-79-21" name="__codelineno-79-21" href="#__codelineno-79-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-79-22" name="__codelineno-79-22" href="#__codelineno-79-22"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-80-1" name="__codelineno-80-1" href="#__codelineno-80-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-80-1" name="__codelineno-80-1" href="#__codelineno-80-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-80-2" name="__codelineno-80-2" href="#__codelineno-80-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-3" name="__codelineno-80-3" href="#__codelineno-80-3"></a><span class="w">    </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-80-4" name="__codelineno-80-4" href="#__codelineno-80-4"></a><span class="p">}</span>
 <a id="__codelineno-80-5" name="__codelineno-80-5" href="#__codelineno-80-5"></a>
-<a id="__codelineno-80-6" name="__codelineno-80-6" href="#__codelineno-80-6"></a><span class="cm">/* Recursively insert node (helper method) */</span>
+<a id="__codelineno-80-6" name="__codelineno-80-6" href="#__codelineno-80-6"></a><span class="cm">/* Insert a node recursively (helper method) */</span>
 <a id="__codelineno-80-7" name="__codelineno-80-7" href="#__codelineno-80-7"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">insertHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-80-8" name="__codelineno-80-8" href="#__codelineno-80-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-80-9" name="__codelineno-80-9" href="#__codelineno-80-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-80-10" name="__codelineno-80-10" href="#__codelineno-80-10"></a><span class="w">    </span><span class="cm">/* 1. Find insertion position and insert node */</span>
+<a id="__codelineno-80-10" name="__codelineno-80-10" href="#__codelineno-80-10"></a><span class="w">    </span><span class="cm">/* 1. Find the insertion position and insert the node */</span>
 <a id="__codelineno-80-11" name="__codelineno-80-11" href="#__codelineno-80-11"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="p">)</span>
 <a id="__codelineno-80-12" name="__codelineno-80-12" href="#__codelineno-80-12"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-80-13" name="__codelineno-80-13" href="#__codelineno-80-13"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="p">)</span>
 <a id="__codelineno-80-14" name="__codelineno-80-14" href="#__codelineno-80-14"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-80-15" name="__codelineno-80-15" href="#__codelineno-80-15"></a><span class="w">    </span><span class="k">else</span>
-<a id="__codelineno-80-16" name="__codelineno-80-16" href="#__codelineno-80-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Duplicate node not inserted, return directly</span>
-<a id="__codelineno-80-17" name="__codelineno-80-17" href="#__codelineno-80-17"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update node height</span>
-<a id="__codelineno-80-18" name="__codelineno-80-18" href="#__codelineno-80-18"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-80-16" name="__codelineno-80-16" href="#__codelineno-80-16"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return without inserting a duplicate node</span>
+<a id="__codelineno-80-17" name="__codelineno-80-17" href="#__codelineno-80-17"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-80-18" name="__codelineno-80-18" href="#__codelineno-80-18"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-80-19" name="__codelineno-80-19" href="#__codelineno-80-19"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-80-20" name="__codelineno-80-20" href="#__codelineno-80-20"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-80-20" name="__codelineno-80-20" href="#__codelineno-80-20"></a><span class="w">    </span><span class="c1">// Return the subtree root</span>
 <a id="__codelineno-80-21" name="__codelineno-80-21" href="#__codelineno-80-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-80-22" name="__codelineno-80-22" href="#__codelineno-80-22"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-81-1" name="__codelineno-81-1" href="#__codelineno-81-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-81-2" name="__codelineno-81-2" href="#__codelineno-81-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-3" name="__codelineno-81-3" href="#__codelineno-81-3"></a><span class="w">    </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">InsertHelper</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-81-4" name="__codelineno-81-4" href="#__codelineno-81-4"></a><span class="p">}</span>
 <a id="__codelineno-81-5" name="__codelineno-81-5" href="#__codelineno-81-5"></a>
-<a id="__codelineno-81-6" name="__codelineno-81-6" href="#__codelineno-81-6"></a><span class="cm">/* Recursively insert node (helper method) */</span>
+<a id="__codelineno-81-6" name="__codelineno-81-6" href="#__codelineno-81-6"></a><span class="cm">/* Insert a node recursively (helper method) */</span>
 <a id="__codelineno-81-7" name="__codelineno-81-7" href="#__codelineno-81-7"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">InsertHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-81-8" name="__codelineno-81-8" href="#__codelineno-81-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-81-9" name="__codelineno-81-9" href="#__codelineno-81-9"></a><span class="w">    </span><span class="cm">/* 1. Find insertion position and insert node */</span>
+<a id="__codelineno-81-9" name="__codelineno-81-9" href="#__codelineno-81-9"></a><span class="w">    </span><span class="cm">/* 1. Find the insertion position and insert the node */</span>
 <a id="__codelineno-81-10" name="__codelineno-81-10" href="#__codelineno-81-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-81-11" name="__codelineno-81-11" href="#__codelineno-81-11"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">InsertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-81-12" name="__codelineno-81-12" href="#__codelineno-81-12"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="nf">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-81-13" name="__codelineno-81-13" href="#__codelineno-81-13"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">InsertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-81-14" name="__codelineno-81-14" href="#__codelineno-81-14"></a><span class="w">    </span><span class="k">else</span>
-<a id="__codelineno-81-15" name="__codelineno-81-15" href="#__codelineno-81-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w">     </span><span class="c1">// Duplicate node not inserted, return directly</span>
-<a id="__codelineno-81-16" name="__codelineno-81-16" href="#__codelineno-81-16"></a><span class="w">    </span><span class="n">UpdateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w">  </span><span class="c1">// Update node height</span>
-<a id="__codelineno-81-17" name="__codelineno-81-17" href="#__codelineno-81-17"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-81-15" name="__codelineno-81-15" href="#__codelineno-81-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w">     </span><span class="c1">// Return without inserting a duplicate node</span>
+<a id="__codelineno-81-16" name="__codelineno-81-16" href="#__codelineno-81-16"></a><span class="w">    </span><span class="n">UpdateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w">  </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-81-17" name="__codelineno-81-17" href="#__codelineno-81-17"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-81-18" name="__codelineno-81-18" href="#__codelineno-81-18"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Rotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-81-19" name="__codelineno-81-19" href="#__codelineno-81-19"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-81-19" name="__codelineno-81-19" href="#__codelineno-81-19"></a><span class="w">    </span><span class="c1">// Return the subtree root</span>
 <a id="__codelineno-81-20" name="__codelineno-81-20" href="#__codelineno-81-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-81-21" name="__codelineno-81-21" href="#__codelineno-81-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-82-1" name="__codelineno-82-1" href="#__codelineno-82-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-82-2" name="__codelineno-82-2" href="#__codelineno-82-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">t</span><span class="w"> </span><span class="o">*</span><span class="nx">aVLTree</span><span class="p">)</span><span class="w"> </span><span class="nx">insert</span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-3" name="__codelineno-82-3" href="#__codelineno-82-3"></a><span class="w">    </span><span class="nx">t</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">insertHelper</span><span class="p">(</span><span class="nx">t</span><span class="p">.</span><span class="nx">root</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span>
 <a id="__codelineno-82-4" name="__codelineno-82-4" href="#__codelineno-82-4"></a><span class="p">}</span>
 <a id="__codelineno-82-5" name="__codelineno-82-5" href="#__codelineno-82-5"></a>
-<a id="__codelineno-82-6" name="__codelineno-82-6" href="#__codelineno-82-6"></a><span class="cm">/* Recursively insert node (helper function) */</span>
+<a id="__codelineno-82-6" name="__codelineno-82-6" href="#__codelineno-82-6"></a><span class="cm">/* Insert a node recursively (helper function) */</span>
 <a id="__codelineno-82-7" name="__codelineno-82-7" href="#__codelineno-82-7"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">t</span><span class="w"> </span><span class="o">*</span><span class="nx">aVLTree</span><span class="p">)</span><span class="w"> </span><span class="nx">insertHelper</span><span class="p">(</span><span class="nx">node</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-8" name="__codelineno-82-8" href="#__codelineno-82-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-9" name="__codelineno-82-9" href="#__codelineno-82-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="nx">val</span><span class="p">)</span>
 <a id="__codelineno-82-10" name="__codelineno-82-10" href="#__codelineno-82-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-82-11" name="__codelineno-82-11" href="#__codelineno-82-11"></a><span class="w">    </span><span class="cm">/* 1. Find insertion position and insert node */</span>
+<a id="__codelineno-82-11" name="__codelineno-82-11" href="#__codelineno-82-11"></a><span class="w">    </span><span class="cm">/* 1. Find the insertion position and insert the node */</span>
 <a id="__codelineno-82-12" name="__codelineno-82-12" href="#__codelineno-82-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Val</span><span class="p">.(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-13" name="__codelineno-82-13" href="#__codelineno-82-13"></a><span class="w">        </span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">insertHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span>
 <a id="__codelineno-82-14" name="__codelineno-82-14" href="#__codelineno-82-14"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Val</span><span class="p">.(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-82-15" name="__codelineno-82-15" href="#__codelineno-82-15"></a><span class="w">        </span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">insertHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span>
 <a id="__codelineno-82-16" name="__codelineno-82-16" href="#__codelineno-82-16"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-82-17" name="__codelineno-82-17" href="#__codelineno-82-17"></a><span class="w">        </span><span class="c1">// Duplicate node not inserted, return directly</span>
+<a id="__codelineno-82-17" name="__codelineno-82-17" href="#__codelineno-82-17"></a><span class="w">        </span><span class="c1">// Return without inserting a duplicate node</span>
 <a id="__codelineno-82-18" name="__codelineno-82-18" href="#__codelineno-82-18"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span>
 <a id="__codelineno-82-19" name="__codelineno-82-19" href="#__codelineno-82-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-82-20" name="__codelineno-82-20" href="#__codelineno-82-20"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-82-20" name="__codelineno-82-20" href="#__codelineno-82-20"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-82-21" name="__codelineno-82-21" href="#__codelineno-82-21"></a><span class="w">    </span><span class="nx">t</span><span class="p">.</span><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span>
-<a id="__codelineno-82-22" name="__codelineno-82-22" href="#__codelineno-82-22"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-82-22" name="__codelineno-82-22" href="#__codelineno-82-22"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-82-23" name="__codelineno-82-23" href="#__codelineno-82-23"></a><span class="w">    </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">rotate</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span>
-<a id="__codelineno-82-24" name="__codelineno-82-24" href="#__codelineno-82-24"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-82-24" name="__codelineno-82-24" href="#__codelineno-82-24"></a><span class="w">    </span><span class="c1">// Return the subtree root</span>
 <a id="__codelineno-82-25" name="__codelineno-82-25" href="#__codelineno-82-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span>
 <a id="__codelineno-82-26" name="__codelineno-82-26" href="#__codelineno-82-26"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.swift</span><pre><span></span><code><a id="__codelineno-83-1" name="__codelineno-83-1" href="#__codelineno-83-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">avl_tree.swift</span><pre><span></span><code><a id="__codelineno-83-1" name="__codelineno-83-1" href="#__codelineno-83-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-83-2" name="__codelineno-83-2" href="#__codelineno-83-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-3" name="__codelineno-83-3" href="#__codelineno-83-3"></a><span class="w">    </span><span class="n">root</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-83-4" name="__codelineno-83-4" href="#__codelineno-83-4"></a><span class="p">}</span>
 <a id="__codelineno-83-5" name="__codelineno-83-5" href="#__codelineno-83-5"></a>
-<a id="__codelineno-83-6" name="__codelineno-83-6" href="#__codelineno-83-6"></a><span class="cm">/* Recursively insert node (helper method) */</span>
+<a id="__codelineno-83-6" name="__codelineno-83-6" href="#__codelineno-83-6"></a><span class="cm">/* Insert a node recursively (helper method) */</span>
 <a id="__codelineno-83-7" name="__codelineno-83-7" href="#__codelineno-83-7"></a><span class="kd">func</span><span class="w"> </span><span class="nf">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-8" name="__codelineno-83-8" href="#__codelineno-83-8"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-83-9" name="__codelineno-83-9" href="#__codelineno-83-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-10" name="__codelineno-83-10" href="#__codelineno-83-10"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-83-11" name="__codelineno-83-11" href="#__codelineno-83-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-83-12" name="__codelineno-83-12" href="#__codelineno-83-12"></a><span class="w">    </span><span class="cm">/* 1. Find insertion position and insert node */</span>
+<a id="__codelineno-83-12" name="__codelineno-83-12" href="#__codelineno-83-12"></a><span class="w">    </span><span class="cm">/* 1. Find the insertion position and insert the node */</span>
 <a id="__codelineno-83-13" name="__codelineno-83-13" href="#__codelineno-83-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-14" name="__codelineno-83-14" href="#__codelineno-83-14"></a><span class="w">        </span><span class="n">node</span><span class="p">?.</span><span class="kr">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-83-15" name="__codelineno-83-15" href="#__codelineno-83-15"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-83-16" name="__codelineno-83-16" href="#__codelineno-83-16"></a><span class="w">        </span><span class="n">node</span><span class="p">?.</span><span class="kr">right</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">right</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-83-17" name="__codelineno-83-17" href="#__codelineno-83-17"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-83-18" name="__codelineno-83-18" href="#__codelineno-83-18"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1">// Duplicate node not inserted, return directly</span>
+<a id="__codelineno-83-18" name="__codelineno-83-18" href="#__codelineno-83-18"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1">// Return without inserting a duplicate node</span>
 <a id="__codelineno-83-19" name="__codelineno-83-19" href="#__codelineno-83-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-83-20" name="__codelineno-83-20" href="#__codelineno-83-20"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="c1">// Update node height</span>
-<a id="__codelineno-83-21" name="__codelineno-83-21" href="#__codelineno-83-21"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-83-20" name="__codelineno-83-20" href="#__codelineno-83-20"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-83-21" name="__codelineno-83-21" href="#__codelineno-83-21"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-83-22" name="__codelineno-83-22" href="#__codelineno-83-22"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-83-23" name="__codelineno-83-23" href="#__codelineno-83-23"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-83-23" name="__codelineno-83-23" href="#__codelineno-83-23"></a><span class="w">    </span><span class="c1">// Return the subtree root</span>
 <a id="__codelineno-83-24" name="__codelineno-83-24" href="#__codelineno-83-24"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-83-25" name="__codelineno-83-25" href="#__codelineno-83-25"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-84-1" name="__codelineno-84-1" href="#__codelineno-84-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-84-1" name="__codelineno-84-1" href="#__codelineno-84-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-84-2" name="__codelineno-84-2" href="#__codelineno-84-2"></a><span class="nx">insert</span><span class="p">(</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-3" name="__codelineno-84-3" href="#__codelineno-84-3"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#insertHelper</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-84-4" name="__codelineno-84-4" href="#__codelineno-84-4"></a><span class="p">}</span>
 <a id="__codelineno-84-5" name="__codelineno-84-5" href="#__codelineno-84-5"></a>
-<a id="__codelineno-84-6" name="__codelineno-84-6" href="#__codelineno-84-6"></a><span class="cm">/* Recursively insert node (helper method) */</span>
+<a id="__codelineno-84-6" name="__codelineno-84-6" href="#__codelineno-84-6"></a><span class="cm">/* Insert a node recursively (helper method) */</span>
 <a id="__codelineno-84-7" name="__codelineno-84-7" href="#__codelineno-84-7"></a><span class="n">#insertHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-84-8" name="__codelineno-84-8" href="#__codelineno-84-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="nx">val</span><span class="p">);</span>
-<a id="__codelineno-84-9" name="__codelineno-84-9" href="#__codelineno-84-9"></a><span class="w">    </span><span class="cm">/* 1. Find insertion position and insert node */</span>
+<a id="__codelineno-84-9" name="__codelineno-84-9" href="#__codelineno-84-9"></a><span class="w">    </span><span class="cm">/* 1. Find the insertion position and insert the node */</span>
 <a id="__codelineno-84-10" name="__codelineno-84-10" href="#__codelineno-84-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#insertHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-84-11" name="__codelineno-84-11" href="#__codelineno-84-11"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span>
 <a id="__codelineno-84-12" name="__codelineno-84-12" href="#__codelineno-84-12"></a><span class="w">        </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#insertHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
-<a id="__codelineno-84-13" name="__codelineno-84-13" href="#__codelineno-84-13"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Duplicate node not inserted, return directly</span>
-<a id="__codelineno-84-14" name="__codelineno-84-14" href="#__codelineno-84-14"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">#updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update node height</span>
-<a id="__codelineno-84-15" name="__codelineno-84-15" href="#__codelineno-84-15"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-84-13" name="__codelineno-84-13" href="#__codelineno-84-13"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return without inserting a duplicate node</span>
+<a id="__codelineno-84-14" name="__codelineno-84-14" href="#__codelineno-84-14"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">#updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-84-15" name="__codelineno-84-15" href="#__codelineno-84-15"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-84-16" name="__codelineno-84-16" href="#__codelineno-84-16"></a><span class="w">    </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#rotate</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
-<a id="__codelineno-84-17" name="__codelineno-84-17" href="#__codelineno-84-17"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-84-17" name="__codelineno-84-17" href="#__codelineno-84-17"></a><span class="w">    </span><span class="c1">// Return the root of the subtree</span>
 <a id="__codelineno-84-18" name="__codelineno-84-18" href="#__codelineno-84-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-84-19" name="__codelineno-84-19" href="#__codelineno-84-19"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-85-1" name="__codelineno-85-1" href="#__codelineno-85-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-85-1" name="__codelineno-85-1" href="#__codelineno-85-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-85-2" name="__codelineno-85-2" href="#__codelineno-85-2"></a><span class="nx">insert</span><span class="p">(</span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-3" name="__codelineno-85-3" href="#__codelineno-85-3"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">insertHelper</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-85-4" name="__codelineno-85-4" href="#__codelineno-85-4"></a><span class="p">}</span>
 <a id="__codelineno-85-5" name="__codelineno-85-5" href="#__codelineno-85-5"></a>
-<a id="__codelineno-85-6" name="__codelineno-85-6" href="#__codelineno-85-6"></a><span class="cm">/* Recursively insert node (helper method) */</span>
+<a id="__codelineno-85-6" name="__codelineno-85-6" href="#__codelineno-85-6"></a><span class="cm">/* Insert a node recursively (helper method) */</span>
 <a id="__codelineno-85-7" name="__codelineno-85-7" href="#__codelineno-85-7"></a><span class="nx">insertHelper</span><span class="p">(</span><span class="nx">node</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-8" name="__codelineno-85-8" href="#__codelineno-85-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="nx">val</span><span class="p">);</span>
-<a id="__codelineno-85-9" name="__codelineno-85-9" href="#__codelineno-85-9"></a><span class="w">    </span><span class="cm">/* 1. Find insertion position and insert node */</span>
+<a id="__codelineno-85-9" name="__codelineno-85-9" href="#__codelineno-85-9"></a><span class="w">    </span><span class="cm">/* 1. Find the insertion position and insert the node */</span>
 <a id="__codelineno-85-10" name="__codelineno-85-10" href="#__codelineno-85-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-11" name="__codelineno-85-11" href="#__codelineno-85-11"></a><span class="w">        </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">insertHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-85-12" name="__codelineno-85-12" href="#__codelineno-85-12"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-85-13" name="__codelineno-85-13" href="#__codelineno-85-13"></a><span class="w">        </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">insertHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-85-14" name="__codelineno-85-14" href="#__codelineno-85-14"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-85-15" name="__codelineno-85-15" href="#__codelineno-85-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Duplicate node not inserted, return directly</span>
+<a id="__codelineno-85-15" name="__codelineno-85-15" href="#__codelineno-85-15"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return without inserting a duplicate node</span>
 <a id="__codelineno-85-16" name="__codelineno-85-16" href="#__codelineno-85-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-85-17" name="__codelineno-85-17" href="#__codelineno-85-17"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update node height</span>
-<a id="__codelineno-85-18" name="__codelineno-85-18" href="#__codelineno-85-18"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-85-17" name="__codelineno-85-17" href="#__codelineno-85-17"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-85-18" name="__codelineno-85-18" href="#__codelineno-85-18"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-85-19" name="__codelineno-85-19" href="#__codelineno-85-19"></a><span class="w">    </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">rotate</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
-<a id="__codelineno-85-20" name="__codelineno-85-20" href="#__codelineno-85-20"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-85-20" name="__codelineno-85-20" href="#__codelineno-85-20"></a><span class="w">    </span><span class="c1">// Return the root of the subtree</span>
 <a id="__codelineno-85-21" name="__codelineno-85-21" href="#__codelineno-85-21"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-85-22" name="__codelineno-85-22" href="#__codelineno-85-22"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.dart</span><pre><span></span><code><a id="__codelineno-86-1" name="__codelineno-86-1" href="#__codelineno-86-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">avl_tree.dart</span><pre><span></span><code><a id="__codelineno-86-1" name="__codelineno-86-1" href="#__codelineno-86-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-86-2" name="__codelineno-86-2" href="#__codelineno-86-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-3" name="__codelineno-86-3" href="#__codelineno-86-3"></a><span class="w">  </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-86-4" name="__codelineno-86-4" href="#__codelineno-86-4"></a><span class="p">}</span>
 <a id="__codelineno-86-5" name="__codelineno-86-5" href="#__codelineno-86-5"></a>
-<a id="__codelineno-86-6" name="__codelineno-86-6" href="#__codelineno-86-6"></a><span class="cm">/* Recursively insert node (helper method) */</span>
+<a id="__codelineno-86-6" name="__codelineno-86-6" href="#__codelineno-86-6"></a><span class="cm">/* Insert a node recursively (helper method) */</span>
 <a id="__codelineno-86-7" name="__codelineno-86-7" href="#__codelineno-86-7"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-86-8" name="__codelineno-86-8" href="#__codelineno-86-8"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-86-9" name="__codelineno-86-9" href="#__codelineno-86-9"></a><span class="w">  </span><span class="cm">/* 1. Find insertion position and insert node */</span>
+<a id="__codelineno-86-9" name="__codelineno-86-9" href="#__codelineno-86-9"></a><span class="w">  </span><span class="cm">/* 1. Find the insertion position and insert the node */</span>
 <a id="__codelineno-86-10" name="__codelineno-86-10" href="#__codelineno-86-10"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-86-11" name="__codelineno-86-11" href="#__codelineno-86-11"></a><span class="w">    </span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-86-12" name="__codelineno-86-12" href="#__codelineno-86-12"></a><span class="w">  </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-86-13" name="__codelineno-86-13" href="#__codelineno-86-13"></a><span class="w">    </span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-86-14" name="__codelineno-86-14" href="#__codelineno-86-14"></a><span class="w">  </span><span class="k">else</span>
-<a id="__codelineno-86-15" name="__codelineno-86-15" href="#__codelineno-86-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Duplicate node not inserted, return directly</span>
-<a id="__codelineno-86-16" name="__codelineno-86-16" href="#__codelineno-86-16"></a><span class="w">  </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update node height</span>
-<a id="__codelineno-86-17" name="__codelineno-86-17" href="#__codelineno-86-17"></a><span class="w">  </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-86-15" name="__codelineno-86-15" href="#__codelineno-86-15"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span><span class="w"> </span><span class="c1">// Return without inserting a duplicate node</span>
+<a id="__codelineno-86-16" name="__codelineno-86-16" href="#__codelineno-86-16"></a><span class="w">  </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-86-17" name="__codelineno-86-17" href="#__codelineno-86-17"></a><span class="w">  </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-86-18" name="__codelineno-86-18" href="#__codelineno-86-18"></a><span class="w">  </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-86-19" name="__codelineno-86-19" href="#__codelineno-86-19"></a><span class="w">  </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-86-19" name="__codelineno-86-19" href="#__codelineno-86-19"></a><span class="w">  </span><span class="c1">// Return the root of the subtree</span>
 <a id="__codelineno-86-20" name="__codelineno-86-20" href="#__codelineno-86-20"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-86-21" name="__codelineno-86-21" href="#__codelineno-86-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.rs</span><pre><span></span><code><a id="__codelineno-87-1" name="__codelineno-87-1" href="#__codelineno-87-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">avl_tree.rs</span><pre><span></span><code><a id="__codelineno-87-1" name="__codelineno-87-1" href="#__codelineno-87-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-87-2" name="__codelineno-87-2" href="#__codelineno-87-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-3" name="__codelineno-87-3" href="#__codelineno-87-3"></a><span class="w">    </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">insert_helper</span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="p">.</span><span class="n">clone</span><span class="p">(),</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-87-4" name="__codelineno-87-4" href="#__codelineno-87-4"></a><span class="p">}</span>
 <a id="__codelineno-87-5" name="__codelineno-87-5" href="#__codelineno-87-5"></a>
-<a id="__codelineno-87-6" name="__codelineno-87-6" href="#__codelineno-87-6"></a><span class="cm">/* Recursively insert node (helper method) */</span>
+<a id="__codelineno-87-6" name="__codelineno-87-6" href="#__codelineno-87-6"></a><span class="cm">/* Insert a node recursively (helper method) */</span>
 <a id="__codelineno-87-7" name="__codelineno-87-7" href="#__codelineno-87-7"></a><span class="k">fn</span><span class="w"> </span><span class="nf">insert_helper</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="nc">OptionTreeNodeRc</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">OptionTreeNodeRc</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-8" name="__codelineno-87-8" href="#__codelineno-87-8"></a><span class="w">    </span><span class="k">match</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-9" name="__codelineno-87-9" href="#__codelineno-87-9"></a><span class="w">        </span><span class="nb">Some</span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-87-10" name="__codelineno-87-10" href="#__codelineno-87-10"></a><span class="w">            </span><span class="cm">/* 1. Find insertion position and insert node */</span>
+<a id="__codelineno-87-10" name="__codelineno-87-10" href="#__codelineno-87-10"></a><span class="w">            </span><span class="cm">/* 1. Find the insertion position and insert the node */</span>
 <a id="__codelineno-87-11" name="__codelineno-87-11" href="#__codelineno-87-11"></a><span class="w">            </span><span class="k">match</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-87-12" name="__codelineno-87-12" href="#__codelineno-87-12"></a><span class="w">                </span><span class="kd">let</span><span class="w"> </span><span class="n">node_val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-87-13" name="__codelineno-87-13" href="#__codelineno-87-13"></a><span class="w">                </span><span class="n">node_val</span>
@@ -6725,14 +6797,14 @@
 <a id="__codelineno-87-23" name="__codelineno-87-23" href="#__codelineno-87-23"></a><span class="w">                    </span><span class="n">node</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">insert_helper</span><span class="p">(</span><span class="n">right</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-87-24" name="__codelineno-87-24" href="#__codelineno-87-24"></a><span class="w">                </span><span class="p">}</span>
 <a id="__codelineno-87-25" name="__codelineno-87-25" href="#__codelineno-87-25"></a><span class="w">                </span><span class="n">Ordering</span><span class="p">::</span><span class="n">Equal</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-87-26" name="__codelineno-87-26" href="#__codelineno-87-26"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Duplicate node not inserted, return directly</span>
+<a id="__codelineno-87-26" name="__codelineno-87-26" href="#__codelineno-87-26"></a><span class="w">                    </span><span class="k">return</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Return without inserting a duplicate node</span>
 <a id="__codelineno-87-27" name="__codelineno-87-27" href="#__codelineno-87-27"></a><span class="w">                </span><span class="p">}</span>
 <a id="__codelineno-87-28" name="__codelineno-87-28" href="#__codelineno-87-28"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-87-29" name="__codelineno-87-29" href="#__codelineno-87-29"></a><span class="w">            </span><span class="bp">Self</span><span class="p">::</span><span class="n">update_height</span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">clone</span><span class="p">()));</span><span class="w"> </span><span class="c1">// Update node height</span>
+<a id="__codelineno-87-29" name="__codelineno-87-29" href="#__codelineno-87-29"></a><span class="w">            </span><span class="bp">Self</span><span class="p">::</span><span class="n">update_height</span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">clone</span><span class="p">()));</span><span class="w"> </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-87-30" name="__codelineno-87-30" href="#__codelineno-87-30"></a>
-<a id="__codelineno-87-31" name="__codelineno-87-31" href="#__codelineno-87-31"></a><span class="w">            </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-87-31" name="__codelineno-87-31" href="#__codelineno-87-31"></a><span class="w">            </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-87-32" name="__codelineno-87-32" href="#__codelineno-87-32"></a><span class="w">            </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">rotate</span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)).</span><span class="n">unwrap</span><span class="p">();</span>
-<a id="__codelineno-87-33" name="__codelineno-87-33" href="#__codelineno-87-33"></a><span class="w">            </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-87-33" name="__codelineno-87-33" href="#__codelineno-87-33"></a><span class="w">            </span><span class="c1">// Return the root of the subtree</span>
 <a id="__codelineno-87-34" name="__codelineno-87-34" href="#__codelineno-87-34"></a><span class="w">            </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-87-35" name="__codelineno-87-35" href="#__codelineno-87-35"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-87-36" name="__codelineno-87-36" href="#__codelineno-87-36"></a><span class="w">        </span><span class="nb">None</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">TreeNode</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">val</span><span class="p">)),</span>
@@ -6741,101 +6813,101 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-88-1" name="__codelineno-88-1" href="#__codelineno-88-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-88-1" name="__codelineno-88-1" href="#__codelineno-88-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-88-2" name="__codelineno-88-2" href="#__codelineno-88-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">AVLTree</span><span class="w"> </span><span class="o">*</span><span class="n">tree</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-3" name="__codelineno-88-3" href="#__codelineno-88-3"></a><span class="w">    </span><span class="n">tree</span><span class="o">-&gt;</span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">tree</span><span class="o">-&gt;</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-88-4" name="__codelineno-88-4" href="#__codelineno-88-4"></a><span class="p">}</span>
 <a id="__codelineno-88-5" name="__codelineno-88-5" href="#__codelineno-88-5"></a>
-<a id="__codelineno-88-6" name="__codelineno-88-6" href="#__codelineno-88-6"></a><span class="cm">/* Recursively insert node (helper function) */</span>
+<a id="__codelineno-88-6" name="__codelineno-88-6" href="#__codelineno-88-6"></a><span class="cm">/* Recursively insert a node (helper function) */</span>
 <a id="__codelineno-88-7" name="__codelineno-88-7" href="#__codelineno-88-7"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">insertHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-8" name="__codelineno-88-8" href="#__codelineno-88-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-9" name="__codelineno-88-9" href="#__codelineno-88-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-88-10" name="__codelineno-88-10" href="#__codelineno-88-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-88-11" name="__codelineno-88-11" href="#__codelineno-88-11"></a><span class="w">    </span><span class="cm">/* 1. Find insertion position and insert node */</span>
+<a id="__codelineno-88-11" name="__codelineno-88-11" href="#__codelineno-88-11"></a><span class="w">    </span><span class="cm">/* 1. Find the insertion position and insert the node */</span>
 <a id="__codelineno-88-12" name="__codelineno-88-12" href="#__codelineno-88-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-13" name="__codelineno-88-13" href="#__codelineno-88-13"></a><span class="w">        </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-88-14" name="__codelineno-88-14" href="#__codelineno-88-14"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-88-15" name="__codelineno-88-15" href="#__codelineno-88-15"></a><span class="w">        </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-88-16" name="__codelineno-88-16" href="#__codelineno-88-16"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-88-17" name="__codelineno-88-17" href="#__codelineno-88-17"></a><span class="w">        </span><span class="c1">// Duplicate node not inserted, return directly</span>
+<a id="__codelineno-88-17" name="__codelineno-88-17" href="#__codelineno-88-17"></a><span class="w">        </span><span class="c1">// Return without inserting a duplicate node</span>
 <a id="__codelineno-88-18" name="__codelineno-88-18" href="#__codelineno-88-18"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-88-19" name="__codelineno-88-19" href="#__codelineno-88-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-88-20" name="__codelineno-88-20" href="#__codelineno-88-20"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-88-20" name="__codelineno-88-20" href="#__codelineno-88-20"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-88-21" name="__codelineno-88-21" href="#__codelineno-88-21"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-88-22" name="__codelineno-88-22" href="#__codelineno-88-22"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-88-22" name="__codelineno-88-22" href="#__codelineno-88-22"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-88-23" name="__codelineno-88-23" href="#__codelineno-88-23"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-88-24" name="__codelineno-88-24" href="#__codelineno-88-24"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-88-24" name="__codelineno-88-24" href="#__codelineno-88-24"></a><span class="w">    </span><span class="c1">// Return the root of the subtree</span>
 <a id="__codelineno-88-25" name="__codelineno-88-25" href="#__codelineno-88-25"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-88-26" name="__codelineno-88-26" href="#__codelineno-88-26"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.kt</span><pre><span></span><code><a id="__codelineno-89-1" name="__codelineno-89-1" href="#__codelineno-89-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">avl_tree.kt</span><pre><span></span><code><a id="__codelineno-89-1" name="__codelineno-89-1" href="#__codelineno-89-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-89-2" name="__codelineno-89-2" href="#__codelineno-89-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-89-3" name="__codelineno-89-3" href="#__codelineno-89-3"></a><span class="w">    </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">)</span>
 <a id="__codelineno-89-4" name="__codelineno-89-4" href="#__codelineno-89-4"></a><span class="p">}</span>
 <a id="__codelineno-89-5" name="__codelineno-89-5" href="#__codelineno-89-5"></a>
-<a id="__codelineno-89-6" name="__codelineno-89-6" href="#__codelineno-89-6"></a><span class="cm">/* Recursively insert node (helper method) */</span>
+<a id="__codelineno-89-6" name="__codelineno-89-6" href="#__codelineno-89-6"></a><span class="cm">/* Insert a node recursively (helper method) */</span>
 <a id="__codelineno-89-7" name="__codelineno-89-7" href="#__codelineno-89-7"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">insertHelper</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-89-8" name="__codelineno-89-8" href="#__codelineno-89-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-89-9" name="__codelineno-89-9" href="#__codelineno-89-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">_val</span><span class="p">)</span>
 <a id="__codelineno-89-10" name="__codelineno-89-10" href="#__codelineno-89-10"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span>
-<a id="__codelineno-89-11" name="__codelineno-89-11" href="#__codelineno-89-11"></a><span class="w">    </span><span class="cm">/* 1. Find insertion position and insert node */</span>
+<a id="__codelineno-89-11" name="__codelineno-89-11" href="#__codelineno-89-11"></a><span class="w">    </span><span class="cm">/* 1. Find the insertion position and insert the node */</span>
 <a id="__codelineno-89-12" name="__codelineno-89-12" href="#__codelineno-89-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">_val</span><span class="p">)</span>
 <a id="__codelineno-89-13" name="__codelineno-89-13" href="#__codelineno-89-13"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">)</span>
 <a id="__codelineno-89-14" name="__codelineno-89-14" href="#__codelineno-89-14"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">_val</span><span class="p">)</span>
 <a id="__codelineno-89-15" name="__codelineno-89-15" href="#__codelineno-89-15"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insertHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">)</span>
 <a id="__codelineno-89-16" name="__codelineno-89-16" href="#__codelineno-89-16"></a><span class="w">    </span><span class="k">else</span>
-<a id="__codelineno-89-17" name="__codelineno-89-17" href="#__codelineno-89-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1">// Duplicate node not inserted, return directly</span>
-<a id="__codelineno-89-18" name="__codelineno-89-18" href="#__codelineno-89-18"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="c1">// Update node height</span>
-<a id="__codelineno-89-19" name="__codelineno-89-19" href="#__codelineno-89-19"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-89-17" name="__codelineno-89-17" href="#__codelineno-89-17"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="c1">// Return without inserting a duplicate node</span>
+<a id="__codelineno-89-18" name="__codelineno-89-18" href="#__codelineno-89-18"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-89-19" name="__codelineno-89-19" href="#__codelineno-89-19"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-89-20" name="__codelineno-89-20" href="#__codelineno-89-20"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-89-21" name="__codelineno-89-21" href="#__codelineno-89-21"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-89-21" name="__codelineno-89-21" href="#__codelineno-89-21"></a><span class="w">    </span><span class="c1">// Return the subtree root</span>
 <a id="__codelineno-89-22" name="__codelineno-89-22" href="#__codelineno-89-22"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-89-23" name="__codelineno-89-23" href="#__codelineno-89-23"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.rb</span><pre><span></span><code><a id="__codelineno-90-1" name="__codelineno-90-1" href="#__codelineno-90-1"></a><span class="c1">### Insert node ###</span>
+<div class="highlight"><span class="filename">avl_tree.rb</span><pre><span></span><code><a id="__codelineno-90-1" name="__codelineno-90-1" href="#__codelineno-90-1"></a><span class="c1">### Insert a node ###</span>
 <a id="__codelineno-90-2" name="__codelineno-90-2" href="#__codelineno-90-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-90-3" name="__codelineno-90-3" href="#__codelineno-90-3"></a><span class="w">  </span><span class="vi">@root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insert_helper</span><span class="p">(</span><span class="vi">@root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-90-4" name="__codelineno-90-4" href="#__codelineno-90-4"></a><span class="k">end</span>
 <a id="__codelineno-90-5" name="__codelineno-90-5" href="#__codelineno-90-5"></a>
-<a id="__codelineno-90-6" name="__codelineno-90-6" href="#__codelineno-90-6"></a><span class="c1">### Recursively insert node (helper method) ###</span>
+<a id="__codelineno-90-6" name="__codelineno-90-6" href="#__codelineno-90-6"></a><span class="c1">### Insert a node recursively (helper method)###</span>
 <a id="__codelineno-90-7" name="__codelineno-90-7" href="#__codelineno-90-7"></a><span class="k">def</span><span class="w"> </span><span class="nf">insert_helper</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-90-8" name="__codelineno-90-8" href="#__codelineno-90-8"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="no">TreeNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">nil?</span>
-<a id="__codelineno-90-9" name="__codelineno-90-9" href="#__codelineno-90-9"></a><span class="w">  </span><span class="c1"># 1. Find insertion position and insert node</span>
+<a id="__codelineno-90-9" name="__codelineno-90-9" href="#__codelineno-90-9"></a><span class="w">  </span><span class="c1"># 1. Find the insertion position and insert the node</span>
 <a id="__codelineno-90-10" name="__codelineno-90-10" href="#__codelineno-90-10"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-90-11" name="__codelineno-90-11" href="#__codelineno-90-11"></a><span class="w">    </span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insert_helper</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-90-12" name="__codelineno-90-12" href="#__codelineno-90-12"></a><span class="w">  </span><span class="k">elsif</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-90-13" name="__codelineno-90-13" href="#__codelineno-90-13"></a><span class="w">    </span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">insert_helper</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-90-14" name="__codelineno-90-14" href="#__codelineno-90-14"></a><span class="w">  </span><span class="k">else</span>
-<a id="__codelineno-90-15" name="__codelineno-90-15" href="#__codelineno-90-15"></a><span class="w">    </span><span class="c1"># Duplicate node not inserted, return directly</span>
+<a id="__codelineno-90-15" name="__codelineno-90-15" href="#__codelineno-90-15"></a><span class="w">    </span><span class="c1"># Return without inserting a duplicate node</span>
 <a id="__codelineno-90-16" name="__codelineno-90-16" href="#__codelineno-90-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-90-17" name="__codelineno-90-17" href="#__codelineno-90-17"></a><span class="w">  </span><span class="k">end</span>
-<a id="__codelineno-90-18" name="__codelineno-90-18" href="#__codelineno-90-18"></a><span class="w">  </span><span class="c1"># Update node height</span>
+<a id="__codelineno-90-18" name="__codelineno-90-18" href="#__codelineno-90-18"></a><span class="w">  </span><span class="c1"># Update the node height</span>
 <a id="__codelineno-90-19" name="__codelineno-90-19" href="#__codelineno-90-19"></a><span class="w">  </span><span class="n">update_height</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-90-20" name="__codelineno-90-20" href="#__codelineno-90-20"></a><span class="w">  </span><span class="c1"># 2. Perform rotation operation to restore balance to this subtree</span>
+<a id="__codelineno-90-20" name="__codelineno-90-20" href="#__codelineno-90-20"></a><span class="w">  </span><span class="c1"># 2. Perform rotations to restore balance in this subtree</span>
 <a id="__codelineno-90-21" name="__codelineno-90-21" href="#__codelineno-90-21"></a><span class="w">  </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-90-22" name="__codelineno-90-22" href="#__codelineno-90-22"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h3 id="2-node-removal">2. &nbsp; Node Removal<a class="headerlink" href="#2-node-removal" title="Permanent link">&para;</a></h3>
-<p>Similarly, on the basis of the binary search tree's node removal method, rotation operations need to be performed from bottom to top to restore balance to all unbalanced nodes. The code is as follows:</p>
+<h3 id="2-deleting-a-node">2. &nbsp; Deleting a node<a class="headerlink" href="#2-deleting-a-node" title="Permanent link">&para;</a></h3>
+<p>Similarly, extend binary-search-tree deletion with rotations from the bottom up to restore balance at every affected node:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="9:13"><input checked="checked" id="__tabbed_9_1" name="__tabbed_9" type="radio" /><input id="__tabbed_9_2" name="__tabbed_9" type="radio" /><input id="__tabbed_9_3" name="__tabbed_9" type="radio" /><input id="__tabbed_9_4" name="__tabbed_9" type="radio" /><input id="__tabbed_9_5" name="__tabbed_9" type="radio" /><input id="__tabbed_9_6" name="__tabbed_9" type="radio" /><input id="__tabbed_9_7" name="__tabbed_9" type="radio" /><input id="__tabbed_9_8" name="__tabbed_9" type="radio" /><input id="__tabbed_9_9" name="__tabbed_9" type="radio" /><input id="__tabbed_9_10" name="__tabbed_9" type="radio" /><input id="__tabbed_9_11" name="__tabbed_9" type="radio" /><input id="__tabbed_9_12" name="__tabbed_9" type="radio" /><input id="__tabbed_9_13" name="__tabbed_9" type="radio" /><div class="tabbed-labels"><label for="__tabbed_9_1">Python</label><label for="__tabbed_9_2">C++</label><label for="__tabbed_9_3">Java</label><label for="__tabbed_9_4">C#</label><label for="__tabbed_9_5">Go</label><label for="__tabbed_9_6">Swift</label><label for="__tabbed_9_7">JS</label><label for="__tabbed_9_8">TS</label><label for="__tabbed_9_9">Dart</label><label for="__tabbed_9_10">Rust</label><label for="__tabbed_9_11">C</label><label for="__tabbed_9_12">Kotlin</label><label for="__tabbed_9_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">avl_tree.py</span><pre><span></span><code><a id="__codelineno-91-1" name="__codelineno-91-1" href="#__codelineno-91-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-91-2" name="__codelineno-91-2" href="#__codelineno-91-2"></a><span class="w">    </span><span class="sd">"""Delete node"""</span>
+<a id="__codelineno-91-2" name="__codelineno-91-2" href="#__codelineno-91-2"></a><span class="w">    </span><span class="sd">"""Remove a node"""</span>
 <a id="__codelineno-91-3" name="__codelineno-91-3" href="#__codelineno-91-3"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">_root</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">remove_helper</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_root</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-91-4" name="__codelineno-91-4" href="#__codelineno-91-4"></a>
 <a id="__codelineno-91-5" name="__codelineno-91-5" href="#__codelineno-91-5"></a><span class="k">def</span><span class="w"> </span><span class="nf">remove_helper</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">node</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-91-6" name="__codelineno-91-6" href="#__codelineno-91-6"></a><span class="w">    </span><span class="sd">"""Recursively delete node (helper method)"""</span>
+<a id="__codelineno-91-6" name="__codelineno-91-6" href="#__codelineno-91-6"></a><span class="w">    </span><span class="sd">"""Remove a node recursively (helper method)"""</span>
 <a id="__codelineno-91-7" name="__codelineno-91-7" href="#__codelineno-91-7"></a>    <span class="k">if</span> <span class="n">node</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-91-8" name="__codelineno-91-8" href="#__codelineno-91-8"></a>        <span class="k">return</span> <span class="kc">None</span>
-<a id="__codelineno-91-9" name="__codelineno-91-9" href="#__codelineno-91-9"></a>    <span class="c1"># 1. Find node and delete</span>
+<a id="__codelineno-91-9" name="__codelineno-91-9" href="#__codelineno-91-9"></a>    <span class="c1"># 1. Find and remove the node</span>
 <a id="__codelineno-91-10" name="__codelineno-91-10" href="#__codelineno-91-10"></a>    <span class="k">if</span> <span class="n">val</span> <span class="o">&lt;</span> <span class="n">node</span><span class="o">.</span><span class="n">val</span><span class="p">:</span>
 <a id="__codelineno-91-11" name="__codelineno-91-11" href="#__codelineno-91-11"></a>        <span class="n">node</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">remove_helper</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-91-12" name="__codelineno-91-12" href="#__codelineno-91-12"></a>    <span class="k">elif</span> <span class="n">val</span> <span class="o">&gt;</span> <span class="n">node</span><span class="o">.</span><span class="n">val</span><span class="p">:</span>
@@ -6843,36 +6915,36 @@
 <a id="__codelineno-91-14" name="__codelineno-91-14" href="#__codelineno-91-14"></a>    <span class="k">else</span><span class="p">:</span>
 <a id="__codelineno-91-15" name="__codelineno-91-15" href="#__codelineno-91-15"></a>        <span class="k">if</span> <span class="n">node</span><span class="o">.</span><span class="n">left</span> <span class="ow">is</span> <span class="kc">None</span> <span class="ow">or</span> <span class="n">node</span><span class="o">.</span><span class="n">right</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-91-16" name="__codelineno-91-16" href="#__codelineno-91-16"></a>            <span class="n">child</span> <span class="o">=</span> <span class="n">node</span><span class="o">.</span><span class="n">left</span> <span class="ow">or</span> <span class="n">node</span><span class="o">.</span><span class="n">right</span>
-<a id="__codelineno-91-17" name="__codelineno-91-17" href="#__codelineno-91-17"></a>            <span class="c1"># Number of child nodes = 0, delete node directly and return</span>
+<a id="__codelineno-91-17" name="__codelineno-91-17" href="#__codelineno-91-17"></a>            <span class="c1"># Number of children = 0: remove node and return</span>
 <a id="__codelineno-91-18" name="__codelineno-91-18" href="#__codelineno-91-18"></a>            <span class="k">if</span> <span class="n">child</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-91-19" name="__codelineno-91-19" href="#__codelineno-91-19"></a>                <span class="k">return</span> <span class="kc">None</span>
-<a id="__codelineno-91-20" name="__codelineno-91-20" href="#__codelineno-91-20"></a>            <span class="c1"># Number of child nodes = 1, delete node directly</span>
+<a id="__codelineno-91-20" name="__codelineno-91-20" href="#__codelineno-91-20"></a>            <span class="c1"># Number of children = 1: remove node directly</span>
 <a id="__codelineno-91-21" name="__codelineno-91-21" href="#__codelineno-91-21"></a>            <span class="k">else</span><span class="p">:</span>
 <a id="__codelineno-91-22" name="__codelineno-91-22" href="#__codelineno-91-22"></a>                <span class="n">node</span> <span class="o">=</span> <span class="n">child</span>
 <a id="__codelineno-91-23" name="__codelineno-91-23" href="#__codelineno-91-23"></a>        <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-91-24" name="__codelineno-91-24" href="#__codelineno-91-24"></a>            <span class="c1"># Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it</span>
+<a id="__codelineno-91-24" name="__codelineno-91-24" href="#__codelineno-91-24"></a>            <span class="c1"># Number of children = 2: remove the inorder successor and use its value to replace the current node value</span>
 <a id="__codelineno-91-25" name="__codelineno-91-25" href="#__codelineno-91-25"></a>            <span class="n">temp</span> <span class="o">=</span> <span class="n">node</span><span class="o">.</span><span class="n">right</span>
 <a id="__codelineno-91-26" name="__codelineno-91-26" href="#__codelineno-91-26"></a>            <span class="k">while</span> <span class="n">temp</span><span class="o">.</span><span class="n">left</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-91-27" name="__codelineno-91-27" href="#__codelineno-91-27"></a>                <span class="n">temp</span> <span class="o">=</span> <span class="n">temp</span><span class="o">.</span><span class="n">left</span>
 <a id="__codelineno-91-28" name="__codelineno-91-28" href="#__codelineno-91-28"></a>            <span class="n">node</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">remove_helper</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="p">,</span> <span class="n">temp</span><span class="o">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-91-29" name="__codelineno-91-29" href="#__codelineno-91-29"></a>            <span class="n">node</span><span class="o">.</span><span class="n">val</span> <span class="o">=</span> <span class="n">temp</span><span class="o">.</span><span class="n">val</span>
-<a id="__codelineno-91-30" name="__codelineno-91-30" href="#__codelineno-91-30"></a>    <span class="c1"># Update node height</span>
+<a id="__codelineno-91-30" name="__codelineno-91-30" href="#__codelineno-91-30"></a>    <span class="c1"># Update the node height</span>
 <a id="__codelineno-91-31" name="__codelineno-91-31" href="#__codelineno-91-31"></a>    <span class="bp">self</span><span class="o">.</span><span class="n">update_height</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-91-32" name="__codelineno-91-32" href="#__codelineno-91-32"></a>    <span class="c1"># 2. Perform rotation operation to restore balance to this subtree</span>
+<a id="__codelineno-91-32" name="__codelineno-91-32" href="#__codelineno-91-32"></a>    <span class="c1"># 2. Perform rotations to restore balance in this subtree</span>
 <a id="__codelineno-91-33" name="__codelineno-91-33" href="#__codelineno-91-33"></a>    <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-92-1" name="__codelineno-92-1" href="#__codelineno-92-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">avl_tree.cpp</span><pre><span></span><code><a id="__codelineno-92-1" name="__codelineno-92-1" href="#__codelineno-92-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-92-2" name="__codelineno-92-2" href="#__codelineno-92-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-92-3" name="__codelineno-92-3" href="#__codelineno-92-3"></a><span class="w">    </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-92-4" name="__codelineno-92-4" href="#__codelineno-92-4"></a><span class="p">}</span>
 <a id="__codelineno-92-5" name="__codelineno-92-5" href="#__codelineno-92-5"></a>
-<a id="__codelineno-92-6" name="__codelineno-92-6" href="#__codelineno-92-6"></a><span class="cm">/* Recursively delete node (helper method) */</span>
+<a id="__codelineno-92-6" name="__codelineno-92-6" href="#__codelineno-92-6"></a><span class="cm">/* Remove a node recursively (helper method) */</span>
 <a id="__codelineno-92-7" name="__codelineno-92-7" href="#__codelineno-92-7"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">removeHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-92-8" name="__codelineno-92-8" href="#__codelineno-92-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span>
 <a id="__codelineno-92-9" name="__codelineno-92-9" href="#__codelineno-92-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">;</span>
-<a id="__codelineno-92-10" name="__codelineno-92-10" href="#__codelineno-92-10"></a><span class="w">    </span><span class="cm">/* 1. Find node and delete */</span>
+<a id="__codelineno-92-10" name="__codelineno-92-10" href="#__codelineno-92-10"></a><span class="w">    </span><span class="cm">/* 1. Find and remove the node */</span>
 <a id="__codelineno-92-11" name="__codelineno-92-11" href="#__codelineno-92-11"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-92-12" name="__codelineno-92-12" href="#__codelineno-92-12"></a><span class="w">        </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-92-13" name="__codelineno-92-13" href="#__codelineno-92-13"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span>
@@ -6880,18 +6952,18 @@
 <a id="__codelineno-92-15" name="__codelineno-92-15" href="#__codelineno-92-15"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-92-16" name="__codelineno-92-16" href="#__codelineno-92-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-92-17" name="__codelineno-92-17" href="#__codelineno-92-17"></a><span class="w">            </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-92-18" name="__codelineno-92-18" href="#__codelineno-92-18"></a><span class="w">            </span><span class="c1">// Number of child nodes = 0, delete node directly and return</span>
+<a id="__codelineno-92-18" name="__codelineno-92-18" href="#__codelineno-92-18"></a><span class="w">            </span><span class="c1">// Number of children = 0: remove node and return</span>
 <a id="__codelineno-92-19" name="__codelineno-92-19" href="#__codelineno-92-19"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">child</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-92-20" name="__codelineno-92-20" href="#__codelineno-92-20"></a><span class="w">                </span><span class="k">delete</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-92-21" name="__codelineno-92-21" href="#__codelineno-92-21"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">;</span>
 <a id="__codelineno-92-22" name="__codelineno-92-22" href="#__codelineno-92-22"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-92-23" name="__codelineno-92-23" href="#__codelineno-92-23"></a><span class="w">            </span><span class="c1">// Number of child nodes = 1, delete node directly</span>
+<a id="__codelineno-92-23" name="__codelineno-92-23" href="#__codelineno-92-23"></a><span class="w">            </span><span class="c1">// Number of children = 1: remove node directly</span>
 <a id="__codelineno-92-24" name="__codelineno-92-24" href="#__codelineno-92-24"></a><span class="w">            </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-92-25" name="__codelineno-92-25" href="#__codelineno-92-25"></a><span class="w">                </span><span class="k">delete</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-92-26" name="__codelineno-92-26" href="#__codelineno-92-26"></a><span class="w">                </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-92-27" name="__codelineno-92-27" href="#__codelineno-92-27"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-92-28" name="__codelineno-92-28" href="#__codelineno-92-28"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-92-29" name="__codelineno-92-29" href="#__codelineno-92-29"></a><span class="w">            </span><span class="c1">// Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it</span>
+<a id="__codelineno-92-29" name="__codelineno-92-29" href="#__codelineno-92-29"></a><span class="w">            </span><span class="c1">// Number of children = 2: remove the inorder successor and use its value to replace the current node value</span>
 <a id="__codelineno-92-30" name="__codelineno-92-30" href="#__codelineno-92-30"></a><span class="w">            </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-92-31" name="__codelineno-92-31" href="#__codelineno-92-31"></a><span class="w">            </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">temp</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-92-32" name="__codelineno-92-32" href="#__codelineno-92-32"></a><span class="w">                </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">;</span>
@@ -6901,25 +6973,25 @@
 <a id="__codelineno-92-36" name="__codelineno-92-36" href="#__codelineno-92-36"></a><span class="w">            </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tempVal</span><span class="p">;</span>
 <a id="__codelineno-92-37" name="__codelineno-92-37" href="#__codelineno-92-37"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-92-38" name="__codelineno-92-38" href="#__codelineno-92-38"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-92-39" name="__codelineno-92-39" href="#__codelineno-92-39"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update node height</span>
-<a id="__codelineno-92-40" name="__codelineno-92-40" href="#__codelineno-92-40"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-92-39" name="__codelineno-92-39" href="#__codelineno-92-39"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-92-40" name="__codelineno-92-40" href="#__codelineno-92-40"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-92-41" name="__codelineno-92-41" href="#__codelineno-92-41"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-92-42" name="__codelineno-92-42" href="#__codelineno-92-42"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-92-42" name="__codelineno-92-42" href="#__codelineno-92-42"></a><span class="w">    </span><span class="c1">// Return the root of the subtree</span>
 <a id="__codelineno-92-43" name="__codelineno-92-43" href="#__codelineno-92-43"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-92-44" name="__codelineno-92-44" href="#__codelineno-92-44"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-93-1" name="__codelineno-93-1" href="#__codelineno-93-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">avl_tree.java</span><pre><span></span><code><a id="__codelineno-93-1" name="__codelineno-93-1" href="#__codelineno-93-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-93-2" name="__codelineno-93-2" href="#__codelineno-93-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-93-3" name="__codelineno-93-3" href="#__codelineno-93-3"></a><span class="w">    </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-93-4" name="__codelineno-93-4" href="#__codelineno-93-4"></a><span class="p">}</span>
 <a id="__codelineno-93-5" name="__codelineno-93-5" href="#__codelineno-93-5"></a>
-<a id="__codelineno-93-6" name="__codelineno-93-6" href="#__codelineno-93-6"></a><span class="cm">/* Recursively delete node (helper method) */</span>
+<a id="__codelineno-93-6" name="__codelineno-93-6" href="#__codelineno-93-6"></a><span class="cm">/* Remove a node recursively (helper method) */</span>
 <a id="__codelineno-93-7" name="__codelineno-93-7" href="#__codelineno-93-7"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">removeHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-93-8" name="__codelineno-93-8" href="#__codelineno-93-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-93-9" name="__codelineno-93-9" href="#__codelineno-93-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-93-10" name="__codelineno-93-10" href="#__codelineno-93-10"></a><span class="w">    </span><span class="cm">/* 1. Find node and delete */</span>
+<a id="__codelineno-93-10" name="__codelineno-93-10" href="#__codelineno-93-10"></a><span class="w">    </span><span class="cm">/* 1. Find and remove the node */</span>
 <a id="__codelineno-93-11" name="__codelineno-93-11" href="#__codelineno-93-11"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="p">)</span>
 <a id="__codelineno-93-12" name="__codelineno-93-12" href="#__codelineno-93-12"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-93-13" name="__codelineno-93-13" href="#__codelineno-93-13"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="p">)</span>
@@ -6927,14 +6999,14 @@
 <a id="__codelineno-93-15" name="__codelineno-93-15" href="#__codelineno-93-15"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-93-16" name="__codelineno-93-16" href="#__codelineno-93-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-93-17" name="__codelineno-93-17" href="#__codelineno-93-17"></a><span class="w">            </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">;</span>
-<a id="__codelineno-93-18" name="__codelineno-93-18" href="#__codelineno-93-18"></a><span class="w">            </span><span class="c1">// Number of child nodes = 0, delete node directly and return</span>
+<a id="__codelineno-93-18" name="__codelineno-93-18" href="#__codelineno-93-18"></a><span class="w">            </span><span class="c1">// Number of children = 0: remove node and return</span>
 <a id="__codelineno-93-19" name="__codelineno-93-19" href="#__codelineno-93-19"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">child</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-93-20" name="__codelineno-93-20" href="#__codelineno-93-20"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-93-21" name="__codelineno-93-21" href="#__codelineno-93-21"></a><span class="w">            </span><span class="c1">// Number of child nodes = 1, delete node directly</span>
+<a id="__codelineno-93-21" name="__codelineno-93-21" href="#__codelineno-93-21"></a><span class="w">            </span><span class="c1">// Number of children = 1: remove node directly</span>
 <a id="__codelineno-93-22" name="__codelineno-93-22" href="#__codelineno-93-22"></a><span class="w">            </span><span class="k">else</span>
 <a id="__codelineno-93-23" name="__codelineno-93-23" href="#__codelineno-93-23"></a><span class="w">                </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-93-24" name="__codelineno-93-24" href="#__codelineno-93-24"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-93-25" name="__codelineno-93-25" href="#__codelineno-93-25"></a><span class="w">            </span><span class="c1">// Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it</span>
+<a id="__codelineno-93-25" name="__codelineno-93-25" href="#__codelineno-93-25"></a><span class="w">            </span><span class="c1">// Number of children = 2: remove the inorder successor and use its value to replace the current node value</span>
 <a id="__codelineno-93-26" name="__codelineno-93-26" href="#__codelineno-93-26"></a><span class="w">            </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">;</span>
 <a id="__codelineno-93-27" name="__codelineno-93-27" href="#__codelineno-93-27"></a><span class="w">            </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">temp</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-93-28" name="__codelineno-93-28" href="#__codelineno-93-28"></a><span class="w">                </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">.</span><span class="na">left</span><span class="p">;</span>
@@ -6943,24 +7015,24 @@
 <a id="__codelineno-93-31" name="__codelineno-93-31" href="#__codelineno-93-31"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">.</span><span class="na">val</span><span class="p">;</span>
 <a id="__codelineno-93-32" name="__codelineno-93-32" href="#__codelineno-93-32"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-93-33" name="__codelineno-93-33" href="#__codelineno-93-33"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-93-34" name="__codelineno-93-34" href="#__codelineno-93-34"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update node height</span>
-<a id="__codelineno-93-35" name="__codelineno-93-35" href="#__codelineno-93-35"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-93-34" name="__codelineno-93-34" href="#__codelineno-93-34"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-93-35" name="__codelineno-93-35" href="#__codelineno-93-35"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-93-36" name="__codelineno-93-36" href="#__codelineno-93-36"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-93-37" name="__codelineno-93-37" href="#__codelineno-93-37"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-93-37" name="__codelineno-93-37" href="#__codelineno-93-37"></a><span class="w">    </span><span class="c1">// Return the subtree root</span>
 <a id="__codelineno-93-38" name="__codelineno-93-38" href="#__codelineno-93-38"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-93-39" name="__codelineno-93-39" href="#__codelineno-93-39"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-94-1" name="__codelineno-94-1" href="#__codelineno-94-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">avl_tree.cs</span><pre><span></span><code><a id="__codelineno-94-1" name="__codelineno-94-1" href="#__codelineno-94-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-94-2" name="__codelineno-94-2" href="#__codelineno-94-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-94-3" name="__codelineno-94-3" href="#__codelineno-94-3"></a><span class="w">    </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">RemoveHelper</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-94-4" name="__codelineno-94-4" href="#__codelineno-94-4"></a><span class="p">}</span>
 <a id="__codelineno-94-5" name="__codelineno-94-5" href="#__codelineno-94-5"></a>
-<a id="__codelineno-94-6" name="__codelineno-94-6" href="#__codelineno-94-6"></a><span class="cm">/* Recursively delete node (helper method) */</span>
+<a id="__codelineno-94-6" name="__codelineno-94-6" href="#__codelineno-94-6"></a><span class="cm">/* Remove a node recursively (helper method) */</span>
 <a id="__codelineno-94-7" name="__codelineno-94-7" href="#__codelineno-94-7"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">RemoveHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-94-8" name="__codelineno-94-8" href="#__codelineno-94-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
-<a id="__codelineno-94-9" name="__codelineno-94-9" href="#__codelineno-94-9"></a><span class="w">    </span><span class="cm">/* 1. Find node and delete */</span>
+<a id="__codelineno-94-9" name="__codelineno-94-9" href="#__codelineno-94-9"></a><span class="w">    </span><span class="cm">/* 1. Find and remove the node */</span>
 <a id="__codelineno-94-10" name="__codelineno-94-10" href="#__codelineno-94-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-94-11" name="__codelineno-94-11" href="#__codelineno-94-11"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">RemoveHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-94-12" name="__codelineno-94-12" href="#__codelineno-94-12"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="nf">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
@@ -6968,14 +7040,14 @@
 <a id="__codelineno-94-14" name="__codelineno-94-14" href="#__codelineno-94-14"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-94-15" name="__codelineno-94-15" href="#__codelineno-94-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-94-16" name="__codelineno-94-16" href="#__codelineno-94-16"></a><span class="w">            </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">??</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-94-17" name="__codelineno-94-17" href="#__codelineno-94-17"></a><span class="w">            </span><span class="c1">// Number of child nodes = 0, delete node directly and return</span>
+<a id="__codelineno-94-17" name="__codelineno-94-17" href="#__codelineno-94-17"></a><span class="w">            </span><span class="c1">// Number of children = 0: remove node and return</span>
 <a id="__codelineno-94-18" name="__codelineno-94-18" href="#__codelineno-94-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">child</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span>
 <a id="__codelineno-94-19" name="__codelineno-94-19" href="#__codelineno-94-19"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
-<a id="__codelineno-94-20" name="__codelineno-94-20" href="#__codelineno-94-20"></a><span class="w">            </span><span class="c1">// Number of child nodes = 1, delete node directly</span>
+<a id="__codelineno-94-20" name="__codelineno-94-20" href="#__codelineno-94-20"></a><span class="w">            </span><span class="c1">// Number of children = 1: remove node directly</span>
 <a id="__codelineno-94-21" name="__codelineno-94-21" href="#__codelineno-94-21"></a><span class="w">            </span><span class="k">else</span>
 <a id="__codelineno-94-22" name="__codelineno-94-22" href="#__codelineno-94-22"></a><span class="w">                </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-94-23" name="__codelineno-94-23" href="#__codelineno-94-23"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-94-24" name="__codelineno-94-24" href="#__codelineno-94-24"></a><span class="w">            </span><span class="c1">// Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it</span>
+<a id="__codelineno-94-24" name="__codelineno-94-24" href="#__codelineno-94-24"></a><span class="w">            </span><span class="c1">// Number of children = 2: remove the inorder successor and use its value to replace the current node value</span>
 <a id="__codelineno-94-25" name="__codelineno-94-25" href="#__codelineno-94-25"></a><span class="w">            </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-94-26" name="__codelineno-94-26" href="#__codelineno-94-26"></a><span class="w">            </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">temp</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-94-27" name="__codelineno-94-27" href="#__codelineno-94-27"></a><span class="w">                </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">.</span><span class="n">left</span><span class="p">;</span>
@@ -6984,26 +7056,26 @@
 <a id="__codelineno-94-30" name="__codelineno-94-30" href="#__codelineno-94-30"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-94-31" name="__codelineno-94-31" href="#__codelineno-94-31"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-94-32" name="__codelineno-94-32" href="#__codelineno-94-32"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-94-33" name="__codelineno-94-33" href="#__codelineno-94-33"></a><span class="w">    </span><span class="n">UpdateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w">  </span><span class="c1">// Update node height</span>
-<a id="__codelineno-94-34" name="__codelineno-94-34" href="#__codelineno-94-34"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-94-33" name="__codelineno-94-33" href="#__codelineno-94-33"></a><span class="w">    </span><span class="n">UpdateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w">  </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-94-34" name="__codelineno-94-34" href="#__codelineno-94-34"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-94-35" name="__codelineno-94-35" href="#__codelineno-94-35"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Rotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-94-36" name="__codelineno-94-36" href="#__codelineno-94-36"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-94-36" name="__codelineno-94-36" href="#__codelineno-94-36"></a><span class="w">    </span><span class="c1">// Return the subtree root</span>
 <a id="__codelineno-94-37" name="__codelineno-94-37" href="#__codelineno-94-37"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-94-38" name="__codelineno-94-38" href="#__codelineno-94-38"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-95-1" name="__codelineno-95-1" href="#__codelineno-95-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">avl_tree.go</span><pre><span></span><code><a id="__codelineno-95-1" name="__codelineno-95-1" href="#__codelineno-95-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-95-2" name="__codelineno-95-2" href="#__codelineno-95-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">t</span><span class="w"> </span><span class="o">*</span><span class="nx">aVLTree</span><span class="p">)</span><span class="w"> </span><span class="nx">remove</span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-95-3" name="__codelineno-95-3" href="#__codelineno-95-3"></a><span class="w">    </span><span class="nx">t</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">removeHelper</span><span class="p">(</span><span class="nx">t</span><span class="p">.</span><span class="nx">root</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span>
 <a id="__codelineno-95-4" name="__codelineno-95-4" href="#__codelineno-95-4"></a><span class="p">}</span>
 <a id="__codelineno-95-5" name="__codelineno-95-5" href="#__codelineno-95-5"></a>
-<a id="__codelineno-95-6" name="__codelineno-95-6" href="#__codelineno-95-6"></a><span class="cm">/* Recursively remove node (helper function) */</span>
+<a id="__codelineno-95-6" name="__codelineno-95-6" href="#__codelineno-95-6"></a><span class="cm">/* Remove a node recursively (helper function) */</span>
 <a id="__codelineno-95-7" name="__codelineno-95-7" href="#__codelineno-95-7"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">t</span><span class="w"> </span><span class="o">*</span><span class="nx">aVLTree</span><span class="p">)</span><span class="w"> </span><span class="nx">removeHelper</span><span class="p">(</span><span class="nx">node</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-95-8" name="__codelineno-95-8" href="#__codelineno-95-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-95-9" name="__codelineno-95-9" href="#__codelineno-95-9"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-95-10" name="__codelineno-95-10" href="#__codelineno-95-10"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-95-11" name="__codelineno-95-11" href="#__codelineno-95-11"></a><span class="w">    </span><span class="cm">/* 1. Find node and delete */</span>
+<a id="__codelineno-95-11" name="__codelineno-95-11" href="#__codelineno-95-11"></a><span class="w">    </span><span class="cm">/* 1. Find and remove the node */</span>
 <a id="__codelineno-95-12" name="__codelineno-95-12" href="#__codelineno-95-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Val</span><span class="p">.(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-95-13" name="__codelineno-95-13" href="#__codelineno-95-13"></a><span class="w">        </span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">removeHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span>
 <a id="__codelineno-95-14" name="__codelineno-95-14" href="#__codelineno-95-14"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Val</span><span class="p">.(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7015,14 +7087,14 @@
 <a id="__codelineno-95-20" name="__codelineno-95-20" href="#__codelineno-95-20"></a><span class="w">                </span><span class="nx">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span>
 <a id="__codelineno-95-21" name="__codelineno-95-21" href="#__codelineno-95-21"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-95-22" name="__codelineno-95-22" href="#__codelineno-95-22"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">child</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-95-23" name="__codelineno-95-23" href="#__codelineno-95-23"></a><span class="w">                </span><span class="c1">// Number of child nodes = 0, delete node directly and return</span>
+<a id="__codelineno-95-23" name="__codelineno-95-23" href="#__codelineno-95-23"></a><span class="w">                </span><span class="c1">// Number of children = 0: remove node and return</span>
 <a id="__codelineno-95-24" name="__codelineno-95-24" href="#__codelineno-95-24"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-95-25" name="__codelineno-95-25" href="#__codelineno-95-25"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-95-26" name="__codelineno-95-26" href="#__codelineno-95-26"></a><span class="w">                </span><span class="c1">// Number of child nodes = 1, delete node directly</span>
+<a id="__codelineno-95-26" name="__codelineno-95-26" href="#__codelineno-95-26"></a><span class="w">                </span><span class="c1">// Number of children = 1: remove node directly</span>
 <a id="__codelineno-95-27" name="__codelineno-95-27" href="#__codelineno-95-27"></a><span class="w">                </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span>
 <a id="__codelineno-95-28" name="__codelineno-95-28" href="#__codelineno-95-28"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-95-29" name="__codelineno-95-29" href="#__codelineno-95-29"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-95-30" name="__codelineno-95-30" href="#__codelineno-95-30"></a><span class="w">            </span><span class="c1">// Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it</span>
+<a id="__codelineno-95-30" name="__codelineno-95-30" href="#__codelineno-95-30"></a><span class="w">            </span><span class="c1">// Number of children = 2: remove the inorder successor and use its value to replace the current node value</span>
 <a id="__codelineno-95-31" name="__codelineno-95-31" href="#__codelineno-95-31"></a><span class="w">            </span><span class="nx">temp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span>
 <a id="__codelineno-95-32" name="__codelineno-95-32" href="#__codelineno-95-32"></a><span class="w">            </span><span class="k">for</span><span class="w"> </span><span class="nx">temp</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-95-33" name="__codelineno-95-33" href="#__codelineno-95-33"></a><span class="w">                </span><span class="nx">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">.</span><span class="nx">Left</span>
@@ -7031,28 +7103,28 @@
 <a id="__codelineno-95-36" name="__codelineno-95-36" href="#__codelineno-95-36"></a><span class="w">            </span><span class="nx">node</span><span class="p">.</span><span class="nx">Val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">.</span><span class="nx">Val</span>
 <a id="__codelineno-95-37" name="__codelineno-95-37" href="#__codelineno-95-37"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-95-38" name="__codelineno-95-38" href="#__codelineno-95-38"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-95-39" name="__codelineno-95-39" href="#__codelineno-95-39"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-95-39" name="__codelineno-95-39" href="#__codelineno-95-39"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-95-40" name="__codelineno-95-40" href="#__codelineno-95-40"></a><span class="w">    </span><span class="nx">t</span><span class="p">.</span><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span>
-<a id="__codelineno-95-41" name="__codelineno-95-41" href="#__codelineno-95-41"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-95-41" name="__codelineno-95-41" href="#__codelineno-95-41"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-95-42" name="__codelineno-95-42" href="#__codelineno-95-42"></a><span class="w">    </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">t</span><span class="p">.</span><span class="nx">rotate</span><span class="p">(</span><span class="nx">node</span><span class="p">)</span>
-<a id="__codelineno-95-43" name="__codelineno-95-43" href="#__codelineno-95-43"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-95-43" name="__codelineno-95-43" href="#__codelineno-95-43"></a><span class="w">    </span><span class="c1">// Return the subtree root</span>
 <a id="__codelineno-95-44" name="__codelineno-95-44" href="#__codelineno-95-44"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span>
 <a id="__codelineno-95-45" name="__codelineno-95-45" href="#__codelineno-95-45"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.swift</span><pre><span></span><code><a id="__codelineno-96-1" name="__codelineno-96-1" href="#__codelineno-96-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">avl_tree.swift</span><pre><span></span><code><a id="__codelineno-96-1" name="__codelineno-96-1" href="#__codelineno-96-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-96-2" name="__codelineno-96-2" href="#__codelineno-96-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-96-3" name="__codelineno-96-3" href="#__codelineno-96-3"></a><span class="w">    </span><span class="n">root</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-96-4" name="__codelineno-96-4" href="#__codelineno-96-4"></a><span class="p">}</span>
 <a id="__codelineno-96-5" name="__codelineno-96-5" href="#__codelineno-96-5"></a>
-<a id="__codelineno-96-6" name="__codelineno-96-6" href="#__codelineno-96-6"></a><span class="cm">/* Recursively delete node (helper method) */</span>
+<a id="__codelineno-96-6" name="__codelineno-96-6" href="#__codelineno-96-6"></a><span class="cm">/* Remove a node recursively (helper method) */</span>
 <a id="__codelineno-96-7" name="__codelineno-96-7" href="#__codelineno-96-7"></a><span class="kd">func</span><span class="w"> </span><span class="nf">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-96-8" name="__codelineno-96-8" href="#__codelineno-96-8"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-96-9" name="__codelineno-96-9" href="#__codelineno-96-9"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-96-10" name="__codelineno-96-10" href="#__codelineno-96-10"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-96-11" name="__codelineno-96-11" href="#__codelineno-96-11"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-96-12" name="__codelineno-96-12" href="#__codelineno-96-12"></a><span class="w">    </span><span class="cm">/* 1. Find node and delete */</span>
+<a id="__codelineno-96-12" name="__codelineno-96-12" href="#__codelineno-96-12"></a><span class="w">    </span><span class="cm">/* 1. Find and remove the node */</span>
 <a id="__codelineno-96-13" name="__codelineno-96-13" href="#__codelineno-96-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-96-14" name="__codelineno-96-14" href="#__codelineno-96-14"></a><span class="w">        </span><span class="n">node</span><span class="p">?.</span><span class="kr">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-96-15" name="__codelineno-96-15" href="#__codelineno-96-15"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="p">{</span>
@@ -7060,16 +7132,16 @@
 <a id="__codelineno-96-17" name="__codelineno-96-17" href="#__codelineno-96-17"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-96-18" name="__codelineno-96-18" href="#__codelineno-96-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">left</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">right</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-96-19" name="__codelineno-96-19" href="#__codelineno-96-19"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="nv">child</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">left</span><span class="w"> </span><span class="p">??</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">right</span>
-<a id="__codelineno-96-20" name="__codelineno-96-20" href="#__codelineno-96-20"></a><span class="w">            </span><span class="c1">// Number of child nodes = 0, delete node directly and return</span>
+<a id="__codelineno-96-20" name="__codelineno-96-20" href="#__codelineno-96-20"></a><span class="w">            </span><span class="c1">// Number of children = 0: remove node and return</span>
 <a id="__codelineno-96-21" name="__codelineno-96-21" href="#__codelineno-96-21"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-96-22" name="__codelineno-96-22" href="#__codelineno-96-22"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="kc">nil</span>
 <a id="__codelineno-96-23" name="__codelineno-96-23" href="#__codelineno-96-23"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-96-24" name="__codelineno-96-24" href="#__codelineno-96-24"></a><span class="w">            </span><span class="c1">// Number of child nodes = 1, delete node directly</span>
+<a id="__codelineno-96-24" name="__codelineno-96-24" href="#__codelineno-96-24"></a><span class="w">            </span><span class="c1">// Number of children = 1: remove node directly</span>
 <a id="__codelineno-96-25" name="__codelineno-96-25" href="#__codelineno-96-25"></a><span class="w">            </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-96-26" name="__codelineno-96-26" href="#__codelineno-96-26"></a><span class="w">                </span><span class="n">node</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">child</span>
 <a id="__codelineno-96-27" name="__codelineno-96-27" href="#__codelineno-96-27"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-96-28" name="__codelineno-96-28" href="#__codelineno-96-28"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-96-29" name="__codelineno-96-29" href="#__codelineno-96-29"></a><span class="w">            </span><span class="c1">// Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it</span>
+<a id="__codelineno-96-29" name="__codelineno-96-29" href="#__codelineno-96-29"></a><span class="w">            </span><span class="c1">// Number of children = 2: remove the inorder successor and use its value to replace the current node value</span>
 <a id="__codelineno-96-30" name="__codelineno-96-30" href="#__codelineno-96-30"></a><span class="w">            </span><span class="kd">var</span><span class="w"> </span><span class="nv">temp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span><span class="p">?.</span><span class="kr">right</span>
 <a id="__codelineno-96-31" name="__codelineno-96-31" href="#__codelineno-96-31"></a><span class="w">            </span><span class="k">while</span><span class="w"> </span><span class="n">temp</span><span class="p">?.</span><span class="kr">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-96-32" name="__codelineno-96-32" href="#__codelineno-96-32"></a><span class="w">                </span><span class="n">temp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">temp</span><span class="p">?.</span><span class="kr">left</span>
@@ -7078,36 +7150,36 @@
 <a id="__codelineno-96-35" name="__codelineno-96-35" href="#__codelineno-96-35"></a><span class="w">            </span><span class="n">node</span><span class="p">?.</span><span class="n">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">temp</span><span class="p">!.</span><span class="n">val</span>
 <a id="__codelineno-96-36" name="__codelineno-96-36" href="#__codelineno-96-36"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-96-37" name="__codelineno-96-37" href="#__codelineno-96-37"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-96-38" name="__codelineno-96-38" href="#__codelineno-96-38"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="c1">// Update node height</span>
-<a id="__codelineno-96-39" name="__codelineno-96-39" href="#__codelineno-96-39"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-96-38" name="__codelineno-96-38" href="#__codelineno-96-38"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-96-39" name="__codelineno-96-39" href="#__codelineno-96-39"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-96-40" name="__codelineno-96-40" href="#__codelineno-96-40"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-96-41" name="__codelineno-96-41" href="#__codelineno-96-41"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-96-41" name="__codelineno-96-41" href="#__codelineno-96-41"></a><span class="w">    </span><span class="c1">// Return the subtree root</span>
 <a id="__codelineno-96-42" name="__codelineno-96-42" href="#__codelineno-96-42"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-96-43" name="__codelineno-96-43" href="#__codelineno-96-43"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-97-1" name="__codelineno-97-1" href="#__codelineno-97-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">avl_tree.js</span><pre><span></span><code><a id="__codelineno-97-1" name="__codelineno-97-1" href="#__codelineno-97-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-97-2" name="__codelineno-97-2" href="#__codelineno-97-2"></a><span class="nx">remove</span><span class="p">(</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-97-3" name="__codelineno-97-3" href="#__codelineno-97-3"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#removeHelper</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-97-4" name="__codelineno-97-4" href="#__codelineno-97-4"></a><span class="p">}</span>
 <a id="__codelineno-97-5" name="__codelineno-97-5" href="#__codelineno-97-5"></a>
-<a id="__codelineno-97-6" name="__codelineno-97-6" href="#__codelineno-97-6"></a><span class="cm">/* Recursively delete node (helper method) */</span>
+<a id="__codelineno-97-6" name="__codelineno-97-6" href="#__codelineno-97-6"></a><span class="cm">/* Remove a node recursively (helper method) */</span>
 <a id="__codelineno-97-7" name="__codelineno-97-7" href="#__codelineno-97-7"></a><span class="n">#removeHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-97-8" name="__codelineno-97-8" href="#__codelineno-97-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-97-9" name="__codelineno-97-9" href="#__codelineno-97-9"></a><span class="w">    </span><span class="cm">/* 1. Find node and delete */</span>
+<a id="__codelineno-97-9" name="__codelineno-97-9" href="#__codelineno-97-9"></a><span class="w">    </span><span class="cm">/* 1. Find and remove the node */</span>
 <a id="__codelineno-97-10" name="__codelineno-97-10" href="#__codelineno-97-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#removeHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-97-11" name="__codelineno-97-11" href="#__codelineno-97-11"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span>
 <a id="__codelineno-97-12" name="__codelineno-97-12" href="#__codelineno-97-12"></a><span class="w">        </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#removeHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-97-13" name="__codelineno-97-13" href="#__codelineno-97-13"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-97-14" name="__codelineno-97-14" href="#__codelineno-97-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-97-15" name="__codelineno-97-15" href="#__codelineno-97-15"></a><span class="w">            </span><span class="kd">const</span><span class="w"> </span><span class="nx">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
-<a id="__codelineno-97-16" name="__codelineno-97-16" href="#__codelineno-97-16"></a><span class="w">            </span><span class="c1">// Number of child nodes = 0, delete node directly and return</span>
+<a id="__codelineno-97-16" name="__codelineno-97-16" href="#__codelineno-97-16"></a><span class="w">            </span><span class="c1">// Number of children = 0: remove node and return</span>
 <a id="__codelineno-97-17" name="__codelineno-97-17" href="#__codelineno-97-17"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">child</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-97-18" name="__codelineno-97-18" href="#__codelineno-97-18"></a><span class="w">            </span><span class="c1">// Number of child nodes = 1, delete node directly</span>
+<a id="__codelineno-97-18" name="__codelineno-97-18" href="#__codelineno-97-18"></a><span class="w">            </span><span class="c1">// Number of children = 1: remove node directly</span>
 <a id="__codelineno-97-19" name="__codelineno-97-19" href="#__codelineno-97-19"></a><span class="w">            </span><span class="k">else</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span><span class="p">;</span>
 <a id="__codelineno-97-20" name="__codelineno-97-20" href="#__codelineno-97-20"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-97-21" name="__codelineno-97-21" href="#__codelineno-97-21"></a><span class="w">            </span><span class="c1">// Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it</span>
+<a id="__codelineno-97-21" name="__codelineno-97-21" href="#__codelineno-97-21"></a><span class="w">            </span><span class="c1">// Number of children = 2: remove the inorder successor and use its value to replace the current node value</span>
 <a id="__codelineno-97-22" name="__codelineno-97-22" href="#__codelineno-97-22"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="nx">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
 <a id="__codelineno-97-23" name="__codelineno-97-23" href="#__codelineno-97-23"></a><span class="w">            </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">temp</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-97-24" name="__codelineno-97-24" href="#__codelineno-97-24"></a><span class="w">                </span><span class="nx">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">.</span><span class="nx">left</span><span class="p">;</span>
@@ -7116,24 +7188,24 @@
 <a id="__codelineno-97-27" name="__codelineno-97-27" href="#__codelineno-97-27"></a><span class="w">            </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-97-28" name="__codelineno-97-28" href="#__codelineno-97-28"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-97-29" name="__codelineno-97-29" href="#__codelineno-97-29"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-97-30" name="__codelineno-97-30" href="#__codelineno-97-30"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">#updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update node height</span>
-<a id="__codelineno-97-31" name="__codelineno-97-31" href="#__codelineno-97-31"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-97-30" name="__codelineno-97-30" href="#__codelineno-97-30"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="n">#updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-97-31" name="__codelineno-97-31" href="#__codelineno-97-31"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-97-32" name="__codelineno-97-32" href="#__codelineno-97-32"></a><span class="w">    </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">#rotate</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
-<a id="__codelineno-97-33" name="__codelineno-97-33" href="#__codelineno-97-33"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-97-33" name="__codelineno-97-33" href="#__codelineno-97-33"></a><span class="w">    </span><span class="c1">// Return the root of the subtree</span>
 <a id="__codelineno-97-34" name="__codelineno-97-34" href="#__codelineno-97-34"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-97-35" name="__codelineno-97-35" href="#__codelineno-97-35"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-98-1" name="__codelineno-98-1" href="#__codelineno-98-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">avl_tree.ts</span><pre><span></span><code><a id="__codelineno-98-1" name="__codelineno-98-1" href="#__codelineno-98-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-98-2" name="__codelineno-98-2" href="#__codelineno-98-2"></a><span class="nx">remove</span><span class="p">(</span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-98-3" name="__codelineno-98-3" href="#__codelineno-98-3"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">removeHelper</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-98-4" name="__codelineno-98-4" href="#__codelineno-98-4"></a><span class="p">}</span>
 <a id="__codelineno-98-5" name="__codelineno-98-5" href="#__codelineno-98-5"></a>
-<a id="__codelineno-98-6" name="__codelineno-98-6" href="#__codelineno-98-6"></a><span class="cm">/* Recursively delete node (helper method) */</span>
+<a id="__codelineno-98-6" name="__codelineno-98-6" href="#__codelineno-98-6"></a><span class="cm">/* Remove a node recursively (helper method) */</span>
 <a id="__codelineno-98-7" name="__codelineno-98-7" href="#__codelineno-98-7"></a><span class="nx">removeHelper</span><span class="p">(</span><span class="nx">node</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-98-8" name="__codelineno-98-8" href="#__codelineno-98-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-98-9" name="__codelineno-98-9" href="#__codelineno-98-9"></a><span class="w">    </span><span class="cm">/* 1. Find node and delete */</span>
+<a id="__codelineno-98-9" name="__codelineno-98-9" href="#__codelineno-98-9"></a><span class="w">    </span><span class="cm">/* 1. Find and remove the node */</span>
 <a id="__codelineno-98-10" name="__codelineno-98-10" href="#__codelineno-98-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-98-11" name="__codelineno-98-11" href="#__codelineno-98-11"></a><span class="w">        </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">removeHelper</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-98-12" name="__codelineno-98-12" href="#__codelineno-98-12"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7141,15 +7213,15 @@
 <a id="__codelineno-98-14" name="__codelineno-98-14" href="#__codelineno-98-14"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-98-15" name="__codelineno-98-15" href="#__codelineno-98-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-98-16" name="__codelineno-98-16" href="#__codelineno-98-16"></a><span class="w">            </span><span class="kd">const</span><span class="w"> </span><span class="nx">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">node.left</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">node.right</span><span class="p">;</span>
-<a id="__codelineno-98-17" name="__codelineno-98-17" href="#__codelineno-98-17"></a><span class="w">            </span><span class="c1">// Number of child nodes = 0, delete node directly and return</span>
+<a id="__codelineno-98-17" name="__codelineno-98-17" href="#__codelineno-98-17"></a><span class="w">            </span><span class="c1">// Number of children = 0: remove node and return</span>
 <a id="__codelineno-98-18" name="__codelineno-98-18" href="#__codelineno-98-18"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">child</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-98-19" name="__codelineno-98-19" href="#__codelineno-98-19"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
 <a id="__codelineno-98-20" name="__codelineno-98-20" href="#__codelineno-98-20"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-98-21" name="__codelineno-98-21" href="#__codelineno-98-21"></a><span class="w">                </span><span class="c1">// Number of child nodes = 1, delete node directly</span>
+<a id="__codelineno-98-21" name="__codelineno-98-21" href="#__codelineno-98-21"></a><span class="w">                </span><span class="c1">// Number of children = 1: remove node directly</span>
 <a id="__codelineno-98-22" name="__codelineno-98-22" href="#__codelineno-98-22"></a><span class="w">                </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span><span class="p">;</span>
 <a id="__codelineno-98-23" name="__codelineno-98-23" href="#__codelineno-98-23"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-98-24" name="__codelineno-98-24" href="#__codelineno-98-24"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-98-25" name="__codelineno-98-25" href="#__codelineno-98-25"></a><span class="w">            </span><span class="c1">// Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it</span>
+<a id="__codelineno-98-25" name="__codelineno-98-25" href="#__codelineno-98-25"></a><span class="w">            </span><span class="c1">// Number of children = 2: remove the inorder successor and use its value to replace the current node value</span>
 <a id="__codelineno-98-26" name="__codelineno-98-26" href="#__codelineno-98-26"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="nx">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
 <a id="__codelineno-98-27" name="__codelineno-98-27" href="#__codelineno-98-27"></a><span class="w">            </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">temp</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-98-28" name="__codelineno-98-28" href="#__codelineno-98-28"></a><span class="w">                </span><span class="nx">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">.</span><span class="nx">left</span><span class="p">;</span>
@@ -7158,24 +7230,24 @@
 <a id="__codelineno-98-31" name="__codelineno-98-31" href="#__codelineno-98-31"></a><span class="w">            </span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">temp</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-98-32" name="__codelineno-98-32" href="#__codelineno-98-32"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-98-33" name="__codelineno-98-33" href="#__codelineno-98-33"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-98-34" name="__codelineno-98-34" href="#__codelineno-98-34"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update node height</span>
-<a id="__codelineno-98-35" name="__codelineno-98-35" href="#__codelineno-98-35"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-98-34" name="__codelineno-98-34" href="#__codelineno-98-34"></a><span class="w">    </span><span class="k">this</span><span class="p">.</span><span class="nx">updateHeight</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-98-35" name="__codelineno-98-35" href="#__codelineno-98-35"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-98-36" name="__codelineno-98-36" href="#__codelineno-98-36"></a><span class="w">    </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">rotate</span><span class="p">(</span><span class="nx">node</span><span class="p">);</span>
-<a id="__codelineno-98-37" name="__codelineno-98-37" href="#__codelineno-98-37"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-98-37" name="__codelineno-98-37" href="#__codelineno-98-37"></a><span class="w">    </span><span class="c1">// Return the root of the subtree</span>
 <a id="__codelineno-98-38" name="__codelineno-98-38" href="#__codelineno-98-38"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-98-39" name="__codelineno-98-39" href="#__codelineno-98-39"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.dart</span><pre><span></span><code><a id="__codelineno-99-1" name="__codelineno-99-1" href="#__codelineno-99-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">avl_tree.dart</span><pre><span></span><code><a id="__codelineno-99-1" name="__codelineno-99-1" href="#__codelineno-99-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-99-2" name="__codelineno-99-2" href="#__codelineno-99-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-99-3" name="__codelineno-99-3" href="#__codelineno-99-3"></a><span class="w">  </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-99-4" name="__codelineno-99-4" href="#__codelineno-99-4"></a><span class="p">}</span>
 <a id="__codelineno-99-5" name="__codelineno-99-5" href="#__codelineno-99-5"></a>
-<a id="__codelineno-99-6" name="__codelineno-99-6" href="#__codelineno-99-6"></a><span class="cm">/* Recursively delete node (helper method) */</span>
+<a id="__codelineno-99-6" name="__codelineno-99-6" href="#__codelineno-99-6"></a><span class="cm">/* Remove a node recursively (helper method) */</span>
 <a id="__codelineno-99-7" name="__codelineno-99-7" href="#__codelineno-99-7"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-99-8" name="__codelineno-99-8" href="#__codelineno-99-8"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-99-9" name="__codelineno-99-9" href="#__codelineno-99-9"></a><span class="w">  </span><span class="cm">/* 1. Find node and delete */</span>
+<a id="__codelineno-99-9" name="__codelineno-99-9" href="#__codelineno-99-9"></a><span class="w">  </span><span class="cm">/* 1. Find and remove the node */</span>
 <a id="__codelineno-99-10" name="__codelineno-99-10" href="#__codelineno-99-10"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-99-11" name="__codelineno-99-11" href="#__codelineno-99-11"></a><span class="w">    </span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-99-12" name="__codelineno-99-12" href="#__codelineno-99-12"></a><span class="w">  </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
@@ -7183,14 +7255,14 @@
 <a id="__codelineno-99-14" name="__codelineno-99-14" href="#__codelineno-99-14"></a><span class="w">  </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-99-15" name="__codelineno-99-15" href="#__codelineno-99-15"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-99-16" name="__codelineno-99-16" href="#__codelineno-99-16"></a><span class="w">      </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">??</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-99-17" name="__codelineno-99-17" href="#__codelineno-99-17"></a><span class="w">      </span><span class="c1">// Number of child nodes = 0, delete node directly and return</span>
+<a id="__codelineno-99-17" name="__codelineno-99-17" href="#__codelineno-99-17"></a><span class="w">      </span><span class="c1">// Number of children = 0: remove node and return</span>
 <a id="__codelineno-99-18" name="__codelineno-99-18" href="#__codelineno-99-18"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">child</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-99-19" name="__codelineno-99-19" href="#__codelineno-99-19"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-99-20" name="__codelineno-99-20" href="#__codelineno-99-20"></a><span class="w">      </span><span class="c1">// Number of child nodes = 1, delete node directly</span>
+<a id="__codelineno-99-20" name="__codelineno-99-20" href="#__codelineno-99-20"></a><span class="w">      </span><span class="c1">// Number of children = 1: remove node directly</span>
 <a id="__codelineno-99-21" name="__codelineno-99-21" href="#__codelineno-99-21"></a><span class="w">      </span><span class="k">else</span>
 <a id="__codelineno-99-22" name="__codelineno-99-22" href="#__codelineno-99-22"></a><span class="w">        </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-99-23" name="__codelineno-99-23" href="#__codelineno-99-23"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-99-24" name="__codelineno-99-24" href="#__codelineno-99-24"></a><span class="w">      </span><span class="c1">// Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it</span>
+<a id="__codelineno-99-24" name="__codelineno-99-24" href="#__codelineno-99-24"></a><span class="w">      </span><span class="c1">// Number of children = 2: remove the inorder successor and use its value to replace the current node value</span>
 <a id="__codelineno-99-25" name="__codelineno-99-25" href="#__codelineno-99-25"></a><span class="w">      </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-99-26" name="__codelineno-99-26" href="#__codelineno-99-26"></a><span class="w">      </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">temp</span><span class="o">!</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-99-27" name="__codelineno-99-27" href="#__codelineno-99-27"></a><span class="w">        </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">.</span><span class="n">left</span><span class="p">;</span>
@@ -7199,25 +7271,25 @@
 <a id="__codelineno-99-30" name="__codelineno-99-30" href="#__codelineno-99-30"></a><span class="w">      </span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-99-31" name="__codelineno-99-31" href="#__codelineno-99-31"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-99-32" name="__codelineno-99-32" href="#__codelineno-99-32"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-99-33" name="__codelineno-99-33" href="#__codelineno-99-33"></a><span class="w">  </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update node height</span>
-<a id="__codelineno-99-34" name="__codelineno-99-34" href="#__codelineno-99-34"></a><span class="w">  </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-99-33" name="__codelineno-99-33" href="#__codelineno-99-33"></a><span class="w">  </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span><span class="w"> </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-99-34" name="__codelineno-99-34" href="#__codelineno-99-34"></a><span class="w">  </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-99-35" name="__codelineno-99-35" href="#__codelineno-99-35"></a><span class="w">  </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-99-36" name="__codelineno-99-36" href="#__codelineno-99-36"></a><span class="w">  </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-99-36" name="__codelineno-99-36" href="#__codelineno-99-36"></a><span class="w">  </span><span class="c1">// Return the root of the subtree</span>
 <a id="__codelineno-99-37" name="__codelineno-99-37" href="#__codelineno-99-37"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-99-38" name="__codelineno-99-38" href="#__codelineno-99-38"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.rs</span><pre><span></span><code><a id="__codelineno-100-1" name="__codelineno-100-1" href="#__codelineno-100-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">avl_tree.rs</span><pre><span></span><code><a id="__codelineno-100-1" name="__codelineno-100-1" href="#__codelineno-100-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-100-2" name="__codelineno-100-2" href="#__codelineno-100-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-100-3" name="__codelineno-100-3" href="#__codelineno-100-3"></a><span class="w">    </span><span class="bp">Self</span><span class="p">::</span><span class="n">remove_helper</span><span class="p">(</span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="p">.</span><span class="n">clone</span><span class="p">(),</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-100-4" name="__codelineno-100-4" href="#__codelineno-100-4"></a><span class="p">}</span>
 <a id="__codelineno-100-5" name="__codelineno-100-5" href="#__codelineno-100-5"></a>
-<a id="__codelineno-100-6" name="__codelineno-100-6" href="#__codelineno-100-6"></a><span class="cm">/* Recursively delete node (helper method) */</span>
+<a id="__codelineno-100-6" name="__codelineno-100-6" href="#__codelineno-100-6"></a><span class="cm">/* Remove a node recursively (helper method) */</span>
 <a id="__codelineno-100-7" name="__codelineno-100-7" href="#__codelineno-100-7"></a><span class="k">fn</span><span class="w"> </span><span class="nf">remove_helper</span><span class="p">(</span><span class="n">node</span><span class="p">:</span><span class="w"> </span><span class="nc">OptionTreeNodeRc</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">OptionTreeNodeRc</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-100-8" name="__codelineno-100-8" href="#__codelineno-100-8"></a><span class="w">    </span><span class="k">match</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-100-9" name="__codelineno-100-9" href="#__codelineno-100-9"></a><span class="w">        </span><span class="nb">Some</span><span class="p">(</span><span class="k">mut</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-100-10" name="__codelineno-100-10" href="#__codelineno-100-10"></a><span class="w">            </span><span class="cm">/* 1. Find node and delete */</span>
+<a id="__codelineno-100-10" name="__codelineno-100-10" href="#__codelineno-100-10"></a><span class="w">            </span><span class="cm">/* 1. Find and remove the node */</span>
 <a id="__codelineno-100-11" name="__codelineno-100-11" href="#__codelineno-100-11"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-100-12" name="__codelineno-100-12" href="#__codelineno-100-12"></a><span class="w">                </span><span class="kd">let</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-100-13" name="__codelineno-100-13" href="#__codelineno-100-13"></a><span class="w">                </span><span class="n">node</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">remove_helper</span><span class="p">(</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
@@ -7231,15 +7303,15 @@
 <a id="__codelineno-100-21" name="__codelineno-100-21" href="#__codelineno-100-21"></a><span class="w">                    </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">()</span>
 <a id="__codelineno-100-22" name="__codelineno-100-22" href="#__codelineno-100-22"></a><span class="w">                </span><span class="p">};</span>
 <a id="__codelineno-100-23" name="__codelineno-100-23" href="#__codelineno-100-23"></a><span class="w">                </span><span class="k">match</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-100-24" name="__codelineno-100-24" href="#__codelineno-100-24"></a><span class="w">                    </span><span class="c1">// Number of child nodes = 0, delete node directly and return</span>
+<a id="__codelineno-100-24" name="__codelineno-100-24" href="#__codelineno-100-24"></a><span class="w">                    </span><span class="c1">// Number of children = 0: remove node and return</span>
 <a id="__codelineno-100-25" name="__codelineno-100-25" href="#__codelineno-100-25"></a><span class="w">                    </span><span class="nb">None</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-100-26" name="__codelineno-100-26" href="#__codelineno-100-26"></a><span class="w">                        </span><span class="k">return</span><span class="w"> </span><span class="nb">None</span><span class="p">;</span>
 <a id="__codelineno-100-27" name="__codelineno-100-27" href="#__codelineno-100-27"></a><span class="w">                    </span><span class="p">}</span>
-<a id="__codelineno-100-28" name="__codelineno-100-28" href="#__codelineno-100-28"></a><span class="w">                    </span><span class="c1">// Number of child nodes = 1, delete node directly</span>
+<a id="__codelineno-100-28" name="__codelineno-100-28" href="#__codelineno-100-28"></a><span class="w">                    </span><span class="c1">// Number of children = 1: remove node directly</span>
 <a id="__codelineno-100-29" name="__codelineno-100-29" href="#__codelineno-100-29"></a><span class="w">                    </span><span class="nb">Some</span><span class="p">(</span><span class="n">child</span><span class="p">)</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">,</span>
 <a id="__codelineno-100-30" name="__codelineno-100-30" href="#__codelineno-100-30"></a><span class="w">                </span><span class="p">}</span>
 <a id="__codelineno-100-31" name="__codelineno-100-31" href="#__codelineno-100-31"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-100-32" name="__codelineno-100-32" href="#__codelineno-100-32"></a><span class="w">                </span><span class="c1">// Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it</span>
+<a id="__codelineno-100-32" name="__codelineno-100-32" href="#__codelineno-100-32"></a><span class="w">                </span><span class="c1">// Number of children = 2: remove the inorder successor and use its value to replace the current node value</span>
 <a id="__codelineno-100-33" name="__codelineno-100-33" href="#__codelineno-100-33"></a><span class="w">                </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">().</span><span class="n">unwrap</span><span class="p">();</span>
 <a id="__codelineno-100-34" name="__codelineno-100-34" href="#__codelineno-100-34"></a><span class="w">                </span><span class="k">loop</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-100-35" name="__codelineno-100-35" href="#__codelineno-100-35"></a><span class="w">                    </span><span class="kd">let</span><span class="w"> </span><span class="n">temp_left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
@@ -7252,11 +7324,11 @@
 <a id="__codelineno-100-42" name="__codelineno-100-42" href="#__codelineno-100-42"></a><span class="w">                </span><span class="n">node</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">remove_helper</span><span class="p">(</span><span class="n">right</span><span class="p">,</span><span class="w"> </span><span class="n">temp</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-100-43" name="__codelineno-100-43" href="#__codelineno-100-43"></a><span class="w">                </span><span class="n">node</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-100-44" name="__codelineno-100-44" href="#__codelineno-100-44"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-100-45" name="__codelineno-100-45" href="#__codelineno-100-45"></a><span class="w">            </span><span class="bp">Self</span><span class="p">::</span><span class="n">update_height</span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">clone</span><span class="p">()));</span><span class="w"> </span><span class="c1">// Update node height</span>
+<a id="__codelineno-100-45" name="__codelineno-100-45" href="#__codelineno-100-45"></a><span class="w">            </span><span class="bp">Self</span><span class="p">::</span><span class="n">update_height</span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">clone</span><span class="p">()));</span><span class="w"> </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-100-46" name="__codelineno-100-46" href="#__codelineno-100-46"></a>
-<a id="__codelineno-100-47" name="__codelineno-100-47" href="#__codelineno-100-47"></a><span class="w">            </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-100-47" name="__codelineno-100-47" href="#__codelineno-100-47"></a><span class="w">            </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-100-48" name="__codelineno-100-48" href="#__codelineno-100-48"></a><span class="w">            </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">Self</span><span class="p">::</span><span class="n">rotate</span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)).</span><span class="n">unwrap</span><span class="p">();</span>
-<a id="__codelineno-100-49" name="__codelineno-100-49" href="#__codelineno-100-49"></a><span class="w">            </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-100-49" name="__codelineno-100-49" href="#__codelineno-100-49"></a><span class="w">            </span><span class="c1">// Return the root of the subtree</span>
 <a id="__codelineno-100-50" name="__codelineno-100-50" href="#__codelineno-100-50"></a><span class="w">            </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-100-51" name="__codelineno-100-51" href="#__codelineno-100-51"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-100-52" name="__codelineno-100-52" href="#__codelineno-100-52"></a><span class="w">        </span><span class="nb">None</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="nb">None</span><span class="p">,</span>
@@ -7265,19 +7337,19 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-101-1" name="__codelineno-101-1" href="#__codelineno-101-1"></a><span class="cm">/* Remove node */</span>
-<a id="__codelineno-101-2" name="__codelineno-101-2" href="#__codelineno-101-2"></a><span class="c1">// Cannot use remove keyword here due to stdio.h inclusion</span>
+<div class="highlight"><span class="filename">avl_tree.c</span><pre><span></span><code><a id="__codelineno-101-1" name="__codelineno-101-1" href="#__codelineno-101-1"></a><span class="cm">/* Remove a node */</span>
+<a id="__codelineno-101-2" name="__codelineno-101-2" href="#__codelineno-101-2"></a><span class="c1">// The name remove conflicts with the function declared in stdio.h</span>
 <a id="__codelineno-101-3" name="__codelineno-101-3" href="#__codelineno-101-3"></a><span class="kt">void</span><span class="w"> </span><span class="nf">removeItem</span><span class="p">(</span><span class="n">AVLTree</span><span class="w"> </span><span class="o">*</span><span class="n">tree</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-101-4" name="__codelineno-101-4" href="#__codelineno-101-4"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">tree</span><span class="o">-&gt;</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-101-5" name="__codelineno-101-5" href="#__codelineno-101-5"></a><span class="p">}</span>
 <a id="__codelineno-101-6" name="__codelineno-101-6" href="#__codelineno-101-6"></a>
-<a id="__codelineno-101-7" name="__codelineno-101-7" href="#__codelineno-101-7"></a><span class="cm">/* Recursively remove node (helper function) */</span>
+<a id="__codelineno-101-7" name="__codelineno-101-7" href="#__codelineno-101-7"></a><span class="cm">/* Recursively remove a node (helper function) */</span>
 <a id="__codelineno-101-8" name="__codelineno-101-8" href="#__codelineno-101-8"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">removeHelper</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-101-9" name="__codelineno-101-9" href="#__codelineno-101-9"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">child</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">grandChild</span><span class="p">;</span>
 <a id="__codelineno-101-10" name="__codelineno-101-10" href="#__codelineno-101-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-101-11" name="__codelineno-101-11" href="#__codelineno-101-11"></a><span class="w">        </span><span class="k">return</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-101-12" name="__codelineno-101-12" href="#__codelineno-101-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-101-13" name="__codelineno-101-13" href="#__codelineno-101-13"></a><span class="w">    </span><span class="cm">/* 1. Find node and delete */</span>
+<a id="__codelineno-101-13" name="__codelineno-101-13" href="#__codelineno-101-13"></a><span class="w">    </span><span class="cm">/* 1. Find and remove the node */</span>
 <a id="__codelineno-101-14" name="__codelineno-101-14" href="#__codelineno-101-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-101-15" name="__codelineno-101-15" href="#__codelineno-101-15"></a><span class="w">        </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-101-16" name="__codelineno-101-16" href="#__codelineno-101-16"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -7288,15 +7360,15 @@
 <a id="__codelineno-101-21" name="__codelineno-101-21" href="#__codelineno-101-21"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-101-22" name="__codelineno-101-22" href="#__codelineno-101-22"></a><span class="w">                </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-101-23" name="__codelineno-101-23" href="#__codelineno-101-23"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-101-24" name="__codelineno-101-24" href="#__codelineno-101-24"></a><span class="w">            </span><span class="c1">// Number of child nodes = 0, delete node directly and return</span>
+<a id="__codelineno-101-24" name="__codelineno-101-24" href="#__codelineno-101-24"></a><span class="w">            </span><span class="c1">// Number of children = 0: remove node and return</span>
 <a id="__codelineno-101-25" name="__codelineno-101-25" href="#__codelineno-101-25"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">child</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-101-26" name="__codelineno-101-26" href="#__codelineno-101-26"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
 <a id="__codelineno-101-27" name="__codelineno-101-27" href="#__codelineno-101-27"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-101-28" name="__codelineno-101-28" href="#__codelineno-101-28"></a><span class="w">                </span><span class="c1">// Number of child nodes = 1, delete node directly</span>
+<a id="__codelineno-101-28" name="__codelineno-101-28" href="#__codelineno-101-28"></a><span class="w">                </span><span class="c1">// Number of children = 1: remove node directly</span>
 <a id="__codelineno-101-29" name="__codelineno-101-29" href="#__codelineno-101-29"></a><span class="w">                </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-101-30" name="__codelineno-101-30" href="#__codelineno-101-30"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-101-31" name="__codelineno-101-31" href="#__codelineno-101-31"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-101-32" name="__codelineno-101-32" href="#__codelineno-101-32"></a><span class="w">            </span><span class="c1">// Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it</span>
+<a id="__codelineno-101-32" name="__codelineno-101-32" href="#__codelineno-101-32"></a><span class="w">            </span><span class="c1">// Number of children = 2: remove the inorder successor and use its value to replace the current node value</span>
 <a id="__codelineno-101-33" name="__codelineno-101-33" href="#__codelineno-101-33"></a><span class="w">            </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-101-34" name="__codelineno-101-34" href="#__codelineno-101-34"></a><span class="w">            </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">temp</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-101-35" name="__codelineno-101-35" href="#__codelineno-101-35"></a><span class="w">                </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">;</span>
@@ -7306,25 +7378,25 @@
 <a id="__codelineno-101-39" name="__codelineno-101-39" href="#__codelineno-101-39"></a><span class="w">            </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tempVal</span><span class="p">;</span>
 <a id="__codelineno-101-40" name="__codelineno-101-40" href="#__codelineno-101-40"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-101-41" name="__codelineno-101-41" href="#__codelineno-101-41"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-101-42" name="__codelineno-101-42" href="#__codelineno-101-42"></a><span class="w">    </span><span class="c1">// Update node height</span>
+<a id="__codelineno-101-42" name="__codelineno-101-42" href="#__codelineno-101-42"></a><span class="w">    </span><span class="c1">// Update the node height</span>
 <a id="__codelineno-101-43" name="__codelineno-101-43" href="#__codelineno-101-43"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-101-44" name="__codelineno-101-44" href="#__codelineno-101-44"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-101-44" name="__codelineno-101-44" href="#__codelineno-101-44"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-101-45" name="__codelineno-101-45" href="#__codelineno-101-45"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
-<a id="__codelineno-101-46" name="__codelineno-101-46" href="#__codelineno-101-46"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-101-46" name="__codelineno-101-46" href="#__codelineno-101-46"></a><span class="w">    </span><span class="c1">// Return the root of the subtree</span>
 <a id="__codelineno-101-47" name="__codelineno-101-47" href="#__codelineno-101-47"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
 <a id="__codelineno-101-48" name="__codelineno-101-48" href="#__codelineno-101-48"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.kt</span><pre><span></span><code><a id="__codelineno-102-1" name="__codelineno-102-1" href="#__codelineno-102-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">avl_tree.kt</span><pre><span></span><code><a id="__codelineno-102-1" name="__codelineno-102-1" href="#__codelineno-102-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-102-2" name="__codelineno-102-2" href="#__codelineno-102-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-102-3" name="__codelineno-102-3" href="#__codelineno-102-3"></a><span class="w">    </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">)</span>
 <a id="__codelineno-102-4" name="__codelineno-102-4" href="#__codelineno-102-4"></a><span class="p">}</span>
 <a id="__codelineno-102-5" name="__codelineno-102-5" href="#__codelineno-102-5"></a>
-<a id="__codelineno-102-6" name="__codelineno-102-6" href="#__codelineno-102-6"></a><span class="cm">/* Recursively delete node (helper method) */</span>
+<a id="__codelineno-102-6" name="__codelineno-102-6" href="#__codelineno-102-6"></a><span class="cm">/* Remove a node recursively (helper method) */</span>
 <a id="__codelineno-102-7" name="__codelineno-102-7" href="#__codelineno-102-7"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">removeHelper</span><span class="p">(</span><span class="n">n</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="n">TreeNode?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-102-8" name="__codelineno-102-8" href="#__codelineno-102-8"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">?:</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span>
-<a id="__codelineno-102-9" name="__codelineno-102-9" href="#__codelineno-102-9"></a><span class="w">    </span><span class="cm">/* 1. Find node and delete */</span>
+<a id="__codelineno-102-9" name="__codelineno-102-9" href="#__codelineno-102-9"></a><span class="w">    </span><span class="cm">/* 1. Find and remove the node */</span>
 <a id="__codelineno-102-10" name="__codelineno-102-10" href="#__codelineno-102-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">_val</span><span class="p">)</span>
 <a id="__codelineno-102-11" name="__codelineno-102-11" href="#__codelineno-102-11"></a><span class="w">        </span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">removeHelper</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">,</span><span class="w"> </span><span class="n">_val</span><span class="p">)</span>
 <a id="__codelineno-102-12" name="__codelineno-102-12" href="#__codelineno-102-12"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">_val</span><span class="p">)</span>
@@ -7335,14 +7407,14 @@
 <a id="__codelineno-102-17" name="__codelineno-102-17" href="#__codelineno-102-17"></a><span class="w">                </span><span class="n">node</span><span class="p">.</span><span class="na">left</span>
 <a id="__codelineno-102-18" name="__codelineno-102-18" href="#__codelineno-102-18"></a><span class="w">            </span><span class="k">else</span>
 <a id="__codelineno-102-19" name="__codelineno-102-19" href="#__codelineno-102-19"></a><span class="w">                </span><span class="n">node</span><span class="p">.</span><span class="na">right</span>
-<a id="__codelineno-102-20" name="__codelineno-102-20" href="#__codelineno-102-20"></a><span class="w">            </span><span class="c1">// Number of child nodes = 0, delete node directly and return</span>
+<a id="__codelineno-102-20" name="__codelineno-102-20" href="#__codelineno-102-20"></a><span class="w">            </span><span class="c1">// Number of children = 0: remove node and return</span>
 <a id="__codelineno-102-21" name="__codelineno-102-21" href="#__codelineno-102-21"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">child</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-102-22" name="__codelineno-102-22" href="#__codelineno-102-22"></a><span class="w">                </span><span class="k">return</span><span class="w"> </span><span class="kc">null</span>
-<a id="__codelineno-102-23" name="__codelineno-102-23" href="#__codelineno-102-23"></a><span class="w">            </span><span class="c1">// Number of child nodes = 1, delete node directly</span>
+<a id="__codelineno-102-23" name="__codelineno-102-23" href="#__codelineno-102-23"></a><span class="w">            </span><span class="c1">// Number of children = 1: remove node directly</span>
 <a id="__codelineno-102-24" name="__codelineno-102-24" href="#__codelineno-102-24"></a><span class="w">            </span><span class="k">else</span>
 <a id="__codelineno-102-25" name="__codelineno-102-25" href="#__codelineno-102-25"></a><span class="w">                </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span>
 <a id="__codelineno-102-26" name="__codelineno-102-26" href="#__codelineno-102-26"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-102-27" name="__codelineno-102-27" href="#__codelineno-102-27"></a><span class="w">            </span><span class="c1">// Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it</span>
+<a id="__codelineno-102-27" name="__codelineno-102-27" href="#__codelineno-102-27"></a><span class="w">            </span><span class="c1">// Number of children = 2: remove the inorder successor and use its value to replace the current node value</span>
 <a id="__codelineno-102-28" name="__codelineno-102-28" href="#__codelineno-102-28"></a><span class="w">            </span><span class="kd">var</span><span class="w"> </span><span class="nv">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="na">right</span>
 <a id="__codelineno-102-29" name="__codelineno-102-29" href="#__codelineno-102-29"></a><span class="w">            </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">temp</span><span class="o">!!</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-102-30" name="__codelineno-102-30" href="#__codelineno-102-30"></a><span class="w">                </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">.</span><span class="na">left</span>
@@ -7351,24 +7423,24 @@
 <a id="__codelineno-102-33" name="__codelineno-102-33" href="#__codelineno-102-33"></a><span class="w">            </span><span class="n">node</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="p">.</span><span class="na">_val</span>
 <a id="__codelineno-102-34" name="__codelineno-102-34" href="#__codelineno-102-34"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-102-35" name="__codelineno-102-35" href="#__codelineno-102-35"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-102-36" name="__codelineno-102-36" href="#__codelineno-102-36"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="c1">// Update node height</span>
-<a id="__codelineno-102-37" name="__codelineno-102-37" href="#__codelineno-102-37"></a><span class="w">    </span><span class="cm">/* 2. Perform rotation operation to restore balance to this subtree */</span>
+<a id="__codelineno-102-36" name="__codelineno-102-36" href="#__codelineno-102-36"></a><span class="w">    </span><span class="n">updateHeight</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="c1">// Update the node height</span>
+<a id="__codelineno-102-37" name="__codelineno-102-37" href="#__codelineno-102-37"></a><span class="w">    </span><span class="cm">/* 2. Perform rotations to restore balance in this subtree */</span>
 <a id="__codelineno-102-38" name="__codelineno-102-38" href="#__codelineno-102-38"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-102-39" name="__codelineno-102-39" href="#__codelineno-102-39"></a><span class="w">    </span><span class="c1">// Return root node of subtree</span>
+<a id="__codelineno-102-39" name="__codelineno-102-39" href="#__codelineno-102-39"></a><span class="w">    </span><span class="c1">// Return the subtree root</span>
 <a id="__codelineno-102-40" name="__codelineno-102-40" href="#__codelineno-102-40"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">node</span>
 <a id="__codelineno-102-41" name="__codelineno-102-41" href="#__codelineno-102-41"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">avl_tree.rb</span><pre><span></span><code><a id="__codelineno-103-1" name="__codelineno-103-1" href="#__codelineno-103-1"></a><span class="c1">### Delete node ###</span>
+<div class="highlight"><span class="filename">avl_tree.rb</span><pre><span></span><code><a id="__codelineno-103-1" name="__codelineno-103-1" href="#__codelineno-103-1"></a><span class="c1">### Remove a node ###</span>
 <a id="__codelineno-103-2" name="__codelineno-103-2" href="#__codelineno-103-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-103-3" name="__codelineno-103-3" href="#__codelineno-103-3"></a><span class="w">  </span><span class="vi">@root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">remove_helper</span><span class="p">(</span><span class="vi">@root</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-103-4" name="__codelineno-103-4" href="#__codelineno-103-4"></a><span class="k">end</span>
 <a id="__codelineno-103-5" name="__codelineno-103-5" href="#__codelineno-103-5"></a>
-<a id="__codelineno-103-6" name="__codelineno-103-6" href="#__codelineno-103-6"></a><span class="c1">### Recursively delete node (helper method) ###</span>
+<a id="__codelineno-103-6" name="__codelineno-103-6" href="#__codelineno-103-6"></a><span class="c1">### Remove a node recursively (helper method)###</span>
 <a id="__codelineno-103-7" name="__codelineno-103-7" href="#__codelineno-103-7"></a><span class="k">def</span><span class="w"> </span><span class="nf">remove_helper</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-103-8" name="__codelineno-103-8" href="#__codelineno-103-8"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">nil?</span>
-<a id="__codelineno-103-9" name="__codelineno-103-9" href="#__codelineno-103-9"></a><span class="w">  </span><span class="c1"># 1. Find node and delete</span>
+<a id="__codelineno-103-9" name="__codelineno-103-9" href="#__codelineno-103-9"></a><span class="w">  </span><span class="c1"># 1. Find and remove the node</span>
 <a id="__codelineno-103-10" name="__codelineno-103-10" href="#__codelineno-103-10"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-103-11" name="__codelineno-103-11" href="#__codelineno-103-11"></a><span class="w">    </span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">remove_helper</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-103-12" name="__codelineno-103-12" href="#__codelineno-103-12"></a><span class="w">  </span><span class="k">elsif</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">val</span>
@@ -7376,12 +7448,12 @@
 <a id="__codelineno-103-14" name="__codelineno-103-14" href="#__codelineno-103-14"></a><span class="w">  </span><span class="k">else</span>
 <a id="__codelineno-103-15" name="__codelineno-103-15" href="#__codelineno-103-15"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="o">.</span><span class="n">nil?</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-103-16" name="__codelineno-103-16" href="#__codelineno-103-16"></a><span class="w">      </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">right</span>
-<a id="__codelineno-103-17" name="__codelineno-103-17" href="#__codelineno-103-17"></a><span class="w">      </span><span class="c1"># Number of child nodes = 0, delete node directly and return</span>
+<a id="__codelineno-103-17" name="__codelineno-103-17" href="#__codelineno-103-17"></a><span class="w">      </span><span class="c1"># Number of children = 0: remove node and return</span>
 <a id="__codelineno-103-18" name="__codelineno-103-18" href="#__codelineno-103-18"></a><span class="w">      </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">child</span><span class="o">.</span><span class="n">nil?</span>
-<a id="__codelineno-103-19" name="__codelineno-103-19" href="#__codelineno-103-19"></a><span class="w">      </span><span class="c1"># Number of child nodes = 1, delete node directly</span>
+<a id="__codelineno-103-19" name="__codelineno-103-19" href="#__codelineno-103-19"></a><span class="w">      </span><span class="c1"># Number of children = 1: remove node directly</span>
 <a id="__codelineno-103-20" name="__codelineno-103-20" href="#__codelineno-103-20"></a><span class="w">      </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span>
 <a id="__codelineno-103-21" name="__codelineno-103-21" href="#__codelineno-103-21"></a><span class="w">    </span><span class="k">else</span>
-<a id="__codelineno-103-22" name="__codelineno-103-22" href="#__codelineno-103-22"></a><span class="w">      </span><span class="c1"># Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it</span>
+<a id="__codelineno-103-22" name="__codelineno-103-22" href="#__codelineno-103-22"></a><span class="w">      </span><span class="c1"># Number of children = 2: remove the inorder successor and use its value to replace the current node value</span>
 <a id="__codelineno-103-23" name="__codelineno-103-23" href="#__codelineno-103-23"></a><span class="w">      </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">right</span>
 <a id="__codelineno-103-24" name="__codelineno-103-24" href="#__codelineno-103-24"></a><span class="w">      </span><span class="k">while</span><span class="w"> </span><span class="o">!</span><span class="n">temp</span><span class="o">.</span><span class="n">left</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-103-25" name="__codelineno-103-25" href="#__codelineno-103-25"></a><span class="w">        </span><span class="n">temp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="o">.</span><span class="n">left</span>
@@ -7390,22 +7462,22 @@
 <a id="__codelineno-103-28" name="__codelineno-103-28" href="#__codelineno-103-28"></a><span class="w">      </span><span class="n">node</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">temp</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-103-29" name="__codelineno-103-29" href="#__codelineno-103-29"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-103-30" name="__codelineno-103-30" href="#__codelineno-103-30"></a><span class="w">  </span><span class="k">end</span>
-<a id="__codelineno-103-31" name="__codelineno-103-31" href="#__codelineno-103-31"></a><span class="w">  </span><span class="c1"># Update node height</span>
+<a id="__codelineno-103-31" name="__codelineno-103-31" href="#__codelineno-103-31"></a><span class="w">  </span><span class="c1"># Update the node height</span>
 <a id="__codelineno-103-32" name="__codelineno-103-32" href="#__codelineno-103-32"></a><span class="w">  </span><span class="n">update_height</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
-<a id="__codelineno-103-33" name="__codelineno-103-33" href="#__codelineno-103-33"></a><span class="w">  </span><span class="c1"># 2. Perform rotation operation to restore balance to this subtree</span>
+<a id="__codelineno-103-33" name="__codelineno-103-33" href="#__codelineno-103-33"></a><span class="w">  </span><span class="c1"># 2. Perform rotations to restore balance in this subtree</span>
 <a id="__codelineno-103-34" name="__codelineno-103-34" href="#__codelineno-103-34"></a><span class="w">  </span><span class="n">rotate</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
 <a id="__codelineno-103-35" name="__codelineno-103-35" href="#__codelineno-103-35"></a><span class="k">end</span>
 </code></pre></div>
 </div>
 </div>
 </div>
-<h3 id="3-node-search">3. &nbsp; Node Search<a class="headerlink" href="#3-node-search" title="Permanent link">&para;</a></h3>
-<p>The node search operation in AVL trees is consistent with that in binary search trees, and will not be elaborated here.</p>
-<h2 id="754-typical-applications-of-avl-trees">7.5.4 &nbsp; Typical Applications of AVL Trees<a class="headerlink" href="#754-typical-applications-of-avl-trees" title="Permanent link">&para;</a></h2>
+<h3 id="3-finding-a-node">3. &nbsp; Finding a node<a class="headerlink" href="#3-finding-a-node" title="Permanent link">&para;</a></h3>
+<p>Lookup is identical to binary-search-tree lookup, so we will not repeat it here.</p>
+<h2 id="754-typical-applications">7.5.4 &nbsp; Typical applications<a class="headerlink" href="#754-typical-applications" title="Permanent link">&para;</a></h2>
 <ul>
-<li>Organizing and storing large-scale data, suitable for scenarios with high-frequency searches and low-frequency insertions and deletions.</li>
-<li>Used to build index systems in databases.</li>
-<li>Red-black trees are also a common type of balanced binary search tree. Compared to AVL trees, red-black trees have more relaxed balance conditions, require fewer rotation operations for node insertion and deletion, and have higher average efficiency for node addition and deletion operations.</li>
+<li>Organizing and storing large datasets, particularly when lookups are frequent and insertions and deletions are less frequent.</li>
+<li>Building database indexing systems.</li>
+<li>Red-black trees are another common kind of balanced binary search tree. Their balance requirements are looser than those of AVL trees, so insertion and deletion need fewer rotations and are more efficient on average.</li>
 </ul>
 
 <!-- Source file information -->
@@ -7429,7 +7501,7 @@ aria-label="Footer"
   <a
     href="../binary_search_tree/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 7.4 Binary Search Tree"
+    aria-label="Previous: 7.4   Binary search trees"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -7441,7 +7513,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        7.4 Binary Search Tree
+        7.4 &nbsp; Binary search trees
       </div>
     </div>
   </a>
@@ -7453,7 +7525,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 7.6 Summary"
+    aria-label="Next: 7.6   Summary"
     rel="next"
   >
     <div class="md-footer__title">
@@ -7461,7 +7533,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        7.6 Summary
+        7.6 &nbsp; Summary
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -7478,7 +7550,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_tree/binary_search_tree/index.html b/en/chapter_tree/binary_search_tree/index.html
index bfddeae96..eaee6c9a9 100644
--- a/en/chapter_tree/binary_search_tree/index.html
+++ b/en/chapter_tree/binary_search_tree/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>7.4 Binary Search Tree - Hello Algo</title>
+        <title>7.4 Binary search trees - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
     <div data-md-component="skip">
 
 
-        <a href="#74-binary-search-tree" class="md-skip">
+        <a href="#74-binary-search-trees" class="md-skip">
           Skip to content
         </a>
 
@@ -119,7 +119,7 @@
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             d="M480 32c0-12.9-7.8-24.6-19.8-29.6s-25.7-2.2-34.9 6.9L381.7 53c-48 48-113.1 75-181 75H192 160 64c-35.3 0-64 28.7-64 64v96c0 35.3 28.7 64 64 64l0 128c0 17.7 14.3 32 32 32h64c17.7 0 32-14.3 32-32V352l8.7 0c67.9 0 133 27 181 75l43.6 43.6c9.2 9.2 22.9 11.9 34.9 6.9s19.8-16.6 19.8-29.6V300.4c18.6-8.8 32-32.5 32-60.4s-13.4-51.6-32-60.4V32zm-64 76.7V240 371.3C357.2 317.8 280.5 288 200.7 288H192V192h8.7c79.8 0 156.5-29.8 215.3-83.3z" />
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              7.4 &nbsp; Binary Search Tree
+              7.4 &nbsp; Binary search trees
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="teal"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
 
-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
   </span>
@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
@@ -1049,7 +1049,7 @@
   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1898,7 +1898,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1919,7 +1919,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1940,7 +1940,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1967,7 +1967,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1994,7 +1994,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2030,7 +2030,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2047,7 +2047,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2073,22 +2073,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#741-operations-on-a-binary-search-tree" class="md-nav__link">
+  <a href="#741-binary-search-tree-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.4.1 &nbsp; Operations on a Binary Search Tree
+        7.4.1 &nbsp; Binary search tree operations
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.4.1 Operations on a Binary Search Tree">
+    <nav class="md-nav" aria-label="7.4.1 Binary search tree operations">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-searching-for-a-node" class="md-nav__link">
+  <a href="#1-finding-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Searching for a Node
+        1. &nbsp; Finding a node
       </span>
     </span>
   </a>
@@ -2099,7 +2099,7 @@
   <a href="#2-inserting-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Inserting a Node
+        2. &nbsp; Inserting a node
       </span>
     </span>
   </a>
@@ -2107,10 +2107,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-removing-a-node" class="md-nav__link">
+  <a href="#3-deleting-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Removing a Node
+        3. &nbsp; Deleting a node
       </span>
     </span>
   </a>
@@ -2118,10 +2118,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#4-inorder-traversal-is-ordered" class="md-nav__link">
+  <a href="#4-inorder-traversal-produces-sorted-data" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4. &nbsp; Inorder Traversal Is Ordered
+        4. &nbsp; Inorder traversal produces sorted data
       </span>
     </span>
   </a>
@@ -2134,10 +2134,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#742-efficiency-of-binary-search-trees" class="md-nav__link">
+  <a href="#742-efficiency" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.4.2 &nbsp; Efficiency of Binary Search Trees
+        7.4.2 &nbsp; Efficiency
       </span>
     </span>
   </a>
@@ -2145,10 +2145,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#743-common-applications-of-binary-search-trees" class="md-nav__link">
+  <a href="#743-common-applications" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.4.3 &nbsp; Common Applications of Binary Search Trees
+        7.4.3 &nbsp; Common applications
       </span>
     </span>
   </a>
@@ -2178,7 +2178,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2205,7 +2205,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2232,7 +2232,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2296,7 +2296,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2317,7 +2317,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2338,7 +2338,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2365,7 +2365,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2392,7 +2392,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2419,7 +2419,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2446,7 +2446,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2510,7 +2510,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2531,7 +2531,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2552,7 +2552,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2579,7 +2579,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2606,7 +2606,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2633,7 +2633,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2660,7 +2660,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2728,7 +2728,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2749,7 +2749,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2770,7 +2770,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2797,7 +2797,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2824,7 +2824,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2851,7 +2851,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2878,7 +2878,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2905,7 +2905,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2932,7 +2932,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -3010,7 +3010,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3031,7 +3031,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3052,7 +3052,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3079,7 +3079,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3106,7 +3106,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3133,7 +3133,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3160,7 +3160,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3187,7 +3187,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3214,7 +3214,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3241,7 +3241,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3268,7 +3268,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3295,7 +3295,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3322,7 +3322,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3349,7 +3349,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3415,7 +3415,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3436,7 +3436,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3457,7 +3457,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3484,7 +3484,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3511,7 +3511,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3538,7 +3538,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3565,7 +3565,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3592,7 +3592,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3658,7 +3658,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3700,7 +3700,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3754,7 +3754,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3781,7 +3781,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3808,7 +3808,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3835,7 +3835,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3905,7 +3905,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3926,7 +3926,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3947,7 +3947,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3974,7 +3974,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -4001,7 +4001,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4028,7 +4028,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4055,7 +4055,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4082,7 +4082,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4109,7 +4109,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4136,7 +4136,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4202,7 +4202,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4223,7 +4223,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4244,7 +4244,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4271,7 +4271,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4325,7 +4325,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4439,7 +4439,7 @@
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4481,7 +4481,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4508,7 +4508,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4535,7 +4535,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4619,6 +4619,78 @@
 
 
 
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@@ -4660,22 +4732,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#741-operations-on-a-binary-search-tree" class="md-nav__link">
+  <a href="#741-binary-search-tree-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.4.1 &nbsp; Operations on a Binary Search Tree
+        7.4.1 &nbsp; Binary search tree operations
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.4.1 Operations on a Binary Search Tree">
+    <nav class="md-nav" aria-label="7.4.1 Binary search tree operations">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-searching-for-a-node" class="md-nav__link">
+  <a href="#1-finding-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Searching for a Node
+        1. &nbsp; Finding a node
       </span>
     </span>
   </a>
@@ -4686,7 +4758,7 @@
   <a href="#2-inserting-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Inserting a Node
+        2. &nbsp; Inserting a node
       </span>
     </span>
   </a>
@@ -4694,10 +4766,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-removing-a-node" class="md-nav__link">
+  <a href="#3-deleting-a-node" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Removing a Node
+        3. &nbsp; Deleting a node
       </span>
     </span>
   </a>
@@ -4705,10 +4777,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#4-inorder-traversal-is-ordered" class="md-nav__link">
+  <a href="#4-inorder-traversal-produces-sorted-data" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4. &nbsp; Inorder Traversal Is Ordered
+        4. &nbsp; Inorder traversal produces sorted data
       </span>
     </span>
   </a>
@@ -4721,10 +4793,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#742-efficiency-of-binary-search-trees" class="md-nav__link">
+  <a href="#742-efficiency" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.4.2 &nbsp; Efficiency of Binary Search Trees
+        7.4.2 &nbsp; Efficiency
       </span>
     </span>
   </a>
@@ -4732,10 +4804,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#743-common-applications-of-binary-search-trees" class="md-nav__link">
+  <a href="#743-common-applications" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.4.3 &nbsp; Common Applications of Binary Search Trees
+        7.4.3 &nbsp; Common applications
       </span>
     </span>
   </a>
@@ -4779,28 +4851,28 @@
 
 
 <!-- Page content -->
-<h1 id="74-binary-search-tree">7.4 &nbsp; Binary Search Tree<a class="headerlink" href="#74-binary-search-tree" title="Permanent link">&para;</a></h1>
-<p>As shown in Figure 7-16, a <u>binary search tree</u> satisfies the following conditions.</p>
+<h1 id="74-binary-search-trees">7.4 &nbsp; Binary search trees<a class="headerlink" href="#74-binary-search-trees" title="Permanent link">&para;</a></h1>
+<p>A <u>binary search tree</u> satisfies the following conditions, as shown below.</p>
 <ol>
-<li>For the root node, the value of all nodes in the left subtree <span class="arithmatex">\(&lt;\)</span> the value of the root node <span class="arithmatex">\(&lt;\)</span> the value of all nodes in the right subtree.</li>
-<li>The left and right subtrees of any node are also binary search trees, i.e., they satisfy condition <code>1.</code> as well.</li>
+<li>Every value in the left subtree is <span class="arithmatex">\(&lt;\)</span> the root value, which is <span class="arithmatex">\(&lt;\)</span> every value in the right subtree.</li>
+<li>The left and right subtrees of every node are also binary search trees and satisfy condition <code>1.</code>.</li>
 </ol>
-<p><img alt="Binary search tree" class="animation-figure" src="../binary_search_tree.assets/binary_search_tree.png" /></p>
-<p align="center"> Figure 7-16 &nbsp; Binary search tree </p>
+<p><img alt="A binary search tree" class="animation-figure" src="../binary_search_tree.assets/binary_search_tree.png" /></p>
+<p align="center"> Figure 7-16 &nbsp; A binary search tree </p>
 
-<h2 id="741-operations-on-a-binary-search-tree">7.4.1 &nbsp; Operations on a Binary Search Tree<a class="headerlink" href="#741-operations-on-a-binary-search-tree" title="Permanent link">&para;</a></h2>
-<p>We encapsulate the binary search tree as a class <code>BinarySearchTree</code> and declare a member variable <code>root</code> pointing to the tree's root node.</p>
-<h3 id="1-searching-for-a-node">1. &nbsp; Searching for a Node<a class="headerlink" href="#1-searching-for-a-node" title="Permanent link">&para;</a></h3>
-<p>Given a target node value <code>num</code>, we can search according to the properties of the binary search tree. As shown in Figure 7-17, we declare a node <code>cur</code> and start from the binary search tree's root node <code>root</code>, looping to compare <code>cur.val</code> with <code>num</code>.</p>
+<h2 id="741-binary-search-tree-operations">7.4.1 &nbsp; Binary search tree operations<a class="headerlink" href="#741-binary-search-tree-operations" title="Permanent link">&para;</a></h2>
+<p>We encapsulate the tree in a <code>BinarySearchTree</code> class with a member variable <code>root</code> pointing to its root node.</p>
+<h3 id="1-finding-a-node">1. &nbsp; Finding a node<a class="headerlink" href="#1-finding-a-node" title="Permanent link">&para;</a></h3>
+<p>To find a target value <code>num</code>, use the tree's ordering property. As shown below, initialize a node reference <code>cur</code> at <code>root</code> and repeatedly compare <code>cur.val</code> with <code>num</code>.</p>
 <ul>
-<li>If <code>cur.val &lt; num</code>, it means the target node is in <code>cur</code>'s right subtree, thus execute <code>cur = cur.right</code>.</li>
-<li>If <code>cur.val &gt; num</code>, it means the target node is in <code>cur</code>'s left subtree, thus execute <code>cur = cur.left</code>.</li>
-<li>If <code>cur.val = num</code>, it means the target node is found, exit the loop, and return the node.</li>
+<li>If <code>cur.val &lt; num</code>, the target lies in the right subtree, so set <code>cur = cur.right</code>.</li>
+<li>If <code>cur.val &gt; num</code>, the target lies in the left subtree, so set <code>cur = cur.left</code>.</li>
+<li>If <code>cur.val = num</code>, the target has been found; exit the loop and return the node.</li>
 </ul>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:4"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">&lt;1&gt;</label><label for="__tabbed_1_2">&lt;2&gt;</label><label for="__tabbed_1_3">&lt;3&gt;</label><label for="__tabbed_1_4">&lt;4&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Example of searching for a node in a binary search tree" class="animation-figure" src="../binary_search_tree.assets/bst_search_step1.png" /></p>
+<p><img alt="Finding a node in a binary search tree" class="animation-figure" src="../binary_search_tree.assets/bst_search_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="bst_search_step2" class="animation-figure" src="../binary_search_tree.assets/bst_search_step2.png" /></p>
@@ -4813,273 +4885,273 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 7-17 &nbsp; Example of searching for a node in a binary search tree </p>
+<p align="center"> Figure 7-17 &nbsp; Finding a node in a binary search tree </p>
 
-<p>The search operation in a binary search tree follows the same principle as binary search: each round rules out half of the remaining cases. The number of loop iterations is at most the height of the tree. When the tree is balanced, the search takes <span class="arithmatex">\(O(\log n)\)</span> time. The example code is as follows:</p>
+<p>Searching a binary search tree follows the same principle as binary search: each round eliminates half the possibilities. The number of iterations is at most the tree's height, giving <span class="arithmatex">\(O(\log n)\)</span> time when the tree is balanced. An implementation follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_tree.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">search</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Search node"""</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Find a node"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="n">cur</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_root</span>
-<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="c1"># Loop search, exit after passing leaf node</span>
+<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="c1"># Search in a loop, stopping if cur becomes None</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="k">while</span> <span class="n">cur</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="c1"># Target node is in cur's right subtree</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="c1"># If present, the target node is in the right subtree of cur</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>        <span class="k">if</span> <span class="n">cur</span><span class="o">.</span><span class="n">val</span> <span class="o">&lt;</span> <span class="n">num</span><span class="p">:</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>            <span class="n">cur</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">right</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="c1"># Target node is in cur's left subtree</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="c1"># If present, the target node is in the left subtree of cur</span>
 <a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="k">elif</span> <span class="n">cur</span><span class="o">.</span><span class="n">val</span> <span class="o">&gt;</span> <span class="n">num</span><span class="p">:</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>            <span class="n">cur</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">left</span>
-<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="c1"># Found target node, exit loop</span>
+<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>        <span class="c1"># The target node was found; exit the loop</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="k">else</span><span class="p">:</span>
 <a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>            <span class="k">break</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="k">return</span> <span class="n">cur</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Search node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Find a node */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">search</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">;</span>
-<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="c1">// Continue searching while the current node pointer is not null</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">        </span><span class="c1">// Target node is in cur's right subtree</span>
+<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">        </span><span class="c1">// The target node lies in the right subtree of cur</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">        </span><span class="c1">// Target node is in cur's left subtree</span>
+<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">        </span><span class="c1">// The target node lies in the left subtree of cur</span>
 <a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">;</span>
-<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="c1">// Found target node, exit loop</span>
+<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="c1">// The target node was found; exit the loop</span>
 <a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">        </span><span class="k">else</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">    </span><span class="c1">// Return target node</span>
+<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">    </span><span class="c1">// Return the target node, or nullptr if it was not found</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
 <a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Search node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Find a node */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="nf">search</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">;</span>
-<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="c1">// Search in a loop until reaching a null reference beyond a leaf</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">        </span><span class="c1">// Target node is in cur's right subtree</span>
+<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">        </span><span class="c1">// The target node lies in the right subtree of cur</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span><span class="p">;</span>
-<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">        </span><span class="c1">// Target node is in cur's left subtree</span>
+<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">        </span><span class="c1">// The target node lies in the left subtree of cur</span>
 <a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">left</span><span class="p">;</span>
-<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="c1">// Found target node, exit loop</span>
+<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">        </span><span class="c1">// The target node was found; exit the loop</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="k">else</span>
 <a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">    </span><span class="c1">// Return target node</span>
+<a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">    </span><span class="c1">// Return the target node</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
 <a id="__codelineno-2-18" name="__codelineno-2-18" href="#__codelineno-2-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Search node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Find a node */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">Search</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">;</span>
-<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="c1">// Search in a loop, stopping if a null reference is reached</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">        </span><span class="c1">// Target node is in cur's right subtree</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">        </span><span class="c1">// The target node lies in the right subtree of cur</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">            </span><span class="n">cur</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">        </span><span class="c1">// Target node is in cur's left subtree</span>
+<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">        </span><span class="c1">// The target node lies in the left subtree of cur</span>
 <a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="nf">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">left</span><span class="p">;</span>
-<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="c1">// Found target node, exit loop</span>
+<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">        </span><span class="c1">// The target node was found; exit the loop</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="k">else</span>
 <a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">    </span><span class="c1">// Return target node</span>
+<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">    </span><span class="c1">// Return the target node</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
 <a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Search node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Find a node */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">bst</span><span class="w"> </span><span class="o">*</span><span class="nx">binarySearchTree</span><span class="p">)</span><span class="w"> </span><span class="nx">search</span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="nx">node</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">bst</span><span class="p">.</span><span class="nx">root</span>
-<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="c1">// Continue searching while the current node is not nil</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Val</span><span class="p">.(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">            </span><span class="c1">// Target node is in cur's right subtree</span>
+<a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">            </span><span class="c1">// If the target node exists, it lies in the right subtree of node</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">            </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Val</span><span class="p">.(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">            </span><span class="c1">// Target node is in cur's left subtree</span>
+<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">            </span><span class="c1">// If the target node exists, it lies in the left subtree of node</span>
 <a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">            </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">            </span><span class="c1">// Found target node, exit loop</span>
+<a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">            </span><span class="c1">// The target node was found; exit the loop</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="c1">// Return target node</span>
+<a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">    </span><span class="c1">// Return the target node, or nil if it was not found</span>
 <a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">node</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Search node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Find a node */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">search</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">cur</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">root</span>
-<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="c1">// Search in a loop, stopping if cur becomes nil</span>
 <a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">        </span><span class="c1">// Target node is in cur's right subtree</span>
+<a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">        </span><span class="c1">// If present, the target node lies in the right subtree of cur</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">cur</span><span class="p">?.</span><span class="kr">right</span>
 <a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="c1">// Target node is in cur's left subtree</span>
+<a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="c1">// If present, the target node lies in the left subtree of cur</span>
 <a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">cur</span><span class="p">?.</span><span class="kr">left</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="c1">// Found target node, exit loop</span>
+<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">        </span><span class="c1">// The target node was found; exit the loop</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-18" name="__codelineno-5-18" href="#__codelineno-5-18"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">    </span><span class="c1">// Return target node</span>
+<a id="__codelineno-5-19" name="__codelineno-5-19" href="#__codelineno-5-19"></a><span class="w">    </span><span class="c1">// Return the target node, or nil if it was not found</span>
 <a id="__codelineno-5-20" name="__codelineno-5-20" href="#__codelineno-5-20"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">cur</span>
 <a id="__codelineno-5-21" name="__codelineno-5-21" href="#__codelineno-5-21"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Search node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Find a node */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="nx">search</span><span class="p">(</span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="p">;</span>
-<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="c1">// Search in a loop until reaching a null reference beyond a leaf</span>
 <a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">        </span><span class="c1">// Target node is in cur's right subtree</span>
+<a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">        </span><span class="c1">// The target node lies in the right subtree of cur</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
-<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="c1">// Target node is in cur's left subtree</span>
+<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="c1">// The target node lies in the left subtree of cur</span>
 <a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">left</span><span class="p">;</span>
-<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="c1">// Found target node, exit loop</span>
+<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="c1">// The target node was found; exit the loop</span>
 <a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="c1">// Return target node</span>
+<a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="c1">// Return the target node</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">cur</span><span class="p">;</span>
 <a id="__codelineno-6-15" name="__codelineno-6-15" href="#__codelineno-6-15"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Search node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Find a node */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="nx">search</span><span class="p">(</span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="p">;</span>
-<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="c1">// Search in a loop, stopping if a null reference is reached</span>
 <a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">        </span><span class="c1">// Target node is in cur's right subtree</span>
+<a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">        </span><span class="c1">// The target node lies in the right subtree of cur</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
-<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">        </span><span class="c1">// Target node is in cur's left subtree</span>
+<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">        </span><span class="c1">// The target node lies in the left subtree of cur</span>
 <a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">left</span><span class="p">;</span>
-<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="c1">// Found target node, exit loop</span>
+<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="c1">// The target node was found; exit the loop</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="c1">// Return target node</span>
+<a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">    </span><span class="c1">// Return the target node</span>
 <a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">cur</span><span class="p">;</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Search node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Find a node */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">search</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_root</span><span class="p">;</span>
-<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="c1">// Search in a loop, stopping if a null reference is reached</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">    </span><span class="c1">// Target node is in cur's right subtree</span>
+<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">    </span><span class="c1">// The target node lies in the right subtree of cur</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">      </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="c1">// Target node is in cur's left subtree</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="c1">// The target node lies in the left subtree of cur</span>
 <a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span>
 <a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">      </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">left</span><span class="p">;</span>
-<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="c1">// Found target node, exit loop</span>
+<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="c1">// The target node was found; exit the loop</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">    </span><span class="k">else</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">      </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">  </span><span class="c1">// Return target node</span>
+<a id="__codelineno-8-16" name="__codelineno-8-16" href="#__codelineno-8-16"></a><span class="w">  </span><span class="c1">// Return the target node</span>
 <a id="__codelineno-8-17" name="__codelineno-8-17" href="#__codelineno-8-17"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
 <a id="__codelineno-8-18" name="__codelineno-8-18" href="#__codelineno-8-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Search node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Find a node */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">search</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">OptionTreeNodeRc</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
-<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="c1">// Search while cur refers to a node</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">clone</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">        </span><span class="k">match</span><span class="w"> </span><span class="n">num</span><span class="p">.</span><span class="n">cmp</span><span class="p">(</span><span class="o">&amp;</span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">            </span><span class="c1">// Target node is in cur's right subtree</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">            </span><span class="c1">// The target node lies in the right subtree of cur</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">            </span><span class="n">Ordering</span><span class="p">::</span><span class="n">Greater</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">(),</span>
-<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">            </span><span class="c1">// Target node is in cur's left subtree</span>
+<a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">            </span><span class="c1">// The target node lies in the left subtree of cur</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">            </span><span class="n">Ordering</span><span class="p">::</span><span class="n">Less</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">(),</span>
-<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">            </span><span class="c1">// Found target node, exit loop</span>
+<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">            </span><span class="c1">// The target node was found; exit the loop</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">            </span><span class="n">Ordering</span><span class="p">::</span><span class="n">Equal</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="k">break</span><span class="p">,</span>
 <a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a>
-<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">    </span><span class="c1">// Return target node</span>
+<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">    </span><span class="c1">// Return the matching node, or None if no match was found</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">    </span><span class="n">cur</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Search node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.c</span><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Find a node */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="nf">search</span><span class="p">(</span><span class="n">BinarySearchTree</span><span class="w"> </span><span class="o">*</span><span class="n">bst</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">bst</span><span class="o">-&gt;</span><span class="n">root</span><span class="p">;</span>
-<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="c1">// Search in a loop until reaching a null reference beyond a leaf</span>
 <a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">            </span><span class="c1">// Target node is in cur's right subtree</span>
+<a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="w">            </span><span class="c1">// The target node lies in the right subtree of cur</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">            </span><span class="c1">// Target node is in cur's left subtree</span>
+<a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">            </span><span class="c1">// The target node lies in the left subtree of cur</span>
 <a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">            </span><span class="c1">// Found target node, exit loop</span>
+<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">            </span><span class="c1">// The target node was found; exit the loop</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">    </span><span class="c1">// Return target node</span>
+<a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">    </span><span class="c1">// Return the target node</span>
 <a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
 <a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Search node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Find a node */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">search</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">):</span><span class="w"> </span><span class="n">TreeNode?</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span>
-<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="c1">// Continue searching while cur is not null</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">        </span><span class="c1">// Target node is in cur's right subtree</span>
+<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">        </span><span class="c1">// The target node lies in the right subtree of cur</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">        </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">            </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">        </span><span class="c1">// Target node is in cur's left subtree</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">        </span><span class="c1">// The target node lies in the left subtree of cur</span>
 <a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">            </span><span class="n">cur</span><span class="p">.</span><span class="na">left</span>
-<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="c1">// Found target node, exit loop</span>
+<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">        </span><span class="c1">// The target node was found; exit the loop</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="k">else</span>
 <a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="c1">// Return target node</span>
+<a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="c1">// Return the target node</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">cur</span>
 <a id="__codelineno-11-18" name="__codelineno-11-18" href="#__codelineno-11-18"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Search node ###</span>
+<div class="highlight"><span class="filename">binary_search_tree.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Find a node ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">search</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@root</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Loop search, exit after passing leaf node</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Search in a loop, stopping if cur becomes nil</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="o">!</span><span class="n">cur</span><span class="o">.</span><span class="n">nil?</span>
-<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">    </span><span class="c1"># Target node is in cur's right subtree</span>
+<a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">    </span><span class="c1"># The target node lies in the right subtree of cur</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">      </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">right</span>
-<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="c1"># Target node is in cur's left subtree</span>
+<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="c1"># The target node lies in the left subtree of cur</span>
 <a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="k">elsif</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">      </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">left</span>
-<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="c1"># Found target node, exit loop</span>
+<a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">    </span><span class="c1"># The target node was found; exit the loop</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="w">    </span><span class="k">else</span>
 <a id="__codelineno-12-15" name="__codelineno-12-15" href="#__codelineno-12-15"></a><span class="w">      </span><span class="k">break</span>
 <a id="__codelineno-12-16" name="__codelineno-12-16" href="#__codelineno-12-16"></a><span class="w">    </span><span class="k">end</span>
@@ -5091,43 +5163,48 @@
 </div>
 </div>
 </div>
-<h3 id="2-inserting-a-node">2. &nbsp; Inserting a Node<a class="headerlink" href="#2-inserting-a-node" title="Permanent link">&para;</a></h3>
-<p>Given an element <code>num</code> to be inserted, in order to maintain the property of the binary search tree "left subtree &lt; root node &lt; right subtree," the insertion process is as shown in Figure 7-18.</p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%29%3A%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%20%20%20%20%20%20%20%20self.left%20%3D%20None%0A%20%20%20%20%20%20%20%20self.right%20%3D%20None%0A%0A%0Aclass%20BinarySearchTree%3A%0A%20%20%20%20%22%22%22Binary%20search%20tree%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Initialize%20an%20empty%20tree%0A%20%20%20%20%20%20%20%20self._root%20%3D%20None%0A%0A%20%20%20%20def%20search%28self,%20num%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%20%20%20%20%22%22%22Find%20a%20node%22%22%22%0A%20%20%20%20%20%20%20%20cur%20%3D%20self._root%0A%20%20%20%20%20%20%20%20%23%20Search%20in%20a%20loop,%20stopping%20if%20cur%20becomes%20None%0A%20%20%20%20%20%20%20%20while%20cur%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20present,%20the%20target%20node%20is%20in%20the%20right%20subtree%20of%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.right%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20present,%20the%20target%20node%20is%20in%20the%20left%20subtree%20of%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20elif%20cur.val%20%3E%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.left%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20The%20target%20node%20was%20found%3B%20exit%20the%20loop%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20return%20cur%0A%0A%20%20%20%20def%20insert%28self,%20num%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Insert%20a%20node%22%22%22%0A%20%20%20%20%20%20%20%20%23%20If%20the%20tree%20is%20empty,%20initialize%20the%20root%20node%0A%20%20%20%20%20%20%20%20if%20self._root%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self._root%20%3D%20TreeNode%28num%29%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%23%20Search%20in%20a%20loop,%20stopping%20if%20cur%20becomes%20None%0A%20%20%20%20%20%20%20%20cur,%20pre%20%3D%20self._root,%20None%0A%20%20%20%20%20%20%20%20while%20cur%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Return%20immediately%20if%20a%20duplicate%20node%20is%20found%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3D%3D%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%20%20%20%20pre%20%3D%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20The%20insertion%20position%20lies%20in%20the%20right%20subtree%20of%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.right%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20The%20insertion%20position%20lies%20in%20the%20left%20subtree%20of%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.left%0A%20%20%20%20%20%20%20%20%23%20Insert%20a%20node%0A%20%20%20%20%20%20%20%20node%20%3D%20TreeNode%28num%29%0A%20%20%20%20%20%20%20%20if%20pre.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20pre.right%20%3D%20node%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20pre.left%20%3D%20node%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20binary%20search%20tree%0A%20%20%20%20bst%20%3D%20BinarySearchTree%28%29%0A%20%20%20%20nums%20%3D%20%5B4,%202,%206,%201,%203,%205,%207%5D%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20bst.insert%28num%29%0A%0A%20%20%20%20%23%20Find%20a%20node%0A%20%20%20%20node%20%3D%20bst.search%287%29%0A%20%20%20%20print%28%22%5CnFound%20node%20object%3A%20%7B%7D,%20node%20value%20%3D%20%7B%7D%22.format%28node,%20node.val%29%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=162&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%29%3A%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%20%20%20%20%20%20%20%20self.left%20%3D%20None%0A%20%20%20%20%20%20%20%20self.right%20%3D%20None%0A%0A%0Aclass%20BinarySearchTree%3A%0A%20%20%20%20%22%22%22Binary%20search%20tree%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Initialize%20an%20empty%20tree%0A%20%20%20%20%20%20%20%20self._root%20%3D%20None%0A%0A%20%20%20%20def%20search%28self,%20num%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%20%20%20%20%22%22%22Find%20a%20node%22%22%22%0A%20%20%20%20%20%20%20%20cur%20%3D%20self._root%0A%20%20%20%20%20%20%20%20%23%20Search%20in%20a%20loop,%20stopping%20if%20cur%20becomes%20None%0A%20%20%20%20%20%20%20%20while%20cur%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20present,%20the%20target%20node%20is%20in%20the%20right%20subtree%20of%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.right%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20If%20present,%20the%20target%20node%20is%20in%20the%20left%20subtree%20of%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20elif%20cur.val%20%3E%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.left%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20The%20target%20node%20was%20found%3B%20exit%20the%20loop%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20return%20cur%0A%0A%20%20%20%20def%20insert%28self,%20num%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Insert%20a%20node%22%22%22%0A%20%20%20%20%20%20%20%20%23%20If%20the%20tree%20is%20empty,%20initialize%20the%20root%20node%0A%20%20%20%20%20%20%20%20if%20self._root%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self._root%20%3D%20TreeNode%28num%29%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%23%20Search%20in%20a%20loop,%20stopping%20if%20cur%20becomes%20None%0A%20%20%20%20%20%20%20%20cur,%20pre%20%3D%20self._root,%20None%0A%20%20%20%20%20%20%20%20while%20cur%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Return%20immediately%20if%20a%20duplicate%20node%20is%20found%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3D%3D%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%20%20%20%20pre%20%3D%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20The%20insertion%20position%20lies%20in%20the%20right%20subtree%20of%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.right%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20The%20insertion%20position%20lies%20in%20the%20left%20subtree%20of%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.left%0A%20%20%20%20%20%20%20%20%23%20Insert%20a%20node%0A%20%20%20%20%20%20%20%20node%20%3D%20TreeNode%28num%29%0A%20%20%20%20%20%20%20%20if%20pre.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20pre.right%20%3D%20node%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20pre.left%20%3D%20node%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20binary%20search%20tree%0A%20%20%20%20bst%20%3D%20BinarySearchTree%28%29%0A%20%20%20%20nums%20%3D%20%5B4,%202,%206,%201,%203,%205,%207%5D%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20bst.insert%28num%29%0A%0A%20%20%20%20%23%20Find%20a%20node%0A%20%20%20%20node%20%3D%20bst.search%287%29%0A%20%20%20%20print%28%22%5CnFound%20node%20object%3A%20%7B%7D,%20node%20value%20%3D%20%7B%7D%22.format%28node,%20node.val%29%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=162&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="2-inserting-a-node">2. &nbsp; Inserting a node<a class="headerlink" href="#2-inserting-a-node" title="Permanent link">&para;</a></h3>
+<p>To insert <code>num</code> while preserving the ordering “left subtree &lt; root &lt; right subtree,” follow the process below.</p>
 <ol>
-<li><strong>Finding the insertion position</strong>: Similar to the search operation, start from the root node and loop downward searching according to the size relationship between the current node value and <code>num</code>, until passing the leaf node (traversing to <code>None</code>) and then exit the loop.</li>
-<li><strong>Insert the node at that position</strong>: Create a node for <code>num</code> and place it at the <code>None</code> position.</li>
+<li><strong>Find the insertion position</strong>: as in a search, start at the root and move down according to comparisons with <code>num</code>. Stop when the traversal passes a leaf and reaches <code>None</code>.</li>
+<li><strong>Insert at that position</strong>: create a node containing <code>num</code> and place it at the <code>None</code> position.</li>
 </ol>
 <p><img alt="Inserting a node into a binary search tree" class="animation-figure" src="../binary_search_tree.assets/bst_insert.png" /></p>
 <p align="center"> Figure 7-18 &nbsp; Inserting a node into a binary search tree </p>
 
-<p>In the code implementation, note the following two points:</p>
+<p>Keep two details in mind when implementing this.</p>
 <ul>
-<li>Binary search trees do not allow duplicate nodes; otherwise, the tree would no longer satisfy its definition. Therefore, if the node to be inserted already exists in the tree, the insertion is skipped and the function returns directly.</li>
-<li>To implement the node insertion, we need to use node <code>pre</code> to save the node from the previous loop iteration. This way, when traversing to <code>None</code>, we can obtain its parent node, thereby completing the node insertion operation.</li>
+<li>Duplicate values are not allowed under this definition. If the value already exists, return without inserting it.</li>
+<li>Keep a reference <code>pre</code> to the node from the previous iteration. When the traversal reaches <code>None</code>, <code>pre</code> gives its parent so that the new node can be attached.</li>
 </ul>
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 <div class="highlight"><span class="filename">binary_search_tree.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Insert node"""</span>
-<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># If tree is empty, initialize root node</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Insert a node"""</span>
+<a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="c1"># If the tree is empty, initialize the root node</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_root</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">_root</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>        <span class="k">return</span>
-<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>    <span class="c1"># Loop search, exit after passing leaf node</span>
+<a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>    <span class="c1"># Search in a loop, stopping if cur becomes None</span>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>    <span class="n">cur</span><span class="p">,</span> <span class="n">pre</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_root</span><span class="p">,</span> <span class="kc">None</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a>    <span class="k">while</span> <span class="n">cur</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>        <span class="c1"># Found duplicate node, return directly</span>
+<a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a>        <span class="c1"># Return immediately if a duplicate node is found</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a>        <span class="k">if</span> <span class="n">cur</span><span class="o">.</span><span class="n">val</span> <span class="o">==</span> <span class="n">num</span><span class="p">:</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>            <span class="k">return</span>
 <a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>        <span class="n">pre</span> <span class="o">=</span> <span class="n">cur</span>
-<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>        <span class="c1"># Insertion position is in cur's right subtree</span>
+<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>        <span class="c1"># The insertion position lies in the right subtree of cur</span>
 <a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>        <span class="k">if</span> <span class="n">cur</span><span class="o">.</span><span class="n">val</span> <span class="o">&lt;</span> <span class="n">num</span><span class="p">:</span>
 <a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>            <span class="n">cur</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">right</span>
-<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>        <span class="c1"># Insertion position is in cur's left subtree</span>
+<a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>        <span class="c1"># The insertion position lies in the left subtree of cur</span>
 <a id="__codelineno-13-18" name="__codelineno-13-18" href="#__codelineno-13-18"></a>        <span class="k">else</span><span class="p">:</span>
 <a id="__codelineno-13-19" name="__codelineno-13-19" href="#__codelineno-13-19"></a>            <span class="n">cur</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">left</span>
-<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a>    <span class="c1"># Insert node</span>
+<a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a>    <span class="c1"># Insert a node</span>
 <a id="__codelineno-13-21" name="__codelineno-13-21" href="#__codelineno-13-21"></a>    <span class="n">node</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a>    <span class="k">if</span> <span class="n">pre</span><span class="o">.</span><span class="n">val</span> <span class="o">&lt;</span> <span class="n">num</span><span class="p">:</span>
 <a id="__codelineno-13-23" name="__codelineno-13-23" href="#__codelineno-13-23"></a>        <span class="n">pre</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="n">node</span>
@@ -5136,28 +5213,28 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// If tree is empty, initialize root node</span>
+<a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="c1">// If the tree is empty, initialize the root node</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">        </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="k">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">;</span>
-<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="w">    </span><span class="c1">// Continue searching while the current node pointer is not null</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">        </span><span class="c1">// Found duplicate node, return directly</span>
+<a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="w">        </span><span class="c1">// Return immediately if a duplicate node is found</span>
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">        </span><span class="k">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
-<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">        </span><span class="c1">// Insertion position is in cur's right subtree</span>
+<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">        </span><span class="c1">// The insertion position lies in the right subtree of cur</span>
 <a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">        </span><span class="c1">// Insertion position is in cur's left subtree</span>
+<a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">        </span><span class="c1">// The insertion position lies in the left subtree of cur</span>
 <a id="__codelineno-14-19" name="__codelineno-14-19" href="#__codelineno-14-19"></a><span class="w">        </span><span class="k">else</span>
 <a id="__codelineno-14-20" name="__codelineno-14-20" href="#__codelineno-14-20"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-14-21" name="__codelineno-14-21" href="#__codelineno-14-21"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a><span class="w">    </span><span class="c1">// Insert node</span>
+<a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a><span class="w">    </span><span class="c1">// Insert a node</span>
 <a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-14-24" name="__codelineno-14-24" href="#__codelineno-14-24"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">pre</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-14-25" name="__codelineno-14-25" href="#__codelineno-14-25"></a><span class="w">        </span><span class="k">pre</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
@@ -5167,28 +5244,28 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// If tree is empty, initialize root node</span>
+<a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="c1">// If the tree is empty, initialize the root node</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">        </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="w">    </span><span class="c1">// Search in a loop until reaching a null reference beyond a leaf</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">        </span><span class="c1">// Found duplicate node, return directly</span>
+<a id="__codelineno-15-11" name="__codelineno-15-11" href="#__codelineno-15-11"></a><span class="w">        </span><span class="c1">// Return immediately if a duplicate node is found</span>
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">        </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
-<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">        </span><span class="c1">// Insertion position is in cur's right subtree</span>
+<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">        </span><span class="c1">// The insertion position lies in the right subtree of cur</span>
 <a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span><span class="p">;</span>
-<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">        </span><span class="c1">// Insertion position is in cur's left subtree</span>
+<a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">        </span><span class="c1">// The insertion position lies in the left subtree of cur</span>
 <a id="__codelineno-15-19" name="__codelineno-15-19" href="#__codelineno-15-19"></a><span class="w">        </span><span class="k">else</span>
 <a id="__codelineno-15-20" name="__codelineno-15-20" href="#__codelineno-15-20"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">left</span><span class="p">;</span>
 <a id="__codelineno-15-21" name="__codelineno-15-21" href="#__codelineno-15-21"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w">    </span><span class="c1">// Insert node</span>
+<a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="w">    </span><span class="c1">// Insert a node</span>
 <a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-15-24" name="__codelineno-15-24" href="#__codelineno-15-24"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-15-25" name="__codelineno-15-25" href="#__codelineno-15-25"></a><span class="w">        </span><span class="n">pre</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
@@ -5198,29 +5275,29 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// If tree is empty, initialize root node</span>
+<a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="c1">// If the tree is empty, initialize the root node</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">        </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="w">    </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
-<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="w">    </span><span class="c1">// Search in a loop, stopping if a null reference is reached</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">        </span><span class="c1">// Found duplicate node, return directly</span>
+<a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="w">        </span><span class="c1">// Return immediately if a duplicate node is found</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">        </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
-<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">        </span><span class="c1">// Insertion position is in cur's right subtree</span>
+<a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">        </span><span class="c1">// The insertion position lies in the right subtree of cur</span>
 <a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-16-17" name="__codelineno-16-17" href="#__codelineno-16-17"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">        </span><span class="c1">// Insertion position is in cur's left subtree</span>
+<a id="__codelineno-16-18" name="__codelineno-16-18" href="#__codelineno-16-18"></a><span class="w">        </span><span class="c1">// The insertion position lies in the left subtree of cur</span>
 <a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="w">        </span><span class="k">else</span>
 <a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a>
-<a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">    </span><span class="c1">// Insert node</span>
+<a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">    </span><span class="c1">// Insert a node</span>
 <a id="__codelineno-16-24" name="__codelineno-16-24" href="#__codelineno-16-24"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-16-25" name="__codelineno-16-25" href="#__codelineno-16-25"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-26" name="__codelineno-16-26" href="#__codelineno-16-26"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
@@ -5232,17 +5309,17 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">bst</span><span class="w"> </span><span class="o">*</span><span class="nx">binarySearchTree</span><span class="p">)</span><span class="w"> </span><span class="nx">insert</span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="nx">cur</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">bst</span><span class="p">.</span><span class="nx">root</span>
-<a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="c1">// If tree is empty, initialize root node</span>
+<a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">    </span><span class="c1">// If the tree is empty, initialize the root node</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">        </span><span class="nx">bst</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="nx">num</span><span class="p">)</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">    </span><span class="c1">// Node position before the node to be inserted</span>
+<a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">    </span><span class="c1">// Track the parent of the node to insert</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nx">pre</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">nil</span>
-<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="w">    </span><span class="c1">// Continue searching while the current node is not nil</span>
 <a id="__codelineno-17-12" name="__codelineno-17-12" href="#__codelineno-17-12"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-13" name="__codelineno-17-13" href="#__codelineno-17-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">Val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">            </span><span class="k">return</span>
@@ -5254,7 +5331,7 @@
 <a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a><span class="w">            </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">Left</span>
 <a id="__codelineno-17-21" name="__codelineno-17-21" href="#__codelineno-17-21"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-17-22" name="__codelineno-17-22" href="#__codelineno-17-22"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a><span class="w">    </span><span class="c1">// Insert node</span>
+<a id="__codelineno-17-23" name="__codelineno-17-23" href="#__codelineno-17-23"></a><span class="w">    </span><span class="c1">// Insert a node</span>
 <a id="__codelineno-17-24" name="__codelineno-17-24" href="#__codelineno-17-24"></a><span class="w">    </span><span class="nx">node</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="nx">num</span><span class="p">)</span>
 <a id="__codelineno-17-25" name="__codelineno-17-25" href="#__codelineno-17-25"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">pre</span><span class="p">.</span><span class="nx">Val</span><span class="p">.(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-26" name="__codelineno-17-26" href="#__codelineno-17-26"></a><span class="w">        </span><span class="nx">pre</span><span class="p">.</span><span class="nx">Right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span>
@@ -5265,32 +5342,32 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// If tree is empty, initialize root node</span>
+<a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="c1">// If the tree is empty, initialize the root node</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">        </span><span class="n">root</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">cur</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">root</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">pre</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?</span>
-<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="w">    </span><span class="c1">// Search in a loop, stopping if cur becomes nil</span>
 <a id="__codelineno-18-11" name="__codelineno-18-11" href="#__codelineno-18-11"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">        </span><span class="c1">// Found duplicate node, return directly</span>
+<a id="__codelineno-18-12" name="__codelineno-18-12" href="#__codelineno-18-12"></a><span class="w">        </span><span class="c1">// Return immediately if a duplicate node is found</span>
 <a id="__codelineno-18-13" name="__codelineno-18-13" href="#__codelineno-18-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">            </span><span class="k">return</span>
 <a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">        </span><span class="n">pre</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">cur</span>
-<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">        </span><span class="c1">// Insertion position is in cur's right subtree</span>
+<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">        </span><span class="c1">// The insertion position lies in the right subtree of cur</span>
 <a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">cur</span><span class="p">?.</span><span class="kr">right</span>
 <a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a><span class="w">        </span><span class="c1">// Insertion position is in cur's left subtree</span>
+<a id="__codelineno-18-21" name="__codelineno-18-21" href="#__codelineno-18-21"></a><span class="w">        </span><span class="c1">// The insertion position lies in the left subtree of cur</span>
 <a id="__codelineno-18-22" name="__codelineno-18-22" href="#__codelineno-18-22"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-23" name="__codelineno-18-23" href="#__codelineno-18-23"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">cur</span><span class="p">?.</span><span class="kr">left</span>
 <a id="__codelineno-18-24" name="__codelineno-18-24" href="#__codelineno-18-24"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-18-25" name="__codelineno-18-25" href="#__codelineno-18-25"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-26" name="__codelineno-18-26" href="#__codelineno-18-26"></a><span class="w">    </span><span class="c1">// Insert node</span>
+<a id="__codelineno-18-26" name="__codelineno-18-26" href="#__codelineno-18-26"></a><span class="w">    </span><span class="c1">// Insert a node</span>
 <a id="__codelineno-18-27" name="__codelineno-18-27" href="#__codelineno-18-27"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-18-28" name="__codelineno-18-28" href="#__codelineno-18-28"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">pre</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-29" name="__codelineno-18-29" href="#__codelineno-18-29"></a><span class="w">        </span><span class="n">pre</span><span class="p">?.</span><span class="kr">right</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span>
@@ -5301,26 +5378,26 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="nx">insert</span><span class="p">(</span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// If tree is empty, initialize root node</span>
+<a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="c1">// If the tree is empty, initialize the root node</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="nx">num</span><span class="p">);</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="p">,</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="w">        </span><span class="nx">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="w">    </span><span class="c1">// Search in a loop until reaching a null reference beyond a leaf</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">        </span><span class="c1">// Found duplicate node, return directly</span>
+<a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">        </span><span class="c1">// Return immediately if a duplicate node is found</span>
 <a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">        </span><span class="nx">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">;</span>
-<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">        </span><span class="c1">// Insertion position is in cur's right subtree</span>
+<a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">        </span><span class="c1">// The insertion position lies in the right subtree of cur</span>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
-<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">        </span><span class="c1">// Insertion position is in cur's left subtree</span>
+<a id="__codelineno-19-17" name="__codelineno-19-17" href="#__codelineno-19-17"></a><span class="w">        </span><span class="c1">// The insertion position lies in the left subtree of cur</span>
 <a id="__codelineno-19-18" name="__codelineno-19-18" href="#__codelineno-19-18"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">left</span><span class="p">;</span>
 <a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">    </span><span class="c1">// Insert node</span>
+<a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="w">    </span><span class="c1">// Insert a node</span>
 <a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="nx">num</span><span class="p">);</span>
 <a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pre</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="nx">pre</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-19-23" name="__codelineno-19-23" href="#__codelineno-19-23"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="nx">pre</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
@@ -5328,26 +5405,26 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="nx">insert</span><span class="p">(</span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// If tree is empty, initialize root node</span>
+<a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="c1">// If the tree is empty, initialize the root node</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="nx">num</span><span class="p">);</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">cur</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="p">,</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">        </span><span class="nx">pre</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="w">    </span><span class="c1">// Search in a loop, stopping if a null reference is reached</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">        </span><span class="c1">// Found duplicate node, return directly</span>
+<a id="__codelineno-20-12" name="__codelineno-20-12" href="#__codelineno-20-12"></a><span class="w">        </span><span class="c1">// Return immediately if a duplicate node is found</span>
 <a id="__codelineno-20-13" name="__codelineno-20-13" href="#__codelineno-20-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">        </span><span class="nx">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">;</span>
-<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">        </span><span class="c1">// Insertion position is in cur's right subtree</span>
+<a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">        </span><span class="c1">// The insertion position lies in the right subtree of cur</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
-<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">        </span><span class="c1">// Insertion position is in cur's left subtree</span>
+<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">        </span><span class="c1">// The insertion position lies in the left subtree of cur</span>
 <a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">left</span><span class="p">;</span>
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">    </span><span class="c1">// Insert node</span>
+<a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">    </span><span class="c1">// Insert a node</span>
 <a id="__codelineno-20-21" name="__codelineno-20-21" href="#__codelineno-20-21"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="nx">num</span><span class="p">);</span>
 <a id="__codelineno-20-22" name="__codelineno-20-22" href="#__codelineno-20-22"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pre</span><span class="o">!</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="nx">pre</span><span class="o">!</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
 <a id="__codelineno-20-23" name="__codelineno-20-23" href="#__codelineno-20-23"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="nx">pre</span><span class="o">!</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">node</span><span class="p">;</span>
@@ -5355,28 +5432,28 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">insert</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// If tree is empty, initialize root node</span>
+<a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="c1">// If the tree is empty, initialize the root node</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">    </span><span class="n">_root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">_num</span><span class="p">);</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">    </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_root</span><span class="p">;</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="w">  </span><span class="c1">// Search in a loop, stopping if a null reference is reached</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">    </span><span class="c1">// Found duplicate node, return directly</span>
+<a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">    </span><span class="c1">// Return immediately if a duplicate node is found</span>
 <a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">    </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
-<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">    </span><span class="c1">// Insertion position is in cur's right subtree</span>
+<a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">    </span><span class="c1">// The insertion position lies in the right subtree of cur</span>
 <a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span>
 <a id="__codelineno-21-17" name="__codelineno-21-17" href="#__codelineno-21-17"></a><span class="w">      </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">    </span><span class="c1">// Insertion position is in cur's left subtree</span>
+<a id="__codelineno-21-18" name="__codelineno-21-18" href="#__codelineno-21-18"></a><span class="w">    </span><span class="c1">// The insertion position lies in the left subtree of cur</span>
 <a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="w">    </span><span class="k">else</span>
 <a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="w">      </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">  </span><span class="c1">// Insert node</span>
+<a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">  </span><span class="c1">// Insert a node</span>
 <a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">_num</span><span class="p">);</span>
 <a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span>
 <a id="__codelineno-21-25" name="__codelineno-21-25" href="#__codelineno-21-25"></a><span class="w">    </span><span class="n">pre</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
@@ -5386,33 +5463,33 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// If tree is empty, initialize root node</span>
+<a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="w">    </span><span class="c1">// If the tree is empty, initialize the root node</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="p">.</span><span class="n">is_none</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">        </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">TreeNode</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">num</span><span class="p">));</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">None</span><span class="p">;</span>
-<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">    </span><span class="c1">// Search while cur refers to a node</span>
 <a id="__codelineno-22-11" name="__codelineno-22-11" href="#__codelineno-22-11"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">clone</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-12" name="__codelineno-22-12" href="#__codelineno-22-12"></a><span class="w">        </span><span class="k">match</span><span class="w"> </span><span class="n">num</span><span class="p">.</span><span class="n">cmp</span><span class="p">(</span><span class="o">&amp;</span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">            </span><span class="c1">// Found duplicate node, return directly</span>
+<a id="__codelineno-22-13" name="__codelineno-22-13" href="#__codelineno-22-13"></a><span class="w">            </span><span class="c1">// Return immediately if a duplicate node is found</span>
 <a id="__codelineno-22-14" name="__codelineno-22-14" href="#__codelineno-22-14"></a><span class="w">            </span><span class="n">Ordering</span><span class="p">::</span><span class="n">Equal</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="k">return</span><span class="p">,</span>
-<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">            </span><span class="c1">// Insertion position is in cur's right subtree</span>
+<a id="__codelineno-22-15" name="__codelineno-22-15" href="#__codelineno-22-15"></a><span class="w">            </span><span class="c1">// The insertion position lies in the right subtree of cur</span>
 <a id="__codelineno-22-16" name="__codelineno-22-16" href="#__codelineno-22-16"></a><span class="w">            </span><span class="n">Ordering</span><span class="p">::</span><span class="n">Greater</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-17" name="__codelineno-22-17" href="#__codelineno-22-17"></a><span class="w">                </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-22-18" name="__codelineno-22-18" href="#__codelineno-22-18"></a><span class="w">                </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-22-19" name="__codelineno-22-19" href="#__codelineno-22-19"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w">            </span><span class="c1">// Insertion position is in cur's left subtree</span>
+<a id="__codelineno-22-20" name="__codelineno-22-20" href="#__codelineno-22-20"></a><span class="w">            </span><span class="c1">// The insertion position lies in the left subtree of cur</span>
 <a id="__codelineno-22-21" name="__codelineno-22-21" href="#__codelineno-22-21"></a><span class="w">            </span><span class="n">Ordering</span><span class="p">::</span><span class="n">Less</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a><span class="w">                </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w">                </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-22-25" name="__codelineno-22-25" href="#__codelineno-22-25"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-22-26" name="__codelineno-22-26" href="#__codelineno-22-26"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-22-27" name="__codelineno-22-27" href="#__codelineno-22-27"></a><span class="w">    </span><span class="c1">// Insert node</span>
+<a id="__codelineno-22-27" name="__codelineno-22-27" href="#__codelineno-22-27"></a><span class="w">    </span><span class="c1">// Insert a node</span>
 <a id="__codelineno-22-28" name="__codelineno-22-28" href="#__codelineno-22-28"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="n">unwrap</span><span class="p">();</span>
 <a id="__codelineno-22-29" name="__codelineno-22-29" href="#__codelineno-22-29"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">TreeNode</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="n">num</span><span class="p">));</span>
 <a id="__codelineno-22-30" name="__codelineno-22-30" href="#__codelineno-22-30"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="p">{</span>
@@ -5424,30 +5501,30 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">BinarySearchTree</span><span class="w"> </span><span class="o">*</span><span class="n">bst</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// If tree is empty, initialize root node</span>
+<a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="c1">// If the tree is empty, initialize the root node</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">bst</span><span class="o">-&gt;</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">        </span><span class="n">bst</span><span class="o">-&gt;</span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">bst</span><span class="o">-&gt;</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
-<a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="w">    </span><span class="c1">// Search in a loop until reaching a null reference beyond a leaf</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">        </span><span class="c1">// Found duplicate node, return directly</span>
+<a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="w">        </span><span class="c1">// Return immediately if a duplicate node is found</span>
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">            </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">        </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
 <a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w">            </span><span class="c1">// Insertion position is in cur's right subtree</span>
+<a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w">            </span><span class="c1">// The insertion position lies in the right subtree of cur</span>
 <a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-23-19" name="__codelineno-23-19" href="#__codelineno-23-19"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="w">            </span><span class="c1">// Insertion position is in cur's left subtree</span>
+<a id="__codelineno-23-20" name="__codelineno-23-20" href="#__codelineno-23-20"></a><span class="w">            </span><span class="c1">// The insertion position lies in the left subtree of cur</span>
 <a id="__codelineno-23-21" name="__codelineno-23-21" href="#__codelineno-23-21"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">    </span><span class="c1">// Insert node</span>
+<a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">    </span><span class="c1">// Insert a node</span>
 <a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">);</span>
 <a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a><span class="w">        </span><span class="n">pre</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">;</span>
@@ -5458,29 +5535,29 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Insert node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Insert a node */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// If tree is empty, initialize root node</span>
+<a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="c1">// If the tree is empty, initialize the root node</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">        </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">pre</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span>
-<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="w">    </span><span class="c1">// Continue searching while cur is not null</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">        </span><span class="c1">// Found duplicate node, return directly</span>
+<a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">        </span><span class="c1">// Return immediately if a duplicate node is found</span>
 <a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">            </span><span class="k">return</span>
 <a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">        </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span>
-<a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">        </span><span class="c1">// Insertion position is in cur's right subtree</span>
+<a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">        </span><span class="c1">// The insertion position lies in the right subtree of cur</span>
 <a id="__codelineno-24-17" name="__codelineno-24-17" href="#__codelineno-24-17"></a><span class="w">        </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-24-18" name="__codelineno-24-18" href="#__codelineno-24-18"></a><span class="w">            </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span>
-<a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="w">        </span><span class="c1">// Insertion position is in cur's left subtree</span>
+<a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="w">        </span><span class="c1">// The insertion position lies in the left subtree of cur</span>
 <a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="w">        </span><span class="k">else</span>
 <a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="w">            </span><span class="n">cur</span><span class="p">.</span><span class="na">left</span>
 <a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">    </span><span class="c1">// Insert node</span>
+<a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">    </span><span class="c1">// Insert a node</span>
 <a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-24-25" name="__codelineno-24-25" href="#__codelineno-24-25"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="o">?.</span><span class="na">_val</span><span class="o">!!</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-24-26" name="__codelineno-24-26" href="#__codelineno-24-26"></a><span class="w">        </span><span class="n">pre</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
@@ -5490,31 +5567,31 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Insert node ###</span>
+<div class="highlight"><span class="filename">binary_search_tree.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Insert a node ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">insert</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
-<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># If tree is empty, initialize root node</span>
+<a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="c1"># If the tree is empty, initialize the root node</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="vi">@root</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">    </span><span class="vi">@root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">TreeNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">    </span><span class="k">return</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a>
-<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="c1"># Loop search, exit after passing leaf node</span>
+<a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="w">  </span><span class="c1"># Search in a loop, stopping if cur becomes nil</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="w">  </span><span class="n">cur</span><span class="p">,</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@root</span><span class="p">,</span><span class="w"> </span><span class="kp">nil</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="o">!</span><span class="n">cur</span><span class="o">.</span><span class="n">nil?</span>
-<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">    </span><span class="c1"># Found duplicate node, return directly</span>
+<a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="w">    </span><span class="c1"># Return immediately if a duplicate node is found</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a>
 <a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">    </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span>
-<a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">    </span><span class="c1"># Insertion position is in cur's right subtree</span>
+<a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">    </span><span class="c1"># The insertion position lies in the right subtree of cur</span>
 <a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w">      </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">right</span>
-<a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="w">    </span><span class="c1"># Insertion position is in cur's left subtree</span>
+<a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="w">    </span><span class="c1"># The insertion position lies in the left subtree of cur</span>
 <a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a><span class="w">    </span><span class="k">else</span>
 <a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="w">      </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">left</span>
 <a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-25-23" name="__codelineno-25-23" href="#__codelineno-25-23"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-25-24" name="__codelineno-25-24" href="#__codelineno-25-24"></a>
-<a id="__codelineno-25-25" name="__codelineno-25-25" href="#__codelineno-25-25"></a><span class="w">  </span><span class="c1"># Insert node</span>
+<a id="__codelineno-25-25" name="__codelineno-25-25" href="#__codelineno-25-25"></a><span class="w">  </span><span class="c1"># Insert a node</span>
 <a id="__codelineno-25-26" name="__codelineno-25-26" href="#__codelineno-25-26"></a><span class="w">  </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">TreeNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-25-27" name="__codelineno-25-27" href="#__codelineno-25-27"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="n">pre</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-25-28" name="__codelineno-25-28" href="#__codelineno-25-28"></a><span class="w">    </span><span class="n">pre</span><span class="o">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span>
@@ -5526,27 +5603,32 @@
 </div>
 </div>
 </div>
-<p>Similar to searching for a node, inserting a node uses <span class="arithmatex">\(O(\log n)\)</span> time.</p>
-<h3 id="3-removing-a-node">3. &nbsp; Removing a Node<a class="headerlink" href="#3-removing-a-node" title="Permanent link">&para;</a></h3>
-<p>First, find the target node in the binary search tree, then remove it. Similar to node insertion, we need to ensure that after the removal operation is completed, the binary search tree's property of "left subtree <span class="arithmatex">\(&lt;\)</span> root node <span class="arithmatex">\(&lt;\)</span> right subtree" is still maintained. Therefore, depending on the number of child nodes the target node has, we consider three cases: degree <span class="arithmatex">\(0\)</span>, degree <span class="arithmatex">\(1\)</span>, and degree <span class="arithmatex">\(2\)</span>, and perform the corresponding removal operation.</p>
-<p>As shown in Figure 7-19, when the degree of the node to be removed is <span class="arithmatex">\(0\)</span>, it means the node is a leaf node and can be directly removed.</p>
-<p><img alt="Removing a node in a binary search tree (degree 0)" class="animation-figure" src="../binary_search_tree.assets/bst_remove_case1.png" /></p>
-<p align="center"> Figure 7-19 &nbsp; Removing a node in a binary search tree (degree 0) </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%29%3A%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%20%20%20%20%20%20%20%20self.left%20%3D%20None%0A%20%20%20%20%20%20%20%20self.right%20%3D%20None%0A%0A%0Aclass%20BinarySearchTree%3A%0A%20%20%20%20%22%22%22Binary%20search%20tree%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Initialize%20an%20empty%20tree%0A%20%20%20%20%20%20%20%20self._root%20%3D%20None%0A%0A%20%20%20%20def%20insert%28self,%20num%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Insert%20a%20node%22%22%22%0A%20%20%20%20%20%20%20%20%23%20If%20the%20tree%20is%20empty,%20initialize%20the%20root%20node%0A%20%20%20%20%20%20%20%20if%20self._root%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self._root%20%3D%20TreeNode%28num%29%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%23%20Search%20in%20a%20loop,%20stopping%20if%20cur%20becomes%20None%0A%20%20%20%20%20%20%20%20cur,%20pre%20%3D%20self._root,%20None%0A%20%20%20%20%20%20%20%20while%20cur%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Return%20immediately%20if%20a%20duplicate%20node%20is%20found%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3D%3D%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%20%20%20%20pre%20%3D%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20The%20insertion%20position%20lies%20in%20the%20right%20subtree%20of%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.right%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20The%20insertion%20position%20lies%20in%20the%20left%20subtree%20of%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.left%0A%20%20%20%20%20%20%20%20%23%20Insert%20a%20node%0A%20%20%20%20%20%20%20%20node%20%3D%20TreeNode%28num%29%0A%20%20%20%20%20%20%20%20if%20pre.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20pre.right%20%3D%20node%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20pre.left%20%3D%20node%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20binary%20search%20tree%0A%20%20%20%20bst%20%3D%20BinarySearchTree%28%29%0A%20%20%20%20nums%20%3D%20%5B4,%202,%206,%201,%203,%205,%207%5D%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20bst.insert%28num%29%0A%0A%20%20%20%20%23%20Insert%20a%20node%0A%20%20%20%20bst.insert%2816%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=162&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%29%3A%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%20%20%20%20%20%20%20%20self.left%20%3D%20None%0A%20%20%20%20%20%20%20%20self.right%20%3D%20None%0A%0A%0Aclass%20BinarySearchTree%3A%0A%20%20%20%20%22%22%22Binary%20search%20tree%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20%23%20Initialize%20an%20empty%20tree%0A%20%20%20%20%20%20%20%20self._root%20%3D%20None%0A%0A%20%20%20%20def%20insert%28self,%20num%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Insert%20a%20node%22%22%22%0A%20%20%20%20%20%20%20%20%23%20If%20the%20tree%20is%20empty,%20initialize%20the%20root%20node%0A%20%20%20%20%20%20%20%20if%20self._root%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self._root%20%3D%20TreeNode%28num%29%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%23%20Search%20in%20a%20loop,%20stopping%20if%20cur%20becomes%20None%0A%20%20%20%20%20%20%20%20cur,%20pre%20%3D%20self._root,%20None%0A%20%20%20%20%20%20%20%20while%20cur%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Return%20immediately%20if%20a%20duplicate%20node%20is%20found%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3D%3D%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%20%20%20%20pre%20%3D%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20The%20insertion%20position%20lies%20in%20the%20right%20subtree%20of%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.right%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20The%20insertion%20position%20lies%20in%20the%20left%20subtree%20of%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.left%0A%20%20%20%20%20%20%20%20%23%20Insert%20a%20node%0A%20%20%20%20%20%20%20%20node%20%3D%20TreeNode%28num%29%0A%20%20%20%20%20%20%20%20if%20pre.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20pre.right%20%3D%20node%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20pre.left%20%3D%20node%0A%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20binary%20search%20tree%0A%20%20%20%20bst%20%3D%20BinarySearchTree%28%29%0A%20%20%20%20nums%20%3D%20%5B4,%202,%206,%201,%203,%205,%207%5D%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20bst.insert%28num%29%0A%0A%20%20%20%20%23%20Insert%20a%20node%0A%20%20%20%20bst.insert%2816%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=162&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<p>Like lookup, insertion takes <span class="arithmatex">\(O(\log n)\)</span> time.</p>
+<h3 id="3-deleting-a-node">3. &nbsp; Deleting a node<a class="headerlink" href="#3-deleting-a-node" title="Permanent link">&para;</a></h3>
+<p>First locate the target node, then remove it while preserving “left subtree &lt; root &lt; right subtree.” The procedure depends on whether the target has 0, 1, or 2 children.</p>
+<p>As shown below, a node of degree <span class="arithmatex">\(0\)</span> is a leaf and can be removed directly.</p>
+<p><img alt="Deleting a node of degree 0" class="animation-figure" src="../binary_search_tree.assets/bst_remove_case1.png" /></p>
+<p align="center"> Figure 7-19 &nbsp; Deleting a node of degree 0 </p>
 
-<p>As shown in Figure 7-20, when the degree of the node to be removed is <span class="arithmatex">\(1\)</span>, replacing the node to be removed with its child node is sufficient.</p>
-<p><img alt="Removing a node in a binary search tree (degree 1)" class="animation-figure" src="../binary_search_tree.assets/bst_remove_case2.png" /></p>
-<p align="center"> Figure 7-20 &nbsp; Removing a node in a binary search tree (degree 1) </p>
+<p>As shown below, replace a node of degree <span class="arithmatex">\(1\)</span> with its only child.</p>
+<p><img alt="Deleting a node of degree 1" class="animation-figure" src="../binary_search_tree.assets/bst_remove_case2.png" /></p>
+<p align="center"> Figure 7-20 &nbsp; Deleting a node of degree 1 </p>
 
-<p>When the degree of the node to be removed is <span class="arithmatex">\(2\)</span>, we cannot directly remove it; instead, we need to use a node to replace it. To maintain the binary search tree's property of "left subtree <span class="arithmatex">\(&lt;\)</span> root node <span class="arithmatex">\(&lt;\)</span> right subtree," <strong>this node can be either the smallest node in the right subtree or the largest node in the left subtree</strong>.</p>
-<p>Assuming we choose the smallest node in the right subtree, that is, the inorder successor, the removal process is as shown in Figure 7-21.</p>
+<p>A node of degree <span class="arithmatex">\(2\)</span> cannot simply be removed; another node must replace it. To preserve “left subtree <span class="arithmatex">\(&lt;\)</span> root <span class="arithmatex">\(&lt;\)</span> right subtree,” <strong>use either the smallest node in the right subtree or the largest node in the left subtree</strong>.</p>
+<p>Suppose we choose the smallest node in the right subtree, the next node in inorder traversal. The process is shown below.</p>
 <ol>
-<li>Find the next node of the node to be removed in the "inorder traversal sequence," denoted as <code>tmp</code>.</li>
-<li>Replace the value of the node to be removed with the value of <code>tmp</code>, and recursively remove node <code>tmp</code> in the tree.</li>
+<li>Find the target node's successor in inorder traversal and call it <code>tmp</code>.</li>
+<li>Replace the target node's value with <code>tmp</code>'s value, then recursively delete <code>tmp</code> from the tree.</li>
 </ol>
 <div class="tabbed-set tabbed-alternate" data-tabs="4:4"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">&lt;1&gt;</label><label for="__tabbed_4_2">&lt;2&gt;</label><label for="__tabbed_4_3">&lt;3&gt;</label><label for="__tabbed_4_4">&lt;4&gt;</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<p><img alt="Removing a node in a binary search tree (degree 2)" class="animation-figure" src="../binary_search_tree.assets/bst_remove_case3_step1.png" /></p>
+<p><img alt="Deleting a node of degree 2" class="animation-figure" src="../binary_search_tree.assets/bst_remove_case3_step1.png" /></p>
 </div>
 <div class="tabbed-block">
 <p><img alt="bst_remove_case3_step2" class="animation-figure" src="../binary_search_tree.assets/bst_remove_case3_step2.png" /></p>
@@ -5559,259 +5641,259 @@
 </div>
 </div>
 </div>
-<p align="center"> Figure 7-21 &nbsp; Removing a node in a binary search tree (degree 2) </p>
+<p align="center"> Figure 7-21 &nbsp; Deleting a node of degree 2 </p>
 
-<p>The node removal operation also uses <span class="arithmatex">\(O(\log n)\)</span> time, where finding the node to be removed requires <span class="arithmatex">\(O(\log n)\)</span> time, and obtaining the inorder successor node requires <span class="arithmatex">\(O(\log n)\)</span> time. Example code is as follows:</p>
+<p>Deletion also takes <span class="arithmatex">\(O(\log n)\)</span> time: locating the target takes <span class="arithmatex">\(O(\log n)\)</span>, and finding its inorder successor takes <span class="arithmatex">\(O(\log n)\)</span>. An implementation follows:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="5:13"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><input id="__tabbed_5_13" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Python</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Java</label><label for="__tabbed_5_4">C#</label><label for="__tabbed_5_5">Go</label><label for="__tabbed_5_6">Swift</label><label for="__tabbed_5_7">JS</label><label for="__tabbed_5_8">TS</label><label for="__tabbed_5_9">Dart</label><label for="__tabbed_5_10">Rust</label><label for="__tabbed_5_11">C</label><label for="__tabbed_5_12">Kotlin</label><label for="__tabbed_5_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_search_tree.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">num</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
-<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Delete node"""</span>
-<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="c1"># If tree is empty, return directly</span>
+<a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="w">    </span><span class="sd">"""Remove a node"""</span>
+<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a>    <span class="c1"># Return immediately if the tree is empty</span>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a>    <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_root</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a>        <span class="k">return</span>
-<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="c1"># Loop search, exit after passing leaf node</span>
+<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a>    <span class="c1"># Search in a loop, stopping if cur becomes None</span>
 <a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a>    <span class="n">cur</span><span class="p">,</span> <span class="n">pre</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_root</span><span class="p">,</span> <span class="kc">None</span>
 <a id="__codelineno-26-8" name="__codelineno-26-8" href="#__codelineno-26-8"></a>    <span class="k">while</span> <span class="n">cur</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>        <span class="c1"># Found node to delete, exit loop</span>
+<a id="__codelineno-26-9" name="__codelineno-26-9" href="#__codelineno-26-9"></a>        <span class="c1"># The node to remove was found; exit the loop</span>
 <a id="__codelineno-26-10" name="__codelineno-26-10" href="#__codelineno-26-10"></a>        <span class="k">if</span> <span class="n">cur</span><span class="o">.</span><span class="n">val</span> <span class="o">==</span> <span class="n">num</span><span class="p">:</span>
 <a id="__codelineno-26-11" name="__codelineno-26-11" href="#__codelineno-26-11"></a>            <span class="k">break</span>
 <a id="__codelineno-26-12" name="__codelineno-26-12" href="#__codelineno-26-12"></a>        <span class="n">pre</span> <span class="o">=</span> <span class="n">cur</span>
-<a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>        <span class="c1"># Node to delete is in cur's right subtree</span>
+<a id="__codelineno-26-13" name="__codelineno-26-13" href="#__codelineno-26-13"></a>        <span class="c1"># If present, the node to remove is in the right subtree of cur</span>
 <a id="__codelineno-26-14" name="__codelineno-26-14" href="#__codelineno-26-14"></a>        <span class="k">if</span> <span class="n">cur</span><span class="o">.</span><span class="n">val</span> <span class="o">&lt;</span> <span class="n">num</span><span class="p">:</span>
 <a id="__codelineno-26-15" name="__codelineno-26-15" href="#__codelineno-26-15"></a>            <span class="n">cur</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">right</span>
-<a id="__codelineno-26-16" name="__codelineno-26-16" href="#__codelineno-26-16"></a>        <span class="c1"># Node to delete is in cur's left subtree</span>
+<a id="__codelineno-26-16" name="__codelineno-26-16" href="#__codelineno-26-16"></a>        <span class="c1"># If present, the node to remove is in the left subtree of cur</span>
 <a id="__codelineno-26-17" name="__codelineno-26-17" href="#__codelineno-26-17"></a>        <span class="k">else</span><span class="p">:</span>
 <a id="__codelineno-26-18" name="__codelineno-26-18" href="#__codelineno-26-18"></a>            <span class="n">cur</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">left</span>
-<a id="__codelineno-26-19" name="__codelineno-26-19" href="#__codelineno-26-19"></a>    <span class="c1"># If no node to delete, return directly</span>
+<a id="__codelineno-26-19" name="__codelineno-26-19" href="#__codelineno-26-19"></a>    <span class="c1"># Return immediately if the node to remove does not exist</span>
 <a id="__codelineno-26-20" name="__codelineno-26-20" href="#__codelineno-26-20"></a>    <span class="k">if</span> <span class="n">cur</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-26-21" name="__codelineno-26-21" href="#__codelineno-26-21"></a>        <span class="k">return</span>
 <a id="__codelineno-26-22" name="__codelineno-26-22" href="#__codelineno-26-22"></a>
-<a id="__codelineno-26-23" name="__codelineno-26-23" href="#__codelineno-26-23"></a>    <span class="c1"># Number of child nodes = 0 or 1</span>
+<a id="__codelineno-26-23" name="__codelineno-26-23" href="#__codelineno-26-23"></a>    <span class="c1"># Number of children = 0 or 1</span>
 <a id="__codelineno-26-24" name="__codelineno-26-24" href="#__codelineno-26-24"></a>    <span class="k">if</span> <span class="n">cur</span><span class="o">.</span><span class="n">left</span> <span class="ow">is</span> <span class="kc">None</span> <span class="ow">or</span> <span class="n">cur</span><span class="o">.</span><span class="n">right</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-26-25" name="__codelineno-26-25" href="#__codelineno-26-25"></a>        <span class="c1"># When number of child nodes = 0 / 1, child = null / that child node</span>
+<a id="__codelineno-26-25" name="__codelineno-26-25" href="#__codelineno-26-25"></a>        <span class="c1"># With 0 or 1 child, child is None or that child, respectively</span>
 <a id="__codelineno-26-26" name="__codelineno-26-26" href="#__codelineno-26-26"></a>        <span class="n">child</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">left</span> <span class="ow">or</span> <span class="n">cur</span><span class="o">.</span><span class="n">right</span>
-<a id="__codelineno-26-27" name="__codelineno-26-27" href="#__codelineno-26-27"></a>        <span class="c1"># Delete node cur</span>
+<a id="__codelineno-26-27" name="__codelineno-26-27" href="#__codelineno-26-27"></a>        <span class="c1"># Remove node cur</span>
 <a id="__codelineno-26-28" name="__codelineno-26-28" href="#__codelineno-26-28"></a>        <span class="k">if</span> <span class="n">cur</span> <span class="o">!=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_root</span><span class="p">:</span>
 <a id="__codelineno-26-29" name="__codelineno-26-29" href="#__codelineno-26-29"></a>            <span class="k">if</span> <span class="n">pre</span><span class="o">.</span><span class="n">left</span> <span class="o">==</span> <span class="n">cur</span><span class="p">:</span>
 <a id="__codelineno-26-30" name="__codelineno-26-30" href="#__codelineno-26-30"></a>                <span class="n">pre</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">child</span>
 <a id="__codelineno-26-31" name="__codelineno-26-31" href="#__codelineno-26-31"></a>            <span class="k">else</span><span class="p">:</span>
 <a id="__codelineno-26-32" name="__codelineno-26-32" href="#__codelineno-26-32"></a>                <span class="n">pre</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="n">child</span>
 <a id="__codelineno-26-33" name="__codelineno-26-33" href="#__codelineno-26-33"></a>        <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-26-34" name="__codelineno-26-34" href="#__codelineno-26-34"></a>            <span class="c1"># If deleted node is root node, reassign root node</span>
+<a id="__codelineno-26-34" name="__codelineno-26-34" href="#__codelineno-26-34"></a>            <span class="c1"># If the removed node is the root, assign the new root</span>
 <a id="__codelineno-26-35" name="__codelineno-26-35" href="#__codelineno-26-35"></a>            <span class="bp">self</span><span class="o">.</span><span class="n">_root</span> <span class="o">=</span> <span class="n">child</span>
-<a id="__codelineno-26-36" name="__codelineno-26-36" href="#__codelineno-26-36"></a>    <span class="c1"># Number of child nodes = 2</span>
+<a id="__codelineno-26-36" name="__codelineno-26-36" href="#__codelineno-26-36"></a>    <span class="c1"># Number of children = 2</span>
 <a id="__codelineno-26-37" name="__codelineno-26-37" href="#__codelineno-26-37"></a>    <span class="k">else</span><span class="p">:</span>
-<a id="__codelineno-26-38" name="__codelineno-26-38" href="#__codelineno-26-38"></a>        <span class="c1"># Get next node of cur in inorder traversal</span>
+<a id="__codelineno-26-38" name="__codelineno-26-38" href="#__codelineno-26-38"></a>        <span class="c1"># Get the inorder successor of cur</span>
 <a id="__codelineno-26-39" name="__codelineno-26-39" href="#__codelineno-26-39"></a>        <span class="n">tmp</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">=</span> <span class="n">cur</span><span class="o">.</span><span class="n">right</span>
 <a id="__codelineno-26-40" name="__codelineno-26-40" href="#__codelineno-26-40"></a>        <span class="k">while</span> <span class="n">tmp</span><span class="o">.</span><span class="n">left</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-26-41" name="__codelineno-26-41" href="#__codelineno-26-41"></a>            <span class="n">tmp</span> <span class="o">=</span> <span class="n">tmp</span><span class="o">.</span><span class="n">left</span>
-<a id="__codelineno-26-42" name="__codelineno-26-42" href="#__codelineno-26-42"></a>        <span class="c1"># Recursively delete node tmp</span>
+<a id="__codelineno-26-42" name="__codelineno-26-42" href="#__codelineno-26-42"></a>        <span class="c1"># Recursively remove node tmp</span>
 <a id="__codelineno-26-43" name="__codelineno-26-43" href="#__codelineno-26-43"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">tmp</span><span class="o">.</span><span class="n">val</span><span class="p">)</span>
-<a id="__codelineno-26-44" name="__codelineno-26-44" href="#__codelineno-26-44"></a>        <span class="c1"># Replace cur with tmp</span>
+<a id="__codelineno-26-44" name="__codelineno-26-44" href="#__codelineno-26-44"></a>        <span class="c1"># Replace the value of cur with the value of tmp</span>
 <a id="__codelineno-26-45" name="__codelineno-26-45" href="#__codelineno-26-45"></a>        <span class="n">cur</span><span class="o">.</span><span class="n">val</span> <span class="o">=</span> <span class="n">tmp</span><span class="o">.</span><span class="n">val</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="c1">// If tree is empty, return directly</span>
+<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="w">    </span><span class="c1">// Return immediately if the tree is empty</span>
 <a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span>
 <a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="k">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">;</span>
-<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="w">    </span><span class="c1">// Continue searching while the current node pointer is not null</span>
 <a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">        </span><span class="c1">// Found node to delete, exit loop</span>
+<a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="w">        </span><span class="c1">// The node to remove was found; exit the loop</span>
 <a id="__codelineno-27-10" name="__codelineno-27-10" href="#__codelineno-27-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-27-11" name="__codelineno-27-11" href="#__codelineno-27-11"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-27-12" name="__codelineno-27-12" href="#__codelineno-27-12"></a><span class="w">        </span><span class="k">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
-<a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">        </span><span class="c1">// Node to delete is in cur's right subtree</span>
+<a id="__codelineno-27-13" name="__codelineno-27-13" href="#__codelineno-27-13"></a><span class="w">        </span><span class="c1">// The node to remove lies in the right subtree of cur</span>
 <a id="__codelineno-27-14" name="__codelineno-27-14" href="#__codelineno-27-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-27-15" name="__codelineno-27-15" href="#__codelineno-27-15"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-27-16" name="__codelineno-27-16" href="#__codelineno-27-16"></a><span class="w">        </span><span class="c1">// Node to delete is in cur's left subtree</span>
+<a id="__codelineno-27-16" name="__codelineno-27-16" href="#__codelineno-27-16"></a><span class="w">        </span><span class="c1">// The node to remove lies in the left subtree of cur</span>
 <a id="__codelineno-27-17" name="__codelineno-27-17" href="#__codelineno-27-17"></a><span class="w">        </span><span class="k">else</span>
 <a id="__codelineno-27-18" name="__codelineno-27-18" href="#__codelineno-27-18"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-27-19" name="__codelineno-27-19" href="#__codelineno-27-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-27-20" name="__codelineno-27-20" href="#__codelineno-27-20"></a><span class="w">    </span><span class="c1">// If no node to delete, return directly</span>
+<a id="__codelineno-27-20" name="__codelineno-27-20" href="#__codelineno-27-20"></a><span class="w">    </span><span class="c1">// Return immediately if the node to remove does not exist</span>
 <a id="__codelineno-27-21" name="__codelineno-27-21" href="#__codelineno-27-21"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span>
 <a id="__codelineno-27-22" name="__codelineno-27-22" href="#__codelineno-27-22"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-27-23" name="__codelineno-27-23" href="#__codelineno-27-23"></a><span class="w">    </span><span class="c1">// Number of child nodes = 0 or 1</span>
+<a id="__codelineno-27-23" name="__codelineno-27-23" href="#__codelineno-27-23"></a><span class="w">    </span><span class="c1">// Number of children = 0 or 1</span>
 <a id="__codelineno-27-24" name="__codelineno-27-24" href="#__codelineno-27-24"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-25" name="__codelineno-27-25" href="#__codelineno-27-25"></a><span class="w">        </span><span class="c1">// When number of child nodes = 0 / 1, child = nullptr / that child node</span>
+<a id="__codelineno-27-25" name="__codelineno-27-25" href="#__codelineno-27-25"></a><span class="w">        </span><span class="c1">// With 0 or 1 child, child is nullptr or that child, respectively</span>
 <a id="__codelineno-27-26" name="__codelineno-27-26" href="#__codelineno-27-26"></a><span class="w">        </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-27-27" name="__codelineno-27-27" href="#__codelineno-27-27"></a><span class="w">        </span><span class="c1">// Delete node cur</span>
+<a id="__codelineno-27-27" name="__codelineno-27-27" href="#__codelineno-27-27"></a><span class="w">        </span><span class="c1">// Remove node cur</span>
 <a id="__codelineno-27-28" name="__codelineno-27-28" href="#__codelineno-27-28"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-29" name="__codelineno-27-29" href="#__codelineno-27-29"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">pre</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">cur</span><span class="p">)</span>
 <a id="__codelineno-27-30" name="__codelineno-27-30" href="#__codelineno-27-30"></a><span class="w">                </span><span class="k">pre</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-27-31" name="__codelineno-27-31" href="#__codelineno-27-31"></a><span class="w">            </span><span class="k">else</span>
 <a id="__codelineno-27-32" name="__codelineno-27-32" href="#__codelineno-27-32"></a><span class="w">                </span><span class="k">pre</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-27-33" name="__codelineno-27-33" href="#__codelineno-27-33"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-34" name="__codelineno-27-34" href="#__codelineno-27-34"></a><span class="w">            </span><span class="c1">// If deleted node is root node, reassign root node</span>
+<a id="__codelineno-27-34" name="__codelineno-27-34" href="#__codelineno-27-34"></a><span class="w">            </span><span class="c1">// If the removed node is the root, assign the new root</span>
 <a id="__codelineno-27-35" name="__codelineno-27-35" href="#__codelineno-27-35"></a><span class="w">            </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-27-36" name="__codelineno-27-36" href="#__codelineno-27-36"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-27-37" name="__codelineno-27-37" href="#__codelineno-27-37"></a><span class="w">        </span><span class="c1">// Free memory</span>
 <a id="__codelineno-27-38" name="__codelineno-27-38" href="#__codelineno-27-38"></a><span class="w">        </span><span class="k">delete</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
 <a id="__codelineno-27-39" name="__codelineno-27-39" href="#__codelineno-27-39"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-27-40" name="__codelineno-27-40" href="#__codelineno-27-40"></a><span class="w">    </span><span class="c1">// Number of child nodes = 2</span>
+<a id="__codelineno-27-40" name="__codelineno-27-40" href="#__codelineno-27-40"></a><span class="w">    </span><span class="c1">// Number of children = 2</span>
 <a id="__codelineno-27-41" name="__codelineno-27-41" href="#__codelineno-27-41"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-27-42" name="__codelineno-27-42" href="#__codelineno-27-42"></a><span class="w">        </span><span class="c1">// Get next node of cur in inorder traversal</span>
+<a id="__codelineno-27-42" name="__codelineno-27-42" href="#__codelineno-27-42"></a><span class="w">        </span><span class="c1">// Get the inorder successor of cur</span>
 <a id="__codelineno-27-43" name="__codelineno-27-43" href="#__codelineno-27-43"></a><span class="w">        </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-27-44" name="__codelineno-27-44" href="#__codelineno-27-44"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">tmp</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-27-45" name="__codelineno-27-45" href="#__codelineno-27-45"></a><span class="w">            </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-27-46" name="__codelineno-27-46" href="#__codelineno-27-46"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-27-47" name="__codelineno-27-47" href="#__codelineno-27-47"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">tmpVal</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
-<a id="__codelineno-27-48" name="__codelineno-27-48" href="#__codelineno-27-48"></a><span class="w">        </span><span class="c1">// Recursively delete node tmp</span>
+<a id="__codelineno-27-48" name="__codelineno-27-48" href="#__codelineno-27-48"></a><span class="w">        </span><span class="c1">// Recursively remove node tmp</span>
 <a id="__codelineno-27-49" name="__codelineno-27-49" href="#__codelineno-27-49"></a><span class="w">        </span><span class="n">remove</span><span class="p">(</span><span class="n">tmp</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-27-50" name="__codelineno-27-50" href="#__codelineno-27-50"></a><span class="w">        </span><span class="c1">// Replace cur with tmp</span>
+<a id="__codelineno-27-50" name="__codelineno-27-50" href="#__codelineno-27-50"></a><span class="w">        </span><span class="c1">// Replace the value of cur with the value of tmp</span>
 <a id="__codelineno-27-51" name="__codelineno-27-51" href="#__codelineno-27-51"></a><span class="w">        </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmpVal</span><span class="p">;</span>
 <a id="__codelineno-27-52" name="__codelineno-27-52" href="#__codelineno-27-52"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-27-53" name="__codelineno-27-53" href="#__codelineno-27-53"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="c1">// If tree is empty, return directly</span>
+<a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="w">    </span><span class="c1">// Return immediately if the tree is empty</span>
 <a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-28-7" name="__codelineno-28-7" href="#__codelineno-28-7"></a><span class="w">    </span><span class="c1">// Search in a loop until reaching a null reference beyond a leaf</span>
 <a id="__codelineno-28-8" name="__codelineno-28-8" href="#__codelineno-28-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="w">        </span><span class="c1">// Found node to delete, exit loop</span>
+<a id="__codelineno-28-9" name="__codelineno-28-9" href="#__codelineno-28-9"></a><span class="w">        </span><span class="c1">// The node to remove was found; exit the loop</span>
 <a id="__codelineno-28-10" name="__codelineno-28-10" href="#__codelineno-28-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-28-11" name="__codelineno-28-11" href="#__codelineno-28-11"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-28-12" name="__codelineno-28-12" href="#__codelineno-28-12"></a><span class="w">        </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
-<a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">        </span><span class="c1">// Node to delete is in cur's right subtree</span>
+<a id="__codelineno-28-13" name="__codelineno-28-13" href="#__codelineno-28-13"></a><span class="w">        </span><span class="c1">// The node to remove lies in the right subtree of cur</span>
 <a id="__codelineno-28-14" name="__codelineno-28-14" href="#__codelineno-28-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-28-15" name="__codelineno-28-15" href="#__codelineno-28-15"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span><span class="p">;</span>
-<a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a><span class="w">        </span><span class="c1">// Node to delete is in cur's left subtree</span>
+<a id="__codelineno-28-16" name="__codelineno-28-16" href="#__codelineno-28-16"></a><span class="w">        </span><span class="c1">// The node to remove lies in the left subtree of cur</span>
 <a id="__codelineno-28-17" name="__codelineno-28-17" href="#__codelineno-28-17"></a><span class="w">        </span><span class="k">else</span>
 <a id="__codelineno-28-18" name="__codelineno-28-18" href="#__codelineno-28-18"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">left</span><span class="p">;</span>
 <a id="__codelineno-28-19" name="__codelineno-28-19" href="#__codelineno-28-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-28-20" name="__codelineno-28-20" href="#__codelineno-28-20"></a><span class="w">    </span><span class="c1">// If no node to delete, return directly</span>
+<a id="__codelineno-28-20" name="__codelineno-28-20" href="#__codelineno-28-20"></a><span class="w">    </span><span class="c1">// Return immediately if the node to remove does not exist</span>
 <a id="__codelineno-28-21" name="__codelineno-28-21" href="#__codelineno-28-21"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-28-22" name="__codelineno-28-22" href="#__codelineno-28-22"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-28-23" name="__codelineno-28-23" href="#__codelineno-28-23"></a><span class="w">    </span><span class="c1">// Number of child nodes = 0 or 1</span>
+<a id="__codelineno-28-23" name="__codelineno-28-23" href="#__codelineno-28-23"></a><span class="w">    </span><span class="c1">// Number of children = 0 or 1</span>
 <a id="__codelineno-28-24" name="__codelineno-28-24" href="#__codelineno-28-24"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-25" name="__codelineno-28-25" href="#__codelineno-28-25"></a><span class="w">        </span><span class="c1">// When number of child nodes = 0 / 1, child = null / that child node</span>
+<a id="__codelineno-28-25" name="__codelineno-28-25" href="#__codelineno-28-25"></a><span class="w">        </span><span class="c1">// With 0 / 1 children, child is null / the child node, respectively</span>
 <a id="__codelineno-28-26" name="__codelineno-28-26" href="#__codelineno-28-26"></a><span class="w">        </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span><span class="p">;</span>
-<a id="__codelineno-28-27" name="__codelineno-28-27" href="#__codelineno-28-27"></a><span class="w">        </span><span class="c1">// Delete node cur</span>
+<a id="__codelineno-28-27" name="__codelineno-28-27" href="#__codelineno-28-27"></a><span class="w">        </span><span class="c1">// Remove node cur</span>
 <a id="__codelineno-28-28" name="__codelineno-28-28" href="#__codelineno-28-28"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-29" name="__codelineno-28-29" href="#__codelineno-28-29"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">cur</span><span class="p">)</span>
 <a id="__codelineno-28-30" name="__codelineno-28-30" href="#__codelineno-28-30"></a><span class="w">                </span><span class="n">pre</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-28-31" name="__codelineno-28-31" href="#__codelineno-28-31"></a><span class="w">            </span><span class="k">else</span>
 <a id="__codelineno-28-32" name="__codelineno-28-32" href="#__codelineno-28-32"></a><span class="w">                </span><span class="n">pre</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-28-33" name="__codelineno-28-33" href="#__codelineno-28-33"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-34" name="__codelineno-28-34" href="#__codelineno-28-34"></a><span class="w">            </span><span class="c1">// If deleted node is root node, reassign root node</span>
+<a id="__codelineno-28-34" name="__codelineno-28-34" href="#__codelineno-28-34"></a><span class="w">            </span><span class="c1">// If the removed node is the root, assign the new root</span>
 <a id="__codelineno-28-35" name="__codelineno-28-35" href="#__codelineno-28-35"></a><span class="w">            </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-28-36" name="__codelineno-28-36" href="#__codelineno-28-36"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-28-37" name="__codelineno-28-37" href="#__codelineno-28-37"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-28-38" name="__codelineno-28-38" href="#__codelineno-28-38"></a><span class="w">    </span><span class="c1">// Number of child nodes = 2</span>
+<a id="__codelineno-28-38" name="__codelineno-28-38" href="#__codelineno-28-38"></a><span class="w">    </span><span class="c1">// Number of children = 2</span>
 <a id="__codelineno-28-39" name="__codelineno-28-39" href="#__codelineno-28-39"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-28-40" name="__codelineno-28-40" href="#__codelineno-28-40"></a><span class="w">        </span><span class="c1">// Get next node of cur in inorder traversal</span>
+<a id="__codelineno-28-40" name="__codelineno-28-40" href="#__codelineno-28-40"></a><span class="w">        </span><span class="c1">// Get the inorder successor of cur</span>
 <a id="__codelineno-28-41" name="__codelineno-28-41" href="#__codelineno-28-41"></a><span class="w">        </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span><span class="p">;</span>
 <a id="__codelineno-28-42" name="__codelineno-28-42" href="#__codelineno-28-42"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">tmp</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-28-43" name="__codelineno-28-43" href="#__codelineno-28-43"></a><span class="w">            </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="na">left</span><span class="p">;</span>
 <a id="__codelineno-28-44" name="__codelineno-28-44" href="#__codelineno-28-44"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-28-45" name="__codelineno-28-45" href="#__codelineno-28-45"></a><span class="w">        </span><span class="c1">// Recursively delete node tmp</span>
+<a id="__codelineno-28-45" name="__codelineno-28-45" href="#__codelineno-28-45"></a><span class="w">        </span><span class="c1">// Recursively remove node tmp</span>
 <a id="__codelineno-28-46" name="__codelineno-28-46" href="#__codelineno-28-46"></a><span class="w">        </span><span class="n">remove</span><span class="p">(</span><span class="n">tmp</span><span class="p">.</span><span class="na">val</span><span class="p">);</span>
-<a id="__codelineno-28-47" name="__codelineno-28-47" href="#__codelineno-28-47"></a><span class="w">        </span><span class="c1">// Replace cur with tmp</span>
+<a id="__codelineno-28-47" name="__codelineno-28-47" href="#__codelineno-28-47"></a><span class="w">        </span><span class="c1">// Replace the value of cur with the value of tmp</span>
 <a id="__codelineno-28-48" name="__codelineno-28-48" href="#__codelineno-28-48"></a><span class="w">        </span><span class="n">cur</span><span class="p">.</span><span class="na">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="na">val</span><span class="p">;</span>
 <a id="__codelineno-28-49" name="__codelineno-28-49" href="#__codelineno-28-49"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-28-50" name="__codelineno-28-50" href="#__codelineno-28-50"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">Remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="c1">// If tree is empty, return directly</span>
+<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="w">    </span><span class="c1">// Return immediately if the tree is empty</span>
 <a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span>
 <a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="w">    </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">null</span><span class="p">;</span>
-<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="w">    </span><span class="c1">// Search in a loop, stopping if a null reference is reached</span>
 <a id="__codelineno-29-8" name="__codelineno-29-8" href="#__codelineno-29-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">        </span><span class="c1">// Found node to delete, exit loop</span>
+<a id="__codelineno-29-9" name="__codelineno-29-9" href="#__codelineno-29-9"></a><span class="w">        </span><span class="c1">// The node to remove was found; exit the loop</span>
 <a id="__codelineno-29-10" name="__codelineno-29-10" href="#__codelineno-29-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-29-11" name="__codelineno-29-11" href="#__codelineno-29-11"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-29-12" name="__codelineno-29-12" href="#__codelineno-29-12"></a><span class="w">        </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
-<a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">        </span><span class="c1">// Node to delete is in cur's right subtree</span>
+<a id="__codelineno-29-13" name="__codelineno-29-13" href="#__codelineno-29-13"></a><span class="w">        </span><span class="c1">// The node to remove lies in the right subtree of cur</span>
 <a id="__codelineno-29-14" name="__codelineno-29-14" href="#__codelineno-29-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-29-15" name="__codelineno-29-15" href="#__codelineno-29-15"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="w">        </span><span class="c1">// Node to delete is in cur's left subtree</span>
+<a id="__codelineno-29-16" name="__codelineno-29-16" href="#__codelineno-29-16"></a><span class="w">        </span><span class="c1">// The node to remove lies in the left subtree of cur</span>
 <a id="__codelineno-29-17" name="__codelineno-29-17" href="#__codelineno-29-17"></a><span class="w">        </span><span class="k">else</span>
 <a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-29-19" name="__codelineno-29-19" href="#__codelineno-29-19"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-29-20" name="__codelineno-29-20" href="#__codelineno-29-20"></a><span class="w">    </span><span class="c1">// If no node to delete, return directly</span>
+<a id="__codelineno-29-20" name="__codelineno-29-20" href="#__codelineno-29-20"></a><span class="w">    </span><span class="c1">// Return immediately if the node to remove does not exist</span>
 <a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span>
 <a id="__codelineno-29-22" name="__codelineno-29-22" href="#__codelineno-29-22"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-29-23" name="__codelineno-29-23" href="#__codelineno-29-23"></a><span class="w">    </span><span class="c1">// Number of child nodes = 0 or 1</span>
+<a id="__codelineno-29-23" name="__codelineno-29-23" href="#__codelineno-29-23"></a><span class="w">    </span><span class="c1">// Number of children = 0 or 1</span>
 <a id="__codelineno-29-24" name="__codelineno-29-24" href="#__codelineno-29-24"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-25" name="__codelineno-29-25" href="#__codelineno-29-25"></a><span class="w">        </span><span class="c1">// When number of child nodes = 0 / 1, child = null / that child node</span>
+<a id="__codelineno-29-25" name="__codelineno-29-25" href="#__codelineno-29-25"></a><span class="w">        </span><span class="c1">// With 0 / 1 children, child is null / the child node, respectively</span>
 <a id="__codelineno-29-26" name="__codelineno-29-26" href="#__codelineno-29-26"></a><span class="w">        </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">??</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-29-27" name="__codelineno-29-27" href="#__codelineno-29-27"></a><span class="w">        </span><span class="c1">// Delete node cur</span>
+<a id="__codelineno-29-27" name="__codelineno-29-27" href="#__codelineno-29-27"></a><span class="w">        </span><span class="c1">// Remove node cur</span>
 <a id="__codelineno-29-28" name="__codelineno-29-28" href="#__codelineno-29-28"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-29" name="__codelineno-29-29" href="#__codelineno-29-29"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="o">!</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">cur</span><span class="p">)</span>
 <a id="__codelineno-29-30" name="__codelineno-29-30" href="#__codelineno-29-30"></a><span class="w">                </span><span class="n">pre</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-29-31" name="__codelineno-29-31" href="#__codelineno-29-31"></a><span class="w">            </span><span class="k">else</span>
 <a id="__codelineno-29-32" name="__codelineno-29-32" href="#__codelineno-29-32"></a><span class="w">                </span><span class="n">pre</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-29-33" name="__codelineno-29-33" href="#__codelineno-29-33"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-34" name="__codelineno-29-34" href="#__codelineno-29-34"></a><span class="w">            </span><span class="c1">// If deleted node is root node, reassign root node</span>
+<a id="__codelineno-29-34" name="__codelineno-29-34" href="#__codelineno-29-34"></a><span class="w">            </span><span class="c1">// If the removed node is the root, assign the new root</span>
 <a id="__codelineno-29-35" name="__codelineno-29-35" href="#__codelineno-29-35"></a><span class="w">            </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-29-36" name="__codelineno-29-36" href="#__codelineno-29-36"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-29-37" name="__codelineno-29-37" href="#__codelineno-29-37"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-29-38" name="__codelineno-29-38" href="#__codelineno-29-38"></a><span class="w">    </span><span class="c1">// Number of child nodes = 2</span>
+<a id="__codelineno-29-38" name="__codelineno-29-38" href="#__codelineno-29-38"></a><span class="w">    </span><span class="c1">// Number of children = 2</span>
 <a id="__codelineno-29-39" name="__codelineno-29-39" href="#__codelineno-29-39"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-29-40" name="__codelineno-29-40" href="#__codelineno-29-40"></a><span class="w">        </span><span class="c1">// Get next node of cur in inorder traversal</span>
+<a id="__codelineno-29-40" name="__codelineno-29-40" href="#__codelineno-29-40"></a><span class="w">        </span><span class="c1">// Get the inorder successor of cur</span>
 <a id="__codelineno-29-41" name="__codelineno-29-41" href="#__codelineno-29-41"></a><span class="w">        </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-29-42" name="__codelineno-29-42" href="#__codelineno-29-42"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">tmp</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-29-43" name="__codelineno-29-43" href="#__codelineno-29-43"></a><span class="w">            </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-29-44" name="__codelineno-29-44" href="#__codelineno-29-44"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-29-45" name="__codelineno-29-45" href="#__codelineno-29-45"></a><span class="w">        </span><span class="c1">// Recursively delete node tmp</span>
+<a id="__codelineno-29-45" name="__codelineno-29-45" href="#__codelineno-29-45"></a><span class="w">        </span><span class="c1">// Recursively remove node tmp</span>
 <a id="__codelineno-29-46" name="__codelineno-29-46" href="#__codelineno-29-46"></a><span class="w">        </span><span class="n">Remove</span><span class="p">(</span><span class="n">tmp</span><span class="p">.</span><span class="n">val</span><span class="o">!</span><span class="p">.</span><span class="n">Value</span><span class="p">);</span>
-<a id="__codelineno-29-47" name="__codelineno-29-47" href="#__codelineno-29-47"></a><span class="w">        </span><span class="c1">// Replace cur with tmp</span>
+<a id="__codelineno-29-47" name="__codelineno-29-47" href="#__codelineno-29-47"></a><span class="w">        </span><span class="c1">// Replace the value of cur with the value of tmp</span>
 <a id="__codelineno-29-48" name="__codelineno-29-48" href="#__codelineno-29-48"></a><span class="w">        </span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-29-49" name="__codelineno-29-49" href="#__codelineno-29-49"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-29-50" name="__codelineno-29-50" href="#__codelineno-29-50"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="kd">func</span><span class="w"> </span><span class="p">(</span><span class="nx">bst</span><span class="w"> </span><span class="o">*</span><span class="nx">binarySearchTree</span><span class="p">)</span><span class="w"> </span><span class="nx">remove</span><span class="p">(</span><span class="nx">num</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="w">    </span><span class="nx">cur</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">bst</span><span class="p">.</span><span class="nx">root</span>
-<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="c1">// If tree is empty, return directly</span>
+<a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="w">    </span><span class="c1">// Return immediately if the tree is empty</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">    </span><span class="c1">// Node position before the node to be removed</span>
+<a id="__codelineno-30-8" name="__codelineno-30-8" href="#__codelineno-30-8"></a><span class="w">    </span><span class="c1">// Track the parent of the node to remove</span>
 <a id="__codelineno-30-9" name="__codelineno-30-9" href="#__codelineno-30-9"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nx">pre</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">nil</span>
-<a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-30-10" name="__codelineno-30-10" href="#__codelineno-30-10"></a><span class="w">    </span><span class="c1">// Continue searching while the current node is not nil</span>
 <a id="__codelineno-30-11" name="__codelineno-30-11" href="#__codelineno-30-11"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-12" name="__codelineno-30-12" href="#__codelineno-30-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">Val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-13" name="__codelineno-30-13" href="#__codelineno-30-13"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-30-14" name="__codelineno-30-14" href="#__codelineno-30-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-30-15" name="__codelineno-30-15" href="#__codelineno-30-15"></a><span class="w">        </span><span class="nx">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span>
 <a id="__codelineno-30-16" name="__codelineno-30-16" href="#__codelineno-30-16"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">Val</span><span class="p">.(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">num</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></a><span class="w">            </span><span class="c1">// Node to be removed is in right subtree</span>
+<a id="__codelineno-30-17" name="__codelineno-30-17" href="#__codelineno-30-17"></a><span class="w">            </span><span class="c1">// The node to remove is in the right subtree</span>
 <a id="__codelineno-30-18" name="__codelineno-30-18" href="#__codelineno-30-18"></a><span class="w">            </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">Right</span>
 <a id="__codelineno-30-19" name="__codelineno-30-19" href="#__codelineno-30-19"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-20" name="__codelineno-30-20" href="#__codelineno-30-20"></a><span class="w">            </span><span class="c1">// Node to be removed is in left subtree</span>
+<a id="__codelineno-30-20" name="__codelineno-30-20" href="#__codelineno-30-20"></a><span class="w">            </span><span class="c1">// The node to remove is in the left subtree</span>
 <a id="__codelineno-30-21" name="__codelineno-30-21" href="#__codelineno-30-21"></a><span class="w">            </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">Left</span>
 <a id="__codelineno-30-22" name="__codelineno-30-22" href="#__codelineno-30-22"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-30-23" name="__codelineno-30-23" href="#__codelineno-30-23"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-24" name="__codelineno-30-24" href="#__codelineno-30-24"></a><span class="w">    </span><span class="c1">// If no node to delete, return directly</span>
+<a id="__codelineno-30-24" name="__codelineno-30-24" href="#__codelineno-30-24"></a><span class="w">    </span><span class="c1">// Return immediately if the node to remove does not exist</span>
 <a id="__codelineno-30-25" name="__codelineno-30-25" href="#__codelineno-30-25"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-26" name="__codelineno-30-26" href="#__codelineno-30-26"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-30-27" name="__codelineno-30-27" href="#__codelineno-30-27"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-30-28" name="__codelineno-30-28" href="#__codelineno-30-28"></a><span class="w">    </span><span class="c1">// Number of child nodes is 0 or 1</span>
+<a id="__codelineno-30-28" name="__codelineno-30-28" href="#__codelineno-30-28"></a><span class="w">    </span><span class="c1">// The node has 0 or 1 child</span>
 <a id="__codelineno-30-29" name="__codelineno-30-29" href="#__codelineno-30-29"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">Right</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-30" name="__codelineno-30-30" href="#__codelineno-30-30"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nx">child</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">nil</span>
-<a id="__codelineno-30-31" name="__codelineno-30-31" href="#__codelineno-30-31"></a><span class="w">        </span><span class="c1">// Get child node of node to be removed</span>
+<a id="__codelineno-30-31" name="__codelineno-30-31" href="#__codelineno-30-31"></a><span class="w">        </span><span class="c1">// Get the child of the node to remove</span>
 <a id="__codelineno-30-32" name="__codelineno-30-32" href="#__codelineno-30-32"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-33" name="__codelineno-30-33" href="#__codelineno-30-33"></a><span class="w">            </span><span class="nx">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">Left</span>
 <a id="__codelineno-30-34" name="__codelineno-30-34" href="#__codelineno-30-34"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-35" name="__codelineno-30-35" href="#__codelineno-30-35"></a><span class="w">            </span><span class="nx">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">Right</span>
 <a id="__codelineno-30-36" name="__codelineno-30-36" href="#__codelineno-30-36"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-30-37" name="__codelineno-30-37" href="#__codelineno-30-37"></a><span class="w">        </span><span class="c1">// Delete node cur</span>
+<a id="__codelineno-30-37" name="__codelineno-30-37" href="#__codelineno-30-37"></a><span class="w">        </span><span class="c1">// Remove node cur</span>
 <a id="__codelineno-30-38" name="__codelineno-30-38" href="#__codelineno-30-38"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nx">bst</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-39" name="__codelineno-30-39" href="#__codelineno-30-39"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="nx">pre</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-40" name="__codelineno-30-40" href="#__codelineno-30-40"></a><span class="w">                </span><span class="nx">pre</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span>
@@ -5819,58 +5901,58 @@
 <a id="__codelineno-30-42" name="__codelineno-30-42" href="#__codelineno-30-42"></a><span class="w">                </span><span class="nx">pre</span><span class="p">.</span><span class="nx">Right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span>
 <a id="__codelineno-30-43" name="__codelineno-30-43" href="#__codelineno-30-43"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-30-44" name="__codelineno-30-44" href="#__codelineno-30-44"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-45" name="__codelineno-30-45" href="#__codelineno-30-45"></a><span class="w">            </span><span class="c1">// If deleted node is root node, reassign root node</span>
+<a id="__codelineno-30-45" name="__codelineno-30-45" href="#__codelineno-30-45"></a><span class="w">            </span><span class="c1">// If the removed node is the root, assign the new root</span>
 <a id="__codelineno-30-46" name="__codelineno-30-46" href="#__codelineno-30-46"></a><span class="w">            </span><span class="nx">bst</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span>
 <a id="__codelineno-30-47" name="__codelineno-30-47" href="#__codelineno-30-47"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-30-48" name="__codelineno-30-48" href="#__codelineno-30-48"></a><span class="w">        </span><span class="c1">// Number of child nodes is 2</span>
+<a id="__codelineno-30-48" name="__codelineno-30-48" href="#__codelineno-30-48"></a><span class="w">        </span><span class="c1">// The node has 2 children</span>
 <a id="__codelineno-30-49" name="__codelineno-30-49" href="#__codelineno-30-49"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-30-50" name="__codelineno-30-50" href="#__codelineno-30-50"></a><span class="w">        </span><span class="c1">// Get next node of node cur to be removed in in-order traversal</span>
+<a id="__codelineno-30-50" name="__codelineno-30-50" href="#__codelineno-30-50"></a><span class="w">        </span><span class="c1">// Get the inorder successor of cur, the node to remove</span>
 <a id="__codelineno-30-51" name="__codelineno-30-51" href="#__codelineno-30-51"></a><span class="w">        </span><span class="nx">tmp</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">Right</span>
 <a id="__codelineno-30-52" name="__codelineno-30-52" href="#__codelineno-30-52"></a><span class="w">        </span><span class="k">for</span><span class="w"> </span><span class="nx">tmp</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-30-53" name="__codelineno-30-53" href="#__codelineno-30-53"></a><span class="w">            </span><span class="nx">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span><span class="p">.</span><span class="nx">Left</span>
 <a id="__codelineno-30-54" name="__codelineno-30-54" href="#__codelineno-30-54"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-30-55" name="__codelineno-30-55" href="#__codelineno-30-55"></a><span class="w">        </span><span class="c1">// Recursively delete node tmp</span>
+<a id="__codelineno-30-55" name="__codelineno-30-55" href="#__codelineno-30-55"></a><span class="w">        </span><span class="c1">// Recursively remove node tmp</span>
 <a id="__codelineno-30-56" name="__codelineno-30-56" href="#__codelineno-30-56"></a><span class="w">        </span><span class="nx">bst</span><span class="p">.</span><span class="nx">remove</span><span class="p">(</span><span class="nx">tmp</span><span class="p">.</span><span class="nx">Val</span><span class="p">.(</span><span class="kt">int</span><span class="p">))</span>
-<a id="__codelineno-30-57" name="__codelineno-30-57" href="#__codelineno-30-57"></a><span class="w">        </span><span class="c1">// Replace cur with tmp</span>
+<a id="__codelineno-30-57" name="__codelineno-30-57" href="#__codelineno-30-57"></a><span class="w">        </span><span class="c1">// Replace the value of cur with the value of tmp</span>
 <a id="__codelineno-30-58" name="__codelineno-30-58" href="#__codelineno-30-58"></a><span class="w">        </span><span class="nx">cur</span><span class="p">.</span><span class="nx">Val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span><span class="p">.</span><span class="nx">Val</span>
 <a id="__codelineno-30-59" name="__codelineno-30-59" href="#__codelineno-30-59"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-30-60" name="__codelineno-30-60" href="#__codelineno-30-60"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="c1">// If tree is empty, return directly</span>
+<a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="w">    </span><span class="c1">// Return immediately if the tree is empty</span>
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-7" name="__codelineno-31-7" href="#__codelineno-31-7"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">cur</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">root</span>
 <a id="__codelineno-31-8" name="__codelineno-31-8" href="#__codelineno-31-8"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">pre</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?</span>
-<a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-31-9" name="__codelineno-31-9" href="#__codelineno-31-9"></a><span class="w">    </span><span class="c1">// Search in a loop, stopping if cur becomes nil</span>
 <a id="__codelineno-31-10" name="__codelineno-31-10" href="#__codelineno-31-10"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">        </span><span class="c1">// Found node to delete, exit loop</span>
+<a id="__codelineno-31-11" name="__codelineno-31-11" href="#__codelineno-31-11"></a><span class="w">        </span><span class="c1">// The node to remove was found; exit the loop</span>
 <a id="__codelineno-31-12" name="__codelineno-31-12" href="#__codelineno-31-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-13" name="__codelineno-31-13" href="#__codelineno-31-13"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-31-14" name="__codelineno-31-14" href="#__codelineno-31-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-31-15" name="__codelineno-31-15" href="#__codelineno-31-15"></a><span class="w">        </span><span class="n">pre</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">cur</span>
-<a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="w">        </span><span class="c1">// Node to delete is in cur's right subtree</span>
+<a id="__codelineno-31-16" name="__codelineno-31-16" href="#__codelineno-31-16"></a><span class="w">        </span><span class="c1">// If present, the node to remove lies in the right subtree of cur</span>
 <a id="__codelineno-31-17" name="__codelineno-31-17" href="#__codelineno-31-17"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="p">!.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-18" name="__codelineno-31-18" href="#__codelineno-31-18"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">cur</span><span class="p">?.</span><span class="kr">right</span>
 <a id="__codelineno-31-19" name="__codelineno-31-19" href="#__codelineno-31-19"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-31-20" name="__codelineno-31-20" href="#__codelineno-31-20"></a><span class="w">        </span><span class="c1">// Node to delete is in cur's left subtree</span>
+<a id="__codelineno-31-20" name="__codelineno-31-20" href="#__codelineno-31-20"></a><span class="w">        </span><span class="c1">// If present, the node to remove lies in the left subtree of cur</span>
 <a id="__codelineno-31-21" name="__codelineno-31-21" href="#__codelineno-31-21"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-22" name="__codelineno-31-22" href="#__codelineno-31-22"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">cur</span><span class="p">?.</span><span class="kr">left</span>
 <a id="__codelineno-31-23" name="__codelineno-31-23" href="#__codelineno-31-23"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-31-24" name="__codelineno-31-24" href="#__codelineno-31-24"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-31-25" name="__codelineno-31-25" href="#__codelineno-31-25"></a><span class="w">    </span><span class="c1">// If no node to delete, return directly</span>
+<a id="__codelineno-31-25" name="__codelineno-31-25" href="#__codelineno-31-25"></a><span class="w">    </span><span class="c1">// Return immediately if the node to remove does not exist</span>
 <a id="__codelineno-31-26" name="__codelineno-31-26" href="#__codelineno-31-26"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-27" name="__codelineno-31-27" href="#__codelineno-31-27"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-31-28" name="__codelineno-31-28" href="#__codelineno-31-28"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-31-29" name="__codelineno-31-29" href="#__codelineno-31-29"></a><span class="w">    </span><span class="c1">// Number of child nodes = 0 or 1</span>
+<a id="__codelineno-31-29" name="__codelineno-31-29" href="#__codelineno-31-29"></a><span class="w">    </span><span class="c1">// Number of children = 0 or 1</span>
 <a id="__codelineno-31-30" name="__codelineno-31-30" href="#__codelineno-31-30"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="p">?.</span><span class="kr">left</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">cur</span><span class="p">?.</span><span class="kr">right</span><span class="w"> </span><span class="p">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-31" name="__codelineno-31-31" href="#__codelineno-31-31"></a><span class="w">        </span><span class="c1">// When number of child nodes = 0 / 1, child = null / that child node</span>
+<a id="__codelineno-31-31" name="__codelineno-31-31" href="#__codelineno-31-31"></a><span class="w">        </span><span class="c1">// With 0 / 1 children, child is nil / the child node, respectively</span>
 <a id="__codelineno-31-32" name="__codelineno-31-32" href="#__codelineno-31-32"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">child</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">cur</span><span class="p">?.</span><span class="kr">left</span><span class="w"> </span><span class="p">??</span><span class="w"> </span><span class="n">cur</span><span class="p">?.</span><span class="kr">right</span>
-<a id="__codelineno-31-33" name="__codelineno-31-33" href="#__codelineno-31-33"></a><span class="w">        </span><span class="c1">// Delete node cur</span>
+<a id="__codelineno-31-33" name="__codelineno-31-33" href="#__codelineno-31-33"></a><span class="w">        </span><span class="c1">// Remove node cur</span>
 <a id="__codelineno-31-34" name="__codelineno-31-34" href="#__codelineno-31-34"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-35" name="__codelineno-31-35" href="#__codelineno-31-35"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="n">pre</span><span class="p">?.</span><span class="kr">left</span><span class="w"> </span><span class="p">===</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-36" name="__codelineno-31-36" href="#__codelineno-31-36"></a><span class="w">                </span><span class="n">pre</span><span class="p">?.</span><span class="kr">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">child</span>
@@ -5878,209 +5960,209 @@
 <a id="__codelineno-31-38" name="__codelineno-31-38" href="#__codelineno-31-38"></a><span class="w">                </span><span class="n">pre</span><span class="p">?.</span><span class="kr">right</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">child</span>
 <a id="__codelineno-31-39" name="__codelineno-31-39" href="#__codelineno-31-39"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-31-40" name="__codelineno-31-40" href="#__codelineno-31-40"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-41" name="__codelineno-31-41" href="#__codelineno-31-41"></a><span class="w">            </span><span class="c1">// If deleted node is root node, reassign root node</span>
+<a id="__codelineno-31-41" name="__codelineno-31-41" href="#__codelineno-31-41"></a><span class="w">            </span><span class="c1">// If the removed node is the root, assign the new root</span>
 <a id="__codelineno-31-42" name="__codelineno-31-42" href="#__codelineno-31-42"></a><span class="w">            </span><span class="n">root</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">child</span>
 <a id="__codelineno-31-43" name="__codelineno-31-43" href="#__codelineno-31-43"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-31-44" name="__codelineno-31-44" href="#__codelineno-31-44"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-31-45" name="__codelineno-31-45" href="#__codelineno-31-45"></a><span class="w">    </span><span class="c1">// Number of child nodes = 2</span>
+<a id="__codelineno-31-45" name="__codelineno-31-45" href="#__codelineno-31-45"></a><span class="w">    </span><span class="c1">// Number of children = 2</span>
 <a id="__codelineno-31-46" name="__codelineno-31-46" href="#__codelineno-31-46"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-31-47" name="__codelineno-31-47" href="#__codelineno-31-47"></a><span class="w">        </span><span class="c1">// Get next node of cur in inorder traversal</span>
+<a id="__codelineno-31-47" name="__codelineno-31-47" href="#__codelineno-31-47"></a><span class="w">        </span><span class="c1">// Get the inorder successor of cur</span>
 <a id="__codelineno-31-48" name="__codelineno-31-48" href="#__codelineno-31-48"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">tmp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">cur</span><span class="p">?.</span><span class="kr">right</span>
 <a id="__codelineno-31-49" name="__codelineno-31-49" href="#__codelineno-31-49"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="n">tmp</span><span class="p">?.</span><span class="kr">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-31-50" name="__codelineno-31-50" href="#__codelineno-31-50"></a><span class="w">            </span><span class="n">tmp</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">?.</span><span class="kr">left</span>
 <a id="__codelineno-31-51" name="__codelineno-31-51" href="#__codelineno-31-51"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-31-52" name="__codelineno-31-52" href="#__codelineno-31-52"></a><span class="w">        </span><span class="c1">// Recursively delete node tmp</span>
+<a id="__codelineno-31-52" name="__codelineno-31-52" href="#__codelineno-31-52"></a><span class="w">        </span><span class="c1">// Recursively remove node tmp</span>
 <a id="__codelineno-31-53" name="__codelineno-31-53" href="#__codelineno-31-53"></a><span class="w">        </span><span class="n">remove</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="n">tmp</span><span class="p">!.</span><span class="n">val</span><span class="p">)</span>
-<a id="__codelineno-31-54" name="__codelineno-31-54" href="#__codelineno-31-54"></a><span class="w">        </span><span class="c1">// Replace cur with tmp</span>
+<a id="__codelineno-31-54" name="__codelineno-31-54" href="#__codelineno-31-54"></a><span class="w">        </span><span class="c1">// Replace the value of cur with the value of tmp</span>
 <a id="__codelineno-31-55" name="__codelineno-31-55" href="#__codelineno-31-55"></a><span class="w">        </span><span class="n">cur</span><span class="p">?.</span><span class="n">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">!.</span><span class="n">val</span>
 <a id="__codelineno-31-56" name="__codelineno-31-56" href="#__codelineno-31-56"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-31-57" name="__codelineno-31-57" href="#__codelineno-31-57"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="nx">remove</span><span class="p">(</span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="c1">// If tree is empty, return directly</span>
+<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="w">    </span><span class="c1">// Return immediately if the tree is empty</span>
 <a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="p">,</span>
 <a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="w">        </span><span class="nx">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="w">    </span><span class="c1">// Search in a loop until reaching a null reference beyond a leaf</span>
 <a id="__codelineno-32-8" name="__codelineno-32-8" href="#__codelineno-32-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w">        </span><span class="c1">// Found node to delete, exit loop</span>
+<a id="__codelineno-32-9" name="__codelineno-32-9" href="#__codelineno-32-9"></a><span class="w">        </span><span class="c1">// The node to remove was found; exit the loop</span>
 <a id="__codelineno-32-10" name="__codelineno-32-10" href="#__codelineno-32-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-32-11" name="__codelineno-32-11" href="#__codelineno-32-11"></a><span class="w">        </span><span class="nx">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">;</span>
-<a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="w">        </span><span class="c1">// Node to delete is in cur's right subtree</span>
+<a id="__codelineno-32-12" name="__codelineno-32-12" href="#__codelineno-32-12"></a><span class="w">        </span><span class="c1">// The node to remove lies in the right subtree of cur</span>
 <a id="__codelineno-32-13" name="__codelineno-32-13" href="#__codelineno-32-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
-<a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="w">        </span><span class="c1">// Node to delete is in cur's left subtree</span>
+<a id="__codelineno-32-14" name="__codelineno-32-14" href="#__codelineno-32-14"></a><span class="w">        </span><span class="c1">// The node to remove lies in the left subtree of cur</span>
 <a id="__codelineno-32-15" name="__codelineno-32-15" href="#__codelineno-32-15"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">left</span><span class="p">;</span>
 <a id="__codelineno-32-16" name="__codelineno-32-16" href="#__codelineno-32-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-32-17" name="__codelineno-32-17" href="#__codelineno-32-17"></a><span class="w">    </span><span class="c1">// If no node to delete, return directly</span>
+<a id="__codelineno-32-17" name="__codelineno-32-17" href="#__codelineno-32-17"></a><span class="w">    </span><span class="c1">// Return immediately if the node to remove does not exist</span>
 <a id="__codelineno-32-18" name="__codelineno-32-18" href="#__codelineno-32-18"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-32-19" name="__codelineno-32-19" href="#__codelineno-32-19"></a><span class="w">    </span><span class="c1">// Number of child nodes = 0 or 1</span>
+<a id="__codelineno-32-19" name="__codelineno-32-19" href="#__codelineno-32-19"></a><span class="w">    </span><span class="c1">// Number of children = 0 or 1</span>
 <a id="__codelineno-32-20" name="__codelineno-32-20" href="#__codelineno-32-20"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-21" name="__codelineno-32-21" href="#__codelineno-32-21"></a><span class="w">        </span><span class="c1">// When number of child nodes = 0 / 1, child = null / that child node</span>
+<a id="__codelineno-32-21" name="__codelineno-32-21" href="#__codelineno-32-21"></a><span class="w">        </span><span class="c1">// With 0 / 1 children, child is null / the child node, respectively</span>
 <a id="__codelineno-32-22" name="__codelineno-32-22" href="#__codelineno-32-22"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
-<a id="__codelineno-32-23" name="__codelineno-32-23" href="#__codelineno-32-23"></a><span class="w">        </span><span class="c1">// Delete node cur</span>
+<a id="__codelineno-32-23" name="__codelineno-32-23" href="#__codelineno-32-23"></a><span class="w">        </span><span class="c1">// Remove node cur</span>
 <a id="__codelineno-32-24" name="__codelineno-32-24" href="#__codelineno-32-24"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-25" name="__codelineno-32-25" href="#__codelineno-32-25"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pre</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">cur</span><span class="p">)</span><span class="w"> </span><span class="nx">pre</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span><span class="p">;</span>
 <a id="__codelineno-32-26" name="__codelineno-32-26" href="#__codelineno-32-26"></a><span class="w">            </span><span class="k">else</span><span class="w"> </span><span class="nx">pre</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span><span class="p">;</span>
 <a id="__codelineno-32-27" name="__codelineno-32-27" href="#__codelineno-32-27"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-28" name="__codelineno-32-28" href="#__codelineno-32-28"></a><span class="w">            </span><span class="c1">// If deleted node is root node, reassign root node</span>
+<a id="__codelineno-32-28" name="__codelineno-32-28" href="#__codelineno-32-28"></a><span class="w">            </span><span class="c1">// If the removed node is the root, assign the new root</span>
 <a id="__codelineno-32-29" name="__codelineno-32-29" href="#__codelineno-32-29"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span><span class="p">;</span>
 <a id="__codelineno-32-30" name="__codelineno-32-30" href="#__codelineno-32-30"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-32-31" name="__codelineno-32-31" href="#__codelineno-32-31"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-32-32" name="__codelineno-32-32" href="#__codelineno-32-32"></a><span class="w">    </span><span class="c1">// Number of child nodes = 2</span>
+<a id="__codelineno-32-32" name="__codelineno-32-32" href="#__codelineno-32-32"></a><span class="w">    </span><span class="c1">// Number of children = 2</span>
 <a id="__codelineno-32-33" name="__codelineno-32-33" href="#__codelineno-32-33"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-32-34" name="__codelineno-32-34" href="#__codelineno-32-34"></a><span class="w">        </span><span class="c1">// Get next node of cur in inorder traversal</span>
+<a id="__codelineno-32-34" name="__codelineno-32-34" href="#__codelineno-32-34"></a><span class="w">        </span><span class="c1">// Get the inorder successor of cur</span>
 <a id="__codelineno-32-35" name="__codelineno-32-35" href="#__codelineno-32-35"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
 <a id="__codelineno-32-36" name="__codelineno-32-36" href="#__codelineno-32-36"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">tmp</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-32-37" name="__codelineno-32-37" href="#__codelineno-32-37"></a><span class="w">            </span><span class="nx">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span><span class="p">.</span><span class="nx">left</span><span class="p">;</span>
 <a id="__codelineno-32-38" name="__codelineno-32-38" href="#__codelineno-32-38"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-32-39" name="__codelineno-32-39" href="#__codelineno-32-39"></a><span class="w">        </span><span class="c1">// Recursively delete node tmp</span>
+<a id="__codelineno-32-39" name="__codelineno-32-39" href="#__codelineno-32-39"></a><span class="w">        </span><span class="c1">// Recursively remove node tmp</span>
 <a id="__codelineno-32-40" name="__codelineno-32-40" href="#__codelineno-32-40"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">remove</span><span class="p">(</span><span class="nx">tmp</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span>
-<a id="__codelineno-32-41" name="__codelineno-32-41" href="#__codelineno-32-41"></a><span class="w">        </span><span class="c1">// Replace cur with tmp</span>
+<a id="__codelineno-32-41" name="__codelineno-32-41" href="#__codelineno-32-41"></a><span class="w">        </span><span class="c1">// Replace the value of cur with the value of tmp</span>
 <a id="__codelineno-32-42" name="__codelineno-32-42" href="#__codelineno-32-42"></a><span class="w">        </span><span class="nx">cur</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-32-43" name="__codelineno-32-43" href="#__codelineno-32-43"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-32-44" name="__codelineno-32-44" href="#__codelineno-32-44"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="nx">remove</span><span class="p">(</span><span class="nx">num</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="ow">void</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="c1">// If tree is empty, return directly</span>
+<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="w">    </span><span class="c1">// Return immediately if the tree is empty</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="nx">cur</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="p">,</span>
 <a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="w">        </span><span class="nx">pre</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="w">    </span><span class="c1">// Search in a loop, stopping if a null reference is reached</span>
 <a id="__codelineno-33-8" name="__codelineno-33-8" href="#__codelineno-33-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">        </span><span class="c1">// Found node to delete, exit loop</span>
+<a id="__codelineno-33-9" name="__codelineno-33-9" href="#__codelineno-33-9"></a><span class="w">        </span><span class="c1">// The node to remove was found; exit the loop</span>
 <a id="__codelineno-33-10" name="__codelineno-33-10" href="#__codelineno-33-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-33-11" name="__codelineno-33-11" href="#__codelineno-33-11"></a><span class="w">        </span><span class="nx">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">;</span>
-<a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="w">        </span><span class="c1">// Node to delete is in cur's right subtree</span>
+<a id="__codelineno-33-12" name="__codelineno-33-12" href="#__codelineno-33-12"></a><span class="w">        </span><span class="c1">// The node to remove lies in the right subtree of cur</span>
 <a id="__codelineno-33-13" name="__codelineno-33-13" href="#__codelineno-33-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="nx">num</span><span class="p">)</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
-<a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">        </span><span class="c1">// Node to delete is in cur's left subtree</span>
+<a id="__codelineno-33-14" name="__codelineno-33-14" href="#__codelineno-33-14"></a><span class="w">        </span><span class="c1">// The node to remove lies in the left subtree of cur</span>
 <a id="__codelineno-33-15" name="__codelineno-33-15" href="#__codelineno-33-15"></a><span class="w">        </span><span class="k">else</span><span class="w"> </span><span class="nx">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">left</span><span class="p">;</span>
 <a id="__codelineno-33-16" name="__codelineno-33-16" href="#__codelineno-33-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-33-17" name="__codelineno-33-17" href="#__codelineno-33-17"></a><span class="w">    </span><span class="c1">// If no node to delete, return directly</span>
+<a id="__codelineno-33-17" name="__codelineno-33-17" href="#__codelineno-33-17"></a><span class="w">    </span><span class="c1">// Return immediately if the node to remove does not exist</span>
 <a id="__codelineno-33-18" name="__codelineno-33-18" href="#__codelineno-33-18"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-33-19" name="__codelineno-33-19" href="#__codelineno-33-19"></a><span class="w">    </span><span class="c1">// Number of child nodes = 0 or 1</span>
+<a id="__codelineno-33-19" name="__codelineno-33-19" href="#__codelineno-33-19"></a><span class="w">    </span><span class="c1">// Number of children = 0 or 1</span>
 <a id="__codelineno-33-20" name="__codelineno-33-20" href="#__codelineno-33-20"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-21" name="__codelineno-33-21" href="#__codelineno-33-21"></a><span class="w">        </span><span class="c1">// When number of child nodes = 0 / 1, child = null / that child node</span>
+<a id="__codelineno-33-21" name="__codelineno-33-21" href="#__codelineno-33-21"></a><span class="w">        </span><span class="c1">// With 0 / 1 children, child is null / the child node, respectively</span>
 <a id="__codelineno-33-22" name="__codelineno-33-22" href="#__codelineno-33-22"></a><span class="w">        </span><span class="kd">const</span><span class="w"> </span><span class="nx">child</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">=</span>
 <a id="__codelineno-33-23" name="__codelineno-33-23" href="#__codelineno-33-23"></a><span class="w">            </span><span class="nx">cur</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">cur.left</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">cur.right</span><span class="p">;</span>
-<a id="__codelineno-33-24" name="__codelineno-33-24" href="#__codelineno-33-24"></a><span class="w">        </span><span class="c1">// Delete node cur</span>
+<a id="__codelineno-33-24" name="__codelineno-33-24" href="#__codelineno-33-24"></a><span class="w">        </span><span class="c1">// Remove node cur</span>
 <a id="__codelineno-33-25" name="__codelineno-33-25" href="#__codelineno-33-25"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">cur</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-26" name="__codelineno-33-26" href="#__codelineno-33-26"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">pre</span><span class="o">!</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="nx">cur</span><span class="p">)</span><span class="w"> </span><span class="nx">pre</span><span class="o">!</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span><span class="p">;</span>
 <a id="__codelineno-33-27" name="__codelineno-33-27" href="#__codelineno-33-27"></a><span class="w">            </span><span class="k">else</span><span class="w"> </span><span class="nx">pre</span><span class="o">!</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span><span class="p">;</span>
 <a id="__codelineno-33-28" name="__codelineno-33-28" href="#__codelineno-33-28"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-29" name="__codelineno-33-29" href="#__codelineno-33-29"></a><span class="w">            </span><span class="c1">// If deleted node is root node, reassign root node</span>
+<a id="__codelineno-33-29" name="__codelineno-33-29" href="#__codelineno-33-29"></a><span class="w">            </span><span class="c1">// If the removed node is the root, assign the new root</span>
 <a id="__codelineno-33-30" name="__codelineno-33-30" href="#__codelineno-33-30"></a><span class="w">            </span><span class="k">this</span><span class="p">.</span><span class="nx">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">child</span><span class="p">;</span>
 <a id="__codelineno-33-31" name="__codelineno-33-31" href="#__codelineno-33-31"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-33-32" name="__codelineno-33-32" href="#__codelineno-33-32"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-33-33" name="__codelineno-33-33" href="#__codelineno-33-33"></a><span class="w">    </span><span class="c1">// Number of child nodes = 2</span>
+<a id="__codelineno-33-33" name="__codelineno-33-33" href="#__codelineno-33-33"></a><span class="w">    </span><span class="c1">// Number of children = 2</span>
 <a id="__codelineno-33-34" name="__codelineno-33-34" href="#__codelineno-33-34"></a><span class="w">    </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-33-35" name="__codelineno-33-35" href="#__codelineno-33-35"></a><span class="w">        </span><span class="c1">// Get next node of cur in inorder traversal</span>
+<a id="__codelineno-33-35" name="__codelineno-33-35" href="#__codelineno-33-35"></a><span class="w">        </span><span class="c1">// Get the inorder successor of cur</span>
 <a id="__codelineno-33-36" name="__codelineno-33-36" href="#__codelineno-33-36"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">tmp</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">cur</span><span class="p">.</span><span class="nx">right</span><span class="p">;</span>
 <a id="__codelineno-33-37" name="__codelineno-33-37" href="#__codelineno-33-37"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">tmp</span><span class="o">!</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">!==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-33-38" name="__codelineno-33-38" href="#__codelineno-33-38"></a><span class="w">            </span><span class="nx">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span><span class="o">!</span><span class="p">.</span><span class="nx">left</span><span class="p">;</span>
 <a id="__codelineno-33-39" name="__codelineno-33-39" href="#__codelineno-33-39"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-33-40" name="__codelineno-33-40" href="#__codelineno-33-40"></a><span class="w">        </span><span class="c1">// Recursively delete node tmp</span>
+<a id="__codelineno-33-40" name="__codelineno-33-40" href="#__codelineno-33-40"></a><span class="w">        </span><span class="c1">// Recursively remove node tmp</span>
 <a id="__codelineno-33-41" name="__codelineno-33-41" href="#__codelineno-33-41"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">remove</span><span class="p">(</span><span class="nx">tmp</span><span class="o">!</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span>
-<a id="__codelineno-33-42" name="__codelineno-33-42" href="#__codelineno-33-42"></a><span class="w">        </span><span class="c1">// Replace cur with tmp</span>
+<a id="__codelineno-33-42" name="__codelineno-33-42" href="#__codelineno-33-42"></a><span class="w">        </span><span class="c1">// Replace the value of cur with the value of tmp</span>
 <a id="__codelineno-33-43" name="__codelineno-33-43" href="#__codelineno-33-43"></a><span class="w">        </span><span class="nx">cur</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">tmp</span><span class="o">!</span><span class="p">.</span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-33-44" name="__codelineno-33-44" href="#__codelineno-33-44"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-33-45" name="__codelineno-33-45" href="#__codelineno-33-45"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">remove</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="c1">// If tree is empty, return directly</span>
+<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="w">  </span><span class="c1">// Return immediately if the tree is empty</span>
 <a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">_root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_root</span><span class="p">;</span>
 <a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
-<a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">  </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="w">  </span><span class="c1">// Search in a loop, stopping if a null reference is reached</span>
 <a id="__codelineno-34-8" name="__codelineno-34-8" href="#__codelineno-34-8"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w">    </span><span class="c1">// Found node to delete, exit loop</span>
+<a id="__codelineno-34-9" name="__codelineno-34-9" href="#__codelineno-34-9"></a><span class="w">    </span><span class="c1">// The node to remove was found; exit the loop</span>
 <a id="__codelineno-34-10" name="__codelineno-34-10" href="#__codelineno-34-10"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span><span class="w"> </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-34-11" name="__codelineno-34-11" href="#__codelineno-34-11"></a><span class="w">    </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
-<a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="w">    </span><span class="c1">// Node to delete is in cur's right subtree</span>
+<a id="__codelineno-34-12" name="__codelineno-34-12" href="#__codelineno-34-12"></a><span class="w">    </span><span class="c1">// The node to remove lies in the right subtree of cur</span>
 <a id="__codelineno-34-13" name="__codelineno-34-13" href="#__codelineno-34-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">_num</span><span class="p">)</span>
 <a id="__codelineno-34-14" name="__codelineno-34-14" href="#__codelineno-34-14"></a><span class="w">      </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">    </span><span class="c1">// Node to delete is in cur's left subtree</span>
+<a id="__codelineno-34-15" name="__codelineno-34-15" href="#__codelineno-34-15"></a><span class="w">    </span><span class="c1">// The node to remove lies in the left subtree of cur</span>
 <a id="__codelineno-34-16" name="__codelineno-34-16" href="#__codelineno-34-16"></a><span class="w">    </span><span class="k">else</span>
 <a id="__codelineno-34-17" name="__codelineno-34-17" href="#__codelineno-34-17"></a><span class="w">      </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-34-18" name="__codelineno-34-18" href="#__codelineno-34-18"></a><span class="w">  </span><span class="p">}</span>
-<a id="__codelineno-34-19" name="__codelineno-34-19" href="#__codelineno-34-19"></a><span class="w">  </span><span class="c1">// If no node to delete, return directly</span>
+<a id="__codelineno-34-19" name="__codelineno-34-19" href="#__codelineno-34-19"></a><span class="w">  </span><span class="c1">// Return if the node to remove does not exist</span>
 <a id="__codelineno-34-20" name="__codelineno-34-20" href="#__codelineno-34-20"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-34-21" name="__codelineno-34-21" href="#__codelineno-34-21"></a><span class="w">  </span><span class="c1">// Number of child nodes = 0 or 1</span>
+<a id="__codelineno-34-21" name="__codelineno-34-21" href="#__codelineno-34-21"></a><span class="w">  </span><span class="c1">// Number of children = 0 or 1</span>
 <a id="__codelineno-34-22" name="__codelineno-34-22" href="#__codelineno-34-22"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-23" name="__codelineno-34-23" href="#__codelineno-34-23"></a><span class="w">    </span><span class="c1">// When number of child nodes = 0 / 1, child = null / that child node</span>
+<a id="__codelineno-34-23" name="__codelineno-34-23" href="#__codelineno-34-23"></a><span class="w">    </span><span class="c1">// With 0 / 1 children, child is null / the child node, respectively</span>
 <a id="__codelineno-34-24" name="__codelineno-34-24" href="#__codelineno-34-24"></a><span class="w">    </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">??</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-34-25" name="__codelineno-34-25" href="#__codelineno-34-25"></a><span class="w">    </span><span class="c1">// Delete node cur</span>
+<a id="__codelineno-34-25" name="__codelineno-34-25" href="#__codelineno-34-25"></a><span class="w">    </span><span class="c1">// Remove node cur</span>
 <a id="__codelineno-34-26" name="__codelineno-34-26" href="#__codelineno-34-26"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">_root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-27" name="__codelineno-34-27" href="#__codelineno-34-27"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="o">!</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">cur</span><span class="p">)</span>
 <a id="__codelineno-34-28" name="__codelineno-34-28" href="#__codelineno-34-28"></a><span class="w">        </span><span class="n">pre</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-34-29" name="__codelineno-34-29" href="#__codelineno-34-29"></a><span class="w">      </span><span class="k">else</span>
 <a id="__codelineno-34-30" name="__codelineno-34-30" href="#__codelineno-34-30"></a><span class="w">        </span><span class="n">pre</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-34-31" name="__codelineno-34-31" href="#__codelineno-34-31"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-32" name="__codelineno-34-32" href="#__codelineno-34-32"></a><span class="w">      </span><span class="c1">// If deleted node is root node, reassign root node</span>
+<a id="__codelineno-34-32" name="__codelineno-34-32" href="#__codelineno-34-32"></a><span class="w">      </span><span class="c1">// If the removed node is the root, assign the new root</span>
 <a id="__codelineno-34-33" name="__codelineno-34-33" href="#__codelineno-34-33"></a><span class="w">      </span><span class="n">_root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-34-34" name="__codelineno-34-34" href="#__codelineno-34-34"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-34-35" name="__codelineno-34-35" href="#__codelineno-34-35"></a><span class="w">  </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-34-36" name="__codelineno-34-36" href="#__codelineno-34-36"></a><span class="w">    </span><span class="c1">// Number of child nodes = 2</span>
-<a id="__codelineno-34-37" name="__codelineno-34-37" href="#__codelineno-34-37"></a><span class="w">    </span><span class="c1">// Get next node of cur in inorder traversal</span>
+<a id="__codelineno-34-36" name="__codelineno-34-36" href="#__codelineno-34-36"></a><span class="w">    </span><span class="c1">// Number of children = 2</span>
+<a id="__codelineno-34-37" name="__codelineno-34-37" href="#__codelineno-34-37"></a><span class="w">    </span><span class="c1">// Get the inorder successor of cur</span>
 <a id="__codelineno-34-38" name="__codelineno-34-38" href="#__codelineno-34-38"></a><span class="w">    </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-34-39" name="__codelineno-34-39" href="#__codelineno-34-39"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">tmp</span><span class="o">!</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-34-40" name="__codelineno-34-40" href="#__codelineno-34-40"></a><span class="w">      </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-34-41" name="__codelineno-34-41" href="#__codelineno-34-41"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-34-42" name="__codelineno-34-42" href="#__codelineno-34-42"></a><span class="w">    </span><span class="c1">// Recursively delete node tmp</span>
+<a id="__codelineno-34-42" name="__codelineno-34-42" href="#__codelineno-34-42"></a><span class="w">    </span><span class="c1">// Recursively remove node tmp</span>
 <a id="__codelineno-34-43" name="__codelineno-34-43" href="#__codelineno-34-43"></a><span class="w">    </span><span class="n">remove</span><span class="p">(</span><span class="n">tmp</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-34-44" name="__codelineno-34-44" href="#__codelineno-34-44"></a><span class="w">    </span><span class="c1">// Replace cur with tmp</span>
+<a id="__codelineno-34-44" name="__codelineno-34-44" href="#__codelineno-34-44"></a><span class="w">    </span><span class="c1">// Replace the value of cur with the value of tmp</span>
 <a id="__codelineno-34-45" name="__codelineno-34-45" href="#__codelineno-34-45"></a><span class="w">    </span><span class="n">cur</span><span class="p">.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-34-46" name="__codelineno-34-46" href="#__codelineno-34-46"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-34-47" name="__codelineno-34-47" href="#__codelineno-34-47"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="o">&amp;</span><span class="k">mut</span><span class="w"> </span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="c1">// If tree is empty, return directly</span>
+<a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="w">    </span><span class="c1">// Return immediately if the tree is empty</span>
 <a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="p">.</span><span class="n">is_none</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-7" name="__codelineno-35-7" href="#__codelineno-35-7"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-35-8" name="__codelineno-35-8" href="#__codelineno-35-8"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">None</span><span class="p">;</span>
-<a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-35-9" name="__codelineno-35-9" href="#__codelineno-35-9"></a><span class="w">    </span><span class="c1">// Search while cur refers to a node</span>
 <a id="__codelineno-35-10" name="__codelineno-35-10" href="#__codelineno-35-10"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">clone</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-11" name="__codelineno-35-11" href="#__codelineno-35-11"></a><span class="w">        </span><span class="k">match</span><span class="w"> </span><span class="n">num</span><span class="p">.</span><span class="n">cmp</span><span class="p">(</span><span class="o">&amp;</span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="w">            </span><span class="c1">// Found node to delete, exit loop</span>
+<a id="__codelineno-35-12" name="__codelineno-35-12" href="#__codelineno-35-12"></a><span class="w">            </span><span class="c1">// The node to remove was found; exit the loop</span>
 <a id="__codelineno-35-13" name="__codelineno-35-13" href="#__codelineno-35-13"></a><span class="w">            </span><span class="n">Ordering</span><span class="p">::</span><span class="n">Equal</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="k">break</span><span class="p">,</span>
-<a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="w">            </span><span class="c1">// Node to delete is in cur's right subtree</span>
+<a id="__codelineno-35-14" name="__codelineno-35-14" href="#__codelineno-35-14"></a><span class="w">            </span><span class="c1">// The node to remove lies in the right subtree of cur</span>
 <a id="__codelineno-35-15" name="__codelineno-35-15" href="#__codelineno-35-15"></a><span class="w">            </span><span class="n">Ordering</span><span class="p">::</span><span class="n">Greater</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-16" name="__codelineno-35-16" href="#__codelineno-35-16"></a><span class="w">                </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-35-17" name="__codelineno-35-17" href="#__codelineno-35-17"></a><span class="w">                </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-35-18" name="__codelineno-35-18" href="#__codelineno-35-18"></a><span class="w">            </span><span class="p">}</span>
-<a id="__codelineno-35-19" name="__codelineno-35-19" href="#__codelineno-35-19"></a><span class="w">            </span><span class="c1">// Node to delete is in cur's left subtree</span>
+<a id="__codelineno-35-19" name="__codelineno-35-19" href="#__codelineno-35-19"></a><span class="w">            </span><span class="c1">// The node to remove lies in the left subtree of cur</span>
 <a id="__codelineno-35-20" name="__codelineno-35-20" href="#__codelineno-35-20"></a><span class="w">            </span><span class="n">Ordering</span><span class="p">::</span><span class="n">Less</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-21" name="__codelineno-35-21" href="#__codelineno-35-21"></a><span class="w">                </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-35-22" name="__codelineno-35-22" href="#__codelineno-35-22"></a><span class="w">                </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-35-23" name="__codelineno-35-23" href="#__codelineno-35-23"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-35-24" name="__codelineno-35-24" href="#__codelineno-35-24"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-35-25" name="__codelineno-35-25" href="#__codelineno-35-25"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-35-26" name="__codelineno-35-26" href="#__codelineno-35-26"></a><span class="w">    </span><span class="c1">// If no node to delete, return directly</span>
+<a id="__codelineno-35-26" name="__codelineno-35-26" href="#__codelineno-35-26"></a><span class="w">    </span><span class="c1">// Return immediately if the node to remove does not exist</span>
 <a id="__codelineno-35-27" name="__codelineno-35-27" href="#__codelineno-35-27"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">is_none</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-28" name="__codelineno-35-28" href="#__codelineno-35-28"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-35-29" name="__codelineno-35-29" href="#__codelineno-35-29"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-35-30" name="__codelineno-35-30" href="#__codelineno-35-30"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">unwrap</span><span class="p">();</span>
 <a id="__codelineno-35-31" name="__codelineno-35-31" href="#__codelineno-35-31"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="p">(</span><span class="n">left_child</span><span class="p">,</span><span class="w"> </span><span class="n">right_child</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">(),</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-35-32" name="__codelineno-35-32" href="#__codelineno-35-32"></a><span class="w">    </span><span class="k">match</span><span class="w"> </span><span class="p">(</span><span class="n">left_child</span><span class="p">.</span><span class="n">clone</span><span class="p">(),</span><span class="w"> </span><span class="n">right_child</span><span class="p">.</span><span class="n">clone</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-33" name="__codelineno-35-33" href="#__codelineno-35-33"></a><span class="w">        </span><span class="c1">// Number of child nodes = 0 or 1</span>
+<a id="__codelineno-35-33" name="__codelineno-35-33" href="#__codelineno-35-33"></a><span class="w">        </span><span class="c1">// Number of children = 0 or 1</span>
 <a id="__codelineno-35-34" name="__codelineno-35-34" href="#__codelineno-35-34"></a><span class="w">        </span><span class="p">(</span><span class="nb">None</span><span class="p">,</span><span class="w"> </span><span class="nb">None</span><span class="p">)</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">_</span><span class="p">),</span><span class="w"> </span><span class="nb">None</span><span class="p">)</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="p">(</span><span class="nb">None</span><span class="p">,</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">_</span><span class="p">))</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-35" name="__codelineno-35-35" href="#__codelineno-35-35"></a><span class="w">            </span><span class="c1">// When number of child nodes = 0 / 1, child = nullptr / that child node</span>
+<a id="__codelineno-35-35" name="__codelineno-35-35" href="#__codelineno-35-35"></a><span class="w">            </span><span class="c1">// With 0 or 1 child, child is None or that child, respectively</span>
 <a id="__codelineno-35-36" name="__codelineno-35-36" href="#__codelineno-35-36"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">left_child</span><span class="p">.</span><span class="n">or</span><span class="p">(</span><span class="n">right_child</span><span class="p">);</span>
 <a id="__codelineno-35-37" name="__codelineno-35-37" href="#__codelineno-35-37"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="n">unwrap</span><span class="p">();</span>
-<a id="__codelineno-35-38" name="__codelineno-35-38" href="#__codelineno-35-38"></a><span class="w">            </span><span class="c1">// Delete node cur</span>
+<a id="__codelineno-35-38" name="__codelineno-35-38" href="#__codelineno-35-38"></a><span class="w">            </span><span class="c1">// Remove node cur</span>
 <a id="__codelineno-35-39" name="__codelineno-35-39" href="#__codelineno-35-39"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="o">!</span><span class="n">Rc</span><span class="p">::</span><span class="n">ptr_eq</span><span class="p">(</span><span class="o">&amp;</span><span class="n">cur</span><span class="p">,</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="p">.</span><span class="n">as_ref</span><span class="p">().</span><span class="n">unwrap</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-40" name="__codelineno-35-40" href="#__codelineno-35-40"></a><span class="w">                </span><span class="kd">let</span><span class="w"> </span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">pre</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-35-41" name="__codelineno-35-41" href="#__codelineno-35-41"></a><span class="w">                </span><span class="k">if</span><span class="w"> </span><span class="n">left</span><span class="p">.</span><span class="n">is_some</span><span class="p">()</span><span class="w"> </span><span class="o">&amp;&amp;</span><span class="w"> </span><span class="n">Rc</span><span class="p">::</span><span class="n">ptr_eq</span><span class="p">(</span><span class="n">left</span><span class="p">.</span><span class="n">as_ref</span><span class="p">().</span><span class="n">unwrap</span><span class="p">(),</span><span class="w"> </span><span class="o">&amp;</span><span class="n">cur</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
@@ -6089,13 +6171,13 @@
 <a id="__codelineno-35-44" name="__codelineno-35-44" href="#__codelineno-35-44"></a><span class="w">                    </span><span class="n">pre</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-35-45" name="__codelineno-35-45" href="#__codelineno-35-45"></a><span class="w">                </span><span class="p">}</span>
 <a id="__codelineno-35-46" name="__codelineno-35-46" href="#__codelineno-35-46"></a><span class="w">            </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-47" name="__codelineno-35-47" href="#__codelineno-35-47"></a><span class="w">                </span><span class="c1">// If deleted node is root node, reassign root node</span>
+<a id="__codelineno-35-47" name="__codelineno-35-47" href="#__codelineno-35-47"></a><span class="w">                </span><span class="c1">// If the removed node is the root, assign the new root</span>
 <a id="__codelineno-35-48" name="__codelineno-35-48" href="#__codelineno-35-48"></a><span class="w">                </span><span class="bp">self</span><span class="p">.</span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-35-49" name="__codelineno-35-49" href="#__codelineno-35-49"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-35-50" name="__codelineno-35-50" href="#__codelineno-35-50"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-35-51" name="__codelineno-35-51" href="#__codelineno-35-51"></a><span class="w">        </span><span class="c1">// Number of child nodes = 2</span>
+<a id="__codelineno-35-51" name="__codelineno-35-51" href="#__codelineno-35-51"></a><span class="w">        </span><span class="c1">// Number of children = 2</span>
 <a id="__codelineno-35-52" name="__codelineno-35-52" href="#__codelineno-35-52"></a><span class="w">        </span><span class="p">(</span><span class="nb">Some</span><span class="p">(</span><span class="n">_</span><span class="p">),</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">_</span><span class="p">))</span><span class="w"> </span><span class="o">=&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-35-53" name="__codelineno-35-53" href="#__codelineno-35-53"></a><span class="w">            </span><span class="c1">// Get next node of cur in inorder traversal</span>
+<a id="__codelineno-35-53" name="__codelineno-35-53" href="#__codelineno-35-53"></a><span class="w">            </span><span class="c1">// Get the inorder successor of cur</span>
 <a id="__codelineno-35-54" name="__codelineno-35-54" href="#__codelineno-35-54"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">();</span>
 <a id="__codelineno-35-55" name="__codelineno-35-55" href="#__codelineno-35-55"></a><span class="w">            </span><span class="k">while</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="n">clone</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-35-56" name="__codelineno-35-56" href="#__codelineno-35-56"></a><span class="w">                </span><span class="k">if</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">is_some</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
@@ -6105,9 +6187,9 @@
 <a id="__codelineno-35-60" name="__codelineno-35-60" href="#__codelineno-35-60"></a><span class="w">                </span><span class="p">}</span>
 <a id="__codelineno-35-61" name="__codelineno-35-61" href="#__codelineno-35-61"></a><span class="w">            </span><span class="p">}</span>
 <a id="__codelineno-35-62" name="__codelineno-35-62" href="#__codelineno-35-62"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="n">tmp_val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="n">unwrap</span><span class="p">().</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="p">;</span>
-<a id="__codelineno-35-63" name="__codelineno-35-63" href="#__codelineno-35-63"></a><span class="w">            </span><span class="c1">// Recursively delete node tmp</span>
+<a id="__codelineno-35-63" name="__codelineno-35-63" href="#__codelineno-35-63"></a><span class="w">            </span><span class="c1">// Recursively remove node tmp</span>
 <a id="__codelineno-35-64" name="__codelineno-35-64" href="#__codelineno-35-64"></a><span class="w">            </span><span class="bp">self</span><span class="p">.</span><span class="n">remove</span><span class="p">(</span><span class="n">tmp_val</span><span class="p">);</span>
-<a id="__codelineno-35-65" name="__codelineno-35-65" href="#__codelineno-35-65"></a><span class="w">            </span><span class="c1">// Replace cur with tmp</span>
+<a id="__codelineno-35-65" name="__codelineno-35-65" href="#__codelineno-35-65"></a><span class="w">            </span><span class="c1">// Replace the value of cur with the value of tmp</span>
 <a id="__codelineno-35-66" name="__codelineno-35-66" href="#__codelineno-35-66"></a><span class="w">            </span><span class="n">cur</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp_val</span><span class="p">;</span>
 <a id="__codelineno-35-67" name="__codelineno-35-67" href="#__codelineno-35-67"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-35-68" name="__codelineno-35-68" href="#__codelineno-35-68"></a><span class="w">    </span><span class="p">}</span>
@@ -6115,36 +6197,36 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Remove node */</span>
-<a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="c1">// Cannot use remove keyword here due to stdio.h inclusion</span>
+<div class="highlight"><span class="filename">binary_search_tree.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Remove a node */</span>
+<a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="c1">// The name remove conflicts with the function declared in stdio.h</span>
 <a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="kt">void</span><span class="w"> </span><span class="nf">removeItem</span><span class="p">(</span><span class="n">BinarySearchTree</span><span class="w"> </span><span class="o">*</span><span class="n">bst</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="c1">// If tree is empty, return directly</span>
+<a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="w">    </span><span class="c1">// Return immediately if the tree is empty</span>
 <a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">bst</span><span class="o">-&gt;</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span>
 <a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">bst</span><span class="o">-&gt;</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="o">*</span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">;</span>
-<a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="w">    </span><span class="c1">// Search in a loop until reaching a null reference beyond a leaf</span>
 <a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="w">        </span><span class="c1">// Found node to delete, exit loop</span>
+<a id="__codelineno-36-10" name="__codelineno-36-10" href="#__codelineno-36-10"></a><span class="w">        </span><span class="c1">// The node to remove was found; exit the loop</span>
 <a id="__codelineno-36-11" name="__codelineno-36-11" href="#__codelineno-36-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-36-12" name="__codelineno-36-12" href="#__codelineno-36-12"></a><span class="w">            </span><span class="k">break</span><span class="p">;</span>
 <a id="__codelineno-36-13" name="__codelineno-36-13" href="#__codelineno-36-13"></a><span class="w">        </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">;</span>
 <a id="__codelineno-36-14" name="__codelineno-36-14" href="#__codelineno-36-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a><span class="w">            </span><span class="c1">// Node to delete is in right subtree of root</span>
+<a id="__codelineno-36-15" name="__codelineno-36-15" href="#__codelineno-36-15"></a><span class="w">            </span><span class="c1">// The node to remove is in cur's right subtree</span>
 <a id="__codelineno-36-16" name="__codelineno-36-16" href="#__codelineno-36-16"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-36-17" name="__codelineno-36-17" href="#__codelineno-36-17"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-18" name="__codelineno-36-18" href="#__codelineno-36-18"></a><span class="w">            </span><span class="c1">// Node to delete is in left subtree of root</span>
+<a id="__codelineno-36-18" name="__codelineno-36-18" href="#__codelineno-36-18"></a><span class="w">            </span><span class="c1">// The node to remove is in cur's left subtree</span>
 <a id="__codelineno-36-19" name="__codelineno-36-19" href="#__codelineno-36-19"></a><span class="w">            </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-36-20" name="__codelineno-36-20" href="#__codelineno-36-20"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-36-21" name="__codelineno-36-21" href="#__codelineno-36-21"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-36-22" name="__codelineno-36-22" href="#__codelineno-36-22"></a><span class="w">    </span><span class="c1">// If no node to delete, return directly</span>
+<a id="__codelineno-36-22" name="__codelineno-36-22" href="#__codelineno-36-22"></a><span class="w">    </span><span class="c1">// Return immediately if the node to remove does not exist</span>
 <a id="__codelineno-36-23" name="__codelineno-36-23" href="#__codelineno-36-23"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span>
 <a id="__codelineno-36-24" name="__codelineno-36-24" href="#__codelineno-36-24"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-36-25" name="__codelineno-36-25" href="#__codelineno-36-25"></a><span class="w">    </span><span class="c1">// Check if node to delete has children</span>
+<a id="__codelineno-36-25" name="__codelineno-36-25" href="#__codelineno-36-25"></a><span class="w">    </span><span class="c1">// Check whether the node to remove has children</span>
 <a id="__codelineno-36-26" name="__codelineno-36-26" href="#__codelineno-36-26"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-27" name="__codelineno-36-27" href="#__codelineno-36-27"></a><span class="w">        </span><span class="cm">/* Number of child nodes = 0 or 1 */</span>
-<a id="__codelineno-36-28" name="__codelineno-36-28" href="#__codelineno-36-28"></a><span class="w">        </span><span class="c1">// When number of child nodes = 0 / 1, child = nullptr / that child node</span>
+<a id="__codelineno-36-27" name="__codelineno-36-27" href="#__codelineno-36-27"></a><span class="w">        </span><span class="cm">/* Number of children = 0 or 1 */</span>
+<a id="__codelineno-36-28" name="__codelineno-36-28" href="#__codelineno-36-28"></a><span class="w">        </span><span class="c1">// With 0 or 1 child, child is NULL or that child, respectively</span>
 <a id="__codelineno-36-29" name="__codelineno-36-29" href="#__codelineno-36-29"></a><span class="w">        </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
-<a id="__codelineno-36-30" name="__codelineno-36-30" href="#__codelineno-36-30"></a><span class="w">        </span><span class="c1">// Delete node cur</span>
+<a id="__codelineno-36-30" name="__codelineno-36-30" href="#__codelineno-36-30"></a><span class="w">        </span><span class="c1">// Remove node cur</span>
 <a id="__codelineno-36-31" name="__codelineno-36-31" href="#__codelineno-36-31"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">cur</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-32" name="__codelineno-36-32" href="#__codelineno-36-32"></a><span class="w">            </span><span class="n">pre</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span><span class="p">;</span>
 <a id="__codelineno-36-33" name="__codelineno-36-33" href="#__codelineno-36-33"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
@@ -6153,106 +6235,106 @@
 <a id="__codelineno-36-36" name="__codelineno-36-36" href="#__codelineno-36-36"></a><span class="w">        </span><span class="c1">// Free memory</span>
 <a id="__codelineno-36-37" name="__codelineno-36-37" href="#__codelineno-36-37"></a><span class="w">        </span><span class="n">free</span><span class="p">(</span><span class="n">cur</span><span class="p">);</span>
 <a id="__codelineno-36-38" name="__codelineno-36-38" href="#__codelineno-36-38"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-36-39" name="__codelineno-36-39" href="#__codelineno-36-39"></a><span class="w">        </span><span class="cm">/* Number of child nodes = 2 */</span>
-<a id="__codelineno-36-40" name="__codelineno-36-40" href="#__codelineno-36-40"></a><span class="w">        </span><span class="c1">// Get next node of cur in inorder traversal</span>
+<a id="__codelineno-36-39" name="__codelineno-36-39" href="#__codelineno-36-39"></a><span class="w">        </span><span class="cm">/* Number of children = 2 */</span>
+<a id="__codelineno-36-40" name="__codelineno-36-40" href="#__codelineno-36-40"></a><span class="w">        </span><span class="c1">// Get the inorder successor of cur</span>
 <a id="__codelineno-36-41" name="__codelineno-36-41" href="#__codelineno-36-41"></a><span class="w">        </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-36-42" name="__codelineno-36-42" href="#__codelineno-36-42"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">tmp</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-36-43" name="__codelineno-36-43" href="#__codelineno-36-43"></a><span class="w">            </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-36-44" name="__codelineno-36-44" href="#__codelineno-36-44"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-36-45" name="__codelineno-36-45" href="#__codelineno-36-45"></a><span class="w">        </span><span class="kt">int</span><span class="w"> </span><span class="n">tmpVal</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
-<a id="__codelineno-36-46" name="__codelineno-36-46" href="#__codelineno-36-46"></a><span class="w">        </span><span class="c1">// Recursively delete node tmp</span>
+<a id="__codelineno-36-46" name="__codelineno-36-46" href="#__codelineno-36-46"></a><span class="w">        </span><span class="c1">// Recursively remove node tmp</span>
 <a id="__codelineno-36-47" name="__codelineno-36-47" href="#__codelineno-36-47"></a><span class="w">        </span><span class="n">removeItem</span><span class="p">(</span><span class="n">bst</span><span class="p">,</span><span class="w"> </span><span class="n">tmp</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">);</span>
-<a id="__codelineno-36-48" name="__codelineno-36-48" href="#__codelineno-36-48"></a><span class="w">        </span><span class="c1">// Replace cur with tmp</span>
+<a id="__codelineno-36-48" name="__codelineno-36-48" href="#__codelineno-36-48"></a><span class="w">        </span><span class="c1">// Replace the value of cur with the value of tmp</span>
 <a id="__codelineno-36-49" name="__codelineno-36-49" href="#__codelineno-36-49"></a><span class="w">        </span><span class="n">cur</span><span class="o">-&gt;</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmpVal</span><span class="p">;</span>
 <a id="__codelineno-36-50" name="__codelineno-36-50" href="#__codelineno-36-50"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-36-51" name="__codelineno-36-51" href="#__codelineno-36-51"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Remove node */</span>
+<div class="highlight"><span class="filename">binary_search_tree.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="cm">/* Remove a node */</span>
 <a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">num</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="c1">// If tree is empty, return directly</span>
+<a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="w">    </span><span class="c1">// Return immediately if the tree is empty</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span>
 <a id="__codelineno-37-7" name="__codelineno-37-7" href="#__codelineno-37-7"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">pre</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span>
-<a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">    </span><span class="c1">// Loop search, exit after passing leaf node</span>
+<a id="__codelineno-37-8" name="__codelineno-37-8" href="#__codelineno-37-8"></a><span class="w">    </span><span class="c1">// Continue searching while cur is not null</span>
 <a id="__codelineno-37-9" name="__codelineno-37-9" href="#__codelineno-37-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">        </span><span class="c1">// Found node to delete, exit loop</span>
+<a id="__codelineno-37-10" name="__codelineno-37-10" href="#__codelineno-37-10"></a><span class="w">        </span><span class="c1">// The node to remove was found; exit the loop</span>
 <a id="__codelineno-37-11" name="__codelineno-37-11" href="#__codelineno-37-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-37-12" name="__codelineno-37-12" href="#__codelineno-37-12"></a><span class="w">            </span><span class="k">break</span>
 <a id="__codelineno-37-13" name="__codelineno-37-13" href="#__codelineno-37-13"></a><span class="w">        </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span>
-<a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="w">        </span><span class="c1">// Node to delete is in cur's right subtree</span>
+<a id="__codelineno-37-14" name="__codelineno-37-14" href="#__codelineno-37-14"></a><span class="w">        </span><span class="c1">// The node to remove lies in the right subtree of cur</span>
 <a id="__codelineno-37-15" name="__codelineno-37-15" href="#__codelineno-37-15"></a><span class="w">        </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span><span class="p">)</span>
 <a id="__codelineno-37-16" name="__codelineno-37-16" href="#__codelineno-37-16"></a><span class="w">            </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span>
-<a id="__codelineno-37-17" name="__codelineno-37-17" href="#__codelineno-37-17"></a><span class="w">        </span><span class="c1">// Node to delete is in cur's left subtree</span>
+<a id="__codelineno-37-17" name="__codelineno-37-17" href="#__codelineno-37-17"></a><span class="w">        </span><span class="c1">// The node to remove lies in the left subtree of cur</span>
 <a id="__codelineno-37-18" name="__codelineno-37-18" href="#__codelineno-37-18"></a><span class="w">        </span><span class="k">else</span>
 <a id="__codelineno-37-19" name="__codelineno-37-19" href="#__codelineno-37-19"></a><span class="w">            </span><span class="n">cur</span><span class="p">.</span><span class="na">left</span>
 <a id="__codelineno-37-20" name="__codelineno-37-20" href="#__codelineno-37-20"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-37-21" name="__codelineno-37-21" href="#__codelineno-37-21"></a><span class="w">    </span><span class="c1">// If no node to delete, return directly</span>
+<a id="__codelineno-37-21" name="__codelineno-37-21" href="#__codelineno-37-21"></a><span class="w">    </span><span class="c1">// Return immediately if the node to remove does not exist</span>
 <a id="__codelineno-37-22" name="__codelineno-37-22" href="#__codelineno-37-22"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-37-23" name="__codelineno-37-23" href="#__codelineno-37-23"></a><span class="w">        </span><span class="k">return</span>
-<a id="__codelineno-37-24" name="__codelineno-37-24" href="#__codelineno-37-24"></a><span class="w">    </span><span class="c1">// Number of child nodes = 0 or 1</span>
+<a id="__codelineno-37-24" name="__codelineno-37-24" href="#__codelineno-37-24"></a><span class="w">    </span><span class="c1">// Number of children = 0 or 1</span>
 <a id="__codelineno-37-25" name="__codelineno-37-25" href="#__codelineno-37-25"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-26" name="__codelineno-37-26" href="#__codelineno-37-26"></a><span class="w">        </span><span class="c1">// When number of child nodes = 0 / 1, child = null / that child node</span>
+<a id="__codelineno-37-26" name="__codelineno-37-26" href="#__codelineno-37-26"></a><span class="w">        </span><span class="c1">// With 0 / 1 children, child is null / the child node, respectively</span>
 <a id="__codelineno-37-27" name="__codelineno-37-27" href="#__codelineno-37-27"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-37-28" name="__codelineno-37-28" href="#__codelineno-37-28"></a><span class="w">            </span><span class="n">cur</span><span class="p">.</span><span class="na">left</span>
 <a id="__codelineno-37-29" name="__codelineno-37-29" href="#__codelineno-37-29"></a><span class="w">        </span><span class="k">else</span>
 <a id="__codelineno-37-30" name="__codelineno-37-30" href="#__codelineno-37-30"></a><span class="w">            </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span>
-<a id="__codelineno-37-31" name="__codelineno-37-31" href="#__codelineno-37-31"></a><span class="w">        </span><span class="c1">// Delete node cur</span>
+<a id="__codelineno-37-31" name="__codelineno-37-31" href="#__codelineno-37-31"></a><span class="w">        </span><span class="c1">// Remove node cur</span>
 <a id="__codelineno-37-32" name="__codelineno-37-32" href="#__codelineno-37-32"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-33" name="__codelineno-37-33" href="#__codelineno-37-33"></a><span class="w">            </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">pre</span><span class="o">!!</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">cur</span><span class="p">)</span>
 <a id="__codelineno-37-34" name="__codelineno-37-34" href="#__codelineno-37-34"></a><span class="w">                </span><span class="n">pre</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span>
 <a id="__codelineno-37-35" name="__codelineno-37-35" href="#__codelineno-37-35"></a><span class="w">            </span><span class="k">else</span>
 <a id="__codelineno-37-36" name="__codelineno-37-36" href="#__codelineno-37-36"></a><span class="w">                </span><span class="n">pre</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span>
 <a id="__codelineno-37-37" name="__codelineno-37-37" href="#__codelineno-37-37"></a><span class="w">        </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-38" name="__codelineno-37-38" href="#__codelineno-37-38"></a><span class="w">            </span><span class="c1">// If deleted node is root node, reassign root node</span>
+<a id="__codelineno-37-38" name="__codelineno-37-38" href="#__codelineno-37-38"></a><span class="w">            </span><span class="c1">// If the removed node is the root, assign the new root</span>
 <a id="__codelineno-37-39" name="__codelineno-37-39" href="#__codelineno-37-39"></a><span class="w">            </span><span class="n">root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span>
 <a id="__codelineno-37-40" name="__codelineno-37-40" href="#__codelineno-37-40"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-37-41" name="__codelineno-37-41" href="#__codelineno-37-41"></a><span class="w">        </span><span class="c1">// Number of child nodes = 2</span>
+<a id="__codelineno-37-41" name="__codelineno-37-41" href="#__codelineno-37-41"></a><span class="w">        </span><span class="c1">// Number of children = 2</span>
 <a id="__codelineno-37-42" name="__codelineno-37-42" href="#__codelineno-37-42"></a><span class="w">    </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-37-43" name="__codelineno-37-43" href="#__codelineno-37-43"></a><span class="w">        </span><span class="c1">// Get next node of cur in inorder traversal</span>
+<a id="__codelineno-37-43" name="__codelineno-37-43" href="#__codelineno-37-43"></a><span class="w">        </span><span class="c1">// Get the inorder successor of cur</span>
 <a id="__codelineno-37-44" name="__codelineno-37-44" href="#__codelineno-37-44"></a><span class="w">        </span><span class="kd">var</span><span class="w"> </span><span class="nv">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="p">.</span><span class="na">right</span>
 <a id="__codelineno-37-45" name="__codelineno-37-45" href="#__codelineno-37-45"></a><span class="w">        </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">tmp</span><span class="o">!!</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-37-46" name="__codelineno-37-46" href="#__codelineno-37-46"></a><span class="w">            </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="na">left</span>
 <a id="__codelineno-37-47" name="__codelineno-37-47" href="#__codelineno-37-47"></a><span class="w">        </span><span class="p">}</span>
-<a id="__codelineno-37-48" name="__codelineno-37-48" href="#__codelineno-37-48"></a><span class="w">        </span><span class="c1">// Recursively delete node tmp</span>
+<a id="__codelineno-37-48" name="__codelineno-37-48" href="#__codelineno-37-48"></a><span class="w">        </span><span class="c1">// Recursively remove node tmp</span>
 <a id="__codelineno-37-49" name="__codelineno-37-49" href="#__codelineno-37-49"></a><span class="w">        </span><span class="n">remove</span><span class="p">(</span><span class="n">tmp</span><span class="p">.</span><span class="na">_val</span><span class="p">)</span>
-<a id="__codelineno-37-50" name="__codelineno-37-50" href="#__codelineno-37-50"></a><span class="w">        </span><span class="c1">// Replace cur with tmp</span>
+<a id="__codelineno-37-50" name="__codelineno-37-50" href="#__codelineno-37-50"></a><span class="w">        </span><span class="c1">// Replace the value of cur with the value of tmp</span>
 <a id="__codelineno-37-51" name="__codelineno-37-51" href="#__codelineno-37-51"></a><span class="w">        </span><span class="n">cur</span><span class="p">.</span><span class="na">_val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="p">.</span><span class="na">_val</span>
 <a id="__codelineno-37-52" name="__codelineno-37-52" href="#__codelineno-37-52"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-37-53" name="__codelineno-37-53" href="#__codelineno-37-53"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_search_tree.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Delete node ###</span>
+<div class="highlight"><span class="filename">binary_search_tree.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1">### Remove a node ###</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">remove</span><span class="p">(</span><span class="n">num</span><span class="p">)</span>
-<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1"># If tree is empty, return directly</span>
+<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="w">  </span><span class="c1"># Return immediately if the tree is empty</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="vi">@root</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a>
-<a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="c1"># Loop search, exit after passing leaf node</span>
+<a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="w">  </span><span class="c1"># Search in a loop, stopping if cur becomes nil</span>
 <a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="w">  </span><span class="n">cur</span><span class="p">,</span><span class="w"> </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="vi">@root</span><span class="p">,</span><span class="w"> </span><span class="kp">nil</span>
 <a id="__codelineno-38-8" name="__codelineno-38-8" href="#__codelineno-38-8"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="o">!</span><span class="n">cur</span><span class="o">.</span><span class="n">nil?</span>
-<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">    </span><span class="c1"># Found node to delete, exit loop</span>
+<a id="__codelineno-38-9" name="__codelineno-38-9" href="#__codelineno-38-9"></a><span class="w">    </span><span class="c1"># The node to remove was found; exit the loop</span>
 <a id="__codelineno-38-10" name="__codelineno-38-10" href="#__codelineno-38-10"></a><span class="w">    </span><span class="k">break</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-38-11" name="__codelineno-38-11" href="#__codelineno-38-11"></a>
 <a id="__codelineno-38-12" name="__codelineno-38-12" href="#__codelineno-38-12"></a><span class="w">    </span><span class="n">pre</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span>
-<a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">    </span><span class="c1"># Node to delete is in cur's right subtree</span>
+<a id="__codelineno-38-13" name="__codelineno-38-13" href="#__codelineno-38-13"></a><span class="w">    </span><span class="c1"># The node to remove lies in the right subtree of cur</span>
 <a id="__codelineno-38-14" name="__codelineno-38-14" href="#__codelineno-38-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">num</span>
 <a id="__codelineno-38-15" name="__codelineno-38-15" href="#__codelineno-38-15"></a><span class="w">      </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">right</span>
-<a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w">    </span><span class="c1"># Node to delete is in cur's left subtree</span>
+<a id="__codelineno-38-16" name="__codelineno-38-16" href="#__codelineno-38-16"></a><span class="w">    </span><span class="c1"># The node to remove lies in the left subtree of cur</span>
 <a id="__codelineno-38-17" name="__codelineno-38-17" href="#__codelineno-38-17"></a><span class="w">    </span><span class="k">else</span>
 <a id="__codelineno-38-18" name="__codelineno-38-18" href="#__codelineno-38-18"></a><span class="w">      </span><span class="n">cur</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">left</span>
 <a id="__codelineno-38-19" name="__codelineno-38-19" href="#__codelineno-38-19"></a><span class="w">    </span><span class="k">end</span>
 <a id="__codelineno-38-20" name="__codelineno-38-20" href="#__codelineno-38-20"></a><span class="w">  </span><span class="k">end</span>
-<a id="__codelineno-38-21" name="__codelineno-38-21" href="#__codelineno-38-21"></a><span class="w">  </span><span class="c1"># If no node to delete, return directly</span>
+<a id="__codelineno-38-21" name="__codelineno-38-21" href="#__codelineno-38-21"></a><span class="w">  </span><span class="c1"># Return immediately if the node to remove does not exist</span>
 <a id="__codelineno-38-22" name="__codelineno-38-22" href="#__codelineno-38-22"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-38-23" name="__codelineno-38-23" href="#__codelineno-38-23"></a>
-<a id="__codelineno-38-24" name="__codelineno-38-24" href="#__codelineno-38-24"></a><span class="w">  </span><span class="c1"># Number of child nodes = 0 or 1</span>
+<a id="__codelineno-38-24" name="__codelineno-38-24" href="#__codelineno-38-24"></a><span class="w">  </span><span class="c1"># Number of children = 0 or 1</span>
 <a id="__codelineno-38-25" name="__codelineno-38-25" href="#__codelineno-38-25"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">left</span><span class="o">.</span><span class="n">nil?</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">right</span><span class="o">.</span><span class="n">nil?</span>
-<a id="__codelineno-38-26" name="__codelineno-38-26" href="#__codelineno-38-26"></a><span class="w">    </span><span class="c1"># When number of child nodes = 0 / 1, child = null / that child node</span>
+<a id="__codelineno-38-26" name="__codelineno-38-26" href="#__codelineno-38-26"></a><span class="w">    </span><span class="c1"># With 0 / 1 children, child is nil / the child node, respectively</span>
 <a id="__codelineno-38-27" name="__codelineno-38-27" href="#__codelineno-38-27"></a><span class="w">    </span><span class="n">child</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="o">||</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">right</span>
-<a id="__codelineno-38-28" name="__codelineno-38-28" href="#__codelineno-38-28"></a><span class="w">    </span><span class="c1"># Delete node cur</span>
+<a id="__codelineno-38-28" name="__codelineno-38-28" href="#__codelineno-38-28"></a><span class="w">    </span><span class="c1"># Remove node cur</span>
 <a id="__codelineno-38-29" name="__codelineno-38-29" href="#__codelineno-38-29"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="n">cur</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="vi">@root</span>
 <a id="__codelineno-38-30" name="__codelineno-38-30" href="#__codelineno-38-30"></a><span class="w">      </span><span class="k">if</span><span class="w"> </span><span class="n">pre</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="n">cur</span>
 <a id="__codelineno-38-31" name="__codelineno-38-31" href="#__codelineno-38-31"></a><span class="w">        </span><span class="n">pre</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span>
@@ -6260,19 +6342,19 @@
 <a id="__codelineno-38-33" name="__codelineno-38-33" href="#__codelineno-38-33"></a><span class="w">        </span><span class="n">pre</span><span class="o">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span>
 <a id="__codelineno-38-34" name="__codelineno-38-34" href="#__codelineno-38-34"></a><span class="w">      </span><span class="k">end</span>
 <a id="__codelineno-38-35" name="__codelineno-38-35" href="#__codelineno-38-35"></a><span class="w">    </span><span class="k">else</span>
-<a id="__codelineno-38-36" name="__codelineno-38-36" href="#__codelineno-38-36"></a><span class="w">      </span><span class="c1"># If deleted node is root node, reassign root node</span>
+<a id="__codelineno-38-36" name="__codelineno-38-36" href="#__codelineno-38-36"></a><span class="w">      </span><span class="c1"># If the removed node is the root, assign the new root</span>
 <a id="__codelineno-38-37" name="__codelineno-38-37" href="#__codelineno-38-37"></a><span class="w">      </span><span class="vi">@root</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">child</span>
 <a id="__codelineno-38-38" name="__codelineno-38-38" href="#__codelineno-38-38"></a><span class="w">    </span><span class="k">end</span>
-<a id="__codelineno-38-39" name="__codelineno-38-39" href="#__codelineno-38-39"></a><span class="w">  </span><span class="c1"># Number of child nodes = 2</span>
+<a id="__codelineno-38-39" name="__codelineno-38-39" href="#__codelineno-38-39"></a><span class="w">  </span><span class="c1"># Number of children = 2</span>
 <a id="__codelineno-38-40" name="__codelineno-38-40" href="#__codelineno-38-40"></a><span class="w">  </span><span class="k">else</span>
-<a id="__codelineno-38-41" name="__codelineno-38-41" href="#__codelineno-38-41"></a><span class="w">    </span><span class="c1"># Get next node of cur in inorder traversal</span>
+<a id="__codelineno-38-41" name="__codelineno-38-41" href="#__codelineno-38-41"></a><span class="w">    </span><span class="c1"># Get the inorder successor of cur</span>
 <a id="__codelineno-38-42" name="__codelineno-38-42" href="#__codelineno-38-42"></a><span class="w">    </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">cur</span><span class="o">.</span><span class="n">right</span>
 <a id="__codelineno-38-43" name="__codelineno-38-43" href="#__codelineno-38-43"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="o">!</span><span class="n">tmp</span><span class="o">.</span><span class="n">left</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-38-44" name="__codelineno-38-44" href="#__codelineno-38-44"></a><span class="w">      </span><span class="n">tmp</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="o">.</span><span class="n">left</span>
 <a id="__codelineno-38-45" name="__codelineno-38-45" href="#__codelineno-38-45"></a><span class="w">    </span><span class="k">end</span>
-<a id="__codelineno-38-46" name="__codelineno-38-46" href="#__codelineno-38-46"></a><span class="w">    </span><span class="c1"># Recursively delete node tmp</span>
+<a id="__codelineno-38-46" name="__codelineno-38-46" href="#__codelineno-38-46"></a><span class="w">    </span><span class="c1"># Recursively remove node tmp</span>
 <a id="__codelineno-38-47" name="__codelineno-38-47" href="#__codelineno-38-47"></a><span class="w">    </span><span class="n">remove</span><span class="p">(</span><span class="n">tmp</span><span class="o">.</span><span class="n">val</span><span class="p">)</span>
-<a id="__codelineno-38-48" name="__codelineno-38-48" href="#__codelineno-38-48"></a><span class="w">    </span><span class="c1"># Replace cur with tmp</span>
+<a id="__codelineno-38-48" name="__codelineno-38-48" href="#__codelineno-38-48"></a><span class="w">    </span><span class="c1"># Replace the value of cur with the value of tmp</span>
 <a id="__codelineno-38-49" name="__codelineno-38-49" href="#__codelineno-38-49"></a><span class="w">    </span><span class="n">cur</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">tmp</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-38-50" name="__codelineno-38-50" href="#__codelineno-38-50"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-38-51" name="__codelineno-38-51" href="#__codelineno-38-51"></a><span class="k">end</span>
@@ -6280,16 +6362,21 @@
 </div>
 </div>
 </div>
-<h3 id="4-inorder-traversal-is-ordered">4. &nbsp; Inorder Traversal Is Ordered<a class="headerlink" href="#4-inorder-traversal-is-ordered" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 7-22, the inorder traversal of a binary tree follows the "left <span class="arithmatex">\(\rightarrow\)</span> root <span class="arithmatex">\(\rightarrow\)</span> right" traversal order, while the binary search tree satisfies the "left child node <span class="arithmatex">\(&lt;\)</span> root node <span class="arithmatex">\(&lt;\)</span> right child node" size relationship.</p>
-<p>This means that when performing an inorder traversal in a binary search tree, the next smallest node is always traversed first, thus yielding an important property: <strong>The inorder traversal sequence of a binary search tree is ascending</strong>.</p>
-<p>Using the property of inorder traversal being ascending, we can obtain ordered data in a binary search tree in only <span class="arithmatex">\(O(n)\)</span> time, without the need for additional sorting operations, which is very efficient.</p>
-<p><img alt="Inorder traversal sequence of a binary search tree" class="animation-figure" src="../binary_search_tree.assets/bst_inorder_traversal.png" /></p>
-<p align="center"> Figure 7-22 &nbsp; Inorder traversal sequence of a binary search tree </p>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%29%3A%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%20%20%20%20%20%20%20%20self.left%20%3D%20None%0A%20%20%20%20%20%20%20%20self.right%20%3D%20None%0A%0Aclass%20BinarySearchTree%3A%0A%20%20%20%20%22%22%22Binary%20search%20tree%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self._root%20%3D%20None%0A%0A%20%20%20%20def%20insert%28self,%20num%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Insert%20a%20node%22%22%22%0A%20%20%20%20%20%20%20%20if%20self._root%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self._root%20%3D%20TreeNode%28num%29%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20cur,%20pre%20%3D%20self._root,%20None%0A%20%20%20%20%20%20%20%20while%20cur%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3D%3D%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%20%20%20%20pre%20%3D%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.right%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.left%0A%20%20%20%20%20%20%20%20node%20%3D%20TreeNode%28num%29%0A%20%20%20%20%20%20%20%20if%20pre.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20pre.right%20%3D%20node%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20pre.left%20%3D%20node%0A%0A%20%20%20%20def%20remove%28self,%20num%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Remove%20a%20node%22%22%22%0A%20%20%20%20%20%20%20%20if%20self._root%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%23%20Find%20a%20node%0A%20%20%20%20%20%20%20%20cur,%20pre%20%3D%20self._root,%20None%0A%20%20%20%20%20%20%20%20while%20cur%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3D%3D%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20%20%20%20%20pre%20%3D%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.right%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.left%0A%20%20%20%20%20%20%20%20if%20cur%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%0A%20%20%20%20%20%20%20%20%23%20Number%20of%20children%20%3D%200%20or%201%0A%20%20%20%20%20%20%20%20if%20cur.left%20is%20None%20or%20cur.right%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20With%200%20or%201%20child,%20child%20is%20None%20or%20that%20child,%20respectively%0A%20%20%20%20%20%20%20%20%20%20%20%20child%20%3D%20cur.left%20or%20cur.right%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Remove%20node%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur%20%21%3D%20self._root%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20if%20pre.left%20%3D%3D%20cur%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20pre.left%20%3D%20child%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20pre.right%20%3D%20child%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20self._root%20%3D%20child%0A%20%20%20%20%20%20%20%20%23%20Number%20of%20children%20%3D%202%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Get%20the%20inorder%20successor%20of%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20tmp%3A%20TreeNode%20%3D%20cur.right%0A%20%20%20%20%20%20%20%20%20%20%20%20while%20tmp.left%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20tmp%20%3D%20tmp.left%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Recursively%20remove%20node%20tmp%0A%20%20%20%20%20%20%20%20%20%20%20%20self.remove%28tmp.val%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Replace%20the%20value%20of%20cur%20with%20the%20value%20of%20tmp%0A%20%20%20%20%20%20%20%20%20%20%20%20cur.val%20%3D%20tmp.val%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20binary%20search%20tree%0A%20%20%20%20bst%20%3D%20BinarySearchTree%28%29%0A%20%20%20%20nums%20%3D%20%5B4,%202,%206,%201,%203,%205,%207%5D%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20bst.insert%28num%29%0A%0A%20%20%20%20%23%20Remove%20a%20node%0A%20%20%20%20bst.remove%281%29%20%23%20Degree%200%0A%20%20%20%20bst.remove%282%29%20%23%20Degree%201%0A%20%20%20%20bst.remove%284%29%20%23%20Degree%202&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=162&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%29%3A%0A%20%20%20%20%20%20%20%20self.val%20%3D%20val%0A%20%20%20%20%20%20%20%20self.left%20%3D%20None%0A%20%20%20%20%20%20%20%20self.right%20%3D%20None%0A%0Aclass%20BinarySearchTree%3A%0A%20%20%20%20%22%22%22Binary%20search%20tree%22%22%22%0A%0A%20%20%20%20def%20__init__%28self%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Constructor%22%22%22%0A%20%20%20%20%20%20%20%20self._root%20%3D%20None%0A%0A%20%20%20%20def%20insert%28self,%20num%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Insert%20a%20node%22%22%22%0A%20%20%20%20%20%20%20%20if%20self._root%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20self._root%20%3D%20TreeNode%28num%29%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20cur,%20pre%20%3D%20self._root,%20None%0A%20%20%20%20%20%20%20%20while%20cur%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3D%3D%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%20%20%20%20pre%20%3D%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.right%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.left%0A%20%20%20%20%20%20%20%20node%20%3D%20TreeNode%28num%29%0A%20%20%20%20%20%20%20%20if%20pre.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20pre.right%20%3D%20node%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20pre.left%20%3D%20node%0A%0A%20%20%20%20def%20remove%28self,%20num%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20%22%22%22Remove%20a%20node%22%22%22%0A%20%20%20%20%20%20%20%20if%20self._root%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%20%20%20%20%20%20%20%20%23%20Find%20a%20node%0A%20%20%20%20%20%20%20%20cur,%20pre%20%3D%20self._root,%20None%0A%20%20%20%20%20%20%20%20while%20cur%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3D%3D%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20break%0A%20%20%20%20%20%20%20%20%20%20%20%20pre%20%3D%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur.val%20%3C%20num%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.right%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20cur%20%3D%20cur.left%0A%20%20%20%20%20%20%20%20if%20cur%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20return%0A%0A%20%20%20%20%20%20%20%20%23%20Number%20of%20children%20%3D%200%20or%201%0A%20%20%20%20%20%20%20%20if%20cur.left%20is%20None%20or%20cur.right%20is%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20With%200%20or%201%20child,%20child%20is%20None%20or%20that%20child,%20respectively%0A%20%20%20%20%20%20%20%20%20%20%20%20child%20%3D%20cur.left%20or%20cur.right%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Remove%20node%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20if%20cur%20%21%3D%20self._root%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20if%20pre.left%20%3D%3D%20cur%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20pre.left%20%3D%20child%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20pre.right%20%3D%20child%0A%20%20%20%20%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20self._root%20%3D%20child%0A%20%20%20%20%20%20%20%20%23%20Number%20of%20children%20%3D%202%0A%20%20%20%20%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Get%20the%20inorder%20successor%20of%20cur%0A%20%20%20%20%20%20%20%20%20%20%20%20tmp%3A%20TreeNode%20%3D%20cur.right%0A%20%20%20%20%20%20%20%20%20%20%20%20while%20tmp.left%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20tmp%20%3D%20tmp.left%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Recursively%20remove%20node%20tmp%0A%20%20%20%20%20%20%20%20%20%20%20%20self.remove%28tmp.val%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20Replace%20the%20value%20of%20cur%20with%20the%20value%20of%20tmp%0A%20%20%20%20%20%20%20%20%20%20%20%20cur.val%20%3D%20tmp.val%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20binary%20search%20tree%0A%20%20%20%20bst%20%3D%20BinarySearchTree%28%29%0A%20%20%20%20nums%20%3D%20%5B4,%202,%206,%201,%203,%205,%207%5D%0A%20%20%20%20for%20num%20in%20nums%3A%0A%20%20%20%20%20%20%20%20bst.insert%28num%29%0A%0A%20%20%20%20%23%20Remove%20a%20node%0A%20%20%20%20bst.remove%281%29%20%23%20Degree%200%0A%20%20%20%20bst.remove%282%29%20%23%20Degree%201%0A%20%20%20%20bst.remove%284%29%20%23%20Degree%202&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=162&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="4-inorder-traversal-produces-sorted-data">4. &nbsp; Inorder traversal produces sorted data<a class="headerlink" href="#4-inorder-traversal-produces-sorted-data" title="Permanent link">&para;</a></h3>
+<p>As shown below, inorder traversal visits “left <span class="arithmatex">\(\rightarrow\)</span> root <span class="arithmatex">\(\rightarrow\)</span> right,” while a binary search tree obeys “left child <span class="arithmatex">\(&lt;\)</span> root <span class="arithmatex">\(&lt;\)</span> right child.”</p>
+<p>Thus, inorder traversal always visits the next smallest node, giving an important property: <strong>the inorder traversal of a binary search tree is in ascending order</strong>.</p>
+<p>This lets us retrieve sorted data from the tree in just <span class="arithmatex">\(O(n)\)</span> time, with no additional sorting.</p>
+<p><img alt="Inorder traversal of a binary search tree" class="animation-figure" src="../binary_search_tree.assets/bst_inorder_traversal.png" /></p>
+<p align="center"> Figure 7-22 &nbsp; Inorder traversal of a binary search tree </p>
 
-<h2 id="742-efficiency-of-binary-search-trees">7.4.2 &nbsp; Efficiency of Binary Search Trees<a class="headerlink" href="#742-efficiency-of-binary-search-trees" title="Permanent link">&para;</a></h2>
-<p>Given a set of data, we consider using an array or a binary search tree for storage. Observing Table 7-2, all operations in a binary search tree have logarithmic time complexity, providing stable and efficient performance. Arrays are more efficient than binary search trees only in scenarios with high-frequency additions and low-frequency searches and deletions.</p>
-<p align="center"> Table 7-2 &nbsp; Efficiency comparison between arrays and search trees </p>
+<h2 id="742-efficiency">7.4.2 &nbsp; Efficiency<a class="headerlink" href="#742-efficiency" title="Permanent link">&para;</a></h2>
+<p>Suppose we want to store a dataset in an array or a binary search tree. As Table 7-2 shows, the tree's operations take logarithmic time, offering consistent and efficient performance. An array is faster only when insertions are frequent but lookups and deletions are infrequent.</p>
+<p align="center"> Table 7-2 &nbsp; Efficiency of arrays and search trees </p>
 
 <div class="center-table">
 <table>
@@ -6302,33 +6389,33 @@
 </thead>
 <tbody>
 <tr>
-<td>Search element</td>
+<td>Find an element</td>
 <td><span class="arithmatex">\(O(n)\)</span></td>
 <td><span class="arithmatex">\(O(\log n)\)</span></td>
 </tr>
 <tr>
-<td>Insert element</td>
+<td>Insert an element</td>
 <td><span class="arithmatex">\(O(1)\)</span></td>
 <td><span class="arithmatex">\(O(\log n)\)</span></td>
 </tr>
 <tr>
-<td>Remove element</td>
+<td>Delete an element</td>
 <td><span class="arithmatex">\(O(n)\)</span></td>
 <td><span class="arithmatex">\(O(\log n)\)</span></td>
 </tr>
 </tbody>
 </table>
 </div>
-<p>In the ideal case, a binary search tree is balanced, so any node can be found within <span class="arithmatex">\(O(\log n)\)</span> loop iterations.</p>
-<p>However, if we continuously insert and remove nodes in a binary search tree, it may degenerate into a linked list as shown in Figure 7-23, where the time complexity of various operations also degrades to <span class="arithmatex">\(O(n)\)</span>.</p>
-<p><img alt="Degradation of a binary search tree" class="animation-figure" src="../binary_search_tree.assets/bst_degradation.png" /></p>
-<p align="center"> Figure 7-23 &nbsp; Degradation of a binary search tree </p>
+<p>Ideally, the tree is balanced, allowing any node to be found within <span class="arithmatex">\(\log n\)</span> iterations.</p>
+<p>However, repeated insertions and deletions may cause it to degenerate into a linked list, as shown below. The time complexity of its operations then degrades to <span class="arithmatex">\(O(n)\)</span>.</p>
+<p><img alt="Degeneration of a binary search tree" class="animation-figure" src="../binary_search_tree.assets/bst_degradation.png" /></p>
+<p align="center"> Figure 7-23 &nbsp; Degeneration of a binary search tree </p>
 
-<h2 id="743-common-applications-of-binary-search-trees">7.4.3 &nbsp; Common Applications of Binary Search Trees<a class="headerlink" href="#743-common-applications-of-binary-search-trees" title="Permanent link">&para;</a></h2>
+<h2 id="743-common-applications">7.4.3 &nbsp; Common applications<a class="headerlink" href="#743-common-applications" title="Permanent link">&para;</a></h2>
 <ul>
-<li>Used as multi-level indexes in systems to implement efficient search, insertion, and removal operations.</li>
-<li>Serves as the underlying data structure for certain search algorithms.</li>
-<li>Used to store data streams to maintain their ordered state.</li>
+<li>Multilevel indexes in systems, supporting efficient lookup, insertion, and deletion.</li>
+<li>The underlying data structure for certain search algorithms.</li>
+<li>Storing a data stream while maintaining its order.</li>
 </ul>
 
 <!-- Source file information -->
@@ -6352,7 +6439,7 @@ aria-label="Footer"
   <a
     href="../array_representation_of_tree/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 7.3 Array Representation of Binary Trees"
+    aria-label="Previous: 7.3   Array representation of a binary tree"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -6364,7 +6451,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        7.3 Array Representation of Binary Trees
+        7.3 &nbsp; Array representation of a binary tree
       </div>
     </div>
   </a>
@@ -6376,7 +6463,7 @@ aria-label="Footer"
   <a
     href="../avl_tree/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 7.5 AVL Tree *"
+    aria-label="Next: 7.5   AVL trees *"
     rel="next"
   >
     <div class="md-footer__title">
@@ -6384,7 +6471,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        7.5 AVL Tree *
+        7.5 &nbsp; AVL trees *
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -6401,7 +6488,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
   <script
diff --git a/en/chapter_tree/binary_tree/index.html b/en/chapter_tree/binary_tree/index.html
index e8a7824b9..a19aff22d 100644
--- a/en/chapter_tree/binary_tree/index.html
+++ b/en/chapter_tree/binary_tree/index.html
@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>7.1 Binary Tree - Hello Algo</title>
+        <title>7.1 Binary trees - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
-        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Inter:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
-        <style>:root{--md-text-font:"Inter";--md-code-font:"JetBrains Mono"}</style>
+        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto+Sans:300,300i,400,400i,500,500i,700,700i%7CJetBrains+Mono:400,400i,700,700i&amp;display=fallback">
+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
     <div data-md-component="skip">
 
 
-        <a href="#71-binary-tree" class="md-skip">
+        <a href="#71-binary-trees" class="md-skip">
           Skip to content
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
 </div>
 
           </div>
@@ -154,7 +154,7 @@
         <div class="md-header__topic" data-md-component="header-topic">
           <span class="md-ellipsis">
 
-              7.1 &nbsp; Binary Tree
+              7.1 &nbsp; Binary trees
 
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-      <label class="md-header__button md-icon" title="Dark mode" for="__palette_1" hidden>
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-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
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         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
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@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
@@ -365,7 +365,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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@@ -440,7 +440,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
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   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
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@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
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           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
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   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
@@ -1400,7 +1400,7 @@
   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
@@ -1614,7 +1614,7 @@
   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1898,7 +1898,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1919,7 +1919,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1949,7 +1949,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1966,7 +1966,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1992,10 +1992,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#711-common-terminology-of-binary-trees" class="md-nav__link">
+  <a href="#711-common-terminology" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.1.1 &nbsp; Common Terminology of Binary Trees
+        7.1.1 &nbsp; Common terminology
       </span>
     </span>
   </a>
@@ -2003,22 +2003,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#712-basic-operations-of-binary-trees" class="md-nav__link">
+  <a href="#712-basic-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.1.2 &nbsp; Basic Operations of Binary Trees
+        7.1.2 &nbsp; Basic operations
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.1.2 Basic Operations of Binary Trees">
+    <nav class="md-nav" aria-label="7.1.2 Basic operations">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-initializing-a-binary-tree" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Initializing a Binary Tree
+        1. &nbsp; Initializing a binary tree
       </span>
     </span>
   </a>
@@ -2026,10 +2026,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-inserting-and-removing-nodes" class="md-nav__link">
+  <a href="#2-inserting-and-deleting-nodes" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Inserting and Removing Nodes
+        2. &nbsp; Inserting and deleting nodes
       </span>
     </span>
   </a>
@@ -2045,19 +2045,19 @@
   <a href="#713-common-types-of-binary-trees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.1.3 &nbsp; Common Types of Binary Trees
+        7.1.3 &nbsp; Common types of binary trees
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.1.3 Common Types of Binary Trees">
+    <nav class="md-nav" aria-label="7.1.3 Common types of binary trees">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-perfect-binary-tree" class="md-nav__link">
+  <a href="#1-perfect-binary-trees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Perfect Binary Tree
+        1. &nbsp; Perfect binary trees
       </span>
     </span>
   </a>
@@ -2065,10 +2065,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-complete-binary-tree" class="md-nav__link">
+  <a href="#2-complete-binary-trees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Complete Binary Tree
+        2. &nbsp; Complete binary trees
       </span>
     </span>
   </a>
@@ -2076,10 +2076,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-full-binary-tree" class="md-nav__link">
+  <a href="#3-full-binary-trees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Full Binary Tree
+        3. &nbsp; Full binary trees
       </span>
     </span>
   </a>
@@ -2087,10 +2087,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#4-balanced-binary-tree" class="md-nav__link">
+  <a href="#4-balanced-binary-trees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        4. &nbsp; Balanced Binary Tree
+        4. &nbsp; Balanced binary trees
       </span>
     </span>
   </a>
@@ -2103,10 +2103,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#714-degeneration-of-binary-trees" class="md-nav__link">
+  <a href="#714-degeneration-of-a-binary-tree" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.1.4 &nbsp; Degeneration of Binary Trees
+        7.1.4 &nbsp; Degeneration of a binary tree
       </span>
     </span>
   </a>
@@ -2136,7 +2136,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -2163,7 +2163,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2190,7 +2190,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2217,7 +2217,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2244,7 +2244,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2271,7 +2271,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2335,7 +2335,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2356,7 +2356,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2377,7 +2377,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
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@@ -2591,7 +2591,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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@@ -2809,7 +2809,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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@@ -3049,7 +3049,7 @@
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3091,7 +3091,7 @@
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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@@ -3496,7 +3496,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4241,7 +4241,7 @@
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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@@ -4658,6 +4658,78 @@
 
 
 
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     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#711-common-terminology-of-binary-trees" class="md-nav__link">
+  <a href="#711-common-terminology" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.1.1 &nbsp; Common Terminology of Binary Trees
+        7.1.1 &nbsp; Common terminology
       </span>
     </span>
   </a>
@@ -4710,22 +4782,22 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#712-basic-operations-of-binary-trees" class="md-nav__link">
+  <a href="#712-basic-operations" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.1.2 &nbsp; Basic Operations of Binary Trees
+        7.1.2 &nbsp; Basic operations
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.1.2 Basic Operations of Binary Trees">
+    <nav class="md-nav" aria-label="7.1.2 Basic operations">
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   <a href="#1-initializing-a-binary-tree" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Initializing a Binary Tree
+        1. &nbsp; Initializing a binary tree
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     </span>
   </a>
@@ -4733,10 +4805,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-inserting-and-removing-nodes" class="md-nav__link">
+  <a href="#2-inserting-and-deleting-nodes" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Inserting and Removing Nodes
+        2. &nbsp; Inserting and deleting nodes
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     </span>
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@@ -4752,19 +4824,19 @@
   <a href="#713-common-types-of-binary-trees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.1.3 &nbsp; Common Types of Binary Trees
+        7.1.3 &nbsp; Common types of binary trees
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.1.3 Common Types of Binary Trees">
+    <nav class="md-nav" aria-label="7.1.3 Common types of binary trees">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-perfect-binary-tree" class="md-nav__link">
+  <a href="#1-perfect-binary-trees" class="md-nav__link">
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       <span class="md-typeset">
-        1. &nbsp; Perfect Binary Tree
+        1. &nbsp; Perfect binary trees
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   </a>
@@ -4772,10 +4844,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-complete-binary-tree" class="md-nav__link">
+  <a href="#2-complete-binary-trees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Complete Binary Tree
+        2. &nbsp; Complete binary trees
       </span>
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   </a>
@@ -4783,10 +4855,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-full-binary-tree" class="md-nav__link">
+  <a href="#3-full-binary-trees" class="md-nav__link">
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-        3. &nbsp; Full Binary Tree
+        3. &nbsp; Full binary trees
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           <li class="md-nav__item">
-  <a href="#4-balanced-binary-tree" class="md-nav__link">
+  <a href="#4-balanced-binary-trees" class="md-nav__link">
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       <span class="md-typeset">
-        4. &nbsp; Balanced Binary Tree
+        4. &nbsp; Balanced binary trees
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@@ -4810,10 +4882,10 @@
 </li>
 
         <li class="md-nav__item">
-  <a href="#714-degeneration-of-binary-trees" class="md-nav__link">
+  <a href="#714-degeneration-of-a-binary-tree" class="md-nav__link">
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       <span class="md-typeset">
-        7.1.4 &nbsp; Degeneration of Binary Trees
+        7.1.4 &nbsp; Degeneration of a binary tree
       </span>
     </span>
   </a>
@@ -4857,50 +4929,50 @@
 
 
 <!-- Page content -->
-<h1 id="71-binary-tree">7.1 &nbsp; Binary Tree<a class="headerlink" href="#71-binary-tree" title="Permanent link">&para;</a></h1>
-<p>A <u>binary tree</u> is a non-linear data structure that models the hierarchical relationship between "ancestors" and "descendants" and embodies a divide-and-conquer pattern in which each split branches into two. Similar to a linked list, the basic unit of a binary tree is a node, and each node contains a value, a reference to its left child node, and a reference to its right child node.</p>
+<h1 id="71-binary-trees">7.1 &nbsp; Binary trees<a class="headerlink" href="#71-binary-trees" title="Permanent link">&para;</a></h1>
+<p>A <u>binary tree</u> is a nonlinear data structure representing relationships between ancestors and descendants. It embodies the divide-and-conquer idea of splitting in two. Like a linked list, it is built from nodes, each containing a value and references to its left and right children.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="1:13"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><input id="__tabbed_1_13" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Python</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Java</label><label for="__tabbed_1_4">C#</label><label for="__tabbed_1_5">Go</label><label for="__tabbed_1_6">Swift</label><label for="__tabbed_1_7">JS</label><label for="__tabbed_1_8">TS</label><label for="__tabbed_1_9">Dart</label><label for="__tabbed_1_10">Rust</label><label for="__tabbed_1_11">C</label><label for="__tabbed_1_12">Kotlin</label><label for="__tabbed_1_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
 <div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">class</span><span class="w"> </span><span class="nc">TreeNode</span><span class="p">:</span>
-<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Binary tree node"""</span>
+<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Binary tree node class"""</span>
 <a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">int</span><span class="p">):</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">val</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="n">val</span>                <span class="c1"># Node value</span>
-<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">left</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span>  <span class="c1"># Reference to left child node</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># Reference to right child node</span>
+<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">left</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span>  <span class="c1"># Left child reference</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>        <span class="bp">self</span><span class="o">.</span><span class="n">right</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span> <span class="o">=</span> <span class="kc">None</span> <span class="c1"># Right child reference</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Binary tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Binary tree node struct */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="k">struct</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w">          </span><span class="c1">// Node value</span>
-<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">left</span><span class="p">;</span><span class="w">   </span><span class="c1">// Pointer to left child node</span>
-<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">right</span><span class="p">;</span><span class="w">  </span><span class="c1">// Pointer to right child node</span>
+<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">left</span><span class="p">;</span><span class="w">   </span><span class="c1">// Left child pointer</span>
+<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">right</span><span class="p">;</span><span class="w">  </span><span class="c1">// Right child pointer</span>
 <a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="n">TreeNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">val</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="w"> </span><span class="n">left</span><span class="p">(</span><span class="nb">nullptr</span><span class="p">),</span><span class="w"> </span><span class="n">right</span><span class="p">(</span><span class="nb">nullptr</span><span class="p">)</span><span class="w"> </span><span class="p">{}</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="p">};</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Binary tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Binary tree node class */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kd">class</span> <span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w">         </span><span class="c1">// Node value</span>
-<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w">   </span><span class="c1">// Reference to left child node</span>
-<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">right</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to right child node</span>
+<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w">   </span><span class="c1">// Left child reference</span>
+<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">right</span><span class="p">;</span><span class="w">  </span><span class="c1">// Right child reference</span>
 <a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="n">TreeNode</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"> </span><span class="p">}</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Binary tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Binary tree node class */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="k">class</span><span class="w"> </span><span class="nf">TreeNode</span><span class="p">(</span><span class="kt">int?</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="kt">int?</span><span class="w"> </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w">    </span><span class="c1">// Node value</span>
-<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to left child node</span>
-<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference to right child node</span>
+<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w">  </span><span class="c1">// Left child reference</span>
+<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="k">public</span><span class="w"> </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// Right child reference</span>
 <a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Binary tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Binary tree node struct */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">type</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="kd">struct</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="nx">Val</span><span class="w">   </span><span class="kt">int</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">Left</span><span class="w">  </span><span class="o">*</span><span class="nx">TreeNode</span>
@@ -4909,19 +4981,19 @@
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="cm">/* Constructor */</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="kd">func</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="nx">v</span><span class="w"> </span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="o">&amp;</span><span class="nx">TreeNode</span><span class="p">{</span>
-<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">        </span><span class="nx">Left</span><span class="p">:</span><span class="w">  </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="c1">// Pointer to left child node</span>
-<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="nx">Right</span><span class="p">:</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="c1">// Pointer to right child node</span>
+<a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">        </span><span class="nx">Left</span><span class="p">:</span><span class="w">  </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="c1">// Left child pointer</span>
+<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="nx">Right</span><span class="p">:</span><span class="w"> </span><span class="kc">nil</span><span class="p">,</span><span class="w"> </span><span class="c1">// Right child pointer</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="nx">Val</span><span class="p">:</span><span class="w">   </span><span class="nx">v</span><span class="p">,</span><span class="w">   </span><span class="c1">// Node value</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Binary tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Binary tree node class */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">val</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="w"> </span><span class="c1">// Node value</span>
-<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">left</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Reference to left child node</span>
-<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">right</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Reference to right child node</span>
+<a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">left</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Left child reference</span>
+<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">right</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">?</span><span class="w"> </span><span class="c1">// Right child reference</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="kd">init</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="nb">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">        </span><span class="n">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">x</span>
@@ -4930,11 +5002,11 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Binary tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Binary tree node class */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="nx">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Node value</span>
-<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="nx">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// Pointer to left child node</span>
-<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="nx">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// Pointer to right child node</span>
+<a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="nx">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// Left child pointer</span>
+<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="nx">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// Right child pointer</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">val</span><span class="p">,</span><span class="w"> </span><span class="nx">left</span><span class="p">,</span><span class="w"> </span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="mf">0</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">val</span><span class="p">;</span>
 <a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="kc">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="nx">left</span><span class="p">;</span>
@@ -4944,7 +5016,7 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Binary tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Binary tree node class */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">class</span><span class="w"> </span><span class="nx">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="nx">val</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">;</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="nx">left</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">;</span>
@@ -4952,18 +5024,18 @@
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="kr">constructor</span><span class="p">(</span><span class="nx">val?</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">,</span><span class="w"> </span><span class="nx">left?</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">,</span><span class="w"> </span><span class="nx">right?</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">val</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">0</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">val</span><span class="p">;</span><span class="w"> </span><span class="c1">// Node value</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference to left child node</span>
-<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference to right child node</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">left</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">left</span><span class="p">;</span><span class="w"> </span><span class="c1">// Left child reference</span>
+<a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="k">this</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">right</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">undefined</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="nx">null</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="kt">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// Right child reference</span>
 <a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="p">}</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Binary tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Binary tree node class */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w">         </span><span class="c1">// Node value</span>
-<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w">  </span><span class="c1">// Reference to left child node</span>
-<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// Reference to right child node</span>
+<a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">left</span><span class="p">;</span><span class="w">  </span><span class="c1">// Left child reference</span>
+<a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// Right child reference</span>
 <a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="n">TreeNode</span><span class="p">(</span><span class="k">this</span><span class="p">.</span><span class="n">val</span><span class="p">,</span><span class="w"> </span><span class="p">[</span><span class="k">this</span><span class="p">.</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="k">this</span><span class="p">.</span><span class="n">right</span><span class="p">]);</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="p">}</span>
 </code></pre></div>
@@ -4972,11 +5044,11 @@
 <div class="highlight"><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="k">use</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">rc</span><span class="p">::</span><span class="n">Rc</span><span class="p">;</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">use</span><span class="w"> </span><span class="n">std</span><span class="p">::</span><span class="n">cell</span><span class="p">::</span><span class="n">RefCell</span><span class="p">;</span>
 <a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a>
-<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="cm">/* Binary tree node */</span>
+<a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="cm">/* Binary tree node struct */</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="k">struct</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="n">val</span><span class="p">:</span><span class="w"> </span><span class="kt">i32</span><span class="p">,</span><span class="w">                               </span><span class="c1">// Node value</span>
-<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="n">left</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w">    </span><span class="c1">// Reference to left child node</span>
-<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="n">right</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w">   </span><span class="c1">// Reference to right child node</span>
+<a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="n">left</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w">    </span><span class="c1">// Left child reference</span>
+<a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a><span class="w">    </span><span class="n">right</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w">   </span><span class="c1">// Right child reference</span>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="p">}</span>
 <a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a>
 <a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="k">impl</span><span class="w"> </span><span class="n">TreeNode</span><span class="w"> </span><span class="p">{</span>
@@ -4992,12 +5064,12 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Binary tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a><span class="cm">/* Binary tree node struct */</span>
 <a id="__codelineno-10-2" name="__codelineno-10-2" href="#__codelineno-10-2"></a><span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-10-3" name="__codelineno-10-3" href="#__codelineno-10-3"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">val</span><span class="p">;</span><span class="w">                </span><span class="c1">// Node value</span>
 <a id="__codelineno-10-4" name="__codelineno-10-4" href="#__codelineno-10-4"></a><span class="w">    </span><span class="kt">int</span><span class="w"> </span><span class="n">height</span><span class="p">;</span><span class="w">             </span><span class="c1">// Node height</span>
-<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="k">struct</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">left</span><span class="p">;</span><span class="w">  </span><span class="c1">// Pointer to left child node</span>
-<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="k">struct</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// Pointer to right child node</span>
+<a id="__codelineno-10-5" name="__codelineno-10-5" href="#__codelineno-10-5"></a><span class="w">    </span><span class="k">struct</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">left</span><span class="p">;</span><span class="w">  </span><span class="c1">// Left child pointer</span>
+<a id="__codelineno-10-6" name="__codelineno-10-6" href="#__codelineno-10-6"></a><span class="w">    </span><span class="k">struct</span><span class="w"> </span><span class="nc">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">right</span><span class="p">;</span><span class="w"> </span><span class="c1">// Right child pointer</span>
 <a id="__codelineno-10-7" name="__codelineno-10-7" href="#__codelineno-10-7"></a><span class="p">}</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">;</span>
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="cm">/* Constructor */</span>
@@ -5014,10 +5086,10 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Binary tree node */</span>
+<div class="highlight"><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Binary tree node class */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">class</span><span class="w"> </span><span class="nc">TreeNode</span><span class="p">(</span><span class="kd">val</span><span class="w"> </span><span class="nv">_val</span><span class="p">:</span><span class="w"> </span><span class="kt">Int</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w">  </span><span class="c1">// Node value</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">left</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="w">   </span><span class="c1">// Reference to left child node</span>
-<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">right</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="w">  </span><span class="c1">// Reference to right child node</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">left</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="w">   </span><span class="c1">// Left child reference</span>
+<a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">right</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="kc">null</span><span class="w">  </span><span class="c1">// Right child reference</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="p">}</span>
 </code></pre></div>
 </div>
@@ -5025,8 +5097,8 @@
 <div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Binary tree node class ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">class</span><span class="w"> </span><span class="nc">TreeNode</span>
 <a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="kp">attr_accessor</span><span class="w"> </span><span class="ss">:val</span><span class="w">    </span><span class="c1"># Node value</span>
-<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="kp">attr_accessor</span><span class="w"> </span><span class="ss">:left</span><span class="w">   </span><span class="c1"># Reference to left child node</span>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="kp">attr_accessor</span><span class="w"> </span><span class="ss">:right</span><span class="w">  </span><span class="c1"># Reference to right child node</span>
+<a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="kp">attr_accessor</span><span class="w"> </span><span class="ss">:left</span><span class="w">   </span><span class="c1"># Left child reference</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="kp">attr_accessor</span><span class="w"> </span><span class="ss">:right</span><span class="w">  </span><span class="c1"># Right child reference</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="k">def</span><span class="w"> </span><span class="nf">initialize</span><span class="p">(</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="vi">@val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">val</span>
@@ -5036,44 +5108,44 @@
 </div>
 </div>
 </div>
-<p>Each node has two references (pointers), pointing respectively to the <u>left-child node</u> and <u>right-child node</u>. This node is called the <u>parent node</u> of these two child nodes. When given a node of a binary tree, we call the tree formed by this node's left child and all nodes below it the <u>left subtree</u> of this node. Similarly, the <u>right subtree</u> can be defined.</p>
-<p><strong>In a binary tree, every non-leaf node has child nodes and therefore non-empty subtrees.</strong> As shown in Figure 7-1, if "Node 2" is regarded as a parent node, its left and right child nodes are "Node 4" and "Node 5" respectively. The left subtree is formed by "Node 4" and all nodes beneath it, while the right subtree is formed by "Node 5" and all nodes beneath it.</p>
-<p><img alt="Parent Node, child Node, subtree" class="animation-figure" src="../binary_tree.assets/binary_tree_definition.png" /></p>
-<p align="center"> Figure 7-1 &nbsp; Parent Node, child Node, subtree </p>
+<p>Each node has two references or pointers, one to its <u>left child</u> and one to its <u>right child</u>. The node is their <u>parent</u>. Its left child and all nodes below that child form its <u>left subtree</u>; the <u>right subtree</u> is defined similarly.</p>
+<p><strong>Every non-leaf node has children and at least one nonempty subtree</strong>. In Figure 7-1, if node 2 is the parent, its left and right children are nodes 4 and 5. Its left subtree consists of node 4 and everything below it, and its right subtree consists of node 5 and everything below it.</p>
+<p><img alt="Parents, children, and subtrees" class="animation-figure" src="../binary_tree.assets/binary_tree_definition.png" /></p>
+<p align="center"> Figure 7-1 &nbsp; Parents, children, and subtrees </p>
 
-<h2 id="711-common-terminology-of-binary-trees">7.1.1 &nbsp; Common Terminology of Binary Trees<a class="headerlink" href="#711-common-terminology-of-binary-trees" title="Permanent link">&para;</a></h2>
-<p>The commonly used terminology of binary trees is shown in Figure 7-2.</p>
+<h2 id="711-common-terminology">7.1.1 &nbsp; Common terminology<a class="headerlink" href="#711-common-terminology" title="Permanent link">&para;</a></h2>
+<p>Figure 7-2 illustrates common binary-tree terminology.</p>
 <ul>
-<li><u>Root node</u>: The node at the top level of a binary tree, which does not have a parent node.</li>
-<li><u>Leaf node</u>: A node that does not have any child nodes, with both of its pointers pointing to <code>None</code>.</li>
-<li><u>Edge</u>: A line segment that connects two nodes, representing a reference (pointer) between the nodes.</li>
-<li>The <u>level</u> of a node: It increases from top to bottom, with the root node being at level 1.</li>
-<li>The <u>degree</u> of a node: The number of child nodes that a node has. In a binary tree, the degree can be 0, 1, or 2.</li>
-<li>The <u>height</u> of a binary tree: The number of edges from the root node to the farthest leaf node.</li>
-<li>The <u>depth</u> of a node: The number of edges from the root node to the node.</li>
-<li>The <u>height</u> of a node: The number of edges from the farthest leaf node to the node.</li>
+<li><u>Root</u>: the topmost node, which has no parent.</li>
+<li><u>Leaf</u>: a node with no children; both of its child references point to <code>None</code>.</li>
+<li><u>Edge</u>: a connection between two nodes, represented by a reference or pointer.</li>
+<li>A node's <u>level</u>: levels increase from top to bottom, with the root at level 1.</li>
+<li>A node's <u>degree</u>: its number of children, which is 0, 1, or 2 in a binary tree.</li>
+<li>A tree's <u>height</u>: the number of edges from the root to its farthest leaf.</li>
+<li>A node's <u>depth</u>: the number of edges from the root to that node.</li>
+<li>A node's <u>height</u>: the number of edges from that node to its farthest leaf.</li>
 </ul>
-<p><img alt="Common Terminology of Binary Trees" class="animation-figure" src="../binary_tree.assets/binary_tree_terminology.png" /></p>
-<p align="center"> Figure 7-2 &nbsp; Common Terminology of Binary Trees </p>
+<p><img alt="Common binary-tree terminology" class="animation-figure" src="../binary_tree.assets/binary_tree_terminology.png" /></p>
+<p align="center"> Figure 7-2 &nbsp; Common binary-tree terminology </p>
 
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>We usually define "height" and "depth" as the number of edges traversed, but some textbooks and problem statements define them as the number of nodes on the path. In that case, both values are larger by 1.</p>
+<p>Height and depth are usually measured in edges, but some problems or textbooks count nodes instead. Under that convention, add 1 to both height and depth.</p>
 </div>
-<h2 id="712-basic-operations-of-binary-trees">7.1.2 &nbsp; Basic Operations of Binary Trees<a class="headerlink" href="#712-basic-operations-of-binary-trees" title="Permanent link">&para;</a></h2>
-<h3 id="1-initializing-a-binary-tree">1. &nbsp; Initializing a Binary Tree<a class="headerlink" href="#1-initializing-a-binary-tree" title="Permanent link">&para;</a></h3>
-<p>Similar to a linked list, the initialization of a binary tree involves first creating the nodes and then establishing the references (pointers) between them.</p>
+<h2 id="712-basic-operations">7.1.2 &nbsp; Basic operations<a class="headerlink" href="#712-basic-operations" title="Permanent link">&para;</a></h2>
+<h3 id="1-initializing-a-binary-tree">1. &nbsp; Initializing a binary tree<a class="headerlink" href="#1-initializing-a-binary-tree" title="Permanent link">&para;</a></h3>
+<p>As with a linked list, first create the nodes, then connect them with references or pointers.</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="c1"># Initializing a binary tree</span>
-<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="c1"># Initializing nodes</span>
+<div class="highlight"><span class="filename">binary_tree.py</span><pre><span></span><code><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a><span class="c1"># Initialize a binary tree</span>
+<a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="c1"># Initialize nodes</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a><span class="n">n1</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a><span class="n">n2</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a><span class="n">n3</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a><span class="n">n4</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a><span class="n">n5</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="n">val</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
-<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a><span class="c1"># Linking references (pointers) between nodes</span>
+<a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a><span class="c1"># Connect nodes with references (pointers)</span>
 <a id="__codelineno-13-9" name="__codelineno-13-9" href="#__codelineno-13-9"></a><span class="n">n1</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">n2</span>
 <a id="__codelineno-13-10" name="__codelineno-13-10" href="#__codelineno-13-10"></a><span class="n">n1</span><span class="o">.</span><span class="n">right</span> <span class="o">=</span> <span class="n">n3</span>
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a><span class="n">n2</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">n4</span>
@@ -5081,14 +5153,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Initializing a binary tree */</span>
-<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="c1">// Initializing nodes</span>
+<div class="highlight"><span class="filename">binary_tree.cpp</span><pre><span></span><code><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a><span class="cm">/* Initialize a binary tree */</span>
+<a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="c1">// Initialize nodes</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="n">TreeNode</span><span class="o">*</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="n">TreeNode</span><span class="o">*</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
 <a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="n">TreeNode</span><span class="o">*</span><span class="w"> </span><span class="n">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="n">TreeNode</span><span class="o">*</span><span class="w"> </span><span class="n">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="n">TreeNode</span><span class="o">*</span><span class="w"> </span><span class="n">n5</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
-<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="c1">// Linking references (pointers) between nodes</span>
+<a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="c1">// Connect nodes with references (pointers)</span>
 <a id="__codelineno-14-9" name="__codelineno-14-9" href="#__codelineno-14-9"></a><span class="n">n1</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
 <a id="__codelineno-14-10" name="__codelineno-14-10" href="#__codelineno-14-10"></a><span class="n">n1</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n3</span><span class="p">;</span>
 <a id="__codelineno-14-11" name="__codelineno-14-11" href="#__codelineno-14-11"></a><span class="n">n2</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n4</span><span class="p">;</span>
@@ -5096,13 +5168,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="c1">// Initializing nodes</span>
+<div class="highlight"><span class="filename">binary_tree.java</span><pre><span></span><code><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a><span class="c1">// Initialize nodes</span>
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n5</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
-<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="c1">// Linking references (pointers) between nodes</span>
+<a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="c1">// Connect nodes with references (pointers)</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="n">n1</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
 <a id="__codelineno-15-9" name="__codelineno-15-9" href="#__codelineno-15-9"></a><span class="n">n1</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n3</span><span class="p">;</span>
 <a id="__codelineno-15-10" name="__codelineno-15-10" href="#__codelineno-15-10"></a><span class="n">n2</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n4</span><span class="p">;</span>
@@ -5110,14 +5182,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Initializing a binary tree */</span>
-<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="c1">// Initializing nodes</span>
+<div class="highlight"><span class="filename">binary_tree.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Initialize a binary tree */</span>
+<a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="c1">// Initialize nodes</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n5</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
-<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="c1">// Linking references (pointers) between nodes</span>
+<a id="__codelineno-16-8" name="__codelineno-16-8" href="#__codelineno-16-8"></a><span class="c1">// Connect nodes with references (pointers)</span>
 <a id="__codelineno-16-9" name="__codelineno-16-9" href="#__codelineno-16-9"></a><span class="n">n1</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="n">n1</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n3</span><span class="p">;</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="n">n2</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n4</span><span class="p">;</span>
@@ -5125,14 +5197,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Initializing a binary tree */</span>
-<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="c1">// Initializing nodes</span>
+<div class="highlight"><span class="filename">binary_tree.go</span><pre><span></span><code><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a><span class="cm">/* Initialize a binary tree */</span>
+<a id="__codelineno-17-2" name="__codelineno-17-2" href="#__codelineno-17-2"></a><span class="c1">// Initialize nodes</span>
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="nx">n1</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="nx">n2</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="nx">n3</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
 <a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="nx">n4</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="nx">n5</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="c1">// Linking references (pointers) between nodes</span>
+<a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="c1">// Connect nodes with references (pointers)</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span>
 <a id="__codelineno-17-10" name="__codelineno-17-10" href="#__codelineno-17-10"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">Right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n3</span>
 <a id="__codelineno-17-11" name="__codelineno-17-11" href="#__codelineno-17-11"></a><span class="nx">n2</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n4</span>
@@ -5140,13 +5212,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="c1">// Initializing nodes</span>
+<div class="highlight"><span class="filename">binary_tree.swift</span><pre><span></span><code><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a><span class="c1">// Initialize nodes</span>
 <a id="__codelineno-18-2" name="__codelineno-18-2" href="#__codelineno-18-2"></a><span class="kd">let</span><span class="w"> </span><span class="nv">n1</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="kd">let</span><span class="w"> </span><span class="nv">n2</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="kd">let</span><span class="w"> </span><span class="nv">n3</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="mi">3</span><span class="p">)</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="kd">let</span><span class="w"> </span><span class="nv">n4</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="kd">let</span><span class="w"> </span><span class="nv">n5</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="mi">5</span><span class="p">)</span>
-<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="c1">// Linking references (pointers) between nodes</span>
+<a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="c1">// Connect nodes with references (pointers)</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="n">n1</span><span class="p">.</span><span class="kr">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">n2</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="n">n1</span><span class="p">.</span><span class="kr">right</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">n3</span>
 <a id="__codelineno-18-10" name="__codelineno-18-10" href="#__codelineno-18-10"></a><span class="n">n2</span><span class="p">.</span><span class="kr">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">n4</span>
@@ -5154,14 +5226,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Initializing a binary tree */</span>
-<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="c1">// Initializing nodes</span>
+<div class="highlight"><span class="filename">binary_tree.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Initialize a binary tree */</span>
+<a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="c1">// Initialize nodes</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="kd">let</span><span class="w"> </span><span class="nx">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">1</span><span class="p">),</span>
 <a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="nx">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">2</span><span class="p">),</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="nx">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">3</span><span class="p">),</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="nx">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">4</span><span class="p">),</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="nx">n5</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">5</span><span class="p">);</span>
-<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="c1">// Linking references (pointers) between nodes</span>
+<a id="__codelineno-19-8" name="__codelineno-19-8" href="#__codelineno-19-8"></a><span class="c1">// Connect nodes with references (pointers)</span>
 <a id="__codelineno-19-9" name="__codelineno-19-9" href="#__codelineno-19-9"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span><span class="p">;</span>
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n3</span><span class="p">;</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="nx">n2</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n4</span><span class="p">;</span>
@@ -5169,14 +5241,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Initializing a binary tree */</span>
-<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="c1">// Initializing nodes</span>
+<div class="highlight"><span class="filename">binary_tree.ts</span><pre><span></span><code><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a><span class="cm">/* Initialize a binary tree */</span>
+<a id="__codelineno-20-2" name="__codelineno-20-2" href="#__codelineno-20-2"></a><span class="c1">// Initialize nodes</span>
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="kd">let</span><span class="w"> </span><span class="nx">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">1</span><span class="p">),</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">    </span><span class="nx">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">2</span><span class="p">),</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="nx">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">3</span><span class="p">),</span>
 <a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="nx">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">4</span><span class="p">),</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="nx">n5</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">5</span><span class="p">);</span>
-<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="c1">// Linking references (pointers) between nodes</span>
+<a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="c1">// Connect nodes with references (pointers)</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span><span class="p">;</span>
 <a id="__codelineno-20-10" name="__codelineno-20-10" href="#__codelineno-20-10"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n3</span><span class="p">;</span>
 <a id="__codelineno-20-11" name="__codelineno-20-11" href="#__codelineno-20-11"></a><span class="nx">n2</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n4</span><span class="p">;</span>
@@ -5184,14 +5256,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Initializing a binary tree */</span>
-<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="c1">// Initializing nodes</span>
+<div class="highlight"><span class="filename">binary_tree.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Initialize a binary tree */</span>
+<a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="c1">// Initialize nodes</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="m">1</span><span class="p">);</span>
 <a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="m">2</span><span class="p">);</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="m">3</span><span class="p">);</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="m">4</span><span class="p">);</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">n5</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="m">5</span><span class="p">);</span>
-<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="c1">// Linking references (pointers) between nodes</span>
+<a id="__codelineno-21-8" name="__codelineno-21-8" href="#__codelineno-21-8"></a><span class="c1">// Connect nodes with references (pointers)</span>
 <a id="__codelineno-21-9" name="__codelineno-21-9" href="#__codelineno-21-9"></a><span class="n">n1</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="n">n1</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n3</span><span class="p">;</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="n">n2</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n4</span><span class="p">;</span>
@@ -5199,13 +5271,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="c1">// Initializing nodes</span>
+<div class="highlight"><span class="filename">binary_tree.rs</span><pre><span></span><code><a id="__codelineno-22-1" name="__codelineno-22-1" href="#__codelineno-22-1"></a><span class="c1">// Initialize nodes</span>
 <a id="__codelineno-22-2" name="__codelineno-22-2" href="#__codelineno-22-2"></a><span class="kd">let</span><span class="w"> </span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-22-3" name="__codelineno-22-3" href="#__codelineno-22-3"></a><span class="kd">let</span><span class="w"> </span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a><span class="kd">let</span><span class="w"> </span><span class="n">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="kd">let</span><span class="w"> </span><span class="n">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="kd">let</span><span class="w"> </span><span class="n">n5</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
-<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="c1">// Linking references (pointers) between nodes</span>
+<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="c1">// Connect nodes with references (pointers)</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="n">n1</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">n2</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="n">n1</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">n3</span><span class="p">);</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="n">n2</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">n4</span><span class="p">);</span>
@@ -5213,14 +5285,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Initializing a binary tree */</span>
-<a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="c1">// Initializing nodes</span>
+<div class="highlight"><span class="filename">binary_tree.c</span><pre><span></span><code><a id="__codelineno-23-1" name="__codelineno-23-1" href="#__codelineno-23-1"></a><span class="cm">/* Initialize a binary tree */</span>
+<a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="c1">// Initialize nodes</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span>
 <a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="mi">4</span><span class="p">);</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">n5</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="mi">5</span><span class="p">);</span>
-<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="c1">// Linking references (pointers) between nodes</span>
+<a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="c1">// Connect nodes with references (pointers)</span>
 <a id="__codelineno-23-9" name="__codelineno-23-9" href="#__codelineno-23-9"></a><span class="n">n1</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
 <a id="__codelineno-23-10" name="__codelineno-23-10" href="#__codelineno-23-10"></a><span class="n">n1</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n3</span><span class="p">;</span>
 <a id="__codelineno-23-11" name="__codelineno-23-11" href="#__codelineno-23-11"></a><span class="n">n2</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n4</span><span class="p">;</span>
@@ -5228,13 +5300,13 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="c1">// Initializing nodes</span>
+<div class="highlight"><span class="filename">binary_tree.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="c1">// Initialize nodes</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">val</span><span class="w"> </span><span class="nv">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="m">1</span><span class="p">)</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="kd">val</span><span class="w"> </span><span class="nv">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="m">2</span><span class="p">)</span>
 <a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="kd">val</span><span class="w"> </span><span class="nv">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="m">3</span><span class="p">)</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="kd">val</span><span class="w"> </span><span class="nv">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="m">4</span><span class="p">)</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="kd">val</span><span class="w"> </span><span class="nv">n5</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="m">5</span><span class="p">)</span>
-<a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="c1">// Linking references (pointers) between nodes</span>
+<a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="c1">// Connect nodes with references (pointers)</span>
 <a id="__codelineno-24-8" name="__codelineno-24-8" href="#__codelineno-24-8"></a><span class="n">n1</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span>
 <a id="__codelineno-24-9" name="__codelineno-24-9" href="#__codelineno-24-9"></a><span class="n">n1</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n3</span>
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="n">n2</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n4</span>
@@ -5242,14 +5314,14 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1"># Initializing a binary tree</span>
-<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="c1"># Initializing nodes</span>
+<div class="highlight"><span class="filename">binary_tree.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1"># Initialize a binary tree</span>
+<a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="c1"># Initialize nodes</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="n">n1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">TreeNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a><span class="n">n2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">TreeNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
 <a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="n">n3</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">TreeNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="n">n4</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">TreeNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="n">n5</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">TreeNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
-<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="c1"># Linking references (pointers) between nodes</span>
+<a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="c1"># Connect nodes with references (pointers)</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="n">n1</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a><span class="n">n1</span><span class="o">.</span><span class="n">right</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n3</span>
 <a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="n">n2</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n4</span>
@@ -5259,190 +5331,194 @@
 </div>
 </div>
 <details class="pythontutor">
-<summary>Code Visualization</summary>
-<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%8F%89%E6%A0%91%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E8%8A%82%E7%82%B9%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E9%92%88%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%8F%89%E6%A0%91%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E8%8A%82%E7%82%B9%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E9%92%88%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Left%20child%20reference%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20Right%20child%20reference%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20binary%20tree%0A%20%20%20%20%23%20Initialize%20nodes%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20Connect%20nodes%20with%20references%20%28pointers%29%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Left%20child%20reference%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20Right%20child%20reference%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20binary%20tree%0A%20%20%20%20%23%20Initialize%20nodes%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20Connect%20nodes%20with%20references%20%28pointers%29%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=3&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
-<h3 id="2-inserting-and-removing-nodes">2. &nbsp; Inserting and Removing Nodes<a class="headerlink" href="#2-inserting-and-removing-nodes" title="Permanent link">&para;</a></h3>
-<p>Similar to a linked list, inserting and removing nodes in a binary tree can be achieved by modifying pointers. Figure 7-3 provides an example.</p>
-<p><img alt="Inserting and removing nodes in a binary tree" class="animation-figure" src="../binary_tree.assets/binary_tree_add_remove.png" /></p>
-<p align="center"> Figure 7-3 &nbsp; Inserting and removing nodes in a binary tree </p>
+<h3 id="2-inserting-and-deleting-nodes">2. &nbsp; Inserting and deleting nodes<a class="headerlink" href="#2-inserting-and-deleting-nodes" title="Permanent link">&para;</a></h3>
+<p>As with linked lists, insertion and deletion can be implemented by changing pointers. An example is shown below.</p>
+<p><img alt="Inserting and deleting a binary-tree node" class="animation-figure" src="../binary_tree.assets/binary_tree_add_remove.png" /></p>
+<p align="center"> Figure 7-3 &nbsp; Inserting and deleting a binary-tree node </p>
 
 <div class="tabbed-set tabbed-alternate" data-tabs="3:13"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><input id="__tabbed_3_13" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Python</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Java</label><label for="__tabbed_3_4">C#</label><label for="__tabbed_3_5">Go</label><label for="__tabbed_3_6">Swift</label><label for="__tabbed_3_7">JS</label><label for="__tabbed_3_8">TS</label><label for="__tabbed_3_9">Dart</label><label for="__tabbed_3_10">Rust</label><label for="__tabbed_3_11">C</label><label for="__tabbed_3_12">Kotlin</label><label for="__tabbed_3_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="c1"># Inserting and removing nodes</span>
+<div class="highlight"><span class="filename">binary_tree.py</span><pre><span></span><code><a id="__codelineno-26-1" name="__codelineno-26-1" href="#__codelineno-26-1"></a><span class="c1"># Insert and delete a node</span>
 <a id="__codelineno-26-2" name="__codelineno-26-2" href="#__codelineno-26-2"></a><span class="n">p</span> <span class="o">=</span> <span class="n">TreeNode</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a><span class="c1"># Inserting node P between n1 -&gt; n2</span>
+<a id="__codelineno-26-3" name="__codelineno-26-3" href="#__codelineno-26-3"></a><span class="c1"># Insert node P between n1 and n2</span>
 <a id="__codelineno-26-4" name="__codelineno-26-4" href="#__codelineno-26-4"></a><span class="n">n1</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">p</span>
 <a id="__codelineno-26-5" name="__codelineno-26-5" href="#__codelineno-26-5"></a><span class="n">p</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">n2</span>
-<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a><span class="c1"># Removing node P</span>
+<a id="__codelineno-26-6" name="__codelineno-26-6" href="#__codelineno-26-6"></a><span class="c1"># Delete node P</span>
 <a id="__codelineno-26-7" name="__codelineno-26-7" href="#__codelineno-26-7"></a><span class="n">n1</span><span class="o">.</span><span class="n">left</span> <span class="o">=</span> <span class="n">n2</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Inserting and removing nodes */</span>
+<div class="highlight"><span class="filename">binary_tree.cpp</span><pre><span></span><code><a id="__codelineno-27-1" name="__codelineno-27-1" href="#__codelineno-27-1"></a><span class="cm">/* Insert and delete a node */</span>
 <a id="__codelineno-27-2" name="__codelineno-27-2" href="#__codelineno-27-2"></a><span class="n">TreeNode</span><span class="o">*</span><span class="w"> </span><span class="n">P</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
-<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="c1">// Inserting node P between n1 and n2</span>
+<a id="__codelineno-27-3" name="__codelineno-27-3" href="#__codelineno-27-3"></a><span class="c1">// Insert node P between n1 and n2</span>
 <a id="__codelineno-27-4" name="__codelineno-27-4" href="#__codelineno-27-4"></a><span class="n">n1</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">P</span><span class="p">;</span>
 <a id="__codelineno-27-5" name="__codelineno-27-5" href="#__codelineno-27-5"></a><span class="n">P</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
-<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="c1">// Removing node P</span>
+<a id="__codelineno-27-6" name="__codelineno-27-6" href="#__codelineno-27-6"></a><span class="c1">// Delete node P</span>
 <a id="__codelineno-27-7" name="__codelineno-27-7" href="#__codelineno-27-7"></a><span class="n">n1</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
+<a id="__codelineno-27-8" name="__codelineno-27-8" href="#__codelineno-27-8"></a><span class="c1">// Free the memory</span>
+<a id="__codelineno-27-9" name="__codelineno-27-9" href="#__codelineno-27-9"></a><span class="k">delete</span><span class="w"> </span><span class="n">P</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_tree.java</span><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">P</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
-<a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="c1">// Inserting node P between n1 and n2</span>
+<a id="__codelineno-28-2" name="__codelineno-28-2" href="#__codelineno-28-2"></a><span class="c1">// Insert node P between n1 and n2</span>
 <a id="__codelineno-28-3" name="__codelineno-28-3" href="#__codelineno-28-3"></a><span class="n">n1</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">P</span><span class="p">;</span>
 <a id="__codelineno-28-4" name="__codelineno-28-4" href="#__codelineno-28-4"></a><span class="n">P</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
-<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="c1">// Removing node P</span>
+<a id="__codelineno-28-5" name="__codelineno-28-5" href="#__codelineno-28-5"></a><span class="c1">// Delete node P</span>
 <a id="__codelineno-28-6" name="__codelineno-28-6" href="#__codelineno-28-6"></a><span class="n">n1</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Inserting and removing nodes */</span>
+<div class="highlight"><span class="filename">binary_tree.cs</span><pre><span></span><code><a id="__codelineno-29-1" name="__codelineno-29-1" href="#__codelineno-29-1"></a><span class="cm">/* Insert and delete a node */</span>
 <a id="__codelineno-29-2" name="__codelineno-29-2" href="#__codelineno-29-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">P</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
-<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="c1">// Inserting node P between n1 and n2</span>
+<a id="__codelineno-29-3" name="__codelineno-29-3" href="#__codelineno-29-3"></a><span class="c1">// Insert node P between n1 and n2</span>
 <a id="__codelineno-29-4" name="__codelineno-29-4" href="#__codelineno-29-4"></a><span class="n">n1</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">P</span><span class="p">;</span>
 <a id="__codelineno-29-5" name="__codelineno-29-5" href="#__codelineno-29-5"></a><span class="n">P</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
-<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="c1">// Removing node P</span>
+<a id="__codelineno-29-6" name="__codelineno-29-6" href="#__codelineno-29-6"></a><span class="c1">// Delete node P</span>
 <a id="__codelineno-29-7" name="__codelineno-29-7" href="#__codelineno-29-7"></a><span class="n">n1</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Inserting and removing nodes */</span>
-<a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="c1">// Inserting node P between n1 and n2</span>
+<div class="highlight"><span class="filename">binary_tree.go</span><pre><span></span><code><a id="__codelineno-30-1" name="__codelineno-30-1" href="#__codelineno-30-1"></a><span class="cm">/* Insert and delete a node */</span>
+<a id="__codelineno-30-2" name="__codelineno-30-2" href="#__codelineno-30-2"></a><span class="c1">// Insert node P between n1 and n2</span>
 <a id="__codelineno-30-3" name="__codelineno-30-3" href="#__codelineno-30-3"></a><span class="nx">p</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">NewTreeNode</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-30-4" name="__codelineno-30-4" href="#__codelineno-30-4"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">p</span>
 <a id="__codelineno-30-5" name="__codelineno-30-5" href="#__codelineno-30-5"></a><span class="nx">p</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span>
-<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="c1">// Removing node P</span>
+<a id="__codelineno-30-6" name="__codelineno-30-6" href="#__codelineno-30-6"></a><span class="c1">// Delete node P</span>
 <a id="__codelineno-30-7" name="__codelineno-30-7" href="#__codelineno-30-7"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_tree.swift</span><pre><span></span><code><a id="__codelineno-31-1" name="__codelineno-31-1" href="#__codelineno-31-1"></a><span class="kd">let</span><span class="w"> </span><span class="nv">P</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="c1">// Inserting node P between n1 and n2</span>
+<a id="__codelineno-31-2" name="__codelineno-31-2" href="#__codelineno-31-2"></a><span class="c1">// Insert node P between n1 and n2</span>
 <a id="__codelineno-31-3" name="__codelineno-31-3" href="#__codelineno-31-3"></a><span class="n">n1</span><span class="p">.</span><span class="kr">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">P</span>
 <a id="__codelineno-31-4" name="__codelineno-31-4" href="#__codelineno-31-4"></a><span class="n">P</span><span class="p">.</span><span class="kr">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">n2</span>
-<a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="c1">// Removing node P</span>
+<a id="__codelineno-31-5" name="__codelineno-31-5" href="#__codelineno-31-5"></a><span class="c1">// Delete node P</span>
 <a id="__codelineno-31-6" name="__codelineno-31-6" href="#__codelineno-31-6"></a><span class="n">n1</span><span class="p">.</span><span class="kr">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">n2</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Inserting and removing nodes */</span>
+<div class="highlight"><span class="filename">binary_tree.js</span><pre><span></span><code><a id="__codelineno-32-1" name="__codelineno-32-1" href="#__codelineno-32-1"></a><span class="cm">/* Insert and delete a node */</span>
 <a id="__codelineno-32-2" name="__codelineno-32-2" href="#__codelineno-32-2"></a><span class="kd">let</span><span class="w"> </span><span class="nx">P</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
-<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="c1">// Inserting node P between n1 and n2</span>
+<a id="__codelineno-32-3" name="__codelineno-32-3" href="#__codelineno-32-3"></a><span class="c1">// Insert node P between n1 and n2</span>
 <a id="__codelineno-32-4" name="__codelineno-32-4" href="#__codelineno-32-4"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">P</span><span class="p">;</span>
 <a id="__codelineno-32-5" name="__codelineno-32-5" href="#__codelineno-32-5"></a><span class="nx">P</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span><span class="p">;</span>
-<a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="c1">// Removing node P</span>
+<a id="__codelineno-32-6" name="__codelineno-32-6" href="#__codelineno-32-6"></a><span class="c1">// Delete node P</span>
 <a id="__codelineno-32-7" name="__codelineno-32-7" href="#__codelineno-32-7"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Inserting and removing nodes */</span>
+<div class="highlight"><span class="filename">binary_tree.ts</span><pre><span></span><code><a id="__codelineno-33-1" name="__codelineno-33-1" href="#__codelineno-33-1"></a><span class="cm">/* Insert and delete a node */</span>
 <a id="__codelineno-33-2" name="__codelineno-33-2" href="#__codelineno-33-2"></a><span class="kd">const</span><span class="w"> </span><span class="nx">P</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="ow">new</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">(</span><span class="mf">0</span><span class="p">);</span>
-<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="c1">// Inserting node P between n1 and n2</span>
+<a id="__codelineno-33-3" name="__codelineno-33-3" href="#__codelineno-33-3"></a><span class="c1">// Insert node P between n1 and n2</span>
 <a id="__codelineno-33-4" name="__codelineno-33-4" href="#__codelineno-33-4"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">P</span><span class="p">;</span>
 <a id="__codelineno-33-5" name="__codelineno-33-5" href="#__codelineno-33-5"></a><span class="nx">P</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span><span class="p">;</span>
-<a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="c1">// Removing node P</span>
+<a id="__codelineno-33-6" name="__codelineno-33-6" href="#__codelineno-33-6"></a><span class="c1">// Delete node P</span>
 <a id="__codelineno-33-7" name="__codelineno-33-7" href="#__codelineno-33-7"></a><span class="nx">n1</span><span class="p">.</span><span class="nx">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">n2</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Inserting and removing nodes */</span>
+<div class="highlight"><span class="filename">binary_tree.dart</span><pre><span></span><code><a id="__codelineno-34-1" name="__codelineno-34-1" href="#__codelineno-34-1"></a><span class="cm">/* Insert and delete a node */</span>
 <a id="__codelineno-34-2" name="__codelineno-34-2" href="#__codelineno-34-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="n">P</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="m">0</span><span class="p">);</span>
-<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="c1">// Inserting node P between n1 and n2</span>
+<a id="__codelineno-34-3" name="__codelineno-34-3" href="#__codelineno-34-3"></a><span class="c1">// Insert node P between n1 and n2</span>
 <a id="__codelineno-34-4" name="__codelineno-34-4" href="#__codelineno-34-4"></a><span class="n">n1</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">P</span><span class="p">;</span>
 <a id="__codelineno-34-5" name="__codelineno-34-5" href="#__codelineno-34-5"></a><span class="n">P</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
-<a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="c1">// Removing node P</span>
+<a id="__codelineno-34-6" name="__codelineno-34-6" href="#__codelineno-34-6"></a><span class="c1">// Delete node P</span>
 <a id="__codelineno-34-7" name="__codelineno-34-7" href="#__codelineno-34-7"></a><span class="n">n1</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_tree.rs</span><pre><span></span><code><a id="__codelineno-35-1" name="__codelineno-35-1" href="#__codelineno-35-1"></a><span class="kd">let</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">::</span><span class="n">new</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
-<a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="c1">// Inserting node P between n1 and n2</span>
+<a id="__codelineno-35-2" name="__codelineno-35-2" href="#__codelineno-35-2"></a><span class="c1">// Insert node P between n1 and n2</span>
 <a id="__codelineno-35-3" name="__codelineno-35-3" href="#__codelineno-35-3"></a><span class="n">n1</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">p</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
 <a id="__codelineno-35-4" name="__codelineno-35-4" href="#__codelineno-35-4"></a><span class="n">p</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">n2</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
-<a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="c1">// Removing node P</span>
+<a id="__codelineno-35-5" name="__codelineno-35-5" href="#__codelineno-35-5"></a><span class="c1">// Delete node p</span>
 <a id="__codelineno-35-6" name="__codelineno-35-6" href="#__codelineno-35-6"></a><span class="n">n1</span><span class="p">.</span><span class="n">borrow_mut</span><span class="p">().</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">n2</span><span class="p">);</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Inserting and removing nodes */</span>
+<div class="highlight"><span class="filename">binary_tree.c</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* Insert and delete a node */</span>
 <a id="__codelineno-36-2" name="__codelineno-36-2" href="#__codelineno-36-2"></a><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">P</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">newTreeNode</span><span class="p">(</span><span class="mi">0</span><span class="p">);</span>
-<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="c1">// Inserting node P between n1 and n2</span>
+<a id="__codelineno-36-3" name="__codelineno-36-3" href="#__codelineno-36-3"></a><span class="c1">// Insert node P between n1 and n2</span>
 <a id="__codelineno-36-4" name="__codelineno-36-4" href="#__codelineno-36-4"></a><span class="n">n1</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">P</span><span class="p">;</span>
 <a id="__codelineno-36-5" name="__codelineno-36-5" href="#__codelineno-36-5"></a><span class="n">P</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
-<a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="c1">// Removing node P</span>
+<a id="__codelineno-36-6" name="__codelineno-36-6" href="#__codelineno-36-6"></a><span class="c1">// Delete node P</span>
 <a id="__codelineno-36-7" name="__codelineno-36-7" href="#__codelineno-36-7"></a><span class="n">n1</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span><span class="p">;</span>
+<a id="__codelineno-36-8" name="__codelineno-36-8" href="#__codelineno-36-8"></a><span class="c1">// Free the memory</span>
+<a id="__codelineno-36-9" name="__codelineno-36-9" href="#__codelineno-36-9"></a><span class="n">free</span><span class="p">(</span><span class="n">P</span><span class="p">);</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_tree.kt</span><pre><span></span><code><a id="__codelineno-37-1" name="__codelineno-37-1" href="#__codelineno-37-1"></a><span class="kd">val</span><span class="w"> </span><span class="nv">P</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">(</span><span class="m">0</span><span class="p">)</span>
-<a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="c1">// Inserting node P between n1 and n2</span>
+<a id="__codelineno-37-2" name="__codelineno-37-2" href="#__codelineno-37-2"></a><span class="c1">// Insert node P between n1 and n2</span>
 <a id="__codelineno-37-3" name="__codelineno-37-3" href="#__codelineno-37-3"></a><span class="n">n1</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">P</span>
 <a id="__codelineno-37-4" name="__codelineno-37-4" href="#__codelineno-37-4"></a><span class="n">P</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span>
-<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="c1">// Removing node P</span>
+<a id="__codelineno-37-5" name="__codelineno-37-5" href="#__codelineno-37-5"></a><span class="c1">// Delete node P</span>
 <a id="__codelineno-37-6" name="__codelineno-37-6" href="#__codelineno-37-6"></a><span class="n">n1</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1"># Inserting and removing nodes</span>
+<div class="highlight"><span class="filename">binary_tree.rb</span><pre><span></span><code><a id="__codelineno-38-1" name="__codelineno-38-1" href="#__codelineno-38-1"></a><span class="c1"># Insert and delete a node</span>
 <a id="__codelineno-38-2" name="__codelineno-38-2" href="#__codelineno-38-2"></a><span class="n">_p</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="no">TreeNode</span><span class="o">.</span><span class="n">new</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
-<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="c1"># Inserting node _p between n1 and n2</span>
+<a id="__codelineno-38-3" name="__codelineno-38-3" href="#__codelineno-38-3"></a><span class="c1"># Insert node _p between n1 and n2</span>
 <a id="__codelineno-38-4" name="__codelineno-38-4" href="#__codelineno-38-4"></a><span class="n">n1</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">_p</span>
 <a id="__codelineno-38-5" name="__codelineno-38-5" href="#__codelineno-38-5"></a><span class="n">_p</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span>
-<a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="c1"># Removing node _p</span>
+<a id="__codelineno-38-6" name="__codelineno-38-6" href="#__codelineno-38-6"></a><span class="c1"># Delete the node</span>
 <a id="__codelineno-38-7" name="__codelineno-38-7" href="#__codelineno-38-7"></a><span class="n">n1</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n2</span>
 </code></pre></div>
 </div>
 </div>
 </div>
 <details class="pythontutor">
-<summary>Code Visualization</summary>
-<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Code Visualization" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%8F%89%E6%A0%91%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E8%8A%82%E7%82%B9%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E9%92%88%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E4%B8%8E%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%0A%20%20%20%20p%20%3D%20TreeNode%280%29%0A%20%20%20%20%23%20%E5%9C%A8%20n1%20-%3E%20n2%20%E4%B8%AD%E9%97%B4%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20p%0A%20%20%20%20p.left%20%3D%20n2%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20n2&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=37&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
-<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%8F%89%E6%A0%91%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E8%8A%82%E7%82%B9%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E9%92%88%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E4%B8%8E%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%0A%20%20%20%20p%20%3D%20TreeNode%280%29%0A%20%20%20%20%23%20%E5%9C%A8%20n1%20-%3E%20n2%20%E4%B8%AD%E9%97%B4%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20p%0A%20%20%20%20p.left%20%3D%20n2%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20n2&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=37&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen &gt;</a></div></p>
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Left%20child%20reference%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20Right%20child%20reference%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20binary%20tree%0A%20%20%20%20%23%20Initialize%20nodes%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20Connect%20nodes%20with%20references%20%28pointers%29%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5%0A%0A%20%20%20%20%23%20Insert%20and%20delete%20a%20node%0A%20%20%20%20p%20%3D%20TreeNode%280%29%0A%20%20%20%20%23%20Insert%20node%20P%20between%20n1%20and%20n2%0A%20%20%20%20n1.left%20%3D%20p%0A%20%20%20%20p.left%20%3D%20n2%0A%20%20%20%20%23%20Delete%20node%20P%0A%20%20%20%20n1.left%20%3D%20n2&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=37&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Left%20child%20reference%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20Right%20child%20reference%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20binary%20tree%0A%20%20%20%20%23%20Initialize%20nodes%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20Connect%20nodes%20with%20references%20%28pointers%29%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5%0A%0A%20%20%20%20%23%20Insert%20and%20delete%20a%20node%0A%20%20%20%20p%20%3D%20TreeNode%280%29%0A%20%20%20%20%23%20Insert%20node%20P%20between%20n1%20and%20n2%0A%20%20%20%20n1.left%20%3D%20p%0A%20%20%20%20p.left%20%3D%20n2%0A%20%20%20%20%23%20Delete%20node%20P%0A%20%20%20%20n1.left%20%3D%20n2&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=37&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
 </details>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>Keep in mind that inserting a node can alter the original logical structure of a binary tree, while deleting a node usually entails removing that node together with its entire subtree. In practice, insertion and deletion in binary trees are therefore typically implemented as coordinated sequences of operations to achieve a meaningful result.</p>
+<p>Inserting a node may change the tree's logical structure, while deleting a node generally means deleting it and all its subtrees. Meaningful insertion and deletion operations therefore usually require several coordinated steps.</p>
 </div>
-<h2 id="713-common-types-of-binary-trees">7.1.3 &nbsp; Common Types of Binary Trees<a class="headerlink" href="#713-common-types-of-binary-trees" title="Permanent link">&para;</a></h2>
-<h3 id="1-perfect-binary-tree">1. &nbsp; Perfect Binary Tree<a class="headerlink" href="#1-perfect-binary-tree" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 7-4, a <u>perfect binary tree</u> has every level completely filled. In a perfect binary tree, leaf nodes have a degree of <span class="arithmatex">\(0\)</span>, while all other nodes have a degree of <span class="arithmatex">\(2\)</span>. If the tree height is <span class="arithmatex">\(h\)</span>, the total number of nodes is <span class="arithmatex">\(2^{h+1} - 1\)</span>, following a standard exponential pattern that mirrors the common phenomenon of cell division in nature.</p>
+<h2 id="713-common-types-of-binary-trees">7.1.3 &nbsp; Common types of binary trees<a class="headerlink" href="#713-common-types-of-binary-trees" title="Permanent link">&para;</a></h2>
+<h3 id="1-perfect-binary-trees">1. &nbsp; Perfect binary trees<a class="headerlink" href="#1-perfect-binary-trees" title="Permanent link">&para;</a></h3>
+<p>In a <u>perfect binary tree</u>, every level is completely filled, as shown below. Leaves have degree <span class="arithmatex">\(0\)</span>, and all other nodes have degree <span class="arithmatex">\(2\)</span>. A tree of height <span class="arithmatex">\(h\)</span> has <span class="arithmatex">\(2^{h+1} - 1\)</span> nodes, an exponential relationship reminiscent of cell division in nature.</p>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>Please note that in the Chinese community, a perfect binary tree is often referred to as a <u>full binary tree</u>.</p>
+<p>In Chinese-language resources, a perfect binary tree is often called <u>满二叉树</u>.</p>
 </div>
-<p><img alt="Perfect binary tree" class="animation-figure" src="../binary_tree.assets/perfect_binary_tree.png" /></p>
-<p align="center"> Figure 7-4 &nbsp; Perfect binary tree </p>
+<p><img alt="A perfect binary tree" class="animation-figure" src="../binary_tree.assets/perfect_binary_tree.png" /></p>
+<p align="center"> Figure 7-4 &nbsp; A perfect binary tree </p>
 
-<h3 id="2-complete-binary-tree">2. &nbsp; Complete Binary Tree<a class="headerlink" href="#2-complete-binary-tree" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 7-5, a <u>complete binary tree</u> only allows the bottom level to be incompletely filled, and the nodes at the bottom level must be filled continuously from left to right. Note that a perfect binary tree is also a complete binary tree.</p>
-<p><img alt="Complete binary tree" class="animation-figure" src="../binary_tree.assets/complete_binary_tree.png" /></p>
-<p align="center"> Figure 7-5 &nbsp; Complete binary tree </p>
+<h3 id="2-complete-binary-trees">2. &nbsp; Complete binary trees<a class="headerlink" href="#2-complete-binary-trees" title="Permanent link">&para;</a></h3>
+<p>Only the lowest level of a <u>complete binary tree</u> may be incomplete, and its nodes must fill that level consecutively from left to right, as shown below. Every perfect binary tree is also complete.</p>
+<p><img alt="A complete binary tree" class="animation-figure" src="../binary_tree.assets/complete_binary_tree.png" /></p>
+<p align="center"> Figure 7-5 &nbsp; A complete binary tree </p>
 
-<h3 id="3-full-binary-tree">3. &nbsp; Full Binary Tree<a class="headerlink" href="#3-full-binary-tree" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 7-6, in a <u>full binary tree</u>, all nodes except leaf nodes have two child nodes.</p>
-<p><img alt="Full binary tree" class="animation-figure" src="../binary_tree.assets/full_binary_tree.png" /></p>
-<p align="center"> Figure 7-6 &nbsp; Full binary tree </p>
+<h3 id="3-full-binary-trees">3. &nbsp; Full binary trees<a class="headerlink" href="#3-full-binary-trees" title="Permanent link">&para;</a></h3>
+<p>In a <u>full binary tree</u>, every node other than a leaf has exactly two children, as shown below.</p>
+<p><img alt="A full binary tree" class="animation-figure" src="../binary_tree.assets/full_binary_tree.png" /></p>
+<p align="center"> Figure 7-6 &nbsp; A full binary tree </p>
 
-<h3 id="4-balanced-binary-tree">4. &nbsp; Balanced Binary Tree<a class="headerlink" href="#4-balanced-binary-tree" title="Permanent link">&para;</a></h3>
-<p>As shown in Figure 7-7, in a <u>balanced binary tree</u>, the absolute difference between the height of the left and right subtrees of any node does not exceed 1.</p>
-<p><img alt="Balanced binary tree" class="animation-figure" src="../binary_tree.assets/balanced_binary_tree.png" /></p>
-<p align="center"> Figure 7-7 &nbsp; Balanced binary tree </p>
+<h3 id="4-balanced-binary-trees">4. &nbsp; Balanced binary trees<a class="headerlink" href="#4-balanced-binary-trees" title="Permanent link">&para;</a></h3>
+<p>In a <u>balanced binary tree</u>, the heights of the left and right subtrees of every node differ by at most 1, as shown below.</p>
+<p><img alt="A balanced binary tree" class="animation-figure" src="../binary_tree.assets/balanced_binary_tree.png" /></p>
+<p align="center"> Figure 7-7 &nbsp; A balanced binary tree </p>
 
-<h2 id="714-degeneration-of-binary-trees">7.1.4 &nbsp; Degeneration of Binary Trees<a class="headerlink" href="#714-degeneration-of-binary-trees" title="Permanent link">&para;</a></h2>
-<p>Figure 7-8 contrasts the ideal and degenerate structures of binary trees. When every level is filled, the tree becomes a "perfect binary tree"; when all nodes skew to one side, the binary tree degenerates into a "linked list".</p>
+<h2 id="714-degeneration-of-a-binary-tree">7.1.4 &nbsp; Degeneration of a binary tree<a class="headerlink" href="#714-degeneration-of-a-binary-tree" title="Permanent link">&para;</a></h2>
+<p>Figure 7-8 shows ideal and degenerate tree structures. When every level is full, the tree is perfect. When every node lies along one side, the tree degenerates into a linked list.</p>
 <ul>
-<li>A perfect binary tree is the ideal case, fully leveraging the divide-and-conquer advantages of binary trees.</li>
-<li>A linked list represents the other extreme, where all operations become linear operations with time complexity degrading to <span class="arithmatex">\(O(n)\)</span>.</li>
+<li>A perfect binary tree is the ideal case, making full use of divide and conquer.</li>
+<li>A linked list is the opposite extreme: operations become linear, with time complexity degrading to <span class="arithmatex">\(O(n)\)</span>.</li>
 </ul>
-<p><img alt="The Best and Worst Structures of Binary Trees" class="animation-figure" src="../binary_tree.assets/binary_tree_best_worst_cases.png" /></p>
-<p align="center"> Figure 7-8 &nbsp; The Best and Worst Structures of Binary Trees </p>
+<p><img alt="Best and worst binary-tree structures" class="animation-figure" src="../binary_tree.assets/binary_tree_best_worst_cases.png" /></p>
+<p align="center"> Figure 7-8 &nbsp; Best and worst binary-tree structures </p>
 
-<p>As shown in Table 7-1, in the best and worst structures, the binary tree achieves either maximum or minimum values for leaf node count, total number of nodes, and height.</p>
-<p align="center"> Table 7-1 &nbsp; The Best and Worst Structures of Binary Trees </p>
+<p>As Table 7-1 shows, quantities such as leaf count, total node count, and height reach their maximum or minimum values in these best and worst structures.</p>
+<p align="center"> Table 7-1 &nbsp; Best and worst binary-tree structures </p>
 
 <div class="center-table">
 <table>
@@ -5455,22 +5531,22 @@
 </thead>
 <tbody>
 <tr>
-<td>Number of nodes at level <span class="arithmatex">\(i\)</span></td>
+<td>Nodes at level <span class="arithmatex">\(i\)</span></td>
 <td><span class="arithmatex">\(2^{i-1}\)</span></td>
 <td><span class="arithmatex">\(1\)</span></td>
 </tr>
 <tr>
-<td>Number of leaf nodes in a tree with height <span class="arithmatex">\(h\)</span></td>
+<td>Leaves in a tree of height <span class="arithmatex">\(h\)</span></td>
 <td><span class="arithmatex">\(2^h\)</span></td>
 <td><span class="arithmatex">\(1\)</span></td>
 </tr>
 <tr>
-<td>Total number of nodes in a tree with height <span class="arithmatex">\(h\)</span></td>
+<td>Total nodes in a tree of height <span class="arithmatex">\(h\)</span></td>
 <td><span class="arithmatex">\(2^{h+1} - 1\)</span></td>
 <td><span class="arithmatex">\(h + 1\)</span></td>
 </tr>
 <tr>
-<td>Height of a tree with <span class="arithmatex">\(n\)</span> total nodes</td>
+<td>Height of a tree with <span class="arithmatex">\(n\)</span> nodes</td>
 <td><span class="arithmatex">\(\log_2 (n+1) - 1\)</span></td>
 <td><span class="arithmatex">\(n - 1\)</span></td>
 </tr>
@@ -5499,7 +5575,7 @@ aria-label="Footer"
   <a
     href="../"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: Chapter 7. &amp;nbsp; Tree"
+    aria-label="Previous: Chapter 7.   Trees"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -5511,7 +5587,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        Chapter 7. &nbsp; Tree
+        Chapter 7. &nbsp; Trees
       </div>
     </div>
   </a>
@@ -5523,7 +5599,7 @@ aria-label="Footer"
   <a
     href="../binary_tree_traversal/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 7.2 Binary Tree Traversal"
+    aria-label="Next: 7.2   Traversing a binary tree"
     rel="next"
   >
     <div class="md-footer__title">
@@ -5531,7 +5607,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        7.2 Binary Tree Traversal
+        7.2 &nbsp; Traversing a binary tree
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -5548,7 +5624,7 @@ aria-label="Footer"
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-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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diff --git a/en/chapter_tree/binary_tree_traversal/index.html b/en/chapter_tree/binary_tree_traversal/index.html
index 4620f29fa..97f7eed21 100644
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>7.2 Binary Tree Traversal - Hello Algo</title>
+        <title>7.2 Traversing a binary tree - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
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+        <a href="#72-traversing-a-binary-tree" class="md-skip">
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+    <span>Help improve the English edition. See our <a href="https://github.com/krahets/hello-algo/blob/main/en/CONTRIBUTING.md">contribution guidelines</a>.</span>
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   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
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-  0.3 Summary
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   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
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-  1.2 What Is an Algorithm
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-  1.3 Summary
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-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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-  3.2 Basic Data Types
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-  3.4 Character Encoding *
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-  3.6 Exercises
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-  Chapter 4. Arrays and Linked Lists
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-  4.2 Linked List
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-  4.4 Random-Access Memory and Cache *
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-  4.5 Summary
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-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
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-  Chapter 5. Stacks and Queues
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
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   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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@@ -1678,7 +1678,7 @@
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
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   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
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-  7.1 Binary Tree
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.2 Binary Tree Traversal
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+  <a href="#1-implementation" class="md-nav__link">
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-    <nav class="md-nav" aria-label="7.2.2 Preorder, Inorder, and Postorder Traversal">
+    <nav class="md-nav" aria-label="7.2.2 Preorder, inorder, and postorder traversal">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
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+  <a href="#1-implementation_1" class="md-nav__link">
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
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   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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+  Chapter 9. &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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@@ -3005,7 +3005,7 @@
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
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@@ -3047,7 +3047,7 @@
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-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
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-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
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-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
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-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
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-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
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-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
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-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
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-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
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-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
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-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
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-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
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-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
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-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
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-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
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-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
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-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
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-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
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-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
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-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
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-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
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-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
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-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
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-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
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-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
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-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
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-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
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-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
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-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
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-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
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-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
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-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
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-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
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-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
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-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
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-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
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-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
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-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
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-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
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-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
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-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
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-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
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-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
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-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
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   <a href="#721-level-order-traversal" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.2.1 &nbsp; Level-Order Traversal
+        7.2.1 &nbsp; Level-order traversal
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.2.1 Level-Order Traversal">
+    <nav class="md-nav" aria-label="7.2.1 Level-order traversal">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
-  <a href="#1-code-implementation" class="md-nav__link">
+  <a href="#1-implementation" class="md-nav__link">
     <span class="md-ellipsis">
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-        1. &nbsp; Code Implementation
+        1. &nbsp; Implementation
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     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Complexity Analysis
+        2. &nbsp; Complexity analysis
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-        7.2.2 &nbsp; Preorder, Inorder, and Postorder Traversal
+        7.2.2 &nbsp; Preorder, inorder, and postorder traversal
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.2.2 Preorder, Inorder, and Postorder Traversal">
+    <nav class="md-nav" aria-label="7.2.2 Preorder, inorder, and postorder traversal">
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           <li class="md-nav__item">
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+  <a href="#1-implementation_1" class="md-nav__link">
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-        1. &nbsp; Code Implementation
+        1. &nbsp; Implementation
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     <span class="md-ellipsis">
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-        2. &nbsp; Complexity Analysis
+        2. &nbsp; Complexity analysis
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 <!-- Page content -->
-<h1 id="72-binary-tree-traversal">7.2 &nbsp; Binary Tree Traversal<a class="headerlink" href="#72-binary-tree-traversal" title="Permanent link">&para;</a></h1>
-<p>From a physical structure perspective, a tree is a data structure based on linked lists. Hence, its traversal method involves accessing nodes one by one through pointers. However, a tree is a non-linear data structure, which makes traversing a tree more complex than traversing a linked list, requiring the assistance of search algorithms.</p>
-<p>The common traversal methods for binary trees include level-order traversal, pre-order traversal, in-order traversal, and post-order traversal.</p>
-<h2 id="721-level-order-traversal">7.2.1 &nbsp; Level-Order Traversal<a class="headerlink" href="#721-level-order-traversal" title="Permanent link">&para;</a></h2>
-<p>As shown in Figure 7-9, <u>level-order traversal</u> traverses the binary tree from top to bottom, layer by layer. Within each level, it visits nodes from left to right.</p>
-<p>Level-order traversal is essentially <u>breadth-first traversal</u>, also known as <u>breadth-first search (BFS)</u>, which proceeds outward level by level.</p>
+<h1 id="72-traversing-a-binary-tree">7.2 &nbsp; Traversing a binary tree<a class="headerlink" href="#72-traversing-a-binary-tree" title="Permanent link">&para;</a></h1>
+<p>In its physical representation, a tree is a linked data structure, so traversal follows pointers from node to node. Unlike a linked list, however, a tree is nonlinear. Traversing it is more complex and requires a search algorithm.</p>
+<p>Common binary-tree traversals include level-order, preorder, inorder, and postorder traversal.</p>
+<h2 id="721-level-order-traversal">7.2.1 &nbsp; Level-order traversal<a class="headerlink" href="#721-level-order-traversal" title="Permanent link">&para;</a></h2>
+<p>As shown below, <u>level-order traversal</u> visits the tree level by level from top to bottom, moving from left to right within each level.</p>
+<p>Level-order traversal is a form of <u>breadth-first traversal</u>, also called <u>breadth-first search (BFS)</u>. It explores outward one layer at a time.</p>
 <p><img alt="Level-order traversal of a binary tree" class="animation-figure" src="../binary_tree_traversal.assets/binary_tree_bfs.png" /></p>
 <p align="center"> Figure 7-9 &nbsp; Level-order traversal of a binary tree </p>
 
-<h3 id="1-code-implementation">1. &nbsp; Code Implementation<a class="headerlink" href="#1-code-implementation" title="Permanent link">&para;</a></h3>
-<p>Breadth-first traversal is typically implemented with the help of a "queue". The queue follows the "first in, first out" rule, while breadth-first traversal follows the "layer-by-layer progression" rule; the underlying ideas of the two are consistent. The implementation code is as follows:</p>
+<h3 id="1-implementation">1. &nbsp; Implementation<a class="headerlink" href="#1-implementation" title="Permanent link">&para;</a></h3>
+<p>Breadth-first traversal usually uses a queue. A queue's first-in, first-out rule matches the traversal's level-by-level progression. The implementation follows:</p>
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 <div class="highlight"><span class="filename">binary_tree_bfs.py</span><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="k">def</span><span class="w"> </span><span class="nf">level_order</span><span class="p">(</span><span class="n">root</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">|</span> <span class="kc">None</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">list</span><span class="p">[</span><span class="nb">int</span><span class="p">]:</span>
 <a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="w">    </span><span class="sd">"""Level-order traversal"""</span>
-<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Initialize queue, add root node</span>
+<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a>    <span class="c1"># Initialize a queue and enqueue the root</span>
 <a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>    <span class="n">queue</span><span class="p">:</span> <span class="n">deque</span><span class="p">[</span><span class="n">TreeNode</span><span class="p">]</span> <span class="o">=</span> <span class="n">deque</span><span class="p">()</span>
 <a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>    <span class="n">queue</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
-<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Initialize a list to save the traversal sequence</span>
+<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a>    <span class="c1"># Initialize a list to store the traversal sequence</span>
 <a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>    <span class="n">res</span> <span class="o">=</span> <span class="p">[]</span>
 <a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a>    <span class="k">while</span> <span class="n">queue</span><span class="p">:</span>
-<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="n">node</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">=</span> <span class="n">queue</span><span class="o">.</span><span class="n">popleft</span><span class="p">()</span>  <span class="c1"># Dequeue</span>
-<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">val</span><span class="p">)</span>  <span class="c1"># Save node value</span>
+<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>        <span class="n">node</span><span class="p">:</span> <span class="n">TreeNode</span> <span class="o">=</span> <span class="n">queue</span><span class="o">.</span><span class="n">popleft</span><span class="p">()</span>  <span class="c1"># Dequeue from the queue</span>
+<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a>        <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">val</span><span class="p">)</span>  <span class="c1"># Store the node value</span>
 <a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a>        <span class="k">if</span> <span class="n">node</span><span class="o">.</span><span class="n">left</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>            <span class="n">queue</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>  <span class="c1"># Left child node enqueue</span>
+<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a>            <span class="n">queue</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>  <span class="c1"># Enqueue the left child</span>
 <a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a>        <span class="k">if</span> <span class="n">node</span><span class="o">.</span><span class="n">right</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
-<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>            <span class="n">queue</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>  <span class="c1"># Right child node enqueue</span>
+<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a>            <span class="n">queue</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>  <span class="c1"># Enqueue the right child</span>
 <a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a>    <span class="k">return</span> <span class="n">res</span>
 </code></pre></div>
 </div>
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_tree_bfs.cpp</span><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">levelOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Initialize queue, add root node</span>
+<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="w">    </span><span class="c1">// Initialize a queue and enqueue the root</span>
 <a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="w">    </span><span class="n">queue</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*&gt;</span><span class="w"> </span><span class="n">queue</span><span class="p">;</span>
 <a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="w">    </span><span class="n">queue</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
-<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Initialize a list to save the traversal sequence</span>
+<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="w">    </span><span class="c1">// Initialize a list to store the traversal sequence</span>
 <a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="w">    </span><span class="n">vector</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">vec</span><span class="p">;</span>
 <a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">queue</span><span class="p">.</span><span class="n">empty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="w">        </span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">front</span><span class="p">();</span>
-<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="n">queue</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span><span class="w">              </span><span class="c1">// Dequeue</span>
-<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="n">vec</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">);</span><span class="w"> </span><span class="c1">// Save node value</span>
+<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="w">        </span><span class="n">queue</span><span class="p">.</span><span class="n">pop</span><span class="p">();</span><span class="w">              </span><span class="c1">// Dequeue from the queue</span>
+<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="w">        </span><span class="n">vec</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">);</span><span class="w"> </span><span class="c1">// Store the node value</span>
 <a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span>
-<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Left child node enqueue</span>
+<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Enqueue the left child</span>
 <a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span>
-<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Right child node enqueue</span>
+<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Enqueue the right child</span>
 <a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">vec</span><span class="p">;</span>
 <a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="p">}</span>
@@ -4824,18 +4896,18 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_tree_bfs.java</span><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="nf">levelOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Initialize queue, add root node</span>
+<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="w">    </span><span class="c1">// Initialize a queue and enqueue the root</span>
 <a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="w">    </span><span class="n">Queue</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">LinkedList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="w">    </span><span class="n">queue</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
-<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// Initialize a list to save the traversal sequence</span>
+<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="w">    </span><span class="c1">// Initialize a list to store the traversal sequence</span>
 <a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="n">Integer</span><span class="o">&gt;</span><span class="w"> </span><span class="n">list</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="n">ArrayList</span><span class="o">&lt;&gt;</span><span class="p">();</span>
 <a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="o">!</span><span class="n">queue</span><span class="p">.</span><span class="na">isEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">        </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="na">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// Dequeue</span>
-<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="n">list</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="p">);</span><span class="w">           </span><span class="c1">// Save node value</span>
+<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="w">        </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="na">poll</span><span class="p">();</span><span class="w"> </span><span class="c1">// Dequeue from the queue</span>
+<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="w">        </span><span class="n">list</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">val</span><span class="p">);</span><span class="w">           </span><span class="c1">// Store the node value</span>
 <a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
-<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">);</span><span class="w">   </span><span class="c1">// Left child node enqueue</span>
+<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">);</span><span class="w">   </span><span class="c1">// Enqueue the left child</span>
 <a id="__codelineno-2-13" name="__codelineno-2-13" href="#__codelineno-2-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
-<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">);</span><span class="w">  </span><span class="c1">// Right child node enqueue</span>
+<a id="__codelineno-2-14" name="__codelineno-2-14" href="#__codelineno-2-14"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">);</span><span class="w">  </span><span class="c1">// Enqueue the right child</span>
 <a id="__codelineno-2-15" name="__codelineno-2-15" href="#__codelineno-2-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-2-16" name="__codelineno-2-16" href="#__codelineno-2-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">list</span><span class="p">;</span>
 <a id="__codelineno-2-17" name="__codelineno-2-17" href="#__codelineno-2-17"></a><span class="p">}</span>
@@ -4844,18 +4916,18 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_tree_bfs.cs</span><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">LevelOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Initialize queue, add root node</span>
+<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="w">    </span><span class="c1">// Initialize a queue and enqueue the root</span>
 <a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a><span class="w">    </span><span class="n">Queue</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;</span><span class="w"> </span><span class="n">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="p">();</span>
 <a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="w">    </span><span class="n">queue</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
-<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Initialize a list to save the traversal sequence</span>
+<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a><span class="w">    </span><span class="c1">// Initialize a list to store the traversal sequence</span>
 <a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="w">    </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">list</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">queue</span><span class="p">.</span><span class="n">Count</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">        </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">Dequeue</span><span class="p">();</span><span class="w"> </span><span class="c1">// Dequeue</span>
-<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="n">list</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="o">!</span><span class="p">.</span><span class="n">Value</span><span class="p">);</span><span class="w">       </span><span class="c1">// Save node value</span>
+<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a><span class="w">        </span><span class="n">TreeNode</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">Dequeue</span><span class="p">();</span><span class="w"> </span><span class="c1">// Dequeue from the queue</span>
+<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a><span class="w">        </span><span class="n">list</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="o">!</span><span class="p">.</span><span class="n">Value</span><span class="p">);</span><span class="w">       </span><span class="c1">// Store the node value</span>
 <a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">null</span><span class="p">)</span>
-<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">);</span><span class="w">    </span><span class="c1">// Left child node enqueue</span>
+<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">);</span><span class="w">    </span><span class="c1">// Enqueue the left child</span>
 <a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="k">null</span><span class="p">)</span>
-<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">);</span><span class="w">   </span><span class="c1">// Right child node enqueue</span>
+<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="n">Enqueue</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">);</span><span class="w">   </span><span class="c1">// Enqueue the right child</span>
 <a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">list</span><span class="p">;</span>
 <a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a><span class="p">}</span>
@@ -4864,22 +4936,22 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_tree_bfs.go</span><pre><span></span><code><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-4-2" name="__codelineno-4-2" href="#__codelineno-4-2"></a><span class="kd">func</span><span class="w"> </span><span class="nx">levelOrder</span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="p">[]</span><span class="kt">any</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Initialize queue, add root node</span>
+<a id="__codelineno-4-3" name="__codelineno-4-3" href="#__codelineno-4-3"></a><span class="w">    </span><span class="c1">// Initialize a queue and enqueue the root</span>
 <a id="__codelineno-4-4" name="__codelineno-4-4" href="#__codelineno-4-4"></a><span class="w">    </span><span class="nx">queue</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">list</span><span class="p">.</span><span class="nx">New</span><span class="p">()</span>
 <a id="__codelineno-4-5" name="__codelineno-4-5" href="#__codelineno-4-5"></a><span class="w">    </span><span class="nx">queue</span><span class="p">.</span><span class="nx">PushBack</span><span class="p">(</span><span class="nx">root</span><span class="p">)</span>
-<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="c1">// Initialize a slice to save traversal sequence</span>
+<a id="__codelineno-4-6" name="__codelineno-4-6" href="#__codelineno-4-6"></a><span class="w">    </span><span class="c1">// Initialize a slice to store the traversal sequence</span>
 <a id="__codelineno-4-7" name="__codelineno-4-7" href="#__codelineno-4-7"></a><span class="w">    </span><span class="nx">nums</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nb">make</span><span class="p">([]</span><span class="kt">any</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span>
 <a id="__codelineno-4-8" name="__codelineno-4-8" href="#__codelineno-4-8"></a><span class="w">    </span><span class="k">for</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">Len</span><span class="p">()</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">        </span><span class="c1">// Dequeue</span>
+<a id="__codelineno-4-9" name="__codelineno-4-9" href="#__codelineno-4-9"></a><span class="w">        </span><span class="c1">// Dequeue from the queue</span>
 <a id="__codelineno-4-10" name="__codelineno-4-10" href="#__codelineno-4-10"></a><span class="w">        </span><span class="nx">node</span><span class="w"> </span><span class="o">:=</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">Remove</span><span class="p">(</span><span class="nx">queue</span><span class="p">.</span><span class="nx">Front</span><span class="p">()).(</span><span class="o">*</span><span class="nx">TreeNode</span><span class="p">)</span>
-<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="c1">// Save node value</span>
+<a id="__codelineno-4-11" name="__codelineno-4-11" href="#__codelineno-4-11"></a><span class="w">        </span><span class="c1">// Store the node value</span>
 <a id="__codelineno-4-12" name="__codelineno-4-12" href="#__codelineno-4-12"></a><span class="w">        </span><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Val</span><span class="p">)</span>
 <a id="__codelineno-4-13" name="__codelineno-4-13" href="#__codelineno-4-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">            </span><span class="c1">// Left child node enqueue</span>
+<a id="__codelineno-4-14" name="__codelineno-4-14" href="#__codelineno-4-14"></a><span class="w">            </span><span class="c1">// Enqueue the left child</span>
 <a id="__codelineno-4-15" name="__codelineno-4-15" href="#__codelineno-4-15"></a><span class="w">            </span><span class="nx">queue</span><span class="p">.</span><span class="nx">PushBack</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="p">)</span>
 <a id="__codelineno-4-16" name="__codelineno-4-16" href="#__codelineno-4-16"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-4-17" name="__codelineno-4-17" href="#__codelineno-4-17"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">            </span><span class="c1">// Right child node enqueue</span>
+<a id="__codelineno-4-18" name="__codelineno-4-18" href="#__codelineno-4-18"></a><span class="w">            </span><span class="c1">// Enqueue the right child</span>
 <a id="__codelineno-4-19" name="__codelineno-4-19" href="#__codelineno-4-19"></a><span class="w">            </span><span class="nx">queue</span><span class="p">.</span><span class="nx">PushBack</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span><span class="p">)</span>
 <a id="__codelineno-4-20" name="__codelineno-4-20" href="#__codelineno-4-20"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-4-21" name="__codelineno-4-21" href="#__codelineno-4-21"></a><span class="w">    </span><span class="p">}</span>
@@ -4890,18 +4962,18 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_tree_bfs.swift</span><pre><span></span><code><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-5-2" name="__codelineno-5-2" href="#__codelineno-5-2"></a><span class="kd">func</span><span class="w"> </span><span class="nf">levelOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Initialize queue, add root node</span>
+<a id="__codelineno-5-3" name="__codelineno-5-3" href="#__codelineno-5-3"></a><span class="w">    </span><span class="c1">// Initialize a queue and enqueue the root</span>
 <a id="__codelineno-5-4" name="__codelineno-5-4" href="#__codelineno-5-4"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">queue</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="n">TreeNode</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[</span><span class="n">root</span><span class="p">]</span>
-<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="c1">// Initialize a list to save the traversal sequence</span>
+<a id="__codelineno-5-5" name="__codelineno-5-5" href="#__codelineno-5-5"></a><span class="w">    </span><span class="c1">// Initialize a list to store the traversal sequence</span>
 <a id="__codelineno-5-6" name="__codelineno-5-6" href="#__codelineno-5-6"></a><span class="w">    </span><span class="kd">var</span><span class="w"> </span><span class="nv">list</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="nb">Int</span><span class="p">]</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">[]</span>
 <a id="__codelineno-5-7" name="__codelineno-5-7" href="#__codelineno-5-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="o">!</span><span class="n">queue</span><span class="p">.</span><span class="bp">isEmpty</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">removeFirst</span><span class="p">()</span><span class="w"> </span><span class="c1">// Dequeue</span>
-<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="n">list</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="c1">// Save node value</span>
+<a id="__codelineno-5-8" name="__codelineno-5-8" href="#__codelineno-5-8"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">removeFirst</span><span class="p">()</span><span class="w"> </span><span class="c1">// Dequeue from the queue</span>
+<a id="__codelineno-5-9" name="__codelineno-5-9" href="#__codelineno-5-9"></a><span class="w">        </span><span class="n">list</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">)</span><span class="w"> </span><span class="c1">// Store the node value</span>
 <a id="__codelineno-5-10" name="__codelineno-5-10" href="#__codelineno-5-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nv">left</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="kr">left</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="kr">left</span><span class="p">)</span><span class="w"> </span><span class="c1">// Left child node enqueue</span>
+<a id="__codelineno-5-11" name="__codelineno-5-11" href="#__codelineno-5-11"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="kr">left</span><span class="p">)</span><span class="w"> </span><span class="c1">// Enqueue the left child</span>
 <a id="__codelineno-5-12" name="__codelineno-5-12" href="#__codelineno-5-12"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-13" name="__codelineno-5-13" href="#__codelineno-5-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nv">right</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="kr">right</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="kr">right</span><span class="p">)</span><span class="w"> </span><span class="c1">// Right child node enqueue</span>
+<a id="__codelineno-5-14" name="__codelineno-5-14" href="#__codelineno-5-14"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="kr">right</span><span class="p">)</span><span class="w"> </span><span class="c1">// Enqueue the right child</span>
 <a id="__codelineno-5-15" name="__codelineno-5-15" href="#__codelineno-5-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-5-16" name="__codelineno-5-16" href="#__codelineno-5-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-5-17" name="__codelineno-5-17" href="#__codelineno-5-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">list</span>
@@ -4911,15 +4983,15 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_tree_bfs.js</span><pre><span></span><code><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-6-2" name="__codelineno-6-2" href="#__codelineno-6-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">levelOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Initialize queue, add root node</span>
+<a id="__codelineno-6-3" name="__codelineno-6-3" href="#__codelineno-6-3"></a><span class="w">    </span><span class="c1">// Initialize a queue and enqueue the root</span>
 <a id="__codelineno-6-4" name="__codelineno-6-4" href="#__codelineno-6-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="nx">root</span><span class="p">];</span>
-<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="c1">// Initialize a list to save the traversal sequence</span>
+<a id="__codelineno-6-5" name="__codelineno-6-5" href="#__codelineno-6-5"></a><span class="w">    </span><span class="c1">// Initialize a list to store the traversal sequence</span>
 <a id="__codelineno-6-6" name="__codelineno-6-6" href="#__codelineno-6-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">list</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-6-7" name="__codelineno-6-7" href="#__codelineno-6-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">queue</span><span class="p">.</span><span class="nx">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">shift</span><span class="p">();</span><span class="w"> </span><span class="c1">// Dequeue</span>
-<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">        </span><span class="nx">list</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span><span class="w"> </span><span class="c1">// Save node value</span>
-<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">)</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Left child node enqueue</span>
-<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Right child node enqueue</span>
+<a id="__codelineno-6-8" name="__codelineno-6-8" href="#__codelineno-6-8"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">shift</span><span class="p">();</span><span class="w"> </span><span class="c1">// Dequeue from the queue</span>
+<a id="__codelineno-6-9" name="__codelineno-6-9" href="#__codelineno-6-9"></a><span class="w">        </span><span class="nx">list</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span><span class="w"> </span><span class="c1">// Store the node value</span>
+<a id="__codelineno-6-10" name="__codelineno-6-10" href="#__codelineno-6-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">)</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Enqueue the left child</span>
+<a id="__codelineno-6-11" name="__codelineno-6-11" href="#__codelineno-6-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Enqueue the right child</span>
 <a id="__codelineno-6-12" name="__codelineno-6-12" href="#__codelineno-6-12"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-6-13" name="__codelineno-6-13" href="#__codelineno-6-13"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">list</span><span class="p">;</span>
 <a id="__codelineno-6-14" name="__codelineno-6-14" href="#__codelineno-6-14"></a><span class="p">}</span>
@@ -4928,18 +5000,18 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_tree_bfs.ts</span><pre><span></span><code><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-7-2" name="__codelineno-7-2" href="#__codelineno-7-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">levelOrder</span><span class="p">(</span><span class="nx">root</span><span class="o">:</span><span class="w"> </span><span class="kt">TreeNode</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Initialize queue, add root node</span>
+<a id="__codelineno-7-3" name="__codelineno-7-3" href="#__codelineno-7-3"></a><span class="w">    </span><span class="c1">// Initialize a queue and enqueue the root</span>
 <a id="__codelineno-7-4" name="__codelineno-7-4" href="#__codelineno-7-4"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[</span><span class="nx">root</span><span class="p">];</span>
-<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="c1">// Initialize a list to save the traversal sequence</span>
+<a id="__codelineno-7-5" name="__codelineno-7-5" href="#__codelineno-7-5"></a><span class="w">    </span><span class="c1">// Initialize a list to store the traversal sequence</span>
 <a id="__codelineno-7-6" name="__codelineno-7-6" href="#__codelineno-7-6"></a><span class="w">    </span><span class="kd">const</span><span class="w"> </span><span class="nx">list</span><span class="o">:</span><span class="w"> </span><span class="kt">number</span><span class="p">[]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-7-7" name="__codelineno-7-7" href="#__codelineno-7-7"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="nx">queue</span><span class="p">.</span><span class="nx">length</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">shift</span><span class="p">()</span><span class="w"> </span><span class="kr">as</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">;</span><span class="w"> </span><span class="c1">// Dequeue</span>
-<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">        </span><span class="nx">list</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span><span class="w"> </span><span class="c1">// Save node value</span>
+<a id="__codelineno-7-8" name="__codelineno-7-8" href="#__codelineno-7-8"></a><span class="w">        </span><span class="kd">let</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nx">queue</span><span class="p">.</span><span class="nx">shift</span><span class="p">()</span><span class="w"> </span><span class="kr">as</span><span class="w"> </span><span class="nx">TreeNode</span><span class="p">;</span><span class="w"> </span><span class="c1">// Dequeue from the queue</span>
+<a id="__codelineno-7-9" name="__codelineno-7-9" href="#__codelineno-7-9"></a><span class="w">        </span><span class="nx">list</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span><span class="w"> </span><span class="c1">// Store the node value</span>
 <a id="__codelineno-7-10" name="__codelineno-7-10" href="#__codelineno-7-10"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">            </span><span class="nx">queue</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Left child node enqueue</span>
+<a id="__codelineno-7-11" name="__codelineno-7-11" href="#__codelineno-7-11"></a><span class="w">            </span><span class="nx">queue</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Enqueue the left child</span>
 <a id="__codelineno-7-12" name="__codelineno-7-12" href="#__codelineno-7-12"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-7-13" name="__codelineno-7-13" href="#__codelineno-7-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">            </span><span class="nx">queue</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Right child node enqueue</span>
+<a id="__codelineno-7-14" name="__codelineno-7-14" href="#__codelineno-7-14"></a><span class="w">            </span><span class="nx">queue</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Enqueue the right child</span>
 <a id="__codelineno-7-15" name="__codelineno-7-15" href="#__codelineno-7-15"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-7-16" name="__codelineno-7-16" href="#__codelineno-7-16"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-7-17" name="__codelineno-7-17" href="#__codelineno-7-17"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="nx">list</span><span class="p">;</span>
@@ -4949,16 +5021,16 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_tree_bfs.dart</span><pre><span></span><code><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-8-2" name="__codelineno-8-2" href="#__codelineno-8-2"></a><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">levelOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Initialize queue, add root node</span>
+<a id="__codelineno-8-3" name="__codelineno-8-3" href="#__codelineno-8-3"></a><span class="w">  </span><span class="c1">// Initialize a queue and enqueue the root</span>
 <a id="__codelineno-8-4" name="__codelineno-8-4" href="#__codelineno-8-4"></a><span class="w">  </span><span class="n">Queue</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">?&gt;</span><span class="w"> </span><span class="n">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Queue</span><span class="p">();</span>
 <a id="__codelineno-8-5" name="__codelineno-8-5" href="#__codelineno-8-5"></a><span class="w">  </span><span class="n">queue</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">root</span><span class="p">);</span>
-<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="c1">// Initialize a list to save the traversal sequence</span>
+<a id="__codelineno-8-6" name="__codelineno-8-6" href="#__codelineno-8-6"></a><span class="w">  </span><span class="c1">// Initialize a list to store the traversal sequence</span>
 <a id="__codelineno-8-7" name="__codelineno-8-7" href="#__codelineno-8-7"></a><span class="w">  </span><span class="n">List</span><span class="o">&lt;</span><span class="kt">int</span><span class="o">&gt;</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">[];</span>
 <a id="__codelineno-8-8" name="__codelineno-8-8" href="#__codelineno-8-8"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">queue</span><span class="p">.</span><span class="n">isNotEmpty</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">removeFirst</span><span class="p">();</span><span class="w"> </span><span class="c1">// Dequeue</span>
-<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">node</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">);</span><span class="w"> </span><span class="c1">// Save node value</span>
-<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Left child node enqueue</span>
-<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Right child node enqueue</span>
+<a id="__codelineno-8-9" name="__codelineno-8-9" href="#__codelineno-8-9"></a><span class="w">    </span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">removeFirst</span><span class="p">();</span><span class="w"> </span><span class="c1">// Dequeue from the queue</span>
+<a id="__codelineno-8-10" name="__codelineno-8-10" href="#__codelineno-8-10"></a><span class="w">    </span><span class="n">res</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">node</span><span class="o">!</span><span class="p">.</span><span class="n">val</span><span class="p">);</span><span class="w"> </span><span class="c1">// Store the node value</span>
+<a id="__codelineno-8-11" name="__codelineno-8-11" href="#__codelineno-8-11"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">);</span><span class="w"> </span><span class="c1">// Enqueue the left child</span>
+<a id="__codelineno-8-12" name="__codelineno-8-12" href="#__codelineno-8-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">);</span><span class="w"> </span><span class="c1">// Enqueue the right child</span>
 <a id="__codelineno-8-13" name="__codelineno-8-13" href="#__codelineno-8-13"></a><span class="w">  </span><span class="p">}</span>
 <a id="__codelineno-8-14" name="__codelineno-8-14" href="#__codelineno-8-14"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="n">res</span><span class="p">;</span>
 <a id="__codelineno-8-15" name="__codelineno-8-15" href="#__codelineno-8-15"></a><span class="p">}</span>
@@ -4967,20 +5039,20 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_tree_bfs.rs</span><pre><span></span><code><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-9-2" name="__codelineno-9-2" href="#__codelineno-9-2"></a><span class="k">fn</span><span class="w"> </span><span class="nf">level_order</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Initialize queue, add root node</span>
+<a id="__codelineno-9-3" name="__codelineno-9-3" href="#__codelineno-9-3"></a><span class="w">    </span><span class="c1">// Initialize a queue and enqueue the root</span>
 <a id="__codelineno-9-4" name="__codelineno-9-4" href="#__codelineno-9-4"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">que</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">VecDeque</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-9-5" name="__codelineno-9-5" href="#__codelineno-9-5"></a><span class="w">    </span><span class="n">que</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span>
-<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// Initialize a list to save the traversal sequence</span>
+<a id="__codelineno-9-6" name="__codelineno-9-6" href="#__codelineno-9-6"></a><span class="w">    </span><span class="c1">// Initialize a list to store the traversal sequence</span>
 <a id="__codelineno-9-7" name="__codelineno-9-7" href="#__codelineno-9-7"></a><span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="k">mut</span><span class="w"> </span><span class="n">vec</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">Vec</span><span class="p">::</span><span class="n">new</span><span class="p">();</span>
 <a id="__codelineno-9-8" name="__codelineno-9-8" href="#__codelineno-9-8"></a>
 <a id="__codelineno-9-9" name="__codelineno-9-9" href="#__codelineno-9-9"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">que</span><span class="p">.</span><span class="n">pop_front</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="c1">// Dequeue</span>
-<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="n">vec</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="p">);</span><span class="w"> </span><span class="c1">// Save node value</span>
+<a id="__codelineno-9-10" name="__codelineno-9-10" href="#__codelineno-9-10"></a><span class="w">        </span><span class="c1">// Dequeue from the queue</span>
+<a id="__codelineno-9-11" name="__codelineno-9-11" href="#__codelineno-9-11"></a><span class="w">        </span><span class="n">vec</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">val</span><span class="p">);</span><span class="w"> </span><span class="c1">// Store the node value</span>
 <a id="__codelineno-9-12" name="__codelineno-9-12" href="#__codelineno-9-12"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">left</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">left</span><span class="p">.</span><span class="n">as_ref</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">            </span><span class="n">que</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span><span class="w"> </span><span class="c1">// Left child node enqueue</span>
+<a id="__codelineno-9-13" name="__codelineno-9-13" href="#__codelineno-9-13"></a><span class="w">            </span><span class="n">que</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">left</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span><span class="w"> </span><span class="c1">// Enqueue the left child</span>
 <a id="__codelineno-9-14" name="__codelineno-9-14" href="#__codelineno-9-14"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-9-15" name="__codelineno-9-15" href="#__codelineno-9-15"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">right</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">().</span><span class="n">right</span><span class="p">.</span><span class="n">as_ref</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">            </span><span class="n">que</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span><span class="w"> </span><span class="c1">// Right child node enqueue</span>
+<a id="__codelineno-9-16" name="__codelineno-9-16" href="#__codelineno-9-16"></a><span class="w">            </span><span class="n">que</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">right</span><span class="p">.</span><span class="n">clone</span><span class="p">());</span><span class="w"> </span><span class="c1">// Enqueue the right child</span>
 <a id="__codelineno-9-17" name="__codelineno-9-17" href="#__codelineno-9-17"></a><span class="w">        </span><span class="p">};</span>
 <a id="__codelineno-9-18" name="__codelineno-9-18" href="#__codelineno-9-18"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-9-19" name="__codelineno-9-19" href="#__codelineno-9-19"></a><span class="w">    </span><span class="n">vec</span>
@@ -4998,34 +5070,34 @@
 <a id="__codelineno-10-8" name="__codelineno-10-8" href="#__codelineno-10-8"></a>
 <a id="__codelineno-10-9" name="__codelineno-10-9" href="#__codelineno-10-9"></a><span class="w">    </span><span class="cm">/* Auxiliary queue */</span>
 <a id="__codelineno-10-10" name="__codelineno-10-10" href="#__codelineno-10-10"></a><span class="w">    </span><span class="n">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">**</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="p">);</span>
-<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// Queue pointer</span>
+<a id="__codelineno-10-11" name="__codelineno-10-11" href="#__codelineno-10-11"></a><span class="w">    </span><span class="c1">// Queue indices</span>
 <a id="__codelineno-10-12" name="__codelineno-10-12" href="#__codelineno-10-12"></a><span class="w">    </span><span class="n">front</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">rear</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
-<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">    </span><span class="c1">// Add root node</span>
+<a id="__codelineno-10-13" name="__codelineno-10-13" href="#__codelineno-10-13"></a><span class="w">    </span><span class="c1">// Enqueue the root node</span>
 <a id="__codelineno-10-14" name="__codelineno-10-14" href="#__codelineno-10-14"></a><span class="w">    </span><span class="n">queue</span><span class="p">[</span><span class="n">rear</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="p">;</span>
-<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="c1">// Initialize a list to save the traversal sequence</span>
+<a id="__codelineno-10-15" name="__codelineno-10-15" href="#__codelineno-10-15"></a><span class="w">    </span><span class="c1">// Initialize a list to store the traversal sequence</span>
 <a id="__codelineno-10-16" name="__codelineno-10-16" href="#__codelineno-10-16"></a><span class="w">    </span><span class="cm">/* Auxiliary array */</span>
 <a id="__codelineno-10-17" name="__codelineno-10-17" href="#__codelineno-10-17"></a><span class="w">    </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="p">)</span><span class="n">malloc</span><span class="p">(</span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">MAX_SIZE</span><span class="p">);</span>
-<a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">    </span><span class="c1">// Array pointer</span>
+<a id="__codelineno-10-18" name="__codelineno-10-18" href="#__codelineno-10-18"></a><span class="w">    </span><span class="c1">// Array index</span>
 <a id="__codelineno-10-19" name="__codelineno-10-19" href="#__codelineno-10-19"></a><span class="w">    </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>
 <a id="__codelineno-10-20" name="__codelineno-10-20" href="#__codelineno-10-20"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">front</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">rear</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">        </span><span class="c1">// Dequeue</span>
+<a id="__codelineno-10-21" name="__codelineno-10-21" href="#__codelineno-10-21"></a><span class="w">        </span><span class="c1">// Dequeue from the queue</span>
 <a id="__codelineno-10-22" name="__codelineno-10-22" href="#__codelineno-10-22"></a><span class="w">        </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">[</span><span class="n">front</span><span class="o">++</span><span class="p">];</span>
-<a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">        </span><span class="c1">// Save node value</span>
+<a id="__codelineno-10-23" name="__codelineno-10-23" href="#__codelineno-10-23"></a><span class="w">        </span><span class="c1">// Store the node value</span>
 <a id="__codelineno-10-24" name="__codelineno-10-24" href="#__codelineno-10-24"></a><span class="w">        </span><span class="n">arr</span><span class="p">[</span><span class="n">index</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-10-25" name="__codelineno-10-25" href="#__codelineno-10-25"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="w">            </span><span class="c1">// Left child node enqueue</span>
+<a id="__codelineno-10-26" name="__codelineno-10-26" href="#__codelineno-10-26"></a><span class="w">            </span><span class="c1">// Enqueue the left child</span>
 <a id="__codelineno-10-27" name="__codelineno-10-27" href="#__codelineno-10-27"></a><span class="w">            </span><span class="n">queue</span><span class="p">[</span><span class="n">rear</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">;</span>
 <a id="__codelineno-10-28" name="__codelineno-10-28" href="#__codelineno-10-28"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-10-29" name="__codelineno-10-29" href="#__codelineno-10-29"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-10-30" name="__codelineno-10-30" href="#__codelineno-10-30"></a><span class="w">            </span><span class="c1">// Right child node enqueue</span>
+<a id="__codelineno-10-30" name="__codelineno-10-30" href="#__codelineno-10-30"></a><span class="w">            </span><span class="c1">// Enqueue the right child</span>
 <a id="__codelineno-10-31" name="__codelineno-10-31" href="#__codelineno-10-31"></a><span class="w">            </span><span class="n">queue</span><span class="p">[</span><span class="n">rear</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">;</span>
 <a id="__codelineno-10-32" name="__codelineno-10-32" href="#__codelineno-10-32"></a><span class="w">        </span><span class="p">}</span>
 <a id="__codelineno-10-33" name="__codelineno-10-33" href="#__codelineno-10-33"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-10-34" name="__codelineno-10-34" href="#__codelineno-10-34"></a><span class="w">    </span><span class="c1">// Update array length value</span>
+<a id="__codelineno-10-34" name="__codelineno-10-34" href="#__codelineno-10-34"></a><span class="w">    </span><span class="c1">// Update the array length</span>
 <a id="__codelineno-10-35" name="__codelineno-10-35" href="#__codelineno-10-35"></a><span class="w">    </span><span class="o">*</span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span>
 <a id="__codelineno-10-36" name="__codelineno-10-36" href="#__codelineno-10-36"></a><span class="w">    </span><span class="n">arr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">realloc</span><span class="p">(</span><span class="n">arr</span><span class="p">,</span><span class="w"> </span><span class="k">sizeof</span><span class="p">(</span><span class="kt">int</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="o">*</span><span class="n">size</span><span class="p">));</span>
 <a id="__codelineno-10-37" name="__codelineno-10-37" href="#__codelineno-10-37"></a>
-<a id="__codelineno-10-38" name="__codelineno-10-38" href="#__codelineno-10-38"></a><span class="w">    </span><span class="c1">// Free auxiliary array space</span>
+<a id="__codelineno-10-38" name="__codelineno-10-38" href="#__codelineno-10-38"></a><span class="w">    </span><span class="c1">// Free the auxiliary queue array</span>
 <a id="__codelineno-10-39" name="__codelineno-10-39" href="#__codelineno-10-39"></a><span class="w">    </span><span class="n">free</span><span class="p">(</span><span class="n">queue</span><span class="p">);</span>
 <a id="__codelineno-10-40" name="__codelineno-10-40" href="#__codelineno-10-40"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">arr</span><span class="p">;</span>
 <a id="__codelineno-10-41" name="__codelineno-10-41" href="#__codelineno-10-41"></a><span class="p">}</span>
@@ -5034,18 +5106,18 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_tree_bfs.kt</span><pre><span></span><code><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a><span class="cm">/* Level-order traversal */</span>
 <a id="__codelineno-11-2" name="__codelineno-11-2" href="#__codelineno-11-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">levelOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">):</span><span class="w"> </span><span class="n">MutableList</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Initialize queue, add root node</span>
+<a id="__codelineno-11-3" name="__codelineno-11-3" href="#__codelineno-11-3"></a><span class="w">    </span><span class="c1">// Initialize a queue and enqueue the root</span>
 <a id="__codelineno-11-4" name="__codelineno-11-4" href="#__codelineno-11-4"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">LinkedList</span><span class="o">&lt;</span><span class="n">TreeNode?</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-11-5" name="__codelineno-11-5" href="#__codelineno-11-5"></a><span class="w">    </span><span class="n">queue</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
-<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// Initialize a list to save the traversal sequence</span>
+<a id="__codelineno-11-6" name="__codelineno-11-6" href="#__codelineno-11-6"></a><span class="w">    </span><span class="c1">// Initialize a list to store the traversal sequence</span>
 <a id="__codelineno-11-7" name="__codelineno-11-7" href="#__codelineno-11-7"></a><span class="w">    </span><span class="kd">val</span><span class="w"> </span><span class="nv">list</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">mutableListOf</span><span class="o">&lt;</span><span class="kt">Int</span><span class="o">&gt;</span><span class="p">()</span>
 <a id="__codelineno-11-8" name="__codelineno-11-8" href="#__codelineno-11-8"></a><span class="w">    </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="n">queue</span><span class="p">.</span><span class="na">isNotEmpty</span><span class="p">())</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="na">poll</span><span class="p">()</span><span class="w">      </span><span class="c1">// Dequeue</span>
-<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">        </span><span class="n">list</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">node</span><span class="o">?.</span><span class="na">_val</span><span class="o">!!</span><span class="p">)</span><span class="w">       </span><span class="c1">// Save node value</span>
+<a id="__codelineno-11-9" name="__codelineno-11-9" href="#__codelineno-11-9"></a><span class="w">        </span><span class="kd">val</span><span class="w"> </span><span class="nv">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="p">.</span><span class="na">poll</span><span class="p">()</span><span class="w">      </span><span class="c1">// Dequeue from the queue</span>
+<a id="__codelineno-11-10" name="__codelineno-11-10" href="#__codelineno-11-10"></a><span class="w">        </span><span class="n">list</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">node</span><span class="o">?.</span><span class="na">_val</span><span class="o">!!</span><span class="p">)</span><span class="w">       </span><span class="c1">// Store the node value</span>
 <a id="__codelineno-11-11" name="__codelineno-11-11" href="#__codelineno-11-11"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
-<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">)</span><span class="w">   </span><span class="c1">// Left child node enqueue</span>
+<a id="__codelineno-11-12" name="__codelineno-11-12" href="#__codelineno-11-12"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">left</span><span class="p">)</span><span class="w">   </span><span class="c1">// Enqueue the left child</span>
 <a id="__codelineno-11-13" name="__codelineno-11-13" href="#__codelineno-11-13"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="w"> </span><span class="o">!=</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
-<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">)</span><span class="w">  </span><span class="c1">// Right child node enqueue</span>
+<a id="__codelineno-11-14" name="__codelineno-11-14" href="#__codelineno-11-14"></a><span class="w">            </span><span class="n">queue</span><span class="p">.</span><span class="na">offer</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="na">right</span><span class="p">)</span><span class="w">  </span><span class="c1">// Enqueue the right child</span>
 <a id="__codelineno-11-15" name="__codelineno-11-15" href="#__codelineno-11-15"></a><span class="w">    </span><span class="p">}</span>
 <a id="__codelineno-11-16" name="__codelineno-11-16" href="#__codelineno-11-16"></a><span class="w">    </span><span class="k">return</span><span class="w"> </span><span class="n">list</span>
 <a id="__codelineno-11-17" name="__codelineno-11-17" href="#__codelineno-11-17"></a><span class="p">}</span>
@@ -5054,15 +5126,15 @@
 <div class="tabbed-block">
 <div class="highlight"><span class="filename">binary_tree_bfs.rb</span><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">### Level-order traversal ###</span>
 <a id="__codelineno-12-2" name="__codelineno-12-2" href="#__codelineno-12-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">level_order</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
-<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Initialize queue, add root node</span>
+<a id="__codelineno-12-3" name="__codelineno-12-3" href="#__codelineno-12-3"></a><span class="w">  </span><span class="c1"># Initialize a queue and enqueue the root</span>
 <a id="__codelineno-12-4" name="__codelineno-12-4" href="#__codelineno-12-4"></a><span class="w">  </span><span class="n">queue</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[</span><span class="n">root</span><span class="o">]</span>
-<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Initialize a list to save the traversal sequence</span>
+<a id="__codelineno-12-5" name="__codelineno-12-5" href="#__codelineno-12-5"></a><span class="w">  </span><span class="c1"># Initialize a list to store the traversal sequence</span>
 <a id="__codelineno-12-6" name="__codelineno-12-6" href="#__codelineno-12-6"></a><span class="w">  </span><span class="n">res</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">[]</span>
 <a id="__codelineno-12-7" name="__codelineno-12-7" href="#__codelineno-12-7"></a><span class="w">  </span><span class="k">while</span><span class="w"> </span><span class="o">!</span><span class="n">queue</span><span class="o">.</span><span class="n">empty?</span>
-<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="o">.</span><span class="n">shift</span><span class="w"> </span><span class="c1"># Dequeue</span>
-<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="c1"># Save node value</span>
-<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="n">queue</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="o">.</span><span class="n">nil?</span><span class="w"> </span><span class="c1"># Left child node enqueue</span>
-<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="n">queue</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="o">.</span><span class="n">nil?</span><span class="w"> </span><span class="c1"># Right child node enqueue</span>
+<a id="__codelineno-12-8" name="__codelineno-12-8" href="#__codelineno-12-8"></a><span class="w">    </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">queue</span><span class="o">.</span><span class="n">shift</span><span class="w"> </span><span class="c1"># Dequeue from the queue</span>
+<a id="__codelineno-12-9" name="__codelineno-12-9" href="#__codelineno-12-9"></a><span class="w">    </span><span class="n">res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">val</span><span class="w"> </span><span class="c1"># Store the node value</span>
+<a id="__codelineno-12-10" name="__codelineno-12-10" href="#__codelineno-12-10"></a><span class="w">    </span><span class="n">queue</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">left</span><span class="o">.</span><span class="n">nil?</span><span class="w"> </span><span class="c1"># Enqueue the left child</span>
+<a id="__codelineno-12-11" name="__codelineno-12-11" href="#__codelineno-12-11"></a><span class="w">    </span><span class="n">queue</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="w"> </span><span class="k">unless</span><span class="w"> </span><span class="n">node</span><span class="o">.</span><span class="n">right</span><span class="o">.</span><span class="n">nil?</span><span class="w"> </span><span class="c1"># Enqueue the right child</span>
 <a id="__codelineno-12-12" name="__codelineno-12-12" href="#__codelineno-12-12"></a><span class="w">  </span><span class="k">end</span>
 <a id="__codelineno-12-13" name="__codelineno-12-13" href="#__codelineno-12-13"></a><span class="w">  </span><span class="n">res</span>
 <a id="__codelineno-12-14" name="__codelineno-12-14" href="#__codelineno-12-14"></a><span class="k">end</span>
@@ -5070,19 +5142,24 @@
 </div>
 </div>
 </div>
-<h3 id="2-complexity-analysis">2. &nbsp; Complexity Analysis<a class="headerlink" href="#2-complexity-analysis" title="Permanent link">&para;</a></h3>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
+<p><div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" title="Visualize execution" src="https://pythontutor.com/iframe-embed.html#code=from%20collections%20import%20deque%0A%0Aclass%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20left%20child%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20Reference%20to%20the%20right%20child%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%3A%20recursion%22%22%22%0A%20%20%20%20%23%20Return%20None%20if%20the%20index%20is%20out%20of%20bounds%20or%20the%20corresponding%20element%20is%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20Create%20the%20current%20node%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20Recursively%20build%20the%20left%20and%20right%20subtrees%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20level_order%28root%3A%20TreeNode%20%7C%20None%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%22%22%22Level-order%20traversal%22%22%22%0A%20%20%20%20%23%20Initialize%20a%20queue%20and%20enqueue%20the%20root%0A%20%20%20%20queue%3A%20deque%5BTreeNode%5D%20%3D%20deque%28%29%0A%20%20%20%20queue.append%28root%29%0A%20%20%20%20%23%20Initialize%20a%20list%20to%20store%20the%20traversal%20sequence%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20while%20queue%3A%0A%20%20%20%20%20%20%20%20node%3A%20TreeNode%20%3D%20queue.popleft%28%29%20%20%23%20Dequeue%20from%20the%20queue%0A%20%20%20%20%20%20%20%20res.append%28node.val%29%20%20%23%20Store%20the%20node%20value%0A%20%20%20%20%20%20%20%20if%20node.left%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20queue.append%28node.left%29%20%20%23%20Enqueue%20the%20left%20child%0A%20%20%20%20%20%20%20%20if%20node.right%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20queue.append%28node.right%29%20%20%23%20Enqueue%20the%20right%20child%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20binary%20tree%0A%20%20%20%20%23%20Use%20a%20helper%20function%20that%20builds%20a%20binary%20tree%20directly%20from%20an%20array%0A%20%20%20%20root%20%3D%20list_to_tree%28arr%3D%5B1,%202,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20Level-order%20traversal%0A%20%20%20%20res%20%3D%20level_order%28root%29%0A%20%20%20%20print%28%22%5CnNode%20sequence%20from%20level-order%20traversal%20%3D%20%22,%20res%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=127&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=from%20collections%20import%20deque%0A%0Aclass%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20left%20child%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20Reference%20to%20the%20right%20child%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%3A%20recursion%22%22%22%0A%20%20%20%20%23%20Return%20None%20if%20the%20index%20is%20out%20of%20bounds%20or%20the%20corresponding%20element%20is%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20Create%20the%20current%20node%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20Recursively%20build%20the%20left%20and%20right%20subtrees%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20level_order%28root%3A%20TreeNode%20%7C%20None%29%20-%3E%20list%5Bint%5D%3A%0A%20%20%20%20%22%22%22Level-order%20traversal%22%22%22%0A%20%20%20%20%23%20Initialize%20a%20queue%20and%20enqueue%20the%20root%0A%20%20%20%20queue%3A%20deque%5BTreeNode%5D%20%3D%20deque%28%29%0A%20%20%20%20queue.append%28root%29%0A%20%20%20%20%23%20Initialize%20a%20list%20to%20store%20the%20traversal%20sequence%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20while%20queue%3A%0A%20%20%20%20%20%20%20%20node%3A%20TreeNode%20%3D%20queue.popleft%28%29%20%20%23%20Dequeue%20from%20the%20queue%0A%20%20%20%20%20%20%20%20res.append%28node.val%29%20%20%23%20Store%20the%20node%20value%0A%20%20%20%20%20%20%20%20if%20node.left%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20queue.append%28node.left%29%20%20%23%20Enqueue%20the%20left%20child%0A%20%20%20%20%20%20%20%20if%20node.right%20is%20not%20None%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20queue.append%28node.right%29%20%20%23%20Enqueue%20the%20right%20child%0A%20%20%20%20return%20res%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20binary%20tree%0A%20%20%20%20%23%20Use%20a%20helper%20function%20that%20builds%20a%20binary%20tree%20directly%20from%20an%20array%0A%20%20%20%20root%20%3D%20list_to_tree%28arr%3D%5B1,%202,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20Level-order%20traversal%0A%20%20%20%20res%20%3D%20level_order%28root%29%0A%20%20%20%20print%28%22%5CnNode%20sequence%20from%20level-order%20traversal%20%3D%20%22,%20res%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=127&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
+<h3 id="2-complexity-analysis">2. &nbsp; Complexity analysis<a class="headerlink" href="#2-complexity-analysis" title="Permanent link">&para;</a></h3>
 <ul>
-<li><strong>Time complexity is <span class="arithmatex">\(O(n)\)</span></strong>: All nodes are visited once, using <span class="arithmatex">\(O(n)\)</span> time, where <span class="arithmatex">\(n\)</span> is the number of nodes.</li>
-<li><strong>Space complexity is <span class="arithmatex">\(O(n)\)</span></strong>: In the worst case, i.e., a full binary tree, before traversing to the bottom level, the queue contains at most <span class="arithmatex">\((n + 1) / 2\)</span> nodes simultaneously, occupying <span class="arithmatex">\(O(n)\)</span> space.</li>
+<li><strong>Time complexity: <span class="arithmatex">\(O(n)\)</span></strong>. Each of the <span class="arithmatex">\(n\)</span> nodes is visited once, taking <span class="arithmatex">\(O(n)\)</span> time.</li>
+<li><strong>Space complexity: <span class="arithmatex">\(O(n)\)</span></strong>. In the worst case, a perfect binary tree, the queue holds up to <span class="arithmatex">\((n + 1) / 2\)</span> nodes just before the lowest level is visited, requiring <span class="arithmatex">\(O(n)\)</span> space.</li>
 </ul>
-<h2 id="722-preorder-inorder-and-postorder-traversal">7.2.2 &nbsp; Preorder, Inorder, and Postorder Traversal<a class="headerlink" href="#722-preorder-inorder-and-postorder-traversal" title="Permanent link">&para;</a></h2>
-<p>Correspondingly, preorder, inorder, and postorder traversals all belong to <u>depth-first traversal</u>, also known as <u>depth-first search (DFS)</u>, which goes as deep as possible before backtracking.</p>
-<p>Figure 7-10 shows how depth-first traversal works on a binary tree. <strong>Depth-first traversal is like "walking" around the perimeter of the entire binary tree</strong>, encountering three positions at each node, corresponding to preorder, inorder, and postorder traversal.</p>
-<p><img alt="Preorder, inorder, and postorder traversal of a binary tree" class="animation-figure" src="../binary_tree_traversal.assets/binary_tree_dfs.png" /></p>
-<p align="center"> Figure 7-10 &nbsp; Preorder, inorder, and postorder traversal of a binary tree </p>
+<h2 id="722-preorder-inorder-and-postorder-traversal">7.2.2 &nbsp; Preorder, inorder, and postorder traversal<a class="headerlink" href="#722-preorder-inorder-and-postorder-traversal" title="Permanent link">&para;</a></h2>
+<p>Preorder, inorder, and postorder traversal are forms of <u>depth-first traversal</u>, also called <u>depth-first search (DFS)</u>. They follow a path to its end, then backtrack and continue.</p>
+<p>Figure 7-10 illustrates depth-first traversal. <strong>Imagine walking once around the outside of the entire binary tree</strong>. At each node, you encounter three positions corresponding to preorder, inorder, and postorder visits.</p>
+<p><img alt="Preorder, inorder, and postorder traversal of a binary search tree" class="animation-figure" src="../binary_tree_traversal.assets/binary_tree_dfs.png" /></p>
+<p align="center"> Figure 7-10 &nbsp; Preorder, inorder, and postorder traversal of a binary search tree </p>
 
-<h3 id="1-code-implementation_1">1. &nbsp; Code Implementation<a class="headerlink" href="#1-code-implementation_1" title="Permanent link">&para;</a></h3>
-<p>Depth-first search is usually implemented based on recursion:</p>
+<h3 id="1-implementation_1">1. &nbsp; Implementation<a class="headerlink" href="#1-implementation_1" title="Permanent link">&para;</a></h3>
+<p>Depth-first search is usually implemented recursively:</p>
 <div class="tabbed-set tabbed-alternate" data-tabs="2:13"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><input id="__tabbed_2_13" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Python</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Java</label><label for="__tabbed_2_4">C#</label><label for="__tabbed_2_5">Go</label><label for="__tabbed_2_6">Swift</label><label for="__tabbed_2_7">JS</label><label for="__tabbed_2_8">TS</label><label for="__tabbed_2_9">Dart</label><label for="__tabbed_2_10">Rust</label><label for="__tabbed_2_11">C</label><label for="__tabbed_2_12">Kotlin</label><label for="__tabbed_2_13">Ruby</label></div>
 <div class="tabbed-content">
 <div class="tabbed-block">
@@ -5090,7 +5167,7 @@
 <a id="__codelineno-13-2" name="__codelineno-13-2" href="#__codelineno-13-2"></a><span class="w">    </span><span class="sd">"""Preorder traversal"""</span>
 <a id="__codelineno-13-3" name="__codelineno-13-3" href="#__codelineno-13-3"></a>    <span class="k">if</span> <span class="n">root</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-13-4" name="__codelineno-13-4" href="#__codelineno-13-4"></a>        <span class="k">return</span>
-<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Visit priority: root node -&gt; left subtree -&gt; right subtree</span>
+<a id="__codelineno-13-5" name="__codelineno-13-5" href="#__codelineno-13-5"></a>    <span class="c1"># Visit order: root -&gt; left subtree -&gt; right subtree</span>
 <a id="__codelineno-13-6" name="__codelineno-13-6" href="#__codelineno-13-6"></a>    <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-13-7" name="__codelineno-13-7" href="#__codelineno-13-7"></a>    <span class="n">pre_order</span><span class="p">(</span><span class="n">root</span><span class="o">=</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
 <a id="__codelineno-13-8" name="__codelineno-13-8" href="#__codelineno-13-8"></a>    <span class="n">pre_order</span><span class="p">(</span><span class="n">root</span><span class="o">=</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
@@ -5099,7 +5176,7 @@
 <a id="__codelineno-13-11" name="__codelineno-13-11" href="#__codelineno-13-11"></a><span class="w">    </span><span class="sd">"""Inorder traversal"""</span>
 <a id="__codelineno-13-12" name="__codelineno-13-12" href="#__codelineno-13-12"></a>    <span class="k">if</span> <span class="n">root</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-13-13" name="__codelineno-13-13" href="#__codelineno-13-13"></a>        <span class="k">return</span>
-<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>    <span class="c1"># Visit priority: left subtree -&gt; root node -&gt; right subtree</span>
+<a id="__codelineno-13-14" name="__codelineno-13-14" href="#__codelineno-13-14"></a>    <span class="c1"># Visit order: left subtree -&gt; root -&gt; right subtree</span>
 <a id="__codelineno-13-15" name="__codelineno-13-15" href="#__codelineno-13-15"></a>    <span class="n">in_order</span><span class="p">(</span><span class="n">root</span><span class="o">=</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
 <a id="__codelineno-13-16" name="__codelineno-13-16" href="#__codelineno-13-16"></a>    <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-13-17" name="__codelineno-13-17" href="#__codelineno-13-17"></a>    <span class="n">in_order</span><span class="p">(</span><span class="n">root</span><span class="o">=</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
@@ -5108,7 +5185,7 @@
 <a id="__codelineno-13-20" name="__codelineno-13-20" href="#__codelineno-13-20"></a><span class="w">    </span><span class="sd">"""Postorder traversal"""</span>
 <a id="__codelineno-13-21" name="__codelineno-13-21" href="#__codelineno-13-21"></a>    <span class="k">if</span> <span class="n">root</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
 <a id="__codelineno-13-22" name="__codelineno-13-22" href="#__codelineno-13-22"></a>        <span class="k">return</span>
-<a id="__codelineno-13-23" name="__codelineno-13-23" href="#__codelineno-13-23"></a>    <span class="c1"># Visit priority: left subtree -&gt; right subtree -&gt; root node</span>
+<a id="__codelineno-13-23" name="__codelineno-13-23" href="#__codelineno-13-23"></a>    <span class="c1"># Visit order: left subtree -&gt; right subtree -&gt; root</span>
 <a id="__codelineno-13-24" name="__codelineno-13-24" href="#__codelineno-13-24"></a>    <span class="n">post_order</span><span class="p">(</span><span class="n">root</span><span class="o">=</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
 <a id="__codelineno-13-25" name="__codelineno-13-25" href="#__codelineno-13-25"></a>    <span class="n">post_order</span><span class="p">(</span><span class="n">root</span><span class="o">=</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
 <a id="__codelineno-13-26" name="__codelineno-13-26" href="#__codelineno-13-26"></a>    <span class="n">res</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">val</span><span class="p">)</span>
@@ -5119,7 +5196,7 @@
 <a id="__codelineno-14-2" name="__codelineno-14-2" href="#__codelineno-14-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-3" name="__codelineno-14-3" href="#__codelineno-14-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span>
 <a id="__codelineno-14-4" name="__codelineno-14-4" href="#__codelineno-14-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="c1">// Visit priority: root node -&gt; left subtree -&gt; right subtree</span>
+<a id="__codelineno-14-5" name="__codelineno-14-5" href="#__codelineno-14-5"></a><span class="w">    </span><span class="c1">// Visit order: root -&gt; left subtree -&gt; right subtree</span>
 <a id="__codelineno-14-6" name="__codelineno-14-6" href="#__codelineno-14-6"></a><span class="w">    </span><span class="n">vec</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-14-7" name="__codelineno-14-7" href="#__codelineno-14-7"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">);</span>
 <a id="__codelineno-14-8" name="__codelineno-14-8" href="#__codelineno-14-8"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">);</span>
@@ -5129,7 +5206,7 @@
 <a id="__codelineno-14-12" name="__codelineno-14-12" href="#__codelineno-14-12"></a><span class="kt">void</span><span class="w"> </span><span class="nf">inOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-13" name="__codelineno-14-13" href="#__codelineno-14-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span>
 <a id="__codelineno-14-14" name="__codelineno-14-14" href="#__codelineno-14-14"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; root node -&gt; right subtree</span>
+<a id="__codelineno-14-15" name="__codelineno-14-15" href="#__codelineno-14-15"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; root -&gt; right subtree</span>
 <a id="__codelineno-14-16" name="__codelineno-14-16" href="#__codelineno-14-16"></a><span class="w">    </span><span class="n">inOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">);</span>
 <a id="__codelineno-14-17" name="__codelineno-14-17" href="#__codelineno-14-17"></a><span class="w">    </span><span class="n">vec</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-14-18" name="__codelineno-14-18" href="#__codelineno-14-18"></a><span class="w">    </span><span class="n">inOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">);</span>
@@ -5139,7 +5216,7 @@
 <a id="__codelineno-14-22" name="__codelineno-14-22" href="#__codelineno-14-22"></a><span class="kt">void</span><span class="w"> </span><span class="nf">postOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-14-23" name="__codelineno-14-23" href="#__codelineno-14-23"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">nullptr</span><span class="p">)</span>
 <a id="__codelineno-14-24" name="__codelineno-14-24" href="#__codelineno-14-24"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-14-25" name="__codelineno-14-25" href="#__codelineno-14-25"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; right subtree -&gt; root node</span>
+<a id="__codelineno-14-25" name="__codelineno-14-25" href="#__codelineno-14-25"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; right subtree -&gt; root</span>
 <a id="__codelineno-14-26" name="__codelineno-14-26" href="#__codelineno-14-26"></a><span class="w">    </span><span class="n">postOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">);</span>
 <a id="__codelineno-14-27" name="__codelineno-14-27" href="#__codelineno-14-27"></a><span class="w">    </span><span class="n">postOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-14-28" name="__codelineno-14-28" href="#__codelineno-14-28"></a><span class="w">    </span><span class="n">vec</span><span class="p">.</span><span class="n">push_back</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">);</span>
@@ -5151,7 +5228,7 @@
 <a id="__codelineno-15-2" name="__codelineno-15-2" href="#__codelineno-15-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-3" name="__codelineno-15-3" href="#__codelineno-15-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-15-4" name="__codelineno-15-4" href="#__codelineno-15-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="c1">// Visit priority: root node -&gt; left subtree -&gt; right subtree</span>
+<a id="__codelineno-15-5" name="__codelineno-15-5" href="#__codelineno-15-5"></a><span class="w">    </span><span class="c1">// Visit order: root -&gt; left subtree -&gt; right subtree</span>
 <a id="__codelineno-15-6" name="__codelineno-15-6" href="#__codelineno-15-6"></a><span class="w">    </span><span class="n">list</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">val</span><span class="p">);</span>
 <a id="__codelineno-15-7" name="__codelineno-15-7" href="#__codelineno-15-7"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">left</span><span class="p">);</span>
 <a id="__codelineno-15-8" name="__codelineno-15-8" href="#__codelineno-15-8"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">right</span><span class="p">);</span>
@@ -5161,7 +5238,7 @@
 <a id="__codelineno-15-12" name="__codelineno-15-12" href="#__codelineno-15-12"></a><span class="kt">void</span><span class="w"> </span><span class="nf">inOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-13" name="__codelineno-15-13" href="#__codelineno-15-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-15-14" name="__codelineno-15-14" href="#__codelineno-15-14"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; root node -&gt; right subtree</span>
+<a id="__codelineno-15-15" name="__codelineno-15-15" href="#__codelineno-15-15"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; root -&gt; right subtree</span>
 <a id="__codelineno-15-16" name="__codelineno-15-16" href="#__codelineno-15-16"></a><span class="w">    </span><span class="n">inOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">left</span><span class="p">);</span>
 <a id="__codelineno-15-17" name="__codelineno-15-17" href="#__codelineno-15-17"></a><span class="w">    </span><span class="n">list</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">val</span><span class="p">);</span>
 <a id="__codelineno-15-18" name="__codelineno-15-18" href="#__codelineno-15-18"></a><span class="w">    </span><span class="n">inOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">right</span><span class="p">);</span>
@@ -5171,7 +5248,7 @@
 <a id="__codelineno-15-22" name="__codelineno-15-22" href="#__codelineno-15-22"></a><span class="kt">void</span><span class="w"> </span><span class="nf">postOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-15-23" name="__codelineno-15-23" href="#__codelineno-15-23"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span>
 <a id="__codelineno-15-24" name="__codelineno-15-24" href="#__codelineno-15-24"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-15-25" name="__codelineno-15-25" href="#__codelineno-15-25"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; right subtree -&gt; root node</span>
+<a id="__codelineno-15-25" name="__codelineno-15-25" href="#__codelineno-15-25"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; right subtree -&gt; root</span>
 <a id="__codelineno-15-26" name="__codelineno-15-26" href="#__codelineno-15-26"></a><span class="w">    </span><span class="n">postOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">left</span><span class="p">);</span>
 <a id="__codelineno-15-27" name="__codelineno-15-27" href="#__codelineno-15-27"></a><span class="w">    </span><span class="n">postOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">right</span><span class="p">);</span>
 <a id="__codelineno-15-28" name="__codelineno-15-28" href="#__codelineno-15-28"></a><span class="w">    </span><span class="n">list</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">val</span><span class="p">);</span>
@@ -5182,7 +5259,7 @@
 <div class="highlight"><span class="filename">binary_tree_dfs.cs</span><pre><span></span><code><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a><span class="cm">/* Preorder traversal */</span>
 <a id="__codelineno-16-2" name="__codelineno-16-2" href="#__codelineno-16-2"></a><span class="k">void</span><span class="w"> </span><span class="nf">PreOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-3" name="__codelineno-16-3" href="#__codelineno-16-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="c1">// Visit priority: root node -&gt; left subtree -&gt; right subtree</span>
+<a id="__codelineno-16-4" name="__codelineno-16-4" href="#__codelineno-16-4"></a><span class="w">    </span><span class="c1">// Visit order: root -&gt; left subtree -&gt; right subtree</span>
 <a id="__codelineno-16-5" name="__codelineno-16-5" href="#__codelineno-16-5"></a><span class="w">    </span><span class="n">list</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="o">!</span><span class="p">.</span><span class="n">Value</span><span class="p">);</span>
 <a id="__codelineno-16-6" name="__codelineno-16-6" href="#__codelineno-16-6"></a><span class="w">    </span><span class="n">PreOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">left</span><span class="p">);</span>
 <a id="__codelineno-16-7" name="__codelineno-16-7" href="#__codelineno-16-7"></a><span class="w">    </span><span class="n">PreOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">right</span><span class="p">);</span>
@@ -5191,7 +5268,7 @@
 <a id="__codelineno-16-10" name="__codelineno-16-10" href="#__codelineno-16-10"></a><span class="cm">/* Inorder traversal */</span>
 <a id="__codelineno-16-11" name="__codelineno-16-11" href="#__codelineno-16-11"></a><span class="k">void</span><span class="w"> </span><span class="nf">InOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-12" name="__codelineno-16-12" href="#__codelineno-16-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; root node -&gt; right subtree</span>
+<a id="__codelineno-16-13" name="__codelineno-16-13" href="#__codelineno-16-13"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; root -&gt; right subtree</span>
 <a id="__codelineno-16-14" name="__codelineno-16-14" href="#__codelineno-16-14"></a><span class="w">    </span><span class="n">InOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">left</span><span class="p">);</span>
 <a id="__codelineno-16-15" name="__codelineno-16-15" href="#__codelineno-16-15"></a><span class="w">    </span><span class="n">list</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="o">!</span><span class="p">.</span><span class="n">Value</span><span class="p">);</span>
 <a id="__codelineno-16-16" name="__codelineno-16-16" href="#__codelineno-16-16"></a><span class="w">    </span><span class="n">InOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">right</span><span class="p">);</span>
@@ -5200,7 +5277,7 @@
 <a id="__codelineno-16-19" name="__codelineno-16-19" href="#__codelineno-16-19"></a><span class="cm">/* Postorder traversal */</span>
 <a id="__codelineno-16-20" name="__codelineno-16-20" href="#__codelineno-16-20"></a><span class="k">void</span><span class="w"> </span><span class="nf">PostOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-16-21" name="__codelineno-16-21" href="#__codelineno-16-21"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="k">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; right subtree -&gt; root node</span>
+<a id="__codelineno-16-22" name="__codelineno-16-22" href="#__codelineno-16-22"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; right subtree -&gt; root</span>
 <a id="__codelineno-16-23" name="__codelineno-16-23" href="#__codelineno-16-23"></a><span class="w">    </span><span class="n">PostOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">left</span><span class="p">);</span>
 <a id="__codelineno-16-24" name="__codelineno-16-24" href="#__codelineno-16-24"></a><span class="w">    </span><span class="n">PostOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-16-25" name="__codelineno-16-25" href="#__codelineno-16-25"></a><span class="w">    </span><span class="n">list</span><span class="p">.</span><span class="n">Add</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="o">!</span><span class="p">.</span><span class="n">Value</span><span class="p">);</span>
@@ -5213,7 +5290,7 @@
 <a id="__codelineno-17-3" name="__codelineno-17-3" href="#__codelineno-17-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-4" name="__codelineno-17-4" href="#__codelineno-17-4"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-17-5" name="__codelineno-17-5" href="#__codelineno-17-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="c1">// Visit priority: root node -&gt; left subtree -&gt; right subtree</span>
+<a id="__codelineno-17-6" name="__codelineno-17-6" href="#__codelineno-17-6"></a><span class="w">    </span><span class="c1">// Visit order: root -&gt; left subtree -&gt; right subtree</span>
 <a id="__codelineno-17-7" name="__codelineno-17-7" href="#__codelineno-17-7"></a><span class="w">    </span><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Val</span><span class="p">)</span>
 <a id="__codelineno-17-8" name="__codelineno-17-8" href="#__codelineno-17-8"></a><span class="w">    </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="p">)</span>
 <a id="__codelineno-17-9" name="__codelineno-17-9" href="#__codelineno-17-9"></a><span class="w">    </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span><span class="p">)</span>
@@ -5224,7 +5301,7 @@
 <a id="__codelineno-17-14" name="__codelineno-17-14" href="#__codelineno-17-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-15" name="__codelineno-17-15" href="#__codelineno-17-15"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-17-16" name="__codelineno-17-16" href="#__codelineno-17-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; root node -&gt; right subtree</span>
+<a id="__codelineno-17-17" name="__codelineno-17-17" href="#__codelineno-17-17"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; root -&gt; right subtree</span>
 <a id="__codelineno-17-18" name="__codelineno-17-18" href="#__codelineno-17-18"></a><span class="w">    </span><span class="nx">inOrder</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="p">)</span>
 <a id="__codelineno-17-19" name="__codelineno-17-19" href="#__codelineno-17-19"></a><span class="w">    </span><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Val</span><span class="p">)</span>
 <a id="__codelineno-17-20" name="__codelineno-17-20" href="#__codelineno-17-20"></a><span class="w">    </span><span class="nx">inOrder</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span><span class="p">)</span>
@@ -5235,7 +5312,7 @@
 <a id="__codelineno-17-25" name="__codelineno-17-25" href="#__codelineno-17-25"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="nx">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">nil</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-17-26" name="__codelineno-17-26" href="#__codelineno-17-26"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-17-27" name="__codelineno-17-27" href="#__codelineno-17-27"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-17-28" name="__codelineno-17-28" href="#__codelineno-17-28"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; right subtree -&gt; root node</span>
+<a id="__codelineno-17-28" name="__codelineno-17-28" href="#__codelineno-17-28"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; right subtree -&gt; root</span>
 <a id="__codelineno-17-29" name="__codelineno-17-29" href="#__codelineno-17-29"></a><span class="w">    </span><span class="nx">postOrder</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Left</span><span class="p">)</span>
 <a id="__codelineno-17-30" name="__codelineno-17-30" href="#__codelineno-17-30"></a><span class="w">    </span><span class="nx">postOrder</span><span class="p">(</span><span class="nx">node</span><span class="p">.</span><span class="nx">Right</span><span class="p">)</span>
 <a id="__codelineno-17-31" name="__codelineno-17-31" href="#__codelineno-17-31"></a><span class="w">    </span><span class="nx">nums</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">append</span><span class="p">(</span><span class="nx">nums</span><span class="p">,</span><span class="w"> </span><span class="nx">node</span><span class="p">.</span><span class="nx">Val</span><span class="p">)</span>
@@ -5248,7 +5325,7 @@
 <a id="__codelineno-18-3" name="__codelineno-18-3" href="#__codelineno-18-3"></a><span class="w">    </span><span class="k">guard</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nv">root</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-4" name="__codelineno-18-4" href="#__codelineno-18-4"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-18-5" name="__codelineno-18-5" href="#__codelineno-18-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="c1">// Visit priority: root node -&gt; left subtree -&gt; right subtree</span>
+<a id="__codelineno-18-6" name="__codelineno-18-6" href="#__codelineno-18-6"></a><span class="w">    </span><span class="c1">// Visit order: root -&gt; left subtree -&gt; right subtree</span>
 <a id="__codelineno-18-7" name="__codelineno-18-7" href="#__codelineno-18-7"></a><span class="w">    </span><span class="n">list</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-18-8" name="__codelineno-18-8" href="#__codelineno-18-8"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="kr">left</span><span class="p">)</span>
 <a id="__codelineno-18-9" name="__codelineno-18-9" href="#__codelineno-18-9"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="kr">right</span><span class="p">)</span>
@@ -5259,7 +5336,7 @@
 <a id="__codelineno-18-14" name="__codelineno-18-14" href="#__codelineno-18-14"></a><span class="w">    </span><span class="k">guard</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nv">root</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-15" name="__codelineno-18-15" href="#__codelineno-18-15"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-18-16" name="__codelineno-18-16" href="#__codelineno-18-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; root node -&gt; right subtree</span>
+<a id="__codelineno-18-17" name="__codelineno-18-17" href="#__codelineno-18-17"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; root -&gt; right subtree</span>
 <a id="__codelineno-18-18" name="__codelineno-18-18" href="#__codelineno-18-18"></a><span class="w">    </span><span class="n">inOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="kr">left</span><span class="p">)</span>
 <a id="__codelineno-18-19" name="__codelineno-18-19" href="#__codelineno-18-19"></a><span class="w">    </span><span class="n">list</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
 <a id="__codelineno-18-20" name="__codelineno-18-20" href="#__codelineno-18-20"></a><span class="w">    </span><span class="n">inOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="kr">right</span><span class="p">)</span>
@@ -5270,7 +5347,7 @@
 <a id="__codelineno-18-25" name="__codelineno-18-25" href="#__codelineno-18-25"></a><span class="w">    </span><span class="k">guard</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nv">root</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-18-26" name="__codelineno-18-26" href="#__codelineno-18-26"></a><span class="w">        </span><span class="k">return</span>
 <a id="__codelineno-18-27" name="__codelineno-18-27" href="#__codelineno-18-27"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-18-28" name="__codelineno-18-28" href="#__codelineno-18-28"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; right subtree -&gt; root node</span>
+<a id="__codelineno-18-28" name="__codelineno-18-28" href="#__codelineno-18-28"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; right subtree -&gt; root</span>
 <a id="__codelineno-18-29" name="__codelineno-18-29" href="#__codelineno-18-29"></a><span class="w">    </span><span class="n">postOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="kr">left</span><span class="p">)</span>
 <a id="__codelineno-18-30" name="__codelineno-18-30" href="#__codelineno-18-30"></a><span class="w">    </span><span class="n">postOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">root</span><span class="p">.</span><span class="kr">right</span><span class="p">)</span>
 <a id="__codelineno-18-31" name="__codelineno-18-31" href="#__codelineno-18-31"></a><span class="w">    </span><span class="n">list</span><span class="p">.</span><span class="n">append</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="n">val</span><span class="p">)</span>
@@ -5281,7 +5358,7 @@
 <div class="highlight"><span class="filename">binary_tree_dfs.js</span><pre><span></span><code><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a><span class="cm">/* Preorder traversal */</span>
 <a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a><span class="kd">function</span><span class="w"> </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="c1">// Visit priority: root node -&gt; left subtree -&gt; right subtree</span>
+<a id="__codelineno-19-4" name="__codelineno-19-4" href="#__codelineno-19-4"></a><span class="w">    </span><span class="c1">// Visit order: root -&gt; left subtree -&gt; right subtree</span>
 <a id="__codelineno-19-5" name="__codelineno-19-5" href="#__codelineno-19-5"></a><span class="w">    </span><span class="nx">list</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-19-6" name="__codelineno-19-6" href="#__codelineno-19-6"></a><span class="w">    </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">left</span><span class="p">);</span>
 <a id="__codelineno-19-7" name="__codelineno-19-7" href="#__codelineno-19-7"></a><span class="w">    </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">right</span><span class="p">);</span>
@@ -5290,7 +5367,7 @@
 <a id="__codelineno-19-10" name="__codelineno-19-10" href="#__codelineno-19-10"></a><span class="cm">/* Inorder traversal */</span>
 <a id="__codelineno-19-11" name="__codelineno-19-11" href="#__codelineno-19-11"></a><span class="kd">function</span><span class="w"> </span><span class="nx">inOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-12" name="__codelineno-19-12" href="#__codelineno-19-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; root node -&gt; right subtree</span>
+<a id="__codelineno-19-13" name="__codelineno-19-13" href="#__codelineno-19-13"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; root -&gt; right subtree</span>
 <a id="__codelineno-19-14" name="__codelineno-19-14" href="#__codelineno-19-14"></a><span class="w">    </span><span class="nx">inOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">left</span><span class="p">);</span>
 <a id="__codelineno-19-15" name="__codelineno-19-15" href="#__codelineno-19-15"></a><span class="w">    </span><span class="nx">list</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-19-16" name="__codelineno-19-16" href="#__codelineno-19-16"></a><span class="w">    </span><span class="nx">inOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">right</span><span class="p">);</span>
@@ -5299,7 +5376,7 @@
 <a id="__codelineno-19-19" name="__codelineno-19-19" href="#__codelineno-19-19"></a><span class="cm">/* Postorder traversal */</span>
 <a id="__codelineno-19-20" name="__codelineno-19-20" href="#__codelineno-19-20"></a><span class="kd">function</span><span class="w"> </span><span class="nx">postOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-19-21" name="__codelineno-19-21" href="#__codelineno-19-21"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; right subtree -&gt; root node</span>
+<a id="__codelineno-19-22" name="__codelineno-19-22" href="#__codelineno-19-22"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; right subtree -&gt; root</span>
 <a id="__codelineno-19-23" name="__codelineno-19-23" href="#__codelineno-19-23"></a><span class="w">    </span><span class="nx">postOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">left</span><span class="p">);</span>
 <a id="__codelineno-19-24" name="__codelineno-19-24" href="#__codelineno-19-24"></a><span class="w">    </span><span class="nx">postOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">right</span><span class="p">);</span>
 <a id="__codelineno-19-25" name="__codelineno-19-25" href="#__codelineno-19-25"></a><span class="w">    </span><span class="nx">list</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span>
@@ -5312,7 +5389,7 @@
 <a id="__codelineno-20-3" name="__codelineno-20-3" href="#__codelineno-20-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-4" name="__codelineno-20-4" href="#__codelineno-20-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-20-5" name="__codelineno-20-5" href="#__codelineno-20-5"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="c1">// Visit priority: root node -&gt; left subtree -&gt; right subtree</span>
+<a id="__codelineno-20-6" name="__codelineno-20-6" href="#__codelineno-20-6"></a><span class="w">    </span><span class="c1">// Visit order: root -&gt; left subtree -&gt; right subtree</span>
 <a id="__codelineno-20-7" name="__codelineno-20-7" href="#__codelineno-20-7"></a><span class="w">    </span><span class="nx">list</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-20-8" name="__codelineno-20-8" href="#__codelineno-20-8"></a><span class="w">    </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">left</span><span class="p">);</span>
 <a id="__codelineno-20-9" name="__codelineno-20-9" href="#__codelineno-20-9"></a><span class="w">    </span><span class="nx">preOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">right</span><span class="p">);</span>
@@ -5323,7 +5400,7 @@
 <a id="__codelineno-20-14" name="__codelineno-20-14" href="#__codelineno-20-14"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-15" name="__codelineno-20-15" href="#__codelineno-20-15"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-20-16" name="__codelineno-20-16" href="#__codelineno-20-16"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; root node -&gt; right subtree</span>
+<a id="__codelineno-20-17" name="__codelineno-20-17" href="#__codelineno-20-17"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; root -&gt; right subtree</span>
 <a id="__codelineno-20-18" name="__codelineno-20-18" href="#__codelineno-20-18"></a><span class="w">    </span><span class="nx">inOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">left</span><span class="p">);</span>
 <a id="__codelineno-20-19" name="__codelineno-20-19" href="#__codelineno-20-19"></a><span class="w">    </span><span class="nx">list</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span>
 <a id="__codelineno-20-20" name="__codelineno-20-20" href="#__codelineno-20-20"></a><span class="w">    </span><span class="nx">inOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">right</span><span class="p">);</span>
@@ -5334,7 +5411,7 @@
 <a id="__codelineno-20-25" name="__codelineno-20-25" href="#__codelineno-20-25"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="nx">root</span><span class="w"> </span><span class="o">===</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-20-26" name="__codelineno-20-26" href="#__codelineno-20-26"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
 <a id="__codelineno-20-27" name="__codelineno-20-27" href="#__codelineno-20-27"></a><span class="w">    </span><span class="p">}</span>
-<a id="__codelineno-20-28" name="__codelineno-20-28" href="#__codelineno-20-28"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; right subtree -&gt; root node</span>
+<a id="__codelineno-20-28" name="__codelineno-20-28" href="#__codelineno-20-28"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; right subtree -&gt; root</span>
 <a id="__codelineno-20-29" name="__codelineno-20-29" href="#__codelineno-20-29"></a><span class="w">    </span><span class="nx">postOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">left</span><span class="p">);</span>
 <a id="__codelineno-20-30" name="__codelineno-20-30" href="#__codelineno-20-30"></a><span class="w">    </span><span class="nx">postOrder</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">right</span><span class="p">);</span>
 <a id="__codelineno-20-31" name="__codelineno-20-31" href="#__codelineno-20-31"></a><span class="w">    </span><span class="nx">list</span><span class="p">.</span><span class="nx">push</span><span class="p">(</span><span class="nx">root</span><span class="p">.</span><span class="nx">val</span><span class="p">);</span>
@@ -5345,7 +5422,7 @@
 <div class="highlight"><span class="filename">binary_tree_dfs.dart</span><pre><span></span><code><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a><span class="cm">/* Preorder traversal */</span>
 <a id="__codelineno-21-2" name="__codelineno-21-2" href="#__codelineno-21-2"></a><span class="kt">void</span><span class="w"> </span><span class="n">preOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-3" name="__codelineno-21-3" href="#__codelineno-21-3"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="c1">// Visit priority: root node -&gt; left subtree -&gt; right subtree</span>
+<a id="__codelineno-21-4" name="__codelineno-21-4" href="#__codelineno-21-4"></a><span class="w">  </span><span class="c1">// Visit order: root -&gt; left subtree -&gt; right subtree</span>
 <a id="__codelineno-21-5" name="__codelineno-21-5" href="#__codelineno-21-5"></a><span class="w">  </span><span class="n">list</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-21-6" name="__codelineno-21-6" href="#__codelineno-21-6"></a><span class="w">  </span><span class="n">preOrder</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">);</span>
 <a id="__codelineno-21-7" name="__codelineno-21-7" href="#__codelineno-21-7"></a><span class="w">  </span><span class="n">preOrder</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">);</span>
@@ -5354,7 +5431,7 @@
 <a id="__codelineno-21-10" name="__codelineno-21-10" href="#__codelineno-21-10"></a><span class="cm">/* Inorder traversal */</span>
 <a id="__codelineno-21-11" name="__codelineno-21-11" href="#__codelineno-21-11"></a><span class="kt">void</span><span class="w"> </span><span class="n">inOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-12" name="__codelineno-21-12" href="#__codelineno-21-12"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="c1">// Visit priority: left subtree -&gt; root node -&gt; right subtree</span>
+<a id="__codelineno-21-13" name="__codelineno-21-13" href="#__codelineno-21-13"></a><span class="w">  </span><span class="c1">// Visit order: left subtree -&gt; root -&gt; right subtree</span>
 <a id="__codelineno-21-14" name="__codelineno-21-14" href="#__codelineno-21-14"></a><span class="w">  </span><span class="n">inOrder</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">);</span>
 <a id="__codelineno-21-15" name="__codelineno-21-15" href="#__codelineno-21-15"></a><span class="w">  </span><span class="n">list</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-21-16" name="__codelineno-21-16" href="#__codelineno-21-16"></a><span class="w">  </span><span class="n">inOrder</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">);</span>
@@ -5363,7 +5440,7 @@
 <a id="__codelineno-21-19" name="__codelineno-21-19" href="#__codelineno-21-19"></a><span class="cm">/* Postorder traversal */</span>
 <a id="__codelineno-21-20" name="__codelineno-21-20" href="#__codelineno-21-20"></a><span class="kt">void</span><span class="w"> </span><span class="n">postOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="o">?</span><span class="w"> </span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-21-21" name="__codelineno-21-21" href="#__codelineno-21-21"></a><span class="w">  </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">node</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">  </span><span class="c1">// Visit priority: left subtree -&gt; right subtree -&gt; root node</span>
+<a id="__codelineno-21-22" name="__codelineno-21-22" href="#__codelineno-21-22"></a><span class="w">  </span><span class="c1">// Visit order: left subtree -&gt; right subtree -&gt; root</span>
 <a id="__codelineno-21-23" name="__codelineno-21-23" href="#__codelineno-21-23"></a><span class="w">  </span><span class="n">postOrder</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">);</span>
 <a id="__codelineno-21-24" name="__codelineno-21-24" href="#__codelineno-21-24"></a><span class="w">  </span><span class="n">postOrder</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">);</span>
 <a id="__codelineno-21-25" name="__codelineno-21-25" href="#__codelineno-21-25"></a><span class="w">  </span><span class="n">list</span><span class="p">.</span><span class="n">add</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
@@ -5377,7 +5454,7 @@
 <a id="__codelineno-22-4" name="__codelineno-22-4" href="#__codelineno-22-4"></a>
 <a id="__codelineno-22-5" name="__codelineno-22-5" href="#__codelineno-22-5"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-6" name="__codelineno-22-6" href="#__codelineno-22-6"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">            </span><span class="c1">// Visit priority: root node -&gt; left subtree -&gt; right subtree</span>
+<a id="__codelineno-22-7" name="__codelineno-22-7" href="#__codelineno-22-7"></a><span class="w">            </span><span class="c1">// Visit order: root -&gt; left subtree -&gt; right subtree</span>
 <a id="__codelineno-22-8" name="__codelineno-22-8" href="#__codelineno-22-8"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">();</span>
 <a id="__codelineno-22-9" name="__codelineno-22-9" href="#__codelineno-22-9"></a><span class="w">            </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
 <a id="__codelineno-22-10" name="__codelineno-22-10" href="#__codelineno-22-10"></a><span class="w">            </span><span class="n">dfs</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">.</span><span class="n">as_ref</span><span class="p">(),</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
@@ -5395,7 +5472,7 @@
 <a id="__codelineno-22-22" name="__codelineno-22-22" href="#__codelineno-22-22"></a>
 <a id="__codelineno-22-23" name="__codelineno-22-23" href="#__codelineno-22-23"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-24" name="__codelineno-22-24" href="#__codelineno-22-24"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-25" name="__codelineno-22-25" href="#__codelineno-22-25"></a><span class="w">            </span><span class="c1">// Visit priority: left subtree -&gt; root node -&gt; right subtree</span>
+<a id="__codelineno-22-25" name="__codelineno-22-25" href="#__codelineno-22-25"></a><span class="w">            </span><span class="c1">// Visit order: left subtree -&gt; root -&gt; right subtree</span>
 <a id="__codelineno-22-26" name="__codelineno-22-26" href="#__codelineno-22-26"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">();</span>
 <a id="__codelineno-22-27" name="__codelineno-22-27" href="#__codelineno-22-27"></a><span class="w">            </span><span class="n">dfs</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">.</span><span class="n">as_ref</span><span class="p">(),</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-22-28" name="__codelineno-22-28" href="#__codelineno-22-28"></a><span class="w">            </span><span class="n">res</span><span class="p">.</span><span class="n">push</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">val</span><span class="p">);</span>
@@ -5413,7 +5490,7 @@
 <a id="__codelineno-22-40" name="__codelineno-22-40" href="#__codelineno-22-40"></a>
 <a id="__codelineno-22-41" name="__codelineno-22-41" href="#__codelineno-22-41"></a><span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="nf">dfs</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="nb">Option</span><span class="o">&lt;&amp;</span><span class="n">Rc</span><span class="o">&lt;</span><span class="n">RefCell</span><span class="o">&lt;</span><span class="n">TreeNode</span><span class="o">&gt;&gt;&gt;</span><span class="p">,</span><span class="w"> </span><span class="n">res</span><span class="p">:</span><span class="w"> </span><span class="kp">&amp;</span><span class="nc">mut</span><span class="w"> </span><span class="nb">Vec</span><span class="o">&lt;</span><span class="kt">i32</span><span class="o">&gt;</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-22-42" name="__codelineno-22-42" href="#__codelineno-22-42"></a><span class="w">        </span><span class="k">if</span><span class="w"> </span><span class="kd">let</span><span class="w"> </span><span class="nb">Some</span><span class="p">(</span><span class="n">node</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="w"> </span><span class="p">{</span>
-<a id="__codelineno-22-43" name="__codelineno-22-43" href="#__codelineno-22-43"></a><span class="w">            </span><span class="c1">// Visit priority: left subtree -&gt; right subtree -&gt; root node</span>
+<a id="__codelineno-22-43" name="__codelineno-22-43" href="#__codelineno-22-43"></a><span class="w">            </span><span class="c1">// Visit order: left subtree -&gt; right subtree -&gt; root</span>
 <a id="__codelineno-22-44" name="__codelineno-22-44" href="#__codelineno-22-44"></a><span class="w">            </span><span class="kd">let</span><span class="w"> </span><span class="n">node</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">node</span><span class="p">.</span><span class="n">borrow</span><span class="p">();</span>
 <a id="__codelineno-22-45" name="__codelineno-22-45" href="#__codelineno-22-45"></a><span class="w">            </span><span class="n">dfs</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">left</span><span class="p">.</span><span class="n">as_ref</span><span class="p">(),</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
 <a id="__codelineno-22-46" name="__codelineno-22-46" href="#__codelineno-22-46"></a><span class="w">            </span><span class="n">dfs</span><span class="p">(</span><span class="n">node</span><span class="p">.</span><span class="n">right</span><span class="p">.</span><span class="n">as_ref</span><span class="p">(),</span><span class="w"> </span><span class="n">res</span><span class="p">);</span>
@@ -5432,7 +5509,7 @@
 <a id="__codelineno-23-2" name="__codelineno-23-2" href="#__codelineno-23-2"></a><span class="kt">void</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-3" name="__codelineno-23-3" href="#__codelineno-23-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span>
 <a id="__codelineno-23-4" name="__codelineno-23-4" href="#__codelineno-23-4"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="c1">// Visit priority: root node -&gt; left subtree -&gt; right subtree</span>
+<a id="__codelineno-23-5" name="__codelineno-23-5" href="#__codelineno-23-5"></a><span class="w">    </span><span class="c1">// Visit order: root -&gt; left subtree -&gt; right subtree</span>
 <a id="__codelineno-23-6" name="__codelineno-23-6" href="#__codelineno-23-6"></a><span class="w">    </span><span class="n">arr</span><span class="p">[(</span><span class="o">*</span><span class="n">size</span><span class="p">)</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-23-7" name="__codelineno-23-7" href="#__codelineno-23-7"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">);</span>
 <a id="__codelineno-23-8" name="__codelineno-23-8" href="#__codelineno-23-8"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">);</span>
@@ -5442,7 +5519,7 @@
 <a id="__codelineno-23-12" name="__codelineno-23-12" href="#__codelineno-23-12"></a><span class="kt">void</span><span class="w"> </span><span class="nf">inOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-13" name="__codelineno-23-13" href="#__codelineno-23-13"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span>
 <a id="__codelineno-23-14" name="__codelineno-23-14" href="#__codelineno-23-14"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; root node -&gt; right subtree</span>
+<a id="__codelineno-23-15" name="__codelineno-23-15" href="#__codelineno-23-15"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; root -&gt; right subtree</span>
 <a id="__codelineno-23-16" name="__codelineno-23-16" href="#__codelineno-23-16"></a><span class="w">    </span><span class="n">inOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">);</span>
 <a id="__codelineno-23-17" name="__codelineno-23-17" href="#__codelineno-23-17"></a><span class="w">    </span><span class="n">arr</span><span class="p">[(</span><span class="o">*</span><span class="n">size</span><span class="p">)</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
 <a id="__codelineno-23-18" name="__codelineno-23-18" href="#__codelineno-23-18"></a><span class="w">    </span><span class="n">inOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">);</span>
@@ -5452,7 +5529,7 @@
 <a id="__codelineno-23-22" name="__codelineno-23-22" href="#__codelineno-23-22"></a><span class="kt">void</span><span class="w"> </span><span class="nf">postOrder</span><span class="p">(</span><span class="n">TreeNode</span><span class="w"> </span><span class="o">*</span><span class="n">root</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="o">*</span><span class="n">size</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-23-23" name="__codelineno-23-23" href="#__codelineno-23-23"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="nb">NULL</span><span class="p">)</span>
 <a id="__codelineno-23-24" name="__codelineno-23-24" href="#__codelineno-23-24"></a><span class="w">        </span><span class="k">return</span><span class="p">;</span>
-<a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; right subtree -&gt; root node</span>
+<a id="__codelineno-23-25" name="__codelineno-23-25" href="#__codelineno-23-25"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; right subtree -&gt; root</span>
 <a id="__codelineno-23-26" name="__codelineno-23-26" href="#__codelineno-23-26"></a><span class="w">    </span><span class="n">postOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">left</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">);</span>
 <a id="__codelineno-23-27" name="__codelineno-23-27" href="#__codelineno-23-27"></a><span class="w">    </span><span class="n">postOrder</span><span class="p">(</span><span class="n">root</span><span class="o">-&gt;</span><span class="n">right</span><span class="p">,</span><span class="w"> </span><span class="n">size</span><span class="p">);</span>
 <a id="__codelineno-23-28" name="__codelineno-23-28" href="#__codelineno-23-28"></a><span class="w">    </span><span class="n">arr</span><span class="p">[(</span><span class="o">*</span><span class="n">size</span><span class="p">)</span><span class="o">++</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">root</span><span class="o">-&gt;</span><span class="n">val</span><span class="p">;</span>
@@ -5463,7 +5540,7 @@
 <div class="highlight"><span class="filename">binary_tree_dfs.kt</span><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="cm">/* Preorder traversal */</span>
 <a id="__codelineno-24-2" name="__codelineno-24-2" href="#__codelineno-24-2"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-3" name="__codelineno-24-3" href="#__codelineno-24-3"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span>
-<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="c1">// Visit priority: root node -&gt; left subtree -&gt; right subtree</span>
+<a id="__codelineno-24-4" name="__codelineno-24-4" href="#__codelineno-24-4"></a><span class="w">    </span><span class="c1">// Visit order: root -&gt; left subtree -&gt; right subtree</span>
 <a id="__codelineno-24-5" name="__codelineno-24-5" href="#__codelineno-24-5"></a><span class="w">    </span><span class="n">list</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">_val</span><span class="p">)</span>
 <a id="__codelineno-24-6" name="__codelineno-24-6" href="#__codelineno-24-6"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">left</span><span class="p">)</span>
 <a id="__codelineno-24-7" name="__codelineno-24-7" href="#__codelineno-24-7"></a><span class="w">    </span><span class="n">preOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">right</span><span class="p">)</span>
@@ -5472,7 +5549,7 @@
 <a id="__codelineno-24-10" name="__codelineno-24-10" href="#__codelineno-24-10"></a><span class="cm">/* Inorder traversal */</span>
 <a id="__codelineno-24-11" name="__codelineno-24-11" href="#__codelineno-24-11"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">inOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-12" name="__codelineno-24-12" href="#__codelineno-24-12"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span>
-<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; root node -&gt; right subtree</span>
+<a id="__codelineno-24-13" name="__codelineno-24-13" href="#__codelineno-24-13"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; root -&gt; right subtree</span>
 <a id="__codelineno-24-14" name="__codelineno-24-14" href="#__codelineno-24-14"></a><span class="w">    </span><span class="n">inOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">left</span><span class="p">)</span>
 <a id="__codelineno-24-15" name="__codelineno-24-15" href="#__codelineno-24-15"></a><span class="w">    </span><span class="n">list</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">_val</span><span class="p">)</span>
 <a id="__codelineno-24-16" name="__codelineno-24-16" href="#__codelineno-24-16"></a><span class="w">    </span><span class="n">inOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">right</span><span class="p">)</span>
@@ -5481,7 +5558,7 @@
 <a id="__codelineno-24-19" name="__codelineno-24-19" href="#__codelineno-24-19"></a><span class="cm">/* Postorder traversal */</span>
 <a id="__codelineno-24-20" name="__codelineno-24-20" href="#__codelineno-24-20"></a><span class="kd">fun</span><span class="w"> </span><span class="nf">postOrder</span><span class="p">(</span><span class="n">root</span><span class="p">:</span><span class="w"> </span><span class="n">TreeNode?</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
 <a id="__codelineno-24-21" name="__codelineno-24-21" href="#__codelineno-24-21"></a><span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">root</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="kc">null</span><span class="p">)</span><span class="w"> </span><span class="k">return</span>
-<a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">    </span><span class="c1">// Visit priority: left subtree -&gt; right subtree -&gt; root node</span>
+<a id="__codelineno-24-22" name="__codelineno-24-22" href="#__codelineno-24-22"></a><span class="w">    </span><span class="c1">// Visit order: left subtree -&gt; right subtree -&gt; root</span>
 <a id="__codelineno-24-23" name="__codelineno-24-23" href="#__codelineno-24-23"></a><span class="w">    </span><span class="n">postOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">left</span><span class="p">)</span>
 <a id="__codelineno-24-24" name="__codelineno-24-24" href="#__codelineno-24-24"></a><span class="w">    </span><span class="n">postOrder</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">right</span><span class="p">)</span>
 <a id="__codelineno-24-25" name="__codelineno-24-25" href="#__codelineno-24-25"></a><span class="w">    </span><span class="n">list</span><span class="p">.</span><span class="na">add</span><span class="p">(</span><span class="n">root</span><span class="p">.</span><span class="na">_val</span><span class="p">)</span>
@@ -5489,31 +5566,31 @@
 </code></pre></div>
 </div>
 <div class="tabbed-block">
-<div class="highlight"><span class="filename">binary_tree_dfs.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Pre-order traversal ###</span>
+<div class="highlight"><span class="filename">binary_tree_dfs.rb</span><pre><span></span><code><a id="__codelineno-25-1" name="__codelineno-25-1" href="#__codelineno-25-1"></a><span class="c1">### Preorder traversal ###</span>
 <a id="__codelineno-25-2" name="__codelineno-25-2" href="#__codelineno-25-2"></a><span class="k">def</span><span class="w"> </span><span class="nf">pre_order</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-25-3" name="__codelineno-25-3" href="#__codelineno-25-3"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">root</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-25-4" name="__codelineno-25-4" href="#__codelineno-25-4"></a>
-<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># Visit priority: root node -&gt; left subtree -&gt; right subtree</span>
+<a id="__codelineno-25-5" name="__codelineno-25-5" href="#__codelineno-25-5"></a><span class="w">  </span><span class="c1"># Visit order: root -&gt; left subtree -&gt; right subtree</span>
 <a id="__codelineno-25-6" name="__codelineno-25-6" href="#__codelineno-25-6"></a><span class="w">  </span><span class="vg">$res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">root</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-25-7" name="__codelineno-25-7" href="#__codelineno-25-7"></a><span class="w">  </span><span class="n">pre_order</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
 <a id="__codelineno-25-8" name="__codelineno-25-8" href="#__codelineno-25-8"></a><span class="w">  </span><span class="n">pre_order</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
 <a id="__codelineno-25-9" name="__codelineno-25-9" href="#__codelineno-25-9"></a><span class="k">end</span>
 <a id="__codelineno-25-10" name="__codelineno-25-10" href="#__codelineno-25-10"></a>
-<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="c1">### In-order traversal ###</span>
+<a id="__codelineno-25-11" name="__codelineno-25-11" href="#__codelineno-25-11"></a><span class="c1">### Inorder traversal ###</span>
 <a id="__codelineno-25-12" name="__codelineno-25-12" href="#__codelineno-25-12"></a><span class="k">def</span><span class="w"> </span><span class="nf">in_order</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-25-13" name="__codelineno-25-13" href="#__codelineno-25-13"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">root</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-25-14" name="__codelineno-25-14" href="#__codelineno-25-14"></a>
-<a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">  </span><span class="c1"># Visit priority: left subtree -&gt; root node -&gt; right subtree</span>
+<a id="__codelineno-25-15" name="__codelineno-25-15" href="#__codelineno-25-15"></a><span class="w">  </span><span class="c1"># Visit order: left subtree -&gt; root -&gt; right subtree</span>
 <a id="__codelineno-25-16" name="__codelineno-25-16" href="#__codelineno-25-16"></a><span class="w">  </span><span class="n">in_order</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
 <a id="__codelineno-25-17" name="__codelineno-25-17" href="#__codelineno-25-17"></a><span class="w">  </span><span class="vg">$res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">root</span><span class="o">.</span><span class="n">val</span>
 <a id="__codelineno-25-18" name="__codelineno-25-18" href="#__codelineno-25-18"></a><span class="w">  </span><span class="n">in_order</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
 <a id="__codelineno-25-19" name="__codelineno-25-19" href="#__codelineno-25-19"></a><span class="k">end</span>
 <a id="__codelineno-25-20" name="__codelineno-25-20" href="#__codelineno-25-20"></a>
-<a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="c1">### Post-order traversal ###</span>
+<a id="__codelineno-25-21" name="__codelineno-25-21" href="#__codelineno-25-21"></a><span class="c1">### Postorder traversal ###</span>
 <a id="__codelineno-25-22" name="__codelineno-25-22" href="#__codelineno-25-22"></a><span class="k">def</span><span class="w"> </span><span class="nf">post_order</span><span class="p">(</span><span class="n">root</span><span class="p">)</span>
 <a id="__codelineno-25-23" name="__codelineno-25-23" href="#__codelineno-25-23"></a><span class="w">  </span><span class="k">return</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">root</span><span class="o">.</span><span class="n">nil?</span>
 <a id="__codelineno-25-24" name="__codelineno-25-24" href="#__codelineno-25-24"></a>
-<a id="__codelineno-25-25" name="__codelineno-25-25" href="#__codelineno-25-25"></a><span class="w">  </span><span class="c1"># Visit priority: left subtree -&gt; right subtree -&gt; root node</span>
+<a id="__codelineno-25-25" name="__codelineno-25-25" href="#__codelineno-25-25"></a><span class="w">  </span><span class="c1"># Visit order: left subtree -&gt; right subtree -&gt; root</span>
 <a id="__codelineno-25-26" name="__codelineno-25-26" href="#__codelineno-25-26"></a><span class="w">  </span><span class="n">post_order</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">left</span><span class="p">)</span>
 <a id="__codelineno-25-27" name="__codelineno-25-27" href="#__codelineno-25-27"></a><span class="w">  </span><span class="n">post_order</span><span class="p">(</span><span class="n">root</span><span class="o">.</span><span class="n">right</span><span class="p">)</span>
 <a id="__codelineno-25-28" name="__codelineno-25-28" href="#__codelineno-25-28"></a><span class="w">  </span><span class="vg">$res</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="n">root</span><span class="o">.</span><span class="n">val</span>
@@ -5522,14 +5599,19 @@
 </div>
 </div>
 </div>
+<details class="pythontutor">
+<summary>Visualize execution</summary>
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+<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Reference%20to%20the%20left%20child%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20Reference%20to%20the%20right%20child%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%3A%20recursion%22%22%22%0A%20%20%20%20%23%20Return%20None%20if%20the%20index%20is%20out%20of%20bounds%20or%20the%20corresponding%20element%20is%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20Create%20the%20current%20node%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20Recursively%20build%20the%20left%20and%20right%20subtrees%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20%2A%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22Deserialize%20a%20list%20into%20a%20binary%20tree%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20pre_order%28root%3A%20TreeNode%20%7C%20None%29%3A%0A%20%20%20%20%22%22%22Preorder%20traversal%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Visit%20order%3A%20root%20-%3E%20left%20subtree%20-%3E%20right%20subtree%0A%20%20%20%20res.append%28root.val%29%0A%20%20%20%20pre_order%28root%3Droot.left%29%0A%20%20%20%20pre_order%28root%3Droot.right%29%0A%0Adef%20in_order%28root%3A%20TreeNode%20%7C%20None%29%3A%0A%20%20%20%20%22%22%22Inorder%20traversal%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Visit%20order%3A%20left%20subtree%20-%3E%20root%20-%3E%20right%20subtree%0A%20%20%20%20in_order%28root%3Droot.left%29%0A%20%20%20%20res.append%28root.val%29%0A%20%20%20%20in_order%28root%3Droot.right%29%0A%0Adef%20post_order%28root%3A%20TreeNode%20%7C%20None%29%3A%0A%20%20%20%20%22%22%22Postorder%20traversal%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20Visit%20order%3A%20left%20subtree%20-%3E%20right%20subtree%20-%3E%20root%0A%20%20%20%20post_order%28root%3Droot.left%29%0A%20%20%20%20post_order%28root%3Droot.right%29%0A%20%20%20%20res.append%28root.val%29%0A%0A%22%22%22Driver%20code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20the%20binary%20tree%0A%20%20%20%20%23%20Use%20a%20helper%20function%20that%20builds%20a%20binary%20tree%20directly%20from%20an%20array%0A%20%20%20%20root%20%3D%20list_to_tree%28arr%3D%5B1,%202,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20Preorder%20traversal%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20pre_order%28root%29%0A%20%20%20%20print%28%22%5CnNode%20sequence%20from%20preorder%20traversal%20%3D%20%22,%20res%29%0A%0A%20%20%20%20%23%20Inorder%20traversal%0A%20%20%20%20res.clear%28%29%0A%20%20%20%20in_order%28root%29%0A%20%20%20%20print%28%22%5CnNode%20sequence%20from%20inorder%20traversal%20%3D%20%22,%20res%29%0A%0A%20%20%20%20%23%20Postorder%20traversal%0A%20%20%20%20res.clear%28%29%0A%20%20%20%20post_order%28root%29%0A%20%20%20%20print%28%22%5CnNode%20sequence%20from%20postorder%20traversal%20%3D%20%22,%20res%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=129&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">View full screen &gt;</a></div></p>
+</details>
 <div class="admonition tip">
 <p class="admonition-title">Tip</p>
-<p>Depth-first search can also be implemented iteratively, and interested readers can explore this on their own.</p>
+<p>Depth-first search can also be implemented iteratively. Interested readers can explore this approach themselves.</p>
 </div>
-<p>Figure 7-11 shows the recursive process of preorder traversal of a binary tree, which can be divided into two opposite phases: "descending" and "returning".</p>
+<p>Figure 7-11 shows the recursive process for preorder traversal, with two opposing phases: descending into calls and returning from them.</p>
 <ol>
-<li>"Descending" means making a new recursive call, during which the program visits the next node.</li>
-<li>"Returning" means the function call returns, indicating that the current node has been fully processed.</li>
+<li>Descending starts a new function call and visits the next node.</li>
+<li>Returning ends a function call, indicating that the current node has been fully processed.</li>
 </ol>
 <div class="tabbed-set tabbed-alternate" data-tabs="3:11"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">&lt;1&gt;</label><label for="__tabbed_3_2">&lt;2&gt;</label><label for="__tabbed_3_3">&lt;3&gt;</label><label for="__tabbed_3_4">&lt;4&gt;</label><label for="__tabbed_3_5">&lt;5&gt;</label><label for="__tabbed_3_6">&lt;6&gt;</label><label for="__tabbed_3_7">&lt;7&gt;</label><label for="__tabbed_3_8">&lt;8&gt;</label><label for="__tabbed_3_9">&lt;9&gt;</label><label for="__tabbed_3_10">&lt;10&gt;</label><label for="__tabbed_3_11">&lt;11&gt;</label></div>
 <div class="tabbed-content">
@@ -5570,10 +5652,10 @@
 </div>
 <p align="center"> Figure 7-11 &nbsp; The recursive process of preorder traversal </p>
 
-<h3 id="2-complexity-analysis_1">2. &nbsp; Complexity Analysis<a class="headerlink" href="#2-complexity-analysis_1" title="Permanent link">&para;</a></h3>
+<h3 id="2-complexity-analysis_1">2. &nbsp; Complexity analysis<a class="headerlink" href="#2-complexity-analysis_1" title="Permanent link">&para;</a></h3>
 <ul>
-<li><strong>Time complexity is <span class="arithmatex">\(O(n)\)</span></strong>: All nodes are visited once, using <span class="arithmatex">\(O(n)\)</span> time.</li>
-<li><strong>Space complexity is <span class="arithmatex">\(O(n)\)</span></strong>: In the worst case, i.e., the tree degenerates into a linked list, the recursion depth reaches <span class="arithmatex">\(n\)</span>, and the system occupies <span class="arithmatex">\(O(n)\)</span> stack frame space.</li>
+<li><strong>Time complexity: <span class="arithmatex">\(O(n)\)</span></strong>. Every node is visited once, taking <span class="arithmatex">\(O(n)\)</span> time.</li>
+<li><strong>Space complexity: <span class="arithmatex">\(O(n)\)</span></strong>. In the worst case, the tree degenerates into a linked list, recursion reaches depth <span class="arithmatex">\(n\)</span>, and stack frames require <span class="arithmatex">\(O(n)\)</span> space.</li>
 </ul>
 
 <!-- Source file information -->
@@ -5597,7 +5679,7 @@ aria-label="Footer"
   <a
     href="../binary_tree/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 7.1 Binary Tree"
+    aria-label="Previous: 7.1   Binary trees"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -5609,7 +5691,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        7.1 Binary Tree
+        7.1 &nbsp; Binary trees
       </div>
     </div>
   </a>
@@ -5621,7 +5703,7 @@ aria-label="Footer"
   <a
     href="../array_representation_of_tree/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 7.3 Array Representation of Binary Trees"
+    aria-label="Next: 7.3   Array representation of a binary tree"
     rel="next"
   >
     <div class="md-footer__title">
@@ -5629,7 +5711,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        7.3 Array Representation of Binary Trees
+        7.3 &nbsp; Array representation of a binary tree
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -5646,7 +5728,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
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+    <input class="md-option" data-md-color-media="none" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="teal"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
 
-      <label class="md-header__button md-icon" title="Light mode" for="__palette_0" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07zm-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13zm-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87zM19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35zm4.33-2.7 1.15-2.77 2.2 2.54zm1.15-4.96-1.14-2.78 3.34.24zM9.63 18.93l2.77 1.15-2.53 2.19z"/></svg>
       </label>
 
@@ -349,7 +349,7 @@
   <span class="md-ellipsis">
 
 
-  Before Starting
+  Foreword
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Before Starting
+  Foreword
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -419,7 +419,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
 
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-  Chapter 0. Preface
+  Chapter 0. &nbsp; Preface
 
           </label>
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@@ -461,7 +461,7 @@
   <span class="md-ellipsis">
 
 
-  0.1 About This Book
+  0.1 &nbsp; About this book
 
 
   </span>
@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
   </span>
@@ -515,7 +515,7 @@
   <span class="md-ellipsis">
 
 
-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
+  1.1 &nbsp; Algorithms are everywhere
 
 
   </span>
@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
   </span>
@@ -671,7 +671,7 @@
   <span class="md-ellipsis">
 
 
-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
 
   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
@@ -914,7 +914,7 @@
   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
@@ -1076,7 +1076,7 @@
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
   </span>
@@ -1103,7 +1103,7 @@
   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
@@ -1130,7 +1130,7 @@
   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
@@ -1157,7 +1157,7 @@
   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
   </span>
@@ -1244,7 +1244,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
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@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
   </span>
@@ -1464,7 +1464,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
   </span>
@@ -1485,7 +1485,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
@@ -1533,7 +1533,7 @@
   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
   </span>
@@ -1699,7 +1699,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1898,7 +1898,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1919,7 +1919,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1940,7 +1940,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1967,7 +1967,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1994,7 +1994,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2021,7 +2021,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2048,7 +2048,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2075,7 +2075,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2111,7 +2111,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2128,7 +2128,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2154,22 +2154,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#771-concept-review" class="md-nav__link">
+  <a href="#771-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.7.1 &nbsp; Concept Review
+        7.7.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.7.1 Concept Review">
+    <nav class="md-nav" aria-label="7.7.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-complete-full-and-perfect-binary-trees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Complete, Full, and Perfect Binary Trees
+        1. &nbsp; Complete, full, and perfect binary trees
       </span>
     </span>
   </a>
@@ -2180,7 +2180,7 @@
   <a href="#2-three-traversal-orders-for-the-same-tree" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Three Traversal Orders for the Same Tree
+        2. &nbsp; Three traversal orders for the same tree
       </span>
     </span>
   </a>
@@ -2188,10 +2188,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-compare-two-binary-search-trees" class="md-nav__link">
+  <a href="#3-comparing-two-binary-search-trees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Compare Two Binary Search Trees
+        3. &nbsp; Comparing two binary search trees
       </span>
     </span>
   </a>
@@ -2207,19 +2207,19 @@
   <a href="#772-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.7.2 &nbsp; Programming Exercises
+        7.7.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.7.2 Programming Exercises">
+    <nav class="md-nav" aria-label="7.7.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-maximum-depth-of-a-binary-tree" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Maximum Depth of a Binary Tree
+        1. &nbsp; Maximum depth of a binary tree
       </span>
     </span>
   </a>
@@ -2227,10 +2227,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-traverse-a-binary-tree-level-by-level" class="md-nav__link">
+  <a href="#2-traversing-a-binary-tree-level-by-level" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Traverse a Binary Tree Level by Level
+        2. &nbsp; Traversing a binary tree level by level
       </span>
     </span>
   </a>
@@ -2238,10 +2238,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-kth-smallest-element-in-a-binary-search-tree" class="md-nav__link">
+  <a href="#3-the-kth-smallest-element-in-a-binary-search-tree" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Kth Smallest Element in a Binary Search Tree
+        3. &nbsp; The kth smallest element in a binary search tree
       </span>
     </span>
   </a>
@@ -2313,7 +2313,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2334,7 +2334,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2355,7 +2355,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2382,7 +2382,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2409,7 +2409,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2436,7 +2436,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2463,7 +2463,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2527,7 +2527,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2548,7 +2548,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2569,7 +2569,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2596,7 +2596,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2623,7 +2623,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2650,7 +2650,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2677,7 +2677,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2745,7 +2745,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2766,7 +2766,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2787,7 +2787,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2814,7 +2814,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2841,7 +2841,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2868,7 +2868,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2895,7 +2895,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2922,7 +2922,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -3027,7 +3027,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -3048,7 +3048,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3069,7 +3069,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3096,7 +3096,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3123,7 +3123,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3150,7 +3150,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3177,7 +3177,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3204,7 +3204,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3231,7 +3231,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3258,7 +3258,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3285,7 +3285,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3312,7 +3312,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3339,7 +3339,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3366,7 +3366,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3432,7 +3432,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3453,7 +3453,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3474,7 +3474,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3501,7 +3501,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3528,7 +3528,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3555,7 +3555,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3582,7 +3582,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3609,7 +3609,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3675,7 +3675,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
@@ -3696,7 +3696,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3717,7 +3717,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3744,7 +3744,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3771,7 +3771,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3798,7 +3798,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3825,7 +3825,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3852,7 +3852,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3922,7 +3922,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3943,7 +3943,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3964,7 +3964,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3991,7 +3991,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -4018,7 +4018,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -4045,7 +4045,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -4072,7 +4072,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4099,7 +4099,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4126,7 +4126,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4153,7 +4153,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4219,7 +4219,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4240,7 +4240,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4261,7 +4261,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4288,7 +4288,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4315,7 +4315,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4342,7 +4342,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
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-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4456,7 +4456,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4477,7 +4477,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4498,7 +4498,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4525,7 +4525,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4552,7 +4552,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4636,6 +4636,78 @@
 
 
 
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@@ -4677,22 +4749,22 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#771-concept-review" class="md-nav__link">
+  <a href="#771-check-your-understanding" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.7.1 &nbsp; Concept Review
+        7.7.1 &nbsp; Check your understanding
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.7.1 Concept Review">
+    <nav class="md-nav" aria-label="7.7.1 Check your understanding">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-complete-full-and-perfect-binary-trees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Complete, Full, and Perfect Binary Trees
+        1. &nbsp; Complete, full, and perfect binary trees
       </span>
     </span>
   </a>
@@ -4703,7 +4775,7 @@
   <a href="#2-three-traversal-orders-for-the-same-tree" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Three Traversal Orders for the Same Tree
+        2. &nbsp; Three traversal orders for the same tree
       </span>
     </span>
   </a>
@@ -4711,10 +4783,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-compare-two-binary-search-trees" class="md-nav__link">
+  <a href="#3-comparing-two-binary-search-trees" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Compare Two Binary Search Trees
+        3. &nbsp; Comparing two binary search trees
       </span>
     </span>
   </a>
@@ -4730,19 +4802,19 @@
   <a href="#772-programming-exercises" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        7.7.2 &nbsp; Programming Exercises
+        7.7.2 &nbsp; Programming exercises
       </span>
     </span>
   </a>
 
-    <nav class="md-nav" aria-label="7.7.2 Programming Exercises">
+    <nav class="md-nav" aria-label="7.7.2 Programming exercises">
       <ul class="md-nav__list">
 
           <li class="md-nav__item">
   <a href="#1-maximum-depth-of-a-binary-tree" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Maximum Depth of a Binary Tree
+        1. &nbsp; Maximum depth of a binary tree
       </span>
     </span>
   </a>
@@ -4750,10 +4822,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#2-traverse-a-binary-tree-level-by-level" class="md-nav__link">
+  <a href="#2-traversing-a-binary-tree-level-by-level" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        2. &nbsp; Traverse a Binary Tree Level by Level
+        2. &nbsp; Traversing a binary tree level by level
       </span>
     </span>
   </a>
@@ -4761,10 +4833,10 @@
 </li>
 
           <li class="md-nav__item">
-  <a href="#3-kth-smallest-element-in-a-binary-search-tree" class="md-nav__link">
+  <a href="#3-the-kth-smallest-element-in-a-binary-search-tree" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        3. &nbsp; Kth Smallest Element in a Binary Search Tree
+        3. &nbsp; The kth smallest element in a binary search tree
       </span>
     </span>
   </a>
@@ -4814,9 +4886,9 @@
 
 <!-- Page content -->
 <h1 id="77-exercises">7.7 &nbsp; Exercises<a class="headerlink" href="#77-exercises" title="Permanent link">&para;</a></h1>
-<h2 id="771-concept-review">7.7.1 &nbsp; Concept Review<a class="headerlink" href="#771-concept-review" title="Permanent link">&para;</a></h2>
-<h3 id="1-complete-full-and-perfect-binary-trees">1. &nbsp; Complete, Full, and Perfect Binary Trees<a class="headerlink" href="#1-complete-full-and-perfect-binary-trees" title="Permanent link">&para;</a></h3>
-<p>The following two arrays represent binary trees in level order, where <code>None</code> marks an empty position:</p>
+<h2 id="771-check-your-understanding">7.7.1 &nbsp; Check your understanding<a class="headerlink" href="#771-check-your-understanding" title="Permanent link">&para;</a></h2>
+<h3 id="1-complete-full-and-perfect-binary-trees">1. &nbsp; Complete, full, and perfect binary trees<a class="headerlink" href="#1-complete-full-and-perfect-binary-trees" title="Permanent link">&para;</a></h3>
+<p>The following arrays represent binary trees in level order, with <code>None</code> marking an empty position:</p>
 <ul>
 <li>Tree A: <code>[1, 2, 3, 4, 5, 6]</code></li>
 <li>Tree B: <code>[1, 2, 3, None, None, 6, 7]</code></li>
@@ -4825,36 +4897,36 @@
 
 <ol>
 <li>Which tree is a complete binary tree?</li>
-<li>Which tree is a full binary tree, meaning every non-leaf node has two children?</li>
-<li>Is either tree a perfect binary tree? Explain the reason for each tree.</li>
+<li>Which is a full binary tree, meaning that every non-leaf node has two children?</li>
+<li>Is either tree a perfect binary tree? Explain for each tree.</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>Tree A is a complete binary tree. Only its lowest level is not full, and the nodes on that level occupy consecutive positions from left to right.
-    Tree B is not complete because there are empty positions on the left of the lowest level while nodes still appear on the right.</p>
+<p>Tree A is complete. Only its bottom level is not full, and the nodes occupy consecutive positions from left to right.
+    Tree B is not complete: its bottom level has empty positions on the left but still has nodes farther to the right.</p>
 </li>
 <li>
-<p>Tree B is a full binary tree: nodes 1 and 3 each have two children, and all other nodes are leaves.
-    Tree A is not full because node 3 has only one child, its left child 6.</p>
+<p>Tree B is full: nodes 1 and 3 each have two children, and all the other nodes are leaves.
+    Tree A is not full because node 3 has only its left child, 6.</p>
 </li>
 <li>
-<p>Neither tree is perfect because the lowest level of each tree is not completely filled.</p>
+<p>Neither tree is perfect, because neither has a completely filled bottom level.</p>
 </li>
 </ol>
 </details>
-<h3 id="2-three-traversal-orders-for-the-same-tree">2. &nbsp; Three Traversal Orders for the Same Tree<a class="headerlink" href="#2-three-traversal-orders-for-the-same-tree" title="Permanent link">&para;</a></h3>
-<p>Store the array <code>[1, 2, 3, 4, 5, 6, 7]</code> in level order in a complete binary tree.</p>
+<h3 id="2-three-traversal-orders-for-the-same-tree">2. &nbsp; Three traversal orders for the same tree<a class="headerlink" href="#2-three-traversal-orders-for-the-same-tree" title="Permanent link">&para;</a></h3>
+<p>Store the array <code>[1, 2, 3, 4, 5, 6, 7]</code> in a complete binary tree in level order.</p>
 <!-- numbered-subquestions -->
 
 <ol>
 <li>Draw the tree.</li>
 <li>Write its preorder, inorder, and postorder traversal sequences.</li>
-<li>In the inorder sequence, which parts of the tree correspond to the subsequences to the left and right of root node 1?</li>
+<li>In the inorder sequence, which parts of the tree do the sequences to the left and right of root node 1 correspond to?</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
 <p>The tree is:</p>
@@ -4866,17 +4938,17 @@
 </code></pre></div>
 </li>
 <li>
-<p>The preorder traversal is <code>1, 2, 4, 5, 3, 6, 7</code>;
-    the inorder traversal is <code>4, 2, 5, 1, 6, 3, 7</code>;
-    the postorder traversal is <code>4, 5, 2, 6, 7, 3, 1</code>.</p>
+<p>Preorder: <code>1, 2, 4, 5, 3, 6, 7</code>;
+    inorder: <code>4, 2, 5, 1, 6, 3, 7</code>;
+    postorder: <code>4, 5, 2, 6, 7, 3, 1</code>.</p>
 </li>
 <li>
-<p>The sequence <code>4, 2, 5</code> to the left of root node 1 is the inorder traversal of the left subtree;
-    the sequence <code>6, 3, 7</code> to its right is the inorder traversal of the right subtree.</p>
+<p>The sequence <code>4, 2, 5</code> to the left of root node 1 is the inorder traversal of the left subtree.
+    The sequence <code>6, 3, 7</code> to its right is the inorder traversal of the right subtree.</p>
 </li>
 </ol>
 </details>
-<h3 id="3-compare-two-binary-search-trees">3. &nbsp; Compare Two Binary Search Trees<a class="headerlink" href="#3-compare-two-binary-search-trees" title="Permanent link">&para;</a></h3>
+<h3 id="3-comparing-two-binary-search-trees">3. &nbsp; Comparing two binary search trees<a class="headerlink" href="#3-comparing-two-binary-search-trees" title="Permanent link">&para;</a></h3>
 <p>Insert each of the following sequences from left to right into an empty binary search tree:</p>
 <ul>
 <li>Sequence A: <code>[4, 2, 6, 1, 3, 5, 7]</code></li>
@@ -4885,64 +4957,64 @@
 <!-- numbered-subquestions -->
 
 <ol>
-<li>For each tree, write the nodes visited when searching for 7.</li>
-<li>If height is measured by the number of edges from the root node to the farthest leaf node, what is the height of each tree?</li>
-<li>Based on the first two questions, is searching for 7 equally efficient in the two trees? Explain using the trees' shapes and search paths.</li>
+<li>Write the nodes visited when searching for 7 in each tree.</li>
+<li>If height is measured by the number of edges from the root to the farthest leaf, what is each tree's height?</li>
+<li>Based on your answers, do the two trees search for 7 equally efficiently? Explain using their shapes and search paths.</li>
 </ol>
 <details class="success">
-<summary>Answer</summary>
+<summary>Reference answer</summary>
 <ol>
 <li>
-<p>In the tree built from Sequence A, the search path is <code>4 → 6 → 7</code>.
-    In the tree built from Sequence B, the search path is <code>1 → 2 → 3 → 4 → 5 → 6 → 7</code>.</p>
+<p>In the tree built from sequence A, the search path is <code>4 → 6 → 7</code>.
+    In the tree built from sequence B, it is <code>1 → 2 → 3 → 4 → 5 → 6 → 7</code>.</p>
 </li>
 <li>
-<p>Every level of the first tree is full, and its height is 2. The second tree has only right children, and its height is 6.</p>
+<p>The first tree has every level filled and a height of 2. The second tree has only right children and a height of 6.</p>
 </li>
 <li>
-<p>Searching for 7 is not equally efficient in the two trees. The insertion order changes the shape and height of a binary search tree. The search visits only 3 nodes in the first tree
-    but all 7 nodes in the second. The taller the tree, the more nodes may need to be compared along a path in the worst case.</p>
+<p>The searches for 7 are not equally efficient. Insertion order changes the shape and height of a binary search tree. Searching for 7 visits only three nodes in the first tree,
+    but must visit seven in sequence in the second. The taller the tree, the more nodes may need to be compared along a path in the worst case.</p>
 </li>
 </ol>
 </details>
-<h2 id="772-programming-exercises">7.7.2 &nbsp; Programming Exercises<a class="headerlink" href="#772-programming-exercises" title="Permanent link">&para;</a></h2>
-<h3 id="1-maximum-depth-of-a-binary-tree">1. &nbsp; Maximum Depth of a Binary Tree<a class="headerlink" href="#1-maximum-depth-of-a-binary-tree" title="Permanent link">&para;</a></h3>
+<h2 id="772-programming-exercises">7.7.2 &nbsp; Programming exercises<a class="headerlink" href="#772-programming-exercises" title="Permanent link">&para;</a></h2>
+<h3 id="1-maximum-depth-of-a-binary-tree">1. &nbsp; Maximum depth of a binary tree<a class="headerlink" href="#1-maximum-depth-of-a-binary-tree" title="Permanent link">&para;</a></h3>
 <p>You are given the root node <code>root</code> of a binary tree. Each node contains an integer value and references to its left and right children.</p>
-<p>The maximum depth is the <strong>number of nodes</strong> on the path from the root node to the farthest leaf node. Return the maximum depth of the tree; the maximum depth of an empty tree is 0.
+<p>The maximum depth is the <strong>number of nodes</strong> on the path from the root to the farthest leaf. Return the maximum depth of the tree; an empty tree has maximum depth 0.
 Use recursion.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>Depth is measured by the number of nodes in this exercise, so a tree containing only a root node has a maximum depth of 1</li>
-<li>Let the recursive function return the maximum depth of the subtree rooted at the current node</li>
-<li>Return 0 for an empty node; for a nonempty node, return max(depth(left), depth(right)) + 1</li>
+<li>Depth is counted in nodes here. A tree containing only a root has maximum depth 1.</li>
+<li>Have the recursive function return the maximum depth of the subtree rooted at the current node.</li>
+<li>Return 0 for an empty node; otherwise, return max(depth(left), depth(right)) + 1.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/maximum-depth-of-binary-tree/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
-<h3 id="2-traverse-a-binary-tree-level-by-level">2. &nbsp; Traverse a Binary Tree Level by Level<a class="headerlink" href="#2-traverse-a-binary-tree-level-by-level" title="Permanent link">&para;</a></h3>
-<p>Given the root node <code>root</code> of a binary tree, use a queue to visit all nodes level by level from top to bottom and from left to right within each level.</p>
-<p>Return a two-dimensional array: the first subarray stores the values at the root's level, the second stores the values at the next level, and so on.
-If the tree is empty, return an empty array.</p>
+<h3 id="2-traversing-a-binary-tree-level-by-level">2. &nbsp; Traversing a binary tree level by level<a class="headerlink" href="#2-traversing-a-binary-tree-level-by-level" title="Permanent link">&para;</a></h3>
+<p>Given the root node <code>root</code> of a binary tree, use a queue to visit all nodes level by level from top to bottom, visiting each level from left to right.</p>
+<p>Return a two-dimensional array: the first subarray contains the values on the root's level, the second contains those on the next level, and so on.
+Return an empty array if the tree is empty.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>Level-order traversal visits earlier enqueued nodes first, so use a queue</li>
-<li>At the beginning of each round, all nodes currently in the queue belong to the same level</li>
-<li>First record the queue's length, then remove exactly that many nodes and enqueue their children</li>
+<li>Level-order traversal visits earlier enqueued nodes first, so use a queue.</li>
+<li>At the start of each round, the nodes in the queue belong to exactly one level.</li>
+<li>Record the queue's length first, then dequeue that many nodes and enqueue their children.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/binary-tree-level-order-traversal/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
-<h3 id="3-kth-smallest-element-in-a-binary-search-tree">3. &nbsp; Kth Smallest Element in a Binary Search Tree<a class="headerlink" href="#3-kth-smallest-element-in-a-binary-search-tree" title="Permanent link">&para;</a></h3>
+<h3 id="3-the-kth-smallest-element-in-a-binary-search-tree">3. &nbsp; The kth smallest element in a binary search tree<a class="headerlink" href="#3-the-kth-smallest-element-in-a-binary-search-tree" title="Permanent link">&para;</a></h3>
 <p>A binary search tree contains <code>n</code> nodes with distinct values.
-If all node values are arranged from smallest to largest, their positions are numbered starting from 1.</p>
-<p>Given the root node <code>root</code> and an integer <code>k</code> satisfying <code>1 &lt;= k &lt;= n</code>, return the value at position <code>k</code>.
-Find the answer directly during an inorder traversal rather than collecting all node values first.</p>
+If all node values are sorted in ascending order, their positions are numbered starting from 1.</p>
+<p>Given the root node <code>root</code> and an integer <code>k</code> satisfying <code>1 &lt;= k &lt;= n</code>, return the node value in position <code>k</code>.
+Find the answer directly during inorder traversal, without first collecting all node values.</p>
 <details class="tip">
 <summary>Hints</summary>
 <ol>
-<li>An inorder traversal of a binary search tree visits node values from smallest to largest</li>
-<li>Inorder traversal processes the left subtree, the current node, and then the right subtree; increment the count when visiting the current node</li>
-<li>When the count first equals k, the current node's value is the answer, so no further traversal is needed</li>
+<li>An inorder traversal of a binary search tree visits node values in ascending order.</li>
+<li>Inorder traversal processes the left subtree, the current node, and then the right subtree. Increment a counter by 1 when visiting the current node.</li>
+<li>When the counter first reaches k, the current node's value is the answer; there is no need to continue traversing.</li>
 </ol>
 </details>
 <p><a class="rounded-button exercise-button" href="https://leetcode.com/problems/kth-smallest-element-in-a-bst/" rel="noopener noreferrer" target="_blank">LeetCode</a></p>
@@ -4968,7 +5040,7 @@ aria-label="Footer"
   <a
     href="../summary/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 7.6 Summary"
+    aria-label="Previous: 7.6   Summary"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4980,7 +5052,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        7.6 Summary
+        7.6 &nbsp; Summary
       </div>
     </div>
   </a>
@@ -4992,7 +5064,7 @@ aria-label="Footer"
   <a
     href="../../chapter_heap/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: Chapter 8. &amp;nbsp; Heap"
+    aria-label="Next: Chapter 8.   Heaps"
     rel="next"
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     <div class="md-footer__title">
@@ -5000,7 +5072,7 @@ aria-label="Footer"
         Next
       </span>
       <div class="md-ellipsis">
-        Chapter 8. &nbsp; Heap
+        Chapter 8. &nbsp; Heaps
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -5017,7 +5089,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
   <!-- Insert generated snippet here -->
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diff --git a/en/chapter_tree/index.html b/en/chapter_tree/index.html
index 8962c8d30..55c5ce7d5 100644
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@@ -6,7 +6,7 @@
       <meta charset="utf-8">
       <meta name="viewport" content="width=device-width,initial-scale=1">
 
-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
+        <meta name="description" content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click">
 
 
         <meta name="author" content="krahets">
@@ -41,7 +41,7 @@
 
 
 
-        <title>Chapter 7. Tree - Hello Algo</title>
+        <title>Chapter 7. Trees - Hello Algo</title>
 
 
 
@@ -67,8 +67,8 @@
 
 
         <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
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+        <style>:root{--md-text-font:"Noto Sans";--md-code-font:"JetBrains Mono"}</style>
 
 
 
@@ -101,7 +101,7 @@
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-        <a href="#chapter-7-tree" class="md-skip">
+        <a href="#chapter-7-trees" class="md-skip">
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 </div>
 
           </div>
@@ -154,7 +154,7 @@
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+  Foreword
 
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-  Chapter 0. Preface
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@@ -488,7 +488,7 @@
   <span class="md-ellipsis">
 
 
-  0.2 How to Use This Book
+  0.2 &nbsp; How to use this book
 
 
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@@ -515,7 +515,7 @@
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-  0.3 Summary
+  0.3 &nbsp; Summary
 
 
   </span>
@@ -575,7 +575,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 1. Encounter with Algorithms
+  Chapter 1. &nbsp; Getting to know algorithms
 
 
   </span>
@@ -596,7 +596,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 1. Encounter with Algorithms
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@@ -617,7 +617,7 @@
   <span class="md-ellipsis">
 
 
-  1.1 Algorithms Are Everywhere
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@@ -644,7 +644,7 @@
   <span class="md-ellipsis">
 
 
-  1.2 What Is an Algorithm
+  1.2 &nbsp; What is an algorithm?
 
 
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-  1.3 Summary
+  1.3 &nbsp; Summary
 
 
   </span>
@@ -737,7 +737,7 @@
   <span class="md-ellipsis">
 
 
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+  Chapter 2. &nbsp; Complexity analysis
 
 
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-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
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@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
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-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
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+  2.3 &nbsp; Time complexity
 
 
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-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
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   <span class="md-ellipsis">
 
 
-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
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@@ -980,7 +980,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
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-  Chapter 3. Data Structures
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   <span class="md-ellipsis">
 
 
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-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
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-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
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-  3.5 Summary
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-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
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-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
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@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
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-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
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-  4.5 Summary
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
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+  5.1 &nbsp; Stacks
 
 
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-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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+  5.3 &nbsp; Deques
 
 
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-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
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-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
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-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
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-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
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-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
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-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
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-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
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-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
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-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
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-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
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-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
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-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
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-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
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-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
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-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
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-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
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-  Chapter 8. Heap
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-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
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+  8.3 &nbsp; The top-k problem
 
 
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-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
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-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
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-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
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-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
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-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
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-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
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-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
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-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
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-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
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-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
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-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
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-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
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-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
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-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
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-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
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-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
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-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
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-  11.6 Merge Sort
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-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
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-  11.9 Counting Sort
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-  14.6 Edit Distance Problem
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-  14.8 Exercises
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 <!-- Page content -->
-<h1 id="chapter-7-tree">Chapter 7. &nbsp; Tree<a class="headerlink" href="#chapter-7-tree" title="Permanent link">&para;</a></h1>
-<p><img alt="Tree" class="cover-image" src="../assets/covers/chapter_tree.jpg" /></p>
+<h1 id="chapter-7-trees">Chapter 7. &nbsp; Trees<a class="headerlink" href="#chapter-7-trees" title="Permanent link">&para;</a></h1>
+<p><img alt="Trees" class="cover-image" src="../assets/covers/chapter_tree.jpg" /></p>
 <div class="admonition abstract">
 <p class="admonition-title">Abstract</p>
-<p>Towering trees are full of vitality, with deep roots, lush foliage, and sprawling branches.</p>
-<p>They offer a vivid illustration of divide-and-conquer in data structures.</p>
+<p>A towering tree brims with life, its roots deep, its leaves lush, and its branches spreading wide.</p>
+<p>It gives vivid form to the idea of dividing data to conquer a problem.</p>
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 <h2 id="chapter-contents">Chapter contents<a class="headerlink" href="#chapter-contents" title="Permanent link">&para;</a></h2>
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-<li><a href="binary_search_tree/">7.4 &nbsp; Binary Search Tree</a></li>
-<li><a href="avl_tree/">7.5 &nbsp; AVL Tree *</a></li>
+<li><a href="binary_tree/">7.1 &nbsp; Binary trees</a></li>
+<li><a href="binary_tree_traversal/">7.2 &nbsp; Traversing a binary tree</a></li>
+<li><a href="array_representation_of_tree/">7.3 &nbsp; Array representation of a binary tree</a></li>
+<li><a href="binary_search_tree/">7.4 &nbsp; Binary search trees</a></li>
+<li><a href="avl_tree/">7.5 &nbsp; AVL trees *</a></li>
 <li><a href="summary/">7.6 &nbsp; Summary</a></li>
 <li><a href="exercises/">7.7 &nbsp; Exercises</a></li>
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     class="md-footer__link md-footer__link--prev"
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     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 7.1 Binary Tree"
+    aria-label="Next: 7.1   Binary trees"
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+        7.1 &nbsp; Binary trees
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   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
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-        <meta name="description" content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code">
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   </span>
@@ -758,7 +758,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 2. Complexity Analysis
+  Chapter 2. &nbsp; Complexity analysis
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -779,7 +779,7 @@
   <span class="md-ellipsis">
 
 
-  2.1 Algorithm Efficiency Evaluation
+  2.1 &nbsp; Evaluating algorithm efficiency
 
 
   </span>
@@ -806,7 +806,7 @@
   <span class="md-ellipsis">
 
 
-  2.2 Iteration and Recursion
+  2.2 &nbsp; Iteration and recursion
 
 
   </span>
@@ -833,7 +833,7 @@
   <span class="md-ellipsis">
 
 
-  2.3 Time Complexity
+  2.3 &nbsp; Time complexity
 
 
   </span>
@@ -860,7 +860,7 @@
   <span class="md-ellipsis">
 
 
-  2.4 Space Complexity
+  2.4 &nbsp; Space complexity
 
 
   </span>
@@ -887,7 +887,7 @@
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-  2.5 Summary
+  2.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  2.6 Exercises
+  2.6 &nbsp; Exercises
 
 
   </span>
@@ -980,7 +980,7 @@
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-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
 
   </span>
@@ -1001,7 +1001,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 3. Data Structures
+  Chapter 3. &nbsp; Data structures
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1022,7 +1022,7 @@
   <span class="md-ellipsis">
 
 
-  3.1 Classification of Data Structures
+  3.1 &nbsp; Classifying data structures
 
 
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   <span class="md-ellipsis">
 
 
-  3.2 Basic Data Types
+  3.2 &nbsp; Basic data types
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.3 Number Encoding *
+  3.3 &nbsp; Number encoding *
 
 
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   <span class="md-ellipsis">
 
 
-  3.4 Character Encoding *
+  3.4 &nbsp; Character encoding *
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.5 Summary
+  3.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  3.6 Exercises
+  3.6 &nbsp; Exercises
 
 
   </span>
@@ -1223,7 +1223,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 4. Arrays and Linked Lists
+  Chapter 4. &nbsp; Arrays and linked lists
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1265,7 +1265,7 @@
   <span class="md-ellipsis">
 
 
-  4.1 Array
+  4.1 &nbsp; Arrays
 
 
   </span>
@@ -1292,7 +1292,7 @@
   <span class="md-ellipsis">
 
 
-  4.2 Linked List
+  4.2 &nbsp; Linked lists
 
 
   </span>
@@ -1319,7 +1319,7 @@
   <span class="md-ellipsis">
 
 
-  4.3 List
+  4.3 &nbsp; Lists
 
 
   </span>
@@ -1346,7 +1346,7 @@
   <span class="md-ellipsis">
 
 
-  4.4 Random-Access Memory and Cache *
+  4.4 &nbsp; Memory and cache *
 
 
   </span>
@@ -1373,7 +1373,7 @@
   <span class="md-ellipsis">
 
 
-  4.5 Summary
+  4.5 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  4.6 Exercises
+  4.6 &nbsp; Exercises
 
 
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   <span class="md-ellipsis">
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 5. Stacks and Queues
+  Chapter 5. &nbsp; Stacks and queues
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1506,7 +1506,7 @@
   <span class="md-ellipsis">
 
 
-  5.1 Stack
+  5.1 &nbsp; Stacks
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.2 Queue
+  5.2 &nbsp; Queues
 
 
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@@ -1560,7 +1560,7 @@
   <span class="md-ellipsis">
 
 
-  5.3 Deque
+  5.3 &nbsp; Deques
 
 
   </span>
@@ -1587,7 +1587,7 @@
   <span class="md-ellipsis">
 
 
-  5.4 Summary
+  5.4 &nbsp; Summary
 
 
   </span>
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   <span class="md-ellipsis">
 
 
-  5.5 Exercises
+  5.5 &nbsp; Exercises
 
 
   </span>
@@ -1678,7 +1678,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
 
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 6. Hashing
+  Chapter 6. &nbsp; Hash tables
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1720,7 +1720,7 @@
   <span class="md-ellipsis">
 
 
-  6.1 Hash Table
+  6.1 &nbsp; Hash tables
 
 
   </span>
@@ -1747,7 +1747,7 @@
   <span class="md-ellipsis">
 
 
-  6.2 Hash Collision
+  6.2 &nbsp; Hash collisions
 
 
   </span>
@@ -1774,7 +1774,7 @@
   <span class="md-ellipsis">
 
 
-  6.3 Hash Algorithm
+  6.3 &nbsp; Hash algorithms
 
 
   </span>
@@ -1801,7 +1801,7 @@
   <span class="md-ellipsis">
 
 
-  6.4 Summary
+  6.4 &nbsp; Summary
 
 
   </span>
@@ -1828,7 +1828,7 @@
   <span class="md-ellipsis">
 
 
-  6.5 Exercises
+  6.5 &nbsp; Exercises
 
 
   </span>
@@ -1898,7 +1898,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
 
   </span>
@@ -1919,7 +1919,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 7. Tree
+  Chapter 7. &nbsp; Trees
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -1940,7 +1940,7 @@
   <span class="md-ellipsis">
 
 
-  7.1 Binary Tree
+  7.1 &nbsp; Binary trees
 
 
   </span>
@@ -1967,7 +1967,7 @@
   <span class="md-ellipsis">
 
 
-  7.2 Binary Tree Traversal
+  7.2 &nbsp; Traversing a binary tree
 
 
   </span>
@@ -1994,7 +1994,7 @@
   <span class="md-ellipsis">
 
 
-  7.3 Array Representation of Binary Trees
+  7.3 &nbsp; Array representation of a binary tree
 
 
   </span>
@@ -2021,7 +2021,7 @@
   <span class="md-ellipsis">
 
 
-  7.4 Binary Search Tree
+  7.4 &nbsp; Binary search trees
 
 
   </span>
@@ -2048,7 +2048,7 @@
   <span class="md-ellipsis">
 
 
-  7.5 AVL Tree *
+  7.5 &nbsp; AVL trees *
 
 
   </span>
@@ -2084,7 +2084,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2101,7 +2101,7 @@
   <span class="md-ellipsis">
 
 
-  7.6 Summary
+  7.6 &nbsp; Summary
 
 
   </span>
@@ -2127,10 +2127,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
       </span>
     </span>
   </a>
@@ -2171,7 +2171,7 @@
   <span class="md-ellipsis">
 
 
-  7.7 Exercises
+  7.7 &nbsp; Exercises
 
 
   </span>
@@ -2235,7 +2235,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
 
   </span>
@@ -2256,7 +2256,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 8. Heap
+  Chapter 8. &nbsp; Heaps
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2277,7 +2277,7 @@
   <span class="md-ellipsis">
 
 
-  8.1 Heap
+  8.1 &nbsp; Heaps
 
 
   </span>
@@ -2304,7 +2304,7 @@
   <span class="md-ellipsis">
 
 
-  8.2 Heap Construction Operation
+  8.2 &nbsp; Building a heap
 
 
   </span>
@@ -2331,7 +2331,7 @@
   <span class="md-ellipsis">
 
 
-  8.3 Top-k Problem
+  8.3 &nbsp; The top-k problem
 
 
   </span>
@@ -2358,7 +2358,7 @@
   <span class="md-ellipsis">
 
 
-  8.4 Summary
+  8.4 &nbsp; Summary
 
 
   </span>
@@ -2385,7 +2385,7 @@
   <span class="md-ellipsis">
 
 
-  8.5 Exercises
+  8.5 &nbsp; Exercises
 
 
   </span>
@@ -2449,7 +2449,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
 
   </span>
@@ -2470,7 +2470,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 9. Graph
+  Chapter 9. &nbsp; Graphs
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2491,7 +2491,7 @@
   <span class="md-ellipsis">
 
 
-  9.1 Graph
+  9.1 &nbsp; Graphs
 
 
   </span>
@@ -2518,7 +2518,7 @@
   <span class="md-ellipsis">
 
 
-  9.2 Basic Operations on Graphs
+  9.2 &nbsp; Basic graph operations
 
 
   </span>
@@ -2545,7 +2545,7 @@
   <span class="md-ellipsis">
 
 
-  9.3 Graph Traversal
+  9.3 &nbsp; Graph traversal
 
 
   </span>
@@ -2572,7 +2572,7 @@
   <span class="md-ellipsis">
 
 
-  9.4 Summary
+  9.4 &nbsp; Summary
 
 
   </span>
@@ -2599,7 +2599,7 @@
   <span class="md-ellipsis">
 
 
-  9.5 Exercises
+  9.5 &nbsp; Exercises
 
 
   </span>
@@ -2667,7 +2667,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
 
   </span>
@@ -2688,7 +2688,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 10. Searching
+  Chapter 10. &nbsp; Searching
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2709,7 +2709,7 @@
   <span class="md-ellipsis">
 
 
-  10.1 Binary Search
+  10.1 &nbsp; Binary search
 
 
   </span>
@@ -2736,7 +2736,7 @@
   <span class="md-ellipsis">
 
 
-  10.2 Binary Search Insertion Point
+  10.2 &nbsp; Finding an insertion point with binary search
 
 
   </span>
@@ -2763,7 +2763,7 @@
   <span class="md-ellipsis">
 
 
-  10.3 Binary Search Boundaries
+  10.3 &nbsp; Finding boundaries with binary search
 
 
   </span>
@@ -2790,7 +2790,7 @@
   <span class="md-ellipsis">
 
 
-  10.4 Hash Optimization Strategy
+  10.4 &nbsp; Optimizing with hashing
 
 
   </span>
@@ -2817,7 +2817,7 @@
   <span class="md-ellipsis">
 
 
-  10.5 Searching Algorithms Revisited
+  10.5 &nbsp; Search algorithms revisited
 
 
   </span>
@@ -2844,7 +2844,7 @@
   <span class="md-ellipsis">
 
 
-  10.6 Summary
+  10.6 &nbsp; Summary
 
 
   </span>
@@ -2871,7 +2871,7 @@
   <span class="md-ellipsis">
 
 
-  10.7 Exercises
+  10.7 &nbsp; Exercises
 
 
   </span>
@@ -2949,7 +2949,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
 
   </span>
@@ -2970,7 +2970,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 11. Sorting
+  Chapter 11. &nbsp; Sorting
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -2991,7 +2991,7 @@
   <span class="md-ellipsis">
 
 
-  11.1 Sorting Algorithm
+  11.1 &nbsp; Sorting algorithms
 
 
   </span>
@@ -3018,7 +3018,7 @@
   <span class="md-ellipsis">
 
 
-  11.2 Selection Sort
+  11.2 &nbsp; Selection sort
 
 
   </span>
@@ -3045,7 +3045,7 @@
   <span class="md-ellipsis">
 
 
-  11.3 Bubble Sort
+  11.3 &nbsp; Bubble sort
 
 
   </span>
@@ -3072,7 +3072,7 @@
   <span class="md-ellipsis">
 
 
-  11.4 Insertion Sort
+  11.4 &nbsp; Insertion sort
 
 
   </span>
@@ -3099,7 +3099,7 @@
   <span class="md-ellipsis">
 
 
-  11.5 Quick Sort
+  11.5 &nbsp; Quick sort
 
 
   </span>
@@ -3126,7 +3126,7 @@
   <span class="md-ellipsis">
 
 
-  11.6 Merge Sort
+  11.6 &nbsp; Merge sort
 
 
   </span>
@@ -3153,7 +3153,7 @@
   <span class="md-ellipsis">
 
 
-  11.7 Heap Sort
+  11.7 &nbsp; Heap sort
 
 
   </span>
@@ -3180,7 +3180,7 @@
   <span class="md-ellipsis">
 
 
-  11.8 Bucket Sort
+  11.8 &nbsp; Bucket sort
 
 
   </span>
@@ -3207,7 +3207,7 @@
   <span class="md-ellipsis">
 
 
-  11.9 Counting Sort
+  11.9 &nbsp; Counting sort
 
 
   </span>
@@ -3234,7 +3234,7 @@
   <span class="md-ellipsis">
 
 
-  11.10 Radix Sort
+  11.10 &nbsp; Radix sort
 
 
   </span>
@@ -3261,7 +3261,7 @@
   <span class="md-ellipsis">
 
 
-  11.11 Summary
+  11.11 &nbsp; Summary
 
 
   </span>
@@ -3288,7 +3288,7 @@
   <span class="md-ellipsis">
 
 
-  11.12 Exercises
+  11.12 &nbsp; Exercises
 
 
   </span>
@@ -3354,7 +3354,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
 
   </span>
@@ -3375,7 +3375,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 12. Divide and Conquer
+  Chapter 12. &nbsp; Divide and conquer
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3396,7 +3396,7 @@
   <span class="md-ellipsis">
 
 
-  12.1 Divide and Conquer Algorithms
+  12.1 &nbsp; Divide and conquer
 
 
   </span>
@@ -3423,7 +3423,7 @@
   <span class="md-ellipsis">
 
 
-  12.2 Divide and Conquer Search Strategy
+  12.2 &nbsp; Divide-and-conquer search
 
 
   </span>
@@ -3450,7 +3450,7 @@
   <span class="md-ellipsis">
 
 
-  12.3 Building a Binary Tree Problem
+  12.3 &nbsp; Building a binary tree
 
 
   </span>
@@ -3477,7 +3477,7 @@
   <span class="md-ellipsis">
 
 
-  12.4 Tower of Hanoi Problem
+  12.4 &nbsp; Tower of Hanoi
 
 
   </span>
@@ -3504,7 +3504,7 @@
   <span class="md-ellipsis">
 
 
-  12.5 Summary
+  12.5 &nbsp; Summary
 
 
   </span>
@@ -3531,7 +3531,7 @@
   <span class="md-ellipsis">
 
 
-  12.6 Exercises
+  12.6 &nbsp; Exercises
 
 
   </span>
@@ -3597,7 +3597,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
 
   </span>
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             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 13. Backtracking
+  Chapter 13. &nbsp; Backtracking
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3639,7 +3639,7 @@
   <span class="md-ellipsis">
 
 
-  13.1 Backtracking Algorithm
+  13.1 &nbsp; Backtracking algorithms
 
 
   </span>
@@ -3666,7 +3666,7 @@
   <span class="md-ellipsis">
 
 
-  13.2 Permutations Problem
+  13.2 &nbsp; Permutations problem
 
 
   </span>
@@ -3693,7 +3693,7 @@
   <span class="md-ellipsis">
 
 
-  13.3 Subset-Sum Problem
+  13.3 &nbsp; Subset sum problem
 
 
   </span>
@@ -3720,7 +3720,7 @@
   <span class="md-ellipsis">
 
 
-  13.4 N-Queens Problem
+  13.4 &nbsp; N-queens problem
 
 
   </span>
@@ -3747,7 +3747,7 @@
   <span class="md-ellipsis">
 
 
-  13.5 Summary
+  13.5 &nbsp; Summary
 
 
   </span>
@@ -3774,7 +3774,7 @@
   <span class="md-ellipsis">
 
 
-  13.6 Exercises
+  13.6 &nbsp; Exercises
 
 
   </span>
@@ -3844,7 +3844,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
 
   </span>
@@ -3865,7 +3865,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 14. Dynamic Programming
+  Chapter 14. &nbsp; Dynamic programming
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -3886,7 +3886,7 @@
   <span class="md-ellipsis">
 
 
-  14.1 Introduction to Dynamic Programming
+  14.1 &nbsp; Introduction to dynamic programming
 
 
   </span>
@@ -3913,7 +3913,7 @@
   <span class="md-ellipsis">
 
 
-  14.2 Characteristics of Dynamic Programming Problems
+  14.2 &nbsp; Properties of dynamic programming problems
 
 
   </span>
@@ -3940,7 +3940,7 @@
   <span class="md-ellipsis">
 
 
-  14.3 Dynamic Programming Problem-Solving Approach
+  14.3 &nbsp; Solving dynamic programming problems
 
 
   </span>
@@ -3967,7 +3967,7 @@
   <span class="md-ellipsis">
 
 
-  14.4 0-1 Knapsack Problem
+  14.4 &nbsp; 0-1 knapsack problem
 
 
   </span>
@@ -3994,7 +3994,7 @@
   <span class="md-ellipsis">
 
 
-  14.5 Unbounded Knapsack Problem
+  14.5 &nbsp; Unbounded knapsack problem
 
 
   </span>
@@ -4021,7 +4021,7 @@
   <span class="md-ellipsis">
 
 
-  14.6 Edit Distance Problem
+  14.6 &nbsp; Edit distance problem
 
 
   </span>
@@ -4048,7 +4048,7 @@
   <span class="md-ellipsis">
 
 
-  14.7 Summary
+  14.7 &nbsp; Summary
 
 
   </span>
@@ -4075,7 +4075,7 @@
   <span class="md-ellipsis">
 
 
-  14.8 Exercises
+  14.8 &nbsp; Exercises
 
 
   </span>
@@ -4141,7 +4141,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
 
   </span>
@@ -4162,7 +4162,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 15. Greedy
+  Chapter 15. &nbsp; Greedy algorithms
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4183,7 +4183,7 @@
   <span class="md-ellipsis">
 
 
-  15.1 Greedy Algorithm
+  15.1 &nbsp; Greedy algorithms
 
 
   </span>
@@ -4210,7 +4210,7 @@
   <span class="md-ellipsis">
 
 
-  15.2 Fractional Knapsack Problem
+  15.2 &nbsp; Fractional knapsack problem
 
 
   </span>
@@ -4237,7 +4237,7 @@
   <span class="md-ellipsis">
 
 
-  15.3 Maximum Capacity Problem
+  15.3 &nbsp; Maximum capacity problem
 
 
   </span>
@@ -4264,7 +4264,7 @@
   <span class="md-ellipsis">
 
 
-  15.4 Maximum Product Cutting Problem
+  15.4 &nbsp; Maximum product cutting problem
 
 
   </span>
@@ -4291,7 +4291,7 @@
   <span class="md-ellipsis">
 
 
-  15.5 Summary
+  15.5 &nbsp; Summary
 
 
   </span>
@@ -4318,7 +4318,7 @@
   <span class="md-ellipsis">
 
 
-  15.6 Exercises
+  15.6 &nbsp; Exercises
 
 
   </span>
@@ -4378,7 +4378,7 @@
   <span class="md-ellipsis">
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
 
   </span>
@@ -4399,7 +4399,7 @@
             <span class="md-nav__icon md-icon"></span>
 
 
-  Chapter 16. Appendix
+  Chapter 16. &nbsp; Appendix
 
           </label>
           <ul class="md-nav__list" data-md-scrollfix>
@@ -4420,7 +4420,7 @@
   <span class="md-ellipsis">
 
 
-  16.1 Programming Environment Installation
+  16.1 &nbsp; Setting up a programming environment
 
 
   </span>
@@ -4447,7 +4447,7 @@
   <span class="md-ellipsis">
 
 
-  16.2 Contributing Together
+  16.2 &nbsp; Contribute to the book
 
 
   </span>
@@ -4474,7 +4474,7 @@
   <span class="md-ellipsis">
 
 
-  16.3 Glossary
+  16.3 &nbsp; Glossary
 
 
   </span>
@@ -4558,6 +4558,78 @@
 
 
 
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+
+            </a>
+
+          </div>
+
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_20_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_20">
+            <span class="md-nav__icon md-icon"></span>
+
+
+  Print edition (Chinese)
+
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+
+
+
+          </ul>
+        </nav>
+
+    </li>
+
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 </nav>
                   </div>
@@ -4599,10 +4671,10 @@
     <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
 
         <li class="md-nav__item">
-  <a href="#1-key-review" class="md-nav__link">
+  <a href="#1-key-takeaways" class="md-nav__link">
     <span class="md-ellipsis">
       <span class="md-typeset">
-        1. &nbsp; Key Review
+        1. &nbsp; Key takeaways
       </span>
     </span>
   </a>
@@ -4658,42 +4730,42 @@
 
 <!-- Page content -->
 <h1 id="76-summary">7.6 &nbsp; Summary<a class="headerlink" href="#76-summary" title="Permanent link">&para;</a></h1>
-<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
+<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
 <ul>
-<li>A binary tree is a non-linear data structure that embodies the divide-and-conquer logic of splitting into two. Each binary tree node contains a value and two pointers, which point to its left and right child nodes.</li>
-<li>For a certain node in a binary tree, the tree formed by its left (right) child node and all nodes below is called the left (right) subtree of that node.</li>
-<li>Related terminology of binary trees includes root node, leaf node, level, degree, edge, height, and depth.</li>
-<li>The initialization, node insertion, and node removal operations of binary trees are similar to those of linked lists.</li>
-<li>Common types of binary trees include perfect binary trees, complete binary trees, full binary trees, and balanced binary trees. A perfect binary tree is the ideal form, while a linked list represents the worst degenerate case.</li>
-<li>A binary tree can be represented using an array by arranging node values and empty slots in level-order traversal sequence, and implementing pointers based on the index mapping relationship between parent and child nodes.</li>
-<li>Level-order traversal of a binary tree is a breadth-first search method that proceeds level by level, typically implemented using a queue.</li>
-<li>Preorder, inorder, and postorder traversals all belong to depth-first search, which proceeds by going as deep as possible before backtracking, typically using recursion.</li>
-<li>A binary search tree is an efficient data structure for element searching, with search, insertion, and removal operations all having time complexity of <span class="arithmatex">\(O(\log n)\)</span>. When a binary search tree degenerates into a linked list, all time complexities degrade to <span class="arithmatex">\(O(n)\)</span>.</li>
-<li>An AVL tree, also known as a balanced binary search tree, ensures the tree remains balanced after continuous node insertions and removals through rotation operations.</li>
-<li>Rotation operations in AVL trees include right rotation, left rotation, right rotation followed by left rotation, and left rotation followed by right rotation. After inserting or removing nodes, AVL trees perform rotations from bottom to top to restore balance.</li>
+<li>A binary tree is a nonlinear data structure that embodies the divide-and-conquer idea of splitting in two. Each node contains a value and two pointers, one to each child.</li>
+<li>A node's left or right child, together with everything below it, forms its left or right subtree.</li>
+<li>Tree terminology includes root, leaf, level, degree, edge, height, and depth.</li>
+<li>Initializing a binary tree and inserting or deleting nodes resemble linked-list operations.</li>
+<li>Common types include perfect, complete, full, and balanced binary trees. A perfect tree is the ideal form; a linked list is the worst degenerate form.</li>
+<li>A binary tree can be represented by an array of node values and empty positions in level order. Index mappings between parents and children replace pointers.</li>
+<li>Level-order traversal is breadth-first search: it expands outward one layer at a time and usually uses a queue.</li>
+<li>Preorder, inorder, and postorder traversal are depth-first search: they follow paths to their ends before backtracking and usually use recursion.</li>
+<li>A binary search tree supports efficient lookup, insertion, and deletion in <span class="arithmatex">\(O(\log n)\)</span> time. If it degenerates into a linked list, all these operations degrade to <span class="arithmatex">\(O(n)\)</span>.</li>
+<li>An AVL tree is a balanced binary search tree that uses rotations to maintain balance through insertions and deletions.</li>
+<li>AVL rotations include right rotation, left rotation, right-then-left rotation, and left-then-right rotation. After an insertion or deletion, rotations are performed from the bottom up to restore balance.</li>
 </ul>
 <h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
-<p><strong>Q</strong>: For a binary tree with only one node, are both the height of the tree and the depth of the root node <span class="arithmatex">\(0\)</span>?</p>
-<p>Yes, because height and depth are typically defined as the number of edges on the path.</p>
-<p><strong>Q</strong>: The insertion and removal in a binary tree are generally accomplished by a set of operations. What does "a set of operations" refer to here? Does it imply releasing the resources of the child nodes?</p>
-<p>Taking the binary search tree as an example, the operation of removing a node needs to be handled in three different scenarios, each requiring multiple steps of node operations.</p>
-<p><strong>Q</strong>: Why does DFS traversal of binary trees have three orders: preorder, inorder, and postorder, and what are their uses?</p>
-<p>Similar to forward and reverse traversal of arrays, preorder, inorder, and postorder traversals are three methods of binary tree traversal that allow us to obtain a traversal result in a specific order. For example, in a binary search tree, since nodes satisfy the relationship <code>left child node value &lt; root node value &lt; right child node value</code>, we only need to traverse the tree with the priority of "left <span class="arithmatex">\(\rightarrow\)</span> root <span class="arithmatex">\(\rightarrow\)</span> right" to obtain an ordered node sequence.</p>
-<p><strong>Q</strong>: In a right rotation operation handling the relationship between unbalanced nodes <code>node</code>, <code>child</code>, and <code>grand_child</code>, doesn't the connection between <code>node</code> and its parent node get lost after the right rotation?</p>
-<p>We need to view this problem from a recursive perspective. The right rotation operation <code>right_rotate(root)</code> passes in the root node of the subtree and eventually returns the root node of the subtree after rotation with <code>return child</code>. The connection between the subtree's root node and its parent node is completed after the function returns, which is not within the maintenance scope of the right rotation operation.</p>
-<p><strong>Q</strong>: In C++, functions are divided into <code>private</code> and <code>public</code> sections. What considerations are there for this? Why are the <code>height()</code> function and the <code>updateHeight()</code> function placed in <code>public</code> and <code>private</code>, respectively?</p>
-<p>It mainly depends on the method's usage scope. If a method is only used within the class, then it is designed as <code>private</code>. For example, calling <code>updateHeight()</code> alone by the user makes no sense, as it is only a step in insertion or removal operations. However, <code>height()</code> is used to access node height, similar to <code>vector.size()</code>, so it is set to <code>public</code> for ease of use.</p>
-<p><strong>Q</strong>: How do you build a binary search tree from a set of input data? Is the choice of root node very important?</p>
-<p>Yes, the method for building a tree is provided in the <code>build_tree()</code> method in the binary search tree code. As for the choice of root node, we typically sort the input data, then select the middle element as the root node, and recursively build the left and right subtrees. This approach maximizes the tree's balance.</p>
-<p><strong>Q</strong>: In Java, do you always have to use the <code>equals()</code> method for string comparison?</p>
-<p>In Java, for primitive data types, <code>==</code> is used to compare whether the values of two variables are equal. For reference types, the working principles of the two symbols are different.</p>
+<p><strong>Q</strong>: For a binary tree with only one node, are both the tree's height and the root's depth <span class="arithmatex">\(0\)</span>?</p>
+<p>Yes. Height and depth are usually defined by the number of edges traversed.</p>
+<p><strong>Q</strong>: The text says binary-tree insertion and deletion usually involve a sequence of coordinated operations. What does that mean? Does it refer to releasing resources held by child nodes?</p>
+<p>For example, deleting a node from a binary search tree has three cases, and each case requires several steps involving nodes.</p>
+<p><strong>Q</strong>: Why does depth-first traversal have preorder, inorder, and postorder variants? What is each used for?</p>
+<p>Just as an array can be traversed forward or backward, these are three ways to visit a binary tree and obtain a particular ordering. In a binary search tree, values satisfy <code>left child value &lt; root value &lt; right child value</code>, so visiting “left <span class="arithmatex">\(\rightarrow\)</span> root <span class="arithmatex">\(\rightarrow\)</span> right” produces a sorted node sequence.</p>
+<p><strong>Q</strong>: A right rotation adjusts <code>node</code>, <code>child</code>, and <code>grand_child</code>. What happens to the link between <code>node</code> and its parent? Doesn't the rotation break it?</p>
+<p>View the operation recursively. <code>right_rotate(root)</code> receives a subtree's root and ends with <code>return child</code>, returning the new root of that subtree. The connection to its parent is updated after the function returns; it is outside the rotation function's responsibility.</p>
+<p><strong>Q</strong>: In C++, how do you decide which methods are <code>private</code> and which are <code>public</code>? Why is <code>height()</code> public but <code>updateHeight()</code> private?</p>
+<p>The main consideration is where the method is used. A method used only within the class should be <code>private</code>. Calling <code>updateHeight()</code> alone is not meaningful to a user; it is just one step of insertion or deletion. In contrast, <code>height()</code> retrieves a node's height, much like <code>vector.size()</code>, so it is <code>public</code> for convenient access.</p>
+<p><strong>Q</strong>: How do you build a binary search tree from input data? Is choosing the root important?</p>
+<p>Yes. The construction is provided by the <code>build_tree()</code> method in the binary-search-tree code. Usually, we sort the input, choose its middle element as the root, and recursively construct the left and right subtrees. This keeps the tree as balanced as possible.</p>
+<p><strong>Q</strong>: Must strings in Java always be compared with <code>equals()</code>?</p>
+<p>For primitive types in Java, <code>==</code> compares variable values. For reference types, the two forms work differently.</p>
 <ul>
-<li><code>==</code>: Used to compare whether two variables point to the same object, i.e., whether their positions in memory are the same.</li>
-<li><code>equals()</code>: Used to compare whether the values of two objects are equal.</li>
+<li><code>==</code> compares whether two variables refer to the same object, meaning the same location in memory.</li>
+<li><code>equals()</code> compares whether two objects have equal values.</li>
 </ul>
-<p>Therefore, if we want to compare values, we should use <code>equals()</code>. However, strings initialized via <code>String a = "hi"; String b = "hi";</code> are stored in the string constant pool and point to the same object, so <code>a == b</code> can also be used to compare the contents of the two strings.</p>
-<p><strong>Q</strong>: Before reaching the bottom level, is the number of nodes in the queue <span class="arithmatex">\(2^h\)</span> in breadth-first traversal?</p>
-<p>Yes, for example, a full binary tree with height <span class="arithmatex">\(h = 2\)</span> has a total of <span class="arithmatex">\(n = 7\)</span> nodes, then the bottom level has <span class="arithmatex">\(4 = 2^h = (n + 1) / 2\)</span> nodes.</p>
+<p>Use <code>equals()</code> when comparing values. However, strings initialized with <code>String a = "hi"; String b = "hi";</code> are stored in the string pool and refer to the same object, so <code>a == b</code> also works for these strings.</p>
+<p><strong>Q</strong>: Just before breadth-first traversal reaches the lowest level, does the queue contain <span class="arithmatex">\(2^h\)</span> nodes?</p>
+<p>Yes. For example, a perfect binary tree of height <span class="arithmatex">\(h = 2\)</span> has <span class="arithmatex">\(n = 7\)</span> nodes, with <span class="arithmatex">\(4 = 2^h = (n + 1) / 2\)</span> nodes on its lowest level.</p>
 
 <!-- Source file information -->
 
@@ -4716,7 +4788,7 @@ aria-label="Footer"
   <a
     href="../avl_tree/"
     class="md-footer__link md-footer__link--prev"
-    aria-label="Previous: 7.5 AVL Tree *"
+    aria-label="Previous: 7.5   AVL trees *"
     rel="prev"
   >
     <div class="md-footer__button md-icon">
@@ -4728,7 +4800,7 @@ aria-label="Footer"
         Previous
       </span>
       <div class="md-ellipsis">
-        7.5 AVL Tree *
+        7.5 &nbsp; AVL trees *
       </div>
     </div>
   </a>
@@ -4740,7 +4812,7 @@ aria-label="Footer"
   <a
     href="../exercises/"
     class="md-footer__link md-footer__link--next"
-    aria-label="Next: 7.7 Exercises"
+    aria-label="Next: 7.7   Exercises"
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   >
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         Next
       </span>
       <div class="md-ellipsis">
-        7.7 Exercises
+        7.7 &nbsp; Exercises
       </div>
     </div>
     <div class="md-footer__button md-icon">
@@ -4765,7 +4837,7 @@ aria-label="Footer"
   <!-- Check-in button above comments -->
 
 
-  <h5 align="center" id="__comments">Feel free to drop your insights, questions or suggestions</h5>
+  <h5 align="center" id="__comments">Share your insights, questions, or suggestions in the comments</h5>
 
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   <head>
     <meta charset="utf-8"/>
     <meta content="width=device-width,initial-scale=1" name="viewport"/>
-    <meta content="Data structures and algorithms tutorial with animated illustrations and ready-to-run code" name="description"/>
+    <meta content="A data structures and algorithms tutorial with animated illustrations and code you can run with one click" name="description"/>
     <meta content="krahets" name="author"/>
     <link href="https://www.hello-algo.com/en/" rel="canonical"/>
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@@ -137,98 +137,98 @@
           </div>
         </div>
         <div class="hero-div hero-div--en">
-          <img alt="" src="../assets/hero/ground.png" style="position: absolute; width: auto; height: 26.445%; left: 28.211%; top: 54.145%;"/>
-          <img alt="" src="../assets/hero/links.png" style="position: absolute; width: auto; height: 78.751%; left: 10.545%; top: 7.326%;"/>
+          <img alt="" src="assets/hero/ground.png" style="position: absolute; width: auto; height: 26.445%; left: 28.211%; top: 54.145%;"/>
+          <img alt="" src="assets/hero/links.png" style="position: absolute; width: auto; height: 78.751%; left: 10.545%; top: 7.326%;"/>
           <a href="chapter_introduction/">
-            <img alt="" src="../assets/hero/astronaut.png" style="height: 46.673%; left: 35.413%; top: 24.343%;"/>
-            <span style="left: 52.744%; top: 22.319%;">
-              Encounter with Algos
+            <img alt="" src="assets/hero/astronaut.png" style="height: 46.673%; left: 35.413%; top: 24.343%;"/>
+            <span style="left: 52.244%; top: 20.919%;">
+              Getting to know algorithms
             </span>
           </a>
           <a href="chapter_computational_complexity/">
-            <img alt="" src="../assets/hero/chapter_computational_complexity.png" style="height: 12.347%; left: 36.267%; top: 37.653%;"/>
-            <span style="left: 36.244%; top: 34.919%;">
-              Complexity Analysis
+            <img alt="" src="assets/hero/chapter_computational_complexity.png" style="height: 12.347%; left: 36.267%; top: 37.653%;"/>
+            <span style="left: 39.244%; top: 33.919%;">
+              Complexity
             </span>
           </a>
           <a href="chapter_array_and_linkedlist/">
-            <img alt="" src="../assets/hero/chapter_array_and_linkedlist.png" style="height: 22.242%; left: 73.242%; top: 52.481%;"/>
-            <span style="left: 85.897%; top: 76.259%;">
-              Array and Linked List
+            <img alt="" src="assets/hero/chapter_array_and_linkedlist.png" style="height: 22.242%; left: 73.242%; top: 52.481%;"/>
+            <span style="left: 90.897%; top: 76.259%;">
+              Arrays and linked lists
             </span>
           </a>
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-            <span style="left: 80.071%; top: 88.25%;">
-              Stack and Queue
+            <img alt="" src="assets/hero/chapter_stack_and_queue.png" style="height: 14.302%; left: 62.646%; top: 77.875%;"/>
+            <span style="left: 77.571%; top: 91.25%;">
+              Stacks and queues
             </span>
           </a>
           <a href="chapter_hashing/">
-            <img alt="" src="../assets/hero/chapter_hashing.png" style="height: 15.266%; left: 63.281%; top: 27.933%;"/>
-            <span style="left: 68.862%; top: 45.792%;">
-              Hashing
+            <img alt="" src="assets/hero/chapter_hashing.png" style="height: 15.266%; left: 63.281%; top: 27.933%;"/>
+            <span style="left: 68.862%; top: 46.292%;">
+              Hash tables
             </span>
           </a>
           <a href="chapter_tree/">
-            <img alt="" src="../assets/hero/chapter_tree.png" style="height: 19.615%; left: 80.137%; top: 26.678%;"/>
+            <img alt="" src="assets/hero/chapter_tree.png" style="height: 19.615%; left: 80.137%; top: 26.678%;"/>
             <span style="left: 96.159%; top: 44.8%;">
-              Tree
+              Trees
             </span>
           </a>
           <a href="chapter_heap/">
-            <img alt="" src="../assets/hero/chapter_heap.png" style="height: 10.566%; left: 77.226%; top: 11.559%;"/>
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             <span style="left: 88.103%; top: 15.422%;">
-              Heap
+              Heaps
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           <a href="chapter_graph/">
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+            <img alt="" src="assets/hero/chapter_graph.png" style="height: 16.112%; left: 51.854%; top: 5.575%;"/>
             <span style="left: 71.195%; top: 6.503%;">
-              Graph
+              Graphs
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           </a>
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               Searching
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+            <img alt="" src="assets/hero/chapter_sorting.png" style="height: 9.574%; left: 25.409%; top: 40.747%;"/>
+            <span style="left: 28.805%; top: 53.808%;">
               Sorting
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-            <span style="left: 29.42%; top: 6.679%;">
-              Divide and Conquer
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+            <span style="left: 31.42%; top: 8.679%;">
+              Divide and conquer
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-            <span style="left: 8.742%; top: 50.113%;">
+            <img alt="" src="assets/hero/chapter_backtracking.png" style="height: 17.338%; left: 4.875%; top: 32.925%;"/>
+            <span style="left: 4.742%; top: 50.113%;">
               Backtracking
             </span>
           </a>
           <a href="chapter_dynamic_programming/">
-            <img alt="" src="../assets/hero/chapter_dynamic_programming.png" style="height: 15.47%; left: 9.406%; top: 57.472%;"/>
-            <span style="left: 14.561%; top: 75.351%;">
-              Dynamic Programming
+            <img alt="" src="assets/hero/chapter_dynamic_programming.png" style="height: 15.47%; left: 9.406%; top: 57.472%;"/>
+            <span style="left: 8.561%; top: 75.351%;">
+              Dynamic programming
             </span>
           </a>
           <a href="chapter_greedy/">
-            <img alt="" src="../assets/hero/chapter_greedy.png" style="height: 14.127%; left: 23.132%; top: 75.803%;"/>
+            <img alt="" src="assets/hero/chapter_greedy.png" style="height: 14.127%; left: 23.132%; top: 75.803%;"/>
             <span style="left: 21.619%; top: 86.85%;">
-              Greedy
+              Greedy algorithms
             </span>
           </a>
         </div>
         <div class="heading-div">
-          <img alt="" src="https://readme-typing-svg.demolab.com/?font=Roboto&amp;weight=400&amp;duration=3500&amp;pause=2000&amp;color=FFF&amp;center=true&amp;vCenter=true&amp;random=false&amp;width=200&amp;lines=Hello%2C+Algo!" style="height: min(9vh, 12vw);"/>
+          <img alt="" src="https://readme-typing-svg.demolab.com/?font=Noto+Sans&amp;weight=400&amp;duration=3500&amp;pause=2000&amp;color=FFF&amp;center=true&amp;vCenter=true&amp;random=false&amp;width=200&amp;lines=Hello%2C+Algo+!" style="height: min(9vh, 12vw);"/>
           <div style="pointer-events: auto;">
             <p style="margin-top: max(-1vh, -2vw); margin-bottom: min(2vh, 3.5vw);">
-              Data structures and algorithms tutorial with animated illustrations and ready-to-run code
+              A data structures and algorithms tutorial with animated illustrations and code you can run with one click
             </p>
             <a class="rounded-button" href="chapter_hello_algo/">
               <svg viewbox="0 0 512 512" xmlns="http://www.w3.org/2000/svg">
@@ -237,7 +237,7 @@
                 </path>
               </svg>
               <span>
-                Get Started
+                Start reading
               </span>
             </a>
             <a class="rounded-button" href="https://github.com/krahets/hello-algo">
@@ -261,16 +261,32 @@
       <section class="home-div" data-md-color-primary="grey" data-md-color-scheme="slate">
         <div class="section-content">
           <div class="reading-media">
+            <div class="media-block">
+              <a href="chapter_paperbook/">
+                <div style="height: 8.17%;">
+                </div>
+                <img alt="Cover of the Simplified Chinese print edition" class="device-on-hover" src="assets/hero/cover_render.png" style="height: 66.83%;"/>
+                <div class="text-button">
+                  <span>
+                    Print edition (Chinese)
+                  </span>
+                  <svg viewbox="0 0 320 512" xmlns="http://www.w3.org/2000/svg">
+                    <path d="M278.6 233.4c12.5 12.5 12.5 32.8 0 45.3l-160 160c-12.5 12.5-32.8 12.5-45.3 0s-12.5-32.8 0-45.3L210.7 256 73.4 118.6c-12.5-12.5-12.5-32.8 0-45.3s32.8-12.5 45.3 0l160 160z">
+                    </path>
+                  </svg>
+                </div>
+              </a>
+            </div>
             <div class="media-block">
               <a href="chapter_hello_algo/">
                 <div style="height: 4.34%;">
                 </div>
-                <img alt="" class="device-on-hover" src="../assets/hero/web_mac_iphone.png" style="height: 66.31%;"/>
+                <img alt="The English web edition" class="device-on-hover" src="assets/hero/web_mac_iphone.png" style="height: 66.31%;"/>
                 <div style="height: 4.34%;">
                 </div>
                 <div class="text-button">
                   <span>
-                    Read Online
+                    Read online
                   </span>
                   <svg viewbox="0 0 320 512" xmlns="http://www.w3.org/2000/svg">
                     <path d="M278.6 233.4c12.5 12.5 12.5 32.8 0 45.3l-160 160c-12.5 12.5-32.8 12.5-45.3 0s-12.5-32.8 0-45.3L210.7 256 73.4 118.6c-12.5-12.5-12.5-32.8 0-45.3s32.8-12.5 45.3 0l160 160z">
@@ -281,7 +297,7 @@
             </div>
             <div class="media-block">
               <a href="https://github.com/krahets/hello-algo/releases">
-                <img alt="" class="device-on-hover" src="../assets/hero/pdf_ipad.png" style="height: 75%;"/>
+                <img alt="Cover of the published English PDF, version 1.3.0" class="device-on-hover" src="assets/hero/pdf_ipad.png" style="height: 75%;"/>
                 <div class="text-button">
                   <span>
                     Download PDF/EPUB
@@ -295,7 +311,7 @@
             </div>
           </div>
           <p style="margin-top: 2em;">
-            500 animated illustrations, 14 programming languages, and 3,000 community Q&amp;As to help you get started with data structures and algorithms.
+            Get started with data structures and algorithms through 500 animated illustrations, code in 14 programming languages, and 3,000 community questions and answers
           </p>
         </div>
       </section>
@@ -307,18 +323,18 @@
           <div class="intro-container endor-container">
             <div class="intro-text endor-text">
               <p style="margin-bottom: 0;">
-                “An easy-to-understand introduction to data structures and algorithms that encourages readers to engage both mind and hands. Highly recommended for beginners!”
+                “An accessible introduction to data structures and algorithms that encourages readers to learn by thinking and doing. Highly recommended for anyone new to algorithms.”
               </p>
               <p style="font-weight: bold;">
-                —— Junhui Deng, Professor, Department of Computer Science and Technology, Tsinghua University
+                — Junhui Deng, Professor, Department of Computer Science, Tsinghua University
               </p>
             </div>
             <div class="intro-text endor-text">
               <p style="margin-bottom: 0;">
-                “If I had 'Hello Algo' when I was learning data structures and algorithms, it would have been 10 times easier!”
+                “If I had had Hello Algo when I was learning data structures and algorithms, it would have been ten times easier!”
               </p>
               <p style="font-weight: bold;">
-                —— Mu Li, Senior Principal Scientist, Amazon
+                — Mu Li, Senior Principal Scientist, Amazon
               </p>
             </div>
           </div>
@@ -335,11 +351,11 @@
                     </path>
                   </svg>
                   <h3 style="margin-left: 0.5em;">
-                    Animated Illustrations
+                    Animated illustrations
                   </h3>
                 </div>
                 <p style="margin: 0;">
-                  It’s crafted for a smooth learning experience.
+                  Clear explanations and a gradual learning curve
                 </p>
                 <p class="intro-quote">
                   "A picture is worth a thousand words."
@@ -360,11 +376,11 @@
                     </path>
                   </svg>
                   <h3 style="margin-left: 0.5em;">
-                    One-Click Runnable Code
+                    Run code with one click
                   </h3>
                 </div>
                 <p style="margin: 0;">
-                  Run code in multiple languages with one click.
+                  Code in more than ten programming languages, with execution visualizations
                 </p>
                 <p class="intro-quote">
                   "Talk is cheap. Show me the code."
@@ -381,11 +397,11 @@
                     </path>
                   </svg>
                   <h3 style="margin-left: 0.5em;">
-                    Learn Together
+                    Learn together
                   </h3>
                 </div>
                 <p style="margin: 0;">
-                  Ask questions, discuss ideas, and learn with others.
+                  Ask questions, join discussions, and learn with fellow readers
                 </p>
                 <p class="intro-quote">
                   "Learning by teaching."
@@ -406,10 +422,10 @@
             <div class="profile-div">
               <div class="profile-cell">
                 <a href="https://github.com/krahets">
-                  <img alt="Author: yudongjin" class="profile-img" src="../assets/avatar/avatar_yudongjin.jpg"/>
+                  <img alt="Author: yudongjin" class="profile-img" src="assets/avatar/avatar_yudongjin.jpg"/>
                   <br/>
                   <b>
-                    Yudong Jin（@krahets）
+                    Yudong Jin (@krahets)
                   </b>
                 </a>
               </div>
@@ -417,115 +433,15 @@
           </div>
           <div style="margin: 2em auto;">
             <h3>
-              Translators
+              Code reviewers
             </h3>
             <p>
-              The English version of this book was reviewed by the following contributors. We thank them for their time and effort.
-            </p>
-            <div class="profile-div en-translator-profile-div">
-              <div class="profile-cell">
-                <a href="https://github.com/yuelinxin">
-                  <img alt="Reviewer: yuelinxin" class="profile-img" src="../assets/avatar/avatar_yuelinxin.jpg"/>
-                  <br/>
-                  <b>
-                    yuelinxin
-                  </b>
-                </a>
-              </div>
-              <div class="profile-cell">
-                <a href="https://github.com/K3v123">
-                  <img alt="Reviewer: K3v123" class="profile-img" src="../assets/avatar/avatar_K3v123.jpg"/>
-                  <br/>
-                  <b>
-                    K3v123
-                  </b>
-                </a>
-              </div>
-              <div class="profile-cell">
-                <a href="https://github.com/magentaqin">
-                  <img alt="Reviewer: magentaqin" class="profile-img" src="../assets/avatar/avatar_magentaqin.jpg"/>
-                  <br/>
-                  <b>
-                    magentaqin
-                  </b>
-                </a>
-              </div>
-              <div class="profile-cell">
-                <a href="https://github.com/QiLOL">
-                  <img alt="Reviewer: QiLOL" class="profile-img" src="../assets/avatar/avatar_QiLOL.jpg"/>
-                  <br/>
-                  <b>
-                    QiLOL
-                  </b>
-                </a>
-              </div>
-              <div class="profile-cell">
-                <a href="https://github.com/Phoenix0415">
-                  <img alt="Reviewer: Phoenix0415" class="profile-img" src="../assets/avatar/avatar_Phoenix0415.jpg"/>
-                  <br/>
-                  <b>
-                    Phoenix0415
-                  </b>
-                </a>
-              </div>
-              <div class="profile-cell">
-                <a href="https://github.com/SamJin98">
-                  <img alt="Reviewer: SamJin98" class="profile-img" src="../assets/avatar/avatar_SamJin98.jpg"/>
-                  <br/>
-                  <b>
-                    SamJin98
-                  </b>
-                </a>
-              </div>
-              <div class="profile-cell">
-                <a href="https://github.com/yanedie">
-                  <img alt="Reviewer: yanedie" class="profile-img" src="../assets/avatar/avatar_yanedie.jpg"/>
-                  <br/>
-                  <b>
-                    yanedie
-                  </b>
-                </a>
-              </div>
-              <div class="profile-cell">
-                <a href="https://github.com/RafaelCaso">
-                  <img alt="Reviewer: RafaelCaso" class="profile-img" src="../assets/avatar/avatar_RafaelCaso.jpg"/>
-                  <br/>
-                  <b>
-                    RafaelCaso
-                  </b>
-                </a>
-              </div>
-              <div class="profile-cell">
-                <a href="https://github.com/pengchzn">
-                  <img alt="Reviewer: pengchzn" class="profile-img" src="../assets/avatar/avatar_pengchzn.jpg"/>
-                  <br/>
-                  <b>
-                    pengchzn
-                  </b>
-                </a>
-              </div>
-              <div class="profile-cell">
-                <a href="https://github.com/thomasq0">
-                  <img alt="Reviewer: thomasq0" class="profile-img" src="../assets/avatar/avatar_thomasq0.jpg"/>
-                  <br/>
-                  <b>
-                    thomasq0
-                  </b>
-                </a>
-              </div>
-            </div>
-          </div>
-          <div style="margin: 2em auto;">
-            <h3>
-              Code Reviewers
-            </h3>
-            <p>
-              The multilingual code versions of this book were made possible by the following developers. We appreciate their efforts and contributions!
+              The developers below worked together on the code in different programming languages. Thank you for your efforts and contributions!
             </p>
             <div class="profile-div">
               <div class="profile-cell">
                 <a href="https://github.com/coderonion">
-                  <img alt="Reviewer: coderonion" class="profile-img" src="../assets/avatar/avatar_coderonion.jpg"/>
+                  <img alt="Reviewer: coderonion" class="profile-img" src="assets/avatar/avatar_coderonion.jpg"/>
                   <br/>
                   <b>
                     coderonion
@@ -538,7 +454,7 @@
               </div>
               <div class="profile-cell">
                 <a href="https://github.com/curtishd">
-                  <img alt="Reviewer: curtishd" class="profile-img" src="../assets/avatar/avatar_curtishd.jpg"/>
+                  <img alt="Reviewer: curtishd" class="profile-img" src="assets/avatar/avatar_curtishd.jpg"/>
                   <br/>
                   <b>
                     curtishd
@@ -551,7 +467,7 @@
               </div>
               <div class="profile-cell">
                 <a href="https://github.com/Gonglja">
-                  <img alt="Reviewer: Gonglja" class="profile-img" src="../assets/avatar/avatar_Gonglja.jpg"/>
+                  <img alt="Reviewer: Gonglja" class="profile-img" src="assets/avatar/avatar_Gonglja.jpg"/>
                   <br/>
                   <b>
                     Gonglja
@@ -564,7 +480,7 @@
               </div>
               <div class="profile-cell">
                 <a href="https://github.com/gvenusleo">
-                  <img alt="Reviewer: gvenusleo" class="profile-img" src="../assets/avatar/avatar_gvenusleo.jpg"/>
+                  <img alt="Reviewer: gvenusleo" class="profile-img" src="assets/avatar/avatar_gvenusleo.jpg"/>
                   <br/>
                   <b>
                     gvenusleo
@@ -577,7 +493,7 @@
               </div>
               <div class="profile-cell">
                 <a href="https://github.com/hpstory">
-                  <img alt="Reviewer: hpstory" class="profile-img" src="../assets/avatar/avatar_hpstory.jpg"/>
+                  <img alt="Reviewer: hpstory" class="profile-img" src="assets/avatar/avatar_hpstory.jpg"/>
                   <br/>
                   <b>
                     hpstory
@@ -590,7 +506,7 @@
               </div>
               <div class="profile-cell">
                 <a href="https://github.com/justin-tse">
-                  <img alt="Reviewer: justin-tse" class="profile-img" src="../assets/avatar/avatar_justin-tse.jpg"/>
+                  <img alt="Reviewer: justin-tse" class="profile-img" src="assets/avatar/avatar_justin-tse.jpg"/>
                   <br/>
                   <b>
                     justin-tse
@@ -603,7 +519,7 @@
               </div>
               <div class="profile-cell">
                 <a href="https://github.com/khoaxuantu">
-                  <img alt="Reviewer: khoaxuantu" class="profile-img" src="../assets/avatar/avatar_khoaxuantu.jpg"/>
+                  <img alt="Reviewer: khoaxuantu" class="profile-img" src="assets/avatar/avatar_khoaxuantu.jpg"/>
                   <br/>
                   <b>
                     khoaxuantu
@@ -616,7 +532,7 @@
               </div>
               <div class="profile-cell">
                 <a href="https://github.com/krahets">
-                  <img alt="Reviewer: krahets" class="profile-img" src="../assets/avatar/avatar_krahets.jpg"/>
+                  <img alt="Reviewer: krahets" class="profile-img" src="assets/avatar/avatar_krahets.jpg"/>
                   <br/>
                   <b>
                     krahets
@@ -629,7 +545,7 @@
               </div>
               <div class="profile-cell">
                 <a href="https://github.com/night-cruise">
-                  <img alt="Reviewer: night-cruise" class="profile-img" src="../assets/avatar/avatar_night-cruise.jpg"/>
+                  <img alt="Reviewer: night-cruise" class="profile-img" src="assets/avatar/avatar_night-cruise.jpg"/>
                   <br/>
                   <b>
                     night-cruise
@@ -642,7 +558,7 @@
               </div>
               <div class="profile-cell">
                 <a href="https://github.com/nuomi1">
-                  <img alt="Reviewer: nuomi1" class="profile-img" src="../assets/avatar/avatar_nuomi1.jpg"/>
+                  <img alt="Reviewer: nuomi1" class="profile-img" src="assets/avatar/avatar_nuomi1.jpg"/>
                   <br/>
                   <b>
                     nuomi1
@@ -655,7 +571,7 @@
               </div>
               <div class="profile-cell">
                 <a href="https://github.com/Reanon">
-                  <img alt="Reviewer: Reanon" class="profile-img" src="../assets/avatar/avatar_Reanon.jpg"/>
+                  <img alt="Reviewer: Reanon" class="profile-img" src="assets/avatar/avatar_Reanon.jpg"/>
                   <br/>
                   <b>
                     Reanon
@@ -668,7 +584,7 @@
               </div>
               <div class="profile-cell">
                 <a href="https://github.com/rongyi">
-                  <img alt="Reviewer: rongyi" class="profile-img" src="../assets/avatar/avatar_rongyi.jpg"/>
+                  <img alt="Reviewer: rongyi" class="profile-img" src="assets/avatar/avatar_rongyi.jpg"/>
                   <br/>
                   <b>
                     rongyi
@@ -686,7 +602,7 @@
               Contributors
             </h3>
             <p>
-              This book has been refined by the efforts of over 200 contributors. We sincerely thank them for their invaluable time and contributions!
+              More than 200 open-source contributors have helped improve this book. Thank you for your time and effort!
             </p>
             <a href="https://github.com/krahets/hello-algo/graphs/contributors">
               <img alt="Contributors" src="https://contrib.rocks/image?repo=krahets/hello-algo&amp;max=300&amp;columns=12" style="width: 100%; max-width: 38.5em;"/>
@@ -730,7 +646,7 @@
                             </path>
                           </svg>
                           <span class="md-ellipsis">
-                            Before Starting
+                            Foreword
                           </span>
                         </a>
                       </div>
@@ -738,7 +654,7 @@
                         <label class="md-nav__title" for="__nav_1">
                           <span class="md-nav__icon md-icon">
                           </span>
-                          Before Starting
+                          Foreword
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                         <ul class="md-nav__list" data-md-scrollfix="">
                         </ul>
@@ -753,7 +669,7 @@
                             </path>
                           </svg>
                           <span class="md-ellipsis">
-                            Chapter 0. Preface
+                            Chapter 0.   Preface
                           </span>
                         </a>
                         <label class="md-nav__link" for="__nav_2" id="__nav_2_label" tabindex="0">
@@ -765,27 +681,27 @@
                         <label class="md-nav__title" for="__nav_2">
                           <span class="md-nav__icon md-icon">
                           </span>
-                          Chapter 0. Preface
+                          Chapter 0.   Preface
                         </label>
                         <ul class="md-nav__list" data-md-scrollfix="">
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_preface/about_the_book/">
                               <span class="md-ellipsis">
-                                0.1 About This Book
+                                0.1   About this book
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_preface/suggestions/">
                               <span class="md-ellipsis">
-                                0.2 How to Use This Book
+                                0.2   How to use this book
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_preface/summary/">
                               <span class="md-ellipsis">
-                                0.3 Summary
+                                0.3   Summary
                               </span>
                             </a>
                           </li>
@@ -801,7 +717,7 @@
                             </path>
                           </svg>
                           <span class="md-ellipsis">
-                            Chapter 1. Encounter with Algorithms
+                            Chapter 1.   Getting to know algorithms
                           </span>
                         </a>
                         <label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
@@ -813,27 +729,27 @@
                         <label class="md-nav__title" for="__nav_3">
                           <span class="md-nav__icon md-icon">
                           </span>
-                          Chapter 1. Encounter with Algorithms
+                          Chapter 1.   Getting to know algorithms
                         </label>
                         <ul class="md-nav__list" data-md-scrollfix="">
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_introduction/algorithms_are_everywhere/">
                               <span class="md-ellipsis">
-                                1.1 Algorithms Are Everywhere
+                                1.1   Algorithms are everywhere
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_introduction/what_is_dsa/">
                               <span class="md-ellipsis">
-                                1.2 What Is an Algorithm
+                                1.2   What is an algorithm?
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_introduction/summary/">
                               <span class="md-ellipsis">
-                                1.3 Summary
+                                1.3   Summary
                               </span>
                             </a>
                           </li>
@@ -849,7 +765,7 @@
                             </path>
                           </svg>
                           <span class="md-ellipsis">
-                            Chapter 2. Complexity Analysis
+                            Chapter 2.   Complexity analysis
                           </span>
                         </a>
                         <label class="md-nav__link" for="__nav_4" id="__nav_4_label" tabindex="0">
@@ -861,48 +777,48 @@
                         <label class="md-nav__title" for="__nav_4">
                           <span class="md-nav__icon md-icon">
                           </span>
-                          Chapter 2. Complexity Analysis
+                          Chapter 2.   Complexity analysis
                         </label>
                         <ul class="md-nav__list" data-md-scrollfix="">
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_computational_complexity/performance_evaluation/">
                               <span class="md-ellipsis">
-                                2.1 Algorithm Efficiency Evaluation
+                                2.1   Evaluating algorithm efficiency
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_computational_complexity/iteration_and_recursion/">
                               <span class="md-ellipsis">
-                                2.2 Iteration and Recursion
+                                2.2   Iteration and recursion
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_computational_complexity/time_complexity/">
                               <span class="md-ellipsis">
-                                2.3 Time Complexity
+                                2.3   Time complexity
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_computational_complexity/space_complexity/">
                               <span class="md-ellipsis">
-                                2.4 Space Complexity
+                                2.4   Space complexity
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_computational_complexity/summary/">
                               <span class="md-ellipsis">
-                                2.5 Summary
+                                2.5   Summary
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_computational_complexity/exercises/">
                               <span class="md-ellipsis">
-                                2.6 Exercises
+                                2.6   Exercises
                               </span>
                             </a>
                           </li>
@@ -918,7 +834,7 @@
                             </path>
                           </svg>
                           <span class="md-ellipsis">
-                            Chapter 3. Data Structures
+                            Chapter 3.   Data structures
                           </span>
                         </a>
                         <label class="md-nav__link" for="__nav_5" id="__nav_5_label" tabindex="0">
@@ -930,48 +846,48 @@
                         <label class="md-nav__title" for="__nav_5">
                           <span class="md-nav__icon md-icon">
                           </span>
-                          Chapter 3. Data Structures
+                          Chapter 3.   Data structures
                         </label>
                         <ul class="md-nav__list" data-md-scrollfix="">
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_data_structure/classification_of_data_structure/">
                               <span class="md-ellipsis">
-                                3.1 Classification of Data Structures
+                                3.1   Classifying data structures
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_data_structure/basic_data_types/">
                               <span class="md-ellipsis">
-                                3.2 Basic Data Types
+                                3.2   Basic data types
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_data_structure/number_encoding/">
                               <span class="md-ellipsis">
-                                3.3 Number Encoding *
+                                3.3   Number encoding *
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_data_structure/character_encoding/">
                               <span class="md-ellipsis">
-                                3.4 Character Encoding *
+                                3.4   Character encoding *
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_data_structure/summary/">
                               <span class="md-ellipsis">
-                                3.5 Summary
+                                3.5   Summary
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_data_structure/exercises/">
                               <span class="md-ellipsis">
-                                3.6 Exercises
+                                3.6   Exercises
                               </span>
                             </a>
                           </li>
@@ -987,7 +903,7 @@
                             </path>
                           </svg>
                           <span class="md-ellipsis">
-                            Chapter 4. Arrays and Linked Lists
+                            Chapter 4.   Arrays and linked lists
                           </span>
                         </a>
                         <label class="md-nav__link" for="__nav_6" id="__nav_6_label" tabindex="0">
@@ -999,48 +915,48 @@
                         <label class="md-nav__title" for="__nav_6">
                           <span class="md-nav__icon md-icon">
                           </span>
-                          Chapter 4. Arrays and Linked Lists
+                          Chapter 4.   Arrays and linked lists
                         </label>
                         <ul class="md-nav__list" data-md-scrollfix="">
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_array_and_linkedlist/array/">
                               <span class="md-ellipsis">
-                                4.1 Array
+                                4.1   Arrays
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_array_and_linkedlist/linked_list/">
                               <span class="md-ellipsis">
-                                4.2 Linked List
+                                4.2   Linked lists
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_array_and_linkedlist/list/">
                               <span class="md-ellipsis">
-                                4.3 List
+                                4.3   Lists
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_array_and_linkedlist/ram_and_cache/">
                               <span class="md-ellipsis">
-                                4.4 Random-Access Memory and Cache *
+                                4.4   Memory and cache *
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_array_and_linkedlist/summary/">
                               <span class="md-ellipsis">
-                                4.5 Summary
+                                4.5   Summary
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_array_and_linkedlist/exercises/">
                               <span class="md-ellipsis">
-                                4.6 Exercises
+                                4.6   Exercises
                               </span>
                             </a>
                           </li>
@@ -1056,7 +972,7 @@
                             </path>
                           </svg>
                           <span class="md-ellipsis">
-                            Chapter 5. Stacks and Queues
+                            Chapter 5.   Stacks and queues
                           </span>
                         </a>
                         <label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
@@ -1068,41 +984,41 @@
                         <label class="md-nav__title" for="__nav_7">
                           <span class="md-nav__icon md-icon">
                           </span>
-                          Chapter 5. Stacks and Queues
+                          Chapter 5.   Stacks and queues
                         </label>
                         <ul class="md-nav__list" data-md-scrollfix="">
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_stack_and_queue/stack/">
                               <span class="md-ellipsis">
-                                5.1 Stack
+                                5.1   Stacks
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_stack_and_queue/queue/">
                               <span class="md-ellipsis">
-                                5.2 Queue
+                                5.2   Queues
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_stack_and_queue/deque/">
                               <span class="md-ellipsis">
-                                5.3 Deque
+                                5.3   Deques
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_stack_and_queue/summary/">
                               <span class="md-ellipsis">
-                                5.4 Summary
+                                5.4   Summary
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_stack_and_queue/exercises/">
                               <span class="md-ellipsis">
-                                5.5 Exercises
+                                5.5   Exercises
                               </span>
                             </a>
                           </li>
@@ -1118,7 +1034,7 @@
                             </path>
                           </svg>
                           <span class="md-ellipsis">
-                            Chapter 6. Hashing
+                            Chapter 6.   Hash tables
                           </span>
                         </a>
                         <label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
@@ -1130,41 +1046,41 @@
                         <label class="md-nav__title" for="__nav_8">
                           <span class="md-nav__icon md-icon">
                           </span>
-                          Chapter 6. Hashing
+                          Chapter 6.   Hash tables
                         </label>
                         <ul class="md-nav__list" data-md-scrollfix="">
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_hashing/hash_map/">
                               <span class="md-ellipsis">
-                                6.1 Hash Table
+                                6.1   Hash tables
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_hashing/hash_collision/">
                               <span class="md-ellipsis">
-                                6.2 Hash Collision
+                                6.2   Hash collisions
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_hashing/hash_algorithm/">
                               <span class="md-ellipsis">
-                                6.3 Hash Algorithm
+                                6.3   Hash algorithms
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_hashing/summary/">
                               <span class="md-ellipsis">
-                                6.4 Summary
+                                6.4   Summary
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_hashing/exercises/">
                               <span class="md-ellipsis">
-                                6.5 Exercises
+                                6.5   Exercises
                               </span>
                             </a>
                           </li>
@@ -1180,7 +1096,7 @@
                             </path>
                           </svg>
                           <span class="md-ellipsis">
-                            Chapter 7. Tree
+                            Chapter 7.   Trees
                           </span>
                         </a>
                         <label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
@@ -1192,55 +1108,55 @@
                         <label class="md-nav__title" for="__nav_9">
                           <span class="md-nav__icon md-icon">
                           </span>
-                          Chapter 7. Tree
+                          Chapter 7.   Trees
                         </label>
                         <ul class="md-nav__list" data-md-scrollfix="">
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_tree/binary_tree/">
                               <span class="md-ellipsis">
-                                7.1 Binary Tree
+                                7.1   Binary trees
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_tree/binary_tree_traversal/">
                               <span class="md-ellipsis">
-                                7.2 Binary Tree Traversal
+                                7.2   Traversing a binary tree
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_tree/array_representation_of_tree/">
                               <span class="md-ellipsis">
-                                7.3 Array Representation of Binary Trees
+                                7.3   Array representation of a binary tree
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_tree/binary_search_tree/">
                               <span class="md-ellipsis">
-                                7.4 Binary Search Tree
+                                7.4   Binary search trees
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_tree/avl_tree/">
                               <span class="md-ellipsis">
-                                7.5 AVL Tree *
+                                7.5   AVL trees *
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_tree/summary/">
                               <span class="md-ellipsis">
-                                7.6 Summary
+                                7.6   Summary
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_tree/exercises/">
                               <span class="md-ellipsis">
-                                7.7 Exercises
+                                7.7   Exercises
                               </span>
                             </a>
                           </li>
@@ -1256,7 +1172,7 @@
                             </path>
                           </svg>
                           <span class="md-ellipsis">
-                            Chapter 8. Heap
+                            Chapter 8.   Heaps
                           </span>
                         </a>
                         <label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
@@ -1268,41 +1184,41 @@
                         <label class="md-nav__title" for="__nav_10">
                           <span class="md-nav__icon md-icon">
                           </span>
-                          Chapter 8. Heap
+                          Chapter 8.   Heaps
                         </label>
                         <ul class="md-nav__list" data-md-scrollfix="">
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_heap/heap/">
                               <span class="md-ellipsis">
-                                8.1 Heap
+                                8.1   Heaps
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_heap/build_heap/">
                               <span class="md-ellipsis">
-                                8.2 Heap Construction Operation
+                                8.2   Building a heap
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_heap/top_k/">
                               <span class="md-ellipsis">
-                                8.3 Top-k Problem
+                                8.3   The top-k problem
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_heap/summary/">
                               <span class="md-ellipsis">
-                                8.4 Summary
+                                8.4   Summary
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_heap/exercises/">
                               <span class="md-ellipsis">
-                                8.5 Exercises
+                                8.5   Exercises
                               </span>
                             </a>
                           </li>
@@ -1318,7 +1234,7 @@
                             </path>
                           </svg>
                           <span class="md-ellipsis">
-                            Chapter 9. Graph
+                            Chapter 9.   Graphs
                           </span>
                         </a>
                         <label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
@@ -1330,41 +1246,41 @@
                         <label class="md-nav__title" for="__nav_11">
                           <span class="md-nav__icon md-icon">
                           </span>
-                          Chapter 9. Graph
+                          Chapter 9.   Graphs
                         </label>
                         <ul class="md-nav__list" data-md-scrollfix="">
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_graph/graph/">
                               <span class="md-ellipsis">
-                                9.1 Graph
+                                9.1   Graphs
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_graph/graph_operations/">
                               <span class="md-ellipsis">
-                                9.2 Basic Operations on Graphs
+                                9.2   Basic graph operations
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_graph/graph_traversal/">
                               <span class="md-ellipsis">
-                                9.3 Graph Traversal
+                                9.3   Graph traversal
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_graph/summary/">
                               <span class="md-ellipsis">
-                                9.4 Summary
+                                9.4   Summary
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_graph/exercises/">
                               <span class="md-ellipsis">
-                                9.5 Exercises
+                                9.5   Exercises
                               </span>
                             </a>
                           </li>
@@ -1380,7 +1296,7 @@
                             </path>
                           </svg>
                           <span class="md-ellipsis">
-                            Chapter 10. Searching
+                            Chapter 10.   Searching
                           </span>
                         </a>
                         <label class="md-nav__link" for="__nav_12" id="__nav_12_label" tabindex="0">
@@ -1392,55 +1308,55 @@
                         <label class="md-nav__title" for="__nav_12">
                           <span class="md-nav__icon md-icon">
                           </span>
-                          Chapter 10. Searching
+                          Chapter 10.   Searching
                         </label>
                         <ul class="md-nav__list" data-md-scrollfix="">
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_searching/binary_search/">
                               <span class="md-ellipsis">
-                                10.1 Binary Search
+                                10.1   Binary search
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_searching/binary_search_insertion/">
                               <span class="md-ellipsis">
-                                10.2 Binary Search Insertion Point
+                                10.2   Finding an insertion point with binary search
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_searching/binary_search_edge/">
                               <span class="md-ellipsis">
-                                10.3 Binary Search Boundaries
+                                10.3   Finding boundaries with binary search
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_searching/replace_linear_by_hashing/">
                               <span class="md-ellipsis">
-                                10.4 Hash Optimization Strategy
+                                10.4   Optimizing with hashing
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_searching/searching_algorithm_revisited/">
                               <span class="md-ellipsis">
-                                10.5 Searching Algorithms Revisited
+                                10.5   Search algorithms revisited
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_searching/summary/">
                               <span class="md-ellipsis">
-                                10.6 Summary
+                                10.6   Summary
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_searching/exercises/">
                               <span class="md-ellipsis">
-                                10.7 Exercises
+                                10.7   Exercises
                               </span>
                             </a>
                           </li>
@@ -1456,7 +1372,7 @@
                             </path>
                           </svg>
                           <span class="md-ellipsis">
-                            Chapter 11. Sorting
+                            Chapter 11.   Sorting
                           </span>
                         </a>
                         <label class="md-nav__link" for="__nav_13" id="__nav_13_label" tabindex="0">
@@ -1468,90 +1384,90 @@
                         <label class="md-nav__title" for="__nav_13">
                           <span class="md-nav__icon md-icon">
                           </span>
-                          Chapter 11. Sorting
+                          Chapter 11.   Sorting
                         </label>
                         <ul class="md-nav__list" data-md-scrollfix="">
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_sorting/sorting_algorithm/">
                               <span class="md-ellipsis">
-                                11.1 Sorting Algorithm
+                                11.1   Sorting algorithms
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_sorting/selection_sort/">
                               <span class="md-ellipsis">
-                                11.2 Selection Sort
+                                11.2   Selection sort
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_sorting/bubble_sort/">
                               <span class="md-ellipsis">
-                                11.3 Bubble Sort
+                                11.3   Bubble sort
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_sorting/insertion_sort/">
                               <span class="md-ellipsis">
-                                11.4 Insertion Sort
+                                11.4   Insertion sort
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_sorting/quick_sort/">
                               <span class="md-ellipsis">
-                                11.5 Quick Sort
+                                11.5   Quick sort
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_sorting/merge_sort/">
                               <span class="md-ellipsis">
-                                11.6 Merge Sort
+                                11.6   Merge sort
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_sorting/heap_sort/">
                               <span class="md-ellipsis">
-                                11.7 Heap Sort
+                                11.7   Heap sort
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_sorting/bucket_sort/">
                               <span class="md-ellipsis">
-                                11.8 Bucket Sort
+                                11.8   Bucket sort
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_sorting/counting_sort/">
                               <span class="md-ellipsis">
-                                11.9 Counting Sort
+                                11.9   Counting sort
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_sorting/radix_sort/">
                               <span class="md-ellipsis">
-                                11.10 Radix Sort
+                                11.10   Radix sort
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_sorting/summary/">
                               <span class="md-ellipsis">
-                                11.11 Summary
+                                11.11   Summary
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_sorting/exercises/">
                               <span class="md-ellipsis">
-                                11.12 Exercises
+                                11.12   Exercises
                               </span>
                             </a>
                           </li>
@@ -1567,7 +1483,7 @@
                             </path>
                           </svg>
                           <span class="md-ellipsis">
-                            Chapter 12. Divide and Conquer
+                            Chapter 12.   Divide and conquer
                           </span>
                         </a>
                         <label class="md-nav__link" for="__nav_14" id="__nav_14_label" tabindex="0">
@@ -1579,48 +1495,48 @@
                         <label class="md-nav__title" for="__nav_14">
                           <span class="md-nav__icon md-icon">
                           </span>
-                          Chapter 12. Divide and Conquer
+                          Chapter 12.   Divide and conquer
                         </label>
                         <ul class="md-nav__list" data-md-scrollfix="">
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_divide_and_conquer/divide_and_conquer/">
                               <span class="md-ellipsis">
-                                12.1 Divide and Conquer Algorithms
+                                12.1   Divide and conquer
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_divide_and_conquer/binary_search_recur/">
                               <span class="md-ellipsis">
-                                12.2 Divide and Conquer Search Strategy
+                                12.2   Divide-and-conquer search
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_divide_and_conquer/build_binary_tree_problem/">
                               <span class="md-ellipsis">
-                                12.3 Building a Binary Tree Problem
+                                12.3   Building a binary tree
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_divide_and_conquer/hanota_problem/">
                               <span class="md-ellipsis">
-                                12.4 Tower of Hanoi Problem
+                                12.4   Tower of Hanoi
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_divide_and_conquer/summary/">
                               <span class="md-ellipsis">
-                                12.5 Summary
+                                12.5   Summary
                               </span>
                             </a>
                           </li>
                           <li class="md-nav__item">
                             <a class="md-nav__link" href="./chapter_divide_and_conquer/exercises/">
                               <span class="md-ellipsis">
-                                12.6 Exercises
+                                12.6   Exercises
                               </span>
                             </a>
                           </li>
@@ -1636,7 +1552,7 @@
                             </path>
                           </svg>
                           <span class="md-ellipsis">
-                            Chapter 13. Backtracking
+                            Chapter 13.   Backtracking
                           </span>
                         </a>
                         <label class="md-nav__link" for="__nav_15" id="__nav_15_label" tabindex="0">
@@ -1648,48 +1564,48 @@
                         <label class="md-nav__title" for="__nav_15">
                           <span class="md-nav__icon md-icon">
                           </span>
-                          Chapter 13. Backtracking
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-                                13.1 Backtracking Algorithm
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                             <a class="md-nav__link" href="./chapter_backtracking/permutations_problem/">
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-                                13.2 Permutations Problem
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-                                13.3 Subset-Sum Problem
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                             <a class="md-nav__link" href="./chapter_backtracking/n_queens_problem/">
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-                                13.4 N-Queens Problem
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-                                14.5 Unbounded Knapsack Problem
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-                                15.2 Fractional Knapsack Problem
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Arrays and Linked Lists","Chapter 4.   Arrays and Linked Lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/","level":1,"title":"4.1   Array","text":"<p>An array is a linear data structure that stores elements of the same type in contiguous memory space. The position of an element in the array is called the element's index. Figure 4-1 illustrates the main concepts and storage method of arrays.</p> <p></p> <p> Figure 4-1   Array definition and storage method </p>","path":["Chapter 4. Arrays and Linked Lists","4.1   Array"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#411-common-array-operations","level":2,"title":"4.1.1   Common Array Operations","text":"","path":["Chapter 4. Arrays and Linked Lists","4.1   Array"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#1-initializing-arrays","level":3,"title":"1.   Initializing Arrays","text":"<p>We can choose between two array initialization methods based on our needs: with or without initial values. When no initial values are specified, most programming languages initialize array elements to \\(0\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code># Initialize array\narr: list[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> array.cpp<pre><code>/* Initialize array */\n// Stored on stack\nint arr[5] = {};\nint nums[5] = { 1, 3, 2, 5, 4 };\n// Stored on heap (requires manual memory release)\nint* arr1 = new int[5] {};\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n</code></pre> array.java<pre><code>/* Initialize array */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n</code></pre> array.cs<pre><code>/* Initialize array */\nint[] arr = new int[5]; // [ 0, 0, 0, 0, 0 ]\nint[] nums = [1, 3, 2, 5, 4];\n</code></pre> array.go<pre><code>/* Initialize array */\nvar arr [5]int\n// In Go, specifying length ([5]int) creates an array; not specifying length ([]int) creates a slice\n// Since Go's arrays are designed to have their length determined at compile time, only constants can be used to specify the length\n// For convenience in implementing the extend() method, slices are treated as arrays below\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> array.swift<pre><code>/* Initialize array */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n</code></pre> array.js<pre><code>/* Initialize array */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n</code></pre> array.ts<pre><code>/* Initialize array */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> array.dart<pre><code>/* Initialize array */\nList&lt;int&gt; arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> array.rs<pre><code>/* Initialize array */\nlet arr: [i32; 5] = [0; 5]; // [0, 0, 0, 0, 0]\nlet slice: &amp;[i32] = &amp;[0; 5];\n// In Rust, specifying length ([i32; 5]) creates an array; not specifying length (&amp;[i32]) creates a slice\n// Since Rust's arrays are designed to have their length determined at compile time, only constants can be used to specify the length\n// Vector is the type generally used as a dynamic array in Rust\n// For convenience in implementing the extend() method, vectors are treated as arrays below\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> array.c<pre><code>/* Initialize array */\nint arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n</code></pre> array.kt<pre><code>/* Initialize array */\nvar arr = IntArray(5) // { 0, 0, 0, 0, 0 }\nvar nums = intArrayOf(1, 3, 2, 5, 4)\n</code></pre> array.rb<pre><code># Initialize array\narr = Array.new(5, 0)\nnums = [1, 3, 2, 5, 4]\n</code></pre> Code Visualization <p> Full Screen &gt;</p>","path":["Chapter 4. Arrays and Linked Lists","4.1   Array"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#2-accessing-elements","level":3,"title":"2.   Accessing Elements","text":"<p>Array elements are stored in contiguous memory space, which means calculating the memory address of array elements is very easy. Given the array's memory address (the memory address of the first element) and an element's index, we can use the formula shown in Figure 4-2 to calculate the element's memory address and directly access that element.</p> <p></p> <p> Figure 4-2   Memory address calculation for array elements </p> <p>Observing Figure 4-2, we find that the first element of an array has an index of \\(0\\), which may seem counterintuitive since counting from \\(1\\) would be more natural. However, from the perspective of the address calculation formula, an index is essentially an offset from the memory address. The address offset of the first element is \\(0\\), so it is reasonable for its index to be \\(0\\).</p> <p>Accessing elements in an array is highly efficient; we can randomly access any element in the array in \\(O(1)\\) time.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def random_access(nums: list[int]) -&gt; int:\n    \"\"\"Random access to element\"\"\"\n    # Randomly select a number from the interval [0, len(nums)-1]\n    random_index = random.randint(0, len(nums) - 1)\n    # Retrieve and return the random element\n    random_num = nums[random_index]\n    return random_num\n</code></pre> array.cpp<pre><code>/* Random access to element */\nint randomAccess(int *nums, int size) {\n    // Randomly select a number from interval [0, size)\n    int randomIndex = rand() % size;\n    // Retrieve and return the random element\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.java<pre><code>/* Random access to element */\nint randomAccess(int[] nums) {\n    // Randomly select a number in the interval [0, nums.length)\n    int randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n    // Retrieve and return the random element\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.cs<pre><code>/* Random access to element */\nint RandomAccess(int[] nums) {\n    Random random = new();\n    // Randomly select a number in interval [0, nums.Length)\n    int randomIndex = random.Next(nums.Length);\n    // Retrieve and return the random element\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.go<pre><code>/* Random access to element */\nfunc randomAccess(nums []int) (randomNum int) {\n    // Randomly select a number in the interval [0, nums.length)\n    randomIndex := rand.Intn(len(nums))\n    // Retrieve and return the random element\n    randomNum = nums[randomIndex]\n    return\n}\n</code></pre> array.swift<pre><code>/* Random access to element */\nfunc randomAccess(nums: [Int]) -&gt; Int {\n    // Randomly select a number in interval [0, nums.count)\n    let randomIndex = nums.indices.randomElement()!\n    // Retrieve and return the random element\n    let randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.js<pre><code>/* Random access to element */\nfunction randomAccess(nums) {\n    // Randomly select a number in the interval [0, nums.length)\n    const random_index = Math.floor(Math.random() * nums.length);\n    // Retrieve and return the random element\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.ts<pre><code>/* Random access to element */\nfunction randomAccess(nums: number[]): number {\n    // Randomly select a number in the interval [0, nums.length)\n    const random_index = Math.floor(Math.random() * nums.length);\n    // Retrieve and return the random element\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.dart<pre><code>/* Random access to element */\nint randomAccess(List&lt;int&gt; nums) {\n  // Randomly select a number in the interval [0, nums.length)\n  int randomIndex = Random().nextInt(nums.length);\n  // Retrieve and return the random element\n  int randomNum = nums[randomIndex];\n  return randomNum;\n}\n</code></pre> array.rs<pre><code>/* Random access to element */\nfn random_access(nums: &amp;[i32]) -&gt; i32 {\n    // Randomly select a number in interval [0, nums.len())\n    let random_index = rand::thread_rng().gen_range(0..nums.len());\n    // Retrieve and return the random element\n    let random_num = nums[random_index];\n    random_num\n}\n</code></pre> array.c<pre><code>/* Random access to element */\nint randomAccess(int *nums, int size) {\n    // Randomly select a number from interval [0, size)\n    int randomIndex = rand() % size;\n    // Retrieve and return the random element\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.kt<pre><code>/* Random access to element */\nfun randomAccess(nums: IntArray): Int {\n    // Randomly select a number in interval [0, nums.size)\n    val randomIndex = ThreadLocalRandom.current().nextInt(0, nums.size)\n    // Retrieve and return the random element\n    val randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.rb<pre><code>### Random access element ###\ndef random_access(nums)\n  # Randomly select a number in the interval [0, nums.length)\n  random_index = Random.rand(0...nums.length)\n\n  # Retrieve and return the random element\n  nums[random_index]\nend\n</code></pre>","path":["Chapter 4. Arrays and Linked Lists","4.1   Array"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#3-inserting-elements","level":3,"title":"3.   Inserting Elements","text":"<p>Array elements are packed tightly together in memory, with no extra space between them for additional data. As shown in Figure 4-3, if we want to insert an element in the middle of an array, we need to shift all subsequent elements one position to the right and then assign the value at that index.</p> <p></p> <p> Figure 4-3   Example of inserting an element into an array </p> <p>It is worth noting that since the length of an array is fixed, inserting an element will inevitably push the last element out of the array. We will leave the solution to this problem for discussion in the \"List\" chapter.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def insert(nums: list[int], num: int, index: int):\n    \"\"\"Insert element num at index index in the array\"\"\"\n    # Move all elements at and after index index backward by one position\n    for i in range(len(nums) - 1, index, -1):\n        nums[i] = nums[i - 1]\n    # Assign num to the element at index index\n    nums[index] = num\n</code></pre> array.cpp<pre><code>/* Insert element num at index index in the array */\nvoid insert(int *nums, int size, int num, int index) {\n    // Move all elements at and after index index backward by one position\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // Assign num to the element at index index\n    nums[index] = num;\n}\n</code></pre> array.java<pre><code>/* Insert element num at index index in the array */\nvoid insert(int[] nums, int num, int index) {\n    // Move all elements at and after index index backward by one position\n    for (int i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // Assign num to the element at index index\n    nums[index] = num;\n}\n</code></pre> array.cs<pre><code>/* Insert element num at index index in the array */\nvoid Insert(int[] nums, int num, int index) {\n    // Move all elements at and after index index backward by one position\n    for (int i = nums.Length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // Assign num to the element at index index\n    nums[index] = num;\n}\n</code></pre> array.go<pre><code>/* Insert element num at index index in the array */\nfunc insert(nums []int, num int, index int) {\n    // Move all elements at and after index index backward by one position\n    for i := len(nums) - 1; i &gt; index; i-- {\n        nums[i] = nums[i-1]\n    }\n    // Assign num to the element at index index\n    nums[index] = num\n}\n</code></pre> array.swift<pre><code>/* Insert element num at index index in the array */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n    // Move all elements at and after index index backward by one position\n    for i in stride(from: nums.count - 1, to: index, by: -1) {\n        nums[i] = nums[i - 1]\n    }\n    // Assign num to the element at index index\n    nums[index] = num\n}\n</code></pre> array.js<pre><code>/* Insert element num at index index in the array */\nfunction insert(nums, num, index) {\n    // Move all elements at and after index index backward by one position\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // Assign num to the element at index index\n    nums[index] = num;\n}\n</code></pre> array.ts<pre><code>/* Insert element num at index index in the array */\nfunction insert(nums: number[], num: number, index: number): void {\n    // Move all elements at and after index index backward by one position\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // Assign num to the element at index index\n    nums[index] = num;\n}\n</code></pre> array.dart<pre><code>/* Insert element _num at array index index */\nvoid insert(List&lt;int&gt; nums, int _num, int index) {\n  // Move all elements at and after index index backward by one position\n  for (var i = nums.length - 1; i &gt; index; i--) {\n    nums[i] = nums[i - 1];\n  }\n  // Assign _num to element at index\n  nums[index] = _num;\n}\n</code></pre> array.rs<pre><code>/* Insert element num at index index in the array */\nfn insert(nums: &amp;mut [i32], num: i32, index: usize) {\n    // Move all elements at and after index index backward by one position\n    for i in (index + 1..nums.len()).rev() {\n        nums[i] = nums[i - 1];\n    }\n    // Assign num to the element at index index\n    nums[index] = num;\n}\n</code></pre> array.c<pre><code>/* Insert element num at index index in the array */\nvoid insert(int *nums, int size, int num, int index) {\n    // Move all elements at and after index index backward by one position\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // Assign num to the element at index index\n    nums[index] = num;\n}\n</code></pre> array.kt<pre><code>/* Insert element num at index index in the array */\nfun insert(nums: IntArray, num: Int, index: Int) {\n    // Move all elements at and after index index backward by one position\n    for (i in nums.size - 1 downTo index + 1) {\n        nums[i] = nums[i - 1]\n    }\n    // Assign num to the element at index index\n    nums[index] = num\n}\n</code></pre> array.rb<pre><code>### Insert element num at index in array ###\ndef insert(nums, num, index)\n  # Move all elements at and after index index backward by one position\n  for i in (nums.length - 1).downto(index + 1)\n    nums[i] = nums[i - 1]\n  end\n\n  # Assign num to the element at index index\n  nums[index] = num\nend\n</code></pre>","path":["Chapter 4. Arrays and Linked Lists","4.1   Array"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#4-removing-elements","level":3,"title":"4.   Removing Elements","text":"<p>Similarly, as shown in Figure 4-4, to delete the element at index \\(i\\), we need to shift all elements after index \\(i\\) forward by one position.</p> <p></p> <p> Figure 4-4   Example of removing an element from an array </p> <p>Note that after the deletion is complete, the original last element is no longer meaningful, so we do not need to modify it explicitly.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def remove(nums: list[int], index: int):\n    \"\"\"Remove the element at index index\"\"\"\n    # Move all elements after index index forward by one position\n    for i in range(index, len(nums) - 1):\n        nums[i] = nums[i + 1]\n</code></pre> array.cpp<pre><code>/* Remove the element at index index */\nvoid remove(int *nums, int size, int index) {\n    // Move all elements after index index forward by one position\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.java<pre><code>/* Remove the element at index index */\nvoid remove(int[] nums, int index) {\n    // Move all elements after index index forward by one position\n    for (int i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.cs<pre><code>/* Remove the element at index index */\nvoid Remove(int[] nums, int index) {\n    // Move all elements after index index forward by one position\n    for (int i = index; i &lt; nums.Length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.go<pre><code>/* Remove the element at index index */\nfunc remove(nums []int, index int) {\n    // Move all elements after index index forward by one position\n    for i := index; i &lt; len(nums)-1; i++ {\n        nums[i] = nums[i+1]\n    }\n}\n</code></pre> array.swift<pre><code>/* Remove the element at index index */\nfunc remove(nums: inout [Int], index: Int) {\n    // Move all elements after index index forward by one position\n    for i in nums.indices.dropFirst(index).dropLast() {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.js<pre><code>/* Remove the element at index index */\nfunction remove(nums, index) {\n    // Move all elements after index index forward by one position\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.ts<pre><code>/* Remove the element at index index */\nfunction remove(nums: number[], index: number): void {\n    // Move all elements after index index forward by one position\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.dart<pre><code>/* Remove the element at index index */\nvoid remove(List&lt;int&gt; nums, int index) {\n  // Move all elements after index index forward by one position\n  for (var i = index; i &lt; nums.length - 1; i++) {\n    nums[i] = nums[i + 1];\n  }\n}\n</code></pre> array.rs<pre><code>/* Remove the element at index index */\nfn remove(nums: &amp;mut [i32], index: usize) {\n    // Move all elements after index index forward by one position\n    for i in index..nums.len() - 1 {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.c<pre><code>/* Remove the element at index index */\n// Note: stdio.h occupies the remove keyword\nvoid removeItem(int *nums, int size, int index) {\n    // Move all elements after index index forward by one position\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.kt<pre><code>/* Remove the element at index index */\nfun remove(nums: IntArray, index: Int) {\n    // Move all elements after index index forward by one position\n    for (i in index..&lt;nums.size - 1) {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.rb<pre><code>### Delete element at index ###\ndef remove(nums, index)\n  # Move all elements after index index forward by one position\n  for i in index...(nums.length - 1)\n    nums[i] = nums[i + 1]\n  end\nend\n</code></pre> <p>Overall, array insertion and deletion operations have the following drawbacks:</p> <ul> <li>High time complexity: The average time complexity for both insertion and deletion in arrays is \\(O(n)\\), where \\(n\\) is the length of the array.</li> <li>Loss of elements: Since the length of an array is immutable, after inserting an element, elements that exceed the array's length will be lost.</li> <li>Memory waste: We can initialize a relatively long array and use only the front portion, so that any overwritten tail elements are merely unused placeholders, but this wastes some memory space.</li> </ul>","path":["Chapter 4. Arrays and Linked Lists","4.1   Array"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#5-traversing-arrays","level":3,"title":"5.   Traversing Arrays","text":"<p>In most programming languages, we can traverse an array either by index or by directly iterating through each element in the array:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def traverse(nums: list[int]):\n    \"\"\"Traverse array\"\"\"\n    count = 0\n    # Traverse array by index\n    for i in range(len(nums)):\n        count += nums[i]\n    # Direct traversal of array elements\n    for num in nums:\n        count += num\n    # Traverse simultaneously data index and elements\n    for i, num in enumerate(nums):\n        count += nums[i]\n        count += num\n</code></pre> array.cpp<pre><code>/* Traverse array */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // Traverse array by index\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.java<pre><code>/* Traverse array */\nvoid traverse(int[] nums) {\n    int count = 0;\n    // Traverse array by index\n    for (int i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // Direct traversal of array elements\n    for (int num : nums) {\n        count += num;\n    }\n}\n</code></pre> array.cs<pre><code>/* Traverse array */\nvoid Traverse(int[] nums) {\n    int count = 0;\n    // Traverse array by index\n    for (int i = 0; i &lt; nums.Length; i++) {\n        count += nums[i];\n    }\n    // Direct traversal of array elements\n    foreach (int num in nums) {\n        count += num;\n    }\n}\n</code></pre> array.go<pre><code>/* Traverse array */\nfunc traverse(nums []int) {\n    count := 0\n    // Traverse array by index\n    for i := 0; i &lt; len(nums); i++ {\n        count += nums[i]\n    }\n    count = 0\n    // Direct traversal of array elements\n    for _, num := range nums {\n        count += num\n    }\n    // Traverse simultaneously data index and elements\n    for i, num := range nums {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.swift<pre><code>/* Traverse array */\nfunc traverse(nums: [Int]) {\n    var count = 0\n    // Traverse array by index\n    for i in nums.indices {\n        count += nums[i]\n    }\n    // Direct traversal of array elements\n    for num in nums {\n        count += num\n    }\n    // Traverse simultaneously data index and elements\n    for (i, num) in nums.enumerated() {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.js<pre><code>/* Traverse array */\nfunction traverse(nums) {\n    let count = 0;\n    // Traverse array by index\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // Direct traversal of array elements\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.ts<pre><code>/* Traverse array */\nfunction traverse(nums: number[]): void {\n    let count = 0;\n    // Traverse array by index\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // Direct traversal of array elements\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.dart<pre><code>/* Traverse array elements */\nvoid traverse(List&lt;int&gt; nums) {\n  int count = 0;\n  // Traverse array by index\n  for (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n  }\n  // Direct traversal of array elements\n  for (int _num in nums) {\n    count += _num;\n  }\n  // Traverse array using forEach method\n  nums.forEach((_num) {\n    count += _num;\n  });\n}\n</code></pre> array.rs<pre><code>/* Traverse array */\nfn traverse(nums: &amp;[i32]) {\n    let mut _count = 0;\n    // Traverse array by index\n    for i in 0..nums.len() {\n        _count += nums[i];\n    }\n    // Direct traversal of array elements\n    _count = 0;\n    for &amp;num in nums {\n        _count += num;\n    }\n}\n</code></pre> array.c<pre><code>/* Traverse array */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // Traverse array by index\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.kt<pre><code>/* Traverse array */\nfun traverse(nums: IntArray) {\n    var count = 0\n    // Traverse array by index\n    for (i in nums.indices) {\n        count += nums[i]\n    }\n    // Direct traversal of array elements\n    for (j in nums) {\n        count += j\n    }\n}\n</code></pre> array.rb<pre><code>### Traverse array ###\ndef traverse(nums)\n  count = 0\n\n  # Traverse array by index\n  for i in 0...nums.length\n    count += nums[i]\n  end\n\n  # Direct traversal of array elements\n  for num in nums\n    count += num\n  end\nend\n</code></pre>","path":["Chapter 4. Arrays and Linked Lists","4.1   Array"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#6-finding-elements","level":3,"title":"6.   Finding Elements","text":"<p>Finding a specified element in an array requires traversing the array and checking whether the element value matches in each iteration; if it matches, output the corresponding index.</p> <p>Since an array is a linear data structure, the above search operation is called a \"linear search\".</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def find(nums: list[int], target: int) -&gt; int:\n    \"\"\"Find the specified element in the array\"\"\"\n    for i in range(len(nums)):\n        if nums[i] == target:\n            return i\n    return -1\n</code></pre> array.cpp<pre><code>/* Find the specified element in the array */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.java<pre><code>/* Find the specified element in the array */\nint find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.cs<pre><code>/* Find the specified element in the array */\nint Find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.go<pre><code>/* Find the specified element in the array */\nfunc find(nums []int, target int) (index int) {\n    index = -1\n    for i := 0; i &lt; len(nums); i++ {\n        if nums[i] == target {\n            index = i\n            break\n        }\n    }\n    return\n}\n</code></pre> array.swift<pre><code>/* Find the specified element in the array */\nfunc find(nums: [Int], target: Int) -&gt; Int {\n    for i in nums.indices {\n        if nums[i] == target {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> array.js<pre><code>/* Find the specified element in the array */\nfunction find(nums, target) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) return i;\n    }\n    return -1;\n}\n</code></pre> array.ts<pre><code>/* Find the specified element in the array */\nfunction find(nums: number[], target: number): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> array.dart<pre><code>/* Find the specified element in the array */\nint find(List&lt;int&gt; nums, int target) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    if (nums[i] == target) return i;\n  }\n  return -1;\n}\n</code></pre> array.rs<pre><code>/* Find the specified element in the array */\nfn find(nums: &amp;[i32], target: i32) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        if nums[i] == target {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> array.c<pre><code>/* Find the specified element in the array */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.kt<pre><code>/* Find the specified element in the array */\nfun find(nums: IntArray, target: Int): Int {\n    for (i in nums.indices) {\n        if (nums[i] == target)\n            return i\n    }\n    return -1\n}\n</code></pre> array.rb<pre><code>### Find specified element in array ###\ndef find(nums, target)\n  for i in 0...nums.length\n    return i if nums[i] == target\n  end\n\n  -1\nend\n</code></pre>","path":["Chapter 4. Arrays and Linked Lists","4.1   Array"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#7-expanding-arrays","level":3,"title":"7.   Expanding Arrays","text":"<p>In complex system environments, programs cannot guarantee that the memory space after an array is available, making it unsafe to expand the array's capacity. Therefore, in most programming languages, the length of an array is immutable.</p> <p>If we want to expand an array, we need to create a new, larger array and then copy the original array elements to the new array one by one. This is an \\(O(n)\\) operation, which is very time-consuming when the array is large. The code is shown below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def extend(nums: list[int], enlarge: int) -&gt; list[int]:\n    \"\"\"Extend array length\"\"\"\n    # Initialize an array with extended length\n    res = [0] * (len(nums) + enlarge)\n    # Copy all elements from the original array to the new array\n    for i in range(len(nums)):\n        res[i] = nums[i]\n    # Return the extended new array\n    return res\n</code></pre> array.cpp<pre><code>/* Extend array length */\nint *extend(int *nums, int size, int enlarge) {\n    // Initialize an array with extended length\n    int *res = new int[size + enlarge];\n    // Copy all elements from the original array to the new array\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // Free memory\n    delete[] nums;\n    // Return the extended new array\n    return res;\n}\n</code></pre> array.java<pre><code>/* Extend array length */\nint[] extend(int[] nums, int enlarge) {\n    // Initialize an array with extended length\n    int[] res = new int[nums.length + enlarge];\n    // Copy all elements from the original array to the new array\n    for (int i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // Return the extended new array\n    return res;\n}\n</code></pre> array.cs<pre><code>/* Extend array length */\nint[] Extend(int[] nums, int enlarge) {\n    // Initialize an array with extended length\n    int[] res = new int[nums.Length + enlarge];\n    // Copy all elements from the original array to the new array\n    for (int i = 0; i &lt; nums.Length; i++) {\n        res[i] = nums[i];\n    }\n    // Return the extended new array\n    return res;\n}\n</code></pre> array.go<pre><code>/* Extend array length */\nfunc extend(nums []int, enlarge int) []int {\n    // Initialize an array with extended length\n    res := make([]int, len(nums)+enlarge)\n    // Copy all elements from the original array to the new array\n    for i, num := range nums {\n        res[i] = num\n    }\n    // Return the extended new array\n    return res\n}\n</code></pre> array.swift<pre><code>/* Extend array length */\nfunc extend(nums: [Int], enlarge: Int) -&gt; [Int] {\n    // Initialize an array with extended length\n    var res = Array(repeating: 0, count: nums.count + enlarge)\n    // Copy all elements from the original array to the new array\n    for i in nums.indices {\n        res[i] = nums[i]\n    }\n    // Return the extended new array\n    return res\n}\n</code></pre> array.js<pre><code>/* Extend array length */\n// Note: JavaScript's Array is dynamic array, can be directly expanded\n// For learning purposes, this function treats Array as fixed-length array\nfunction extend(nums, enlarge) {\n    // Initialize an array with extended length\n    const res = new Array(nums.length + enlarge).fill(0);\n    // Copy all elements from the original array to the new array\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // Return the extended new array\n    return res;\n}\n</code></pre> array.ts<pre><code>/* Extend array length */\n// Note: TypeScript's Array is dynamic array, can be directly expanded\n// For learning purposes, this function treats Array as fixed-length array\nfunction extend(nums: number[], enlarge: number): number[] {\n    // Initialize an array with extended length\n    const res = new Array(nums.length + enlarge).fill(0);\n    // Copy all elements from the original array to the new array\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // Return the extended new array\n    return res;\n}\n</code></pre> array.dart<pre><code>/* Extend array length */\nList&lt;int&gt; extend(List&lt;int&gt; nums, int enlarge) {\n  // Initialize an array with extended length\n  List&lt;int&gt; res = List.filled(nums.length + enlarge, 0);\n  // Copy all elements from the original array to the new array\n  for (var i = 0; i &lt; nums.length; i++) {\n    res[i] = nums[i];\n  }\n  // Return the extended new array\n  return res;\n}\n</code></pre> array.rs<pre><code>/* Extend array length */\nfn extend(nums: &amp;[i32], enlarge: usize) -&gt; Vec&lt;i32&gt; {\n    // Initialize an array with extended length\n    let mut res: Vec&lt;i32&gt; = vec![0; nums.len() + enlarge];\n    // Copy all elements from original array to new\n    res[0..nums.len()].copy_from_slice(nums);\n\n    // Return the extended new array\n    res\n}\n</code></pre> array.c<pre><code>/* Extend array length */\nint *extend(int *nums, int size, int enlarge) {\n    // Initialize an array with extended length\n    int *res = (int *)malloc(sizeof(int) * (size + enlarge));\n    // Copy all elements from the original array to the new array\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // Initialize expanded space\n    for (int i = size; i &lt; size + enlarge; i++) {\n        res[i] = 0;\n    }\n    // Return the extended new array\n    return res;\n}\n</code></pre> array.kt<pre><code>/* Extend array length */\nfun extend(nums: IntArray, enlarge: Int): IntArray {\n    // Initialize an array with extended length\n    val res = IntArray(nums.size + enlarge)\n    // Copy all elements from the original array to the new array\n    for (i in nums.indices) {\n        res[i] = nums[i]\n    }\n    // Return the extended new array\n    return res\n}\n</code></pre> array.rb<pre><code>### Extend array length ###\n# Note: Ruby's Array is dynamic array, can be directly expanded\n# For learning purposes, this function treats Array as fixed-length array\ndef extend(nums, enlarge)\n  # Initialize an array with extended length\n  res = Array.new(nums.length + enlarge, 0)\n\n  # Copy all elements from the original array to the new array\n  for i in 0...nums.length\n    res[i] = nums[i]\n  end\n\n  # Return the extended new array\n  res\nend\n</code></pre>","path":["Chapter 4. Arrays and Linked Lists","4.1   Array"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#412-advantages-and-limitations-of-arrays","level":2,"title":"4.1.2   Advantages and Limitations of Arrays","text":"<p>Arrays are stored in contiguous memory space with elements of the same type. This approach contains rich prior information that the system can use to optimize the efficiency of data structure operations.</p> <ul> <li>High space efficiency: Arrays allocate contiguous memory blocks for data without additional structural overhead.</li> <li>Support for random access: Arrays allow accessing any element in \\(O(1)\\) time.</li> <li>Cache locality: When accessing array elements, the computer not only loads the element but also caches the surrounding data, thereby leveraging the cache to improve the execution speed of subsequent operations.</li> </ul> <p>Contiguous space storage is a double-edged sword with the following limitations:</p> <ul> <li>Low insertion and deletion efficiency: When an array has many elements, insertion and deletion operations require shifting a large number of elements.</li> <li>Immutable length: After an array is initialized, its length is fixed. Expanding the array requires copying all data to a new array, which is very costly.</li> <li>Space waste: If the allocated size of an array exceeds what is actually needed, the extra space is wasted.</li> </ul>","path":["Chapter 4. Arrays and Linked Lists","4.1   Array"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#413-typical-applications-of-arrays","level":2,"title":"4.1.3   Typical Applications of Arrays","text":"<p>Arrays are a fundamental and common data structure, frequently used in various algorithms and for implementing various complex data structures.</p> <ul> <li>Random access: If we want to randomly sample some items, we can use an array to store them and generate a random sequence to implement random sampling based on indices.</li> <li>Sorting and searching: Arrays are the most commonly used data structure for sorting and searching algorithms. Quick sort, merge sort, binary search, and others are primarily performed on arrays.</li> <li>Lookup tables: When we need to quickly find an element or its corresponding relationship, we can use an array as a lookup table. For example, if we want to implement a mapping from characters to ASCII codes, we can use the ASCII code value of a character as an index, with the corresponding element stored at that position in the array.</li> <li>Machine learning: Neural networks make extensive use of linear algebra operations between vectors, matrices, and tensors, all of which are constructed in the form of arrays. Arrays are the most commonly used data structure in neural network programming.</li> <li>Data structure implementation: Arrays can be used to implement stacks, queues, hash tables, heaps, graphs, and other data structures. For example, the adjacency matrix representation of a graph is essentially a two-dimensional array.</li> </ul>","path":["Chapter 4. Arrays and Linked Lists","4.1   Array"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/","level":1,"title":"4.6   Exercises","text":"","path":["Chapter 4. Arrays and Linked Lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#461-concept-review","level":2,"title":"4.6.1   Concept Review","text":"","path":["Chapter 4. Arrays and Linked Lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#1-how-arrays-and-linked-lists-find-an-element","level":3,"title":"1.   How Arrays and Linked Lists Find an Element","text":"<p>An array and a singly linked list both store <code>[A, B, C, D, E]</code> in order. You now need to access the fourth element, <code>D</code>:</p> <ol> <li>Which index can the array use directly?</li> <li>Starting from the head node <code>A</code>, which nodes are visited in order as you follow <code>next</code>?</li> <li>As the requested element moves farther toward the end, how does the number of steps change for each structure? Which structure is better for repeated access by position, and why?</li> </ol> Answer <ol> <li> <p>With zero-based indexing, the fourth element has index 3, so the array can access <code>arr[3]</code> directly.</p> </li> <li> <p>The singly linked list must start at the head. Its access path is <code>A → B → C → D</code>, requiring three moves through <code>next</code>.</p> </li> <li> <p>An array can locate an element directly from its starting address and index, so access by position has a time complexity of \\(O(1)\\).     To access the \\(k\\)th node, a singly linked list must start at the head node and follow <code>next</code> \\(k-1\\) times,     which takes \\(O(n)\\) time in the worst case.</p> <p>This comparison concerns only access by position; it does not mean that linked lists are slower for every operation.</p> </li> </ol>","path":["Chapter 4. Arrays and Linked Lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#2-how-arrays-and-linked-lists-insert-an-element","level":3,"title":"2.   How Arrays and Linked Lists Insert an Element","text":"<p>An array and a singly linked list both store <code>A, B, C, D</code>. You now need to insert <code>X</code> after <code>B</code>:</p> <ul> <li>The array has capacity 5 and is currently <code>[A, B, C, D, _]</code>.</li> <li>The linked list is <code>A → B → C → D</code>, and you already have a reference to node <code>B</code>.</li> </ul> <ol> <li>Which elements must the array move? Write the array after insertion.</li> <li>In what order should the linked list update <code>X.next</code> and <code>B.next</code>? Write the linked list after insertion.</li> <li>When comparing insertion efficiency, why is it important to state that \"you already have a reference to node B\"?</li> </ol> Answer <ol> <li> <p>The array first moves <code>D</code> one position to the right, then moves <code>C</code> one position to the right, and finally places <code>X</code> at index 2,     producing <code>[A, B, X, C, D]</code>.</p> </li> <li> <p><code>B.next</code> originally points to <code>C</code>. First set <code>X.next = B.next</code>, making <code>X</code> point to <code>C</code>;     then set <code>B.next = X</code>. The result is <code>A → B → X → C → D</code>.     If <code>B.next</code> is overwritten first without saving the original link, <code>C</code> may become unreachable.</p> </li> <li> <p>Once the location of <code>B</code> is known, insertion into the linked list only changes two links and takes \\(O(1)\\) time.     If <code>B</code> must first be found from the head, that search alone may take \\(O(n)\\) time.</p> </li> </ol>","path":["Chapter 4. Arrays and Linked Lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#3-how-a-lists-capacity-grows","level":3,"title":"3.   How a List's Capacity Grows","text":"<p>An array-based list currently contains <code>[A, B, C]</code>, with length <code>size = 3</code> and capacity <code>capacity = 4</code>. When there is not enough capacity, a new array with twice the old capacity is created.</p> <ol> <li>After appending <code>D</code>, what are the list's length and capacity? Is capacity expansion needed?</li> <li>When <code>E</code> is appended next, what does the capacity become? How many existing elements must be copied?</li> <li>The length of the underlying array cannot change. Why does the list's capacity nevertheless appear able to grow?</li> </ol> Answer <ol> <li> <p><code>D</code> fits in the last empty position. The contents are now <code>[A, B, C, D]</code>,     with <code>size = 4</code> and <code>capacity = 4</code>, so no capacity expansion is needed.</p> </li> <li> <p>There is no empty position when <code>E</code> is appended, so a new array with capacity 8 must be created.     The four existing elements <code>A, B, C, D</code> are copied into it before <code>E</code> is added.     The result is <code>size = 5</code> and <code>capacity = 8</code>.</p> </li> <li> <p>The original array itself does not grow. The list creates a larger new array, copies the existing elements,     and then uses the new array as its underlying storage. From the user's perspective, the capacity has grown.</p> </li> </ol>","path":["Chapter 4. Arrays and Linked Lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#462-programming-exercises","level":2,"title":"4.6.2   Programming Exercises","text":"","path":["Chapter 4. Arrays and Linked Lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#1-add-one-to-a-large-integer-stored-as-an-array","level":3,"title":"1.   Add One to a Large Integer Stored as an Array","text":"<p>The array <code>digits</code> stores the decimal digits of a non-negative integer from left to right. For example, <code>[3, 0, 8]</code> represents 308. The number 0 is represented by <code>[0]</code>; for every other input, the first digit is not 0.</p> <p>Simulate decimal column addition to add 1 to this integer, and return the result in the same array format. You may modify <code>digits</code> directly. If a new carry appears at the front, you may return a longer array.</p> Hints <ol> <li>Start with the last digit of the array, just as in ordinary column addition</li> <li>If the current digit is less than 9, add one and return immediately</li> <li>If the current digit is 9, change it to 0; if every digit is 9, place a 1 at the front</li> </ol> <p>LeetCode</p>","path":["Chapter 4. Arrays and Linked Lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#2-reverse-a-singly-linked-list","level":3,"title":"2.   Reverse a Singly Linked List","text":"<p>You are given the head node <code>head</code> of a singly linked list. Each node contains a value and a <code>next</code> reference to the following node.</p> <p>Use an iterative approach to reverse all links between the nodes, and return the new head node. Do not create any new linked list nodes.</p> Hints <ol> <li>First draw three connected nodes and the two pointers prev and cur on paper</li> <li>Before changing cur.next, save the original next node in nxt</li> <li>After reversing cur.next, set prev = cur and then cur = nxt to continue with the next node in the original list</li> </ol> <p>LeetCode</p>","path":["Chapter 4. Arrays and Linked Lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/","level":1,"title":"4.2   Linked List","text":"<p>Memory is a shared resource for all programs. In a complex runtime environment, free memory may be scattered throughout the address space. We know that arrays require contiguous memory, and when an array is very large, the system may not be able to provide such a large contiguous block. This is where the flexibility of linked lists becomes apparent.</p> <p>A linked list is a linear data structure in which each element is a node object, and the nodes are connected through \"references\". A reference records the memory address of the next node, through which the next node can be accessed from the current node.</p> <p>This design allows linked-list nodes to be stored in different locations in memory, and their addresses do not need to be contiguous.</p> <p></p> <p> Figure 4-5   Linked list definition and storage method </p> <p>Observing Figure 4-5, the basic unit of a linked list is a node object. Each node contains two pieces of data: the node's \"value\" and a \"reference\" to the next node.</p> <ul> <li>The first node of a linked list is called the \"head node\", and the last node is called the \"tail node\".</li> <li>The tail node points to \"null\", which is denoted as <code>null</code>, <code>nullptr</code>, and <code>None</code> in Java, C++, and Python, respectively.</li> <li>In languages that support pointers, such as C, C++, Go, and Rust, the aforementioned \"reference\" should be replaced with \"pointer\".</li> </ul> <p>As shown in the following code, a linked list node <code>ListNode</code> contains not only a value but also an additional reference (pointer). Therefore, linked lists occupy more memory space than arrays when storing the same amount of data.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class ListNode:\n    \"\"\"Linked list node class\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val               # Node value\n        self.next: ListNode | None = None # Reference to the next node\n</code></pre> <pre><code>/* Linked list node structure */\nstruct ListNode {\n    int val;         // Node value\n    ListNode *next;  // Pointer to the next node\n    ListNode(int x) : val(x), next(nullptr) {}  // Constructor\n};\n</code></pre> <pre><code>/* Linked list node class */\nclass ListNode {\n    int val;        // Node value\n    ListNode next;  // Reference to the next node\n    ListNode(int x) { val = x; }  // Constructor\n}\n</code></pre> <pre><code>/* Linked list node class */\nclass ListNode(int x) {  // Constructor\n    int val = x;         // Node value\n    ListNode? next;      // Reference to the next node\n}\n</code></pre> <pre><code>/* Linked list node structure */\ntype ListNode struct {\n    Val  int       // Node value\n    Next *ListNode // Pointer to the next node\n}\n\n// NewListNode Constructor, creates a new linked list\nfunc NewListNode(val int) *ListNode {\n    return &amp;ListNode{\n        Val:  val,\n        Next: nil,\n    }\n}\n</code></pre> <pre><code>/* Linked list node class */\nclass ListNode {\n    var val: Int // Node value\n    var next: ListNode? // Reference to the next node\n\n    init(x: Int) { // Constructor\n        val = x\n    }\n}\n</code></pre> <pre><code>/* Linked list node class */\nclass ListNode {\n    constructor(val, next) {\n        this.val = (val === undefined ? 0 : val);       // Node value\n        this.next = (next === undefined ? null : next); // Reference to the next node\n    }\n}\n</code></pre> <pre><code>/* Linked list node class */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    constructor(val?: number, next?: ListNode | null) {\n        this.val = val === undefined ? 0 : val;        // Node value\n        this.next = next === undefined ? null : next;  // Reference to the next node\n    }\n}\n</code></pre> <pre><code>/* Linked list node class */\nclass ListNode {\n  int val; // Node value\n  ListNode? next; // Reference to the next node\n  ListNode(this.val, [this.next]); // Constructor\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* Linked list node class */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // Node value\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // Pointer to the next node\n}\n</code></pre> <pre><code>/* Linked list node structure */\ntypedef struct ListNode {\n    int val;               // Node value\n    struct ListNode *next; // Pointer to the next node\n} ListNode;\n\n/* Constructor */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* Linked list node class */\n// Constructor\nclass ListNode(x: Int) {\n    val _val: Int = x          // Node value\n    var next: ListNode? = null // Reference to the next node\n}\n</code></pre> <pre><code># Linked list node class\nclass ListNode\n  attr_accessor :val  # Node value\n  attr_accessor :next # Reference to the next node\n\n  def initialize(val=0, next_node=nil)\n    @val = val\n    @next = next_node\n  end\nend\n</code></pre>","path":["Chapter 4. Arrays and Linked Lists","4.2   Linked List"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#421-common-linked-list-operations","level":2,"title":"4.2.1   Common Linked List Operations","text":"","path":["Chapter 4. Arrays and Linked Lists","4.2   Linked List"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#1-initializing-a-linked-list","level":3,"title":"1.   Initializing a Linked List","text":"<p>Building a linked list involves two steps: first, initializing each node object; second, constructing the reference relationships between nodes. Once initialization is complete, we can traverse all nodes starting from the head node of the linked list through the reference <code>next</code>.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code># Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# Initialize each node\nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# Build references between nodes\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.cpp<pre><code>/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// Build references between nodes\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.java<pre><code>/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.cs<pre><code>/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nListNode n0 = new(1);\nListNode n1 = new(3);\nListNode n2 = new(2);\nListNode n3 = new(5);\nListNode n4 = new(4);\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.go<pre><code>/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// Build references between nodes\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n</code></pre> linked_list.swift<pre><code>/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// Build references between nodes\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.js<pre><code>/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.ts<pre><code>/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.dart<pre><code>/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\\\n// Initialize each node\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rs<pre><code>/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nlet n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None }));\nlet n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None }));\nlet n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None }));\nlet n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None }));\nlet n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None }));\n\n// Build references between nodes\nn0.borrow_mut().next = Some(n1.clone());\nn1.borrow_mut().next = Some(n2.clone());\nn2.borrow_mut().next = Some(n3.clone());\nn3.borrow_mut().next = Some(n4.clone());\n</code></pre> linked_list.c<pre><code>/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// Build references between nodes\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.kt<pre><code>/* Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize each node\nval n0 = ListNode(1)\nval n1 = ListNode(3)\nval n2 = ListNode(2)\nval n3 = ListNode(5)\nval n4 = ListNode(4)\n// Build references between nodes\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rb<pre><code># Initialize linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# Initialize each node\nn0 = ListNode.new(1)\nn1 = ListNode.new(3)\nn2 = ListNode.new(2)\nn3 = ListNode.new(5)\nn4 = ListNode.new(4)\n# Build references between nodes\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>An array is a single variable; for example, an array <code>nums</code> contains elements <code>nums[0]</code>, <code>nums[1]</code>, and so on. A linked list, by contrast, is composed of multiple independent node objects. We usually use the head node as a stand-in for the entire linked list; for example, the linked list in the above code can be referred to as linked list <code>n0</code>.</p>","path":["Chapter 4. Arrays and Linked Lists","4.2   Linked List"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#2-inserting-a-node","level":3,"title":"2.   Inserting a Node","text":"<p>Inserting a node in a linked list is very easy. As shown in Figure 4-6, suppose we want to insert a new node <code>P</code> between two adjacent nodes <code>n0</code> and <code>n1</code>. We only need to change two node references (pointers), with a time complexity of \\(O(1)\\).</p> <p>In contrast, the time complexity of inserting an element in an array is \\(O(n)\\), which is inefficient when dealing with large amounts of data.</p> <p></p> <p> Figure 4-6   Example of inserting a node into a linked list </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def insert(n0: ListNode, P: ListNode):\n    \"\"\"Insert node P after node n0 in the linked list\"\"\"\n    n1 = n0.next\n    P.next = n1\n    n0.next = P\n</code></pre> linked_list.cpp<pre><code>/* Insert node P after node n0 in the linked list */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.java<pre><code>/* Insert node P after node n0 in the linked list */\nvoid insert(ListNode n0, ListNode P) {\n    ListNode n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.cs<pre><code>/* Insert node P after node n0 in the linked list */\nvoid Insert(ListNode n0, ListNode P) {\n    ListNode? n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.go<pre><code>/* Insert node P after node n0 in the linked list */\nfunc insertNode(n0 *ListNode, P *ListNode) {\n    n1 := n0.Next\n    P.Next = n1\n    n0.Next = P\n}\n</code></pre> linked_list.swift<pre><code>/* Insert node P after node n0 in the linked list */\nfunc insert(n0: ListNode, P: ListNode) {\n    let n1 = n0.next\n    P.next = n1\n    n0.next = P\n}\n</code></pre> linked_list.js<pre><code>/* Insert node P after node n0 in the linked list */\nfunction insert(n0, P) {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.ts<pre><code>/* Insert node P after node n0 in the linked list */\nfunction insert(n0: ListNode, P: ListNode): void {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.dart<pre><code>/* Insert node P after node n0 in the linked list */\nvoid insert(ListNode n0, ListNode P) {\n  ListNode? n1 = n0.next;\n  P.next = n1;\n  n0.next = P;\n}\n</code></pre> linked_list.rs<pre><code>/* Insert node P after node n0 in the linked list */\n#[allow(non_snake_case)]\npub fn insert&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, P: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    let n1 = n0.borrow_mut().next.take();\n    P.borrow_mut().next = n1;\n    n0.borrow_mut().next = Some(P);\n}\n</code></pre> linked_list.c<pre><code>/* Insert node P after node n0 in the linked list */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.kt<pre><code>/* Insert node P after node n0 in the linked list */\nfun insert(n0: ListNode?, p: ListNode?) {\n    val n1 = n0?.next\n    p?.next = n1\n    n0?.next = p\n}\n</code></pre> linked_list.rb<pre><code>### Insert node _p after node n0 in linked list ###\n# Ruby's `p` is a built-in function, `P` is a constant, so use `_p` instead\ndef insert(n0, _p)\n  n1 = n0.next\n  _p.next = n1\n  n0.next = _p\nend\n</code></pre>","path":["Chapter 4. Arrays and Linked Lists","4.2   Linked List"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#3-removing-a-node","level":3,"title":"3.   Removing a Node","text":"<p>As shown in Figure 4-7, removing a node in a linked list is also very convenient. We only need to change one node's reference (pointer).</p> <p>Note that although node <code>P</code> still points to <code>n1</code> after the deletion operation is complete, the linked list can no longer access <code>P</code> when traversing, which means <code>P</code> no longer belongs to this linked list.</p> <p></p> <p> Figure 4-7   Removing a node from a linked list </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def remove(n0: ListNode):\n    \"\"\"Remove the first node after node n0 in the linked list\"\"\"\n    if not n0.next:\n        return\n    # n0 -&gt; P -&gt; n1\n    P = n0.next\n    n1 = P.next\n    n0.next = n1\n</code></pre> linked_list.cpp<pre><code>/* Remove the first node after node n0 in the linked list */\nvoid remove(ListNode *n0) {\n    if (n0-&gt;next == nullptr)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // Free memory\n    delete P;\n}\n</code></pre> linked_list.java<pre><code>/* Remove the first node after node n0 in the linked list */\nvoid remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.cs<pre><code>/* Remove the first node after node n0 in the linked list */\nvoid Remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode? n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.go<pre><code>/* Remove the first node after node n0 in the linked list */\nfunc removeItem(n0 *ListNode) {\n    if n0.Next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    P := n0.Next\n    n1 := P.Next\n    n0.Next = n1\n}\n</code></pre> linked_list.swift<pre><code>/* Remove the first node after node n0 in the linked list */\nfunc remove(n0: ListNode) {\n    if n0.next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    let P = n0.next\n    let n1 = P?.next\n    n0.next = n1\n}\n</code></pre> linked_list.js<pre><code>/* Remove the first node after node n0 in the linked list */\nfunction remove(n0) {\n    if (!n0.next) return;\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.ts<pre><code>/* Remove the first node after node n0 in the linked list */\nfunction remove(n0: ListNode): void {\n    if (!n0.next) {\n        return;\n    }\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.dart<pre><code>/* Remove the first node after node n0 in the linked list */\nvoid remove(ListNode n0) {\n  if (n0.next == null) return;\n  // n0 -&gt; P -&gt; n1\n  ListNode P = n0.next!;\n  ListNode? n1 = P.next;\n  n0.next = n1;\n}\n</code></pre> linked_list.rs<pre><code>/* Remove the first node after node n0 in the linked list */\n#[allow(non_snake_case)]\npub fn remove&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    // n0 -&gt; P -&gt; n1\n    let P = n0.borrow_mut().next.take();\n    if let Some(node) = P {\n        let n1 = node.borrow_mut().next.take();\n        n0.borrow_mut().next = n1;\n    }\n}\n</code></pre> linked_list.c<pre><code>/* Remove the first node after node n0 in the linked list */\n// Note: stdio.h occupies the remove keyword\nvoid removeItem(ListNode *n0) {\n    if (!n0-&gt;next)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // Free memory\n    free(P);\n}\n</code></pre> linked_list.kt<pre><code>/* Remove the first node after node n0 in the linked list */\nfun remove(n0: ListNode?) {\n    if (n0?.next == null)\n        return\n    // n0 -&gt; P -&gt; n1\n    val p = n0.next\n    val n1 = p?.next\n    n0.next = n1\n}\n</code></pre> linked_list.rb<pre><code>### Delete first node after node n0 in linked list ###\ndef remove(n0)\n  return if n0.next.nil?\n\n  # n0 -&gt; remove_node -&gt; n1\n  remove_node = n0.next\n  n1 = remove_node.next\n  n0.next = n1\nend\n</code></pre>","path":["Chapter 4. Arrays and Linked Lists","4.2   Linked List"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#4-accessing-a-node","level":3,"title":"4.   Accessing a Node","text":"<p>Accessing nodes in a linked list is less efficient. As mentioned in the previous section, we can access any element in an array in \\(O(1)\\) time. This is not the case with linked lists. The program needs to start from the head node and traverse backward one by one until the target node is found. That is, accessing the \\(i\\)-th node in a linked list requires \\(i - 1\\) iterations, with a time complexity of \\(O(n)\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def access(head: ListNode, index: int) -&gt; ListNode | None:\n    \"\"\"Access the node at index index in the linked list\"\"\"\n    for _ in range(index):\n        if not head:\n            return None\n        head = head.next\n    return head\n</code></pre> linked_list.cpp<pre><code>/* Access the node at index index in the linked list */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == nullptr)\n            return nullptr;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.java<pre><code>/* Access the node at index index in the linked list */\nListNode access(ListNode head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.cs<pre><code>/* Access the node at index index in the linked list */\nListNode? Access(ListNode? head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.go<pre><code>/* Access the node at index index in the linked list */\nfunc access(head *ListNode, index int) *ListNode {\n    for i := 0; i &lt; index; i++ {\n        if head == nil {\n            return nil\n        }\n        head = head.Next\n    }\n    return head\n}\n</code></pre> linked_list.swift<pre><code>/* Access the node at index index in the linked list */\nfunc access(head: ListNode, index: Int) -&gt; ListNode? {\n    var head: ListNode? = head\n    for _ in 0 ..&lt; index {\n        if head == nil {\n            return nil\n        }\n        head = head?.next\n    }\n    return head\n}\n</code></pre> linked_list.js<pre><code>/* Access the node at index index in the linked list */\nfunction access(head, index) {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.ts<pre><code>/* Access the node at index index in the linked list */\nfunction access(head: ListNode | null, index: number): ListNode | null {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.dart<pre><code>/* Access the node at index index in the linked list */\nListNode? access(ListNode? head, int index) {\n  for (var i = 0; i &lt; index; i++) {\n    if (head == null) return null;\n    head = head.next;\n  }\n  return head;\n}\n</code></pre> linked_list.rs<pre><code>/* Access the node at index index in the linked list */\npub fn access&lt;T&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, index: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n    fn dfs&lt;T&gt;(\n        head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,\n        index: i32,\n    ) -&gt; Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        if index &lt;= 0 {\n            return head.cloned();\n        }\n\n        if let Some(node) = head {\n            dfs(node.borrow().next.as_ref(), index - 1)\n        } else {\n            None\n        }\n    }\n\n    dfs(Some(head).as_ref(), index)\n}\n</code></pre> linked_list.c<pre><code>/* Access the node at index index in the linked list */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == NULL)\n            return NULL;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.kt<pre><code>/* Access the node at index index in the linked list */\nfun access(head: ListNode?, index: Int): ListNode? {\n    var h = head\n    for (i in 0..&lt;index) {\n        if (h == null)\n            return null\n        h = h.next\n    }\n    return h\n}\n</code></pre> linked_list.rb<pre><code>### Access node at index in linked list ###\ndef access(head, index)\n  for i in 0...index\n    return nil if head.nil?\n    head = head.next\n  end\n\n  head\nend\n</code></pre>","path":["Chapter 4. Arrays and Linked Lists","4.2   Linked List"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#5-finding-a-node","level":3,"title":"5.   Finding a Node","text":"<p>Traverse the linked list to find a node with value <code>target</code>, and output the index of that node in the linked list. This process is also a linear search. The code is shown below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def find(head: ListNode, target: int) -&gt; int:\n    \"\"\"Find the first node with value target in the linked list\"\"\"\n    index = 0\n    while head:\n        if head.val == target:\n            return index\n        head = head.next\n        index += 1\n    return -1\n</code></pre> linked_list.cpp<pre><code>/* Find the first node with value target in the linked list */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head != nullptr) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.java<pre><code>/* Find the first node with value target in the linked list */\nint find(ListNode head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.cs<pre><code>/* Find the first node with value target in the linked list */\nint Find(ListNode? head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.go<pre><code>/* Find the first node with value target in the linked list */\nfunc findNode(head *ListNode, target int) int {\n    index := 0\n    for head != nil {\n        if head.Val == target {\n            return index\n        }\n        head = head.Next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.swift<pre><code>/* Find the first node with value target in the linked list */\nfunc find(head: ListNode, target: Int) -&gt; Int {\n    var head: ListNode? = head\n    var index = 0\n    while head != nil {\n        if head?.val == target {\n            return index\n        }\n        head = head?.next\n        index += 1\n    }\n    return -1\n}\n</code></pre> linked_list.js<pre><code>/* Find the first node with value target in the linked list */\nfunction find(head, target) {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.ts<pre><code>/* Find the first node with value target in the linked list */\nfunction find(head: ListNode | null, target: number): number {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.dart<pre><code>/* Find the first node with value target in the linked list */\nint find(ListNode? head, int target) {\n  int index = 0;\n  while (head != null) {\n    if (head.val == target) {\n      return index;\n    }\n    head = head.next;\n    index++;\n  }\n  return -1;\n}\n</code></pre> linked_list.rs<pre><code>/* Find the first node with value target in the linked list */\npub fn find&lt;T: PartialEq&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, target: T) -&gt; i32 {\n    fn find&lt;T: PartialEq&gt;(head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, target: T, idx: i32) -&gt; i32 {\n        if let Some(node) = head {\n            if node.borrow().val == target {\n                return idx;\n            }\n            return find(node.borrow().next.as_ref(), target, idx + 1);\n        } else {\n            -1\n        }\n    }\n\n    find(Some(head).as_ref(), target, 0)\n}\n</code></pre> linked_list.c<pre><code>/* Find the first node with value target in the linked list */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.kt<pre><code>/* Find the first node with value target in the linked list */\nfun find(head: ListNode?, target: Int): Int {\n    var index = 0\n    var h = head\n    while (h != null) {\n        if (h._val == target)\n            return index\n        h = h.next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.rb<pre><code>### Find first node with value target in linked list ###\ndef find(head, target)\n  index = 0\n  while head\n    return index if head.val == target\n    head = head.next\n    index += 1\n  end\n\n  -1\nend\n</code></pre>","path":["Chapter 4. Arrays and Linked Lists","4.2   Linked List"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#422-arrays-vs-linked-lists","level":2,"title":"4.2.2   Arrays vs. Linked Lists","text":"<p>Table 4-1 summarizes the characteristics of arrays and linked lists and compares their operational efficiencies. Since they employ two opposite storage strategies, their various properties and operational efficiencies also exhibit contrasting characteristics.</p> <p> Table 4-1   Comparison of array and linked list efficiencies </p> Array Linked List Storage method Contiguous memory space Scattered memory space Capacity expansion Immutable length Flexible expansion Memory efficiency Elements occupy less memory, but space may be wasted Elements occupy more memory Accessing an element \\(O(1)\\) \\(O(n)\\) Adding an element \\(O(n)\\) \\(O(1)\\) Removing an element \\(O(n)\\) \\(O(1)\\)","path":["Chapter 4. Arrays and Linked Lists","4.2   Linked List"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#423-common-types-of-linked-lists","level":2,"title":"4.2.3   Common Types of Linked Lists","text":"<p>As shown in Figure 4-8, there are three common types of linked lists:</p> <ul> <li>Singly linked list: This is the ordinary linked list introduced earlier. The nodes of a singly linked list contain a value and a reference to the next node. We call the first node the head node and the last node the tail node; the tail node points to <code>None</code>.</li> <li>Circular linked list: If we make the tail node of a singly linked list point to the head node (connecting the tail to the head), we get a circular linked list. In a circular linked list, any node can be viewed as the head node.</li> <li>Doubly linked list: Compared to a singly linked list, a doubly linked list records references in both directions. The node definition of a doubly linked list includes references to both the successor node (next node) and the predecessor node (previous node). Compared to a singly linked list, a doubly linked list is more flexible and can traverse the linked list in both directions, but it also requires more memory space.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class ListNode:\n    \"\"\"Doubly linked list node class\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # Node value\n        self.next: ListNode | None = None  # Reference to the successor node\n        self.prev: ListNode | None = None  # Reference to the predecessor node\n</code></pre> <pre><code>/* Doubly linked list node structure */\nstruct ListNode {\n    int val;         // Node value\n    ListNode *next;  // Pointer to the successor node\n    ListNode *prev;  // Pointer to the predecessor node\n    ListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // Constructor\n};\n</code></pre> <pre><code>/* Doubly linked list node class */\nclass ListNode {\n    int val;        // Node value\n    ListNode next;  // Reference to the successor node\n    ListNode prev;  // Reference to the predecessor node\n    ListNode(int x) { val = x; }  // Constructor\n}\n</code></pre> <pre><code>/* Doubly linked list node class */\nclass ListNode(int x) {  // Constructor\n    int val = x;    // Node value\n    ListNode next;  // Reference to the successor node\n    ListNode prev;  // Reference to the predecessor node\n}\n</code></pre> <pre><code>/* Doubly linked list node structure */\ntype DoublyListNode struct {\n    Val  int             // Node value\n    Next *DoublyListNode // Pointer to the successor node\n    Prev *DoublyListNode // Pointer to the predecessor node\n}\n\n// NewDoublyListNode Initialization\nfunc NewDoublyListNode(val int) *DoublyListNode {\n    return &amp;DoublyListNode{\n        Val:  val,\n        Next: nil,\n        Prev: nil,\n    }\n}\n</code></pre> <pre><code>/* Doubly linked list node class */\nclass ListNode {\n    var val: Int // Node value\n    var next: ListNode? // Reference to the successor node\n    var prev: ListNode? // Reference to the predecessor node\n\n    init(x: Int) { // Constructor\n        val = x\n    }\n}\n</code></pre> <pre><code>/* Doubly linked list node class */\nclass ListNode {\n    constructor(val, next, prev) {\n        this.val = val  ===  undefined ? 0 : val;        // Node value\n        this.next = next  ===  undefined ? null : next;  // Reference to the successor node\n        this.prev = prev  ===  undefined ? null : prev;  // Reference to the predecessor node\n    }\n}\n</code></pre> <pre><code>/* Doubly linked list node class */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    prev: ListNode | null;\n    constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\n        this.val = val  ===  undefined ? 0 : val;        // Node value\n        this.next = next  ===  undefined ? null : next;  // Reference to the successor node\n        this.prev = prev  ===  undefined ? null : prev;  // Reference to the predecessor node\n    }\n}\n</code></pre> <pre><code>/* Doubly linked list node class */\nclass ListNode {\n    int val;        // Node value\n    ListNode? next;  // Reference to the successor node\n    ListNode? prev;  // Reference to the predecessor node\n    ListNode(this.val, [this.next, this.prev]);  // Constructor\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* Doubly linked list node type */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // Node value\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // Pointer to the successor node\n    prev: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // Pointer to the predecessor node\n}\n\n/* Constructor */\nimpl ListNode {\n    fn new(val: i32) -&gt; Self {\n        ListNode {\n            val,\n            next: None,\n            prev: None,\n        }\n    }\n}\n</code></pre> <pre><code>/* Doubly linked list node structure */\ntypedef struct ListNode {\n    int val;               // Node value\n    struct ListNode *next; // Pointer to the successor node\n    struct ListNode *prev; // Pointer to the predecessor node\n} ListNode;\n\n/* Constructor */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    node-&gt;prev = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* Doubly linked list node class */\n// Constructor\nclass ListNode(x: Int) {\n    val _val: Int = x           // Node value\n    var next: ListNode? = null  // Reference to the successor node\n    var prev: ListNode? = null  // Reference to the predecessor node\n}\n</code></pre> <pre><code># Doubly linked list node class\nclass ListNode\n  attr_accessor :val    # Node value\n  attr_accessor :next   # Reference to the successor node\n  attr_accessor :prev   # Reference to the predecessor node\n\n  def initialize(val=0, next_node=nil, prev_node=nil)\n    @val = val\n    @next = next_node\n    @prev = prev_node\n  end\nend\n</code></pre> <p></p> <p> Figure 4-8   Common types of linked lists </p>","path":["Chapter 4. Arrays and Linked Lists","4.2   Linked List"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#424-typical-applications-of-linked-lists","level":2,"title":"4.2.4   Typical Applications of Linked Lists","text":"<p>Singly linked lists are commonly used to implement stacks, queues, hash tables, and graphs.</p> <ul> <li>Stacks and queues: When insertion and deletion operations both occur at one end of the linked list, it exhibits last-in-first-out characteristics, corresponding to a stack. When insertion operations occur at one end of the linked list and deletion operations occur at the other end, it exhibits first-in-first-out characteristics, corresponding to a queue.</li> <li>Hash tables: Separate chaining is one of the mainstream solutions for resolving hash collisions. In this approach, all colliding elements are placed in a linked list.</li> <li>Graphs: An adjacency list is a common way to represent a graph, where each vertex in the graph is associated with a linked list, and each element in the linked list represents another vertex connected to that vertex.</li> </ul> <p>Doubly linked lists are commonly used in scenarios where quick access to the previous and next elements is needed.</p> <ul> <li>Advanced data structures: For example, in red-black trees and B-trees, we need to access the parent node of a node, which can be achieved by saving a reference to the parent node in the node, similar to a doubly linked list.</li> <li>Browser history: In web browsers, when a user clicks the forward or backward button, the browser needs to know the previous and next web pages the user visited. The characteristics of doubly linked lists make this operation simple.</li> <li>LRU algorithm: In cache eviction (LRU) algorithms, we need to quickly find the least recently used data and support quick addition and deletion of nodes. Using a doubly linked list is very suitable for this.</li> </ul> <p>Circular linked lists are commonly used in scenarios that require periodic operations, such as operating system resource scheduling.</p> <ul> <li>Round-robin scheduling algorithm: In operating systems, round-robin scheduling is a common CPU scheduling algorithm that needs to cycle through a set of processes. Each process is assigned a time slice, and when the time slice expires, the CPU switches to the next process. This cyclic operation can be implemented using a circular linked list.</li> <li>Data buffers: In some data buffer implementations, circular linked lists may also be used. For example, in audio and video players, the data stream may be divided into multiple buffer blocks and placed in a circular linked list to achieve seamless playback.</li> </ul>","path":["Chapter 4. Arrays and Linked Lists","4.2   Linked List"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/","level":1,"title":"4.3   List","text":"<p>A list is an abstract data structure concept that represents an ordered collection of elements, supporting operations such as element access, modification, insertion, deletion, and traversal, without requiring users to consider capacity limitations. Lists can be implemented based on linked lists or arrays.</p> <ul> <li>A linked list can naturally be viewed as a list: it supports insertion, deletion, search, and update, and can grow flexibly as needed.</li> <li>An array also supports insertion, deletion, search, and update, but because its length is fixed, it can only be regarded as a list with a capacity limit.</li> </ul> <p>When a list is implemented with an array, its fixed length makes it less practical. This is because we usually cannot determine in advance how much data we need to store, making it difficult to choose an appropriate capacity. If the capacity is too small, it may fail to meet our needs; if it is too large, memory space will be wasted.</p> <p>To solve this problem, we can use a dynamic array to implement a list. It inherits all the advantages of arrays while supporting dynamic resizing during program execution.</p> <p>In fact, the list types provided by the standard libraries of many programming languages are implemented with dynamic arrays, such as <code>list</code> in Python, <code>ArrayList</code> in Java, <code>vector</code> in C++, and <code>List</code> in C#. In the following discussion, we will treat \"list\" and \"dynamic array\" as equivalent concepts.</p>","path":["Chapter 4. Arrays and Linked Lists","4.3   List"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#431-common-list-operations","level":2,"title":"4.3.1   Common List Operations","text":"","path":["Chapter 4. Arrays and Linked Lists","4.3   List"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#1-initialize-a-list","level":3,"title":"1.   Initialize a List","text":"<p>We typically initialize a list in one of two ways: empty or with predefined values:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># Initialize a list\n# Without initial values\nnums1: list[int] = []\n# With initial values\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> list.cpp<pre><code>/* Initialize a list */\n// Note that vector in C++ is equivalent to nums as described in this article\n// Without initial values\nvector&lt;int&gt; nums1;\n// With initial values\nvector&lt;int&gt; nums = { 1, 3, 2, 5, 4 };\n</code></pre> list.java<pre><code>/* Initialize a list */\n// Without initial values\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;();\n// With initial values (note that array elements should use the wrapper class Integer[] instead of int[])\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList&lt;Integer&gt; nums = new ArrayList&lt;&gt;(Arrays.asList(numbers));\n</code></pre> list.cs<pre><code>/* Initialize a list */\n// Without initial values\nList&lt;int&gt; nums1 = [];\n// With initial values\nint[] numbers = [1, 3, 2, 5, 4];\nList&lt;int&gt; nums = [.. numbers];\n</code></pre> list_test.go<pre><code>/* Initialize a list */\n// Without initial values\nnums1 := []int{}\n// With initial values\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> list.swift<pre><code>/* Initialize a list */\n// Without initial values\nlet nums1: [Int] = []\n// With initial values\nvar nums = [1, 3, 2, 5, 4]\n</code></pre> list.js<pre><code>/* Initialize a list */\n// Without initial values\nconst nums1 = [];\n// With initial values\nconst nums = [1, 3, 2, 5, 4];\n</code></pre> list.ts<pre><code>/* Initialize a list */\n// Without initial values\nconst nums1: number[] = [];\n// With initial values\nconst nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> list.dart<pre><code>/* Initialize a list */\n// Without initial values\nList&lt;int&gt; nums1 = [];\n// With initial values\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> list.rs<pre><code>/* Initialize a list */\n// Without initial values\nlet nums1: Vec&lt;i32&gt; = Vec::new();\n// With initial values\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>/* Initialize a list */\n// Without initial values\nvar nums1 = listOf&lt;Int&gt;()\n// With initial values\nvar numbers = arrayOf(1, 3, 2, 5, 4)\nvar nums = numbers.toMutableList()\n</code></pre> list.rb<pre><code># Initialize a list\n# Without initial values\nnums1 = []\n# With initial values\nnums = [1, 3, 2, 5, 4]\n</code></pre> Code Visualization <p> Full Screen &gt;</p>","path":["Chapter 4. Arrays and Linked Lists","4.3   List"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#2-access-elements","level":3,"title":"2.   Access Elements","text":"<p>Since a list is essentially an array, we can access and update elements in \\(O(1)\\) time complexity, which is very efficient.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># Access an element\nnum: int = nums[1]  # Access element at index 1\n\n# Update an element\nnums[1] = 0    # Update element at index 1 to 0\n</code></pre> list.cpp<pre><code>/* Access an element */\nint num = nums[1];  // Access element at index 1\n\n/* Update an element */\nnums[1] = 0;  // Update element at index 1 to 0\n</code></pre> list.java<pre><code>/* Access an element */\nint num = nums.get(1);  // Access element at index 1\n\n/* Update an element */\nnums.set(1, 0);  // Update element at index 1 to 0\n</code></pre> list.cs<pre><code>/* Access an element */\nint num = nums[1];  // Access element at index 1\n\n/* Update an element */\nnums[1] = 0;  // Update element at index 1 to 0\n</code></pre> list_test.go<pre><code>/* Access an element */\nnum := nums[1]  // Access element at index 1\n\n/* Update an element */\nnums[1] = 0     // Update element at index 1 to 0\n</code></pre> list.swift<pre><code>/* Access an element */\nlet num = nums[1] // Access element at index 1\n\n/* Update an element */\nnums[1] = 0 // Update element at index 1 to 0\n</code></pre> list.js<pre><code>/* Access an element */\nconst num = nums[1];  // Access element at index 1\n\n/* Update an element */\nnums[1] = 0;  // Update element at index 1 to 0\n</code></pre> list.ts<pre><code>/* Access an element */\nconst num: number = nums[1];  // Access element at index 1\n\n/* Update an element */\nnums[1] = 0;  // Update element at index 1 to 0\n</code></pre> list.dart<pre><code>/* Access an element */\nint num = nums[1];  // Access element at index 1\n\n/* Update an element */\nnums[1] = 0;  // Update element at index 1 to 0\n</code></pre> list.rs<pre><code>/* Access an element */\nlet num: i32 = nums[1];  // Access element at index 1\n/* Update an element */\nnums[1] = 0;             // Update element at index 1 to 0\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>/* Access an element */\nval num = nums[1]       // Access element at index 1\n/* Update an element */\nnums[1] = 0             // Update element at index 1 to 0\n</code></pre> list.rb<pre><code># Access an element\nnum = nums[1] # Access element at index 1\n# Update an element\nnums[1] = 0 # Update element at index 1 to 0\n</code></pre> Code Visualization <p> Full Screen &gt;</p>","path":["Chapter 4. Arrays and Linked Lists","4.3   List"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#3-insert-and-delete-elements","level":3,"title":"3.   Insert and Delete Elements","text":"<p>Compared to arrays, lists can freely add and delete elements. Adding an element at the end of a list has a time complexity of \\(O(1)\\), but inserting and deleting elements still have the same efficiency as arrays, with a time complexity of \\(O(n)\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># Clear the list\nnums.clear()\n\n# Add elements at the end\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n# Insert an element in the middle\nnums.insert(3, 6)  # Insert number 6 at index 3\n\n# Delete an element\nnums.pop(3)        # Delete element at index 3\n</code></pre> list.cpp<pre><code>/* Clear the list */\nnums.clear();\n\n/* Add elements at the end */\nnums.push_back(1);\nnums.push_back(3);\nnums.push_back(2);\nnums.push_back(5);\nnums.push_back(4);\n\n/* Insert an element in the middle */\nnums.insert(nums.begin() + 3, 6);  // Insert number 6 at index 3\n\n/* Delete an element */\nnums.erase(nums.begin() + 3);      // Delete element at index 3\n</code></pre> list.java<pre><code>/* Clear the list */\nnums.clear();\n\n/* Add elements at the end */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* Insert an element in the middle */\nnums.add(3, 6);  // Insert number 6 at index 3\n\n/* Delete an element */\nnums.remove(3);  // Delete element at index 3\n</code></pre> list.cs<pre><code>/* Clear the list */\nnums.Clear();\n\n/* Add elements at the end */\nnums.Add(1);\nnums.Add(3);\nnums.Add(2);\nnums.Add(5);\nnums.Add(4);\n\n/* Insert an element in the middle */\nnums.Insert(3, 6);  // Insert number 6 at index 3\n\n/* Delete an element */\nnums.RemoveAt(3);  // Delete element at index 3\n</code></pre> list_test.go<pre><code>/* Clear the list */\nnums = nil\n\n/* Add elements at the end */\nnums = append(nums, 1)\nnums = append(nums, 3)\nnums = append(nums, 2)\nnums = append(nums, 5)\nnums = append(nums, 4)\n\n/* Insert an element in the middle */\nnums = append(nums[:3], append([]int{6}, nums[3:]...)...) // Insert number 6 at index 3\n\n/* Delete an element */\nnums = append(nums[:3], nums[4:]...) // Delete element at index 3\n</code></pre> list.swift<pre><code>/* Clear the list */\nnums.removeAll()\n\n/* Add elements at the end */\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n/* Insert an element in the middle */\nnums.insert(6, at: 3) // Insert number 6 at index 3\n\n/* Delete an element */\nnums.remove(at: 3) // Delete element at index 3\n</code></pre> list.js<pre><code>/* Clear the list */\nnums.length = 0;\n\n/* Add elements at the end */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* Insert an element in the middle */\nnums.splice(3, 0, 6); // Insert number 6 at index 3\n\n/* Delete an element */\nnums.splice(3, 1);  // Delete element at index 3\n</code></pre> list.ts<pre><code>/* Clear the list */\nnums.length = 0;\n\n/* Add elements at the end */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* Insert an element in the middle */\nnums.splice(3, 0, 6); // Insert number 6 at index 3\n\n/* Delete an element */\nnums.splice(3, 1);  // Delete element at index 3\n</code></pre> list.dart<pre><code>/* Clear the list */\nnums.clear();\n\n/* Add elements at the end */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* Insert an element in the middle */\nnums.insert(3, 6); // Insert number 6 at index 3\n\n/* Delete an element */\nnums.removeAt(3); // Delete element at index 3\n</code></pre> list.rs<pre><code>/* Clear the list */\nnums.clear();\n\n/* Add elements at the end */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* Insert an element in the middle */\nnums.insert(3, 6);  // Insert number 6 at index 3\n\n/* Delete an element */\nnums.remove(3);    // Delete element at index 3\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>/* Clear the list */\nnums.clear();\n\n/* Add elements at the end */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* Insert an element in the middle */\nnums.add(3, 6);  // Insert number 6 at index 3\n\n/* Delete an element */\nnums.remove(3);  // Delete element at index 3\n</code></pre> list.rb<pre><code># Clear the list\nnums.clear\n\n# Add elements at the end\nnums &lt;&lt; 1\nnums &lt;&lt; 3\nnums &lt;&lt; 2\nnums &lt;&lt; 5\nnums &lt;&lt; 4\n\n# Insert an element in the middle\nnums.insert(3, 6) # Insert number 6 at index 3\n\n# Delete an element\nnums.delete_at(3) # Delete element at index 3\n</code></pre> Code Visualization <p> Full Screen &gt;</p>","path":["Chapter 4. Arrays and Linked Lists","4.3   List"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#4-traverse-a-list","level":3,"title":"4.   Traverse a List","text":"<p>Like arrays, lists can be traversed by index or by directly iterating through elements.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># Traverse the list by index\ncount = 0\nfor i in range(len(nums)):\n    count += nums[i]\n\n# Traverse list elements directly\nfor num in nums:\n    count += num\n</code></pre> list.cpp<pre><code>/* Traverse the list by index */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums[i];\n}\n\n/* Traverse list elements directly */\ncount = 0;\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.java<pre><code>/* Traverse the list by index */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums.get(i);\n}\n\n/* Traverse list elements directly */\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.cs<pre><code>/* Traverse the list by index */\nint count = 0;\nfor (int i = 0; i &lt; nums.Count; i++) {\n    count += nums[i];\n}\n\n/* Traverse list elements directly */\ncount = 0;\nforeach (int num in nums) {\n    count += num;\n}\n</code></pre> list_test.go<pre><code>/* Traverse the list by index */\ncount := 0\nfor i := 0; i &lt; len(nums); i++ {\n    count += nums[i]\n}\n\n/* Traverse list elements directly */\ncount = 0\nfor _, num := range nums {\n    count += num\n}\n</code></pre> list.swift<pre><code>/* Traverse the list by index */\nvar count = 0\nfor i in nums.indices {\n    count += nums[i]\n}\n\n/* Traverse list elements directly */\ncount = 0\nfor num in nums {\n    count += num\n}\n</code></pre> list.js<pre><code>/* Traverse the list by index */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* Traverse list elements directly */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.ts<pre><code>/* Traverse the list by index */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* Traverse list elements directly */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.dart<pre><code>/* Traverse the list by index */\nint count = 0;\nfor (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* Traverse list elements directly */\ncount = 0;\nfor (var num in nums) {\n    count += num;\n}\n</code></pre> list.rs<pre><code>// Traverse the list by index\nlet mut _count = 0;\nfor i in 0..nums.len() {\n    _count += nums[i];\n}\n\n// Traverse list elements directly\n_count = 0;\nfor num in &amp;nums {\n    _count += num;\n}\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>/* Traverse the list by index */\nvar count = 0\nfor (i in nums.indices) {\n    count += nums[i]\n}\n\n/* Traverse list elements directly */\nfor (num in nums) {\n    count += num\n}\n</code></pre> list.rb<pre><code># Traverse the list by index\ncount = 0\nfor i in 0...nums.length\n    count += nums[i]\nend\n\n# Traverse list elements directly\ncount = 0\nfor num in nums\n    count += num\nend\n</code></pre> Code Visualization <p> Full Screen &gt;</p>","path":["Chapter 4. Arrays and Linked Lists","4.3   List"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#5-concatenate-lists","level":3,"title":"5.   Concatenate Lists","text":"<p>Given a new list <code>nums1</code>, we can concatenate it to the end of the original list.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># Concatenate two lists\nnums1: list[int] = [6, 8, 7, 10, 9]\nnums += nums1  # Concatenate list nums1 to the end of nums\n</code></pre> list.cpp<pre><code>/* Concatenate two lists */\nvector&lt;int&gt; nums1 = { 6, 8, 7, 10, 9 };\n// Concatenate list nums1 to the end of nums\nnums.insert(nums.end(), nums1.begin(), nums1.end());\n</code></pre> list.java<pre><code>/* Concatenate two lists */\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nnums.addAll(nums1);  // Concatenate list nums1 to the end of nums\n</code></pre> list.cs<pre><code>/* Concatenate two lists */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.AddRange(nums1);  // Concatenate list nums1 to the end of nums\n</code></pre> list_test.go<pre><code>/* Concatenate two lists */\nnums1 := []int{6, 8, 7, 10, 9}\nnums = append(nums, nums1...)  // Concatenate list nums1 to the end of nums\n</code></pre> list.swift<pre><code>/* Concatenate two lists */\nlet nums1 = [6, 8, 7, 10, 9]\nnums.append(contentsOf: nums1) // Concatenate list nums1 to the end of nums\n</code></pre> list.js<pre><code>/* Concatenate two lists */\nconst nums1 = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // Concatenate list nums1 to the end of nums\n</code></pre> list.ts<pre><code>/* Concatenate two lists */\nconst nums1: number[] = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // Concatenate list nums1 to the end of nums\n</code></pre> list.dart<pre><code>/* Concatenate two lists */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.addAll(nums1);  // Concatenate list nums1 to the end of nums\n</code></pre> list.rs<pre><code>/* Concatenate two lists */\nlet nums1: Vec&lt;i32&gt; = vec![6, 8, 7, 10, 9];\nnums.extend(nums1);\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>/* Concatenate two lists */\nval nums1 = intArrayOf(6, 8, 7, 10, 9).toMutableList()\nnums.addAll(nums1)  // Concatenate list nums1 to the end of nums\n</code></pre> list.rb<pre><code># Concatenate two lists\nnums1 = [6, 8, 7, 10, 9]\nnums += nums1\n</code></pre> Code Visualization <p> Full Screen &gt;</p>","path":["Chapter 4. Arrays and Linked Lists","4.3   List"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#6-sort-a-list","level":3,"title":"6.   Sort a List","text":"<p>After sorting a list, we can use \"binary search\" and \"two-pointer\" algorithms, which are frequently tested in array algorithm problems.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># Sort a list\nnums.sort()  # After sorting, list elements are arranged from smallest to largest\n</code></pre> list.cpp<pre><code>/* Sort a list */\nsort(nums.begin(), nums.end());  // After sorting, list elements are arranged from smallest to largest\n</code></pre> list.java<pre><code>/* Sort a list */\nCollections.sort(nums);  // After sorting, list elements are arranged from smallest to largest\n</code></pre> list.cs<pre><code>/* Sort a list */\nnums.Sort(); // After sorting, list elements are arranged from smallest to largest\n</code></pre> list_test.go<pre><code>/* Sort a list */\nsort.Ints(nums)  // After sorting, list elements are arranged from smallest to largest\n</code></pre> list.swift<pre><code>/* Sort a list */\nnums.sort() // After sorting, list elements are arranged from smallest to largest\n</code></pre> list.js<pre><code>/* Sort a list */\nnums.sort((a, b) =&gt; a - b);  // After sorting, list elements are arranged from smallest to largest\n</code></pre> list.ts<pre><code>/* Sort a list */\nnums.sort((a, b) =&gt; a - b);  // After sorting, list elements are arranged from smallest to largest\n</code></pre> list.dart<pre><code>/* Sort a list */\nnums.sort(); // After sorting, list elements are arranged from smallest to largest\n</code></pre> list.rs<pre><code>/* Sort a list */\nnums.sort(); // After sorting, list elements are arranged from smallest to largest\n</code></pre> list.c<pre><code>// C does not provide built-in dynamic arrays\n</code></pre> list.kt<pre><code>/* Sort a list */\nnums.sort() // After sorting, list elements are arranged from smallest to largest\n</code></pre> list.rb<pre><code># Sort a list\nnums = nums.sort { |a, b| a &lt;=&gt; b } # After sorting, list elements are arranged from smallest to largest\n</code></pre> Code Visualization <p> Full Screen &gt;</p>","path":["Chapter 4. Arrays and Linked Lists","4.3   List"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#432-list-implementation","level":2,"title":"4.3.2   List Implementation","text":"<p>Many programming languages have built-in lists, such as Java, C++, and Python. Their implementations are quite complex, and the parameters are carefully considered, such as initial capacity, expansion multiples, and so on. Interested readers can consult the source code to learn more.</p> <p>To deepen our understanding of how lists work, we attempt to implement a simple list with three key design considerations:</p> <ul> <li>Initial capacity: Select a reasonable initial capacity for the underlying array. In this example, we choose 10 as the initial capacity.</li> <li>Size tracking: Declare a variable <code>size</code> to record the current number of elements in the list and update it in real-time as elements are inserted and deleted. Based on this variable, we can locate the end of the list and determine whether expansion is needed.</li> <li>Expansion mechanism: When the list capacity is full upon inserting an element, we need to expand. We create a larger array based on the expansion multiple and then move all elements from the current array to the new array in order. In this example, we specify that the array should be expanded to 2 times its previous size each time.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_list.py<pre><code>class MyList:\n    \"\"\"List class\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        self._capacity: int = 10  # List capacity\n        self._arr: list[int] = [0] * self._capacity  # Array (stores list elements)\n        self._size: int = 0  # List length (current number of elements)\n        self._extend_ratio: int = 2  # Multiple by which the list capacity is extended each time\n\n    def size(self) -&gt; int:\n        \"\"\"Get list length (current number of elements)\"\"\"\n        return self._size\n\n    def capacity(self) -&gt; int:\n        \"\"\"Get list capacity\"\"\"\n        return self._capacity\n\n    def get(self, index: int) -&gt; int:\n        \"\"\"Access element\"\"\"\n        # If the index is out of bounds, throw an exception, as below\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"Index out of bounds\")\n        return self._arr[index]\n\n    def set(self, num: int, index: int):\n        \"\"\"Update element\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"Index out of bounds\")\n        self._arr[index] = num\n\n    def add(self, num: int):\n        \"\"\"Add element at the end\"\"\"\n        # When the number of elements exceeds capacity, trigger the extension mechanism\n        if self.size() == self.capacity():\n            self.extend_capacity()\n        self._arr[self._size] = num\n        self._size += 1\n\n    def insert(self, num: int, index: int):\n        \"\"\"Insert element in the middle\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"Index out of bounds\")\n        # When the number of elements exceeds capacity, trigger the extension mechanism\n        if self._size == self.capacity():\n            self.extend_capacity()\n        # Move all elements at and after index index backward by one position\n        for j in range(self._size - 1, index - 1, -1):\n            self._arr[j + 1] = self._arr[j]\n        self._arr[index] = num\n        # Update the number of elements\n        self._size += 1\n\n    def remove(self, index: int) -&gt; int:\n        \"\"\"Remove element\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"Index out of bounds\")\n        num = self._arr[index]\n        # Move all elements after index index forward by one position\n        for j in range(index, self._size - 1):\n            self._arr[j] = self._arr[j + 1]\n        # Update the number of elements\n        self._size -= 1\n        # Return the removed element\n        return num\n\n    def extend_capacity(self):\n        \"\"\"Extend list capacity\"\"\"\n        # Create a new array with length _extend_ratio times the original array, and copy the original array to the new array\n        self._arr = self._arr + [0] * self.capacity() * (self._extend_ratio - 1)\n        # Update list capacity\n        self._capacity = len(self._arr)\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"Return list with valid length\"\"\"\n        return self._arr[: self._size]\n</code></pre> my_list.cpp<pre><code>/* List class */\nclass MyList {\n  private:\n    int *arr;             // Array (stores list elements)\n    int arrCapacity = 10; // List capacity\n    int arrSize = 0;      // List length (current number of elements)\n    int extendRatio = 2;   // Multiple by which the list capacity is extended each time\n\n  public:\n    /* Constructor */\n    MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* Destructor */\n    ~MyList() {\n        delete[] arr;\n    }\n\n    /* Get list length (current number of elements)*/\n    int size() {\n        return arrSize;\n    }\n\n    /* Get list capacity */\n    int capacity() {\n        return arrCapacity;\n    }\n\n    /* Update element */\n    int get(int index) {\n        // If the index is out of bounds, throw an exception, as below\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"Index out of bounds\");\n        return arr[index];\n    }\n\n    /* Add elements at the end */\n    void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"Index out of bounds\");\n        arr[index] = num;\n    }\n\n    /* Direct traversal of list elements */\n    void add(int num) {\n        // When the number of elements exceeds capacity, trigger the extension mechanism\n        if (size() == capacity())\n            extendCapacity();\n        arr[size()] = num;\n        // Update the number of elements\n        arrSize++;\n    }\n\n    /* Sort list */\n    void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"Index out of bounds\");\n        // When the number of elements exceeds capacity, trigger the extension mechanism\n        if (size() == capacity())\n            extendCapacity();\n        // Move all elements after index index forward by one position\n        for (int j = size() - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // Update the number of elements\n        arrSize++;\n    }\n\n    /* Remove element */\n    int remove(int index) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"Index out of bounds\");\n        int num = arr[index];\n        // Create a new array with length _extend_ratio times the original array, and copy the original array to the new array\n        for (int j = index; j &lt; size() - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // Update the number of elements\n        arrSize--;\n        // Return the removed element\n        return num;\n    }\n\n    /* Driver Code */\n    void extendCapacity() {\n        // Create a new array with length extendRatio times the original array\n        int newCapacity = capacity() * extendRatio;\n        int *tmp = arr;\n        arr = new int[newCapacity];\n        // Copy all elements from the original array to the new array\n        for (int i = 0; i &lt; size(); i++) {\n            arr[i] = tmp[i];\n        }\n        // Free memory\n        delete[] tmp;\n        arrCapacity = newCapacity;\n    }\n\n    /* Convert list to Vector for printing */\n    vector&lt;int&gt; toVector() {\n        // Elements enqueue\n        vector&lt;int&gt; vec(size());\n        for (int i = 0; i &lt; size(); i++) {\n            vec[i] = arr[i];\n        }\n        return vec;\n    }\n};\n</code></pre> my_list.java<pre><code>/* List class */\nclass MyList {\n    private int[] arr; // Array (stores list elements)\n    private int capacity = 10; // List capacity\n    private int size = 0; // List length (current number of elements)\n    private int extendRatio = 2; // Multiple by which the list capacity is extended each time\n\n    /* Constructor */\n    public MyList() {\n        arr = new int[capacity];\n    }\n\n    /* Get list length (current number of elements) */\n    public int size() {\n        return size;\n    }\n\n    /* Get list capacity */\n    public int capacity() {\n        return capacity;\n    }\n\n    /* Update element */\n    public int get(int index) {\n        // If the index is out of bounds, throw an exception, as below\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"Index out of bounds\");\n        return arr[index];\n    }\n\n    /* Add elements at the end */\n    public void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"Index out of bounds\");\n        arr[index] = num;\n    }\n\n    /* Direct traversal of list elements */\n    public void add(int num) {\n        // When the number of elements exceeds capacity, trigger the extension mechanism\n        if (size == capacity())\n            extendCapacity();\n        arr[size] = num;\n        // Update the number of elements\n        size++;\n    }\n\n    /* Sort list */\n    public void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"Index out of bounds\");\n        // When the number of elements exceeds capacity, trigger the extension mechanism\n        if (size == capacity())\n            extendCapacity();\n        // Move all elements after index index forward by one position\n        for (int j = size - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // Update the number of elements\n        size++;\n    }\n\n    /* Remove element */\n    public int remove(int index) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"Index out of bounds\");\n        int num = arr[index];\n        // Move all elements after index forward by one position\n        for (int j = index; j &lt; size - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // Update the number of elements\n        size--;\n        // Return the removed element\n        return num;\n    }\n\n    /* Driver Code */\n    public void extendCapacity() {\n        // Create a new array with length extendRatio times the original array and copy the original array to the new array\n        arr = Arrays.copyOf(arr, capacity() * extendRatio);\n        // Add elements at the end\n        capacity = arr.length;\n    }\n\n    /* Convert list to array */\n    public int[] toArray() {\n        int size = size();\n        // Elements enqueue\n        int[] arr = new int[size];\n        for (int i = 0; i &lt; size; i++) {\n            arr[i] = get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.cs<pre><code>/* List class */\nclass MyList {\n    private int[] arr;           // Array (stores list elements)\n    private int arrCapacity = 10;    // List capacity\n    private int arrSize = 0;         // List length (current number of elements)\n    private readonly int extendRatio = 2;  // Multiple by which the list capacity is extended each time\n\n    /* Constructor */\n    public MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* Get list length (current number of elements) */\n    public int Size() {\n        return arrSize;\n    }\n\n    /* Get list capacity */\n    public int Capacity() {\n        return arrCapacity;\n    }\n\n    /* Update element */\n    public int Get(int index) {\n        // If the index is out of bounds, throw an exception, as below\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"Index out of bounds\");\n        return arr[index];\n    }\n\n    /* Add elements at the end */\n    public void Set(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"Index out of bounds\");\n        arr[index] = num;\n    }\n\n    /* Direct traversal of list elements */\n    public void Add(int num) {\n        // When the number of elements exceeds capacity, trigger the extension mechanism\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        arr[arrSize] = num;\n        // Update the number of elements\n        arrSize++;\n    }\n\n    /* Sort list */\n    public void Insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"Index out of bounds\");\n        // When the number of elements exceeds capacity, trigger the extension mechanism\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        // Move all elements after index index forward by one position\n        for (int j = arrSize - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // Update the number of elements\n        arrSize++;\n    }\n\n    /* Remove element */\n    public int Remove(int index) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"Index out of bounds\");\n        int num = arr[index];\n        // Move all elements after index forward by one position\n        for (int j = index; j &lt; arrSize - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // Update the number of elements\n        arrSize--;\n        // Return the removed element\n        return num;\n    }\n\n    /* Driver Code */\n    public void ExtendCapacity() {\n        // Create new array of length arrCapacity * extendRatio and copy original array to new array\n        Array.Resize(ref arr, arrCapacity * extendRatio);\n        // Add elements at the end\n        arrCapacity = arr.Length;\n    }\n\n    /* Convert list to array */\n    public int[] ToArray() {\n        // Elements enqueue\n        int[] arr = new int[arrSize];\n        for (int i = 0; i &lt; arrSize; i++) {\n            arr[i] = Get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.go<pre><code>/* List class */\ntype myList struct {\n    arrCapacity int\n    arr         []int\n    arrSize     int\n    extendRatio int\n}\n\n/* Constructor */\nfunc newMyList() *myList {\n    return &amp;myList{\n        arrCapacity: 10,              // List capacity\n        arr:         make([]int, 10), // Array (stores list elements)\n        arrSize:     0,               // List length (current number of elements)\n        extendRatio: 2,               // Multiple by which the list capacity is extended each time\n    }\n}\n\n/* Get list length (current number of elements) */\nfunc (l *myList) size() int {\n    return l.arrSize\n}\n\n/* Get list capacity */\nfunc (l *myList) capacity() int {\n    return l.arrCapacity\n}\n\n/* Update element */\nfunc (l *myList) get(index int) int {\n    // If the index is out of bounds, throw an exception, as below\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"Index out of bounds\")\n    }\n    return l.arr[index]\n}\n\n/* Add elements at the end */\nfunc (l *myList) set(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"Index out of bounds\")\n    }\n    l.arr[index] = num\n}\n\n/* Direct traversal of list elements */\nfunc (l *myList) add(num int) {\n    // When the number of elements exceeds capacity, trigger the extension mechanism\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    l.arr[l.arrSize] = num\n    // Update the number of elements\n    l.arrSize++\n}\n\n/* Sort list */\nfunc (l *myList) insert(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"Index out of bounds\")\n    }\n    // When the number of elements exceeds capacity, trigger the extension mechanism\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    // Move all elements after index index forward by one position\n    for j := l.arrSize - 1; j &gt;= index; j-- {\n        l.arr[j+1] = l.arr[j]\n    }\n    l.arr[index] = num\n    // Update the number of elements\n    l.arrSize++\n}\n\n/* Remove element */\nfunc (l *myList) remove(index int) int {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"Index out of bounds\")\n    }\n    num := l.arr[index]\n    // Create a new array with length _extend_ratio times the original array, and copy the original array to the new array\n    for j := index; j &lt; l.arrSize-1; j++ {\n        l.arr[j] = l.arr[j+1]\n    }\n    // Update the number of elements\n    l.arrSize--\n    // Return the removed element\n    return num\n}\n\n/* Driver Code */\nfunc (l *myList) extendCapacity() {\n    // Create a new array with length extendRatio times the original array and copy the original array to the new array\n    l.arr = append(l.arr, make([]int, l.arrCapacity*(l.extendRatio-1))...)\n    // Add elements at the end\n    l.arrCapacity = len(l.arr)\n}\n\n/* Return list with valid length */\nfunc (l *myList) toArray() []int {\n    // Elements enqueue\n    return l.arr[:l.arrSize]\n}\n</code></pre> my_list.swift<pre><code>/* List class */\nclass MyList {\n    private var arr: [Int] // Array (stores list elements)\n    private var _capacity: Int // List capacity\n    private var _size: Int // List length (current number of elements)\n    private let extendRatio: Int // Multiple by which the list capacity is extended each time\n\n    /* Constructor */\n    init() {\n        _capacity = 10\n        _size = 0\n        extendRatio = 2\n        arr = Array(repeating: 0, count: _capacity)\n    }\n\n    /* Get list length (current number of elements) */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* Get list capacity */\n    func capacity() -&gt; Int {\n        _capacity\n    }\n\n    /* Update element */\n    func get(index: Int) -&gt; Int {\n        // Throw error if index out of bounds, same below\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"Index out of bounds\")\n        }\n        return arr[index]\n    }\n\n    /* Add elements at the end */\n    func set(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"Index out of bounds\")\n        }\n        arr[index] = num\n    }\n\n    /* Direct traversal of list elements */\n    func add(num: Int) {\n        // When the number of elements exceeds capacity, trigger the extension mechanism\n        if size() == capacity() {\n            extendCapacity()\n        }\n        arr[size()] = num\n        // Update the number of elements\n        _size += 1\n    }\n\n    /* Sort list */\n    func insert(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"Index out of bounds\")\n        }\n        // When the number of elements exceeds capacity, trigger the extension mechanism\n        if size() == capacity() {\n            extendCapacity()\n        }\n        // Move all elements after index index forward by one position\n        for j in (index ..&lt; size()).reversed() {\n            arr[j + 1] = arr[j]\n        }\n        arr[index] = num\n        // Update the number of elements\n        _size += 1\n    }\n\n    /* Remove element */\n    @discardableResult\n    func remove(index: Int) -&gt; Int {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"Index out of bounds\")\n        }\n        let num = arr[index]\n        // Move all elements after index forward by one position\n        for j in index ..&lt; (size() - 1) {\n            arr[j] = arr[j + 1]\n        }\n        // Update the number of elements\n        _size -= 1\n        // Return the removed element\n        return num\n    }\n\n    /* Driver Code */\n    func extendCapacity() {\n        // Create a new array with length extendRatio times the original array and copy the original array to the new array\n        arr = arr + Array(repeating: 0, count: capacity() * (extendRatio - 1))\n        // Add elements at the end\n        _capacity = arr.count\n    }\n\n    /* Convert list to array */\n    func toArray() -&gt; [Int] {\n        Array(arr.prefix(size()))\n    }\n}\n</code></pre> my_list.js<pre><code>/* List class */\nclass MyList {\n    #arr = new Array(); // Array (stores list elements)\n    #capacity = 10; // List capacity\n    #size = 0; // List length (current number of elements)\n    #extendRatio = 2; // Multiple by which the list capacity is extended each time\n\n    /* Constructor */\n    constructor() {\n        this.#arr = new Array(this.#capacity);\n    }\n\n    /* Get list length (current number of elements) */\n    size() {\n        return this.#size;\n    }\n\n    /* Get list capacity */\n    capacity() {\n        return this.#capacity;\n    }\n\n    /* Update element */\n    get(index) {\n        // If the index is out of bounds, throw an exception, as below\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('Index out of bounds');\n        return this.#arr[index];\n    }\n\n    /* Add elements at the end */\n    set(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('Index out of bounds');\n        this.#arr[index] = num;\n    }\n\n    /* Direct traversal of list elements */\n    add(num) {\n        // If length equals capacity, need to expand\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // Add new element to end of list\n        this.#arr[this.#size] = num;\n        this.#size++;\n    }\n\n    /* Sort list */\n    insert(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('Index out of bounds');\n        // When the number of elements exceeds capacity, trigger the extension mechanism\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // Move all elements after index index forward by one position\n        for (let j = this.#size - 1; j &gt;= index; j--) {\n            this.#arr[j + 1] = this.#arr[j];\n        }\n        // Update the number of elements\n        this.#arr[index] = num;\n        this.#size++;\n    }\n\n    /* Remove element */\n    remove(index) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('Index out of bounds');\n        let num = this.#arr[index];\n        // Create a new array with length _extend_ratio times the original array, and copy the original array to the new array\n        for (let j = index; j &lt; this.#size - 1; j++) {\n            this.#arr[j] = this.#arr[j + 1];\n        }\n        // Update the number of elements\n        this.#size--;\n        // Return the removed element\n        return num;\n    }\n\n    /* Driver Code */\n    extendCapacity() {\n        // Create a new array with length extendRatio times the original array and copy the original array to the new array\n        this.#arr = this.#arr.concat(\n            new Array(this.capacity() * (this.#extendRatio - 1))\n        );\n        // Add elements at the end\n        this.#capacity = this.#arr.length;\n    }\n\n    /* Convert list to array */\n    toArray() {\n        let size = this.size();\n        // Elements enqueue\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.ts<pre><code>/* List class */\nclass MyList {\n    private arr: Array&lt;number&gt;; // Array (stores list elements)\n    private _capacity: number = 10; // List capacity\n    private _size: number = 0; // List length (current number of elements)\n    private extendRatio: number = 2; // Multiple by which the list capacity is extended each time\n\n    /* Constructor */\n    constructor() {\n        this.arr = new Array(this._capacity);\n    }\n\n    /* Get list length (current number of elements) */\n    public size(): number {\n        return this._size;\n    }\n\n    /* Get list capacity */\n    public capacity(): number {\n        return this._capacity;\n    }\n\n    /* Update element */\n    public get(index: number): number {\n        // If the index is out of bounds, throw an exception, as below\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('Index out of bounds');\n        return this.arr[index];\n    }\n\n    /* Add elements at the end */\n    public set(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('Index out of bounds');\n        this.arr[index] = num;\n    }\n\n    /* Direct traversal of list elements */\n    public add(num: number): void {\n        // If length equals capacity, need to expand\n        if (this._size === this._capacity) this.extendCapacity();\n        // Add new element to end of list\n        this.arr[this._size] = num;\n        this._size++;\n    }\n\n    /* Sort list */\n    public insert(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('Index out of bounds');\n        // When the number of elements exceeds capacity, trigger the extension mechanism\n        if (this._size === this._capacity) {\n            this.extendCapacity();\n        }\n        // Move all elements after index index forward by one position\n        for (let j = this._size - 1; j &gt;= index; j--) {\n            this.arr[j + 1] = this.arr[j];\n        }\n        // Update the number of elements\n        this.arr[index] = num;\n        this._size++;\n    }\n\n    /* Remove element */\n    public remove(index: number): number {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('Index out of bounds');\n        let num = this.arr[index];\n        // Move all elements after index forward by one position\n        for (let j = index; j &lt; this._size - 1; j++) {\n            this.arr[j] = this.arr[j + 1];\n        }\n        // Update the number of elements\n        this._size--;\n        // Return the removed element\n        return num;\n    }\n\n    /* Driver Code */\n    public extendCapacity(): void {\n        // Create new array of length size and copy original array to new array\n        this.arr = this.arr.concat(\n            new Array(this.capacity() * (this.extendRatio - 1))\n        );\n        // Add elements at the end\n        this._capacity = this.arr.length;\n    }\n\n    /* Convert list to array */\n    public toArray(): number[] {\n        let size = this.size();\n        // Elements enqueue\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.dart<pre><code>/* List class */\nclass MyList {\n  late List&lt;int&gt; _arr; // Array (stores list elements)\n  int _capacity = 10; // List capacity\n  int _size = 0; // List length (current number of elements)\n  int _extendRatio = 2; // Multiple by which the list capacity is extended each time\n\n  /* Constructor */\n  MyList() {\n    _arr = List.filled(_capacity, 0);\n  }\n\n  /* Get list length (current number of elements) */\n  int size() =&gt; _size;\n\n  /* Get list capacity */\n  int capacity() =&gt; _capacity;\n\n  /* Update element */\n  int get(int index) {\n    if (index &gt;= _size) throw RangeError('Index out of bounds');\n    return _arr[index];\n  }\n\n  /* Add elements at the end */\n  void set(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('Index out of bounds');\n    _arr[index] = _num;\n  }\n\n  /* Direct traversal of list elements */\n  void add(int _num) {\n    // When the number of elements exceeds capacity, trigger the extension mechanism\n    if (_size == _capacity) extendCapacity();\n    _arr[_size] = _num;\n    // Update the number of elements\n    _size++;\n  }\n\n  /* Sort list */\n  void insert(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('Index out of bounds');\n    // When the number of elements exceeds capacity, trigger the extension mechanism\n    if (_size == _capacity) extendCapacity();\n    // Move all elements after index index forward by one position\n    for (var j = _size - 1; j &gt;= index; j--) {\n      _arr[j + 1] = _arr[j];\n    }\n    _arr[index] = _num;\n    // Update the number of elements\n    _size++;\n  }\n\n  /* Remove element */\n  int remove(int index) {\n    if (index &gt;= _size) throw RangeError('Index out of bounds');\n    int _num = _arr[index];\n    // Move all elements after index forward by one position\n    for (var j = index; j &lt; _size - 1; j++) {\n      _arr[j] = _arr[j + 1];\n    }\n    // Update the number of elements\n    _size--;\n    // Return the removed element\n    return _num;\n  }\n\n  /* Driver Code */\n  void extendCapacity() {\n    // Create new array with length _extendRatio times original array\n    final _newNums = List.filled(_capacity * _extendRatio, 0);\n    // Copy original array to new array\n    List.copyRange(_newNums, 0, _arr);\n    // Update _arr reference\n    _arr = _newNums;\n    // Add elements at the end\n    _capacity = _arr.length;\n  }\n\n  /* Convert list to array */\n  List&lt;int&gt; toArray() {\n    List&lt;int&gt; arr = [];\n    for (var i = 0; i &lt; _size; i++) {\n      arr.add(get(i));\n    }\n    return arr;\n  }\n}\n</code></pre> my_list.rs<pre><code>/* List class */\n#[allow(dead_code)]\nstruct MyList {\n    arr: Vec&lt;i32&gt;,       // Array (stores list elements)\n    capacity: usize,     // List capacity\n    size: usize,         // List length (current number of elements)\n    extend_ratio: usize, // Multiple by which the list capacity is extended each time\n}\n\n#[allow(unused, unused_comparisons)]\nimpl MyList {\n    /* Constructor */\n    pub fn new(capacity: usize) -&gt; Self {\n        let mut vec = vec![0; capacity];\n        Self {\n            arr: vec,\n            capacity,\n            size: 0,\n            extend_ratio: 2,\n        }\n    }\n\n    /* Get list length (current number of elements) */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.size;\n    }\n\n    /* Get list capacity */\n    pub fn capacity(&amp;self) -&gt; usize {\n        return self.capacity;\n    }\n\n    /* Update element */\n    pub fn get(&amp;self, index: usize) -&gt; i32 {\n        // If the index is out of bounds, throw an exception, as below\n        if index &gt;= self.size {\n            panic!(\"Index out of bounds\")\n        };\n        return self.arr[index];\n    }\n\n    /* Add elements at the end */\n    pub fn set(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size {\n            panic!(\"Index out of bounds\")\n        };\n        self.arr[index] = num;\n    }\n\n    /* Direct traversal of list elements */\n    pub fn add(&amp;mut self, num: i32) {\n        // When the number of elements exceeds capacity, trigger the extension mechanism\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        self.arr[self.size] = num;\n        // Update the number of elements\n        self.size += 1;\n    }\n\n    /* Sort list */\n    pub fn insert(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size() {\n            panic!(\"Index out of bounds\")\n        };\n        // When the number of elements exceeds capacity, trigger the extension mechanism\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        // Move all elements after index index forward by one position\n        for j in (index..self.size).rev() {\n            self.arr[j + 1] = self.arr[j];\n        }\n        self.arr[index] = num;\n        // Update the number of elements\n        self.size += 1;\n    }\n\n    /* Remove element */\n    pub fn remove(&amp;mut self, index: usize) -&gt; i32 {\n        if index &gt;= self.size() {\n            panic!(\"Index out of bounds\")\n        };\n        let num = self.arr[index];\n        // Create a new array with length _extend_ratio times the original array, and copy the original array to the new array\n        for j in index..self.size - 1 {\n            self.arr[j] = self.arr[j + 1];\n        }\n        // Update the number of elements\n        self.size -= 1;\n        // Return the removed element\n        return num;\n    }\n\n    /* Driver Code */\n    pub fn extend_capacity(&amp;mut self) {\n        // Create new array with length extend_ratio times original, copy original array to new array\n        let new_capacity = self.capacity * self.extend_ratio;\n        self.arr.resize(new_capacity, 0);\n        // Add elements at the end\n        self.capacity = new_capacity;\n    }\n\n    /* Convert list to array */\n    pub fn to_array(&amp;self) -&gt; Vec&lt;i32&gt; {\n        // Elements enqueue\n        let mut arr = Vec::new();\n        for i in 0..self.size {\n            arr.push(self.get(i));\n        }\n        arr\n    }\n}\n</code></pre> my_list.c<pre><code>/* List class */\ntypedef struct {\n    int *arr;        // Array (stores list elements)\n    int capacity;    // List capacity\n    int size;        // List size\n    int extendRatio; // List expansion multiplier\n} MyList;\n\n/* Constructor */\nMyList *newMyList() {\n    MyList *nums = malloc(sizeof(MyList));\n    nums-&gt;capacity = 10;\n    nums-&gt;arr = malloc(sizeof(int) * nums-&gt;capacity);\n    nums-&gt;size = 0;\n    nums-&gt;extendRatio = 2;\n    return nums;\n}\n\n/* Destructor */\nvoid delMyList(MyList *nums) {\n    free(nums-&gt;arr);\n    free(nums);\n}\n\n/* Get list length */\nint size(MyList *nums) {\n    return nums-&gt;size;\n}\n\n/* Get list capacity */\nint capacity(MyList *nums) {\n    return nums-&gt;capacity;\n}\n\n/* Update element */\nint get(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    return nums-&gt;arr[index];\n}\n\n/* Add elements at the end */\nvoid set(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    nums-&gt;arr[index] = num;\n}\n\n/* Direct traversal of list elements */\nvoid add(MyList *nums, int num) {\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // Expand capacity\n    }\n    nums-&gt;arr[size(nums)] = num;\n    nums-&gt;size++;\n}\n\n/* Sort list */\nvoid insert(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    // When the number of elements exceeds capacity, trigger the extension mechanism\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // Expand capacity\n    }\n    for (int i = size(nums); i &gt; index; --i) {\n        nums-&gt;arr[i] = nums-&gt;arr[i - 1];\n    }\n    nums-&gt;arr[index] = num;\n    nums-&gt;size++;\n}\n\n/* Remove element */\n// Note: stdio.h occupies the remove keyword\nint removeItem(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    int num = nums-&gt;arr[index];\n    for (int i = index; i &lt; size(nums) - 1; i++) {\n        nums-&gt;arr[i] = nums-&gt;arr[i + 1];\n    }\n    nums-&gt;size--;\n    return num;\n}\n\n/* Driver Code */\nvoid extendCapacity(MyList *nums) {\n    // Allocate space first\n    int newCapacity = capacity(nums) * nums-&gt;extendRatio;\n    int *extend = (int *)malloc(sizeof(int) * newCapacity);\n    int *temp = nums-&gt;arr;\n\n    // Copy old data to new data\n    for (int i = 0; i &lt; size(nums); i++)\n        extend[i] = nums-&gt;arr[i];\n\n    // Free old data\n    free(temp);\n\n    // Update new data\n    nums-&gt;arr = extend;\n    nums-&gt;capacity = newCapacity;\n}\n\n/* Convert list to Array for printing */\nint *toArray(MyList *nums) {\n    return nums-&gt;arr;\n}\n</code></pre> my_list.kt<pre><code>/* List class */\nclass MyList {\n    private var arr: IntArray = intArrayOf() // Array (stores list elements)\n    private var capacity: Int = 10 // List capacity\n    private var size: Int = 0 // List length (current number of elements)\n    private var extendRatio: Int = 2 // Multiple by which the list capacity is extended each time\n\n    /* Constructor */\n    init {\n        arr = IntArray(capacity)\n    }\n\n    /* Get list length (current number of elements) */\n    fun size(): Int {\n        return size\n    }\n\n    /* Get list capacity */\n    fun capacity(): Int {\n        return capacity\n    }\n\n    /* Update element */\n    fun get(index: Int): Int {\n        // If the index is out of bounds, throw an exception, as below\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"Index out of bounds\")\n        return arr[index]\n    }\n\n    /* Add elements at the end */\n    fun set(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"Index out of bounds\")\n        arr[index] = num\n    }\n\n    /* Direct traversal of list elements */\n    fun add(num: Int) {\n        // When the number of elements exceeds capacity, trigger the extension mechanism\n        if (size == capacity())\n            extendCapacity()\n        arr[size] = num\n        // Update the number of elements\n        size++\n    }\n\n    /* Sort list */\n    fun insert(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"Index out of bounds\")\n        // When the number of elements exceeds capacity, trigger the extension mechanism\n        if (size == capacity())\n            extendCapacity()\n        // Move all elements after index index forward by one position\n        for (j in size - 1 downTo index)\n            arr[j + 1] = arr[j]\n        arr[index] = num\n        // Update the number of elements\n        size++\n    }\n\n    /* Remove element */\n    fun remove(index: Int): Int {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"Index out of bounds\")\n        val num = arr[index]\n        // Move all elements after index forward by one position\n        for (j in index..&lt;size - 1)\n            arr[j] = arr[j + 1]\n        // Update the number of elements\n        size--\n        // Return the removed element\n        return num\n    }\n\n    /* Driver Code */\n    fun extendCapacity() {\n        // Create a new array with length extendRatio times the original array and copy the original array to the new array\n        arr = arr.copyOf(capacity() * extendRatio)\n        // Add elements at the end\n        capacity = arr.size\n    }\n\n    /* Convert list to array */\n    fun toArray(): IntArray {\n        val size = size()\n        // Elements enqueue\n        val arr = IntArray(size)\n        for (i in 0..&lt;size) {\n            arr[i] = get(i)\n        }\n        return arr\n    }\n}\n</code></pre> my_list.rb<pre><code>### List class ###\nclass MyList\n  attr_reader :size       # Get list length (current number of elements)\n  attr_reader :capacity   # Get list capacity\n\n  ### Constructor ###\n  def initialize\n    @capacity = 10\n    @size = 0\n    @extend_ratio = 2\n    @arr = Array.new(capacity)\n  end\n\n  ### Access element ###\n  def get(index)\n    # If the index is out of bounds, throw an exception, as below\n    raise IndexError, \"Index out of bounds\" if index &lt; 0 || index &gt;= size\n    @arr[index]\n  end\n\n  ### Access element ###\n  def set(index, num)\n    raise IndexError, \"Index out of bounds\" if index &lt; 0 || index &gt;= size\n    @arr[index] = num\n  end\n\n  ### Add element at end ###\n  def add(num)\n    # When the number of elements exceeds capacity, trigger the extension mechanism\n    extend_capacity if size == capacity\n    @arr[size] = num\n\n    # Update the number of elements\n    @size += 1\n  end\n\n  ### Insert element in middle ###\n  def insert(index, num)\n    raise IndexError, \"Index out of bounds\" if index &lt; 0 || index &gt;= size\n\n    # When the number of elements exceeds capacity, trigger the extension mechanism\n    extend_capacity if size == capacity\n\n    # Move all elements after index index forward by one position\n    for j in (size - 1).downto(index)\n      @arr[j + 1] = @arr[j]\n    end\n    @arr[index] = num\n\n    # Update the number of elements\n    @size += 1\n  end\n\n  ### Delete element ###\n  def remove(index)\n    raise IndexError, \"Index out of bounds\" if index &lt; 0 || index &gt;= size\n    num = @arr[index]\n\n    # Move all elements after index forward by one position\n    for j in index...size\n      @arr[j] = @arr[j + 1]\n    end\n\n    # Update the number of elements\n    @size -= 1\n\n    # Return the removed element\n    num\n  end\n\n  ### Expand list capacity ###\n  def extend_capacity\n    # Create new array with length extend_ratio times original, copy original array to new array\n    arr = @arr.dup + Array.new(capacity * (@extend_ratio - 1))\n    # Add elements at the end\n    @capacity = arr.length\n  end\n\n  ### Convert list to array ###\n  def to_array\n    sz = size\n    # Elements enqueue\n    arr = Array.new(sz)\n    for i in 0...sz\n      arr[i] = get(i)\n    end\n    arr\n  end\nend\n</code></pre>","path":["Chapter 4. Arrays and Linked Lists","4.3   List"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/","level":1,"title":"4.4   Random-Access Memory and Cache *","text":"<p>In the first two sections of this chapter, we explored arrays and linked lists, two fundamental and important data structures that represent two physical layouts: \"contiguous storage\" and \"distributed storage\", respectively.</p> <p>In fact, physical structure largely determines the efficiency with which programs utilize memory and cache, which in turn affects the overall performance of algorithmic programs.</p>","path":["Chapter 4. Arrays and Linked Lists","4.4   Random-Access Memory and Cache *"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/#441-computer-storage-devices","level":2,"title":"4.4.1   Computer Storage Devices","text":"<p>Computers include three types of storage devices: hard disk, random-access memory (RAM), and cache memory. The following table shows their different roles and performance characteristics in a computer system.</p> <p> Table 4-2   Computer Storage Devices </p> Hard Disk RAM Cache Purpose Long-term storage of data, including operating systems, programs, and files Temporary storage of currently running programs and data being processed Storage of frequently accessed data and instructions to reduce CPU's accesses to memory Volatility Data is not lost after power-off Data is lost after power-off Data is lost after power-off Capacity Large, on the order of terabytes (TB) Small, on the order of gigabytes (GB) Very small, on the order of megabytes (MB) Speed Slow, hundreds to thousands of MB/s Fast, tens of GB/s Very fast, tens to hundreds of GB/s Cost (CNY/GB) Inexpensive, from a few tenths of a yuan to a few yuan per GB Expensive, from tens to hundreds of yuan per GB Very expensive, effectively bundled with the CPU package <p>We can imagine the computer storage system as a pyramid, as shown in the diagram below. Storage devices closer to the top are faster, have smaller capacity, and are more expensive. This multi-layered design is deliberate, the result of careful consideration by computer scientists and engineers.</p> <ul> <li>Hard disk cannot be easily replaced by RAM. First, data in memory is lost after power-off, making it unsuitable for long-term data storage. Second, memory is tens of times more expensive than hard disk, which makes it difficult to popularize in the consumer market.</li> <li>Cache cannot simultaneously achieve large capacity and high speed. As the capacity of L1, L2, and L3 caches increases, their physical size becomes larger, and the physical distance between them and the CPU core increases, resulting in longer data transmission time and higher element access latency. With current technology, the multi-layered cache structure represents the best balance point between capacity, speed, and cost.</li> </ul> <p></p> <p> Figure 4-9   Computer Storage System </p> <p>Tip</p> <p>The storage hierarchy of computers embodies a delicate balance among speed, capacity, and cost. In fact, such trade-offs are common across all industrial fields, requiring us to find the optimal balance point between different advantages and constraints.</p> <p>In summary, hard disks are used for long-term storage of large amounts of data, RAM is used to temporarily store the data being processed during program execution, and cache is used to store frequently accessed data and instructions, thereby improving program execution efficiency. The three work together to keep the computer system running efficiently.</p> <p>As shown in the diagram below, during program execution, data is read from the hard disk into RAM for CPU computation. Cache can be viewed as part of the CPU. By intelligently loading data from RAM, it provides the CPU with high-speed access to data, significantly improving program execution efficiency and reducing reliance on slower RAM.</p> <p></p> <p> Figure 4-10   Data Flow Among Hard Disk, RAM, and Cache </p>","path":["Chapter 4. Arrays and Linked Lists","4.4   Random-Access Memory and Cache *"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/#442-memory-efficiency-of-data-structures","level":2,"title":"4.4.2   Memory Efficiency of Data Structures","text":"<p>In terms of memory space utilization, arrays and linked lists each have advantages and limitations.</p> <p>On one hand, memory is limited, and the same memory cannot be shared by multiple programs, so we hope data structures can utilize space as efficiently as possible. Array elements are tightly packed and do not require additional space to store references (pointers) between linked list nodes, thus having higher space efficiency. However, arrays need to allocate sufficient contiguous memory space at once, which may lead to memory waste, and array expansion requires additional time and space costs. In comparison, linked lists perform dynamic memory allocation and deallocation on a \"node\" basis, providing greater flexibility.</p> <p>On the other hand, during program execution, as memory is repeatedly allocated and freed, the degree of fragmentation of free memory becomes increasingly severe, leading to reduced memory utilization efficiency. Arrays, due to their contiguous storage approach, are relatively less prone to memory fragmentation. Conversely, linked list elements are distributed in storage, and frequent insertion and deletion operations are more likely to cause memory fragmentation.</p>","path":["Chapter 4. Arrays and Linked Lists","4.4   Random-Access Memory and Cache *"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/#443-cache-efficiency-of-data-structures","level":2,"title":"4.4.3   Cache Efficiency of Data Structures","text":"<p>Although cache has much smaller space capacity than memory, it is much faster than memory and plays a crucial role in program execution speed. Since cache capacity is limited and can only store a small portion of frequently accessed data, when the CPU attempts to access data that is not in the cache, a cache miss occurs, and the CPU must load the required data from the slower memory.</p> <p>Clearly, the fewer \"cache misses,\" the higher the efficiency of CPU data reads and writes, and the better the program performance. We call the proportion of data that the CPU successfully obtains from the cache the cache hit rate, a metric typically used to measure cache efficiency.</p> <p>To achieve the highest efficiency possible, cache employs the following data loading mechanisms.</p> <ul> <li>Cache lines: The cache does not store and load data on a byte-by-byte basis, but rather as cache lines. Compared to byte-by-byte transmission, cache line transmission is more efficient.</li> <li>Prefetching mechanism: The processor attempts to predict data access patterns (e.g., sequential access, fixed-stride jumping access, etc.) and loads data into the cache according to specific patterns, thereby improving hit rate.</li> <li>Spatial locality: If a piece of data is accessed, nearby data may also be accessed in the near future. Therefore, when the cache loads a particular piece of data, it also loads nearby data to improve hit rate.</li> <li>Temporal locality: If a piece of data is accessed, it is likely to be accessed again in the near future. Cache leverages this principle by retaining recently accessed data to improve hit rate.</li> </ul> <p>In fact, arrays and linked lists differ in how efficiently they utilize cache, mainly in the following respects.</p> <ul> <li>Space occupied: Linked-list elements occupy more space than array elements, so less useful data can fit in the cache.</li> <li>Cache lines: Linked list data are scattered throughout memory, while cache loads \"by lines,\" so the proportion of invalid data loaded is higher.</li> <li>Prefetching mechanism: Arrays have more \"predictable\" data access patterns than linked lists, making it easier for the system to guess which data will be loaded next.</li> <li>Spatial locality: Arrays are stored in centralized memory space, so data near loaded data is more likely to be accessed soon.</li> </ul> <p>Overall, arrays have higher cache hit rates, thus they usually outperform linked lists in operation efficiency. This makes data structures implemented based on arrays more popular when solving algorithmic problems.</p> <p>It is important to note that high cache efficiency does not mean arrays are superior to linked lists in all cases. In practical applications, which data structure to choose should be determined based on specific requirements. For example, both arrays and linked lists can implement the \"stack\" data structure (which will be discussed in detail in the next chapter), but they are suitable for different scenarios.</p> <ul> <li>When solving algorithm problems, we tend to prefer stack implementations based on arrays, because they provide higher operation efficiency and the ability of random access, at the cost of needing to pre-allocate a certain amount of memory space for the array.</li> <li>If the data volume is very large, the dynamic nature is high, and the expected size of the stack is difficult to estimate, then a stack implementation based on linked lists is more suitable. Linked lists can distribute large amounts of data across different parts of memory and avoid the additional overhead produced by array expansion.</li> </ul>","path":["Chapter 4. Arrays and Linked Lists","4.4   Random-Access Memory and Cache *"],"tags":[]},{"location":"chapter_array_and_linkedlist/summary/","level":1,"title":"4.5   Summary","text":"","path":["Chapter 4. Arrays and Linked Lists","4.5   Summary"],"tags":[]},{"location":"chapter_array_and_linkedlist/summary/#1-key-review","level":3,"title":"1.   Key Review","text":"<ul> <li>Arrays and linked lists are two fundamental data structures, representing two different ways data can be stored in computer memory: contiguous storage and scattered storage. Their strengths and weaknesses complement each other.</li> <li>Arrays support random access and use less memory; however, inserting and deleting elements is inefficient, and the length is immutable after initialization.</li> <li>Linked lists achieve efficient insertion and deletion of nodes by modifying references (pointers), and can flexibly adjust length; however, node access is inefficient and memory consumption is higher. Common linked list types include singly linked lists, circular linked lists, and doubly linked lists.</li> <li>A list is an ordered collection of elements that supports insertion, deletion, search, and modification, typically implemented based on dynamic arrays. It retains the advantages of arrays while allowing flexible adjustment of length.</li> <li>The emergence of lists has greatly improved the practicality of arrays, but it may also waste some memory space.</li> <li>During program execution, data is primarily stored in memory. Arrays provide higher memory space efficiency, while linked lists offer greater flexibility in memory usage.</li> <li>Caches provide fast data access to the CPU through mechanisms such as cache lines, prefetching, and spatial and temporal locality, significantly improving program execution efficiency.</li> <li>Because arrays have higher cache hit rates, they are generally more efficient than linked lists. When choosing a data structure, appropriate selection should be made based on specific requirements and scenarios.</li> </ul>","path":["Chapter 4. Arrays and Linked Lists","4.5   Summary"],"tags":[]},{"location":"chapter_array_and_linkedlist/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: Does storing an array on the stack versus on the heap affect time efficiency and space efficiency?</p> <p>Arrays stored on the stack and on the heap are both stored in contiguous memory space, so data operation efficiency is basically the same. However, the stack and heap have their own characteristics, leading to the following differences.</p> <ol> <li>Allocation and deallocation efficiency: The stack is a relatively small piece of memory, with allocation automatically handled by the compiler; the heap is relatively larger and can be dynamically allocated in code, more prone to fragmentation. Therefore, allocation and deallocation operations on the heap are usually slower than on the stack.</li> <li>Size limitations: Stack memory is relatively small, and the heap size is generally limited by available memory. Therefore, the heap is more suitable for storing large arrays.</li> <li>Flexibility: The size of an array on the stack must be determined at compile time, while the size of an array on the heap can be determined dynamically at runtime.</li> </ol> <p>Q: Why do arrays require elements of the same type, while linked lists do not emphasize this requirement?</p> <p>Linked lists are composed of nodes, with nodes connected through references (pointers), and each node can store different types of data, such as <code>int</code>, <code>double</code>, <code>string</code>, <code>object</code>, etc.</p> <p>In contrast, array elements must be of the same type so that their positions can be determined by calculating offsets. For example, if an array contains both <code>int</code> and <code>long</code> types, with individual elements occupying 4 bytes and 8 bytes respectively, then the following formula cannot be used to calculate the offset, because the array contains two different \"element sizes\".</p> <pre><code># element address = array base address (address of the first element) + element size * element index\n</code></pre> <p>Q: After deleting node <code>P</code>, do we need to set <code>P.next</code> to <code>None</code>?</p> <p>It is not necessary to modify <code>P.next</code>. From the perspective of the linked list, traversing from the head node to the tail node will no longer encounter <code>P</code>. This means that node <code>P</code> has been removed from the linked list, and it doesn't matter where node <code>P</code> points to at this time—it won't affect the linked list.</p> <p>From an algorithms-and-problem-solving perspective, leaving the pointer connected is fine as long as the program logic is correct. From a standard-library implementation perspective, explicitly disconnecting it is safer and clearer. If it is not disconnected and the deleted node is not reclaimed properly, it may affect the reclamation of successor nodes.</p> <p>Q: In a linked list, the time complexity of insertion and deletion operations is \\(O(1)\\). However, both insertion and deletion require \\(O(n)\\) time to find the element; why isn't the time complexity \\(O(n)\\)?</p> <p>If the element is first found and then deleted, the time complexity is indeed \\(O(n)\\). However, the advantage of \\(O(1)\\) insertion and deletion in linked lists can be demonstrated in other applications. For example, a deque is well-suited for linked list implementation, where we maintain pointer variables always pointing to the head and tail nodes, with each insertion and deletion operation being \\(O(1)\\).</p> <p>Q: In the diagram \"Linked List Definition and Storage Methods\", does the light blue pointer node occupy a single memory address, or does it share equally with the node value?</p> <p>This diagram is a qualitative representation; a quantitative representation requires analysis based on the specific situation.</p> <ul> <li>Different types of node values occupy different amounts of space, such as <code>int</code>, <code>long</code>, <code>double</code>, and instance objects, etc.</li> <li>The amount of memory space occupied by pointer variables depends on the operating system and compilation environment used, usually 8 bytes or 4 bytes.</li> </ul> <p>Q: Is appending an element at the end of a list always \\(O(1)\\)?</p> <p>If appending an element exceeds the list length, the list must first be expanded before adding. The system allocates a new block of memory and moves all elements from the original list to it, in which case the time complexity becomes \\(O(n)\\).</p> <p>Q: \"The emergence of lists has greatly improved the practicality of arrays, but may result in some wasted memory space\"—does this space waste refer to the memory occupied by additional variables such as capacity, length, and expansion factor?</p> <p>This space waste mainly has two aspects: on one hand, lists typically set an initial length, which we may not need to fully utilize; on the other hand, to prevent frequent expansion, expansion generally multiplies by a coefficient, such as \\(\\times 1.5\\). As a result, there will be many empty positions that we typically cannot completely fill.</p> <p>Q: In Python, after initializing <code>n = [1, 2, 3]</code>, the addresses of these 3 elements are contiguous, but initializing <code>m = [2, 1, 3]</code> reveals that each element's id is not continuous; rather, they are the same as those in <code>n</code>. Since the addresses of these elements are not contiguous, is <code>m</code> still an array?</p> <p>If we replace list elements with linked list nodes <code>n = [n1, n2, n3, n4, n5]</code>, usually these 5 node objects are also scattered throughout memory. However, given a list index, we can still obtain the node memory address in \\(O(1)\\) time, thereby accessing the corresponding node. This is because the array stores references to nodes, not the nodes themselves.</p> <p>Unlike many languages, numbers in Python are wrapped as objects, and lists store not the numbers themselves, but references to the numbers. Therefore, we find that the same numbers in two arrays have the same id, and the memory addresses of these numbers need not be contiguous.</p> <p>Q: C++ STL has <code>std::list</code> which has already implemented a doubly linked list, but it seems that some algorithm books don't use it directly. Is there a limitation?</p> <p>On one hand, we often prefer to use arrays for implementing algorithms and only use linked lists when necessary, mainly for two reasons.</p> <ul> <li>Space overhead: Since each element requires two additional pointers (one for the previous element and one for the next element), <code>std::list</code> typically consumes more space than <code>std::vector</code>.</li> <li>Cache unfriendliness: Since data is not stored contiguously, <code>std::list</code> has lower cache utilization. In general, <code>std::vector</code> has better performance.</li> </ul> <p>On the other hand, cases where linked lists are necessary mainly involve binary trees and graphs. Stacks and queues usually use the <code>stack</code> and <code>queue</code> provided by the programming language, rather than linked lists.</p> <p>Q: Does the operation <code>res = [[0]] * n</code> create a 2D list where each <code>[0]</code> is independent?</p> <p>No, they are not independent. In this 2D list, all the <code>[0]</code> are actually references to the same object. If we modify one element, we will find that all corresponding elements change accordingly.</p> <p>If we want each <code>[0]</code> in the 2D list to be independent, we can use <code>res = [[0] for _ in range(n)]</code> to achieve this. The principle of this approach is to initialize \\(n\\) independent <code>[0]</code> list objects.</p> <p>Q: Does the operation <code>res = [0] * n</code> create a list where each integer 0 is independent?</p> <p>In this list, all the integer zeros reference the same object. This is because Python uses a caching mechanism for small integers (typically -5 to 256) to maximize object reuse and improve performance.</p> <p>Although they all reference the same object, we can still modify each element in the list independently. This is because Python integers are \"immutable objects\". When we modify an element, we actually switch that element to reference a different object, rather than changing the original object itself.</p> <p>However, when list elements are \"mutable objects\" (such as lists, dictionaries, or class instances), modifying an element directly changes the object itself, and all elements referencing that object will have the same change.</p>","path":["Chapter 4. Arrays and Linked Lists","4.5   Summary"],"tags":[]},{"location":"chapter_backtracking/","level":1,"title":"Chapter 13.   Backtracking","text":"<p>Abstract</p> <p>We are like explorers in a maze, and may encounter difficulties on the path forward.</p> <p>The power of backtracking allows us to start over, keep trying, and eventually find the exit leading to light.</p>","path":["Chapter 13. Backtracking","Chapter 13.   Backtracking"],"tags":[]},{"location":"chapter_backtracking/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>13.1   Backtracking Algorithm</li> <li>13.2   Permutations Problem</li> <li>13.3   Subset-Sum Problem</li> <li>13.4   N-Queens Problem</li> <li>13.5   Summary</li> <li>13.6   Exercises</li> </ul>","path":["Chapter 13. Backtracking","Chapter 13.   Backtracking"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/","level":1,"title":"13.1   Backtracking Algorithm","text":"<p>The backtracking algorithm is a method for solving problems through exhaustive search. Its core idea is to start from an initial state and exhaustively search all possible solutions. When a correct solution is found, it is recorded. This process continues until a solution is found or all possible choices have been tried without finding a solution.</p> <p>The backtracking algorithm typically employs \"depth-first search\" to traverse the solution space. In the \"Binary Tree\" chapter, we mentioned that preorder, inorder, and postorder traversals all belong to depth-first search. Next, we will construct a backtracking problem using preorder traversal to progressively understand how the backtracking algorithm works.</p> <p>Example 1</p> <p>Given a binary tree, search and record all nodes with value \\(7\\), and return a list of these nodes.</p> <p>For this problem, we perform a preorder traversal of the tree and check whether the current node's value is \\(7\\). If it is, we add the node to the result list <code>res</code>. The relevant implementation is shown in the following figure and code:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_i_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"Preorder traversal: Example 1\"\"\"\n    if root is None:\n        return\n    if root.val == 7:\n        # Record solution\n        res.append(root)\n    pre_order(root.left)\n    pre_order(root.right)\n</code></pre> preorder_traversal_i_compact.cpp<pre><code>/* Preorder traversal: Example 1 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // Record solution\n        res.push_back(root);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.java<pre><code>/* Preorder traversal: Example 1 */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // Record solution\n        res.add(root);\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.cs<pre><code>/* Preorder traversal: Example 1 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // Record solution\n        res.Add(root);\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.go<pre><code>/* Preorder traversal: Example 1 */\nfunc preOrderI(root *TreeNode, res *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    if (root.Val).(int) == 7 {\n        // Record solution\n        *res = append(*res, root)\n    }\n    preOrderI(root.Left, res)\n    preOrderI(root.Right, res)\n}\n</code></pre> preorder_traversal_i_compact.swift<pre><code>/* Preorder traversal: Example 1 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    if root.val == 7 {\n        // Record solution\n        res.append(root)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n</code></pre> preorder_traversal_i_compact.js<pre><code>/* Preorder traversal: Example 1 */\nfunction preOrder(root, res) {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // Record solution\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.ts<pre><code>/* Preorder traversal: Example 1 */\nfunction preOrder(root: TreeNode | null, res: TreeNode[]): void {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // Record solution\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.dart<pre><code>/* Preorder traversal: Example 1 */\nvoid preOrder(TreeNode? root, List&lt;TreeNode&gt; res) {\n  if (root == null) {\n    return;\n  }\n  if (root.val == 7) {\n    // Record solution\n    res.add(root);\n  }\n  preOrder(root.left, res);\n  preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.rs<pre><code>/* Preorder traversal: Example 1 */\nfn pre_order(res: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        if node.borrow().val == 7 {\n            // Record solution\n            res.push(node.clone());\n        }\n        pre_order(res, node.borrow().left.as_ref());\n        pre_order(res, node.borrow().right.as_ref());\n    }\n}\n</code></pre> preorder_traversal_i_compact.c<pre><code>/* Preorder traversal: Example 1 */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // Record solution\n        res[resSize++] = root;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.kt<pre><code>/* Preorder traversal: Example 1 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    if (root._val == 7) {\n        // Record solution\n        res!!.add(root)\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n}\n</code></pre> preorder_traversal_i_compact.rb<pre><code>### Pre-order traversal: example 1 ###\ndef pre_order(root)\n  return unless root\n\n  # Record solution\n  $res &lt;&lt; root if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\nend\n</code></pre> <p></p> <p> Figure 13-1   Search for nodes in preorder traversal </p>","path":["Chapter 13. Backtracking","13.1   Backtracking Algorithm"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1311-attempt-and-backtrack","level":2,"title":"13.1.1   Attempt and Backtrack","text":"<p>The reason it is called a backtracking algorithm is that it employs \"attempt\" and \"backtrack\" strategies when searching the solution space. When the algorithm encounters a state where it cannot continue forward or cannot find a solution that satisfies the constraints, it will undo the previous choice, return to a previous state, and try other possible choices.</p> <p>For Example 1, visiting each node represents an \"attempt\", while skipping over a leaf node or the <code>return</code> that brings the traversal back to the parent node represents a \"backtrack\".</p> <p>It is worth noting that backtracking is not limited to function returns alone. To illustrate this, let's extend Example 1 slightly.</p> <p>Example 2</p> <p>In a binary tree, search all nodes with value \\(7\\), and return the paths from the root node to these nodes.</p> <p>Based on the code from Example 1, we need to use a list <code>path</code> to record the path of visited nodes. When we reach a node with value \\(7\\), we copy <code>path</code> and add it to the result list <code>res</code>. After traversal is complete, <code>res</code> contains all the solutions. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_ii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"Preorder traversal: Example 2\"\"\"\n    if root is None:\n        return\n    # Attempt\n    path.append(root)\n    if root.val == 7:\n        # Record solution\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # Backtrack\n    path.pop()\n</code></pre> preorder_traversal_ii_compact.cpp<pre><code>/* Preorder traversal: Example 2 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    // Attempt\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // Record solution\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // Backtrack\n    path.pop_back();\n}\n</code></pre> preorder_traversal_ii_compact.java<pre><code>/* Preorder traversal: Example 2 */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    // Attempt\n    path.add(root);\n    if (root.val == 7) {\n        // Record solution\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // Backtrack\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_ii_compact.cs<pre><code>/* Preorder traversal: Example 2 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    // Attempt\n    path.Add(root);\n    if (root.val == 7) {\n        // Record solution\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // Backtrack\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_ii_compact.go<pre><code>/* Preorder traversal: Example 2 */\nfunc preOrderII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    // Attempt\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // Record solution\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderII(root.Left, res, path)\n    preOrderII(root.Right, res, path)\n    // Backtrack\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_ii_compact.swift<pre><code>/* Preorder traversal: Example 2 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // Attempt\n    path.append(root)\n    if root.val == 7 {\n        // Record solution\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // Backtrack\n    path.removeLast()\n}\n</code></pre> preorder_traversal_ii_compact.js<pre><code>/* Preorder traversal: Example 2 */\nfunction preOrder(root, path, res) {\n    if (root === null) {\n        return;\n    }\n    // Attempt\n    path.push(root);\n    if (root.val === 7) {\n        // Record solution\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // Backtrack\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.ts<pre><code>/* Preorder traversal: Example 2 */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    if (root === null) {\n        return;\n    }\n    // Attempt\n    path.push(root);\n    if (root.val === 7) {\n        // Record solution\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // Backtrack\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.dart<pre><code>/* Preorder traversal: Example 2 */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null) {\n    return;\n  }\n\n  // Attempt\n  path.add(root);\n  if (root.val == 7) {\n    // Record solution\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // Backtrack\n  path.removeLast();\n}\n</code></pre> preorder_traversal_ii_compact.rs<pre><code>/* Preorder traversal: Example 2 */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        // Attempt\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // Record solution\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.as_ref());\n        pre_order(res, path, node.borrow().right.as_ref());\n        // Backtrack\n        path.pop();\n    }\n}\n</code></pre> preorder_traversal_ii_compact.c<pre><code>/* Preorder traversal: Example 2 */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    // Attempt\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // Record solution\n        for (int i = 0; i &lt; pathSize; ++i) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // Backtrack\n    pathSize--;\n}\n</code></pre> preorder_traversal_ii_compact.kt<pre><code>/* Preorder traversal: Example 2 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    // Attempt\n    path!!.add(root)\n    if (root._val == 7) {\n        // Record solution\n        res!!.add(path!!.toMutableList())\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // Backtrack\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_ii_compact.rb<pre><code>### Pre-order traversal: example 2 ###\ndef pre_order(root)\n  return unless root\n\n  # Attempt\n  $path &lt;&lt; root\n\n  # Record solution\n  $res &lt;&lt; $path.dup if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\n\n  # Backtrack\n  $path.pop\nend\n</code></pre> <p>In each \"attempt\", we record the path by adding the current node to <code>path</code>; before \"backtracking\", we need to remove the node from <code>path</code>, to restore the state before this attempt.</p> <p>Observing the process shown in the following figure, we can understand attempt and backtrack as \"advance\" and \"undo\", two operations that are the reverse of each other.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 13-2   Attempt and backtrack </p>","path":["Chapter 13. Backtracking","13.1   Backtracking Algorithm"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1312-pruning","level":2,"title":"13.1.2   Pruning","text":"<p>Complex backtracking problems usually contain one or more constraints. Constraints can typically be used for \"pruning\".</p> <p>Example 3</p> <p>In a binary tree, search all nodes with value \\(7\\) and return the paths from the root node to these nodes, but require that the paths do not contain nodes with value \\(3\\).</p> <p>To satisfy the above constraints, we need to add pruning operations: during the search process, if we encounter a node with value \\(3\\), we return early and do not continue searching. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_iii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"Preorder traversal: Example 3\"\"\"\n    # Pruning\n    if root is None or root.val == 3:\n        return\n    # Attempt\n    path.append(root)\n    if root.val == 7:\n        # Record solution\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # Backtrack\n    path.pop()\n</code></pre> preorder_traversal_iii_compact.cpp<pre><code>/* Preorder traversal: Example 3 */\nvoid preOrder(TreeNode *root) {\n    // Pruning\n    if (root == nullptr || root-&gt;val == 3) {\n        return;\n    }\n    // Attempt\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // Record solution\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // Backtrack\n    path.pop_back();\n}\n</code></pre> preorder_traversal_iii_compact.java<pre><code>/* Preorder traversal: Example 3 */\nvoid preOrder(TreeNode root) {\n    // Pruning\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // Attempt\n    path.add(root);\n    if (root.val == 7) {\n        // Record solution\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // Backtrack\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_iii_compact.cs<pre><code>/* Preorder traversal: Example 3 */\nvoid PreOrder(TreeNode? root) {\n    // Pruning\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // Attempt\n    path.Add(root);\n    if (root.val == 7) {\n        // Record solution\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // Backtrack\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_iii_compact.go<pre><code>/* Preorder traversal: Example 3 */\nfunc preOrderIII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    // Pruning\n    if root == nil || root.Val == 3 {\n        return\n    }\n    // Attempt\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // Record solution\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderIII(root.Left, res, path)\n    preOrderIII(root.Right, res, path)\n    // Backtrack\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_iii_compact.swift<pre><code>/* Preorder traversal: Example 3 */\nfunc preOrder(root: TreeNode?) {\n    // Pruning\n    guard let root = root, root.val != 3 else {\n        return\n    }\n    // Attempt\n    path.append(root)\n    if root.val == 7 {\n        // Record solution\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // Backtrack\n    path.removeLast()\n}\n</code></pre> preorder_traversal_iii_compact.js<pre><code>/* Preorder traversal: Example 3 */\nfunction preOrder(root, path, res) {\n    // Pruning\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // Attempt\n    path.push(root);\n    if (root.val === 7) {\n        // Record solution\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // Backtrack\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.ts<pre><code>/* Preorder traversal: Example 3 */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // Pruning\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // Attempt\n    path.push(root);\n    if (root.val === 7) {\n        // Record solution\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // Backtrack\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.dart<pre><code>/* Preorder traversal: Example 3 */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null || root.val == 3) {\n    return;\n  }\n\n  // Attempt\n  path.add(root);\n  if (root.val == 7) {\n    // Record solution\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // Backtrack\n  path.removeLast();\n}\n</code></pre> preorder_traversal_iii_compact.rs<pre><code>/* Preorder traversal: Example 3 */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    // Pruning\n    if root.is_none() || root.as_ref().unwrap().borrow().val == 3 {\n        return;\n    }\n    if let Some(node) = root {\n        // Attempt\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // Record solution\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.as_ref());\n        pre_order(res, path, node.borrow().right.as_ref());\n        // Backtrack\n        path.pop();\n    }\n}\n</code></pre> preorder_traversal_iii_compact.c<pre><code>/* Preorder traversal: Example 3 */\nvoid preOrder(TreeNode *root) {\n    // Pruning\n    if (root == NULL || root-&gt;val == 3) {\n        return;\n    }\n    // Attempt\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // Record solution\n        for (int i = 0; i &lt; pathSize; i++) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // Backtrack\n    pathSize--;\n}\n</code></pre> preorder_traversal_iii_compact.kt<pre><code>/* Preorder traversal: Example 3 */\nfun preOrder(root: TreeNode?) {\n    // Pruning\n    if (root == null || root._val == 3) {\n        return\n    }\n    // Attempt\n    path!!.add(root)\n    if (root._val == 7) {\n        // Record solution\n        res!!.add(path!!.toMutableList())\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // Backtrack\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_iii_compact.rb<pre><code>### Pre-order traversal: example 3 ###\ndef pre_order(root)\n  # Pruning\n  return if !root || root.val == 3\n\n  # Attempt\n  $path.append(root)\n\n  # Record solution\n  $res &lt;&lt; $path.dup if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\n\n  # Backtrack\n  $path.pop\nend\n</code></pre> <p>\"Pruning\" is a vivid term. As shown in the following figure, during the search process, we \"prune\" search branches that do not satisfy the constraints, avoiding many meaningless attempts and thus improving search efficiency.</p> <p></p> <p> Figure 13-3   Pruning according to constraints </p>","path":["Chapter 13. Backtracking","13.1   Backtracking Algorithm"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1313-framework-code","level":2,"title":"13.1.3   Framework Code","text":"<p>Next, we attempt to extract a general framework centered on backtracking's \"attempt, backtrack, and pruning\" to improve code generality.</p> <p>In the following framework code, <code>state</code> represents the current state of the problem, and <code>choices</code> represents the choices available in the current state:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def backtrack(state: State, choices: list[choice], res: list[state]):\n    \"\"\"Backtracking algorithm framework\"\"\"\n    # Check if it is a solution\n    if is_solution(state):\n        # Record the solution\n        record_solution(state, res)\n        # Stop searching\n        return\n    # Traverse all choices\n    for choice in choices:\n        # Pruning: check if the choice is valid\n        if is_valid(state, choice):\n            # Attempt: make a choice and update the state\n            make_choice(state, choice)\n            backtrack(state, choices, res)\n            # Backtrack: undo the choice and restore to the previous state\n            undo_choice(state, choice)\n</code></pre> <pre><code>/* Backtracking algorithm framework */\nvoid backtrack(State *state, vector&lt;Choice *&gt; &amp;choices, vector&lt;State *&gt; &amp;res) {\n    // Check if it is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res);\n        // Stop searching\n        return;\n    }\n    // Traverse all choices\n    for (Choice choice : choices) {\n        // Pruning: check if the choice is valid\n        if (isValid(state, choice)) {\n            // Attempt: make a choice and update the state\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // Backtrack: undo the choice and restore to the previous state\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking algorithm framework */\nvoid backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // Check if it is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res);\n        // Stop searching\n        return;\n    }\n    // Traverse all choices\n    for (Choice choice : choices) {\n        // Pruning: check if the choice is valid\n        if (isValid(state, choice)) {\n            // Attempt: make a choice and update the state\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // Backtrack: undo the choice and restore to the previous state\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking algorithm framework */\nvoid Backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // Check if it is a solution\n    if (IsSolution(state)) {\n        // Record the solution\n        RecordSolution(state, res);\n        // Stop searching\n        return;\n    }\n    // Traverse all choices\n    foreach (Choice choice in choices) {\n        // Pruning: check if the choice is valid\n        if (IsValid(state, choice)) {\n            // Attempt: make a choice and update the state\n            MakeChoice(state, choice);\n            Backtrack(state, choices, res);\n            // Backtrack: undo the choice and restore to the previous state\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking algorithm framework */\nfunc backtrack(state *State, choices []Choice, res *[]State) {\n    // Check if it is a solution\n    if isSolution(state) {\n        // Record the solution\n        recordSolution(state, res)\n        // Stop searching\n        return\n    }\n    // Traverse all choices\n    for _, choice := range choices {\n        // Pruning: check if the choice is valid\n        if isValid(state, choice) {\n            // Attempt: make a choice and update the state\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // Backtrack: undo the choice and restore to the previous state\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking algorithm framework */\nfunc backtrack(state: inout State, choices: [Choice], res: inout [State]) {\n    // Check if it is a solution\n    if isSolution(state: state) {\n        // Record the solution\n        recordSolution(state: state, res: &amp;res)\n        // Stop searching\n        return\n    }\n    // Traverse all choices\n    for choice in choices {\n        // Pruning: check if the choice is valid\n        if isValid(state: state, choice: choice) {\n            // Attempt: make a choice and update the state\n            makeChoice(state: &amp;state, choice: choice)\n            backtrack(state: &amp;state, choices: choices, res: &amp;res)\n            // Backtrack: undo the choice and restore to the previous state\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking algorithm framework */\nfunction backtrack(state, choices, res) {\n    // Check if it is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res);\n        // Stop searching\n        return;\n    }\n    // Traverse all choices\n    for (let choice of choices) {\n        // Pruning: check if the choice is valid\n        if (isValid(state, choice)) {\n            // Attempt: make a choice and update the state\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // Backtrack: undo the choice and restore to the previous state\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking algorithm framework */\nfunction backtrack(state: State, choices: Choice[], res: State[]): void {\n    // Check if it is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res);\n        // Stop searching\n        return;\n    }\n    // Traverse all choices\n    for (let choice of choices) {\n        // Pruning: check if the choice is valid\n        if (isValid(state, choice)) {\n            // Attempt: make a choice and update the state\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // Backtrack: undo the choice and restore to the previous state\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking algorithm framework */\nvoid backtrack(State state, List&lt;Choice&gt;, List&lt;State&gt; res) {\n  // Check if it is a solution\n  if (isSolution(state)) {\n    // Record the solution\n    recordSolution(state, res);\n    // Stop searching\n    return;\n  }\n  // Traverse all choices\n  for (Choice choice in choices) {\n    // Pruning: check if the choice is valid\n    if (isValid(state, choice)) {\n      // Attempt: make a choice and update the state\n      makeChoice(state, choice);\n      backtrack(state, choices, res);\n      // Backtrack: undo the choice and restore to the previous state\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> <pre><code>/* Backtracking algorithm framework */\nfn backtrack(state: &amp;mut State, choices: &amp;Vec&lt;Choice&gt;, res: &amp;mut Vec&lt;State&gt;) {\n    // Check if it is a solution\n    if is_solution(state) {\n        // Record the solution\n        record_solution(state, res);\n        // Stop searching\n        return;\n    }\n    // Traverse all choices\n    for choice in choices {\n        // Pruning: check if the choice is valid\n        if is_valid(state, choice) {\n            // Attempt: make a choice and update the state\n            make_choice(state, choice);\n            backtrack(state, choices, res);\n            // Backtrack: undo the choice and restore to the previous state\n            undo_choice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking algorithm framework */\nvoid backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) {\n    // Check if it is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res, numRes);\n        // Stop searching\n        return;\n    }\n    // Traverse all choices\n    for (int i = 0; i &lt; numChoices; i++) {\n        // Pruning: check if the choice is valid\n        if (isValid(state, &amp;choices[i])) {\n            // Attempt: make a choice and update the state\n            makeChoice(state, &amp;choices[i]);\n            backtrack(state, choices, numChoices, res, numRes);\n            // Backtrack: undo the choice and restore to the previous state\n            undoChoice(state, &amp;choices[i]);\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking algorithm framework */\nfun backtrack(state: State?, choices: List&lt;Choice?&gt;, res: List&lt;State?&gt;?) {\n    // Check if it is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res)\n        // Stop searching\n        return\n    }\n    // Traverse all choices\n    for (choice in choices) {\n        // Pruning: check if the choice is valid\n        if (isValid(state, choice)) {\n            // Attempt: make a choice and update the state\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // Backtrack: undo the choice and restore to the previous state\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>### Backtracking algorithm framework ###\ndef backtrack(state, choices, res)\n    # Check if it is a solution\n    if is_solution?(state)\n        # Record the solution\n        record_solution(state, res)\n        return\n    end\n\n    # Traverse all choices\n    for choice in choices\n        # Pruning: check if the choice is valid\n        if is_valid?(state, choice)\n            # Attempt: make a choice and update the state\n            make_choice(state, choice)\n            backtrack(state, choices, res)\n            # Backtrack: undo the choice and restore to the previous state\n            undo_choice(state, choice)\n        end\n    end\nend\n</code></pre> <p>Next, we solve Example 3 based on the framework code. The state <code>state</code> is the node traversal path, the choices <code>choices</code> are the left and right child nodes of the current node, and the result <code>res</code> is a list of paths:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_iii_template.py<pre><code>def is_solution(state: list[TreeNode]) -&gt; bool:\n    \"\"\"Check if the current state is a solution\"\"\"\n    return state and state[-1].val == 7\n\ndef record_solution(state: list[TreeNode], res: list[list[TreeNode]]):\n    \"\"\"Record solution\"\"\"\n    res.append(list(state))\n\ndef is_valid(state: list[TreeNode], choice: TreeNode) -&gt; bool:\n    \"\"\"Check if the choice is valid under the current state\"\"\"\n    return choice is not None and choice.val != 3\n\ndef make_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"Update state\"\"\"\n    state.append(choice)\n\ndef undo_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"Restore state\"\"\"\n    state.pop()\n\ndef backtrack(\n    state: list[TreeNode], choices: list[TreeNode], res: list[list[TreeNode]]\n):\n    \"\"\"Backtracking algorithm: Example 3\"\"\"\n    # Check if it is a solution\n    if is_solution(state):\n        # Record solution\n        record_solution(state, res)\n    # Traverse all choices\n    for choice in choices:\n        # Pruning: check if the choice is valid\n        if is_valid(state, choice):\n            # Attempt: make choice, update state\n            make_choice(state, choice)\n            # Proceed to the next round of selection\n            backtrack(state, [choice.left, choice.right], res)\n            # Backtrack: undo choice, restore to previous state\n            undo_choice(state, choice)\n</code></pre> preorder_traversal_iii_template.cpp<pre><code>/* Check if the current state is a solution */\nbool isSolution(vector&lt;TreeNode *&gt; &amp;state) {\n    return !state.empty() &amp;&amp; state.back()-&gt;val == 7;\n}\n\n/* Record solution */\nvoid recordSolution(vector&lt;TreeNode *&gt; &amp;state, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    res.push_back(state);\n}\n\n/* Check if the choice is valid under the current state */\nbool isValid(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    return choice != nullptr &amp;&amp; choice-&gt;val != 3;\n}\n\n/* Update state */\nvoid makeChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.push_back(choice);\n}\n\n/* Restore state */\nvoid undoChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.pop_back();\n}\n\n/* Backtracking algorithm: Example 3 */\nvoid backtrack(vector&lt;TreeNode *&gt; &amp;state, vector&lt;TreeNode *&gt; &amp;choices, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    // Check if it is a solution\n    if (isSolution(state)) {\n        // Record solution\n        recordSolution(state, res);\n    }\n    // Traverse all choices\n    for (TreeNode *choice : choices) {\n        // Pruning: check if the choice is valid\n        if (isValid(state, choice)) {\n            // Attempt: make choice, update state\n            makeChoice(state, choice);\n            // Proceed to the next round of selection\n            vector&lt;TreeNode *&gt; nextChoices{choice-&gt;left, choice-&gt;right};\n            backtrack(state, nextChoices, res);\n            // Backtrack: undo choice, restore to previous state\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.java<pre><code>/* Check if the current state is a solution */\nboolean isSolution(List&lt;TreeNode&gt; state) {\n    return !state.isEmpty() &amp;&amp; state.get(state.size() - 1).val == 7;\n}\n\n/* Record solution */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.add(new ArrayList&lt;&gt;(state));\n}\n\n/* Check if the choice is valid under the current state */\nboolean isValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* Update state */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.add(choice);\n}\n\n/* Restore state */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.remove(state.size() - 1);\n}\n\n/* Backtracking algorithm: Example 3 */\nvoid backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // Check if it is a solution\n    if (isSolution(state)) {\n        // Record solution\n        recordSolution(state, res);\n    }\n    // Traverse all choices\n    for (TreeNode choice : choices) {\n        // Pruning: check if the choice is valid\n        if (isValid(state, choice)) {\n            // Attempt: make choice, update state\n            makeChoice(state, choice);\n            // Proceed to the next round of selection\n            backtrack(state, Arrays.asList(choice.left, choice.right), res);\n            // Backtrack: undo choice, restore to previous state\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.cs<pre><code>/* Check if the current state is a solution */\nbool IsSolution(List&lt;TreeNode&gt; state) {\n    return state.Count != 0 &amp;&amp; state[^1].val == 7;\n}\n\n/* Record solution */\nvoid RecordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.Add(new List&lt;TreeNode&gt;(state));\n}\n\n/* Check if the choice is valid under the current state */\nbool IsValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* Update state */\nvoid MakeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.Add(choice);\n}\n\n/* Restore state */\nvoid UndoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.RemoveAt(state.Count - 1);\n}\n\n/* Backtracking algorithm: Example 3 */\nvoid Backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // Check if it is a solution\n    if (IsSolution(state)) {\n        // Record solution\n        RecordSolution(state, res);\n    }\n    // Traverse all choices\n    foreach (TreeNode choice in choices) {\n        // Pruning: check if the choice is valid\n        if (IsValid(state, choice)) {\n            // Attempt: make choice, update state\n            MakeChoice(state, choice);\n            // Proceed to the next round of selection\n            Backtrack(state, [choice.left!, choice.right!], res);\n            // Backtrack: undo choice, restore to previous state\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.go<pre><code>/* Check if the current state is a solution */\nfunc isSolution(state *[]*TreeNode) bool {\n    return len(*state) != 0 &amp;&amp; (*state)[len(*state)-1].Val == 7\n}\n\n/* Record solution */\nfunc recordSolution(state *[]*TreeNode, res *[][]*TreeNode) {\n    *res = append(*res, append([]*TreeNode{}, *state...))\n}\n\n/* Check if the choice is valid under the current state */\nfunc isValid(state *[]*TreeNode, choice *TreeNode) bool {\n    return choice != nil &amp;&amp; choice.Val != 3\n}\n\n/* Update state */\nfunc makeChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = append(*state, choice)\n}\n\n/* Restore state */\nfunc undoChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = (*state)[:len(*state)-1]\n}\n\n/* Backtracking algorithm: Example 3 */\nfunc backtrackIII(state *[]*TreeNode, choices *[]*TreeNode, res *[][]*TreeNode) {\n    // Check if it is a solution\n    if isSolution(state) {\n        // Record solution\n        recordSolution(state, res)\n    }\n    // Traverse all choices\n    for _, choice := range *choices {\n        // Pruning: check if the choice is valid\n        if isValid(state, choice) {\n            // Attempt: make choice, update state\n            makeChoice(state, choice)\n            // Proceed to the next round of selection\n            temp := make([]*TreeNode, 0)\n            temp = append(temp, choice.Left, choice.Right)\n            backtrackIII(state, &amp;temp, res)\n            // Backtrack: undo choice, restore to previous state\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.swift<pre><code>/* Check if the current state is a solution */\nfunc isSolution(state: [TreeNode]) -&gt; Bool {\n    !state.isEmpty &amp;&amp; state.last!.val == 7\n}\n\n/* Record solution */\nfunc recordSolution(state: [TreeNode], res: inout [[TreeNode]]) {\n    res.append(state)\n}\n\n/* Check if the choice is valid under the current state */\nfunc isValid(state: [TreeNode], choice: TreeNode?) -&gt; Bool {\n    choice != nil &amp;&amp; choice!.val != 3\n}\n\n/* Update state */\nfunc makeChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.append(choice)\n}\n\n/* Restore state */\nfunc undoChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.removeLast()\n}\n\n/* Backtracking algorithm: Example 3 */\nfunc backtrack(state: inout [TreeNode], choices: [TreeNode], res: inout [[TreeNode]]) {\n    // Check if it is a solution\n    if isSolution(state: state) {\n        recordSolution(state: state, res: &amp;res)\n    }\n    // Traverse all choices\n    for choice in choices {\n        // Pruning: check if the choice is valid\n        if isValid(state: state, choice: choice) {\n            // Attempt: make choice, update state\n            makeChoice(state: &amp;state, choice: choice)\n            // Proceed to the next round of selection\n            backtrack(state: &amp;state, choices: [choice.left, choice.right].compactMap { $0 }, res: &amp;res)\n            // Backtrack: undo choice, restore to previous state\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.js<pre><code>/* Check if the current state is a solution */\nfunction isSolution(state) {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* Record solution */\nfunction recordSolution(state, res) {\n    res.push([...state]);\n}\n\n/* Check if the choice is valid under the current state */\nfunction isValid(state, choice) {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* Update state */\nfunction makeChoice(state, choice) {\n    state.push(choice);\n}\n\n/* Restore state */\nfunction undoChoice(state) {\n    state.pop();\n}\n\n/* Backtracking algorithm: Example 3 */\nfunction backtrack(state, choices, res) {\n    // Check if it is a solution\n    if (isSolution(state)) {\n        // Record solution\n        recordSolution(state, res);\n    }\n    // Traverse all choices\n    for (const choice of choices) {\n        // Pruning: check if the choice is valid\n        if (isValid(state, choice)) {\n            // Attempt: make choice, update state\n            makeChoice(state, choice);\n            // Proceed to the next round of selection\n            backtrack(state, [choice.left, choice.right], res);\n            // Backtrack: undo choice, restore to previous state\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.ts<pre><code>/* Check if the current state is a solution */\nfunction isSolution(state: TreeNode[]): boolean {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* Record solution */\nfunction recordSolution(state: TreeNode[], res: TreeNode[][]): void {\n    res.push([...state]);\n}\n\n/* Check if the choice is valid under the current state */\nfunction isValid(state: TreeNode[], choice: TreeNode): boolean {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* Update state */\nfunction makeChoice(state: TreeNode[], choice: TreeNode): void {\n    state.push(choice);\n}\n\n/* Restore state */\nfunction undoChoice(state: TreeNode[]): void {\n    state.pop();\n}\n\n/* Backtracking algorithm: Example 3 */\nfunction backtrack(\n    state: TreeNode[],\n    choices: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // Check if it is a solution\n    if (isSolution(state)) {\n        // Record solution\n        recordSolution(state, res);\n    }\n    // Traverse all choices\n    for (const choice of choices) {\n        // Pruning: check if the choice is valid\n        if (isValid(state, choice)) {\n            // Attempt: make choice, update state\n            makeChoice(state, choice);\n            // Proceed to the next round of selection\n            backtrack(state, [choice.left, choice.right], res);\n            // Backtrack: undo choice, restore to previous state\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.dart<pre><code>/* Check if the current state is a solution */\nbool isSolution(List&lt;TreeNode&gt; state) {\n  return state.isNotEmpty &amp;&amp; state.last.val == 7;\n}\n\n/* Record solution */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n  res.add(List.from(state));\n}\n\n/* Check if the choice is valid under the current state */\nbool isValid(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* Update state */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.add(choice!);\n}\n\n/* Restore state */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.removeLast();\n}\n\n/* Backtracking algorithm: Example 3 */\nvoid backtrack(\n  List&lt;TreeNode&gt; state,\n  List&lt;TreeNode?&gt; choices,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  // Check if it is a solution\n  if (isSolution(state)) {\n    // Record solution\n    recordSolution(state, res);\n  }\n  // Traverse all choices\n  for (TreeNode? choice in choices) {\n    // Pruning: check if the choice is valid\n    if (isValid(state, choice)) {\n      // Attempt: make choice, update state\n      makeChoice(state, choice);\n      // Proceed to the next round of selection\n      backtrack(state, [choice!.left, choice.right], res);\n      // Backtrack: undo choice, restore to previous state\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> preorder_traversal_iii_template.rs<pre><code>/* Check if the current state is a solution */\nfn is_solution(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\n    return !state.is_empty() &amp;&amp; state.last().unwrap().borrow().val == 7;\n}\n\n/* Record solution */\nfn record_solution(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    res.push(state.clone());\n}\n\n/* Check if the choice is valid under the current state */\nfn is_valid(_: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\n    return choice.is_some() &amp;&amp; choice.unwrap().borrow().val != 3;\n}\n\n/* Update state */\nfn make_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.push(choice);\n}\n\n/* Restore state */\nfn undo_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, _: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.pop();\n}\n\n/* Backtracking algorithm: Example 3 */\nfn backtrack(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    choices: &amp;Vec&lt;Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    // Check if it is a solution\n    if is_solution(state) {\n        // Record solution\n        record_solution(state, res);\n    }\n    // Traverse all choices\n    for &amp;choice in choices.iter() {\n        // Pruning: check if the choice is valid\n        if is_valid(state, choice) {\n            // Attempt: make choice, update state\n            make_choice(state, choice.unwrap().clone());\n            // Proceed to the next round of selection\n            backtrack(\n                state,\n                &amp;vec![\n                    choice.unwrap().borrow().left.as_ref(),\n                    choice.unwrap().borrow().right.as_ref(),\n                ],\n                res,\n            );\n            // Backtrack: undo choice, restore to previous state\n            undo_choice(state, choice.unwrap().clone());\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.c<pre><code>/* Check if the current state is a solution */\nbool isSolution(void) {\n    return pathSize &gt; 0 &amp;&amp; path[pathSize - 1]-&gt;val == 7;\n}\n\n/* Record solution */\nvoid recordSolution(void) {\n    for (int i = 0; i &lt; pathSize; i++) {\n        res[resSize][i] = path[i];\n    }\n    resSize++;\n}\n\n/* Check if the choice is valid under the current state */\nbool isValid(TreeNode *choice) {\n    return choice != NULL &amp;&amp; choice-&gt;val != 3;\n}\n\n/* Update state */\nvoid makeChoice(TreeNode *choice) {\n    path[pathSize++] = choice;\n}\n\n/* Restore state */\nvoid undoChoice(void) {\n    pathSize--;\n}\n\n/* Backtracking algorithm: Example 3 */\nvoid backtrack(TreeNode *choices[2]) {\n    // Check if it is a solution\n    if (isSolution()) {\n        // Record solution\n        recordSolution();\n    }\n    // Traverse all choices\n    for (int i = 0; i &lt; 2; i++) {\n        TreeNode *choice = choices[i];\n        // Pruning: check if the choice is valid\n        if (isValid(choice)) {\n            // Attempt: make choice, update state\n            makeChoice(choice);\n            // Proceed to the next round of selection\n            TreeNode *nextChoices[2] = {choice-&gt;left, choice-&gt;right};\n            backtrack(nextChoices);\n            // Backtrack: undo choice, restore to previous state\n            undoChoice();\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.kt<pre><code>/* Check if the current state is a solution */\nfun isSolution(state: MutableList&lt;TreeNode?&gt;): Boolean {\n    return state.isNotEmpty() &amp;&amp; state[state.size - 1]?._val == 7\n}\n\n/* Record solution */\nfun recordSolution(state: MutableList&lt;TreeNode?&gt;?, res: MutableList&lt;MutableList&lt;TreeNode?&gt;?&gt;) {\n    res.add(state!!.toMutableList())\n}\n\n/* Check if the choice is valid under the current state */\nfun isValid(state: MutableList&lt;TreeNode?&gt;?, choice: TreeNode?): Boolean {\n    return choice != null &amp;&amp; choice._val != 3\n}\n\n/* Update state */\nfun makeChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.add(choice)\n}\n\n/* Restore state */\nfun undoChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.removeLast()\n}\n\n/* Backtracking algorithm: Example 3 */\nfun backtrack(\n    state: MutableList&lt;TreeNode?&gt;,\n    choices: MutableList&lt;TreeNode?&gt;,\n    res: MutableList&lt;MutableList&lt;TreeNode?&gt;?&gt;\n) {\n    // Check if it is a solution\n    if (isSolution(state)) {\n        // Record solution\n        recordSolution(state, res)\n    }\n    // Traverse all choices\n    for (choice in choices) {\n        // Pruning: check if the choice is valid\n        if (isValid(state, choice)) {\n            // Attempt: make choice, update state\n            makeChoice(state, choice)\n            // Proceed to the next round of selection\n            backtrack(state, mutableListOf(choice!!.left, choice.right), res)\n            // Backtrack: undo choice, restore to previous state\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.rb<pre><code>### Check if current state is solution ###\ndef is_solution?(state)\n  !state.empty? &amp;&amp; state.last.val == 7\nend\n\n### Record solution ###\ndef record_solution(state, res)\n  res &lt;&lt; state.dup\nend\n\n### Check if choice is valid in current state ###\ndef is_valid?(state, choice)\n  choice &amp;&amp; choice.val != 3\nend\n\n### Update state ###\ndef make_choice(state, choice)\n  state &lt;&lt; choice\nend\n\n### Restore state ###\ndef undo_choice(state, choice)\n  state.pop\nend\n\n### Backtracking: example 3 ###\ndef backtrack(state, choices, res)\n  # Check if it is a solution\n  record_solution(state, res) if is_solution?(state)\n\n  # Traverse all choices\n  for choice in choices\n    # Pruning: check if the choice is valid\n    if is_valid?(state, choice)\n      # Attempt: make choice, update state\n      make_choice(state, choice)\n      # Proceed to the next round of selection\n      backtrack(state, [choice.left, choice.right], res)\n      # Backtrack: undo choice, restore to previous state\n      undo_choice(state, choice)\n    end\n  end\nend\n</code></pre> <p>As per the problem statement, we should continue searching after finding a node with value \\(7\\). Therefore, we need to remove the <code>return</code> statement after recording the solution. The following figure compares the search process with and without the <code>return</code> statement.</p> <p></p> <p> Figure 13-4   Comparison of search process with and without return statement </p> <p>Compared to code based on preorder traversal, code based on the backtracking algorithm framework appears more verbose, but is more general. In fact, many backtracking problems can be solved within this framework. We only need to define <code>state</code> and <code>choices</code> for the specific problem and implement each method in the framework.</p>","path":["Chapter 13. Backtracking","13.1   Backtracking Algorithm"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1314-common-terminology","level":2,"title":"13.1.4   Common Terminology","text":"<p>To analyze algorithmic problems more clearly, we summarize the meanings of common terminology used in backtracking algorithms and provide corresponding examples from Example 3, as shown in the following table.</p> <p> Table 13-1   Common Backtracking Algorithm Terminology </p> Term Definition Example 3 Solution (solution) A solution is an answer that satisfies the specific conditions of a problem; there may be one or more solutions All paths from root to nodes with value \\(7\\) that satisfy the constraint Constraint (constraint) A constraint is a condition in the problem that limits the feasibility of solutions, typically used for pruning Paths do not contain nodes with value \\(3\\) State (state) State represents the situation of a problem at a certain moment, including the choices already made The currently visited node path, i.e., the <code>path</code> list of nodes Attempt (attempt) An attempt is the process of exploring the solution space according to available choices, including making choices, updating state, and checking if it is a solution Recursively visit left (right) child nodes, add nodes to <code>path</code>, check if node value is \\(7\\) Backtrack (backtracking) Backtracking refers to undoing previous choices and returning to a previous state when encountering a state that does not satisfy constraints Stop searching when passing over leaf nodes, ending node visits, or encountering nodes with value \\(3\\); function returns Pruning (pruning) Pruning is a method of avoiding meaningless search paths according to problem characteristics and constraints, which can improve search efficiency When encountering a node with value \\(3\\), do not continue searching <p>Tip</p> <p>The concepts of problem, solution, state, etc. are universal and appear in divide-and-conquer, backtracking, dynamic programming, greedy algorithms, and others.</p>","path":["Chapter 13. Backtracking","13.1   Backtracking Algorithm"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1315-advantages-and-limitations","level":2,"title":"13.1.5   Advantages and Limitations","text":"<p>The backtracking algorithm is essentially a depth-first search algorithm that tries all possible solutions until it finds one that satisfies the conditions. The advantage of this approach is that it can find all possible solutions, and with reasonable pruning operations, it achieves high efficiency.</p> <p>However, when dealing with large-scale or complex problems, the running efficiency of the backtracking algorithm may be unacceptable.</p> <ul> <li>Time: The backtracking algorithm usually needs to traverse all possibilities in the state space, and the time complexity can reach exponential or factorial order.</li> <li>Space: During recursive calls, the current state needs to be saved (such as paths, auxiliary variables used for pruning, etc.), and when the depth is large, the space requirement can become very large.</li> </ul> <p>Nevertheless, the backtracking algorithm is still the best solution for certain search problems and constraint satisfaction problems. For these problems, since we cannot predict which choices will generate valid solutions, we must traverse all possible choices. In this case, the key is how to optimize efficiency. There are two common efficiency optimization methods.</p> <ul> <li>Pruning: Avoid searching paths that are guaranteed not to produce solutions, thereby saving time and space.</li> <li>Heuristic search: Introduce certain strategies or estimation values during the search process to prioritize searching paths that are most likely to produce valid solutions.</li> </ul>","path":["Chapter 13. Backtracking","13.1   Backtracking Algorithm"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1316-typical-backtracking-examples","level":2,"title":"13.1.6   Typical Backtracking Examples","text":"<p>The backtracking algorithm can be used to solve many search problems, constraint satisfaction problems, and combinatorial optimization problems.</p> <p>Search problems: The goal of these problems is to find solutions that satisfy specific conditions.</p> <ul> <li>Permutation problem: Given a set, find all possible permutations and combinations.</li> <li>Subset sum problem: Given a set and a target sum, find all subsets in the set whose elements sum to the target.</li> <li>Tower of Hanoi: Given three pegs and a series of disks of different sizes, move all disks from one peg to another, moving only one disk at a time, and never placing a larger disk on a smaller disk.</li> </ul> <p>Constraint satisfaction problems: The goal of these problems is to find solutions that satisfy all constraints.</p> <ul> <li>N-Queens: Place \\(n\\) queens on an \\(n \\times n\\) chessboard such that they do not attack each other.</li> <li>Sudoku: Fill numbers \\(1\\) to \\(9\\) in a \\(9 \\times 9\\) grid such that each row, column, and \\(3 \\times 3\\) subgrid contains no repeated digits.</li> <li>Graph coloring: Given an undirected graph, color each vertex with the minimum number of colors such that adjacent vertices have different colors.</li> </ul> <p>Combinatorial optimization problems: The goal of these problems is to find an optimal solution that satisfies certain conditions in a combinatorial space.</p> <ul> <li>0-1 Knapsack: Given a set of items and a knapsack, each item has a value and weight. Under the knapsack capacity constraint, select items to maximize total value.</li> <li>Traveling Salesman Problem: Starting from a point in a graph, visit all other points exactly once and return to the starting point, finding the shortest path.</li> <li>Maximum Clique: Given an undirected graph, find the largest complete subgraph, i.e., a subgraph where any two vertices are connected by an edge.</li> </ul> <p>Note that for many combinatorial optimization problems, backtracking is not the optimal solution.</p> <ul> <li>The 0-1 Knapsack problem is usually solved using dynamic programming to achieve higher time efficiency.</li> <li>The Traveling Salesman Problem is a famous NP-Hard problem; common solutions include genetic algorithms and ant colony algorithms.</li> <li>The Maximum Clique problem is a classical problem in graph theory and can be solved using heuristic algorithms such as greedy algorithms.</li> </ul>","path":["Chapter 13. Backtracking","13.1   Backtracking Algorithm"],"tags":[]},{"location":"chapter_backtracking/exercises/","level":1,"title":"13.6   Exercises","text":"","path":["Chapter 13. Backtracking","13.6   Exercises"],"tags":[]},{"location":"chapter_backtracking/exercises/#1361-concept-review","level":2,"title":"13.6.1   Concept Review","text":"","path":["Chapter 13. Backtracking","13.6   Exercises"],"tags":[]},{"location":"chapter_backtracking/exercises/#1-will-this-permutation-algorithm-miss-results","level":3,"title":"1.   Will This Permutation Algorithm Miss Results?","text":"<p>A backtracking algorithm tries to generate all permutations using <code>1, 2, 3</code> in that order. Each time it chooses a number <code>x</code>, it:</p> <ol> <li>Appends <code>x</code> to the current path.</li> <li>Marks <code>x</code> as \"used.\"</li> <li>Recursively fills the next position.</li> </ol> <p>When the recursive call returns, the student removes only <code>x</code> from the end of the path and then tries the next number.</p> <ol> <li>Which permutation does the algorithm generate first? Can it still generate all 6 permutations?</li> <li>Before returning to the previous level, is removing only the last number from the path enough? If not, what else must be done? Explain why.</li> </ol> Answer <ol> <li> <p>It first generates <code>[1, 2, 3]</code>, but it cannot generate all the permutations. Although the path becomes shorter when calls return,     the markers for 1, 2, and 3 all remain \"used,\" leaving no available number for later branches.</p> </li> <li> <p>It is not enough. After removing <code>x</code> from the end of the path, the algorithm must also mark <code>x</code> as \"unused\" again.     The current path and the used markers together describe the search state. A choice changes both of them, so backtracking must restore both     before another branch can choose <code>x</code> again.</p> </li> </ol>","path":["Chapter 13. Backtracking","13.6   Exercises"],"tags":[]},{"location":"chapter_backtracking/exercises/#2-does-the-order-of-choosing-numbers-matter","level":3,"title":"2.   Does the Order of Choosing Numbers Matter?","text":"<p>You are given the sorted array <code>[2, 3, 5]</code> and target value 5. Each number may be chosen repeatedly. The algorithm requires the numbers along each search path to appear only in nondecreasing order.</p> <ol> <li>Which distinct combinations can be found?</li> <li>Why is there no need to search for the same group of numbers in different orders? What does the nondecreasing-order restriction accomplish?</li> <li>Suppose the current path is <code>[3]</code>, the remaining amount is 2, and the next candidate is 3. Why can the algorithm stop checking all later candidates at this level?</li> </ol> Answer <ol> <li> <p>The distinct combinations are <code>[2, 3]</code> and <code>[5]</code>.</p> </li> <li> <p>This exercise treats <code>[2, 3]</code> and <code>[3, 2]</code> as the same combination; the order in which numbers are selected does not create a different answer.     Requiring numbers in a path to appear in nondecreasing order lets the search skip duplicate arrangements such as <code>[3, 2]</code>.</p> </li> <li> <p>The remaining amount is 2, while candidate 3 is already greater than 2. Because the array is sorted,     all later candidates are even larger and cannot be added to the current combination, so the algorithm can stop checking this level immediately.</p> </li> </ol>","path":["Chapter 13. Backtracking","13.6   Exercises"],"tags":[]},{"location":"chapter_backtracking/exercises/#3-where-can-the-next-queen-be-placed","level":3,"title":"3.   Where Can the Next Queen Be Placed?","text":"<p>Place queens row by row on a <code>4 × 4</code> chessboard, with both row and column indices starting at 0. Queens have already been placed at <code>(0, 1)</code> and <code>(1, 3)</code>. The next queen must be placed in row 2.</p> <ol> <li>Which columns are ruled out because they already contain a queen?</li> <li>Among the remaining columns, which positions are ruled out because they share a diagonal with an existing queen?</li> <li>Which positions in row 2 remain available to try?</li> </ol> Answer <ol> <li> <p>Columns 1 and 3 already contain queens, so positions <code>(2, 1)</code> and <code>(2, 3)</code> are ruled out.</p> </li> <li> <p>Among the remaining positions, <code>(2, 2)</code> lies on the same diagonal as <code>(1, 3)</code>, so it is also ruled out.     Position <code>(2, 0)</code> shares neither a column nor a diagonal with either existing queen.</p> </li> <li> <p>The only position to try in row 2 is <code>(2, 0)</code>.</p> <p>This step shows only that the current placement is valid. If the board cannot be completed later, the algorithm must still backtrack and try an earlier alternative.</p> </li> </ol>","path":["Chapter 13. Backtracking","13.6   Exercises"],"tags":[]},{"location":"chapter_backtracking/exercises/#1362-programming-exercises","level":2,"title":"13.6.2   Programming Exercises","text":"","path":["Chapter 13. Backtracking","13.6   Exercises"],"tags":[]},{"location":"chapter_backtracking/exercises/#1-permutations-of-distinct-elements","level":3,"title":"1.   Permutations of Distinct Elements","text":"<p>The integer array <code>nums</code> contains at least one element, and all its elements are distinct. List every possible order formed by using each element exactly once, and return each order as an array. The permutations may appear in any order in the result. Use backtracking, with a Boolean array recording whether the element at each position has already been selected for the current permutation.</p> Hints <ol> <li>The recursion depth indicates which position of the permutation is currently being filled</li> <li>At each level, try only elements that have not yet been used</li> <li>When the path's length equals the length of <code>nums</code>, add a copy of the path to the answer</li> </ol> <p>LeetCode</p>","path":["Chapter 13. Backtracking","13.6   Exercises"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/","level":1,"title":"13.4   N-Queens Problem","text":"<p>Question</p> <p>According to the rules of chess, a queen can attack any piece in the same row, column, or diagonal. Given \\(n\\) queens and an \\(n \\times n\\) chessboard, find an arrangement such that no two queens can attack each other.</p> <p>As shown in Figure 13-15, when \\(n = 4\\), there are two solutions that can be found. From the perspective of the backtracking algorithm, an \\(n \\times n\\) chessboard has \\(n^2\\) squares, which provide all the choices <code>choices</code>. During the process of placing queens one by one, the chessboard state changes continuously, and the chessboard at each moment represents the state <code>state</code>.</p> <p></p> <p> Figure 13-15   Solution to the 4-queens problem </p> <p>Figure 13-16 illustrates the three constraints of this problem: multiple queens cannot be in the same row, the same column, or on the same diagonal. It is worth noting that diagonals are divided into two types: the main diagonal <code>\\</code> and the anti-diagonal <code>/</code>.</p> <p></p> <p> Figure 13-16   Constraints of the n-queens problem </p>","path":["Chapter 13. Backtracking","13.4   N-Queens Problem"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/#1-row-by-row-placement-strategy","level":3,"title":"1.   Row-By-Row Placement Strategy","text":"<p>Since both the number of queens and the number of rows on the chessboard are \\(n\\), we can easily derive a conclusion: each row of the chessboard allows one and only one queen to be placed.</p> <p>This means we can adopt a row-by-row placement strategy: starting from the first row, place one queen in each row until the last row is completed.</p> <p>Figure 13-17 shows the row-by-row placement process for the 4-queens problem. Due to space limitations, the figure only expands one search branch of the first row, and all schemes that violate the column or diagonal constraints are pruned.</p> <p></p> <p> Figure 13-17   Row-by-row placement strategy </p> <p>Essentially, the row-by-row placement strategy serves a pruning function, as it avoids all search branches where multiple queens appear in the same row.</p>","path":["Chapter 13. Backtracking","13.4   N-Queens Problem"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/#2-column-and-diagonal-pruning","level":3,"title":"2.   Column and Diagonal Pruning","text":"<p>To satisfy the column constraint, we can use a boolean array <code>cols</code> of length \\(n\\) to record whether each column has a queen. Before each placement decision, we use <code>cols</code> to prune columns that already have queens, and dynamically update the state of <code>cols</code> during backtracking.</p> <p>Tip</p> <p>Please note that the origin of the matrix is located in the upper-left corner, where the row index increases from top to bottom, and the column index increases from left to right.</p> <p>So how do we handle diagonal constraints? Consider a square on the chessboard with row and column indices \\((row, col)\\). If we select a specific main diagonal in the matrix, we find that all squares on that diagonal have the same difference between their row and column indices, meaning that \\(row - col\\) is a constant value for all squares on the main diagonal.</p> <p>In other words, if two squares satisfy \\(row_1 - col_1 = row_2 - col_2\\), they must be on the same main diagonal. Using this pattern, we can use the array <code>diags1</code> shown in Figure 13-18 to record whether there is a queen on each main diagonal.</p> <p>Similarly, for all squares on an anti-diagonal, the sum \\(row + col\\) is a constant value. We can likewise use the array <code>diags2</code> to handle anti-diagonal constraints.</p> <p></p> <p> Figure 13-18   Handling column and diagonal constraints </p>","path":["Chapter 13. Backtracking","13.4   N-Queens Problem"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/#3-code-implementation","level":3,"title":"3.   Code Implementation","text":"<p>Please note that in an \\(n \\times n\\) square matrix, the range of \\(row - col\\) is \\([-n + 1, n - 1]\\), and the range of \\(row + col\\) is \\([0, 2n - 2]\\). Therefore, the number of both main diagonals and anti-diagonals is \\(2n - 1\\), meaning the length of both arrays <code>diags1</code> and <code>diags2</code> is \\(2n - 1\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby n_queens.py<pre><code>def backtrack(\n    row: int,\n    n: int,\n    state: list[list[str]],\n    res: list[list[list[str]]],\n    cols: list[bool],\n    diags1: list[bool],\n    diags2: list[bool],\n):\n    \"\"\"Backtracking algorithm: N queens\"\"\"\n    # When all rows are placed, record the solution\n    if row == n:\n        res.append([list(row) for row in state])\n        return\n    # Traverse all columns\n    for col in range(n):\n        # Calculate the main diagonal and anti-diagonal corresponding to this cell\n        diag1 = row - col + n - 1\n        diag2 = row + col\n        # Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell\n        if not cols[col] and not diags1[diag1] and not diags2[diag2]:\n            # Attempt: place the queen in this cell\n            state[row][col] = \"Q\"\n            cols[col] = diags1[diag1] = diags2[diag2] = True\n            # Place the next row\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            # Backtrack: restore this cell to an empty cell\n            state[row][col] = \"#\"\n            cols[col] = diags1[diag1] = diags2[diag2] = False\n\ndef n_queens(n: int) -&gt; list[list[list[str]]]:\n    \"\"\"Solve N queens\"\"\"\n    # Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell\n    state = [[\"#\" for _ in range(n)] for _ in range(n)]\n    cols = [False] * n  # Record whether there is a queen in the column\n    diags1 = [False] * (2 * n - 1)  # Record whether there is a queen on the main diagonal\n    diags2 = [False] * (2 * n - 1)  # Record whether there is a queen on the anti-diagonal\n    res = []\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n</code></pre> n_queens.cpp<pre><code>/* Backtracking algorithm: N queens */\nvoid backtrack(int row, int n, vector&lt;vector&lt;string&gt;&gt; &amp;state, vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; &amp;res, vector&lt;bool&gt; &amp;cols,\n               vector&lt;bool&gt; &amp;diags1, vector&lt;bool&gt; &amp;diags2) {\n    // When all rows are placed, record the solution\n    if (row == n) {\n        res.push_back(state);\n        return;\n    }\n    // Traverse all columns\n    for (int col = 0; col &lt; n; col++) {\n        // Calculate the main diagonal and anti-diagonal corresponding to this cell\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // Attempt: place the queen in this cell\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // Place the next row\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // Backtrack: restore this cell to an empty cell\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* Solve N queens */\nvector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; nQueens(int n) {\n    // Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell\n    vector&lt;vector&lt;string&gt;&gt; state(n, vector&lt;string&gt;(n, \"#\"));\n    vector&lt;bool&gt; cols(n, false);           // Record whether there is a queen in the column\n    vector&lt;bool&gt; diags1(2 * n - 1, false); // Record whether there is a queen on the main diagonal\n    vector&lt;bool&gt; diags2(2 * n - 1, false); // Record whether there is a queen on the anti-diagonal\n    vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; res;\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.java<pre><code>/* Backtracking algorithm: N queens */\nvoid backtrack(int row, int n, List&lt;List&lt;String&gt;&gt; state, List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n        boolean[] cols, boolean[] diags1, boolean[] diags2) {\n    // When all rows are placed, record the solution\n    if (row == n) {\n        List&lt;List&lt;String&gt;&gt; copyState = new ArrayList&lt;&gt;();\n        for (List&lt;String&gt; sRow : state) {\n            copyState.add(new ArrayList&lt;&gt;(sRow));\n        }\n        res.add(copyState);\n        return;\n    }\n    // Traverse all columns\n    for (int col = 0; col &lt; n; col++) {\n        // Calculate the main diagonal and anti-diagonal corresponding to this cell\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // Attempt: place the queen in this cell\n            state.get(row).set(col, \"Q\");\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // Place the next row\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // Backtrack: restore this cell to an empty cell\n            state.get(row).set(col, \"#\");\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* Solve N queens */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n    // Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell\n    List&lt;List&lt;String&gt;&gt; state = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;String&gt; row = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            row.add(\"#\");\n        }\n        state.add(row);\n    }\n    boolean[] cols = new boolean[n]; // Record whether there is a queen in the column\n    boolean[] diags1 = new boolean[2 * n - 1]; // Record whether there is a queen on the main diagonal\n    boolean[] diags2 = new boolean[2 * n - 1]; // Record whether there is a queen on the anti-diagonal\n    List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = new ArrayList&lt;&gt;();\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.cs<pre><code>/* Backtracking algorithm: N queens */\nvoid Backtrack(int row, int n, List&lt;List&lt;string&gt;&gt; state, List&lt;List&lt;List&lt;string&gt;&gt;&gt; res,\n        bool[] cols, bool[] diags1, bool[] diags2) {\n    // When all rows are placed, record the solution\n    if (row == n) {\n        List&lt;List&lt;string&gt;&gt; copyState = [];\n        foreach (List&lt;string&gt; sRow in state) {\n            copyState.Add(new List&lt;string&gt;(sRow));\n        }\n        res.Add(copyState);\n        return;\n    }\n    // Traverse all columns\n    for (int col = 0; col &lt; n; col++) {\n        // Calculate the main diagonal and anti-diagonal corresponding to this cell\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // Attempt: place the queen in this cell\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // Place the next row\n            Backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // Backtrack: restore this cell to an empty cell\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* Solve N queens */\nList&lt;List&lt;List&lt;string&gt;&gt;&gt; NQueens(int n) {\n    // Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell\n    List&lt;List&lt;string&gt;&gt; state = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;string&gt; row = [];\n        for (int j = 0; j &lt; n; j++) {\n            row.Add(\"#\");\n        }\n        state.Add(row);\n    }\n    bool[] cols = new bool[n]; // Record whether there is a queen in the column\n    bool[] diags1 = new bool[2 * n - 1]; // Record whether there is a queen on the main diagonal\n    bool[] diags2 = new bool[2 * n - 1]; // Record whether there is a queen on the anti-diagonal\n    List&lt;List&lt;List&lt;string&gt;&gt;&gt; res = [];\n\n    Backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.go<pre><code>/* Backtracking algorithm: N queens */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n    // When all rows are placed, record the solution\n    if row == n {\n        newState := make([][]string, len(*state))\n        for i, _ := range newState {\n            newState[i] = make([]string, len((*state)[0]))\n            copy(newState[i], (*state)[i])\n\n        }\n        *res = append(*res, newState)\n        return\n    }\n    // Traverse all columns\n    for col := 0; col &lt; n; col++ {\n        // Calculate the main diagonal and anti-diagonal corresponding to this cell\n        diag1 := row - col + n - 1\n        diag2 := row + col\n        // Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell\n        if !(*cols)[col] &amp;&amp; !(*diags1)[diag1] &amp;&amp; !(*diags2)[diag2] {\n            // Attempt: place the queen in this cell\n            (*state)[row][col] = \"Q\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n            // Place the next row\n            backtrack(row+1, n, state, res, cols, diags1, diags2)\n            // Backtrack: restore this cell to an empty cell\n            (*state)[row][col] = \"#\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n        }\n    }\n}\n\n/* Solve N queens */\nfunc nQueens(n int) [][][]string {\n    // Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell\n    state := make([][]string, n)\n    for i := 0; i &lt; n; i++ {\n        row := make([]string, n)\n        for i := 0; i &lt; n; i++ {\n            row[i] = \"#\"\n        }\n        state[i] = row\n    }\n    // Record whether there is a queen in the column\n    cols := make([]bool, n)\n    diags1 := make([]bool, 2*n-1)\n    diags2 := make([]bool, 2*n-1)\n    res := make([][][]string, 0)\n    backtrack(0, n, &amp;state, &amp;res, &amp;cols, &amp;diags1, &amp;diags2)\n    return res\n}\n</code></pre> n_queens.swift<pre><code>/* Backtracking algorithm: N queens */\nfunc backtrack(row: Int, n: Int, state: inout [[String]], res: inout [[[String]]], cols: inout [Bool], diags1: inout [Bool], diags2: inout [Bool]) {\n    // When all rows are placed, record the solution\n    if row == n {\n        res.append(state)\n        return\n    }\n    // Traverse all columns\n    for col in 0 ..&lt; n {\n        // Calculate the main diagonal and anti-diagonal corresponding to this cell\n        let diag1 = row - col + n - 1\n        let diag2 = row + col\n        // Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // Attempt: place the queen in this cell\n            state[row][col] = \"Q\"\n            cols[col] = true\n            diags1[diag1] = true\n            diags2[diag2] = true\n            // Place the next row\n            backtrack(row: row + 1, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n            // Backtrack: restore this cell to an empty cell\n            state[row][col] = \"#\"\n            cols[col] = false\n            diags1[diag1] = false\n            diags2[diag2] = false\n        }\n    }\n}\n\n/* Solve N queens */\nfunc nQueens(n: Int) -&gt; [[[String]]] {\n    // Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell\n    var state = Array(repeating: Array(repeating: \"#\", count: n), count: n)\n    var cols = Array(repeating: false, count: n) // Record whether there is a queen in the column\n    var diags1 = Array(repeating: false, count: 2 * n - 1) // Record whether there is a queen on the main diagonal\n    var diags2 = Array(repeating: false, count: 2 * n - 1) // Record whether there is a queen on the anti-diagonal\n    var res: [[[String]]] = []\n\n    backtrack(row: 0, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n\n    return res\n}\n</code></pre> n_queens.js<pre><code>/* Backtracking algorithm: N queens */\nfunction backtrack(row, n, state, res, cols, diags1, diags2) {\n    // When all rows are placed, record the solution\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // Traverse all columns\n    for (let col = 0; col &lt; n; col++) {\n        // Calculate the main diagonal and anti-diagonal corresponding to this cell\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // Attempt: place the queen in this cell\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // Place the next row\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // Backtrack: restore this cell to an empty cell\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* Solve N queens */\nfunction nQueens(n) {\n    // Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // Record whether there is a queen in the column\n    const diags1 = Array(2 * n - 1).fill(false); // Record whether there is a queen on the main diagonal\n    const diags2 = Array(2 * n - 1).fill(false); // Record whether there is a queen on the anti-diagonal\n    const res = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.ts<pre><code>/* Backtracking algorithm: N queens */\nfunction backtrack(\n    row: number,\n    n: number,\n    state: string[][],\n    res: string[][][],\n    cols: boolean[],\n    diags1: boolean[],\n    diags2: boolean[]\n): void {\n    // When all rows are placed, record the solution\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // Traverse all columns\n    for (let col = 0; col &lt; n; col++) {\n        // Calculate the main diagonal and anti-diagonal corresponding to this cell\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // Attempt: place the queen in this cell\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // Place the next row\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // Backtrack: restore this cell to an empty cell\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* Solve N queens */\nfunction nQueens(n: number): string[][][] {\n    // Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // Record whether there is a queen in the column\n    const diags1 = Array(2 * n - 1).fill(false); // Record whether there is a queen on the main diagonal\n    const diags2 = Array(2 * n - 1).fill(false); // Record whether there is a queen on the anti-diagonal\n    const res: string[][][] = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.dart<pre><code>/* Backtracking algorithm: N queens */\nvoid backtrack(\n  int row,\n  int n,\n  List&lt;List&lt;String&gt;&gt; state,\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n  List&lt;bool&gt; cols,\n  List&lt;bool&gt; diags1,\n  List&lt;bool&gt; diags2,\n) {\n  // When all rows are placed, record the solution\n  if (row == n) {\n    List&lt;List&lt;String&gt;&gt; copyState = [];\n    for (List&lt;String&gt; sRow in state) {\n      copyState.add(List.from(sRow));\n    }\n    res.add(copyState);\n    return;\n  }\n  // Traverse all columns\n  for (int col = 0; col &lt; n; col++) {\n    // Calculate the main diagonal and anti-diagonal corresponding to this cell\n    int diag1 = row - col + n - 1;\n    int diag2 = row + col;\n    // Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell\n    if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n      // Attempt: place the queen in this cell\n      state[row][col] = \"Q\";\n      cols[col] = true;\n      diags1[diag1] = true;\n      diags2[diag2] = true;\n      // Place the next row\n      backtrack(row + 1, n, state, res, cols, diags1, diags2);\n      // Backtrack: restore this cell to an empty cell\n      state[row][col] = \"#\";\n      cols[col] = false;\n      diags1[diag1] = false;\n      diags2[diag2] = false;\n    }\n  }\n}\n\n/* Solve N queens */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n  // Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell\n  List&lt;List&lt;String&gt;&gt; state = List.generate(n, (index) =&gt; List.filled(n, \"#\"));\n  List&lt;bool&gt; cols = List.filled(n, false); // Record whether there is a queen in the column\n  List&lt;bool&gt; diags1 = List.filled(2 * n - 1, false); // Record whether there is a queen on the main diagonal\n  List&lt;bool&gt; diags2 = List.filled(2 * n - 1, false); // Record whether there is a queen on the anti-diagonal\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = [];\n\n  backtrack(0, n, state, res, cols, diags1, diags2);\n\n  return res;\n}\n</code></pre> n_queens.rs<pre><code>/* Backtracking algorithm: N queens */\nfn backtrack(\n    row: usize,\n    n: usize,\n    state: &amp;mut Vec&lt;Vec&lt;String&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt;,\n    cols: &amp;mut [bool],\n    diags1: &amp;mut [bool],\n    diags2: &amp;mut [bool],\n) {\n    // When all rows are placed, record the solution\n    if row == n {\n        res.push(state.clone());\n        return;\n    }\n    // Traverse all columns\n    for col in 0..n {\n        // Calculate the main diagonal and anti-diagonal corresponding to this cell\n        let diag1 = row + n - 1 - col;\n        let diag2 = row + col;\n        // Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // Attempt: place the queen in this cell\n            state[row][col] = \"Q\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (true, true, true);\n            // Place the next row\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // Backtrack: restore this cell to an empty cell\n            state[row][col] = \"#\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (false, false, false);\n        }\n    }\n}\n\n/* Solve N queens */\nfn n_queens(n: usize) -&gt; Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; {\n    // Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell\n    let mut state: Vec&lt;Vec&lt;String&gt;&gt; = vec![vec![\"#\".to_string(); n]; n];\n    let mut cols = vec![false; n]; // Record whether there is a queen in the column\n    let mut diags1 = vec![false; 2 * n - 1]; // Record whether there is a queen on the main diagonal\n    let mut diags2 = vec![false; 2 * n - 1]; // Record whether there is a queen on the anti-diagonal\n    let mut res: Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; = Vec::new();\n\n    backtrack(\n        0,\n        n,\n        &amp;mut state,\n        &amp;mut res,\n        &amp;mut cols,\n        &amp;mut diags1,\n        &amp;mut diags2,\n    );\n\n    res\n}\n</code></pre> n_queens.c<pre><code>/* Backtracking algorithm: N queens */\nvoid backtrack(int row, int n, char state[MAX_SIZE][MAX_SIZE], char ***res, int *resSize, bool cols[MAX_SIZE],\n               bool diags1[2 * MAX_SIZE - 1], bool diags2[2 * MAX_SIZE - 1]) {\n    // When all rows are placed, record the solution\n    if (row == n) {\n        res[*resSize] = (char **)malloc(sizeof(char *) * n);\n        for (int i = 0; i &lt; n; ++i) {\n            res[*resSize][i] = (char *)malloc(sizeof(char) * (n + 1));\n            strcpy(res[*resSize][i], state[i]);\n        }\n        (*resSize)++;\n        return;\n    }\n    // Traverse all columns\n    for (int col = 0; col &lt; n; col++) {\n        // Calculate the main diagonal and anti-diagonal corresponding to this cell\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // Attempt: place the queen in this cell\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // Place the next row\n            backtrack(row + 1, n, state, res, resSize, cols, diags1, diags2);\n            // Backtrack: restore this cell to an empty cell\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* Solve N queens */\nchar ***nQueens(int n, int *returnSize) {\n    char state[MAX_SIZE][MAX_SIZE];\n    // Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell\n    for (int i = 0; i &lt; n; ++i) {\n        for (int j = 0; j &lt; n; ++j) {\n            state[i][j] = '#';\n        }\n        state[i][n] = '\\0';\n    }\n    bool cols[MAX_SIZE] = {false};           // Record whether there is a queen in the column\n    bool diags1[2 * MAX_SIZE - 1] = {false}; // Record whether there is a queen on the main diagonal\n    bool diags2[2 * MAX_SIZE - 1] = {false}; // Record whether there is a queen on the anti-diagonal\n\n    char ***res = (char ***)malloc(sizeof(char **) * MAX_SIZE);\n    *returnSize = 0;\n    backtrack(0, n, state, res, returnSize, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.kt<pre><code>/* Backtracking algorithm: N queens */\nfun backtrack(\n    row: Int,\n    n: Int,\n    state: MutableList&lt;MutableList&lt;String&gt;&gt;,\n    res: MutableList&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt;,\n    cols: BooleanArray,\n    diags1: BooleanArray,\n    diags2: BooleanArray\n) {\n    // When all rows are placed, record the solution\n    if (row == n) {\n        val copyState = mutableListOf&lt;MutableList&lt;String&gt;&gt;()\n        for (sRow in state) {\n            copyState.add(sRow.toMutableList())\n        }\n        res.add(copyState)\n        return\n    }\n    // Traverse all columns\n    for (col in 0..&lt;n) {\n        // Calculate the main diagonal and anti-diagonal corresponding to this cell\n        val diag1 = row - col + n - 1\n        val diag2 = row + col\n        // Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // Attempt: place the queen in this cell\n            state[row][col] = \"Q\"\n            diags2[diag2] = true\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n            // Place the next row\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            // Backtrack: restore this cell to an empty cell\n            state[row][col] = \"#\"\n            diags2[diag2] = false\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n        }\n    }\n}\n\n/* Solve N queens */\nfun nQueens(n: Int): MutableList&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt; {\n    // Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell\n    val state = mutableListOf&lt;MutableList&lt;String&gt;&gt;()\n    for (i in 0..&lt;n) {\n        val row = mutableListOf&lt;String&gt;()\n        for (j in 0..&lt;n) {\n            row.add(\"#\")\n        }\n        state.add(row)\n    }\n    val cols = BooleanArray(n) // Record whether there is a queen in the column\n    val diags1 = BooleanArray(2 * n - 1) // Record whether there is a queen on the main diagonal\n    val diags2 = BooleanArray(2 * n - 1) // Record whether there is a queen on the anti-diagonal\n    val res = mutableListOf&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt;()\n\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n}\n</code></pre> n_queens.rb<pre><code>### Backtracking: n queens ###\ndef backtrack(row, n, state, res, cols, diags1, diags2)\n  # When all rows are placed, record the solution\n  if row == n\n    res &lt;&lt; state.map { |row| row.dup }\n    return\n  end\n\n  # Traverse all columns\n  for col in 0...n\n    # Calculate the main diagonal and anti-diagonal corresponding to this cell\n    diag1 = row - col + n - 1\n    diag2 = row + col\n    # Pruning: do not allow queens to exist in the column, main diagonal, and anti-diagonal of this cell\n    if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]\n      # Attempt: place the queen in this cell\n      state[row][col] = \"Q\"\n      cols[col] = diags1[diag1] = diags2[diag2] = true\n      # Place the next row\n      backtrack(row + 1, n, state, res, cols, diags1, diags2)\n      # Backtrack: restore this cell to an empty cell\n      state[row][col] = \"#\"\n      cols[col] = diags1[diag1] = diags2[diag2] = false\n    end\n  end\nend\n\n### Solve n queens ###\ndef n_queens(n)\n  # Initialize an n*n chessboard, where 'Q' represents a queen and '#' represents an empty cell\n  state = Array.new(n) { Array.new(n, \"#\") }\n  cols = Array.new(n, false) # Record whether there is a queen in the column\n  diags1 = Array.new(2 * n - 1, false) # Record whether there is a queen on the main diagonal\n  diags2 = Array.new(2 * n - 1, false) # Record whether there is a queen on the anti-diagonal\n  res = []\n  backtrack(0, n, state, res, cols, diags1, diags2)\n\n  res\nend\n</code></pre> <p>Placing \\(n\\) queens row by row, considering the column constraint, from the first row to the last row there are \\(n\\), \\(n-1\\), \\(\\dots\\), \\(2\\), \\(1\\) choices, using \\(O(n!)\\) time. When recording a solution, it is necessary to copy the matrix <code>state</code> and add it to <code>res</code>, and the copy operation uses \\(O(n^2)\\) time. Therefore, the overall time complexity is \\(O(n! \\cdot n^2)\\). In practice, pruning based on diagonal constraints can also significantly reduce the search space, so the search efficiency is often better than the time complexity mentioned above.</p> <p>The array <code>state</code> uses \\(O(n^2)\\) space, and the arrays <code>cols</code>, <code>diags1</code>, and <code>diags2</code> each use \\(O(n)\\) space. The maximum recursion depth is \\(n\\), using \\(O(n)\\) stack frame space. Therefore, the space complexity is \\(O(n^2)\\).</p>","path":["Chapter 13. Backtracking","13.4   N-Queens Problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/","level":1,"title":"13.2   Permutations Problem","text":"<p>The permutations problem is a classic application of backtracking algorithms. It is defined as finding all possible arrangements of elements in a given collection (such as an array or string).</p> <p>Table 13-2 shows several example datasets, including input arrays and their corresponding permutations.</p> <p> Table 13-2   Permutations Examples </p> Input Array All Permutations \\([1]\\) \\([1]\\) \\([1, 2]\\) \\([1, 2], [2, 1]\\) \\([1, 2, 3]\\) \\([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\\)","path":["Chapter 13. Backtracking","13.2   Permutations Problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1321-case-with-distinct-elements","level":2,"title":"13.2.1   Case with Distinct Elements","text":"<p>Question</p> <p>Given an integer array with no duplicate elements, return all possible permutations.</p> <p>From the perspective of backtracking algorithms, we can imagine the process of generating permutations as the result of a series of choices. Suppose the input array is \\([1, 2, 3]\\). If we first choose \\(1\\), then choose \\(3\\), and finally choose \\(2\\), we obtain the permutation \\([1, 3, 2]\\). Backtracking means undoing a choice and then trying other choices.</p> <p>From the perspective of backtracking code, the candidate set <code>choices</code> consists of all elements in the input array, and the state <code>state</code> is the elements that have been chosen so far. Note that each element can only be chosen once, therefore all elements in <code>state</code> should be unique.</p> <p>As shown in Figure 13-5, we can unfold the search process into a recursion tree, where each node in the tree represents the current state <code>state</code>. Starting from the root node, after three rounds of choices, we reach a leaf node, and each leaf node corresponds to a permutation.</p> <p></p> <p> Figure 13-5   Recursion tree of permutations </p>","path":["Chapter 13. Backtracking","13.2   Permutations Problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1-pruning-duplicate-choices","level":3,"title":"1.   Pruning Duplicate Choices","text":"<p>To ensure that each element is chosen only once, we consider introducing a boolean array <code>selected</code>, where <code>selected[i]</code> indicates whether <code>choices[i]</code> has been chosen. We implement the following pruning operation based on it.</p> <ul> <li>After making a choice <code>choices[i]</code>, we set <code>selected[i]</code> to \\(\\text{True}\\), indicating that it has been chosen.</li> <li>When traversing the candidate list <code>choices</code>, we skip all nodes that have been chosen, which is pruning.</li> </ul> <p>As shown in Figure 13-6, suppose we choose \\(1\\) in the first round, \\(3\\) in the second round, and \\(2\\) in the third round. Then we need to prune the branch of element \\(1\\) in the second round and prune the branches of elements \\(1\\) and \\(3\\) in the third round.</p> <p></p> <p> Figure 13-6   Pruning example of permutations </p> <p>Observing the above figure, we find that this pruning operation reduces the search space size from \\(O(n^n)\\) to \\(O(n!)\\).</p>","path":["Chapter 13. Backtracking","13.2   Permutations Problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#2-code-implementation","level":3,"title":"2.   Code Implementation","text":"<p>After understanding the above information, we can fill in the blanks in the template code. To shorten the overall code, we do not implement each function in the template separately, but instead unfold them in the <code>backtrack()</code> function:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby permutations_i.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"Backtracking algorithm: Permutations I\"\"\"\n    # When the state length equals the number of elements, record the solution\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # Traverse all choices\n    for i, choice in enumerate(choices):\n        # Pruning: do not allow repeated selection of elements\n        if not selected[i]:\n            # Attempt: make choice, update state\n            selected[i] = True\n            state.append(choice)\n            # Proceed to the next round of selection\n            backtrack(state, choices, selected, res)\n            # Backtrack: undo choice, restore to previous state\n            selected[i] = False\n            state.pop()\n\ndef permutations_i(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"Permutations I\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_i.cpp<pre><code>/* Backtracking algorithm: Permutations I */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // When the state length equals the number of elements, record the solution\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // Traverse all choices\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // Pruning: do not allow repeated selection of elements\n        if (!selected[i]) {\n            // Attempt: make choice, update state\n            selected[i] = true;\n            state.push_back(choice);\n            // Proceed to the next round of selection\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* Permutations I */\nvector&lt;vector&lt;int&gt;&gt; permutationsI(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_i.java<pre><code>/* Backtracking algorithm: Permutations I */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // When the state length equals the number of elements, record the solution\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // Traverse all choices\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // Pruning: do not allow repeated selection of elements\n        if (!selected[i]) {\n            // Attempt: make choice, update state\n            selected[i] = true;\n            state.add(choice);\n            // Proceed to the next round of selection\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* Permutations I */\nList&lt;List&lt;Integer&gt;&gt; permutationsI(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_i.cs<pre><code>/* Backtracking algorithm: Permutations I */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // When the state length equals the number of elements, record the solution\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // Traverse all choices\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // Pruning: do not allow repeated selection of elements\n        if (!selected[i]) {\n            // Attempt: make choice, update state\n            selected[i] = true;\n            state.Add(choice);\n            // Proceed to the next round of selection\n            Backtrack(state, choices, selected, res);\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* Permutations I */\nList&lt;List&lt;int&gt;&gt; PermutationsI(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_i.go<pre><code>/* Backtracking algorithm: Permutations I */\nfunc backtrackI(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // When the state length equals the number of elements, record the solution\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // Traverse all choices\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // Pruning: do not allow repeated selection of elements\n        if !(*selected)[i] {\n            // Attempt: make choice, update state\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // Proceed to the next round of selection\n            backtrackI(state, choices, selected, res)\n            // Backtrack: undo choice, restore to previous state\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* Permutations I */\nfunc permutationsI(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackI(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_i.swift<pre><code>/* Backtracking algorithm: Permutations I */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // When the state length equals the number of elements, record the solution\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // Traverse all choices\n    for (i, choice) in choices.enumerated() {\n        // Pruning: do not allow repeated selection of elements\n        if !selected[i] {\n            // Attempt: make choice, update state\n            selected[i] = true\n            state.append(choice)\n            // Proceed to the next round of selection\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* Permutations I */\nfunc permutationsI(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_i.js<pre><code>/* Backtracking algorithm: Permutations I */\nfunction backtrack(state, choices, selected, res) {\n    // When the state length equals the number of elements, record the solution\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // Traverse all choices\n    choices.forEach((choice, i) =&gt; {\n        // Pruning: do not allow repeated selection of elements\n        if (!selected[i]) {\n            // Attempt: make choice, update state\n            selected[i] = true;\n            state.push(choice);\n            // Proceed to the next round of selection\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* Permutations I */\nfunction permutationsI(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.ts<pre><code>/* Backtracking algorithm: Permutations I */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // When the state length equals the number of elements, record the solution\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // Traverse all choices\n    choices.forEach((choice, i) =&gt; {\n        // Pruning: do not allow repeated selection of elements\n        if (!selected[i]) {\n            // Attempt: make choice, update state\n            selected[i] = true;\n            state.push(choice);\n            // Proceed to the next round of selection\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* Permutations I */\nfunction permutationsI(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.dart<pre><code>/* Backtracking algorithm: Permutations I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // When the state length equals the number of elements, record the solution\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // Traverse all choices\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // Pruning: do not allow repeated selection of elements\n    if (!selected[i]) {\n      // Attempt: make choice, update state\n      selected[i] = true;\n      state.add(choice);\n      // Proceed to the next round of selection\n      backtrack(state, choices, selected, res);\n      // Backtrack: undo choice, restore to previous state\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* Permutations I */\nList&lt;List&lt;int&gt;&gt; permutationsI(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_i.rs<pre><code>/* Backtracking algorithm: Permutations I */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // When the state length equals the number of elements, record the solution\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // Traverse all choices\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // Pruning: do not allow repeated selection of elements\n        if !selected[i] {\n            // Attempt: make choice, update state\n            selected[i] = true;\n            state.push(choice);\n            // Proceed to the next round of selection\n            backtrack(state.clone(), choices, selected, res);\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false;\n            state.pop();\n        }\n    }\n}\n\n/* Permutations I */\nfn permutations_i(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new(); // State (subset)\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_i.c<pre><code>/* Backtracking algorithm: Permutations I */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // When the state length equals the number of elements, record the solution\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // Traverse all choices\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // Pruning: do not allow repeated selection of elements\n        if (!selected[i]) {\n            // Attempt: make choice, update state\n            selected[i] = true;\n            state[stateSize] = choice;\n            // Proceed to the next round of selection\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false;\n        }\n    }\n}\n\n/* Permutations I */\nint **permutationsI(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_i.kt<pre><code>/* Backtracking algorithm: Permutations I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // When the state length equals the number of elements, record the solution\n    if (state.size == choices.size) {\n        res.add(state.toMutableList())\n        return\n    }\n    // Traverse all choices\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // Pruning: do not allow repeated selection of elements\n        if (!selected[i]) {\n            // Attempt: make choice, update state\n            selected[i] = true\n            state.add(choice)\n            // Proceed to the next round of selection\n            backtrack(state, choices, selected, res)\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* Permutations I */\nfun permutationsI(nums: IntArray): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;()\n    backtrack(mutableListOf(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_i.rb<pre><code>### Backtracking: permutations I ###\ndef backtrack(state, choices, selected, res)\n  # When the state length equals the number of elements, record the solution\n  if state.length == choices.length\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # Traverse all choices\n  choices.each_with_index do |choice, i|\n    # Pruning: do not allow repeated selection of elements\n    unless selected[i]\n      # Attempt: make choice, update state\n      selected[i] = true\n      state &lt;&lt; choice\n      # Proceed to the next round of selection\n      backtrack(state, choices, selected, res)\n      # Backtrack: undo choice, restore to previous state\n      selected[i] = false\n      state.pop\n    end\n  end\nend\n\n### Permutations I ###\ndef permutations_i(nums)\n  res = []\n  backtrack([], nums, Array.new(nums.length, false), res)\n  res\nend\n</code></pre>","path":["Chapter 13. Backtracking","13.2   Permutations Problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1322-case-with-duplicate-elements","level":2,"title":"13.2.2   Case with Duplicate Elements","text":"<p>Question</p> <p>Given an integer array that may contain duplicate elements, return all unique permutations.</p> <p>Suppose the input array is \\([1, 1, 2]\\). To distinguish the two duplicate elements \\(1\\), we denote the second \\(1\\) as \\(\\hat{1}\\).</p> <p>As shown in Figure 13-7, half of the permutations generated by the above method are duplicates.</p> <p></p> <p> Figure 13-7   Duplicate permutations </p> <p>So how do we remove duplicate permutations? The most direct approach is to use a hash set to directly deduplicate the permutation results. However, this is not elegant because the search branches that generate duplicate permutations are unnecessary and should be identified and pruned early, which can further improve algorithm efficiency.</p>","path":["Chapter 13. Backtracking","13.2   Permutations Problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1-pruning-equal-elements","level":3,"title":"1.   Pruning Equal Elements","text":"<p>Observe Figure 13-8. In the first round, choosing \\(1\\) or choosing \\(\\hat{1}\\) is equivalent. All permutations generated under these two choices are duplicates. Therefore, we should prune \\(\\hat{1}\\).</p> <p>Similarly, after choosing \\(2\\) in the first round, the \\(1\\) and \\(\\hat{1}\\) in the second round also produce duplicate branches, so the second round's \\(\\hat{1}\\) should also be pruned.</p> <p>Essentially, our goal is to ensure that multiple equal elements are chosen only once in a certain round of choices.</p> <p></p> <p> Figure 13-8   Pruning duplicate permutations </p>","path":["Chapter 13. Backtracking","13.2   Permutations Problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#2-code-implementation_1","level":3,"title":"2.   Code Implementation","text":"<p>Building on the code from the previous problem, we initialize a hash set <code>duplicated</code> in each round of choices to record which elements have already been tried in that round, and prune equal elements:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby permutations_ii.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"Backtracking algorithm: Permutations II\"\"\"\n    # When the state length equals the number of elements, record the solution\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # Traverse all choices\n    duplicated = set[int]()\n    for i, choice in enumerate(choices):\n        # Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements\n        if not selected[i] and choice not in duplicated:\n            # Attempt: make choice, update state\n            duplicated.add(choice)  # Record the selected element value\n            selected[i] = True\n            state.append(choice)\n            # Proceed to the next round of selection\n            backtrack(state, choices, selected, res)\n            # Backtrack: undo choice, restore to previous state\n            selected[i] = False\n            state.pop()\n\ndef permutations_ii(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"Permutations II\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_ii.cpp<pre><code>/* Backtracking algorithm: Permutations II */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // When the state length equals the number of elements, record the solution\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // Traverse all choices\n    unordered_set&lt;int&gt; duplicated;\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements\n        if (!selected[i] &amp;&amp; duplicated.find(choice) == duplicated.end()) {\n            // Attempt: make choice, update state\n            duplicated.emplace(choice); // Record the selected element value\n            selected[i] = true;\n            state.push_back(choice);\n            // Proceed to the next round of selection\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* Permutations II */\nvector&lt;vector&lt;int&gt;&gt; permutationsII(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_ii.java<pre><code>/* Backtracking algorithm: Permutations II */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // When the state length equals the number of elements, record the solution\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // Traverse all choices\n    Set&lt;Integer&gt; duplicated = new HashSet&lt;Integer&gt;();\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // Attempt: make choice, update state\n            duplicated.add(choice); // Record the selected element value\n            selected[i] = true;\n            state.add(choice);\n            // Proceed to the next round of selection\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* Permutations II */\nList&lt;List&lt;Integer&gt;&gt; permutationsII(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_ii.cs<pre><code>/* Backtracking algorithm: Permutations II */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // When the state length equals the number of elements, record the solution\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // Traverse all choices\n    HashSet&lt;int&gt; duplicated = [];\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements\n        if (!selected[i] &amp;&amp; !duplicated.Contains(choice)) {\n            // Attempt: make choice, update state\n            duplicated.Add(choice); // Record the selected element value\n            selected[i] = true;\n            state.Add(choice);\n            // Proceed to the next round of selection\n            Backtrack(state, choices, selected, res);\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* Permutations II */\nList&lt;List&lt;int&gt;&gt; PermutationsII(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_ii.go<pre><code>/* Backtracking algorithm: Permutations II */\nfunc backtrackII(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // When the state length equals the number of elements, record the solution\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // Traverse all choices\n    duplicated := make(map[int]struct{}, 0)\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements\n        if _, ok := duplicated[choice]; !ok &amp;&amp; !(*selected)[i] {\n            // Attempt: make choice, update state\n            // Record the selected element value\n            duplicated[choice] = struct{}{}\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // Proceed to the next round of selection\n            backtrackII(state, choices, selected, res)\n            // Backtrack: undo choice, restore to previous state\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* Permutations II */\nfunc permutationsII(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackII(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_ii.swift<pre><code>/* Backtracking algorithm: Permutations II */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // When the state length equals the number of elements, record the solution\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // Traverse all choices\n    var duplicated: Set&lt;Int&gt; = []\n    for (i, choice) in choices.enumerated() {\n        // Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements\n        if !selected[i], !duplicated.contains(choice) {\n            // Attempt: make choice, update state\n            duplicated.insert(choice) // Record the selected element value\n            selected[i] = true\n            state.append(choice)\n            // Proceed to the next round of selection\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* Permutations II */\nfunc permutationsII(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_ii.js<pre><code>/* Backtracking algorithm: Permutations II */\nfunction backtrack(state, choices, selected, res) {\n    // When the state length equals the number of elements, record the solution\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // Traverse all choices\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // Attempt: make choice, update state\n            duplicated.add(choice); // Record the selected element value\n            selected[i] = true;\n            state.push(choice);\n            // Proceed to the next round of selection\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* Permutations II */\nfunction permutationsII(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.ts<pre><code>/* Backtracking algorithm: Permutations II */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // When the state length equals the number of elements, record the solution\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // Traverse all choices\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // Attempt: make choice, update state\n            duplicated.add(choice); // Record the selected element value\n            selected[i] = true;\n            state.push(choice);\n            // Proceed to the next round of selection\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* Permutations II */\nfunction permutationsII(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.dart<pre><code>/* Backtracking algorithm: Permutations II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // When the state length equals the number of elements, record the solution\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // Traverse all choices\n  Set&lt;int&gt; duplicated = {};\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements\n    if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n      // Attempt: make choice, update state\n      duplicated.add(choice); // Record the selected element value\n      selected[i] = true;\n      state.add(choice);\n      // Proceed to the next round of selection\n      backtrack(state, choices, selected, res);\n      // Backtrack: undo choice, restore to previous state\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* Permutations II */\nList&lt;List&lt;int&gt;&gt; permutationsII(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_ii.rs<pre><code>/* Backtracking algorithm: Permutations II */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // When the state length equals the number of elements, record the solution\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // Traverse all choices\n    let mut duplicated = HashSet::&lt;i32&gt;::new();\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements\n        if !selected[i] &amp;&amp; !duplicated.contains(&amp;choice) {\n            // Attempt: make choice, update state\n            duplicated.insert(choice); // Record the selected element value\n            selected[i] = true;\n            state.push(choice);\n            // Proceed to the next round of selection\n            backtrack(state.clone(), choices, selected, res);\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false;\n            state.pop();\n        }\n    }\n}\n\n/* Permutations II */\nfn permutations_ii(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new();\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_ii.c<pre><code>/* Backtracking algorithm: Permutations II */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // When the state length equals the number of elements, record the solution\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // Traverse all choices\n    bool duplicated[MAX_SIZE] = {false};\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements\n        if (!selected[i] &amp;&amp; !duplicated[choice]) {\n            // Attempt: make choice, update state\n            duplicated[choice] = true; // Record the selected element value\n            selected[i] = true;\n            state[stateSize] = choice;\n            // Proceed to the next round of selection\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false;\n        }\n    }\n}\n\n/* Permutations II */\nint **permutationsII(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_ii.kt<pre><code>/* Backtracking algorithm: Permutations II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // When the state length equals the number of elements, record the solution\n    if (state.size == choices.size) {\n        res.add(state.toMutableList())\n        return\n    }\n    // Traverse all choices\n    val duplicated = HashSet&lt;Int&gt;()\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // Attempt: make choice, update state\n            duplicated.add(choice) // Record the selected element value\n            selected[i] = true\n            state.add(choice)\n            // Proceed to the next round of selection\n            backtrack(state, choices, selected, res)\n            // Backtrack: undo choice, restore to previous state\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* Permutations II */\nfun permutationsII(nums: IntArray): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;()\n    backtrack(mutableListOf(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_ii.rb<pre><code>### Backtracking: permutations II ###\ndef backtrack(state, choices, selected, res)\n  # When the state length equals the number of elements, record the solution\n  if state.length == choices.length\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # Traverse all choices\n  duplicated = Set.new\n  choices.each_with_index do |choice, i|\n    # Pruning: do not allow repeated selection of elements and do not allow repeated selection of equal elements\n    if !selected[i] &amp;&amp; !duplicated.include?(choice)\n      # Attempt: make choice, update state\n      duplicated.add(choice)\n      selected[i] = true\n      state &lt;&lt; choice\n      # Proceed to the next round of selection\n      backtrack(state, choices, selected, res)\n      # Backtrack: undo choice, restore to previous state\n      selected[i] = false\n      state.pop\n    end\n  end\nend\n\n### Permutations II ###\ndef permutations_ii(nums)\n  res = []\n  backtrack([], nums, Array.new(nums.length, false), res)\n  res\nend\n</code></pre> <p>Assuming elements are pairwise distinct, there are \\(n!\\) (factorial) permutations of \\(n\\) elements. When recording results, we need to copy a list of length \\(n\\), using \\(O(n)\\) time. Therefore, the time complexity is \\(O(n! \\cdot n)\\).</p> <p>The maximum recursion depth is \\(n\\), using \\(O(n)\\) stack frame space. <code>selected</code> uses \\(O(n)\\) space. At most \\(n\\) <code>duplicated</code> sets exist simultaneously, using \\(O(n^2)\\) space. Therefore, the space complexity is \\(O(n^2)\\).</p>","path":["Chapter 13. Backtracking","13.2   Permutations Problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#3-comparison-of-two-pruning-methods","level":3,"title":"3.   Comparison of Two Pruning Methods","text":"<p>Note that although both <code>selected</code> and <code>duplicated</code> are used for pruning, they have different objectives.</p> <ul> <li>Pruning duplicate choices: There is only one <code>selected</code> throughout the entire search process. It records which elements are included in the current state, and its purpose is to prevent an element from appearing repeatedly in <code>state</code>.</li> <li>Pruning equal elements: Each round of choices (each <code>backtrack</code> function call) contains a <code>duplicated</code> set. It records which elements have been chosen in this round's iteration (the <code>for</code> loop), and its purpose is to ensure that equal elements are chosen only once.</li> </ul> <p>Figure 13-9 shows the effective scope of the two pruning conditions. Note that each node in the tree represents a choice, and the nodes on the path from the root to a leaf node form a permutation.</p> <p></p> <p> Figure 13-9   Effective scope of two pruning conditions </p>","path":["Chapter 13. Backtracking","13.2   Permutations Problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/","level":1,"title":"13.3   Subset-Sum Problem","text":"","path":["Chapter 13. Backtracking","13.3   Subset-Sum Problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1331-without-duplicate-elements","level":2,"title":"13.3.1   Without Duplicate Elements","text":"<p>Question</p> <p>Given a positive integer array <code>nums</code> and a target positive integer <code>target</code>, find all possible combinations where the sum of elements in the combination equals <code>target</code>. The given array has no duplicate elements, and each element can be selected multiple times. Return these combinations in list form, where the list should not contain duplicate combinations.</p> <p>For example, given the set \\(\\{3, 4, 5\\}\\) and target integer \\(9\\), the solutions are \\(\\{3, 3, 3\\}, \\{4, 5\\}\\). Note the following two points:</p> <ul> <li>Elements in the input set can be selected repeatedly without limit.</li> <li>Subsets do not distinguish element order; for example, \\(\\{4, 5\\}\\) and \\(\\{5, 4\\}\\) are the same subset.</li> </ul>","path":["Chapter 13. Backtracking","13.3   Subset-Sum Problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1-using-the-permutation-solution-as-a-reference","level":3,"title":"1.   Using the Permutation Solution as a Reference","text":"<p>Similar to the permutation problem, we can view the process of generating subsets as the result of a series of choices and update the running sum during the selection process. When the sum equals <code>target</code>, we record the subset in the result list.</p> <p>Unlike the permutation problem, elements in this problem can be selected any number of times, so we do not need to use a <code>selected</code> boolean list to track whether an element has already been selected. With a few small changes to the permutation code, we obtain an initial solution:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby subset_sum_i_naive.py<pre><code>def backtrack(\n    state: list[int],\n    target: int,\n    total: int,\n    choices: list[int],\n    res: list[list[int]],\n):\n    \"\"\"Backtracking algorithm: Subset sum I\"\"\"\n    # When the subset sum equals target, record the solution\n    if total == target:\n        res.append(list(state))\n        return\n    # Traverse all choices\n    for i in range(len(choices)):\n        # Pruning: if the subset sum exceeds target, skip this choice\n        if total + choices[i] &gt; target:\n            continue\n        # Attempt: make choice, update element sum total\n        state.append(choices[i])\n        # Proceed to the next round of selection\n        backtrack(state, target, total + choices[i], choices, res)\n        # Backtrack: undo choice, restore to previous state\n        state.pop()\n\ndef subset_sum_i_naive(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"Solve subset sum I (including duplicate subsets)\"\"\"\n    state = []  # State (subset)\n    total = 0  # Subset sum\n    res = []  # Result list (subset list)\n    backtrack(state, target, total, nums, res)\n    return res\n</code></pre> subset_sum_i_naive.cpp<pre><code>/* Backtracking algorithm: Subset sum I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, int total, vector&lt;int&gt; &amp;choices, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // When the subset sum equals target, record the solution\n    if (total == target) {\n        res.push_back(state);\n        return;\n    }\n    // Traverse all choices\n    for (size_t i = 0; i &lt; choices.size(); i++) {\n        // Pruning: if the subset sum exceeds target, skip this choice\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // Attempt: make choice, update element sum total\n        state.push_back(choices[i]);\n        // Proceed to the next round of selection\n        backtrack(state, target, total + choices[i], choices, res);\n        // Backtrack: undo choice, restore to previous state\n        state.pop_back();\n    }\n}\n\n/* Solve subset sum I (including duplicate subsets) */\nvector&lt;vector&lt;int&gt;&gt; subsetSumINaive(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;       // State (subset)\n    int total = 0;           // Subset sum\n    vector&lt;vector&lt;int&gt;&gt; res; // Result list (subset list)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.java<pre><code>/* Backtracking algorithm: Subset sum I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int total, int[] choices, List&lt;List&lt;Integer&gt;&gt; res) {\n    // When the subset sum equals target, record the solution\n    if (total == target) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // Traverse all choices\n    for (int i = 0; i &lt; choices.length; i++) {\n        // Pruning: if the subset sum exceeds target, skip this choice\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // Attempt: make choice, update element sum total\n        state.add(choices[i]);\n        // Proceed to the next round of selection\n        backtrack(state, target, total + choices[i], choices, res);\n        // Backtrack: undo choice, restore to previous state\n        state.remove(state.size() - 1);\n    }\n}\n\n/* Solve subset sum I (including duplicate subsets) */\nList&lt;List&lt;Integer&gt;&gt; subsetSumINaive(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // State (subset)\n    int total = 0; // Subset sum\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // Result list (subset list)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.cs<pre><code>/* Backtracking algorithm: Subset sum I */\nvoid Backtrack(List&lt;int&gt; state, int target, int total, int[] choices, List&lt;List&lt;int&gt;&gt; res) {\n    // When the subset sum equals target, record the solution\n    if (total == target) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // Traverse all choices\n    for (int i = 0; i &lt; choices.Length; i++) {\n        // Pruning: if the subset sum exceeds target, skip this choice\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // Attempt: make choice, update element sum total\n        state.Add(choices[i]);\n        // Proceed to the next round of selection\n        Backtrack(state, target, total + choices[i], choices, res);\n        // Backtrack: undo choice, restore to previous state\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* Solve subset sum I (including duplicate subsets) */\nList&lt;List&lt;int&gt;&gt; SubsetSumINaive(int[] nums, int target) {\n    List&lt;int&gt; state = []; // State (subset)\n    int total = 0; // Subset sum\n    List&lt;List&lt;int&gt;&gt; res = []; // Result list (subset list)\n    Backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.go<pre><code>/* Backtracking algorithm: Subset sum I */\nfunc backtrackSubsetSumINaive(total, target int, state, choices *[]int, res *[][]int) {\n    // When the subset sum equals target, record the solution\n    if target == total {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // Traverse all choices\n    for i := 0; i &lt; len(*choices); i++ {\n        // Pruning: if the subset sum exceeds target, skip this choice\n        if total+(*choices)[i] &gt; target {\n            continue\n        }\n        // Attempt: make choice, update element sum total\n        *state = append(*state, (*choices)[i])\n        // Proceed to the next round of selection\n        backtrackSubsetSumINaive(total+(*choices)[i], target, state, choices, res)\n        // Backtrack: undo choice, restore to previous state\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* Solve subset sum I (including duplicate subsets) */\nfunc subsetSumINaive(nums []int, target int) [][]int {\n    state := make([]int, 0) // State (subset)\n    total := 0              // Subset sum\n    res := make([][]int, 0) // Result list (subset list)\n    backtrackSubsetSumINaive(total, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.swift<pre><code>/* Backtracking algorithm: Subset sum I */\nfunc backtrack(state: inout [Int], target: Int, total: Int, choices: [Int], res: inout [[Int]]) {\n    // When the subset sum equals target, record the solution\n    if total == target {\n        res.append(state)\n        return\n    }\n    // Traverse all choices\n    for i in choices.indices {\n        // Pruning: if the subset sum exceeds target, skip this choice\n        if total + choices[i] &gt; target {\n            continue\n        }\n        // Attempt: make choice, update element sum total\n        state.append(choices[i])\n        // Proceed to the next round of selection\n        backtrack(state: &amp;state, target: target, total: total + choices[i], choices: choices, res: &amp;res)\n        // Backtrack: undo choice, restore to previous state\n        state.removeLast()\n    }\n}\n\n/* Solve subset sum I (including duplicate subsets) */\nfunc subsetSumINaive(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // State (subset)\n    let total = 0 // Subset sum\n    var res: [[Int]] = [] // Result list (subset list)\n    backtrack(state: &amp;state, target: target, total: total, choices: nums, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.js<pre><code>/* Backtracking algorithm: Subset sum I */\nfunction backtrack(state, target, total, choices, res) {\n    // When the subset sum equals target, record the solution\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // Traverse all choices\n    for (let i = 0; i &lt; choices.length; i++) {\n        // Pruning: if the subset sum exceeds target, skip this choice\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // Attempt: make choice, update element sum total\n        state.push(choices[i]);\n        // Proceed to the next round of selection\n        backtrack(state, target, total + choices[i], choices, res);\n        // Backtrack: undo choice, restore to previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum I (including duplicate subsets) */\nfunction subsetSumINaive(nums, target) {\n    const state = []; // State (subset)\n    const total = 0; // Subset sum\n    const res = []; // Result list (subset list)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.ts<pre><code>/* Backtracking algorithm: Subset sum I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    total: number,\n    choices: number[],\n    res: number[][]\n): void {\n    // When the subset sum equals target, record the solution\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // Traverse all choices\n    for (let i = 0; i &lt; choices.length; i++) {\n        // Pruning: if the subset sum exceeds target, skip this choice\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // Attempt: make choice, update element sum total\n        state.push(choices[i]);\n        // Proceed to the next round of selection\n        backtrack(state, target, total + choices[i], choices, res);\n        // Backtrack: undo choice, restore to previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum I (including duplicate subsets) */\nfunction subsetSumINaive(nums: number[], target: number): number[][] {\n    const state = []; // State (subset)\n    const total = 0; // Subset sum\n    const res = []; // Result list (subset list)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.dart<pre><code>/* Backtracking algorithm: Subset sum I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  int total,\n  List&lt;int&gt; choices,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // When the subset sum equals target, record the solution\n  if (total == target) {\n    res.add(List.from(state));\n    return;\n  }\n  // Traverse all choices\n  for (int i = 0; i &lt; choices.length; i++) {\n    // Pruning: if the subset sum exceeds target, skip this choice\n    if (total + choices[i] &gt; target) {\n      continue;\n    }\n    // Attempt: make choice, update element sum total\n    state.add(choices[i]);\n    // Proceed to the next round of selection\n    backtrack(state, target, total + choices[i], choices, res);\n    // Backtrack: undo choice, restore to previous state\n    state.removeLast();\n  }\n}\n\n/* Solve subset sum I (including duplicate subsets) */\nList&lt;List&lt;int&gt;&gt; subsetSumINaive(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // State (subset)\n  int total = 0; // Sum of elements\n  List&lt;List&lt;int&gt;&gt; res = []; // Result list (subset list)\n  backtrack(state, target, total, nums, res);\n  return res;\n}\n</code></pre> subset_sum_i_naive.rs<pre><code>/* Backtracking algorithm: Subset sum I */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    total: i32,\n    choices: &amp;[i32],\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // When the subset sum equals target, record the solution\n    if total == target {\n        res.push(state.clone());\n        return;\n    }\n    // Traverse all choices\n    for i in 0..choices.len() {\n        // Pruning: if the subset sum exceeds target, skip this choice\n        if total + choices[i] &gt; target {\n            continue;\n        }\n        // Attempt: make choice, update element sum total\n        state.push(choices[i]);\n        // Proceed to the next round of selection\n        backtrack(state, target, total + choices[i], choices, res);\n        // Backtrack: undo choice, restore to previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum I (including duplicate subsets) */\nfn subset_sum_i_naive(nums: &amp;[i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // State (subset)\n    let total = 0; // Subset sum\n    let mut res = Vec::new(); // Result list (subset list)\n    backtrack(&amp;mut state, target, total, nums, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i_naive.c<pre><code>/* Backtracking algorithm: Subset sum I */\nvoid backtrack(int target, int total, int *choices, int choicesSize) {\n    // When the subset sum equals target, record the solution\n    if (total == target) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // Traverse all choices\n    for (int i = 0; i &lt; choicesSize; i++) {\n        // Pruning: if the subset sum exceeds target, skip this choice\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // Attempt: make choice, update element sum total\n        state[stateSize++] = choices[i];\n        // Proceed to the next round of selection\n        backtrack(target, total + choices[i], choices, choicesSize);\n        // Backtrack: undo choice, restore to previous state\n        stateSize--;\n    }\n}\n\n/* Solve subset sum I (including duplicate subsets) */\nvoid subsetSumINaive(int *nums, int numsSize, int target) {\n    resSize = 0; // Initialize solution count to 0\n    backtrack(target, 0, nums, numsSize);\n}\n</code></pre> subset_sum_i_naive.kt<pre><code>/* Backtracking algorithm: Subset sum I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    total: Int,\n    choices: IntArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // When the subset sum equals target, record the solution\n    if (total == target) {\n        res.add(state.toMutableList())\n        return\n    }\n    // Traverse all choices\n    for (i in choices.indices) {\n        // Pruning: if the subset sum exceeds target, skip this choice\n        if (total + choices[i] &gt; target) {\n            continue\n        }\n        // Attempt: make choice, update element sum total\n        state.add(choices[i])\n        // Proceed to the next round of selection\n        backtrack(state, target, total + choices[i], choices, res)\n        // Backtrack: undo choice, restore to previous state\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* Solve subset sum I (including duplicate subsets) */\nfun subsetSumINaive(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // State (subset)\n    val total = 0 // Subset sum\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // Result list (subset list)\n    backtrack(state, target, total, nums, res)\n    return res\n}\n</code></pre> subset_sum_i_naive.rb<pre><code>### Backtracking: subset sum I ###\ndef backtrack(state, target, total, choices, res)\n  # When the subset sum equals target, record the solution\n  if total == target\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # Traverse all choices\n  for i in 0...choices.length\n    # Pruning: if the subset sum exceeds target, skip this choice\n    next if total + choices[i] &gt; target\n    # Attempt: make choice, update element sum total\n    state &lt;&lt; choices[i]\n    # Proceed to the next round of selection\n    backtrack(state, target, total + choices[i], choices, res)\n    # Backtrack: undo choice, restore to previous state\n    state.pop\n  end\nend\n\n### Solve subset sum I (with duplicate subsets) ###\ndef subset_sum_i_naive(nums, target)\n  state = [] # State (subset)\n  total = 0 # Subset sum\n  res = [] # Result list (subset list)\n  backtrack(state, target, total, nums, res)\n  res\nend\n</code></pre> <p>Running the above code on array \\([3, 4, 5]\\) with target value \\(9\\) produces \\([3, 3, 3], [4, 5], [5, 4]\\). Although we successfully found all subsets that sum to \\(9\\), there are duplicate subsets \\([4, 5]\\) and \\([5, 4]\\).</p> <p>This is because the search process distinguishes the order of selections, but subsets do not distinguish selection order. As shown in Figure 13-10, selecting 4 first and then 5 versus selecting 5 first and then 4 are different branches, but they correspond to the same subset.</p> <p></p> <p> Figure 13-10   Subset search and boundary pruning </p> <p>To eliminate duplicate subsets, one straightforward idea is to deduplicate the result list. However, this approach is very inefficient for two reasons:</p> <ul> <li>When there are many array elements, especially when <code>target</code> is large, the search process generates many duplicate subsets.</li> <li>Comparing subsets (arrays) is very time-consuming, requiring sorting the arrays first, then comparing each element in them.</li> </ul>","path":["Chapter 13. Backtracking","13.3   Subset-Sum Problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#2-pruning-duplicate-subsets","level":3,"title":"2.   Pruning Duplicate Subsets","text":"<p>We consider deduplication through pruning during the search process. Observing Figure 13-11, duplicate subsets occur when array elements are selected in different orders, as in the following cases:</p> <ol> <li>When the first and second rounds select \\(3\\) and \\(4\\) respectively, all subsets containing these two elements are generated, denoted as \\([3, 4, \\dots]\\).</li> <li>Afterward, when the first round selects \\(4\\), the second round should skip \\(3\\), because the subset \\([4, 3, \\dots]\\) generated by this choice is an exact duplicate of the subset generated in step <code>1.</code></li> </ol> <p>In the search process, each level's choices are tried from left to right, so the rightmost branches are pruned more.</p> <ol> <li>The first two rounds select \\(3\\) and \\(5\\), generating subset \\([3, 5, \\dots]\\).</li> <li>The first two rounds select \\(4\\) and \\(5\\), generating subset \\([4, 5, \\dots]\\).</li> <li>If the first round selects \\(5\\), the second round should skip \\(3\\) and \\(4\\), because subsets \\([5, 3, \\dots]\\) and \\([5, 4, \\dots]\\) are exact duplicates of the subsets described in steps <code>1.</code> and <code>2.</code></li> </ol> <p></p> <p> Figure 13-11   Different selection orders leading to duplicate subsets </p> <p>In summary, given an input array \\([x_1, x_2, \\dots, x_n]\\), let the selection sequence in the search process be \\([x_{i_1}, x_{i_2}, \\dots, x_{i_m}]\\). This selection sequence must satisfy \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\); any selection sequence that does not satisfy this condition will cause duplicates and should be pruned.</p>","path":["Chapter 13. Backtracking","13.3   Subset-Sum Problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#3-code-implementation","level":3,"title":"3.   Code Implementation","text":"<p>To implement this pruning, we initialize a variable <code>start</code> to indicate the starting point of traversal. After making choice \\(x_{i}\\), set the next round to start traversal from index \\(i\\). This ensures that the selection sequence satisfies \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\), guaranteeing subset uniqueness.</p> <p>In addition, we have made the following two optimizations to the code:</p> <ul> <li>Before starting the search, first sort the array <code>nums</code>. When traversing all choices, end the loop immediately when the subset sum exceeds <code>target</code>, because subsequent elements are larger, and their subset sums must exceed <code>target</code>.</li> <li>Omit the element sum variable <code>total</code> and use subtraction on <code>target</code> to track the sum of elements. Record the solution when <code>target</code> equals \\(0\\).</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby subset_sum_i.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"Backtracking algorithm: Subset sum I\"\"\"\n    # When the subset sum equals target, record the solution\n    if target == 0:\n        res.append(list(state))\n        return\n    # Traverse all choices\n    # Pruning 2: start traversing from start to avoid generating duplicate subsets\n    for i in range(start, len(choices)):\n        # Pruning 1: if the subset sum exceeds target, end the loop directly\n        # This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if target - choices[i] &lt; 0:\n            break\n        # Attempt: make choice, update target, start\n        state.append(choices[i])\n        # Proceed to the next round of selection\n        backtrack(state, target - choices[i], choices, i, res)\n        # Backtrack: undo choice, restore to previous state\n        state.pop()\n\ndef subset_sum_i(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"Solve subset sum I\"\"\"\n    state = []  # State (subset)\n    nums.sort()  # Sort nums\n    start = 0  # Start point for traversal\n    res = []  # Result list (subset list)\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_i.cpp<pre><code>/* Backtracking algorithm: Subset sum I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // When the subset sum equals target, record the solution\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    for (int i = start; i &lt; choices.size(); i++) {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Attempt: make choice, update target, start\n        state.push_back(choices[i]);\n        // Proceed to the next round of selection\n        backtrack(state, target - choices[i], choices, i, res);\n        // Backtrack: undo choice, restore to previous state\n        state.pop_back();\n    }\n}\n\n/* Solve subset sum I */\nvector&lt;vector&lt;int&gt;&gt; subsetSumI(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // State (subset)\n    sort(nums.begin(), nums.end()); // Sort nums\n    int start = 0;                  // Start point for traversal\n    vector&lt;vector&lt;int&gt;&gt; res;        // Result list (subset list)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.java<pre><code>/* Backtracking algorithm: Subset sum I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // When the subset sum equals target, record the solution\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    for (int i = start; i &lt; choices.length; i++) {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Attempt: make choice, update target, start\n        state.add(choices[i]);\n        // Proceed to the next round of selection\n        backtrack(state, target - choices[i], choices, i, res);\n        // Backtrack: undo choice, restore to previous state\n        state.remove(state.size() - 1);\n    }\n}\n\n/* Solve subset sum I */\nList&lt;List&lt;Integer&gt;&gt; subsetSumI(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // State (subset)\n    Arrays.sort(nums); // Sort nums\n    int start = 0; // Start point for traversal\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // Result list (subset list)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.cs<pre><code>/* Backtracking algorithm: Subset sum I */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // When the subset sum equals target, record the solution\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    for (int i = start; i &lt; choices.Length; i++) {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Attempt: make choice, update target, start\n        state.Add(choices[i]);\n        // Proceed to the next round of selection\n        Backtrack(state, target - choices[i], choices, i, res);\n        // Backtrack: undo choice, restore to previous state\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* Solve subset sum I */\nList&lt;List&lt;int&gt;&gt; SubsetSumI(int[] nums, int target) {\n    List&lt;int&gt; state = []; // State (subset)\n    Array.Sort(nums); // Sort nums\n    int start = 0; // Start point for traversal\n    List&lt;List&lt;int&gt;&gt; res = []; // Result list (subset list)\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.go<pre><code>/* Backtracking algorithm: Subset sum I */\nfunc backtrackSubsetSumI(start, target int, state, choices *[]int, res *[][]int) {\n    // When the subset sum equals target, record the solution\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    for i := start; i &lt; len(*choices); i++ {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // Attempt: make choice, update target, start\n        *state = append(*state, (*choices)[i])\n        // Proceed to the next round of selection\n        backtrackSubsetSumI(i, target-(*choices)[i], state, choices, res)\n        // Backtrack: undo choice, restore to previous state\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* Solve subset sum I */\nfunc subsetSumI(nums []int, target int) [][]int {\n    state := make([]int, 0) // State (subset)\n    sort.Ints(nums)         // Sort nums\n    start := 0              // Start point for traversal\n    res := make([][]int, 0) // Result list (subset list)\n    backtrackSubsetSumI(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.swift<pre><code>/* Backtracking algorithm: Subset sum I */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // When the subset sum equals target, record the solution\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    for i in choices.indices.dropFirst(start) {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // Attempt: make choice, update target, start\n        state.append(choices[i])\n        // Proceed to the next round of selection\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i, res: &amp;res)\n        // Backtrack: undo choice, restore to previous state\n        state.removeLast()\n    }\n}\n\n/* Solve subset sum I */\nfunc subsetSumI(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // State (subset)\n    let nums = nums.sorted() // Sort nums\n    let start = 0 // Start point for traversal\n    var res: [[Int]] = [] // Result list (subset list)\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.js<pre><code>/* Backtracking algorithm: Subset sum I */\nfunction backtrack(state, target, choices, start, res) {\n    // When the subset sum equals target, record the solution\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    for (let i = start; i &lt; choices.length; i++) {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Attempt: make choice, update target, start\n        state.push(choices[i]);\n        // Proceed to the next round of selection\n        backtrack(state, target - choices[i], choices, i, res);\n        // Backtrack: undo choice, restore to previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum I */\nfunction subsetSumI(nums, target) {\n    const state = []; // State (subset)\n    nums.sort((a, b) =&gt; a - b); // Sort nums\n    const start = 0; // Start point for traversal\n    const res = []; // Result list (subset list)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.ts<pre><code>/* Backtracking algorithm: Subset sum I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // When the subset sum equals target, record the solution\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    for (let i = start; i &lt; choices.length; i++) {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Attempt: make choice, update target, start\n        state.push(choices[i]);\n        // Proceed to the next round of selection\n        backtrack(state, target - choices[i], choices, i, res);\n        // Backtrack: undo choice, restore to previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum I */\nfunction subsetSumI(nums: number[], target: number): number[][] {\n    const state = []; // State (subset)\n    nums.sort((a, b) =&gt; a - b); // Sort nums\n    const start = 0; // Start point for traversal\n    const res = []; // Result list (subset list)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.dart<pre><code>/* Backtracking algorithm: Subset sum I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // When the subset sum equals target, record the solution\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // Traverse all choices\n  // Pruning 2: start traversing from start to avoid generating duplicate subsets\n  for (int i = start; i &lt; choices.length; i++) {\n    // Pruning 1: if the subset sum exceeds target, end the loop directly\n    // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // Attempt: make choice, update target, start\n    state.add(choices[i]);\n    // Proceed to the next round of selection\n    backtrack(state, target - choices[i], choices, i, res);\n    // Backtrack: undo choice, restore to previous state\n    state.removeLast();\n  }\n}\n\n/* Solve subset sum I */\nList&lt;List&lt;int&gt;&gt; subsetSumI(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // State (subset)\n  nums.sort(); // Sort nums\n  int start = 0; // Start point for traversal\n  List&lt;List&lt;int&gt;&gt; res = []; // Result list (subset list)\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_i.rs<pre><code>/* Backtracking algorithm: Subset sum I */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // When the subset sum equals target, record the solution\n    if target == 0 {\n        res.push(state.clone());\n        return;\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    for i in start..choices.len() {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // Attempt: make choice, update target, start\n        state.push(choices[i]);\n        // Proceed to the next round of selection\n        backtrack(state, target - choices[i], choices, i, res);\n        // Backtrack: undo choice, restore to previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum I */\nfn subset_sum_i(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // State (subset)\n    nums.sort(); // Sort nums\n    let start = 0; // Start point for traversal\n    let mut res = Vec::new(); // Result list (subset list)\n    backtrack(&amp;mut state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i.c<pre><code>/* Backtracking algorithm: Subset sum I */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // When the subset sum equals target, record the solution\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; ++i) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    for (int i = start; i &lt; choicesSize; i++) {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Attempt: make choice, update target, start\n        state[stateSize] = choices[i];\n        stateSize++;\n        // Proceed to the next round of selection\n        backtrack(target - choices[i], choices, choicesSize, i);\n        // Backtrack: undo choice, restore to previous state\n        stateSize--;\n    }\n}\n\n/* Solve subset sum I */\nvoid subsetSumI(int *nums, int numsSize, int target) {\n    qsort(nums, numsSize, sizeof(int), cmp); // Sort nums\n    int start = 0;                           // Start point for traversal\n    backtrack(target, nums, numsSize, start);\n}\n</code></pre> subset_sum_i.kt<pre><code>/* Backtracking algorithm: Subset sum I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // When the subset sum equals target, record the solution\n    if (target == 0) {\n        res.add(state.toMutableList())\n        return\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    for (i in start..&lt;choices.size) {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // Attempt: make choice, update target, start\n        state.add(choices[i])\n        // Proceed to the next round of selection\n        backtrack(state, target - choices[i], choices, i, res)\n        // Backtrack: undo choice, restore to previous state\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* Solve subset sum I */\nfun subsetSumI(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // State (subset)\n    nums.sort() // Sort nums\n    val start = 0 // Start point for traversal\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // Result list (subset list)\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_i.rb<pre><code>### Backtracking: subset sum I ###\ndef backtrack(state, target, choices, start, res)\n  # When the subset sum equals target, record the solution\n  if target.zero?\n    res &lt;&lt; state.dup\n    return\n  end\n  # Traverse all choices\n  # Pruning 2: start traversing from start to avoid generating duplicate subsets\n  for i in start...choices.length\n    # Pruning 1: if the subset sum exceeds target, end the loop directly\n    # This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n    break if target - choices[i] &lt; 0\n    # Attempt: make choice, update target, start\n    state &lt;&lt; choices[i]\n    # Proceed to the next round of selection\n    backtrack(state, target - choices[i], choices, i, res)\n    # Backtrack: undo choice, restore to previous state\n    state.pop\n  end\nend\n\n### Solve subset sum I ###\ndef subset_sum_i(nums, target)\n  state = [] # State (subset)\n  nums.sort! # Sort nums\n  start = 0 # Start point for traversal\n  res = [] # Result list (subset list)\n  backtrack(state, target, nums, start, res)\n  res\nend\n</code></pre> <p>Figure 13-12 shows the complete backtracking process produced by running the above code on array \\([3, 4, 5]\\) with target value \\(9\\).</p> <p></p> <p> Figure 13-12   Subset-sum I backtracking process </p>","path":["Chapter 13. Backtracking","13.3   Subset-Sum Problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1332-with-duplicate-elements-in-array","level":2,"title":"13.3.2   With Duplicate Elements in Array","text":"<p>Question</p> <p>Given a positive integer array <code>nums</code> and a target positive integer <code>target</code>, find all possible combinations where the sum of elements in the combination equals <code>target</code>. The given array may contain duplicate elements, and each element can be selected at most once. Return these combinations in list form, where the list should not contain duplicate combinations.</p> <p>Compared to the previous problem, the input array in this problem may contain duplicate elements, which introduces a new issue. For example, given array \\([4, \\hat{4}, 5]\\) and target value \\(9\\), the output of the existing code is \\([4, 5], [\\hat{4}, 5]\\), which contains duplicate subsets.</p> <p>The reason for this duplication is that equal elements are selected multiple times in a certain round. In Figure 13-13, the first round has three choices, two of which are \\(4\\), creating two duplicate search branches that output duplicate subsets. Similarly, the two \\(4\\)'s in the second round also produce duplicate subsets.</p> <p></p> <p> Figure 13-13   Duplicate subsets caused by equal elements </p>","path":["Chapter 13. Backtracking","13.3   Subset-Sum Problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1-pruning-equal-elements","level":3,"title":"1.   Pruning Equal Elements","text":"<p>To solve this problem, we need to limit equal elements to be selected only once in each round. The implementation is quite clever: since the array is already sorted, equal elements are adjacent. This means that in a given round of selection, if the current element equals the element to its left, then the same value has already been chosen in this round, so we skip the current element directly.</p> <p>At the same time, this problem specifies that each array element can only be selected once. Fortunately, we can also use the variable <code>start</code> to satisfy this constraint: after making choice \\(x_{i}\\), set the next round to start traversal from index \\(i + 1\\) onwards. This both eliminates duplicate subsets and avoids selecting elements multiple times.</p>","path":["Chapter 13. Backtracking","13.3   Subset-Sum Problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#2-code-implementation","level":3,"title":"2.   Code Implementation","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby subset_sum_ii.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"Backtracking algorithm: Subset sum II\"\"\"\n    # When the subset sum equals target, record the solution\n    if target == 0:\n        res.append(list(state))\n        return\n    # Traverse all choices\n    # Pruning 2: start traversing from start to avoid generating duplicate subsets\n    # Pruning 3: start traversing from start to avoid repeatedly selecting the same element\n    for i in range(start, len(choices)):\n        # Pruning 1: if the subset sum exceeds target, end the loop directly\n        # This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if target - choices[i] &lt; 0:\n            break\n        # Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly\n        if i &gt; start and choices[i] == choices[i - 1]:\n            continue\n        # Attempt: make choice, update target, start\n        state.append(choices[i])\n        # Proceed to the next round of selection\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        # Backtrack: undo choice, restore to previous state\n        state.pop()\n\ndef subset_sum_ii(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"Solve subset sum II\"\"\"\n    state = []  # State (subset)\n    nums.sort()  # Sort nums\n    start = 0  # Start point for traversal\n    res = []  # Result list (subset list)\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_ii.cpp<pre><code>/* Backtracking algorithm: Subset sum II */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // When the subset sum equals target, record the solution\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    // Pruning 3: start traversing from start to avoid repeatedly selecting the same element\n    for (int i = start; i &lt; choices.size(); i++) {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // Attempt: make choice, update target, start\n        state.push_back(choices[i]);\n        // Proceed to the next round of selection\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // Backtrack: undo choice, restore to previous state\n        state.pop_back();\n    }\n}\n\n/* Solve subset sum II */\nvector&lt;vector&lt;int&gt;&gt; subsetSumII(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // State (subset)\n    sort(nums.begin(), nums.end()); // Sort nums\n    int start = 0;                  // Start point for traversal\n    vector&lt;vector&lt;int&gt;&gt; res;        // Result list (subset list)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.java<pre><code>/* Backtracking algorithm: Subset sum II */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // When the subset sum equals target, record the solution\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    // Pruning 3: start traversing from start to avoid repeatedly selecting the same element\n    for (int i = start; i &lt; choices.length; i++) {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // Attempt: make choice, update target, start\n        state.add(choices[i]);\n        // Proceed to the next round of selection\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // Backtrack: undo choice, restore to previous state\n        state.remove(state.size() - 1);\n    }\n}\n\n/* Solve subset sum II */\nList&lt;List&lt;Integer&gt;&gt; subsetSumII(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // State (subset)\n    Arrays.sort(nums); // Sort nums\n    int start = 0; // Start point for traversal\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // Result list (subset list)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.cs<pre><code>/* Backtracking algorithm: Subset sum II */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // When the subset sum equals target, record the solution\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    // Pruning 3: start traversing from start to avoid repeatedly selecting the same element\n    for (int i = start; i &lt; choices.Length; i++) {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // Attempt: make choice, update target, start\n        state.Add(choices[i]);\n        // Proceed to the next round of selection\n        Backtrack(state, target - choices[i], choices, i + 1, res);\n        // Backtrack: undo choice, restore to previous state\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* Solve subset sum II */\nList&lt;List&lt;int&gt;&gt; SubsetSumII(int[] nums, int target) {\n    List&lt;int&gt; state = []; // State (subset)\n    Array.Sort(nums); // Sort nums\n    int start = 0; // Start point for traversal\n    List&lt;List&lt;int&gt;&gt; res = []; // Result list (subset list)\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.go<pre><code>/* Backtracking algorithm: Subset sum II */\nfunc backtrackSubsetSumII(start, target int, state, choices *[]int, res *[][]int) {\n    // When the subset sum equals target, record the solution\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    // Pruning 3: start traversing from start to avoid repeatedly selecting the same element\n    for i := start; i &lt; len(*choices); i++ {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly\n        if i &gt; start &amp;&amp; (*choices)[i] == (*choices)[i-1] {\n            continue\n        }\n        // Attempt: make choice, update target, start\n        *state = append(*state, (*choices)[i])\n        // Proceed to the next round of selection\n        backtrackSubsetSumII(i+1, target-(*choices)[i], state, choices, res)\n        // Backtrack: undo choice, restore to previous state\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* Solve subset sum II */\nfunc subsetSumII(nums []int, target int) [][]int {\n    state := make([]int, 0) // State (subset)\n    sort.Ints(nums)         // Sort nums\n    start := 0              // Start point for traversal\n    res := make([][]int, 0) // Result list (subset list)\n    backtrackSubsetSumII(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.swift<pre><code>/* Backtracking algorithm: Subset sum II */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // When the subset sum equals target, record the solution\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    // Pruning 3: start traversing from start to avoid repeatedly selecting the same element\n    for i in choices.indices.dropFirst(start) {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly\n        if i &gt; start, choices[i] == choices[i - 1] {\n            continue\n        }\n        // Attempt: make choice, update target, start\n        state.append(choices[i])\n        // Proceed to the next round of selection\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i + 1, res: &amp;res)\n        // Backtrack: undo choice, restore to previous state\n        state.removeLast()\n    }\n}\n\n/* Solve subset sum II */\nfunc subsetSumII(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // State (subset)\n    let nums = nums.sorted() // Sort nums\n    let start = 0 // Start point for traversal\n    var res: [[Int]] = [] // Result list (subset list)\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.js<pre><code>/* Backtracking algorithm: Subset sum II */\nfunction backtrack(state, target, choices, start, res) {\n    // When the subset sum equals target, record the solution\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    // Pruning 3: start traversing from start to avoid repeatedly selecting the same element\n    for (let i = start; i &lt; choices.length; i++) {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // Attempt: make choice, update target, start\n        state.push(choices[i]);\n        // Proceed to the next round of selection\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // Backtrack: undo choice, restore to previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum II */\nfunction subsetSumII(nums, target) {\n    const state = []; // State (subset)\n    nums.sort((a, b) =&gt; a - b); // Sort nums\n    const start = 0; // Start point for traversal\n    const res = []; // Result list (subset list)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.ts<pre><code>/* Backtracking algorithm: Subset sum II */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // When the subset sum equals target, record the solution\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    // Pruning 3: start traversing from start to avoid repeatedly selecting the same element\n    for (let i = start; i &lt; choices.length; i++) {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // Attempt: make choice, update target, start\n        state.push(choices[i]);\n        // Proceed to the next round of selection\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // Backtrack: undo choice, restore to previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum II */\nfunction subsetSumII(nums: number[], target: number): number[][] {\n    const state = []; // State (subset)\n    nums.sort((a, b) =&gt; a - b); // Sort nums\n    const start = 0; // Start point for traversal\n    const res = []; // Result list (subset list)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.dart<pre><code>/* Backtracking algorithm: Subset sum II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // When the subset sum equals target, record the solution\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // Traverse all choices\n  // Pruning 2: start traversing from start to avoid generating duplicate subsets\n  // Pruning 3: start traversing from start to avoid repeatedly selecting the same element\n  for (int i = start; i &lt; choices.length; i++) {\n    // Pruning 1: if the subset sum exceeds target, end the loop directly\n    // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly\n    if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n      continue;\n    }\n    // Attempt: make choice, update target, start\n    state.add(choices[i]);\n    // Proceed to the next round of selection\n    backtrack(state, target - choices[i], choices, i + 1, res);\n    // Backtrack: undo choice, restore to previous state\n    state.removeLast();\n  }\n}\n\n/* Solve subset sum II */\nList&lt;List&lt;int&gt;&gt; subsetSumII(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // State (subset)\n  nums.sort(); // Sort nums\n  int start = 0; // Start point for traversal\n  List&lt;List&lt;int&gt;&gt; res = []; // Result list (subset list)\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_ii.rs<pre><code>/* Backtracking algorithm: Subset sum II */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // When the subset sum equals target, record the solution\n    if target == 0 {\n        res.push(state.clone());\n        return;\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    // Pruning 3: start traversing from start to avoid repeatedly selecting the same element\n    for i in start..choices.len() {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly\n        if i &gt; start &amp;&amp; choices[i] == choices[i - 1] {\n            continue;\n        }\n        // Attempt: make choice, update target, start\n        state.push(choices[i]);\n        // Proceed to the next round of selection\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // Backtrack: undo choice, restore to previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum II */\nfn subset_sum_ii(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // State (subset)\n    nums.sort(); // Sort nums\n    let start = 0; // Start point for traversal\n    let mut res = Vec::new(); // Result list (subset list)\n    backtrack(&amp;mut state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_ii.c<pre><code>/* Backtracking algorithm: Subset sum II */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // When the subset sum equals target, record the solution\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    // Pruning 3: start traversing from start to avoid repeatedly selecting the same element\n    for (int i = start; i &lt; choicesSize; i++) {\n        // Pruning 1: Skip if subset sum exceeds target\n        if (target - choices[i] &lt; 0) {\n            continue;\n        }\n        // Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // Attempt: make choice, update target, start\n        state[stateSize] = choices[i];\n        stateSize++;\n        // Proceed to the next round of selection\n        backtrack(target - choices[i], choices, choicesSize, i + 1);\n        // Backtrack: undo choice, restore to previous state\n        stateSize--;\n    }\n}\n\n/* Solve subset sum II */\nvoid subsetSumII(int *nums, int numsSize, int target) {\n    // Sort nums\n    qsort(nums, numsSize, sizeof(int), cmp);\n    // Start backtracking\n    backtrack(target, nums, numsSize, 0);\n}\n</code></pre> subset_sum_ii.kt<pre><code>/* Backtracking algorithm: Subset sum II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // When the subset sum equals target, record the solution\n    if (target == 0) {\n        res.add(state.toMutableList())\n        return\n    }\n    // Traverse all choices\n    // Pruning 2: start traversing from start to avoid generating duplicate subsets\n    // Pruning 3: start traversing from start to avoid repeatedly selecting the same element\n    for (i in start..&lt;choices.size) {\n        // Pruning 1: if the subset sum exceeds target, end the loop directly\n        // This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue\n        }\n        // Attempt: make choice, update target, start\n        state.add(choices[i])\n        // Proceed to the next round of selection\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        // Backtrack: undo choice, restore to previous state\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* Solve subset sum II */\nfun subsetSumII(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // State (subset)\n    nums.sort() // Sort nums\n    val start = 0 // Start point for traversal\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // Result list (subset list)\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_ii.rb<pre><code>### Backtracking: subset sum II ###\ndef backtrack(state, target, choices, start, res)\n  # When the subset sum equals target, record the solution\n  if target.zero?\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # Traverse all choices\n  # Pruning 2: start traversing from start to avoid generating duplicate subsets\n  # Pruning 3: start traversing from start to avoid repeatedly selecting the same element\n  for i in start...choices.length\n    # Pruning 1: if the subset sum exceeds target, end the loop directly\n    # This is because the array is sorted, and later elements are larger, so the subset sum will definitely exceed target\n    break if target - choices[i] &lt; 0\n    # Pruning 4: if this element equals the left element, it means this search branch is duplicate, skip it directly\n    next if i &gt; start &amp;&amp; choices[i] == choices[i - 1]\n    # Attempt: make choice, update target, start\n    state &lt;&lt; choices[i]\n    # Proceed to the next round of selection\n    backtrack(state, target - choices[i], choices, i + 1, res)\n    # Backtrack: undo choice, restore to previous state\n    state.pop\n  end\nend\n\n### Solve subset sum II ###\ndef subset_sum_ii(nums, target)\n  state = [] # State (subset)\n  nums.sort! # Sort nums\n  start = 0 # Start point for traversal\n  res = [] # Result list (subset list)\n  backtrack(state, target, nums, start, res)\n  res\nend\n</code></pre> <p>Figure 13-14 shows the backtracking process for array \\([4, 4, 5]\\) with target value \\(9\\), which includes four types of pruning operations. Combine the illustration with the code comments to understand the entire search process and how each pruning operation works.</p> <p></p> <p> Figure 13-14   Subset-sum II backtracking process </p>","path":["Chapter 13. Backtracking","13.3   Subset-Sum Problem"],"tags":[]},{"location":"chapter_backtracking/summary/","level":1,"title":"13.5   Summary","text":"","path":["Chapter 13. Backtracking","13.5   Summary"],"tags":[]},{"location":"chapter_backtracking/summary/#1-key-review","level":3,"title":"1.   Key Review","text":"<ul> <li>The backtracking algorithm is fundamentally an exhaustive search method. It finds solutions that meet specified conditions by performing a depth-first traversal of the solution space. During the search process, when a solution satisfying the conditions is found, it is recorded. The search ends either after finding all solutions or when the traversal is complete.</li> <li>The backtracking algorithm search process consists of two parts: attempting and backtracking. It tries various choices through depth-first search. When encountering situations that violate constraints, it reverts the previous choice, returns to the previous state, and continues exploring other options. Attempting and backtracking are operations in opposite directions.</li> <li>Backtracking problems typically contain multiple constraints, which can be utilized to implement pruning operations. Pruning can terminate unnecessary search branches early, significantly improving search efficiency.</li> <li>The backtracking algorithm is primarily used to solve search problems and constraint satisfaction problems. While combinatorial optimization problems can be solved with backtracking, there are often more efficient or better-performing solutions available.</li> <li>The permutation problem aims to find all possible permutations of elements in a given set. We use an array to record whether each element has been selected, thereby pruning search branches that attempt to select the same element repeatedly, ensuring each element is selected exactly once.</li> <li>In the permutation problem, if the set contains duplicate elements, the final result will contain duplicate permutations. We need to impose a constraint so that equal elements can only be selected once per round, which is typically achieved using a hash set.</li> <li>The subset-sum problem aims to find all subsets of a given set that sum to a target value. Since the set is unordered but the search process outputs results in all orders, duplicate subsets are generated. We sort the data before backtracking and use a variable to indicate the starting point of each round's traversal, thereby pruning search branches that generate duplicate subsets.</li> <li>For the subset-sum problem, equal elements in the array produce duplicate subsets. We leverage the precondition that the array is sorted by checking whether adjacent elements are equal to implement pruning, ensuring that equal elements can only be selected once per round.</li> <li>The \\(n\\) queens problem aims to find placements of \\(n\\) queens on an \\(n \\times n\\) chessboard such that no two queens can attack each other. The constraints of this problem include row constraints, column constraints, and main and anti-diagonal constraints. To satisfy row constraints, we adopt a row-by-row placement strategy, ensuring exactly one queen is placed in each row.</li> <li>The handling of column constraints and diagonal constraints is similar. For column constraints, we use an array to record whether each column has a queen, thereby indicating whether a selected cell is valid. For diagonal constraints, we use two arrays to separately record whether queens exist on each main or anti-diagonal. The challenge lies in finding the row-column index pattern that characterizes cells on the same main (anti-)diagonal.</li> </ul>","path":["Chapter 13. Backtracking","13.5   Summary"],"tags":[]},{"location":"chapter_backtracking/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: How can we understand the relationship between backtracking and recursion?</p> <p>Overall, backtracking is an algorithmic strategy, while recursion is better viewed as a tool.</p> <ul> <li>Backtracking is typically implemented with recursion. However, backtracking is only one application of recursion, specifically its use in search problems.</li> <li>The structure of recursion reflects a problem-solving paradigm based on decomposing a problem into subproblems, and it is commonly used in divide-and-conquer, backtracking, and dynamic programming (memoized recursion).</li> </ul>","path":["Chapter 13. Backtracking","13.5   Summary"],"tags":[]},{"location":"chapter_computational_complexity/","level":1,"title":"Chapter 2.   Complexity Analysis","text":"<p>Abstract</p> <p>Complexity analysis is like a space-time guide in the vast universe of algorithms.</p> <p>It leads us to explore deeply within the two dimensions of time and space, seeking more elegant solutions.</p>","path":["Chapter 2. Complexity Analysis","Chapter 2.   Complexity Analysis"],"tags":[]},{"location":"chapter_computational_complexity/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>2.1   Algorithm Efficiency Evaluation</li> <li>2.2   Iteration and Recursion</li> <li>2.3   Time Complexity</li> <li>2.4   Space Complexity</li> <li>2.5   Summary</li> <li>2.6   Exercises</li> </ul>","path":["Chapter 2. Complexity Analysis","Chapter 2.   Complexity Analysis"],"tags":[]},{"location":"chapter_computational_complexity/exercises/","level":1,"title":"2.6   Exercises","text":"","path":["Chapter 2. Complexity Analysis","2.6   Exercises"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#261-concept-review","level":2,"title":"2.6.1   Concept Review","text":"","path":["Chapter 2. Complexity Analysis","2.6   Exercises"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#1-time-and-space-complexity-of-iteration-and-recursion","level":3,"title":"1.   Time and Space Complexity of Iteration and Recursion","text":"<p>The two functions below both calculate \\(1 + 2 + \\dots + n\\) (assume \\(n \\ge 1\\)). Set <code>n</code> to 4, answer the questions by following the program's actual execution order, and then compare the efficiency of the two approaches.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby complexity_exercises.py<pre><code>def sum_iter(n: int) -&gt; int:\n    \"\"\"Iterative summation\"\"\"\n    res = 0\n    for i in range(1, n + 1):\n        res += i\n    return res\n\ndef sum_recur(n: int) -&gt; int:\n    \"\"\"Recursive summation\"\"\"\n    if n == 1:\n        return 1\n    return n + sum_recur(n - 1)\n</code></pre> complexity_exercises.cpp<pre><code>/* Iterative summation */\nint sumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; ++i) {\n        res += i;\n    }\n    return res;\n}\n\n/* Recursive summation */\nint sumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.java<pre><code>/* Iterative summation */\nint sumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Recursive summation */\nint sumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.cs<pre><code>/* Iterative summation */\nint SumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Recursive summation */\nint SumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + SumRecur(n - 1);\n}\n</code></pre> complexity_exercises.go<pre><code>/* Iterative summation */\nfunc sumIter(n int) int {\n    res := 0\n    for i := 1; i &lt;= n; i++ {\n        res += i\n    }\n    return res\n}\n\n/* Recursive summation */\nfunc sumRecur(n int) int {\n    if n == 1 {\n        return 1\n    }\n    return n + sumRecur(n-1)\n}\n</code></pre> complexity_exercises.swift<pre><code>/* Iterative summation */\nfunc sumIter(n: Int) -&gt; Int {\n    var res = 0\n    for i in 1 ... n {\n        res += i\n    }\n    return res\n}\n\n/* Recursive summation */\nfunc sumRecur(n: Int) -&gt; Int {\n    if n == 1 {\n        return 1\n    }\n    return n + sumRecur(n: n - 1)\n}\n</code></pre> complexity_exercises.js<pre><code>/* Iterative summation */\nfunction sumIter(n) {\n    let res = 0;\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Recursive summation */\nfunction sumRecur(n) {\n    if (n === 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.ts<pre><code>/* Iterative summation */\nfunction sumIter(n: number): number {\n    let res = 0;\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Recursive summation */\nfunction sumRecur(n: number): number {\n    if (n === 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.dart<pre><code>/* Iterative summation */\nint sumIter(int n) {\n  int res = 0;\n  for (int i = 1; i &lt;= n; i++) {\n    res += i;\n  }\n  return res;\n}\n\n/* Recursive summation */\nint sumRecur(int n) {\n  if (n == 1) {\n    return 1;\n  }\n  return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.rs<pre><code>/* Iterative summation */\nfn sum_iter(n: i32) -&gt; i32 {\n    let mut res = 0;\n    for i in 1..=n {\n        res += i;\n    }\n    res\n}\n\n/* Recursive summation */\nfn sum_recur(n: i32) -&gt; i32 {\n    if n == 1 {\n        return 1;\n    }\n    n + sum_recur(n - 1)\n}\n</code></pre> complexity_exercises.c<pre><code>/* Iterative summation */\nint sumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Recursive summation */\nint sumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.kt<pre><code>/* Iterative summation */\nfun sumIter(n: Int): Int {\n    var res = 0\n    for (i in 1..n) {\n        res += i\n    }\n    return res\n}\n\n/* Recursive summation */\nfun sumRecur(n: Int): Int {\n    if (n == 1) {\n        return 1\n    }\n    return n + sumRecur(n - 1)\n}\n</code></pre> complexity_exercises.rb<pre><code>### Iterative summation ###\ndef sum_iter(n)\n  res = 0\n  for i in 1..n\n    res += i\n  end\n  res\nend\n\n### Recursive summation ###\ndef sum_recur(n)\n  return 1 if n == 1\n\n  n + sum_recur(n - 1)\nend\n</code></pre> <ol> <li>When the iterative function runs with <code>n = 4</code>, what is the value of the accumulator <code>res</code> after each loop iteration?</li> <li>When the recursive function runs with <code>n = 4</code>, which values does the argument <code>n</code> take in order? As the calls return from the deepest level, how is the result obtained?</li> <li>What are the time and space complexities of the two approaches? Explain your reasoning using the execution processes from Questions 1 and 2.</li> </ol> Answer <ol> <li> <p>The loop variable <code>i</code> takes the values <code>1, 2, 3, 4</code>. After each iteration, <code>res</code> becomes     <code>1, 3, 6, 10</code>, respectively, so the iterative function returns 10.</p> </li> <li> <p>The argument <code>n</code> takes the values <code>4 → 3 → 2 → 1</code>.     The deepest call returns 1. The remaining calls then obtain <code>2 + 1 = 3</code>, <code>3 + 3 = 6</code>, and <code>4 + 6 = 10</code>, in that order.     At the deepest point, all four function calls are still unfinished.</p> </li> <li> <p>Both functions perform a number of loop iterations or calls proportional to \\(n\\), so both have a time complexity of \\(O(n)\\).     Their space complexities differ. The iterative version uses only a constant number of variables, so its space complexity is \\(O(1)\\).     In the recursive version, earlier calls must wait for a result before returning, so the call stack holds up to \\(n\\) calls at the same time.     Its space complexity is \\(O(n)\\).</p> <p>When analyzing space complexity, remember to include the space used by recursive calls as well as the variables written in the code.</p> </li> </ol>","path":["Chapter 2. Complexity Analysis","2.6   Exercises"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#2-time-complexity-of-three-code-fragments","level":3,"title":"2.   Time Complexity of Three Code Fragments","text":"<p>Each of the following code fragments takes a positive integer \\(n\\) as input. Order them from lowest to highest time complexity, and give the complexity of each one.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby complexity_exercises.py<pre><code>def linear_loop(n: int) -&gt; int:\n    \"\"\"Linear loop\"\"\"\n    res = 0\n    for i in range(n):\n        res += i\n    return res\n\ndef quadratic_loop(n: int) -&gt; int:\n    \"\"\"Quadratic loop\"\"\"\n    res = 0\n    for i in range(n):\n        for j in range(i, n):\n            res += j\n    return res\n\ndef logarithmic_loop(n: int) -&gt; int:\n    \"\"\"Logarithmic loop\"\"\"\n    while n &gt; 1:\n        n //= 2\n    return n\n</code></pre> complexity_exercises.cpp<pre><code>/* Linear loop */\nint linearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; ++i) {\n        res += i;\n    }\n    return res;\n}\n\n/* Quadratic loop */\nint quadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; ++i) {\n        for (int j = i; j &lt; n; ++j) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* Logarithmic loop */\nint logarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.java<pre><code>/* Linear loop */\nint linearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Quadratic loop */\nint quadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* Logarithmic loop */\nint logarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.cs<pre><code>/* Linear loop */\nint LinearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Quadratic loop */\nint QuadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* Logarithmic loop */\nint LogarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.go<pre><code>/* Linear loop */\nfunc linearLoop(n int) int {\n    res := 0\n    for i := 0; i &lt; n; i++ {\n        res += i\n    }\n    return res\n}\n\n/* Quadratic loop */\nfunc quadraticLoop(n int) int {\n    res := 0\n    for i := 0; i &lt; n; i++ {\n        for j := i; j &lt; n; j++ {\n            res += j\n        }\n    }\n    return res\n}\n\n/* Logarithmic loop */\nfunc logarithmicLoop(n int) int {\n    for n &gt; 1 {\n        n /= 2\n    }\n    return n\n}\n</code></pre> complexity_exercises.swift<pre><code>/* Linear loop */\nfunc linearLoop(n: Int) -&gt; Int {\n    var res = 0\n    for i in 0 ..&lt; n {\n        res += i\n    }\n    return res\n}\n\n/* Quadratic loop */\nfunc quadraticLoop(n: Int) -&gt; Int {\n    var res = 0\n    for i in 0 ..&lt; n {\n        for j in i ..&lt; n {\n            res += j\n        }\n    }\n    return res\n}\n\n/* Logarithmic loop */\nfunc logarithmicLoop(n: Int) -&gt; Int {\n    var n = n\n    while n &gt; 1 {\n        n /= 2\n    }\n    return n\n}\n</code></pre> complexity_exercises.js<pre><code>/* Linear loop */\nfunction linearLoop(n) {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Quadratic loop */\nfunction quadraticLoop(n) {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* Logarithmic loop */\nfunction logarithmicLoop(n) {\n    while (n &gt; 1) {\n        n = Math.floor(n / 2);\n    }\n    return n;\n}\n</code></pre> complexity_exercises.ts<pre><code>/* Linear loop */\nfunction linearLoop(n: number): number {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Quadratic loop */\nfunction quadraticLoop(n: number): number {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* Logarithmic loop */\nfunction logarithmicLoop(n: number): number {\n    while (n &gt; 1) {\n        n = Math.floor(n / 2);\n    }\n    return n;\n}\n</code></pre> complexity_exercises.dart<pre><code>/* Linear loop */\nint linearLoop(int n) {\n  int res = 0;\n  for (int i = 0; i &lt; n; i++) {\n    res += i;\n  }\n  return res;\n}\n\n/* Quadratic loop */\nint quadraticLoop(int n) {\n  int res = 0;\n  for (int i = 0; i &lt; n; i++) {\n    for (int j = i; j &lt; n; j++) {\n      res += j;\n    }\n  }\n  return res;\n}\n\n/* Logarithmic loop */\nint logarithmicLoop(int n) {\n  while (n &gt; 1) {\n    n ~/= 2;\n  }\n  return n;\n}\n</code></pre> complexity_exercises.rs<pre><code>/* Linear loop */\nfn linear_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    for i in 0..n {\n        res += i;\n    }\n    res\n}\n\n/* Quadratic loop */\nfn quadratic_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    for i in 0..n {\n        for j in i..n {\n            res += j;\n        }\n    }\n    res\n}\n\n/* Logarithmic loop */\nfn logarithmic_loop(mut n: i32) -&gt; i32 {\n    while n &gt; 1 {\n        n /= 2;\n    }\n    n\n}\n</code></pre> complexity_exercises.c<pre><code>/* Linear loop */\nint linearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Quadratic loop */\nint quadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* Logarithmic loop */\nint logarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.kt<pre><code>/* Linear loop */\nfun linearLoop(n: Int): Int {\n    var res = 0\n    for (i in 0 until n) {\n        res += i\n    }\n    return res\n}\n\n/* Quadratic loop */\nfun quadraticLoop(n: Int): Int {\n    var res = 0\n    for (i in 0 until n) {\n        for (j in i until n) {\n            res += j\n        }\n    }\n    return res\n}\n\n/* Logarithmic loop */\nfun logarithmicLoop(n: Int): Int {\n    var value = n\n    while (value &gt; 1) {\n        value /= 2\n    }\n    return value\n}\n</code></pre> complexity_exercises.rb<pre><code>### Linear loop ###\ndef linear_loop(n)\n  res = 0\n  for i in 0...n\n    res += i\n  end\n  res\nend\n\n### Quadratic loop ###\ndef quadratic_loop(n)\n  res = 0\n  for i in 0...n\n    for j in i...n\n      res += j\n    end\n  end\n  res\nend\n\n### Logarithmic loop ###\ndef logarithmic_loop(n)\n  n /= 2 while n &gt; 1\n  n\nend\n</code></pre> Answer <p>From lowest to highest, the order is Fragment 3 with \\(O(\\log n)\\), Fragment 1 with \\(O(n)\\), and Fragment 2 with \\(O(n^2)\\). Fragment 3 halves \\(n\\) in each iteration, so it runs about \\(\\log_2 n\\) times. The loop in Fragment 1 runs exactly \\(n\\) times. The inner loop in Fragment 2 runs \\(n,n-1,\\dots,1\\) times, for a total of \\(n(n+1)/2\\), so its time complexity is quadratic.</p>","path":["Chapter 2. Complexity Analysis","2.6   Exercises"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#3-which-reversal-uses-less-space","level":3,"title":"3.   Which Reversal Uses Less Space?","text":"<p>There are two ways to reverse all the elements in the array <code>nums</code>:</p> <ol> <li>Create a new array <code>res</code> of the same length, copy the elements into it in reverse order, and return it.</li> <li> <p>Move two indices <code>i</code> and <code>j</code> inward from the beginning and end, swapping <code>nums[i]</code> and <code>nums[j]</code> at each step.</p> <p>What is the space complexity of each approach? Which one is an \"in-place\" operation?</p> </li> </ol> Answer <ol> <li> <p>This approach needs an auxiliary array with the same length as the input, so its space complexity is \\(O(n)\\).</p> </li> <li> <p>This approach uses only two index variables,     so its space complexity is \\(O(1)\\). It is an in-place operation.</p> <p>Note that an in-place reversal changes the input array, so it should be preferred only when modifying the input is allowed. If the original array must be kept, the copying cost of the first approach is unavoidable.</p> </li> </ol>","path":["Chapter 2. Complexity Analysis","2.6   Exercises"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#262-programming-exercises","level":2,"title":"2.6.2   Programming Exercises","text":"","path":["Chapter 2. Complexity Analysis","2.6   Exercises"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#1-fibonacci-number","level":3,"title":"1.   Fibonacci Number","text":"<p>The Fibonacci sequence is defined by \\(F(0)=0\\), \\(F(1)=1\\), and, for \\(n\\ge2\\), \\(F(n)=F(n-1)+F(n-2)\\).</p> <p>Given a non-negative integer <code>n</code>, use a loop to calculate and return \\(F(n)\\). Do not use recursion.</p> Hints <ol> <li>Handle the cases where n is 0 or 1 separately</li> <li>Only the previous two terms are needed to calculate the next term; there is no need to store the entire sequence</li> <li>When updating the two variables, take care not to overwrite an old value before it is used</li> </ol> <p>LeetCode</p>","path":["Chapter 2. Complexity Analysis","2.6   Exercises"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/","level":1,"title":"2.2   Iteration and Recursion","text":"<p>In algorithms, repeatedly executing a task is very common and closely related to complexity analysis. Therefore, before introducing time complexity and space complexity, let's first understand how to implement repeated task execution in programs, namely the two basic program control structures: iteration and recursion.</p>","path":["Chapter 2. Complexity Analysis","2.2   Iteration and Recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#221-iteration","level":2,"title":"2.2.1   Iteration","text":"<p>Iteration is a control structure for repeatedly executing a task. In iteration, a program repeatedly executes a segment of code under certain conditions until those conditions are no longer satisfied.</p>","path":["Chapter 2. Complexity Analysis","2.2   Iteration and Recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for-loop","level":3,"title":"1.   For Loop","text":"<p>The <code>for</code> loop is one of the most common forms of iteration, suitable for use when the number of iterations is known in advance.</p> <p>The following function implements the summation \\(1 + 2 + \\dots + n\\) using a <code>for</code> loop, with the result stored in the variable <code>res</code>. Note that in Python, <code>range(a, b)</code> corresponds to a \"left-closed, right-open\" interval, with the traversal range being \\(a, a + 1, \\dots, b-1\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def for_loop(n: int) -&gt; int:\n    \"\"\"for loop\"\"\"\n    res = 0\n    # Sum 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        res += i\n    return res\n</code></pre> iteration.cpp<pre><code>/* for loop */\nint forLoop(int n) {\n    int res = 0;\n    // Sum 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* for loop */\nint forLoop(int n) {\n    int res = 0;\n    // Sum 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* for loop */\nint ForLoop(int n) {\n    int res = 0;\n    // Sum 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* for loop */\nfunc forLoop(n int) int {\n    res := 0\n    // Sum 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* for loop */\nfunc forLoop(n: Int) -&gt; Int {\n    var res = 0\n    // Sum 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* for loop */\nfunction forLoop(n) {\n    let res = 0;\n    // Sum 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* for loop */\nfunction forLoop(n: number): number {\n    let res = 0;\n    // Sum 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* for loop */\nint forLoop(int n) {\n  int res = 0;\n  // Sum 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    res += i;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* for loop */\nfn for_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    // Sum 1, 2, ..., n-1, n\n    for i in 1..=n {\n        res += i;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* for loop */\nint forLoop(int n) {\n    int res = 0;\n    // Sum 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* for loop */\nfun forLoop(n: Int): Int {\n    var res = 0\n    // Sum 1, 2, ..., n-1, n\n    for (i in 1..n) {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### for loop ###\ndef for_loop(n)\n  res = 0\n\n  # Sum 1, 2, ..., n-1, n\n  for i in 1..n\n    res += i\n  end\n\n  res\nend\n</code></pre> <p>Figure 2-1 shows the flowchart of this summation function.</p> <p></p> <p> Figure 2-1   Flowchart of the summation function </p> <p>The number of operations in this summation function is proportional to the input data size \\(n\\), or has a \"linear relationship\". In fact, time complexity describes precisely this \"linear relationship\". Related content will be introduced in detail in the next section.</p>","path":["Chapter 2. Complexity Analysis","2.2   Iteration and Recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while-loop","level":3,"title":"2.   While Loop","text":"<p>Similar to the <code>for</code> loop, the <code>while</code> loop is also a method for implementing iteration. In a <code>while</code> loop, the program first checks the condition in each round; if the condition is true, it continues execution, otherwise it ends the loop.</p> <p>Below we use a <code>while</code> loop to implement the summation \\(1 + 2 + \\dots + n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def while_loop(n: int) -&gt; int:\n    \"\"\"while loop\"\"\"\n    res = 0\n    i = 1  # Initialize condition variable\n    # Sum 1, 2, ..., n-1, n\n    while i &lt;= n:\n        res += i\n        i += 1  # Update condition variable\n    return res\n</code></pre> iteration.cpp<pre><code>/* while loop */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // Initialize condition variable\n    // Sum 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // Update condition variable\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while loop */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // Initialize condition variable\n    // Sum 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // Update condition variable\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while loop */\nint WhileLoop(int n) {\n    int res = 0;\n    int i = 1; // Initialize condition variable\n    // Sum 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i += 1; // Update condition variable\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while loop */\nfunc whileLoop(n int) int {\n    res := 0\n    // Initialize condition variable\n    i := 1\n    // Sum 1, 2, ..., n-1, n\n    for i &lt;= n {\n        res += i\n        // Update condition variable\n        i++\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while loop */\nfunc whileLoop(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // Initialize condition variable\n    // Sum 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i\n        i += 1 // Update condition variable\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while loop */\nfunction whileLoop(n) {\n    let res = 0;\n    let i = 1; // Initialize condition variable\n    // Sum 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // Update condition variable\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while loop */\nfunction whileLoop(n: number): number {\n    let res = 0;\n    let i = 1; // Initialize condition variable\n    // Sum 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // Update condition variable\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while loop */\nint whileLoop(int n) {\n  int res = 0;\n  int i = 1; // Initialize condition variable\n  // Sum 1, 2, ..., n-1, n\n  while (i &lt;= n) {\n    res += i;\n    i++; // Update condition variable\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while loop */\nfn while_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // Initialize condition variable\n\n    // Sum 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i;\n        i += 1; // Update condition variable\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while loop */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // Initialize condition variable\n    // Sum 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // Update condition variable\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while loop */\nfun whileLoop(n: Int): Int {\n    var res = 0\n    var i = 1 // Initialize condition variable\n    // Sum 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i\n        i++ // Update condition variable\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### while loop ###\ndef while_loop(n)\n  res = 0\n  i = 1 # Initialize condition variable\n\n  # Sum 1, 2, ..., n-1, n\n  while i &lt;= n\n    res += i\n    i += 1 # Update condition variable\n  end\n\n  res\nend\n</code></pre> <p>The <code>while</code> loop has greater flexibility than the <code>for</code> loop. In a <code>while</code> loop, we can freely design the initialization and update steps of the condition variable.</p> <p>For example, in the following code, the condition variable \\(i\\) is updated twice per round, which is not convenient to implement using a <code>for</code> loop:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def while_loop_ii(n: int) -&gt; int:\n    \"\"\"while loop (two updates)\"\"\"\n    res = 0\n    i = 1  # Initialize condition variable\n    # Sum 1, 4, 10, ...\n    while i &lt;= n:\n        res += i\n        # Update condition variable\n        i += 1\n        i *= 2\n    return res\n</code></pre> iteration.cpp<pre><code>/* while loop (two updates) */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // Initialize condition variable\n    // Sum 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // Update condition variable\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while loop (two updates) */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // Initialize condition variable\n    // Sum 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // Update condition variable\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while loop (two updates) */\nint WhileLoopII(int n) {\n    int res = 0;\n    int i = 1; // Initialize condition variable\n    // Sum 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // Update condition variable\n        i += 1; \n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while loop (two updates) */\nfunc whileLoopII(n int) int {\n    res := 0\n    // Initialize condition variable\n    i := 1\n    // Sum 1, 4, 10, ...\n    for i &lt;= n {\n        res += i\n        // Update condition variable\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while loop (two updates) */\nfunc whileLoopII(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // Initialize condition variable\n    // Sum 1, 4, 10, ...\n    while i &lt;= n {\n        res += i\n        // Update condition variable\n        i += 1\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while loop (two updates) */\nfunction whileLoopII(n) {\n    let res = 0;\n    let i = 1; // Initialize condition variable\n    // Sum 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // Update condition variable\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while loop (two updates) */\nfunction whileLoopII(n: number): number {\n    let res = 0;\n    let i = 1; // Initialize condition variable\n    // Sum 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // Update condition variable\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while loop (two updates) */\nint whileLoopII(int n) {\n  int res = 0;\n  int i = 1; // Initialize condition variable\n  // Sum 1, 4, 10, ...\n  while (i &lt;= n) {\n    res += i;\n    // Update condition variable\n    i++;\n    i *= 2;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while loop (two updates) */\nfn while_loop_ii(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // Initialize condition variable\n\n    // Sum 1, 4, 10, ...\n    while i &lt;= n {\n        res += i;\n        // Update condition variable\n        i += 1;\n        i *= 2;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while loop (two updates) */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // Initialize condition variable\n    // Sum 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // Update condition variable\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while loop (two updates) */\nfun whileLoopII(n: Int): Int {\n    var res = 0\n    var i = 1 // Initialize condition variable\n    // Sum 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i\n        // Update condition variable\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### while loop (two updates) ###\ndef while_loop_ii(n)\n  res = 0\n  i = 1 # Initialize condition variable\n\n  # Sum 1, 4, 10, ...\n  while i &lt;= n\n    res += i\n    # Update condition variable\n    i += 1\n    i *= 2\n  end\n\n  res\nend\n</code></pre> <p>Overall, <code>for</code> loops have more compact code, while <code>while</code> loops are more flexible; both can implement iterative structures. The choice of which to use should be determined based on the requirements of the specific problem.</p>","path":["Chapter 2. Complexity Analysis","2.2   Iteration and Recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#3-nested-loops","level":3,"title":"3.   Nested Loops","text":"<p>We can nest one loop structure inside another. Below is an example using <code>for</code> loops:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def nested_for_loop(n: int) -&gt; str:\n    \"\"\"Nested for loop\"\"\"\n    res = \"\"\n    # Loop i = 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        # Loop j = 1, 2, ..., n-1, n\n        for j in range(1, n + 1):\n            res += f\"({i}, {j}), \"\n    return res\n</code></pre> iteration.cpp<pre><code>/* Nested for loop */\nstring nestedForLoop(int n) {\n    ostringstream res;\n    // Loop i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        // Loop j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; ++j) {\n            res &lt;&lt; \"(\" &lt;&lt; i &lt;&lt; \", \" &lt;&lt; j &lt;&lt; \"), \";\n        }\n    }\n    return res.str();\n}\n</code></pre> iteration.java<pre><code>/* Nested for loop */\nString nestedForLoop(int n) {\n    StringBuilder res = new StringBuilder();\n    // Loop i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // Loop j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.append(\"(\" + i + \", \" + j + \"), \");\n        }\n    }\n    return res.toString();\n}\n</code></pre> iteration.cs<pre><code>/* Nested for loop */\nstring NestedForLoop(int n) {\n    StringBuilder res = new();\n    // Loop i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // Loop j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.Append($\"({i}, {j}), \");\n        }\n    }\n    return res.ToString();\n}\n</code></pre> iteration.go<pre><code>/* Nested for loop */\nfunc nestedForLoop(n int) string {\n    res := \"\"\n    // Loop i = 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= n; j++ {\n            // Loop j = 1, 2, ..., n-1, n\n            res += fmt.Sprintf(\"(%d, %d), \", i, j)\n        }\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* Nested for loop */\nfunc nestedForLoop(n: Int) -&gt; String {\n    var res = \"\"\n    // Loop i = 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        // Loop j = 1, 2, ..., n-1, n\n        for j in 1 ... n {\n            res.append(\"(\\(i), \\(j)), \")\n        }\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* Nested for loop */\nfunction nestedForLoop(n) {\n    let res = '';\n    // Loop i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // Loop j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* Nested for loop */\nfunction nestedForLoop(n: number): string {\n    let res = '';\n    // Loop i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // Loop j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* Nested for loop */\nString nestedForLoop(int n) {\n  String res = \"\";\n  // Loop i = 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    // Loop j = 1, 2, ..., n-1, n\n    for (int j = 1; j &lt;= n; j++) {\n      res += \"($i, $j), \";\n    }\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* Nested for loop */\nfn nested_for_loop(n: i32) -&gt; String {\n    let mut res = vec![];\n    // Loop i = 1, 2, ..., n-1, n\n    for i in 1..=n {\n        // Loop j = 1, 2, ..., n-1, n\n        for j in 1..=n {\n            res.push(format!(\"({}, {}), \", i, j));\n        }\n    }\n    res.join(\"\")\n}\n</code></pre> iteration.c<pre><code>/* Nested for loop */\nchar *nestedForLoop(int n) {\n    // n * n is the number of points, \"(i, j), \" string max length is 6+10*2, plus extra space for null character \\0\n    int size = n * n * 26 + 1;\n    char *res = malloc(size * sizeof(char));\n    // Loop i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // Loop j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            char tmp[26];\n            snprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\n            strncat(res, tmp, size - strlen(res) - 1);\n        }\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* Nested for loop */\nfun nestedForLoop(n: Int): String {\n    val res = StringBuilder()\n    // Loop i = 1, 2, ..., n-1, n\n    for (i in 1..n) {\n        // Loop j = 1, 2, ..., n-1, n\n        for (j in 1..n) {\n            res.append(\" ($i, $j), \")\n        }\n    }\n    return res.toString()\n}\n</code></pre> iteration.rb<pre><code>### Nested for loop ###\ndef nested_for_loop(n)\n  res = \"\"\n\n  # Loop i = 1, 2, ..., n-1, n\n  for i in 1..n\n    # Loop j = 1, 2, ..., n-1, n\n    for j in 1..n\n      res += \"(#{i}, #{j}), \"\n    end\n  end\n\n  res\nend\n</code></pre> <p>Figure 2-2 shows the flowchart of this nested loop.</p> <p></p> <p> Figure 2-2   Flowchart of nested loops </p> <p>In this case, the number of operations of the function is proportional to \\(n^2\\), or the algorithm's running time has a \"quadratic relationship\" with the input data size \\(n\\).</p> <p>We can continue adding nested loops, where each additional level of nesting can be viewed as an increase in dimensionality, raising the time complexity to a \"cubic relationship\", a \"quartic relationship\", and so on.</p>","path":["Chapter 2. Complexity Analysis","2.2   Iteration and Recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#222-recursion","level":2,"title":"2.2.2   Recursion","text":"<p>Recursion is an algorithmic strategy that solves problems by having a function call itself. It mainly consists of two phases.</p> <ol> <li>Descend: The program continuously calls itself deeper, usually passing in smaller or more simplified parameters, until reaching a \"termination condition\".</li> <li>Ascend: After triggering the \"termination condition\", the program returns layer by layer from the deepest recursive function, aggregating the result of each layer.</li> </ol> <p>From an implementation perspective, recursive code mainly consists of three elements.</p> <ol> <li>Termination condition: Used to determine when to switch from \"descending\" to \"ascending\".</li> <li>Recursive call: Corresponds to \"descending\", where the function calls itself, usually with smaller or more simplified parameters.</li> <li>Return result: Corresponds to \"ascending\", returning the result of the current recursion level to the previous layer.</li> </ol> <p>Observe the following code. We only need to call the function <code>recur(n)</code> to complete the calculation of \\(1 + 2 + \\dots + n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def recur(n: int) -&gt; int:\n    \"\"\"Recursion\"\"\"\n    # Termination condition\n    if n == 1:\n        return 1\n    # Recurse: recursive call\n    res = recur(n - 1)\n    # Return: return result\n    return n + res\n</code></pre> recursion.cpp<pre><code>/* Recursion */\nint recur(int n) {\n    // Termination condition\n    if (n == 1)\n        return 1;\n    // Recurse: recursive call\n    int res = recur(n - 1);\n    // Return: return result\n    return n + res;\n}\n</code></pre> recursion.java<pre><code>/* Recursion */\nint recur(int n) {\n    // Termination condition\n    if (n == 1)\n        return 1;\n    // Recurse: recursive call\n    int res = recur(n - 1);\n    // Return: return result\n    return n + res;\n}\n</code></pre> recursion.cs<pre><code>/* Recursion */\nint Recur(int n) {\n    // Termination condition\n    if (n == 1)\n        return 1;\n    // Recurse: recursive call\n    int res = Recur(n - 1);\n    // Return: return result\n    return n + res;\n}\n</code></pre> recursion.go<pre><code>/* Recursion */\nfunc recur(n int) int {\n    // Termination condition\n    if n == 1 {\n        return 1\n    }\n    // Recurse: recursive call\n    res := recur(n - 1)\n    // Return: return result\n    return n + res\n}\n</code></pre> recursion.swift<pre><code>/* Recursion */\nfunc recur(n: Int) -&gt; Int {\n    // Termination condition\n    if n == 1 {\n        return 1\n    }\n    // Recurse: recursive call\n    let res = recur(n: n - 1)\n    // Return: return result\n    return n + res\n}\n</code></pre> recursion.js<pre><code>/* Recursion */\nfunction recur(n) {\n    // Termination condition\n    if (n === 1) return 1;\n    // Recurse: recursive call\n    const res = recur(n - 1);\n    // Return: return result\n    return n + res;\n}\n</code></pre> recursion.ts<pre><code>/* Recursion */\nfunction recur(n: number): number {\n    // Termination condition\n    if (n === 1) return 1;\n    // Recurse: recursive call\n    const res = recur(n - 1);\n    // Return: return result\n    return n + res;\n}\n</code></pre> recursion.dart<pre><code>/* Recursion */\nint recur(int n) {\n  // Termination condition\n  if (n == 1) return 1;\n  // Recurse: recursive call\n  int res = recur(n - 1);\n  // Return: return result\n  return n + res;\n}\n</code></pre> recursion.rs<pre><code>/* Recursion */\nfn recur(n: i32) -&gt; i32 {\n    // Termination condition\n    if n == 1 {\n        return 1;\n    }\n    // Recurse: recursive call\n    let res = recur(n - 1);\n    // Return: return result\n    n + res\n}\n</code></pre> recursion.c<pre><code>/* Recursion */\nint recur(int n) {\n    // Termination condition\n    if (n == 1)\n        return 1;\n    // Recurse: recursive call\n    int res = recur(n - 1);\n    // Return: return result\n    return n + res;\n}\n</code></pre> recursion.kt<pre><code>/* Recursion */\nfun recur(n: Int): Int {\n    // Termination condition\n    if (n == 1)\n        return 1\n    // Descend: recursive call\n    val res = recur(n - 1)\n    // Return: return result\n    return n + res\n}\n</code></pre> recursion.rb<pre><code>### Recursion ###\ndef recur(n)\n  # Termination condition\n  return 1 if n == 1\n  # Recurse: recursive call\n  res = recur(n - 1)\n  # Return: return result\n  n + res\nend\n</code></pre> <p>Figure 2-3 shows the recursive process of this function.</p> <p></p> <p> Figure 2-3   Recursive process of the summation function </p> <p>Although from a computational perspective, iteration and recursion can achieve the same results, they represent two completely different paradigms for thinking about and solving problems.</p> <ul> <li>Iteration: Solves problems \"bottom-up\". Starting from the most basic steps, these steps are then repeatedly executed or accumulated until the task is complete.</li> <li>Recursion: Solves problems \"top-down\". The original problem is decomposed into smaller subproblems that have the same form as the original problem. These subproblems continue to be decomposed into even smaller subproblems until reaching the base case (where the solution is known).</li> </ul> <p>Taking the above summation function as an example, let the problem be \\(f(n) = 1 + 2 + \\dots + n\\).</p> <ul> <li>Iteration: Simulates the summation process in a loop, traversing from \\(1\\) to \\(n\\), performing the summation operation in each round to obtain \\(f(n)\\).</li> <li>Recursion: Decomposes the problem into the subproblem \\(f(n) = n + f(n-1)\\), continuously decomposing (recursively) until terminating at the base case \\(f(1) = 1\\).</li> </ul>","path":["Chapter 2. Complexity Analysis","2.2   Iteration and Recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-call-stack","level":3,"title":"1.   Call Stack","text":"<p>Each time a recursive function calls itself, the system allocates memory for the newly invoked function to store local variables, call addresses, and other information. This leads to two consequences.</p> <ul> <li>The function's context data is stored in a memory area called \"stack frame space\", which is not released until the function returns. Therefore, recursion usually consumes more memory space than iteration.</li> <li>Recursive function calls incur additional overhead. Therefore, recursion is usually less time-efficient than loops.</li> </ul> <p>As shown in Figure 2-4, before the termination condition is triggered, there are \\(n\\) unreturned recursive functions existing simultaneously, with a recursion depth of \\(n\\).</p> <p></p> <p> Figure 2-4   Recursion call depth </p> <p>In practice, the recursion depth allowed by programming languages is usually limited, and excessively deep recursion may lead to stack overflow errors.</p>","path":["Chapter 2. Complexity Analysis","2.2   Iteration and Recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-tail-recursion","level":3,"title":"2.   Tail Recursion","text":"<p>Interestingly, if a function makes the recursive call as the very last step before returning, the compiler or interpreter may optimize it so that its space efficiency is comparable to iteration. This case is called tail recursion.</p> <ul> <li>Regular recursion: When a function returns to the previous level, it needs to continue executing code, so the system needs to save the context of the previous layer's call.</li> <li>Tail recursion: The recursive call is the last operation before the function returns, meaning that after returning to the previous level, there is no need to continue executing other operations, so the system does not need to save the context of the previous layer's function.</li> </ul> <p>Taking the calculation of \\(1 + 2 + \\dots + n\\) as an example, we can set the result variable <code>res</code> as a function parameter to implement tail recursion:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def tail_recur(n, res):\n    \"\"\"Tail recursion\"\"\"\n    # Termination condition\n    if n == 0:\n        return res\n    # Tail recursive call\n    return tail_recur(n - 1, res + n)\n</code></pre> recursion.cpp<pre><code>/* Tail recursion */\nint tailRecur(int n, int res) {\n    // Termination condition\n    if (n == 0)\n        return res;\n    // Tail recursive call\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.java<pre><code>/* Tail recursion */\nint tailRecur(int n, int res) {\n    // Termination condition\n    if (n == 0)\n        return res;\n    // Tail recursive call\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.cs<pre><code>/* Tail recursion */\nint TailRecur(int n, int res) {\n    // Termination condition\n    if (n == 0)\n        return res;\n    // Tail recursive call\n    return TailRecur(n - 1, res + n);\n}\n</code></pre> recursion.go<pre><code>/* Tail recursion */\nfunc tailRecur(n int, res int) int {\n    // Termination condition\n    if n == 0 {\n        return res\n    }\n    // Tail recursive call\n    return tailRecur(n-1, res+n)\n}\n</code></pre> recursion.swift<pre><code>/* Tail recursion */\nfunc tailRecur(n: Int, res: Int) -&gt; Int {\n    // Termination condition\n    if n == 0 {\n        return res\n    }\n    // Tail recursive call\n    return tailRecur(n: n - 1, res: res + n)\n}\n</code></pre> recursion.js<pre><code>/* Tail recursion */\nfunction tailRecur(n, res) {\n    // Termination condition\n    if (n === 0) return res;\n    // Tail recursive call\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.ts<pre><code>/* Tail recursion */\nfunction tailRecur(n: number, res: number): number {\n    // Termination condition\n    if (n === 0) return res;\n    // Tail recursive call\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.dart<pre><code>/* Tail recursion */\nint tailRecur(int n, int res) {\n  // Termination condition\n  if (n == 0) return res;\n  // Tail recursive call\n  return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.rs<pre><code>/* Tail recursion */\nfn tail_recur(n: i32, res: i32) -&gt; i32 {\n    // Termination condition\n    if n == 0 {\n        return res;\n    }\n    // Tail recursive call\n    tail_recur(n - 1, res + n)\n}\n</code></pre> recursion.c<pre><code>/* Tail recursion */\nint tailRecur(int n, int res) {\n    // Termination condition\n    if (n == 0)\n        return res;\n    // Tail recursive call\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.kt<pre><code>/* Tail recursion */\ntailrec fun tailRecur(n: Int, res: Int): Int {\n    // Add tailrec keyword to enable tail recursion optimization\n    // Termination condition\n    if (n == 0)\n        return res\n    // Tail recursive call\n    return tailRecur(n - 1, res + n)\n}\n</code></pre> recursion.rb<pre><code>### Tail recursion ###\ndef tail_recur(n, res)\n  # Termination condition\n  return res if n == 0\n  # Tail recursive call\n  tail_recur(n - 1, res + n)\nend\n</code></pre> <p>The execution process of tail recursion is shown in Figure 2-5. Comparing regular recursion and tail recursion, the summation operation is performed at different points.</p> <ul> <li>Regular recursion: The summation operation is performed during the \"ascending\" process, requiring an additional summation operation after each layer returns.</li> <li>Tail recursion: The summation operation is performed during the \"descending\" process; the \"ascending\" process only needs to return layer by layer.</li> </ul> <p></p> <p> Figure 2-5   Tail recursion process </p> <p>Tip</p> <p>Please note that many compilers or interpreters do not support tail recursion optimization. For example, Python does not support tail recursion optimization by default, so even if a function is in tail recursive form, it may still encounter stack overflow issues.</p>","path":["Chapter 2. Complexity Analysis","2.2   Iteration and Recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#3-recursion-tree","level":3,"title":"3.   Recursion Tree","text":"<p>When dealing with algorithmic problems related to \"divide and conquer\", recursion often provides a more intuitive approach and more readable code than iteration. Taking the \"Fibonacci sequence\" as an example.</p> <p>Question</p> <p>Given a Fibonacci sequence \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\), find the \\(n\\)-th number in the sequence.</p> <p>Let the \\(n\\)-th number of the Fibonacci sequence be \\(f(n)\\). Two conclusions can be easily obtained.</p> <ul> <li>The first two numbers of the sequence are \\(f(1) = 0\\) and \\(f(2) = 1\\).</li> <li>Each number in the sequence is the sum of the previous two numbers, i.e., \\(f(n) = f(n - 1) + f(n - 2)\\).</li> </ul> <p>Following the recurrence relation to make recursive calls, with the first two numbers as termination conditions, we can write the recursive code. Calling <code>fib(n)</code> will give us the \\(n\\)-th number of the Fibonacci sequence:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def fib(n: int) -&gt; int:\n    \"\"\"Fibonacci sequence: recursion\"\"\"\n    # Termination condition f(1) = 0, f(2) = 1\n    if n == 1 or n == 2:\n        return n - 1\n    # Recursive call f(n) = f(n-1) + f(n-2)\n    res = fib(n - 1) + fib(n - 2)\n    # Return result f(n)\n    return res\n</code></pre> recursion.cpp<pre><code>/* Fibonacci sequence: recursion */\nint fib(int n) {\n    // Termination condition f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // Recursive call f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // Return result f(n)\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* Fibonacci sequence: recursion */\nint fib(int n) {\n    // Termination condition f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // Recursive call f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // Return result f(n)\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* Fibonacci sequence: recursion */\nint Fib(int n) {\n    // Termination condition f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // Recursive call f(n) = f(n-1) + f(n-2)\n    int res = Fib(n - 1) + Fib(n - 2);\n    // Return result f(n)\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* Fibonacci sequence: recursion */\nfunc fib(n int) int {\n    // Termination condition f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // Recursive call f(n) = f(n-1) + f(n-2)\n    res := fib(n-1) + fib(n-2)\n    // Return result f(n)\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* Fibonacci sequence: recursion */\nfunc fib(n: Int) -&gt; Int {\n    // Termination condition f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // Recursive call f(n) = f(n-1) + f(n-2)\n    let res = fib(n: n - 1) + fib(n: n - 2)\n    // Return result f(n)\n    return res\n}\n</code></pre> recursion.js<pre><code>/* Fibonacci sequence: recursion */\nfunction fib(n) {\n    // Termination condition f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // Recursive call f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // Return result f(n)\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* Fibonacci sequence: recursion */\nfunction fib(n: number): number {\n    // Termination condition f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // Recursive call f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // Return result f(n)\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* Fibonacci sequence: recursion */\nint fib(int n) {\n  // Termination condition f(1) = 0, f(2) = 1\n  if (n == 1 || n == 2) return n - 1;\n  // Recursive call f(n) = f(n-1) + f(n-2)\n  int res = fib(n - 1) + fib(n - 2);\n  // Return result f(n)\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* Fibonacci sequence: recursion */\nfn fib(n: i32) -&gt; i32 {\n    // Termination condition f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1;\n    }\n    // Recursive call f(n) = f(n-1) + f(n-2)\n    let res = fib(n - 1) + fib(n - 2);\n    // Return result\n    res\n}\n</code></pre> recursion.c<pre><code>/* Fibonacci sequence: recursion */\nint fib(int n) {\n    // Termination condition f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // Recursive call f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // Return result f(n)\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* Fibonacci sequence: recursion */\nfun fib(n: Int): Int {\n    // Termination condition f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1\n    // Recursive call f(n) = f(n-1) + f(n-2)\n    val res = fib(n - 1) + fib(n - 2)\n    // Return result f(n)\n    return res\n}\n</code></pre> recursion.rb<pre><code>### Fibonacci sequence: recursion ###\ndef fib(n)\n  # Termination condition f(1) = 0, f(2) = 1\n  return n - 1 if n == 1 || n == 2\n  # Recursive call f(n) = f(n-1) + f(n-2)\n  res = fib(n - 1) + fib(n - 2)\n  # Return result f(n)\n  res\nend\n</code></pre> <p>Observing the above code, we make two recursive calls within the function, meaning that one call produces two call branches. As shown in Figure 2-6, this repeated recursive calling eventually produces a recursion tree with \\(n\\) levels.</p> <p></p> <p> Figure 2-6   Recursion tree of the Fibonacci sequence </p> <p>Fundamentally, recursion embodies the paradigm of \"decomposing a problem into smaller subproblems\", and this divide-and-conquer strategy is crucial.</p> <ul> <li>From an algorithmic perspective, many important algorithmic strategies such as searching, sorting, backtracking, divide and conquer, and dynamic programming directly or indirectly apply this way of thinking.</li> <li>From a data structure perspective, recursion is naturally suited for handling problems related to linked lists, trees, and graphs, because they are well-suited for analysis using divide-and-conquer thinking.</li> </ul>","path":["Chapter 2. Complexity Analysis","2.2   Iteration and Recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#223-comparison-of-the-two","level":2,"title":"2.2.3   Comparison of the Two","text":"<p>Summarizing the above content, as shown in Table 2-1, iteration and recursion differ in implementation, performance, and applicability.</p> <p> Table 2-1   Comparison of iteration and recursion characteristics </p> Iteration Recursion Implementation Loop structure Function calls itself Time efficiency Generally more efficient, no function call overhead Each function call incurs overhead Memory usage Usually uses a fixed amount of memory space Accumulated function calls may use a large amount of stack frame space Suitable problems Suitable for simple loop tasks, with intuitive and readable code Suitable for subproblem decomposition, such as trees, graphs, divide and conquer, backtracking, etc., with concise and clear code structure <p>Tip</p> <p>If you find the following content difficult to understand, you can review it after reading the \"Stack\" chapter.</p> <p>What is the intrinsic relationship between iteration and recursion? Taking the above recursive function as an example, the summation operation is performed during the \"ascending\" phase of recursion. This means that the function called first actually completes its summation operation last, and this working mechanism is similar to the \"last-in, first-out\" principle of stacks.</p> <p>In fact, recursive terminology such as \"call stack\" and \"stack frame space\" already hints at the close relationship between recursion and stacks.</p> <ol> <li>Descend: When a function is called, the system allocates a new stack frame on the \"call stack\" for that function to store the function's local variables, parameters, return address, and other data.</li> <li>Ascend: When the function completes execution and returns, the corresponding stack frame is removed from the \"call stack\", restoring the execution environment of the previous function.</li> </ol> <p>Therefore, we can use an explicit stack to simulate the behavior of the call stack, thus transforming recursion into iterative form:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def for_loop_recur(n: int) -&gt; int:\n    \"\"\"Simulate recursion using iteration\"\"\"\n    # Use an explicit stack to simulate the system call stack\n    stack = []\n    res = 0\n    # Recurse: recursive call\n    for i in range(n, 0, -1):\n        # Simulate \"recurse\" with \"push\"\n        stack.append(i)\n    # Return: return result\n    while stack:\n        # Simulate \"return\" with \"pop\"\n        res += stack.pop()\n    # res = 1+2+3+...+n\n    return res\n</code></pre> recursion.cpp<pre><code>/* Simulate recursion using iteration */\nint forLoopRecur(int n) {\n    // Use an explicit stack to simulate the system call stack\n    stack&lt;int&gt; stack;\n    int res = 0;\n    // Recurse: recursive call\n    for (int i = n; i &gt; 0; i--) {\n        // Simulate \"recurse\" with \"push\"\n        stack.push(i);\n    }\n    // Return: return result\n    while (!stack.empty()) {\n        // Simulate \"return\" with \"pop\"\n        res += stack.top();\n        stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* Simulate recursion using iteration */\nint forLoopRecur(int n) {\n    // Use an explicit stack to simulate the system call stack\n    Stack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n    int res = 0;\n    // Recurse: recursive call\n    for (int i = n; i &gt; 0; i--) {\n        // Simulate \"recurse\" with \"push\"\n        stack.push(i);\n    }\n    // Return: return result\n    while (!stack.isEmpty()) {\n        // Simulate \"return\" with \"pop\"\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* Simulate recursion using iteration */\nint ForLoopRecur(int n) {\n    // Use an explicit stack to simulate the system call stack\n    Stack&lt;int&gt; stack = new();\n    int res = 0;\n    // Recurse: recursive call\n    for (int i = n; i &gt; 0; i--) {\n        // Simulate \"recurse\" with \"push\"\n        stack.Push(i);\n    }\n    // Return: return result\n    while (stack.Count &gt; 0) {\n        // Simulate \"return\" with \"pop\"\n        res += stack.Pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* Simulate recursion using iteration */\nfunc forLoopRecur(n int) int {\n    // Use an explicit stack to simulate the system call stack\n    stack := list.New()\n    res := 0\n    // Recurse: recursive call\n    for i := n; i &gt; 0; i-- {\n        // Simulate \"recurse\" with \"push\"\n        stack.PushBack(i)\n    }\n    // Return: return result\n    for stack.Len() != 0 {\n        // Simulate \"return\" with \"pop\"\n        res += stack.Back().Value.(int)\n        stack.Remove(stack.Back())\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* Simulate recursion using iteration */\nfunc forLoopRecur(n: Int) -&gt; Int {\n    // Use an explicit stack to simulate the system call stack\n    var stack: [Int] = []\n    var res = 0\n    // Recurse: recursive call\n    for i in (1 ... n).reversed() {\n        // Simulate \"recurse\" with \"push\"\n        stack.append(i)\n    }\n    // Return: return result\n    while !stack.isEmpty {\n        // Simulate \"return\" with \"pop\"\n        res += stack.removeLast()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.js<pre><code>/* Simulate recursion using iteration */\nfunction forLoopRecur(n) {\n    // Use an explicit stack to simulate the system call stack\n    const stack = [];\n    let res = 0;\n    // Recurse: recursive call\n    for (let i = n; i &gt; 0; i--) {\n        // Simulate \"recurse\" with \"push\"\n        stack.push(i);\n    }\n    // Return: return result\n    while (stack.length) {\n        // Simulate \"return\" with \"pop\"\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* Simulate recursion using iteration */\nfunction forLoopRecur(n: number): number {\n    // Use an explicit stack to simulate the system call stack\n    const stack: number[] = [];\n    let res: number = 0;\n    // Recurse: recursive call\n    for (let i = n; i &gt; 0; i--) {\n        // Simulate \"recurse\" with \"push\"\n        stack.push(i);\n    }\n    // Return: return result\n    while (stack.length) {\n        // Simulate \"return\" with \"pop\"\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* Simulate recursion using iteration */\nint forLoopRecur(int n) {\n  // Use an explicit stack to simulate the system call stack\n  List&lt;int&gt; stack = [];\n  int res = 0;\n  // Recurse: recursive call\n  for (int i = n; i &gt; 0; i--) {\n    // Simulate \"recurse\" with \"push\"\n    stack.add(i);\n  }\n  // Return: return result\n  while (!stack.isEmpty) {\n    // Simulate \"return\" with \"pop\"\n    res += stack.removeLast();\n  }\n  // res = 1+2+3+...+n\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* Simulate recursion using iteration */\nfn for_loop_recur(n: i32) -&gt; i32 {\n    // Use an explicit stack to simulate the system call stack\n    let mut stack = Vec::new();\n    let mut res = 0;\n    // Recurse: recursive call\n    for i in (1..=n).rev() {\n        // Simulate \"recurse\" with \"push\"\n        stack.push(i);\n    }\n    // Return: return result\n    while !stack.is_empty() {\n        // Simulate \"return\" with \"pop\"\n        res += stack.pop().unwrap();\n    }\n    // res = 1+2+3+...+n\n    res\n}\n</code></pre> recursion.c<pre><code>/* Simulate recursion using iteration */\nint forLoopRecur(int n) {\n    int stack[1000]; // Use a large array to simulate stack\n    int top = -1;    // Stack top index\n    int res = 0;\n    // Recurse: recursive call\n    for (int i = n; i &gt; 0; i--) {\n        // Simulate \"recurse\" with \"push\"\n        stack[1 + top++] = i;\n    }\n    // Return: return result\n    while (top &gt;= 0) {\n        // Simulate \"return\" with \"pop\"\n        res += stack[top--];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* Simulate recursion using iteration */\nfun forLoopRecur(n: Int): Int {\n    // Use an explicit stack to simulate the system call stack\n    val stack = Stack&lt;Int&gt;()\n    var res = 0\n    // Descend: recursive call\n    for (i in n downTo 0) {\n        // Simulate \"recurse\" with \"push\"\n        stack.push(i)\n    }\n    // Return: return result\n    while (stack.isNotEmpty()) {\n        // Simulate \"return\" with \"pop\"\n        res += stack.pop()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.rb<pre><code>### Use iteration to simulate recursion ###\ndef for_loop_recur(n)\n  # Use an explicit stack to simulate the system call stack\n  stack = []\n  res = 0\n\n  # Recurse: recursive call\n  for i in n.downto(0)\n    # Simulate \"recurse\" with \"push\"\n    stack &lt;&lt; i\n  end\n  # Return: return result\n  while !stack.empty?\n    res += stack.pop\n  end\n\n  # res = 1+2+3+...+n\n  res\nend\n</code></pre> <p>Observing the above code, when recursion is transformed into iteration, the code becomes more complex. Although iteration and recursion can be converted into each other in many cases, it may not be worthwhile to do so for the following two reasons.</p> <ul> <li>The transformed code may be more difficult to understand and less readable.</li> <li>For some complex problems, simulating the behavior of the system call stack can be very difficult.</li> </ul> <p>In summary, choosing between iteration and recursion depends on the nature of the specific problem. In programming practice, it is crucial to weigh the pros and cons of both and choose the appropriate method based on the context.</p>","path":["Chapter 2. Complexity Analysis","2.2   Iteration and Recursion"],"tags":[]},{"location":"chapter_computational_complexity/performance_evaluation/","level":1,"title":"2.1   Algorithm Efficiency Evaluation","text":"<p>In algorithm design, we pursue the following two levels of objectives sequentially.</p> <ol> <li>Finding a solution to the problem: The algorithm must reliably obtain the correct solution within the specified input range.</li> <li>Seeking the optimal solution: Multiple solutions may exist for the same problem, and we hope to find an algorithm that is as efficient as possible.</li> </ol> <p>In other words, under the premise of being able to solve the problem, algorithm efficiency has become the primary evaluation criterion for measuring the quality of algorithms. It includes the following two dimensions.</p> <ul> <li>Time efficiency: The length of time the algorithm runs.</li> <li>Space efficiency: The size of memory space the algorithm occupies.</li> </ul> <p>In short, our goal is to design data structures and algorithms that are \"both fast and memory-efficient\". Effectively evaluating algorithm efficiency is crucial, because only in this way can we compare various algorithms and guide the algorithm design and optimization process.</p> <p>Efficiency evaluation methods are mainly divided into two types: actual testing and theoretical estimation.</p>","path":["Chapter 2. Complexity Analysis","2.1   Algorithm Efficiency Evaluation"],"tags":[]},{"location":"chapter_computational_complexity/performance_evaluation/#211-actual-testing","level":2,"title":"2.1.1   Actual Testing","text":"<p>Suppose we now have algorithm <code>A</code> and algorithm <code>B</code>, both of which can solve the same problem, and we need to compare their efficiency. The most direct method is to run them on a computer and measure their running time and memory usage. This evaluation approach can reflect real-world behavior, but it also has considerable limitations.</p> <p>On one hand, it is difficult to eliminate interference factors from the testing environment. Hardware configuration affects algorithmic performance. For example, if an algorithm has a high degree of parallelism, it is more suitable for running on multi-core CPUs; if an algorithm performs memory-intensive operations, it will benefit more from high-performance memory. In other words, the test results of an algorithm on different machines may be inconsistent. This means we need to test on various machines and calculate average efficiency, which is impractical.</p> <p>On the other hand, conducting complete testing is very resource-intensive. As the input data volume changes, the algorithm will exhibit different efficiencies. For example, when the input data volume is small, the running time of algorithm <code>A</code> is shorter than algorithm <code>B</code>; but when the input data volume is large, the test results may be exactly the opposite. Therefore, to obtain convincing conclusions, we need to test input data of various scales, which requires a large amount of computational resources.</p>","path":["Chapter 2. Complexity Analysis","2.1   Algorithm Efficiency Evaluation"],"tags":[]},{"location":"chapter_computational_complexity/performance_evaluation/#212-theoretical-estimation","level":2,"title":"2.1.2   Theoretical Estimation","text":"<p>Since actual testing has considerable limitations, we can consider evaluating algorithm efficiency through theoretical calculation. This estimation method is called asymptotic complexity analysis, or complexity analysis for short.</p> <p>Complexity analysis can reflect the relationship between the time and space resources required for algorithm execution and the input data scale. It describes the growth trend of the time and space required for algorithm execution as the input data scale increases. This definition is a bit cumbersome, so we can break it down into three key points to understand.</p> <ul> <li>\"Time and space resources\" correspond to time complexity and space complexity, respectively.</li> <li>\"As the input data scale increases\" means that complexity reflects the relationship between algorithm running efficiency and input data scale.</li> <li>\"Growth trend of time and space\" indicates that complexity analysis focuses not on the specific values of running time or occupied space, but on how \"fast\" time or space grows.</li> </ul> <p>Complexity analysis overcomes the drawbacks of the actual testing method, reflected in the following aspects.</p> <ul> <li>It does not need to actually run the code, making it more environmentally friendly and energy-efficient.</li> <li>It is independent of the testing environment, and the analysis results are applicable to all running platforms.</li> <li>It can reflect algorithm efficiency at different data volumes, especially algorithm performance at large data volumes.</li> </ul> <p>Tip</p> <p>If you are still confused about the concept of complexity, don't worry—we will introduce it in detail in subsequent chapters.</p> <p>Complexity analysis provides us with a \"ruler\" for evaluating algorithm efficiency, allowing us to measure the time and space resources required to execute a certain algorithm and compare the efficiency between different algorithms.</p> <p>Complexity is a mathematical concept that may feel abstract and challenging for beginners. From this perspective, complexity analysis may not be the most suitable topic to introduce first. However, when we discuss the characteristics of a certain data structure or algorithm, it is difficult to avoid analyzing its running speed and space usage.</p> <p>In summary, it is recommended that before diving deep into data structures and algorithms, you first establish a preliminary understanding of complexity analysis so that you can analyze the complexity of simple algorithms.</p>","path":["Chapter 2. Complexity Analysis","2.1   Algorithm Efficiency Evaluation"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/","level":1,"title":"2.4   Space Complexity","text":"<p>Space complexity measures the growth trend of memory space occupied by an algorithm as the data size increases. This concept is very similar to time complexity, except that \"running time\" is replaced with \"occupied memory space\".</p>","path":["Chapter 2. Complexity Analysis","2.4   Space Complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#241-algorithm-related-space","level":2,"title":"2.4.1   Algorithm-Related Space","text":"<p>The memory space used by an algorithm during execution mainly includes the following types.</p> <ul> <li>Input space: Used to store the input data of the algorithm.</li> <li>Temporary space: Used to store variables, objects, function contexts, and other data during the algorithm's execution.</li> <li>Output space: Used to store the output data of the algorithm.</li> </ul> <p>In general, the scope of space complexity statistics is \"temporary space\" plus \"output space\".</p> <p>Temporary space can be further divided into three parts.</p> <ul> <li>Temporary data: Used to save various constants, variables, objects, etc., during the algorithm's execution.</li> <li>Stack frame space: Used to save the context data of called functions. The system creates a stack frame at the top of the stack each time a function is called, and the stack frame space is released after the function returns.</li> <li>Instruction space: Used to save compiled program instructions, which are usually ignored in actual statistics.</li> </ul> <p>When analyzing the space complexity of a program, we usually consider three parts: temporary data, stack frame space, and output data, as shown in the following figure.</p> <p></p> <p> Figure 2-15   Algorithm-related space </p> <p>The related code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class Node:\n    \"\"\"Class\"\"\"\n    def __init__(self, x: int):\n        self.val: int = x              # Node value\n        self.next: Node | None = None  # Reference to the next node\n\ndef function() -&gt; int:\n    \"\"\"Function\"\"\"\n    # Perform some operations...\n    return 0\n\ndef algorithm(n) -&gt; int:  # Input data\n    A = 0                 # Temporary data (constant, usually represented by uppercase letters)\n    b = 0                 # Temporary data (variable)\n    node = Node(0)        # Temporary data (object)\n    c = function()        # Stack frame space (function call)\n    return A + b + c      # Output data\n</code></pre> <pre><code>/* Structure */\nstruct Node {\n    int val;\n    Node *next;\n    Node(int x) : val(x), next(nullptr) {}\n};\n\n/* Function */\nint func() {\n    // Perform some operations...\n    return 0;\n}\n\nint algorithm(int n) {        // Input data\n    const int a = 0;          // Temporary data (constant)\n    int b = 0;                // Temporary data (variable)\n    Node* node = new Node(0); // Temporary data (object)\n    int c = func();           // Stack frame space (function call)\n    return a + b + c;         // Output data\n}\n</code></pre> <pre><code>/* Class */\nclass Node {\n    int val;\n    Node next;\n    Node(int x) { val = x; }\n}\n\n/* Function */\nint function() {\n    // Perform some operations...\n    return 0;\n}\n\nint algorithm(int n) {        // Input data\n    final int a = 0;          // Temporary data (constant)\n    int b = 0;                // Temporary data (variable)\n    Node node = new Node(0);  // Temporary data (object)\n    int c = function();       // Stack frame space (function call)\n    return a + b + c;         // Output data\n}\n</code></pre> <pre><code>/* Class */\nclass Node(int x) {\n    int val = x;\n    Node next;\n}\n\n/* Function */\nint Function() {\n    // Perform some operations...\n    return 0;\n}\n\nint Algorithm(int n) {        // Input data\n    const int a = 0;          // Temporary data (constant)\n    int b = 0;                // Temporary data (variable)\n    Node node = new(0);       // Temporary data (object)\n    int c = Function();       // Stack frame space (function call)\n    return a + b + c;         // Output data\n}\n</code></pre> <pre><code>/* Structure */\ntype node struct {\n    val  int\n    next *node\n}\n\n/* Create node structure */\nfunc newNode(val int) *node {\n    return &amp;node{val: val}\n}\n\n/* Function */\nfunc function() int {\n    // Perform some operations...\n    return 0\n}\n\nfunc algorithm(n int) int { // Input data\n    const a = 0             // Temporary data (constant)\n    b := 0                  // Temporary data (variable)\n    newNode(0)              // Temporary data (object)\n    c := function()         // Stack frame space (function call)\n    return a + b + c        // Output data\n}\n</code></pre> <pre><code>/* Class */\nclass Node {\n    var val: Int\n    var next: Node?\n\n    init(x: Int) {\n        val = x\n    }\n}\n\n/* Function */\nfunc function() -&gt; Int {\n    // Perform some operations...\n    return 0\n}\n\nfunc algorithm(n: Int) -&gt; Int { // Input data\n    let a = 0             // Temporary data (constant)\n    var b = 0             // Temporary data (variable)\n    let node = Node(x: 0) // Temporary data (object)\n    let c = function()    // Stack frame space (function call)\n    return a + b + c      // Output data\n}\n</code></pre> <pre><code>/* Class */\nclass Node {\n    val;\n    next;\n    constructor(val) {\n        this.val = val === undefined ? 0 : val; // Node value\n        this.next = null;                       // Reference to the next node\n    }\n}\n\n/* Function */\nfunction constFunc() {\n    // Perform some operations\n    return 0;\n}\n\nfunction algorithm(n) {       // Input data\n    const a = 0;              // Temporary data (constant)\n    let b = 0;                // Temporary data (variable)\n    const node = new Node(0); // Temporary data (object)\n    const c = constFunc();    // Stack frame space (function call)\n    return a + b + c;         // Output data\n}\n</code></pre> <pre><code>/* Class */\nclass Node {\n    val: number;\n    next: Node | null;\n    constructor(val?: number) {\n        this.val = val === undefined ? 0 : val; // Node value\n        this.next = null;                       // Reference to the next node\n    }\n}\n\n/* Function */\nfunction constFunc(): number {\n    // Perform some operations\n    return 0;\n}\n\nfunction algorithm(n: number): number { // Input data\n    const a = 0;                        // Temporary data (constant)\n    let b = 0;                          // Temporary data (variable)\n    const node = new Node(0);           // Temporary data (object)\n    const c = constFunc();              // Stack frame space (function call)\n    return a + b + c;                   // Output data\n}\n</code></pre> <pre><code>/* Class */\nclass Node {\n  int val;\n  Node next;\n  Node(this.val, [this.next]);\n}\n\n/* Function */\nint function() {\n  // Perform some operations...\n  return 0;\n}\n\nint algorithm(int n) {  // Input data\n  const int a = 0;      // Temporary data (constant)\n  int b = 0;            // Temporary data (variable)\n  Node node = Node(0);  // Temporary data (object)\n  int c = function();   // Stack frame space (function call)\n  return a + b + c;     // Output data\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* Structure */\nstruct Node {\n    val: i32,\n    next: Option&lt;Rc&lt;RefCell&lt;Node&gt;&gt;&gt;,\n}\n\n/* Create Node structure */\nimpl Node {\n    fn new(val: i32) -&gt; Self {\n        Self { val: val, next: None }\n    }\n}\n\n/* Function */\nfn function() -&gt; i32 {\n    // Perform some operations...\n    return 0;\n}\n\nfn algorithm(n: i32) -&gt; i32 {       // Input data\n    const a: i32 = 0;               // Temporary data (constant)\n    let mut b = 0;                  // Temporary data (variable)\n    let node = Node::new(0);        // Temporary data (object)\n    let c = function();             // Stack frame space (function call)\n    return a + b + c;               // Output data\n}\n</code></pre> <pre><code>/* Function */\nint func() {\n    // Perform some operations...\n    return 0;\n}\n\nint algorithm(int n) { // Input data\n    const int a = 0;   // Temporary data (constant)\n    int b = 0;         // Temporary data (variable)\n    int c = func();    // Stack frame space (function call)\n    return a + b + c;  // Output data\n}\n</code></pre> <pre><code>/* Class */\nclass Node(var _val: Int) {\n    var next: Node? = null\n}\n\n/* Function */\nfun function(): Int {\n    // Perform some operations...\n    return 0\n}\n\nfun algorithm(n: Int): Int { // Input data\n    val a = 0                // Temporary data (constant)\n    var b = 0                // Temporary data (variable)\n    val node = Node(0)       // Temporary data (object)\n    val c = function()       // Stack frame space (function call)\n    return a + b + c         // Output data\n}\n</code></pre> <pre><code>### Class ###\nclass Node\n    attr_accessor :val      # Node value\n    attr_accessor :next     # Reference to the next node\n\n    def initialize(x)\n        @val = x\n    end\nend\n\n### Function ###\ndef function\n    # Perform some operations...\n    0\nend\n\n### Algorithm ###\ndef algorithm(n)        # Input data\n    a = 0               # Temporary data (constant)\n    b = 0               # Temporary data (variable)\n    node = Node.new(0)  # Temporary data (object)\n    c = function        # Stack frame space (function call)\n    a + b + c           # Output data\nend\n</code></pre>","path":["Chapter 2. Complexity Analysis","2.4   Space Complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#242-calculation-method","level":2,"title":"2.4.2   Calculation Method","text":"<p>The calculation method for space complexity is roughly the same as for time complexity, except that what we measure changes from the \"number of operations\" to the \"amount of space used\".</p> <p>Unlike time complexity, we usually only focus on the worst-case space complexity. This is because memory space is a hard requirement, and we must ensure that sufficient memory space is reserved for all input data.</p> <p>Observe the following code. Here, \"worst case\" in worst-case space complexity has two meanings.</p> <ol> <li>Based on the worst input data: When \\(n &lt; 10\\), the space complexity is \\(O(1)\\); but when \\(n &gt; 10\\), the initialized array <code>nums</code> occupies \\(O(n)\\) space, so the worst-case space complexity is \\(O(n)\\).</li> <li>Based on the peak memory during algorithm execution: For example, before executing the last line, the program occupies \\(O(1)\\) space; when initializing the array <code>nums</code>, the program occupies \\(O(n)\\) space, so the worst-case space complexity is \\(O(n)\\).</li> </ol> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def algorithm(n: int):\n    a = 0               # O(1)\n    b = [0] * 10000     # O(1)\n    if n &gt; 10:\n        nums = [0] * n  # O(n)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    vector&lt;int&gt; b(10000);    // O(1)\n    if (n &gt; 10)\n        vector&lt;int&gt; nums(n); // O(n)\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10)\n        int[] nums = new int[n]; // O(n)\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10) {\n        int[] nums = new int[n]; // O(n)\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 0                      // O(1)\n    b := make([]int, 10000)     // O(1)\n    var nums []int\n    if n &gt; 10 {\n        nums := make([]int, n)  // O(n)\n    }\n    fmt.Println(a, b, nums)\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    let a = 0 // O(1)\n    let b = Array(repeating: 0, count: 10000) // O(1)\n    if n &gt; 10 {\n        let nums = Array(repeating: 0, count: n) // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 0;                            // O(1)\n  List&lt;int&gt; b = List.filled(10000, 0);  // O(1)\n  if (n &gt; 10) {\n    List&lt;int&gt; nums = List.filled(n, 0); // O(n)\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let a = 0;                              // O(1)\n    let b = [0; 10000];                     // O(1)\n    if n &gt; 10 {\n        let nums = vec![0; n as usize];     // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    int b[10000];            // O(1)\n    if (n &gt; 10)\n        int nums[n] = {0};   // O(n)\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    val a = 0                    // O(1)\n    val b = IntArray(10000)      // O(1)\n    if (n &gt; 10) {\n        val nums = IntArray(n)   // O(n)\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 0                           # O(1)\n    b = Array.new(10000)            # O(1)\n    nums = Array.new(n) if n &gt; 10   # O(n)\nend\n</code></pre> <p>In recursive functions, it is necessary to count the stack frame space. Observe the following code:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def function() -&gt; int:\n    # Perform some operations\n    return 0\n\ndef loop(n: int):\n    \"\"\"Loop has space complexity of O(1)\"\"\"\n    for _ in range(n):\n        function()\n\ndef recur(n: int):\n    \"\"\"Recursion has space complexity of O(n)\"\"\"\n    if n == 1:\n        return\n    return recur(n - 1)\n</code></pre> <pre><code>int func() {\n    // Perform some operations\n    return 0;\n}\n/* Loop has space complexity of O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* Recursion has space complexity of O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n    // Perform some operations\n    return 0;\n}\n/* Loop has space complexity of O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n/* Recursion has space complexity of O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    recur(n - 1);\n}\n</code></pre> <pre><code>int Function() {\n    // Perform some operations\n    return 0;\n}\n/* Loop has space complexity of O(1) */\nvoid Loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n/* Recursion has space complexity of O(n) */\nint Recur(int n) {\n    if (n == 1) return 1;\n    return Recur(n - 1);\n}\n</code></pre> <pre><code>func function() int {\n    // Perform some operations\n    return 0\n}\n\n/* Loop has space complexity of O(1) */\nfunc loop(n int) {\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n}\n\n/* Recursion has space complexity of O(n) */\nfunc recur(n int) {\n    if n == 1 {\n        return\n    }\n    recur(n - 1)\n}\n</code></pre> <pre><code>@discardableResult\nfunc function() -&gt; Int {\n    // Perform some operations\n    return 0\n}\n\n/* Loop has space complexity of O(1) */\nfunc loop(n: Int) {\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n\n/* Recursion has space complexity of O(n) */\nfunc recur(n: Int) {\n    if n == 1 {\n        return\n    }\n    recur(n: n - 1)\n}\n</code></pre> <pre><code>function constFunc() {\n    // Perform some operations\n    return 0;\n}\n/* Loop has space complexity of O(1) */\nfunction loop(n) {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* Recursion has space complexity of O(n) */\nfunction recur(n) {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>function constFunc(): number {\n    // Perform some operations\n    return 0;\n}\n/* Loop has space complexity of O(1) */\nfunction loop(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* Recursion has space complexity of O(n) */\nfunction recur(n: number): void {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n  // Perform some operations\n  return 0;\n}\n/* Loop has space complexity of O(1) */\nvoid loop(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n/* Recursion has space complexity of O(n) */\nvoid recur(int n) {\n  if (n == 1) return;\n  recur(n - 1);\n}\n</code></pre> <pre><code>fn function() -&gt; i32 {\n    // Perform some operations\n    return 0;\n}\n/* Loop has space complexity of O(1) */\nfn loop(n: i32) {\n    for i in 0..n {\n        function();\n    }\n}\n/* Recursion has space complexity of O(n) */\nfn recur(n: i32) {\n    if n == 1 {\n        return;\n    }\n    recur(n - 1);\n}\n</code></pre> <pre><code>int func() {\n    // Perform some operations\n    return 0;\n}\n/* Loop has space complexity of O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* Recursion has space complexity of O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    recur(n - 1);\n}\n</code></pre> <pre><code>fun function(): Int {\n    // Perform some operations\n    return 0\n}\n/* Loop has space complexity of O(1) */\nfun loop(n: Int) {\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n/* Recursion has space complexity of O(n) */\nfun recur(n: Int) {\n    if (n == 1) return\n    return recur(n - 1)\n}\n</code></pre> <pre><code>def function\n    # Perform some operations\n    0\nend\n\n### Loop has space complexity of O(1) ###\ndef loop(n)\n    (0...n).each { function }\nend\n\n### Recursion has space complexity of O(n) ###\ndef recur(n)\n    return if n == 1\n    recur(n - 1)\nend\n</code></pre> <p>The time complexity of both functions <code>loop()</code> and <code>recur()</code> is \\(O(n)\\), but their space complexities are different.</p> <ul> <li>The function <code>loop()</code> calls <code>function()</code> \\(n\\) times in a loop. In each iteration, <code>function()</code> returns and releases its stack frame space, so the space complexity remains \\(O(1)\\).</li> <li>The recursive function <code>recur()</code> has \\(n\\) unreturned <code>recur()</code> instances existing simultaneously during execution, thus occupying \\(O(n)\\) stack frame space.</li> </ul>","path":["Chapter 2. Complexity Analysis","2.4   Space Complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#243-common-types","level":2,"title":"2.4.3   Common Types","text":"<p>Let the input data size be \\(n\\). The following figure shows common types of space complexity (arranged from low to high).</p> \\[ \\begin{aligned} &amp; O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \\newline &amp; \\text{Constant} &lt; \\text{Logarithmic} &lt; \\text{Linear} &lt; \\text{Quadratic} &lt; \\text{Exponential} \\end{aligned} \\] <p></p> <p> Figure 2-16   Common types of space complexity </p>","path":["Chapter 2. Complexity Analysis","2.4   Space Complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#1-constant-order-o1","level":3,"title":"1.   Constant Order \\(O(1)\\)","text":"<p>Constant order is common for constants, variables, and objects whose number is independent of the input data size \\(n\\).</p> <p>It should be noted that memory occupied by initializing variables or calling functions in a loop is released when entering the next iteration, so it does not accumulate space, and the space complexity remains \\(O(1)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def function() -&gt; int:\n    \"\"\"Function\"\"\"\n    # Perform some operations\n    return 0\n\ndef constant(n: int):\n    \"\"\"Constant order\"\"\"\n    # Constants, variables, objects occupy O(1) space\n    a = 0\n    nums = [0] * 10000\n    node = ListNode(0)\n    # Variables in the loop occupy O(1) space\n    for _ in range(n):\n        c = 0\n    # Functions in the loop occupy O(1) space\n    for _ in range(n):\n        function()\n</code></pre> space_complexity.cpp<pre><code>/* Function */\nint func() {\n    // Perform some operations\n    return 0;\n}\n\n/* Constant order */\nvoid constant(int n) {\n    // Constants, variables, objects occupy O(1) space\n    const int a = 0;\n    int b = 0;\n    vector&lt;int&gt; nums(10000);\n    ListNode node(0);\n    // Variables in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // Functions in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* Function */\nint function() {\n    // Perform some operations\n    return 0;\n}\n\n/* Constant order */\nvoid constant(int n) {\n    // Constants, variables, objects occupy O(1) space\n    final int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new ListNode(0);\n    // Variables in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // Functions in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* Function */\nint Function() {\n    // Perform some operations\n    return 0;\n}\n\n/* Constant order */\nvoid Constant(int n) {\n    // Constants, variables, objects occupy O(1) space\n    int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new(0);\n    // Variables in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // Functions in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* Function */\nfunc function() int {\n    // Perform some operations...\n    return 0\n}\n\n/* Constant order */\nfunc spaceConstant(n int) {\n    // Constants, variables, objects occupy O(1) space\n    const a = 0\n    b := 0\n    nums := make([]int, 10000)\n    node := newNode(0)\n    // Variables in the loop occupy O(1) space\n    var c int\n    for i := 0; i &lt; n; i++ {\n        c = 0\n    }\n    // Functions in the loop occupy O(1) space\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n    b += 0\n    c += 0\n    nums[0] = 0\n    node.val = 0\n}\n</code></pre> space_complexity.swift<pre><code>/* Function */\n@discardableResult\nfunc function() -&gt; Int {\n    // Perform some operations\n    return 0\n}\n\n/* Constant order */\nfunc constant(n: Int) {\n    // Constants, variables, objects occupy O(1) space\n    let a = 0\n    var b = 0\n    let nums = Array(repeating: 0, count: 10000)\n    let node = ListNode(x: 0)\n    // Variables in the loop occupy O(1) space\n    for _ in 0 ..&lt; n {\n        let c = 0\n    }\n    // Functions in the loop occupy O(1) space\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n</code></pre> space_complexity.js<pre><code>/* Function */\nfunction constFunc() {\n    // Perform some operations\n    return 0;\n}\n\n/* Constant order */\nfunction constant(n) {\n    // Constants, variables, objects occupy O(1) space\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // Variables in the loop occupy O(1) space\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // Functions in the loop occupy O(1) space\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* Function */\nfunction constFunc(): number {\n    // Perform some operations\n    return 0;\n}\n\n/* Constant order */\nfunction constant(n: number): void {\n    // Constants, variables, objects occupy O(1) space\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // Variables in the loop occupy O(1) space\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // Functions in the loop occupy O(1) space\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* Function */\nint function() {\n  // Perform some operations\n  return 0;\n}\n\n/* Constant order */\nvoid constant(int n) {\n  // Constants, variables, objects occupy O(1) space\n  final int a = 0;\n  int b = 0;\n  List&lt;int&gt; nums = List.filled(10000, 0);\n  ListNode node = ListNode(0);\n  // Variables in the loop occupy O(1) space\n  for (var i = 0; i &lt; n; i++) {\n    int c = 0;\n  }\n  // Functions in the loop occupy O(1) space\n  for (var i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* Function */\nfn function() -&gt; i32 {\n    // Perform some operations\n    return 0;\n}\n\n/* Constant order */\n#[allow(unused)]\nfn constant(n: i32) {\n    // Constants, variables, objects occupy O(1) space\n    const A: i32 = 0;\n    let b = 0;\n    let nums = vec![0; 10000];\n    let node = ListNode::new(0);\n    // Variables in the loop occupy O(1) space\n    for i in 0..n {\n        let c = 0;\n    }\n    // Functions in the loop occupy O(1) space\n    for i in 0..n {\n        function();\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* Function */\nint func() {\n    // Perform some operations\n    return 0;\n}\n\n/* Constant order */\nvoid constant(int n) {\n    // Constants, variables, objects occupy O(1) space\n    const int a = 0;\n    int b = 0;\n    int nums[1000];\n    ListNode *node = newListNode(0);\n    free(node);\n    // Variables in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // Functions in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* Function */\nfun function(): Int {\n    // Perform some operations\n    return 0\n}\n\n/* Constant order */\nfun constant(n: Int) {\n    // Constants, variables, objects occupy O(1) space\n    val a = 0\n    var b = 0\n    val nums = Array(10000) { 0 }\n    val node = ListNode(0)\n    // Variables in the loop occupy O(1) space\n    for (i in 0..&lt;n) {\n        val c = 0\n    }\n    // Functions in the loop occupy O(1) space\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### Function ###\ndef function\n  # Perform some operations\n  0\nend\n\n### Constant time ###\ndef constant(n)\n  # Constants, variables, objects occupy O(1) space\n  a = 0\n  nums = [0] * 10000\n  node = ListNode.new\n\n  # Variables in the loop occupy O(1) space\n  (0...n).each { c = 0 }\n  # Functions in the loop occupy O(1) space\n  (0...n).each { function }\nend\n</code></pre>","path":["Chapter 2. Complexity Analysis","2.4   Space Complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#2-linear-order-on","level":3,"title":"2.   Linear Order \\(O(n)\\)","text":"<p>Linear order is common in arrays, linked lists, stacks, queues, etc., where the number of elements is proportional to \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def linear(n: int):\n    \"\"\"Linear order\"\"\"\n    # A list of length n occupies O(n) space\n    nums = [0] * n\n    # A hash table of length n occupies O(n) space\n    hmap = dict[int, str]()\n    for i in range(n):\n        hmap[i] = str(i)\n</code></pre> space_complexity.cpp<pre><code>/* Linear order */\nvoid linear(int n) {\n    // Array of length n uses O(n) space\n    vector&lt;int&gt; nums(n);\n    // A list of length n occupies O(n) space\n    vector&lt;ListNode&gt; nodes;\n    for (int i = 0; i &lt; n; i++) {\n        nodes.push_back(ListNode(i));\n    }\n    // A hash table of length n occupies O(n) space\n    unordered_map&lt;int, string&gt; map;\n    for (int i = 0; i &lt; n; i++) {\n        map[i] = to_string(i);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* Linear order */\nvoid linear(int n) {\n    // Array of length n uses O(n) space\n    int[] nums = new int[n];\n    // A list of length n occupies O(n) space\n    List&lt;ListNode&gt; nodes = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        nodes.add(new ListNode(i));\n    }\n    // A hash table of length n occupies O(n) space\n    Map&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        map.put(i, String.valueOf(i));\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* Linear order */\nvoid Linear(int n) {\n    // Array of length n uses O(n) space\n    int[] nums = new int[n];\n    // A list of length n occupies O(n) space\n    List&lt;ListNode&gt; nodes = [];\n    for (int i = 0; i &lt; n; i++) {\n        nodes.Add(new ListNode(i));\n    }\n    // A hash table of length n occupies O(n) space\n    Dictionary&lt;int, string&gt; map = [];\n    for (int i = 0; i &lt; n; i++) {\n        map.Add(i, i.ToString());\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* Linear order */\nfunc spaceLinear(n int) {\n    // Array of length n uses O(n) space\n    _ = make([]int, n)\n    // A list of length n occupies O(n) space\n    var nodes []*node\n    for i := 0; i &lt; n; i++ {\n        nodes = append(nodes, newNode(i))\n    }\n    // A hash table of length n occupies O(n) space\n    m := make(map[int]string, n)\n    for i := 0; i &lt; n; i++ {\n        m[i] = strconv.Itoa(i)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* Linear order */\nfunc linear(n: Int) {\n    // Array of length n uses O(n) space\n    let nums = Array(repeating: 0, count: n)\n    // A list of length n occupies O(n) space\n    let nodes = (0 ..&lt; n).map { ListNode(x: $0) }\n    // A hash table of length n occupies O(n) space\n    let map = Dictionary(uniqueKeysWithValues: (0 ..&lt; n).map { ($0, \"\\($0)\") })\n}\n</code></pre> space_complexity.js<pre><code>/* Linear order */\nfunction linear(n) {\n    // Array of length n uses O(n) space\n    const nums = new Array(n);\n    // A list of length n occupies O(n) space\n    const nodes = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // A hash table of length n occupies O(n) space\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* Linear order */\nfunction linear(n: number): void {\n    // Array of length n uses O(n) space\n    const nums = new Array(n);\n    // A list of length n occupies O(n) space\n    const nodes: ListNode[] = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // A hash table of length n occupies O(n) space\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* Linear order */\nvoid linear(int n) {\n  // Array of length n uses O(n) space\n  List&lt;int&gt; nums = List.filled(n, 0);\n  // A list of length n occupies O(n) space\n  List&lt;ListNode&gt; nodes = [];\n  for (var i = 0; i &lt; n; i++) {\n    nodes.add(ListNode(i));\n  }\n  // A hash table of length n occupies O(n) space\n  Map&lt;int, String&gt; map = HashMap();\n  for (var i = 0; i &lt; n; i++) {\n    map.putIfAbsent(i, () =&gt; i.toString());\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* Linear order */\n#[allow(unused)]\nfn linear(n: i32) {\n    // Array of length n uses O(n) space\n    let mut nums = vec![0; n as usize];\n    // A list of length n occupies O(n) space\n    let mut nodes = Vec::new();\n    for i in 0..n {\n        nodes.push(ListNode::new(i))\n    }\n    // A hash table of length n occupies O(n) space\n    let mut map = HashMap::new();\n    for i in 0..n {\n        map.insert(i, i.to_string());\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* Hash table */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // Implemented using uthash.h\n} HashTable;\n\n/* Linear order */\nvoid linear(int n) {\n    // Array of length n uses O(n) space\n    int *nums = malloc(sizeof(int) * n);\n    free(nums);\n\n    // A list of length n occupies O(n) space\n    ListNode **nodes = malloc(sizeof(ListNode *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        nodes[i] = newListNode(i);\n    }\n    // Memory release\n    for (int i = 0; i &lt; n; i++) {\n        free(nodes[i]);\n    }\n    free(nodes);\n\n    // A hash table of length n occupies O(n) space\n    HashTable *h = NULL;\n    for (int i = 0; i &lt; n; i++) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = i;\n        tmp-&gt;val = i;\n        HASH_ADD_INT(h, key, tmp);\n    }\n\n    // Memory release\n    HashTable *curr, *tmp;\n    HASH_ITER(hh, h, curr, tmp) {\n        HASH_DEL(h, curr);\n        free(curr);\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* Linear order */\nfun linear(n: Int) {\n    // Array of length n uses O(n) space\n    val nums = Array(n) { 0 }\n    // A list of length n occupies O(n) space\n    val nodes = mutableListOf&lt;ListNode&gt;()\n    for (i in 0..&lt;n) {\n        nodes.add(ListNode(i))\n    }\n    // A hash table of length n occupies O(n) space\n    val map = mutableMapOf&lt;Int, String&gt;()\n    for (i in 0..&lt;n) {\n        map[i] = i.toString()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### Linear time ###\ndef linear(n)\n  # A list of length n occupies O(n) space\n  nums = Array.new(n, 0)\n\n  # A hash table of length n occupies O(n) space\n  hmap = {}\n  for i in 0...n\n    hmap[i] = i.to_s\n  end\nend\n</code></pre> <p>As shown in the following figure, the recursion depth of this function is \\(n\\), meaning that there are \\(n\\) unreturned <code>linear_recur()</code> functions existing simultaneously, using \\(O(n)\\) stack frame space:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def linear_recur(n: int):\n    \"\"\"Linear order (recursive implementation)\"\"\"\n    print(\"Recursion n =\", n)\n    if n == 1:\n        return\n    linear_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* Linear order (recursive implementation) */\nvoid linearRecur(int n) {\n    cout &lt;&lt; \"Recursion n = \" &lt;&lt; n &lt;&lt; endl;\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* Linear order (recursive implementation) */\nvoid linearRecur(int n) {\n    System.out.println(\"Recursion n = \" + n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* Linear order (recursive implementation) */\nvoid LinearRecur(int n) {\n    Console.WriteLine(\"Recursion n = \" + n);\n    if (n == 1) return;\n    LinearRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* Linear order (recursive implementation) */\nfunc spaceLinearRecur(n int) {\n    fmt.Println(\"Recursion n =\", n)\n    if n == 1 {\n        return\n    }\n    spaceLinearRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* Linear order (recursive implementation) */\nfunc linearRecur(n: Int) {\n    print(\"Recursion n = \\(n)\")\n    if n == 1 {\n        return\n    }\n    linearRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* Linear order (recursive implementation) */\nfunction linearRecur(n) {\n    console.log(`Recursion n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* Linear order (recursive implementation) */\nfunction linearRecur(n: number): void {\n    console.log(`Recursion n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* Linear order (recursive implementation) */\nvoid linearRecur(int n) {\n  print('Recursion n = $n');\n  if (n == 1) return;\n  linearRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* Linear order (recursive implementation) */\nfn linear_recur(n: i32) {\n    println!(\"Recursion n = {}\", n);\n    if n == 1 {\n        return;\n    };\n    linear_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* Linear order (recursive implementation) */\nvoid linearRecur(int n) {\n    printf(\"Recursion n = %d\\r\\n\", n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.kt<pre><code>/* Linear order (recursive implementation) */\nfun linearRecur(n: Int) {\n    println(\"Recursion n = $n\")\n    if (n == 1)\n        return\n    linearRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>### Linear space (recursive) ###\ndef linear_recur(n)\n  puts \"Recursion n = #{n}\"\n  return if n == 1\n  linear_recur(n - 1)\nend\n</code></pre> <p></p> <p> Figure 2-17   Linear order space complexity generated by recursive function </p>","path":["Chapter 2. Complexity Analysis","2.4   Space Complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#3-quadratic-order-on2","level":3,"title":"3.   Quadratic Order \\(O(n^2)\\)","text":"<p>Quadratic order is common in matrices and graphs, where the number of elements is quadratically related to \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def quadratic(n: int):\n    \"\"\"Quadratic order\"\"\"\n    # A 2D list occupies O(n^2) space\n    num_matrix = [[0] * n for _ in range(n)]\n</code></pre> space_complexity.cpp<pre><code>/* Exponential order */\nvoid quadratic(int n) {\n    // 2D list uses O(n^2) space\n    vector&lt;vector&lt;int&gt;&gt; numMatrix;\n    for (int i = 0; i &lt; n; i++) {\n        vector&lt;int&gt; tmp;\n        for (int j = 0; j &lt; n; j++) {\n            tmp.push_back(0);\n        }\n        numMatrix.push_back(tmp);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* Exponential order */\nvoid quadratic(int n) {\n    // Matrix uses O(n^2) space\n    int[][] numMatrix = new int[n][n];\n    // 2D list uses O(n^2) space\n    List&lt;List&lt;Integer&gt;&gt; numList = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            tmp.add(0);\n        }\n        numList.add(tmp);\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* Exponential order */\nvoid Quadratic(int n) {\n    // Matrix uses O(n^2) space\n    int[,] numMatrix = new int[n, n];\n    // 2D list uses O(n^2) space\n    List&lt;List&lt;int&gt;&gt; numList = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;int&gt; tmp = [];\n        for (int j = 0; j &lt; n; j++) {\n            tmp.Add(0);\n        }\n        numList.Add(tmp);\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* Exponential order */\nfunc spaceQuadratic(n int) {\n    // Matrix uses O(n^2) space\n    numMatrix := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        numMatrix[i] = make([]int, n)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* Exponential order */\nfunc quadratic(n: Int) {\n    // 2D list uses O(n^2) space\n    let numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n</code></pre> space_complexity.js<pre><code>/* Exponential order */\nfunction quadratic(n) {\n    // Matrix uses O(n^2) space\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // 2D list uses O(n^2) space\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* Exponential order */\nfunction quadratic(n: number): void {\n    // Matrix uses O(n^2) space\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // 2D list uses O(n^2) space\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* Exponential order */\nvoid quadratic(int n) {\n  // Matrix uses O(n^2) space\n  List&lt;List&lt;int&gt;&gt; numMatrix = List.generate(n, (_) =&gt; List.filled(n, 0));\n  // 2D list uses O(n^2) space\n  List&lt;List&lt;int&gt;&gt; numList = [];\n  for (var i = 0; i &lt; n; i++) {\n    List&lt;int&gt; tmp = [];\n    for (int j = 0; j &lt; n; j++) {\n      tmp.add(0);\n    }\n    numList.add(tmp);\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* Exponential order */\n#[allow(unused)]\nfn quadratic(n: i32) {\n    // Matrix uses O(n^2) space\n    let num_matrix = vec![vec![0; n as usize]; n as usize];\n    // 2D list uses O(n^2) space\n    let mut num_list = Vec::new();\n    for i in 0..n {\n        let mut tmp = Vec::new();\n        for j in 0..n {\n            tmp.push(0);\n        }\n        num_list.push(tmp);\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* Exponential order */\nvoid quadratic(int n) {\n    // 2D list uses O(n^2) space\n    int **numMatrix = malloc(sizeof(int *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        int *tmp = malloc(sizeof(int) * n);\n        for (int j = 0; j &lt; n; j++) {\n            tmp[j] = 0;\n        }\n        numMatrix[i] = tmp;\n    }\n\n    // Memory release\n    for (int i = 0; i &lt; n; i++) {\n        free(numMatrix[i]);\n    }\n    free(numMatrix);\n}\n</code></pre> space_complexity.kt<pre><code>/* Exponential order */\nfun quadratic(n: Int) {\n    // Matrix uses O(n^2) space\n    val numMatrix = arrayOfNulls&lt;Array&lt;Int&gt;?&gt;(n)\n    // 2D list uses O(n^2) space\n    val numList = mutableListOf&lt;MutableList&lt;Int&gt;&gt;()\n    for (i in 0..&lt;n) {\n        val tmp = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            tmp.add(0)\n        }\n        numList.add(tmp)\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### Quadratic time ###\ndef quadratic(n)\n  # 2D list uses O(n^2) space\n  Array.new(n) { Array.new(n, 0) }\nend\n</code></pre> <p>As shown in the following figure, the recursion depth of this function is \\(n\\), and an array is initialized in each recursive function with lengths of \\(n\\), \\(n-1\\), \\(\\dots\\), \\(2\\), \\(1\\), with an average length of \\(n / 2\\), thus occupying \\(O(n^2)\\) space overall:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def quadratic_recur(n: int) -&gt; int:\n    \"\"\"Quadratic order (recursive implementation)\"\"\"\n    if n &lt;= 0:\n        return 0\n    # Array nums length is n, n-1, ..., 2, 1\n    nums = [0] * n\n    return quadratic_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* Quadratic order (recursive implementation) */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    vector&lt;int&gt; nums(n);\n    cout &lt;&lt; \"In recursion n = \" &lt;&lt; n &lt;&lt; \", nums length = \" &lt;&lt; nums.size() &lt;&lt; endl;\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* Quadratic order (recursive implementation) */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    // Array nums has length n, n-1, ..., 2, 1\n    int[] nums = new int[n];\n    System.out.println(\"In recursion n = \" + n + \", nums length = \" + nums.length);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* Quadratic order (recursive implementation) */\nint QuadraticRecur(int n) {\n    if (n &lt;= 0) return 0;\n    int[] nums = new int[n];\n    Console.WriteLine(\"Recursion n = \" + n + \", nums length = \" + nums.Length);\n    return QuadraticRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* Quadratic order (recursive implementation) */\nfunc spaceQuadraticRecur(n int) int {\n    if n &lt;= 0 {\n        return 0\n    }\n    nums := make([]int, n)\n    fmt.Printf(\"In recursion n = %d, nums length = %d \\n\", n, len(nums))\n    return spaceQuadraticRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* Quadratic order (recursive implementation) */\n@discardableResult\nfunc quadraticRecur(n: Int) -&gt; Int {\n    if n &lt;= 0 {\n        return 0\n    }\n    // Array nums has length n, n-1, ..., 2, 1\n    let nums = Array(repeating: 0, count: n)\n    print(\"In recursion n = \\(n), nums length = \\(nums.count)\")\n    return quadraticRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* Quadratic order (recursive implementation) */\nfunction quadraticRecur(n) {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`In recursion n = ${n}, nums length = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* Quadratic order (recursive implementation) */\nfunction quadraticRecur(n: number): number {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`In recursion n = ${n}, nums length = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* Quadratic order (recursive implementation) */\nint quadraticRecur(int n) {\n  if (n &lt;= 0) return 0;\n  List&lt;int&gt; nums = List.filled(n, 0);\n  print('In recursion n = $n, nums length = ${nums.length}');\n  return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* Quadratic order (recursive implementation) */\nfn quadratic_recur(n: i32) -&gt; i32 {\n    if n &lt;= 0 {\n        return 0;\n    };\n    // Array nums has length n, n-1, ..., 2, 1\n    let nums = vec![0; n as usize];\n    println!(\"In recursion n = {}, nums length = {}\", n, nums.len());\n    return quadratic_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* Quadratic order (recursive implementation) */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    int *nums = malloc(sizeof(int) * n);\n    printf(\"In recursion n = %d, nums length = %d\\r\\n\", n, n);\n    int res = quadraticRecur(n - 1);\n    free(nums);\n    return res;\n}\n</code></pre> space_complexity.kt<pre><code>/* Quadratic order (recursive implementation) */\ntailrec fun quadraticRecur(n: Int): Int {\n    if (n &lt;= 0)\n        return 0\n    // Array nums has length n, n-1, ..., 2, 1\n    val nums = Array(n) { 0 }\n    println(\"In recursion n = $n, nums length = ${nums.size}\")\n    return quadraticRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>### Quadratic space (recursive) ###\ndef quadratic_recur(n)\n  return 0 unless n &gt; 0\n\n  # Array nums has length n, n-1, ..., 2, 1\n  nums = Array.new(n, 0)\n  quadratic_recur(n - 1)\nend\n</code></pre> <p></p> <p> Figure 2-18   Quadratic order space complexity generated by recursive function </p>","path":["Chapter 2. Complexity Analysis","2.4   Space Complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#4-exponential-order-o2n","level":3,"title":"4.   Exponential Order \\(O(2^n)\\)","text":"<p>Exponential order is common in binary trees. Observe the following figure: a \"full binary tree\" with \\(n\\) levels has \\(2^n - 1\\) nodes, occupying \\(O(2^n)\\) space:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def build_tree(n: int) -&gt; TreeNode | None:\n    \"\"\"Exponential order (build full binary tree)\"\"\"\n    if n == 0:\n        return None\n    root = TreeNode(0)\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n    return root\n</code></pre> space_complexity.cpp<pre><code>/* Driver Code */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return nullptr;\n    TreeNode *root = new TreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.java<pre><code>/* Driver Code */\nTreeNode buildTree(int n) {\n    if (n == 0)\n        return null;\n    TreeNode root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.cs<pre><code>/* Driver Code */\nTreeNode? BuildTree(int n) {\n    if (n == 0) return null;\n    TreeNode root = new(0) {\n        left = BuildTree(n - 1),\n        right = BuildTree(n - 1)\n    };\n    return root;\n}\n</code></pre> space_complexity.go<pre><code>/* Driver Code */\nfunc buildTree(n int) *TreeNode {\n    if n == 0 {\n        return nil\n    }\n    root := NewTreeNode(0)\n    root.Left = buildTree(n - 1)\n    root.Right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.swift<pre><code>/* Driver Code */\nfunc buildTree(n: Int) -&gt; TreeNode? {\n    if n == 0 {\n        return nil\n    }\n    let root = TreeNode(x: 0)\n    root.left = buildTree(n: n - 1)\n    root.right = buildTree(n: n - 1)\n    return root\n}\n</code></pre> space_complexity.js<pre><code>/* Driver Code */\nfunction buildTree(n) {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.ts<pre><code>/* Driver Code */\nfunction buildTree(n: number): TreeNode | null {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.dart<pre><code>/* Driver Code */\nTreeNode? buildTree(int n) {\n  if (n == 0) return null;\n  TreeNode root = TreeNode(0);\n  root.left = buildTree(n - 1);\n  root.right = buildTree(n - 1);\n  return root;\n}\n</code></pre> space_complexity.rs<pre><code>/* Driver Code */\nfn build_tree(n: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    if n == 0 {\n        return None;\n    };\n    let root = TreeNode::new(0);\n    root.borrow_mut().left = build_tree(n - 1);\n    root.borrow_mut().right = build_tree(n - 1);\n    return Some(root);\n}\n</code></pre> space_complexity.c<pre><code>/* Driver Code */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return NULL;\n    TreeNode *root = newTreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.kt<pre><code>/* Driver Code */\nfun buildTree(n: Int): TreeNode? {\n    if (n == 0)\n        return null\n    val root = TreeNode(0)\n    root.left = buildTree(n - 1)\n    root.right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.rb<pre><code>### Exponential space (build full binary tree) ###\ndef build_tree(n)\n  return if n == 0\n\n  TreeNode.new.tap do |root|\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n  end\nend\n</code></pre> <p></p> <p> Figure 2-19   Exponential order space complexity generated by full binary tree </p>","path":["Chapter 2. Complexity Analysis","2.4   Space Complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#5-logarithmic-order-olog-n","level":3,"title":"5.   Logarithmic Order \\(O(\\log n)\\)","text":"<p>Logarithmic order is common in divide-and-conquer algorithms. For example, merge sort: given an input array of length \\(n\\), each recursion divides the array in half from the midpoint, forming a recursion tree of height \\(\\log n\\), using \\(O(\\log n)\\) stack frame space.</p> <p>Another example is converting a number to a string. Given a positive integer \\(n\\), it has \\(\\lfloor \\log_{10} n \\rfloor + 1\\) digits, i.e., the corresponding string length is \\(\\lfloor \\log_{10} n \\rfloor + 1\\), so the space complexity is \\(O(\\log_{10} n + 1) = O(\\log n)\\).</p>","path":["Chapter 2. Complexity Analysis","2.4   Space Complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#244-trading-time-for-space","level":2,"title":"2.4.4   Trading Time for Space","text":"<p>Ideally, we hope that both the time complexity and space complexity of an algorithm can reach optimal. However, in practice, optimizing both time complexity and space complexity simultaneously is usually very difficult.</p> <p>Reducing time complexity usually comes at the cost of increasing space complexity, and vice versa. Sacrificing memory space to improve execution speed is called \"trading space for time\"; the reverse is called \"trading time for space\".</p> <p>The choice of which approach depends on which aspect we value more. In most cases, time is more precious than space, so \"trading space for time\" is usually the more common strategy. Of course, when the data volume is very large, controlling space complexity is also very important.</p>","path":["Chapter 2. Complexity Analysis","2.4   Space Complexity"],"tags":[]},{"location":"chapter_computational_complexity/summary/","level":1,"title":"2.5   Summary","text":"","path":["Chapter 2. Complexity Analysis","2.5   Summary"],"tags":[]},{"location":"chapter_computational_complexity/summary/#1-key-review","level":3,"title":"1.   Key Review","text":"<p>Algorithm Efficiency Assessment</p> <ul> <li>Time efficiency and space efficiency are the two primary evaluation metrics for measuring algorithm performance.</li> <li>We can evaluate algorithm efficiency through actual testing, but it is difficult to eliminate the influence of the testing environment, and it consumes substantial computational resources.</li> <li>Complexity analysis can overcome the limitations of actual testing. Its results apply across running platforms, and it can reveal algorithm efficiency under different data scales.</li> </ul> <p>Time Complexity</p> <ul> <li>Time complexity is used to measure the trend of algorithm runtime as data volume increases. It can effectively evaluate algorithm efficiency, but it may be less informative in certain situations, such as when the input data volume is small or when time complexities are identical, making it impossible to precisely compare algorithm efficiency.</li> <li>Worst-case time complexity is represented using Big \\(O\\) notation, corresponding to the asymptotic upper bound of a function, reflecting the growth level of the number of operations \\(T(n)\\) as \\(n\\) approaches positive infinity.</li> <li>Deriving time complexity involves two steps: first, counting the number of operations, then determining the asymptotic upper bound.</li> <li>Common time complexities arranged from low to high include \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n \\log n)\\), \\(O(n^2)\\), \\(O(2^n)\\), and \\(O(n!)\\).</li> <li>The time complexity of some algorithms is not fixed, but rather depends on the distribution of input data. Time complexity is divided into worst-case, best-case, and average-case time complexity. Best-case time complexity is rarely used because input data generally needs to satisfy strict conditions to achieve the best case.</li> <li>Average time complexity reflects the algorithm's runtime efficiency under random data input, and is closest to the algorithm's performance in practical applications. Calculating average time complexity requires analyzing the input data distribution and the resulting mathematical expectation.</li> </ul> <p>Space Complexity</p> <ul> <li>Space complexity serves a similar purpose to time complexity, used to measure the trend of algorithm memory usage as data volume increases.</li> <li>The memory space related to algorithm execution can be divided into input space, temporary space, and output space. Typically, input space is not included in space complexity calculations. Temporary space can be divided into temporary data, stack frame space, and instruction space, where stack frame space usually affects space complexity only in recursive functions.</li> <li>We typically only focus on worst-case space complexity, which is the space complexity of an algorithm under worst-case input data and worst-case runtime.</li> <li>Common space complexities arranged from low to high include \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n^2)\\), and \\(O(2^n)\\).</li> </ul>","path":["Chapter 2. Complexity Analysis","2.5   Summary"],"tags":[]},{"location":"chapter_computational_complexity/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: Is the space complexity of tail recursion \\(O(1)\\)?</p> <p>Theoretically, the space complexity of tail recursive functions can be optimized to \\(O(1)\\). However, most programming languages (such as Java, Python, C++, Go, C#, etc.) do not support automatic tail recursion optimization, so the space complexity is generally considered to be \\(O(n)\\).</p> <p>Q: What is the difference between the terms function and method?</p> <p>A function can be executed independently, with all parameters passed explicitly. A method is associated with an object, is implicitly bound to the object that invokes it, and can operate on data contained in class instances.</p> <p>The following examples use several common programming languages for illustration.</p> <ul> <li>C is a procedural programming language without object-oriented concepts, so it only has functions. However, we can simulate object-oriented programming by creating structures (struct), and functions associated with structures are equivalent to methods in other programming languages.</li> <li>Java and C# are object-oriented programming languages where code blocks (methods) are typically part of a class. Static methods behave like functions because they are bound to the class and cannot access specific instance variables.</li> <li>C++ and Python support both procedural programming (functions) and object-oriented programming (methods).</li> </ul> <p>Q: Does the diagram for \"common space complexity types\" reflect the absolute size of occupied space?</p> <p>No, the diagram shows space complexity, which reflects growth trends rather than the absolute size of occupied space.</p> <p>Assuming \\(n = 8\\), you might find that the values of each curve do not correspond to the functions. This is because each curve contains a constant term used to compress the value range into a visually comfortable range.</p> <p>In practice, because we generally do not know the \"constant-term\" cost of each method, we usually cannot choose the optimal solution for cases like \\(n = 8\\) based on complexity alone. But for \\(n = 8^5\\), the choice is straightforward, because the growth trend already dominates.</p> <p>Q: Are there situations where algorithms are designed to sacrifice time (or space) based on actual use cases?</p> <p>In practical applications, most situations choose to sacrifice space for time. For example, with database indexes, we typically choose to build B+ trees or hash indexes, occupying substantial memory space in exchange for efficient queries of \\(O(\\log n)\\) or even \\(O(1)\\).</p> <p>In scenarios where space resources are precious, time may be sacrificed for space. For example, in embedded development, device memory is precious, and engineers may forgo using hash tables and choose to use array sequential search to save memory usage, at the cost of slower searches.</p>","path":["Chapter 2. Complexity Analysis","2.5   Summary"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/","level":1,"title":"2.3   Time Complexity","text":"<p>Runtime can intuitively and accurately reflect the efficiency of an algorithm. If we want to accurately estimate the runtime of a piece of code, how should we proceed?</p> <ol> <li>Determine the running platform, including hardware configuration, programming language, system environment, etc., as these factors all affect code execution efficiency.</li> <li>Evaluate the runtime required for various computational operations, for example, an addition operation <code>+</code> requires 1 ns, a multiplication operation <code>*</code> requires 10 ns, a print operation <code>print()</code> requires 5 ns, etc.</li> <li>Count all computational operations in the code, and sum the execution times of all operations to obtain the runtime.</li> </ol> <p>For example, in the following code, the input data size is \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># On a certain running platform\ndef algorithm(n: int):\n    a = 2      # 1 ns\n    a = a + 1  # 1 ns\n    a = a * 2  # 10 ns\n    # Loop n times\n    for _ in range(n):  # 1 ns\n        print(0)        # 5 ns\n</code></pre> <pre><code>// On a certain running platform\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        cout &lt;&lt; 0 &lt;&lt; endl;         // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a certain running platform\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        System.out.println(0);     // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a certain running platform\nvoid Algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        Console.WriteLine(0);      // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a certain running platform\nfunc algorithm(n int) {\n    a := 2     // 1 ns\n    a = a + 1  // 1 ns\n    a = a * 2  // 10 ns\n    // Loop n times\n    for i := 0; i &lt; n; i++ {  // 1 ns\n        fmt.Println(a)        // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a certain running platform\nfunc algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // Loop n times\n    for _ in 0 ..&lt; n { // 1 ns\n        print(0) // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a certain running platform\nfunction algorithm(n) {\n    var a = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // Loop n times\n    for(let i = 0; i &lt; n; i++) { // 1 ns\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a certain running platform\nfunction algorithm(n: number): void {\n    var a: number = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // Loop n times\n    for(let i = 0; i &lt; n; i++) { // 1 ns\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a certain running platform\nvoid algorithm(int n) {\n  int a = 2; // 1 ns\n  a = a + 1; // 1 ns\n  a = a * 2; // 10 ns\n  // Loop n times\n  for (int i = 0; i &lt; n; i++) { // 1 ns\n    print(0); // 5 ns\n  }\n}\n</code></pre> <pre><code>// On a certain running platform\nfn algorithm(n: i32) {\n    let mut a = 2;      // 1 ns\n    a = a + 1;          // 1 ns\n    a = a * 2;          // 10 ns\n    // Loop n times\n    for _ in 0..n {     // 1 ns\n        println!(\"{}\", 0);  // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a certain running platform\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {   // 1 ns\n        printf(\"%d\", 0);            // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a certain running platform\nfun algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // Loop n times\n    for (i in 0..&lt;n) {  // 1 ns\n        println(0)      // 5 ns\n    }\n}\n</code></pre> <pre><code># On a certain running platform\ndef algorithm(n)\n    a = 2       # 1 ns\n    a = a + 1   # 1 ns\n    a = a * 2   # 10 ns\n    # Loop n times\n    (0...n).each do # 1 ns\n        puts 0      # 5 ns\n    end\nend\n</code></pre> <p>According to the above method, the algorithm's runtime can be obtained as \\((6n + 12)\\) ns:</p> \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] <p>In reality, however, trying to count an algorithm's exact runtime is neither practical nor realistic. First, we do not want to tie the estimated time to the running platform, because algorithms need to run on many different platforms. Second, it is difficult to know the runtime of each type of operation, which makes the estimation process extremely difficult.</p>","path":["Chapter 2. Complexity Analysis","2.3   Time Complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#231-counting-time-growth-trends","level":2,"title":"2.3.1   Counting Time Growth Trends","text":"<p>Time complexity analysis does not count the algorithm's runtime, but rather counts the growth trend of the algorithm's runtime as the data volume increases.</p> <p>The concept of \"time growth trend\" is rather abstract; let us understand it through an example. Suppose the input data size is \\(n\\), and given three algorithms <code>A</code>, <code>B</code>, and <code>C</code>:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># Time complexity of algorithm A: constant order\ndef algorithm_A(n: int):\n    print(0)\n# Time complexity of algorithm B: linear order\ndef algorithm_B(n: int):\n    for _ in range(n):\n        print(0)\n# Time complexity of algorithm C: constant order\ndef algorithm_C(n: int):\n    for _ in range(1000000):\n        print(0)\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nvoid algorithm_A(int n) {\n    cout &lt;&lt; 0 &lt;&lt; endl;\n}\n// Time complexity of algorithm B: linear order\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nvoid algorithm_A(int n) {\n    System.out.println(0);\n}\n// Time complexity of algorithm B: linear order\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        System.out.println(0);\n    }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        System.out.println(0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nvoid AlgorithmA(int n) {\n    Console.WriteLine(0);\n}\n// Time complexity of algorithm B: linear order\nvoid AlgorithmB(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Console.WriteLine(0);\n    }\n}\n// Time complexity of algorithm C: constant order\nvoid AlgorithmC(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        Console.WriteLine(0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfunc algorithm_A(n int) {\n    fmt.Println(0)\n}\n// Time complexity of algorithm B: linear order\nfunc algorithm_B(n int) {\n    for i := 0; i &lt; n; i++ {\n        fmt.Println(0)\n    }\n}\n// Time complexity of algorithm C: constant order\nfunc algorithm_C(n int) {\n    for i := 0; i &lt; 1000000; i++ {\n        fmt.Println(0)\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfunc algorithmA(n: Int) {\n    print(0)\n}\n\n// Time complexity of algorithm B: linear order\nfunc algorithmB(n: Int) {\n    for _ in 0 ..&lt; n {\n        print(0)\n    }\n}\n\n// Time complexity of algorithm C: constant order\nfunc algorithmC(n: Int) {\n    for _ in 0 ..&lt; 1_000_000 {\n        print(0)\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfunction algorithm_A(n) {\n    console.log(0);\n}\n// Time complexity of algorithm B: linear order\nfunction algorithm_B(n) {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// Time complexity of algorithm C: constant order\nfunction algorithm_C(n) {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfunction algorithm_A(n: number): void {\n    console.log(0);\n}\n// Time complexity of algorithm B: linear order\nfunction algorithm_B(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// Time complexity of algorithm C: constant order\nfunction algorithm_C(n: number): void {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nvoid algorithmA(int n) {\n  print(0);\n}\n// Time complexity of algorithm B: linear order\nvoid algorithmB(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    print(0);\n  }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithmC(int n) {\n  for (int i = 0; i &lt; 1000000; i++) {\n    print(0);\n  }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfn algorithm_A(n: i32) {\n    println!(\"{}\", 0);\n}\n// Time complexity of algorithm B: linear order\nfn algorithm_B(n: i32) {\n    for _ in 0..n {\n        println!(\"{}\", 0);\n    }\n}\n// Time complexity of algorithm C: constant order\nfn algorithm_C(n: i32) {\n    for _ in 0..1000000 {\n        println!(\"{}\", 0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nvoid algorithm_A(int n) {\n    printf(\"%d\", 0);\n}\n// Time complexity of algorithm B: linear order\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        printf(\"%d\", 0);\n    }\n}\n// Time complexity of algorithm C: constant order\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        printf(\"%d\", 0);\n    }\n}\n</code></pre> <pre><code>// Time complexity of algorithm A: constant order\nfun algoritm_A(n: Int) {\n    println(0)\n}\n// Time complexity of algorithm B: linear order\nfun algorithm_B(n: Int) {\n    for (i in 0..&lt;n){\n        println(0)\n    }\n}\n// Time complexity of algorithm C: constant order\nfun algorithm_C(n: Int) {\n    for (i in 0..&lt;1000000) {\n        println(0)\n    }\n}\n</code></pre> <pre><code># Time complexity of algorithm A: constant order\ndef algorithm_A(n)\n    puts 0\nend\n\n# Time complexity of algorithm B: linear order\ndef algorithm_B(n)\n    (0...n).each { puts 0 }\nend\n\n# Time complexity of algorithm C: constant order\ndef algorithm_C(n)\n    (0...1_000_000).each { puts 0 }\nend\n</code></pre> <p>Figure 2-7 shows the time complexity of the above three algorithm functions.</p> <ul> <li>Algorithm <code>A</code> has only \\(1\\) print operation, and the algorithm's runtime does not grow as \\(n\\) increases. We call the time complexity of this algorithm \"constant order\".</li> <li>In algorithm <code>B</code>, the print operation needs to loop \\(n\\) times, and the algorithm's runtime grows linearly as \\(n\\) increases. The time complexity of this algorithm is called \"linear order\".</li> <li>In algorithm <code>C</code>, the print operation needs to loop \\(1000000\\) times. Although the runtime is very long, it is independent of the input data size \\(n\\). Therefore, the time complexity of <code>C</code> is the same as <code>A</code>, still \"constant order\".</li> </ul> <p></p> <p> Figure 2-7   Time growth trends of algorithms A, B, and C </p> <p>Compared to directly counting the algorithm's runtime, what are the characteristics of time complexity analysis?</p> <ul> <li>Time complexity can effectively evaluate algorithm efficiency. For example, the runtime of algorithm <code>B</code> grows linearly; when \\(n &gt; 1\\) it is slower than algorithm <code>A</code>, and when \\(n &gt; 1000000\\) it is slower than algorithm <code>C</code>. In fact, as long as the input data size \\(n\\) is sufficiently large, an algorithm with \"constant order\" complexity will always be superior to one with \"linear order\" complexity, which is precisely the meaning of time growth trend.</li> <li>The derivation method for time complexity is simpler. Obviously, the running platform and the types of computational operations are both unrelated to the growth trend of the algorithm's runtime. Therefore, in time complexity analysis, we can simply treat the execution time of all computational operations as the same \"unit time\", reducing \"tracking the runtime of each operation\" to \"counting the number of operations\", which greatly reduces the difficulty of estimation.</li> <li>Time complexity also has certain limitations. For example, although algorithms <code>A</code> and <code>C</code> have the same time complexity, their actual runtimes differ significantly. Similarly, although algorithm <code>B</code> has a higher time complexity than <code>C</code>, when the input data size \\(n\\) is small, algorithm <code>B</code> is clearly superior to algorithm <code>C</code>. In such cases, it is often difficult to judge the efficiency of algorithms based solely on time complexity. Of course, despite the above issues, complexity analysis remains the most effective and commonly used method for evaluating algorithm efficiency.</li> </ul>","path":["Chapter 2. Complexity Analysis","2.3   Time Complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#232-asymptotic-upper-bound-of-functions","level":2,"title":"2.3.2   Asymptotic Upper Bound of Functions","text":"<p>Given a function with input size \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def algorithm(n: int):\n    a = 1      # +1\n    a = a + 1  # +1\n    a = a * 2  # +1\n    # Loop n times\n    for i in range(n):  # +1\n        print(0)        # +1\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) { // +1 (i++ is executed each round)\n        cout &lt;&lt; 0 &lt;&lt; endl;    // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) { // +1 (i++ is executed each round)\n        System.out.println(0);    // +1\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {   // +1 (i++ is executed each round)\n        Console.WriteLine(0);   // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1      // +1\n    a = a + 1   // +1\n    a = a * 2   // +1\n    // Loop n times\n    for i := 0; i &lt; n; i++ {   // +1\n        fmt.Println(a)         // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // Loop n times\n    for _ in 0 ..&lt; n { // +1\n        print(0) // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    var a = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // Loop n times\n    for(let i = 0; i &lt; n; i++){ // +1 (i++ is executed each round)\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void{\n    var a: number = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // Loop n times\n    for(let i = 0; i &lt; n; i++){ // +1 (i++ is executed each round)\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +1\n  a = a + 1; // +1\n  a = a * 2; // +1\n  // Loop n times\n  for (int i = 0; i &lt; n; i++) { // +1 (i++ is executed each round)\n    print(0); // +1\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;   // +1\n    a = a + 1;      // +1\n    a = a * 2;      // +1\n\n    // Loop n times\n    for _ in 0..n { // +1 (i++ is executed each round)\n        println!(\"{}\", 0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Loop n times\n    for (int i = 0; i &lt; n; i++) {   // +1 (i++ is executed each round)\n        printf(\"%d\", 0);            // +1\n    }\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // Loop n times\n    for (i in 0..&lt;n) { // +1 (i++ is executed each round)\n        println(0) // +1\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 1       # +1\n    a = a + 1   # +1\n    a = a * 2   # +1\n    # Loop n times\n    (0...n).each do # +1\n        puts 0      # +1\n    end\nend\n</code></pre> <p>Let the number of operations of the algorithm be a function of the input data size \\(n\\), denoted as \\(T(n)\\). Then the number of operations of the above function is:</p> \\[ T(n) = 3 + 2n \\] <p>\\(T(n)\\) is a linear function, indicating that its runtime growth trend is linear, and therefore its time complexity is linear order.</p> <p>We denote the time complexity of linear order as \\(O(n)\\). This mathematical symbol is called big-\\(O\\) notation, representing the asymptotic upper bound of the function \\(T(n)\\).</p> <p>Time complexity analysis essentially calculates the asymptotic upper bound of \"the number of operations \\(T(n)\\)\", which has a clear mathematical definition.</p> <p>Asymptotic upper bound of functions</p> <p>If there exist positive real numbers \\(c\\) and \\(n_0\\) such that for all \\(n &gt; n_0\\), we have \\(T(n) \\leq c \\cdot f(n)\\), then \\(f(n)\\) can be considered as an asymptotic upper bound of \\(T(n)\\), denoted as \\(T(n) = O(f(n))\\).</p> <p>As shown in Figure 2-8, calculating the asymptotic upper bound is to find a function \\(f(n)\\) such that when \\(n\\) tends to infinity, \\(T(n)\\) and \\(f(n)\\) are at the same growth level, differing only by a constant coefficient \\(c\\).</p> <p></p> <p> Figure 2-8   Asymptotic upper bound of a function </p>","path":["Chapter 2. Complexity Analysis","2.3   Time Complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#233-derivation-method","level":2,"title":"2.3.3   Derivation Method","text":"<p>The idea of an asymptotic upper bound is somewhat mathematical. If you feel you haven't fully understood it, don't worry. We can first master the derivation method, and gradually grasp its mathematical meaning through continuous practice.</p> <p>According to the definition, after determining \\(f(n)\\), we can obtain the time complexity \\(O(f(n))\\). So how do we determine the asymptotic upper bound \\(f(n)\\)? Overall, it is divided into two steps: first count the number of operations, then determine the asymptotic upper bound.</p>","path":["Chapter 2. Complexity Analysis","2.3   Time Complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#1-step-1-count-the-number-of-operations","level":3,"title":"1.   Step 1: Count the Number of Operations","text":"<p>For code, count from top to bottom line by line. However, since the constant coefficient \\(c\\) in \\(c \\cdot f(n)\\) above can be of any size, coefficients and constant terms in the number of operations \\(T(n)\\) can all be ignored. According to this principle, the following counting simplification techniques can be summarized.</p> <ol> <li>Ignore constants in \\(T(n)\\). Because they are all independent of \\(n\\), they do not affect time complexity.</li> <li>Omit all coefficients. For example, looping \\(2n\\) times, \\(5n + 1\\) times, etc., can all be simplified as \\(n\\) times, because the coefficient before \\(n\\) does not affect time complexity.</li> <li>Use multiplication for nested loops. The total number of operations equals the product of the number of operations in the outer and inner loops, with each layer of loop still able to apply techniques <code>1.</code> and <code>2.</code> separately.</li> </ol> <p>Given a function, we can use the above techniques to count the number of operations:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def algorithm(n: int):\n    a = 1      # +0 (Technique 1)\n    a = a + n  # +0 (Technique 1)\n    # +n (Technique 2)\n    for i in range(5 * n + 1):\n        print(0)\n    # +n*n (Technique 3)\n    for i in range(2 * n):\n        for j in range(n + 1):\n            print(0)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0 (Technique 1)\n    a = a + n;  // +0 (Technique 1)\n    // +n (Technique 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n    // +n*n (Technique 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            cout &lt;&lt; 0 &lt;&lt; endl;\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0 (Technique 1)\n    a = a + n;  // +0 (Technique 1)\n    // +n (Technique 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        System.out.println(0);\n    }\n    // +n*n (Technique 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            System.out.println(0);\n        }\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +0 (Technique 1)\n    a = a + n;  // +0 (Technique 1)\n    // +n (Technique 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        Console.WriteLine(0);\n    }\n    // +n*n (Technique 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            Console.WriteLine(0);\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1     // +0 (Technique 1)\n    a = a + n  // +0 (Technique 1)\n    // +n (Technique 2)\n    for i := 0; i &lt; 5 * n + 1; i++ {\n        fmt.Println(0)\n    }\n    // +n*n (Technique 3)\n    for i := 0; i &lt; 2 * n; i++ {\n        for j := 0; j &lt; n + 1; j++ {\n            fmt.Println(0)\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +0 (Technique 1)\n    a = a + n // +0 (Technique 1)\n    // +n (Technique 2)\n    for _ in 0 ..&lt; (5 * n + 1) {\n        print(0)\n    }\n    // +n*n (Technique 3)\n    for _ in 0 ..&lt; (2 * n) {\n        for _ in 0 ..&lt; (n + 1) {\n            print(0)\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    let a = 1;  // +0 (Technique 1)\n    a = a + n;  // +0 (Technique 1)\n    // +n (Technique 2)\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n (Technique 3)\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    let a = 1;  // +0 (Technique 1)\n    a = a + n;  // +0 (Technique 1)\n    // +n (Technique 2)\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n (Technique 3)\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +0 (Technique 1)\n  a = a + n; // +0 (Technique 1)\n  // +n (Technique 2)\n  for (int i = 0; i &lt; 5 * n + 1; i++) {\n    print(0);\n  }\n  // +n*n (Technique 3)\n  for (int i = 0; i &lt; 2 * n; i++) {\n    for (int j = 0; j &lt; n + 1; j++) {\n      print(0);\n    }\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;     // +0 (Technique 1)\n    a = a + n;        // +0 (Technique 1)\n\n    // +n (Technique 2)\n    for i in 0..(5 * n + 1) {\n        println!(\"{}\", 0);\n    }\n\n    // +n*n (Technique 3)\n    for i in 0..(2 * n) {\n        for j in 0..(n + 1) {\n            println!(\"{}\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0 (Technique 1)\n    a = a + n;  // +0 (Technique 1)\n    // +n (Technique 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        printf(\"%d\", 0);\n    }\n    // +n*n (Technique 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            printf(\"%d\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1   // +0 (Technique 1)\n    a = a + n   // +0 (Technique 1)\n    // +n (Technique 2)\n    for (i in 0..&lt;5 * n + 1) {\n        println(0)\n    }\n    // +n*n (Technique 3)\n    for (i in 0..&lt;2 * n) {\n        for (j in 0..&lt;n + 1) {\n            println(0)\n        }\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 1       # +0 (Technique 1)\n    a = a + n   # +0 (Technique 1)\n    # +n (Technique 2)\n    (0...(5 * n + 1)).each do { puts 0 }\n    # +n*n (Technique 3)\n    (0...(2 * n)).each do\n        (0...(n + 1)).each do { puts 0 }\n    end\nend\n</code></pre> <p>The following formula shows the counting results before and after using the above techniques; both derive a time complexity of \\(O(n^2)\\).</p> \\[ \\begin{aligned} T(n) &amp; = 2n(n + 1) + (5n + 1) + 2 &amp; \\text{Complete count (-.-|||)} \\newline &amp; = 2n^2 + 7n + 3 \\newline T(n) &amp; = n^2 + n &amp; \\text{Simplified count (o.O)} \\end{aligned} \\]","path":["Chapter 2. Complexity Analysis","2.3   Time Complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#2-step-2-determine-the-asymptotic-upper-bound","level":3,"title":"2.   Step 2: Determine the Asymptotic Upper Bound","text":"<p>Time complexity is determined by the highest-order term in \\(T(n)\\). This is because as \\(n\\) tends to infinity, the highest-order term will play a dominant role, and the influence of other terms can be ignored.</p> <p>Table 2-2 shows some examples, where some exaggerated values are used to emphasize the conclusion that \"coefficients cannot shake the order\". When \\(n\\) tends to infinity, these constants become insignificant.</p> <p> Table 2-2   Time complexities corresponding to different numbers of operations </p> Number of Operations \\(T(n)\\) Time Complexity \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)","path":["Chapter 2. Complexity Analysis","2.3   Time Complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#234-common-types","level":2,"title":"2.3.4   Common Types","text":"<p>Let the input data size be \\(n\\). Common time complexity types are shown in Figure 2-9 (arranged in order from low to high).</p> \\[ \\begin{aligned} &amp; O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n \\log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!) \\newline &amp; \\text{Constant} &lt; \\text{Logarithmic} &lt; \\text{Linear} &lt; \\text{Linearithmic} &lt; \\text{Quadratic} &lt; \\text{Exponential} &lt; \\text{Factorial} \\end{aligned} \\] <p></p> <p> Figure 2-9   Common time complexity types </p>","path":["Chapter 2. Complexity Analysis","2.3   Time Complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#1-constant-order-o1","level":3,"title":"1.   Constant Order \\(O(1)\\)","text":"<p>The number of operations in constant order is independent of the input data size \\(n\\), meaning it does not change as \\(n\\) changes.</p> <p>In the following function, although the value of <code>size</code> may be large, it is independent of the input data size \\(n\\), so the time complexity remains \\(O(1)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def constant(n: int) -&gt; int:\n    \"\"\"Constant order\"\"\"\n    count = 0\n    size = 100000\n    for _ in range(size):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Constant order */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Constant order */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Constant order */\nint Constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Constant order */\nfunc constant(n int) int {\n    count := 0\n    size := 100000\n    for i := 0; i &lt; size; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Constant order */\nfunc constant(n: Int) -&gt; Int {\n    var count = 0\n    let size = 100_000\n    for _ in 0 ..&lt; size {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Constant order */\nfunction constant(n) {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Constant order */\nfunction constant(n: number): number {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Constant order */\nint constant(int n) {\n  int count = 0;\n  int size = 100000;\n  for (var i = 0; i &lt; size; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Constant order */\nfn constant(n: i32) -&gt; i32 {\n    _ = n;\n    let mut count = 0;\n    let size = 100_000;\n    for _ in 0..size {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Constant order */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Constant order */\nfun constant(n: Int): Int {\n    var count = 0\n    val size = 100000\n    for (i in 0..&lt;size)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Constant time ###\ndef constant(n)\n  count = 0\n  size = 100000\n\n  (0...size).each { count += 1 }\n\n  count\nend\n</code></pre>","path":["Chapter 2. Complexity Analysis","2.3   Time Complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#2-linear-order-on","level":3,"title":"2.   Linear Order \\(O(n)\\)","text":"<p>The number of operations in linear order grows linearly relative to the input data size \\(n\\). Linear order typically appears in single-layer loops:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def linear(n: int) -&gt; int:\n    \"\"\"Linear order\"\"\"\n    count = 0\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Linear order */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Linear order */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Linear order */\nint Linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Linear order */\nfunc linear(n int) int {\n    count := 0\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Linear order */\nfunc linear(n: Int) -&gt; Int {\n    var count = 0\n    for _ in 0 ..&lt; n {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Linear order */\nfunction linear(n) {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Linear order */\nfunction linear(n: number): number {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Linear order */\nint linear(int n) {\n  int count = 0;\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Linear order */\nfn linear(n: i32) -&gt; i32 {\n    let mut count = 0;\n    for _ in 0..n {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Linear order */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Linear order */\nfun linear(n: Int): Int {\n    var count = 0\n    for (i in 0..&lt;n)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Linear time ###\ndef linear(n)\n  count = 0\n  (0...n).each { count += 1 }\n  count\nend\n</code></pre> <p>Operations such as traversing arrays and traversing linked lists have a time complexity of \\(O(n)\\), where \\(n\\) is the length of the array or linked list:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def array_traversal(nums: list[int]) -&gt; int:\n    \"\"\"Linear order (traversing array)\"\"\"\n    count = 0\n    # Number of iterations is proportional to the array length\n    for num in nums:\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Linear order (traversing array) */\nint arrayTraversal(vector&lt;int&gt; &amp;nums) {\n    int count = 0;\n    // Number of iterations is proportional to the array length\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Linear order (traversing array) */\nint arrayTraversal(int[] nums) {\n    int count = 0;\n    // Number of iterations is proportional to the array length\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Linear order (traversing array) */\nint ArrayTraversal(int[] nums) {\n    int count = 0;\n    // Number of iterations is proportional to the array length\n    foreach (int num in nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Linear order (traversing array) */\nfunc arrayTraversal(nums []int) int {\n    count := 0\n    // Number of iterations is proportional to the array length\n    for range nums {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Linear order (traversing array) */\nfunc arrayTraversal(nums: [Int]) -&gt; Int {\n    var count = 0\n    // Number of iterations is proportional to the array length\n    for _ in nums {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Linear order (traversing array) */\nfunction arrayTraversal(nums) {\n    let count = 0;\n    // Number of iterations is proportional to the array length\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Linear order (traversing array) */\nfunction arrayTraversal(nums: number[]): number {\n    let count = 0;\n    // Number of iterations is proportional to the array length\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Linear order (traversing array) */\nint arrayTraversal(List&lt;int&gt; nums) {\n  int count = 0;\n  // Number of iterations is proportional to the array length\n  for (var _num in nums) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Linear order (traversing array) */\nfn array_traversal(nums: &amp;[i32]) -&gt; i32 {\n    let mut count = 0;\n    // Number of iterations is proportional to the array length\n    for _ in nums {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Linear order (traversing array) */\nint arrayTraversal(int *nums, int n) {\n    int count = 0;\n    // Number of iterations is proportional to the array length\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Linear order (traversing array) */\nfun arrayTraversal(nums: IntArray): Int {\n    var count = 0\n    // Number of iterations is proportional to the array length\n    for (num in nums) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Linear time (array traversal) ###\ndef array_traversal(nums)\n  count = 0\n\n  # Number of iterations is proportional to the array length\n  for num in nums\n    count += 1\n  end\n\n  count\nend\n</code></pre> <p>It is worth noting that the input data size \\(n\\) should be determined according to the type of input data. For example, in the first example, the variable \\(n\\) is the input data size; in the second example, the array length \\(n\\) is the data size.</p>","path":["Chapter 2. Complexity Analysis","2.3   Time Complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#3-quadratic-order-on2","level":3,"title":"3.   Quadratic Order \\(O(n^2)\\)","text":"<p>The number of operations in quadratic order grows quadratically relative to the input data size \\(n\\). Quadratic order typically appears in nested loops, where both the outer and inner loops have a time complexity of \\(O(n)\\), resulting in an overall time complexity of \\(O(n^2)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def quadratic(n: int) -&gt; int:\n    \"\"\"Quadratic order\"\"\"\n    count = 0\n    # Number of iterations is quadratically related to the data size n\n    for i in range(n):\n        for j in range(n):\n            count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Exponential order */\nint quadratic(int n) {\n    int count = 0;\n    // Number of iterations is quadratically related to the data size n\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Exponential order */\nint quadratic(int n) {\n    int count = 0;\n    // Number of iterations is quadratically related to the data size n\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Exponential order */\nint Quadratic(int n) {\n    int count = 0;\n    // Number of iterations is quadratically related to the data size n\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Exponential order */\nfunc quadratic(n int) int {\n    count := 0\n    // Number of iterations is quadratically related to the data size n\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; n; j++ {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Exponential order */\nfunc quadratic(n: Int) -&gt; Int {\n    var count = 0\n    // Number of iterations is quadratically related to the data size n\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; n {\n            count += 1\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Exponential order */\nfunction quadratic(n) {\n    let count = 0;\n    // Number of iterations is quadratically related to the data size n\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Exponential order */\nfunction quadratic(n: number): number {\n    let count = 0;\n    // Number of iterations is quadratically related to the data size n\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Exponential order */\nint quadratic(int n) {\n  int count = 0;\n  // Number of iterations is quadratically related to the data size n\n  for (int i = 0; i &lt; n; i++) {\n    for (int j = 0; j &lt; n; j++) {\n      count++;\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Exponential order */\nfn quadratic(n: i32) -&gt; i32 {\n    let mut count = 0;\n    // Number of iterations is quadratically related to the data size n\n    for _ in 0..n {\n        for _ in 0..n {\n            count += 1;\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Exponential order */\nint quadratic(int n) {\n    int count = 0;\n    // Number of iterations is quadratically related to the data size n\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Exponential order */\nfun quadratic(n: Int): Int {\n    var count = 0\n    // Number of iterations is quadratically related to the data size n\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;n) {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Quadratic time ###\ndef quadratic(n)\n  count = 0\n\n  # Number of iterations is quadratically related to the data size n\n  for i in 0...n\n    for j in 0...n\n      count += 1\n    end\n  end\n\n  count\nend\n</code></pre> <p>Figure 2-10 compares constant order, linear order, and quadratic order time complexities.</p> <p></p> <p> Figure 2-10   Time complexities of constant, linear, and quadratic orders </p> <p>Taking bubble sort as an example, the outer loop executes \\(n - 1\\) times, and the inner loop executes \\(n-1\\), \\(n-2\\), \\(\\dots\\), \\(2\\), \\(1\\) times, averaging \\(n / 2\\) times, resulting in a time complexity of \\(O((n - 1) n / 2) = O(n^2)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def bubble_sort(nums: list[int]) -&gt; int:\n    \"\"\"Quadratic order (bubble sort)\"\"\"\n    count = 0  # Counter\n    # Outer loop: unsorted range is [0, i]\n    for i in range(len(nums) - 1, 0, -1):\n        # Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # Swap nums[j] and nums[j + 1]\n                tmp: int = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3  # Element swap includes 3 unit operations\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Quadratic order (bubble sort) */\nint bubbleSort(vector&lt;int&gt; &amp;nums) {\n    int count = 0; // Counter\n    // Outer loop: unsorted range is [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // Element swap includes 3 unit operations\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Quadratic order (bubble sort) */\nint bubbleSort(int[] nums) {\n    int count = 0; // Counter\n    // Outer loop: unsorted range is [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // Element swap includes 3 unit operations\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Quadratic order (bubble sort) */\nint BubbleSort(int[] nums) {\n    int count = 0;  // Counter\n    // Outer loop: unsorted range is [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                count += 3;  // Element swap includes 3 unit operations\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Quadratic order (bubble sort) */\nfunc bubbleSort(nums []int) int {\n    count := 0 // Counter\n    // Outer loop: unsorted range is [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // Swap nums[j] and nums[j + 1]\n                tmp := nums[j]\n                nums[j] = nums[j+1]\n                nums[j+1] = tmp\n                count += 3 // Element swap includes 3 unit operations\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Quadratic order (bubble sort) */\nfunc bubbleSort(nums: inout [Int]) -&gt; Int {\n    var count = 0 // Counter\n    // Outer loop: unsorted range is [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3 // Element swap includes 3 unit operations\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Quadratic order (bubble sort) */\nfunction bubbleSort(nums) {\n    let count = 0; // Counter\n    // Outer loop: unsorted range is [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // Element swap includes 3 unit operations\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Quadratic order (bubble sort) */\nfunction bubbleSort(nums: number[]): number {\n    let count = 0; // Counter\n    // Outer loop: unsorted range is [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // Element swap includes 3 unit operations\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Quadratic order (bubble sort) */\nint bubbleSort(List&lt;int&gt; nums) {\n  int count = 0; // Counter\n  // Outer loop: unsorted range is [0, i]\n  for (var i = nums.length - 1; i &gt; 0; i--) {\n    // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n    for (var j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // Swap nums[j] and nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        count += 3; // Element swap includes 3 unit operations\n      }\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Quadratic order (bubble sort) */\nfn bubble_sort(nums: &amp;mut [i32]) -&gt; i32 {\n    let mut count = 0; // Counter\n\n    // Outer loop: unsorted range is [0, i]\n    for i in (1..nums.len()).rev() {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // Element swap includes 3 unit operations\n            }\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Quadratic order (bubble sort) */\nint bubbleSort(int *nums, int n) {\n    int count = 0; // Counter\n    // Outer loop: unsorted range is [0, i]\n    for (int i = n - 1; i &gt; 0; i--) {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // Element swap includes 3 unit operations\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Quadratic order (bubble sort) */\nfun bubbleSort(nums: IntArray): Int {\n    var count = 0 // Counter\n    // Outer loop: unsorted range is [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n                count += 3 // Element swap includes 3 unit operations\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Quadratic time (bubble sort) ###\ndef bubble_sort(nums)\n  count = 0  # Counter\n\n  # Outer loop: unsorted range is [0, i]\n  for i in (nums.length - 1).downto(0)\n    # Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # Swap nums[j] and nums[j + 1]\n        tmp = nums[j]\n        nums[j] = nums[j + 1]\n        nums[j + 1] = tmp\n        count += 3 # Element swap includes 3 unit operations\n      end\n    end\n  end\n\n  count\nend\n</code></pre>","path":["Chapter 2. Complexity Analysis","2.3   Time Complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#4-exponential-order-o2n","level":3,"title":"4.   Exponential Order \\(O(2^n)\\)","text":"<p>Biological \"cell division\" is a typical example of exponential order growth: the initial state is \\(1\\) cell, after one round of division it becomes \\(2\\), after two rounds it becomes \\(4\\), and so on; after \\(n\\) rounds of division there are \\(2^n\\) cells.</p> <p>Figure 2-11 and the following code simulate the cell division process, with a time complexity of \\(O(2^n)\\). Note that the input \\(n\\) represents the number of division rounds, and the return value <code>count</code> represents the total number of divisions.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def exponential(n: int) -&gt; int:\n    \"\"\"Exponential order (loop implementation)\"\"\"\n    count = 0\n    base = 1\n    # Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in range(n):\n        for _ in range(base):\n            count += 1\n        base *= 2\n    # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Exponential order (loop implementation) */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Exponential order (loop implementation) */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Exponential order (loop implementation) */\nint Exponential(int n) {\n    int count = 0, bas = 1;\n    // Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Exponential order (loop implementation) */\nfunc exponential(n int) int {\n    count, base := 0, 1\n    // Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; base; j++ {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Exponential order (loop implementation) */\nfunc exponential(n: Int) -&gt; Int {\n    var count = 0\n    var base = 1\n    // Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; base {\n            count += 1\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Exponential order (loop implementation) */\nfunction exponential(n) {\n    let count = 0,\n        base = 1;\n    // Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Exponential order (loop implementation) */\nfunction exponential(n: number): number {\n    let count = 0,\n        base = 1;\n    // Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Exponential order (loop implementation) */\nint exponential(int n) {\n  int count = 0, base = 1;\n  // Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)\n  for (var i = 0; i &lt; n; i++) {\n    for (var j = 0; j &lt; base; j++) {\n      count++;\n    }\n    base *= 2;\n  }\n  // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Exponential order (loop implementation) */\nfn exponential(n: i32) -&gt; i32 {\n    let mut count = 0;\n    let mut base = 1;\n    // Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0..n {\n        for _ in 0..base {\n            count += 1\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Exponential order (loop implementation) */\nint exponential(int n) {\n    int count = 0;\n    int bas = 1;\n    // Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Exponential order (loop implementation) */\nfun exponential(n: Int): Int {\n    var count = 0\n    var base = 1\n    // Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;base) {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Exponential time (iterative) ###\ndef exponential(n)\n  count, base = 0, 1\n\n  # Cells divide into two every round, forming sequence 1, 2, 4, 8, ..., 2^(n-1)\n  (0...n).each do\n    (0...base).each { count += 1 }\n    base *= 2\n  end\n\n  # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  count\nend\n</code></pre> <p></p> <p> Figure 2-11   Time complexity of exponential order </p> <p>In actual algorithms, exponential order often appears in recursive functions. For example, in the following code, it recursively splits in two, stopping after \\(n\\) splits:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def exp_recur(n: int) -&gt; int:\n    \"\"\"Exponential order (recursive implementation)\"\"\"\n    if n == 1:\n        return 1\n    return exp_recur(n - 1) + exp_recur(n - 1) + 1\n</code></pre> time_complexity.cpp<pre><code>/* Exponential order (recursive implementation) */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* Exponential order (recursive implementation) */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* Exponential order (recursive implementation) */\nint ExpRecur(int n) {\n    if (n == 1) return 1;\n    return ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* Exponential order (recursive implementation) */\nfunc expRecur(n int) int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n-1) + expRecur(n-1) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* Exponential order (recursive implementation) */\nfunc expRecur(n: Int) -&gt; Int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* Exponential order (recursive implementation) */\nfunction expRecur(n) {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* Exponential order (recursive implementation) */\nfunction expRecur(n: number): number {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* Exponential order (recursive implementation) */\nint expRecur(int n) {\n  if (n == 1) return 1;\n  return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* Exponential order (recursive implementation) */\nfn exp_recur(n: i32) -&gt; i32 {\n    if n == 1 {\n        return 1;\n    }\n    exp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* Exponential order (recursive implementation) */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* Exponential order (recursive implementation) */\nfun expRecur(n: Int): Int {\n    if (n == 1) {\n        return 1\n    }\n    return expRecur(n - 1) + expRecur(n - 1) + 1\n}\n</code></pre> time_complexity.rb<pre><code>### Exponential time (recursive) ###\ndef exp_recur(n)\n  return 1 if n == 1\n  exp_recur(n - 1) + exp_recur(n - 1) + 1\nend\n</code></pre> <p>Exponential order growth is very rapid and is common in exhaustive methods (brute force search, backtracking, etc.). For problems with large data scales, exponential order is unacceptable and typically requires dynamic programming or greedy algorithms to solve.</p>","path":["Chapter 2. Complexity Analysis","2.3   Time Complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#5-logarithmic-order-olog-n","level":3,"title":"5.   Logarithmic Order \\(O(\\log n)\\)","text":"<p>In contrast to exponential order, logarithmic order reflects the situation of \"reducing to half each round\". Let the input data size be \\(n\\). Since it is reduced to half each round, the number of loops is \\(\\log_2 n\\), which is the inverse function of \\(2^n\\).</p> <p>Figure 2-12 and the following code simulate the process of \"reducing to half each round\", with a time complexity of \\(O(\\log_2 n)\\), abbreviated as \\(O(\\log n)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def logarithmic(n: int) -&gt; int:\n    \"\"\"Logarithmic order (loop implementation)\"\"\"\n    count = 0\n    while n &gt; 1:\n        n = n / 2\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Logarithmic order (loop implementation) */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Logarithmic order (loop implementation) */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Logarithmic order (loop implementation) */\nint Logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n /= 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Logarithmic order (loop implementation) */\nfunc logarithmic(n int) int {\n    count := 0\n    for n &gt; 1 {\n        n = n / 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Logarithmic order (loop implementation) */\nfunc logarithmic(n: Int) -&gt; Int {\n    var count = 0\n    var n = n\n    while n &gt; 1 {\n        n = n / 2\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Logarithmic order (loop implementation) */\nfunction logarithmic(n) {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Logarithmic order (loop implementation) */\nfunction logarithmic(n: number): number {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Logarithmic order (loop implementation) */\nint logarithmic(int n) {\n  int count = 0;\n  while (n &gt; 1) {\n    n = n ~/ 2;\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Logarithmic order (loop implementation) */\nfn logarithmic(mut n: i32) -&gt; i32 {\n    let mut count = 0;\n    while n &gt; 1 {\n        n = n / 2;\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Logarithmic order (loop implementation) */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Logarithmic order (loop implementation) */\nfun logarithmic(n: Int): Int {\n    var n1 = n\n    var count = 0\n    while (n1 &gt; 1) {\n        n1 /= 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Logarithmic time (iterative) ###\ndef logarithmic(n)\n  count = 0\n\n  while n &gt; 1\n    n /= 2\n    count += 1\n  end\n\n  count\nend\n</code></pre> <p></p> <p> Figure 2-12   Time complexity of logarithmic order </p> <p>Like exponential order, logarithmic order also commonly appears in recursive functions. The following code forms a recursion tree of height \\(\\log_2 n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def log_recur(n: int) -&gt; int:\n    \"\"\"Logarithmic order (recursive implementation)\"\"\"\n    if n &lt;= 1:\n        return 0\n    return log_recur(n / 2) + 1\n</code></pre> time_complexity.cpp<pre><code>/* Logarithmic order (recursive implementation) */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* Logarithmic order (recursive implementation) */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* Logarithmic order (recursive implementation) */\nint LogRecur(int n) {\n    if (n &lt;= 1) return 0;\n    return LogRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* Logarithmic order (recursive implementation) */\nfunc logRecur(n int) int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n/2) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* Logarithmic order (recursive implementation) */\nfunc logRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n: n / 2) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* Logarithmic order (recursive implementation) */\nfunction logRecur(n) {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* Logarithmic order (recursive implementation) */\nfunction logRecur(n: number): number {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* Logarithmic order (recursive implementation) */\nint logRecur(int n) {\n  if (n &lt;= 1) return 0;\n  return logRecur(n ~/ 2) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* Logarithmic order (recursive implementation) */\nfn log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 0;\n    }\n    log_recur(n / 2) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* Logarithmic order (recursive implementation) */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* Logarithmic order (recursive implementation) */\nfun logRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 0\n    return logRecur(n / 2) + 1\n}\n</code></pre> time_complexity.rb<pre><code>### Logarithmic time (recursive) ###\ndef log_recur(n)\n  return 0 unless n &gt; 1\n  log_recur(n / 2) + 1\nend\n</code></pre> <p>Logarithmic order commonly appears in algorithms based on the divide-and-conquer strategy, reflecting the idea of repeatedly splitting a problem and simplifying it. It grows slowly and is the ideal time complexity second only to constant order.</p> <p>What is the base of \\(O(\\log n)\\)?</p> <p>To be precise, \"dividing into \\(m\\)\" corresponds to a time complexity of \\(O(\\log_m n)\\). And through the logarithmic base change formula, we can obtain time complexities with different bases that are equal:</p> \\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\] <p>That is to say, the base \\(m\\) can be converted without affecting the complexity. Therefore, we usually omit the base \\(m\\) and denote logarithmic order simply as \\(O(\\log n)\\).</p>","path":["Chapter 2. Complexity Analysis","2.3   Time Complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#6-linearithmic-order-on-log-n","level":3,"title":"6.   Linearithmic Order \\(O(n \\log n)\\)","text":"<p>Linearithmic order commonly appears in nested loops, where the time complexities of the two layers of loops are \\(O(\\log n)\\) and \\(O(n)\\) respectively. The relevant code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def linear_log_recur(n: int) -&gt; int:\n    \"\"\"Linearithmic order\"\"\"\n    if n &lt;= 1:\n        return 1\n    # Divide into two, the scale of subproblems is reduced by half\n    count = linear_log_recur(n // 2) + linear_log_recur(n // 2)\n    # Current subproblem contains n operations\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Linearithmic order */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Linearithmic order */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Linearithmic order */\nint LinearLogRecur(int n) {\n    if (n &lt;= 1) return 1;\n    int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Linearithmic order */\nfunc linearLogRecur(n int) int {\n    if n &lt;= 1 {\n        return 1\n    }\n    count := linearLogRecur(n/2) + linearLogRecur(n/2)\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Linearithmic order */\nfunc linearLogRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 1\n    }\n    var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\n    for _ in stride(from: 0, to: n, by: 1) {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Linearithmic order */\nfunction linearLogRecur(n) {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Linearithmic order */\nfunction linearLogRecur(n: number): number {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Linearithmic order */\nint linearLogRecur(int n) {\n  if (n &lt;= 1) return 1;\n  int count = linearLogRecur(n ~/ 2) + linearLogRecur(n ~/ 2);\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Linearithmic order */\nfn linear_log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 1;\n    }\n    let mut count = linear_log_recur(n / 2) + linear_log_recur(n / 2);\n    for _ in 0..n {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> time_complexity.c<pre><code>/* Linearithmic order */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Linearithmic order */\nfun linearLogRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 1\n    var count = linearLogRecur(n / 2) + linearLogRecur(n / 2)\n    for (i in 0..&lt;n) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Linearithmic time ###\ndef linear_log_recur(n)\n  return 1 unless n &gt; 1\n\n  count = linear_log_recur(n / 2) + linear_log_recur(n / 2)\n  (0...n).each { count += 1 }\n\n  count\nend\n</code></pre> <p>Figure 2-13 shows how linearithmic order is generated. Each level of the binary tree has a total of \\(n\\) operations, and the tree has \\(\\log_2 n + 1\\) levels, resulting in a time complexity of \\(O(n \\log n)\\).</p> <p></p> <p> Figure 2-13   Time complexity of linearithmic order </p> <p>Mainstream sorting algorithms typically have a time complexity of \\(O(n \\log n)\\), such as quicksort, merge sort, and heap sort.</p>","path":["Chapter 2. Complexity Analysis","2.3   Time Complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#7-factorial-order-on","level":3,"title":"7.   Factorial Order \\(O(n!)\\)","text":"<p>Factorial order corresponds to the mathematical \"permutation\" problem. Given \\(n\\) distinct elements, find all possible permutation schemes; the number of schemes is:</p> \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\] <p>Factorials are typically implemented using recursion. As shown in Figure 2-14 and the following code, the first level splits into \\(n\\) branches, the second level splits into \\(n - 1\\) branches, and so on, until the \\(n\\)-th level when splitting stops:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def factorial_recur(n: int) -&gt; int:\n    \"\"\"Factorial order (recursive implementation)\"\"\"\n    if n == 0:\n        return 1\n    count = 0\n    # Split from 1 into n\n    for _ in range(n):\n        count += factorial_recur(n - 1)\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Factorial order (recursive implementation) */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // Split from 1 into n\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Factorial order (recursive implementation) */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // Split from 1 into n\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Factorial order (recursive implementation) */\nint FactorialRecur(int n) {\n    if (n == 0) return 1;\n    int count = 0;\n    // Split from 1 into n\n    for (int i = 0; i &lt; n; i++) {\n        count += FactorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Factorial order (recursive implementation) */\nfunc factorialRecur(n int) int {\n    if n == 0 {\n        return 1\n    }\n    count := 0\n    // Split from 1 into n\n    for i := 0; i &lt; n; i++ {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Factorial order (recursive implementation) */\nfunc factorialRecur(n: Int) -&gt; Int {\n    if n == 0 {\n        return 1\n    }\n    var count = 0\n    // Split from 1 into n\n    for _ in 0 ..&lt; n {\n        count += factorialRecur(n: n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Factorial order (recursive implementation) */\nfunction factorialRecur(n) {\n    if (n === 0) return 1;\n    let count = 0;\n    // Split from 1 into n\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Factorial order (recursive implementation) */\nfunction factorialRecur(n: number): number {\n    if (n === 0) return 1;\n    let count = 0;\n    // Split from 1 into n\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Factorial order (recursive implementation) */\nint factorialRecur(int n) {\n  if (n == 0) return 1;\n  int count = 0;\n  // Split from 1 into n\n  for (var i = 0; i &lt; n; i++) {\n    count += factorialRecur(n - 1);\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Factorial order (recursive implementation) */\nfn factorial_recur(n: i32) -&gt; i32 {\n    if n == 0 {\n        return 1;\n    }\n    let mut count = 0;\n    // Split from 1 into n\n    for _ in 0..n {\n        count += factorial_recur(n - 1);\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Factorial order (recursive implementation) */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Factorial order (recursive implementation) */\nfun factorialRecur(n: Int): Int {\n    if (n == 0)\n        return 1\n    var count = 0\n    // Split from 1 into n\n    for (i in 0..&lt;n) {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Factorial time (recursive) ###\ndef factorial_recur(n)\n  return 1 if n == 0\n\n  count = 0\n  # Split from 1 into n\n  (0...n).each { count += factorial_recur(n - 1) }\n\n  count\nend\n</code></pre> <p></p> <p> Figure 2-14   Time complexity of factorial order </p> <p>Note that because when \\(n \\geq 4\\) we always have \\(n! &gt; 2^n\\), factorial order grows faster than exponential order, and is also unacceptable for large \\(n\\).</p>","path":["Chapter 2. Complexity Analysis","2.3   Time Complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#235-worst-best-and-average-time-complexities","level":2,"title":"2.3.5   Worst, Best, and Average Time Complexities","text":"<p>The time efficiency of an algorithm is often not fixed, but is related to the distribution of the input data. Suppose we input an array <code>nums</code> of length \\(n\\), where <code>nums</code> consists of numbers from \\(1\\) to \\(n\\), with each number appearing only once, but the element order is randomly shuffled. The task is to return the index of element \\(1\\). We can draw the following conclusions.</p> <ul> <li>When <code>nums = [?, ?, ..., 1]</code>, i.e., when the last element is \\(1\\), it requires a complete traversal of the array, reaching worst-case time complexity \\(O(n)\\).</li> <li>When <code>nums = [1, ?, ?, ...]</code>, i.e., when the first element is \\(1\\), no matter how long the array is, there is no need to continue traversing, reaching best-case time complexity \\(\\Omega(1)\\).</li> </ul> <p>The \"worst-case time complexity\" corresponds to the function's asymptotic upper bound, denoted using big-\\(O\\) notation. Correspondingly, the \"best-case time complexity\" corresponds to the function's asymptotic lower bound, denoted using \\(\\Omega\\) notation:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby worst_best_time_complexity.py<pre><code>def random_numbers(n: int) -&gt; list[int]:\n    \"\"\"Generate an array with elements: 1, 2, ..., n, shuffled in order\"\"\"\n    # Generate array nums =: 1, 2, 3, ..., n\n    nums = [i for i in range(1, n + 1)]\n    # Randomly shuffle array elements\n    random.shuffle(nums)\n    return nums\n\ndef find_one(nums: list[int]) -&gt; int:\n    \"\"\"Find the index of number 1 in array nums\"\"\"\n    for i in range(len(nums)):\n        # When element 1 is at the head of the array, best time complexity O(1) is achieved\n        # When element 1 is at the tail of the array, worst time complexity O(n) is achieved\n        if nums[i] == 1:\n            return i\n    return -1\n</code></pre> worst_best_time_complexity.cpp<pre><code>/* Generate an array with elements { 1, 2, ..., n }, order shuffled */\nvector&lt;int&gt; randomNumbers(int n) {\n    vector&lt;int&gt; nums(n);\n    // Generate array nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // Use system time to generate random seed\n    unsigned seed = chrono::system_clock::now().time_since_epoch().count();\n    // Randomly shuffle array elements\n    shuffle(nums.begin(), nums.end(), default_random_engine(seed));\n    return nums;\n}\n\n/* Find the index of number 1 in array nums */\nint findOne(vector&lt;int&gt; &amp;nums) {\n    for (int i = 0; i &lt; nums.size(); i++) {\n        // When element 1 is at the head of the array, best time complexity O(1) is achieved\n        // When element 1 is at the tail of the array, worst time complexity O(n) is achieved\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.java<pre><code>/* Generate an array with elements { 1, 2, ..., n }, order shuffled */\nint[] randomNumbers(int n) {\n    Integer[] nums = new Integer[n];\n    // Generate array nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // Randomly shuffle array elements\n    Collections.shuffle(Arrays.asList(nums));\n    // Integer[] -&gt; int[]\n    int[] res = new int[n];\n    for (int i = 0; i &lt; n; i++) {\n        res[i] = nums[i];\n    }\n    return res;\n}\n\n/* Find the index of number 1 in array nums */\nint findOne(int[] nums) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        // When element 1 is at the head of the array, best time complexity O(1) is achieved\n        // When element 1 is at the tail of the array, worst time complexity O(n) is achieved\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.cs<pre><code>/* Generate an array with elements { 1, 2, ..., n }, order shuffled */\nint[] RandomNumbers(int n) {\n    int[] nums = new int[n];\n    // Generate array nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n\n    // Randomly shuffle array elements\n    for (int i = 0; i &lt; nums.Length; i++) {\n        int index = new Random().Next(i, nums.Length);\n        (nums[i], nums[index]) = (nums[index], nums[i]);\n    }\n    return nums;\n}\n\n/* Find the index of number 1 in array nums */\nint FindOne(int[] nums) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        // When element 1 is at the head of the array, best time complexity O(1) is achieved\n        // When element 1 is at the tail of the array, worst time complexity O(n) is achieved\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.go<pre><code>/* Generate an array with elements { 1, 2, ..., n }, order shuffled */\nfunc randomNumbers(n int) []int {\n    nums := make([]int, n)\n    // Generate array nums = { 1, 2, 3, ..., n }\n    for i := 0; i &lt; n; i++ {\n        nums[i] = i + 1\n    }\n    // Randomly shuffle array elements\n    rand.Shuffle(len(nums), func(i, j int) {\n        nums[i], nums[j] = nums[j], nums[i]\n    })\n    return nums\n}\n\n/* Find the index of number 1 in array nums */\nfunc findOne(nums []int) int {\n    for i := 0; i &lt; len(nums); i++ {\n        // When element 1 is at the head of the array, best time complexity O(1) is achieved\n        // When element 1 is at the tail of the array, worst time complexity O(n) is achieved\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.swift<pre><code>/* Generate an array with elements { 1, 2, ..., n }, order shuffled */\nfunc randomNumbers(n: Int) -&gt; [Int] {\n    // Generate array nums = { 1, 2, 3, ..., n }\n    var nums = Array(1 ... n)\n    // Randomly shuffle array elements\n    nums.shuffle()\n    return nums\n}\n\n/* Find the index of number 1 in array nums */\nfunc findOne(nums: [Int]) -&gt; Int {\n    for i in nums.indices {\n        // When element 1 is at the head of the array, best time complexity O(1) is achieved\n        // When element 1 is at the tail of the array, worst time complexity O(n) is achieved\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.js<pre><code>/* Generate an array with elements { 1, 2, ..., n }, order shuffled */\nfunction randomNumbers(n) {\n    const nums = Array(n);\n    // Generate array nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // Randomly shuffle array elements\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* Find the index of number 1 in array nums */\nfunction findOne(nums) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // When element 1 is at the head of the array, best time complexity O(1) is achieved\n        // When element 1 is at the tail of the array, worst time complexity O(n) is achieved\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.ts<pre><code>/* Generate an array with elements { 1, 2, ..., n }, order shuffled */\nfunction randomNumbers(n: number): number[] {\n    const nums = Array(n);\n    // Generate array nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // Randomly shuffle array elements\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* Find the index of number 1 in array nums */\nfunction findOne(nums: number[]): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // When element 1 is at the head of the array, best time complexity O(1) is achieved\n        // When element 1 is at the tail of the array, worst time complexity O(n) is achieved\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.dart<pre><code>/* Generate an array with elements { 1, 2, ..., n }, order shuffled */\nList&lt;int&gt; randomNumbers(int n) {\n  final nums = List.filled(n, 0);\n  // Generate array nums = { 1, 2, 3, ..., n }\n  for (var i = 0; i &lt; n; i++) {\n    nums[i] = i + 1;\n  }\n  // Randomly shuffle array elements\n  nums.shuffle();\n\n  return nums;\n}\n\n/* Find the index of number 1 in array nums */\nint findOne(List&lt;int&gt; nums) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    // When element 1 is at the head of the array, best time complexity O(1) is achieved\n    // When element 1 is at the tail of the array, worst time complexity O(n) is achieved\n    if (nums[i] == 1) return i;\n  }\n\n  return -1;\n}\n</code></pre> worst_best_time_complexity.rs<pre><code>/* Generate an array with elements { 1, 2, ..., n }, order shuffled */\nfn random_numbers(n: i32) -&gt; Vec&lt;i32&gt; {\n    // Generate array nums = { 1, 2, 3, ..., n }\n    let mut nums = (1..=n).collect::&lt;Vec&lt;i32&gt;&gt;();\n    // Randomly shuffle array elements\n    nums.shuffle(&amp;mut thread_rng());\n    nums\n}\n\n/* Find the index of number 1 in array nums */\nfn find_one(nums: &amp;[i32]) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        // When element 1 is at the head of the array, best time complexity O(1) is achieved\n        // When element 1 is at the tail of the array, worst time complexity O(n) is achieved\n        if nums[i] == 1 {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> worst_best_time_complexity.c<pre><code>/* Generate an array with elements { 1, 2, ..., n }, order shuffled */\nint *randomNumbers(int n) {\n    // Allocate heap memory (create 1D variable-length array: n elements of type int)\n    int *nums = (int *)malloc(n * sizeof(int));\n    // Generate array nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // Randomly shuffle array elements\n    for (int i = n - 1; i &gt; 0; i--) {\n        int j = rand() % (i + 1);\n        int temp = nums[i];\n        nums[i] = nums[j];\n        nums[j] = temp;\n    }\n    return nums;\n}\n\n/* Find the index of number 1 in array nums */\nint findOne(int *nums, int n) {\n    for (int i = 0; i &lt; n; i++) {\n        // When element 1 is at the head of the array, best time complexity O(1) is achieved\n        // When element 1 is at the tail of the array, worst time complexity O(n) is achieved\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.kt<pre><code>/* Generate an array with elements { 1, 2, ..., n }, order shuffled */\nfun randomNumbers(n: Int): Array&lt;Int?&gt; {\n    val nums = IntArray(n)\n    // Generate array nums = { 1, 2, 3, ..., n }\n    for (i in 0..&lt;n) {\n        nums[i] = i + 1\n    }\n    // Randomly shuffle array elements\n    nums.shuffle()\n    val res = arrayOfNulls&lt;Int&gt;(n)\n    for (i in 0..&lt;n) {\n        res[i] = nums[i]\n    }\n    return res\n}\n\n/* Find the index of number 1 in array nums */\nfun findOne(nums: Array&lt;Int?&gt;): Int {\n    for (i in nums.indices) {\n        // When element 1 is at the head of the array, best time complexity O(1) is achieved\n        // When element 1 is at the tail of the array, worst time complexity O(n) is achieved\n        if (nums[i] == 1)\n            return i\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.rb<pre><code>### Generate array with elements: 1, 2, ..., n, shuffled ###\ndef random_numbers(n)\n  # Generate array nums =: 1, 2, 3, ..., n\n  nums = Array.new(n) { |i| i + 1 }\n  # Randomly shuffle array elements\n  nums.shuffle!\nend\n\n### Find index of number 1 in array nums ###\ndef find_one(nums)\n  for i in 0...nums.length\n    # When element 1 is at the head of the array, best time complexity O(1) is achieved\n    # When element 1 is at the tail of the array, worst time complexity O(n) is achieved\n    return i if nums[i] == 1\n  end\n\n  -1\nend\n</code></pre> <p>It is worth noting that we rarely use best-case time complexity in practice, because it can usually only be achieved with a very small probability and may be somewhat misleading. The worst-case time complexity is more practical because it gives a safety value for efficiency, allowing us to use the algorithm with confidence.</p> <p>From the above example, we can see that both worst-case and best-case time complexities arise only under particular input distributions, which may occur with very low probability and may not truly reflect the algorithm's running efficiency. In contrast, average time complexity can reflect the algorithm's running efficiency under random input data, denoted using the \\(\\Theta\\) notation.</p> <p>For some algorithms, we can simply derive the average case under random data distribution. For example, in the above example, since the input array is shuffled, the probability of element \\(1\\) appearing at any index is equal, so the algorithm's average number of loops is half the array length \\(n / 2\\), giving an average time complexity of \\(\\Theta(n / 2) = \\Theta(n)\\).</p> <p>But for more complex algorithms, calculating average time complexity is often quite difficult, because it is hard to analyze the overall mathematical expectation under data distribution. In this case, we usually use worst-case time complexity as the criterion for judging algorithm efficiency.</p> <p>Why is the \\(\\Theta\\) symbol rarely seen?</p> <p>This may be because the \\(O\\) symbol is too catchy, so we often use it to represent average time complexity. But strictly speaking, this practice is not standard. In this book and other materials, if you encounter expressions like \"average time complexity \\(O(n)\\)\", please understand it directly as \\(\\Theta(n)\\).</p>","path":["Chapter 2. Complexity Analysis","2.3   Time Complexity"],"tags":[]},{"location":"chapter_data_structure/","level":1,"title":"Chapter 3.   Data Structures","text":"<p>Abstract</p> <p>Data structures are like a sturdy and diverse framework.</p> <p>It provides a blueprint for the orderly organization of data, upon which algorithms come to life.</p>","path":["Chapter 3. Data Structures","Chapter 3.   Data Structures"],"tags":[]},{"location":"chapter_data_structure/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>3.1   Classification of Data Structures</li> <li>3.2   Basic Data Types</li> <li>3.3   Number Encoding *</li> <li>3.4   Character Encoding *</li> <li>3.5   Summary</li> <li>3.6   Exercises</li> </ul>","path":["Chapter 3. Data Structures","Chapter 3.   Data Structures"],"tags":[]},{"location":"chapter_data_structure/basic_data_types/","level":1,"title":"3.2   Basic Data Types","text":"<p>When we talk about data stored in computers, we think of various forms such as text, images, videos, audio, 3D models, and more. Although these kinds of data are organized in different ways, they are all composed of various basic data types.</p> <p>Basic data types are types that the CPU can directly operate on, and they are directly used in algorithms, mainly including the following.</p> <ul> <li>Integer types <code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>.</li> <li>Floating-point types <code>float</code>, <code>double</code>, used to represent decimal numbers.</li> <li>Character type <code>char</code>, used to represent letters, punctuation marks, and even emojis in various languages.</li> <li>Boolean type <code>bool</code>, used to represent \"yes\" and \"no\" judgments.</li> </ul> <p>Basic data types are stored in binary form in computers. A binary digit is one bit. In most modern operating systems, \\(1\\) byte consists of \\(8\\) bits.</p> <p>The range of values for basic data types depends on the size of the space they occupy. Below is an example using Java.</p> <ul> <li>Integer type <code>byte</code> occupies \\(1\\) byte = \\(8\\) bits, and can represent \\(2^{8}\\) numbers.</li> <li>Integer type <code>int</code> occupies \\(4\\) bytes = \\(32\\) bits, and can represent \\(2^{32}\\) numbers.</li> </ul> <p>The following table lists the space occupied, value ranges, and default values of various basic data types in Java. You don't need to memorize this table; a general understanding is sufficient, and you can refer to it when needed.</p> <p> Table 3-1   Space occupied and value ranges of basic data types </p> Type Symbol Space Occupied Minimum Value Maximum Value Default Value Integer <code>byte</code> 1 byte \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) \\(0\\) <code>short</code> 2 bytes \\(-2^{15}\\) \\(2^{15} - 1\\) \\(0\\) <code>int</code> 4 bytes \\(-2^{31}\\) \\(2^{31} - 1\\) \\(0\\) <code>long</code> 8 bytes \\(-2^{63}\\) \\(2^{63} - 1\\) \\(0\\) Float <code>float</code> 4 bytes \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0\\text{f}\\) <code>double</code> 8 bytes \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) \\(0.0\\) Character <code>char</code> 2 bytes \\(0\\) \\(2^{16} - 1\\) \\(0\\) Boolean <code>bool</code> 1 byte \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\) <p>Please note that Table 3-1 applies specifically to Java's basic data types. Each programming language has its own type definitions, and their space usage, value ranges, and default values may vary.</p> <ul> <li>In Python, the integer type <code>int</code> can be of any size, limited only by available memory; the floating-point type <code>float</code> is double-precision 64-bit; there is no <code>char</code> type, a single character is actually a string <code>str</code> of length 1.</li> <li>C and C++ do not explicitly specify the size of basic data types, which varies by implementation and platform. The above table follows the LP64 data model, which is used in Unix 64-bit operating systems including Linux and macOS.</li> <li>The size of character <code>char</code> is 1 byte in C and C++, and in most programming languages it depends on the specific character encoding method, as detailed in the \"Character Encoding\" section.</li> <li>Even though representing a boolean value requires only 1 bit (\\(0\\) or \\(1\\)), it is usually stored as 1 byte in memory. This is because modern computer CPUs typically use 1 byte as the minimum addressable memory unit.</li> </ul> <p>So, what is the relationship between basic data types and data structures? We know that data structures are ways of organizing and storing data in computers. Here, the emphasis is on the \"structure\", not the \"data\".</p> <p>If we want to represent \"a row of numbers\", we naturally think of using an array. This is because the linear structure of an array can represent the adjacency and order relationships of numbers, but whether the stored content is integer <code>int</code>, floating-point <code>float</code>, or character <code>char</code> is unrelated to the \"data structure\".</p> <p>In other words, basic data types provide the \"content type\" of data, while data structures provide the \"organization method\" of data. For example, in the following code, we use the same data structure (array) to store and represent different basic data types, including <code>int</code>, <code>float</code>, <code>char</code>, <code>bool</code>, etc.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># Initialize arrays using various basic data types\nnumbers: list[int] = [0] * 5\ndecimals: list[float] = [0.0] * 5\n# In Python, characters are actually strings of length 1\ncharacters: list[str] = ['0'] * 5\nbools: list[bool] = [False] * 5\n# Python lists can freely store various basic data types and object references\ndata = [0, 0.0, 'a', False, ListNode(0)]\n</code></pre> <pre><code>// Initialize arrays using various basic data types\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool bools[5];\n</code></pre> <pre><code>// Initialize arrays using various basic data types\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] bools = new boolean[5];\n</code></pre> <pre><code>// Initialize arrays using various basic data types\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] bools = new bool[5];\n</code></pre> <pre><code>// Initialize arrays using various basic data types\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar bools = [5]bool{}\n</code></pre> <pre><code>// Initialize arrays using various basic data types\nlet numbers = Array(repeating: 0, count: 5)\nlet decimals = Array(repeating: 0.0, count: 5)\nlet characters: [Character] = Array(repeating: \"a\", count: 5)\nlet bools = Array(repeating: false, count: 5)\n</code></pre> <pre><code>// JavaScript arrays can freely store various basic data types and objects\nconst array = [0, 0.0, 'a', false];\n</code></pre> <pre><code>// Initialize arrays using various basic data types\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst bools: boolean[] = [];\n</code></pre> <pre><code>// Initialize arrays using various basic data types\nList&lt;int&gt; numbers = List.filled(5, 0);\nList&lt;double&gt; decimals = List.filled(5, 0.0);\nList&lt;String&gt; characters = List.filled(5, 'a');\nList&lt;bool&gt; bools = List.filled(5, false);\n</code></pre> <pre><code>// Initialize arrays using various basic data types\nlet numbers: Vec&lt;i32&gt; = vec![0; 5];\nlet decimals: Vec&lt;f32&gt; = vec![0.0; 5];\nlet characters: Vec&lt;char&gt; = vec!['0'; 5];\nlet bools: Vec&lt;bool&gt; = vec![false; 5];\n</code></pre> <pre><code>// Initialize arrays using various basic data types\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool bools[10];\n</code></pre> <pre><code>// Initialize arrays using various basic data types\nval numbers = IntArray(5)\nval decinals = FloatArray(5)\nval characters = CharArray(5)\nval bools = BooleanArray(5)\n</code></pre> <pre><code># Ruby lists can freely store various basic data types and object references\ndata = [0, 0.0, 'a', false, ListNode(0)]\n</code></pre> Visualized Execution <p> Full Screen &gt;</p>","path":["Chapter 3. Data Structures","3.2   Basic Data Types"],"tags":[]},{"location":"chapter_data_structure/character_encoding/","level":1,"title":"3.4   Character Encoding *","text":"<p>In computers, all data is stored in binary form, and character <code>char</code> is no exception. To represent characters, we need to establish a \"character set\" that defines a one-to-one correspondence between each character and binary numbers. With a character set, computers can convert binary numbers to characters by looking up the table.</p>","path":["Chapter 3. Data Structures","3.4   Character Encoding *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#341-ascii-character-set","level":2,"title":"3.4.1   ASCII Character Set","text":"<p>ASCII code is the earliest character set, with the full name American Standard Code for Information Interchange. It uses 7 binary bits (the lower 7 bits of one byte) to represent a character, and can represent a maximum of 128 different characters. As shown in Figure 3-6, ASCII code includes uppercase and lowercase English letters, numbers 0 ~ 9, some punctuation marks, and some control characters (such as newline and tab).</p> <p></p> <p> Figure 3-6   ASCII code </p> <p>However, ASCII code can only represent English. With the globalization of computers, a character set called EASCII that can represent more languages emerged. It expands from the 7-bit basis of ASCII to 8 bits, and can represent 256 different characters.</p> <p>Worldwide, a batch of EASCII character sets suitable for different regions have appeared successively. The first 128 characters of these character sets are unified as ASCII code, and the last 128 characters are defined differently to adapt to the needs of different languages.</p>","path":["Chapter 3. Data Structures","3.4   Character Encoding *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#342-gbk-character-set","level":2,"title":"3.4.2   GBK Character Set","text":"<p>Later, people found that EASCII still could not provide enough characters for many languages. For example, there are nearly one hundred thousand Chinese characters, and several thousand are used in everyday life. In 1980, the China National Standardization Administration released the GB2312 character set, which included 6,763 Chinese characters, basically meeting the needs of computer processing for Chinese.</p> <p>However, GB2312 cannot handle some rare characters and traditional Chinese characters. The GBK character set is an extension based on GB2312, which includes a total of 21,886 Chinese characters. In the GBK encoding scheme, ASCII characters are represented using one byte, and Chinese characters are represented using two bytes.</p>","path":["Chapter 3. Data Structures","3.4   Character Encoding *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#343-unicode-character-set","level":2,"title":"3.4.3   Unicode Character Set","text":"<p>With the vigorous development of computer technology, character sets and encoding standards flourished, which brought many problems. On the one hand, these character sets generally only define characters for specific languages and cannot work normally in multilingual environments. On the other hand, multiple character set standards exist for the same language, and if two computers use different encoding standards, garbled characters will appear during information transmission.</p> <p>Researchers of that era thought: If a sufficiently complete character set were released to include all languages and symbols in the world, wouldn't that solve problems in cross-language environments and eliminate garbled text? Driven by this idea, a large and comprehensive character set, Unicode, was born.</p> <p>Unicode, or Unified Code, can theoretically accommodate over one million characters. It is committed to including characters from around the world into a unified character set, providing a universal character set to handle and display various language texts, reducing garbled character problems caused by different encoding standards. Since its release in 1991, Unicode has continuously expanded to include new languages and characters. As of September 2022, Unicode has included 149,186 characters, including characters, symbols, and even emojis from various languages.</p> <p>As a universal character set, Unicode essentially assigns each character a unique \"code point\" (character identifier), whose range is U+0000 to U+10FFFF, forming a unified character numbering space. However, Unicode does not specify how to store these character code points in computers. We can't help but ask: when Unicode code points of multiple lengths appear simultaneously in a text, how does the system parse the characters? For example, given an encoding with a length of 2 bytes, how does the system determine whether it is one 2-byte character or two 1-byte characters?</p> <p>For the above problem, a straightforward solution is to store all characters as equal-length encodings. As shown in Figure 3-7, each character in \"Hello\" occupies 1 byte, and each character in \"算法\" (algorithm) occupies 2 bytes. We can encode all characters in \"Hello 算法\" as 2 bytes in length by padding the high bits with 0. In this way, the system can parse one character every 2 bytes and restore the content of this phrase.</p> <p></p> <p> Figure 3-7   Unicode encoding example </p> <p>However, ASCII code has already proven to us that encoding English only requires 1 byte. If the above scheme is adopted, the size of English text will be twice that under ASCII encoding, which is very wasteful of memory space. Therefore, we need a more efficient Unicode encoding method.</p>","path":["Chapter 3. Data Structures","3.4   Character Encoding *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#344-utf-8-encoding","level":2,"title":"3.4.4   UTF-8 Encoding","text":"<p>Currently, UTF-8 has become the most widely used Unicode encoding method internationally. It is a variable-length encoding that uses 1 to 4 bytes to represent a character, depending on the complexity of the character. ASCII characters only require 1 byte, Latin and Greek letters require 2 bytes, commonly used Chinese characters require 3 bytes, and some other rare characters require 4 bytes.</p> <p>The encoding rules of UTF-8 are not complicated and can be divided into the following two cases.</p> <ul> <li>For 1-byte characters, set the highest bit to \\(0\\), and set the remaining 7 bits to the Unicode code point. It is worth noting that ASCII characters occupy the first 128 code points in the Unicode character set. That is to say, UTF-8 encoding is backward compatible with ASCII code. This means we can use UTF-8 to parse very old ASCII code text.</li> <li>For characters with a length of \\(n\\) bytes (where \\(n &gt; 1\\)), set the highest \\(n\\) bits of the first byte to \\(1\\), and set the \\((n + 1)\\)-th bit to \\(0\\); starting from the second byte, set the highest 2 bits of each byte to \\(10\\); use all remaining bits to fill in the Unicode code point of the character.</li> </ul> <p>Figure 3-8 shows the UTF-8 encoding corresponding to \"Hello 算法\". It can be observed that since the highest \\(n\\) bits are all set to \\(1\\), the system can determine that the character length is \\(n\\) by counting the leading \\(1\\) bits.</p> <p>But why set the highest 2 bits of all other bytes to \\(10\\)? In fact, this \\(10\\) can serve as a check symbol. Assuming the system starts parsing text from an incorrect byte, the \\(10\\) at the beginning of the byte can help the system quickly determine an anomaly.</p> <p>The reason for using \\(10\\) as a check symbol is that under UTF-8 encoding rules, it is impossible for a character's highest two bits to be \\(10\\). This conclusion can be proven by contradiction: assuming the highest two bits of a character are \\(10\\), it means the length of the character is \\(1\\), corresponding to ASCII code. However, the highest bit of ASCII code should be \\(0\\), which contradicts the assumption.</p> <p></p> <p> Figure 3-8   UTF-8 encoding example </p> <p>In addition to UTF-8, common encoding methods also include the following two.</p> <ul> <li>UTF-16 encoding: Uses 2 or 4 bytes to represent a character. All ASCII characters and commonly used non-English characters are represented with 2 bytes; a few characters need to use 4 bytes. For 2-byte characters, UTF-16 encoding is equal to the Unicode code point.</li> <li>UTF-32 encoding: Every character uses 4 bytes. This means that UTF-32 takes up more space than UTF-8 and UTF-16, especially for text with a high proportion of ASCII characters.</li> </ul> <p>From the perspective of storage space occupation, using UTF-8 to represent English characters is very efficient because it only requires 1 byte; using UTF-16 encoding for some non-English characters (such as Chinese) will be more efficient because it only requires 2 bytes, while UTF-8 may require 3 bytes.</p> <p>From a compatibility perspective, UTF-8 has the best universality, and many tools and libraries support UTF-8 first.</p>","path":["Chapter 3. Data Structures","3.4   Character Encoding *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#345-character-encoding-in-programming-languages","level":2,"title":"3.4.5   Character Encoding in Programming Languages","text":"<p>For many programming languages in the past, strings during program execution used internal encodings such as UTF-16 or UTF-32. Under these representations, we can often treat strings like arrays during processing, and this approach has the following advantages.</p> <ul> <li>Random access: UTF-16 encoded strings can be easily accessed randomly. UTF-8 is a variable-length encoding. To find the \\(i\\)-th character, we need to traverse from the beginning of the string to the \\(i\\)-th character, which requires \\(O(n)\\) time.</li> <li>Character counting: Similar to random access, calculating the length of a UTF-16 encoded string is also an \\(O(1)\\) operation. However, calculating the length of a UTF-8 encoded string requires traversing the entire string.</li> <li>String operations: Many string operations (such as splitting, joining, inserting, deleting, etc.) on UTF-16 encoded strings are easier to perform. Performing these operations on UTF-8 encoded strings usually requires additional calculations to ensure that invalid UTF-8 encoding is not generated.</li> </ul> <p>In fact, the design of character encoding schemes for programming languages is a very interesting topic involving many factors.</p> <ul> <li>Java's <code>String</code> type uses UTF-16 encoding, with each character occupying 2 bytes. This is because at the beginning of Java language design, people believed that 16 bits were sufficient to represent all possible characters. However, this was an incorrect judgment. Later, the Unicode specification expanded beyond 16 bits, so characters in Java may now be represented by a pair of 16-bit values (called \"surrogate pairs\").</li> <li>The strings of JavaScript and TypeScript use UTF-16 encoding for reasons similar to Java. When Netscape first introduced the JavaScript language in 1995, Unicode was still in its early stages of development, and at that time, using 16-bit encoding was sufficient to represent all Unicode characters.</li> <li>C# uses UTF-16 encoding mainly because the .NET platform was designed by Microsoft, and many of Microsoft's technologies (including the Windows operating system) extensively use UTF-16 encoding.</li> </ul> <p>Due to the underestimation of character quantities by the above programming languages, they had to adopt the \"surrogate pair\" method to represent Unicode characters with lengths exceeding 16 bits. This is a reluctant compromise. On the one hand, in strings containing surrogate pairs, one character may occupy 2 bytes or 4 bytes, thus losing the advantage of fixed-length encoding. On the other hand, handling surrogate pairs requires additional code, which increases the complexity and difficulty of debugging in programming.</p> <p>For the above reasons, some programming languages have proposed different encoding schemes.</p> <ul> <li>Python's <code>str</code> uses Unicode encoding and adopts a flexible string representation where the stored character length depends on the largest Unicode code point in the string. If all characters in the string are ASCII characters, each character occupies 1 byte; if there are characters exceeding the ASCII range but all within the Basic Multilingual Plane (BMP), each character occupies 2 bytes; if there are characters exceeding the BMP, each character occupies 4 bytes.</li> <li>Go language's <code>string</code> type uses UTF-8 encoding internally. Go language also provides the <code>rune</code> type, which is used to represent a single Unicode code point.</li> <li>Rust language's <code>str</code> and <code>String</code> types use UTF-8 encoding internally. Rust also provides the <code>char</code> type for representing a single Unicode code point.</li> </ul> <p>It should be noted that the above discussion is about how strings are stored in programming languages, which is different from how strings are stored in files or transmitted over networks. In file storage or network transmission, we usually encode strings into UTF-8 format to achieve optimal compatibility and space efficiency.</p>","path":["Chapter 3. Data Structures","3.4   Character Encoding *"],"tags":[]},{"location":"chapter_data_structure/classification_of_data_structure/","level":1,"title":"3.1   Classification of Data Structures","text":"<p>Common data structures include arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graphs. They can be classified from two dimensions: \"logical structure\" and \"physical structure\".</p>","path":["Chapter 3. Data Structures","3.1   Classification of Data Structures"],"tags":[]},{"location":"chapter_data_structure/classification_of_data_structure/#311-logical-structure-linear-and-non-linear","level":2,"title":"3.1.1   Logical Structure: Linear and Non-Linear","text":"<p>Logical structure reveals the logical relationships between data elements. In arrays and linked lists, data is arranged in a certain order, embodying linear relationships between elements; while in trees, data is arranged hierarchically from top to bottom, showing parent-descendant relationships; graphs are composed of nodes and edges, reflecting complex network relationships.</p> <p>As shown in Figure 3-1, logical structures can be divided into two major categories: \"linear\" and \"non-linear\". Linear structures are more intuitive, indicating that data is linearly arranged in logical relationships; non-linear structures are the opposite, arranged non-linearly.</p> <ul> <li>Linear data structures: Arrays, linked lists, stacks, queues, hash tables, where elements have a one-to-one sequential relationship.</li> <li>Non-linear data structures: Trees, heaps, graphs, hash tables.</li> </ul> <p>Non-linear data structures can be further divided into tree structures and network structures.</p> <ul> <li>Tree structures: Trees, heaps, hash tables, where elements have a one-to-many relationship.</li> <li>Network structures: Graphs, where elements have a many-to-many relationship.</li> </ul> <p></p> <p> Figure 3-1   Linear and non-linear data structures </p>","path":["Chapter 3. Data Structures","3.1   Classification of Data Structures"],"tags":[]},{"location":"chapter_data_structure/classification_of_data_structure/#312-physical-structure-contiguous-and-dispersed","level":2,"title":"3.1.2   Physical Structure: Contiguous and Dispersed","text":"<p>When an algorithm program runs, the data being processed is mainly stored in memory. Figure 3-2 shows a computer memory stick, where each black square contains a memory space. We can imagine memory as a huge Excel spreadsheet, where each cell can store a certain amount of data.</p> <p>The system accesses data at the target location through memory addresses. As shown in Figure 3-2, the computer assigns a number to each cell in the spreadsheet according to specific rules, ensuring that each memory space has a unique memory address. With these addresses, the program can access data in memory.</p> <p></p> <p> Figure 3-2   Memory stick, memory space, memory address </p> <p>Tip</p> <p>It should be noted that comparing memory to an Excel spreadsheet is only a simplified analogy. The actual workings of memory are much more complex, involving concepts such as address space, memory management, cache mechanisms, virtual memory, and physical memory.</p> <p>Memory is a shared resource for all programs. When a block of memory is occupied by a program, it usually cannot be used by other programs at the same time. Therefore, in the design of data structures and algorithms, memory resources are an important consideration. For example, the peak memory occupied by an algorithm should not exceed the remaining free memory of the system; if there is a lack of contiguous large memory blocks, then the data structure chosen must be able to be stored in dispersed memory spaces.</p> <p>As shown in Figure 3-3, physical structure reflects the way data is stored in computer memory. It can be divided into contiguous-space storage (arrays) and dispersed-space storage (linked lists). At a low level, physical structure determines how data is accessed, updated, inserted, and deleted. These two physical structures exhibit complementary characteristics in terms of time efficiency and space efficiency.</p> <p></p> <p> Figure 3-3   Contiguous space storage and dispersed space storage </p> <p>It is worth noting that all data structures are implemented based on arrays, linked lists, or a combination of both. For example, stacks and queues can be implemented using either arrays or linked lists; while the implementation of hash tables may include both arrays and linked lists.</p> <ul> <li>Can be implemented based on arrays: Stacks, queues, hash tables, trees, heaps, graphs, matrices, tensors (arrays with dimensions \\(\\geq 3\\)), etc.</li> <li>Can be implemented based on linked lists: Stacks, queues, hash tables, trees, heaps, graphs, etc.</li> </ul> <p>After initialization, linked lists can still adjust their length during program execution, so they are also called \"dynamic data structures\". After initialization, the length of arrays cannot be changed, so they are also called \"static data structures\". It is worth noting that arrays can change length by reallocating memory, thus retaining a limited degree of flexibility.</p> <p>Tip</p> <p>If you find it difficult to understand physical structure, it is recommended to read the next chapter first, and then review this section.</p>","path":["Chapter 3. Data Structures","3.1   Classification of Data Structures"],"tags":[]},{"location":"chapter_data_structure/exercises/","level":1,"title":"3.6   Exercises","text":"","path":["Chapter 3. Data Structures","3.6   Exercises"],"tags":[]},{"location":"chapter_data_structure/exercises/#361-concept-review","level":2,"title":"3.6.1   Concept Review","text":"","path":["Chapter 3. Data Structures","3.6   Exercises"],"tags":[]},{"location":"chapter_data_structure/exercises/#1-data-relationships-in-everyday-situations","level":3,"title":"1.   Data Relationships in Everyday Situations","text":"<p>Based on the relationships among the data, choose one of \"linear structure,\" \"tree structure,\" and \"network structure\" for each situation below, and explain why:</p> <ol> <li>Students stand in a line; we consider only each student's immediate neighbors in front of and behind them.</li> <li>A school is organized in levels: \"school → grade → class.\"</li> <li>City roads connect many intersections. One intersection may lead to several others, and the roads may form cycles.</li> </ol> Answer <ol> <li> <p>This is a linear structure. Except for the first and last students, each person is adjacent to exactly one person in front and one behind, so the relationships extend along a single line.</p> </li> <li> <p>This is a tree structure. Each class belongs to one grade, and each grade belongs to the school, so the relationships form levels from top to bottom.</p> </li> <li> <p>This is a network structure. An intersection can connect to several other intersections, and the routes can form cycles, so they cannot be arranged in a single order or a strict hierarchy.</p> <p>To identify a structure, first examine the relationships among the elements rather than how much space it uses in memory.</p> </li> </ol>","path":["Chapter 3. Data Structures","3.6   Exercises"],"tags":[]},{"location":"chapter_data_structure/exercises/#2-storing-a-logical-order-in-memory","level":3,"title":"2.   Storing a Logical Order in Memory","text":"<p>Consider two simplified memory layouts for storing the logical order <code>A → B → C</code>:</p> <ul> <li>Layout A: <code>A, B, C</code> are stored in memory cells numbered <code>20, 21, 22</code>, respectively.</li> <li>Layout B: <code>A, B, C</code> are stored in memory cells numbered <code>20, 7, 31</code>, respectively. <code>A</code> records the location of <code>B</code>, and <code>B</code> records the location of <code>C</code>.</li> </ul> <ol> <li>Which layout uses contiguous-space storage, and which uses dispersed-space storage?</li> <li>Which layout is more like an array, and which is more like a linked list?</li> <li>The memory-cell numbers in Layout B are not in increasing order. Why can it still represent the logical order <code>A → B → C</code>?</li> </ol> Answer <ol> <li> <p>Layout A uses consecutive memory cells, so it uses contiguous-space storage. The nodes in Layout B are scattered across different locations, so it uses dispersed-space storage.</p> </li> <li> <p>Layout A is more like an array, and Layout B is more like a linked list.</p> </li> <li> <p>The logical order is determined by the links recorded between nodes, not by the numerical order of the memory-cell numbers.     The location stored by <code>A</code> leads to <code>B</code>, and the location stored by <code>B</code> then leads to <code>C</code>, so <code>A, B, C</code> can still be visited in order.</p> <p>This also shows that logical structure and physical structure are two different ways of viewing the same data.</p> </li> </ol>","path":["Chapter 3. Data Structures","3.6   Exercises"],"tags":[]},{"location":"chapter_data_structure/exercises/#3-data-types-and-structures-in-homework-records","level":3,"title":"3.   Data Types and Structures in Homework Records","text":"<p>A study group records, in seating order, whether each of four students submitted their homework:</p> <p><code>[true, false, true, true]</code></p> <ol> <li>Which basic data type is suitable for each element?</li> <li>The four elements are arranged in a line by seating order. What logical structure does this use?</li> <li>Suppose the group later records each student's homework score as <code>[90, 0, 85, 100]</code>. Does this change the data's \"content type\" or its \"organization\"? Explain why.</li> </ol> Answer <ol> <li> <p>Each element represents only \"yes\" or \"no,\" so the Boolean type <code>bool</code> is suitable.</p> </li> <li> <p>The elements are arranged in seating order, forming a linear structure that can be stored in an array.</p> </li> <li> <p>The content type changes: the elements change from Boolean values to integers. The organization does not change.     The data is still arranged in a line by seating order and can still be stored in an array, which is a linear structure.</p> <p>A basic data type describes \"what is stored,\" while a data structure describes \"how the data is organized.\"</p> </li> </ol>","path":["Chapter 3. Data Structures","3.6   Exercises"],"tags":[]},{"location":"chapter_data_structure/exercises/#362-programming-exercises","level":2,"title":"3.6.2   Programming Exercises","text":"","path":["Chapter 3. Data Structures","3.6   Exercises"],"tags":[]},{"location":"chapter_data_structure/exercises/#1-count-the-1s-in-a-binary-representation","level":3,"title":"1.   Count the 1s in a Binary Representation","text":"<p>Given a non-negative integer <code>n</code>, count the number of 1s in its binary representation.</p> <p>Use bitwise operations. Do not convert the binary representation to a string or use a built-in function that directly counts 1s.</p> Hints <ol> <li>n &amp; 1 extracts the rightmost bit of n, which tells you whether that bit is 1</li> <li>Shifting right by one discards the current rightmost bit; most languages use the operator &gt;&gt;</li> <li>After implementing the method that checks and shifts one bit at a time, observe that n &amp; (n - 1) turns the rightmost 1 in n into 0</li> </ol> <p>LeetCode</p>","path":["Chapter 3. Data Structures","3.6   Exercises"],"tags":[]},{"location":"chapter_data_structure/number_encoding/","level":1,"title":"3.3   Number Encoding *","text":"<p>Tip</p> <p>In this book, chapters marked with an asterisk * are optional readings. If you are short on time or find them challenging, you may skip these initially and return to them after completing the essential chapters.</p>","path":["Chapter 3. Data Structures","3.3   Number Encoding *"],"tags":[]},{"location":"chapter_data_structure/number_encoding/#331-sign-magnitude-1s-complement-and-2s-complement","level":2,"title":"3.3.1   Sign-Magnitude, 1's Complement, and 2's Complement","text":"<p>In the table from the previous section, we found that all integer types can represent one more negative number than positive numbers. For example, the <code>byte</code> range is \\([-128, 127]\\). This phenomenon is counterintuitive, and its underlying cause lies in sign-magnitude, 1's complement, and 2's complement representations.</p> <p>First, it should be noted that numbers are stored in computers in the form of \"2's complement\". Before analyzing the reasons for this, let's first define these three concepts.</p> <ul> <li>Sign-magnitude: We treat the highest bit of the binary representation of a number as the sign bit, where \\(0\\) represents a positive number and \\(1\\) represents a negative number, and the remaining bits represent the value of the number.</li> <li>1's complement: The 1's complement of a positive number is the same as its sign-magnitude. For a negative number, the 1's complement is obtained by inverting all bits except the sign bit of its sign-magnitude.</li> <li>2's complement: The 2's complement of a positive number is the same as its sign-magnitude. For a negative number, the 2's complement is obtained by adding \\(1\\) to its 1's complement.</li> </ul> <p>Figure 3-4 shows the conversion methods among sign-magnitude, 1's complement, and 2's complement.</p> <p></p> <p> Figure 3-4   Conversions among sign-magnitude, 1's complement, and 2's complement </p> <p>Sign-magnitude, although the most intuitive, has some limitations. On one hand, the sign-magnitude of negative numbers cannot be directly used in operations. For example, calculating \\(1 + (-2)\\) in sign-magnitude yields \\(-3\\), which is clearly incorrect.</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 + 1000 \\; 0010 \\newline &amp; = 1000 \\; 0011 \\newline &amp; \\rightarrow -3 \\end{aligned} \\] <p>To solve this problem, computers introduced 1's complement. If we first convert sign-magnitude to 1's complement and calculate \\(1 + (-2)\\) in 1's complement, then convert the result back to sign-magnitude, we can obtain the correct result of \\(-1\\).</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 \\; \\text{(Sign-magnitude)} + 1000 \\; 0010 \\; \\text{(Sign-magnitude)} \\newline &amp; = 0000 \\; 0001 \\; \\text{(1's complement)} + 1111  \\; 1101 \\; \\text{(1's complement)} \\newline &amp; = 1111 \\; 1110 \\; \\text{(1's complement)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(Sign-magnitude)} \\newline &amp; \\rightarrow -1 \\end{aligned} \\] <p>On the other hand, the sign-magnitude of the number zero has two representations, \\(+0\\) and \\(-0\\). This means that the number zero corresponds to two different binary encodings, which may cause ambiguity. For example, in conditional judgments, if we don't distinguish between positive zero and negative zero, it may lead to incorrect judgment results. If we want to handle the ambiguity of positive and negative zero, we need to introduce additional judgment operations, which may reduce the computational efficiency of the computer.</p> \\[ \\begin{aligned} +0 &amp; \\rightarrow 0000 \\; 0000 \\newline -0 &amp; \\rightarrow 1000 \\; 0000 \\end{aligned} \\] <p>Like sign-magnitude, 1's complement also has the problem of positive and negative zero ambiguity. Therefore, computers further introduced 2's complement. Let's first observe the conversion process of negative zero from sign-magnitude to 1's complement to 2's complement:</p> \\[ \\begin{aligned} -0 \\rightarrow \\; &amp; 1000 \\; 0000 \\; \\text{(Sign-magnitude)} \\newline = \\; &amp; 1111 \\; 1111 \\; \\text{(1's complement)} \\newline = 1 \\; &amp; 0000 \\; 0000 \\; \\text{(2's complement)} \\newline \\end{aligned} \\] <p>Adding \\(1\\) to the 1's complement of negative zero produces a carry, but since the <code>byte</code> type has a length of only 8 bits, the \\(1\\) that overflows to the 9<sup>th</sup> bit is discarded. That is to say, the 2's complement of negative zero is \\(0000 \\; 0000\\), which is the same as the 2's complement of positive zero. This means that in 2's complement representation, there is only one zero, and the positive and negative zero ambiguity is thus resolved.</p> <p>One last question remains: the range of the <code>byte</code> type is \\([-128, 127]\\), so where does the extra negative number \\(-128\\) come from? We notice that all integers in the interval \\([-127, +127]\\) have corresponding sign-magnitude, 1's complement, and 2's complement, and sign-magnitude and 2's complement can be converted to each other.</p> <p>However, the 2's complement \\(1000 \\; 0000\\) is an exception, and it does not have a corresponding sign-magnitude. According to the conversion method, we get that the sign-magnitude of this 2's complement is \\(0000 \\; 0000\\). This is clearly contradictory because this sign-magnitude represents the number \\(0\\), and its 2's complement should be itself. The computer specifies that this special 2's complement \\(1000 \\; 0000\\) represents \\(-128\\). In fact, the result of calculating \\((-1) + (-127)\\) in 2's complement is \\(-128\\).</p> \\[ \\begin{aligned} &amp; (-127) + (-1) \\newline &amp; \\rightarrow 1111 \\; 1111 \\; \\text{(Sign-magnitude)} + 1000 \\; 0001 \\; \\text{(Sign-magnitude)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(1's complement)} + 1111  \\; 1110 \\; \\text{(1's complement)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(2's complement)} + 1111  \\; 1111 \\; \\text{(2's complement)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(2's complement)} \\newline &amp; \\rightarrow -128 \\end{aligned} \\] <p>You may have noticed that all the above calculations are addition operations. This hints at an important fact: the hardware circuits inside computers are mainly designed based on addition operations. This is because addition operations are simpler to implement in hardware compared to other operations (such as multiplication, division, and subtraction), easier to parallelize, and have faster operation speeds.</p> <p>Please note that this does not mean that computers can only perform addition. By combining addition with some basic logical operations, computers can implement various other mathematical operations. For example, calculating the subtraction \\(a - b\\) can be converted to calculating the addition \\(a + (-b)\\); calculating multiplication and division can be converted to calculating multiple additions or subtractions.</p> <p>We can now summarize why computers use 2's complement: with 2's complement representation, computers can use the same circuits and operations to handle the addition of positive and negative numbers, without designing special hardware circuits for subtraction or separately handling the ambiguity of positive and negative zero. This greatly simplifies hardware design and improves efficiency.</p> <p>The design of 2's complement is very ingenious. Due to space limitations, we will stop here. Interested readers are encouraged to explore further.</p>","path":["Chapter 3. Data Structures","3.3   Number Encoding *"],"tags":[]},{"location":"chapter_data_structure/number_encoding/#332-floating-point-number-encoding","level":2,"title":"3.3.2   Floating-Point Number Encoding","text":"<p>Careful readers may have noticed: <code>int</code> and <code>float</code> have the same length, both are 4 bytes, but why does <code>float</code> have a much larger range than <code>int</code>? This is very counterintuitive because it stands to reason that <code>float</code> needs to represent decimals, so the range should be smaller.</p> <p>In fact, this is because floating-point number <code>float</code> uses a different representation method. Let's denote a 32-bit binary number as:</p> \\[ b_{31} b_{30} b_{29} \\ldots b_2 b_1 b_0 \\] <p>According to the IEEE 754 standard, a 32-bit <code>float</code> consists of the following three parts.</p> <ul> <li>Sign bit \\(\\mathrm{S}\\): occupies 1 bit, corresponding to \\(b_{31}\\).</li> <li>Exponent bit \\(\\mathrm{E}\\): occupies 8 bits, corresponding to \\(b_{30} b_{29} \\ldots b_{23}\\).</li> <li>Fraction bit \\(\\mathrm{N}\\): occupies 23 bits, corresponding to \\(b_{22} b_{21} \\ldots b_0\\).</li> </ul> <p>The calculation method for the value corresponding to the binary <code>float</code> is:</p> \\[ \\text {val} = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\] <p>Converted to decimal, the calculation formula is:</p> \\[ \\text {val}=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\] <p>The range of each component is:</p> \\[ \\begin{aligned} \\mathrm{S} \\in &amp; \\{ 0, 1\\}, \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = &amp; (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\] <p></p> <p> Figure 3-5   Calculation example of float under IEEE 754 standard </p> <p>Observing Figure 3-5, given example data \\(\\mathrm{S} = 0\\), \\(\\mathrm{E} = 124\\), \\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\), we have:</p> \\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\] <p>Now we can answer the initial question: the representation of <code>float</code> includes an exponent bit, resulting in a range far greater than <code>int</code>. According to the above calculation, the maximum positive number that <code>float</code> can represent is \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\), and the minimum negative number can be obtained by switching the sign bit.</p> <p>Although floating-point number <code>float</code> expands the range, its side effect is sacrificing precision. The integer type <code>int</code> uses all 32 bits to represent numbers, and the numbers are evenly distributed; however, due to the existence of the exponent bit, the larger the value of floating-point number <code>float</code>, the larger the difference between two adjacent numbers tends to be.</p> <p>As shown in Table 3-2, exponent bits \\(\\mathrm{E} = 0\\) and \\(\\mathrm{E} = 255\\) have special meanings, used to represent zero, infinity, \\(\\mathrm{NaN}\\), etc.</p> <p> Table 3-2   Meaning of exponent bits </p> Exponent Bit E Fraction Bit \\(\\mathrm{N} = 0\\) Fraction Bit \\(\\mathrm{N} \\ne 0\\) Calculation Formula \\(0\\) \\(\\pm 0\\) Subnormal Number \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) Normal Number Normal Number \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\) <p>It is worth noting that subnormal numbers significantly improve the precision of floating-point numbers. The smallest positive normal number is \\(2^{-126}\\), and the smallest positive subnormal number is \\(2^{-126} \\times 2^{-23}\\).</p> <p>Double-precision <code>double</code> also uses a representation method similar to <code>float</code>, which will not be elaborated here.</p>","path":["Chapter 3. Data Structures","3.3   Number Encoding *"],"tags":[]},{"location":"chapter_data_structure/summary/","level":1,"title":"3.5   Summary","text":"","path":["Chapter 3. Data Structures","3.5   Summary"],"tags":[]},{"location":"chapter_data_structure/summary/#1-key-review","level":3,"title":"1.   Key Review","text":"<ul> <li>Data structures can be classified from two perspectives: logical structure and physical structure. Logical structure describes the logical relationships between data elements, while physical structure describes how data is stored in computer memory.</li> <li>Common logical structures include linear, tree-like, and network structures. We typically classify data structures as linear (arrays, linked lists, stacks, queues) and non-linear (trees, graphs, heaps) based on their logical structure. The implementation of hash tables may involve both linear and non-linear data structures.</li> <li>When a program runs, data is stored in computer memory. Each memory space has a corresponding memory address, and the program accesses data through these memory addresses.</li> <li>Physical structures are primarily divided into contiguous space storage (arrays) and dispersed space storage (linked lists). All data structures are implemented using arrays, linked lists, or a combination of both.</li> <li>Basic data types in computers include integers <code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>, floating-point numbers <code>float</code>, <code>double</code>, characters <code>char</code>, and booleans <code>bool</code>. Their value ranges depend on the size of space they occupy and their representation method.</li> <li>Sign-magnitude, 1's complement, and 2's complement are three methods for encoding numbers in computers, and they can be converted into each other. The most significant bit of sign-magnitude is the sign bit, and the remaining bits represent the value of the number.</li> <li>Integers are stored in computers in 2's complement form. Under 2's complement representation, computers can treat the addition of positive and negative numbers uniformly, without needing to design special hardware circuits for subtraction, and there is no ambiguity of positive and negative zero.</li> <li>The encoding of floating-point numbers consists of 1 sign bit, 8 exponent bits, and 23 fraction bits. Due to the exponent bits, the range of floating-point numbers is much larger than that of integers, at the cost of sacrificing precision.</li> <li>ASCII is the earliest English character set, with a length of 1 byte, containing a total of 128 characters. GBK is a commonly used Chinese character set, containing over 20,000 Chinese characters. Unicode is committed to providing a complete character set standard, collecting characters from various languages around the world, thereby solving the garbled text problem caused by inconsistent character encoding methods.</li> <li>UTF-8 is the most popular Unicode encoding method and has excellent compatibility. It is a variable-length encoding method with good scalability, effectively improving storage space efficiency. UTF-16 and UTF-32 are common Unicode encoding methods. When encoding Chinese characters, UTF-16 occupies less space than UTF-8. Programming languages such as Java and C# use UTF-16 encoding by default.</li> </ul>","path":["Chapter 3. Data Structures","3.5   Summary"],"tags":[]},{"location":"chapter_data_structure/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: Why do hash tables contain both linear and non-linear data structures?</p> <p>The underlying structure of a hash table is an array. To resolve hash collisions, we may use \"chaining\" (discussed in the subsequent \"Hash Collision\" section): each bucket in the array points to a linked list, which may be converted to a tree (usually a red-black tree) when the list length exceeds a certain threshold.</p> <p>From a storage perspective, the underlying structure of a hash table is an array, where each bucket slot may contain a value, a linked list, or a tree. Therefore, hash tables may contain both linear data structures (arrays, linked lists) and non-linear data structures (trees).</p> <p>Q: Is the length of the <code>char</code> type 1 byte?</p> <p>The length of the <code>char</code> type is determined by the encoding method used by the programming language. For example, Java, JavaScript, TypeScript, and C# all use UTF-16 encoding (to store Unicode code points), so the <code>char</code> type has a length of 2 bytes.</p> <p>Q: Is there ambiguity in referring to array-based data structures as \"static data structures\"? Stacks can also perform \"dynamic\" operations such as push and pop.</p> <p>Stacks can indeed implement dynamic data operations, but the data structure is still \"static\" (fixed length). Although array-based data structures can dynamically add or remove elements, their capacity is fixed. If the data volume exceeds the pre-allocated size, a new larger array needs to be created, and the contents of the old array must be copied to the new array.</p> <p>Q: When constructing a stack (queue), its size is not specified. Why are they \"static data structures\"?</p> <p>In high-level programming languages, we do not need to manually specify the initial capacity of a stack (queue); the class handles this automatically. For example, the initial capacity of Java's <code>ArrayList</code> is typically 10. Additionally, the expansion operation is also automatically implemented. See the subsequent \"List\" section for details.</p> <p>Q: The method of converting sign-magnitude to 2's complement is \"first negate then add 1\". So converting 2's complement to sign-magnitude should be the inverse operation \"first subtract 1 then negate\". However, 2's complement can also be converted to sign-magnitude through \"first negate then add 1\". Why is this?</p> <p>This is because the mutual conversion between sign-magnitude and 2's complement is actually the process of computing the \"complement\". Let us first define the complement: assuming \\(a + b = c\\), then we say that \\(a\\) is the complement of \\(b\\) to \\(c\\), and conversely, \\(b\\) is the complement of \\(a\\) to \\(c\\).</p> <p>Given an \\(n = 4\\) bit binary number \\(0010\\), if we treat this number as sign-magnitude (ignoring the sign bit), then its 2's complement can be obtained through \"first negate then add 1\":</p> \\[ 0010 \\rightarrow 1101 \\rightarrow 1110 \\] <p>We find that the sum of sign-magnitude and 2's complement is \\(0010 + 1110 = 10000\\), which means the 2's complement \\(1110\\) is the \"complement\" of sign-magnitude \\(0010\\) to \\(10000\\). This means the above \"first negate then add 1\" is actually the process of computing the complement to \\(10000\\).</p> <p>So, what is the \"complement\" of 2's complement \\(1110\\) to \\(10000\\)? We can still use \"first negate then add 1\" to obtain it:</p> \\[ 1110 \\rightarrow 0001 \\rightarrow 0010 \\] <p>In other words, sign-magnitude and 2's complement are each other's \"complement\" to \\(10000\\), so \"sign-magnitude to 2's complement\" and \"2's complement to sign-magnitude\" can be implemented using the same operation (first negate then add 1).</p> <p>Of course, we can also use the inverse operation to find the sign-magnitude of 2's complement \\(1110\\), that is, \"first subtract 1 then negate\":</p> \\[ 1110 \\rightarrow 1101 \\rightarrow 0010 \\] <p>In summary, both \"first negate then add 1\" and \"first subtract 1 then negate\" are computing the complement to \\(10000\\), and they are equivalent.</p> <p>Essentially, the \"negate\" operation is actually finding the complement to \\(1111\\) (because \"sign-magnitude + 1's complement = 1111\" always holds); and adding 1 to the 1's complement yields the 2's complement, which is the complement to \\(10000\\).</p> <p>The above uses \\(n = 4\\) as an example, and it can be generalized to binary numbers of any number of bits.</p>","path":["Chapter 3. Data Structures","3.5   Summary"],"tags":[]},{"location":"chapter_divide_and_conquer/","level":1,"title":"Chapter 12.   Divide and Conquer","text":"<p>Abstract</p> <p>Difficult problems are decomposed layer by layer, with each decomposition making them simpler.</p> <p>Divide and conquer reveals an important truth: start with what is simple, and nothing remains complex.</p>","path":["Chapter 12. Divide and Conquer","Chapter 12.   Divide and Conquer"],"tags":[]},{"location":"chapter_divide_and_conquer/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>12.1   Divide and Conquer Algorithms</li> <li>12.2   Divide and Conquer Search Strategy</li> <li>12.3   Building a Binary Tree Problem</li> <li>12.4   Tower of Hanoi Problem</li> <li>12.5   Summary</li> <li>12.6   Exercises</li> </ul>","path":["Chapter 12. Divide and Conquer","Chapter 12.   Divide and Conquer"],"tags":[]},{"location":"chapter_divide_and_conquer/binary_search_recur/","level":1,"title":"12.2   Divide and Conquer Search Strategy","text":"<p>We have already learned that search algorithms are divided into two major categories.</p> <ul> <li>Brute-force search: Implemented by traversing the data structure, with a time complexity of \\(O(n)\\).</li> <li>Adaptive search: Leverages specific data organization or prior information, with time complexity reaching \\(O(\\log n)\\) or even \\(O(1)\\).</li> </ul> <p>In fact, search algorithms with time complexity of \\(O(\\log n)\\) are typically implemented based on the divide and conquer strategy, such as binary search and trees.</p> <ul> <li>Each step of binary search divides the problem (searching for a target element in an array) into a smaller problem (searching for the target element in half of the array), continuing until the array is empty or the target element is found.</li> <li>Trees are representative of the divide and conquer idea. In data structures such as binary search trees, AVL trees, and heaps, the time complexity of various operations is \\(O(\\log n)\\).</li> </ul> <p>The divide and conquer strategy of binary search is as follows.</p> <ul> <li>The problem can be decomposed: Binary search recursively decomposes the original problem (searching in an array) into subproblems (searching in half of the array), achieved by comparing the middle element with the target element.</li> <li>Subproblems are independent: In binary search, each round only processes one subproblem, which is not affected by other subproblems.</li> <li>Solutions of subproblems do not need to be merged: Binary search aims to find a specific element, so there is no need to merge the solutions of subproblems. When a subproblem is solved, the original problem is also solved.</li> </ul> <p>Divide and conquer can improve search efficiency because brute-force search can only eliminate one option per round, while divide and conquer search can eliminate half of the options per round.</p>","path":["Chapter 12. Divide and Conquer","12.2   Divide and Conquer Search Strategy"],"tags":[]},{"location":"chapter_divide_and_conquer/binary_search_recur/#1-implementing-binary-search-based-on-divide-and-conquer","level":3,"title":"1.   Implementing Binary Search Based on Divide and Conquer","text":"<p>In previous sections, binary search was implemented based on iteration. Now we implement it based on divide and conquer (recursion).</p> <p>Question</p> <p>Given a sorted array <code>nums</code> of length \\(n\\), where all elements are unique, find <code>target</code>.</p> <p>From a divide and conquer perspective, we denote the subproblem corresponding to the search interval \\([i, j]\\) as \\(f(i, j)\\).</p> <p>Starting from the original problem \\(f(0, n-1)\\), perform binary search through the following steps.</p> <ol> <li>Calculate the midpoint \\(m\\) of the search interval \\([i, j]\\), and use it to eliminate half of the search interval.</li> <li>Recursively solve the subproblem reduced by half in size, which could be \\(f(i, m-1)\\) or \\(f(m+1, j)\\).</li> <li>Repeat steps <code>1.</code> and <code>2.</code> until <code>target</code> is found, or return when the interval is empty.</li> </ol> <p>Figure 12-4 shows the divide and conquer process of binary search for element \\(6\\) in an array.</p> <p></p> <p> Figure 12-4   Divide and conquer process of binary search </p> <p>In the implementation code, we declare a recursive function <code>dfs()</code> to solve the problem \\(f(i, j)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_recur.py<pre><code>def dfs(nums: list[int], target: int, i: int, j: int) -&gt; int:\n    \"\"\"Binary search: problem f(i, j)\"\"\"\n    # If the interval is empty, it means there is no target element, return -1\n    if i &gt; j:\n        return -1\n    # Calculate the midpoint index m\n    m = (i + j) // 2\n    if nums[m] &lt; target:\n        # Recursion subproblem f(m+1, j)\n        return dfs(nums, target, m + 1, j)\n    elif nums[m] &gt; target:\n        # Recursion subproblem f(i, m-1)\n        return dfs(nums, target, i, m - 1)\n    else:\n        # Found the target element, return its index\n        return m\n\ndef binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"Binary search\"\"\"\n    n = len(nums)\n    # Solve the problem f(0, n-1)\n    return dfs(nums, target, 0, n - 1)\n</code></pre> binary_search_recur.cpp<pre><code>/* Binary search: problem f(i, j) */\nint dfs(vector&lt;int&gt; &amp;nums, int target, int i, int j) {\n    // If the interval is empty, it means there is no target element, return -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // Calculate the midpoint index m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // Recursion subproblem f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // Recursion subproblem f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // Found the target element, return its index\n        return m;\n    }\n}\n\n/* Binary search */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    int n = nums.size();\n    // Solve the problem f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.java<pre><code>/* Binary search: problem f(i, j) */\nint dfs(int[] nums, int target, int i, int j) {\n    // If the interval is empty, it means there is no target element, return -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // Calculate the midpoint index m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // Recursion subproblem f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // Recursion subproblem f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // Found the target element, return its index\n        return m;\n    }\n}\n\n/* Binary search */\nint binarySearch(int[] nums, int target) {\n    int n = nums.length;\n    // Solve the problem f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.cs<pre><code>/* Binary search: problem f(i, j) */\nint DFS(int[] nums, int target, int i, int j) {\n    // If the interval is empty, it means there is no target element, return -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // Calculate the midpoint index m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // Recursion subproblem f(m+1, j)\n        return DFS(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // Recursion subproblem f(i, m-1)\n        return DFS(nums, target, i, m - 1);\n    } else {\n        // Found the target element, return its index\n        return m;\n    }\n}\n\n/* Binary search */\nint BinarySearch(int[] nums, int target) {\n    int n = nums.Length;\n    // Solve the problem f(0, n-1)\n    return DFS(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.go<pre><code>/* Binary search: problem f(i, j) */\nfunc dfs(nums []int, target, i, j int) int {\n    // If interval is empty, indicating no target element, return -1\n    if i &gt; j {\n        return -1\n    }\n    // Calculate midpoint index\n    m := i + ((j - i) &gt;&gt; 1)\n    // Compare midpoint with target element\n    if nums[m] &lt; target {\n        // If smaller, recurse on right half of array\n        // Recursion subproblem f(m+1, j)\n        return dfs(nums, target, m+1, j)\n    } else if nums[m] &gt; target {\n        // If larger, recurse on left half of array\n        // Recursion subproblem f(i, m-1)\n        return dfs(nums, target, i, m-1)\n    } else {\n        // Found the target element, return its index\n        return m\n    }\n}\n\n/* Binary search */\nfunc binarySearch(nums []int, target int) int {\n    n := len(nums)\n    return dfs(nums, target, 0, n-1)\n}\n</code></pre> binary_search_recur.swift<pre><code>/* Binary search: problem f(i, j) */\nfunc dfs(nums: [Int], target: Int, i: Int, j: Int) -&gt; Int {\n    // If the interval is empty, it means there is no target element, return -1\n    if i &gt; j {\n        return -1\n    }\n    // Calculate the midpoint index m\n    let m = (i + j) / 2\n    if nums[m] &lt; target {\n        // Recursion subproblem f(m+1, j)\n        return dfs(nums: nums, target: target, i: m + 1, j: j)\n    } else if nums[m] &gt; target {\n        // Recursion subproblem f(i, m-1)\n        return dfs(nums: nums, target: target, i: i, j: m - 1)\n    } else {\n        // Found the target element, return its index\n        return m\n    }\n}\n\n/* Binary search */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // Solve the problem f(0, n-1)\n    dfs(nums: nums, target: target, i: nums.startIndex, j: nums.endIndex - 1)\n}\n</code></pre> binary_search_recur.js<pre><code>/* Binary search: problem f(i, j) */\nfunction dfs(nums, target, i, j) {\n    // If the interval is empty, it means there is no target element, return -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // Calculate the midpoint index m\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // Recursion subproblem f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // Recursion subproblem f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // Found the target element, return its index\n        return m;\n    }\n}\n\n/* Binary search */\nfunction binarySearch(nums, target) {\n    const n = nums.length;\n    // Solve the problem f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.ts<pre><code>/* Binary search: problem f(i, j) */\nfunction dfs(nums: number[], target: number, i: number, j: number): number {\n    // If the interval is empty, it means there is no target element, return -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // Calculate the midpoint index m\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // Recursion subproblem f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // Recursion subproblem f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // Found the target element, return its index\n        return m;\n    }\n}\n\n/* Binary search */\nfunction binarySearch(nums: number[], target: number): number {\n    const n = nums.length;\n    // Solve the problem f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.dart<pre><code>/* Binary search: problem f(i, j) */\nint dfs(List&lt;int&gt; nums, int target, int i, int j) {\n  // If the interval is empty, it means there is no target element, return -1\n  if (i &gt; j) {\n    return -1;\n  }\n  // Calculate the midpoint index m\n  int m = (i + j) ~/ 2;\n  if (nums[m] &lt; target) {\n    // Recursion subproblem f(m+1, j)\n    return dfs(nums, target, m + 1, j);\n  } else if (nums[m] &gt; target) {\n    // Recursion subproblem f(i, m-1)\n    return dfs(nums, target, i, m - 1);\n  } else {\n    // Found the target element, return its index\n    return m;\n  }\n}\n\n/* Binary search */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  int n = nums.length;\n  // Solve the problem f(0, n-1)\n  return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.rs<pre><code>/* Binary search: problem f(i, j) */\nfn dfs(nums: &amp;[i32], target: i32, i: i32, j: i32) -&gt; i32 {\n    // If the interval is empty, it means there is no target element, return -1\n    if i &gt; j {\n        return -1;\n    }\n    let m: i32 = i + (j - i) / 2;\n    if nums[m as usize] &lt; target {\n        // Recursion subproblem f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if nums[m as usize] &gt; target {\n        // Recursion subproblem f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // Found the target element, return its index\n        return m;\n    }\n}\n\n/* Binary search */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let n = nums.len() as i32;\n    // Solve the problem f(0, n-1)\n    dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.c<pre><code>/* Binary search: problem f(i, j) */\nint dfs(int nums[], int target, int i, int j) {\n    // If the interval is empty, it means there is no target element, return -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // Calculate the midpoint index m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // Recursion subproblem f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // Recursion subproblem f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // Found the target element, return its index\n        return m;\n    }\n}\n\n/* Binary search */\nint binarySearch(int nums[], int target, int numsSize) {\n    int n = numsSize;\n    // Solve the problem f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.kt<pre><code>/* Binary search: problem f(i, j) */\nfun dfs(\n    nums: IntArray,\n    target: Int,\n    i: Int,\n    j: Int\n): Int {\n    // If the interval is empty, it means there is no target element, return -1\n    if (i &gt; j) {\n        return -1\n    }\n    // Calculate the midpoint index m\n    val m = (i + j) / 2\n    return if (nums[m] &lt; target) {\n        // Recursion subproblem f(m+1, j)\n        dfs(nums, target, m + 1, j)\n    } else if (nums[m] &gt; target) {\n        // Recursion subproblem f(i, m-1)\n        dfs(nums, target, i, m - 1)\n    } else {\n        // Found the target element, return its index\n        m\n    }\n}\n\n/* Binary search */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    val n = nums.size\n    // Solve the problem f(0, n-1)\n    return dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.rb<pre><code>### Binary search: problem f(i, j) ###\ndef dfs(nums, target, i, j)\n  # If the interval is empty, it means there is no target element, return -1\n  return -1 if i &gt; j\n\n  # Calculate the midpoint index m\n  m = (i + j) / 2\n\n  if nums[m] &lt; target\n    # Recursion subproblem f(m+1, j)\n    return dfs(nums, target, m + 1, j)\n  elsif nums[m] &gt; target\n    # Recursion subproblem f(i, m-1)\n    return dfs(nums, target, i, m - 1)\n  else\n    # Found the target element, return its index\n    return m\n  end\nend\n\n### Binary search ###\ndef binary_search(nums, target)\n  n = nums.length\n  # Solve the problem f(0, n-1)\n  dfs(nums, target, 0, n - 1)\nend\n</code></pre>","path":["Chapter 12. Divide and Conquer","12.2   Divide and Conquer Search Strategy"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/","level":1,"title":"12.3   Building a Binary Tree Problem","text":"<p>Question</p> <p>Given the preorder traversal <code>preorder</code> and inorder traversal <code>inorder</code> of a binary tree, construct the binary tree and return the root node of the binary tree. Assume there are no duplicate node values in the binary tree (as shown in Figure 12-5).</p> <p></p> <p> Figure 12-5   Example data for building a binary tree </p>","path":["Chapter 12. Divide and Conquer","12.3   Building a Binary Tree Problem"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#1-determining-if-it-is-a-divide-and-conquer-problem","level":3,"title":"1.   Determining If It Is a Divide and Conquer Problem","text":"<p>The original problem is defined as constructing a binary tree from <code>preorder</code> and <code>inorder</code>, which is a typical divide and conquer problem.</p> <ul> <li>The problem can be decomposed: From a divide and conquer perspective, we can divide the original problem into two subproblems: constructing the left subtree and constructing the right subtree, plus one operation: initializing the root node. For each subtree (subproblem), we can still reuse the above division method, dividing it into smaller subtrees (subproblems) until the smallest subproblem (empty subtree) is reached.</li> <li>Subproblems are independent: The left and right subtrees are independent of each other; there is no overlap between them. When constructing the left subtree, we only need to focus on the parts of the inorder and preorder traversals corresponding to the left subtree. The same applies to the right subtree.</li> <li>Solutions of subproblems can be merged: Once we have the left and right subtrees (solutions of subproblems), we can link them to the root node to obtain the solution to the original problem.</li> </ul>","path":["Chapter 12. Divide and Conquer","12.3   Building a Binary Tree Problem"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#2-how-to-divide-subtrees","level":3,"title":"2.   How to Divide Subtrees","text":"<p>Based on the above analysis, this problem can be solved using divide and conquer, but how do we divide the left and right subtrees through the preorder traversal <code>preorder</code> and inorder traversal <code>inorder</code>?</p> <p>According to the definition, both <code>preorder</code> and <code>inorder</code> can be divided into three parts.</p> <ul> <li>Preorder traversal: <code>[ Root Node | Left Subtree | Right Subtree ]</code>, for example, the tree in Figure 12-5 corresponds to <code>[ 3 | 9 | 2 1 7 ]</code>.</li> <li>Inorder traversal: <code>[ Left Subtree | Root Node | Right Subtree ]</code>, for example, the tree in Figure 12-5 corresponds to <code>[ 9 | 3 | 1 2 7 ]</code>.</li> </ul> <p>Using the data from the figure above as an example, we can obtain the division results through the steps shown in Figure 12-6.</p> <ol> <li>The first element 3 in the preorder traversal is the value of the root node.</li> <li>Find the index of root node 3 in <code>inorder</code>, and use this index to divide <code>inorder</code> into <code>[ 9 | 3 | 1 2 7 ]</code>.</li> <li>Based on the division result of <code>inorder</code>, it is easy to determine that the left and right subtrees have 1 and 3 nodes respectively, allowing us to divide <code>preorder</code> into <code>[ 3 | 9 | 2 1 7 ]</code>.</li> </ol> <p></p> <p> Figure 12-6   Dividing subtrees in preorder and inorder traversals </p>","path":["Chapter 12. Divide and Conquer","12.3   Building a Binary Tree Problem"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#3-describing-subtree-intervals-based-on-variables","level":3,"title":"3.   Describing Subtree Intervals Based on Variables","text":"<p>Based on the above division method, we have obtained the index intervals of the root node, left subtree, and right subtree in <code>preorder</code> and <code>inorder</code>. To describe these index intervals, we need to use several index variables.</p> <ul> <li>Denote the index of the current tree's root node in <code>preorder</code> as \\(i\\).</li> <li>Denote the index of the current tree's root node in <code>inorder</code> as \\(m\\).</li> <li>Denote the index interval of the current tree in <code>inorder</code> as \\([l, r]\\).</li> </ul> <p>As shown in Table 12-1, through these variables we can represent the index of the root node in <code>preorder</code> and the index intervals of the subtrees in <code>inorder</code>.</p> <p> Table 12-1   Indices of root node and subtrees in preorder and inorder traversals </p> Root node index in <code>preorder</code> Subtree index interval in <code>inorder</code> Current tree \\(i\\) \\([l, r]\\) Left subtree \\(i + 1\\) \\([l, m-1]\\) Right subtree \\(i + 1 + (m - l)\\) \\([m+1, r]\\) <p>Please note that \\((m-l)\\) in the right subtree root node index means \"the number of nodes in the left subtree\". It is recommended to understand this in conjunction with Figure 12-7.</p> <p></p> <p> Figure 12-7   Index interval representation of root node and left and right subtrees </p>","path":["Chapter 12. Divide and Conquer","12.3   Building a Binary Tree Problem"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#4-code-implementation","level":3,"title":"4.   Code Implementation","text":"<p>To improve the efficiency of querying \\(m\\), we use a hash table <code>hmap</code> to store the mapping from elements in the <code>inorder</code> array to their indices:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby build_tree.py<pre><code>def dfs(\n    preorder: list[int],\n    inorder_map: dict[int, int],\n    i: int,\n    l: int,\n    r: int,\n) -&gt; TreeNode | None:\n    \"\"\"Build binary tree: divide and conquer\"\"\"\n    # Terminate when the subtree interval is empty\n    if r - l &lt; 0:\n        return None\n    # Initialize the root node\n    root = TreeNode(preorder[i])\n    # Query m to divide the left and right subtrees\n    m = inorder_map[preorder[i]]\n    # Subproblem: build the left subtree\n    root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n    # Subproblem: build the right subtree\n    root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n    # Return the root node\n    return root\n\ndef build_tree(preorder: list[int], inorder: list[int]) -&gt; TreeNode | None:\n    \"\"\"Build binary tree\"\"\"\n    # Initialize hash map, storing the mapping from inorder elements to indices\n    inorder_map = {val: i for i, val in enumerate(inorder)}\n    root = dfs(preorder, inorder_map, 0, 0, len(inorder) - 1)\n    return root\n</code></pre> build_tree.cpp<pre><code>/* Build binary tree: divide and conquer */\nTreeNode *dfs(vector&lt;int&gt; &amp;preorder, unordered_map&lt;int, int&gt; &amp;inorderMap, int i, int l, int r) {\n    // Terminate when the subtree interval is empty\n    if (r - l &lt; 0)\n        return NULL;\n    // Initialize the root node\n    TreeNode *root = new TreeNode(preorder[i]);\n    // Query m to divide the left and right subtrees\n    int m = inorderMap[preorder[i]];\n    // Subproblem: build the left subtree\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // Subproblem: build the right subtree\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // Return the root node\n    return root;\n}\n\n/* Build binary tree */\nTreeNode *buildTree(vector&lt;int&gt; &amp;preorder, vector&lt;int&gt; &amp;inorder) {\n    // Initialize hash map, storing the mapping from inorder elements to indices\n    unordered_map&lt;int, int&gt; inorderMap;\n    for (int i = 0; i &lt; inorder.size(); i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorder.size() - 1);\n    return root;\n}\n</code></pre> build_tree.java<pre><code>/* Build binary tree: divide and conquer */\nTreeNode dfs(int[] preorder, Map&lt;Integer, Integer&gt; inorderMap, int i, int l, int r) {\n    // Terminate when the subtree interval is empty\n    if (r - l &lt; 0)\n        return null;\n    // Initialize the root node\n    TreeNode root = new TreeNode(preorder[i]);\n    // Query m to divide the left and right subtrees\n    int m = inorderMap.get(preorder[i]);\n    // Subproblem: build the left subtree\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // Subproblem: build the right subtree\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // Return the root node\n    return root;\n}\n\n/* Build binary tree */\nTreeNode buildTree(int[] preorder, int[] inorder) {\n    // Initialize hash map, storing the mapping from inorder elements to indices\n    Map&lt;Integer, Integer&gt; inorderMap = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; inorder.length; i++) {\n        inorderMap.put(inorder[i], i);\n    }\n    TreeNode root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.cs<pre><code>/* Build binary tree: divide and conquer */\nTreeNode? DFS(int[] preorder, Dictionary&lt;int, int&gt; inorderMap, int i, int l, int r) {\n    // Terminate when the subtree interval is empty\n    if (r - l &lt; 0)\n        return null;\n    // Initialize the root node\n    TreeNode root = new(preorder[i]);\n    // Query m to divide the left and right subtrees\n    int m = inorderMap[preorder[i]];\n    // Subproblem: build the left subtree\n    root.left = DFS(preorder, inorderMap, i + 1, l, m - 1);\n    // Subproblem: build the right subtree\n    root.right = DFS(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // Return the root node\n    return root;\n}\n\n/* Build binary tree */\nTreeNode? BuildTree(int[] preorder, int[] inorder) {\n    // Initialize hash map, storing the mapping from inorder elements to indices\n    Dictionary&lt;int, int&gt; inorderMap = [];\n    for (int i = 0; i &lt; inorder.Length; i++) {\n        inorderMap.TryAdd(inorder[i], i);\n    }\n    TreeNode? root = DFS(preorder, inorderMap, 0, 0, inorder.Length - 1);\n    return root;\n}\n</code></pre> build_tree.go<pre><code>/* Build binary tree: divide and conquer */\nfunc dfsBuildTree(preorder []int, inorderMap map[int]int, i, l, r int) *TreeNode {\n    // Terminate when the subtree interval is empty\n    if r-l &lt; 0 {\n        return nil\n    }\n    // Initialize the root node\n    root := NewTreeNode(preorder[i])\n    // Query m to divide the left and right subtrees\n    m := inorderMap[preorder[i]]\n    // Subproblem: build the left subtree\n    root.Left = dfsBuildTree(preorder, inorderMap, i+1, l, m-1)\n    // Subproblem: build the right subtree\n    root.Right = dfsBuildTree(preorder, inorderMap, i+1+m-l, m+1, r)\n    // Return the root node\n    return root\n}\n\n/* Build binary tree */\nfunc buildTree(preorder, inorder []int) *TreeNode {\n    // Initialize hash map, storing the mapping from inorder elements to indices\n    inorderMap := make(map[int]int, len(inorder))\n    for i := 0; i &lt; len(inorder); i++ {\n        inorderMap[inorder[i]] = i\n    }\n\n    root := dfsBuildTree(preorder, inorderMap, 0, 0, len(inorder)-1)\n    return root\n}\n</code></pre> build_tree.swift<pre><code>/* Build binary tree: divide and conquer */\nfunc dfs(preorder: [Int], inorderMap: [Int: Int], i: Int, l: Int, r: Int) -&gt; TreeNode? {\n    // Terminate when the subtree interval is empty\n    if r - l &lt; 0 {\n        return nil\n    }\n    // Initialize the root node\n    let root = TreeNode(x: preorder[i])\n    // Query m to divide the left and right subtrees\n    let m = inorderMap[preorder[i]]!\n    // Subproblem: build the left subtree\n    root.left = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1, l: l, r: m - 1)\n    // Subproblem: build the right subtree\n    root.right = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1 + m - l, l: m + 1, r: r)\n    // Return the root node\n    return root\n}\n\n/* Build binary tree */\nfunc buildTree(preorder: [Int], inorder: [Int]) -&gt; TreeNode? {\n    // Initialize hash map, storing the mapping from inorder elements to indices\n    let inorderMap = inorder.enumerated().reduce(into: [:]) { $0[$1.element] = $1.offset }\n    return dfs(preorder: preorder, inorderMap: inorderMap, i: inorder.startIndex, l: inorder.startIndex, r: inorder.endIndex - 1)\n}\n</code></pre> build_tree.js<pre><code>/* Build binary tree: divide and conquer */\nfunction dfs(preorder, inorderMap, i, l, r) {\n    // Terminate when the subtree interval is empty\n    if (r - l &lt; 0) return null;\n    // Initialize the root node\n    const root = new TreeNode(preorder[i]);\n    // Query m to divide the left and right subtrees\n    const m = inorderMap.get(preorder[i]);\n    // Subproblem: build the left subtree\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // Subproblem: build the right subtree\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // Return the root node\n    return root;\n}\n\n/* Build binary tree */\nfunction buildTree(preorder, inorder) {\n    // Initialize hash map, storing the mapping from inorder elements to indices\n    let inorderMap = new Map();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.ts<pre><code>/* Build binary tree: divide and conquer */\nfunction dfs(\n    preorder: number[],\n    inorderMap: Map&lt;number, number&gt;,\n    i: number,\n    l: number,\n    r: number\n): TreeNode | null {\n    // Terminate when the subtree interval is empty\n    if (r - l &lt; 0) return null;\n    // Initialize the root node\n    const root: TreeNode = new TreeNode(preorder[i]);\n    // Query m to divide the left and right subtrees\n    const m = inorderMap.get(preorder[i]);\n    // Subproblem: build the left subtree\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // Subproblem: build the right subtree\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // Return the root node\n    return root;\n}\n\n/* Build binary tree */\nfunction buildTree(preorder: number[], inorder: number[]): TreeNode | null {\n    // Initialize hash map, storing the mapping from inorder elements to indices\n    let inorderMap = new Map&lt;number, number&gt;();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.dart<pre><code>/* Build binary tree: divide and conquer */\nTreeNode? dfs(\n  List&lt;int&gt; preorder,\n  Map&lt;int, int&gt; inorderMap,\n  int i,\n  int l,\n  int r,\n) {\n  // Terminate when the subtree interval is empty\n  if (r - l &lt; 0) {\n    return null;\n  }\n  // Initialize the root node\n  TreeNode? root = TreeNode(preorder[i]);\n  // Query m to divide the left and right subtrees\n  int m = inorderMap[preorder[i]]!;\n  // Subproblem: build the left subtree\n  root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n  // Subproblem: build the right subtree\n  root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n  // Return the root node\n  return root;\n}\n\n/* Build binary tree */\nTreeNode? buildTree(List&lt;int&gt; preorder, List&lt;int&gt; inorder) {\n  // Initialize hash map, storing the mapping from inorder elements to indices\n  Map&lt;int, int&gt; inorderMap = {};\n  for (int i = 0; i &lt; inorder.length; i++) {\n    inorderMap[inorder[i]] = i;\n  }\n  TreeNode? root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n  return root;\n}\n</code></pre> build_tree.rs<pre><code>/* Build binary tree: divide and conquer */\nfn dfs(\n    preorder: &amp;[i32],\n    inorder_map: &amp;HashMap&lt;i32, i32&gt;,\n    i: i32,\n    l: i32,\n    r: i32,\n) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // Terminate when the subtree interval is empty\n    if r - l &lt; 0 {\n        return None;\n    }\n    // Initialize the root node\n    let root = TreeNode::new(preorder[i as usize]);\n    // Query m to divide the left and right subtrees\n    let m = inorder_map.get(&amp;preorder[i as usize]).unwrap();\n    // Subproblem: build the left subtree\n    root.borrow_mut().left = dfs(preorder, inorder_map, i + 1, l, m - 1);\n    // Subproblem: build the right subtree\n    root.borrow_mut().right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r);\n    // Return the root node\n    Some(root)\n}\n\n/* Build binary tree */\nfn build_tree(preorder: &amp;[i32], inorder: &amp;[i32]) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // Initialize hash map, storing the mapping from inorder elements to indices\n    let mut inorder_map: HashMap&lt;i32, i32&gt; = HashMap::new();\n    for i in 0..inorder.len() {\n        inorder_map.insert(inorder[i], i as i32);\n    }\n    let root = dfs(preorder, &amp;inorder_map, 0, 0, inorder.len() as i32 - 1);\n    root\n}\n</code></pre> build_tree.c<pre><code>/* Build binary tree: divide and conquer */\nTreeNode *dfs(int *preorder, int *inorderMap, int i, int l, int r, int size) {\n    // Terminate when the subtree interval is empty\n    if (r - l &lt; 0)\n        return NULL;\n    // Initialize the root node\n    TreeNode *root = (TreeNode *)malloc(sizeof(TreeNode));\n    root-&gt;val = preorder[i];\n    root-&gt;left = NULL;\n    root-&gt;right = NULL;\n    // Query m to divide the left and right subtrees\n    int m = inorderMap[preorder[i]];\n    // Subproblem: build the left subtree\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1, size);\n    // Subproblem: build the right subtree\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r, size);\n    // Return the root node\n    return root;\n}\n\n/* Build binary tree */\nTreeNode *buildTree(int *preorder, int preorderSize, int *inorder, int inorderSize) {\n    // Initialize hash map, storing the mapping from inorder elements to indices\n    int *inorderMap = (int *)malloc(sizeof(int) * MAX_SIZE);\n    for (int i = 0; i &lt; inorderSize; i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorderSize - 1, inorderSize);\n    free(inorderMap);\n    return root;\n}\n</code></pre> build_tree.kt<pre><code>/* Build binary tree: divide and conquer */\nfun dfs(\n    preorder: IntArray,\n    inorderMap: Map&lt;Int?, Int?&gt;,\n    i: Int,\n    l: Int,\n    r: Int\n): TreeNode? {\n    // Terminate when the subtree interval is empty\n    if (r - l &lt; 0) return null\n    // Initialize the root node\n    val root = TreeNode(preorder[i])\n    // Query m to divide the left and right subtrees\n    val m = inorderMap[preorder[i]]!!\n    // Subproblem: build the left subtree\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1)\n    // Subproblem: build the right subtree\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r)\n    // Return the root node\n    return root\n}\n\n/* Build binary tree */\nfun buildTree(preorder: IntArray, inorder: IntArray): TreeNode? {\n    // Initialize hash map, storing the mapping from inorder elements to indices\n    val inorderMap = HashMap&lt;Int?, Int?&gt;()\n    for (i in inorder.indices) {\n        inorderMap[inorder[i]] = i\n    }\n    val root = dfs(preorder, inorderMap, 0, 0, inorder.size - 1)\n    return root\n}\n</code></pre> build_tree.rb<pre><code>### Build binary tree: divide and conquer ###\ndef dfs(preorder, inorder_map, i, l, r)\n  # Terminate when the subtree interval is empty\n  return if r - l &lt; 0\n\n  # Initialize the root node\n  root = TreeNode.new(preorder[i])\n  # Query m to divide the left and right subtrees\n  m = inorder_map[preorder[i]]\n  # Subproblem: build the left subtree\n  root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n  # Subproblem: build the right subtree\n  root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n\n  # Return the root node\n  root\nend\n\n### Build binary tree ###\ndef build_tree(preorder, inorder)\n  # Initialize hash map, storing the mapping from inorder elements to indices\n  inorder_map = {}\n  inorder.each_with_index { |val, i| inorder_map[val] = i }\n  dfs(preorder, inorder_map, 0, 0, inorder.length - 1)\nend\n</code></pre> <p>Figure 12-8 shows the recursive process of building the binary tree. Each node is established during the downward \"recursion\" process, while each edge (reference) is established during the upward \"return\" process.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 12-8   Recursive process of building a binary tree </p> <p>The division results of the preorder traversal <code>preorder</code> and inorder traversal <code>inorder</code> within each recursive function are shown in Figure 12-9.</p> <p></p> <p> Figure 12-9   Division results in each recursive function </p> <p>Let the number of nodes in the tree be \\(n\\). Initializing each node (executing one recursive function <code>dfs()</code>) takes \\(O(1)\\) time. Therefore, the overall time complexity is \\(O(n)\\).</p> <p>The hash table stores the mapping from <code>inorder</code> elements to their indices, with a space complexity of \\(O(n)\\). In the worst case, when the binary tree degenerates into a linked list, the recursion depth reaches \\(n\\), using \\(O(n)\\) stack frame space. Therefore, the overall space complexity is \\(O(n)\\).</p>","path":["Chapter 12. Divide and Conquer","12.3   Building a Binary Tree Problem"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/","level":1,"title":"12.1   Divide and Conquer Algorithms","text":"<p>Divide and conquer is a very important and common algorithmic strategy. Divide and conquer is typically implemented based on recursion, consisting of two steps: \"divide\" and \"conquer\".</p> <ol> <li>Divide (partition phase): Recursively divide the original problem into two or more subproblems until the smallest subproblem is reached.</li> <li>Conquer (merge phase): Starting from the smallest subproblems with known solutions, merge the solutions of subproblems from bottom to top to construct the solution to the original problem.</li> </ol> <p>As shown in Figure 12-1, \"merge sort\" is one of the typical applications of the divide and conquer strategy.</p> <ol> <li>Divide: Recursively divide the original array (original problem) into two subarrays (subproblems) until the subarray has only one element (smallest subproblem).</li> <li>Conquer: Merge the sorted subarrays (solutions to subproblems) from bottom to top to obtain a sorted original array (solution to the original problem).</li> </ol> <p></p> <p> Figure 12-1   Divide and conquer strategy of merge sort </p>","path":["Chapter 12. Divide and Conquer","12.1   Divide and Conquer Algorithms"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1211-how-to-determine-divide-and-conquer-problems","level":2,"title":"12.1.1   How to Determine Divide and Conquer Problems","text":"<p>Whether a problem is suitable for solving with divide and conquer can usually be determined based on the following criteria.</p> <ol> <li>The problem can be decomposed: The original problem can be divided into smaller, similar subproblems, and can be recursively divided in the same way.</li> <li>Subproblems are independent: There is no overlap between subproblems, they are independent of each other and can be solved independently.</li> <li>Solutions of subproblems can be merged: The solution to the original problem is obtained by merging the solutions of subproblems.</li> </ol> <p>Clearly, merge sort satisfies these three criteria.</p> <ol> <li>The problem can be decomposed: Recursively divide the array (original problem) into two subarrays (subproblems).</li> <li>Subproblems are independent: Each subarray can be sorted independently (subproblems can be solved independently).</li> <li>Solutions of subproblems can be merged: Two sorted subarrays (solutions of subproblems) can be merged into one sorted array (solution of the original problem).</li> </ol>","path":["Chapter 12. Divide and Conquer","12.1   Divide and Conquer Algorithms"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1212-improving-efficiency-through-divide-and-conquer","level":2,"title":"12.1.2   Improving Efficiency Through Divide and Conquer","text":"<p>Divide and conquer can not only effectively solve algorithmic problems, but can often also improve algorithmic efficiency. In sorting algorithms, quick sort, merge sort, and heap sort are faster than selection, bubble, and insertion sort because they apply the divide and conquer strategy.</p> <p>This raises the question: Why can divide and conquer improve algorithm efficiency, and what is the underlying logic? In other words, why is dividing a large problem into multiple subproblems, solving the subproblems, and merging their solutions more efficient than directly solving the original problem? This question can be discussed from two aspects: operation count and parallel computation.</p>","path":["Chapter 12. Divide and Conquer","12.1   Divide and Conquer Algorithms"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1-operation-count-optimization","level":3,"title":"1.   Operation Count Optimization","text":"<p>Taking \"bubble sort\" as an example, processing an array of length \\(n\\) requires \\(O(n^2)\\) time. Suppose we divide the array at the midpoint into two subarrays, as shown in Figure 12-2. The division requires \\(O(n)\\) time, sorting each subarray requires \\(O((n / 2)^2)\\) time, and merging the two subarrays requires \\(O(n)\\) time, resulting in an overall time complexity of:</p> \\[ O(n + (\\frac{n}{2})^2 \\times 2 + n) = O(\\frac{n^2}{2} + 2n) \\] <p></p> <p> Figure 12-2   Bubble sort before and after array division </p> <p>Next, we compute the following inequality, where the left and right sides represent the total number of operations before and after division, respectively:</p> \\[ \\begin{aligned} n^2 &amp; &gt; \\frac{n^2}{2} + 2n \\newline n^2 - \\frac{n^2}{2} - 2n &amp; &gt; 0 \\newline n(n - 4) &amp; &gt; 0 \\end{aligned} \\] <p>This means that when \\(n &gt; 4\\), the number of operations after division is smaller, and sorting efficiency should be higher. Note that the time complexity after division is still quadratic \\(O(n^2)\\), but the constant term in the complexity has become smaller.</p> <p>Going further, what if we continuously divide the subarrays from their midpoints into two subarrays until the subarrays have only one element? This approach is actually \"merge sort\", with a time complexity of \\(O(n \\log n)\\).</p> <p>Thinking further, what if we set multiple division points and evenly divide the original array into \\(k\\) subarrays? This situation is very similar to \"bucket sort\", which is well-suited for sorting massive amounts of data, with a theoretical time complexity of \\(O(n + k)\\).</p>","path":["Chapter 12. Divide and Conquer","12.1   Divide and Conquer Algorithms"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#2-parallel-computation-optimization","level":3,"title":"2.   Parallel Computation Optimization","text":"<p>We know that the subproblems generated by divide and conquer are independent of each other, so they can typically be solved in parallel. This means divide and conquer can not only reduce the time complexity of algorithms, but is also amenable to parallel optimization by the operating system.</p> <p>Parallel optimization is particularly effective in multi-core or multi-processor environments, as the system can simultaneously handle multiple subproblems, making fuller use of computing resources and significantly reducing overall runtime.</p> <p>For example, in the \"bucket sort\" shown in Figure 12-3, we evenly distribute massive data into various buckets, and the sorting tasks for all buckets can be distributed to various computing units. After completion, the results are merged.</p> <p></p> <p> Figure 12-3   Parallel computation in bucket sort </p>","path":["Chapter 12. Divide and Conquer","12.1   Divide and Conquer Algorithms"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1213-common-applications-of-divide-and-conquer","level":2,"title":"12.1.3   Common Applications of Divide and Conquer","text":"<p>On the one hand, divide and conquer can be used to solve many classic algorithmic problems.</p> <ul> <li>Finding the closest pair of points: This algorithm first divides the point set into two parts, then finds the closest pair of points in each part separately, and finally finds the closest pair of points that spans both parts.</li> <li>Large integer multiplication: For example, the Karatsuba algorithm, which decomposes large integer multiplication into several smaller integer multiplications and additions.</li> <li>Matrix multiplication: For example, the Strassen algorithm, which decomposes large matrix multiplication into multiple small matrix multiplications and additions.</li> <li>Tower of Hanoi problem: The Tower of Hanoi problem can be solved through recursion, which is a typical application of the divide and conquer strategy.</li> <li>Solving inversion pairs: In a sequence, if a preceding number is greater than a following number, these two numbers form an inversion pair. Solving the inversion pair problem can utilize the divide and conquer approach with the help of merge sort.</li> </ul> <p>On the other hand, divide and conquer is widely applied in the design of algorithms and data structures.</p> <ul> <li>Binary search: Binary search divides a sorted array into two parts from the midpoint index, then decides which half to eliminate based on the comparison result between the target value and the middle element value, and performs the same binary-search step on the remaining interval.</li> <li>Merge sort: Already introduced at the beginning of this section, no further elaboration needed.</li> <li>Quick sort: Quick sort selects a pivot value, then divides the array into two subarrays, one with elements smaller than the pivot and the other with elements larger than the pivot, then performs the same division operation on these two parts until the subarrays have only one element.</li> <li>Bucket sort: The basic idea of bucket sort is to scatter data into multiple buckets, then sort the elements within each bucket, and finally extract the elements from each bucket in sequence to obtain a sorted array.</li> <li>Trees: For example, binary search trees, AVL trees, red-black trees, B-trees, B+ trees, etc. Their search, insertion, and deletion operations can all be viewed as applications of the divide and conquer strategy.</li> <li>Heaps: A heap is a special complete binary tree, and its various operations, such as insertion, deletion, and heapify, actually imply the divide and conquer idea.</li> <li>Hash tables: Although hash tables do not directly apply divide and conquer, some methods for resolving hash collisions indirectly apply the divide and conquer strategy. For example, long linked lists in chaining may be converted to red-black trees to improve lookup efficiency.</li> </ul> <p>It can be seen that divide and conquer is a \"quietly pervasive\" algorithmic idea, embedded in various algorithms and data structures.</p>","path":["Chapter 12. Divide and Conquer","12.1   Divide and Conquer Algorithms"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/","level":1,"title":"12.6   Exercises","text":"","path":["Chapter 12. Divide and Conquer","12.6   Exercises"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1261-concept-review","level":2,"title":"12.6.1   Concept Review","text":"","path":["Chapter 12. Divide and Conquer","12.6   Exercises"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1-which-tasks-are-suitable-for-divide-and-conquer","level":3,"title":"1.   Which Tasks Are Suitable for Divide and Conquer?","text":"<p>A student wants to solve each task below by \"dividing it into two halves, solving each half separately, and then combining the results.\" Classify each task as \"suitable for divide and conquer,\" \"can use divide and conquer, but it will not reduce the total work,\" or \"the two halves cannot be solved independently,\" and explain why.</p> <ol> <li>Sort an unsorted array.</li> <li>Find the maximum value in an array.</li> <li>Execute a sequence of <code>push(x)</code> and <code>pop()</code> stack operations in order, and output the element returned by each <code>pop()</code>.</li> </ol> Answer <ol> <li>Suitable: divide the array in half, sort each half independently, and merge them in \\(O(n)\\) time. This is exactly merge sort.</li> <li>Divide and conquer can be used, but it does not reduce the total work. The two halves still require examining all \\(n\\) elements in total,     so the time complexity remains \\(O(n)\\), just like a direct scan.</li> <li>The two halves cannot be solved independently. The stack's contents at the beginning of the second half depend on the result of executing the first half,     so the two halves cannot be completed without knowing each other's results.</li> </ol>","path":["Chapter 12. Divide and Conquer","12.6   Exercises"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#2-how-exponentiation-by-squaring-reduces-computation","level":3,"title":"2.   How Exponentiation by Squaring Reduces Computation","text":"<p>The recursive function below uses divide and conquer to calculate \\(x^n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby fast_power.py<pre><code>def fast_pow(x: int, n: int) -&gt; int:\n    \"\"\"Exponentiation by squaring\"\"\"\n    if n == 0:\n        return 1\n    half = fast_pow(x, n // 2)\n    if n % 2 == 0:\n        return half * half\n    return half * half * x\n</code></pre> fast_power.cpp<pre><code>/* Exponentiation by squaring */\nint fastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = fastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.java<pre><code>/* Exponentiation by squaring */\nint fastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = fastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.cs<pre><code>/* Exponentiation by squaring */\nint FastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = FastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.go<pre><code>/* Exponentiation by squaring */\nfunc fastPow(x, n int) int {\n    if n == 0 {\n        return 1\n    }\n    half := fastPow(x, n/2)\n    if n%2 == 0 {\n        return half * half\n    }\n    return half * half * x\n}\n</code></pre> fast_power.swift<pre><code>/* Exponentiation by squaring */\nfunc fastPow(x: Int, n: Int) -&gt; Int {\n    if n == 0 {\n        return 1\n    }\n    let half = fastPow(x: x, n: n / 2)\n    if n % 2 == 0 {\n        return half * half\n    }\n    return half * half * x\n}\n</code></pre> fast_power.js<pre><code>/* Exponentiation by squaring */\nfunction fastPow(x, n) {\n    if (n === 0) {\n        return 1;\n    }\n    const half = fastPow(x, Math.floor(n / 2));\n    if (n % 2 === 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.ts<pre><code>/* Exponentiation by squaring */\nfunction fastPow(x: number, n: number): number {\n    if (n === 0) {\n        return 1;\n    }\n    const half = fastPow(x, Math.floor(n / 2));\n    if (n % 2 === 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.dart<pre><code>/* Exponentiation by squaring */\nint fastPow(int x, int n) {\n  if (n == 0) {\n    return 1;\n  }\n  int half = fastPow(x, n ~/ 2);\n  if (n % 2 == 0) {\n    return half * half;\n  }\n  return half * half * x;\n}\n</code></pre> fast_power.rs<pre><code>/* Exponentiation by squaring */\nfn fast_pow(x: i32, n: i32) -&gt; i32 {\n    if n == 0 {\n        return 1;\n    }\n    let half = fast_pow(x, n / 2);\n    if n % 2 == 0 {\n        return half * half;\n    }\n    half * half * x\n}\n</code></pre> fast_power.c<pre><code>/* Exponentiation by squaring */\nint fastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = fastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.kt<pre><code>/* Exponentiation by squaring */\nfun fastPow(x: Int, n: Int): Int {\n    if (n == 0) {\n        return 1\n    }\n    val half = fastPow(x, n / 2)\n    if (n % 2 == 0) {\n        return half * half\n    }\n    return half * half * x\n}\n</code></pre> fast_power.rb<pre><code>### Exponentiation by squaring ###\ndef fast_pow(x, n)\n  return 1 if n == 0\n\n  half = fast_pow(x, n / 2)\n  return half * half if n.even?\n\n  half * half * x\nend\n</code></pre> <p>Set <code>x = 3</code> and <code>n = 5</code>, and use this function to calculate the result:</p> <ol> <li>As the recursive calls proceed, which values does the argument <code>n</code> take in order?</li> <li>Starting from the deepest call, what value does each level return?</li> <li>Why should the recursive result be stored in <code>half</code> instead of calling the same subproblem once on each side of the multiplication?</li> </ol> Answer <ol> <li> <p>The argument takes the values <code>5 → 2 → 1 → 0</code>. The exponent is halved at each step until the base case is reached.</p> </li> <li> <p>When <code>n = 0</code>, the function returns 1. When <code>n = 1</code>, it returns \\(1×1×3=3\\).     When <code>n = 2</code>, it returns \\(3×3=9\\). When <code>n = 5</code>, it returns \\(9×9×3=243\\).</p> </li> <li> <p>If the same subproblem were called once on each side of the multiplication, the two recursive calls would perform exactly the same calculation.     Storing the result in <code>half</code> means that each level makes only one recursive call, so the recursion depth is about \\(\\log n\\).     Making two calls would cause a great deal of repeated computation.</p> </li> </ol>","path":["Chapter 12. Divide and Conquer","12.6   Exercises"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#3-split-traversal-sequences-into-left-and-right-subtrees","level":3,"title":"3.   Split Traversal Sequences into Left and Right Subtrees","text":"<p>A binary tree has no duplicate nodes. Its preorder and inorder traversals are:</p> <ul> <li>Preorder: <code>[A, B, D, E, C]</code></li> <li>Inorder: <code>[D, B, E, A, C]</code></li> </ul> <p>Split the sequences only once, at the root. You do not need to continue recursively or draw the whole tree:</p> <ol> <li>Which node is the root?</li> <li>Which subsequences of the inorder traversal correspond to the left and right subtrees?</li> <li>Which subsequences of the preorder traversal correspond to the left and right subtrees? Which nodes are the root's direct children?</li> </ol> Answer <ol> <li> <p>The first node in a preorder traversal is the root, so the root is <code>A</code>.</p> </li> <li> <p><code>A</code> divides the inorder traversal into two parts: <code>[D, B, E]</code> for the left subtree and <code>[C]</code> for the right subtree.</p> </li> <li> <p>The left subtree contains 3 nodes, so the 3 preorder elements after the root <code>A</code> belong to the left subtree,     giving <code>[B, D, E]</code>. The remaining <code>[C]</code> belongs to the right subtree.     Therefore, the root's left child is <code>B</code>, and its right child is <code>C</code>.</p> </li> </ol>","path":["Chapter 12. Divide and Conquer","12.6   Exercises"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1262-programming-exercises","level":2,"title":"12.6.2   Programming Exercises","text":"","path":["Chapter 12. Divide and Conquer","12.6   Exercises"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1-exponentiation-by-squaring","level":3,"title":"1.   Exponentiation by Squaring","text":"<p>Given a real number <code>x</code> and an integer <code>n</code>, calculate \\(x^n\\) without calling the language's built-in power function. Use recursive divide and conquer: halve the exponent at each step and reuse the result of the subproblem already calculated. This exercise defines \\(x^0=1\\), including when <code>x = 0</code>. When <code>n &lt; 0</code>, <code>x != 0</code> is guaranteed, and the answer can be transformed into \\((1/x)^{-n}\\).</p> Hints <ol> <li>When n is 0, the answer is 1</li> <li>After calculating x to the power n // 2, store the result in half rather than making the recursive call a second time</li> <li>When n &lt; 0, first change x to 1 / x and then change n to -n; in C++ or Java, first convert n to a 64-bit integer to avoid overflow when negating the smallest 32-bit integer</li> </ol> <p>LeetCode</p>","path":["Chapter 12. Divide and Conquer","12.6   Exercises"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/","level":1,"title":"12.4   Tower of Hanoi Problem","text":"<p>In merge sort and building binary trees, we decompose the original problem into two subproblems, each half the size of the original problem. However, for the Tower of Hanoi problem, we adopt a different decomposition strategy.</p> <p>Question</p> <p>Given three pillars, denoted as <code>A</code>, <code>B</code>, and <code>C</code>. Initially, pillar <code>A</code> has \\(n\\) discs stacked on it, arranged from top to bottom in ascending order of size. Our task is to move these \\(n\\) discs to pillar <code>C</code> while maintaining their original order (as shown in Figure 12-10). The following rules must be followed when moving the discs.</p> <ol> <li>A disc can only be taken from the top of one pillar and placed on top of another pillar.</li> <li>Only one disc can be moved at a time.</li> <li>A smaller disc must always be on top of a larger disc.</li> </ol> <p></p> <p> Figure 12-10   Example of the Tower of Hanoi problem </p> <p>We denote the Tower of Hanoi problem of size \\(i\\) as \\(f(i)\\). For example, \\(f(3)\\) represents moving \\(3\\) discs from <code>A</code> to <code>C</code>.</p>","path":["Chapter 12. Divide and Conquer","12.4   Tower of Hanoi Problem"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/#1-considering-the-base-cases","level":3,"title":"1.   Considering the Base Cases","text":"<p>As shown in Figure 12-11, for problem \\(f(1)\\), when there is only one disc, we can move it directly from <code>A</code> to <code>C</code>.</p> &lt;1&gt;&lt;2&gt; <p></p> <p></p> <p> Figure 12-11   Solution for a problem of size 1 </p> <p>As shown in Figure 12-12, for problem \\(f(2)\\), when there are two discs, since we must always keep the smaller disc on top of the larger disc, we need to use <code>B</code> to assist in the move.</p> <ol> <li>First, move the smaller disc from <code>A</code> to <code>B</code>.</li> <li>Then move the larger disc from <code>A</code> to <code>C</code>.</li> <li>Finally, move the smaller disc from <code>B</code> to <code>C</code>.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 12-12   Solution for a problem of size 2 </p> <p>The process of solving problem \\(f(2)\\) can be summarized as: moving two discs from <code>A</code> to <code>C</code> with the help of <code>B</code>. Here, <code>C</code> is called the target pillar, and <code>B</code> is called the buffer pillar.</p>","path":["Chapter 12. Divide and Conquer","12.4   Tower of Hanoi Problem"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/#2-subproblem-decomposition","level":3,"title":"2.   Subproblem Decomposition","text":"<p>For problem \\(f(3)\\), when there are three discs, the situation becomes slightly more complex.</p> <p>Since we already know the solutions to \\(f(1)\\) and \\(f(2)\\), we can think from a divide and conquer perspective, treating the top two discs on <code>A</code> as a whole, and execute the steps shown in Figure 12-13. This successfully moves the three discs from <code>A</code> to <code>C</code>.</p> <ol> <li>Let <code>B</code> be the target pillar and <code>C</code> be the buffer pillar, and move two discs from <code>A</code> to <code>B</code>.</li> <li>Move the remaining disc from <code>A</code> directly to <code>C</code>.</li> <li>Let <code>C</code> be the target pillar and <code>A</code> be the buffer pillar, and move two discs from <code>B</code> to <code>C</code>.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 12-13   Solution for a problem of size 3 </p> <p>Essentially, we divide problem \\(f(3)\\) into two subproblems \\(f(2)\\) and one subproblem \\(f(1)\\). By solving these three subproblems in order, the original problem is solved. This shows that the subproblems are independent and their solutions can be merged.</p> <p>From this, we can summarize the divide and conquer strategy for solving the Tower of Hanoi problem shown in Figure 12-14: divide the original problem \\(f(n)\\) into two subproblems \\(f(n-1)\\) and one subproblem \\(f(1)\\), and solve these three subproblems in the following order.</p> <ol> <li>Move \\(n-1\\) discs from <code>A</code> to <code>B</code> with the help of <code>C</code>.</li> <li>Move the remaining \\(1\\) disc directly from <code>A</code> to <code>C</code>.</li> <li>Move \\(n-1\\) discs from <code>B</code> to <code>C</code> with the help of <code>A</code>.</li> </ol> <p>For these two subproblems \\(f(n-1)\\), we can recursively divide them in the same way until reaching the smallest subproblem \\(f(1)\\). The solution to \\(f(1)\\) is known and requires only one move operation.</p> <p></p> <p> Figure 12-14   Divide and conquer strategy for solving the Tower of Hanoi problem </p>","path":["Chapter 12. Divide and Conquer","12.4   Tower of Hanoi Problem"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/#3-code-implementation","level":3,"title":"3.   Code Implementation","text":"<p>In the code, we declare a recursive function <code>dfs(i, src, buf, tar)</code>, whose purpose is to move the top \\(i\\) discs from pillar <code>src</code> to target pillar <code>tar</code> with the help of buffer pillar <code>buf</code>:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hanota.py<pre><code>def move(src: list[int], tar: list[int]):\n    \"\"\"Move a disk\"\"\"\n    # Take out a disk from the top of src\n    pan = src.pop()\n    # Place the disk on top of tar\n    tar.append(pan)\n\ndef dfs(i: int, src: list[int], buf: list[int], tar: list[int]):\n    \"\"\"Solve the Tower of Hanoi problem f(i)\"\"\"\n    # If there is only one disk left in src, move it directly to tar\n    if i == 1:\n        move(src, tar)\n        return\n    # Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, tar, buf)\n    # Subproblem f(1): move the remaining disk from src to tar\n    move(src, tar)\n    # Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, src, tar)\n\ndef solve_hanota(A: list[int], B: list[int], C: list[int]):\n    \"\"\"Solve the Tower of Hanoi problem\"\"\"\n    n = len(A)\n    # Move the top n disks from A to C using B\n    dfs(n, A, B, C)\n</code></pre> hanota.cpp<pre><code>/* Move a disk */\nvoid move(vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;tar) {\n    // Take out a disk from the top of src\n    int pan = src.back();\n    src.pop_back();\n    // Place the disk on top of tar\n    tar.push_back(pan);\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nvoid dfs(int i, vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;buf, vector&lt;int&gt; &amp;tar) {\n    // If there is only one disk left in src, move it directly to tar\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, tar, buf);\n    // Subproblem f(1): move the remaining disk from src to tar\n    move(src, tar);\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, src, tar);\n}\n\n/* Solve the Tower of Hanoi problem */\nvoid solveHanota(vector&lt;int&gt; &amp;A, vector&lt;int&gt; &amp;B, vector&lt;int&gt; &amp;C) {\n    int n = A.size();\n    // Move the top n disks from A to C using B\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.java<pre><code>/* Move a disk */\nvoid move(List&lt;Integer&gt; src, List&lt;Integer&gt; tar) {\n    // Take out a disk from the top of src\n    Integer pan = src.remove(src.size() - 1);\n    // Place the disk on top of tar\n    tar.add(pan);\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nvoid dfs(int i, List&lt;Integer&gt; src, List&lt;Integer&gt; buf, List&lt;Integer&gt; tar) {\n    // If there is only one disk left in src, move it directly to tar\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, tar, buf);\n    // Subproblem f(1): move the remaining disk from src to tar\n    move(src, tar);\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, src, tar);\n}\n\n/* Solve the Tower of Hanoi problem */\nvoid solveHanota(List&lt;Integer&gt; A, List&lt;Integer&gt; B, List&lt;Integer&gt; C) {\n    int n = A.size();\n    // Move the top n disks from A to C using B\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.cs<pre><code>/* Move a disk */\nvoid Move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n    // Take out a disk from the top of src\n    int pan = src[^1];\n    src.RemoveAt(src.Count - 1);\n    // Place the disk on top of tar\n    tar.Add(pan);\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nvoid DFS(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n    // If there is only one disk left in src, move it directly to tar\n    if (i == 1) {\n        Move(src, tar);\n        return;\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    DFS(i - 1, src, tar, buf);\n    // Subproblem f(1): move the remaining disk from src to tar\n    Move(src, tar);\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    DFS(i - 1, buf, src, tar);\n}\n\n/* Solve the Tower of Hanoi problem */\nvoid SolveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n    int n = A.Count;\n    // Move the top n disks from A to C using B\n    DFS(n, A, B, C);\n}\n</code></pre> hanota.go<pre><code>/* Move a disk */\nfunc move(src, tar *list.List) {\n    // Take out a disk from the top of src\n    pan := src.Back()\n    // Place the disk on top of tar\n    tar.PushBack(pan.Value)\n    // Remove top disk from src\n    src.Remove(pan)\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nfunc dfsHanota(i int, src, buf, tar *list.List) {\n    // If there is only one disk left in src, move it directly to tar\n    if i == 1 {\n        move(src, tar)\n        return\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfsHanota(i-1, src, tar, buf)\n    // Subproblem f(1): move the remaining disk from src to tar\n    move(src, tar)\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfsHanota(i-1, buf, src, tar)\n}\n\n/* Solve the Tower of Hanoi problem */\nfunc solveHanota(A, B, C *list.List) {\n    n := A.Len()\n    // Move the top n disks from A to C using B\n    dfsHanota(n, A, B, C)\n}\n</code></pre> hanota.swift<pre><code>/* Move a disk */\nfunc move(src: inout [Int], tar: inout [Int]) {\n    // Take out a disk from the top of src\n    let pan = src.popLast()!\n    // Place the disk on top of tar\n    tar.append(pan)\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nfunc dfs(i: Int, src: inout [Int], buf: inout [Int], tar: inout [Int]) {\n    // If there is only one disk left in src, move it directly to tar\n    if i == 1 {\n        move(src: &amp;src, tar: &amp;tar)\n        return\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i: i - 1, src: &amp;src, buf: &amp;tar, tar: &amp;buf)\n    // Subproblem f(1): move the remaining disk from src to tar\n    move(src: &amp;src, tar: &amp;tar)\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i: i - 1, src: &amp;buf, buf: &amp;src, tar: &amp;tar)\n}\n\n/* Solve the Tower of Hanoi problem */\nfunc solveHanota(A: inout [Int], B: inout [Int], C: inout [Int]) {\n    let n = A.count\n    // The tail of the list is the top of the rod\n    // Move top n disks from src to C using B\n    dfs(i: n, src: &amp;A, buf: &amp;B, tar: &amp;C)\n}\n</code></pre> hanota.js<pre><code>/* Move a disk */\nfunction move(src, tar) {\n    // Take out a disk from the top of src\n    const pan = src.pop();\n    // Place the disk on top of tar\n    tar.push(pan);\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nfunction dfs(i, src, buf, tar) {\n    // If there is only one disk left in src, move it directly to tar\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, tar, buf);\n    // Subproblem f(1): move the remaining disk from src to tar\n    move(src, tar);\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, src, tar);\n}\n\n/* Solve the Tower of Hanoi problem */\nfunction solveHanota(A, B, C) {\n    const n = A.length;\n    // Move the top n disks from A to C using B\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.ts<pre><code>/* Move a disk */\nfunction move(src: number[], tar: number[]): void {\n    // Take out a disk from the top of src\n    const pan = src.pop();\n    // Place the disk on top of tar\n    tar.push(pan);\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nfunction dfs(i: number, src: number[], buf: number[], tar: number[]): void {\n    // If there is only one disk left in src, move it directly to tar\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, tar, buf);\n    // Subproblem f(1): move the remaining disk from src to tar\n    move(src, tar);\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, src, tar);\n}\n\n/* Solve the Tower of Hanoi problem */\nfunction solveHanota(A: number[], B: number[], C: number[]): void {\n    const n = A.length;\n    // Move the top n disks from A to C using B\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.dart<pre><code>/* Move a disk */\nvoid move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n  // Take out a disk from the top of src\n  int pan = src.removeLast();\n  // Place the disk on top of tar\n  tar.add(pan);\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nvoid dfs(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n  // If there is only one disk left in src, move it directly to tar\n  if (i == 1) {\n    move(src, tar);\n    return;\n  }\n  // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n  dfs(i - 1, src, tar, buf);\n  // Subproblem f(1): move the remaining disk from src to tar\n  move(src, tar);\n  // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n  dfs(i - 1, buf, src, tar);\n}\n\n/* Solve the Tower of Hanoi problem */\nvoid solveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n  int n = A.length;\n  // Move the top n disks from A to C using B\n  dfs(n, A, B, C);\n}\n</code></pre> hanota.rs<pre><code>/* Move a disk */\nfn move_pan(src: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // Take out a disk from the top of src\n    let pan = src.pop().unwrap();\n    // Place the disk on top of tar\n    tar.push(pan);\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nfn dfs(i: i32, src: &amp;mut Vec&lt;i32&gt;, buf: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // If there is only one disk left in src, move it directly to tar\n    if i == 1 {\n        move_pan(src, tar);\n        return;\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, tar, buf);\n    // Subproblem f(1): move the remaining disk from src to tar\n    move_pan(src, tar);\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, src, tar);\n}\n\n/* Solve the Tower of Hanoi problem */\nfn solve_hanota(A: &amp;mut Vec&lt;i32&gt;, B: &amp;mut Vec&lt;i32&gt;, C: &amp;mut Vec&lt;i32&gt;) {\n    let n = A.len() as i32;\n    // Move the top n disks from A to C using B\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.c<pre><code>/* Move a disk */\nvoid move(int *src, int *srcSize, int *tar, int *tarSize) {\n    // Take out a disk from the top of src\n    int pan = src[*srcSize - 1];\n    src[*srcSize - 1] = 0;\n    (*srcSize)--;\n    // Place the disk on top of tar\n    tar[*tarSize] = pan;\n    (*tarSize)++;\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nvoid dfs(int i, int *src, int *srcSize, int *buf, int *bufSize, int *tar, int *tarSize) {\n    // If there is only one disk left in src, move it directly to tar\n    if (i == 1) {\n        move(src, srcSize, tar, tarSize);\n        return;\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, srcSize, tar, tarSize, buf, bufSize);\n    // Subproblem f(1): move the remaining disk from src to tar\n    move(src, srcSize, tar, tarSize);\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, bufSize, src, srcSize, tar, tarSize);\n}\n\n/* Solve the Tower of Hanoi problem */\nvoid solveHanota(int *A, int *ASize, int *B, int *BSize, int *C, int *CSize) {\n    // Move the top n disks from A to C using B\n    dfs(*ASize, A, ASize, B, BSize, C, CSize);\n}\n</code></pre> hanota.kt<pre><code>/* Move a disk */\nfun move(src: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // Take out a disk from the top of src\n    val pan = src.removeAt(src.size - 1)\n    // Place the disk on top of tar\n    tar.add(pan)\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nfun dfs(i: Int, src: MutableList&lt;Int&gt;, buf: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // If there is only one disk left in src, move it directly to tar\n    if (i == 1) {\n        move(src, tar)\n        return\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, tar, buf)\n    // Subproblem f(1): move the remaining disk from src to tar\n    move(src, tar)\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, src, tar)\n}\n\n/* Solve the Tower of Hanoi problem */\nfun solveHanota(A: MutableList&lt;Int&gt;, B: MutableList&lt;Int&gt;, C: MutableList&lt;Int&gt;) {\n    val n = A.size\n    // Move the top n disks from A to C using B\n    dfs(n, A, B, C)\n}\n</code></pre> hanota.rb<pre><code>### Move one disk ###\ndef move(src, tar)\n  # Take out a disk from the top of src\n  pan = src.pop\n  # Place the disk on top of tar\n  tar &lt;&lt; pan\nend\n\n### Solve Tower of Hanoi f(i) ###\ndef dfs(i, src, buf, tar)\n  # If there is only one disk left in src, move it directly to tar\n  if i == 1\n    move(src, tar)\n    return\n  end\n\n  # Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n  dfs(i - 1, src, tar, buf)\n  # Subproblem f(1): move the remaining disk from src to tar\n  move(src, tar)\n  # Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n  dfs(i - 1, buf, src, tar)\nend\n\n### Solve Tower of Hanoi ###\ndef solve_hanota(_A, _B, _C)\n  n = _A.length\n  # Move the top n disks from A to C using B\n  dfs(n, _A, _B, _C)\nend\n</code></pre> <p>As shown in Figure 12-15, the Tower of Hanoi problem forms a recursion tree of height \\(n\\), where each node represents a subproblem corresponding to an invocation of the <code>dfs()</code> function, therefore the time complexity is \\(O(2^n)\\) and the space complexity is \\(O(n)\\).</p> <p></p> <p> Figure 12-15   Recursion tree of the Tower of Hanoi problem </p> <p>Quote</p> <p>The Tower of Hanoi problem originates from an ancient legend. In a temple in ancient India, monks had three tall diamond pillars and \\(64\\) golden discs of different sizes. The monks continuously moved the discs, believing that when the last disc was correctly placed, the world would come to an end.</p> <p>However, even if the monks moved one disc per second, it would take approximately \\(2^{64} \\approx 1.84×10^{19}\\) seconds, which is about \\(585\\) billion years, far exceeding current estimates of the age of the universe. Therefore, if this legend is true, we should not need to worry about the end of the world.</p>","path":["Chapter 12. Divide and Conquer","12.4   Tower of Hanoi Problem"],"tags":[]},{"location":"chapter_divide_and_conquer/summary/","level":1,"title":"12.5   Summary","text":"","path":["Chapter 12. Divide and Conquer","12.5   Summary"],"tags":[]},{"location":"chapter_divide_and_conquer/summary/#1-key-review","level":3,"title":"1.   Key Review","text":"<ul> <li>Divide and conquer is a common algorithm design strategy consisting of two phases, divide (partition) and conquer (merge), and is typically implemented recursively.</li> <li>The criteria for determining whether a problem is a divide and conquer problem include: whether the problem can be decomposed, whether subproblems are independent, and whether subproblems can be merged.</li> <li>Merge sort is a typical application of the divide and conquer strategy. It recursively divides an array into two equal-length subarrays until only one element remains, then merges them layer by layer to complete the sorting.</li> <li>Introducing the divide and conquer strategy can often improve algorithm efficiency. On one hand, it reduces the number of operations; on the other hand, it makes parallel optimization by the system easier.</li> <li>Divide and conquer can solve many algorithmic problems and is also widely used in data structures and algorithm design, making it ubiquitous.</li> <li>Compared to brute-force search, adaptive search is more efficient. Search algorithms with time complexity of \\(O(\\log n)\\) are typically implemented based on the divide and conquer strategy.</li> <li>Binary search is another typical application of divide and conquer. It does not include the step of merging solutions of subproblems. We can implement binary search through recursive divide and conquer.</li> <li>In the problem of building a binary tree, building the tree (original problem) can be divided into building the left subtree and right subtree (subproblems), which can be achieved by dividing the index intervals of the preorder and inorder traversals.</li> <li>In the Tower of Hanoi problem, a problem of size \\(n\\) can be divided into two subproblems of size \\(n-1\\) and one subproblem of size \\(1\\). After solving these three subproblems in order, the original problem is solved.</li> </ul>","path":["Chapter 12. Divide and Conquer","12.5   Summary"],"tags":[]},{"location":"chapter_dynamic_programming/","level":1,"title":"Chapter 14.   Dynamic Programming","text":"<p>Abstract</p> <p>Streams flow into rivers, rivers flow into the sea.</p> <p>Dynamic programming combines solutions to small problems into the answer to a large problem, leading us step by step to the other shore of problem-solving.</p>","path":["Chapter 14. Dynamic Programming","Chapter 14.   Dynamic Programming"],"tags":[]},{"location":"chapter_dynamic_programming/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>14.1   Introduction to Dynamic Programming</li> <li>14.2   Characteristics of Dynamic Programming Problems</li> <li>14.3   Dynamic Programming Problem-Solving Approach</li> <li>14.4   0-1 Knapsack Problem</li> <li>14.5   Unbounded Knapsack Problem</li> <li>14.6   Edit Distance Problem</li> <li>14.7   Summary</li> <li>14.8   Exercises</li> </ul>","path":["Chapter 14. Dynamic Programming","Chapter 14.   Dynamic Programming"],"tags":[]},{"location":"chapter_dynamic_programming/dp_problem_features/","level":1,"title":"14.2   Characteristics of Dynamic Programming Problems","text":"<p>In the previous section, we learned how dynamic programming solves the original problem by decomposing it into subproblems. In fact, subproblem decomposition is a general algorithmic approach, with different emphases in divide and conquer, dynamic programming, and backtracking.</p> <ul> <li>Divide and conquer algorithms recursively divide the original problem into multiple independent subproblems until the smallest subproblems are reached, and merge the solutions to the subproblems during backtracking to ultimately obtain the solution to the original problem.</li> <li>Dynamic programming also recursively decomposes problems, but the main difference from divide and conquer algorithms is that subproblems in dynamic programming are interdependent, and many overlapping subproblems appear during the decomposition process.</li> <li>Backtracking algorithms enumerate all possible solutions through trial and error, and avoid unnecessary search branches through pruning. The solution to the original problem consists of a series of decision steps, and we can regard the subsequence before each decision step as a subproblem.</li> </ul> <p>In fact, dynamic programming is commonly used to solve optimization problems, which not only contain overlapping subproblems but also have two other major characteristics: optimal substructure and no aftereffects.</p>","path":["Chapter 14. Dynamic Programming","14.2   Characteristics of Dynamic Programming Problems"],"tags":[]},{"location":"chapter_dynamic_programming/dp_problem_features/#1421-optimal-substructure","level":2,"title":"14.2.1   Optimal Substructure","text":"<p>We make a slight modification to the stair climbing problem to make it more suitable for demonstrating the concept of optimal substructure.</p> <p>Climbing stairs with minimum cost</p> <p>Given a staircase, you can climb \\(1\\) or \\(2\\) steps at a time, and each step is labeled with a non-negative integer representing the cost of stepping on it. Given a non-negative integer array \\(cost\\), where \\(cost[i]\\) represents the cost of the \\(i\\)-th step and \\(cost[0]\\) is the ground (starting point), what is the minimum cost required to reach the top?</p> <p>As shown in Figure 14-6, if the costs of the \\(1\\)st, \\(2\\)nd, and \\(3\\)rd steps are \\(1\\), \\(10\\), and \\(1\\) respectively, then climbing from the ground to the \\(3\\)rd step requires a minimum cost of \\(2\\).</p> <p></p> <p> Figure 14-6   Minimum cost to climb to the 3rd step </p> <p>Let \\(dp[i]\\) be the accumulated cost of climbing to the \\(i\\)-th step. Since the \\(i\\)-th step can only come from the \\(i-1\\)-th or \\(i-2\\)-th step, \\(dp[i]\\) can only equal \\(dp[i-1] + cost[i]\\) or \\(dp[i-2] + cost[i]\\). To minimize the cost, we should choose the smaller of the two:</p> \\[ dp[i] = \\min(dp[i-1], dp[i-2]) + cost[i] \\] <p>This leads us to the meaning of optimal substructure: the optimal solution to the original problem is constructed from the optimal solutions to the subproblems.</p> <p>This problem clearly has optimal substructure: we select the better one from the optimal solutions to the two subproblems \\(dp[i-1]\\) and \\(dp[i-2]\\), and use it to construct the optimal solution to the original problem \\(dp[i]\\).</p> <p>So, does the stair climbing problem from the previous section have optimal substructure? Its goal is to find the number of ways, which seems to be a counting problem, but if we change the question: \"Find the maximum number of ways\". We surprisingly discover that although the problem before and after modification are equivalent, the optimal substructure has emerged: the maximum number of ways for the \\(n\\)-th step equals the sum of the maximum number of ways for the \\(n-1\\)-th and \\(n-2\\)-th steps. Therefore, the interpretation of optimal substructure is quite flexible and will have different meanings in different problems.</p> <p>According to the state transition equation and the initial states \\(dp[1] = cost[1]\\) and \\(dp[2] = cost[2]\\), we can obtain the dynamic programming code:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp(cost: list[int]) -&gt; int:\n    \"\"\"Minimum cost climbing stairs: Dynamic programming\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    # Initialize dp table, used to store solutions to subproblems\n    dp = [0] * (n + 1)\n    # Initial state: preset the solution to the smallest subproblem\n    dp[1], dp[2] = cost[1], cost[2]\n    # State transition: gradually solve larger subproblems from smaller ones\n    for i in range(3, n + 1):\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    return dp[n]\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* Minimum cost climbing stairs: Dynamic programming */\nint minCostClimbingStairsDP(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // Initialize dp table, used to store solutions to subproblems\n    vector&lt;int&gt; dp(n + 1);\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* Minimum cost climbing stairs: Dynamic programming */\nint minCostClimbingStairsDP(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // Initialize dp table, used to store solutions to subproblems\n    int[] dp = new int[n + 1];\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* Minimum cost climbing stairs: Dynamic programming */\nint MinCostClimbingStairsDP(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // Initialize dp table, used to store solutions to subproblems\n    int[] dp = new int[n + 1];\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.Min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* Minimum cost climbing stairs: Dynamic programming */\nfunc minCostClimbingStairsDP(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    dp := make([]int, n+1)\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // State transition: gradually solve larger subproblems from smaller ones\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = min(dp[i-1], dp[i-2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* Minimum cost climbing stairs: Dynamic programming */\nfunc minCostClimbingStairsDP(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    var dp = Array(repeating: 0, count: n + 1)\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // State transition: gradually solve larger subproblems from smaller ones\n    for i in 3 ... n {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* Minimum cost climbing stairs: Dynamic programming */\nfunction minCostClimbingStairsDP(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    const dp = new Array(n + 1);\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* Minimum cost climbing stairs: Dynamic programming */\nfunction minCostClimbingStairsDP(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    const dp = new Array(n + 1);\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* Minimum cost climbing stairs: Dynamic programming */\nint minCostClimbingStairsDP(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  // Initialize dp table, used to store solutions to subproblems\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // Initial state: preset the solution to the smallest subproblem\n  dp[1] = cost[1];\n  dp[2] = cost[2];\n  // State transition: gradually solve larger subproblems from smaller ones\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n  }\n  return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* Minimum cost climbing stairs: Dynamic programming */\nfn min_cost_climbing_stairs_dp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    let mut dp = vec![-1; n + 1];\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // State transition: gradually solve larger subproblems from smaller ones\n    for i in 3..=n {\n        dp[i] = cmp::min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* Minimum cost climbing stairs: Dynamic programming */\nint minCostClimbingStairsDP(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // Initialize dp table, used to store solutions to subproblems\n    int *dp = calloc(n + 1, sizeof(int));\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = myMin(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    int res = dp[n];\n    // Free memory\n    free(dp);\n    return res;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* Minimum cost climbing stairs: Dynamic programming */\nfun minCostClimbingStairsDP(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    // Initialize dp table, used to store solutions to subproblems\n    val dp = IntArray(n + 1)\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (i in 3..n) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>### Minimum cost climbing stairs: DP ###\ndef min_cost_climbing_stairs_dp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  # Initialize dp table, used to store solutions to subproblems\n  dp = Array.new(n + 1, 0)\n  # Initial state: preset the solution to the smallest subproblem\n  dp[1], dp[2] = cost[1], cost[2]\n  # State transition: gradually solve larger subproblems from smaller ones\n  (3...(n + 1)).each { |i| dp[i] = [dp[i - 1], dp[i - 2]].min + cost[i] }\n  dp[n]\nend\n</code></pre> <p>Figure 14-7 shows the dynamic programming process for the above code.</p> <p></p> <p> Figure 14-7   Dynamic programming process for climbing stairs with minimum cost </p> <p>This problem can also be space-optimized, compressing from one dimension to zero, reducing the space complexity from \\(O(n)\\) to \\(O(1)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp_comp(cost: list[int]) -&gt; int:\n    \"\"\"Minimum cost climbing stairs: Space-optimized dynamic programming\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    a, b = cost[1], cost[2]\n    for i in range(3, n + 1):\n        a, b = b, min(a, b) + cost[i]\n    return b\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* Minimum cost climbing stairs: Space-optimized dynamic programming */\nint minCostClimbingStairsDPComp(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* Minimum cost climbing stairs: Space-optimized dynamic programming */\nint minCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* Minimum cost climbing stairs: Space-optimized dynamic programming */\nint MinCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.Min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* Minimum cost climbing stairs: Space-optimized dynamic programming */\nfunc minCostClimbingStairsDPComp(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // Initial state: preset the solution to the smallest subproblem\n    a, b := cost[1], cost[2]\n    // State transition: gradually solve larger subproblems from smaller ones\n    for i := 3; i &lt;= n; i++ {\n        tmp := b\n        b = min(a, tmp) + cost[i]\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* Minimum cost climbing stairs: Space-optimized dynamic programming */\nfunc minCostClimbingStairsDPComp(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    var (a, b) = (cost[1], cost[2])\n    for i in 3 ... n {\n        (a, b) = (b, min(a, b) + cost[i])\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* Minimum cost climbing stairs: Space-optimized dynamic programming */\nfunction minCostClimbingStairsDPComp(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* Minimum cost climbing stairs: Space-optimized dynamic programming */\nfunction minCostClimbingStairsDPComp(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* Minimum cost climbing stairs: Space-optimized dynamic programming */\nint minCostClimbingStairsDPComp(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  int a = cost[1], b = cost[2];\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = min(a, tmp) + cost[i];\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* Minimum cost climbing stairs: Space-optimized dynamic programming */\nfn min_cost_climbing_stairs_dp_comp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    };\n    let (mut a, mut b) = (cost[1], cost[2]);\n    for i in 3..=n {\n        let tmp = b;\n        b = cmp::min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    b\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* Minimum cost climbing stairs: Space-optimized dynamic programming */\nint minCostClimbingStairsDPComp(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = myMin(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* Minimum cost climbing stairs: Space-optimized dynamic programming */\nfun minCostClimbingStairsDPComp(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    var a = cost[1]\n    var b = cost[2]\n    for (i in 3..n) {\n        val tmp = b\n        b = min(a, tmp) + cost[i]\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>### Minimum cost climbing stairs: DP ###\ndef min_cost_climbing_stairs_dp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  # Initialize dp table, used to store solutions to subproblems\n  dp = Array.new(n + 1, 0)\n  # Initial state: preset the solution to the smallest subproblem\n  dp[1], dp[2] = cost[1], cost[2]\n  # State transition: gradually solve larger subproblems from smaller ones\n  (3...(n + 1)).each { |i| dp[i] = [dp[i - 1], dp[i - 2]].min + cost[i] }\n  dp[n]\nend\n\n# Minimum cost climbing stairs: Space-optimized dynamic programming\ndef min_cost_climbing_stairs_dp_comp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  a, b = cost[1], cost[2]\n  (3...(n + 1)).each { |i| a, b = b, [a, b].min + cost[i] }\n  b\nend\n</code></pre>","path":["Chapter 14. Dynamic Programming","14.2   Characteristics of Dynamic Programming Problems"],"tags":[]},{"location":"chapter_dynamic_programming/dp_problem_features/#1422-no-aftereffects","level":2,"title":"14.2.2   No Aftereffects","text":"<p>No aftereffects is one of the important characteristics that enable dynamic programming to solve problems effectively. Its definition is: given a certain state, its future development is only related to the current state and has nothing to do with all past states.</p> <p>Taking the stair climbing problem as an example, given state \\(i\\), it will develop into states \\(i+1\\) and \\(i+2\\), corresponding to jumping \\(1\\) step and jumping \\(2\\) steps, respectively. When making these two choices, we do not need to consider the states before state \\(i\\), as they have no effect on the future of state \\(i\\).</p> <p>However, if we add a constraint to the stair climbing problem, the situation changes.</p> <p>Climbing stairs with constraint</p> <p>Given a staircase with \\(n\\) steps, where you can climb \\(1\\) or \\(2\\) steps at a time, but you cannot jump \\(1\\) step in two consecutive rounds. How many ways are there to climb to the top?</p> <p>As shown in Figure 14-8, there are only \\(2\\) feasible ways to climb to the \\(3\\)rd step. The path with three consecutive \\(1\\)-step jumps does not satisfy the constraint and is therefore discarded.</p> <p></p> <p> Figure 14-8   Number of ways to climb to the 3rd step with constraint </p> <p>In this problem, if the previous round was a jump of \\(1\\) step, then the next round must jump \\(2\\) steps. This means that the next choice cannot be determined solely by the current state (current stair step number), but also depends on the previous state (the stair step number from the previous round).</p> <p>It is not difficult to see that this problem no longer satisfies no aftereffects, and the state transition equation \\(dp[i] = dp[i-1] + dp[i-2]\\) also fails, because \\(dp[i-1]\\) represents jumping \\(1\\) step in this round, but it includes many solutions where \"the previous round was a jump of \\(1\\) step\", which cannot be directly counted in \\(dp[i]\\) to satisfy the constraint.</p> <p>For this reason, we need to expand the state definition: state \\([i, j]\\) represents being on the \\(i\\)-th step with the previous round having jumped \\(j\\) steps, where \\(j \\in \\{1, 2\\}\\). This state definition effectively distinguishes whether the previous round was a jump of \\(1\\) step or \\(2\\) steps, allowing us to determine where the current state came from.</p> <ul> <li>When the previous round jumped \\(1\\) step, the round before that could only choose to jump \\(2\\) steps, i.e., \\(dp[i, 1]\\) can only transition from \\(dp[i-1, 2]\\).</li> <li>When the previous round jumped \\(2\\) steps, the round before that could choose to jump \\(1\\) step or \\(2\\) steps, i.e., \\(dp[i, 2]\\) can transition from \\(dp[i-2, 1]\\) or \\(dp[i-2, 2]\\).</li> </ul> <p>As shown in Figure 14-9, under this definition, \\(dp[i, j]\\) represents the number of ways for state \\([i, j]\\). The state transition equation is then:</p> \\[ \\begin{cases} dp[i, 1] = dp[i-1, 2] \\\\ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] \\end{cases} \\] <p></p> <p> Figure 14-9   Recurrence relation considering constraints </p> <p>Finally, return \\(dp[n, 1] + dp[n, 2]\\), where the sum of the two represents the total number of ways to climb to the \\(n\\)-th step:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_constraint_dp.py<pre><code>def climbing_stairs_constraint_dp(n: int) -&gt; int:\n    \"\"\"Climbing stairs with constraint: Dynamic programming\"\"\"\n    if n == 1 or n == 2:\n        return 1\n    # Initialize dp table, used to store solutions to subproblems\n    dp = [[0] * 3 for _ in range(n + 1)]\n    # Initial state: preset the solution to the smallest subproblem\n    dp[1][1], dp[1][2] = 1, 0\n    dp[2][1], dp[2][2] = 0, 1\n    # State transition: gradually solve larger subproblems from smaller ones\n    for i in range(3, n + 1):\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    return dp[n][1] + dp[n][2]\n</code></pre> climbing_stairs_constraint_dp.cpp<pre><code>/* Climbing stairs with constraint: Dynamic programming */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(3, 0));\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.java<pre><code>/* Climbing stairs with constraint: Dynamic programming */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    int[][] dp = new int[n + 1][3];\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.cs<pre><code>/* Climbing stairs with constraint: Dynamic programming */\nint ClimbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    int[,] dp = new int[n + 1, 3];\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1, 1] = 1;\n    dp[1, 2] = 0;\n    dp[2, 1] = 0;\n    dp[2, 2] = 1;\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i, 1] = dp[i - 1, 2];\n        dp[i, 2] = dp[i - 2, 1] + dp[i - 2, 2];\n    }\n    return dp[n, 1] + dp[n, 2];\n}\n</code></pre> climbing_stairs_constraint_dp.go<pre><code>/* Climbing stairs with constraint: Dynamic programming */\nfunc climbingStairsConstraintDP(n int) int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    dp := make([][3]int, n+1)\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // State transition: gradually solve larger subproblems from smaller ones\n    for i := 3; i &lt;= n; i++ {\n        dp[i][1] = dp[i-1][2]\n        dp[i][2] = dp[i-2][1] + dp[i-2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.swift<pre><code>/* Climbing stairs with constraint: Dynamic programming */\nfunc climbingStairsConstraintDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    var dp = Array(repeating: Array(repeating: 0, count: 3), count: n + 1)\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // State transition: gradually solve larger subproblems from smaller ones\n    for i in 3 ... n {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.js<pre><code>/* Climbing stairs with constraint: Dynamic programming */\nfunction climbingStairsConstraintDP(n) {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    const dp = Array.from(new Array(n + 1), () =&gt; new Array(3));\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.ts<pre><code>/* Climbing stairs with constraint: Dynamic programming */\nfunction climbingStairsConstraintDP(n: number): number {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(3));\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.dart<pre><code>/* Climbing stairs with constraint: Dynamic programming */\nint climbingStairsConstraintDP(int n) {\n  if (n == 1 || n == 2) {\n    return 1;\n  }\n  // Initialize dp table, used to store solutions to subproblems\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(3, 0));\n  // Initial state: preset the solution to the smallest subproblem\n  dp[1][1] = 1;\n  dp[1][2] = 0;\n  dp[2][1] = 0;\n  dp[2][2] = 1;\n  // State transition: gradually solve larger subproblems from smaller ones\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i][1] = dp[i - 1][2];\n    dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n  }\n  return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.rs<pre><code>/* Climbing stairs with constraint: Dynamic programming */\nfn climbing_stairs_constraint_dp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return 1;\n    };\n    // Initialize dp table, used to store solutions to subproblems\n    let mut dp = vec![vec![-1; 3]; n + 1];\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // State transition: gradually solve larger subproblems from smaller ones\n    for i in 3..=n {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.c<pre><code>/* Climbing stairs with constraint: Dynamic programming */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(3, sizeof(int));\n    }\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    int res = dp[n][1] + dp[n][2];\n    // Free memory\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> climbing_stairs_constraint_dp.kt<pre><code>/* Climbing stairs with constraint: Dynamic programming */\nfun climbingStairsConstraintDP(n: Int): Int {\n    if (n == 1 || n == 2) {\n        return 1\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    val dp = Array(n + 1) { IntArray(3) }\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (i in 3..n) {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.rb<pre><code>### Climbing stairs with constraint: DP ###\ndef climbing_stairs_constraint_dp(n)\n  return 1 if n == 1 || n == 2\n\n  # Initialize dp table, used to store solutions to subproblems\n  dp = Array.new(n + 1) { Array.new(3, 0) }\n  # Initial state: preset the solution to the smallest subproblem\n  dp[1][1], dp[1][2] = 1, 0\n  dp[2][1], dp[2][2] = 0, 1\n  # State transition: gradually solve larger subproblems from smaller ones\n  for i in 3...(n + 1)\n    dp[i][1] = dp[i - 1][2]\n    dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n  end\n\n  dp[n][1] + dp[n][2]\nend\n</code></pre> <p>In the above case, since we only need to consider one more preceding state, we can still make the problem satisfy no aftereffects by expanding the state definition. However, some problems have very severe \"aftereffects\".</p> <p>Climbing stairs with obstacle generation</p> <p>Given a staircase with \\(n\\) steps, where you can climb \\(1\\) or \\(2\\) steps at a time. Whenever you reach the \\(i\\)-th step, the system automatically places an obstacle on the \\(2i\\)-th step, and no subsequent round is allowed to jump to the \\(2i\\)-th step. For example, if the first two rounds jump to the \\(2\\)nd and \\(3\\)rd steps, then afterwards you cannot jump to the \\(4\\)th and \\(6\\)th steps. How many ways are there to climb to the top?</p> <p>In this problem, the next jump depends on all past states, because each jump places obstacles on higher steps, affecting future jumps. For such problems, dynamic programming is often difficult to solve.</p> <p>In fact, many complex combinatorial optimization problems (such as the traveling salesman problem) do not satisfy no aftereffects. For such problems, we usually use other methods, such as heuristic search, genetic algorithms, and reinforcement learning, to obtain usable locally optimal solutions within a limited time.</p>","path":["Chapter 14. Dynamic Programming","14.2   Characteristics of Dynamic Programming Problems"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/","level":1,"title":"14.3   Dynamic Programming Problem-Solving Approach","text":"<p>The previous two sections introduced the main characteristics of dynamic programming problems. Next, let us explore two more practical issues together.</p> <ol> <li>How to determine whether a problem is a dynamic programming problem?</li> <li>What is the complete process for solving a dynamic programming problem, and where should we start?</li> </ol>","path":["Chapter 14. Dynamic Programming","14.3   Dynamic Programming Problem-Solving Approach"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1431-problem-identification","level":2,"title":"14.3.1   Problem Identification","text":"<p>Generally speaking, if a problem contains overlapping subproblems, optimal substructure, and satisfies no aftereffects, then it is usually suitable for solving with dynamic programming. However, it is difficult to directly extract these characteristics from the problem description. Therefore, we usually relax the conditions and first observe whether the problem is suitable for solving with backtracking (exhaustive search).</p> <p>Problems suitable for solving with backtracking usually satisfy the \"decision tree model\", which means the problem can be described using a tree structure, where each node represents a decision and each path represents a sequence of decisions.</p> <p>In other words, if a problem contains an explicit concept of decisions, and the solution is generated through a series of decisions, then it satisfies the decision tree model and can usually be solved using backtracking.</p> <p>On this basis, dynamic programming problems also have some positive indicators.</p> <ul> <li>The problem contains descriptions such as maximum (minimum) or most (least), indicating optimization.</li> <li>The problem's state can be represented using a list, multi-dimensional matrix, or tree, and a state has a recurrence relation with its surrounding states.</li> </ul> <p>Correspondingly, there are also some negative indicators.</p> <ul> <li>The goal of the problem is to find all possible solutions, rather than finding the optimal solution.</li> <li>The problem description has obvious permutation and combination characteristics, requiring the return of specific multiple solutions.</li> </ul> <p>If a problem satisfies the decision tree model and has relatively obvious positive indicators, we can assume it is a dynamic programming problem and verify that assumption during the solving process.</p>","path":["Chapter 14. Dynamic Programming","14.3   Dynamic Programming Problem-Solving Approach"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1432-problem-solving-steps","level":2,"title":"14.3.2   Problem-Solving Steps","text":"<p>The problem-solving process for dynamic programming varies depending on the nature and difficulty of the problem, but generally follows these steps: describe decisions, define states, establish the \\(dp\\) table, derive state transition equations, determine boundary conditions, etc.</p> <p>To illustrate the problem-solving steps more vividly, we use a classic problem \"minimum path sum\" as an example.</p> <p>Question</p> <p>Given an \\(n \\times m\\) two-dimensional grid <code>grid</code> in which each cell contains a non-negative integer representing its cost, a robot starts from the top-left cell and can only move down or right at each step until reaching the bottom-right cell. Return the minimum path sum from the top-left to the bottom-right.</p> <p>Figure 14-10 shows an example where the minimum path sum for the given grid is \\(13\\).</p> <p></p> <p> Figure 14-10   Minimum path sum example data </p> <p>Step 1: Think about the decisions in each round, define the state, and thus obtain the \\(dp\\) table</p> <p>The decision in each round of this problem is to move one step down or right from the current cell. Let the row and column indices of the current cell be \\([i, j]\\). After moving down or right, the indices become \\([i+1, j]\\) or \\([i, j+1]\\). Therefore, the state should include two variables, the row index and column index, denoted as \\([i, j]\\).</p> <p>State \\([i, j]\\) corresponds to the subproblem: the minimum path sum from the starting point \\([0, 0]\\) to \\([i, j]\\), denoted as \\(dp[i, j]\\).</p> <p>From this, we obtain the two-dimensional \\(dp\\) matrix shown in Figure 14-11, whose size is the same as the input grid \\(grid\\).</p> <p></p> <p> Figure 14-11   State definition and dp table </p> <p>Note</p> <p>The dynamic programming and backtracking processes can be described as a sequence of decisions, and the state consists of all decision variables. It should contain all variables describing the progress of problem-solving, and should contain sufficient information to derive the next state.</p> <p>Each state corresponds to a subproblem, and we define a \\(dp\\) table to store the solutions to all subproblems. Each independent variable of the state is a dimension of the \\(dp\\) table. Essentially, the \\(dp\\) table is a mapping between states and solutions to subproblems.</p> <p>Step 2: Identify the optimal substructure, and then derive the state transition equation</p> <p>For state \\([i, j]\\), it can only transition from the cell above \\([i-1, j]\\) or the cell to the left \\([i, j-1]\\). Therefore, the optimal substructure is: the minimum path sum to reach \\([i, j]\\) is determined by the smaller of the minimum path sums of \\([i, j-1]\\) and \\([i-1, j]\\).</p> <p>Based on the above analysis, the state transition equation shown in Figure 14-12 can be derived:</p> \\[ dp[i, j] = \\min(dp[i-1, j], dp[i, j-1]) + grid[i, j] \\] <p></p> <p> Figure 14-12   Optimal substructure and state transition equation </p> <p>Note</p> <p>Based on the defined \\(dp\\) table, think about the relationship between the original problem and subproblems, and find the method to construct the optimal solution to the original problem from the optimal solutions to the subproblems, which is the optimal substructure.</p> <p>Once we identify the optimal substructure, we can use it to construct the state transition equation.</p> <p>Step 3: Determine boundary conditions and state transition order</p> <p>In this problem, states in the first row can only come from the state to their left, and states in the first column can only come from the state above them. Therefore, the first row \\(i = 0\\) and first column \\(j = 0\\) are boundary conditions.</p> <p>As shown in Figure 14-13, since each cell transitions from the cell to its left and the cell above it, we use loops to traverse the matrix, with the outer loop traversing rows and the inner loop traversing columns.</p> <p></p> <p> Figure 14-13   Boundary conditions and state transition order </p> <p>Note</p> <p>Boundary conditions in dynamic programming are used to initialize the \\(dp\\) table, while in search they are used for pruning.</p> <p>The core of state transition order is to ensure that when computing the solution to the current problem, all the smaller subproblems it depends on have already been computed correctly.</p> <p>Based on the above analysis, we can directly write the dynamic programming code. However, subproblem decomposition is a top-down approach, so implementing in the order \"brute force search \\(\\rightarrow\\) memoization \\(\\rightarrow\\) dynamic programming\" is more aligned with thinking habits.</p>","path":["Chapter 14. Dynamic Programming","14.3   Dynamic Programming Problem-Solving Approach"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1-method-1-brute-force-search","level":3,"title":"1.   Method 1: Brute Force Search","text":"<p>Starting from state \\([i, j]\\), we continuously decompose it into smaller states \\([i-1, j]\\) and \\([i, j-1]\\). The recursive function includes the following elements.</p> <ul> <li>Recursive parameters: state \\([i, j]\\).</li> <li>Return value: minimum path sum from \\([0, 0]\\) to \\([i, j]\\), which is \\(dp[i, j]\\).</li> <li>Termination condition: when \\(i = 0\\) and \\(j = 0\\), return cost \\(grid[0, 0]\\).</li> <li>Pruning: when \\(i &lt; 0\\) or \\(j &lt; 0\\), the index is out of bounds, return cost \\(+\\infty\\), representing infeasibility.</li> </ul> <p>The implementation code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dfs(grid: list[list[int]], i: int, j: int) -&gt; int:\n    \"\"\"Minimum path sum: Brute-force search\"\"\"\n    # If it's the top-left cell, terminate the search\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # If row or column index is out of bounds, return +∞ cost\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)\n    up = min_path_sum_dfs(grid, i - 1, j)\n    left = min_path_sum_dfs(grid, i, j - 1)\n    # Return the minimum path cost from top-left to (i, j)\n    return min(left, up) + grid[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* Minimum path sum: Brute-force search */\nint minPathSumDFS(vector&lt;vector&lt;int&gt;&gt; &amp;grid, int i, int j) {\n    // If it's the top-left cell, terminate the search\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // Return the minimum path cost from top-left to (i, j)\n    return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.java<pre><code>/* Minimum path sum: Brute-force search */\nint minPathSumDFS(int[][] grid, int i, int j) {\n    // If it's the top-left cell, terminate the search\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // Return the minimum path cost from top-left to (i, j)\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* Minimum path sum: Brute-force search */\nint MinPathSumDFS(int[][] grid, int i, int j) {\n    // If it's the top-left cell, terminate the search\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)\n    int up = MinPathSumDFS(grid, i - 1, j);\n    int left = MinPathSumDFS(grid, i, j - 1);\n    // Return the minimum path cost from top-left to (i, j)\n    return Math.Min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* Minimum path sum: Brute-force search */\nfunc minPathSumDFS(grid [][]int, i, j int) int {\n    // If it's the top-left cell, terminate the search\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)\n    up := minPathSumDFS(grid, i-1, j)\n    left := minPathSumDFS(grid, i, j-1)\n    // Return the minimum path cost from top-left to (i, j)\n    return int(math.Min(float64(left), float64(up))) + grid[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* Minimum path sum: Brute-force search */\nfunc minPathSumDFS(grid: [[Int]], i: Int, j: Int) -&gt; Int {\n    // If it's the top-left cell, terminate the search\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)\n    let up = minPathSumDFS(grid: grid, i: i - 1, j: j)\n    let left = minPathSumDFS(grid: grid, i: i, j: j - 1)\n    // Return the minimum path cost from top-left to (i, j)\n    return min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* Minimum path sum: Brute-force search */\nfunction minPathSumDFS(grid, i, j) {\n    // If it's the top-left cell, terminate the search\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // Return the minimum path cost from top-left to (i, j)\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* Minimum path sum: Brute-force search */\nfunction minPathSumDFS(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // If it's the top-left cell, terminate the search\n    if (i === 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // Return the minimum path cost from top-left to (i, j)\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* Minimum path sum: Brute-force search */\nint minPathSumDFS(List&lt;List&lt;int&gt;&gt; grid, int i, int j) {\n  // If it's the top-left cell, terminate the search\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // If row or column index is out of bounds, return +∞ cost\n  if (i &lt; 0 || j &lt; 0) {\n    // In Dart, int type is fixed-range integer, no value representing \"infinity\"\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)\n  int up = minPathSumDFS(grid, i - 1, j);\n  int left = minPathSumDFS(grid, i, j - 1);\n  // Return the minimum path cost from top-left to (i, j)\n  return min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* Minimum path sum: Brute-force search */\nfn min_path_sum_dfs(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // If it's the top-left cell, terminate the search\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)\n    let up = min_path_sum_dfs(grid, i - 1, j);\n    let left = min_path_sum_dfs(grid, i, j - 1);\n    // Return the minimum path cost from top-left to (i, j)\n    std::cmp::min(left, up) + grid[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* Minimum path sum: Brute-force search */\nint minPathSumDFS(int grid[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // If it's the top-left cell, terminate the search\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // Return the minimum path cost from top-left to (i, j)\n    return myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.kt<pre><code>/* Minimum path sum: Brute-force search */\nfun minPathSumDFS(grid: Array&lt;IntArray&gt;, i: Int, j: Int): Int {\n    // If it's the top-left cell, terminate the search\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)\n    val up = minPathSumDFS(grid, i - 1, j)\n    val left = minPathSumDFS(grid, i, j - 1)\n    // Return the minimum path cost from top-left to (i, j)\n    return min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.rb<pre><code>### Minimum path sum: brute force search ###\ndef min_path_sum_dfs(grid, i, j)\n  # If it's the top-left cell, terminate the search\n  return grid[i][j] if i == 0 &amp;&amp; j == 0\n  # If row or column index is out of bounds, return +∞ cost\n  return Float::INFINITY if i &lt; 0 || j &lt; 0\n  # Calculate the minimum path cost from top-left to (i-1, j) and (i, j-1)\n  up = min_path_sum_dfs(grid, i - 1, j)\n  left = min_path_sum_dfs(grid, i, j - 1)\n  # Return the minimum path cost from top-left to (i, j)\n  [left, up].min + grid[i][j]\nend\n</code></pre> <p>Figure 14-14 shows the recursion tree rooted at \\(dp[2, 1]\\), which includes some overlapping subproblems whose number will increase sharply as the size of grid <code>grid</code> grows.</p> <p>Essentially, the reason for overlapping subproblems is: there are multiple paths from the top-left corner to reach a certain cell.</p> <p></p> <p> Figure 14-14   Brute force search recursion tree </p> <p>Each state has two choices, down and right, so the total number of steps from the top-left corner to the bottom-right corner is \\(m + n - 2\\), giving a worst-case time complexity of \\(O(2^{m + n})\\), where \\(n\\) and \\(m\\) are the number of rows and columns of the grid, respectively. Note that this calculation does not account for situations near the grid boundaries, where only one choice remains when reaching the grid boundary, so the actual number of paths will be somewhat less.</p>","path":["Chapter 14. Dynamic Programming","14.3   Dynamic Programming Problem-Solving Approach"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#2-method-2-memoization","level":3,"title":"2.   Method 2: Memoization","text":"<p>We introduce a memo list <code>mem</code> of the same size as grid <code>grid</code> to record the solutions to subproblems and prune overlapping subproblems:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dfs_mem(\n    grid: list[list[int]], mem: list[list[int]], i: int, j: int\n) -&gt; int:\n    \"\"\"Minimum path sum: Memoization search\"\"\"\n    # If it's the top-left cell, terminate the search\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # If row or column index is out of bounds, return +∞ cost\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # If there's a record, return it directly\n    if mem[i][j] != -1:\n        return mem[i][j]\n    # Minimum path cost for left and upper cells\n    up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n    left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n    # Record and return the minimum path cost from top-left to (i, j)\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* Minimum path sum: Memoization search */\nint minPathSumDFSMem(vector&lt;vector&lt;int&gt;&gt; &amp;grid, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int j) {\n    // If it's the top-left cell, terminate the search\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // If there's a record, return it directly\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // Minimum path cost for left and upper cells\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // Record and return the minimum path cost from top-left to (i, j)\n    mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.java<pre><code>/* Minimum path sum: Memoization search */\nint minPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // If it's the top-left cell, terminate the search\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // If there's a record, return it directly\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // Minimum path cost for left and upper cells\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // Record and return the minimum path cost from top-left to (i, j)\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* Minimum path sum: Memoization search */\nint MinPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // If it's the top-left cell, terminate the search\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // If there's a record, return it directly\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // Minimum path cost for left and upper cells\n    int up = MinPathSumDFSMem(grid, mem, i - 1, j);\n    int left = MinPathSumDFSMem(grid, mem, i, j - 1);\n    // Record and return the minimum path cost from top-left to (i, j)\n    mem[i][j] = Math.Min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* Minimum path sum: Memoization search */\nfunc minPathSumDFSMem(grid, mem [][]int, i, j int) int {\n    // If it's the top-left cell, terminate the search\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // If there's a record, return it directly\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // Minimum path cost for left and upper cells\n    up := minPathSumDFSMem(grid, mem, i-1, j)\n    left := minPathSumDFSMem(grid, mem, i, j-1)\n    // Record and return the minimum path cost from top-left to (i, j)\n    mem[i][j] = int(math.Min(float64(left), float64(up))) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* Minimum path sum: Memoization search */\nfunc minPathSumDFSMem(grid: [[Int]], mem: inout [[Int]], i: Int, j: Int) -&gt; Int {\n    // If it's the top-left cell, terminate the search\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // If there's a record, return it directly\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // Minimum path cost for left and upper cells\n    let up = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i - 1, j: j)\n    let left = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i, j: j - 1)\n    // Record and return the minimum path cost from top-left to (i, j)\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* Minimum path sum: Memoization search */\nfunction minPathSumDFSMem(grid, mem, i, j) {\n    // If it's the top-left cell, terminate the search\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // If there's a record, return it directly\n    if (mem[i][j] !== -1) {\n        return mem[i][j];\n    }\n    // Minimum path cost for left and upper cells\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // Record and return the minimum path cost from top-left to (i, j)\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* Minimum path sum: Memoization search */\nfunction minPathSumDFSMem(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // If it's the top-left cell, terminate the search\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // If there's a record, return it directly\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // Minimum path cost for left and upper cells\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // Record and return the minimum path cost from top-left to (i, j)\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* Minimum path sum: Memoization search */\nint minPathSumDFSMem(List&lt;List&lt;int&gt;&gt; grid, List&lt;List&lt;int&gt;&gt; mem, int i, int j) {\n  // If it's the top-left cell, terminate the search\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // If row or column index is out of bounds, return +∞ cost\n  if (i &lt; 0 || j &lt; 0) {\n    // In Dart, int type is fixed-range integer, no value representing \"infinity\"\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // If there's a record, return it directly\n  if (mem[i][j] != -1) {\n    return mem[i][j];\n  }\n  // Minimum path cost for left and upper cells\n  int up = minPathSumDFSMem(grid, mem, i - 1, j);\n  int left = minPathSumDFSMem(grid, mem, i, j - 1);\n  // Record and return the minimum path cost from top-left to (i, j)\n  mem[i][j] = min(left, up) + grid[i][j];\n  return mem[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* Minimum path sum: Memoization search */\nfn min_path_sum_dfs_mem(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // If it's the top-left cell, terminate the search\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // If there's a record, return it directly\n    if mem[i as usize][j as usize] != -1 {\n        return mem[i as usize][j as usize];\n    }\n    // Minimum path cost for left and upper cells\n    let up = min_path_sum_dfs_mem(grid, mem, i - 1, j);\n    let left = min_path_sum_dfs_mem(grid, mem, i, j - 1);\n    // Record and return the minimum path cost from top-left to (i, j)\n    mem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];\n    mem[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* Minimum path sum: Memoization search */\nint minPathSumDFSMem(int grid[MAX_SIZE][MAX_SIZE], int mem[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // If it's the top-left cell, terminate the search\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // If there's a record, return it directly\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // Minimum path cost for left and upper cells\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // Record and return the minimum path cost from top-left to (i, j)\n    mem[i][j] = myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.kt<pre><code>/* Minimum path sum: Memoization search */\nfun minPathSumDFSMem(\n    grid: Array&lt;IntArray&gt;,\n    mem: Array&lt;IntArray&gt;,\n    i: Int,\n    j: Int\n): Int {\n    // If it's the top-left cell, terminate the search\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // If row or column index is out of bounds, return +∞ cost\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // If there's a record, return it directly\n    if (mem[i][j] != -1) {\n        return mem[i][j]\n    }\n    // Minimum path cost for left and upper cells\n    val up = minPathSumDFSMem(grid, mem, i - 1, j)\n    val left = minPathSumDFSMem(grid, mem, i, j - 1)\n    // Record and return the minimum path cost from top-left to (i, j)\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.rb<pre><code>### Minimum path sum: memoization search ###\ndef min_path_sum_dfs_mem(grid, mem, i, j)\n  # If it's the top-left cell, terminate the search\n  return grid[0][0] if i == 0 &amp;&amp; j == 0\n  # If row or column index is out of bounds, return +∞ cost\n  return Float::INFINITY if i &lt; 0 || j &lt; 0\n  # If there's a record, return it directly\n  return mem[i][j] if mem[i][j] != -1\n  # Minimum path cost for left and upper cells\n  up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n  left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n  # Record and return the minimum path cost from top-left to (i, j)\n  mem[i][j] = [left, up].min + grid[i][j]\nend\n</code></pre> <p>As shown in Figure 14-15, after introducing memoization, all subproblem solutions only need to be computed once, so the time complexity depends on the total number of states, which is the grid size \\(O(nm)\\).</p> <p></p> <p> Figure 14-15   Memoization recursion tree </p>","path":["Chapter 14. Dynamic Programming","14.3   Dynamic Programming Problem-Solving Approach"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#3-method-3-dynamic-programming","level":3,"title":"3.   Method 3: Dynamic Programming","text":"<p>Implement the dynamic programming solution based on iteration, as shown in the code below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dp(grid: list[list[int]]) -&gt; int:\n    \"\"\"Minimum path sum: Dynamic programming\"\"\"\n    n, m = len(grid), len(grid[0])\n    # Initialize dp table\n    dp = [[0] * m for _ in range(n)]\n    dp[0][0] = grid[0][0]\n    # State transition: first row\n    for j in range(1, m):\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    # State transition: first column\n    for i in range(1, n):\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    # State transition: rest of the rows and columns\n    for i in range(1, n):\n        for j in range(1, m):\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n    return dp[n - 1][m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* Minimum path sum: Dynamic programming */\nint minPathSumDP(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // Initialize dp table\n    vector&lt;vector&lt;int&gt;&gt; dp(n, vector&lt;int&gt;(m));\n    dp[0][0] = grid[0][0];\n    // State transition: first row\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // State transition: first column\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // State transition: rest of the rows and columns\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* Minimum path sum: Dynamic programming */\nint minPathSumDP(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // Initialize dp table\n    int[][] dp = new int[n][m];\n    dp[0][0] = grid[0][0];\n    // State transition: first row\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // State transition: first column\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // State transition: rest of the rows and columns\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* Minimum path sum: Dynamic programming */\nint MinPathSumDP(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // Initialize dp table\n    int[,] dp = new int[n, m];\n    dp[0, 0] = grid[0][0];\n    // State transition: first row\n    for (int j = 1; j &lt; m; j++) {\n        dp[0, j] = dp[0, j - 1] + grid[0][j];\n    }\n    // State transition: first column\n    for (int i = 1; i &lt; n; i++) {\n        dp[i, 0] = dp[i - 1, 0] + grid[i][0];\n    }\n    // State transition: rest of the rows and columns\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i, j] = Math.Min(dp[i, j - 1], dp[i - 1, j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1, m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* Minimum path sum: Dynamic programming */\nfunc minPathSumDP(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // Initialize dp table\n    dp := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        dp[i] = make([]int, m)\n    }\n    dp[0][0] = grid[0][0]\n    // State transition: first row\n    for j := 1; j &lt; m; j++ {\n        dp[0][j] = dp[0][j-1] + grid[0][j]\n    }\n    // State transition: first column\n    for i := 1; i &lt; n; i++ {\n        dp[i][0] = dp[i-1][0] + grid[i][0]\n    }\n    // State transition: rest of the rows and columns\n    for i := 1; i &lt; n; i++ {\n        for j := 1; j &lt; m; j++ {\n            dp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(dp[i-1][j]))) + grid[i][j]\n        }\n    }\n    return dp[n-1][m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* Minimum path sum: Dynamic programming */\nfunc minPathSumDP(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // Initialize dp table\n    var dp = Array(repeating: Array(repeating: 0, count: m), count: n)\n    dp[0][0] = grid[0][0]\n    // State transition: first row\n    for j in 1 ..&lt; m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // State transition: first column\n    for i in 1 ..&lt; n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // State transition: rest of the rows and columns\n    for i in 1 ..&lt; n {\n        for j in 1 ..&lt; m {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* Minimum path sum: Dynamic programming */\nfunction minPathSumDP(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // Initialize dp table\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // State transition: first row\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // State transition: first column\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // State transition: rest of the rows and columns\n    for (let i = 1; i &lt; n; i++) {\n        for (let j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* Minimum path sum: Dynamic programming */\nfunction minPathSumDP(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // Initialize dp table\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // State transition: first row\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // State transition: first column\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // State transition: rest of the rows and columns\n    for (let i = 1; i &lt; n; i++) {\n        for (let j: number = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* Minimum path sum: Dynamic programming */\nint minPathSumDP(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // Initialize dp table\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n, (i) =&gt; List.filled(m, 0));\n  dp[0][0] = grid[0][0];\n  // State transition: first row\n  for (int j = 1; j &lt; m; j++) {\n    dp[0][j] = dp[0][j - 1] + grid[0][j];\n  }\n  // State transition: first column\n  for (int i = 1; i &lt; n; i++) {\n    dp[i][0] = dp[i - 1][0] + grid[i][0];\n  }\n  // State transition: rest of the rows and columns\n  for (int i = 1; i &lt; n; i++) {\n    for (int j = 1; j &lt; m; j++) {\n      dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n    }\n  }\n  return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* Minimum path sum: Dynamic programming */\nfn min_path_sum_dp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // Initialize dp table\n    let mut dp = vec![vec![0; m]; n];\n    dp[0][0] = grid[0][0];\n    // State transition: first row\n    for j in 1..m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // State transition: first column\n    for i in 1..n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // State transition: rest of the rows and columns\n    for i in 1..n {\n        for j in 1..m {\n            dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* Minimum path sum: Dynamic programming */\nint minPathSumDP(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // Initialize dp table\n    int **dp = malloc(n * sizeof(int *));\n    for (int i = 0; i &lt; n; i++) {\n        dp[i] = calloc(m, sizeof(int));\n    }\n    dp[0][0] = grid[0][0];\n    // State transition: first row\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // State transition: first column\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // State transition: rest of the rows and columns\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = myMin(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    int res = dp[n - 1][m - 1];\n    // Free memory\n    for (int i = 0; i &lt; n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* Minimum path sum: Dynamic programming */\nfun minPathSumDP(grid: Array&lt;IntArray&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // Initialize dp table\n    val dp = Array(n) { IntArray(m) }\n    dp[0][0] = grid[0][0]\n    // State transition: first row\n    for (j in 1..&lt;m) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // State transition: first column\n    for (i in 1..&lt;n) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // State transition: rest of the rows and columns\n    for (i in 1..&lt;n) {\n        for (j in 1..&lt;m) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>### Minimum path sum: dynamic programming ###\ndef min_path_sum_dp(grid)\n  n, m = grid.length, grid.first.length\n  # Initialize dp table\n  dp = Array.new(n) { Array.new(m, 0) }\n  dp[0][0] = grid[0][0]\n  # State transition: first row\n  (1...m).each { |j| dp[0][j] = dp[0][j - 1] + grid[0][j] }\n  # State transition: first column\n  (1...n).each { |i| dp[i][0] = dp[i - 1][0] + grid[i][0] }\n  # State transition: rest of the rows and columns\n  for i in 1...n\n    for j in 1...m\n      dp[i][j] = [dp[i][j - 1], dp[i - 1][j]].min + grid[i][j]\n    end\n  end\n  dp[n -1][m -1]\nend\n</code></pre> <p>Figure 14-16 shows the state transition process for minimum path sum, which traverses the entire grid, thus the time complexity is \\(O(nm)\\).</p> <p>The array <code>dp</code> has size \\(n \\times m\\), thus the space complexity is \\(O(nm)\\).</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 14-16   Dynamic programming process for minimum path sum </p>","path":["Chapter 14. Dynamic Programming","14.3   Dynamic Programming Problem-Solving Approach"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#4-space-optimization","level":3,"title":"4.   Space Optimization","text":"<p>Since each cell is only related to the cell to its left and the cell above it, we can use a single-row array to implement the \\(dp\\) table.</p> <p>Note that since the array <code>dp</code> can only represent the state of one row, we cannot initialize the first column state in advance, but rather update it when traversing each row:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dp_comp(grid: list[list[int]]) -&gt; int:\n    \"\"\"Minimum path sum: Space-optimized dynamic programming\"\"\"\n    n, m = len(grid), len(grid[0])\n    # Initialize dp table\n    dp = [0] * m\n    # State transition: first row\n    dp[0] = grid[0][0]\n    for j in range(1, m):\n        dp[j] = dp[j - 1] + grid[0][j]\n    # State transition: rest of the rows\n    for i in range(1, n):\n        # State transition: first column\n        dp[0] = dp[0] + grid[i][0]\n        # State transition: rest of the columns\n        for j in range(1, m):\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n    return dp[m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* Minimum path sum: Space-optimized dynamic programming */\nint minPathSumDPComp(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // Initialize dp table\n    vector&lt;int&gt; dp(m);\n    // State transition: first row\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // State transition: rest of the rows\n    for (int i = 1; i &lt; n; i++) {\n        // State transition: first column\n        dp[0] = dp[0] + grid[i][0];\n        // State transition: rest of the columns\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* Minimum path sum: Space-optimized dynamic programming */\nint minPathSumDPComp(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // Initialize dp table\n    int[] dp = new int[m];\n    // State transition: first row\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // State transition: rest of the rows\n    for (int i = 1; i &lt; n; i++) {\n        // State transition: first column\n        dp[0] = dp[0] + grid[i][0];\n        // State transition: rest of the columns\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* Minimum path sum: Space-optimized dynamic programming */\nint MinPathSumDPComp(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // Initialize dp table\n    int[] dp = new int[m];\n    dp[0] = grid[0][0];\n    // State transition: first row\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // State transition: rest of the rows\n    for (int i = 1; i &lt; n; i++) {\n        // State transition: first column\n        dp[0] = dp[0] + grid[i][0];\n        // State transition: rest of the columns\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.Min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* Minimum path sum: Space-optimized dynamic programming */\nfunc minPathSumDPComp(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // Initialize dp table\n    dp := make([]int, m)\n    // State transition: first row\n    dp[0] = grid[0][0]\n    for j := 1; j &lt; m; j++ {\n        dp[j] = dp[j-1] + grid[0][j]\n    }\n    // State transition: rest of the rows and columns\n    for i := 1; i &lt; n; i++ {\n        // State transition: first column\n        dp[0] = dp[0] + grid[i][0]\n        // State transition: rest of the columns\n        for j := 1; j &lt; m; j++ {\n            dp[j] = int(math.Min(float64(dp[j-1]), float64(dp[j]))) + grid[i][j]\n        }\n    }\n    return dp[m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* Minimum path sum: Space-optimized dynamic programming */\nfunc minPathSumDPComp(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // Initialize dp table\n    var dp = Array(repeating: 0, count: m)\n    // State transition: first row\n    dp[0] = grid[0][0]\n    for j in 1 ..&lt; m {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // State transition: rest of the rows\n    for i in 1 ..&lt; n {\n        // State transition: first column\n        dp[0] = dp[0] + grid[i][0]\n        // State transition: rest of the columns\n        for j in 1 ..&lt; m {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* Minimum path sum: Space-optimized dynamic programming */\nfunction minPathSumDPComp(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // Initialize dp table\n    const dp = new Array(m);\n    // State transition: first row\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // State transition: rest of the rows\n    for (let i = 1; i &lt; n; i++) {\n        // State transition: first column\n        dp[0] = dp[0] + grid[i][0];\n        // State transition: rest of the columns\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* Minimum path sum: Space-optimized dynamic programming */\nfunction minPathSumDPComp(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // Initialize dp table\n    const dp = new Array(m);\n    // State transition: first row\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // State transition: rest of the rows\n    for (let i = 1; i &lt; n; i++) {\n        // State transition: first column\n        dp[0] = dp[0] + grid[i][0];\n        // State transition: rest of the columns\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* Minimum path sum: Space-optimized dynamic programming */\nint minPathSumDPComp(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // Initialize dp table\n  List&lt;int&gt; dp = List.filled(m, 0);\n  dp[0] = grid[0][0];\n  for (int j = 1; j &lt; m; j++) {\n    dp[j] = dp[j - 1] + grid[0][j];\n  }\n  // State transition: rest of the rows\n  for (int i = 1; i &lt; n; i++) {\n    // State transition: first column\n    dp[0] = dp[0] + grid[i][0];\n    // State transition: rest of the columns\n    for (int j = 1; j &lt; m; j++) {\n      dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n    }\n  }\n  return dp[m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* Minimum path sum: Space-optimized dynamic programming */\nfn min_path_sum_dp_comp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // Initialize dp table\n    let mut dp = vec![0; m];\n    // State transition: first row\n    dp[0] = grid[0][0];\n    for j in 1..m {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // State transition: rest of the rows\n    for i in 1..n {\n        // State transition: first column\n        dp[0] = dp[0] + grid[i][0];\n        // State transition: rest of the columns\n        for j in 1..m {\n            dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    dp[m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* Minimum path sum: Space-optimized dynamic programming */\nint minPathSumDPComp(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // Initialize dp table\n    int *dp = calloc(m, sizeof(int));\n    // State transition: first row\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // State transition: rest of the rows\n    for (int i = 1; i &lt; n; i++) {\n        // State transition: first column\n        dp[0] = dp[0] + grid[i][0];\n        // State transition: rest of the columns\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = myMin(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    int res = dp[m - 1];\n    // Free memory\n    free(dp);\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* Minimum path sum: Space-optimized dynamic programming */\nfun minPathSumDPComp(grid: Array&lt;IntArray&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // Initialize dp table\n    val dp = IntArray(m)\n    // State transition: first row\n    dp[0] = grid[0][0]\n    for (j in 1..&lt;m) {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // State transition: rest of the rows\n    for (i in 1..&lt;n) {\n        // State transition: first column\n        dp[0] = dp[0] + grid[i][0]\n        // State transition: rest of the columns\n        for (j in 1..&lt;m) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>### Minimum path sum: space-optimized DP ###\ndef min_path_sum_dp_comp(grid)\n  n, m = grid.length, grid.first.length\n  # Initialize dp table\n  dp = Array.new(m, 0)\n  # State transition: first row\n  dp[0] = grid[0][0]\n  (1...m).each { |j| dp[j] = dp[j - 1] + grid[0][j] }\n  # State transition: rest of the rows\n  for i in 1...n\n    # State transition: first column\n    dp[0] = dp[0] + grid[i][0]\n    # State transition: rest of the columns\n    (1...m).each { |j| dp[j] = [dp[j - 1], dp[j]].min + grid[i][j] }\n  end\n  dp[m - 1]\nend\n</code></pre>","path":["Chapter 14. Dynamic Programming","14.3   Dynamic Programming Problem-Solving Approach"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/","level":1,"title":"14.6   Edit Distance Problem","text":"<p>Edit distance, also known as Levenshtein distance, refers to the minimum number of edits required to transform one string into another, commonly used in information retrieval and natural language processing to measure the similarity between two sequences.</p> <p>Question</p> <p>Given two strings \\(s\\) and \\(t\\), return the minimum number of edits required to transform \\(s\\) into \\(t\\).</p> <p>You can perform three types of edit operations on a string: insert a character, delete a character, or replace a character with any other character.</p> <p>As shown in Figure 14-27, transforming <code>kitten</code> into <code>sitting</code> requires 3 edits, including 2 replacements and 1 insertion; transforming <code>hello</code> into <code>algo</code> requires 3 steps, including 2 replacements and 1 deletion.</p> <p></p> <p> Figure 14-27   Example data for edit distance </p> <p>The edit distance problem can be naturally explained using the decision tree model. Strings correspond to tree nodes, and each edit operation corresponds to an edge in the tree.</p> <p>As shown in Figure 14-28, without restricting operations, each node can branch into many edges, with each edge corresponding to one operation, meaning there are many possible paths to transform <code>hello</code> into <code>algo</code>.</p> <p>From the perspective of the decision tree, the goal of this problem is to find the shortest path between node <code>hello</code> and node <code>algo</code>.</p> <p></p> <p> Figure 14-28   Representing edit distance problem based on decision tree model </p>","path":["Chapter 14. Dynamic Programming","14.6   Edit Distance Problem"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/#1-dynamic-programming-approach","level":3,"title":"1.   Dynamic Programming Approach","text":"<p>Step 1: Think about the decisions in each round, define the state, and thus obtain the \\(dp\\) table</p> <p>Each round of decision involves performing one edit operation on string \\(s\\).</p> <p>We want the problem size to gradually decrease during the editing process so that we can construct subproblems. Let the lengths of strings \\(s\\) and \\(t\\) be \\(n\\) and \\(m\\) respectively. We first consider the tail characters of the two strings, \\(s[n-1]\\) and \\(t[m-1]\\).</p> <ul> <li>If \\(s[n-1]\\) and \\(t[m-1]\\) are the same, we can skip them and directly consider \\(s[n-2]\\) and \\(t[m-2]\\).</li> <li>If \\(s[n-1]\\) and \\(t[m-1]\\) are different, we need to perform one edit on \\(s\\) (insert, delete, or replace) to make the tail characters of the two strings the same, allowing us to skip them and consider a smaller-scale problem.</li> </ul> <p>In other words, each round of decision (edit operation) we make on string \\(s\\) will change the remaining characters to be matched in \\(s\\) and \\(t\\). Therefore, the state is the \\(i\\)-th and \\(j\\)-th characters currently being considered in \\(s\\) and \\(t\\), denoted as \\([i, j]\\).</p> <p>State \\([i, j]\\) corresponds to the subproblem: the minimum number of edits required to change the first \\(i\\) characters of \\(s\\) into the first \\(j\\) characters of \\(t\\).</p> <p>From this, we obtain a two-dimensional \\(dp\\) table of size \\((n+1) \\times (m+1)\\).</p> <p>Step 2: Identify the optimal substructure, and then derive the state transition equation</p> <p>Consider subproblem \\(dp[i, j]\\), where the tail characters of the corresponding two strings are \\(s[i-1]\\) and \\(t[j-1]\\), which can be divided into the three cases shown in Figure 14-29 based on different edit operations.</p> <ol> <li>Insert \\(t[j-1]\\) after \\(s[i-1]\\), then the remaining subproblem is \\(dp[i, j-1]\\).</li> <li>Delete \\(s[i-1]\\), then the remaining subproblem is \\(dp[i-1, j]\\).</li> <li>Replace \\(s[i-1]\\) with \\(t[j-1]\\), then the remaining subproblem is \\(dp[i-1, j-1]\\).</li> </ol> <p></p> <p> Figure 14-29   State transition for edit distance </p> <p>Based on the above analysis, we obtain the optimal substructure: the minimum number of edits for \\(dp[i, j]\\) equals the minimum of \\(dp[i, j-1]\\), \\(dp[i-1, j]\\), and \\(dp[i-1, j-1]\\), plus the current edit cost of \\(1\\). The corresponding state transition equation is:</p> \\[ dp[i, j] = \\min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1 \\] <p>Please note that when \\(s[i-1]\\) and \\(t[j-1]\\) are the same, no edit is required for the current character, in which case the state transition equation is:</p> \\[ dp[i, j] = dp[i-1, j-1] \\] <p>Step 3: Determine boundary conditions and state transition order</p> <p>When both strings are empty, the number of edit steps is \\(0\\), i.e., \\(dp[0, 0] = 0\\). When \\(s\\) is empty but \\(t\\) is not, the minimum number of edit steps equals the length of \\(t\\), i.e., the first row \\(dp[0, j] = j\\). When \\(s\\) is not empty but \\(t\\) is empty, the minimum number of edit steps equals the length of \\(s\\), i.e., the first column \\(dp[i, 0] = i\\).</p> <p>Observing the state transition equation, the solution \\(dp[i, j]\\) depends on solutions to the left, above, and upper-left, so the entire \\(dp\\) table can be traversed in order through two nested loops.</p>","path":["Chapter 14. Dynamic Programming","14.6   Edit Distance Problem"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/#2-code-implementation","level":3,"title":"2.   Code Implementation","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby edit_distance.py<pre><code>def edit_distance_dp(s: str, t: str) -&gt; int:\n    \"\"\"Edit distance: Dynamic programming\"\"\"\n    n, m = len(s), len(t)\n    dp = [[0] * (m + 1) for _ in range(n + 1)]\n    # State transition: first row and first column\n    for i in range(1, n + 1):\n        dp[i][0] = i\n    for j in range(1, m + 1):\n        dp[0][j] = j\n    # State transition: rest of the rows and columns\n    for i in range(1, n + 1):\n        for j in range(1, m + 1):\n            if s[i - 1] == t[j - 1]:\n                # If two characters are equal, skip both characters\n                dp[i][j] = dp[i - 1][j - 1]\n            else:\n                # Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[i][j] = min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1\n    return dp[n][m]\n</code></pre> edit_distance.cpp<pre><code>/* Edit distance: Dynamic programming */\nint editDistanceDP(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(m + 1, 0));\n    // State transition: first row and first column\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // State transition: rest of the rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // If two characters are equal, skip both characters\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.java<pre><code>/* Edit distance: Dynamic programming */\nint editDistanceDP(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[][] dp = new int[n + 1][m + 1];\n    // State transition: first row and first column\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // State transition: rest of the rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // If two characters are equal, skip both characters\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[i][j] = Math.min(Math.min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.cs<pre><code>/* Edit distance: Dynamic programming */\nint EditDistanceDP(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[,] dp = new int[n + 1, m + 1];\n    // State transition: first row and first column\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i, 0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0, j] = j;\n    }\n    // State transition: rest of the rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // If two characters are equal, skip both characters\n                dp[i, j] = dp[i - 1, j - 1];\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[i, j] = Math.Min(Math.Min(dp[i, j - 1], dp[i - 1, j]), dp[i - 1, j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n, m];\n}\n</code></pre> edit_distance.go<pre><code>/* Edit distance: Dynamic programming */\nfunc editDistanceDP(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, m+1)\n    }\n    // State transition: first row and first column\n    for i := 1; i &lt;= n; i++ {\n        dp[i][0] = i\n    }\n    for j := 1; j &lt;= m; j++ {\n        dp[0][j] = j\n    }\n    // State transition: rest of the rows and columns\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= m; j++ {\n            if s[i-1] == t[j-1] {\n                // If two characters are equal, skip both characters\n                dp[i][j] = dp[i-1][j-1]\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[i][j] = MinInt(MinInt(dp[i][j-1], dp[i-1][j]), dp[i-1][j-1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.swift<pre><code>/* Edit distance: Dynamic programming */\nfunc editDistanceDP(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: Array(repeating: 0, count: m + 1), count: n + 1)\n    // State transition: first row and first column\n    for i in 1 ... n {\n        dp[i][0] = i\n    }\n    for j in 1 ... m {\n        dp[0][j] = j\n    }\n    // State transition: rest of the rows and columns\n    for i in 1 ... n {\n        for j in 1 ... m {\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // If two characters are equal, skip both characters\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.js<pre><code>/* Edit distance: Dynamic programming */\nfunction editDistanceDP(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(m + 1).fill(0));\n    // State transition: first row and first column\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // State transition: rest of the rows and columns\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // If two characters are equal, skip both characters\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.ts<pre><code>/* Edit distance: Dynamic programming */\nfunction editDistanceDP(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: m + 1 }, () =&gt; 0)\n    );\n    // State transition: first row and first column\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // State transition: rest of the rows and columns\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // If two characters are equal, skip both characters\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.dart<pre><code>/* Edit distance: Dynamic programming */\nint editDistanceDP(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (_) =&gt; List.filled(m + 1, 0));\n  // State transition: first row and first column\n  for (int i = 1; i &lt;= n; i++) {\n    dp[i][0] = i;\n  }\n  for (int j = 1; j &lt;= m; j++) {\n    dp[0][j] = j;\n  }\n  // State transition: rest of the rows and columns\n  for (int i = 1; i &lt;= n; i++) {\n    for (int j = 1; j &lt;= m; j++) {\n      if (s[i - 1] == t[j - 1]) {\n        // If two characters are equal, skip both characters\n        dp[i][j] = dp[i - 1][j - 1];\n      } else {\n        // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n        dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n      }\n    }\n  }\n  return dp[n][m];\n}\n</code></pre> edit_distance.rs<pre><code>/* Edit distance: Dynamic programming */\nfn edit_distance_dp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![vec![0; m + 1]; n + 1];\n    // State transition: first row and first column\n    for i in 1..=n {\n        dp[i][0] = i as i32;\n    }\n    for j in 1..m {\n        dp[0][j] = j as i32;\n    }\n    // State transition: rest of the rows and columns\n    for i in 1..=n {\n        for j in 1..=m {\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // If two characters are equal, skip both characters\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[i][j] =\n                    std::cmp::min(std::cmp::min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    dp[n][m]\n}\n</code></pre> edit_distance.c<pre><code>/* Edit distance: Dynamic programming */\nint editDistanceDP(char *s, char *t, int n, int m) {\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(m + 1, sizeof(int));\n    }\n    // State transition: first row and first column\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // State transition: rest of the rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // If two characters are equal, skip both characters\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[i][j] = myMin(myMin(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    int res = dp[n][m];\n    // Free memory\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* Edit distance: Dynamic programming */\nfun editDistanceDP(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = Array(n + 1) { IntArray(m + 1) }\n    // State transition: first row and first column\n    for (i in 1..n) {\n        dp[i][0] = i\n    }\n    for (j in 1..m) {\n        dp[0][j] = j\n    }\n    // State transition: rest of the rows and columns\n    for (i in 1..n) {\n        for (j in 1..m) {\n            if (s[i - 1] == t[j - 1]) {\n                // If two characters are equal, skip both characters\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.rb<pre><code>### Edit distance: dynamic programming ###\ndef edit_distance_dp(s, t)\n  n, m = s.length, t.length\n  dp = Array.new(n + 1) { Array.new(m + 1, 0) }\n  # State transition: first row and first column\n  (1...(n + 1)).each { |i| dp[i][0] = i }\n  (1...(m + 1)).each { |j| dp[0][j] = j }\n  # State transition: rest of the rows and columns\n  for i in 1...(n + 1)\n    for j in 1...(m +1)\n      if s[i - 1] == t[j - 1]\n        # If two characters are equal, skip both characters\n        dp[i][j] = dp[i - 1][j - 1]\n      else\n        # Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n        dp[i][j] = [dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]].min + 1\n      end\n    end\n  end\n  dp[n][m]\nend\n</code></pre> <p>As shown in Figure 14-30, the state transition process for the edit distance problem is very similar to that of the knapsack problem; both can be viewed as the process of filling a two-dimensional grid.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 14-30   Dynamic programming process for edit distance </p>","path":["Chapter 14. Dynamic Programming","14.6   Edit Distance Problem"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/#3-space-optimization","level":3,"title":"3.   Space Optimization","text":"<p>Since \\(dp[i, j]\\) depends on the states above \\(dp[i-1, j]\\), to the left \\(dp[i, j-1]\\), and at the upper-left \\(dp[i-1, j-1]\\), forward traversal will lose the upper-left state \\(dp[i-1, j-1]\\), while reverse traversal cannot construct \\(dp[i, j-1]\\) in advance, so neither traversal order is suitable.</p> <p>For this reason, we can use a variable <code>leftup</code> to temporarily store the upper-left solution \\(dp[i-1, j-1]\\), so we only need to consider the solutions to the left and above. This situation is the same as in the unbounded knapsack problem, so we can use forward traversal. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby edit_distance.py<pre><code>def edit_distance_dp_comp(s: str, t: str) -&gt; int:\n    \"\"\"Edit distance: Space-optimized dynamic programming\"\"\"\n    n, m = len(s), len(t)\n    dp = [0] * (m + 1)\n    # State transition: first row\n    for j in range(1, m + 1):\n        dp[j] = j\n    # State transition: rest of the rows\n    for i in range(1, n + 1):\n        # State transition: first column\n        leftup = dp[0]  # Temporarily store dp[i-1, j-1]\n        dp[0] += 1\n        # State transition: rest of the columns\n        for j in range(1, m + 1):\n            temp = dp[j]\n            if s[i - 1] == t[j - 1]:\n                # If two characters are equal, skip both characters\n                dp[j] = leftup\n            else:\n                # Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[j] = min(dp[j - 1], dp[j], leftup) + 1\n            leftup = temp  # Update for next round's dp[i-1, j-1]\n    return dp[m]\n</code></pre> edit_distance.cpp<pre><code>/* Edit distance: Space-optimized dynamic programming */\nint editDistanceDPComp(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;int&gt; dp(m + 1, 0);\n    // State transition: first row\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // State transition: rest of the rows\n    for (int i = 1; i &lt;= n; i++) {\n        // State transition: first column\n        int leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n        dp[0] = i;\n        // State transition: rest of the columns\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // If two characters are equal, skip both characters\n                dp[j] = leftup;\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // Update for next round's dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.java<pre><code>/* Edit distance: Space-optimized dynamic programming */\nint editDistanceDPComp(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[] dp = new int[m + 1];\n    // State transition: first row\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // State transition: rest of the rows\n    for (int i = 1; i &lt;= n; i++) {\n        // State transition: first column\n        int leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n        dp[0] = i;\n        // State transition: rest of the columns\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // If two characters are equal, skip both characters\n                dp[j] = leftup;\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[j] = Math.min(Math.min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // Update for next round's dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.cs<pre><code>/* Edit distance: Space-optimized dynamic programming */\nint EditDistanceDPComp(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[] dp = new int[m + 1];\n    // State transition: first row\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // State transition: rest of the rows\n    for (int i = 1; i &lt;= n; i++) {\n        // State transition: first column\n        int leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n        dp[0] = i;\n        // State transition: rest of the columns\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // If two characters are equal, skip both characters\n                dp[j] = leftup;\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[j] = Math.Min(Math.Min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // Update for next round's dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.go<pre><code>/* Edit distance: Space-optimized dynamic programming */\nfunc editDistanceDPComp(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([]int, m+1)\n    // State transition: first row\n    for j := 1; j &lt;= m; j++ {\n        dp[j] = j\n    }\n    // State transition: rest of the rows\n    for i := 1; i &lt;= n; i++ {\n        // State transition: first column\n        leftUp := dp[0] // Temporarily store dp[i-1, j-1]\n        dp[0] = i\n        // State transition: rest of the columns\n        for j := 1; j &lt;= m; j++ {\n            temp := dp[j]\n            if s[i-1] == t[j-1] {\n                // If two characters are equal, skip both characters\n                dp[j] = leftUp\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[j] = MinInt(MinInt(dp[j-1], dp[j]), leftUp) + 1\n            }\n            leftUp = temp // Update for next round's dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.swift<pre><code>/* Edit distance: Space-optimized dynamic programming */\nfunc editDistanceDPComp(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: 0, count: m + 1)\n    // State transition: first row\n    for j in 1 ... m {\n        dp[j] = j\n    }\n    // State transition: rest of the rows\n    for i in 1 ... n {\n        // State transition: first column\n        var leftup = dp[0] // Temporarily store dp[i-1, j-1]\n        dp[0] = i\n        // State transition: rest of the columns\n        for j in 1 ... m {\n            let temp = dp[j]\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // If two characters are equal, skip both characters\n                dp[j] = leftup\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n            }\n            leftup = temp // Update for next round's dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.js<pre><code>/* Edit distance: Space-optimized dynamic programming */\nfunction editDistanceDPComp(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // State transition: first row\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // State transition: rest of the rows\n    for (let i = 1; i &lt;= n; i++) {\n        // State transition: first column\n        let leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n        dp[0] = i;\n        // State transition: rest of the columns\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // If two characters are equal, skip both characters\n                dp[j] = leftup;\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // Update for next round's dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.ts<pre><code>/* Edit distance: Space-optimized dynamic programming */\nfunction editDistanceDPComp(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // State transition: first row\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // State transition: rest of the rows\n    for (let i = 1; i &lt;= n; i++) {\n        // State transition: first column\n        let leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n        dp[0] = i;\n        // State transition: rest of the columns\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // If two characters are equal, skip both characters\n                dp[j] = leftup;\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // Update for next round's dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.dart<pre><code>/* Edit distance: Space-optimized dynamic programming */\nint editDistanceDPComp(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;int&gt; dp = List.filled(m + 1, 0);\n  // State transition: first row\n  for (int j = 1; j &lt;= m; j++) {\n    dp[j] = j;\n  }\n  // State transition: rest of the rows\n  for (int i = 1; i &lt;= n; i++) {\n    // State transition: first column\n    int leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n    dp[0] = i;\n    // State transition: rest of the columns\n    for (int j = 1; j &lt;= m; j++) {\n      int temp = dp[j];\n      if (s[i - 1] == t[j - 1]) {\n        // If two characters are equal, skip both characters\n        dp[j] = leftup;\n      } else {\n        // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n        dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n      }\n      leftup = temp; // Update for next round's dp[i-1, j-1]\n    }\n  }\n  return dp[m];\n}\n</code></pre> edit_distance.rs<pre><code>/* Edit distance: Space-optimized dynamic programming */\nfn edit_distance_dp_comp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![0; m + 1];\n    // State transition: first row\n    for j in 1..m {\n        dp[j] = j as i32;\n    }\n    // State transition: rest of the rows\n    for i in 1..=n {\n        // State transition: first column\n        let mut leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n        dp[0] = i as i32;\n        // State transition: rest of the columns\n        for j in 1..=m {\n            let temp = dp[j];\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // If two characters are equal, skip both characters\n                dp[j] = leftup;\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[j] = std::cmp::min(std::cmp::min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // Update for next round's dp[i-1, j-1]\n        }\n    }\n    dp[m]\n}\n</code></pre> edit_distance.c<pre><code>/* Edit distance: Space-optimized dynamic programming */\nint editDistanceDPComp(char *s, char *t, int n, int m) {\n    int *dp = calloc(m + 1, sizeof(int));\n    // State transition: first row\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // State transition: rest of the rows\n    for (int i = 1; i &lt;= n; i++) {\n        // State transition: first column\n        int leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n        dp[0] = i;\n        // State transition: rest of the columns\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // If two characters are equal, skip both characters\n                dp[j] = leftup;\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[j] = myMin(myMin(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // Update for next round's dp[i-1, j-1]\n        }\n    }\n    int res = dp[m];\n    // Free memory\n    free(dp);\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* Edit distance: Space-optimized dynamic programming */\nfun editDistanceDPComp(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = IntArray(m + 1)\n    // State transition: first row\n    for (j in 1..m) {\n        dp[j] = j\n    }\n    // State transition: rest of the rows\n    for (i in 1..n) {\n        // State transition: first column\n        var leftup = dp[0] // Temporarily store dp[i-1, j-1]\n        dp[0] = i\n        // State transition: rest of the columns\n        for (j in 1..m) {\n            val temp = dp[j]\n            if (s[i - 1] == t[j - 1]) {\n                // If two characters are equal, skip both characters\n                dp[j] = leftup\n            } else {\n                // Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n            }\n            leftup = temp // Update for next round's dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.rb<pre><code>### Edit distance: space-optimized DP ###\ndef edit_distance_dp_comp(s, t)\n  n, m = s.length, t.length\n  dp = Array.new(m + 1, 0)\n  # State transition: first row\n  (1...(m + 1)).each { |j| dp[j] = j }\n  # State transition: rest of the rows\n  for i in 1...(n + 1)\n    # State transition: first column\n    leftup = dp.first # Temporarily store dp[i-1, j-1]\n    dp[0] += 1\n    # State transition: rest of the columns\n    for j in 1...(m + 1)\n      temp = dp[j]\n      if s[i - 1] == t[j - 1]\n        # If two characters are equal, skip both characters\n        dp[j] = leftup\n      else\n        # Minimum edit steps = minimum edit steps of insert, delete, replace + 1\n        dp[j] = [dp[j - 1], dp[j], leftup].min + 1\n      end\n      leftup = temp # Update for next round's dp[i-1, j-1]\n    end\n  end\n  dp[m]\nend\n</code></pre>","path":["Chapter 14. Dynamic Programming","14.6   Edit Distance Problem"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/","level":1,"title":"14.8   Exercises","text":"","path":["Chapter 14. Dynamic Programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1481-concept-review","level":2,"title":"14.8.1   Concept Review","text":"","path":["Chapter 14. Dynamic Programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1-when-is-dynamic-programming-appropriate","level":3,"title":"1.   When Is Dynamic Programming Appropriate?","text":"<p>A student says, \"Whenever a recurrence can be written, dynamic programming should be used.\" For each task below, decide whether dynamic programming, backtracking, or a loop or mathematical formula without a <code>dp</code> table is more appropriate. Give one key reason.</p> <ol> <li>Using coin denominations <code>[1, 3, 4]</code>, make an amount of 6 with the fewest coins. Each denomination may be used repeatedly.</li> <li>Output all permutations of <code>[1, 2, 3]</code>.</li> <li> <p>Calculate \\(1 + 2 + \\dots + n\\).</p> <p>For the task you consider suitable for dynamic programming, also state what <code>dp[i]</code> represents.</p> </li> </ol> Answer <ol> <li> <p>Dynamic programming is suitable. Let <code>dp[i]</code> be the minimum number of coins needed to make amount <code>i</code>.     For every coin <code>c</code> that does not exceed <code>i</code>, <code>dp[i-c] + 1</code> is a candidate answer,     and the minimum of these candidates is chosen. Different choices repeatedly encounter the same amounts, and an optimal solution for a larger amount can be built from optimal solutions for smaller amounts.     The answer for amount 6 is 2, using <code>3 + 3</code>.</p> </li> <li> <p>Backtracking is suitable. The task requires generating all 6 permutations one by one. Backtracking can systematically make a choice, continue searching,     undo the choice, and then try another branch. Regardless of the method, actually outputting every permutation requires enumerating them.</p> </li> <li> <p>A loop or the arithmetic-series formula is sufficient. Although the recurrence <code>S(i) = S(i-1) + i</code> can be written, calculating <code>S(i)</code> depends on only one smaller value, <code>S(i-1)</code>.     Each partial sum needs to be calculated only once, so there are no repeated subproblems and no need for a <code>dp</code> table. \"A recurrence can be written\" does not mean \"dynamic programming is needed.\"</p> </li> </ol>","path":["Chapter 14. Dynamic Programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#2-calculating-one-cell-in-a-knapsack-table","level":3,"title":"2.   Calculating One Cell in a Knapsack Table","text":"<p>Consider this 0-1 knapsack problem: item weights <code>wgt = [1, 2, 3]</code>, values <code>val = [5, 11, 15]</code>, and knapsack capacity 4. <code>dp[i][c]</code> is the maximum value obtainable using only the first \\(i\\) items with a knapsack capacity limit of \\(c\\); the knapsack does not have to be filled exactly.</p> <p>Calculate only the state <code>dp[3][4]</code>. You are given <code>dp[2][4] = 16</code> and <code>dp[2][1] = 5</code>:</p> <ol> <li>If the third item is not selected, what is the candidate value?</li> <li>If the third item is selected, how much capacity remains, and what is the candidate value?</li> <li>What should <code>dp[3][4]</code> be? Which items does this value correspond to selecting?</li> </ol> Answer <ol> <li> <p>If the third item is not selected, keep the result from the first two items. The candidate value is <code>dp[2][4] = 16</code>.</p> </li> <li> <p>The third item has weight 3, so capacity \\(4-3=1\\) remains after it is placed in the knapsack. The candidate value is     <code>dp[2][1] + 15 = 5 + 15 = 20</code>.</p> </li> <li> <p>Comparing 16 and 20 gives <code>dp[3][4] = 20</code>. This corresponds to selecting the first and third items,     whose total weight is \\(1+3=4\\) and total value is \\(5+15=20\\).</p> <p>This state calculation demonstrates one \"select or do not select\" comparison in the 0-1 knapsack problem.</p> </li> </ol>","path":["Chapter 14. Dynamic Programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#3-in-which-order-should-knapsack-capacities-be-updated","level":3,"title":"3.   In Which Order Should Knapsack Capacities Be Updated?","text":"<p>A 0-1 knapsack problem has only one item, with weight 2 and value 5, and the knapsack has capacity 4. The item can be selected at most once. The initial one-dimensional array is <code>dp = [0, 0, 0, 0, 0]</code>.</p> <p>A student processes the item by updating capacities from 2 to 4:</p> <ul> <li>After updating <code>dp[2]</code>, its value is 5.</li> <li>After updating <code>dp[3]</code>, its value is also 5.</li> <li>When updating <code>dp[4]</code>, the student uses the newly obtained <code>dp[2]</code>, producing <code>dp[4] = 10</code>.</li> </ul> <ol> <li>Is <code>dp[4] = 10</code> correct? Why or why not?</li> <li>Given that each item may be selected at most once, what should <code>dp[4]</code> be?</li> <li>When processing each item, should capacities be updated from largest to smallest or from smallest to largest? What problem does this avoid?</li> </ol> Answer <ol> <li> <p>The result is incorrect. A value of 10 is equivalent to placing the item with value 5 into the knapsack twice,     violating the condition that each item may be selected at most once.</p> </li> <li> <p>The knapsack can contain at most this one item, so the correct value of <code>dp[4]</code> is 5.</p> </li> <li> <p>Capacities should be updated from largest to smallest, in the order 4, 3, 2.     Then, when calculating <code>dp[c]</code>, the value read from <code>dp[c-2]</code> still comes from before the current item was processed,     preventing the current item from being reused during the same round.</p> </li> </ol>","path":["Chapter 14. Dynamic Programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1482-programming-exercises","level":2,"title":"14.8.2   Programming Exercises","text":"","path":["Chapter 14. Dynamic Programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1-number-of-ways-to-climb-stairs","level":3,"title":"1.   Number of Ways to Climb Stairs","text":"<p>A staircase has <code>n</code> steps. Each move climbs either 1 or 2 steps, and you must land exactly on step <code>n</code>. Calculate the number of distinct ways to reach the top. Assume <code>n &gt;= 1</code>; ways are distinguished only by their sequence of 1-step and 2-step moves. Use a one-dimensional dynamic programming array. For now, do not use the space optimization that keeps only two states.</p> Hints <ol> <li>The last move to step i can cover only 1 or 2 steps</li> <li>Therefore, dp[i] = dp[i-1] + dp[i-2]</li> <li>Handle the cases where n is 1 or 2 first, then fill the table starting from step 3</li> </ol> <p>LeetCode</p>","path":["Chapter 14. Dynamic Programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#2-0-1-knapsack","level":3,"title":"2.   0-1 Knapsack","text":"<p>You are given equal-length arrays <code>wgt</code> and <code>val</code>. Item <code>i</code> has positive integer weight <code>wgt[i]</code> and non-negative integer value <code>val[i]</code>. The knapsack capacity <code>cap</code> is a non-negative integer. Each item may be selected at most once. Find the maximum total value that can be placed in the knapsack without exceeding <code>cap</code>. Use one-dimensional dynamic programming.</p> Hints <ol> <li>Initialize an array dp of length cap + 1, where dp[c] is the maximum value for a capacity limit of c</li> <li>When processing item i, compare dp[c], which does not select it, with dp[c-wgt[i]] + val[i], which does</li> <li>Update capacities from largest to smallest to avoid selecting the current item repeatedly in the same round</li> </ol>","path":["Chapter 14. Dynamic Programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/","level":1,"title":"14.1   Introduction to Dynamic Programming","text":"<p>Dynamic programming is an important algorithmic paradigm that decomposes a problem into a series of smaller subproblems and avoids redundant computation by storing the solutions to subproblems, thereby significantly improving time efficiency.</p> <p>In this section, we start with a classic example, first presenting its brute force backtracking solution, observing the overlapping subproblems within it, and then gradually deriving a more efficient dynamic programming solution.</p> <p>Climbing stairs</p> <p>Given a staircase with \\(n\\) steps, where you can climb \\(1\\) or \\(2\\) steps at a time, how many different ways are there to reach the top?</p> <p>As shown in Figure 14-1, for a \\(3\\)-step staircase, there are \\(3\\) different ways to reach the top.</p> <p></p> <p> Figure 14-1   Number of ways to reach the 3rd step </p> <p>The goal of this problem is to determine the number of ways, so we can consider using backtracking to enumerate all possibilities. Specifically, imagine climbing stairs as a multi-round selection process: starting from the ground, choosing to go up \\(1\\) or \\(2\\) steps in each round, incrementing the count by \\(1\\) whenever the top of the stairs is reached, and pruning when exceeding the top. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_backtrack.py<pre><code>def backtrack(choices: list[int], state: int, n: int, res: list[int]) -&gt; int:\n    \"\"\"Backtracking\"\"\"\n    # When climbing to the n-th stair, add 1 to the solution count\n    if state == n:\n        res[0] += 1\n    # Traverse all choices\n    for choice in choices:\n        # Pruning: not allowed to go beyond the n-th stair\n        if state + choice &gt; n:\n            continue\n        # Attempt: make a choice, update state\n        backtrack(choices, state + choice, n, res)\n        # Backtrack\n\ndef climbing_stairs_backtrack(n: int) -&gt; int:\n    \"\"\"Climbing stairs: Backtracking\"\"\"\n    choices = [1, 2]  # Can choose to climb up 1 or 2 stairs\n    state = 0  # Start climbing from the 0-th stair\n    res = [0]  # Use res[0] to record the solution count\n    backtrack(choices, state, n, res)\n    return res[0]\n</code></pre> climbing_stairs_backtrack.cpp<pre><code>/* Backtracking */\nvoid backtrack(vector&lt;int&gt; &amp;choices, int state, int n, vector&lt;int&gt; &amp;res) {\n    // When climbing to the n-th stair, add 1 to the solution count\n    if (state == n)\n        res[0]++;\n    // Traverse all choices\n    for (auto &amp;choice : choices) {\n        // Pruning: not allowed to go beyond the n-th stair\n        if (state + choice &gt; n)\n            continue;\n        // Attempt: make choice, update state\n        backtrack(choices, state + choice, n, res);\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: Backtracking */\nint climbingStairsBacktrack(int n) {\n    vector&lt;int&gt; choices = {1, 2}; // Can choose to climb up 1 or 2 stairs\n    int state = 0;                // Start climbing from the 0-th stair\n    vector&lt;int&gt; res = {0};        // Use res[0] to record the solution count\n    backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.java<pre><code>/* Backtracking */\nvoid backtrack(List&lt;Integer&gt; choices, int state, int n, List&lt;Integer&gt; res) {\n    // When climbing to the n-th stair, add 1 to the solution count\n    if (state == n)\n        res.set(0, res.get(0) + 1);\n    // Traverse all choices\n    for (Integer choice : choices) {\n        // Pruning: not allowed to go beyond the n-th stair\n        if (state + choice &gt; n)\n            continue;\n        // Attempt: make choice, update state\n        backtrack(choices, state + choice, n, res);\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: Backtracking */\nint climbingStairsBacktrack(int n) {\n    List&lt;Integer&gt; choices = Arrays.asList(1, 2); // Can choose to climb up 1 or 2 stairs\n    int state = 0; // Start climbing from the 0-th stair\n    List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n    res.add(0); // Use res[0] to record the solution count\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.cs<pre><code>/* Backtracking */\nvoid Backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n    // When climbing to the n-th stair, add 1 to the solution count\n    if (state == n)\n        res[0]++;\n    // Traverse all choices\n    foreach (int choice in choices) {\n        // Pruning: not allowed to go beyond the n-th stair\n        if (state + choice &gt; n)\n            continue;\n        // Attempt: make choice, update state\n        Backtrack(choices, state + choice, n, res);\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: Backtracking */\nint ClimbingStairsBacktrack(int n) {\n    List&lt;int&gt; choices = [1, 2]; // Can choose to climb up 1 or 2 stairs\n    int state = 0; // Start climbing from the 0-th stair\n    List&lt;int&gt; res = [0]; // Use res[0] to record the solution count\n    Backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.go<pre><code>/* Backtracking */\nfunc backtrack(choices []int, state, n int, res []int) {\n    // When climbing to the n-th stair, add 1 to the solution count\n    if state == n {\n        res[0] = res[0] + 1\n    }\n    // Traverse all choices\n    for _, choice := range choices {\n        // Pruning: not allowed to go beyond the n-th stair\n        if state+choice &gt; n {\n            continue\n        }\n        // Attempt: make choice, update state\n        backtrack(choices, state+choice, n, res)\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: Backtracking */\nfunc climbingStairsBacktrack(n int) int {\n    // Can choose to climb up 1 or 2 stairs\n    choices := []int{1, 2}\n    // Start climbing from the 0-th stair\n    state := 0\n    res := make([]int, 1)\n    // Use res[0] to record the solution count\n    res[0] = 0\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.swift<pre><code>/* Backtracking */\nfunc backtrack(choices: [Int], state: Int, n: Int, res: inout [Int]) {\n    // When climbing to the n-th stair, add 1 to the solution count\n    if state == n {\n        res[0] += 1\n    }\n    // Traverse all choices\n    for choice in choices {\n        // Pruning: not allowed to go beyond the n-th stair\n        if state + choice &gt; n {\n            continue\n        }\n        // Attempt: make choice, update state\n        backtrack(choices: choices, state: state + choice, n: n, res: &amp;res)\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: Backtracking */\nfunc climbingStairsBacktrack(n: Int) -&gt; Int {\n    let choices = [1, 2] // Can choose to climb up 1 or 2 stairs\n    let state = 0 // Start climbing from the 0-th stair\n    var res: [Int] = []\n    res.append(0) // Use res[0] to record the solution count\n    backtrack(choices: choices, state: state, n: n, res: &amp;res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.js<pre><code>/* Backtracking */\nfunction backtrack(choices, state, n, res) {\n    // When climbing to the n-th stair, add 1 to the solution count\n    if (state === n) res.set(0, res.get(0) + 1);\n    // Traverse all choices\n    for (const choice of choices) {\n        // Pruning: not allowed to go beyond the n-th stair\n        if (state + choice &gt; n) continue;\n        // Attempt: make choice, update state\n        backtrack(choices, state + choice, n, res);\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: Backtracking */\nfunction climbingStairsBacktrack(n) {\n    const choices = [1, 2]; // Can choose to climb up 1 or 2 stairs\n    const state = 0; // Start climbing from the 0-th stair\n    const res = new Map();\n    res.set(0, 0); // Use res[0] to record the solution count\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.ts<pre><code>/* Backtracking */\nfunction backtrack(\n    choices: number[],\n    state: number,\n    n: number,\n    res: Map&lt;0, any&gt;\n): void {\n    // When climbing to the n-th stair, add 1 to the solution count\n    if (state === n) res.set(0, res.get(0) + 1);\n    // Traverse all choices\n    for (const choice of choices) {\n        // Pruning: not allowed to go beyond the n-th stair\n        if (state + choice &gt; n) continue;\n        // Attempt: make choice, update state\n        backtrack(choices, state + choice, n, res);\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: Backtracking */\nfunction climbingStairsBacktrack(n: number): number {\n    const choices = [1, 2]; // Can choose to climb up 1 or 2 stairs\n    const state = 0; // Start climbing from the 0-th stair\n    const res = new Map();\n    res.set(0, 0); // Use res[0] to record the solution count\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.dart<pre><code>/* Backtracking */\nvoid backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n  // When climbing to the n-th stair, add 1 to the solution count\n  if (state == n) {\n    res[0]++;\n  }\n  // Traverse all choices\n  for (int choice in choices) {\n    // Pruning: not allowed to go beyond the n-th stair\n    if (state + choice &gt; n) continue;\n    // Attempt: make choice, update state\n    backtrack(choices, state + choice, n, res);\n    // Backtrack\n  }\n}\n\n/* Climbing stairs: Backtracking */\nint climbingStairsBacktrack(int n) {\n  List&lt;int&gt; choices = [1, 2]; // Can choose to climb up 1 or 2 stairs\n  int state = 0; // Start climbing from the 0-th stair\n  List&lt;int&gt; res = [];\n  res.add(0); // Use res[0] to record the solution count\n  backtrack(choices, state, n, res);\n  return res[0];\n}\n</code></pre> climbing_stairs_backtrack.rs<pre><code>/* Backtracking */\nfn backtrack(choices: &amp;[i32], state: i32, n: i32, res: &amp;mut [i32]) {\n    // When climbing to the n-th stair, add 1 to the solution count\n    if state == n {\n        res[0] = res[0] + 1;\n    }\n    // Traverse all choices\n    for &amp;choice in choices {\n        // Pruning: not allowed to go beyond the n-th stair\n        if state + choice &gt; n {\n            continue;\n        }\n        // Attempt: make choice, update state\n        backtrack(choices, state + choice, n, res);\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: Backtracking */\nfn climbing_stairs_backtrack(n: usize) -&gt; i32 {\n    let choices = vec![1, 2]; // Can choose to climb up 1 or 2 stairs\n    let state = 0; // Start climbing from the 0-th stair\n    let mut res = Vec::new();\n    res.push(0); // Use res[0] to record the solution count\n    backtrack(&amp;choices, state, n as i32, &amp;mut res);\n    res[0]\n}\n</code></pre> climbing_stairs_backtrack.c<pre><code>/* Backtracking */\nvoid backtrack(int *choices, int state, int n, int *res, int len) {\n    // When climbing to the n-th stair, add 1 to the solution count\n    if (state == n)\n        res[0]++;\n    // Traverse all choices\n    for (int i = 0; i &lt; len; i++) {\n        int choice = choices[i];\n        // Pruning: not allowed to go beyond the n-th stair\n        if (state + choice &gt; n)\n            continue;\n        // Attempt: make choice, update state\n        backtrack(choices, state + choice, n, res, len);\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: Backtracking */\nint climbingStairsBacktrack(int n) {\n    int choices[2] = {1, 2}; // Can choose to climb up 1 or 2 stairs\n    int state = 0;           // Start climbing from the 0-th stair\n    int *res = (int *)malloc(sizeof(int));\n    *res = 0; // Use res[0] to record the solution count\n    int len = sizeof(choices) / sizeof(int);\n    backtrack(choices, state, n, res, len);\n    int result = *res;\n    free(res);\n    return result;\n}\n</code></pre> climbing_stairs_backtrack.kt<pre><code>/* Backtracking */\nfun backtrack(\n    choices: MutableList&lt;Int&gt;,\n    state: Int,\n    n: Int,\n    res: MutableList&lt;Int&gt;\n) {\n    // When climbing to the n-th stair, add 1 to the solution count\n    if (state == n)\n        res[0] = res[0] + 1\n    // Traverse all choices\n    for (choice in choices) {\n        // Pruning: not allowed to go beyond the n-th stair\n        if (state + choice &gt; n) continue\n        // Attempt: make choice, update state\n        backtrack(choices, state + choice, n, res)\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: Backtracking */\nfun climbingStairsBacktrack(n: Int): Int {\n    val choices = mutableListOf(1, 2) // Can choose to climb up 1 or 2 stairs\n    val state = 0 // Start climbing from the 0-th stair\n    val res = mutableListOf&lt;Int&gt;()\n    res.add(0) // Use res[0] to record the solution count\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.rb<pre><code>### Backtracking ###\ndef backtrack(choices, state, n, res)\n  # When climbing to the n-th stair, add 1 to the solution count\n  res[0] += 1 if state == n\n  # Traverse all choices\n  for choice in choices\n    # Pruning: not allowed to go beyond the n-th stair\n    next if state + choice &gt; n\n\n    # Attempt: make choice, update state\n    backtrack(choices, state + choice, n, res)\n  end\n  # Backtrack\nend\n\n### Climbing stairs: backtracking ###\ndef climbing_stairs_backtrack(n)\n  choices = [1, 2] # Can choose to climb up 1 or 2 stairs\n  state = 0 # Start climbing from the 0-th stair\n  res = [0] # Use res[0] to record the solution count\n  backtrack(choices, state, n, res)\n  res.first\nend\n</code></pre>","path":["Chapter 14. Dynamic Programming","14.1   Introduction to Dynamic Programming"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1411-method-1-brute-force-search","level":2,"title":"14.1.1   Method 1: Brute Force Search","text":"<p>Backtracking algorithms typically do not explicitly decompose problems, but rather treat solving the problem as a series of decision steps, searching for all possible solutions through trial and pruning.</p> <p>We can try to analyze this problem from the perspective of problem decomposition. Let the number of ways to climb to the \\(i\\)-th step be \\(dp[i]\\), then \\(dp[i]\\) is the original problem, and its subproblems include:</p> \\[ dp[i-1], dp[i-2], \\dots, dp[2], dp[1] \\] <p>Since we can only go up \\(1\\) or \\(2\\) steps in each round, when we stand on the \\(i\\)-th step, we could only have been on the \\(i-1\\)-th or \\(i-2\\)-th step in the previous round. In other words, we can only reach the \\(i\\)-th step from the \\(i-1\\)-th or \\(i-2\\)-th step.</p> <p>This leads to an important conclusion: the number of ways to climb to the \\(i-1\\)-th step plus the number of ways to climb to the \\(i-2\\)-th step equals the number of ways to climb to the \\(i\\)-th step. The formula is as follows:</p> \\[ dp[i] = dp[i-1] + dp[i-2] \\] <p>This means that in the stair climbing problem, there exists a recurrence relation among the subproblems, and the solution to the original problem can be constructed from the solutions to the subproblems. Figure 14-2 illustrates this recurrence relation.</p> <p></p> <p> Figure 14-2   Recurrence relation for the number of ways </p> <p>We can obtain a brute force search solution based on the recurrence formula. Starting from \\(dp[n]\\), recursively decompose a larger problem into the sum of two smaller problems, until reaching the smallest subproblems \\(dp[1]\\) and \\(dp[2]\\) and returning. Among them, the solutions to the smallest subproblems are known, namely \\(dp[1] = 1\\) and \\(dp[2] = 2\\), representing \\(1\\) and \\(2\\) ways to climb to the \\(1\\)st and \\(2\\)nd steps, respectively.</p> <p>Observe the following code: like standard backtracking code, it also uses depth-first search but is more concise:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dfs.py<pre><code>def dfs(i: int) -&gt; int:\n    \"\"\"Search\"\"\"\n    # Known dp[1] and dp[2], return them\n    if i == 1 or i == 2:\n        return i\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1) + dfs(i - 2)\n    return count\n\ndef climbing_stairs_dfs(n: int) -&gt; int:\n    \"\"\"Climbing stairs: Search\"\"\"\n    return dfs(n)\n</code></pre> climbing_stairs_dfs.cpp<pre><code>/* Search */\nint dfs(int i) {\n    // Known dp[1] and dp[2], return them\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* Climbing stairs: Search */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.java<pre><code>/* Search */\nint dfs(int i) {\n    // Known dp[1] and dp[2], return them\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* Climbing stairs: Search */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.cs<pre><code>/* Search */\nint DFS(int i) {\n    // Known dp[1] and dp[2], return them\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1) + DFS(i - 2);\n    return count;\n}\n\n/* Climbing stairs: Search */\nint ClimbingStairsDFS(int n) {\n    return DFS(n);\n}\n</code></pre> climbing_stairs_dfs.go<pre><code>/* Search */\nfunc dfs(i int) int {\n    // Known dp[1] and dp[2], return them\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfs(i-1) + dfs(i-2)\n    return count\n}\n\n/* Climbing stairs: Search */\nfunc climbingStairsDFS(n int) int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.swift<pre><code>/* Search */\nfunc dfs(i: Int) -&gt; Int {\n    // Known dp[1] and dp[2], return them\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1) + dfs(i: i - 2)\n    return count\n}\n\n/* Climbing stairs: Search */\nfunc climbingStairsDFS(n: Int) -&gt; Int {\n    dfs(i: n)\n}\n</code></pre> climbing_stairs_dfs.js<pre><code>/* Search */\nfunction dfs(i) {\n    // Known dp[1] and dp[2], return them\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* Climbing stairs: Search */\nfunction climbingStairsDFS(n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.ts<pre><code>/* Search */\nfunction dfs(i: number): number {\n    // Known dp[1] and dp[2], return them\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* Climbing stairs: Search */\nfunction climbingStairsDFS(n: number): number {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.dart<pre><code>/* Search */\nint dfs(int i) {\n  // Known dp[1] and dp[2], return them\n  if (i == 1 || i == 2) return i;\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1) + dfs(i - 2);\n  return count;\n}\n\n/* Climbing stairs: Search */\nint climbingStairsDFS(int n) {\n  return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.rs<pre><code>/* Search */\nfn dfs(i: usize) -&gt; i32 {\n    // Known dp[1] and dp[2], return them\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1) + dfs(i - 2);\n    count\n}\n\n/* Climbing stairs: Search */\nfn climbing_stairs_dfs(n: usize) -&gt; i32 {\n    dfs(n)\n}\n</code></pre> climbing_stairs_dfs.c<pre><code>/* Search */\nint dfs(int i) {\n    // Known dp[1] and dp[2], return them\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* Climbing stairs: Search */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.kt<pre><code>/* Search */\nfun dfs(i: Int): Int {\n    // Known dp[1] and dp[2], return them\n    if (i == 1 || i == 2) return i\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1) + dfs(i - 2)\n    return count\n}\n\n/* Climbing stairs: Search */\nfun climbingStairsDFS(n: Int): Int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.rb<pre><code>### Search ###\ndef dfs(i)\n  # Known dp[1] and dp[2], return them\n  return i if i == 1 || i == 2\n  # dp[i] = dp[i-1] + dp[i-2]\n  dfs(i - 1) + dfs(i - 2)\nend\n\n### Climbing stairs: search ###\ndef climbing_stairs_dfs(n)\n  dfs(n)\nend\n</code></pre> <p>Figure 14-3 shows the recursion tree formed by brute force search. For the problem \\(dp[n]\\), the depth of its recursion tree is \\(n\\), with a time complexity of \\(O(2^n)\\). Exponential growth is explosive; if we input a relatively large \\(n\\), the wait can be very long.</p> <p></p> <p> Figure 14-3   Recursion tree for climbing stairs </p> <p>Observing the above figure, the exponential time complexity is caused by \"overlapping subproblems\". For example, \\(dp[9]\\) is decomposed into \\(dp[8]\\) and \\(dp[7]\\), and \\(dp[8]\\) is decomposed into \\(dp[7]\\) and \\(dp[6]\\), both of which contain the subproblem \\(dp[7]\\).</p> <p>And so on, subproblems contain smaller overlapping subproblems, ad infinitum. The vast majority of computational resources are wasted on these overlapping subproblems.</p>","path":["Chapter 14. Dynamic Programming","14.1   Introduction to Dynamic Programming"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1412-method-2-memoization","level":2,"title":"14.1.2   Method 2: Memoization","text":"<p>To improve algorithm efficiency, we want all overlapping subproblems to be computed only once. For this purpose, we declare an array <code>mem</code> to record the solution to each subproblem and prune overlapping subproblems during the search process.</p> <ol> <li>When computing \\(dp[i]\\) for the first time, we record it in <code>mem[i]</code> for later use.</li> <li>When we need to compute \\(dp[i]\\) again, we can directly retrieve the result from <code>mem[i]</code>, thereby avoiding redundant computation of that subproblem.</li> </ol> <p>The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dfs_mem.py<pre><code>def dfs(i: int, mem: list[int]) -&gt; int:\n    \"\"\"Memoization search\"\"\"\n    # Known dp[1] and dp[2], return them\n    if i == 1 or i == 2:\n        return i\n    # If record dp[i] exists, return it directly\n    if mem[i] != -1:\n        return mem[i]\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    # Record dp[i]\n    mem[i] = count\n    return count\n\ndef climbing_stairs_dfs_mem(n: int) -&gt; int:\n    \"\"\"Climbing stairs: Memoization search\"\"\"\n    # mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record\n    mem = [-1] * (n + 1)\n    return dfs(n, mem)\n</code></pre> climbing_stairs_dfs_mem.cpp<pre><code>/* Memoization search */\nint dfs(int i, vector&lt;int&gt; &amp;mem) {\n    // Known dp[1] and dp[2], return them\n    if (i == 1 || i == 2)\n        return i;\n    // If record dp[i] exists, return it directly\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // Record dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* Climbing stairs: Memoization search */\nint climbingStairsDFSMem(int n) {\n    // mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record\n    vector&lt;int&gt; mem(n + 1, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.java<pre><code>/* Memoization search */\nint dfs(int i, int[] mem) {\n    // Known dp[1] and dp[2], return them\n    if (i == 1 || i == 2)\n        return i;\n    // If record dp[i] exists, return it directly\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // Record dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* Climbing stairs: Memoization search */\nint climbingStairsDFSMem(int n) {\n    // mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record\n    int[] mem = new int[n + 1];\n    Arrays.fill(mem, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.cs<pre><code>/* Memoization search */\nint DFS(int i, int[] mem) {\n    // Known dp[1] and dp[2], return them\n    if (i == 1 || i == 2)\n        return i;\n    // If record dp[i] exists, return it directly\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1, mem) + DFS(i - 2, mem);\n    // Record dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* Climbing stairs: Memoization search */\nint ClimbingStairsDFSMem(int n) {\n    // mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record\n    int[] mem = new int[n + 1];\n    Array.Fill(mem, -1);\n    return DFS(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.go<pre><code>/* Memoization search */\nfunc dfsMem(i int, mem []int) int {\n    // Known dp[1] and dp[2], return them\n    if i == 1 || i == 2 {\n        return i\n    }\n    // If record dp[i] exists, return it directly\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfsMem(i-1, mem) + dfsMem(i-2, mem)\n    // Record dp[i]\n    mem[i] = count\n    return count\n}\n\n/* Climbing stairs: Memoization search */\nfunc climbingStairsDFSMem(n int) int {\n    // mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record\n    mem := make([]int, n+1)\n    for i := range mem {\n        mem[i] = -1\n    }\n    return dfsMem(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.swift<pre><code>/* Memoization search */\nfunc dfs(i: Int, mem: inout [Int]) -&gt; Int {\n    // Known dp[1] and dp[2], return them\n    if i == 1 || i == 2 {\n        return i\n    }\n    // If record dp[i] exists, return it directly\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1, mem: &amp;mem) + dfs(i: i - 2, mem: &amp;mem)\n    // Record dp[i]\n    mem[i] = count\n    return count\n}\n\n/* Climbing stairs: Memoization search */\nfunc climbingStairsDFSMem(n: Int) -&gt; Int {\n    // mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record\n    var mem = Array(repeating: -1, count: n + 1)\n    return dfs(i: n, mem: &amp;mem)\n}\n</code></pre> climbing_stairs_dfs_mem.js<pre><code>/* Memoization search */\nfunction dfs(i, mem) {\n    // Known dp[1] and dp[2], return them\n    if (i === 1 || i === 2) return i;\n    // If record dp[i] exists, return it directly\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // Record dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* Climbing stairs: Memoization search */\nfunction climbingStairsDFSMem(n) {\n    // mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.ts<pre><code>/* Memoization search */\nfunction dfs(i: number, mem: number[]): number {\n    // Known dp[1] and dp[2], return them\n    if (i === 1 || i === 2) return i;\n    // If record dp[i] exists, return it directly\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // Record dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* Climbing stairs: Memoization search */\nfunction climbingStairsDFSMem(n: number): number {\n    // mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.dart<pre><code>/* Memoization search */\nint dfs(int i, List&lt;int&gt; mem) {\n  // Known dp[1] and dp[2], return them\n  if (i == 1 || i == 2) return i;\n  // If record dp[i] exists, return it directly\n  if (mem[i] != -1) return mem[i];\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n  // Record dp[i]\n  mem[i] = count;\n  return count;\n}\n\n/* Climbing stairs: Memoization search */\nint climbingStairsDFSMem(int n) {\n  // mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record\n  List&lt;int&gt; mem = List.filled(n + 1, -1);\n  return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.rs<pre><code>/* Memoization search */\nfn dfs(i: usize, mem: &amp;mut [i32]) -&gt; i32 {\n    // Known dp[1] and dp[2], return them\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // If record dp[i] exists, return it directly\n    if mem[i] != -1 {\n        return mem[i];\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // Record dp[i]\n    mem[i] = count;\n    count\n}\n\n/* Climbing stairs: Memoization search */\nfn climbing_stairs_dfs_mem(n: usize) -&gt; i32 {\n    // mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record\n    let mut mem = vec![-1; n + 1];\n    dfs(n, &amp;mut mem)\n}\n</code></pre> climbing_stairs_dfs_mem.c<pre><code>/* Memoization search */\nint dfs(int i, int *mem) {\n    // Known dp[1] and dp[2], return them\n    if (i == 1 || i == 2)\n        return i;\n    // If record dp[i] exists, return it directly\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // Record dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* Climbing stairs: Memoization search */\nint climbingStairsDFSMem(int n) {\n    // mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record\n    int *mem = (int *)malloc((n + 1) * sizeof(int));\n    for (int i = 0; i &lt;= n; i++) {\n        mem[i] = -1;\n    }\n    int result = dfs(n, mem);\n    free(mem);\n    return result;\n}\n</code></pre> climbing_stairs_dfs_mem.kt<pre><code>/* Memoization search */\nfun dfs(i: Int, mem: IntArray): Int {\n    // Known dp[1] and dp[2], return them\n    if (i == 1 || i == 2) return i\n    // If record dp[i] exists, return it directly\n    if (mem[i] != -1) return mem[i]\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    // Record dp[i]\n    mem[i] = count\n    return count\n}\n\n/* Climbing stairs: Memoization search */\nfun climbingStairsDFSMem(n: Int): Int {\n    // mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record\n    val mem = IntArray(n + 1)\n    mem.fill(-1)\n    return dfs(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.rb<pre><code>### Memoization search ###\ndef dfs(i, mem)\n  # Known dp[1] and dp[2], return them\n  return i if i == 1 || i == 2\n  # If record dp[i] exists, return it directly\n  return mem[i] if mem[i] != -1\n\n  # dp[i] = dp[i-1] + dp[i-2]\n  count = dfs(i - 1, mem) + dfs(i - 2, mem)\n  # Record dp[i]\n  mem[i] = count\nend\n\n### Climbing stairs: memoization search ###\ndef climbing_stairs_dfs_mem(n)\n  # mem[i] records the total number of solutions to climb to the i-th stair, -1 means no record\n  mem = Array.new(n + 1, -1)\n  dfs(n, mem)\nend\n</code></pre> <p>Observe Figure 14-4: after memoization, all overlapping subproblems need to be computed only once, reducing the time complexity to \\(O(n)\\), which is a tremendous leap.</p> <p></p> <p> Figure 14-4   Recursion tree with memoization </p>","path":["Chapter 14. Dynamic Programming","14.1   Introduction to Dynamic Programming"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1413-method-3-dynamic-programming","level":2,"title":"14.1.3   Method 3: Dynamic Programming","text":"<p>Memoization is a \"top-down\" method: we start from the original problem (root node), recursively decompose larger subproblems into smaller ones, until reaching the smallest known subproblems (leaf nodes). Afterward, by backtracking, we collect the solutions to the subproblems layer by layer to construct the solution to the original problem.</p> <p>In contrast, dynamic programming is a \"bottom-up\" method: starting from the solutions to the smallest subproblems, iteratively constructing solutions to larger subproblems until obtaining the solution to the original problem.</p> <p>Since dynamic programming does not include a backtracking process, it only requires loop iteration for implementation and does not need recursion. In the following code, we initialize an array <code>dp</code> to store the solutions to subproblems, which serves the same recording function as the array <code>mem</code> in memoization:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dp.py<pre><code>def climbing_stairs_dp(n: int) -&gt; int:\n    \"\"\"Climbing stairs: Dynamic programming\"\"\"\n    if n == 1 or n == 2:\n        return n\n    # Initialize dp table, used to store solutions to subproblems\n    dp = [0] * (n + 1)\n    # Initial state: preset the solution to the smallest subproblem\n    dp[1], dp[2] = 1, 2\n    # State transition: gradually solve larger subproblems from smaller ones\n    for i in range(3, n + 1):\n        dp[i] = dp[i - 1] + dp[i - 2]\n    return dp[n]\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* Climbing stairs: Dynamic programming */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // Initialize dp table, used to store solutions to subproblems\n    vector&lt;int&gt; dp(n + 1);\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = 1;\n    dp[2] = 2;\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* Climbing stairs: Dynamic programming */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // Initialize dp table, used to store solutions to subproblems\n    int[] dp = new int[n + 1];\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = 1;\n    dp[2] = 2;\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* Climbing stairs: Dynamic programming */\nint ClimbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // Initialize dp table, used to store solutions to subproblems\n    int[] dp = new int[n + 1];\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = 1;\n    dp[2] = 2;\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* Climbing stairs: Dynamic programming */\nfunc climbingStairsDP(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    dp := make([]int, n+1)\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = 1\n    dp[2] = 2\n    // State transition: gradually solve larger subproblems from smaller ones\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = dp[i-1] + dp[i-2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* Climbing stairs: Dynamic programming */\nfunc climbingStairsDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    var dp = Array(repeating: 0, count: n + 1)\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = 1\n    dp[2] = 2\n    // State transition: gradually solve larger subproblems from smaller ones\n    for i in 3 ... n {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* Climbing stairs: Dynamic programming */\nfunction climbingStairsDP(n) {\n    if (n === 1 || n === 2) return n;\n    // Initialize dp table, used to store solutions to subproblems\n    const dp = new Array(n + 1).fill(-1);\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = 1;\n    dp[2] = 2;\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* Climbing stairs: Dynamic programming */\nfunction climbingStairsDP(n: number): number {\n    if (n === 1 || n === 2) return n;\n    // Initialize dp table, used to store solutions to subproblems\n    const dp = new Array(n + 1).fill(-1);\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = 1;\n    dp[2] = 2;\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* Climbing stairs: Dynamic programming */\nint climbingStairsDP(int n) {\n  if (n == 1 || n == 2) return n;\n  // Initialize dp table, used to store solutions to subproblems\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // Initial state: preset the solution to the smallest subproblem\n  dp[1] = 1;\n  dp[2] = 2;\n  // State transition: gradually solve larger subproblems from smaller ones\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = dp[i - 1] + dp[i - 2];\n  }\n  return dp[n];\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* Climbing stairs: Dynamic programming */\nfn climbing_stairs_dp(n: usize) -&gt; i32 {\n    // Known dp[1] and dp[2], return them\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    // Initialize dp table, used to store solutions to subproblems\n    let mut dp = vec![-1; n + 1];\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = 1;\n    dp[2] = 2;\n    // State transition: gradually solve larger subproblems from smaller ones\n    for i in 3..=n {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    dp[n]\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* Climbing stairs: Dynamic programming */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // Initialize dp table, used to store solutions to subproblems\n    int *dp = (int *)malloc((n + 1) * sizeof(int));\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = 1;\n    dp[2] = 2;\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    int result = dp[n];\n    free(dp);\n    return result;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* Climbing stairs: Dynamic programming */\nfun climbingStairsDP(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    // Initialize dp table, used to store solutions to subproblems\n    val dp = IntArray(n + 1)\n    // Initial state: preset the solution to the smallest subproblem\n    dp[1] = 1\n    dp[2] = 2\n    // State transition: gradually solve larger subproblems from smaller ones\n    for (i in 3..n) {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>### Climbing stairs: dynamic programming ###\ndef climbing_stairs_dp(n)\n  return n  if n == 1 || n == 2\n\n  # Initialize dp table, used to store solutions to subproblems\n  dp = Array.new(n + 1, 0)\n  # Initial state: preset the solution to the smallest subproblem\n  dp[1], dp[2] = 1, 2\n  # State transition: gradually solve larger subproblems from smaller ones\n  (3...(n + 1)).each { |i| dp[i] = dp[i - 1] + dp[i - 2] }\n\n  dp[n]\nend\n</code></pre> <p>Figure 14-5 simulates the execution process of the above code.</p> <p></p> <p> Figure 14-5   Dynamic programming process for climbing stairs </p> <p>Like backtracking algorithms, dynamic programming also uses the \"state\" concept to represent specific stages of problem solving, with each state corresponding to a subproblem and its corresponding local optimal solution. For example, the state in the stair climbing problem is defined as the current stair step number \\(i\\).</p> <p>Based on the above content, we can summarize the commonly used terminology in dynamic programming.</p> <ul> <li>The array <code>dp</code> is called the dp table, where \\(dp[i]\\) represents the solution to the subproblem corresponding to state \\(i\\).</li> <li>The states corresponding to the smallest subproblems (the \\(1\\)st and \\(2\\)nd steps) are called initial states.</li> <li>The recurrence formula \\(dp[i] = dp[i-1] + dp[i-2]\\) is called the state transition equation.</li> </ul>","path":["Chapter 14. Dynamic Programming","14.1   Introduction to Dynamic Programming"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1414-space-optimization","level":2,"title":"14.1.4   Space Optimization","text":"<p>Observant readers may have noticed that since \\(dp[i]\\) is only related to \\(dp[i-1]\\) and \\(dp[i-2]\\), we do not need to use an array <code>dp</code> to store the solutions to all subproblems, and can instead use two variables that roll forward. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dp.py<pre><code>def climbing_stairs_dp_comp(n: int) -&gt; int:\n    \"\"\"Climbing stairs: Space-optimized dynamic programming\"\"\"\n    if n == 1 or n == 2:\n        return n\n    a, b = 1, 2\n    for _ in range(3, n + 1):\n        a, b = b, a + b\n    return b\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* Climbing stairs: Space-optimized dynamic programming */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* Climbing stairs: Space-optimized dynamic programming */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* Climbing stairs: Space-optimized dynamic programming */\nint ClimbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* Climbing stairs: Space-optimized dynamic programming */\nfunc climbingStairsDPComp(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    a, b := 1, 2\n    // State transition: gradually solve larger subproblems from smaller ones\n    for i := 3; i &lt;= n; i++ {\n        a, b = b, a+b\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* Climbing stairs: Space-optimized dynamic programming */\nfunc climbingStairsDPComp(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    var a = 1\n    var b = 2\n    for _ in 3 ... n {\n        (a, b) = (b, a + b)\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* Climbing stairs: Space-optimized dynamic programming */\nfunction climbingStairsDPComp(n) {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* Climbing stairs: Space-optimized dynamic programming */\nfunction climbingStairsDPComp(n: number): number {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* Climbing stairs: Space-optimized dynamic programming */\nint climbingStairsDPComp(int n) {\n  if (n == 1 || n == 2) return n;\n  int a = 1, b = 2;\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = a + b;\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* Climbing stairs: Space-optimized dynamic programming */\nfn climbing_stairs_dp_comp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    let (mut a, mut b) = (1, 2);\n    for _ in 3..=n {\n        let tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    b\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* Climbing stairs: Space-optimized dynamic programming */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* Climbing stairs: Space-optimized dynamic programming */\nfun climbingStairsDPComp(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    var a = 1\n    var b = 2\n    for (i in 3..n) {\n        val temp = b\n        b += a\n        a = temp\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>### Climbing stairs: space-optimized DP ###\ndef climbing_stairs_dp_comp(n)\n  return n if n == 1 || n == 2\n\n  a, b = 1, 2\n  (3...(n + 1)).each { a, b = b, a + b }\n\n  b\nend\n</code></pre> <p>As the above code shows, by eliminating the space occupied by the array <code>dp</code>, the space complexity is reduced from \\(O(n)\\) to \\(O(1)\\).</p> <p>In dynamic programming problems, the current state often depends only on a limited number of preceding states, allowing us to retain only the necessary states and save memory space through \"dimension reduction\". This space optimization technique is called \"rolling variable\" or \"rolling array\".</p>","path":["Chapter 14. Dynamic Programming","14.1   Introduction to Dynamic Programming"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/","level":1,"title":"14.4   0-1 Knapsack Problem","text":"<p>The knapsack problem is an excellent introductory problem for dynamic programming and is one of the most common problem forms in dynamic programming. It has many variants, such as the 0-1 knapsack problem, the unbounded knapsack problem, and the multiple knapsack problem.</p> <p>In this section, we will first solve the most common 0-1 knapsack problem.</p> <p>Question</p> <p>Given \\(n\\) items and a knapsack with capacity \\(cap\\), where the weight and value of the \\(i\\)-th item are \\(wgt[i-1]\\) and \\(val[i-1]\\), respectively. Each item can be selected at most once. What is the maximum value that can fit in the knapsack under the capacity limit?</p> <p>Observe Figure 14-17. Since item number \\(i\\) starts counting from \\(1\\) and array indices start from \\(0\\), item \\(i\\) corresponds to weight \\(wgt[i-1]\\) and value \\(val[i-1]\\).</p> <p></p> <p> Figure 14-17   Example data for 0-1 knapsack </p> <p>We can view the 0-1 knapsack problem as a process consisting of \\(n\\) rounds of decisions, where for each item there are two decisions: not putting it in and putting it in, thus the problem satisfies the decision tree model.</p> <p>The goal of this problem is to find \"the maximum value that can be placed in the knapsack within the capacity limit\", so it is more likely to be a dynamic programming problem.</p> <p>Step 1: Think about the decisions in each round, define the state, and thus obtain the \\(dp\\) table</p> <p>For each item, if not placed in the knapsack, the knapsack capacity remains unchanged; if placed in, the knapsack capacity decreases. From this, we can derive the state definition: current item number \\(i\\) and knapsack capacity \\(c\\), denoted as \\([i, c]\\).</p> <p>State \\([i, c]\\) corresponds to the subproblem: the maximum value among the first \\(i\\) items in a knapsack of capacity \\(c\\), denoted as \\(dp[i, c]\\).</p> <p>What we need to find is \\(dp[n, cap]\\), so we need a two-dimensional \\(dp\\) table of size \\((n+1) \\times (cap+1)\\).</p> <p>Step 2: Identify the optimal substructure, and then derive the state transition equation</p> <p>After making the decision for item \\(i\\), what remains is the subproblem of the first \\(i-1\\) items, which can be divided into the following two cases.</p> <ul> <li>Not putting item \\(i\\): The knapsack capacity remains unchanged, and the state changes to \\([i-1, c]\\).</li> <li>Putting item \\(i\\): The knapsack capacity decreases by \\(wgt[i-1]\\), the value increases by \\(val[i-1]\\), and the state changes to \\([i-1, c-wgt[i-1]]\\).</li> </ul> <p>The above analysis reveals the optimal substructure of this problem: the maximum value \\(dp[i, c]\\) equals the greater of the values obtained by not putting item \\(i\\) into the knapsack and by putting it into the knapsack. From this, the state transition equation can be derived:</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1]) \\] <p>Note that if the weight of the current item \\(wgt[i - 1]\\) exceeds the remaining knapsack capacity \\(c\\), then the only option is not to put it in the knapsack.</p> <p>Step 3: Determine boundary conditions and state transition order</p> <p>When there are no items or the knapsack capacity is \\(0\\), the maximum value is \\(0\\), i.e., the first column \\(dp[i, 0]\\) and the first row \\(dp[0, c]\\) are both equal to \\(0\\).</p> <p>The current state \\([i, c]\\) transitions from the state above \\([i-1, c]\\) and the upper-left state \\([i-1, c-wgt[i-1]]\\), so we can traverse the entire \\(dp\\) table in forward order using two nested loops.</p> <p>Based on the above analysis, we will next implement the brute force search, memoization, and dynamic programming solutions in order.</p>","path":["Chapter 14. Dynamic Programming","14.4   0-1 Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#1-method-1-brute-force-search","level":3,"title":"1.   Method 1: Brute Force Search","text":"<p>The search code includes the following elements.</p> <ul> <li>Recursive parameters: state \\([i, c]\\).</li> <li>Return value: solution to the subproblem \\(dp[i, c]\\).</li> <li>Termination condition: when there are no items left (\\(i = 0\\)) or the remaining knapsack capacity is \\(0\\), terminate the recursion and return value \\(0\\).</li> <li>Pruning: if the weight of the current item exceeds the remaining knapsack capacity, only the option of not putting it in is available.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dfs(wgt: list[int], val: list[int], i: int, c: int) -&gt; int:\n    \"\"\"0-1 knapsack: Brute-force search\"\"\"\n    # If all items have been selected or knapsack has no remaining capacity, return value 0\n    if i == 0 or c == 0:\n        return 0\n    # If exceeds knapsack capacity, can only choose not to put it in\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs(wgt, val, i - 1, c)\n    # Calculate the maximum value of not putting in and putting in item i\n    no = knapsack_dfs(wgt, val, i - 1, c)\n    yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # Return the larger value of the two options\n    return max(no, yes)\n</code></pre> knapsack.cpp<pre><code>/* 0-1 knapsack: Brute-force search */\nint knapsackDFS(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int i, int c) {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Return the larger value of the two options\n    return max(no, yes);\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 knapsack: Brute-force search */\nint knapsackDFS(int[] wgt, int[] val, int i, int c) {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Return the larger value of the two options\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 knapsack: Brute-force search */\nint KnapsackDFS(int[] weight, int[] val, int i, int c) {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFS(weight, val, i - 1, c);\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    int no = KnapsackDFS(weight, val, i - 1, c);\n    int yes = KnapsackDFS(weight, val, i - 1, c - weight[i - 1]) + val[i - 1];\n    // Return the larger value of the two options\n    return Math.Max(no, yes);\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 knapsack: Brute-force search */\nfunc knapsackDFS(wgt, val []int, i, c int) int {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if wgt[i-1] &gt; c {\n        return knapsackDFS(wgt, val, i-1, c)\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    no := knapsackDFS(wgt, val, i-1, c)\n    yes := knapsackDFS(wgt, val, i-1, c-wgt[i-1]) + val[i-1]\n    // Return the larger value of the two options\n    return int(math.Max(float64(no), float64(yes)))\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 knapsack: Brute-force search */\nfunc knapsackDFS(wgt: [Int], val: [Int], i: Int, c: Int) -&gt; Int {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if wgt[i - 1] &gt; c {\n        return knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    let no = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    let yes = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // Return the larger value of the two options\n    return max(no, yes)\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 knapsack: Brute-force search */\nfunction knapsackDFS(wgt, val, i, c) {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Return the larger value of the two options\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 knapsack: Brute-force search */\nfunction knapsackDFS(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    i: number,\n    c: number\n): number {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Return the larger value of the two options\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 knapsack: Brute-force search */\nint knapsackDFS(List&lt;int&gt; wgt, List&lt;int&gt; val, int i, int c) {\n  // If all items have been selected or knapsack has no remaining capacity, return value 0\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // If exceeds knapsack capacity, can only choose not to put it in\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFS(wgt, val, i - 1, c);\n  }\n  // Calculate the maximum value of not putting in and putting in item i\n  int no = knapsackDFS(wgt, val, i - 1, c);\n  int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // Return the larger value of the two options\n  return max(no, yes);\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 knapsack: Brute-force search */\nfn knapsack_dfs(wgt: &amp;[i32], val: &amp;[i32], i: usize, c: usize) -&gt; i32 {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs(wgt, val, i - 1, c);\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    let no = knapsack_dfs(wgt, val, i - 1, c);\n    let yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // Return the larger value of the two options\n    std::cmp::max(no, yes)\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 knapsack: Brute-force search */\nint knapsackDFS(int wgt[], int val[], int i, int c) {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Return the larger value of the two options\n    return myMax(no, yes);\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 knapsack: Brute-force search */\nfun knapsackDFS(\n    wgt: IntArray,\n    _val: IntArray,\n    i: Int,\n    c: Int\n): Int {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, _val, i - 1, c)\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    val no = knapsackDFS(wgt, _val, i - 1, c)\n    val yes = knapsackDFS(wgt, _val, i - 1, c - wgt[i - 1]) + _val[i - 1]\n    // Return the larger value of the two options\n    return max(no, yes)\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 knapsack: brute force search ###\ndef knapsack_dfs(wgt, val, i, c)\n  # If all items have been selected or knapsack has no remaining capacity, return value 0\n  return 0 if i == 0 || c == 0\n  # If exceeds knapsack capacity, can only choose not to put it in\n  return knapsack_dfs(wgt, val, i - 1, c) if wgt[i - 1] &gt; c\n  # Calculate the maximum value of not putting in and putting in item i\n  no = knapsack_dfs(wgt, val, i - 1, c)\n  yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n  # Return the larger value of the two options\n  [no, yes].max\nend\n</code></pre> <p>As shown in Figure 14-18, since each item generates two search branches, excluding it and including it, the time complexity is \\(O(2^n)\\).</p> <p>Observing the recursion tree, it is easy to see overlapping subproblems, such as \\(dp[1, 10]\\). When there are many items, large knapsack capacity, and especially many items with the same weight, the number of overlapping subproblems will increase significantly.</p> <p></p> <p> Figure 14-18   Brute force search recursion tree for 0-1 knapsack problem </p>","path":["Chapter 14. Dynamic Programming","14.4   0-1 Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#2-method-2-memoization","level":3,"title":"2.   Method 2: Memoization","text":"<p>To ensure that overlapping subproblems are only computed once, we use a memo list <code>mem</code> to record the solutions to subproblems, where <code>mem[i][c]</code> corresponds to \\(dp[i, c]\\).</p> <p>After introducing memoization, the time complexity depends on the number of subproblems, which is \\(O(n \\times cap)\\). The implementation code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dfs_mem(\n    wgt: list[int], val: list[int], mem: list[list[int]], i: int, c: int\n) -&gt; int:\n    \"\"\"0-1 knapsack: Memoization search\"\"\"\n    # If all items have been selected or knapsack has no remaining capacity, return value 0\n    if i == 0 or c == 0:\n        return 0\n    # If there's a record, return it directly\n    if mem[i][c] != -1:\n        return mem[i][c]\n    # If exceeds knapsack capacity, can only choose not to put it in\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    # Calculate the maximum value of not putting in and putting in item i\n    no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # Record and return the larger value of the two options\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 knapsack: Memoization search */\nint knapsackDFSMem(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int c) {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If there's a record, return it directly\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Record and return the larger value of the two options\n    mem[i][c] = max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 knapsack: Memoization search */\nint knapsackDFSMem(int[] wgt, int[] val, int[][] mem, int i, int c) {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If there's a record, return it directly\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Record and return the larger value of the two options\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 knapsack: Memoization search */\nint KnapsackDFSMem(int[] weight, int[] val, int[][] mem, int i, int c) {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If there's a record, return it directly\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFSMem(weight, val, mem, i - 1, c);\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    int no = KnapsackDFSMem(weight, val, mem, i - 1, c);\n    int yes = KnapsackDFSMem(weight, val, mem, i - 1, c - weight[i - 1]) + val[i - 1];\n    // Record and return the larger value of the two options\n    mem[i][c] = Math.Max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 knapsack: Memoization search */\nfunc knapsackDFSMem(wgt, val []int, mem [][]int, i, c int) int {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // If there's a record, return it directly\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if wgt[i-1] &gt; c {\n        return knapsackDFSMem(wgt, val, mem, i-1, c)\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    no := knapsackDFSMem(wgt, val, mem, i-1, c)\n    yes := knapsackDFSMem(wgt, val, mem, i-1, c-wgt[i-1]) + val[i-1]\n    // Return the larger value of the two options\n    mem[i][c] = int(math.Max(float64(no), float64(yes)))\n    return mem[i][c]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 knapsack: Memoization search */\nfunc knapsackDFSMem(wgt: [Int], val: [Int], mem: inout [[Int]], i: Int, c: Int) -&gt; Int {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // If there's a record, return it directly\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if wgt[i - 1] &gt; c {\n        return knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    let no = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    let yes = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // Record and return the larger value of the two options\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 knapsack: Memoization search */\nfunction knapsackDFSMem(wgt, val, mem, i, c) {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // If there's a record, return it directly\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Record and return the larger value of the two options\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 knapsack: Memoization search */\nfunction knapsackDFSMem(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    c: number\n): number {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // If there's a record, return it directly\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Record and return the larger value of the two options\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 knapsack: Memoization search */\nint knapsackDFSMem(\n  List&lt;int&gt; wgt,\n  List&lt;int&gt; val,\n  List&lt;List&lt;int&gt;&gt; mem,\n  int i,\n  int c,\n) {\n  // If all items have been selected or knapsack has no remaining capacity, return value 0\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // If there's a record, return it directly\n  if (mem[i][c] != -1) {\n    return mem[i][c];\n  }\n  // If exceeds knapsack capacity, can only choose not to put it in\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFSMem(wgt, val, mem, i - 1, c);\n  }\n  // Calculate the maximum value of not putting in and putting in item i\n  int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n  int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // Record and return the larger value of the two options\n  mem[i][c] = max(no, yes);\n  return mem[i][c];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 knapsack: Memoization search */\nfn knapsack_dfs_mem(wgt: &amp;[i32], val: &amp;[i32], mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: usize, c: usize) -&gt; i32 {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // If there's a record, return it directly\n    if mem[i][c] != -1 {\n        return mem[i][c];\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    let no = knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    let yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // Record and return the larger value of the two options\n    mem[i][c] = std::cmp::max(no, yes);\n    mem[i][c]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 knapsack: Memoization search */\nint knapsackDFSMem(int wgt[], int val[], int memCols, int **mem, int i, int c) {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If there's a record, return it directly\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    int no = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Record and return the larger value of the two options\n    mem[i][c] = myMax(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 knapsack: Memoization search */\nfun knapsackDFSMem(\n    wgt: IntArray,\n    _val: IntArray,\n    mem: Array&lt;IntArray&gt;,\n    i: Int,\n    c: Int\n): Int {\n    // If all items have been selected or knapsack has no remaining capacity, return value 0\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // If there's a record, return it directly\n    if (mem[i][c] != -1) {\n        return mem[i][c]\n    }\n    // If exceeds knapsack capacity, can only choose not to put it in\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, _val, mem, i - 1, c)\n    }\n    // Calculate the maximum value of not putting in and putting in item i\n    val no = knapsackDFSMem(wgt, _val, mem, i - 1, c)\n    val yes = knapsackDFSMem(wgt, _val, mem, i - 1, c - wgt[i - 1]) + _val[i - 1]\n    // Record and return the larger value of the two options\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 knapsack: memoization search ###\ndef knapsack_dfs_mem(wgt, val, mem, i, c)\n  # If all items have been selected or knapsack has no remaining capacity, return value 0\n  return 0 if i == 0 || c == 0\n  # If there's a record, return it directly\n  return mem[i][c] if mem[i][c] != -1\n  # If exceeds knapsack capacity, can only choose not to put it in\n  return knapsack_dfs_mem(wgt, val, mem, i - 1, c) if wgt[i - 1] &gt; c\n  # Calculate the maximum value of not putting in and putting in item i\n  no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n  yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n  # Record and return the larger value of the two options\n  mem[i][c] = [no, yes].max\nend\n</code></pre> <p>Figure 14-19 shows the search branches pruned in memoization.</p> <p></p> <p> Figure 14-19   Memoization recursion tree for 0-1 knapsack problem </p>","path":["Chapter 14. Dynamic Programming","14.4   0-1 Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#3-method-3-dynamic-programming","level":3,"title":"3.   Method 3: Dynamic Programming","text":"<p>Dynamic programming is essentially the process of filling the \\(dp\\) table during state transitions. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 knapsack: Dynamic programming\"\"\"\n    n = len(wgt)\n    # Initialize dp table\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # State transition\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # The larger value between not selecting and selecting item i\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 knapsack: Dynamic programming */\nint knapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // Initialize dp table\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 knapsack: Dynamic programming */\nint knapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // Initialize dp table\n    int[][] dp = new int[n + 1][cap + 1];\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 knapsack: Dynamic programming */\nint KnapsackDP(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // Initialize dp table\n    int[,] dp = new int[n + 1, cap + 1];\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (weight[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i, c] = Math.Max(dp[i - 1, c - weight[i - 1]] + val[i - 1], dp[i - 1, c]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 knapsack: Dynamic programming */\nfunc knapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // Initialize dp table\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // State transition\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i-1][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 knapsack: Dynamic programming */\nfunc knapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // Initialize dp table\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // State transition\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 knapsack: Dynamic programming */\nfunction knapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // Initialize dp table\n    const dp = Array(n + 1)\n        .fill(0)\n        .map(() =&gt; Array(cap + 1).fill(0));\n    // State transition\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 knapsack: Dynamic programming */\nfunction knapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // Initialize dp table\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // State transition\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 knapsack: Dynamic programming */\nint knapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // Initialize dp table\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // State transition\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // If exceeds knapsack capacity, don't select item i\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // The larger value between not selecting and selecting item i\n        dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 knapsack: Dynamic programming */\nfn knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // Initialize dp table\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // State transition\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = std::cmp::max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1] as usize] + val[i - 1],\n                );\n            }\n        }\n    }\n    dp[n][cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 knapsack: Dynamic programming */\nint knapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // Initialize dp table\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = myMax(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // Free memory\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 knapsack: Dynamic programming */\nfun knapsackDP(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // Initialize dp table\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // State transition\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 knapsack: dynamic programming ###\ndef knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # Initialize dp table\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # State transition\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # If exceeds knapsack capacity, don't select item i\n        dp[i][c] = dp[i - 1][c]\n      else\n        # The larger value between not selecting and selecting item i\n        dp[i][c] = [dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n</code></pre> <p>As shown in Figure 14-20, both time complexity and space complexity are determined by the size of the array <code>dp</code>, which is \\(O(n \\times cap)\\).</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 14-20   Dynamic programming process for 0-1 knapsack problem </p>","path":["Chapter 14. Dynamic Programming","14.4   0-1 Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#4-space-optimization","level":3,"title":"4.   Space Optimization","text":"<p>Since each state is only related to the state in the row above it, we can use two arrays rolling forward to reduce the space complexity from \\(O(n \\times cap)\\) to \\(O(cap)\\).</p> <p>Further thinking, can we achieve space optimization using just one array? Observing, we can see that each state is transferred from the cell directly above or the cell in the upper-left. If there is only one array, when we start traversing row \\(i\\), that array still stores the state of row \\(i-1\\).</p> <ul> <li>If using forward traversal, then when traversing to \\(dp[i, j]\\), the values in the upper-left \\(dp[i-1, 1]\\) ~ \\(dp[i-1, j-1]\\) may have already been overwritten, thus preventing correct state transition.</li> <li>If using reverse traversal, there will be no overwriting issue, and state transition can proceed correctly.</li> </ul> <p>Figure 14-21 shows the transition process from row \\(i = 1\\) to row \\(i = 2\\) using a single array. Please consider the difference between forward and reverse traversal.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 14-21   Space-optimized dynamic programming process for 0-1 knapsack </p> <p>In the code implementation, we simply need to delete the first dimension \\(i\\) of the array <code>dp</code> and change the inner loop to reverse traversal:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 knapsack: Space-optimized dynamic programming\"\"\"\n    n = len(wgt)\n    # Initialize dp table\n    dp = [0] * (cap + 1)\n    # State transition\n    for i in range(1, n + 1):\n        # Traverse in reverse order\n        for c in range(cap, 0, -1):\n            if wgt[i - 1] &gt; c:\n                # If exceeds knapsack capacity, don't select item i\n                dp[c] = dp[c]\n            else:\n                # The larger value between not selecting and selecting item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 knapsack: Space-optimized dynamic programming */\nint knapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // Initialize dp table\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        // Traverse in reverse order\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // The larger value between not selecting and selecting item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 knapsack: Space-optimized dynamic programming */\nint knapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // Initialize dp table\n    int[] dp = new int[cap + 1];\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        // Traverse in reverse order\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // The larger value between not selecting and selecting item i\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 knapsack: Space-optimized dynamic programming */\nint KnapsackDPComp(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // Initialize dp table\n    int[] dp = new int[cap + 1];\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        // Traverse in reverse order\n        for (int c = cap; c &gt; 0; c--) {\n            if (weight[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[c] = dp[c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[c] = Math.Max(dp[c], dp[c - weight[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 knapsack: Space-optimized dynamic programming */\nfunc knapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // Initialize dp table\n    dp := make([]int, cap+1)\n    // State transition\n    for i := 1; i &lt;= n; i++ {\n        // Traverse in reverse order\n        for c := cap; c &gt;= 1; c-- {\n            if wgt[i-1] &lt;= c {\n                // The larger value between not selecting and selecting item i\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 knapsack: Space-optimized dynamic programming */\nfunc knapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // Initialize dp table\n    var dp = Array(repeating: 0, count: cap + 1)\n    // State transition\n    for i in 1 ... n {\n        // Traverse in reverse order\n        for c in (1 ... cap).reversed() {\n            if wgt[i - 1] &lt;= c {\n                // The larger value between not selecting and selecting item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 knapsack: Space-optimized dynamic programming */\nfunction knapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // Initialize dp table\n    const dp = Array(cap + 1).fill(0);\n    // State transition\n    for (let i = 1; i &lt;= n; i++) {\n        // Traverse in reverse order\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // The larger value between not selecting and selecting item i\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 knapsack: Space-optimized dynamic programming */\nfunction knapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // Initialize dp table\n    const dp = Array(cap + 1).fill(0);\n    // State transition\n    for (let i = 1; i &lt;= n; i++) {\n        // Traverse in reverse order\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // The larger value between not selecting and selecting item i\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 knapsack: Space-optimized dynamic programming */\nint knapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // Initialize dp table\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // State transition\n  for (int i = 1; i &lt;= n; i++) {\n    // Traverse in reverse order\n    for (int c = cap; c &gt;= 1; c--) {\n      if (wgt[i - 1] &lt;= c) {\n        // The larger value between not selecting and selecting item i\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 knapsack: Space-optimized dynamic programming */\nfn knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // Initialize dp table\n    let mut dp = vec![0; cap + 1];\n    // State transition\n    for i in 1..=n {\n        // Traverse in reverse order\n        for c in (1..=cap).rev() {\n            if wgt[i - 1] &lt;= c as i32 {\n                // The larger value between not selecting and selecting item i\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 knapsack: Space-optimized dynamic programming */\nint knapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // Initialize dp table\n    int *dp = calloc(cap + 1, sizeof(int));\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        // Traverse in reverse order\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // The larger value between not selecting and selecting item i\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // Free memory\n    free(dp);\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 knapsack: Space-optimized dynamic programming */\nfun knapsackDPComp(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // Initialize dp table\n    val dp = IntArray(cap + 1)\n    // State transition\n    for (i in 1..n) {\n        // Traverse in reverse order\n        for (c in cap downTo 1) {\n            if (wgt[i - 1] &lt;= c) {\n                // The larger value between not selecting and selecting item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 knapsack: space-optimized DP ###\ndef knapsack_dp_comp(wgt, val, cap)\n  n = wgt.length\n  # Initialize dp table\n  dp = Array.new(cap + 1, 0)\n  # State transition\n  for i in 1...(n + 1)\n    # Traverse in reverse order\n    for c in cap.downto(1)\n      if wgt[i - 1] &gt; c\n        # If exceeds knapsack capacity, don't select item i\n        dp[c] = dp[c]\n      else\n        # The larger value between not selecting and selecting item i\n        dp[c] = [dp[c], dp[c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[cap]\nend\n</code></pre>","path":["Chapter 14. Dynamic Programming","14.4   0-1 Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/summary/","level":1,"title":"14.7   Summary","text":"","path":["Chapter 14. Dynamic Programming","14.7   Summary"],"tags":[]},{"location":"chapter_dynamic_programming/summary/#1-key-points","level":3,"title":"1.   Key Points","text":"<ul> <li>Dynamic programming decomposes problems and avoids redundant computation by storing the solutions to subproblems, thereby significantly improving computational efficiency.</li> <li>Without considering time constraints, all dynamic programming problems can be solved using backtracking (brute force search), but the recursion tree contains a large number of overlapping subproblems, resulting in extremely low efficiency. By introducing a memo list, we can store the solutions to all computed subproblems, ensuring that overlapping subproblems are only computed once.</li> <li>Memoization is a top-down recursive solution, while the corresponding dynamic programming is a bottom-up iterative solution, similar to \"filling in a table\". Since the current state only depends on certain local states, we can eliminate one dimension of the \\(dp\\) table to reduce space complexity.</li> <li>Subproblem decomposition is a general algorithmic approach, with different properties in divide and conquer, dynamic programming, and backtracking.</li> <li>Dynamic programming problems have three major characteristics: overlapping subproblems, optimal substructure, and no aftereffects.</li> <li>If the optimal solution to the original problem can be constructed from the optimal solutions to the subproblems, then it has optimal substructure.</li> <li>No aftereffects means that for a given state, its future development is only related to that state and has nothing to do with all past states. Many combinatorial optimization problems do not satisfy this property and cannot be solved efficiently using dynamic programming.</li> </ul> <p>Knapsack problem</p> <ul> <li>The knapsack problem is one of the most typical dynamic programming problems, with variants such as the 0-1 knapsack, unbounded knapsack, and multiple knapsack.</li> <li>The state definition for the 0-1 knapsack is the maximum value achievable using the first \\(i\\) items with a knapsack capacity of \\(c\\). Based on the two decisions of not putting an item in the knapsack and putting it in, the optimal substructure can be identified and the state transition equation constructed. In space optimization, since each state depends on the state directly above and to the upper-left, the list needs to be traversed in reverse order to avoid overwriting the upper-left state.</li> <li>The unbounded knapsack problem has no limit on the selection quantity of each type of item, so the state transition for choosing to put in an item differs from the 0-1 knapsack problem. Since the state depends on the state directly above and directly to the left, space optimization should use forward traversal.</li> <li>The coin change problem is a variant of the unbounded knapsack problem. It changes from seeking the \"maximum\" value to seeking the \"minimum\" number of coins, so \\(\\max()\\) in the state transition equation should be changed to \\(\\min()\\). It changes from seeking \"not exceeding\" the knapsack capacity to seeking \"exactly\" making up the target amount, so \\(amt + 1\\) is used to represent the invalid solution of \"unable to make up the target amount\".</li> <li>Coin change problem II changes from seeking the \"minimum number of coins\" to seeking the \"number of coin combinations\", so the state transition equation correspondingly changes from \\(\\min()\\) to a summation operator.</li> </ul> <p>Edit distance problem</p> <ul> <li>Edit distance (Levenshtein distance) is used to measure the similarity between two strings, defined as the minimum number of edit steps from one string to another, with edit operations including insert, delete, and replace.</li> <li>The state definition for the edit distance problem is the minimum number of edit steps required to change the first \\(i\\) characters of \\(s\\) into the first \\(j\\) characters of \\(t\\). When \\(s[i-1] \\ne t[j-1]\\), there are three decisions: insert, delete, replace, each with corresponding remaining subproblems. From this, the optimal substructure can be identified and the state transition equation constructed. When \\(s[i-1] = t[j-1]\\), no edit is required for the current character.</li> <li>In edit distance, the state depends on the state directly above, directly to the left, and to the upper-left, so after space optimization, neither forward nor reverse traversal can correctly perform state transitions. For this reason, we use a variable to temporarily store the upper-left state, thus transforming to a situation equivalent to the unbounded knapsack problem, allowing for forward traversal after space optimization.</li> </ul>","path":["Chapter 14. Dynamic Programming","14.7   Summary"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/","level":1,"title":"14.5   Unbounded Knapsack Problem","text":"<p>In this section, we first solve another common knapsack problem: the unbounded knapsack, and then explore a special case of it: the coin change problem.</p>","path":["Chapter 14. Dynamic Programming","14.5   Unbounded Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1451-unbounded-knapsack-problem","level":2,"title":"14.5.1   Unbounded Knapsack Problem","text":"<p>Question</p> <p>Given \\(n\\) items, where the weight of the \\(i\\)-th item is \\(wgt[i-1]\\) and its value is \\(val[i-1]\\), and a knapsack with capacity \\(cap\\). Each item can be selected multiple times. What is the maximum value that can be placed in the knapsack within the capacity limit? An example is shown in Figure 14-22.</p> <p></p> <p> Figure 14-22   Example data for unbounded knapsack problem </p>","path":["Chapter 14. Dynamic Programming","14.5   Unbounded Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1-dynamic-programming-approach","level":3,"title":"1.   Dynamic Programming Approach","text":"<p>The unbounded knapsack problem is very similar to the 0-1 knapsack problem, differing only in that there is no limit on the number of times an item can be selected.</p> <ul> <li>In the 0-1 knapsack problem, there is only one of each type of item, so after placing item \\(i\\) in the knapsack, we can only choose from the first \\(i-1\\) items.</li> <li>In the unbounded knapsack problem, the quantity of each type of item is unlimited, so after placing item \\(i\\) in the knapsack, we can still choose from the first \\(i\\) items.</li> </ul> <p>Under the rules of the unbounded knapsack problem, the changes in state \\([i, c]\\) are divided into two cases.</p> <ul> <li>Not putting item \\(i\\): Same as the 0-1 knapsack problem, transfer to \\([i-1, c]\\).</li> <li>Putting item \\(i\\): Different from the 0-1 knapsack problem, transfer to \\([i, c-wgt[i-1]]\\).</li> </ul> <p>Thus, the state transition equation becomes:</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1]) \\]","path":["Chapter 14. Dynamic Programming","14.5   Unbounded Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2-code-implementation","level":3,"title":"2.   Code Implementation","text":"<p>Comparing the code for the two problems, there is one change in state transition from \\(i-1\\) to \\(i\\), with everything else identical:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"Unbounded knapsack: Dynamic programming\"\"\"\n    n = len(wgt)\n    # Initialize dp table\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # State transition\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # The larger value between not selecting and selecting item i\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* Unbounded knapsack: Dynamic programming */\nint unboundedKnapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // Initialize dp table\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* Unbounded knapsack: Dynamic programming */\nint unboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // Initialize dp table\n    int[][] dp = new int[n + 1][cap + 1];\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* Unbounded knapsack: Dynamic programming */\nint UnboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // Initialize dp table\n    int[,] dp = new int[n + 1, cap + 1];\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i, c] = Math.Max(dp[i - 1, c], dp[i, c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* Unbounded knapsack: Dynamic programming */\nfunc unboundedKnapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // Initialize dp table\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // State transition\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* Unbounded knapsack: Dynamic programming */\nfunc unboundedKnapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // Initialize dp table\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // State transition\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* Unbounded knapsack: Dynamic programming */\nfunction unboundedKnapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // Initialize dp table\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // State transition\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* Unbounded knapsack: Dynamic programming */\nfunction unboundedKnapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // Initialize dp table\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // State transition\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* Unbounded knapsack: Dynamic programming */\nint unboundedKnapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // Initialize dp table\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // State transition\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // If exceeds knapsack capacity, don't select item i\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // The larger value between not selecting and selecting item i\n        dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* Unbounded knapsack: Dynamic programming */\nfn unbounded_knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // Initialize dp table\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // State transition\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i][c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* Unbounded knapsack: Dynamic programming */\nint unboundedKnapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // Initialize dp table\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = myMax(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // Free memory\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* Unbounded knapsack: Dynamic programming */\nfun unboundedKnapsackDP(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // Initialize dp table\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // State transition\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>### Unbounded knapsack: dynamic programming ###\ndef unbounded_knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # Initialize dp table\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # State transition\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # If exceeds knapsack capacity, don't select item i\n        dp[i][c] = dp[i - 1][c]\n      else\n        # The larger value between not selecting and selecting item i\n        dp[i][c] = [dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n</code></pre>","path":["Chapter 14. Dynamic Programming","14.5   Unbounded Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3-space-optimization","level":3,"title":"3.   Space Optimization","text":"<p>Since the current state is transferred from states on the left and above, after space optimization, each row in the \\(dp\\) table should be traversed in forward order.</p> <p>This traversal order is exactly opposite to the 0-1 knapsack. Please refer to Figure 14-23 to understand the difference between the two.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 14-23   Space-optimized dynamic programming process for unbounded knapsack problem </p> <p>The code implementation is relatively simple, just delete the first dimension of the array <code>dp</code>:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"Unbounded knapsack: Space-optimized dynamic programming\"\"\"\n    n = len(wgt)\n    # Initialize dp table\n    dp = [0] * (cap + 1)\n    # State transition\n    for i in range(1, n + 1):\n        # Traverse in forward order\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # If exceeds knapsack capacity, don't select item i\n                dp[c] = dp[c]\n            else:\n                # The larger value between not selecting and selecting item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* Unbounded knapsack: Space-optimized dynamic programming */\nint unboundedKnapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // Initialize dp table\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[c] = dp[c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* Unbounded knapsack: Space-optimized dynamic programming */\nint unboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // Initialize dp table\n    int[] dp = new int[cap + 1];\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[c] = dp[c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* Unbounded knapsack: Space-optimized dynamic programming */\nint UnboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // Initialize dp table\n    int[] dp = new int[cap + 1];\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[c] = dp[c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[c] = Math.Max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* Unbounded knapsack: Space-optimized dynamic programming */\nfunc unboundedKnapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // Initialize dp table\n    dp := make([]int, cap+1)\n    // State transition\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // If exceeds knapsack capacity, don't select item i\n                dp[c] = dp[c]\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* Unbounded knapsack: Space-optimized dynamic programming */\nfunc unboundedKnapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // Initialize dp table\n    var dp = Array(repeating: 0, count: cap + 1)\n    // State transition\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // If exceeds knapsack capacity, don't select item i\n                dp[c] = dp[c]\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* Unbounded knapsack: Space-optimized dynamic programming */\nfunction unboundedKnapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // Initialize dp table\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // State transition\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[c] = dp[c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* Unbounded knapsack: Space-optimized dynamic programming */\nfunction unboundedKnapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // Initialize dp table\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // State transition\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[c] = dp[c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* Unbounded knapsack: Space-optimized dynamic programming */\nint unboundedKnapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // Initialize dp table\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // State transition\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // If exceeds knapsack capacity, don't select item i\n        dp[c] = dp[c];\n      } else {\n        // The larger value between not selecting and selecting item i\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* Unbounded knapsack: Space-optimized dynamic programming */\nfn unbounded_knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // Initialize dp table\n    let mut dp = vec![0; cap + 1];\n    // State transition\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // If exceeds knapsack capacity, don't select item i\n                dp[c] = dp[c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* Unbounded knapsack: Space-optimized dynamic programming */\nint unboundedKnapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // Initialize dp table\n    int *dp = calloc(cap + 1, sizeof(int));\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[c] = dp[c];\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // Free memory\n    free(dp);\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* Unbounded knapsack: Space-optimized dynamic programming */\nfun unboundedKnapsackDPComp(\n    wgt: IntArray,\n    _val: IntArray,\n    cap: Int\n): Int {\n    val n = wgt.size\n    // Initialize dp table\n    val dp = IntArray(cap + 1)\n    // State transition\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // If exceeds knapsack capacity, don't select item i\n                dp[c] = dp[c]\n            } else {\n                // The larger value between not selecting and selecting item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>### Unbounded knapsack: space-optimized DP ###\ndef unbounded_knapsack_dp_comp(wgt, val, cap)\n  n = wgt.length\n  # Initialize dp table\n  dp = Array.new(cap + 1, 0)\n  # State transition\n  for i in 1...(n + 1)\n    # Traverse in forward order\n    for c in 1...(cap + 1)\n      if wgt[i -1] &gt; c\n        # If exceeds knapsack capacity, don't select item i\n        dp[c] = dp[c]\n      else\n        # The larger value between not selecting and selecting item i\n        dp[c] = [dp[c], dp[c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[cap]\nend\n</code></pre>","path":["Chapter 14. Dynamic Programming","14.5   Unbounded Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1452-coin-change-problem","level":2,"title":"14.5.2   Coin Change Problem","text":"<p>The knapsack problem represents a large class of dynamic programming problems and has many variants, such as the coin change problem.</p> <p>Question</p> <p>Given \\(n\\) types of coins, where the denomination of the \\(i\\)-th type of coin is \\(coins[i - 1]\\), and the target amount is \\(amt\\). Each type of coin can be selected multiple times. What is the minimum number of coins needed to make up the target amount? If it is impossible to make up the target amount, return \\(-1\\). An example is shown in Figure 14-24.</p> <p></p> <p> Figure 14-24   Example data for coin change problem </p>","path":["Chapter 14. Dynamic Programming","14.5   Unbounded Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1-dynamic-programming-approach_1","level":3,"title":"1.   Dynamic Programming Approach","text":"<p>The coin change problem can be viewed as a special case of the unbounded knapsack problem, with the following connections and differences.</p> <ul> <li>The two problems can be converted to each other: \"item\" corresponds to \"coin\", \"item weight\" corresponds to \"coin denomination\", and \"knapsack capacity\" corresponds to \"target amount\".</li> <li>The optimization goals are opposite: the unbounded knapsack problem aims to maximize item value, while the coin change problem aims to minimize the number of coins.</li> <li>The unbounded knapsack problem seeks solutions \"not exceeding\" the knapsack capacity, while the coin change problem seeks solutions that \"exactly\" make up the target amount.</li> </ul> <p>Step 1: Think about the decisions in each round, define the state, and thus obtain the \\(dp\\) table</p> <p>State \\([i, a]\\) corresponds to the subproblem: the minimum number of coins among the first \\(i\\) types of coins that can make up amount \\(a\\), denoted as \\(dp[i, a]\\).</p> <p>The two-dimensional \\(dp\\) table has size \\((n+1) \\times (amt+1)\\).</p> <p>Step 2: Identify the optimal substructure, and then derive the state transition equation</p> <p>This problem differs from the unbounded knapsack problem in the following two aspects regarding the state transition equation.</p> <ul> <li>This problem seeks the minimum value, so the operator \\(\\max()\\) needs to be changed to \\(\\min()\\).</li> <li>The optimization target is the number of coins rather than item value, so when a coin is selected, simply add \\(1\\).</li> </ul> \\[ dp[i, a] = \\min(dp[i-1, a], dp[i, a - coins[i-1]] + 1) \\] <p>Step 3: Determine boundary conditions and state transition order</p> <p>When the target amount is \\(0\\), the minimum number of coins needed to make it up is \\(0\\), so all \\(dp[i, 0]\\) in the first column equal \\(0\\).</p> <p>When there are no coins, it is impossible to make up any amount \\(&gt; 0\\), which is an invalid solution. To enable the \\(\\min()\\) function in the state transition equation to identify and filter out invalid solutions, we consider using \\(+ \\infty\\) to represent them, i.e., set all \\(dp[0, a]\\) in the first row to \\(+ \\infty\\).</p>","path":["Chapter 14. Dynamic Programming","14.5   Unbounded Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2-code-implementation_1","level":3,"title":"2.   Code Implementation","text":"<p>Most programming languages do not provide a \\(+ \\infty\\) variable, and can only use the maximum value of integer type <code>int</code> as a substitute. However, this can lead to integer overflow: the \\(+ 1\\) operation in the state transition equation may cause overflow.</p> <p>For this reason, we use the number \\(amt + 1\\) to represent invalid solutions, because the maximum number of coins needed to make up \\(amt\\) is at most \\(amt\\). Before returning, check whether \\(dp[n, amt]\\) equals \\(amt + 1\\); if so, return \\(-1\\), indicating that the target amount cannot be made up. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change.py<pre><code>def coin_change_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"Coin change: Dynamic programming\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # Initialize dp table\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # State transition: first row and first column\n    for a in range(1, amt + 1):\n        dp[0][a] = MAX\n    # State transition: rest of the rows and columns\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # The smaller value between not selecting and selecting coin i\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n    return dp[n][amt] if dp[n][amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* Coin change: Dynamic programming */\nint coinChangeDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // Initialize dp table\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // State transition: first row and first column\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // State transition: rest of the rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* Coin change: Dynamic programming */\nint coinChangeDP(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // Initialize dp table\n    int[][] dp = new int[n + 1][amt + 1];\n    // State transition: first row and first column\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // State transition: rest of the rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* Coin change: Dynamic programming */\nint CoinChangeDP(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // Initialize dp table\n    int[,] dp = new int[n + 1, amt + 1];\n    // State transition: first row and first column\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0, a] = MAX;\n    }\n    // State transition: rest of the rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[i, a] = Math.Min(dp[i - 1, a], dp[i, a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n, amt] != MAX ? dp[n, amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* Coin change: Dynamic programming */\nfunc coinChangeDP(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // Initialize dp table\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // State transition: first row and first column\n    for a := 1; a &lt;= amt; a++ {\n        dp[0][a] = max\n    }\n    // State transition: rest of the rows and columns\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[i][a] = int(math.Min(float64(dp[i-1][a]), float64(dp[i][a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* Coin change: Dynamic programming */\nfunc coinChangeDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // Initialize dp table\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // State transition: first row and first column\n    for a in 1 ... amt {\n        dp[0][a] = MAX\n    }\n    // State transition: rest of the rows and columns\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* Coin change: Dynamic programming */\nfunction coinChangeDP(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // Initialize dp table\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // State transition: first row and first column\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // State transition: rest of the rows and columns\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* Coin change: Dynamic programming */\nfunction coinChangeDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // Initialize dp table\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // State transition: first row and first column\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // State transition: rest of the rows and columns\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* Coin change: Dynamic programming */\nint coinChangeDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // Initialize dp table\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // State transition: first row and first column\n  for (int a = 1; a &lt;= amt; a++) {\n    dp[0][a] = MAX;\n  }\n  // State transition: rest of the rows and columns\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // If exceeds target amount, don't select coin i\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // The smaller value between not selecting and selecting coin i\n        dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* Coin change: Dynamic programming */\nfn coin_change_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // Initialize dp table\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // State transition: first row and first column\n    for a in 1..=amt {\n        dp[0][a] = max;\n    }\n    // State transition: rest of the rows and columns\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[i][a] = std::cmp::min(dp[i - 1][a], dp[i][a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* Coin change: Dynamic programming */\nint coinChangeDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // Initialize dp table\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // State transition: first row and first column\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // State transition: rest of the rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[i][a] = myMin(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[n][amt] != MAX ? dp[n][amt] : -1;\n    // Free memory\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* Coin change: Dynamic programming */\nfun coinChangeDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // Initialize dp table\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // State transition: first row and first column\n    for (a in 1..amt) {\n        dp[0][a] = MAX\n    }\n    // State transition: rest of the rows and columns\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return if (dp[n][amt] != MAX) dp[n][amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>### Coin change: dynamic programming ###\ndef coin_change_dp(coins, amt)\n  n = coins.length\n  _MAX = amt + 1\n  # Initialize dp table\n  dp = Array.new(n + 1) { Array.new(amt + 1, 0) }\n  # State transition: first row and first column\n  (1...(amt + 1)).each { |a| dp[0][a] = _MAX }\n  # State transition: rest of the rows and columns\n  for i in 1...(n + 1)\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # If exceeds target amount, don't select coin i\n        dp[i][a] = dp[i - 1][a]\n      else\n        # The smaller value between not selecting and selecting coin i\n        dp[i][a] = [dp[i - 1][a], dp[i][a - coins[i - 1]] + 1].min\n      end\n    end\n  end\n  dp[n][amt] != _MAX ? dp[n][amt] : -1\nend\n</code></pre> <p>Figure 14-25 shows the dynamic programming process for coin change, which is very similar to the unbounded knapsack problem.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 14-25   Dynamic programming process for coin change problem </p>","path":["Chapter 14. Dynamic Programming","14.5   Unbounded Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3-space-optimization_1","level":3,"title":"3.   Space Optimization","text":"<p>The space optimization for the coin change problem is handled in the same way as the unbounded knapsack problem:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change.py<pre><code>def coin_change_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"Coin change: Space-optimized dynamic programming\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # Initialize dp table\n    dp = [MAX] * (amt + 1)\n    dp[0] = 0\n    # State transition\n    for i in range(1, n + 1):\n        # Traverse in forward order\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # If exceeds target amount, don't select coin i\n                dp[a] = dp[a]\n            else:\n                # The smaller value between not selecting and selecting coin i\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n    return dp[amt] if dp[amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* Coin change: Space-optimized dynamic programming */\nint coinChangeDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // Initialize dp table\n    vector&lt;int&gt; dp(amt + 1, MAX);\n    dp[0] = 0;\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a];\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* Coin change: Space-optimized dynamic programming */\nint coinChangeDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // Initialize dp table\n    int[] dp = new int[amt + 1];\n    Arrays.fill(dp, MAX);\n    dp[0] = 0;\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a];\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* Coin change: Space-optimized dynamic programming */\nint CoinChangeDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // Initialize dp table\n    int[] dp = new int[amt + 1];\n    Array.Fill(dp, MAX);\n    dp[0] = 0;\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a];\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[a] = Math.Min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* Coin change: Dynamic programming */\nfunc coinChangeDPComp(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // Initialize dp table\n    dp := make([]int, amt+1)\n    for i := 1; i &lt;= amt; i++ {\n        dp[i] = max\n    }\n    // State transition\n    for i := 1; i &lt;= n; i++ {\n        // Traverse in forward order\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a]\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[a] = int(math.Min(float64(dp[a]), float64(dp[a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* Coin change: Space-optimized dynamic programming */\nfunc coinChangeDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // Initialize dp table\n    var dp = Array(repeating: MAX, count: amt + 1)\n    dp[0] = 0\n    // State transition\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a]\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* Coin change: Space-optimized dynamic programming */\nfunction coinChangeDPComp(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // Initialize dp table\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // State transition\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a];\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* Coin change: Space-optimized dynamic programming */\nfunction coinChangeDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // Initialize dp table\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // State transition\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a];\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* Coin change: Space-optimized dynamic programming */\nint coinChangeDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // Initialize dp table\n  List&lt;int&gt; dp = List.filled(amt + 1, MAX);\n  dp[0] = 0;\n  // State transition\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // If exceeds target amount, don't select coin i\n        dp[a] = dp[a];\n      } else {\n        // The smaller value between not selecting and selecting coin i\n        dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* Coin change: Space-optimized dynamic programming */\nfn coin_change_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // Initialize dp table\n    let mut dp = vec![0; amt + 1];\n    dp.fill(max);\n    dp[0] = 0;\n    // State transition\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a];\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[a] = std::cmp::min(dp[a], dp[a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* Coin change: Space-optimized dynamic programming */\nint coinChangeDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // Initialize dp table\n    int *dp = malloc((amt + 1) * sizeof(int));\n    for (int j = 1; j &lt;= amt; j++) {\n        dp[j] = MAX;\n    } \n    dp[0] = 0;\n\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a];\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[a] = myMin(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[amt] != MAX ? dp[amt] : -1;\n    // Free memory\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* Coin change: Space-optimized dynamic programming */\nfun coinChangeDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // Initialize dp table\n    val dp = IntArray(amt + 1)\n    dp.fill(MAX)\n    dp[0] = 0\n    // State transition\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a]\n            } else {\n                // The smaller value between not selecting and selecting coin i\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return if (dp[amt] != MAX) dp[amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>### Coin change: space-optimized DP ###\ndef coin_change_dp_comp(coins, amt)\n  n = coins.length\n  _MAX = amt + 1\n  # Initialize dp table\n  dp = Array.new(amt + 1, _MAX)\n  dp[0] = 0\n  # State transition\n  for i in 1...(n + 1)\n    # Traverse in forward order\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # If exceeds target amount, don't select coin i\n        dp[a] = dp[a]\n      else\n        # The smaller value between not selecting and selecting coin i\n        dp[a] = [dp[a], dp[a - coins[i - 1]] + 1].min\n      end\n    end\n  end\n  dp[amt] != _MAX ? dp[amt] : -1\nend\n</code></pre>","path":["Chapter 14. Dynamic Programming","14.5   Unbounded Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1453-coin-change-problem-ii","level":2,"title":"14.5.3   Coin Change Problem II","text":"<p>Question</p> <p>Given \\(n\\) types of coins, where the denomination of the \\(i\\)-th type of coin is \\(coins[i - 1]\\), and the target amount is \\(amt\\). Each type of coin can be selected multiple times. What is the number of coin combinations that can make up the target amount? An example is shown in Figure 14-26.</p> <p></p> <p> Figure 14-26   Example data for coin change problem II </p>","path":["Chapter 14. Dynamic Programming","14.5   Unbounded Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1-dynamic-programming-approach_2","level":3,"title":"1.   Dynamic Programming Approach","text":"<p>Compared to the previous problem, this problem's goal is to find the number of combinations, so the subproblem becomes: the number of combinations among the first \\(i\\) types of coins that can make up amount \\(a\\). The \\(dp\\) table remains a two-dimensional matrix of size \\((n+1) \\times (amt + 1)\\).</p> <p>The number of combinations for the current state equals the sum of the combinations from not selecting the current coin and selecting the current coin. The state transition equation is:</p> \\[ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]] \\] <p>When the target amount is \\(0\\), no coins need to be selected to make up the target amount, so all \\(dp[i, 0]\\) in the first column should be initialized to \\(1\\). When there are no coins, it is impossible to make up any amount \\(&gt;0\\), so all \\(dp[0, a]\\) in the first row equal \\(0\\).</p>","path":["Chapter 14. Dynamic Programming","14.5   Unbounded Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2-code-implementation_2","level":3,"title":"2.   Code Implementation","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change_ii.py<pre><code>def coin_change_ii_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"Coin change II: Dynamic programming\"\"\"\n    n = len(coins)\n    # Initialize dp table\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # Initialize first column\n    for i in range(n + 1):\n        dp[i][0] = 1\n    # State transition\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # Sum of the two options: not selecting and selecting coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n    return dp[n][amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* Coin change II: Dynamic programming */\nint coinChangeIIDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // Initialize dp table\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // Initialize first column\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* Coin change II: Dynamic programming */\nint coinChangeIIDP(int[] coins, int amt) {\n    int n = coins.length;\n    // Initialize dp table\n    int[][] dp = new int[n + 1][amt + 1];\n    // Initialize first column\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* Coin change II: Dynamic programming */\nint CoinChangeIIDP(int[] coins, int amt) {\n    int n = coins.Length;\n    // Initialize dp table\n    int[,] dp = new int[n + 1, amt + 1];\n    // Initialize first column\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i, 0] = 1;\n    }\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[i, a] = dp[i - 1, a] + dp[i, a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n, amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* Coin change II: Dynamic programming */\nfunc coinChangeIIDP(coins []int, amt int) int {\n    n := len(coins)\n    // Initialize dp table\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // Initialize first column\n    for i := 0; i &lt;= n; i++ {\n        dp[i][0] = 1\n    }\n    // State transition: rest of the rows and columns\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[i][a] = dp[i-1][a] + dp[i][a-coins[i-1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* Coin change II: Dynamic programming */\nfunc coinChangeIIDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // Initialize dp table\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // Initialize first column\n    for i in 0 ... n {\n        dp[i][0] = 1\n    }\n    // State transition\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* Coin change II: Dynamic programming */\nfunction coinChangeIIDP(coins, amt) {\n    const n = coins.length;\n    // Initialize dp table\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // Initialize first column\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // State transition\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* Coin change II: Dynamic programming */\nfunction coinChangeIIDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // Initialize dp table\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // Initialize first column\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // State transition\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* Coin change II: Dynamic programming */\nint coinChangeIIDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // Initialize dp table\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // Initialize first column\n  for (int i = 0; i &lt;= n; i++) {\n    dp[i][0] = 1;\n  }\n  // State transition\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // If exceeds target amount, don't select coin i\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // Sum of the two options: not selecting and selecting coin i\n        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[n][amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* Coin change II: Dynamic programming */\nfn coin_change_ii_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // Initialize dp table\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // Initialize first column\n    for i in 0..=n {\n        dp[i][0] = 1;\n    }\n    // State transition\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[n][amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* Coin change II: Dynamic programming */\nint coinChangeIIDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // Initialize dp table\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // Initialize first column\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[n][amt];\n    // Free memory\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* Coin change II: Dynamic programming */\nfun coinChangeIIDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // Initialize dp table\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // Initialize first column\n    for (i in 0..n) {\n        dp[i][0] = 1\n    }\n    // State transition\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>### Coin change II: dynamic programming ###\ndef coin_change_ii_dp(coins, amt)\n  n = coins.length\n  # Initialize dp table\n  dp = Array.new(n + 1) { Array.new(amt + 1, 0) }\n  # Initialize first column\n  (0...(n + 1)).each { |i| dp[i][0] = 1 }\n  # State transition\n  for i in 1...(n + 1)\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # If exceeds target amount, don't select coin i\n        dp[i][a] = dp[i - 1][a]\n      else\n        # Sum of the two options: not selecting and selecting coin i\n        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n      end\n    end\n  end\n  dp[n][amt]\nend\n</code></pre>","path":["Chapter 14. Dynamic Programming","14.5   Unbounded Knapsack Problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3-space-optimization_2","level":3,"title":"3.   Space Optimization","text":"<p>The space optimization is handled in the same way, just delete the coin dimension:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change_ii.py<pre><code>def coin_change_ii_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"Coin change II: Space-optimized dynamic programming\"\"\"\n    n = len(coins)\n    # Initialize dp table\n    dp = [0] * (amt + 1)\n    dp[0] = 1\n    # State transition\n    for i in range(1, n + 1):\n        # Traverse in forward order\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # If exceeds target amount, don't select coin i\n                dp[a] = dp[a]\n            else:\n                # Sum of the two options: not selecting and selecting coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n    return dp[amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* Coin change II: Space-optimized dynamic programming */\nint coinChangeIIDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // Initialize dp table\n    vector&lt;int&gt; dp(amt + 1, 0);\n    dp[0] = 1;\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a];\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* Coin change II: Space-optimized dynamic programming */\nint coinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    // Initialize dp table\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a];\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* Coin change II: Space-optimized dynamic programming */\nint CoinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    // Initialize dp table\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a];\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* Coin change II: Space-optimized dynamic programming */\nfunc coinChangeIIDPComp(coins []int, amt int) int {\n    n := len(coins)\n    // Initialize dp table\n    dp := make([]int, amt+1)\n    dp[0] = 1\n    // State transition\n    for i := 1; i &lt;= n; i++ {\n        // Traverse in forward order\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a]\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[a] = dp[a] + dp[a-coins[i-1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* Coin change II: Space-optimized dynamic programming */\nfunc coinChangeIIDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // Initialize dp table\n    var dp = Array(repeating: 0, count: amt + 1)\n    dp[0] = 1\n    // State transition\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a]\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* Coin change II: Space-optimized dynamic programming */\nfunction coinChangeIIDPComp(coins, amt) {\n    const n = coins.length;\n    // Initialize dp table\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // State transition\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a];\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* Coin change II: Space-optimized dynamic programming */\nfunction coinChangeIIDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // Initialize dp table\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // State transition\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a];\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* Coin change II: Space-optimized dynamic programming */\nint coinChangeIIDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // Initialize dp table\n  List&lt;int&gt; dp = List.filled(amt + 1, 0);\n  dp[0] = 1;\n  // State transition\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // If exceeds target amount, don't select coin i\n        dp[a] = dp[a];\n      } else {\n        // Sum of the two options: not selecting and selecting coin i\n        dp[a] = dp[a] + dp[a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* Coin change II: Space-optimized dynamic programming */\nfn coin_change_ii_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // Initialize dp table\n    let mut dp = vec![0; amt + 1];\n    dp[0] = 1;\n    // State transition\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a];\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* Coin change II: Space-optimized dynamic programming */\nint coinChangeIIDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // Initialize dp table\n    int *dp = calloc(amt + 1, sizeof(int));\n    dp[0] = 1;\n    // State transition\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a];\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[amt];\n    // Free memory\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* Coin change II: Space-optimized dynamic programming */\nfun coinChangeIIDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // Initialize dp table\n    val dp = IntArray(amt + 1)\n    dp[0] = 1\n    // State transition\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // If exceeds target amount, don't select coin i\n                dp[a] = dp[a]\n            } else {\n                // Sum of the two options: not selecting and selecting coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>### Coin change II: space-optimized DP ###\ndef coin_change_ii_dp_comp(coins, amt)\n  n = coins.length\n  # Initialize dp table\n  dp = Array.new(amt + 1, 0)\n  dp[0] = 1\n  # State transition\n  for i in 1...(n + 1)\n    # Traverse in forward order\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # If exceeds target amount, don't select coin i\n        dp[a] = dp[a]\n      else\n        # Sum of the two options: not selecting and selecting coin i\n        dp[a] = dp[a] + dp[a - coins[i - 1]]\n      end\n    end\n  end\n  dp[amt]\nend\n</code></pre>","path":["Chapter 14. Dynamic Programming","14.5   Unbounded Knapsack Problem"],"tags":[]},{"location":"chapter_graph/","level":1,"title":"Chapter 9.   Graph","text":"<p>Abstract</p> <p>In the journey of life, we are like nodes, connected by countless invisible edges.</p> <p>Each encounter and parting leaves a unique mark on this vast network graph.</p>","path":["Chapter 9. Graph","Chapter 9.   Graph"],"tags":[]},{"location":"chapter_graph/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>9.1   Graph</li> <li>9.2   Basic Operations on Graphs</li> <li>9.3   Graph Traversal</li> <li>9.4   Summary</li> <li>9.5   Exercises</li> </ul>","path":["Chapter 9. Graph","Chapter 9.   Graph"],"tags":[]},{"location":"chapter_graph/exercises/","level":1,"title":"9.5   Exercises","text":"","path":["Chapter 9. Graph","9.5   Exercises"],"tags":[]},{"location":"chapter_graph/exercises/#951-concept-review","level":2,"title":"9.5.1   Concept Review","text":"","path":["Chapter 9. Graph","9.5   Exercises"],"tags":[]},{"location":"chapter_graph/exercises/#1-represent-the-same-graph-in-two-ways","level":3,"title":"1.   Represent the Same Graph in Two Ways","text":"<p>An undirected graph has four vertices, <code>A, B, C, D</code>, and the edges <code>A-B, A-C, B-C, C-D</code>.</p> <ol> <li>Write its adjacency list.</li> <li>Fill in its adjacency matrix using only 0s and 1s.</li> <li>To determine whether <code>A</code> and <code>D</code> are directly connected, which graph representation requires checking only one stored entry?</li> <li>If a graph has many vertices but few edges, which representation usually uses less space?</li> </ol> Answer <ol> <li> <p>The adjacency list is:</p> <pre><code>A: B, C\nB: A, C\nC: A, B, D\nD: C\n</code></pre> </li> <li> <p>The adjacency matrix is:</p> A B C D A 0 1 1 0 B 1 0 1 0 C 1 1 0 1 D 0 0 1 0 </li> <li> <p>In an adjacency matrix, you can directly check row <code>A</code>, column <code>D</code>, making it well suited to determining whether any two vertices are directly connected.</p> </li> <li> <p>When a graph has many vertices but few edges, an adjacency list records only the edges that actually exist. It usually uses less space than an adjacency matrix, which reserves a position for every pair of vertices.</p> </li> </ol>","path":["Chapter 9. Graph","9.5   Exercises"],"tags":[]},{"location":"chapter_graph/exercises/#2-breadth-first-and-depth-first-traversal-orders","level":3,"title":"2.   Breadth-First and Depth-First Traversal Orders","text":"<p>An undirected graph has vertices <code>A, B, C, D, E</code> and edges <code>A-B, A-C, B-D, C-D, D-E</code>.</p> <p>Start at A. Whenever there are several unvisited adjacent vertices, choose them in alphabetical order:</p> <ol> <li>Write the visit order of breadth-first traversal (BFS).</li> <li>Write the visit order of recursive depth-first traversal (DFS).</li> <li>Why must both traversals record which vertices have already been visited?</li> </ol> Answer <ol> <li> <p>The BFS visit order is <code>A, B, C, D, E</code>. It first visits B and C, which are one edge away from A,     and then visits the more distant D and E.</p> </li> <li> <p>The DFS visit order is <code>A, B, D, C, E</code>. It repeatedly enters an unvisited adjacent vertex,     first following <code>A → B → D → C</code>. When C has no new adjacent vertex, it returns to D and then visits E.</p> </li> <li> <p>The graph contains a cycle, such as <code>A-B-D-C-A</code>. Without recording visited vertices,     a traversal could repeatedly visit the same vertices around the cycle and fail to terminate normally.</p> </li> </ol>","path":["Chapter 9. Graph","9.5   Exercises"],"tags":[]},{"location":"chapter_graph/exercises/#3-can-one-bfs-visit-the-entire-graph","level":3,"title":"3.   Can One BFS Visit the Entire Graph?","text":"<p>An undirected graph has vertices <code>A, B, C, D, E, F</code> and only the edges <code>A-B, B-C, D-E</code>.</p> <ol> <li>Which vertices can one BFS starting at A visit?</li> <li>Based on Question 1, has this BFS visited every vertex in the graph? Why or why not?</li> <li>Suppose you scan all vertices in alphabetical order and start a new BFS whenever you reach an unvisited vertex.     What is the starting vertex of each BFS? Into how many mutually disconnected parts (connected components) is the graph divided?</li> </ol> Answer <ol> <li> <p>Starting from A, the traversal can visit only <code>A, B, C</code>.</p> </li> <li> <p>It has not visited every vertex. <code>D, E</code> form another connected part, while F is an isolated vertex.     None of them has a path to A, so they cannot be reached from A.</p> </li> <li> <p>The three BFS traversals start at <code>A, D, F</code>, and visit     <code>{A, B, C}</code>, <code>{D, E}</code>, and <code>{F}</code>, respectively. Therefore, the graph has 3 connected components.</p> </li> </ol>","path":["Chapter 9. Graph","9.5   Exercises"],"tags":[]},{"location":"chapter_graph/exercises/#952-programming-exercises","level":2,"title":"9.5.2   Programming Exercises","text":"","path":["Chapter 9. Graph","9.5   Exercises"],"tags":[]},{"location":"chapter_graph/exercises/#1-determine-whether-a-path-exists-in-an-undirected-graph","level":3,"title":"1.   Determine Whether a Path Exists in an Undirected Graph","text":"<p>You are given an undirected graph with \\(n\\) vertices numbered from \\(0\\) to \\(n-1\\). Each entry <code>[u, v]</code> in the array <code>edges</code> represents an undirected edge between vertices <code>u</code> and <code>v</code>.</p> <p>You are also given a starting vertex <code>source</code> and a destination vertex <code>destination</code>. First build an adjacency list from <code>edges</code>, then use BFS or DFS to determine whether a path exists from <code>source</code> to <code>destination</code>. Return <code>true</code> if one exists and <code>false</code> otherwise. The graph may contain cycles and may be disconnected.</p> Hints <ol> <li>Add every undirected edge in both directions</li> <li>The graph may contain cycles, so you must record which vertices have already been visited</li> <li>Starting from source, return true if you encounter destination; if the traversal ends without reaching it, return false</li> </ol> <p>LeetCode</p>","path":["Chapter 9. Graph","9.5   Exercises"],"tags":[]},{"location":"chapter_graph/graph/","level":1,"title":"9.1   Graph","text":"<p>A graph is a nonlinear data structure consisting of vertices and edges. We can abstractly represent a graph \\(G\\) as a set of vertices \\(V\\) and a set of edges \\(E\\). The following example shows a graph containing 5 vertices and 7 edges.</p> \\[ \\begin{aligned} V &amp; = \\{ 1, 2, 3, 4, 5 \\} \\newline E &amp; = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G &amp; = \\{ V, E \\} \\newline \\end{aligned} \\] <p>If we view vertices as nodes and edges as references (pointers) connecting them, we can regard a graph as an extension of the linked list data structure. As shown in Figure 9-1, compared to linear relationships (linked lists) and divide-and-conquer relationships (trees), network relationships (graphs) have a higher degree of freedom and are therefore more complex.</p> <p></p> <p> Figure 9-1   Relationships among linked lists, trees, and graphs </p>","path":["Chapter 9. Graph","9.1   Graph"],"tags":[]},{"location":"chapter_graph/graph/#911-common-types-and-terminology-of-graphs","level":2,"title":"9.1.1   Common Types and Terminology of Graphs","text":"<p>Graphs can be divided into undirected graphs and directed graphs based on whether edges have direction, as shown in Figure 9-2.</p> <ul> <li>In undirected graphs, edges represent a \"bidirectional\" connection between two vertices, such as friendships on WeChat or QQ.</li> <li>In directed graphs, edges have directionality, meaning edges \\(A \\rightarrow B\\) and \\(A \\leftarrow B\\) are independent of each other, such as following and follower relationships on Weibo or TikTok.</li> </ul> <p></p> <p> Figure 9-2   Directed and undirected graphs </p> <p>Graphs can be divided into connected graphs and disconnected graphs based on whether all vertices are connected, as shown in Figure 9-3.</p> <ul> <li>For connected graphs, starting from any vertex, all other vertices can be reached.</li> <li>For disconnected graphs, starting from a certain vertex, at least one vertex cannot be reached.</li> </ul> <p></p> <p> Figure 9-3   Connected and disconnected graphs </p> <p>We can also add a \"weight\" variable to edges, resulting in weighted graphs as shown in Figure 9-4. For example, in mobile games like \"Honor of Kings\", the system calculates the \"intimacy\" between players based on how long they have played together, and such intimacy networks can be represented using weighted graphs.</p> <p></p> <p> Figure 9-4   Weighted and unweighted graphs </p> <p>Graph data structures include the following commonly used terms.</p> <ul> <li>Adjacency: When two vertices are connected by an edge, these two vertices are said to be \"adjacent\". In Figure 9-4, the adjacent vertices of vertex 1 are vertices 2, 3, and 5.</li> <li>Path: The sequence of edges from vertex A to vertex B is called a \"path\" from A to B. In Figure 9-4, the edge sequence 1-5-2-4 is a path from vertex 1 to vertex 4.</li> <li>Degree: The number of edges a vertex has. For directed graphs, in-degree indicates how many edges point to the vertex, and out-degree indicates how many edges leave the vertex.</li> </ul>","path":["Chapter 9. Graph","9.1   Graph"],"tags":[]},{"location":"chapter_graph/graph/#912-representation-of-graphs","level":2,"title":"9.1.2   Representation of Graphs","text":"<p>Common representations of graphs include \"adjacency matrices\" and \"adjacency lists\". The following uses undirected graphs as examples.</p>","path":["Chapter 9. Graph","9.1   Graph"],"tags":[]},{"location":"chapter_graph/graph/#1-adjacency-matrix","level":3,"title":"1.   Adjacency Matrix","text":"<p>Given a graph with \\(n\\) vertices, an adjacency matrix uses an \\(n \\times n\\) matrix to represent the graph, where each row (column) represents a vertex, and matrix elements represent edges, using \\(1\\) or \\(0\\) to indicate whether an edge exists between two vertices.</p> <p>As shown in Figure 9-5, let the adjacency matrix be \\(M\\) and the vertex list be \\(V\\). Then matrix element \\(M[i, j] = 1\\) indicates that an edge exists between vertex \\(V[i]\\) and vertex \\(V[j]\\), whereas \\(M[i, j] = 0\\) indicates no edge between the two vertices.</p> <p></p> <p> Figure 9-5   Adjacency matrix representation of a graph </p> <p>Adjacency matrices have the following properties.</p> <ul> <li>In simple graphs, vertices cannot connect to themselves, so the elements on the main diagonal of the adjacency matrix are meaningless.</li> <li>For undirected graphs, edges in both directions are equivalent, so the adjacency matrix is symmetric about the main diagonal.</li> <li>Replacing the \\(1\\) and \\(0\\) entries in the adjacency matrix with weights allows it to represent weighted graphs.</li> </ul> <p>When using adjacency matrices to represent graphs, we can directly access matrix elements to obtain edges, resulting in highly efficient addition, deletion, lookup, and modification operations, all with a time complexity of \\(O(1)\\). However, the space complexity of the matrix is \\(O(n^2)\\), which consumes significant memory.</p>","path":["Chapter 9. Graph","9.1   Graph"],"tags":[]},{"location":"chapter_graph/graph/#2-adjacency-list","level":3,"title":"2.   Adjacency List","text":"<p>An adjacency list uses \\(n\\) linked lists to represent a graph, with linked list nodes representing vertices. The \\(i\\)-th linked list corresponds to vertex \\(i\\) and stores all adjacent vertices of that vertex (vertices connected to that vertex). Figure 9-6 shows an example of a graph stored using an adjacency list.</p> <p></p> <p> Figure 9-6   Adjacency list representation of a graph </p> <p>Adjacency lists only store edges that actually exist, and the total number of edges is typically much less than \\(n^2\\), making them more space-efficient. However, finding edges in an adjacency list requires traversing the linked list, so it is less time-efficient than an adjacency matrix.</p> <p>As shown in Figure 9-6, the structure of adjacency lists is very similar to separate chaining in hash tables, so we can use similar methods to improve efficiency. For example, when a linked list becomes long, it can be converted into an AVL tree or red-black tree, improving the time complexity from \\(O(n)\\) to \\(O(\\log n)\\); it can also be converted into a hash table, reducing the time complexity to \\(O(1)\\).</p>","path":["Chapter 9. Graph","9.1   Graph"],"tags":[]},{"location":"chapter_graph/graph/#913-common-applications-of-graphs","level":2,"title":"9.1.3   Common Applications of Graphs","text":"<p>As shown in Table 9-1, many real-world systems can be modeled using graphs, and corresponding problems can be reduced to graph computation problems.</p> <p> Table 9-1   Common graphs in real life </p> Vertices Edges Graph Computation Problem Social network Users Friend relationships Potential friend recommendation Subway lines Stations Connectivity between stations Shortest route recommendation Solar system Celestial bodies Gravitational forces between celestial bodies Planetary orbit calculation","path":["Chapter 9. Graph","9.1   Graph"],"tags":[]},{"location":"chapter_graph/graph_operations/","level":1,"title":"9.2   Basic Operations on Graphs","text":"<p>Basic operations on graphs can be divided into operations on \"edges\" and operations on \"vertices\". Their implementations differ depending on whether the graph is represented as an \"adjacency matrix\" or an \"adjacency list\".</p>","path":["Chapter 9. Graph","9.2   Basic Operations on Graphs"],"tags":[]},{"location":"chapter_graph/graph_operations/#921-implementation-based-on-adjacency-matrix","level":2,"title":"9.2.1   Implementation Based on Adjacency Matrix","text":"<p>Given an undirected graph with \\(n\\) vertices, the various operations are implemented as shown in Figure 9-7.</p> <ul> <li>Adding or removing an edge: Directly modify the specified edge in the adjacency matrix, using \\(O(1)\\) time. Since it is an undirected graph, both directions of the edge need to be updated simultaneously.</li> <li>Adding a vertex: Add a row and a column at the end of the adjacency matrix and fill them all with \\(0\\)s, using \\(O(n)\\) time.</li> <li>Removing a vertex: Delete a row and a column in the adjacency matrix. The worst case occurs when removing the first row and column, requiring \\((n-1)^2\\) elements to be \"moved up and to the left\", thus using \\(O(n^2)\\) time.</li> <li>Initialization: Given \\(n\\) vertices, initialize a vertex list <code>vertices</code> of length \\(n\\), using \\(O(n)\\) time; initialize an adjacency matrix <code>adjMat</code> of size \\(n \\times n\\), using \\(O(n^2)\\) time.</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-7   Initialization, adding and removing edges, adding and removing vertices in adjacency matrix </p> <p>The following is the implementation code for graphs represented using an adjacency matrix:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_adjacency_matrix.py<pre><code>class GraphAdjMat:\n    \"\"\"Undirected graph class based on adjacency matrix\"\"\"\n\n    def __init__(self, vertices: list[int], edges: list[list[int]]):\n        \"\"\"Constructor\"\"\"\n        # Vertex list, where the element represents the \"vertex value\" and the index represents the \"vertex index\"\n        self.vertices: list[int] = []\n        # Adjacency matrix, where the row and column indices correspond to the \"vertex index\"\n        self.adj_mat: list[list[int]] = []\n        # Add vertices\n        for val in vertices:\n            self.add_vertex(val)\n        # Add edges\n        # Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices\n        for e in edges:\n            self.add_edge(e[0], e[1])\n\n    def size(self) -&gt; int:\n        \"\"\"Get the number of vertices\"\"\"\n        return len(self.vertices)\n\n    def add_vertex(self, val: int):\n        \"\"\"Add vertex\"\"\"\n        n = self.size()\n        # Add the value of the new vertex to the vertex list\n        self.vertices.append(val)\n        # Add a row to the adjacency matrix\n        new_row = [0] * n\n        self.adj_mat.append(new_row)\n        # Add a column to the adjacency matrix\n        for row in self.adj_mat:\n            row.append(0)\n\n    def remove_vertex(self, index: int):\n        \"\"\"Remove vertex\"\"\"\n        if index &gt;= self.size():\n            raise IndexError()\n        # Remove the vertex at index from the vertex list\n        self.vertices.pop(index)\n        # Remove the row at index from the adjacency matrix\n        self.adj_mat.pop(index)\n        # Remove the column at index from the adjacency matrix\n        for row in self.adj_mat:\n            row.pop(index)\n\n    def add_edge(self, i: int, j: int):\n        \"\"\"Add edge\"\"\"\n        # Parameters i, j correspond to the vertices element indices\n        # Handle index out of bounds and equality\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        # In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)\n        self.adj_mat[i][j] = 1\n        self.adj_mat[j][i] = 1\n\n    def remove_edge(self, i: int, j: int):\n        \"\"\"Remove edge\"\"\"\n        # Parameters i, j correspond to the vertices element indices\n        # Handle index out of bounds and equality\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        self.adj_mat[i][j] = 0\n        self.adj_mat[j][i] = 0\n\n    def print(self):\n        \"\"\"Print adjacency matrix\"\"\"\n        print(\"Vertex list =\", self.vertices)\n        print(\"Adjacency matrix =\")\n        print_matrix(self.adj_mat)\n</code></pre> graph_adjacency_matrix.cpp<pre><code>/* Undirected graph class based on adjacency matrix */\nclass GraphAdjMat {\n    vector&lt;int&gt; vertices;       // Vertex list, where the element represents the \"vertex value\" and the index represents the \"vertex index\"\n    vector&lt;vector&lt;int&gt;&gt; adjMat; // Adjacency matrix, where the row and column indices correspond to the \"vertex index\"\n\n  public:\n    /* Constructor */\n    GraphAdjMat(const vector&lt;int&gt; &amp;vertices, const vector&lt;vector&lt;int&gt;&gt; &amp;edges) {\n        // Add vertex\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // Add edge\n        // Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices\n        for (const vector&lt;int&gt; &amp;edge : edges) {\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    int size() const {\n        return vertices.size();\n    }\n\n    /* Add vertex */\n    void addVertex(int val) {\n        int n = size();\n        // Add the value of the new vertex to the vertex list\n        vertices.push_back(val);\n        // Add a row to the adjacency matrix\n        adjMat.emplace_back(vector&lt;int&gt;(n, 0));\n        // Add a column to the adjacency matrix\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.push_back(0);\n        }\n    }\n\n    /* Remove vertex */\n    void removeVertex(int index) {\n        if (index &gt;= size()) {\n            throw out_of_range(\"Vertex does not exist\");\n        }\n        // Remove the vertex at index from the vertex list\n        vertices.erase(vertices.begin() + index);\n        // Remove the row at index from the adjacency matrix\n        adjMat.erase(adjMat.begin() + index);\n        // Remove the column at index from the adjacency matrix\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.erase(row.begin() + index);\n        }\n    }\n\n    /* Add edge */\n    // Parameters i, j correspond to the vertices element indices\n    void addEdge(int i, int j) {\n        // Handle index out of bounds and equality\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"Vertex does not exist\");\n        }\n        // In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* Remove edge */\n    // Parameters i, j correspond to the vertices element indices\n    void removeEdge(int i, int j) {\n        // Handle index out of bounds and equality\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"Vertex does not exist\");\n        }\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* Print adjacency matrix */\n    void print() {\n        cout &lt;&lt; \"Vertex list = \";\n        printVector(vertices);\n        cout &lt;&lt; \"Adjacency matrix =\" &lt;&lt; endl;\n        printVectorMatrix(adjMat);\n    }\n};\n</code></pre> graph_adjacency_matrix.java<pre><code>/* Undirected graph class based on adjacency matrix */\nclass GraphAdjMat {\n    List&lt;Integer&gt; vertices; // Vertex list, where the element represents the \"vertex value\" and the index represents the \"vertex index\"\n    List&lt;List&lt;Integer&gt;&gt; adjMat; // Adjacency matrix, where the row and column indices correspond to the \"vertex index\"\n\n    /* Constructor */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = new ArrayList&lt;&gt;();\n        this.adjMat = new ArrayList&lt;&gt;();\n        // Add vertex\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // Add edge\n        // Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices\n        for (int[] e : edges) {\n            addEdge(e[0], e[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    public int size() {\n        return vertices.size();\n    }\n\n    /* Add vertex */\n    public void addVertex(int val) {\n        int n = size();\n        // Add the value of the new vertex to the vertex list\n        vertices.add(val);\n        // Add a row to the adjacency matrix\n        List&lt;Integer&gt; newRow = new ArrayList&lt;&gt;(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.add(0);\n        }\n        adjMat.add(newRow);\n        // Add a column to the adjacency matrix\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.add(0);\n        }\n    }\n\n    /* Remove vertex */\n    public void removeVertex(int index) {\n        if (index &gt;= size())\n            throw new IndexOutOfBoundsException();\n        // Remove the vertex at index from the vertex list\n        vertices.remove(index);\n        // Remove the row at index from the adjacency matrix\n        adjMat.remove(index);\n        // Remove the column at index from the adjacency matrix\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.remove(index);\n        }\n    }\n\n    /* Add edge */\n    // Parameters i, j correspond to the vertices element indices\n    public void addEdge(int i, int j) {\n        // Handle index out of bounds and equality\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        // In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)\n        adjMat.get(i).set(j, 1);\n        adjMat.get(j).set(i, 1);\n    }\n\n    /* Remove edge */\n    // Parameters i, j correspond to the vertices element indices\n    public void removeEdge(int i, int j) {\n        // Handle index out of bounds and equality\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        adjMat.get(i).set(j, 0);\n        adjMat.get(j).set(i, 0);\n    }\n\n    /* Print adjacency matrix */\n    public void print() {\n        System.out.print(\"Vertex list = \");\n        System.out.println(vertices);\n        System.out.println(\"Adjacency matrix =\");\n        PrintUtil.printMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.cs<pre><code>/* Undirected graph class based on adjacency matrix */\nclass GraphAdjMat {\n    List&lt;int&gt; vertices;     // Vertex list, where the element represents the \"vertex value\" and the index represents the \"vertex index\"\n    List&lt;List&lt;int&gt;&gt; adjMat; // Adjacency matrix, where the row and column indices correspond to the \"vertex index\"\n\n    /* Constructor */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // Add vertex\n        foreach (int val in vertices) {\n            AddVertex(val);\n        }\n        // Add edge\n        // Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices\n        foreach (int[] e in edges) {\n            AddEdge(e[0], e[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    int Size() {\n        return vertices.Count;\n    }\n\n    /* Add vertex */\n    public void AddVertex(int val) {\n        int n = Size();\n        // Add the value of the new vertex to the vertex list\n        vertices.Add(val);\n        // Add a row to the adjacency matrix\n        List&lt;int&gt; newRow = new(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.Add(0);\n        }\n        adjMat.Add(newRow);\n        // Add a column to the adjacency matrix\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.Add(0);\n        }\n    }\n\n    /* Remove vertex */\n    public void RemoveVertex(int index) {\n        if (index &gt;= Size())\n            throw new IndexOutOfRangeException();\n        // Remove the vertex at index from the vertex list\n        vertices.RemoveAt(index);\n        // Remove the row at index from the adjacency matrix\n        adjMat.RemoveAt(index);\n        // Remove the column at index from the adjacency matrix\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.RemoveAt(index);\n        }\n    }\n\n    /* Add edge */\n    // Parameters i, j correspond to the vertices element indices\n    public void AddEdge(int i, int j) {\n        // Handle index out of bounds and equality\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        // In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* Remove edge */\n    // Parameters i, j correspond to the vertices element indices\n    public void RemoveEdge(int i, int j) {\n        // Handle index out of bounds and equality\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* Print adjacency matrix */\n    public void Print() {\n        Console.Write(\"Vertex list = \");\n        PrintUtil.PrintList(vertices);\n        Console.WriteLine(\"Adjacency matrix =\");\n        PrintUtil.PrintMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.go<pre><code>/* Undirected graph class based on adjacency matrix */\ntype graphAdjMat struct {\n    // Vertex list, where the element represents the \"vertex value\" and the index represents the \"vertex index\"\n    vertices []int\n    // Adjacency matrix, where the row and column indices correspond to the \"vertex index\"\n    adjMat [][]int\n}\n\n/* Constructor */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n    // Add vertex\n    n := len(vertices)\n    adjMat := make([][]int, n)\n    for i := range adjMat {\n        adjMat[i] = make([]int, n)\n    }\n    // Initialize graph\n    g := &amp;graphAdjMat{\n        vertices: vertices,\n        adjMat:   adjMat,\n    }\n    // Add edge\n    // Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices\n    for i := range edges {\n        g.addEdge(edges[i][0], edges[i][1])\n    }\n    return g\n}\n\n/* Get the number of vertices */\nfunc (g *graphAdjMat) size() int {\n    return len(g.vertices)\n}\n\n/* Add vertex */\nfunc (g *graphAdjMat) addVertex(val int) {\n    n := g.size()\n    // Add the value of the new vertex to the vertex list\n    g.vertices = append(g.vertices, val)\n    // Add a row to the adjacency matrix\n    newRow := make([]int, n)\n    g.adjMat = append(g.adjMat, newRow)\n    // Add a column to the adjacency matrix\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i], 0)\n    }\n}\n\n/* Remove vertex */\nfunc (g *graphAdjMat) removeVertex(index int) {\n    if index &gt;= g.size() {\n        return\n    }\n    // Remove the vertex at index from the vertex list\n    g.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n    // Remove the row at index from the adjacency matrix\n    g.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n    // Remove the column at index from the adjacency matrix\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n    }\n}\n\n/* Add edge */\n// Parameters i, j correspond to the vertices element indices\nfunc (g *graphAdjMat) addEdge(i, j int) {\n    // Handle index out of bounds and equality\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    // In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)\n    g.adjMat[i][j] = 1\n    g.adjMat[j][i] = 1\n}\n\n/* Remove edge */\n// Parameters i, j correspond to the vertices element indices\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n    // Handle index out of bounds and equality\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    g.adjMat[i][j] = 0\n    g.adjMat[j][i] = 0\n}\n\n/* Print adjacency matrix */\nfunc (g *graphAdjMat) print() {\n    fmt.Printf(\"\\tVertex list = %v\\n\", g.vertices)\n    fmt.Printf(\"\\tAdjacency matrix = \\n\")\n    for i := range g.adjMat {\n        fmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n    }\n}\n</code></pre> graph_adjacency_matrix.swift<pre><code>/* Undirected graph class based on adjacency matrix */\nclass GraphAdjMat {\n    private var vertices: [Int] // Vertex list, where the element represents the \"vertex value\" and the index represents the \"vertex index\"\n    private var adjMat: [[Int]] // Adjacency matrix, where the row and column indices correspond to the \"vertex index\"\n\n    /* Constructor */\n    init(vertices: [Int], edges: [[Int]]) {\n        self.vertices = []\n        adjMat = []\n        // Add vertex\n        for val in vertices {\n            addVertex(val: val)\n        }\n        // Add edge\n        // Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices\n        for e in edges {\n            addEdge(i: e[0], j: e[1])\n        }\n    }\n\n    /* Get the number of vertices */\n    func size() -&gt; Int {\n        vertices.count\n    }\n\n    /* Add vertex */\n    func addVertex(val: Int) {\n        let n = size()\n        // Add the value of the new vertex to the vertex list\n        vertices.append(val)\n        // Add a row to the adjacency matrix\n        let newRow = Array(repeating: 0, count: n)\n        adjMat.append(newRow)\n        // Add a column to the adjacency matrix\n        for i in adjMat.indices {\n            adjMat[i].append(0)\n        }\n    }\n\n    /* Remove vertex */\n    func removeVertex(index: Int) {\n        if index &gt;= size() {\n            fatalError(\"Out of bounds\")\n        }\n        // Remove the vertex at index from the vertex list\n        vertices.remove(at: index)\n        // Remove the row at index from the adjacency matrix\n        adjMat.remove(at: index)\n        // Remove the column at index from the adjacency matrix\n        for i in adjMat.indices {\n            adjMat[i].remove(at: index)\n        }\n    }\n\n    /* Add edge */\n    // Parameters i, j correspond to the vertices element indices\n    func addEdge(i: Int, j: Int) {\n        // Handle index out of bounds and equality\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"Out of bounds\")\n        }\n        // In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* Remove edge */\n    // Parameters i, j correspond to the vertices element indices\n    func removeEdge(i: Int, j: Int) {\n        // Handle index out of bounds and equality\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"Out of bounds\")\n        }\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* Print adjacency matrix */\n    func print() {\n        Swift.print(\"Vertex list = \", terminator: \"\")\n        Swift.print(vertices)\n        Swift.print(\"Adjacency matrix =\")\n        PrintUtil.printMatrix(matrix: adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.js<pre><code>/* Undirected graph class based on adjacency matrix */\nclass GraphAdjMat {\n    vertices; // Vertex list, where the element represents the \"vertex value\" and the index represents the \"vertex index\"\n    adjMat; // Adjacency matrix, where the row and column indices correspond to the \"vertex index\"\n\n    /* Constructor */\n    constructor(vertices, edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // Add vertex\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // Add edge\n        // Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    size() {\n        return this.vertices.length;\n    }\n\n    /* Add vertex */\n    addVertex(val) {\n        const n = this.size();\n        // Add the value of the new vertex to the vertex list\n        this.vertices.push(val);\n        // Add a row to the adjacency matrix\n        const newRow = [];\n        for (let j = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // Add a column to the adjacency matrix\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* Remove vertex */\n    removeVertex(index) {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // Remove the vertex at index from the vertex list\n        this.vertices.splice(index, 1);\n\n        // Remove the row at index from the adjacency matrix\n        this.adjMat.splice(index, 1);\n        // Remove the column at index from the adjacency matrix\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* Add edge */\n    // Parameters i, j correspond to the vertices element indices\n    addEdge(i, j) {\n        // Handle index out of bounds and equality\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // In undirected graph, adjacency matrix is symmetric about main diagonal, i.e., satisfies (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* Remove edge */\n    // Parameters i, j correspond to the vertices element indices\n    removeEdge(i, j) {\n        // Handle index out of bounds and equality\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* Print adjacency matrix */\n    print() {\n        console.log('Vertex list = ', this.vertices);\n        console.log('Adjacency matrix =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.ts<pre><code>/* Undirected graph class based on adjacency matrix */\nclass GraphAdjMat {\n    vertices: number[]; // Vertex list, where the element represents the \"vertex value\" and the index represents the \"vertex index\"\n    adjMat: number[][]; // Adjacency matrix, where the row and column indices correspond to the \"vertex index\"\n\n    /* Constructor */\n    constructor(vertices: number[], edges: number[][]) {\n        this.vertices = [];\n        this.adjMat = [];\n        // Add vertex\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // Add edge\n        // Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    size(): number {\n        return this.vertices.length;\n    }\n\n    /* Add vertex */\n    addVertex(val: number): void {\n        const n: number = this.size();\n        // Add the value of the new vertex to the vertex list\n        this.vertices.push(val);\n        // Add a row to the adjacency matrix\n        const newRow: number[] = [];\n        for (let j: number = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // Add a column to the adjacency matrix\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* Remove vertex */\n    removeVertex(index: number): void {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // Remove the vertex at index from the vertex list\n        this.vertices.splice(index, 1);\n\n        // Remove the row at index from the adjacency matrix\n        this.adjMat.splice(index, 1);\n        // Remove the column at index from the adjacency matrix\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* Add edge */\n    // Parameters i, j correspond to the vertices element indices\n    addEdge(i: number, j: number): void {\n        // Handle index out of bounds and equality\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // In undirected graph, adjacency matrix is symmetric about main diagonal, i.e., satisfies (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* Remove edge */\n    // Parameters i, j correspond to the vertices element indices\n    removeEdge(i: number, j: number): void {\n        // Handle index out of bounds and equality\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* Print adjacency matrix */\n    print(): void {\n        console.log('Vertex list = ', this.vertices);\n        console.log('Adjacency matrix =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.dart<pre><code>/* Undirected graph class based on adjacency matrix */\nclass GraphAdjMat {\n  List&lt;int&gt; vertices = []; // Vertex elements, elements represent \"vertex values\", indices represent \"vertex indices\"\n  List&lt;List&lt;int&gt;&gt; adjMat = []; // Adjacency matrix, where the row and column indices correspond to the \"vertex index\"\n\n  /* Constructor */\n  GraphAdjMat(List&lt;int&gt; vertices, List&lt;List&lt;int&gt;&gt; edges) {\n    this.vertices = [];\n    this.adjMat = [];\n    // Add vertex\n    for (int val in vertices) {\n      addVertex(val);\n    }\n    // Add edge\n    // Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices\n    for (List&lt;int&gt; e in edges) {\n      addEdge(e[0], e[1]);\n    }\n  }\n\n  /* Get the number of vertices */\n  int size() {\n    return vertices.length;\n  }\n\n  /* Add vertex */\n  void addVertex(int val) {\n    int n = size();\n    // Add the value of the new vertex to the vertex list\n    vertices.add(val);\n    // Add a row to the adjacency matrix\n    List&lt;int&gt; newRow = List.filled(n, 0, growable: true);\n    adjMat.add(newRow);\n    // Add a column to the adjacency matrix\n    for (List&lt;int&gt; row in adjMat) {\n      row.add(0);\n    }\n  }\n\n  /* Remove vertex */\n  void removeVertex(int index) {\n    if (index &gt;= size()) {\n      throw IndexError;\n    }\n    // Remove the vertex at index from the vertex list\n    vertices.removeAt(index);\n    // Remove the row at index from the adjacency matrix\n    adjMat.removeAt(index);\n    // Remove the column at index from the adjacency matrix\n    for (List&lt;int&gt; row in adjMat) {\n      row.removeAt(index);\n    }\n  }\n\n  /* Add edge */\n  // Parameters i, j correspond to the vertices element indices\n  void addEdge(int i, int j) {\n    // Handle index out of bounds and equality\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    // In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)\n    adjMat[i][j] = 1;\n    adjMat[j][i] = 1;\n  }\n\n  /* Remove edge */\n  // Parameters i, j correspond to the vertices element indices\n  void removeEdge(int i, int j) {\n    // Handle index out of bounds and equality\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    adjMat[i][j] = 0;\n    adjMat[j][i] = 0;\n  }\n\n  /* Print adjacency matrix */\n  void printAdjMat() {\n    print(\"Vertex list = $vertices\");\n    print(\"Adjacency matrix = \");\n    printMatrix(adjMat);\n  }\n}\n</code></pre> graph_adjacency_matrix.rs<pre><code>/* Undirected graph type based on adjacency matrix */\npub struct GraphAdjMat {\n    // Vertex list, where the element represents the \"vertex value\" and the index represents the \"vertex index\"\n    pub vertices: Vec&lt;i32&gt;,\n    // Adjacency matrix, where the row and column indices correspond to the \"vertex index\"\n    pub adj_mat: Vec&lt;Vec&lt;i32&gt;&gt;,\n}\n\nimpl GraphAdjMat {\n    /* Constructor */\n    pub fn new(vertices: Vec&lt;i32&gt;, edges: Vec&lt;[usize; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjMat {\n            vertices: vec![],\n            adj_mat: vec![],\n        };\n        // Add vertex\n        for val in vertices {\n            graph.add_vertex(val);\n        }\n        // Add edge\n        // Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices\n        for edge in edges {\n            graph.add_edge(edge[0], edge[1])\n        }\n\n        graph\n    }\n\n    /* Get the number of vertices */\n    pub fn size(&amp;self) -&gt; usize {\n        self.vertices.len()\n    }\n\n    /* Add vertex */\n    pub fn add_vertex(&amp;mut self, val: i32) {\n        let n = self.size();\n        // Add the value of the new vertex to the vertex list\n        self.vertices.push(val);\n        // Add a row to the adjacency matrix\n        self.adj_mat.push(vec![0; n]);\n        // Add a column to the adjacency matrix\n        for row in self.adj_mat.iter_mut() {\n            row.push(0);\n        }\n    }\n\n    /* Remove vertex */\n    pub fn remove_vertex(&amp;mut self, index: usize) {\n        if index &gt;= self.size() {\n            panic!(\"index error\")\n        }\n        // Remove the vertex at index from the vertex list\n        self.vertices.remove(index);\n        // Remove the row at index from the adjacency matrix\n        self.adj_mat.remove(index);\n        // Remove the column at index from the adjacency matrix\n        for row in self.adj_mat.iter_mut() {\n            row.remove(index);\n        }\n    }\n\n    /* Add edge */\n    pub fn add_edge(&amp;mut self, i: usize, j: usize) {\n        // Parameters i, j correspond to the vertices element indices\n        // Handle index out of bounds and equality\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        // In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)\n        self.adj_mat[i][j] = 1;\n        self.adj_mat[j][i] = 1;\n    }\n\n    /* Remove edge */\n    // Parameters i, j correspond to the vertices element indices\n    pub fn remove_edge(&amp;mut self, i: usize, j: usize) {\n        // Parameters i, j correspond to the vertices element indices\n        // Handle index out of bounds and equality\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        self.adj_mat[i][j] = 0;\n        self.adj_mat[j][i] = 0;\n    }\n\n    /* Print adjacency matrix */\n    pub fn print(&amp;self) {\n        println!(\"Vertex list = {:?}\", self.vertices);\n        println!(\"Adjacency matrix =\");\n        println!(\"[\");\n        for row in &amp;self.adj_mat {\n            println!(\"  {:?},\", row);\n        }\n        println!(\"]\")\n    }\n}\n</code></pre> graph_adjacency_matrix.c<pre><code>/* Undirected graph structure based on adjacency matrix */\ntypedef struct {\n    int vertices[MAX_SIZE];\n    int adjMat[MAX_SIZE][MAX_SIZE];\n    int size;\n} GraphAdjMat;\n\n/* Constructor */\nGraphAdjMat *newGraphAdjMat() {\n    GraphAdjMat *graph = (GraphAdjMat *)malloc(sizeof(GraphAdjMat));\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        for (int j = 0; j &lt; MAX_SIZE; j++) {\n            graph-&gt;adjMat[i][j] = 0;\n        }\n    }\n    return graph;\n}\n\n/* Destructor */\nvoid delGraphAdjMat(GraphAdjMat *graph) {\n    free(graph);\n}\n\n/* Add vertex */\nvoid addVertex(GraphAdjMat *graph, int val) {\n    if (graph-&gt;size == MAX_SIZE) {\n        fprintf(stderr, \"Graph vertex count has reached maximum\\n\");\n        return;\n    }\n    // Add nth vertex and zero nth row and column\n    int n = graph-&gt;size;\n    graph-&gt;vertices[n] = val;\n    for (int i = 0; i &lt;= n; i++) {\n        graph-&gt;adjMat[n][i] = graph-&gt;adjMat[i][n] = 0;\n    }\n    graph-&gt;size++;\n}\n\n/* Remove vertex */\nvoid removeVertex(GraphAdjMat *graph, int index) {\n    if (index &lt; 0 || index &gt;= graph-&gt;size) {\n        fprintf(stderr, \"Vertex index out of bounds\\n\");\n        return;\n    }\n    // Remove the vertex at index from the vertex list\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        graph-&gt;vertices[i] = graph-&gt;vertices[i + 1];\n    }\n    // Remove the row at index from the adjacency matrix\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        for (int j = 0; j &lt; graph-&gt;size; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i + 1][j];\n        }\n    }\n    // Remove the column at index from the adjacency matrix\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        for (int j = index; j &lt; graph-&gt;size - 1; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i][j + 1];\n        }\n    }\n    graph-&gt;size--;\n}\n\n/* Add edge */\n// Parameters i, j correspond to the vertices element indices\nvoid addEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"Edge index out of bounds or equal\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 1;\n    graph-&gt;adjMat[j][i] = 1;\n}\n\n/* Remove edge */\n// Parameters i, j correspond to the vertices element indices\nvoid removeEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"Edge index out of bounds or equal\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 0;\n    graph-&gt;adjMat[j][i] = 0;\n}\n\n/* Print adjacency matrix */\nvoid printGraphAdjMat(GraphAdjMat *graph) {\n    printf(\"Vertex list = \");\n    printArray(graph-&gt;vertices, graph-&gt;size);\n    printf(\"Adjacency matrix =\\n\");\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        printArray(graph-&gt;adjMat[i], graph-&gt;size);\n    }\n}\n</code></pre> graph_adjacency_matrix.kt<pre><code>/* Undirected graph class based on adjacency matrix */\nclass GraphAdjMat(vertices: IntArray, edges: Array&lt;IntArray&gt;) {\n    val vertices = mutableListOf&lt;Int&gt;() // Vertex list, where the element represents the \"vertex value\" and the index represents the \"vertex index\"\n    val adjMat = mutableListOf&lt;MutableList&lt;Int&gt;&gt;() // Adjacency matrix, where the row and column indices correspond to the \"vertex index\"\n\n    /* Constructor */\n    init {\n        // Add vertex\n        for (vertex in vertices) {\n            addVertex(vertex)\n        }\n        // Add edge\n        // Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices\n        for (edge in edges) {\n            addEdge(edge[0], edge[1])\n        }\n    }\n\n    /* Get the number of vertices */\n    fun size(): Int {\n        return vertices.size\n    }\n\n    /* Add vertex */\n    fun addVertex(_val: Int) {\n        val n = size()\n        // Add the value of the new vertex to the vertex list\n        vertices.add(_val)\n        // Add a row to the adjacency matrix\n        val newRow = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            newRow.add(0)\n        }\n        adjMat.add(newRow)\n        // Add a column to the adjacency matrix\n        for (row in adjMat) {\n            row.add(0)\n        }\n    }\n\n    /* Remove vertex */\n    fun removeVertex(index: Int) {\n        if (index &gt;= size())\n            throw IndexOutOfBoundsException()\n        // Remove the vertex at index from the vertex list\n        vertices.removeAt(index)\n        // Remove the row at index from the adjacency matrix\n        adjMat.removeAt(index)\n        // Remove the column at index from the adjacency matrix\n        for (row in adjMat) {\n            row.removeAt(index)\n        }\n    }\n\n    /* Add edge */\n    // Parameters i, j correspond to the vertices element indices\n    fun addEdge(i: Int, j: Int) {\n        // Handle index out of bounds and equality\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw IndexOutOfBoundsException()\n        // In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* Remove edge */\n    // Parameters i, j correspond to the vertices element indices\n    fun removeEdge(i: Int, j: Int) {\n        // Handle index out of bounds and equality\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw IndexOutOfBoundsException()\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* Print adjacency matrix */\n    fun print() {\n        print(\"Vertex list = \")\n        println(vertices)\n        println(\"Adjacency matrix =\")\n        printMatrix(adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.rb<pre><code>### Undirected graph class based on adjacency matrix ###\nclass GraphAdjMat\n  def initialize(vertices, edges)\n    ### Constructor ###\n    # Vertex list, where the element represents the \"vertex value\" and the index represents the \"vertex index\"\n    @vertices = []\n    # Adjacency matrix, where the row and column indices correspond to the \"vertex index\"\n    @adj_mat = []\n    # Add vertex\n    vertices.each { |val| add_vertex(val) }\n    # Add edge\n    # Note that the edges elements represent vertex indices, i.e., corresponding to the vertices element indices\n    edges.each { |e| add_edge(e[0], e[1]) }\n  end\n\n  ### Get number of vertices ###\n  def size\n    @vertices.length\n  end\n\n  ### Add vertex ###\n  def add_vertex(val)\n    n = size\n    # Add the value of the new vertex to the vertex list\n    @vertices &lt;&lt; val\n    # Add a row to the adjacency matrix\n    new_row = Array.new(n, 0)\n    @adj_mat &lt;&lt; new_row\n    # Add a column to the adjacency matrix\n    @adj_mat.each { |row| row &lt;&lt; 0 }\n  end\n\n  ### Delete vertex ###\n  def remove_vertex(index)\n    raise IndexError if index &gt;= size\n\n    # Remove the vertex at index from the vertex list\n    @vertices.delete_at(index)\n    # Remove the row at index from the adjacency matrix\n    @adj_mat.delete_at(index)\n    # Remove the column at index from the adjacency matrix\n    @adj_mat.each { |row| row.delete_at(index) }\n  end\n\n  ### Add edge ###\n  def add_edge(i, j)\n    # Parameters i, j correspond to the vertices element indices\n    # Handle index out of bounds and equality\n    if i &lt; 0 || j &lt; 0 || i &gt;= size || j &gt;= size || i == j\n      raise IndexError\n    end\n    # In an undirected graph, the adjacency matrix is symmetric about the main diagonal, i.e., (i, j) == (j, i)\n    @adj_mat[i][j] = 1\n    @adj_mat[j][i] = 1\n  end\n\n  ### Delete edge ###\n  def remove_edge(i, j)\n    # Parameters i, j correspond to the vertices element indices\n    # Handle index out of bounds and equality\n    if i &lt; 0 || j &lt; 0 || i &gt;= size || j &gt;= size || i == j\n      raise IndexError\n    end\n    @adj_mat[i][j] = 0\n    @adj_mat[j][i] = 0\n  end\n\n  ### Print adjacency matrix ###\n  def __print__\n    puts \"Vertex list = #{@vertices}\"\n    puts 'Adjacency matrix ='\n    print_matrix(@adj_mat)\n  end\nend\n</code></pre>","path":["Chapter 9. Graph","9.2   Basic Operations on Graphs"],"tags":[]},{"location":"chapter_graph/graph_operations/#922-implementation-based-on-adjacency-list","level":2,"title":"9.2.2   Implementation Based on Adjacency List","text":"<p>Given an undirected graph with a total of \\(n\\) vertices and \\(m\\) edges, the various operations can be implemented as shown in Figure 9-8.</p> <ul> <li>Adding an edge: Add the edge at the end of the corresponding vertex's linked list, using \\(O(1)\\) time. Since it is an undirected graph, edges in both directions need to be added simultaneously.</li> <li>Removing an edge: Find and remove the specified edge in the corresponding vertex's linked list, using \\(O(m)\\) time. In an undirected graph, edges in both directions need to be removed simultaneously.</li> <li>Adding a vertex: Add a linked list to the adjacency list, with the new vertex as the head node, using \\(O(1)\\) time.</li> <li>Removing a vertex: Traverse the entire adjacency list and remove all edges containing the specified vertex, using \\(O(n + m)\\) time.</li> <li>Initialization: Create \\(n\\) vertices and \\(2m\\) edges in the adjacency list, using \\(O(n + m)\\) time.</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-8   Initialization, adding and removing edges, adding and removing vertices in adjacency list </p> <p>The following code shows the adjacency list implementation. Compared with Figure 9-8, the actual code differs in the following ways.</p> <ul> <li>For convenience in adding and removing vertices, and to simplify the code, we use lists (dynamic arrays) instead of linked lists.</li> <li>A hash table is used to store the adjacency list, where <code>key</code> is the vertex instance and <code>value</code> is the list (linked list) of adjacent vertices for that vertex.</li> </ul> <p>Additionally, we use the <code>Vertex</code> class to represent vertices in the adjacency list for the following reason: if we used list indices to distinguish different vertices, as with adjacency matrices, then to delete the vertex at index \\(i\\), we would need to traverse the entire adjacency list and decrement all indices greater than \\(i\\) by \\(1\\), which is very inefficient. However, if each vertex is a unique <code>Vertex</code> instance, deleting one vertex does not require modifying the others.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_adjacency_list.py<pre><code>class GraphAdjList:\n    \"\"\"Undirected graph class based on adjacency list\"\"\"\n\n    def __init__(self, edges: list[list[Vertex]]):\n        \"\"\"Constructor\"\"\"\n        # Adjacency list, key: vertex, value: all adjacent vertices of that vertex\n        self.adj_list = dict[Vertex, list[Vertex]]()\n        # Add all vertices and edges\n        for edge in edges:\n            self.add_vertex(edge[0])\n            self.add_vertex(edge[1])\n            self.add_edge(edge[0], edge[1])\n\n    def size(self) -&gt; int:\n        \"\"\"Get the number of vertices\"\"\"\n        return len(self.adj_list)\n\n    def add_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"Add edge\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # Add edge vet1 - vet2\n        self.adj_list[vet1].append(vet2)\n        self.adj_list[vet2].append(vet1)\n\n    def remove_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"Remove edge\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # Remove edge vet1 - vet2\n        self.adj_list[vet1].remove(vet2)\n        self.adj_list[vet2].remove(vet1)\n\n    def add_vertex(self, vet: Vertex):\n        \"\"\"Add vertex\"\"\"\n        if vet in self.adj_list:\n            return\n        # Add a new linked list in the adjacency list\n        self.adj_list[vet] = []\n\n    def remove_vertex(self, vet: Vertex):\n        \"\"\"Remove vertex\"\"\"\n        if vet not in self.adj_list:\n            raise ValueError()\n        # Remove the linked list corresponding to vertex vet in the adjacency list\n        self.adj_list.pop(vet)\n        # Traverse the linked lists of other vertices and remove all edges containing vet\n        for vertex in self.adj_list:\n            if vet in self.adj_list[vertex]:\n                self.adj_list[vertex].remove(vet)\n\n    def print(self):\n        \"\"\"Print adjacency list\"\"\"\n        print(\"Adjacency list =\")\n        for vertex in self.adj_list:\n            tmp = [v.val for v in self.adj_list[vertex]]\n            print(f\"{vertex.val}: {tmp},\")\n</code></pre> graph_adjacency_list.cpp<pre><code>/* Undirected graph class based on adjacency list */\nclass GraphAdjList {\n  public:\n    // Adjacency list, key: vertex, value: all adjacent vertices of that vertex\n    unordered_map&lt;Vertex *, vector&lt;Vertex *&gt;&gt; adjList;\n\n    /* Remove specified node from vector */\n    void remove(vector&lt;Vertex *&gt; &amp;vec, Vertex *vet) {\n        for (int i = 0; i &lt; vec.size(); i++) {\n            if (vec[i] == vet) {\n                vec.erase(vec.begin() + i);\n                break;\n            }\n        }\n    }\n\n    /* Constructor */\n    GraphAdjList(const vector&lt;vector&lt;Vertex *&gt;&gt; &amp;edges) {\n        // Add all vertices and edges\n        for (const vector&lt;Vertex *&gt; &amp;edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    int size() {\n        return adjList.size();\n    }\n\n    /* Add edge */\n    void addEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"Vertex does not exist\");\n        // Add edge vet1 - vet2\n        adjList[vet1].push_back(vet2);\n        adjList[vet2].push_back(vet1);\n    }\n\n    /* Remove edge */\n    void removeEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"Vertex does not exist\");\n        // Remove edge vet1 - vet2\n        remove(adjList[vet1], vet2);\n        remove(adjList[vet2], vet1);\n    }\n\n    /* Add vertex */\n    void addVertex(Vertex *vet) {\n        if (adjList.count(vet))\n            return;\n        // Add a new linked list in the adjacency list\n        adjList[vet] = vector&lt;Vertex *&gt;();\n    }\n\n    /* Remove vertex */\n    void removeVertex(Vertex *vet) {\n        if (!adjList.count(vet))\n            throw invalid_argument(\"Vertex does not exist\");\n        // Remove the linked list corresponding to vertex vet in the adjacency list\n        adjList.erase(vet);\n        // Traverse the linked lists of other vertices and remove all edges containing vet\n        for (auto &amp;adj : adjList) {\n            remove(adj.second, vet);\n        }\n    }\n\n    /* Print adjacency list */\n    void print() {\n        cout &lt;&lt; \"Adjacency list =\" &lt;&lt; endl;\n        for (auto &amp;adj : adjList) {\n            const auto &amp;key = adj.first;\n            const auto &amp;vec = adj.second;\n            cout &lt;&lt; key-&gt;val &lt;&lt; \": \";\n            printVector(vetsToVals(vec));\n        }\n    }\n};\n</code></pre> graph_adjacency_list.java<pre><code>/* Undirected graph class based on adjacency list */\nclass GraphAdjList {\n    // Adjacency list, key: vertex, value: all adjacent vertices of that vertex\n    Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* Constructor */\n    public GraphAdjList(Vertex[][] edges) {\n        this.adjList = new HashMap&lt;&gt;();\n        // Add all vertices and edges\n        for (Vertex[] edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    public int size() {\n        return adjList.size();\n    }\n\n    /* Add edge */\n    public void addEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // Add edge vet1 - vet2\n        adjList.get(vet1).add(vet2);\n        adjList.get(vet2).add(vet1);\n    }\n\n    /* Remove edge */\n    public void removeEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // Remove edge vet1 - vet2\n        adjList.get(vet1).remove(vet2);\n        adjList.get(vet2).remove(vet1);\n    }\n\n    /* Add vertex */\n    public void addVertex(Vertex vet) {\n        if (adjList.containsKey(vet))\n            return;\n        // Add a new linked list in the adjacency list\n        adjList.put(vet, new ArrayList&lt;&gt;());\n    }\n\n    /* Remove vertex */\n    public void removeVertex(Vertex vet) {\n        if (!adjList.containsKey(vet))\n            throw new IllegalArgumentException();\n        // Remove the linked list corresponding to vertex vet in the adjacency list\n        adjList.remove(vet);\n        // Traverse the linked lists of other vertices and remove all edges containing vet\n        for (List&lt;Vertex&gt; list : adjList.values()) {\n            list.remove(vet);\n        }\n    }\n\n    /* Print adjacency list */\n    public void print() {\n        System.out.println(\"Adjacency list =\");\n        for (Map.Entry&lt;Vertex, List&lt;Vertex&gt;&gt; pair : adjList.entrySet()) {\n            List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n            for (Vertex vertex : pair.getValue())\n                tmp.add(vertex.val);\n            System.out.println(pair.getKey().val + \": \" + tmp + \",\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.cs<pre><code>/* Undirected graph class based on adjacency list */\nclass GraphAdjList {\n    // Adjacency list, key: vertex, value: all adjacent vertices of that vertex\n    public Dictionary&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* Constructor */\n    public GraphAdjList(Vertex[][] edges) {\n        adjList = [];\n        // Add all vertices and edges\n        foreach (Vertex[] edge in edges) {\n            AddVertex(edge[0]);\n            AddVertex(edge[1]);\n            AddEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    int Size() {\n        return adjList.Count;\n    }\n\n    /* Add edge */\n    public void AddEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // Add edge vet1 - vet2\n        adjList[vet1].Add(vet2);\n        adjList[vet2].Add(vet1);\n    }\n\n    /* Remove edge */\n    public void RemoveEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // Remove edge vet1 - vet2\n        adjList[vet1].Remove(vet2);\n        adjList[vet2].Remove(vet1);\n    }\n\n    /* Add vertex */\n    public void AddVertex(Vertex vet) {\n        if (adjList.ContainsKey(vet))\n            return;\n        // Add a new linked list in the adjacency list\n        adjList.Add(vet, []);\n    }\n\n    /* Remove vertex */\n    public void RemoveVertex(Vertex vet) {\n        if (!adjList.ContainsKey(vet))\n            throw new InvalidOperationException();\n        // Remove the linked list corresponding to vertex vet in the adjacency list\n        adjList.Remove(vet);\n        // Traverse the linked lists of other vertices and remove all edges containing vet\n        foreach (List&lt;Vertex&gt; list in adjList.Values) {\n            list.Remove(vet);\n        }\n    }\n\n    /* Print adjacency list */\n    public void Print() {\n        Console.WriteLine(\"Adjacency list =\");\n        foreach (KeyValuePair&lt;Vertex, List&lt;Vertex&gt;&gt; pair in adjList) {\n            List&lt;int&gt; tmp = [];\n            foreach (Vertex vertex in pair.Value)\n                tmp.Add(vertex.val);\n            Console.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.go<pre><code>/* Undirected graph class based on adjacency list */\ntype graphAdjList struct {\n    // Adjacency list, key: vertex, value: all adjacent vertices of that vertex\n    adjList map[Vertex][]Vertex\n}\n\n/* Constructor */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\n    g := &amp;graphAdjList{\n        adjList: make(map[Vertex][]Vertex),\n    }\n    // Add all vertices and edges\n    for _, edge := range edges {\n        g.addVertex(edge[0])\n        g.addVertex(edge[1])\n        g.addEdge(edge[0], edge[1])\n    }\n    return g\n}\n\n/* Get the number of vertices */\nfunc (g *graphAdjList) size() int {\n    return len(g.adjList)\n}\n\n/* Add edge */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // Add edge vet1 - vet2, add anonymous struct{},\n    g.adjList[vet1] = append(g.adjList[vet1], vet2)\n    g.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n\n/* Remove edge */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // Remove edge vet1 - vet2\n    g.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2)\n    g.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1)\n}\n\n/* Add vertex */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if ok {\n        return\n    }\n    // Add a new linked list in the adjacency list\n    g.adjList[vet] = make([]Vertex, 0)\n}\n\n/* Remove vertex */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if !ok {\n        panic(\"error\")\n    }\n    // Remove the linked list corresponding to vertex vet in the adjacency list\n    delete(g.adjList, vet)\n    // Traverse the linked lists of other vertices and remove all edges containing vet\n    for v, list := range g.adjList {\n        g.adjList[v] = DeleteSliceElms(list, vet)\n    }\n}\n\n/* Print adjacency list */\nfunc (g *graphAdjList) print() {\n    var builder strings.Builder\n    fmt.Printf(\"Adjacency list = \\n\")\n    for k, v := range g.adjList {\n        builder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\n        for _, vet := range v {\n            builder.WriteString(strconv.Itoa(vet.Val) + \" \")\n        }\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> graph_adjacency_list.swift<pre><code>/* Undirected graph class based on adjacency list */\nclass GraphAdjList {\n    // Adjacency list, key: vertex, value: all adjacent vertices of that vertex\n    public private(set) var adjList: [Vertex: [Vertex]]\n\n    /* Constructor */\n    public init(edges: [[Vertex]]) {\n        adjList = [:]\n        // Add all vertices and edges\n        for edge in edges {\n            addVertex(vet: edge[0])\n            addVertex(vet: edge[1])\n            addEdge(vet1: edge[0], vet2: edge[1])\n        }\n    }\n\n    /* Get the number of vertices */\n    public func size() -&gt; Int {\n        adjList.count\n    }\n\n    /* Add edge */\n    public func addEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"Invalid parameter\")\n        }\n        // Add edge vet1 - vet2\n        adjList[vet1]?.append(vet2)\n        adjList[vet2]?.append(vet1)\n    }\n\n    /* Remove edge */\n    public func removeEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"Invalid parameter\")\n        }\n        // Remove edge vet1 - vet2\n        adjList[vet1]?.removeAll { $0 == vet2 }\n        adjList[vet2]?.removeAll { $0 == vet1 }\n    }\n\n    /* Add vertex */\n    public func addVertex(vet: Vertex) {\n        if adjList[vet] != nil {\n            return\n        }\n        // Add a new linked list in the adjacency list\n        adjList[vet] = []\n    }\n\n    /* Remove vertex */\n    public func removeVertex(vet: Vertex) {\n        if adjList[vet] == nil {\n            fatalError(\"Invalid parameter\")\n        }\n        // Remove the linked list corresponding to vertex vet in the adjacency list\n        adjList.removeValue(forKey: vet)\n        // Traverse the linked lists of other vertices and remove all edges containing vet\n        for key in adjList.keys {\n            adjList[key]?.removeAll { $0 == vet }\n        }\n    }\n\n    /* Print adjacency list */\n    public func print() {\n        Swift.print(\"Adjacency list =\")\n        for (vertex, list) in adjList {\n            let list = list.map { $0.val }\n            Swift.print(\"\\(vertex.val): \\(list),\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.js<pre><code>/* Undirected graph class based on adjacency list */\nclass GraphAdjList {\n    // Adjacency list, key: vertex, value: all adjacent vertices of that vertex\n    adjList;\n\n    /* Constructor */\n    constructor(edges) {\n        this.adjList = new Map();\n        // Add all vertices and edges\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    size() {\n        return this.adjList.size;\n    }\n\n    /* Add edge */\n    addEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // Add edge vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* Remove edge */\n    removeEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2 ||\n            this.adjList.get(vet1).indexOf(vet2) === -1\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // Remove edge vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* Add vertex */\n    addVertex(vet) {\n        if (this.adjList.has(vet)) return;\n        // Add a new linked list in the adjacency list\n        this.adjList.set(vet, []);\n    }\n\n    /* Remove vertex */\n    removeVertex(vet) {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // Remove the linked list corresponding to vertex vet in the adjacency list\n        this.adjList.delete(vet);\n        // Traverse the linked lists of other vertices and remove all edges containing vet\n        for (const set of this.adjList.values()) {\n            const index = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* Print adjacency list */\n    print() {\n        console.log('Adjacency list =');\n        for (const [key, value] of this.adjList) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.ts<pre><code>/* Undirected graph class based on adjacency list */\nclass GraphAdjList {\n    // Adjacency list, key: vertex, value: all adjacent vertices of that vertex\n    adjList: Map&lt;Vertex, Vertex[]&gt;;\n\n    /* Constructor */\n    constructor(edges: Vertex[][]) {\n        this.adjList = new Map();\n        // Add all vertices and edges\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    size(): number {\n        return this.adjList.size;\n    }\n\n    /* Add edge */\n    addEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // Add edge vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* Remove edge */\n    removeEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2 ||\n            this.adjList.get(vet1).indexOf(vet2) === -1\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // Remove edge vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* Add vertex */\n    addVertex(vet: Vertex): void {\n        if (this.adjList.has(vet)) return;\n        // Add a new linked list in the adjacency list\n        this.adjList.set(vet, []);\n    }\n\n    /* Remove vertex */\n    removeVertex(vet: Vertex): void {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // Remove the linked list corresponding to vertex vet in the adjacency list\n        this.adjList.delete(vet);\n        // Traverse the linked lists of other vertices and remove all edges containing vet\n        for (const set of this.adjList.values()) {\n            const index: number = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* Print adjacency list */\n    print(): void {\n        console.log('Adjacency list =');\n        for (const [key, value] of this.adjList.entries()) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.dart<pre><code>/* Undirected graph class based on adjacency list */\nclass GraphAdjList {\n  // Adjacency list, key: vertex, value: all adjacent vertices of that vertex\n  Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList = {};\n\n  /* Constructor */\n  GraphAdjList(List&lt;List&lt;Vertex&gt;&gt; edges) {\n    for (List&lt;Vertex&gt; edge in edges) {\n      addVertex(edge[0]);\n      addVertex(edge[1]);\n      addEdge(edge[0], edge[1]);\n    }\n  }\n\n  /* Get the number of vertices */\n  int size() {\n    return adjList.length;\n  }\n\n  /* Add edge */\n  void addEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // Add edge vet1 - vet2\n    adjList[vet1]!.add(vet2);\n    adjList[vet2]!.add(vet1);\n  }\n\n  /* Remove edge */\n  void removeEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // Remove edge vet1 - vet2\n    adjList[vet1]!.remove(vet2);\n    adjList[vet2]!.remove(vet1);\n  }\n\n  /* Add vertex */\n  void addVertex(Vertex vet) {\n    if (adjList.containsKey(vet)) return;\n    // Add a new linked list in the adjacency list\n    adjList[vet] = [];\n  }\n\n  /* Remove vertex */\n  void removeVertex(Vertex vet) {\n    if (!adjList.containsKey(vet)) {\n      throw ArgumentError;\n    }\n    // Remove the linked list corresponding to vertex vet in the adjacency list\n    adjList.remove(vet);\n    // Traverse the linked lists of other vertices and remove all edges containing vet\n    adjList.forEach((key, value) {\n      value.remove(vet);\n    });\n  }\n\n  /* Print adjacency list */\n  void printAdjList() {\n    print(\"Adjacency list =\");\n    adjList.forEach((key, value) {\n      List&lt;int&gt; tmp = [];\n      for (Vertex vertex in value) {\n        tmp.add(vertex.val);\n      }\n      print(\"${key.val}: $tmp,\");\n    });\n  }\n}\n</code></pre> graph_adjacency_list.rs<pre><code>/* Undirected graph type based on adjacency list */\npub struct GraphAdjList {\n    // Adjacency list, key: vertex, value: all adjacent vertices of that vertex\n    pub adj_list: HashMap&lt;Vertex, Vec&lt;Vertex&gt;&gt;, // maybe HashSet&lt;Vertex&gt; for value part is better?\n}\n\nimpl GraphAdjList {\n    /* Constructor */\n    pub fn new(edges: Vec&lt;[Vertex; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjList {\n            adj_list: HashMap::new(),\n        };\n        // Add all vertices and edges\n        for edge in edges {\n            graph.add_vertex(edge[0]);\n            graph.add_vertex(edge[1]);\n            graph.add_edge(edge[0], edge[1]);\n        }\n\n        graph\n    }\n\n    /* Get the number of vertices */\n    #[allow(unused)]\n    pub fn size(&amp;self) -&gt; usize {\n        self.adj_list.len()\n    }\n\n    /* Add edge */\n    pub fn add_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if vet1 == vet2 {\n            panic!(\"value error\");\n        }\n        // Add edge vet1 - vet2\n        self.adj_list.entry(vet1).or_default().push(vet2);\n        self.adj_list.entry(vet2).or_default().push(vet1);\n    }\n\n    /* Remove edge */\n    #[allow(unused)]\n    pub fn remove_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if vet1 == vet2 {\n            panic!(\"value error\");\n        }\n        // Remove edge vet1 - vet2\n        self.adj_list\n            .entry(vet1)\n            .and_modify(|v| v.retain(|&amp;e| e != vet2));\n        self.adj_list\n            .entry(vet2)\n            .and_modify(|v| v.retain(|&amp;e| e != vet1));\n    }\n\n    /* Add vertex */\n    pub fn add_vertex(&amp;mut self, vet: Vertex) {\n        if self.adj_list.contains_key(&amp;vet) {\n            return;\n        }\n        // Add a new linked list in the adjacency list\n        self.adj_list.insert(vet, vec![]);\n    }\n\n    /* Remove vertex */\n    #[allow(unused)]\n    pub fn remove_vertex(&amp;mut self, vet: Vertex) {\n        // Remove the linked list corresponding to vertex vet in the adjacency list\n        self.adj_list.remove(&amp;vet);\n        // Traverse the linked lists of other vertices and remove all edges containing vet\n        for list in self.adj_list.values_mut() {\n            list.retain(|&amp;v| v != vet);\n        }\n    }\n\n    /* Print adjacency list */\n    pub fn print(&amp;self) {\n        println!(\"Adjacency list =\");\n        for (vertex, list) in &amp;self.adj_list {\n            let list = list.iter().map(|vertex| vertex.val).collect::&lt;Vec&lt;i32&gt;&gt;();\n            println!(\"{}: {:?},\", vertex.val, list);\n        }\n    }\n}\n</code></pre> graph_adjacency_list.c<pre><code>/* Node structure */\ntypedef struct AdjListNode {\n    Vertex *vertex;           // Vertex\n    struct AdjListNode *next; // Successor node\n} AdjListNode;\n\n/* Find node corresponding to vertex */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* Add edge helper function */\nvoid addEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *node = (AdjListNode *)malloc(sizeof(AdjListNode));\n    node-&gt;vertex = vet;\n    // Head insertion\n    node-&gt;next = head-&gt;next;\n    head-&gt;next = node;\n}\n\n/* Remove edge helper function */\nvoid removeEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *pre = head;\n    AdjListNode *cur = head-&gt;next;\n    // Search for node corresponding to vet in list\n    while (cur != NULL &amp;&amp; cur-&gt;vertex != vet) {\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n    if (cur == NULL)\n        return;\n    // Remove node corresponding to vet from list\n    pre-&gt;next = cur-&gt;next;\n    // Free memory\n    free(cur);\n}\n\n/* Undirected graph class based on adjacency list */\ntypedef struct {\n    AdjListNode *heads[MAX_SIZE]; // Node array\n    int size;                     // Node count\n} GraphAdjList;\n\n/* Constructor */\nGraphAdjList *newGraphAdjList() {\n    GraphAdjList *graph = (GraphAdjList *)malloc(sizeof(GraphAdjList));\n    if (!graph) {\n        return NULL;\n    }\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        graph-&gt;heads[i] = NULL;\n    }\n    return graph;\n}\n\n/* Destructor */\nvoid delGraphAdjList(GraphAdjList *graph) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        AdjListNode *cur = graph-&gt;heads[i];\n        while (cur != NULL) {\n            AdjListNode *next = cur-&gt;next;\n            if (cur != graph-&gt;heads[i]) {\n                free(cur);\n            }\n            cur = next;\n        }\n        free(graph-&gt;heads[i]-&gt;vertex);\n        free(graph-&gt;heads[i]);\n    }\n    free(graph);\n}\n\n/* Find node corresponding to vertex */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* Add edge */\nvoid addEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL &amp;&amp; head1 != head2);\n    // Add edge vet1 - vet2\n    addEdgeHelper(head1, vet2);\n    addEdgeHelper(head2, vet1);\n}\n\n/* Remove edge */\nvoid removeEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL);\n    // Remove edge vet1 - vet2\n    removeEdgeHelper(head1, head2-&gt;vertex);\n    removeEdgeHelper(head2, head1-&gt;vertex);\n}\n\n/* Add vertex */\nvoid addVertex(GraphAdjList *graph, Vertex *vet) {\n    assert(graph != NULL &amp;&amp; graph-&gt;size &lt; MAX_SIZE);\n    AdjListNode *head = (AdjListNode *)malloc(sizeof(AdjListNode));\n    head-&gt;vertex = vet;\n    head-&gt;next = NULL;\n    // Add a new linked list in the adjacency list\n    graph-&gt;heads[graph-&gt;size++] = head;\n}\n\n/* Remove vertex */\nvoid removeVertex(GraphAdjList *graph, Vertex *vet) {\n    AdjListNode *node = findNode(graph, vet);\n    assert(node != NULL);\n    // Remove the linked list corresponding to vertex vet in the adjacency list\n    AdjListNode *cur = node, *pre = NULL;\n    while (cur) {\n        pre = cur;\n        cur = cur-&gt;next;\n        free(pre);\n    }\n    // Traverse the linked lists of other vertices and remove all edges containing vet\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        cur = graph-&gt;heads[i];\n        pre = NULL;\n        while (cur) {\n            pre = cur;\n            cur = cur-&gt;next;\n            if (cur &amp;&amp; cur-&gt;vertex == vet) {\n                pre-&gt;next = cur-&gt;next;\n                free(cur);\n                break;\n            }\n        }\n    }\n    // Move vertices after this vertex forward to fill gap\n    int i;\n    for (i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i] == node)\n            break;\n    }\n    for (int j = i; j &lt; graph-&gt;size - 1; j++) {\n        graph-&gt;heads[j] = graph-&gt;heads[j + 1];\n    }\n    graph-&gt;size--;\n    free(vet);\n}\n</code></pre> graph_adjacency_list.kt<pre><code>/* Undirected graph class based on adjacency list */\nclass GraphAdjList(edges: Array&lt;Array&lt;Vertex?&gt;&gt;) {\n    // Adjacency list, key: vertex, value: all adjacent vertices of that vertex\n    val adjList = HashMap&lt;Vertex, MutableList&lt;Vertex&gt;&gt;()\n\n    /* Constructor */\n    init {\n        // Add all vertices and edges\n        for (edge in edges) {\n            addVertex(edge[0]!!)\n            addVertex(edge[1]!!)\n            addEdge(edge[0]!!, edge[1]!!)\n        }\n    }\n\n    /* Get the number of vertices */\n    fun size(): Int {\n        return adjList.size\n    }\n\n    /* Add edge */\n    fun addEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // Add edge vet1 - vet2\n        adjList[vet1]?.add(vet2)\n        adjList[vet2]?.add(vet1)\n    }\n\n    /* Remove edge */\n    fun removeEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // Remove edge vet1 - vet2\n        adjList[vet1]?.remove(vet2)\n        adjList[vet2]?.remove(vet1)\n    }\n\n    /* Add vertex */\n    fun addVertex(vet: Vertex) {\n        if (adjList.containsKey(vet))\n            return\n        // Add a new linked list in the adjacency list\n        adjList[vet] = mutableListOf()\n    }\n\n    /* Remove vertex */\n    fun removeVertex(vet: Vertex) {\n        if (!adjList.containsKey(vet))\n            throw IllegalArgumentException()\n        // Remove the linked list corresponding to vertex vet in the adjacency list\n        adjList.remove(vet)\n        // Traverse the linked lists of other vertices and remove all edges containing vet\n        for (list in adjList.values) {\n            list.remove(vet)\n        }\n    }\n\n    /* Print adjacency list */\n    fun print() {\n        println(\"Adjacency list =\")\n        for (pair in adjList.entries) {\n            val tmp = mutableListOf&lt;Int&gt;()\n            for (vertex in pair.value) {\n                tmp.add(vertex._val)\n            }\n            println(\"${pair.key._val}: $tmp,\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.rb<pre><code>### Undirected graph class based on adjacency list ###\nclass GraphAdjList\n  attr_reader :adj_list\n\n  ### Constructor ###\n  def initialize(edges)\n    # Adjacency list, key: vertex, value: all adjacent vertices of that vertex\n    @adj_list = {}\n    # Add all vertices and edges\n    for edge in edges\n      add_vertex(edge[0])\n      add_vertex(edge[1])\n      add_edge(edge[0], edge[1])\n    end\n  end\n\n  ### Get number of vertices ###\n  def size\n    @adj_list.length\n  end\n\n  ### Add edge ###\n  def add_edge(vet1, vet2)\n    raise ArgumentError if !@adj_list.include?(vet1) || !@adj_list.include?(vet2)\n\n    @adj_list[vet1] &lt;&lt; vet2\n    @adj_list[vet2] &lt;&lt; vet1\n  end\n\n  ### Delete edge ###\n  def remove_edge(vet1, vet2)\n    raise ArgumentError if !@adj_list.include?(vet1) || !@adj_list.include?(vet2)\n\n    # Remove edge vet1 - vet2\n    @adj_list[vet1].delete(vet2)\n    @adj_list[vet2].delete(vet1)\n  end\n\n  ### Add vertex ###\n  def add_vertex(vet)\n    return if @adj_list.include?(vet)\n\n    # Add a new linked list in the adjacency list\n    @adj_list[vet] = []\n  end\n\n  ### Delete vertex ###\n  def remove_vertex(vet)\n    raise ArgumentError unless @adj_list.include?(vet)\n\n    # Remove the linked list corresponding to vertex vet in the adjacency list\n    @adj_list.delete(vet)\n    # Traverse the linked lists of other vertices and remove all edges containing vet\n    for vertex in @adj_list\n      @adj_list[vertex.first].delete(vet) if @adj_list[vertex.first].include?(vet)\n    end\n  end\n\n  ### Print adjacency list ###\n  def __print__\n    puts 'Adjacency list ='\n    for vertex in @adj_list\n      tmp = @adj_list[vertex.first].map { |v| v.val }\n      puts \"#{vertex.first.val}: #{tmp},\"\n    end\n  end\nend\n</code></pre>","path":["Chapter 9. Graph","9.2   Basic Operations on Graphs"],"tags":[]},{"location":"chapter_graph/graph_operations/#923-efficiency-comparison","level":2,"title":"9.2.3   Efficiency Comparison","text":"<p>Assuming the graph has \\(n\\) vertices and \\(m\\) edges, Table 9-2 compares the time efficiency and space efficiency of adjacency matrices and adjacency lists. Note that the adjacency list (linked list) corresponds to the implementation used in this section, while the adjacency list (hash table) refers specifically to the implementation where all linked lists are replaced with hash tables.</p> <p> Table 9-2   Comparison of adjacency matrix and adjacency list </p> Adjacency matrix Adjacency list (linked list) Adjacency list (hash table) Determine adjacency \\(O(1)\\) \\(O(n)\\) \\(O(1)\\) Add an edge \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) Remove an edge \\(O(1)\\) \\(O(n)\\) \\(O(1)\\) Add a vertex \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) Remove a vertex \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) Memory space usage \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\) <p>Observing Table 9-2, it appears that the adjacency list (hash table) has the best time efficiency and space efficiency. However, in practice, operating on edges in the adjacency matrix is more efficient, requiring only a single array access or assignment operation. Overall, adjacency matrices embody the principle of \"trading space for time\", while adjacency lists embody \"trading time for space\".</p>","path":["Chapter 9. Graph","9.2   Basic Operations on Graphs"],"tags":[]},{"location":"chapter_graph/graph_traversal/","level":1,"title":"9.3   Graph Traversal","text":"<p>Trees represent \"one-to-many\" relationships, while graphs have a higher degree of freedom and can represent any \"many-to-many\" relationships. Therefore, we can view trees as a special case of graphs. Clearly, tree traversal operations are also a special case of graph traversal operations.</p> <p>Both graphs and trees require the application of search algorithms to implement traversal operations. Graph traversal methods can also be divided into two types: breadth-first traversal and depth-first traversal.</p>","path":["Chapter 9. Graph","9.3   Graph Traversal"],"tags":[]},{"location":"chapter_graph/graph_traversal/#931-breadth-first-search","level":2,"title":"9.3.1   Breadth-First Search","text":"<p>Breadth-first search proceeds from near to far: starting from a given node, it always visits the nearest vertices first and expands outward layer by layer. As shown in Figure 9-9, starting from the top-left vertex, first traverse all adjacent vertices of that vertex, then traverse all adjacent vertices of the next vertex, and so on, until all vertices have been visited.</p> <p></p> <p> Figure 9-9   Breadth-first search of a graph </p>","path":["Chapter 9. Graph","9.3   Graph Traversal"],"tags":[]},{"location":"chapter_graph/graph_traversal/#1-algorithm-implementation","level":3,"title":"1.   Algorithm Implementation","text":"<p>BFS is typically implemented with the help of a queue, as shown in the code below. The queue has a \"first in, first out\" property, which aligns with the BFS idea of \"near to far\".</p> <ol> <li>Add the starting vertex <code>startVet</code> to the queue and begin the loop.</li> <li>In each iteration of the loop, pop the vertex at the front of the queue and record it as visited, then add all adjacent vertices of that vertex to the back of the queue.</li> <li>Repeat step <code>2.</code> until all vertices have been visited.</li> </ol> <p>To prevent revisiting vertices, we use a hash set <code>visited</code> to record which nodes have been visited.</p> <p>Tip</p> <p>A hash set can be viewed as a hash table that stores only <code>key</code> without storing <code>value</code>. It supports insertion, deletion, lookup, and update operations on <code>key</code> in \\(O(1)\\) time. Based on the uniqueness of <code>key</code>, hash sets are typically used for data deduplication and similar scenarios.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_bfs.py<pre><code>def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"Breadth-first traversal\"\"\"\n    # Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\n    # Vertex traversal sequence\n    res = []\n    # Hash set for recording vertices that have been visited\n    visited = set[Vertex]([start_vet])\n    # Queue used to implement BFS\n    que = deque[Vertex]([start_vet])\n    # Starting from vertex vet, loop until all vertices are visited\n    while len(que) &gt; 0:\n        vet = que.popleft()  # Dequeue the front vertex\n        res.append(vet)  # Record visited vertex\n        # Traverse all adjacent vertices of this vertex\n        for adj_vet in graph.adj_list[vet]:\n            if adj_vet in visited:\n                continue  # Skip vertices that have been visited\n            que.append(adj_vet)  # Only enqueue unvisited vertices\n            visited.add(adj_vet)  # Mark this vertex as visited\n    # Return vertex traversal sequence\n    return res\n</code></pre> graph_bfs.cpp<pre><code>/* Breadth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nvector&lt;Vertex *&gt; graphBFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // Vertex traversal sequence\n    vector&lt;Vertex *&gt; res;\n    // Hash set for recording vertices that have been visited\n    unordered_set&lt;Vertex *&gt; visited = {startVet};\n    // Queue used to implement BFS\n    queue&lt;Vertex *&gt; que;\n    que.push(startVet);\n    // Starting from vertex vet, loop until all vertices are visited\n    while (!que.empty()) {\n        Vertex *vet = que.front();\n        que.pop();          // Dequeue the front vertex\n        res.push_back(vet); // Record visited vertex\n        // Traverse all adjacent vertices of this vertex\n        for (auto adjVet : graph.adjList[vet]) {\n            if (visited.count(adjVet))\n                continue;            // Skip vertices that have been visited\n            que.push(adjVet);        // Only enqueue unvisited vertices\n            visited.emplace(adjVet); // Mark this vertex as visited\n        }\n    }\n    // Return vertex traversal sequence\n    return res;\n}\n</code></pre> graph_bfs.java<pre><code>/* Breadth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n    // Vertex traversal sequence\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // Hash set for recording vertices that have been visited\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    visited.add(startVet);\n    // Queue used to implement BFS\n    Queue&lt;Vertex&gt; que = new LinkedList&lt;&gt;();\n    que.offer(startVet);\n    // Starting from vertex vet, loop until all vertices are visited\n    while (!que.isEmpty()) {\n        Vertex vet = que.poll(); // Dequeue the front vertex\n        res.add(vet);            // Record visited vertex\n        // Traverse all adjacent vertices of this vertex\n        for (Vertex adjVet : graph.adjList.get(vet)) {\n            if (visited.contains(adjVet))\n                continue;        // Skip vertices that have been visited\n            que.offer(adjVet);   // Only enqueue unvisited vertices\n            visited.add(adjVet); // Mark this vertex as visited\n        }\n    }\n    // Return vertex traversal sequence\n    return res;\n}\n</code></pre> graph_bfs.cs<pre><code>/* Breadth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nList&lt;Vertex&gt; GraphBFS(GraphAdjList graph, Vertex startVet) {\n    // Vertex traversal sequence\n    List&lt;Vertex&gt; res = [];\n    // Hash set for recording vertices that have been visited\n    HashSet&lt;Vertex&gt; visited = [startVet];\n    // Queue used to implement BFS\n    Queue&lt;Vertex&gt; que = new();\n    que.Enqueue(startVet);\n    // Starting from vertex vet, loop until all vertices are visited\n    while (que.Count &gt; 0) {\n        Vertex vet = que.Dequeue(); // Dequeue the front vertex\n        res.Add(vet);               // Record visited vertex\n        foreach (Vertex adjVet in graph.adjList[vet]) {\n            if (visited.Contains(adjVet)) {\n                continue;          // Skip vertices that have been visited\n            }\n            que.Enqueue(adjVet);   // Only enqueue unvisited vertices\n            visited.Add(adjVet);   // Mark this vertex as visited\n        }\n    }\n\n    // Return vertex traversal sequence\n    return res;\n}\n</code></pre> graph_bfs.go<pre><code>/* Breadth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // Vertex traversal sequence\n    res := make([]Vertex, 0)\n    // Hash set for recording vertices that have been visited\n    visited := make(map[Vertex]struct{})\n    visited[startVet] = struct{}{}\n    // Queue used to implement BFS, using slice to simulate queue\n    queue := make([]Vertex, 0)\n    queue = append(queue, startVet)\n    // Starting from vertex vet, loop until all vertices are visited\n    for len(queue) &gt; 0 {\n        // Dequeue the front vertex\n        vet := queue[0]\n        queue = queue[1:]\n        // Record visited vertex\n        res = append(res, vet)\n        // Traverse all adjacent vertices of this vertex\n        for _, adjVet := range g.adjList[vet] {\n            _, isExist := visited[adjVet]\n            // Only enqueue unvisited vertices\n            if !isExist {\n                queue = append(queue, adjVet)\n                visited[adjVet] = struct{}{}\n            }\n        }\n    }\n    // Return vertex traversal sequence\n    return res\n}\n</code></pre> graph_bfs.swift<pre><code>/* Breadth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // Vertex traversal sequence\n    var res: [Vertex] = []\n    // Hash set for recording vertices that have been visited\n    var visited: Set&lt;Vertex&gt; = [startVet]\n    // Queue used to implement BFS\n    var que: [Vertex] = [startVet]\n    // Starting from vertex vet, loop until all vertices are visited\n    while !que.isEmpty {\n        let vet = que.removeFirst() // Dequeue the front vertex\n        res.append(vet) // Record visited vertex\n        // Traverse all adjacent vertices of this vertex\n        for adjVet in graph.adjList[vet] ?? [] {\n            if visited.contains(adjVet) {\n                continue // Skip vertices that have been visited\n            }\n            que.append(adjVet) // Only enqueue unvisited vertices\n            visited.insert(adjVet) // Mark this vertex as visited\n        }\n    }\n    // Return vertex traversal sequence\n    return res\n}\n</code></pre> graph_bfs.js<pre><code>/* Breadth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nfunction graphBFS(graph, startVet) {\n    // Vertex traversal sequence\n    const res = [];\n    // Hash set for recording vertices that have been visited\n    const visited = new Set();\n    visited.add(startVet);\n    // Queue used to implement BFS\n    const que = [startVet];\n    // Starting from vertex vet, loop until all vertices are visited\n    while (que.length) {\n        const vet = que.shift(); // Dequeue the front vertex\n        res.push(vet); // Record visited vertex\n        // Traverse all adjacent vertices of this vertex\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // Skip vertices that have been visited\n            }\n            que.push(adjVet); // Only enqueue unvisited vertices\n            visited.add(adjVet); // Mark this vertex as visited\n        }\n    }\n    // Return vertex traversal sequence\n    return res;\n}\n</code></pre> graph_bfs.ts<pre><code>/* Breadth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // Vertex traversal sequence\n    const res: Vertex[] = [];\n    // Hash set for recording vertices that have been visited\n    const visited: Set&lt;Vertex&gt; = new Set();\n    visited.add(startVet);\n    // Queue used to implement BFS\n    const que = [startVet];\n    // Starting from vertex vet, loop until all vertices are visited\n    while (que.length) {\n        const vet = que.shift(); // Dequeue the front vertex\n        res.push(vet); // Record visited vertex\n        // Traverse all adjacent vertices of this vertex\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // Skip vertices that have been visited\n            }\n            que.push(adjVet); // Only enqueue unvisited\n            visited.add(adjVet); // Mark this vertex as visited\n        }\n    }\n    // Return vertex traversal sequence\n    return res;\n}\n</code></pre> graph_bfs.dart<pre><code>/* Breadth-first traversal */\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n  // Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\n  // Vertex traversal sequence\n  List&lt;Vertex&gt; res = [];\n  // Hash set for recording vertices that have been visited\n  Set&lt;Vertex&gt; visited = {};\n  visited.add(startVet);\n  // Queue used to implement BFS\n  Queue&lt;Vertex&gt; que = Queue();\n  que.add(startVet);\n  // Starting from vertex vet, loop until all vertices are visited\n  while (que.isNotEmpty) {\n    Vertex vet = que.removeFirst(); // Dequeue the front vertex\n    res.add(vet); // Record visited vertex\n    // Traverse all adjacent vertices of this vertex\n    for (Vertex adjVet in graph.adjList[vet]!) {\n      if (visited.contains(adjVet)) {\n        continue; // Skip vertices that have been visited\n      }\n      que.add(adjVet); // Only enqueue unvisited vertices\n      visited.add(adjVet); // Mark this vertex as visited\n    }\n  }\n  // Return vertex traversal sequence\n  return res;\n}\n</code></pre> graph_bfs.rs<pre><code>/* Breadth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nfn graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // Vertex traversal sequence\n    let mut res = vec![];\n    // Hash set for recording vertices that have been visited\n    let mut visited = HashSet::new();\n    visited.insert(start_vet);\n    // Queue used to implement BFS\n    let mut que = VecDeque::new();\n    que.push_back(start_vet);\n    // Starting from vertex vet, loop until all vertices are visited\n    while let Some(vet) = que.pop_front() {\n        res.push(vet); // Record visited vertex\n\n        // Traverse all adjacent vertices of this vertex\n        if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n            for &amp;adj_vet in adj_vets {\n                if visited.contains(&amp;adj_vet) {\n                    continue; // Skip vertices that have been visited\n                }\n                que.push_back(adj_vet); // Only enqueue unvisited vertices\n                visited.insert(adj_vet); // Mark this vertex as visited\n            }\n        }\n    }\n    // Return vertex traversal sequence\n    res\n}\n</code></pre> graph_bfs.c<pre><code>/* Node queue structure */\ntypedef struct {\n    Vertex *vertices[MAX_SIZE];\n    int front, rear, size;\n} Queue;\n\n/* Constructor */\nQueue *newQueue() {\n    Queue *q = (Queue *)malloc(sizeof(Queue));\n    q-&gt;front = q-&gt;rear = q-&gt;size = 0;\n    return q;\n}\n\n/* Check if the queue is empty */\nint isEmpty(Queue *q) {\n    return q-&gt;size == 0;\n}\n\n/* Enqueue operation */\nvoid enqueue(Queue *q, Vertex *vet) {\n    q-&gt;vertices[q-&gt;rear] = vet;\n    q-&gt;rear = (q-&gt;rear + 1) % MAX_SIZE;\n    q-&gt;size++;\n}\n\n/* Dequeue operation */\nVertex *dequeue(Queue *q) {\n    Vertex *vet = q-&gt;vertices[q-&gt;front];\n    q-&gt;front = (q-&gt;front + 1) % MAX_SIZE;\n    q-&gt;size--;\n    return vet;\n}\n\n/* Check if vertex has been visited */\nint isVisited(Vertex **visited, int size, Vertex *vet) {\n    // Traverse to find node using O(n) time\n    for (int i = 0; i &lt; size; i++) {\n        if (visited[i] == vet)\n            return 1;\n    }\n    return 0;\n}\n\n/* Breadth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nvoid graphBFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize, Vertex **visited, int *visitedSize) {\n    // Queue used to implement BFS\n    Queue *queue = newQueue();\n    enqueue(queue, startVet);\n    visited[(*visitedSize)++] = startVet;\n    // Starting from vertex vet, loop until all vertices are visited\n    while (!isEmpty(queue)) {\n        Vertex *vet = dequeue(queue); // Dequeue the front vertex\n        res[(*resSize)++] = vet;      // Record visited vertex\n        // Traverse all adjacent vertices of this vertex\n        AdjListNode *node = findNode(graph, vet);\n        while (node != NULL) {\n            // Skip vertices that have been visited\n            if (!isVisited(visited, *visitedSize, node-&gt;vertex)) {\n                enqueue(queue, node-&gt;vertex);             // Only enqueue unvisited vertices\n                visited[(*visitedSize)++] = node-&gt;vertex; // Mark this vertex as visited\n            }\n            node = node-&gt;next;\n        }\n    }\n    // Free memory\n    free(queue);\n}\n</code></pre> graph_bfs.kt<pre><code>/* Breadth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nfun graphBFS(graph: GraphAdjList, startVet: Vertex): MutableList&lt;Vertex?&gt; {\n    // Vertex traversal sequence\n    val res = mutableListOf&lt;Vertex?&gt;()\n    // Hash set for recording vertices that have been visited\n    val visited = HashSet&lt;Vertex&gt;()\n    visited.add(startVet)\n    // Queue used to implement BFS\n    val que = LinkedList&lt;Vertex&gt;()\n    que.offer(startVet)\n    // Starting from vertex vet, loop until all vertices are visited\n    while (!que.isEmpty()) {\n        val vet = que.poll() // Dequeue the front vertex\n        res.add(vet)         // Record visited vertex\n        // Traverse all adjacent vertices of this vertex\n        for (adjVet in graph.adjList[vet]!!) {\n            if (visited.contains(adjVet))\n                continue        // Skip vertices that have been visited\n            que.offer(adjVet)   // Only enqueue unvisited vertices\n            visited.add(adjVet) // Mark this vertex as visited\n        }\n    }\n    // Return vertex traversal sequence\n    return res\n}\n</code></pre> graph_bfs.rb<pre><code>### Breadth-first traversal ###\ndef graph_bfs(graph, start_vet)\n  # Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\n  # Vertex traversal sequence\n  res = []\n  # Hash set for recording vertices that have been visited\n  visited = Set.new([start_vet])\n  # Queue used to implement BFS\n  que = [start_vet]\n  # Starting from vertex vet, loop until all vertices are visited\n  while que.length &gt; 0\n    vet = que.shift # Dequeue the front vertex\n    res &lt;&lt; vet # Record visited vertex\n    # Traverse all adjacent vertices of this vertex\n    for adj_vet in graph.adj_list[vet]\n      next if visited.include?(adj_vet) # Skip vertices that have been visited\n      que &lt;&lt; adj_vet # Only enqueue unvisited vertices\n      visited.add(adj_vet) # Mark this vertex as visited\n    end\n  end\n  # Return vertex traversal sequence\n  res\nend\n</code></pre> <p>The code is relatively abstract; it is recommended to refer to Figure 9-10 to deepen understanding.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-10   Steps of breadth-first search of a graph </p> <p>Is the breadth-first traversal sequence unique?</p> <p>Not unique. Breadth-first search only requires traversing in a \"near to far\" order, and the traversal order of vertices at the same distance can be arbitrarily shuffled. Taking Figure 9-10 as an example, the visit order of vertices \\(1\\) and \\(3\\) can be swapped, as can the visit order of vertices \\(2\\), \\(4\\), and \\(6\\).</p>","path":["Chapter 9. Graph","9.3   Graph Traversal"],"tags":[]},{"location":"chapter_graph/graph_traversal/#2-complexity-analysis","level":3,"title":"2.   Complexity Analysis","text":"<p>Time complexity: All vertices will be enqueued and dequeued once, using \\(O(|V|)\\) time; in the process of traversing adjacent vertices, since it is an undirected graph, all edges will be visited \\(2\\) times, using \\(O(2|E|)\\) time; overall using \\(O(|V| + |E|)\\) time.</p> <p>Space complexity: The list <code>res</code>, hash set <code>visited</code>, and queue <code>que</code> can contain at most \\(|V|\\) vertices, using \\(O(|V|)\\) space.</p>","path":["Chapter 9. Graph","9.3   Graph Traversal"],"tags":[]},{"location":"chapter_graph/graph_traversal/#932-depth-first-search","level":2,"title":"9.3.2   Depth-First Search","text":"<p>Depth-first search is a traversal method that prioritizes going as far as possible, then backtracks when no path remains. As shown in Figure 9-11, starting from the top-left vertex, visit an adjacent vertex of the current vertex, continuing until reaching a dead end, then return and continue going as far as possible before returning again, and so on, until all vertices have been traversed.</p> <p></p> <p> Figure 9-11   Depth-first search of a graph </p>","path":["Chapter 9. Graph","9.3   Graph Traversal"],"tags":[]},{"location":"chapter_graph/graph_traversal/#1-algorithm-implementation_1","level":3,"title":"1.   Algorithm Implementation","text":"<p>This \"go as far as possible then return\" algorithm paradigm is typically implemented using recursion. Similar to breadth-first search, in depth-first search we also need a hash set <code>visited</code> to record visited vertices and avoid revisiting.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_dfs.py<pre><code>def dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n    \"\"\"Depth-first traversal helper function\"\"\"\n    res.append(vet)  # Record visited vertex\n    visited.add(vet)  # Mark this vertex as visited\n    # Traverse all adjacent vertices of this vertex\n    for adjVet in graph.adj_list[vet]:\n        if adjVet in visited:\n            continue  # Skip vertices that have been visited\n        # Recursively visit adjacent vertices\n        dfs(graph, visited, res, adjVet)\n\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"Depth-first traversal\"\"\"\n    # Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\n    # Vertex traversal sequence\n    res = []\n    # Hash set for recording vertices that have been visited\n    visited = set[Vertex]()\n    dfs(graph, visited, res, start_vet)\n    return res\n</code></pre> graph_dfs.cpp<pre><code>/* Depth-first traversal helper function */\nvoid dfs(GraphAdjList &amp;graph, unordered_set&lt;Vertex *&gt; &amp;visited, vector&lt;Vertex *&gt; &amp;res, Vertex *vet) {\n    res.push_back(vet);   // Record visited vertex\n    visited.emplace(vet); // Mark this vertex as visited\n    // Traverse all adjacent vertices of this vertex\n    for (Vertex *adjVet : graph.adjList[vet]) {\n        if (visited.count(adjVet))\n            continue; // Skip vertices that have been visited\n        // Recursively visit adjacent vertices\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* Depth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nvector&lt;Vertex *&gt; graphDFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // Vertex traversal sequence\n    vector&lt;Vertex *&gt; res;\n    // Hash set for recording vertices that have been visited\n    unordered_set&lt;Vertex *&gt; visited;\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.java<pre><code>/* Depth-first traversal helper function */\nvoid dfs(GraphAdjList graph, Set&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.add(vet);     // Record visited vertex\n    visited.add(vet); // Mark this vertex as visited\n    // Traverse all adjacent vertices of this vertex\n    for (Vertex adjVet : graph.adjList.get(vet)) {\n        if (visited.contains(adjVet))\n            continue; // Skip vertices that have been visited\n        // Recursively visit adjacent vertices\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* Depth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n    // Vertex traversal sequence\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // Hash set for recording vertices that have been visited\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.cs<pre><code>/* Depth-first traversal helper function */\nvoid DFS(GraphAdjList graph, HashSet&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.Add(vet);     // Record visited vertex\n    visited.Add(vet); // Mark this vertex as visited\n    // Traverse all adjacent vertices of this vertex\n    foreach (Vertex adjVet in graph.adjList[vet]) {\n        if (visited.Contains(adjVet)) {\n            continue; // Skip vertices that have been visited\n        }\n        // Recursively visit adjacent vertices\n        DFS(graph, visited, res, adjVet);\n    }\n}\n\n/* Depth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nList&lt;Vertex&gt; GraphDFS(GraphAdjList graph, Vertex startVet) {\n    // Vertex traversal sequence\n    List&lt;Vertex&gt; res = [];\n    // Hash set for recording vertices that have been visited\n    HashSet&lt;Vertex&gt; visited = [];\n    DFS(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.go<pre><code>/* Depth-first traversal helper function */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n    // append operation returns a new reference, must reassign original reference to new slice's reference\n    *res = append(*res, vet)\n    visited[vet] = struct{}{}\n    // Traverse all adjacent vertices of this vertex\n    for _, adjVet := range g.adjList[vet] {\n        _, isExist := visited[adjVet]\n        // Recursively visit adjacent vertices\n        if !isExist {\n            dfs(g, visited, res, adjVet)\n        }\n    }\n}\n\n/* Depth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // Vertex traversal sequence\n    res := make([]Vertex, 0)\n    // Hash set for recording vertices that have been visited\n    visited := make(map[Vertex]struct{})\n    dfs(g, visited, &amp;res, startVet)\n    // Return vertex traversal sequence\n    return res\n}\n</code></pre> graph_dfs.swift<pre><code>/* Depth-first traversal helper function */\nfunc dfs(graph: GraphAdjList, visited: inout Set&lt;Vertex&gt;, res: inout [Vertex], vet: Vertex) {\n    res.append(vet) // Record visited vertex\n    visited.insert(vet) // Mark this vertex as visited\n    // Traverse all adjacent vertices of this vertex\n    for adjVet in graph.adjList[vet] ?? [] {\n        if visited.contains(adjVet) {\n            continue // Skip vertices that have been visited\n        }\n        // Recursively visit adjacent vertices\n        dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: adjVet)\n    }\n}\n\n/* Depth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // Vertex traversal sequence\n    var res: [Vertex] = []\n    // Hash set for recording vertices that have been visited\n    var visited: Set&lt;Vertex&gt; = []\n    dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: startVet)\n    return res\n}\n</code></pre> graph_dfs.js<pre><code>/* Depth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nfunction dfs(graph, visited, res, vet) {\n    res.push(vet); // Record visited vertex\n    visited.add(vet); // Mark this vertex as visited\n    // Traverse all adjacent vertices of this vertex\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // Skip vertices that have been visited\n        }\n        // Recursively visit adjacent vertices\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* Depth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nfunction graphDFS(graph, startVet) {\n    // Vertex traversal sequence\n    const res = [];\n    // Hash set for recording vertices that have been visited\n    const visited = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.ts<pre><code>/* Depth-first traversal helper function */\nfunction dfs(\n    graph: GraphAdjList,\n    visited: Set&lt;Vertex&gt;,\n    res: Vertex[],\n    vet: Vertex\n): void {\n    res.push(vet); // Record visited vertex\n    visited.add(vet); // Mark this vertex as visited\n    // Traverse all adjacent vertices of this vertex\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // Skip vertices that have been visited\n        }\n        // Recursively visit adjacent vertices\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* Depth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // Vertex traversal sequence\n    const res: Vertex[] = [];\n    // Hash set for recording vertices that have been visited\n    const visited: Set&lt;Vertex&gt; = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.dart<pre><code>/* Depth-first traversal helper function */\nvoid dfs(\n  GraphAdjList graph,\n  Set&lt;Vertex&gt; visited,\n  List&lt;Vertex&gt; res,\n  Vertex vet,\n) {\n  res.add(vet); // Record visited vertex\n  visited.add(vet); // Mark this vertex as visited\n  // Traverse all adjacent vertices of this vertex\n  for (Vertex adjVet in graph.adjList[vet]!) {\n    if (visited.contains(adjVet)) {\n      continue; // Skip vertices that have been visited\n    }\n    // Recursively visit adjacent vertices\n    dfs(graph, visited, res, adjVet);\n  }\n}\n\n/* Depth-first traversal */\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n  // Vertex traversal sequence\n  List&lt;Vertex&gt; res = [];\n  // Hash set for recording vertices that have been visited\n  Set&lt;Vertex&gt; visited = {};\n  dfs(graph, visited, res, startVet);\n  return res;\n}\n</code></pre> graph_dfs.rs<pre><code>/* Depth-first traversal helper function */\nfn dfs(graph: &amp;GraphAdjList, visited: &amp;mut HashSet&lt;Vertex&gt;, res: &amp;mut Vec&lt;Vertex&gt;, vet: Vertex) {\n    res.push(vet); // Record visited vertex\n    visited.insert(vet); // Mark this vertex as visited\n                         // Traverse all adjacent vertices of this vertex\n    if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n        for &amp;adj_vet in adj_vets {\n            if visited.contains(&amp;adj_vet) {\n                continue; // Skip vertices that have been visited\n            }\n            // Recursively visit adjacent vertices\n            dfs(graph, visited, res, adj_vet);\n        }\n    }\n}\n\n/* Depth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nfn graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // Vertex traversal sequence\n    let mut res = vec![];\n    // Hash set for recording vertices that have been visited\n    let mut visited = HashSet::new();\n    dfs(&amp;graph, &amp;mut visited, &amp;mut res, start_vet);\n\n    res\n}\n</code></pre> graph_dfs.c<pre><code>/* Check if vertex has been visited */\nint isVisited(Vertex **res, int size, Vertex *vet) {\n    // Traverse to find node using O(n) time\n    for (int i = 0; i &lt; size; i++) {\n        if (res[i] == vet) {\n            return 1;\n        }\n    }\n    return 0;\n}\n\n/* Depth-first traversal helper function */\nvoid dfs(GraphAdjList *graph, Vertex **res, int *resSize, Vertex *vet) {\n    // Record visited vertex\n    res[(*resSize)++] = vet;\n    // Traverse all adjacent vertices of this vertex\n    AdjListNode *node = findNode(graph, vet);\n    while (node != NULL) {\n        // Skip vertices that have been visited\n        if (!isVisited(res, *resSize, node-&gt;vertex)) {\n            // Recursively visit adjacent vertices\n            dfs(graph, res, resSize, node-&gt;vertex);\n        }\n        node = node-&gt;next;\n    }\n}\n\n/* Depth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nvoid graphDFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize) {\n    dfs(graph, res, resSize, startVet);\n}\n</code></pre> graph_dfs.kt<pre><code>/* Depth-first traversal helper function */\nfun dfs(\n    graph: GraphAdjList,\n    visited: MutableSet&lt;Vertex?&gt;,\n    res: MutableList&lt;Vertex?&gt;,\n    vet: Vertex?\n) {\n    res.add(vet)     // Record visited vertex\n    visited.add(vet) // Mark this vertex as visited\n    // Traverse all adjacent vertices of this vertex\n    for (adjVet in graph.adjList[vet]!!) {\n        if (visited.contains(adjVet))\n            continue  // Skip vertices that have been visited\n        // Recursively visit adjacent vertices\n        dfs(graph, visited, res, adjVet)\n    }\n}\n\n/* Depth-first traversal */\n// Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\nfun graphDFS(graph: GraphAdjList, startVet: Vertex?): MutableList&lt;Vertex?&gt; {\n    // Vertex traversal sequence\n    val res = mutableListOf&lt;Vertex?&gt;()\n    // Hash set for recording vertices that have been visited\n    val visited = HashSet&lt;Vertex?&gt;()\n    dfs(graph, visited, res, startVet)\n    return res\n}\n</code></pre> graph_dfs.rb<pre><code>### Depth-first traversal helper function ###\ndef dfs(graph, visited, res, vet)\n  res &lt;&lt; vet # Record visited vertex\n  visited.add(vet) # Mark this vertex as visited\n  # Traverse all adjacent vertices of this vertex\n  for adj_vet in graph.adj_list[vet]\n    next if visited.include?(adj_vet) # Skip vertices that have been visited\n    # Recursively visit adjacent vertices\n    dfs(graph, visited, res, adj_vet)\n  end\nend\n\n### Depth-first traversal ###\ndef graph_dfs(graph, start_vet)\n  # Use adjacency list to represent the graph, in order to obtain all adjacent vertices of a specified vertex\n  # Vertex traversal sequence\n  res = []\n  # Hash set for recording vertices that have been visited\n  visited = Set.new\n  dfs(graph, visited, res, start_vet)\n  res\nend\n</code></pre> <p>The algorithm flow of depth-first search is shown in Figure 9-12.</p> <ul> <li>Straight dashed lines represent downward recursion, indicating that a new recursive method has been initiated to visit a new vertex.</li> <li>Curved dashed lines represent upward backtracking, indicating that this recursive call has returned to the point where it was made.</li> </ul> <p>To deepen understanding, it is recommended to combine Figure 9-12 with the code to mentally simulate (or draw out) the entire DFS process, including when each recursive call begins and when it returns.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-12   Steps of depth-first search of a graph </p> <p>Is the depth-first traversal sequence unique?</p> <p>Similar to breadth-first search, depth-first traversal sequences are also not unique. Given a vertex, any exploration direction may be chosen first; that is, the order of adjacent vertices can be arbitrarily rearranged and still constitute depth-first search.</p> <p>Taking tree traversal as an example, \"root \\(\\rightarrow\\) left \\(\\rightarrow\\) right\", \"left \\(\\rightarrow\\) root \\(\\rightarrow\\) right\", and \"left \\(\\rightarrow\\) right \\(\\rightarrow\\) root\" correspond to pre-order, in-order, and post-order traversals, respectively. They represent three different traversal priorities, yet all three belong to depth-first search.</p>","path":["Chapter 9. Graph","9.3   Graph Traversal"],"tags":[]},{"location":"chapter_graph/graph_traversal/#2-complexity-analysis_1","level":3,"title":"2.   Complexity Analysis","text":"<p>Time complexity: All vertices will be visited \\(1\\) time, using \\(O(|V|)\\) time; all edges will be visited \\(2\\) times, using \\(O(2|E|)\\) time; overall using \\(O(|V| + |E|)\\) time.</p> <p>Space complexity: The list <code>res</code> and hash set <code>visited</code> can contain at most \\(|V|\\) vertices, and the maximum recursion depth is \\(|V|\\), therefore using \\(O(|V|)\\) space.</p>","path":["Chapter 9. Graph","9.3   Graph Traversal"],"tags":[]},{"location":"chapter_graph/summary/","level":1,"title":"9.4   Summary","text":"","path":["Chapter 9. Graph","9.4   Summary"],"tags":[]},{"location":"chapter_graph/summary/#1-key-review","level":3,"title":"1.   Key Review","text":"<ul> <li>Graphs consist of vertices and edges and can be represented as a set of vertices and a set of edges.</li> <li>Compared with the linear relationships modeled by linked lists and the divide-and-conquer relationships modeled by trees, the network relationships modeled by graphs offer much greater flexibility and are therefore more complex.</li> <li>In directed graphs, edges have direction; in connected graphs, every vertex is reachable from any other vertex; and in weighted graphs, each edge carries a weight.</li> <li>Adjacency matrices use matrices to represent graphs, where each row (column) represents a vertex, and matrix elements represent edges, using \\(1\\) or \\(0\\) to indicate whether two vertices have an edge or not. Adjacency matrices are highly efficient for addition, deletion, lookup, and modification operations, but consume significant space.</li> <li>Adjacency lists use multiple linked lists to represent a graph: the \\(i\\)-th linked list corresponds to vertex \\(i\\) and stores all vertices adjacent to it. Compared with adjacency matrices, adjacency lists use less space, but edge lookups are less efficient because the linked list must be traversed.</li> <li>When linked lists in adjacency lists become too long, they can be converted to red-black trees or hash tables, thereby improving lookup efficiency.</li> <li>From an algorithmic perspective, adjacency matrices embody \"trading space for time\", while adjacency lists embody \"trading time for space\".</li> <li>Graphs can be used to model various real-world systems, such as social networks and subway lines.</li> <li>Trees are a special case of graphs, and tree traversal is a special case of graph traversal.</li> <li>Breadth-first search in graphs explores from near to far, expanding layer by layer, and is typically implemented with a queue.</li> <li>Depth-first search in graphs follows a path as deep as possible and backtracks when it can go no farther, and is commonly implemented with recursion.</li> </ul>","path":["Chapter 9. Graph","9.4   Summary"],"tags":[]},{"location":"chapter_graph/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: Is a path defined as a sequence of vertices or a sequence of edges?</p> <p>The definitions in different language versions of Wikipedia are inconsistent: the English version states \"a path is a sequence of edges\", while the Chinese version states \"a path is a sequence of vertices\". The following is the original English text: In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.</p> <p>In this text, a path is viewed as a sequence of edges, not a sequence of vertices. This is because there may be multiple edges connecting two vertices, in which case each edge corresponds to a path.</p> <p>Q: In a disconnected graph, will there be unreachable vertices?</p> <p>In a disconnected graph, if you start from one vertex, at least one other vertex will be unreachable. To traverse a disconnected graph, you need multiple starting points so that all connected components are covered.</p> <p>Q: In an adjacency list, is there any required ordering for the vertices adjacent to a given vertex?</p> <p>They can appear in any order. In practice, however, they may need to be sorted according to specific rules, such as the order in which vertices were added or the order of vertex values, which helps when quickly finding a vertex with some extreme value.</p>","path":["Chapter 9. Graph","9.4   Summary"],"tags":[]},{"location":"chapter_greedy/","level":1,"title":"Chapter 15.   Greedy","text":"<p>Abstract</p> <p>Sunflowers turn toward the sun, always seeking the fullest growth possible.</p> <p>Through successive simple choices, greedy strategies gradually lead to the optimal solution.</p>","path":["Chapter 15. Greedy","Chapter 15.   Greedy"],"tags":[]},{"location":"chapter_greedy/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>15.1   Greedy Algorithm</li> <li>15.2   Fractional Knapsack Problem</li> <li>15.3   Maximum Capacity Problem</li> <li>15.4   Maximum Product Cutting Problem</li> <li>15.5   Summary</li> <li>15.6   Exercises</li> </ul>","path":["Chapter 15. Greedy","Chapter 15.   Greedy"],"tags":[]},{"location":"chapter_greedy/exercises/","level":1,"title":"15.6   Exercises","text":"","path":["Chapter 15. Greedy","15.6   Exercises"],"tags":[]},{"location":"chapter_greedy/exercises/#1561-concept-review","level":2,"title":"15.6.1   Concept Review","text":"","path":["Chapter 15. Greedy","15.6   Exercises"],"tags":[]},{"location":"chapter_greedy/exercises/#1-is-choosing-the-largest-coin-always-best","level":3,"title":"1.   Is Choosing the Largest Coin Always Best?","text":"<p>The coin denominations are <code>[1, 7, 10]</code>, and the target amount is 14.</p> <ol> <li>Follow the rule \"always choose the largest denomination that does not exceed the remaining amount,\" and write the coins selected.</li> <li>Is there a solution using fewer coins? If so, give one; otherwise, explain why not.</li> <li>Does this example show that the greedy strategy is correct for every set of coin denominations?</li> </ol> Answer <ol> <li> <p>The greedy strategy selects <code>10 + 1 + 1 + 1 + 1</code>, using 5 coins in total.</p> </li> <li> <p>There is a solution using fewer coins: <code>7 + 7</code>, which uses only 2 coins.</p> </li> <li> <p>No. This counterexample shows that, for arbitrary coin denominations, repeatedly choosing the largest currently available denomination does not necessarily minimize the number of coins.     The largest immediate choice may prevent a better combination later.</p> </li> </ol>","path":["Chapter 15. Greedy","15.6   Exercises"],"tags":[]},{"location":"chapter_greedy/exercises/#2-which-item-should-go-into-the-knapsack-first","level":3,"title":"2.   Which Item Should Go into the Knapsack First?","text":"<p>A knapsack with a capacity of 4 kilograms can hold the following items. A fraction of an item may be taken, and the value obtained is proportional to its weight:</p> <ul> <li>Item A: weight 4 kilograms, value 20.</li> <li>Item B: weight 3 kilograms, value 18.</li> </ul> <ol> <li>What is the value per kilogram of each item? Which item should be placed in the knapsack first?</li> <li>Fill the knapsack using the greedy strategy for the fractional knapsack problem. What is the final value?</li> <li>When items can be divided and the knapsack limits total weight, should items be compared by total value or by value per kilogram? Why?</li> </ol> Answer <ol> <li> <p>A has value <code>20 ÷ 4 = 5</code> per kilogram, while B has value <code>18 ÷ 3 = 6</code> per kilogram,     so B, with the higher value per unit weight, should be placed in the knapsack first.</p> </li> <li> <p>First take all of B, using 3 kilograms of capacity and gaining a value of 18. With 1 kilogram of capacity remaining,     take 1 kilogram of A, gaining a value of 5. The final value is <code>18 + 5 = 23</code>.</p> </li> <li> <p>The knapsack limits total weight, and items can be divided, so they should be compared by value per unit weight.     Although A has a higher total value, its value per kilogram is lower than B's. Filling the knapsack with A first would yield only a value of 20.</p> </li> </ol>","path":["Chapter 15. Greedy","15.6   Exercises"],"tags":[]},{"location":"chapter_greedy/exercises/#3-which-pointer-should-move-next","level":3,"title":"3.   Which Pointer Should Move Next?","text":"<p>The partition heights are <code>[1, 8, 6, 2, 5]</code>. Use two pointers, one at each end, to find the maximum capacity. The capacity equals \"the height of the shorter partition × the distance between the partitions' indices.\"</p> <ol> <li>Initially, the left pointer is at index 0 and the right pointer is at index 4. What is the current capacity? Which pointer should move next?</li> <li>After making the move chosen in Question 1, at which indices are the two pointers? What is the capacity now? Which pointer should move next?</li> <li>For the current pair of partitions, you could move either the pointer at the shorter partition or the pointer at the taller partition. Which move could still produce a greater capacity, and why?</li> </ol> Answer <ol> <li> <p>The current capacity is <code>min(1, 5) × (4 - 0) = 4</code>. The left partition is shorter, so move the left pointer.</p> </li> <li> <p>After the left pointer moves, the pointers are at indices 1 and 4. The current capacity is     <code>min(8, 5) × (4 - 1) = 15</code>. The right partition is shorter, so move the right pointer next.</p> </li> <li> <p>Moving the pointer at the shorter partition is the only move that could still produce a greater capacity. If the taller partition's pointer moves, the distance certainly decreases while the height remains limited by the unmoved shorter partition,     so the capacity can only stay the same or decrease. Only by moving the shorter partition can a taller partition possibly be found.</p> </li> </ol>","path":["Chapter 15. Greedy","15.6   Exercises"],"tags":[]},{"location":"chapter_greedy/exercises/#1562-programming-exercises","level":2,"title":"15.6.2   Programming Exercises","text":"","path":["Chapter 15. Greedy","15.6   Exercises"],"tags":[]},{"location":"chapter_greedy/exercises/#1-fractional-knapsack","level":3,"title":"1.   Fractional Knapsack","text":"<p>You are given equal-length arrays <code>wgt</code> and <code>val</code>, where <code>wgt[i] &gt; 0</code> and <code>val[i] &gt;= 0</code>. The knapsack has capacity <code>cap &gt;= 0</code>. There is only one of each item, but any fraction of an item may be placed in the knapsack. The value obtained is proportional to the fraction of the item's total weight that is included. Use a greedy algorithm and return the maximum total value the knapsack can hold as a real number.</p> Hints <ol> <li>First calculate each item's value per unit weight as val[i] / wgt[i], keeping the fractional part of the division</li> <li>Place items with higher value per unit weight into the knapsack first</li> <li>If the remaining capacity is less than the current item's weight, take exactly the fraction that fills the knapsack and stop</li> </ol>","path":["Chapter 15. Greedy","15.6   Exercises"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/","level":1,"title":"15.2   Fractional Knapsack Problem","text":"<p>Question</p> <p>Given \\(n\\) items, where the weight of the \\(i\\)-th item is \\(wgt[i-1]\\) and its value is \\(val[i-1]\\), and a knapsack with capacity \\(cap\\). Each item can be selected only once, but a fraction of an item may be selected, with its value proportional to the selected weight. What is the maximum total value that can be placed in the knapsack under the capacity constraint? An example is shown in Figure 15-3.</p> <p></p> <p> Figure 15-3   Example data for the fractional knapsack problem </p> <p>The fractional knapsack problem is very similar overall to the 0-1 knapsack problem, with states including the current item \\(i\\) and capacity \\(c\\), and the goal being to maximize value under the limited knapsack capacity.</p> <p>The difference is that this problem allows selecting only a fraction of an item. As shown in Figure 15-4, we can split an item arbitrarily and compute its value in proportion to the selected weight.</p> <ol> <li>For item \\(i\\), its value per unit weight is \\(val[i-1] / wgt[i-1]\\), referred to as unit value.</li> <li>Suppose we put a portion of item \\(i\\) with weight \\(w\\) into the knapsack, then the value added to the knapsack is \\(w \\times val[i-1] / wgt[i-1]\\).</li> </ol> <p></p> <p> Figure 15-4   Value of items per unit weight </p>","path":["Chapter 15. Greedy","15.2   Fractional Knapsack Problem"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/#1-greedy-strategy-determination","level":3,"title":"1.   Greedy Strategy Determination","text":"<p>Maximizing the total value in the knapsack essentially means prioritizing items with higher value per unit weight. From this observation, we can derive the greedy strategy shown in Figure 15-5.</p> <ol> <li>Sort items by unit value from high to low.</li> <li>Iterate through all items, greedily selecting the item with the highest unit value in each round.</li> <li>If the remaining knapsack capacity is insufficient, use a portion of the current item to fill the knapsack.</li> </ol> <p></p> <p> Figure 15-5   Greedy strategy for the fractional knapsack problem </p>","path":["Chapter 15. Greedy","15.2   Fractional Knapsack Problem"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/#2-code-implementation","level":3,"title":"2.   Code Implementation","text":"<p>We define an <code>Item</code> class so that items can be sorted by unit value. We then iterate through the sorted items greedily, stopping once the knapsack is full and returning the result:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby fractional_knapsack.py<pre><code>class Item:\n    \"\"\"Item\"\"\"\n\n    def __init__(self, w: int, v: int):\n        self.w = w  # Item weight\n        self.v = v  # Item value\n\ndef fractional_knapsack(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"Fractional knapsack: Greedy algorithm\"\"\"\n    # Create item list with two attributes: weight, value\n    items = [Item(w, v) for w, v in zip(wgt, val)]\n    # Sort by unit value item.v / item.w from high to low\n    items.sort(key=lambda item: item.v / item.w, reverse=True)\n    # Loop for greedy selection\n    res = 0\n    for item in items:\n        if item.w &lt;= cap:\n            # If remaining capacity is sufficient, put the entire current item into the knapsack\n            res += item.v\n            cap -= item.w\n        else:\n            # If remaining capacity is insufficient, put part of the current item into the knapsack\n            res += (item.v / item.w) * cap\n            # No remaining capacity, so break out of the loop\n            break\n    return res\n</code></pre> fractional_knapsack.cpp<pre><code>/* Item */\nclass Item {\n  public:\n    int w; // Item weight\n    int v; // Item value\n\n    Item(int w, int v) : w(w), v(v) {\n    }\n};\n\n/* Fractional knapsack: Greedy algorithm */\ndouble fractionalKnapsack(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    // Create item list with two attributes: weight, value\n    vector&lt;Item&gt; items;\n    for (int i = 0; i &lt; wgt.size(); i++) {\n        items.push_back(Item(wgt[i], val[i]));\n    }\n    // Sort by unit value item.v / item.w from high to low\n    sort(items.begin(), items.end(), [](Item &amp;a, Item &amp;b) { return (double)a.v / a.w &gt; (double)b.v / b.w; });\n    // Loop for greedy selection\n    double res = 0;\n    for (auto &amp;item : items) {\n        if (item.w &lt;= cap) {\n            // If remaining capacity is sufficient, put the entire current item into the knapsack\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // If remaining capacity is insufficient, put part of the current item into the knapsack\n            res += (double)item.v / item.w * cap;\n            // No remaining capacity, so break out of the loop\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.java<pre><code>/* Item */\nclass Item {\n    int w; // Item weight\n    int v; // Item value\n\n    public Item(int w, int v) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* Fractional knapsack: Greedy algorithm */\ndouble fractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // Create item list with two attributes: weight, value\n    Item[] items = new Item[wgt.length];\n    for (int i = 0; i &lt; wgt.length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // Sort by unit value item.v / item.w from high to low\n    Arrays.sort(items, Comparator.comparingDouble(item -&gt; -((double) item.v / item.w)));\n    // Loop for greedy selection\n    double res = 0;\n    for (Item item : items) {\n        if (item.w &lt;= cap) {\n            // If remaining capacity is sufficient, put the entire current item into the knapsack\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // If remaining capacity is insufficient, put part of the current item into the knapsack\n            res += (double) item.v / item.w * cap;\n            // No remaining capacity, so break out of the loop\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.cs<pre><code>/* Item */\nclass Item(int w, int v) {\n    public int w = w; // Item weight\n    public int v = v; // Item value\n}\n\n/* Fractional knapsack: Greedy algorithm */\ndouble FractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // Create item list with two attributes: weight, value\n    Item[] items = new Item[wgt.Length];\n    for (int i = 0; i &lt; wgt.Length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // Sort by unit value item.v / item.w from high to low\n    Array.Sort(items, (x, y) =&gt; (y.v / y.w).CompareTo(x.v / x.w));\n    // Loop for greedy selection\n    double res = 0;\n    foreach (Item item in items) {\n        if (item.w &lt;= cap) {\n            // If remaining capacity is sufficient, put the entire current item into the knapsack\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // If remaining capacity is insufficient, put part of the current item into the knapsack\n            res += (double)item.v / item.w * cap;\n            // No remaining capacity, so break out of the loop\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.go<pre><code>/* Item */\ntype Item struct {\n    w int // Item weight\n    v int // Item value\n}\n\n/* Fractional knapsack: Greedy algorithm */\nfunc fractionalKnapsack(wgt []int, val []int, cap int) float64 {\n    // Create item list with two attributes: weight, value\n    items := make([]Item, len(wgt))\n    for i := 0; i &lt; len(wgt); i++ {\n        items[i] = Item{wgt[i], val[i]}\n    }\n    // Sort by unit value item.v / item.w from high to low\n    sort.Slice(items, func(i, j int) bool {\n        return float64(items[i].v)/float64(items[i].w) &gt; float64(items[j].v)/float64(items[j].w)\n    })\n    // Loop for greedy selection\n    res := 0.0\n    for _, item := range items {\n        if item.w &lt;= cap {\n            // If remaining capacity is sufficient, put the entire current item into the knapsack\n            res += float64(item.v)\n            cap -= item.w\n        } else {\n            // If remaining capacity is insufficient, put part of the current item into the knapsack\n            res += float64(item.v) / float64(item.w) * float64(cap)\n            // No remaining capacity, so break out of the loop\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.swift<pre><code>/* Item */\nclass Item {\n    var w: Int // Item weight\n    var v: Int // Item value\n\n    init(w: Int, v: Int) {\n        self.w = w\n        self.v = v\n    }\n}\n\n/* Fractional knapsack: Greedy algorithm */\nfunc fractionalKnapsack(wgt: [Int], val: [Int], cap: Int) -&gt; Double {\n    // Create item list with two attributes: weight, value\n    var items = zip(wgt, val).map { Item(w: $0, v: $1) }\n    // Sort by unit value item.v / item.w from high to low\n    items.sort { -(Double($0.v) / Double($0.w)) &lt; -(Double($1.v) / Double($1.w)) }\n    // Loop for greedy selection\n    var res = 0.0\n    var cap = cap\n    for item in items {\n        if item.w &lt;= cap {\n            // If remaining capacity is sufficient, put the entire current item into the knapsack\n            res += Double(item.v)\n            cap -= item.w\n        } else {\n            // If remaining capacity is insufficient, put part of the current item into the knapsack\n            res += Double(item.v) / Double(item.w) * Double(cap)\n            // No remaining capacity, so break out of the loop\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.js<pre><code>/* Item */\nclass Item {\n    constructor(w, v) {\n        this.w = w; // Item weight\n        this.v = v; // Item value\n    }\n}\n\n/* Fractional knapsack: Greedy algorithm */\nfunction fractionalKnapsack(wgt, val, cap) {\n    // Create item list with two attributes: weight, value\n    const items = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // Sort by unit value item.v / item.w from high to low\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // Loop for greedy selection\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // If remaining capacity is sufficient, put the entire current item into the knapsack\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // If remaining capacity is insufficient, put part of the current item into the knapsack\n            res += (item.v / item.w) * cap;\n            // No remaining capacity, so break out of the loop\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.ts<pre><code>/* Item */\nclass Item {\n    w: number; // Item weight\n    v: number; // Item value\n\n    constructor(w: number, v: number) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* Fractional knapsack: Greedy algorithm */\nfunction fractionalKnapsack(wgt: number[], val: number[], cap: number): number {\n    // Create item list with two attributes: weight, value\n    const items: Item[] = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // Sort by unit value item.v / item.w from high to low\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // Loop for greedy selection\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // If remaining capacity is sufficient, put the entire current item into the knapsack\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // If remaining capacity is insufficient, put part of the current item into the knapsack\n            res += (item.v / item.w) * cap;\n            // No remaining capacity, so break out of the loop\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.dart<pre><code>/* Item */\nclass Item {\n  int w; // Item weight\n  int v; // Item value\n\n  Item(this.w, this.v);\n}\n\n/* Fractional knapsack: Greedy algorithm */\ndouble fractionalKnapsack(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  // Create item list with two attributes: weight, value\n  List&lt;Item&gt; items = List.generate(wgt.length, (i) =&gt; Item(wgt[i], val[i]));\n  // Sort by unit value item.v / item.w from high to low\n  items.sort((a, b) =&gt; (b.v / b.w).compareTo(a.v / a.w));\n  // Loop for greedy selection\n  double res = 0;\n  for (Item item in items) {\n    if (item.w &lt;= cap) {\n      // If remaining capacity is sufficient, put the entire current item into the knapsack\n      res += item.v;\n      cap -= item.w;\n    } else {\n      // If remaining capacity is insufficient, put part of the current item into the knapsack\n      res += item.v / item.w * cap;\n      // No remaining capacity, so break out of the loop\n      break;\n    }\n  }\n  return res;\n}\n</code></pre> fractional_knapsack.rs<pre><code>/* Item */\nstruct Item {\n    w: i32, // Item weight\n    v: i32, // Item value\n}\n\nimpl Item {\n    fn new(w: i32, v: i32) -&gt; Self {\n        Self { w, v }\n    }\n}\n\n/* Fractional knapsack: Greedy algorithm */\nfn fractional_knapsack(wgt: &amp;[i32], val: &amp;[i32], mut cap: i32) -&gt; f64 {\n    // Create item list with two attributes: weight, value\n    let mut items = wgt\n        .iter()\n        .zip(val.iter())\n        .map(|(&amp;w, &amp;v)| Item::new(w, v))\n        .collect::&lt;Vec&lt;Item&gt;&gt;();\n    // Sort by unit value item.v / item.w from high to low\n    items.sort_by(|a, b| {\n        (b.v as f64 / b.w as f64)\n            .partial_cmp(&amp;(a.v as f64 / a.w as f64))\n            .unwrap()\n    });\n    // Loop for greedy selection\n    let mut res = 0.0;\n    for item in &amp;items {\n        if item.w &lt;= cap {\n            // If remaining capacity is sufficient, put the entire current item into the knapsack\n            res += item.v as f64;\n            cap -= item.w;\n        } else {\n            // If remaining capacity is insufficient, put part of the current item into the knapsack\n            res += item.v as f64 / item.w as f64 * cap as f64;\n            // No remaining capacity, so break out of the loop\n            break;\n        }\n    }\n    res\n}\n</code></pre> fractional_knapsack.c<pre><code>/* Item */\ntypedef struct {\n    int w; // Item weight\n    int v; // Item value\n} Item;\n\n/* Fractional knapsack: Greedy algorithm */\nfloat fractionalKnapsack(int wgt[], int val[], int itemCount, int cap) {\n    // Create item list with two attributes: weight, value\n    Item *items = malloc(sizeof(Item) * itemCount);\n    for (int i = 0; i &lt; itemCount; i++) {\n        items[i] = (Item){.w = wgt[i], .v = val[i]};\n    }\n    // Sort by unit value item.v / item.w from high to low\n    qsort(items, (size_t)itemCount, sizeof(Item), sortByValueDensity);\n    // Loop for greedy selection\n    float res = 0.0;\n    for (int i = 0; i &lt; itemCount; i++) {\n        if (items[i].w &lt;= cap) {\n            // If remaining capacity is sufficient, put the entire current item into the knapsack\n            res += items[i].v;\n            cap -= items[i].w;\n        } else {\n            // If remaining capacity is insufficient, put part of the current item into the knapsack\n            res += (float)cap / items[i].w * items[i].v;\n            cap = 0;\n            break;\n        }\n    }\n    free(items);\n    return res;\n}\n</code></pre> fractional_knapsack.kt<pre><code>/* Item */\nclass Item(\n    val w: Int, // Item\n    val v: Int  // Item value\n)\n\n/* Fractional knapsack: Greedy algorithm */\nfun fractionalKnapsack(wgt: IntArray, _val: IntArray, c: Int): Double {\n    // Create item list with two attributes: weight, value\n    var cap = c\n    val items = arrayOfNulls&lt;Item&gt;(wgt.size)\n    for (i in wgt.indices) {\n        items[i] = Item(wgt[i], _val[i])\n    }\n    // Sort by unit value item.v / item.w from high to low\n    items.sortBy { item: Item? -&gt; -(item!!.v.toDouble() / item.w) }\n    // Loop for greedy selection\n    var res = 0.0\n    for (item in items) {\n        if (item!!.w &lt;= cap) {\n            // If remaining capacity is sufficient, put the entire current item into the knapsack\n            res += item.v\n            cap -= item.w\n        } else {\n            // If remaining capacity is insufficient, put part of the current item into the knapsack\n            res += item.v.toDouble() / item.w * cap\n            // No remaining capacity, so break out of the loop\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.rb<pre><code>### Item ###\nclass Item\n  attr_accessor :w # Item weight\n  attr_accessor :v # Item value\n\n  def initialize(w, v)\n    @w = w\n    @v = v\n  end\nend\n\n### Fractional knapsack: greedy ###\ndef fractional_knapsack(wgt, val, cap)\n  # Create item list with two attributes: weight, value\n  items = wgt.each_with_index.map { |w, i| Item.new(w, val[i]) }\n  # Sort by unit value item.v / item.w from high to low\n  items.sort! { |a, b| (b.v.to_f / b.w) &lt;=&gt; (a.v.to_f / a.w) }\n  # Loop for greedy selection\n  res = 0\n  for item in items\n    if item.w &lt;= cap\n      # If remaining capacity is sufficient, put the entire current item into the knapsack\n      res += item.v\n      cap -= item.w\n    else\n      # If remaining capacity is insufficient, put part of the current item into the knapsack\n      res += (item.v.to_f / item.w) * cap\n      # No remaining capacity, so break out of the loop\n      break\n    end\n  end\n  res\nend\n</code></pre> <p>Built-in sorting algorithms usually take \\(O(n \\log n)\\) time, and their space complexity is usually \\(O(\\log n)\\) or \\(O(n)\\), depending on the specific implementation of the programming language.</p> <p>Apart from sorting, in the worst case the entire item list needs to be traversed, therefore the time complexity is \\(O(n)\\), where \\(n\\) is the number of items.</p> <p>Since an <code>Item</code> object list is initialized, the space complexity is \\(O(n)\\).</p>","path":["Chapter 15. Greedy","15.2   Fractional Knapsack Problem"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/#3-correctness-proof","level":3,"title":"3.   Correctness Proof","text":"<p>We use proof by contradiction. Suppose item \\(x\\) has the highest unit value, and some algorithm produces an optimal value <code>res</code>, but the resulting solution does not include item \\(x\\).</p> <p>Now remove one unit of weight from any item in the knapsack and replace it with one unit of weight from item \\(x\\). Since item \\(x\\) has the highest unit value, the total value after the replacement must be greater than <code>res</code>. This contradicts the assumption that <code>res</code> is optimal, proving that any optimal solution must include item \\(x\\).</p> <p>We can construct the same contradiction for the other items in the solution as well. In summary, items with higher unit value are always the better choice, which proves that the greedy strategy is effective.</p> <p>As shown in Figure 15-6, if we treat item weight and unit value as the horizontal and vertical axes of a two-dimensional chart, then the fractional knapsack problem can be viewed as \"finding the maximum area enclosed within a bounded interval on the horizontal axis.\" This analogy helps explain the effectiveness of the greedy strategy from a geometric perspective.</p> <p></p> <p> Figure 15-6   Geometric representation of the fractional knapsack problem </p>","path":["Chapter 15. Greedy","15.2   Fractional Knapsack Problem"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/","level":1,"title":"15.1   Greedy Algorithm","text":"<p>Greedy algorithm is a common approach to solving optimization problems. Its basic idea is to choose the option that appears best at each decision stage, that is, to greedily make locally optimal decisions in the hope of obtaining a globally optimal solution. Greedy algorithms are simple and efficient, and are widely used in many practical problems.</p> <p>Greedy algorithms and dynamic programming are both commonly used to solve optimization problems. They share some similarities, such as both relying on the optimal substructure property, but they work differently.</p> <ul> <li>Dynamic programming considers all previous decisions when making the current decision, and uses solutions to past subproblems to construct the solution to the current subproblem.</li> <li>Greedy algorithms do not consider past decisions, but instead make greedy choices moving forward, continually reducing the problem size until the problem is solved.</li> </ul> <p>We will first understand how greedy algorithms work through the example problem \"coin change.\" This problem was already introduced in the \"Complete Knapsack Problem\" chapter, so it should already be familiar to you.</p> <p>Question</p> <p>Given \\(n\\) types of coins, where the denomination of the \\(i\\)-th type is \\(coins[i - 1]\\), a target amount \\(amt\\), and an unlimited number of coins of each type, what is the minimum number of coins needed to make up the target amount? If the target amount cannot be made up, return \\(-1\\).</p> <p>The greedy strategy for this problem is shown in Figure 15-1. Given a target amount, we greedily choose the coin that does not exceed it and is closest to it, repeating this step until the target amount is made up.</p> <p></p> <p> Figure 15-1   Greedy strategy for coin change </p> <p>The implementation code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change_greedy.py<pre><code>def coin_change_greedy(coins: list[int], amt: int) -&gt; int:\n    \"\"\"Coin change: Greedy algorithm\"\"\"\n    # Assume coins list is sorted\n    i = len(coins) - 1\n    count = 0\n    # Loop to make greedy choices until no remaining amount\n    while amt &gt; 0:\n        # Find the coin that is less than and closest to the remaining amount\n        while i &gt; 0 and coins[i] &gt; amt:\n            i -= 1\n        # Choose coins[i]\n        amt -= coins[i]\n        count += 1\n    # If no feasible solution is found, return -1\n    return count if amt == 0 else -1\n</code></pre> coin_change_greedy.cpp<pre><code>/* Coin change: Greedy algorithm */\nint coinChangeGreedy(vector&lt;int&gt; &amp;coins, int amt) {\n    // Assume coins list is sorted\n    int i = coins.size() - 1;\n    int count = 0;\n    // Loop to make greedy choices until no remaining amount\n    while (amt &gt; 0) {\n        // Find the coin that is less than and closest to the remaining amount\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // Choose coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // If no feasible solution is found, return -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.java<pre><code>/* Coin change: Greedy algorithm */\nint coinChangeGreedy(int[] coins, int amt) {\n    // Assume coins list is sorted\n    int i = coins.length - 1;\n    int count = 0;\n    // Loop to make greedy choices until no remaining amount\n    while (amt &gt; 0) {\n        // Find the coin that is less than and closest to the remaining amount\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // Choose coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // If no feasible solution is found, return -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.cs<pre><code>/* Coin change: Greedy algorithm */\nint CoinChangeGreedy(int[] coins, int amt) {\n    // Assume coins list is sorted\n    int i = coins.Length - 1;\n    int count = 0;\n    // Loop to make greedy choices until no remaining amount\n    while (amt &gt; 0) {\n        // Find the coin that is less than and closest to the remaining amount\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // Choose coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // If no feasible solution is found, return -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.go<pre><code>/* Coin change: Greedy algorithm */\nfunc coinChangeGreedy(coins []int, amt int) int {\n    // Assume coins list is sorted\n    i := len(coins) - 1\n    count := 0\n    // Loop to make greedy choices until no remaining amount\n    for amt &gt; 0 {\n        // Find the coin that is less than and closest to the remaining amount\n        for i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i--\n        }\n        // Choose coins[i]\n        amt -= coins[i]\n        count++\n    }\n    // If no feasible solution is found, return -1\n    if amt != 0 {\n        return -1\n    }\n    return count\n}\n</code></pre> coin_change_greedy.swift<pre><code>/* Coin change: Greedy algorithm */\nfunc coinChangeGreedy(coins: [Int], amt: Int) -&gt; Int {\n    // Assume coins list is sorted\n    var i = coins.count - 1\n    var count = 0\n    var amt = amt\n    // Loop to make greedy choices until no remaining amount\n    while amt &gt; 0 {\n        // Find the coin that is less than and closest to the remaining amount\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1\n        }\n        // Choose coins[i]\n        amt -= coins[i]\n        count += 1\n    }\n    // If no feasible solution is found, return -1\n    return amt == 0 ? count : -1\n}\n</code></pre> coin_change_greedy.js<pre><code>/* Coin change: Greedy algorithm */\nfunction coinChangeGreedy(coins, amt) {\n    // Assume coins array is sorted\n    let i = coins.length - 1;\n    let count = 0;\n    // Loop to make greedy choices until no remaining amount\n    while (amt &gt; 0) {\n        // Find the coin that is less than and closest to the remaining amount\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // Choose coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // If no feasible solution is found, return -1\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.ts<pre><code>/* Coin change: Greedy algorithm */\nfunction coinChangeGreedy(coins: number[], amt: number): number {\n    // Assume coins array is sorted\n    let i = coins.length - 1;\n    let count = 0;\n    // Loop to make greedy choices until no remaining amount\n    while (amt &gt; 0) {\n        // Find the coin that is less than and closest to the remaining amount\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // Choose coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // If no feasible solution is found, return -1\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.dart<pre><code>/* Coin change: Greedy algorithm */\nint coinChangeGreedy(List&lt;int&gt; coins, int amt) {\n  // Assume coins list is sorted\n  int i = coins.length - 1;\n  int count = 0;\n  // Loop to make greedy choices until no remaining amount\n  while (amt &gt; 0) {\n    // Find the coin that is less than and closest to the remaining amount\n    while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n      i--;\n    }\n    // Choose coins[i]\n    amt -= coins[i];\n    count++;\n  }\n  // If no feasible solution is found, return -1\n  return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.rs<pre><code>/* Coin change: Greedy algorithm */\nfn coin_change_greedy(coins: &amp;[i32], mut amt: i32) -&gt; i32 {\n    // Assume coins list is sorted\n    let mut i = coins.len() - 1;\n    let mut count = 0;\n    // Loop to make greedy choices until no remaining amount\n    while amt &gt; 0 {\n        // Find the coin that is less than and closest to the remaining amount\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1;\n        }\n        // Choose coins[i]\n        amt -= coins[i];\n        count += 1;\n    }\n    // If no feasible solution is found, return -1\n    if amt == 0 {\n        count\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change_greedy.c<pre><code>/* Coin change: Greedy algorithm */\nint coinChangeGreedy(int *coins, int size, int amt) {\n    // Assume coins list is sorted\n    int i = size - 1;\n    int count = 0;\n    // Loop to make greedy choices until no remaining amount\n    while (amt &gt; 0) {\n        // Find the coin that is less than and closest to the remaining amount\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // Choose coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // If no feasible solution is found, return -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.kt<pre><code>/* Coin change: Greedy algorithm */\nfun coinChangeGreedy(coins: IntArray, amt: Int): Int {\n    // Assume coins list is sorted\n    var am = amt\n    var i = coins.size - 1\n    var count = 0\n    // Loop to make greedy choices until no remaining amount\n    while (am &gt; 0) {\n        // Find the coin that is less than and closest to the remaining amount\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; am) {\n            i--\n        }\n        // Choose coins[i]\n        am -= coins[i]\n        count++\n    }\n    // If no feasible solution is found, return -1\n    return if (am == 0) count else -1\n}\n</code></pre> coin_change_greedy.rb<pre><code>### Coin change: greedy ###\ndef coin_change_greedy(coins, amt)\n  # Assume coins list is sorted\n  i = coins.length - 1\n  count = 0\n  # Loop to make greedy choices until no remaining amount\n  while amt &gt; 0\n    # Find the coin that is less than and closest to the remaining amount\n    while i &gt; 0 &amp;&amp; coins[i] &gt; amt\n      i -= 1\n    end\n    # Choose coins[i]\n    amt -= coins[i]\n    count += 1\n  end\n  # Return -1 if no solution found\n  amt == 0 ? count : -1\nend\n</code></pre> <p>You may find yourself exclaiming, \"So clean!\" The greedy algorithm solves the coin change problem in only about ten lines of code.</p>","path":["Chapter 15. Greedy","15.1   Greedy Algorithm"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1511-advantages-and-limitations-of-greedy-algorithms","level":2,"title":"15.1.1   Advantages and Limitations of Greedy Algorithms","text":"<p>Greedy algorithms are not only straightforward to apply and easy to implement, but are also usually very efficient. In the code above, if the smallest coin denomination is \\(\\min(coins)\\), the greedy selection loop runs at most \\(amt / \\min(coins)\\) times, giving a time complexity of \\(O(amt / \\min(coins))\\). This is an order of magnitude lower than the time complexity of the dynamic programming solution, \\(O(n \\times amt)\\).</p> <p>However, for some coin denomination sets, greedy algorithms cannot find the optimal solution. Figure 15-2 shows two examples.</p> <ul> <li>Positive example \\(coins = [1, 5, 10, 20, 50, 100]\\): With this coin set, the greedy algorithm can find the optimal solution for any \\(amt\\).</li> <li>Counterexample \\(coins = [1, 20, 50]\\): Suppose \\(amt = 60\\). The greedy algorithm can only find the combination \\(50 + 1 \\times 10\\), using \\(11\\) coins in total, whereas dynamic programming can find the optimal solution \\(20 + 20 + 20\\) using only \\(3\\) coins.</li> <li>Counterexample \\(coins = [1, 49, 50]\\): Suppose \\(amt = 98\\). The greedy algorithm can only find the combination \\(50 + 1 \\times 48\\), using \\(49\\) coins in total, whereas dynamic programming can find the optimal solution \\(49 + 49\\) using only \\(2\\) coins.</li> </ul> <p></p> <p> Figure 15-2   Examples where greedy algorithms cannot find the optimal solution </p> <p>In other words, for the coin change problem, greedy algorithms cannot guarantee a globally optimal solution and may even produce very poor results. This problem is better solved with dynamic programming.</p> <p>In general, greedy algorithms are applicable in the following two situations.</p> <ol> <li>The optimal solution can be guaranteed: In this case, greedy algorithms are often the best choice because they tend to be more efficient than backtracking and dynamic programming.</li> <li>An approximately optimal solution can be found: Greedy algorithms are also useful in this case. For many complex problems, finding the global optimal solution is very difficult, so efficiently finding a suboptimal solution is already a very good outcome.</li> </ol>","path":["Chapter 15. Greedy","15.1   Greedy Algorithm"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1512-characteristics-of-greedy-algorithms","level":2,"title":"15.1.2   Characteristics of Greedy Algorithms","text":"<p>So the question arises: what kind of problems are suitable for solving with greedy algorithms? Or in other words, under what conditions can greedy algorithms guarantee finding the optimal solution?</p> <p>Compared to dynamic programming, the conditions for using greedy algorithms are stricter, mainly focusing on two properties of the problem.</p> <ul> <li>Greedy choice property: Only when locally optimal choices can always lead to a globally optimal solution can greedy algorithms guarantee obtaining the optimal solution.</li> <li>Optimal substructure: The optimal solution to the original problem contains the optimal solutions to subproblems.</li> </ul> <p>Optimal substructure has already been introduced in the \"Dynamic Programming\" chapter, so we won't elaborate on it here. It's worth noting that the optimal substructure of some problems is not obvious, but they can still be solved using greedy algorithms.</p> <p>We mainly explore methods for determining the greedy choice property. Although its description seems relatively simple, in practice, for many problems, proving the greedy choice property is not easy.</p> <p>For example, in the coin change problem, although we can easily provide counterexamples to disprove the greedy choice property, proving that it holds is much harder. If asked, under what conditions can a coin set be solved using a greedy algorithm? We often can only rely on intuition or examples to give a vague answer, and it is difficult to provide a rigorous mathematical proof.</p> <p>Quote</p> <p>There is a paper that presents an \\(O(n^3)\\) algorithm for determining whether a coin set can be solved optimally by a greedy algorithm for any amount.</p> <p>Pearson, D. A polynomial-time algorithm for the change-making problem[J]. Operations Research Letters, 2005, 33(3): 231-234.</p>","path":["Chapter 15. Greedy","15.1   Greedy Algorithm"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1513-steps-for-solving-problems-with-greedy-algorithms","level":2,"title":"15.1.3   Steps for Solving Problems with Greedy Algorithms","text":"<p>The general process for solving greedy problems can be divided into the following three steps.</p> <ol> <li>Problem analysis: Sort out and understand the characteristics of the problem, including state definitions, optimization objectives, and constraints. This step also appears in backtracking and dynamic programming.</li> <li>Determine the greedy strategy: Decide how to make a greedy choice at each step. This strategy should reduce the problem size step by step and ultimately solve the entire problem.</li> <li>Correctness proof: It is usually necessary to prove that the problem has both greedy choice property and optimal substructure. This step may require mathematical tools such as induction or proof by contradiction.</li> </ol> <p>Determining the greedy strategy is the core step in solving such problems, but it may not be easy in practice, mainly for the following reasons.</p> <ul> <li>Greedy strategies vary greatly from problem to problem. For many problems, the greedy strategy is fairly intuitive and can be derived through rough reasoning and experimentation. For some complex problems, however, the greedy strategy may be deeply hidden, which strongly tests one's problem-solving experience and algorithmic ability.</li> <li>Some greedy strategies are highly deceptive. We may confidently design a greedy strategy, write the solution code, and submit it, only to find that some test cases fail. This is because the designed greedy strategy is only \"partially correct,\" as exemplified by the coin change problem discussed above.</li> </ul> <p>To ensure correctness, we should give a rigorous mathematical proof of the greedy strategy, usually using proof by contradiction or mathematical induction.</p> <p>However, correctness proofs can also be difficult. If we have no clear direction, we usually resort to debugging against test cases, revising and validating the greedy strategy step by step.</p>","path":["Chapter 15. Greedy","15.1   Greedy Algorithm"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1514-typical-problems-solved-by-greedy-algorithms","level":2,"title":"15.1.4   Typical Problems Solved by Greedy Algorithms","text":"<p>Greedy algorithms are often applied to optimization problems that satisfy greedy choice property and optimal substructure. Below are some typical greedy algorithm problems.</p> <ul> <li>Coin change problem: With certain coin combinations, greedy algorithms can always obtain the optimal solution.</li> <li>Interval scheduling problem: Suppose you have some tasks, each taking place during a period of time, and your goal is to complete as many tasks as possible. If you always choose the task that ends earliest, then the greedy algorithm can obtain the optimal solution.</li> <li>Fractional knapsack problem: Given a set of items and a carrying capacity, your goal is to select a set of items such that the total weight does not exceed the carrying capacity and the total value is maximized. If you always choose the item with the highest value-to-weight ratio (value / weight), then the greedy algorithm can obtain the optimal solution in some cases.</li> <li>Stock trading problem: Given a set of historical stock prices, you can make multiple trades, but if you already hold stocks, you cannot buy again before selling, and the goal is to obtain the maximum profit.</li> <li>Huffman coding: Huffman coding is a greedy algorithm used for lossless data compression. By constructing a Huffman tree and always merging the two nodes with the lowest frequency, the resulting Huffman tree has the minimum weighted path length (encoding length).</li> <li>Dijkstra's algorithm: For graphs with non-negative edge weights, it is a greedy algorithm for solving the shortest path problem from a given source vertex to all other vertices.</li> </ul>","path":["Chapter 15. Greedy","15.1   Greedy Algorithm"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/","level":1,"title":"15.3   Max Capacity Problem","text":"<p>Question</p> <p>Given an array \\(ht\\), where each element represents the height of a vertical partition. Any two partitions in the array, together with the space between them, can form a container.</p> <p>The capacity of the container equals the product of its height and width (that is, its area), where the height is determined by the shorter partition and the width is the difference between the array indices of the two partitions.</p> <p>Select two partitions in the array such that the capacity of the resulting container is maximized, and return that maximum capacity. An example is shown in Figure 15-7.</p> <p></p> <p> Figure 15-7   Example data for the max capacity problem </p> <p>The container is formed by any two partitions, so the state of this problem is the indices of the two partitions, denoted by \\([i, j]\\).</p> <p>According to the problem statement, capacity equals height multiplied by width, where the height is determined by the shorter partition and the width is the difference between the array indices of the two partitions. Let the capacity be \\(cap[i, j]\\); then we obtain the following formula:</p> \\[ cap[i, j] = \\min(ht[i], ht[j]) \\times (j - i) \\] <p>Let the array length be \\(n\\). Then the number of ways to choose two partitions (that is, the total number of states) is \\(C_n^2 = \\frac{n(n - 1)}{2}\\). The most straightforward approach is to exhaustively enumerate all states to find the maximum capacity, which has a time complexity of \\(O(n^2)\\).</p>","path":["Chapter 15. Greedy","15.3   Max Capacity Problem"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/#1-greedy-strategy-determination","level":3,"title":"1.   Greedy Strategy Determination","text":"<p>This problem has a more efficient solution. As shown in Figure 15-8, consider a state \\([i, j]\\) where \\(i &lt; j\\) and \\(ht[i] &lt; ht[j]\\). In this case, \\(i\\) is the shorter partition and \\(j\\) is the taller partition.</p> <p></p> <p> Figure 15-8   Initial state </p> <p>As shown in Figure 15-9, if we now move the taller partition \\(j\\) inward toward the shorter partition \\(i\\), the capacity will definitely decrease.</p> <p>This is because after moving the taller partition \\(j\\), the width \\(j-i\\) definitely decreases. Since the height is determined by the shorter partition, the height can only stay the same (\\(i\\) remains the shorter partition) or decrease (\\(j\\) becomes the shorter partition after being moved).</p> <p></p> <p> Figure 15-9   State after moving the long partition inward </p> <p>Conversely, only by moving the shorter partition \\(i\\) inward can the capacity possibly increase. Although the width will definitely decrease, the height may increase (the moved partition at \\(i\\) may be taller). For example, in Figure 15-10, the area increases after moving the shorter partition.</p> <p></p> <p> Figure 15-10   State after moving the short partition inward </p> <p>From this, we can derive the greedy strategy for this problem: initialize two pointers at the two ends, and in each round move the pointer corresponding to the shorter partition inward until the two pointers meet.</p> <p>Figure 15-11 shows the execution process of the greedy strategy.</p> <ol> <li>In the initial state, pointers \\(i\\) and \\(j\\) are at both ends of the array.</li> <li>Calculate the capacity of the current state \\(cap[i, j]\\), and update the maximum capacity.</li> <li>Compare the heights of partitions \\(i\\) and \\(j\\), and move the pointer corresponding to the shorter partition inward by one position.</li> <li>Repeat steps <code>2.</code> and <code>3.</code> until \\(i\\) and \\(j\\) meet.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 15-11   Greedy process for the max capacity problem </p>","path":["Chapter 15. Greedy","15.3   Max Capacity Problem"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/#2-code-implementation","level":3,"title":"2.   Code Implementation","text":"<p>The code runs for at most \\(n\\) rounds, so the time complexity is \\(O(n)\\).</p> <p>Variables \\(i\\), \\(j\\), and \\(res\\) use only a constant amount of extra space, so the space complexity is \\(O(1)\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby max_capacity.py<pre><code>def max_capacity(ht: list[int]) -&gt; int:\n    \"\"\"Max capacity: Greedy algorithm\"\"\"\n    # Initialize i, j to be at both ends of the array\n    i, j = 0, len(ht) - 1\n    # Initial max capacity is 0\n    res = 0\n    # Loop for greedy selection until the two boards meet\n    while i &lt; j:\n        # Update max capacity\n        cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        # Move the shorter board inward\n        if ht[i] &lt; ht[j]:\n            i += 1\n        else:\n            j -= 1\n    return res\n</code></pre> max_capacity.cpp<pre><code>/* Max capacity: Greedy algorithm */\nint maxCapacity(vector&lt;int&gt; &amp;ht) {\n    // Initialize i, j to be at both ends of the array\n    int i = 0, j = ht.size() - 1;\n    // Initial max capacity is 0\n    int res = 0;\n    // Loop for greedy selection until the two boards meet\n    while (i &lt; j) {\n        // Update max capacity\n        int cap = min(ht[i], ht[j]) * (j - i);\n        res = max(res, cap);\n        // Move the shorter board inward\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.java<pre><code>/* Max capacity: Greedy algorithm */\nint maxCapacity(int[] ht) {\n    // Initialize i, j to be at both ends of the array\n    int i = 0, j = ht.length - 1;\n    // Initial max capacity is 0\n    int res = 0;\n    // Loop for greedy selection until the two boards meet\n    while (i &lt; j) {\n        // Update max capacity\n        int cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // Move the shorter board inward\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.cs<pre><code>/* Max capacity: Greedy algorithm */\nint MaxCapacity(int[] ht) {\n    // Initialize i, j to be at both ends of the array\n    int i = 0, j = ht.Length - 1;\n    // Initial max capacity is 0\n    int res = 0;\n    // Loop for greedy selection until the two boards meet\n    while (i &lt; j) {\n        // Update max capacity\n        int cap = Math.Min(ht[i], ht[j]) * (j - i);\n        res = Math.Max(res, cap);\n        // Move the shorter board inward\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.go<pre><code>/* Max capacity: Greedy algorithm */\nfunc maxCapacity(ht []int) int {\n    // Initialize i, j to be at both ends of the array\n    i, j := 0, len(ht)-1\n    // Initial max capacity is 0\n    res := 0\n    // Loop for greedy selection until the two boards meet\n    for i &lt; j {\n        // Update max capacity\n        capacity := int(math.Min(float64(ht[i]), float64(ht[j]))) * (j - i)\n        res = int(math.Max(float64(res), float64(capacity)))\n        // Move the shorter board inward\n        if ht[i] &lt; ht[j] {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.swift<pre><code>/* Max capacity: Greedy algorithm */\nfunc maxCapacity(ht: [Int]) -&gt; Int {\n    // Initialize i, j to be at both ends of the array\n    var i = ht.startIndex, j = ht.endIndex - 1\n    // Initial max capacity is 0\n    var res = 0\n    // Loop for greedy selection until the two boards meet\n    while i &lt; j {\n        // Update max capacity\n        let cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        // Move the shorter board inward\n        if ht[i] &lt; ht[j] {\n            i += 1\n        } else {\n            j -= 1\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.js<pre><code>/* Max capacity: Greedy algorithm */\nfunction maxCapacity(ht) {\n    // Initialize i, j to be at both ends of the array\n    let i = 0,\n        j = ht.length - 1;\n    // Initial max capacity is 0\n    let res = 0;\n    // Loop for greedy selection until the two boards meet\n    while (i &lt; j) {\n        // Update max capacity\n        const cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // Move the shorter board inward\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.ts<pre><code>/* Max capacity: Greedy algorithm */\nfunction maxCapacity(ht: number[]): number {\n    // Initialize i, j to be at both ends of the array\n    let i = 0,\n        j = ht.length - 1;\n    // Initial max capacity is 0\n    let res = 0;\n    // Loop for greedy selection until the two boards meet\n    while (i &lt; j) {\n        // Update max capacity\n        const cap: number = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // Move the shorter board inward\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.dart<pre><code>/* Max capacity: Greedy algorithm */\nint maxCapacity(List&lt;int&gt; ht) {\n  // Initialize i, j to be at both ends of the array\n  int i = 0, j = ht.length - 1;\n  // Initial max capacity is 0\n  int res = 0;\n  // Loop for greedy selection until the two boards meet\n  while (i &lt; j) {\n    // Update max capacity\n    int cap = min(ht[i], ht[j]) * (j - i);\n    res = max(res, cap);\n    // Move the shorter board inward\n    if (ht[i] &lt; ht[j]) {\n      i++;\n    } else {\n      j--;\n    }\n  }\n  return res;\n}\n</code></pre> max_capacity.rs<pre><code>/* Max capacity: Greedy algorithm */\nfn max_capacity(ht: &amp;[i32]) -&gt; i32 {\n    // Initialize i, j to be at both ends of the array\n    let mut i = 0;\n    let mut j = ht.len() - 1;\n    // Initial max capacity is 0\n    let mut res = 0;\n    // Loop for greedy selection until the two boards meet\n    while i &lt; j {\n        // Update max capacity\n        let cap = std::cmp::min(ht[i], ht[j]) * (j - i) as i32;\n        res = std::cmp::max(res, cap);\n        // Move the shorter board inward\n        if ht[i] &lt; ht[j] {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    res\n}\n</code></pre> max_capacity.c<pre><code>/* Max capacity: Greedy algorithm */\nint maxCapacity(int ht[], int htLength) {\n    // Initialize i, j to be at both ends of the array\n    int i = 0;\n    int j = htLength - 1;\n    // Initial max capacity is 0\n    int res = 0;\n    // Loop for greedy selection until the two boards meet\n    while (i &lt; j) {\n        // Update max capacity\n        int capacity = myMin(ht[i], ht[j]) * (j - i);\n        res = myMax(res, capacity);\n        // Move the shorter board inward\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.kt<pre><code>/* Max capacity: Greedy algorithm */\nfun maxCapacity(ht: IntArray): Int {\n    // Initialize i, j to be at both ends of the array\n    var i = 0\n    var j = ht.size - 1\n    // Initial max capacity is 0\n    var res = 0\n    // Loop for greedy selection until the two boards meet\n    while (i &lt; j) {\n        // Update max capacity\n        val cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        // Move the shorter board inward\n        if (ht[i] &lt; ht[j]) {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.rb<pre><code>### Maximum capacity: greedy ###\ndef max_capacity(ht)\n  # Initialize i, j to be at both ends of the array\n  i, j = 0, ht.length - 1\n  # Initial max capacity is 0\n  res = 0\n\n  # Loop for greedy selection until the two boards meet\n  while i &lt; j\n    # Update max capacity\n    cap = [ht[i], ht[j]].min * (j - i)\n    res = [res, cap].max\n    # Move the shorter board inward\n    if ht[i] &lt; ht[j]\n      i += 1\n    else\n      j -= 1\n    end\n  end\n\n  res\nend\n</code></pre>","path":["Chapter 15. Greedy","15.3   Max Capacity Problem"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/#3-correctness-proof","level":3,"title":"3.   Correctness Proof","text":"<p>The reason greedy is faster than exhaustive enumeration is that each round of greedy selection \"skips\" some states.</p> <p>For example, in state \\(cap[i, j]\\), suppose \\(i\\) is the shorter partition and \\(j\\) is the taller partition. If we greedily move the shorter partition \\(i\\) inward by one position, the states shown in Figure 15-12 will be \"skipped.\" This means that their capacities can no longer be checked later.</p> \\[ cap[i, i+1], cap[i, i+2], \\dots, cap[i, j-2], cap[i, j-1] \\] <p></p> <p> Figure 15-12   States skipped by moving the short partition </p> <p>A closer look shows that these skipped states are exactly the states obtained by moving the taller partition \\(j\\) inward. We have already proven that moving the taller partition inward will definitely decrease the capacity. Therefore, none of the skipped states can be the optimal solution, so skipping them does not cause us to miss the optimum.</p> <p>The above analysis shows that moving the shorter partition is a \"safe\" operation, and that the greedy strategy is effective.</p>","path":["Chapter 15. Greedy","15.3   Max Capacity Problem"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/","level":1,"title":"15.4   Maximum Product Cutting Problem","text":"<p>Question</p> <p>Given a positive integer \\(n\\), split it into the sum of at least two positive integers and find the maximum product of the resulting integers, as shown in Figure 15-13.</p> <p></p> <p> Figure 15-13   Problem definition of max product cutting </p> <p>Suppose we split \\(n\\) into \\(m\\) integer factors, where the \\(i\\)-th factor is denoted as \\(n_i\\), that is</p> \\[ n = \\sum_{i=1}^{m}n_i \\] <p>The goal of this problem is to find the maximum product of all integer factors, namely</p> \\[ \\max(\\prod_{i=1}^{m}n_i) \\] <p>We need to determine how many parts \\(m\\) there should be and what each \\(n_i\\) should be.</p>","path":["Chapter 15. Greedy","15.4   Maximum Product Cutting Problem"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/#1-determining-the-greedy-strategy","level":3,"title":"1.   Determining the Greedy Strategy","text":"<p>As a rule of thumb, the product of two integers is often greater than their sum. Suppose we split off a factor of \\(2\\) from \\(n\\); the resulting product is \\(2(n-2)\\). We compare this product with \\(n\\):</p> \\[ \\begin{aligned} 2(n-2) &amp; \\geq n \\newline 2n - n - 4 &amp; \\geq 0 \\newline n &amp; \\geq 4 \\end{aligned} \\] <p>As shown in Figure 15-14, when \\(n \\geq 4\\), splitting out a \\(2\\) will increase the product, which indicates that integers greater than or equal to \\(4\\) should all be split.</p> <p>Greedy strategy one: If the splitting scheme contains a factor \\(\\geq 4\\), it should be split further. The final splitting scheme should contain only the factors \\(1\\), \\(2\\), and \\(3\\).</p> <p></p> <p> Figure 15-14   Splitting causes product to increase </p> <p>Next, consider which factor is optimal. Among the three factors \\(1\\), \\(2\\), and \\(3\\), clearly \\(1\\) is the worst, because \\(1 \\times (n-1) &lt; n\\) always holds, meaning splitting out \\(1\\) will actually decrease the product.</p> <p>As shown in Figure 15-15, when \\(n = 6\\), we have \\(3 \\times 3 &gt; 2 \\times 2 \\times 2\\). This means that splitting out \\(3\\) is better than splitting out \\(2\\).</p> <p>Greedy strategy two: In the splitting scheme, there should be at most two \\(2\\)s, because three \\(2\\)s can always be replaced by two \\(3\\)s to obtain a larger product.</p> <p></p> <p> Figure 15-15   Optimal splitting factor </p> <p>In summary, the following greedy strategies can be derived.</p> <ol> <li>Input integer \\(n\\), continuously split out factor \\(3\\) until the remainder is \\(0\\), \\(1\\), or \\(2\\).</li> <li>When the remainder is \\(0\\), it means \\(n\\) is a multiple of \\(3\\), so no further action is needed.</li> <li>When the remainder is \\(2\\), do not split it further; keep it as is.</li> <li>When the remainder is \\(1\\), since \\(2 \\times 2 &gt; 1 \\times 3\\), replace the final \\(3\\) and the remaining \\(1\\) with two \\(2\\)s.</li> </ol>","path":["Chapter 15. Greedy","15.4   Maximum Product Cutting Problem"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/#2-code-implementation","level":3,"title":"2.   Code Implementation","text":"<p>As shown in Figure 15-16, we do not need loops to split the integer. Instead, we use integer division to obtain the number of \\(3\\)s, denoted by \\(a\\), and the modulo operation to obtain the remainder \\(b\\), giving:</p> \\[ n = 3 a + b \\] <p>Please note that for the edge case of \\(n \\leq 3\\), a \\(1\\) must be split out, with product \\(1 \\times (n - 1)\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby max_product_cutting.py<pre><code>def max_product_cutting(n: int) -&gt; int:\n    \"\"\"Max product cutting: Greedy algorithm\"\"\"\n    # When n &lt;= 3, must cut out a 1\n    if n &lt;= 3:\n        return 1 * (n - 1)\n    # Greedily cut out 3, a is the number of 3s, b is the remainder\n    a, b = n // 3, n % 3\n    if b == 1:\n        # When the remainder is 1, convert a pair of 1 * 3 to 2 * 2\n        return int(math.pow(3, a - 1)) * 2 * 2\n    if b == 2:\n        # When the remainder is 2, do nothing\n        return int(math.pow(3, a)) * 2\n    # When the remainder is 0, do nothing\n    return int(math.pow(3, a))\n</code></pre> max_product_cutting.cpp<pre><code>/* Max product cutting: Greedy algorithm */\nint maxProductCutting(int n) {\n    // When n &lt;= 3, must cut out a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // Greedily cut out 3, a is the number of 3s, b is the remainder\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // When the remainder is 1, convert a pair of 1 * 3 to 2 * 2\n        return (int)pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // When the remainder is 2, do nothing\n        return (int)pow(3, a) * 2;\n    }\n    // When the remainder is 0, do nothing\n    return (int)pow(3, a);\n}\n</code></pre> max_product_cutting.java<pre><code>/* Max product cutting: Greedy algorithm */\nint maxProductCutting(int n) {\n    // When n &lt;= 3, must cut out a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // Greedily cut out 3, a is the number of 3s, b is the remainder\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // When the remainder is 1, convert a pair of 1 * 3 to 2 * 2\n        return (int) Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // When the remainder is 2, do nothing\n        return (int) Math.pow(3, a) * 2;\n    }\n    // When the remainder is 0, do nothing\n    return (int) Math.pow(3, a);\n}\n</code></pre> max_product_cutting.cs<pre><code>/* Max product cutting: Greedy algorithm */\nint MaxProductCutting(int n) {\n    // When n &lt;= 3, must cut out a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // Greedily cut out 3, a is the number of 3s, b is the remainder\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // When the remainder is 1, convert a pair of 1 * 3 to 2 * 2\n        return (int)Math.Pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // When the remainder is 2, do nothing\n        return (int)Math.Pow(3, a) * 2;\n    }\n    // When the remainder is 0, do nothing\n    return (int)Math.Pow(3, a);\n}\n</code></pre> max_product_cutting.go<pre><code>/* Max product cutting: Greedy algorithm */\nfunc maxProductCutting(n int) int {\n    // When n &lt;= 3, must cut out a 1\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // Greedily cut out 3, a is the number of 3s, b is the remainder\n    a := n / 3\n    b := n % 3\n    if b == 1 {\n        // When the remainder is 1, convert a pair of 1 * 3 to 2 * 2\n        return int(math.Pow(3, float64(a-1))) * 2 * 2\n    }\n    if b == 2 {\n        // When the remainder is 2, do nothing\n        return int(math.Pow(3, float64(a))) * 2\n    }\n    // When the remainder is 0, do nothing\n    return int(math.Pow(3, float64(a)))\n}\n</code></pre> max_product_cutting.swift<pre><code>/* Max product cutting: Greedy algorithm */\nfunc maxProductCutting(n: Int) -&gt; Int {\n    // When n &lt;= 3, must cut out a 1\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // Greedily cut out 3, a is the number of 3s, b is the remainder\n    let a = n / 3\n    let b = n % 3\n    if b == 1 {\n        // When the remainder is 1, convert a pair of 1 * 3 to 2 * 2\n        return pow(3, a - 1) * 2 * 2\n    }\n    if b == 2 {\n        // When the remainder is 2, do nothing\n        return pow(3, a) * 2\n    }\n    // When the remainder is 0, do nothing\n    return pow(3, a)\n}\n</code></pre> max_product_cutting.js<pre><code>/* Max product cutting: Greedy algorithm */\nfunction maxProductCutting(n) {\n    // When n &lt;= 3, must cut out a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // Greedily cut out 3, a is the number of 3s, b is the remainder\n    let a = Math.floor(n / 3);\n    let b = n % 3;\n    if (b === 1) {\n        // When the remainder is 1, convert a pair of 1 * 3 to 2 * 2\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // When the remainder is 2, do nothing\n        return Math.pow(3, a) * 2;\n    }\n    // When the remainder is 0, do nothing\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.ts<pre><code>/* Max product cutting: Greedy algorithm */\nfunction maxProductCutting(n: number): number {\n    // When n &lt;= 3, must cut out a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // Greedily cut out 3, a is the number of 3s, b is the remainder\n    let a: number = Math.floor(n / 3);\n    let b: number = n % 3;\n    if (b === 1) {\n        // When the remainder is 1, convert a pair of 1 * 3 to 2 * 2\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // When the remainder is 2, do nothing\n        return Math.pow(3, a) * 2;\n    }\n    // When the remainder is 0, do nothing\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.dart<pre><code>/* Max product cutting: Greedy algorithm */\nint maxProductCutting(int n) {\n  // When n &lt;= 3, must cut out a 1\n  if (n &lt;= 3) {\n    return 1 * (n - 1);\n  }\n  // Greedily cut out 3, a is the number of 3s, b is the remainder\n  int a = n ~/ 3;\n  int b = n % 3;\n  if (b == 1) {\n    // When the remainder is 1, convert a pair of 1 * 3 to 2 * 2\n    return (pow(3, a - 1) * 2 * 2).toInt();\n  }\n  if (b == 2) {\n    // When the remainder is 2, do nothing\n    return (pow(3, a) * 2).toInt();\n  }\n  // When the remainder is 0, do nothing\n  return pow(3, a).toInt();\n}\n</code></pre> max_product_cutting.rs<pre><code>/* Max product cutting: Greedy algorithm */\nfn max_product_cutting(n: i32) -&gt; i32 {\n    // When n &lt;= 3, must cut out a 1\n    if n &lt;= 3 {\n        return 1 * (n - 1);\n    }\n    // Greedily cut out 3, a is the number of 3s, b is the remainder\n    let a = n / 3;\n    let b = n % 3;\n    if b == 1 {\n        // When the remainder is 1, convert a pair of 1 * 3 to 2 * 2\n        3_i32.pow(a as u32 - 1) * 2 * 2\n    } else if b == 2 {\n        // When the remainder is 2, do nothing\n        3_i32.pow(a as u32) * 2\n    } else {\n        // When the remainder is 0, do nothing\n        3_i32.pow(a as u32)\n    }\n}\n</code></pre> max_product_cutting.c<pre><code>/* Max product cutting: Greedy algorithm */\nint maxProductCutting(int n) {\n    // When n &lt;= 3, must cut out a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // Greedily cut out 3, a is the number of 3s, b is the remainder\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // When the remainder is 1, convert a pair of 1 * 3 to 2 * 2\n        return pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // When the remainder is 2, do nothing\n        return pow(3, a) * 2;\n    }\n    // When the remainder is 0, do nothing\n    return pow(3, a);\n}\n</code></pre> max_product_cutting.kt<pre><code>/* Max product cutting: Greedy algorithm */\nfun maxProductCutting(n: Int): Int {\n    // When n &lt;= 3, must cut out a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1)\n    }\n    // Greedily cut out 3, a is the number of 3s, b is the remainder\n    val a = n / 3\n    val b = n % 3\n    if (b == 1) {\n        // When the remainder is 1, convert a pair of 1 * 3 to 2 * 2\n        return 3.0.pow((a - 1)).toInt() * 2 * 2\n    }\n    if (b == 2) {\n        // When the remainder is 2, do nothing\n        return 3.0.pow(a).toInt() * 2 * 2\n    }\n    // When the remainder is 0, do nothing\n    return 3.0.pow(a).toInt()\n}\n</code></pre> max_product_cutting.rb<pre><code>### Maximum cutting product: greedy ###\ndef max_product_cutting(n)\n  # When n &lt;= 3, must cut out a 1\n  return 1 * (n - 1) if n &lt;= 3\n  # Greedily cut out 3, a is the number of 3s, b is the remainder\n  a, b = n / 3, n % 3\n  # When the remainder is 1, convert a pair of 1 * 3 to 2 * 2\n  return (3.pow(a - 1) * 2 * 2).to_i if b == 1\n  # When the remainder is 2, do nothing\n  return (3.pow(a) * 2).to_i if b == 2\n  # When the remainder is 0, do nothing\n  3.pow(a).to_i\nend\n</code></pre> <p></p> <p> Figure 15-16   Calculation method for max product cutting </p> <p>The time complexity depends on how exponentiation is implemented in the programming language. Taking Python as an example, there are three commonly used ways to compute powers.</p> <ul> <li>Both the operator <code>**</code> and the function <code>pow()</code> have time complexity \\(O(\\log⁡ a)\\).</li> <li>The function <code>math.pow()</code> internally calls the C library's <code>pow()</code> function, which performs floating-point exponentiation, with time complexity \\(O(1)\\).</li> </ul> <p>Variables \\(a\\) and \\(b\\) use a constant amount of extra space, therefore the space complexity is \\(O(1)\\).</p>","path":["Chapter 15. Greedy","15.4   Maximum Product Cutting Problem"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/#3-correctness-proof","level":3,"title":"3.   Correctness Proof","text":"<p>We use proof by contradiction and consider only the case where \\(n \\geq 4\\).</p> <ol> <li>All factors \\(\\leq 3\\): Suppose the optimal splitting scheme includes a factor \\(x \\geq 4\\). Then it can be further split into \\(2(x-2)\\) to obtain a larger (or equal) product. This contradicts the assumption.</li> <li>The splitting scheme does not contain \\(1\\): Suppose the optimal splitting scheme includes a factor of \\(1\\). Then it can be merged into another factor to obtain a larger product. This contradicts the assumption.</li> <li>The splitting scheme contains at most two \\(2\\)s: Suppose the optimal splitting scheme includes three \\(2\\)s. Then they can be replaced by two \\(3\\)s, yielding a larger product. This contradicts the assumption.</li> </ol>","path":["Chapter 15. Greedy","15.4   Maximum Product Cutting Problem"],"tags":[]},{"location":"chapter_greedy/summary/","level":1,"title":"15.5   Summary","text":"","path":["Chapter 15. Greedy","15.5   Summary"],"tags":[]},{"location":"chapter_greedy/summary/#1-key-review","level":3,"title":"1.   Key Review","text":"<ul> <li>Greedy algorithms are typically used to solve optimization problems. The principle is to make locally optimal decisions at each decision stage in hopes of obtaining a globally optimal solution.</li> <li>Greedy algorithms iteratively make one greedy choice after another, transforming the problem into a smaller subproblem in each round, until the problem is solved.</li> <li>Greedy algorithms are not only simple to implement, but also have high problem-solving efficiency. Compared to dynamic programming, greedy algorithms typically have lower time complexity.</li> <li>In the coin change problem, for certain coin combinations, greedy algorithms can guarantee finding the optimal solution; for other coin combinations, however, greedy algorithms may find very poor solutions.</li> <li>Problems suitable for solving with greedy algorithms have two major properties: greedy choice property and optimal substructure. The greedy choice property represents the effectiveness of the greedy strategy.</li> <li>For some complex problems, proving the greedy choice property is not simple. Relatively speaking, disproving it is easier, such as in the coin change problem.</li> <li>Solving greedy problems mainly consists of three steps: problem analysis, determining the greedy strategy, and correctness proof. Among these, determining the greedy strategy is the core step, and correctness proof is often the main difficulty.</li> <li>The fractional knapsack problem, based on the 0-1 knapsack problem, allows selecting fractions of items, and therefore can be solved using greedy algorithms. The correctness of the greedy strategy can be proven using proof by contradiction.</li> <li>The max capacity problem can be solved using exhaustive enumeration with time complexity \\(O(n^2)\\). By designing a greedy strategy to move the shorter side inward in each round, the time complexity can be optimized to \\(O(n)\\).</li> <li>In the max product cutting problem, we successively derive two greedy strategies: integers \\(\\geq 4\\) should all continue to be split, and the optimal splitting factor is \\(3\\). The code includes exponentiation operations, and the time complexity depends on the implementation method of exponentiation, typically being \\(O(1)\\) or \\(O(\\log n)\\).</li> </ul>","path":["Chapter 15. Greedy","15.5   Summary"],"tags":[]},{"location":"chapter_hashing/","level":1,"title":"Chapter 6.   Hashing","text":"<p>Abstract</p> <p>In the world of computing, a hash table is like a clever librarian.</p> <p>It knows how to compute call numbers, allowing it to quickly locate the desired book.</p>","path":["Chapter 6. Hashing","Chapter 6.   Hashing"],"tags":[]},{"location":"chapter_hashing/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>6.1   Hash Table</li> <li>6.2   Hash Collision</li> <li>6.3   Hash Algorithm</li> <li>6.4   Summary</li> <li>6.5   Exercises</li> </ul>","path":["Chapter 6. Hashing","Chapter 6.   Hashing"],"tags":[]},{"location":"chapter_hashing/exercises/","level":1,"title":"6.5   Exercises","text":"","path":["Chapter 6. Hashing","6.5   Exercises"],"tags":[]},{"location":"chapter_hashing/exercises/#651-concept-review","level":2,"title":"6.5.1   Concept Review","text":"","path":["Chapter 6. Hashing","6.5   Exercises"],"tags":[]},{"location":"chapter_hashing/exercises/#1-searching-after-a-hash-collision","level":3,"title":"1.   Searching After a Hash Collision","text":"<p>A hash table has 5 buckets and uses the hash function \\(h(x)=x \\bmod 5\\). When a collision occurs, elements are placed one after another in a list within that bucket. Insert <code>[1, 6, 11, 7]</code> in order:</p> <ol> <li>Write the contents of buckets 0–4.</li> <li>When searching for 6, which bucket is checked first, and which elements are examined in order?</li> <li>Based on the bucket contents from Question 1, do later insertions overwrite earlier ones? Explain using this collision-resolution method.</li> </ol> Answer <ol> <li> <p>Because \\(1\\bmod5=6\\bmod5=11\\bmod5=1\\), while \\(7\\bmod5=2\\), the buckets are:</p> <pre><code>0: []\n1: [1, 6, 11]\n2: [7]\n3: []\n4: []\n</code></pre> </li> <li> <p>A search for 6 first goes to bucket 1, then compares 1 and 6 in order. The target is found on the second comparison.</p> </li> <li> <p>Equal hash values mean only that the elements go into the same bucket, not that the elements are equal. Separate chaining keeps all colliding elements in the bucket     and compares them one at a time during a search, so 1, 6, and 11 do not overwrite one another.</p> </li> </ol>","path":["Chapter 6. Hashing","6.5   Exercises"],"tags":[]},{"location":"chapter_hashing/exercises/#2-where-do-elements-go-after-a-hash-table-expands","level":3,"title":"2.   Where Do Elements Go After a Hash Table Expands?","text":"<p>A hash table using separate chaining originally has 5 buckets and the hash function \\(h(x)=x\\bmod5\\). The keys <code>[1, 6, 11]</code> are all in bucket 1.</p> <p>The table is now expanded to 7 buckets, and the hash function becomes \\(h(x)=x\\bmod7\\):</p> <ol> <li>Calculate the new bucket number for 1, 6, and 11.</li> <li>Which buckets contain elements after the expansion?</li> <li>During the expansion, can the list from the old bucket 1 simply be copied into the new bucket 1? Explain using the results from Questions 1 and 2.</li> </ol> Answer <ol> <li> <p>The new bucket numbers are:</p> <ul> <li>\\(1\\bmod7=1\\);</li> <li>\\(6\\bmod7=6\\);</li> <li>\\(11\\bmod7=4\\).</li> </ul> </li> <li> <p>Bucket 1 stores 1, bucket 4 stores 11, and bucket 6 stores 6. The three keys are no longer crowded into the same bucket.</p> </li> <li> <p>The list cannot be copied as is. A bucket number is calculated by taking the key modulo the number of buckets. When the bucket count changes from 5 to 7, a key's bucket number may change,     so the position of every key must be recalculated. If the old bucket 1 were copied unchanged, later searches using the new formula would go to bucket 6 for 6 and bucket 4 for 11     and would fail to find them.</p> </li> </ol>","path":["Chapter 6. Hashing","6.5   Exercises"],"tags":[]},{"location":"chapter_hashing/exercises/#3-can-11-still-be-found-after-deleting-6","level":3,"title":"3.   Can 11 Still Be Found After Deleting 6?","text":"<p>A hash table has 5 positions with indices <code>0–4</code> and uses the hash function \\(h(x)=x\\bmod5\\). When a collision occurs, it searches to the right from the index produced by the hash function for the first empty position.</p> <p>Insert <code>[1, 6, 11]</code> in order:</p> <ol> <li>At which index is each number ultimately stored?</li> <li>When searching for 11, which indices are examined in order?</li> <li>Suppose deleting 6 changes its position directly to an \"unused empty position,\" and a search stops whenever it reaches an empty position.     What happens when searching for 11 afterward? Is this search result correct? If there is a problem, how can it be avoided?</li> </ol> Answer <ol> <li> <p>The number 1 is stored at index 1. The number 6 also maps to index 1, so after the collision it is stored at index 2.     The number 11 also starts at index 1, skips the occupied indices 1 and 2, and is ultimately stored at index 3.</p> </li> <li> <p>A search for 11 examines indices <code>1, 2, 3</code> in order and finds it at index 3.</p> </li> <li> <p>If index 2 is changed to mean \"never used,\" a search for 11 checks index 1 and then stops at index 2,     incorrectly concluding that 11 is absent. Deletion should leave a \"deleted\" marker.     A search that reaches this marker continues to the next index (wrapping from index 4 to index 0), while a later insertion may still reuse the position.</p> </li> </ol>","path":["Chapter 6. Hashing","6.5   Exercises"],"tags":[]},{"location":"chapter_hashing/exercises/#652-programming-exercises","level":2,"title":"6.5.2   Programming Exercises","text":"","path":["Chapter 6. Hashing","6.5   Exercises"],"tags":[]},{"location":"chapter_hashing/exercises/#1-compare-the-character-counts-of-two-strings","level":3,"title":"1.   Compare the Character Counts of Two Strings","text":"<p>Given two strings <code>s</code> and <code>t</code> containing only lowercase English letters, you may rearrange the characters in <code>s</code> in any order, but you may not add, remove, or replace characters.</p> <p>Determine whether the rearranged string can form <code>t</code>. Return <code>true</code> if it can and <code>false</code> otherwise. Use a hash table to record how many times each letter occurs. Do not sort the characters in the strings.</p> Hints <ol> <li>If the two strings have different lengths, they cannot contain each character the same number of times</li> <li>Use a hash table to record the count of each letter; increment the corresponding count while scanning s</li> <li>Decrement the corresponding count while scanning t; the strings contain each character the same number of times only if every count is 0 at the end</li> </ol> <p>LeetCode</p>","path":["Chapter 6. Hashing","6.5   Exercises"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/","level":1,"title":"6.3   Hash Algorithm","text":"<p>The previous two sections introduced the working principle of hash tables and the methods to handle hash collisions. However, both open addressing and separate chaining can only ensure that the hash table functions normally when hash collisions occur, but cannot reduce the frequency of hash collisions.</p> <p>If hash collisions occur too frequently, the performance of the hash table will deteriorate drastically. As shown in Figure 6-8, for a separate chaining hash table, in the ideal case, the key-value pairs are evenly distributed across the buckets, achieving optimal query efficiency; in the worst case, all key-value pairs are stored in the same bucket, degrading the time complexity to \\(O(n)\\).</p> <p></p> <p> Figure 6-8   Ideal and worst cases of hash collisions </p> <p>The distribution of key-value pairs is determined by the hash function. Recall the steps of the hash function: first compute the hash value, then take it modulo the array length:</p> <pre><code>index = hash(key) % capacity\n</code></pre> <p>Observing the above formula, when the hash table capacity <code>capacity</code> is fixed, the hash algorithm <code>hash()</code> determines the output value, thereby determining the distribution of key-value pairs in the hash table.</p> <p>This means that, to reduce the probability of hash collisions, we should focus on the design of the hash algorithm <code>hash()</code>.</p>","path":["Chapter 6. Hashing","6.3   Hash Algorithm"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#631-goals-of-hash-algorithms","level":2,"title":"6.3.1   Goals of Hash Algorithms","text":"<p>To build a hash table that is both fast and robust, a hash algorithm should have the following properties:</p> <ul> <li>Determinism: For the same input, the hash algorithm should always produce the same output. Only then can the hash table be reliable.</li> <li>High efficiency: The process of computing the hash value should be fast enough. The smaller the computational overhead, the more practical the hash table.</li> <li>Uniform distribution: The hash algorithm should ensure that key-value pairs are evenly distributed in the hash table. The more uniform the distribution, the lower the probability of hash collisions.</li> </ul> <p>In fact, hash algorithms are not only used to implement hash tables but are also widely applied in other fields.</p> <ul> <li>Password storage: To protect the security of user passwords, systems usually do not store the plaintext passwords but rather the hash values of the passwords. When a user enters a password, the system calculates the hash value of the input and compares it with the stored hash value. If they match, the password is considered correct.</li> <li>Data integrity check: The data sender can calculate the hash value of the data and send it along; the receiver can recalculate the hash value of the received data and compare it with the received hash value. If they match, the data is considered intact.</li> </ul> <p>For cryptographic applications, hash algorithms need stronger security properties to prevent reverse engineering, such as inferring the original password from a hash value.</p> <ul> <li>Unidirectionality: It should be impossible to deduce any information about the input data from the hash value.</li> <li>Collision resistance: It should be extremely difficult to find two different inputs that produce the same hash value.</li> <li>Avalanche effect: Minor changes in the input should lead to significant and unpredictable changes in the output.</li> </ul> <p>Note that \"uniform distribution\" and \"collision resistance\" are two independent concepts. Satisfying uniform distribution does not necessarily mean collision resistance. For example, under random input <code>key</code>, the hash function <code>key % 100</code> can produce a uniformly distributed output. However, this hash algorithm is too simple, and all <code>key</code> with the same last two digits will have the same output, making it easy to deduce a usable <code>key</code> from the hash value, thereby cracking the password.</p>","path":["Chapter 6. Hashing","6.3   Hash Algorithm"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#632-design-of-hash-algorithms","level":2,"title":"6.3.2   Design of Hash Algorithms","text":"<p>The design of hash algorithms is a complex issue that requires consideration of many factors. However, for some less demanding scenarios, we can also design some simple hash algorithms.</p> <ul> <li>Additive hash: Add up the ASCII codes of each character in the input and use the total sum as the hash value.</li> <li>Multiplicative hash: Leverage the low correlation introduced by multiplication: multiply by a constant at each step and accumulate the ASCII codes of the characters into the hash value.</li> <li>XOR hash: Accumulate the hash value by XORing each element of the input data.</li> <li>Rotating hash: Accumulate the ASCII code of each character into a hash value, performing a rotation operation on the hash value before each accumulation.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby simple_hash.py<pre><code>def add_hash(key: str) -&gt; int:\n    \"\"\"Additive hash\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash += ord(c)\n    return hash % modulus\n\ndef mul_hash(key: str) -&gt; int:\n    \"\"\"Multiplicative hash\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = 31 * hash + ord(c)\n    return hash % modulus\n\ndef xor_hash(key: str) -&gt; int:\n    \"\"\"XOR hash\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash ^= ord(c)\n    return hash % modulus\n\ndef rot_hash(key: str) -&gt; int:\n    \"\"\"Rotational hash\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ ord(c)\n    return hash % modulus\n</code></pre> simple_hash.cpp<pre><code>/* Additive hash */\nint addHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* Multiplicative hash */\nint mulHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (31 * hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* XOR hash */\nint xorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash ^= (int)c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* Rotational hash */\nint rotHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.java<pre><code>/* Additive hash */\nint addHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* Multiplicative hash */\nint mulHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (31 * hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* XOR hash */\nint xorHash(String key) {\n    int hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash ^= (int) c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* Rotational hash */\nint rotHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n</code></pre> simple_hash.cs<pre><code>/* Additive hash */\nint AddHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* Multiplicative hash */\nint MulHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (31 * hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* XOR hash */\nint XorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash ^= c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* Rotational hash */\nint RotHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.go<pre><code>/* Additive hash */\nfunc addHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* Multiplicative hash */\nfunc mulHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (31*hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* XOR hash */\nfunc xorHash(key string) int {\n    hash := 0\n    modulus := 1000000007\n    for _, b := range []byte(key) {\n        fmt.Println(int(b))\n        hash ^= int(b)\n        hash = (31*hash + int(b)) % modulus\n    }\n    return hash &amp; modulus\n}\n\n/* Rotational hash */\nfunc rotHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ int64(b)) % modulus\n    }\n    return int(hash)\n}\n</code></pre> simple_hash.swift<pre><code>/* Additive hash */\nfunc addHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* Multiplicative hash */\nfunc mulHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (31 * hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* XOR hash */\nfunc xorHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash ^= Int(scalar.value)\n        }\n    }\n    return hash &amp; MODULUS\n}\n\n/* Rotational hash */\nfunc rotHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n</code></pre> simple_hash.js<pre><code>/* Additive hash */\nfunction addHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* Multiplicative hash */\nfunction mulHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* XOR hash */\nfunction xorHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash % MODULUS;\n}\n\n/* Rotational hash */\nfunction rotHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.ts<pre><code>/* Additive hash */\nfunction addHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* Multiplicative hash */\nfunction mulHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* XOR hash */\nfunction xorHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash % MODULUS;\n}\n\n/* Rotational hash */\nfunction rotHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.dart<pre><code>/* Additive hash */\nint addHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* Multiplicative hash */\nint mulHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (31 * hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* XOR hash */\nint xorHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash ^= key.codeUnitAt(i);\n  }\n  return hash &amp; MODULUS;\n}\n\n/* Rotational hash */\nint rotHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n</code></pre> simple_hash.rs<pre><code>/* Additive hash */\nfn add_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* Multiplicative hash */\nfn mul_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (31 * hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* XOR hash */\nfn xor_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash ^= c as i64;\n    }\n\n    (hash &amp; MODULUS) as i32\n}\n\n/* Rotational hash */\nfn rot_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n</code></pre> simple_hash.c<pre><code>/* Additive hash */\nint addHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* Multiplicative hash */\nint mulHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (31 * hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* XOR hash */\nint xorHash(char *key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash ^= (unsigned char)key[i];\n    }\n    return hash &amp; MODULUS;\n}\n\n/* Rotational hash */\nint rotHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (unsigned char)key[i]) % MODULUS;\n    }\n\n    return (int)hash;\n}\n</code></pre> simple_hash.kt<pre><code>/* Additive hash */\nfun addHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = (hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* Multiplicative hash */\nfun mulHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = (31 * hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* XOR hash */\nfun xorHash(key: String): Int {\n    var hash = 0\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = hash xor c.code\n    }\n    return hash and MODULUS\n}\n\n/* Rotational hash */\nfun rotHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = ((hash shl 4) xor (hash shr 28) xor c.code.toLong()) % MODULUS\n    }\n    return hash.toInt()\n}\n</code></pre> simple_hash.rb<pre><code>### Additive hash ###\ndef add_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash += c.ord }\n\n  hash % modulus\nend\n\n### Multiplicative hash ###\ndef mul_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash = 31 * hash + c.ord }\n\n  hash % modulus\nend\n\n### XOR hash ###\ndef xor_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash ^= c.ord }\n\n  hash % modulus\nend\n\n### Rotational hash ###\ndef rot_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.ord }\n\n  hash % modulus\nend\n</code></pre> <p>We can observe that the final step of each hash algorithm is to take the result modulo the large prime \\(1000000007\\), ensuring that the hash value stays within a suitable range. This naturally raises a question: why emphasize using a prime modulus, and what are the drawbacks of using a composite modulus?</p> <p>In short: using a large prime as the modulus helps maximize the uniformity of hash values. Because a prime shares no common factors with other numbers, it can reduce periodic patterns introduced by the modulo operation and thus mitigate hash collisions.</p> <p>For example, suppose we choose the composite number \\(9\\) as the modulus, which can be divided by \\(3\\), then all <code>key</code> divisible by \\(3\\) will be mapped to hash values \\(0\\), \\(3\\), \\(6\\).</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 9 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\dots \\} \\end{aligned} \\] <p>If the input <code>key</code> values happen to follow this kind of arithmetic progression, the hash values will cluster, worsening hash collisions. Now suppose we replace <code>modulus</code> with the prime number \\(13\\). Because <code>key</code> and <code>modulus</code> share no common factors, the output hash values become much more evenly distributed.</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 13 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\dots \\} \\end{aligned} \\] <p>It is worth noting that if the <code>key</code> is guaranteed to be randomly and uniformly distributed, then choosing a prime number or a composite number as the modulus can both produce uniformly distributed hash values. However, when the distribution of <code>key</code> has some periodicity, modulo a composite number is more likely to result in clustering.</p> <p>In summary, we usually choose a prime number as the modulus, and this prime number should be large enough to eliminate periodic patterns as much as possible, enhancing the robustness of the hash algorithm.</p>","path":["Chapter 6. Hashing","6.3   Hash Algorithm"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#633-common-hash-algorithms","level":2,"title":"6.3.3   Common Hash Algorithms","text":"<p>It is easy to see that the simple hash algorithms introduced above are fairly \"fragile\" and fall far short of the design goals of hash algorithms. For example, because addition and XOR are commutative, additive hash and XOR hash cannot distinguish strings with the same characters in a different order, which may worsen hash collisions and introduce security risks.</p> <p>In practice, we usually use some standard hash algorithms, such as MD5, SHA-1, SHA-2, and SHA-3. They can map input data of any length to a fixed-length hash value.</p> <p>Over the past century, hash algorithms have been in a continuous process of upgrading and optimization. Some researchers strive to improve the performance of hash algorithms, while others, including hackers, are dedicated to finding security issues in hash algorithms. Table 6-2 shows hash algorithms commonly used in practical applications.</p> <ul> <li>MD5 and SHA-1 have been successfully attacked multiple times and are thus abandoned in various security applications.</li> <li>SHA-2 series, especially SHA-256, is one of the most secure hash algorithms to date, with no successful attacks reported, hence commonly used in various security applications and protocols.</li> <li>SHA-3 has lower implementation costs and higher computational efficiency compared to SHA-2, but its current usage coverage is not as extensive as the SHA-2 series.</li> </ul> <p> Table 6-2   Common hash algorithms </p> MD5 SHA-1 SHA-2 SHA-3 Release Year 1992 1995 2002 2008 Output Length 128 bit 160 bit 256/512 bit 224/256/384/512 bit Hash Collisions Frequent Frequent Rare Rare Security Level Low, has been successfully attacked Low, has been successfully attacked High High Applications Abandoned, still used for data integrity checks Abandoned Cryptocurrency transaction verification, digital signatures, etc. Can be used to replace SHA-2","path":["Chapter 6. Hashing","6.3   Hash Algorithm"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#634-hash-values-in-data-structures","level":2,"title":"6.3.4   Hash Values in Data Structures","text":"<p>We know that hash table keys can be integers, floating-point numbers, strings, and other data types. Programming languages usually provide built-in hash algorithms for these types to compute bucket indices in a hash table. Taking Python as an example, we can call the <code>hash()</code> function to compute hash values for various data types.</p> <ul> <li>The hash values of integers and booleans are their own values.</li> <li>The calculation of hash values for floating-point numbers and strings is more complex, and interested readers are encouraged to study this on their own.</li> <li>The hash value of a tuple is obtained by hashing each of its elements and combining those results into a single hash value.</li> <li>An object's hash value is typically generated from its memory address. By overriding the object's hash method, it can instead be generated from the object's contents.</li> </ul> <p>Tip</p> <p>Be aware that the definition and methods of the built-in hash value calculation functions in different programming languages vary.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby built_in_hash.py<pre><code>num = 3\nhash_num = hash(num)\n# Hash value of integer 3 is 3\n\nbol = True\nhash_bol = hash(bol)\n# Hash value of boolean True is 1\n\ndec = 3.14159\nhash_dec = hash(dec)\n# Hash value of decimal 3.14159 is 326484311674566659\n\nstr = \"Hello 算法\"\nhash_str = hash(str)\n# Hash value of string \"Hello 算法\" is 4617003410720528961\n\ntup = (12836, \"小哈\")\nhash_tup = hash(tup)\n# Hash value of tuple (12836, '小哈') is 1029005403108185979\n\nobj = ListNode(0)\nhash_obj = hash(obj)\n# Hash value of ListNode object at 0x1058fd810 is 274267521\n</code></pre> built_in_hash.cpp<pre><code>int num = 3;\nsize_t hashNum = hash&lt;int&gt;()(num);\n// Hash value of integer 3 is 3\n\nbool bol = true;\nsize_t hashBol = hash&lt;bool&gt;()(bol);\n// Hash value of boolean 1 is 1\n\ndouble dec = 3.14159;\nsize_t hashDec = hash&lt;double&gt;()(dec);\n// Hash value of decimal 3.14159 is 4614256650576692846\n\nstring str = \"Hello 算法\";\nsize_t hashStr = hash&lt;string&gt;()(str);\n// Hash value of string \"Hello 算法\" is 15466937326284535026\n\n// In C++, built-in std::hash() only provides hash values for basic data types\n// Hash values for arrays and objects need to be implemented separately\n</code></pre> built_in_hash.java<pre><code>int num = 3;\nint hashNum = Integer.hashCode(num);\n// Hash value of integer 3 is 3\n\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// Hash value of boolean true is 1231\n\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// Hash value of decimal 3.14159 is -1340954729\n\nString str = \"Hello 算法\";\nint hashStr = str.hashCode();\n// Hash value of string \"Hello 算法\" is -727081396\n\nObject[] arr = { 12836, \"小哈\" };\nint hashTup = Arrays.hashCode(arr);\n// Hash value of array [12836, 小哈] is 1151158\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// Hash value of ListNode object utils.ListNode@7dc5e7b4 is 2110121908\n</code></pre> built_in_hash.cs<pre><code>int num = 3;\nint hashNum = num.GetHashCode();\n// Hash value of integer 3 is 3;\n\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// Hash value of boolean true is 1;\n\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// Hash value of decimal 3.14159 is -1340954729;\n\nstring str = \"Hello 算法\";\nint hashStr = str.GetHashCode();\n// Hash value of string \"Hello 算法\" is -586107568;\n\nobject[] arr = [12836, \"小哈\"];\nint hashTup = arr.GetHashCode();\n// Hash value of array [12836, 小哈] is 42931033;\n\nListNode obj = new(0);\nint hashObj = obj.GetHashCode();\n// Hash value of ListNode object 0 is 39053774;\n</code></pre> built_in_hash.go<pre><code>// Go does not provide built-in hash code functions\n</code></pre> built_in_hash.swift<pre><code>let num = 3\nlet hashNum = num.hashValue\n// Hash value of integer 3 is 9047044699613009734\n\nlet bol = true\nlet hashBol = bol.hashValue\n// Hash value of boolean true is -4431640247352757451\n\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// Hash value of decimal 3.14159 is -2465384235396674631\n\nlet str = \"Hello 算法\"\nlet hashStr = str.hashValue\n// Hash value of string \"Hello 算法\" is -7850626797806988787\n\nlet arr = [AnyHashable(12836), AnyHashable(\"小哈\")]\nlet hashTup = arr.hashValue\n// Hash value of array [AnyHashable(12836), AnyHashable(\"小哈\")] is -2308633508154532996\n\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// Hash value of ListNode object utils.ListNode is -2434780518035996159\n</code></pre> built_in_hash.js<pre><code>// JavaScript does not provide built-in hash code functions\n</code></pre> built_in_hash.ts<pre><code>// TypeScript does not provide built-in hash code functions\n</code></pre> built_in_hash.dart<pre><code>int num = 3;\nint hashNum = num.hashCode;\n// Hash value of integer 3 is 34803\n\nbool bol = true;\nint hashBol = bol.hashCode;\n// Hash value of boolean true is 1231\n\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// Hash value of decimal 3.14159 is 2570631074981783\n\nString str = \"Hello 算法\";\nint hashStr = str.hashCode;\n// Hash value of string \"Hello 算法\" is 468167534\n\nList arr = [12836, \"小哈\"];\nint hashArr = arr.hashCode;\n// Hash value of array [12836, 小哈] is 976512528\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// Hash value of ListNode object Instance of 'ListNode' is 1033450432\n</code></pre> built_in_hash.rs<pre><code>use std::collections::hash_map::DefaultHasher;\nuse std::hash::{Hash, Hasher};\n\nlet num = 3;\nlet mut num_hasher = DefaultHasher::new();\nnum.hash(&amp;mut num_hasher);\nlet hash_num = num_hasher.finish();\n// Hash value of integer 3 is 568126464209439262\n\nlet bol = true;\nlet mut bol_hasher = DefaultHasher::new();\nbol.hash(&amp;mut bol_hasher);\nlet hash_bol = bol_hasher.finish();\n// Hash value of boolean true is 4952851536318644461\n\nlet dec: f32 = 3.14159;\nlet mut dec_hasher = DefaultHasher::new();\ndec.to_bits().hash(&amp;mut dec_hasher);\nlet hash_dec = dec_hasher.finish();\n// Hash value of decimal 3.14159 is 2566941990314602357\n\nlet str = \"Hello 算法\";\nlet mut str_hasher = DefaultHasher::new();\nstr.hash(&amp;mut str_hasher);\nlet hash_str = str_hasher.finish();\n// Hash value of string \"Hello 算法\" is 16092673739211250988\n\nlet arr = (&amp;12836, &amp;\"小哈\");\nlet mut tup_hasher = DefaultHasher::new();\narr.hash(&amp;mut tup_hasher);\nlet hash_tup = tup_hasher.finish();\n// Hash value of tuple (12836, \"小哈\") is 1885128010422702749\n\nlet node = ListNode::new(42);\nlet mut hasher = DefaultHasher::new();\nnode.borrow().val.hash(&amp;mut hasher);\nlet hash = hasher.finish();\n// Hash value of ListNode object RefCell { value: ListNode { val: 42, next: None } } is 15387811073369036852\n</code></pre> built_in_hash.c<pre><code>// C does not provide built-in hash code functions\n</code></pre> built_in_hash.kt<pre><code>val num = 3\nval hashNum = num.hashCode()\n// Hash value of integer 3 is 3\n\nval bol = true\nval hashBol = bol.hashCode()\n// Hash value of boolean true is 1231\n\nval dec = 3.14159\nval hashDec = dec.hashCode()\n// Hash value of decimal 3.14159 is -1340954729\n\nval str = \"Hello 算法\"\nval hashStr = str.hashCode()\n// Hash value of string \"Hello 算法\" is -727081396\n\nval arr = arrayOf&lt;Any&gt;(12836, \"小哈\")\nval hashTup = arr.hashCode()\n// Hash value of array [12836, 小哈] is 189568618\n\nval obj = ListNode(0)\nval hashObj = obj.hashCode()\n// Hash value of ListNode object utils.ListNode@1d81eb93 is 495053715\n</code></pre> built_in_hash.rb<pre><code>num = 3\nhash_num = num.hash\n# Hash value of integer 3 is -4385856518450339636\n\nbol = true\nhash_bol = bol.hash\n# Hash value of boolean true is -1617938112149317027\n\ndec = 3.14159\nhash_dec = dec.hash\n# Hash value of decimal 3.14159 is -1479186995943067893\n\nstr = \"Hello 算法\"\nhash_str = str.hash\n# Hash value of string \"Hello 算法\" is -4075943250025831763\n\ntup = [12836, '小哈']\nhash_tup = tup.hash\n# Hash value of tuple (12836, '小哈') is 1999544809202288822\n\nobj = ListNode.new(0)\nhash_obj = obj.hash\n# Hash value of ListNode object #&lt;ListNode:0x000078133140ab70&gt; is 4302940560806366381\n</code></pre> Visualized Execution <p> Full Screen &gt;</p> <p>In many programming languages, only immutable objects can serve as the <code>key</code> in a hash table. If we use a list (dynamic array) as a <code>key</code>, when the contents of the list change, its hash value also changes, and we would no longer be able to find the original <code>value</code> in the hash table.</p> <p>Although the member variables of a custom object (such as a linked list node) are mutable, it is hashable. This is because the hash value of an object is usually generated based on its memory address, and even if the contents of the object change, the memory address remains the same, so the hash value remains unchanged.</p> <p>You might have noticed that the hash values output in different consoles are different. This is because the Python interpreter adds a random salt to the string hash function each time it starts up. This approach effectively prevents HashDoS attacks and enhances the security of the hash algorithm.</p>","path":["Chapter 6. Hashing","6.3   Hash Algorithm"],"tags":[]},{"location":"chapter_hashing/hash_collision/","level":1,"title":"6.2   Hash Collision","text":"<p>The previous section mentioned that, in most cases, the input space of a hash function is much larger than the output space, so theoretically, hash collisions are inevitable. For example, if the input space is all integers and the output space is the array capacity size, then multiple integers will inevitably be mapped to the same bucket index.</p> <p>Hash collisions can lead to incorrect query results, severely impacting the usability of the hash table. To address this issue, whenever a hash collision occurs, we can perform hash table expansion until the collision disappears. This approach is simple, straightforward, and effective, but it is very inefficient because hash table expansion involves a large amount of data migration and hash value recalculation. To improve efficiency, we can adopt the following strategies:</p> <ol> <li>Improve the hash table data structure so that the hash table can function normally when hash collisions occur.</li> <li>Only expand when necessary, that is, only when hash collisions are severe.</li> </ol> <p>The main approaches to improving a hash table's structure are separate chaining and open addressing.</p>","path":["Chapter 6. Hashing","6.2   Hash Collision"],"tags":[]},{"location":"chapter_hashing/hash_collision/#621-separate-chaining","level":2,"title":"6.2.1   Separate Chaining","text":"<p>In the original hash table, each bucket can store only one key-value pair. Separate chaining replaces the single element in each bucket with a linked list, treating each key-value pair as a node and storing all colliding key-value pairs in the same list. Figure 6-5 shows an example of a separate chaining hash table.</p> <p></p> <p> Figure 6-5   Separate chaining hash table </p> <p>In a hash table implemented with separate chaining, the basic operations work as follows:</p> <ul> <li>Querying elements: Input <code>key</code>, compute the bucket index using the hash function, access the head of the corresponding linked list, and traverse the list while comparing keys until the target key-value pair is found.</li> <li>Adding elements: First use the hash function to locate the corresponding linked list, then insert the node (key-value pair) into the list.</li> <li>Deleting elements: Use the hash function to locate the corresponding linked list, then traverse it to find and delete the target node.</li> </ul> <p>Separate chaining has the following limitations:</p> <ul> <li>Increased Space Usage: The linked list contains node pointers, which consume more memory space than arrays.</li> <li>Reduced Query Efficiency: This is because linear traversal of the linked list is required to find the corresponding element.</li> </ul> <p>The code below provides a simple implementation of a separate chaining hash table, with two things to note:</p> <ul> <li>Lists (dynamic arrays) are used instead of linked lists to simplify the code. In this setup, the hash table (array) contains multiple buckets, each of which is a list.</li> <li>This implementation includes a hash table expansion method. When the load factor exceeds \\(\\frac{2}{3}\\), we expand the hash table to \\(2\\) times its original size.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map_chaining.py<pre><code>class HashMapChaining:\n    \"\"\"Hash table with separate chaining\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        self.size = 0  # Number of key-value pairs\n        self.capacity = 4  # Hash table capacity\n        self.load_thres = 2.0 / 3.0  # Load factor threshold for triggering expansion\n        self.extend_ratio = 2  # Expansion multiplier\n        self.buckets = [[] for _ in range(self.capacity)]  # Bucket array\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"Hash function\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"Load factor\"\"\"\n        return self.size / self.capacity\n\n    def get(self, key: int) -&gt; str | None:\n        \"\"\"Query operation\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # Traverse bucket, if key is found, return corresponding val\n        for pair in bucket:\n            if pair.key == key:\n                return pair.val\n        # If key is not found, return None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"Add operation\"\"\"\n        # When load factor exceeds threshold, perform expansion\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # Traverse bucket, if specified key is encountered, update corresponding val and return\n        for pair in bucket:\n            if pair.key == key:\n                pair.val = val\n                return\n        # If key does not exist, append key-value pair to the end\n        pair = Pair(key, val)\n        bucket.append(pair)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"Remove operation\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # Traverse bucket and remove key-value pair from it\n        for pair in bucket:\n            if pair.key == key:\n                bucket.remove(pair)\n                self.size -= 1\n                break\n\n    def extend(self):\n        \"\"\"Expand hash table\"\"\"\n        # Temporarily store the original hash table\n        buckets = self.buckets\n        # Initialize expanded new hash table\n        self.capacity *= self.extend_ratio\n        self.buckets = [[] for _ in range(self.capacity)]\n        self.size = 0\n        # Move key-value pairs from original hash table to new hash table\n        for bucket in buckets:\n            for pair in bucket:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"Print hash table\"\"\"\n        for bucket in self.buckets:\n            res = []\n            for pair in bucket:\n                res.append(str(pair.key) + \" -&gt; \" + pair.val)\n            print(res)\n</code></pre> hash_map_chaining.cpp<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n  private:\n    int size;                       // Number of key-value pairs\n    int capacity;                   // Hash table capacity\n    double loadThres;               // Load factor threshold for triggering expansion\n    int extendRatio;                // Expansion multiplier\n    vector&lt;vector&lt;Pair *&gt;&gt; buckets; // Bucket array\n\n  public:\n    /* Constructor */\n    HashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3.0), extendRatio(2) {\n        buckets.resize(capacity);\n    }\n\n    /* Destructor */\n    ~HashMapChaining() {\n        for (auto &amp;bucket : buckets) {\n            for (Pair *pair : bucket) {\n                // Free memory\n                delete pair;\n            }\n        }\n    }\n\n    /* Hash function */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* Load factor */\n    double loadFactor() {\n        return (double)size / (double)capacity;\n    }\n\n    /* Query operation */\n    string get(int key) {\n        int index = hashFunc(key);\n        // Traverse bucket, if key is found, return corresponding val\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                return pair-&gt;val;\n            }\n        }\n        // Return empty string if key not found\n        return \"\";\n    }\n\n    /* Add operation */\n    void put(int key, string val) {\n        // When load factor exceeds threshold, perform expansion\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        // Traverse bucket, if specified key is encountered, update corresponding val and return\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                pair-&gt;val = val;\n                return;\n            }\n        }\n        // If key does not exist, append key-value pair to the end\n        buckets[index].push_back(new Pair(key, val));\n        size++;\n    }\n\n    /* Remove operation */\n    void remove(int key) {\n        int index = hashFunc(key);\n        auto &amp;bucket = buckets[index];\n        // Traverse bucket and remove key-value pair from it\n        for (int i = 0; i &lt; bucket.size(); i++) {\n            if (bucket[i]-&gt;key == key) {\n                Pair *tmp = bucket[i];\n                bucket.erase(bucket.begin() + i); // Remove key-value pair from it\n                delete tmp;                       // Free memory\n                size--;\n                return;\n            }\n        }\n    }\n\n    /* Expand hash table */\n    void extend() {\n        // Temporarily store the original hash table\n        vector&lt;vector&lt;Pair *&gt;&gt; bucketsTmp = buckets;\n        // Initialize expanded new hash table\n        capacity *= extendRatio;\n        buckets.clear();\n        buckets.resize(capacity);\n        size = 0;\n        // Move key-value pairs from original hash table to new hash table\n        for (auto &amp;bucket : bucketsTmp) {\n            for (Pair *pair : bucket) {\n                put(pair-&gt;key, pair-&gt;val);\n                // Free memory\n                delete pair;\n            }\n        }\n    }\n\n    /* Print hash table */\n    void print() {\n        for (auto &amp;bucket : buckets) {\n            cout &lt;&lt; \"[\";\n            for (Pair *pair : bucket) {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; \", \";\n            }\n            cout &lt;&lt; \"]\\n\";\n        }\n    }\n};\n</code></pre> hash_map_chaining.java<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n    int size; // Number of key-value pairs\n    int capacity; // Hash table capacity\n    double loadThres; // Load factor threshold for triggering expansion\n    int extendRatio; // Expansion multiplier\n    List&lt;List&lt;Pair&gt;&gt; buckets; // Bucket array\n\n    /* Constructor */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n    }\n\n    /* Hash function */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* Load factor */\n    double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* Query operation */\n    String get(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // Traverse bucket, if key is found, return corresponding val\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // If key is not found, return null\n        return null;\n    }\n\n    /* Add operation */\n    void put(int key, String val) {\n        // When load factor exceeds threshold, perform expansion\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // Traverse bucket, if specified key is encountered, update corresponding val and return\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // If key does not exist, append key-value pair to the end\n        Pair pair = new Pair(key, val);\n        bucket.add(pair);\n        size++;\n    }\n\n    /* Remove operation */\n    void remove(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // Traverse bucket and remove key-value pair from it\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* Expand hash table */\n    void extend() {\n        // Temporarily store the original hash table\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // Initialize expanded new hash table\n        capacity *= extendRatio;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n        size = 0;\n        // Move key-value pairs from original hash table to new hash table\n        for (List&lt;Pair&gt; bucket : bucketsTmp) {\n            for (Pair pair : bucket) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print hash table */\n    void print() {\n        for (List&lt;Pair&gt; bucket : buckets) {\n            List&lt;String&gt; res = new ArrayList&lt;&gt;();\n            for (Pair pair : bucket) {\n                res.add(pair.key + \" -&gt; \" + pair.val);\n            }\n            System.out.println(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.cs<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n    int size; // Number of key-value pairs\n    int capacity; // Hash table capacity\n    double loadThres; // Load factor threshold for triggering expansion\n    int extendRatio; // Expansion multiplier\n    List&lt;List&lt;Pair&gt;&gt; buckets; // Bucket array\n\n    /* Constructor */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n    }\n\n    /* Hash function */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* Load factor */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* Query operation */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        // Traverse bucket, if key is found, return corresponding val\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // If key is not found, return null\n        return null;\n    }\n\n    /* Add operation */\n    public void Put(int key, string val) {\n        // When load factor exceeds threshold, perform expansion\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        int index = HashFunc(key);\n        // Traverse bucket, if specified key is encountered, update corresponding val and return\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // If key does not exist, append key-value pair to the end\n        buckets[index].Add(new Pair(key, val));\n        size++;\n    }\n\n    /* Remove operation */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // Traverse bucket and remove key-value pair from it\n        foreach (Pair pair in buckets[index].ToList()) {\n            if (pair.key == key) {\n                buckets[index].Remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* Expand hash table */\n    void Extend() {\n        // Temporarily store the original hash table\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // Initialize expanded new hash table\n        capacity *= extendRatio;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n        size = 0;\n        // Move key-value pairs from original hash table to new hash table\n        foreach (List&lt;Pair&gt; bucket in bucketsTmp) {\n            foreach (Pair pair in bucket) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print hash table */\n    public void Print() {\n        foreach (List&lt;Pair&gt; bucket in buckets) {\n            List&lt;string&gt; res = [];\n            foreach (Pair pair in bucket) {\n                res.Add(pair.key + \" -&gt; \" + pair.val);\n            }\n            foreach (string kv in res) {\n                Console.WriteLine(kv);\n            }\n        }\n    }\n}\n</code></pre> hash_map_chaining.go<pre><code>/* Hash table with separate chaining */\ntype hashMapChaining struct {\n    size        int      // Number of key-value pairs\n    capacity    int      // Hash table capacity\n    loadThres   float64  // Load factor threshold for triggering expansion\n    extendRatio int      // Expansion multiplier\n    buckets     [][]pair // Bucket array\n}\n\n/* Constructor */\nfunc newHashMapChaining() *hashMapChaining {\n    buckets := make([][]pair, 4)\n    for i := 0; i &lt; 4; i++ {\n        buckets[i] = make([]pair, 0)\n    }\n    return &amp;hashMapChaining{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     buckets,\n    }\n}\n\n/* Hash function */\nfunc (m *hashMapChaining) hashFunc(key int) int {\n    return key % m.capacity\n}\n\n/* Load factor */\nfunc (m *hashMapChaining) loadFactor() float64 {\n    return float64(m.size) / float64(m.capacity)\n}\n\n/* Query operation */\nfunc (m *hashMapChaining) get(key int) string {\n    idx := m.hashFunc(key)\n    bucket := m.buckets[idx]\n    // Traverse bucket, if key is found, return corresponding val\n    for _, p := range bucket {\n        if p.key == key {\n            return p.val\n        }\n    }\n    // Return empty string if key not found\n    return \"\"\n}\n\n/* Add operation */\nfunc (m *hashMapChaining) put(key int, val string) {\n    // When load factor exceeds threshold, perform expansion\n    if m.loadFactor() &gt; m.loadThres {\n        m.extend()\n    }\n    idx := m.hashFunc(key)\n    // Traverse bucket, if specified key is encountered, update corresponding val and return\n    for i := range m.buckets[idx] {\n        if m.buckets[idx][i].key == key {\n            m.buckets[idx][i].val = val\n            return\n        }\n    }\n    // If key does not exist, append key-value pair to the end\n    p := pair{\n        key: key,\n        val: val,\n    }\n    m.buckets[idx] = append(m.buckets[idx], p)\n    m.size += 1\n}\n\n/* Remove operation */\nfunc (m *hashMapChaining) remove(key int) {\n    idx := m.hashFunc(key)\n    // Traverse bucket and remove key-value pair from it\n    for i, p := range m.buckets[idx] {\n        if p.key == key {\n            // Slice deletion\n            m.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\n            m.size -= 1\n            break\n        }\n    }\n}\n\n/* Expand hash table */\nfunc (m *hashMapChaining) extend() {\n    // Temporarily store the original hash table\n    tmpBuckets := make([][]pair, len(m.buckets))\n    for i := 0; i &lt; len(m.buckets); i++ {\n        tmpBuckets[i] = make([]pair, len(m.buckets[i]))\n        copy(tmpBuckets[i], m.buckets[i])\n    }\n    // Initialize expanded new hash table\n    m.capacity *= m.extendRatio\n    m.buckets = make([][]pair, m.capacity)\n    for i := 0; i &lt; m.capacity; i++ {\n        m.buckets[i] = make([]pair, 0)\n    }\n    m.size = 0\n    // Move key-value pairs from original hash table to new hash table\n    for _, bucket := range tmpBuckets {\n        for _, p := range bucket {\n            m.put(p.key, p.val)\n        }\n    }\n}\n\n/* Print hash table */\nfunc (m *hashMapChaining) print() {\n    var builder strings.Builder\n\n    for _, bucket := range m.buckets {\n        builder.WriteString(\"[\")\n        for _, p := range bucket {\n            builder.WriteString(strconv.Itoa(p.key) + \" -&gt; \" + p.val + \" \")\n        }\n        builder.WriteString(\"]\")\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> hash_map_chaining.swift<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n    var size: Int // Number of key-value pairs\n    var capacity: Int // Hash table capacity\n    var loadThres: Double // Load factor threshold for triggering expansion\n    var extendRatio: Int // Expansion multiplier\n    var buckets: [[Pair]] // Bucket array\n\n    /* Constructor */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: [], count: capacity)\n    }\n\n    /* Hash function */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* Load factor */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* Query operation */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // Traverse bucket, if key is found, return corresponding val\n        for pair in bucket {\n            if pair.key == key {\n                return pair.val\n            }\n        }\n        // Return nil if key not found\n        return nil\n    }\n\n    /* Add operation */\n    func put(key: Int, val: String) {\n        // When load factor exceeds threshold, perform expansion\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // Traverse bucket, if specified key is encountered, update corresponding val and return\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val\n                return\n            }\n        }\n        // If key does not exist, append key-value pair to the end\n        let pair = Pair(key: key, val: val)\n        buckets[index].append(pair)\n        size += 1\n    }\n\n    /* Remove operation */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // Traverse bucket and remove key-value pair from it\n        for (pairIndex, pair) in bucket.enumerated() {\n            if pair.key == key {\n                buckets[index].remove(at: pairIndex)\n                size -= 1\n                break\n            }\n        }\n    }\n\n    /* Expand hash table */\n    func extend() {\n        // Temporarily store the original hash table\n        let bucketsTmp = buckets\n        // Initialize expanded new hash table\n        capacity *= extendRatio\n        buckets = Array(repeating: [], count: capacity)\n        size = 0\n        // Move key-value pairs from original hash table to new hash table\n        for bucket in bucketsTmp {\n            for pair in bucket {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* Print hash table */\n    func print() {\n        for bucket in buckets {\n            let res = bucket.map { \"\\($0.key) -&gt; \\($0.val)\" }\n            Swift.print(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.js<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n    #size; // Number of key-value pairs\n    #capacity; // Hash table capacity\n    #loadThres; // Load factor threshold for triggering expansion\n    #extendRatio; // Expansion multiplier\n    #buckets; // Bucket array\n\n    /* Constructor */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* Hash function */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* Load factor */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* Query operation */\n    get(key) {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // Traverse bucket, if key is found, return corresponding val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // If key is not found, return null\n        return null;\n    }\n\n    /* Add operation */\n    put(key, val) {\n        // When load factor exceeds threshold, perform expansion\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // Traverse bucket, if specified key is encountered, update corresponding val and return\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // If key does not exist, append key-value pair to the end\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* Remove operation */\n    remove(key) {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // Traverse bucket and remove key-value pair from it\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* Expand hash table */\n    #extend() {\n        // Temporarily store the original hash table\n        const bucketsTmp = this.#buckets;\n        // Initialize expanded new hash table\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // Move key-value pairs from original hash table to new hash table\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print hash table */\n    print() {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.ts<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n    #size: number; // Number of key-value pairs\n    #capacity: number; // Hash table capacity\n    #loadThres: number; // Load factor threshold for triggering expansion\n    #extendRatio: number; // Expansion multiplier\n    #buckets: Pair[][]; // Bucket array\n\n    /* Constructor */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* Hash function */\n    #hashFunc(key: number): number {\n        return key % this.#capacity;\n    }\n\n    /* Load factor */\n    #loadFactor(): number {\n        return this.#size / this.#capacity;\n    }\n\n    /* Query operation */\n    get(key: number): string | null {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // Traverse bucket, if key is found, return corresponding val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // If key is not found, return null\n        return null;\n    }\n\n    /* Add operation */\n    put(key: number, val: string): void {\n        // When load factor exceeds threshold, perform expansion\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // Traverse bucket, if specified key is encountered, update corresponding val and return\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // If key does not exist, append key-value pair to the end\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* Remove operation */\n    remove(key: number): void {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // Traverse bucket and remove key-value pair from it\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* Expand hash table */\n    #extend(): void {\n        // Temporarily store the original hash table\n        const bucketsTmp = this.#buckets;\n        // Initialize expanded new hash table\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // Move key-value pairs from original hash table to new hash table\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print hash table */\n    print(): void {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.dart<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n  late int size; // Number of key-value pairs\n  late int capacity; // Hash table capacity\n  late double loadThres; // Load factor threshold for triggering expansion\n  late int extendRatio; // Expansion multiplier\n  late List&lt;List&lt;Pair&gt;&gt; buckets; // Bucket array\n\n  /* Constructor */\n  HashMapChaining() {\n    size = 0;\n    capacity = 4;\n    loadThres = 2.0 / 3.0;\n    extendRatio = 2;\n    buckets = List.generate(capacity, (_) =&gt; []);\n  }\n\n  /* Hash function */\n  int hashFunc(int key) {\n    return key % capacity;\n  }\n\n  /* Load factor */\n  double loadFactor() {\n    return size / capacity;\n  }\n\n  /* Query operation */\n  String? get(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // Traverse bucket, if key is found, return corresponding val\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        return pair.val;\n      }\n    }\n    // If key is not found, return null\n    return null;\n  }\n\n  /* Add operation */\n  void put(int key, String val) {\n    // When load factor exceeds threshold, perform expansion\n    if (loadFactor() &gt; loadThres) {\n      extend();\n    }\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // Traverse bucket, if specified key is encountered, update corresponding val and return\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        pair.val = val;\n        return;\n      }\n    }\n    // If key does not exist, append key-value pair to the end\n    Pair pair = Pair(key, val);\n    bucket.add(pair);\n    size++;\n  }\n\n  /* Remove operation */\n  void remove(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // Traverse bucket and remove key-value pair from it\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        bucket.remove(pair);\n        size--;\n        break;\n      }\n    }\n  }\n\n  /* Expand hash table */\n  void extend() {\n    // Temporarily store the original hash table\n    List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n    // Initialize expanded new hash table\n    capacity *= extendRatio;\n    buckets = List.generate(capacity, (_) =&gt; []);\n    size = 0;\n    // Move key-value pairs from original hash table to new hash table\n    for (List&lt;Pair&gt; bucket in bucketsTmp) {\n      for (Pair pair in bucket) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* Print hash table */\n  void printHashMap() {\n    for (List&lt;Pair&gt; bucket in buckets) {\n      List&lt;String&gt; res = [];\n      for (Pair pair in bucket) {\n        res.add(\"${pair.key} -&gt; ${pair.val}\");\n      }\n      print(res);\n    }\n  }\n}\n</code></pre> hash_map_chaining.rs<pre><code>/* Hash table with separate chaining */\nstruct HashMapChaining {\n    size: usize,\n    capacity: usize,\n    load_thres: f32,\n    extend_ratio: usize,\n    buckets: Vec&lt;Vec&lt;Pair&gt;&gt;,\n}\n\nimpl HashMapChaining {\n    /* Constructor */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![vec![]; 4],\n        }\n    }\n\n    /* Hash function */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % self.capacity\n    }\n\n    /* Load factor */\n    fn load_factor(&amp;self) -&gt; f32 {\n        self.size as f32 / self.capacity as f32\n    }\n\n    /* Remove operation */\n    fn remove(&amp;mut self, key: i32) -&gt; Option&lt;String&gt; {\n        let index = self.hash_func(key);\n\n        // Traverse bucket and remove key-value pair from it\n        for (i, p) in self.buckets[index].iter_mut().enumerate() {\n            if p.key == key {\n                let pair = self.buckets[index].remove(i);\n                self.size -= 1;\n                return Some(pair.val);\n            }\n        }\n\n        // If key is not found, return None\n        None\n    }\n\n    /* Expand hash table */\n    fn extend(&amp;mut self) {\n        // Temporarily store the original hash table\n        let buckets_tmp = std::mem::take(&amp;mut self.buckets);\n\n        // Initialize expanded new hash table\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![Vec::new(); self.capacity as usize];\n        self.size = 0;\n\n        // Move key-value pairs from original hash table to new hash table\n        for bucket in buckets_tmp {\n            for pair in bucket {\n                self.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print hash table */\n    fn print(&amp;self) {\n        for bucket in &amp;self.buckets {\n            let mut res = Vec::new();\n            for pair in bucket {\n                res.push(format!(\"{} -&gt; {}\", pair.key, pair.val));\n            }\n            println!(\"{:?}\", res);\n        }\n    }\n\n    /* Add operation */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // When load factor exceeds threshold, perform expansion\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n\n        let index = self.hash_func(key);\n\n        // Traverse bucket, if specified key is encountered, update corresponding val and return\n        for pair in self.buckets[index].iter_mut() {\n            if pair.key == key {\n                pair.val = val;\n                return;\n            }\n        }\n\n        // If key does not exist, append key-value pair to the end\n        let pair = Pair { key, val };\n        self.buckets[index].push(pair);\n        self.size += 1;\n    }\n\n    /* Query operation */\n    fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        let index = self.hash_func(key);\n\n        // Traverse bucket, if key is found, return corresponding val\n        for pair in self.buckets[index].iter() {\n            if pair.key == key {\n                return Some(&amp;pair.val);\n            }\n        }\n\n        // If key is not found, return None\n        None\n    }\n}\n</code></pre> hash_map_chaining.c<pre><code>/* Linked list node */\ntypedef struct Node {\n    Pair *pair;\n    struct Node *next;\n} Node;\n\n/* Hash table with separate chaining */\ntypedef struct {\n    int size;         // Number of key-value pairs\n    int capacity;     // Hash table capacity\n    double loadThres; // Load factor threshold for triggering expansion\n    int extendRatio;  // Expansion multiplier\n    Node **buckets;   // Bucket array\n} HashMapChaining;\n\n/* Constructor */\nHashMapChaining *newHashMapChaining() {\n    HashMapChaining *hashMap = (HashMapChaining *)malloc(sizeof(HashMapChaining));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    return hashMap;\n}\n\n/* Destructor */\nvoid delHashMapChaining(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        while (cur) {\n            Node *tmp = cur;\n            cur = cur-&gt;next;\n            free(tmp-&gt;pair);\n            free(tmp);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap);\n}\n\n/* Hash function */\nint hashFunc(HashMapChaining *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* Load factor */\ndouble loadFactor(HashMapChaining *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* Query operation */\nchar *get(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    // Traverse bucket, if key is found, return corresponding val\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            return cur-&gt;pair-&gt;val;\n        }\n        cur = cur-&gt;next;\n    }\n    return \"\"; // Return empty string if key not found\n}\n\n/* Add operation */\nvoid put(HashMapChaining *hashMap, int key, const char *val) {\n    // When load factor exceeds threshold, perform expansion\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    int index = hashFunc(hashMap, key);\n    // Traverse bucket, if specified key is encountered, update corresponding val and return\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            strcpy(cur-&gt;pair-&gt;val, val); // If specified key is found, update corresponding val and return\n            return;\n        }\n        cur = cur-&gt;next;\n    }\n    // If key not found, add key-value pair to list head\n    Pair *newPair = (Pair *)malloc(sizeof(Pair));\n    newPair-&gt;key = key;\n    strcpy(newPair-&gt;val, val);\n    Node *newNode = (Node *)malloc(sizeof(Node));\n    newNode-&gt;pair = newPair;\n    newNode-&gt;next = hashMap-&gt;buckets[index];\n    hashMap-&gt;buckets[index] = newNode;\n    hashMap-&gt;size++;\n}\n\n/* Expand hash table */\nvoid extend(HashMapChaining *hashMap) {\n    // Temporarily store the original hash table\n    int oldCapacity = hashMap-&gt;capacity;\n    Node **oldBuckets = hashMap-&gt;buckets;\n    // Initialize expanded new hash table\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    hashMap-&gt;size = 0;\n    // Move key-value pairs from original hash table to new hash table\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Node *cur = oldBuckets[i];\n        while (cur) {\n            put(hashMap, cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            Node *temp = cur;\n            cur = cur-&gt;next;\n            // Free memory\n            free(temp-&gt;pair);\n            free(temp);\n        }\n    }\n\n    free(oldBuckets);\n}\n\n/* Remove operation */\nvoid removeItem(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    Node *cur = hashMap-&gt;buckets[index];\n    Node *pre = NULL;\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            // Remove key-value pair from it\n            if (pre) {\n                pre-&gt;next = cur-&gt;next;\n            } else {\n                hashMap-&gt;buckets[index] = cur-&gt;next;\n            }\n            // Free memory\n            free(cur-&gt;pair);\n            free(cur);\n            hashMap-&gt;size--;\n            return;\n        }\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n}\n\n/* Print hash table */\nvoid print(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        printf(\"[\");\n        while (cur) {\n            printf(\"%d -&gt; %s, \", cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            cur = cur-&gt;next;\n        }\n        printf(\"]\\n\");\n    }\n}\n</code></pre> hash_map_chaining.kt<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n    var size: Int // Number of key-value pairs\n    var capacity: Int // Hash table capacity\n    val loadThres: Double // Load factor threshold for triggering expansion\n    val extendRatio: Int // Expansion multiplier\n    var buckets: MutableList&lt;MutableList&lt;Pair&gt;&gt; // Bucket array\n\n    /* Constructor */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n    }\n\n    /* Hash function */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* Load factor */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* Query operation */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // Traverse bucket, if key is found, return corresponding val\n        for (pair in bucket) {\n            if (pair.key == key) return pair._val\n        }\n        // If key is not found, return null\n        return null\n    }\n\n    /* Add operation */\n    fun put(key: Int, _val: String) {\n        // When load factor exceeds threshold, perform expansion\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // Traverse bucket, if specified key is encountered, update corresponding val and return\n        for (pair in bucket) {\n            if (pair.key == key) {\n                pair._val = _val\n                return\n            }\n        }\n        // If key does not exist, append key-value pair to the end\n        val pair = Pair(key, _val)\n        bucket.add(pair)\n        size++\n    }\n\n    /* Remove operation */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // Traverse bucket and remove key-value pair from it\n        for (pair in bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair)\n                size--\n                break\n            }\n        }\n    }\n\n    /* Expand hash table */\n    fun extend() {\n        // Temporarily store the original hash table\n        val bucketsTmp = buckets\n        // Initialize expanded new hash table\n        capacity *= extendRatio\n        // mutablelist has no fixed size\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n        size = 0\n        // Move key-value pairs from original hash table to new hash table\n        for (bucket in bucketsTmp) {\n            for (pair in bucket) {\n                put(pair.key, pair._val)\n            }\n        }\n    }\n\n    /* Print hash table */\n    fun print() {\n        for (bucket in buckets) {\n            val res = mutableListOf&lt;String&gt;()\n            for (pair in bucket) {\n                val k = pair.key\n                val v = pair._val\n                res.add(\"$k -&gt; $v\")\n            }\n            println(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.rb<pre><code>### Hash map with chaining ###\nclass HashMapChaining\n  ### Constructor ###\n  def initialize\n    @size = 0 # Number of key-value pairs\n    @capacity = 4 # Hash table capacity\n    @load_thres = 2.0 / 3.0 # Load factor threshold for triggering expansion\n    @extend_ratio = 2 # Expansion multiplier\n    @buckets = Array.new(@capacity) { [] } # Bucket array\n  end\n\n  ### Hash function ###\n  def hash_func(key)\n    key % @capacity\n  end\n\n  ### Load factor ###\n  def load_factor\n    @size / @capacity\n  end\n\n  ### Query operation ###\n  def get(key)\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # Traverse bucket, if key is found, return corresponding val\n    for pair in bucket\n      return pair.val if pair.key == key\n    end\n    # Return nil if key not found\n    nil\n  end\n\n  ### Add operation ###\n  def put(key, val)\n    # When load factor exceeds threshold, perform expansion\n    extend if load_factor &gt; @load_thres\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # Traverse bucket, if specified key is encountered, update corresponding val and return\n    for pair in bucket\n      if pair.key == key\n        pair.val = val\n        return\n      end\n    end\n    # If key does not exist, append key-value pair to the end\n    pair = Pair.new(key, val)\n    bucket &lt;&lt; pair\n    @size += 1\n  end\n\n  ### Delete operation ###\n  def remove(key)\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # Traverse bucket and remove key-value pair from it\n    for pair in bucket\n      if pair.key == key\n        bucket.delete(pair)\n        @size -= 1\n        break\n      end\n    end\n  end\n\n  ### Expand hash table ###\n  def extend\n    # Temporarily store original hash table\n    buckets = @buckets\n    # Initialize expanded new hash table\n    @capacity *= @extend_ratio\n    @buckets = Array.new(@capacity) { [] }\n    @size = 0\n    # Move key-value pairs from original hash table to new hash table\n    for bucket in buckets\n      for pair in bucket\n        put(pair.key, pair.val)\n      end\n    end\n  end\n\n  ### Print hash table ###\n  def print\n    for bucket in @buckets\n      res = []\n      for pair in bucket\n        res &lt;&lt; \"#{pair.key} -&gt; #{pair.val}\"\n      end\n      pp res\n    end\n  end\nend\n</code></pre> <p>It's worth noting that when the linked list becomes very long, the query time \\(O(n)\\) is poor. In this case, the linked list can be converted into an AVL tree or a red-black tree, reducing the time complexity of lookups to \\(O(\\log n)\\).</p>","path":["Chapter 6. Hashing","6.2   Hash Collision"],"tags":[]},{"location":"chapter_hashing/hash_collision/#622-open-addressing","level":2,"title":"6.2.2   Open Addressing","text":"<p>Open addressing does not introduce additional data structures. Instead, it handles hash collisions through repeated probing. Common probing strategies include linear probing, quadratic probing, and multiple hashing.</p> <p>Let's use linear probing as an example to introduce the mechanism of open addressing hash tables.</p>","path":["Chapter 6. Hashing","6.2   Hash Collision"],"tags":[]},{"location":"chapter_hashing/hash_collision/#1-linear-probing","level":3,"title":"1.   Linear Probing","text":"<p>Linear probing uses a fixed step size to probe sequentially, so its operations differ somewhat from those of an ordinary hash table.</p> <ul> <li>Inserting elements: Compute the bucket index using the hash function. If the bucket is already occupied, continue probing forward from the collision position with a fixed step size (usually \\(1\\)) until an empty bucket is found, then insert the element there.</li> <li>Searching for elements: If a collision occurs, continue probing forward with the same step size until the corresponding element is found and return its <code>value</code>; if an empty bucket is encountered, the target element is not in the hash table, so return <code>None</code>.</li> </ul> <p>Figure 6-6 shows the distribution of key-value pairs in an open-addressing hash table that uses linear probing. Under this hash function, keys with the same last two digits are mapped to the same bucket. Linear probing then places them in that bucket and the subsequent buckets.</p> <p></p> <p> Figure 6-6   Distribution of key-value pairs in open addressing (linear probing) hash table </p> <p>However, linear probing is prone to clustering. Specifically, the longer a contiguous occupied region in the array becomes, the more likely new collisions are to occur within that region. This in turn makes the cluster grow even further, creating a vicious cycle that gradually degrades the efficiency of insertion, deletion, lookup, and update operations.</p> <p>It's important to note that we cannot directly delete elements from an open-addressing hash table. Deleting an element creates an empty bucket <code>None</code> in the array. During lookup, once linear probing reaches that empty bucket, it stops, which means any elements stored farther along the probe sequence become unreachable. As a result, the program may incorrectly conclude that those elements do not exist, as shown in Figure 6-7.</p> <p></p> <p> Figure 6-7   Query issues caused by deletion in open addressing </p> <p>To solve this problem, we can adopt lazy deletion: instead of directly removing an element from the hash table, use a constant <code>TOMBSTONE</code> to mark the bucket. Under this mechanism, both <code>None</code> and <code>TOMBSTONE</code> denote buckets that can accept key-value pairs. The difference is that when linear probing encounters <code>TOMBSTONE</code>, it must continue probing, because key-value pairs may still exist farther along the sequence.</p> <p>However, lazy deletion may accelerate hash-table performance degradation. Each deletion leaves behind a marker, and as the number of <code>TOMBSTONE</code> entries grows, search time increases as well, because linear probing may need to skip over multiple tombstones before finding the target element.</p> <p>To address this, we can record the index of the first <code>TOMBSTONE</code> encountered during linear probing and swap the found target element into that position. The benefit is that each query or insertion can move elements closer to their ideal positions, that is, closer to where probing begins, which improves lookup efficiency.</p> <p>The code below implements an open addressing (linear probing) hash table with lazy deletion. To make better use of the hash table space, we treat the hash table as a \"circular array\". When going beyond the end of the array, we return to the beginning and continue traversing.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map_open_addressing.py<pre><code>class HashMapOpenAddressing:\n    \"\"\"Hash table with open addressing\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        self.size = 0  # Number of key-value pairs\n        self.capacity = 4  # Hash table capacity\n        self.load_thres = 2.0 / 3.0  # Load factor threshold for triggering expansion\n        self.extend_ratio = 2  # Expansion multiplier\n        self.buckets: list[Pair | None] = [None] * self.capacity  # Bucket array\n        self.TOMBSTONE = Pair(-1, \"-1\")  # Removal marker\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"Hash function\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"Load factor\"\"\"\n        return self.size / self.capacity\n\n    def find_bucket(self, key: int) -&gt; int:\n        \"\"\"Search for bucket index corresponding to key\"\"\"\n        index = self.hash_func(key)\n        first_tombstone = -1\n        # Linear probing, break when encountering an empty bucket\n        while self.buckets[index] is not None:\n            # If key is encountered, return the corresponding bucket index\n            if self.buckets[index].key == key:\n                # If a removal marker was encountered before, move the key-value pair to that index\n                if first_tombstone != -1:\n                    self.buckets[first_tombstone] = self.buckets[index]\n                    self.buckets[index] = self.TOMBSTONE\n                    return first_tombstone  # Return the moved bucket index\n                return index  # Return bucket index\n            # Record the first removal marker encountered\n            if first_tombstone == -1 and self.buckets[index] is self.TOMBSTONE:\n                first_tombstone = index\n            # Calculate bucket index, wrap around to the head if past the tail\n            index = (index + 1) % self.capacity\n        # If key does not exist, return the index for insertion\n        return index if first_tombstone == -1 else first_tombstone\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"Query operation\"\"\"\n        # Search for bucket index corresponding to key\n        index = self.find_bucket(key)\n        # If key-value pair is found, return corresponding val\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            return self.buckets[index].val\n        # If key-value pair does not exist, return None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"Add operation\"\"\"\n        # When load factor exceeds threshold, perform expansion\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        # Search for bucket index corresponding to key\n        index = self.find_bucket(key)\n        # If key-value pair is found, overwrite val and return\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index].val = val\n            return\n        # If key-value pair does not exist, add the key-value pair\n        self.buckets[index] = Pair(key, val)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"Remove operation\"\"\"\n        # Search for bucket index corresponding to key\n        index = self.find_bucket(key)\n        # If key-value pair is found, overwrite it with removal marker\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index] = self.TOMBSTONE\n            self.size -= 1\n\n    def extend(self):\n        \"\"\"Expand hash table\"\"\"\n        # Temporarily store the original hash table\n        buckets_tmp = self.buckets\n        # Initialize expanded new hash table\n        self.capacity *= self.extend_ratio\n        self.buckets = [None] * self.capacity\n        self.size = 0\n        # Move key-value pairs from original hash table to new hash table\n        for pair in buckets_tmp:\n            if pair not in [None, self.TOMBSTONE]:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"Print hash table\"\"\"\n        for pair in self.buckets:\n            if pair is None:\n                print(\"None\")\n            elif pair is self.TOMBSTONE:\n                print(\"TOMBSTONE\")\n            else:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> hash_map_open_addressing.cpp<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n  private:\n    int size;                             // Number of key-value pairs\n    int capacity = 4;                     // Hash table capacity\n    const double loadThres = 2.0 / 3.0;     // Load factor threshold for triggering expansion\n    const int extendRatio = 2;            // Expansion multiplier\n    vector&lt;Pair *&gt; buckets;               // Bucket array\n    Pair *TOMBSTONE = new Pair(-1, \"-1\"); // Removal marker\n\n  public:\n    /* Constructor */\n    HashMapOpenAddressing() : size(0), buckets(capacity, nullptr) {\n    }\n\n    /* Destructor */\n    ~HashMapOpenAddressing() {\n        for (Pair *pair : buckets) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                delete pair;\n            }\n        }\n        delete TOMBSTONE;\n    }\n\n    /* Hash function */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* Load factor */\n    double loadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* Search for bucket index corresponding to key */\n    int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // Linear probing, break when encountering an empty bucket\n        while (buckets[index] != nullptr) {\n            // If key is encountered, return the corresponding bucket index\n            if (buckets[index]-&gt;key == key) {\n                // If a removal marker was encountered before, move the key-value pair to that index\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // Return the moved bucket index\n                }\n                return index; // Return bucket index\n            }\n            // Record the first removal marker encountered\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // Calculate bucket index, wrap around to the head if past the tail\n            index = (index + 1) % capacity;\n        }\n        // If key does not exist, return the index for insertion\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* Query operation */\n    string get(int key) {\n        // Search for bucket index corresponding to key\n        int index = findBucket(key);\n        // If key-value pair is found, return corresponding val\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]-&gt;val;\n        }\n        // Return empty string if key-value pair does not exist\n        return \"\";\n    }\n\n    /* Add operation */\n    void put(int key, string val) {\n        // When load factor exceeds threshold, perform expansion\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // Search for bucket index corresponding to key\n        int index = findBucket(key);\n        // If key-value pair is found, overwrite val and return\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]-&gt;val = val;\n            return;\n        }\n        // If key-value pair does not exist, add the key-value pair\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* Remove operation */\n    void remove(int key) {\n        // Search for bucket index corresponding to key\n        int index = findBucket(key);\n        // If key-value pair is found, overwrite it with removal marker\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            delete buckets[index];\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* Expand hash table */\n    void extend() {\n        // Temporarily store the original hash table\n        vector&lt;Pair *&gt; bucketsTmp = buckets;\n        // Initialize expanded new hash table\n        capacity *= extendRatio;\n        buckets = vector&lt;Pair *&gt;(capacity, nullptr);\n        size = 0;\n        // Move key-value pairs from original hash table to new hash table\n        for (Pair *pair : bucketsTmp) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                put(pair-&gt;key, pair-&gt;val);\n                delete pair;\n            }\n        }\n    }\n\n    /* Print hash table */\n    void print() {\n        for (Pair *pair : buckets) {\n            if (pair == nullptr) {\n                cout &lt;&lt; \"nullptr\" &lt;&lt; endl;\n            } else if (pair == TOMBSTONE) {\n                cout &lt;&lt; \"TOMBSTONE\" &lt;&lt; endl;\n            } else {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; endl;\n            }\n        }\n    }\n};\n</code></pre> hash_map_open_addressing.java<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n    private int size; // Number of key-value pairs\n    private int capacity = 4; // Hash table capacity\n    private final double loadThres = 2.0 / 3.0; // Load factor threshold for triggering expansion\n    private final int extendRatio = 2; // Expansion multiplier\n    private Pair[] buckets; // Bucket array\n    private final Pair TOMBSTONE = new Pair(-1, \"-1\"); // Removal marker\n\n    /* Constructor */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* Hash function */\n    private int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* Load factor */\n    private double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* Search for bucket index corresponding to key */\n    private int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // Linear probing, break when encountering an empty bucket\n        while (buckets[index] != null) {\n            // If key is encountered, return the corresponding bucket index\n            if (buckets[index].key == key) {\n                // If a removal marker was encountered before, move the key-value pair to that index\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // Return the moved bucket index\n                }\n                return index; // Return bucket index\n            }\n            // Record the first removal marker encountered\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // Calculate bucket index, wrap around to the head if past the tail\n            index = (index + 1) % capacity;\n        }\n        // If key does not exist, return the index for insertion\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* Query operation */\n    public String get(int key) {\n        // Search for bucket index corresponding to key\n        int index = findBucket(key);\n        // If key-value pair is found, return corresponding val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // If key-value pair does not exist, return null\n        return null;\n    }\n\n    /* Add operation */\n    public void put(int key, String val) {\n        // When load factor exceeds threshold, perform expansion\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // Search for bucket index corresponding to key\n        int index = findBucket(key);\n        // If key-value pair is found, overwrite val and return\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // If key-value pair does not exist, add the key-value pair\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* Remove operation */\n    public void remove(int key) {\n        // Search for bucket index corresponding to key\n        int index = findBucket(key);\n        // If key-value pair is found, overwrite it with removal marker\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* Expand hash table */\n    private void extend() {\n        // Temporarily store the original hash table\n        Pair[] bucketsTmp = buckets;\n        // Initialize expanded new hash table\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // Move key-value pairs from original hash table to new hash table\n        for (Pair pair : bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print hash table */\n    public void print() {\n        for (Pair pair : buckets) {\n            if (pair == null) {\n                System.out.println(\"null\");\n            } else if (pair == TOMBSTONE) {\n                System.out.println(\"TOMBSTONE\");\n            } else {\n                System.out.println(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.cs<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n    int size; // Number of key-value pairs\n    int capacity = 4; // Hash table capacity\n    double loadThres = 2.0 / 3.0; // Load factor threshold for triggering expansion\n    int extendRatio = 2; // Expansion multiplier\n    Pair[] buckets; // Bucket array\n    Pair TOMBSTONE = new(-1, \"-1\"); // Removal marker\n\n    /* Constructor */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* Hash function */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* Load factor */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* Search for bucket index corresponding to key */\n    int FindBucket(int key) {\n        int index = HashFunc(key);\n        int firstTombstone = -1;\n        // Linear probing, break when encountering an empty bucket\n        while (buckets[index] != null) {\n            // If key is encountered, return the corresponding bucket index\n            if (buckets[index].key == key) {\n                // If a removal marker was encountered before, move the key-value pair to that index\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // Return the moved bucket index\n                }\n                return index; // Return bucket index\n            }\n            // Record the first removal marker encountered\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // Calculate bucket index, wrap around to the head if past the tail\n            index = (index + 1) % capacity;\n        }\n        // If key does not exist, return the index for insertion\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* Query operation */\n    public string? Get(int key) {\n        // Search for bucket index corresponding to key\n        int index = FindBucket(key);\n        // If key-value pair is found, return corresponding val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // If key-value pair does not exist, return null\n        return null;\n    }\n\n    /* Add operation */\n    public void Put(int key, string val) {\n        // When load factor exceeds threshold, perform expansion\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        // Search for bucket index corresponding to key\n        int index = FindBucket(key);\n        // If key-value pair is found, overwrite val and return\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // If key-value pair does not exist, add the key-value pair\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* Remove operation */\n    public void Remove(int key) {\n        // Search for bucket index corresponding to key\n        int index = FindBucket(key);\n        // If key-value pair is found, overwrite it with removal marker\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* Expand hash table */\n    void Extend() {\n        // Temporarily store the original hash table\n        Pair[] bucketsTmp = buckets;\n        // Initialize expanded new hash table\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // Move key-value pairs from original hash table to new hash table\n        foreach (Pair pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print hash table */\n    public void Print() {\n        foreach (Pair pair in buckets) {\n            if (pair == null) {\n                Console.WriteLine(\"null\");\n            } else if (pair == TOMBSTONE) {\n                Console.WriteLine(\"TOMBSTONE\");\n            } else {\n                Console.WriteLine(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.go<pre><code>/* Hash table with open addressing */\ntype hashMapOpenAddressing struct {\n    size        int     // Number of key-value pairs\n    capacity    int     // Hash table capacity\n    loadThres   float64 // Load factor threshold for triggering expansion\n    extendRatio int     // Expansion multiplier\n    buckets     []*pair // Bucket array\n    TOMBSTONE   *pair   // Removal marker\n}\n\n/* Constructor */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\n    return &amp;hashMapOpenAddressing{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     make([]*pair, 4),\n        TOMBSTONE:   &amp;pair{-1, \"-1\"},\n    }\n}\n\n/* Hash function */\nfunc (h *hashMapOpenAddressing) hashFunc(key int) int {\n    return key % h.capacity // Calculate hash value based on key\n}\n\n/* Load factor */\nfunc (h *hashMapOpenAddressing) loadFactor() float64 {\n    return float64(h.size) / float64(h.capacity) // Calculate current load factor\n}\n\n/* Search for bucket index corresponding to key */\nfunc (h *hashMapOpenAddressing) findBucket(key int) int {\n    index := h.hashFunc(key) // Get initial index\n    firstTombstone := -1     // Record position of first TOMBSTONE encountered\n    for h.buckets[index] != nil {\n        if h.buckets[index].key == key {\n            if firstTombstone != -1 {\n                // If a removal marker was encountered before, move the key-value pair to that index\n                h.buckets[firstTombstone] = h.buckets[index]\n                h.buckets[index] = h.TOMBSTONE\n                return firstTombstone // Return the moved bucket index\n            }\n            return index // Return found index\n        }\n        if firstTombstone == -1 &amp;&amp; h.buckets[index] == h.TOMBSTONE {\n            firstTombstone = index // Record position of first deletion marker encountered\n        }\n        index = (index + 1) % h.capacity // Linear probing, wrap around to head if past tail\n    }\n    // If key does not exist, return the index for insertion\n    if firstTombstone != -1 {\n        return firstTombstone\n    }\n    return index\n}\n\n/* Query operation */\nfunc (h *hashMapOpenAddressing) get(key int) string {\n    index := h.findBucket(key) // Search for bucket index corresponding to key\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        return h.buckets[index].val // If key-value pair is found, return corresponding val\n    }\n    return \"\" // Return \"\" if key-value pair does not exist\n}\n\n/* Add operation */\nfunc (h *hashMapOpenAddressing) put(key int, val string) {\n    if h.loadFactor() &gt; h.loadThres {\n        h.extend() // When load factor exceeds threshold, perform expansion\n    }\n    index := h.findBucket(key) // Search for bucket index corresponding to key\n    if h.buckets[index] == nil || h.buckets[index] == h.TOMBSTONE {\n        h.buckets[index] = &amp;pair{key, val} // If key-value pair does not exist, add the key-value pair\n        h.size++\n    } else {\n        h.buckets[index].val = val // If key-value pair found, overwrite val\n    }\n}\n\n/* Remove operation */\nfunc (h *hashMapOpenAddressing) remove(key int) {\n    index := h.findBucket(key) // Search for bucket index corresponding to key\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        h.buckets[index] = h.TOMBSTONE // If key-value pair is found, overwrite it with removal marker\n        h.size--\n    }\n}\n\n/* Expand hash table */\nfunc (h *hashMapOpenAddressing) extend() {\n    oldBuckets := h.buckets               // Temporarily store the original hash table\n    h.capacity *= h.extendRatio           // Update capacity\n    h.buckets = make([]*pair, h.capacity) // Initialize expanded new hash table\n    h.size = 0                            // Reset size\n    // Move key-value pairs from original hash table to new hash table\n    for _, pair := range oldBuckets {\n        if pair != nil &amp;&amp; pair != h.TOMBSTONE {\n            h.put(pair.key, pair.val)\n        }\n    }\n}\n\n/* Print hash table */\nfunc (h *hashMapOpenAddressing) print() {\n    for _, pair := range h.buckets {\n        if pair == nil {\n            fmt.Println(\"nil\")\n        } else if pair == h.TOMBSTONE {\n            fmt.Println(\"TOMBSTONE\")\n        } else {\n            fmt.Printf(\"%d -&gt; %s\\n\", pair.key, pair.val)\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.swift<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n    var size: Int // Number of key-value pairs\n    var capacity: Int // Hash table capacity\n    var loadThres: Double // Load factor threshold for triggering expansion\n    var extendRatio: Int // Expansion multiplier\n    var buckets: [Pair?] // Bucket array\n    var TOMBSTONE: Pair // Removal marker\n\n    /* Constructor */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: nil, count: capacity)\n        TOMBSTONE = Pair(key: -1, val: \"-1\")\n    }\n\n    /* Hash function */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* Load factor */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* Search for bucket index corresponding to key */\n    func findBucket(key: Int) -&gt; Int {\n        var index = hashFunc(key: key)\n        var firstTombstone = -1\n        // Linear probing, break when encountering an empty bucket\n        while buckets[index] != nil {\n            // If key is encountered, return the corresponding bucket index\n            if buckets[index]!.key == key {\n                // If a removal marker was encountered before, move the key-value pair to that index\n                if firstTombstone != -1 {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // Return the moved bucket index\n                }\n                return index // Return bucket index\n            }\n            // Record the first removal marker encountered\n            if firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE {\n                firstTombstone = index\n            }\n            // Calculate bucket index, wrap around to the head if past the tail\n            index = (index + 1) % capacity\n        }\n        // If key does not exist, return the index for insertion\n        return firstTombstone == -1 ? index : firstTombstone\n    }\n\n    /* Query operation */\n    func get(key: Int) -&gt; String? {\n        // Search for bucket index corresponding to key\n        let index = findBucket(key: key)\n        // If key-value pair is found, return corresponding val\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            return buckets[index]!.val\n        }\n        // If key-value pair does not exist, return null\n        return nil\n    }\n\n    /* Add operation */\n    func put(key: Int, val: String) {\n        // When load factor exceeds threshold, perform expansion\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        // Search for bucket index corresponding to key\n        let index = findBucket(key: key)\n        // If key-value pair is found, overwrite val and return\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index]!.val = val\n            return\n        }\n        // If key-value pair does not exist, add the key-value pair\n        buckets[index] = Pair(key: key, val: val)\n        size += 1\n    }\n\n    /* Remove operation */\n    func remove(key: Int) {\n        // Search for bucket index corresponding to key\n        let index = findBucket(key: key)\n        // If key-value pair is found, overwrite it with removal marker\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index] = TOMBSTONE\n            size -= 1\n        }\n    }\n\n    /* Expand hash table */\n    func extend() {\n        // Temporarily store the original hash table\n        let bucketsTmp = buckets\n        // Initialize expanded new hash table\n        capacity *= extendRatio\n        buckets = Array(repeating: nil, count: capacity)\n        size = 0\n        // Move key-value pairs from original hash table to new hash table\n        for pair in bucketsTmp {\n            if let pair, pair != TOMBSTONE {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* Print hash table */\n    func print() {\n        for pair in buckets {\n            if pair == nil {\n                Swift.print(\"null\")\n            } else if pair == TOMBSTONE {\n                Swift.print(\"TOMBSTONE\")\n            } else {\n                Swift.print(\"\\(pair!.key) -&gt; \\(pair!.val)\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.js<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n    #size; // Number of key-value pairs\n    #capacity; // Hash table capacity\n    #loadThres; // Load factor threshold for triggering expansion\n    #extendRatio; // Expansion multiplier\n    #buckets; // Bucket array\n    #TOMBSTONE; // Removal marker\n\n    /* Constructor */\n    constructor() {\n        this.#size = 0; // Number of key-value pairs\n        this.#capacity = 4; // Hash table capacity\n        this.#loadThres = 2.0 / 3.0; // Load factor threshold for triggering expansion\n        this.#extendRatio = 2; // Expansion multiplier\n        this.#buckets = Array(this.#capacity).fill(null); // Bucket array\n        this.#TOMBSTONE = new Pair(-1, '-1'); // Removal marker\n    }\n\n    /* Hash function */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* Load factor */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* Search for bucket index corresponding to key */\n    #findBucket(key) {\n        let index = this.#hashFunc(key);\n        let firstTombstone = -1;\n        // Linear probing, break when encountering an empty bucket\n        while (this.#buckets[index] !== null) {\n            // If key is encountered, return the corresponding bucket index\n            if (this.#buckets[index].key === key) {\n                // If a removal marker was encountered before, move the key-value pair to that index\n                if (firstTombstone !== -1) {\n                    this.#buckets[firstTombstone] = this.#buckets[index];\n                    this.#buckets[index] = this.#TOMBSTONE;\n                    return firstTombstone; // Return the moved bucket index\n                }\n                return index; // Return bucket index\n            }\n            // Record the first removal marker encountered\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.#buckets[index] === this.#TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // Calculate bucket index, wrap around to the head if past the tail\n            index = (index + 1) % this.#capacity;\n        }\n        // If key does not exist, return the index for insertion\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* Query operation */\n    get(key) {\n        // Search for bucket index corresponding to key\n        const index = this.#findBucket(key);\n        // If key-value pair is found, return corresponding val\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            return this.#buckets[index].val;\n        }\n        // If key-value pair does not exist, return null\n        return null;\n    }\n\n    /* Add operation */\n    put(key, val) {\n        // When load factor exceeds threshold, perform expansion\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        // Search for bucket index corresponding to key\n        const index = this.#findBucket(key);\n        // If key-value pair is found, overwrite val and return\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index].val = val;\n            return;\n        }\n        // If key-value pair does not exist, add the key-value pair\n        this.#buckets[index] = new Pair(key, val);\n        this.#size++;\n    }\n\n    /* Remove operation */\n    remove(key) {\n        // Search for bucket index corresponding to key\n        const index = this.#findBucket(key);\n        // If key-value pair is found, overwrite it with removal marker\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index] = this.#TOMBSTONE;\n            this.#size--;\n        }\n    }\n\n    /* Expand hash table */\n    #extend() {\n        // Temporarily store the original hash table\n        const bucketsTmp = this.#buckets;\n        // Initialize expanded new hash table\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = Array(this.#capacity).fill(null);\n        this.#size = 0;\n        // Move key-value pairs from original hash table to new hash table\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.#TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print hash table */\n    print() {\n        for (const pair of this.#buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.#TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.ts<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n    private size: number; // Number of key-value pairs\n    private capacity: number; // Hash table capacity\n    private loadThres: number; // Load factor threshold for triggering expansion\n    private extendRatio: number; // Expansion multiplier\n    private buckets: Array&lt;Pair | null&gt;; // Bucket array\n    private TOMBSTONE: Pair; // Removal marker\n\n    /* Constructor */\n    constructor() {\n        this.size = 0; // Number of key-value pairs\n        this.capacity = 4; // Hash table capacity\n        this.loadThres = 2.0 / 3.0; // Load factor threshold for triggering expansion\n        this.extendRatio = 2; // Expansion multiplier\n        this.buckets = Array(this.capacity).fill(null); // Bucket array\n        this.TOMBSTONE = new Pair(-1, '-1'); // Removal marker\n    }\n\n    /* Hash function */\n    private hashFunc(key: number): number {\n        return key % this.capacity;\n    }\n\n    /* Load factor */\n    private loadFactor(): number {\n        return this.size / this.capacity;\n    }\n\n    /* Search for bucket index corresponding to key */\n    private findBucket(key: number): number {\n        let index = this.hashFunc(key);\n        let firstTombstone = -1;\n        // Linear probing, break when encountering an empty bucket\n        while (this.buckets[index] !== null) {\n            // If key is encountered, return the corresponding bucket index\n            if (this.buckets[index]!.key === key) {\n                // If a removal marker was encountered before, move the key-value pair to that index\n                if (firstTombstone !== -1) {\n                    this.buckets[firstTombstone] = this.buckets[index];\n                    this.buckets[index] = this.TOMBSTONE;\n                    return firstTombstone; // Return the moved bucket index\n                }\n                return index; // Return bucket index\n            }\n            // Record the first removal marker encountered\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.buckets[index] === this.TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // Calculate bucket index, wrap around to the head if past the tail\n            index = (index + 1) % this.capacity;\n        }\n        // If key does not exist, return the index for insertion\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* Query operation */\n    get(key: number): string | null {\n        // Search for bucket index corresponding to key\n        const index = this.findBucket(key);\n        // If key-value pair is found, return corresponding val\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            return this.buckets[index]!.val;\n        }\n        // If key-value pair does not exist, return null\n        return null;\n    }\n\n    /* Add operation */\n    put(key: number, val: string): void {\n        // When load factor exceeds threshold, perform expansion\n        if (this.loadFactor() &gt; this.loadThres) {\n            this.extend();\n        }\n        // Search for bucket index corresponding to key\n        const index = this.findBucket(key);\n        // If key-value pair is found, overwrite val and return\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index]!.val = val;\n            return;\n        }\n        // If key-value pair does not exist, add the key-value pair\n        this.buckets[index] = new Pair(key, val);\n        this.size++;\n    }\n\n    /* Remove operation */\n    remove(key: number): void {\n        // Search for bucket index corresponding to key\n        const index = this.findBucket(key);\n        // If key-value pair is found, overwrite it with removal marker\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index] = this.TOMBSTONE;\n            this.size--;\n        }\n    }\n\n    /* Expand hash table */\n    private extend(): void {\n        // Temporarily store the original hash table\n        const bucketsTmp = this.buckets;\n        // Initialize expanded new hash table\n        this.capacity *= this.extendRatio;\n        this.buckets = Array(this.capacity).fill(null);\n        this.size = 0;\n        // Move key-value pairs from original hash table to new hash table\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print hash table */\n    print(): void {\n        for (const pair of this.buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.dart<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n  late int _size; // Number of key-value pairs\n  int _capacity = 4; // Hash table capacity\n  double _loadThres = 2.0 / 3.0; // Load factor threshold for triggering expansion\n  int _extendRatio = 2; // Expansion multiplier\n  late List&lt;Pair?&gt; _buckets; // Bucket array\n  Pair _TOMBSTONE = Pair(-1, \"-1\"); // Removal marker\n\n  /* Constructor */\n  HashMapOpenAddressing() {\n    _size = 0;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n  }\n\n  /* Hash function */\n  int hashFunc(int key) {\n    return key % _capacity;\n  }\n\n  /* Load factor */\n  double loadFactor() {\n    return _size / _capacity;\n  }\n\n  /* Search for bucket index corresponding to key */\n  int findBucket(int key) {\n    int index = hashFunc(key);\n    int firstTombstone = -1;\n    // Linear probing, break when encountering an empty bucket\n    while (_buckets[index] != null) {\n      // If key is encountered, return the corresponding bucket index\n      if (_buckets[index]!.key == key) {\n        // If a removal marker was encountered before, move the key-value pair to that index\n        if (firstTombstone != -1) {\n          _buckets[firstTombstone] = _buckets[index];\n          _buckets[index] = _TOMBSTONE;\n          return firstTombstone; // Return the moved bucket index\n        }\n        return index; // Return bucket index\n      }\n      // Record the first removal marker encountered\n      if (firstTombstone == -1 &amp;&amp; _buckets[index] == _TOMBSTONE) {\n        firstTombstone = index;\n      }\n      // Calculate bucket index, wrap around to the head if past the tail\n      index = (index + 1) % _capacity;\n    }\n    // If key does not exist, return the index for insertion\n    return firstTombstone == -1 ? index : firstTombstone;\n  }\n\n  /* Query operation */\n  String? get(int key) {\n    // Search for bucket index corresponding to key\n    int index = findBucket(key);\n    // If key-value pair is found, return corresponding val\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      return _buckets[index]!.val;\n    }\n    // If key-value pair does not exist, return null\n    return null;\n  }\n\n  /* Add operation */\n  void put(int key, String val) {\n    // When load factor exceeds threshold, perform expansion\n    if (loadFactor() &gt; _loadThres) {\n      extend();\n    }\n    // Search for bucket index corresponding to key\n    int index = findBucket(key);\n    // If key-value pair is found, overwrite val and return\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index]!.val = val;\n      return;\n    }\n    // If key-value pair does not exist, add the key-value pair\n    _buckets[index] = new Pair(key, val);\n    _size++;\n  }\n\n  /* Remove operation */\n  void remove(int key) {\n    // Search for bucket index corresponding to key\n    int index = findBucket(key);\n    // If key-value pair is found, overwrite it with removal marker\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index] = _TOMBSTONE;\n      _size--;\n    }\n  }\n\n  /* Expand hash table */\n  void extend() {\n    // Temporarily store the original hash table\n    List&lt;Pair?&gt; bucketsTmp = _buckets;\n    // Initialize expanded new hash table\n    _capacity *= _extendRatio;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n    _size = 0;\n    // Move key-value pairs from original hash table to new hash table\n    for (Pair? pair in bucketsTmp) {\n      if (pair != null &amp;&amp; pair != _TOMBSTONE) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* Print hash table */\n  void printHashMap() {\n    for (Pair? pair in _buckets) {\n      if (pair == null) {\n        print(\"null\");\n      } else if (pair == _TOMBSTONE) {\n        print(\"TOMBSTONE\");\n      } else {\n        print(\"${pair.key} -&gt; ${pair.val}\");\n      }\n    }\n  }\n}\n</code></pre> hash_map_open_addressing.rs<pre><code>/* Hash table with open addressing */\nstruct HashMapOpenAddressing {\n    size: usize,                // Number of key-value pairs\n    capacity: usize,            // Hash table capacity\n    load_thres: f64,            // Load factor threshold for triggering expansion\n    extend_ratio: usize,        // Expansion multiplier\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;, // Bucket array\n    TOMBSTONE: Option&lt;Pair&gt;,    // Removal marker\n}\n\nimpl HashMapOpenAddressing {\n    /* Constructor */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![None; 4],\n            TOMBSTONE: Some(Pair {\n                key: -1,\n                val: \"-1\".to_string(),\n            }),\n        }\n    }\n\n    /* Hash function */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        (key % self.capacity as i32) as usize\n    }\n\n    /* Load factor */\n    fn load_factor(&amp;self) -&gt; f64 {\n        self.size as f64 / self.capacity as f64\n    }\n\n    /* Search for bucket index corresponding to key */\n    fn find_bucket(&amp;mut self, key: i32) -&gt; usize {\n        let mut index = self.hash_func(key);\n        let mut first_tombstone = -1;\n        // Linear probing, break when encountering an empty bucket\n        while self.buckets[index].is_some() {\n            // If key is found, return corresponding bucket index\n            if self.buckets[index].as_ref().unwrap().key == key {\n                // If deletion marker was encountered before, move key-value pair to that index\n                if first_tombstone != -1 {\n                    self.buckets[first_tombstone as usize] = self.buckets[index].take();\n                    self.buckets[index] = self.TOMBSTONE.clone();\n                    return first_tombstone as usize; // Return the moved bucket index\n                }\n                return index; // Return bucket index\n            }\n            // Record the first removal marker encountered\n            if first_tombstone == -1 &amp;&amp; self.buckets[index] == self.TOMBSTONE {\n                first_tombstone = index as i32;\n            }\n            // Calculate bucket index, wrap around to the head if past the tail\n            index = (index + 1) % self.capacity;\n        }\n        // If key does not exist, return the index for insertion\n        if first_tombstone == -1 {\n            index\n        } else {\n            first_tombstone as usize\n        }\n    }\n\n    /* Query operation */\n    fn get(&amp;mut self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        // Search for bucket index corresponding to key\n        let index = self.find_bucket(key);\n        // If key-value pair is found, return corresponding val\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            return self.buckets[index].as_ref().map(|pair| &amp;pair.val as &amp;str);\n        }\n        // If key-value pair does not exist, return null\n        None\n    }\n\n    /* Add operation */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // When load factor exceeds threshold, perform expansion\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n        // Search for bucket index corresponding to key\n        let index = self.find_bucket(key);\n        // If key-value pair is found, overwrite val and return\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index].as_mut().unwrap().val = val;\n            return;\n        }\n        // If key-value pair does not exist, add the key-value pair\n        self.buckets[index] = Some(Pair { key, val });\n        self.size += 1;\n    }\n\n    /* Remove operation */\n    fn remove(&amp;mut self, key: i32) {\n        // Search for bucket index corresponding to key\n        let index = self.find_bucket(key);\n        // If key-value pair is found, overwrite it with removal marker\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index] = self.TOMBSTONE.clone();\n            self.size -= 1;\n        }\n    }\n\n    /* Expand hash table */\n    fn extend(&amp;mut self) {\n        // Temporarily store the original hash table\n        let buckets_tmp = self.buckets.clone();\n        // Initialize expanded new hash table\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![None; self.capacity];\n        self.size = 0;\n\n        // Move key-value pairs from original hash table to new hash table\n        for pair in buckets_tmp {\n            if pair.is_none() || pair == self.TOMBSTONE {\n                continue;\n            }\n            let pair = pair.unwrap();\n\n            self.put(pair.key, pair.val);\n        }\n    }\n    /* Print hash table */\n    fn print(&amp;self) {\n        for pair in &amp;self.buckets {\n            if pair.is_none() {\n                println!(\"null\");\n            } else if pair == &amp;self.TOMBSTONE {\n                println!(\"TOMBSTONE\");\n            } else {\n                let pair = pair.as_ref().unwrap();\n                println!(\"{} -&gt; {}\", pair.key, pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.c<pre><code>/* Hash table with open addressing */\ntypedef struct {\n    int size;         // Number of key-value pairs\n    int capacity;     // Hash table capacity\n    double loadThres; // Load factor threshold for triggering expansion\n    int extendRatio;  // Expansion multiplier\n    Pair **buckets;   // Bucket array\n    Pair *TOMBSTONE;  // Removal marker\n} HashMapOpenAddressing;\n\n/* Constructor */\nHashMapOpenAddressing *newHashMapOpenAddressing() {\n    HashMapOpenAddressing *hashMap = (HashMapOpenAddressing *)malloc(sizeof(HashMapOpenAddressing));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Pair **)calloc(hashMap-&gt;capacity, sizeof(Pair *));\n    hashMap-&gt;TOMBSTONE = (Pair *)malloc(sizeof(Pair));\n    hashMap-&gt;TOMBSTONE-&gt;key = -1;\n    hashMap-&gt;TOMBSTONE-&gt;val = \"-1\";\n\n    return hashMap;\n}\n\n/* Destructor */\nvoid delHashMapOpenAddressing(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap-&gt;TOMBSTONE);\n    free(hashMap);\n}\n\n/* Hash function */\nint hashFunc(HashMapOpenAddressing *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* Load factor */\ndouble loadFactor(HashMapOpenAddressing *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* Search for bucket index corresponding to key */\nint findBucket(HashMapOpenAddressing *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    int firstTombstone = -1;\n    // Linear probing, break when encountering an empty bucket\n    while (hashMap-&gt;buckets[index] != NULL) {\n        // If key is encountered, return the corresponding bucket index\n        if (hashMap-&gt;buckets[index]-&gt;key == key) {\n            // If a removal marker was encountered before, move the key-value pair to that index\n            if (firstTombstone != -1) {\n                hashMap-&gt;buckets[firstTombstone] = hashMap-&gt;buckets[index];\n                hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n                return firstTombstone; // Return the moved bucket index\n            }\n            return index; // Return bucket index\n        }\n        // Record the first removal marker encountered\n        if (firstTombstone == -1 &amp;&amp; hashMap-&gt;buckets[index] == hashMap-&gt;TOMBSTONE) {\n            firstTombstone = index;\n        }\n        // Calculate bucket index, wrap around to the head if past the tail\n        index = (index + 1) % hashMap-&gt;capacity;\n    }\n    // If key does not exist, return the index for insertion\n    return firstTombstone == -1 ? index : firstTombstone;\n}\n\n/* Query operation */\nchar *get(HashMapOpenAddressing *hashMap, int key) {\n    // Search for bucket index corresponding to key\n    int index = findBucket(hashMap, key);\n    // If key-value pair is found, return corresponding val\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        return hashMap-&gt;buckets[index]-&gt;val;\n    }\n    // Return empty string if key-value pair does not exist\n    return \"\";\n}\n\n/* Add operation */\nvoid put(HashMapOpenAddressing *hashMap, int key, char *val) {\n    // When load factor exceeds threshold, perform expansion\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    // Search for bucket index corresponding to key\n    int index = findBucket(hashMap, key);\n    // If key-value pair is found, overwrite val and return\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        free(hashMap-&gt;buckets[index]-&gt;val);\n        hashMap-&gt;buckets[index]-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n        strcpy(hashMap-&gt;buckets[index]-&gt;val, val);\n        hashMap-&gt;buckets[index]-&gt;val[strlen(val)] = '\\0';\n        return;\n    }\n    // If key-value pair does not exist, add the key-value pair\n    Pair *pair = (Pair *)malloc(sizeof(Pair));\n    pair-&gt;key = key;\n    pair-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n    strcpy(pair-&gt;val, val);\n    pair-&gt;val[strlen(val)] = '\\0';\n\n    hashMap-&gt;buckets[index] = pair;\n    hashMap-&gt;size++;\n}\n\n/* Remove operation */\nvoid removeItem(HashMapOpenAddressing *hashMap, int key) {\n    // Search for bucket index corresponding to key\n    int index = findBucket(hashMap, key);\n    // If key-value pair is found, overwrite it with removal marker\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        Pair *pair = hashMap-&gt;buckets[index];\n        free(pair-&gt;val);\n        free(pair);\n        hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n        hashMap-&gt;size--;\n    }\n}\n\n/* Expand hash table */\nvoid extend(HashMapOpenAddressing *hashMap) {\n    // Temporarily store the original hash table\n    Pair **bucketsTmp = hashMap-&gt;buckets;\n    int oldCapacity = hashMap-&gt;capacity;\n    // Initialize expanded new hash table\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Pair **)calloc(hashMap-&gt;capacity, sizeof(Pair *));\n    hashMap-&gt;size = 0;\n    // Move key-value pairs from original hash table to new hash table\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Pair *pair = bucketsTmp[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            put(hashMap, pair-&gt;key, pair-&gt;val);\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(bucketsTmp);\n}\n\n/* Print hash table */\nvoid print(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair == NULL) {\n            printf(\"NULL\\n\");\n        } else if (pair == hashMap-&gt;TOMBSTONE) {\n            printf(\"TOMBSTONE\\n\");\n        } else {\n            printf(\"%d -&gt; %s\\n\", pair-&gt;key, pair-&gt;val);\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.kt<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n    private var size: Int               // Number of key-value pairs\n    private var capacity: Int           // Hash table capacity\n    private val loadThres: Double       // Load factor threshold for triggering expansion\n    private val extendRatio: Int        // Expansion multiplier\n    private var buckets: Array&lt;Pair?&gt;   // Bucket array\n    private val TOMBSTONE: Pair         // Removal marker\n\n    /* Constructor */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = arrayOfNulls(capacity)\n        TOMBSTONE = Pair(-1, \"-1\")\n    }\n\n    /* Hash function */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* Load factor */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* Search for bucket index corresponding to key */\n    fun findBucket(key: Int): Int {\n        var index = hashFunc(key)\n        var firstTombstone = -1\n        // Linear probing, break when encountering an empty bucket\n        while (buckets[index] != null) {\n            // If key is encountered, return the corresponding bucket index\n            if (buckets[index]?.key == key) {\n                // If a removal marker was encountered before, move the key-value pair to that index\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // Return the moved bucket index\n                }\n                return index // Return bucket index\n            }\n            // Record the first removal marker encountered\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index\n            }\n            // Calculate bucket index, wrap around to the head if past the tail\n            index = (index + 1) % capacity\n        }\n        // If key does not exist, return the index for insertion\n        return if (firstTombstone == -1) index else firstTombstone\n    }\n\n    /* Query operation */\n    fun get(key: Int): String? {\n        // Search for bucket index corresponding to key\n        val index = findBucket(key)\n        // If key-value pair is found, return corresponding val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]?._val\n        }\n        // If key-value pair does not exist, return null\n        return null\n    }\n\n    /* Add operation */\n    fun put(key: Int, _val: String) {\n        // When load factor exceeds threshold, perform expansion\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        // Search for bucket index corresponding to key\n        val index = findBucket(key)\n        // If key-value pair is found, overwrite val and return\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]!!._val = _val\n            return\n        }\n        // If key-value pair does not exist, add the key-value pair\n        buckets[index] = Pair(key, _val)\n        size++\n    }\n\n    /* Remove operation */\n    fun remove(key: Int) {\n        // Search for bucket index corresponding to key\n        val index = findBucket(key)\n        // If key-value pair is found, overwrite it with removal marker\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE\n            size--\n        }\n    }\n\n    /* Expand hash table */\n    fun extend() {\n        // Temporarily store the original hash table\n        val bucketsTmp = buckets\n        // Initialize expanded new hash table\n        capacity *= extendRatio\n        buckets = arrayOfNulls(capacity)\n        size = 0\n        // Move key-value pairs from original hash table to new hash table\n        for (pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair._val)\n            }\n        }\n    }\n\n    /* Print hash table */\n    fun print() {\n        for (pair in buckets) {\n            if (pair == null) {\n                println(\"null\")\n            } else if (pair == TOMBSTONE) {\n                println(\"TOMESTOME\")\n            } else {\n                println(\"${pair.key} -&gt; ${pair._val}\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.rb<pre><code>### Hash map with open addressing ###\nclass HashMapOpenAddressing\n  TOMBSTONE = Pair.new(-1, '-1') # Removal marker\n\n  ### Constructor ###\n  def initialize\n    @size = 0 # Number of key-value pairs\n    @capacity = 4 # Hash table capacity\n    @load_thres = 2.0 / 3.0 # Load factor threshold for triggering expansion\n    @extend_ratio = 2 # Expansion multiplier\n    @buckets = Array.new(@capacity) # Bucket array\n  end\n\n  ### Hash function ###\n  def hash_func(key)\n    key % @capacity\n  end\n\n  ### Load factor ###\n  def load_factor\n    @size / @capacity\n  end\n\n  ### Search bucket index for key ###\n  def find_bucket(key)\n    index = hash_func(key)\n    first_tombstone = -1\n    # Linear probing, break when encountering an empty bucket\n    while !@buckets[index].nil?\n      # If key is encountered, return the corresponding bucket index\n      if @buckets[index].key == key\n        # If a removal marker was encountered before, move the key-value pair to that index\n        if first_tombstone != -1\n          @buckets[first_tombstone] = @buckets[index]\n          @buckets[index] = TOMBSTONE\n          return first_tombstone # Return the moved bucket index\n        end\n        return index # Return bucket index\n      end\n      # Record the first removal marker encountered\n      first_tombstone = index if first_tombstone == -1 &amp;&amp; @buckets[index] == TOMBSTONE\n      # Calculate bucket index, wrap around to the head if past the tail\n      index = (index + 1) % @capacity\n    end\n    # If key does not exist, return the index for insertion\n    first_tombstone == -1 ? index : first_tombstone\n  end\n\n  ### Query operation ###\n  def get(key)\n    # Search for bucket index corresponding to key\n    index = find_bucket(key)\n    # If key-value pair is found, return corresponding val\n    return @buckets[index].val unless [nil, TOMBSTONE].include?(@buckets[index])\n    # Return nil if key-value pair does not exist\n    nil\n  end\n\n  ### Add operation ###\n  def put(key, val)\n    # When load factor exceeds threshold, perform expansion\n    extend if load_factor &gt; @load_thres\n    # Search for bucket index corresponding to key\n    index = find_bucket(key)\n    # If key-value pair found, overwrite val and return\n    unless [nil, TOMBSTONE].include?(@buckets[index])\n      @buckets[index].val = val\n      return\n    end\n    # If key-value pair does not exist, add the key-value pair\n    @buckets[index] = Pair.new(key, val)\n    @size += 1\n  end\n\n  ### Delete operation ###\n  def remove(key)\n    # Search for bucket index corresponding to key\n    index = find_bucket(key)\n    # If key-value pair is found, overwrite it with removal marker\n    unless [nil, TOMBSTONE].include?(@buckets[index])\n      @buckets[index] = TOMBSTONE\n      @size -= 1\n    end\n  end\n\n  ### Expand hash table ###\n  def extend\n    # Temporarily store the original hash table\n    buckets_tmp = @buckets\n    # Initialize expanded new hash table\n    @capacity *= @extend_ratio\n    @buckets = Array.new(@capacity)\n    @size = 0\n    # Move key-value pairs from original hash table to new hash table\n    for pair in buckets_tmp\n      put(pair.key, pair.val) unless [nil, TOMBSTONE].include?(pair)\n    end\n  end\n\n  ### Print hash table ###\n  def print\n    for pair in @buckets\n      if pair.nil?\n        puts \"Nil\"\n      elsif pair == TOMBSTONE\n        puts \"TOMBSTONE\"\n      else\n        puts \"#{pair.key} -&gt; #{pair.val}\"\n      end\n    end\n  end\nend\n</code></pre>","path":["Chapter 6. Hashing","6.2   Hash Collision"],"tags":[]},{"location":"chapter_hashing/hash_collision/#2-quadratic-probing","level":3,"title":"2.   Quadratic Probing","text":"<p>Quadratic probing is similar to linear probing and is one of the common strategies for open addressing. When a collision occurs, quadratic probing does not simply skip a fixed number of steps but skips a number of steps equal to the \"square of the number of probes\", i.e., \\(1, 4, 9, \\dots\\) steps.</p> <p>Quadratic probing has the following advantages:</p> <ul> <li>Quadratic probing attempts to alleviate the clustering effect of linear probing by skipping distances equal to the square of the probe count.</li> <li>Quadratic probing skips larger distances to find empty positions, which helps to distribute data more evenly.</li> </ul> <p>However, quadratic probing is not perfect:</p> <ul> <li>Clustering still exists, i.e., some positions are more likely to be occupied than others.</li> <li>Due to the growth of squares, quadratic probing may not probe the entire hash table, meaning that even if there are empty buckets in the hash table, quadratic probing may not be able to access them.</li> </ul>","path":["Chapter 6. Hashing","6.2   Hash Collision"],"tags":[]},{"location":"chapter_hashing/hash_collision/#3-multiple-hashing","level":3,"title":"3.   Multiple Hashing","text":"<p>As the name suggests, multiple hashing uses multiple hash functions \\(f_1(x)\\), \\(f_2(x)\\), \\(f_3(x)\\), \\(\\dots\\) for probing.</p> <ul> <li>Inserting elements: If hash function \\(f_1(x)\\) encounters a conflict, try \\(f_2(x)\\), and so on, until an empty position is found and the element is inserted.</li> <li>Searching for elements: Search in the same order of hash functions until the target element is found and return it; if an empty position is encountered or all hash functions have been tried, it indicates the element is not in the hash table, then return <code>None</code>.</li> </ul> <p>Compared with linear probing, multiple hashing is less prone to clustering, but using multiple hash functions introduces additional computational overhead.</p> <p>Tip</p> <p>Please note that hash tables based on open addressing, including linear probing, quadratic probing, and multiple hashing, all have the problem that elements cannot be deleted directly.</p>","path":["Chapter 6. Hashing","6.2   Hash Collision"],"tags":[]},{"location":"chapter_hashing/hash_collision/#623-choice-of-programming-languages","level":2,"title":"6.2.3   Choice of Programming Languages","text":"<p>Different programming languages adopt different hash table implementation strategies. Here are a few examples:</p> <ul> <li>Python uses open addressing. The <code>dict</code> dictionary uses pseudo-random numbers for probing.</li> <li>Java uses separate chaining. Since JDK 1.8, when the array length in <code>HashMap</code> reaches 64 and the length of a linked list reaches 8, the linked list is converted to a red-black tree to improve search performance.</li> <li>Starting with Go 1.24, the built-in <code>map</code> uses an open-addressed implementation based on Swiss Tables by default. Go 1.23 and earlier use separate chaining, with up to 8 key-value pairs per bucket and overflow buckets linked when that capacity is exceeded.</li> </ul>","path":["Chapter 6. Hashing","6.2   Hash Collision"],"tags":[]},{"location":"chapter_hashing/hash_map/","level":1,"title":"6.1   Hash Table","text":"<p>A hash table, also known as a hash map, stores mappings from keys <code>key</code> to values <code>value</code>, enabling efficient lookups. Specifically, given a key <code>key</code>, we can retrieve the corresponding value <code>value</code> from a hash table in \\(O(1)\\) time.</p> <p>As shown below, suppose we have \\(n\\) students, each with two pieces of information: a name and a student ID. If we want to support the query \"given a student ID, return the corresponding name,\" we can use the hash table shown below.</p> <p></p> <p> Figure 6-1   Abstract representation of a hash table </p> <p>In addition to hash tables, arrays and linked lists can also implement query functionality. Their efficiency comparison is shown in the following table.</p> <ul> <li>Adding elements: Simply add elements to the end of the array (linked list), using \\(O(1)\\) time.</li> <li>Querying elements: Since the array (linked list) is unordered, all elements need to be traversed, using \\(O(n)\\) time.</li> <li>Deleting elements: The element must first be located, then deleted from the array (linked list), using \\(O(n)\\) time.</li> </ul> <p> Table 6-1   Comparison of element query efficiency </p> Array Linked List Hash Table Find element \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) Add element \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) Delete element \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) <p>As we can see, insertion, deletion, lookup, and update operations in a hash table all have time complexity \\(O(1)\\), making hash tables highly efficient.</p>","path":["Chapter 6. Hashing","6.1   Hash Table"],"tags":[]},{"location":"chapter_hashing/hash_map/#611-common-hash-table-operations","level":2,"title":"6.1.1   Common Hash Table Operations","text":"<p>Common operations on hash tables include: initialization, query operations, adding key-value pairs, and deleting key-value pairs. Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map.py<pre><code># Initialize hash table\nhmap: dict = {}\n\n# Add operation\n# Add key-value pair (key, value) to hash table\nhmap[12836] = \"XiaoHa\"\nhmap[15937] = \"XiaoLuo\"\nhmap[16750] = \"XiaoSuan\"\nhmap[13276] = \"XiaoFa\"\nhmap[10583] = \"XiaoYa\"\n\n# Query operation\n# Input key into hash table to get value\nname: str = hmap[15937]\n\n# Delete operation\n# Delete key-value pair (key, value) from hash table\nhmap.pop(10583)\n</code></pre> hash_map.cpp<pre><code>/* Initialize hash table */\nunordered_map&lt;int, string&gt; map;\n\n/* Add operation */\n// Add key-value pair (key, value) to hash table\nmap[12836] = \"XiaoHa\";\nmap[15937] = \"XiaoLuo\";\nmap[16750] = \"XiaoSuan\";\nmap[13276] = \"XiaoFa\";\nmap[10583] = \"XiaoYa\";\n\n/* Query operation */\n// Input key into hash table to get value\nstring name = map[15937];\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.erase(10583);\n</code></pre> hash_map.java<pre><code>/* Initialize hash table */\nMap&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n\n/* Add operation */\n// Add key-value pair (key, value) to hash table\nmap.put(12836, \"XiaoHa\");\nmap.put(15937, \"XiaoLuo\");\nmap.put(16750, \"XiaoSuan\");\nmap.put(13276, \"XiaoFa\");\nmap.put(10583, \"XiaoYa\");\n\n/* Query operation */\n// Input key into hash table to get value\nString name = map.get(15937);\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.remove(10583);\n</code></pre> hash_map.cs<pre><code>/* Initialize hash table */\nDictionary&lt;int, string&gt; map = new() {\n    /* Add operation */\n    // Add key-value pair (key, value) to hash table\n    { 12836, \"XiaoHa\" },\n    { 15937, \"XiaoLuo\" },\n    { 16750, \"XiaoSuan\" },\n    { 13276, \"XiaoFa\" },\n    { 10583, \"XiaoYa\" }\n};\n\n/* Query operation */\n// Input key into hash table to get value\nstring name = map[15937];\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.Remove(10583);\n</code></pre> hash_map_test.go<pre><code>/* Initialize hash table */\nhmap := make(map[int]string)\n\n/* Add operation */\n// Add key-value pair (key, value) to hash table\nhmap[12836] = \"XiaoHa\"\nhmap[15937] = \"XiaoLuo\"\nhmap[16750] = \"XiaoSuan\"\nhmap[13276] = \"XiaoFa\"\nhmap[10583] = \"XiaoYa\"\n\n/* Query operation */\n// Input key into hash table to get value\nname := hmap[15937]\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\ndelete(hmap, 10583)\n</code></pre> hash_map.swift<pre><code>/* Initialize hash table */\nvar map: [Int: String] = [:]\n\n/* Add operation */\n// Add key-value pair (key, value) to hash table\nmap[12836] = \"XiaoHa\"\nmap[15937] = \"XiaoLuo\"\nmap[16750] = \"XiaoSuan\"\nmap[13276] = \"XiaoFa\"\nmap[10583] = \"XiaoYa\"\n\n/* Query operation */\n// Input key into hash table to get value\nlet name = map[15937]!\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.removeValue(forKey: 10583)\n</code></pre> hash_map.js<pre><code>/* Initialize hash table */\nconst map = new Map();\n/* Add operation */\n// Add key-value pair (key, value) to hash table\nmap.set(12836, 'XiaoHa');\nmap.set(15937, 'XiaoLuo');\nmap.set(16750, 'XiaoSuan');\nmap.set(13276, 'XiaoFa');\nmap.set(10583, 'XiaoYa');\n\n/* Query operation */\n// Input key into hash table to get value\nlet name = map.get(15937);\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.delete(10583);\n</code></pre> hash_map.ts<pre><code>/* Initialize hash table */\nconst map = new Map&lt;number, string&gt;();\n/* Add operation */\n// Add key-value pair (key, value) to hash table\nmap.set(12836, 'XiaoHa');\nmap.set(15937, 'XiaoLuo');\nmap.set(16750, 'XiaoSuan');\nmap.set(13276, 'XiaoFa');\nmap.set(10583, 'XiaoYa');\nconsole.info('\\nAfter adding, hash table is\\nKey -&gt; Value');\nconsole.info(map);\n\n/* Query operation */\n// Input key into hash table to get value\nlet name = map.get(15937);\nconsole.info('\\nInput student ID 15937, queried name ' + name);\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.delete(10583);\nconsole.info('\\nAfter deleting 10583, hash table is\\nKey -&gt; Value');\nconsole.info(map);\n</code></pre> hash_map.dart<pre><code>/* Initialize hash table */\nMap&lt;int, String&gt; map = {};\n\n/* Add operation */\n// Add key-value pair (key, value) to hash table\nmap[12836] = \"XiaoHa\";\nmap[15937] = \"XiaoLuo\";\nmap[16750] = \"XiaoSuan\";\nmap[13276] = \"XiaoFa\";\nmap[10583] = \"XiaoYa\";\n\n/* Query operation */\n// Input key into hash table to get value\nString name = map[15937];\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.remove(10583);\n</code></pre> hash_map.rs<pre><code>use std::collections::HashMap;\n\n/* Initialize hash table */\nlet mut map: HashMap&lt;i32, String&gt; = HashMap::new();\n\n/* Add operation */\n// Add key-value pair (key, value) to hash table\nmap.insert(12836, \"XiaoHa\".to_string());\nmap.insert(15937, \"XiaoLuo\".to_string());\nmap.insert(16750, \"XiaoSuan\".to_string());\nmap.insert(13276, \"XiaoFa\".to_string());\nmap.insert(10583, \"XiaoYa\".to_string());\n\n/* Query operation */\n// Input key into hash table to get value\nlet _name: Option&lt;&amp;String&gt; = map.get(&amp;15937);\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nlet _removed_value: Option&lt;String&gt; = map.remove(&amp;10583);\n</code></pre> hash_map.c<pre><code>// C does not provide a built-in hash table\n</code></pre> hash_map.kt<pre><code>/* Initialize hash table */\nval map = HashMap&lt;Int,String&gt;()\n\n/* Add operation */\n// Add key-value pair (key, value) to hash table\nmap[12836] = \"XiaoHa\"\nmap[15937] = \"XiaoLuo\"\nmap[16750] = \"XiaoSuan\"\nmap[13276] = \"XiaoFa\"\nmap[10583] = \"XiaoYa\"\n\n/* Query operation */\n// Input key into hash table to get value\nval name = map[15937]\n\n/* Delete operation */\n// Delete key-value pair (key, value) from hash table\nmap.remove(10583)\n</code></pre> hash_map.rb<pre><code># Initialize hash table\nhmap = {}\n\n# Add operation\n# Add key-value pair (key, value) to hash table\nhmap[12836] = \"XiaoHa\"\nhmap[15937] = \"XiaoLuo\"\nhmap[16750] = \"XiaoSuan\"\nhmap[13276] = \"XiaoFa\"\nhmap[10583] = \"XiaoYa\"\n\n# Query operation\n# Input key into hash table to get value\nname = hmap[15937]\n\n# Delete operation\n# Delete key-value pair (key, value) from hash table\nhmap.delete(10583)\n</code></pre> Visualized Execution <p> Full Screen &gt;</p> <p>There are three common ways to traverse a hash table: traversing key-value pairs, traversing keys, and traversing values. Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map.py<pre><code># Traverse hash table\n# Traverse key-value pairs key-&gt;value\nfor key, value in hmap.items():\n    print(key, \"-&gt;\", value)\n# Traverse keys only\nfor key in hmap.keys():\n    print(key)\n# Traverse values only\nfor value in hmap.values():\n    print(value)\n</code></pre> hash_map.cpp<pre><code>/* Traverse hash table */\n// Traverse key-value pairs key-&gt;value\nfor (auto kv: map) {\n    cout &lt;&lt; kv.first &lt;&lt; \" -&gt; \" &lt;&lt; kv.second &lt;&lt; endl;\n}\n// Traverse using iterator key-&gt;value\nfor (auto iter = map.begin(); iter != map.end(); iter++) {\n    cout &lt;&lt; iter-&gt;first &lt;&lt; \"-&gt;\" &lt;&lt; iter-&gt;second &lt;&lt; endl;\n}\n</code></pre> hash_map.java<pre><code>/* Traverse hash table */\n// Traverse key-value pairs key-&gt;value\nfor (Map.Entry&lt;Integer, String&gt; kv: map.entrySet()) {\n    System.out.println(kv.getKey() + \" -&gt; \" + kv.getValue());\n}\n// Traverse keys only\nfor (int key: map.keySet()) {\n    System.out.println(key);\n}\n// Traverse values only\nfor (String val: map.values()) {\n    System.out.println(val);\n}\n</code></pre> hash_map.cs<pre><code>/* Traverse hash table */\n// Traverse key-value pairs Key-&gt;Value\nforeach (var kv in map) {\n    Console.WriteLine(kv.Key + \" -&gt; \" + kv.Value);\n}\n// Traverse keys only\nforeach (int key in map.Keys) {\n    Console.WriteLine(key);\n}\n// Traverse values only\nforeach (string val in map.Values) {\n    Console.WriteLine(val);\n}\n</code></pre> hash_map_test.go<pre><code>/* Traverse hash table */\n// Traverse key-value pairs key-&gt;value\nfor key, value := range hmap {\n    fmt.Println(key, \"-&gt;\", value)\n}\n// Traverse keys only\nfor key := range hmap {\n    fmt.Println(key)\n}\n// Traverse values only\nfor _, value := range hmap {\n    fmt.Println(value)\n}\n</code></pre> hash_map.swift<pre><code>/* Traverse hash table */\n// Traverse key-value pairs Key-&gt;Value\nfor (key, value) in map {\n    print(\"\\(key) -&gt; \\(value)\")\n}\n// Traverse keys only\nfor key in map.keys {\n    print(key)\n}\n// Traverse values only\nfor value in map.values {\n    print(value)\n}\n</code></pre> hash_map.js<pre><code>/* Traverse hash table */\nconsole.info('\\nTraverse key-value pairs Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\nTraverse keys only Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\nTraverse values only Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.ts<pre><code>/* Traverse hash table */\nconsole.info('\\nTraverse key-value pairs Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\nTraverse keys only Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\nTraverse values only Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.dart<pre><code>/* Traverse hash table */\n// Traverse key-value pairs Key-&gt;Value\nmap.forEach((key, value) {\n  print('$key -&gt; $value');\n});\n\n// Traverse keys only\nmap.keys.forEach((key) {\n  print(key);\n});\n\n// Traverse values only\nmap.values.forEach((value) {\n  print(value);\n});\n</code></pre> hash_map.rs<pre><code>/* Traverse hash table */\n// Traverse key-value pairs Key-&gt;Value\nfor (key, value) in &amp;map {\n    println!(\"{key} -&gt; {value}\");\n}\n\n// Traverse keys only\nfor key in map.keys() {\n    println!(\"{key}\");\n}\n\n// Traverse values only\nfor value in map.values() {\n    println!(\"{value}\");\n}\n</code></pre> hash_map.c<pre><code>// C does not provide a built-in hash table\n</code></pre> hash_map.kt<pre><code>/* Traverse hash table */\n// Traverse key-value pairs key-&gt;value\nfor ((key, value) in map) {\n    println(\"$key -&gt; $value\")\n}\n// Traverse keys only\nfor (key in map.keys) {\n    println(key)\n}\n// Traverse values only\nfor (_val in map.values) {\n    println(_val)\n}\n</code></pre> hash_map.rb<pre><code># Traverse hash table\n# Traverse key-value pairs key-&gt;value\nhmap.entries.each { |key, value| puts \"#{key} -&gt; #{value}\" }\n\n# Traverse keys only\nhmap.keys.each { |key| puts key }\n\n# Traverse values only\nhmap.values.each { |val| puts val }\n</code></pre> Visualized Execution <p> Full Screen &gt;</p>","path":["Chapter 6. Hashing","6.1   Hash Table"],"tags":[]},{"location":"chapter_hashing/hash_map/#612-simple-hash-table-implementation","level":2,"title":"6.1.2   Simple Hash Table Implementation","text":"<p>Let's start with the simplest case: implementing a hash table with just an array. In a hash table, each empty slot in the array is called a bucket, and each bucket can store one key-value pair. A lookup therefore consists of finding the bucket for <code>key</code> and reading the <code>value</code> stored there.</p> <p>So how do we find the right bucket for a given <code>key</code>? We do this with a hash function. A hash function maps a larger input space to a smaller output space. In a hash table, the input space is the set of all <code>key</code>s, and the output space is the set of all buckets (array indices). In other words, given a <code>key</code>, the hash function tells us where the corresponding key-value pair should be stored in the array.</p> <p>Given a <code>key</code>, computing the bucket index involves the following two steps:</p> <ol> <li>Use a hash algorithm <code>hash()</code> to compute a hash value.</li> <li>Take that hash value modulo the number of buckets (array length), <code>capacity</code>, to obtain the bucket (array index) <code>index</code> corresponding to the <code>key</code>.</li> </ol> <pre><code>index = hash(key) % capacity\n</code></pre> <p>We can then use <code>index</code> to access the corresponding bucket in the hash table and retrieve the <code>value</code>.</p> <p>Suppose the array length is <code>capacity = 100</code> and the hash algorithm is <code>hash(key) = key</code>. Then the hash function is <code>key % 100</code>. Figure 6-2 illustrates how this hash function works, using student ID as <code>key</code> and name as <code>value</code>.</p> <p></p> <p> Figure 6-2   Working principle of hash function </p> <p>The following code implements a simple hash table. Here, we encapsulate <code>key</code> and <code>value</code> into a class <code>Pair</code> to represent a key-value pair.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_hash_map.py<pre><code>class Pair:\n    \"\"\"Key-value pair\"\"\"\n\n    def __init__(self, key: int, val: str):\n        self.key = key\n        self.val = val\n\nclass ArrayHashMap:\n    \"\"\"Hash table based on array implementation\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        # Initialize array with 100 buckets\n        self.buckets: list[Pair | None] = [None] * 100\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"Hash function\"\"\"\n        index = key % 100\n        return index\n\n    def get(self, key: int) -&gt; str | None:\n        \"\"\"Query operation\"\"\"\n        index: int = self.hash_func(key)\n        pair: Pair = self.buckets[index]\n        if pair is None:\n            return None\n        return pair.val\n\n    def put(self, key: int, val: str):\n        \"\"\"Add and update operation\"\"\"\n        pair = Pair(key, val)\n        index: int = self.hash_func(key)\n        self.buckets[index] = pair\n\n    def remove(self, key: int):\n        \"\"\"Remove operation\"\"\"\n        index: int = self.hash_func(key)\n        # Set to None to represent removal\n        self.buckets[index] = None\n\n    def entry_set(self) -&gt; list[Pair]:\n        \"\"\"Get all key-value pairs\"\"\"\n        result: list[Pair] = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair)\n        return result\n\n    def key_set(self) -&gt; list[int]:\n        \"\"\"Get all keys\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.key)\n        return result\n\n    def value_set(self) -&gt; list[str]:\n        \"\"\"Get all values\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.val)\n        return result\n\n    def print(self):\n        \"\"\"Print hash table\"\"\"\n        for pair in self.buckets:\n            if pair is not None:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> array_hash_map.cpp<pre><code>/* Key-value pair */\nstruct Pair {\n  public:\n    int key;\n    string val;\n    Pair(int key, string val) {\n        this-&gt;key = key;\n        this-&gt;val = val;\n    }\n};\n\n/* Hash table based on array implementation */\nclass ArrayHashMap {\n  private:\n    vector&lt;Pair *&gt; buckets;\n\n  public:\n    ArrayHashMap() {\n        // Initialize array with 100 buckets\n        buckets = vector&lt;Pair *&gt;(100);\n    }\n\n    ~ArrayHashMap() {\n        // Free memory\n        for (const auto &amp;bucket : buckets) {\n            delete bucket;\n        }\n        buckets.clear();\n    }\n\n    /* Hash function */\n    int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* Query operation */\n    string get(int key) {\n        int index = hashFunc(key);\n        Pair *pair = buckets[index];\n        if (pair == nullptr)\n            return \"\";\n        return pair-&gt;val;\n    }\n\n    /* Add operation */\n    void put(int key, string val) {\n        Pair *pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* Remove operation */\n    void remove(int key) {\n        int index = hashFunc(key);\n        // Free memory and set to nullptr\n        delete buckets[index];\n        buckets[index] = nullptr;\n    }\n\n    /* Get all key-value pairs */\n    vector&lt;Pair *&gt; pairSet() {\n        vector&lt;Pair *&gt; pairSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                pairSet.push_back(pair);\n            }\n        }\n        return pairSet;\n    }\n\n    /* Get all keys */\n    vector&lt;int&gt; keySet() {\n        vector&lt;int&gt; keySet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                keySet.push_back(pair-&gt;key);\n            }\n        }\n        return keySet;\n    }\n\n    /* Get all values */\n    vector&lt;string&gt; valueSet() {\n        vector&lt;string&gt; valueSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                valueSet.push_back(pair-&gt;val);\n            }\n        }\n        return valueSet;\n    }\n\n    /* Print hash table */\n    void print() {\n        for (Pair *kv : pairSet()) {\n            cout &lt;&lt; kv-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; kv-&gt;val &lt;&lt; endl;\n        }\n    }\n};\n</code></pre> array_hash_map.java<pre><code>/* Key-value pair */\nclass Pair {\n    public int key;\n    public String val;\n\n    public Pair(int key, String val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* Hash table based on array implementation */\nclass ArrayHashMap {\n    private List&lt;Pair&gt; buckets;\n\n    public ArrayHashMap() {\n        // Initialize array with 100 buckets\n        buckets = new ArrayList&lt;&gt;();\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.add(null);\n        }\n    }\n\n    /* Hash function */\n    private int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* Query operation */\n    public String get(int key) {\n        int index = hashFunc(key);\n        Pair pair = buckets.get(index);\n        if (pair == null)\n            return null;\n        return pair.val;\n    }\n\n    /* Add operation */\n    public void put(int key, String val) {\n        Pair pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets.set(index, pair);\n    }\n\n    /* Remove operation */\n    public void remove(int key) {\n        int index = hashFunc(key);\n        // Set to null to represent deletion\n        buckets.set(index, null);\n    }\n\n    /* Get all key-value pairs */\n    public List&lt;Pair&gt; pairSet() {\n        List&lt;Pair&gt; pairSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                pairSet.add(pair);\n        }\n        return pairSet;\n    }\n\n    /* Get all keys */\n    public List&lt;Integer&gt; keySet() {\n        List&lt;Integer&gt; keySet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                keySet.add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* Get all values */\n    public List&lt;String&gt; valueSet() {\n        List&lt;String&gt; valueSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                valueSet.add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* Print hash table */\n    public void print() {\n        for (Pair kv : pairSet()) {\n            System.out.println(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.cs<pre><code>/* Key-value pair int-&gt;string */\nclass Pair(int key, string val) {\n    public int key = key;\n    public string val = val;\n}\n\n/* Hash table based on array implementation */\nclass ArrayHashMap {\n    List&lt;Pair?&gt; buckets;\n    public ArrayHashMap() {\n        // Initialize array with 100 buckets\n        buckets = [];\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.Add(null);\n        }\n    }\n\n    /* Hash function */\n    int HashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* Query operation */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        Pair? pair = buckets[index];\n        if (pair == null) return null;\n        return pair.val;\n    }\n\n    /* Add operation */\n    public void Put(int key, string val) {\n        Pair pair = new(key, val);\n        int index = HashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* Remove operation */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // Set to null to represent deletion\n        buckets[index] = null;\n    }\n\n    /* Get all key-value pairs */\n    public List&lt;Pair&gt; PairSet() {\n        List&lt;Pair&gt; pairSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                pairSet.Add(pair);\n        }\n        return pairSet;\n    }\n\n    /* Get all keys */\n    public List&lt;int&gt; KeySet() {\n        List&lt;int&gt; keySet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                keySet.Add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* Get all values */\n    public List&lt;string&gt; ValueSet() {\n        List&lt;string&gt; valueSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                valueSet.Add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* Print hash table */\n    public void Print() {\n        foreach (Pair kv in PairSet()) {\n            Console.WriteLine(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.go<pre><code>/* Key-value pair */\ntype pair struct {\n    key int\n    val string\n}\n\n/* Hash table based on array implementation */\ntype arrayHashMap struct {\n    buckets []*pair\n}\n\n/* Initialize hash table */\nfunc newArrayHashMap() *arrayHashMap {\n    // Initialize array with 100 buckets\n    buckets := make([]*pair, 100)\n    return &amp;arrayHashMap{buckets: buckets}\n}\n\n/* Hash function */\nfunc (a *arrayHashMap) hashFunc(key int) int {\n    index := key % 100\n    return index\n}\n\n/* Query operation */\nfunc (a *arrayHashMap) get(key int) string {\n    index := a.hashFunc(key)\n    pair := a.buckets[index]\n    if pair == nil {\n        return \"Not Found\"\n    }\n    return pair.val\n}\n\n/* Add operation */\nfunc (a *arrayHashMap) put(key int, val string) {\n    pair := &amp;pair{key: key, val: val}\n    index := a.hashFunc(key)\n    a.buckets[index] = pair\n}\n\n/* Remove operation */\nfunc (a *arrayHashMap) remove(key int) {\n    index := a.hashFunc(key)\n    // Set to nil to delete\n    a.buckets[index] = nil\n}\n\n/* Get all key pairs */\nfunc (a *arrayHashMap) pairSet() []*pair {\n    var pairs []*pair\n    for _, pair := range a.buckets {\n        if pair != nil {\n            pairs = append(pairs, pair)\n        }\n    }\n    return pairs\n}\n\n/* Get all keys */\nfunc (a *arrayHashMap) keySet() []int {\n    var keys []int\n    for _, pair := range a.buckets {\n        if pair != nil {\n            keys = append(keys, pair.key)\n        }\n    }\n    return keys\n}\n\n/* Get all values */\nfunc (a *arrayHashMap) valueSet() []string {\n    var values []string\n    for _, pair := range a.buckets {\n        if pair != nil {\n            values = append(values, pair.val)\n        }\n    }\n    return values\n}\n\n/* Print hash table */\nfunc (a *arrayHashMap) print() {\n    for _, pair := range a.buckets {\n        if pair != nil {\n            fmt.Println(pair.key, \"-&gt;\", pair.val)\n        }\n    }\n}\n</code></pre> array_hash_map.swift<pre><code>/* Key-value pair */\nclass Pair: Equatable {\n    public var key: Int\n    public var val: String\n\n    public init(key: Int, val: String) {\n        self.key = key\n        self.val = val\n    }\n\n    public static func == (lhs: Pair, rhs: Pair) -&gt; Bool {\n        lhs.key == rhs.key &amp;&amp; lhs.val == rhs.val\n    }\n}\n\n/* Hash table based on array implementation */\nclass ArrayHashMap {\n    private var buckets: [Pair?]\n\n    init() {\n        // Initialize array with 100 buckets\n        buckets = Array(repeating: nil, count: 100)\n    }\n\n    /* Hash function */\n    private func hashFunc(key: Int) -&gt; Int {\n        let index = key % 100\n        return index\n    }\n\n    /* Query operation */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let pair = buckets[index]\n        return pair?.val\n    }\n\n    /* Add operation */\n    func put(key: Int, val: String) {\n        let pair = Pair(key: key, val: val)\n        let index = hashFunc(key: key)\n        buckets[index] = pair\n    }\n\n    /* Remove operation */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        // Set to nil to delete\n        buckets[index] = nil\n    }\n\n    /* Get all key-value pairs */\n    func pairSet() -&gt; [Pair] {\n        buckets.compactMap { $0 }\n    }\n\n    /* Get all keys */\n    func keySet() -&gt; [Int] {\n        buckets.compactMap { $0?.key }\n    }\n\n    /* Get all values */\n    func valueSet() -&gt; [String] {\n        buckets.compactMap { $0?.val }\n    }\n\n    /* Print hash table */\n    func print() {\n        for pair in pairSet() {\n            Swift.print(\"\\(pair.key) -&gt; \\(pair.val)\")\n        }\n    }\n}\n</code></pre> array_hash_map.js<pre><code>/* Key-value pair Number -&gt; String */\nclass Pair {\n    constructor(key, val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* Hash table based on array implementation */\nclass ArrayHashMap {\n    #buckets;\n    constructor() {\n        // Initialize array with 100 buckets\n        this.#buckets = new Array(100).fill(null);\n    }\n\n    /* Hash function */\n    #hashFunc(key) {\n        return key % 100;\n    }\n\n    /* Query operation */\n    get(key) {\n        let index = this.#hashFunc(key);\n        let pair = this.#buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* Add operation */\n    set(key, val) {\n        let index = this.#hashFunc(key);\n        this.#buckets[index] = new Pair(key, val);\n    }\n\n    /* Remove operation */\n    delete(key) {\n        let index = this.#hashFunc(key);\n        // Set to null to represent deletion\n        this.#buckets[index] = null;\n    }\n\n    /* Get all key-value pairs */\n    entries() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* Get all keys */\n    keys() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* Get all values */\n    values() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* Print hash table */\n    print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.ts<pre><code>/* Key-value pair Number -&gt; String */\nclass Pair {\n    public key: number;\n    public val: string;\n\n    constructor(key: number, val: string) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* Hash table based on array implementation */\nclass ArrayHashMap {\n    private readonly buckets: (Pair | null)[];\n\n    constructor() {\n        // Initialize array with 100 buckets\n        this.buckets = new Array(100).fill(null);\n    }\n\n    /* Hash function */\n    private hashFunc(key: number): number {\n        return key % 100;\n    }\n\n    /* Query operation */\n    public get(key: number): string | null {\n        let index = this.hashFunc(key);\n        let pair = this.buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* Add operation */\n    public set(key: number, val: string) {\n        let index = this.hashFunc(key);\n        this.buckets[index] = new Pair(key, val);\n    }\n\n    /* Remove operation */\n    public delete(key: number) {\n        let index = this.hashFunc(key);\n        // Set to null to represent deletion\n        this.buckets[index] = null;\n    }\n\n    /* Get all key-value pairs */\n    public entries(): (Pair | null)[] {\n        let arr: (Pair | null)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* Get all keys */\n    public keys(): (number | undefined)[] {\n        let arr: (number | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* Get all values */\n    public values(): (string | undefined)[] {\n        let arr: (string | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* Print hash table */\n    public print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.dart<pre><code>/* Key-value pair */\nclass Pair {\n  int key;\n  String val;\n  Pair(this.key, this.val);\n}\n\n/* Hash table based on array implementation */\nclass ArrayHashMap {\n  late List&lt;Pair?&gt; _buckets;\n\n  ArrayHashMap() {\n    // Initialize array with 100 buckets\n    _buckets = List.filled(100, null);\n  }\n\n  /* Hash function */\n  int _hashFunc(int key) {\n    final int index = key % 100;\n    return index;\n  }\n\n  /* Query operation */\n  String? get(int key) {\n    final int index = _hashFunc(key);\n    final Pair? pair = _buckets[index];\n    if (pair == null) {\n      return null;\n    }\n    return pair.val;\n  }\n\n  /* Add operation */\n  void put(int key, String val) {\n    final Pair pair = Pair(key, val);\n    final int index = _hashFunc(key);\n    _buckets[index] = pair;\n  }\n\n  /* Remove operation */\n  void remove(int key) {\n    final int index = _hashFunc(key);\n    _buckets[index] = null;\n  }\n\n  /* Get all key-value pairs */\n  List&lt;Pair&gt; pairSet() {\n    List&lt;Pair&gt; pairSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        pairSet.add(pair);\n      }\n    }\n    return pairSet;\n  }\n\n  /* Get all keys */\n  List&lt;int&gt; keySet() {\n    List&lt;int&gt; keySet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        keySet.add(pair.key);\n      }\n    }\n    return keySet;\n  }\n\n  /* Get all values */\n  List&lt;String&gt; values() {\n    List&lt;String&gt; valueSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        valueSet.add(pair.val);\n      }\n    }\n    return valueSet;\n  }\n\n  /* Print hash table */\n  void printHashMap() {\n    for (final Pair kv in pairSet()) {\n      print(\"${kv.key} -&gt; ${kv.val}\");\n    }\n  }\n}\n</code></pre> array_hash_map.rs<pre><code>/* Key-value pair */\n#[derive(Debug, Clone, PartialEq)]\npub struct Pair {\n    pub key: i32,\n    pub val: String,\n}\n\n/* Hash table based on array implementation */\npub struct ArrayHashMap {\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;,\n}\n\nimpl ArrayHashMap {\n    pub fn new() -&gt; ArrayHashMap {\n        // Initialize array with 100 buckets\n        Self {\n            buckets: vec![None; 100],\n        }\n    }\n\n    /* Hash function */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % 100\n    }\n\n    /* Query operation */\n    pub fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;String&gt; {\n        let index = self.hash_func(key);\n        self.buckets[index].as_ref().map(|pair| &amp;pair.val)\n    }\n\n    /* Add operation */\n    pub fn put(&amp;mut self, key: i32, val: &amp;str) {\n        let index = self.hash_func(key);\n        self.buckets[index] = Some(Pair {\n            key,\n            val: val.to_string(),\n        });\n    }\n\n    /* Remove operation */\n    pub fn remove(&amp;mut self, key: i32) {\n        let index = self.hash_func(key);\n        // Set to None to represent removal\n        self.buckets[index] = None;\n    }\n\n    /* Get all key-value pairs */\n    pub fn entry_set(&amp;self) -&gt; Vec&lt;&amp;Pair&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref())\n            .collect()\n    }\n\n    /* Get all keys */\n    pub fn key_set(&amp;self) -&gt; Vec&lt;&amp;i32&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.key))\n            .collect()\n    }\n\n    /* Get all values */\n    pub fn value_set(&amp;self) -&gt; Vec&lt;&amp;String&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.val))\n            .collect()\n    }\n\n    /* Print hash table */\n    pub fn print(&amp;self) {\n        for pair in self.entry_set() {\n            println!(\"{} -&gt; {}\", pair.key, pair.val);\n        }\n    }\n}\n</code></pre> array_hash_map.c<pre><code>/* Key-value pair int-&gt;string */\ntypedef struct {\n    int key;\n    char *val;\n} Pair;\n\n/* Hash table based on array implementation */\ntypedef struct {\n    Pair *buckets[MAX_SIZE];\n} ArrayHashMap;\n\n/* Constructor */\nArrayHashMap *newArrayHashMap() {\n    ArrayHashMap *hmap = malloc(sizeof(ArrayHashMap));\n    for (int i=0; i &lt; MAX_SIZE; i++) {\n        hmap-&gt;buckets[i] = NULL;\n    }\n    return hmap;\n}\n\n/* Destructor */\nvoid delArrayHashMap(ArrayHashMap *hmap) {\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            free(hmap-&gt;buckets[i]-&gt;val);\n            free(hmap-&gt;buckets[i]);\n        }\n    }\n    free(hmap);\n}\n\n/* Add operation */\nvoid put(ArrayHashMap *hmap, const int key, const char *val) {\n    Pair *Pair = malloc(sizeof(Pair));\n    Pair-&gt;key = key;\n    Pair-&gt;val = malloc(strlen(val) + 1);\n    strcpy(Pair-&gt;val, val);\n\n    int index = hashFunc(key);\n    hmap-&gt;buckets[index] = Pair;\n}\n\n/* Remove operation */\nvoid removeItem(ArrayHashMap *hmap, const int key) {\n    int index = hashFunc(key);\n    free(hmap-&gt;buckets[index]-&gt;val);\n    free(hmap-&gt;buckets[index]);\n    hmap-&gt;buckets[index] = NULL;\n}\n\n/* Get all key-value pairs */\nvoid pairSet(ArrayHashMap *hmap, MapSet *set) {\n    Pair *entries;\n    int i = 0, index = 0;\n    int total = 0;\n    /* Count valid key-value pairs */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    entries = malloc(sizeof(Pair) * total);\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            entries[index].key = hmap-&gt;buckets[i]-&gt;key;\n            entries[index].val = malloc(strlen(hmap-&gt;buckets[i]-&gt;val) + 1);\n            strcpy(entries[index].val, hmap-&gt;buckets[i]-&gt;val);\n            index++;\n        }\n    }\n    set-&gt;set = entries;\n    set-&gt;len = total;\n}\n\n/* Get all keys */\nvoid keySet(ArrayHashMap *hmap, MapSet *set) {\n    int *keys;\n    int i = 0, index = 0;\n    int total = 0;\n    /* Count valid key-value pairs */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    keys = malloc(total * sizeof(int));\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            keys[index] = hmap-&gt;buckets[i]-&gt;key;\n            index++;\n        }\n    }\n    set-&gt;set = keys;\n    set-&gt;len = total;\n}\n\n/* Get all values */\nvoid valueSet(ArrayHashMap *hmap, MapSet *set) {\n    char **vals;\n    int i = 0, index = 0;\n    int total = 0;\n    /* Count valid key-value pairs */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    vals = malloc(total * sizeof(char *));\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            vals[index] = hmap-&gt;buckets[i]-&gt;val;\n            index++;\n        }\n    }\n    set-&gt;set = vals;\n    set-&gt;len = total;\n}\n\n/* Print hash table */\nvoid print(ArrayHashMap *hmap) {\n    int i;\n    MapSet set;\n    pairSet(hmap, &amp;set);\n    Pair *entries = (Pair *)set.set;\n    for (i = 0; i &lt; set.len; i++) {\n        printf(\"%d -&gt; %s\\n\", entries[i].key, entries[i].val);\n    }\n    free(set.set);\n}\n</code></pre> array_hash_map.kt<pre><code>/* Key-value pair */\nclass Pair(\n    var key: Int,\n    var _val: String\n)\n\n/* Hash table based on array implementation */\nclass ArrayHashMap {\n    // Initialize array with 100 buckets\n    private val buckets = arrayOfNulls&lt;Pair&gt;(100)\n\n    /* Hash function */\n    fun hashFunc(key: Int): Int {\n        val index = key % 100\n        return index\n    }\n\n    /* Query operation */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val pair = buckets[index] ?: return null\n        return pair._val\n    }\n\n    /* Add operation */\n    fun put(key: Int, _val: String) {\n        val pair = Pair(key, _val)\n        val index = hashFunc(key)\n        buckets[index] = pair\n    }\n\n    /* Remove operation */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        // Set to null to represent deletion\n        buckets[index] = null\n    }\n\n    /* Get all key-value pairs */\n    fun pairSet(): MutableList&lt;Pair&gt; {\n        val pairSet = mutableListOf&lt;Pair&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                pairSet.add(pair)\n        }\n        return pairSet\n    }\n\n    /* Get all keys */\n    fun keySet(): MutableList&lt;Int&gt; {\n        val keySet = mutableListOf&lt;Int&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                keySet.add(pair.key)\n        }\n        return keySet\n    }\n\n    /* Get all values */\n    fun valueSet(): MutableList&lt;String&gt; {\n        val valueSet = mutableListOf&lt;String&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                valueSet.add(pair._val)\n        }\n        return valueSet\n    }\n\n    /* Print hash table */\n    fun print() {\n        for (kv in pairSet()) {\n            val key = kv.key\n            val _val = kv._val\n            println(\"$key -&gt; $_val\")\n        }\n    }\n}\n</code></pre> array_hash_map.rb<pre><code>### Key-value pair ###\nclass Pair\n  attr_accessor :key, :val\n\n  def initialize(key, val)\n    @key = key\n    @val = val\n  end\nend\n\n### Hash map based on array ###\nclass ArrayHashMap\n  ### Constructor ###\n  def initialize\n    # Initialize array with 100 buckets\n    @buckets = Array.new(100)\n  end\n\n  ### Hash function ###\n  def hash_func(key)\n    index = key % 100\n  end\n\n  ### Query operation ###\n  def get(key)\n    index = hash_func(key)\n    pair = @buckets[index]\n\n    return if pair.nil?\n    pair.val\n  end\n\n  ### Add operation ###\n  def put(key, val)\n    pair = Pair.new(key, val)\n    index = hash_func(key)\n    @buckets[index] = pair\n  end\n\n  ### Delete operation ###\n  def remove(key)\n    index = hash_func(key)\n    # Set to nil to delete\n    @buckets[index] = nil\n  end\n\n  ### Get all key-value pairs ###\n  def entry_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair unless pair.nil? }\n    result\n  end\n\n  ### Get all keys ###\n  def key_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair.key unless pair.nil? }\n    result\n  end\n\n  ### Get all values ###\n  def value_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair.val unless pair.nil? }\n    result\n  end\n\n  ### Print hash table ###\n  def print\n    @buckets.each { |pair| puts \"#{pair.key} -&gt; #{pair.val}\" unless pair.nil? }\n  end\nend\n</code></pre>","path":["Chapter 6. Hashing","6.1   Hash Table"],"tags":[]},{"location":"chapter_hashing/hash_map/#613-hash-collision-and-resizing","level":2,"title":"6.1.3   Hash Collision and Resizing","text":"<p>Fundamentally, a hash function maps the input space consisting of all <code>key</code>s to the output space consisting of all array indices, and the input space is often much larger than the output space. Therefore, in theory, different inputs must sometimes map to the same output.</p> <p>For the hash function in the above example, when the input <code>key</code>s have the same last two digits, the hash function produces the same output. For example, when querying two students with IDs 12836 and 20336, we get:</p> <pre><code>12836 % 100 = 36\n20336 % 100 = 36\n</code></pre> <p>As shown below, two student IDs now point to the same name, which is clearly incorrect. We call this situation, where multiple inputs map to the same output, a hash collision.</p> <p></p> <p> Figure 6-3   Hash collision example </p> <p>It's easy to see that the larger the hash table capacity \\(n\\), the lower the probability that multiple <code>key</code>s will be assigned to the same bucket, and the fewer collisions. Therefore, we can reduce hash collisions by expanding the hash table.</p> <p>As shown in Figure 6-4, before expansion, the key-value pairs <code>(136, A)</code> and <code>(236, D)</code> collided, but after expansion, the collision disappears.</p> <p></p> <p> Figure 6-4   Hash table resizing </p> <p>Like resizing an array, resizing a hash table requires migrating all key-value pairs from the original table to the new table, which is expensive. In addition, because the hash table capacity <code>capacity</code> changes, we must recompute the storage location of every key-value pair using the hash function, which further increases the cost of resizing. For this reason, programming languages typically reserve a sufficiently large hash table capacity to avoid frequent resizing.</p> <p>The load factor is an important concept in hash tables. It is defined as the number of elements in the hash table divided by the number of buckets and is used to measure the severity of hash collisions. It is also commonly used as a threshold for triggering hash table resizing. For example, in Java, when the load factor exceeds \\(0.75\\), the system expands the hash table to twice its original size.</p>","path":["Chapter 6. Hashing","6.1   Hash Table"],"tags":[]},{"location":"chapter_hashing/summary/","level":1,"title":"6.4   Summary","text":"","path":["Chapter 6. Hashing","6.4   Summary"],"tags":[]},{"location":"chapter_hashing/summary/#1-key-review","level":3,"title":"1.   Key Review","text":"<ul> <li>Given an input <code>key</code>, a hash table can retrieve the corresponding <code>value</code> in \\(O(1)\\) time, which is highly efficient.</li> <li>Common hash table operations include querying, adding key-value pairs, deleting key-value pairs, and traversing the hash table.</li> <li>The hash function maps a <code>key</code> to an array index, allowing access to the corresponding bucket and retrieval of the <code>value</code>.</li> <li>Two different keys may end up with the same array index after hashing, leading to erroneous query results. This phenomenon is known as hash collision.</li> <li>The larger the capacity of the hash table, the lower the probability of hash collisions. Therefore, hash table expansion can mitigate hash collisions. Similar to array expansion, hash table expansion is costly.</li> <li>The load factor, defined as the number of elements divided by the number of buckets, reflects the severity of hash collisions and is often used as a condition to trigger hash table expansion.</li> <li>Separate chaining addresses hash collisions by storing all colliding elements in the same linked list. However, excessively long linked lists can reduce query efficiency, which can be improved by further converting the linked lists into red-black trees.</li> <li>Open addressing handles hash collisions through multiple probing. Linear probing uses a fixed step size but cannot delete elements and is prone to clustering. Double hashing uses multiple hash functions for probing, which reduces clustering compared to linear probing but increases computational overhead.</li> <li>Different programming languages adopt various hash table implementations. For example, Java's <code>HashMap</code> uses separate chaining, while Python's <code>dict</code> employs open addressing.</li> <li>In hash tables, we desire hash algorithms with determinism, high efficiency, and uniform distribution. In cryptography, hash algorithms should also possess collision resistance and the avalanche effect.</li> <li>Hash algorithms typically use large prime numbers as moduli to maximize the uniform distribution of hash values and reduce hash collisions.</li> <li>Common hash algorithms include MD5, SHA-1, SHA-2, and SHA-3. MD5 is often used for file integrity checks, while SHA-2 is commonly used in secure applications and protocols.</li> <li>Programming languages usually provide built-in hash algorithms for data types to calculate bucket indices in hash tables. Generally, only immutable objects are hashable.</li> </ul>","path":["Chapter 6. Hashing","6.4   Summary"],"tags":[]},{"location":"chapter_hashing/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: When does the time complexity of a hash table degrade to \\(O(n)\\)?</p> <p>The time complexity of a hash table can degrade to \\(O(n)\\) when hash collisions are severe. When the hash function is well-designed, the capacity is set appropriately, and collisions are evenly distributed, the time complexity is \\(O(1)\\). We usually consider the time complexity to be \\(O(1)\\) when using built-in hash tables in programming languages.</p> <p>Q: Why not use the hash function \\(f(x) = x\\)? This would eliminate collisions.</p> <p>Under the hash function \\(f(x) = x\\), each element corresponds to a unique bucket index, which is equivalent to an array. However, the input space is usually much larger than the output space (array length), so the last step of a hash function is often to take the modulo of the array length. In other words, the goal of a hash table is to map a larger state space to a smaller one while providing \\(O(1)\\) query efficiency.</p> <p>Q: Why can hash tables be more efficient than arrays, linked lists, or binary trees, even though hash tables are implemented using these structures?</p> <p>Firstly, hash tables have higher time efficiency but lower space efficiency. A significant portion of memory in hash tables remains unused.</p> <p>Secondly, hash tables are only more time-efficient in specific use cases. If a feature can be implemented with the same time complexity using an array or a linked list, it's usually faster than using a hash table. This is because the computation of the hash function incurs overhead, making the constant factor in the time complexity larger.</p> <p>Lastly, the time complexity of hash tables can degrade. For example, in separate chaining, we perform search operations in a linked list or red-black tree, which still risks degrading to \\(O(n)\\) time.</p> <p>Q: Does double hashing also have the flaw of not being able to delete elements directly? Can space marked as deleted be reused?</p> <p>Double hashing is a form of open addressing, and all open addressing methods have the drawback of not being able to delete elements directly; they require marking elements as deleted. Marked spaces can be reused. When inserting new elements into the hash table, and the hash function points to a position marked as deleted, that position can be used by the new element. This maintains the probing sequence of the hash table while ensuring efficient use of space.</p> <p>Q: Why do hash collisions occur during the search process in linear probing?</p> <p>During the search process, the hash function points to the corresponding bucket and key-value pair. If the <code>key</code> doesn't match, it indicates a hash collision. Therefore, linear probing will search downward at a predetermined step size until the correct key-value pair is found or the search fails.</p> <p>Q: Why can expanding a hash table alleviate hash collisions?</p> <p>The last step of a hash function often involves taking the modulo of the array length \\(n\\), to keep the output within the array index range. When expanding, the array length \\(n\\) changes, and the indices corresponding to the keys may also change. Keys that were previously mapped to the same bucket might be distributed across multiple buckets after expansion, thereby mitigating hash collisions.</p> <p>Q: If the goal is efficient access, why not just use an array directly?</p> <p>When the <code>key</code> values are continuous integers within a small range, an array is indeed a simple and efficient choice. But when the <code>key</code> is of another type, such as a string, we need a hash function to map the <code>key</code> to an array index and then store the element in a bucket array. That structure is precisely what a hash table is.</p>","path":["Chapter 6. Hashing","6.4   Summary"],"tags":[]},{"location":"chapter_heap/","level":1,"title":"Chapter 8.   Heap","text":"<p>Abstract</p> <p>Heaps are like mountain peaks, rising layer upon layer, each with a distinct shape.</p> <p>The peaks rise and fall at varying heights, yet the tallest peak always catches the eye first.</p>","path":["Chapter 8. Heap","Chapter 8.   Heap"],"tags":[]},{"location":"chapter_heap/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>8.1   Heap</li> <li>8.2   Heap Construction Operation</li> <li>8.3   Top-k Problem</li> <li>8.4   Summary</li> <li>8.5   Exercises</li> </ul>","path":["Chapter 8. Heap","Chapter 8.   Heap"],"tags":[]},{"location":"chapter_heap/build_heap/","level":1,"title":"8.2   Heap Construction Operation","text":"<p>In some cases, we want to build a heap using all elements of a list, and this process is called \"heap construction operation.\"</p>","path":["Chapter 8. Heap","8.2   Heap Construction Operation"],"tags":[]},{"location":"chapter_heap/build_heap/#821-implementing-with-element-insertion","level":2,"title":"8.2.1   Implementing with Element Insertion","text":"<p>We first create an empty heap, then iterate through the list, performing the \"element insertion operation\" on each element in sequence. This means appending the element to the end of the heap and then performing \"bottom-to-top\" heapify on that element.</p> <p>Each time an element is inserted into the heap, the heap's length increases by one. Since nodes are added to the binary tree sequentially from top to bottom, the heap is constructed \"from top to bottom.\"</p> <p>Given \\(n\\) elements, each element's insertion operation takes \\(O(\\log{n})\\) time, so the time complexity of this heap construction method is \\(O(n \\log n)\\).</p>","path":["Chapter 8. Heap","8.2   Heap Construction Operation"],"tags":[]},{"location":"chapter_heap/build_heap/#822-implementing-through-heapify-traversal","level":2,"title":"8.2.2   Implementing Through Heapify Traversal","text":"<p>In fact, we can implement a more efficient heap construction method in two steps.</p> <ol> <li>Add all elements of the list as-is to the heap, at which point the heap property is not yet satisfied.</li> <li>Traverse the heap in reverse order (reverse of level-order traversal), performing \"top-to-bottom heapify\" on each non-leaf node in sequence.</li> </ol> <p>After heapifying a node, the subtree rooted at that node becomes a valid sub-heap. Since we traverse in reverse order, the heap is constructed \"from bottom to top.\"</p> <p>The reason for choosing reverse-order traversal is that it ensures the subtrees beneath the current node are already valid sub-heaps, so heapifying the current node is effective.</p> <p>It's worth noting that since leaf nodes have no children, they are naturally valid sub-heaps and do not require heapification. As shown in the code below, the last non-leaf node is the parent of the last node; we start from that node and heapify while traversing in reverse order:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def __init__(self, nums: list[int]):\n    \"\"\"Constructor, build heap based on input list\"\"\"\n    # Add list elements to heap as is\n    self.max_heap = nums\n    # Heapify all nodes except leaf nodes\n    for i in range(self.parent(self.size() - 1), -1, -1):\n        self.sift_down(i)\n</code></pre> my_heap.cpp<pre><code>/* Constructor, build heap based on input list */\nMaxHeap(vector&lt;int&gt; nums) {\n    // Add list elements to heap as is\n    maxHeap = nums;\n    // Heapify all nodes except leaf nodes\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.java<pre><code>/* Constructor, build heap based on input list */\nMaxHeap(List&lt;Integer&gt; nums) {\n    // Add list elements to heap as is\n    maxHeap = new ArrayList&lt;&gt;(nums);\n    // Heapify all nodes except leaf nodes\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* Constructor, build heap from input list */\nMaxHeap(IEnumerable&lt;int&gt; nums) {\n    // Add list elements to heap as is\n    maxHeap = new List&lt;int&gt;(nums);\n    // Heapify all nodes except leaf nodes\n    var size = Parent(this.Size() - 1);\n    for (int i = size; i &gt;= 0; i--) {\n        SiftDown(i);\n    }\n}\n</code></pre> my_heap.go<pre><code>/* Constructor, build heap from slice */\nfunc newMaxHeap(nums []any) *maxHeap {\n    // Add list elements to heap as is\n    h := &amp;maxHeap{data: nums}\n    for i := h.parent(len(h.data) - 1); i &gt;= 0; i-- {\n        // Heapify all nodes except leaf nodes\n        h.siftDown(i)\n    }\n    return h\n}\n</code></pre> my_heap.swift<pre><code>/* Constructor, build heap based on input list */\ninit(nums: [Int]) {\n    // Add list elements to heap as is\n    maxHeap = nums\n    // Heapify all nodes except leaf nodes\n    for i in (0 ... parent(i: size() - 1)).reversed() {\n        siftDown(i: i)\n    }\n}\n</code></pre> my_heap.js<pre><code>/* Constructor, build empty heap or build heap from input list */\nconstructor(nums) {\n    // Add list elements to heap as is\n    this.#maxHeap = nums === undefined ? [] : [...nums];\n    // Heapify all nodes except leaf nodes\n    for (let i = this.#parent(this.size() - 1); i &gt;= 0; i--) {\n        this.#siftDown(i);\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* Constructor, build empty heap or build heap from input list */\nconstructor(nums?: number[]) {\n    // Add list elements to heap as is\n    this.maxHeap = nums === undefined ? [] : [...nums];\n    // Heapify all nodes except leaf nodes\n    for (let i = this.parent(this.size() - 1); i &gt;= 0; i--) {\n        this.siftDown(i);\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* Constructor, build heap based on input list */\nMaxHeap(List&lt;int&gt; nums) {\n  // Add list elements to heap as is\n  _maxHeap = nums;\n  // Heapify all nodes except leaf nodes\n  for (int i = _parent(size() - 1); i &gt;= 0; i--) {\n    siftDown(i);\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* Constructor, build heap based on input list */\nfn new(nums: Vec&lt;i32&gt;) -&gt; Self {\n    // Add list elements to heap as is\n    let mut heap = MaxHeap { max_heap: nums };\n    // Heapify all nodes except leaf nodes\n    for i in (0..=Self::parent(heap.size() - 1)).rev() {\n        heap.sift_down(i);\n    }\n    heap\n}\n</code></pre> my_heap.c<pre><code>/* Constructor, build heap from slice */\nMaxHeap *newMaxHeap(int nums[], int size) {\n    // Push all elements to heap\n    MaxHeap *maxHeap = (MaxHeap *)malloc(sizeof(MaxHeap));\n    maxHeap-&gt;size = size;\n    memcpy(maxHeap-&gt;data, nums, size * sizeof(int));\n    for (int i = parent(maxHeap, size - 1); i &gt;= 0; i--) {\n        // Heapify all nodes except leaf nodes\n        siftDown(maxHeap, i);\n    }\n    return maxHeap;\n}\n</code></pre> my_heap.kt<pre><code>/* Max heap */\nclass MaxHeap(nums: MutableList&lt;Int&gt;?) {\n    // Use list instead of array, no need to consider capacity expansion\n    private val maxHeap = mutableListOf&lt;Int&gt;()\n\n    /* Constructor, build heap based on input list */\n    init {\n        // Add list elements to heap as is\n        maxHeap.addAll(nums!!)\n        // Heapify all nodes except leaf nodes\n        for (i in parent(size() - 1) downTo 0) {\n            siftDown(i)\n        }\n    }\n\n    /* Get index of left child node */\n    private fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* Get index of right child node */\n    private fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* Get index of parent node */\n    private fun parent(i: Int): Int {\n        return (i - 1) / 2 // Floor division\n    }\n\n    /* Swap elements */\n    private fun swap(i: Int, j: Int) {\n        val temp = maxHeap[i]\n        maxHeap[i] = maxHeap[j]\n        maxHeap[j] = temp\n    }\n\n    /* Get heap size */\n    fun size(): Int {\n        return maxHeap.size\n    }\n\n    /* Check if heap is empty */\n    fun isEmpty(): Boolean {\n        /* Check if heap is empty */\n        return size() == 0\n    }\n\n    /* Access top element */\n    fun peek(): Int {\n        return maxHeap[0]\n    }\n\n    /* Element enters heap */\n    fun push(_val: Int) {\n        // Add node\n        maxHeap.add(_val)\n        // Heapify from bottom to top\n        siftUp(size() - 1)\n    }\n\n    /* Starting from node i, heapify from bottom to top */\n    private fun siftUp(it: Int) {\n        // Kotlin function parameters are immutable, so create temporary variable\n        var i = it\n        while (true) {\n            // Get parent node of node i\n            val p = parent(i)\n            // When \"crossing root node\" or \"node needs no repair\", end heapify\n            if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n            // Swap two nodes\n            swap(i, p)\n            // Loop upward heapify\n            i = p\n        }\n    }\n\n    /* Element exits heap */\n    fun pop(): Int {\n        // Handle empty case\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // Delete node\n        swap(0, size() - 1)\n        // Remove node\n        val _val = maxHeap.removeAt(size() - 1)\n        // Return top element\n        siftDown(0)\n        // Return heap top element\n        return _val\n    }\n\n    /* Starting from node i, heapify from top to bottom */\n    private fun siftDown(it: Int) {\n        // Kotlin function parameters are immutable, so create temporary variable\n        var i = it\n        while (true) {\n            // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n            val l = left(i)\n            val r = right(i)\n            var ma = i\n            if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n            if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n            // Swap two nodes\n            if (ma == i) break\n            // Swap two nodes\n            swap(i, ma)\n            // Loop downwards heapification\n            i = ma\n        }\n    }\n\n    /* Driver Code */\n    fun print() {\n        val queue = PriorityQueue { a: Int, b: Int -&gt; b - a }\n        queue.addAll(maxHeap)\n        printHeap(queue)\n    }\n}\n</code></pre> my_heap.rb<pre><code>### Constructor, build heap from input list ###\ndef initialize(nums)\n  # Add list elements to heap as is\n  @max_heap = nums\n  # Heapify all nodes except leaf nodes\n  parent(size - 1).downto(0) do |i|\n    sift_down(i)\n  end\nend\n</code></pre>","path":["Chapter 8. Heap","8.2   Heap Construction Operation"],"tags":[]},{"location":"chapter_heap/build_heap/#823-complexity-analysis","level":2,"title":"8.2.3   Complexity Analysis","text":"<p>Next, let's attempt to derive the time complexity of this second heap construction method.</p> <ul> <li>Assuming the complete binary tree has \\(n\\) nodes, then the number of leaf nodes is \\((n + 1) / 2\\), where \\(/\\) is floor division. Therefore, the number of nodes that need heapification is \\(n / 2\\).</li> <li>In the top-to-bottom heapify process, each node can sink at most to a leaf node, so the maximum number of iterations is the height of the binary tree, \\(\\log n\\).</li> </ul> <p>Multiplying these two together, we get a time complexity of \\(O(n \\log n)\\) for the heap construction process. However, this estimate is not accurate because it doesn't account for the property that binary trees have far more nodes at lower levels than at upper levels.</p> <p>Let's perform a more accurate calculation. To simplify the analysis, assume a \"perfect binary tree\" with \\(n\\) nodes and height \\(h\\); this assumption does not affect the correctness of the result.</p> <p></p> <p> Figure 8-5   Node count at each level of a perfect binary tree </p> <p>As shown in Figure 8-5, the maximum number of iterations for a node's \"top-to-bottom heapify\" equals the distance from that node to a leaf node, which is precisely the node's height. Therefore, we can sum the \"number of nodes \\(\\times\\) node height\" at each level to obtain the total number of heapify iterations for all nodes.</p> \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{(h-1)}\\times1 \\] <p>Simplifying the expression above requires some high-school sequence algebra. First, multiply \\(T(h)\\) by \\(2\\) to get:</p> \\[ \\begin{aligned} T(h) &amp; = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{h-1}\\times1 \\newline 2 T(h) &amp; = 2^1h + 2^2(h-1) + 2^3(h-2) + \\dots + 2^{h}\\times1 \\newline \\end{aligned} \\] <p>Using subtraction of shifted sums, subtract the first equation \\(T(h)\\) from the second equation \\(2 T(h)\\) to get:</p> \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\dots + 2^{h-1} + 2^h \\] <p>Observing the above expression, we find that \\(T(h)\\) is a geometric series, which can be calculated directly using the sum formula, yielding a time complexity of:</p> \\[ \\begin{aligned} T(h) &amp; = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline &amp; = 2^{h+1} - h - 2 \\newline &amp; = O(2^h) \\end{aligned} \\] <p>Furthermore, a perfect binary tree with height \\(h\\) has \\(n = 2^{h+1} - 1\\) nodes, so the complexity is \\(O(2^h) = O(n)\\). This derivation shows that the time complexity of building a heap from an input list is \\(O(n)\\), which is highly efficient.</p>","path":["Chapter 8. Heap","8.2   Heap Construction Operation"],"tags":[]},{"location":"chapter_heap/exercises/","level":1,"title":"8.5   Exercises","text":"","path":["Chapter 8. Heap","8.5   Exercises"],"tags":[]},{"location":"chapter_heap/exercises/#851-concept-review","level":2,"title":"8.5.1   Concept Review","text":"","path":["Chapter 8. Heap","8.5   Exercises"],"tags":[]},{"location":"chapter_heap/exercises/#1-how-does-the-heap-change-after-inserting-10","level":3,"title":"1.   How Does the Heap Change After Inserting 10?","text":"<p>The array <code>[9, 7, 8, 3, 5]</code> represents a max heap. Now insert the number 10.</p> <ol> <li>First append 10 to the end of the array. What is the value of its parent node?</li> <li>Starting from the new node, perform bottom-to-top heapify and write the array after each swap.</li> <li>What is the final top element of the heap? How many swaps occur in total?</li> </ol> Answer <ol> <li> <p>After 10 is appended, its index is 5, so its parent's index is     \\(\\lfloor(5-1)/2\\rfloor=2\\). The parent node's value is 8.</p> </li> <li> <p>Since 10 is greater than 8, the array after the first swap is <code>[9, 7, 10, 3, 5, 8]</code>.     Since 10 is also greater than its parent 9, the array after the second swap is <code>[10, 7, 9, 3, 5, 8]</code>.     The value 10 has now reached the root node, so heapification is complete.</p> </li> <li> <p>The final top element is 10, and 2 swaps occur in total.</p> </li> </ol>","path":["Chapter 8. Heap","8.5   Exercises"],"tags":[]},{"location":"chapter_heap/exercises/#2-check-parentchild-relationships-in-a-min-heap","level":3,"title":"2.   Check Parent–Child Relationships in a Min Heap","text":"<p>The array <code>[1, 4, 3, 7, 6, 2]</code> represents a complete binary tree. In a min heap, every parent node must be no greater than its children. For index \\(i\\), the left and right child indices are \\(2i+1\\) and \\(2i+2\\), respectively.</p> <ol> <li>What are the indices and values of the children of index 2?</li> <li>The node at index 2 has value 3. Does it violate the min-heap rule with its child? If so, which two elements should be swapped?</li> <li>Based on your answer to Question 2, write the array after the swap if the rule is violated; otherwise, explain why no swap is needed. Finally, check the remaining parent–child relationships.</li> </ol> Answer <ol> <li> <p>The left child of index 2 is at index 5 and has value 2. The right child's index is 6, but the array has length 6, so the right child does not exist.</p> </li> <li> <p>The parent value 3 is greater than the child value 2, violating the min-heap rule. The elements at indices 2 and 5 should be swapped.</p> </li> <li> <p>After the swap, the array is <code>[1, 4, 2, 7, 6, 3]</code>. Check each relationship:     <code>1 ≤ 4</code>, <code>1 ≤ 2</code>; <code>4 ≤ 7</code>, <code>4 ≤ 6</code>; and <code>2 ≤ 3</code>.     Every parent is now no greater than its children, so the min-heap rule is satisfied.</p> </li> </ol>","path":["Chapter 8. Heap","8.5   Exercises"],"tags":[]},{"location":"chapter_heap/exercises/#3-keep-the-three-largest-numbers-with-a-min-heap","level":3,"title":"3.   Keep the Three Largest Numbers with a Min Heap","text":"<p>To keep the 3 largest numbers from the data stream <code>[4, 1, 7, 3, 8]</code>, maintain a min heap containing no more than 3 elements.</p> <p>First insert the first 3 numbers into the min heap in order. Once the heap is full, for each new number: if it is greater than the top element, remove the top and insert the new number; otherwise, leave the heap unchanged.</p> <p>After each number is read, write the numbers kept in the heap and the top element. Write the kept numbers as a set; you do not need to give their order in the heap array.</p> Answer <p>The result after each number is read is:</p> Number read Numbers kept Top 4 <code>{4}</code> 4 1 <code>{1, 4}</code> 1 7 <code>{1, 4, 7}</code> 1 3 <code>{3, 4, 7}</code> 3 8 <code>{4, 7, 8}</code> 4 <p>Once the heap is full, its top is the smallest of the numbers currently kept. A new number replaces the top only when it is greater than the top. The final set <code>{4, 7, 8}</code> contains exactly the 3 largest numbers.</p>","path":["Chapter 8. Heap","8.5   Exercises"],"tags":[]},{"location":"chapter_heap/exercises/#852-programming-exercises","level":2,"title":"8.5.2   Programming Exercises","text":"","path":["Chapter 8. Heap","8.5   Exercises"],"tags":[]},{"location":"chapter_heap/exercises/#1-find-the-kth-largest-element-in-an-array","level":3,"title":"1.   Find the Kth Largest Element in an Array","text":"<p>Given an integer array <code>nums</code> and an integer \\(k\\), where \\(1 \\le k \\le n\\) and \\(n\\) is the array's length, return the element that would appear at position \\(k\\) if the array were arranged from largest to smallest.</p> <p>Count duplicate elements separately. For example, the second-largest element of <code>[5, 5, 2]</code> is still 5. Use a min heap containing no more than \\(k\\) elements.</p> Hints <ol> <li>The kth largest element is the smallest among the k largest numbers</li> <li>Insert each number into the min heap, and remove the smallest value whenever the heap's size exceeds k</li> <li>After the traversal, the heap contains the k largest numbers, and its top is the answer</li> </ol> <p>LeetCode</p>","path":["Chapter 8. Heap","8.5   Exercises"],"tags":[]},{"location":"chapter_heap/heap/","level":1,"title":"8.1   Heap","text":"<p>A heap is a complete binary tree that satisfies specific conditions and can be mainly categorized into two types, as shown in Figure 8-1.</p> <ul> <li>min heap: The value of any node \\(\\leq\\) the values of its child nodes.</li> <li>max heap: The value of any node \\(\\geq\\) the values of its child nodes.</li> </ul> <p></p> <p> Figure 8-1   Min heap and max heap </p> <p>As a special case of a complete binary tree, heaps have the following characteristics.</p> <ul> <li>The bottom layer nodes are filled from left to right, and nodes in other layers are fully filled.</li> <li>We call the root node of the binary tree the \"heap top\" and the bottom-rightmost node the \"heap bottom.\"</li> <li>For max heaps (min heaps), the value of the heap top element (root node) is the largest (smallest).</li> </ul>","path":["Chapter 8. Heap","8.1   Heap"],"tags":[]},{"location":"chapter_heap/heap/#811-common-heap-operations","level":2,"title":"8.1.1   Common Heap Operations","text":"<p>It should be noted that many programming languages provide a priority queue, an abstract data structure defined as a queue whose elements are ordered by priority.</p> <p>In fact, heaps are typically used to implement priority queues, with max heaps corresponding to priority queues where elements are dequeued in descending order. From a usage perspective, we can regard \"priority queue\" and \"heap\" as equivalent data structures. Therefore, this book does not make a special distinction between the two and uniformly refers to them as \"heap.\"</p> <p>Common heap operations are shown in Table 8-1, and method names need to be determined based on the programming language.</p> <p> Table 8-1   Efficiency of Heap Operations </p> Method name Description Time complexity <code>push()</code> Insert an element into the heap \\(O(\\log n)\\) <code>pop()</code> Remove the heap top element \\(O(\\log n)\\) <code>peek()</code> Access the heap top element (max/min value for max/min heap) \\(O(1)\\) <code>size()</code> Get the number of elements in the heap \\(O(1)\\) <code>isEmpty()</code> Check if the heap is empty \\(O(1)\\) <p>In practical applications, we can directly use the heap class (or priority queue class) provided by programming languages.</p> <p>Similar to \"ascending order\" and \"descending order\" in sorting algorithms, we can implement conversion between \"min heap\" and \"max heap\" by setting a <code>flag</code> or modifying the <code>Comparator</code>. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby heap.py<pre><code># Initialize a min heap\nmin_heap, flag = [], 1\n# Initialize a max heap\nmax_heap, flag = [], -1\n\n# Python's heapq module implements a min heap by default\n# Consider negating elements before pushing them to the heap, which inverts the size relationship and thus implements a max heap\n# In this example, flag = 1 corresponds to a min heap, flag = -1 corresponds to a max heap\n\n# Push elements into the heap\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n\n# Get the heap top element\npeek: int = flag * max_heap[0] # 5\n\n# Remove the heap top element\n# The removed elements will form a descending sequence\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n\n# Get the heap size\nsize: int = len(max_heap)\n\n# Check if the heap is empty\nis_empty: bool = not max_heap\n\n# Build a heap from an input list\nmin_heap: list[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n</code></pre> heap.cpp<pre><code>/* Initialize a heap */\n// Initialize a min heap\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap;\n// Initialize a max heap\npriority_queue&lt;int, vector&lt;int&gt;, less&lt;int&gt;&gt; maxHeap;\n\n/* Push elements into the heap */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n\n/* Get the heap top element */\nint peek = maxHeap.top(); // 5\n\n/* Remove the heap top element */\n// The removed elements will form a descending sequence\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n\n/* Get the heap size */\nint size = maxHeap.size();\n\n/* Check if the heap is empty */\nbool isEmpty = maxHeap.empty();\n\n/* Build a heap from an input list */\nvector&lt;int&gt; input{1, 3, 2, 5, 4};\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap(input.begin(), input.end());\n</code></pre> heap.java<pre><code>/* Initialize a heap */\n// Initialize a min heap\nQueue&lt;Integer&gt; minHeap = new PriorityQueue&lt;&gt;();\n// Initialize a max heap (use lambda expression to modify Comparator)\nQueue&lt;Integer&gt; maxHeap = new PriorityQueue&lt;&gt;((a, b) -&gt; b - a);\n\n/* Push elements into the heap */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n\n/* Get the heap top element */\nint peek = maxHeap.peek(); // 5\n\n/* Remove the heap top element */\n// The removed elements will form a descending sequence\npeek = maxHeap.poll(); // 5\npeek = maxHeap.poll(); // 4\npeek = maxHeap.poll(); // 3\npeek = maxHeap.poll(); // 2\npeek = maxHeap.poll(); // 1\n\n/* Get the heap size */\nint size = maxHeap.size();\n\n/* Check if the heap is empty */\nboolean isEmpty = maxHeap.isEmpty();\n\n/* Build a heap from an input list */\nminHeap = new PriorityQueue&lt;&gt;(Arrays.asList(1, 3, 2, 5, 4));\n</code></pre> heap.cs<pre><code>/* Initialize a heap */\n// Initialize a min heap\nPriorityQueue&lt;int, int&gt; minHeap = new();\n// Initialize a max heap (use lambda expression to modify Comparer)\nPriorityQueue&lt;int, int&gt; maxHeap = new(Comparer&lt;int&gt;.Create((x, y) =&gt; y.CompareTo(x)));\n\n/* Push elements into the heap */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n\n/* Get the heap top element */\nint peek = maxHeap.Peek();//5\n\n/* Remove the heap top element */\n// The removed elements will form a descending sequence\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n\n/* Get the heap size */\nint size = maxHeap.Count;\n\n/* Check if the heap is empty */\nbool isEmpty = maxHeap.Count == 0;\n\n/* Build a heap from an input list */\nminHeap = new PriorityQueue&lt;int, int&gt;([(1, 1), (3, 3), (2, 2), (5, 5), (4, 4)]);\n</code></pre> heap.go<pre><code>// In Go, we can construct a max heap of integers by implementing heap.Interface\n// Implementing heap.Interface also requires implementing sort.Interface\ntype intHeap []any\n\n// Push implements the heap.Interface method for pushing an element into the heap\nfunc (h *intHeap) Push(x any) {\n    // Push and Pop use pointer receiver as parameters\n    // because they not only adjust the slice contents but also modify the slice length\n    *h = append(*h, x.(int))\n}\n\n// Pop implements the heap.Interface method for popping the heap top element\nfunc (h *intHeap) Pop() any {\n    // The element to be removed is stored at the end\n    last := (*h)[len(*h)-1]\n    *h = (*h)[:len(*h)-1]\n    return last\n}\n\n// Len is a sort.Interface method\nfunc (h *intHeap) Len() int {\n    return len(*h)\n}\n\n// Less is a sort.Interface method\nfunc (h *intHeap) Less(i, j int) bool {\n    // To implement a min heap, change this to a less-than sign\n    return (*h)[i].(int) &gt; (*h)[j].(int)\n}\n\n// Swap is a sort.Interface method\nfunc (h *intHeap) Swap(i, j int) {\n    (*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n\n// Top gets the heap top element\nfunc (h *intHeap) Top() any {\n    return (*h)[0]\n}\n\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n    /* Initialize a heap */\n    // Initialize a max heap\n    maxHeap := &amp;intHeap{}\n    heap.Init(maxHeap)\n    /* Push elements into the heap */\n    // Call heap.Interface methods to add elements\n    heap.Push(maxHeap, 1)\n    heap.Push(maxHeap, 3)\n    heap.Push(maxHeap, 2)\n    heap.Push(maxHeap, 4)\n    heap.Push(maxHeap, 5)\n\n    /* Get the heap top element */\n    top := maxHeap.Top()\n    fmt.Printf(\"Heap top element is %d\\n\", top)\n\n    /* Remove the heap top element */\n    // Call heap.Interface methods to remove elements\n    heap.Pop(maxHeap) // 5\n    heap.Pop(maxHeap) // 4\n    heap.Pop(maxHeap) // 3\n    heap.Pop(maxHeap) // 2\n    heap.Pop(maxHeap) // 1\n\n    /* Get the heap size */\n    size := len(*maxHeap)\n    fmt.Printf(\"Number of heap elements is %d\\n\", size)\n\n    /* Check if the heap is empty */\n    isEmpty := len(*maxHeap) == 0\n    fmt.Printf(\"Is the heap empty? %t\\n\", isEmpty)\n}\n</code></pre> heap.swift<pre><code>/* Initialize a heap */\n// Swift's Heap type supports both max heaps and min heaps, and requires importing swift-collections\nvar heap = Heap&lt;Int&gt;()\n\n/* Push elements into the heap */\nheap.insert(1)\nheap.insert(3)\nheap.insert(2)\nheap.insert(5)\nheap.insert(4)\n\n/* Get the heap top element */\nvar peek = heap.max()!\n\n/* Remove the heap top element */\npeek = heap.removeMax() // 5\npeek = heap.removeMax() // 4\npeek = heap.removeMax() // 3\npeek = heap.removeMax() // 2\npeek = heap.removeMax() // 1\n\n/* Get the heap size */\nlet size = heap.count\n\n/* Check if the heap is empty */\nlet isEmpty = heap.isEmpty\n\n/* Build a heap from an input list */\nlet heap2 = Heap([1, 3, 2, 5, 4])\n</code></pre> heap.js<pre><code>// JavaScript does not provide a built-in Heap class\n</code></pre> heap.ts<pre><code>// TypeScript does not provide a built-in Heap class\n</code></pre> heap.dart<pre><code>// Dart does not provide a built-in Heap class\n</code></pre> heap.rs<pre><code>use std::collections::BinaryHeap;\nuse std::cmp::Reverse;\n\n/* Initialize a heap */\n// Initialize a min heap\nlet mut min_heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n// Initialize a max heap\nlet mut max_heap = BinaryHeap::new();\n\n/* Push elements into the heap */\nmax_heap.push(1);\nmax_heap.push(3);\nmax_heap.push(2);\nmax_heap.push(5);\nmax_heap.push(4);\n\n/* Get the heap top element */\nlet peek = max_heap.peek().unwrap();  // 5\n\n/* Remove the heap top element */\n// The removed elements will form a descending sequence\nlet peek = max_heap.pop().unwrap();   // 5\nlet peek = max_heap.pop().unwrap();   // 4\nlet peek = max_heap.pop().unwrap();   // 3\nlet peek = max_heap.pop().unwrap();   // 2\nlet peek = max_heap.pop().unwrap();   // 1\n\n/* Get the heap size */\nlet size = max_heap.len();\n\n/* Check if the heap is empty */\nlet is_empty = max_heap.is_empty();\n\n/* Build a heap from an input list */\nlet min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]);\n</code></pre> heap.c<pre><code>// C does not provide a built-in Heap class\n</code></pre> heap.kt<pre><code>/* Initialize a heap */\n// Initialize a min heap\nvar minHeap = PriorityQueue&lt;Int&gt;()\n// Initialize a max heap (use lambda expression to modify Comparator)\nval maxHeap = PriorityQueue { a: Int, b: Int -&gt; b - a }\n\n/* Push elements into the heap */\nmaxHeap.offer(1)\nmaxHeap.offer(3)\nmaxHeap.offer(2)\nmaxHeap.offer(5)\nmaxHeap.offer(4)\n\n/* Get the heap top element */\nvar peek = maxHeap.peek() // 5\n\n/* Remove the heap top element */\n// The removed elements will form a descending sequence\npeek = maxHeap.poll() // 5\npeek = maxHeap.poll() // 4\npeek = maxHeap.poll() // 3\npeek = maxHeap.poll() // 2\npeek = maxHeap.poll() // 1\n\n/* Get the heap size */\nval size = maxHeap.size\n\n/* Check if the heap is empty */\nval isEmpty = maxHeap.isEmpty()\n\n/* Build a heap from an input list */\nminHeap = PriorityQueue(mutableListOf(1, 3, 2, 5, 4))\n</code></pre> heap.rb<pre><code># Ruby does not provide a built-in Heap class\n</code></pre> Code Visualization <p> Full Screen &gt;</p>","path":["Chapter 8. Heap","8.1   Heap"],"tags":[]},{"location":"chapter_heap/heap/#812-implementation-of-the-heap","level":2,"title":"8.1.2   Implementation of the Heap","text":"<p>The following implementation is for a max heap. To convert it to a min heap, simply reverse all comparison logic related to ordering (for example, replace \\(\\geq\\) with \\(\\leq\\)). Interested readers are encouraged to implement this on their own.</p>","path":["Chapter 8. Heap","8.1   Heap"],"tags":[]},{"location":"chapter_heap/heap/#1-heap-storage-and-representation","level":3,"title":"1.   Heap Storage and Representation","text":"<p>As mentioned in the \"Binary Tree\" chapter, complete binary trees are well-suited for array representation. Since heaps are a type of complete binary tree, we will use arrays to store heaps.</p> <p>When representing a binary tree with an array, elements represent node values, and indexes represent node positions in the binary tree. Parent-child relationships are represented through index-mapping formulas.</p> <p>As shown in Figure 8-2, given an index \\(i\\), the index of its left child is \\(2i + 1\\), the index of its right child is \\(2i + 2\\), and the index of its parent is \\((i - 1) / 2\\) (floor division). When an index is out of bounds, it indicates a null node or that the node does not exist.</p> <p></p> <p> Figure 8-2   Representation and storage of heaps </p> <p>We can encapsulate the index mapping formula into functions for convenient subsequent use:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def left(self, i: int) -&gt; int:\n    \"\"\"Get index of left child node\"\"\"\n    return 2 * i + 1\n\ndef right(self, i: int) -&gt; int:\n    \"\"\"Get index of right child node\"\"\"\n    return 2 * i + 2\n\ndef parent(self, i: int) -&gt; int:\n    \"\"\"Get index of parent node\"\"\"\n    return (i - 1) // 2  # Floor division\n</code></pre> my_heap.cpp<pre><code>/* Get index of left child node */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* Get index of right child node */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* Get index of parent node */\nint parent(int i) {\n    return (i - 1) / 2; // Floor division\n}\n</code></pre> my_heap.java<pre><code>/* Get index of left child node */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* Get index of right child node */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* Get index of parent node */\nint parent(int i) {\n    return (i - 1) / 2; // Floor division\n}\n</code></pre> my_heap.cs<pre><code>/* Get index of left child node */\nint Left(int i) {\n    return 2 * i + 1;\n}\n\n/* Get index of right child node */\nint Right(int i) {\n    return 2 * i + 2;\n}\n\n/* Get index of parent node */\nint Parent(int i) {\n    return (i - 1) / 2; // Floor division\n}\n</code></pre> my_heap.go<pre><code>/* Get index of left child node */\nfunc (h *maxHeap) left(i int) int {\n    return 2*i + 1\n}\n\n/* Get index of right child node */\nfunc (h *maxHeap) right(i int) int {\n    return 2*i + 2\n}\n\n/* Get index of parent node */\nfunc (h *maxHeap) parent(i int) int {\n    // Floor division\n    return (i - 1) / 2\n}\n</code></pre> my_heap.swift<pre><code>/* Get index of left child node */\nfunc left(i: Int) -&gt; Int {\n    2 * i + 1\n}\n\n/* Get index of right child node */\nfunc right(i: Int) -&gt; Int {\n    2 * i + 2\n}\n\n/* Get index of parent node */\nfunc parent(i: Int) -&gt; Int {\n    (i - 1) / 2 // Floor division\n}\n</code></pre> my_heap.js<pre><code>/* Get index of left child node */\n#left(i) {\n    return 2 * i + 1;\n}\n\n/* Get index of right child node */\n#right(i) {\n    return 2 * i + 2;\n}\n\n/* Get index of parent node */\n#parent(i) {\n    return Math.floor((i - 1) / 2); // Floor division\n}\n</code></pre> my_heap.ts<pre><code>/* Get index of left child node */\nleft(i: number): number {\n    return 2 * i + 1;\n}\n\n/* Get index of right child node */\nright(i: number): number {\n    return 2 * i + 2;\n}\n\n/* Get index of parent node */\nparent(i: number): number {\n    return Math.floor((i - 1) / 2); // Floor division\n}\n</code></pre> my_heap.dart<pre><code>/* Get index of left child node */\nint _left(int i) {\n  return 2 * i + 1;\n}\n\n/* Get index of right child node */\nint _right(int i) {\n  return 2 * i + 2;\n}\n\n/* Get index of parent node */\nint _parent(int i) {\n  return (i - 1) ~/ 2; // Floor division\n}\n</code></pre> my_heap.rs<pre><code>/* Get index of left child node */\nfn left(i: usize) -&gt; usize {\n    2 * i + 1\n}\n\n/* Get index of right child node */\nfn right(i: usize) -&gt; usize {\n    2 * i + 2\n}\n\n/* Get index of parent node */\nfn parent(i: usize) -&gt; usize {\n    (i - 1) / 2 // Floor division\n}\n</code></pre> my_heap.c<pre><code>/* Get index of left child node */\nint left(MaxHeap *maxHeap, int i) {\n    return 2 * i + 1;\n}\n\n/* Get index of right child node */\nint right(MaxHeap *maxHeap, int i) {\n    return 2 * i + 2;\n}\n\n/* Get index of parent node */\nint parent(MaxHeap *maxHeap, int i) {\n    return (i - 1) / 2; // Round down\n}\n</code></pre> my_heap.kt<pre><code>/* Get index of left child node */\nfun left(i: Int): Int {\n    return 2 * i + 1\n}\n\n/* Get index of right child node */\nfun right(i: Int): Int {\n    return 2 * i + 2\n}\n\n/* Get index of parent node */\nfun parent(i: Int): Int {\n    return (i - 1) / 2 // Floor division\n}\n</code></pre> my_heap.rb<pre><code>### Get left child index ###\ndef left(i)\n  2 * i + 1\nend\n\n### Get right child index ###\ndef right(i)\n  2 * i + 2\nend\n\n### Get parent node index ###\ndef parent(i)\n  (i - 1) / 2     # Floor division\nend\n</code></pre>","path":["Chapter 8. Heap","8.1   Heap"],"tags":[]},{"location":"chapter_heap/heap/#2-accessing-the-heap-top-element","level":3,"title":"2.   Accessing the Heap Top Element","text":"<p>The heap top element is the root node of the binary tree, which is also the first element of the list:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def peek(self) -&gt; int:\n    \"\"\"Access top element\"\"\"\n    return self.max_heap[0]\n</code></pre> my_heap.cpp<pre><code>/* Access top element */\nint peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.java<pre><code>/* Access top element */\nint peek() {\n    return maxHeap.get(0);\n}\n</code></pre> my_heap.cs<pre><code>/* Access top element */\nint Peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.go<pre><code>/* Access top element */\nfunc (h *maxHeap) peek() any {\n    return h.data[0]\n}\n</code></pre> my_heap.swift<pre><code>/* Access top element */\nfunc peek() -&gt; Int {\n    maxHeap[0]\n}\n</code></pre> my_heap.js<pre><code>/* Access top element */\npeek() {\n    return this.#maxHeap[0];\n}\n</code></pre> my_heap.ts<pre><code>/* Access top element */\npeek(): number {\n    return this.maxHeap[0];\n}\n</code></pre> my_heap.dart<pre><code>/* Access top element */\nint peek() {\n  return _maxHeap[0];\n}\n</code></pre> my_heap.rs<pre><code>/* Access top element */\nfn peek(&amp;self) -&gt; Option&lt;i32&gt; {\n    self.max_heap.first().copied()\n}\n</code></pre> my_heap.c<pre><code>/* Access top element */\nint peek(MaxHeap *maxHeap) {\n    return maxHeap-&gt;data[0];\n}\n</code></pre> my_heap.kt<pre><code>/* Access top element */\nfun peek(): Int {\n    return maxHeap[0]\n}\n</code></pre> my_heap.rb<pre><code>### Access heap top element ###\ndef peek\n  @max_heap[0]\nend\n</code></pre>","path":["Chapter 8. Heap","8.1   Heap"],"tags":[]},{"location":"chapter_heap/heap/#3-inserting-an-element-into-the-heap","level":3,"title":"3.   Inserting an Element Into the Heap","text":"<p>Given an element <code>val</code>, we first add it to the bottom of the heap. After insertion, because <code>val</code> may be larger than other elements in the heap, the heap property may be violated. Therefore, we need to restore the heap property along the path from the inserted node to the root. This operation is called heapify.</p> <p>Starting from the inserted node, perform heapify from bottom to top. As shown in Figure 8-3, we compare the inserted node with its parent, and if the inserted node is larger, we swap them. We continue this process from bottom to top until we move past the root or reach a node that no longer needs to be swapped.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 8-3   Steps of inserting an element into the heap </p> <p>Given a total of \\(n\\) nodes, the tree height is \\(O(\\log n)\\). Thus, the number of loop iterations in the heapify operation is at most \\(O(\\log n)\\), making the time complexity of the element insertion operation \\(O(\\log n)\\). The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def push(self, val: int):\n    \"\"\"Element enters heap\"\"\"\n    # Add node\n    self.max_heap.append(val)\n    # Heapify from bottom to top\n    self.sift_up(self.size() - 1)\n\ndef sift_up(self, i: int):\n    \"\"\"Starting from node i, heapify from bottom to top\"\"\"\n    while True:\n        # Get parent node of node i\n        p = self.parent(i)\n        # When \"crossing root node\" or \"node needs no repair\", end heapify\n        if p &lt; 0 or self.max_heap[i] &lt;= self.max_heap[p]:\n            break\n        # Swap two nodes\n        self.swap(i, p)\n        # Loop upward heapify\n        i = p\n</code></pre> my_heap.cpp<pre><code>/* Element enters heap */\nvoid push(int val) {\n    // Add node\n    maxHeap.push_back(val);\n    // Heapify from bottom to top\n    siftUp(size() - 1);\n}\n\n/* Starting from node i, heapify from bottom to top */\nvoid siftUp(int i) {\n    while (true) {\n        // Get parent node of node i\n        int p = parent(i);\n        // When \"crossing root node\" or \"node needs no repair\", end heapify\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // Swap two nodes\n        swap(maxHeap[i], maxHeap[p]);\n        // Loop upward heapify\n        i = p;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* Element enters heap */\nvoid push(int val) {\n    // Add node\n    maxHeap.add(val);\n    // Heapify from bottom to top\n    siftUp(size() - 1);\n}\n\n/* Starting from node i, heapify from bottom to top */\nvoid siftUp(int i) {\n    while (true) {\n        // Get parent node of node i\n        int p = parent(i);\n        // When \"crossing root node\" or \"node needs no repair\", end heapify\n        if (p &lt; 0 || maxHeap.get(i) &lt;= maxHeap.get(p))\n            break;\n        // Swap two nodes\n        swap(i, p);\n        // Loop upward heapify\n        i = p;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* Element enters heap */\nvoid Push(int val) {\n    // Add node\n    maxHeap.Add(val);\n    // Heapify from bottom to top\n    SiftUp(Size() - 1);\n}\n\n/* Starting from node i, heapify from bottom to top */\nvoid SiftUp(int i) {\n    while (true) {\n        // Get parent node of node i\n        int p = Parent(i);\n        // If 'past root node' or 'node needs no repair', end heapify\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // Swap two nodes\n        Swap(i, p);\n        // Loop upward heapify\n        i = p;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* Element enters heap */\nfunc (h *maxHeap) push(val any) {\n    // Add node\n    h.data = append(h.data, val)\n    // Heapify from bottom to top\n    h.siftUp(len(h.data) - 1)\n}\n\n/* Starting from node i, heapify from bottom to top */\nfunc (h *maxHeap) siftUp(i int) {\n    for true {\n        // Get parent node of node i\n        p := h.parent(i)\n        // When \"crossing root node\" or \"node needs no repair\", end heapify\n        if p &lt; 0 || h.data[i].(int) &lt;= h.data[p].(int) {\n            break\n        }\n        // Swap two nodes\n        h.swap(i, p)\n        // Loop upward heapify\n        i = p\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* Element enters heap */\nfunc push(val: Int) {\n    // Add node\n    maxHeap.append(val)\n    // Heapify from bottom to top\n    siftUp(i: size() - 1)\n}\n\n/* Starting from node i, heapify from bottom to top */\nfunc siftUp(i: Int) {\n    var i = i\n    while true {\n        // Get parent node of node i\n        let p = parent(i: i)\n        // When \"crossing root node\" or \"node needs no repair\", end heapify\n        if p &lt; 0 || maxHeap[i] &lt;= maxHeap[p] {\n            break\n        }\n        // Swap two nodes\n        swap(i: i, j: p)\n        // Loop upward heapify\n        i = p\n    }\n}\n</code></pre> my_heap.js<pre><code>/* Element enters heap */\npush(val) {\n    // Add node\n    this.#maxHeap.push(val);\n    // Heapify from bottom to top\n    this.#siftUp(this.size() - 1);\n}\n\n/* Starting from node i, heapify from bottom to top */\n#siftUp(i) {\n    while (true) {\n        // Get parent node of node i\n        const p = this.#parent(i);\n        // When \"crossing root node\" or \"node needs no repair\", end heapify\n        if (p &lt; 0 || this.#maxHeap[i] &lt;= this.#maxHeap[p]) break;\n        // Swap two nodes\n        this.#swap(i, p);\n        // Loop upward heapify\n        i = p;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* Element enters heap */\npush(val: number): void {\n    // Add node\n    this.maxHeap.push(val);\n    // Heapify from bottom to top\n    this.siftUp(this.size() - 1);\n}\n\n/* Starting from node i, heapify from bottom to top */\nsiftUp(i: number): void {\n    while (true) {\n        // Get parent node of node i\n        const p = this.parent(i);\n        // When \"crossing root node\" or \"node needs no repair\", end heapify\n        if (p &lt; 0 || this.maxHeap[i] &lt;= this.maxHeap[p]) break;\n        // Swap two nodes\n        this.swap(i, p);\n        // Loop upward heapify\n        i = p;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* Element enters heap */\nvoid push(int val) {\n  // Add node\n  _maxHeap.add(val);\n  // Heapify from bottom to top\n  siftUp(size() - 1);\n}\n\n/* Starting from node i, heapify from bottom to top */\nvoid siftUp(int i) {\n  while (true) {\n    // Get parent node of node i\n    int p = _parent(i);\n    // When \"crossing root node\" or \"node needs no repair\", end heapify\n    if (p &lt; 0 || _maxHeap[i] &lt;= _maxHeap[p]) {\n      break;\n    }\n    // Swap two nodes\n    _swap(i, p);\n    // Loop upward heapify\n    i = p;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* Element enters heap */\nfn push(&amp;mut self, val: i32) {\n    // Add node\n    self.max_heap.push(val);\n    // Heapify from bottom to top\n    self.sift_up(self.size() - 1);\n}\n\n/* Starting from node i, heapify from bottom to top */\nfn sift_up(&amp;mut self, mut i: usize) {\n    loop {\n        // Node i is already the heap root, end heapification\n        if i == 0 {\n            break;\n        }\n        // Get parent node of node i\n        let p = Self::parent(i);\n        // When \"node needs no repair\", end heapification\n        if self.max_heap[i] &lt;= self.max_heap[p] {\n            break;\n        }\n        // Swap two nodes\n        self.swap(i, p);\n        // Loop upward heapify\n        i = p;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* Element enters heap */\nvoid push(MaxHeap *maxHeap, int val) {\n    // By default, should not add this many nodes\n    if (maxHeap-&gt;size == MAX_SIZE) {\n        printf(\"heap is full!\");\n        return;\n    }\n    // Add node\n    maxHeap-&gt;data[maxHeap-&gt;size] = val;\n    maxHeap-&gt;size++;\n\n    // Heapify from bottom to top\n    siftUp(maxHeap, maxHeap-&gt;size - 1);\n}\n\n/* Starting from node i, heapify from bottom to top */\nvoid siftUp(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // Get parent node of node i\n        int p = parent(maxHeap, i);\n        // When \"crossing root node\" or \"node needs no repair\", end heapify\n        if (p &lt; 0 || maxHeap-&gt;data[i] &lt;= maxHeap-&gt;data[p]) {\n            break;\n        }\n        // Swap two nodes\n        swap(maxHeap, i, p);\n        // Loop upward heapify\n        i = p;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* Element enters heap */\nfun push(_val: Int) {\n    // Add node\n    maxHeap.add(_val)\n    // Heapify from bottom to top\n    siftUp(size() - 1)\n}\n\n/* Starting from node i, heapify from bottom to top */\nfun siftUp(it: Int) {\n    // Kotlin function parameters are immutable, so create temporary variable\n    var i = it\n    while (true) {\n        // Get parent node of node i\n        val p = parent(i)\n        // When \"crossing root node\" or \"node needs no repair\", end heapify\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n        // Swap two nodes\n        swap(i, p)\n        // Loop upward heapify\n        i = p\n    }\n}\n</code></pre> my_heap.rb<pre><code>### Push element to heap ###\ndef push(val)\n  # Add node\n  @max_heap &lt;&lt; val\n  # Heapify from bottom to top\n  sift_up(size - 1)\nend\n\n### Heapify from node i, bottom to top ###\ndef sift_up(i)\n  loop do\n    # Get parent node of node i\n    p = parent(i)\n    # When \"crossing root node\" or \"node needs no repair\", end heapify\n    break if p &lt; 0 || @max_heap[i] &lt;= @max_heap[p]\n    # Swap two nodes\n    swap(i, p)\n    # Loop upward heapify\n    i = p\n  end\nend\n</code></pre>","path":["Chapter 8. Heap","8.1   Heap"],"tags":[]},{"location":"chapter_heap/heap/#4-removing-the-heap-top-element","level":3,"title":"4.   Removing the Heap Top Element","text":"<p>The heap top element is the root node of the binary tree, which is the first element of the list. If we directly remove the first element from the list, all node indexes in the binary tree would change, making subsequent repair with heapify difficult. To minimize changes in element indexes, we use the following steps.</p> <ol> <li>Swap the heap top element with the heap bottom element (swap the root node with the rightmost leaf node).</li> <li>After swapping, remove the heap bottom from the list (note that since we've swapped, we're actually removing the original heap top element).</li> <li>Starting from the root node, perform heapify from top to bottom.</li> </ol> <p>As shown in Figure 8-4, the direction of \"top-to-bottom heapify\" is opposite to \"bottom-to-top heapify\". We compare the root node's value with its two children and swap it with the largest child. Then loop this operation until we pass a leaf node or encounter a node that doesn't need swapping.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 8-4   Steps of removing the heap top element </p> <p>Similar to the element insertion operation, the time complexity of the heap top element removal operation is also \\(O(\\log n)\\). The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def pop(self) -&gt; int:\n    \"\"\"Element exits heap\"\"\"\n    # Handle empty case\n    if self.is_empty():\n        raise IndexError(\"Heap is empty\")\n    # Swap root node with rightmost leaf node (swap first element with last element)\n    self.swap(0, self.size() - 1)\n    # Delete node\n    val = self.max_heap.pop()\n    # Heapify from top to bottom\n    self.sift_down(0)\n    # Return top element\n    return val\n\ndef sift_down(self, i: int):\n    \"\"\"Starting from node i, heapify from top to bottom\"\"\"\n    while True:\n        # Find node with largest value among i, l, r, denoted as ma\n        l, r, ma = self.left(i), self.right(i), i\n        if l &lt; self.size() and self.max_heap[l] &gt; self.max_heap[ma]:\n            ma = l\n        if r &lt; self.size() and self.max_heap[r] &gt; self.max_heap[ma]:\n            ma = r\n        # If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        if ma == i:\n            break\n        # Swap two nodes\n        self.swap(i, ma)\n        # Loop downward heapify\n        i = ma\n</code></pre> my_heap.cpp<pre><code>/* Element exits heap */\nvoid pop() {\n    // Handle empty case\n    if (isEmpty()) {\n        throw out_of_range(\"Heap is empty\");\n    }\n    // Delete node\n    swap(maxHeap[0], maxHeap[size() - 1]);\n    // Remove node\n    maxHeap.pop_back();\n    // Return top element\n    siftDown(0);\n}\n\n/* Starting from node i, heapify from top to bottom */\nvoid siftDown(int i) {\n    while (true) {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // Swap two nodes\n        if (ma == i)\n            break;\n        swap(maxHeap[i], maxHeap[ma]);\n        // Loop downwards heapification\n        i = ma;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* Element exits heap */\nint pop() {\n    // Handle empty case\n    if (isEmpty())\n        throw new IndexOutOfBoundsException();\n    // Delete node\n    swap(0, size() - 1);\n    // Remove node\n    int val = maxHeap.remove(size() - 1);\n    // Return top element\n    siftDown(0);\n    // Return heap top element\n    return val;\n}\n\n/* Starting from node i, heapify from top to bottom */\nvoid siftDown(int i) {\n    while (true) {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap.get(l) &gt; maxHeap.get(ma))\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap.get(r) &gt; maxHeap.get(ma))\n            ma = r;\n        // Swap two nodes\n        if (ma == i)\n            break;\n        // Swap two nodes\n        swap(i, ma);\n        // Loop downwards heapification\n        i = ma;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* Element exits heap */\nint Pop() {\n    // Handle empty case\n    if (IsEmpty())\n        throw new IndexOutOfRangeException();\n    // Delete node\n    Swap(0, Size() - 1);\n    // Remove node\n    int val = maxHeap.Last();\n    maxHeap.RemoveAt(Size() - 1);\n    // Return top element\n    SiftDown(0);\n    // Return heap top element\n    return val;\n}\n\n/* Starting from node i, heapify from top to bottom */\nvoid SiftDown(int i) {\n    while (true) {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        int l = Left(i), r = Right(i), ma = i;\n        if (l &lt; Size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; Size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // If 'node i is largest' or 'past leaf node', end heapify\n        if (ma == i) break;\n        // Swap two nodes\n        Swap(i, ma);\n        // Loop downwards heapification\n        i = ma;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* Element exits heap */\nfunc (h *maxHeap) pop() any {\n    // Handle empty case\n    if h.isEmpty() {\n        fmt.Println(\"error\")\n        return nil\n    }\n    // Delete node\n    h.swap(0, h.size()-1)\n    // Remove node\n    val := h.data[len(h.data)-1]\n    h.data = h.data[:len(h.data)-1]\n    // Return top element\n    h.siftDown(0)\n\n    // Return heap top element\n    return val\n}\n\n/* Starting from node i, heapify from top to bottom */\nfunc (h *maxHeap) siftDown(i int) {\n    for true {\n        // Find node with maximum value among nodes i, l, r, denoted as max\n        l, r, max := h.left(i), h.right(i), i\n        if l &lt; h.size() &amp;&amp; h.data[l].(int) &gt; h.data[max].(int) {\n            max = l\n        }\n        if r &lt; h.size() &amp;&amp; h.data[r].(int) &gt; h.data[max].(int) {\n            max = r\n        }\n        // Swap two nodes\n        if max == i {\n            break\n        }\n        // Swap two nodes\n        h.swap(i, max)\n        // Loop downwards heapification\n        i = max\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* Element exits heap */\nfunc pop() -&gt; Int {\n    // Handle empty case\n    if isEmpty() {\n        fatalError(\"Heap is empty\")\n    }\n    // Delete node\n    swap(i: 0, j: size() - 1)\n    // Remove node\n    let val = maxHeap.remove(at: size() - 1)\n    // Return top element\n    siftDown(i: 0)\n    // Return heap top element\n    return val\n}\n\n/* Starting from node i, heapify from top to bottom */\nfunc siftDown(i: Int) {\n    var i = i\n    while true {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        let l = left(i: i)\n        let r = right(i: i)\n        var ma = i\n        if l &lt; size(), maxHeap[l] &gt; maxHeap[ma] {\n            ma = l\n        }\n        if r &lt; size(), maxHeap[r] &gt; maxHeap[ma] {\n            ma = r\n        }\n        // Swap two nodes\n        if ma == i {\n            break\n        }\n        // Swap two nodes\n        swap(i: i, j: ma)\n        // Loop downwards heapification\n        i = ma\n    }\n}\n</code></pre> my_heap.js<pre><code>/* Element exits heap */\npop() {\n    // Handle empty case\n    if (this.isEmpty()) throw new Error('Heap is empty');\n    // Delete node\n    this.#swap(0, this.size() - 1);\n    // Remove node\n    const val = this.#maxHeap.pop();\n    // Return top element\n    this.#siftDown(0);\n    // Return heap top element\n    return val;\n}\n\n/* Starting from node i, heapify from top to bottom */\n#siftDown(i) {\n    while (true) {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        const l = this.#left(i),\n            r = this.#right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.#maxHeap[l] &gt; this.#maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.#maxHeap[r] &gt; this.#maxHeap[ma]) ma = r;\n        // Swap two nodes\n        if (ma === i) break;\n        // Swap two nodes\n        this.#swap(i, ma);\n        // Loop downwards heapification\n        i = ma;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* Element exits heap */\npop(): number {\n    // Handle empty case\n    if (this.isEmpty()) throw new RangeError('Heap is empty.');\n    // Delete node\n    this.swap(0, this.size() - 1);\n    // Remove node\n    const val = this.maxHeap.pop();\n    // Return top element\n    this.siftDown(0);\n    // Return heap top element\n    return val;\n}\n\n/* Starting from node i, heapify from top to bottom */\nsiftDown(i: number): void {\n    while (true) {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        const l = this.left(i),\n            r = this.right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.maxHeap[l] &gt; this.maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.maxHeap[r] &gt; this.maxHeap[ma]) ma = r;\n        // Swap two nodes\n        if (ma === i) break;\n        // Swap two nodes\n        this.swap(i, ma);\n        // Loop downwards heapification\n        i = ma;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* Element exits heap */\nint pop() {\n  // Handle empty case\n  if (isEmpty()) throw Exception('Heap is empty');\n  // Delete node\n  _swap(0, size() - 1);\n  // Remove node\n  int val = _maxHeap.removeLast();\n  // Return top element\n  siftDown(0);\n  // Return heap top element\n  return val;\n}\n\n/* Starting from node i, heapify from top to bottom */\nvoid siftDown(int i) {\n  while (true) {\n    // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n    int l = _left(i);\n    int r = _right(i);\n    int ma = i;\n    if (l &lt; size() &amp;&amp; _maxHeap[l] &gt; _maxHeap[ma]) ma = l;\n    if (r &lt; size() &amp;&amp; _maxHeap[r] &gt; _maxHeap[ma]) ma = r;\n    // Swap two nodes\n    if (ma == i) break;\n    // Swap two nodes\n    _swap(i, ma);\n    // Loop downwards heapification\n    i = ma;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* Element exits heap */\nfn pop(&amp;mut self) -&gt; i32 {\n    // Handle empty case\n    if self.is_empty() {\n        panic!(\"index out of bounds\");\n    }\n    // Delete node\n    self.swap(0, self.size() - 1);\n    // Remove node\n    let val = self.max_heap.pop().unwrap();\n    // Return top element\n    self.sift_down(0);\n    // Return heap top element\n    val\n}\n\n/* Starting from node i, heapify from top to bottom */\nfn sift_down(&amp;mut self, mut i: usize) {\n    loop {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        let (l, r, mut ma) = (Self::left(i), Self::right(i), i);\n        if l &lt; self.size() &amp;&amp; self.max_heap[l] &gt; self.max_heap[ma] {\n            ma = l;\n        }\n        if r &lt; self.size() &amp;&amp; self.max_heap[r] &gt; self.max_heap[ma] {\n            ma = r;\n        }\n        // Swap two nodes\n        if ma == i {\n            break;\n        }\n        // Swap two nodes\n        self.swap(i, ma);\n        // Loop downwards heapification\n        i = ma;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* Element exits heap */\nint pop(MaxHeap *maxHeap) {\n    // Handle empty case\n    if (isEmpty(maxHeap)) {\n        printf(\"heap is empty!\");\n        return INT_MAX;\n    }\n    // Delete node\n    swap(maxHeap, 0, size(maxHeap) - 1);\n    // Remove node\n    int val = maxHeap-&gt;data[maxHeap-&gt;size - 1];\n    maxHeap-&gt;size--;\n    // Return top element\n    siftDown(maxHeap, 0);\n\n    // Return heap top element\n    return val;\n}\n\n/* Starting from node i, heapify from top to bottom */\nvoid siftDown(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // Find node with maximum value among nodes i, l, r, denoted as max\n        int l = left(maxHeap, i);\n        int r = right(maxHeap, i);\n        int max = i;\n        if (l &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[l] &gt; maxHeap-&gt;data[max]) {\n            max = l;\n        }\n        if (r &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[r] &gt; maxHeap-&gt;data[max]) {\n            max = r;\n        }\n        // Swap two nodes\n        if (max == i) {\n            break;\n        }\n        // Swap two nodes\n        swap(maxHeap, i, max);\n        // Loop downwards heapification\n        i = max;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* Element exits heap */\nfun pop(): Int {\n    // Handle empty case\n    if (isEmpty()) throw IndexOutOfBoundsException()\n    // Delete node\n    swap(0, size() - 1)\n    // Remove node\n    val _val = maxHeap.removeAt(size() - 1)\n    // Return top element\n    siftDown(0)\n    // Return heap top element\n    return _val\n}\n\n/* Starting from node i, heapify from top to bottom */\nfun siftDown(it: Int) {\n    // Kotlin function parameters are immutable, so create temporary variable\n    var i = it\n    while (true) {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        val l = left(i)\n        val r = right(i)\n        var ma = i\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n        // Swap two nodes\n        if (ma == i) break\n        // Swap two nodes\n        swap(i, ma)\n        // Loop downwards heapification\n        i = ma\n    }\n}\n</code></pre> my_heap.rb<pre><code>### Pop element from heap ###\ndef pop\n  # Handle empty case\n  raise IndexError, \"Heap is empty\" if is_empty?\n  # Delete node\n  swap(0, size - 1)\n  # Remove node\n  val = @max_heap.pop\n  # Return top element\n  sift_down(0)\n  # Return heap top element\n  val\nend\n\n### Heapify from node i, top to bottom ###\ndef sift_down(i)\n  loop do\n    # If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n    l, r, ma = left(i), right(i), i\n    ma = l if l &lt; size &amp;&amp; @max_heap[l] &gt; @max_heap[ma]\n    ma = r if r &lt; size &amp;&amp; @max_heap[r] &gt; @max_heap[ma]\n\n    # Swap two nodes\n    break if ma == i\n\n    # Swap two nodes\n    swap(i, ma)\n    # Loop downwards heapification\n    i = ma\n  end\nend\n</code></pre>","path":["Chapter 8. Heap","8.1   Heap"],"tags":[]},{"location":"chapter_heap/heap/#813-common-applications-of-heaps","level":2,"title":"8.1.3   Common Applications of Heaps","text":"<ul> <li>Priority queue: Heaps are typically the preferred data structure for implementing priority queues. The time complexity of both enqueue and dequeue operations is \\(O(\\log n)\\), and heap construction has a time complexity of \\(O(n)\\), making these operations highly efficient.</li> <li>Heap sort: Given a set of data, we can build a heap with them and then continuously perform element removal operations to obtain sorted data. However, we usually use a more elegant approach to implement heap sort, as detailed in the \"Heap Sort\" chapter.</li> <li>Getting the largest \\(k\\) elements: This is a classic algorithm problem and also a typical application, such as selecting the top 10 trending news items for Weibo Hot Search or the top 10 best-selling products.</li> </ul>","path":["Chapter 8. Heap","8.1   Heap"],"tags":[]},{"location":"chapter_heap/summary/","level":1,"title":"8.4   Summary","text":"","path":["Chapter 8. Heap","8.4   Summary"],"tags":[]},{"location":"chapter_heap/summary/#1-key-review","level":3,"title":"1.   Key Review","text":"<ul> <li>A heap is a complete binary tree. Depending on the property it satisfies, it can be classified as either a max heap or a min heap. The top element of a max heap (min heap) is the largest (smallest) element.</li> <li>A priority queue is a queue in which elements are dequeued according to priority, and it is typically implemented using a heap.</li> <li>Common heap operations and their corresponding time complexities include inserting an element \\(O(\\log n)\\), removing the top element \\(O(\\log n)\\), and accessing the top element \\(O(1)\\).</li> <li>Complete binary trees are well-suited for array representation, so we typically use arrays to store heaps.</li> <li>Heapify operations are used to maintain the heap property and are employed in both element insertion and removal operations.</li> <li>Building a heap from \\(n\\) input elements can be optimized to \\(O(n)\\), which is highly efficient.</li> <li>Top-k is a classic algorithmic problem that can be solved efficiently using a heap, with a time complexity of \\(O(n \\log k)\\).</li> </ul>","path":["Chapter 8. Heap","8.4   Summary"],"tags":[]},{"location":"chapter_heap/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: Does the term \"heap\" in data structures mean the same thing as \"heap\" in memory management?</p> <p>They are not the same concept; they simply share the same name. In computer systems, the heap is part of dynamic memory allocation, and programs can use it to store data at runtime. A program can request a certain amount of heap memory to store complex structures such as objects and arrays. When the data is no longer needed, the program must release that memory to prevent memory leaks. Compared with stack memory, heap memory requires more careful management and use; improper handling can lead to problems such as memory leaks and dangling pointers.</p>","path":["Chapter 8. Heap","8.4   Summary"],"tags":[]},{"location":"chapter_heap/top_k/","level":1,"title":"8.3   Top-k Problem","text":"<p>Question</p> <p>Given an unordered array <code>nums</code> of length \\(n\\), return the largest \\(k\\) elements in the array.</p> <p>For this problem, we will first introduce two relatively straightforward solutions, followed by a more efficient heap-based solution.</p>","path":["Chapter 8. Heap","8.3   Top-k Problem"],"tags":[]},{"location":"chapter_heap/top_k/#831-method-1-iterative-selection","level":2,"title":"8.3.1   Method 1: Iterative Selection","text":"<p>We can perform \\(k\\) rounds of traversal as shown in Figure 8-6, extracting the \\(1^{st}\\), \\(2^{nd}\\), \\(\\dots\\), \\(k^{th}\\) largest elements in each round, with a time complexity of \\(O(nk)\\).</p> <p>This method is only suitable when \\(k \\ll n\\), because when \\(k\\) is close to \\(n\\), the time complexity approaches \\(O(n^2)\\), making it very inefficient.</p> <p></p> <p> Figure 8-6   Traversing to find the largest k elements </p> <p>Tip</p> <p>When \\(k = n\\), we can obtain a complete sorted sequence, which is equivalent to the \"selection sort\" algorithm.</p>","path":["Chapter 8. Heap","8.3   Top-k Problem"],"tags":[]},{"location":"chapter_heap/top_k/#832-method-2-sorting","level":2,"title":"8.3.2   Method 2: Sorting","text":"<p>As shown in Figure 8-7, we can first sort the array <code>nums</code>, then return the rightmost \\(k\\) elements, with a time complexity of \\(O(n \\log n)\\).</p> <p>Clearly, this method does more work than necessary, because we only need to find the largest \\(k\\) elements rather than sort the other elements.</p> <p></p> <p> Figure 8-7   Sorting to find the largest k elements </p>","path":["Chapter 8. Heap","8.3   Top-k Problem"],"tags":[]},{"location":"chapter_heap/top_k/#833-method-3-heap","level":2,"title":"8.3.3   Method 3: Heap","text":"<p>We can solve the Top-k problem more efficiently with a heap, as shown in Figure 8-8.</p> <ol> <li>Initialize a min heap, where the heap top element is the smallest.</li> <li>First, insert the first \\(k\\) elements of the array into the heap in sequence.</li> <li>Starting from the \\((k + 1)^{th}\\) element, if the current element is greater than the heap top element, remove the heap top element and insert the current element into the heap.</li> <li>After traversal is complete, the heap contains the largest \\(k\\) elements.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 8-8   Finding the largest k elements using a heap </p> <p>Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby top_k.py<pre><code>def top_k_heap(nums: list[int], k: int) -&gt; list[int]:\n    \"\"\"Find the largest k elements in array based on heap\"\"\"\n    # Initialize min heap\n    heap = []\n    # Enter the first k elements of array into heap\n    for i in range(k):\n        heapq.heappush(heap, nums[i])\n    # Starting from the (k+1)th element, maintain heap length as k\n    for i in range(k, len(nums)):\n        # If current element is greater than top element, top element exits heap, current element enters heap\n        if nums[i] &gt; heap[0]:\n            heapq.heappop(heap)\n            heapq.heappush(heap, nums[i])\n    return heap\n</code></pre> top_k.cpp<pre><code>/* Find the largest k elements in array based on heap */\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; topKHeap(vector&lt;int&gt; &amp;nums, int k) {\n    // Python's heapq module implements min heap by default\n    priority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; heap;\n    // Enter the first k elements of array into heap\n    for (int i = 0; i &lt; k; i++) {\n        heap.push(nums[i]);\n    }\n    // Starting from the (k+1)th element, maintain heap length as k\n    for (int i = k; i &lt; nums.size(); i++) {\n        // If current element is greater than top element, top element exits heap, current element enters heap\n        if (nums[i] &gt; heap.top()) {\n            heap.pop();\n            heap.push(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.java<pre><code>/* Find the largest k elements in array based on heap */\nQueue&lt;Integer&gt; topKHeap(int[] nums, int k) {\n    // Python's heapq module implements min heap by default\n    Queue&lt;Integer&gt; heap = new PriorityQueue&lt;Integer&gt;();\n    // Enter the first k elements of array into heap\n    for (int i = 0; i &lt; k; i++) {\n        heap.offer(nums[i]);\n    }\n    // Starting from the (k+1)th element, maintain heap length as k\n    for (int i = k; i &lt; nums.length; i++) {\n        // If current element is greater than top element, top element exits heap, current element enters heap\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll();\n            heap.offer(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.cs<pre><code>/* Find the largest k elements in array based on heap */\nPriorityQueue&lt;int, int&gt; TopKHeap(int[] nums, int k) {\n    // Python's heapq module implements min heap by default\n    PriorityQueue&lt;int, int&gt; heap = new();\n    // Enter the first k elements of array into heap\n    for (int i = 0; i &lt; k; i++) {\n        heap.Enqueue(nums[i], nums[i]);\n    }\n    // Starting from the (k+1)th element, maintain heap length as k\n    for (int i = k; i &lt; nums.Length; i++) {\n        // If current element is greater than top element, top element exits heap, current element enters heap\n        if (nums[i] &gt; heap.Peek()) {\n            heap.Dequeue();\n            heap.Enqueue(nums[i], nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.go<pre><code>/* Find the largest k elements in array based on heap */\nfunc topKHeap(nums []int, k int) *minHeap {\n    // Python's heapq module implements min heap by default\n    h := &amp;minHeap{}\n    heap.Init(h)\n    // Enter the first k elements of array into heap\n    for i := 0; i &lt; k; i++ {\n        heap.Push(h, nums[i])\n    }\n    // Starting from the (k+1)th element, maintain heap length as k\n    for i := k; i &lt; len(nums); i++ {\n        // If current element is greater than top element, top element exits heap, current element enters heap\n        if nums[i] &gt; h.Top().(int) {\n            heap.Pop(h)\n            heap.Push(h, nums[i])\n        }\n    }\n    return h\n}\n</code></pre> top_k.swift<pre><code>/* Find the largest k elements in array based on heap */\nfunc topKHeap(nums: [Int], k: Int) -&gt; [Int] {\n    // Initialize min heap and build heap with first k elements\n    var heap = Heap(nums.prefix(k))\n    // Starting from the (k+1)th element, maintain heap length as k\n    for i in nums.indices.dropFirst(k) {\n        // If current element is greater than top element, top element exits heap, current element enters heap\n        if nums[i] &gt; heap.min()! {\n            _ = heap.removeMin()\n            heap.insert(nums[i])\n        }\n    }\n    return heap.unordered\n}\n</code></pre> top_k.js<pre><code>/* Element enters heap */\nfunction pushMinHeap(maxHeap, val) {\n    // Negate element\n    maxHeap.push(-val);\n}\n\n/* Element exits heap */\nfunction popMinHeap(maxHeap) {\n    // Negate element\n    return -maxHeap.pop();\n}\n\n/* Access top element */\nfunction peekMinHeap(maxHeap) {\n    // Negate element\n    return -maxHeap.peek();\n}\n\n/* Extract elements from heap */\nfunction getMinHeap(maxHeap) {\n    // Negate element\n    return maxHeap.getMaxHeap().map((num) =&gt; -num);\n}\n\n/* Find the largest k elements in array based on heap */\nfunction topKHeap(nums, k) {\n    // Python's heapq module implements min heap by default\n    // Note: We negate all heap elements to simulate min heap using max heap\n    const maxHeap = new MaxHeap([]);\n    // Enter the first k elements of array into heap\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // Starting from the (k+1)th element, maintain heap length as k\n    for (let i = k; i &lt; nums.length; i++) {\n        // If current element is greater than top element, top element exits heap, current element enters heap\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // Return elements in heap\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.ts<pre><code>/* Element enters heap */\nfunction pushMinHeap(maxHeap: MaxHeap, val: number): void {\n    // Negate element\n    maxHeap.push(-val);\n}\n\n/* Element exits heap */\nfunction popMinHeap(maxHeap: MaxHeap): number {\n    // Negate element\n    return -maxHeap.pop();\n}\n\n/* Access top element */\nfunction peekMinHeap(maxHeap: MaxHeap): number {\n    // Negate element\n    return -maxHeap.peek();\n}\n\n/* Extract elements from heap */\nfunction getMinHeap(maxHeap: MaxHeap): number[] {\n    // Negate element\n    return maxHeap.getMaxHeap().map((num: number) =&gt; -num);\n}\n\n/* Find the largest k elements in array based on heap */\nfunction topKHeap(nums: number[], k: number): number[] {\n    // Python's heapq module implements min heap by default\n    // Note: We negate all heap elements to simulate min heap using max heap\n    const maxHeap = new MaxHeap([]);\n    // Enter the first k elements of array into heap\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // Starting from the (k+1)th element, maintain heap length as k\n    for (let i = k; i &lt; nums.length; i++) {\n        // If current element is greater than top element, top element exits heap, current element enters heap\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // Return elements in heap\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.dart<pre><code>/* Find the largest k elements in array based on heap */\nMinHeap topKHeap(List&lt;int&gt; nums, int k) {\n  // Initialize min heap, push first k elements of array to heap\n  MinHeap heap = MinHeap(nums.sublist(0, k));\n  // Starting from the (k+1)th element, maintain heap length as k\n  for (int i = k; i &lt; nums.length; i++) {\n    // If current element is greater than top element, top element exits heap, current element enters heap\n    if (nums[i] &gt; heap.peek()) {\n      heap.pop();\n      heap.push(nums[i]);\n    }\n  }\n  return heap;\n}\n</code></pre> top_k.rs<pre><code>/* Find the largest k elements in array based on heap */\nfn top_k_heap(nums: Vec&lt;i32&gt;, k: usize) -&gt; BinaryHeap&lt;Reverse&lt;i32&gt;&gt; {\n    // BinaryHeap is a max heap, use Reverse to negate elements to implement min heap\n    let mut heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n    // Enter the first k elements of array into heap\n    for &amp;num in nums.iter().take(k) {\n        heap.push(Reverse(num));\n    }\n    // Starting from the (k+1)th element, maintain heap length as k\n    for &amp;num in nums.iter().skip(k) {\n        // If current element is greater than top element, top element exits heap, current element enters heap\n        if num &gt; heap.peek().unwrap().0 {\n            heap.pop();\n            heap.push(Reverse(num));\n        }\n    }\n    heap\n}\n</code></pre> top_k.c<pre><code>/* Element enters heap */\nvoid pushMinHeap(MaxHeap *maxHeap, int val) {\n    // Negate element\n    push(maxHeap, -val);\n}\n\n/* Element exits heap */\nint popMinHeap(MaxHeap *maxHeap) {\n    // Negate element\n    return -pop(maxHeap);\n}\n\n/* Access top element */\nint peekMinHeap(MaxHeap *maxHeap) {\n    // Negate element\n    return -peek(maxHeap);\n}\n\n/* Extract elements from heap */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // Negate all heap elements and store in res array\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n/* Extract elements from heap */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // Negate all heap elements and store in res array\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n// Function to find k largest elements in array using heap\nint *topKHeap(int *nums, int sizeNums, int k) {\n    // Python's heapq module implements min heap by default\n    // Note: We negate all heap elements to simulate min heap using max heap\n    int *empty = (int *)malloc(0);\n    MaxHeap *maxHeap = newMaxHeap(empty, 0);\n    // Enter the first k elements of array into heap\n    for (int i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // Starting from the (k+1)th element, maintain heap length as k\n    for (int i = k; i &lt; sizeNums; i++) {\n        // If current element is greater than top element, top element exits heap, current element enters heap\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    int *res = getMinHeap(maxHeap);\n    // Free memory\n    delMaxHeap(maxHeap);\n    return res;\n}\n</code></pre> top_k.kt<pre><code>/* Find the largest k elements in array based on heap */\nfun topKHeap(nums: IntArray, k: Int): Queue&lt;Int&gt; {\n    // Python's heapq module implements min heap by default\n    val heap = PriorityQueue&lt;Int&gt;()\n    // Enter the first k elements of array into heap\n    for (i in 0..&lt;k) {\n        heap.offer(nums[i])\n    }\n    // Starting from the (k+1)th element, maintain heap length as k\n    for (i in k..&lt;nums.size) {\n        // If current element is greater than top element, top element exits heap, current element enters heap\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll()\n            heap.offer(nums[i])\n        }\n    }\n    return heap\n}\n</code></pre> top_k.rb<pre><code>### Find largest k elements in array using heap ###\ndef top_k_heap(nums, k)\n  # Python's heapq module implements min heap by default\n  # Note: We negate all heap elements to simulate min heap using max heap\n  max_heap = MaxHeap.new([])\n\n  # Enter the first k elements of array into heap\n  for i in 0...k\n    push_min_heap(max_heap, nums[i])\n  end\n\n  # Starting from the (k+1)th element, maintain heap length as k\n  for i in k...nums.length\n    # If current element is greater than top element, top element exits heap, current element enters heap\n    if nums[i] &gt; peek_min_heap(max_heap)\n      pop_min_heap(max_heap)\n      push_min_heap(max_heap, nums[i])\n    end\n  end\n\n  get_min_heap(max_heap)\nend\n</code></pre> <p>A total of \\(n\\) rounds of heap insertions and removals are performed, with the heap's maximum length being \\(k\\), so the time complexity is \\(O(n \\log k)\\). This method is very efficient; when \\(k\\) is small, the time complexity approaches \\(O(n)\\); when \\(k\\) is large, the time complexity does not exceed \\(O(n \\log n)\\).</p> <p>Additionally, this method is well suited to dynamic data streams. As new data arrives, we can continuously maintain the elements in the heap, enabling dynamic updates to the largest \\(k\\) elements.</p>","path":["Chapter 8. Heap","8.3   Top-k Problem"],"tags":[]},{"location":"chapter_hello_algo/","level":1,"title":"Preface","text":"<p>A few years ago, I shared the \"Sword for Offer\" problem solutions on LeetCode, receiving encouragement and support from many readers. During interactions with readers, the most frequently asked question I encountered was \"how to get started with algorithms.\" Gradually, I developed a keen interest in this question.</p> <p>Diving straight into problem-solving seems to be the most popular approach—it's simple, direct, and effective. However, problem-solving is like playing Minesweeper: those with strong self-learning abilities can successfully defuse the mines one by one, while those with insufficient foundations may end up bruised and battered, retreating step by step in frustration. Reading through textbooks is also a common practice, but for job seekers, graduation theses, resume submissions, and preparations for written tests and interviews have already consumed most of their energy, making working through thick books an arduous challenge.</p> <p>If you're facing similar struggles, then it's fortunate that this book has \"found\" you. This book is my answer to this question—even if it may not be the optimal solution, it is at least a positive attempt. While this book alone won't directly land you a job offer, it will guide you through the \"landscape\" of data structures and algorithms, help you understand the shapes, sizes, and distributions of different \"mines,\" and enable you to master various \"mine-clearing methods.\" With these skills, I believe you can tackle problems and read technical literature more confidently, gradually building a complete knowledge system.</p> <p>I deeply agree with Professor Feynman's words: \"Knowledge isn't free. You have to pay attention.\" In this sense, this book is not entirely \"free.\" In order to live up to the precious \"attention\" you invest in this book, I will do my utmost and devote my greatest \"attention\" to completing this work.</p> <p>I'm keenly aware of the limits of my knowledge and experience. Although the content of this book has been refined over a period of time, there are certainly still many errors, and I sincerely welcome critiques and corrections from teachers and fellow students.</p> <p></p> Hello, Algorithms! <p>The advent of computers has brought tremendous changes to the world. With their high-speed computing capabilities and excellent programmability, they have become the ideal medium for executing algorithms and processing data. Whether it's the realistic graphics in video games, the intelligent decision-making in autonomous driving, AlphaGo's brilliant Go matches, or ChatGPT's natural interactions, these applications are all striking demonstrations of algorithms at work on computers.</p> <p>In fact, before the advent of computers, algorithms and data structures already existed in every corner of the world. Early algorithms were relatively simple, such as ancient counting methods and tool-making procedures. As civilization progressed, algorithms gradually became more refined and complex. From the ingenious craftsmanship of master artisans, to industrial products that liberate productive forces, to the scientific laws governing the operation of the universe, behind almost every ordinary or astonishing thing lies ingenious algorithmic thinking.</p> <p>Similarly, data structures are everywhere: from large-scale social networks to small subway systems, many systems can be modeled as \"graphs\"; from a nation to a family, the primary organizational forms of society exhibit characteristics of \"trees\"; winter clothing is like a \"stack,\" where the first item put on is the last to be taken off; a badminton tube is like a \"queue,\" with items inserted at one end and retrieved from the other; a dictionary is like a \"hash table,\" enabling quick lookup of target entries.</p> <p>This book aims to help readers understand the core concepts of algorithms and data structures through clear and accessible animated illustrations and runnable code examples, and to implement them in code. Building on this foundation, the book endeavors to reveal the vivid manifestations of algorithms in the complex world and showcase the beauty of algorithms. I hope this book can be of help to you!</p>","path":["Before Starting","Preface"],"tags":[]},{"location":"chapter_introduction/","level":1,"title":"Chapter 1.   Encounter with Algorithms","text":"<p>Abstract</p> <p>A young girl dances gracefully, intertwined with data, her skirt flowing with the melody of algorithms.</p> <p>She invites you to dance with her. Follow her steps closely and enter the world of algorithms, full of logic and beauty.</p>","path":["Chapter 1. Encounter with Algorithms","Chapter 1.   Encounter with Algorithms"],"tags":[]},{"location":"chapter_introduction/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>1.1   Algorithms Are Everywhere</li> <li>1.2   What Is an Algorithm</li> <li>1.3   Summary</li> </ul>","path":["Chapter 1. Encounter with Algorithms","Chapter 1.   Encounter with Algorithms"],"tags":[]},{"location":"chapter_introduction/algorithms_are_everywhere/","level":1,"title":"1.1   Algorithms Are Everywhere","text":"<p>When we hear the term \"algorithm,\" we naturally think of mathematics. However, many algorithms do not involve complex mathematics but rely more on basic logic, which can be seen everywhere in our daily lives.</p> <p>Before we formally explore algorithms, here's an interesting fact worth sharing: you have already learned many algorithms without realizing it, and you are used to applying them in daily life. Let me give a few specific examples to illustrate this point.</p> <p>Example 1: Looking Up a Dictionary. In an English dictionary, words are listed alphabetically. Assuming we're searching for a word that starts with the letter \\(r\\), this is typically done in the following way:</p> <ol> <li>Open the dictionary to about halfway and check the first word on that page; suppose it starts with the letter \\(m\\).</li> <li>Since \\(r\\) comes after \\(m\\) in the alphabet, the first half can be ignored and the search space is narrowed down to the second half.</li> <li>Repeat steps <code>1.</code> and <code>2.</code> until you find the page where the word starts with \\(r\\).</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 1-1   Process of looking up a dictionary </p> <p>Looking up a dictionary, an essential skill for elementary school students is actually the famous \"Binary Search\" algorithm. From a data structure perspective, we can consider the dictionary as a sorted \"array\"; from an algorithmic perspective, the series of actions taken to look up a word in the dictionary can be viewed as the algorithm \"Binary Search.\"</p> <p>Example 2: Organizing Playing Cards. When playing cards, we need to arrange the cards in our hands in ascending order, as shown in the following process.</p> <ol> <li>Divide the playing cards into \"ordered\" and \"unordered\" sections, assuming initially the leftmost card is already in order.</li> <li>Take out a card from the unordered section and insert it into the correct position in the ordered section; after this, the leftmost two cards are in order.</li> <li>Repeat step <code>2</code> until all cards are in order.</li> </ol> <p></p> <p> Figure 1-2   Process of sorting a deck of cards </p> <p>The above method of organizing playing cards is essentially the \"Insertion Sort\" algorithm, which is very efficient for small datasets. Many programming languages' built-in sorting implementations use insertion sort internally.</p> <p>Example 3: Making Change. Assume making a purchase of \\(69\\) at a supermarket. If you give the cashier \\(100\\), they will need to provide you with \\(31\\) in change. This process can be clearly understood as illustrated in Figure 1-3.</p> <ol> <li>The available denominations smaller than \\(31\\) are \\(1\\), \\(5\\), \\(10\\), and \\(20\\).</li> <li>Take out the largest \\(20\\) from the options, leaving \\(31 - 20 = 11\\).</li> <li>Take out the largest \\(10\\) from the remaining options, leaving \\(11 - 10 = 1\\).</li> <li>Take out the largest \\(1\\) from the remaining options, leaving \\(1 - 1 = 0\\).</li> <li>Complete change-making, the solution is \\(20 + 10 + 1 = 31\\).</li> </ol> <p></p> <p> Figure 1-3   Process of making change </p> <p>In the steps above, we choose what seems to be the best option at each stage by using the largest denomination available, which leads to an effective way to make change. From a data structures and algorithms perspective, this approach is known as a \"Greedy\" algorithm.</p> <p>From cooking a meal to interstellar travel, almost all problem-solving involves algorithms. The advent of computers allows us to store data structures in memory and write code to call the CPU and GPU to execute algorithms. In this way, we can transfer real-life problems to computers and solve various complex issues in a more efficient way.</p> <p>Tip</p> <p>If concepts such as data structures, algorithms, arrays, and binary search still feel only half-familiar, keep reading. This book will guide you into the world of data structures and algorithms.</p>","path":["Chapter 1. Encounter with Algorithms","1.1   Algorithms Are Everywhere"],"tags":[]},{"location":"chapter_introduction/summary/","level":1,"title":"1.3   Summary","text":"","path":["Chapter 1. Encounter with Algorithms","1.3   Summary"],"tags":[]},{"location":"chapter_introduction/summary/#1-key-review","level":3,"title":"1.   Key Review","text":"<ul> <li>Algorithms are ubiquitous in daily life and are not some distant, esoteric body of knowledge. In fact, we have already learned many algorithms unconsciously and use them to solve problems big and small in life.</li> <li>The principle of looking up a dictionary is consistent with the binary search algorithm. Binary search embodies the important algorithmic idea of divide and conquer.</li> <li>The process of organizing playing cards is very similar to the insertion sort algorithm. Insertion sort is suitable for sorting small datasets.</li> <li>The steps of making change are essentially a greedy algorithm, where the best choice is made at each step based on the current situation.</li> <li>An algorithm is a set of instructions or operational steps that solves a specific problem within a finite amount of time, while a data structure is a way of organizing and storing data in a computer.</li> <li>Data structures and algorithms are closely connected. Data structures are the foundation of algorithms, and algorithms breathe life into data structures.</li> <li>We can compare data structures and algorithms to assembling building blocks. The blocks represent data, the way they are shaped and connected represents the data structure, and the steps used to assemble them correspond to the algorithm.</li> </ul>","path":["Chapter 1. Encounter with Algorithms","1.3   Summary"],"tags":[]},{"location":"chapter_introduction/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: As a programmer, I have never used algorithms to solve problems in my daily work. Common algorithms are already encapsulated by programming languages and can be used directly. Does this mean that the problems in our work have not yet reached the level where algorithms are needed?</p> <p>If we compare specific work skills to \"techniques\" in martial arts, then fundamental subjects should be more like \"internal skills\".</p> <p>I believe the significance of learning algorithms (and other fundamental subjects) is not that you will need to implement them from scratch at work, but that the knowledge you gain enables you to make sound professional judgments when solving problems, thereby improving the overall quality of your work. Here is a simple example. Every programming language has a built-in sorting function:</p> <ul> <li>If we have not studied data structures and algorithms, we might simply feed any given data to this sorting function. It runs smoothly with good performance, and there doesn't seem to be any problem.</li> <li>But if we have studied algorithms, we would know that the time complexity of the built-in sorting function is \\(O(n \\log n)\\). However, if the given data consists of integers with a fixed number of digits (such as student IDs), we can use the more efficient \"radix sort\", reducing the time complexity to \\(O(nk)\\), where \\(k\\) is the number of digits. When the data volume is very large, the saved running time can create significant value (reduced costs, improved experience, etc.).</li> </ul> <p>In engineering, many problems are difficult to solve optimally, and many others are only solved \"well enough.\" The difficulty of a problem depends, on the one hand, on the nature of the problem itself and, on the other hand, on the knowledge of the person examining it. The more complete a person's knowledge and the more experience they have, the deeper their analysis will be, and the more elegantly the problem can be solved.</p> <p>Q: Now that we let AI agents write almost all our code, do we still need to learn algorithms?</p> <p>To quote Mu Li: “An agent is an amplifier of ability—it multiplies your ability by 10, rather than adding 10 to it.”</p> <p>Since AI amplifies our abilities, the abilities being multiplied matter too. I think two abilities are especially worth developing. The first is communication, particularly the ability to work with AI: making goals clear, providing the necessary context, and correcting course in time. Explaining a problem to AI also helps us think it through ourselves. The second is foundational knowledge. AI can fill in the details, but we need to understand the basic concepts ourselves to ask the right questions, judge whether answers are correct, and turn its output into useful results.</p> <p>One distinctive thing about AI is that, unlike earlier tools, it does not have clearly defined functional boundaries. It is a little like water, taking different shapes in different people's hands. The greater a person's abilities and the better their taste, the more pronounced AI's amplifying effect becomes.</p>","path":["Chapter 1. Encounter with Algorithms","1.3   Summary"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/","level":1,"title":"1.2   What Is an Algorithm","text":"","path":["Chapter 1. Encounter with Algorithms","1.2   What Is an Algorithm"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/#121-algorithm-definition","level":2,"title":"1.2.1   Algorithm Definition","text":"<p>An algorithm is a set of instructions or operational steps that solves a specific problem within a finite amount of time. It has the following characteristics.</p> <ul> <li>The problem is well-defined, with clear input and output definitions.</li> <li>It is feasible and can be completed with finite steps, time, and memory.</li> <li>Each step has a definite meaning, and under the same input and operating conditions, the output is always the same.</li> </ul>","path":["Chapter 1. Encounter with Algorithms","1.2   What Is an Algorithm"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/#122-data-structure-definition","level":2,"title":"1.2.2   Data Structure Definition","text":"<p>A data structure is a way of organizing and storing data, including the data itself, the relationships between data elements, and the methods used to operate on them. It has the following design objectives.</p> <ul> <li>Occupy as little space as possible to save computer memory.</li> <li>Data operations should be as fast as possible, covering data access, addition, deletion, update, etc.</li> <li>Provide a concise data representation and logical information so that algorithms can run efficiently.</li> </ul> <p>Data structure design is a process full of trade-offs. If we want to achieve improvements in one aspect, we often need to make compromises in another aspect. Here are two examples.</p> <ul> <li>Compared to arrays, linked lists are more convenient for data addition and deletion operations but sacrifice data access speed.</li> <li>Compared to linked lists, graphs provide richer logical information but require larger memory space.</li> </ul>","path":["Chapter 1. Encounter with Algorithms","1.2   What Is an Algorithm"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/#123-the-relationship-between-data-structures-and-algorithms","level":2,"title":"1.2.3   The Relationship Between Data Structures and Algorithms","text":"<p>As shown in Figure 1-4, data structures and algorithms are highly related and tightly coupled, specifically manifested in the following three aspects.</p> <ul> <li>Data structures are the foundation of algorithms. Data structures provide algorithms with structured storage of data and methods for operating on data.</li> <li>Algorithms breathe life into data structures. Data structures themselves only store data information; combined with algorithms, they can solve specific problems.</li> <li>Algorithms can usually be implemented based on different data structures, but execution efficiency may vary greatly. Choosing the appropriate data structure is key.</li> </ul> <p></p> <p> Figure 1-4   The relationship between data structures and algorithms </p> <p>Data structures and algorithms are like assembling building blocks as shown in Figure 1-5. A set of building blocks, in addition to containing many parts, also comes with detailed assembly instructions. By following the instructions step by step, we can assemble an exquisite building block model.</p> <p></p> <p> Figure 1-5   Assembling blocks </p> <p>The detailed correspondence between the two is shown in Table 1-1.</p> <p> Table 1-1   Comparing data structures and algorithms to assembling building blocks </p> Data structures and algorithms Assembling building blocks Input data Unassembled building blocks Data structure Organization form of building blocks, including shape, size, connection method, etc. Algorithm A series of operational steps to assemble the blocks into the target form Output data Building block model <p>It is worth noting that data structures and algorithms are independent of programming languages. That is why this book can provide implementations in multiple programming languages.</p> <p>Conventional abbreviation</p> <p>In actual discussions, we usually abbreviate \"data structures and algorithms\" as \"algorithms\". For example, the well-known LeetCode algorithm problems actually examine knowledge of both data structures and algorithms.</p>","path":["Chapter 1. Encounter with Algorithms","1.2   What Is an Algorithm"],"tags":[]},{"location":"chapter_preface/","level":1,"title":"Chapter 0.   Preface","text":"<p>Abstract</p> <p>Algorithms are like a beautiful symphony, each line of code flows like a melody.</p> <p>May this book gently resonate in your mind, leaving a unique and profound melody.</p>","path":["Chapter 0. Preface","Chapter 0.   Preface"],"tags":[]},{"location":"chapter_preface/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>0.1   About This Book</li> <li>0.2   How to Use This Book</li> <li>0.3   Summary</li> </ul>","path":["Chapter 0. Preface","Chapter 0.   Preface"],"tags":[]},{"location":"chapter_preface/about_the_book/","level":1,"title":"0.1   About This Book","text":"<p>This project aims to create an open-source, free, beginner-friendly introductory tutorial on data structures and algorithms.</p> <ul> <li>The entire book uses animated illustrations, with clear and easy-to-understand content and a smooth learning curve, guiding beginners through the landscape of data structures and algorithms.</li> <li>The source code can be run with one click, helping readers improve their programming skills through practice and understand how algorithms work and the underlying implementation of data structures.</li> <li>We encourage readers to learn from each other, and everyone is welcome to ask questions and share insights in the comments section, making progress together through discussion and exchange.</li> </ul>","path":["Chapter 0. Preface","0.1   About This Book"],"tags":[]},{"location":"chapter_preface/about_the_book/#011-target-audience","level":2,"title":"0.1.1   Target Audience","text":"<p>If you are an algorithm beginner who has never studied algorithms, or if you already have some problem-solving experience but only a hazy understanding of data structures and algorithms, then this book is tailor-made for you!</p> <p>If you have already accumulated a certain amount of problem-solving experience and are familiar with most question types, this book can help you review and organize your algorithm knowledge system, and the repository's source code can be used as a \"problem-solving toolkit\" or \"algorithm dictionary.\"</p> <p>If you are an algorithm \"expert,\" we look forward to receiving your valuable suggestions, or joining us as a contributor.</p> <p>Prerequisites</p> <p>You need basic programming knowledge in at least one language and the ability to read and write simple code.</p>","path":["Chapter 0. Preface","0.1   About This Book"],"tags":[]},{"location":"chapter_preface/about_the_book/#012-content-structure","level":2,"title":"0.1.2   Content Structure","text":"<p>The main content of this book is shown in Figure 0-1.</p> <ul> <li>Complexity analysis: Evaluation dimensions and methods for data structures and algorithms. Methods for calculating time complexity and space complexity, common types, examples, etc.</li> <li>Data structures: Classification methods for basic data types and data structures. Definitions, advantages and disadvantages, common operations, common types, typical applications, implementation methods, and more for data structures such as arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graphs.</li> <li>Algorithms: The definition, advantages and disadvantages, efficiency, application scenarios, problem-solving steps, and example problems of algorithms such as searching, sorting, divide and conquer, backtracking, dynamic programming, and greedy algorithms.</li> </ul> <p></p> <p> Figure 0-1   Main content of this book </p>","path":["Chapter 0. Preface","0.1   About This Book"],"tags":[]},{"location":"chapter_preface/about_the_book/#013-acknowledgements","level":2,"title":"0.1.3   Acknowledgements","text":"<p>This book has been continuously improved through the joint efforts of many contributors in the open-source community. Thanks to every contributor who invested time and effort, they are (in the order automatically generated by GitHub): krahets, coderonion, Gonglja, nuomi1, Reanon, justin-tse, hpstory, danielsss, curtishd, night-cruise, S-N-O-R-L-A-X, rongyi, msk397, gvenusleo, khoaxuantu, rivertwilight, K3v123, gyt95, zhuoqinyue, yuelinxin, Zuoxun, mingXta, Phoenix0415, FangYuan33, GN-Yu, longsizhuo, pengchzn, QiLOL, Cathay-Chen, guowei-gong, xBLACKICEx, IsChristina, JoseHung, qualifier1024, hello-ikun, magentaqin, Guanngxu, thomasq0, sunshinesDL, L-Super, Transmigration-zhou, WSL0809, Slone123c, lhxsm, yuan0221, what-is-me, theNefelibatas, Shyam-Chen, sangxiaai, longranger2, codeberg-user, xiongsp, JeffersonHuang, prinpal, seven1240, Wonderdch, malone6, xiaomiusa87, gaofer, bluebean-cloud, a16su, SamJin98, hongyun-robot, nanlei, XiaChuerwu, yd-j, iron-irax, mgisr, steventimes, junminhong, heshuyue, danny900714, Nigh, Dr-XYZ, MolDuM, XC-Zero, reeswell, PXG-XPG, NI-SW, Horbin-Magician, Enlightenus, YangXuanyi, xjr7670, beatrix-chan, DullSword, qq909244296, iStig, boloboloda, hts0000, gledfish, fbigm, echo1937, jiaxianhua, wenjianmin, keshida, kilikilikid, lclc6, lwbaptx, linyejoe2, liuxjerry, szu17dmy, dshlstarr, Yucao-cy, coderlef, czruby, bongbongbakudan, beintentional, ZongYangL, ZhongYuuu, ZhongGuanbin, hezhizhen, linzeyan, ZJKung, JTCPOWI, KawaiiAsh, luluxia, xb534, ztkuaikuai, yw-1021, ElaBosak233, baagod, zhouLion, yishangzhang, yi427, yanedie, yabo083, weibk, wangwang105, th1nk3r-ing, tao363, 4yDX3906, syd168, sslmj2020, smilelsb, siqyka, selear, sdshaoda, Xi-Row, popozhu, nuquist19, noobcodemaker, XiaoK29, chadyi, lyl625760, lucaswangdev, llql1211, 0130w, shanghai-Jerry, EJackYang, Javesun99, eltociear, lipusheng, KNChiu, BlindTerran, ShiMaRing, lovelock, FreddieLi, FloranceYeh, fanchenggang, gltianwen, goerll, nedchu, curly210102, CuB3y0nd, KraHsu, CarrotDLaw, youshaoXG, bubble9um, Asashishi, Asa0oo0o0o, fanenr, eagleanurag, akshiterate, 52coder, foursevenlove, KorsChen, hopkings2008, yang-le, realwujing, Evilrabbit520, Umer-Jahangir, Turing-1024-Lee, Suremotoo, paoxiaomooo, Chieko-Seren, Senrian, Allen-Scai, 19santosh99, ymmmas, Risuntsy, Richard-Zhang1019, RafaelCaso, qingpeng9802, primexiao, Urbaner3, codetypess, nidhoggfgg, MwumLi, CreatorMetaSky, martinx, ZnYang2018, hugtyftg, logan-qiu, psychelzh, Kunchen-Luo, Keynman, and KeiichiKasai.</p> <p>The code review work for this book was completed by coderonion, curtishd, Gonglja, gvenusleo, hpstory, justin-tse, khoaxuantu, krahets, night-cruise, nuomi1, Reanon and rongyi (in alphabetical order). Thanks to them for the time and effort they put in; they helped keep the code consistent and standardized across the different language versions.</p> <p>The English version of this book was reviewed by yuelinxin, K3v123, magentaqin, QiLOL, Phoenix0415, SamJin98, yanedie, RafaelCaso, pengchzn and thomasq0; the Japanese version was reviewed by eltociear; the Russian version was reviewed by И. А. Шевкун and Yuyan Huang; and the Traditional Chinese version was reviewed by Shyam-Chen and Dr-XYZ. Thanks to their contributions, this book is able to serve a broader readership, and we are deeply grateful to them.</p> <p>The ePub ebook generation tool for this book was developed by zhongfq. We thank him for his contribution, which provides readers with a more flexible way to read.</p> <p>During the creation of this book, I received help from many people.</p> <ul> <li>Thanks to my mentor at the company, Dr. Li Xi, who encouraged me to \"take action quickly\" during a conversation, strengthening my determination to write this book;</li> <li>Thanks to my girlfriend Bubble, the first reader of this book, who provided many valuable suggestions from the perspective of an algorithm beginner, making this book more approachable for beginners;</li> <li>Thanks to Tengbao, Qibao, and Feibao for coming up with a creative name for this book, evoking everyone's fond memories of writing their first line of code \"Hello World!\";</li> <li>Thanks to Xiaoquan for providing professional help in intellectual property rights, which played an important role in the improvement of this open-source book;</li> <li>Thanks to Sutong for designing the beautiful cover and logo for this book, and for patiently revising them many times at my perfectionist insistence;</li> <li>Thanks to @squidfunk for the typesetting suggestions, as well as for developing the open-source documentation theme Material-for-MkDocs.</li> </ul> <p>During the writing process, I read many textbooks and articles on data structures and algorithms. These works served as excellent models for this book and helped ensure the accuracy and quality of its content. I would like to thank all the teachers and predecessors for their outstanding contributions!</p> <p>This book advocates a hands-on approach to learning, and in this respect I was deeply inspired by Dive into Deep Learning. I highly recommend this excellent work to all readers.</p> <p>Heartfelt thanks to my parents. It is your support and encouragement that gave me the opportunity to pursue this enjoyable project.</p>","path":["Chapter 0. Preface","0.1   About This Book"],"tags":[]},{"location":"chapter_preface/suggestions/","level":1,"title":"0.2   How to Use This Book","text":"<p>Tip</p> <p>For the best reading experience, it is recommended that you read through this section.</p>","path":["Chapter 0. Preface","0.2   How to Use This Book"],"tags":[]},{"location":"chapter_preface/suggestions/#021-writing-style-conventions","level":2,"title":"0.2.1   Writing Style Conventions","text":"<ul> <li>Sections marked with <code>*</code> after the title are optional and somewhat more challenging. If you're short on time, you can skip them on your first pass.</li> <li>Technical terms are shown in bold (in the print and PDF editions) or underlined (in the web edition), such as array. They are worth remembering, as they will help when reading technical literature.</li> <li>Key content and summary statements will be bolded, and such text deserves special attention.</li> <li>Words and phrases with specific meanings will be marked with \"quotation marks\" to avoid ambiguity.</li> <li>When terminology differs across programming languages, this book follows Python conventions; for example, it uses <code>None</code> to represent \"null\".</li> <li>This book partially relaxes conventional programming-language comment styles in favor of a more compact layout. Comments are mainly divided into three types: title comments, content comments, and multi-line comments.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>\"\"\"Title comment, used to label functions, classes, test cases, etc.\"\"\"\n\n# Content comment, used to explain code in detail\n\n\"\"\"\nMulti-line\ncomment\n\"\"\"\n</code></pre> <pre><code>/* Title comment, used to label functions, classes, test cases, etc. */\n\n// Content comment, used to explain code in detail\n\n/**\n * Multi-line\n * comment\n */\n</code></pre> <pre><code>/* Title comment, used to label functions, classes, test cases, etc. */\n\n// Content comment, used to explain code in detail\n\n/**\n * Multi-line\n * comment\n */\n</code></pre> <pre><code>/* Title comment, used to label functions, classes, test cases, etc. */\n\n// Content comment, used to explain code in detail\n\n/**\n * Multi-line\n * comment\n */\n</code></pre> <pre><code>/* Title comment, used to label functions, classes, test cases, etc. */\n\n// Content comment, used to explain code in detail\n\n/**\n * Multi-line\n * comment\n */\n</code></pre> <pre><code>/* Title comment, used to label functions, classes, test cases, etc. */\n\n// Content comment, used to explain code in detail\n\n/**\n * Multi-line\n * comment\n */\n</code></pre> <pre><code>/* Title comment, used to label functions, classes, test cases, etc. */\n\n// Content comment, used to explain code in detail\n\n/**\n * Multi-line\n * comment\n */\n</code></pre> <pre><code>/* Title comment, used to label functions, classes, test cases, etc. */\n\n// Content comment, used to explain code in detail\n\n/**\n * Multi-line\n * comment\n */\n</code></pre> <pre><code>/* Title comment, used to label functions, classes, test cases, etc. */\n\n// Content comment, used to explain code in detail\n\n/**\n * Multi-line\n * comment\n */\n</code></pre> <pre><code>/* Title comment, used to label functions, classes, test cases, etc. */\n\n// Content comment, used to explain code in detail\n\n// Multi-line\n// comment\n</code></pre> <pre><code>/* Title comment, used to label functions, classes, test cases, etc. */\n\n// Content comment, used to explain code in detail\n\n/**\n * Multi-line\n * comment\n */\n</code></pre> <pre><code>/* Title comment, used to label functions, classes, test cases, etc. */\n\n// Content comment, used to explain code in detail\n\n/**\n * Multi-line\n * comment\n */\n</code></pre> <pre><code>### Title comment, used to label functions, classes, test cases, etc. ###\n\n# Content comment, used to explain code in detail\n\n# Multi-line\n# comment\n</code></pre>","path":["Chapter 0. Preface","0.2   How to Use This Book"],"tags":[]},{"location":"chapter_preface/suggestions/#022-learning-efficiently-with-animated-illustrations","level":2,"title":"0.2.2   Learning Efficiently with Animated Illustrations","text":"<p>Compared with plain text, videos and images have higher information density and a clearer structure, making them easier to understand. In this book, key concepts and challenging topics are presented mainly through animated illustrations, with text serving as explanation and supplement.</p> <p>If, while reading this book, you encounter an animated illustration like the one shown below, treat the illustration as primary and the text as supplementary, and use both together to understand the content.</p> <p></p> <p> Figure 0-2   Example of animated illustrations </p>","path":["Chapter 0. Preface","0.2   How to Use This Book"],"tags":[]},{"location":"chapter_preface/suggestions/#023-deepening-understanding-through-code-practice","level":2,"title":"0.2.3   Deepening Understanding Through Code Practice","text":"<p>The accompanying code for this book is hosted in the GitHub repository. As shown in Figure 0-3, the source code comes with test cases and can be run with one click.</p> <p>If time permits, it is recommended that you type out the code yourself. If you have limited study time, please at least read through and run all the code.</p> <p>Compared with simply reading code, writing it yourself often brings greater rewards. Hands-on practice is where real learning happens.</p> <p></p> <p> Figure 0-3   Example of running code </p> <p>Getting the code running mainly involves three preliminary steps.</p> <p>Step 1: Install the local programming environment. Please follow the tutorial in the appendix. If it is already installed, you can skip this step.</p> <p>Step 2: Clone or download the code repository. Visit the GitHub repository. If you have already installed Git, you can clone this repository with the following command:</p> <pre><code>git clone https://github.com/krahets/hello-algo.git\n</code></pre> <p>Alternatively, you can click the \"Download ZIP\" button shown below to download a ZIP archive of the repository directly and then extract it locally.</p> <p></p> <p> Figure 0-4   Clone repository and download code </p> <p>Step 3: Run the source code. As shown in Figure 0-5, for code blocks with file names at the top, we can find the corresponding source code files in the <code>codes</code> folder of the repository. The source code files can be run with one click, which will help you save unnecessary debugging time and allow you to focus on learning content.</p> <p></p> <p> Figure 0-5   Code blocks and corresponding source code files </p> <p>In addition to running code locally, the web version also supports visual execution of Python code (implemented based on pythontutor). As shown in Figure 0-6, you can click \"Visual Run\" below the code block to expand the view and observe the execution process of the algorithm code; you can also click \"Full Screen View\" for a better viewing experience.</p> <p></p> <p> Figure 0-6   Visual running of Python code </p>","path":["Chapter 0. Preface","0.2   How to Use This Book"],"tags":[]},{"location":"chapter_preface/suggestions/#024-growing-together-through-questions-and-discussions","level":2,"title":"0.2.4   Growing Together Through Questions and Discussions","text":"<p>When reading this book, please do not skip over points that you still do not fully understand. Feel free to ask your questions in the comments section, and my friends and I will do our best to answer them, usually within two days.</p> <p>As shown in Figure 0-7, the web version has a comments section at the bottom of each chapter. I encourage you to pay close attention to the discussions there. On the one hand, you can learn about the problems that others encounter, thereby filling gaps in your own understanding and prompting deeper thought. On the other hand, I hope you will generously answer other readers' questions, share your insights, and help others improve.</p> <p></p> <p> Figure 0-7   Example of comments section </p>","path":["Chapter 0. Preface","0.2   How to Use This Book"],"tags":[]},{"location":"chapter_preface/suggestions/#025-algorithm-learning-roadmap","level":2,"title":"0.2.5   Algorithm Learning Roadmap","text":"<p>Overall, we can divide the process of learning data structures and algorithms into three stages.</p> <ol> <li>Stage 1: Algorithm introduction. We need to familiarize ourselves with the characteristics and usage of various data structures, and learn the principles, processes, uses, and efficiency of different algorithms.</li> <li>Stage 2: Practice algorithm problems. It is recommended to start with popular problems and solve at least 100 of them first, so that you become familiar with mainstream algorithm questions. When you first begin practicing problems, \"knowledge forgetting\" may feel like a challenge, but rest assured, this is very normal. We can review problems according to the \"Ebbinghaus forgetting curve\", and after 3-5 rounds of repetition, they usually stick firmly in memory. For recommended problem lists and practice plans, please see this GitHub repository.</li> <li>Stage 3: Building a knowledge system. In terms of learning, we can read algorithm column articles, problem-solving frameworks, and algorithm textbooks to continuously enrich our knowledge system. In terms of practicing problems, we can try advanced problem-solving strategies, such as categorization by topic, one problem multiple solutions, one solution multiple problems, etc. Related problem-solving insights can be found in various communities.</li> </ol> <p>As shown in Figure 0-8, the content of this book mainly covers \"Stage 1\", aiming to help you more efficiently carry out Stage 2 and Stage 3 learning.</p> <p></p> <p> Figure 0-8   Algorithm learning roadmap </p>","path":["Chapter 0. Preface","0.2   How to Use This Book"],"tags":[]},{"location":"chapter_preface/summary/","level":1,"title":"0.3   Summary","text":"","path":["Chapter 0. Preface","0.3   Summary"],"tags":[]},{"location":"chapter_preface/summary/#1-key-review","level":3,"title":"1.   Key Review","text":"<ul> <li>The main audience of this book is algorithm beginners. If you already have some background, this book can help you systematically review algorithm knowledge, and the source code in the book can also be used as a \"problem-solving toolkit.\"</li> <li>The content of the book mainly includes three parts: complexity analysis, data structures, and algorithms, covering most topics in this field.</li> <li>For algorithm novices, reading an introductory book during the initial learning stage is crucial, as it can help you avoid many detours.</li> <li>The animated illustrations in the book are usually used to introduce key concepts and challenging topics. When reading this book, you should pay more attention to these topics.</li> <li>Practice is the best way to learn programming. It is strongly recommended to run the source code and type the code yourself.</li> <li>The web version of this book has a comments section for each chapter, where you are welcome to share your questions and insights at any time.</li> </ul>","path":["Chapter 0. Preface","0.3   Summary"],"tags":[]},{"location":"chapter_reference/","level":1,"title":"References","text":"<p>[1] Thomas H. Cormen, et al. Introduction to Algorithms (3<sup>rd</sup> Edition).</p> <p>[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1<sup>st</sup> Edition).</p> <p>[3] Robert Sedgewick, et al. Algorithms (4<sup>th</sup> Edition).</p> <p>[4] Yan Weimin. Data Structures (C Language Version).</p> <p>[5] Deng Junhui. Data Structures (C++ Language Version, Third Edition).</p> <p>[6] Mark Allen Weiss, translated by Chen Yue. Data Structures and Algorithm Analysis in Java (Third Edition).</p> <p>[7] Cheng Jie. Data Structures in Plain Language.</p> <p>[8] Wang Zheng. The Beauty of Data Structures and Algorithms.</p> <p>[9] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6<sup>th</sup> Edition).</p> <p>[10] Aston Zhang, et al. Dive into Deep Learning.</p>","path":["References"],"tags":[]},{"location":"chapter_searching/","level":1,"title":"Chapter 10.   Searching","text":"<p>Abstract</p> <p>Searching is an adventure into the unknown, where we may need to traverse every corner of the mysterious space, or we may be able to quickly lock onto the target.</p> <p>In this journey of discovery, each exploration may yield an unexpected answer.</p>","path":["Chapter 10. Searching","Chapter 10.   Searching"],"tags":[]},{"location":"chapter_searching/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>10.1   Binary Search</li> <li>10.2   Binary Search Insertion Point</li> <li>10.3   Binary Search Boundaries</li> <li>10.4   Hash Optimization Strategy</li> <li>10.5   Searching Algorithms Revisited</li> <li>10.6   Summary</li> <li>10.7   Exercises</li> </ul>","path":["Chapter 10. Searching","Chapter 10.   Searching"],"tags":[]},{"location":"chapter_searching/binary_search/","level":1,"title":"10.1   Binary Search","text":"<p>Binary search is an efficient search algorithm based on the divide-and-conquer strategy. It leverages the sorted order of the data to reduce the search range by half in each round until the target element is found or the search interval becomes empty.</p> <p>Question</p> <p>Given an array <code>nums</code> of length \\(n\\) with elements arranged in ascending order and no duplicates, search for and return the index of element <code>target</code> in the array. If the array does not contain the element, return \\(-1\\). An example is shown in Figure 10-1.</p> <p></p> <p> Figure 10-1   Binary search example data </p> <p>As shown in Figure 10-2, we first initialize pointers \\(i = 0\\) and \\(j = n - 1\\), pointing to the first and last elements of the array respectively, representing the search interval \\([0, n - 1]\\). Note that square brackets denote a closed interval, which includes the boundary values themselves.</p> <p>Next, perform the following two steps in a loop:</p> <ol> <li>Calculate the midpoint index \\(m = \\lfloor {(i + j) / 2} \\rfloor\\), where \\(\\lfloor \\: \\rfloor\\) denotes the floor operation.</li> <li>Compare <code>nums[m]</code> and <code>target</code>, which results in three cases:<ol> <li>When <code>nums[m] &lt; target</code>, it indicates that <code>target</code> is in the interval \\([m + 1, j]\\), so execute \\(i = m + 1\\).</li> <li>When <code>nums[m] &gt; target</code>, it indicates that <code>target</code> is in the interval \\([i, m - 1]\\), so execute \\(j = m - 1\\).</li> <li>When <code>nums[m] = target</code>, it indicates that <code>target</code> has been found, so return index \\(m\\).</li> </ol> </li> </ol> <p>If the array does not contain the target element, the search interval will eventually become empty. In this case, return \\(-1\\).</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 10-2   Binary search process </p> <p>It's worth noting that since both \\(i\\) and \\(j\\) are of <code>int</code> type, \\(i + j\\) may exceed the range of the <code>int</code> type. To avoid integer overflow, we typically use the formula \\(m = \\lfloor {i + (j - i) / 2} \\rfloor\\) to calculate the midpoint.</p> <p>The code is shown below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search.py<pre><code>def binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"Binary search (closed interval)\"\"\"\n    # Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array\n    i, j = 0, len(nums) - 1\n    # Loop, exit when the search interval is empty (empty when i &gt; j)\n    while i &lt;= j:\n        # In theory, Python numbers can be infinitely large (depending on memory size), no need to consider large number overflow\n        m = (i + j) // 2  # Calculate midpoint index m\n        if nums[m] &lt; target:\n            i = m + 1  # This means target is in the interval [m+1, j]\n        elif nums[m] &gt; target:\n            j = m - 1  # This means target is in the interval [i, m-1]\n        else:\n            return m  # Found the target element, return its index\n    return -1  # Target element not found, return -1\n</code></pre> binary_search.cpp<pre><code>/* Binary search (closed interval on both sides) */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    // Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array\n    int i = 0, j = nums.size() - 1;\n    // Loop, exit when the search interval is empty (empty when i &gt; j)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target)    // This means target is in the interval [m+1, j]\n            i = m + 1;\n        else if (nums[m] &gt; target) // This means target is in the interval [i, m-1]\n            j = m - 1;\n        else // Found the target element, return its index\n            return m;\n    }\n    // Target element not found, return -1\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* Binary search (closed interval on both sides) */\nint binarySearch(int[] nums, int target) {\n    // Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array\n    int i = 0, j = nums.length - 1;\n    // Loop, exit when the search interval is empty (empty when i &gt; j)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) // This means target is in the interval [m+1, j]\n            i = m + 1;\n        else if (nums[m] &gt; target) // This means target is in the interval [i, m-1]\n            j = m - 1;\n        else // Found the target element, return its index\n            return m;\n    }\n    // Target element not found, return -1\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* Binary search (closed interval on both sides) */\nint BinarySearch(int[] nums, int target) {\n    // Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array\n    int i = 0, j = nums.Length - 1;\n    // Loop, exit when the search interval is empty (empty when i &gt; j)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2;   // Calculate the midpoint index m\n        if (nums[m] &lt; target)      // This means target is in the interval [m+1, j]\n            i = m + 1;\n        else if (nums[m] &gt; target) // This means target is in the interval [i, m-1]\n            j = m - 1;\n        else                       // Found the target element, return its index\n            return m;\n    }\n    // Target element not found, return -1\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* Binary search (closed interval on both sides) */\nfunc binarySearch(nums []int, target int) int {\n    // Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array\n    i, j := 0, len(nums)-1\n    // Loop, exit when the search interval is empty (empty when i &gt; j)\n    for i &lt;= j {\n        m := i + (j-i)/2      // Calculate the midpoint index m\n        if nums[m] &lt; target { // This means target is in the interval [m+1, j]\n            i = m + 1\n        } else if nums[m] &gt; target { // This means target is in the interval [i, m-1]\n            j = m - 1\n        } else { // Found the target element, return its index\n            return m\n        }\n    }\n    // Target element not found, return -1\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* Binary search (closed interval on both sides) */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    // Loop, exit when the search interval is empty (empty when i &gt; j)\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // Calculate the midpoint index m\n        if nums[m] &lt; target { // This means target is in the interval [m+1, j]\n            i = m + 1\n        } else if nums[m] &gt; target { // This means target is in the interval [i, m-1]\n            j = m - 1\n        } else { // Found the target element, return its index\n            return m\n        }\n    }\n    // Target element not found, return -1\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* Binary search (closed interval on both sides) */\nfunction binarySearch(nums, target) {\n    // Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array\n    let i = 0,\n        j = nums.length - 1;\n    // Loop, exit when the search interval is empty (empty when i &gt; j)\n    while (i &lt;= j) {\n        // Calculate midpoint index m, use parseInt() to round down\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // This means target is in the interval [m+1, j]\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // This means target is in the interval [i, m-1]\n            j = m - 1;\n        else return m; // Found the target element, return its index\n    }\n    // Target element not found, return -1\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* Binary search (closed interval on both sides) */\nfunction binarySearch(nums: number[], target: number): number {\n    // Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array\n    let i = 0,\n        j = nums.length - 1;\n    // Loop, exit when the search interval is empty (empty when i &gt; j)\n    while (i &lt;= j) {\n        // Calculate the midpoint index m\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // This means target is in the interval [m+1, j]\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // This means target is in the interval [i, m-1]\n            j = m - 1;\n        } else {\n            // Found the target element, return its index\n            return m;\n        }\n    }\n    return -1; // Target element not found, return -1\n}\n</code></pre> binary_search.dart<pre><code>/* Binary search (closed interval on both sides) */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  // Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array\n  int i = 0, j = nums.length - 1;\n  // Loop, exit when the search interval is empty (empty when i &gt; j)\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // Calculate the midpoint index m\n    if (nums[m] &lt; target) {\n      // This means target is in the interval [m+1, j]\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // This means target is in the interval [i, m-1]\n      j = m - 1;\n    } else {\n      // Found the target element, return its index\n      return m;\n    }\n  }\n  // Target element not found, return -1\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* Binary search (closed interval on both sides) */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array\n    let mut i = 0;\n    let mut j = nums.len() as i32 - 1;\n    // Loop, exit when the search interval is empty (empty when i &gt; j)\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // Calculate the midpoint index m\n        if nums[m as usize] &lt; target {\n            // This means target is in the interval [m+1, j]\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // This means target is in the interval [i, m-1]\n            j = m - 1;\n        } else {\n            // Found the target element, return its index\n            return m;\n        }\n    }\n    // Target element not found, return -1\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* Binary search (closed interval on both sides) */\nint binarySearch(int *nums, int len, int target) {\n    // Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array\n    int i = 0, j = len - 1;\n    // Loop, exit when the search interval is empty (empty when i &gt; j)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target)    // This means target is in the interval [m+1, j]\n            i = m + 1;\n        else if (nums[m] &gt; target) // This means target is in the interval [i, m-1]\n            j = m - 1;\n        else // Found the target element, return its index\n            return m;\n    }\n    // Target element not found, return -1\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* Binary search (closed interval on both sides) */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    // Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array\n    var i = 0\n    var j = nums.size - 1\n    // Loop, exit when the search interval is empty (empty when i &gt; j)\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // Calculate the midpoint index m\n        if (nums[m] &lt; target) // This means target is in the interval [m+1, j]\n            i = m + 1\n        else if (nums[m] &gt; target) // This means target is in the interval [i, m-1]\n            j = m - 1\n        else  // Found the target element, return its index\n            return m\n    }\n    // Target element not found, return -1\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>### Binary search (closed interval) ###\ndef binary_search(nums, target)\n  # Initialize closed interval [0, n-1], i.e., i, j point to the first and last elements of the array\n  i, j = 0, nums.length - 1\n\n  # Loop, exit when the search interval is empty (empty when i &gt; j)\n  while i &lt;= j\n    # In theory, Ruby numbers can be infinitely large (limited by memory), no need to consider overflow\n    m = (i + j) / 2   # Calculate the midpoint index m\n\n    if nums[m] &lt; target\n      i = m + 1 # This means target is in the interval [m+1, j]\n    elsif nums[m] &gt; target\n      j = m - 1 # This means target is in the interval [i, m-1]\n    else\n      return m  # Found the target element, return its index\n    end\n  end\n\n  -1  # Target element not found, return -1\nend\n</code></pre> <p>Time complexity is \\(O(\\log n)\\): In the binary search loop, the interval is reduced by half each round, so the number of iterations is \\(\\log_2 n\\).</p> <p>Space complexity is \\(O(1)\\): Pointers \\(i\\) and \\(j\\) use constant-size space.</p>","path":["Chapter 10. Searching","10.1   Binary Search"],"tags":[]},{"location":"chapter_searching/binary_search/#1011-interval-representation-methods","level":2,"title":"10.1.1   Interval Representation Methods","text":"<p>In addition to the closed interval mentioned above, another common interval representation is the \"left-closed right-open\" interval, defined as \\([0, n)\\), meaning that the left boundary is inclusive while the right boundary is exclusive. Under this representation, the interval \\([i, j)\\) is empty when \\(i = j\\).</p> <p>We can implement a binary search algorithm with the same functionality based on this representation:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search.py<pre><code>def binary_search_lcro(nums: list[int], target: int) -&gt; int:\n    \"\"\"Binary search (left-closed right-open interval)\"\"\"\n    # Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1\n    i, j = 0, len(nums)\n    # Loop, exit when the search interval is empty (empty when i = j)\n    while i &lt; j:\n        m = (i + j) // 2  # Calculate midpoint index m\n        if nums[m] &lt; target:\n            i = m + 1  # This means target is in the interval [m+1, j)\n        elif nums[m] &gt; target:\n            j = m  # This means target is in the interval [i, m)\n        else:\n            return m  # Found the target element, return its index\n    return -1  # Target element not found, return -1\n</code></pre> binary_search.cpp<pre><code>/* Binary search (left-closed right-open interval) */\nint binarySearchLCRO(vector&lt;int&gt; &amp;nums, int target) {\n    // Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1\n    int i = 0, j = nums.size();\n    // Loop, exit when the search interval is empty (empty when i = j)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target)    // This means target is in the interval [m+1, j)\n            i = m + 1;\n        else if (nums[m] &gt; target) // This means target is in the interval [i, m)\n            j = m;\n        else // Found the target element, return its index\n            return m;\n    }\n    // Target element not found, return -1\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* Binary search (left-closed right-open interval) */\nint binarySearchLCRO(int[] nums, int target) {\n    // Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1\n    int i = 0, j = nums.length;\n    // Loop, exit when the search interval is empty (empty when i = j)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) // This means target is in the interval [m+1, j)\n            i = m + 1;\n        else if (nums[m] &gt; target) // This means target is in the interval [i, m)\n            j = m;\n        else // Found the target element, return its index\n            return m;\n    }\n    // Target element not found, return -1\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* Binary search (left-closed right-open interval) */\nint BinarySearchLCRO(int[] nums, int target) {\n    // Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1\n    int i = 0, j = nums.Length;\n    // Loop, exit when the search interval is empty (empty when i = j)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2;   // Calculate the midpoint index m\n        if (nums[m] &lt; target)      // This means target is in the interval [m+1, j)\n            i = m + 1;\n        else if (nums[m] &gt; target) // This means target is in the interval [i, m)\n            j = m;\n        else                       // Found the target element, return its index\n            return m;\n    }\n    // Target element not found, return -1\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* Binary search (left-closed right-open interval) */\nfunc binarySearchLCRO(nums []int, target int) int {\n    // Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1\n    i, j := 0, len(nums)\n    // Loop, exit when the search interval is empty (empty when i = j)\n    for i &lt; j {\n        m := i + (j-i)/2      // Calculate the midpoint index m\n        if nums[m] &lt; target { // This means target is in the interval [m+1, j)\n            i = m + 1\n        } else if nums[m] &gt; target { // This means target is in the interval [i, m)\n            j = m\n        } else { // Found the target element, return its index\n            return m\n        }\n    }\n    // Target element not found, return -1\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* Binary search (left-closed right-open interval) */\nfunc binarySearchLCRO(nums: [Int], target: Int) -&gt; Int {\n    // Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1\n    var i = nums.startIndex\n    var j = nums.endIndex\n    // Loop, exit when the search interval is empty (empty when i = j)\n    while i &lt; j {\n        let m = i + (j - i) / 2 // Calculate the midpoint index m\n        if nums[m] &lt; target { // This means target is in the interval [m+1, j)\n            i = m + 1\n        } else if nums[m] &gt; target { // This means target is in the interval [i, m)\n            j = m\n        } else { // Found the target element, return its index\n            return m\n        }\n    }\n    // Target element not found, return -1\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* Binary search (left-closed right-open interval) */\nfunction binarySearchLCRO(nums, target) {\n    // Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1\n    let i = 0,\n        j = nums.length;\n    // Loop, exit when the search interval is empty (empty when i = j)\n    while (i &lt; j) {\n        // Calculate midpoint index m, use parseInt() to round down\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // This means target is in the interval [m+1, j)\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // This means target is in the interval [i, m)\n            j = m;\n        // Found the target element, return its index\n        else return m;\n    }\n    // Target element not found, return -1\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* Binary search (left-closed right-open interval) */\nfunction binarySearchLCRO(nums: number[], target: number): number {\n    // Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1\n    let i = 0,\n        j = nums.length;\n    // Loop, exit when the search interval is empty (empty when i = j)\n    while (i &lt; j) {\n        // Calculate the midpoint index m\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // This means target is in the interval [m+1, j)\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // This means target is in the interval [i, m)\n            j = m;\n        } else {\n            // Found the target element, return its index\n            return m;\n        }\n    }\n    return -1; // Target element not found, return -1\n}\n</code></pre> binary_search.dart<pre><code>/* Binary search (left-closed right-open interval) */\nint binarySearchLCRO(List&lt;int&gt; nums, int target) {\n  // Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1\n  int i = 0, j = nums.length;\n  // Loop, exit when the search interval is empty (empty when i = j)\n  while (i &lt; j) {\n    int m = i + (j - i) ~/ 2; // Calculate the midpoint index m\n    if (nums[m] &lt; target) {\n      // This means target is in the interval [m+1, j)\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // This means target is in the interval [i, m)\n      j = m;\n    } else {\n      // Found the target element, return its index\n      return m;\n    }\n  }\n  // Target element not found, return -1\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* Binary search (left-closed right-open interval) */\nfn binary_search_lcro(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1\n    let mut i = 0;\n    let mut j = nums.len() as i32;\n    // Loop, exit when the search interval is empty (empty when i = j)\n    while i &lt; j {\n        let m = i + (j - i) / 2; // Calculate the midpoint index m\n        if nums[m as usize] &lt; target {\n            // This means target is in the interval [m+1, j)\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // This means target is in the interval [i, m)\n            j = m;\n        } else {\n            // Found the target element, return its index\n            return m;\n        }\n    }\n    // Target element not found, return -1\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* Binary search (left-closed right-open interval) */\nint binarySearchLCRO(int *nums, int len, int target) {\n    // Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1\n    int i = 0, j = len;\n    // Loop, exit when the search interval is empty (empty when i = j)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target)    // This means target is in the interval [m+1, j)\n            i = m + 1;\n        else if (nums[m] &gt; target) // This means target is in the interval [i, m)\n            j = m;\n        else // Found the target element, return its index\n            return m;\n    }\n    // Target element not found, return -1\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* Binary search (left-closed right-open interval) */\nfun binarySearchLCRO(nums: IntArray, target: Int): Int {\n    // Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1\n    var i = 0\n    var j = nums.size\n    // Loop, exit when the search interval is empty (empty when i = j)\n    while (i &lt; j) {\n        val m = i + (j - i) / 2 // Calculate the midpoint index m\n        if (nums[m] &lt; target) // This means target is in the interval [m+1, j)\n            i = m + 1\n        else if (nums[m] &gt; target) // This means target is in the interval [i, m)\n            j = m\n        else  // Found the target element, return its index\n            return m\n    }\n    // Target element not found, return -1\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>### Binary search (left-closed right-open interval) ###\ndef binary_search_lcro(nums, target)\n  # Initialize left-closed right-open interval [0, n), i.e., i, j point to the first element and last element+1\n  i, j = 0, nums.length\n\n  # Loop, exit when the search interval is empty (empty when i = j)\n  while i &lt; j\n    # Calculate the midpoint index m\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # This means target is in the interval [m+1, j)\n    elsif nums[m] &gt; target\n      j = m - 1 # This means target is in the interval [i, m)\n    else\n      return m  # Found the target element, return its index\n    end\n  end\n\n  -1  # Target element not found, return -1\nend\n</code></pre> <p>As shown in Figure 10-3, under the two interval representations, the initialization, loop condition, and interval narrowing operations of the binary search algorithm are all different.</p> <p>Since both the left and right boundaries in the \"closed interval\" representation are defined as closed, the operations to narrow the interval through pointers \\(i\\) and \\(j\\) are also symmetric. This makes it less error-prone, so the \"closed interval\" approach is generally recommended.</p> <p></p> <p> Figure 10-3   Two interval definitions </p>","path":["Chapter 10. Searching","10.1   Binary Search"],"tags":[]},{"location":"chapter_searching/binary_search/#1012-advantages-and-limitations","level":2,"title":"10.1.2   Advantages and Limitations","text":"<p>Binary search offers good performance in both time and space.</p> <ul> <li>Binary search has high time efficiency. With large data volumes, the logarithmic time complexity has significant advantages. For example, when the data size \\(n = 2^{20}\\), linear search requires \\(2^{20} = 1048576\\) iterations, while binary search only needs \\(\\log_2 2^{20} = 20\\) iterations.</li> <li>Binary search requires no extra space. Compared to searching algorithms that require additional space (such as hash-based search), binary search is more space-efficient.</li> </ul> <p>However, binary search is not suitable for all situations, mainly for the following reasons:</p> <ul> <li>Binary search is only applicable to sorted data. If the input data is unsorted, sorting specifically to use binary search would be counterproductive, as sorting algorithms typically have a time complexity of \\(O(n \\log n)\\), which is higher than both linear search and binary search. For scenarios with frequent element insertions, keeping the array sorted requires inserting elements at specific positions with a time complexity of \\(O(n)\\), which is also very expensive.</li> <li>Binary search is only applicable to arrays. Binary search requires non-contiguous, jump-style access to elements, and this kind of access is inefficient in linked lists, making it unsuitable for linked lists or linked-list-based data structures.</li> <li>For small data volumes, linear search performs better. In linear search, each round requires only 1 comparison operation; while in binary search, it requires 1 addition, 1 division, 1-3 comparison operations, and 1 addition (subtraction), totaling 4-6 unit operations. Therefore, when the data volume \\(n\\) is small, linear search is actually faster than binary search.</li> </ul>","path":["Chapter 10. Searching","10.1   Binary Search"],"tags":[]},{"location":"chapter_searching/binary_search_edge/","level":1,"title":"10.3   Binary Search Boundaries","text":"","path":["Chapter 10. Searching","10.3   Binary Search Boundaries"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#1031-finding-the-left-boundary","level":2,"title":"10.3.1   Finding the Left Boundary","text":"<p>Question</p> <p>Given a sorted array <code>nums</code> of length \\(n\\) that may contain duplicate elements, return the index of the leftmost occurrence of <code>target</code>. If the array does not contain <code>target</code>, return \\(-1\\).</p> <p>Recall the method for finding the insertion point with binary search. After the search completes, \\(i\\) points to the leftmost <code>target</code>, so finding the insertion point is essentially finding the index of the leftmost <code>target</code>.</p> <p>Consider implementing the left boundary search using the insertion point finding function. Note that the array may not contain <code>target</code>, which could result in the following two cases:</p> <ul> <li>The insertion point index \\(i\\) is out of bounds.</li> <li>The element <code>nums[i]</code> is not equal to <code>target</code>.</li> </ul> <p>When either of these situations occurs, simply return \\(-1\\). The code is shown below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_edge.py<pre><code>def binary_search_left_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"Binary search for the leftmost target\"\"\"\n    # Equivalent to finding the insertion point of target\n    i = binary_search_insertion(nums, target)\n    # Target not found, return -1\n    if i == len(nums) or nums[i] != target:\n        return -1\n    # Found target, return index i\n    return i\n</code></pre> binary_search_edge.cpp<pre><code>/* Binary search for the leftmost target */\nint binarySearchLeftEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // Equivalent to finding the insertion point of target\n    int i = binarySearchInsertion(nums, target);\n    // Target not found, return -1\n    if (i == nums.size() || nums[i] != target) {\n        return -1;\n    }\n    // Found target, return index i\n    return i;\n}\n</code></pre> binary_search_edge.java<pre><code>/* Binary search for the leftmost target */\nint binarySearchLeftEdge(int[] nums, int target) {\n    // Equivalent to finding the insertion point of target\n    int i = binary_search_insertion.binarySearchInsertion(nums, target);\n    // Target not found, return -1\n    if (i == nums.length || nums[i] != target) {\n        return -1;\n    }\n    // Found target, return index i\n    return i;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* Binary search for the leftmost target */\nint BinarySearchLeftEdge(int[] nums, int target) {\n    // Equivalent to finding the insertion point of target\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target);\n    // Target not found, return -1\n    if (i == nums.Length || nums[i] != target) {\n        return -1;\n    }\n    // Found target, return index i\n    return i;\n}\n</code></pre> binary_search_edge.go<pre><code>/* Binary search for the leftmost target */\nfunc binarySearchLeftEdge(nums []int, target int) int {\n    // Equivalent to finding the insertion point of target\n    i := binarySearchInsertion(nums, target)\n    // Target not found, return -1\n    if i == len(nums) || nums[i] != target {\n        return -1\n    }\n    // Found target, return index i\n    return i\n}\n</code></pre> binary_search_edge.swift<pre><code>/* Binary search for the leftmost target */\nfunc binarySearchLeftEdge(nums: [Int], target: Int) -&gt; Int {\n    // Equivalent to finding the insertion point of target\n    let i = binarySearchInsertion(nums: nums, target: target)\n    // Target not found, return -1\n    if i == nums.endIndex || nums[i] != target {\n        return -1\n    }\n    // Found target, return index i\n    return i\n}\n</code></pre> binary_search_edge.js<pre><code>/* Binary search for the leftmost target */\nfunction binarySearchLeftEdge(nums, target) {\n    // Equivalent to finding the insertion point of target\n    const i = binarySearchInsertion(nums, target);\n    // Target not found, return -1\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // Found target, return index i\n    return i;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* Binary search for the leftmost target */\nfunction binarySearchLeftEdge(nums: Array&lt;number&gt;, target: number): number {\n    // Equivalent to finding the insertion point of target\n    const i = binarySearchInsertion(nums, target);\n    // Target not found, return -1\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // Found target, return index i\n    return i;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* Binary search for the leftmost target */\nint binarySearchLeftEdge(List&lt;int&gt; nums, int target) {\n  // Equivalent to finding the insertion point of target\n  int i = binarySearchInsertion(nums, target);\n  // Target not found, return -1\n  if (i == nums.length || nums[i] != target) {\n    return -1;\n  }\n  // Found target, return index i\n  return i;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* Binary search for the leftmost target */\nfn binary_search_left_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // Equivalent to finding the insertion point of target\n    let i = binary_search_insertion(nums, target);\n    // Target not found, return -1\n    if i == nums.len() as i32 || nums[i as usize] != target {\n        return -1;\n    }\n    // Found target, return index i\n    i\n}\n</code></pre> binary_search_edge.c<pre><code>/* Binary search for the leftmost target */\nint binarySearchLeftEdge(int *nums, int numSize, int target) {\n    // Equivalent to finding the insertion point of target\n    int i = binarySearchInsertion(nums, numSize, target);\n    // Target not found, return -1\n    if (i == numSize || nums[i] != target) {\n        return -1;\n    }\n    // Found target, return index i\n    return i;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* Binary search for the leftmost target */\nfun binarySearchLeftEdge(nums: IntArray, target: Int): Int {\n    // Equivalent to finding the insertion point of target\n    val i = binarySearchInsertion(nums, target)\n    // Target not found, return -1\n    if (i == nums.size || nums[i] != target) {\n        return -1\n    }\n    // Found target, return index i\n    return i\n}\n</code></pre> binary_search_edge.rb<pre><code>### Binary search leftmost target ###\ndef binary_search_left_edge(nums, target)\n  # Equivalent to finding the insertion point of target\n  i = binary_search_insertion(nums, target)\n\n  # Target not found, return -1\n  return -1 if i == nums.length || nums[i] != target\n\n  i # Found target, return index i\nend\n</code></pre>","path":["Chapter 10. Searching","10.3   Binary Search Boundaries"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#1032-finding-the-right-boundary","level":2,"title":"10.3.2   Finding the Right Boundary","text":"<p>So how do we find the rightmost <code>target</code>? The most direct approach is to modify the code and replace the pointer shrinking operation in the <code>nums[m] == target</code> case. The code is omitted here; interested readers can implement it themselves.</p> <p>Below we introduce two more clever methods.</p>","path":["Chapter 10. Searching","10.3   Binary Search Boundaries"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#1-reusing-left-boundary-search","level":3,"title":"1.   Reusing Left Boundary Search","text":"<p>In fact, we can use the function for finding the leftmost <code>target</code> to find the rightmost <code>target</code>. The specific method is: convert finding the rightmost <code>target</code> into finding the leftmost <code>target + 1</code>.</p> <p>As shown in Figure 10-7, after the search completes, the pointer \\(i\\) points to the leftmost <code>target + 1</code> (if it exists), while \\(j\\) points to the rightmost <code>target</code>, so we can return \\(j\\).</p> <p></p> <p> Figure 10-7   Converting right boundary search to left boundary search </p> <p>Note that the returned insertion point is \\(i\\), so we need to subtract \\(1\\) from it to obtain \\(j\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_edge.py<pre><code>def binary_search_right_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"Binary search for the rightmost target\"\"\"\n    # Convert to finding the leftmost target + 1\n    i = binary_search_insertion(nums, target + 1)\n    # j points to the rightmost target, i points to the first element greater than target\n    j = i - 1\n    # Target not found, return -1\n    if j == -1 or nums[j] != target:\n        return -1\n    # Found target, return index j\n    return j\n</code></pre> binary_search_edge.cpp<pre><code>/* Binary search for the rightmost target */\nint binarySearchRightEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // Convert to finding the leftmost target + 1\n    int i = binarySearchInsertion(nums, target + 1);\n    // j points to the rightmost target, i points to the first element greater than target\n    int j = i - 1;\n    // Target not found, return -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // Found target, return index j\n    return j;\n}\n</code></pre> binary_search_edge.java<pre><code>/* Binary search for the rightmost target */\nint binarySearchRightEdge(int[] nums, int target) {\n    // Convert to finding the leftmost target + 1\n    int i = binary_search_insertion.binarySearchInsertion(nums, target + 1);\n    // j points to the rightmost target, i points to the first element greater than target\n    int j = i - 1;\n    // Target not found, return -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // Found target, return index j\n    return j;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* Binary search for the rightmost target */\nint BinarySearchRightEdge(int[] nums, int target) {\n    // Convert to finding the leftmost target + 1\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target + 1);\n    // j points to the rightmost target, i points to the first element greater than target\n    int j = i - 1;\n    // Target not found, return -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // Found target, return index j\n    return j;\n}\n</code></pre> binary_search_edge.go<pre><code>/* Binary search for the rightmost target */\nfunc binarySearchRightEdge(nums []int, target int) int {\n    // Convert to finding the leftmost target + 1\n    i := binarySearchInsertion(nums, target+1)\n    // j points to the rightmost target, i points to the first element greater than target\n    j := i - 1\n    // Target not found, return -1\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // Found target, return index j\n    return j\n}\n</code></pre> binary_search_edge.swift<pre><code>/* Binary search for the rightmost target */\nfunc binarySearchRightEdge(nums: [Int], target: Int) -&gt; Int {\n    // Convert to finding the leftmost target + 1\n    let i = binarySearchInsertion(nums: nums, target: target + 1)\n    // j points to the rightmost target, i points to the first element greater than target\n    let j = i - 1\n    // Target not found, return -1\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // Found target, return index j\n    return j\n}\n</code></pre> binary_search_edge.js<pre><code>/* Binary search for the rightmost target */\nfunction binarySearchRightEdge(nums, target) {\n    // Convert to finding the leftmost target + 1\n    const i = binarySearchInsertion(nums, target + 1);\n    // j points to the rightmost target, i points to the first element greater than target\n    const j = i - 1;\n    // Target not found, return -1\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // Found target, return index j\n    return j;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* Binary search for the rightmost target */\nfunction binarySearchRightEdge(nums: Array&lt;number&gt;, target: number): number {\n    // Convert to finding the leftmost target + 1\n    const i = binarySearchInsertion(nums, target + 1);\n    // j points to the rightmost target, i points to the first element greater than target\n    const j = i - 1;\n    // Target not found, return -1\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // Found target, return index j\n    return j;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* Binary search for the rightmost target */\nint binarySearchRightEdge(List&lt;int&gt; nums, int target) {\n  // Convert to finding the leftmost target + 1\n  int i = binarySearchInsertion(nums, target + 1);\n  // j points to the rightmost target, i points to the first element greater than target\n  int j = i - 1;\n  // Target not found, return -1\n  if (j == -1 || nums[j] != target) {\n    return -1;\n  }\n  // Found target, return index j\n  return j;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* Binary search for the rightmost target */\nfn binary_search_right_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // Convert to finding the leftmost target + 1\n    let i = binary_search_insertion(nums, target + 1);\n    // j points to the rightmost target, i points to the first element greater than target\n    let j = i - 1;\n    // Target not found, return -1\n    if j == -1 || nums[j as usize] != target {\n        return -1;\n    }\n    // Found target, return index j\n    j\n}\n</code></pre> binary_search_edge.c<pre><code>/* Binary search for the rightmost target */\nint binarySearchRightEdge(int *nums, int numSize, int target) {\n    // Convert to finding the leftmost target + 1\n    int i = binarySearchInsertion(nums, numSize, target + 1);\n    // j points to the rightmost target, i points to the first element greater than target\n    int j = i - 1;\n    // Target not found, return -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // Found target, return index j\n    return j;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* Binary search for the rightmost target */\nfun binarySearchRightEdge(nums: IntArray, target: Int): Int {\n    // Convert to finding the leftmost target + 1\n    val i = binarySearchInsertion(nums, target + 1)\n    // j points to the rightmost target, i points to the first element greater than target\n    val j = i - 1\n    // Target not found, return -1\n    if (j == -1 || nums[j] != target) {\n        return -1\n    }\n    // Found target, return index j\n    return j\n}\n</code></pre> binary_search_edge.rb<pre><code>### Binary search rightmost target ###\ndef binary_search_right_edge(nums, target)\n  # Convert to finding the leftmost target + 1\n  i = binary_search_insertion(nums, target + 1)\n\n  # j points to the rightmost target, i points to the first element greater than target\n  j = i - 1\n\n  # Target not found, return -1\n  return -1 if j == -1 || nums[j] != target\n\n  j # Found target, return index j\nend\n</code></pre>","path":["Chapter 10. Searching","10.3   Binary Search Boundaries"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#2-converting-to-element-search","level":3,"title":"2.   Converting to Element Search","text":"<p>We know that when the array does not contain <code>target</code>, \\(i\\) and \\(j\\) will eventually point to the first element greater than <code>target</code> and the rightmost element less than <code>target</code>, respectively.</p> <p>Therefore, as shown in Figure 10-8, we can construct an element that does not exist in the array to find the left and right boundaries.</p> <ul> <li>Finding the leftmost <code>target</code>: This can be converted to finding <code>target - 0.5</code> and returning the pointer \\(i\\).</li> <li>Finding the rightmost <code>target</code>: This can be converted to finding <code>target + 0.5</code> and returning the pointer \\(j\\).</li> </ul> <p></p> <p> Figure 10-8   Converting boundary search to element search </p> <p>The code is omitted here, but the following two points are worth noting:</p> <ul> <li>Since the given array does not contain decimal values, we do not need to worry about how to handle equality.</li> <li>Because this method introduces decimals, the variable <code>target</code> in the function needs to be changed to a floating-point type (Python does not require this change).</li> </ul>","path":["Chapter 10. Searching","10.3   Binary Search Boundaries"],"tags":[]},{"location":"chapter_searching/binary_search_insertion/","level":1,"title":"10.2   Binary Search Insertion Point","text":"<p>Binary search can be used not only to search for target elements, but also to solve many variant problems, such as finding the insertion position of a target element.</p>","path":["Chapter 10. Searching","10.2   Binary Search Insertion Point"],"tags":[]},{"location":"chapter_searching/binary_search_insertion/#1021-case-without-duplicate-elements","level":2,"title":"10.2.1   Case Without Duplicate Elements","text":"<p>Question</p> <p>Given a sorted array <code>nums</code> of length \\(n\\) and an element <code>target</code>, where the array contains no duplicate elements, insert <code>target</code> into <code>nums</code> while maintaining its sorted order. If <code>target</code> already exists in the array, insert it to its left. Return the index of <code>target</code> after insertion. An example is shown below.</p> <p></p> <p> Figure 10-4   Binary search insertion point example data </p> <p>If we want to reuse the binary search code from the previous section, we need to answer the following two questions.</p> <p>Question 1: When the array contains <code>target</code>, is the insertion point index the same as that element's index?</p> <p>The problem requires inserting <code>target</code> to the left of equal elements, which means the newly inserted <code>target</code> replaces the position of the original <code>target</code>. In other words, when the array contains <code>target</code>, the insertion point index is the index of that <code>target</code>.</p> <p>Question 2: When the array does not contain <code>target</code>, what is the insertion point index?</p> <p>To analyze this further, consider the binary search process: when <code>nums[m] &lt; target</code>, \\(i\\) moves, meaning that pointer \\(i\\) is approaching elements greater than or equal to <code>target</code>. Similarly, pointer \\(j\\) is always approaching elements less than or equal to <code>target</code>.</p> <p>Therefore, when the binary search ends, \\(i\\) must point to the first element greater than <code>target</code>, and \\(j\\) must point to the rightmost element less than <code>target</code>. It follows that when the array does not contain <code>target</code>, the insertion index is \\(i\\). The code is shown below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_insertion.py<pre><code>def binary_search_insertion_simple(nums: list[int], target: int) -&gt; int:\n    \"\"\"Binary search for insertion point (no duplicate elements)\"\"\"\n    i, j = 0, len(nums) - 1  # Initialize closed interval [0, n-1]\n    while i &lt;= j:\n        m = (i + j) // 2  # Calculate midpoint index m\n        if nums[m] &lt; target:\n            i = m + 1  # target is in the interval [m+1, j]\n        elif nums[m] &gt; target:\n            j = m - 1  # target is in the interval [i, m-1]\n        else:\n            return m  # Found target, return insertion point m\n    # Target not found, return insertion point i\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* Binary search for insertion point (no duplicate elements) */\nint binarySearchInsertionSimple(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // Initialize closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target is in the interval [m+1, j]\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target is in the interval [i, m-1]\n        } else {\n            return m; // Found target, return insertion point m\n        }\n    }\n    // Target not found, return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* Binary search for insertion point (no duplicate elements) */\nint binarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // Initialize closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target is in the interval [m+1, j]\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target is in the interval [i, m-1]\n        } else {\n            return m; // Found target, return insertion point m\n        }\n    }\n    // Target not found, return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* Binary search for insertion point (no duplicate elements) */\nint BinarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // Initialize closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target is in the interval [m+1, j]\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target is in the interval [i, m-1]\n        } else {\n            return m; // Found target, return insertion point m\n        }\n    }\n    // Target not found, return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* Binary search for insertion point (no duplicate elements) */\nfunc binarySearchInsertionSimple(nums []int, target int) int {\n    // Initialize closed interval [0, n-1]\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // Calculate the midpoint index m\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target is in the interval [m+1, j]\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target is in the interval [i, m-1]\n            j = m - 1\n        } else {\n            // Found target, return insertion point m\n            return m\n        }\n    }\n    // Target not found, return insertion point i\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* Binary search for insertion point (no duplicate elements) */\nfunc binarySearchInsertionSimple(nums: [Int], target: Int) -&gt; Int {\n    // Initialize closed interval [0, n-1]\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // Calculate the midpoint index m\n        if nums[m] &lt; target {\n            i = m + 1 // target is in the interval [m+1, j]\n        } else if nums[m] &gt; target {\n            j = m - 1 // target is in the interval [i, m-1]\n        } else {\n            return m // Found target, return insertion point m\n        }\n    }\n    // Target not found, return insertion point i\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* Binary search for insertion point (no duplicate elements) */\nfunction binarySearchInsertionSimple(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // Initialize closed interval [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // Calculate midpoint index m, use Math.floor() to round down\n        if (nums[m] &lt; target) {\n            i = m + 1; // target is in the interval [m+1, j]\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target is in the interval [i, m-1]\n        } else {\n            return m; // Found target, return insertion point m\n        }\n    }\n    // Target not found, return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* Binary search for insertion point (no duplicate elements) */\nfunction binarySearchInsertionSimple(\n    nums: Array&lt;number&gt;,\n    target: number\n): number {\n    let i = 0,\n        j = nums.length - 1; // Initialize closed interval [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // Calculate midpoint index m, use Math.floor() to round down\n        if (nums[m] &lt; target) {\n            i = m + 1; // target is in the interval [m+1, j]\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target is in the interval [i, m-1]\n        } else {\n            return m; // Found target, return insertion point m\n        }\n    }\n    // Target not found, return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* Binary search for insertion point (no duplicate elements) */\nint binarySearchInsertionSimple(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // Initialize closed interval [0, n-1]\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // Calculate the midpoint index m\n    if (nums[m] &lt; target) {\n      i = m + 1; // target is in the interval [m+1, j]\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target is in the interval [i, m-1]\n    } else {\n      return m; // Found target, return insertion point m\n    }\n  }\n  // Target not found, return insertion point i\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* Binary search for insertion point (no duplicate elements) */\nfn binary_search_insertion_simple(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // Initialize closed interval [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // Calculate the midpoint index m\n        if nums[m as usize] &lt; target {\n            i = m + 1; // target is in the interval [m+1, j]\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // target is in the interval [i, m-1]\n        } else {\n            return m;\n        }\n    }\n    // Target not found, return insertion point i\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* Binary search for insertion point (no duplicate elements) */\nint binarySearchInsertionSimple(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // Initialize closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target is in the interval [m+1, j]\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target is in the interval [i, m-1]\n        } else {\n            return m; // Found target, return insertion point m\n        }\n    }\n    // Target not found, return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* Binary search for insertion point (no duplicate elements) */\nfun binarySearchInsertionSimple(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // Initialize closed interval [0, n-1]\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1 // target is in the interval [m+1, j]\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // target is in the interval [i, m-1]\n        } else {\n            return m // Found target, return insertion point m\n        }\n    }\n    // Target not found, return insertion point i\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>### Binary search insertion point (no duplicates) ###\ndef binary_search_insertion_simple(nums, target)\n  # Initialize closed interval [0, n-1]\n  i, j = 0, nums.length - 1\n\n  while i &lt;= j\n    # Calculate the midpoint index m\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # target is in the interval [m+1, j]\n    elsif nums[m] &gt; target\n      j = m - 1 # target is in the interval [i, m-1]\n    else\n      return m  # Found target, return insertion point m\n    end\n  end\n\n  i # Target not found, return insertion point i\nend\n</code></pre>","path":["Chapter 10. Searching","10.2   Binary Search Insertion Point"],"tags":[]},{"location":"chapter_searching/binary_search_insertion/#1022-case-with-duplicate-elements","level":2,"title":"10.2.2   Case with Duplicate Elements","text":"<p>Question</p> <p>Based on the previous problem, assume the array may contain duplicate elements, with everything else remaining the same.</p> <p>Suppose there are multiple <code>target</code> elements in the array. Ordinary binary search can only return the index of one <code>target</code>, and cannot determine how many <code>target</code> elements are to the left and right of that element.</p> <p>The problem requires inserting the target element at the leftmost position, so we need to find the index of the leftmost <code>target</code> in the array. A straightforward initial approach is to follow the steps shown in Figure 10-5:</p> <ol> <li>Perform binary search to obtain the index of any <code>target</code>, denoted as \\(k\\).</li> <li>Starting from index \\(k\\), perform linear traversal to the left, and return when the leftmost <code>target</code> is found.</li> </ol> <p></p> <p> Figure 10-5   Linear search for insertion point of duplicate elements </p> <p>Although this method works, it includes linear search, resulting in a time complexity of \\(O(n)\\). When the array contains many duplicate <code>target</code> elements, this method is very inefficient.</p> <p>Now consider extending the binary search code. As shown in Figure 10-6, the overall process remains unchanged: in each iteration, we first compute the midpoint index \\(m\\), then compare <code>target</code> with <code>nums[m]</code>, leading to the following cases:</p> <ul> <li>When <code>nums[m] &lt; target</code> or <code>nums[m] &gt; target</code>, it means <code>target</code> has not been found yet, so use the standard interval-shrinking operation of binary search to move pointers \\(i\\) and \\(j\\) closer to <code>target</code>.</li> <li>When <code>nums[m] == target</code>, it means elements less than <code>target</code> are in the interval \\([i, m - 1]\\), so use \\(j = m - 1\\) to shrink the interval, thereby moving pointer \\(j\\) closer to elements less than <code>target</code>.</li> </ul> <p>After the loop completes, \\(i\\) points to the leftmost <code>target</code>, and \\(j\\) points to the rightmost element less than <code>target</code>, so index \\(i\\) is the insertion point.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 10-6   Steps for binary search insertion point of duplicate elements </p> <p>Observe the following code: the branches <code>nums[m] &gt; target</code> and <code>nums[m] == target</code> perform the same operation, so they can be merged.</p> <p>Even so, we can still keep the conditional branches expanded, as the logic is clearer and more readable.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_insertion.py<pre><code>def binary_search_insertion(nums: list[int], target: int) -&gt; int:\n    \"\"\"Binary search for insertion point (with duplicate elements)\"\"\"\n    i, j = 0, len(nums) - 1  # Initialize closed interval [0, n-1]\n    while i &lt;= j:\n        m = (i + j) // 2  # Calculate midpoint index m\n        if nums[m] &lt; target:\n            i = m + 1  # target is in the interval [m+1, j]\n        elif nums[m] &gt; target:\n            j = m - 1  # target is in the interval [i, m-1]\n        else:\n            j = m - 1  # The rightmost element less than target is in the interval [i, m-1]\n    # Return insertion point i\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* Binary search for insertion point (with duplicate elements) */\nint binarySearchInsertion(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // Initialize closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target is in the interval [m+1, j]\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target is in the interval [i, m-1]\n        } else {\n            j = m - 1; // The rightmost element less than target is in the interval [i, m-1]\n        }\n    }\n    // Return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* Binary search for insertion point (with duplicate elements) */\nint binarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // Initialize closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target is in the interval [m+1, j]\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target is in the interval [i, m-1]\n        } else {\n            j = m - 1; // The rightmost element less than target is in the interval [i, m-1]\n        }\n    }\n    // Return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* Binary search for insertion point (with duplicate elements) */\nint BinarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // Initialize closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target is in the interval [m+1, j]\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target is in the interval [i, m-1]\n        } else {\n            j = m - 1; // The rightmost element less than target is in the interval [i, m-1]\n        }\n    }\n    // Return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* Binary search for insertion point (with duplicate elements) */\nfunc binarySearchInsertion(nums []int, target int) int {\n    // Initialize closed interval [0, n-1]\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // Calculate the midpoint index m\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target is in the interval [m+1, j]\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target is in the interval [i, m-1]\n            j = m - 1\n        } else {\n            // The rightmost element less than target is in the interval [i, m-1]\n            j = m - 1\n        }\n    }\n    // Return insertion point i\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* Binary search for insertion point (with duplicate elements) */\nfunc binarySearchInsertion(nums: [Int], target: Int) -&gt; Int {\n    // Initialize closed interval [0, n-1]\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // Calculate the midpoint index m\n        if nums[m] &lt; target {\n            i = m + 1 // target is in the interval [m+1, j]\n        } else if nums[m] &gt; target {\n            j = m - 1 // target is in the interval [i, m-1]\n        } else {\n            j = m - 1 // The rightmost element less than target is in the interval [i, m-1]\n        }\n    }\n    // Return insertion point i\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* Binary search for insertion point (with duplicate elements) */\nfunction binarySearchInsertion(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // Initialize closed interval [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // Calculate midpoint index m, use Math.floor() to round down\n        if (nums[m] &lt; target) {\n            i = m + 1; // target is in the interval [m+1, j]\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target is in the interval [i, m-1]\n        } else {\n            j = m - 1; // The rightmost element less than target is in the interval [i, m-1]\n        }\n    }\n    // Return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* Binary search for insertion point (with duplicate elements) */\nfunction binarySearchInsertion(nums: Array&lt;number&gt;, target: number): number {\n    let i = 0,\n        j = nums.length - 1; // Initialize closed interval [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // Calculate midpoint index m, use Math.floor() to round down\n        if (nums[m] &lt; target) {\n            i = m + 1; // target is in the interval [m+1, j]\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target is in the interval [i, m-1]\n        } else {\n            j = m - 1; // The rightmost element less than target is in the interval [i, m-1]\n        }\n    }\n    // Return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* Binary search for insertion point (with duplicate elements) */\nint binarySearchInsertion(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // Initialize closed interval [0, n-1]\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // Calculate the midpoint index m\n    if (nums[m] &lt; target) {\n      i = m + 1; // target is in the interval [m+1, j]\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target is in the interval [i, m-1]\n    } else {\n      j = m - 1; // The rightmost element less than target is in the interval [i, m-1]\n    }\n  }\n  // Return insertion point i\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* Binary search for insertion point (with duplicate elements) */\npub fn binary_search_insertion(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // Initialize closed interval [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // Calculate the midpoint index m\n        if nums[m as usize] &lt; target {\n            i = m + 1; // target is in the interval [m+1, j]\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // target is in the interval [i, m-1]\n        } else {\n            j = m - 1; // The rightmost element less than target is in the interval [i, m-1]\n        }\n    }\n    // Return insertion point i\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* Binary search for insertion point (with duplicate elements) */\nint binarySearchInsertion(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // Initialize closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target is in the interval [m+1, j]\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target is in the interval [i, m-1]\n        } else {\n            j = m - 1; // The rightmost element less than target is in the interval [i, m-1]\n        }\n    }\n    // Return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* Binary search for insertion point (with duplicate elements) */\nfun binarySearchInsertion(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // Initialize closed interval [0, n-1]\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1 // target is in the interval [m+1, j]\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // target is in the interval [i, m-1]\n        } else {\n            j = m - 1 // The rightmost element less than target is in the interval [i, m-1]\n        }\n    }\n    // Return insertion point i\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>### Binary search insertion point (with duplicates) ###\ndef binary_search_insertion(nums, target)\n  # Initialize closed interval [0, n-1]\n  i, j = 0, nums.length - 1\n\n  while i &lt;= j\n    # Calculate the midpoint index m\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # target is in the interval [m+1, j]\n    elsif nums[m] &gt; target\n      j = m - 1 # target is in the interval [i, m-1]\n    else\n      j = m - 1 # The rightmost element less than target is in the interval [i, m-1]\n    end\n  end\n\n  i # Return insertion point i\nend\n</code></pre> <p>Tip</p> <p>The code in this section uses the \"closed interval\" approach throughout. Interested readers can implement the \"left-closed, right-open\" approach themselves.</p> <p>Overall, binary search is simply a matter of setting separate search targets for pointers \\(i\\) and \\(j\\). The target may be a specific element (such as <code>target</code>) or a range of elements (such as elements less than <code>target</code>).</p> <p>With each iteration of binary search, pointers \\(i\\) and \\(j\\) gradually approach their preset targets. Ultimately, they either find the answer or stop after crossing the boundary.</p>","path":["Chapter 10. Searching","10.2   Binary Search Insertion Point"],"tags":[]},{"location":"chapter_searching/exercises/","level":1,"title":"10.7   Exercises","text":"","path":["Chapter 10. Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/exercises/#1071-concept-review","level":2,"title":"10.7.1   Concept Review","text":"","path":["Chapter 10. Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/exercises/#1-how-binary-search-narrows-the-search-interval","level":3,"title":"1.   How Binary Search Narrows the Search Interval","text":"<p>Search for 16 in the sorted array <code>[2, 5, 8, 12, 16, 23, 38]</code>. Use the closed interval <code>[i, j]</code> and calculate the midpoint as \\(m=i+(j-i)/2\\), rounded down.</p> <p>For each round, write <code>(i, j, m)</code>, the middle element, and how the interval is narrowed next, until the target is found.</p> Answer <p>The search proceeds as follows:</p> Round <code>(i, j, m)</code> Middle element Next step 1 <code>(0, 6, 3)</code> 12 <code>12 &lt; 16</code>, set <code>i = 4</code> 2 <code>(4, 6, 5)</code> 23 <code>23 &gt; 16</code>, set <code>j = 4</code> 3 <code>(4, 4, 4)</code> 16 Target found; return index 4 <p>Because the array is sorted, when the middle value is smaller than the target, the middle and everything to its left can be excluded. When the middle value is greater than the target, the middle and everything to its right can be excluded.</p>","path":["Chapter 10. Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/exercises/#2-left-and-right-boundaries-of-duplicate-elements","level":3,"title":"2.   Left and Right Boundaries of Duplicate Elements","text":"<p>Search for the number 2 in the array <code>[1, 2, 2, 2, 4, 6]</code>. A student uses binary search, returns immediately after finding the target at index 2, and says, \"Index 2 is the left boundary of the number 2.\"</p> <ol> <li>Is the student's statement correct? What are the left and right boundaries of 2? Explain why.</li> <li>When searching for the left boundary, if the middle element equals the target, which side should be searched next?</li> <li>When searching for the right boundary, which side should be searched next? State only the direction; you do not need to write the complete search process.</li> </ol> Answer <ol> <li> <p>The student's statement is incorrect. Returning as soon as a 2 is found guarantees only that some occurrence of 2 was found, not the leftmost or rightmost one.     Here, the left boundary is index 1 and the right boundary is index 3.</p> </li> <li> <p>When searching for the left boundary, continue searching on the left even if the middle element equals 2.     For example, with a closed interval, set <code>j = m - 1</code>.</p> </li> <li> <p>When searching for the right boundary, continue searching on the right after the middle element equals 2.     For example, set <code>i = m + 1</code>.</p> </li> </ol>","path":["Chapter 10. Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/exercises/#3-choosing-a-search-method-for-different-data","level":3,"title":"3.   Choosing a Search Method for Different Data","text":"<p>For each situation below, choose an appropriate method from \"linear search,\" \"binary search,\" and \"hash table,\" and explain why:</p> <ol> <li>Repeatedly search among \\(10^7\\) integers that are already sorted and never change, without building any additional data structure.</li> <li>Repeatedly test whether a key exists in a collection with frequent insertions and deletions. The collection need not remain sorted, and no range searches are needed.</li> <li>Search an unsorted array for a value only once.</li> </ol> Answer <ol> <li>Binary search: the data is sorted and static, so searching takes \\(O(\\log n)\\) time without extra space.</li> <li>Hash table: when the hash function distributes keys fairly evenly among the buckets, insertion, deletion, and lookup by key all take \\(O(1)\\) time on average.</li> <li> <p>Scan directly from beginning to end. When searching only once, sorting the array or building a hash table still requires processing the entire array first,     so neither reduces the total work for this single task.</p> <p>The choice depends on whether the data is sorted, whether an additional structure is allowed, how many searches are needed, and which operations must be supported.</p> </li> </ol>","path":["Chapter 10. Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/exercises/#1072-programming-exercises","level":2,"title":"10.7.2   Programming Exercises","text":"","path":["Chapter 10. Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/exercises/#1-binary-search-in-a-sorted-array","level":3,"title":"1.   Binary Search in a Sorted Array","text":"<p>Given an integer array <code>nums</code> in strictly increasing order and a target value <code>target</code>, use binary search to find <code>target</code>. If it exists, return its array index; otherwise, return -1.</p> Hints <ol> <li>The initial interval is left = 0 and right = n - 1; it is nonempty while left &lt;= right</li> <li>Calculate the midpoint with mid = left + (right - left) // 2</li> <li>If <code>nums[mid]</code> is less than <code>target</code>, move the left boundary to <code>mid + 1</code>; if <code>nums[mid]</code> is greater than <code>target</code>, move the right boundary to <code>mid - 1</code>; if they are equal, return immediately</li> </ol> <p>LeetCode</p>","path":["Chapter 10. Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/exercises/#2-insertion-point-in-a-sorted-array","level":3,"title":"2.   Insertion Point in a Sorted Array","text":"<p>You are given an integer array <code>nums</code> in strictly increasing order and a target value <code>target</code>.</p> <p>Return the index of <code>target</code> if it is already in the array. Otherwise, return the insertion point at which <code>target</code> can be inserted while keeping the array in strictly increasing order. The insertion point may be 0 or may equal the array's length. Use binary search.</p> Hints <ol> <li>The answer may be 0 or the array length n</li> <li>With a closed interval, if <code>nums[mid]</code> is greater than or equal to <code>target</code>, set <code>right = mid - 1</code> and continue checking farther left; otherwise, set <code>left = mid + 1</code></li> <li>When the loop ends, left is the insertion point</li> </ol> <p>LeetCode</p>","path":["Chapter 10. Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/replace_linear_by_hashing/","level":1,"title":"10.4   Hash Optimization Strategy","text":"<p>In algorithm problems, we often reduce the time complexity of algorithms by replacing linear search with hash-based search. Let's use an algorithm problem to deepen our understanding.</p> <p>Question</p> <p>Given an integer array <code>nums</code> and a target value <code>target</code>, find two elements in the array whose sum is <code>target</code>, and return their indices. Any solution will do.</p>","path":["Chapter 10. Searching","10.4   Hash Optimization Strategy"],"tags":[]},{"location":"chapter_searching/replace_linear_by_hashing/#1041-linear-search-trading-time-for-space","level":2,"title":"10.4.1   Linear Search: Trading Time for Space","text":"<p>Consider directly traversing all possible combinations. As shown in Figure 10-9, we use nested loops and check in each iteration whether the sum of two integers is <code>target</code>. If so, return their indices.</p> <p></p> <p> Figure 10-9   Linear search solution for two sum </p> <p>The code is shown below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby two_sum.py<pre><code>def two_sum_brute_force(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"Method 1: Brute force enumeration\"\"\"\n    # Two nested loops, time complexity is O(n^2)\n    for i in range(len(nums) - 1):\n        for j in range(i + 1, len(nums)):\n            if nums[i] + nums[j] == target:\n                return [i, j]\n    return []\n</code></pre> two_sum.cpp<pre><code>/* Method 1: Brute force enumeration */\nvector&lt;int&gt; twoSumBruteForce(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // Two nested loops, time complexity is O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return {i, j};\n        }\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* Method 1: Brute force enumeration */\nint[] twoSumBruteForce(int[] nums, int target) {\n    int size = nums.length;\n    // Two nested loops, time complexity is O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return new int[] { i, j };\n        }\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* Method 1: Brute force enumeration */\nint[] TwoSumBruteForce(int[] nums, int target) {\n    int size = nums.Length;\n    // Two nested loops, time complexity is O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return [i, j];\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* Method 1: Brute force enumeration */\nfunc twoSumBruteForce(nums []int, target int) []int {\n    size := len(nums)\n    // Two nested loops, time complexity is O(n^2)\n    for i := 0; i &lt; size-1; i++ {\n        for j := i + 1; j &lt; size; j++ {\n            if nums[i]+nums[j] == target {\n                return []int{i, j}\n            }\n        }\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* Method 1: Brute force enumeration */\nfunc twoSumBruteForce(nums: [Int], target: Int) -&gt; [Int] {\n    // Two nested loops, time complexity is O(n^2)\n    for i in nums.indices.dropLast() {\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[i] + nums[j] == target {\n                return [i, j]\n            }\n        }\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* Method 1: Brute force enumeration */\nfunction twoSumBruteForce(nums, target) {\n    const n = nums.length;\n    // Two nested loops, time complexity is O(n^2)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* Method 1: Brute force enumeration */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\n    const n = nums.length;\n    // Two nested loops, time complexity is O(n^2)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* Method 1: Brute force enumeration */\nList&lt;int&gt; twoSumBruteForce(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // Two nested loops, time complexity is O(n^2)\n  for (var i = 0; i &lt; size - 1; i++) {\n    for (var j = i + 1; j &lt; size; j++) {\n      if (nums[i] + nums[j] == target) return [i, j];\n    }\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* Method 1: Brute force enumeration */\npub fn two_sum_brute_force(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    let size = nums.len();\n    // Two nested loops, time complexity is O(n^2)\n    for i in 0..size - 1 {\n        for j in i + 1..size {\n            if nums[i] + nums[j] == target {\n                return Some(vec![i as i32, j as i32]);\n            }\n        }\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* Method 1: Brute force enumeration */\nint *twoSumBruteForce(int *nums, int numsSize, int target, int *returnSize) {\n    for (int i = 0; i &lt; numsSize; ++i) {\n        for (int j = i + 1; j &lt; numsSize; ++j) {\n            if (nums[i] + nums[j] == target) {\n                int *res = malloc(sizeof(int) * 2);\n                res[0] = i, res[1] = j;\n                *returnSize = 2;\n                return res;\n            }\n        }\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* Method 1: Brute force enumeration */\nfun twoSumBruteForce(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // Two nested loops, time complexity is O(n^2)\n    for (i in 0..&lt;size - 1) {\n        for (j in i + 1..&lt;size) {\n            if (nums[i] + nums[j] == target) return intArrayOf(i, j)\n        }\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>### Method 1: Brute force enumeration ###\ndef two_sum_brute_force(nums, target)\n  # Two nested loops, time complexity is O(n^2)\n  for i in 0...(nums.length - 1)\n    for j in (i + 1)...nums.length\n      return [i, j] if nums[i] + nums[j] == target\n    end\n  end\n\n  []\nend\n</code></pre> <p>This method has a time complexity of \\(O(n^2)\\) and a space complexity of \\(O(1)\\), making it very time-consuming on large inputs.</p>","path":["Chapter 10. Searching","10.4   Hash Optimization Strategy"],"tags":[]},{"location":"chapter_searching/replace_linear_by_hashing/#1042-hash-based-search-trading-space-for-time","level":2,"title":"10.4.2   Hash-Based Search: Trading Space for Time","text":"<p>Consider using a hash table whose keys are array elements and whose values are their indices. Traverse the array and perform the steps shown in Figure 10-10 in each iteration:</p> <ol> <li>Check if the number <code>target - nums[i]</code> is in the hash table. If so, directly return the indices of these two elements.</li> <li>Add the key-value pair <code>nums[i]</code> and index <code>i</code> to the hash table.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> Figure 10-10   Hash table solution for two sum </p> <p>The implementation is shown below and requires only a single loop:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby two_sum.py<pre><code>def two_sum_hash_table(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"Method 2: Auxiliary hash table\"\"\"\n    # Auxiliary hash table, space complexity is O(n)\n    dic = {}\n    # Single loop, time complexity is O(n)\n    for i in range(len(nums)):\n        if target - nums[i] in dic:\n            return [dic[target - nums[i]], i]\n        dic[nums[i]] = i\n    return []\n</code></pre> two_sum.cpp<pre><code>/* Method 2: Auxiliary hash table */\nvector&lt;int&gt; twoSumHashTable(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // Auxiliary hash table, space complexity is O(n)\n    unordered_map&lt;int, int&gt; dic;\n    // Single loop, time complexity is O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.find(target - nums[i]) != dic.end()) {\n            return {dic[target - nums[i]], i};\n        }\n        dic.emplace(nums[i], i);\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* Method 2: Auxiliary hash table */\nint[] twoSumHashTable(int[] nums, int target) {\n    int size = nums.length;\n    // Auxiliary hash table, space complexity is O(n)\n    Map&lt;Integer, Integer&gt; dic = new HashMap&lt;&gt;();\n    // Single loop, time complexity is O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.containsKey(target - nums[i])) {\n            return new int[] { dic.get(target - nums[i]), i };\n        }\n        dic.put(nums[i], i);\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* Method 2: Auxiliary hash table */\nint[] TwoSumHashTable(int[] nums, int target) {\n    int size = nums.Length;\n    // Auxiliary hash table, space complexity is O(n)\n    Dictionary&lt;int, int&gt; dic = [];\n    // Single loop, time complexity is O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.ContainsKey(target - nums[i])) {\n            return [dic[target - nums[i]], i];\n        }\n        dic.Add(nums[i], i);\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* Method 2: Auxiliary hash table */\nfunc twoSumHashTable(nums []int, target int) []int {\n    // Auxiliary hash table, space complexity is O(n)\n    hashTable := map[int]int{}\n    // Single loop, time complexity is O(n)\n    for idx, val := range nums {\n        if preIdx, ok := hashTable[target-val]; ok {\n            return []int{preIdx, idx}\n        }\n        hashTable[val] = idx\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* Method 2: Auxiliary hash table */\nfunc twoSumHashTable(nums: [Int], target: Int) -&gt; [Int] {\n    // Auxiliary hash table, space complexity is O(n)\n    var dic: [Int: Int] = [:]\n    // Single loop, time complexity is O(n)\n    for i in nums.indices {\n        if let j = dic[target - nums[i]] {\n            return [j, i]\n        }\n        dic[nums[i]] = i\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* Method 2: Auxiliary hash table */\nfunction twoSumHashTable(nums, target) {\n    // Auxiliary hash table, space complexity is O(n)\n    let m = {};\n    // Single loop, time complexity is O(n)\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (m[target - nums[i]] !== undefined) {\n            return [m[target - nums[i]], i];\n        } else {\n            m[nums[i]] = i;\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* Method 2: Auxiliary hash table */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n    // Auxiliary hash table, space complexity is O(n)\n    let m: Map&lt;number, number&gt; = new Map();\n    // Single loop, time complexity is O(n)\n    for (let i = 0; i &lt; nums.length; i++) {\n        let index = m.get(target - nums[i]);\n        if (index !== undefined) {\n            return [index, i];\n        } else {\n            m.set(nums[i], i);\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* Method 2: Auxiliary hash table */\nList&lt;int&gt; twoSumHashTable(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // Auxiliary hash table, space complexity is O(n)\n  Map&lt;int, int&gt; dic = HashMap();\n  // Single loop, time complexity is O(n)\n  for (var i = 0; i &lt; size; i++) {\n    if (dic.containsKey(target - nums[i])) {\n      return [dic[target - nums[i]]!, i];\n    }\n    dic.putIfAbsent(nums[i], () =&gt; i);\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* Method 2: Auxiliary hash table */\npub fn two_sum_hash_table(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    // Auxiliary hash table, space complexity is O(n)\n    let mut dic = HashMap::new();\n    // Single loop, time complexity is O(n)\n    for (i, num) in nums.iter().enumerate() {\n        match dic.get(&amp;(target - num)) {\n            Some(v) =&gt; return Some(vec![*v as i32, i as i32]),\n            None =&gt; dic.insert(num, i as i32),\n        };\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* Hash table */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // Implemented using uthash.h\n} HashTable;\n\n/* Hash table lookup */\nHashTable *find(HashTable *h, int key) {\n    HashTable *tmp;\n    HASH_FIND_INT(h, &amp;key, tmp);\n    return tmp;\n}\n\n/* Hash table element insertion */\nvoid insert(HashTable **h, int key, int val) {\n    HashTable *t = find(*h, key);\n    if (t == NULL) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = key, tmp-&gt;val = val;\n        HASH_ADD_INT(*h, key, tmp);\n    } else {\n        t-&gt;val = val;\n    }\n}\n\n/* Method 2: Auxiliary hash table */\nint *twoSumHashTable(int *nums, int numsSize, int target, int *returnSize) {\n    HashTable *hashtable = NULL;\n    for (int i = 0; i &lt; numsSize; i++) {\n        HashTable *t = find(hashtable, target - nums[i]);\n        if (t != NULL) {\n            int *res = malloc(sizeof(int) * 2);\n            res[0] = t-&gt;val, res[1] = i;\n            *returnSize = 2;\n            return res;\n        }\n        insert(&amp;hashtable, nums[i], i);\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* Method 2: Auxiliary hash table */\nfun twoSumHashTable(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // Auxiliary hash table, space complexity is O(n)\n    val dic = HashMap&lt;Int, Int&gt;()\n    // Single loop, time complexity is O(n)\n    for (i in 0..&lt;size) {\n        if (dic.containsKey(target - nums[i])) {\n            return intArrayOf(dic[target - nums[i]]!!, i)\n        }\n        dic[nums[i]] = i\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>### Method 2: Auxiliary hash table ###\ndef two_sum_hash_table(nums, target)\n  # Auxiliary hash table, space complexity is O(n)\n  dic = {}\n  # Single loop, time complexity is O(n)\n  for i in 0...nums.length\n    return [dic[target - nums[i]], i] if dic.has_key?(target - nums[i])\n\n    dic[nums[i]] = i\n  end\n\n  []\nend\n</code></pre> <p>This method reduces the time complexity from \\(O(n^2)\\) to \\(O(n)\\) through hash-based search, greatly improving runtime efficiency.</p> <p>Since an additional hash table needs to be maintained, the space complexity is \\(O(n)\\). Nevertheless, this method offers a more balanced overall time-space trade-off, making it the optimal solution to this problem.</p>","path":["Chapter 10. Searching","10.4   Hash Optimization Strategy"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/","level":1,"title":"10.5   Searching Algorithms Revisited","text":"<p>Searching algorithms are used to search for one or a group of elements that meet specific conditions in data structures (such as arrays, linked lists, trees, or graphs).</p> <p>Searching algorithms can be divided into the following two categories based on their implementation approach:</p> <ul> <li>Locating target elements by traversing the data structure, such as traversing arrays, linked lists, trees, and graphs.</li> <li>Achieving efficient element lookup by leveraging the way data is organized or prior information about the data, such as binary search, hash-based search, and binary search tree search.</li> </ul> <p>As these topics have already been introduced in earlier chapters, searching algorithms should already be familiar to us. In this section, we revisit them from a more systematic perspective.</p>","path":["Chapter 10. Searching","10.5   Searching Algorithms Revisited"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/#1051-brute-force-search","level":2,"title":"10.5.1   Brute-Force Search","text":"<p>Brute-force search locates target elements by traversing each element of the data structure.</p> <ul> <li>\"Linear search\" is applicable to linear data structures such as arrays and linked lists. It starts from one end of the data structure and accesses elements one by one until the target element is found or the other end is reached without finding the target element.</li> <li>\"Breadth-first search\" and \"depth-first search\" are two traversal strategies for graphs and trees. Breadth-first search starts from the initial node and searches layer by layer, visiting nodes from near to far. Depth-first search starts from the initial node, follows a path to the end, then backtracks and tries other paths until the entire data structure is traversed.</li> </ul> <p>The advantage of brute-force search is that it is simple and has good generality, requiring no data preprocessing or additional data structures.</p> <p>However, the time complexity of such algorithms is \\(O(n)\\), where \\(n\\) is the number of elements, so performance is poor when dealing with large amounts of data.</p>","path":["Chapter 10. Searching","10.5   Searching Algorithms Revisited"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/#1052-adaptive-search","level":2,"title":"10.5.2   Adaptive Search","text":"<p>Adaptive search leverages properties of the data itself (such as sorted order) to optimize the search process and locate target elements more efficiently.</p> <ul> <li>\"Binary search\" uses the orderliness of data to achieve efficient searching, applicable only to arrays.</li> <li>\"Hash-based search\" uses hash tables to store searchable data as key-value pairs, thereby enabling efficient queries.</li> <li>\"Tree search\" operates on specific tree structures (such as binary search trees), quickly ruling out nodes by comparing node values to locate the target element.</li> </ul> <p>The advantage of such algorithms is high efficiency, with time complexity reaching \\(O(\\log n)\\) or even \\(O(1)\\).</p> <p>However, using these algorithms often requires data preprocessing. For example, binary search requires pre-sorting the array, while hash-based search and tree search both require additional data structures, and maintaining these data structures also requires extra time and space overhead.</p> <p>Tip</p> <p>Adaptive search algorithms are often called lookup algorithms, mainly used to quickly retrieve target elements in specific data structures.</p>","path":["Chapter 10. Searching","10.5   Searching Algorithms Revisited"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/#1053-search-method-selection","level":2,"title":"10.5.3   Search Method Selection","text":"<p>Given a dataset of size \\(n\\), we can use linear search, binary search, tree search, hash-based search, and other methods to search for the target element. The working principles of each method are shown in Figure 10-11.</p> <p></p> <p> Figure 10-11   Multiple search strategies </p> <p>The efficiency and characteristics of these methods are summarized in Table 10-1.</p> <p> Table 10-1   Comparison of search algorithm efficiency </p> Linear search Binary search Tree search Hash-based search Search element \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) Insert element \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) Delete element \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) Extra space \\(O(1)\\) \\(O(1)\\) \\(O(n)\\) \\(O(n)\\) Data preprocessing / Sorting \\(O(n \\log n)\\) Tree building \\(O(n \\log n)\\) Hash table building \\(O(n)\\) Data ordered Unordered Ordered Ordered Unordered <p>The choice of search algorithm also depends on data volume, search performance requirements, data query and update frequency, etc.</p> <p>Linear search</p> <ul> <li>Good generality, requiring no data preprocessing operations. If we need to query the data only once, the preprocessing required by the other three methods can take longer than the linear search itself.</li> <li>Suitable for small data volumes, where time complexity has less impact on efficiency.</li> <li>Suitable for scenarios with high data update frequency, as this method does not require any additional data maintenance.</li> </ul> <p>Binary search</p> <ul> <li>Suitable for large datasets, with stable performance and a worst-case time complexity of \\(O(\\log n)\\).</li> <li>Data volume cannot be too large, as storing arrays requires contiguous memory space.</li> <li>Not suitable for scenarios with frequent data insertion and deletion, as maintaining a sorted array has high overhead.</li> </ul> <p>Hash-based search</p> <ul> <li>Suitable for scenarios with high query performance requirements, with an average time complexity of \\(O(1)\\).</li> <li>Not suitable for scenarios requiring ordered data or range searches, as hash tables cannot maintain the data in sorted order.</li> <li>High dependence on hash functions and hash collision handling strategies, with significant risk of performance degradation.</li> <li>Not suitable for excessively large data volumes, as hash tables require extra space to minimize collisions and thus provide good query performance.</li> </ul> <p>Tree search</p> <ul> <li>Suitable for massive datasets, as tree nodes are stored non-contiguously in memory.</li> <li>Suitable for scenarios that require maintaining ordered data or performing range searches.</li> <li>During continuous node insertion and deletion, binary search trees may become skewed, degrading time complexity to \\(O(n)\\).</li> <li>If AVL trees or red-black trees are used, all operations can consistently run in \\(O(\\log n)\\) time, though maintaining tree balance adds extra overhead.</li> </ul>","path":["Chapter 10. Searching","10.5   Searching Algorithms Revisited"],"tags":[]},{"location":"chapter_searching/summary/","level":1,"title":"10.6   Summary","text":"","path":["Chapter 10. Searching","10.6   Summary"],"tags":[]},{"location":"chapter_searching/summary/#1-key-review","level":3,"title":"1.   Key Review","text":"<ul> <li>Binary search relies on ordered data and searches by repeatedly halving the search interval. It requires the input data to be sorted and applies only to arrays or array-based data structures.</li> <li>Brute-force search locates data by traversing the data structure. Linear search applies to arrays and linked lists, while breadth-first search and depth-first search apply to graphs and trees. These algorithms are broadly applicable and require no data preprocessing, but their relatively high time complexity is \\(O(n)\\).</li> <li>Hash-based search, tree search, and binary search are efficient search methods that can quickly locate target elements in specific data structures. Such algorithms are highly efficient with time complexity reaching \\(O(\\log n)\\) or even \\(O(1)\\), but typically require additional data structures.</li> <li>In practice, we need to analyze factors such as data scale, search performance requirements, and data query and update frequency to choose the appropriate search method.</li> <li>Linear search is suitable for small datasets or data that is updated frequently; binary search is suitable for large sorted datasets; hash-based search is suitable when high query efficiency is required and range queries are unnecessary; tree search is suitable for large dynamic datasets that must maintain order and support range queries.</li> <li>Replacing linear search with hash-based search is a commonly used strategy to optimize runtime, reducing time complexity from \\(O(n)\\) to \\(O(1)\\).</li> </ul>","path":["Chapter 10. Searching","10.6   Summary"],"tags":[]},{"location":"chapter_sorting/","level":1,"title":"Chapter 11.   Sorting","text":"<p>Abstract</p> <p>Sorting is like a magic key that transforms chaos into order, enabling us to understand and process data more efficiently.</p> <p>From simple ascending order to more complex classification schemes, sorting reveals the harmonious beauty of data.</p>","path":["Chapter 11. Sorting","Chapter 11.   Sorting"],"tags":[]},{"location":"chapter_sorting/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>11.1   Sorting Algorithm</li> <li>11.2   Selection Sort</li> <li>11.3   Bubble Sort</li> <li>11.4   Insertion Sort</li> <li>11.5   Quick Sort</li> <li>11.6   Merge Sort</li> <li>11.7   Heap Sort</li> <li>11.8   Bucket Sort</li> <li>11.9   Counting Sort</li> <li>11.10   Radix Sort</li> <li>11.11   Summary</li> <li>11.12   Exercises</li> </ul>","path":["Chapter 11. Sorting","Chapter 11.   Sorting"],"tags":[]},{"location":"chapter_sorting/bubble_sort/","level":1,"title":"11.3   Bubble Sort","text":"<p>Bubble sort sorts an array by continuously comparing and swapping adjacent elements. This process resembles bubbles rising from the bottom to the top, hence the name bubble sort.</p> <p>As shown in Figure 11-4, the bubbling process can be simulated using element swaps: starting from the leftmost end of the array and traversing to the right, compare each pair of adjacent elements, and if \"left element &gt; right element\", swap them. After the traversal is complete, the largest element is moved to the rightmost end of the array.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 11-4   Simulating bubble sort using element swaps </p>","path":["Chapter 11. Sorting","11.3   Bubble Sort"],"tags":[]},{"location":"chapter_sorting/bubble_sort/#1131-algorithm-flow","level":2,"title":"11.3.1   Algorithm Flow","text":"<p>Assume the array has length \\(n\\). The steps of bubble sort are shown in Figure 11-5.</p> <ol> <li>First, perform \"bubbling\" on \\(n\\) elements, swapping the largest element of the array to its correct position.</li> <li>Next, perform \"bubbling\" on the remaining \\(n - 1\\) elements, swapping the second largest element to its correct position.</li> <li>And so on. After \\(n - 1\\) rounds of \"bubbling\", the largest \\(n - 1\\) elements have all been swapped to their correct positions.</li> <li>The only remaining element must be the smallest element, requiring no sorting, so the array sorting is complete.</li> </ol> <p></p> <p> Figure 11-5   Bubble sort flow </p> <p>Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby bubble_sort.py<pre><code>def bubble_sort(nums: list[int]):\n    \"\"\"Bubble sort\"\"\"\n    n = len(nums)\n    # Outer loop: unsorted interval is [0, i]\n    for i in range(n - 1, 0, -1):\n        # Inner loop: swap the largest element in the unsorted interval [0, i] to the rightmost end of the interval\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # Swap nums[j] and nums[j + 1]\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n</code></pre> bubble_sort.cpp<pre><code>/* Bubble sort */\nvoid bubbleSort(vector&lt;int&gt; &amp;nums) {\n    // Outer loop: unsorted range is [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                // Using std::swap() function here\n                swap(nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* Bubble sort */\nvoid bubbleSort(int[] nums) {\n    // Outer loop: unsorted range is [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* Bubble sort */\nvoid BubbleSort(int[] nums) {\n    // Outer loop: unsorted range is [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* Bubble sort */\nfunc bubbleSort(nums []int) {\n    // Outer loop: unsorted range is [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // Swap nums[j] and nums[j + 1]\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* Bubble sort */\nfunc bubbleSort(nums: inout [Int]) {\n    // Outer loop: unsorted range is [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // Swap nums[j] and nums[j + 1]\n                nums.swapAt(j, j + 1)\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* Bubble sort */\nfunction bubbleSort(nums) {\n    // Outer loop: unsorted range is [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* Bubble sort */\nfunction bubbleSort(nums: number[]): void {\n    // Outer loop: unsorted range is [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* Bubble sort */\nvoid bubbleSort(List&lt;int&gt; nums) {\n  // Outer loop: unsorted range is [0, i]\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // Swap nums[j] and nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n      }\n    }\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* Bubble sort */\nfn bubble_sort(nums: &amp;mut [i32]) {\n    // Outer loop: unsorted range is [0, i]\n    for i in (1..nums.len()).rev() {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // Swap nums[j] and nums[j + 1]\n                nums.swap(j, j + 1);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* Bubble sort */\nvoid bubbleSort(int nums[], int size) {\n    // Outer loop: unsorted range is [0, i]\n    for (int i = size - 1; i &gt; 0; i--) {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* Bubble sort */\nfun bubbleSort(nums: IntArray) {\n    // Outer loop: unsorted range is [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>### Bubble sort ###\ndef bubble_sort(nums)\n  n = nums.length\n  # Outer loop: unsorted range is [0, i]\n  for i in (n - 1).downto(1)\n    # Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # Swap nums[j] and nums[j + 1]\n        nums[j], nums[j + 1] = nums[j + 1], nums[j]\n      end\n    end\n  end\nend\n</code></pre>","path":["Chapter 11. Sorting","11.3   Bubble Sort"],"tags":[]},{"location":"chapter_sorting/bubble_sort/#1132-efficiency-optimization","level":2,"title":"11.3.2   Efficiency Optimization","text":"<p>We can observe that if no swaps occur during a round of \"bubbling\", the array is already sorted and the algorithm can return immediately. Therefore, we can add a flag <code>flag</code> to detect this situation and terminate as soon as it occurs.</p> <p>After this optimization, the worst-case and average-case time complexities of bubble sort remain \\(O(n^2)\\); however, when the input array is already sorted, the best-case time complexity becomes \\(O(n)\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby bubble_sort.py<pre><code>def bubble_sort_with_flag(nums: list[int]):\n    \"\"\"Bubble sort (flag optimization)\"\"\"\n    n = len(nums)\n    # Outer loop: unsorted interval is [0, i]\n    for i in range(n - 1, 0, -1):\n        flag = False  # Initialize flag\n        # Inner loop: swap the largest element in the unsorted interval [0, i] to the rightmost end of the interval\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # Swap nums[j] and nums[j + 1]\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n                flag = True  # Record element swap\n        if not flag:\n            break  # No elements were swapped in this round of \"bubbling\", exit directly\n</code></pre> bubble_sort.cpp<pre><code>/* Bubble sort (flag optimization)*/\nvoid bubbleSortWithFlag(vector&lt;int&gt; &amp;nums) {\n    // Outer loop: unsorted range is [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        bool flag = false; // Initialize flag\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                // Using std::swap() function here\n                swap(nums[j], nums[j + 1]);\n                flag = true; // Record element swap\n            }\n        }\n        if (!flag)\n            break; // No elements were swapped in this round of \"bubbling\", exit directly\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* Bubble sort (flag optimization) */\nvoid bubbleSortWithFlag(int[] nums) {\n    // Outer loop: unsorted range is [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        boolean flag = false; // Initialize flag\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // Record element swap\n            }\n        }\n        if (!flag)\n            break; // No elements were swapped in this round of \"bubbling\", exit directly\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* Bubble sort (flag optimization) */\nvoid BubbleSortWithFlag(int[] nums) {\n    // Outer loop: unsorted range is [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        bool flag = false; // Initialize flag\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                flag = true;  // Record element swap\n            }\n        }\n        if (!flag) break;     // No elements were swapped in this round of \"bubbling\", exit directly\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* Bubble sort (flag optimization) */\nfunc bubbleSortWithFlag(nums []int) {\n    // Outer loop: unsorted range is [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        flag := false // Initialize flag\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // Swap nums[j] and nums[j + 1]\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n                flag = true // Record element swap\n            }\n        }\n        if flag == false { // No elements were swapped in this round of \"bubbling\", exit directly\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* Bubble sort (flag optimization) */\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n    // Outer loop: unsorted range is [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        var flag = false // Initialize flag\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // Swap nums[j] and nums[j + 1]\n                nums.swapAt(j, j + 1)\n                flag = true // Record element swap\n            }\n        }\n        if !flag { // No elements were swapped in this round of \"bubbling\", exit directly\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* Bubble sort (flag optimization) */\nfunction bubbleSortWithFlag(nums) {\n    // Outer loop: unsorted range is [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // Initialize flag\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // Record element swap\n            }\n        }\n        if (!flag) break; // No elements were swapped in this round of \"bubbling\", exit directly\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* Bubble sort (flag optimization) */\nfunction bubbleSortWithFlag(nums: number[]): void {\n    // Outer loop: unsorted range is [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // Initialize flag\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // Record element swap\n            }\n        }\n        if (!flag) break; // No elements were swapped in this round of \"bubbling\", exit directly\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* Bubble sort (flag optimization) */\nvoid bubbleSortWithFlag(List&lt;int&gt; nums) {\n  // Outer loop: unsorted range is [0, i]\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    bool flag = false; // Initialize flag\n    // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // Swap nums[j] and nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        flag = true; // Record element swap\n      }\n    }\n    if (!flag) break; // No elements were swapped in this round of \"bubbling\", exit directly\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* Bubble sort (flag optimization) */\nfn bubble_sort_with_flag(nums: &amp;mut [i32]) {\n    // Outer loop: unsorted range is [0, i]\n    for i in (1..nums.len()).rev() {\n        let mut flag = false; // Initialize flag\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // Swap nums[j] and nums[j + 1]\n                nums.swap(j, j + 1);\n                flag = true; // Record element swap\n            }\n        }\n        if !flag {\n            break; // No elements were swapped in this round of \"bubbling\", exit directly\n        };\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* Bubble sort (flag optimization) */\nvoid bubbleSortWithFlag(int nums[], int size) {\n    // Outer loop: unsorted range is [0, i]\n    for (int i = size - 1; i &gt; 0; i--) {\n        bool flag = false;\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n                flag = true;\n            }\n        }\n        if (!flag)\n            break;\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* Bubble sort (flag optimization) */\nfun bubbleSortWithFlag(nums: IntArray) {\n    // Outer loop: unsorted range is [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        var flag = false // Initialize flag\n        // Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n                flag = true // Record element swap\n            }\n        }\n        if (!flag) break // No elements were swapped in this round of \"bubbling\", exit directly\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>### Bubble sort (flag optimization) ###\ndef bubble_sort_with_flag(nums)\n  n = nums.length\n  # Outer loop: unsorted range is [0, i]\n  for i in (n - 1).downto(1)\n    flag = false # Initialize flag\n\n    # Inner loop: swap the largest element in the unsorted range [0, i] to the rightmost end of that range\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # Swap nums[j] and nums[j + 1]\n        nums[j], nums[j + 1] = nums[j + 1], nums[j]\n        flag = true # Record element swap\n      end\n    end\n\n    break unless flag # No elements were swapped in this round of \"bubbling\", exit directly\n  end\nend\n</code></pre>","path":["Chapter 11. Sorting","11.3   Bubble Sort"],"tags":[]},{"location":"chapter_sorting/bubble_sort/#1133-algorithm-characteristics","level":2,"title":"11.3.3   Algorithm Characteristics","text":"<ul> <li>Time complexity is \\(O(n^2)\\); adaptive: In successive rounds of \"bubbling\", the traversed portion of the array has lengths \\(n - 1\\), \\(n - 2\\), \\(\\dots\\), \\(2\\), \\(1\\), for a total of \\((n - 1) n / 2\\). After introducing the <code>flag</code> optimization, the best-case time complexity can reach \\(O(n)\\).</li> <li>Space complexity of \\(O(1)\\), in-place sorting: Pointers \\(i\\) and \\(j\\) use a constant amount of extra space.</li> <li>Stable sorting: Equal elements are not swapped during \"bubbling\".</li> </ul>","path":["Chapter 11. Sorting","11.3   Bubble Sort"],"tags":[]},{"location":"chapter_sorting/bucket_sort/","level":1,"title":"11.8   Bucket Sort","text":"<p>The sorting algorithms discussed earlier are all comparison-based sorting algorithms, which sort by comparing the relative order of elements. The worst-case time complexity of such algorithms has a lower bound of \\(\\Omega(n \\log n)\\). Next, we will explore several non-comparison sorting algorithms, whose time complexity can be linear.</p> <p>Bucket sort is a typical application of the divide-and-conquer strategy. It works by creating a sequence of ordered buckets, each corresponding to a data range, and distributing the data evenly among them. The elements within each bucket are then sorted separately. Finally, all buckets are merged in order.</p>","path":["Chapter 11. Sorting","11.8   Bucket Sort"],"tags":[]},{"location":"chapter_sorting/bucket_sort/#1181-algorithm-flow","level":2,"title":"11.8.1   Algorithm Flow","text":"<p>Consider an array of length \\(n\\), whose elements are floating-point numbers in the range \\([0, 1)\\). The flow of bucket sort is shown in Figure 11-13.</p> <ol> <li>Initialize \\(k\\) buckets and distribute the \\(n\\) elements into the \\(k\\) buckets.</li> <li>Sort each bucket separately (here we use the built-in sorting function of the programming language).</li> <li>Merge the results in order from smallest to largest bucket.</li> </ol> <p></p> <p> Figure 11-13   Bucket sort algorithm flow </p> <p>The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby bucket_sort.py<pre><code>def bucket_sort(nums: list[float]):\n    \"\"\"Bucket sort\"\"\"\n    # Initialize k = n/2 buckets, expected to allocate 2 elements per bucket\n    k = len(nums) // 2\n    buckets = [[] for _ in range(k)]\n    # 1. Distribute array elements into various buckets\n    for num in nums:\n        # Input data range is [0, 1), use num * k to map to index range [0, k-1]\n        i = int(num * k)\n        # Add num to bucket i\n        buckets[i].append(num)\n    # 2. Sort each bucket\n    for bucket in buckets:\n        # Use built-in sorting function, can also replace with other sorting algorithms\n        bucket.sort()\n    # 3. Traverse buckets to merge results\n    i = 0\n    for bucket in buckets:\n        for num in bucket:\n            nums[i] = num\n            i += 1\n</code></pre> bucket_sort.cpp<pre><code>/* Bucket sort */\nvoid bucketSort(vector&lt;float&gt; &amp;nums) {\n    // Initialize k = n/2 buckets, expected to allocate 2 elements per bucket\n    int k = nums.size() / 2;\n    vector&lt;vector&lt;float&gt;&gt; buckets(k);\n    // 1. Distribute array elements into various buckets\n    for (float num : nums) {\n        // Input data range is [0, 1), use num * k to map to index range [0, k-1]\n        int i = num * k;\n        // Add num to bucket bucket_idx\n        buckets[i].push_back(num);\n    }\n    // 2. Sort each bucket\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        // Use built-in sorting function, can also replace with other sorting algorithms\n        sort(bucket.begin(), bucket.end());\n    }\n    // 3. Traverse buckets to merge results\n    int i = 0;\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.java<pre><code>/* Bucket sort */\nvoid bucketSort(float[] nums) {\n    // Initialize k = n/2 buckets, expected to allocate 2 elements per bucket\n    int k = nums.length / 2;\n    List&lt;List&lt;Float&gt;&gt; buckets = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; k; i++) {\n        buckets.add(new ArrayList&lt;&gt;());\n    }\n    // 1. Distribute array elements into various buckets\n    for (float num : nums) {\n        // Input data range is [0, 1), use num * k to map to index range [0, k-1]\n        int i = (int) (num * k);\n        // Add num to bucket i\n        buckets.get(i).add(num);\n    }\n    // 2. Sort each bucket\n    for (List&lt;Float&gt; bucket : buckets) {\n        // Use built-in sorting function, can also replace with other sorting algorithms\n        Collections.sort(bucket);\n    }\n    // 3. Traverse buckets to merge results\n    int i = 0;\n    for (List&lt;Float&gt; bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.cs<pre><code>/* Bucket sort */\nvoid BucketSort(float[] nums) {\n    // Initialize k = n/2 buckets, expected to allocate 2 elements per bucket\n    int k = nums.Length / 2;\n    List&lt;List&lt;float&gt;&gt; buckets = [];\n    for (int i = 0; i &lt; k; i++) {\n        buckets.Add([]);\n    }\n    // 1. Distribute array elements into various buckets\n    foreach (float num in nums) {\n        // Input data range is [0, 1), use num * k to map to index range [0, k-1]\n        int i = (int)(num * k);\n        // Add num to bucket i\n        buckets[i].Add(num);\n    }\n    // 2. Sort each bucket\n    foreach (List&lt;float&gt; bucket in buckets) {\n        // Use built-in sorting function, can also replace with other sorting algorithms\n        bucket.Sort();\n    }\n    // 3. Traverse buckets to merge results\n    int j = 0;\n    foreach (List&lt;float&gt; bucket in buckets) {\n        foreach (float num in bucket) {\n            nums[j++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.go<pre><code>/* Bucket sort */\nfunc bucketSort(nums []float64) {\n    // Initialize k = n/2 buckets, expected to allocate 2 elements per bucket\n    k := len(nums) / 2\n    buckets := make([][]float64, k)\n    for i := 0; i &lt; k; i++ {\n        buckets[i] = make([]float64, 0)\n    }\n    // 1. Distribute array elements into various buckets\n    for _, num := range nums {\n        // Input data range is [0, 1), use num * k to map to index range [0, k-1]\n        i := int(num * float64(k))\n        // Add num to bucket i\n        buckets[i] = append(buckets[i], num)\n    }\n    // 2. Sort each bucket\n    for i := 0; i &lt; k; i++ {\n        // Use built-in slice sorting function, can also be replaced with other sorting algorithms\n        sort.Float64s(buckets[i])\n    }\n    // 3. Traverse buckets to merge results\n    i := 0\n    for _, bucket := range buckets {\n        for _, num := range bucket {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> bucket_sort.swift<pre><code>/* Bucket sort */\nfunc bucketSort(nums: inout [Double]) {\n    // Initialize k = n/2 buckets, expected to allocate 2 elements per bucket\n    let k = nums.count / 2\n    var buckets = (0 ..&lt; k).map { _ in [Double]() }\n    // 1. Distribute array elements into various buckets\n    for num in nums {\n        // Input data range is [0, 1), use num * k to map to index range [0, k-1]\n        let i = Int(num * Double(k))\n        // Add num to bucket i\n        buckets[i].append(num)\n    }\n    // 2. Sort each bucket\n    for i in buckets.indices {\n        // Use built-in sorting function, can also replace with other sorting algorithms\n        buckets[i].sort()\n    }\n    // 3. Traverse buckets to merge results\n    var i = nums.startIndex\n    for bucket in buckets {\n        for num in bucket {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> bucket_sort.js<pre><code>/* Bucket sort */\nfunction bucketSort(nums) {\n    // Initialize k = n/2 buckets, expected to allocate 2 elements per bucket\n    const k = nums.length / 2;\n    const buckets = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. Distribute array elements into various buckets\n    for (const num of nums) {\n        // Input data range is [0, 1), use num * k to map to index range [0, k-1]\n        const i = Math.floor(num * k);\n        // Add num to bucket i\n        buckets[i].push(num);\n    }\n    // 2. Sort each bucket\n    for (const bucket of buckets) {\n        // Use built-in sorting function, can also replace with other sorting algorithms\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. Traverse buckets to merge results\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.ts<pre><code>/* Bucket sort */\nfunction bucketSort(nums: number[]): void {\n    // Initialize k = n/2 buckets, expected to allocate 2 elements per bucket\n    const k = nums.length / 2;\n    const buckets: number[][] = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. Distribute array elements into various buckets\n    for (const num of nums) {\n        // Input data range is [0, 1), use num * k to map to index range [0, k-1]\n        const i = Math.floor(num * k);\n        // Add num to bucket i\n        buckets[i].push(num);\n    }\n    // 2. Sort each bucket\n    for (const bucket of buckets) {\n        // Use built-in sorting function, can also replace with other sorting algorithms\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. Traverse buckets to merge results\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.dart<pre><code>/* Bucket sort */\nvoid bucketSort(List&lt;double&gt; nums) {\n  // Initialize k = n/2 buckets, expected to allocate 2 elements per bucket\n  int k = nums.length ~/ 2;\n  List&lt;List&lt;double&gt;&gt; buckets = List.generate(k, (index) =&gt; []);\n\n  // 1. Distribute array elements into various buckets\n  for (double _num in nums) {\n    // Input data range is [0, 1), use _num * k to map to index range [0, k-1]\n    int i = (_num * k).toInt();\n    // Add _num to bucket bucket_idx\n    buckets[i].add(_num);\n  }\n  // 2. Sort each bucket\n  for (List&lt;double&gt; bucket in buckets) {\n    bucket.sort();\n  }\n  // 3. Traverse buckets to merge results\n  int i = 0;\n  for (List&lt;double&gt; bucket in buckets) {\n    for (double _num in bucket) {\n      nums[i++] = _num;\n    }\n  }\n}\n</code></pre> bucket_sort.rs<pre><code>/* Bucket sort */\nfn bucket_sort(nums: &amp;mut [f64]) {\n    // Initialize k = n/2 buckets, expected to allocate 2 elements per bucket\n    let k = nums.len() / 2;\n    let mut buckets = vec![vec![]; k];\n    // 1. Distribute array elements into various buckets\n    for &amp;num in nums.iter() {\n        // Input data range is [0, 1), use num * k to map to index range [0, k-1]\n        let i = (num * k as f64) as usize;\n        // Add num to bucket i\n        buckets[i].push(num);\n    }\n    // 2. Sort each bucket\n    for bucket in &amp;mut buckets {\n        // Use built-in sorting function, can also replace with other sorting algorithms\n        bucket.sort_by(|a, b| a.partial_cmp(b).unwrap());\n    }\n    // 3. Traverse buckets to merge results\n    let mut i = 0;\n    for bucket in buckets.iter() {\n        for &amp;num in bucket.iter() {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> bucket_sort.c<pre><code>/* Bucket sort */\nvoid bucketSort(float nums[], int n) {\n    int k = n / 2;                                 // Initialize k = n/2 buckets\n    int *sizes = malloc(k * sizeof(int));          // Record each bucket's size\n    float **buckets = malloc(k * sizeof(float *)); // Array of dynamic arrays (buckets)\n    // Pre-allocate sufficient space for each bucket\n    for (int i = 0; i &lt; k; ++i) {\n        buckets[i] = (float *)malloc(n * sizeof(float));\n        sizes[i] = 0;\n    }\n    // 1. Distribute array elements into various buckets\n    for (int i = 0; i &lt; n; ++i) {\n        int idx = (int)(nums[i] * k);\n        buckets[idx][sizes[idx]++] = nums[i];\n    }\n    // 2. Sort each bucket\n    for (int i = 0; i &lt; k; ++i) {\n        qsort(buckets[i], sizes[i], sizeof(float), compare);\n    }\n    // 3. Merge sorted buckets\n    int idx = 0;\n    for (int i = 0; i &lt; k; ++i) {\n        for (int j = 0; j &lt; sizes[i]; ++j) {\n            nums[idx++] = buckets[i][j];\n        }\n        // Free memory\n        free(buckets[i]);\n    }\n}\n</code></pre> bucket_sort.kt<pre><code>/* Bucket sort */\nfun bucketSort(nums: FloatArray) {\n    // Initialize k = n/2 buckets, expected to allocate 2 elements per bucket\n    val k = nums.size / 2\n    val buckets = mutableListOf&lt;MutableList&lt;Float&gt;&gt;()\n    for (i in 0..&lt;k) {\n        buckets.add(mutableListOf())\n    }\n    // 1. Distribute array elements into various buckets\n    for (num in nums) {\n        // Input data range is [0, 1), use num * k to map to index range [0, k-1]\n        val i = (num * k).toInt()\n        // Add num to bucket i\n        buckets[i].add(num)\n    }\n    // 2. Sort each bucket\n    for (bucket in buckets) {\n        // Use built-in sorting function, can also replace with other sorting algorithms\n        bucket.sort()\n    }\n    // 3. Traverse buckets to merge results\n    var i = 0\n    for (bucket in buckets) {\n        for (num in bucket) {\n            nums[i++] = num\n        }\n    }\n}\n</code></pre> bucket_sort.rb<pre><code>### Bucket sort ###\ndef bucket_sort(nums)\n  # Initialize k = n/2 buckets, expected to allocate 2 elements per bucket\n  k = nums.length / 2\n  buckets = Array.new(k) { [] }\n\n  # 1. Distribute array elements into various buckets\n  nums.each do |num|\n    # Input data range is [0, 1), use num * k to map to index range [0, k-1]\n    i = (num * k).to_i\n    # Add num to bucket i\n    buckets[i] &lt;&lt; num\n  end\n\n  # 2. Sort each bucket\n  buckets.each do |bucket|\n    # Use built-in sorting function, can also replace with other sorting algorithms\n    bucket.sort!\n  end\n\n  # 3. Traverse buckets to merge results\n  i = 0\n  buckets.each do |bucket|\n    bucket.each do |num|\n      nums[i] = num\n      i += 1\n    end\n  end\nend\n</code></pre>","path":["Chapter 11. Sorting","11.8   Bucket Sort"],"tags":[]},{"location":"chapter_sorting/bucket_sort/#1182-algorithm-characteristics","level":2,"title":"11.8.2   Algorithm Characteristics","text":"<p>Bucket sort is suitable for processing very large datasets. For example, suppose the input contains 1 million elements, and limited memory prevents the system from loading all of them at once. In that case, the data can be divided into 1000 buckets, each bucket can be sorted separately, and the results can then be merged.</p> <ul> <li>Time complexity is \\(O(n + k)\\): Assuming the elements are evenly distributed across the buckets, each bucket contains \\(\\frac{n}{k}\\) elements. If sorting a single bucket takes \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) time, then sorting all buckets takes \\(O(n \\log\\frac{n}{k})\\) time. When the number of buckets \\(k\\) is relatively large, the time complexity approaches \\(O(n)\\). Merging the results requires traversing all buckets and elements, which takes \\(O(n + k)\\) time. In the worst case, all data is placed into a single bucket, and sorting that bucket takes \\(O(n^2)\\) time.</li> <li>Space complexity is \\(O(n + k)\\), and bucket sort is not in-place: It requires extra space for \\(k\\) buckets and a total of \\(n\\) elements.</li> <li>Whether bucket sort is stable depends on whether the algorithm for sorting elements within buckets is stable.</li> </ul>","path":["Chapter 11. Sorting","11.8   Bucket Sort"],"tags":[]},{"location":"chapter_sorting/bucket_sort/#1183-how-to-achieve-even-distribution","level":2,"title":"11.8.3   How to Achieve Even Distribution","text":"<p>In theory, bucket sort can achieve \\(O(n)\\) time complexity. The key is to distribute the elements evenly across the buckets, because real-world data is often not uniformly distributed. For example, suppose we want to divide all products on Taobao evenly into 10 buckets by price range, but the price distribution is uneven: there are many products priced below 100 yuan and very few priced above 1000 yuan. If the price range is divided evenly into 10 intervals, the numbers of products in the buckets will differ greatly.</p> <p>To achieve a more even distribution, we can first choose a rough boundary and partition the data into 3 buckets. After that, buckets containing more products can be further divided into 3 buckets until the numbers of elements in all buckets are roughly equal.</p> <p>As shown in Figure 11-14, this method essentially builds a recursion tree whose goal is to make the leaf nodes as balanced as possible. Of course, the data does not have to be split into 3 buckets in every round; the specific partitioning strategy can be chosen flexibly based on the characteristics of the data.</p> <p></p> <p> Figure 11-14   Recursively dividing buckets </p> <p>If we know the probability distribution of product prices in advance, we can set the price boundaries for each bucket according to that distribution. Notably, the data distribution does not need to be measured exactly; it can also be approximated with a probability model chosen to fit the characteristics of the data.</p> <p>As shown in Figure 11-15, we assume that product prices follow a normal distribution, which allows us to reasonably set price intervals to evenly distribute products to each bucket.</p> <p></p> <p> Figure 11-15   Dividing buckets based on probability distribution </p>","path":["Chapter 11. Sorting","11.8   Bucket Sort"],"tags":[]},{"location":"chapter_sorting/counting_sort/","level":1,"title":"11.9   Counting Sort","text":"<p>Counting sort sorts by counting the occurrences of elements and is typically applied to integer arrays.</p>","path":["Chapter 11. Sorting","11.9   Counting Sort"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1191-simple-implementation","level":2,"title":"11.9.1   Simple Implementation","text":"<p>Let's start with a simple example. Given an array <code>nums</code> of length \\(n\\), where the elements are all \"non-negative integers\", the overall flow of counting sort is shown in Figure 11-16.</p> <ol> <li>Traverse the array to find the largest number, denoted as \\(m\\), and then create an auxiliary array <code>counter</code> of length \\(m + 1\\).</li> <li>Use <code>counter</code> to count how many times each number appears in <code>nums</code>, where <code>counter[num]</code> stores the number of occurrences of <code>num</code>. This is simple: traverse <code>nums</code> (denote the current number by <code>num</code>) and increment <code>counter[num]</code> by \\(1\\) each time.</li> <li>Because the indices of <code>counter</code> are naturally ordered, the numbers are effectively already sorted. Next, traverse <code>counter</code> and write the numbers back into <code>nums</code> in ascending order according to their occurrence counts.</li> </ol> <p></p> <p> Figure 11-16   Counting sort flow </p> <p>The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby counting_sort.py<pre><code>def counting_sort_naive(nums: list[int]):\n    \"\"\"Counting sort\"\"\"\n    # Simple implementation, cannot be used for sorting objects\n    # 1. Count the maximum element m in the array\n    m = 0\n    for num in nums:\n        m = max(m, num)\n    # 2. Count the occurrence of each number\n    # counter[num] represents the occurrence of num\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. Traverse counter, filling each element back into the original array nums\n    i = 0\n    for num in range(m + 1):\n        for _ in range(counter[num]):\n            nums[i] = num\n            i += 1\n</code></pre> counting_sort.cpp<pre><code>/* Counting sort */\n// Simple implementation, cannot be used for sorting objects\nvoid countingSortNaive(vector&lt;int&gt; &amp;nums) {\n    // 1. Count the maximum element m in the array\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. Traverse counter, filling each element back into the original array nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.java<pre><code>/* Counting sort */\n// Simple implementation, cannot be used for sorting objects\nvoid countingSortNaive(int[] nums) {\n    // 1. Count the maximum element m in the array\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. Traverse counter, filling each element back into the original array nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* Counting sort */\n// Simple implementation, cannot be used for sorting objects\nvoid CountingSortNaive(int[] nums) {\n    // 1. Count the maximum element m in the array\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. Traverse counter, filling each element back into the original array nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* Counting sort */\n// Simple implementation, cannot be used for sorting objects\nfunc countingSortNaive(nums []int) {\n    // 1. Count the maximum element m in the array\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. Traverse counter, filling each element back into the original array nums\n    for i, num := 0, 0; num &lt; m+1; num++ {\n        for j := 0; j &lt; counter[num]; j++ {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.swift<pre><code>/* Counting sort */\n// Simple implementation, cannot be used for sorting objects\nfunc countingSortNaive(nums: inout [Int]) {\n    // 1. Count the maximum element m in the array\n    let m = nums.max()!\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. Traverse counter, filling each element back into the original array nums\n    var i = 0\n    for num in 0 ..&lt; m + 1 {\n        for _ in 0 ..&lt; counter[num] {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* Counting sort */\n// Simple implementation, cannot be used for sorting objects\nfunction countingSortNaive(nums) {\n    // 1. Count the maximum element m in the array\n    let m = Math.max(...nums);\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. Traverse counter, filling each element back into the original array nums\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* Counting sort */\n// Simple implementation, cannot be used for sorting objects\nfunction countingSortNaive(nums: number[]): void {\n    // 1. Count the maximum element m in the array\n    let m: number = Math.max(...nums);\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. Traverse counter, filling each element back into the original array nums\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* Counting sort */\n// Simple implementation, cannot be used for sorting objects\nvoid countingSortNaive(List&lt;int&gt; nums) {\n  // 1. Count the maximum element m in the array\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. Count the occurrence of each number\n  // counter[_num] represents occurrence count of _num\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. Traverse counter, filling each element back into the original array nums\n  int i = 0;\n  for (int _num = 0; _num &lt; m + 1; _num++) {\n    for (int j = 0; j &lt; counter[_num]; j++, i++) {\n      nums[i] = _num;\n    }\n  }\n}\n</code></pre> counting_sort.rs<pre><code>/* Counting sort */\n// Simple implementation, cannot be used for sorting objects\nfn counting_sort_naive(nums: &amp;mut [i32]) {\n    // 1. Count the maximum element m in the array\n    let m = *nums.iter().max().unwrap();\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    let mut counter = vec![0; m as usize + 1];\n    for &amp;num in nums.iter() {\n        counter[num as usize] += 1;\n    }\n    // 3. Traverse counter, filling each element back into the original array nums\n    let mut i = 0;\n    for num in 0..m + 1 {\n        for _ in 0..counter[num as usize] {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> counting_sort.c<pre><code>/* Counting sort */\n// Simple implementation, cannot be used for sorting objects\nvoid countingSortNaive(int nums[], int size) {\n    // 1. Count the maximum element m in the array\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    int *counter = calloc(m + 1, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. Traverse counter, filling each element back into the original array nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n    // 4. Free memory\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* Counting sort */\n// Simple implementation, cannot be used for sorting objects\nfun countingSortNaive(nums: IntArray) {\n    // 1. Count the maximum element m in the array\n    var m = 0\n    for (num in nums) {\n        m = max(m, num)\n    }\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. Traverse counter, filling each element back into the original array nums\n    var i = 0\n    for (num in 0..&lt;m + 1) {\n        var j = 0\n        while (j &lt; counter[num]) {\n            nums[i] = num\n            j++\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.rb<pre><code>### Counting sort ###\ndef counting_sort_naive(nums)\n  # Simple implementation, cannot be used for sorting objects\n  # 1. Count the maximum element m in the array\n  m = 0\n  nums.each { |num| m = [m, num].max }\n  # 2. Count the occurrence of each number\n  # counter[num] represents the occurrence of num\n  counter = Array.new(m + 1, 0)\n  nums.each { |num| counter[num] += 1 }\n  # 3. Traverse counter, filling each element back into the original array nums\n  i = 0\n  for num in 0...(m + 1)\n    (0...counter[num]).each do\n      nums[i] = num\n      i += 1\n    end\n  end\nend\n</code></pre> <p>Connection between counting sort and bucket sort</p> <p>From the perspective of bucket sort, each index of the counting array <code>counter</code> can be viewed as a bucket, and the counting process can be seen as distributing elements into their corresponding buckets. Essentially, counting sort is a special case of bucket sort for integer data.</p>","path":["Chapter 11. Sorting","11.9   Counting Sort"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1192-complete-implementation","level":2,"title":"11.9.2   Complete Implementation","text":"<p>Observant readers may have noticed that if the input consists of objects, step <code>3.</code> above no longer works. Suppose the input consists of product objects and we want to sort them by price (a member variable of the class); the above algorithm can only produce the sorted order of the prices themselves.</p> <p>So how can we obtain the sorted order of the original data? We first compute the prefix sums of <code>counter</code>. As the name suggests, the prefix sum at index <code>i</code>, <code>prefix[i]</code>, equals the sum of the elements from index <code>0</code> through <code>i</code>:</p> \\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\] <p>The prefix sum has a clear interpretation: <code>prefix[num] - 1</code> gives the index of the last occurrence of element <code>num</code> in the result array <code>res</code>. This information is crucial because it tells us where each element should be placed in the result array. Next, we traverse the original array <code>nums</code> in reverse, and for each element <code>num</code>, perform the following two steps.</p> <ol> <li>Place <code>num</code> at index <code>prefix[num] - 1</code> of the array <code>res</code>.</li> <li>Decrease the prefix sum <code>prefix[num]</code> by \\(1\\) to get the index for the next placement of <code>num</code>.</li> </ol> <p>After the traversal is complete, the array <code>res</code> contains the sorted result, and finally <code>res</code> is used to overwrite the original array <code>nums</code>. The complete counting sort flow is shown in Figure 11-17.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 11-17   Counting sort steps </p> <p>The counting sort implementation is shown below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby counting_sort.py<pre><code>def counting_sort(nums: list[int]):\n    \"\"\"Counting sort\"\"\"\n    # Complete implementation, can sort objects and is a stable sort\n    # 1. Count the maximum element m in the array\n    m = max(nums)\n    # 2. Count the occurrence of each number\n    # counter[num] represents the occurrence of num\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. Calculate the prefix sum of counter, converting \"occurrence count\" to \"tail index\"\n    # counter[num]-1 is the last index where num appears in res\n    for i in range(m):\n        counter[i + 1] += counter[i]\n    # 4. Traverse nums in reverse order, placing each element into the result array res\n    # Initialize the array res to record results\n    n = len(nums)\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        num = nums[i]\n        res[counter[num] - 1] = num  # Place num at the corresponding index\n        counter[num] -= 1  # Decrement the prefix sum by 1, getting the next index to place num\n    # Use result array res to overwrite the original array nums\n    for i in range(n):\n        nums[i] = res[i]\n</code></pre> counting_sort.cpp<pre><code>/* Counting sort */\n// Complete implementation, can sort objects and is a stable sort\nvoid countingSort(vector&lt;int&gt; &amp;nums) {\n    // 1. Count the maximum element m in the array\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. Calculate the prefix sum of counter, converting \"occurrence count\" to \"tail index\"\n    // counter[num]-1 is the last index where num appears in res\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. Traverse nums in reverse order, placing each element into the result array res\n    // Initialize the array res to record results\n    int n = nums.size();\n    vector&lt;int&gt; res(n);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // Place num at the corresponding index\n        counter[num]--;              // Decrement the prefix sum by 1, getting the next index to place num\n    }\n    // Use result array res to overwrite the original array nums\n    nums = res;\n}\n</code></pre> counting_sort.java<pre><code>/* Counting sort */\n// Complete implementation, can sort objects and is a stable sort\nvoid countingSort(int[] nums) {\n    // 1. Count the maximum element m in the array\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. Calculate the prefix sum of counter, converting \"occurrence count\" to \"tail index\"\n    // counter[num]-1 is the last index where num appears in res\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. Traverse nums in reverse order, placing each element into the result array res\n    // Initialize the array res to record results\n    int n = nums.length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // Place num at the corresponding index\n        counter[num]--; // Decrement the prefix sum by 1, getting the next index to place num\n    }\n    // Use result array res to overwrite the original array nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* Counting sort */\n// Complete implementation, can sort objects and is a stable sort\nvoid CountingSort(int[] nums) {\n    // 1. Count the maximum element m in the array\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. Calculate the prefix sum of counter, converting \"occurrence count\" to \"tail index\"\n    // counter[num]-1 is the last index where num appears in res\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. Traverse nums in reverse order, placing each element into the result array res\n    // Initialize the array res to record results\n    int n = nums.Length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // Place num at the corresponding index\n        counter[num]--; // Decrement the prefix sum by 1, getting the next index to place num\n    }\n    // Use result array res to overwrite the original array nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* Counting sort */\n// Complete implementation, can sort objects and is a stable sort\nfunc countingSort(nums []int) {\n    // 1. Count the maximum element m in the array\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. Calculate the prefix sum of counter, converting \"occurrence count\" to \"tail index\"\n    // counter[num]-1 is the last index where num appears in res\n    for i := 0; i &lt; m; i++ {\n        counter[i+1] += counter[i]\n    }\n    // 4. Traverse nums in reverse order, placing each element into the result array res\n    // Initialize the array res to record results\n    n := len(nums)\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        num := nums[i]\n        // Place num at the corresponding index\n        res[counter[num]-1] = num\n        // Decrement the prefix sum by 1, getting the next index to place num\n        counter[num]--\n    }\n    // Use result array res to overwrite the original array nums\n    copy(nums, res)\n}\n</code></pre> counting_sort.swift<pre><code>/* Counting sort */\n// Complete implementation, can sort objects and is a stable sort\nfunc countingSort(nums: inout [Int]) {\n    // 1. Count the maximum element m in the array\n    let m = nums.max()!\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. Calculate the prefix sum of counter, converting \"occurrence count\" to \"tail index\"\n    // counter[num]-1 is the last index where num appears in res\n    for i in 0 ..&lt; m {\n        counter[i + 1] += counter[i]\n    }\n    // 4. Traverse nums in reverse order, placing each element into the result array res\n    // Initialize the array res to record results\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let num = nums[i]\n        res[counter[num] - 1] = num // Place num at the corresponding index\n        counter[num] -= 1 // Decrement the prefix sum by 1, getting the next index to place num\n    }\n    // Use result array res to overwrite the original array nums\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* Counting sort */\n// Complete implementation, can sort objects and is a stable sort\nfunction countingSort(nums) {\n    // 1. Count the maximum element m in the array\n    let m = Math.max(...nums);\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. Calculate the prefix sum of counter, converting \"occurrence count\" to \"tail index\"\n    // counter[num]-1 is the last index where num appears in res\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. Traverse nums in reverse order, placing each element into the result array res\n    // Initialize the array res to record results\n    const n = nums.length;\n    const res = new Array(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // Place num at the corresponding index\n        counter[num]--; // Decrement the prefix sum by 1, getting the next index to place num\n    }\n    // Use result array res to overwrite the original array nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* Counting sort */\n// Complete implementation, can sort objects and is a stable sort\nfunction countingSort(nums: number[]): void {\n    // 1. Count the maximum element m in the array\n    let m: number = Math.max(...nums);\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. Calculate the prefix sum of counter, converting \"occurrence count\" to \"tail index\"\n    // counter[num]-1 is the last index where num appears in res\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. Traverse nums in reverse order, placing each element into the result array res\n    // Initialize the array res to record results\n    const n = nums.length;\n    const res: number[] = new Array&lt;number&gt;(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // Place num at the corresponding index\n        counter[num]--; // Decrement the prefix sum by 1, getting the next index to place num\n    }\n    // Use result array res to overwrite the original array nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* Counting sort */\n// Complete implementation, can sort objects and is a stable sort\nvoid countingSort(List&lt;int&gt; nums) {\n  // 1. Count the maximum element m in the array\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. Count the occurrence of each number\n  // counter[_num] represents occurrence count of _num\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. Calculate the prefix sum of counter, converting \"occurrence count\" to \"tail index\"\n  // That is, counter[_num]-1 is the last occurrence index of _num in res\n  for (int i = 0; i &lt; m; i++) {\n    counter[i + 1] += counter[i];\n  }\n  // 4. Traverse nums in reverse order, placing each element into the result array res\n  // Initialize the array res to record results\n  int n = nums.length;\n  List&lt;int&gt; res = List.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int _num = nums[i];\n    res[counter[_num] - 1] = _num; // Place _num at corresponding index\n    counter[_num]--; // Decrement prefix sum by 1 to get next placement index for _num\n  }\n  // Use result array res to overwrite the original array nums\n  nums.setAll(0, res);\n}\n</code></pre> counting_sort.rs<pre><code>/* Counting sort */\n// Complete implementation, can sort objects and is a stable sort\nfn counting_sort(nums: &amp;mut [i32]) {\n    // 1. Count the maximum element m in the array\n    let m = *nums.iter().max().unwrap() as usize;\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    let mut counter = vec![0; m + 1];\n    for &amp;num in nums.iter() {\n        counter[num as usize] += 1;\n    }\n    // 3. Calculate the prefix sum of counter, converting \"occurrence count\" to \"tail index\"\n    // counter[num]-1 is the last index where num appears in res\n    for i in 0..m {\n        counter[i + 1] += counter[i];\n    }\n    // 4. Traverse nums in reverse order, placing each element into the result array res\n    // Initialize the array res to record results\n    let n = nums.len();\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let num = nums[i];\n        res[counter[num as usize] - 1] = num; // Place num at the corresponding index\n        counter[num as usize] -= 1; // Decrement the prefix sum by 1, getting the next index to place num\n    }\n    // Use result array res to overwrite the original array nums\n    nums.copy_from_slice(&amp;res)\n}\n</code></pre> counting_sort.c<pre><code>/* Counting sort */\n// Complete implementation, can sort objects and is a stable sort\nvoid countingSort(int nums[], int size) {\n    // 1. Count the maximum element m in the array\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    int *counter = calloc(m, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. Calculate the prefix sum of counter, converting \"occurrence count\" to \"tail index\"\n    // counter[num]-1 is the last index where num appears in res\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. Traverse nums in reverse order, placing each element into the result array res\n    // Initialize the array res to record results\n    int *res = malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // Place num at the corresponding index\n        counter[num]--;              // Decrement the prefix sum by 1, getting the next index to place num\n    }\n    // Use result array res to overwrite the original array nums\n    memcpy(nums, res, size * sizeof(int));\n    // 5. Free memory\n    free(res);\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* Counting sort */\n// Complete implementation, can sort objects and is a stable sort\nfun countingSort(nums: IntArray) {\n    // 1. Count the maximum element m in the array\n    var m = 0\n    for (num in nums) {\n        m = max(m, num)\n    }\n    // 2. Count the occurrence of each number\n    // counter[num] represents the occurrence of num\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. Calculate the prefix sum of counter, converting \"occurrence count\" to \"tail index\"\n    // counter[num]-1 is the last index where num appears in res\n    for (i in 0..&lt;m) {\n        counter[i + 1] += counter[i]\n    }\n    // 4. Traverse nums in reverse order, placing each element into the result array res\n    // Initialize the array res to record results\n    val n = nums.size\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val num = nums[i]\n        res[counter[num] - 1] = num // Place num at the corresponding index\n        counter[num]-- // Decrement the prefix sum by 1, getting the next index to place num\n    }\n    // Use result array res to overwrite the original array nums\n    for (i in 0..&lt;n) {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.rb<pre><code>### Counting sort ###\ndef counting_sort(nums)\n  # Complete implementation, can sort objects and is a stable sort\n  # 1. Count the maximum element m in the array\n  m = nums.max\n  # 2. Count the occurrence of each number\n  # counter[num] represents the occurrence of num\n  counter = Array.new(m + 1, 0)\n  nums.each { |num| counter[num] += 1 }\n  # 3. Calculate the prefix sum of counter, converting \"occurrence count\" to \"tail index\"\n  # counter[num]-1 is the last index where num appears in res\n  (0...m).each { |i| counter[i + 1] += counter[i] }\n  # 4. Traverse nums in reverse, fill elements into result array res\n  # Initialize the array res to record results\n  n = nums.length\n  res = Array.new(n, 0)\n  (n - 1).downto(0).each do |i|\n    num = nums[i]\n    res[counter[num] - 1] = num # Place num at the corresponding index\n    counter[num] -= 1 # Decrement the prefix sum by 1, getting the next index to place num\n  end\n  # Use result array res to overwrite the original array nums\n  (0...n).each { |i| nums[i] = res[i] }\nend\n</code></pre>","path":["Chapter 11. Sorting","11.9   Counting Sort"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1193-algorithm-characteristics","level":2,"title":"11.9.3   Algorithm Characteristics","text":"<ul> <li>Time complexity is \\(O(n + m)\\), and counting sort is non-adaptive: Traversing <code>nums</code> and <code>counter</code> both takes linear time. In general, when \\(n \\gg m\\), the time complexity approaches \\(O(n)\\).</li> <li>Space complexity of \\(O(n + m)\\), non-in-place sorting: Uses arrays <code>res</code> and <code>counter</code> of lengths \\(n\\) and \\(m\\) respectively.</li> <li>Stable sorting: Since elements are filled into <code>res</code> in a \"right-to-left\" order, traversing <code>nums</code> in reverse can avoid changing the relative positions of equal elements, thereby achieving stable sorting. In fact, traversing <code>nums</code> in forward order can also yield correct sorting results, but the result would be unstable.</li> </ul>","path":["Chapter 11. Sorting","11.9   Counting Sort"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1194-limitations","level":2,"title":"11.9.4   Limitations","text":"<p>At this point, you might think counting sort is quite ingenious because it achieves efficient sorting simply by counting occurrences. However, the prerequisites for using counting sort are fairly restrictive.</p> <p>Counting sort is only applicable to non-negative integers. To apply it to other types of data, you must ensure that they can be converted to non-negative integers without changing the relative ordering of the elements. For example, for an integer array containing negative numbers, you can first add a constant to every number to shift them into the non-negative range, and then shift them back after sorting.</p> <p>Counting sort is well suited to cases with many elements but a small value range. For example, in the above scenario, \\(m\\) cannot be too large; otherwise, it consumes too much space. And when \\(n \\ll m\\), counting sort takes \\(O(m)\\) time, which may be slower than sorting algorithms with \\(O(n \\log n)\\) time complexity.</p>","path":["Chapter 11. Sorting","11.9   Counting Sort"],"tags":[]},{"location":"chapter_sorting/exercises/","level":1,"title":"11.12   Exercises","text":"","path":["Chapter 11. Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/exercises/#11121-concept-review","level":2,"title":"11.12.1   Concept Review","text":"","path":["Chapter 11. Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/exercises/#1-the-first-few-rounds-of-selection-sort-and-bubble-sort","level":3,"title":"1.   The First Few Rounds of Selection Sort and Bubble Sort","text":"<p>Given the array <code>[4, 2, 5, 1, 3]</code>, sort it in ascending order in both parts below.</p> <ol> <li>Simulate the first two rounds of selection sort. Write the array after each round and identify which positions are now fixed.</li> <li>Simulate the first round of bubble sort. Write the resulting array and the number of swaps, and identify which position is now fixed.</li> </ol> Answer <ol> <li> <p>The first two rounds are:</p> Round Array Explanation 1 <code>[1, 2, 5, 4, 3]</code> The smallest element, 1, is swapped with the first element 2 <code>[1, 2, 5, 4, 3]</code> The value 2 is already at index 1, so no swap is needed <p>The first two positions are now fixed. Later rounds need to find the smallest element only within <code>[5, 4, 3]</code>.</p> </li> <li> <p>Compare adjacent elements in order: swap 4 and 2; do not swap 4 and 5; swap 5 and 1; then swap 5 and 3.     The result is <code>[2, 4, 1, 3, 5]</code>, after 3 swaps. The largest element, 5, has moved to the end of the array, so the last position is now fixed.</p> </li> </ol>","path":["Chapter 11. Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/exercises/#2-can-equal-elements-change-their-relative-order","level":3,"title":"2.   Can Equal Elements Change Their Relative Order?","text":"<p>In the array \\([2_a, 2_b, 1]\\), \\(2_a\\) and \\(2_b\\) have equal values, but their subscripts mark their original order.</p> <ol> <li>Write the array after the first round of selection sort. Has the relative order of \\(2_a\\) and \\(2_b\\) changed?</li> <li>Write the array after the first round of bubble sort. Has the relative order of \\(2_a\\) and \\(2_b\\) changed?</li> <li>Based on the first two questions, explain how the two sorting algorithms differ in preserving the original order of equal elements.</li> </ol> Answer <ol> <li> <p>In its first round, selection sort selects the smallest element, 1, and swaps it with the first element, \\(2_a\\), producing     \\([1, 2_b, 2_a]\\). The relative order has changed because \\(2_a\\) has moved behind \\(2_b\\).</p> </li> <li> <p>Bubble sort first compares \\(2_a\\) and \\(2_b\\). Because they are equal, it does not swap them. It then compares \\(2_b\\) and 1 and swaps them,     producing \\([2_a, 1, 2_b]\\) after the first round. \\(2_a\\) is still before \\(2_b\\), so their relative order has not changed.</p> </li> <li> <p>In this example, selection sort changes the original order of equal elements. Bubble sort swaps adjacent elements only when the left element is greater than the right one.     Equal elements are not swapped, so their original relative order is preserved.</p> </li> </ol>","path":["Chapter 11. Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/exercises/#3-compare-counting-sort-and-radix-sort","level":3,"title":"3.   Compare Counting Sort and Radix Sort","text":"<p>A school needs to sort many student ID numbers, each exactly 8 digits long. Answer the following questions:</p> <ol> <li>How many rounds does radix sort need when it starts with the least significant digit?</li> <li>If the student IDs are treated directly as integers for counting sort, why would the count array need many entries that are never used?</li> <li>Based on the first two questions, which would you choose for sorting many fixed-length 8-digit student IDs: counting sort or radix sort? Why?</li> </ol> Answer <ol> <li> <p>A student ID has 8 digits, so 8 rounds are needed from the least significant to the most significant digit. Each round groups values only by 0–9.</p> </li> <li> <p>Direct counting would require an entry for every possible 8-digit value, but only a small fraction of those values are actually assigned to students.     Most entries in the count array would remain 0.</p> </li> <li> <p>Radix sort is the better choice. It uses the facts that the length is fixed and each digit has only 10 possible values, requiring only 8 rounds of stable grouping.     Counting sort, if it used the entire 8-digit ID as an integer index, would require count-array entries for many values that never occur.</p> </li> </ol>","path":["Chapter 11. Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/exercises/#11122-programming-exercises","level":2,"title":"11.12.2   Programming Exercises","text":"","path":["Chapter 11. Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/exercises/#1-sort-an-array-with-merge-sort","level":3,"title":"1.   Sort an Array with Merge Sort","text":"<p>Given an integer array <code>nums</code>, implement merge sort yourself, arrange its elements in nondecreasing order, and return the result. Do not call the language's built-in sorting function.</p> Hints <ol> <li>An interval of length at most 1 is already sorted</li> <li>Divide the interval in half at its midpoint and recursively sort both halves</li> <li>Use two pointers to merge the two sorted halves, then write the result back into the original array</li> </ol> <p>LeetCode</p>","path":["Chapter 11. Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/exercises/#2-sort-an-integer-array-with-counting-sort","level":3,"title":"2.   Sort an Integer Array with Counting Sort","text":"<p>You are given an integer array <code>nums</code> and a non-negative integer \\(K\\). Every element of the array is between \\(0\\) and \\(K\\).</p> <p>Implement counting sort, write the result back into <code>nums</code> in nondecreasing order, and return <code>nums</code>. Do not determine the order by comparing elements, and do not call the language's built-in sorting function.</p> Hints <ol> <li>Because every element is between 0 and K, use each element's value directly as an index in the counting array</li> <li>Scan nums once and increment the count at the corresponding position</li> <li>Then scan the counting array from 0 to K; if the value x occurs a certain number of times, write x into nums that many consecutive times</li> </ol>","path":["Chapter 11. Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/heap_sort/","level":1,"title":"11.7   Heap Sort","text":"<p>Tip</p> <p>Before reading this section, please ensure you have completed the \"Heap\" chapter.</p> <p>Heap sort is an efficient sorting algorithm based on the heap data structure. We can implement heap sort using the heap construction and element removal operations introduced earlier.</p> <ol> <li>Input the array and build a min-heap, at which point the smallest element is at the heap top.</li> <li>Continuously perform element removal operations and record the removed elements in order to obtain a sequence sorted in ascending order.</li> </ol> <p>Although the above method is feasible, it requires an additional array to save the popped elements, which is quite wasteful of space. In practice, we usually use a more elegant implementation method.</p>","path":["Chapter 11. Sorting","11.7   Heap Sort"],"tags":[]},{"location":"chapter_sorting/heap_sort/#1171-algorithm-flow","level":2,"title":"11.7.1   Algorithm Flow","text":"<p>Assume the array length is \\(n\\). The flow of heap sort is shown in Figure 11-12.</p> <ol> <li>Input the array and build a max-heap. After completion, the largest element is at the heap top.</li> <li>Swap the heap top element (first element) with the heap bottom element (last element). After the swap is complete, reduce the heap length by \\(1\\) and increase the count of sorted elements by \\(1\\).</li> <li>Starting from the heap top element, perform a top-to-bottom heapify operation (sift down). After heapify is complete, the heap property is restored.</li> <li>Repeat steps <code>2.</code> and <code>3.</code> After \\(n - 1\\) rounds, the array is sorted.</li> </ol> <p>Tip</p> <p>In fact, the element removal operation also includes steps <code>2.</code> and <code>3.</code>, with the additional step of removing the element.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 11-12   Heap sort steps </p> <p>In the code below, we use the same <code>sift_down()</code> function for top-to-bottom heapify as in the \"Heap\" chapter. It is worth noting that since the heap length decreases as the largest element is extracted, we need to add a length parameter \\(n\\) to <code>sift_down()</code> to specify the current effective length of the heap. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby heap_sort.py<pre><code>def sift_down(nums: list[int], n: int, i: int):\n    \"\"\"Heap length is n, start heapifying node i, from top to bottom\"\"\"\n    while True:\n        # Determine the largest node among i, l, r, noted as ma\n        l = 2 * i + 1\n        r = 2 * i + 2\n        ma = i\n        if l &lt; n and nums[l] &gt; nums[ma]:\n            ma = l\n        if r &lt; n and nums[r] &gt; nums[ma]:\n            ma = r\n        # If node i is the largest or indices l, r are out of bounds, no further heapification needed, break\n        if ma == i:\n            break\n        # Swap two nodes\n        nums[i], nums[ma] = nums[ma], nums[i]\n        # Loop downwards heapification\n        i = ma\n\ndef heap_sort(nums: list[int]):\n    \"\"\"Heap sort\"\"\"\n    # Build heap operation: heapify all nodes except leaves\n    for i in range(len(nums) // 2 - 1, -1, -1):\n        sift_down(nums, len(nums), i)\n    # Extract the largest element from the heap and repeat for n-1 rounds\n    for i in range(len(nums) - 1, 0, -1):\n        # Swap the root node with the rightmost leaf node (swap the first element with the last element)\n        nums[0], nums[i] = nums[i], nums[0]\n        # Start heapifying the root node, from top to bottom\n        sift_down(nums, i, 0)\n</code></pre> heap_sort.cpp<pre><code>/* Heap length is n, start heapifying node i, from top to bottom */\nvoid siftDown(vector&lt;int&gt; &amp;nums, int n, int i) {\n    while (true) {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // Swap two nodes\n        if (ma == i) {\n            break;\n        }\n        // Swap two nodes\n        swap(nums[i], nums[ma]);\n        // Loop downwards heapification\n        i = ma;\n    }\n}\n\n/* Heap sort */\nvoid heapSort(vector&lt;int&gt; &amp;nums) {\n    // Build heap operation: heapify all nodes except leaves\n    for (int i = nums.size() / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, nums.size(), i);\n    }\n    // Extract the largest element from the heap and repeat for n-1 rounds\n    for (int i = nums.size() - 1; i &gt; 0; --i) {\n        // Delete node\n        swap(nums[0], nums[i]);\n        // Start heapifying the root node, from top to bottom\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.java<pre><code>/* Heap length is n, start heapifying node i, from top to bottom */\nvoid siftDown(int[] nums, int n, int i) {\n    while (true) {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // Swap two nodes\n        if (ma == i)\n            break;\n        // Swap two nodes\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // Loop downwards heapification\n        i = ma;\n    }\n}\n\n/* Heap sort */\nvoid heapSort(int[] nums) {\n    // Build heap operation: heapify all nodes except leaves\n    for (int i = nums.length / 2 - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // Extract the largest element from the heap and repeat for n-1 rounds\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // Delete node\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // Start heapifying the root node, from top to bottom\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.cs<pre><code>/* Heap length is n, start heapifying node i, from top to bottom */\nvoid SiftDown(int[] nums, int n, int i) {\n    while (true) {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // Swap two nodes\n        if (ma == i)\n            break;\n        // Swap two nodes\n        (nums[ma], nums[i]) = (nums[i], nums[ma]);\n        // Loop downwards heapification\n        i = ma;\n    }\n}\n\n/* Heap sort */\nvoid HeapSort(int[] nums) {\n    // Build heap operation: heapify all nodes except leaves\n    for (int i = nums.Length / 2 - 1; i &gt;= 0; i--) {\n        SiftDown(nums, nums.Length, i);\n    }\n    // Extract the largest element from the heap and repeat for n-1 rounds\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // Delete node\n        (nums[i], nums[0]) = (nums[0], nums[i]);\n        // Start heapifying the root node, from top to bottom\n        SiftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.go<pre><code>/* Heap length is n, start heapifying node i, from top to bottom */\nfunc siftDown(nums *[]int, n, i int) {\n    for true {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        l := 2*i + 1\n        r := 2*i + 2\n        ma := i\n        if l &lt; n &amp;&amp; (*nums)[l] &gt; (*nums)[ma] {\n            ma = l\n        }\n        if r &lt; n &amp;&amp; (*nums)[r] &gt; (*nums)[ma] {\n            ma = r\n        }\n        // Swap two nodes\n        if ma == i {\n            break\n        }\n        // Swap two nodes\n        (*nums)[i], (*nums)[ma] = (*nums)[ma], (*nums)[i]\n        // Loop downwards heapification\n        i = ma\n    }\n}\n\n/* Heap sort */\nfunc heapSort(nums *[]int) {\n    // Build heap operation: heapify all nodes except leaves\n    for i := len(*nums)/2 - 1; i &gt;= 0; i-- {\n        siftDown(nums, len(*nums), i)\n    }\n    // Extract the largest element from the heap and repeat for n-1 rounds\n    for i := len(*nums) - 1; i &gt; 0; i-- {\n        // Delete node\n        (*nums)[0], (*nums)[i] = (*nums)[i], (*nums)[0]\n        // Start heapifying the root node, from top to bottom\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.swift<pre><code>/* Heap length is n, start heapifying node i, from top to bottom */\nfunc siftDown(nums: inout [Int], n: Int, i: Int) {\n    var i = i\n    while true {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        let l = 2 * i + 1\n        let r = 2 * i + 2\n        var ma = i\n        if l &lt; n, nums[l] &gt; nums[ma] {\n            ma = l\n        }\n        if r &lt; n, nums[r] &gt; nums[ma] {\n            ma = r\n        }\n        // Swap two nodes\n        if ma == i {\n            break\n        }\n        // Swap two nodes\n        nums.swapAt(i, ma)\n        // Loop downwards heapification\n        i = ma\n    }\n}\n\n/* Heap sort */\nfunc heapSort(nums: inout [Int]) {\n    // Build heap operation: heapify all nodes except leaves\n    for i in stride(from: nums.count / 2 - 1, through: 0, by: -1) {\n        siftDown(nums: &amp;nums, n: nums.count, i: i)\n    }\n    // Extract the largest element from the heap and repeat for n-1 rounds\n    for i in nums.indices.dropFirst().reversed() {\n        // Delete node\n        nums.swapAt(0, i)\n        // Start heapifying the root node, from top to bottom\n        siftDown(nums: &amp;nums, n: i, i: 0)\n    }\n}\n</code></pre> heap_sort.js<pre><code>/* Heap length is n, start heapifying node i, from top to bottom */\nfunction siftDown(nums, n, i) {\n    while (true) {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // Swap two nodes\n        if (ma === i) {\n            break;\n        }\n        // Swap two nodes\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // Loop downwards heapification\n        i = ma;\n    }\n}\n\n/* Heap sort */\nfunction heapSort(nums) {\n    // Build heap operation: heapify all nodes except leaves\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // Extract the largest element from the heap and repeat for n-1 rounds\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // Delete node\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // Start heapifying the root node, from top to bottom\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.ts<pre><code>/* Heap length is n, start heapifying node i, from top to bottom */\nfunction siftDown(nums: number[], n: number, i: number): void {\n    while (true) {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // Swap two nodes\n        if (ma === i) {\n            break;\n        }\n        // Swap two nodes\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // Loop downwards heapification\n        i = ma;\n    }\n}\n\n/* Heap sort */\nfunction heapSort(nums: number[]): void {\n    // Build heap operation: heapify all nodes except leaves\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // Extract the largest element from the heap and repeat for n-1 rounds\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // Delete node\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // Start heapifying the root node, from top to bottom\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.dart<pre><code>/* Heap length is n, start heapifying node i, from top to bottom */\nvoid siftDown(List&lt;int&gt; nums, int n, int i) {\n  while (true) {\n    // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n    int l = 2 * i + 1;\n    int r = 2 * i + 2;\n    int ma = i;\n    if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) ma = l;\n    if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) ma = r;\n    // Swap two nodes\n    if (ma == i) break;\n    // Swap two nodes\n    int temp = nums[i];\n    nums[i] = nums[ma];\n    nums[ma] = temp;\n    // Loop downwards heapification\n    i = ma;\n  }\n}\n\n/* Heap sort */\nvoid heapSort(List&lt;int&gt; nums) {\n  // Build heap operation: heapify all nodes except leaves\n  for (int i = nums.length ~/ 2 - 1; i &gt;= 0; i--) {\n    siftDown(nums, nums.length, i);\n  }\n  // Extract the largest element from the heap and repeat for n-1 rounds\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // Delete node\n    int tmp = nums[0];\n    nums[0] = nums[i];\n    nums[i] = tmp;\n    // Start heapifying the root node, from top to bottom\n    siftDown(nums, i, 0);\n  }\n}\n</code></pre> heap_sort.rs<pre><code>/* Heap length is n, start heapifying node i, from top to bottom */\nfn sift_down(nums: &amp;mut [i32], n: usize, mut i: usize) {\n    loop {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let mut ma = i;\n        if l &lt; n &amp;&amp; nums[l] &gt; nums[ma] {\n            ma = l;\n        }\n        if r &lt; n &amp;&amp; nums[r] &gt; nums[ma] {\n            ma = r;\n        }\n        // Swap two nodes\n        if ma == i {\n            break;\n        }\n        // Swap two nodes\n        nums.swap(i, ma);\n        // Loop downwards heapification\n        i = ma;\n    }\n}\n\n/* Heap sort */\nfn heap_sort(nums: &amp;mut [i32]) {\n    // Build heap operation: heapify all nodes except leaves\n    for i in (0..nums.len() / 2).rev() {\n        sift_down(nums, nums.len(), i);\n    }\n    // Extract the largest element from the heap and repeat for n-1 rounds\n    for i in (1..nums.len()).rev() {\n        // Delete node\n        nums.swap(0, i);\n        // Start heapifying the root node, from top to bottom\n        sift_down(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.c<pre><code>/* Heap length is n, start heapifying node i, from top to bottom */\nvoid siftDown(int nums[], int n, int i) {\n    while (1) {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // Swap two nodes\n        if (ma == i) {\n            break;\n        }\n        // Swap two nodes\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // Loop downwards heapification\n        i = ma;\n    }\n}\n\n/* Heap sort */\nvoid heapSort(int nums[], int n) {\n    // Build heap operation: heapify all nodes except leaves\n    for (int i = n / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, n, i);\n    }\n    // Extract the largest element from the heap and repeat for n-1 rounds\n    for (int i = n - 1; i &gt; 0; --i) {\n        // Delete node\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // Start heapifying the root node, from top to bottom\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.kt<pre><code>/* Heap length is n, start heapifying node i, from top to bottom */\nfun siftDown(nums: IntArray, n: Int, li: Int) {\n    var i = li\n    while (true) {\n        // If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n        val l = 2 * i + 1\n        val r = 2 * i + 2\n        var ma = i\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) \n            ma = l\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) \n            ma = r\n        // Swap two nodes\n        if (ma == i) \n            break\n        // Swap two nodes\n        val temp = nums[i]\n        nums[i] = nums[ma]\n        nums[ma] = temp\n        // Loop downwards heapification\n        i = ma\n    }\n}\n\n/* Heap sort */\nfun heapSort(nums: IntArray) {\n    // Build heap operation: heapify all nodes except leaves\n    for (i in nums.size / 2 - 1 downTo 0) {\n        siftDown(nums, nums.size, i)\n    }\n    // Extract the largest element from the heap and repeat for n-1 rounds\n    for (i in nums.size - 1 downTo 1) {\n        // Delete node\n        val temp = nums[0]\n        nums[0] = nums[i]\n        nums[i] = temp\n        // Start heapifying the root node, from top to bottom\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.rb<pre><code>### Heap length is n, heapify from node i, top to bottom ###\ndef sift_down(nums, n, i)\n  while true\n    # If node i is largest or indices l, r are out of bounds, no need to continue heapify, break\n    l = 2 * i + 1\n    r = 2 * i + 2\n    ma = i\n    ma = l if l &lt; n &amp;&amp; nums[l] &gt; nums[ma]\n    ma = r if r &lt; n &amp;&amp; nums[r] &gt; nums[ma]\n    # Swap two nodes\n    break if ma == i\n    # Swap two nodes\n    nums[i], nums[ma] = nums[ma], nums[i]\n    # Loop downwards heapification\n    i = ma\n  end\nend\n\n### Heap sort ###\ndef heap_sort(nums)\n  # Build heap operation: heapify all nodes except leaves\n  (nums.length / 2 - 1).downto(0) do |i|\n    sift_down(nums, nums.length, i)\n  end\n  # Extract the largest element from the heap and repeat for n-1 rounds\n  (nums.length - 1).downto(1) do |i|\n    # Delete node\n    nums[0], nums[i] = nums[i], nums[0]\n    # Start heapifying the root node, from top to bottom\n    sift_down(nums, i, 0)\n  end\nend\n</code></pre>","path":["Chapter 11. Sorting","11.7   Heap Sort"],"tags":[]},{"location":"chapter_sorting/heap_sort/#1172-algorithm-characteristics","level":2,"title":"11.7.2   Algorithm Characteristics","text":"<ul> <li>Time complexity is \\(O(n \\log n)\\); heap sort is non-adaptive: Heap construction takes \\(O(n)\\) time. Extracting the largest element from the heap takes \\(O(\\log n)\\) time, and this is repeated for a total of \\(n - 1\\) rounds.</li> <li>Space complexity is \\(O(1)\\); heap sort is in-place: A few pointer variables use \\(O(1)\\) space. Element swapping and heapify are both performed on the original array.</li> <li>Unstable sorting: When swapping the heap top element and heap bottom element, the relative positions of equal elements may change.</li> </ul>","path":["Chapter 11. Sorting","11.7   Heap Sort"],"tags":[]},{"location":"chapter_sorting/insertion_sort/","level":1,"title":"11.4   Insertion Sort","text":"<p>Insertion sort is a simple sorting algorithm that works very similarly to the process of manually sorting a deck of cards.</p> <p>Specifically, we select a base element from the unsorted portion, compare it one by one with the elements in the sorted portion to its left, and insert it into the correct position.</p> <p>Figure 11-6 illustrates how an element is inserted into an array. Let the base element be <code>base</code>. We need to shift all elements between the target index and <code>base</code> one position to the right, and then assign <code>base</code> to the target index.</p> <p></p> <p> Figure 11-6   Single insertion operation </p>","path":["Chapter 11. Sorting","11.4   Insertion Sort"],"tags":[]},{"location":"chapter_sorting/insertion_sort/#1141-algorithm-flow","level":2,"title":"11.4.1   Algorithm Flow","text":"<p>The overall flow of insertion sort is shown in Figure 11-7.</p> <ol> <li>Initially, the first element of the array is already sorted.</li> <li>Select the second element of the array as <code>base</code>, and after inserting it into the correct position, the first 2 elements of the array are sorted.</li> <li>Select the third element as <code>base</code>, and after inserting it into the correct position, the first 3 elements of the array are sorted.</li> <li>And so on. In the last round, select the last element as <code>base</code>, and after inserting it into the correct position, all elements are sorted.</li> </ol> <p></p> <p> Figure 11-7   Insertion sort flow </p> <p>Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby insertion_sort.py<pre><code>def insertion_sort(nums: list[int]):\n    \"\"\"Insertion sort\"\"\"\n    # Outer loop: sorted interval is [0, i-1]\n    for i in range(1, len(nums)):\n        base = nums[i]\n        j = i - 1\n        # Inner loop: insert base into the correct position within the sorted interval [0, i-1]\n        while j &gt;= 0 and nums[j] &gt; base:\n            nums[j + 1] = nums[j]  # Move nums[j] to the right by one position\n            j -= 1\n        nums[j + 1] = base  # Assign base to the correct position\n</code></pre> insertion_sort.cpp<pre><code>/* Insertion sort */\nvoid insertionSort(vector&lt;int&gt; &amp;nums) {\n    // Outer loop: sorted interval is [0, i-1]\n    for (int i = 1; i &lt; nums.size(); i++) {\n        int base = nums[i], j = i - 1;\n        // Inner loop: insert base into the correct position within the sorted interval [0, i-1]\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // Move nums[j] to the right by one position\n            j--;\n        }\n        nums[j + 1] = base; // Assign base to the correct position\n    }\n}\n</code></pre> insertion_sort.java<pre><code>/* Insertion sort */\nvoid insertionSort(int[] nums) {\n    // Outer loop: sorted interval is [0, i-1]\n    for (int i = 1; i &lt; nums.length; i++) {\n        int base = nums[i], j = i - 1;\n        // Inner loop: insert base into the correct position within the sorted interval [0, i-1]\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // Move nums[j] to the right by one position\n            j--;\n        }\n        nums[j + 1] = base;        // Assign base to the correct position\n    }\n}\n</code></pre> insertion_sort.cs<pre><code>/* Insertion sort */\nvoid InsertionSort(int[] nums) {\n    // Outer loop: sorted interval is [0, i-1]\n    for (int i = 1; i &lt; nums.Length; i++) {\n        int bas = nums[i], j = i - 1;\n        // Inner loop: insert base into the correct position within the sorted interval [0, i-1]\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; bas) {\n            nums[j + 1] = nums[j]; // Move nums[j] to the right by one position\n            j--;\n        }\n        nums[j + 1] = bas;         // Assign base to the correct position\n    }\n}\n</code></pre> insertion_sort.go<pre><code>/* Insertion sort */\nfunc insertionSort(nums []int) {\n    // Outer loop: sorted interval is [0, i-1]\n    for i := 1; i &lt; len(nums); i++ {\n        base := nums[i]\n        j := i - 1\n        // Inner loop: insert base into the correct position within the sorted interval [0, i-1]\n        for j &gt;= 0 &amp;&amp; nums[j] &gt; base {\n            nums[j+1] = nums[j] // Move nums[j] to the right by one position\n            j--\n        }\n        nums[j+1] = base // Assign base to the correct position\n    }\n}\n</code></pre> insertion_sort.swift<pre><code>/* Insertion sort */\nfunc insertionSort(nums: inout [Int]) {\n    // Outer loop: sorted interval is [0, i-1]\n    for i in nums.indices.dropFirst() {\n        let base = nums[i]\n        var j = i - 1\n        // Inner loop: insert base into the correct position within the sorted interval [0, i-1]\n        while j &gt;= 0, nums[j] &gt; base {\n            nums[j + 1] = nums[j] // Move nums[j] to the right by one position\n            j -= 1\n        }\n        nums[j + 1] = base // Assign base to the correct position\n    }\n}\n</code></pre> insertion_sort.js<pre><code>/* Insertion sort */\nfunction insertionSort(nums) {\n    // Outer loop: sorted interval is [0, i-1]\n    for (let i = 1; i &lt; nums.length; i++) {\n        let base = nums[i],\n            j = i - 1;\n        // Inner loop: insert base into the correct position within the sorted interval [0, i-1]\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // Move nums[j] to the right by one position\n            j--;\n        }\n        nums[j + 1] = base; // Assign base to the correct position\n    }\n}\n</code></pre> insertion_sort.ts<pre><code>/* Insertion sort */\nfunction insertionSort(nums: number[]): void {\n    // Outer loop: sorted interval is [0, i-1]\n    for (let i = 1; i &lt; nums.length; i++) {\n        const base = nums[i];\n        let j = i - 1;\n        // Inner loop: insert base into the correct position within the sorted interval [0, i-1]\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // Move nums[j] to the right by one position\n            j--;\n        }\n        nums[j + 1] = base; // Assign base to the correct position\n    }\n}\n</code></pre> insertion_sort.dart<pre><code>/* Insertion sort */\nvoid insertionSort(List&lt;int&gt; nums) {\n  // Outer loop: sorted interval is [0, i-1]\n  for (int i = 1; i &lt; nums.length; i++) {\n    int base = nums[i], j = i - 1;\n    // Inner loop: insert base into the correct position within the sorted interval [0, i-1]\n    while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n      nums[j + 1] = nums[j]; // Move nums[j] to the right by one position\n      j--;\n    }\n    nums[j + 1] = base; // Assign base to the correct position\n  }\n}\n</code></pre> insertion_sort.rs<pre><code>/* Insertion sort */\nfn insertion_sort(nums: &amp;mut [i32]) {\n    // Outer loop: sorted interval is [0, i-1]\n    for i in 1..nums.len() {\n        let (base, mut j) = (nums[i], (i - 1) as i32);\n        // Inner loop: insert base into the correct position within the sorted interval [0, i-1]\n        while j &gt;= 0 &amp;&amp; nums[j as usize] &gt; base {\n            nums[(j + 1) as usize] = nums[j as usize]; // Move nums[j] to the right by one position\n            j -= 1;\n        }\n        nums[(j + 1) as usize] = base; // Assign base to the correct position\n    }\n}\n</code></pre> insertion_sort.c<pre><code>/* Insertion sort */\nvoid insertionSort(int nums[], int size) {\n    // Outer loop: sorted interval is [0, i-1]\n    for (int i = 1; i &lt; size; i++) {\n        int base = nums[i], j = i - 1;\n        // Inner loop: insert base into the correct position within the sorted interval [0, i-1]\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            // Move nums[j] to the right by one position\n            nums[j + 1] = nums[j];\n            j--;\n        }\n        // Assign base to the correct position\n        nums[j + 1] = base;\n    }\n}\n</code></pre> insertion_sort.kt<pre><code>/* Insertion sort */\nfun insertionSort(nums: IntArray) {\n    // Outer loop: sorted elements are 1, 2, ..., n\n    for (i in nums.indices) {\n        val base = nums[i]\n        var j = i - 1\n        // Inner loop: insert base into the correct position within the sorted interval [0, i-1]\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j] // Move nums[j] to the right by one position\n            j--\n        }\n        nums[j + 1] = base        // Assign base to the correct position\n    }\n}\n</code></pre> insertion_sort.rb<pre><code>### Insertion sort ###\ndef insertion_sort(nums)\n  n = nums.length\n  # Outer loop: sorted interval is [0, i-1]\n  for i in 1...n\n    base = nums[i]\n    j = i - 1\n    # Inner loop: insert base into the correct position within the sorted interval [0, i-1]\n    while j &gt;= 0 &amp;&amp; nums[j] &gt; base\n      nums[j + 1] = nums[j] # Move nums[j] to the right by one position\n      j -= 1\n    end\n    nums[j + 1] = base # Assign base to the correct position\n  end\nend\n</code></pre>","path":["Chapter 11. Sorting","11.4   Insertion Sort"],"tags":[]},{"location":"chapter_sorting/insertion_sort/#1142-algorithm-characteristics","level":2,"title":"11.4.2   Algorithm Characteristics","text":"<ul> <li>Time complexity of \\(O(n^2)\\), adaptive sorting: In the worst case, the insertion operations require \\(n - 1\\), \\(n-2\\), \\(\\dots\\), \\(2\\), and \\(1\\) iterations, respectively, summing to \\((n - 1) n / 2\\), so the time complexity is \\(O(n^2)\\). When the data is already sorted, each insertion operation terminates early. When the input array is completely sorted, insertion sort achieves its best-case time complexity of \\(O(n)\\).</li> <li>Space complexity of \\(O(1)\\), in-place sorting: Pointers \\(i\\) and \\(j\\) use a constant amount of extra space.</li> <li>Stable sorting: During insertion, we place elements to the right of equal elements, so their relative order is unchanged.</li> </ul>","path":["Chapter 11. Sorting","11.4   Insertion Sort"],"tags":[]},{"location":"chapter_sorting/insertion_sort/#1143-advantages-of-insertion-sort","level":2,"title":"11.4.3   Advantages of Insertion Sort","text":"<p>The time complexity of insertion sort is \\(O(n^2)\\), while the time complexity of quick sort, which we will learn about next, is \\(O(n \\log n)\\). Although insertion sort has a higher time complexity, it is usually faster on small datasets.</p> <p>This conclusion is similar to the one about when linear search and binary search are applicable. Algorithms such as quick sort, with \\(O(n \\log n)\\) complexity, are divide-and-conquer sorting algorithms and often involve more primitive operations. When the dataset is small, the values of \\(n^2\\) and \\(n \\log n\\) are relatively close, so asymptotic complexity does not dominate; instead, the number of primitive operations per round becomes the deciding factor.</p> <p>In fact, the built-in sorting functions of many programming languages (such as Java) use insertion sort. The general idea is: for large arrays, use divide-and-conquer sorting algorithms such as quick sort; for short arrays, use insertion sort directly.</p> <p>Although bubble sort, selection sort, and insertion sort all have a time complexity of \\(O(n^2)\\), in actual situations, insertion sort is used significantly more frequently than bubble sort and selection sort, mainly for the following reasons.</p> <ul> <li>Bubble sort is implemented through element swaps, which require a temporary variable and involve 3 primitive operations; insertion sort is implemented through element assignment and requires only 1 primitive operation. Therefore, bubble sort usually has higher computational overhead than insertion sort.</li> <li>Selection sort has a time complexity of \\(O(n^2)\\) in any case. If given a set of partially ordered data, insertion sort is usually more efficient than selection sort.</li> <li>Selection sort is unstable and cannot be applied to multi-level sorting.</li> </ul>","path":["Chapter 11. Sorting","11.4   Insertion Sort"],"tags":[]},{"location":"chapter_sorting/merge_sort/","level":1,"title":"11.6   Merge Sort","text":"<p>Merge sort is a sorting algorithm based on a divide-and-conquer strategy, consisting of the \"divide\" and \"merge\" phases shown in Figure 11-10.</p> <ol> <li>Divide phase: Recursively split the array at the midpoint, reducing the problem of sorting a long array to the problem of sorting shorter arrays.</li> <li>Merge phase: When a sub-array has length 1, stop dividing and start merging, continuously combining the shorter sorted sub-arrays on the left and right into a longer sorted array until the process is complete.</li> </ol> <p></p> <p> Figure 11-10   Divide and merge phases of merge sort </p>","path":["Chapter 11. Sorting","11.6   Merge Sort"],"tags":[]},{"location":"chapter_sorting/merge_sort/#1161-algorithm-flow","level":2,"title":"11.6.1   Algorithm Flow","text":"<p>As shown in Figure 11-11, the \"divide phase\" recursively splits the array from the midpoint into two sub-arrays from top to bottom.</p> <ol> <li>Calculate the array midpoint <code>mid</code>, recursively divide the left sub-array (interval <code>[left, mid]</code>) and right sub-array (interval <code>[mid + 1, right]</code>).</li> <li>Repeat step <code>1.</code> recursively until a sub-array has length 1.</li> </ol> <p>The \"merge phase\" merges the left and right sub-arrays into a sorted array from bottom to top. Note that merging starts from sub-arrays of length 1, so every sub-array involved in this phase is already sorted.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 11-11   Merge sort steps </p> <p>The recursive order of merge sort is consistent with the post-order traversal of a binary tree.</p> <ul> <li>Post-order traversal: First recursively traverse the left subtree, then recursively traverse the right subtree, and finally process the root node.</li> <li>Merge sort: First recursively process the left sub-array, then recursively process the right sub-array, and finally perform the merge.</li> </ul> <p>The implementation of merge sort is shown in the code below. Note that the interval to be merged in <code>nums</code> is <code>[left, right]</code>, while the corresponding interval in <code>tmp</code> is <code>[0, right - left]</code>.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby merge_sort.py<pre><code>def merge(nums: list[int], left: int, mid: int, right: int):\n    \"\"\"Merge left subarray and right subarray\"\"\"\n    # Left subarray interval is [left, mid], right subarray interval is [mid+1, right]\n    # Create a temporary array tmp to store the merged results\n    tmp = [0] * (right - left + 1)\n    # Initialize the start indices of the left and right subarrays\n    i, j, k = left, mid + 1, 0\n    # While both subarrays still have elements, compare and copy the smaller element into the temporary array\n    while i &lt;= mid and j &lt;= right:\n        if nums[i] &lt;= nums[j]:\n            tmp[k] = nums[i]\n            i += 1\n        else:\n            tmp[k] = nums[j]\n            j += 1\n        k += 1\n    # Copy the remaining elements of the left and right subarrays into the temporary array\n    while i &lt;= mid:\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    while j &lt;= right:\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    # Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval\n    for k in range(0, len(tmp)):\n        nums[left + k] = tmp[k]\n\ndef merge_sort(nums: list[int], left: int, right: int):\n    \"\"\"Merge sort\"\"\"\n    # Termination condition\n    if left &gt;= right:\n        return  # Terminate recursion when subarray length is 1\n    # Divide and conquer stage\n    mid = (left + right) // 2  # Calculate midpoint\n    merge_sort(nums, left, mid)  # Recursively process the left subarray\n    merge_sort(nums, mid + 1, right)  # Recursively process the right subarray\n    # Merge stage\n    merge(nums, left, mid, right)\n</code></pre> merge_sort.cpp<pre><code>/* Merge left subarray and right subarray */\nvoid merge(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    // Left subarray interval is [left, mid], right subarray interval is [mid+1, right]\n    // Create a temporary array tmp to store the merged results\n    vector&lt;int&gt; tmp(right - left + 1);\n    // Initialize the start indices of the left and right subarrays\n    int i = left, j = mid + 1, k = 0;\n    // While both subarrays still have elements, compare and copy the smaller element into the temporary array\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval\n    for (k = 0; k &lt; tmp.size(); k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* Merge sort */\nvoid mergeSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // Termination condition\n    if (left &gt;= right)\n        return; // Terminate recursion when subarray length is 1\n    // Divide and conquer stage\n    int mid = left + (right - left) / 2;    // Calculate midpoint\n    mergeSort(nums, left, mid);      // Recursively process the left subarray\n    mergeSort(nums, mid + 1, right); // Recursively process the right subarray\n    // Merge stage\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.java<pre><code>/* Merge left subarray and right subarray */\nvoid merge(int[] nums, int left, int mid, int right) {\n    // Left subarray interval is [left, mid], right subarray interval is [mid+1, right]\n    // Create a temporary array tmp to store the merged results\n    int[] tmp = new int[right - left + 1];\n    // Initialize the start indices of the left and right subarrays\n    int i = left, j = mid + 1, k = 0;\n    // While both subarrays still have elements, compare and copy the smaller element into the temporary array\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* Merge sort */\nvoid mergeSort(int[] nums, int left, int right) {\n    // Termination condition\n    if (left &gt;= right)\n        return; // Terminate recursion when subarray length is 1\n    // Divide and conquer stage\n    int mid = left + (right - left) / 2; // Calculate midpoint\n    mergeSort(nums, left, mid); // Recursively process the left subarray\n    mergeSort(nums, mid + 1, right); // Recursively process the right subarray\n    // Merge stage\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.cs<pre><code>/* Merge left subarray and right subarray */\nvoid Merge(int[] nums, int left, int mid, int right) {\n    // Left subarray interval is [left, mid], right subarray interval is [mid+1, right]\n    // Create a temporary array tmp to store the merged results\n    int[] tmp = new int[right - left + 1];\n    // Initialize the start indices of the left and right subarrays\n    int i = left, j = mid + 1, k = 0;\n    // While both subarrays still have elements, compare and copy the smaller element into the temporary array\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval\n    for (k = 0; k &lt; tmp.Length; ++k) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* Merge sort */\nvoid MergeSort(int[] nums, int left, int right) {\n    // Termination condition\n    if (left &gt;= right) return;       // Terminate recursion when subarray length is 1\n    // Divide and conquer stage\n    int mid = left + (right - left) / 2;    // Calculate midpoint\n    MergeSort(nums, left, mid);      // Recursively process the left subarray\n    MergeSort(nums, mid + 1, right); // Recursively process the right subarray\n    // Merge stage\n    Merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.go<pre><code>/* Merge left subarray and right subarray */\nfunc merge(nums []int, left, mid, right int) {\n    // Left subarray interval is [left, mid], right subarray interval is [mid+1, right]\n    // Create a temporary array tmp to store the merged results\n    tmp := make([]int, right-left+1)\n    // Initialize the start indices of the left and right subarrays\n    i, j, k := left, mid+1, 0\n    // While both subarrays still have elements, compare and copy the smaller element into the temporary array\n    for i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i++\n        } else {\n            tmp[k] = nums[j]\n            j++\n        }\n        k++\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    for i &lt;= mid {\n        tmp[k] = nums[i]\n        i++\n        k++\n    }\n    for j &lt;= right {\n        tmp[k] = nums[j]\n        j++\n        k++\n    }\n    // Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval\n    for k := 0; k &lt; len(tmp); k++ {\n        nums[left+k] = tmp[k]\n    }\n}\n\n/* Merge sort */\nfunc mergeSort(nums []int, left, right int) {\n    // Termination condition\n    if left &gt;= right {\n        return\n    }\n    // Divide and conquer stage\n    mid := left + (right - left) / 2\n    mergeSort(nums, left, mid)\n    mergeSort(nums, mid+1, right)\n    // Merge stage\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.swift<pre><code>/* Merge left subarray and right subarray */\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n    // Left subarray interval is [left, mid], right subarray interval is [mid+1, right]\n    // Create a temporary array tmp to store the merged results\n    var tmp = Array(repeating: 0, count: right - left + 1)\n    // Initialize the start indices of the left and right subarrays\n    var i = left, j = mid + 1, k = 0\n    // While both subarrays still have elements, compare and copy the smaller element into the temporary array\n    while i &lt;= mid, j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i += 1\n        } else {\n            tmp[k] = nums[j]\n            j += 1\n        }\n        k += 1\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while i &lt;= mid {\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    }\n    // Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval\n    for k in tmp.indices {\n        nums[left + k] = tmp[k]\n    }\n}\n\n/* Merge sort */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n    // Termination condition\n    if left &gt;= right { // Terminate recursion when subarray length is 1\n        return\n    }\n    // Divide and conquer stage\n    let mid = left + (right - left) / 2 // Calculate midpoint\n    mergeSort(nums: &amp;nums, left: left, right: mid) // Recursively process the left subarray\n    mergeSort(nums: &amp;nums, left: mid + 1, right: right) // Recursively process the right subarray\n    // Merge stage\n    merge(nums: &amp;nums, left: left, mid: mid, right: right)\n}\n</code></pre> merge_sort.js<pre><code>/* Merge left subarray and right subarray */\nfunction merge(nums, left, mid, right) {\n    // Left subarray interval is [left, mid], right subarray interval is [mid+1, right]\n    // Create a temporary array tmp to store the merged results\n    const tmp = new Array(right - left + 1);\n    // Initialize the start indices of the left and right subarrays\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // While both subarrays still have elements, compare and copy the smaller element into the temporary array\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* Merge sort */\nfunction mergeSort(nums, left, right) {\n    // Termination condition\n    if (left &gt;= right) return; // Terminate recursion when subarray length is 1\n    // Divide and conquer stage\n    let mid = Math.floor(left + (right - left) / 2); // Calculate midpoint\n    mergeSort(nums, left, mid); // Recursively process the left subarray\n    mergeSort(nums, mid + 1, right); // Recursively process the right subarray\n    // Merge stage\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.ts<pre><code>/* Merge left subarray and right subarray */\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n    // Left subarray interval is [left, mid], right subarray interval is [mid+1, right]\n    // Create a temporary array tmp to store the merged results\n    const tmp = new Array(right - left + 1);\n    // Initialize the start indices of the left and right subarrays\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // While both subarrays still have elements, compare and copy the smaller element into the temporary array\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* Merge sort */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n    // Termination condition\n    if (left &gt;= right) return; // Terminate recursion when subarray length is 1\n    // Divide and conquer stage\n    let mid = Math.floor(left + (right - left) / 2); // Calculate midpoint\n    mergeSort(nums, left, mid); // Recursively process the left subarray\n    mergeSort(nums, mid + 1, right); // Recursively process the right subarray\n    // Merge stage\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.dart<pre><code>/* Merge left subarray and right subarray */\nvoid merge(List&lt;int&gt; nums, int left, int mid, int right) {\n  // Left subarray interval is [left, mid], right subarray interval is [mid+1, right]\n  // Create a temporary array tmp to store the merged results\n  List&lt;int&gt; tmp = List.filled(right - left + 1, 0);\n  // Initialize the start indices of the left and right subarrays\n  int i = left, j = mid + 1, k = 0;\n  // While both subarrays still have elements, compare and copy the smaller element into the temporary array\n  while (i &lt;= mid &amp;&amp; j &lt;= right) {\n    if (nums[i] &lt;= nums[j])\n      tmp[k++] = nums[i++];\n    else\n      tmp[k++] = nums[j++];\n  }\n  // Copy the remaining elements of the left and right subarrays into the temporary array\n  while (i &lt;= mid) {\n    tmp[k++] = nums[i++];\n  }\n  while (j &lt;= right) {\n    tmp[k++] = nums[j++];\n  }\n  // Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval\n  for (k = 0; k &lt; tmp.length; k++) {\n    nums[left + k] = tmp[k];\n  }\n}\n\n/* Merge sort */\nvoid mergeSort(List&lt;int&gt; nums, int left, int right) {\n  // Termination condition\n  if (left &gt;= right) return; // Terminate recursion when subarray length is 1\n  // Divide and conquer stage\n  int mid = left + (right - left) ~/ 2; // Calculate midpoint\n  mergeSort(nums, left, mid); // Recursively process the left subarray\n  mergeSort(nums, mid + 1, right); // Recursively process the right subarray\n  // Merge stage\n  merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.rs<pre><code>/* Merge left subarray and right subarray */\nfn merge(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) {\n    // Left subarray interval is [left, mid], right subarray interval is [mid+1, right]\n    // Create a temporary array tmp to store the merged results\n    let tmp_size = right - left + 1;\n    let mut tmp = vec![0; tmp_size];\n    // Initialize the start indices of the left and right subarrays\n    let (mut i, mut j, mut k) = (left, mid + 1, 0);\n    // While both subarrays still have elements, compare and copy the smaller element into the temporary array\n    while i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i];\n            i += 1;\n        } else {\n            tmp[k] = nums[j];\n            j += 1;\n        }\n        k += 1;\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while i &lt;= mid {\n        tmp[k] = nums[i];\n        k += 1;\n        i += 1;\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j];\n        k += 1;\n        j += 1;\n    }\n    // Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval\n    for k in 0..tmp_size {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* Merge sort */\nfn merge_sort(nums: &amp;mut [i32], left: usize, right: usize) {\n    // Termination condition\n    if left &gt;= right {\n        return; // Terminate recursion when subarray length is 1\n    }\n\n    // Divide and conquer stage\n    let mid = left + (right - left) / 2; // Calculate midpoint\n    merge_sort(nums, left, mid); // Recursively process the left subarray\n    merge_sort(nums, mid + 1, right); // Recursively process the right subarray\n\n    // Merge stage\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.c<pre><code>/* Merge left subarray and right subarray */\nvoid merge(int *nums, int left, int mid, int right) {\n    // Left subarray interval is [left, mid], right subarray interval is [mid+1, right]\n    // Create a temporary array tmp to store the merged results\n    int tmpSize = right - left + 1;\n    int *tmp = (int *)malloc(tmpSize * sizeof(int));\n    // Initialize the start indices of the left and right subarrays\n    int i = left, j = mid + 1, k = 0;\n    // While both subarrays still have elements, compare and copy the smaller element into the temporary array\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval\n    for (k = 0; k &lt; tmpSize; ++k) {\n        nums[left + k] = tmp[k];\n    }\n    // Free memory\n    free(tmp);\n}\n\n/* Merge sort */\nvoid mergeSort(int *nums, int left, int right) {\n    // Termination condition\n    if (left &gt;= right)\n        return; // Terminate recursion when subarray length is 1\n    // Divide and conquer stage\n    int mid = left + (right - left) / 2;    // Calculate midpoint\n    mergeSort(nums, left, mid);      // Recursively process the left subarray\n    mergeSort(nums, mid + 1, right); // Recursively process the right subarray\n    // Merge stage\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.kt<pre><code>/* Merge left subarray and right subarray */\nfun merge(nums: IntArray, left: Int, mid: Int, right: Int) {\n    // Left subarray interval is [left, mid], right subarray interval is [mid+1, right]\n    // Create a temporary array tmp to store the merged results\n    val tmp = IntArray(right - left + 1)\n    // Initialize the start indices of the left and right subarrays\n    var i = left\n    var j = mid + 1\n    var k = 0\n    // While both subarrays still have elements, compare and copy the smaller element into the temporary array\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++]\n        else\n            tmp[k++] = nums[j++]\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++]\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++]\n    }\n    // Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval\n    for (l in tmp.indices) {\n        nums[left + l] = tmp[l]\n    }\n}\n\n/* Merge sort */\nfun mergeSort(nums: IntArray, left: Int, right: Int) {\n    // Termination condition\n    if (left &gt;= right) return  // Terminate recursion when subarray length is 1\n    // Divide and conquer stage\n    val mid = left + (right - left) / 2 // Calculate midpoint\n    mergeSort(nums, left, mid) // Recursively process the left subarray\n    mergeSort(nums, mid + 1, right) // Recursively process the right subarray\n    // Merge stage\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.rb<pre><code>### Merge left and right subarrays ###\ndef merge(nums, left, mid, right)\n  # Left subarray interval is [left, mid], right subarray interval is [mid+1, right]\n  # Create temporary array tmp to store merged result\n  tmp = Array.new(right - left + 1, 0)\n  # Initialize the start indices of the left and right subarrays\n  i, j, k = left, mid + 1, 0\n  # While both subarrays still have elements, compare and copy the smaller element into the temporary array\n  while i &lt;= mid &amp;&amp; j &lt;= right\n    if nums[i] &lt;= nums[j]\n      tmp[k] = nums[i]\n      i += 1\n    else\n      tmp[k] = nums[j]\n      j += 1\n    end\n    k += 1\n  end\n  # Copy the remaining elements of the left and right subarrays into the temporary array\n  while i &lt;= mid\n    tmp[k] = nums[i]\n    i += 1\n    k += 1\n  end\n  while j &lt;= right\n    tmp[k] = nums[j]\n    j += 1\n    k += 1\n  end\n  # Copy the elements from the temporary array tmp back to the original array nums at the corresponding interval\n  (0...tmp.length).each do |k|\n    nums[left + k] = tmp[k]\n  end\nend\n\n### Merge sort ###\ndef merge_sort(nums, left, right)\n  # Termination condition\n  # Terminate recursion when subarray length is 1\n  return if left &gt;= right\n  # Divide and conquer stage\n  mid = left + (right - left) / 2 # Calculate midpoint\n  merge_sort(nums, left, mid) # Recursively process the left subarray\n  merge_sort(nums, mid + 1, right) # Recursively process the right subarray\n  # Merge stage\n  merge(nums, left, mid, right)\nend\n</code></pre>","path":["Chapter 11. Sorting","11.6   Merge Sort"],"tags":[]},{"location":"chapter_sorting/merge_sort/#1162-algorithm-characteristics","level":2,"title":"11.6.2   Algorithm Characteristics","text":"<ul> <li>Time complexity is \\(O(n \\log n)\\); merge sort is non-adaptive: The divide phase produces a recursion tree of height \\(\\log n\\), and the total number of operations performed during merging at each level is \\(n\\), so the overall time complexity is \\(O(n \\log n)\\).</li> <li>Space complexity is \\(O(n)\\); merge sort is not in-place: The recursion depth is \\(\\log n\\), which uses \\(O(\\log n)\\) stack-frame space. The merge operation requires an auxiliary array, which uses \\(O(n)\\) additional space.</li> <li>Stable sort: During merging, the relative order of equal elements remains unchanged.</li> </ul>","path":["Chapter 11. Sorting","11.6   Merge Sort"],"tags":[]},{"location":"chapter_sorting/merge_sort/#1163-linked-list-sorting","level":2,"title":"11.6.3   Linked List Sorting","text":"<p>For linked lists, merge sort has significant advantages over other sorting algorithms, and it can reduce the space complexity of the sorting task to \\(O(1)\\).</p> <ul> <li>Divide phase: Iteration can be used instead of recursion to split the linked list, thereby eliminating the stack-frame space used by recursion.</li> <li>Merge phase: In linked lists, node insertion and deletion require only pointer updates, so the merge phase (merging two short sorted linked lists into one longer sorted linked list) does not require creating an additional linked list.</li> </ul> <p>The specific implementation details are quite complex, and interested readers can consult related materials for learning.</p>","path":["Chapter 11. Sorting","11.6   Merge Sort"],"tags":[]},{"location":"chapter_sorting/quick_sort/","level":1,"title":"11.5   Quick Sort","text":"<p>Quick sort is an efficient and widely used sorting algorithm based on the divide-and-conquer strategy.</p> <p>The core operation of quick sort is \"sentinel partitioning\", whose goal is to select an element as the \"pivot\", move all elements smaller than the pivot to its left, and move all elements larger than the pivot to its right. Specifically, the process is shown in Figure 11-8.</p> <ol> <li>Select the leftmost element as the pivot, and initialize two pointers <code>i</code> and <code>j</code> at the two ends of the array.</li> <li>Enter a loop. In each round, use <code>i</code> (<code>j</code>) to find the first element larger (smaller) than the pivot, and then swap the two elements.</li> <li>Repeat step <code>2.</code> until <code>i</code> and <code>j</code> meet, then swap the pivot into the boundary position between the two sub-arrays.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 11-8   Sentinel partitioning steps </p> <p>After sentinel partitioning, the original array is divided into three parts: the left sub-array, the pivot, and the right sub-array, such that \"any element in the left sub-array \\(\\leq\\) the pivot \\(\\leq\\) any element in the right sub-array\". Therefore, we only need to sort the two sub-arrays next.</p> <p>Divide-and-conquer strategy of quick sort</p> <p>The essence of sentinel partitioning is to simplify the sorting problem of a longer array into the sorting problems of two shorter arrays.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"Sentinel partition\"\"\"\n    # Use nums[left] as the pivot\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # Search from right to left for the first element smaller than the pivot\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # Search from left to right for the first element greater than the pivot\n        # Swap elements\n        nums[i], nums[j] = nums[j], nums[i]\n    # Swap the pivot to the boundary between the two subarrays\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # Return the index of the pivot\n</code></pre> quick_sort.cpp<pre><code>/* Sentinel partition */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;                // Search from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;                // Search from left to right for the first element greater than the pivot\n        swap(nums[i], nums[j]); // Swap these two elements\n    }\n    swap(nums[i], nums[left]);  // Swap the pivot to the boundary between the two subarrays\n    return i;                   // Return the index of the pivot\n}\n</code></pre> quick_sort.java<pre><code>/* Swap elements */\nvoid swap(int[] nums, int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* Sentinel partition */\nint partition(int[] nums, int left, int right) {\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // Search from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // Search from left to right for the first element greater than the pivot\n        swap(nums, i, j); // Swap these two elements\n    }\n    swap(nums, i, left);  // Swap the pivot to the boundary between the two subarrays\n    return i;             // Return the index of the pivot\n}\n</code></pre> quick_sort.cs<pre><code>/* Swap elements */\nvoid Swap(int[] nums, int i, int j) {\n    (nums[j], nums[i]) = (nums[i], nums[j]);\n}\n\n/* Sentinel partition */\nint Partition(int[] nums, int left, int right) {\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // Search from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // Search from left to right for the first element greater than the pivot\n        Swap(nums, i, j); // Swap these two elements\n    }\n    Swap(nums, i, left);  // Swap the pivot to the boundary between the two subarrays\n    return i;             // Return the index of the pivot\n}\n</code></pre> quick_sort.go<pre><code>/* Sentinel partition */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n    // Use nums[left] as the pivot\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- // Search from right to left for the first element smaller than the pivot\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ // Search from left to right for the first element greater than the pivot\n        }\n        // Swap elements\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    // Swap the pivot to the boundary between the two subarrays\n    nums[i], nums[left] = nums[left], nums[i]\n    return i // Return the index of the pivot\n}\n</code></pre> quick_sort.swift<pre><code>/* Sentinel partition */\nfunc partition(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // Use nums[left] as the pivot\n    var i = left\n    var j = right\n    while i &lt; j {\n        while i &lt; j, nums[j] &gt;= nums[left] {\n            j -= 1 // Search from right to left for the first element smaller than the pivot\n        }\n        while i &lt; j, nums[i] &lt;= nums[left] {\n            i += 1 // Search from left to right for the first element greater than the pivot\n        }\n        nums.swapAt(i, j) // Swap these two elements\n    }\n    nums.swapAt(i, left) // Swap the pivot to the boundary between the two subarrays\n    return i // Return the index of the pivot\n}\n</code></pre> quick_sort.js<pre><code>/* Swap elements */\nswap(nums, i, j) {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* Sentinel partition */\npartition(nums, left, right) {\n    // Use nums[left] as the pivot\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // Search from right to left for the first element smaller than the pivot\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // Search from left to right for the first element greater than the pivot\n        }\n        // Swap elements\n        this.swap(nums, i, j); // Swap these two elements\n    }\n    this.swap(nums, i, left); // Swap the pivot to the boundary between the two subarrays\n    return i; // Return the index of the pivot\n}\n</code></pre> quick_sort.ts<pre><code>/* Swap elements */\nswap(nums: number[], i: number, j: number): void {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* Sentinel partition */\npartition(nums: number[], left: number, right: number): number {\n    // Use nums[left] as the pivot\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // Search from right to left for the first element smaller than the pivot\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // Search from left to right for the first element greater than the pivot\n        }\n        // Swap elements\n        this.swap(nums, i, j); // Swap these two elements\n    }\n    this.swap(nums, i, left); // Swap the pivot to the boundary between the two subarrays\n    return i; // Return the index of the pivot\n}\n</code></pre> quick_sort.dart<pre><code>/* Swap elements */\nvoid _swap(List&lt;int&gt; nums, int i, int j) {\n  int tmp = nums[i];\n  nums[i] = nums[j];\n  nums[j] = tmp;\n}\n\n/* Sentinel partition */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // Use nums[left] as the pivot\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // Search from right to left for the first element smaller than the pivot\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // Search from left to right for the first element greater than the pivot\n    _swap(nums, i, j); // Swap these two elements\n  }\n  _swap(nums, i, left); // Swap the pivot to the boundary between the two subarrays\n  return i; // Return the index of the pivot\n}\n</code></pre> quick_sort.rs<pre><code>/* Sentinel partition */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // Use nums[left] as the pivot\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // Search from right to left for the first element smaller than the pivot\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // Search from left to right for the first element greater than the pivot\n        }\n        nums.swap(i, j); // Swap these two elements\n    }\n    nums.swap(i, left); // Swap the pivot to the boundary between the two subarrays\n    i // Return the index of the pivot\n}\n</code></pre> quick_sort.c<pre><code>/* Swap elements */\nvoid swap(int nums[], int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* Sentinel partition */\nint partition(int nums[], int left, int right) {\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j--; // Search from right to left for the first element smaller than the pivot\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i++; // Search from left to right for the first element greater than the pivot\n        }\n        // Swap these two elements\n        swap(nums, i, j);\n    }\n    // Swap the pivot to the boundary between the two subarrays\n    swap(nums, i, left);\n    // Return the index of the pivot\n    return i;\n}\n</code></pre> quick_sort.kt<pre><code>/* Swap elements */\nfun swap(nums: IntArray, i: Int, j: Int) {\n    val temp = nums[i]\n    nums[i] = nums[j]\n    nums[j] = temp\n}\n\n/* Sentinel partition */\nfun partition(nums: IntArray, left: Int, right: Int): Int {\n    // Use nums[left] as the pivot\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--           // Search from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++           // Search from left to right for the first element greater than the pivot\n        swap(nums, i, j)  // Swap these two elements\n    }\n    swap(nums, i, left)   // Swap the pivot to the boundary between the two subarrays\n    return i              // Return the index of the pivot\n}\n</code></pre> quick_sort.rb<pre><code>### Sentinel partition ###\ndef partition(nums, left, right)\n  # Use nums[left] as the pivot\n  i, j = left, right\n  while i &lt; j\n    while i &lt; j &amp;&amp; nums[j] &gt;= nums[left]\n      j -= 1 # Search from right to left for the first element smaller than the pivot\n    end\n    while i &lt; j &amp;&amp; nums[i] &lt;= nums[left]\n      i += 1 # Search from left to right for the first element greater than the pivot\n    end\n    # Swap elements\n    nums[i], nums[j] = nums[j], nums[i]\n  end\n  # Swap the pivot to the boundary between the two subarrays\n  nums[i], nums[left] = nums[left], nums[i]\n  i # Return the index of the pivot\nend\n</code></pre>","path":["Chapter 11. Sorting","11.5   Quick Sort"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1151-algorithm-flow","level":2,"title":"11.5.1   Algorithm Flow","text":"<p>The overall flow of quick sort is shown in Figure 11-9.</p> <ol> <li>First, perform one \"sentinel partitioning\" on the original array to obtain the unsorted left sub-array and right sub-array.</li> <li>Then, recursively perform \"sentinel partitioning\" on the left sub-array and right sub-array respectively.</li> <li>Continue recursively until the sub-array length is 1, at which point sorting of the entire array is complete.</li> </ol> <p></p> <p> Figure 11-9   Quick sort flow </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"Quick sort\"\"\"\n    # Terminate recursion when subarray length is 1\n    if left &gt;= right:\n        return\n    # Sentinel partition\n    pivot = self.partition(nums, left, right)\n    # Recursively process the left subarray and right subarray\n    self.quick_sort(nums, left, pivot - 1)\n    self.quick_sort(nums, pivot + 1, right)\n</code></pre> quick_sort.cpp<pre><code>/* Quick sort */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // Terminate recursion when subarray length is 1\n    if (left &gt;= right)\n        return;\n    // Sentinel partition\n    int pivot = partition(nums, left, right);\n    // Recursively process the left subarray and right subarray\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.java<pre><code>/* Quick sort */\nvoid quickSort(int[] nums, int left, int right) {\n    // Terminate recursion when subarray length is 1\n    if (left &gt;= right)\n        return;\n    // Sentinel partition\n    int pivot = partition(nums, left, right);\n    // Recursively process the left subarray and right subarray\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.cs<pre><code>/* Quick sort */\nvoid QuickSort(int[] nums, int left, int right) {\n    // Terminate recursion when subarray length is 1\n    if (left &gt;= right)\n        return;\n    // Sentinel partition\n    int pivot = Partition(nums, left, right);\n    // Recursively process the left subarray and right subarray\n    QuickSort(nums, left, pivot - 1);\n    QuickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.go<pre><code>/* Quick sort */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n    // Terminate recursion when subarray length is 1\n    if left &gt;= right {\n        return\n    }\n    // Sentinel partition\n    pivot := q.partition(nums, left, right)\n    // Recursively process the left subarray and right subarray\n    q.quickSort(nums, left, pivot-1)\n    q.quickSort(nums, pivot+1, right)\n}\n</code></pre> quick_sort.swift<pre><code>/* Quick sort */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n    // Terminate recursion when subarray length is 1\n    if left &gt;= right {\n        return\n    }\n    // Sentinel partition\n    let pivot = partition(nums: &amp;nums, left: left, right: right)\n    // Recursively process the left subarray and right subarray\n    quickSort(nums: &amp;nums, left: left, right: pivot - 1)\n    quickSort(nums: &amp;nums, left: pivot + 1, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* Quick sort */\nquickSort(nums, left, right) {\n    // Terminate recursion when subarray length is 1\n    if (left &gt;= right) return;\n    // Sentinel partition\n    const pivot = this.partition(nums, left, right);\n    // Recursively process the left subarray and right subarray\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.ts<pre><code>/* Quick sort */\nquickSort(nums: number[], left: number, right: number): void {\n    // Terminate recursion when subarray length is 1\n    if (left &gt;= right) {\n        return;\n    }\n    // Sentinel partition\n    const pivot = this.partition(nums, left, right);\n    // Recursively process the left subarray and right subarray\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.dart<pre><code>/* Quick sort */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // Terminate recursion when subarray length is 1\n  if (left &gt;= right) return;\n  // Sentinel partition\n  int pivot = _partition(nums, left, right);\n  // Recursively process the left subarray and right subarray\n  quickSort(nums, left, pivot - 1);\n  quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.rs<pre><code>/* Quick sort */\npub fn quick_sort(left: i32, right: i32, nums: &amp;mut [i32]) {\n    // Terminate recursion when subarray length is 1\n    if left &gt;= right {\n        return;\n    }\n    // Sentinel partition\n    let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n    // Recursively process the left subarray and right subarray\n    Self::quick_sort(left, pivot - 1, nums);\n    Self::quick_sort(pivot + 1, right, nums);\n}\n</code></pre> quick_sort.c<pre><code>/* Quick sort */\nvoid quickSort(int nums[], int left, int right) {\n    // Terminate recursion when subarray length is 1\n    if (left &gt;= right) {\n        return;\n    }\n    // Sentinel partition\n    int pivot = partition(nums, left, right);\n    // Recursively process the left subarray and right subarray\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.kt<pre><code>/* Quick sort */\nfun quickSort(nums: IntArray, left: Int, right: Int) {\n    // Terminate recursion when subarray length is 1\n    if (left &gt;= right) return\n    // Sentinel partition\n    val pivot = partition(nums, left, right)\n    // Recursively process the left subarray and right subarray\n    quickSort(nums, left, pivot - 1)\n    quickSort(nums, pivot + 1, right)\n}\n</code></pre> quick_sort.rb<pre><code>### Quick sort class ###\ndef quick_sort(nums, left, right)\n  # Recurse when subarray length is not 1\n  if left &lt; right\n    # Sentinel partition\n    pivot = partition(nums, left, right)\n    # Recursively process the left subarray and right subarray\n    quick_sort(nums, left, pivot - 1)\n    quick_sort(nums, pivot + 1, right)\n  end\n  nums\nend\n</code></pre>","path":["Chapter 11. Sorting","11.5   Quick Sort"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1152-algorithm-characteristics","level":2,"title":"11.5.2   Algorithm Characteristics","text":"<ul> <li>Time complexity of \\(O(n \\log n)\\), non-adaptive sorting: On average, sentinel partitioning produces \\(\\log n\\) recursive levels, and the total number of loop iterations across each level is \\(n\\), so the overall time complexity is \\(O(n \\log n)\\). In the worst case, each round of sentinel partitioning splits an array of length \\(n\\) into sub-arrays of lengths \\(0\\) and \\(n - 1\\). The recursion depth then reaches \\(n\\), with \\(n\\) loop iterations at each level, yielding an overall time complexity of \\(O(n^2)\\).</li> <li>Space complexity of \\(O(n)\\), in-place sorting: In the case where the input array is completely reversed, the worst recursive depth reaches \\(n\\), using \\(O(n)\\) stack frame space. The sorting operation is performed on the original array without the aid of an additional array.</li> <li>Unstable sorting: In the last step of sentinel partitioning, the pivot may be swapped to the right of an equal element.</li> </ul>","path":["Chapter 11. Sorting","11.5   Quick Sort"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1153-why-is-quick-sort-fast","level":2,"title":"11.5.3   Why Is Quick Sort Fast","text":"<p>As the name suggests, quick sort has clear efficiency advantages. Although its average time complexity is the same as that of \"merge sort\" and \"heap sort\", quick sort is usually faster in practice for the following reasons.</p> <ul> <li>The worst case is unlikely to occur: Although the worst-case time complexity of quick sort is \\(O(n^2)\\) and its performance is less predictable than that of merge sort, quick sort runs in \\(O(n \\log n)\\) time in the vast majority of cases.</li> <li>High cache efficiency: During sentinel partitioning, the system can load the entire sub-array into cache, so accessing elements is relatively efficient. By contrast, algorithms such as \"heap sort\" require non-contiguous access to elements and therefore do not enjoy this advantage.</li> <li>Small constant factors: Among the three algorithms above, quick sort performs the fewest comparisons, assignments, and swaps in total. This is similar to why \"insertion sort\" is faster than \"bubble sort\".</li> </ul>","path":["Chapter 11. Sorting","11.5   Quick Sort"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1154-pivot-optimization","level":2,"title":"11.5.4   Pivot Optimization","text":"<p>Quick sort can become less time-efficient for certain inputs. Consider an extreme example in which the input array is in completely descending order. Because we choose the leftmost element as the pivot, once sentinel partitioning is complete, the pivot is swapped to the far right of the array, leaving a left sub-array of length \\(n - 1\\) and a right sub-array of length \\(0\\). If this continues recursively, each round of sentinel partitioning produces one sub-array of length \\(0\\), the divide-and-conquer strategy breaks down, and quick sort degenerates into an approximation of \"bubble sort\".</p> <p>To reduce the chance of this happening, we can optimize the pivot selection strategy used in sentinel partitioning. For example, we can choose a pivot at random. However, if we are unlucky and repeatedly pick poor pivots, performance can still be unsatisfactory.</p> <p>It should be noted that programming languages usually generate \"pseudo-random numbers\". If we construct a specific test case against a pseudo-random sequence, quick sort can still suffer degraded performance.</p> <p>To improve further, we can choose three candidate elements from the array, usually the first, last, and middle elements, and use the median of the three as the pivot. This greatly increases the chance that the pivot is \"neither too small nor too large\". We can also choose more candidate elements to further improve the robustness of the algorithm. With this method, the probability that the time complexity degrades to \\(O(n^2)\\) is significantly reduced.</p> <p>Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def median_three(self, nums: list[int], left: int, mid: int, right: int) -&gt; int:\n    \"\"\"Select the median of three candidate elements\"\"\"\n    l, m, r = nums[left], nums[mid], nums[right]\n    if (l &lt;= m &lt;= r) or (r &lt;= m &lt;= l):\n        return mid  # m is between l and r\n    if (m &lt;= l &lt;= r) or (r &lt;= l &lt;= m):\n        return left  # l is between m and r\n    return right\n\ndef partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"Sentinel partition (median of three)\"\"\"\n    # Use nums[left] as the pivot\n    med = self.median_three(nums, left, (left + right) // 2, right)\n    # Swap the median to the array's leftmost position\n    nums[left], nums[med] = nums[med], nums[left]\n    # Use nums[left] as the pivot\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # Search from right to left for the first element smaller than the pivot\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # Search from left to right for the first element greater than the pivot\n        # Swap elements\n        nums[i], nums[j] = nums[j], nums[i]\n    # Swap the pivot to the boundary between the two subarrays\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # Return the index of the pivot\n</code></pre> quick_sort.cpp<pre><code>/* Select the median of three candidate elements */\nint medianThree(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m is between l and r\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l is between m and r\n    return right;\n}\n\n/* Sentinel partition (median of three) */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // Select the median of three candidate elements\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // Swap the median to the array's leftmost position\n    swap(nums[left], nums[med]);\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;                // Search from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;                // Search from left to right for the first element greater than the pivot\n        swap(nums[i], nums[j]); // Swap these two elements\n    }\n    swap(nums[i], nums[left]);  // Swap the pivot to the boundary between the two subarrays\n    return i;                   // Return the index of the pivot\n}\n</code></pre> quick_sort.java<pre><code>/* Select the median of three candidate elements */\nint medianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m is between l and r\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l is between m and r\n    return right;\n}\n\n/* Sentinel partition (median of three) */\nint partition(int[] nums, int left, int right) {\n    // Select the median of three candidate elements\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // Swap the median to the array's leftmost position\n    swap(nums, left, med);\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // Search from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // Search from left to right for the first element greater than the pivot\n        swap(nums, i, j); // Swap these two elements\n    }\n    swap(nums, i, left);  // Swap the pivot to the boundary between the two subarrays\n    return i;             // Return the index of the pivot\n}\n</code></pre> quick_sort.cs<pre><code>/* Select the median of three candidate elements */\nint MedianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m is between l and r\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l is between m and r\n    return right;\n}\n\n/* Sentinel partition (median of three) */\nint Partition(int[] nums, int left, int right) {\n    // Select the median of three candidate elements\n    int med = MedianThree(nums, left, (left + right) / 2, right);\n    // Swap the median to the array's leftmost position\n    Swap(nums, left, med);\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // Search from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // Search from left to right for the first element greater than the pivot\n        Swap(nums, i, j); // Swap these two elements\n    }\n    Swap(nums, i, left);  // Swap the pivot to the boundary between the two subarrays\n    return i;             // Return the index of the pivot\n}\n</code></pre> quick_sort.go<pre><code>/* Select the median of three candidate elements */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n    l, m, r := nums[left], nums[mid], nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m is between l and r\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l is between m and r\n    }\n    return right\n}\n\n/* Sentinel partition (median of three) */\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n    // Use nums[left] as the pivot\n    med := q.medianThree(nums, left, (left+right)/2, right)\n    // Swap the median to the array's leftmost position\n    nums[left], nums[med] = nums[med], nums[left]\n    // Use nums[left] as the pivot\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- // Search from right to left for the first element smaller than the pivot\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ // Search from left to right for the first element greater than the pivot\n        }\n        // Swap elements\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    // Swap the pivot to the boundary between the two subarrays\n    nums[i], nums[left] = nums[left], nums[i]\n    return i // Return the index of the pivot\n}\n</code></pre> quick_sort.swift<pre><code>/* Select the median of three candidate elements */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -&gt; Int {\n    let l = nums[left]\n    let m = nums[mid]\n    let r = nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m is between l and r\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l is between m and r\n    }\n    return right\n}\n\n/* Sentinel partition (median of three) */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // Select the median of three candidate elements\n    let med = medianThree(nums: nums, left: left, mid: left + (right - left) / 2, right: right)\n    // Swap the median to the array's leftmost position\n    nums.swapAt(left, med)\n    return partition(nums: &amp;nums, left: left, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* Select the median of three candidate elements */\nmedianThree(nums, left, mid, right) {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m is between l and r\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l is between m and r\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* Sentinel partition (median of three) */\npartition(nums, left, right) {\n    // Select the median of three candidate elements\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // Swap the median to the array's leftmost position\n    this.swap(nums, left, med);\n    // Use nums[left] as the pivot\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // Search from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // Search from left to right for the first element greater than the pivot\n        this.swap(nums, i, j); // Swap these two elements\n    }\n    this.swap(nums, i, left); // Swap the pivot to the boundary between the two subarrays\n    return i; // Return the index of the pivot\n}\n</code></pre> quick_sort.ts<pre><code>/* Select the median of three candidate elements */\nmedianThree(\n    nums: number[],\n    left: number,\n    mid: number,\n    right: number\n): number {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m is between l and r\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l is between m and r\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* Sentinel partition (median of three) */\npartition(nums: number[], left: number, right: number): number {\n    // Select the median of three candidate elements\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // Swap the median to the array's leftmost position\n    this.swap(nums, left, med);\n    // Use nums[left] as the pivot\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j--; // Search from right to left for the first element smaller than the pivot\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i++; // Search from left to right for the first element greater than the pivot\n        }\n        this.swap(nums, i, j); // Swap these two elements\n    }\n    this.swap(nums, i, left); // Swap the pivot to the boundary between the two subarrays\n    return i; // Return the index of the pivot\n}\n</code></pre> quick_sort.dart<pre><code>/* Select the median of three candidate elements */\nint _medianThree(List&lt;int&gt; nums, int left, int mid, int right) {\n  int l = nums[left], m = nums[mid], r = nums[right];\n  if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n    return mid; // m is between l and r\n  if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n    return left; // l is between m and r\n  return right;\n}\n\n/* Sentinel partition (median of three) */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // Select the median of three candidate elements\n  int med = _medianThree(nums, left, (left + right) ~/ 2, right);\n  // Swap the median to the array's leftmost position\n  _swap(nums, left, med);\n  // Use nums[left] as the pivot\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // Search from right to left for the first element smaller than the pivot\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // Search from left to right for the first element greater than the pivot\n    _swap(nums, i, j); // Swap these two elements\n  }\n  _swap(nums, i, left); // Swap the pivot to the boundary between the two subarrays\n  return i; // Return the index of the pivot\n}\n</code></pre> quick_sort.rs<pre><code>/* Select the median of three candidate elements */\nfn median_three(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) -&gt; usize {\n    let (l, m, r) = (nums[left], nums[mid], nums[right]);\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid; // m is between l and r\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left; // l is between m and r\n    }\n    right\n}\n\n/* Sentinel partition (median of three) */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // Select the median of three candidate elements\n    let med = Self::median_three(nums, left, (left + right) / 2, right);\n    // Swap the median to the array's leftmost position\n    nums.swap(left, med);\n    // Use nums[left] as the pivot\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // Search from right to left for the first element smaller than the pivot\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // Search from left to right for the first element greater than the pivot\n        }\n        nums.swap(i, j); // Swap these two elements\n    }\n    nums.swap(i, left); // Swap the pivot to the boundary between the two subarrays\n    i // Return the index of the pivot\n}\n</code></pre> quick_sort.c<pre><code>/* Select the median of three candidate elements */\nint medianThree(int nums[], int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m is between l and r\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l is between m and r\n    return right;\n}\n\n/* Sentinel partition (median of three) */\nint partitionMedian(int nums[], int left, int right) {\n    // Select the median of three candidate elements\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // Swap the median to the array's leftmost position\n    swap(nums, left, med);\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--; // Search from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // Search from left to right for the first element greater than the pivot\n        swap(nums, i, j); // Swap these two elements\n    }\n    swap(nums, i, left); // Swap the pivot to the boundary between the two subarrays\n    return i;            // Return the index of the pivot\n}\n</code></pre> quick_sort.kt<pre><code>/* Select the median of three candidate elements */\nfun medianThree(nums: IntArray, left: Int, mid: Int, right: Int): Int {\n    val l = nums[left]\n    val m = nums[mid]\n    val r = nums[right]\n    if ((m in l..r) || (m in r..l))\n        return mid  // m is between l and r\n    if ((l in m..r) || (l in r..m))\n        return left // l is between m and r\n    return right\n}\n\n/* Sentinel partition (median of three) */\nfun partitionMedian(nums: IntArray, left: Int, right: Int): Int {\n    // Select the median of three candidate elements\n    val med = medianThree(nums, left, (left + right) / 2, right)\n    // Swap the median to the array's leftmost position\n    swap(nums, left, med)\n    // Use nums[left] as the pivot\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--                      // Search from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++                      // Search from left to right for the first element greater than the pivot\n        swap(nums, i, j)             // Swap these two elements\n    }\n    swap(nums, i, left)              // Swap the pivot to the boundary between the two subarrays\n    return i                         // Return the index of the pivot\n}\n</code></pre> quick_sort.rb<pre><code>### Select median of three candidate elements ###\ndef median_three(nums, left, mid, right)\n  # Select the median of three candidate elements\n  _l, _m, _r = nums[left], nums[mid], nums[right]\n  # m is between l and r\n  return mid if (_l &lt;= _m &amp;&amp; _m &lt;= _r) || (_r &lt;= _m &amp;&amp; _m &lt;= _l)\n  # l is between m and r\n  return left if (_m &lt;= _l &amp;&amp; _l &lt;= _r) || (_r &lt;= _l &amp;&amp; _l &lt;= _m)\n  return right\nend\n\n### Sentinel partition (median of three) ###\ndef partition(nums, left, right)\n  ### Use nums[left] as pivot\n  med = median_three(nums, left, (left + right) / 2, right)\n  # Swap median to leftmost position of array\n  nums[left], nums[med] = nums[med], nums[left]\n  i, j = left, right\n  while i &lt; j\n    while i &lt; j &amp;&amp; nums[j] &gt;= nums[left]\n      j -= 1 # Search from right to left for the first element smaller than the pivot\n    end\n    while i &lt; j &amp;&amp; nums[i] &lt;= nums[left]\n      i += 1 # Search from left to right for the first element greater than the pivot\n    end\n    # Swap elements\n    nums[i], nums[j] = nums[j], nums[i]\n  end\n  # Swap the pivot to the boundary between the two subarrays\n  nums[i], nums[left] = nums[left], nums[i]\n  i # Return the index of the pivot\nend\n</code></pre>","path":["Chapter 11. Sorting","11.5   Quick Sort"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1155-recursive-depth-optimization","level":2,"title":"11.5.5   Recursive Depth Optimization","text":"<p>Quick sort may also use more space for certain inputs. Consider a fully sorted input array. Let the length of the current sub-array in the recursion be \\(m\\). Each round of sentinel partitioning produces a left sub-array of length \\(0\\) and a right sub-array of length \\(m - 1\\), which means each recursive call reduces the problem size by only one element. The recursion tree can therefore reach a height of \\(n - 1\\), requiring \\(O(n)\\) stack-frame space.</p> <p>To prevent stack frames from accumulating, we can compare the lengths of the two sub-arrays after each round of sentinel partitioning, and recurse only on the shorter one. Because the shorter sub-array has length at most \\(n / 2\\), this method ensures that the recursion depth does not exceed \\(\\log n\\), reducing the worst-case space complexity to \\(O(\\log n)\\). The code is shown below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"Quick sort (recursion depth optimization)\"\"\"\n    # Terminate when subarray length is 1\n    while left &lt; right:\n        # Sentinel partition operation\n        pivot = self.partition(nums, left, right)\n        # Perform quick sort on the shorter of the two subarrays\n        if pivot - left &lt; right - pivot:\n            self.quick_sort(nums, left, pivot - 1)  # Recursively sort the left subarray\n            left = pivot + 1  # Remaining unsorted interval is [pivot + 1, right]\n        else:\n            self.quick_sort(nums, pivot + 1, right)  # Recursively sort the right subarray\n            right = pivot - 1  # Remaining unsorted interval is [left, pivot - 1]\n</code></pre> quick_sort.cpp<pre><code>/* Quick sort (recursion depth optimization) */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // Terminate when subarray length is 1\n    while (left &lt; right) {\n        // Sentinel partition operation\n        int pivot = partition(nums, left, right);\n        // Perform quick sort on the shorter of the two subarrays\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // Recursively sort the left subarray\n            left = pivot + 1;                 // Remaining unsorted interval is [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right); // Recursively sort the right subarray\n            right = pivot - 1;                 // Remaining unsorted interval is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.java<pre><code>/* Quick sort (recursion depth optimization) */\nvoid quickSort(int[] nums, int left, int right) {\n    // Terminate when subarray length is 1\n    while (left &lt; right) {\n        // Sentinel partition operation\n        int pivot = partition(nums, left, right);\n        // Perform quick sort on the shorter of the two subarrays\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // Recursively sort the left subarray\n            left = pivot + 1; // Remaining unsorted interval is [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right); // Recursively sort the right subarray\n            right = pivot - 1; // Remaining unsorted interval is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.cs<pre><code>/* Quick sort (recursion depth optimization) */\nvoid QuickSort(int[] nums, int left, int right) {\n    // Terminate when subarray length is 1\n    while (left &lt; right) {\n        // Sentinel partition operation\n        int pivot = Partition(nums, left, right);\n        // Perform quick sort on the shorter of the two subarrays\n        if (pivot - left &lt; right - pivot) {\n            QuickSort(nums, left, pivot - 1);  // Recursively sort the left subarray\n            left = pivot + 1;  // Remaining unsorted interval is [pivot + 1, right]\n        } else {\n            QuickSort(nums, pivot + 1, right); // Recursively sort the right subarray\n            right = pivot - 1; // Remaining unsorted interval is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.go<pre><code>/* Quick sort (recursion depth optimization) */\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n    // Terminate when subarray length is 1\n    for left &lt; right {\n        // Sentinel partition operation\n        pivot := q.partition(nums, left, right)\n        // Perform quick sort on the shorter of the two subarrays\n        if pivot-left &lt; right-pivot {\n            q.quickSort(nums, left, pivot-1) // Recursively sort the left subarray\n            left = pivot + 1                 // Remaining unsorted interval is [pivot + 1, right]\n        } else {\n            q.quickSort(nums, pivot+1, right) // Recursively sort the right subarray\n            right = pivot - 1                 // Remaining unsorted interval is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.swift<pre><code>/* Quick sort (recursion depth optimization) */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\n    var left = left\n    var right = right\n    // Terminate when subarray length is 1\n    while left &lt; right {\n        // Sentinel partition operation\n        let pivot = partition(nums: &amp;nums, left: left, right: right)\n        // Perform quick sort on the shorter of the two subarrays\n        if (pivot - left) &lt; (right - pivot) {\n            quickSortTailCall(nums: &amp;nums, left: left, right: pivot - 1) // Recursively sort the left subarray\n            left = pivot + 1 // Remaining unsorted interval is [pivot + 1, right]\n        } else {\n            quickSortTailCall(nums: &amp;nums, left: pivot + 1, right: right) // Recursively sort the right subarray\n            right = pivot - 1 // Remaining unsorted interval is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.js<pre><code>/* Quick sort (recursion depth optimization) */\nquickSort(nums, left, right) {\n    // Terminate when subarray length is 1\n    while (left &lt; right) {\n        // Sentinel partition operation\n        let pivot = this.partition(nums, left, right);\n        // Perform quick sort on the shorter of the two subarrays\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // Recursively sort the left subarray\n            left = pivot + 1; // Remaining unsorted interval is [pivot + 1, right]\n        } else {\n            this.quickSort(nums, pivot + 1, right); // Recursively sort the right subarray\n            right = pivot - 1; // Remaining unsorted interval is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.ts<pre><code>/* Quick sort (recursion depth optimization) */\nquickSort(nums: number[], left: number, right: number): void {\n    // Terminate when subarray length is 1\n    while (left &lt; right) {\n        // Sentinel partition operation\n        let pivot = this.partition(nums, left, right);\n        // Perform quick sort on the shorter of the two subarrays\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // Recursively sort the left subarray\n            left = pivot + 1; // Remaining unsorted interval is [pivot + 1, right]\n        } else {\n            this.quickSort(nums, pivot + 1, right); // Recursively sort the right subarray\n            right = pivot - 1; // Remaining unsorted interval is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.dart<pre><code>/* Quick sort (recursion depth optimization) */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // Terminate when subarray length is 1\n  while (left &lt; right) {\n    // Sentinel partition operation\n    int pivot = _partition(nums, left, right);\n    // Perform quick sort on the shorter of the two subarrays\n    if (pivot - left &lt; right - pivot) {\n      quickSort(nums, left, pivot - 1); // Recursively sort the left subarray\n      left = pivot + 1; // Remaining unsorted interval is [pivot + 1, right]\n    } else {\n      quickSort(nums, pivot + 1, right); // Recursively sort the right subarray\n      right = pivot - 1; // Remaining unsorted interval is [left, pivot - 1]\n    }\n  }\n}\n</code></pre> quick_sort.rs<pre><code>/* Quick sort (recursion depth optimization) */\npub fn quick_sort(mut left: i32, mut right: i32, nums: &amp;mut [i32]) {\n    // Terminate when subarray length is 1\n    while left &lt; right {\n        // Sentinel partition operation\n        let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n        // Perform quick sort on the shorter of the two subarrays\n        if pivot - left &lt; right - pivot {\n            Self::quick_sort(left, pivot - 1, nums); // Recursively sort the left subarray\n            left = pivot + 1; // Remaining unsorted interval is [pivot + 1, right]\n        } else {\n            Self::quick_sort(pivot + 1, right, nums); // Recursively sort the right subarray\n            right = pivot - 1; // Remaining unsorted interval is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.c<pre><code>/* Quick sort (recursion depth optimization) */\nvoid quickSortTailCall(int nums[], int left, int right) {\n    // Terminate when subarray length is 1\n    while (left &lt; right) {\n        // Sentinel partition operation\n        int pivot = partition(nums, left, right);\n        // Perform quick sort on the shorter of the two subarrays\n        if (pivot - left &lt; right - pivot) {\n            // Recursively sort the left subarray\n            quickSortTailCall(nums, left, pivot - 1);\n            // Remaining unsorted interval is [pivot + 1, right]\n            left = pivot + 1;\n        } else {\n            // Recursively sort the right subarray\n            quickSortTailCall(nums, pivot + 1, right);\n            // Remaining unsorted interval is [left, pivot - 1]\n            right = pivot - 1;\n        }\n    }\n}\n</code></pre> quick_sort.kt<pre><code>/* Quick sort (recursion depth optimization) */\nfun quickSortTailCall(nums: IntArray, left: Int, right: Int) {\n    // Terminate when subarray length is 1\n    var l = left\n    var r = right\n    while (l &lt; r) {\n        // Sentinel partition operation\n        val pivot = partition(nums, l, r)\n        // Perform quick sort on the shorter of the two subarrays\n        if (pivot - l &lt; r - pivot) {\n            quickSort(nums, l, pivot - 1) // Recursively sort the left subarray\n            l = pivot + 1 // Remaining unsorted interval is [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, r) // Recursively sort the right subarray\n            r = pivot - 1 // Remaining unsorted interval is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.rb<pre><code>### Quick sort (recursion depth optimization) ###\ndef quick_sort(nums, left, right)\n  # Recurse when subarray length is not 1\n  while left &lt; right\n    # Sentinel partition\n    pivot = partition(nums, left, right)\n    # Perform quick sort on the shorter of the two subarrays\n    if pivot - left &lt; right - pivot\n      quick_sort(nums, left, pivot - 1)\n      left = pivot + 1 # Remaining unsorted interval is [pivot + 1, right]\n    else\n      quick_sort(nums, pivot + 1, right)\n      right = pivot - 1 # Remaining unsorted interval is [left, pivot - 1]\n    end\n  end\nend\n</code></pre>","path":["Chapter 11. Sorting","11.5   Quick Sort"],"tags":[]},{"location":"chapter_sorting/radix_sort/","level":1,"title":"11.10   Radix Sort","text":"<p>The previous section introduced counting sort, which is suitable when the number of items \\(n\\) is large but the value range \\(m\\) is small. Suppose we need to sort \\(n = 10^6\\) student IDs, each of which is an 8-digit number. Then the value range \\(m = 10^8\\) is very large. Using counting sort would require a large amount of memory, whereas radix sort avoids this problem.</p> <p>Radix sort is based on the same core idea as counting sort: it also sorts by counting occurrences. Building on this, radix sort exploits the positional relationship among digits and sorts them one digit at a time to obtain the final result.</p>","path":["Chapter 11. Sorting","11.10   Radix Sort"],"tags":[]},{"location":"chapter_sorting/radix_sort/#11101-algorithm-flow","level":2,"title":"11.10.1   Algorithm Flow","text":"<p>Taking student ID data as an example, assume the lowest digit is the \\(1\\)st digit and the highest digit is the \\(8\\)th digit. The flow of radix sort is shown in Figure 11-18.</p> <ol> <li>Initialize the digit \\(k = 1\\).</li> <li>Perform \"counting sort\" on the \\(k\\)th digit of the student IDs. After completion, the data will be sorted from smallest to largest according to the \\(k\\)th digit.</li> <li>Increase \\(k\\) by \\(1\\), then return to step <code>2.</code> and continue iterating until all digits are sorted, at which point the process ends.</li> </ol> <p></p> <p> Figure 11-18   Radix sort algorithm flow </p> <p>Next, let us look at the code. For a number \\(x\\) in base \\(d\\), its \\(k\\)th digit \\(x_k\\) can be obtained with the following formula:</p> \\[ x_k = \\lfloor\\frac{x}{d^{k-1}}\\rfloor \\bmod d \\] <p>Here, \\(\\lfloor a \\rfloor\\) denotes rounding the floating-point number \\(a\\) down, and \\(\\bmod \\: d\\) denotes taking the remainder modulo \\(d\\). For student ID data, \\(d = 10\\) and \\(k \\in [1, 8]\\).</p> <p>Additionally, we need to slightly modify the counting sort code to make it sort based on the \\(k\\)th digit of the number:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby radix_sort.py<pre><code>def digit(num: int, exp: int) -&gt; int:\n    \"\"\"Get the k-th digit of element num, where exp = 10^(k-1)\"\"\"\n    # Passing exp instead of k can avoid repeated expensive exponentiation here\n    return (num // exp) % 10\n\ndef counting_sort_digit(nums: list[int], exp: int):\n    \"\"\"Counting sort (based on nums k-th digit)\"\"\"\n    # Decimal digit range is 0~9, therefore need a bucket array of length 10\n    counter = [0] * 10\n    n = len(nums)\n    # Count the occurrence of digits 0~9\n    for i in range(n):\n        d = digit(nums[i], exp)  # Get the k-th digit of nums[i], noted as d\n        counter[d] += 1  # Count the occurrence of digit d\n    # Calculate prefix sum, converting \"occurrence count\" into \"array index\"\n    for i in range(1, 10):\n        counter[i] += counter[i - 1]\n    # Traverse in reverse, based on bucket statistics, place each element into res\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        d = digit(nums[i], exp)\n        j = counter[d] - 1  # Get the index j for d in the array\n        res[j] = nums[i]  # Place the current element at index j\n        counter[d] -= 1  # Decrease the count of d by 1\n    # Use result to overwrite the original array nums\n    for i in range(n):\n        nums[i] = res[i]\n\ndef radix_sort(nums: list[int]):\n    \"\"\"Radix sort\"\"\"\n    # Get the maximum element of the array, used to determine the maximum number of digits\n    m = max(nums)\n    # Traverse from the lowest to the highest digit\n    exp = 1\n    while exp &lt;= m:\n        # Perform counting sort on the k-th digit of array elements\n        # k = 1 -&gt; exp = 1\n        # k = 2 -&gt; exp = 10\n        # i.e., exp = 10^(k-1)\n        counting_sort_digit(nums, exp)\n        exp *= 10\n</code></pre> radix_sort.cpp<pre><code>/* Get the k-th digit of element num, where exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // Passing exp instead of k can avoid repeated expensive exponentiation here\n    return (num / exp) % 10;\n}\n\n/* Counting sort (based on nums k-th digit) */\nvoid countingSortDigit(vector&lt;int&gt; &amp;nums, int exp) {\n    // Decimal digit range is 0~9, therefore need a bucket array of length 10\n    vector&lt;int&gt; counter(10, 0);\n    int n = nums.size();\n    // Count the occurrence of digits 0~9\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // Get the k-th digit of nums[i], noted as d\n        counter[d]++;                // Count the occurrence of digit d\n    }\n    // Calculate prefix sum, converting \"occurrence count\" into \"array index\"\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // Traverse in reverse, based on bucket statistics, place each element into res\n    vector&lt;int&gt; res(n, 0);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // Get the index j for d in the array\n        res[j] = nums[i];       // Place the current element at index j\n        counter[d]--;           // Decrease the count of d by 1\n    }\n    // Use result to overwrite the original array nums\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* Radix sort */\nvoid radixSort(vector&lt;int&gt; &amp;nums) {\n    // Get the maximum element of the array, used to determine the maximum number of digits\n    int m = *max_element(nums.begin(), nums.end());\n    // Traverse from the lowest to the highest digit\n    for (int exp = 1; exp &lt;= m; exp *= 10)\n        // Perform counting sort on the k-th digit of array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // i.e., exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.java<pre><code>/* Get the k-th digit of element num, where exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // Passing exp instead of k can avoid repeated expensive exponentiation here\n    return (num / exp) % 10;\n}\n\n/* Counting sort (based on nums k-th digit) */\nvoid countingSortDigit(int[] nums, int exp) {\n    // Decimal digit range is 0~9, therefore need a bucket array of length 10\n    int[] counter = new int[10];\n    int n = nums.length;\n    // Count the occurrence of digits 0~9\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // Get the k-th digit of nums[i], noted as d\n        counter[d]++;                // Count the occurrence of digit d\n    }\n    // Calculate prefix sum, converting \"occurrence count\" into \"array index\"\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // Traverse in reverse, based on bucket statistics, place each element into res\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // Get the index j for d in the array\n        res[j] = nums[i];       // Place the current element at index j\n        counter[d]--;           // Decrease the count of d by 1\n    }\n    // Use result to overwrite the original array nums\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* Radix sort */\nvoid radixSort(int[] nums) {\n    // Get the maximum element of the array, used to determine the maximum number of digits\n    int m = Integer.MIN_VALUE;\n    for (int num : nums)\n        if (num &gt; m)\n            m = num;\n    // Traverse from the lowest to the highest digit\n    for (int exp = 1; exp &lt;= m; exp *= 10) {\n        // Perform counting sort on the k-th digit of array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // i.e., exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.cs<pre><code>/* Get the k-th digit of element num, where exp = 10^(k-1) */\nint Digit(int num, int exp) {\n    // Passing exp instead of k can avoid repeated expensive exponentiation here\n    return (num / exp) % 10;\n}\n\n/* Counting sort (based on nums k-th digit) */\nvoid CountingSortDigit(int[] nums, int exp) {\n    // Decimal digit range is 0~9, therefore need a bucket array of length 10\n    int[] counter = new int[10];\n    int n = nums.Length;\n    // Count the occurrence of digits 0~9\n    for (int i = 0; i &lt; n; i++) {\n        int d = Digit(nums[i], exp); // Get the k-th digit of nums[i], noted as d\n        counter[d]++;                // Count the occurrence of digit d\n    }\n    // Calculate prefix sum, converting \"occurrence count\" into \"array index\"\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // Traverse in reverse, based on bucket statistics, place each element into res\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = Digit(nums[i], exp);\n        int j = counter[d] - 1; // Get the index j for d in the array\n        res[j] = nums[i];       // Place the current element at index j\n        counter[d]--;           // Decrease the count of d by 1\n    }\n    // Use result to overwrite the original array nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* Radix sort */\nvoid RadixSort(int[] nums) {\n    // Get the maximum element of the array, used to determine the maximum number of digits\n    int m = int.MinValue;\n    foreach (int num in nums) {\n        if (num &gt; m) m = num;\n    }\n    // Traverse from the lowest to the highest digit\n    for (int exp = 1; exp &lt;= m; exp *= 10) {\n        // Perform counting sort on the k-th digit of array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // i.e., exp = 10^(k-1)\n        CountingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.go<pre><code>/* Get the k-th digit of element num, where exp = 10^(k-1) */\nfunc digit(num, exp int) int {\n    // Passing exp instead of k can avoid repeated expensive exponentiation here\n    return (num / exp) % 10\n}\n\n/* Counting sort (based on nums k-th digit) */\nfunc countingSortDigit(nums []int, exp int) {\n    // Decimal digit range is 0~9, therefore need a bucket array of length 10\n    counter := make([]int, 10)\n    n := len(nums)\n    // Count the occurrence of digits 0~9\n    for i := 0; i &lt; n; i++ {\n        d := digit(nums[i], exp) // Get the k-th digit of nums[i], noted as d\n        counter[d]++             // Count the occurrence of digit d\n    }\n    // Calculate prefix sum, converting \"occurrence count\" into \"array index\"\n    for i := 1; i &lt; 10; i++ {\n        counter[i] += counter[i-1]\n    }\n    // Traverse in reverse, based on bucket statistics, place each element into res\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        d := digit(nums[i], exp)\n        j := counter[d] - 1 // Get the index j for d in the array\n        res[j] = nums[i]    // Place the current element at index j\n        counter[d]--        // Decrease the count of d by 1\n    }\n    // Use result to overwrite the original array nums\n    for i := 0; i &lt; n; i++ {\n        nums[i] = res[i]\n    }\n}\n\n/* Radix sort */\nfunc radixSort(nums []int) {\n    // Get the maximum element of the array, used to determine the maximum number of digits\n    max := math.MinInt\n    for _, num := range nums {\n        if num &gt; max {\n            max = num\n        }\n    }\n    // Traverse from the lowest to the highest digit\n    for exp := 1; max &gt;= exp; exp *= 10 {\n        // Perform counting sort on the k-th digit of array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // i.e., exp = 10^(k-1)\n        countingSortDigit(nums, exp)\n    }\n}\n</code></pre> radix_sort.swift<pre><code>/* Get the k-th digit of element num, where exp = 10^(k-1) */\nfunc digit(num: Int, exp: Int) -&gt; Int {\n    // Passing exp instead of k can avoid repeated expensive exponentiation here\n    (num / exp) % 10\n}\n\n/* Counting sort (based on nums k-th digit) */\nfunc countingSortDigit(nums: inout [Int], exp: Int) {\n    // Decimal digit range is 0~9, therefore need a bucket array of length 10\n    var counter = Array(repeating: 0, count: 10)\n    // Count the occurrence of digits 0~9\n    for i in nums.indices {\n        let d = digit(num: nums[i], exp: exp) // Get the k-th digit of nums[i], noted as d\n        counter[d] += 1 // Count the occurrence of digit d\n    }\n    // Calculate prefix sum, converting \"occurrence count\" into \"array index\"\n    for i in 1 ..&lt; 10 {\n        counter[i] += counter[i - 1]\n    }\n    // Traverse in reverse, based on bucket statistics, place each element into res\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let d = digit(num: nums[i], exp: exp)\n        let j = counter[d] - 1 // Get the index j for d in the array\n        res[j] = nums[i] // Place the current element at index j\n        counter[d] -= 1 // Decrease the count of d by 1\n    }\n    // Use result to overwrite the original array nums\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n\n/* Radix sort */\nfunc radixSort(nums: inout [Int]) {\n    // Get the maximum element of the array, used to determine the maximum number of digits\n    var m = Int.min\n    for num in nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // Traverse from the lowest to the highest digit\n    for exp in sequence(first: 1, next: { m &gt;= ($0 * 10) ? $0 * 10 : nil }) {\n        // Perform counting sort on the k-th digit of array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // i.e., exp = 10^(k-1)\n        countingSortDigit(nums: &amp;nums, exp: exp)\n    }\n}\n</code></pre> radix_sort.js<pre><code>/* Get the k-th digit of element num, where exp = 10^(k-1) */\nfunction digit(num, exp) {\n    // Passing exp instead of k can avoid repeated expensive exponentiation here\n    return Math.floor(num / exp) % 10;\n}\n\n/* Counting sort (based on nums k-th digit) */\nfunction countingSortDigit(nums, exp) {\n    // Decimal digit range is 0~9, therefore need a bucket array of length 10\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // Count the occurrence of digits 0~9\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // Get the k-th digit of nums[i], noted as d\n        counter[d]++; // Count the occurrence of digit d\n    }\n    // Calculate prefix sum, converting \"occurrence count\" into \"array index\"\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // Traverse in reverse, based on bucket statistics, place each element into res\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // Get the index j for d in the array\n        res[j] = nums[i]; // Place the current element at index j\n        counter[d]--; // Decrease the count of d by 1\n    }\n    // Use result to overwrite the original array nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* Radix sort */\nfunction radixSort(nums) {\n    // Get the maximum element of the array, used to determine the maximum number of digits\n    let m = Math.max(... nums);\n    // Traverse from the lowest to the highest digit\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // Perform counting sort on the k-th digit of array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // i.e., exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.ts<pre><code>/* Get the k-th digit of element num, where exp = 10^(k-1) */\nfunction digit(num: number, exp: number): number {\n    // Passing exp instead of k can avoid repeated expensive exponentiation here\n    return Math.floor(num / exp) % 10;\n}\n\n/* Counting sort (based on nums k-th digit) */\nfunction countingSortDigit(nums: number[], exp: number): void {\n    // Decimal digit range is 0~9, therefore need a bucket array of length 10\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // Count the occurrence of digits 0~9\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // Get the k-th digit of nums[i], noted as d\n        counter[d]++; // Count the occurrence of digit d\n    }\n    // Calculate prefix sum, converting \"occurrence count\" into \"array index\"\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // Traverse in reverse, based on bucket statistics, place each element into res\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // Get the index j for d in the array\n        res[j] = nums[i]; // Place the current element at index j\n        counter[d]--; // Decrease the count of d by 1\n    }\n    // Use result to overwrite the original array nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* Radix sort */\nfunction radixSort(nums: number[]): void {\n    // Get the maximum element of the array, used to determine the maximum number of digits\n    let m: number = Math.max(... nums);\n    // Traverse from the lowest to the highest digit\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // Perform counting sort on the k-th digit of array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // i.e., exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.dart<pre><code>/* Get k-th digit of element _num, where exp = 10^(k-1) */\nint digit(int _num, int exp) {\n  // Passing exp instead of k can avoid repeated expensive exponentiation here\n  return (_num ~/ exp) % 10;\n}\n\n/* Counting sort (based on nums k-th digit) */\nvoid countingSortDigit(List&lt;int&gt; nums, int exp) {\n  // Decimal digit range is 0~9, therefore need a bucket array of length 10\n  List&lt;int&gt; counter = List&lt;int&gt;.filled(10, 0);\n  int n = nums.length;\n  // Count the occurrence of digits 0~9\n  for (int i = 0; i &lt; n; i++) {\n    int d = digit(nums[i], exp); // Get the k-th digit of nums[i], noted as d\n    counter[d]++; // Count the occurrence of digit d\n  }\n  // Calculate prefix sum, converting \"occurrence count\" into \"array index\"\n  for (int i = 1; i &lt; 10; i++) {\n    counter[i] += counter[i - 1];\n  }\n  // Traverse in reverse, based on bucket statistics, place each element into res\n  List&lt;int&gt; res = List&lt;int&gt;.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int d = digit(nums[i], exp);\n    int j = counter[d] - 1; // Get the index j for d in the array\n    res[j] = nums[i]; // Place the current element at index j\n    counter[d]--; // Decrease the count of d by 1\n  }\n  // Use result to overwrite the original array nums\n  for (int i = 0; i &lt; n; i++) nums[i] = res[i];\n}\n\n/* Radix sort */\nvoid radixSort(List&lt;int&gt; nums) {\n  // Get the maximum element of the array, used to determine the maximum number of digits\n  // In Dart, int length is 64 bits\n  int m = -1 &lt;&lt; 63;\n  for (int _num in nums) if (_num &gt; m) m = _num;\n  // Traverse from the lowest to the highest digit\n  for (int exp = 1; exp &lt;= m; exp *= 10)\n    // Perform counting sort on the k-th digit of array elements\n    // k = 1 -&gt; exp = 1\n    // k = 2 -&gt; exp = 10\n    // i.e., exp = 10^(k-1)\n    countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.rs<pre><code>/* Get the k-th digit of element num, where exp = 10^(k-1) */\nfn digit(num: i32, exp: i32) -&gt; usize {\n    // Passing exp instead of k can avoid repeated expensive exponentiation here\n    return ((num / exp) % 10) as usize;\n}\n\n/* Counting sort (based on nums k-th digit) */\nfn counting_sort_digit(nums: &amp;mut [i32], exp: i32) {\n    // Decimal digit range is 0~9, therefore need a bucket array of length 10\n    let mut counter = [0; 10];\n    let n = nums.len();\n    // Count the occurrence of digits 0~9\n    for i in 0..n {\n        let d = digit(nums[i], exp); // Get the k-th digit of nums[i], noted as d\n        counter[d] += 1; // Count the occurrence of digit d\n    }\n    // Calculate prefix sum, converting \"occurrence count\" into \"array index\"\n    for i in 1..10 {\n        counter[i] += counter[i - 1];\n    }\n    // Traverse in reverse, based on bucket statistics, place each element into res\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let d = digit(nums[i], exp);\n        let j = counter[d] - 1; // Get the index j for d in the array\n        res[j] = nums[i]; // Place the current element at index j\n        counter[d] -= 1; // Decrease the count of d by 1\n    }\n    // Use result to overwrite the original array nums\n    nums.copy_from_slice(&amp;res);\n}\n\n/* Radix sort */\nfn radix_sort(nums: &amp;mut [i32]) {\n    // Get the maximum element of the array, used to determine the maximum number of digits\n    let m = *nums.into_iter().max().unwrap();\n    // Traverse from the lowest to the highest digit\n    let mut exp = 1;\n    while exp &lt;= m {\n        counting_sort_digit(nums, exp);\n        exp *= 10;\n    }\n}\n</code></pre> radix_sort.c<pre><code>/* Get the k-th digit of element num, where exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // Passing exp instead of k can avoid repeated expensive exponentiation here\n    return (num / exp) % 10;\n}\n\n/* Counting sort (based on nums k-th digit) */\nvoid countingSortDigit(int nums[], int size, int exp) {\n    // Decimal digit range is 0~9, therefore need a bucket array of length 10\n    int *counter = (int *)malloc((sizeof(int) * 10));\n    memset(counter, 0, sizeof(int) * 10); // Initialize to 0 to support subsequent memory release\n    // Count the occurrence of digits 0~9\n    for (int i = 0; i &lt; size; i++) {\n        // Get the k-th digit of nums[i], noted as d\n        int d = digit(nums[i], exp);\n        // Count the occurrence of digit d\n        counter[d]++;\n    }\n    // Calculate prefix sum, converting \"occurrence count\" into \"array index\"\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // Traverse in reverse, based on bucket statistics, place each element into res\n    int *res = (int *)malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // Get the index j for d in the array\n        res[j] = nums[i];       // Place the current element at index j\n        counter[d]--;           // Decrease the count of d by 1\n    }\n    // Use result to overwrite the original array nums\n    for (int i = 0; i &lt; size; i++) {\n        nums[i] = res[i];\n    }\n    // Free memory\n    free(res);\n    free(counter);\n}\n\n/* Radix sort */\nvoid radixSort(int nums[], int size) {\n    // Get the maximum element of the array, used to determine the maximum number of digits\n    int max = INT32_MIN;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; max) {\n            max = nums[i];\n        }\n    }\n    // Traverse from the lowest to the highest digit\n    for (int exp = 1; max &gt;= exp; exp *= 10)\n        // Perform counting sort on the k-th digit of array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // i.e., exp = 10^(k-1)\n        countingSortDigit(nums, size, exp);\n}\n</code></pre> radix_sort.kt<pre><code>/* Get the k-th digit of element num, where exp = 10^(k-1) */\nfun digit(num: Int, exp: Int): Int {\n    // Passing exp instead of k can avoid repeated expensive exponentiation here\n    return (num / exp) % 10\n}\n\n/* Counting sort (based on nums k-th digit) */\nfun countingSortDigit(nums: IntArray, exp: Int) {\n    // Decimal digit range is 0~9, therefore need a bucket array of length 10\n    val counter = IntArray(10)\n    val n = nums.size\n    // Count the occurrence of digits 0~9\n    for (i in 0..&lt;n) {\n        val d = digit(nums[i], exp) // Get the k-th digit of nums[i], noted as d\n        counter[d]++                // Count the occurrence of digit d\n    }\n    // Calculate prefix sum, converting \"occurrence count\" into \"array index\"\n    for (i in 1..9) {\n        counter[i] += counter[i - 1]\n    }\n    // Traverse in reverse, based on bucket statistics, place each element into res\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val d = digit(nums[i], exp)\n        val j = counter[d] - 1 // Get the index j for d in the array\n        res[j] = nums[i]       // Place the current element at index j\n        counter[d]--           // Decrease the count of d by 1\n    }\n    // Use result to overwrite the original array nums\n    for (i in 0..&lt;n)\n        nums[i] = res[i]\n}\n\n/* Radix sort */\nfun radixSort(nums: IntArray) {\n    // Get the maximum element of the array, used to determine the maximum number of digits\n    var m = Int.MIN_VALUE\n    for (num in nums) if (num &gt; m) m = num\n    var exp = 1\n    // Traverse from the lowest to the highest digit\n    while (exp &lt;= m) {\n        // Perform counting sort on the k-th digit of array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // i.e., exp = 10^(k-1)\n        countingSortDigit(nums, exp)\n        exp *= 10\n    }\n}\n</code></pre> radix_sort.rb<pre><code>### Get k-th digit of element num, where exp = 10^(k-1) ###\ndef digit(num, exp)\n  # Passing exp instead of k avoids expensive exponentiation calculations\n  (num / exp) % 10\nend\n\n### Counting sort (sort by k-th digit of nums) ###\ndef counting_sort_digit(nums, exp)\n  # Decimal digit range is 0~9, therefore need a bucket array of length 10\n  counter = Array.new(10, 0)\n  n = nums.length\n  # Count the occurrence of digits 0~9\n  for i in 0...n\n    d = digit(nums[i], exp) # Get the k-th digit of nums[i], noted as d\n    counter[d] += 1 # Count the occurrence of digit d\n  end\n  # Calculate prefix sum, converting \"occurrence count\" into \"array index\"\n  (1...10).each { |i| counter[i] += counter[i - 1] }\n  # Traverse in reverse, based on bucket statistics, place each element into res\n  res = Array.new(n, 0)\n  for i in (n - 1).downto(0)\n    d = digit(nums[i], exp)\n    j = counter[d] - 1 # Get the index j for d in the array\n    res[j] = nums[i] # Place the current element at index j\n    counter[d] -= 1 # Decrease the count of d by 1\n  end\n  # Use result to overwrite the original array nums\n  (0...n).each { |i| nums[i] = res[i] }\nend\n\n### Radix sort ###\ndef radix_sort(nums)\n  # Get the maximum element of the array, used to determine the maximum number of digits\n  m = nums.max\n  # Traverse from the lowest to the highest digit\n  exp = 1\n  while exp &lt;= m\n    # Perform counting sort on the k-th digit of array elements\n    # k = 1 -&gt; exp = 1\n    # k = 2 -&gt; exp = 10\n    # i.e., exp = 10^(k-1)\n    counting_sort_digit(nums, exp)\n    exp *= 10\n  end\nend\n</code></pre> <p>Why start sorting from the lowest digit?</p> <p>In successive sorting passes, a later pass overrides the result of an earlier one. For example, if the first pass yields \\(a &lt; b\\) but the second yields \\(a &gt; b\\), then the result of the second pass prevails. Because higher-order digits have higher priority than lower-order digits, we should sort the lower digits first and then the higher digits.</p>","path":["Chapter 11. Sorting","11.10   Radix Sort"],"tags":[]},{"location":"chapter_sorting/radix_sort/#11102-algorithm-characteristics","level":2,"title":"11.10.2   Algorithm Characteristics","text":"<p>Compared with counting sort, radix sort is suitable for larger value ranges, but only when the data can be represented with a fixed number of digits and that digit count is not too large. For example, floating-point numbers are not well suited to radix sort because the digit count \\(k\\) can be too large, potentially leading to time complexity \\(O(nk) \\gg O(n^2)\\).</p> <ul> <li>Time complexity of \\(O(nk)\\), non-adaptive sorting: Let the number of items be \\(n\\), let the values be represented in base \\(d\\), and let the maximum number of digits be \\(k\\). Counting sort on one digit takes \\(O(n + d)\\) time, so sorting all \\(k\\) digits takes \\(O((n + d)k)\\) time. In practice, \\(d\\) and \\(k\\) are usually relatively small, so the overall time complexity approaches \\(O(n)\\).</li> <li>Space complexity of \\(O(n + d)\\), non-in-place sorting: Same as counting sort, radix sort requires auxiliary arrays <code>res</code> and <code>counter</code> of lengths \\(n\\) and \\(d\\).</li> <li>Stable sort: When counting sort is stable, radix sort is also stable; when counting sort is unstable, radix sort cannot guarantee correct sorting results.</li> </ul>","path":["Chapter 11. Sorting","11.10   Radix Sort"],"tags":[]},{"location":"chapter_sorting/selection_sort/","level":1,"title":"11.2   Selection Sort","text":"<p>Selection sort works very simply: in each round, it selects the smallest element from the unsorted interval and places it at the end of the sorted interval.</p> <p>Assume the array has length \\(n\\). The procedure of selection sort is shown in Figure 11-2.</p> <ol> <li>Initially, all elements are unsorted, i.e., the unsorted (index) interval is \\([0, n-1]\\).</li> <li>Select the smallest element in the interval \\([0, n-1]\\) and swap it with the element at index \\(0\\). After completion, the first element of the array is sorted.</li> <li>Select the smallest element in the interval \\([1, n-1]\\) and swap it with the element at index \\(1\\). After completion, the first 2 elements of the array are sorted.</li> <li>And so on. After \\(n - 1\\) rounds of selection and swapping, the first \\(n - 1\\) elements of the array are sorted.</li> <li>The only remaining element must be the largest, so no further sorting is needed and the array is sorted.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 11-2   Selection sort steps </p> <p>In the code, we use \\(k\\) to track the smallest element within the unsorted interval:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby selection_sort.py<pre><code>def selection_sort(nums: list[int]):\n    \"\"\"Selection sort\"\"\"\n    n = len(nums)\n    # Outer loop: unsorted interval is [i, n-1]\n    for i in range(n - 1):\n        # Inner loop: find the smallest element within the unsorted interval\n        k = i\n        for j in range(i + 1, n):\n            if nums[j] &lt; nums[k]:\n                k = j  # Record the index of the smallest element\n        # Swap the smallest element with the first element of the unsorted interval\n        nums[i], nums[k] = nums[k], nums[i]\n</code></pre> selection_sort.cpp<pre><code>/* Selection sort */\nvoid selectionSort(vector&lt;int&gt; &amp;nums) {\n    int n = nums.size();\n    // Outer loop: unsorted interval is [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // Inner loop: find the smallest element within the unsorted interval\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // Record the index of the smallest element\n        }\n        // Swap the smallest element with the first element of the unsorted interval\n        swap(nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.java<pre><code>/* Selection sort */\nvoid selectionSort(int[] nums) {\n    int n = nums.length;\n    // Outer loop: unsorted interval is [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // Inner loop: find the smallest element within the unsorted interval\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // Record the index of the smallest element\n        }\n        // Swap the smallest element with the first element of the unsorted interval\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.cs<pre><code>/* Selection sort */\nvoid SelectionSort(int[] nums) {\n    int n = nums.Length;\n    // Outer loop: unsorted interval is [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // Inner loop: find the smallest element within the unsorted interval\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // Record the index of the smallest element\n        }\n        // Swap the smallest element with the first element of the unsorted interval\n        (nums[k], nums[i]) = (nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.go<pre><code>/* Selection sort */\nfunc selectionSort(nums []int) {\n    n := len(nums)\n    // Outer loop: unsorted interval is [i, n-1]\n    for i := 0; i &lt; n-1; i++ {\n        // Inner loop: find the smallest element within the unsorted interval\n        k := i\n        for j := i + 1; j &lt; n; j++ {\n            if nums[j] &lt; nums[k] {\n                // Record the index of the smallest element\n                k = j\n            }\n        }\n        // Swap the smallest element with the first element of the unsorted interval\n        nums[i], nums[k] = nums[k], nums[i]\n\n    }\n}\n</code></pre> selection_sort.swift<pre><code>/* Selection sort */\nfunc selectionSort(nums: inout [Int]) {\n    // Outer loop: unsorted interval is [i, n-1]\n    for i in nums.indices.dropLast() {\n        // Inner loop: find the smallest element within the unsorted interval\n        var k = i\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[j] &lt; nums[k] {\n                k = j // Record the index of the smallest element\n            }\n        }\n        // Swap the smallest element with the first element of the unsorted interval\n        nums.swapAt(i, k)\n    }\n}\n</code></pre> selection_sort.js<pre><code>/* Selection sort */\nfunction selectionSort(nums) {\n    let n = nums.length;\n    // Outer loop: unsorted interval is [i, n-1]\n    for (let i = 0; i &lt; n - 1; i++) {\n        // Inner loop: find the smallest element within the unsorted interval\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // Record the index of the smallest element\n            }\n        }\n        // Swap the smallest element with the first element of the unsorted interval\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.ts<pre><code>/* Selection sort */\nfunction selectionSort(nums: number[]): void {\n    let n = nums.length;\n    // Outer loop: unsorted interval is [i, n-1]\n    for (let i = 0; i &lt; n - 1; i++) {\n        // Inner loop: find the smallest element within the unsorted interval\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // Record the index of the smallest element\n            }\n        }\n        // Swap the smallest element with the first element of the unsorted interval\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.dart<pre><code>/* Selection sort */\nvoid selectionSort(List&lt;int&gt; nums) {\n  int n = nums.length;\n  // Outer loop: unsorted interval is [i, n-1]\n  for (int i = 0; i &lt; n - 1; i++) {\n    // Inner loop: find the smallest element within the unsorted interval\n    int k = i;\n    for (int j = i + 1; j &lt; n; j++) {\n      if (nums[j] &lt; nums[k]) k = j; // Record the index of the smallest element\n    }\n    // Swap the smallest element with the first element of the unsorted interval\n    int temp = nums[i];\n    nums[i] = nums[k];\n    nums[k] = temp;\n  }\n}\n</code></pre> selection_sort.rs<pre><code>/* Selection sort */\nfn selection_sort(nums: &amp;mut [i32]) {\n    if nums.is_empty() {\n        return;\n    }\n    let n = nums.len();\n    // Outer loop: unsorted interval is [i, n-1]\n    for i in 0..n - 1 {\n        // Inner loop: find the smallest element within the unsorted interval\n        let mut k = i;\n        for j in i + 1..n {\n            if nums[j] &lt; nums[k] {\n                k = j; // Record the index of the smallest element\n            }\n        }\n        // Swap the smallest element with the first element of the unsorted interval\n        nums.swap(i, k);\n    }\n}\n</code></pre> selection_sort.c<pre><code>/* Selection sort */\nvoid selectionSort(int nums[], int n) {\n    // Outer loop: unsorted interval is [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // Inner loop: find the smallest element within the unsorted interval\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // Record the index of the smallest element\n        }\n        // Swap the smallest element with the first element of the unsorted interval\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.kt<pre><code>/* Selection sort */\nfun selectionSort(nums: IntArray) {\n    val n = nums.size\n    // Outer loop: unsorted interval is [i, n-1]\n    for (i in 0..&lt;n - 1) {\n        var k = i\n        // Inner loop: find the smallest element within the unsorted interval\n        for (j in i + 1..&lt;n) {\n            if (nums[j] &lt; nums[k])\n                k = j // Record the index of the smallest element\n        }\n        // Swap the smallest element with the first element of the unsorted interval\n        val temp = nums[i]\n        nums[i] = nums[k]\n        nums[k] = temp\n    }\n}\n</code></pre> selection_sort.rb<pre><code>### Selection sort ###\ndef selection_sort(nums)\n  n = nums.length\n  # Outer loop: unsorted interval is [i, n-1]\n  for i in 0...(n - 1)\n    # Inner loop: find the smallest element within the unsorted interval\n    k = i\n    for j in (i + 1)...n\n      if nums[j] &lt; nums[k]\n        k = j # Record the index of the smallest element\n      end\n    end\n    # Swap the smallest element with the first element of the unsorted interval\n    nums[i], nums[k] = nums[k], nums[i]\n  end\nend\n</code></pre>","path":["Chapter 11. Sorting","11.2   Selection Sort"],"tags":[]},{"location":"chapter_sorting/selection_sort/#1121-algorithm-characteristics","level":2,"title":"11.2.1   Algorithm Characteristics","text":"<ul> <li>Time complexity \\(O(n^2)\\), non-adaptive sorting: The outer loop has \\(n - 1\\) rounds in total. The inner loop runs \\(n - 1\\) times in the first round and \\(1\\) time in the last round. Thus, it runs \\(n - 1\\), \\(n - 2\\), \\(\\dots\\), \\(2\\), and \\(1\\) times across the rounds, summing to \\(\\frac{n(n - 1)}{2}\\).</li> <li>Space complexity \\(O(1)\\), in-place sorting: Pointers \\(i\\) and \\(j\\) use a constant amount of extra space.</li> <li>Unstable sorting: As shown in Figure 11-3, element <code>nums[i]</code> may be swapped to the right of an element equal to it, causing a change in their relative order.</li> </ul> <p> Figure 11-3   Selection sort non-stability example </p>","path":["Chapter 11. Sorting","11.2   Selection Sort"],"tags":[]},{"location":"chapter_sorting/sorting_algorithm/","level":1,"title":"11.1   Sorting Algorithm","text":"<p>A sorting algorithm arranges a set of data in a specific order. Sorting algorithms have extensive applications because ordered data can usually be searched, analyzed, and processed more efficiently.</p> <p>As shown in Figure 11-1, the data being sorted can be integers, floating-point numbers, characters, strings, and so on. The sorting rule can be defined as needed, such as numerical order, ASCII order, or a custom rule.</p> <p></p> <p> Figure 11-1   Data type and criterion examples </p>","path":["Chapter 11. Sorting","11.1   Sorting Algorithm"],"tags":[]},{"location":"chapter_sorting/sorting_algorithm/#1111-evaluation-dimensions","level":2,"title":"11.1.1   Evaluation Dimensions","text":"<p>Execution efficiency: We expect the time complexity of sorting algorithms to be as low as possible, with a smaller total number of operations (reducing the constant factor in time complexity). For large data volumes, execution efficiency is particularly important.</p> <p>In-place property: As the name implies, in-place sorting achieves sorting by operating directly on the original array without requiring additional auxiliary arrays, thus saving memory. Typically, in-place sorting involves fewer data movement operations and runs faster.</p> <p>Stability: Stable sorting ensures that the relative order of equal elements in the array does not change after sorting is completed.</p> <p>Stable sorting is a necessary condition for multi-level sorting scenarios. Suppose we have a table storing student information, where column 1 and column 2 are name and age, respectively. In this case, unstable sorting may cause the ordered nature of the input data to be lost:</p> <pre><code># The input data is sorted by name\n# (name, age)\n  ('A', 19)\n  ('B', 18)\n  ('C', 21)\n  ('D', 19)\n  ('E', 23)\n\n# Suppose we use an unstable sorting algorithm to sort the list by age.\n# In the result, the relative positions of ('D', 19) and ('A', 19) change,\n# so the property that the input data is sorted by name is lost.\n  ('B', 18)\n  ('D', 19)\n  ('A', 19)\n  ('C', 21)\n  ('E', 23)\n</code></pre> <p>Adaptability: Adaptive sorting can utilize the existing order information in the input data to reduce the amount of computation, achieving better time efficiency. The best-case time complexity of adaptive sorting algorithms is typically better than the average time complexity.</p> <p>Comparison-based or non-comparison: Comparison-based sorting relies on comparison operators (\\(&lt;\\), \\(=\\), \\(&gt;\\)) to determine the relative order of elements, thereby sorting the entire array. Its worst-case time complexity has a lower bound of \\(\\Omega(n \\log n)\\). Non-comparison sorting does not use comparison operators and can achieve a time complexity of \\(O(n)\\), but its versatility is relatively limited.</p>","path":["Chapter 11. Sorting","11.1   Sorting Algorithm"],"tags":[]},{"location":"chapter_sorting/sorting_algorithm/#1112-ideal-sorting-algorithm","level":2,"title":"11.1.2   Ideal Sorting Algorithm","text":"<p>Fast, in-place, stable, adaptive, and broadly applicable. Clearly, no sorting algorithm has been discovered to date that combines all of these characteristics. Therefore, when selecting a sorting algorithm, it is necessary to decide based on the specific characteristics of the data and the requirements of the problem.</p> <p>Next, we will examine various sorting algorithms and analyze their advantages and disadvantages based on the evaluation dimensions above.</p>","path":["Chapter 11. Sorting","11.1   Sorting Algorithm"],"tags":[]},{"location":"chapter_sorting/summary/","level":1,"title":"11.11   Summary","text":"","path":["Chapter 11. Sorting","11.11   Summary"],"tags":[]},{"location":"chapter_sorting/summary/#1-key-review","level":3,"title":"1.   Key Review","text":"<ul> <li>Bubble sort achieves sorting by swapping adjacent elements. By adding a flag to enable early return, we can optimize the best-case time complexity of bubble sort to \\(O(n)\\).</li> <li>In each round, insertion sort inserts an element from the unsorted portion into its correct position in the sorted portion. Although insertion sort has a time complexity of \\(O(n^2)\\), it remains very popular for small sorting tasks because each operation is relatively lightweight.</li> <li>Quick sort relies on sentinel partitioning. In sentinel partitioning, repeatedly choosing the worst possible pivot can degrade the time complexity to \\(O(n^2)\\). Choosing a median-based pivot or a random pivot can reduce the probability of this degradation. By recursing on the shorter subarray first, we can effectively reduce the recursion depth and optimize the space complexity to \\(O(\\log n)\\).</li> <li>Merge sort includes two phases: divide and merge, which typically embody the divide-and-conquer strategy. In merge sort, sorting an array requires creating auxiliary arrays, with a space complexity of \\(O(n)\\); however, the space complexity of sorting a linked list can be optimized to \\(O(1)\\).</li> <li>Bucket sort consists of three steps: distributing data into buckets, sorting within buckets, and merging results. It also embodies the divide-and-conquer strategy and is suitable for very large data volumes. The key to bucket sort is distributing data evenly.</li> <li>Counting sort is a special case of bucket sort, which achieves sorting by counting the number of occurrences of data. Counting sort is suitable for situations where the data volume is large but the data range is limited, and requires that data can be converted to positive integers.</li> <li>Radix sort achieves data sorting by sorting digit by digit, requiring that data can be represented as fixed-digit numbers.</li> <li>Overall, we hope to find a sorting algorithm that is efficient, stable, in-place, and adaptive. However, as with other data structures and algorithms, no sorting algorithm can satisfy all of these criteria at the same time. In practice, we need to choose the appropriate sorting algorithm based on the characteristics of the data.</li> <li>Figure 11-19 compares mainstream sorting algorithms in terms of efficiency, stability, in-place property, and adaptability.</li> </ul> <p> Figure 11-19   Sorting algorithm comparison </p>","path":["Chapter 11. Sorting","11.11   Summary"],"tags":[]},{"location":"chapter_sorting/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: In what situations is the stability of sorting algorithms necessary?</p> <p>In reality, we may sort based on a certain attribute of objects. For example, students have two attributes: name and height. We want to implement multi-level sorting: first sort by name to get <code>(A, 180) (B, 185) (C, 170) (D, 170)</code>; then sort by height. Because the sorting algorithm is unstable, we may get <code>(D, 170) (C, 170) (A, 180) (B, 185)</code>.</p> <p>We can see that students D and C have swapped positions, destroying the ordering by name, which is not what we want.</p> <p>Q: Can the order of \"searching from right to left\" and \"searching from left to right\" in sentinel partitioning be swapped?</p> <p>No. When we use the leftmost element as the pivot, we must first \"search from right to left\" and then \"search from left to right\". This conclusion is somewhat counterintuitive; let's analyze the reason.</p> <p>The last step of sentinel partitioning <code>partition()</code> is to swap <code>nums[left]</code> and <code>nums[i]</code>. After the swap is complete, the elements to the left of the pivot are all <code>&lt;=</code> the pivot, which requires that <code>nums[left] &gt;= nums[i]</code> must hold before the last swap. Suppose we first \"search from left to right\", then if we cannot find an element larger than the pivot, we will exit the loop when <code>i == j</code>, at which point it may be that <code>nums[j] == nums[i] &gt; nums[left]</code>. In other words, the last swap operation will swap an element larger than the pivot to the leftmost end of the array, causing sentinel partitioning to fail.</p> <p>For example, given the array <code>[0, 0, 0, 0, 1]</code>, if we first \"search from left to right\", the array after sentinel partitioning is <code>[1, 0, 0, 0, 0]</code>, which is incorrect.</p> <p>By the same reasoning, if we select <code>nums[right]</code> as the pivot, the order is reversed: we must first \"search from left to right\".</p> <p>Q: Regarding the optimization of recursion depth in quick sort, why can selecting the shorter array ensure that the recursion depth does not exceed \\(\\log n\\)?</p> <p>Recursion depth is the number of recursive calls that have not yet returned. Each round of sentinel partitioning divides the original array into two sub-arrays. After this optimization, the sub-array selected for further recursion is at most half the length of the original array. In the worst case, if it is always half as long, the final recursion depth is \\(\\log n\\).</p> <p>Reviewing the original quick sort, we may continuously recurse on the longer array. In the worst case, it would be \\(n\\), \\(n - 1\\), \\(\\dots\\), \\(2\\), \\(1\\), with a recursion depth of \\(n\\). Recursion depth optimization can avoid this situation.</p> <p>Q: When all elements in the array are equal, is the time complexity of quick sort \\(O(n^2)\\)? How should this degenerate case be handled?</p> <p>Yes. In this case, the array can be partitioned into three parts through sentinel partitioning: less than, equal to, and greater than the pivot. We then recurse only on the less-than and greater-than parts. With this approach, an array whose elements are all equal can be sorted in just one round of sentinel partitioning.</p> <p>Q: Why is the worst-case time complexity of bucket sort \\(O(n^2)\\)?</p> <p>In the worst case, all elements are distributed into the same bucket. If we use an \\(O(n^2)\\) algorithm to sort these elements, the time complexity will be \\(O(n^2)\\).</p>","path":["Chapter 11. Sorting","11.11   Summary"],"tags":[]},{"location":"chapter_stack_and_queue/","level":1,"title":"Chapter 5.   Stacks and Queues","text":"<p>Abstract</p> <p>A stack is like cats piled on top of one another, while a queue is like cats lining up.</p> <p>They represent the logical relationships of LIFO (Last In, First Out) and FIFO (First In, First Out), respectively.</p>","path":["Chapter 5. Stacks and Queues","Chapter 5.   Stacks and Queues"],"tags":[]},{"location":"chapter_stack_and_queue/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>5.1   Stack</li> <li>5.2   Queue</li> <li>5.3   Deque</li> <li>5.4   Summary</li> <li>5.5   Exercises</li> </ul>","path":["Chapter 5. Stacks and Queues","Chapter 5.   Stacks and Queues"],"tags":[]},{"location":"chapter_stack_and_queue/deque/","level":1,"title":"5.3   Deque","text":"<p>In a queue, we can only remove elements from the front or add elements at the rear. As shown in Figure 5-7, a double-ended queue (deque) provides greater flexibility, allowing elements to be added or removed at both the front and the rear.</p> <p></p> <p> Figure 5-7   Operations of deque </p>","path":["Chapter 5. Stacks and Queues","5.3   Deque"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#531-common-deque-operations","level":2,"title":"5.3.1   Common Deque Operations","text":"<p>The common operations on a deque are shown in Table 5-3. The specific method names depend on the programming language used.</p> <p> Table 5-3   Efficiency of Deque Operations </p> Method Description Time Complexity <code>push_first()</code> Add element to front \\(O(1)\\) <code>push_last()</code> Add element to rear \\(O(1)\\) <code>pop_first()</code> Remove front element \\(O(1)\\) <code>pop_last()</code> Remove rear element \\(O(1)\\) <code>peek_first()</code> Access front element \\(O(1)\\) <code>peek_last()</code> Access rear element \\(O(1)\\) <p>Similarly, we can directly use the deque classes provided by the programming language:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby deque.py<pre><code>from collections import deque\n\n# Initialize deque\ndeq: deque[int] = deque()\n\n# Enqueue elements\ndeq.append(2)      # Add to rear\ndeq.append(5)\ndeq.append(4)\ndeq.appendleft(3)  # Add to front\ndeq.appendleft(1)\n\n# Access elements\nfront: int = deq[0]  # Front element\nrear: int = deq[-1]  # Rear element\n\n# Dequeue elements\npop_front: int = deq.popleft()  # Front element dequeue\npop_rear: int = deq.pop()       # Rear element dequeue\n\n# Get deque length\nsize: int = len(deq)\n\n# Check if deque is empty\nis_empty: bool = len(deq) == 0\n</code></pre> deque.cpp<pre><code>/* Initialize deque */\ndeque&lt;int&gt; deque;\n\n/* Enqueue elements */\ndeque.push_back(2);   // Add to rear\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // Add to front\ndeque.push_front(1);\n\n/* Access elements */\nint front = deque.front(); // Front element\nint back = deque.back();   // Rear element\n\n/* Dequeue elements */\ndeque.pop_front();  // Front element dequeue\ndeque.pop_back();   // Rear element dequeue\n\n/* Get deque length */\nint size = deque.size();\n\n/* Check if deque is empty */\nbool empty = deque.empty();\n</code></pre> deque.java<pre><code>/* Initialize deque */\nDeque&lt;Integer&gt; deque = new LinkedList&lt;&gt;();\n\n/* Enqueue elements */\ndeque.offerLast(2);   // Add to rear\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // Add to front\ndeque.offerFirst(1);\n\n/* Access elements */\nint peekFirst = deque.peekFirst();  // Front element\nint peekLast = deque.peekLast();    // Rear element\n\n/* Dequeue elements */\nint popFirst = deque.pollFirst();  // Front element dequeue\nint popLast = deque.pollLast();    // Rear element dequeue\n\n/* Get deque length */\nint size = deque.size();\n\n/* Check if deque is empty */\nboolean isEmpty = deque.isEmpty();\n</code></pre> deque.cs<pre><code>/* Initialize deque */\n// In C#, use LinkedList as a deque\nLinkedList&lt;int&gt; deque = new();\n\n/* Enqueue elements */\ndeque.AddLast(2);   // Add to rear\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // Add to front\ndeque.AddFirst(1);\n\n/* Access elements */\nint peekFirst = deque.First.Value;  // Front element\nint peekLast = deque.Last.Value;    // Rear element\n\n/* Dequeue elements */\ndeque.RemoveFirst();  // Front element dequeue\ndeque.RemoveLast();   // Rear element dequeue\n\n/* Get deque length */\nint size = deque.Count;\n\n/* Check if deque is empty */\nbool isEmpty = deque.Count == 0;\n</code></pre> deque_test.go<pre><code>/* Initialize deque */\n// In Go, use list as a deque\ndeque := list.New()\n\n/* Enqueue elements */\ndeque.PushBack(2)      // Add to rear\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // Add to front\ndeque.PushFront(1)\n\n/* Access elements */\nfront := deque.Front() // Front element\nrear := deque.Back()   // Rear element\n\n/* Dequeue elements */\ndeque.Remove(front)    // Front element dequeue\ndeque.Remove(rear)     // Rear element dequeue\n\n/* Get deque length */\nsize := deque.Len()\n\n/* Check if deque is empty */\nisEmpty := deque.Len() == 0\n</code></pre> deque.swift<pre><code>/* Initialize deque */\n// Swift does not have a built-in deque class, can use Array as a deque\nvar deque: [Int] = []\n\n/* Enqueue elements */\ndeque.append(2) // Add to rear\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // Add to front\ndeque.insert(1, at: 0)\n\n/* Access elements */\nlet peekFirst = deque.first! // Front element\nlet peekLast = deque.last! // Rear element\n\n/* Dequeue elements */\n// When using Array simulation, popFirst has O(n) complexity\nlet popFirst = deque.removeFirst() // Front element dequeue\nlet popLast = deque.removeLast() // Rear element dequeue\n\n/* Get deque length */\nlet size = deque.count\n\n/* Check if deque is empty */\nlet isEmpty = deque.isEmpty\n</code></pre> deque.js<pre><code>/* Initialize deque */\n// JavaScript does not have a built-in deque, can only use Array as a deque\nconst deque = [];\n\n/* Enqueue elements */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// Please note that since it's an array, unshift() has O(n) time complexity\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* Access elements */\nconst peekFirst = deque[0];\nconst peekLast = deque[deque.length - 1];\n\n/* Dequeue elements */\n// Please note that since it's an array, shift() has O(n) time complexity\nconst popFront = deque.shift();\nconst popBack = deque.pop();\n\n/* Get deque length */\nconst size = deque.length;\n\n/* Check if deque is empty */\nconst isEmpty = size === 0;\n</code></pre> deque.ts<pre><code>/* Initialize deque */\n// TypeScript does not have a built-in deque, can only use Array as a deque\nconst deque: number[] = [];\n\n/* Enqueue elements */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// Please note that since it's an array, unshift() has O(n) time complexity\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* Access elements */\nconst peekFirst: number = deque[0];\nconst peekLast: number = deque[deque.length - 1];\n\n/* Dequeue elements */\n// Please note that since it's an array, shift() has O(n) time complexity\nconst popFront: number = deque.shift() as number;\nconst popBack: number = deque.pop() as number;\n\n/* Get deque length */\nconst size: number = deque.length;\n\n/* Check if deque is empty */\nconst isEmpty: boolean = size === 0;\n</code></pre> deque.dart<pre><code>/* Initialize deque */\n// In Dart, Queue is defined as a deque\nQueue&lt;int&gt; deque = Queue&lt;int&gt;();\n\n/* Enqueue elements */\ndeque.addLast(2);  // Add to rear\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // Add to front\ndeque.addFirst(1);\n\n/* Access elements */\nint peekFirst = deque.first; // Front element\nint peekLast = deque.last;   // Rear element\n\n/* Dequeue elements */\nint popFirst = deque.removeFirst(); // Front element dequeue\nint popLast = deque.removeLast();   // Rear element dequeue\n\n/* Get deque length */\nint size = deque.length;\n\n/* Check if deque is empty */\nbool isEmpty = deque.isEmpty;\n</code></pre> deque.rs<pre><code>/* Initialize deque */\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* Enqueue elements */\ndeque.push_back(2);  // Add to rear\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // Add to front\ndeque.push_front(1);\n\n/* Access elements */\nif let Some(front) = deque.front() { // Front element\n}\nif let Some(rear) = deque.back() {   // Rear element\n}\n\n/* Dequeue elements */\nif let Some(pop_front) = deque.pop_front() { // Front element dequeue\n}\nif let Some(pop_rear) = deque.pop_back() {   // Rear element dequeue\n}\n\n/* Get deque length */\nlet size = deque.len();\n\n/* Check if deque is empty */\nlet is_empty = deque.is_empty();\n</code></pre> deque.c<pre><code>// C does not provide a built-in deque\n</code></pre> deque.kt<pre><code>/* Initialize deque */\nval deque = LinkedList&lt;Int&gt;()\n\n/* Enqueue elements */\ndeque.offerLast(2)  // Add to rear\ndeque.offerLast(5)\ndeque.offerLast(4)\ndeque.offerFirst(3) // Add to front\ndeque.offerFirst(1)\n\n/* Access elements */\nval peekFirst = deque.peekFirst() // Front element\nval peekLast = deque.peekLast()   // Rear element\n\n/* Dequeue elements */\nval popFirst = deque.pollFirst() // Front element dequeue\nval popLast = deque.pollLast()   // Rear element dequeue\n\n/* Get deque length */\nval size = deque.size\n\n/* Check if deque is empty */\nval isEmpty = deque.isEmpty()\n</code></pre> deque.rb<pre><code># Initialize deque\n# Ruby does not have a built-in deque, can only use Array as a deque\ndeque = []\n\n# Enqueue elements\ndeque &lt;&lt; 2\ndeque &lt;&lt; 5\ndeque &lt;&lt; 4\n# Please note that since it's an array, Array#unshift has O(n) time complexity\ndeque.unshift(3)\ndeque.unshift(1)\n\n# Access elements\npeek_first = deque.first\npeek_last = deque.last\n\n# Dequeue elements\n# Please note that since it's an array, Array#shift has O(n) time complexity\npop_front = deque.shift\npop_back = deque.pop\n\n# Get deque length\nsize = deque.length\n\n# Check if deque is empty\nis_empty = size.zero?\n</code></pre> Code Visualization <p> Full Screen &gt;</p>","path":["Chapter 5. Stacks and Queues","5.3   Deque"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#532-deque-implementation","level":2,"title":"5.3.2   Deque Implementation *","text":"<p>The implementation of a deque is similar to that of a queue. You can choose either a linked list or an array as the underlying data structure.</p>","path":["Chapter 5. Stacks and Queues","5.3   Deque"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#1-doubly-linked-list-implementation","level":3,"title":"1.   Doubly Linked List Implementation","text":"<p>Reviewing the previous section, we used a regular singly linked list to implement a queue because it conveniently allows deleting the head node (corresponding to dequeue) and adding new nodes after the tail node (corresponding to enqueue).</p> <p>For a deque, both the front and rear can perform enqueue and dequeue operations. In other words, a deque needs to implement operations in the opposite direction as well. For this reason, we use a \"doubly linked list\" as the underlying data structure for the deque.</p> <p>As shown in Figure 5-8, we treat the head and tail nodes of the doubly linked list as the front and rear of the deque, implementing functionality to add and remove nodes at both ends.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 5-8   Enqueue and dequeue operations in linked list implementation of deque </p> <p>The implementation code is shown below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linkedlist_deque.py<pre><code>class ListNode:\n    \"\"\"Doubly linked list node\"\"\"\n\n    def __init__(self, val: int):\n        \"\"\"Constructor\"\"\"\n        self.val: int = val\n        self.next: ListNode | None = None  # Successor node reference\n        self.prev: ListNode | None = None  # Predecessor node reference\n\nclass LinkedListDeque:\n    \"\"\"Double-ended queue based on doubly linked list implementation\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        self._front: ListNode | None = None  # Head node front\n        self._rear: ListNode | None = None  # Tail node rear\n        self._size: int = 0  # Length of the double-ended queue\n\n    def size(self) -&gt; int:\n        \"\"\"Get the length of the double-ended queue\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"Check if the double-ended queue is empty\"\"\"\n        return self._size == 0\n\n    def push(self, num: int, is_front: bool):\n        \"\"\"Enqueue operation\"\"\"\n        node = ListNode(num)\n        # If the linked list is empty, make both front and rear point to node\n        if self.is_empty():\n            self._front = self._rear = node\n        # Front of the queue enqueue operation\n        elif is_front:\n            # Add node to the head of the linked list\n            self._front.prev = node\n            node.next = self._front\n            self._front = node  # Update head node\n        # Rear of the queue enqueue operation\n        else:\n            # Add node to the tail of the linked list\n            self._rear.next = node\n            node.prev = self._rear\n            self._rear = node  # Update tail node\n        self._size += 1  # Update queue length\n\n    def push_first(self, num: int):\n        \"\"\"Front of the queue enqueue\"\"\"\n        self.push(num, True)\n\n    def push_last(self, num: int):\n        \"\"\"Rear of the queue enqueue\"\"\"\n        self.push(num, False)\n\n    def pop(self, is_front: bool) -&gt; int:\n        \"\"\"Dequeue operation\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Double-ended queue is empty\")\n        # When the deque has one element, front and rear point to the same node\n        if self._size == 1:\n            val: int = self._front.val\n            self._front = None\n            self._rear = None\n        # Front of the queue dequeue operation\n        elif is_front:\n            val: int = self._front.val  # Temporarily store head node value\n            # Delete head node\n            fnext: ListNode | None = self._front.next\n            fnext.prev = None\n            self._front.next = None\n            self._front = fnext  # Update head node\n        # Rear of the queue dequeue operation\n        else:\n            val: int = self._rear.val  # Temporarily store tail node value\n            # Delete tail node\n            rprev: ListNode | None = self._rear.prev\n            rprev.next = None\n            self._rear.prev = None\n            self._rear = rprev  # Update tail node\n        self._size -= 1  # Update queue length\n        return val\n\n    def pop_first(self) -&gt; int:\n        \"\"\"Front of the queue dequeue\"\"\"\n        return self.pop(True)\n\n    def pop_last(self) -&gt; int:\n        \"\"\"Rear of the queue dequeue\"\"\"\n        return self.pop(False)\n\n    def peek_first(self) -&gt; int:\n        \"\"\"Access front of the queue element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Double-ended queue is empty\")\n        return self._front.val\n\n    def peek_last(self) -&gt; int:\n        \"\"\"Access rear of the queue element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Double-ended queue is empty\")\n        return self._rear.val\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"Return array for printing\"\"\"\n        node = self._front\n        res = [0] * self.size()\n        for i in range(self.size()):\n            res[i] = node.val\n            node = node.next\n        return res\n</code></pre> linkedlist_deque.cpp<pre><code>/* Doubly linked list node */\nstruct DoublyListNode {\n    int val;              // Node value\n    DoublyListNode *next; // Successor node pointer\n    DoublyListNode *prev; // Predecessor node pointer\n    DoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n    }\n};\n\n/* Double-ended queue based on doubly linked list implementation */\nclass LinkedListDeque {\n  private:\n    DoublyListNode *front, *rear; // Head node front, tail node rear\n    int queSize = 0;              // Length of the double-ended queue\n\n  public:\n    /* Constructor */\n    LinkedListDeque() : front(nullptr), rear(nullptr) {\n    }\n\n    /* Destructor */\n    ~LinkedListDeque() {\n        // Traverse linked list to delete nodes and free memory\n        DoublyListNode *pre, *cur = front;\n        while (cur != nullptr) {\n            pre = cur;\n            cur = cur-&gt;next;\n            delete pre;\n        }\n    }\n\n    /* Get the length of the double-ended queue */\n    int size() {\n        return queSize;\n    }\n\n    /* Check if the double-ended queue is empty */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* Enqueue operation */\n    void push(int num, bool isFront) {\n        DoublyListNode *node = new DoublyListNode(num);\n        // If the linked list is empty, make both front and rear point to node\n        if (isEmpty())\n            front = rear = node;\n        // Front of the queue enqueue operation\n        else if (isFront) {\n            // Add node to the head of the linked list\n            front-&gt;prev = node;\n            node-&gt;next = front;\n            front = node; // Update head node\n        // Rear of the queue enqueue operation\n        } else {\n            // Add node to the tail of the linked list\n            rear-&gt;next = node;\n            node-&gt;prev = rear;\n            rear = node; // Update tail node\n        }\n        queSize++; // Update queue length\n    }\n\n    /* Front of the queue enqueue */\n    void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* Rear of the queue enqueue */\n    void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* Dequeue operation */\n    int pop(bool isFront) {\n        if (isEmpty())\n            throw out_of_range(\"Queue is empty\");\n        int val;\n        // When the deque has one element, front and rear point to the same node\n        if (queSize == 1) {\n            val = front-&gt;val;\n            delete front; // Free the shared node once\n            front = nullptr;\n            rear = nullptr;\n        }\n        // Temporarily store head node value\n        else if (isFront) {\n            val = front-&gt;val; // Delete head node\n            // Delete head node\n            DoublyListNode *fNext = front-&gt;next;\n            fNext-&gt;prev = nullptr;\n            front-&gt;next = nullptr;\n            delete front;\n            front = fNext; // Update head node\n        // Temporarily store tail node value\n        } else {\n            val = rear-&gt;val; // Delete tail node\n            // Update tail node\n            DoublyListNode *rPrev = rear-&gt;prev;\n            rPrev-&gt;next = nullptr;\n            rear-&gt;prev = nullptr;\n            delete rear;\n            rear = rPrev; // Update tail node\n        }\n        queSize--; // Update queue length\n        return val;\n    }\n\n    /* Rear of the queue dequeue */\n    int popFirst() {\n        return pop(true);\n    }\n\n    /* Access rear of the queue element */\n    int popLast() {\n        return pop(false);\n    }\n\n    /* Return list for printing */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"Deque is empty\");\n        return front-&gt;val;\n    }\n\n    /* Driver Code */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"Deque is empty\");\n        return rear-&gt;val;\n    }\n\n    /* Return array for printing */\n    vector&lt;int&gt; toVector() {\n        DoublyListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_deque.java<pre><code>/* Doubly linked list node */\nclass ListNode {\n    int val; // Node value\n    ListNode next; // Successor node reference\n    ListNode prev; // Predecessor node reference\n\n    ListNode(int val) {\n        this.val = val;\n        prev = next = null;\n    }\n}\n\n/* Double-ended queue based on doubly linked list implementation */\nclass LinkedListDeque {\n    private ListNode front, rear; // Head node front, tail node rear\n    private int queSize = 0; // Length of the double-ended queue\n\n    public LinkedListDeque() {\n        front = rear = null;\n    }\n\n    /* Get the length of the double-ended queue */\n    public int size() {\n        return queSize;\n    }\n\n    /* Check if the double-ended queue is empty */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* Enqueue operation */\n    private void push(int num, boolean isFront) {\n        ListNode node = new ListNode(num);\n        // If the linked list is empty, make both front and rear point to node\n        if (isEmpty())\n            front = rear = node;\n        // Front of the queue enqueue operation\n        else if (isFront) {\n            // Add node to the head of the linked list\n            front.prev = node;\n            node.next = front;\n            front = node; // Update head node\n        // Rear of the queue enqueue operation\n        } else {\n            // Add node to the tail of the linked list\n            rear.next = node;\n            node.prev = rear;\n            rear = node; // Update tail node\n        }\n        queSize++; // Update queue length\n    }\n\n    /* Front of the queue enqueue */\n    public void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* Rear of the queue enqueue */\n    public void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* Dequeue operation */\n    private int pop(boolean isFront) {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        int val;\n        // When the deque has one element, front and rear point to the same node\n        if (queSize == 1) {\n            val = front.val;\n            front = null;\n            rear = null;\n        }\n        // Temporarily store head node value\n        else if (isFront) {\n            val = front.val; // Delete head node\n            // Delete head node\n            ListNode fNext = front.next;\n            fNext.prev = null;\n            front.next = null;\n            front = fNext; // Update head node\n        // Temporarily store tail node value\n        } else {\n            val = rear.val; // Delete tail node\n            // Update tail node\n            ListNode rPrev = rear.prev;\n            rPrev.next = null;\n            rear.prev = null;\n            rear = rPrev; // Update tail node\n        }\n        queSize--; // Update queue length\n        return val;\n    }\n\n    /* Rear of the queue dequeue */\n    public int popFirst() {\n        return pop(true);\n    }\n\n    /* Access rear of the queue element */\n    public int popLast() {\n        return pop(false);\n    }\n\n    /* Return list for printing */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* Driver Code */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return rear.val;\n    }\n\n    /* Return array for printing */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.cs<pre><code>/* Doubly linked list node */\nclass ListNode(int val) {\n    public int val = val;       // Node value\n    public ListNode? next = null; // Successor node reference\n    public ListNode? prev = null; // Predecessor node reference\n}\n\n/* Double-ended queue based on doubly linked list implementation */\nclass LinkedListDeque {\n    ListNode? front, rear; // Head node front, tail node rear\n    int queSize = 0;      // Length of the double-ended queue\n\n    public LinkedListDeque() {\n        front = null;\n        rear = null;\n    }\n\n    /* Get the length of the double-ended queue */\n    public int Size() {\n        return queSize;\n    }\n\n    /* Check if the double-ended queue is empty */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* Enqueue operation */\n    void Push(int num, bool isFront) {\n        ListNode node = new(num);\n        // If the linked list is empty, make both front and rear point to node\n        if (IsEmpty()) {\n            front = node;\n            rear = node;\n        }\n        // Front of the queue enqueue operation\n        else if (isFront) {\n            // Add node to the head of the linked list\n            front!.prev = node;\n            node.next = front;\n            front = node; // Update head node\n        }\n        // Rear of the queue enqueue operation\n        else {\n            // Add node to the tail of the linked list\n            rear!.next = node;\n            node.prev = rear;\n            rear = node;  // Update tail node\n        }\n\n        queSize++; // Update queue length\n    }\n\n    /* Front of the queue enqueue */\n    public void PushFirst(int num) {\n        Push(num, true);\n    }\n\n    /* Rear of the queue enqueue */\n    public void PushLast(int num) {\n        Push(num, false);\n    }\n\n    /* Dequeue operation */\n    int? Pop(bool isFront) {\n        if (IsEmpty())\n            throw new Exception();\n        int? val;\n        // When the deque has one element, front and rear point to the same node\n        if (queSize == 1) {\n            val = front!.val;\n            front = null;\n            rear = null;\n        }\n        // Temporarily store head node value\n        else if (isFront) {\n            val = front!.val; // Delete head node\n            // Delete head node\n            ListNode? fNext = front?.next;\n            fNext!.prev = null;\n            front.next = null;\n            front = fNext;   // Update head node\n        }\n        // Temporarily store tail node value\n        else {\n            val = rear!.val;  // Delete tail node\n            // Update tail node\n            ListNode? rPrev = rear?.prev;\n            rPrev!.next = null;\n            rear.prev = null;\n            rear = rPrev;    // Update tail node\n        }\n\n        queSize--; // Update queue length\n        return val;\n    }\n\n    /* Rear of the queue dequeue */\n    public int? PopFirst() {\n        return Pop(true);\n    }\n\n    /* Access rear of the queue element */\n    public int? PopLast() {\n        return Pop(false);\n    }\n\n    /* Return list for printing */\n    public int? PeekFirst() {\n        if (IsEmpty())\n            throw new Exception();\n        return front?.val;\n    }\n\n    /* Driver Code */\n    public int? PeekLast() {\n        if (IsEmpty())\n            throw new Exception();\n        return rear?.val;\n    }\n\n    /* Return array for printing */\n    public int?[] ToArray() {\n        ListNode? node = front;\n        int?[] res = new int?[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node?.val;\n            node = node?.next;\n        }\n\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.go<pre><code>/* Double-ended queue based on doubly linked list implementation */\ntype linkedListDeque struct {\n    // Use built-in package list\n    data *list.List\n}\n\n/* Initialize deque */\nfunc newLinkedListDeque() *linkedListDeque {\n    return &amp;linkedListDeque{\n        data: list.New(),\n    }\n}\n\n/* Front element enqueue */\nfunc (s *linkedListDeque) pushFirst(value any) {\n    s.data.PushFront(value)\n}\n\n/* Rear element enqueue */\nfunc (s *linkedListDeque) pushLast(value any) {\n    s.data.PushBack(value)\n}\n\n/* Check if the double-ended queue is empty */\nfunc (s *linkedListDeque) popFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* Rear element dequeue */\nfunc (s *linkedListDeque) popLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* Return list for printing */\nfunc (s *linkedListDeque) peekFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* Driver Code */\nfunc (s *linkedListDeque) peekLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* Get the length of the queue */\nfunc (s *linkedListDeque) size() int {\n    return s.data.Len()\n}\n\n/* Check if the queue is empty */\nfunc (s *linkedListDeque) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* Get List for printing */\nfunc (s *linkedListDeque) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_deque.swift<pre><code>/* Doubly linked list node */\nclass ListNode {\n    var val: Int // Node value\n    var next: ListNode? // Successor node reference\n    weak var prev: ListNode? // Predecessor node reference\n\n    init(val: Int) {\n        self.val = val\n    }\n}\n\n/* Double-ended queue based on doubly linked list implementation */\nclass LinkedListDeque {\n    private var front: ListNode? // Head node front\n    private var rear: ListNode? // Tail node rear\n    private var _size: Int // Length of the double-ended queue\n\n    init() {\n        _size = 0\n    }\n\n    /* Get the length of the double-ended queue */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* Check if the double-ended queue is empty */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* Enqueue operation */\n    private func push(num: Int, isFront: Bool) {\n        let node = ListNode(val: num)\n        // If the linked list is empty, make both front and rear point to node\n        if isEmpty() {\n            front = node\n            rear = node\n        }\n        // Front of the queue enqueue operation\n        else if isFront {\n            // Add node to the head of the linked list\n            front?.prev = node\n            node.next = front\n            front = node // Update head node\n        }\n        // Rear of the queue enqueue operation\n        else {\n            // Add node to the tail of the linked list\n            rear?.next = node\n            node.prev = rear\n            rear = node // Update tail node\n        }\n        _size += 1 // Update queue length\n    }\n\n    /* Front of the queue enqueue */\n    func pushFirst(num: Int) {\n        push(num: num, isFront: true)\n    }\n\n    /* Rear of the queue enqueue */\n    func pushLast(num: Int) {\n        push(num: num, isFront: false)\n    }\n\n    /* Dequeue operation */\n    private func pop(isFront: Bool) -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Deque is empty\")\n        }\n        let val: Int\n        // When the deque has one element, front and rear point to the same node\n        if _size == 1 {\n            val = front!.val\n            front = nil\n            rear = nil\n        }\n        // Temporarily store head node value\n        else if isFront {\n            val = front!.val // Delete head node\n            // Delete head node\n            let fNext = front?.next\n            fNext?.prev = nil\n            front?.next = nil\n            front = fNext // Update head node\n        }\n        // Temporarily store tail node value\n        else {\n            val = rear!.val // Delete tail node\n            // Update tail node\n            let rPrev = rear?.prev\n            rPrev?.next = nil\n            rear?.prev = nil\n            rear = rPrev // Update tail node\n        }\n        _size -= 1 // Update queue length\n        return val\n    }\n\n    /* Rear of the queue dequeue */\n    func popFirst() -&gt; Int {\n        pop(isFront: true)\n    }\n\n    /* Access rear of the queue element */\n    func popLast() -&gt; Int {\n        pop(isFront: false)\n    }\n\n    /* Return list for printing */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Deque is empty\")\n        }\n        return front!.val\n    }\n\n    /* Driver Code */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Deque is empty\")\n        }\n        return rear!.val\n    }\n\n    /* Return array for printing */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.js<pre><code>/* Doubly linked list node */\nclass ListNode {\n    prev; // Predecessor node reference (pointer)\n    next; // Successor node reference (pointer)\n    val; // Node value\n\n    constructor(val) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* Double-ended queue based on doubly linked list implementation */\nclass LinkedListDeque {\n    #front; // Head node front\n    #rear; // Tail node rear\n    #queSize; // Length of the double-ended queue\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n        this.#queSize = 0;\n    }\n\n    /* Rear of the queue enqueue operation */\n    pushLast(val) {\n        const node = new ListNode(val);\n        // If the linked list is empty, make both front and rear point to node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // Add node to the tail of the linked list\n            this.#rear.next = node;\n            node.prev = this.#rear;\n            this.#rear = node; // Update tail node\n        }\n        this.#queSize++;\n    }\n\n    /* Front of the queue enqueue operation */\n    pushFirst(val) {\n        const node = new ListNode(val);\n        // If the linked list is empty, make both front and rear point to node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // Add node to the head of the linked list\n            this.#front.prev = node;\n            node.next = this.#front;\n            this.#front = node; // Update head node\n        }\n        this.#queSize++;\n    }\n\n    /* Temporarily store tail node value */\n    popLast() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#rear.val; // Store tail node value\n        // When the deque has one element, front and rear point to the same node\n        if (this.#queSize === 1) {\n            this.#front = null;\n            this.#rear = null;\n        }\n        // Update tail node\n        else {\n            let temp = this.#rear.prev;\n            temp.next = null;\n            this.#rear.prev = null;\n            this.#rear = temp; // Update tail node\n        }\n        this.#queSize--;\n        return value;\n    }\n\n    /* Temporarily store head node value */\n    popFirst() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#front.val; // Store the head node value\n        // When the deque has one element, front and rear point to the same node\n        if (this.#queSize === 1) {\n            this.#front = null;\n            this.#rear = null;\n        }\n        // Delete head node\n        else {\n            let temp = this.#front.next;\n            temp.prev = null;\n            this.#front.next = null;\n            this.#front = temp; // Update head node\n        }\n        this.#queSize--;\n        return value;\n    }\n\n    /* Driver Code */\n    peekLast() {\n        return this.#queSize === 0 ? null : this.#rear.val;\n    }\n\n    /* Return list for printing */\n    peekFirst() {\n        return this.#queSize === 0 ? null : this.#front.val;\n    }\n\n    /* Get the length of the double-ended queue */\n    size() {\n        return this.#queSize;\n    }\n\n    /* Check if the double-ended queue is empty */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* Print deque */\n    print() {\n        const arr = [];\n        let temp = this.#front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.ts<pre><code>/* Doubly linked list node */\nclass ListNode {\n    prev: ListNode; // Predecessor node reference (pointer)\n    next: ListNode; // Successor node reference (pointer)\n    val: number; // Node value\n\n    constructor(val: number) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* Double-ended queue based on doubly linked list implementation */\nclass LinkedListDeque {\n    private front: ListNode; // Head node front\n    private rear: ListNode; // Tail node rear\n    private queSize: number; // Length of the double-ended queue\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n        this.queSize = 0;\n    }\n\n    /* Rear of the queue enqueue operation */\n    pushLast(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // If the linked list is empty, make both front and rear point to node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // Add node to the tail of the linked list\n            this.rear.next = node;\n            node.prev = this.rear;\n            this.rear = node; // Update tail node\n        }\n        this.queSize++;\n    }\n\n    /* Front of the queue enqueue operation */\n    pushFirst(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // If the linked list is empty, make both front and rear point to node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // Add node to the head of the linked list\n            this.front.prev = node;\n            node.next = this.front;\n            this.front = node; // Update head node\n        }\n        this.queSize++;\n    }\n\n    /* Temporarily store tail node value */\n    popLast(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.rear!.val; // Store tail node value\n        // When the deque has one element, front and rear point to the same node\n        if (this.queSize === 1) {\n            this.front = null;\n            this.rear = null;\n        }\n        // Update tail node\n        else {\n            let temp: ListNode = this.rear!.prev;\n            temp!.next = null;\n            this.rear!.prev = null;\n            this.rear = temp; // Update tail node\n        }\n        this.queSize--;\n        return value;\n    }\n\n    /* Temporarily store head node value */\n    popFirst(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.front.val; // Store the head node value\n        // When the deque has one element, front and rear point to the same node\n        if (this.queSize === 1) {\n            this.front = null;\n            this.rear = null;\n        }\n        // Delete head node\n        else {\n            let temp: ListNode = this.front!.next;\n            temp!.prev = null;\n            this.front!.next = null;\n            this.front = temp; // Update head node\n        }\n        this.queSize--;\n        return value;\n    }\n\n    /* Driver Code */\n    peekLast(): number {\n        return this.queSize === 0 ? null : this.rear.val;\n    }\n\n    /* Return list for printing */\n    peekFirst(): number {\n        return this.queSize === 0 ? null : this.front.val;\n    }\n\n    /* Get the length of the double-ended queue */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* Check if the double-ended queue is empty */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* Print deque */\n    print(): void {\n        const arr: number[] = [];\n        let temp: ListNode = this.front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.dart<pre><code>/* Doubly linked list node */\nclass ListNode {\n  int val; // Node value\n  ListNode? next; // Successor node reference\n  ListNode? prev; // Predecessor node reference\n\n  ListNode(this.val, {this.next, this.prev});\n}\n\n/* Deque implemented based on doubly linked list */\nclass LinkedListDeque {\n  late ListNode? _front; // Head node _front\n  late ListNode? _rear; // Tail node _rear\n  int _queSize = 0; // Length of the double-ended queue\n\n  LinkedListDeque() {\n    this._front = null;\n    this._rear = null;\n  }\n\n  /* Get deque length */\n  int size() {\n    return this._queSize;\n  }\n\n  /* Check if the double-ended queue is empty */\n  bool isEmpty() {\n    return size() == 0;\n  }\n\n  /* Enqueue operation */\n  void push(int _num, bool isFront) {\n    final ListNode node = ListNode(_num);\n    if (isEmpty()) {\n      // If list is empty, let both _front and _rear point to node\n      _front = _rear = node;\n    } else if (isFront) {\n      // Front of the queue enqueue operation\n      // Add node to the head of the linked list\n      _front!.prev = node;\n      node.next = _front;\n      _front = node; // Update head node\n    } else {\n      // Rear of the queue enqueue operation\n      // Add node to the tail of the linked list\n      _rear!.next = node;\n      node.prev = _rear;\n      _rear = node; // Update tail node\n    }\n    _queSize++; // Update queue length\n  }\n\n  /* Front of the queue enqueue */\n  void pushFirst(int _num) {\n    push(_num, true);\n  }\n\n  /* Rear of the queue enqueue */\n  void pushLast(int _num) {\n    push(_num, false);\n  }\n\n  /* Dequeue operation */\n  int? pop(bool isFront) {\n    // If queue is empty, return null directly\n    if (isEmpty()) {\n      return null;\n    }\n    final int val;\n    // When the deque has one element, front and rear point to the same node\n    if (_queSize == 1) {\n      val = _front!.val;\n      _front = null;\n      _rear = null;\n    }\n    // Temporarily store head node value\n    else if (isFront) {\n      val = _front!.val; // Delete head node\n      // Delete head node\n      ListNode? fNext = _front!.next;\n      fNext!.prev = null;\n      _front!.next = null;\n      _front = fNext; // Update head node\n    // Temporarily store tail node value\n    } else {\n      val = _rear!.val; // Delete tail node\n      // Update tail node\n      ListNode? rPrev = _rear!.prev;\n      rPrev!.next = null;\n      _rear!.prev = null;\n      _rear = rPrev; // Update tail node\n    }\n    _queSize--; // Update queue length\n    return val;\n  }\n\n  /* Rear of the queue dequeue */\n  int? popFirst() {\n    return pop(true);\n  }\n\n  /* Access rear of the queue element */\n  int? popLast() {\n    return pop(false);\n  }\n\n  /* Return list for printing */\n  int? peekFirst() {\n    return _front?.val;\n  }\n\n  /* Driver Code */\n  int? peekLast() {\n    return _rear?.val;\n  }\n\n  /* Return array for printing */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; _queSize; i++) {\n      res.add(node!.val);\n      node = node.next;\n    }\n    return res;\n  }\n}\n</code></pre> linkedlist_deque.rs<pre><code>/* Doubly linked list node */\npub struct ListNode&lt;T&gt; {\n    pub val: T,                                 // Node value\n    pub next: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // Successor node pointer\n    pub prev: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // Predecessor node pointer\n}\n\nimpl&lt;T&gt; ListNode&lt;T&gt; {\n    pub fn new(val: T) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n        Rc::new(RefCell::new(ListNode {\n            val,\n            next: None,\n            prev: None,\n        }))\n    }\n}\n\n/* Double-ended queue based on doubly linked list implementation */\n#[allow(dead_code)]\npub struct LinkedListDeque&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // Head node front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // Tail node rear\n    que_size: usize,                         // Length of the double-ended queue\n}\n\nimpl&lt;T: Copy&gt; LinkedListDeque&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* Get the length of the double-ended queue */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* Check if the double-ended queue is empty */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.que_size == 0;\n    }\n\n    /* Enqueue operation */\n    fn push(&amp;mut self, num: T, is_front: bool) {\n        let node = ListNode::new(num);\n        // Front of the queue enqueue operation\n        if is_front {\n            match self.front.take() {\n                // If the linked list is empty, make both front and rear point to node\n                None =&gt; {\n                    self.rear = Some(node.clone());\n                    self.front = Some(node);\n                }\n                // Add node to the head of the linked list\n                Some(old_front) =&gt; {\n                    old_front.borrow_mut().prev = Some(node.clone());\n                    node.borrow_mut().next = Some(old_front);\n                    self.front = Some(node); // Update head node\n                }\n            }\n        }\n        // Rear of the queue enqueue operation\n        else {\n            match self.rear.take() {\n                // If the linked list is empty, make both front and rear point to node\n                None =&gt; {\n                    self.front = Some(node.clone());\n                    self.rear = Some(node);\n                }\n                // Add node to the tail of the linked list\n                Some(old_rear) =&gt; {\n                    old_rear.borrow_mut().next = Some(node.clone());\n                    node.borrow_mut().prev = Some(old_rear);\n                    self.rear = Some(node); // Update tail node\n                }\n            }\n        }\n        self.que_size += 1; // Update queue length\n    }\n\n    /* Front of the queue enqueue */\n    pub fn push_first(&amp;mut self, num: T) {\n        self.push(num, true);\n    }\n\n    /* Rear of the queue enqueue */\n    pub fn push_last(&amp;mut self, num: T) {\n        self.push(num, false);\n    }\n\n    /* Dequeue operation */\n    fn pop(&amp;mut self, is_front: bool) -&gt; Option&lt;T&gt; {\n        // If queue is empty, return None directly\n        if self.is_empty() {\n            return None;\n        };\n        // Temporarily store head node value\n        if is_front {\n            self.front.take().map(|old_front| {\n                match old_front.borrow_mut().next.take() {\n                    Some(new_front) =&gt; {\n                        new_front.borrow_mut().prev.take();\n                        self.front = Some(new_front); // Update head node\n                    }\n                    None =&gt; {\n                        self.rear.take();\n                    }\n                }\n                self.que_size -= 1; // Update queue length\n                old_front.borrow().val\n            })\n        }\n        // Temporarily store tail node value\n        else {\n            self.rear.take().map(|old_rear| {\n                match old_rear.borrow_mut().prev.take() {\n                    Some(new_rear) =&gt; {\n                        new_rear.borrow_mut().next.take();\n                        self.rear = Some(new_rear); // Update tail node\n                    }\n                    None =&gt; {\n                        self.front.take();\n                    }\n                }\n                self.que_size -= 1; // Update queue length\n                old_rear.borrow().val\n            })\n        }\n    }\n\n    /* Rear of the queue dequeue */\n    pub fn pop_first(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(true);\n    }\n\n    /* Access rear of the queue element */\n    pub fn pop_last(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(false);\n    }\n\n    /* Return list for printing */\n    pub fn peek_first(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* Driver Code */\n    pub fn peek_last(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.rear.as_ref()\n    }\n\n    /* Return array for printing */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        let mut res: Vec&lt;T&gt; = Vec::new();\n        fn recur&lt;T: Copy&gt;(cur: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, res: &amp;mut Vec&lt;T&gt;) {\n            if let Some(cur) = cur {\n                res.push(cur.borrow().val);\n                recur(cur.borrow().next.as_ref(), res);\n            }\n        }\n\n        recur(head, &amp;mut res);\n        res\n    }\n}\n</code></pre> linkedlist_deque.c<pre><code>/* Doubly linked list node */\ntypedef struct DoublyListNode {\n    int val;                     // Node value\n    struct DoublyListNode *next; // Successor node\n    struct DoublyListNode *prev; // Predecessor node\n} DoublyListNode;\n\n/* Constructor */\nDoublyListNode *newDoublyListNode(int num) {\n    DoublyListNode *new = (DoublyListNode *)malloc(sizeof(DoublyListNode));\n    new-&gt;val = num;\n    new-&gt;next = NULL;\n    new-&gt;prev = NULL;\n    return new;\n}\n\n/* Destructor */\nvoid delDoublyListNode(DoublyListNode *node) {\n    free(node);\n}\n\n/* Double-ended queue based on doubly linked list implementation */\ntypedef struct {\n    DoublyListNode *front, *rear; // Head node front, tail node rear\n    int queSize;                  // Length of the double-ended queue\n} LinkedListDeque;\n\n/* Constructor */\nLinkedListDeque *newLinkedListDeque() {\n    LinkedListDeque *deque = (LinkedListDeque *)malloc(sizeof(LinkedListDeque));\n    deque-&gt;front = NULL;\n    deque-&gt;rear = NULL;\n    deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* Destructor */\nvoid delLinkedListdeque(LinkedListDeque *deque) {\n    // Free all nodes\n    for (int i = 0; i &lt; deque-&gt;queSize &amp;&amp; deque-&gt;front != NULL; i++) {\n        DoublyListNode *tmp = deque-&gt;front;\n        deque-&gt;front = deque-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // Free deque structure\n    free(deque);\n}\n\n/* Get the length of the queue */\nint size(LinkedListDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* Check if the queue is empty */\nbool empty(LinkedListDeque *deque) {\n    return (size(deque) == 0);\n}\n\n/* Enqueue */\nvoid push(LinkedListDeque *deque, int num, bool isFront) {\n    DoublyListNode *node = newDoublyListNode(num);\n    // If list is empty, set both front and rear to node\n    if (empty(deque)) {\n        deque-&gt;front = deque-&gt;rear = node;\n    }\n    // Front of the queue enqueue operation\n    else if (isFront) {\n        // Add node to the head of the linked list\n        deque-&gt;front-&gt;prev = node;\n        node-&gt;next = deque-&gt;front;\n        deque-&gt;front = node; // Update head node\n    }\n    // Rear of the queue enqueue operation\n    else {\n        // Add node to the tail of the linked list\n        deque-&gt;rear-&gt;next = node;\n        node-&gt;prev = deque-&gt;rear;\n        deque-&gt;rear = node;\n    }\n    deque-&gt;queSize++; // Update queue length\n}\n\n/* Front of the queue enqueue */\nvoid pushFirst(LinkedListDeque *deque, int num) {\n    push(deque, num, true);\n}\n\n/* Rear of the queue enqueue */\nvoid pushLast(LinkedListDeque *deque, int num) {\n    push(deque, num, false);\n}\n\n/* Return list for printing */\nint peekFirst(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;front);\n    return deque-&gt;front-&gt;val;\n}\n\n/* Driver Code */\nint peekLast(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;rear);\n    return deque-&gt;rear-&gt;val;\n}\n\n/* Dequeue */\nint pop(LinkedListDeque *deque, bool isFront) {\n    if (empty(deque))\n        return -1;\n    int val;\n    // When the deque has one element, front and rear point to the same node\n    if (deque-&gt;queSize == 1) {\n        val = deque-&gt;front-&gt;val;\n        delDoublyListNode(deque-&gt;front); // Free the shared node once\n        deque-&gt;front = NULL;\n        deque-&gt;rear = NULL;\n    }\n    // Temporarily store head node value\n    else if (isFront) {\n        val = peekFirst(deque); // Delete head node\n        DoublyListNode *fNext = deque-&gt;front-&gt;next;\n        fNext-&gt;prev = NULL;\n        deque-&gt;front-&gt;next = NULL;\n        delDoublyListNode(deque-&gt;front);\n        deque-&gt;front = fNext; // Update head node\n    }\n    // Temporarily store tail node value\n    else {\n        val = peekLast(deque); // Delete tail node\n        DoublyListNode *rPrev = deque-&gt;rear-&gt;prev;\n        rPrev-&gt;next = NULL;\n        deque-&gt;rear-&gt;prev = NULL;\n        delDoublyListNode(deque-&gt;rear);\n        deque-&gt;rear = rPrev; // Update tail node\n    }\n    deque-&gt;queSize--; // Update queue length\n    return val;\n}\n\n/* Rear of the queue dequeue */\nint popFirst(LinkedListDeque *deque) {\n    return pop(deque, true);\n}\n\n/* Access rear of the queue element */\nint popLast(LinkedListDeque *deque) {\n    return pop(deque, false);\n}\n\n/* Print queue */\nvoid printLinkedListDeque(LinkedListDeque *deque) {\n    int *arr = malloc(sizeof(int) * deque-&gt;queSize);\n    // Copy data from list to array\n    int i;\n    DoublyListNode *node;\n    for (i = 0, node = deque-&gt;front; i &lt; deque-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, deque-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_deque.kt<pre><code>/* Doubly linked list node */\nclass ListNode(var _val: Int) {\n    // Node value\n    var next: ListNode? = null // Successor node reference\n    var prev: ListNode? = null // Predecessor node reference\n}\n\n/* Double-ended queue based on doubly linked list implementation */\nclass LinkedListDeque {\n    private var front: ListNode? = null // Head node front\n    private var rear: ListNode? = null // Tail node rear\n    private var queSize: Int = 0 // Length of the double-ended queue\n\n    /* Get the length of the double-ended queue */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* Check if the double-ended queue is empty */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* Enqueue operation */\n    fun push(num: Int, isFront: Boolean) {\n        val node = ListNode(num)\n        // If the linked list is empty, make both front and rear point to node\n        if (isEmpty()) {\n            rear = node\n            front = rear\n            // Front of the queue enqueue operation\n        } else if (isFront) {\n            // Add node to the head of the linked list\n            front?.prev = node\n            node.next = front\n            front = node // Update head node\n            // Rear of the queue enqueue operation\n        } else {\n            // Add node to the tail of the linked list\n            rear?.next = node\n            node.prev = rear\n            rear = node // Update tail node\n        }\n        queSize++ // Update queue length\n    }\n\n    /* Front of the queue enqueue */\n    fun pushFirst(num: Int) {\n        push(num, true)\n    }\n\n    /* Rear of the queue enqueue */\n    fun pushLast(num: Int) {\n        push(num, false)\n    }\n\n    /* Dequeue operation */\n    fun pop(isFront: Boolean): Int {\n        if (isEmpty()) \n            throw IndexOutOfBoundsException()\n        val _val: Int\n        // When the deque has one element, front and rear point to the same node\n        if (queSize == 1) {\n            _val = front!!._val\n            front = null\n            rear = null\n        }\n        // Temporarily store head node value\n        else if (isFront) {\n            _val = front!!._val // Delete head node\n            // Delete head node\n            val fNext = front!!.next\n            fNext!!.prev = null\n            front!!.next = null\n            front = fNext // Update head node\n            // Temporarily store tail node value\n        } else {\n            _val = rear!!._val // Delete tail node\n            // Update tail node\n            val rPrev = rear!!.prev\n            rPrev!!.next = null\n            rear!!.prev = null\n            rear = rPrev // Update tail node\n        }\n        queSize-- // Update queue length\n        return _val\n    }\n\n    /* Rear of the queue dequeue */\n    fun popFirst(): Int {\n        return pop(true)\n    }\n\n    /* Access rear of the queue element */\n    fun popLast(): Int {\n        return pop(false)\n    }\n\n    /* Return list for printing */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!._val\n    }\n\n    /* Driver Code */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return rear!!._val\n    }\n\n    /* Return array for printing */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!._val\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.rb<pre><code>=begin\nFile: linkedlist_deque.rb\nCreated Time: 2024-04-06\nAuthor: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)\n=end\n\n### Doubly linked list node\nclass ListNode\n  attr_accessor :val\n  attr_accessor :next # Successor node reference\n  attr_accessor :prev # Predecessor node reference\n\n  ### Constructor ###\n  def initialize(val)\n    @val = val\n  end\nend\n\n### Deque based on doubly linked list ###\nclass LinkedListDeque\n  ### Get deque length ###\n  attr_reader :size\n\n  ### Constructor ###\n  def initialize\n    @front = nil  # Head node front\n    @rear = nil   # Tail node rear\n    @size = 0     # Length of the double-ended queue\n  end\n\n  ### Check if deque is empty ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### Enqueue operation ###\n  def push(num, is_front)\n    node = ListNode.new(num)\n    # If list is empty, set both front and rear to node\n    if is_empty?\n      @front = @rear = node\n    # Front of the queue enqueue operation\n    elsif is_front\n      # Add node to the head of the linked list\n      @front.prev = node\n      node.next = @front\n      @front = node # Update head node\n    # Rear of the queue enqueue operation\n    else\n      # Add node to the tail of the linked list\n      @rear.next = node\n      node.prev = @rear\n      @rear = node # Update tail node\n    end\n    @size += 1 # Update queue length\n  end\n\n  ### Enqueue at front ###\n  def push_first(num)\n    push(num, true)\n  end\n\n  ### Enqueue at rear ###\n  def push_last(num)\n    push(num, false)\n  end\n\n  ### Dequeue operation ###\n  def pop(is_front)\n    raise IndexError, 'Deque is empty' if is_empty?\n\n    # When the deque has one element, front and rear point to the same node\n    if @size == 1\n      val = @front.val\n      @front = nil\n      @rear = nil\n    # Temporarily store head node value\n    elsif is_front\n      val = @front.val # Delete head node\n      # Delete head node\n      fnext = @front.next\n      fnext.prev = nil\n      @front.next = nil\n      @front = fnext # Update head node\n    # Temporarily store tail node value\n    else\n      val = @rear.val # Delete tail node\n      # Update tail node\n      rprev = @rear.prev\n      rprev.next = nil\n      @rear.prev = nil\n      @rear = rprev # Update tail node\n    end\n    @size -= 1 # Update queue length\n\n    val\n  end\n\n  ### Dequeue from front ###\n  def pop_first\n    pop(true)\n  end\n\n  ### Dequeue from front ###\n  def pop_last\n    pop(false)\n  end\n\n  ### Access front element ###\n  def peek_first\n    raise IndexError, 'Deque is empty' if is_empty?\n\n    @front.val\n  end\n\n  ### Access rear element ###\n  def peek_last\n    raise IndexError, 'Deque is empty' if is_empty?\n\n    @rear.val\n  end\n\n  ### Return array for printing ###\n  def to_array\n    node = @front\n    res = Array.new(size, 0)\n    for i in 0...size\n      res[i] = node.val\n      node = node.next\n    end\n    res\n  end\nend\n</code></pre>","path":["Chapter 5. Stacks and Queues","5.3   Deque"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#2-array-implementation","level":3,"title":"2.   Array Implementation","text":"<p>As shown in Figure 5-9, similar to implementing a queue based on an array, we can also use a circular array to implement a deque.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 5-9   Enqueue and dequeue operations in array implementation of deque </p> <p>Based on the queue implementation, we only need to add methods for \"enqueue at front\" and \"dequeue from rear\":</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_deque.py<pre><code>class ArrayDeque:\n    \"\"\"Double-ended queue based on circular array implementation\"\"\"\n\n    def __init__(self, capacity: int):\n        \"\"\"Constructor\"\"\"\n        self._nums: list[int] = [0] * capacity\n        self._front: int = 0\n        self._size: int = 0\n\n    def capacity(self) -&gt; int:\n        \"\"\"Get the capacity of the double-ended queue\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"Get the length of the double-ended queue\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"Check if the double-ended queue is empty\"\"\"\n        return self._size == 0\n\n    def index(self, i: int) -&gt; int:\n        \"\"\"Calculate circular array index\"\"\"\n        # Use modulo operation to wrap the array head and tail together\n        # When i passes the tail of the array, return to the head\n        # When i passes the head of the array, return to the tail\n        return (i + self.capacity()) % self.capacity()\n\n    def push_first(self, num: int):\n        \"\"\"Front of the queue enqueue\"\"\"\n        if self._size == self.capacity():\n            print(\"Double-ended queue is full\")\n            return\n        # Front pointer moves one position to the left\n        # Use modulo operation to wrap front around to the tail after passing the head of the array\n        self._front = self.index(self._front - 1)\n        # Add num to the front of the queue\n        self._nums[self._front] = num\n        self._size += 1\n\n    def push_last(self, num: int):\n        \"\"\"Rear of the queue enqueue\"\"\"\n        if self._size == self.capacity():\n            print(\"Double-ended queue is full\")\n            return\n        # Calculate rear pointer, points to rear index + 1\n        rear = self.index(self._front + self._size)\n        # Add num to the rear of the queue\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop_first(self) -&gt; int:\n        \"\"\"Front of the queue dequeue\"\"\"\n        num = self.peek_first()\n        # Front pointer moves one position backward\n        self._front = self.index(self._front + 1)\n        self._size -= 1\n        return num\n\n    def pop_last(self) -&gt; int:\n        \"\"\"Rear of the queue dequeue\"\"\"\n        num = self.peek_last()\n        self._size -= 1\n        return num\n\n    def peek_first(self) -&gt; int:\n        \"\"\"Access front of the queue element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Double-ended queue is empty\")\n        return self._nums[self._front]\n\n    def peek_last(self) -&gt; int:\n        \"\"\"Access rear of the queue element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Double-ended queue is empty\")\n        # Calculate tail element index\n        last = self.index(self._front + self._size - 1)\n        return self._nums[last]\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"Return array for printing\"\"\"\n        # Only convert list elements within the valid length range\n        res = []\n        for i in range(self._size):\n            res.append(self._nums[self.index(self._front + i)])\n        return res\n</code></pre> array_deque.cpp<pre><code>/* Double-ended queue based on circular array implementation */\nclass ArrayDeque {\n  private:\n    vector&lt;int&gt; nums; // Array for storing double-ended queue elements\n    int front;        // Front pointer, points to the front of the queue element\n    int queSize;      // Double-ended queue length\n\n  public:\n    /* Constructor */\n    ArrayDeque(int capacity) {\n        nums.resize(capacity);\n        front = queSize = 0;\n    }\n\n    /* Get the capacity of the double-ended queue */\n    int capacity() {\n        return nums.size();\n    }\n\n    /* Get the length of the double-ended queue */\n    int size() {\n        return queSize;\n    }\n\n    /* Check if the double-ended queue is empty */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* Calculate circular array index */\n    int index(int i) {\n        // Use modulo operation to wrap the array head and tail together\n        // When i passes the tail of the array, return to the head\n        // When i passes the head of the array, return to the tail\n        return (i + capacity()) % capacity();\n    }\n\n    /* Front of the queue enqueue */\n    void pushFirst(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"Double-ended queue is full\" &lt;&lt; endl;\n            return;\n        }\n        // Use modulo operation to wrap front around to the tail after passing the head of the array\n        // Add num to the front of the queue\n        front = index(front - 1);\n        // Add num to front of queue\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* Rear of the queue enqueue */\n    void pushLast(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"Double-ended queue is full\" &lt;&lt; endl;\n            return;\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        int rear = index(front + queSize);\n        // Front pointer moves one position backward\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* Rear of the queue dequeue */\n    int popFirst() {\n        int num = peekFirst();\n        // Move front pointer backward by one position\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* Access rear of the queue element */\n    int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"Deque is empty\");\n        return nums[front];\n    }\n\n    /* Driver Code */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"Deque is empty\");\n        // Initialize double-ended queue\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* Return array for printing */\n    vector&lt;int&gt; toVector() {\n        // Elements enqueue\n        vector&lt;int&gt; res(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n};\n</code></pre> array_deque.java<pre><code>/* Double-ended queue based on circular array implementation */\nclass ArrayDeque {\n    private int[] nums; // Array for storing double-ended queue elements\n    private int front; // Front pointer, points to the front of the queue element\n    private int queSize; // Double-ended queue length\n\n    /* Constructor */\n    public ArrayDeque(int capacity) {\n        this.nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* Get the capacity of the double-ended queue */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* Get the length of the double-ended queue */\n    public int size() {\n        return queSize;\n    }\n\n    /* Check if the double-ended queue is empty */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* Calculate circular array index */\n    private int index(int i) {\n        // Use modulo operation to wrap the array head and tail together\n        // When i passes the tail of the array, return to the head\n        // When i passes the head of the array, return to the tail\n        return (i + capacity()) % capacity();\n    }\n\n    /* Front of the queue enqueue */\n    public void pushFirst(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"Double-ended queue is full\");\n            return;\n        }\n        // Use modulo operation to wrap front around to the tail after passing the head of the array\n        // Add num to the front of the queue\n        front = index(front - 1);\n        // Add num to front of queue\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* Rear of the queue enqueue */\n    public void pushLast(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"Double-ended queue is full\");\n            return;\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        int rear = index(front + queSize);\n        // Front pointer moves one position backward\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* Rear of the queue dequeue */\n    public int popFirst() {\n        int num = peekFirst();\n        // Move front pointer backward by one position\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* Access rear of the queue element */\n    public int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* Driver Code */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        // Initialize double-ended queue\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* Return array for printing */\n    public int[] toArray() {\n        // Elements enqueue\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.cs<pre><code>/* Double-ended queue based on circular array implementation */\nclass ArrayDeque {\n    int[] nums;  // Array for storing double-ended queue elements\n    int front;   // Front pointer, points to the front of the queue element\n    int queSize; // Double-ended queue length\n\n    /* Constructor */\n    public ArrayDeque(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* Get the capacity of the double-ended queue */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* Get the length of the double-ended queue */\n    public int Size() {\n        return queSize;\n    }\n\n    /* Check if the double-ended queue is empty */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* Calculate circular array index */\n    int Index(int i) {\n        // Use modulo operation to wrap the array head and tail together\n        // When i passes the tail of the array, return to the head\n        // When i passes the head of the array, return to the tail\n        return (i + Capacity()) % Capacity();\n    }\n\n    /* Front of the queue enqueue */\n    public void PushFirst(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"Double-ended queue is full\");\n            return;\n        }\n        // Use modulo operation to wrap front around to the tail after passing the head of the array\n        // Add num to the front of the queue\n        front = Index(front - 1);\n        // Add num to front of queue\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* Rear of the queue enqueue */\n    public void PushLast(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"Double-ended queue is full\");\n            return;\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        int rear = Index(front + queSize);\n        // Front pointer moves one position backward\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* Rear of the queue dequeue */\n    public int PopFirst() {\n        int num = PeekFirst();\n        // Move front pointer backward by one position\n        front = Index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* Access rear of the queue element */\n    public int PopLast() {\n        int num = PeekLast();\n        queSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    public int PeekFirst() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        return nums[front];\n    }\n\n    /* Driver Code */\n    public int PeekLast() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        // Initialize double-ended queue\n        int last = Index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* Return array for printing */\n    public int[] ToArray() {\n        // Elements enqueue\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[Index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.go<pre><code>/* Double-ended queue based on circular array implementation */\ntype arrayDeque struct {\n    nums        []int // Array for storing double-ended queue elements\n    front       int   // Front pointer, points to the front of the queue element\n    queSize     int   // Double-ended queue length\n    queCapacity int   // Queue capacity (maximum number of elements)\n}\n\n/* Access front of the queue element */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\n    return &amp;arrayDeque{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* Get the length of the double-ended queue */\nfunc (q *arrayDeque) size() int {\n    return q.queSize\n}\n\n/* Check if the double-ended queue is empty */\nfunc (q *arrayDeque) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* Calculate circular array index */\nfunc (q *arrayDeque) index(i int) int {\n    // Use modulo operation to wrap the array head and tail together\n    // When i passes the tail of the array, return to the head\n    // When i passes the head of the array, return to the tail\n    return (i + q.queCapacity) % q.queCapacity\n}\n\n/* Front of the queue enqueue */\nfunc (q *arrayDeque) pushFirst(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"Double-ended queue is full\")\n        return\n    }\n    // Use modulo operation to wrap front around to the tail after passing the head of the array\n    // Add num to the front of the queue\n    q.front = q.index(q.front - 1)\n    // Add num to front of queue\n    q.nums[q.front] = num\n    q.queSize++\n}\n\n/* Rear of the queue enqueue */\nfunc (q *arrayDeque) pushLast(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"Double-ended queue is full\")\n        return\n    }\n    // Use modulo operation to wrap rear around to the head after passing the tail of the array\n    rear := q.index(q.front + q.queSize)\n    // Front pointer moves one position backward\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* Rear of the queue dequeue */\nfunc (q *arrayDeque) popFirst() any {\n    num := q.peekFirst()\n    if num == nil {\n        return nil\n    }\n    // Move front pointer backward by one position\n    q.front = q.index(q.front + 1)\n    q.queSize--\n    return num\n}\n\n/* Access rear of the queue element */\nfunc (q *arrayDeque) popLast() any {\n    num := q.peekLast()\n    if num == nil {\n        return nil\n    }\n    q.queSize--\n    return num\n}\n\n/* Return list for printing */\nfunc (q *arrayDeque) peekFirst() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* Driver Code */\nfunc (q *arrayDeque) peekLast() any {\n    if q.isEmpty() {\n        return nil\n    }\n    // Initialize double-ended queue\n    last := q.index(q.front + q.queSize - 1)\n    return q.nums[last]\n}\n\n/* Get Slice for printing */\nfunc (q *arrayDeque) toSlice() []int {\n    // Elements enqueue\n    res := make([]int, q.queSize)\n    for i, j := 0, q.front; i &lt; q.queSize; i++ {\n        res[i] = q.nums[q.index(j)]\n        j++\n    }\n    return res\n}\n</code></pre> array_deque.swift<pre><code>/* Double-ended queue based on circular array implementation */\nclass ArrayDeque {\n    private var nums: [Int] // Array for storing double-ended queue elements\n    private var front: Int // Front pointer, points to the front of the queue element\n    private var _size: Int // Double-ended queue length\n\n    /* Constructor */\n    init(capacity: Int) {\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* Get the capacity of the double-ended queue */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* Get the length of the double-ended queue */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* Check if the double-ended queue is empty */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* Calculate circular array index */\n    private func index(i: Int) -&gt; Int {\n        // Use modulo operation to wrap the array head and tail together\n        // When i passes the tail of the array, return to the head\n        // When i passes the head of the array, return to the tail\n        (i + capacity()) % capacity()\n    }\n\n    /* Front of the queue enqueue */\n    func pushFirst(num: Int) {\n        if size() == capacity() {\n            print(\"Double-ended queue is full\")\n            return\n        }\n        // Use modulo operation to wrap front around to the tail after passing the head of the array\n        // Add num to the front of the queue\n        front = index(i: front - 1)\n        // Add num to front of queue\n        nums[front] = num\n        _size += 1\n    }\n\n    /* Rear of the queue enqueue */\n    func pushLast(num: Int) {\n        if size() == capacity() {\n            print(\"Double-ended queue is full\")\n            return\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        let rear = index(i: front + size())\n        // Front pointer moves one position backward\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* Rear of the queue dequeue */\n    func popFirst() -&gt; Int {\n        let num = peekFirst()\n        // Move front pointer backward by one position\n        front = index(i: front + 1)\n        _size -= 1\n        return num\n    }\n\n    /* Access rear of the queue element */\n    func popLast() -&gt; Int {\n        let num = peekLast()\n        _size -= 1\n        return num\n    }\n\n    /* Return list for printing */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Deque is empty\")\n        }\n        return nums[front]\n    }\n\n    /* Driver Code */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Deque is empty\")\n        }\n        // Initialize double-ended queue\n        let last = index(i: front + size() - 1)\n        return nums[last]\n    }\n\n    /* Return array for printing */\n    func toArray() -&gt; [Int] {\n        // Elements enqueue\n        (front ..&lt; front + size()).map { nums[index(i: $0)] }\n    }\n}\n</code></pre> array_deque.js<pre><code>/* Double-ended queue based on circular array implementation */\nclass ArrayDeque {\n    #nums; // Array for storing double-ended queue elements\n    #front; // Front pointer, points to the front of the queue element\n    #queSize; // Double-ended queue length\n\n    /* Constructor */\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n        this.#front = 0;\n        this.#queSize = 0;\n    }\n\n    /* Get the capacity of the double-ended queue */\n    capacity() {\n        return this.#nums.length;\n    }\n\n    /* Get the length of the double-ended queue */\n    size() {\n        return this.#queSize;\n    }\n\n    /* Check if the double-ended queue is empty */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* Calculate circular array index */\n    index(i) {\n        // Use modulo operation to wrap the array head and tail together\n        // When i passes the tail of the array, return to the head\n        // When i passes the head of the array, return to the tail\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* Front of the queue enqueue */\n    pushFirst(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('Double-ended queue is full');\n            return;\n        }\n        // Use modulo operation to wrap front around to the tail after passing the head of the array\n        // Add num to the front of the queue\n        this.#front = this.index(this.#front - 1);\n        // Add num to front of queue\n        this.#nums[this.#front] = num;\n        this.#queSize++;\n    }\n\n    /* Rear of the queue enqueue */\n    pushLast(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('Double-ended queue is full');\n            return;\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        const rear = this.index(this.#front + this.#queSize);\n        // Front pointer moves one position backward\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* Rear of the queue dequeue */\n    popFirst() {\n        const num = this.peekFirst();\n        // Move front pointer backward by one position\n        this.#front = this.index(this.#front + 1);\n        this.#queSize--;\n        return num;\n    }\n\n    /* Access rear of the queue element */\n    popLast() {\n        const num = this.peekLast();\n        this.#queSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    peekFirst() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.#nums[this.#front];\n    }\n\n    /* Driver Code */\n    peekLast() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // Initialize double-ended queue\n        const last = this.index(this.#front + this.#queSize - 1);\n        return this.#nums[last];\n    }\n\n    /* Return array for printing */\n    toArray() {\n        // Elements enqueue\n        const res = [];\n        for (let i = 0, j = this.#front; i &lt; this.#queSize; i++, j++) {\n            res[i] = this.#nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.ts<pre><code>/* Double-ended queue based on circular array implementation */\nclass ArrayDeque {\n    private nums: number[]; // Array for storing double-ended queue elements\n    private front: number; // Front pointer, points to the front of the queue element\n    private queSize: number; // Double-ended queue length\n\n    /* Constructor */\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = 0;\n        this.queSize = 0;\n    }\n\n    /* Get the capacity of the double-ended queue */\n    capacity(): number {\n        return this.nums.length;\n    }\n\n    /* Get the length of the double-ended queue */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* Check if the double-ended queue is empty */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* Calculate circular array index */\n    index(i: number): number {\n        // Use modulo operation to wrap the array head and tail together\n        // When i passes the tail of the array, return to the head\n        // When i passes the head of the array, return to the tail\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* Front of the queue enqueue */\n    pushFirst(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('Double-ended queue is full');\n            return;\n        }\n        // Use modulo operation to wrap front around to the tail after passing the head of the array\n        // Add num to the front of the queue\n        this.front = this.index(this.front - 1);\n        // Add num to front of queue\n        this.nums[this.front] = num;\n        this.queSize++;\n    }\n\n    /* Rear of the queue enqueue */\n    pushLast(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('Double-ended queue is full');\n            return;\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        const rear: number = this.index(this.front + this.queSize);\n        // Front pointer moves one position backward\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* Rear of the queue dequeue */\n    popFirst(): number {\n        const num: number = this.peekFirst();\n        // Move front pointer backward by one position\n        this.front = this.index(this.front + 1);\n        this.queSize--;\n        return num;\n    }\n\n    /* Access rear of the queue element */\n    popLast(): number {\n        const num: number = this.peekLast();\n        this.queSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    peekFirst(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.nums[this.front];\n    }\n\n    /* Driver Code */\n    peekLast(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // Initialize double-ended queue\n        const last = this.index(this.front + this.queSize - 1);\n        return this.nums[last];\n    }\n\n    /* Return array for printing */\n    toArray(): number[] {\n        // Elements enqueue\n        const res: number[] = [];\n        for (let i = 0, j = this.front; i &lt; this.queSize; i++, j++) {\n            res[i] = this.nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.dart<pre><code>/* Double-ended queue based on circular array implementation */\nclass ArrayDeque {\n  late List&lt;int&gt; _nums; // Array for storing double-ended queue elements\n  late int _front; // Front pointer, points to the front of the queue element\n  late int _queSize; // Double-ended queue length\n\n  /* Constructor */\n  ArrayDeque(int capacity) {\n    this._nums = List.filled(capacity, 0);\n    this._front = this._queSize = 0;\n  }\n\n  /* Get the capacity of the double-ended queue */\n  int capacity() {\n    return _nums.length;\n  }\n\n  /* Get the length of the double-ended queue */\n  int size() {\n    return _queSize;\n  }\n\n  /* Check if the double-ended queue is empty */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* Calculate circular array index */\n  int index(int i) {\n    // Use modulo operation to wrap the array head and tail together\n    // When i passes the tail of the array, return to the head\n    // When i passes the head of the array, return to the tail\n    return (i + capacity()) % capacity();\n  }\n\n  /* Front of the queue enqueue */\n  void pushFirst(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"Double-ended queue is full\");\n    }\n    // Use modulo operation to wrap front around to the tail after passing the head of the array\n    // Use modulo operation to wrap _front from array head back to tail\n    _front = index(_front - 1);\n    // Add _num to queue front\n    _nums[_front] = _num;\n    _queSize++;\n  }\n\n  /* Rear of the queue enqueue */\n  void pushLast(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"Double-ended queue is full\");\n    }\n    // Use modulo operation to wrap rear around to the head after passing the tail of the array\n    int rear = index(_front + _queSize);\n    // Add _num to queue rear\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* Rear of the queue dequeue */\n  int popFirst() {\n    int _num = peekFirst();\n    // Move front pointer right by one\n    _front = index(_front + 1);\n    _queSize--;\n    return _num;\n  }\n\n  /* Access rear of the queue element */\n  int popLast() {\n    int _num = peekLast();\n    _queSize--;\n    return _num;\n  }\n\n  /* Return list for printing */\n  int peekFirst() {\n    if (isEmpty()) {\n      throw Exception(\"Deque is empty\");\n    }\n    return _nums[_front];\n  }\n\n  /* Driver Code */\n  int peekLast() {\n    if (isEmpty()) {\n      throw Exception(\"Deque is empty\");\n    }\n    // Initialize double-ended queue\n    int last = index(_front + _queSize - 1);\n    return _nums[last];\n  }\n\n  /* Return array for printing */\n  List&lt;int&gt; toArray() {\n    // Elements enqueue\n    List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[index(j)];\n    }\n    return res;\n  }\n}\n</code></pre> array_deque.rs<pre><code>/* Double-ended queue based on circular array implementation */\nstruct ArrayDeque&lt;T&gt; {\n    nums: Vec&lt;T&gt;,    // Array for storing double-ended queue elements\n    front: usize,    // Front pointer, points to the front of the queue element\n    que_size: usize, // Double-ended queue length\n}\n\nimpl&lt;T: Copy + Default&gt; ArrayDeque&lt;T&gt; {\n    /* Constructor */\n    pub fn new(capacity: usize) -&gt; Self {\n        Self {\n            nums: vec![T::default(); capacity],\n            front: 0,\n            que_size: 0,\n        }\n    }\n\n    /* Get the capacity of the double-ended queue */\n    pub fn capacity(&amp;self) -&gt; usize {\n        self.nums.len()\n    }\n\n    /* Get the length of the double-ended queue */\n    pub fn size(&amp;self) -&gt; usize {\n        self.que_size\n    }\n\n    /* Check if the double-ended queue is empty */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* Calculate circular array index */\n    fn index(&amp;self, i: i32) -&gt; usize {\n        // Use modulo operation to wrap the array head and tail together\n        // When i passes the tail of the array, return to the head\n        // When i passes the head of the array, return to the tail\n        ((i + self.capacity() as i32) % self.capacity() as i32) as usize\n    }\n\n    /* Front of the queue enqueue */\n    pub fn push_first(&amp;mut self, num: T) {\n        if self.que_size == self.capacity() {\n            println!(\"Double-ended queue is full\");\n            return;\n        }\n        // Use modulo operation to wrap front around to the tail after passing the head of the array\n        // Add num to the front of the queue\n        self.front = self.index(self.front as i32 - 1);\n        // Add num to front of queue\n        self.nums[self.front] = num;\n        self.que_size += 1;\n    }\n\n    /* Rear of the queue enqueue */\n    pub fn push_last(&amp;mut self, num: T) {\n        if self.que_size == self.capacity() {\n            println!(\"Double-ended queue is full\");\n            return;\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        let rear = self.index(self.front as i32 + self.que_size as i32);\n        // Front pointer moves one position backward\n        self.nums[rear] = num;\n        self.que_size += 1;\n    }\n\n    /* Rear of the queue dequeue */\n    fn pop_first(&amp;mut self) -&gt; T {\n        let num = self.peek_first();\n        // Move front pointer backward by one position\n        self.front = self.index(self.front as i32 + 1);\n        self.que_size -= 1;\n        num\n    }\n\n    /* Access rear of the queue element */\n    fn pop_last(&amp;mut self) -&gt; T {\n        let num = self.peek_last();\n        self.que_size -= 1;\n        num\n    }\n\n    /* Return list for printing */\n    fn peek_first(&amp;self) -&gt; T {\n        if self.is_empty() {\n            panic!(\"Deque is empty\")\n        };\n        self.nums[self.front]\n    }\n\n    /* Driver Code */\n    fn peek_last(&amp;self) -&gt; T {\n        if self.is_empty() {\n            panic!(\"Deque is empty\")\n        };\n        // Initialize double-ended queue\n        let last = self.index(self.front as i32 + self.que_size as i32 - 1);\n        self.nums[last]\n    }\n\n    /* Return array for printing */\n    fn to_array(&amp;self) -&gt; Vec&lt;T&gt; {\n        // Elements enqueue\n        let mut res = vec![T::default(); self.que_size];\n        let mut j = self.front;\n        for i in 0..self.que_size {\n            res[i] = self.nums[self.index(j as i32)];\n            j += 1;\n        }\n        res\n    }\n}\n</code></pre> array_deque.c<pre><code>/* Double-ended queue based on circular array implementation */\ntypedef struct {\n    int *nums;       // Array for storing queue elements\n    int front;       // Front pointer, points to the front of the queue element\n    int queSize;     // Rear pointer, points to rear + 1\n    int queCapacity; // Queue capacity\n} ArrayDeque;\n\n/* Constructor */\nArrayDeque *newArrayDeque(int capacity) {\n    ArrayDeque *deque = (ArrayDeque *)malloc(sizeof(ArrayDeque));\n    // Initialize array\n    deque-&gt;queCapacity = capacity;\n    deque-&gt;nums = (int *)malloc(sizeof(int) * deque-&gt;queCapacity);\n    deque-&gt;front = deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* Destructor */\nvoid delArrayDeque(ArrayDeque *deque) {\n    free(deque-&gt;nums);\n    free(deque);\n}\n\n/* Get the capacity of the double-ended queue */\nint capacity(ArrayDeque *deque) {\n    return deque-&gt;queCapacity;\n}\n\n/* Get the length of the double-ended queue */\nint size(ArrayDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* Check if the double-ended queue is empty */\nbool empty(ArrayDeque *deque) {\n    return deque-&gt;queSize == 0;\n}\n\n/* Calculate circular array index */\nint dequeIndex(ArrayDeque *deque, int i) {\n    // Use modulo operation to wrap the array head and tail together\n    // When i exceeds array end, wrap to head\n    // When i passes the head of the array, return to the tail\n    return ((i + capacity(deque)) % capacity(deque));\n}\n\n/* Front of the queue enqueue */\nvoid pushFirst(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"Deque is full\\r\\n\");\n        return;\n    }\n    // Use modulo operation to wrap front around to the tail after passing the head of the array\n    // Use modulo to wrap front from array head to rear\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front - 1);\n    // Add num to queue front\n    deque-&gt;nums[deque-&gt;front] = num;\n    deque-&gt;queSize++;\n}\n\n/* Rear of the queue enqueue */\nvoid pushLast(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"Deque is full\\r\\n\");\n        return;\n    }\n    // Use modulo operation to wrap rear around to the head after passing the tail of the array\n    int rear = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize);\n    // Front pointer moves one position backward\n    deque-&gt;nums[rear] = num;\n    deque-&gt;queSize++;\n}\n\n/* Return list for printing */\nint peekFirst(ArrayDeque *deque) {\n    // Access error: Deque is empty\n    assert(empty(deque) == 0);\n    return deque-&gt;nums[deque-&gt;front];\n}\n\n/* Driver Code */\nint peekLast(ArrayDeque *deque) {\n    // Access error: Deque is empty\n    assert(empty(deque) == 0);\n    int last = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize - 1);\n    return deque-&gt;nums[last];\n}\n\n/* Rear of the queue dequeue */\nint popFirst(ArrayDeque *deque) {\n    int num = peekFirst(deque);\n    // Move front pointer backward by one position\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front + 1);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* Access rear of the queue element */\nint popLast(ArrayDeque *deque) {\n    int num = peekLast(deque);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* Return array for printing */\nint *toArray(ArrayDeque *deque, int *queSize) {\n    *queSize = deque-&gt;queSize;\n    int *res = (int *)calloc(deque-&gt;queSize, sizeof(int));\n    int j = deque-&gt;front;\n    for (int i = 0; i &lt; deque-&gt;queSize; i++) {\n        res[i] = deque-&gt;nums[j % deque-&gt;queCapacity];\n        j++;\n    }\n    return res;\n}\n</code></pre> array_deque.kt<pre><code>/* Constructor */\nclass ArrayDeque(capacity: Int) {\n    private var nums: IntArray = IntArray(capacity) // Array for storing double-ended queue elements\n    private var front: Int = 0 // Front pointer, points to the front of the queue element\n    private var queSize: Int = 0 // Double-ended queue length\n\n    /* Get the capacity of the double-ended queue */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* Get the length of the double-ended queue */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* Check if the double-ended queue is empty */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* Calculate circular array index */\n    private fun index(i: Int): Int {\n        // Use modulo operation to wrap the array head and tail together\n        // When i passes the tail of the array, return to the head\n        // When i passes the head of the array, return to the tail\n        return (i + capacity()) % capacity()\n    }\n\n    /* Front of the queue enqueue */\n    fun pushFirst(num: Int) {\n        if (queSize == capacity()) {\n            println(\"Double-ended queue is full\")\n            return\n        }\n        // Use modulo operation to wrap front around to the tail after passing the head of the array\n        // Add num to the front of the queue\n        front = index(front - 1)\n        // Add num to front of queue\n        nums[front] = num\n        queSize++\n    }\n\n    /* Rear of the queue enqueue */\n    fun pushLast(num: Int) {\n        if (queSize == capacity()) {\n            println(\"Double-ended queue is full\")\n            return\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        val rear = index(front + queSize)\n        // Front pointer moves one position backward\n        nums[rear] = num\n        queSize++\n    }\n\n    /* Rear of the queue dequeue */\n    fun popFirst(): Int {\n        val num = peekFirst()\n        // Move front pointer backward by one position\n        front = index(front + 1)\n        queSize--\n        return num\n    }\n\n    /* Access rear of the queue element */\n    fun popLast(): Int {\n        val num = peekLast()\n        queSize--\n        return num\n    }\n\n    /* Return list for printing */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* Driver Code */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // Initialize double-ended queue\n        val last = index(front + queSize - 1)\n        return nums[last]\n    }\n\n    /* Return array for printing */\n    fun toArray(): IntArray {\n        // Elements enqueue\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[index(j)]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_deque.rb<pre><code>### Deque based on circular array ###\nclass ArrayDeque\n  ### Get deque length ###\n  attr_reader :size\n\n  ### Constructor ###\n  def initialize(capacity)\n    @nums = Array.new(capacity, 0)\n    @front = 0\n    @size = 0\n  end\n\n  ### Get deque capacity ###\n  def capacity\n    @nums.length\n  end\n\n  ### Check if deque is empty ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### Enqueue at front ###\n  def push_first(num)\n    if size == capacity\n      puts 'Double-ended queue is full'\n      return\n    end\n\n    # Use modulo operation to wrap front around to the tail after passing the head of the array\n    # Add num to the front of the queue\n    @front = index(@front - 1)\n    # Add num to front of queue\n    @nums[@front] = num\n    @size += 1\n  end\n\n  ### Enqueue at rear ###\n  def push_last(num)\n    if size == capacity\n      puts 'Double-ended queue is full'\n      return\n    end\n\n    # Use modulo operation to wrap rear around to the head after passing the tail of the array\n    rear = index(@front + size)\n    # Front pointer moves one position backward\n    @nums[rear] = num\n    @size += 1\n  end\n\n  ### Dequeue from front ###\n  def pop_first\n    num = peek_first\n    # Move front pointer backward by one position\n    @front = index(@front + 1)\n    @size -= 1\n    num\n  end\n\n  ### Dequeue from rear ###\n  def pop_last\n    num = peek_last\n    @size -= 1\n    num\n  end\n\n  ### Access front element ###\n  def peek_first\n    raise IndexError, 'Deque is empty' if is_empty?\n\n    @nums[@front]\n  end\n\n  ### Access rear element ###\n  def peek_last\n    raise IndexError, 'Deque is empty' if is_empty?\n\n    # Initialize double-ended queue\n    last = index(@front + size - 1)\n    @nums[last]\n  end\n\n  ### Return array for printing ###\n  def to_array\n    # Elements enqueue\n    res = []\n    for i in 0...size\n      res &lt;&lt; @nums[index(@front + i)]\n    end\n    res\n  end\n\n  private\n\n  ### Calculate circular array index ###\n  def index(i)\n    # Use modulo operation to wrap the array head and tail together\n    # When i passes the tail of the array, return to the head\n    # When i passes the head of the array, return to the tail\n    (i + capacity) % capacity\n  end\nend\n</code></pre>","path":["Chapter 5. Stacks and Queues","5.3   Deque"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#533-deque-applications","level":2,"title":"5.3.3   Deque Applications","text":"<p>A deque combines the logic of both stacks and queues. Therefore, it can implement all application scenarios of both, while providing greater flexibility.</p> <p>We know that the \"undo\" function in software is typically implemented using a stack: the system pushes each change operation onto the stack and then implements undo through pop. However, considering system resource limitations, software usually limits the number of undo steps (for example, only allowing 50 steps to be saved). When the stack length exceeds 50, the software needs to perform a deletion operation at the bottom of the stack (front of the queue). But a stack cannot implement this functionality, so a deque is needed to replace the stack. Note that the core logic of \"undo\" still follows the LIFO principle of a stack; it's just that the deque can more flexibly implement some additional logic.</p>","path":["Chapter 5. Stacks and Queues","5.3   Deque"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/","level":1,"title":"5.5   Exercises","text":"","path":["Chapter 5. Stacks and Queues","5.5   Exercises"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#551-concept-review","level":2,"title":"5.5.1   Concept Review","text":"","path":["Chapter 5. Stacks and Queues","5.5   Exercises"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#1-which-element-leaves-a-stack-or-queue-first","level":3,"title":"1.   Which Element Leaves a Stack or Queue First?","text":"<p>Prepare an empty stack <code>S</code> and an empty queue <code>Q</code>. Perform the same sequence of operations on each one:</p> <p>Step 1: Add <code>A</code>. Step 2: Add <code>B</code>. Step 3: Remove and record one element. Step 4: Add <code>C</code>. Step 5: Keep removing and recording elements until the container is empty.</p> <p>Write the order in which elements are removed from <code>S</code> and from <code>Q</code>. Explain the difference using \"last-in-first-out\" or \"first-in-first-out.\"</p> Answer <p>The removal order for stack <code>S</code> is <code>B, C, A</code>. After the elements <code>A, B</code> are added, the most recently added element, <code>B</code>, is popped first. After <code>C</code> is added, <code>C, A</code> are popped in that order. This is \"last-in-first-out.\"</p> <p>The removal order for queue <code>Q</code> is <code>A, B, C</code>. After the elements <code>A, B</code> are added, the earliest added element, <code>A</code>, is removed first. After <code>C</code> is added, <code>B, C</code> are removed in that order. This is \"first-in-first-out.\"</p>","path":["Chapter 5. Stacks and Queues","5.5   Exercises"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#2-what-happens-when-the-rear-passes-the-end-of-the-array","level":3,"title":"2.   What Happens When the Rear Passes the End of the Array?","text":"<p>A queue is implemented with a circular array of length 5, whose indices are <code>0–4</code>. Currently, <code>front = 3</code> and <code>size = 2</code>; <code>A, B</code> are stored at indices 3 and 4, respectively.</p> <ol> <li>When <code>C</code> is enqueued, at which index should <code>C</code> be stored? What is <code>size</code> after the enqueue?</li> <li>Next, dequeue once. Which element is removed? What are the new values of <code>front</code> and <code>size</code>?</li> <li>What is the logical order from the front to the rear now? Does dequeuing require moving the other elements in the array? Why or why not?</li> </ol> Answer <ol> <li> <p>The new element's position is     <code>(front + size) % 5 = (3 + 2) % 5 = 0</code>,     so <code>C</code> is stored at index 0. After the enqueue, <code>size = 3</code>.</p> </li> <li> <p>Dequeuing removes the current front element, <code>A</code>. The new front index is     <code>(3 + 1) % 5 = 4</code>, so <code>front = 4</code> and <code>size = 2</code>.</p> </li> <li> <p>The logical order of valid elements is <code>B, C</code>, with <code>B</code> at index 4 and <code>C</code> at index 0.     Dequeuing requires only changing <code>front</code> and <code>size</code>. The circular array uses the remainder operation to wrap the index back to the beginning,     so there is no need to shift all the other elements forward.</p> </li> </ol>","path":["Chapter 5. Stacks and Queues","5.5   Exercises"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#3-operations-at-both-ends-of-a-deque","level":3,"title":"3.   Operations at Both Ends of a Deque","text":"<p>Here, <code>push_first</code> adds an element at the front, <code>push_last</code> adds one at the rear, <code>pop_first</code> removes one from the front, and <code>pop_last</code> removes one from the rear.</p> <p>Perform the following operations on an empty deque <code>deq</code>:</p> <ol> <li><code>push_last(A)</code></li> <li><code>push_last(B)</code></li> <li><code>push_first(C)</code></li> <li><code>pop_last()</code></li> <li><code>push_last(D)</code></li> <li><code>pop_first()</code></li> </ol> <ol> <li>Which elements are returned by the two pop operations?</li> <li>After all operations are complete, which elements remain from front to rear?</li> <li>Examine the six operations. Can a queue that allows insertion only at the rear and removal only at the front perform all of them? If not, identify the operations it cannot perform. Then state whether a deque can perform them and explain why.</li> </ol> Answer <p>After the first three steps, the deque from front to rear is <code>[C, A, B]</code>.</p> <ol> <li> <p><code>pop_last()</code> removes <code>B</code>. After <code>D</code> is added, the deque is <code>[C, A, D]</code>,     and <code>pop_first()</code> then removes <code>C</code>.</p> </li> <li> <p><code>[A, D]</code> remains.</p> </li> <li> <p>A queue that permits insertion only at the rear and removal only at the front cannot perform all the operations.     Step 3, <code>push_first(C)</code>, requires insertion at the front, and Step 4, <code>pop_last()</code>, requires removal at the rear. Both are outside the operations supported by such a queue.     A deque permits insertion and removal at both ends, so it can perform all six operations.</p> </li> </ol>","path":["Chapter 5. Stacks and Queues","5.5   Exercises"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#552-programming-exercises","level":2,"title":"5.5.2   Programming Exercises","text":"","path":["Chapter 5. Stacks and Queues","5.5   Exercises"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#1-check-a-bracket-sequence","level":3,"title":"1.   Check a Bracket Sequence","text":"<p>Given a string <code>s</code> containing only the three types of brackets <code>()</code>, <code>[]</code>, and <code>{}</code>, use a stack to determine whether the string is valid.</p> <p>A valid sequence must satisfy both conditions: every closing bracket matches the type of the most recent unmatched opening bracket, and no unmatched opening bracket remains after the traversal. Return a Boolean value for the result.</p> Hints <ol> <li>You can create a mapping from each closing bracket to its matching opening bracket</li> <li>When you encounter a closing bracket, first check whether the stack is empty, and then check whether the top matches</li> <li>The stack must also be empty after the traversal</li> </ol> <p>LeetCode</p>","path":["Chapter 5. Stacks and Queues","5.5   Exercises"],"tags":[]},{"location":"chapter_stack_and_queue/queue/","level":1,"title":"5.2   Queue","text":"<p>A queue is a linear data structure that follows the First In, First Out (FIFO) rule. As the name suggests, it models people lining up: newcomers continuously join the rear of the queue, while the people at the front leave one by one.</p> <p>As shown in Figure 5-4, we call the front of the queue the \"front\" and the end the \"rear.\" The operation of adding an element to the rear is called \"enqueue,\" and the operation of removing the front element is called \"dequeue.\"</p> <p></p> <p> Figure 5-4   FIFO rule of queue </p>","path":["Chapter 5. Stacks and Queues","5.2   Queue"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#521-common-queue-operations","level":2,"title":"5.2.1   Common Queue Operations","text":"<p>The common operations on a queue are shown in Table 5-2. Note that method names may vary across programming languages. Here, we use the same naming convention as for stacks.</p> <p> Table 5-2   Efficiency of Queue Operations </p> Method Description Time Complexity <code>push()</code> Enqueue element, add element to rear \\(O(1)\\) <code>pop()</code> Dequeue front element \\(O(1)\\) <code>peek()</code> Access front element \\(O(1)\\) <p>We can directly use the queue classes provided by the programming language:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby queue.py<pre><code>from collections import deque\n\n# Initialize queue\n# In Python, we generally use the deque class as a queue\n# Although queue.Queue() is a pure queue class, it is not very user-friendly, so it is not recommended\nque: deque[int] = deque()\n\n# Enqueue elements\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n\n# Access front element\nfront: int = que[0]\n\n# Dequeue element\npop: int = que.popleft()\n\n# Get queue length\nsize: int = len(que)\n\n# Check if queue is empty\nis_empty: bool = len(que) == 0\n</code></pre> queue.cpp<pre><code>/* Initialize queue */\nqueue&lt;int&gt; queue;\n\n/* Enqueue elements */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* Access front element */\nint front = queue.front();\n\n/* Dequeue element */\nqueue.pop();\n\n/* Get queue length */\nint size = queue.size();\n\n/* Check if queue is empty */\nbool empty = queue.empty();\n</code></pre> queue.java<pre><code>/* Initialize queue */\nQueue&lt;Integer&gt; queue = new LinkedList&lt;&gt;();\n\n/* Enqueue elements */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n\n/* Access front element */\nint peek = queue.peek();\n\n/* Dequeue element */\nint pop = queue.poll();\n\n/* Get queue length */\nint size = queue.size();\n\n/* Check if queue is empty */\nboolean isEmpty = queue.isEmpty();\n</code></pre> queue.cs<pre><code>/* Initialize queue */\nQueue&lt;int&gt; queue = new();\n\n/* Enqueue elements */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n\n/* Access front element */\nint peek = queue.Peek();\n\n/* Dequeue element */\nint pop = queue.Dequeue();\n\n/* Get queue length */\nint size = queue.Count;\n\n/* Check if queue is empty */\nbool isEmpty = queue.Count == 0;\n</code></pre> queue_test.go<pre><code>/* Initialize queue */\n// In Go, use list as a queue\nqueue := list.New()\n\n/* Enqueue elements */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n\n/* Access front element */\npeek := queue.Front()\n\n/* Dequeue element */\npop := queue.Front()\nqueue.Remove(pop)\n\n/* Get queue length */\nsize := queue.Len()\n\n/* Check if queue is empty */\nisEmpty := queue.Len() == 0\n</code></pre> queue.swift<pre><code>/* Initialize queue */\n// Swift does not have a built-in queue class, can use Array as a queue\nvar queue: [Int] = []\n\n/* Enqueue elements */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n\n/* Access front element */\nlet peek = queue.first!\n\n/* Dequeue element */\n// Since it's an array, removeFirst has O(n) complexity\nlet pop = queue.removeFirst()\n\n/* Get queue length */\nlet size = queue.count\n\n/* Check if queue is empty */\nlet isEmpty = queue.isEmpty\n</code></pre> queue.js<pre><code>/* Initialize queue */\n// JavaScript does not have a built-in queue, can use Array as a queue\nconst queue = [];\n\n/* Enqueue elements */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* Access front element */\nconst peek = queue[0];\n\n/* Dequeue element */\n// The underlying structure is an array, so shift() has O(n) time complexity\nconst pop = queue.shift();\n\n/* Get queue length */\nconst size = queue.length;\n\n/* Check if queue is empty */\nconst empty = queue.length === 0;\n</code></pre> queue.ts<pre><code>/* Initialize queue */\n// TypeScript does not have a built-in queue, can use Array as a queue\nconst queue: number[] = [];\n\n/* Enqueue elements */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* Access front element */\nconst peek = queue[0];\n\n/* Dequeue element */\n// The underlying structure is an array, so shift() has O(n) time complexity\nconst pop = queue.shift();\n\n/* Get queue length */\nconst size = queue.length;\n\n/* Check if queue is empty */\nconst empty = queue.length === 0;\n</code></pre> queue.dart<pre><code>/* Initialize queue */\n// In Dart, the Queue class is a deque and can also be used as a queue\nQueue&lt;int&gt; queue = Queue();\n\n/* Enqueue elements */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n\n/* Access front element */\nint peek = queue.first;\n\n/* Dequeue element */\nint pop = queue.removeFirst();\n\n/* Get queue length */\nint size = queue.length;\n\n/* Check if queue is empty */\nbool isEmpty = queue.isEmpty;\n</code></pre> queue.rs<pre><code>/* Initialize deque */\n// In Rust, use deque as a regular queue\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* Enqueue elements */\ndeque.push_back(1);\ndeque.push_back(3);\ndeque.push_back(2);\ndeque.push_back(5);\ndeque.push_back(4);\n\n/* Access front element */\nif let Some(front) = deque.front() {\n}\n\n/* Dequeue element */\nif let Some(pop) = deque.pop_front() {\n}\n\n/* Get queue length */\nlet size = deque.len();\n\n/* Check if queue is empty */\nlet is_empty = deque.is_empty();\n</code></pre> queue.c<pre><code>// C does not provide a built-in queue\n</code></pre> queue.kt<pre><code>/* Initialize queue */\nval queue = LinkedList&lt;Int&gt;()\n\n/* Enqueue elements */\nqueue.offer(1)\nqueue.offer(3)\nqueue.offer(2)\nqueue.offer(5)\nqueue.offer(4)\n\n/* Access front element */\nval peek = queue.peek()\n\n/* Dequeue element */\nval pop = queue.poll()\n\n/* Get queue length */\nval size = queue.size\n\n/* Check if queue is empty */\nval isEmpty = queue.isEmpty()\n</code></pre> queue.rb<pre><code># Initialize queue\n# Ruby's built-in queue (Thread::Queue) does not have peek and traversal methods, can use Array as a queue\nqueue = []\n\n# Enqueue elements\nqueue.push(1)\nqueue.push(3)\nqueue.push(2)\nqueue.push(5)\nqueue.push(4)\n\n# Access front element\npeek = queue.first\n\n# Dequeue element\n# Please note that since it's an array, Array#shift has O(n) time complexity\npop = queue.shift\n\n# Get queue length\nsize = queue.length\n\n# Check if queue is empty\nis_empty = queue.empty?\n</code></pre> Code Visualization <p> Full Screen &gt;</p>","path":["Chapter 5. Stacks and Queues","5.2   Queue"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#522-queue-implementation","level":2,"title":"5.2.2   Queue Implementation","text":"<p>To implement a queue, we need a data structure that allows adding elements at one end and removing elements at the other end. Both linked lists and arrays meet this requirement.</p>","path":["Chapter 5. Stacks and Queues","5.2   Queue"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#1-linked-list-implementation","level":3,"title":"1.   Linked List Implementation","text":"<p>As shown in Figure 5-5, we can treat the \"head node\" and \"tail node\" of a linked list as the \"front\" and \"rear\" of the queue, respectively, with the rule that nodes can only be added at the rear and removed from the front.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> Figure 5-5   Enqueue and dequeue operations in linked list implementation of queue </p> <p>Below is the code for implementing a queue using a linked list:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linkedlist_queue.py<pre><code>class LinkedListQueue:\n    \"\"\"Queue based on linked list implementation\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        self._front: ListNode | None = None  # Head node front\n        self._rear: ListNode | None = None  # Tail node rear\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"Get the length of the queue\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"Check if the queue is empty\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"Enqueue\"\"\"\n        # Add num after the tail node\n        node = ListNode(num)\n        # If the queue is empty, make both front and rear point to the node\n        if self._front is None:\n            self._front = node\n            self._rear = node\n        # If the queue is not empty, add the node after the tail node\n        else:\n            self._rear.next = node\n            self._rear = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"Dequeue\"\"\"\n        num = self.peek()\n        # Delete head node\n        self._front = self._front.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"Access front of the queue element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Queue is empty\")\n        return self._front.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"Convert to list for printing\"\"\"\n        queue = []\n        temp = self._front\n        while temp:\n            queue.append(temp.val)\n            temp = temp.next\n        return queue\n</code></pre> linkedlist_queue.cpp<pre><code>/* Queue based on linked list implementation */\nclass LinkedListQueue {\n  private:\n    ListNode *front, *rear; // Head node front, tail node rear\n    int queSize;\n\n  public:\n    LinkedListQueue() {\n        front = nullptr;\n        rear = nullptr;\n        queSize = 0;\n    }\n\n    ~LinkedListQueue() {\n        // Traverse linked list to delete nodes and free memory\n        freeMemoryLinkedList(front);\n    }\n\n    /* Get the length of the queue */\n    int size() {\n        return queSize;\n    }\n\n    /* Check if the queue is empty */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* Enqueue */\n    void push(int num) {\n        // Add num after the tail node\n        ListNode *node = new ListNode(num);\n        // If the queue is empty, make both front and rear point to the node\n        if (front == nullptr) {\n            front = node;\n            rear = node;\n        }\n        // If the queue is not empty, add the node after the tail node\n        else {\n            rear-&gt;next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* Dequeue */\n    int pop() {\n        int num = peek();\n        // Delete head node\n        ListNode *tmp = front;\n        front = front-&gt;next;\n        // Free memory\n        delete tmp;\n        queSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    int peek() {\n        if (size() == 0)\n            throw out_of_range(\"Queue is empty\");\n        return front-&gt;val;\n    }\n\n    /* Convert linked list to Vector and return */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_queue.java<pre><code>/* Queue based on linked list implementation */\nclass LinkedListQueue {\n    private ListNode front, rear; // Head node front, tail node rear\n    private int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* Get the length of the queue */\n    public int size() {\n        return queSize;\n    }\n\n    /* Check if the queue is empty */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* Enqueue */\n    public void push(int num) {\n        // Add num after the tail node\n        ListNode node = new ListNode(num);\n        // If the queue is empty, make both front and rear point to the node\n        if (front == null) {\n            front = node;\n            rear = node;\n        // If the queue is not empty, add the node after the tail node\n        } else {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* Dequeue */\n    public int pop() {\n        int num = peek();\n        // Delete head node\n        front = front.next;\n        queSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* Convert linked list to Array and return */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.cs<pre><code>/* Queue based on linked list implementation */\nclass LinkedListQueue {\n    ListNode? front, rear;  // Head node front, tail node rear\n    int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* Get the length of the queue */\n    public int Size() {\n        return queSize;\n    }\n\n    /* Check if the queue is empty */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* Enqueue */\n    public void Push(int num) {\n        // Add num after the tail node\n        ListNode node = new(num);\n        // If the queue is empty, make both front and rear point to the node\n        if (front == null) {\n            front = node;\n            rear = node;\n            // If the queue is not empty, add the node after the tail node\n        } else if (rear != null) {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* Dequeue */\n    public int Pop() {\n        int num = Peek();\n        // Delete head node\n        front = front?.next;\n        queSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return front!.val;\n    }\n\n    /* Convert linked list to Array and return */\n    public int[] ToArray() {\n        if (front == null)\n            return [];\n\n        ListNode? node = front;\n        int[] res = new int[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.go<pre><code>/* Queue based on linked list implementation */\ntype linkedListQueue struct {\n    // Use built-in package list to implement queue\n    data *list.List\n}\n\n/* Access front of the queue element */\nfunc newLinkedListQueue() *linkedListQueue {\n    return &amp;linkedListQueue{\n        data: list.New(),\n    }\n}\n\n/* Enqueue */\nfunc (s *linkedListQueue) push(value any) {\n    s.data.PushBack(value)\n}\n\n/* Dequeue */\nfunc (s *linkedListQueue) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* Return list for printing */\nfunc (s *linkedListQueue) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* Get the length of the queue */\nfunc (s *linkedListQueue) size() int {\n    return s.data.Len()\n}\n\n/* Check if the queue is empty */\nfunc (s *linkedListQueue) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* Get List for printing */\nfunc (s *linkedListQueue) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_queue.swift<pre><code>/* Queue based on linked list implementation */\nclass LinkedListQueue {\n    private var front: ListNode? // Head node\n    private var rear: ListNode? // Tail node\n    private var _size: Int\n\n    init() {\n        _size = 0\n    }\n\n    /* Get the length of the queue */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* Check if the queue is empty */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* Enqueue */\n    func push(num: Int) {\n        // Add num after the tail node\n        let node = ListNode(x: num)\n        // If the queue is empty, make both front and rear point to the node\n        if front == nil {\n            front = node\n            rear = node\n        }\n        // If the queue is not empty, add the node after the tail node\n        else {\n            rear?.next = node\n            rear = node\n        }\n        _size += 1\n    }\n\n    /* Dequeue */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // Delete head node\n        front = front?.next\n        _size -= 1\n        return num\n    }\n\n    /* Return list for printing */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Queue is empty\")\n        }\n        return front!.val\n    }\n\n    /* Convert linked list to Array and return */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.js<pre><code>/* Queue based on linked list implementation */\nclass LinkedListQueue {\n    #front; // Front node #front\n    #rear; // Rear node #rear\n    #queSize = 0;\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n    }\n\n    /* Get the length of the queue */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* Check if the queue is empty */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* Enqueue */\n    push(num) {\n        // Add num after the tail node\n        const node = new ListNode(num);\n        // If the queue is empty, make both front and rear point to the node\n        if (!this.#front) {\n            this.#front = node;\n            this.#rear = node;\n            // If the queue is not empty, add the node after the tail node\n        } else {\n            this.#rear.next = node;\n            this.#rear = node;\n        }\n        this.#queSize++;\n    }\n\n    /* Dequeue */\n    pop() {\n        const num = this.peek();\n        // Delete head node\n        this.#front = this.#front.next;\n        this.#queSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    peek() {\n        if (this.size === 0) throw new Error('Queue is empty');\n        return this.#front.val;\n    }\n\n    /* Convert linked list to Array and return */\n    toArray() {\n        let node = this.#front;\n        const res = new Array(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.ts<pre><code>/* Queue based on linked list implementation */\nclass LinkedListQueue {\n    private front: ListNode | null; // Head node front\n    private rear: ListNode | null; // Tail node rear\n    private queSize: number = 0;\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n    }\n\n    /* Get the length of the queue */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* Check if the queue is empty */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* Enqueue */\n    push(num: number): void {\n        // Add num after the tail node\n        const node = new ListNode(num);\n        // If the queue is empty, make both front and rear point to the node\n        if (!this.front) {\n            this.front = node;\n            this.rear = node;\n            // If the queue is not empty, add the node after the tail node\n        } else {\n            this.rear!.next = node;\n            this.rear = node;\n        }\n        this.queSize++;\n    }\n\n    /* Dequeue */\n    pop(): number {\n        const num = this.peek();\n        if (!this.front) throw new Error('Queue is empty');\n        // Delete head node\n        this.front = this.front.next;\n        this.queSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    peek(): number {\n        if (this.size === 0) throw new Error('Queue is empty');\n        return this.front!.val;\n    }\n\n    /* Convert linked list to Array and return */\n    toArray(): number[] {\n        let node = this.front;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.dart<pre><code>/* Queue based on linked list implementation */\nclass LinkedListQueue {\n  ListNode? _front; // Head node _front\n  ListNode? _rear; // Tail node _rear\n  int _queSize = 0; // Queue length\n\n  LinkedListQueue() {\n    _front = null;\n    _rear = null;\n  }\n\n  /* Get the length of the queue */\n  int size() {\n    return _queSize;\n  }\n\n  /* Check if the queue is empty */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* Enqueue */\n  void push(int _num) {\n    // Add _num after tail node\n    final node = ListNode(_num);\n    // If the queue is empty, make both front and rear point to the node\n    if (_front == null) {\n      _front = node;\n      _rear = node;\n    } else {\n      // If the queue is not empty, add the node after the tail node\n      _rear!.next = node;\n      _rear = node;\n    }\n    _queSize++;\n  }\n\n  /* Dequeue */\n  int pop() {\n    final int _num = peek();\n    // Delete head node\n    _front = _front!.next;\n    _queSize--;\n    return _num;\n  }\n\n  /* Return list for printing */\n  int peek() {\n    if (_queSize == 0) {\n      throw Exception('Queue is empty');\n    }\n    return _front!.val;\n  }\n\n  /* Convert linked list to Array and return */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; queue = [];\n    while (node != null) {\n      queue.add(node.val);\n      node = node.next;\n    }\n    return queue;\n  }\n}\n</code></pre> linkedlist_queue.rs<pre><code>/* Queue based on linked list implementation */\n#[allow(dead_code)]\npub struct LinkedListQueue&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // Head node front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // Tail node rear\n    que_size: usize,                         // Queue length\n}\n\nimpl&lt;T: Copy&gt; LinkedListQueue&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* Get the length of the queue */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* Check if the queue is empty */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.que_size == 0;\n    }\n\n    /* Enqueue */\n    pub fn push(&amp;mut self, num: T) {\n        // Add num after the tail node\n        let new_rear = ListNode::new(num);\n        match self.rear.take() {\n            // If the queue is not empty, add the node after the tail node\n            Some(old_rear) =&gt; {\n                old_rear.borrow_mut().next = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n            // If the queue is empty, make both front and rear point to the node\n            None =&gt; {\n                self.front = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n        }\n        self.que_size += 1;\n    }\n\n    /* Dequeue */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.front.take().map(|old_front| {\n            match old_front.borrow_mut().next.take() {\n                Some(new_front) =&gt; {\n                    self.front = Some(new_front);\n                }\n                None =&gt; {\n                    self.rear.take();\n                }\n            }\n            self.que_size -= 1;\n            old_front.borrow().val\n        })\n    }\n\n    /* Return list for printing */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* Convert linked list to Array and return */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        let mut res: Vec&lt;T&gt; = Vec::new();\n\n        fn recur&lt;T: Copy&gt;(cur: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, res: &amp;mut Vec&lt;T&gt;) {\n            if let Some(cur) = cur {\n                res.push(cur.borrow().val);\n                recur(cur.borrow().next.as_ref(), res);\n            }\n        }\n\n        recur(head, &amp;mut res);\n\n        res\n    }\n}\n</code></pre> linkedlist_queue.c<pre><code>/* Queue based on linked list implementation */\ntypedef struct {\n    ListNode *front, *rear;\n    int queSize;\n} LinkedListQueue;\n\n/* Constructor */\nLinkedListQueue *newLinkedListQueue() {\n    LinkedListQueue *queue = (LinkedListQueue *)malloc(sizeof(LinkedListQueue));\n    queue-&gt;front = NULL;\n    queue-&gt;rear = NULL;\n    queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* Destructor */\nvoid delLinkedListQueue(LinkedListQueue *queue) {\n    // Free all nodes\n    while (queue-&gt;front != NULL) {\n        ListNode *tmp = queue-&gt;front;\n        queue-&gt;front = queue-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // Free queue structure\n    free(queue);\n}\n\n/* Get the length of the queue */\nint size(LinkedListQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* Check if the queue is empty */\nbool empty(LinkedListQueue *queue) {\n    return (size(queue) == 0);\n}\n\n/* Enqueue */\nvoid push(LinkedListQueue *queue, int num) {\n    // Add node at tail\n    ListNode *node = newListNode(num);\n    // If the queue is empty, make both front and rear point to the node\n    if (queue-&gt;front == NULL) {\n        queue-&gt;front = node;\n        queue-&gt;rear = node;\n    }\n    // If the queue is not empty, add the node after the tail node\n    else {\n        queue-&gt;rear-&gt;next = node;\n        queue-&gt;rear = node;\n    }\n    queue-&gt;queSize++;\n}\n\n/* Return list for printing */\nint peek(LinkedListQueue *queue) {\n    assert(size(queue) &amp;&amp; queue-&gt;front);\n    return queue-&gt;front-&gt;val;\n}\n\n/* Dequeue */\nint pop(LinkedListQueue *queue) {\n    int num = peek(queue);\n    ListNode *tmp = queue-&gt;front;\n    queue-&gt;front = queue-&gt;front-&gt;next;\n    free(tmp);\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* Print queue */\nvoid printLinkedListQueue(LinkedListQueue *queue) {\n    int *arr = malloc(sizeof(int) * queue-&gt;queSize);\n    // Copy data from list to array\n    int i;\n    ListNode *node;\n    for (i = 0, node = queue-&gt;front; i &lt; queue-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, queue-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_queue.kt<pre><code>/* Queue based on linked list implementation */\nclass LinkedListQueue(\n    // Head node front, tail node rear\n    private var front: ListNode? = null,\n    private var rear: ListNode? = null,\n    private var queSize: Int = 0\n) {\n\n    /* Get the length of the queue */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* Check if the queue is empty */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* Enqueue */\n    fun push(num: Int) {\n        // Add num after the tail node\n        val node = ListNode(num)\n        // If the queue is empty, make both front and rear point to the node\n        if (front == null) {\n            front = node\n            rear = node\n            // If the queue is not empty, add the node after the tail node\n        } else {\n            rear?.next = node\n            rear = node\n        }\n        queSize++\n    }\n\n    /* Dequeue */\n    fun pop(): Int {\n        val num = peek()\n        // Delete head node\n        front = front?.next\n        queSize--\n        return num\n    }\n\n    /* Return list for printing */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!._val\n    }\n\n    /* Convert linked list to Array and return */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!._val\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.rb<pre><code>### Queue based on linked list ###\nclass LinkedListQueue\n  ### Get queue length ###\n  attr_reader :size\n\n  ### Constructor ###\n  def initialize\n    @front = nil  # Head node front\n    @rear = nil   # Tail node rear\n    @size = 0\n  end\n\n  ### Check if queue is empty ###\n  def is_empty?\n    @front.nil?\n  end\n\n  ### Enqueue ###\n  def push(num)\n    # Add num after the tail node\n    node = ListNode.new(num)\n\n    # If queue is empty, set both front and rear to this node\n    if @front.nil?\n      @front = node\n      @rear = node\n    # If queue is not empty, add this node after rear\n    else\n      @rear.next = node\n      @rear = node\n    end\n\n    @size += 1\n  end\n\n  ### Dequeue ###\n  def pop\n    num = peek\n    # Delete head node\n    @front = @front.next\n    @size -= 1\n    num\n  end\n\n  ### Access front element ###\n  def peek\n    raise IndexError, 'Queue is empty' if is_empty?\n\n    @front.val\n  end\n\n  ### Convert linked list to Array and return ###\n  def to_array\n    queue = []\n    temp = @front\n    while temp\n      queue &lt;&lt; temp.val\n      temp = temp.next\n    end\n    queue\n  end\nend\n</code></pre>","path":["Chapter 5. Stacks and Queues","5.2   Queue"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#2-array-implementation","level":3,"title":"2.   Array Implementation","text":"<p>Deleting the first element in an array has a time complexity of \\(O(n)\\), which would make the dequeue operation inefficient. However, we can use the following clever method to avoid this problem.</p> <p>We can use a variable <code>front</code> to point to the index of the front element and maintain a variable <code>size</code> to record the queue length. We define <code>rear = front + size</code>, which calculates the position right after the rear element.</p> <p>Based on this design, the valid interval containing elements in the array is <code>[front, rear - 1]</code>. The implementation methods for various operations are shown in Figure 5-6:</p> <ul> <li>Enqueue operation: Assign the input element to the <code>rear</code> index and increase <code>size</code> by 1.</li> <li>Dequeue operation: Simply increase <code>front</code> by 1 and decrease <code>size</code> by 1.</li> </ul> <p>As you can see, both enqueue and dequeue operations require only one operation, with a time complexity of \\(O(1)\\).</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> Figure 5-6   Enqueue and dequeue operations in array implementation of queue </p> <p>You may notice a problem: as we continuously enqueue and dequeue, both <code>front</code> and <code>rear</code> move to the right. When they reach the end of the array, they cannot continue moving. To solve this problem, we can treat the array as a \"circular array\" with head and tail connected.</p> <p>For a circular array, we need to let <code>front</code> or <code>rear</code> wrap around to the beginning of the array when they cross the end. This periodic pattern can be implemented using the \"modulo operation,\" as shown in the code below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_queue.py<pre><code>class ArrayQueue:\n    \"\"\"Queue based on circular array implementation\"\"\"\n\n    def __init__(self, size: int):\n        \"\"\"Constructor\"\"\"\n        self._nums: list[int] = [0] * size  # Array for storing queue elements\n        self._front: int = 0  # Front pointer, points to the front of the queue element\n        self._size: int = 0  # Queue length\n\n    def capacity(self) -&gt; int:\n        \"\"\"Get the capacity of the queue\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"Get the length of the queue\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"Check if the queue is empty\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"Enqueue\"\"\"\n        if self._size == self.capacity():\n            raise IndexError(\"Queue is full\")\n        # Calculate rear pointer, points to rear index + 1\n        # Use modulo operation to wrap rear around to the head after passing the tail of the array\n        rear: int = (self._front + self._size) % self.capacity()\n        # Add num to the rear of the queue\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"Dequeue\"\"\"\n        num: int = self.peek()\n        # Front pointer moves one position backward, if it passes the tail, return to the head of the array\n        self._front = (self._front + 1) % self.capacity()\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"Access front of the queue element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Queue is empty\")\n        return self._nums[self._front]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"Return list for printing\"\"\"\n        res = [0] * self.size()\n        j: int = self._front\n        for i in range(self.size()):\n            res[i] = self._nums[(j % self.capacity())]\n            j += 1\n        return res\n</code></pre> array_queue.cpp<pre><code>/* Queue based on circular array implementation */\nclass ArrayQueue {\n  private:\n    int *nums;       // Array for storing queue elements\n    int front;       // Front pointer, points to the front of the queue element\n    int queSize;     // Queue length\n    int queCapacity; // Queue capacity\n\n  public:\n    ArrayQueue(int capacity) {\n        // Initialize array\n        nums = new int[capacity];\n        queCapacity = capacity;\n        front = queSize = 0;\n    }\n\n    ~ArrayQueue() {\n        delete[] nums;\n    }\n\n    /* Get the capacity of the queue */\n    int capacity() {\n        return queCapacity;\n    }\n\n    /* Get the length of the queue */\n    int size() {\n        return queSize;\n    }\n\n    /* Check if the queue is empty */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* Enqueue */\n    void push(int num) {\n        if (queSize == queCapacity) {\n            cout &lt;&lt; \"Queue is full\" &lt;&lt; endl;\n            return;\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        // Add num to the rear of the queue\n        int rear = (front + queSize) % queCapacity;\n        // Front pointer moves one position backward\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* Dequeue */\n    int pop() {\n        int num = peek();\n        // Move front pointer backward by one position, if it passes the tail, return to array head\n        front = (front + 1) % queCapacity;\n        queSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    int peek() {\n        if (isEmpty())\n            throw out_of_range(\"Queue is empty\");\n        return nums[front];\n    }\n\n    /* Convert array to Vector and return */\n    vector&lt;int&gt; toVector() {\n        // Elements enqueue\n        vector&lt;int&gt; arr(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            arr[i] = nums[j % queCapacity];\n        }\n        return arr;\n    }\n};\n</code></pre> array_queue.java<pre><code>/* Queue based on circular array implementation */\nclass ArrayQueue {\n    private int[] nums; // Array for storing queue elements\n    private int front; // Front pointer, points to the front of the queue element\n    private int queSize; // Queue length\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* Get the capacity of the queue */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* Get the length of the queue */\n    public int size() {\n        return queSize;\n    }\n\n    /* Check if the queue is empty */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* Enqueue */\n    public void push(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"Queue is full\");\n            return;\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        // Add num to the rear of the queue\n        int rear = (front + queSize) % capacity();\n        // Front pointer moves one position backward\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* Dequeue */\n    public int pop() {\n        int num = peek();\n        // Move front pointer backward by one position, if it passes the tail, return to array head\n        front = (front + 1) % capacity();\n        queSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* Return array */\n    public int[] toArray() {\n        // Elements enqueue\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.cs<pre><code>/* Queue based on circular array implementation */\nclass ArrayQueue {\n    int[] nums;  // Array for storing queue elements\n    int front;   // Front pointer, points to the front of the queue element\n    int queSize; // Queue length\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* Get the capacity of the queue */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* Get the length of the queue */\n    public int Size() {\n        return queSize;\n    }\n\n    /* Check if the queue is empty */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* Enqueue */\n    public void Push(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"Queue is full\");\n            return;\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        // Add num to the rear of the queue\n        int rear = (front + queSize) % Capacity();\n        // Front pointer moves one position backward\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* Dequeue */\n    public int Pop() {\n        int num = Peek();\n        // Move front pointer backward by one position, if it passes the tail, return to array head\n        front = (front + 1) % Capacity();\n        queSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return nums[front];\n    }\n\n    /* Return array */\n    public int[] ToArray() {\n        // Elements enqueue\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % this.Capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.go<pre><code>/* Queue based on circular array implementation */\ntype arrayQueue struct {\n    nums        []int // Array for storing queue elements\n    front       int   // Front pointer, points to the front of the queue element\n    queSize     int   // Queue length\n    queCapacity int   // Queue capacity (maximum number of elements)\n}\n\n/* Access front of the queue element */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\n    return &amp;arrayQueue{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* Get the length of the queue */\nfunc (q *arrayQueue) size() int {\n    return q.queSize\n}\n\n/* Check if the queue is empty */\nfunc (q *arrayQueue) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* Enqueue */\nfunc (q *arrayQueue) push(num int) {\n    // When rear == queCapacity, queue is full\n    if q.queSize == q.queCapacity {\n        return\n    }\n    // Use modulo operation to wrap rear around to the head after passing the tail of the array\n    // Add num to the rear of the queue\n    rear := (q.front + q.queSize) % q.queCapacity\n    // Front pointer moves one position backward\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* Dequeue */\nfunc (q *arrayQueue) pop() any {\n    num := q.peek()\n    if num == nil {\n        return nil\n    }\n\n    // Move front pointer backward by one position, if it passes the tail, return to array head\n    q.front = (q.front + 1) % q.queCapacity\n    q.queSize--\n    return num\n}\n\n/* Return list for printing */\nfunc (q *arrayQueue) peek() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* Get Slice for printing */\nfunc (q *arrayQueue) toSlice() []int {\n    rear := (q.front + q.queSize)\n    if rear &gt;= q.queCapacity {\n        rear %= q.queCapacity\n        return append(q.nums[q.front:], q.nums[:rear]...)\n    }\n    return q.nums[q.front:rear]\n}\n</code></pre> array_queue.swift<pre><code>/* Queue based on circular array implementation */\nclass ArrayQueue {\n    private var nums: [Int] // Array for storing queue elements\n    private var front: Int // Front pointer, points to the front of the queue element\n    private var _size: Int // Queue length\n\n    init(capacity: Int) {\n        // Initialize array\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* Get the capacity of the queue */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* Get the length of the queue */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* Check if the queue is empty */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* Enqueue */\n    func push(num: Int) {\n        if size() == capacity() {\n            print(\"Queue is full\")\n            return\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        // Add num to the rear of the queue\n        let rear = (front + size()) % capacity()\n        // Front pointer moves one position backward\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* Dequeue */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // Move front pointer backward by one position, if it passes the tail, return to array head\n        front = (front + 1) % capacity()\n        _size -= 1\n        return num\n    }\n\n    /* Return list for printing */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Queue is empty\")\n        }\n        return nums[front]\n    }\n\n    /* Return array */\n    func toArray() -&gt; [Int] {\n        // Elements enqueue\n        (front ..&lt; front + size()).map { nums[$0 % capacity()] }\n    }\n}\n</code></pre> array_queue.js<pre><code>/* Queue based on circular array implementation */\nclass ArrayQueue {\n    #nums; // Array for storing queue elements\n    #front = 0; // Front pointer, points to the front of the queue element\n    #queSize = 0; // Queue length\n\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n    }\n\n    /* Get the capacity of the queue */\n    get capacity() {\n        return this.#nums.length;\n    }\n\n    /* Get the length of the queue */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* Check if the queue is empty */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* Enqueue */\n    push(num) {\n        if (this.size === this.capacity) {\n            console.log('Queue is full');\n            return;\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        // Add num to the rear of the queue\n        const rear = (this.#front + this.size) % this.capacity;\n        // Front pointer moves one position backward\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* Dequeue */\n    pop() {\n        const num = this.peek();\n        // Move front pointer backward by one position, if it passes the tail, return to array head\n        this.#front = (this.#front + 1) % this.capacity;\n        this.#queSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    peek() {\n        if (this.isEmpty()) throw new Error('Queue is empty');\n        return this.#nums[this.#front];\n    }\n\n    /* Return Array */\n    toArray() {\n        // Elements enqueue\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.#front; i &lt; this.size; i++, j++) {\n            arr[i] = this.#nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.ts<pre><code>/* Queue based on circular array implementation */\nclass ArrayQueue {\n    private nums: number[]; // Array for storing queue elements\n    private front: number; // Front pointer, points to the front of the queue element\n    private queSize: number; // Queue length\n\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = this.queSize = 0;\n    }\n\n    /* Get the capacity of the queue */\n    get capacity(): number {\n        return this.nums.length;\n    }\n\n    /* Get the length of the queue */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* Check if the queue is empty */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* Enqueue */\n    push(num: number): void {\n        if (this.size === this.capacity) {\n            console.log('Queue is full');\n            return;\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        // Add num to the rear of the queue\n        const rear = (this.front + this.queSize) % this.capacity;\n        // Front pointer moves one position backward\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* Dequeue */\n    pop(): number {\n        const num = this.peek();\n        // Move front pointer backward by one position, if it passes the tail, return to array head\n        this.front = (this.front + 1) % this.capacity;\n        this.queSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    peek(): number {\n        if (this.isEmpty()) throw new Error('Queue is empty');\n        return this.nums[this.front];\n    }\n\n    /* Return Array */\n    toArray(): number[] {\n        // Elements enqueue\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.front; i &lt; this.size; i++, j++) {\n            arr[i] = this.nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.dart<pre><code>/* Queue based on circular array implementation */\nclass ArrayQueue {\n  late List&lt;int&gt; _nums; // Array for storing queue elements\n  late int _front; // Front pointer, points to the front of the queue element\n  late int _queSize; // Queue length\n\n  ArrayQueue(int capacity) {\n    _nums = List.filled(capacity, 0);\n    _front = _queSize = 0;\n  }\n\n  /* Get the capacity of the queue */\n  int capaCity() {\n    return _nums.length;\n  }\n\n  /* Get the length of the queue */\n  int size() {\n    return _queSize;\n  }\n\n  /* Check if the queue is empty */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* Enqueue */\n  void push(int _num) {\n    if (_queSize == capaCity()) {\n      throw Exception(\"Queue is full\");\n    }\n    // Use modulo operation to wrap rear around to the head after passing the tail of the array\n    // Add num to the rear of the queue\n    int rear = (_front + _queSize) % capaCity();\n    // Add _num to queue rear\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* Dequeue */\n  int pop() {\n    int _num = peek();\n    // Move front pointer backward by one position, if it passes the tail, return to array head\n    _front = (_front + 1) % capaCity();\n    _queSize--;\n    return _num;\n  }\n\n  /* Return list for printing */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"Queue is empty\");\n    }\n    return _nums[_front];\n  }\n\n  /* Return Array */\n  List&lt;int&gt; toArray() {\n    // Elements enqueue\n    final List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[j % capaCity()];\n    }\n    return res;\n  }\n}\n</code></pre> array_queue.rs<pre><code>/* Queue based on circular array implementation */\nstruct ArrayQueue&lt;T&gt; {\n    nums: Vec&lt;T&gt;,      // Array for storing queue elements\n    front: i32,        // Front pointer, points to the front of the queue element\n    que_size: i32,     // Queue length\n    que_capacity: i32, // Queue capacity\n}\n\nimpl&lt;T: Copy + Default&gt; ArrayQueue&lt;T&gt; {\n    /* Constructor */\n    fn new(capacity: i32) -&gt; ArrayQueue&lt;T&gt; {\n        ArrayQueue {\n            nums: vec![T::default(); capacity as usize],\n            front: 0,\n            que_size: 0,\n            que_capacity: capacity,\n        }\n    }\n\n    /* Get the capacity of the queue */\n    fn capacity(&amp;self) -&gt; i32 {\n        self.que_capacity\n    }\n\n    /* Get the length of the queue */\n    fn size(&amp;self) -&gt; i32 {\n        self.que_size\n    }\n\n    /* Check if the queue is empty */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* Enqueue */\n    fn push(&amp;mut self, num: T) {\n        if self.que_size == self.capacity() {\n            println!(\"Queue is full\");\n            return;\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        // Add num to the rear of the queue\n        let rear = (self.front + self.que_size) % self.que_capacity;\n        // Front pointer moves one position backward\n        self.nums[rear as usize] = num;\n        self.que_size += 1;\n    }\n\n    /* Dequeue */\n    fn pop(&amp;mut self) -&gt; T {\n        let num = self.peek();\n        // Move front pointer backward by one position, if it passes the tail, return to array head\n        self.front = (self.front + 1) % self.que_capacity;\n        self.que_size -= 1;\n        num\n    }\n\n    /* Return list for printing */\n    fn peek(&amp;self) -&gt; T {\n        if self.is_empty() {\n            panic!(\"index out of bounds\");\n        }\n        self.nums[self.front as usize]\n    }\n\n    /* Return array */\n    fn to_vector(&amp;self) -&gt; Vec&lt;T&gt; {\n        let cap = self.que_capacity;\n        let mut j = self.front;\n        let mut arr = vec![T::default(); cap as usize];\n        for i in 0..self.que_size {\n            arr[i as usize] = self.nums[(j % cap) as usize];\n            j += 1;\n        }\n        arr\n    }\n}\n</code></pre> array_queue.c<pre><code>/* Queue based on circular array implementation */\ntypedef struct {\n    int *nums;       // Array for storing queue elements\n    int front;       // Front pointer, points to the front of the queue element\n    int queSize;     // Current number of elements in the queue\n    int queCapacity; // Queue capacity\n} ArrayQueue;\n\n/* Constructor */\nArrayQueue *newArrayQueue(int capacity) {\n    ArrayQueue *queue = (ArrayQueue *)malloc(sizeof(ArrayQueue));\n    // Initialize array\n    queue-&gt;queCapacity = capacity;\n    queue-&gt;nums = (int *)malloc(sizeof(int) * queue-&gt;queCapacity);\n    queue-&gt;front = queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* Destructor */\nvoid delArrayQueue(ArrayQueue *queue) {\n    free(queue-&gt;nums);\n    free(queue);\n}\n\n/* Get the capacity of the queue */\nint capacity(ArrayQueue *queue) {\n    return queue-&gt;queCapacity;\n}\n\n/* Get the length of the queue */\nint size(ArrayQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* Check if the queue is empty */\nbool empty(ArrayQueue *queue) {\n    return queue-&gt;queSize == 0;\n}\n\n/* Return list for printing */\nint peek(ArrayQueue *queue) {\n    assert(size(queue) != 0);\n    return queue-&gt;nums[queue-&gt;front];\n}\n\n/* Enqueue */\nvoid push(ArrayQueue *queue, int num) {\n    if (size(queue) == capacity(queue)) {\n        printf(\"Queue is full\\r\\n\");\n        return;\n    }\n    // Use modulo operation to wrap rear around to the head after passing the tail of the array\n    // Add num to the rear of the queue\n    int rear = (queue-&gt;front + queue-&gt;queSize) % queue-&gt;queCapacity;\n    // Front pointer moves one position backward\n    queue-&gt;nums[rear] = num;\n    queue-&gt;queSize++;\n}\n\n/* Dequeue */\nint pop(ArrayQueue *queue) {\n    int num = peek(queue);\n    // Move front pointer backward by one position, if it passes the tail, return to array head\n    queue-&gt;front = (queue-&gt;front + 1) % queue-&gt;queCapacity;\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* Return array for printing */\nint *toArray(ArrayQueue *queue, int *queSize) {\n    *queSize = queue-&gt;queSize;\n    int *res = (int *)calloc(queue-&gt;queSize, sizeof(int));\n    int j = queue-&gt;front;\n    for (int i = 0; i &lt; queue-&gt;queSize; i++) {\n        res[i] = queue-&gt;nums[j % queue-&gt;queCapacity];\n        j++;\n    }\n    return res;\n}\n</code></pre> array_queue.kt<pre><code>/* Queue based on circular array implementation */\nclass ArrayQueue(capacity: Int) {\n    private val nums: IntArray = IntArray(capacity) // Array for storing queue elements\n    private var front: Int = 0 // Front pointer, points to the front of the queue element\n    private var queSize: Int = 0 // Queue length\n\n    /* Get the capacity of the queue */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* Get the length of the queue */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* Check if the queue is empty */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* Enqueue */\n    fun push(num: Int) {\n        if (queSize == capacity()) {\n            println(\"Queue is full\")\n            return\n        }\n        // Use modulo operation to wrap rear around to the head after passing the tail of the array\n        // Add num to the rear of the queue\n        val rear = (front + queSize) % capacity()\n        // Front pointer moves one position backward\n        nums[rear] = num\n        queSize++\n    }\n\n    /* Dequeue */\n    fun pop(): Int {\n        val num = peek()\n        // Move front pointer backward by one position, if it passes the tail, return to array head\n        front = (front + 1) % capacity()\n        queSize--\n        return num\n    }\n\n    /* Return list for printing */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* Return array */\n    fun toArray(): IntArray {\n        // Elements enqueue\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[j % capacity()]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_queue.rb<pre><code>### Queue based on circular array ###\nclass ArrayQueue\n  ### Get queue length ###\n  attr_reader :size\n\n  ### Constructor ###\n  def initialize(size)\n    @nums = Array.new(size, 0) # Array for storing queue elements\n    @front = 0 # Front pointer, points to the front of the queue element\n    @size = 0 # Queue length\n  end\n\n  ### Get queue capacity ###\n  def capacity\n    @nums.length\n  end\n\n  ### Check if queue is empty ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### Enqueue ###\n  def push(num)\n    raise IndexError, 'Queue is full' if size == capacity\n\n    # Use modulo operation to wrap rear around to the head after passing the tail of the array\n    # Add num to the rear of the queue\n    rear = (@front + size) % capacity\n    # Front pointer moves one position backward\n    @nums[rear] = num\n    @size += 1\n  end\n\n  ### Dequeue ###\n  def pop\n    num = peek\n    # Move front pointer backward by one position, if it passes the tail, return to array head\n    @front = (@front + 1) % capacity\n    @size -= 1\n    num\n  end\n\n  ### Access front element ###\n  def peek\n    raise IndexError, 'Queue is empty' if is_empty?\n\n    @nums[@front]\n  end\n\n  ### Return list for printing ###\n  def to_array\n    res = Array.new(size, 0)\n    j = @front\n\n    for i in 0...size\n      res[i] = @nums[j % capacity]\n      j += 1\n    end\n\n    res\n  end\nend\n</code></pre> <p>The queue implemented above still has limitations: its length is immutable. However, this problem is not difficult to solve. We can replace the array with a dynamic array to introduce an expansion mechanism. Interested readers can try to implement this themselves.</p> <p>The comparison conclusions for the two implementations are consistent with those for stacks and will not be repeated here.</p>","path":["Chapter 5. Stacks and Queues","5.2   Queue"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#523-typical-applications-of-queue","level":2,"title":"5.2.3   Typical Applications of Queue","text":"<ul> <li>Taobao orders. After shoppers place orders, the orders are added to a queue, and the system subsequently processes the orders in the queue according to their sequence. During Double Eleven, massive orders are generated in a short time, and high concurrency becomes a key challenge that engineers need to tackle.</li> <li>Various to-do tasks. Any scenario that needs to implement \"first come, first served\" functionality, such as a printer's task queue or a restaurant's order queue, can effectively maintain the processing order using queues.</li> </ul>","path":["Chapter 5. Stacks and Queues","5.2   Queue"],"tags":[]},{"location":"chapter_stack_and_queue/stack/","level":1,"title":"5.1   Stack","text":"<p>A stack is a linear data structure that follows the Last In, First Out (LIFO) principle.</p> <p>We can compare a stack to a pile of plates on a table. If we specify that only one plate can be moved at a time, then to get the bottom plate, we must first remove the plates above it one by one. If we replace the plates with various types of elements (such as integers, characters, objects, etc.), we get the stack data structure.</p> <p>As shown in Figure 5-1, we call the top of the stacked elements the \"top\" and the bottom the \"bottom.\" The operation of adding an element to the top is called \"push,\" and the operation of removing the top element is called \"pop.\"</p> <p></p> <p> Figure 5-1   LIFO rule of stack </p>","path":["Chapter 5. Stacks and Queues","5.1   Stack"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#511-common-stack-operations","level":2,"title":"5.1.1   Common Stack Operations","text":"<p>The common operations on a stack are shown in Table 5-1. The specific method names depend on the programming language used. Here, we use the common naming convention of <code>push()</code>, <code>pop()</code>, and <code>peek()</code>.</p> <p> Table 5-1   Efficiency of Stack Operations </p> Method Description Time Complexity <code>push()</code> Push element onto stack (add to top) \\(O(1)\\) <code>pop()</code> Pop top element from stack \\(O(1)\\) <code>peek()</code> Access top element \\(O(1)\\) <p>Typically, we can directly use the built-in stack class provided by the programming language. However, some languages may not provide a dedicated stack class. In such cases, we can use the language's \"array\" or \"linked list\" as a stack and simply avoid using operations unrelated to stack behavior.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby stack.py<pre><code># Initialize stack\n# Python does not have a built-in stack class, can use list as a stack\nstack: list[int] = []\n\n# Push elements\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n# Access top element\npeek: int = stack[-1]\n\n# Pop element\npop: int = stack.pop()\n\n# Get stack length\nsize: int = len(stack)\n\n# Check if empty\nis_empty: bool = len(stack) == 0\n</code></pre> stack.cpp<pre><code>/* Initialize stack */\nstack&lt;int&gt; stack;\n\n/* Push elements */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access top element */\nint top = stack.top();\n\n/* Pop element */\nstack.pop(); // No return value\n\n/* Get stack length */\nint size = stack.size();\n\n/* Check if empty */\nbool empty = stack.empty();\n</code></pre> stack.java<pre><code>/* Initialize stack */\nStack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n\n/* Push elements */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access top element */\nint peek = stack.peek();\n\n/* Pop element */\nint pop = stack.pop();\n\n/* Get stack length */\nint size = stack.size();\n\n/* Check if empty */\nboolean isEmpty = stack.isEmpty();\n</code></pre> stack.cs<pre><code>/* Initialize stack */\nStack&lt;int&gt; stack = new();\n\n/* Push elements */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n\n/* Access top element */\nint peek = stack.Peek();\n\n/* Pop element */\nint pop = stack.Pop();\n\n/* Get stack length */\nint size = stack.Count;\n\n/* Check if empty */\nbool isEmpty = stack.Count == 0;\n</code></pre> stack_test.go<pre><code>/* Initialize stack */\n// In Go, it is recommended to use Slice as a stack\nvar stack []int\n\n/* Push elements */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n\n/* Access top element */\npeek := stack[len(stack)-1]\n\n/* Pop element */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n\n/* Get stack length */\nsize := len(stack)\n\n/* Check if empty */\nisEmpty := len(stack) == 0\n</code></pre> stack.swift<pre><code>/* Initialize stack */\n// Swift does not have a built-in stack class, can use Array as a stack\nvar stack: [Int] = []\n\n/* Push elements */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n/* Access top element */\nlet peek = stack.last!\n\n/* Pop element */\nlet pop = stack.removeLast()\n\n/* Get stack length */\nlet size = stack.count\n\n/* Check if empty */\nlet isEmpty = stack.isEmpty\n</code></pre> stack.js<pre><code>/* Initialize stack */\n// JavaScript does not have a built-in stack class, can use Array as a stack\nconst stack = [];\n\n/* Push elements */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access top element */\nconst peek = stack[stack.length-1];\n\n/* Pop element */\nconst pop = stack.pop();\n\n/* Get stack length */\nconst size = stack.length;\n\n/* Check if empty */\nconst is_empty = stack.length === 0;\n</code></pre> stack.ts<pre><code>/* Initialize stack */\n// TypeScript does not have a built-in stack class, can use Array as a stack\nconst stack: number[] = [];\n\n/* Push elements */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access top element */\nconst peek = stack[stack.length - 1];\n\n/* Pop element */\nconst pop = stack.pop();\n\n/* Get stack length */\nconst size = stack.length;\n\n/* Check if empty */\nconst is_empty = stack.length === 0;\n</code></pre> stack.dart<pre><code>/* Initialize stack */\n// Dart does not have a built-in stack class, can use List as a stack\nList&lt;int&gt; stack = [];\n\n/* Push elements */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n\n/* Access top element */\nint peek = stack.last;\n\n/* Pop element */\nint pop = stack.removeLast();\n\n/* Get stack length */\nint size = stack.length;\n\n/* Check if empty */\nbool isEmpty = stack.isEmpty;\n</code></pre> stack.rs<pre><code>/* Initialize stack */\n// Use Vec as a stack\nlet mut stack: Vec&lt;i32&gt; = Vec::new();\n\n/* Push elements */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access top element */\nlet top = stack.last().unwrap();\n\n/* Pop element */\nlet pop = stack.pop().unwrap();\n\n/* Get stack length */\nlet size = stack.len();\n\n/* Check if empty */\nlet is_empty = stack.is_empty();\n</code></pre> stack.c<pre><code>// C does not provide a built-in stack\n</code></pre> stack.kt<pre><code>/* Initialize stack */\nval stack = Stack&lt;Int&gt;()\n\n/* Push elements */\nstack.push(1)\nstack.push(3)\nstack.push(2)\nstack.push(5)\nstack.push(4)\n\n/* Access top element */\nval peek = stack.peek()\n\n/* Pop element */\nval pop = stack.pop()\n\n/* Get stack length */\nval size = stack.size\n\n/* Check if empty */\nval isEmpty = stack.isEmpty()\n</code></pre> stack.rb<pre><code># Initialize stack\n# Ruby does not have a built-in stack class, can use Array as a stack\nstack = []\n\n# Push elements\nstack &lt;&lt; 1\nstack &lt;&lt; 3\nstack &lt;&lt; 2\nstack &lt;&lt; 5\nstack &lt;&lt; 4\n\n# Access top element\npeek = stack.last\n\n# Pop element\npop = stack.pop\n\n# Get stack length\nsize = stack.length\n\n# Check if empty\nis_empty = stack.empty?\n</code></pre> Code Visualization <p> Full Screen &gt;</p>","path":["Chapter 5. Stacks and Queues","5.1   Stack"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#512-stack-implementation","level":2,"title":"5.1.2   Stack Implementation","text":"<p>To gain a deeper understanding of how a stack operates, let's try implementing a stack class ourselves.</p> <p>A stack follows the LIFO principle, so we can only add or remove elements at the top. However, both arrays and linked lists allow adding and removing elements at any position. Therefore, a stack can be viewed as a restricted array or linked list. In other words, we can \"shield\" some irrelevant operations of arrays or linked lists so that their external logic conforms to the characteristics of a stack.</p>","path":["Chapter 5. Stacks and Queues","5.1   Stack"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#1-linked-list-implementation","level":3,"title":"1.   Linked List Implementation","text":"<p>When implementing a stack using a linked list, we can treat the head node of the linked list as the top of the stack and the tail node as the base.</p> <p>As shown in Figure 5-2, for the push operation, we simply insert an element at the head of the linked list. This node insertion method is called the \"head insertion method.\" For the pop operation, we just need to remove the head node from the linked list.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> Figure 5-2   Push and pop operations in linked list implementation of stack </p> <p>Below is sample code for implementing a stack based on a linked list:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linkedlist_stack.py<pre><code>class LinkedListStack:\n    \"\"\"Stack based on linked list implementation\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        self._peek: ListNode | None = None\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"Get the length of the stack\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"Check if the stack is empty\"\"\"\n        return self._size == 0\n\n    def push(self, val: int):\n        \"\"\"Push\"\"\"\n        node = ListNode(val)\n        node.next = self._peek\n        self._peek = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"Pop\"\"\"\n        num = self.peek()\n        self._peek = self._peek.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"Access top of the stack element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Stack is empty\")\n        return self._peek.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"Convert to list for printing\"\"\"\n        arr = []\n        node = self._peek\n        while node:\n            arr.append(node.val)\n            node = node.next\n        arr.reverse()\n        return arr\n</code></pre> linkedlist_stack.cpp<pre><code>/* Stack based on linked list implementation */\nclass LinkedListStack {\n  private:\n    ListNode *stackTop; // Use head node as stack top\n    int stkSize;        // Stack length\n\n  public:\n    LinkedListStack() {\n        stackTop = nullptr;\n        stkSize = 0;\n    }\n\n    ~LinkedListStack() {\n        // Traverse linked list to delete nodes and free memory\n        freeMemoryLinkedList(stackTop);\n    }\n\n    /* Get the length of the stack */\n    int size() {\n        return stkSize;\n    }\n\n    /* Check if the stack is empty */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* Push */\n    void push(int num) {\n        ListNode *node = new ListNode(num);\n        node-&gt;next = stackTop;\n        stackTop = node;\n        stkSize++;\n    }\n\n    /* Pop */\n    int pop() {\n        int num = top();\n        ListNode *tmp = stackTop;\n        stackTop = stackTop-&gt;next;\n        // Free memory\n        delete tmp;\n        stkSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"Stack is empty\");\n        return stackTop-&gt;val;\n    }\n\n    /* Convert List to Array and return */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = stackTop;\n        vector&lt;int&gt; res(size());\n        for (int i = res.size() - 1; i &gt;= 0; i--) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_stack.java<pre><code>/* Stack based on linked list implementation */\nclass LinkedListStack {\n    private ListNode stackPeek; // Use head node as stack top\n    private int stkSize = 0; // Stack length\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* Get the length of the stack */\n    public int size() {\n        return stkSize;\n    }\n\n    /* Check if the stack is empty */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* Push */\n    public void push(int num) {\n        ListNode node = new ListNode(num);\n        node.next = stackPeek;\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* Pop */\n    public int pop() {\n        int num = peek();\n        stackPeek = stackPeek.next;\n        stkSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stackPeek.val;\n    }\n\n    /* Convert List to Array and return */\n    public int[] toArray() {\n        ListNode node = stackPeek;\n        int[] res = new int[size()];\n        for (int i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.cs<pre><code>/* Stack based on linked list implementation */\nclass LinkedListStack {\n    ListNode? stackPeek;  // Use head node as stack top\n    int stkSize = 0;   // Stack length\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* Get the length of the stack */\n    public int Size() {\n        return stkSize;\n    }\n\n    /* Check if the stack is empty */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* Push */\n    public void Push(int num) {\n        ListNode node = new(num) {\n            next = stackPeek\n        };\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* Pop */\n    public int Pop() {\n        int num = Peek();\n        stackPeek = stackPeek!.next;\n        stkSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stackPeek!.val;\n    }\n\n    /* Convert List to Array and return */\n    public int[] ToArray() {\n        if (stackPeek == null)\n            return [];\n\n        ListNode? node = stackPeek;\n        int[] res = new int[Size()];\n        for (int i = res.Length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.go<pre><code>/* Stack based on linked list implementation */\ntype linkedListStack struct {\n    // Use built-in package list to implement stack\n    data *list.List\n}\n\n/* Access top of the stack element */\nfunc newLinkedListStack() *linkedListStack {\n    return &amp;linkedListStack{\n        data: list.New(),\n    }\n}\n\n/* Push */\nfunc (s *linkedListStack) push(value int) {\n    s.data.PushBack(value)\n}\n\n/* Pop */\nfunc (s *linkedListStack) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* Return list for printing */\nfunc (s *linkedListStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* Get the length of the stack */\nfunc (s *linkedListStack) size() int {\n    return s.data.Len()\n}\n\n/* Check if the stack is empty */\nfunc (s *linkedListStack) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* Get List for printing */\nfunc (s *linkedListStack) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_stack.swift<pre><code>/* Stack based on linked list implementation */\nclass LinkedListStack {\n    private var _peek: ListNode? // Use head node as stack top\n    private var _size: Int // Stack length\n\n    init() {\n        _size = 0\n    }\n\n    /* Get the length of the stack */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* Check if the stack is empty */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* Push */\n    func push(num: Int) {\n        let node = ListNode(x: num)\n        node.next = _peek\n        _peek = node\n        _size += 1\n    }\n\n    /* Pop */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        _peek = _peek?.next\n        _size -= 1\n        return num\n    }\n\n    /* Return list for printing */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Stack is empty\")\n        }\n        return _peek!.val\n    }\n\n    /* Convert List to Array and return */\n    func toArray() -&gt; [Int] {\n        var node = _peek\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices.reversed() {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.js<pre><code>/* Stack based on linked list implementation */\nclass LinkedListStack {\n    #stackPeek; // Use head node as stack top\n    #stkSize = 0; // Stack length\n\n    constructor() {\n        this.#stackPeek = null;\n    }\n\n    /* Get the length of the stack */\n    get size() {\n        return this.#stkSize;\n    }\n\n    /* Check if the stack is empty */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* Push */\n    push(num) {\n        const node = new ListNode(num);\n        node.next = this.#stackPeek;\n        this.#stackPeek = node;\n        this.#stkSize++;\n    }\n\n    /* Pop */\n    pop() {\n        const num = this.peek();\n        this.#stackPeek = this.#stackPeek.next;\n        this.#stkSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    peek() {\n        if (!this.#stackPeek) throw new Error('Stack is empty');\n        return this.#stackPeek.val;\n    }\n\n    /* Convert linked list to Array and return */\n    toArray() {\n        let node = this.#stackPeek;\n        const res = new Array(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.ts<pre><code>/* Stack based on linked list implementation */\nclass LinkedListStack {\n    private stackPeek: ListNode | null; // Use head node as stack top\n    private stkSize: number = 0; // Stack length\n\n    constructor() {\n        this.stackPeek = null;\n    }\n\n    /* Get the length of the stack */\n    get size(): number {\n        return this.stkSize;\n    }\n\n    /* Check if the stack is empty */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* Push */\n    push(num: number): void {\n        const node = new ListNode(num);\n        node.next = this.stackPeek;\n        this.stackPeek = node;\n        this.stkSize++;\n    }\n\n    /* Pop */\n    pop(): number {\n        const num = this.peek();\n        if (!this.stackPeek) throw new Error('Stack is empty');\n        this.stackPeek = this.stackPeek.next;\n        this.stkSize--;\n        return num;\n    }\n\n    /* Return list for printing */\n    peek(): number {\n        if (!this.stackPeek) throw new Error('Stack is empty');\n        return this.stackPeek.val;\n    }\n\n    /* Convert linked list to Array and return */\n    toArray(): number[] {\n        let node = this.stackPeek;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.dart<pre><code>/* Stack implemented based on linked list class */\nclass LinkedListStack {\n  ListNode? _stackPeek; // Use head node as stack top\n  int _stkSize = 0; // Stack length\n\n  LinkedListStack() {\n    _stackPeek = null;\n  }\n\n  /* Get the length of the stack */\n  int size() {\n    return _stkSize;\n  }\n\n  /* Check if the stack is empty */\n  bool isEmpty() {\n    return _stkSize == 0;\n  }\n\n  /* Push */\n  void push(int _num) {\n    final ListNode node = ListNode(_num);\n    node.next = _stackPeek;\n    _stackPeek = node;\n    _stkSize++;\n  }\n\n  /* Pop */\n  int pop() {\n    final int _num = peek();\n    _stackPeek = _stackPeek!.next;\n    _stkSize--;\n    return _num;\n  }\n\n  /* Return list for printing */\n  int peek() {\n    if (_stackPeek == null) {\n      throw Exception(\"Stack is empty\");\n    }\n    return _stackPeek!.val;\n  }\n\n  /* Convert linked list to List and return */\n  List&lt;int&gt; toList() {\n    ListNode? node = _stackPeek;\n    List&lt;int&gt; list = [];\n    while (node != null) {\n      list.add(node.val);\n      node = node.next;\n    }\n    list = list.reversed.toList();\n    return list;\n  }\n}\n</code></pre> linkedlist_stack.rs<pre><code>/* Stack based on linked list implementation */\n#[allow(dead_code)]\npub struct LinkedListStack&lt;T&gt; {\n    stack_peek: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // Use head node as stack top\n    stk_size: usize,                              // Stack length\n}\n\nimpl&lt;T: Copy&gt; LinkedListStack&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            stack_peek: None,\n            stk_size: 0,\n        }\n    }\n\n    /* Get the length of the stack */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.stk_size;\n    }\n\n    /* Check if the stack is empty */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* Push */\n    pub fn push(&amp;mut self, num: T) {\n        let node = ListNode::new(num);\n        node.borrow_mut().next = self.stack_peek.take();\n        self.stack_peek = Some(node);\n        self.stk_size += 1;\n    }\n\n    /* Pop */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack_peek.take().map(|old_head| {\n            self.stack_peek = old_head.borrow_mut().next.take();\n            self.stk_size -= 1;\n\n            old_head.borrow().val\n        })\n    }\n\n    /* Return list for printing */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.stack_peek.as_ref()\n    }\n\n    /* Convert List to Array and return */\n    pub fn to_array(&amp;self) -&gt; Vec&lt;T&gt; {\n        fn _to_array&lt;T: Sized + Copy&gt;(head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n            if let Some(node) = head {\n                let mut nums = _to_array(node.borrow().next.as_ref());\n                nums.push(node.borrow().val);\n                return nums;\n            }\n            return Vec::new();\n        }\n\n        _to_array(self.peek())\n    }\n}\n</code></pre> linkedlist_stack.c<pre><code>/* Stack based on linked list implementation */\ntypedef struct {\n    ListNode *top; // Use head node as stack top\n    int size;      // Stack length\n} LinkedListStack;\n\n/* Constructor */\nLinkedListStack *newLinkedListStack() {\n    LinkedListStack *s = malloc(sizeof(LinkedListStack));\n    s-&gt;top = NULL;\n    s-&gt;size = 0;\n    return s;\n}\n\n/* Destructor */\nvoid delLinkedListStack(LinkedListStack *s) {\n    while (s-&gt;top) {\n        ListNode *n = s-&gt;top-&gt;next;\n        free(s-&gt;top);\n        s-&gt;top = n;\n    }\n    free(s);\n}\n\n/* Get the length of the stack */\nint size(LinkedListStack *s) {\n    return s-&gt;size;\n}\n\n/* Check if the stack is empty */\nbool isEmpty(LinkedListStack *s) {\n    return size(s) == 0;\n}\n\n/* Push */\nvoid push(LinkedListStack *s, int num) {\n    ListNode *node = (ListNode *)malloc(sizeof(ListNode));\n    node-&gt;next = s-&gt;top; // Update new node's pointer field\n    node-&gt;val = num;     // Update new node's data field\n    s-&gt;top = node;       // Update stack top\n    s-&gt;size++;           // Update stack size\n}\n\n/* Return list for printing */\nint peek(LinkedListStack *s) {\n    if (s-&gt;size == 0) {\n        printf(\"Stack is empty\\n\");\n        return INT_MAX;\n    }\n    return s-&gt;top-&gt;val;\n}\n\n/* Pop */\nint pop(LinkedListStack *s) {\n    int val = peek(s);\n    ListNode *tmp = s-&gt;top;\n    s-&gt;top = s-&gt;top-&gt;next;\n    // Free memory\n    free(tmp);\n    s-&gt;size--;\n    return val;\n}\n</code></pre> linkedlist_stack.kt<pre><code>/* Stack based on linked list implementation */\nclass LinkedListStack(\n    private var stackPeek: ListNode? = null, // Use head node as stack top\n    private var stkSize: Int = 0 // Stack length\n) {\n\n    /* Get the length of the stack */\n    fun size(): Int {\n        return stkSize\n    }\n\n    /* Check if the stack is empty */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* Push */\n    fun push(num: Int) {\n        val node = ListNode(num)\n        node.next = stackPeek\n        stackPeek = node\n        stkSize++\n    }\n\n    /* Pop */\n    fun pop(): Int? {\n        val num = peek()\n        stackPeek = stackPeek?.next\n        stkSize--\n        return num\n    }\n\n    /* Return list for printing */\n    fun peek(): Int? {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stackPeek?._val\n    }\n\n    /* Convert List to Array and return */\n    fun toArray(): IntArray {\n        var node = stackPeek\n        val res = IntArray(size())\n        for (i in res.size - 1 downTo 0) {\n            res[i] = node?._val!!\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.rb<pre><code>### Stack based on linked list ###\nclass LinkedListStack\n  attr_reader :size\n\n  ### Constructor ###\n  def initialize\n    @size = 0\n  end\n\n  ### Check if stack is empty ###\n  def is_empty?\n    @peek.nil?\n  end\n\n  ### Push ###\n  def push(val)\n    node = ListNode.new(val)\n    node.next = @peek\n    @peek = node\n    @size += 1\n  end\n\n  ### Pop ###\n  def pop\n    num = peek\n    @peek = @peek.next\n    @size -= 1\n    num\n  end\n\n  ### Access top element ###\n  def peek\n    raise IndexError, 'Stack is empty' if is_empty?\n\n    @peek.val\n  end\n\n  ### Convert linked list to Array and return ###\n  def to_array\n    arr = []\n    node = @peek\n    while node\n      arr &lt;&lt; node.val\n      node = node.next\n    end\n    arr.reverse\n  end\nend\n</code></pre>","path":["Chapter 5. Stacks and Queues","5.1   Stack"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#2-array-implementation","level":3,"title":"2.   Array Implementation","text":"<p>When implementing a stack using an array, we can treat the end of the array as the top of the stack. As shown in Figure 5-3, push and pop operations correspond to adding and removing elements at the end of the array, both with a time complexity of \\(O(1)\\).</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> Figure 5-3   Push and pop operations in array implementation of stack </p> <p>Since elements pushed onto the stack may increase continuously, we can use a dynamic array, which eliminates the need to handle array expansion ourselves. Here is the sample code:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_stack.py<pre><code>class ArrayStack:\n    \"\"\"Stack based on array implementation\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        self._stack: list[int] = []\n\n    def size(self) -&gt; int:\n        \"\"\"Get the length of the stack\"\"\"\n        return len(self._stack)\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"Check if the stack is empty\"\"\"\n        return self.size() == 0\n\n    def push(self, item: int):\n        \"\"\"Push\"\"\"\n        self._stack.append(item)\n\n    def pop(self) -&gt; int:\n        \"\"\"Pop\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Stack is empty\")\n        return self._stack.pop()\n\n    def peek(self) -&gt; int:\n        \"\"\"Access top of the stack element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Stack is empty\")\n        return self._stack[-1]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"Return list for printing\"\"\"\n        return self._stack\n</code></pre> array_stack.cpp<pre><code>/* Stack based on array implementation */\nclass ArrayStack {\n  private:\n    vector&lt;int&gt; stack;\n\n  public:\n    /* Get the length of the stack */\n    int size() {\n        return stack.size();\n    }\n\n    /* Check if the stack is empty */\n    bool isEmpty() {\n        return stack.size() == 0;\n    }\n\n    /* Push */\n    void push(int num) {\n        stack.push_back(num);\n    }\n\n    /* Pop */\n    int pop() {\n        int num = top();\n        stack.pop_back();\n        return num;\n    }\n\n    /* Return list for printing */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"Stack is empty\");\n        return stack.back();\n    }\n\n    /* Return Vector */\n    vector&lt;int&gt; toVector() {\n        return stack;\n    }\n};\n</code></pre> array_stack.java<pre><code>/* Stack based on array implementation */\nclass ArrayStack {\n    private ArrayList&lt;Integer&gt; stack;\n\n    public ArrayStack() {\n        // Initialize list (dynamic array)\n        stack = new ArrayList&lt;&gt;();\n    }\n\n    /* Get the length of the stack */\n    public int size() {\n        return stack.size();\n    }\n\n    /* Check if the stack is empty */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* Push */\n    public void push(int num) {\n        stack.add(num);\n    }\n\n    /* Pop */\n    public int pop() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.remove(size() - 1);\n    }\n\n    /* Return list for printing */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.get(size() - 1);\n    }\n\n    /* Convert List to Array and return */\n    public Object[] toArray() {\n        return stack.toArray();\n    }\n}\n</code></pre> array_stack.cs<pre><code>/* Stack based on array implementation */\nclass ArrayStack {\n    List&lt;int&gt; stack;\n    public ArrayStack() {\n        // Initialize list (dynamic array)\n        stack = [];\n    }\n\n    /* Get the length of the stack */\n    public int Size() {\n        return stack.Count;\n    }\n\n    /* Check if the stack is empty */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* Push */\n    public void Push(int num) {\n        stack.Add(num);\n    }\n\n    /* Pop */\n    public int Pop() {\n        if (IsEmpty())\n            throw new Exception();\n        var val = Peek();\n        stack.RemoveAt(Size() - 1);\n        return val;\n    }\n\n    /* Return list for printing */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stack[Size() - 1];\n    }\n\n    /* Convert List to Array and return */\n    public int[] ToArray() {\n        return [.. stack];\n    }\n}\n</code></pre> array_stack.go<pre><code>/* Stack based on array implementation */\ntype arrayStack struct {\n    data []int // Data\n}\n\n/* Access top of the stack element */\nfunc newArrayStack() *arrayStack {\n    return &amp;arrayStack{\n        // Set stack length to 0, capacity to 16\n        data: make([]int, 0, 16),\n    }\n}\n\n/* Stack length */\nfunc (s *arrayStack) size() int {\n    return len(s.data)\n}\n\n/* Is stack empty */\nfunc (s *arrayStack) isEmpty() bool {\n    return s.size() == 0\n}\n\n/* Push */\nfunc (s *arrayStack) push(v int) {\n    // Slice will automatically expand\n    s.data = append(s.data, v)\n}\n\n/* Pop */\nfunc (s *arrayStack) pop() any {\n    val := s.peek()\n    s.data = s.data[:len(s.data)-1]\n    return val\n}\n\n/* Get stack top element */\nfunc (s *arrayStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    val := s.data[len(s.data)-1]\n    return val\n}\n\n/* Get Slice for printing */\nfunc (s *arrayStack) toSlice() []int {\n    return s.data\n}\n</code></pre> array_stack.swift<pre><code>/* Stack based on array implementation */\nclass ArrayStack {\n    private var stack: [Int]\n\n    init() {\n        // Initialize list (dynamic array)\n        stack = []\n    }\n\n    /* Get the length of the stack */\n    func size() -&gt; Int {\n        stack.count\n    }\n\n    /* Check if the stack is empty */\n    func isEmpty() -&gt; Bool {\n        stack.isEmpty\n    }\n\n    /* Push */\n    func push(num: Int) {\n        stack.append(num)\n    }\n\n    /* Pop */\n    @discardableResult\n    func pop() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Stack is empty\")\n        }\n        return stack.removeLast()\n    }\n\n    /* Return list for printing */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Stack is empty\")\n        }\n        return stack.last!\n    }\n\n    /* Convert List to Array and return */\n    func toArray() -&gt; [Int] {\n        stack\n    }\n}\n</code></pre> array_stack.js<pre><code>/* Stack based on array implementation */\nclass ArrayStack {\n    #stack;\n    constructor() {\n        this.#stack = [];\n    }\n\n    /* Get the length of the stack */\n    get size() {\n        return this.#stack.length;\n    }\n\n    /* Check if the stack is empty */\n    isEmpty() {\n        return this.#stack.length === 0;\n    }\n\n    /* Push */\n    push(num) {\n        this.#stack.push(num);\n    }\n\n    /* Pop */\n    pop() {\n        if (this.isEmpty()) throw new Error('Stack is empty');\n        return this.#stack.pop();\n    }\n\n    /* Return list for printing */\n    top() {\n        if (this.isEmpty()) throw new Error('Stack is empty');\n        return this.#stack[this.#stack.length - 1];\n    }\n\n    /* Return Array */\n    toArray() {\n        return this.#stack;\n    }\n}\n</code></pre> array_stack.ts<pre><code>/* Stack based on array implementation */\nclass ArrayStack {\n    private stack: number[];\n    constructor() {\n        this.stack = [];\n    }\n\n    /* Get the length of the stack */\n    get size(): number {\n        return this.stack.length;\n    }\n\n    /* Check if the stack is empty */\n    isEmpty(): boolean {\n        return this.stack.length === 0;\n    }\n\n    /* Push */\n    push(num: number): void {\n        this.stack.push(num);\n    }\n\n    /* Pop */\n    pop(): number | undefined {\n        if (this.isEmpty()) throw new Error('Stack is empty');\n        return this.stack.pop();\n    }\n\n    /* Return list for printing */\n    top(): number | undefined {\n        if (this.isEmpty()) throw new Error('Stack is empty');\n        return this.stack[this.stack.length - 1];\n    }\n\n    /* Return Array */\n    toArray() {\n        return this.stack;\n    }\n}\n</code></pre> array_stack.dart<pre><code>/* Stack based on array implementation */\nclass ArrayStack {\n  late List&lt;int&gt; _stack;\n  ArrayStack() {\n    _stack = [];\n  }\n\n  /* Get the length of the stack */\n  int size() {\n    return _stack.length;\n  }\n\n  /* Check if the stack is empty */\n  bool isEmpty() {\n    return _stack.isEmpty;\n  }\n\n  /* Push */\n  void push(int _num) {\n    _stack.add(_num);\n  }\n\n  /* Pop */\n  int pop() {\n    if (isEmpty()) {\n      throw Exception(\"Stack is empty\");\n    }\n    return _stack.removeLast();\n  }\n\n  /* Return list for printing */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"Stack is empty\");\n    }\n    return _stack.last;\n  }\n\n  /* Convert stack to Array and return */\n  List&lt;int&gt; toArray() =&gt; _stack;\n}\n</code></pre> array_stack.rs<pre><code>/* Stack based on array implementation */\nstruct ArrayStack&lt;T&gt; {\n    stack: Vec&lt;T&gt;,\n}\n\nimpl&lt;T&gt; ArrayStack&lt;T&gt; {\n    /* Access top of the stack element */\n    fn new() -&gt; ArrayStack&lt;T&gt; {\n        ArrayStack::&lt;T&gt; {\n            stack: Vec::&lt;T&gt;::new(),\n        }\n    }\n\n    /* Get the length of the stack */\n    fn size(&amp;self) -&gt; usize {\n        self.stack.len()\n    }\n\n    /* Check if the stack is empty */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.size() == 0\n    }\n\n    /* Push */\n    fn push(&amp;mut self, num: T) {\n        self.stack.push(num);\n    }\n\n    /* Pop */\n    fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack.pop()\n    }\n\n    /* Return list for printing */\n    fn peek(&amp;self) -&gt; Option&lt;&amp;T&gt; {\n        if self.is_empty() {\n            panic!(\"Stack is empty\")\n        };\n        self.stack.last()\n    }\n\n    /* Return &amp;Vec */\n    fn to_array(&amp;self) -&gt; &amp;Vec&lt;T&gt; {\n        &amp;self.stack\n    }\n}\n</code></pre> array_stack.c<pre><code>/* Stack based on array implementation */\ntypedef struct {\n    int *data;\n    int size;\n} ArrayStack;\n\n/* Constructor */\nArrayStack *newArrayStack() {\n    ArrayStack *stack = malloc(sizeof(ArrayStack));\n    // Initialize with large capacity to avoid expansion\n    stack-&gt;data = malloc(sizeof(int) * MAX_SIZE);\n    stack-&gt;size = 0;\n    return stack;\n}\n\n/* Destructor */\nvoid delArrayStack(ArrayStack *stack) {\n    free(stack-&gt;data);\n    free(stack);\n}\n\n/* Get the length of the stack */\nint size(ArrayStack *stack) {\n    return stack-&gt;size;\n}\n\n/* Check if the stack is empty */\nbool isEmpty(ArrayStack *stack) {\n    return stack-&gt;size == 0;\n}\n\n/* Push */\nvoid push(ArrayStack *stack, int num) {\n    if (stack-&gt;size == MAX_SIZE) {\n        printf(\"Stack is full\\n\");\n        return;\n    }\n    stack-&gt;data[stack-&gt;size] = num;\n    stack-&gt;size++;\n}\n\n/* Return list for printing */\nint peek(ArrayStack *stack) {\n    if (stack-&gt;size == 0) {\n        printf(\"Stack is empty\\n\");\n        return INT_MAX;\n    }\n    return stack-&gt;data[stack-&gt;size - 1];\n}\n\n/* Pop */\nint pop(ArrayStack *stack) {\n    int val = peek(stack);\n    stack-&gt;size--;\n    return val;\n}\n</code></pre> array_stack.kt<pre><code>/* Stack based on array implementation */\nclass ArrayStack {\n    // Initialize list (dynamic array)\n    private val stack = mutableListOf&lt;Int&gt;()\n\n    /* Get the length of the stack */\n    fun size(): Int {\n        return stack.size\n    }\n\n    /* Check if the stack is empty */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* Push */\n    fun push(num: Int) {\n        stack.add(num)\n    }\n\n    /* Pop */\n    fun pop(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack.removeAt(size() - 1)\n    }\n\n    /* Return list for printing */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack[size() - 1]\n    }\n\n    /* Convert List to Array and return */\n    fun toArray(): Array&lt;Any&gt; {\n        return stack.toTypedArray()\n    }\n}\n</code></pre> array_stack.rb<pre><code>### Stack based on array ###\nclass ArrayStack\n  ### Constructor ###\n  def initialize\n    @stack = []\n  end\n\n  ### Get stack length ###\n  def size\n    @stack.length\n  end\n\n  ### Check if stack is empty ###\n  def is_empty?\n    @stack.empty?\n  end\n\n  ### Push ###\n  def push(item)\n    @stack &lt;&lt; item\n  end\n\n  ### Pop ###\n  def pop\n    raise IndexError, 'Stack is empty' if is_empty?\n\n    @stack.pop\n  end\n\n  ### Access top element ###\n  def peek\n    raise IndexError, 'Stack is empty' if is_empty?\n\n    @stack.last\n  end\n\n  ### Return list for printing ###\n  def to_array\n    @stack\n  end\nend\n</code></pre>","path":["Chapter 5. Stacks and Queues","5.1   Stack"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#513-comparison-of-the-two-implementations","level":2,"title":"5.1.3   Comparison of the Two Implementations","text":"<p>Supported Operations</p> <p>Both implementations support all operations defined by the stack. The array implementation additionally supports random access, but this goes beyond the stack definition and is generally not used.</p> <p>Time Efficiency</p> <p>In the array-based implementation, both push and pop operations occur in pre-allocated contiguous memory, which has good cache locality and is therefore more efficient. However, if pushing exceeds the array capacity, it triggers an expansion mechanism, causing the time complexity of that particular push operation to become \\(O(n)\\).</p> <p>In the linked list-based implementation, list expansion is very flexible, and there is no issue of reduced efficiency due to array expansion. However, the push operation requires initializing a node object and modifying pointers, so it is relatively less efficient. Nevertheless, if the pushed elements are already node objects, the initialization step can be omitted, thereby improving efficiency.</p> <p>In summary, when the elements pushed and popped are basic data types such as <code>int</code> or <code>double</code>, we can draw the following conclusions:</p> <ul> <li>The array-based stack implementation has reduced efficiency when expansion is triggered, but since expansion is an infrequent operation, the average efficiency is higher.</li> <li>The linked list-based stack implementation can provide more stable efficiency performance.</li> </ul> <p>Space Efficiency</p> <p>When initializing a list, the system allocates an \"initial capacity\" that may exceed the actual need. Additionally, the expansion mechanism typically expands at a specific ratio (e.g., 2x), and the capacity after expansion may also exceed actual needs. Therefore, the array-based stack implementation may cause some space wastage.</p> <p>However, since linked list nodes need to store additional pointers, the space occupied by linked list nodes is relatively large.</p> <p>In summary, we cannot simply determine which implementation is more memory-efficient and need to analyze the specific situation.</p>","path":["Chapter 5. Stacks and Queues","5.1   Stack"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#514-typical-applications-of-stack","level":2,"title":"5.1.4   Typical Applications of Stack","text":"<ul> <li>Back and forward in browsers, undo and redo in software. Every time we open a new webpage, the browser pushes the previous page onto the stack, allowing us to return to the previous page via the back operation. The back operation is essentially performing a pop. To support both back and forward, two stacks are needed to work together.</li> <li>Program memory management. Each time a function is called, the system adds a stack frame to the top of the stack to record the function's context information. During recursion, the downward recursive phase continuously performs push operations, while the upward backtracking phase continuously performs pop operations.</li> </ul>","path":["Chapter 5. Stacks and Queues","5.1   Stack"],"tags":[]},{"location":"chapter_stack_and_queue/summary/","level":1,"title":"5.4   Summary","text":"","path":["Chapter 5. Stacks and Queues","5.4   Summary"],"tags":[]},{"location":"chapter_stack_and_queue/summary/#1-key-review","level":3,"title":"1.   Key Review","text":"<ul> <li>A stack is a data structure that follows the LIFO principle and can be implemented using arrays or linked lists.</li> <li>In terms of time efficiency, the array implementation of a stack has higher average efficiency, but during expansion, the time complexity of a single push operation degrades to \\(O(n)\\). In contrast, the linked-list implementation of a stack offers more stable performance.</li> <li>In terms of space efficiency, the array implementation of a stack may lead to some degree of space wastage. However, it should be noted that the memory space occupied by linked list nodes is larger than that of array elements.</li> <li>A queue is a data structure that follows the FIFO principle and can also be implemented using arrays or linked lists. The conclusions regarding time efficiency and space efficiency comparisons for queues are similar to those for stacks mentioned above.</li> <li>A deque is a queue with greater flexibility that allows adding and removing elements at both ends.</li> </ul>","path":["Chapter 5. Stacks and Queues","5.4   Summary"],"tags":[]},{"location":"chapter_stack_and_queue/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: Is the browser's forward and backward functionality implemented with a doubly linked list?</p> <p>The browser's forward and backward behavior is essentially an application of a \"stack.\" When a user visits a new page, that page is added to the top of the stack; when the user clicks the back button, that page is popped from the top of the stack. A deque can conveniently support some additional operations, as mentioned in the \"Deque\" section.</p> <p>Q: After popping from the stack, do we need to free the memory of the popped node?</p> <p>If the popped node will still be needed later, then memory does not need to be freed. If it won't be used afterward, languages like Java and Python have automatic garbage collection, so manual memory deallocation is not required; in C and C++, manual memory deallocation is necessary.</p> <p>Q: A deque seems like two stacks joined together. What is its purpose?</p> <p>A deque is like a combination of a stack and a queue, or two stacks joined together. It combines the logic of both, so it can support all applications of stacks and queues while offering greater flexibility.</p> <p>Q: How are undo and redo specifically implemented?</p> <p>Use two stacks: stack <code>A</code> for undo and stack <code>B</code> for redo.</p> <ol> <li>Whenever the user performs an operation, push this operation onto stack <code>A</code> and clear stack <code>B</code>.</li> <li>When the user performs \"undo,\" pop the most recent operation from stack <code>A</code> and push it onto stack <code>B</code>.</li> <li>When the user performs \"redo,\" pop the most recent operation from stack <code>B</code> and push it onto stack <code>A</code>.</li> </ol>","path":["Chapter 5. Stacks and Queues","5.4   Summary"],"tags":[]},{"location":"chapter_tree/","level":1,"title":"Chapter 7.   Tree","text":"<p>Abstract</p> <p>Towering trees are full of vitality, with deep roots, lush foliage, and sprawling branches.</p> <p>They offer a vivid illustration of divide-and-conquer in data structures.</p>","path":["Chapter 7. Tree","Chapter 7.   Tree"],"tags":[]},{"location":"chapter_tree/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>7.1   Binary Tree</li> <li>7.2   Binary Tree Traversal</li> <li>7.3   Array Representation of Binary Trees</li> <li>7.4   Binary Search Tree</li> <li>7.5   AVL Tree *</li> <li>7.6   Summary</li> <li>7.7   Exercises</li> </ul>","path":["Chapter 7. Tree","Chapter 7.   Tree"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/","level":1,"title":"7.3   Array Representation of Binary Trees","text":"<p>In the linked-list representation, the storage unit of a binary tree is a node <code>TreeNode</code>, and nodes are connected by pointers. The previous section introduced the basic operations of binary trees in this representation.</p> <p>So, can we use an array to represent a binary tree? The answer is yes.</p>","path":["Chapter 7. Tree","7.3   Array Representation of Binary Trees"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/#731-representing-perfect-binary-trees","level":2,"title":"7.3.1   Representing Perfect Binary Trees","text":"<p>Let's analyze a simple case first. Given a perfect binary tree, we store all nodes in an array according to the order of level-order traversal, where each node corresponds to a unique array index.</p> <p>Based on the characteristics of level-order traversal, we can derive a \"mapping formula\" between parent node index and child node indices: If a node's index is \\(i\\), then its left child index is \\(2i + 1\\) and its right child index is \\(2i + 2\\). Figure 7-12 shows the mapping relationships between various node indices.</p> <p></p> <p> Figure 7-12   Array representation of a perfect binary tree </p> <p>The mapping formula plays a role similar to the node references (pointers) in linked lists. Given any node in the array, we can access its left (right) child node using the mapping formula.</p>","path":["Chapter 7. Tree","7.3   Array Representation of Binary Trees"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/#732-representing-any-binary-tree","level":2,"title":"7.3.2   Representing Any Binary Tree","text":"<p>Perfect binary trees are a special case; in the middle levels of a binary tree, there are typically many <code>None</code> values. Since the level-order traversal sequence does not include these <code>None</code> values, we cannot infer the number and distribution of <code>None</code> values based on this sequence alone. This means multiple binary tree structures can correspond to the same level-order traversal sequence.</p> <p>As shown in Figure 7-13, given a non-perfect binary tree, the above method of array representation fails.</p> <p></p> <p> Figure 7-13   Level-order traversal sequence corresponds to multiple binary tree possibilities </p> <p>To solve this problem, we can explicitly write out all <code>None</code> values in the level-order traversal sequence. As shown in Figure 7-14, once we do this, the level-order traversal sequence can uniquely represent a binary tree. Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># Array representation of a binary tree\n# Using None to represent empty slots\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using the maximum integer value INT_MAX to mark empty slots\nvector&lt;int&gt; tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using the Integer wrapper class allows for using null to mark empty slots\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using nullable int (int?) allows for using null to mark empty slots\nint?[] tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using an any type slice, allowing for nil to mark empty slots\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using optional Int (Int?) allows for using nil to mark empty slots\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using null to represent empty slots\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using null to represent empty slots\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using nullable int (int?) allows for using null to mark empty slots\nList&lt;int?&gt; tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using None to mark empty slots\nlet tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using the maximum int value to mark empty slots, therefore, node values must not be INT_MAX\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Using null to represent empty slots\nval tree = arrayOf( 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 )\n</code></pre> <pre><code>### Array representation of a binary tree ###\n# Using nil to represent empty slots\ntree = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <p></p> <p> Figure 7-14   Array representation of an arbitrary binary tree </p> <p>It's worth noting that complete binary trees are very well-suited for array representation. Recalling the definition of a complete binary tree, <code>None</code> only appears at the bottom level and towards the right, meaning all <code>None</code> values must appear at the end of the level-order traversal sequence.</p> <p>This means that when using an array to represent a complete binary tree, it's possible to omit storing all <code>None</code> values, which is very convenient. Figure 7-15 gives an example.</p> <p></p> <p> Figure 7-15   Array representation of a complete binary tree </p> <p>The following code implements a binary tree using an array representation, including the following operations:</p> <ul> <li>Given a node, obtain its value, left (right) child node, and parent node.</li> <li>Obtain the preorder, inorder, postorder, and level-order traversal sequences.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_binary_tree.py<pre><code>class ArrayBinaryTree:\n    \"\"\"Binary tree class represented by array\"\"\"\n\n    def __init__(self, arr: list[int | None]):\n        \"\"\"Constructor\"\"\"\n        self._tree = list(arr)\n\n    def size(self):\n        \"\"\"List capacity\"\"\"\n        return len(self._tree)\n\n    def val(self, i: int) -&gt; int | None:\n        \"\"\"Get value of node at index i\"\"\"\n        # If index is out of bounds, return None, representing empty position\n        if i &lt; 0 or i &gt;= self.size():\n            return None\n        return self._tree[i]\n\n    def left(self, i: int) -&gt; int | None:\n        \"\"\"Get index of left child node of node at index i\"\"\"\n        return 2 * i + 1\n\n    def right(self, i: int) -&gt; int | None:\n        \"\"\"Get index of right child node of node at index i\"\"\"\n        return 2 * i + 2\n\n    def parent(self, i: int) -&gt; int | None:\n        \"\"\"Get index of parent node of node at index i\"\"\"\n        return (i - 1) // 2\n\n    def level_order(self) -&gt; list[int]:\n        \"\"\"Level-order traversal\"\"\"\n        self.res = []\n        # Traverse array directly\n        for i in range(self.size()):\n            if self.val(i) is not None:\n                self.res.append(self.val(i))\n        return self.res\n\n    def dfs(self, i: int, order: str):\n        \"\"\"Depth-first traversal\"\"\"\n        if self.val(i) is None:\n            return\n        # Preorder traversal\n        if order == \"pre\":\n            self.res.append(self.val(i))\n        self.dfs(self.left(i), order)\n        # Inorder traversal\n        if order == \"in\":\n            self.res.append(self.val(i))\n        self.dfs(self.right(i), order)\n        # Postorder traversal\n        if order == \"post\":\n            self.res.append(self.val(i))\n\n    def pre_order(self) -&gt; list[int]:\n        \"\"\"Preorder traversal\"\"\"\n        self.res = []\n        self.dfs(0, order=\"pre\")\n        return self.res\n\n    def in_order(self) -&gt; list[int]:\n        \"\"\"Inorder traversal\"\"\"\n        self.res = []\n        self.dfs(0, order=\"in\")\n        return self.res\n\n    def post_order(self) -&gt; list[int]:\n        \"\"\"Postorder traversal\"\"\"\n        self.res = []\n        self.dfs(0, order=\"post\")\n        return self.res\n</code></pre> array_binary_tree.cpp<pre><code>/* Binary tree class represented by array */\nclass ArrayBinaryTree {\n  public:\n    /* Constructor */\n    ArrayBinaryTree(vector&lt;int&gt; arr) {\n        tree = arr;\n    }\n\n    /* List capacity */\n    int size() {\n        return tree.size();\n    }\n\n    /* Get value of node at index i */\n    int val(int i) {\n        // Return INT_MAX if index out of bounds, representing empty position\n        if (i &lt; 0 || i &gt;= size())\n            return INT_MAX;\n        return tree[i];\n    }\n\n    /* Get index of left child node of node at index i */\n    int left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* Get index of right child node of node at index i */\n    int right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* Get index of parent node of node at index i */\n    int parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* Level-order traversal */\n    vector&lt;int&gt; levelOrder() {\n        vector&lt;int&gt; res;\n        // Traverse array directly\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != INT_MAX)\n                res.push_back(val(i));\n        }\n        return res;\n    }\n\n    /* Preorder traversal */\n    vector&lt;int&gt; preOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* Inorder traversal */\n    vector&lt;int&gt; inOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* Postorder traversal */\n    vector&lt;int&gt; postOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"post\", res);\n        return res;\n    }\n\n  private:\n    vector&lt;int&gt; tree;\n\n    /* Depth-first traversal */\n    void dfs(int i, string order, vector&lt;int&gt; &amp;res) {\n        // If empty position, return\n        if (val(i) == INT_MAX)\n            return;\n        // Preorder traversal\n        if (order == \"pre\")\n            res.push_back(val(i));\n        dfs(left(i), order, res);\n        // Inorder traversal\n        if (order == \"in\")\n            res.push_back(val(i));\n        dfs(right(i), order, res);\n        // Postorder traversal\n        if (order == \"post\")\n            res.push_back(val(i));\n    }\n};\n</code></pre> array_binary_tree.java<pre><code>/* Binary tree class represented by array */\nclass ArrayBinaryTree {\n    private List&lt;Integer&gt; tree;\n\n    /* Constructor */\n    public ArrayBinaryTree(List&lt;Integer&gt; arr) {\n        tree = new ArrayList&lt;&gt;(arr);\n    }\n\n    /* List capacity */\n    public int size() {\n        return tree.size();\n    }\n\n    /* Get value of node at index i */\n    public Integer val(int i) {\n        // If index out of bounds, return null to represent empty position\n        if (i &lt; 0 || i &gt;= size())\n            return null;\n        return tree.get(i);\n    }\n\n    /* Get index of left child node of node at index i */\n    public Integer left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* Get index of right child node of node at index i */\n    public Integer right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* Get index of parent node of node at index i */\n    public Integer parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* Level-order traversal */\n    public List&lt;Integer&gt; levelOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        // Traverse array directly\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != null)\n                res.add(val(i));\n        }\n        return res;\n    }\n\n    /* Depth-first traversal */\n    private void dfs(Integer i, String order, List&lt;Integer&gt; res) {\n        // If empty position, return\n        if (val(i) == null)\n            return;\n        // Preorder traversal\n        if (\"pre\".equals(order))\n            res.add(val(i));\n        dfs(left(i), order, res);\n        // Inorder traversal\n        if (\"in\".equals(order))\n            res.add(val(i));\n        dfs(right(i), order, res);\n        // Postorder traversal\n        if (\"post\".equals(order))\n            res.add(val(i));\n    }\n\n    /* Preorder traversal */\n    public List&lt;Integer&gt; preOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* Inorder traversal */\n    public List&lt;Integer&gt; inOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* Postorder traversal */\n    public List&lt;Integer&gt; postOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.cs<pre><code>/* Binary tree class represented by array */\nclass ArrayBinaryTree(List&lt;int?&gt; arr) {\n    List&lt;int?&gt; tree = new(arr);\n\n    /* List capacity */\n    public int Size() {\n        return tree.Count;\n    }\n\n    /* Get value of node at index i */\n    public int? Val(int i) {\n        // If index out of bounds, return null to represent empty position\n        if (i &lt; 0 || i &gt;= Size())\n            return null;\n        return tree[i];\n    }\n\n    /* Get index of left child node of node at index i */\n    public int Left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* Get index of right child node of node at index i */\n    public int Right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* Get index of parent node of node at index i */\n    public int Parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* Level-order traversal */\n    public List&lt;int&gt; LevelOrder() {\n        List&lt;int&gt; res = [];\n        // Traverse array directly\n        for (int i = 0; i &lt; Size(); i++) {\n            if (Val(i).HasValue)\n                res.Add(Val(i)!.Value);\n        }\n        return res;\n    }\n\n    /* Depth-first traversal */\n    void DFS(int i, string order, List&lt;int&gt; res) {\n        // If empty position, return\n        if (!Val(i).HasValue)\n            return;\n        // Preorder traversal\n        if (order == \"pre\")\n            res.Add(Val(i)!.Value);\n        DFS(Left(i), order, res);\n        // Inorder traversal\n        if (order == \"in\")\n            res.Add(Val(i)!.Value);\n        DFS(Right(i), order, res);\n        // Postorder traversal\n        if (order == \"post\")\n            res.Add(Val(i)!.Value);\n    }\n\n    /* Preorder traversal */\n    public List&lt;int&gt; PreOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"pre\", res);\n        return res;\n    }\n\n    /* Inorder traversal */\n    public List&lt;int&gt; InOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"in\", res);\n        return res;\n    }\n\n    /* Postorder traversal */\n    public List&lt;int&gt; PostOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.go<pre><code>/* Binary tree class represented by array */\ntype arrayBinaryTree struct {\n    tree []any\n}\n\n/* Constructor */\nfunc newArrayBinaryTree(arr []any) *arrayBinaryTree {\n    return &amp;arrayBinaryTree{\n        tree: arr,\n    }\n}\n\n/* List capacity */\nfunc (abt *arrayBinaryTree) size() int {\n    return len(abt.tree)\n}\n\n/* Get value of node at index i */\nfunc (abt *arrayBinaryTree) val(i int) any {\n    // If index out of bounds, return null to represent empty position\n    if i &lt; 0 || i &gt;= abt.size() {\n        return nil\n    }\n    return abt.tree[i]\n}\n\n/* Get index of left child node of node at index i */\nfunc (abt *arrayBinaryTree) left(i int) int {\n    return 2*i + 1\n}\n\n/* Get index of right child node of node at index i */\nfunc (abt *arrayBinaryTree) right(i int) int {\n    return 2*i + 2\n}\n\n/* Get index of parent node of node at index i */\nfunc (abt *arrayBinaryTree) parent(i int) int {\n    return (i - 1) / 2\n}\n\n/* Level-order traversal */\nfunc (abt *arrayBinaryTree) levelOrder() []any {\n    var res []any\n    // Traverse array directly\n    for i := 0; i &lt; abt.size(); i++ {\n        if abt.val(i) != nil {\n            res = append(res, abt.val(i))\n        }\n    }\n    return res\n}\n\n/* Depth-first traversal */\nfunc (abt *arrayBinaryTree) dfs(i int, order string, res *[]any) {\n    // If empty position, return\n    if abt.val(i) == nil {\n        return\n    }\n    // Preorder traversal\n    if order == \"pre\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.left(i), order, res)\n    // Inorder traversal\n    if order == \"in\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.right(i), order, res)\n    // Postorder traversal\n    if order == \"post\" {\n        *res = append(*res, abt.val(i))\n    }\n}\n\n/* Preorder traversal */\nfunc (abt *arrayBinaryTree) preOrder() []any {\n    var res []any\n    abt.dfs(0, \"pre\", &amp;res)\n    return res\n}\n\n/* Inorder traversal */\nfunc (abt *arrayBinaryTree) inOrder() []any {\n    var res []any\n    abt.dfs(0, \"in\", &amp;res)\n    return res\n}\n\n/* Postorder traversal */\nfunc (abt *arrayBinaryTree) postOrder() []any {\n    var res []any\n    abt.dfs(0, \"post\", &amp;res)\n    return res\n}\n</code></pre> array_binary_tree.swift<pre><code>/* Binary tree class represented by array */\nclass ArrayBinaryTree {\n    private var tree: [Int?]\n\n    /* Constructor */\n    init(arr: [Int?]) {\n        tree = arr\n    }\n\n    /* List capacity */\n    func size() -&gt; Int {\n        tree.count\n    }\n\n    /* Get value of node at index i */\n    func val(i: Int) -&gt; Int? {\n        // If index out of bounds, return null to represent empty position\n        if i &lt; 0 || i &gt;= size() {\n            return nil\n        }\n        return tree[i]\n    }\n\n    /* Get index of left child node of node at index i */\n    func left(i: Int) -&gt; Int {\n        2 * i + 1\n    }\n\n    /* Get index of right child node of node at index i */\n    func right(i: Int) -&gt; Int {\n        2 * i + 2\n    }\n\n    /* Get index of parent node of node at index i */\n    func parent(i: Int) -&gt; Int {\n        (i - 1) / 2\n    }\n\n    /* Level-order traversal */\n    func levelOrder() -&gt; [Int] {\n        var res: [Int] = []\n        // Traverse array directly\n        for i in 0 ..&lt; size() {\n            if let val = val(i: i) {\n                res.append(val)\n            }\n        }\n        return res\n    }\n\n    /* Depth-first traversal */\n    private func dfs(i: Int, order: String, res: inout [Int]) {\n        // If empty position, return\n        guard let val = val(i: i) else {\n            return\n        }\n        // Preorder traversal\n        if order == \"pre\" {\n            res.append(val)\n        }\n        dfs(i: left(i: i), order: order, res: &amp;res)\n        // Inorder traversal\n        if order == \"in\" {\n            res.append(val)\n        }\n        dfs(i: right(i: i), order: order, res: &amp;res)\n        // Postorder traversal\n        if order == \"post\" {\n            res.append(val)\n        }\n    }\n\n    /* Preorder traversal */\n    func preOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"pre\", res: &amp;res)\n        return res\n    }\n\n    /* Inorder traversal */\n    func inOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"in\", res: &amp;res)\n        return res\n    }\n\n    /* Postorder traversal */\n    func postOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"post\", res: &amp;res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.js<pre><code>/* Binary tree class represented by array */\nclass ArrayBinaryTree {\n    #tree;\n\n    /* Constructor */\n    constructor(arr) {\n        this.#tree = arr;\n    }\n\n    /* List capacity */\n    size() {\n        return this.#tree.length;\n    }\n\n    /* Get value of node at index i */\n    val(i) {\n        // If index out of bounds, return null to represent empty position\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* Get index of left child node of node at index i */\n    left(i) {\n        return 2 * i + 1;\n    }\n\n    /* Get index of right child node of node at index i */\n    right(i) {\n        return 2 * i + 2;\n    }\n\n    /* Get index of parent node of node at index i */\n    parent(i) {\n        return Math.floor((i - 1) / 2); // Floor division\n    }\n\n    /* Level-order traversal */\n    levelOrder() {\n        let res = [];\n        // Traverse array directly\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* Depth-first traversal */\n    #dfs(i, order, res) {\n        // If empty position, return\n        if (this.val(i) === null) return;\n        // Preorder traversal\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // Inorder traversal\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // Postorder traversal\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* Preorder traversal */\n    preOrder() {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* Inorder traversal */\n    inOrder() {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* Postorder traversal */\n    postOrder() {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.ts<pre><code>/* Binary tree class represented by array */\nclass ArrayBinaryTree {\n    #tree: (number | null)[];\n\n    /* Constructor */\n    constructor(arr: (number | null)[]) {\n        this.#tree = arr;\n    }\n\n    /* List capacity */\n    size(): number {\n        return this.#tree.length;\n    }\n\n    /* Get value of node at index i */\n    val(i: number): number | null {\n        // If index out of bounds, return null to represent empty position\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* Get index of left child node of node at index i */\n    left(i: number): number {\n        return 2 * i + 1;\n    }\n\n    /* Get index of right child node of node at index i */\n    right(i: number): number {\n        return 2 * i + 2;\n    }\n\n    /* Get index of parent node of node at index i */\n    parent(i: number): number {\n        return Math.floor((i - 1) / 2); // Floor division\n    }\n\n    /* Level-order traversal */\n    levelOrder(): number[] {\n        let res = [];\n        // Traverse array directly\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* Depth-first traversal */\n    #dfs(i: number, order: Order, res: (number | null)[]): void {\n        // If empty position, return\n        if (this.val(i) === null) return;\n        // Preorder traversal\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // Inorder traversal\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // Postorder traversal\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* Preorder traversal */\n    preOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* Inorder traversal */\n    inOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* Postorder traversal */\n    postOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.dart<pre><code>/* Binary tree class represented by array */\nclass ArrayBinaryTree {\n  late List&lt;int?&gt; _tree;\n\n  /* Constructor */\n  ArrayBinaryTree(this._tree);\n\n  /* List capacity */\n  int size() {\n    return _tree.length;\n  }\n\n  /* Get value of node at index i */\n  int? val(int i) {\n    // If index out of bounds, return null to represent empty position\n    if (i &lt; 0 || i &gt;= size()) {\n      return null;\n    }\n    return _tree[i];\n  }\n\n  /* Get index of left child node of node at index i */\n  int? left(int i) {\n    return 2 * i + 1;\n  }\n\n  /* Get index of right child node of node at index i */\n  int? right(int i) {\n    return 2 * i + 2;\n  }\n\n  /* Get index of parent node of node at index i */\n  int? parent(int i) {\n    return (i - 1) ~/ 2;\n  }\n\n  /* Level-order traversal */\n  List&lt;int&gt; levelOrder() {\n    List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; size(); i++) {\n      if (val(i) != null) {\n        res.add(val(i)!);\n      }\n    }\n    return res;\n  }\n\n  /* Depth-first traversal */\n  void dfs(int i, String order, List&lt;int?&gt; res) {\n    // If empty position, return\n    if (val(i) == null) {\n      return;\n    }\n    // Preorder traversal\n    if (order == 'pre') {\n      res.add(val(i));\n    }\n    dfs(left(i)!, order, res);\n    // Inorder traversal\n    if (order == 'in') {\n      res.add(val(i));\n    }\n    dfs(right(i)!, order, res);\n    // Postorder traversal\n    if (order == 'post') {\n      res.add(val(i));\n    }\n  }\n\n  /* Preorder traversal */\n  List&lt;int?&gt; preOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'pre', res);\n    return res;\n  }\n\n  /* Inorder traversal */\n  List&lt;int?&gt; inOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'in', res);\n    return res;\n  }\n\n  /* Postorder traversal */\n  List&lt;int?&gt; postOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'post', res);\n    return res;\n  }\n}\n</code></pre> array_binary_tree.rs<pre><code>/* Binary tree class represented by array */\nstruct ArrayBinaryTree {\n    tree: Vec&lt;Option&lt;i32&gt;&gt;,\n}\n\nimpl ArrayBinaryTree {\n    /* Constructor */\n    fn new(arr: Vec&lt;Option&lt;i32&gt;&gt;) -&gt; Self {\n        Self { tree: arr }\n    }\n\n    /* List capacity */\n    fn size(&amp;self) -&gt; i32 {\n        self.tree.len() as i32\n    }\n\n    /* Get value of node at index i */\n    fn val(&amp;self, i: i32) -&gt; Option&lt;i32&gt; {\n        // If index is out of bounds, return None, representing empty position\n        if i &lt; 0 || i &gt;= self.size() {\n            None\n        } else {\n            self.tree[i as usize]\n        }\n    }\n\n    /* Get index of left child node of node at index i */\n    fn left(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 1\n    }\n\n    /* Get index of right child node of node at index i */\n    fn right(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 2\n    }\n\n    /* Get index of parent node of node at index i */\n    fn parent(&amp;self, i: i32) -&gt; i32 {\n        (i - 1) / 2\n    }\n\n    /* Level-order traversal */\n    fn level_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        self.tree.iter().filter_map(|&amp;x| x).collect()\n    }\n\n    /* Depth-first traversal */\n    fn dfs(&amp;self, i: i32, order: &amp;'static str, res: &amp;mut Vec&lt;i32&gt;) {\n        if self.val(i).is_none() {\n            return;\n        }\n        let val = self.val(i).unwrap();\n        // Preorder traversal\n        if order == \"pre\" {\n            res.push(val);\n        }\n        self.dfs(self.left(i), order, res);\n        // Inorder traversal\n        if order == \"in\" {\n            res.push(val);\n        }\n        self.dfs(self.right(i), order, res);\n        // Postorder traversal\n        if order == \"post\" {\n            res.push(val);\n        }\n    }\n\n    /* Preorder traversal */\n    fn pre_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"pre\", &amp;mut res);\n        res\n    }\n\n    /* Inorder traversal */\n    fn in_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"in\", &amp;mut res);\n        res\n    }\n\n    /* Postorder traversal */\n    fn post_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"post\", &amp;mut res);\n        res\n    }\n}\n</code></pre> array_binary_tree.c<pre><code>/* Binary tree structure in array representation */\ntypedef struct {\n    int *tree;\n    int size;\n} ArrayBinaryTree;\n\n/* Constructor */\nArrayBinaryTree *newArrayBinaryTree(int *arr, int arrSize) {\n    ArrayBinaryTree *abt = (ArrayBinaryTree *)malloc(sizeof(ArrayBinaryTree));\n    abt-&gt;tree = malloc(sizeof(int) * arrSize);\n    memcpy(abt-&gt;tree, arr, sizeof(int) * arrSize);\n    abt-&gt;size = arrSize;\n    return abt;\n}\n\n/* Destructor */\nvoid delArrayBinaryTree(ArrayBinaryTree *abt) {\n    free(abt-&gt;tree);\n    free(abt);\n}\n\n/* List capacity */\nint size(ArrayBinaryTree *abt) {\n    return abt-&gt;size;\n}\n\n/* Get value of node at index i */\nint val(ArrayBinaryTree *abt, int i) {\n    // Return INT_MAX if index out of bounds, representing empty position\n    if (i &lt; 0 || i &gt;= size(abt))\n        return INT_MAX;\n    return abt-&gt;tree[i];\n}\n\n/* Level-order traversal */\nint *levelOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    // Traverse array directly\n    for (int i = 0; i &lt; size(abt); i++) {\n        if (val(abt, i) != INT_MAX)\n            res[index++] = val(abt, i);\n    }\n    *returnSize = index;\n    return res;\n}\n\n/* Depth-first traversal */\nvoid dfs(ArrayBinaryTree *abt, int i, char *order, int *res, int *index) {\n    // If empty position, return\n    if (val(abt, i) == INT_MAX)\n        return;\n    // Preorder traversal\n    if (strcmp(order, \"pre\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, left(i), order, res, index);\n    // Inorder traversal\n    if (strcmp(order, \"in\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, right(i), order, res, index);\n    // Postorder traversal\n    if (strcmp(order, \"post\") == 0)\n        res[(*index)++] = val(abt, i);\n}\n\n/* Preorder traversal */\nint *preOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"pre\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* Inorder traversal */\nint *inOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"in\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* Postorder traversal */\nint *postOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"post\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n</code></pre> array_binary_tree.kt<pre><code>/* Binary tree class represented by array */\nclass ArrayBinaryTree(val tree: MutableList&lt;Int?&gt;) {\n    /* List capacity */\n    fun size(): Int {\n        return tree.size\n    }\n\n    /* Get value of node at index i */\n    fun _val(i: Int): Int? {\n        // If index out of bounds, return null to represent empty position\n        if (i &lt; 0 || i &gt;= size()) return null\n        return tree[i]\n    }\n\n    /* Get index of left child node of node at index i */\n    fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* Get index of right child node of node at index i */\n    fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* Get index of parent node of node at index i */\n    fun parent(i: Int): Int {\n        return (i - 1) / 2\n    }\n\n    /* Level-order traversal */\n    fun levelOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        // Traverse array directly\n        for (i in 0..&lt;size()) {\n            if (_val(i) != null)\n                res.add(_val(i))\n        }\n        return res\n    }\n\n    /* Depth-first traversal */\n    fun dfs(i: Int, order: String, res: MutableList&lt;Int?&gt;) {\n        // If empty position, return\n        if (_val(i) == null)\n            return\n        // Preorder traversal\n        if (\"pre\" == order)\n            res.add(_val(i))\n        dfs(left(i), order, res)\n        // Inorder traversal\n        if (\"in\" == order)\n            res.add(_val(i))\n        dfs(right(i), order, res)\n        // Postorder traversal\n        if (\"post\" == order)\n            res.add(_val(i))\n    }\n\n    /* Preorder traversal */\n    fun preOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"pre\", res)\n        return res\n    }\n\n    /* Inorder traversal */\n    fun inOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"in\", res)\n        return res\n    }\n\n    /* Postorder traversal */\n    fun postOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"post\", res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.rb<pre><code>### Array representation of binary tree class ###\nclass ArrayBinaryTree\n  ### Constructor ###\n  def initialize(arr)\n    @tree = arr.to_a\n  end\n\n  ### List capacity ###\n  def size\n    @tree.length\n  end\n\n  ### Get value of node at index i ###\n  def val(i)\n    # Return nil if index out of bounds, representing empty position\n    return if i &lt; 0 || i &gt;= size\n\n    @tree[i]\n  end\n\n  ### Get left child index of node at index i ###\n  def left(i)\n    2 * i + 1\n  end\n\n  ### Get right child index of node at index i ###\n  def right(i)\n    2 * i + 2\n  end\n\n  ### Get parent node index of node at index i ###\n  def parent(i)\n    (i - 1) / 2\n  end\n\n  ### Level-order traversal ###\n  def level_order\n    @res = []\n\n    # Traverse array directly\n    for i in 0...size\n      @res &lt;&lt; val(i) unless val(i).nil?\n    end\n\n    @res\n  end\n\n  ### Depth-first traversal ###\n  def dfs(i, order)\n    return if val(i).nil?\n    # Preorder traversal\n    @res &lt;&lt; val(i) if order == :pre\n    dfs(left(i), order)\n    # Inorder traversal\n    @res &lt;&lt; val(i) if order == :in\n    dfs(right(i), order)\n    # Postorder traversal\n    @res &lt;&lt; val(i) if order == :post\n  end\n\n  ### Pre-order traversal ###\n  def pre_order\n    @res = []\n    dfs(0, :pre)\n    @res\n  end\n\n  ### In-order traversal ###\n  def in_order\n    @res = []\n    dfs(0, :in)\n    @res\n  end\n\n  ### Post-order traversal ###\n  def post_order\n    @res = []\n    dfs(0, :post)\n    @res\n  end\nend\n</code></pre>","path":["Chapter 7. Tree","7.3   Array Representation of Binary Trees"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/#733-advantages-and-limitations","level":2,"title":"7.3.3   Advantages and Limitations","text":"<p>The array representation of binary trees has the following advantages:</p> <ul> <li>Arrays are stored in contiguous memory space, which is cache-friendly, allowing faster access and traversal.</li> <li>It does not require storing pointers, which saves space.</li> <li>It allows random access to nodes.</li> </ul> <p>However, the array representation also has some limitations:</p> <ul> <li>Array storage requires contiguous memory space, so it is not suitable for storing trees with a large amount of data.</li> <li>Adding or removing nodes requires array insertion and deletion operations, which have lower efficiency.</li> <li>When there are many <code>None</code> values in the binary tree, the proportion of node data contained in the array is low, leading to lower space utilization.</li> </ul>","path":["Chapter 7. Tree","7.3   Array Representation of Binary Trees"],"tags":[]},{"location":"chapter_tree/avl_tree/","level":1,"title":"7.5   AVL Tree *","text":"<p>In the \"Binary Search Tree\" section, we mentioned that after multiple insertion and removal operations, a binary search tree may degenerate into a linked list. In this case, the time complexity of all operations degrades from \\(O(\\log n)\\) to \\(O(n)\\).</p> <p>As shown in Figure 7-24, after two node removal operations, this binary search tree will degrade into a linked list.</p> <p></p> <p> Figure 7-24   Degradation of an AVL tree after removing nodes </p> <p>For example, in the perfect binary tree shown in Figure 7-25, after inserting two nodes, the tree will lean heavily to the left, and the time complexity of search operations will also degrade.</p> <p></p> <p> Figure 7-25   Degradation of an AVL tree after inserting nodes </p> <p>In 1962, G. M. Adelson-Velsky and E. M. Landis proposed the AVL tree in their paper \"An algorithm for the organization of information\". The paper describes a series of operations that prevent an AVL tree from degenerating as nodes are inserted and removed, thereby keeping the time complexity of various operations at \\(O(\\log n)\\). In other words, in scenarios that require frequent insertion, deletion, lookup, and update operations, AVL trees can maintain consistently efficient performance and therefore have strong practical value.</p>","path":["Chapter 7. Tree","7.5   AVL Tree *"],"tags":[]},{"location":"chapter_tree/avl_tree/#751-common-terminology-in-avl-trees","level":2,"title":"7.5.1   Common Terminology in AVL Trees","text":"<p>An AVL tree is both a binary search tree and a balanced binary tree, simultaneously satisfying all the properties of these two types of binary trees, hence it is a balanced binary search tree.</p>","path":["Chapter 7. Tree","7.5   AVL Tree *"],"tags":[]},{"location":"chapter_tree/avl_tree/#1-node-height","level":3,"title":"1.   Node Height","text":"<p>Since the operations related to AVL trees require obtaining node heights, we need to add a <code>height</code> variable to the node class:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class TreeNode:\n    \"\"\"AVL tree node\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                 # Node value\n        self.height: int = 0                # Node height\n        self.left: TreeNode | None = None   # Left child reference\n        self.right: TreeNode | None = None  # Right child reference\n</code></pre> <pre><code>/* AVL tree node */\nstruct TreeNode {\n    int val{};          // Node value\n    int height = 0;     // Node height\n    TreeNode *left{};   // Left child\n    TreeNode *right{};  // Right child\n    TreeNode() = default;\n    explicit TreeNode(int x) : val(x){}\n};\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode {\n    public int val;        // Node value\n    public int height;     // Node height\n    public TreeNode left;  // Left child\n    public TreeNode right; // Right child\n    public TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode(int? x) {\n    public int? val = x;    // Node value\n    public int height;      // Node height\n    public TreeNode? left;  // Left child reference\n    public TreeNode? right; // Right child reference\n}\n</code></pre> <pre><code>/* AVL tree node */\ntype TreeNode struct {\n    Val    int       // Node value\n    Height int       // Node height\n    Left   *TreeNode // Left child reference\n    Right  *TreeNode // Right child reference\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode {\n    var val: Int // Node value\n    var height: Int // Node height\n    var left: TreeNode? // Left child\n    var right: TreeNode? // Right child\n\n    init(x: Int) {\n        val = x\n        height = 0\n    }\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode {\n    val; // Node value\n    height; // Node height\n    left; // Left child pointer\n    right; // Right child pointer\n    constructor(val, left, right, height) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode {\n    val: number;            // Node value\n    height: number;         // Node height\n    left: TreeNode | null;  // Left child pointer\n    right: TreeNode | null; // Right child pointer\n    constructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height; \n        this.left = left === undefined ? null : left; \n        this.right = right === undefined ? null : right; \n    }\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode {\n  int val;         // Node value\n  int height;      // Node height\n  TreeNode? left;  // Left child\n  TreeNode? right; // Right child\n  TreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* AVL tree node */\nstruct TreeNode {\n    val: i32,                               // Node value\n    height: i32,                            // Node height\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // Left child\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // Right child\n}\n\nimpl TreeNode {\n    /* Constructor */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            height: 0,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* AVL tree node */\ntypedef struct TreeNode {\n    int val;\n    int height;\n    struct TreeNode *left;\n    struct TreeNode *right;\n} TreeNode;\n\n/* Constructor */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* AVL tree node */\nclass TreeNode(val _val: Int) {  // Node value\n    var height: Int = 0          // Node height\n    var left: TreeNode? = null   // Left child\n    var right: TreeNode? = null  // Right child\n}\n</code></pre> <pre><code>### AVL tree node class ###\nclass TreeNode\n  attr_accessor :val    # Node value\n  attr_accessor :height # Node height\n  attr_accessor :left   # Left child reference\n  attr_accessor :right  # Right child reference\n\n  def initialize(val)\n    @val = val\n    @height = 0\n  end\nend\n</code></pre> <p>The \"node height\" refers to the distance from that node to its farthest leaf node, i.e., the number of edges on the path. It is important to note that the height of a leaf node is \\(0\\), and the height of a null node is \\(-1\\). We will create two utility functions for getting and updating the height of a node:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def height(self, node: TreeNode | None) -&gt; int:\n    \"\"\"Get node height\"\"\"\n    # Empty node height is -1, leaf node height is 0\n    if node is not None:\n        return node.height\n    return -1\n\ndef update_height(self, node: TreeNode | None):\n    \"\"\"Update node height\"\"\"\n    # Node height equals the height of the tallest subtree + 1\n    node.height = max([self.height(node.left), self.height(node.right)]) + 1\n</code></pre> avl_tree.cpp<pre><code>/* Get node height */\nint height(TreeNode *node) {\n    // Empty node height is -1, leaf node height is 0\n    return node == nullptr ? -1 : node-&gt;height;\n}\n\n/* Update node height */\nvoid updateHeight(TreeNode *node) {\n    // Node height equals the height of the tallest subtree + 1\n    node-&gt;height = max(height(node-&gt;left), height(node-&gt;right)) + 1;\n}\n</code></pre> avl_tree.java<pre><code>/* Get node height */\nint height(TreeNode node) {\n    // Empty node height is -1, leaf node height is 0\n    return node == null ? -1 : node.height;\n}\n\n/* Update node height */\nvoid updateHeight(TreeNode node) {\n    // Node height equals the height of the tallest subtree + 1\n    node.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.cs<pre><code>/* Get node height */\nint Height(TreeNode? node) {\n    // Empty node height is -1, leaf node height is 0\n    return node == null ? -1 : node.height;\n}\n\n/* Update node height */\nvoid UpdateHeight(TreeNode node) {\n    // Node height equals the height of the tallest subtree + 1\n    node.height = Math.Max(Height(node.left), Height(node.right)) + 1;\n}\n</code></pre> avl_tree.go<pre><code>/* Get node height */\nfunc (t *aVLTree) height(node *TreeNode) int {\n    // Empty node height is -1, leaf node height is 0\n    if node != nil {\n        return node.Height\n    }\n    return -1\n}\n\n/* Update node height */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\n    lh := t.height(node.Left)\n    rh := t.height(node.Right)\n    // Node height equals the height of the tallest subtree + 1\n    if lh &gt; rh {\n        node.Height = lh + 1\n    } else {\n        node.Height = rh + 1\n    }\n}\n</code></pre> avl_tree.swift<pre><code>/* Get node height */\nfunc height(node: TreeNode?) -&gt; Int {\n    // Empty node height is -1, leaf node height is 0\n    node?.height ?? -1\n}\n\n/* Update node height */\nfunc updateHeight(node: TreeNode?) {\n    // Node height equals the height of the tallest subtree + 1\n    node?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n</code></pre> avl_tree.js<pre><code>/* Get node height */\nheight(node) {\n    // Empty node height is -1, leaf node height is 0\n    return node === null ? -1 : node.height;\n}\n\n/* Update node height */\n#updateHeight(node) {\n    // Node height equals the height of the tallest subtree + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.ts<pre><code>/* Get node height */\nheight(node: TreeNode): number {\n    // Empty node height is -1, leaf node height is 0\n    return node === null ? -1 : node.height;\n}\n\n/* Update node height */\nupdateHeight(node: TreeNode): void {\n    // Node height equals the height of the tallest subtree + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.dart<pre><code>/* Get node height */\nint height(TreeNode? node) {\n  // Empty node height is -1, leaf node height is 0\n  return node == null ? -1 : node.height;\n}\n\n/* Update node height */\nvoid updateHeight(TreeNode? node) {\n  // Node height equals the height of the tallest subtree + 1\n  node!.height = max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.rs<pre><code>/* Get node height */\nfn height(node: OptionTreeNodeRc) -&gt; i32 {\n    // Empty node height is -1, leaf node height is 0\n    match node {\n        Some(node) =&gt; node.borrow().height,\n        None =&gt; -1,\n    }\n}\n\n/* Update node height */\nfn update_height(node: OptionTreeNodeRc) {\n    if let Some(node) = node {\n        let left = node.borrow().left.clone();\n        let right = node.borrow().right.clone();\n        // Node height equals the height of the tallest subtree + 1\n        node.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* Get node height */\nint height(TreeNode *node) {\n    // Empty node height is -1, leaf node height is 0\n    if (node != NULL) {\n        return node-&gt;height;\n    }\n    return -1;\n}\n\n/* Update node height */\nvoid updateHeight(TreeNode *node) {\n    int lh = height(node-&gt;left);\n    int rh = height(node-&gt;right);\n    // Node height equals the height of the tallest subtree + 1\n    if (lh &gt; rh) {\n        node-&gt;height = lh + 1;\n    } else {\n        node-&gt;height = rh + 1;\n    }\n}\n</code></pre> avl_tree.kt<pre><code>/* Get node height */\nfun height(node: TreeNode?): Int {\n    // Empty node height is -1, leaf node height is 0\n    return node?.height ?: -1\n}\n\n/* Update node height */\nfun updateHeight(node: TreeNode?) {\n    // Node height equals the height of the tallest subtree + 1\n    node?.height = max(height(node?.left), height(node?.right)) + 1\n}\n</code></pre> avl_tree.rb<pre><code>### Get node height ###\ndef height(node)\n  # Empty node height is -1, leaf node height is 0\n  return node.height unless node.nil?\n\n  -1\nend\n\n### Update node height ###\ndef update_height(node)\n  # Node height equals the height of the tallest subtree + 1\n  node.height = [height(node.left), height(node.right)].max + 1\nend\n</code></pre>","path":["Chapter 7. Tree","7.5   AVL Tree *"],"tags":[]},{"location":"chapter_tree/avl_tree/#2-node-balance-factor","level":3,"title":"2.   Node Balance Factor","text":"<p>The balance factor of a node is defined as the height of the node's left subtree minus the height of its right subtree, and the balance factor of a null node is defined as \\(0\\). We also encapsulate the function to obtain the node's balance factor for convenient subsequent use:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def balance_factor(self, node: TreeNode | None) -&gt; int:\n    \"\"\"Get balance factor\"\"\"\n    # Empty node balance factor is 0\n    if node is None:\n        return 0\n    # Node balance factor = left subtree height - right subtree height\n    return self.height(node.left) - self.height(node.right)\n</code></pre> avl_tree.cpp<pre><code>/* Get balance factor */\nint balanceFactor(TreeNode *node) {\n    // Empty node balance factor is 0\n    if (node == nullptr)\n        return 0;\n    // Node balance factor = left subtree height - right subtree height\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.java<pre><code>/* Get balance factor */\nint balanceFactor(TreeNode node) {\n    // Empty node balance factor is 0\n    if (node == null)\n        return 0;\n    // Node balance factor = left subtree height - right subtree height\n    return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.cs<pre><code>/* Get balance factor */\nint BalanceFactor(TreeNode? node) {\n    // Empty node balance factor is 0\n    if (node == null) return 0;\n    // Node balance factor = left subtree height - right subtree height\n    return Height(node.left) - Height(node.right);\n}\n</code></pre> avl_tree.go<pre><code>/* Get balance factor */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n    // Empty node balance factor is 0\n    if node == nil {\n        return 0\n    }\n    // Node balance factor = left subtree height - right subtree height\n    return t.height(node.Left) - t.height(node.Right)\n}\n</code></pre> avl_tree.swift<pre><code>/* Get balance factor */\nfunc balanceFactor(node: TreeNode?) -&gt; Int {\n    // Empty node balance factor is 0\n    guard let node = node else { return 0 }\n    // Node balance factor = left subtree height - right subtree height\n    return height(node: node.left) - height(node: node.right)\n}\n</code></pre> avl_tree.js<pre><code>/* Get balance factor */\nbalanceFactor(node) {\n    // Empty node balance factor is 0\n    if (node === null) return 0;\n    // Node balance factor = left subtree height - right subtree height\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.ts<pre><code>/* Get balance factor */\nbalanceFactor(node: TreeNode): number {\n    // Empty node balance factor is 0\n    if (node === null) return 0;\n    // Node balance factor = left subtree height - right subtree height\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.dart<pre><code>/* Get balance factor */\nint balanceFactor(TreeNode? node) {\n  // Empty node balance factor is 0\n  if (node == null) return 0;\n  // Node balance factor = left subtree height - right subtree height\n  return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.rs<pre><code>/* Get balance factor */\nfn balance_factor(node: OptionTreeNodeRc) -&gt; i32 {\n    match node {\n        // Empty node balance factor is 0\n        None =&gt; 0,\n        // Node balance factor = left subtree height - right subtree height\n        Some(node) =&gt; {\n            Self::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())\n        }\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* Get balance factor */\nint balanceFactor(TreeNode *node) {\n    // Empty node balance factor is 0\n    if (node == NULL) {\n        return 0;\n    }\n    // Node balance factor = left subtree height - right subtree height\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.kt<pre><code>/* Get balance factor */\nfun balanceFactor(node: TreeNode?): Int {\n    // Empty node balance factor is 0\n    if (node == null) return 0\n    // Node balance factor = left subtree height - right subtree height\n    return height(node.left) - height(node.right)\n}\n</code></pre> avl_tree.rb<pre><code>### Get balance factor ###\ndef balance_factor(node)\n  # Empty node balance factor is 0\n  return 0 if node.nil?\n\n  # Node balance factor = left subtree height - right subtree height\n  height(node.left) - height(node.right)\nend\n</code></pre> <p>Tip</p> <p>Let the balance factor be \\(f\\), then the balance factor of any node in an AVL tree satisfies \\(-1 \\le f \\le 1\\).</p>","path":["Chapter 7. Tree","7.5   AVL Tree *"],"tags":[]},{"location":"chapter_tree/avl_tree/#752-rotations-in-avl-trees","level":2,"title":"7.5.2   Rotations in AVL Trees","text":"<p>The characteristic of AVL trees lies in the \"rotation\" operation, which can restore balance to unbalanced nodes without affecting the inorder traversal sequence of the binary tree. In other words, rotation operations can both maintain the property of a \"binary search tree\" and make the tree return to a \"balanced binary tree\".</p> <p>We call nodes with a balance factor absolute value \\(&gt; 1\\) \"unbalanced nodes\". Depending on the imbalance situation, rotation operations are divided into four types: right rotation, left rotation, right rotation then left rotation, and left rotation then right rotation. Below we describe these rotation operations in detail.</p>","path":["Chapter 7. Tree","7.5   AVL Tree *"],"tags":[]},{"location":"chapter_tree/avl_tree/#1-right-rotation","level":3,"title":"1.   Right Rotation","text":"<p>As shown in Figure 7-26, the value below the node is the balance factor. From bottom to top, the first unbalanced node in the binary tree is \"node 3\". We focus on the subtree with this unbalanced node as the root, denoting the node as <code>node</code> and its left child as <code>child</code>, and perform a \"right rotation\" operation. After the right rotation is completed, the subtree regains balance and still maintains the properties of a binary search tree.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 7-26   Steps of right rotation </p> <p>As shown in Figure 7-27, when the <code>child</code> node has a right child (denoted as <code>grand_child</code>), a step needs to be added in the right rotation: set <code>grand_child</code> as the left child of <code>node</code>.</p> <p></p> <p> Figure 7-27   Right rotation with grand_child </p> <p>\"Right rotation\" is a figurative term; in practice, it is achieved by modifying node pointers, as shown in the following code:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def right_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"Right rotation operation\"\"\"\n    child = node.left\n    grand_child = child.right\n    # Using child as pivot, rotate node to the right\n    child.right = node\n    node.left = grand_child\n    # Update node height\n    self.update_height(node)\n    self.update_height(child)\n    # Return root node of subtree after rotation\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* Right rotation operation */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;left;\n    TreeNode *grandChild = child-&gt;right;\n    // Using child as pivot, rotate node to the right\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // Update node height\n    updateHeight(node);\n    updateHeight(child);\n    // Return root node of subtree after rotation\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* Right rotation operation */\nTreeNode rightRotate(TreeNode node) {\n    TreeNode child = node.left;\n    TreeNode grandChild = child.right;\n    // Using child as pivot, rotate node to the right\n    child.right = node;\n    node.left = grandChild;\n    // Update node height\n    updateHeight(node);\n    updateHeight(child);\n    // Return root node of subtree after rotation\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* Right rotation operation */\nTreeNode? RightRotate(TreeNode? node) {\n    TreeNode? child = node?.left;\n    TreeNode? grandChild = child?.right;\n    // Using child as pivot, rotate node to the right\n    child.right = node;\n    node.left = grandChild;\n    // Update node height\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // Return root node of subtree after rotation\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* Right rotation operation */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\n    child := node.Left\n    grandChild := child.Right\n    // Using child as pivot, rotate node to the right\n    child.Right = node\n    node.Left = grandChild\n    // Update node height\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // Return root node of subtree after rotation\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* Right rotation operation */\nfunc rightRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.left\n    let grandChild = child?.right\n    // Using child as pivot, rotate node to the right\n    child?.right = node\n    node?.left = grandChild\n    // Update node height\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // Return root node of subtree after rotation\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* Right rotation operation */\n#rightRotate(node) {\n    const child = node.left;\n    const grandChild = child.right;\n    // Using child as pivot, rotate node to the right\n    child.right = node;\n    node.left = grandChild;\n    // Update node height\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // Return root node of subtree after rotation\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* Right rotation operation */\nrightRotate(node: TreeNode): TreeNode {\n    const child = node.left;\n    const grandChild = child.right;\n    // Using child as pivot, rotate node to the right\n    child.right = node;\n    node.left = grandChild;\n    // Update node height\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // Return root node of subtree after rotation\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* Right rotation operation */\nTreeNode? rightRotate(TreeNode? node) {\n  TreeNode? child = node!.left;\n  TreeNode? grandChild = child!.right;\n  // Using child as pivot, rotate node to the right\n  child.right = node;\n  node.left = grandChild;\n  // Update node height\n  updateHeight(node);\n  updateHeight(child);\n  // Return root node of subtree after rotation\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* Right rotation operation */\nfn right_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().left.clone().unwrap();\n            let grand_child = child.borrow().right.clone();\n            // Using child as pivot, rotate node to the right\n            child.borrow_mut().right = Some(node.clone());\n            node.borrow_mut().left = grand_child;\n            // Update node height\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // Return root node of subtree after rotation\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* Right rotation operation */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;left;\n    grandChild = child-&gt;right;\n    // Using child as pivot, rotate node to the right\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // Update node height\n    updateHeight(node);\n    updateHeight(child);\n    // Return root node of subtree after rotation\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* Right rotation operation */\nfun rightRotate(node: TreeNode?): TreeNode {\n    val child = node!!.left\n    val grandChild = child!!.right\n    // Using child as pivot, rotate node to the right\n    child.right = node\n    node.left = grandChild\n    // Update node height\n    updateHeight(node)\n    updateHeight(child)\n    // Return root node of subtree after rotation\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>### Right rotation ###\ndef right_rotate(node)\n  child = node.left\n  grand_child = child.right\n  # Using child as pivot, rotate node to the right\n  child.right = node\n  node.left = grand_child\n  # Update node height\n  update_height(node)\n  update_height(child)\n  # Return root node of subtree after rotation\n  child\nend\n</code></pre>","path":["Chapter 7. Tree","7.5   AVL Tree *"],"tags":[]},{"location":"chapter_tree/avl_tree/#2-left-rotation","level":3,"title":"2.   Left Rotation","text":"<p>Correspondingly, if considering the \"mirror\" of the above unbalanced binary tree, the \"left rotation\" operation shown in Figure 7-28 needs to be performed.</p> <p></p> <p> Figure 7-28   Left rotation operation </p> <p>Similarly, as shown in Figure 7-29, when the <code>child</code> node has a left child (denoted as <code>grand_child</code>), a step needs to be added in the left rotation: set <code>grand_child</code> as the right child of <code>node</code>.</p> <p></p> <p> Figure 7-29   Left rotation with grand_child </p> <p>It can be observed that right rotation and left rotation operations are mirror symmetric in logic, and the two imbalance cases they solve are also symmetric. Based on symmetry, we only need to replace all <code>left</code> in the right rotation implementation code with <code>right</code>, and all <code>right</code> with <code>left</code>, to obtain the left rotation implementation code:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def left_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"Left rotation operation\"\"\"\n    child = node.right\n    grand_child = child.left\n    # Using child as pivot, rotate node to the left\n    child.left = node\n    node.right = grand_child\n    # Update node height\n    self.update_height(node)\n    self.update_height(child)\n    # Return root node of subtree after rotation\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* Left rotation operation */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;right;\n    TreeNode *grandChild = child-&gt;left;\n    // Using child as pivot, rotate node to the left\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // Update node height\n    updateHeight(node);\n    updateHeight(child);\n    // Return root node of subtree after rotation\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* Left rotation operation */\nTreeNode leftRotate(TreeNode node) {\n    TreeNode child = node.right;\n    TreeNode grandChild = child.left;\n    // Using child as pivot, rotate node to the left\n    child.left = node;\n    node.right = grandChild;\n    // Update node height\n    updateHeight(node);\n    updateHeight(child);\n    // Return root node of subtree after rotation\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* Left rotation operation */\nTreeNode? LeftRotate(TreeNode? node) {\n    TreeNode? child = node?.right;\n    TreeNode? grandChild = child?.left;\n    // Using child as pivot, rotate node to the left\n    child.left = node;\n    node.right = grandChild;\n    // Update node height\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // Return root node of subtree after rotation\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* Left rotation operation */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\n    child := node.Right\n    grandChild := child.Left\n    // Using child as pivot, rotate node to the left\n    child.Left = node\n    node.Right = grandChild\n    // Update node height\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // Return root node of subtree after rotation\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* Left rotation operation */\nfunc leftRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.right\n    let grandChild = child?.left\n    // Using child as pivot, rotate node to the left\n    child?.left = node\n    node?.right = grandChild\n    // Update node height\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // Return root node of subtree after rotation\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* Left rotation operation */\n#leftRotate(node) {\n    const child = node.right;\n    const grandChild = child.left;\n    // Using child as pivot, rotate node to the left\n    child.left = node;\n    node.right = grandChild;\n    // Update node height\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // Return root node of subtree after rotation\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* Left rotation operation */\nleftRotate(node: TreeNode): TreeNode {\n    const child = node.right;\n    const grandChild = child.left;\n    // Using child as pivot, rotate node to the left\n    child.left = node;\n    node.right = grandChild;\n    // Update node height\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // Return root node of subtree after rotation\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* Left rotation operation */\nTreeNode? leftRotate(TreeNode? node) {\n  TreeNode? child = node!.right;\n  TreeNode? grandChild = child!.left;\n  // Using child as pivot, rotate node to the left\n  child.left = node;\n  node.right = grandChild;\n  // Update node height\n  updateHeight(node);\n  updateHeight(child);\n  // Return root node of subtree after rotation\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* Left rotation operation */\nfn left_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().right.clone().unwrap();\n            let grand_child = child.borrow().left.clone();\n            // Using child as pivot, rotate node to the left\n            child.borrow_mut().left = Some(node.clone());\n            node.borrow_mut().right = grand_child;\n            // Update node height\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // Return root node of subtree after rotation\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* Left rotation operation */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;right;\n    grandChild = child-&gt;left;\n    // Using child as pivot, rotate node to the left\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // Update node height\n    updateHeight(node);\n    updateHeight(child);\n    // Return root node of subtree after rotation\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* Left rotation operation */\nfun leftRotate(node: TreeNode?): TreeNode {\n    val child = node!!.right\n    val grandChild = child!!.left\n    // Using child as pivot, rotate node to the left\n    child.left = node\n    node.right = grandChild\n    // Update node height\n    updateHeight(node)\n    updateHeight(child)\n    // Return root node of subtree after rotation\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>### Left rotation ###\ndef left_rotate(node)\n  child = node.right\n  grand_child = child.left\n  # Using child as pivot, rotate node to the left\n  child.left = node\n  node.right = grand_child\n  # Update node height\n  update_height(node)\n  update_height(child)\n  # Return root node of subtree after rotation\n  child\nend\n</code></pre>","path":["Chapter 7. Tree","7.5   AVL Tree *"],"tags":[]},{"location":"chapter_tree/avl_tree/#3-left-rotation-then-right-rotation","level":3,"title":"3.   Left Rotation Then Right Rotation","text":"<p>For the unbalanced node 3 in Figure 7-30, using either left rotation or right rotation alone cannot restore the subtree to balance. In this case, a \"left rotation\" needs to be performed on <code>child</code> first, followed by a \"right rotation\" on <code>node</code>.</p> <p></p> <p> Figure 7-30   Left-right rotation </p>","path":["Chapter 7. Tree","7.5   AVL Tree *"],"tags":[]},{"location":"chapter_tree/avl_tree/#4-right-rotation-then-left-rotation","level":3,"title":"4.   Right Rotation Then Left Rotation","text":"<p>As shown in Figure 7-31, for the mirror case of the above unbalanced binary tree, a \"right rotation\" needs to be performed on <code>child</code> first, then a \"left rotation\" on <code>node</code>.</p> <p></p> <p> Figure 7-31   Right-left rotation </p>","path":["Chapter 7. Tree","7.5   AVL Tree *"],"tags":[]},{"location":"chapter_tree/avl_tree/#5-choice-of-rotation","level":3,"title":"5.   Choice of Rotation","text":"<p>The four imbalances shown in Figure 7-32 correspond one-to-one with the above cases, requiring right rotation, left rotation then right rotation, right rotation then left rotation, and left rotation operations respectively.</p> <p></p> <p> Figure 7-32   The four rotation cases of AVL tree </p> <p>As shown in Table 7-3, we determine which case the unbalanced node belongs to by judging the signs of the balance factor of the unbalanced node and the balance factor of its taller-side child node.</p> <p> Table 7-3   Conditions for Choosing Among the Four Rotation Cases </p> Balance factor of the unbalanced node Balance factor of the child node Rotation method to apply \\(&gt; 1\\) (left-leaning tree) \\(\\geq 0\\) Right rotation \\(&gt; 1\\) (left-leaning tree) \\(&lt;0\\) Left rotation then right rotation \\(&lt; -1\\) (right-leaning tree) \\(\\leq 0\\) Left rotation \\(&lt; -1\\) (right-leaning tree) \\(&gt;0\\) Right rotation then left rotation <p>For ease of use, we encapsulate the rotation operations into a function. With this function, we can perform rotations for various imbalance situations, restoring balance to unbalanced nodes. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"Perform rotation operation to restore balance to this subtree\"\"\"\n    # Get balance factor of node\n    balance_factor = self.balance_factor(node)\n    # Left-leaning tree\n    if balance_factor &gt; 1:\n        if self.balance_factor(node.left) &gt;= 0:\n            # Right rotation\n            return self.right_rotate(node)\n        else:\n            # First left rotation then right rotation\n            node.left = self.left_rotate(node.left)\n            return self.right_rotate(node)\n    # Right-leaning tree\n    elif balance_factor &lt; -1:\n        if self.balance_factor(node.right) &lt;= 0:\n            # Left rotation\n            return self.left_rotate(node)\n        else:\n            # First right rotation then left rotation\n            node.right = self.right_rotate(node.right)\n            return self.left_rotate(node)\n    # Balanced tree, no rotation needed, return directly\n    return node\n</code></pre> avl_tree.cpp<pre><code>/* Perform rotation operation to restore balance to this subtree */\nTreeNode *rotate(TreeNode *node) {\n    // Get balance factor of node\n    int _balanceFactor = balanceFactor(node);\n    // Left-leaning tree\n    if (_balanceFactor &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // Right rotation\n            return rightRotate(node);\n        } else {\n            // First left rotation then right rotation\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // Right-leaning tree\n    if (_balanceFactor &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // Left rotation\n            return leftRotate(node);\n        } else {\n            // First right rotation then left rotation\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // Balanced tree, no rotation needed, return directly\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* Perform rotation operation to restore balance to this subtree */\nTreeNode rotate(TreeNode node) {\n    // Get balance factor of node\n    int balanceFactor = balanceFactor(node);\n    // Left-leaning tree\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // Right rotation\n            return rightRotate(node);\n        } else {\n            // First left rotation then right rotation\n            node.left = leftRotate(node.left);\n            return rightRotate(node);\n        }\n    }\n    // Right-leaning tree\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // Left rotation\n            return leftRotate(node);\n        } else {\n            // First right rotation then left rotation\n            node.right = rightRotate(node.right);\n            return leftRotate(node);\n        }\n    }\n    // Balanced tree, no rotation needed, return directly\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* Perform rotation operation to restore balance to this subtree */\nTreeNode? Rotate(TreeNode? node) {\n    // Get balance factor of node\n    int balanceFactorInt = BalanceFactor(node);\n    // Left-leaning tree\n    if (balanceFactorInt &gt; 1) {\n        if (BalanceFactor(node?.left) &gt;= 0) {\n            // Right rotation\n            return RightRotate(node);\n        } else {\n            // First left rotation then right rotation\n            node!.left = LeftRotate(node!.left);\n            return RightRotate(node);\n        }\n    }\n    // Right-leaning tree\n    if (balanceFactorInt &lt; -1) {\n        if (BalanceFactor(node?.right) &lt;= 0) {\n            // Left rotation\n            return LeftRotate(node);\n        } else {\n            // First right rotation then left rotation\n            node!.right = RightRotate(node!.right);\n            return LeftRotate(node);\n        }\n    }\n    // Balanced tree, no rotation needed, return directly\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* Perform rotation operation to restore balance to this subtree */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n    // Get balance factor of node\n    // Go recommends short variables, here bf refers to t.balanceFactor\n    bf := t.balanceFactor(node)\n    // Left-leaning tree\n    if bf &gt; 1 {\n        if t.balanceFactor(node.Left) &gt;= 0 {\n            // Right rotation\n            return t.rightRotate(node)\n        } else {\n            // First left rotation then right rotation\n            node.Left = t.leftRotate(node.Left)\n            return t.rightRotate(node)\n        }\n    }\n    // Right-leaning tree\n    if bf &lt; -1 {\n        if t.balanceFactor(node.Right) &lt;= 0 {\n            // Left rotation\n            return t.leftRotate(node)\n        } else {\n            // First right rotation then left rotation\n            node.Right = t.rightRotate(node.Right)\n            return t.leftRotate(node)\n        }\n    }\n    // Balanced tree, no rotation needed, return directly\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* Perform rotation operation to restore balance to this subtree */\nfunc rotate(node: TreeNode?) -&gt; TreeNode? {\n    // Get balance factor of node\n    let balanceFactor = balanceFactor(node: node)\n    // Left-leaning tree\n    if balanceFactor &gt; 1 {\n        if self.balanceFactor(node: node?.left) &gt;= 0 {\n            // Right rotation\n            return rightRotate(node: node)\n        } else {\n            // First left rotation then right rotation\n            node?.left = leftRotate(node: node?.left)\n            return rightRotate(node: node)\n        }\n    }\n    // Right-leaning tree\n    if balanceFactor &lt; -1 {\n        if self.balanceFactor(node: node?.right) &lt;= 0 {\n            // Left rotation\n            return leftRotate(node: node)\n        } else {\n            // First right rotation then left rotation\n            node?.right = rightRotate(node: node?.right)\n            return leftRotate(node: node)\n        }\n    }\n    // Balanced tree, no rotation needed, return directly\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* Perform rotation operation to restore balance to this subtree */\n#rotate(node) {\n    // Get balance factor of node\n    const balanceFactor = this.balanceFactor(node);\n    // Left-leaning tree\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // Right rotation\n            return this.#rightRotate(node);\n        } else {\n            // First left rotation then right rotation\n            node.left = this.#leftRotate(node.left);\n            return this.#rightRotate(node);\n        }\n    }\n    // Right-leaning tree\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // Left rotation\n            return this.#leftRotate(node);\n        } else {\n            // First right rotation then left rotation\n            node.right = this.#rightRotate(node.right);\n            return this.#leftRotate(node);\n        }\n    }\n    // Balanced tree, no rotation needed, return directly\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* Perform rotation operation to restore balance to this subtree */\nrotate(node: TreeNode): TreeNode {\n    // Get balance factor of node\n    const balanceFactor = this.balanceFactor(node);\n    // Left-leaning tree\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // Right rotation\n            return this.rightRotate(node);\n        } else {\n            // First left rotation then right rotation\n            node.left = this.leftRotate(node.left);\n            return this.rightRotate(node);\n        }\n    }\n    // Right-leaning tree\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // Left rotation\n            return this.leftRotate(node);\n        } else {\n            // First right rotation then left rotation\n            node.right = this.rightRotate(node.right);\n            return this.leftRotate(node);\n        }\n    }\n    // Balanced tree, no rotation needed, return directly\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* Perform rotation operation to restore balance to this subtree */\nTreeNode? rotate(TreeNode? node) {\n  // Get balance factor of node\n  int factor = balanceFactor(node);\n  // Left-leaning tree\n  if (factor &gt; 1) {\n    if (balanceFactor(node!.left) &gt;= 0) {\n      // Right rotation\n      return rightRotate(node);\n    } else {\n      // First left rotation then right rotation\n      node.left = leftRotate(node.left);\n      return rightRotate(node);\n    }\n  }\n  // Right-leaning tree\n  if (factor &lt; -1) {\n    if (balanceFactor(node!.right) &lt;= 0) {\n      // Left rotation\n      return leftRotate(node);\n    } else {\n      // First right rotation then left rotation\n      node.right = rightRotate(node.right);\n      return leftRotate(node);\n    }\n  }\n  // Balanced tree, no rotation needed, return directly\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* Perform rotation operation to restore balance to this subtree */\nfn rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    // Get balance factor of node\n    let balance_factor = Self::balance_factor(node.clone());\n    // Left-leaning tree\n    if balance_factor &gt; 1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().left.clone()) &gt;= 0 {\n            // Right rotation\n            Self::right_rotate(Some(node))\n        } else {\n            // First left rotation then right rotation\n            let left = node.borrow().left.clone();\n            node.borrow_mut().left = Self::left_rotate(left);\n            Self::right_rotate(Some(node))\n        }\n    }\n    // Right-leaning tree\n    else if balance_factor &lt; -1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().right.clone()) &lt;= 0 {\n            // Left rotation\n            Self::left_rotate(Some(node))\n        } else {\n            // First right rotation then left rotation\n            let right = node.borrow().right.clone();\n            node.borrow_mut().right = Self::right_rotate(right);\n            Self::left_rotate(Some(node))\n        }\n    } else {\n        // Balanced tree, no rotation needed, return directly\n        node\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* Perform rotation operation to restore balance to this subtree */\nTreeNode *rotate(TreeNode *node) {\n    // Get balance factor of node\n    int bf = balanceFactor(node);\n    // Left-leaning tree\n    if (bf &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // Right rotation\n            return rightRotate(node);\n        } else {\n            // First left rotation then right rotation\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // Right-leaning tree\n    if (bf &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // Left rotation\n            return leftRotate(node);\n        } else {\n            // First right rotation then left rotation\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // Balanced tree, no rotation needed, return directly\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* Perform rotation operation to restore balance to this subtree */\nfun rotate(node: TreeNode): TreeNode {\n    // Get balance factor of node\n    val balanceFactor = balanceFactor(node)\n    // Left-leaning tree\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // Right rotation\n            return rightRotate(node)\n        } else {\n            // First left rotation then right rotation\n            node.left = leftRotate(node.left)\n            return rightRotate(node)\n        }\n    }\n    // Right-leaning tree\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // Left rotation\n            return leftRotate(node)\n        } else {\n            // First right rotation then left rotation\n            node.right = rightRotate(node.right)\n            return leftRotate(node)\n        }\n    }\n    // Balanced tree, no rotation needed, return directly\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### Perform rotation to rebalance subtree ###\ndef rotate(node)\n  # Get balance factor of node\n  balance_factor = balance_factor(node)\n  # Left-heavy tree\n  if balance_factor &gt; 1\n    if balance_factor(node.left) &gt;= 0\n      # Right rotation\n      return right_rotate(node)\n    else\n      # First left rotation then right rotation\n      node.left = left_rotate(node.left)\n      return right_rotate(node)\n    end\n  # Right-heavy tree\n  elsif balance_factor &lt; -1\n    if balance_factor(node.right) &lt;= 0\n      # Left rotation\n      return left_rotate(node)\n    else\n      # First right rotation then left rotation\n      node.right = right_rotate(node.right)\n      return left_rotate(node)\n    end\n  end\n  # Balanced tree, no rotation needed, return directly\n  node\nend\n</code></pre>","path":["Chapter 7. Tree","7.5   AVL Tree *"],"tags":[]},{"location":"chapter_tree/avl_tree/#753-common-operations-in-avl-trees","level":2,"title":"7.5.3   Common Operations in AVL Trees","text":"","path":["Chapter 7. Tree","7.5   AVL Tree *"],"tags":[]},{"location":"chapter_tree/avl_tree/#1-node-insertion","level":3,"title":"1.   Node Insertion","text":"<p>The node insertion operation in AVL trees is similar in principle to that in binary search trees. The only difference is that after inserting a node in an AVL tree, a series of unbalanced nodes may appear on the path from that node to the root. Therefore, we need to start from this node and perform rotation operations from bottom to top, restoring balance to all unbalanced nodes. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def insert(self, val):\n    \"\"\"Insert node\"\"\"\n    self._root = self.insert_helper(self._root, val)\n\ndef insert_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode:\n    \"\"\"Recursively insert node (helper method)\"\"\"\n    if node is None:\n        return TreeNode(val)\n    # 1. Find insertion position and insert node\n    if val &lt; node.val:\n        node.left = self.insert_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.insert_helper(node.right, val)\n    else:\n        # Duplicate node not inserted, return directly\n        return node\n    # Update node height\n    self.update_height(node)\n    # 2. Perform rotation operation to restore balance to this subtree\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* Insert node */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* Recursively insert node (helper method) */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return new TreeNode(val);\n    /* 1. Find insertion position and insert node */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    else\n        return node;    // Duplicate node not inserted, return directly\n    updateHeight(node); // Update node height\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = rotate(node);\n    // Return root node of subtree\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* Insert node */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* Recursively insert node (helper method) */\nTreeNode insertHelper(TreeNode node, int val) {\n    if (node == null)\n        return new TreeNode(val);\n    /* 1. Find insertion position and insert node */\n    if (val &lt; node.val)\n        node.left = insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = insertHelper(node.right, val);\n    else\n        return node; // Duplicate node not inserted, return directly\n    updateHeight(node); // Update node height\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = rotate(node);\n    // Return root node of subtree\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* Insert node */\nvoid Insert(int val) {\n    root = InsertHelper(root, val);\n}\n\n/* Recursively insert node (helper method) */\nTreeNode? InsertHelper(TreeNode? node, int val) {\n    if (node == null) return new TreeNode(val);\n    /* 1. Find insertion position and insert node */\n    if (val &lt; node.val)\n        node.left = InsertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = InsertHelper(node.right, val);\n    else\n        return node;     // Duplicate node not inserted, return directly\n    UpdateHeight(node);  // Update node height\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = Rotate(node);\n    // Return root node of subtree\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* Insert node */\nfunc (t *aVLTree) insert(val int) {\n    t.root = t.insertHelper(t.root, val)\n}\n\n/* Recursively insert node (helper function) */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return NewTreeNode(val)\n    }\n    /* 1. Find insertion position and insert node */\n    if val &lt; node.Val.(int) {\n        node.Left = t.insertHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.insertHelper(node.Right, val)\n    } else {\n        // Duplicate node not inserted, return directly\n        return node\n    }\n    // Update node height\n    t.updateHeight(node)\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = t.rotate(node)\n    // Return root node of subtree\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* Insert node */\nfunc insert(val: Int) {\n    root = insertHelper(node: root, val: val)\n}\n\n/* Recursively insert node (helper method) */\nfunc insertHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return TreeNode(x: val)\n    }\n    /* 1. Find insertion position and insert node */\n    if val &lt; node!.val {\n        node?.left = insertHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = insertHelper(node: node?.right, val: val)\n    } else {\n        return node // Duplicate node not inserted, return directly\n    }\n    updateHeight(node: node) // Update node height\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = rotate(node: node)\n    // Return root node of subtree\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* Insert node */\ninsert(val) {\n    this.root = this.#insertHelper(this.root, val);\n}\n\n/* Recursively insert node (helper method) */\n#insertHelper(node, val) {\n    if (node === null) return new TreeNode(val);\n    /* 1. Find insertion position and insert node */\n    if (val &lt; node.val) node.left = this.#insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#insertHelper(node.right, val);\n    else return node; // Duplicate node not inserted, return directly\n    this.#updateHeight(node); // Update node height\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = this.#rotate(node);\n    // Return root node of subtree\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* Insert node */\ninsert(val: number): void {\n    this.root = this.insertHelper(this.root, val);\n}\n\n/* Recursively insert node (helper method) */\ninsertHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return new TreeNode(val);\n    /* 1. Find insertion position and insert node */\n    if (val &lt; node.val) {\n        node.left = this.insertHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.insertHelper(node.right, val);\n    } else {\n        return node; // Duplicate node not inserted, return directly\n    }\n    this.updateHeight(node); // Update node height\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = this.rotate(node);\n    // Return root node of subtree\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* Insert node */\nvoid insert(int val) {\n  root = insertHelper(root, val);\n}\n\n/* Recursively insert node (helper method) */\nTreeNode? insertHelper(TreeNode? node, int val) {\n  if (node == null) return TreeNode(val);\n  /* 1. Find insertion position and insert node */\n  if (val &lt; node.val)\n    node.left = insertHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = insertHelper(node.right, val);\n  else\n    return node; // Duplicate node not inserted, return directly\n  updateHeight(node); // Update node height\n  /* 2. Perform rotation operation to restore balance to this subtree */\n  node = rotate(node);\n  // Return root node of subtree\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* Insert node */\nfn insert(&amp;mut self, val: i32) {\n    self.root = Self::insert_helper(self.root.clone(), val);\n}\n\n/* Recursively insert node (helper method) */\nfn insert_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. Find insertion position and insert node */\n            match {\n                let node_val = node.borrow().val;\n                node_val\n            }\n            .cmp(&amp;val)\n            {\n                Ordering::Greater =&gt; {\n                    let left = node.borrow().left.clone();\n                    node.borrow_mut().left = Self::insert_helper(left, val);\n                }\n                Ordering::Less =&gt; {\n                    let right = node.borrow().right.clone();\n                    node.borrow_mut().right = Self::insert_helper(right, val);\n                }\n                Ordering::Equal =&gt; {\n                    return Some(node); // Duplicate node not inserted, return directly\n                }\n            }\n            Self::update_height(Some(node.clone())); // Update node height\n\n            /* 2. Perform rotation operation to restore balance to this subtree */\n            node = Self::rotate(Some(node)).unwrap();\n            // Return root node of subtree\n            Some(node)\n        }\n        None =&gt; Some(TreeNode::new(val)),\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* Insert node */\nvoid insert(AVLTree *tree, int val) {\n    tree-&gt;root = insertHelper(tree-&gt;root, val);\n}\n\n/* Recursively insert node (helper function) */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == NULL) {\n        return newTreeNode(val);\n    }\n    /* 1. Find insertion position and insert node */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    } else {\n        // Duplicate node not inserted, return directly\n        return node;\n    }\n    // Update node height\n    updateHeight(node);\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = rotate(node);\n    // Return root node of subtree\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* Insert node */\nfun insert(_val: Int) {\n    root = insertHelper(root, _val)\n}\n\n/* Recursively insert node (helper method) */\nfun insertHelper(n: TreeNode?, _val: Int): TreeNode {\n    if (n == null)\n        return TreeNode(_val)\n    var node = n\n    /* 1. Find insertion position and insert node */\n    if (_val &lt; node._val)\n        node.left = insertHelper(node.left, _val)\n    else if (_val &gt; node._val)\n        node.right = insertHelper(node.right, _val)\n    else\n        return node // Duplicate node not inserted, return directly\n    updateHeight(node) // Update node height\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = rotate(node)\n    // Return root node of subtree\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### Insert node ###\ndef insert(val)\n  @root = insert_helper(@root, val)\nend\n\n### Recursively insert node (helper method) ###\ndef insert_helper(node, val)\n  return TreeNode.new(val) if node.nil?\n  # 1. Find insertion position and insert node\n  if val &lt; node.val\n    node.left = insert_helper(node.left, val)\n  elsif val &gt; node.val\n    node.right = insert_helper(node.right, val)\n  else\n    # Duplicate node not inserted, return directly\n    return node\n  end\n  # Update node height\n  update_height(node)\n  # 2. Perform rotation operation to restore balance to this subtree\n  rotate(node)\nend\n</code></pre>","path":["Chapter 7. Tree","7.5   AVL Tree *"],"tags":[]},{"location":"chapter_tree/avl_tree/#2-node-removal","level":3,"title":"2.   Node Removal","text":"<p>Similarly, on the basis of the binary search tree's node removal method, rotation operations need to be performed from bottom to top to restore balance to all unbalanced nodes. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def remove(self, val: int):\n    \"\"\"Delete node\"\"\"\n    self._root = self.remove_helper(self._root, val)\n\ndef remove_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode | None:\n    \"\"\"Recursively delete node (helper method)\"\"\"\n    if node is None:\n        return None\n    # 1. Find node and delete\n    if val &lt; node.val:\n        node.left = self.remove_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.remove_helper(node.right, val)\n    else:\n        if node.left is None or node.right is None:\n            child = node.left or node.right\n            # Number of child nodes = 0, delete node directly and return\n            if child is None:\n                return None\n            # Number of child nodes = 1, delete node directly\n            else:\n                node = child\n        else:\n            # Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it\n            temp = node.right\n            while temp.left is not None:\n                temp = temp.left\n            node.right = self.remove_helper(node.right, temp.val)\n            node.val = temp.val\n    # Update node height\n    self.update_height(node)\n    # 2. Perform rotation operation to restore balance to this subtree\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* Remove node */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* Recursively delete node (helper method) */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return nullptr;\n    /* 1. Find node and delete */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    else {\n        if (node-&gt;left == nullptr || node-&gt;right == nullptr) {\n            TreeNode *child = node-&gt;left != nullptr ? node-&gt;left : node-&gt;right;\n            // Number of child nodes = 0, delete node directly and return\n            if (child == nullptr) {\n                delete node;\n                return nullptr;\n            }\n            // Number of child nodes = 1, delete node directly\n            else {\n                delete node;\n                node = child;\n            }\n        } else {\n            // Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != nullptr) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    updateHeight(node); // Update node height\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = rotate(node);\n    // Return root node of subtree\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* Remove node */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* Recursively delete node (helper method) */\nTreeNode removeHelper(TreeNode node, int val) {\n    if (node == null)\n        return null;\n    /* 1. Find node and delete */\n    if (val &lt; node.val)\n        node.left = removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = removeHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode child = node.left != null ? node.left : node.right;\n            // Number of child nodes = 0, delete node directly and return\n            if (child == null)\n                return null;\n            // Number of child nodes = 1, delete node directly\n            else\n                node = child;\n        } else {\n            // Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it\n            TreeNode temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    updateHeight(node); // Update node height\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = rotate(node);\n    // Return root node of subtree\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* Remove node */\nvoid Remove(int val) {\n    root = RemoveHelper(root, val);\n}\n\n/* Recursively delete node (helper method) */\nTreeNode? RemoveHelper(TreeNode? node, int val) {\n    if (node == null) return null;\n    /* 1. Find node and delete */\n    if (val &lt; node.val)\n        node.left = RemoveHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = RemoveHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode? child = node.left ?? node.right;\n            // Number of child nodes = 0, delete node directly and return\n            if (child == null)\n                return null;\n            // Number of child nodes = 1, delete node directly\n            else\n                node = child;\n        } else {\n            // Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it\n            TreeNode? temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = RemoveHelper(node.right, temp.val!.Value);\n            node.val = temp.val;\n        }\n    }\n    UpdateHeight(node);  // Update node height\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = Rotate(node);\n    // Return root node of subtree\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* Remove node */\nfunc (t *aVLTree) remove(val int) {\n    t.root = t.removeHelper(t.root, val)\n}\n\n/* Recursively remove node (helper function) */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return nil\n    }\n    /* 1. Find node and delete */\n    if val &lt; node.Val.(int) {\n        node.Left = t.removeHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.removeHelper(node.Right, val)\n    } else {\n        if node.Left == nil || node.Right == nil {\n            child := node.Left\n            if node.Right != nil {\n                child = node.Right\n            }\n            if child == nil {\n                // Number of child nodes = 0, delete node directly and return\n                return nil\n            } else {\n                // Number of child nodes = 1, delete node directly\n                node = child\n            }\n        } else {\n            // Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it\n            temp := node.Right\n            for temp.Left != nil {\n                temp = temp.Left\n            }\n            node.Right = t.removeHelper(node.Right, temp.Val.(int))\n            node.Val = temp.Val\n        }\n    }\n    // Update node height\n    t.updateHeight(node)\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = t.rotate(node)\n    // Return root node of subtree\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* Remove node */\nfunc remove(val: Int) {\n    root = removeHelper(node: root, val: val)\n}\n\n/* Recursively delete node (helper method) */\nfunc removeHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return nil\n    }\n    /* 1. Find node and delete */\n    if val &lt; node!.val {\n        node?.left = removeHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = removeHelper(node: node?.right, val: val)\n    } else {\n        if node?.left == nil || node?.right == nil {\n            let child = node?.left ?? node?.right\n            // Number of child nodes = 0, delete node directly and return\n            if child == nil {\n                return nil\n            }\n            // Number of child nodes = 1, delete node directly\n            else {\n                node = child\n            }\n        } else {\n            // Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it\n            var temp = node?.right\n            while temp?.left != nil {\n                temp = temp?.left\n            }\n            node?.right = removeHelper(node: node?.right, val: temp!.val)\n            node?.val = temp!.val\n        }\n    }\n    updateHeight(node: node) // Update node height\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = rotate(node: node)\n    // Return root node of subtree\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* Remove node */\nremove(val) {\n    this.root = this.#removeHelper(this.root, val);\n}\n\n/* Recursively delete node (helper method) */\n#removeHelper(node, val) {\n    if (node === null) return null;\n    /* 1. Find node and delete */\n    if (val &lt; node.val) node.left = this.#removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#removeHelper(node.right, val);\n    else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // Number of child nodes = 0, delete node directly and return\n            if (child === null) return null;\n            // Number of child nodes = 1, delete node directly\n            else node = child;\n        } else {\n            // Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.#removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.#updateHeight(node); // Update node height\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = this.#rotate(node);\n    // Return root node of subtree\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* Remove node */\nremove(val: number): void {\n    this.root = this.removeHelper(this.root, val);\n}\n\n/* Recursively delete node (helper method) */\nremoveHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return null;\n    /* 1. Find node and delete */\n    if (val &lt; node.val) {\n        node.left = this.removeHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.removeHelper(node.right, val);\n    } else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // Number of child nodes = 0, delete node directly and return\n            if (child === null) {\n                return null;\n            } else {\n                // Number of child nodes = 1, delete node directly\n                node = child;\n            }\n        } else {\n            // Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.updateHeight(node); // Update node height\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = this.rotate(node);\n    // Return root node of subtree\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* Remove node */\nvoid remove(int val) {\n  root = removeHelper(root, val);\n}\n\n/* Recursively delete node (helper method) */\nTreeNode? removeHelper(TreeNode? node, int val) {\n  if (node == null) return null;\n  /* 1. Find node and delete */\n  if (val &lt; node.val)\n    node.left = removeHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = removeHelper(node.right, val);\n  else {\n    if (node.left == null || node.right == null) {\n      TreeNode? child = node.left ?? node.right;\n      // Number of child nodes = 0, delete node directly and return\n      if (child == null)\n        return null;\n      // Number of child nodes = 1, delete node directly\n      else\n        node = child;\n    } else {\n      // Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it\n      TreeNode? temp = node.right;\n      while (temp!.left != null) {\n        temp = temp.left;\n      }\n      node.right = removeHelper(node.right, temp.val);\n      node.val = temp.val;\n    }\n  }\n  updateHeight(node); // Update node height\n  /* 2. Perform rotation operation to restore balance to this subtree */\n  node = rotate(node);\n  // Return root node of subtree\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* Remove node */\nfn remove(&amp;self, val: i32) {\n    Self::remove_helper(self.root.clone(), val);\n}\n\n/* Recursively delete node (helper method) */\nfn remove_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. Find node and delete */\n            if val &lt; node.borrow().val {\n                let left = node.borrow().left.clone();\n                node.borrow_mut().left = Self::remove_helper(left, val);\n            } else if val &gt; node.borrow().val {\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, val);\n            } else if node.borrow().left.is_none() || node.borrow().right.is_none() {\n                let child = if node.borrow().left.is_some() {\n                    node.borrow().left.clone()\n                } else {\n                    node.borrow().right.clone()\n                };\n                match child {\n                    // Number of child nodes = 0, delete node directly and return\n                    None =&gt; {\n                        return None;\n                    }\n                    // Number of child nodes = 1, delete node directly\n                    Some(child) =&gt; node = child,\n                }\n            } else {\n                // Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it\n                let mut temp = node.borrow().right.clone().unwrap();\n                loop {\n                    let temp_left = temp.borrow().left.clone();\n                    if temp_left.is_none() {\n                        break;\n                    }\n                    temp = temp_left.unwrap();\n                }\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);\n                node.borrow_mut().val = temp.borrow().val;\n            }\n            Self::update_height(Some(node.clone())); // Update node height\n\n            /* 2. Perform rotation operation to restore balance to this subtree */\n            node = Self::rotate(Some(node)).unwrap();\n            // Return root node of subtree\n            Some(node)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* Remove node */\n// Cannot use remove keyword here due to stdio.h inclusion\nvoid removeItem(AVLTree *tree, int val) {\n    TreeNode *root = removeHelper(tree-&gt;root, val);\n}\n\n/* Recursively remove node (helper function) */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    TreeNode *child, *grandChild;\n    if (node == NULL) {\n        return NULL;\n    }\n    /* 1. Find node and delete */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    } else {\n        if (node-&gt;left == NULL || node-&gt;right == NULL) {\n            child = node-&gt;left;\n            if (node-&gt;right != NULL) {\n                child = node-&gt;right;\n            }\n            // Number of child nodes = 0, delete node directly and return\n            if (child == NULL) {\n                return NULL;\n            } else {\n                // Number of child nodes = 1, delete node directly\n                node = child;\n            }\n        } else {\n            // Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != NULL) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    // Update node height\n    updateHeight(node);\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = rotate(node);\n    // Return root node of subtree\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* Remove node */\nfun remove(_val: Int) {\n    root = removeHelper(root, _val)\n}\n\n/* Recursively delete node (helper method) */\nfun removeHelper(n: TreeNode?, _val: Int): TreeNode? {\n    var node = n ?: return null\n    /* 1. Find node and delete */\n    if (_val &lt; node._val)\n        node.left = removeHelper(node.left, _val)\n    else if (_val &gt; node._val)\n        node.right = removeHelper(node.right, _val)\n    else {\n        if (node.left == null || node.right == null) {\n            val child = if (node.left != null)\n                node.left\n            else\n                node.right\n            // Number of child nodes = 0, delete node directly and return\n            if (child == null)\n                return null\n            // Number of child nodes = 1, delete node directly\n            else\n                node = child\n        } else {\n            // Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it\n            var temp = node.right\n            while (temp!!.left != null) {\n                temp = temp.left\n            }\n            node.right = removeHelper(node.right, temp._val)\n            node._val = temp._val\n        }\n    }\n    updateHeight(node) // Update node height\n    /* 2. Perform rotation operation to restore balance to this subtree */\n    node = rotate(node)\n    // Return root node of subtree\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### Delete node ###\ndef remove(val)\n  @root = remove_helper(@root, val)\nend\n\n### Recursively delete node (helper method) ###\ndef remove_helper(node, val)\n  return if node.nil?\n  # 1. Find node and delete\n  if val &lt; node.val\n    node.left = remove_helper(node.left, val)\n  elsif val &gt; node.val\n    node.right = remove_helper(node.right, val)\n  else\n    if node.left.nil? || node.right.nil?\n      child = node.left || node.right\n      # Number of child nodes = 0, delete node directly and return\n      return if child.nil?\n      # Number of child nodes = 1, delete node directly\n      node = child\n    else\n      # Number of child nodes = 2, delete the next node in inorder traversal and replace current node with it\n      temp = node.right\n      while !temp.left.nil?\n        temp = temp.left\n      end\n      node.right = remove_helper(node.right, temp.val)\n      node.val = temp.val\n    end\n  end\n  # Update node height\n  update_height(node)\n  # 2. Perform rotation operation to restore balance to this subtree\n  rotate(node)\nend\n</code></pre>","path":["Chapter 7. Tree","7.5   AVL Tree *"],"tags":[]},{"location":"chapter_tree/avl_tree/#3-node-search","level":3,"title":"3.   Node Search","text":"<p>The node search operation in AVL trees is consistent with that in binary search trees, and will not be elaborated here.</p>","path":["Chapter 7. Tree","7.5   AVL Tree *"],"tags":[]},{"location":"chapter_tree/avl_tree/#754-typical-applications-of-avl-trees","level":2,"title":"7.5.4   Typical Applications of AVL Trees","text":"<ul> <li>Organizing and storing large-scale data, suitable for scenarios with high-frequency searches and low-frequency insertions and deletions.</li> <li>Used to build index systems in databases.</li> <li>Red-black trees are also a common type of balanced binary search tree. Compared to AVL trees, red-black trees have more relaxed balance conditions, require fewer rotation operations for node insertion and deletion, and have higher average efficiency for node addition and deletion operations.</li> </ul>","path":["Chapter 7. Tree","7.5   AVL Tree *"],"tags":[]},{"location":"chapter_tree/binary_search_tree/","level":1,"title":"7.4   Binary Search Tree","text":"<p>As shown in Figure 7-16, a binary search tree satisfies the following conditions.</p> <ol> <li>For the root node, the value of all nodes in the left subtree \\(&lt;\\) the value of the root node \\(&lt;\\) the value of all nodes in the right subtree.</li> <li>The left and right subtrees of any node are also binary search trees, i.e., they satisfy condition <code>1.</code> as well.</li> </ol> <p></p> <p> Figure 7-16   Binary search tree </p>","path":["Chapter 7. Tree","7.4   Binary Search Tree"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#741-operations-on-a-binary-search-tree","level":2,"title":"7.4.1   Operations on a Binary Search Tree","text":"<p>We encapsulate the binary search tree as a class <code>BinarySearchTree</code> and declare a member variable <code>root</code> pointing to the tree's root node.</p>","path":["Chapter 7. Tree","7.4   Binary Search Tree"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#1-searching-for-a-node","level":3,"title":"1.   Searching for a Node","text":"<p>Given a target node value <code>num</code>, we can search according to the properties of the binary search tree. As shown in Figure 7-17, we declare a node <code>cur</code> and start from the binary search tree's root node <code>root</code>, looping to compare <code>cur.val</code> with <code>num</code>.</p> <ul> <li>If <code>cur.val &lt; num</code>, it means the target node is in <code>cur</code>'s right subtree, thus execute <code>cur = cur.right</code>.</li> <li>If <code>cur.val &gt; num</code>, it means the target node is in <code>cur</code>'s left subtree, thus execute <code>cur = cur.left</code>.</li> <li>If <code>cur.val = num</code>, it means the target node is found, exit the loop, and return the node.</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 7-17   Example of searching for a node in a binary search tree </p> <p>The search operation in a binary search tree follows the same principle as binary search: each round rules out half of the remaining cases. The number of loop iterations is at most the height of the tree. When the tree is balanced, the search takes \\(O(\\log n)\\) time. The example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_tree.py<pre><code>def search(self, num: int) -&gt; TreeNode | None:\n    \"\"\"Search node\"\"\"\n    cur = self._root\n    # Loop search, exit after passing leaf node\n    while cur is not None:\n        # Target node is in cur's right subtree\n        if cur.val &lt; num:\n            cur = cur.right\n        # Target node is in cur's left subtree\n        elif cur.val &gt; num:\n            cur = cur.left\n        # Found target node, exit loop\n        else:\n            break\n    return cur\n</code></pre> binary_search_tree.cpp<pre><code>/* Search node */\nTreeNode *search(int num) {\n    TreeNode *cur = root;\n    // Loop search, exit after passing leaf node\n    while (cur != nullptr) {\n        // Target node is in cur's right subtree\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // Target node is in cur's left subtree\n        else if (cur-&gt;val &gt; num)\n            cur = cur-&gt;left;\n        // Found target node, exit loop\n        else\n            break;\n    }\n    // Return target node\n    return cur;\n}\n</code></pre> binary_search_tree.java<pre><code>/* Search node */\nTreeNode search(int num) {\n    TreeNode cur = root;\n    // Loop search, exit after passing leaf node\n    while (cur != null) {\n        // Target node is in cur's right subtree\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // Target node is in cur's left subtree\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // Found target node, exit loop\n        else\n            break;\n    }\n    // Return target node\n    return cur;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* Search node */\nTreeNode? Search(int num) {\n    TreeNode? cur = root;\n    // Loop search, exit after passing leaf node\n    while (cur != null) {\n        // Target node is in cur's right subtree\n        if (cur.val &lt; num) cur =\n            cur.right;\n        // Target node is in cur's left subtree\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // Found target node, exit loop\n        else\n            break;\n    }\n    // Return target node\n    return cur;\n}\n</code></pre> binary_search_tree.go<pre><code>/* Search node */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\n    node := bst.root\n    // Loop search, exit after passing leaf node\n    for node != nil {\n        if node.Val.(int) &lt; num {\n            // Target node is in cur's right subtree\n            node = node.Right\n        } else if node.Val.(int) &gt; num {\n            // Target node is in cur's left subtree\n            node = node.Left\n        } else {\n            // Found target node, exit loop\n            break\n        }\n    }\n    // Return target node\n    return node\n}\n</code></pre> binary_search_tree.swift<pre><code>/* Search node */\nfunc search(num: Int) -&gt; TreeNode? {\n    var cur = root\n    // Loop search, exit after passing leaf node\n    while cur != nil {\n        // Target node is in cur's right subtree\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // Target node is in cur's left subtree\n        else if cur!.val &gt; num {\n            cur = cur?.left\n        }\n        // Found target node, exit loop\n        else {\n            break\n        }\n    }\n    // Return target node\n    return cur\n}\n</code></pre> binary_search_tree.js<pre><code>/* Search node */\nsearch(num) {\n    let cur = this.root;\n    // Loop search, exit after passing leaf node\n    while (cur !== null) {\n        // Target node is in cur's right subtree\n        if (cur.val &lt; num) cur = cur.right;\n        // Target node is in cur's left subtree\n        else if (cur.val &gt; num) cur = cur.left;\n        // Found target node, exit loop\n        else break;\n    }\n    // Return target node\n    return cur;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* Search node */\nsearch(num: number): TreeNode | null {\n    let cur = this.root;\n    // Loop search, exit after passing leaf node\n    while (cur !== null) {\n        // Target node is in cur's right subtree\n        if (cur.val &lt; num) cur = cur.right;\n        // Target node is in cur's left subtree\n        else if (cur.val &gt; num) cur = cur.left;\n        // Found target node, exit loop\n        else break;\n    }\n    // Return target node\n    return cur;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* Search node */\nTreeNode? search(int _num) {\n  TreeNode? cur = _root;\n  // Loop search, exit after passing leaf node\n  while (cur != null) {\n    // Target node is in cur's right subtree\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // Target node is in cur's left subtree\n    else if (cur.val &gt; _num)\n      cur = cur.left;\n    // Found target node, exit loop\n    else\n      break;\n  }\n  // Return target node\n  return cur;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* Search node */\npub fn search(&amp;self, num: i32) -&gt; OptionTreeNodeRc {\n    let mut cur = self.root.clone();\n    // Loop search, exit after passing leaf node\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // Target node is in cur's right subtree\n            Ordering::Greater =&gt; cur = node.borrow().right.clone(),\n            // Target node is in cur's left subtree\n            Ordering::Less =&gt; cur = node.borrow().left.clone(),\n            // Found target node, exit loop\n            Ordering::Equal =&gt; break,\n        }\n    }\n\n    // Return target node\n    cur\n}\n</code></pre> binary_search_tree.c<pre><code>/* Search node */\nTreeNode *search(BinarySearchTree *bst, int num) {\n    TreeNode *cur = bst-&gt;root;\n    // Loop search, exit after passing leaf node\n    while (cur != NULL) {\n        if (cur-&gt;val &lt; num) {\n            // Target node is in cur's right subtree\n            cur = cur-&gt;right;\n        } else if (cur-&gt;val &gt; num) {\n            // Target node is in cur's left subtree\n            cur = cur-&gt;left;\n        } else {\n            // Found target node, exit loop\n            break;\n        }\n    }\n    // Return target node\n    return cur;\n}\n</code></pre> binary_search_tree.kt<pre><code>/* Search node */\nfun search(num: Int): TreeNode? {\n    var cur = root\n    // Loop search, exit after passing leaf node\n    while (cur != null) {\n        // Target node is in cur's right subtree\n        cur = if (cur._val &lt; num)\n            cur.right\n        // Target node is in cur's left subtree\n        else if (cur._val &gt; num)\n            cur.left\n        // Found target node, exit loop\n        else\n            break\n    }\n    // Return target node\n    return cur\n}\n</code></pre> binary_search_tree.rb<pre><code>### Search node ###\ndef search(num)\n  cur = @root\n\n  # Loop search, exit after passing leaf node\n  while !cur.nil?\n    # Target node is in cur's right subtree\n    if cur.val &lt; num\n      cur = cur.right\n    # Target node is in cur's left subtree\n    elsif cur.val &gt; num\n      cur = cur.left\n    # Found target node, exit loop\n    else\n      break\n    end\n  end\n\n  cur\nend\n</code></pre>","path":["Chapter 7. Tree","7.4   Binary Search Tree"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#2-inserting-a-node","level":3,"title":"2.   Inserting a Node","text":"<p>Given an element <code>num</code> to be inserted, in order to maintain the property of the binary search tree \"left subtree &lt; root node &lt; right subtree,\" the insertion process is as shown in Figure 7-18.</p> <ol> <li>Finding the insertion position: Similar to the search operation, start from the root node and loop downward searching according to the size relationship between the current node value and <code>num</code>, until passing the leaf node (traversing to <code>None</code>) and then exit the loop.</li> <li>Insert the node at that position: Create a node for <code>num</code> and place it at the <code>None</code> position.</li> </ol> <p></p> <p> Figure 7-18   Inserting a node into a binary search tree </p> <p>In the code implementation, note the following two points:</p> <ul> <li>Binary search trees do not allow duplicate nodes; otherwise, the tree would no longer satisfy its definition. Therefore, if the node to be inserted already exists in the tree, the insertion is skipped and the function returns directly.</li> <li>To implement the node insertion, we need to use node <code>pre</code> to save the node from the previous loop iteration. This way, when traversing to <code>None</code>, we can obtain its parent node, thereby completing the node insertion operation.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_tree.py<pre><code>def insert(self, num: int):\n    \"\"\"Insert node\"\"\"\n    # If tree is empty, initialize root node\n    if self._root is None:\n        self._root = TreeNode(num)\n        return\n    # Loop search, exit after passing leaf node\n    cur, pre = self._root, None\n    while cur is not None:\n        # Found duplicate node, return directly\n        if cur.val == num:\n            return\n        pre = cur\n        # Insertion position is in cur's right subtree\n        if cur.val &lt; num:\n            cur = cur.right\n        # Insertion position is in cur's left subtree\n        else:\n            cur = cur.left\n    # Insert node\n    node = TreeNode(num)\n    if pre.val &lt; num:\n        pre.right = node\n    else:\n        pre.left = node\n</code></pre> binary_search_tree.cpp<pre><code>/* Insert node */\nvoid insert(int num) {\n    // If tree is empty, initialize root node\n    if (root == nullptr) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode *cur = root, *pre = nullptr;\n    // Loop search, exit after passing leaf node\n    while (cur != nullptr) {\n        // Found duplicate node, return directly\n        if (cur-&gt;val == num)\n            return;\n        pre = cur;\n        // Insertion position is in cur's right subtree\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // Insertion position is in cur's left subtree\n        else\n            cur = cur-&gt;left;\n    }\n    // Insert node\n    TreeNode *node = new TreeNode(num);\n    if (pre-&gt;val &lt; num)\n        pre-&gt;right = node;\n    else\n        pre-&gt;left = node;\n}\n</code></pre> binary_search_tree.java<pre><code>/* Insert node */\nvoid insert(int num) {\n    // If tree is empty, initialize root node\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode cur = root, pre = null;\n    // Loop search, exit after passing leaf node\n    while (cur != null) {\n        // Found duplicate node, return directly\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // Insertion position is in cur's right subtree\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // Insertion position is in cur's left subtree\n        else\n            cur = cur.left;\n    }\n    // Insert node\n    TreeNode node = new TreeNode(num);\n    if (pre.val &lt; num)\n        pre.right = node;\n    else\n        pre.left = node;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* Insert node */\nvoid Insert(int num) {\n    // If tree is empty, initialize root node\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode? cur = root, pre = null;\n    // Loop search, exit after passing leaf node\n    while (cur != null) {\n        // Found duplicate node, return directly\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // Insertion position is in cur's right subtree\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // Insertion position is in cur's left subtree\n        else\n            cur = cur.left;\n    }\n\n    // Insert node\n    TreeNode node = new(num);\n    if (pre != null) {\n        if (pre.val &lt; num)\n            pre.right = node;\n        else\n            pre.left = node;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* Insert node */\nfunc (bst *binarySearchTree) insert(num int) {\n    cur := bst.root\n    // If tree is empty, initialize root node\n    if cur == nil {\n        bst.root = NewTreeNode(num)\n        return\n    }\n    // Node position before the node to be inserted\n    var pre *TreeNode = nil\n    // Loop search, exit after passing leaf node\n    for cur != nil {\n        if cur.Val == num {\n            return\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            cur = cur.Right\n        } else {\n            cur = cur.Left\n        }\n    }\n    // Insert node\n    node := NewTreeNode(num)\n    if pre.Val.(int) &lt; num {\n        pre.Right = node\n    } else {\n        pre.Left = node\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* Insert node */\nfunc insert(num: Int) {\n    // If tree is empty, initialize root node\n    if root == nil {\n        root = TreeNode(x: num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // Loop search, exit after passing leaf node\n    while cur != nil {\n        // Found duplicate node, return directly\n        if cur!.val == num {\n            return\n        }\n        pre = cur\n        // Insertion position is in cur's right subtree\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // Insertion position is in cur's left subtree\n        else {\n            cur = cur?.left\n        }\n    }\n    // Insert node\n    let node = TreeNode(x: num)\n    if pre!.val &lt; num {\n        pre?.right = node\n    } else {\n        pre?.left = node\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* Insert node */\ninsert(num) {\n    // If tree is empty, initialize root node\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur = this.root,\n        pre = null;\n    // Loop search, exit after passing leaf node\n    while (cur !== null) {\n        // Found duplicate node, return directly\n        if (cur.val === num) return;\n        pre = cur;\n        // Insertion position is in cur's right subtree\n        if (cur.val &lt; num) cur = cur.right;\n        // Insertion position is in cur's left subtree\n        else cur = cur.left;\n    }\n    // Insert node\n    const node = new TreeNode(num);\n    if (pre.val &lt; num) pre.right = node;\n    else pre.left = node;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* Insert node */\ninsert(num: number): void {\n    // If tree is empty, initialize root node\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // Loop search, exit after passing leaf node\n    while (cur !== null) {\n        // Found duplicate node, return directly\n        if (cur.val === num) return;\n        pre = cur;\n        // Insertion position is in cur's right subtree\n        if (cur.val &lt; num) cur = cur.right;\n        // Insertion position is in cur's left subtree\n        else cur = cur.left;\n    }\n    // Insert node\n    const node = new TreeNode(num);\n    if (pre!.val &lt; num) pre!.right = node;\n    else pre!.left = node;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* Insert node */\nvoid insert(int _num) {\n  // If tree is empty, initialize root node\n  if (_root == null) {\n    _root = TreeNode(_num);\n    return;\n  }\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // Loop search, exit after passing leaf node\n  while (cur != null) {\n    // Found duplicate node, return directly\n    if (cur.val == _num) return;\n    pre = cur;\n    // Insertion position is in cur's right subtree\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // Insertion position is in cur's left subtree\n    else\n      cur = cur.left;\n  }\n  // Insert node\n  TreeNode? node = TreeNode(_num);\n  if (pre!.val &lt; _num)\n    pre.right = node;\n  else\n    pre.left = node;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* Insert node */\npub fn insert(&amp;mut self, num: i32) {\n    // If tree is empty, initialize root node\n    if self.root.is_none() {\n        self.root = Some(TreeNode::new(num));\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // Loop search, exit after passing leaf node\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // Found duplicate node, return directly\n            Ordering::Equal =&gt; return,\n            // Insertion position is in cur's right subtree\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // Insertion position is in cur's left subtree\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // Insert node\n    let pre = pre.unwrap();\n    let node = Some(TreeNode::new(num));\n    if num &gt; pre.borrow().val {\n        pre.borrow_mut().right = node;\n    } else {\n        pre.borrow_mut().left = node;\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* Insert node */\nvoid insert(BinarySearchTree *bst, int num) {\n    // If tree is empty, initialize root node\n    if (bst-&gt;root == NULL) {\n        bst-&gt;root = newTreeNode(num);\n        return;\n    }\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // Loop search, exit after passing leaf node\n    while (cur != NULL) {\n        // Found duplicate node, return directly\n        if (cur-&gt;val == num) {\n            return;\n        }\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // Insertion position is in cur's right subtree\n            cur = cur-&gt;right;\n        } else {\n            // Insertion position is in cur's left subtree\n            cur = cur-&gt;left;\n        }\n    }\n    // Insert node\n    TreeNode *node = newTreeNode(num);\n    if (pre-&gt;val &lt; num) {\n        pre-&gt;right = node;\n    } else {\n        pre-&gt;left = node;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* Insert node */\nfun insert(num: Int) {\n    // If tree is empty, initialize root node\n    if (root == null) {\n        root = TreeNode(num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode? = null\n    // Loop search, exit after passing leaf node\n    while (cur != null) {\n        // Found duplicate node, return directly\n        if (cur._val == num)\n            return\n        pre = cur\n        // Insertion position is in cur's right subtree\n        cur = if (cur._val &lt; num)\n            cur.right\n        // Insertion position is in cur's left subtree\n        else\n            cur.left\n    }\n    // Insert node\n    val node = TreeNode(num)\n    if (pre?._val!! &lt; num)\n        pre.right = node\n    else\n        pre.left = node\n}\n</code></pre> binary_search_tree.rb<pre><code>### Insert node ###\ndef insert(num)\n  # If tree is empty, initialize root node\n  if @root.nil?\n    @root = TreeNode.new(num)\n    return\n  end\n\n  # Loop search, exit after passing leaf node\n  cur, pre = @root, nil\n  while !cur.nil?\n    # Found duplicate node, return directly\n    return if cur.val == num\n\n    pre = cur\n    # Insertion position is in cur's right subtree\n    if cur.val &lt; num\n      cur = cur.right\n    # Insertion position is in cur's left subtree\n    else\n      cur = cur.left\n    end\n  end\n\n  # Insert node\n  node = TreeNode.new(num)\n  if pre.val &lt; num\n    pre.right = node\n  else\n    pre.left = node\n  end\nend\n</code></pre> <p>Similar to searching for a node, inserting a node uses \\(O(\\log n)\\) time.</p>","path":["Chapter 7. Tree","7.4   Binary Search Tree"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#3-removing-a-node","level":3,"title":"3.   Removing a Node","text":"<p>First, find the target node in the binary search tree, then remove it. Similar to node insertion, we need to ensure that after the removal operation is completed, the binary search tree's property of \"left subtree \\(&lt;\\) root node \\(&lt;\\) right subtree\" is still maintained. Therefore, depending on the number of child nodes the target node has, we consider three cases: degree \\(0\\), degree \\(1\\), and degree \\(2\\), and perform the corresponding removal operation.</p> <p>As shown in Figure 7-19, when the degree of the node to be removed is \\(0\\), it means the node is a leaf node and can be directly removed.</p> <p></p> <p> Figure 7-19   Removing a node in a binary search tree (degree 0) </p> <p>As shown in Figure 7-20, when the degree of the node to be removed is \\(1\\), replacing the node to be removed with its child node is sufficient.</p> <p></p> <p> Figure 7-20   Removing a node in a binary search tree (degree 1) </p> <p>When the degree of the node to be removed is \\(2\\), we cannot directly remove it; instead, we need to use a node to replace it. To maintain the binary search tree's property of \"left subtree \\(&lt;\\) root node \\(&lt;\\) right subtree,\" this node can be either the smallest node in the right subtree or the largest node in the left subtree.</p> <p>Assuming we choose the smallest node in the right subtree, that is, the inorder successor, the removal process is as shown in Figure 7-21.</p> <ol> <li>Find the next node of the node to be removed in the \"inorder traversal sequence,\" denoted as <code>tmp</code>.</li> <li>Replace the value of the node to be removed with the value of <code>tmp</code>, and recursively remove node <code>tmp</code> in the tree.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 7-21   Removing a node in a binary search tree (degree 2) </p> <p>The node removal operation also uses \\(O(\\log n)\\) time, where finding the node to be removed requires \\(O(\\log n)\\) time, and obtaining the inorder successor node requires \\(O(\\log n)\\) time. Example code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_tree.py<pre><code>def remove(self, num: int):\n    \"\"\"Delete node\"\"\"\n    # If tree is empty, return directly\n    if self._root is None:\n        return\n    # Loop search, exit after passing leaf node\n    cur, pre = self._root, None\n    while cur is not None:\n        # Found node to delete, exit loop\n        if cur.val == num:\n            break\n        pre = cur\n        # Node to delete is in cur's right subtree\n        if cur.val &lt; num:\n            cur = cur.right\n        # Node to delete is in cur's left subtree\n        else:\n            cur = cur.left\n    # If no node to delete, return directly\n    if cur is None:\n        return\n\n    # Number of child nodes = 0 or 1\n    if cur.left is None or cur.right is None:\n        # When number of child nodes = 0 / 1, child = null / that child node\n        child = cur.left or cur.right\n        # Delete node cur\n        if cur != self._root:\n            if pre.left == cur:\n                pre.left = child\n            else:\n                pre.right = child\n        else:\n            # If deleted node is root node, reassign root node\n            self._root = child\n    # Number of child nodes = 2\n    else:\n        # Get next node of cur in inorder traversal\n        tmp: TreeNode = cur.right\n        while tmp.left is not None:\n            tmp = tmp.left\n        # Recursively delete node tmp\n        self.remove(tmp.val)\n        # Replace cur with tmp\n        cur.val = tmp.val\n</code></pre> binary_search_tree.cpp<pre><code>/* Remove node */\nvoid remove(int num) {\n    // If tree is empty, return directly\n    if (root == nullptr)\n        return;\n    TreeNode *cur = root, *pre = nullptr;\n    // Loop search, exit after passing leaf node\n    while (cur != nullptr) {\n        // Found node to delete, exit loop\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        // Node to delete is in cur's right subtree\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // Node to delete is in cur's left subtree\n        else\n            cur = cur-&gt;left;\n    }\n    // If no node to delete, return directly\n    if (cur == nullptr)\n        return;\n    // Number of child nodes = 0 or 1\n    if (cur-&gt;left == nullptr || cur-&gt;right == nullptr) {\n        // When number of child nodes = 0 / 1, child = nullptr / that child node\n        TreeNode *child = cur-&gt;left != nullptr ? cur-&gt;left : cur-&gt;right;\n        // Delete node cur\n        if (cur != root) {\n            if (pre-&gt;left == cur)\n                pre-&gt;left = child;\n            else\n                pre-&gt;right = child;\n        } else {\n            // If deleted node is root node, reassign root node\n            root = child;\n        }\n        // Free memory\n        delete cur;\n    }\n    // Number of child nodes = 2\n    else {\n        // Get next node of cur in inorder traversal\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != nullptr) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // Recursively delete node tmp\n        remove(tmp-&gt;val);\n        // Replace cur with tmp\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.java<pre><code>/* Remove node */\nvoid remove(int num) {\n    // If tree is empty, return directly\n    if (root == null)\n        return;\n    TreeNode cur = root, pre = null;\n    // Loop search, exit after passing leaf node\n    while (cur != null) {\n        // Found node to delete, exit loop\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // Node to delete is in cur's right subtree\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // Node to delete is in cur's left subtree\n        else\n            cur = cur.left;\n    }\n    // If no node to delete, return directly\n    if (cur == null)\n        return;\n    // Number of child nodes = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // When number of child nodes = 0 / 1, child = null / that child node\n        TreeNode child = cur.left != null ? cur.left : cur.right;\n        // Delete node cur\n        if (cur != root) {\n            if (pre.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // If deleted node is root node, reassign root node\n            root = child;\n        }\n    }\n    // Number of child nodes = 2\n    else {\n        // Get next node of cur in inorder traversal\n        TreeNode tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // Recursively delete node tmp\n        remove(tmp.val);\n        // Replace cur with tmp\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.cs<pre><code>/* Remove node */\nvoid Remove(int num) {\n    // If tree is empty, return directly\n    if (root == null)\n        return;\n    TreeNode? cur = root, pre = null;\n    // Loop search, exit after passing leaf node\n    while (cur != null) {\n        // Found node to delete, exit loop\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // Node to delete is in cur's right subtree\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // Node to delete is in cur's left subtree\n        else\n            cur = cur.left;\n    }\n    // If no node to delete, return directly\n    if (cur == null)\n        return;\n    // Number of child nodes = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // When number of child nodes = 0 / 1, child = null / that child node\n        TreeNode? child = cur.left ?? cur.right;\n        // Delete node cur\n        if (cur != root) {\n            if (pre!.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // If deleted node is root node, reassign root node\n            root = child;\n        }\n    }\n    // Number of child nodes = 2\n    else {\n        // Get next node of cur in inorder traversal\n        TreeNode? tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // Recursively delete node tmp\n        Remove(tmp.val!.Value);\n        // Replace cur with tmp\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* Remove node */\nfunc (bst *binarySearchTree) remove(num int) {\n    cur := bst.root\n    // If tree is empty, return directly\n    if cur == nil {\n        return\n    }\n    // Node position before the node to be removed\n    var pre *TreeNode = nil\n    // Loop search, exit after passing leaf node\n    for cur != nil {\n        if cur.Val == num {\n            break\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            // Node to be removed is in right subtree\n            cur = cur.Right\n        } else {\n            // Node to be removed is in left subtree\n            cur = cur.Left\n        }\n    }\n    // If no node to delete, return directly\n    if cur == nil {\n        return\n    }\n    // Number of child nodes is 0 or 1\n    if cur.Left == nil || cur.Right == nil {\n        var child *TreeNode = nil\n        // Get child node of node to be removed\n        if cur.Left != nil {\n            child = cur.Left\n        } else {\n            child = cur.Right\n        }\n        // Delete node cur\n        if cur != bst.root {\n            if pre.Left == cur {\n                pre.Left = child\n            } else {\n                pre.Right = child\n            }\n        } else {\n            // If deleted node is root node, reassign root node\n            bst.root = child\n        }\n        // Number of child nodes is 2\n    } else {\n        // Get next node of node cur to be removed in in-order traversal\n        tmp := cur.Right\n        for tmp.Left != nil {\n            tmp = tmp.Left\n        }\n        // Recursively delete node tmp\n        bst.remove(tmp.Val.(int))\n        // Replace cur with tmp\n        cur.Val = tmp.Val\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* Remove node */\nfunc remove(num: Int) {\n    // If tree is empty, return directly\n    if root == nil {\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // Loop search, exit after passing leaf node\n    while cur != nil {\n        // Found node to delete, exit loop\n        if cur!.val == num {\n            break\n        }\n        pre = cur\n        // Node to delete is in cur's right subtree\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // Node to delete is in cur's left subtree\n        else {\n            cur = cur?.left\n        }\n    }\n    // If no node to delete, return directly\n    if cur == nil {\n        return\n    }\n    // Number of child nodes = 0 or 1\n    if cur?.left == nil || cur?.right == nil {\n        // When number of child nodes = 0 / 1, child = null / that child node\n        let child = cur?.left ?? cur?.right\n        // Delete node cur\n        if cur !== root {\n            if pre?.left === cur {\n                pre?.left = child\n            } else {\n                pre?.right = child\n            }\n        } else {\n            // If deleted node is root node, reassign root node\n            root = child\n        }\n    }\n    // Number of child nodes = 2\n    else {\n        // Get next node of cur in inorder traversal\n        var tmp = cur?.right\n        while tmp?.left != nil {\n            tmp = tmp?.left\n        }\n        // Recursively delete node tmp\n        remove(num: tmp!.val)\n        // Replace cur with tmp\n        cur?.val = tmp!.val\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* Remove node */\nremove(num) {\n    // If tree is empty, return directly\n    if (this.root === null) return;\n    let cur = this.root,\n        pre = null;\n    // Loop search, exit after passing leaf node\n    while (cur !== null) {\n        // Found node to delete, exit loop\n        if (cur.val === num) break;\n        pre = cur;\n        // Node to delete is in cur's right subtree\n        if (cur.val &lt; num) cur = cur.right;\n        // Node to delete is in cur's left subtree\n        else cur = cur.left;\n    }\n    // If no node to delete, return directly\n    if (cur === null) return;\n    // Number of child nodes = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // When number of child nodes = 0 / 1, child = null / that child node\n        const child = cur.left !== null ? cur.left : cur.right;\n        // Delete node cur\n        if (cur !== this.root) {\n            if (pre.left === cur) pre.left = child;\n            else pre.right = child;\n        } else {\n            // If deleted node is root node, reassign root node\n            this.root = child;\n        }\n    }\n    // Number of child nodes = 2\n    else {\n        // Get next node of cur in inorder traversal\n        let tmp = cur.right;\n        while (tmp.left !== null) {\n            tmp = tmp.left;\n        }\n        // Recursively delete node tmp\n        this.remove(tmp.val);\n        // Replace cur with tmp\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.ts<pre><code>/* Remove node */\nremove(num: number): void {\n    // If tree is empty, return directly\n    if (this.root === null) return;\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // Loop search, exit after passing leaf node\n    while (cur !== null) {\n        // Found node to delete, exit loop\n        if (cur.val === num) break;\n        pre = cur;\n        // Node to delete is in cur's right subtree\n        if (cur.val &lt; num) cur = cur.right;\n        // Node to delete is in cur's left subtree\n        else cur = cur.left;\n    }\n    // If no node to delete, return directly\n    if (cur === null) return;\n    // Number of child nodes = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // When number of child nodes = 0 / 1, child = null / that child node\n        const child: TreeNode | null =\n            cur.left !== null ? cur.left : cur.right;\n        // Delete node cur\n        if (cur !== this.root) {\n            if (pre!.left === cur) pre!.left = child;\n            else pre!.right = child;\n        } else {\n            // If deleted node is root node, reassign root node\n            this.root = child;\n        }\n    }\n    // Number of child nodes = 2\n    else {\n        // Get next node of cur in inorder traversal\n        let tmp: TreeNode | null = cur.right;\n        while (tmp!.left !== null) {\n            tmp = tmp!.left;\n        }\n        // Recursively delete node tmp\n        this.remove(tmp!.val);\n        // Replace cur with tmp\n        cur.val = tmp!.val;\n    }\n}\n</code></pre> binary_search_tree.dart<pre><code>/* Remove node */\nvoid remove(int _num) {\n  // If tree is empty, return directly\n  if (_root == null) return;\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // Loop search, exit after passing leaf node\n  while (cur != null) {\n    // Found node to delete, exit loop\n    if (cur.val == _num) break;\n    pre = cur;\n    // Node to delete is in cur's right subtree\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // Node to delete is in cur's left subtree\n    else\n      cur = cur.left;\n  }\n  // If no node to delete, return directly\n  if (cur == null) return;\n  // Number of child nodes = 0 or 1\n  if (cur.left == null || cur.right == null) {\n    // When number of child nodes = 0 / 1, child = null / that child node\n    TreeNode? child = cur.left ?? cur.right;\n    // Delete node cur\n    if (cur != _root) {\n      if (pre!.left == cur)\n        pre.left = child;\n      else\n        pre.right = child;\n    } else {\n      // If deleted node is root node, reassign root node\n      _root = child;\n    }\n  } else {\n    // Number of child nodes = 2\n    // Get next node of cur in inorder traversal\n    TreeNode? tmp = cur.right;\n    while (tmp!.left != null) {\n      tmp = tmp.left;\n    }\n    // Recursively delete node tmp\n    remove(tmp.val);\n    // Replace cur with tmp\n    cur.val = tmp.val;\n  }\n}\n</code></pre> binary_search_tree.rs<pre><code>/* Remove node */\npub fn remove(&amp;mut self, num: i32) {\n    // If tree is empty, return directly\n    if self.root.is_none() {\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // Loop search, exit after passing leaf node\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // Found node to delete, exit loop\n            Ordering::Equal =&gt; break,\n            // Node to delete is in cur's right subtree\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // Node to delete is in cur's left subtree\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // If no node to delete, return directly\n    if cur.is_none() {\n        return;\n    }\n    let cur = cur.unwrap();\n    let (left_child, right_child) = (cur.borrow().left.clone(), cur.borrow().right.clone());\n    match (left_child.clone(), right_child.clone()) {\n        // Number of child nodes = 0 or 1\n        (None, None) | (Some(_), None) | (None, Some(_)) =&gt; {\n            // When number of child nodes = 0 / 1, child = nullptr / that child node\n            let child = left_child.or(right_child);\n            let pre = pre.unwrap();\n            // Delete node cur\n            if !Rc::ptr_eq(&amp;cur, self.root.as_ref().unwrap()) {\n                let left = pre.borrow().left.clone();\n                if left.is_some() &amp;&amp; Rc::ptr_eq(left.as_ref().unwrap(), &amp;cur) {\n                    pre.borrow_mut().left = child;\n                } else {\n                    pre.borrow_mut().right = child;\n                }\n            } else {\n                // If deleted node is root node, reassign root node\n                self.root = child;\n            }\n        }\n        // Number of child nodes = 2\n        (Some(_), Some(_)) =&gt; {\n            // Get next node of cur in inorder traversal\n            let mut tmp = cur.borrow().right.clone();\n            while let Some(node) = tmp.clone() {\n                if node.borrow().left.is_some() {\n                    tmp = node.borrow().left.clone();\n                } else {\n                    break;\n                }\n            }\n            let tmp_val = tmp.unwrap().borrow().val;\n            // Recursively delete node tmp\n            self.remove(tmp_val);\n            // Replace cur with tmp\n            cur.borrow_mut().val = tmp_val;\n        }\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* Remove node */\n// Cannot use remove keyword here due to stdio.h inclusion\nvoid removeItem(BinarySearchTree *bst, int num) {\n    // If tree is empty, return directly\n    if (bst-&gt;root == NULL)\n        return;\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // Loop search, exit after passing leaf node\n    while (cur != NULL) {\n        // Found node to delete, exit loop\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // Node to delete is in right subtree of root\n            cur = cur-&gt;right;\n        } else {\n            // Node to delete is in left subtree of root\n            cur = cur-&gt;left;\n        }\n    }\n    // If no node to delete, return directly\n    if (cur == NULL)\n        return;\n    // Check if node to delete has children\n    if (cur-&gt;left == NULL || cur-&gt;right == NULL) {\n        /* Number of child nodes = 0 or 1 */\n        // When number of child nodes = 0 / 1, child = nullptr / that child node\n        TreeNode *child = cur-&gt;left != NULL ? cur-&gt;left : cur-&gt;right;\n        // Delete node cur\n        if (pre-&gt;left == cur) {\n            pre-&gt;left = child;\n        } else {\n            pre-&gt;right = child;\n        }\n        // Free memory\n        free(cur);\n    } else {\n        /* Number of child nodes = 2 */\n        // Get next node of cur in inorder traversal\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != NULL) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // Recursively delete node tmp\n        removeItem(bst, tmp-&gt;val);\n        // Replace cur with tmp\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* Remove node */\nfun remove(num: Int) {\n    // If tree is empty, return directly\n    if (root == null)\n        return\n    var cur = root\n    var pre: TreeNode? = null\n    // Loop search, exit after passing leaf node\n    while (cur != null) {\n        // Found node to delete, exit loop\n        if (cur._val == num)\n            break\n        pre = cur\n        // Node to delete is in cur's right subtree\n        cur = if (cur._val &lt; num)\n            cur.right\n        // Node to delete is in cur's left subtree\n        else\n            cur.left\n    }\n    // If no node to delete, return directly\n    if (cur == null)\n        return\n    // Number of child nodes = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // When number of child nodes = 0 / 1, child = null / that child node\n        val child = if (cur.left != null)\n            cur.left\n        else\n            cur.right\n        // Delete node cur\n        if (cur != root) {\n            if (pre!!.left == cur)\n                pre.left = child\n            else\n                pre.right = child\n        } else {\n            // If deleted node is root node, reassign root node\n            root = child\n        }\n        // Number of child nodes = 2\n    } else {\n        // Get next node of cur in inorder traversal\n        var tmp = cur.right\n        while (tmp!!.left != null) {\n            tmp = tmp.left\n        }\n        // Recursively delete node tmp\n        remove(tmp._val)\n        // Replace cur with tmp\n        cur._val = tmp._val\n    }\n}\n</code></pre> binary_search_tree.rb<pre><code>### Delete node ###\ndef remove(num)\n  # If tree is empty, return directly\n  return if @root.nil?\n\n  # Loop search, exit after passing leaf node\n  cur, pre = @root, nil\n  while !cur.nil?\n    # Found node to delete, exit loop\n    break if cur.val == num\n\n    pre = cur\n    # Node to delete is in cur's right subtree\n    if cur.val &lt; num\n      cur = cur.right\n    # Node to delete is in cur's left subtree\n    else\n      cur = cur.left\n    end\n  end\n  # If no node to delete, return directly\n  return if cur.nil?\n\n  # Number of child nodes = 0 or 1\n  if cur.left.nil? || cur.right.nil?\n    # When number of child nodes = 0 / 1, child = null / that child node\n    child = cur.left || cur.right\n    # Delete node cur\n    if cur != @root\n      if pre.left == cur\n        pre.left = child\n      else\n        pre.right = child\n      end\n    else\n      # If deleted node is root node, reassign root node\n      @root = child\n    end\n  # Number of child nodes = 2\n  else\n    # Get next node of cur in inorder traversal\n    tmp = cur.right\n    while !tmp.left.nil?\n      tmp = tmp.left\n    end\n    # Recursively delete node tmp\n    remove(tmp.val)\n    # Replace cur with tmp\n    cur.val = tmp.val\n  end\nend\n</code></pre>","path":["Chapter 7. Tree","7.4   Binary Search Tree"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#4-inorder-traversal-is-ordered","level":3,"title":"4.   Inorder Traversal Is Ordered","text":"<p>As shown in Figure 7-22, the inorder traversal of a binary tree follows the \"left \\(\\rightarrow\\) root \\(\\rightarrow\\) right\" traversal order, while the binary search tree satisfies the \"left child node \\(&lt;\\) root node \\(&lt;\\) right child node\" size relationship.</p> <p>This means that when performing an inorder traversal in a binary search tree, the next smallest node is always traversed first, thus yielding an important property: The inorder traversal sequence of a binary search tree is ascending.</p> <p>Using the property of inorder traversal being ascending, we can obtain ordered data in a binary search tree in only \\(O(n)\\) time, without the need for additional sorting operations, which is very efficient.</p> <p></p> <p> Figure 7-22   Inorder traversal sequence of a binary search tree </p>","path":["Chapter 7. Tree","7.4   Binary Search Tree"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#742-efficiency-of-binary-search-trees","level":2,"title":"7.4.2   Efficiency of Binary Search Trees","text":"<p>Given a set of data, we consider using an array or a binary search tree for storage. Observing Table 7-2, all operations in a binary search tree have logarithmic time complexity, providing stable and efficient performance. Arrays are more efficient than binary search trees only in scenarios with high-frequency additions and low-frequency searches and deletions.</p> <p> Table 7-2   Efficiency comparison between arrays and search trees </p> Unsorted array Binary search tree Search element \\(O(n)\\) \\(O(\\log n)\\) Insert element \\(O(1)\\) \\(O(\\log n)\\) Remove element \\(O(n)\\) \\(O(\\log n)\\) <p>In the ideal case, a binary search tree is balanced, so any node can be found within \\(O(\\log n)\\) loop iterations.</p> <p>However, if we continuously insert and remove nodes in a binary search tree, it may degenerate into a linked list as shown in Figure 7-23, where the time complexity of various operations also degrades to \\(O(n)\\).</p> <p></p> <p> Figure 7-23   Degradation of a binary search tree </p>","path":["Chapter 7. Tree","7.4   Binary Search Tree"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#743-common-applications-of-binary-search-trees","level":2,"title":"7.4.3   Common Applications of Binary Search Trees","text":"<ul> <li>Used as multi-level indexes in systems to implement efficient search, insertion, and removal operations.</li> <li>Serves as the underlying data structure for certain search algorithms.</li> <li>Used to store data streams to maintain their ordered state.</li> </ul>","path":["Chapter 7. Tree","7.4   Binary Search Tree"],"tags":[]},{"location":"chapter_tree/binary_tree/","level":1,"title":"7.1   Binary Tree","text":"<p>A binary tree is a non-linear data structure that models the hierarchical relationship between \"ancestors\" and \"descendants\" and embodies a divide-and-conquer pattern in which each split branches into two. Similar to a linked list, the basic unit of a binary tree is a node, and each node contains a value, a reference to its left child node, and a reference to its right child node.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class TreeNode:\n    \"\"\"Binary tree node\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # Node value\n        self.left: TreeNode | None = None  # Reference to left child node\n        self.right: TreeNode | None = None # Reference to right child node\n</code></pre> <pre><code>/* Binary tree node */\nstruct TreeNode {\n    int val;          // Node value\n    TreeNode *left;   // Pointer to left child node\n    TreeNode *right;  // Pointer to right child node\n    TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode {\n    int val;         // Node value\n    TreeNode left;   // Reference to left child node\n    TreeNode right;  // Reference to right child node\n    TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode(int? x) {\n    public int? val = x;    // Node value\n    public TreeNode? left;  // Reference to left child node\n    public TreeNode? right; // Reference to right child node\n}\n</code></pre> <pre><code>/* Binary tree node */\ntype TreeNode struct {\n    Val   int\n    Left  *TreeNode\n    Right *TreeNode\n}\n/* Constructor */\nfunc NewTreeNode(v int) *TreeNode {\n    return &amp;TreeNode{\n        Left:  nil, // Pointer to left child node\n        Right: nil, // Pointer to right child node\n        Val:   v,   // Node value\n    }\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode {\n    var val: Int // Node value\n    var left: TreeNode? // Reference to left child node\n    var right: TreeNode? // Reference to right child node\n\n    init(x: Int) {\n        val = x\n    }\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode {\n    val; // Node value\n    left; // Pointer to left child node\n    right; // Pointer to right child node\n    constructor(val, left, right) {\n        this.val = val === undefined ? 0 : val;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode {\n    val: number;\n    left: TreeNode | null;\n    right: TreeNode | null;\n\n    constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val; // Node value\n        this.left = left === undefined ? null : left; // Reference to left child node\n        this.right = right === undefined ? null : right; // Reference to right child node\n    }\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode {\n  int val;         // Node value\n  TreeNode? left;  // Reference to left child node\n  TreeNode? right; // Reference to right child node\n  TreeNode(this.val, [this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* Binary tree node */\nstruct TreeNode {\n    val: i32,                               // Node value\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // Reference to left child node\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // Reference to right child node\n}\n\nimpl TreeNode {\n    /* Constructor */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* Binary tree node */\ntypedef struct TreeNode {\n    int val;                // Node value\n    int height;             // Node height\n    struct TreeNode *left;  // Pointer to left child node\n    struct TreeNode *right; // Pointer to right child node\n} TreeNode;\n\n/* Constructor */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* Binary tree node */\nclass TreeNode(val _val: Int) {  // Node value\n    var left: TreeNode? = null   // Reference to left child node\n    var right: TreeNode? = null  // Reference to right child node\n}\n</code></pre> <pre><code>### Binary tree node class ###\nclass TreeNode\n  attr_accessor :val    # Node value\n  attr_accessor :left   # Reference to left child node\n  attr_accessor :right  # Reference to right child node\n\n  def initialize(val)\n    @val = val\n  end\nend\n</code></pre> <p>Each node has two references (pointers), pointing respectively to the left-child node and right-child node. This node is called the parent node of these two child nodes. When given a node of a binary tree, we call the tree formed by this node's left child and all nodes below it the left subtree of this node. Similarly, the right subtree can be defined.</p> <p>In a binary tree, every non-leaf node has child nodes and therefore non-empty subtrees. As shown in Figure 7-1, if \"Node 2\" is regarded as a parent node, its left and right child nodes are \"Node 4\" and \"Node 5\" respectively. The left subtree is formed by \"Node 4\" and all nodes beneath it, while the right subtree is formed by \"Node 5\" and all nodes beneath it.</p> <p></p> <p> Figure 7-1   Parent Node, child Node, subtree </p>","path":["Chapter 7. Tree","7.1   Binary Tree"],"tags":[]},{"location":"chapter_tree/binary_tree/#711-common-terminology-of-binary-trees","level":2,"title":"7.1.1   Common Terminology of Binary Trees","text":"<p>The commonly used terminology of binary trees is shown in Figure 7-2.</p> <ul> <li>Root node: The node at the top level of a binary tree, which does not have a parent node.</li> <li>Leaf node: A node that does not have any child nodes, with both of its pointers pointing to <code>None</code>.</li> <li>Edge: A line segment that connects two nodes, representing a reference (pointer) between the nodes.</li> <li>The level of a node: It increases from top to bottom, with the root node being at level 1.</li> <li>The degree of a node: The number of child nodes that a node has. In a binary tree, the degree can be 0, 1, or 2.</li> <li>The height of a binary tree: The number of edges from the root node to the farthest leaf node.</li> <li>The depth of a node: The number of edges from the root node to the node.</li> <li>The height of a node: The number of edges from the farthest leaf node to the node.</li> </ul> <p></p> <p> Figure 7-2   Common Terminology of Binary Trees </p> <p>Tip</p> <p>We usually define \"height\" and \"depth\" as the number of edges traversed, but some textbooks and problem statements define them as the number of nodes on the path. In that case, both values are larger by 1.</p>","path":["Chapter 7. Tree","7.1   Binary Tree"],"tags":[]},{"location":"chapter_tree/binary_tree/#712-basic-operations-of-binary-trees","level":2,"title":"7.1.2   Basic Operations of Binary Trees","text":"","path":["Chapter 7. Tree","7.1   Binary Tree"],"tags":[]},{"location":"chapter_tree/binary_tree/#1-initializing-a-binary-tree","level":3,"title":"1.   Initializing a Binary Tree","text":"<p>Similar to a linked list, the initialization of a binary tree involves first creating the nodes and then establishing the references (pointers) between them.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree.py<pre><code># Initializing a binary tree\n# Initializing nodes\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# Linking references (pointers) between nodes\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.cpp<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// Linking references (pointers) between nodes\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.java<pre><code>// Initializing nodes\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// Linking references (pointers) between nodes\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.cs<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nTreeNode n1 = new(1);\nTreeNode n2 = new(2);\nTreeNode n3 = new(3);\nTreeNode n4 = new(4);\nTreeNode n5 = new(5);\n// Linking references (pointers) between nodes\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.go<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// Linking references (pointers) between nodes\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n</code></pre> binary_tree.swift<pre><code>// Initializing nodes\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// Linking references (pointers) between nodes\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.js<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// Linking references (pointers) between nodes\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.ts<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// Linking references (pointers) between nodes\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.dart<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// Linking references (pointers) between nodes\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.rs<pre><code>// Initializing nodes\nlet n1 = TreeNode::new(1);\nlet n2 = TreeNode::new(2);\nlet n3 = TreeNode::new(3);\nlet n4 = TreeNode::new(4);\nlet n5 = TreeNode::new(5);\n// Linking references (pointers) between nodes\nn1.borrow_mut().left = Some(n2.clone());\nn1.borrow_mut().right = Some(n3);\nn2.borrow_mut().left = Some(n4);\nn2.borrow_mut().right = Some(n5);\n</code></pre> binary_tree.c<pre><code>/* Initializing a binary tree */\n// Initializing nodes\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// Linking references (pointers) between nodes\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.kt<pre><code>// Initializing nodes\nval n1 = TreeNode(1)\nval n2 = TreeNode(2)\nval n3 = TreeNode(3)\nval n4 = TreeNode(4)\nval n5 = TreeNode(5)\n// Linking references (pointers) between nodes\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.rb<pre><code># Initializing a binary tree\n# Initializing nodes\nn1 = TreeNode.new(1)\nn2 = TreeNode.new(2)\nn3 = TreeNode.new(3)\nn4 = TreeNode.new(4)\nn5 = TreeNode.new(5)\n# Linking references (pointers) between nodes\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> Code Visualization <p> Full Screen &gt;</p>","path":["Chapter 7. Tree","7.1   Binary Tree"],"tags":[]},{"location":"chapter_tree/binary_tree/#2-inserting-and-removing-nodes","level":3,"title":"2.   Inserting and Removing Nodes","text":"<p>Similar to a linked list, inserting and removing nodes in a binary tree can be achieved by modifying pointers. Figure 7-3 provides an example.</p> <p></p> <p> Figure 7-3   Inserting and removing nodes in a binary tree </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree.py<pre><code># Inserting and removing nodes\np = TreeNode(0)\n# Inserting node P between n1 -&gt; n2\nn1.left = p\np.left = n2\n# Removing node P\nn1.left = n2\n</code></pre> binary_tree.cpp<pre><code>/* Inserting and removing nodes */\nTreeNode* P = new TreeNode(0);\n// Inserting node P between n1 and n2\nn1-&gt;left = P;\nP-&gt;left = n2;\n// Removing node P\nn1-&gt;left = n2;\n</code></pre> binary_tree.java<pre><code>TreeNode P = new TreeNode(0);\n// Inserting node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Removing node P\nn1.left = n2;\n</code></pre> binary_tree.cs<pre><code>/* Inserting and removing nodes */\nTreeNode P = new(0);\n// Inserting node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Removing node P\nn1.left = n2;\n</code></pre> binary_tree.go<pre><code>/* Inserting and removing nodes */\n// Inserting node P between n1 and n2\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// Removing node P\nn1.Left = n2\n</code></pre> binary_tree.swift<pre><code>let P = TreeNode(x: 0)\n// Inserting node P between n1 and n2\nn1.left = P\nP.left = n2\n// Removing node P\nn1.left = n2\n</code></pre> binary_tree.js<pre><code>/* Inserting and removing nodes */\nlet P = new TreeNode(0);\n// Inserting node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Removing node P\nn1.left = n2;\n</code></pre> binary_tree.ts<pre><code>/* Inserting and removing nodes */\nconst P = new TreeNode(0);\n// Inserting node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Removing node P\nn1.left = n2;\n</code></pre> binary_tree.dart<pre><code>/* Inserting and removing nodes */\nTreeNode P = new TreeNode(0);\n// Inserting node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Removing node P\nn1.left = n2;\n</code></pre> binary_tree.rs<pre><code>let p = TreeNode::new(0);\n// Inserting node P between n1 and n2\nn1.borrow_mut().left = Some(p.clone());\np.borrow_mut().left = Some(n2.clone());\n// Removing node P\nn1.borrow_mut().left = Some(n2);\n</code></pre> binary_tree.c<pre><code>/* Inserting and removing nodes */\nTreeNode *P = newTreeNode(0);\n// Inserting node P between n1 and n2\nn1-&gt;left = P;\nP-&gt;left = n2;\n// Removing node P\nn1-&gt;left = n2;\n</code></pre> binary_tree.kt<pre><code>val P = TreeNode(0)\n// Inserting node P between n1 and n2\nn1.left = P\nP.left = n2\n// Removing node P\nn1.left = n2\n</code></pre> binary_tree.rb<pre><code># Inserting and removing nodes\n_p = TreeNode.new(0)\n# Inserting node _p between n1 and n2\nn1.left = _p\n_p.left = n2\n# Removing node _p\nn1.left = n2\n</code></pre> Code Visualization <p> Full Screen &gt;</p> <p>Tip</p> <p>Keep in mind that inserting a node can alter the original logical structure of a binary tree, while deleting a node usually entails removing that node together with its entire subtree. In practice, insertion and deletion in binary trees are therefore typically implemented as coordinated sequences of operations to achieve a meaningful result.</p>","path":["Chapter 7. Tree","7.1   Binary Tree"],"tags":[]},{"location":"chapter_tree/binary_tree/#713-common-types-of-binary-trees","level":2,"title":"7.1.3   Common Types of Binary Trees","text":"","path":["Chapter 7. Tree","7.1   Binary Tree"],"tags":[]},{"location":"chapter_tree/binary_tree/#1-perfect-binary-tree","level":3,"title":"1.   Perfect Binary Tree","text":"<p>As shown in Figure 7-4, a perfect binary tree has every level completely filled. In a perfect binary tree, leaf nodes have a degree of \\(0\\), while all other nodes have a degree of \\(2\\). If the tree height is \\(h\\), the total number of nodes is \\(2^{h+1} - 1\\), following a standard exponential pattern that mirrors the common phenomenon of cell division in nature.</p> <p>Tip</p> <p>Please note that in the Chinese community, a perfect binary tree is often referred to as a full binary tree.</p> <p></p> <p> Figure 7-4   Perfect binary tree </p>","path":["Chapter 7. Tree","7.1   Binary Tree"],"tags":[]},{"location":"chapter_tree/binary_tree/#2-complete-binary-tree","level":3,"title":"2.   Complete Binary Tree","text":"<p>As shown in Figure 7-5, a complete binary tree only allows the bottom level to be incompletely filled, and the nodes at the bottom level must be filled continuously from left to right. Note that a perfect binary tree is also a complete binary tree.</p> <p></p> <p> Figure 7-5   Complete binary tree </p>","path":["Chapter 7. Tree","7.1   Binary Tree"],"tags":[]},{"location":"chapter_tree/binary_tree/#3-full-binary-tree","level":3,"title":"3.   Full Binary Tree","text":"<p>As shown in Figure 7-6, in a full binary tree, all nodes except leaf nodes have two child nodes.</p> <p></p> <p> Figure 7-6   Full binary tree </p>","path":["Chapter 7. Tree","7.1   Binary Tree"],"tags":[]},{"location":"chapter_tree/binary_tree/#4-balanced-binary-tree","level":3,"title":"4.   Balanced Binary Tree","text":"<p>As shown in Figure 7-7, in a balanced binary tree, the absolute difference between the height of the left and right subtrees of any node does not exceed 1.</p> <p></p> <p> Figure 7-7   Balanced binary tree </p>","path":["Chapter 7. Tree","7.1   Binary Tree"],"tags":[]},{"location":"chapter_tree/binary_tree/#714-degeneration-of-binary-trees","level":2,"title":"7.1.4   Degeneration of Binary Trees","text":"<p>Figure 7-8 contrasts the ideal and degenerate structures of binary trees. When every level is filled, the tree becomes a \"perfect binary tree\"; when all nodes skew to one side, the binary tree degenerates into a \"linked list\".</p> <ul> <li>A perfect binary tree is the ideal case, fully leveraging the divide-and-conquer advantages of binary trees.</li> <li>A linked list represents the other extreme, where all operations become linear operations with time complexity degrading to \\(O(n)\\).</li> </ul> <p></p> <p> Figure 7-8   The Best and Worst Structures of Binary Trees </p> <p>As shown in Table 7-1, in the best and worst structures, the binary tree achieves either maximum or minimum values for leaf node count, total number of nodes, and height.</p> <p> Table 7-1   The Best and Worst Structures of Binary Trees </p> Perfect binary tree Linked list Number of nodes at level \\(i\\) \\(2^{i-1}\\) \\(1\\) Number of leaf nodes in a tree with height \\(h\\) \\(2^h\\) \\(1\\) Total number of nodes in a tree with height \\(h\\) \\(2^{h+1} - 1\\) \\(h + 1\\) Height of a tree with \\(n\\) total nodes \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)","path":["Chapter 7. Tree","7.1   Binary Tree"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/","level":1,"title":"7.2   Binary Tree Traversal","text":"<p>From a physical structure perspective, a tree is a data structure based on linked lists. Hence, its traversal method involves accessing nodes one by one through pointers. However, a tree is a non-linear data structure, which makes traversing a tree more complex than traversing a linked list, requiring the assistance of search algorithms.</p> <p>The common traversal methods for binary trees include level-order traversal, pre-order traversal, in-order traversal, and post-order traversal.</p>","path":["Chapter 7. Tree","7.2   Binary Tree Traversal"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#721-level-order-traversal","level":2,"title":"7.2.1   Level-Order Traversal","text":"<p>As shown in Figure 7-9, level-order traversal traverses the binary tree from top to bottom, layer by layer. Within each level, it visits nodes from left to right.</p> <p>Level-order traversal is essentially breadth-first traversal, also known as breadth-first search (BFS), which proceeds outward level by level.</p> <p></p> <p> Figure 7-9   Level-order traversal of a binary tree </p>","path":["Chapter 7. Tree","7.2   Binary Tree Traversal"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#1-code-implementation","level":3,"title":"1.   Code Implementation","text":"<p>Breadth-first traversal is typically implemented with the help of a \"queue\". The queue follows the \"first in, first out\" rule, while breadth-first traversal follows the \"layer-by-layer progression\" rule; the underlying ideas of the two are consistent. The implementation code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree_bfs.py<pre><code>def level_order(root: TreeNode | None) -&gt; list[int]:\n    \"\"\"Level-order traversal\"\"\"\n    # Initialize queue, add root node\n    queue: deque[TreeNode] = deque()\n    queue.append(root)\n    # Initialize a list to save the traversal sequence\n    res = []\n    while queue:\n        node: TreeNode = queue.popleft()  # Dequeue\n        res.append(node.val)  # Save node value\n        if node.left is not None:\n            queue.append(node.left)  # Left child node enqueue\n        if node.right is not None:\n            queue.append(node.right)  # Right child node enqueue\n    return res\n</code></pre> binary_tree_bfs.cpp<pre><code>/* Level-order traversal */\nvector&lt;int&gt; levelOrder(TreeNode *root) {\n    // Initialize queue, add root node\n    queue&lt;TreeNode *&gt; queue;\n    queue.push(root);\n    // Initialize a list to save the traversal sequence\n    vector&lt;int&gt; vec;\n    while (!queue.empty()) {\n        TreeNode *node = queue.front();\n        queue.pop();              // Dequeue\n        vec.push_back(node-&gt;val); // Save node value\n        if (node-&gt;left != nullptr)\n            queue.push(node-&gt;left); // Left child node enqueue\n        if (node-&gt;right != nullptr)\n            queue.push(node-&gt;right); // Right child node enqueue\n    }\n    return vec;\n}\n</code></pre> binary_tree_bfs.java<pre><code>/* Level-order traversal */\nList&lt;Integer&gt; levelOrder(TreeNode root) {\n    // Initialize queue, add root node\n    Queue&lt;TreeNode&gt; queue = new LinkedList&lt;&gt;();\n    queue.add(root);\n    // Initialize a list to save the traversal sequence\n    List&lt;Integer&gt; list = new ArrayList&lt;&gt;();\n    while (!queue.isEmpty()) {\n        TreeNode node = queue.poll(); // Dequeue\n        list.add(node.val);           // Save node value\n        if (node.left != null)\n            queue.offer(node.left);   // Left child node enqueue\n        if (node.right != null)\n            queue.offer(node.right);  // Right child node enqueue\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.cs<pre><code>/* Level-order traversal */\nList&lt;int&gt; LevelOrder(TreeNode root) {\n    // Initialize queue, add root node\n    Queue&lt;TreeNode&gt; queue = new();\n    queue.Enqueue(root);\n    // Initialize a list to save the traversal sequence\n    List&lt;int&gt; list = [];\n    while (queue.Count != 0) {\n        TreeNode node = queue.Dequeue(); // Dequeue\n        list.Add(node.val!.Value);       // Save node value\n        if (node.left != null)\n            queue.Enqueue(node.left);    // Left child node enqueue\n        if (node.right != null)\n            queue.Enqueue(node.right);   // Right child node enqueue\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.go<pre><code>/* Level-order traversal */\nfunc levelOrder(root *TreeNode) []any {\n    // Initialize queue, add root node\n    queue := list.New()\n    queue.PushBack(root)\n    // Initialize a slice to save traversal sequence\n    nums := make([]any, 0)\n    for queue.Len() &gt; 0 {\n        // Dequeue\n        node := queue.Remove(queue.Front()).(*TreeNode)\n        // Save node value\n        nums = append(nums, node.Val)\n        if node.Left != nil {\n            // Left child node enqueue\n            queue.PushBack(node.Left)\n        }\n        if node.Right != nil {\n            // Right child node enqueue\n            queue.PushBack(node.Right)\n        }\n    }\n    return nums\n}\n</code></pre> binary_tree_bfs.swift<pre><code>/* Level-order traversal */\nfunc levelOrder(root: TreeNode) -&gt; [Int] {\n    // Initialize queue, add root node\n    var queue: [TreeNode] = [root]\n    // Initialize a list to save the traversal sequence\n    var list: [Int] = []\n    while !queue.isEmpty {\n        let node = queue.removeFirst() // Dequeue\n        list.append(node.val) // Save node value\n        if let left = node.left {\n            queue.append(left) // Left child node enqueue\n        }\n        if let right = node.right {\n            queue.append(right) // Right child node enqueue\n        }\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.js<pre><code>/* Level-order traversal */\nfunction levelOrder(root) {\n    // Initialize queue, add root node\n    const queue = [root];\n    // Initialize a list to save the traversal sequence\n    const list = [];\n    while (queue.length) {\n        let node = queue.shift(); // Dequeue\n        list.push(node.val); // Save node value\n        if (node.left) queue.push(node.left); // Left child node enqueue\n        if (node.right) queue.push(node.right); // Right child node enqueue\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.ts<pre><code>/* Level-order traversal */\nfunction levelOrder(root: TreeNode | null): number[] {\n    // Initialize queue, add root node\n    const queue = [root];\n    // Initialize a list to save the traversal sequence\n    const list: number[] = [];\n    while (queue.length) {\n        let node = queue.shift() as TreeNode; // Dequeue\n        list.push(node.val); // Save node value\n        if (node.left) {\n            queue.push(node.left); // Left child node enqueue\n        }\n        if (node.right) {\n            queue.push(node.right); // Right child node enqueue\n        }\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.dart<pre><code>/* Level-order traversal */\nList&lt;int&gt; levelOrder(TreeNode? root) {\n  // Initialize queue, add root node\n  Queue&lt;TreeNode?&gt; queue = Queue();\n  queue.add(root);\n  // Initialize a list to save the traversal sequence\n  List&lt;int&gt; res = [];\n  while (queue.isNotEmpty) {\n    TreeNode? node = queue.removeFirst(); // Dequeue\n    res.add(node!.val); // Save node value\n    if (node.left != null) queue.add(node.left); // Left child node enqueue\n    if (node.right != null) queue.add(node.right); // Right child node enqueue\n  }\n  return res;\n}\n</code></pre> binary_tree_bfs.rs<pre><code>/* Level-order traversal */\nfn level_order(root: &amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    // Initialize queue, add root node\n    let mut que = VecDeque::new();\n    que.push_back(root.clone());\n    // Initialize a list to save the traversal sequence\n    let mut vec = Vec::new();\n\n    while let Some(node) = que.pop_front() {\n        // Dequeue\n        vec.push(node.borrow().val); // Save node value\n        if let Some(left) = node.borrow().left.as_ref() {\n            que.push_back(left.clone()); // Left child node enqueue\n        }\n        if let Some(right) = node.borrow().right.as_ref() {\n            que.push_back(right.clone()); // Right child node enqueue\n        };\n    }\n    vec\n}\n</code></pre> binary_tree_bfs.c<pre><code>/* Level-order traversal */\nint *levelOrder(TreeNode *root, int *size) {\n    /* Auxiliary queue */\n    int front, rear;\n    int index, *arr;\n    TreeNode *node;\n    TreeNode **queue;\n\n    /* Auxiliary queue */\n    queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);\n    // Queue pointer\n    front = 0, rear = 0;\n    // Add root node\n    queue[rear++] = root;\n    // Initialize a list to save the traversal sequence\n    /* Auxiliary array */\n    arr = (int *)malloc(sizeof(int) * MAX_SIZE);\n    // Array pointer\n    index = 0;\n    while (front &lt; rear) {\n        // Dequeue\n        node = queue[front++];\n        // Save node value\n        arr[index++] = node-&gt;val;\n        if (node-&gt;left != NULL) {\n            // Left child node enqueue\n            queue[rear++] = node-&gt;left;\n        }\n        if (node-&gt;right != NULL) {\n            // Right child node enqueue\n            queue[rear++] = node-&gt;right;\n        }\n    }\n    // Update array length value\n    *size = index;\n    arr = realloc(arr, sizeof(int) * (*size));\n\n    // Free auxiliary array space\n    free(queue);\n    return arr;\n}\n</code></pre> binary_tree_bfs.kt<pre><code>/* Level-order traversal */\nfun levelOrder(root: TreeNode?): MutableList&lt;Int&gt; {\n    // Initialize queue, add root node\n    val queue = LinkedList&lt;TreeNode?&gt;()\n    queue.add(root)\n    // Initialize a list to save the traversal sequence\n    val list = mutableListOf&lt;Int&gt;()\n    while (queue.isNotEmpty()) {\n        val node = queue.poll()      // Dequeue\n        list.add(node?._val!!)       // Save node value\n        if (node.left != null)\n            queue.offer(node.left)   // Left child node enqueue\n        if (node.right != null)\n            queue.offer(node.right)  // Right child node enqueue\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.rb<pre><code>### Level-order traversal ###\ndef level_order(root)\n  # Initialize queue, add root node\n  queue = [root]\n  # Initialize a list to save the traversal sequence\n  res = []\n  while !queue.empty?\n    node = queue.shift # Dequeue\n    res &lt;&lt; node.val # Save node value\n    queue &lt;&lt; node.left unless node.left.nil? # Left child node enqueue\n    queue &lt;&lt; node.right unless node.right.nil? # Right child node enqueue\n  end\n  res\nend\n</code></pre>","path":["Chapter 7. Tree","7.2   Binary Tree Traversal"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#2-complexity-analysis","level":3,"title":"2.   Complexity Analysis","text":"<ul> <li>Time complexity is \\(O(n)\\): All nodes are visited once, using \\(O(n)\\) time, where \\(n\\) is the number of nodes.</li> <li>Space complexity is \\(O(n)\\): In the worst case, i.e., a full binary tree, before traversing to the bottom level, the queue contains at most \\((n + 1) / 2\\) nodes simultaneously, occupying \\(O(n)\\) space.</li> </ul>","path":["Chapter 7. Tree","7.2   Binary Tree Traversal"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#722-preorder-inorder-and-postorder-traversal","level":2,"title":"7.2.2   Preorder, Inorder, and Postorder Traversal","text":"<p>Correspondingly, preorder, inorder, and postorder traversals all belong to depth-first traversal, also known as depth-first search (DFS), which goes as deep as possible before backtracking.</p> <p>Figure 7-10 shows how depth-first traversal works on a binary tree. Depth-first traversal is like \"walking\" around the perimeter of the entire binary tree, encountering three positions at each node, corresponding to preorder, inorder, and postorder traversal.</p> <p></p> <p> Figure 7-10   Preorder, inorder, and postorder traversal of a binary tree </p>","path":["Chapter 7. Tree","7.2   Binary Tree Traversal"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#1-code-implementation_1","level":3,"title":"1.   Code Implementation","text":"<p>Depth-first search is usually implemented based on recursion:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree_dfs.py<pre><code>def pre_order(root: TreeNode | None):\n    \"\"\"Preorder traversal\"\"\"\n    if root is None:\n        return\n    # Visit priority: root node -&gt; left subtree -&gt; right subtree\n    res.append(root.val)\n    pre_order(root=root.left)\n    pre_order(root=root.right)\n\ndef in_order(root: TreeNode | None):\n    \"\"\"Inorder traversal\"\"\"\n    if root is None:\n        return\n    # Visit priority: left subtree -&gt; root node -&gt; right subtree\n    in_order(root=root.left)\n    res.append(root.val)\n    in_order(root=root.right)\n\ndef post_order(root: TreeNode | None):\n    \"\"\"Postorder traversal\"\"\"\n    if root is None:\n        return\n    # Visit priority: left subtree -&gt; right subtree -&gt; root node\n    post_order(root=root.left)\n    post_order(root=root.right)\n    res.append(root.val)\n</code></pre> binary_tree_dfs.cpp<pre><code>/* Preorder traversal */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // Visit priority: root node -&gt; left subtree -&gt; right subtree\n    vec.push_back(root-&gt;val);\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n\n/* Inorder traversal */\nvoid inOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // Visit priority: left subtree -&gt; root node -&gt; right subtree\n    inOrder(root-&gt;left);\n    vec.push_back(root-&gt;val);\n    inOrder(root-&gt;right);\n}\n\n/* Postorder traversal */\nvoid postOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // Visit priority: left subtree -&gt; right subtree -&gt; root node\n    postOrder(root-&gt;left);\n    postOrder(root-&gt;right);\n    vec.push_back(root-&gt;val);\n}\n</code></pre> binary_tree_dfs.java<pre><code>/* Preorder traversal */\nvoid preOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // Visit priority: root node -&gt; left subtree -&gt; right subtree\n    list.add(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* Inorder traversal */\nvoid inOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // Visit priority: left subtree -&gt; root node -&gt; right subtree\n    inOrder(root.left);\n    list.add(root.val);\n    inOrder(root.right);\n}\n\n/* Postorder traversal */\nvoid postOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // Visit priority: left subtree -&gt; right subtree -&gt; root node\n    postOrder(root.left);\n    postOrder(root.right);\n    list.add(root.val);\n}\n</code></pre> binary_tree_dfs.cs<pre><code>/* Preorder traversal */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) return;\n    // Visit priority: root node -&gt; left subtree -&gt; right subtree\n    list.Add(root.val!.Value);\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n\n/* Inorder traversal */\nvoid InOrder(TreeNode? root) {\n    if (root == null) return;\n    // Visit priority: left subtree -&gt; root node -&gt; right subtree\n    InOrder(root.left);\n    list.Add(root.val!.Value);\n    InOrder(root.right);\n}\n\n/* Postorder traversal */\nvoid PostOrder(TreeNode? root) {\n    if (root == null) return;\n    // Visit priority: left subtree -&gt; right subtree -&gt; root node\n    PostOrder(root.left);\n    PostOrder(root.right);\n    list.Add(root.val!.Value);\n}\n</code></pre> binary_tree_dfs.go<pre><code>/* Preorder traversal */\nfunc preOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // Visit priority: root node -&gt; left subtree -&gt; right subtree\n    nums = append(nums, node.Val)\n    preOrder(node.Left)\n    preOrder(node.Right)\n}\n\n/* Inorder traversal */\nfunc inOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // Visit priority: left subtree -&gt; root node -&gt; right subtree\n    inOrder(node.Left)\n    nums = append(nums, node.Val)\n    inOrder(node.Right)\n}\n\n/* Postorder traversal */\nfunc postOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // Visit priority: left subtree -&gt; right subtree -&gt; root node\n    postOrder(node.Left)\n    postOrder(node.Right)\n    nums = append(nums, node.Val)\n}\n</code></pre> binary_tree_dfs.swift<pre><code>/* Preorder traversal */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // Visit priority: root node -&gt; left subtree -&gt; right subtree\n    list.append(root.val)\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n\n/* Inorder traversal */\nfunc inOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // Visit priority: left subtree -&gt; root node -&gt; right subtree\n    inOrder(root: root.left)\n    list.append(root.val)\n    inOrder(root: root.right)\n}\n\n/* Postorder traversal */\nfunc postOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // Visit priority: left subtree -&gt; right subtree -&gt; root node\n    postOrder(root: root.left)\n    postOrder(root: root.right)\n    list.append(root.val)\n}\n</code></pre> binary_tree_dfs.js<pre><code>/* Preorder traversal */\nfunction preOrder(root) {\n    if (root === null) return;\n    // Visit priority: root node -&gt; left subtree -&gt; right subtree\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* Inorder traversal */\nfunction inOrder(root) {\n    if (root === null) return;\n    // Visit priority: left subtree -&gt; root node -&gt; right subtree\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* Postorder traversal */\nfunction postOrder(root) {\n    if (root === null) return;\n    // Visit priority: left subtree -&gt; right subtree -&gt; root node\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.ts<pre><code>/* Preorder traversal */\nfunction preOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // Visit priority: root node -&gt; left subtree -&gt; right subtree\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* Inorder traversal */\nfunction inOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // Visit priority: left subtree -&gt; root node -&gt; right subtree\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* Postorder traversal */\nfunction postOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // Visit priority: left subtree -&gt; right subtree -&gt; root node\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.dart<pre><code>/* Preorder traversal */\nvoid preOrder(TreeNode? node) {\n  if (node == null) return;\n  // Visit priority: root node -&gt; left subtree -&gt; right subtree\n  list.add(node.val);\n  preOrder(node.left);\n  preOrder(node.right);\n}\n\n/* Inorder traversal */\nvoid inOrder(TreeNode? node) {\n  if (node == null) return;\n  // Visit priority: left subtree -&gt; root node -&gt; right subtree\n  inOrder(node.left);\n  list.add(node.val);\n  inOrder(node.right);\n}\n\n/* Postorder traversal */\nvoid postOrder(TreeNode? node) {\n  if (node == null) return;\n  // Visit priority: left subtree -&gt; right subtree -&gt; root node\n  postOrder(node.left);\n  postOrder(node.right);\n  list.add(node.val);\n}\n</code></pre> binary_tree_dfs.rs<pre><code>/* Preorder traversal */\nfn pre_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // Visit priority: root node -&gt; left subtree -&gt; right subtree\n            let node = node.borrow();\n            res.push(node.val);\n            dfs(node.left.as_ref(), res);\n            dfs(node.right.as_ref(), res);\n        }\n    }\n    dfs(root, &amp;mut result);\n\n    result\n}\n\n/* Inorder traversal */\nfn in_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // Visit priority: left subtree -&gt; root node -&gt; right subtree\n            let node = node.borrow();\n            dfs(node.left.as_ref(), res);\n            res.push(node.val);\n            dfs(node.right.as_ref(), res);\n        }\n    }\n    dfs(root, &amp;mut result);\n\n    result\n}\n\n/* Postorder traversal */\nfn post_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // Visit priority: left subtree -&gt; right subtree -&gt; root node\n            let node = node.borrow();\n            dfs(node.left.as_ref(), res);\n            dfs(node.right.as_ref(), res);\n            res.push(node.val);\n        }\n    }\n\n    dfs(root, &amp;mut result);\n\n    result\n}\n</code></pre> binary_tree_dfs.c<pre><code>/* Preorder traversal */\nvoid preOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // Visit priority: root node -&gt; left subtree -&gt; right subtree\n    arr[(*size)++] = root-&gt;val;\n    preOrder(root-&gt;left, size);\n    preOrder(root-&gt;right, size);\n}\n\n/* Inorder traversal */\nvoid inOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // Visit priority: left subtree -&gt; root node -&gt; right subtree\n    inOrder(root-&gt;left, size);\n    arr[(*size)++] = root-&gt;val;\n    inOrder(root-&gt;right, size);\n}\n\n/* Postorder traversal */\nvoid postOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // Visit priority: left subtree -&gt; right subtree -&gt; root node\n    postOrder(root-&gt;left, size);\n    postOrder(root-&gt;right, size);\n    arr[(*size)++] = root-&gt;val;\n}\n</code></pre> binary_tree_dfs.kt<pre><code>/* Preorder traversal */\nfun preOrder(root: TreeNode?) {\n    if (root == null) return\n    // Visit priority: root node -&gt; left subtree -&gt; right subtree\n    list.add(root._val)\n    preOrder(root.left)\n    preOrder(root.right)\n}\n\n/* Inorder traversal */\nfun inOrder(root: TreeNode?) {\n    if (root == null) return\n    // Visit priority: left subtree -&gt; root node -&gt; right subtree\n    inOrder(root.left)\n    list.add(root._val)\n    inOrder(root.right)\n}\n\n/* Postorder traversal */\nfun postOrder(root: TreeNode?) {\n    if (root == null) return\n    // Visit priority: left subtree -&gt; right subtree -&gt; root node\n    postOrder(root.left)\n    postOrder(root.right)\n    list.add(root._val)\n}\n</code></pre> binary_tree_dfs.rb<pre><code>### Pre-order traversal ###\ndef pre_order(root)\n  return if root.nil?\n\n  # Visit priority: root node -&gt; left subtree -&gt; right subtree\n  $res &lt;&lt; root.val\n  pre_order(root.left)\n  pre_order(root.right)\nend\n\n### In-order traversal ###\ndef in_order(root)\n  return if root.nil?\n\n  # Visit priority: left subtree -&gt; root node -&gt; right subtree\n  in_order(root.left)\n  $res &lt;&lt; root.val\n  in_order(root.right)\nend\n\n### Post-order traversal ###\ndef post_order(root)\n  return if root.nil?\n\n  # Visit priority: left subtree -&gt; right subtree -&gt; root node\n  post_order(root.left)\n  post_order(root.right)\n  $res &lt;&lt; root.val\nend\n</code></pre> <p>Tip</p> <p>Depth-first search can also be implemented iteratively, and interested readers can explore this on their own.</p> <p>Figure 7-11 shows the recursive process of preorder traversal of a binary tree, which can be divided into two opposite phases: \"descending\" and \"returning\".</p> <ol> <li>\"Descending\" means making a new recursive call, during which the program visits the next node.</li> <li>\"Returning\" means the function call returns, indicating that the current node has been fully processed.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 7-11   The recursive process of preorder traversal </p>","path":["Chapter 7. Tree","7.2   Binary Tree Traversal"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#2-complexity-analysis_1","level":3,"title":"2.   Complexity Analysis","text":"<ul> <li>Time complexity is \\(O(n)\\): All nodes are visited once, using \\(O(n)\\) time.</li> <li>Space complexity is \\(O(n)\\): In the worst case, i.e., the tree degenerates into a linked list, the recursion depth reaches \\(n\\), and the system occupies \\(O(n)\\) stack frame space.</li> </ul>","path":["Chapter 7. Tree","7.2   Binary Tree Traversal"],"tags":[]},{"location":"chapter_tree/exercises/","level":1,"title":"7.7   Exercises","text":"","path":["Chapter 7. Tree","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#771-concept-review","level":2,"title":"7.7.1   Concept Review","text":"","path":["Chapter 7. Tree","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#1-complete-full-and-perfect-binary-trees","level":3,"title":"1.   Complete, Full, and Perfect Binary Trees","text":"<p>The following two arrays represent binary trees in level order, where <code>None</code> marks an empty position:</p> <ul> <li>Tree A: <code>[1, 2, 3, 4, 5, 6]</code></li> <li>Tree B: <code>[1, 2, 3, None, None, 6, 7]</code></li> </ul> <ol> <li>Which tree is a complete binary tree?</li> <li>Which tree is a full binary tree, meaning every non-leaf node has two children?</li> <li>Is either tree a perfect binary tree? Explain the reason for each tree.</li> </ol> Answer <ol> <li> <p>Tree A is a complete binary tree. Only its lowest level is not full, and the nodes on that level occupy consecutive positions from left to right.     Tree B is not complete because there are empty positions on the left of the lowest level while nodes still appear on the right.</p> </li> <li> <p>Tree B is a full binary tree: nodes 1 and 3 each have two children, and all other nodes are leaves.     Tree A is not full because node 3 has only one child, its left child 6.</p> </li> <li> <p>Neither tree is perfect because the lowest level of each tree is not completely filled.</p> </li> </ol>","path":["Chapter 7. Tree","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#2-three-traversal-orders-for-the-same-tree","level":3,"title":"2.   Three Traversal Orders for the Same Tree","text":"<p>Store the array <code>[1, 2, 3, 4, 5, 6, 7]</code> in level order in a complete binary tree.</p> <ol> <li>Draw the tree.</li> <li>Write its preorder, inorder, and postorder traversal sequences.</li> <li>In the inorder sequence, which parts of the tree correspond to the subsequences to the left and right of root node 1?</li> </ol> Answer <ol> <li> <p>The tree is:</p> <pre><code>      1\n    /   \\\n   2     3\n  / \\   / \\\n 4   5 6   7\n</code></pre> </li> <li> <p>The preorder traversal is <code>1, 2, 4, 5, 3, 6, 7</code>;     the inorder traversal is <code>4, 2, 5, 1, 6, 3, 7</code>;     the postorder traversal is <code>4, 5, 2, 6, 7, 3, 1</code>.</p> </li> <li> <p>The sequence <code>4, 2, 5</code> to the left of root node 1 is the inorder traversal of the left subtree;     the sequence <code>6, 3, 7</code> to its right is the inorder traversal of the right subtree.</p> </li> </ol>","path":["Chapter 7. Tree","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#3-compare-two-binary-search-trees","level":3,"title":"3.   Compare Two Binary Search Trees","text":"<p>Insert each of the following sequences from left to right into an empty binary search tree:</p> <ul> <li>Sequence A: <code>[4, 2, 6, 1, 3, 5, 7]</code></li> <li>Sequence B: <code>[1, 2, 3, 4, 5, 6, 7]</code></li> </ul> <ol> <li>For each tree, write the nodes visited when searching for 7.</li> <li>If height is measured by the number of edges from the root node to the farthest leaf node, what is the height of each tree?</li> <li>Based on the first two questions, is searching for 7 equally efficient in the two trees? Explain using the trees' shapes and search paths.</li> </ol> Answer <ol> <li> <p>In the tree built from Sequence A, the search path is <code>4 → 6 → 7</code>.     In the tree built from Sequence B, the search path is <code>1 → 2 → 3 → 4 → 5 → 6 → 7</code>.</p> </li> <li> <p>Every level of the first tree is full, and its height is 2. The second tree has only right children, and its height is 6.</p> </li> <li> <p>Searching for 7 is not equally efficient in the two trees. The insertion order changes the shape and height of a binary search tree. The search visits only 3 nodes in the first tree     but all 7 nodes in the second. The taller the tree, the more nodes may need to be compared along a path in the worst case.</p> </li> </ol>","path":["Chapter 7. Tree","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#772-programming-exercises","level":2,"title":"7.7.2   Programming Exercises","text":"","path":["Chapter 7. Tree","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#1-maximum-depth-of-a-binary-tree","level":3,"title":"1.   Maximum Depth of a Binary Tree","text":"<p>You are given the root node <code>root</code> of a binary tree. Each node contains an integer value and references to its left and right children.</p> <p>The maximum depth is the number of nodes on the path from the root node to the farthest leaf node. Return the maximum depth of the tree; the maximum depth of an empty tree is 0. Use recursion.</p> Hints <ol> <li>Depth is measured by the number of nodes in this exercise, so a tree containing only a root node has a maximum depth of 1</li> <li>Let the recursive function return the maximum depth of the subtree rooted at the current node</li> <li>Return 0 for an empty node; for a nonempty node, return max(depth(left), depth(right)) + 1</li> </ol> <p>LeetCode</p>","path":["Chapter 7. Tree","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#2-traverse-a-binary-tree-level-by-level","level":3,"title":"2.   Traverse a Binary Tree Level by Level","text":"<p>Given the root node <code>root</code> of a binary tree, use a queue to visit all nodes level by level from top to bottom and from left to right within each level.</p> <p>Return a two-dimensional array: the first subarray stores the values at the root's level, the second stores the values at the next level, and so on. If the tree is empty, return an empty array.</p> Hints <ol> <li>Level-order traversal visits earlier enqueued nodes first, so use a queue</li> <li>At the beginning of each round, all nodes currently in the queue belong to the same level</li> <li>First record the queue's length, then remove exactly that many nodes and enqueue their children</li> </ol> <p>LeetCode</p>","path":["Chapter 7. Tree","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#3-kth-smallest-element-in-a-binary-search-tree","level":3,"title":"3.   Kth Smallest Element in a Binary Search Tree","text":"<p>A binary search tree contains <code>n</code> nodes with distinct values. If all node values are arranged from smallest to largest, their positions are numbered starting from 1.</p> <p>Given the root node <code>root</code> and an integer <code>k</code> satisfying <code>1 &lt;= k &lt;= n</code>, return the value at position <code>k</code>. Find the answer directly during an inorder traversal rather than collecting all node values first.</p> Hints <ol> <li>An inorder traversal of a binary search tree visits node values from smallest to largest</li> <li>Inorder traversal processes the left subtree, the current node, and then the right subtree; increment the count when visiting the current node</li> <li>When the count first equals k, the current node's value is the answer, so no further traversal is needed</li> </ol> <p>LeetCode</p>","path":["Chapter 7. Tree","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/summary/","level":1,"title":"7.6   Summary","text":"","path":["Chapter 7. Tree","7.6   Summary"],"tags":[]},{"location":"chapter_tree/summary/#1-key-review","level":3,"title":"1.   Key Review","text":"<ul> <li>A binary tree is a non-linear data structure that embodies the divide-and-conquer logic of splitting into two. Each binary tree node contains a value and two pointers, which point to its left and right child nodes.</li> <li>For a certain node in a binary tree, the tree formed by its left (right) child node and all nodes below is called the left (right) subtree of that node.</li> <li>Related terminology of binary trees includes root node, leaf node, level, degree, edge, height, and depth.</li> <li>The initialization, node insertion, and node removal operations of binary trees are similar to those of linked lists.</li> <li>Common types of binary trees include perfect binary trees, complete binary trees, full binary trees, and balanced binary trees. A perfect binary tree is the ideal form, while a linked list represents the worst degenerate case.</li> <li>A binary tree can be represented using an array by arranging node values and empty slots in level-order traversal sequence, and implementing pointers based on the index mapping relationship between parent and child nodes.</li> <li>Level-order traversal of a binary tree is a breadth-first search method that proceeds level by level, typically implemented using a queue.</li> <li>Preorder, inorder, and postorder traversals all belong to depth-first search, which proceeds by going as deep as possible before backtracking, typically using recursion.</li> <li>A binary search tree is an efficient data structure for element searching, with search, insertion, and removal operations all having time complexity of \\(O(\\log n)\\). When a binary search tree degenerates into a linked list, all time complexities degrade to \\(O(n)\\).</li> <li>An AVL tree, also known as a balanced binary search tree, ensures the tree remains balanced after continuous node insertions and removals through rotation operations.</li> <li>Rotation operations in AVL trees include right rotation, left rotation, right rotation followed by left rotation, and left rotation followed by right rotation. After inserting or removing nodes, AVL trees perform rotations from bottom to top to restore balance.</li> </ul>","path":["Chapter 7. Tree","7.6   Summary"],"tags":[]},{"location":"chapter_tree/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: For a binary tree with only one node, are both the height of the tree and the depth of the root node \\(0\\)?</p> <p>Yes, because height and depth are typically defined as the number of edges on the path.</p> <p>Q: The insertion and removal in a binary tree are generally accomplished by a set of operations. What does \"a set of operations\" refer to here? Does it imply releasing the resources of the child nodes?</p> <p>Taking the binary search tree as an example, the operation of removing a node needs to be handled in three different scenarios, each requiring multiple steps of node operations.</p> <p>Q: Why does DFS traversal of binary trees have three orders: preorder, inorder, and postorder, and what are their uses?</p> <p>Similar to forward and reverse traversal of arrays, preorder, inorder, and postorder traversals are three methods of binary tree traversal that allow us to obtain a traversal result in a specific order. For example, in a binary search tree, since nodes satisfy the relationship <code>left child node value &lt; root node value &lt; right child node value</code>, we only need to traverse the tree with the priority of \"left \\(\\rightarrow\\) root \\(\\rightarrow\\) right\" to obtain an ordered node sequence.</p> <p>Q: In a right rotation operation handling the relationship between unbalanced nodes <code>node</code>, <code>child</code>, and <code>grand_child</code>, doesn't the connection between <code>node</code> and its parent node get lost after the right rotation?</p> <p>We need to view this problem from a recursive perspective. The right rotation operation <code>right_rotate(root)</code> passes in the root node of the subtree and eventually returns the root node of the subtree after rotation with <code>return child</code>. The connection between the subtree's root node and its parent node is completed after the function returns, which is not within the maintenance scope of the right rotation operation.</p> <p>Q: In C++, functions are divided into <code>private</code> and <code>public</code> sections. What considerations are there for this? Why are the <code>height()</code> function and the <code>updateHeight()</code> function placed in <code>public</code> and <code>private</code>, respectively?</p> <p>It mainly depends on the method's usage scope. If a method is only used within the class, then it is designed as <code>private</code>. For example, calling <code>updateHeight()</code> alone by the user makes no sense, as it is only a step in insertion or removal operations. However, <code>height()</code> is used to access node height, similar to <code>vector.size()</code>, so it is set to <code>public</code> for ease of use.</p> <p>Q: How do you build a binary search tree from a set of input data? Is the choice of root node very important?</p> <p>Yes, the method for building a tree is provided in the <code>build_tree()</code> method in the binary search tree code. As for the choice of root node, we typically sort the input data, then select the middle element as the root node, and recursively build the left and right subtrees. This approach maximizes the tree's balance.</p> <p>Q: In Java, do you always have to use the <code>equals()</code> method for string comparison?</p> <p>In Java, for primitive data types, <code>==</code> is used to compare whether the values of two variables are equal. For reference types, the working principles of the two symbols are different.</p> <ul> <li><code>==</code>: Used to compare whether two variables point to the same object, i.e., whether their positions in memory are the same.</li> <li><code>equals()</code>: Used to compare whether the values of two objects are equal.</li> </ul> <p>Therefore, if we want to compare values, we should use <code>equals()</code>. However, strings initialized via <code>String a = \"hi\"; String b = \"hi\";</code> are stored in the string constant pool and point to the same object, so <code>a == b</code> can also be used to compare the contents of the two strings.</p> <p>Q: Before reaching the bottom level, is the number of nodes in the queue \\(2^h\\) in breadth-first traversal?</p> <p>Yes, for example, a full binary tree with height \\(h = 2\\) has a total of \\(n = 7\\) nodes, then the bottom level has \\(4 = 2^h = (n + 1) / 2\\) nodes.</p>","path":["Chapter 7. Tree","7.6   Summary"],"tags":[]}]}
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If you find typos, broken links, missing content, ambiguous wording, unclear explanations, or poorly organized passages, please help us correct them so that we can provide a better learning resource.</p> <p>To thank everyone for their generous contributions to the open-source community, the GitHub IDs of all contributors are displayed on the home pages of the book's repository, website, and PDF edition.</p> <p>The beauty of open source</p> <p>Printed books often have long intervals between print runs, making their content difficult to update.</p> <p>In this open-source book, updates can be made within days or even hours.</p>","path":["Chapter 16.   Appendix","16.2   Contribute to the book"],"tags":[]},{"location":"chapter_appendix/contribution/#1-small-edits","level":3,"title":"1.   Small edits","text":"<p>As shown in Figure 16-3, each page has an edit icon in its upper-right corner. Follow these steps to edit text or code.</p> <ol> <li>Click the edit icon. If you are prompted to fork the repository, agree to do so.</li> <li>Edit the Markdown source file, check that the content is correct, and keep the formatting as consistent as possible.</li> <li>Enter a description of your changes at the bottom of the page, then click \"Propose file change\". On the next page, click \"Create pull request\" to submit your pull request.</li> </ol> <p></p> <p> Figure 16-3   Page edit button </p> <p>Images cannot be edited directly. Please describe the problem in a new issue or a comment, and we will redraw and replace the image as soon as possible.</p>","path":["Chapter 16.   Appendix","16.2   Contribute to the book"],"tags":[]},{"location":"chapter_appendix/contribution/#2-creating-content","level":3,"title":"2.   Creating content","text":"<p>If you would like to contribute to this open-source project, for example by translating code into another programming language or expanding an article, follow this pull request workflow.</p> <ol> <li>Sign in to GitHub and fork the book's code repository to your own account.</li> <li>Open your fork's repository page and use <code>git clone</code> to clone it to your computer.</li> <li>Create your content locally and test it thoroughly to verify that the code is correct.</li> <li>Commit your local changes, then push them to your remote repository.</li> <li>Refresh the repository page and click \"Create pull request\" to submit your pull request.</li> </ol>","path":["Chapter 16.   Appendix","16.2   Contribute to the book"],"tags":[]},{"location":"chapter_appendix/contribution/#3-deploying-with-docker","level":3,"title":"3.   Deploying with Docker","text":"<p>Run the following Docker command in the <code>hello-algo</code> root directory to access the English edition at <code>http://localhost:8000/en/</code>:</p> <pre><code>docker-compose up -d\n</code></pre> <p>Use the following command to remove the deployment:</p> <pre><code>docker-compose down\n</code></pre>","path":["Chapter 16.   Appendix","16.2   Contribute to the book"],"tags":[]},{"location":"chapter_appendix/installation/","level":1,"title":"16.1   Setting up a programming environment","text":"","path":["Chapter 16.   Appendix","16.1   Setting up a programming environment"],"tags":[]},{"location":"chapter_appendix/installation/#1611-install-an-ide","level":2,"title":"16.1.1   Install an IDE","text":"<p>We recommend VS Code, a lightweight, open-source integrated development environment (IDE). Visit the VS Code website, then download and install the version for your operating system.</p> <p></p> <p> Figure 16-1   Downloading VS Code from the official website </p> <p>VS Code has an extensive ecosystem of extensions that support running and debugging code in most programming languages. For example, installing the Python Extension Pack lets you debug Python code. The installation steps are shown in Figure 16-2.</p> <p></p> <p> Figure 16-2   Installing VS Code extensions </p>","path":["Chapter 16.   Appendix","16.1   Setting up a programming environment"],"tags":[]},{"location":"chapter_appendix/installation/#1612-install-a-language-environment","level":2,"title":"16.1.2   Install a language environment","text":"","path":["Chapter 16.   Appendix","16.1   Setting up a programming environment"],"tags":[]},{"location":"chapter_appendix/installation/#1-python","level":3,"title":"1.   Python","text":"<ol> <li>Download and install Miniconda3. Python 3.10 or later is required.</li> <li>Search for <code>python</code> in the VS Code Extensions Marketplace and install Python Extension Pack.</li> <li>(Optional) Run <code>pip install black</code> in a terminal to install a code formatter.</li> </ol>","path":["Chapter 16.   Appendix","16.1   Setting up a programming environment"],"tags":[]},{"location":"chapter_appendix/installation/#2-cc","level":3,"title":"2.   C/C++","text":"<ol> <li>On Windows, install MinGW (setup guide). macOS comes with Clang, so no installation is needed.</li> <li>Search for <code>c++</code> in the VS Code Extensions Marketplace and install C/C++ Extension Pack.</li> <li>(Optional) Open Settings, search for the <code>Clang_format_fallback Style</code> formatting option, and set it to <code>{ BasedOnStyle: Microsoft, BreakBeforeBraces: Attach }</code>.</li> </ol>","path":["Chapter 16.   Appendix","16.1   Setting up a programming environment"],"tags":[]},{"location":"chapter_appendix/installation/#3-java","level":3,"title":"3.   Java","text":"<ol> <li>Download and install OpenJDK (a version later than JDK 9 is required).</li> <li>Search for <code>java</code> in the VS Code Extensions Marketplace and install Extension Pack for Java.</li> </ol>","path":["Chapter 16.   Appendix","16.1   Setting up a programming environment"],"tags":[]},{"location":"chapter_appendix/installation/#4-c","level":3,"title":"4.   C#","text":"<ol> <li>Download and install .NET 8.0.</li> <li>Search for <code>C# Dev Kit</code> in the VS Code Extensions Marketplace and install C# Dev Kit (setup guide).</li> <li>You can also use Visual Studio (installation guide).</li> </ol>","path":["Chapter 16.   Appendix","16.1   Setting up a programming environment"],"tags":[]},{"location":"chapter_appendix/installation/#5-go","level":3,"title":"5.   Go","text":"<ol> <li>Download and install Go.</li> <li>Search for <code>go</code> in the VS Code Extensions Marketplace and install Go.</li> <li>Press <code>Ctrl + Shift + P</code> to open the Command Palette, enter go, select <code>Go: Install/Update Tools</code>, then select and install all the tools.</li> </ol>","path":["Chapter 16.   Appendix","16.1   Setting up a programming environment"],"tags":[]},{"location":"chapter_appendix/installation/#6-swift","level":3,"title":"6.   Swift","text":"<ol> <li>Download and install Swift.</li> <li>Search for <code>swift</code> in the VS Code Extensions Marketplace and install Swift for Visual Studio Code.</li> </ol>","path":["Chapter 16.   Appendix","16.1   Setting up a programming environment"],"tags":[]},{"location":"chapter_appendix/installation/#7-javascript","level":3,"title":"7.   JavaScript","text":"<ol> <li>Download and install Node.js.</li> <li>(Optional) Search for <code>Prettier</code> in the VS Code Extensions Marketplace and install this code formatter.</li> </ol>","path":["Chapter 16.   Appendix","16.1   Setting up a programming environment"],"tags":[]},{"location":"chapter_appendix/installation/#8-typescript","level":3,"title":"8.   TypeScript","text":"<ol> <li>Follow the JavaScript environment setup steps.</li> <li>Install TypeScript Execute (tsx).</li> <li>Search for <code>typescript</code> in the VS Code Extensions Marketplace and install Pretty TypeScript Errors.</li> </ol>","path":["Chapter 16.   Appendix","16.1   Setting up a programming environment"],"tags":[]},{"location":"chapter_appendix/installation/#9-dart","level":3,"title":"9.   Dart","text":"<ol> <li>Download and install Dart.</li> <li>Search for <code>dart</code> in the VS Code Extensions Marketplace and install Dart.</li> </ol>","path":["Chapter 16.   Appendix","16.1   Setting up a programming environment"],"tags":[]},{"location":"chapter_appendix/installation/#10-rust","level":3,"title":"10.   Rust","text":"<ol> <li>Download and install Rust.</li> <li>Search for <code>rust</code> in the VS Code Extensions Marketplace and install rust-analyzer.</li> </ol>","path":["Chapter 16.   Appendix","16.1   Setting up a programming environment"],"tags":[]},{"location":"chapter_appendix/terminology/","level":1,"title":"16.3   Glossary","text":"<p>Table 16-1 lists important terms used in this book. Familiarity with their English names will help when reading technical literature in English.</p> <p> Table 16-1   Key terms in data structures and algorithms </p> English Simplified Chinese algorithm 算法 data structure 数据结构 code 代码 file 文件 function 函数 method 方法 variable 变量 asymptotic complexity analysis 渐近复杂度分析 time complexity 时间复杂度 space complexity 空间复杂度 loop 循环 iteration 迭代 recursion 递归 tail recursion 尾递归 recursion tree 递归树 big-\\(O\\) notation 大 \\(O\\) 记号 asymptotic upper bound 渐近上界 sign-magnitude 原码 one's complement 反码 two's complement 补码 array 数组 index 索引 linked list 链表 linked list node, list node 链表节点 head node 头节点 tail node 尾节点 list 列表 dynamic array 动态数组 hard disk 硬盘 random-access memory (RAM) 内存 cache memory 缓存 cache miss 缓存未命中 cache hit rate 缓存命中率 stack 栈 top of the stack 栈顶 bottom of the stack 栈底 queue 队列 double-ended queue 双向队列 front of the queue 队首 rear of the queue 队尾 hash table 哈希表 hash set 哈希集合 bucket 桶 hash function 哈希函数 hash collision 哈希冲突 load factor 负载因子 separate chaining 链式地址 open addressing 开放寻址 linear probing 线性探测 lazy deletion 懒删除 binary tree 二叉树 tree node 树节点 left-child node 左子节点 right-child node 右子节点 parent node 父节点 left subtree 左子树 right subtree 右子树 root node 根节点 leaf node 叶节点 edge 边 level 层 degree 度 height 高度 depth 深度 perfect binary tree 完美二叉树 complete binary tree 完全二叉树 full binary tree 完满二叉树 balanced binary tree 平衡二叉树 binary search tree 二叉搜索树 AVL tree AVL 树 red-black tree 红黑树 level-order traversal 层序遍历 breadth-first traversal 广度优先遍历 depth-first traversal 深度优先遍历 preorder traversal 前序遍历 inorder traversal 中序遍历 postorder traversal 后序遍历 balanced binary search tree 平衡二叉搜索树 balance factor 平衡因子 heap 堆 max heap 大顶堆 min heap 小顶堆 priority queue 优先队列 heapify 堆化 top-\\(k\\) problem Top-\\(k\\) 问题 graph 图 vertex 顶点 undirected graph 无向图 directed graph 有向图 connected graph 连通图 disconnected graph 非连通图 weighted graph 有权图 adjacency 邻接 path 路径 in-degree 入度 out-degree 出度 adjacency matrix 邻接矩阵 adjacency list 邻接表 breadth-first search 广度优先搜索 depth-first search 深度优先搜索 binary search 二分查找 searching algorithm 搜索算法 sorting algorithm 排序算法 selection sort 选择排序 bubble sort 冒泡排序 insertion sort 插入排序 quick sort 快速排序 merge sort 归并排序 heap sort 堆排序 bucket sort 桶排序 counting sort 计数排序 radix sort 基数排序 divide and conquer 分治 Tower of Hanoi problem 汉诺塔问题 backtracking algorithm 回溯算法 constraint 约束 solution 解 state 状态 pruning 剪枝 permutations problem 全排列问题 subset-sum problem 子集和问题 \\(n\\)-queens problem \\(n\\) 皇后问题 dynamic programming 动态规划 initial state 初始状态 state transition equation 状态转移方程 knapsack problem 背包问题 edit distance problem 编辑距离问题 greedy algorithm 贪心算法","path":["Chapter 16.   Appendix","16.3   Glossary"],"tags":[]},{"location":"chapter_array_and_linkedlist/","level":1,"title":"Chapter 4.   Arrays and linked lists","text":"<p>Abstract</p> <p>The world of data structures is like a solid brick wall.</p> <p>In an array, the bricks line up neatly, each right beside the next. In a linked list, the bricks are scattered, with connecting vines weaving freely through the gaps.</p>","path":["Chapter 4.   Arrays and linked lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>4.1   Arrays</li> <li>4.2   Linked lists</li> <li>4.3   Lists</li> <li>4.4   Memory and cache *</li> <li>4.5   Summary</li> <li>4.6   Exercises</li> </ul>","path":["Chapter 4.   Arrays and linked lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/","level":1,"title":"4.1   Arrays","text":"<p>An array is a linear data structure that stores elements of the same type in contiguous memory. An element's position in the array is its index. Figure 4-1 shows the main concepts and storage layout.</p> <p></p> <p> Figure 4-1   Array definition and storage layout </p>","path":["Chapter 4.   Arrays and linked lists","4.1   Arrays"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#411-common-array-operations","level":2,"title":"4.1.1   Common array operations","text":"","path":["Chapter 4.   Arrays and linked lists","4.1   Arrays"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#1-initializing-an-array","level":3,"title":"1.   Initializing an array","text":"<p>We can initialize an array with or without explicit values. When values are not specified, most programming languages initialize the elements to \\(0\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code># Initialize arrays\narr: list[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> array.cpp<pre><code>/* Initialize arrays */\n// Stored on the stack\nint arr[5] = {};\nint nums[5] = { 1, 3, 2, 5, 4 };\n// Stored on the heap (memory must be freed manually)\nint* arr1 = new int[5] {};\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n</code></pre> array.java<pre><code>/* Initialize arrays */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n</code></pre> array.cs<pre><code>/* Initialize arrays */\nint[] arr = new int[5]; // [ 0, 0, 0, 0, 0 ]\nint[] nums = [1, 3, 2, 5, 4];\n</code></pre> array.go<pre><code>/* Initialize arrays */\nvar arr [5]int\n// In Go, [5]int is an array with a specified length; []int is a slice\n// Go array lengths are determined at compile time and must be constants\n// To make extend() easier to implement, we treat slices as arrays below\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> array.swift<pre><code>/* Initialize arrays */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n</code></pre> array.js<pre><code>/* Initialize arrays */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n</code></pre> array.ts<pre><code>/* Initialize arrays */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> array.dart<pre><code>/* Initialize arrays */\nList&lt;int&gt; arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> array.rs<pre><code>/* Initialize arrays */\nlet arr: [i32; 5] = [0; 5]; // [0, 0, 0, 0, 0]\nlet slice: &amp;[i32] = &amp;[0; 5];\n// In Rust, [i32; 5] is an array with a specified length; &amp;[i32] is a slice\n// Rust array lengths are determined at compile time and must be constants\n// Vector is Rust's usual dynamic array type\n// To make extend() easier to implement, we treat a vector as an array below\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> array.c<pre><code>/* Initialize arrays */\nint arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n</code></pre> array.kt<pre><code>/* Initialize arrays */\nvar arr = IntArray(5) // { 0, 0, 0, 0, 0 }\nvar nums = intArrayOf(1, 3, 2, 5, 4)\n</code></pre> array.rb<pre><code># Initialize arrays\narr = Array.new(5, 0)\nnums = [1, 3, 2, 5, 4]\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.1   Arrays"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#2-accessing-an-element","level":3,"title":"2.   Accessing an element","text":"<p>Because array elements occupy contiguous memory, their addresses are easy to calculate. Given the array's address (the address of its first element) and an index, the formula in Figure 4-2 gives the element's address, allowing direct access.</p> <p></p> <p> Figure 4-2   Calculating an array element's memory address </p> <p>The first element has index \\(0\\), which may seem unintuitive because counting from \\(1\\) feels more natural. However, an index essentially describes an offset from the starting memory address. The first element's offset is \\(0\\), so index \\(0\\) makes sense.</p> <p>Array access is very efficient: any element can be accessed directly in \\(O(1)\\) time.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def random_access(nums: list[int]) -&gt; int:\n    \"\"\"Access an element at a random index\"\"\"\n    # Choose a random number in the range [0, len(nums)-1]\n    random_index = random.randint(0, len(nums) - 1)\n    # Get and return the randomly selected element\n    random_num = nums[random_index]\n    return random_num\n</code></pre> array.cpp<pre><code>/* Access an element at a random index */\nint randomAccess(int *nums, int size) {\n    // Choose a random number in [0, size)\n    int randomIndex = rand() % size;\n    // Get and return the randomly selected element\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.java<pre><code>/* Access an element at a random index */\nint randomAccess(int[] nums) {\n    // Choose a random number in the interval [0, nums.length)\n    int randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n    // Get and return the randomly selected element\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.cs<pre><code>/* Access an element at a random index */\nint RandomAccess(int[] nums) {\n    Random random = new();\n    // Randomly select a number from [0, nums.Length)\n    int randomIndex = random.Next(nums.Length);\n    // Get and return the randomly selected element\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.go<pre><code>/* Access an element at a random index */\nfunc randomAccess(nums []int) (randomNum int) {\n    // Choose a random number in the interval [0, len(nums))\n    randomIndex := rand.Intn(len(nums))\n    // Get and return the randomly selected element\n    randomNum = nums[randomIndex]\n    return\n}\n</code></pre> array.swift<pre><code>/* Access an element at a random index */\nfunc randomAccess(nums: [Int]) -&gt; Int {\n    // Pick a random number in the interval [0, nums.count)\n    let randomIndex = nums.indices.randomElement()!\n    // Get and return the randomly selected element\n    let randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.js<pre><code>/* Access an element at a random index */\nfunction randomAccess(nums) {\n    // Choose a random index in the interval [0, nums.length)\n    const random_index = Math.floor(Math.random() * nums.length);\n    // Get and return the randomly selected element\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.ts<pre><code>/* Access an element at a random index */\nfunction randomAccess(nums: number[]): number {\n    // Choose a random index in the interval [0, nums.length)\n    const random_index = Math.floor(Math.random() * nums.length);\n    // Get and return the randomly selected element\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.dart<pre><code>/* Access an element at a random index */\nint randomAccess(List&lt;int&gt; nums) {\n  // Choose a random index in the interval [0, nums.length)\n  int randomIndex = Random().nextInt(nums.length);\n  // Get and return the randomly selected element\n  int randomNum = nums[randomIndex];\n  return randomNum;\n}\n</code></pre> array.rs<pre><code>/* Access an element at a random index */\nfn random_access(nums: &amp;[i32]) -&gt; i32 {\n    // Choose a random index in [0, nums.len())\n    let random_index = rand::thread_rng().gen_range(0..nums.len());\n    // Get and return the randomly selected element\n    let random_num = nums[random_index];\n    random_num\n}\n</code></pre> array.c<pre><code>/* Access an element at a random index */\nint randomAccess(int *nums, int size) {\n    // Choose a random number in [0, size)\n    int randomIndex = rand() % size;\n    // Get and return the randomly selected element\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.kt<pre><code>/* Access an element at a random index */\nfun randomAccess(nums: IntArray): Int {\n    // Randomly select a number in [0, nums.size)\n    val randomIndex = ThreadLocalRandom.current().nextInt(0, nums.size)\n    // Get and return the randomly selected element\n    val randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.rb<pre><code>### Access an element at a random index ###\ndef random_access(nums)\n  # Choose a random index in the interval [0, nums.length)\n  random_index = Random.rand(0...nums.length)\n\n  # Get and return the randomly selected element\n  nums[random_index]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.1   Arrays"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#3-inserting-an-element","level":3,"title":"3.   Inserting an element","text":"<p>Array elements sit next to one another, with no room between them for more data. As shown in Figure 4-3, inserting into the middle requires shifting every element at or after the insertion point one position toward the end, then assigning the new element at the desired index.</p> <p></p> <p> Figure 4-3   Inserting an element into an array </p> <p>Because an array's length is fixed, inserting an element causes the last element to be lost. We will discuss the solution in the section on lists.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def insert(nums: list[int], num: int, index: int):\n    \"\"\"Insert element num at index `index` in the array\"\"\"\n    # Shift all elements at and after index `index` one position to the right\n    for i in range(len(nums) - 1, index, -1):\n        nums[i] = nums[i - 1]\n    # Assign num to the element at index `index`\n    nums[index] = num\n</code></pre> array.cpp<pre><code>/* Insert element num at index `index` in the array */\nvoid insert(int *nums, int size, int num, int index) {\n    // Shift all elements at and after index `index` one position to the right\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // Assign num to the element at index `index`\n    nums[index] = num;\n}\n</code></pre> array.java<pre><code>/* Insert element num at index `index` in the array */\nvoid insert(int[] nums, int num, int index) {\n    // Shift all elements at and after index `index` one position to the right\n    for (int i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // Assign num to the element at index `index`\n    nums[index] = num;\n}\n</code></pre> array.cs<pre><code>/* Insert element num at index `index` in the array */\nvoid Insert(int[] nums, int num, int index) {\n    // Shift all elements at and after index `index` one position to the right\n    for (int i = nums.Length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // Assign num to the element at index `index`\n    nums[index] = num;\n}\n</code></pre> array.go<pre><code>/* Insert element num at index `index` in the array */\nfunc insert(nums []int, num int, index int) {\n    // Shift all elements at and after index `index` one position to the right\n    for i := len(nums) - 1; i &gt; index; i-- {\n        nums[i] = nums[i-1]\n    }\n    // Assign num to the element at index `index`\n    nums[index] = num\n}\n</code></pre> array.swift<pre><code>/* Insert element num at index `index` in the array */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n    // Shift all elements at and after index `index` one position to the right\n    for i in stride(from: nums.count - 1, to: index, by: -1) {\n        nums[i] = nums[i - 1]\n    }\n    // Assign num to the element at index `index`\n    nums[index] = num\n}\n</code></pre> array.js<pre><code>/* Insert element num at index `index` in the array */\nfunction insert(nums, num, index) {\n    // Shift all elements at and after index `index` one position to the right\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // Assign num to the element at index `index`\n    nums[index] = num;\n}\n</code></pre> array.ts<pre><code>/* Insert element num at index `index` in the array */\nfunction insert(nums: number[], num: number, index: number): void {\n    // Shift all elements at and after index `index` one position to the right\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // Assign num to the element at index `index`\n    nums[index] = num;\n}\n</code></pre> array.dart<pre><code>/* Insert element _num at index in the array */\nvoid insert(List&lt;int&gt; nums, int _num, int index) {\n  // Shift all elements at and after index `index` one position to the right\n  for (var i = nums.length - 1; i &gt; index; i--) {\n    nums[i] = nums[i - 1];\n  }\n  // Assign _num to the element at index\n  nums[index] = _num;\n}\n</code></pre> array.rs<pre><code>/* Insert element num at index `index` in the array */\nfn insert(nums: &amp;mut [i32], num: i32, index: usize) {\n    // Shift all elements at and after index `index` one position to the right\n    for i in (index + 1..nums.len()).rev() {\n        nums[i] = nums[i - 1];\n    }\n    // Assign num to the element at index `index`\n    nums[index] = num;\n}\n</code></pre> array.c<pre><code>/* Insert element num at index `index` in the array */\nvoid insert(int *nums, int size, int num, int index) {\n    // Shift all elements at and after index `index` one position to the right\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // Assign num to the element at index `index`\n    nums[index] = num;\n}\n</code></pre> array.kt<pre><code>/* Insert element num at index `index` in the array */\nfun insert(nums: IntArray, num: Int, index: Int) {\n    // Shift all elements at and after index `index` one position to the right\n    for (i in nums.size - 1 downTo index + 1) {\n        nums[i] = nums[i - 1]\n    }\n    // Assign num to the element at index `index`\n    nums[index] = num\n}\n</code></pre> array.rb<pre><code>### Insert element num at index `index` in the array ###\ndef insert(nums, num, index)\n  # Shift all elements at and after index `index` one position to the right\n  for i in (nums.length - 1).downto(index + 1)\n    nums[i] = nums[i - 1]\n  end\n\n  # Assign num to the element at index `index`\n  nums[index] = num\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.1   Arrays"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#4-removing-an-element","level":3,"title":"4.   Removing an element","text":"<p>Similarly, as shown in Figure 4-4, removing the element at index \\(i\\) requires shifting every element after \\(i\\) one position toward the beginning.</p> <p></p> <p> Figure 4-4   Removing an element from an array </p> <p>After removal, the old final entry is no longer meaningful, so we do not need to change it explicitly.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def remove(nums: list[int], index: int):\n    \"\"\"Remove the element at index `index`\"\"\"\n    # Shift all elements after index `index` one position to the left\n    for i in range(index, len(nums) - 1):\n        nums[i] = nums[i + 1]\n</code></pre> array.cpp<pre><code>/* Remove the element at index `index` */\nvoid remove(int *nums, int size, int index) {\n    // Shift all elements after index `index` one position to the left\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.java<pre><code>/* Remove the element at index `index` */\nvoid remove(int[] nums, int index) {\n    // Shift all elements after index `index` one position to the left\n    for (int i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.cs<pre><code>/* Remove the element at index `index` */\nvoid Remove(int[] nums, int index) {\n    // Shift all elements after index `index` one position to the left\n    for (int i = index; i &lt; nums.Length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.go<pre><code>/* Remove the element at index `index` */\nfunc remove(nums []int, index int) {\n    // Shift all elements after index `index` one position to the left\n    for i := index; i &lt; len(nums)-1; i++ {\n        nums[i] = nums[i+1]\n    }\n}\n</code></pre> array.swift<pre><code>/* Remove the element at index `index` */\nfunc remove(nums: inout [Int], index: Int) {\n    // Shift all elements after index `index` one position to the left\n    for i in nums.indices.dropFirst(index).dropLast() {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.js<pre><code>/* Remove the element at index `index` */\nfunction remove(nums, index) {\n    // Shift all elements after index `index` one position to the left\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.ts<pre><code>/* Remove the element at index `index` */\nfunction remove(nums: number[], index: number): void {\n    // Shift all elements after index `index` one position to the left\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.dart<pre><code>/* Remove the element at index `index` */\nvoid remove(List&lt;int&gt; nums, int index) {\n  // Shift all elements after index `index` one position to the left\n  for (var i = index; i &lt; nums.length - 1; i++) {\n    nums[i] = nums[i + 1];\n  }\n}\n</code></pre> array.rs<pre><code>/* Remove the element at index `index` */\nfn remove(nums: &amp;mut [i32], index: usize) {\n    // Shift all elements after index `index` one position to the left\n    for i in index..nums.len() - 1 {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.c<pre><code>/* Remove the element at index `index` */\n// Note: stdio.h already declares a function named remove\nvoid removeItem(int *nums, int size, int index) {\n    // Shift all elements after index `index` one position to the left\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.kt<pre><code>/* Remove the element at index `index` */\nfun remove(nums: IntArray, index: Int) {\n    // Shift all elements after index `index` one position to the left\n    for (i in index..&lt;nums.size - 1) {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.rb<pre><code>### Remove the element at index `index` ###\ndef remove(nums, index)\n  # Shift all elements after index `index` one position to the left\n  for i in index...(nums.length - 1)\n    nums[i] = nums[i + 1]\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Overall, inserting and removing array elements have these drawbacks.</p> <ul> <li>High time complexity: Both operations take \\(O(n)\\) time on average, where \\(n\\) is the array length.</li> <li>Lost elements: Since the length is fixed, insertion loses any element pushed beyond the array's bounds.</li> <li>Wasted memory: We can allocate a longer array and use only its beginning, so any final entries lost during insertion are unused. However, this leaves some memory wasted.</li> </ul>","path":["Chapter 4.   Arrays and linked lists","4.1   Arrays"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#5-traversing-an-array","level":3,"title":"5.   Traversing an array","text":"<p>Most programming languages let us traverse an array either by index or by visiting its elements directly:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def traverse(nums: list[int]):\n    \"\"\"Traverse the array\"\"\"\n    count = 0\n    # Traverse the array by index\n    for i in range(len(nums)):\n        count += nums[i]\n    # Iterate directly over the array elements\n    for num in nums:\n        count += num\n    # Iterate over both the indices and the elements\n    for i, num in enumerate(nums):\n        count += nums[i]\n        count += num\n</code></pre> array.cpp<pre><code>/* Traverse the array */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // Traverse the array by index\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.java<pre><code>/* Traverse the array */\nvoid traverse(int[] nums) {\n    int count = 0;\n    // Traverse the array by index\n    for (int i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // Iterate directly over the array elements\n    for (int num : nums) {\n        count += num;\n    }\n}\n</code></pre> array.cs<pre><code>/* Traverse the array */\nvoid Traverse(int[] nums) {\n    int count = 0;\n    // Traverse the array by index\n    for (int i = 0; i &lt; nums.Length; i++) {\n        count += nums[i];\n    }\n    // Iterate directly over the array elements\n    foreach (int num in nums) {\n        count += num;\n    }\n}\n</code></pre> array.go<pre><code>/* Traverse the array */\nfunc traverse(nums []int) {\n    count := 0\n    // Traverse the array by index\n    for i := 0; i &lt; len(nums); i++ {\n        count += nums[i]\n    }\n    count = 0\n    // Iterate directly over the array elements\n    for _, num := range nums {\n        count += num\n    }\n    // Iterate over both the indices and the elements\n    for i, num := range nums {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.swift<pre><code>/* Traverse the array */\nfunc traverse(nums: [Int]) {\n    var count = 0\n    // Traverse the array by index\n    for i in nums.indices {\n        count += nums[i]\n    }\n    // Iterate directly over the array elements\n    for num in nums {\n        count += num\n    }\n    // Iterate over both the indices and the elements\n    for (i, num) in nums.enumerated() {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.js<pre><code>/* Traverse the array */\nfunction traverse(nums) {\n    let count = 0;\n    // Traverse the array by index\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // Iterate directly over the array elements\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.ts<pre><code>/* Traverse the array */\nfunction traverse(nums: number[]): void {\n    let count = 0;\n    // Traverse the array by index\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // Iterate directly over the array elements\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.dart<pre><code>/* Traverse the array elements */\nvoid traverse(List&lt;int&gt; nums) {\n  int count = 0;\n  // Traverse the array by index\n  for (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n  }\n  // Iterate directly over the array elements\n  for (int _num in nums) {\n    count += _num;\n  }\n  // Traverse the array using forEach\n  nums.forEach((_num) {\n    count += _num;\n  });\n}\n</code></pre> array.rs<pre><code>/* Traverse the array */\nfn traverse(nums: &amp;[i32]) {\n    let mut _count = 0;\n    // Traverse the array by index\n    for i in 0..nums.len() {\n        _count += nums[i];\n    }\n    // Iterate directly over the array elements\n    _count = 0;\n    for &amp;num in nums {\n        _count += num;\n    }\n}\n</code></pre> array.c<pre><code>/* Traverse the array */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // Traverse the array by index\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.kt<pre><code>/* Traverse the array */\nfun traverse(nums: IntArray) {\n    var count = 0\n    // Traverse the array by index\n    for (i in nums.indices) {\n        count += nums[i]\n    }\n    // Iterate directly over the array elements\n    for (j in nums) {\n        count += j\n    }\n}\n</code></pre> array.rb<pre><code>### Traverse the array ###\ndef traverse(nums)\n  count = 0\n\n  # Traverse the array by index\n  for i in 0...nums.length\n    count += nums[i]\n  end\n\n  # Iterate directly over the array elements\n  for num in nums\n    count += num\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.1   Arrays"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#6-finding-an-element","level":3,"title":"6.   Finding an element","text":"<p>To find a specified element, traverse the array and compare each value with the target. Return the index when a match is found.</p> <p>Because an array is a linear data structure, this is called linear search.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def find(nums: list[int], target: int) -&gt; int:\n    \"\"\"Find a specified element in the array\"\"\"\n    for i in range(len(nums)):\n        if nums[i] == target:\n            return i\n    return -1\n</code></pre> array.cpp<pre><code>/* Find a specified element in the array */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.java<pre><code>/* Find a specified element in the array */\nint find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.cs<pre><code>/* Find a specified element in the array */\nint Find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.go<pre><code>/* Find a specified element in the array */\nfunc find(nums []int, target int) (index int) {\n    index = -1\n    for i := 0; i &lt; len(nums); i++ {\n        if nums[i] == target {\n            index = i\n            break\n        }\n    }\n    return\n}\n</code></pre> array.swift<pre><code>/* Find a specified element in the array */\nfunc find(nums: [Int], target: Int) -&gt; Int {\n    for i in nums.indices {\n        if nums[i] == target {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> array.js<pre><code>/* Find a specified element in the array */\nfunction find(nums, target) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) return i;\n    }\n    return -1;\n}\n</code></pre> array.ts<pre><code>/* Find a specified element in the array */\nfunction find(nums: number[], target: number): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> array.dart<pre><code>/* Find a specified element in the array */\nint find(List&lt;int&gt; nums, int target) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    if (nums[i] == target) return i;\n  }\n  return -1;\n}\n</code></pre> array.rs<pre><code>/* Find a specified element in the array */\nfn find(nums: &amp;[i32], target: i32) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        if nums[i] == target {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> array.c<pre><code>/* Find a specified element in the array */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.kt<pre><code>/* Find a specified element in the array */\nfun find(nums: IntArray, target: Int): Int {\n    for (i in nums.indices) {\n        if (nums[i] == target)\n            return i\n    }\n    return -1\n}\n</code></pre> array.rb<pre><code>### Find a specified element in the array ###\ndef find(nums, target)\n  for i in 0...nums.length\n    return i if nums[i] == target\n  end\n\n  -1\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.1   Arrays"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#7-expanding-an-array","level":3,"title":"7.   Expanding an array","text":"<p>In a complex system, a program cannot ensure that memory immediately after an array is available, so extending it in place is unsafe. Therefore, in most programming languages, an array's length is fixed.</p> <p>To expand an array, create a larger one and copy every element into it. This takes \\(O(n)\\) time and can be expensive for a large array. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def extend(nums: list[int], enlarge: int) -&gt; list[int]:\n    \"\"\"Extend the array length\"\"\"\n    # Initialize an array with the extended length\n    res = [0] * (len(nums) + enlarge)\n    # Copy all elements from the original array to the new array\n    for i in range(len(nums)):\n        res[i] = nums[i]\n    # Return the new, extended array\n    return res\n</code></pre> array.cpp<pre><code>/* Extend the array length */\nint *extend(int *nums, int size, int enlarge) {\n    // Initialize an array with the extended length\n    int *res = new int[size + enlarge];\n    // Copy all elements from the original array to the new array\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // Free memory\n    delete[] nums;\n    // Return the new, extended array\n    return res;\n}\n</code></pre> array.java<pre><code>/* Extend the array length */\nint[] extend(int[] nums, int enlarge) {\n    // Initialize an array with the extended length\n    int[] res = new int[nums.length + enlarge];\n    // Copy all elements from the original array to the new array\n    for (int i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // Return the new, extended array\n    return res;\n}\n</code></pre> array.cs<pre><code>/* Extend the array length */\nint[] Extend(int[] nums, int enlarge) {\n    // Initialize an array with the extended length\n    int[] res = new int[nums.Length + enlarge];\n    // Copy all elements from the original array to the new array\n    for (int i = 0; i &lt; nums.Length; i++) {\n        res[i] = nums[i];\n    }\n    // Return the new, extended array\n    return res;\n}\n</code></pre> array.go<pre><code>/* Extend the array length */\nfunc extend(nums []int, enlarge int) []int {\n    // Initialize an array with the extended length\n    res := make([]int, len(nums)+enlarge)\n    // Copy all elements from the original array to the new array\n    for i, num := range nums {\n        res[i] = num\n    }\n    // Return the new, extended array\n    return res\n}\n</code></pre> array.swift<pre><code>/* Extend the array length */\nfunc extend(nums: [Int], enlarge: Int) -&gt; [Int] {\n    // Initialize an array with the extended length\n    var res = Array(repeating: 0, count: nums.count + enlarge)\n    // Copy all elements from the original array to the new array\n    for i in nums.indices {\n        res[i] = nums[i]\n    }\n    // Return the new, extended array\n    return res\n}\n</code></pre> array.js<pre><code>/* Extend the array length */\n// JavaScript Array is a dynamic array and can be extended directly\n// For learning purposes, this function treats Array as a fixed-length array\nfunction extend(nums, enlarge) {\n    // Initialize an array with the extended length\n    const res = new Array(nums.length + enlarge).fill(0);\n    // Copy all elements from the original array to the new array\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // Return the new, extended array\n    return res;\n}\n</code></pre> array.ts<pre><code>/* Extend the array length */\n// Note: TypeScript Array is a dynamic array and can grow directly\n// For learning purposes, this function treats Array as a fixed-length array\nfunction extend(nums: number[], enlarge: number): number[] {\n    // Initialize an array with the extended length\n    const res = new Array(nums.length + enlarge).fill(0);\n    // Copy all elements from the original array to the new array\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // Return the new, extended array\n    return res;\n}\n</code></pre> array.dart<pre><code>/* Extend the array length */\nList&lt;int&gt; extend(List&lt;int&gt; nums, int enlarge) {\n  // Initialize an array with the extended length\n  List&lt;int&gt; res = List.filled(nums.length + enlarge, 0);\n  // Copy all elements from the original array to the new array\n  for (var i = 0; i &lt; nums.length; i++) {\n    res[i] = nums[i];\n  }\n  // Return the new, extended array\n  return res;\n}\n</code></pre> array.rs<pre><code>/* Extend the array length */\nfn extend(nums: &amp;[i32], enlarge: usize) -&gt; Vec&lt;i32&gt; {\n    // Initialize an array with the extended length\n    let mut res: Vec&lt;i32&gt; = vec![0; nums.len() + enlarge];\n    // Copy all elements from the original array to the new one\n    res[0..nums.len()].copy_from_slice(nums);\n\n    // Return the new, extended array\n    res\n}\n</code></pre> array.c<pre><code>/* Extend the array length */\nint *extend(int *nums, int size, int enlarge) {\n    // Initialize an array with the extended length\n    int *res = (int *)malloc(sizeof(int) * (size + enlarge));\n    // Copy all elements from the original array to the new array\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // Initialize the newly added space\n    for (int i = size; i &lt; size + enlarge; i++) {\n        res[i] = 0;\n    }\n    // Return the new, extended array\n    return res;\n}\n</code></pre> array.kt<pre><code>/* Extend the array length */\nfun extend(nums: IntArray, enlarge: Int): IntArray {\n    // Initialize an array with the extended length\n    val res = IntArray(nums.size + enlarge)\n    // Copy all elements from the original array to the new array\n    for (i in nums.indices) {\n        res[i] = nums[i]\n    }\n    // Return the new, extended array\n    return res\n}\n</code></pre> array.rb<pre><code>### Extend the array length ###\n# Ruby Array is a dynamic array that can grow directly\n# For learning purposes, this function treats Array as a fixed-length array\ndef extend(nums, enlarge)\n  # Initialize an array with the extended length\n  res = Array.new(nums.length + enlarge, 0)\n\n  # Copy all elements from the original array to the new array\n  for i in 0...nums.length\n    res[i] = nums[i]\n  end\n\n  # Return the new, extended array\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.1   Arrays"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#412-strengths-and-limitations-of-arrays","level":2,"title":"4.1.2   Strengths and limitations of arrays","text":"<p>An array stores elements of one type in contiguous memory. These known properties give the system opportunities to optimize operations.</p> <ul> <li>Space efficiency: Data occupies a contiguous block with no extra structural overhead.</li> <li>Random access: Any element can be accessed in \\(O(1)\\) time.</li> <li>Cache locality: Accessing an element loads not only that element but also nearby data into the cache, speeding up later operations.</li> </ul> <p>Contiguous storage also has limitations.</p> <ul> <li>Slow insertion and deletion: With many elements, these operations require substantial shifting.</li> <li>Fixed length: Once initialized, an array cannot change length; expansion requires copying all data into a new array.</li> <li>Wasted space: Any allocated space beyond actual needs is unused.</li> </ul>","path":["Chapter 4.   Arrays and linked lists","4.1   Arrays"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#413-typical-applications-of-arrays","level":2,"title":"4.1.3   Typical applications of arrays","text":"<p>Arrays are fundamental, widely used data structures. They appear throughout algorithms and underpin more complex structures.</p> <ul> <li>Random access: To draw random samples, store data in an array and generate random indices to select elements.</li> <li>Sorting and searching: Arrays are the most common structures for these algorithms. Quick sort, merge sort, and binary search operate primarily on arrays.</li> <li>Lookup tables: An array can provide fast access to an element or a mapping. For example, a character-to-ASCII mapping can use a character's ASCII value as an index and store its corresponding element there.</li> <li>Machine learning: Neural networks make extensive use of linear algebra on vectors, matrices, and tensors, all built from arrays. Arrays are the most common data structure in neural network programming.</li> <li>Implementing data structures: Arrays can implement stacks, queues, hash tables, heaps, graphs, and more. A graph's adjacency matrix, for example, is a two-dimensional array.</li> </ul>","path":["Chapter 4.   Arrays and linked lists","4.1   Arrays"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/","level":1,"title":"4.6   Exercises","text":"","path":["Chapter 4.   Arrays and linked lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#461-check-your-understanding","level":2,"title":"4.6.1   Check your understanding","text":"","path":["Chapter 4.   Arrays and linked lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#1-finding-elements-in-arrays-and-linked-lists","level":3,"title":"1.   Finding elements in arrays and linked lists","text":"<p>An array and a singly linked list both store <code>[A, B, C, D, E]</code> in order. We want to read the fourth element, <code>D</code>:</p> <ol> <li>Which index can we use to access it directly in the array?</li> <li>Starting at the head node <code>A</code>, which nodes must the singly linked list visit by following <code>next</code>?</li> <li>As the requested element moves farther toward the end, how does the number of steps change for each structure? Which is better suited to repeated access by position, and why?</li> </ol> Reference answer <ol> <li> <p>With zero-based indexing, the fourth element has index 3, so the array can read <code>arr[3]</code> directly.</p> </li> <li> <p>The singly linked list must start at the head and follow the path <code>A → B → C → D</code>, advancing along <code>next</code> three times.</p> </li> <li> <p>An array can locate an element directly from its starting address and the index, giving time complexity \\(O(1)\\) for access by position.     To access the \\(k\\)-th node of a singly linked list, we must start at the head and follow <code>next</code> \\(k-1\\) times,     taking \\(O(n)\\) time in the worst case.</p> <p>This comparison concerns access by position only; it does not mean that linked lists are slower for every operation.</p> </li> </ol>","path":["Chapter 4.   Arrays and linked lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#2-inserting-elements-into-arrays-and-linked-lists","level":3,"title":"2.   Inserting elements into arrays and linked lists","text":"<p>An array and a singly linked list both store <code>A, B, C, D</code>. We want to insert <code>X</code> after <code>B</code>:</p> <ul> <li>The array has capacity 5 and currently contains <code>[A, B, C, D, _]</code>.</li> <li>The linked list is <code>A → B → C → D</code>, and we already have a reference to node <code>B</code>.</li> </ul> <ol> <li>Which elements must the array move? Write out the array after insertion.</li> <li>In what order should the linked list update <code>X.next</code> and <code>B.next</code>? Write out the resulting list.</li> <li>Why must a comparison of insertion efficiency specify that we already have a reference to node B?</li> </ol> Reference answer <ol> <li> <p>First move <code>D</code> one position to the right, then move <code>C</code> one position to the right, and finally put <code>X</code> at index 2.     The result is <code>[A, B, X, C, D]</code>.</p> </li> <li> <p>Initially, <code>B.next</code> points to <code>C</code>. First set <code>X.next = B.next</code>, making <code>X</code> point to <code>C</code>.     Then set <code>B.next = X</code>. The result is <code>A → B → X → C → D</code>.     If we overwrite <code>B.next</code> first without saving the original link, we may lose access to <code>C</code>.</p> </li> <li> <p>Once the location of <code>B</code> is known, insertion into the linked list requires changing only two links and takes \\(O(1)\\) time.     If we first need to search for <code>B</code> from the head, that search itself may take \\(O(n)\\) time.</p> </li> </ol>","path":["Chapter 4.   Arrays and linked lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#3-how-does-a-lists-capacity-grow","level":3,"title":"3.   How does a list's capacity grow?","text":"<p>An array-based list currently contains <code>[A, B, C]</code>, with length <code>size = 3</code> and capacity <code>capacity = 4</code>. When capacity runs out, the new array is allocated with twice the previous capacity.</p> <ol> <li>After appending <code>D</code>, what are the list's length and capacity? Does it need to grow?</li> <li>When <code>E</code> is appended next, what does the capacity become? How many existing elements must be copied?</li> <li>If the length of the underlying array cannot change, why does the list's capacity appear to grow?</li> </ol> Reference answer <ol> <li> <p><code>D</code> fits in the last empty position. The contents become <code>[A, B, C, D]</code>,     with <code>size = 4</code> and <code>capacity = 4</code>. No capacity increase is needed.</p> </li> <li> <p>Appending <code>E</code> finds no empty position, so a new array of capacity 8 is created.     The four existing elements <code>A, B, C, D</code> are copied into it, and then <code>E</code> is added.     The list now has <code>size = 5</code> and <code>capacity = 8</code>.</p> </li> <li> <p>The original array itself does not become longer. The list creates a larger array, copies the existing elements,     and switches to using the new array for storage. From the user's perspective, its capacity has grown.</p> </li> </ol>","path":["Chapter 4.   Arrays and linked lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#462-programming-exercises","level":2,"title":"4.6.2   Programming exercises","text":"","path":["Chapter 4.   Arrays and linked lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#1-adding-one-to-a-large-integer-stored-in-an-array","level":3,"title":"1.   Adding one to a large integer stored in an array","text":"<p>The array <code>digits</code> stores the digits of a nonnegative integer from left to right. For example, <code>[3, 0, 8]</code> represents 308. Zero is represented by <code>[0]</code>; all other inputs have a nonzero first digit.</p> <p>Simulate the usual digit-by-digit method for decimal addition to add 1 to this integer, and return the result in the same array format. You may modify <code>digits</code> directly. If a carry creates a new leading digit, you may return a longer array.</p> Hints <ol> <li>Start at the last digit of the array, just as you would when adding by hand.</li> <li>If the current digit is less than 9, increment it and return immediately.</li> <li>If the current digit is 9, change it to 0. If all digits are 9, add a leading 1.</li> </ol> <p>LeetCode</p>","path":["Chapter 4.   Arrays and linked lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#2-reversing-a-singly-linked-list","level":3,"title":"2.   Reversing a singly linked list","text":"<p>You are given the head node <code>head</code> of a singly linked list. Each node contains a value and a <code>next</code> reference to the following node.</p> <p>Use iteration to reverse every link between nodes, and return the new head. Do not create any new linked-list nodes.</p> Hints <ol> <li>First draw three connected nodes and the two pointers prev and cur on paper.</li> <li>Before changing cur.next, save the original next node in nxt.</li> <li>After reversing cur.next, set prev = cur and then cur = nxt to continue with the next node in the original list.</li> </ol> <p>LeetCode</p>","path":["Chapter 4.   Arrays and linked lists","4.6   Exercises"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/","level":1,"title":"4.2   Linked lists","text":"<p>Memory is a resource used by all programs, and in a complex system, free space may be scattered throughout it. An array requires contiguous storage, which may be unavailable for a very large array. This is where a linked list's flexibility helps.</p> <p>A linked list is a linear data structure whose elements are node objects connected by references. Each reference stores the next node's memory address, allowing us to reach it from the current node.</p> <p>This design lets nodes occupy scattered memory locations; their addresses need not be contiguous.</p> <p></p> <p> Figure 4-5   Linked list definition and storage layout </p> <p>As shown above, the basic unit of a linked list is a node object. Each node holds a value and a reference to the next node.</p> <ul> <li>The first node is called the head, and the last is called the tail.</li> <li>The tail points to a null value: <code>null</code> in Java, <code>nullptr</code> in C++, and <code>None</code> in Python.</li> <li>In languages supporting pointers, such as C, C++, Go, and Rust, these references are expressed as pointers.</li> </ul> <p>As the code below shows, each <code>ListNode</code> stores a reference (pointer) in addition to its value. Thus, a linked list uses more memory than an array for the same amount of data.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class ListNode:\n    \"\"\"Linked list node class\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val               # Node value\n        self.next: ListNode | None = None # Reference to the next node\n</code></pre> <pre><code>/* Linked list node structure */\nstruct ListNode {\n    int val;         // Node value\n    ListNode *next;  // Pointer to the next node\n    ListNode(int x) : val(x), next(nullptr) {}  // Constructor\n};\n</code></pre> <pre><code>/* Linked list node class */\nclass ListNode {\n    int val;        // Node value\n    ListNode next;  // Reference to the next node\n    ListNode(int x) { val = x; }  // Constructor\n}\n</code></pre> <pre><code>/* Linked list node class */\nclass ListNode(int x) {  // Constructor\n    int val = x;         // Node value\n    ListNode? next;      // Reference to the next node\n}\n</code></pre> <pre><code>/* Linked list node structure */\ntype ListNode struct {\n    Val  int       // Node value\n    Next *ListNode // Pointer to the next node\n}\n\n// NewListNode constructor: create a new linked list\nfunc NewListNode(val int) *ListNode {\n    return &amp;ListNode{\n        Val:  val,\n        Next: nil,\n    }\n}\n</code></pre> <pre><code>/* Linked list node class */\nclass ListNode {\n    var val: Int // Node value\n    var next: ListNode? // Reference to the next node\n\n    init(x: Int) { // Constructor\n        val = x\n    }\n}\n</code></pre> <pre><code>/* Linked list node class */\nclass ListNode {\n    constructor(val, next) {\n        this.val = (val === undefined ? 0 : val);       // Node value\n        this.next = (next === undefined ? null : next); // Reference to the next node\n    }\n}\n</code></pre> <pre><code>/* Linked list node class */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    constructor(val?: number, next?: ListNode | null) {\n        this.val = val === undefined ? 0 : val;        // Node value\n        this.next = next === undefined ? null : next;  // Reference to the next node\n    }\n}\n</code></pre> <pre><code>/* Linked list node class */\nclass ListNode {\n  int val; // Node value\n  ListNode? next; // Reference to the next node\n  ListNode(this.val, [this.next]); // Constructor\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* Linked list node class */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // Node value\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // Pointer to the next node\n}\n</code></pre> <pre><code>/* Linked list node structure */\ntypedef struct ListNode {\n    int val;               // Node value\n    struct ListNode *next; // Pointer to the next node\n} ListNode;\n\n/* Constructor */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* Linked list node class */\n// Constructor\nclass ListNode(x: Int) {\n    val _val: Int = x          // Node value\n    var next: ListNode? = null // Reference to the next node\n}\n</code></pre> <pre><code># Linked list node class\nclass ListNode\n  attr_accessor :val  # Node value\n  attr_accessor :next # Reference to the next node\n\n  def initialize(val=0, next_node=nil)\n    @val = val\n    @next = next_node\n  end\nend\n</code></pre>","path":["Chapter 4.   Arrays and linked lists","4.2   Linked lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#421-common-linked-list-operations","level":2,"title":"4.2.1   Common linked list operations","text":"","path":["Chapter 4.   Arrays and linked lists","4.2   Linked lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#1-initializing-a-linked-list","level":3,"title":"1.   Initializing a linked list","text":"<p>Building a linked list takes two steps: initialize the node objects, then connect them with references. Afterward, start at the head and follow <code>next</code> references to visit every node in order.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code># Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# Initialize the nodes\nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# Link the nodes with references\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.cpp<pre><code>/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize the nodes\nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// Link the nodes with references\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.java<pre><code>/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize the nodes\nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// Link the nodes with references\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.cs<pre><code>/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize the nodes\nListNode n0 = new(1);\nListNode n1 = new(3);\nListNode n2 = new(2);\nListNode n3 = new(5);\nListNode n4 = new(4);\n// Link the nodes with references\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.go<pre><code>/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize the nodes\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// Link the nodes with references\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n</code></pre> linked_list.swift<pre><code>/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize the nodes\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// Link the nodes with references\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.js<pre><code>/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize the nodes\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// Link the nodes with references\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.ts<pre><code>/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize the nodes\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// Link the nodes with references\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.dart<pre><code>/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\\\n// Initialize the nodes\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// Link the nodes with references\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rs<pre><code>/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize the nodes\nlet n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None }));\nlet n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None }));\nlet n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None }));\nlet n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None }));\nlet n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None }));\n\n// Link the nodes with references\nn0.borrow_mut().next = Some(n1.clone());\nn1.borrow_mut().next = Some(n2.clone());\nn2.borrow_mut().next = Some(n3.clone());\nn3.borrow_mut().next = Some(n4.clone());\n</code></pre> linked_list.c<pre><code>/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize the nodes\nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// Link the nodes with references\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.kt<pre><code>/* Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// Initialize the nodes\nval n0 = ListNode(1)\nval n1 = ListNode(3)\nval n2 = ListNode(2)\nval n3 = ListNode(5)\nval n4 = ListNode(4)\n// Link the nodes with references\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rb<pre><code># Initialize the linked list 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# Initialize the nodes\nn0 = ListNode.new(1)\nn1 = ListNode.new(3)\nn2 = ListNode.new(2)\nn3 = ListNode.new(5)\nn4 = ListNode.new(4)\n# Link the nodes with references\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>An array is a single variable, such as <code>nums</code>, containing entries <code>nums[0]</code>, <code>nums[1]</code>, and so on. A linked list consists of separate node objects. We usually refer to the list by its head node; for example, the list above can be called <code>n0</code>.</p>","path":["Chapter 4.   Arrays and linked lists","4.2   Linked lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#2-inserting-a-node","level":3,"title":"2.   Inserting a node","text":"<p>Inserting a node is straightforward. As shown in Figure 4-6, inserting a new node <code>P</code> between adjacent nodes <code>n0</code> and <code>n1</code> requires changing only two references (pointers), taking \\(O(1)\\) time.</p> <p>In contrast, inserting an array element takes \\(O(n)\\) time and is slower for large datasets.</p> <p></p> <p> Figure 4-6   Inserting a node into a linked list </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def insert(n0: ListNode, P: ListNode):\n    \"\"\"Insert node P after node n0 in the linked list\"\"\"\n    n1 = n0.next\n    P.next = n1\n    n0.next = P\n</code></pre> linked_list.cpp<pre><code>/* Insert node P after node n0 in the linked list */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.java<pre><code>/* Insert node P after node n0 in the linked list */\nvoid insert(ListNode n0, ListNode P) {\n    ListNode n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.cs<pre><code>/* Insert node P after node n0 in the linked list */\nvoid Insert(ListNode n0, ListNode P) {\n    ListNode? n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.go<pre><code>/* Insert node P after node n0 in the linked list */\nfunc insertNode(n0 *ListNode, P *ListNode) {\n    n1 := n0.Next\n    P.Next = n1\n    n0.Next = P\n}\n</code></pre> linked_list.swift<pre><code>/* Insert node P after node n0 in the linked list */\nfunc insert(n0: ListNode, P: ListNode) {\n    let n1 = n0.next\n    P.next = n1\n    n0.next = P\n}\n</code></pre> linked_list.js<pre><code>/* Insert node P after node n0 in the linked list */\nfunction insert(n0, P) {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.ts<pre><code>/* Insert node P after node n0 in the linked list */\nfunction insert(n0: ListNode, P: ListNode): void {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.dart<pre><code>/* Insert node P after node n0 in the linked list */\nvoid insert(ListNode n0, ListNode P) {\n  ListNode? n1 = n0.next;\n  P.next = n1;\n  n0.next = P;\n}\n</code></pre> linked_list.rs<pre><code>/* Insert node P after node n0 in the linked list */\n#[allow(non_snake_case)]\npub fn insert&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, P: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    let n1 = n0.borrow_mut().next.take();\n    P.borrow_mut().next = n1;\n    n0.borrow_mut().next = Some(P);\n}\n</code></pre> linked_list.c<pre><code>/* Insert node P after node n0 in the linked list */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.kt<pre><code>/* Insert node p after node n0 in the linked list */\nfun insert(n0: ListNode?, p: ListNode?) {\n    val n1 = n0?.next\n    p?.next = n1\n    n0?.next = p\n}\n</code></pre> linked_list.rb<pre><code>### Insert node _p after node n0 in the linked list ###\n# In Ruby, `p` is a built-in method and `P` denotes a constant, so use `_p` instead\ndef insert(n0, _p)\n  n1 = n0.next\n  _p.next = n1\n  n0.next = _p\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.2   Linked lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#3-removing-a-node","level":3,"title":"3.   Removing a node","text":"<p>Removing a node is also straightforward: as shown in Figure 4-7, only one reference (pointer) needs to change.</p> <p>Although <code>P</code> still points to <code>n1</code> after removal, traversing the list can no longer reach <code>P</code>. It therefore no longer belongs to the list.</p> <p></p> <p> Figure 4-7   Removing a linked list node </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def remove(n0: ListNode):\n    \"\"\"Remove the first node after node n0 in the linked list\"\"\"\n    if not n0.next:\n        return\n    # n0 -&gt; P -&gt; n1\n    P = n0.next\n    n1 = P.next\n    n0.next = n1\n</code></pre> linked_list.cpp<pre><code>/* Remove the first node after node n0 in the linked list */\nvoid remove(ListNode *n0) {\n    if (n0-&gt;next == nullptr)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // Free memory\n    delete P;\n}\n</code></pre> linked_list.java<pre><code>/* Remove the first node after node n0 in the linked list */\nvoid remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.cs<pre><code>/* Remove the first node after node n0 in the linked list */\nvoid Remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode? n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.go<pre><code>/* Remove the first node after node n0 in the linked list */\nfunc removeItem(n0 *ListNode) {\n    if n0.Next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    P := n0.Next\n    n1 := P.Next\n    n0.Next = n1\n}\n</code></pre> linked_list.swift<pre><code>/* Remove the first node after node n0 in the linked list */\nfunc remove(n0: ListNode) {\n    if n0.next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    let P = n0.next\n    let n1 = P?.next\n    n0.next = n1\n}\n</code></pre> linked_list.js<pre><code>/* Remove the first node after node n0 in the linked list */\nfunction remove(n0) {\n    if (!n0.next) return;\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.ts<pre><code>/* Remove the first node after node n0 in the linked list */\nfunction remove(n0: ListNode): void {\n    if (!n0.next) {\n        return;\n    }\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.dart<pre><code>/* Remove the first node after node n0 in the linked list */\nvoid remove(ListNode n0) {\n  if (n0.next == null) return;\n  // n0 -&gt; P -&gt; n1\n  ListNode P = n0.next!;\n  ListNode? n1 = P.next;\n  n0.next = n1;\n}\n</code></pre> linked_list.rs<pre><code>/* Remove the first node after node n0 in the linked list */\n#[allow(non_snake_case)]\npub fn remove&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    // n0 -&gt; P -&gt; n1\n    let P = n0.borrow_mut().next.take();\n    if let Some(node) = P {\n        let n1 = node.borrow_mut().next.take();\n        n0.borrow_mut().next = n1;\n    }\n}\n</code></pre> linked_list.c<pre><code>/* Remove the first node after node n0 in the linked list */\n// Note: stdio.h already declares a function named remove\nvoid removeItem(ListNode *n0) {\n    if (!n0-&gt;next)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // Free memory\n    free(P);\n}\n</code></pre> linked_list.kt<pre><code>/* Remove the first node after node n0 in the linked list */\nfun remove(n0: ListNode?) {\n    if (n0?.next == null)\n        return\n    // n0 -&gt; P -&gt; n1\n    val p = n0.next\n    val n1 = p?.next\n    n0.next = n1\n}\n</code></pre> linked_list.rb<pre><code>### Remove the first node after node n0 in the linked list ###\ndef remove(n0)\n  return if n0.next.nil?\n\n  # n0 -&gt; remove_node -&gt; n1\n  remove_node = n0.next\n  n1 = remove_node.next\n  n0.next = n1\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.2   Linked lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#4-accessing-a-node","level":3,"title":"4.   Accessing a node","text":"<p>Accessing a node in a linked list is relatively slow. An array provides access to any element in \\(O(1)\\) time, but a linked list must be traversed from its head until the desired node is reached. Accessing the \\(i\\)-th node requires \\(i - 1\\) iterations, giving \\(O(n)\\) time complexity.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def access(head: ListNode, index: int) -&gt; ListNode | None:\n    \"\"\"Access the node at index `index` in the linked list\"\"\"\n    for _ in range(index):\n        if not head:\n            return None\n        head = head.next\n    return head\n</code></pre> linked_list.cpp<pre><code>/* Access the node at index `index` in the linked list */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == nullptr)\n            return nullptr;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.java<pre><code>/* Access the node at index `index` in the linked list */\nListNode access(ListNode head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.cs<pre><code>/* Access the node at index `index` in the linked list */\nListNode? Access(ListNode? head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.go<pre><code>/* Access the node at index `index` in the linked list */\nfunc access(head *ListNode, index int) *ListNode {\n    for i := 0; i &lt; index; i++ {\n        if head == nil {\n            return nil\n        }\n        head = head.Next\n    }\n    return head\n}\n</code></pre> linked_list.swift<pre><code>/* Access the node at index `index` in the linked list */\nfunc access(head: ListNode, index: Int) -&gt; ListNode? {\n    var head: ListNode? = head\n    for _ in 0 ..&lt; index {\n        if head == nil {\n            return nil\n        }\n        head = head?.next\n    }\n    return head\n}\n</code></pre> linked_list.js<pre><code>/* Access the node at index `index` in the linked list */\nfunction access(head, index) {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.ts<pre><code>/* Access the node at index `index` in the linked list */\nfunction access(head: ListNode | null, index: number): ListNode | null {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.dart<pre><code>/* Access the node at index `index` in the linked list */\nListNode? access(ListNode? head, int index) {\n  for (var i = 0; i &lt; index; i++) {\n    if (head == null) return null;\n    head = head.next;\n  }\n  return head;\n}\n</code></pre> linked_list.rs<pre><code>/* Access the node at index `index` in the linked list */\npub fn access&lt;T&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, index: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n    fn dfs&lt;T&gt;(\n        head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,\n        index: i32,\n    ) -&gt; Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        if index &lt;= 0 {\n            return head.cloned();\n        }\n\n        if let Some(node) = head {\n            dfs(node.borrow().next.as_ref(), index - 1)\n        } else {\n            None\n        }\n    }\n\n    dfs(Some(head).as_ref(), index)\n}\n</code></pre> linked_list.c<pre><code>/* Access the node at index `index` in the linked list */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == NULL)\n            return NULL;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.kt<pre><code>/* Access the node at index `index` in the linked list */\nfun access(head: ListNode?, index: Int): ListNode? {\n    var h = head\n    for (i in 0..&lt;index) {\n        if (h == null)\n            return null\n        h = h.next\n    }\n    return h\n}\n</code></pre> linked_list.rb<pre><code>### Access the node at index `index` in the linked list ###\ndef access(head, index)\n  for i in 0...index\n    return nil if head.nil?\n    head = head.next\n  end\n\n  head\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.2   Linked lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#5-finding-a-node","level":3,"title":"5.   Finding a node","text":"<p>Traverse the list to find a node with value <code>target</code>, then return its index. This is also linear search. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def find(head: ListNode, target: int) -&gt; int:\n    \"\"\"Find the first node with value target in the linked list\"\"\"\n    index = 0\n    while head:\n        if head.val == target:\n            return index\n        head = head.next\n        index += 1\n    return -1\n</code></pre> linked_list.cpp<pre><code>/* Find the first node with value target in the linked list */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head != nullptr) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.java<pre><code>/* Find the first node with value target in the linked list */\nint find(ListNode head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.cs<pre><code>/* Find the first node with value target in the linked list */\nint Find(ListNode? head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.go<pre><code>/* Find the first node with value target in the linked list */\nfunc findNode(head *ListNode, target int) int {\n    index := 0\n    for head != nil {\n        if head.Val == target {\n            return index\n        }\n        head = head.Next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.swift<pre><code>/* Find the first node with value target in the linked list */\nfunc find(head: ListNode, target: Int) -&gt; Int {\n    var head: ListNode? = head\n    var index = 0\n    while head != nil {\n        if head?.val == target {\n            return index\n        }\n        head = head?.next\n        index += 1\n    }\n    return -1\n}\n</code></pre> linked_list.js<pre><code>/* Find the first node with value target in the linked list */\nfunction find(head, target) {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.ts<pre><code>/* Find the first node with value target in the linked list */\nfunction find(head: ListNode | null, target: number): number {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.dart<pre><code>/* Find the first node with value target in the linked list */\nint find(ListNode? head, int target) {\n  int index = 0;\n  while (head != null) {\n    if (head.val == target) {\n      return index;\n    }\n    head = head.next;\n    index++;\n  }\n  return -1;\n}\n</code></pre> linked_list.rs<pre><code>/* Find the first node with value target in the linked list */\npub fn find&lt;T: PartialEq&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, target: T) -&gt; i32 {\n    fn find&lt;T: PartialEq&gt;(head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, target: T, idx: i32) -&gt; i32 {\n        if let Some(node) = head {\n            if node.borrow().val == target {\n                return idx;\n            }\n            return find(node.borrow().next.as_ref(), target, idx + 1);\n        } else {\n            -1\n        }\n    }\n\n    find(Some(head).as_ref(), target, 0)\n}\n</code></pre> linked_list.c<pre><code>/* Find the first node with value target in the linked list */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.kt<pre><code>/* Find the first node with value target in the linked list */\nfun find(head: ListNode?, target: Int): Int {\n    var index = 0\n    var h = head\n    while (h != null) {\n        if (h._val == target)\n            return index\n        h = h.next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.rb<pre><code>### Find the first node with value target in the linked list ###\ndef find(head, target)\n  index = 0\n  while head\n    return index if head.val == target\n    head = head.next\n    index += 1\n  end\n\n  -1\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.2   Linked lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#422-arrays-versus-linked-lists","level":2,"title":"4.2.2   Arrays versus linked lists","text":"<p>Table 4-1 compares their properties and operation costs. Their contrasting storage strategies give them contrasting strengths.</p> <p> Table 4-1   Comparing array and linked list efficiency </p> Array Linked list Storage Contiguous memory Scattered memory Expansion Fixed length Flexible expansion Memory efficiency Less memory per element, but may waste space More memory per element Access an element \\(O(1)\\) \\(O(n)\\) Add an element \\(O(n)\\) \\(O(1)\\) Remove an element \\(O(n)\\) \\(O(1)\\)","path":["Chapter 4.   Arrays and linked lists","4.2   Linked lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#423-common-types-of-linked-lists","level":2,"title":"4.2.3   Common types of linked lists","text":"<p>Figure 4-8 shows three common types.</p> <ul> <li>Singly linked list: The ordinary linked list introduced above. Each node stores a value and a reference to the next node. The first and last nodes are the head and tail, and the tail points to <code>None</code>.</li> <li>Circular linked list: Connecting a singly linked list's tail to its head forms a circular list. Any node can then be treated as the head.</li> <li>Doubly linked list: Each node stores references (pointers) in both directions, to its successor and predecessor. This makes it more flexible and allows traversal in either direction, at the cost of additional memory.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class ListNode:\n    \"\"\"Doubly linked list node class\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # Node value\n        self.next: ListNode | None = None  # Reference to the successor\n        self.prev: ListNode | None = None  # Reference to the predecessor\n</code></pre> <pre><code>/* Doubly linked list node structure */\nstruct ListNode {\n    int val;         // Node value\n    ListNode *next;  // Pointer to the successor\n    ListNode *prev;  // Pointer to the predecessor\n    ListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // Constructor\n};\n</code></pre> <pre><code>/* Doubly linked list node class */\nclass ListNode {\n    int val;        // Node value\n    ListNode next;  // Reference to the successor\n    ListNode prev;  // Reference to the predecessor\n    ListNode(int x) { val = x; }  // Constructor\n}\n</code></pre> <pre><code>/* Doubly linked list node class */\nclass ListNode(int x) {  // Constructor\n    int val = x;    // Node value\n    ListNode next;  // Reference to the successor\n    ListNode prev;  // Reference to the predecessor\n}\n</code></pre> <pre><code>/* Doubly linked list node structure */\ntype DoublyListNode struct {\n    Val  int             // Node value\n    Next *DoublyListNode // Pointer to the successor\n    Prev *DoublyListNode // Pointer to the predecessor\n}\n\n// Initialize a node with NewDoublyListNode\nfunc NewDoublyListNode(val int) *DoublyListNode {\n    return &amp;DoublyListNode{\n        Val:  val,\n        Next: nil,\n        Prev: nil,\n    }\n}\n</code></pre> <pre><code>/* Doubly linked list node class */\nclass ListNode {\n    var val: Int // Node value\n    var next: ListNode? // Reference to the successor\n    var prev: ListNode? // Reference to the predecessor\n\n    init(x: Int) { // Constructor\n        val = x\n    }\n}\n</code></pre> <pre><code>/* Doubly linked list node class */\nclass ListNode {\n    constructor(val, next, prev) {\n        this.val = val  ===  undefined ? 0 : val;        // Node value\n        this.next = next  ===  undefined ? null : next;  // Reference to the successor\n        this.prev = prev  ===  undefined ? null : prev;  // Reference to the predecessor\n    }\n}\n</code></pre> <pre><code>/* Doubly linked list node class */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    prev: ListNode | null;\n    constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\n        this.val = val  ===  undefined ? 0 : val;        // Node value\n        this.next = next  ===  undefined ? null : next;  // Reference to the successor\n        this.prev = prev  ===  undefined ? null : prev;  // Reference to the predecessor\n    }\n}\n</code></pre> <pre><code>/* Doubly linked list node class */\nclass ListNode {\n    int val;        // Node value\n    ListNode? next;  // Reference to the successor\n    ListNode? prev;  // Reference to the predecessor\n    ListNode(this.val, [this.next, this.prev]);  // Constructor\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* Doubly linked list node type */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // Node value\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // Pointer to the successor\n    prev: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // Pointer to the predecessor\n}\n\n/* Constructor */\nimpl ListNode {\n    fn new(val: i32) -&gt; Self {\n        ListNode {\n            val,\n            next: None,\n            prev: None,\n        }\n    }\n}\n</code></pre> <pre><code>/* Doubly linked list node structure */\ntypedef struct ListNode {\n    int val;               // Node value\n    struct ListNode *next; // Pointer to the successor\n    struct ListNode *prev; // Pointer to the predecessor\n} ListNode;\n\n/* Constructor */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    node-&gt;prev = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* Doubly linked list node class */\n// Constructor\nclass ListNode(x: Int) {\n    val _val: Int = x           // Node value\n    var next: ListNode? = null  // Reference to the successor\n    var prev: ListNode? = null  // Reference to the predecessor\n}\n</code></pre> <pre><code># Doubly linked list node class\nclass ListNode\n  attr_accessor :val    # Node value\n  attr_accessor :next   # Reference to the successor\n  attr_accessor :prev   # Reference to the predecessor\n\n  def initialize(val=0, next_node=nil, prev_node=nil)\n    @val = val\n    @next = next_node\n    @prev = prev_node\n  end\nend\n</code></pre> <p></p> <p> Figure 4-8   Common types of linked lists </p>","path":["Chapter 4.   Arrays and linked lists","4.2   Linked lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#424-typical-applications-of-linked-lists","level":2,"title":"4.2.4   Typical applications of linked lists","text":"<p>Singly linked lists commonly implement stacks, queues, hash tables, and graphs.</p> <ul> <li>Stacks and queues: Inserting and removing at the same end gives last-in, first-out behavior, forming a stack. Inserting at one end and removing at the other gives first-in, first-out behavior, forming a queue.</li> <li>Hash tables: Separate chaining is a common collision-handling strategy that places colliding elements in a linked list.</li> <li>Graphs: An adjacency list associates each vertex with a linked list whose elements represent the other vertices connected to it.</li> </ul> <p>Doubly linked lists are useful when both the previous and next elements must be found quickly.</p> <ul> <li>Advanced data structures: In structures such as red-black trees and B-trees, a node may need access to its parent. Storing a parent reference is similar to the backward link in a doubly linked list.</li> <li>Browser history: The forward and back buttons require access to the next and previous visited pages. A doubly linked list makes this straightforward.</li> <li>LRU caching: Least recently used eviction needs fast access to the least recently used data and efficient insertion and removal, making doubly linked lists a good fit.</li> </ul> <p>Circular linked lists suit cyclic operations, such as operating-system resource scheduling.</p> <ul> <li>Round-robin scheduling: This common CPU scheduling algorithm cycles through processes, giving each a time slice before moving to the next. A circular linked list can implement this cycle.</li> <li>Data buffers: Some buffers use circular linked lists. Audio and video players, for example, may divide a stream into buffer blocks arranged in a circular list for seamless playback.</li> </ul>","path":["Chapter 4.   Arrays and linked lists","4.2   Linked lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/","level":1,"title":"4.3   Lists","text":"<p>A list is an abstract data structure representing an ordered collection. It supports access, modification, insertion, removal, and traversal without requiring users to manage capacity limits. Lists can be implemented with linked lists or arrays.</p> <ul> <li>A linked list naturally serves as a list, supporting these operations and flexible dynamic growth.</li> <li>An array also supports these operations, but its fixed length makes it a list with a length limit.</li> </ul> <p>When an array implements a list, its fixed length limits the list's usefulness. We often cannot predict how much data we will store. Too small an allocation may not meet our needs; too large an allocation wastes memory.</p> <p>A dynamic array solves this problem. It retains the advantages of an array while allowing capacity to grow during execution.</p> <p>Many standard-library lists use dynamic arrays, including Python's <code>list</code>, Java's <code>ArrayList</code>, C++'s <code>vector</code>, and C#'s <code>List</code>. In the discussion that follows, we treat lists and dynamic arrays as equivalent concepts.</p>","path":["Chapter 4.   Arrays and linked lists","4.3   Lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#431-common-list-operations","level":2,"title":"4.3.1   Common list operations","text":"","path":["Chapter 4.   Arrays and linked lists","4.3   Lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#1-initializing-a-list","level":3,"title":"1.   Initializing a list","text":"<p>We usually initialize a list either without elements or with explicit initial values:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># Initialize lists\n# Without initial values\nnums1: list[int] = []\n# With initial values\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> list.cpp<pre><code>/* Initialize lists */\n// In C++, vector is the list type used for nums here\n// Without initial values\nvector&lt;int&gt; nums1;\n// With initial values\nvector&lt;int&gt; nums = { 1, 3, 2, 5, 4 };\n</code></pre> list.java<pre><code>/* Initialize lists */\n// Without initial values\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;();\n// With initial values (use Integer[], the boxed counterpart of int[])\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList&lt;Integer&gt; nums = new ArrayList&lt;&gt;(Arrays.asList(numbers));\n</code></pre> list.cs<pre><code>/* Initialize lists */\n// Without initial values\nList&lt;int&gt; nums1 = [];\n// With initial values\nint[] numbers = [1, 3, 2, 5, 4];\nList&lt;int&gt; nums = [.. numbers];\n</code></pre> list_test.go<pre><code>/* Initialize lists */\n// Without initial values\nnums1 := []int{}\n// With initial values\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> list.swift<pre><code>/* Initialize lists */\n// Without initial values\nlet nums1: [Int] = []\n// With initial values\nvar nums = [1, 3, 2, 5, 4]\n</code></pre> list.js<pre><code>/* Initialize lists */\n// Without initial values\nconst nums1 = [];\n// With initial values\nconst nums = [1, 3, 2, 5, 4];\n</code></pre> list.ts<pre><code>/* Initialize lists */\n// Without initial values\nconst nums1: number[] = [];\n// With initial values\nconst nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> list.dart<pre><code>/* Initialize lists */\n// Without initial values\nList&lt;int&gt; nums1 = [];\n// With initial values\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> list.rs<pre><code>/* Initialize lists */\n// Without initial values\nlet nums1: Vec&lt;i32&gt; = Vec::new();\n// With initial values\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> list.c<pre><code>// C does not provide a built-in dynamic array\n</code></pre> list.kt<pre><code>/* Initialize lists */\n// Without initial values\nvar nums1 = listOf&lt;Int&gt;()\n// With initial values\nvar numbers = arrayOf(1, 3, 2, 5, 4)\nvar nums = numbers.toMutableList()\n</code></pre> list.rb<pre><code># Initialize lists\n# Without initial values\nnums1 = []\n# With initial values\nnums = [1, 3, 2, 5, 4]\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.3   Lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#2-accessing-an-element","level":3,"title":"2.   Accessing an element","text":"<p>Since the list is based on an array, accessing and updating an element are efficient \\(O(1)\\) operations.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># Access an element\nnum: int = nums[1]  # Access the element at index 1\n\n# Update an element\nnums[1] = 0    # Set the element at index 1 to 0\n</code></pre> list.cpp<pre><code>/* Access an element */\nint num = nums[1];  // Access the element at index 1\n\n/* Update an element */\nnums[1] = 0;  // Set the element at index 1 to 0\n</code></pre> list.java<pre><code>/* Access an element */\nint num = nums.get(1);  // Access the element at index 1\n\n/* Update an element */\nnums.set(1, 0);  // Set the element at index 1 to 0\n</code></pre> list.cs<pre><code>/* Access an element */\nint num = nums[1];  // Access the element at index 1\n\n/* Update an element */\nnums[1] = 0;  // Set the element at index 1 to 0\n</code></pre> list_test.go<pre><code>/* Access an element */\nnum := nums[1]  // Access the element at index 1\n\n/* Update an element */\nnums[1] = 0     // Set the element at index 1 to 0\n</code></pre> list.swift<pre><code>/* Access an element */\nlet num = nums[1] // Access the element at index 1\n\n/* Update an element */\nnums[1] = 0 // Set the element at index 1 to 0\n</code></pre> list.js<pre><code>/* Access an element */\nconst num = nums[1];  // Access the element at index 1\n\n/* Update an element */\nnums[1] = 0;  // Set the element at index 1 to 0\n</code></pre> list.ts<pre><code>/* Access an element */\nconst num: number = nums[1];  // Access the element at index 1\n\n/* Update an element */\nnums[1] = 0;  // Set the element at index 1 to 0\n</code></pre> list.dart<pre><code>/* Access an element */\nint num = nums[1];  // Access the element at index 1\n\n/* Update an element */\nnums[1] = 0;  // Set the element at index 1 to 0\n</code></pre> list.rs<pre><code>/* Access an element */\nlet num: i32 = nums[1];  // Access the element at index 1\n/* Update an element */\nnums[1] = 0;             // Set the element at index 1 to 0\n</code></pre> list.c<pre><code>// C does not provide a built-in dynamic array\n</code></pre> list.kt<pre><code>/* Access an element */\nval num = nums[1]       // Access the element at index 1\n/* Update an element */\nnums[1] = 0             // Set the element at index 1 to 0\n</code></pre> list.rb<pre><code># Access an element\nnum = nums[1] # Access the element at index 1\n# Update an element\nnums[1] = 0 # Set the element at index 1 to 0\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.3   Lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#3-inserting-and-removing-elements","level":3,"title":"3.   Inserting and removing elements","text":"<p>Unlike a fixed array, a list can freely add and remove elements. Appending takes \\(O(1)\\) time, while insertion and removal retain the array's \\(O(n)\\) time complexity.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># Clear the list\nnums.clear()\n\n# Append elements\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n# Insert an element in the middle\nnums.insert(3, 6)  # Insert 6 at index 3\n\n# Remove an element\nnums.pop(3)        # Remove the element at index 3\n</code></pre> list.cpp<pre><code>/* Clear the list */\nnums.clear();\n\n/* Append elements */\nnums.push_back(1);\nnums.push_back(3);\nnums.push_back(2);\nnums.push_back(5);\nnums.push_back(4);\n\n/* Insert an element in the middle */\nnums.insert(nums.begin() + 3, 6);  // Insert 6 at index 3\n\n/* Remove an element */\nnums.erase(nums.begin() + 3);      // Remove the element at index 3\n</code></pre> list.java<pre><code>/* Clear the list */\nnums.clear();\n\n/* Append elements */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* Insert an element in the middle */\nnums.add(3, 6);  // Insert 6 at index 3\n\n/* Remove an element */\nnums.remove(3);  // Remove the element at index 3\n</code></pre> list.cs<pre><code>/* Clear the list */\nnums.Clear();\n\n/* Append elements */\nnums.Add(1);\nnums.Add(3);\nnums.Add(2);\nnums.Add(5);\nnums.Add(4);\n\n/* Insert an element in the middle */\nnums.Insert(3, 6);  // Insert 6 at index 3\n\n/* Remove an element */\nnums.RemoveAt(3);  // Remove the element at index 3\n</code></pre> list_test.go<pre><code>/* Clear the list */\nnums = nil\n\n/* Append elements */\nnums = append(nums, 1)\nnums = append(nums, 3)\nnums = append(nums, 2)\nnums = append(nums, 5)\nnums = append(nums, 4)\n\n/* Insert an element in the middle */\nnums = append(nums[:3], append([]int{6}, nums[3:]...)...) // Insert 6 at index 3\n\n/* Remove an element */\nnums = append(nums[:3], nums[4:]...) // Remove the element at index 3\n</code></pre> list.swift<pre><code>/* Clear the list */\nnums.removeAll()\n\n/* Append elements */\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n/* Insert an element in the middle */\nnums.insert(6, at: 3) // Insert 6 at index 3\n\n/* Remove an element */\nnums.remove(at: 3) // Remove the element at index 3\n</code></pre> list.js<pre><code>/* Clear the list */\nnums.length = 0;\n\n/* Append elements */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* Insert an element in the middle */\nnums.splice(3, 0, 6); // Insert 6 at index 3\n\n/* Remove an element */\nnums.splice(3, 1);  // Remove the element at index 3\n</code></pre> list.ts<pre><code>/* Clear the list */\nnums.length = 0;\n\n/* Append elements */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* Insert an element in the middle */\nnums.splice(3, 0, 6); // Insert 6 at index 3\n\n/* Remove an element */\nnums.splice(3, 1);  // Remove the element at index 3\n</code></pre> list.dart<pre><code>/* Clear the list */\nnums.clear();\n\n/* Append elements */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* Insert an element in the middle */\nnums.insert(3, 6); // Insert 6 at index 3\n\n/* Remove an element */\nnums.removeAt(3); // Remove the element at index 3\n</code></pre> list.rs<pre><code>/* Clear the list */\nnums.clear();\n\n/* Append elements */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* Insert an element in the middle */\nnums.insert(3, 6);  // Insert 6 at index 3\n\n/* Remove an element */\nnums.remove(3);    // Remove the element at index 3\n</code></pre> list.c<pre><code>// C does not provide a built-in dynamic array\n</code></pre> list.kt<pre><code>/* Clear the list */\nnums.clear();\n\n/* Append elements */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* Insert an element in the middle */\nnums.add(3, 6);  // Insert 6 at index 3\n\n/* Remove an element */\nnums.remove(3);  // Remove the element at index 3\n</code></pre> list.rb<pre><code># Clear the list\nnums.clear\n\n# Append elements\nnums &lt;&lt; 1\nnums &lt;&lt; 3\nnums &lt;&lt; 2\nnums &lt;&lt; 5\nnums &lt;&lt; 4\n\n# Insert an element in the middle\nnums.insert(3, 6) # Insert 6 at index 3\n\n# Remove an element\nnums.delete_at(3) # Remove the element at index 3\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.3   Lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#4-traversing-a-list","level":3,"title":"4.   Traversing a list","text":"<p>Like an array, a list can be traversed by index or by visiting its elements directly.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># Traverse the list by index\ncount = 0\nfor i in range(len(nums)):\n    count += nums[i]\n\n# Traverse list elements directly\nfor num in nums:\n    count += num\n</code></pre> list.cpp<pre><code>/* Traverse the list by index */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums[i];\n}\n\n/* Traverse list elements directly */\ncount = 0;\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.java<pre><code>/* Traverse the list by index */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums.get(i);\n}\n\n/* Traverse list elements directly */\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.cs<pre><code>/* Traverse the list by index */\nint count = 0;\nfor (int i = 0; i &lt; nums.Count; i++) {\n    count += nums[i];\n}\n\n/* Traverse list elements directly */\ncount = 0;\nforeach (int num in nums) {\n    count += num;\n}\n</code></pre> list_test.go<pre><code>/* Traverse the list by index */\ncount := 0\nfor i := 0; i &lt; len(nums); i++ {\n    count += nums[i]\n}\n\n/* Traverse list elements directly */\ncount = 0\nfor _, num := range nums {\n    count += num\n}\n</code></pre> list.swift<pre><code>/* Traverse the list by index */\nvar count = 0\nfor i in nums.indices {\n    count += nums[i]\n}\n\n/* Traverse list elements directly */\ncount = 0\nfor num in nums {\n    count += num\n}\n</code></pre> list.js<pre><code>/* Traverse the list by index */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* Traverse list elements directly */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.ts<pre><code>/* Traverse the list by index */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* Traverse list elements directly */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.dart<pre><code>/* Traverse the list by index */\nint count = 0;\nfor (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* Traverse list elements directly */\ncount = 0;\nfor (var num in nums) {\n    count += num;\n}\n</code></pre> list.rs<pre><code>// Traverse the list by index\nlet mut _count = 0;\nfor i in 0..nums.len() {\n    _count += nums[i];\n}\n\n// Traverse list elements directly\n_count = 0;\nfor num in &amp;nums {\n    _count += num;\n}\n</code></pre> list.c<pre><code>// C does not provide a built-in dynamic array\n</code></pre> list.kt<pre><code>/* Traverse the list by index */\nvar count = 0\nfor (i in nums.indices) {\n    count += nums[i]\n}\n\n/* Traverse list elements directly */\nfor (num in nums) {\n    count += num\n}\n</code></pre> list.rb<pre><code># Traverse the list by index\ncount = 0\nfor i in 0...nums.length\n    count += nums[i]\nend\n\n# Traverse list elements directly\ncount = 0\nfor num in nums\n    count += num\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.3   Lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#5-concatenating-lists","level":3,"title":"5.   Concatenating lists","text":"<p>Given another list <code>nums1</code>, we can append its elements to the end of the original list.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># Concatenate two lists\nnums1: list[int] = [6, 8, 7, 10, 9]\nnums += nums1  # Append nums1 to nums\n</code></pre> list.cpp<pre><code>/* Concatenate two lists */\nvector&lt;int&gt; nums1 = { 6, 8, 7, 10, 9 };\n// Append nums1 to nums\nnums.insert(nums.end(), nums1.begin(), nums1.end());\n</code></pre> list.java<pre><code>/* Concatenate two lists */\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nnums.addAll(nums1);  // Append nums1 to nums\n</code></pre> list.cs<pre><code>/* Concatenate two lists */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.AddRange(nums1);  // Append nums1 to nums\n</code></pre> list_test.go<pre><code>/* Concatenate two lists */\nnums1 := []int{6, 8, 7, 10, 9}\nnums = append(nums, nums1...)  // Append nums1 to nums\n</code></pre> list.swift<pre><code>/* Concatenate two lists */\nlet nums1 = [6, 8, 7, 10, 9]\nnums.append(contentsOf: nums1) // Append nums1 to nums\n</code></pre> list.js<pre><code>/* Concatenate two lists */\nconst nums1 = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // Append nums1 to nums\n</code></pre> list.ts<pre><code>/* Concatenate two lists */\nconst nums1: number[] = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // Append nums1 to nums\n</code></pre> list.dart<pre><code>/* Concatenate two lists */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.addAll(nums1);  // Append nums1 to nums\n</code></pre> list.rs<pre><code>/* Concatenate two lists */\nlet nums1: Vec&lt;i32&gt; = vec![6, 8, 7, 10, 9];\nnums.extend(nums1);\n</code></pre> list.c<pre><code>// C does not provide a built-in dynamic array\n</code></pre> list.kt<pre><code>/* Concatenate two lists */\nval nums1 = intArrayOf(6, 8, 7, 10, 9).toMutableList()\nnums.addAll(nums1)  // Append nums1 to nums\n</code></pre> list.rb<pre><code># Concatenate two lists\nnums1 = [6, 8, 7, 10, 9]\nnums += nums1\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.3   Lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#6-sorting-a-list","level":3,"title":"6.   Sorting a list","text":"<p>Once the list is sorted, we can apply binary search and two-pointer algorithms, which often appear in array problems.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># Sort the list\nnums.sort()  # After sorting, elements are in ascending order\n</code></pre> list.cpp<pre><code>/* Sort the list */\nsort(nums.begin(), nums.end());  // After sorting, elements are in ascending order\n</code></pre> list.java<pre><code>/* Sort the list */\nCollections.sort(nums);  // After sorting, elements are in ascending order\n</code></pre> list.cs<pre><code>/* Sort the list */\nnums.Sort(); // After sorting, elements are in ascending order\n</code></pre> list_test.go<pre><code>/* Sort the list */\nsort.Ints(nums)  // After sorting, elements are in ascending order\n</code></pre> list.swift<pre><code>/* Sort the list */\nnums.sort() // After sorting, elements are in ascending order\n</code></pre> list.js<pre><code>/* Sort the list */\nnums.sort((a, b) =&gt; a - b);  // After sorting, elements are in ascending order\n</code></pre> list.ts<pre><code>/* Sort the list */\nnums.sort((a, b) =&gt; a - b);  // After sorting, elements are in ascending order\n</code></pre> list.dart<pre><code>/* Sort the list */\nnums.sort(); // After sorting, elements are in ascending order\n</code></pre> list.rs<pre><code>/* Sort the list */\nnums.sort(); // After sorting, elements are in ascending order\n</code></pre> list.c<pre><code>// C does not provide a built-in dynamic array\n</code></pre> list.kt<pre><code>/* Sort the list */\nnums.sort() // After sorting, elements are in ascending order\n</code></pre> list.rb<pre><code># Sort the list\nnums = nums.sort { |a, b| a &lt;=&gt; b } # After sorting, elements are in ascending order\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.3   Lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#432-implementing-a-list","level":2,"title":"4.3.2   Implementing a list","text":"<p>Many languages, including Java, C++, and Python, provide lists. Their implementations are sophisticated, with carefully chosen parameters such as initial capacity and growth factor. Interested readers can study their source code.</p> <p>To understand how a list works, let's implement a simplified version with three key design choices.</p> <ul> <li>Initial capacity: Choose a reasonable starting array capacity. We use 10 here.</li> <li>Element count: Maintain <code>size</code>, the number of elements, updating it on insertion and removal. It identifies the list's end and tells us when expansion is needed.</li> <li>Growth mechanism: If insertion finds the capacity full, allocate a larger array using a growth factor and copy all current elements into it. Here, capacity doubles each time.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_list.py<pre><code>class MyList:\n    \"\"\"List class\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        self._capacity: int = 10  # List capacity\n        self._arr: list[int] = [0] * self._capacity  # Array (stores the list elements)\n        self._size: int = 0  # List length (current number of elements)\n        self._extend_ratio: int = 2  # Factor by which the list capacity grows each time\n\n    def size(self) -&gt; int:\n        \"\"\"Get the list length (current number of elements)\"\"\"\n        return self._size\n\n    def capacity(self) -&gt; int:\n        \"\"\"Get the list capacity\"\"\"\n        return self._capacity\n\n    def get(self, index: int) -&gt; int:\n        \"\"\"Access an element\"\"\"\n        # Raise an exception if the index is out of bounds; the same applies below\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"Index out of bounds\")\n        return self._arr[index]\n\n    def set(self, num: int, index: int):\n        \"\"\"Update an element\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"Index out of bounds\")\n        self._arr[index] = num\n\n    def add(self, num: int):\n        \"\"\"Append an element at the end\"\"\"\n        # Grow the capacity when adding an element would exceed it\n        if self.size() == self.capacity():\n            self.extend_capacity()\n        self._arr[self._size] = num\n        self._size += 1\n\n    def insert(self, num: int, index: int):\n        \"\"\"Insert an element in the middle\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"Index out of bounds\")\n        # Grow the capacity when adding an element would exceed it\n        if self._size == self.capacity():\n            self.extend_capacity()\n        # Shift all elements at and after index `index` one position to the right\n        for j in range(self._size - 1, index - 1, -1):\n            self._arr[j + 1] = self._arr[j]\n        self._arr[index] = num\n        # Update the number of elements\n        self._size += 1\n\n    def remove(self, index: int) -&gt; int:\n        \"\"\"Remove an element\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"Index out of bounds\")\n        num = self._arr[index]\n        # Shift all elements after index `index` one position to the left\n        for j in range(index, self._size - 1):\n            self._arr[j] = self._arr[j + 1]\n        # Update the number of elements\n        self._size -= 1\n        # Return the removed element\n        return num\n\n    def extend_capacity(self):\n        \"\"\"Grow the list capacity\"\"\"\n        # Create a new array with _extend_ratio times the original length and copy the original array into it\n        self._arr = self._arr + [0] * self.capacity() * (self._extend_ratio - 1)\n        # Update the list capacity\n        self._capacity = len(self._arr)\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"Return the portion of the list that contains elements\"\"\"\n        return self._arr[: self._size]\n</code></pre> my_list.cpp<pre><code>/* List class */\nclass MyList {\n  private:\n    int *arr;             // Array (stores the list elements)\n    int arrCapacity = 10; // List capacity\n    int arrSize = 0;      // List length (current number of elements)\n    int extendRatio = 2;   // Factor by which the list capacity grows each time\n\n  public:\n    /* Constructor */\n    MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* Destructor */\n    ~MyList() {\n        delete[] arr;\n    }\n\n    /* Get the list length (current number of elements)*/\n    int size() {\n        return arrSize;\n    }\n\n    /* Get the list capacity */\n    int capacity() {\n        return arrCapacity;\n    }\n\n    /* Access an element */\n    int get(int index) {\n        // Raise an exception if the index is out of bounds; the same applies below\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"Index out of bounds\");\n        return arr[index];\n    }\n\n    /* Update an element */\n    void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"Index out of bounds\");\n        arr[index] = num;\n    }\n\n    /* Append an element at the end */\n    void add(int num) {\n        // Grow the capacity when adding an element would exceed it\n        if (size() == capacity())\n            extendCapacity();\n        arr[size()] = num;\n        // Update the number of elements\n        arrSize++;\n    }\n\n    /* Insert an element in the middle */\n    void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"Index out of bounds\");\n        // Grow the capacity when adding an element would exceed it\n        if (size() == capacity())\n            extendCapacity();\n        // Shift all elements at and after index `index` one position to the right\n        for (int j = size() - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // Update the number of elements\n        arrSize++;\n    }\n\n    /* Remove an element */\n    int remove(int index) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"Index out of bounds\");\n        int num = arr[index];\n        // Shift all elements after index `index` one position to the left\n        for (int j = index; j &lt; size() - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // Update the number of elements\n        arrSize--;\n        // Return the removed element\n        return num;\n    }\n\n    /* Grow the list capacity */\n    void extendCapacity() {\n        // Create a new array with extendRatio times the original array's length\n        int newCapacity = capacity() * extendRatio;\n        int *tmp = arr;\n        arr = new int[newCapacity];\n        // Copy all elements from the original array to the new array\n        for (int i = 0; i &lt; size(); i++) {\n            arr[i] = tmp[i];\n        }\n        // Free memory\n        delete[] tmp;\n        arrCapacity = newCapacity;\n    }\n\n    /* Convert the list to a vector for printing */\n    vector&lt;int&gt; toVector() {\n        // Convert only the list elements within the occupied range\n        vector&lt;int&gt; vec(size());\n        for (int i = 0; i &lt; size(); i++) {\n            vec[i] = arr[i];\n        }\n        return vec;\n    }\n};\n</code></pre> my_list.java<pre><code>/* List class */\nclass MyList {\n    private int[] arr; // Array (stores the list elements)\n    private int capacity = 10; // List capacity\n    private int size = 0; // List length (current number of elements)\n    private int extendRatio = 2; // Factor by which the list capacity grows each time\n\n    /* Constructor */\n    public MyList() {\n        arr = new int[capacity];\n    }\n\n    /* Get the list length (current number of elements) */\n    public int size() {\n        return size;\n    }\n\n    /* Get the list capacity */\n    public int capacity() {\n        return capacity;\n    }\n\n    /* Access an element */\n    public int get(int index) {\n        // Throw an exception if the index is out of bounds; the same applies below\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"Index out of bounds\");\n        return arr[index];\n    }\n\n    /* Update an element */\n    public void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"Index out of bounds\");\n        arr[index] = num;\n    }\n\n    /* Append at the end */\n    public void add(int num) {\n        // Grow the capacity when adding an element would exceed it\n        if (size == capacity())\n            extendCapacity();\n        arr[size] = num;\n        // Update the number of elements\n        size++;\n    }\n\n    /* Insert an element in the middle */\n    public void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"Index out of bounds\");\n        // Grow the capacity when adding an element would exceed it\n        if (size == capacity())\n            extendCapacity();\n        // Shift all elements at and after index `index` one position to the right\n        for (int j = size - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // Update the number of elements\n        size++;\n    }\n\n    /* Remove an element */\n    public int remove(int index) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"Index out of bounds\");\n        int num = arr[index];\n        // Shift all elements after index `index` one position to the left\n        for (int j = index; j &lt; size - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // Update the number of elements\n        size--;\n        // Return the removed element\n        return num;\n    }\n\n    /* Grow the list capacity */\n    public void extendCapacity() {\n        // Create an array extendRatio times as long as the original and copy the original array into it\n        arr = Arrays.copyOf(arr, capacity() * extendRatio);\n        // Update the list capacity\n        capacity = arr.length;\n    }\n\n    /* Convert the list to an array */\n    public int[] toArray() {\n        int size = size();\n        // Convert only the list elements within the occupied range\n        int[] arr = new int[size];\n        for (int i = 0; i &lt; size; i++) {\n            arr[i] = get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.cs<pre><code>/* List class */\nclass MyList {\n    private int[] arr;           // Array (stores the list elements)\n    private int arrCapacity = 10;    // List capacity\n    private int arrSize = 0;         // List length (current number of elements)\n    private readonly int extendRatio = 2;  // Factor by which the list capacity grows each time\n\n    /* Constructor */\n    public MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* Get the list length (current number of elements)*/\n    public int Size() {\n        return arrSize;\n    }\n\n    /* Get the list capacity */\n    public int Capacity() {\n        return arrCapacity;\n    }\n\n    /* Access an element */\n    public int Get(int index) {\n        // Throw an exception if the index is out of bounds; the same applies below\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"Index out of bounds\");\n        return arr[index];\n    }\n\n    /* Update an element */\n    public void Set(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"Index out of bounds\");\n        arr[index] = num;\n    }\n\n    /* Append at the end */\n    public void Add(int num) {\n        // Grow the capacity when adding an element would exceed it\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        arr[arrSize] = num;\n        // Update the number of elements\n        arrSize++;\n    }\n\n    /* Insert an element in the middle */\n    public void Insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"Index out of bounds\");\n        // Grow the capacity when adding an element would exceed it\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        // Shift all elements at and after index `index` one position to the right\n        for (int j = arrSize - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // Update the number of elements\n        arrSize++;\n    }\n\n    /* Remove an element */\n    public int Remove(int index) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"Index out of bounds\");\n        int num = arr[index];\n        // Shift all elements after index `index` one position to the left\n        for (int j = index; j &lt; arrSize - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // Update the number of elements\n        arrSize--;\n        // Return the removed element\n        return num;\n    }\n\n    /* Grow the list capacity */\n    public void ExtendCapacity() {\n        // Create an array of length arrCapacity * extendRatio and copy the original array into it\n        Array.Resize(ref arr, arrCapacity * extendRatio);\n        // Update the list capacity\n        arrCapacity = arr.Length;\n    }\n\n    /* Convert the list to an array */\n    public int[] ToArray() {\n        // Convert only the list elements within the occupied range\n        int[] arr = new int[arrSize];\n        for (int i = 0; i &lt; arrSize; i++) {\n            arr[i] = Get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.go<pre><code>/* List type */\ntype myList struct {\n    arrCapacity int\n    arr         []int\n    arrSize     int\n    extendRatio int\n}\n\n/* Constructor */\nfunc newMyList() *myList {\n    return &amp;myList{\n        arrCapacity: 10,              // List capacity\n        arr:         make([]int, 10), // Array (stores the list elements)\n        arrSize:     0,               // List length (current number of elements)\n        extendRatio: 2,               // Factor by which the list capacity grows each time\n    }\n}\n\n/* Get the list length (current number of elements) */\nfunc (l *myList) size() int {\n    return l.arrSize\n}\n\n/*  Get the list capacity */\nfunc (l *myList) capacity() int {\n    return l.arrCapacity\n}\n\n/* Access an element */\nfunc (l *myList) get(index int) int {\n    // Panic if the index is out of bounds; the same applies below\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"Index out of bounds\")\n    }\n    return l.arr[index]\n}\n\n/* Update an element */\nfunc (l *myList) set(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"Index out of bounds\")\n    }\n    l.arr[index] = num\n}\n\n/* Append at the end */\nfunc (l *myList) add(num int) {\n    // Grow the capacity when adding an element would exceed it\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    l.arr[l.arrSize] = num\n    // Update the number of elements\n    l.arrSize++\n}\n\n/* Insert an element in the middle */\nfunc (l *myList) insert(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"Index out of bounds\")\n    }\n    // Grow the capacity when adding an element would exceed it\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    // Shift all elements at and after index `index` one position to the right\n    for j := l.arrSize - 1; j &gt;= index; j-- {\n        l.arr[j+1] = l.arr[j]\n    }\n    l.arr[index] = num\n    // Update the number of elements\n    l.arrSize++\n}\n\n/* Remove an element */\nfunc (l *myList) remove(index int) int {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"Index out of bounds\")\n    }\n    num := l.arr[index]\n    // Shift all elements after index `index` one position to the left\n    for j := index; j &lt; l.arrSize-1; j++ {\n        l.arr[j] = l.arr[j+1]\n    }\n    // Update the number of elements\n    l.arrSize--\n    // Return the removed element\n    return num\n}\n\n/* Grow the list capacity */\nfunc (l *myList) extendCapacity() {\n    // Extend the backing slice to extendRatio times its original length, preserving the existing elements\n    l.arr = append(l.arr, make([]int, l.arrCapacity*(l.extendRatio-1))...)\n    // Update the list capacity\n    l.arrCapacity = len(l.arr)\n}\n\n/* Return the portion of the list that contains elements */\nfunc (l *myList) toArray() []int {\n    // Return a slice containing only the occupied elements\n    return l.arr[:l.arrSize]\n}\n</code></pre> my_list.swift<pre><code>/* List class */\nclass MyList {\n    private var arr: [Int] // Array (stores the list elements)\n    private var _capacity: Int // List capacity\n    private var _size: Int // List length (current number of elements)\n    private let extendRatio: Int // Factor by which the list capacity grows each time\n\n    /* Constructor */\n    init() {\n        _capacity = 10\n        _size = 0\n        extendRatio = 2\n        arr = Array(repeating: 0, count: _capacity)\n    }\n\n    /* Get the list length (current number of elements)*/\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* Get the list capacity */\n    func capacity() -&gt; Int {\n        _capacity\n    }\n\n    /* Access an element */\n    func get(index: Int) -&gt; Int {\n        // An out-of-bounds index triggers a fatal error; the same check is used below\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"Index out of bounds\")\n        }\n        return arr[index]\n    }\n\n    /* Update an element */\n    func set(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"Index out of bounds\")\n        }\n        arr[index] = num\n    }\n\n    /* Append at the end */\n    func add(num: Int) {\n        // Grow the capacity when adding an element would exceed it\n        if size() == capacity() {\n            extendCapacity()\n        }\n        arr[size()] = num\n        // Update the number of elements\n        _size += 1\n    }\n\n    /* Insert an element in the middle */\n    func insert(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"Index out of bounds\")\n        }\n        // Grow the capacity when adding an element would exceed it\n        if size() == capacity() {\n            extendCapacity()\n        }\n        // Shift all elements at and after index `index` one position to the right\n        for j in (index ..&lt; size()).reversed() {\n            arr[j + 1] = arr[j]\n        }\n        arr[index] = num\n        // Update the number of elements\n        _size += 1\n    }\n\n    /* Remove an element */\n    @discardableResult\n    func remove(index: Int) -&gt; Int {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"Index out of bounds\")\n        }\n        let num = arr[index]\n        // Shift all elements after index `index` one position to the left\n        for j in index ..&lt; (size() - 1) {\n            arr[j] = arr[j + 1]\n        }\n        // Update the number of elements\n        _size -= 1\n        // Return the removed element\n        return num\n    }\n\n    /* Grow the list capacity */\n    func extendCapacity() {\n        // Create an array extendRatio times as long as the original and copy the original array into it\n        arr = arr + Array(repeating: 0, count: capacity() * (extendRatio - 1))\n        // Update the list capacity\n        _capacity = arr.count\n    }\n\n    /* Convert the list to an array */\n    func toArray() -&gt; [Int] {\n        Array(arr.prefix(size()))\n    }\n}\n</code></pre> my_list.js<pre><code>/* List class */\nclass MyList {\n    #arr = new Array(); // Array (stores the list elements)\n    #capacity = 10; // List capacity\n    #size = 0; // List length (current number of elements)\n    #extendRatio = 2; // Factor by which the list capacity grows each time\n\n    /* Constructor */\n    constructor() {\n        this.#arr = new Array(this.#capacity);\n    }\n\n    /* Get the list length (current number of elements)*/\n    size() {\n        return this.#size;\n    }\n\n    /* Get the list capacity */\n    capacity() {\n        return this.#capacity;\n    }\n\n    /* Access an element */\n    get(index) {\n        // Throw an exception if the index is out of bounds; the same applies below\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('Index out of bounds');\n        return this.#arr[index];\n    }\n\n    /* Update an element */\n    set(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('Index out of bounds');\n        this.#arr[index] = num;\n    }\n\n    /* Append an element at the end */\n    add(num) {\n        // Grow the capacity if the length equals the capacity\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // Append the new element to the end of the list\n        this.#arr[this.#size] = num;\n        this.#size++;\n    }\n\n    /* Insert an element in the middle */\n    insert(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('Index out of bounds');\n        // Grow the capacity when adding an element would exceed it\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // Shift all elements at and after index `index` one position to the right\n        for (let j = this.#size - 1; j &gt;= index; j--) {\n            this.#arr[j + 1] = this.#arr[j];\n        }\n        // Update the number of elements\n        this.#arr[index] = num;\n        this.#size++;\n    }\n\n    /* Remove an element */\n    remove(index) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('Index out of bounds');\n        let num = this.#arr[index];\n        // Shift all elements after index `index` one position to the left\n        for (let j = index; j &lt; this.#size - 1; j++) {\n            this.#arr[j] = this.#arr[j + 1];\n        }\n        // Update the number of elements\n        this.#size--;\n        // Return the removed element\n        return num;\n    }\n\n    /* Grow the list capacity */\n    extendCapacity() {\n        // Create an array extendRatio times the original length and copy the original array into it\n        this.#arr = this.#arr.concat(\n            new Array(this.capacity() * (this.#extendRatio - 1))\n        );\n        // Update the list capacity\n        this.#capacity = this.#arr.length;\n    }\n\n    /* Convert the list to an array */\n    toArray() {\n        let size = this.size();\n        // Convert only the list elements within the occupied range\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.ts<pre><code>/* List class */\nclass MyList {\n    private arr: Array&lt;number&gt;; // Array (stores the list elements)\n    private _capacity: number = 10; // List capacity\n    private _size: number = 0; // List length (current number of elements)\n    private extendRatio: number = 2; // Factor by which the list capacity grows each time\n\n    /* Constructor */\n    constructor() {\n        this.arr = new Array(this._capacity);\n    }\n\n    /* Get the list length (current number of elements)*/\n    public size(): number {\n        return this._size;\n    }\n\n    /* Get the list capacity */\n    public capacity(): number {\n        return this._capacity;\n    }\n\n    /* Access an element */\n    public get(index: number): number {\n        // Throw an exception if the index is out of bounds; the same applies below\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('Index out of bounds');\n        return this.arr[index];\n    }\n\n    /* Update an element */\n    public set(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('Index out of bounds');\n        this.arr[index] = num;\n    }\n\n    /* Append an element at the end */\n    public add(num: number): void {\n        // Grow the capacity if the length equals the capacity\n        if (this._size === this._capacity) this.extendCapacity();\n        // Append the new element to the end of the list\n        this.arr[this._size] = num;\n        this._size++;\n    }\n\n    /* Insert an element in the middle */\n    public insert(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('Index out of bounds');\n        // Grow the capacity when adding an element would exceed it\n        if (this._size === this._capacity) {\n            this.extendCapacity();\n        }\n        // Shift all elements at and after index `index` one position to the right\n        for (let j = this._size - 1; j &gt;= index; j--) {\n            this.arr[j + 1] = this.arr[j];\n        }\n        // Update the number of elements\n        this.arr[index] = num;\n        this._size++;\n    }\n\n    /* Remove an element */\n    public remove(index: number): number {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('Index out of bounds');\n        let num = this.arr[index];\n        // Shift all elements after index one position to the left\n        for (let j = index; j &lt; this._size - 1; j++) {\n            this.arr[j] = this.arr[j + 1];\n        }\n        // Update the number of elements\n        this._size--;\n        // Return the removed element\n        return num;\n    }\n\n    /* Grow the list capacity */\n    public extendCapacity(): void {\n        // Create a larger array by appending empty slots to a copy of the original array\n        this.arr = this.arr.concat(\n            new Array(this.capacity() * (this.extendRatio - 1))\n        );\n        // Update the list capacity\n        this._capacity = this.arr.length;\n    }\n\n    /* Convert the list to an array */\n    public toArray(): number[] {\n        let size = this.size();\n        // Convert only the list elements within the occupied range\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.dart<pre><code>/* List class */\nclass MyList {\n  late List&lt;int&gt; _arr; // Array (stores the list elements)\n  int _capacity = 10; // List capacity\n  int _size = 0; // List length (current number of elements)\n  int _extendRatio = 2; // Factor by which the list capacity grows each time\n\n  /* Constructor */\n  MyList() {\n    _arr = List.filled(_capacity, 0);\n  }\n\n  /* Get the list length (current number of elements)*/\n  int size() =&gt; _size;\n\n  /* Get the list capacity */\n  int capacity() =&gt; _capacity;\n\n  /* Access an element */\n  int get(int index) {\n    if (index &gt;= _size) throw RangeError('Index out of bounds');\n    return _arr[index];\n  }\n\n  /* Update an element */\n  void set(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('Index out of bounds');\n    _arr[index] = _num;\n  }\n\n  /* Append an element at the end */\n  void add(int _num) {\n    // Grow the capacity when adding an element would exceed it\n    if (_size == _capacity) extendCapacity();\n    _arr[_size] = _num;\n    // Update the number of elements\n    _size++;\n  }\n\n  /* Insert an element in the middle */\n  void insert(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('Index out of bounds');\n    // Grow the capacity when adding an element would exceed it\n    if (_size == _capacity) extendCapacity();\n    // Shift all elements at and after index `index` one position to the right\n    for (var j = _size - 1; j &gt;= index; j--) {\n      _arr[j + 1] = _arr[j];\n    }\n    _arr[index] = _num;\n    // Update the number of elements\n    _size++;\n  }\n\n  /* Remove an element */\n  int remove(int index) {\n    if (index &gt;= _size) throw RangeError('Index out of bounds');\n    int _num = _arr[index];\n    // Shift all elements after index one position to the left\n    for (var j = index; j &lt; _size - 1; j++) {\n      _arr[j] = _arr[j + 1];\n    }\n    // Update the number of elements\n    _size--;\n    // Return the removed element\n    return _num;\n  }\n\n  /* Grow the list capacity */\n  void extendCapacity() {\n    // Create a new array with _extendRatio times the length of the original array\n    final _newNums = List.filled(_capacity * _extendRatio, 0);\n    // Copy the original array into the new array\n    List.copyRange(_newNums, 0, _arr);\n    // Update the reference _arr\n    _arr = _newNums;\n    // Update the list capacity\n    _capacity = _arr.length;\n  }\n\n  /* Convert the list to an array */\n  List&lt;int&gt; toArray() {\n    List&lt;int&gt; arr = [];\n    for (var i = 0; i &lt; _size; i++) {\n      arr.add(get(i));\n    }\n    return arr;\n  }\n}\n</code></pre> my_list.rs<pre><code>/* List type */\n#[allow(dead_code)]\nstruct MyList {\n    arr: Vec&lt;i32&gt;,       // Array (stores the list elements)\n    capacity: usize,     // List capacity\n    size: usize,         // List length (current number of elements)\n    extend_ratio: usize, // Factor by which the list capacity grows each time\n}\n\n#[allow(unused, unused_comparisons)]\nimpl MyList {\n    /* Constructor */\n    pub fn new(capacity: usize) -&gt; Self {\n        let mut vec = vec![0; capacity];\n        Self {\n            arr: vec,\n            capacity,\n            size: 0,\n            extend_ratio: 2,\n        }\n    }\n\n    /* Get the list length (current number of elements)*/\n    pub fn size(&amp;self) -&gt; usize {\n        return self.size;\n    }\n\n    /* Get the list capacity */\n    pub fn capacity(&amp;self) -&gt; usize {\n        return self.capacity;\n    }\n\n    /* Access an element */\n    pub fn get(&amp;self, index: usize) -&gt; i32 {\n        // Panic if the index is out of bounds; the same applies below\n        if index &gt;= self.size {\n            panic!(\"Index out of bounds\")\n        };\n        return self.arr[index];\n    }\n\n    /* Update an element */\n    pub fn set(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size {\n            panic!(\"Index out of bounds\")\n        };\n        self.arr[index] = num;\n    }\n\n    /* Append an element at the end */\n    pub fn add(&amp;mut self, num: i32) {\n        // Grow the capacity when adding an element would exceed it\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        self.arr[self.size] = num;\n        // Update the number of elements\n        self.size += 1;\n    }\n\n    /* Insert an element in the middle */\n    pub fn insert(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size() {\n            panic!(\"Index out of bounds\")\n        };\n        // Grow the capacity when adding an element would exceed it\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        // Shift all elements at and after index `index` one position to the right\n        for j in (index..self.size).rev() {\n            self.arr[j + 1] = self.arr[j];\n        }\n        self.arr[index] = num;\n        // Update the number of elements\n        self.size += 1;\n    }\n\n    /* Remove an element */\n    pub fn remove(&amp;mut self, index: usize) -&gt; i32 {\n        if index &gt;= self.size() {\n            panic!(\"Index out of bounds\")\n        };\n        let num = self.arr[index];\n        // Shift all elements after index `index` one position to the left\n        for j in index..self.size - 1 {\n            self.arr[j] = self.arr[j + 1];\n        }\n        // Update the number of elements\n        self.size -= 1;\n        // Return the removed element\n        return num;\n    }\n\n    /* Grow the list capacity */\n    pub fn extend_capacity(&amp;mut self) {\n        // Grow the backing vector to extend_ratio times its original length, preserving its elements\n        let new_capacity = self.capacity * self.extend_ratio;\n        self.arr.resize(new_capacity, 0);\n        // Update the list capacity\n        self.capacity = new_capacity;\n    }\n\n    /* Copy the list elements into a vector */\n    pub fn to_array(&amp;self) -&gt; Vec&lt;i32&gt; {\n        // Convert only the list elements within the occupied range\n        let mut arr = Vec::new();\n        for i in 0..self.size {\n            arr.push(self.get(i));\n        }\n        arr\n    }\n}\n</code></pre> my_list.c<pre><code>/* List structure */\ntypedef struct {\n    int *arr;        // Array (stores the list elements)\n    int capacity;    // List capacity\n    int size;        // List size\n    int extendRatio; // List capacity growth factor\n} MyList;\n\n/* Constructor */\nMyList *newMyList() {\n    MyList *nums = malloc(sizeof(MyList));\n    nums-&gt;capacity = 10;\n    nums-&gt;arr = malloc(sizeof(int) * nums-&gt;capacity);\n    nums-&gt;size = 0;\n    nums-&gt;extendRatio = 2;\n    return nums;\n}\n\n/* Destructor */\nvoid delMyList(MyList *nums) {\n    free(nums-&gt;arr);\n    free(nums);\n}\n\n/* Get the list size */\nint size(MyList *nums) {\n    return nums-&gt;size;\n}\n\n/* Get the list capacity */\nint capacity(MyList *nums) {\n    return nums-&gt;capacity;\n}\n\n/* Access an element */\nint get(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    return nums-&gt;arr[index];\n}\n\n/* Update an element */\nvoid set(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    nums-&gt;arr[index] = num;\n}\n\n/* Append an element at the end */\nvoid add(MyList *nums, int num) {\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // Grow the capacity\n    }\n    nums-&gt;arr[size(nums)] = num;\n    nums-&gt;size++;\n}\n\n/* Insert an element in the middle */\nvoid insert(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    // Grow the capacity when adding an element would exceed it\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // Grow the capacity\n    }\n    for (int i = size(nums); i &gt; index; --i) {\n        nums-&gt;arr[i] = nums-&gt;arr[i - 1];\n    }\n    nums-&gt;arr[index] = num;\n    nums-&gt;size++;\n}\n\n/* Remove an element */\n// Note: stdio.h already declares a function named remove\nint removeItem(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    int num = nums-&gt;arr[index];\n    for (int i = index; i &lt; size(nums) - 1; i++) {\n        nums-&gt;arr[i] = nums-&gt;arr[i + 1];\n    }\n    nums-&gt;size--;\n    return num;\n}\n\n/* Grow the list capacity */\nvoid extendCapacity(MyList *nums) {\n    // Allocate new space first\n    int newCapacity = capacity(nums) * nums-&gt;extendRatio;\n    int *extend = (int *)malloc(sizeof(int) * newCapacity);\n    int *temp = nums-&gt;arr;\n\n    // Copy the old data into the new array\n    for (int i = 0; i &lt; size(nums); i++)\n        extend[i] = nums-&gt;arr[i];\n\n    // Free the old array\n    free(temp);\n\n    // Update the array pointer and capacity\n    nums-&gt;arr = extend;\n    nums-&gt;capacity = newCapacity;\n}\n\n/* Convert the list to an array for printing */\nint *toArray(MyList *nums) {\n    return nums-&gt;arr;\n}\n</code></pre> my_list.kt<pre><code>/* List class */\nclass MyList {\n    private var arr: IntArray = intArrayOf() // Array (stores the list elements)\n    private var capacity: Int = 10 // List capacity\n    private var size: Int = 0 // List length (current number of elements)\n    private var extendRatio: Int = 2 // Factor by which the list capacity grows each time\n\n    /* Constructor */\n    init {\n        arr = IntArray(capacity)\n    }\n\n    /* Get the list length (current number of elements) */\n    fun size(): Int {\n        return size\n    }\n\n    /* Get the list capacity */\n    fun capacity(): Int {\n        return capacity\n    }\n\n    /* Access an element */\n    fun get(index: Int): Int {\n        // Throw an exception if the index is out of bounds; the same applies below\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"Index out of bounds\")\n        return arr[index]\n    }\n\n    /* Update an element */\n    fun set(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"Index out of bounds\")\n        arr[index] = num\n    }\n\n    /* Append at the end */\n    fun add(num: Int) {\n        // Grow the capacity when adding an element would exceed it\n        if (size == capacity())\n            extendCapacity()\n        arr[size] = num\n        // Update the number of elements\n        size++\n    }\n\n    /* Insert an element in the middle */\n    fun insert(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"Index out of bounds\")\n        // Grow the capacity when adding an element would exceed it\n        if (size == capacity())\n            extendCapacity()\n        // Shift all elements at and after index `index` one position to the right\n        for (j in size - 1 downTo index)\n            arr[j + 1] = arr[j]\n        arr[index] = num\n        // Update the number of elements\n        size++\n    }\n\n    /* Remove an element */\n    fun remove(index: Int): Int {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"Index out of bounds\")\n        val num = arr[index]\n        // Shift all elements after index `index` one position to the left\n        for (j in index..&lt;size - 1)\n            arr[j] = arr[j + 1]\n        // Update the number of elements\n        size--\n        // Return the removed element\n        return num\n    }\n\n    /* Grow the list capacity */\n    fun extendCapacity() {\n        // Create an array extendRatio times as long as the original and copy the original array into it\n        arr = arr.copyOf(capacity() * extendRatio)\n        // Update the list capacity\n        capacity = arr.size\n    }\n\n    /* Convert the list to an array */\n    fun toArray(): IntArray {\n        val size = size()\n        // Convert only the list elements within the occupied range\n        val arr = IntArray(size)\n        for (i in 0..&lt;size) {\n            arr[i] = get(i)\n        }\n        return arr\n    }\n}\n</code></pre> my_list.rb<pre><code>### List class ###\nclass MyList\n  attr_reader :size       # Get the list length (current number of elements)\n  attr_reader :capacity   # Get the list capacity\n\n  ### Constructor ###\n  def initialize\n    @capacity = 10\n    @size = 0\n    @extend_ratio = 2\n    @arr = Array.new(capacity)\n  end\n\n  ### Access an element ###\n  def get(index)\n    # Raise an exception if the index is out of bounds; the same applies below\n    raise IndexError, \"Index out of bounds\" if index &lt; 0 || index &gt;= size\n    @arr[index]\n  end\n\n  ### Update an element ###\n  def set(index, num)\n    raise IndexError, \"Index out of bounds\" if index &lt; 0 || index &gt;= size\n    @arr[index] = num\n  end\n\n  ### Append an element at the end ###\n  def add(num)\n    # Grow the capacity when adding an element would exceed it\n    extend_capacity if size == capacity\n    @arr[size] = num\n\n    # Update the number of elements\n    @size += 1\n  end\n\n  ### Insert an element in the middle ###\n  def insert(index, num)\n    raise IndexError, \"Index out of bounds\" if index &lt; 0 || index &gt;= size\n\n    # Grow the capacity when adding an element would exceed it\n    extend_capacity if size == capacity\n\n    # Shift all elements at and after index `index` one position to the right\n    for j in (size - 1).downto(index)\n      @arr[j + 1] = @arr[j]\n    end\n    @arr[index] = num\n\n    # Update the number of elements\n    @size += 1\n  end\n\n  ### Remove an element ###\n  def remove(index)\n    raise IndexError, \"Index out of bounds\" if index &lt; 0 || index &gt;= size\n    num = @arr[index]\n\n    # Shift all elements after index one position to the left\n    for j in index...size\n      @arr[j] = @arr[j + 1]\n    end\n\n    # Update the number of elements\n    @size -= 1\n\n    # Return the removed element\n    num\n  end\n\n  ### Grow the list capacity ###\n  def extend_capacity\n    # Duplicate the original array and append capacity * (extend_ratio - 1) empty slots\n    arr = @arr.dup + Array.new(capacity * (@extend_ratio - 1))\n    # Update the list capacity\n    @capacity = arr.length\n  end\n\n  ### Convert the list to an array ###\n  def to_array\n    sz = size\n    # Convert only the list elements within the occupied range\n    arr = Array.new(sz)\n    for i in 0...sz\n      arr[i] = get(i)\n    end\n    arr\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 4.   Arrays and linked lists","4.3   Lists"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/","level":1,"title":"4.4   Memory and cache *","text":"<p>The first two sections of this chapter introduced arrays and linked lists, two fundamental structures representing contiguous and scattered storage, respectively.</p> <p>Physical structure largely determines how efficiently a program uses memory and cache, affecting its overall performance.</p>","path":["Chapter 4.   Arrays and linked lists","4.4   Memory and cache *"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/#441-computer-storage-devices","level":2,"title":"4.4.1   Computer storage devices","text":"<p>Computers have three types of storage: disk storage, random-access memory (RAM), and cache memory. Table 4-2 compares their roles and performance characteristics.</p> <p> Table 4-2   Computer storage devices </p> Disk storage RAM Cache Purpose Long-term storage of the operating system, programs, files, and other data Temporary storage of running programs and data being processed Stores frequently accessed data and instructions to reduce CPU accesses to RAM Volatility Retains data without power Loses data without power Loses data without power Capacity Large, on the TB scale Smaller, on the GB scale Very small, on the MB scale Speed Slower, hundreds to thousands of MB/s Faster, tens of GB/s Very fast, tens to hundreds of GB/s Price (Chinese yuan) Inexpensive, fractions of a yuan to several yuan per GB More expensive, tens to hundreds of yuan per GB Very expensive, priced as part of the CPU <p>We can picture computer storage as the pyramid in Figure 4-9. Nearer the top, storage is faster, smaller, and more expensive. This hierarchy is a deliberate design by computer scientists and engineers.</p> <ul> <li>RAM cannot readily replace disk storage. It loses data when power is removed, making it unsuitable for long-term storage. It also costs tens of times more than disk storage, making such a replacement impractical for consumers.</li> <li>Cache capacity and speed are difficult to maximize together. From L1 to L2 to L3, increasing capacity means greater physical size and distance from the CPU core, increasing transfer time and access latency. With current technology, a multilevel cache offers the best balance of capacity, speed, and cost.</li> </ul> <p></p> <p> Figure 4-9   Computer storage hierarchy </p> <p>Tip</p> <p>The storage hierarchy balances speed, capacity, and cost. Such tradeoffs occur throughout industry, requiring us to find the best balance among competing strengths and limitations.</p> <p>In general, disks store large amounts of data long term, RAM temporarily holds data being processed, and caches hold frequently accessed data and instructions to speed up execution. The three work together to keep the computer efficient.</p> <p>As shown in Figure 4-10, a running program reads data from disk into RAM for the CPU to process. The cache can be viewed as part of the CPU: it intelligently loads data from RAM and provides faster access, improving execution speed and reducing dependence on slower memory.</p> <p></p> <p> Figure 4-10   Data flow between disk, RAM, and cache </p>","path":["Chapter 4.   Arrays and linked lists","4.4   Memory and cache *"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/#442-memory-efficiency-of-data-structures","level":2,"title":"4.4.2   Memory efficiency of data structures","text":"<p>Arrays and linked lists each have strengths and limitations in memory use.</p> <p>On one hand, memory is finite, and the same region cannot be shared by multiple programs, so data structures should use it efficiently. Arrays pack elements tightly without extra references (pointers) between nodes, making them more space-efficient. However, they require a sufficiently large contiguous allocation, which can waste memory, and expansion costs extra time and space. Linked lists instead allocate and reclaim memory node by node, providing greater flexibility.</p> <p>On the other hand, repeated allocation and deallocation increasingly fragment free memory, reducing its usability. Arrays' contiguous storage is less prone to causing fragmentation. Linked lists store elements in scattered locations, so frequent insertions and deletions are more likely to fragment memory.</p>","path":["Chapter 4.   Arrays and linked lists","4.4   Memory and cache *"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/#443-cache-efficiency-of-data-structures","level":2,"title":"4.4.3   Cache efficiency of data structures","text":"<p>A cache is much smaller than RAM but far faster, making it crucial to execution speed. Its limited capacity holds only a small portion of frequently accessed data. If the CPU requests data not in the cache, a cache miss occurs, and the data must be fetched from slower RAM.</p> <p>Clearly, fewer cache misses mean more efficient CPU reads and writes and better program performance. The fraction of accesses successfully served by the cache is the cache hit rate, a common measure of cache efficiency.</p> <p>Caches use the following mechanisms to improve efficiency.</p> <ul> <li>Cache lines: Data is stored and loaded in cache-line units rather than one byte at a time, making transfers more efficient.</li> <li>Prefetching: The processor tries to predict access patterns, such as sequential access or fixed strides, and loads matching data into the cache to improve the hit rate.</li> <li>Spatial locality: Accessing data suggests that nearby data may soon be needed, so the cache loads neighboring data as well.</li> <li>Temporal locality: Recently accessed data is likely to be accessed again soon, so keeping it in the cache improves the hit rate.</li> </ul> <p>Arrays and linked lists use the cache differently, mainly in the following ways.</p> <ul> <li>Storage size: Linked list elements require more space, leaving less room for useful data in the cache.</li> <li>Cache lines: Linked list data is scattered, so loading a whole cache line brings in a larger proportion of unneeded data.</li> <li>Prefetching: Array access patterns are more predictable, making it easier to anticipate the next data needed.</li> <li>Spatial locality: Array elements are stored together, so data near a loaded element is more likely to be accessed soon.</li> </ul> <p>Overall, arrays have higher cache hit rates and usually faster operations than linked lists. This makes array-based structures popular for algorithmic problems.</p> <p>However, better cache efficiency does not make arrays superior in every situation. Choose a structure based on the requirements. Both arrays and linked lists can implement a stack, discussed in the next chapter, but suit different situations.</p> <ul> <li>For algorithm exercises, an array-based stack is often preferable because of faster operations and random access, at the cost of allocating some memory in advance.</li> <li>If the dataset is very large, changes frequently, and the stack's size is hard to predict, a linked-list-based stack may be more suitable. It can distribute data throughout memory and avoid the extra cost of array expansion.</li> </ul>","path":["Chapter 4.   Arrays and linked lists","4.4   Memory and cache *"],"tags":[]},{"location":"chapter_array_and_linkedlist/summary/","level":1,"title":"4.5   Summary","text":"","path":["Chapter 4.   Arrays and linked lists","4.5   Summary"],"tags":[]},{"location":"chapter_array_and_linkedlist/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<ul> <li>Arrays and linked lists are fundamental structures representing contiguous and scattered memory storage. Their strengths are complementary.</li> <li>Arrays support random access and use less memory, but insertion and deletion are slow, and their length is fixed after initialization.</li> <li>Linked lists efficiently insert and remove nodes by changing references (pointers), and their length is flexible. However, accessing nodes is slow and they use more memory. Common forms include singly linked, circular, and doubly linked lists.</li> <li>A list is an ordered collection supporting insertion, removal, access, and modification. It is usually implemented as a dynamic array, retaining array advantages while allowing its length to change.</li> <li>Lists make arrays much more practical, though they may waste some memory.</li> <li>During execution, data is mainly stored in RAM. Arrays use memory more efficiently, while linked lists offer more flexible allocation.</li> <li>Cache lines, prefetching, spatial locality, and temporal locality provide fast CPU access to data, substantially improving execution speed.</li> <li>Arrays usually outperform linked lists because of higher cache hit rates. Choose a structure according to the actual requirements and setting.</li> </ul>","path":["Chapter 4.   Arrays and linked lists","4.5   Summary"],"tags":[]},{"location":"chapter_array_and_linkedlist/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: Does storing an array on the stack or heap affect time and space efficiency?</p> <p>Both use contiguous memory, so data operations have essentially the same efficiency. However, the stack and heap differ in these ways.</p> <ol> <li>Allocation and deallocation: Stack memory is relatively small and allocated automatically by the compiler. The heap is larger, supports dynamic allocation in code, and is more prone to fragmentation. Heap allocation and deallocation are therefore usually slower.</li> <li>Size limits: Stack space is relatively small, while the heap is generally limited by available memory, making it better suited to large arrays.</li> <li>Flexibility: Stack array sizes must be known at compile time, while heap array sizes can be determined at runtime.</li> </ol> <p>Q: Why must array elements have the same type, while this requirement is not emphasized for linked lists?</p> <p>Linked lists consist of nodes connected by references (pointers). Different nodes can hold different types, such as <code>int</code>, <code>double</code>, <code>string</code>, and <code>object</code>.</p> <p>Array elements must have the same type so that their positions can be calculated from offsets. If an array contained both <code>int</code> and <code>long</code>, occupying 4 and 8 bytes respectively, the following formula would not work because there would be two element sizes.</p> <pre><code># Element address = array address (first element's address) + element size * element index\n</code></pre> <p>Q: After removing node <code>P</code>, must we set <code>P.next</code> to <code>None</code>?</p> <p>Leaving <code>P.next</code> unchanged is acceptable. A traversal from the list's head to its tail no longer encounters <code>P</code>, so it has been removed. Whatever <code>P</code> points to no longer affects the list.</p> <p>For algorithm exercises, leaving it connected is fine as long as the logic is correct. For a standard library, disconnecting it is safer and clearer: if the removed node is not reclaimed properly, its reference may prevent its successor from being reclaimed.</p> <p>Q: Insertion and deletion in a linked list take \\(O(1)\\) time, but locating the element first takes \\(O(n)\\). Why is the overall complexity not \\(O(n)\\)?</p> <p>Finding an element and then deleting it does take \\(O(n)\\) time. However, \\(O(1)\\) insertion and deletion are useful in other settings. A deque, for example, can use a linked list with pointers maintained at the head and tail, giving \\(O(1)\\) operations.</p> <p>Q: In the linked list storage diagram, does the light-blue pointer occupy a separate memory location, or share half the node's space with its value?</p> <p>The diagram is qualitative. Exact sizes depend on the circumstances.</p> <ul> <li>Node values of different types, such as <code>int</code>, <code>long</code>, <code>double</code>, and objects, occupy different amounts of space.</li> <li>A pointer's size depends on the operating system and compilation environment, and is usually 8 or 4 bytes.</li> </ul> <p>Q: Does appending to a list always take \\(O(1)\\) time?</p> <p>If appending exceeds the available capacity, the list must first expand. The system allocates new memory and copies all existing elements, making that operation \\(O(n)\\).</p> <p>Q: When lists waste memory, does this mean the extra variables for capacity, length, and growth factor?</p> <p>The main waste has two sources. First, the initial allocation may exceed what we need. Second, to avoid frequent resizing, capacity usually grows by a factor such as \\(\\times 1.5\\), leaving empty slots that may never all be filled.</p> <p>Q: In Python, after <code>n = [1, 2, 3]</code>, the three elements seem to have consecutive addresses. But in <code>m = [2, 1, 3]</code>, their IDs are not consecutive and match those in <code>n</code>. Is <code>m</code> still an array?</p> <p>Consider storing linked list nodes instead: <code>n = [n1, n2, n3, n4, n5]</code>. These five objects are usually scattered in memory, yet an index still gives a node's address in \\(O(1)\\) time. The array stores references to the nodes, not the nodes themselves.</p> <p>Unlike many languages, Python also wraps numbers in objects. A list stores references to those objects rather than the numbers themselves. Thus, the same numbers in the two arrays can have the same IDs, and their addresses need not be contiguous.</p> <p>Q: C++ STL's <code>std::list</code> already implements a doubly linked list. Why do algorithm books rarely use it directly? Does it have limitations?</p> <p>First, we often favor arrays and use linked lists only when needed, mainly for two reasons.</p> <ul> <li>Space overhead: Each element needs two extra pointers, to its predecessor and successor, so <code>std::list</code> usually consumes more space than <code>std::vector</code>.</li> <li>Poor cache locality: Its data is not contiguous, making cache use less efficient. In general, <code>std::vector</code> performs better.</li> </ul> <p>Second, linked structures are mainly needed for binary trees and graphs. For stacks and queues, we usually use the language's provided <code>stack</code> and <code>queue</code> rather than a linked list directly.</p> <p>Q: Does <code>res = [[0]] * n</code> create a two-dimensional list with independent <code>[0]</code> entries?</p> <p>No. Every <code>[0]</code> is a reference to the same object. Modifying an element through one reference changes the corresponding element seen through all the others.</p> <p>To create independent <code>[0]</code> lists, use <code>res = [[0] for _ in range(n)]</code>. This initializes \\(n\\) separate list objects.</p> <p>Q: Does <code>res = [0] * n</code> create independent integer zeros?</p> <p>All zeros in the list refer to the same object. Python caches small integers, usually -5 through 256, to maximize reuse and improve performance.</p> <p>Even so, each list entry can be changed independently because Python integers are immutable. Assigning a new value switches the entry to a different object reference instead of modifying the original object.</p> <p>When an element is mutable, such as a list, dictionary, or class instance, modifying the object itself is visible through every reference to it.</p>","path":["Chapter 4.   Arrays and linked lists","4.5   Summary"],"tags":[]},{"location":"chapter_backtracking/","level":1,"title":"Chapter 13.   Backtracking","text":"<p>Abstract</p> <p>Like explorers in a maze, we may encounter obstacles as we move forward.</p> <p>Backtracking lets us start again, try new paths, and eventually find our way out into the light.</p>","path":["Chapter 13.   Backtracking"],"tags":[]},{"location":"chapter_backtracking/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>13.1   Backtracking algorithms</li> <li>13.2   Permutations problem</li> <li>13.3   Subset sum problem</li> <li>13.4   N-queens problem</li> <li>13.5   Summary</li> <li>13.6   Exercises</li> </ul>","path":["Chapter 13.   Backtracking"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/","level":1,"title":"13.1   Backtracking algorithms","text":"<p>A backtracking algorithm solves a problem through exhaustive search. Starting from an initial state, it explores possible solutions by brute force, recording valid solutions as it finds them, until a solution is found or every possible choice has been tried without success.</p> <p>Backtracking usually traverses the solution space through depth-first search. In the binary tree chapter, we learned that preorder, inorder, and postorder traversal are all forms of depth-first search. We will use preorder traversal to build a backtracking problem and explore the approach step by step.</p> <p>Example 1</p> <p>Given a binary tree, find and record every node with value \\(7\\). Return a list of these nodes.</p> <p>Traverse the tree in preorder and check whether the current node's value is \\(7\\). If so, add that node's value to the result list <code>res</code>. The code and figure below show the process:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_i_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"Preorder traversal: example 1\"\"\"\n    if root is None:\n        return\n    if root.val == 7:\n        # Record the solution\n        res.append(root)\n    pre_order(root.left)\n    pre_order(root.right)\n</code></pre> preorder_traversal_i_compact.cpp<pre><code>/* Preorder traversal: example 1 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // Record the solution\n        res.push_back(root);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.java<pre><code>/* Preorder traversal: example 1 */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // Record the solution\n        res.add(root);\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.cs<pre><code>/* Preorder traversal: example 1 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // Record the solution\n        res.Add(root);\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.go<pre><code>/* Preorder traversal: example 1 */\nfunc preOrderI(root *TreeNode, res *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    if (root.Val).(int) == 7 {\n        // Record the solution\n        *res = append(*res, root)\n    }\n    preOrderI(root.Left, res)\n    preOrderI(root.Right, res)\n}\n</code></pre> preorder_traversal_i_compact.swift<pre><code>/* Preorder traversal: example 1 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    if root.val == 7 {\n        // Record the solution\n        res.append(root)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n</code></pre> preorder_traversal_i_compact.js<pre><code>/* Preorder traversal: example 1 */\nfunction preOrder(root, res) {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // Record the solution\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.ts<pre><code>/* Preorder traversal: example 1 */\nfunction preOrder(root: TreeNode | null, res: TreeNode[]): void {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // Record the solution\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.dart<pre><code>/* Preorder traversal: example 1 */\nvoid preOrder(TreeNode? root, List&lt;TreeNode&gt; res) {\n  if (root == null) {\n    return;\n  }\n  if (root.val == 7) {\n    // Record the solution\n    res.add(root);\n  }\n  preOrder(root.left, res);\n  preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.rs<pre><code>/* Preorder traversal: example 1 */\nfn pre_order(res: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        if node.borrow().val == 7 {\n            // Record the solution\n            res.push(node.clone());\n        }\n        pre_order(res, node.borrow().left.as_ref());\n        pre_order(res, node.borrow().right.as_ref());\n    }\n}\n</code></pre> preorder_traversal_i_compact.c<pre><code>/* Preorder traversal: example 1 */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // Record the solution\n        res[resSize++] = root;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.kt<pre><code>/* Preorder traversal: example 1 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    if (root._val == 7) {\n        // Record the solution\n        res!!.add(root)\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n}\n</code></pre> preorder_traversal_i_compact.rb<pre><code>### Preorder traversal: example 1 ###\ndef pre_order(root)\n  return unless root\n\n  # Record the solution\n  $res &lt;&lt; root if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p></p> <p> Figure 13-1   Finding nodes during preorder traversal </p>","path":["Chapter 13.   Backtracking","13.1   Backtracking algorithms"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1311-trying-choices-and-backtracking","level":2,"title":"13.1.1   Trying choices and backtracking","text":"<p>Backtracking gets its name from its pattern of trying choices and undoing them while searching the solution space. When the search reaches a state that cannot advance or lead to a valid solution, it undoes the last choice, returns to the previous state, and tries other possibilities.</p> <p>In Example 1, visiting a node is a trial, while a <code>return</code> after passing a leaf or returning to a parent is backtracking.</p> <p>Backtracking involves more than returning from a function. To see why, let's extend Example 1.</p> <p>Example 2</p> <p>Find every node with value \\(7\\) in a binary tree, and return the paths from the root to those nodes.</p> <p>Extend the first solution with a list <code>path</code> recording the current path of visited nodes. When a node with value \\(7\\) is reached, append a copy of <code>path</code> to the result list <code>res</code>. After traversal, <code>res</code> contains all solutions. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_ii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"Preorder traversal: example 2\"\"\"\n    if root is None:\n        return\n    # Try\n    path.append(root)\n    if root.val == 7:\n        # Record the solution\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # Backtrack\n    path.pop()\n</code></pre> preorder_traversal_ii_compact.cpp<pre><code>/* Preorder traversal: example 2 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    // Try\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // Record the solution\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // Backtrack\n    path.pop_back();\n}\n</code></pre> preorder_traversal_ii_compact.java<pre><code>/* Preorder traversal: example 2 */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    // Try\n    path.add(root);\n    if (root.val == 7) {\n        // Record the solution\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // Backtrack\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_ii_compact.cs<pre><code>/* Preorder traversal: example 2 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    // Try\n    path.Add(root);\n    if (root.val == 7) {\n        // Record the solution\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // Backtrack\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_ii_compact.go<pre><code>/* Preorder traversal: example 2 */\nfunc preOrderII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    // Try\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // Record the solution\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderII(root.Left, res, path)\n    preOrderII(root.Right, res, path)\n    // Backtrack\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_ii_compact.swift<pre><code>/* Preorder traversal: example 2 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // Try\n    path.append(root)\n    if root.val == 7 {\n        // Record the solution\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // Backtrack\n    path.removeLast()\n}\n</code></pre> preorder_traversal_ii_compact.js<pre><code>/* Preorder traversal: example 2 */\nfunction preOrder(root, path, res) {\n    if (root === null) {\n        return;\n    }\n    // Try\n    path.push(root);\n    if (root.val === 7) {\n        // Record the solution\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // Backtrack\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.ts<pre><code>/* Preorder traversal: example 2 */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    if (root === null) {\n        return;\n    }\n    // Try\n    path.push(root);\n    if (root.val === 7) {\n        // Record the solution\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // Backtrack\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.dart<pre><code>/* Preorder traversal: example 2 */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null) {\n    return;\n  }\n\n  // Try\n  path.add(root);\n  if (root.val == 7) {\n    // Record the solution\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // Backtrack\n  path.removeLast();\n}\n</code></pre> preorder_traversal_ii_compact.rs<pre><code>/* Preorder traversal: example 2 */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        // Try\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // Record the solution\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.as_ref());\n        pre_order(res, path, node.borrow().right.as_ref());\n        // Backtrack\n        path.pop();\n    }\n}\n</code></pre> preorder_traversal_ii_compact.c<pre><code>/* Preorder traversal: example 2 */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    // Try\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // Record the solution\n        for (int i = 0; i &lt; pathSize; ++i) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // Backtrack\n    pathSize--;\n}\n</code></pre> preorder_traversal_ii_compact.kt<pre><code>/* Preorder traversal: example 2 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    // Try\n    path!!.add(root)\n    if (root._val == 7) {\n        // Record the solution\n        res!!.add(path!!.toMutableList())\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // Backtrack\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_ii_compact.rb<pre><code>### Preorder traversal: example 2 ###\ndef pre_order(root)\n  return unless root\n\n  # Try\n  $path &lt;&lt; root\n\n  # Record the solution\n  $res &lt;&lt; $path.dup if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\n\n  # Backtrack\n  $path.pop\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Each time we try a node, append it to <code>path</code>. Before backtracking, pop that node from <code>path</code> to restore the state from before the trial.</p> <p>As Figure 13-2 illustrates, trying a choice and backtracking are like moving forward and undoing that move: they are inverse operations.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 13-2   Trying choices and backtracking </p>","path":["Chapter 13.   Backtracking","13.1   Backtracking algorithms"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1312-pruning","level":2,"title":"13.1.2   Pruning","text":"<p>Complex backtracking problems usually have one or more constraints, which can often be used for pruning.</p> <p>Example 3</p> <p>Find every node with value \\(7\\) in a binary tree and return the paths from the root to those nodes, requiring that no path contain a node with value \\(3\\).</p> <p>To enforce this constraint, add a pruning step: when a node with value \\(3\\) is encountered, return immediately without searching farther. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_iii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"Preorder traversal: example 3\"\"\"\n    # Prune\n    if root is None or root.val == 3:\n        return\n    # Try\n    path.append(root)\n    if root.val == 7:\n        # Record the solution\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # Backtrack\n    path.pop()\n</code></pre> preorder_traversal_iii_compact.cpp<pre><code>/* Preorder traversal: example 3 */\nvoid preOrder(TreeNode *root) {\n    // Prune\n    if (root == nullptr || root-&gt;val == 3) {\n        return;\n    }\n    // Try\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // Record the solution\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // Backtrack\n    path.pop_back();\n}\n</code></pre> preorder_traversal_iii_compact.java<pre><code>/* Preorder traversal: example 3 */\nvoid preOrder(TreeNode root) {\n    // Prune\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // Try\n    path.add(root);\n    if (root.val == 7) {\n        // Record the solution\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // Backtrack\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_iii_compact.cs<pre><code>/* Preorder traversal: example 3 */\nvoid PreOrder(TreeNode? root) {\n    // Prune\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // Try\n    path.Add(root);\n    if (root.val == 7) {\n        // Record the solution\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // Backtrack\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_iii_compact.go<pre><code>/* Preorder traversal: example 3 */\nfunc preOrderIII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    // Prune\n    if root == nil || root.Val == 3 {\n        return\n    }\n    // Try\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // Record the solution\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderIII(root.Left, res, path)\n    preOrderIII(root.Right, res, path)\n    // Backtrack\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_iii_compact.swift<pre><code>/* Preorder traversal: example 3 */\nfunc preOrder(root: TreeNode?) {\n    // Prune\n    guard let root = root, root.val != 3 else {\n        return\n    }\n    // Try\n    path.append(root)\n    if root.val == 7 {\n        // Record the solution\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // Backtrack\n    path.removeLast()\n}\n</code></pre> preorder_traversal_iii_compact.js<pre><code>/* Preorder traversal: example 3 */\nfunction preOrder(root, path, res) {\n    // Prune\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // Try\n    path.push(root);\n    if (root.val === 7) {\n        // Record the solution\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // Backtrack\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.ts<pre><code>/* Preorder traversal: example 3 */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // Prune\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // Try\n    path.push(root);\n    if (root.val === 7) {\n        // Record the solution\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // Backtrack\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.dart<pre><code>/* Preorder traversal: example 3 */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null || root.val == 3) {\n    return;\n  }\n\n  // Try\n  path.add(root);\n  if (root.val == 7) {\n    // Record the solution\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // Backtrack\n  path.removeLast();\n}\n</code></pre> preorder_traversal_iii_compact.rs<pre><code>/* Preorder traversal: example 3 */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    // Prune\n    if root.is_none() || root.as_ref().unwrap().borrow().val == 3 {\n        return;\n    }\n    if let Some(node) = root {\n        // Try\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // Record the solution\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.as_ref());\n        pre_order(res, path, node.borrow().right.as_ref());\n        // Backtrack\n        path.pop();\n    }\n}\n</code></pre> preorder_traversal_iii_compact.c<pre><code>/* Preorder traversal: example 3 */\nvoid preOrder(TreeNode *root) {\n    // Prune\n    if (root == NULL || root-&gt;val == 3) {\n        return;\n    }\n    // Try\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // Record the solution\n        for (int i = 0; i &lt; pathSize; i++) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // Backtrack\n    pathSize--;\n}\n</code></pre> preorder_traversal_iii_compact.kt<pre><code>/* Preorder traversal: example 3 */\nfun preOrder(root: TreeNode?) {\n    // Prune\n    if (root == null || root._val == 3) {\n        return\n    }\n    // Try\n    path!!.add(root)\n    if (root._val == 7) {\n        // Record the solution\n        res!!.add(path!!.toMutableList())\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // Backtrack\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_iii_compact.rb<pre><code>### Preorder traversal: example 3 ###\ndef pre_order(root)\n  # Prune\n  return if !root || root.val == 3\n\n  # Try\n  $path.append(root)\n\n  # Record the solution\n  $res &lt;&lt; $path.dup if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\n\n  # Backtrack\n  $path.pop\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Pruning is an apt name. As shown in Figure 13-3, we cut off search branches that violate the constraints, avoiding pointless trials and improving efficiency.</p> <p></p> <p> Figure 13-3   Pruning according to constraints </p>","path":["Chapter 13.   Backtracking","13.1   Backtracking algorithms"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1313-a-code-template","level":2,"title":"13.1.3   A code template","text":"<p>We will now extract the core pattern of trying, undoing, and pruning into a more general backtracking template.</p> <p>In the following template, <code>state</code> is the current problem state, and <code>choices</code> contains the choices available from that state:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def backtrack(state: State, choices: list[choice], res: list[state]):\n    \"\"\"Backtracking template\"\"\"\n    # Check whether the state is a solution\n    if is_solution(state):\n        # Record the solution\n        record_solution(state, res)\n        # Stop searching this branch\n        return\n    # Try every choice\n    for choice in choices:\n        # Prune: check whether the choice is valid\n        if is_valid(state, choice):\n            # Try: make the choice and update the state\n            make_choice(state, choice)\n            backtrack(state, choices, res)\n            # Backtrack: undo the choice and restore the previous state\n            undo_choice(state, choice)\n</code></pre> <pre><code>/* Backtracking template */\nvoid backtrack(State *state, vector&lt;Choice *&gt; &amp;choices, vector&lt;State *&gt; &amp;res) {\n    // Check whether the state is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res);\n        // Stop searching this branch\n        return;\n    }\n    // Try every choice\n    for (Choice choice : choices) {\n        // Prune: check whether the choice is valid\n        if (isValid(state, choice)) {\n            // Try: make the choice and update the state\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking template */\nvoid backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // Check whether the state is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res);\n        // Stop searching this branch\n        return;\n    }\n    // Try every choice\n    for (Choice choice : choices) {\n        // Prune: check whether the choice is valid\n        if (isValid(state, choice)) {\n            // Try: make the choice and update the state\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking template */\nvoid Backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // Check whether the state is a solution\n    if (IsSolution(state)) {\n        // Record the solution\n        RecordSolution(state, res);\n        // Stop searching this branch\n        return;\n    }\n    // Try every choice\n    foreach (Choice choice in choices) {\n        // Prune: check whether the choice is valid\n        if (IsValid(state, choice)) {\n            // Try: make the choice and update the state\n            MakeChoice(state, choice);\n            Backtrack(state, choices, res);\n            // Backtrack: undo the choice and restore the previous state\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking template */\nfunc backtrack(state *State, choices []Choice, res *[]State) {\n    // Check whether the state is a solution\n    if isSolution(state) {\n        // Record the solution\n        recordSolution(state, res)\n        // Stop searching this branch\n        return\n    }\n    // Try every choice\n    for _, choice := range choices {\n        // Prune: check whether the choice is valid\n        if isValid(state, choice) {\n            // Try: make the choice and update the state\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking template */\nfunc backtrack(state: inout State, choices: [Choice], res: inout [State]) {\n    // Check whether the state is a solution\n    if isSolution(state: state) {\n        // Record the solution\n        recordSolution(state: state, res: &amp;res)\n        // Stop searching this branch\n        return\n    }\n    // Try every choice\n    for choice in choices {\n        // Prune: check whether the choice is valid\n        if isValid(state: state, choice: choice) {\n            // Try: make the choice and update the state\n            makeChoice(state: &amp;state, choice: choice)\n            backtrack(state: &amp;state, choices: choices, res: &amp;res)\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking template */\nfunction backtrack(state, choices, res) {\n    // Check whether the state is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res);\n        // Stop searching this branch\n        return;\n    }\n    // Try every choice\n    for (let choice of choices) {\n        // Prune: check whether the choice is valid\n        if (isValid(state, choice)) {\n            // Try: make the choice and update the state\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking template */\nfunction backtrack(state: State, choices: Choice[], res: State[]): void {\n    // Check whether the state is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res);\n        // Stop searching this branch\n        return;\n    }\n    // Try every choice\n    for (let choice of choices) {\n        // Prune: check whether the choice is valid\n        if (isValid(state, choice)) {\n            // Try: make the choice and update the state\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking template */\nvoid backtrack(State state, List&lt;Choice&gt;, List&lt;State&gt; res) {\n  // Check whether the state is a solution\n  if (isSolution(state)) {\n    // Record the solution\n    recordSolution(state, res);\n    // Stop searching this branch\n    return;\n  }\n  // Try every choice\n  for (Choice choice in choices) {\n    // Prune: check whether the choice is valid\n    if (isValid(state, choice)) {\n      // Try: make the choice and update the state\n      makeChoice(state, choice);\n      backtrack(state, choices, res);\n      // Backtrack: undo the choice and restore the previous state\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> <pre><code>/* Backtracking template */\nfn backtrack(state: &amp;mut State, choices: &amp;Vec&lt;Choice&gt;, res: &amp;mut Vec&lt;State&gt;) {\n    // Check whether the state is a solution\n    if is_solution(state) {\n        // Record the solution\n        record_solution(state, res);\n        // Stop searching this branch\n        return;\n    }\n    // Try every choice\n    for choice in choices {\n        // Prune: check whether the choice is valid\n        if is_valid(state, choice) {\n            // Try: make the choice and update the state\n            make_choice(state, choice);\n            backtrack(state, choices, res);\n            // Backtrack: undo the choice and restore the previous state\n            undo_choice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking template */\nvoid backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) {\n    // Check whether the state is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res, numRes);\n        // Stop searching this branch\n        return;\n    }\n    // Try every choice\n    for (int i = 0; i &lt; numChoices; i++) {\n        // Prune: check whether the choice is valid\n        if (isValid(state, &amp;choices[i])) {\n            // Try: make the choice and update the state\n            makeChoice(state, &amp;choices[i]);\n            backtrack(state, choices, numChoices, res, numRes);\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice(state, &amp;choices[i]);\n        }\n    }\n}\n</code></pre> <pre><code>/* Backtracking template */\nfun backtrack(state: State?, choices: List&lt;Choice?&gt;, res: List&lt;State?&gt;?) {\n    // Check whether the state is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res)\n        // Stop searching this branch\n        return\n    }\n    // Try every choice\n    for (choice in choices) {\n        // Prune: check whether the choice is valid\n        if (isValid(state, choice)) {\n            // Try: make the choice and update the state\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>### Backtracking template ###\ndef backtrack(state, choices, res)\n    # Check whether the state is a solution\n    if is_solution?(state)\n        # Record the solution\n        record_solution(state, res)\n        return\n    end\n\n    # Try every choice\n    for choice in choices\n        # Prune: check whether the choice is valid\n        if is_valid?(state, choice)\n            # Try: make the choice and update the state\n            make_choice(state, choice)\n            backtrack(state, choices, res)\n            # Backtrack: undo the choice and restore the previous state\n            undo_choice(state, choice)\n        end\n    end\nend\n</code></pre> <p>Now apply this template to Example 3. The <code>state</code> is the current path of visited nodes, <code>choices</code> consists of the current node's left and right children, and <code>res</code> is the list of paths:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_iii_template.py<pre><code>def is_solution(state: list[TreeNode]) -&gt; bool:\n    \"\"\"Check whether the current state is a solution\"\"\"\n    return state and state[-1].val == 7\n\ndef record_solution(state: list[TreeNode], res: list[list[TreeNode]]):\n    \"\"\"Record the solution\"\"\"\n    res.append(list(state))\n\ndef is_valid(state: list[TreeNode], choice: TreeNode) -&gt; bool:\n    \"\"\"Check whether the choice is valid in the current state\"\"\"\n    return choice is not None and choice.val != 3\n\ndef make_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"Update the state\"\"\"\n    state.append(choice)\n\ndef undo_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"Restore the state\"\"\"\n    state.pop()\n\ndef backtrack(\n    state: list[TreeNode], choices: list[TreeNode], res: list[list[TreeNode]]\n):\n    \"\"\"Backtracking: example 3\"\"\"\n    # Check whether this is a solution\n    if is_solution(state):\n        # Record the solution\n        record_solution(state, res)\n    # Iterate over all choices\n    for choice in choices:\n        # Prune: check whether the choice is valid\n        if is_valid(state, choice):\n            # Try: make a choice and update the state\n            make_choice(state, choice)\n            # Proceed to the next choice\n            backtrack(state, [choice.left, choice.right], res)\n            # Backtrack: undo the choice and restore the previous state\n            undo_choice(state, choice)\n</code></pre> preorder_traversal_iii_template.cpp<pre><code>/* Check whether the current state is a solution */\nbool isSolution(vector&lt;TreeNode *&gt; &amp;state) {\n    return !state.empty() &amp;&amp; state.back()-&gt;val == 7;\n}\n\n/* Record the solution */\nvoid recordSolution(vector&lt;TreeNode *&gt; &amp;state, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    res.push_back(state);\n}\n\n/* Check whether the choice is valid in the current state */\nbool isValid(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    return choice != nullptr &amp;&amp; choice-&gt;val != 3;\n}\n\n/* Update the state */\nvoid makeChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.push_back(choice);\n}\n\n/* Restore the state */\nvoid undoChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.pop_back();\n}\n\n/* Backtracking: example 3 */\nvoid backtrack(vector&lt;TreeNode *&gt; &amp;state, vector&lt;TreeNode *&gt; &amp;choices, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    // Check whether this is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res);\n    }\n    // Iterate over all choices\n    for (TreeNode *choice : choices) {\n        // Prune: check whether the choice is valid\n        if (isValid(state, choice)) {\n            // Try: make a choice and update the state\n            makeChoice(state, choice);\n            // Proceed to the next choice\n            vector&lt;TreeNode *&gt; nextChoices{choice-&gt;left, choice-&gt;right};\n            backtrack(state, nextChoices, res);\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.java<pre><code>/* Check whether the current state is a solution */\nboolean isSolution(List&lt;TreeNode&gt; state) {\n    return !state.isEmpty() &amp;&amp; state.get(state.size() - 1).val == 7;\n}\n\n/* Record the solution */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.add(new ArrayList&lt;&gt;(state));\n}\n\n/* Check whether the choice is valid in the current state */\nboolean isValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* Update the state */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.add(choice);\n}\n\n/* Restore the state */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.remove(state.size() - 1);\n}\n\n/* Backtracking: example 3 */\nvoid backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // Check whether this is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res);\n    }\n    // Iterate over all choices\n    for (TreeNode choice : choices) {\n        // Prune: check whether the choice is valid\n        if (isValid(state, choice)) {\n            // Try: make a choice and update the state\n            makeChoice(state, choice);\n            // Proceed to the next choice\n            backtrack(state, Arrays.asList(choice.left, choice.right), res);\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.cs<pre><code>/* Check whether the current state is a solution */\nbool IsSolution(List&lt;TreeNode&gt; state) {\n    return state.Count != 0 &amp;&amp; state[^1].val == 7;\n}\n\n/* Record the solution */\nvoid RecordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.Add(new List&lt;TreeNode&gt;(state));\n}\n\n/* Check whether the choice is valid in the current state */\nbool IsValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* Update the state */\nvoid MakeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.Add(choice);\n}\n\n/* Restore the state */\nvoid UndoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.RemoveAt(state.Count - 1);\n}\n\n/* Backtracking: example 3 */\nvoid Backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // Check whether this is a solution\n    if (IsSolution(state)) {\n        // Record the solution\n        RecordSolution(state, res);\n    }\n    // Iterate over all choices\n    foreach (TreeNode choice in choices) {\n        // Prune: check whether the choice is valid\n        if (IsValid(state, choice)) {\n            // Try: make a choice and update the state\n            MakeChoice(state, choice);\n            // Proceed to the next choice\n            Backtrack(state, [choice.left!, choice.right!], res);\n            // Backtrack: undo the choice and restore the previous state\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.go<pre><code>/* Check whether the current state is a solution */\nfunc isSolution(state *[]*TreeNode) bool {\n    return len(*state) != 0 &amp;&amp; (*state)[len(*state)-1].Val == 7\n}\n\n/* Record the solution */\nfunc recordSolution(state *[]*TreeNode, res *[][]*TreeNode) {\n    *res = append(*res, append([]*TreeNode{}, *state...))\n}\n\n/* Check whether the choice is valid in the current state */\nfunc isValid(state *[]*TreeNode, choice *TreeNode) bool {\n    return choice != nil &amp;&amp; choice.Val != 3\n}\n\n/* Update the state */\nfunc makeChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = append(*state, choice)\n}\n\n/* Restore the state */\nfunc undoChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = (*state)[:len(*state)-1]\n}\n\n/* Backtracking: example 3 */\nfunc backtrackIII(state *[]*TreeNode, choices *[]*TreeNode, res *[][]*TreeNode) {\n    // Check whether this is a solution\n    if isSolution(state) {\n        // Record the solution\n        recordSolution(state, res)\n    }\n    // Iterate over all choices\n    for _, choice := range *choices {\n        // Prune: check whether the choice is valid\n        if isValid(state, choice) {\n            // Try: make a choice and update the state\n            makeChoice(state, choice)\n            // Proceed to the next choice\n            temp := make([]*TreeNode, 0)\n            temp = append(temp, choice.Left, choice.Right)\n            backtrackIII(state, &amp;temp, res)\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.swift<pre><code>/* Check whether the current state is a solution */\nfunc isSolution(state: [TreeNode]) -&gt; Bool {\n    !state.isEmpty &amp;&amp; state.last!.val == 7\n}\n\n/* Record the solution */\nfunc recordSolution(state: [TreeNode], res: inout [[TreeNode]]) {\n    res.append(state)\n}\n\n/* Check whether the choice is valid in the current state */\nfunc isValid(state: [TreeNode], choice: TreeNode?) -&gt; Bool {\n    choice != nil &amp;&amp; choice!.val != 3\n}\n\n/* Update the state */\nfunc makeChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.append(choice)\n}\n\n/* Restore the state */\nfunc undoChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.removeLast()\n}\n\n/* Backtracking: example 3 */\nfunc backtrack(state: inout [TreeNode], choices: [TreeNode], res: inout [[TreeNode]]) {\n    // Check whether this is a solution\n    if isSolution(state: state) {\n        recordSolution(state: state, res: &amp;res)\n    }\n    // Iterate over all choices\n    for choice in choices {\n        // Prune: check whether the choice is valid\n        if isValid(state: state, choice: choice) {\n            // Try: make a choice and update the state\n            makeChoice(state: &amp;state, choice: choice)\n            // Proceed to the next choice\n            backtrack(state: &amp;state, choices: [choice.left, choice.right].compactMap { $0 }, res: &amp;res)\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.js<pre><code>/* Check whether the current state is a solution */\nfunction isSolution(state) {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* Record the solution */\nfunction recordSolution(state, res) {\n    res.push([...state]);\n}\n\n/* Check whether the choice is valid in the current state */\nfunction isValid(state, choice) {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* Update the state */\nfunction makeChoice(state, choice) {\n    state.push(choice);\n}\n\n/* Restore the state */\nfunction undoChoice(state) {\n    state.pop();\n}\n\n/* Backtracking: example 3 */\nfunction backtrack(state, choices, res) {\n    // Check whether this is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res);\n    }\n    // Iterate over all choices\n    for (const choice of choices) {\n        // Prune: check whether the choice is valid\n        if (isValid(state, choice)) {\n            // Try: make a choice and update the state\n            makeChoice(state, choice);\n            // Proceed to the next choice\n            backtrack(state, [choice.left, choice.right], res);\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.ts<pre><code>/* Check whether the current state is a solution */\nfunction isSolution(state: TreeNode[]): boolean {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* Record the solution */\nfunction recordSolution(state: TreeNode[], res: TreeNode[][]): void {\n    res.push([...state]);\n}\n\n/* Check whether the choice is valid in the current state */\nfunction isValid(state: TreeNode[], choice: TreeNode): boolean {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* Update the state */\nfunction makeChoice(state: TreeNode[], choice: TreeNode): void {\n    state.push(choice);\n}\n\n/* Restore the state */\nfunction undoChoice(state: TreeNode[]): void {\n    state.pop();\n}\n\n/* Backtracking: example 3 */\nfunction backtrack(\n    state: TreeNode[],\n    choices: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // Check whether this is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res);\n    }\n    // Iterate over all choices\n    for (const choice of choices) {\n        // Prune: check whether the choice is valid\n        if (isValid(state, choice)) {\n            // Try: make a choice and update the state\n            makeChoice(state, choice);\n            // Proceed to the next choice\n            backtrack(state, [choice.left, choice.right], res);\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.dart<pre><code>/* Check whether the current state is a solution */\nbool isSolution(List&lt;TreeNode&gt; state) {\n  return state.isNotEmpty &amp;&amp; state.last.val == 7;\n}\n\n/* Record the solution */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n  res.add(List.from(state));\n}\n\n/* Check whether the choice is valid in the current state */\nbool isValid(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* Update the state */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.add(choice!);\n}\n\n/* Restore the state */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.removeLast();\n}\n\n/* Backtracking: example 3 */\nvoid backtrack(\n  List&lt;TreeNode&gt; state,\n  List&lt;TreeNode?&gt; choices,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  // Check whether this is a solution\n  if (isSolution(state)) {\n    // Record the solution\n    recordSolution(state, res);\n  }\n  // Iterate over all choices\n  for (TreeNode? choice in choices) {\n    // Prune: check whether the choice is valid\n    if (isValid(state, choice)) {\n      // Try: make a choice and update the state\n      makeChoice(state, choice);\n      // Proceed to the next choice\n      backtrack(state, [choice!.left, choice.right], res);\n      // Backtrack: undo the choice and restore the previous state\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> preorder_traversal_iii_template.rs<pre><code>/* Check whether the current state is a solution */\nfn is_solution(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\n    return !state.is_empty() &amp;&amp; state.last().unwrap().borrow().val == 7;\n}\n\n/* Record the solution */\nfn record_solution(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    res.push(state.clone());\n}\n\n/* Check whether the choice is valid in the current state */\nfn is_valid(_: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\n    return choice.is_some() &amp;&amp; choice.unwrap().borrow().val != 3;\n}\n\n/* Update the state */\nfn make_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.push(choice);\n}\n\n/* Restore the state */\nfn undo_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, _: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.pop();\n}\n\n/* Backtracking: example 3 */\nfn backtrack(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    choices: &amp;Vec&lt;Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    // Check whether this is a solution\n    if is_solution(state) {\n        // Record the solution\n        record_solution(state, res);\n    }\n    // Iterate over all choices\n    for &amp;choice in choices.iter() {\n        // Prune: check whether the choice is valid\n        if is_valid(state, choice) {\n            // Try: make a choice and update the state\n            make_choice(state, choice.unwrap().clone());\n            // Proceed to the next choice\n            backtrack(\n                state,\n                &amp;vec![\n                    choice.unwrap().borrow().left.as_ref(),\n                    choice.unwrap().borrow().right.as_ref(),\n                ],\n                res,\n            );\n            // Backtrack: undo the choice and restore the previous state\n            undo_choice(state, choice.unwrap().clone());\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.c<pre><code>/* Check whether the current state is a solution */\nbool isSolution(void) {\n    return pathSize &gt; 0 &amp;&amp; path[pathSize - 1]-&gt;val == 7;\n}\n\n/* Record the solution */\nvoid recordSolution(void) {\n    for (int i = 0; i &lt; pathSize; i++) {\n        res[resSize][i] = path[i];\n    }\n    resSize++;\n}\n\n/* Check whether the choice is valid in the current state */\nbool isValid(TreeNode *choice) {\n    return choice != NULL &amp;&amp; choice-&gt;val != 3;\n}\n\n/* Update the state */\nvoid makeChoice(TreeNode *choice) {\n    path[pathSize++] = choice;\n}\n\n/* Restore the state */\nvoid undoChoice(void) {\n    pathSize--;\n}\n\n/* Backtracking: example 3 */\nvoid backtrack(TreeNode *choices[2]) {\n    // Check whether this is a solution\n    if (isSolution()) {\n        // Record the solution\n        recordSolution();\n    }\n    // Iterate over all choices\n    for (int i = 0; i &lt; 2; i++) {\n        TreeNode *choice = choices[i];\n        // Prune: check whether the choice is valid\n        if (isValid(choice)) {\n            // Try: make a choice and update the state\n            makeChoice(choice);\n            // Proceed to the next choice\n            TreeNode *nextChoices[2] = {choice-&gt;left, choice-&gt;right};\n            backtrack(nextChoices);\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice();\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.kt<pre><code>/* Check whether the current state is a solution */\nfun isSolution(state: MutableList&lt;TreeNode?&gt;): Boolean {\n    return state.isNotEmpty() &amp;&amp; state[state.size - 1]?._val == 7\n}\n\n/* Record the solution */\nfun recordSolution(state: MutableList&lt;TreeNode?&gt;?, res: MutableList&lt;MutableList&lt;TreeNode?&gt;?&gt;) {\n    res.add(state!!.toMutableList())\n}\n\n/* Check whether the choice is valid in the current state */\nfun isValid(state: MutableList&lt;TreeNode?&gt;?, choice: TreeNode?): Boolean {\n    return choice != null &amp;&amp; choice._val != 3\n}\n\n/* Update the state */\nfun makeChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.add(choice)\n}\n\n/* Restore the state */\nfun undoChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.removeLast()\n}\n\n/* Backtracking: example 3 */\nfun backtrack(\n    state: MutableList&lt;TreeNode?&gt;,\n    choices: MutableList&lt;TreeNode?&gt;,\n    res: MutableList&lt;MutableList&lt;TreeNode?&gt;?&gt;\n) {\n    // Check whether this is a solution\n    if (isSolution(state)) {\n        // Record the solution\n        recordSolution(state, res)\n    }\n    // Iterate over all choices\n    for (choice in choices) {\n        // Prune: check whether the choice is valid\n        if (isValid(state, choice)) {\n            // Try: make a choice and update the state\n            makeChoice(state, choice)\n            // Proceed to the next choice\n            backtrack(state, mutableListOf(choice!!.left, choice.right), res)\n            // Backtrack: undo the choice and restore the previous state\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.rb<pre><code>### Check whether the current state is a solution ###\ndef is_solution?(state)\n  !state.empty? &amp;&amp; state.last.val == 7\nend\n\n### Record the solution ###\ndef record_solution(state, res)\n  res &lt;&lt; state.dup\nend\n\n### Check whether the choice is valid in the current state ###\ndef is_valid?(state, choice)\n  choice &amp;&amp; choice.val != 3\nend\n\n### Update the state ###\ndef make_choice(state, choice)\n  state &lt;&lt; choice\nend\n\n### Restore the state ###\ndef undo_choice(state, choice)\n  state.pop\nend\n\n### Backtracking: example 3 ###\ndef backtrack(state, choices, res)\n  # Check whether this is a solution\n  record_solution(state, res) if is_solution?(state)\n\n  # Iterate over all choices\n  for choice in choices\n    # Prune: check whether the choice is valid\n    if is_valid?(state, choice)\n      # Try: make a choice and update the state\n      make_choice(state, choice)\n      # Proceed to the next choice\n      backtrack(state, [choice.left, choice.right], res)\n      # Backtrack: undo the choice and restore the previous state\n      undo_choice(state, choice)\n    end\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>The problem requires us to keep searching after finding a node with value \\(7\\), so remove the <code>return</code> after recording a solution. Figure 13-4 compares the searches with and without that <code>return</code>.</p> <p></p> <p> Figure 13-4   Comparing searches with and without return </p> <p>The template-based implementation is more verbose than the preorder version, but more general. Many backtracking problems can be solved within this template: define <code>state</code> and <code>choices</code> for the problem, then implement each helper method.</p>","path":["Chapter 13.   Backtracking","13.1   Backtracking algorithms"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1314-common-terminology","level":2,"title":"13.1.4   Common terminology","text":"<p>Table 13-1 summarizes common backtracking terms and illustrates them with Example 3, helping us analyze problems more clearly.</p> <p> Table 13-1   Common backtracking terminology </p> Term Definition Example 3 Solution An answer satisfying the problem's conditions; there may be one or more All paths from the root to a node valued \\(7\\) that satisfy the constraint Constraint A condition restricting which solutions are feasible, often used for pruning The path contains no node valued \\(3\\) State The situation at a particular point, including the choices already made The current path of visited nodes, stored in <code>path</code> Trial Exploring the solution space by making a choice, updating the state, and checking for a solution Recursively visit a left or right child, append it to <code>path</code>, and check whether its value is \\(7\\) Backtracking Undoing an earlier choice and returning to the previous state when a constraint cannot be met Stop searching and return after passing a leaf, finishing a node, or encountering value \\(3\\) Pruning Using problem properties and constraints to avoid pointless search paths and improve efficiency Stop searching when a node valued \\(3\\) is encountered <p>Tip</p> <p>Concepts such as problems, solutions, and states are shared by divide and conquer, backtracking, dynamic programming, greedy algorithms, and other approaches.</p>","path":["Chapter 13.   Backtracking","13.1   Backtracking algorithms"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1315-strengths-and-limitations","level":2,"title":"13.1.5   Strengths and limitations","text":"<p>Backtracking is fundamentally a depth-first search that tries possible solutions until it finds ones meeting the conditions. It can find all possible solutions and can be highly efficient with effective pruning.</p> <p>However, for large or complex problems, backtracking may be unacceptably slow.</p> <ul> <li>Time: Backtracking often explores every possibility in the state space, leading to exponential or factorial time complexity.</li> <li>Space: Recursive calls must preserve state, such as the path and auxiliary pruning variables. Deep recursion can therefore require substantial space.</li> </ul> <p>Even so, backtracking remains the best approach for some search and constraint satisfaction problems. When we cannot predict which choices lead to valid solutions, we must explore every possibility. The key is then to improve efficiency, commonly through two techniques.</p> <ul> <li>Pruning: Avoid paths that cannot produce a solution, saving time and space.</li> <li>Heuristic search: Use strategies or estimates to explore paths most likely to yield valid solutions first.</li> </ul>","path":["Chapter 13.   Backtracking","13.1   Backtracking algorithms"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1316-typical-backtracking-problems","level":2,"title":"13.1.6   Typical backtracking problems","text":"<p>Backtracking can solve many search, constraint satisfaction, and combinatorial optimization problems.</p> <p>Search problems seek solutions satisfying specified conditions.</p> <ul> <li>Permutations: Given a collection, find every possible ordering of its elements.</li> <li>Subset sum: Given a collection and a target sum, find every subset whose sum equals the target.</li> <li>Tower of Hanoi: Given three rods and disks of different sizes, move all disks from one rod to another, moving only one at a time and never placing a larger disk on a smaller one.</li> </ul> <p>Constraint satisfaction problems seek solutions meeting all constraints.</p> <ul> <li>\\(n\\)-queens: Place \\(n\\) queens on an \\(n \\times n\\) chessboard so that no two attack each other.</li> <li>Sudoku: Fill a \\(9 \\times 9\\) grid with digits \\(1\\) ~ \\(9\\) so that no digit repeats in any row, column, or \\(3 \\times 3\\) subgrid.</li> <li>Graph coloring: Color an undirected graph's vertices with as few colors as possible, ensuring adjacent vertices have different colors.</li> </ul> <p>Combinatorial optimization problems seek an optimal solution satisfying conditions within a space of combinations.</p> <ul> <li>0-1 knapsack: Given items with values and weights and a knapsack, choose items to maximize total value within the capacity limit.</li> <li>Traveling salesperson: Starting at a vertex in a graph, visit every other vertex exactly once and return to the start along the shortest possible route.</li> <li>Maximum clique: Given an undirected graph, find its largest complete subgraph, in which every pair of vertices is connected by an edge.</li> </ul> <p>For many combinatorial optimization problems, backtracking is not the best approach.</p> <ul> <li>0-1 knapsack is usually solved more efficiently with dynamic programming.</li> <li>The traveling salesperson problem is a well-known NP-hard problem. Common approaches include genetic algorithms and ant colony algorithms.</li> <li>Maximum clique is a classic graph theory problem that can be approached with heuristics such as greedy algorithms.</li> </ul>","path":["Chapter 13.   Backtracking","13.1   Backtracking algorithms"],"tags":[]},{"location":"chapter_backtracking/exercises/","level":1,"title":"13.6   Exercises","text":"","path":["Chapter 13.   Backtracking","13.6   Exercises"],"tags":[]},{"location":"chapter_backtracking/exercises/#1361-check-your-understanding","level":2,"title":"13.6.1   Check your understanding","text":"","path":["Chapter 13.   Backtracking","13.6   Exercises"],"tags":[]},{"location":"chapter_backtracking/exercises/#1-will-this-permutation-algorithm-miss-results","level":3,"title":"1.   Will this permutation algorithm miss results?","text":"<p>A backtracking algorithm tries the numbers <code>1, 2, 3</code> in order to generate all permutations. Whenever it chooses a number <code>x</code>, it:</p> <ol> <li>Appends <code>x</code> to the current path.</li> <li>Marks <code>x</code> as used.</li> <li>Recursively fills the next position.</li> </ol> <p>After the recursive call returns, the student removes only <code>x</code> from the end of the path, then continues trying the next number.</p> <ol> <li>Which permutation does the algorithm produce first? Can it still produce all six permutations?</li> <li>Before returning to the previous recursion level, is removing the last number from the path enough? If not, what else is needed? Explain.</li> </ol> Reference answer <ol> <li> <p>It first produces <code>[1, 2, 3]</code>, but cannot generate every permutation. Although the path becomes shorter as calls return,     the numbers 1, 2, and 3 remain marked as used, leaving later branches with no available numbers.</p> </li> <li> <p>No. After removing <code>x</code> from the end of the path, we must also mark <code>x</code> as unused again.     The current path and the used markers together describe the search state. Choosing a number changes both, so backtracking must restore both     to allow other branches to choose <code>x</code> again.</p> </li> </ol>","path":["Chapter 13.   Backtracking","13.6   Exercises"],"tags":[]},{"location":"chapter_backtracking/exercises/#2-does-the-order-of-choosing-numbers-matter","level":3,"title":"2.   Does the order of choosing numbers matter?","text":"<p>Given the sorted array <code>[2, 3, 5]</code> and target value 5, each number may be chosen repeatedly. The algorithm requires the numbers along each search path to appear in nondecreasing order.</p> <ol> <li>Which distinct combinations can be obtained?</li> <li>Why is it unnecessary to search the same set of numbers in different orders? What does the nondecreasing-order restriction achieve?</li> <li>The current path is <code>[3]</code>, the remaining sum is 2, and the next candidate is 3. Why can we stop checking all further candidates at this level?</li> </ol> Reference answer <ol> <li> <p>The distinct combinations are <code>[2, 3]</code> and <code>[5]</code>.</p> </li> <li> <p>This problem treats <code>[2, 3]</code> and <code>[3, 2]</code> as the same combination: selection order does not distinguish answers.     Requiring the path's numbers to appear in nondecreasing order lets the search skip duplicate combinations such as <code>[3, 2]</code> directly.</p> </li> <li> <p>The remaining sum is 2, and candidate 3 already exceeds it. Since the array is sorted,     all candidates after 3 are larger still and cannot join the current combination. We can therefore stop checking this level.</p> </li> </ol>","path":["Chapter 13.   Backtracking","13.6   Exercises"],"tags":[]},{"location":"chapter_backtracking/exercises/#3-where-can-the-next-queen-be-placed","level":3,"title":"3.   Where can the next queen be placed?","text":"<p>Place queens row by row on a <code>4 × 4</code> board, with both row and column indices starting at 0. Queens are already at <code>(0, 1)</code> and <code>(1, 3)</code>. The next queen must be placed in row 2.</p> <ol> <li>Which columns are ruled out because they already contain a queen?</li> <li>Among the remaining columns, which positions are ruled out by sharing a diagonal with a queen?</li> <li>Which positions in row 2 remain available to try?</li> </ol> Reference answer <ol> <li> <p>Columns 1 and 3 already contain queens, ruling out positions <code>(2, 1)</code> and <code>(2, 3)</code>.</p> </li> <li> <p>Of the remaining positions, <code>(2, 2)</code> shares a diagonal with <code>(1, 3)</code>, so it is also ruled out.     Position <code>(2, 0)</code> shares neither a column nor a diagonal with either existing queen.</p> </li> <li> <p>The only position to try in row 2 is <code>(2, 0)</code>.</p> <p>This step establishes only that the current placement is valid. If the board cannot be completed later, we must still backtrack and try other earlier choices.</p> </li> </ol>","path":["Chapter 13.   Backtracking","13.6   Exercises"],"tags":[]},{"location":"chapter_backtracking/exercises/#1362-programming-exercises","level":2,"title":"13.6.2   Programming exercises","text":"","path":["Chapter 13.   Backtracking","13.6   Exercises"],"tags":[]},{"location":"chapter_backtracking/exercises/#1-all-permutations-of-distinct-elements","level":3,"title":"1.   All permutations of distinct elements","text":"<p>The integer array <code>nums</code> contains at least one element, and all its elements are distinct. List every possible ordering that uses each element exactly once, returning each ordering as an array. The permutations may appear in any order in the result. Use backtracking and a Boolean array to record whether the element at each position is already included in the current permutation.</p> Hints <ol> <li>The recursion depth indicates which position in the permutation is being filled.</li> <li>At each level, try only unused elements.</li> <li>When the path length reaches the length of <code>nums</code>, add a copy of the path to the answer.</li> </ol> <p>LeetCode</p>","path":["Chapter 13.   Backtracking","13.6   Exercises"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/","level":1,"title":"13.4   N-queens problem","text":"<p>Question</p> <p>In chess, a queen can attack any piece in the same row, column, or diagonal. Given \\(n\\) queens and an \\(n \\times n\\) chessboard, find all arrangements in which no two queens can attack each other.</p> <p>As shown in Figure 13-15, there are two solutions when \\(n = 4\\). From a backtracking perspective, the \\(n \\times n\\) board has \\(n^2\\) squares, which provide all possible <code>choices</code>. As we place queens one by one, the board changes; its arrangement at any moment is the current <code>state</code>.</p> <p></p> <p> Figure 13-15   Solutions to the 4-queens problem </p> <p>Figure 13-16 shows the three constraints: no two queens may share a row, column, or diagonal. Diagonals run in two directions: the main-diagonal direction <code>\\</code> and the antidiagonal direction <code>/</code>.</p> <p></p> <p> Figure 13-16   Constraints of the n-queens problem </p>","path":["Chapter 13.   Backtracking","13.4   N-queens problem"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/#1-placing-queens-row-by-row","level":3,"title":"1.   Placing queens row by row","text":"<p>There are as many queens as rows, namely \\(n\\), so we can immediately conclude that each row must contain exactly one queen.</p> <p>We can therefore place queens row by row: start at the first row and place one queen in each row through the last.</p> <p>Figure 13-17 illustrates this process for 4 queens. To fit the illustration, only one branch from the first row is expanded, and all arrangements violating the column or diagonal constraints are pruned.</p> <p></p> <p> Figure 13-17   Placing queens row by row </p> <p>In essence, placing queens row by row is itself a form of pruning: it eliminates every search branch that places multiple queens in the same row.</p>","path":["Chapter 13.   Backtracking","13.4   N-queens problem"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/#2-pruning-by-columns-and-diagonals","level":3,"title":"2.   Pruning by columns and diagonals","text":"<p>To enforce the column constraint, use a Boolean array <code>cols</code> of length \\(n\\) to record whether each column contains a queen. Before placing a queen, consult <code>cols</code> to prune columns already occupied, and update <code>cols</code> as the search proceeds and backtracks.</p> <p>Tip</p> <p>The matrix starts at its top-left corner. Row indices increase downward, and column indices increase to the right.</p> <p>How do we enforce the diagonal constraints? Let \\((row, col)\\) be the row and column indices of a square. Along any diagonal in the main-diagonal direction, the row index minus the column index is the same: \\(row - col\\) is constant along that diagonal.</p> <p>Thus, if two squares satisfy \\(row_1 - col_1 = row_2 - col_2\\), they lie on the same main-direction diagonal. Using this property, the array <code>diags1</code> in Figure 13-18 records whether each such diagonal contains a queen.</p> <p>Similarly, \\(row + col\\) is constant along each antidiagonal. We can use an array <code>diags2</code> to enforce the antidiagonal constraint.</p> <p></p> <p> Figure 13-18   Enforcing column and diagonal constraints </p>","path":["Chapter 13.   Backtracking","13.4   N-queens problem"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/#3-code-implementation","level":3,"title":"3.   Code implementation","text":"<p>In a square matrix of order \\(n\\), \\(row - col\\) ranges over \\([-n + 1, n - 1]\\), and \\(row + col\\) ranges over \\([0, 2n - 2]\\). There are therefore \\(2n - 1\\) diagonals in each direction, so both <code>diags1</code> and <code>diags2</code> have length \\(2n - 1\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby n_queens.py<pre><code>def backtrack(\n    row: int,\n    n: int,\n    state: list[list[str]],\n    res: list[list[list[str]]],\n    cols: list[bool],\n    diags1: list[bool],\n    diags2: list[bool],\n):\n    \"\"\"Backtracking: n-queens\"\"\"\n    # Record a solution when a queen has been placed in every row\n    if row == n:\n        res.append([list(row) for row in state])\n        return\n    # Iterate over all columns\n    for col in range(n):\n        # Calculate the main-diagonal and anti-diagonal indices for this square\n        diag1 = row - col + n - 1\n        diag2 = row + col\n        # Prune: the square must have no queen in its column, main diagonal, or anti-diagonal\n        if not cols[col] and not diags1[diag1] and not diags2[diag2]:\n            # Try: place a queen on this square\n            state[row][col] = \"Q\"\n            cols[col] = diags1[diag1] = diags2[diag2] = True\n            # Place a queen in the next row\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            # Backtrack: restore this square to empty\n            state[row][col] = \"#\"\n            cols[col] = diags1[diag1] = diags2[diag2] = False\n\ndef n_queens(n: int) -&gt; list[list[list[str]]]:\n    \"\"\"Solve the n-queens problem\"\"\"\n    # Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square\n    state = [[\"#\" for _ in range(n)] for _ in range(n)]\n    cols = [False] * n  # Track whether a column contains a queen\n    diags1 = [False] * (2 * n - 1)  # Track whether a main diagonal contains a queen\n    diags2 = [False] * (2 * n - 1)  # Track whether an anti-diagonal contains a queen\n    res = []\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n</code></pre> n_queens.cpp<pre><code>/* Backtracking: n-queens */\nvoid backtrack(int row, int n, vector&lt;vector&lt;string&gt;&gt; &amp;state, vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; &amp;res, vector&lt;bool&gt; &amp;cols,\n               vector&lt;bool&gt; &amp;diags1, vector&lt;bool&gt; &amp;diags2) {\n    // Record a solution when a queen has been placed in every row\n    if (row == n) {\n        res.push_back(state);\n        return;\n    }\n    // Iterate over all columns\n    for (int col = 0; col &lt; n; col++) {\n        // Calculate the main-diagonal and anti-diagonal indices for this square\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // Prune: the square must have no queen in its column, main diagonal, or anti-diagonal\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // Try: place a queen on this square\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // Place a queen in the next row\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // Backtrack: restore this square to empty\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* Solve the n-queens problem */\nvector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; nQueens(int n) {\n    // Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square\n    vector&lt;vector&lt;string&gt;&gt; state(n, vector&lt;string&gt;(n, \"#\"));\n    vector&lt;bool&gt; cols(n, false);           // Track whether a column contains a queen\n    vector&lt;bool&gt; diags1(2 * n - 1, false); // Track whether a main diagonal contains a queen\n    vector&lt;bool&gt; diags2(2 * n - 1, false); // Track whether an anti-diagonal contains a queen\n    vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; res;\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.java<pre><code>/* Backtracking: n-queens */\nvoid backtrack(int row, int n, List&lt;List&lt;String&gt;&gt; state, List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n        boolean[] cols, boolean[] diags1, boolean[] diags2) {\n    // Record a solution when a queen has been placed in every row\n    if (row == n) {\n        List&lt;List&lt;String&gt;&gt; copyState = new ArrayList&lt;&gt;();\n        for (List&lt;String&gt; sRow : state) {\n            copyState.add(new ArrayList&lt;&gt;(sRow));\n        }\n        res.add(copyState);\n        return;\n    }\n    // Iterate over all columns\n    for (int col = 0; col &lt; n; col++) {\n        // Calculate the main-diagonal and anti-diagonal indices for this square\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // Prune: the square must have no queen in its column, main diagonal, or anti-diagonal\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // Try: place a queen on this square\n            state.get(row).set(col, \"Q\");\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // Place a queen in the next row\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // Backtrack: restore this square to empty\n            state.get(row).set(col, \"#\");\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* Solve the n-queens problem */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n    // Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square\n    List&lt;List&lt;String&gt;&gt; state = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;String&gt; row = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            row.add(\"#\");\n        }\n        state.add(row);\n    }\n    boolean[] cols = new boolean[n]; // Track whether a column contains a queen\n    boolean[] diags1 = new boolean[2 * n - 1]; // Track whether a main diagonal contains a queen\n    boolean[] diags2 = new boolean[2 * n - 1]; // Track whether an anti-diagonal contains a queen\n    List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = new ArrayList&lt;&gt;();\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.cs<pre><code>/* Backtracking: n-queens */\nvoid Backtrack(int row, int n, List&lt;List&lt;string&gt;&gt; state, List&lt;List&lt;List&lt;string&gt;&gt;&gt; res,\n        bool[] cols, bool[] diags1, bool[] diags2) {\n    // Record a solution when a queen has been placed in every row\n    if (row == n) {\n        List&lt;List&lt;string&gt;&gt; copyState = [];\n        foreach (List&lt;string&gt; sRow in state) {\n            copyState.Add(new List&lt;string&gt;(sRow));\n        }\n        res.Add(copyState);\n        return;\n    }\n    // Iterate over all columns\n    for (int col = 0; col &lt; n; col++) {\n        // Calculate the main-diagonal and anti-diagonal indices for this square\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // Prune: the square must have no queen in its column, main diagonal, or anti-diagonal\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // Try: place a queen on this square\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // Place a queen in the next row\n            Backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // Backtrack: restore this square to empty\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* Solve the n-queens problem */\nList&lt;List&lt;List&lt;string&gt;&gt;&gt; NQueens(int n) {\n    // Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square\n    List&lt;List&lt;string&gt;&gt; state = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;string&gt; row = [];\n        for (int j = 0; j &lt; n; j++) {\n            row.Add(\"#\");\n        }\n        state.Add(row);\n    }\n    bool[] cols = new bool[n]; // Track whether a column contains a queen\n    bool[] diags1 = new bool[2 * n - 1]; // Track whether a main diagonal contains a queen\n    bool[] diags2 = new bool[2 * n - 1]; // Track whether an anti-diagonal contains a queen\n    List&lt;List&lt;List&lt;string&gt;&gt;&gt; res = [];\n\n    Backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.go<pre><code>/* Backtracking: n-queens */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n    // Record a solution when a queen has been placed in every row\n    if row == n {\n        newState := make([][]string, len(*state))\n        for i, _ := range newState {\n            newState[i] = make([]string, len((*state)[0]))\n            copy(newState[i], (*state)[i])\n\n        }\n        *res = append(*res, newState)\n        return\n    }\n    // Iterate over all columns\n    for col := 0; col &lt; n; col++ {\n        // Calculate the main-diagonal and anti-diagonal indices for this square\n        diag1 := row - col + n - 1\n        diag2 := row + col\n        // Prune: the square must have no queen in its column, main diagonal, or anti-diagonal\n        if !(*cols)[col] &amp;&amp; !(*diags1)[diag1] &amp;&amp; !(*diags2)[diag2] {\n            // Try: place a queen on this square\n            (*state)[row][col] = \"Q\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n            // Place a queen in the next row\n            backtrack(row+1, n, state, res, cols, diags1, diags2)\n            // Backtrack: restore this square to empty\n            (*state)[row][col] = \"#\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n        }\n    }\n}\n\n/* Solve the n-queens problem */\nfunc nQueens(n int) [][][]string {\n    // Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square\n    state := make([][]string, n)\n    for i := 0; i &lt; n; i++ {\n        row := make([]string, n)\n        for i := 0; i &lt; n; i++ {\n            row[i] = \"#\"\n        }\n        state[i] = row\n    }\n    // Track whether a column contains a queen\n    cols := make([]bool, n)\n    diags1 := make([]bool, 2*n-1)\n    diags2 := make([]bool, 2*n-1)\n    res := make([][][]string, 0)\n    backtrack(0, n, &amp;state, &amp;res, &amp;cols, &amp;diags1, &amp;diags2)\n    return res\n}\n</code></pre> n_queens.swift<pre><code>/* Backtracking: n-queens */\nfunc backtrack(row: Int, n: Int, state: inout [[String]], res: inout [[[String]]], cols: inout [Bool], diags1: inout [Bool], diags2: inout [Bool]) {\n    // Record a solution when a queen has been placed in every row\n    if row == n {\n        res.append(state)\n        return\n    }\n    // Iterate over all columns\n    for col in 0 ..&lt; n {\n        // Calculate the main-diagonal and anti-diagonal indices for this square\n        let diag1 = row - col + n - 1\n        let diag2 = row + col\n        // Prune: the square must have no queen in its column, main diagonal, or anti-diagonal\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // Try: place a queen on this square\n            state[row][col] = \"Q\"\n            cols[col] = true\n            diags1[diag1] = true\n            diags2[diag2] = true\n            // Place a queen in the next row\n            backtrack(row: row + 1, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n            // Backtrack: restore this square to empty\n            state[row][col] = \"#\"\n            cols[col] = false\n            diags1[diag1] = false\n            diags2[diag2] = false\n        }\n    }\n}\n\n/* Solve the n-queens problem */\nfunc nQueens(n: Int) -&gt; [[[String]]] {\n    // Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square\n    var state = Array(repeating: Array(repeating: \"#\", count: n), count: n)\n    var cols = Array(repeating: false, count: n) // Track whether a column contains a queen\n    var diags1 = Array(repeating: false, count: 2 * n - 1) // Track whether a main diagonal contains a queen\n    var diags2 = Array(repeating: false, count: 2 * n - 1) // Track whether an anti-diagonal contains a queen\n    var res: [[[String]]] = []\n\n    backtrack(row: 0, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n\n    return res\n}\n</code></pre> n_queens.js<pre><code>/* Backtracking: n-queens */\nfunction backtrack(row, n, state, res, cols, diags1, diags2) {\n    // Record a solution when a queen has been placed in every row\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // Iterate over all columns\n    for (let col = 0; col &lt; n; col++) {\n        // Calculate the main-diagonal and anti-diagonal indices for this square\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // Prune: the square must have no queen in its column, main diagonal, or anti-diagonal\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // Try: place a queen on this square\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // Place a queen in the next row\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // Backtrack: restore this square to empty\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* Solve the n-queens problem */\nfunction nQueens(n) {\n    // Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // Track whether a column contains a queen\n    const diags1 = Array(2 * n - 1).fill(false); // Track whether a main diagonal contains a queen\n    const diags2 = Array(2 * n - 1).fill(false); // Track whether an anti-diagonal contains a queen\n    const res = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.ts<pre><code>/* Backtracking: n-queens */\nfunction backtrack(\n    row: number,\n    n: number,\n    state: string[][],\n    res: string[][][],\n    cols: boolean[],\n    diags1: boolean[],\n    diags2: boolean[]\n): void {\n    // Record a solution when a queen has been placed in every row\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // Iterate over all columns\n    for (let col = 0; col &lt; n; col++) {\n        // Calculate the main-diagonal and anti-diagonal indices for this square\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // Prune: the square must have no queen in its column, main diagonal, or anti-diagonal\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // Try: place a queen on this square\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // Place a queen in the next row\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // Backtrack: restore this square to empty\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* Solve the n-queens problem */\nfunction nQueens(n: number): string[][][] {\n    // Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // Track whether a column contains a queen\n    const diags1 = Array(2 * n - 1).fill(false); // Track whether a main diagonal contains a queen\n    const diags2 = Array(2 * n - 1).fill(false); // Track whether an anti-diagonal contains a queen\n    const res: string[][][] = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.dart<pre><code>/* Backtracking: n-queens */\nvoid backtrack(\n  int row,\n  int n,\n  List&lt;List&lt;String&gt;&gt; state,\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n  List&lt;bool&gt; cols,\n  List&lt;bool&gt; diags1,\n  List&lt;bool&gt; diags2,\n) {\n  // Record a solution when a queen has been placed in every row\n  if (row == n) {\n    List&lt;List&lt;String&gt;&gt; copyState = [];\n    for (List&lt;String&gt; sRow in state) {\n      copyState.add(List.from(sRow));\n    }\n    res.add(copyState);\n    return;\n  }\n  // Iterate over all columns\n  for (int col = 0; col &lt; n; col++) {\n    // Calculate the main-diagonal and anti-diagonal indices for this square\n    int diag1 = row - col + n - 1;\n    int diag2 = row + col;\n    // Prune: the square must have no queen in its column, main diagonal, or anti-diagonal\n    if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n      // Try: place a queen on this square\n      state[row][col] = \"Q\";\n      cols[col] = true;\n      diags1[diag1] = true;\n      diags2[diag2] = true;\n      // Place a queen in the next row\n      backtrack(row + 1, n, state, res, cols, diags1, diags2);\n      // Backtrack: restore this square to empty\n      state[row][col] = \"#\";\n      cols[col] = false;\n      diags1[diag1] = false;\n      diags2[diag2] = false;\n    }\n  }\n}\n\n/* Solve the n-queens problem */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n  // Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square\n  List&lt;List&lt;String&gt;&gt; state = List.generate(n, (index) =&gt; List.filled(n, \"#\"));\n  List&lt;bool&gt; cols = List.filled(n, false); // Track whether a column contains a queen\n  List&lt;bool&gt; diags1 = List.filled(2 * n - 1, false); // Track whether a main diagonal contains a queen\n  List&lt;bool&gt; diags2 = List.filled(2 * n - 1, false); // Track whether an anti-diagonal contains a queen\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = [];\n\n  backtrack(0, n, state, res, cols, diags1, diags2);\n\n  return res;\n}\n</code></pre> n_queens.rs<pre><code>/* Backtracking: n-queens */\nfn backtrack(\n    row: usize,\n    n: usize,\n    state: &amp;mut Vec&lt;Vec&lt;String&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt;,\n    cols: &amp;mut [bool],\n    diags1: &amp;mut [bool],\n    diags2: &amp;mut [bool],\n) {\n    // Record a solution when a queen has been placed in every row\n    if row == n {\n        res.push(state.clone());\n        return;\n    }\n    // Iterate over all columns\n    for col in 0..n {\n        // Calculate the main-diagonal and anti-diagonal indices for this square\n        let diag1 = row + n - 1 - col;\n        let diag2 = row + col;\n        // Prune: the square must have no queen in its column, main diagonal, or anti-diagonal\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // Try: place a queen on this square\n            state[row][col] = \"Q\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (true, true, true);\n            // Place a queen in the next row\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // Backtrack: restore this square to empty\n            state[row][col] = \"#\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (false, false, false);\n        }\n    }\n}\n\n/* Solve the n-queens problem */\nfn n_queens(n: usize) -&gt; Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; {\n    // Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square\n    let mut state: Vec&lt;Vec&lt;String&gt;&gt; = vec![vec![\"#\".to_string(); n]; n];\n    let mut cols = vec![false; n]; // Track whether a column contains a queen\n    let mut diags1 = vec![false; 2 * n - 1]; // Track whether a main diagonal contains a queen\n    let mut diags2 = vec![false; 2 * n - 1]; // Track whether an anti-diagonal contains a queen\n    let mut res: Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; = Vec::new();\n\n    backtrack(\n        0,\n        n,\n        &amp;mut state,\n        &amp;mut res,\n        &amp;mut cols,\n        &amp;mut diags1,\n        &amp;mut diags2,\n    );\n\n    res\n}\n</code></pre> n_queens.c<pre><code>/* Backtracking: n-queens */\nvoid backtrack(int row, int n, char state[MAX_SIZE][MAX_SIZE], char ***res, int *resSize, bool cols[MAX_SIZE],\n               bool diags1[2 * MAX_SIZE - 1], bool diags2[2 * MAX_SIZE - 1]) {\n    // Record a solution when a queen has been placed in every row\n    if (row == n) {\n        res[*resSize] = (char **)malloc(sizeof(char *) * n);\n        for (int i = 0; i &lt; n; ++i) {\n            res[*resSize][i] = (char *)malloc(sizeof(char) * (n + 1));\n            strcpy(res[*resSize][i], state[i]);\n        }\n        (*resSize)++;\n        return;\n    }\n    // Iterate over all columns\n    for (int col = 0; col &lt; n; col++) {\n        // Calculate the main-diagonal and anti-diagonal indices for this square\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // Prune: the square must have no queen in its column, main diagonal, or anti-diagonal\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // Try: place a queen on this square\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // Place a queen in the next row\n            backtrack(row + 1, n, state, res, resSize, cols, diags1, diags2);\n            // Backtrack: restore this square to empty\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* Solve the n-queens problem */\nchar ***nQueens(int n, int *returnSize) {\n    char state[MAX_SIZE][MAX_SIZE];\n    // Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square\n    for (int i = 0; i &lt; n; ++i) {\n        for (int j = 0; j &lt; n; ++j) {\n            state[i][j] = '#';\n        }\n        state[i][n] = '\\0';\n    }\n    bool cols[MAX_SIZE] = {false};           // Track whether a column contains a queen\n    bool diags1[2 * MAX_SIZE - 1] = {false}; // Track whether a main diagonal contains a queen\n    bool diags2[2 * MAX_SIZE - 1] = {false}; // Track whether an anti-diagonal contains a queen\n\n    char ***res = (char ***)malloc(sizeof(char **) * MAX_SIZE);\n    *returnSize = 0;\n    backtrack(0, n, state, res, returnSize, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.kt<pre><code>/* Backtracking: n-queens */\nfun backtrack(\n    row: Int,\n    n: Int,\n    state: MutableList&lt;MutableList&lt;String&gt;&gt;,\n    res: MutableList&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt;,\n    cols: BooleanArray,\n    diags1: BooleanArray,\n    diags2: BooleanArray\n) {\n    // Record a solution when a queen has been placed in every row\n    if (row == n) {\n        val copyState = mutableListOf&lt;MutableList&lt;String&gt;&gt;()\n        for (sRow in state) {\n            copyState.add(sRow.toMutableList())\n        }\n        res.add(copyState)\n        return\n    }\n    // Iterate over all columns\n    for (col in 0..&lt;n) {\n        // Calculate the main-diagonal and anti-diagonal indices for this square\n        val diag1 = row - col + n - 1\n        val diag2 = row + col\n        // Prune: the square must have no queen in its column, main diagonal, or anti-diagonal\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // Try: place a queen on this square\n            state[row][col] = \"Q\"\n            diags2[diag2] = true\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n            // Place a queen in the next row\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            // Backtrack: restore this square to empty\n            state[row][col] = \"#\"\n            diags2[diag2] = false\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n        }\n    }\n}\n\n/* Solve the n-queens problem */\nfun nQueens(n: Int): MutableList&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt; {\n    // Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square\n    val state = mutableListOf&lt;MutableList&lt;String&gt;&gt;()\n    for (i in 0..&lt;n) {\n        val row = mutableListOf&lt;String&gt;()\n        for (j in 0..&lt;n) {\n            row.add(\"#\")\n        }\n        state.add(row)\n    }\n    val cols = BooleanArray(n) // Track whether a column contains a queen\n    val diags1 = BooleanArray(2 * n - 1) // Track whether a main diagonal contains a queen\n    val diags2 = BooleanArray(2 * n - 1) // Track whether an anti-diagonal contains a queen\n    val res = mutableListOf&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt;()\n\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n}\n</code></pre> n_queens.rb<pre><code>### Backtracking: n-queens ###\ndef backtrack(row, n, state, res, cols, diags1, diags2)\n  # Record a solution when a queen has been placed in every row\n  if row == n\n    res &lt;&lt; state.map { |row| row.dup }\n    return\n  end\n\n  # Iterate over all columns\n  for col in 0...n\n    # Calculate the main-diagonal and anti-diagonal indices for this square\n    diag1 = row - col + n - 1\n    diag2 = row + col\n    # Prune: the square must have no queen in its column, main diagonal, or anti-diagonal\n    if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]\n      # Try: place a queen on this square\n      state[row][col] = \"Q\"\n      cols[col] = diags1[diag1] = diags2[diag2] = true\n      # Place a queen in the next row\n      backtrack(row + 1, n, state, res, cols, diags1, diags2)\n      # Backtrack: restore this square to empty\n      state[row][col] = \"#\"\n      cols[col] = diags1[diag1] = diags2[diag2] = false\n    end\n  end\nend\n\n### Solve the n-queens problem ###\ndef n_queens(n)\n  # Initialize an n*n board, where 'Q' represents a queen and '#' represents an empty square\n  state = Array.new(n) { Array.new(n, \"#\") }\n  cols = Array.new(n, false) # Track whether a column contains a queen\n  diags1 = Array.new(2 * n - 1, false) # Track whether a main diagonal contains a queen\n  diags2 = Array.new(2 * n - 1, false) # Track whether an anti-diagonal contains a queen\n  res = []\n  backtrack(0, n, state, res, cols, diags1, diags2)\n\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>We place a queen in each of \\(n\\) rows. Considering the column constraint, the successive rows have \\(n\\), \\(n-1\\), \\(\\dots\\), \\(2\\), and \\(1\\) choices, giving \\(O(n!)\\) time. Recording a solution requires copying the matrix <code>state</code> and appending the copy to <code>res</code>, which takes \\(O(n^2)\\) time. The overall time complexity is therefore \\(O(n! \\cdot n^2)\\). In practice, diagonal pruning greatly reduces the search space, so the search is often faster than this bound suggests.</p> <p>The array <code>state</code> uses \\(O(n^2)\\) space, while <code>cols</code>, <code>diags1</code>, and <code>diags2</code> each use \\(O(n)\\) space. The maximum recursion depth is \\(n\\), requiring \\(O(n)\\) space for stack frames. The space complexity is therefore \\(O(n^2)\\).</p>","path":["Chapter 13.   Backtracking","13.4   N-queens problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/","level":1,"title":"13.2   Permutations problem","text":"<p>Generating permutations is a classic application of backtracking. Given a collection, such as an array or a string, the task is to find every possible ordering of its elements.</p> <p>Table 13-2 gives several input arrays and all their permutations.</p> <p> Table 13-2   Examples of permutations </p> Input array All permutations \\([1]\\) \\([1]\\) \\([1, 2]\\) \\([1, 2], [2, 1]\\) \\([1, 2, 3]\\) \\([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\\)","path":["Chapter 13.   Backtracking","13.2   Permutations problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1321-distinct-elements","level":2,"title":"13.2.1   Distinct elements","text":"<p>Question</p> <p>Given an integer array with no duplicate elements, return all possible permutations.</p> <p>From a backtracking perspective, we can view a permutation as the result of a sequence of choices. For input \\([1, 2, 3]\\), choosing \\(1\\), then \\(3\\), then \\(2\\) produces \\([1, 3, 2]\\). Backtracking undoes a choice so that other choices can be tried.</p> <p>In the code, <code>choices</code> contains all the input elements, and <code>state</code> contains the elements chosen so far. Each element may be chosen only once, so every element in <code>state</code> must be unique.</p> <p>As shown in Figure 13-5, we can expand the search into a recursion tree, where each node represents the current <code>state</code>. Starting from the root, three rounds of choices lead to a leaf, and each leaf corresponds to one permutation.</p> <p></p> <p> Figure 13-5   Recursion tree for permutations </p>","path":["Chapter 13.   Backtracking","13.2   Permutations problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1-pruning-repeated-selections","level":3,"title":"1.   Pruning repeated selections","text":"<p>To ensure that each element is chosen only once, introduce a Boolean array <code>selected</code>, where <code>selected[i]</code> indicates whether <code>choices[i]</code> has been chosen. Use it for the following pruning steps.</p> <ul> <li>After choosing <code>choices[i]</code>, set <code>selected[i]</code> to \\(\\text{True}\\) to mark it as selected.</li> <li>When iterating over <code>choices</code>, skip all selected elements, pruning their branches.</li> </ul> <p>As shown in Figure 13-6, if we choose 1 first, 3 second, and 2 third, we must prune the branch for 1 in the second round and the branches for 1 and 3 in the third round.</p> <p></p> <p> Figure 13-6   Example of pruning permutations </p> <p>This pruning reduces the search space from \\(O(n^n)\\) to \\(O(n!)\\).</p>","path":["Chapter 13.   Backtracking","13.2   Permutations problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#2-code-implementation","level":3,"title":"2.   Code implementation","text":"<p>With these details established, we can fill in the backtracking template. To keep the code short, we inline the template's helper functions into <code>backtrack()</code>:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby permutations_i.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"Backtracking: permutations I\"\"\"\n    # Record a solution when the state length equals the number of elements\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # Iterate over all choices\n    for i, choice in enumerate(choices):\n        # Prune: do not choose an element more than once\n        if not selected[i]:\n            # Try: make a choice and update the state\n            selected[i] = True\n            state.append(choice)\n            # Proceed to the next choice\n            backtrack(state, choices, selected, res)\n            # Backtrack: undo the choice and restore the previous state\n            selected[i] = False\n            state.pop()\n\ndef permutations_i(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"Permutations I\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_i.cpp<pre><code>/* Backtracking: permutations I */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // Record a solution when the state length equals the number of elements\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // Iterate over all choices\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // Prune: do not choose an element more than once\n        if (!selected[i]) {\n            // Try: make a choice and update the state\n            selected[i] = true;\n            state.push_back(choice);\n            // Proceed to the next choice\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* Permutations I */\nvector&lt;vector&lt;int&gt;&gt; permutationsI(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_i.java<pre><code>/* Backtracking: permutations I */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // Record a solution when the state length equals the number of elements\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // Iterate over all choices\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // Prune: do not choose an element more than once\n        if (!selected[i]) {\n            // Try: make a choice and update the state\n            selected[i] = true;\n            state.add(choice);\n            // Proceed to the next choice\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* Permutations I */\nList&lt;List&lt;Integer&gt;&gt; permutationsI(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_i.cs<pre><code>/* Backtracking: permutations I */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // Record a solution when the state length equals the number of elements\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // Iterate over all choices\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // Prune: do not choose an element more than once\n        if (!selected[i]) {\n            // Try: make a choice and update the state\n            selected[i] = true;\n            state.Add(choice);\n            // Proceed to the next choice\n            Backtrack(state, choices, selected, res);\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* Permutations I */\nList&lt;List&lt;int&gt;&gt; PermutationsI(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_i.go<pre><code>/* Backtracking: permutations I */\nfunc backtrackI(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // Record a solution when the state length equals the number of elements\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // Iterate over all choices\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // Prune: do not choose an element more than once\n        if !(*selected)[i] {\n            // Try: make a choice and update the state\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // Proceed to the next choice\n            backtrackI(state, choices, selected, res)\n            // Backtrack: undo the choice and restore the previous state\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* Permutations I */\nfunc permutationsI(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackI(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_i.swift<pre><code>/* Backtracking: permutations I */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // Record a solution when the state length equals the number of elements\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // Iterate over all choices\n    for (i, choice) in choices.enumerated() {\n        // Prune: do not choose an element more than once\n        if !selected[i] {\n            // Try: make a choice and update the state\n            selected[i] = true\n            state.append(choice)\n            // Proceed to the next choice\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* Permutations I */\nfunc permutationsI(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_i.js<pre><code>/* Backtracking: permutations I */\nfunction backtrack(state, choices, selected, res) {\n    // Record a solution when the state length equals the number of elements\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // Iterate over all choices\n    choices.forEach((choice, i) =&gt; {\n        // Prune: do not choose an element more than once\n        if (!selected[i]) {\n            // Try: make a choice and update the state\n            selected[i] = true;\n            state.push(choice);\n            // Proceed to the next choice\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* Permutations I */\nfunction permutationsI(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.ts<pre><code>/* Backtracking: permutations I */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // Record a solution when the state length equals the number of elements\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // Iterate over all choices\n    choices.forEach((choice, i) =&gt; {\n        // Prune: do not choose an element more than once\n        if (!selected[i]) {\n            // Try: make a choice and update the state\n            selected[i] = true;\n            state.push(choice);\n            // Proceed to the next choice\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* Permutations I */\nfunction permutationsI(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.dart<pre><code>/* Backtracking: permutations I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // Record a solution when the state length equals the number of elements\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // Iterate over all choices\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // Prune: do not choose an element more than once\n    if (!selected[i]) {\n      // Try: make a choice and update the state\n      selected[i] = true;\n      state.add(choice);\n      // Proceed to the next choice\n      backtrack(state, choices, selected, res);\n      // Backtrack: undo the choice and restore the previous state\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* Permutations I */\nList&lt;List&lt;int&gt;&gt; permutationsI(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_i.rs<pre><code>/* Backtracking: permutations I */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // Record a solution when the state length equals the number of elements\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // Iterate over all choices\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // Prune: do not choose an element more than once\n        if !selected[i] {\n            // Try: make a choice and update the state\n            selected[i] = true;\n            state.push(choice);\n            // Proceed to the next choice\n            backtrack(state.clone(), choices, selected, res);\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false;\n            state.pop();\n        }\n    }\n}\n\n/* Permutations I */\nfn permutations_i(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new(); // Result list (permutations)\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_i.c<pre><code>/* Backtracking: permutations I */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // Record a solution when the state length equals the number of elements\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // Iterate over all choices\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // Prune: do not choose an element more than once\n        if (!selected[i]) {\n            // Try: make a choice and update the state\n            selected[i] = true;\n            state[stateSize] = choice;\n            // Proceed to the next choice\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false;\n        }\n    }\n}\n\n/* Permutations I */\nint **permutationsI(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_i.kt<pre><code>/* Backtracking: permutations I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // Record a solution when the state length equals the number of elements\n    if (state.size == choices.size) {\n        res.add(state.toMutableList())\n        return\n    }\n    // Iterate over all choices\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // Prune: do not choose an element more than once\n        if (!selected[i]) {\n            // Try: make a choice and update the state\n            selected[i] = true\n            state.add(choice)\n            // Proceed to the next choice\n            backtrack(state, choices, selected, res)\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* Permutations I */\nfun permutationsI(nums: IntArray): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;()\n    backtrack(mutableListOf(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_i.rb<pre><code>### Backtracking: permutations I ###\ndef backtrack(state, choices, selected, res)\n  # Record a solution when the state length equals the number of elements\n  if state.length == choices.length\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # Iterate over all choices\n  choices.each_with_index do |choice, i|\n    # Prune: do not choose an element more than once\n    unless selected[i]\n      # Try: make a choice and update the state\n      selected[i] = true\n      state &lt;&lt; choice\n      # Proceed to the next choice\n      backtrack(state, choices, selected, res)\n      # Backtrack: undo the choice and restore the previous state\n      selected[i] = false\n      state.pop\n    end\n  end\nend\n\n### Permutations I ###\ndef permutations_i(nums)\n  res = []\n  backtrack([], nums, Array.new(nums.length, false), res)\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 13.   Backtracking","13.2   Permutations problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1322-duplicate-elements","level":2,"title":"13.2.2   Duplicate elements","text":"<p>Question</p> <p>Given an integer array that may contain duplicate elements, return all distinct permutations.</p> <p>Suppose the input is \\([1, 1, 2]\\). To distinguish the two occurrences of \\(1\\), denote the second \\(1\\) by \\(\\hat{1}\\).</p> <p>As shown in Figure 13-7, half the permutations generated by the previous method are duplicates.</p> <p></p> <p> Figure 13-7   Duplicate permutations </p> <p>How can we remove these duplicates? A straightforward approach is to use a hash set to deduplicate the results. However, this is inelegant: branches that generate duplicate permutations are unnecessary and should be identified and pruned in advance, improving efficiency.</p>","path":["Chapter 13.   Backtracking","13.2   Permutations problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1-pruning-equal-elements","level":3,"title":"1.   Pruning equal elements","text":"<p>In Figure 13-8, choosing \\(1\\) or \\(\\hat{1}\\) in the first round is equivalent: every permutation produced under one choice is duplicated under the other. We should therefore prune \\(\\hat{1}\\).</p> <p>Similarly, after choosing \\(2\\) in the first round, choosing either \\(1\\) or \\(\\hat{1}\\) in the second produces duplicate branches. Prune the second-round choice of \\(\\hat{1}\\) as well.</p> <p>In essence, within any one round of choices, we want to try only one of the elements with the same value.</p> <p></p> <p> Figure 13-8   Pruning duplicate permutations </p>","path":["Chapter 13.   Backtracking","13.2   Permutations problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#2-code-implementation_1","level":3,"title":"2.   Code implementation","text":"<p>Building on the previous solution, create a hash set <code>duplicated</code> in each round to record the values already tried in that round and prune repeated values:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby permutations_ii.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"Backtracking: permutations II\"\"\"\n    # Record a solution when the state length equals the number of elements\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # Iterate over all choices\n    duplicated = set[int]()\n    for i, choice in enumerate(choices):\n        # Prune: do not reuse an element or choose the same value again at this depth\n        if not selected[i] and choice not in duplicated:\n            # Try: make a choice and update the state\n            duplicated.add(choice)  # Record the chosen value\n            selected[i] = True\n            state.append(choice)\n            # Proceed to the next choice\n            backtrack(state, choices, selected, res)\n            # Backtrack: undo the choice and restore the previous state\n            selected[i] = False\n            state.pop()\n\ndef permutations_ii(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"Permutations II\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_ii.cpp<pre><code>/* Backtracking: permutations II */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // Record a solution when the state length equals the number of elements\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // Iterate over all choices\n    unordered_set&lt;int&gt; duplicated;\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // Prune: do not reuse an element or choose the same value again at this depth\n        if (!selected[i] &amp;&amp; duplicated.find(choice) == duplicated.end()) {\n            // Try: make a choice and update the state\n            duplicated.emplace(choice); // Record the chosen value\n            selected[i] = true;\n            state.push_back(choice);\n            // Proceed to the next choice\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* Permutations II */\nvector&lt;vector&lt;int&gt;&gt; permutationsII(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_ii.java<pre><code>/* Backtracking: permutations II */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // Record a solution when the state length equals the number of elements\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // Iterate over all choices\n    Set&lt;Integer&gt; duplicated = new HashSet&lt;Integer&gt;();\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // Prune: do not reuse an element or choose the same value again at this depth\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // Try: make a choice and update the state\n            duplicated.add(choice); // Record the chosen value\n            selected[i] = true;\n            state.add(choice);\n            // Proceed to the next choice\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* Permutations II */\nList&lt;List&lt;Integer&gt;&gt; permutationsII(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_ii.cs<pre><code>/* Backtracking: permutations II */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // Record a solution when the state length equals the number of elements\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // Iterate over all choices\n    HashSet&lt;int&gt; duplicated = [];\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // Prune: do not reuse an element or choose the same value again at this depth\n        if (!selected[i] &amp;&amp; !duplicated.Contains(choice)) {\n            // Try: make a choice and update the state\n            duplicated.Add(choice); // Record the chosen value\n            selected[i] = true;\n            state.Add(choice);\n            // Proceed to the next choice\n            Backtrack(state, choices, selected, res);\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* Permutations II */\nList&lt;List&lt;int&gt;&gt; PermutationsII(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_ii.go<pre><code>/* Backtracking: permutations II */\nfunc backtrackII(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // Record a solution when the state length equals the number of elements\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // Iterate over all choices\n    duplicated := make(map[int]struct{}, 0)\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // Prune: do not reuse an element or choose the same value again at this depth\n        if _, ok := duplicated[choice]; !ok &amp;&amp; !(*selected)[i] {\n            // Try: make a choice and update the state\n            // Record the chosen value\n            duplicated[choice] = struct{}{}\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // Proceed to the next choice\n            backtrackII(state, choices, selected, res)\n            // Backtrack: undo the choice and restore the previous state\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* Permutations II */\nfunc permutationsII(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackII(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_ii.swift<pre><code>/* Backtracking: permutations II */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // Record a solution when the state length equals the number of elements\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // Iterate over all choices\n    var duplicated: Set&lt;Int&gt; = []\n    for (i, choice) in choices.enumerated() {\n        // Prune: do not reuse an element or choose the same value again at this depth\n        if !selected[i], !duplicated.contains(choice) {\n            // Try: make a choice and update the state\n            duplicated.insert(choice) // Record the chosen value\n            selected[i] = true\n            state.append(choice)\n            // Proceed to the next choice\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* Permutations II */\nfunc permutationsII(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_ii.js<pre><code>/* Backtracking: permutations II */\nfunction backtrack(state, choices, selected, res) {\n    // Record a solution when the state length equals the number of elements\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // Iterate over all choices\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // Prune: do not reuse an element or choose the same value again at this depth\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // Try: make a choice and update the state\n            duplicated.add(choice); // Record the chosen value\n            selected[i] = true;\n            state.push(choice);\n            // Proceed to the next choice\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* Permutations II */\nfunction permutationsII(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.ts<pre><code>/* Backtracking: permutations II */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // Record a solution when the state length equals the number of elements\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // Iterate over all choices\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // Prune: do not reuse an element or choose the same value again at this depth\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // Try: make a choice and update the state\n            duplicated.add(choice); // Record the chosen value\n            selected[i] = true;\n            state.push(choice);\n            // Proceed to the next choice\n            backtrack(state, choices, selected, res);\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* Permutations II */\nfunction permutationsII(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.dart<pre><code>/* Backtracking: permutations II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // Record a solution when the state length equals the number of elements\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // Iterate over all choices\n  Set&lt;int&gt; duplicated = {};\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // Prune: do not reuse an element or choose the same value again at this depth\n    if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n      // Try: make a choice and update the state\n      duplicated.add(choice); // Record the chosen value\n      selected[i] = true;\n      state.add(choice);\n      // Proceed to the next choice\n      backtrack(state, choices, selected, res);\n      // Backtrack: undo the choice and restore the previous state\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* Permutations II */\nList&lt;List&lt;int&gt;&gt; permutationsII(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_ii.rs<pre><code>/* Backtracking: permutations II */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // Record a solution when the state length equals the number of elements\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // Iterate over all choices\n    let mut duplicated = HashSet::&lt;i32&gt;::new();\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // Prune: do not reuse an element or choose the same value again at this depth\n        if !selected[i] &amp;&amp; !duplicated.contains(&amp;choice) {\n            // Try: make a choice and update the state\n            duplicated.insert(choice); // Record the chosen value\n            selected[i] = true;\n            state.push(choice);\n            // Proceed to the next choice\n            backtrack(state.clone(), choices, selected, res);\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false;\n            state.pop();\n        }\n    }\n}\n\n/* Permutations II */\nfn permutations_ii(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new();\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_ii.c<pre><code>/* Backtracking: permutations II */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // Record a solution when the state length equals the number of elements\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // Iterate over all choices\n    bool duplicated[MAX_SIZE] = {false};\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // Prune: do not reuse an element or choose the same value again at this depth\n        if (!selected[i] &amp;&amp; !duplicated[choice]) {\n            // Try: make a choice and update the state\n            duplicated[choice] = true; // Record the chosen value\n            selected[i] = true;\n            state[stateSize] = choice;\n            // Proceed to the next choice\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false;\n        }\n    }\n}\n\n/* Permutations II */\nint **permutationsII(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_ii.kt<pre><code>/* Backtracking: permutations II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // Record a solution when the state length equals the number of elements\n    if (state.size == choices.size) {\n        res.add(state.toMutableList())\n        return\n    }\n    // Iterate over all choices\n    val duplicated = HashSet&lt;Int&gt;()\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // Prune: do not reuse an element or choose the same value again at this depth\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // Try: make a choice and update the state\n            duplicated.add(choice) // Record the chosen value\n            selected[i] = true\n            state.add(choice)\n            // Proceed to the next choice\n            backtrack(state, choices, selected, res)\n            // Backtrack: undo the choice and restore the previous state\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* Permutations II */\nfun permutationsII(nums: IntArray): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;()\n    backtrack(mutableListOf(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_ii.rb<pre><code>### Backtracking: permutations II ###\ndef backtrack(state, choices, selected, res)\n  # Record a solution when the state length equals the number of elements\n  if state.length == choices.length\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # Iterate over all choices\n  duplicated = Set.new\n  choices.each_with_index do |choice, i|\n    # Prune: do not reuse an element or choose the same value again at this depth\n    if !selected[i] &amp;&amp; !duplicated.include?(choice)\n      # Try: make a choice and update the state\n      duplicated.add(choice)\n      selected[i] = true\n      state &lt;&lt; choice\n      # Proceed to the next choice\n      backtrack(state, choices, selected, res)\n      # Backtrack: undo the choice and restore the previous state\n      selected[i] = false\n      state.pop\n    end\n  end\nend\n\n### Permutations II ###\ndef permutations_ii(nums)\n  res = []\n  backtrack([], nums, Array.new(nums.length, false), res)\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>If all elements are distinct, \\(n\\) elements have \\(n!\\) permutations, where the exclamation mark denotes factorial. Recording each result requires copying a list of length \\(n\\), taking \\(O(n)\\) time. The time complexity is therefore \\(O(n!n)\\).</p> <p>The maximum recursion depth is \\(n\\), requiring \\(O(n)\\) space for stack frames. The array <code>selected</code> uses \\(O(n)\\) space. At most \\(n\\) <code>duplicated</code> sets exist at once, using \\(O(n^2)\\) space. The space complexity is therefore \\(O(n^2)\\).</p>","path":["Chapter 13.   Backtracking","13.2   Permutations problem"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#3-comparing-the-two-pruning-methods","level":3,"title":"3.   Comparing the two pruning methods","text":"<p>Although both <code>selected</code> and <code>duplicated</code> are used for pruning, they serve different purposes.</p> <ul> <li>Pruning repeated selections: There is only one <code>selected</code> array throughout the search. It records which elements are in the current state, preventing the same element from appearing twice in <code>state</code>.</li> <li>Pruning equal elements: Each round of choices (each call to <code>backtrack</code>) has its own <code>duplicated</code> set. It records the values already chosen in that round's <code>for</code> loop, ensuring that equal elements are tried only once in that round.</li> </ul> <p>Figure 13-9 shows the scope of each pruning condition. Each node represents a choice, and the nodes along a path from root to leaf form a permutation.</p> <p></p> <p> Figure 13-9   Scope of the two pruning conditions </p>","path":["Chapter 13.   Backtracking","13.2   Permutations problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/","level":1,"title":"13.3   Subset sum problem","text":"","path":["Chapter 13.   Backtracking","13.3   Subset sum problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1331-distinct-elements","level":2,"title":"13.3.1   Distinct elements","text":"<p>Question</p> <p>Given an array of positive integers <code>nums</code> and a positive target integer <code>target</code>, find all combinations whose elements sum to <code>target</code>. The array has no duplicate elements, and each element may be chosen multiple times. Return the combinations as a list containing no duplicates.</p> <p>For example, with input \\(\\{3, 4, 5\\}\\) and target \\(9\\), the solutions are \\(\\{3, 3, 3\\}, \\{4, 5\\}\\). Keep two points in mind.</p> <ul> <li>Each input element may be chosen an unlimited number of times.</li> <li>Element order does not matter: \\(\\{4, 5\\}\\) and \\(\\{5, 4\\}\\) represent the same subset.</li> </ul>","path":["Chapter 13.   Backtracking","13.3   Subset sum problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1-starting-from-the-permutations-solution","level":3,"title":"1.   Starting from the permutations solution","text":"<p>As with permutations, we can view generating a subset as a sequence of choices. Update the sum as we make choices, and record the subset when the sum equals <code>target</code>.</p> <p>Unlike permutations, each element may be chosen an unlimited number of times, so we do not need a Boolean <code>selected</code> list. A small modification to the permutations code gives an initial solution:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby subset_sum_i_naive.py<pre><code>def backtrack(\n    state: list[int],\n    target: int,\n    total: int,\n    choices: list[int],\n    res: list[list[int]],\n):\n    \"\"\"Backtracking: subset sum I\"\"\"\n    # Record a solution when the subset sum equals target\n    if total == target:\n        res.append(list(state))\n        return\n    # Iterate over all choices\n    for i in range(len(choices)):\n        # Prune: skip this choice if the subset sum would exceed target\n        if total + choices[i] &gt; target:\n            continue\n        # Try: make a choice and update the sum total\n        state.append(choices[i])\n        # Proceed to the next choice\n        backtrack(state, target, total + choices[i], choices, res)\n        # Backtrack: undo the choice and restore the previous state\n        state.pop()\n\ndef subset_sum_i_naive(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"Solve subset sum I (with duplicate subsets)\"\"\"\n    state = []  # State (subset)\n    total = 0  # Subset sum\n    res = []  # Result list (list of subsets)\n    backtrack(state, target, total, nums, res)\n    return res\n</code></pre> subset_sum_i_naive.cpp<pre><code>/* Backtracking: subset sum I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, int total, vector&lt;int&gt; &amp;choices, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // Record a solution when the subset sum equals target\n    if (total == target) {\n        res.push_back(state);\n        return;\n    }\n    // Iterate over all choices\n    for (size_t i = 0; i &lt; choices.size(); i++) {\n        // Prune: skip this choice if the subset sum would exceed target\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // Try: make a choice and update the sum total\n        state.push_back(choices[i]);\n        // Proceed to the next choice\n        backtrack(state, target, total + choices[i], choices, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.pop_back();\n    }\n}\n\n/* Solve subset sum I (with duplicate subsets) */\nvector&lt;vector&lt;int&gt;&gt; subsetSumINaive(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;       // State (subset)\n    int total = 0;           // Subset sum\n    vector&lt;vector&lt;int&gt;&gt; res; // Result list (list of subsets)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.java<pre><code>/* Backtracking: subset sum I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int total, int[] choices, List&lt;List&lt;Integer&gt;&gt; res) {\n    // Record a solution when the subset sum equals target\n    if (total == target) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // Iterate over all choices\n    for (int i = 0; i &lt; choices.length; i++) {\n        // Prune: skip this choice if the subset sum would exceed target\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // Try: make a choice and update the sum total\n        state.add(choices[i]);\n        // Proceed to the next choice\n        backtrack(state, target, total + choices[i], choices, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.remove(state.size() - 1);\n    }\n}\n\n/* Solve subset sum I (with duplicate subsets) */\nList&lt;List&lt;Integer&gt;&gt; subsetSumINaive(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // State (subset)\n    int total = 0; // Subset sum\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // Result list (list of subsets)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.cs<pre><code>/* Backtracking: subset sum I */\nvoid Backtrack(List&lt;int&gt; state, int target, int total, int[] choices, List&lt;List&lt;int&gt;&gt; res) {\n    // Record a solution when the subset sum equals target\n    if (total == target) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // Iterate over all choices\n    for (int i = 0; i &lt; choices.Length; i++) {\n        // Prune: skip this choice if the subset sum would exceed target\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // Try: make a choice and update the sum total\n        state.Add(choices[i]);\n        // Proceed to the next choice\n        Backtrack(state, target, total + choices[i], choices, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* Solve subset sum I (with duplicate subsets) */\nList&lt;List&lt;int&gt;&gt; SubsetSumINaive(int[] nums, int target) {\n    List&lt;int&gt; state = []; // State (subset)\n    int total = 0; // Subset sum\n    List&lt;List&lt;int&gt;&gt; res = []; // Result list (list of subsets)\n    Backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.go<pre><code>/* Backtracking: subset sum I */\nfunc backtrackSubsetSumINaive(total, target int, state, choices *[]int, res *[][]int) {\n    // Record a solution when the subset sum equals target\n    if target == total {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // Iterate over all choices\n    for i := 0; i &lt; len(*choices); i++ {\n        // Prune: skip this choice if the subset sum would exceed target\n        if total+(*choices)[i] &gt; target {\n            continue\n        }\n        // Try: make a choice and update the sum total\n        *state = append(*state, (*choices)[i])\n        // Proceed to the next choice\n        backtrackSubsetSumINaive(total+(*choices)[i], target, state, choices, res)\n        // Backtrack: undo the choice and restore the previous state\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* Solve subset sum I (with duplicate subsets) */\nfunc subsetSumINaive(nums []int, target int) [][]int {\n    state := make([]int, 0) // State (subset)\n    total := 0              // Subset sum\n    res := make([][]int, 0) // Result list (list of subsets)\n    backtrackSubsetSumINaive(total, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.swift<pre><code>/* Backtracking: subset sum I */\nfunc backtrack(state: inout [Int], target: Int, total: Int, choices: [Int], res: inout [[Int]]) {\n    // Record a solution when the subset sum equals target\n    if total == target {\n        res.append(state)\n        return\n    }\n    // Iterate over all choices\n    for i in choices.indices {\n        // Prune: skip this choice if the subset sum would exceed target\n        if total + choices[i] &gt; target {\n            continue\n        }\n        // Try: make a choice and update the sum total\n        state.append(choices[i])\n        // Proceed to the next choice\n        backtrack(state: &amp;state, target: target, total: total + choices[i], choices: choices, res: &amp;res)\n        // Backtrack: undo the choice and restore the previous state\n        state.removeLast()\n    }\n}\n\n/* Solve subset sum I (with duplicate subsets) */\nfunc subsetSumINaive(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // State (subset)\n    let total = 0 // Subset sum\n    var res: [[Int]] = [] // Result list (list of subsets)\n    backtrack(state: &amp;state, target: target, total: total, choices: nums, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.js<pre><code>/* Backtracking: subset sum I */\nfunction backtrack(state, target, total, choices, res) {\n    // Record a solution when the subset sum equals target\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // Iterate over all choices\n    for (let i = 0; i &lt; choices.length; i++) {\n        // Prune: skip this choice if the subset sum would exceed target\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // Try: make a choice and update the sum total\n        state.push(choices[i]);\n        // Proceed to the next choice\n        backtrack(state, target, total + choices[i], choices, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum I (with duplicate subsets) */\nfunction subsetSumINaive(nums, target) {\n    const state = []; // State (subset)\n    const total = 0; // Subset sum\n    const res = []; // Result list (list of subsets)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.ts<pre><code>/* Backtracking: subset sum I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    total: number,\n    choices: number[],\n    res: number[][]\n): void {\n    // Record a solution when the subset sum equals target\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // Iterate over all choices\n    for (let i = 0; i &lt; choices.length; i++) {\n        // Prune: skip this choice if the subset sum would exceed target\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // Try: make a choice and update the sum total\n        state.push(choices[i]);\n        // Proceed to the next choice\n        backtrack(state, target, total + choices[i], choices, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum I (with duplicate subsets) */\nfunction subsetSumINaive(nums: number[], target: number): number[][] {\n    const state = []; // State (subset)\n    const total = 0; // Subset sum\n    const res = []; // Result list (list of subsets)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.dart<pre><code>/* Backtracking: subset sum I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  int total,\n  List&lt;int&gt; choices,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // Record a solution when the subset sum equals target\n  if (total == target) {\n    res.add(List.from(state));\n    return;\n  }\n  // Iterate over all choices\n  for (int i = 0; i &lt; choices.length; i++) {\n    // Prune: skip this choice if the subset sum would exceed target\n    if (total + choices[i] &gt; target) {\n      continue;\n    }\n    // Try: make a choice and update the sum total\n    state.add(choices[i]);\n    // Proceed to the next choice\n    backtrack(state, target, total + choices[i], choices, res);\n    // Backtrack: undo the choice and restore the previous state\n    state.removeLast();\n  }\n}\n\n/* Solve subset sum I (with duplicate subsets) */\nList&lt;List&lt;int&gt;&gt; subsetSumINaive(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // State (subset)\n  int total = 0; // Sum of the elements\n  List&lt;List&lt;int&gt;&gt; res = []; // Result list (list of subsets)\n  backtrack(state, target, total, nums, res);\n  return res;\n}\n</code></pre> subset_sum_i_naive.rs<pre><code>/* Backtracking: subset sum I */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    total: i32,\n    choices: &amp;[i32],\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // Record a solution when the subset sum equals target\n    if total == target {\n        res.push(state.clone());\n        return;\n    }\n    // Iterate over all choices\n    for i in 0..choices.len() {\n        // Prune: skip this choice if the subset sum would exceed target\n        if total + choices[i] &gt; target {\n            continue;\n        }\n        // Try: make a choice and update the sum total\n        state.push(choices[i]);\n        // Proceed to the next choice\n        backtrack(state, target, total + choices[i], choices, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum I (with duplicate subsets) */\nfn subset_sum_i_naive(nums: &amp;[i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // State (subset)\n    let total = 0; // Subset sum\n    let mut res = Vec::new(); // Result list (list of subsets)\n    backtrack(&amp;mut state, target, total, nums, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i_naive.c<pre><code>/* Backtracking: subset sum I */\nvoid backtrack(int target, int total, int *choices, int choicesSize) {\n    // Record a solution when the subset sum equals target\n    if (total == target) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // Iterate over all choices\n    for (int i = 0; i &lt; choicesSize; i++) {\n        // Prune: skip this choice if the subset sum would exceed target\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // Try: make a choice and update the sum total\n        state[stateSize++] = choices[i];\n        // Proceed to the next choice\n        backtrack(target, total + choices[i], choices, choicesSize);\n        // Backtrack: undo the choice and restore the previous state\n        stateSize--;\n    }\n}\n\n/* Solve subset sum I (with duplicate subsets) */\nvoid subsetSumINaive(int *nums, int numsSize, int target) {\n    resSize = 0; // Initialize the solution count to 0\n    backtrack(target, 0, nums, numsSize);\n}\n</code></pre> subset_sum_i_naive.kt<pre><code>/* Backtracking: subset sum I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    total: Int,\n    choices: IntArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // Record a solution when the subset sum equals target\n    if (total == target) {\n        res.add(state.toMutableList())\n        return\n    }\n    // Iterate over all choices\n    for (i in choices.indices) {\n        // Prune: skip this choice if the subset sum would exceed target\n        if (total + choices[i] &gt; target) {\n            continue\n        }\n        // Try: make a choice and update the sum total\n        state.add(choices[i])\n        // Proceed to the next choice\n        backtrack(state, target, total + choices[i], choices, res)\n        // Backtrack: undo the choice and restore the previous state\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* Solve subset sum I (with duplicate subsets) */\nfun subsetSumINaive(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // State (subset)\n    val total = 0 // Subset sum\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // Result list (list of subsets)\n    backtrack(state, target, total, nums, res)\n    return res\n}\n</code></pre> subset_sum_i_naive.rb<pre><code>### Backtracking: subset sum I ###\ndef backtrack(state, target, total, choices, res)\n  # Record a solution when the subset sum equals target\n  if total == target\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # Iterate over all choices\n  for i in 0...choices.length\n    # Prune: skip this choice if the subset sum would exceed target\n    next if total + choices[i] &gt; target\n    # Try: make a choice and update the sum total\n    state &lt;&lt; choices[i]\n    # Proceed to the next choice\n    backtrack(state, target, total + choices[i], choices, res)\n    # Backtrack: undo the choice and restore the previous state\n    state.pop\n  end\nend\n\n### Solve subset sum I (with duplicate subsets)###\ndef subset_sum_i_naive(nums, target)\n  state = [] # State (subset)\n  total = 0 # Subset sum\n  res = [] # Result list (list of subsets)\n  backtrack(state, target, total, nums, res)\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>For input array \\([3, 4, 5]\\) and target \\(9\\), this code returns \\([3, 3, 3], [4, 5], [5, 4]\\). It finds all subsets summing to \\(9\\), but includes the duplicate subsets \\([4, 5]\\) and \\([5, 4]\\).</p> <p>The search distinguishes the order of choices, whereas subsets do not. As shown in Figure 13-10, choosing \\(4\\) then \\(5\\) and choosing \\(5\\) then \\(4\\) follow different branches but produce the same subset.</p> <p></p> <p> Figure 13-10   Subset search and pruning when the sum exceeds the target </p> <p>One straightforward way to remove duplicate subsets is to deduplicate the result list. However, this is inefficient for two reasons.</p> <ul> <li>With many array elements, especially when <code>target</code> is large, the search generates many duplicate subsets.</li> <li>Comparing subsets (arrays) is expensive: we must sort them first, then compare their elements one by one.</li> </ul>","path":["Chapter 13.   Backtracking","13.3   Subset sum problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#2-pruning-duplicate-subsets","level":3,"title":"2.   Pruning duplicate subsets","text":"<p>Instead, we can eliminate duplicates by pruning during the search. As shown in Figure 13-11, duplicates arise when array elements are chosen in different orders, as in the following example.</p> <ol> <li>Choosing \\(3\\) and then \\(4\\) in the first two rounds generates all subsets beginning with these choices, denoted by \\([3, 4, \\dots]\\).</li> <li>Later, after choosing \\(4\\) first, we should skip \\(3\\) in the second round, because subsets \\([4, 3, \\dots]\\) duplicate those generated in step <code>1.</code>.</li> </ol> <p>At each level of the search, choices are tried from left to right, so branches farther to the right have more choices pruned.</p> <ol> <li>Choosing \\(3\\) and then \\(5\\) generates subsets \\([3, 5, \\dots]\\).</li> <li>Choosing \\(4\\) and then \\(5\\) generates subsets \\([4, 5, \\dots]\\).</li> <li>If \\(5\\) is chosen first, skip both \\(3\\) and \\(4\\) in the second round, because \\([5, 3, \\dots]\\) and \\([5, 4, \\dots]\\) duplicate the subsets described in steps <code>1.</code> and <code>2.</code>.</li> </ol> <p></p> <p> Figure 13-11   Duplicate subsets caused by different choice orders </p> <p>In general, for input \\([x_1, x_2, \\dots, x_n]\\), a sequence of choices \\([x_{i_1}, x_{i_2}, \\dots, x_{i_m}]\\) must satisfy \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\). Any sequence violating this condition creates a duplicate and should be pruned.</p>","path":["Chapter 13.   Backtracking","13.3   Subset sum problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#3-code-implementation","level":3,"title":"3.   Code implementation","text":"<p>To implement this pruning, introduce <code>start</code> to specify where iteration begins. After choosing \\(x_{i}\\), begin the next round at index \\(i\\). This enforces \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) and ensures that each subset is unique.</p> <p>We also make two further improvements.</p> <ul> <li>Sort <code>nums</code> before starting the search. While iterating over choices, stop the loop as soon as the subset sum exceeds <code>target</code>, because all later elements are larger and would also exceed it.</li> <li>Remove the sum variable <code>total</code> and track the sum by subtracting from <code>target</code>. Record a solution when <code>target</code> reaches \\(0\\).</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby subset_sum_i.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"Backtracking: subset sum I\"\"\"\n    # Record a solution when the remaining target is zero\n    if target == 0:\n        res.append(list(state))\n        return\n    # Iterate over all choices\n    # Pruning 2: iterate from start to avoid generating duplicate subsets\n    for i in range(start, len(choices)):\n        # Pruning 1: exit the loop if this choice exceeds the remaining target\n        # The array is sorted, so later elements are at least as large and also exceed the remaining target\n        if target - choices[i] &lt; 0:\n            break\n        # Try: make a choice and update target and start\n        state.append(choices[i])\n        # Proceed to the next choice\n        backtrack(state, target - choices[i], choices, i, res)\n        # Backtrack: undo the choice and restore the previous state\n        state.pop()\n\ndef subset_sum_i(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"Solve subset sum I\"\"\"\n    state = []  # State (subset)\n    nums.sort()  # Sort nums\n    start = 0  # Starting index for iteration\n    res = []  # Result list (list of subsets)\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_i.cpp<pre><code>/* Backtracking: subset sum I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // Record a solution when the remaining target reaches 0\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    for (int i = start; i &lt; choices.size(); i++) {\n        // Pruning 1: exit the loop if this choice exceeds the remaining target\n        // The array is sorted, so later elements are at least as large and also exceed the remaining target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Try: make a choice and update target and start\n        state.push_back(choices[i]);\n        // Proceed to the next choice\n        backtrack(state, target - choices[i], choices, i, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.pop_back();\n    }\n}\n\n/* Solve subset sum I */\nvector&lt;vector&lt;int&gt;&gt; subsetSumI(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // State (subset)\n    sort(nums.begin(), nums.end()); // Sort nums\n    int start = 0;                  // Starting index for iteration\n    vector&lt;vector&lt;int&gt;&gt; res;        // Result list (list of subsets)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.java<pre><code>/* Backtracking: subset sum I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // Record a solution when the remaining target is zero, so the subset sums to the original target\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    for (int i = start; i &lt; choices.length; i++) {\n        // Pruning 1: exit the loop if the next element exceeds the remaining target\n        // The array is sorted, so later elements are at least as large and also exceed the remaining target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Try: make a choice and update target and start\n        state.add(choices[i]);\n        // Proceed to the next choice\n        backtrack(state, target - choices[i], choices, i, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.remove(state.size() - 1);\n    }\n}\n\n/* Solve subset sum I */\nList&lt;List&lt;Integer&gt;&gt; subsetSumI(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // State (subset)\n    Arrays.sort(nums); // Sort nums\n    int start = 0; // Starting index for iteration\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // Result list (list of subsets)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.cs<pre><code>/* Backtracking: subset sum I */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // Record a solution when the subset reaches the original target (remaining target is 0)\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    for (int i = start; i &lt; choices.Length; i++) {\n        // Pruning 1: exit the loop if this choice would exceed the remaining target\n        // The array is sorted, so later elements are at least as large and would also exceed the remaining target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Try: make a choice and update target and start\n        state.Add(choices[i]);\n        // Proceed to the next choice\n        Backtrack(state, target - choices[i], choices, i, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* Solve subset sum I */\nList&lt;List&lt;int&gt;&gt; SubsetSumI(int[] nums, int target) {\n    List&lt;int&gt; state = []; // State (subset)\n    Array.Sort(nums); // Sort nums\n    int start = 0; // Starting index for iteration\n    List&lt;List&lt;int&gt;&gt; res = []; // Result list (list of subsets)\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.go<pre><code>/* Backtracking: subset sum I */\nfunc backtrackSubsetSumI(start, target int, state, choices *[]int, res *[][]int) {\n    // Record a solution when the subset reaches the original target sum (remaining target is zero)\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    for i := start; i &lt; len(*choices); i++ {\n        // Pruning 1: exit the loop if this choice exceeds the remaining target\n        // The array is sorted, so later elements are at least as large and also exceed the remaining target\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // Try: make a choice and update target and start\n        *state = append(*state, (*choices)[i])\n        // Proceed to the next choice\n        backtrackSubsetSumI(i, target-(*choices)[i], state, choices, res)\n        // Backtrack: undo the choice and restore the previous state\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* Solve subset sum I */\nfunc subsetSumI(nums []int, target int) [][]int {\n    state := make([]int, 0) // State (subset)\n    sort.Ints(nums)         // Sort nums\n    start := 0              // Starting index for iteration\n    res := make([][]int, 0) // Result list (list of subsets)\n    backtrackSubsetSumI(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.swift<pre><code>/* Backtracking: subset sum I */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // Record a solution when the remaining target reaches zero\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    for i in choices.indices.dropFirst(start) {\n        // Pruning 1: exit the loop if this choice exceeds the remaining target\n        // The array is sorted, so later elements are at least as large and also exceed the remaining target\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // Try: make a choice and update target and start\n        state.append(choices[i])\n        // Proceed to the next choice\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i, res: &amp;res)\n        // Backtrack: undo the choice and restore the previous state\n        state.removeLast()\n    }\n}\n\n/* Solve subset sum I */\nfunc subsetSumI(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // State (subset)\n    let nums = nums.sorted() // Sort nums\n    let start = 0 // Starting index for iteration\n    var res: [[Int]] = [] // Result list (list of subsets)\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.js<pre><code>/* Backtracking: subset sum I */\nfunction backtrack(state, target, choices, start, res) {\n    // Record a solution when the remaining target is zero, so the subset sum equals the original target\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    for (let i = start; i &lt; choices.length; i++) {\n        // Pruning 1: exit the loop if the next choice exceeds the remaining target\n        // The array is sorted, so later elements are at least as large and would also exceed the remaining target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Try: make a choice and update target and start\n        state.push(choices[i]);\n        // Proceed to the next choice\n        backtrack(state, target - choices[i], choices, i, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum I */\nfunction subsetSumI(nums, target) {\n    const state = []; // State (subset)\n    nums.sort((a, b) =&gt; a - b); // Sort nums\n    const start = 0; // Starting index for iteration\n    const res = []; // Result list (list of subsets)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.ts<pre><code>/* Backtracking: subset sum I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // Record a solution when the subset reaches the original target (remaining target is 0)\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    for (let i = start; i &lt; choices.length; i++) {\n        // Pruning 1: exit the loop if this choice would exceed the remaining target\n        // The array is sorted, so later elements are at least as large and would also exceed the remaining target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Try: make a choice and update target and start\n        state.push(choices[i]);\n        // Proceed to the next choice\n        backtrack(state, target - choices[i], choices, i, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum I */\nfunction subsetSumI(nums: number[], target: number): number[][] {\n    const state = []; // State (subset)\n    nums.sort((a, b) =&gt; a - b); // Sort nums\n    const start = 0; // Starting index for iteration\n    const res = []; // Result list (list of subsets)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.dart<pre><code>/* Backtracking: subset sum I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // Record a solution when the subset sum reaches the original target (the remaining target is 0)\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // Iterate over all choices\n  // Pruning 2: iterate from start to avoid generating duplicate subsets\n  for (int i = start; i &lt; choices.length; i++) {\n    // Pruning 1: exit the loop if this choice exceeds the remaining target\n    // The array is sorted, so later elements are at least as large and also exceed the remaining target\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // Try: make a choice and update target and start\n    state.add(choices[i]);\n    // Proceed to the next choice\n    backtrack(state, target - choices[i], choices, i, res);\n    // Backtrack: undo the choice and restore the previous state\n    state.removeLast();\n  }\n}\n\n/* Solve subset sum I */\nList&lt;List&lt;int&gt;&gt; subsetSumI(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // State (subset)\n  nums.sort(); // Sort nums\n  int start = 0; // Starting index for iteration\n  List&lt;List&lt;int&gt;&gt; res = []; // Result list (list of subsets)\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_i.rs<pre><code>/* Backtracking: subset sum I */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // Record a solution when the remaining target is zero\n    if target == 0 {\n        res.push(state.clone());\n        return;\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    for i in start..choices.len() {\n        // Pruning 1: exit the loop if the next element exceeds the remaining target\n        // The array is sorted, so later elements are at least as large and also exceed the remaining target\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // Try: make a choice and update target and start\n        state.push(choices[i]);\n        // Proceed to the next choice\n        backtrack(state, target - choices[i], choices, i, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum I */\nfn subset_sum_i(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // State (subset)\n    nums.sort(); // Sort nums\n    let start = 0; // Starting index for iteration\n    let mut res = Vec::new(); // Result list (list of subsets)\n    backtrack(&amp;mut state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i.c<pre><code>/* Backtracking: subset sum I */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // Record a solution when the remaining target is 0\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; ++i) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    for (int i = start; i &lt; choicesSize; i++) {\n        // Pruning 1: exit the loop if the choice exceeds the remaining target\n        // The array is sorted, so later elements are at least as large and also exceed the remaining target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Try: make a choice and update target and start\n        state[stateSize] = choices[i];\n        stateSize++;\n        // Proceed to the next choice\n        backtrack(target - choices[i], choices, choicesSize, i);\n        // Backtrack: undo the choice and restore the previous state\n        stateSize--;\n    }\n}\n\n/* Solve subset sum I */\nvoid subsetSumI(int *nums, int numsSize, int target) {\n    qsort(nums, numsSize, sizeof(int), cmp); // Sort nums\n    int start = 0;                           // Starting index for iteration\n    backtrack(target, nums, numsSize, start);\n}\n</code></pre> subset_sum_i.kt<pre><code>/* Backtracking: subset sum I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // Record a solution when the subset reaches the original target (remaining target is 0)\n    if (target == 0) {\n        res.add(state.toMutableList())\n        return\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    for (i in start..&lt;choices.size) {\n        // Pruning 1: exit the loop if this choice would exceed the remaining target\n        // The array is sorted, so later elements are at least as large and would also exceed the remaining target\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // Try: make a choice and update target and start\n        state.add(choices[i])\n        // Proceed to the next choice\n        backtrack(state, target - choices[i], choices, i, res)\n        // Backtrack: undo the choice and restore the previous state\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* Solve subset sum I */\nfun subsetSumI(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // State (subset)\n    nums.sort() // Sort nums\n    val start = 0 // Starting index for iteration\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // Result list (list of subsets)\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_i.rb<pre><code>### Backtracking: subset sum I ###\ndef backtrack(state, target, choices, start, res)\n  # Record a solution when the remaining target reaches zero\n  if target.zero?\n    res &lt;&lt; state.dup\n    return\n  end\n  # Iterate over all choices\n  # Pruning 2: iterate from start to avoid generating duplicate subsets\n  for i in start...choices.length\n    # Pruning 1: exit the loop if this choice would exceed the remaining target\n    # The array is sorted, so later elements are at least as large and would also exceed the remaining target\n    break if target - choices[i] &lt; 0\n    # Try: make a choice and update target and start\n    state &lt;&lt; choices[i]\n    # Proceed to the next choice\n    backtrack(state, target - choices[i], choices, i, res)\n    # Backtrack: undo the choice and restore the previous state\n    state.pop\n  end\nend\n\n### Solve subset sum I ###\ndef subset_sum_i(nums, target)\n  state = [] # State (subset)\n  nums.sort! # Sort nums\n  start = 0 # Starting index for iteration\n  res = [] # Result list (list of subsets)\n  backtrack(state, target, nums, start, res)\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 13-12 shows the complete backtracking process for input \\([3, 4, 5]\\) and target \\(9\\).</p> <p></p> <p> Figure 13-12   Backtracking for subset sum I </p>","path":["Chapter 13.   Backtracking","13.3   Subset sum problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1332-duplicate-elements","level":2,"title":"13.3.2   Duplicate elements","text":"<p>Question</p> <p>Given an array of positive integers <code>nums</code> and a positive target integer <code>target</code>, find all combinations whose elements sum to <code>target</code>. The array may contain duplicate elements, and each element may be chosen only once. Return the combinations as a list containing no duplicates.</p> <p>Unlike the previous problem, the input may contain duplicate elements, which introduces a new issue. For array \\([4, \\hat{4}, 5]\\) and target \\(9\\), for example, the existing code produces \\([4, 5], [\\hat{4}, 5]\\): duplicate subsets.</p> <p>These duplicates occur because equal elements are tried more than once in a single round. In Figure 13-13, the first round has three choices, two of which are \\(4\\). They generate duplicate branches and therefore duplicate subsets. The two occurrences of \\(4\\) in the second round cause the same problem.</p> <p></p> <p> Figure 13-13   Duplicate subsets caused by equal elements </p>","path":["Chapter 13.   Backtracking","13.3   Subset sum problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1-pruning-equal-elements","level":3,"title":"1.   Pruning equal elements","text":"<p>To solve this, try each value only once in each round. The implementation uses the fact that equal elements are adjacent in a sorted array. Within a round, if an element equals the preceding element, that value has already been tried, so skip the current element.</p> <p>In addition, each array element may be chosen only once. Fortunately, <code>start</code> can enforce this constraint too: after choosing \\(x_{i}\\), begin the next round at index \\(i + 1\\). This both eliminates duplicate subsets and prevents reusing an element.</p>","path":["Chapter 13.   Backtracking","13.3   Subset sum problem"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#2-code-implementation","level":3,"title":"2.   Code implementation","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby subset_sum_ii.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"Backtracking: subset sum II\"\"\"\n    # Record a solution when the remaining target is zero\n    if target == 0:\n        res.append(list(state))\n        return\n    # Iterate over all choices\n    # Pruning 2: iterate from start to avoid generating duplicate subsets\n    # Pruning 3: start after the previously chosen index to avoid choosing the same element again\n    for i in range(start, len(choices)):\n        # Pruning 1: exit the loop if this choice exceeds the remaining target\n        # The array is sorted, so later elements are at least as large and also exceed the remaining target\n        if target - choices[i] &lt; 0:\n            break\n        # Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it\n        if i &gt; start and choices[i] == choices[i - 1]:\n            continue\n        # Try: make a choice and update target and start\n        state.append(choices[i])\n        # Proceed to the next choice\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        # Backtrack: undo the choice and restore the previous state\n        state.pop()\n\ndef subset_sum_ii(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"Solve subset sum II\"\"\"\n    state = []  # State (subset)\n    nums.sort()  # Sort nums\n    start = 0  # Starting index for iteration\n    res = []  # Result list (list of subsets)\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_ii.cpp<pre><code>/* Backtracking: subset sum II */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // Record a solution when the remaining target reaches 0\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    // Pruning 3: iterate from start to avoid choosing the same element again\n    for (int i = start; i &lt; choices.size(); i++) {\n        // Pruning 1: exit the loop if this choice exceeds the remaining target\n        // The array is sorted, so later elements are at least as large and also exceed the remaining target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // Try: make a choice and update target and start\n        state.push_back(choices[i]);\n        // Proceed to the next choice\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.pop_back();\n    }\n}\n\n/* Solve subset sum II */\nvector&lt;vector&lt;int&gt;&gt; subsetSumII(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // State (subset)\n    sort(nums.begin(), nums.end()); // Sort nums\n    int start = 0;                  // Starting index for iteration\n    vector&lt;vector&lt;int&gt;&gt; res;        // Result list (list of subsets)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.java<pre><code>/* Backtracking: subset sum II */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // Record a solution when the remaining target is zero, so the subset sums to the original target\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    // Pruning 3: iterate from start to avoid choosing the same element again\n    for (int i = start; i &lt; choices.length; i++) {\n        // Pruning 1: exit the loop if the next element exceeds the remaining target\n        // The array is sorted, so later elements are at least as large and also exceed the remaining target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // Try: make a choice and update target and start\n        state.add(choices[i]);\n        // Proceed to the next choice\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.remove(state.size() - 1);\n    }\n}\n\n/* Solve subset sum II */\nList&lt;List&lt;Integer&gt;&gt; subsetSumII(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // State (subset)\n    Arrays.sort(nums); // Sort nums\n    int start = 0; // Starting index for iteration\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // Result list (list of subsets)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.cs<pre><code>/* Backtracking: subset sum II */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // Record a solution when the subset reaches the original target (remaining target is 0)\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    // Pruning 3: start after the previously chosen index to avoid choosing the same element again\n    for (int i = start; i &lt; choices.Length; i++) {\n        // Pruning 1: exit the loop if this choice would exceed the remaining target\n        // The array is sorted, so later elements are at least as large and would also exceed the remaining target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // Try: make a choice and update target and start\n        state.Add(choices[i]);\n        // Proceed to the next choice\n        Backtrack(state, target - choices[i], choices, i + 1, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* Solve subset sum II */\nList&lt;List&lt;int&gt;&gt; SubsetSumII(int[] nums, int target) {\n    List&lt;int&gt; state = []; // State (subset)\n    Array.Sort(nums); // Sort nums\n    int start = 0; // Starting index for iteration\n    List&lt;List&lt;int&gt;&gt; res = []; // Result list (list of subsets)\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.go<pre><code>/* Backtracking: subset sum II */\nfunc backtrackSubsetSumII(start, target int, state, choices *[]int, res *[][]int) {\n    // Record a solution when the subset reaches the original target sum (remaining target is zero)\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    // Pruning 3: start after the previously chosen index to avoid choosing the same element again\n    for i := start; i &lt; len(*choices); i++ {\n        // Pruning 1: exit the loop if this choice exceeds the remaining target\n        // The array is sorted, so later elements are at least as large and also exceed the remaining target\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it\n        if i &gt; start &amp;&amp; (*choices)[i] == (*choices)[i-1] {\n            continue\n        }\n        // Try: make a choice and update target and start\n        *state = append(*state, (*choices)[i])\n        // Proceed to the next choice\n        backtrackSubsetSumII(i+1, target-(*choices)[i], state, choices, res)\n        // Backtrack: undo the choice and restore the previous state\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* Solve subset sum II */\nfunc subsetSumII(nums []int, target int) [][]int {\n    state := make([]int, 0) // State (subset)\n    sort.Ints(nums)         // Sort nums\n    start := 0              // Starting index for iteration\n    res := make([][]int, 0) // Result list (list of subsets)\n    backtrackSubsetSumII(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.swift<pre><code>/* Backtracking: subset sum II */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // Record a solution when the remaining target reaches zero\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    // Pruning 3: start after the previously chosen index to avoid choosing the same element again\n    for i in choices.indices.dropFirst(start) {\n        // Pruning 1: exit the loop if this choice exceeds the remaining target\n        // The array is sorted, so later elements are at least as large and also exceed the remaining target\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it\n        if i &gt; start, choices[i] == choices[i - 1] {\n            continue\n        }\n        // Try: make a choice and update target and start\n        state.append(choices[i])\n        // Proceed to the next choice\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i + 1, res: &amp;res)\n        // Backtrack: undo the choice and restore the previous state\n        state.removeLast()\n    }\n}\n\n/* Solve subset sum II */\nfunc subsetSumII(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // State (subset)\n    let nums = nums.sorted() // Sort nums\n    let start = 0 // Starting index for iteration\n    var res: [[Int]] = [] // Result list (list of subsets)\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.js<pre><code>/* Backtracking: subset sum II */\nfunction backtrack(state, target, choices, start, res) {\n    // Record a solution when the remaining target is zero, so the subset sum equals the original target\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    // Pruning 3: start after the previously chosen index to avoid choosing the same element again\n    for (let i = start; i &lt; choices.length; i++) {\n        // Pruning 1: exit the loop if the next choice exceeds the remaining target\n        // The array is sorted, so later elements are at least as large and would also exceed the remaining target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // Try: make a choice and update target and start\n        state.push(choices[i]);\n        // Proceed to the next choice\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum II */\nfunction subsetSumII(nums, target) {\n    const state = []; // State (subset)\n    nums.sort((a, b) =&gt; a - b); // Sort nums\n    const start = 0; // Starting index for iteration\n    const res = []; // Result list (list of subsets)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.ts<pre><code>/* Backtracking: subset sum II */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // Record a solution when the subset reaches the original target (remaining target is 0)\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    // Pruning 3: start after the previously chosen index to avoid choosing the same element again\n    for (let i = start; i &lt; choices.length; i++) {\n        // Pruning 1: exit the loop if this choice would exceed the remaining target\n        // The array is sorted, so later elements are at least as large and would also exceed the remaining target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // Try: make a choice and update target and start\n        state.push(choices[i]);\n        // Proceed to the next choice\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum II */\nfunction subsetSumII(nums: number[], target: number): number[][] {\n    const state = []; // State (subset)\n    nums.sort((a, b) =&gt; a - b); // Sort nums\n    const start = 0; // Starting index for iteration\n    const res = []; // Result list (list of subsets)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.dart<pre><code>/* Backtracking: subset sum II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // Record a solution when the subset sum reaches the original target (the remaining target is 0)\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // Iterate over all choices\n  // Pruning 2: iterate from start to avoid generating duplicate subsets\n  // Pruning 3: start after the previously chosen index to avoid choosing the same element again\n  for (int i = start; i &lt; choices.length; i++) {\n    // Pruning 1: exit the loop if this choice exceeds the remaining target\n    // The array is sorted, so later elements are at least as large and also exceed the remaining target\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it\n    if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n      continue;\n    }\n    // Try: make a choice and update target and start\n    state.add(choices[i]);\n    // Proceed to the next choice\n    backtrack(state, target - choices[i], choices, i + 1, res);\n    // Backtrack: undo the choice and restore the previous state\n    state.removeLast();\n  }\n}\n\n/* Solve subset sum II */\nList&lt;List&lt;int&gt;&gt; subsetSumII(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // State (subset)\n  nums.sort(); // Sort nums\n  int start = 0; // Starting index for iteration\n  List&lt;List&lt;int&gt;&gt; res = []; // Result list (list of subsets)\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_ii.rs<pre><code>/* Backtracking: subset sum II */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // Record a solution when the remaining target is zero\n    if target == 0 {\n        res.push(state.clone());\n        return;\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    // Pruning 3: iterate from start to avoid choosing the same element again\n    for i in start..choices.len() {\n        // Pruning 1: exit the loop if the next element exceeds the remaining target\n        // The array is sorted, so later elements are at least as large and also exceed the remaining target\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it\n        if i &gt; start &amp;&amp; choices[i] == choices[i - 1] {\n            continue;\n        }\n        // Try: make a choice and update target and start\n        state.push(choices[i]);\n        // Proceed to the next choice\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // Backtrack: undo the choice and restore the previous state\n        state.pop();\n    }\n}\n\n/* Solve subset sum II */\nfn subset_sum_ii(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // State (subset)\n    nums.sort(); // Sort nums\n    let start = 0; // Starting index for iteration\n    let mut res = Vec::new(); // Result list (list of subsets)\n    backtrack(&amp;mut state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_ii.c<pre><code>/* Backtracking: subset sum II */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // Record a solution when the remaining target is 0\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    // Pruning 3: iterate from start to avoid choosing the same element again\n    for (int i = start; i &lt; choicesSize; i++) {\n        // Pruning 1: skip this choice if it exceeds the remaining target\n        if (target - choices[i] &lt; 0) {\n            continue;\n        }\n        // Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // Try: make a choice and update target and start\n        state[stateSize] = choices[i];\n        stateSize++;\n        // Proceed to the next choice\n        backtrack(target - choices[i], choices, choicesSize, i + 1);\n        // Backtrack: undo the choice and restore the previous state\n        stateSize--;\n    }\n}\n\n/* Solve subset sum II */\nvoid subsetSumII(int *nums, int numsSize, int target) {\n    // Sort nums\n    qsort(nums, numsSize, sizeof(int), cmp);\n    // Start backtracking\n    backtrack(target, nums, numsSize, 0);\n}\n</code></pre> subset_sum_ii.kt<pre><code>/* Backtracking: subset sum II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // Record a solution when the subset reaches the original target (remaining target is 0)\n    if (target == 0) {\n        res.add(state.toMutableList())\n        return\n    }\n    // Iterate over all choices\n    // Pruning 2: iterate from start to avoid generating duplicate subsets\n    // Pruning 3: start after the previously chosen index to avoid choosing the same element again\n    for (i in start..&lt;choices.size) {\n        // Pruning 1: exit the loop if this choice would exceed the remaining target\n        // The array is sorted, so later elements are at least as large and would also exceed the remaining target\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue\n        }\n        // Try: make a choice and update target and start\n        state.add(choices[i])\n        // Proceed to the next choice\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        // Backtrack: undo the choice and restore the previous state\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* Solve subset sum II */\nfun subsetSumII(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // State (subset)\n    nums.sort() // Sort nums\n    val start = 0 // Starting index for iteration\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // Result list (list of subsets)\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_ii.rb<pre><code>### Backtracking: subset sum II ###\ndef backtrack(state, target, choices, start, res)\n  # Record a solution when the remaining target reaches zero\n  if target.zero?\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # Iterate over all choices\n  # Pruning 2: iterate from start to avoid generating duplicate subsets\n  # Pruning 3: start after the previously chosen index to avoid choosing the same element again\n  for i in start...choices.length\n    # Pruning 1: exit the loop if this choice would exceed the remaining target\n    # The array is sorted, so later elements are at least as large and would also exceed the remaining target\n    break if target - choices[i] &lt; 0\n    # Pruning 4: if this element equals its left neighbor at the same search depth, the branch is a duplicate; skip it\n    next if i &gt; start &amp;&amp; choices[i] == choices[i - 1]\n    # Try: make a choice and update target and start\n    state &lt;&lt; choices[i]\n    # Proceed to the next choice\n    backtrack(state, target - choices[i], choices, i + 1, res)\n    # Backtrack: undo the choice and restore the previous state\n    state.pop\n  end\nend\n\n### Solve subset sum II ###\ndef subset_sum_ii(nums, target)\n  state = [] # State (subset)\n  nums.sort! # Sort nums\n  start = 0 # Starting index for iteration\n  res = [] # Result list (list of subsets)\n  backtrack(state, target, nums, start, res)\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 13-14 shows backtracking for array \\([4, 4, 5]\\) and target \\(9\\), including four types of pruning. Follow the illustration alongside the code comments to understand the full search and how each pruning rule works.</p> <p></p> <p> Figure 13-14   Backtracking for subset sum II </p>","path":["Chapter 13.   Backtracking","13.3   Subset sum problem"],"tags":[]},{"location":"chapter_backtracking/summary/","level":1,"title":"13.5   Summary","text":"","path":["Chapter 13.   Backtracking","13.5   Summary"],"tags":[]},{"location":"chapter_backtracking/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<ul> <li>Backtracking is fundamentally exhaustive search. It traverses the solution space depth first to find solutions meeting the requirements, records them when found, and continues until all solutions are found or traversal is complete.</li> <li>Backtracking alternates between trying a choice and undoing it. It explores choices through depth-first search; when a constraint is violated, it undoes the last choice, restores the previous state, and tries another choice. Trying and undoing are opposite operations.</li> <li>Backtracking problems usually have multiple constraints that enable pruning. Pruning stops unnecessary branches early and can greatly improve search efficiency.</li> <li>Backtracking is mainly used for search and constraint satisfaction problems. It can also solve combinatorial optimization problems, though more efficient or more effective approaches often exist.</li> <li>The permutations problem asks for every possible ordering of a collection's elements. An array tracks whether each element has been selected, pruning branches that select the same element again and ensuring each is used only once.</li> <li>If the input contains equal elements, permutation generation produces duplicate results. We must try each value only once per round, usually by using a hash set.</li> <li>The subset sum problem asks for all subsets of a collection whose sum equals a target. Subsets disregard order, but the search generates all orderings, producing duplicates. Sorting the data before backtracking and using a variable to mark each round's starting index allows us to prune duplicate-generating branches.</li> <li>Equal elements also produce duplicate subsets. Because the array is sorted, we can prune by comparing adjacent elements, ensuring that each value is tried only once per round.</li> <li>The \\(n\\)-queens problem asks for arrangements of \\(n\\) queens on an \\(n \\times n\\) chessboard such that no two queens can attack each other. It imposes row, column, and both diagonal constraints. Placing queens row by row ensures exactly one queen per row.</li> <li>Column and diagonal constraints are handled similarly. An array records whether each column contains a queen, indicating whether a chosen square is valid. Two more arrays track queens on diagonals in each direction. The main challenge is identifying the relationship between row and column indices for squares on the same diagonal.</li> </ul>","path":["Chapter 13.   Backtracking","13.5   Summary"],"tags":[]},{"location":"chapter_backtracking/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: How are backtracking and recursion related?</p> <p>Broadly speaking, backtracking is an algorithmic strategy, while recursion is a tool.</p> <ul> <li>Backtracking is usually implemented recursively. It is one application of recursion, specifically to search problems.</li> <li>Recursion expresses the idea of decomposing a problem into subproblems. It is commonly used in divide and conquer, backtracking, and dynamic programming (memoized recursion).</li> </ul>","path":["Chapter 13.   Backtracking","13.5   Summary"],"tags":[]},{"location":"chapter_computational_complexity/","level":1,"title":"Chapter 2.   Complexity analysis","text":"<p>Abstract</p> <p>Complexity analysis is our guide to time and space in the vast universe of algorithms.</p> <p>It leads us to explore these two dimensions in depth as we search for more elegant solutions.</p>","path":["Chapter 2.   Complexity analysis"],"tags":[]},{"location":"chapter_computational_complexity/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>2.1   Evaluating algorithm efficiency</li> <li>2.2   Iteration and recursion</li> <li>2.3   Time complexity</li> <li>2.4   Space complexity</li> <li>2.5   Summary</li> <li>2.6   Exercises</li> </ul>","path":["Chapter 2.   Complexity analysis"],"tags":[]},{"location":"chapter_computational_complexity/exercises/","level":1,"title":"2.6   Exercises","text":"","path":["Chapter 2.   Complexity analysis","2.6   Exercises"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#261-check-your-understanding","level":2,"title":"2.6.1   Check your understanding","text":"","path":["Chapter 2.   Complexity analysis","2.6   Exercises"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#1-time-and-space-in-iteration-and-recursion","level":3,"title":"1.   Time and space in iteration and recursion","text":"<p>Both functions below compute \\(1 + 2 + \\dots + n\\) (assume \\(n \\ge 1\\)). Set <code>n</code> to 4, follow the program's execution order to answer the questions, and then compare the efficiency of the two approaches.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby complexity_exercises.py<pre><code>def sum_iter(n: int) -&gt; int:\n    \"\"\"Sum using iteration\"\"\"\n    res = 0\n    for i in range(1, n + 1):\n        res += i\n    return res\n\ndef sum_recur(n: int) -&gt; int:\n    \"\"\"Sum using recursion\"\"\"\n    if n == 1:\n        return 1\n    return n + sum_recur(n - 1)\n</code></pre> complexity_exercises.cpp<pre><code>/* Sum using iteration */\nint sumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; ++i) {\n        res += i;\n    }\n    return res;\n}\n\n/* Sum using recursion */\nint sumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.java<pre><code>/* Sum using iteration */\nint sumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Sum using recursion */\nint sumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.cs<pre><code>/* Sum using iteration */\nint SumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Sum using recursion */\nint SumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + SumRecur(n - 1);\n}\n</code></pre> complexity_exercises.go<pre><code>/* Sum using iteration */\nfunc sumIter(n int) int {\n    res := 0\n    for i := 1; i &lt;= n; i++ {\n        res += i\n    }\n    return res\n}\n\n/* Sum using recursion */\nfunc sumRecur(n int) int {\n    if n == 1 {\n        return 1\n    }\n    return n + sumRecur(n-1)\n}\n</code></pre> complexity_exercises.swift<pre><code>/* Sum using iteration */\nfunc sumIter(n: Int) -&gt; Int {\n    var res = 0\n    for i in 1 ... n {\n        res += i\n    }\n    return res\n}\n\n/* Sum using recursion */\nfunc sumRecur(n: Int) -&gt; Int {\n    if n == 1 {\n        return 1\n    }\n    return n + sumRecur(n: n - 1)\n}\n</code></pre> complexity_exercises.js<pre><code>/* Sum using iteration */\nfunction sumIter(n) {\n    let res = 0;\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Sum using recursion */\nfunction sumRecur(n) {\n    if (n === 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.ts<pre><code>/* Sum using iteration */\nfunction sumIter(n: number): number {\n    let res = 0;\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Sum using recursion */\nfunction sumRecur(n: number): number {\n    if (n === 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.dart<pre><code>/* Sum using iteration */\nint sumIter(int n) {\n  int res = 0;\n  for (int i = 1; i &lt;= n; i++) {\n    res += i;\n  }\n  return res;\n}\n\n/* Sum using recursion */\nint sumRecur(int n) {\n  if (n == 1) {\n    return 1;\n  }\n  return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.rs<pre><code>/* Sum using iteration */\nfn sum_iter(n: i32) -&gt; i32 {\n    let mut res = 0;\n    for i in 1..=n {\n        res += i;\n    }\n    res\n}\n\n/* Sum using recursion */\nfn sum_recur(n: i32) -&gt; i32 {\n    if n == 1 {\n        return 1;\n    }\n    n + sum_recur(n - 1)\n}\n</code></pre> complexity_exercises.c<pre><code>/* Sum using iteration */\nint sumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Sum using recursion */\nint sumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.kt<pre><code>/* Sum using iteration */\nfun sumIter(n: Int): Int {\n    var res = 0\n    for (i in 1..n) {\n        res += i\n    }\n    return res\n}\n\n/* Sum using recursion */\nfun sumRecur(n: Int): Int {\n    if (n == 1) {\n        return 1\n    }\n    return n + sumRecur(n - 1)\n}\n</code></pre> complexity_exercises.rb<pre><code>### Sum using iteration ###\ndef sum_iter(n)\n  res = 0\n  for i in 1..n\n    res += i\n  end\n  res\nend\n\n### Sum using recursion ###\ndef sum_recur(n)\n  return 1 if n == 1\n\n  n + sum_recur(n - 1)\nend\n</code></pre> <ol> <li>When the iterative function runs with <code>n = 4</code>, what is the value of the accumulator <code>res</code> at the end of each iteration?</li> <li>When the recursive function runs with <code>n = 4</code>, what values does the parameter <code>n</code> take, in order? How is the result computed as the calls return, starting from the deepest call?</li> <li>What are the time and space complexities of each approach? Explain using the execution steps from questions 1 and 2.</li> </ol> Reference answer <ol> <li> <p>The loop variable <code>i</code> takes the values <code>1, 2, 3, 4</code> in order. At the end of each iteration, <code>res</code> becomes     <code>1, 3, 6, 10</code>, respectively, so the iterative function returns 10.</p> </li> <li> <p>The parameter <code>n</code> follows the sequence <code>4 → 3 → 2 → 1</code>.     The deepest call returns 1. The remaining calls then compute <code>2 + 1 = 3</code>, <code>3 + 3 = 6</code>, and <code>4 + 6 = 10</code>, in order.     At the deepest point, all four function calls are still active.</p> </li> <li> <p>Both functions perform a number of iterations or calls proportional to \\(n\\), so both have time complexity \\(O(n)\\).     Their space complexities differ: the iterative version uses only a constant number of variables, giving \\(O(1)\\).     In the recursive version, earlier calls must wait for their results until the base case is reached. The call stack therefore holds up to \\(n\\) calls at once,     giving space complexity \\(O(n)\\).</p> <p>When analyzing space complexity, consider the space used by recursive calls as well as the variables in the code.</p> </li> </ol>","path":["Chapter 2.   Complexity analysis","2.6   Exercises"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#2-time-complexity-of-three-code-snippets","level":3,"title":"2.   Time complexity of three code snippets","text":"<p>Each of the following three code snippets takes a positive integer \\(n\\) as input. Rank them from lowest to highest time complexity, and state each complexity.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby complexity_exercises.py<pre><code>def linear_loop(n: int) -&gt; int:\n    \"\"\"Linear-time loop\"\"\"\n    res = 0\n    for i in range(n):\n        res += i\n    return res\n\ndef quadratic_loop(n: int) -&gt; int:\n    \"\"\"Quadratic-time loop\"\"\"\n    res = 0\n    for i in range(n):\n        for j in range(i, n):\n            res += j\n    return res\n\ndef logarithmic_loop(n: int) -&gt; int:\n    \"\"\"Logarithmic-time loop\"\"\"\n    while n &gt; 1:\n        n //= 2\n    return n\n</code></pre> complexity_exercises.cpp<pre><code>/* Linear-time loop */\nint linearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; ++i) {\n        res += i;\n    }\n    return res;\n}\n\n/* Quadratic-time loop */\nint quadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; ++i) {\n        for (int j = i; j &lt; n; ++j) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* Logarithmic-time loop */\nint logarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.java<pre><code>/* Linear-time loop */\nint linearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Quadratic-time loop */\nint quadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* Logarithmic-time loop */\nint logarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.cs<pre><code>/* Linear-time loop */\nint LinearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Quadratic-time loop */\nint QuadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* Logarithmic-time loop */\nint LogarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.go<pre><code>/* Linear-time loop */\nfunc linearLoop(n int) int {\n    res := 0\n    for i := 0; i &lt; n; i++ {\n        res += i\n    }\n    return res\n}\n\n/* Quadratic-time loop */\nfunc quadraticLoop(n int) int {\n    res := 0\n    for i := 0; i &lt; n; i++ {\n        for j := i; j &lt; n; j++ {\n            res += j\n        }\n    }\n    return res\n}\n\n/* Logarithmic-time loop */\nfunc logarithmicLoop(n int) int {\n    for n &gt; 1 {\n        n /= 2\n    }\n    return n\n}\n</code></pre> complexity_exercises.swift<pre><code>/* Linear-time loop */\nfunc linearLoop(n: Int) -&gt; Int {\n    var res = 0\n    for i in 0 ..&lt; n {\n        res += i\n    }\n    return res\n}\n\n/* Quadratic-time loop */\nfunc quadraticLoop(n: Int) -&gt; Int {\n    var res = 0\n    for i in 0 ..&lt; n {\n        for j in i ..&lt; n {\n            res += j\n        }\n    }\n    return res\n}\n\n/* Logarithmic-time loop */\nfunc logarithmicLoop(n: Int) -&gt; Int {\n    var n = n\n    while n &gt; 1 {\n        n /= 2\n    }\n    return n\n}\n</code></pre> complexity_exercises.js<pre><code>/* Linear-time loop */\nfunction linearLoop(n) {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Quadratic-time loop */\nfunction quadraticLoop(n) {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* Logarithmic-time loop */\nfunction logarithmicLoop(n) {\n    while (n &gt; 1) {\n        n = Math.floor(n / 2);\n    }\n    return n;\n}\n</code></pre> complexity_exercises.ts<pre><code>/* Linear-time loop */\nfunction linearLoop(n: number): number {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Quadratic-time loop */\nfunction quadraticLoop(n: number): number {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* Logarithmic-time loop */\nfunction logarithmicLoop(n: number): number {\n    while (n &gt; 1) {\n        n = Math.floor(n / 2);\n    }\n    return n;\n}\n</code></pre> complexity_exercises.dart<pre><code>/* Linear-time loop */\nint linearLoop(int n) {\n  int res = 0;\n  for (int i = 0; i &lt; n; i++) {\n    res += i;\n  }\n  return res;\n}\n\n/* Quadratic-time loop */\nint quadraticLoop(int n) {\n  int res = 0;\n  for (int i = 0; i &lt; n; i++) {\n    for (int j = i; j &lt; n; j++) {\n      res += j;\n    }\n  }\n  return res;\n}\n\n/* Logarithmic-time loop */\nint logarithmicLoop(int n) {\n  while (n &gt; 1) {\n    n ~/= 2;\n  }\n  return n;\n}\n</code></pre> complexity_exercises.rs<pre><code>/* Linear-time loop */\nfn linear_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    for i in 0..n {\n        res += i;\n    }\n    res\n}\n\n/* Quadratic-time loop */\nfn quadratic_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    for i in 0..n {\n        for j in i..n {\n            res += j;\n        }\n    }\n    res\n}\n\n/* Logarithmic-time loop */\nfn logarithmic_loop(mut n: i32) -&gt; i32 {\n    while n &gt; 1 {\n        n /= 2;\n    }\n    n\n}\n</code></pre> complexity_exercises.c<pre><code>/* Linear-time loop */\nint linearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* Quadratic-time loop */\nint quadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* Logarithmic-time loop */\nint logarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.kt<pre><code>/* Linear-time loop */\nfun linearLoop(n: Int): Int {\n    var res = 0\n    for (i in 0 until n) {\n        res += i\n    }\n    return res\n}\n\n/* Quadratic-time loop */\nfun quadraticLoop(n: Int): Int {\n    var res = 0\n    for (i in 0 until n) {\n        for (j in i until n) {\n            res += j\n        }\n    }\n    return res\n}\n\n/* Logarithmic-time loop */\nfun logarithmicLoop(n: Int): Int {\n    var value = n\n    while (value &gt; 1) {\n        value /= 2\n    }\n    return value\n}\n</code></pre> complexity_exercises.rb<pre><code>### Linear-time loop ###\ndef linear_loop(n)\n  res = 0\n  for i in 0...n\n    res += i\n  end\n  res\nend\n\n### Quadratic-time loop ###\ndef quadratic_loop(n)\n  res = 0\n  for i in 0...n\n    for j in i...n\n      res += j\n    end\n  end\n  res\nend\n\n### Logarithmic-time loop ###\ndef logarithmic_loop(n)\n  n /= 2 while n &gt; 1\n  n\nend\n</code></pre> Reference answer <p>From lowest to highest: snippet 3, \\(O(\\log n)\\); snippet 1, \\(O(n)\\); snippet 2, \\(O(n^2)\\). Snippet 3 halves \\(n\\) at each iteration, so it runs approximately \\(\\log_2 n\\) times. Snippet 1 runs its loop exactly \\(n\\) times. In snippet 2, the inner loop runs \\(n,n-1,\\dots,1\\) times in successive iterations, for a total of \\(n(n+1)/2\\), giving quadratic time complexity.</p>","path":["Chapter 2.   Complexity analysis","2.6   Exercises"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#3-which-reversal-method-uses-less-space","level":3,"title":"3.   Which reversal method uses less space?","text":"<p>There are two ways to reverse all the elements in an array <code>nums</code>:</p> <ol> <li>Create an array <code>res</code> of the same length, copy the elements into it in reverse order, and return it.</li> <li> <p>Move two indices, <code>i</code> and <code>j</code>, inward from the beginning and end, swapping <code>nums[i]</code> with <code>nums[j]</code> in pairs.</p> <p>What is the space complexity of each approach? Which operation is in-place?</p> </li> </ol> Reference answer <ol> <li> <p>An auxiliary array as long as the input is needed, giving space complexity \\(O(n)\\).</p> </li> <li> <p>Only two index variables are used,     giving space complexity \\(O(1)\\). This is an in-place operation.</p> <p>Keep in mind that in-place reversal modifies the input array. Prefer it only when modifying the input is allowed. If the original array must be preserved, the copying cost of approach 1 is unavoidable.</p> </li> </ol>","path":["Chapter 2.   Complexity analysis","2.6   Exercises"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#262-programming-exercises","level":2,"title":"2.6.2   Programming exercises","text":"","path":["Chapter 2.   Complexity analysis","2.6   Exercises"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#1-fibonacci-number","level":3,"title":"1.   Fibonacci number","text":"<p>The Fibonacci sequence satisfies \\(F(0)=0\\), \\(F(1)=1\\), and, for \\(n\\ge2\\), \\(F(n)=F(n-1)+F(n-2)\\).</p> <p>Given a nonnegative integer <code>n</code>, use a loop to compute and return \\(F(n)\\) without recursion.</p> Hints <ol> <li>Handle n equal to 0 or 1 separately first.</li> <li>Computing the next term requires only the previous two terms; there is no need to store the entire sequence.</li> <li>When updating the two variables, take care not to overwrite an old value before you have finished using it.</li> </ol> <p>LeetCode</p>","path":["Chapter 2.   Complexity analysis","2.6   Exercises"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/","level":1,"title":"2.2   Iteration and recursion","text":"<p>Algorithms often repeat tasks, and repetition is closely tied to complexity analysis. Before introducing time and space complexity, let's examine two basic ways to repeat work in a program: iteration and recursion.</p>","path":["Chapter 2.   Complexity analysis","2.2   Iteration and recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#221-iteration","level":2,"title":"2.2.1   Iteration","text":"<p>Iteration is a control structure for repeating a task. A program repeatedly executes a block of code while a condition holds, stopping when it no longer does.</p>","path":["Chapter 2.   Complexity analysis","2.2   Iteration and recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for-loops","level":3,"title":"1.   For loops","text":"<p>A <code>for</code> loop is one of the most common forms of iteration, especially when the number of iterations is known in advance.</p> <p>The following function uses a <code>for</code> loop to compute \\(1 + 2 + \\dots + n\\), storing the result in <code>res</code>. In Python, <code>range(a, b)</code> includes the left endpoint but excludes the right, so it visits \\(a, a + 1, \\dots, b-1\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def for_loop(n: int) -&gt; int:\n    \"\"\"for loop\"\"\"\n    res = 0\n    # Sum 1, 2, ..., n-1, n in a loop\n    for i in range(1, n + 1):\n        res += i\n    return res\n</code></pre> iteration.cpp<pre><code>/* for loop */\nint forLoop(int n) {\n    int res = 0;\n    // Sum 1, 2, ..., n-1, n in a loop\n    for (int i = 1; i &lt;= n; ++i) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* for loop */\nint forLoop(int n) {\n    int res = 0;\n    // Sum 1, 2, ..., n-1, n in a loop\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* for loop */\nint ForLoop(int n) {\n    int res = 0;\n    // Sum 1, 2, ..., n-1, n in a loop\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* for loop */\nfunc forLoop(n int) int {\n    res := 0\n    // Sum 1, 2, ..., n-1, n in a loop\n    for i := 1; i &lt;= n; i++ {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* for loop */\nfunc forLoop(n: Int) -&gt; Int {\n    var res = 0\n    // Sum 1, 2, ..., n-1, n in a loop\n    for i in 1 ... n {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* for loop */\nfunction forLoop(n) {\n    let res = 0;\n    // Sum 1, 2, ..., n-1, n in a loop\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* for loop */\nfunction forLoop(n: number): number {\n    let res = 0;\n    // Sum 1, 2, ..., n-1, n in a loop\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* for loop */\nint forLoop(int n) {\n  int res = 0;\n  // Sum 1, 2, ..., n-1, n in a loop\n  for (int i = 1; i &lt;= n; i++) {\n    res += i;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* for loop */\nfn for_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    // Sum 1, 2, ..., n-1, n in a loop\n    for i in 1..=n {\n        res += i;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* for loop */\nint forLoop(int n) {\n    int res = 0;\n    // Sum 1, 2, ..., n-1, n in a loop\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* for loop */\nfun forLoop(n: Int): Int {\n    var res = 0\n    // Sum 1, 2, ..., n-1, n in a loop\n    for (i in 1..n) {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### for loop ###\ndef for_loop(n)\n  res = 0\n\n  # Sum 1, 2, ..., n-1, n in a loop\n  for i in 1..n\n    res += i\n  end\n\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 2-1 is a flowchart of this summation function.</p> <p></p> <p> Figure 2-1   Flowchart of the summation function </p> <p>Its operation count is proportional to the input size \\(n\\): the relationship is linear. Time complexity describes this relationship, as the next section will explain.</p>","path":["Chapter 2.   Complexity analysis","2.2   Iteration and recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while-loops","level":3,"title":"2.   While loops","text":"<p>Like <code>for</code>, a <code>while</code> loop implements iteration. Before each iteration, it checks a condition: if true, execution continues; otherwise, the loop ends.</p> <p>Here is a <code>while</code> loop computing \\(1 + 2 + \\dots + n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def while_loop(n: int) -&gt; int:\n    \"\"\"while loop\"\"\"\n    res = 0\n    i = 1  # Initialize the loop control variable\n    # Sum 1, 2, ..., n-1, n in a loop\n    while i &lt;= n:\n        res += i\n        i += 1  # Update the loop control variable\n    return res\n</code></pre> iteration.cpp<pre><code>/* while loop */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // Initialize the loop control variable\n    // Sum 1, 2, ..., n-1, n in a loop\n    while (i &lt;= n) {\n        res += i;\n        i++; // Update the loop control variable\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while loop */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // Initialize the loop control variable\n    // Sum 1, 2, ..., n-1, n in a loop\n    while (i &lt;= n) {\n        res += i;\n        i++; // Update the loop control variable\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while loop */\nint WhileLoop(int n) {\n    int res = 0;\n    int i = 1; // Initialize the loop control variable\n    // Sum 1, 2, ..., n-1, n in a loop\n    while (i &lt;= n) {\n        res += i;\n        i += 1; // Update the loop control variable\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while loop */\nfunc whileLoop(n int) int {\n    res := 0\n    // Initialize the loop control variable\n    i := 1\n    // Sum 1, 2, ..., n-1, n in a loop\n    for i &lt;= n {\n        res += i\n        // Update the loop control variable\n        i++\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while loop */\nfunc whileLoop(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // Initialize the loop control variable\n    // Sum 1, 2, ..., n-1, n in a loop\n    while i &lt;= n {\n        res += i\n        i += 1 // Update the loop control variable\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while loop */\nfunction whileLoop(n) {\n    let res = 0;\n    let i = 1; // Initialize the loop control variable\n    // Sum 1, 2, ..., n-1, n in a loop\n    while (i &lt;= n) {\n        res += i;\n        i++; // Update the loop control variable\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while loop */\nfunction whileLoop(n: number): number {\n    let res = 0;\n    let i = 1; // Initialize the loop control variable\n    // Sum 1, 2, ..., n-1, n in a loop\n    while (i &lt;= n) {\n        res += i;\n        i++; // Update the loop control variable\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while loop */\nint whileLoop(int n) {\n  int res = 0;\n  int i = 1; // Initialize the loop control variable\n  // Sum 1, 2, ..., n-1, n in a loop\n  while (i &lt;= n) {\n    res += i;\n    i++; // Update the loop control variable\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while loop */\nfn while_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // Initialize the loop control variable\n\n    // Sum 1, 2, ..., n-1, n in a loop\n    while i &lt;= n {\n        res += i;\n        i += 1; // Update the loop control variable\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while loop */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // Initialize the loop control variable\n    // Sum 1, 2, ..., n-1, n in a loop\n    while (i &lt;= n) {\n        res += i;\n        i++; // Update the loop control variable\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while loop */\nfun whileLoop(n: Int): Int {\n    var res = 0\n    var i = 1 // Initialize the loop control variable\n    // Sum 1, 2, ..., n-1, n in a loop\n    while (i &lt;= n) {\n        res += i\n        i++ // Update the loop control variable\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### while loop ###\ndef while_loop(n)\n  res = 0\n  i = 1 # Initialize the loop control variable\n\n  # Sum 1, 2, ..., n-1, n in a loop\n  while i &lt;= n\n    res += i\n    i += 1 # Update the loop control variable\n  end\n\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>A <code>while</code> loop offers more flexibility than a <code>for</code> loop in how its condition variables are initialized and updated.</p> <p>For example, the following code updates the condition variable \\(i\\) twice per iteration, which is less convenient to express with a <code>for</code> loop:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def while_loop_ii(n: int) -&gt; int:\n    \"\"\"while loop (two updates)\"\"\"\n    res = 0\n    i = 1  # Initialize the loop control variable\n    # Sum 1, 4, 10, ... in a loop\n    while i &lt;= n:\n        res += i\n        # Update the loop control variable\n        i += 1\n        i *= 2\n    return res\n</code></pre> iteration.cpp<pre><code>/* while loop (two updates) */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // Initialize the loop control variable\n    // Sum 1, 4, 10, ... in a loop\n    while (i &lt;= n) {\n        res += i;\n        // Update the loop control variable\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while loop (two updates) */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // Initialize the loop control variable\n    // Sum 1, 4, 10, ... in a loop\n    while (i &lt;= n) {\n        res += i;\n        // Update the loop control variable\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while loop (two updates) */\nint WhileLoopII(int n) {\n    int res = 0;\n    int i = 1; // Initialize the loop control variable\n    // Sum 1, 4, 10, ... in a loop\n    while (i &lt;= n) {\n        res += i;\n        // Update the loop control variable\n        i += 1; \n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while loop (two updates) */\nfunc whileLoopII(n int) int {\n    res := 0\n    // Initialize the loop control variable\n    i := 1\n    // Sum 1, 4, 10, ... in a loop\n    for i &lt;= n {\n        res += i\n        // Update the loop control variable\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while loop (two updates) */\nfunc whileLoopII(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // Initialize the loop control variable\n    // Sum 1, 4, 10, ... in a loop\n    while i &lt;= n {\n        res += i\n        // Update the loop control variable\n        i += 1\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while loop (two updates) */\nfunction whileLoopII(n) {\n    let res = 0;\n    let i = 1; // Initialize the loop control variable\n    // Sum 1, 4, 10, ... in a loop\n    while (i &lt;= n) {\n        res += i;\n        // Update the loop control variable\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while loop (two updates) */\nfunction whileLoopII(n: number): number {\n    let res = 0;\n    let i = 1; // Initialize the loop control variable\n    // Sum 1, 4, 10, ... in a loop\n    while (i &lt;= n) {\n        res += i;\n        // Update the loop control variable\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while loop (two updates) */\nint whileLoopII(int n) {\n  int res = 0;\n  int i = 1; // Initialize the loop control variable\n  // Sum 1, 4, 10, ... in a loop\n  while (i &lt;= n) {\n    res += i;\n    // Update the loop control variable\n    i++;\n    i *= 2;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while loop (two updates) */\nfn while_loop_ii(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // Initialize the loop control variable\n\n    // Sum 1, 4, 10, ... in a loop\n    while i &lt;= n {\n        res += i;\n        // Update the loop control variable\n        i += 1;\n        i *= 2;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while loop (two updates) */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // Initialize the loop control variable\n    // Sum 1, 4, 10, ... in a loop\n    while (i &lt;= n) {\n        res += i;\n        // Update the loop control variable\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while loop (two updates) */\nfun whileLoopII(n: Int): Int {\n    var res = 0\n    var i = 1 // Initialize the loop control variable\n    // Sum 1, 4, 10, ... in a loop\n    while (i &lt;= n) {\n        res += i\n        // Update the loop control variable\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### while loop (two updates)###\ndef while_loop_ii(n)\n  res = 0\n  i = 1 # Initialize the loop control variable\n\n  # Sum 1, 4, 10, ... in a loop\n  while i &lt;= n\n    res += i\n    # Update the loop control variable\n    i += 1\n    i *= 2\n  end\n\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>In general, <code>for</code> loops are more compact, while <code>while</code> loops are more flexible. Both implement iteration; choose according to the problem's needs.</p>","path":["Chapter 2.   Complexity analysis","2.2   Iteration and recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#3-nested-loops","level":3,"title":"3.   Nested loops","text":"<p>We can place one loop inside another, as in the following <code>for</code>-loop example:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def nested_for_loop(n: int) -&gt; str:\n    \"\"\"Nested for loops\"\"\"\n    res = \"\"\n    # Loop over i = 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        # Loop over j = 1, 2, ..., n-1, n\n        for j in range(1, n + 1):\n            res += f\"({i}, {j}), \"\n    return res\n</code></pre> iteration.cpp<pre><code>/* Nested for loops */\nstring nestedForLoop(int n) {\n    ostringstream res;\n    // Loop over i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        // Loop over j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; ++j) {\n            res &lt;&lt; \"(\" &lt;&lt; i &lt;&lt; \", \" &lt;&lt; j &lt;&lt; \"), \";\n        }\n    }\n    return res.str();\n}\n</code></pre> iteration.java<pre><code>/* Nested for loops */\nString nestedForLoop(int n) {\n    StringBuilder res = new StringBuilder();\n    // Loop over i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // Loop over j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.append(\"(\" + i + \", \" + j + \"), \");\n        }\n    }\n    return res.toString();\n}\n</code></pre> iteration.cs<pre><code>/* Nested for loops */\nstring NestedForLoop(int n) {\n    StringBuilder res = new();\n    // Loop over i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // Loop over j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.Append($\"({i}, {j}), \");\n        }\n    }\n    return res.ToString();\n}\n</code></pre> iteration.go<pre><code>/* Nested for loops */\nfunc nestedForLoop(n int) string {\n    res := \"\"\n    // Loop over i = 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= n; j++ {\n            // Loop over j = 1, 2, ..., n-1, n\n            res += fmt.Sprintf(\"(%d, %d), \", i, j)\n        }\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* Nested for loops */\nfunc nestedForLoop(n: Int) -&gt; String {\n    var res = \"\"\n    // Loop over i = 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        // Loop over j = 1, 2, ..., n-1, n\n        for j in 1 ... n {\n            res.append(\"(\\(i), \\(j)), \")\n        }\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* Nested for loops */\nfunction nestedForLoop(n) {\n    let res = '';\n    // Loop over i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // Loop over j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* Nested for loops */\nfunction nestedForLoop(n: number): string {\n    let res = '';\n    // Loop over i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // Loop over j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* Nested for loops */\nString nestedForLoop(int n) {\n  String res = \"\";\n  // Loop over i = 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    // Loop over j = 1, 2, ..., n-1, n\n    for (int j = 1; j &lt;= n; j++) {\n      res += \"($i, $j), \";\n    }\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* Nested for loops */\nfn nested_for_loop(n: i32) -&gt; String {\n    let mut res = vec![];\n    // Loop over i = 1, 2, ..., n-1, n\n    for i in 1..=n {\n        // Loop over j = 1, 2, ..., n-1, n\n        for j in 1..=n {\n            res.push(format!(\"({}, {}), \", i, j));\n        }\n    }\n    res.join(\"\")\n}\n</code></pre> iteration.c<pre><code>/* Nested for loops */\nchar *nestedForLoop(int n) {\n    // There are n * n points; each \"(i, j), \" takes at most 6+10*2 characters, plus space for the final null character \\0\n    int size = n * n * 26 + 1;\n    char *res = malloc(size * sizeof(char));\n    // Loop over i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // Loop over j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            char tmp[26];\n            snprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\n            strncat(res, tmp, size - strlen(res) - 1);\n        }\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* Nested for loops */\nfun nestedForLoop(n: Int): String {\n    val res = StringBuilder()\n    // Loop over i = 1, 2, ..., n-1, n\n    for (i in 1..n) {\n        // Loop over j = 1, 2, ..., n-1, n\n        for (j in 1..n) {\n            res.append(\" ($i, $j), \")\n        }\n    }\n    return res.toString()\n}\n</code></pre> iteration.rb<pre><code>### Nested for loops ###\ndef nested_for_loop(n)\n  res = \"\"\n\n  # Loop over i = 1, 2, ..., n-1, n\n  for i in 1..n\n    # Loop over j = 1, 2, ..., n-1, n\n    for j in 1..n\n      res += \"(#{i}, #{j}), \"\n    end\n  end\n\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 2-2 shows the nested loop's flowchart.</p> <p></p> <p> Figure 2-2   Flowchart of nested loops </p> <p>Here, the operation count is proportional to \\(n^2\\): runtime has a quadratic relationship with input size \\(n\\).</p> <p>Adding further nesting increases the dimension each time, producing cubic, fourth-power, and higher-order time complexity.</p>","path":["Chapter 2.   Complexity analysis","2.2   Iteration and recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#222-recursion","level":2,"title":"2.2.2   Recursion","text":"<p>Recursion is an algorithmic strategy in which a function calls itself to solve a problem. It has two main phases.</p> <ol> <li>Descent: The function calls itself more deeply, usually with smaller or simpler arguments, until a base case is reached.</li> <li>Return: Once the base case is reached, calls return from the deepest level upward, combining results along the way.</li> </ol> <p>In implementation, recursive code has three main elements.</p> <ol> <li>Base case: Determines when descent stops and returning begins.</li> <li>Recursive call: The function calls itself, usually with smaller or simpler arguments.</li> <li>Returned result: The current level returns its result to the previous level.</li> </ol> <p>In the following code, calling <code>recur(n)</code> computes \\(1 + 2 + \\dots + n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def recur(n: int) -&gt; int:\n    \"\"\"Recursion\"\"\"\n    # Base case\n    if n == 1:\n        return 1\n    # Recurse: make a recursive call\n    res = recur(n - 1)\n    # Unwind: return the result\n    return n + res\n</code></pre> recursion.cpp<pre><code>/* Recursion */\nint recur(int n) {\n    // Base case\n    if (n == 1)\n        return 1;\n    // Recurse: make a recursive call\n    int res = recur(n - 1);\n    // Unwind: return the result\n    return n + res;\n}\n</code></pre> recursion.java<pre><code>/* Recursion */\nint recur(int n) {\n    // Base case\n    if (n == 1)\n        return 1;\n    // Recurse: make a recursive call\n    int res = recur(n - 1);\n    // Unwind: return the result\n    return n + res;\n}\n</code></pre> recursion.cs<pre><code>/* Recursion */\nint Recur(int n) {\n    // Base case\n    if (n == 1)\n        return 1;\n    // Recurse: make a recursive call\n    int res = Recur(n - 1);\n    // Unwind: return the result\n    return n + res;\n}\n</code></pre> recursion.go<pre><code>/* Recursion */\nfunc recur(n int) int {\n    // Base case\n    if n == 1 {\n        return 1\n    }\n    // Recurse: make a recursive call\n    res := recur(n - 1)\n    // Unwind: return the result\n    return n + res\n}\n</code></pre> recursion.swift<pre><code>/* Recursion */\nfunc recur(n: Int) -&gt; Int {\n    // Base case\n    if n == 1 {\n        return 1\n    }\n    // Recurse: make a recursive call\n    let res = recur(n: n - 1)\n    // Unwind: return the result\n    return n + res\n}\n</code></pre> recursion.js<pre><code>/* Recursion */\nfunction recur(n) {\n    // Base case\n    if (n === 1) return 1;\n    // Recurse: make a recursive call\n    const res = recur(n - 1);\n    // Unwind: return the result\n    return n + res;\n}\n</code></pre> recursion.ts<pre><code>/* Recursion */\nfunction recur(n: number): number {\n    // Base case\n    if (n === 1) return 1;\n    // Recurse: make a recursive call\n    const res = recur(n - 1);\n    // Unwind: return the result\n    return n + res;\n}\n</code></pre> recursion.dart<pre><code>/* Recursion */\nint recur(int n) {\n  // Base case\n  if (n == 1) return 1;\n  // Recurse: make a recursive call\n  int res = recur(n - 1);\n  // Unwind: return the result\n  return n + res;\n}\n</code></pre> recursion.rs<pre><code>/* Recursion */\nfn recur(n: i32) -&gt; i32 {\n    // Base case\n    if n == 1 {\n        return 1;\n    }\n    // Recurse: make a recursive call\n    let res = recur(n - 1);\n    // Unwind: return the result\n    n + res\n}\n</code></pre> recursion.c<pre><code>/* Recursion */\nint recur(int n) {\n    // Base case\n    if (n == 1)\n        return 1;\n    // Recurse: make a recursive call\n    int res = recur(n - 1);\n    // Unwind: return the result\n    return n + res;\n}\n</code></pre> recursion.kt<pre><code>/* Recursion */\nfun recur(n: Int): Int {\n    // Base case\n    if (n == 1)\n        return 1\n    // Descending: recursive calls\n    val res = recur(n - 1)\n    // Unwinding: return the result\n    return n + res\n}\n</code></pre> recursion.rb<pre><code>### Recursion ###\ndef recur(n)\n  # Base case\n  return 1 if n == 1\n  # Recurse: make a recursive call\n  res = recur(n - 1)\n  # Unwind: return the result\n  n + res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 2-3 illustrates the recursive process.</p> <p></p> <p> Figure 2-3   Recursive process of the summation function </p> <p>Iteration and recursion can produce the same answer, but represent two different ways of thinking about and solving a problem.</p> <ul> <li>Iteration works bottom up, starting with basic steps and repeatedly performing or accumulating them until the task is complete.</li> <li>Recursion works top down, dividing the original problem into smaller subproblems of the same form, then repeating the division until reaching base cases with known solutions.</li> </ul> <p>For the summation example, let \\(f(n) = 1 + 2 + \\dots + n\\).</p> <ul> <li>Iteration: Loop from \\(1\\) to \\(n\\), accumulating the sum at each step to obtain \\(f(n)\\).</li> <li>Recursion: Decompose the problem as \\(f(n) = n + f(n-1)\\) and keep doing so recursively until the base case \\(f(1) = 1\\).</li> </ul>","path":["Chapter 2.   Complexity analysis","2.2   Iteration and recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-the-call-stack","level":3,"title":"1.   The call stack","text":"<p>Each time a recursive function calls itself, the system allocates memory for the new call's local variables, return address, and other information. This has two consequences.</p> <ul> <li>The call's context is stored in a stack frame until it returns. Recursion therefore usually uses more memory than iteration.</li> <li>Each recursive call adds overhead. Recursion is therefore usually slower than a loop.</li> </ul> <p>As shown in Figure 2-4, before the base case is reached, \\(n\\) recursive calls remain active at once: the recursion depth is \\(n\\).</p> <p></p> <p> Figure 2-4   Depth of recursive calls </p> <p>Programming languages usually limit recursion depth, and excessively deep recursion may cause a stack overflow.</p>","path":["Chapter 2.   Complexity analysis","2.2   Iteration and recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-tail-recursion","level":3,"title":"2.   Tail recursion","text":"<p>Interestingly, if the recursive call is the last operation before a function returns, a compiler or interpreter can optimize it to use as little space as iteration. This is called tail recursion.</p> <ul> <li>Ordinary recursion: After a call returns, its caller must continue executing code, so the caller's context must be preserved.</li> <li>Tail recursion: Nothing remains to be done after the recursive call returns, so the caller's context need not be retained.</li> </ul> <p>For \\(1 + 2 + \\dots + n\\), passing the accumulated result <code>res</code> as a parameter gives a tail-recursive solution:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def tail_recur(n, res):\n    \"\"\"Tail recursion\"\"\"\n    # Base case\n    if n == 0:\n        return res\n    # Tail-recursive call\n    return tail_recur(n - 1, res + n)\n</code></pre> recursion.cpp<pre><code>/* Tail recursion */\nint tailRecur(int n, int res) {\n    // Base case\n    if (n == 0)\n        return res;\n    // Tail-recursive call\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.java<pre><code>/* Tail recursion */\nint tailRecur(int n, int res) {\n    // Base case\n    if (n == 0)\n        return res;\n    // Tail-recursive call\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.cs<pre><code>/* Tail recursion */\nint TailRecur(int n, int res) {\n    // Base case\n    if (n == 0)\n        return res;\n    // Tail-recursive call\n    return TailRecur(n - 1, res + n);\n}\n</code></pre> recursion.go<pre><code>/* Tail recursion */\nfunc tailRecur(n int, res int) int {\n    // Base case\n    if n == 0 {\n        return res\n    }\n    // Tail-recursive call\n    return tailRecur(n-1, res+n)\n}\n</code></pre> recursion.swift<pre><code>/* Tail recursion */\nfunc tailRecur(n: Int, res: Int) -&gt; Int {\n    // Base case\n    if n == 0 {\n        return res\n    }\n    // Tail-recursive call\n    return tailRecur(n: n - 1, res: res + n)\n}\n</code></pre> recursion.js<pre><code>/* Tail recursion */\nfunction tailRecur(n, res) {\n    // Base case\n    if (n === 0) return res;\n    // Tail-recursive call\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.ts<pre><code>/* Tail recursion */\nfunction tailRecur(n: number, res: number): number {\n    // Base case\n    if (n === 0) return res;\n    // Tail-recursive call\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.dart<pre><code>/* Tail recursion */\nint tailRecur(int n, int res) {\n  // Base case\n  if (n == 0) return res;\n  // Tail-recursive call\n  return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.rs<pre><code>/* Tail recursion */\nfn tail_recur(n: i32, res: i32) -&gt; i32 {\n    // Base case\n    if n == 0 {\n        return res;\n    }\n    // Tail-recursive call\n    tail_recur(n - 1, res + n)\n}\n</code></pre> recursion.c<pre><code>/* Tail recursion */\nint tailRecur(int n, int res) {\n    // Base case\n    if (n == 0)\n        return res;\n    // Tail-recursive call\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.kt<pre><code>/* Tail recursion */\ntailrec fun tailRecur(n: Int, res: Int): Int {\n    // Add the tailrec modifier to enable tail-call optimization\n    // Base case\n    if (n == 0)\n        return res\n    // Tail-recursive call\n    return tailRecur(n - 1, res + n)\n}\n</code></pre> recursion.rb<pre><code>### Tail recursion ###\ndef tail_recur(n, res)\n  # Base case\n  return res if n == 0\n  # Tail-recursive call\n  tail_recur(n - 1, res + n)\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 2-5 illustrates tail recursion. Ordinary and tail recursion perform their additions at different points.</p> <ul> <li>Ordinary recursion: Addition occurs during the return phase, after each deeper call returns.</li> <li>Tail recursion: Addition occurs during descent; the return phase simply returns through the levels.</li> </ul> <p></p> <p> Figure 2-5   Tail-recursive summation </p> <p>Tip</p> <p>Many compilers and interpreters do not support tail-recursion optimization. Python, for example, does not optimize it by default, so even a tail-recursive function can run into a stack overflow.</p>","path":["Chapter 2.   Complexity analysis","2.2   Iteration and recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#3-recursion-trees","level":3,"title":"3.   Recursion trees","text":"<p>For divide-and-conquer problems, recursion often provides a more intuitive approach and more readable code than iteration. Consider the Fibonacci sequence.</p> <p>Question</p> <p>Given the Fibonacci sequence \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\), find its \\(n\\)-th number.</p> <p>Let \\(f(n)\\) be its \\(n\\)-th number. Two facts follow.</p> <ul> <li>The first two numbers are \\(f(1) = 0\\) and \\(f(2) = 1\\).</li> <li>Every later number is the sum of the two preceding numbers: \\(f(n) = f(n - 1) + f(n - 2)\\).</li> </ul> <p>Use the recurrence for recursive calls and the first two numbers as base cases. Calling <code>fib(n)</code> then returns the \\(n\\)-th Fibonacci number:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def fib(n: int) -&gt; int:\n    \"\"\"Fibonacci sequence: recursion\"\"\"\n    # Base cases: f(1) = 0, f(2) = 1\n    if n == 1 or n == 2:\n        return n - 1\n    # Recursive calls: f(n) = f(n-1) + f(n-2)\n    res = fib(n - 1) + fib(n - 2)\n    # Return the result f(n)\n    return res\n</code></pre> recursion.cpp<pre><code>/* Fibonacci sequence: recursion */\nint fib(int n) {\n    // Base cases: f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // Recursive calls: f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // Return the result f(n)\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* Fibonacci sequence: recursion */\nint fib(int n) {\n    // Base cases: f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // Recursive calls: f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // Return the result f(n)\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* Fibonacci sequence: recursion */\nint Fib(int n) {\n    // Base cases: f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // Recursive calls: f(n) = f(n-1) + f(n-2)\n    int res = Fib(n - 1) + Fib(n - 2);\n    // Return the result f(n)\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* Fibonacci sequence: recursion */\nfunc fib(n int) int {\n    // Base cases: f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // Recursive calls: f(n) = f(n-1) + f(n-2)\n    res := fib(n-1) + fib(n-2)\n    // Return the result f(n)\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* Fibonacci sequence: recursion */\nfunc fib(n: Int) -&gt; Int {\n    // Base cases: f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // Recursive calls: f(n) = f(n-1) + f(n-2)\n    let res = fib(n: n - 1) + fib(n: n - 2)\n    // Return the result f(n)\n    return res\n}\n</code></pre> recursion.js<pre><code>/* Fibonacci sequence: recursion */\nfunction fib(n) {\n    // Base cases: f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // Recursive calls: f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // Return the result f(n)\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* Fibonacci sequence: recursion */\nfunction fib(n: number): number {\n    // Base cases: f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // Recursive calls: f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // Return the result f(n)\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* Fibonacci sequence: recursion */\nint fib(int n) {\n  // Base cases: f(1) = 0, f(2) = 1\n  if (n == 1 || n == 2) return n - 1;\n  // Recursive calls: f(n) = f(n-1) + f(n-2)\n  int res = fib(n - 1) + fib(n - 2);\n  // Return the result f(n)\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* Fibonacci sequence: recursion */\nfn fib(n: i32) -&gt; i32 {\n    // Base cases: f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1;\n    }\n    // Recursive calls: f(n) = f(n-1) + f(n-2)\n    let res = fib(n - 1) + fib(n - 2);\n    // Return the result\n    res\n}\n</code></pre> recursion.c<pre><code>/* Fibonacci sequence: recursion */\nint fib(int n) {\n    // Base cases: f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // Recursive calls: f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // Return the result f(n)\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* Fibonacci sequence: recursion */\nfun fib(n: Int): Int {\n    // Base cases: f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1\n    // Recursive calls: f(n) = f(n-1) + f(n-2)\n    val res = fib(n - 1) + fib(n - 2)\n    // Return the result f(n)\n    return res\n}\n</code></pre> recursion.rb<pre><code>### Fibonacci sequence: recursion ###\ndef fib(n)\n  # Base cases: f(1) = 0, f(2) = 1\n  return n - 1 if n == 1 || n == 2\n  # Recursive calls: f(n) = f(n-1) + f(n-2)\n  res = fib(n - 1) + fib(n - 2)\n  # Return the result f(n)\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>The function makes two recursive calls, so each call creates two branches. Continuing this process produces a recursion tree with \\(n\\) levels, as shown in Figure 2-6.</p> <p></p> <p> Figure 2-6   Recursion tree for the Fibonacci sequence </p> <p>Recursion embodies the idea of breaking a problem into smaller subproblems, a crucial divide-and-conquer strategy.</p> <ul> <li>In algorithms, searching, sorting, backtracking, divide and conquer, dynamic programming, and many other techniques use this idea directly or indirectly.</li> <li>In data structures, recursion naturally handles linked lists, trees, and graphs because divide-and-conquer reasoning suits them well.</li> </ul>","path":["Chapter 2.   Complexity analysis","2.2   Iteration and recursion"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#223-comparing-iteration-and-recursion","level":2,"title":"2.2.3   Comparing iteration and recursion","text":"<p>Table 2-1 summarizes their differences in implementation, performance, and suitability.</p> <p> Table 2-1   Comparing iteration and recursion </p> Iteration Recursion Implementation Loops A function calls itself Time efficiency Usually high, with no function-call overhead Each call adds overhead Memory use Usually a fixed amount of memory Accumulated calls may require many stack frames Suitable problems Simple repetitive tasks, with straightforward, readable code Subproblem decomposition, including trees, graphs, divide and conquer, and backtracking; concise, clear structure <p>Tip</p> <p>If the following discussion feels difficult, return to it after reading the section on stacks.</p> <p>How are iteration and recursion connected? In the summation example, additions occur during the return phase. The first function called is the last to finish its addition, matching a stack's last-in, first-out behavior.</p> <p>The terms call stack and stack frame already suggest this close relationship.</p> <ol> <li>Descent: A function call adds a new stack frame to the call stack, holding local variables, parameters, the return address, and other data.</li> <li>Return: When the function finishes, its frame is removed, restoring the caller's execution context.</li> </ol> <p>Thus, an explicit stack can simulate the call stack, converting recursion into iteration:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def for_loop_recur(n: int) -&gt; int:\n    \"\"\"Simulate recursion using iteration\"\"\"\n    # Use an explicit stack to simulate the system call stack\n    stack = []\n    res = 0\n    # Simulate the recursive-call phase\n    for i in range(n, 0, -1):\n        # Simulate recursive calls by pushing onto the stack\n        stack.append(i)\n    # Simulate unwinding and accumulating the result\n    while stack:\n        # Simulate unwinding by popping from the stack\n        res += stack.pop()\n    # res = 1+2+3+...+n\n    return res\n</code></pre> recursion.cpp<pre><code>/* Simulate recursion using iteration */\nint forLoopRecur(int n) {\n    // Use an explicit stack to simulate the system call stack\n    stack&lt;int&gt; stack;\n    int res = 0;\n    // Recurse: make a recursive call\n    for (int i = n; i &gt; 0; i--) {\n        // Simulate recursive calls by pushing onto the stack\n        stack.push(i);\n    }\n    // Unwind: return the result\n    while (!stack.empty()) {\n        // Simulate unwinding by popping from the stack\n        res += stack.top();\n        stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* Simulate recursion using iteration */\nint forLoopRecur(int n) {\n    // Use an explicit stack to simulate the system call stack\n    Stack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n    int res = 0;\n    // Recurse: make a recursive call\n    for (int i = n; i &gt; 0; i--) {\n        // Simulate recursive calls by pushing onto the stack\n        stack.push(i);\n    }\n    // Unwind: return the result\n    while (!stack.isEmpty()) {\n        // Simulate unwinding by popping from the stack\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* Simulate recursion using iteration */\nint ForLoopRecur(int n) {\n    // Use an explicit stack to simulate the system call stack\n    Stack&lt;int&gt; stack = new();\n    int res = 0;\n    // Recurse: make a recursive call\n    for (int i = n; i &gt; 0; i--) {\n        // Simulate recursive calls by pushing onto the stack\n        stack.Push(i);\n    }\n    // Unwind: return the result\n    while (stack.Count &gt; 0) {\n        // Simulate unwinding by popping from the stack\n        res += stack.Pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* Simulate recursion using iteration */\nfunc forLoopRecur(n int) int {\n    // Use an explicit stack to simulate the system call stack\n    stack := list.New()\n    res := 0\n    // Descending: simulate recursive calls\n    for i := n; i &gt; 0; i-- {\n        // Simulate recursive calls by pushing onto the stack\n        stack.PushBack(i)\n    }\n    // Unwinding: simulate returning results\n    for stack.Len() != 0 {\n        // Simulate unwinding by popping from the stack\n        res += stack.Back().Value.(int)\n        stack.Remove(stack.Back())\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* Simulate recursion using iteration */\nfunc forLoopRecur(n: Int) -&gt; Int {\n    // Use an explicit stack to simulate the system call stack\n    var stack: [Int] = []\n    var res = 0\n    // Descending: simulate recursive calls\n    for i in (1 ... n).reversed() {\n        // Simulate recursive calls by pushing onto the stack\n        stack.append(i)\n    }\n    // Unwinding: simulate returning results\n    while !stack.isEmpty {\n        // Simulate unwinding by popping from the stack\n        res += stack.removeLast()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.js<pre><code>/* Simulate recursion using iteration */\nfunction forLoopRecur(n) {\n    // Use an explicit stack to simulate the system call stack\n    const stack = [];\n    let res = 0;\n    // Recurse: make a recursive call\n    for (let i = n; i &gt; 0; i--) {\n        // Simulate recursive calls by pushing onto the stack\n        stack.push(i);\n    }\n    // Unwind: return the result\n    while (stack.length) {\n        // Simulate unwinding by popping from the stack\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* Simulate recursion using iteration */\nfunction forLoopRecur(n: number): number {\n    // Use an explicit stack to simulate the system call stack \n    const stack: number[] = [];\n    let res: number = 0;\n    // Recurse: make a recursive call\n    for (let i = n; i &gt; 0; i--) {\n        // Simulate recursive calls by pushing onto the stack\n        stack.push(i);\n    }\n    // Unwind: return the result\n    while (stack.length) {\n        // Simulate unwinding by popping from the stack\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* Simulate recursion using iteration */\nint forLoopRecur(int n) {\n  // Use an explicit stack to simulate the system call stack\n  List&lt;int&gt; stack = [];\n  int res = 0;\n  // Recurse: make a recursive call\n  for (int i = n; i &gt; 0; i--) {\n    // Simulate recursive calls by pushing onto the stack\n    stack.add(i);\n  }\n  // Unwind: return the result\n  while (!stack.isEmpty) {\n    // Simulate unwinding by popping from the stack\n    res += stack.removeLast();\n  }\n  // res = 1+2+3+...+n\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* Simulate recursion using iteration */\nfn for_loop_recur(n: i32) -&gt; i32 {\n    // Use an explicit stack to simulate the system call stack\n    let mut stack = Vec::new();\n    let mut res = 0;\n    // Recurse: make a recursive call\n    for i in (1..=n).rev() {\n        // Simulate recursive calls by pushing onto the stack\n        stack.push(i);\n    }\n    // Unwind: return the result\n    while !stack.is_empty() {\n        // Simulate unwinding by popping from the stack\n        res += stack.pop().unwrap();\n    }\n    // res = 1+2+3+...+n\n    res\n}\n</code></pre> recursion.c<pre><code>/* Simulate recursion using iteration */\nint forLoopRecur(int n) {\n    int stack[1000]; // Use a large array to simulate a stack\n    int top = -1;    // Top-of-stack index\n    int res = 0;\n    // Recurse: make a recursive call\n    for (int i = n; i &gt; 0; i--) {\n        // Simulate recursive calls by pushing onto the stack\n        stack[1 + top++] = i;\n    }\n    // Unwind: return the result\n    while (top &gt;= 0) {\n        // Simulate unwinding by popping from the stack\n        res += stack[top--];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* Simulate recursion using iteration */\nfun forLoopRecur(n: Int): Int {\n    // Use an explicit stack to simulate the system call stack\n    val stack = Stack&lt;Int&gt;()\n    var res = 0\n    // Descending: recursive calls\n    for (i in n downTo 0) {\n        // Simulate recursive calls by pushing onto the stack\n        stack.push(i)\n    }\n    // Unwinding: return the result\n    while (stack.isNotEmpty()) {\n        // Simulate unwinding by popping from the stack\n        res += stack.pop()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.rb<pre><code>### Simulate recursion using iteration ###\ndef for_loop_recur(n)\n  # Use an explicit stack to simulate the system call stack\n  stack = []\n  res = 0\n\n  # Recurse: make a recursive call\n  for i in n.downto(0)\n    # Simulate recursive calls by pushing onto the stack\n    stack &lt;&lt; i\n  end\n  # Unwind: return the result\n  while !stack.empty?\n    res += stack.pop\n  end\n\n  # res = 1+2+3+...+n\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>The resulting iterative code is more complicated. Although iteration and recursion can often be converted into each other, doing so is not always worthwhile.</p> <ul> <li>The converted code may be harder to understand and less readable.</li> <li>Simulating the call stack can be very difficult for complex problems.</li> </ul> <p>The choice between iteration and recursion depends on the problem. In practice, weigh their strengths and weaknesses and choose the approach that fits the situation.</p>","path":["Chapter 2.   Complexity analysis","2.2   Iteration and recursion"],"tags":[]},{"location":"chapter_computational_complexity/performance_evaluation/","level":1,"title":"2.1   Evaluating algorithm efficiency","text":"<p>Algorithm design pursues two goals in sequence.</p> <ol> <li>Find a solution: The algorithm must reliably produce correct results across the specified input range.</li> <li>Find the best solution: When several approaches exist, we want the most efficient algorithm possible.</li> </ol> <p>Once correctness is established, efficiency becomes a primary measure of an algorithm's quality. It has two dimensions.</p> <ul> <li>Time efficiency: How long the algorithm takes to run.</li> <li>Space efficiency: How much memory it uses.</li> </ul> <p>Our goal is to design data structures and algorithms that are both fast and economical with memory. Evaluating efficiency effectively lets us compare algorithms and guide their design and optimization.</p> <p>There are two main evaluation methods: practical testing and theoretical estimation.</p>","path":["Chapter 2.   Complexity analysis","2.1   Evaluating algorithm efficiency"],"tags":[]},{"location":"chapter_computational_complexity/performance_evaluation/#211-practical-testing","level":2,"title":"2.1.1   Practical testing","text":"<p>Suppose algorithms <code>A</code> and <code>B</code> solve the same problem. The most direct comparison is to run both on a computer and measure runtime and memory use. This reflects actual behavior, but has substantial limitations.</p> <p>First, it is difficult to remove the influence of the test environment. Hardware affects performance: a highly parallel algorithm benefits from a multicore CPU, while a memory-intensive algorithm benefits from faster RAM. Results may differ across machines, so a fair average would require testing on many kinds of hardware, which is impractical.</p> <p>Second, comprehensive testing consumes substantial resources. Efficiency changes with input size. Algorithm <code>A</code> may be faster than <code>B</code> for small inputs, while the reverse holds for large inputs. A convincing conclusion requires testing many input sizes, consuming considerable computing resources.</p>","path":["Chapter 2.   Complexity analysis","2.1   Evaluating algorithm efficiency"],"tags":[]},{"location":"chapter_computational_complexity/performance_evaluation/#212-theoretical-estimation","level":2,"title":"2.1.2   Theoretical estimation","text":"<p>Given these limitations, we can estimate efficiency through calculation alone. This is called asymptotic complexity analysis, or simply complexity analysis.</p> <p>Complexity analysis relates an algorithm's time and space requirements to its input size. It describes how runtime and memory use grow as the input gets larger. We can unpack this definition into three points.</p> <ul> <li>Time and space resources correspond to time complexity and space complexity.</li> <li>Growth in input size means complexity relates efficiency to the amount of input data.</li> <li>Growth trends mean we study how quickly time or space increases, rather than its exact value.</li> </ul> <p>Complexity analysis overcomes limitations of practical testing in several ways.</p> <ul> <li>It does not require running code, saving energy and computing resources.</li> <li>It is independent of the test environment, so its conclusions apply across platforms.</li> <li>It reveals efficiency at different input sizes, especially for large inputs.</li> </ul> <p>Tip</p> <p>If complexity still feels unclear, do not worry. The following sections will explain it in detail.</p> <p>Complexity analysis gives us a yardstick for evaluating the time and space an algorithm needs and comparing different algorithms.</p> <p>As a mathematical concept, complexity can be abstract and challenging for beginners. This makes it a difficult topic to introduce first. Yet discussing a data structure or algorithm's properties inevitably involves runtime and memory use.</p> <p>Before studying data structures and algorithms in depth, build an initial understanding of complexity analysis so that you can analyze simple algorithms.</p>","path":["Chapter 2.   Complexity analysis","2.1   Evaluating algorithm efficiency"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/","level":1,"title":"2.4   Space complexity","text":"<p>Space complexity measures how an algorithm's memory use grows as the input gets larger. It is much like time complexity, with memory usage replacing runtime.</p>","path":["Chapter 2.   Complexity analysis","2.4   Space complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#241-space-used-by-an-algorithm","level":2,"title":"2.4.1   Space used by an algorithm","text":"<p>The memory an algorithm uses during execution falls into these main categories.</p> <ul> <li>Input space: Stores the input data.</li> <li>Temporary space: Stores variables, objects, function contexts, and other working data.</li> <li>Output space: Stores the output data.</li> </ul> <p>Space complexity generally counts temporary space plus output space.</p> <p>Temporary space can be divided further into three parts.</p> <ul> <li>Temporary data: Constants, variables, objects, and other data used during execution.</li> <li>Stack frames: Context data for function calls. Each call creates a frame at the top of the stack, which is released when the function returns.</li> <li>Instruction space: Compiled program instructions, usually ignored in the analysis.</li> </ul> <p>We usually count temporary data, stack frames, and output data when analyzing space complexity, as shown in Figure 2-15.</p> <p></p> <p> Figure 2-15   Space used by an algorithm </p> <p>The corresponding code is:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class Node:\n    \"\"\"Class\"\"\"\n    def __init__(self, x: int):\n        self.val: int = x              # Node value\n        self.next: Node | None = None  # Reference to the next node\n\ndef function() -&gt; int:\n    \"\"\"Function\"\"\"\n    # Perform some operations...\n    return 0\n\ndef algorithm(n) -&gt; int:  # Input data\n    A = 0                 # Temporary data (constant, conventionally named in uppercase)\n    b = 0                 # Temporary data (variable)\n    node = Node(0)        # Temporary data (object)\n    c = function()        # Stack frame (function call)\n    return A + b + c      # Output data\n</code></pre> <pre><code>/* Structure */\nstruct Node {\n    int val;\n    Node *next;\n    Node(int x) : val(x), next(nullptr) {}\n};\n\n/* Function */\nint func() {\n    // Perform some operations...\n    return 0;\n}\n\nint algorithm(int n) {        // Input data\n    const int a = 0;          // Temporary data (constant)\n    int b = 0;                // Temporary data (variable)\n    Node* node = new Node(0); // Temporary data (object)\n    int c = func();           // Stack frame (function call)\n    return a + b + c;         // Output data\n}\n</code></pre> <pre><code>/* Class */\nclass Node {\n    int val;\n    Node next;\n    Node(int x) { val = x; }\n}\n\n/* Function */\nint function() {\n    // Perform some operations...\n    return 0;\n}\n\nint algorithm(int n) {        // Input data\n    final int a = 0;          // Temporary data (constant)\n    int b = 0;                // Temporary data (variable)\n    Node node = new Node(0);  // Temporary data (object)\n    int c = function();       // Stack frame (function call)\n    return a + b + c;         // Output data\n}\n</code></pre> <pre><code>/* Class */\nclass Node(int x) {\n    int val = x;\n    Node next;\n}\n\n/* Function */\nint Function() {\n    // Perform some operations...\n    return 0;\n}\n\nint Algorithm(int n) {        // Input data\n    const int a = 0;          // Temporary data (constant)\n    int b = 0;                // Temporary data (variable)\n    Node node = new(0);       // Temporary data (object)\n    int c = Function();       // Stack frame (function call)\n    return a + b + c;         // Output data\n}\n</code></pre> <pre><code>/* Structure */\ntype node struct {\n    val  int\n    next *node\n}\n\n/* Create a node structure  */\nfunc newNode(val int) *node {\n    return &amp;node{val: val}\n}\n\n/* Function */\nfunc function() int {\n    // Perform some operations...\n    return 0\n}\n\nfunc algorithm(n int) int { // Input data\n    const a = 0             // Temporary data (constant)\n    b := 0                  // Temporary data (variable)\n    newNode(0)              // Temporary data (object)\n    c := function()         // Stack frame (function call)\n    return a + b + c        // Output data\n}\n</code></pre> <pre><code>/* Class */\nclass Node {\n    var val: Int\n    var next: Node?\n\n    init(x: Int) {\n        val = x\n    }\n}\n\n/* Function */\nfunc function() -&gt; Int {\n    // Perform some operations...\n    return 0\n}\n\nfunc algorithm(n: Int) -&gt; Int { // Input data\n    let a = 0             // Temporary data (constant)\n    var b = 0             // Temporary data (variable)\n    let node = Node(x: 0) // Temporary data (object)\n    let c = function()    // Stack frame (function call)\n    return a + b + c      // Output data\n}\n</code></pre> <pre><code>/* Class */\nclass Node {\n    val;\n    next;\n    constructor(val) {\n        this.val = val === undefined ? 0 : val; // Node value\n        this.next = null;                       // Reference to the next node\n    }\n}\n\n/* Function */\nfunction constFunc() {\n    // Perform some operations\n    return 0;\n}\n\nfunction algorithm(n) {       // Input data\n    const a = 0;              // Temporary data (constant)\n    let b = 0;                // Temporary data (variable)\n    const node = new Node(0); // Temporary data (object)\n    const c = constFunc();    // Stack frame (function call)\n    return a + b + c;         // Output data\n}\n</code></pre> <pre><code>/* Class */\nclass Node {\n    val: number;\n    next: Node | null;\n    constructor(val?: number) {\n        this.val = val === undefined ? 0 : val; // Node value\n        this.next = null;                       // Reference to the next node\n    }\n}\n\n/* Function */\nfunction constFunc(): number {\n    // Perform some operations\n    return 0;\n}\n\nfunction algorithm(n: number): number { // Input data\n    const a = 0;                        // Temporary data (constant)\n    let b = 0;                          // Temporary data (variable)\n    const node = new Node(0);           // Temporary data (object)\n    const c = constFunc();              // Stack frame (function call)\n    return a + b + c;                   // Output data\n}\n</code></pre> <pre><code>/* Class */\nclass Node {\n  int val;\n  Node next;\n  Node(this.val, [this.next]);\n}\n\n/* Function */\nint function() {\n  // Perform some operations...\n  return 0;\n}\n\nint algorithm(int n) {  // Input data\n  const int a = 0;      // Temporary data (constant)\n  int b = 0;            // Temporary data (variable)\n  Node node = Node(0);  // Temporary data (object)\n  int c = function();   // Stack frame (function call)\n  return a + b + c;     // Output data\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* Structure */\nstruct Node {\n    val: i32,\n    next: Option&lt;Rc&lt;RefCell&lt;Node&gt;&gt;&gt;,\n}\n\n/* Create a Node structure */\nimpl Node {\n    fn new(val: i32) -&gt; Self {\n        Self { val: val, next: None }\n    }\n}\n\n/* Function */\nfn function() -&gt; i32 {      \n    // Perform some operations...\n    return 0;\n}\n\nfn algorithm(n: i32) -&gt; i32 {       // Input data\n    const a: i32 = 0;               // Temporary data (constant)\n    let mut b = 0;                  // Temporary data (variable)\n    let node = Node::new(0);        // Temporary data (object)\n    let c = function();             // Stack frame (function call)\n    return a + b + c;               // Output data\n}\n</code></pre> <pre><code>/* Function */\nint func() {\n    // Perform some operations...\n    return 0;\n}\n\nint algorithm(int n) { // Input data\n    const int a = 0;   // Temporary data (constant)\n    int b = 0;         // Temporary data (variable)\n    int c = func();    // Stack frame (function call)\n    return a + b + c;  // Output data\n}\n</code></pre> <pre><code>/* Class */\nclass Node(var _val: Int) {\n    var next: Node? = null\n}\n\n/* Function */\nfun function(): Int {\n    // Perform some operations...\n    return 0\n}\n\nfun algorithm(n: Int): Int { // Input data\n    val a = 0                // Temporary data (constant)\n    var b = 0                // Temporary data (variable)\n    val node = Node(0)       // Temporary data (object)\n    val c = function()       // Stack frame (function call)\n    return a + b + c         // Output data\n}\n</code></pre> <pre><code>### Class ###\nclass Node\n    attr_accessor :val      # Node value\n    attr_accessor :next     # Reference to the next node\n\n    def initialize(x)\n        @val = x\n    end\nend\n\n### Function ###\ndef function\n    # Perform some operations...\n    0\nend\n\n### Algorithm ###\ndef algorithm(n)        # Input data\n    a = 0               # Temporary data (constant)\n    b = 0               # Temporary data (variable)\n    node = Node.new(0)  # Temporary data (object)\n    c = function        # Stack frame (function call)\n    a + b + c           # Output data\nend\n</code></pre>","path":["Chapter 2.   Complexity analysis","2.4   Space complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#242-deriving-space-complexity","level":2,"title":"2.4.2   Deriving space complexity","text":"<p>The approach is much like time complexity analysis, except that we count memory used rather than operations performed.</p> <p>Unlike time complexity, we usually focus only on worst-case space complexity. Memory is a hard requirement: enough must be available for every valid input.</p> <p>In the following code, \"worst case\" has two meanings.</p> <ol> <li>The worst input: When \\(n &lt; 10\\), space complexity is \\(O(1)\\). When \\(n &gt; 10\\), initializing <code>nums</code> requires \\(O(n)\\) space, so worst-case space complexity is \\(O(n)\\).</li> <li>Peak memory during execution: Before the final line, the program uses \\(O(1)\\) space. Initializing <code>nums</code> raises usage to \\(O(n)\\), so worst-case space complexity is \\(O(n)\\).</li> </ol> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def algorithm(n: int):\n    a = 0               # O(1)\n    b = [0] * 10000     # O(1)\n    if n &gt; 10:\n        nums = [0] * n  # O(n)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    vector&lt;int&gt; b(10000);    // O(1)\n    if (n &gt; 10)\n        vector&lt;int&gt; nums(n); // O(n)\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10)\n        int[] nums = new int[n]; // O(n)\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10) {\n        int[] nums = new int[n]; // O(n)\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 0                      // O(1)\n    b := make([]int, 10000)     // O(1)\n    var nums []int\n    if n &gt; 10 {\n        nums := make([]int, n)  // O(n)\n    }\n    fmt.Println(a, b, nums)\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    let a = 0 // O(1)\n    let b = Array(repeating: 0, count: 10000) // O(1)\n    if n &gt; 10 {\n        let nums = Array(repeating: 0, count: n) // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 0;                            // O(1)\n  List&lt;int&gt; b = List.filled(10000, 0);  // O(1)\n  if (n &gt; 10) {\n    List&lt;int&gt; nums = List.filled(n, 0); // O(n)\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let a = 0;                              // O(1)\n    let b = [0; 10000];                     // O(1)\n    if n &gt; 10 {\n        let nums = vec![0; n as usize];     // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    int b[10000];            // O(1)\n    if (n &gt; 10)\n        int nums[n] = {0};   // O(n)\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    val a = 0                    // O(1)\n    val b = IntArray(10000)      // O(1)\n    if (n &gt; 10) {\n        val nums = IntArray(n)   // O(n)\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 0                           # O(1)\n    b = Array.new(10000)            # O(1)\n    nums = Array.new(n) if n &gt; 10   # O(n)\nend\n</code></pre> <p>For recursive functions, remember to count stack frames. Consider the following code:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def function() -&gt; int:\n    # Perform some operations\n    return 0\n\ndef loop(n: int):\n    \"\"\"The loop uses O(1) space\"\"\"\n    for _ in range(n):\n        function()\n\ndef recur(n: int):\n    \"\"\"The recursion uses O(n) space\"\"\"\n    if n == 1:\n        return\n    return recur(n - 1)\n</code></pre> <pre><code>int func() {\n    // Perform some operations\n    return 0;\n}\n/* The loop uses O(1) space */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* The recursion uses O(n) space */\nvoid recur(int n) {\n    if (n == 1) return;\n    recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n    // Perform some operations\n    return 0;\n}\n/* The loop uses O(1) space */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n/* The recursion uses O(n) space */\nvoid recur(int n) {\n    if (n == 1) return;\n    recur(n - 1);\n}\n</code></pre> <pre><code>int Function() {\n    // Perform some operations\n    return 0;\n}\n/* The loop uses O(1) space */\nvoid Loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n/* The recursion uses O(n) space */\nint Recur(int n) {\n    if (n == 1) return 1;\n    return Recur(n - 1);\n}\n</code></pre> <pre><code>func function() int {\n    // Perform some operations\n    return 0\n}\n\n/* The loop uses O(1) space */\nfunc loop(n int) {\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n}\n\n/* The recursion uses O(n) space */\nfunc recur(n int) {\n    if n == 1 {\n        return\n    }\n    recur(n - 1)\n}\n</code></pre> <pre><code>@discardableResult\nfunc function() -&gt; Int {\n    // Perform some operations\n    return 0\n}\n\n/* The loop uses O(1) space */\nfunc loop(n: Int) {\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n\n/* The recursion uses O(n) space */\nfunc recur(n: Int) {\n    if n == 1 {\n        return\n    }\n    recur(n: n - 1)\n}\n</code></pre> <pre><code>function constFunc() {\n    // Perform some operations\n    return 0;\n}\n/* The loop uses O(1) space */\nfunction loop(n) {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* The recursion uses O(n) space */\nfunction recur(n) {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>function constFunc(): number {\n    // Perform some operations\n    return 0;\n}\n/* The loop uses O(1) space */\nfunction loop(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* The recursion uses O(n) space */\nfunction recur(n: number): void {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n  // Perform some operations\n  return 0;\n}\n/* The loop uses O(1) space */\nvoid loop(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n/* The recursion uses O(n) space */\nvoid recur(int n) {\n  if (n == 1) return;\n  recur(n - 1);\n}\n</code></pre> <pre><code>fn function() -&gt; i32 {\n    // Perform some operations\n    return 0;\n}\n/* The loop uses O(1) space */\nfn loop(n: i32) {\n    for i in 0..n {\n        function();\n    }\n}\n/* The recursion uses O(n) space */\nfn recur(n: i32) {\n    if n == 1 {\n        return;\n    }\n    recur(n - 1);\n}\n</code></pre> <pre><code>int func() {\n    // Perform some operations\n    return 0;\n}\n/* The loop uses O(1) space */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* The recursion uses O(n) space */\nvoid recur(int n) {\n    if (n == 1) return;\n    recur(n - 1);\n}\n</code></pre> <pre><code>fun function(): Int {\n    // Perform some operations\n    return 0\n}\n/* The loop uses O(1) space */\nfun loop(n: Int) {\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n/* The recursion uses O(n) space */\nfun recur(n: Int) {\n    if (n == 1) return\n    return recur(n - 1)\n}\n</code></pre> <pre><code>def function\n    # Perform some operations\n    0\nend\n\n### The loop uses O(1) space ###\ndef loop(n)\n    (0...n).each { function }\nend\n\n### The recursion uses O(n) space ###\ndef recur(n)\n    return if n == 1\n    recur(n - 1)\nend\n</code></pre> <p>Both <code>loop()</code> and <code>recur()</code> take \\(O(n)\\) time, but their space complexities differ.</p> <ul> <li><code>loop()</code> calls <code>function()</code> \\(n\\) times. Each call returns and releases its stack frame before the next iteration, so space complexity remains \\(O(1)\\).</li> <li>During <code>recur()</code>, \\(n\\) calls can remain active at once, using \\(O(n)\\) space for stack frames.</li> </ul>","path":["Chapter 2.   Complexity analysis","2.4   Space complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#243-common-complexity-classes","level":2,"title":"2.4.3   Common complexity classes","text":"<p>For input size \\(n\\), Figure 2-16 shows common space complexities in increasing order of growth.</p> \\[ \\begin{aligned} &amp; O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \\newline &amp; \\text{Constant} &lt; \\text{Logarithmic} &lt; \\text{Linear} &lt; \\text{Quadratic} &lt; \\text{Exponential} \\end{aligned} \\] <p></p> <p> Figure 2-16   Common space complexity classes </p>","path":["Chapter 2.   Complexity analysis","2.4   Space complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#1-constant-space-o1","level":3,"title":"1.   Constant space \\(O(1)\\)","text":"<p>Constant space commonly occurs when the number of constants, variables, and objects is independent of input size \\(n\\).</p> <p>Memory for variables initialized or functions called within a loop is released by the next iteration, so it does not accumulate. Space complexity remains \\(O(1)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def function() -&gt; int:\n    \"\"\"Function\"\"\"\n    # Perform some operations\n    return 0\n\ndef constant(n: int):\n    \"\"\"Constant complexity\"\"\"\n    # Constants, variables, and objects occupy O(1) space\n    a = 0\n    nums = [0] * 10000\n    node = ListNode(0)\n    # Variables in the loop occupy O(1) space\n    for _ in range(n):\n        c = 0\n    # Function calls in the loop occupy O(1) space\n    for _ in range(n):\n        function()\n</code></pre> space_complexity.cpp<pre><code>/* Function */\nint func() {\n    // Perform some operations\n    return 0;\n}\n\n/* Constant complexity */\nvoid constant(int n) {\n    // Constants, variables, and objects occupy O(1) space\n    const int a = 0;\n    int b = 0;\n    vector&lt;int&gt; nums(10000);\n    ListNode node(0);\n    // Variables in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // Function calls in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* Function */\nint function() {\n    // Perform some operations\n    return 0;\n}\n\n/* Constant complexity */\nvoid constant(int n) {\n    // Constants, variables, and objects occupy O(1) space\n    final int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new ListNode(0);\n    // Variables in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // Function calls in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* Function */\nint Function() {\n    // Perform some operations\n    return 0;\n}\n\n/* Constant complexity */\nvoid Constant(int n) {\n    // Constants, variables, and objects occupy O(1) space\n    int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new(0);\n    // Variables in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // Function calls in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* Function */\nfunc function() int {\n    // Perform some operations...\n    return 0\n}\n\n/* Constant complexity */\nfunc spaceConstant(n int) {\n    // Constants, variables, and objects occupy O(1) space\n    const a = 0\n    b := 0\n    nums := make([]int, 10000)\n    node := newNode(0)\n    // Variables in the loop occupy O(1) space\n    var c int\n    for i := 0; i &lt; n; i++ {\n        c = 0\n    }\n    // Function calls in the loop occupy O(1) space\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n    b += 0\n    c += 0\n    nums[0] = 0\n    node.val = 0\n}\n</code></pre> space_complexity.swift<pre><code>/* Function */\n@discardableResult\nfunc function() -&gt; Int {\n    // Perform some operations\n    return 0\n}\n\n/* Constant complexity */\nfunc constant(n: Int) {\n    // Constants, variables, and objects occupy O(1) space\n    let a = 0\n    var b = 0\n    let nums = Array(repeating: 0, count: 10000)\n    let node = ListNode(x: 0)\n    // Variables in the loop occupy O(1) space\n    for _ in 0 ..&lt; n {\n        let c = 0\n    }\n    // Function calls in the loop occupy O(1) space\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n</code></pre> space_complexity.js<pre><code>/* Function */\nfunction constFunc() {\n    // Perform some operations\n    return 0;\n}\n\n/* Constant complexity */\nfunction constant(n) {\n    // Constants, variables, and objects occupy O(1) space\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // Variables in the loop occupy O(1) space\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // Function calls in the loop occupy O(1) space\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* Function */\nfunction constFunc(): number {\n    // Perform some operations\n    return 0;\n}\n\n/* Constant complexity */\nfunction constant(n: number): void {\n    // Constants, variables, and objects occupy O(1) space\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // Variables in the loop occupy O(1) space\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // Function calls in the loop occupy O(1) space\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* Function */\nint function() {\n  // Perform some operations\n  return 0;\n}\n\n/* Constant complexity */\nvoid constant(int n) {\n  // Constants, variables, and objects occupy O(1) space\n  final int a = 0;\n  int b = 0;\n  List&lt;int&gt; nums = List.filled(10000, 0);\n  ListNode node = ListNode(0);\n  // Variables in the loop occupy O(1) space\n  for (var i = 0; i &lt; n; i++) {\n    int c = 0;\n  }\n  // Function calls in the loop occupy O(1) space\n  for (var i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* Function */\nfn function() -&gt; i32 {\n    // Perform some operations\n    return 0;\n}\n\n/* Constant complexity */\n#[allow(unused)]\nfn constant(n: i32) {\n    // Constants, variables, and objects occupy O(1) space\n    const A: i32 = 0;\n    let b = 0;\n    let nums = vec![0; 10000];\n    let node = ListNode::new(0);\n    // Variables in the loop occupy O(1) space\n    for i in 0..n {\n        let c = 0;\n    }\n    // Function calls in the loop occupy O(1) space\n    for i in 0..n {\n        function();\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* Function */\nint func() {\n    // Perform some operations\n    return 0;\n}\n\n/* Constant complexity */\nvoid constant(int n) {\n    // Constants, variables, and objects occupy O(1) space\n    const int a = 0;\n    int b = 0;\n    int nums[1000];\n    ListNode *node = newListNode(0);\n    free(node);\n    // Variables in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // Function calls in the loop occupy O(1) space\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* Function */\nfun function(): Int {\n    // Perform some operations\n    return 0\n}\n\n/* Constant complexity */\nfun constant(n: Int) {\n    // Constants, variables, and objects occupy O(1) space\n    val a = 0\n    var b = 0\n    val nums = Array(10000) { 0 }\n    val node = ListNode(0)\n    // Variables in the loop occupy O(1) space\n    for (i in 0..&lt;n) {\n        val c = 0\n    }\n    // Function calls in the loop occupy O(1) space\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### Function ###\ndef function\n  # Perform some operations\n  0\nend\n\n### Constant complexity ###\ndef constant(n)\n  # Constants, variables, and objects occupy O(1) space\n  a = 0\n  nums = [0] * 10000\n  node = ListNode.new\n\n  # Variables in the loop occupy O(1) space\n  (0...n).each { c = 0 }\n  # Function calls in the loop occupy O(1) space\n  (0...n).each { function }\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 2.   Complexity analysis","2.4   Space complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#2-linear-space-on","level":3,"title":"2.   Linear space \\(O(n)\\)","text":"<p>Linear space commonly occurs in arrays, linked lists, stacks, and queues whose element counts are proportional to \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def linear(n: int):\n    \"\"\"Linear complexity\"\"\"\n    # A list of length n occupies O(n) space\n    nums = [0] * n\n    # A hash table with n entries occupies O(n) space\n    hmap = dict[int, str]()\n    for i in range(n):\n        hmap[i] = str(i)\n</code></pre> space_complexity.cpp<pre><code>/* Linear complexity */\nvoid linear(int n) {\n    // An array of length n uses O(n) space\n    vector&lt;int&gt; nums(n);\n    // A list of length n occupies O(n) space\n    vector&lt;ListNode&gt; nodes;\n    for (int i = 0; i &lt; n; i++) {\n        nodes.push_back(ListNode(i));\n    }\n    // A hash table with n entries occupies O(n) space\n    unordered_map&lt;int, string&gt; map;\n    for (int i = 0; i &lt; n; i++) {\n        map[i] = to_string(i);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* Linear complexity */\nvoid linear(int n) {\n    // An array of length n uses O(n) space\n    int[] nums = new int[n];\n    // A list of length n occupies O(n) space\n    List&lt;ListNode&gt; nodes = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        nodes.add(new ListNode(i));\n    }\n    // A hash table with n entries occupies O(n) space\n    Map&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        map.put(i, String.valueOf(i));\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* Linear complexity */\nvoid Linear(int n) {\n    // An array of length n uses O(n) space\n    int[] nums = new int[n];\n    // A list of length n occupies O(n) space\n    List&lt;ListNode&gt; nodes = [];\n    for (int i = 0; i &lt; n; i++) {\n        nodes.Add(new ListNode(i));\n    }\n    // A hash table with n entries occupies O(n) space\n    Dictionary&lt;int, string&gt; map = [];\n    for (int i = 0; i &lt; n; i++) {\n        map.Add(i, i.ToString());\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* Linear complexity */\nfunc spaceLinear(n int) {\n    // An array of length n uses O(n) space\n    _ = make([]int, n)\n    // A list of length n occupies O(n) space\n    var nodes []*node\n    for i := 0; i &lt; n; i++ {\n        nodes = append(nodes, newNode(i))\n    }\n    // A hash table with n entries occupies O(n) space\n    m := make(map[int]string, n)\n    for i := 0; i &lt; n; i++ {\n        m[i] = strconv.Itoa(i)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* Linear complexity */\nfunc linear(n: Int) {\n    // An array of length n uses O(n) space\n    let nums = Array(repeating: 0, count: n)\n    // A list of length n occupies O(n) space\n    let nodes = (0 ..&lt; n).map { ListNode(x: $0) }\n    // A hash table with n entries occupies O(n) space\n    let map = Dictionary(uniqueKeysWithValues: (0 ..&lt; n).map { ($0, \"\\($0)\") })\n}\n</code></pre> space_complexity.js<pre><code>/* Linear complexity */\nfunction linear(n) {\n    // An array of length n uses O(n) space\n    const nums = new Array(n);\n    // A list of length n occupies O(n) space\n    const nodes = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // A hash table with n entries occupies O(n) space\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* Linear complexity */\nfunction linear(n: number): void {\n    // An array of length n uses O(n) space\n    const nums = new Array(n);\n    // A list of length n occupies O(n) space\n    const nodes: ListNode[] = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // A hash table with n entries occupies O(n) space\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* Linear complexity */\nvoid linear(int n) {\n  // An array of length n uses O(n) space\n  List&lt;int&gt; nums = List.filled(n, 0);\n  // A list of length n occupies O(n) space\n  List&lt;ListNode&gt; nodes = [];\n  for (var i = 0; i &lt; n; i++) {\n    nodes.add(ListNode(i));\n  }\n  // A hash table with n entries occupies O(n) space\n  Map&lt;int, String&gt; map = HashMap();\n  for (var i = 0; i &lt; n; i++) {\n    map.putIfAbsent(i, () =&gt; i.toString());\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* Linear complexity */\n#[allow(unused)]\nfn linear(n: i32) {\n    // An array of length n uses O(n) space\n    let mut nums = vec![0; n as usize];\n    // A list of length n occupies O(n) space\n    let mut nodes = Vec::new();\n    for i in 0..n {\n        nodes.push(ListNode::new(i))\n    }\n    // A hash table with n entries occupies O(n) space\n    let mut map = HashMap::new();\n    for i in 0..n {\n        map.insert(i, i.to_string());\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* Hash table */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // Implemented using uthash.h\n} HashTable;\n\n/* Linear complexity */\nvoid linear(int n) {\n    // An array of length n uses O(n) space\n    int *nums = malloc(sizeof(int) * n);\n    free(nums);\n\n    // A list of length n occupies O(n) space\n    ListNode **nodes = malloc(sizeof(ListNode *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        nodes[i] = newListNode(i);\n    }\n    // Free memory\n    for (int i = 0; i &lt; n; i++) {\n        free(nodes[i]);\n    }\n    free(nodes);\n\n    // A hash table with n entries occupies O(n) space\n    HashTable *h = NULL;\n    for (int i = 0; i &lt; n; i++) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = i;\n        tmp-&gt;val = i;\n        HASH_ADD_INT(h, key, tmp);\n    }\n\n    // Free memory\n    HashTable *curr, *tmp;\n    HASH_ITER(hh, h, curr, tmp) {\n        HASH_DEL(h, curr);\n        free(curr);\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* Linear complexity */\nfun linear(n: Int) {\n    // An array of length n uses O(n) space\n    val nums = Array(n) { 0 }\n    // A list of length n occupies O(n) space\n    val nodes = mutableListOf&lt;ListNode&gt;()\n    for (i in 0..&lt;n) {\n        nodes.add(ListNode(i))\n    }\n    // A hash table with n entries occupies O(n) space\n    val map = mutableMapOf&lt;Int, String&gt;()\n    for (i in 0..&lt;n) {\n        map[i] = i.toString()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### Linear complexity ###\ndef linear(n)\n  # A list of length n occupies O(n) space\n  nums = Array.new(n, 0)\n\n  # A hash table with n entries occupies O(n) space\n  hmap = {}\n  for i in 0...n\n    hmap[i] = i.to_s\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>As shown in Figure 2-17, this function has recursion depth \\(n\\), with \\(n\\) active calls to <code>linear_recur()</code> at once, using \\(O(n)\\) space for stack frames:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def linear_recur(n: int):\n    \"\"\"Linear complexity (recursive implementation)\"\"\"\n    print(\"Recursion: n =\", n)\n    if n == 1:\n        return\n    linear_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* Linear complexity (recursive implementation) */\nvoid linearRecur(int n) {\n    cout &lt;&lt; \"Recursion: n = \" &lt;&lt; n &lt;&lt; endl;\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* Linear complexity (recursive implementation) */\nvoid linearRecur(int n) {\n    System.out.println(\"Recursive call: n = \" + n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* Linear complexity (recursive implementation) */\nvoid LinearRecur(int n) {\n    Console.WriteLine(\"Recursive call: n = \" + n);\n    if (n == 1) return;\n    LinearRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* Linear complexity (recursive implementation) */\nfunc spaceLinearRecur(n int) {\n    fmt.Println(\"Recursive call: n =\", n)\n    if n == 1 {\n        return\n    }\n    spaceLinearRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* Linear complexity (recursive implementation) */\nfunc linearRecur(n: Int) {\n    print(\"Recursion: n = \\(n)\")\n    if n == 1 {\n        return\n    }\n    linearRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* Linear complexity (recursive implementation) */\nfunction linearRecur(n) {\n    console.log(`Recursive call with n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* Linear complexity (recursive implementation) */\nfunction linearRecur(n: number): void {\n    console.log(`Recursive call with n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* Linear complexity (recursive implementation) */\nvoid linearRecur(int n) {\n  print('Recursive call: n = $n');\n  if (n == 1) return;\n  linearRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* Linear complexity (recursive implementation) */\nfn linear_recur(n: i32) {\n    println!(\"Recursive call n = {}\", n);\n    if n == 1 {\n        return;\n    };\n    linear_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* Linear complexity (recursive implementation) */\nvoid linearRecur(int n) {\n    printf(\"Recursive call n = %d\\r\\n\", n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.kt<pre><code>/* Linear complexity (recursive implementation) */\nfun linearRecur(n: Int) {\n    println(\"Recursive call: n = $n\")\n    if (n == 1)\n        return\n    linearRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>### Linear complexity (recursive implementation)###\ndef linear_recur(n)\n  puts \"Recursion: n = #{n}\"\n  return if n == 1\n  linear_recur(n - 1)\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p></p> <p> Figure 2-17   Linear space complexity from recursion </p>","path":["Chapter 2.   Complexity analysis","2.4   Space complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#3-quadratic-space-on2","level":3,"title":"3.   Quadratic space \\(O(n^2)\\)","text":"<p>Quadratic space commonly occurs in matrices and graphs whose element counts grow with the square of \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def quadratic(n: int):\n    \"\"\"Quadratic complexity\"\"\"\n    # The two-dimensional list occupies O(n^2) space\n    num_matrix = [[0] * n for _ in range(n)]\n</code></pre> space_complexity.cpp<pre><code>/* Quadratic complexity */\nvoid quadratic(int n) {\n    // The two-dimensional list occupies O(n^2) space\n    vector&lt;vector&lt;int&gt;&gt; numMatrix;\n    for (int i = 0; i &lt; n; i++) {\n        vector&lt;int&gt; tmp;\n        for (int j = 0; j &lt; n; j++) {\n            tmp.push_back(0);\n        }\n        numMatrix.push_back(tmp);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* Quadratic complexity */\nvoid quadratic(int n) {\n    // The matrix uses O(n^2) space\n    int[][] numMatrix = new int[n][n];\n    // The two-dimensional list occupies O(n^2) space\n    List&lt;List&lt;Integer&gt;&gt; numList = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            tmp.add(0);\n        }\n        numList.add(tmp);\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* Quadratic complexity */\nvoid Quadratic(int n) {\n    // The matrix uses O(n^2) space\n    int[,] numMatrix = new int[n, n];\n    // The two-dimensional list occupies O(n^2) space\n    List&lt;List&lt;int&gt;&gt; numList = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;int&gt; tmp = [];\n        for (int j = 0; j &lt; n; j++) {\n            tmp.Add(0);\n        }\n        numList.Add(tmp);\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* Quadratic complexity */\nfunc spaceQuadratic(n int) {\n    // The matrix uses O(n^2) space\n    numMatrix := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        numMatrix[i] = make([]int, n)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* Quadratic complexity */\nfunc quadratic(n: Int) {\n    // A fully materialized two-dimensional list uses O(n^2) space; Swift may share repeated row storage\n    let numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n</code></pre> space_complexity.js<pre><code>/* Quadratic complexity */\nfunction quadratic(n) {\n    // The matrix uses O(n^2) space\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // The two-dimensional list occupies O(n^2) space\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* Quadratic complexity */\nfunction quadratic(n: number): void {\n    // The matrix uses O(n^2) space\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // The two-dimensional list occupies O(n^2) space\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* Quadratic complexity */\nvoid quadratic(int n) {\n  // The matrix uses O(n^2) space\n  List&lt;List&lt;int&gt;&gt; numMatrix = List.generate(n, (_) =&gt; List.filled(n, 0));\n  // The two-dimensional list occupies O(n^2) space\n  List&lt;List&lt;int&gt;&gt; numList = [];\n  for (var i = 0; i &lt; n; i++) {\n    List&lt;int&gt; tmp = [];\n    for (int j = 0; j &lt; n; j++) {\n      tmp.add(0);\n    }\n    numList.add(tmp);\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* Quadratic complexity */\n#[allow(unused)]\nfn quadratic(n: i32) {\n    // The matrix uses O(n^2) space\n    let num_matrix = vec![vec![0; n as usize]; n as usize];\n    // The two-dimensional list occupies O(n^2) space\n    let mut num_list = Vec::new();\n    for i in 0..n {\n        let mut tmp = Vec::new();\n        for j in 0..n {\n            tmp.push(0);\n        }\n        num_list.push(tmp);\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* Quadratic complexity */\nvoid quadratic(int n) {\n    // The two-dimensional list occupies O(n^2) space\n    int **numMatrix = malloc(sizeof(int *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        int *tmp = malloc(sizeof(int) * n);\n        for (int j = 0; j &lt; n; j++) {\n            tmp[j] = 0;\n        }\n        numMatrix[i] = tmp;\n    }\n\n    // Free memory\n    for (int i = 0; i &lt; n; i++) {\n        free(numMatrix[i]);\n    }\n    free(numMatrix);\n}\n</code></pre> space_complexity.kt<pre><code>/* Quadratic complexity */\nfun quadratic(n: Int) {\n    // A full n*n matrix uses O(n^2) space; only its outer array is allocated here\n    val numMatrix = arrayOfNulls&lt;Array&lt;Int&gt;?&gt;(n)\n    // The two-dimensional list occupies O(n^2) space\n    val numList = mutableListOf&lt;MutableList&lt;Int&gt;&gt;()\n    for (i in 0..&lt;n) {\n        val tmp = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            tmp.add(0)\n        }\n        numList.add(tmp)\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### Quadratic complexity ###\ndef quadratic(n)\n  # The two-dimensional list occupies O(n^2) space\n  Array.new(n) { Array.new(n, 0) }\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>As shown in Figure 2-18, the recursion depth is \\(n\\), and each call initializes an array. Their lengths are \\(n\\), \\(n-1\\), \\(\\dots\\), \\(2\\), and \\(1\\), averaging \\(n / 2\\), so total space is \\(O(n^2)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def quadratic_recur(n: int) -&gt; int:\n    \"\"\"Quadratic complexity (recursive implementation)\"\"\"\n    if n &lt;= 0:\n        return 0\n    # The lengths of array nums are n, n-1, ..., 2, 1\n    nums = [0] * n\n    return quadratic_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* Quadratic complexity (recursive implementation) */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    vector&lt;int&gt; nums(n);\n    cout &lt;&lt; \"Recursion: n = \" &lt;&lt; n &lt;&lt; \" , length of nums = \" &lt;&lt; nums.size() &lt;&lt; endl;\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* Quadratic complexity (recursive implementation) */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    // The lengths of array nums are n, n-1, ..., 2, 1\n    int[] nums = new int[n];\n    System.out.println(\"Recursive call: n = \" + n + \" with nums length = \" + nums.length);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* Quadratic complexity (recursive implementation) */\nint QuadraticRecur(int n) {\n    if (n &lt;= 0) return 0;\n    int[] nums = new int[n];\n    Console.WriteLine(\"Recursive call: n = \" + n + \" with nums length = \" + nums.Length);\n    return QuadraticRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* Quadratic complexity (recursive implementation) */\nfunc spaceQuadraticRecur(n int) int {\n    if n &lt;= 0 {\n        return 0\n    }\n    nums := make([]int, n)\n    fmt.Printf(\"Recursive call n = %d: length of nums = %d \\n\", n, len(nums))\n    return spaceQuadraticRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* Quadratic complexity (recursive implementation) */\n@discardableResult\nfunc quadraticRecur(n: Int) -&gt; Int {\n    if n &lt;= 0 {\n        return 0\n    }\n    // The lengths of array nums are n, n-1, ..., 2, 1\n    let nums = Array(repeating: 0, count: n)\n    print(\"At recursion n = \\(n), nums has length \\(nums.count)\")\n    return quadraticRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* Quadratic complexity (recursive implementation) */\nfunction quadraticRecur(n) {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`In the recursive call with n = ${n}, nums has length ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* Quadratic complexity (recursive implementation) */\nfunction quadraticRecur(n: number): number {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`In the recursive call with n = ${n}, nums has length ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* Quadratic complexity (recursive implementation) */\nint quadraticRecur(int n) {\n  if (n &lt;= 0) return 0;\n  List&lt;int&gt; nums = List.filled(n, 0);\n  print('In the recursive call with n = $n, length of nums = ${nums.length}');\n  return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* Quadratic complexity (recursive implementation) */\nfn quadratic_recur(n: i32) -&gt; i32 {\n    if n &lt;= 0 {\n        return 0;\n    };\n    // The lengths of array nums are n, n-1, ..., 2, 1\n    let nums = vec![0; n as usize];\n    println!(\"In recursive call n = {}, nums has length {}\", n, nums.len());\n    return quadratic_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* Quadratic complexity (recursive implementation) */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    int *nums = malloc(sizeof(int) * n);\n    printf(\"In recursive call n = %d, nums has length = %d\\r\\n\", n, n);\n    int res = quadraticRecur(n - 1);\n    free(nums);\n    return res;\n}\n</code></pre> space_complexity.kt<pre><code>/* Quadratic allocation total (recursive example; tailrec removes recursive stack growth) */\ntailrec fun quadraticRecur(n: Int): Int {\n    if (n &lt;= 0)\n        return 0\n    // The lengths of array nums are n, n-1, ..., 2, 1\n    val nums = Array(n) { 0 }\n    println(\"In the recursive call with n = $n, nums has length ${nums.size}\")\n    return quadraticRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>### Quadratic complexity (recursive implementation)###\ndef quadratic_recur(n)\n  return 0 unless n &gt; 0\n\n  # The lengths of array nums are n, n-1, ..., 2, 1\n  nums = Array.new(n, 0)\n  quadratic_recur(n - 1)\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p></p> <p> Figure 2-18   Quadratic space complexity from recursion </p>","path":["Chapter 2.   Complexity analysis","2.4   Space complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#4-exponential-space-o2n","level":3,"title":"4.   Exponential space \\(O(2^n)\\)","text":"<p>Exponential space commonly occurs in binary trees. In Figure 2-19, a perfect binary tree with \\(n\\) levels contains \\(2^n - 1\\) nodes and uses \\(O(2^n)\\) space:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def build_tree(n: int) -&gt; TreeNode | None:\n    \"\"\"Exponential complexity (building a perfect binary tree)\"\"\"\n    if n == 0:\n        return None\n    root = TreeNode(0)\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n    return root\n</code></pre> space_complexity.cpp<pre><code>/* Exponential complexity (building a perfect binary tree) */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return nullptr;\n    TreeNode *root = new TreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.java<pre><code>/* Exponential complexity (building a perfect binary tree) */\nTreeNode buildTree(int n) {\n    if (n == 0)\n        return null;\n    TreeNode root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.cs<pre><code>/* Exponential complexity (building a perfect binary tree) */\nTreeNode? BuildTree(int n) {\n    if (n == 0) return null;\n    TreeNode root = new(0) {\n        left = BuildTree(n - 1),\n        right = BuildTree(n - 1)\n    };\n    return root;\n}\n</code></pre> space_complexity.go<pre><code>/* Exponential complexity (building a perfect binary tree) */\nfunc buildTree(n int) *TreeNode {\n    if n == 0 {\n        return nil\n    }\n    root := NewTreeNode(0)\n    root.Left = buildTree(n - 1)\n    root.Right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.swift<pre><code>/* Exponential complexity (building a perfect binary tree) */\nfunc buildTree(n: Int) -&gt; TreeNode? {\n    if n == 0 {\n        return nil\n    }\n    let root = TreeNode(x: 0)\n    root.left = buildTree(n: n - 1)\n    root.right = buildTree(n: n - 1)\n    return root\n}\n</code></pre> space_complexity.js<pre><code>/* Exponential complexity (building a perfect binary tree) */\nfunction buildTree(n) {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.ts<pre><code>/* Exponential complexity (building a perfect binary tree) */\nfunction buildTree(n: number): TreeNode | null {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.dart<pre><code>/* Exponential complexity (building a perfect binary tree) */\nTreeNode? buildTree(int n) {\n  if (n == 0) return null;\n  TreeNode root = TreeNode(0);\n  root.left = buildTree(n - 1);\n  root.right = buildTree(n - 1);\n  return root;\n}\n</code></pre> space_complexity.rs<pre><code>/* Exponential complexity (building a perfect binary tree) */\nfn build_tree(n: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    if n == 0 {\n        return None;\n    };\n    let root = TreeNode::new(0);\n    root.borrow_mut().left = build_tree(n - 1);\n    root.borrow_mut().right = build_tree(n - 1);\n    return Some(root);\n}\n</code></pre> space_complexity.c<pre><code>/* Exponential complexity (building a perfect binary tree) */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return NULL;\n    TreeNode *root = newTreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.kt<pre><code>/* Exponential complexity (building a perfect binary tree) */\nfun buildTree(n: Int): TreeNode? {\n    if (n == 0)\n        return null\n    val root = TreeNode(0)\n    root.left = buildTree(n - 1)\n    root.right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.rb<pre><code>### Exponential complexity (building a perfect binary tree)###\ndef build_tree(n)\n  return if n == 0\n\n  TreeNode.new.tap do |root|\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p></p> <p> Figure 2-19   Exponential space used by a perfect binary tree </p>","path":["Chapter 2.   Complexity analysis","2.4   Space complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#5-logarithmic-space-olog-n","level":3,"title":"5.   Logarithmic space \\(O(\\log n)\\)","text":"<p>Logarithmic space often appears in divide-and-conquer algorithms. For example, merge sort splits an array of length \\(n\\) in half at each recursive step, producing a recursion tree of height \\(\\log n\\) and using \\(O(\\log n)\\) space for stack frames.</p> <p>Another example is converting a number to a string. A positive integer \\(n\\) has \\(\\lfloor \\log_{10} n \\rfloor + 1\\) digits, giving a string of length \\(\\lfloor \\log_{10} n \\rfloor + 1\\). The space complexity is therefore \\(O(\\log_{10} n + 1) = O(\\log n)\\).</p>","path":["Chapter 2.   Complexity analysis","2.4   Space complexity"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#244-balancing-time-and-space","level":2,"title":"2.4.4   Balancing time and space","text":"<p>Ideally, we would minimize both time and space complexity. In practice, optimizing both simultaneously is usually difficult.</p> <p>Reducing time complexity often increases space complexity, and vice versa. Using more memory to run faster is called trading space for time; accepting slower execution to use less memory is trading time for space.</p> <p>The choice depends on which resource matters more. Time is usually more valuable than space, so trading space for time is common. For very large datasets, however, keeping space complexity under control is also crucial.</p>","path":["Chapter 2.   Complexity analysis","2.4   Space complexity"],"tags":[]},{"location":"chapter_computational_complexity/summary/","level":1,"title":"2.5   Summary","text":"","path":["Chapter 2.   Complexity analysis","2.5   Summary"],"tags":[]},{"location":"chapter_computational_complexity/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<p>Evaluating algorithm efficiency</p> <ul> <li>Time and space efficiency are the two main measures of algorithm quality.</li> <li>Practical testing measures efficiency, but cannot easily remove environmental effects and consumes substantial computing resources.</li> <li>Complexity analysis avoids those limitations, applies across platforms, and reveals efficiency at different input sizes.</li> </ul> <p>Time complexity</p> <ul> <li>Time complexity describes how runtime grows with input size. It is useful for evaluating efficiency, but may not distinguish algorithms accurately for small inputs or equal complexity classes.</li> <li>Worst-case time complexity is expressed with big-\\(O\\) notation, an asymptotic upper bound describing the growth of the operation count \\(T(n)\\) as \\(n\\) tends to positive infinity.</li> <li>Deriving time complexity takes two steps: count operations, then determine the asymptotic upper bound.</li> <li>Common complexities, from slower to faster growth, include \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n \\log n)\\), \\(O(n^2)\\), \\(O(2^n)\\), and \\(O(n!)\\).</li> <li>Some algorithms' complexity depends on the input's arrangement. We distinguish worst-case, best-case, and average-case complexity. Best-case complexity is rarely used because reaching it generally requires strict input conditions.</li> <li>Average-case complexity describes performance on random inputs and is closest to typical practical behavior. Calculating it requires an input distribution and its mathematical expectation.</li> </ul> <p>Space complexity</p> <ul> <li>Like time complexity, space complexity describes a growth trend: how memory usage increases with input size.</li> <li>Memory used during execution includes input, temporary, and output space. Input space is usually excluded from space complexity. Temporary space includes temporary data, stack frames, and instructions; stack frames usually affect complexity only in recursive functions.</li> <li>We usually consider worst-case space complexity, measuring memory under the worst input and at the moment of maximum usage.</li> <li>Common space complexities, from slower to faster growth, include \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n^2)\\), and \\(O(2^n)\\).</li> </ul>","path":["Chapter 2.   Complexity analysis","2.5   Summary"],"tags":[]},{"location":"chapter_computational_complexity/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: Does tail recursion have \\(O(1)\\) space complexity?</p> <p>In theory, tail recursion can be optimized to \\(O(1)\\) space. However, most languages, including Java, Python, C++, Go, and C#, do not automatically optimize tail recursion, so its space complexity is usually treated as \\(O(n)\\).</p> <p>Q: What is the difference between a function and a method?</p> <p>A function can run independently, with all arguments passed explicitly. A method is associated with an object, receives that object implicitly when called, and can operate on the instance's data.</p> <p>Here are examples from several common languages.</p> <ul> <li>C is procedural and has no built-in object-oriented model, so it has functions only. We can simulate object-oriented programming with structures (<code>struct</code>); functions associated with a structure then act like methods in other languages.</li> <li>Java and C# are object-oriented, and executable blocks (methods) are usually part of a class. Static methods behave like functions: they belong to the class and cannot access a particular instance's variables directly.</li> <li>C++ and Python support both procedural programming with functions and object-oriented programming with methods.</li> </ul> <p>Q: Does the common space complexity chart show absolute memory usage?</p> <p>No. It illustrates growth trends rather than absolute amounts of memory.</p> <p>For \\(n = 8\\), the plotted values may not match the corresponding functions exactly. Each curve includes a constant term that adjusts it to a visually convenient range.</p> <p>In practice, these constant terms are usually unknown, so complexity alone cannot generally identify the best solution for \\(n = 8\\). For \\(n = 8^5\\), the choice is easier because growth rates dominate.</p> <p>Q: Do algorithms sometimes trade time for space, or space for time, depending on the application?</p> <p>Most practical applications trade space for time. Database indexes, for example, often use B+ trees or hash indexes, consuming substantial memory to achieve \\(O(\\log n)\\) or even \\(O(1)\\) query time.</p> <p>When memory is scarce, the reverse tradeoff may be useful. In embedded development, an engineer might replace a hash table with sequential array search to save memory, accepting slower lookups.</p>","path":["Chapter 2.   Complexity analysis","2.5   Summary"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/","level":1,"title":"2.3   Time complexity","text":"<p>Runtime gives a direct, accurate picture of an algorithm's efficiency. How might we predict the runtime of a piece of code precisely?</p> <ol> <li>Specify the execution platform, including hardware, programming language, and system environment, all of which affect performance.</li> <li>Determine the time required by each operation: for example, addition <code>+</code> might take 1 ns, multiplication <code>*</code> 10 ns, and <code>print()</code> 5 ns.</li> <li>Count every operation in the code and sum their execution times to obtain the total runtime.</li> </ol> <p>For example, consider the following code with input size \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># On a particular execution platform\ndef algorithm(n: int):\n    a = 2      # 1 ns\n    a = a + 1  # 1 ns\n    a = a * 2  # 10 ns\n    # Repeat n times\n    for _ in range(n):  # 1 ns\n        print(0)        # 5 ns\n</code></pre> <pre><code>// On a particular execution platform\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Repeat n times\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        cout &lt;&lt; 0 &lt;&lt; endl;         // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a particular execution platform\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Repeat n times\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        System.out.println(0);     // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a particular execution platform\nvoid Algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Repeat n times\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        Console.WriteLine(0);      // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a particular execution platform\nfunc algorithm(n int) {\n    a := 2     // 1 ns\n    a = a + 1  // 1 ns\n    a = a * 2  // 10 ns\n    // Repeat n times\n    for i := 0; i &lt; n; i++ {  // 1 ns\n        fmt.Println(a)        // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a particular execution platform\nfunc algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // Repeat n times\n    for _ in 0 ..&lt; n { // 1 ns\n        print(0) // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a particular execution platform\nfunction algorithm(n) {\n    var a = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // Repeat n times\n    for(let i = 0; i &lt; n; i++) { // 1 ns\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a particular execution platform\nfunction algorithm(n: number): void {\n    var a: number = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // Repeat n times\n    for(let i = 0; i &lt; n; i++) { // 1 ns\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a particular execution platform\nvoid algorithm(int n) {\n  int a = 2; // 1 ns\n  a = a + 1; // 1 ns\n  a = a * 2; // 10 ns\n  // Repeat n times\n  for (int i = 0; i &lt; n; i++) { // 1 ns\n    print(0); // 5 ns\n  }\n}\n</code></pre> <pre><code>// On a particular execution platform\nfn algorithm(n: i32) {\n    let mut a = 2;      // 1 ns\n    a = a + 1;          // 1 ns\n    a = a * 2;          // 10 ns\n    // Repeat n times\n    for _ in 0..n {     // 1 ns\n        println!(\"{}\", 0);  // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a particular execution platform\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // Repeat n times\n    for (int i = 0; i &lt; n; i++) {   // 1 ns\n        printf(\"%d\", 0);            // 5 ns\n    }\n}\n</code></pre> <pre><code>// On a particular execution platform\nfun algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // Repeat n times\n    for (i in 0..&lt;n) {  // 1 ns\n        println(0)      // 5 ns\n    }\n}\n</code></pre> <pre><code># On a particular execution platform\ndef algorithm(n)\n    a = 2       # 1 ns\n    a = a + 1   # 1 ns\n    a = a * 2   # 10 ns\n    # Repeat n times\n    (0...n).each do # 1 ns\n        puts 0      # 5 ns\n    end\nend\n</code></pre> <p>This gives a runtime of \\((6n + 12)\\) ns:</p> \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] <p>In practice, calculating runtime this way is neither suitable nor realistic. We do not want estimates tied to one platform, since algorithms run on many platforms. Also, the time taken by each operation is difficult to determine, making accurate prediction very hard.</p>","path":["Chapter 2.   Complexity analysis","2.3   Time complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#231-measuring-growth-in-runtime","level":2,"title":"2.3.1   Measuring growth in runtime","text":"<p>Time complexity analysis measures how runtime grows as the input gets larger, rather than the runtime itself.</p> <p>To make this abstract idea concrete, consider three algorithms, <code>A</code>, <code>B</code>, and <code>C</code>, with input size \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># Algorithm A: constant time complexity\ndef algorithm_A(n: int):\n    print(0)\n# Algorithm B: linear time complexity\ndef algorithm_B(n: int):\n    for _ in range(n):\n        print(0)\n# Algorithm C: constant time complexity\ndef algorithm_C(n: int):\n    for _ in range(1000000):\n        print(0)\n</code></pre> <pre><code>// Algorithm A: constant time complexity\nvoid algorithm_A(int n) {\n    cout &lt;&lt; 0 &lt;&lt; endl;\n}\n// Algorithm B: linear time complexity\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n// Algorithm C: constant time complexity\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n</code></pre> <pre><code>// Algorithm A: constant time complexity\nvoid algorithm_A(int n) {\n    System.out.println(0);\n}\n// Algorithm B: linear time complexity\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        System.out.println(0);\n    }\n}\n// Algorithm C: constant time complexity\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        System.out.println(0);\n    }\n}\n</code></pre> <pre><code>// Algorithm A: constant time complexity\nvoid AlgorithmA(int n) {\n    Console.WriteLine(0);\n}\n// Algorithm B: linear time complexity\nvoid AlgorithmB(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Console.WriteLine(0);\n    }\n}\n// Algorithm C: constant time complexity\nvoid AlgorithmC(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        Console.WriteLine(0);\n    }\n}\n</code></pre> <pre><code>// Algorithm A: constant time complexity\nfunc algorithm_A(n int) {\n    fmt.Println(0)\n}\n// Algorithm B: linear time complexity\nfunc algorithm_B(n int) {\n    for i := 0; i &lt; n; i++ {\n        fmt.Println(0)\n    }\n}\n// Algorithm C: constant time complexity\nfunc algorithm_C(n int) {\n    for i := 0; i &lt; 1000000; i++ {\n        fmt.Println(0)\n    }\n}\n</code></pre> <pre><code>// Algorithm A: constant time complexity\nfunc algorithmA(n: Int) {\n    print(0)\n}\n\n// Algorithm B: linear time complexity\nfunc algorithmB(n: Int) {\n    for _ in 0 ..&lt; n {\n        print(0)\n    }\n}\n\n// Algorithm C: constant time complexity\nfunc algorithmC(n: Int) {\n    for _ in 0 ..&lt; 1_000_000 {\n        print(0)\n    }\n}\n</code></pre> <pre><code>// Algorithm A: constant time complexity\nfunction algorithm_A(n) {\n    console.log(0);\n}\n// Algorithm B: linear time complexity\nfunction algorithm_B(n) {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// Algorithm C: constant time complexity\nfunction algorithm_C(n) {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// Algorithm A: constant time complexity\nfunction algorithm_A(n: number): void {\n    console.log(0);\n}\n// Algorithm B: linear time complexity\nfunction algorithm_B(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// Algorithm C: constant time complexity\nfunction algorithm_C(n: number): void {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// Algorithm A: constant time complexity\nvoid algorithmA(int n) {\n  print(0);\n}\n// Algorithm B: linear time complexity\nvoid algorithmB(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    print(0);\n  }\n}\n// Algorithm C: constant time complexity\nvoid algorithmC(int n) {\n  for (int i = 0; i &lt; 1000000; i++) {\n    print(0);\n  }\n}\n</code></pre> <pre><code>// Algorithm A: constant time complexity\nfn algorithm_A(n: i32) {\n    println!(\"{}\", 0);\n}\n// Algorithm B: linear time complexity\nfn algorithm_B(n: i32) {\n    for _ in 0..n {\n        println!(\"{}\", 0);\n    }\n}\n// Algorithm C: constant time complexity\nfn algorithm_C(n: i32) {\n    for _ in 0..1000000 {\n        println!(\"{}\", 0);\n    }\n}\n</code></pre> <pre><code>// Algorithm A: constant time complexity\nvoid algorithm_A(int n) {\n    printf(\"%d\", 0);\n}\n// Algorithm B: linear time complexity\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        printf(\"%d\", 0);\n    }\n}\n// Algorithm C: constant time complexity\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        printf(\"%d\", 0);\n    }\n}\n</code></pre> <pre><code>// Algorithm A: constant time complexity\nfun algoritm_A(n: Int) {\n    println(0)\n}\n// Algorithm B: linear time complexity\nfun algorithm_B(n: Int) {\n    for (i in 0..&lt;n){\n        println(0)\n    }\n}\n// Algorithm C: constant time complexity\nfun algorithm_C(n: Int) {\n    for (i in 0..&lt;1000000) {\n        println(0)\n    }\n}\n</code></pre> <pre><code># Algorithm A: constant time complexity\ndef algorithm_A(n)\n    puts 0\nend\n\n# Algorithm B: linear time complexity\ndef algorithm_B(n)\n    (0...n).each { puts 0 }\nend\n\n# Algorithm C: constant time complexity\ndef algorithm_C(n)\n    (0...1_000_000).each { puts 0 }\nend\n</code></pre> <p>Figure 2-7 shows their time complexities.</p> <ul> <li>Algorithm <code>A</code> performs \\(1\\) print operation. Its runtime does not grow with \\(n\\), so it has constant time complexity.</li> <li>Algorithm <code>B</code> prints \\(n\\) times. Its runtime grows linearly with \\(n\\), so it has linear time complexity.</li> <li>Algorithm <code>C</code> prints \\(1000000\\) times. It may take a long time, but that time is independent of input size \\(n\\). Its complexity is therefore constant, like <code>A</code>.</li> </ul> <p></p> <p> Figure 2-7   Runtime growth of algorithms A, B, and C </p> <p>How does time complexity analysis differ from calculating runtime directly?</p> <ul> <li>Time complexity evaluates efficiency effectively. Algorithm <code>B</code> grows linearly, becoming slower than <code>A</code> when \\(n &gt; 1\\) and slower than <code>C</code> when \\(n &gt; 1000000\\). With sufficiently large input size \\(n\\), a constant-time algorithm will outperform a linear-time one. This is what a growth trend tells us.</li> <li>Time complexity is easier to derive. The execution platform and kinds of elementary operations do not determine the growth trend. We can treat each elementary operation as taking the same unit of time, reducing runtime estimation to operation counting and greatly simplifying analysis.</li> <li>Time complexity has limitations. Algorithms <code>A</code> and <code>C</code> have the same complexity but very different runtimes. Although <code>B</code> has higher complexity than <code>C</code>, it is much faster for small \\(n\\). Complexity alone may not settle such comparisons. Even so, it remains the most effective and widely used way to evaluate algorithm efficiency.</li> </ul>","path":["Chapter 2.   Complexity analysis","2.3   Time complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#232-asymptotic-upper-bounds","level":2,"title":"2.3.2   Asymptotic upper bounds","text":"<p>Consider this function with input size \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def algorithm(n: int):\n    a = 1      # +1\n    a = a + 1  # +1\n    a = a * 2  # +1\n    # Repeat n times\n    for i in range(n):  # +1\n        print(0)        # +1\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Repeat n times\n    for (int i = 0; i &lt; n; i++) { // +1 (i ++ executes each iteration)\n        cout &lt;&lt; 0 &lt;&lt; endl;    // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Repeat n times\n    for (int i = 0; i &lt; n; i++) { // +1 (i ++ executes each iteration)\n        System.out.println(0);    // +1\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Repeat n times\n    for (int i = 0; i &lt; n; i++) {   // +1 (i ++ executes each iteration)\n        Console.WriteLine(0);   // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1      // +1\n    a = a + 1   // +1\n    a = a * 2   // +1\n    // Repeat n times\n    for i := 0; i &lt; n; i++ {   // +1\n        fmt.Println(a)         // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // Repeat n times\n    for _ in 0 ..&lt; n { // +1\n        print(0) // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    var a = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // Repeat n times\n    for(let i = 0; i &lt; n; i++){ // +1 (i ++ executes each iteration)\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void{\n    var a: number = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // Repeat n times\n    for(let i = 0; i &lt; n; i++){ // +1 (i ++ executes each iteration)\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +1\n  a = a + 1; // +1\n  a = a * 2; // +1\n  // Repeat n times\n  for (int i = 0; i &lt; n; i++) { // +1 (i ++ executes each iteration)\n    print(0); // +1\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;   // +1\n    a = a + 1;      // +1\n    a = a * 2;      // +1\n\n    // Repeat n times\n    for _ in 0..n { // +1 (i ++ executes each iteration)\n        println!(\"{}\", 0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // Repeat n times\n    for (int i = 0; i &lt; n; i++) {   // +1 (i ++ executes each iteration)\n        printf(\"%d\", 0);            // +1\n    }\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // Repeat n times\n    for (i in 0..&lt;n) { // +1 (i ++ executes each iteration)\n        println(0) // +1\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 1       # +1\n    a = a + 1   # +1\n    a = a * 2   # +1\n    # Repeat n times\n    (0...n).each do # +1\n        puts 0      # +1\n    end\nend\n</code></pre> <p>Let \\(T(n)\\) be the operation count as a function of input size \\(n\\). For the function above:</p> \\[ T(n) = 3 + 2n \\] <p>Since \\(T(n)\\) is a first-degree function, runtime grows linearly, giving linear time complexity.</p> <p>We write linear time complexity as \\(O(n)\\). This is called big-\\(O\\) notation, and it gives an asymptotic upper bound on \\(T(n)\\).</p> <p>Time complexity analysis essentially finds an asymptotic upper bound on the operation count \\(T(n)\\). It has a precise mathematical definition.</p> <p>Asymptotic upper bound</p> <p>If a positive real constant \\(c\\) and a real number \\(n_0\\) exist such that \\(T(n) \\leq c \\cdot f(n)\\) for all \\(n &gt; n_0\\), then \\(f(n)\\) is an asymptotic upper bound on \\(T(n)\\), written \\(T(n) = O(f(n))\\).</p> <p>As shown in Figure 2-8, we seek a function \\(f(n)\\) such that, as \\(n\\) tends to infinity, \\(T(n)\\) and \\(f(n)\\) have the same order of growth, differing only by a constant factor \\(c\\).</p> <p></p> <p> Figure 2-8   Asymptotic upper bound of a function </p>","path":["Chapter 2.   Complexity analysis","2.3   Time complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#233-how-to-derive-time-complexity","level":2,"title":"2.3.3   How to derive time complexity","text":"<p>The mathematics of asymptotic bounds may feel unfamiliar. Do not worry if it is not fully clear yet: learn the practical method first, and its meaning will become clearer with practice.</p> <p>Once we identify \\(f(n)\\), the time complexity is \\(O(f(n))\\). Finding the asymptotic upper bound \\(f(n)\\) takes two steps: count operations, then determine the bound.</p>","path":["Chapter 2.   Complexity analysis","2.3   Time complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#1-step-1-count-operations","level":3,"title":"1.   Step 1: Count operations","text":"<p>We can count code operations line by line. Since the constant factor \\(c\\) in \\(c \\cdot f(n)\\) may be arbitrarily large, coefficients and constant terms in \\(T(n)\\) can be ignored. This gives three useful shortcuts.</p> <ol> <li>Ignore constant terms in \\(T(n)\\). They are independent of \\(n\\) and do not affect complexity.</li> <li>Drop coefficients. For example, \\(2n\\) or \\(5n + 1\\) iterations can be counted simply as \\(n\\), because the coefficient of \\(n\\) does not affect complexity.</li> <li>Multiply for nested loops. Multiply the outer and inner iteration counts, applying shortcuts <code>1.</code> and <code>2.</code> to each level.</li> </ol> <p>We can apply these shortcuts to the following function:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def algorithm(n: int):\n    a = 1      # +0 (shortcut 1)\n    a = a + n  # +0 (shortcut 1)\n    # +n (shortcut 2)\n    for i in range(5 * n + 1):\n        print(0)\n    # +n*n (shortcut 3)\n    for i in range(2 * n):\n        for j in range(n + 1):\n            print(0)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0 (shortcut 1)\n    a = a + n;  // +0 (shortcut 1)\n    // +n (shortcut 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n    // +n*n (shortcut 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            cout &lt;&lt; 0 &lt;&lt; endl;\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0 (shortcut 1)\n    a = a + n;  // +0 (shortcut 1)\n    // +n (shortcut 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        System.out.println(0);\n    }\n    // +n*n (shortcut 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            System.out.println(0);\n        }\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +0 (shortcut 1)\n    a = a + n;  // +0 (shortcut 1)\n    // +n (shortcut 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        Console.WriteLine(0);\n    }\n    // +n*n (shortcut 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            Console.WriteLine(0);\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1     // +0 (shortcut 1)\n    a = a + n  // +0 (shortcut 1)\n    // +n (shortcut 2)\n    for i := 0; i &lt; 5 * n + 1; i++ {\n        fmt.Println(0)\n    }\n    // +n*n (shortcut 3)\n    for i := 0; i &lt; 2 * n; i++ {\n        for j := 0; j &lt; n + 1; j++ {\n            fmt.Println(0)\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +0 (shortcut 1)\n    a = a + n // +0 (shortcut 1)\n    // +n (shortcut 2)\n    for _ in 0 ..&lt; (5 * n + 1) {\n        print(0)\n    }\n    // +n*n (shortcut 3)\n    for _ in 0 ..&lt; (2 * n) {\n        for _ in 0 ..&lt; (n + 1) {\n            print(0)\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    let a = 1;  // +0 (shortcut 1)\n    a = a + n;  // +0 (shortcut 1)\n    // +n (shortcut 2)\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n (shortcut 3)\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    let a = 1;  // +0 (shortcut 1)\n    a = a + n;  // +0 (shortcut 1)\n    // +n (shortcut 2)\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n (shortcut 3)\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +0 (shortcut 1)\n  a = a + n; // +0 (shortcut 1)\n  // +n (shortcut 2)\n  for (int i = 0; i &lt; 5 * n + 1; i++) {\n    print(0);\n  }\n  // +n*n (shortcut 3)\n  for (int i = 0; i &lt; 2 * n; i++) {\n    for (int j = 0; j &lt; n + 1; j++) {\n      print(0);\n    }\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;     // +0 (shortcut 1)\n    a = a + n;        // +0 (shortcut 1)\n\n    // +n (shortcut 2)\n    for i in 0..(5 * n + 1) {\n        println!(\"{}\", 0);\n    }\n\n    // +n*n (shortcut 3)\n    for i in 0..(2 * n) {\n        for j in 0..(n + 1) {\n            println!(\"{}\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0 (shortcut 1)\n    a = a + n;  // +0 (shortcut 1)\n    // +n (shortcut 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        printf(\"%d\", 0);\n    }\n    // +n*n (shortcut 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            printf(\"%d\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1   // +0 (shortcut 1)\n    a = a + n   // +0 (shortcut 1)\n    // +n (shortcut 2)\n    for (i in 0..&lt;5 * n + 1) {\n        println(0)\n    }\n    // +n*n (shortcut 3)\n    for (i in 0..&lt;2 * n) {\n        for (j in 0..&lt;n + 1) {\n            println(0)\n        }\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 1       # +0 (shortcut 1)\n    a = a + n   # +0 (shortcut 1)\n    # +n (shortcut 2)\n    (0...(5 * n + 1)).each do { puts 0 }\n    # +n*n (shortcut 3)\n    (0...(2 * n)).each do\n        (0...(n + 1)).each do { puts 0 }\n    end\nend\n</code></pre> <p>The formulas below compare full and simplified counts. Both give time complexity \\(O(n^2)\\).</p> \\[ \\begin{aligned} T(n) &amp; = 2n(n + 1) + (5n + 1) + 2 &amp; \\text{Full count (-.-|||)} \\newline &amp; = 2n^2 + 7n + 3 \\newline T(n) &amp; = n^2 + n &amp; \\text{Shortcut count (o.O)} \\end{aligned} \\]","path":["Chapter 2.   Complexity analysis","2.3   Time complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#2-step-2-determine-the-asymptotic-upper-bound","level":3,"title":"2.   Step 2: Determine the asymptotic upper bound","text":"<p>The highest-order term in \\(T(n)\\) determines time complexity. As \\(n\\) tends to infinity, that term dominates, and the others become negligible.</p> <p>Table 2-2 includes deliberately extreme coefficients to emphasize that coefficients cannot overcome growth order. As \\(n\\) tends to infinity, those constants become insignificant.</p> <p> Table 2-2   Operation counts and their time complexities </p> Operation count \\(T(n)\\) Time complexity \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)","path":["Chapter 2.   Complexity analysis","2.3   Time complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#234-common-complexity-classes","level":2,"title":"2.3.4   Common complexity classes","text":"<p>For input size \\(n\\), common time complexities are shown below in increasing order of growth.</p> \\[ \\begin{aligned} &amp; O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n \\log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!) \\newline &amp; \\text{Constant} &lt; \\text{Logarithmic} &lt; \\text{Linear} &lt; \\text{Linearithmic} &lt; \\text{Quadratic} &lt; \\text{Exponential} &lt; \\text{Factorial} \\end{aligned} \\] <p></p> <p> Figure 2-9   Common time complexity classes </p>","path":["Chapter 2.   Complexity analysis","2.3   Time complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#1-constant-time-o1","level":3,"title":"1.   Constant time \\(O(1)\\)","text":"<p>A constant operation count is independent of input size \\(n\\): changing \\(n\\) does not change the count.</p> <p>In the following function, <code>size</code> may be large, but it is independent of input size \\(n\\), so the time complexity remains \\(O(1)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def constant(n: int) -&gt; int:\n    \"\"\"Constant complexity\"\"\"\n    count = 0\n    size = 100000\n    for _ in range(size):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Constant complexity */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Constant complexity */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Constant complexity */\nint Constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Constant complexity */\nfunc constant(n int) int {\n    count := 0\n    size := 100000\n    for i := 0; i &lt; size; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Constant complexity */\nfunc constant(n: Int) -&gt; Int {\n    var count = 0\n    let size = 100_000\n    for _ in 0 ..&lt; size {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Constant complexity */\nfunction constant(n) {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Constant complexity */\nfunction constant(n: number): number {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Constant complexity */\nint constant(int n) {\n  int count = 0;\n  int size = 100000;\n  for (var i = 0; i &lt; size; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Constant complexity */\nfn constant(n: i32) -&gt; i32 {\n    _ = n;\n    let mut count = 0;\n    let size = 100_000;\n    for _ in 0..size {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Constant complexity */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Constant complexity */\nfun constant(n: Int): Int {\n    var count = 0\n    val size = 100000\n    for (i in 0..&lt;size)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Constant complexity ###\ndef constant(n)\n  count = 0\n  size = 100000\n\n  (0...size).each { count += 1 }\n\n  count\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 2.   Complexity analysis","2.3   Time complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#2-linear-time-on","level":3,"title":"2.   Linear time \\(O(n)\\)","text":"<p>A linear operation count grows proportionally to input size \\(n\\). It commonly appears in a single loop:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def linear(n: int) -&gt; int:\n    \"\"\"Linear complexity\"\"\"\n    count = 0\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Linear complexity */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Linear complexity */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Linear complexity */\nint Linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Linear complexity */\nfunc linear(n int) int {\n    count := 0\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Linear complexity */\nfunc linear(n: Int) -&gt; Int {\n    var count = 0\n    for _ in 0 ..&lt; n {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Linear complexity */\nfunction linear(n) {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Linear complexity */\nfunction linear(n: number): number {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Linear complexity */\nint linear(int n) {\n  int count = 0;\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Linear complexity */\nfn linear(n: i32) -&gt; i32 {\n    let mut count = 0;\n    for _ in 0..n {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Linear complexity */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Linear complexity */\nfun linear(n: Int): Int {\n    var count = 0\n    for (i in 0..&lt;n)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Linear complexity ###\ndef linear(n)\n  count = 0\n  (0...n).each { count += 1 }\n  count\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Traversing an array or linked list takes \\(O(n)\\) time, where \\(n\\) is its length:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def array_traversal(nums: list[int]) -&gt; int:\n    \"\"\"Linear complexity (array traversal)\"\"\"\n    count = 0\n    # The number of iterations is proportional to the array length\n    for num in nums:\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Linear complexity (array traversal) */\nint arrayTraversal(vector&lt;int&gt; &amp;nums) {\n    int count = 0;\n    // The number of iterations is proportional to the array length\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Linear complexity (array traversal) */\nint arrayTraversal(int[] nums) {\n    int count = 0;\n    // The number of iterations is proportional to the array length\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Linear complexity (array traversal) */\nint ArrayTraversal(int[] nums) {\n    int count = 0;\n    // The number of iterations is proportional to the array length\n    foreach (int num in nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Linear complexity (array traversal) */\nfunc arrayTraversal(nums []int) int {\n    count := 0\n    // The number of iterations is proportional to the array length\n    for range nums {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Linear complexity (array traversal) */\nfunc arrayTraversal(nums: [Int]) -&gt; Int {\n    var count = 0\n    // The number of iterations is proportional to the array length\n    for _ in nums {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Linear complexity (array traversal) */\nfunction arrayTraversal(nums) {\n    let count = 0;\n    // The number of iterations is proportional to the array length\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Linear complexity (array traversal) */\nfunction arrayTraversal(nums: number[]): number {\n    let count = 0;\n    // The number of iterations is proportional to the array length\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Linear complexity (array traversal) */\nint arrayTraversal(List&lt;int&gt; nums) {\n  int count = 0;\n  // The number of iterations is proportional to the array length\n  for (var _num in nums) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Linear complexity (array traversal) */\nfn array_traversal(nums: &amp;[i32]) -&gt; i32 {\n    let mut count = 0;\n    // The number of iterations is proportional to the array length\n    for _ in nums {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Linear complexity (array traversal) */\nint arrayTraversal(int *nums, int n) {\n    int count = 0;\n    // The number of iterations is proportional to the array length\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Linear complexity (array traversal) */\nfun arrayTraversal(nums: IntArray): Int {\n    var count = 0\n    // The number of iterations is proportional to the array length\n    for (num in nums) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Linear complexity (array traversal)###\ndef array_traversal(nums)\n  count = 0\n\n  # The number of iterations is proportional to the array length\n  for num in nums\n    count += 1\n  end\n\n  count\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>The meaning of input size \\(n\\) depends on the input type. In the first example, it is the variable \\(n\\); in the second, it is the array length \\(n\\).</p>","path":["Chapter 2.   Complexity analysis","2.3   Time complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#3-quadratic-time-on2","level":3,"title":"3.   Quadratic time \\(O(n^2)\\)","text":"<p>A quadratic operation count grows with the square of input size \\(n\\). It commonly appears in nested loops where both levels take \\(O(n)\\) time, giving \\(O(n^2)\\) overall:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def quadratic(n: int) -&gt; int:\n    \"\"\"Quadratic complexity\"\"\"\n    count = 0\n    # The number of iterations is proportional to the square of the input size n\n    for i in range(n):\n        for j in range(n):\n            count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Quadratic complexity */\nint quadratic(int n) {\n    int count = 0;\n    // The number of iterations is proportional to the square of the input size n\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Quadratic complexity */\nint quadratic(int n) {\n    int count = 0;\n    // The number of iterations is proportional to the square of the input size n\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Quadratic complexity */\nint Quadratic(int n) {\n    int count = 0;\n    // The number of iterations is proportional to the square of the input size n\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Quadratic complexity */\nfunc quadratic(n int) int {\n    count := 0\n    // The number of iterations is proportional to the square of the input size n\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; n; j++ {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Quadratic complexity */\nfunc quadratic(n: Int) -&gt; Int {\n    var count = 0\n    // The number of iterations is proportional to the square of the input size n\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; n {\n            count += 1\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Quadratic complexity */\nfunction quadratic(n) {\n    let count = 0;\n    // The number of iterations is proportional to the square of the input size n\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Quadratic complexity */\nfunction quadratic(n: number): number {\n    let count = 0;\n    // The number of iterations is proportional to the square of the input size n\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Quadratic complexity */\nint quadratic(int n) {\n  int count = 0;\n  // The number of iterations is proportional to the square of the input size n\n  for (int i = 0; i &lt; n; i++) {\n    for (int j = 0; j &lt; n; j++) {\n      count++;\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Quadratic complexity */\nfn quadratic(n: i32) -&gt; i32 {\n    let mut count = 0;\n    // The number of iterations is proportional to the square of the input size n\n    for _ in 0..n {\n        for _ in 0..n {\n            count += 1;\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Quadratic complexity */\nint quadratic(int n) {\n    int count = 0;\n    // The number of iterations is proportional to the square of the input size n\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Quadratic complexity */\nfun quadratic(n: Int): Int {\n    var count = 0\n    // The number of iterations is proportional to the square of the input size n\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;n) {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Quadratic complexity ###\ndef quadratic(n)\n  count = 0\n\n  # The number of iterations is proportional to the square of the input size n\n  for i in 0...n\n    for j in 0...n\n      count += 1\n    end\n  end\n\n  count\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 2-10 compares constant, linear, and quadratic time complexity.</p> <p></p> <p> Figure 2-10   Constant, linear, and quadratic time complexity </p> <p>In bubble sort, for example, the outer loop runs \\(n - 1\\) times, while the inner loop runs \\(n-1\\), \\(n-2\\), \\(\\dots\\), \\(2\\), and \\(1\\) times, averaging \\(n / 2\\). The resulting complexity is \\(O((n - 1) n / 2) = O(n^2)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def bubble_sort(nums: list[int]) -&gt; int:\n    \"\"\"Quadratic complexity (bubble sort)\"\"\"\n    count = 0  # Counter\n    # Outer loop: the unsorted range is [0, i]\n    for i in range(len(nums) - 1, 0, -1):\n        # Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # Swap nums[j] and nums[j + 1]\n                tmp: int = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3  # Swapping two elements takes 3 elementary operations\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Quadratic complexity (bubble sort) */\nint bubbleSort(vector&lt;int&gt; &amp;nums) {\n    int count = 0; // Counter\n    // Outer loop: the unsorted range is [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // Swapping two elements takes 3 elementary operations\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Quadratic complexity (bubble sort) */\nint bubbleSort(int[] nums) {\n    int count = 0; // Counter\n    // Outer loop: the unsorted range is [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // Swapping two elements takes 3 elementary operations\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Quadratic complexity (bubble sort) */\nint BubbleSort(int[] nums) {\n    int count = 0;  // Counter\n    // Outer loop: the unsorted range is [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                count += 3;  // Swapping two elements takes 3 elementary operations\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Quadratic complexity (bubble sort) */\nfunc bubbleSort(nums []int) int {\n    count := 0 // Counter\n    // Outer loop: the unsorted range is [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // Swap nums[j] and nums[j + 1]\n                tmp := nums[j]\n                nums[j] = nums[j+1]\n                nums[j+1] = tmp\n                count += 3 // Swapping two elements takes 3 elementary operations\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Quadratic complexity (bubble sort) */\nfunc bubbleSort(nums: inout [Int]) -&gt; Int {\n    var count = 0 // Counter\n    // Outer loop: the unsorted range is [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3 // Swapping two elements takes 3 elementary operations\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Quadratic complexity (bubble sort) */\nfunction bubbleSort(nums) {\n    let count = 0; // Counter\n    // Outer loop: the unsorted range is [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // Swapping two elements takes 3 elementary operations\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Quadratic complexity (bubble sort) */\nfunction bubbleSort(nums: number[]): number {\n    let count = 0; // Counter\n    // Outer loop: the unsorted range is [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // Swapping two elements takes 3 elementary operations\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Quadratic complexity (bubble sort) */\nint bubbleSort(List&lt;int&gt; nums) {\n  int count = 0; // Counter\n  // Outer loop: the unsorted range is [0, i]\n  for (var i = nums.length - 1; i &gt; 0; i--) {\n    // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n    for (var j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // Swap nums[j] and nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        count += 3; // Swapping two elements takes 3 elementary operations\n      }\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Quadratic complexity (bubble sort) */\nfn bubble_sort(nums: &amp;mut [i32]) -&gt; i32 {\n    let mut count = 0; // Counter\n\n    // Outer loop: the unsorted range is [0, i]\n    for i in (1..nums.len()).rev() {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // Swapping two elements takes 3 elementary operations\n            }\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Quadratic complexity (bubble sort) */\nint bubbleSort(int *nums, int n) {\n    int count = 0; // Counter\n    // Outer loop: the unsorted range is [0, i]\n    for (int i = n - 1; i &gt; 0; i--) {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // Swapping two elements takes 3 elementary operations\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Quadratic complexity (bubble sort) */\nfun bubbleSort(nums: IntArray): Int {\n    var count = 0 // Counter\n    // Outer loop: the unsorted range is [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n                count += 3 // Swapping two elements takes 3 elementary operations\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Quadratic complexity (bubble sort)###\ndef bubble_sort(nums)\n  count = 0  # Counter\n\n  # Outer loop: the unsorted range is [0, i]\n  for i in (nums.length - 1).downto(0)\n    # Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # Swap nums[j] and nums[j + 1]\n        tmp = nums[j]\n        nums[j] = nums[j + 1]\n        nums[j + 1] = tmp\n        count += 3 # Swapping two elements takes 3 elementary operations\n      end\n    end\n  end\n\n  count\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 2.   Complexity analysis","2.3   Time complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#4-exponential-time-o2n","level":3,"title":"4.   Exponential time \\(O(2^n)\\)","text":"<p>Cell division is a classic example of exponential growth. Starting with \\(1\\) cell, one round produces \\(2\\), two rounds produce \\(4\\), and after \\(n\\) rounds there are \\(2^n\\) cells.</p> <p>The figure and code below simulate cell division in \\(O(2^n)\\) time. The input \\(n\\) is the number of rounds, while the returned <code>count</code> is the total number of individual divisions.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def exponential(n: int) -&gt; int:\n    \"\"\"Exponential complexity (iterative implementation)\"\"\"\n    count = 0\n    base = 1\n    # Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in range(n):\n        for _ in range(base):\n            count += 1\n        base *= 2\n    # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Exponential complexity (iterative implementation) */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Exponential complexity (iterative implementation) */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Exponential complexity (iterative implementation) */\nint Exponential(int n) {\n    int count = 0, bas = 1;\n    // Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Exponential complexity (iterative implementation)*/\nfunc exponential(n int) int {\n    count, base := 0, 1\n    // Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; base; j++ {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Exponential complexity (iterative implementation) */\nfunc exponential(n: Int) -&gt; Int {\n    var count = 0\n    var base = 1\n    // Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; base {\n            count += 1\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Exponential complexity (iterative implementation) */\nfunction exponential(n) {\n    let count = 0,\n        base = 1;\n    // Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Exponential complexity (iterative implementation) */\nfunction exponential(n: number): number {\n    let count = 0,\n        base = 1;\n    // Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Exponential complexity (iterative implementation) */\nint exponential(int n) {\n  int count = 0, base = 1;\n  // Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)\n  for (var i = 0; i &lt; n; i++) {\n    for (var j = 0; j &lt; base; j++) {\n      count++;\n    }\n    base *= 2;\n  }\n  // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Exponential complexity (iterative implementation) */\nfn exponential(n: i32) -&gt; i32 {\n    let mut count = 0;\n    let mut base = 1;\n    // Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0..n {\n        for _ in 0..base {\n            count += 1\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Exponential complexity (iterative implementation) */\nint exponential(int n) {\n    int count = 0;\n    int bas = 1;\n    // Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Exponential complexity (iterative implementation) */\nfun exponential(n: Int): Int {\n    var count = 0\n    var base = 1\n    // Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;base) {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Exponential complexity (iterative implementation)###\ndef exponential(n)\n  count, base = 0, 1\n\n  # Each cell divides into two per round, giving the sequence 1, 2, 4, 8, ..., 2^(n-1)\n  (0...n).each do\n    (0...base).each { count += 1 }\n    base *= 2\n  end\n\n  # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  count\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p></p> <p> Figure 2-11   Exponential time complexity </p> <p>Exponential time often appears in recursive algorithms. The following function branches into two calls at each level, stopping after \\(n\\) levels:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def exp_recur(n: int) -&gt; int:\n    \"\"\"Exponential complexity (recursive implementation)\"\"\"\n    if n == 1:\n        return 1\n    return exp_recur(n - 1) + exp_recur(n - 1) + 1\n</code></pre> time_complexity.cpp<pre><code>/* Exponential complexity (recursive implementation) */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* Exponential complexity (recursive implementation) */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* Exponential complexity (recursive implementation) */\nint ExpRecur(int n) {\n    if (n == 1) return 1;\n    return ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* Exponential complexity (recursive implementation)*/\nfunc expRecur(n int) int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n-1) + expRecur(n-1) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* Exponential complexity (recursive implementation) */\nfunc expRecur(n: Int) -&gt; Int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* Exponential complexity (recursive implementation) */\nfunction expRecur(n) {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* Exponential complexity (recursive implementation) */\nfunction expRecur(n: number): number {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* Exponential complexity (recursive implementation) */\nint expRecur(int n) {\n  if (n == 1) return 1;\n  return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* Exponential complexity (recursive implementation) */\nfn exp_recur(n: i32) -&gt; i32 {\n    if n == 1 {\n        return 1;\n    }\n    exp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* Exponential complexity (recursive implementation) */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* Exponential complexity (recursive implementation) */\nfun expRecur(n: Int): Int {\n    if (n == 1) {\n        return 1\n    }\n    return expRecur(n - 1) + expRecur(n - 1) + 1\n}\n</code></pre> time_complexity.rb<pre><code>### Exponential complexity (recursive implementation)###\ndef exp_recur(n)\n  return 1 if n == 1\n  exp_recur(n - 1) + exp_recur(n - 1) + 1\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Exponential growth is very rapid and common in exhaustive approaches such as brute-force search and backtracking. It is unacceptable for large inputs, which usually require approaches such as dynamic programming or greedy algorithms.</p>","path":["Chapter 2.   Complexity analysis","2.3   Time complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#5-logarithmic-time-olog-n","level":3,"title":"5.   Logarithmic time \\(O(\\log n)\\)","text":"<p>Logarithmic growth corresponds to repeatedly halving a quantity, the reverse of exponential growth. Starting with input size \\(n\\), halving each round gives \\(\\log_2 n\\) iterations, the inverse of \\(2^n\\).</p> <p>The figure and code below simulate repeated halving, taking \\(O(\\log_2 n)\\) time, usually written \\(O(\\log n)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def logarithmic(n: int) -&gt; int:\n    \"\"\"Logarithmic complexity (iterative implementation)\"\"\"\n    count = 0\n    while n &gt; 1:\n        n = n / 2\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Logarithmic complexity (iterative implementation) */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Logarithmic complexity (iterative implementation) */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Logarithmic complexity (iterative implementation) */\nint Logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n /= 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Logarithmic complexity (iterative implementation)*/\nfunc logarithmic(n int) int {\n    count := 0\n    for n &gt; 1 {\n        n = n / 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Logarithmic complexity (iterative implementation) */\nfunc logarithmic(n: Int) -&gt; Int {\n    var count = 0\n    var n = n\n    while n &gt; 1 {\n        n = n / 2\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Logarithmic complexity (iterative implementation) */\nfunction logarithmic(n) {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Logarithmic complexity (iterative implementation) */\nfunction logarithmic(n: number): number {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Logarithmic complexity (iterative implementation) */\nint logarithmic(int n) {\n  int count = 0;\n  while (n &gt; 1) {\n    n = n ~/ 2;\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Logarithmic complexity (iterative implementation) */\nfn logarithmic(mut n: i32) -&gt; i32 {\n    let mut count = 0;\n    while n &gt; 1 {\n        n = n / 2;\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Logarithmic complexity (iterative implementation) */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Logarithmic complexity (iterative implementation) */\nfun logarithmic(n: Int): Int {\n    var n1 = n\n    var count = 0\n    while (n1 &gt; 1) {\n        n1 /= 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Logarithmic complexity (iterative implementation)###\ndef logarithmic(n)\n  count = 0\n\n  while n &gt; 1\n    n /= 2\n    count += 1\n  end\n\n  count\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p></p> <p> Figure 2-12   Logarithmic time complexity </p> <p>Like exponential growth, logarithmic growth often occurs in recursion. The following code produces a recursion tree of height \\(\\log_2 n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def log_recur(n: int) -&gt; int:\n    \"\"\"Logarithmic complexity (recursive implementation)\"\"\"\n    if n &lt;= 1:\n        return 0\n    return log_recur(n / 2) + 1\n</code></pre> time_complexity.cpp<pre><code>/* Logarithmic complexity (recursive implementation) */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* Logarithmic complexity (recursive implementation) */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* Logarithmic complexity (recursive implementation) */\nint LogRecur(int n) {\n    if (n &lt;= 1) return 0;\n    return LogRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* Logarithmic complexity (recursive implementation)*/\nfunc logRecur(n int) int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n/2) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* Logarithmic complexity (recursive implementation) */\nfunc logRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n: n / 2) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* Logarithmic complexity (recursive implementation) */\nfunction logRecur(n) {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* Logarithmic complexity (recursive implementation) */\nfunction logRecur(n: number): number {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* Logarithmic complexity (recursive implementation) */\nint logRecur(int n) {\n  if (n &lt;= 1) return 0;\n  return logRecur(n ~/ 2) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* Logarithmic complexity (recursive implementation) */\nfn log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 0;\n    }\n    log_recur(n / 2) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* Logarithmic complexity (recursive implementation) */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* Logarithmic complexity (recursive implementation) */\nfun logRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 0\n    return logRecur(n / 2) + 1\n}\n</code></pre> time_complexity.rb<pre><code>### Logarithmic complexity (recursive implementation)###\ndef log_recur(n)\n  return 0 unless n &gt; 1\n  log_recur(n / 2) + 1\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Logarithmic time often arises in divide-and-conquer algorithms, reflecting the idea of dividing a problem into smaller, simpler parts. It grows slowly and is highly desirable, second only to constant time among these classes.</p> <p>What is the base of \\(O(\\log n)\\)?</p> <p>More precisely, reducing the size by a factor of \\(m\\) gives \\(O(\\log_m n)\\). The change-of-base formula shows that different fixed bases yield the same complexity:</p> \\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\] <p>Thus, changing the base \\(m\\) does not affect the complexity. We usually omit \\(m\\) and simply write \\(O(\\log n)\\).</p>","path":["Chapter 2.   Complexity analysis","2.3   Time complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#6-linearithmic-time-on-log-n","level":3,"title":"6.   Linearithmic time \\(O(n \\log n)\\)","text":"<p>Linearithmic time often appears when one level of a nested process takes \\(O(\\log n)\\) and another takes \\(O(n)\\). An example follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def linear_log_recur(n: int) -&gt; int:\n    \"\"\"Linearithmic complexity\"\"\"\n    if n &lt;= 1:\n        return 1\n    # Split into two subproblems, each half the original size\n    count = linear_log_recur(n // 2) + linear_log_recur(n // 2)\n    # The current subproblem includes n operations\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Linearithmic complexity */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Linearithmic complexity */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Linearithmic complexity */\nint LinearLogRecur(int n) {\n    if (n &lt;= 1) return 1;\n    int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Linearithmic complexity */\nfunc linearLogRecur(n int) int {\n    if n &lt;= 1 {\n        return 1\n    }\n    count := linearLogRecur(n/2) + linearLogRecur(n/2)\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Linearithmic complexity */\nfunc linearLogRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 1\n    }\n    var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\n    for _ in stride(from: 0, to: n, by: 1) {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Linearithmic complexity */\nfunction linearLogRecur(n) {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Linearithmic complexity */\nfunction linearLogRecur(n: number): number {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Linearithmic complexity */\nint linearLogRecur(int n) {\n  if (n &lt;= 1) return 1;\n  int count = linearLogRecur(n ~/ 2) + linearLogRecur(n ~/ 2);\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Linearithmic complexity */\nfn linear_log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 1;\n    }\n    let mut count = linear_log_recur(n / 2) + linear_log_recur(n / 2);\n    for _ in 0..n {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> time_complexity.c<pre><code>/* Linearithmic complexity */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Linearithmic complexity */\nfun linearLogRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 1\n    var count = linearLogRecur(n / 2) + linearLogRecur(n / 2)\n    for (i in 0..&lt;n) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Linearithmic complexity ###\ndef linear_log_recur(n)\n  return 1 unless n &gt; 1\n\n  count = linear_log_recur(n / 2) + linear_log_recur(n / 2)\n  (0...n).each { count += 1 }\n\n  count\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 2-13 shows how linearithmic growth arises. Each level of the binary tree performs \\(n\\) operations, and there are \\(\\log_2 n + 1\\) levels, giving \\(O(n \\log n)\\) time.</p> <p></p> <p> Figure 2-13   Linearithmic time complexity </p> <p>Common sorting algorithms, including quick sort, merge sort, and heap sort, typically take \\(O(n \\log n)\\) time.</p>","path":["Chapter 2.   Complexity analysis","2.3   Time complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#7-factorial-time-on","level":3,"title":"7.   Factorial time \\(O(n!)\\)","text":"<p>Factorial growth arises when generating every permutation of \\(n\\) distinct elements. The number of permutations is:</p> \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\] <p>Algorithms with factorial time complexity are usually implemented recursively. As the figure and code below show, each call creates \\(n\\) branches at the first level, \\(n - 1\\) at the second, and so on until level \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def factorial_recur(n: int) -&gt; int:\n    \"\"\"Factorial complexity (recursive implementation)\"\"\"\n    if n == 0:\n        return 1\n    count = 0\n    # Split one problem into n subproblems\n    for _ in range(n):\n        count += factorial_recur(n - 1)\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* Factorial complexity (recursive implementation) */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // Split one problem into n subproblems\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* Factorial complexity (recursive implementation) */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // Split one problem into n subproblems\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* Factorial complexity (recursive implementation) */\nint FactorialRecur(int n) {\n    if (n == 0) return 1;\n    int count = 0;\n    // Split one problem into n subproblems\n    for (int i = 0; i &lt; n; i++) {\n        count += FactorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* Factorial complexity (recursive implementation) */\nfunc factorialRecur(n int) int {\n    if n == 0 {\n        return 1\n    }\n    count := 0\n    // Split one problem into n subproblems\n    for i := 0; i &lt; n; i++ {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* Factorial complexity (recursive implementation) */\nfunc factorialRecur(n: Int) -&gt; Int {\n    if n == 0 {\n        return 1\n    }\n    var count = 0\n    // Split one problem into n subproblems\n    for _ in 0 ..&lt; n {\n        count += factorialRecur(n: n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* Factorial complexity (recursive implementation) */\nfunction factorialRecur(n) {\n    if (n === 0) return 1;\n    let count = 0;\n    // Split one problem into n subproblems\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* Factorial complexity (recursive implementation) */\nfunction factorialRecur(n: number): number {\n    if (n === 0) return 1;\n    let count = 0;\n    // Split one problem into n subproblems\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* Factorial complexity (recursive implementation) */\nint factorialRecur(int n) {\n  if (n == 0) return 1;\n  int count = 0;\n  // Split one problem into n subproblems\n  for (var i = 0; i &lt; n; i++) {\n    count += factorialRecur(n - 1);\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* Factorial complexity (recursive implementation) */\nfn factorial_recur(n: i32) -&gt; i32 {\n    if n == 0 {\n        return 1;\n    }\n    let mut count = 0;\n    // Split one problem into n subproblems\n    for _ in 0..n {\n        count += factorial_recur(n - 1);\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* Factorial complexity (recursive implementation) */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* Factorial complexity (recursive implementation) */\nfun factorialRecur(n: Int): Int {\n    if (n == 0)\n        return 1\n    var count = 0\n    // Split one problem into n subproblems\n    for (i in 0..&lt;n) {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### Factorial complexity (recursive implementation)###\ndef factorial_recur(n)\n  return 1 if n == 0\n\n  count = 0\n  # Split one problem into n subproblems\n  (0...n).each { count += factorial_recur(n - 1) }\n\n  count\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p></p> <p> Figure 2-14   Factorial time complexity </p> <p>Since \\(n! &gt; 2^n\\) whenever \\(n \\geq 4\\), factorial growth is faster than exponential growth and is also unacceptable for large \\(n\\).</p>","path":["Chapter 2.   Complexity analysis","2.3   Time complexity"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#235-worst-case-best-case-and-average-case-time-complexity","level":2,"title":"2.3.5   Worst-case, best-case, and average-case time complexity","text":"<p>An algorithm's efficiency often depends on how its input data is arranged. Suppose <code>nums</code> is an array of length \\(n\\) containing every integer from \\(1\\) through \\(n\\) exactly once in a random order. The task is to return the index of \\(1\\). We can conclude the following.</p> <ul> <li>If <code>nums = [?, ?, ..., 1]</code>, with \\(1\\) last, we must traverse the whole array, giving worst-case time complexity \\(O(n)\\).</li> <li>If <code>nums = [1, ?, ?, ...]</code>, with \\(1\\) first, no further traversal is needed regardless of array length, giving best-case time complexity \\(\\Omega(1)\\).</li> </ul> <p>Worst-case time complexity corresponds to an asymptotic upper bound, expressed with big-\\(O\\) notation. Best-case time complexity corresponds to an asymptotic lower bound, expressed with \\(\\Omega\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby worst_best_time_complexity.py<pre><code>def random_numbers(n: int) -&gt; list[int]:\n    \"\"\"Generate an array containing 1, 2, ..., n in shuffled order\"\"\"\n    # Generate the array nums = 1, 2, 3, ..., n\n    nums = [i for i in range(1, n + 1)]\n    # Randomly shuffle the array elements\n    random.shuffle(nums)\n    return nums\n\ndef find_one(nums: list[int]) -&gt; int:\n    \"\"\"Find the index of the number 1 in array nums\"\"\"\n    for i in range(len(nums)):\n        # When 1 is at the start of the array, the best-case time complexity is O(1)\n        # When 1 is at the end of the array, the worst-case time complexity is O(n)\n        if nums[i] == 1:\n            return i\n    return -1\n</code></pre> worst_best_time_complexity.cpp<pre><code>/* Generate an array containing { 1, 2, ..., n } in shuffled order */\nvector&lt;int&gt; randomNumbers(int n) {\n    vector&lt;int&gt; nums(n);\n    // Generate the array nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // Use the system time to generate a random seed\n    unsigned seed = chrono::system_clock::now().time_since_epoch().count();\n    // Randomly shuffle the array elements\n    shuffle(nums.begin(), nums.end(), default_random_engine(seed));\n    return nums;\n}\n\n/* Find the index of the number 1 in array nums */\nint findOne(vector&lt;int&gt; &amp;nums) {\n    for (int i = 0; i &lt; nums.size(); i++) {\n        // When 1 is at the start of the array, the best-case time complexity is O(1)\n        // When 1 is at the end of the array, the worst-case time complexity is O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.java<pre><code>/* Generate an array containing { 1, 2, ..., n } in shuffled order */\nint[] randomNumbers(int n) {\n    Integer[] nums = new Integer[n];\n    // Generate the array nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // Randomly shuffle the array elements\n    Collections.shuffle(Arrays.asList(nums));\n    // Integer[] -&gt; int[]\n    int[] res = new int[n];\n    for (int i = 0; i &lt; n; i++) {\n        res[i] = nums[i];\n    }\n    return res;\n}\n\n/* Find the index of the number 1 in array nums */\nint findOne(int[] nums) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        // When 1 is at the start of the array, the best-case time complexity is O(1)\n        // When 1 is at the end of the array, the worst-case time complexity is O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.cs<pre><code>/* Generate an array containing { 1, 2, ..., n } in shuffled order */\nint[] RandomNumbers(int n) {\n    int[] nums = new int[n];\n    // Generate the array nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n\n    // Randomly shuffle the array elements\n    for (int i = 0; i &lt; nums.Length; i++) {\n        int index = new Random().Next(i, nums.Length);\n        (nums[i], nums[index]) = (nums[index], nums[i]);\n    }\n    return nums;\n}\n\n/* Find the index of the number 1 in array nums */\nint FindOne(int[] nums) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        // When 1 is at the start of the array, the best-case time complexity is O(1)\n        // When 1 is at the end of the array, the worst-case time complexity is O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.go<pre><code>/* Generate an array containing { 1, 2, ..., n } in shuffled order */\nfunc randomNumbers(n int) []int {\n    nums := make([]int, n)\n    // Generate the array nums = { 1, 2, 3, ..., n }\n    for i := 0; i &lt; n; i++ {\n        nums[i] = i + 1\n    }\n    // Randomly shuffle the array elements\n    rand.Shuffle(len(nums), func(i, j int) {\n        nums[i], nums[j] = nums[j], nums[i]\n    })\n    return nums\n}\n\n/* Find the index of the number 1 in array nums */\nfunc findOne(nums []int) int {\n    for i := 0; i &lt; len(nums); i++ {\n        // When 1 is at the start of the array, the best-case time complexity is O(1)\n        // When 1 is at the end of the array, the worst-case time complexity is O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.swift<pre><code>/* Generate an array containing { 1, 2, ..., n } in shuffled order */\nfunc randomNumbers(n: Int) -&gt; [Int] {\n    // Generate the array nums = { 1, 2, 3, ..., n }\n    var nums = Array(1 ... n)\n    // Randomly shuffle the array elements\n    nums.shuffle()\n    return nums\n}\n\n/* Find the index of the number 1 in array nums */\nfunc findOne(nums: [Int]) -&gt; Int {\n    for i in nums.indices {\n        // When 1 is at the start of the array, the best-case time complexity is O(1)\n        // When 1 is at the end of the array, the worst-case time complexity is O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.js<pre><code>/* Generate an array containing { 1, 2, ..., n } in shuffled order */\nfunction randomNumbers(n) {\n    const nums = Array(n);\n    // Generate the array nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // Randomly shuffle the array elements\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* Find the index of the number 1 in array nums */\nfunction findOne(nums) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // When 1 is at the start of the array, the best-case time complexity is O(1)\n        // When 1 is at the end of the array, the worst-case time complexity is O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.ts<pre><code>/* Generate an array containing { 1, 2, ..., n } in shuffled order */\nfunction randomNumbers(n: number): number[] {\n    const nums = Array(n);\n    // Generate the array nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // Randomly shuffle the array elements\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* Find the index of the number 1 in array nums */\nfunction findOne(nums: number[]): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // When 1 is at the start of the array, the best-case time complexity is O(1)\n        // When 1 is at the end of the array, the worst-case time complexity is O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.dart<pre><code>/* Generate an array containing { 1, 2, ..., n } in shuffled order */\nList&lt;int&gt; randomNumbers(int n) {\n  final nums = List.filled(n, 0);\n  // Generate the array nums = { 1, 2, 3, ..., n }\n  for (var i = 0; i &lt; n; i++) {\n    nums[i] = i + 1;\n  }\n  // Randomly shuffle the array elements\n  nums.shuffle();\n\n  return nums;\n}\n\n/* Find the index of the number 1 in array nums */\nint findOne(List&lt;int&gt; nums) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    // When 1 is at the start of the array, the best-case time complexity is O(1)\n    // When 1 is at the end of the array, the worst-case time complexity is O(n)\n    if (nums[i] == 1) return i;\n  }\n\n  return -1;\n}\n</code></pre> worst_best_time_complexity.rs<pre><code>/* Generate an array containing { 1, 2, ..., n } in shuffled order */\nfn random_numbers(n: i32) -&gt; Vec&lt;i32&gt; {\n    // Generate the array nums = { 1, 2, 3, ..., n }\n    let mut nums = (1..=n).collect::&lt;Vec&lt;i32&gt;&gt;();\n    // Randomly shuffle the array elements\n    nums.shuffle(&amp;mut thread_rng());\n    nums\n}\n\n/* Find the index of the number 1 in array nums */\nfn find_one(nums: &amp;[i32]) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        // When 1 is at the start of the array, the best-case time complexity is O(1)\n        // When 1 is at the end of the array, the worst-case time complexity is O(n)\n        if nums[i] == 1 {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> worst_best_time_complexity.c<pre><code>/* Generate an array containing { 1, 2, ..., n } in shuffled order */\nint *randomNumbers(int n) {\n    // Allocate heap memory for a one-dimensional array of n int elements\n    int *nums = (int *)malloc(n * sizeof(int));\n    // Generate the array nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // Randomly shuffle the array elements\n    for (int i = n - 1; i &gt; 0; i--) {\n        int j = rand() % (i + 1);\n        int temp = nums[i];\n        nums[i] = nums[j];\n        nums[j] = temp;\n    }\n    return nums;\n}\n\n/* Find the index of the number 1 in array nums */\nint findOne(int *nums, int n) {\n    for (int i = 0; i &lt; n; i++) {\n        // When 1 is at the start of the array, the best-case time complexity is O(1)\n        // When 1 is at the end of the array, the worst-case time complexity is O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.kt<pre><code>/* Generate an array containing { 1, 2, ..., n } in shuffled order */\nfun randomNumbers(n: Int): Array&lt;Int?&gt; {\n    val nums = IntArray(n)\n    // Generate the array nums = { 1, 2, 3, ..., n }\n    for (i in 0..&lt;n) {\n        nums[i] = i + 1\n    }\n    // Randomly shuffle the array elements\n    nums.shuffle()\n    val res = arrayOfNulls&lt;Int&gt;(n)\n    for (i in 0..&lt;n) {\n        res[i] = nums[i]\n    }\n    return res\n}\n\n/* Find the index of the number 1 in array nums */\nfun findOne(nums: Array&lt;Int?&gt;): Int {\n    for (i in nums.indices) {\n        // When 1 is at the start of the array, the best-case time complexity is O(1)\n        // When 1 is at the end of the array, the worst-case time complexity is O(n)\n        if (nums[i] == 1)\n            return i\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.rb<pre><code>### Generate an array containing 1, 2, ..., n in shuffled order ###\ndef random_numbers(n)\n  # Generate the array nums = 1, 2, 3, ..., n\n  nums = Array.new(n) { |i| i + 1 }\n  # Randomly shuffle the array elements\n  nums.shuffle!\nend\n\n### Find the index of the number 1 in array nums ###\ndef find_one(nums)\n  for i in 0...nums.length\n    # When 1 is at the start of the array, the best-case time complexity is O(1)\n    # When 1 is at the end of the array, the worst-case time complexity is O(n)\n    return i if nums[i] == 1\n  end\n\n  -1\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Best-case complexity is rarely used in practice because it may occur only with low probability and can be misleading. Worst-case complexity is more useful because it gives a safe performance bound, allowing us to use the algorithm with confidence.</p> <p>In this example, both the worst and best cases occur only for particular arrangements, perhaps with low probability, and may not reflect ordinary performance. Average-case complexity describes performance on random inputs and is expressed with \\(\\Theta\\).</p> <p>For some algorithms, the average case is easy to derive. In the example above, \\(1\\) is equally likely to occur at any index in the shuffled array. The average number of iterations is half the array length, \\(n / 2\\), giving \\(\\Theta(n / 2) = \\Theta(n)\\).</p> <p>For more complex algorithms, average-case complexity can be difficult to calculate because the overall expectation under the input distribution is hard to derive. In such cases, we usually judge efficiency by worst-case complexity.</p> <p>Why is \\(\\Theta\\) rarely seen?</p> <p>Perhaps because \\(O\\) is so familiar, it is often used for average-case complexity too. Strictly speaking, this usage is not standard. In this book and other resources, read a statement such as \"average-case time complexity \\(O(n)\\)\" as \\(\\Theta(n)\\).</p>","path":["Chapter 2.   Complexity analysis","2.3   Time complexity"],"tags":[]},{"location":"chapter_data_structure/","level":1,"title":"Chapter 3.   Data structures","text":"<p>Abstract</p> <p>Data structures provide a framework that is both sturdy and versatile.</p> <p>They offer a blueprint for organizing data, a foundation on which algorithms come to life.</p>","path":["Chapter 3.   Data structures"],"tags":[]},{"location":"chapter_data_structure/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>3.1   Classifying data structures</li> <li>3.2   Basic data types</li> <li>3.3   Number encoding *</li> <li>3.4   Character encoding *</li> <li>3.5   Summary</li> <li>3.6   Exercises</li> </ul>","path":["Chapter 3.   Data structures"],"tags":[]},{"location":"chapter_data_structure/basic_data_types/","level":1,"title":"3.2   Basic data types","text":"<p>Computer data comes in many forms: text, images, video, audio, 3D models, and more. Although organized differently, all these forms are built from basic data types.</p> <p>Basic data types are types on which the CPU can operate directly and which algorithms use directly. The main types include the following.</p> <ul> <li>Integer types: <code>byte</code>, <code>short</code>, <code>int</code>, and <code>long</code>.</li> <li>Floating-point types: <code>float</code> and <code>double</code>, used to represent fractional numbers.</li> <li>The character type <code>char</code>, used for letters in different languages, punctuation, and even emoji.</li> <li>The Boolean type <code>bool</code>, used for yes-or-no values.</li> </ul> <p>Basic data types are stored in binary form. One binary digit is \\(1\\) bit. In the vast majority of modern operating systems, \\(1\\) byte consists of \\(8\\) bits.</p> <p>A basic data type's value range depends on the space it occupies. Taking Java as an example:</p> <ul> <li>An integer <code>byte</code> occupies \\(1\\) byte = \\(8\\) bits and can represent \\(2^{8}\\) distinct values.</li> <li>An integer <code>int</code> occupies \\(4\\) bytes = \\(32\\) bits and can represent \\(2^{32}\\) distinct values.</li> </ul> <p>Table 3-1 lists storage sizes, value ranges, and default values for Java's basic data types. There is no need to memorize it; understand the general picture and refer back when needed.</p> <p> Table 3-1   Storage sizes and value ranges of basic data types </p> Type Name Storage size Minimum value Maximum value Default value Integer <code>byte</code> 1 byte \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) \\(0\\) <code>short</code> 2 bytes \\(-2^{15}\\) \\(2^{15} - 1\\) \\(0\\) <code>int</code> 4 bytes \\(-2^{31}\\) \\(2^{31} - 1\\) \\(0\\) <code>long</code> 8 bytes \\(-2^{63}\\) \\(2^{63} - 1\\) \\(0\\) Floating-point <code>float</code> 4 bytes \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0\\text{f}\\) <code>double</code> 8 bytes \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) \\(0.0\\) Character <code>char</code> 2 bytes \\(0\\) \\(2^{16} - 1\\) \\(0\\) Boolean <code>bool</code> 1 byte \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\) <p>Table 3-1 describes Java's basic data types. Each programming language defines its own types, whose sizes, value ranges, and default values may differ.</p> <ul> <li>In Python, an <code>int</code> can be arbitrarily large, limited only by available memory. A <code>float</code> is a 64-bit double-precision number. There is no <code>char</code> type: a single character is a <code>str</code> of length 1.</li> <li>C and C++ do not fix the sizes of basic data types; they vary by implementation and platform. Table 3-1 follows the LP64 data model, used by 64-bit Unix operating systems including Linux and macOS.</li> <li>A <code>char</code> occupies 1 byte in C and C++. In most programming languages, character size depends on the encoding used; see the character encoding section.</li> <li>Although a Boolean needs only 1 bit (\\(0\\) or \\(1\\)), it usually occupies 1 byte in memory because modern CPUs typically use a byte as the smallest addressable memory unit.</li> </ul> <p>How do basic data types relate to data structures? A data structure is a way to organize and store data in a computer. The emphasis is on structure, rather than the data itself.</p> <p>To represent a row of numbers, an array comes naturally to mind. Its linear structure captures adjacency and ordering, regardless of whether the stored values are integers (<code>int</code>), fractional numbers (<code>float</code>), or characters (<code>char</code>).</p> <p>In other words, basic data types specify what kind of data is stored, while data structures specify how it is organized. The following code uses the same data structure, an array, for different basic types, including <code>int</code>, <code>float</code>, <code>char</code>, and <code>bool</code>.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># Initialize arrays with different basic data types\nnumbers: list[int] = [0] * 5\ndecimals: list[float] = [0.0] * 5\n# A Python character is a string of length 1\ncharacters: list[str] = ['0'] * 5\nbools: list[bool] = [False] * 5\n# Python lists can hold different basic data types and object references\ndata = [0, 0.0, 'a', False, ListNode(0)]\n</code></pre> <pre><code>// Initialize arrays with different basic data types\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool bools[5];\n</code></pre> <pre><code>// Initialize arrays with different basic data types\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] bools = new boolean[5];\n</code></pre> <pre><code>// Initialize arrays with different basic data types\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] bools = new bool[5];\n</code></pre> <pre><code>// Initialize arrays with different basic data types\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar bools = [5]bool{}\n</code></pre> <pre><code>// Initialize arrays with different basic data types\nlet numbers = Array(repeating: 0, count: 5)\nlet decimals = Array(repeating: 0.0, count: 5)\nlet characters: [Character] = Array(repeating: \"a\", count: 5)\nlet bools = Array(repeating: false, count: 5)\n</code></pre> <pre><code>// JavaScript arrays can hold different basic data types and objects\nconst array = [0, 0.0, 'a', false];\n</code></pre> <pre><code>// Initialize arrays with different basic data types\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst bools: boolean[] = [];\n</code></pre> <pre><code>// Initialize arrays with different basic data types\nList&lt;int&gt; numbers = List.filled(5, 0);\nList&lt;double&gt; decimals = List.filled(5, 0.0);\nList&lt;String&gt; characters = List.filled(5, 'a');\nList&lt;bool&gt; bools = List.filled(5, false);\n</code></pre> <pre><code>// Initialize arrays with different basic data types\nlet numbers: Vec&lt;i32&gt; = vec![0; 5];\nlet decimals: Vec&lt;f32&gt; = vec![0.0; 5];\nlet characters: Vec&lt;char&gt; = vec!['0'; 5];\nlet bools: Vec&lt;bool&gt; = vec![false; 5];\n</code></pre> <pre><code>// Initialize arrays with different basic data types\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool bools[10];\n</code></pre> <pre><code>// Initialize arrays with different basic data types\nval numbers = IntArray(5)\nval decinals = FloatArray(5)\nval characters = CharArray(5)\nval bools = BooleanArray(5)\n</code></pre> <pre><code># Ruby arrays can hold different basic data types and object references\ndata = [0, 0.0, 'a', false, ListNode(0)]\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 3.   Data structures","3.2   Basic data types"],"tags":[]},{"location":"chapter_data_structure/character_encoding/","level":1,"title":"3.4   Character encoding *","text":"<p>All computer data is stored as binary numbers, including characters of type <code>char</code>. To represent characters, we need a character set that assigns each character a unique binary number. The computer can then convert numbers to characters by looking them up.</p>","path":["Chapter 3.   Data structures","3.4   Character encoding *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#341-ascii-character-set","level":2,"title":"3.4.1   ASCII character set","text":"<p>ASCII, short for American Standard Code for Information Interchange, was the earliest character set. It uses 7 bits (the lower 7 bits of a byte) per character, allowing up to 128 characters. As shown in Figure 3-6, ASCII includes uppercase and lowercase English letters, digits 0 ~ 9, punctuation, and control characters such as newlines and tabs.</p> <p></p> <p> Figure 3-6   ASCII codes </p> <p>However, ASCII can represent only English text. As computers spread worldwide, EASCII character sets emerged to support more languages. Extending ASCII from 7 bits to 8 allowed 256 distinct characters.</p> <p>Different regions developed their own EASCII character sets. The first 128 characters remained identical to ASCII, while the remaining 128 varied to meet different languages' needs.</p>","path":["Chapter 3.   Data structures","3.4   Character encoding *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#342-gbk-character-set","level":2,"title":"3.4.2   GBK character set","text":"<p>Eventually, even EASCII proved too small for many languages. Chinese, for example, has nearly 100,000 characters, with thousands used daily. In 1980, China's National Bureau of Standards released GB2312, containing 6,763 Chinese characters and meeting most basic needs for processing Chinese text.</p> <p>GB2312 could not handle some rare and traditional characters. GBK extended it to 21,886 Chinese characters. In GBK encoding, ASCII characters use one byte, and Chinese characters use two.</p>","path":["Chapter 3.   Data structures","3.4   Character encoding *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#343-unicode-character-set","level":2,"title":"3.4.3   Unicode character set","text":"<p>As computing grew, character sets and encoding standards proliferated, creating problems. Most covered only particular languages and did not work well in multilingual settings. Multiple standards also existed for the same language, so exchanging text between computers using different encodings could produce garbled text.</p> <p>Researchers asked: could a sufficiently comprehensive character set, containing all the world's languages and symbols, solve both multilingual compatibility and garbled-text problems? This idea led to Unicode.</p> <p>Unicode, whose Chinese name means \"unified code\", can theoretically accommodate over a million characters. It aims to bring the world's characters into one universal set for processing and displaying different languages, reducing garbled text caused by incompatible standards. Since its release in 1991, it has continued adding languages and characters. As of September 2022, Unicode contained 149,186 characters, including letters, symbols, and emoji.</p> <p>As a universal character set, Unicode assigns each character a unique code point, or character number, from U+0000 to U+10FFFF. It does not, by itself, specify how these code points are stored in a computer. How can a system parse text containing code points of different lengths? For example, how does it distinguish one two-byte character from two one-byte characters?</p> <p>One straightforward solution is to store every character in a fixed-width encoding. In Figure 3-7, each character in \"Hello\" occupies 1 byte, while each in \"算法\" occupies 2 bytes. Padding the high bits with zeros lets us encode every character in \"Hello 算法\" with 2 bytes. The system can then read one character every 2 bytes to recover the phrase.</p> <p></p> <p> Figure 3-7   Example of Unicode encoding </p> <p>ASCII has already shown that English text needs only 1 byte per character. This scheme would double its storage compared with ASCII, wasting memory. We therefore need a more efficient Unicode encoding.</p>","path":["Chapter 3.   Data structures","3.4   Character encoding *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#344-utf-8-encoding","level":2,"title":"3.4.4   UTF-8 encoding","text":"<p>UTF-8 is now the most widely used Unicode encoding worldwide. It is variable-length, using 1 to 4 bytes per character depending on the character. ASCII characters need 1 byte, Latin and Greek letters need 2, common Chinese characters need 3, and some less common characters need 4.</p> <p>UTF-8's rules have two cases.</p> <ul> <li>For a one-byte character, set the highest bit to \\(0\\) and the remaining 7 bits to its Unicode code point. ASCII occupies the first 128 Unicode code points, so UTF-8 is backward-compatible with ASCII. UTF-8 can therefore decode old ASCII text.</li> <li>For an \\(n\\)-byte character, where \\(n &gt; 1\\), set the first byte's leading \\(n\\) bits to \\(1\\) and the next bit, \\(n + 1\\), to \\(0\\). Starting with the second byte, set the leading 2 bits of every byte to \\(10\\). Fill the remaining bits with the character's Unicode code point.</li> </ul> <p>Figure 3-8 shows UTF-8 for \"Hello算法\". Since the leading \\(n\\) bits are \\(1\\), the system can determine the character length \\(n\\) by counting the leading \\(1\\) bits.</p> <p>Why do all remaining bytes begin with \\(10\\)? This \\(10\\) acts as a check marker. If decoding starts at the wrong byte, the leading \\(10\\) helps the system quickly detect the problem.</p> <p>The prefix \\(10\\) works as a marker because no character's first byte can begin with \\(10\\) under UTF-8's rules. To see this by contradiction, suppose it did begin with \\(10\\). The single leading one would imply a length of \\(1\\) byte, corresponding to ASCII, but an ASCII byte must begin with \\(0\\), a contradiction.</p> <p></p> <p> Figure 3-8   Example of UTF-8 encoding </p> <p>Two other common encodings are:</p> <ul> <li>UTF-16: Uses 2 or 4 bytes per character. ASCII and common non-English characters use 2 bytes; some characters require 4. For a two-byte character, its UTF-16 encoding equals its Unicode code point.</li> <li>UTF-32: Uses 4 bytes for every character. It therefore takes more space than UTF-8 or UTF-16, especially for text containing mostly ASCII characters.</li> </ul> <p>In terms of storage, UTF-8 is efficient for English characters because each uses only 1 byte. UTF-16 is more efficient for some non-English characters, such as Chinese, using 2 bytes where UTF-8 may require 3.</p> <p>In terms of compatibility, UTF-8 is the most widely supported, and many tools and libraries prioritize it.</p>","path":["Chapter 3.   Data structures","3.4   Character encoding *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#345-character-encoding-in-programming-languages","level":2,"title":"3.4.5   Character encoding in programming languages","text":"<p>Historically, most programming languages have stored strings at runtime using fixed-width encodings such as UTF-16 or UTF-32. Fixed-width encoding lets us treat a string as an array, with the following advantages.</p> <ul> <li>Random access: UTF-16 strings support straightforward random access. With variable-length UTF-8, finding the \\(i\\)-th character requires traversing from the beginning to the \\(i\\)-th character, taking \\(O(n)\\) time.</li> <li>Character counting: Similarly, finding the length of a UTF-16 string is an \\(O(1)\\) operation, while finding the character count of a UTF-8 string requires traversing the whole string.</li> <li>String operations: Operations such as splitting, concatenation, insertion, and deletion are easier on UTF-16 strings. With UTF-8, they usually require extra work to avoid producing invalid encodings.</li> </ul> <p>The design of string representations is an interesting subject involving many factors.</p> <ul> <li>Java's <code>String</code> uses UTF-16, with 2 bytes per character. When Java was designed, 16 bits were thought sufficient for every character. This turned out to be incorrect: Unicode later expanded beyond 16 bits, so a Java character may now require a pair of 16-bit values called a surrogate pair.</li> <li>JavaScript and TypeScript use UTF-16 for similar historical reasons. When Netscape introduced JavaScript in 1995, Unicode was still young, and 16 bits could represent all its characters at the time.</li> <li>C# uses UTF-16 largely because Microsoft designed .NET, and many Microsoft technologies, including Windows, use UTF-16 extensively.</li> </ul> <p>Having underestimated the number of characters, these languages had to adopt surrogate pairs for Unicode characters beyond 16 bits. This was a compromise. A character in such a string may occupy 2 or 4 bytes, losing the benefit of fixed width, and handling surrogate pairs requires additional code, making programming and debugging more complex.</p> <p>For these reasons, some languages use different representations.</p> <ul> <li>Python's <code>str</code> uses Unicode with a flexible representation whose character width depends on the largest code point in the string. If all characters are ASCII, each uses 1 byte. If some exceed ASCII but all lie within the Basic Multilingual Plane (BMP), each uses 2 bytes. If any lie outside the BMP, each uses 4 bytes.</li> <li>Go's <code>string</code> uses UTF-8 internally. Go also provides <code>rune</code> to represent a single Unicode code point.</li> <li>Rust's <code>str</code> and <code>String</code> use UTF-8 internally, while <code>char</code> represents a single Unicode code point.</li> </ul> <p>These descriptions concern string storage within programming languages. How strings are stored in files or transmitted over a network is a separate question. For file storage and network transmission, strings are usually encoded as UTF-8 for compatibility and space efficiency.</p>","path":["Chapter 3.   Data structures","3.4   Character encoding *"],"tags":[]},{"location":"chapter_data_structure/classification_of_data_structure/","level":1,"title":"3.1   Classifying data structures","text":"<p>Common data structures include arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graphs. They can be classified by their logical structure and physical structure.</p>","path":["Chapter 3.   Data structures","3.1   Classifying data structures"],"tags":[]},{"location":"chapter_data_structure/classification_of_data_structure/#311-logical-structure-linear-and-nonlinear","level":2,"title":"3.1.1   Logical structure: linear and nonlinear","text":"<p>Logical structure describes the relationships between data elements. Arrays and linked lists arrange data in sequence, expressing linear relationships. Trees arrange data hierarchically from top to bottom, expressing ancestry and descent. Graphs consist of vertices and edges that express complex network relationships.</p> <p>As shown in Figure 3-1, logical structures fall into two broad categories: linear and nonlinear. A linear structure arranges data in a logical sequence; a nonlinear structure does not.</p> <ul> <li>Linear data structures: Arrays, linked lists, stacks, queues, and hash tables. Elements have one-to-one sequential relationships.</li> <li>Nonlinear data structures: Trees, heaps, graphs, and hash tables.</li> </ul> <p>Nonlinear data structures can be divided further into tree and network structures.</p> <ul> <li>Tree structures: Trees, heaps, and hash tables, with one-to-many relationships.</li> <li>Network structures: Graphs, with many-to-many relationships.</li> </ul> <p></p> <p> Figure 3-1   Linear and nonlinear data structures </p>","path":["Chapter 3.   Data structures","3.1   Classifying data structures"],"tags":[]},{"location":"chapter_data_structure/classification_of_data_structure/#312-physical-structure-contiguous-and-scattered-storage","level":2,"title":"3.1.2   Physical structure: contiguous and scattered storage","text":"<p>While an algorithm runs, the data it processes is mainly stored in memory. Figure 3-2 shows a memory module, with each black chip containing storage space. We can imagine memory as a huge Excel spreadsheet, where each cell holds a certain amount of data.</p> <p>The system uses memory addresses to access data at particular locations. As shown in Figure 3-2, the computer assigns each cell a number according to specific rules, giving every memory location a unique address. Programs use these addresses to access data.</p> <p></p> <p> Figure 3-2   Memory module, storage space, and memory addresses </p> <p>Tip</p> <p>The Excel spreadsheet analogy is a simplification. Real memory is more complex and involves address spaces, memory management, caching, virtual memory, and physical memory.</p> <p>Memory is a resource used by all programs. When a program occupies a region of memory, other programs usually cannot use that region at the same time. Memory resources are therefore an important consideration when designing data structures and algorithms. For example, an algorithm's peak memory usage should not exceed the system's available memory. If no large contiguous region is available, the chosen data structure must support scattered storage.</p> <p>As shown in Figure 3-3, physical structure describes how data is stored in computer memory: either contiguously (arrays) or in scattered locations (linked lists). It determines how data can be accessed, updated, inserted, and deleted. The two storage forms have complementary strengths in time and space efficiency.</p> <p></p> <p> Figure 3-3   Contiguous and scattered storage </p> <p>All data structures are implemented using arrays, linked lists, or a combination of the two. For example, stacks and queues can use either arrays or linked lists, while hash tables may use both.</p> <ul> <li>Array-based implementations: Stacks, queues, hash tables, trees, heaps, graphs, matrices, tensors (arrays with dimension \\(\\geq 3\\)), and more.</li> <li>Linked-list-based implementations: Stacks, queues, hash tables, trees, heaps, graphs, and more.</li> </ul> <p>A linked list can change length during execution after initialization, so it is called a dynamic data structure. An array's length is fixed after initialization, so it is called a static data structure. However, reallocating memory can change an array's length, providing a degree of dynamic behavior.</p> <p>Tip</p> <p>If physical structure feels difficult to understand, read the next chapter first and then return to this section.</p>","path":["Chapter 3.   Data structures","3.1   Classifying data structures"],"tags":[]},{"location":"chapter_data_structure/exercises/","level":1,"title":"3.6   Exercises","text":"","path":["Chapter 3.   Data structures","3.6   Exercises"],"tags":[]},{"location":"chapter_data_structure/exercises/#361-check-your-understanding","level":2,"title":"3.6.1   Check your understanding","text":"","path":["Chapter 3.   Data structures","3.6   Exercises"],"tags":[]},{"location":"chapter_data_structure/exercises/#1-relationships-between-data-in-everyday-situations","level":3,"title":"1.   Relationships between data in everyday situations","text":"<p>For each situation below, choose a linear, tree, or network structure based on the relationships between the data, and explain your choice:</p> <ol> <li>Students stand in a line, each concerned only with the people immediately in front of and behind them.</li> <li>A school is organized in a hierarchy: school → grade → class.</li> <li>City roads connect intersections. An intersection can lead to several others, and the roads may form cycles.</li> </ol> Reference answer <ol> <li> <p>A linear structure. Apart from the first and last people, each person has exactly one neighbor in front and one behind, forming a line of relationships.</p> </li> <li> <p>A tree structure. Each class belongs to a grade, and each grade belongs to the school, forming a hierarchy from top to bottom.</p> </li> <li> <p>A network structure. An intersection can connect to several others, and routes can form cycles. These relationships cannot be arranged in a single sequence or a strict hierarchy.</p> <p>To identify a structure, first examine the relationships between its elements, rather than how much memory it occupies.</p> </li> </ol>","path":["Chapter 3.   Data structures","3.6   Exercises"],"tags":[]},{"location":"chapter_data_structure/exercises/#2-storing-a-logical-sequence-in-memory","level":3,"title":"2.   Storing a logical sequence in memory","text":"<p>Consider two simplified memory layouts for storing the logical sequence <code>A → B → C</code>:</p> <ul> <li>Layout A: <code>A, B, C</code> are stored in memory cells numbered <code>20, 21, 22</code>, respectively.</li> <li>Layout B: <code>A, B, C</code> are stored in cells numbered <code>20, 7, 31</code>, respectively. <code>A</code> records the location of <code>B</code>, and <code>B</code> records the location of <code>C</code>.</li> </ul> <ol> <li>Which layout uses contiguous storage, and which uses noncontiguous storage?</li> <li>Which layout is more like an array, and which is more like a linked list?</li> <li>The cell numbers in layout B are not in ascending order. Why can it still represent the logical sequence <code>A → B → C</code>?</li> </ol> Reference answer <ol> <li> <p>Layout A uses consecutive memory cells, so its storage is contiguous. In layout B, nodes occupy separate locations, so its storage is noncontiguous.</p> </li> <li> <p>Layout A is more like an array; layout B is more like a linked list.</p> </li> <li> <p>The logical order is determined by the links recorded between nodes, not by the numerical order of their memory cell numbers.     The location stored in <code>A</code> leads to <code>B</code>, and the location stored in <code>B</code> leads to <code>C</code>, so we can still visit <code>A, B, C</code> in sequence.</p> <p>This also shows that logical and physical structure are two ways of looking at the same data.</p> </li> </ol>","path":["Chapter 3.   Data structures","3.6   Exercises"],"tags":[]},{"location":"chapter_data_structure/exercises/#3-data-types-and-structures-in-homework-records","level":3,"title":"3.   Data types and structures in homework records","text":"<p>A study group records whether four students have submitted their homework, in seating order:</p> <p><code>[true, false, true, true]</code></p> <ol> <li>Which basic data type is suitable for each element?</li> <li>These four elements form a sequence in seating order. What logical structure does this use?</li> <li>If the group later records homework scores instead, as <code>[90, 0, 85, 100]</code>, has the type of content changed, or the way it is organized? Explain.</li> </ol> Reference answer <ol> <li> <p>Each element represents only yes or no, so the Boolean type <code>bool</code> is appropriate.</p> </li> <li> <p>The elements form a linear structure in seating order and can be stored in an array.</p> </li> <li> <p>The content type has changed: the elements are now integers instead of Boolean values. Their organization has not changed.     The data still forms a sequence in seating order and can still use the linear structure of an array.</p> <p>A basic data type describes what is stored; a data structure describes how the data is organized.</p> </li> </ol>","path":["Chapter 3.   Data structures","3.6   Exercises"],"tags":[]},{"location":"chapter_data_structure/exercises/#362-programming-exercises","level":2,"title":"3.6.2   Programming exercises","text":"","path":["Chapter 3.   Data structures","3.6   Exercises"],"tags":[]},{"location":"chapter_data_structure/exercises/#1-counting-ones-in-a-binary-representation","level":3,"title":"1.   Counting ones in a binary representation","text":"<p>Given a nonnegative integer <code>n</code>, count the ones in its binary representation.</p> <p>Use bitwise operations. Do not convert the binary representation to a string or use a built-in function that counts the ones directly.</p> Hints <ol> <li>n &amp; 1 extracts the rightmost bit of n, allowing you to check whether it is 1.</li> <li>Shifting right by one bit discards the current rightmost bit; most languages use the &gt;&gt; operator.</li> <li>After implementing the method that checks each bit and shifts right, observe that n &amp; (n - 1) changes the rightmost 1 in n to 0.</li> </ol> <p>LeetCode</p>","path":["Chapter 3.   Data structures","3.6   Exercises"],"tags":[]},{"location":"chapter_data_structure/number_encoding/","level":1,"title":"3.3   Number encoding *","text":"<p>Tip</p> <p>Sections marked with * in their titles are optional. If you are short on time or find them difficult, you can skip them initially and return after completing the required sections.</p>","path":["Chapter 3.   Data structures","3.3   Number encoding *"],"tags":[]},{"location":"chapter_data_structure/number_encoding/#331-sign-magnitude-ones-complement-and-twos-complement","level":2,"title":"3.3.1   Sign-magnitude, one's complement, and two's complement","text":"<p>The previous table showed that every integer type represents one more negative value than positive value. For example, <code>byte</code> ranges over \\([-128, 127]\\). Understanding this counterintuitive fact requires sign-magnitude, one's complement, and two's complement.</p> <p>First, integers are stored in two's complement form. Before explaining why, let's define the three representations.</p> <ul> <li>Sign-magnitude: The highest bit is the sign: \\(0\\) for positive and \\(1\\) for negative. The remaining bits represent the magnitude.</li> <li>One's complement: A positive number has the same representation as sign-magnitude. For a negative number, invert every bit except the sign bit of its sign-magnitude representation.</li> <li>Two's complement: A positive number again has the same representation as sign-magnitude. For a negative number, add \\(1\\) to its one's complement.</li> </ul> <p>Figure 3-4 shows how to convert among these representations.</p> <p></p> <p> Figure 3-4   Converting between sign-magnitude, one's complement, and two's complement </p> <p>Although sign-magnitude is intuitive, it has limitations. First, negative sign-magnitude values cannot be used directly in ordinary binary arithmetic. For example, directly adding \\(1 + (-2)\\) in this representation gives \\(-3\\), which is clearly wrong.</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 + 1000 \\; 0010 \\newline &amp; = 1000 \\; 0011 \\newline &amp; \\rightarrow -3 \\end{aligned} \\] <p>To address this, computers introduced one's complement. Convert the operands to one's complement, calculate \\(1 + (-2)\\), and convert the result back to sign-magnitude to obtain the correct answer, \\(-1\\).</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 \\; \\text{(sign-magnitude)} + 1000 \\; 0010 \\; \\text{(sign-magnitude)} \\newline &amp; = 0000 \\; 0001 \\; \\text{(one's complement)} + 1111  \\; 1101 \\; \\text{(one's complement)} \\newline &amp; = 1111 \\; 1110 \\; \\text{(one's complement)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(sign-magnitude)} \\newline &amp; \\rightarrow -1 \\end{aligned} \\] <p>Second, sign-magnitude has two representations of zero, \\(+0\\) and \\(-0\\). Two binary encodings for zero can create ambiguity. Conditional checks may be wrong if they do not handle both forms, while special handling requires extra checks that can reduce efficiency.</p> \\[ \\begin{aligned} +0 &amp; \\rightarrow 0000 \\; 0000 \\newline -0 &amp; \\rightarrow 1000 \\; 0000 \\end{aligned} \\] <p>One's complement also has positive and negative zero, so computers introduced two's complement. Consider converting negative zero through all three representations:</p> \\[ \\begin{aligned} -0 \\rightarrow \\; &amp; 1000 \\; 0000 \\; \\text{(sign-magnitude)} \\newline = \\; &amp; 1111 \\; 1111 \\; \\text{(one's complement)} \\newline = 1 \\; &amp; 0000 \\; 0000 \\; \\text{(two's complement)} \\newline \\end{aligned} \\] <p>Adding \\(1\\) to the one's complement of negative zero produces a carry. A <code>byte</code> has only 8 bits, so the \\(1\\) carried into the ninth bit is discarded. Negative zero's two's complement is therefore \\(0000 \\; 0000\\), the same as positive zero. Two's complement has only one zero, eliminating the ambiguity.</p> <p>One question remains: if <code>byte</code> ranges over \\([-128, 127]\\), where does the extra negative value \\(-128\\) come from? Every integer in \\([-127, +127]\\) has sign-magnitude, one's complement, and two's complement representations, and its sign-magnitude and two's complement forms can be converted into each other.</p> <p>However, the two's complement value \\(1000 \\; 0000\\) is an exception with no corresponding sign-magnitude form. Applying the conversion gives \\(0000 \\; 0000\\), but that represents \\(0\\), whose two's complement should be itself. The special pattern \\(1000 \\; 0000\\) is defined to represent \\(-128\\). Indeed, two's complement addition of \\((-1) + (-127)\\) gives \\(-128\\).</p> \\[ \\begin{aligned} &amp; (-127) + (-1) \\newline &amp; \\rightarrow 1111 \\; 1111 \\; \\text{(sign-magnitude)} + 1000 \\; 0001 \\; \\text{(sign-magnitude)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(one's complement)} + 1111  \\; 1110 \\; \\text{(one's complement)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(two's complement)} + 1111  \\; 1111 \\; \\text{(two's complement)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(two's complement)} \\newline &amp; \\rightarrow -128 \\end{aligned} \\] <p>All the calculations above use addition. This points to an important fact: computer arithmetic circuits are largely built around addition. Compared with operations such as multiplication, division, and subtraction, addition is simpler to implement in hardware, easier to parallelize, and faster.</p> <p>This does not mean computers can only add. Combining addition with basic logical operations makes other arithmetic possible. For example, subtraction \\(a - b\\) becomes addition \\(a + (-b)\\), while multiplication and division can be expressed as repeated additions or subtractions.</p> <p>We can now summarize why computers use two's complement. It lets the same circuitry and operations handle both positive and negative addition, without dedicated subtraction circuitry or special handling for positive and negative zero. This simplifies hardware and improves efficiency.</p> <p>Two's complement is an elegant design. Space limits our discussion here, but interested readers are encouraged to explore it further.</p>","path":["Chapter 3.   Data structures","3.3   Number encoding *"],"tags":[]},{"location":"chapter_data_structure/number_encoding/#332-floating-point-encoding","level":2,"title":"3.3.2   Floating-point encoding","text":"<p>Both <code>int</code> and <code>float</code> occupy 4 bytes, yet <code>float</code> has a much wider range. This seems surprising: since <code>float</code> also represents fractions, should its range not be smaller?</p> <p>The reason is that <code>float</code> uses a different representation. Write a 32-bit binary number as:</p> \\[ b_{31} b_{30} b_{29} \\ldots b_2 b_1 b_0 \\] <p>Under IEEE 754, a 32-bit <code>float</code> has three parts.</p> <ul> <li>Sign \\(\\mathrm{S}\\): 1 bit, corresponding to \\(b_{31}\\).</li> <li>Exponent \\(\\mathrm{E}\\): 8 bits, corresponding to \\(b_{30} b_{29} \\ldots b_{23}\\).</li> <li>Fraction \\(\\mathrm{N}\\): 23 bits, corresponding to \\(b_{22} b_{21} \\ldots b_0\\).</li> </ul> <p>The value represented by the binary <code>float</code> is:</p> \\[ \\text {val} = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\] <p>In decimal notation, this becomes:</p> \\[ \\text {val}=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\] <p>The components have these ranges:</p> \\[ \\begin{aligned} \\mathrm{S} \\in &amp; \\{ 0, 1\\}, \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = &amp; (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\] <p></p> <p> Figure 3-5   Example of an IEEE 754 float calculation </p> <p>In Figure 3-5, \\(\\mathrm{S} = 0\\), \\(\\mathrm{E} = 124\\), and \\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\), giving:</p> \\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\] <p>We can now answer the original question: the exponent gives <code>float</code> a much wider range than <code>int</code>. The largest positive <code>float</code> is \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\); changing the sign bit gives the most negative value.</p> <p>The wider range of <code>float</code> comes at the cost of precision. An <code>int</code> uses all 32 bits to represent evenly spaced integer values. Because <code>float</code> includes an exponent, the gap between adjacent representable values tends to grow as their magnitude increases.</p> <p>As shown in Table 3-2, exponent values \\(\\mathrm{E} = 0\\) and \\(\\mathrm{E} = 255\\) have special meanings, representing zero, infinity, and \\(\\mathrm{NaN}\\), among other values.</p> <p> Table 3-2   Meanings of exponent values </p> Exponent E Fraction \\(\\mathrm{N} = 0\\) Fraction \\(\\mathrm{N} \\ne 0\\) Formula \\(0\\) \\(\\pm 0\\) Subnormal number \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) Normal number Normal number \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\) <p>Subnormal numbers substantially improve precision near zero. The smallest positive normal number is \\(2^{-126}\\), and the smallest positive subnormal number is \\(2^{-126} \\times 2^{-23}\\).</p> <p>Double-precision <code>double</code> uses a similar representation, which we will not detail here.</p>","path":["Chapter 3.   Data structures","3.3   Number encoding *"],"tags":[]},{"location":"chapter_data_structure/summary/","level":1,"title":"3.5   Summary","text":"","path":["Chapter 3.   Data structures","3.5   Summary"],"tags":[]},{"location":"chapter_data_structure/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<ul> <li>Data structures can be classified by logical and physical structure. Logical structure describes relationships between data elements, while physical structure describes how data is stored in memory.</li> <li>Common logical structures are linear, tree, and network structures. We usually classify data structures as linear (arrays, linked lists, stacks, queues) or nonlinear (trees, graphs, heaps). A hash table implementation may contain both linear and nonlinear structures.</li> <li>During execution, data resides in computer memory. Each location has a memory address, which programs use to access its data.</li> <li>The two main physical structures are contiguous storage (arrays) and scattered storage (linked lists). All data structures are implemented using arrays, linked lists, or both.</li> <li>Basic data types include integers (<code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>), floating-point numbers (<code>float</code>, <code>double</code>), characters (<code>char</code>), and Booleans (<code>bool</code>). Their value ranges depend on storage size and representation.</li> <li>Sign-magnitude, one's complement, and two's complement are three ways to encode numbers, and can be converted into one another. In sign-magnitude representation, the highest bit is the sign and the remaining bits give the magnitude.</li> <li>Integers are stored in two's complement. This lets the computer handle positive and negative addition uniformly, avoids separate hardware circuitry for subtraction, and eliminates the ambiguity of positive and negative zero.</li> <li>A floating-point encoding consists of 1 sign bit, 8 exponent bits, and 23 fraction bits. The exponent provides a much wider range than integers, at the cost of precision.</li> <li>ASCII was the earliest English character set. It uses 1 byte and contains 128 characters. GBK is a common Chinese character set with over 20,000 Chinese characters. Unicode aims to provide a comprehensive character standard for the world's languages, avoiding garbled text caused by incompatible encodings.</li> <li>UTF-8 is the most popular Unicode encoding and is widely compatible. Its variable length makes it extensible and space-efficient. UTF-16 and UTF-32 use fixed lengths. For Chinese text, UTF-16 uses less space than UTF-8. Languages such as Java and C# use UTF-16 by default.</li> </ul>","path":["Chapter 3.   Data structures","3.5   Summary"],"tags":[]},{"location":"chapter_data_structure/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: Why can a hash table contain both linear and nonlinear data structures?</p> <p>A hash table is built on an array. To handle hash collisions, it may use separate chaining, discussed later in the Hash collisions section: each array bucket points to a linked list, which may be converted to a tree (usually a red-black tree) if it grows beyond a threshold.</p> <p>From a storage perspective, each bucket in the underlying array may contain a value, a linked list, or a tree. A hash table may therefore contain both linear structures (arrays and linked lists) and nonlinear structures (trees).</p> <p>Q: Does <code>char</code> occupy 1 byte?</p> <p>The size of <code>char</code> depends on the programming language's encoding. For example, Java, JavaScript, TypeScript, and C# use UTF-16 to store Unicode code points, so their <code>char</code> type occupies 2 bytes.</p> <p>Q: Is calling array-based structures static misleading? A stack supports dynamic operations such as pushing and popping.</p> <p>A stack does support dynamic data operations, but its underlying structure remains static in length. Although array-based structures can add and remove elements dynamically, their capacity is fixed. When the data exceeds the allocated capacity, a larger array must be created and the old contents copied into it.</p> <p>Q: When we create a stack or queue without specifying its size, why is it still called a static data structure?</p> <p>In high-level languages, the class automatically chooses an initial capacity for a stack or queue. For example, Java's <code>ArrayList</code> typically starts with a capacity of 10. Resizing is also automatic; see the later section on lists.</p> <p>Q: Converting sign-magnitude to two's complement uses \"invert, then add 1\". Its inverse should be \"subtract 1, then invert\", so why does \"invert, then add 1\" also work in the reverse direction?</p> <p>These conversions are really computations of a complement. If \\(a + b = c\\), we call \\(a\\) the complement of \\(b\\) with respect to \\(c\\), and conversely \\(b\\) the complement of \\(a\\) with respect to \\(c\\).</p> <p>Consider the \\(n = 4\\)-bit binary number \\(0010\\). Treating it as sign-magnitude and disregarding the sign bit, we obtain its two's complement by inverting and adding 1:</p> \\[ 0010 \\rightarrow 1101 \\rightarrow 1110 \\] <p>The two representations sum to \\(0010 + 1110 = 10000\\). Thus, \\(1110\\) is the complement of \\(0010\\) with respect to \\(10000\\). The \"invert, then add 1\" operation computes the complement with respect to \\(10000\\).</p> <p>What is the complement of \\(1110\\) with respect to \\(10000\\)? We can again invert and add 1:</p> \\[ 1110 \\rightarrow 0001 \\rightarrow 0010 \\] <p>The two representations are complements of each other with respect to \\(10000\\), so conversion in either direction can use the same operation: invert, then add 1.</p> <p>Of course, we can also recover the sign-magnitude form of \\(1110\\) by the inverse operation: subtract 1, then invert.</p> \\[ 1110 \\rightarrow 1101 \\rightarrow 0010 \\] <p>Both \"invert, then add 1\" and \"subtract 1, then invert\" compute the complement with respect to \\(10000\\), so they are equivalent.</p> <p>In essence, inverting computes the complement with respect to \\(1111\\), since <code>sign-magnitude + one's complement = 1111</code>. Adding 1 to the one's complement then gives the complement with respect to \\(10000\\).</p> <p>The example uses \\(n = 4\\), but the reasoning extends to binary numbers of any bit width.</p>","path":["Chapter 3.   Data structures","3.5   Summary"],"tags":[]},{"location":"chapter_divide_and_conquer/","level":1,"title":"Chapter 12.   Divide and conquer","text":"<p>Abstract</p> <p>A difficult problem is broken down layer by layer, becoming simpler with each division.</p> <p>Divide and conquer reveals an important truth: start with the simple parts, and complexity becomes manageable.</p>","path":["Chapter 12.   Divide and conquer"],"tags":[]},{"location":"chapter_divide_and_conquer/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>12.1   Divide and conquer</li> <li>12.2   Divide-and-conquer search</li> <li>12.3   Building a binary tree</li> <li>12.4   Tower of Hanoi</li> <li>12.5   Summary</li> <li>12.6   Exercises</li> </ul>","path":["Chapter 12.   Divide and conquer"],"tags":[]},{"location":"chapter_divide_and_conquer/binary_search_recur/","level":1,"title":"12.2   Divide-and-conquer search","text":"<p>We have learned that search algorithms fall into two broad categories.</p> <ul> <li>Brute-force search: Traverses a data structure, taking \\(O(n)\\) time.</li> <li>Adaptive search: Uses a particular organization of data or prior information to achieve \\(O(\\log n)\\) or even \\(O(1)\\) time.</li> </ul> <p>In fact, search algorithms that run in \\(O(\\log n)\\) time are usually based on divide and conquer, as seen in binary search and trees.</p> <ul> <li>Each step of binary search reduces the problem of finding a target in an array to a smaller problem of finding it in half of the array. This continues until the array is empty or the target is found.</li> <li>Trees exemplify divide and conquer. In data structures such as binary search trees, AVL trees, and heaps, various operations take \\(O(\\log n)\\) time.</li> </ul> <p>Binary search uses divide and conquer in the following way.</p> <ul> <li>The problem can be divided: By comparing the middle element with the target, binary search recursively reduces the original problem of searching an array to the subproblem of searching half of that array.</li> <li>The subproblems are independent: Binary search handles only one subproblem at each step, unaffected by other subproblems.</li> <li>The subproblem solutions need not be combined: Binary search seeks a particular element, so there is no need to combine solutions. Solving the subproblem also solves the original problem.</li> </ul> <p>Divide and conquer makes searching faster because brute-force search eliminates only one option per step, whereas divide-and-conquer search can eliminate half the options.</p>","path":["Chapter 12.   Divide and conquer","12.2   Divide-and-conquer search"],"tags":[]},{"location":"chapter_divide_and_conquer/binary_search_recur/#1-implementing-binary-search-with-divide-and-conquer","level":3,"title":"1.   Implementing binary search with divide and conquer","text":"<p>Earlier chapters implemented binary search iteratively. We will now implement it recursively using divide and conquer.</p> <p>Question</p> <p>Given a sorted array <code>nums</code> of length \\(n\\) with distinct elements, find the element <code>target</code>.</p> <p>From a divide-and-conquer perspective, let \\(f(i, j)\\) denote the subproblem for the search interval \\([i, j]\\).</p> <p>Starting with the original problem \\(f(0, n-1)\\), perform binary search as follows.</p> <ol> <li>Calculate the midpoint \\(m\\) of the search interval \\([i, j]\\) and use it to eliminate half the interval.</li> <li>Recursively solve the subproblem of half the size, either \\(f(i, m-1)\\) or \\(f(m+1, j)\\).</li> <li>Repeat steps <code>1.</code> and <code>2.</code>, returning when <code>target</code> is found or the interval is empty.</li> </ol> <p>Figure 12-4 shows the divide-and-conquer process for finding \\(6\\) in an array using binary search.</p> <p></p> <p> Figure 12-4   Divide-and-conquer process for binary search </p> <p>In the implementation, we define a recursive function <code>dfs()</code> to solve \\(f(i, j)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_recur.py<pre><code>def dfs(nums: list[int], target: int, i: int, j: int) -&gt; int:\n    \"\"\"Binary search: problem f(i, j)\"\"\"\n    # If the interval is empty, the target is absent; return -1\n    if i &gt; j:\n        return -1\n    # Calculate the midpoint index m\n    m = (i + j) // 2\n    if nums[m] &lt; target:\n        # Recurse on subproblem f(m+1, j)\n        return dfs(nums, target, m + 1, j)\n    elif nums[m] &gt; target:\n        # Recurse on subproblem f(i, m-1)\n        return dfs(nums, target, i, m - 1)\n    else:\n        # The target element was found; return its index\n        return m\n\ndef binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"Binary search\"\"\"\n    n = len(nums)\n    # Solve problem f(0, n-1)\n    return dfs(nums, target, 0, n - 1)\n</code></pre> binary_search_recur.cpp<pre><code>/* Binary search: problem f(i, j) */\nint dfs(vector&lt;int&gt; &amp;nums, int target, int i, int j) {\n    // If the interval is empty, the target is absent; return -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // Calculate the midpoint index m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // Recurse on subproblem f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // Recurse on subproblem f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // The target element was found; return its index\n        return m;\n    }\n}\n\n/* Binary search */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    int n = nums.size();\n    // Solve problem f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.java<pre><code>/* Binary search: problem f(i, j) */\nint dfs(int[] nums, int target, int i, int j) {\n    // If the interval is empty, the target is absent; return -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // Calculate the midpoint index m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // Recurse on subproblem f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // Recurse on subproblem f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // The target element was found; return its index\n        return m;\n    }\n}\n\n/* Binary search */\nint binarySearch(int[] nums, int target) {\n    int n = nums.length;\n    // Solve problem f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.cs<pre><code>/* Binary search: problem f(i, j) */\nint DFS(int[] nums, int target, int i, int j) {\n    // If the interval is empty, the target is absent; return -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // Calculate the midpoint index m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // Recurse on subproblem f(m+1, j)\n        return DFS(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // Recurse on subproblem f(i, m-1)\n        return DFS(nums, target, i, m - 1);\n    } else {\n        // The target element was found; return its index\n        return m;\n    }\n}\n\n/* Binary search */\nint BinarySearch(int[] nums, int target) {\n    int n = nums.Length;\n    // Solve problem f(0, n-1)\n    return DFS(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.go<pre><code>/* Binary search: problem f(i, j) */\nfunc dfs(nums []int, target, i, j int) int {\n    // If the interval is empty, the target is absent; return -1\n    if i &gt; j {\n        return -1\n    }\n    //    Calculate the midpoint index\n    m := i + ((j - i) &gt;&gt; 1)\n    //Compare the midpoint element with the target\n    if nums[m] &lt; target {\n        // If it is smaller, recurse on the right half\n        // Recurse on subproblem f(m+1, j)\n        return dfs(nums, target, m+1, j)\n    } else if nums[m] &gt; target {\n        // If it is larger, recurse on the left half\n        // Recurse on subproblem f(i, m-1)\n        return dfs(nums, target, i, m-1)\n    } else {\n        // The target element was found; return its index\n        return m\n    }\n}\n\n/* Binary search */\nfunc binarySearch(nums []int, target int) int {\n    n := len(nums)\n    return dfs(nums, target, 0, n-1)\n}\n</code></pre> binary_search_recur.swift<pre><code>/* Binary search: problem f(i, j) */\nfunc dfs(nums: [Int], target: Int, i: Int, j: Int) -&gt; Int {\n    // If the interval is empty, the target is absent; return -1\n    if i &gt; j {\n        return -1\n    }\n    // Calculate the midpoint index m\n    let m = (i + j) / 2\n    if nums[m] &lt; target {\n        // Recurse on subproblem f(m+1, j)\n        return dfs(nums: nums, target: target, i: m + 1, j: j)\n    } else if nums[m] &gt; target {\n        // Recurse on subproblem f(i, m-1)\n        return dfs(nums: nums, target: target, i: i, j: m - 1)\n    } else {\n        // The target element was found; return its index\n        return m\n    }\n}\n\n/* Binary search */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // Solve problem f(0, n-1)\n    dfs(nums: nums, target: target, i: nums.startIndex, j: nums.endIndex - 1)\n}\n</code></pre> binary_search_recur.js<pre><code>/* Binary search: problem f(i, j) */\nfunction dfs(nums, target, i, j) {\n    // If the interval is empty, the target is absent; return -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // Calculate the midpoint index m\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // Recurse on subproblem f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // Recurse on subproblem f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // The target element was found; return its index\n        return m;\n    }\n}\n\n/* Binary search */\nfunction binarySearch(nums, target) {\n    const n = nums.length;\n    // Solve problem f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.ts<pre><code>/* Binary search: problem f(i, j) */\nfunction dfs(nums: number[], target: number, i: number, j: number): number {\n    // If the interval is empty, the target is absent; return -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // Calculate the midpoint index m\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // Recurse on subproblem f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // Recurse on subproblem f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // The target element was found; return its index\n        return m;\n    }\n}\n\n/* Binary search */\nfunction binarySearch(nums: number[], target: number): number {\n    const n = nums.length;\n    // Solve problem f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.dart<pre><code>/* Binary search: problem f(i, j) */\nint dfs(List&lt;int&gt; nums, int target, int i, int j) {\n  // If the interval is empty, the target is absent; return -1\n  if (i &gt; j) {\n    return -1;\n  }\n  // Calculate the midpoint index m\n  int m = (i + j) ~/ 2;\n  if (nums[m] &lt; target) {\n    // Recurse on subproblem f(m+1, j)\n    return dfs(nums, target, m + 1, j);\n  } else if (nums[m] &gt; target) {\n    // Recurse on subproblem f(i, m-1)\n    return dfs(nums, target, i, m - 1);\n  } else {\n    // The target element was found; return its index\n    return m;\n  }\n}\n\n/* Binary search */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  int n = nums.length;\n  // Solve problem f(0, n-1)\n  return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.rs<pre><code>/* Binary search: problem f(i, j) */\nfn dfs(nums: &amp;[i32], target: i32, i: i32, j: i32) -&gt; i32 {\n    // If the interval is empty, the target is absent; return -1\n    if i &gt; j {\n        return -1;\n    }\n    let m: i32 = i + (j - i) / 2;\n    if nums[m as usize] &lt; target {\n        // Recurse on subproblem f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if nums[m as usize] &gt; target {\n        // Recurse on subproblem f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // The target element was found; return its index\n        return m;\n    }\n}\n\n/* Binary search */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let n = nums.len() as i32;\n    // Solve problem f(0, n-1)\n    dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.c<pre><code>/* Binary search: problem f(i, j) */\nint dfs(int nums[], int target, int i, int j) {\n    // If the interval is empty, the target is absent; return -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // Calculate the midpoint index m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // Recurse on subproblem f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // Recurse on subproblem f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // The target element was found; return its index\n        return m;\n    }\n}\n\n/* Binary search */\nint binarySearch(int nums[], int target, int numsSize) {\n    int n = numsSize;\n    // Solve problem f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.kt<pre><code>/* Binary search: problem f(i, j) */\nfun dfs(\n    nums: IntArray,\n    target: Int,\n    i: Int,\n    j: Int\n): Int {\n    // If the interval is empty, the target is absent; return -1\n    if (i &gt; j) {\n        return -1\n    }\n    // Calculate the midpoint index m\n    val m = (i + j) / 2\n    return if (nums[m] &lt; target) {\n        // Recurse on subproblem f(m+1, j)\n        dfs(nums, target, m + 1, j)\n    } else if (nums[m] &gt; target) {\n        // Recurse on subproblem f(i, m-1)\n        dfs(nums, target, i, m - 1)\n    } else {\n        // The target element was found; return its index\n        m\n    }\n}\n\n/* Binary search */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    val n = nums.size\n    // Solve problem f(0, n-1)\n    return dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.rb<pre><code>### Binary search: problem f(i, j) ###\ndef dfs(nums, target, i, j)\n  # If the interval is empty, the target is absent; return -1\n  return -1 if i &gt; j\n\n  # Calculate the midpoint index m\n  m = (i + j) / 2\n\n  if nums[m] &lt; target\n    # Recurse on subproblem f(m+1, j)\n    return dfs(nums, target, m + 1, j)\n  elsif nums[m] &gt; target\n    # Recurse on subproblem f(i, m-1)\n    return dfs(nums, target, i, m - 1)\n  else\n    # The target element was found; return its index\n    return m\n  end\nend\n\n### Binary search ###\ndef binary_search(nums, target)\n  n = nums.length\n  # Solve problem f(0, n-1)\n  dfs(nums, target, 0, n - 1)\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 12.   Divide and conquer","12.2   Divide-and-conquer search"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/","level":1,"title":"12.3   Building a binary tree","text":"<p>Question</p> <p>Given a binary tree's preorder traversal <code>preorder</code> and inorder traversal <code>inorder</code>, reconstruct the tree and return its root node. Assume that all node values are distinct, as shown in Figure 12-5.</p> <p></p> <p> Figure 12-5   Example input for reconstructing a binary tree </p>","path":["Chapter 12.   Divide and conquer","12.3   Building a binary tree"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#1-identifying-a-divide-and-conquer-problem","level":3,"title":"1.   Identifying a divide-and-conquer problem","text":"<p>Reconstructing a binary tree from <code>preorder</code> and <code>inorder</code> is a typical divide-and-conquer problem.</p> <ul> <li>The problem can be divided: We can split the original problem into two subproblems, building the left subtree and building the right subtree, plus one operation: initializing the root node. Each subtree can be divided in the same way into smaller subtrees, until we reach the smallest subproblem, an empty subtree.</li> <li>The subproblems are independent: The left and right subtrees are independent and do not overlap. To build the left subtree, we need only the portions of the preorder and inorder traversals corresponding to it. The same applies to the right subtree.</li> <li>The subproblem solutions can be combined: Once we have built the left and right subtrees, we can attach them to the root to obtain the solution to the original problem.</li> </ul>","path":["Chapter 12.   Divide and conquer","12.3   Building a binary tree"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#2-dividing-the-subtrees","level":3,"title":"2.   Dividing the subtrees","text":"<p>The analysis above shows that divide and conquer applies, but how do we use <code>preorder</code> and <code>inorder</code> to separate the left and right subtrees?</p> <p>By definition, both <code>preorder</code> and <code>inorder</code> consist of three parts.</p> <ul> <li>Preorder: <code>[ root | left subtree | right subtree ]</code>. For the tree above, this is <code>[ 3 | 9 | 2 1 7 ]</code>.</li> <li>Inorder: <code>[ left subtree | root | right subtree ]</code>. For the tree above, this is <code>[ 9 | 3 | 1 2 7 ]</code>.</li> </ul> <p>Using the example above, we can divide the traversals with the steps shown in Figure 12-6.</p> <ol> <li>The first element in the preorder traversal, 3, is the root value.</li> <li>Find the index of root value 3 in <code>inorder</code>, then use it to divide <code>inorder</code> into <code>[ 9 | 3 | 1 2 7 ]</code>.</li> <li>This division of <code>inorder</code> tells us that the left and right subtrees have 1 and 3 nodes, respectively, allowing us to divide <code>preorder</code> into <code>[ 3 | 9 | 2 1 7 ]</code>.</li> </ol> <p></p> <p> Figure 12-6   Separating subtrees in preorder and inorder traversals </p>","path":["Chapter 12.   Divide and conquer","12.3   Building a binary tree"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#3-describing-subtree-intervals-with-variables","level":3,"title":"3.   Describing subtree intervals with variables","text":"<p>Using this method, we have identified the index intervals for the root, left subtree, and right subtree in <code>preorder</code> and <code>inorder</code>. We need a few pointer variables to describe these intervals.</p> <ul> <li>Let \\(i\\) be the index of the current tree's root in <code>preorder</code>.</li> <li>Let \\(m\\) be the index of the current tree's root in <code>inorder</code>.</li> <li>Let \\([l, r]\\) be the interval occupied by the current tree in <code>inorder</code>.</li> </ul> <p>As shown in Table 12-1, these variables describe the root index in <code>preorder</code> and each subtree's interval in <code>inorder</code>.</p> <p> Table 12-1   Root and subtree indices in preorder and inorder traversals </p> Root index in <code>preorder</code> Subtree interval in <code>inorder</code> Current tree \\(i\\) \\([l, r]\\) Left subtree \\(i + 1\\) \\([l, m-1]\\) Right subtree \\(i + 1 + (m - l)\\) \\([m+1, r]\\) <p>The term \\((m-l)\\) in the right subtree's root index is the number of nodes in the left subtree. Figure 12-7 helps illustrate this.</p> <p></p> <p> Figure 12-7   Index intervals for the root and the left and right subtrees </p>","path":["Chapter 12.   Divide and conquer","12.3   Building a binary tree"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#4-code-implementation","level":3,"title":"4.   Code implementation","text":"<p>To find \\(m\\) efficiently, we use a hash table <code>hmap</code> mapping each element of <code>inorder</code> to its index:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby build_tree.py<pre><code>def dfs(\n    preorder: list[int],\n    inorder_map: dict[int, int],\n    i: int,\n    l: int,\n    r: int,\n) -&gt; TreeNode | None:\n    \"\"\"Build a binary tree: divide and conquer\"\"\"\n    # Stop when the subtree interval is empty\n    if r - l &lt; 0:\n        return None\n    # Initialize the root node\n    root = TreeNode(preorder[i])\n    # Look up m to separate the left and right subtrees\n    m = inorder_map[preorder[i]]\n    # Subproblem: build the left subtree\n    root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n    # Subproblem: build the right subtree\n    root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n    # Return the root node\n    return root\n\ndef build_tree(preorder: list[int], inorder: list[int]) -&gt; TreeNode | None:\n    \"\"\"Build a binary tree\"\"\"\n    # Initialize a hash table mapping each element in inorder to its index\n    inorder_map = {val: i for i, val in enumerate(inorder)}\n    root = dfs(preorder, inorder_map, 0, 0, len(inorder) - 1)\n    return root\n</code></pre> build_tree.cpp<pre><code>/* Build a binary tree: divide and conquer */\nTreeNode *dfs(vector&lt;int&gt; &amp;preorder, unordered_map&lt;int, int&gt; &amp;inorderMap, int i, int l, int r) {\n    // Stop when the subtree interval is empty\n    if (r - l &lt; 0)\n        return NULL;\n    // Initialize the root node\n    TreeNode *root = new TreeNode(preorder[i]);\n    // Look up m to separate the left and right subtrees\n    int m = inorderMap[preorder[i]];\n    // Subproblem: build the left subtree\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // Subproblem: build the right subtree\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // Return the root node\n    return root;\n}\n\n/* Build a binary tree */\nTreeNode *buildTree(vector&lt;int&gt; &amp;preorder, vector&lt;int&gt; &amp;inorder) {\n    // Initialize a hash table mapping each element in inorder to its index\n    unordered_map&lt;int, int&gt; inorderMap;\n    for (int i = 0; i &lt; inorder.size(); i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorder.size() - 1);\n    return root;\n}\n</code></pre> build_tree.java<pre><code>/* Build a binary tree: divide and conquer */\nTreeNode dfs(int[] preorder, Map&lt;Integer, Integer&gt; inorderMap, int i, int l, int r) {\n    // Stop when the subtree interval is empty\n    if (r - l &lt; 0)\n        return null;\n    // Initialize the root node\n    TreeNode root = new TreeNode(preorder[i]);\n    // Look up m to separate the left and right subtrees\n    int m = inorderMap.get(preorder[i]);\n    // Subproblem: build the left subtree\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // Subproblem: build the right subtree\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // Return the root node\n    return root;\n}\n\n/* Build a binary tree */\nTreeNode buildTree(int[] preorder, int[] inorder) {\n    // Initialize a hash table mapping each element in inorder to its index\n    Map&lt;Integer, Integer&gt; inorderMap = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; inorder.length; i++) {\n        inorderMap.put(inorder[i], i);\n    }\n    TreeNode root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.cs<pre><code>/* Build a binary tree: divide and conquer */\nTreeNode? DFS(int[] preorder, Dictionary&lt;int, int&gt; inorderMap, int i, int l, int r) {\n    // Stop when the subtree interval is empty\n    if (r - l &lt; 0)\n        return null;\n    // Initialize the root node\n    TreeNode root = new(preorder[i]);\n    // Look up m to separate the left and right subtrees\n    int m = inorderMap[preorder[i]];\n    // Subproblem: build the left subtree\n    root.left = DFS(preorder, inorderMap, i + 1, l, m - 1);\n    // Subproblem: build the right subtree\n    root.right = DFS(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // Return the root node\n    return root;\n}\n\n/* Build a binary tree */\nTreeNode? BuildTree(int[] preorder, int[] inorder) {\n    // Initialize a hash table mapping each element in inorder to its index\n    Dictionary&lt;int, int&gt; inorderMap = [];\n    for (int i = 0; i &lt; inorder.Length; i++) {\n        inorderMap.TryAdd(inorder[i], i);\n    }\n    TreeNode? root = DFS(preorder, inorderMap, 0, 0, inorder.Length - 1);\n    return root;\n}\n</code></pre> build_tree.go<pre><code>/* Build a binary tree: divide and conquer */\nfunc dfsBuildTree(preorder []int, inorderMap map[int]int, i, l, r int) *TreeNode {\n    // Stop when the subtree interval is empty\n    if r-l &lt; 0 {\n        return nil\n    }\n    // Initialize the root node\n    root := NewTreeNode(preorder[i])\n    // Look up m to separate the left and right subtrees\n    m := inorderMap[preorder[i]]\n    // Subproblem: build the left subtree\n    root.Left = dfsBuildTree(preorder, inorderMap, i+1, l, m-1)\n    // Subproblem: build the right subtree\n    root.Right = dfsBuildTree(preorder, inorderMap, i+1+m-l, m+1, r)\n    // Return the root node\n    return root\n}\n\n/* Build a binary tree */\nfunc buildTree(preorder, inorder []int) *TreeNode {\n    // Initialize a hash table mapping each element in inorder to its index\n    inorderMap := make(map[int]int, len(inorder))\n    for i := 0; i &lt; len(inorder); i++ {\n        inorderMap[inorder[i]] = i\n    }\n\n    root := dfsBuildTree(preorder, inorderMap, 0, 0, len(inorder)-1)\n    return root\n}\n</code></pre> build_tree.swift<pre><code>/* Build a binary tree: divide and conquer */\nfunc dfs(preorder: [Int], inorderMap: [Int: Int], i: Int, l: Int, r: Int) -&gt; TreeNode? {\n    // Stop when the subtree interval is empty\n    if r - l &lt; 0 {\n        return nil\n    }\n    // Initialize the root node\n    let root = TreeNode(x: preorder[i])\n    // Look up m to separate the left and right subtrees\n    let m = inorderMap[preorder[i]]!\n    // Subproblem: build the left subtree\n    root.left = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1, l: l, r: m - 1)\n    // Subproblem: build the right subtree\n    root.right = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1 + m - l, l: m + 1, r: r)\n    // Return the root node\n    return root\n}\n\n/* Build a binary tree */\nfunc buildTree(preorder: [Int], inorder: [Int]) -&gt; TreeNode? {\n    // Initialize a hash table mapping each element in inorder to its index\n    let inorderMap = inorder.enumerated().reduce(into: [:]) { $0[$1.element] = $1.offset }\n    return dfs(preorder: preorder, inorderMap: inorderMap, i: inorder.startIndex, l: inorder.startIndex, r: inorder.endIndex - 1)\n}\n</code></pre> build_tree.js<pre><code>/* Build a binary tree: divide and conquer */\nfunction dfs(preorder, inorderMap, i, l, r) {\n    // Stop when the subtree interval is empty\n    if (r - l &lt; 0) return null;\n    // Initialize the root node\n    const root = new TreeNode(preorder[i]);\n    // Look up m to separate the left and right subtrees\n    const m = inorderMap.get(preorder[i]);\n    // Subproblem: build the left subtree\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // Subproblem: build the right subtree\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // Return the root node\n    return root;\n}\n\n/* Build a binary tree */\nfunction buildTree(preorder, inorder) {\n    // Initialize a hash table mapping each element in inorder to its index\n    let inorderMap = new Map();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.ts<pre><code>/* Build a binary tree: divide and conquer */\nfunction dfs(\n    preorder: number[],\n    inorderMap: Map&lt;number, number&gt;,\n    i: number,\n    l: number,\n    r: number\n): TreeNode | null {\n    // Stop when the subtree interval is empty\n    if (r - l &lt; 0) return null;\n    // Initialize the root node\n    const root: TreeNode = new TreeNode(preorder[i]);\n    // Look up m to separate the left and right subtrees\n    const m = inorderMap.get(preorder[i]);\n    // Subproblem: build the left subtree\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // Subproblem: build the right subtree\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // Return the root node\n    return root;\n}\n\n/* Build a binary tree */\nfunction buildTree(preorder: number[], inorder: number[]): TreeNode | null {\n    // Initialize a hash table mapping each element in inorder to its index\n    let inorderMap = new Map&lt;number, number&gt;();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.dart<pre><code>/* Build a binary tree: divide and conquer */\nTreeNode? dfs(\n  List&lt;int&gt; preorder,\n  Map&lt;int, int&gt; inorderMap,\n  int i,\n  int l,\n  int r,\n) {\n  // Stop when the subtree interval is empty\n  if (r - l &lt; 0) {\n    return null;\n  }\n  // Initialize the root node\n  TreeNode? root = TreeNode(preorder[i]);\n  // Look up m to separate the left and right subtrees\n  int m = inorderMap[preorder[i]]!;\n  // Subproblem: build the left subtree\n  root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n  // Subproblem: build the right subtree\n  root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n  // Return the root node\n  return root;\n}\n\n/* Build a binary tree */\nTreeNode? buildTree(List&lt;int&gt; preorder, List&lt;int&gt; inorder) {\n  // Initialize a hash table mapping each element in inorder to its index\n  Map&lt;int, int&gt; inorderMap = {};\n  for (int i = 0; i &lt; inorder.length; i++) {\n    inorderMap[inorder[i]] = i;\n  }\n  TreeNode? root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n  return root;\n}\n</code></pre> build_tree.rs<pre><code>/* Build a binary tree: divide and conquer */\nfn dfs(\n    preorder: &amp;[i32],\n    inorder_map: &amp;HashMap&lt;i32, i32&gt;,\n    i: i32,\n    l: i32,\n    r: i32,\n) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // Stop when the subtree interval is empty\n    if r - l &lt; 0 {\n        return None;\n    }\n    // Initialize the root node\n    let root = TreeNode::new(preorder[i as usize]);\n    // Look up m to separate the left and right subtrees\n    let m = inorder_map.get(&amp;preorder[i as usize]).unwrap();\n    // Subproblem: build the left subtree\n    root.borrow_mut().left = dfs(preorder, inorder_map, i + 1, l, m - 1);\n    // Subproblem: build the right subtree\n    root.borrow_mut().right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r);\n    // Return the root node\n    Some(root)\n}\n\n/* Build a binary tree */\nfn build_tree(preorder: &amp;[i32], inorder: &amp;[i32]) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // Initialize a hash table mapping each element in inorder to its index\n    let mut inorder_map: HashMap&lt;i32, i32&gt; = HashMap::new();\n    for i in 0..inorder.len() {\n        inorder_map.insert(inorder[i], i as i32);\n    }\n    let root = dfs(preorder, &amp;inorder_map, 0, 0, inorder.len() as i32 - 1);\n    root\n}\n</code></pre> build_tree.c<pre><code>/* Build a binary tree: divide and conquer */\nTreeNode *dfs(int *preorder, int *inorderMap, int i, int l, int r, int size) {\n    // Stop when the subtree interval is empty\n    if (r - l &lt; 0)\n        return NULL;\n    // Initialize the root node\n    TreeNode *root = (TreeNode *)malloc(sizeof(TreeNode));\n    root-&gt;val = preorder[i];\n    root-&gt;left = NULL;\n    root-&gt;right = NULL;\n    // Look up m to separate the left and right subtrees\n    int m = inorderMap[preorder[i]];\n    // Subproblem: build the left subtree\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1, size);\n    // Subproblem: build the right subtree\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r, size);\n    // Return the root node\n    return root;\n}\n\n/* Build a binary tree */\nTreeNode *buildTree(int *preorder, int preorderSize, int *inorder, int inorderSize) {\n    // Initialize a lookup array mapping inorder values to their indices\n    int *inorderMap = (int *)malloc(sizeof(int) * MAX_SIZE);\n    for (int i = 0; i &lt; inorderSize; i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorderSize - 1, inorderSize);\n    free(inorderMap);\n    return root;\n}\n</code></pre> build_tree.kt<pre><code>/* Build a binary tree: divide and conquer */\nfun dfs(\n    preorder: IntArray,\n    inorderMap: Map&lt;Int?, Int?&gt;,\n    i: Int,\n    l: Int,\n    r: Int\n): TreeNode? {\n    // Stop when the subtree interval is empty\n    if (r - l &lt; 0) return null\n    // Initialize the root node\n    val root = TreeNode(preorder[i])\n    // Look up m to separate the left and right subtrees\n    val m = inorderMap[preorder[i]]!!\n    // Subproblem: build the left subtree\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1)\n    // Subproblem: build the right subtree\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r)\n    // Return the root node\n    return root\n}\n\n/* Build a binary tree */\nfun buildTree(preorder: IntArray, inorder: IntArray): TreeNode? {\n    // Initialize a hash table mapping each element in inorder to its index\n    val inorderMap = HashMap&lt;Int?, Int?&gt;()\n    for (i in inorder.indices) {\n        inorderMap[inorder[i]] = i\n    }\n    val root = dfs(preorder, inorderMap, 0, 0, inorder.size - 1)\n    return root\n}\n</code></pre> build_tree.rb<pre><code>### Build a binary tree: divide and conquer ###\ndef dfs(preorder, inorder_map, i, l, r)\n  # Stop when the subtree interval is empty\n  return if r - l &lt; 0\n\n  # Initialize the root node\n  root = TreeNode.new(preorder[i])\n  # Look up m to separate the left and right subtrees\n  m = inorder_map[preorder[i]]\n  # Subproblem: build the left subtree\n  root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n  # Subproblem: build the right subtree\n  root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n\n  # Return the root node\n  root\nend\n\n### Build a binary tree ###\ndef build_tree(preorder, inorder)\n  # Initialize a hash table mapping each element in inorder to its index\n  inorder_map = {}\n  inorder.each_with_index { |val, i| inorder_map[val] = i }\n  dfs(preorder, inorder_map, 0, 0, inorder.length - 1)\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 12-8 shows the recursive reconstruction process. Nodes are created during recursive descent, while edges (references) are established as the recursive calls return.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 12-8   Recursive process for building a binary tree </p> <p>Figure 12-9 shows how <code>preorder</code> and <code>inorder</code> are divided in each recursive call.</p> <p></p> <p> Figure 12-9   Division within each recursive call </p> <p>For a tree with \\(n\\) nodes, initializing each node (executing one call to <code>dfs()</code>) takes \\(O(1)\\) time. The overall time complexity is therefore \\(O(n)\\).</p> <p>The hash table maps <code>inorder</code> elements to indices, requiring \\(O(n)\\) space. In the worst case, the binary tree degenerates into a linked list, and the recursion depth reaches \\(n\\), using \\(O(n)\\) space for stack frames. The overall space complexity is therefore \\(O(n)\\).</p>","path":["Chapter 12.   Divide and conquer","12.3   Building a binary tree"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/","level":1,"title":"12.1   Divide and conquer","text":"<p>Divide and conquer is a fundamental and widely used algorithmic strategy. Usually implemented recursively, it has two stages: dividing the problem and combining its solutions.</p> <ol> <li>Divide (the partitioning stage): Recursively split the original problem into two or more subproblems until the smallest subproblems are reached.</li> <li>Conquer (the combining stage): Starting with the known solutions to the smallest subproblems, combine solutions from the bottom up to construct the solution to the original problem.</li> </ol> <p>As shown in Figure 12-1, merge sort is a classic application of divide and conquer.</p> <ol> <li>Divide: Recursively split the original array (the original problem) into two subarrays (subproblems) until each subarray contains just one element (the smallest subproblem).</li> <li>Conquer: Merge the sorted subarrays (subproblem solutions) from the bottom up to obtain the sorted original array (the solution to the original problem).</li> </ol> <p></p> <p> Figure 12-1   Divide-and-conquer strategy in merge sort </p>","path":["Chapter 12.   Divide and conquer","12.1   Divide and conquer"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1211-recognizing-divide-and-conquer-problems","level":2,"title":"12.1.1   Recognizing divide-and-conquer problems","text":"<p>The following criteria usually help determine whether divide and conquer is suitable for a problem.</p> <ol> <li>The problem can be divided: The original problem can be split into smaller, similar subproblems, which can in turn be divided recursively in the same way.</li> <li>The subproblems are independent: They neither overlap nor depend on one another, so they can be solved independently.</li> <li>The subproblem solutions can be combined: Combining them yields the solution to the original problem.</li> </ol> <p>Merge sort clearly satisfies all three criteria.</p> <ol> <li>The problem can be divided: Recursively split an array (the original problem) into two subarrays (subproblems).</li> <li>The subproblems are independent: Each subarray can be sorted independently.</li> <li>The subproblem solutions can be combined: Merge the two sorted subarrays (subproblem solutions) into a single sorted array (the solution to the original problem).</li> </ol>","path":["Chapter 12.   Divide and conquer","12.1   Divide and conquer"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1212-improving-efficiency-with-divide-and-conquer","level":2,"title":"12.1.2   Improving efficiency with divide and conquer","text":"<p>Divide and conquer not only solves algorithmic problems effectively, but often improves efficiency as well. Quick sort, merge sort, and heap sort outperform selection sort, bubble sort, and insertion sort because they use this strategy.</p> <p>This raises a question: why does divide and conquer improve efficiency, and what is the underlying reason? Why is dividing a large problem, solving its subproblems, and combining their solutions faster than solving it directly? We can examine this from two perspectives: the number of operations and parallel computation.</p>","path":["Chapter 12.   Divide and conquer","12.1   Divide and conquer"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1-reducing-the-number-of-operations","level":3,"title":"1.   Reducing the number of operations","text":"<p>Consider bubble sort, which takes \\(O(n^2)\\) time to sort an array of length \\(n\\). Suppose we split the array at its midpoint, as shown in Figure 12-2. Splitting takes \\(O(n)\\) time, sorting each half takes \\(O((n / 2)^2)\\) time, and merging the halves takes \\(O(n)\\) time. The total time complexity is:</p> \\[ O(n + (\\frac{n}{2})^2 \\times 2 + n) = O(\\frac{n^2}{2} + 2n) \\] <p></p> <p> Figure 12-2   Bubble sort before and after dividing the array </p> <p>Now consider the following inequality, whose left and right sides represent the total operation counts before and after division:</p> \\[ \\begin{aligned} n^2 &amp; &gt; \\frac{n^2}{2} + 2n \\newline n^2 - \\frac{n^2}{2} - 2n &amp; &gt; 0 \\newline n(n - 4) &amp; &gt; 0 \\end{aligned} \\] <p>When \\(n &gt; 4\\), dividing the array reduces the number of operations and should make sorting faster. The time complexity is still quadratic, \\(O(n^2)\\); only the constant factor has decreased.</p> <p>What if we keep splitting each subarray into two at its midpoint until every subarray contains just one element? This is precisely the idea behind merge sort, which takes \\(O(n \\log n)\\) time.</p> <p>What if we introduce more split points, dividing the original array evenly into \\(k\\) subarrays? This resembles bucket sort, which is well suited to sorting large amounts of data and can theoretically achieve \\(O(n + k)\\) time.</p>","path":["Chapter 12.   Divide and conquer","12.1   Divide and conquer"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#2-using-parallel-computation","level":3,"title":"2.   Using parallel computation","text":"<p>The subproblems produced by divide and conquer are independent, so they can usually be solved in parallel. Thus, divide and conquer can both reduce time complexity and help the operating system parallelize the work.</p> <p>Parallelization is especially effective on multicore or multiprocessor systems. Multiple subproblems can be processed simultaneously, making fuller use of computing resources and substantially reducing total runtime.</p> <p>For example, in the bucket sort shown in Figure 12-3, a large dataset is evenly distributed among buckets. The sorting tasks for different buckets can then be assigned to different computing units, and their results combined afterward.</p> <p></p> <p> Figure 12-3   Parallel computation in bucket sort </p>","path":["Chapter 12.   Divide and conquer","12.1   Divide and conquer"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1213-common-applications-of-divide-and-conquer","level":2,"title":"12.1.3   Common applications of divide and conquer","text":"<p>Divide and conquer solves many classic algorithmic problems.</p> <ul> <li>Closest pair of points: Divide a set of points into two parts, find the closest pair within each part, then find the closest pair spanning the two parts.</li> <li>Large integer multiplication: Karatsuba's algorithm, for example, breaks multiplication of large integers into several smaller integer multiplications and additions.</li> <li>Matrix multiplication: Strassen's algorithm, for example, breaks multiplication of large matrices into multiple smaller matrix multiplications and additions.</li> <li>Tower of Hanoi: This problem can be solved recursively and is a classic application of divide and conquer.</li> <li>Counting inversions: In a sequence, an earlier number and a later number form an inversion if the earlier one is larger. Inversions can be counted using divide and conquer with merge sort.</li> </ul> <p>Divide and conquer is also widely used in designing algorithms and data structures.</p> <ul> <li>Binary search: Split a sorted array at its middle index, compare the middle element with the target to decide which half to discard, then repeat the same operation on the remaining interval.</li> <li>Merge sort: This was described at the start of the section.</li> <li>Quick sort: Choose a pivot and divide the array into two subarrays, one containing elements smaller than the pivot and the other containing larger elements. Repeat the division on both parts until each subarray contains just one element.</li> <li>Bucket sort: Distribute data among multiple buckets, sort the elements in each bucket, and then take the elements from the buckets in order to obtain a sorted array.</li> <li>Trees: Operations such as search, insertion, and deletion in binary search trees, AVL trees, red-black trees, B-trees, and B+ trees can be viewed as applications of divide and conquer.</li> <li>Heaps: A heap is a special kind of complete binary tree. Its operations, such as insertion, deletion, and heapification, implicitly use divide and conquer.</li> <li>Hash tables: Although hash tables do not directly use divide and conquer, some collision-handling strategies use it indirectly. For example, long linked lists in separate chaining may be converted into red-black trees to speed up lookups.</li> </ul> <p>As we can see, divide and conquer quietly underlies many algorithms and data structures, often without being explicitly named.</p>","path":["Chapter 12.   Divide and conquer","12.1   Divide and conquer"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/","level":1,"title":"12.6   Exercises","text":"","path":["Chapter 12.   Divide and conquer","12.6   Exercises"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1261-check-your-understanding","level":2,"title":"12.6.1   Check your understanding","text":"","path":["Chapter 12.   Divide and conquer","12.6   Exercises"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1-which-tasks-suit-divide-and-conquer","level":3,"title":"1.   Which tasks suit divide and conquer?","text":"<p>A student wants to solve the following tasks by splitting each in half, solving the halves separately, and combining the results. For each task, decide whether divide and conquer is suitable, is possible but does not reduce the total work, or is unsuitable because the halves cannot be solved independently. Explain.</p> <ol> <li>Sort an unsorted array.</li> <li>Find the maximum value in an array.</li> <li>Execute a sequence of stack operations <code>push(x)</code> and <code>pop()</code> in order, outputting the element returned by each <code>pop()</code>.</li> </ol> Reference answer <ol> <li>Suitable: split in half, sort the halves independently, and merge in \\(O(n)\\) time. This is exactly merge sort.</li> <li>Possible, but without reducing the total work: the two halves must still examine all \\(n\\) elements between them,     taking \\(O(n)\\) time, just like a direct scan.</li> <li>The halves cannot be solved independently: the stack contents at the start of the second half depend on the results of the first half.     The second half therefore cannot run independently of the first half's result.</li> </ol>","path":["Chapter 12.   Divide and conquer","12.6   Exercises"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#2-how-fast-exponentiation-reduces-computation","level":3,"title":"2.   How fast exponentiation reduces computation","text":"<p>The following recursive function uses divide and conquer to compute \\(x^n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby fast_power.py<pre><code>def fast_pow(x: int, n: int) -&gt; int:\n    \"\"\"Fast exponentiation\"\"\"\n    if n == 0:\n        return 1\n    half = fast_pow(x, n // 2)\n    if n % 2 == 0:\n        return half * half\n    return half * half * x\n</code></pre> fast_power.cpp<pre><code>/* Fast exponentiation */\nint fastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = fastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.java<pre><code>/* Fast exponentiation */\nint fastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = fastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.cs<pre><code>/* Fast exponentiation */\nint FastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = FastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.go<pre><code>/* Fast exponentiation */\nfunc fastPow(x, n int) int {\n    if n == 0 {\n        return 1\n    }\n    half := fastPow(x, n/2)\n    if n%2 == 0 {\n        return half * half\n    }\n    return half * half * x\n}\n</code></pre> fast_power.swift<pre><code>/* Fast exponentiation */\nfunc fastPow(x: Int, n: Int) -&gt; Int {\n    if n == 0 {\n        return 1\n    }\n    let half = fastPow(x: x, n: n / 2)\n    if n % 2 == 0 {\n        return half * half\n    }\n    return half * half * x\n}\n</code></pre> fast_power.js<pre><code>/* Fast exponentiation */\nfunction fastPow(x, n) {\n    if (n === 0) {\n        return 1;\n    }\n    const half = fastPow(x, Math.floor(n / 2));\n    if (n % 2 === 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.ts<pre><code>/* Fast exponentiation */\nfunction fastPow(x: number, n: number): number {\n    if (n === 0) {\n        return 1;\n    }\n    const half = fastPow(x, Math.floor(n / 2));\n    if (n % 2 === 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.dart<pre><code>/* Fast exponentiation */\nint fastPow(int x, int n) {\n  if (n == 0) {\n    return 1;\n  }\n  int half = fastPow(x, n ~/ 2);\n  if (n % 2 == 0) {\n    return half * half;\n  }\n  return half * half * x;\n}\n</code></pre> fast_power.rs<pre><code>/* Fast exponentiation */\nfn fast_pow(x: i32, n: i32) -&gt; i32 {\n    if n == 0 {\n        return 1;\n    }\n    let half = fast_pow(x, n / 2);\n    if n % 2 == 0 {\n        return half * half;\n    }\n    half * half * x\n}\n</code></pre> fast_power.c<pre><code>/* Fast exponentiation */\nint fastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = fastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.kt<pre><code>/* Fast exponentiation */\nfun fastPow(x: Int, n: Int): Int {\n    if (n == 0) {\n        return 1\n    }\n    val half = fastPow(x, n / 2)\n    if (n % 2 == 0) {\n        return half * half\n    }\n    return half * half * x\n}\n</code></pre> fast_power.rb<pre><code>### Fast exponentiation ###\ndef fast_pow(x, n)\n  return 1 if n == 0\n\n  half = fast_pow(x, n / 2)\n  return half * half if n.even?\n\n  half * half * x\nend\n</code></pre> <p>Run it with <code>x = 3</code> and <code>n = 5</code>:</p> <ol> <li>What values does <code>n</code> take in successive recursive calls?</li> <li>Starting with the deepest call, what value does each call return?</li> <li>Why should the recursive result be saved in <code>half</code> instead of making the same recursive call on each side of the multiplication?</li> </ol> Reference answer <ol> <li> <p>The parameter follows <code>5 → 2 → 1 → 0</code>, halving the exponent each time until the base case is reached.</p> </li> <li> <p>When <code>n = 0</code>, the return value is 1. When <code>n = 1</code>, it is \\(1×1×3=3\\).     When <code>n = 2</code>, it is \\(3×3=9\\); when <code>n = 5</code>, it is \\(9×9×3=243\\).</p> </li> <li> <p>Calling the same subproblem on both sides of the multiplication repeats exactly the same computation.     Saving the result in <code>half</code> means each level makes only one recursive call, with recursion depth approximately \\(\\log n\\).     Making two calls would cause substantial repeated computation.</p> </li> </ol>","path":["Chapter 12.   Divide and conquer","12.6   Exercises"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#3-splitting-traversal-sequences-into-left-and-right-subtrees","level":3,"title":"3.   Splitting traversal sequences into left and right subtrees","text":"<p>A binary tree with distinct node values has these preorder and inorder traversals:</p> <ul> <li>Preorder: <code>[A, B, D, E, C]</code></li> <li>Inorder: <code>[D, B, E, A, C]</code></li> </ul> <p>Split only at the root level; there is no need to recurse further or draw the entire tree.</p> <ol> <li>What is the root node?</li> <li>Which parts of the inorder sequence belong to the left and right subtrees?</li> <li>Which parts of the preorder sequence belong to the left and right subtrees? What are the root's immediate children?</li> </ol> Reference answer <ol> <li> <p>The first node in preorder is the root, so the root is <code>A</code>.</p> </li> <li> <p><code>A</code> splits the inorder sequence into two parts: <code>[D, B, E]</code> for the left subtree and <code>[C]</code> for the right subtree.</p> </li> <li> <p>The left subtree has three nodes, so the three preorder elements after root <code>A</code> belong to the left subtree:     <code>[B, D, E]</code>. The remaining <code>[C]</code> belongs to the right subtree.     Thus, the root's left child is <code>B</code>, and its right child is <code>C</code>.</p> </li> </ol>","path":["Chapter 12.   Divide and conquer","12.6   Exercises"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1262-programming-exercises","level":2,"title":"12.6.2   Programming exercises","text":"","path":["Chapter 12.   Divide and conquer","12.6   Exercises"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1-fast-exponentiation","level":3,"title":"1.   Fast exponentiation","text":"<p>Given a real number <code>x</code> and an integer <code>n</code>, compute \\(x^n\\) without calling a built-in power function. Use recursive divide and conquer: halve the exponent at each step and reuse the computed subproblem result. For this problem, \\(x^0=1\\), including when <code>x = 0</code>. When <code>n &lt; 0</code>, <code>x != 0</code> is guaranteed, and the answer can be expressed as \\((1/x)^{-n}\\).</p> Hints <ol> <li>When n is 0, the answer is 1.</li> <li>After computing x to the power n // 2, save it in half instead of making the recursive call a second time.</li> <li>When n &lt; 0, first replace x with 1 / x and n with -n. In C++ or Java, you can convert n to a 64-bit integer first to avoid overflow when negating the smallest 32-bit integer.</li> </ol> <p>LeetCode</p>","path":["Chapter 12.   Divide and conquer","12.6   Exercises"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/","level":1,"title":"12.4   Tower of Hanoi","text":"<p>In merge sort and binary tree reconstruction, we divided the original problem into two subproblems, each half its size. The Tower of Hanoi uses a different division strategy.</p> <p>Question</p> <p>Given three rods called <code>A</code>, <code>B</code>, and <code>C</code>, initially rod <code>A</code> holds \\(n\\) disks arranged from smallest at the top to largest at the bottom. Move all \\(n\\) disks to rod <code>C</code> while preserving their order, as shown in Figure 12-10. The following rules apply.</p> <ol> <li>A disk can only be removed from the top of one rod and placed on top of another.</li> <li>Only one disk may be moved at a time.</li> <li>A smaller disk must always be above a larger disk.</li> </ol> <p></p> <p> Figure 12-10   Example of the Tower of Hanoi </p> <p>Let \\(f(i)\\) denote a Tower of Hanoi problem of size \\(i\\). For example, \\(f(3)\\) means moving \\(3\\) disks from <code>A</code> to <code>C</code>.</p>","path":["Chapter 12.   Divide and conquer","12.4   Tower of Hanoi"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/#1-considering-the-base-cases","level":3,"title":"1.   Considering the base cases","text":"<p>For \\(f(1)\\), with just one disk, simply move it directly from <code>A</code> to <code>C</code>, as shown in Figure 12-11.</p> &lt;1&gt;&lt;2&gt; <p></p> <p></p> <p> Figure 12-11   Solution for a problem of size 1 </p> <p>For \\(f(2)\\), with two disks, we need to use <code>B</code> because the smaller disk must remain above the larger one, as shown in Figure 12-12.</p> <ol> <li>Move the smaller top disk from <code>A</code> to <code>B</code>.</li> <li>Move the larger disk from <code>A</code> to <code>C</code>.</li> <li>Move the smaller disk from <code>B</code> to <code>C</code>.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 12-12   Solution for a problem of size 2 </p> <p>We can summarize the solution to \\(f(2)\\) as moving two disks from <code>A</code> to <code>C</code> using <code>B</code>. Here, <code>C</code> is the destination rod and <code>B</code> is the auxiliary rod.</p>","path":["Chapter 12.   Divide and conquer","12.4   Tower of Hanoi"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/#2-dividing-the-problem","level":3,"title":"2.   Dividing the problem","text":"<p>For \\(f(3)\\), with three disks, the situation is a little more complicated.</p> <p>Since we know how to solve \\(f(1)\\) and \\(f(2)\\), we can use divide and conquer: treat the top two disks on <code>A</code> as a group and follow the steps in Figure 12-13. This moves all three disks from <code>A</code> to <code>C</code>.</p> <ol> <li>Use <code>B</code> as the destination and <code>C</code> as the auxiliary rod to move two disks from <code>A</code> to <code>B</code>.</li> <li>Move the remaining disk directly from <code>A</code> to <code>C</code>.</li> <li>Use <code>C</code> as the destination and <code>A</code> as the auxiliary rod to move two disks from <code>B</code> to <code>C</code>.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 12-13   Solution for a problem of size 3 </p> <p>In essence, we have divided \\(f(3)\\) into two \\(f(2)\\) subproblems and one \\(f(1)\\) subproblem. Solving these three subproblems in order solves the original problem. This shows that the subproblems are independent and their solutions can be combined.</p> <p>We can now state the divide-and-conquer strategy for the Tower of Hanoi, illustrated in Figure 12-14: divide \\(f(n)\\) into two subproblems \\(f(n-1)\\) and one subproblem \\(f(1)\\), then solve them in this order.</p> <ol> <li>Move \\(n-1\\) disks from <code>A</code> to <code>B</code> using <code>C</code>.</li> <li>Move the remaining \\(1\\) disk directly from <code>A</code> to <code>C</code>.</li> <li>Move \\(n-1\\) disks from <code>B</code> to <code>C</code> using <code>A</code>.</li> </ol> <p>Each of the two \\(f(n-1)\\) subproblems can be divided recursively in the same way until we reach the smallest subproblem, \\(f(1)\\). The solution to \\(f(1)\\) is a single move.</p> <p></p> <p> Figure 12-14   Divide-and-conquer strategy for the Tower of Hanoi </p>","path":["Chapter 12.   Divide and conquer","12.4   Tower of Hanoi"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/#3-code-implementation","level":3,"title":"3.   Code implementation","text":"<p>We define a recursive function <code>dfs(i, src, buf, tar)</code> that moves the top \\(i\\) disks from rod <code>src</code> to destination rod <code>tar</code> using auxiliary rod <code>buf</code>:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hanota.py<pre><code>def move(src: list[int], tar: list[int]):\n    \"\"\"Move one disk\"\"\"\n    # Take a disk from the top of src\n    pan = src.pop()\n    # Place the disk on top of tar\n    tar.append(pan)\n\ndef dfs(i: int, src: list[int], buf: list[int], tar: list[int]):\n    \"\"\"Solve the Tower of Hanoi problem f(i)\"\"\"\n    # If this subproblem has only one disk to move, move it directly from src to tar\n    if i == 1:\n        move(src, tar)\n        return\n    # Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, tar, buf)\n    # Subproblem f(1): move the remaining disk of this subproblem from src to tar\n    move(src, tar)\n    # Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, src, tar)\n\ndef solve_hanota(A: list[int], B: list[int], C: list[int]):\n    \"\"\"Solve the Tower of Hanoi problem\"\"\"\n    n = len(A)\n    # Move the top n disks from A to C using B\n    dfs(n, A, B, C)\n</code></pre> hanota.cpp<pre><code>/* Move one disk */\nvoid move(vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;tar) {\n    // Take a disk from the top of src\n    int pan = src.back();\n    src.pop_back();\n    // Place the disk on top of tar\n    tar.push_back(pan);\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nvoid dfs(int i, vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;buf, vector&lt;int&gt; &amp;tar) {\n    // If only one disk remains to be moved from src, move it directly to tar\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, tar, buf);\n    // Subproblem f(1): move the remaining disk of this subproblem from src to tar\n    move(src, tar);\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, src, tar);\n}\n\n/* Solve the Tower of Hanoi problem */\nvoid solveHanota(vector&lt;int&gt; &amp;A, vector&lt;int&gt; &amp;B, vector&lt;int&gt; &amp;C) {\n    int n = A.size();\n    // Move the top n disks from A to C using B\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.java<pre><code>/* Move one disk */\nvoid move(List&lt;Integer&gt; src, List&lt;Integer&gt; tar) {\n    // Take a disk from the top of src\n    Integer pan = src.remove(src.size() - 1);\n    // Place the disk on top of tar\n    tar.add(pan);\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nvoid dfs(int i, List&lt;Integer&gt; src, List&lt;Integer&gt; buf, List&lt;Integer&gt; tar) {\n    // If only one disk remains to be moved from src, move it directly to tar\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, tar, buf);\n    // Subproblem f(1): move the remaining disk of this subproblem from src to tar\n    move(src, tar);\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, src, tar);\n}\n\n/* Solve the Tower of Hanoi problem */\nvoid solveHanota(List&lt;Integer&gt; A, List&lt;Integer&gt; B, List&lt;Integer&gt; C) {\n    int n = A.size();\n    // Move the top n disks from A to C using B\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.cs<pre><code>/* Move one disk */\nvoid Move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n    // Take a disk from the top of src\n    int pan = src[^1];\n    src.RemoveAt(src.Count - 1);\n    // Place the disk on top of tar\n    tar.Add(pan);\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nvoid DFS(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n    // If only one disk in this subproblem remains on src, move it directly to tar\n    if (i == 1) {\n        Move(src, tar);\n        return;\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    DFS(i - 1, src, tar, buf);\n    // Subproblem f(1): move the remaining disk of this subproblem from src to tar\n    Move(src, tar);\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    DFS(i - 1, buf, src, tar);\n}\n\n/* Solve the Tower of Hanoi problem */\nvoid SolveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n    int n = A.Count;\n    // Move the top n disks from A to C using B\n    DFS(n, A, B, C);\n}\n</code></pre> hanota.go<pre><code>/* Move one disk */\nfunc move(src, tar *list.List) {\n    // Take a disk from the top of src\n    pan := src.Back()\n    // Place the disk on top of tar\n    tar.PushBack(pan.Value)\n    // Remove the disk from the top of src\n    src.Remove(pan)\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nfunc dfsHanota(i int, src, buf, tar *list.List) {\n    // If only one disk in this subproblem remains on src, move it directly to tar\n    if i == 1 {\n        move(src, tar)\n        return\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfsHanota(i-1, src, tar, buf)\n    // Subproblem f(1): move the remaining disk of this subproblem from src to tar\n    move(src, tar)\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfsHanota(i-1, buf, src, tar)\n}\n\n/* Solve the Tower of Hanoi problem */\nfunc solveHanota(A, B, C *list.List) {\n    n := A.Len()\n    // Move the top n disks from A to C using B\n    dfsHanota(n, A, B, C)\n}\n</code></pre> hanota.swift<pre><code>/* Move one disk */\nfunc move(src: inout [Int], tar: inout [Int]) {\n    // Take a disk from the top of src\n    let pan = src.popLast()!\n    // Place the disk on top of tar\n    tar.append(pan)\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nfunc dfs(i: Int, src: inout [Int], buf: inout [Int], tar: inout [Int]) {\n    // If only one disk in this subproblem remains on src, move it directly to tar\n    if i == 1 {\n        move(src: &amp;src, tar: &amp;tar)\n        return\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i: i - 1, src: &amp;src, buf: &amp;tar, tar: &amp;buf)\n    // Subproblem f(1): move the remaining disk of this subproblem from src to tar\n    move(src: &amp;src, tar: &amp;tar)\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i: i - 1, src: &amp;buf, buf: &amp;src, tar: &amp;tar)\n}\n\n/* Solve the Tower of Hanoi problem */\nfunc solveHanota(A: inout [Int], B: inout [Int], C: inout [Int]) {\n    let n = A.count\n    // The end of each list represents the top of the peg\n    // Move the top n disks from A to C using B\n    dfs(i: n, src: &amp;A, buf: &amp;B, tar: &amp;C)\n}\n</code></pre> hanota.js<pre><code>/* Move one disk */\nfunction move(src, tar) {\n    // Take a disk from the top of src\n    const pan = src.pop();\n    // Place the disk on top of tar\n    tar.push(pan);\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nfunction dfs(i, src, buf, tar) {\n    // If only one disk remains to be moved from src, move it directly to tar\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, tar, buf);\n    // Subproblem f(1): move the remaining disk of this subproblem from src to tar\n    move(src, tar);\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, src, tar);\n}\n\n/* Solve the Tower of Hanoi problem */\nfunction solveHanota(A, B, C) {\n    const n = A.length;\n    // Move the top n disks from A to C using B\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.ts<pre><code>/* Move one disk */\nfunction move(src: number[], tar: number[]): void {\n    // Take a disk from the top of src\n    const pan = src.pop();\n    // Place the disk on top of tar\n    tar.push(pan);\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nfunction dfs(i: number, src: number[], buf: number[], tar: number[]): void {\n    // If only one disk remains to be moved from src, move it directly to tar\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, tar, buf);\n    // Subproblem f(1): move the remaining disk of this subproblem from src to tar\n    move(src, tar);\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, src, tar);\n}\n\n/* Solve the Tower of Hanoi problem */\nfunction solveHanota(A: number[], B: number[], C: number[]): void {\n    const n = A.length;\n    // Move the top n disks from A to C using B\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.dart<pre><code>/* Move one disk */\nvoid move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n  // Take a disk from the top of src\n  int pan = src.removeLast();\n  // Place the disk on top of tar\n  tar.add(pan);\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nvoid dfs(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n  // If only one disk remains to be moved from src, move it directly to tar\n  if (i == 1) {\n    move(src, tar);\n    return;\n  }\n  // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n  dfs(i - 1, src, tar, buf);\n  // Subproblem f(1): move the remaining disk of this subproblem from src to tar\n  move(src, tar);\n  // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n  dfs(i - 1, buf, src, tar);\n}\n\n/* Solve the Tower of Hanoi problem */\nvoid solveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n  int n = A.length;\n  // Move the top n disks from A to C using B\n  dfs(n, A, B, C);\n}\n</code></pre> hanota.rs<pre><code>/* Move one disk */\nfn move_pan(src: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // Take a disk from the top of src\n    let pan = src.pop().unwrap();\n    // Place the disk on top of tar\n    tar.push(pan);\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nfn dfs(i: i32, src: &amp;mut Vec&lt;i32&gt;, buf: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // If only one disk remains to be moved from src, move it directly to tar\n    if i == 1 {\n        move_pan(src, tar);\n        return;\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, tar, buf);\n    // Subproblem f(1): move the remaining disk of this subproblem from src to tar\n    move_pan(src, tar);\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, src, tar);\n}\n\n/* Solve the Tower of Hanoi problem */\nfn solve_hanota(A: &amp;mut Vec&lt;i32&gt;, B: &amp;mut Vec&lt;i32&gt;, C: &amp;mut Vec&lt;i32&gt;) {\n    let n = A.len() as i32;\n    // Move the top n disks from A to C using B\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.c<pre><code>/* Move one disk */\nvoid move(int *src, int *srcSize, int *tar, int *tarSize) {\n    // Take a disk from the top of src\n    int pan = src[*srcSize - 1];\n    src[*srcSize - 1] = 0;\n    (*srcSize)--;\n    // Place the disk on top of tar\n    tar[*tarSize] = pan;\n    (*tarSize)++;\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nvoid dfs(int i, int *src, int *srcSize, int *buf, int *bufSize, int *tar, int *tarSize) {\n    // If only one disk remains to be moved from src, move it directly to tar\n    if (i == 1) {\n        move(src, srcSize, tar, tarSize);\n        return;\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, srcSize, tar, tarSize, buf, bufSize);\n    // Subproblem f(1): move the remaining disk of this subproblem from src to tar\n    move(src, srcSize, tar, tarSize);\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, bufSize, src, srcSize, tar, tarSize);\n}\n\n/* Solve the Tower of Hanoi problem */\nvoid solveHanota(int *A, int *ASize, int *B, int *BSize, int *C, int *CSize) {\n    // Move the top n disks from A to C using B\n    dfs(*ASize, A, ASize, B, BSize, C, CSize);\n}\n</code></pre> hanota.kt<pre><code>/* Move one disk */\nfun move(src: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // Take a disk from the top of src\n    val pan = src.removeAt(src.size - 1)\n    // Place the disk on top of tar\n    tar.add(pan)\n}\n\n/* Solve the Tower of Hanoi problem f(i) */\nfun dfs(i: Int, src: MutableList&lt;Int&gt;, buf: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // If only one disk in this subproblem remains on src, move it directly to tar\n    if (i == 1) {\n        move(src, tar)\n        return\n    }\n    // Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n    dfs(i - 1, src, tar, buf)\n    // Subproblem f(1): move the remaining disk of this subproblem from src to tar\n    move(src, tar)\n    // Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n    dfs(i - 1, buf, src, tar)\n}\n\n/* Solve the Tower of Hanoi problem */\nfun solveHanota(A: MutableList&lt;Int&gt;, B: MutableList&lt;Int&gt;, C: MutableList&lt;Int&gt;) {\n    val n = A.size\n    // Move the top n disks from A to C using B\n    dfs(n, A, B, C)\n}\n</code></pre> hanota.rb<pre><code>### Move one disk ###\ndef move(src, tar)\n  # Take a disk from the top of src\n  pan = src.pop\n  # Place the disk on top of tar\n  tar &lt;&lt; pan\nend\n\n### Solve the Tower of Hanoi problem f(i) ###\ndef dfs(i, src, buf, tar)\n  # If only one disk remains to be moved from src, move it directly to tar\n  if i == 1\n    move(src, tar)\n    return\n  end\n\n  # Subproblem f(i-1): move the top i-1 disks from src to buf using tar\n  dfs(i - 1, src, tar, buf)\n  # Subproblem f(1): move the remaining disk in the current subproblem from src to tar\n  move(src, tar)\n  # Subproblem f(i-1): move the top i-1 disks from buf to tar using src\n  dfs(i - 1, buf, src, tar)\nend\n\n### Solve the Tower of Hanoi problem ###\ndef solve_hanota(_A, _B, _C)\n  n = _A.length\n  # Move the top n disks from A to C using B\n  dfs(n, _A, _B, _C)\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>As shown in Figure 12-15, the Tower of Hanoi forms a recursion tree of height \\(n\\). Each node represents a subproblem and corresponds to a call to <code>dfs()</code>. The time complexity is therefore \\(O(2^n)\\), and the space complexity is \\(O(n)\\).</p> <p></p> <p> Figure 12-15   Recursion tree for the Tower of Hanoi </p> <p>Quote</p> <p>The Tower of Hanoi comes from an ancient legend. In a temple in ancient India, monks had three tall diamond rods and \\(64\\) gold disks of different sizes. They continually moved the disks, believing that the world would end when the last disk was placed in its correct position.</p> <p>Even at one move per second, however, this would take about \\(2^{64} \\approx 1.84×10^{19}\\) seconds, or roughly \\(585\\) billion years, far longer than current estimates of the age of the universe. If the legend is true, we should have little reason to worry about the end of the world.</p>","path":["Chapter 12.   Divide and conquer","12.4   Tower of Hanoi"],"tags":[]},{"location":"chapter_divide_and_conquer/summary/","level":1,"title":"12.5   Summary","text":"","path":["Chapter 12.   Divide and conquer","12.5   Summary"],"tags":[]},{"location":"chapter_divide_and_conquer/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<ul> <li>Divide and conquer is a common algorithm design strategy with two stages: dividing the problem and combining solutions. It is usually implemented recursively.</li> <li>To recognize a divide-and-conquer problem, ask whether it can be divided, whether the subproblems are independent, and whether their solutions can be combined.</li> <li>Merge sort is a classic example. It recursively divides an array into two equally sized subarrays until each contains one element, then merges them level by level to complete the sort.</li> <li>Divide and conquer often improves efficiency by reducing the number of operations and making it easier to parallelize the work.</li> <li>Divide and conquer solves many algorithmic problems and appears widely in the design of data structures and algorithms.</li> <li>Adaptive search is more efficient than brute-force search. Search algorithms with \\(O(\\log n)\\) time complexity are usually based on divide and conquer.</li> <li>Binary search is another classic example, with no step for combining subproblem solutions. It can be implemented through recursive divide and conquer.</li> <li>Reconstructing a binary tree can be divided into building its left and right subtrees. This is achieved by partitioning index intervals in the preorder and inorder traversals.</li> <li>A Tower of Hanoi problem of size \\(n\\) can be divided into two subproblems of size \\(n-1\\) and one of size \\(1\\). Solving these three subproblems in order solves the original problem.</li> </ul>","path":["Chapter 12.   Divide and conquer","12.5   Summary"],"tags":[]},{"location":"chapter_dynamic_programming/","level":1,"title":"Chapter 14.   Dynamic programming","text":"<p>Abstract</p> <p>Streams flow into rivers, and rivers flow into the sea.</p> <p>Dynamic programming brings together solutions to small problems to answer a larger one, guiding us step by step to the other shore.</p>","path":["Chapter 14.   Dynamic programming"],"tags":[]},{"location":"chapter_dynamic_programming/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>14.1   Introduction to dynamic programming</li> <li>14.2   Properties of dynamic programming problems</li> <li>14.3   Solving dynamic programming problems</li> <li>14.4   0-1 knapsack problem</li> <li>14.5   Unbounded knapsack problem</li> <li>14.6   Edit distance problem</li> <li>14.7   Summary</li> <li>14.8   Exercises</li> </ul>","path":["Chapter 14.   Dynamic programming"],"tags":[]},{"location":"chapter_dynamic_programming/dp_problem_features/","level":1,"title":"14.2   Properties of dynamic programming problems","text":"<p>The previous section showed how dynamic programming solves a problem by decomposing it into subproblems. Decomposition is a general algorithmic idea, but divide and conquer, dynamic programming, and backtracking use it differently.</p> <ul> <li>Divide and conquer recursively splits a problem into independent subproblems until the smallest ones are reached, then combines their solutions as the recursion returns to solve the original problem.</li> <li>Dynamic programming also decomposes problems recursively. Its main difference from divide and conquer is that its subproblems depend on one another, and many overlapping subproblems appear during decomposition.</li> <li>Backtracking enumerates possible solutions by trying choices and undoing them, using pruning to avoid unnecessary branches. A solution consists of a sequence of decisions, and the prefix before each decision can be viewed as a subproblem.</li> </ul> <p>Dynamic programming is often used for optimization problems. Besides overlapping subproblems, these problems have two key properties: optimal substructure and the no-aftereffect property.</p>","path":["Chapter 14.   Dynamic programming","14.2   Properties of dynamic programming problems"],"tags":[]},{"location":"chapter_dynamic_programming/dp_problem_features/#1421-optimal-substructure","level":2,"title":"14.2.1   Optimal substructure","text":"<p>Let's modify the climbing stairs problem slightly to illustrate optimal substructure more clearly.</p> <p>Minimum cost to climb stairs</p> <p>Given a staircase, you can climb \\(1\\) or \\(2\\) steps at a time. Each step has a nonnegative integer indicating the cost paid on that step. A nonnegative integer array \\(cost\\) gives the costs: \\(cost[i]\\) is the cost of step \\(i\\), and \\(cost[0]\\) refers to the ground (the starting point). What is the minimum cost of reaching the top?</p> <p>As shown in Figure 14-6, if steps \\(1\\), \\(2\\), and \\(3\\) cost \\(1\\), \\(10\\), and \\(1\\), respectively, the minimum cost of reaching step \\(3\\) from the ground is \\(2\\).</p> <p></p> <p> Figure 14-6   Minimum cost of reaching step 3 </p> <p>Let \\(dp[i]\\) be the total cost of reaching step \\(i\\). Since step \\(i\\) can only be reached from step \\(i - 1\\) or \\(i - 2\\), \\(dp[i]\\) must be either \\(dp[i - 1] + cost[i]\\) or \\(dp[i - 2] + cost[i]\\). To minimize the cost, choose the smaller of the two:</p> \\[ dp[i] = \\min(dp[i-1], dp[i-2]) + cost[i] \\] <p>This introduces the meaning of optimal substructure: an optimal solution to the original problem is constructed from optimal solutions to its subproblems.</p> <p>This problem clearly has optimal substructure: choose the better of \\(dp[i-1]\\) and \\(dp[i-2]\\) and use it to construct the optimal solution \\(dp[i]\\).</p> <p>Does the climbing stairs problem from the previous section also have optimal substructure? Its goal is to count solutions, so it seems to be a counting problem. But rephrase the goal as finding the maximum number of solutions, and although the problem remains equivalent, optimal substructure becomes apparent: the maximum number of ways to reach step \\(n\\) is the sum of the maximum numbers for steps \\(n-1\\) and \\(n-2\\). The interpretation of optimal substructure is therefore flexible and can vary by problem.</p> <p>The state transition equation, together with the initial states \\(dp[1] = cost[1]\\) and \\(dp[2] = cost[2]\\), gives the following dynamic programming code:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp(cost: list[int]) -&gt; int:\n    \"\"\"Minimum cost to climb stairs: dynamic programming\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    # Initialize the dp table to store subproblem solutions\n    dp = [0] * (n + 1)\n    # Initial states: set the solutions to the smallest subproblems\n    dp[1], dp[2] = cost[1], cost[2]\n    # State transitions: solve progressively larger subproblems using smaller ones\n    for i in range(3, n + 1):\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    return dp[n]\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* Minimum cost to climb stairs: dynamic programming */\nint minCostClimbingStairsDP(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // Initialize the dp table to store subproblem solutions\n    vector&lt;int&gt; dp(n + 1);\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* Minimum cost to climb stairs: dynamic programming */\nint minCostClimbingStairsDP(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // Initialize the dp table to store subproblem solutions\n    int[] dp = new int[n + 1];\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* Minimum cost to climb stairs: dynamic programming */\nint MinCostClimbingStairsDP(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // Initialize the dp table to store subproblem solutions\n    int[] dp = new int[n + 1];\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.Min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* Minimum cost to climb stairs: dynamic programming */\nfunc minCostClimbingStairsDP(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // Initialize the dp table to store subproblem solutions\n    dp := make([]int, n+1)\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = min(dp[i-1], dp[i-2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* Minimum cost to climb stairs: dynamic programming */\nfunc minCostClimbingStairsDP(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    // Initialize the dp table to store subproblem solutions\n    var dp = Array(repeating: 0, count: n + 1)\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for i in 3 ... n {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* Minimum cost to climb stairs: dynamic programming */\nfunction minCostClimbingStairsDP(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // Initialize the dp table to store subproblem solutions\n    const dp = new Array(n + 1);\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* Minimum cost to climb stairs: dynamic programming */\nfunction minCostClimbingStairsDP(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // Initialize the dp table to store subproblem solutions\n    const dp = new Array(n + 1);\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* Minimum cost to climb stairs: dynamic programming */\nint minCostClimbingStairsDP(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  // Initialize the dp table to store subproblem solutions\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // Initial states: set the solutions to the smallest subproblems\n  dp[1] = cost[1];\n  dp[2] = cost[2];\n  // State transitions: solve progressively larger subproblems using smaller ones\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n  }\n  return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* Minimum cost to climb stairs: dynamic programming */\nfn min_cost_climbing_stairs_dp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    }\n    // Initialize the dp table to store subproblem solutions\n    let mut dp = vec![-1; n + 1];\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for i in 3..=n {\n        dp[i] = cmp::min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* Minimum cost to climb stairs: dynamic programming */\nint minCostClimbingStairsDP(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // Initialize the dp table to store subproblem solutions\n    int *dp = calloc(n + 1, sizeof(int));\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = myMin(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    int res = dp[n];\n    // Free memory\n    free(dp);\n    return res;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* Minimum cost to climb stairs: dynamic programming */\nfun minCostClimbingStairsDP(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    // Initialize the dp table to store subproblem solutions\n    val dp = IntArray(n + 1)\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (i in 3..n) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>### Minimum cost to climb stairs: dynamic programming ###\ndef min_cost_climbing_stairs_dp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  # Initialize the dp table to store subproblem solutions\n  dp = Array.new(n + 1, 0)\n  # Initial states: set the solutions to the smallest subproblems\n  dp[1], dp[2] = cost[1], cost[2]\n  # State transitions: solve progressively larger subproblems using smaller ones\n  (3...(n + 1)).each { |i| dp[i] = [dp[i - 1], dp[i - 2]].min + cost[i] }\n  dp[n]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 14-7 illustrates this dynamic programming process.</p> <p></p> <p> Figure 14-7   Dynamic programming for the minimum climbing cost </p> <p>We can also reduce the state storage from a one-dimensional array to scalar variables, lowering the space complexity from \\(O(n)\\) to \\(O(1)\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp_comp(cost: list[int]) -&gt; int:\n    \"\"\"Minimum cost to climb stairs: space-optimized dynamic programming\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    a, b = cost[1], cost[2]\n    for i in range(3, n + 1):\n        a, b = b, min(a, b) + cost[i]\n    return b\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* Minimum cost to climb stairs: space-optimized dynamic programming */\nint minCostClimbingStairsDPComp(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* Minimum cost to climb stairs: space-optimized dynamic programming */\nint minCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* Minimum cost to climb stairs: space-optimized dynamic programming */\nint MinCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.Min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* Minimum cost to climb stairs: space-optimized dynamic programming */\nfunc minCostClimbingStairsDPComp(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // Initial states: set the solutions to the smallest subproblems\n    a, b := cost[1], cost[2]\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for i := 3; i &lt;= n; i++ {\n        tmp := b\n        b = min(a, tmp) + cost[i]\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* Minimum cost to climb stairs: space-optimized dynamic programming */\nfunc minCostClimbingStairsDPComp(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    var (a, b) = (cost[1], cost[2])\n    for i in 3 ... n {\n        (a, b) = (b, min(a, b) + cost[i])\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* Minimum cost to climb stairs: space-optimized dynamic programming */\nfunction minCostClimbingStairsDPComp(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* Minimum cost to climb stairs: space-optimized dynamic programming */\nfunction minCostClimbingStairsDPComp(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* Minimum cost to climb stairs: space-optimized dynamic programming */\nint minCostClimbingStairsDPComp(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  int a = cost[1], b = cost[2];\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = min(a, tmp) + cost[i];\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* Minimum cost to climb stairs: space-optimized dynamic programming */\nfn min_cost_climbing_stairs_dp_comp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    };\n    let (mut a, mut b) = (cost[1], cost[2]);\n    for i in 3..=n {\n        let tmp = b;\n        b = cmp::min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    b\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* Minimum cost to climb stairs: space-optimized dynamic programming */\nint minCostClimbingStairsDPComp(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = myMin(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* Minimum cost to climb stairs: space-optimized dynamic programming */\nfun minCostClimbingStairsDPComp(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    var a = cost[1]\n    var b = cost[2]\n    for (i in 3..n) {\n        val tmp = b\n        b = min(a, tmp) + cost[i]\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>### Minimum cost to climb stairs: dynamic programming ###\ndef min_cost_climbing_stairs_dp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  # Initialize the dp table to store subproblem solutions\n  dp = Array.new(n + 1, 0)\n  # Initial states: set the solutions to the smallest subproblems\n  dp[1], dp[2] = cost[1], cost[2]\n  # State transitions: solve progressively larger subproblems using smaller ones\n  (3...(n + 1)).each { |i| dp[i] = [dp[i - 1], dp[i - 2]].min + cost[i] }\n  dp[n]\nend\n\n# Minimum cost to climb stairs: space-optimized dynamic programming\ndef min_cost_climbing_stairs_dp_comp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  a, b = cost[1], cost[2]\n  (3...(n + 1)).each { |i| a, b = b, [a, b].min + cost[i] }\n  b\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 14.   Dynamic programming","14.2   Properties of dynamic programming problems"],"tags":[]},{"location":"chapter_dynamic_programming/dp_problem_features/#1422-no-aftereffect-property","level":2,"title":"14.2.2   No-aftereffect property","text":"<p>The no-aftereffect property is central to effective dynamic programming: given a fully specified state, its future depends only on that state, not on the sequence of states that led to it.</p> <p>In the climbing stairs problem, state \\(i\\) can lead to \\(i+1\\) or \\(i+2\\), corresponding to moves of \\(1\\) or \\(2\\) steps. Neither choice requires us to consider states before \\(i\\): they do not affect the future of state \\(i\\).</p> <p>Adding a constraint changes the situation.</p> <p>Climbing stairs with a constraint</p> <p>Given a staircase with \\(n\\) steps, you may climb \\(1\\) or \\(2\\) steps at a time, but may not make two consecutive moves of \\(1\\) step. How many ways are there to reach the top?</p> <p>As shown in Figure 14-8, only \\(2\\) ways remain to reach step \\(3\\): taking \\(1\\) step three times in a row violates the constraint and is discarded.</p> <p></p> <p> Figure 14-8   Number of ways to reach step 3 with the constraint </p> <p>If the previous move climbed \\(1\\) step, the next must climb \\(2\\) steps. Thus, the next choice cannot be determined by the current state (the current step number) alone; it also depends on the preceding state (the previous step number).</p> <p>The no-aftereffect property no longer holds with this state definition, and \\(dp[i] = dp[i-1] + dp[i-2]\\) is no longer valid. Using \\(dp[i-1]\\) means taking \\(1\\) step now, but it includes many ways in which the preceding move also took \\(1\\) step. We cannot simply add all of \\(dp[i-1]\\) to \\(dp[i]\\) without violating the constraint.</p> <p>We therefore expand the state definition: state \\([i, j]\\) means we are on step \\(i\\) and the previous move climbed \\(j\\) steps, where \\(j \\in \\{1, 2\\}\\). This distinguishes whether the preceding move climbed \\(1\\) or \\(2\\) steps, allowing us to determine how the current state could have been reached.</p> <ul> <li>If the previous move climbed \\(1\\) step, the move before it must have climbed \\(2\\) steps. Thus, \\(dp[i, 1]\\) can only come from \\(dp[i-1, 2]\\).</li> <li>If the previous move climbed \\(2\\) steps, the move before it could have climbed \\(1\\) or \\(2\\) steps. Thus, \\(dp[i, 2]\\) can come from \\(dp[i-2, 1]\\) or \\(dp[i-2, 2]\\).</li> </ul> <p>As shown in Figure 14-9, \\(dp[i, j]\\) now counts the ways to reach state \\([i, j]\\). The state transition equations are:</p> \\[ \\begin{cases} dp[i, 1] = dp[i-1, 2] \\\\ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] \\end{cases} \\] <p></p> <p> Figure 14-9   Recurrence with the constraint </p> <p>Finally, return \\(dp[n, 1] + dp[n, 2]\\), the total number of ways to reach step \\(n\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_constraint_dp.py<pre><code>def climbing_stairs_constraint_dp(n: int) -&gt; int:\n    \"\"\"Climbing stairs with constraints: dynamic programming\"\"\"\n    if n == 1 or n == 2:\n        return 1\n    # Initialize the dp table to store subproblem solutions\n    dp = [[0] * 3 for _ in range(n + 1)]\n    # Initial states: set the solutions to the smallest subproblems\n    dp[1][1], dp[1][2] = 1, 0\n    dp[2][1], dp[2][2] = 0, 1\n    # State transitions: solve progressively larger subproblems using smaller ones\n    for i in range(3, n + 1):\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    return dp[n][1] + dp[n][2]\n</code></pre> climbing_stairs_constraint_dp.cpp<pre><code>/* Climbing stairs with constraints: dynamic programming */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // Initialize the dp table to store subproblem solutions\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(3, 0));\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.java<pre><code>/* Climbing stairs with constraints: dynamic programming */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // Initialize the dp table to store subproblem solutions\n    int[][] dp = new int[n + 1][3];\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.cs<pre><code>/* Climbing stairs with constraints: dynamic programming */\nint ClimbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // Initialize the dp table to store subproblem solutions\n    int[,] dp = new int[n + 1, 3];\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1, 1] = 1;\n    dp[1, 2] = 0;\n    dp[2, 1] = 0;\n    dp[2, 2] = 1;\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i, 1] = dp[i - 1, 2];\n        dp[i, 2] = dp[i - 2, 1] + dp[i - 2, 2];\n    }\n    return dp[n, 1] + dp[n, 2];\n}\n</code></pre> climbing_stairs_constraint_dp.go<pre><code>/* Climbing stairs with constraints: dynamic programming */\nfunc climbingStairsConstraintDP(n int) int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // Initialize the dp table to store subproblem solutions\n    dp := make([][3]int, n+1)\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for i := 3; i &lt;= n; i++ {\n        dp[i][1] = dp[i-1][2]\n        dp[i][2] = dp[i-2][1] + dp[i-2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.swift<pre><code>/* Climbing stairs with constraints: dynamic programming */\nfunc climbingStairsConstraintDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // Initialize the dp table to store subproblem solutions\n    var dp = Array(repeating: Array(repeating: 0, count: 3), count: n + 1)\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for i in 3 ... n {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.js<pre><code>/* Climbing stairs with constraints: dynamic programming */\nfunction climbingStairsConstraintDP(n) {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // Initialize the dp table to store subproblem solutions\n    const dp = Array.from(new Array(n + 1), () =&gt; new Array(3));\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.ts<pre><code>/* Climbing stairs with constraints: dynamic programming */\nfunction climbingStairsConstraintDP(n: number): number {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // Initialize the dp table to store subproblem solutions\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(3));\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.dart<pre><code>/* Climbing stairs with constraints: dynamic programming */\nint climbingStairsConstraintDP(int n) {\n  if (n == 1 || n == 2) {\n    return 1;\n  }\n  // Initialize the dp table to store subproblem solutions\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(3, 0));\n  // Initial states: set the solutions to the smallest subproblems\n  dp[1][1] = 1;\n  dp[1][2] = 0;\n  dp[2][1] = 0;\n  dp[2][2] = 1;\n  // State transitions: solve progressively larger subproblems using smaller ones\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i][1] = dp[i - 1][2];\n    dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n  }\n  return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.rs<pre><code>/* Climbing stairs with constraints: dynamic programming */\nfn climbing_stairs_constraint_dp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return 1;\n    };\n    // Initialize the dp table to store subproblem solutions\n    let mut dp = vec![vec![-1; 3]; n + 1];\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for i in 3..=n {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.c<pre><code>/* Climbing stairs with constraints: dynamic programming */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // Initialize the dp table to store subproblem solutions\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(3, sizeof(int));\n    }\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    int res = dp[n][1] + dp[n][2];\n    // Free memory\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> climbing_stairs_constraint_dp.kt<pre><code>/* Climbing stairs with constraints: dynamic programming */\nfun climbingStairsConstraintDP(n: Int): Int {\n    if (n == 1 || n == 2) {\n        return 1\n    }\n    // Initialize the dp table to store subproblem solutions\n    val dp = Array(n + 1) { IntArray(3) }\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (i in 3..n) {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.rb<pre><code>### Climbing stairs with constraints: dynamic programming ###\ndef climbing_stairs_constraint_dp(n)\n  return 1 if n == 1 || n == 2\n\n  # Initialize the dp table to store subproblem solutions\n  dp = Array.new(n + 1) { Array.new(3, 0) }\n  # Initial states: set the solutions to the smallest subproblems\n  dp[1][1], dp[1][2] = 1, 0\n  dp[2][1], dp[2][2] = 0, 1\n  # State transitions: solve progressively larger subproblems using smaller ones\n  for i in 3...(n + 1)\n    dp[i][1] = dp[i - 1][2]\n    dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n  end\n\n  dp[n][1] + dp[n][2]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>In this example, only one preceding state mattered, so expanding the state definition restored the no-aftereffect property. Some problems, however, depend much more strongly on their history.</p> <p>Climbing stairs that generate obstacles</p> <p>Given a staircase with \\(n\\) steps, you may climb \\(1\\) or \\(2\\) steps at a time. Whenever you reach step \\(i\\), the system places an obstacle on step \\(2i\\), and you may never land on step \\(2i\\) afterward. For example, if your first two moves land on steps \\(2\\) and \\(3\\), then steps \\(4\\) and \\(6\\) are blocked for all later moves. How many ways are there to reach the top?</p> <p>Here, each next move depends on all previous states, because every move places an obstacle higher up and affects future moves. Such problems are often difficult to solve with dynamic programming.</p> <p>Many complex combinatorial optimization problems, such as the traveling salesperson problem, do not satisfy the no-aftereffect property. For such problems, we often choose other methods, such as heuristic search, genetic algorithms, or reinforcement learning, to obtain useful locally optimal solutions within a limited time.</p>","path":["Chapter 14.   Dynamic programming","14.2   Properties of dynamic programming problems"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/","level":1,"title":"14.3   Solving dynamic programming problems","text":"<p>The previous two sections introduced the main properties of dynamic programming problems. We will now explore two practical questions.</p> <ol> <li>How can we recognize a dynamic programming problem?</li> <li>Where should we start, and what are the steps for solving one?</li> </ol>","path":["Chapter 14.   Dynamic programming","14.3   Solving dynamic programming problems"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1431-recognizing-the-problem","level":2,"title":"14.3.1   Recognizing the problem","text":"<p>In general, a problem is suitable for dynamic programming if it has overlapping subproblems, optimal substructure, and the no-aftereffect property. These can be difficult to identify directly from a problem statement, so we usually start with a broader test: could backtracking (exhaustive search) solve it?</p> <p>Problems suited to backtracking usually fit a decision tree model. They can be represented as a tree in which each node is a decision and each path is a sequence of decisions.</p> <p>In other words, if a problem has clear choices and a solution is produced through a sequence of them, it fits the decision tree model and can usually be solved by backtracking.</p> <p>Beyond this, some signs make dynamic programming more promising.</p> <ul> <li>The problem asks for an optimum, such as a maximum, minimum, greatest number, or smallest number.</li> <li>States can be represented by a list, a multidimensional matrix, or a tree, and a recurrence relates each state to nearby states.</li> </ul> <p>Other signs make it less promising.</p> <ul> <li>The goal is to find all possible solutions rather than an optimal one.</li> <li>The problem explicitly involves permutations or combinations and asks for multiple concrete arrangements.</li> </ul> <p>If a problem fits the decision tree model and has clear positive signs, we can tentatively treat it as a dynamic programming problem and verify that assumption while solving it.</p>","path":["Chapter 14.   Dynamic programming","14.3   Solving dynamic programming problems"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1432-steps-for-solving-the-problem","level":2,"title":"14.3.2   Steps for solving the problem","text":"<p>The process varies with the problem's nature and difficulty, but usually includes describing the decisions, defining states, creating a \\(dp\\) table, deriving a state transition equation, and identifying boundary conditions.</p> <p>To make these steps concrete, we will use the classic minimum path sum problem.</p> <p>Question</p> <p>Given an \\(n \\times m\\) grid <code>grid</code>, each cell contains a nonnegative integer representing its cost. A robot starts at the top-left cell and can move one cell down or right at a time until it reaches the bottom-right cell. Return the minimum path sum from the top-left to the bottom-right.</p> <p>In the example shown in Figure 14-10, the minimum path sum is \\(13\\).</p> <p></p> <p> Figure 14-10   Example input for minimum path sum </p> <p>Step 1: Identify each decision, define the state, and establish the \\(dp\\) table</p> <p>Each decision moves one cell down or right. If the current cell has indices \\([i, j]\\), the next indices are \\([i+1, j]\\) or \\([i, j+1]\\). The state therefore contains two variables, the row and column indices, and is denoted by \\([i, j]\\).</p> <p>State \\([i, j]\\) corresponds to the subproblem of finding the minimum path sum from the starting point \\([0, 0]\\) to \\([i, j]\\). Its solution is \\(dp[i, j]\\).</p> <p>This gives the two-dimensional \\(dp\\) matrix shown in Figure 14-11, with the same dimensions as the input \\(grid\\).</p> <p></p> <p> Figure 14-11   State definition and dp table </p> <p>Note</p> <p>Dynamic programming and backtracking can be described as sequences of decisions. A state consists of the decision variables and should contain every variable needed to describe progress, with enough information to derive the next state.</p> <p>Each state corresponds to a subproblem. A \\(dp\\) table stores the solutions to all subproblems, with one dimension of the \\(dp\\) table for each independent state variable. In essence, the \\(dp\\) table maps states to subproblem solutions.</p> <p>Step 2: Identify optimal substructure and derive the state transition equation</p> <p>State \\([i, j]\\) can only be reached from the cell above, \\([i-1, j]\\), or the cell to the left, \\([i, j-1]\\). The optimal substructure is therefore that the minimum path sum to \\([i, j]\\) is determined by the smaller of the minimum path sums to \\([i, j-1]\\) and \\([i-1, j]\\).</p> <p>This gives the state transition equation illustrated in Figure 14-12:</p> \\[ dp[i, j] = \\min(dp[i-1, j], dp[i, j-1]) + grid[i, j] \\] <p></p> <p> Figure 14-12   Optimal substructure and state transition equation </p> <p>Note</p> <p>Using the defined \\(dp\\) table, examine the relationship between the original problem and its subproblems. Find how to construct an optimal solution from optimal subproblem solutions: this is the optimal substructure.</p> <p>Once we identify optimal substructure, we can use it to build the state transition equation.</p> <p>Step 3: Determine the boundary conditions and evaluation order</p> <p>A state in the first row can only come from its left, and a state in the first column can only come from above. Thus, the first row \\(i = 0\\) and first column \\(j = 0\\) define the boundary conditions.</p> <p>As shown in Figure 14-13, each cell depends on the cells to its left and above. We therefore traverse the matrix with an outer loop over rows and an inner loop over columns.</p> <p></p> <p> Figure 14-13   Boundary conditions and evaluation order </p> <p>Note</p> <p>Boundary conditions initialize the \\(dp\\) table in dynamic programming and support pruning in search.</p> <p>The key to choosing an evaluation order is ensuring that all smaller subproblems needed by the current state have already been solved correctly.</p> <p>We could now write the dynamic programming solution directly. However, decomposing a problem is a top-down way of thinking, so implementing \"brute-force search \\(\\rightarrow\\) memoized search \\(\\rightarrow\\) dynamic programming\" often follows our reasoning more naturally.</p>","path":["Chapter 14.   Dynamic programming","14.3   Solving dynamic programming problems"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1-method-1-brute-force-search","level":3,"title":"1.   Method 1: Brute-force search","text":"<p>Starting from \\([i, j]\\), recursively decompose the problem into smaller states \\([i-1, j]\\) and \\([i, j-1]\\). The recursive function has the following elements.</p> <ul> <li>Parameters: The state \\([i, j]\\).</li> <li>Return value: The minimum path sum \\(dp[i, j]\\) from \\([0, 0]\\) to \\([i, j]\\).</li> <li>Base case: When \\(i = 0\\) and \\(j = 0\\), return \\(grid[0, 0]\\).</li> <li>Pruning: If \\(i &lt; 0\\) or \\(j &lt; 0\\), the indices are out of bounds. Return cost \\(+\\infty\\) to mark the path as infeasible.</li> </ul> <p>The implementation is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dfs(grid: list[list[int]], i: int, j: int) -&gt; int:\n    \"\"\"Minimum path sum: brute-force search\"\"\"\n    # Stop searching at the top-left cell\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # If a row or column index is out of bounds, return a cost of +∞\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)\n    up = min_path_sum_dfs(grid, i - 1, j)\n    left = min_path_sum_dfs(grid, i, j - 1)\n    # Return the minimum path cost from the top-left corner to (i, j)\n    return min(left, up) + grid[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* Minimum path sum: brute-force search */\nint minPathSumDFS(vector&lt;vector&lt;int&gt;&gt; &amp;grid, int i, int j) {\n    // Stop searching at the top-left cell\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If a row or column index is out of bounds, return a cost of +∞\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // Return the minimum path cost from the top-left corner to (i, j)\n    return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.java<pre><code>/* Minimum path sum: brute-force search */\nint minPathSumDFS(int[][] grid, int i, int j) {\n    // Stop searching at the top-left cell\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If a row or column index is out of bounds, return a cost of +∞\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // Return the minimum path cost from the top-left corner to (i, j)\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* Minimum path sum: brute-force search */\nint MinPathSumDFS(int[][] grid, int i, int j) {\n    // Stop searching at the top-left cell\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If a row or column index is out of bounds, return a cost of +∞\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)\n    int up = MinPathSumDFS(grid, i - 1, j);\n    int left = MinPathSumDFS(grid, i, j - 1);\n    // Return the minimum path cost from the top-left corner to (i, j)\n    return Math.Min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* Minimum path sum: brute-force search */\nfunc minPathSumDFS(grid [][]int, i, j int) int {\n    // Stop searching at the top-left cell\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // If a row or column index is out of bounds, return math.MaxInt to represent an infinite cost\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)\n    up := minPathSumDFS(grid, i-1, j)\n    left := minPathSumDFS(grid, i, j-1)\n    // Return the minimum path cost from the top-left corner to (i, j)\n    return int(math.Min(float64(left), float64(up))) + grid[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* Minimum path sum: brute-force search */\nfunc minPathSumDFS(grid: [[Int]], i: Int, j: Int) -&gt; Int {\n    // Stop searching at the top-left cell\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // If a row or column index is out of bounds, return Int.max to represent an infinite cost\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)\n    let up = minPathSumDFS(grid: grid, i: i - 1, j: j)\n    let left = minPathSumDFS(grid: grid, i: i, j: j - 1)\n    // Return the minimum path cost from the top-left corner to (i, j)\n    return min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* Minimum path sum: brute-force search */\nfunction minPathSumDFS(grid, i, j) {\n    // Stop searching at the top-left cell\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // If a row or column index is out of bounds, return a cost of +∞\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // Return the minimum path cost from the top-left corner to (i, j)\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* Minimum path sum: brute-force search */\nfunction minPathSumDFS(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // Stop searching at the top-left cell\n    if (i === 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If a row or column index is out of bounds, return a cost of +∞\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // Return the minimum path cost from the top-left corner to (i, j)\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* Minimum path sum: brute-force search */\nint minPathSumDFS(List&lt;List&lt;int&gt;&gt; grid, int i, int j) {\n  // Stop searching at the top-left cell\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // If a row or column index is out of bounds, return a large cost representing +∞\n  if (i &lt; 0 || j &lt; 0) {\n    // Dart int has no infinity value; use 2^31 as a sentinel for this example\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)\n  int up = minPathSumDFS(grid, i - 1, j);\n  int left = minPathSumDFS(grid, i, j - 1);\n  // Return the minimum path cost from the top-left corner to (i, j)\n  return min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* Minimum path sum: brute-force search */\nfn min_path_sum_dfs(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // Stop searching at the top-left cell\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // If a row or column index is out of bounds, return i32::MAX to represent infinite cost\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)\n    let up = min_path_sum_dfs(grid, i - 1, j);\n    let left = min_path_sum_dfs(grid, i, j - 1);\n    // Return the minimum path cost from the top-left corner to (i, j)\n    std::cmp::min(left, up) + grid[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* Minimum path sum: brute-force search */\nint minPathSumDFS(int grid[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // Stop searching at the top-left cell\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If a row or column index is out of bounds, return a cost of +∞\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // Return the minimum path cost from the top-left corner to (i, j)\n    return myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.kt<pre><code>/* Minimum path sum: brute-force search */\nfun minPathSumDFS(grid: Array&lt;IntArray&gt;, i: Int, j: Int): Int {\n    // Stop searching at the top-left cell\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // If a row or column index is out of bounds, return Int.MAX_VALUE as an infinite-cost sentinel\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)\n    val up = minPathSumDFS(grid, i - 1, j)\n    val left = minPathSumDFS(grid, i, j - 1)\n    // Return the minimum path cost from the top-left corner to (i, j)\n    return min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.rb<pre><code>### Minimum path sum: brute-force search ###\ndef min_path_sum_dfs(grid, i, j)\n  # Stop searching at the top-left cell\n  return grid[i][j] if i == 0 &amp;&amp; j == 0\n  # If a row or column index is out of bounds, return a cost of +∞\n  return Float::INFINITY if i &lt; 0 || j &lt; 0\n  # Calculate the minimum path costs from the top-left corner to (i-1, j) and (i, j-1)\n  up = min_path_sum_dfs(grid, i - 1, j)\n  left = min_path_sum_dfs(grid, i, j - 1)\n  # Return the minimum path cost from the top-left corner to (i, j)\n  [left, up].min + grid[i][j]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 14-14 shows the recursion tree rooted at \\(dp[2, 1]\\). It includes overlapping subproblems, whose number grows rapidly with the size of <code>grid</code>.</p> <p>The underlying reason for overlapping subproblems is that multiple paths can reach the same cell from the top-left corner.</p> <p></p> <p> Figure 14-14   Recursion tree for brute-force search </p> <p>Each state offers two choices, down and right, and reaching the bottom-right corner takes \\(m + n - 2\\) moves. The worst-case time complexity is therefore \\(O(2^{m + n})\\), where \\(n\\) and \\(m\\) are the numbers of rows and columns. This estimate ignores the grid boundaries, where only one choice remains, so the actual number of paths is smaller.</p>","path":["Chapter 14.   Dynamic programming","14.3   Solving dynamic programming problems"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#2-method-2-memoized-search","level":3,"title":"2.   Method 2: Memoized search","text":"<p>Introduce a memoization table <code>mem</code> with the same dimensions as <code>grid</code> to store subproblem solutions and prune repeated subproblems:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dfs_mem(\n    grid: list[list[int]], mem: list[list[int]], i: int, j: int\n) -&gt; int:\n    \"\"\"Minimum path sum: memoized search\"\"\"\n    # Stop searching at the top-left cell\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # If a row or column index is out of bounds, return a cost of +∞\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # If a result has already been recorded, return it directly\n    if mem[i][j] != -1:\n        return mem[i][j]\n    # Minimum path costs to the cells on the left and above\n    up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n    left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n    # Record and return the minimum path cost from the top-left corner to (i, j)\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* Minimum path sum: memoized search */\nint minPathSumDFSMem(vector&lt;vector&lt;int&gt;&gt; &amp;grid, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int j) {\n    // Stop searching at the top-left cell\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If a row or column index is out of bounds, return a cost of +∞\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // If a result has already been recorded, return it directly\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // Minimum path costs to the cells on the left and above\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // Record and return the minimum path cost from the top-left corner to (i, j)\n    mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.java<pre><code>/* Minimum path sum: memoized search */\nint minPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // Stop searching at the top-left cell\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If a row or column index is out of bounds, return a cost of +∞\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // If a result has already been recorded, return it directly\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // Minimum path costs to the cells on the left and above\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // Record and return the minimum path cost from the top-left corner to (i, j)\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* Minimum path sum: memoized search */\nint MinPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // Stop searching at the top-left cell\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If a row or column index is out of bounds, return a cost of +∞\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // If a result has already been recorded, return it directly\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // Minimum path costs to the cells on the left and above\n    int up = MinPathSumDFSMem(grid, mem, i - 1, j);\n    int left = MinPathSumDFSMem(grid, mem, i, j - 1);\n    // Record and return the minimum path cost from the top-left corner to (i, j)\n    mem[i][j] = Math.Min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* Minimum path sum: memoized search */\nfunc minPathSumDFSMem(grid, mem [][]int, i, j int) int {\n    // Stop searching at the top-left cell\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // If a row or column index is out of bounds, return math.MaxInt to represent an infinite cost\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // If a result has already been recorded, return it directly\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // Minimum path costs to the cells on the left and above\n    up := minPathSumDFSMem(grid, mem, i-1, j)\n    left := minPathSumDFSMem(grid, mem, i, j-1)\n    // Record and return the minimum path cost from the top-left corner to (i, j)\n    mem[i][j] = int(math.Min(float64(left), float64(up))) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* Minimum path sum: memoized search */\nfunc minPathSumDFSMem(grid: [[Int]], mem: inout [[Int]], i: Int, j: Int) -&gt; Int {\n    // Stop searching at the top-left cell\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // If a row or column index is out of bounds, return Int.max to represent an infinite cost\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // If a result has already been recorded, return it directly\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // Minimum path costs to the cells on the left and above\n    let up = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i - 1, j: j)\n    let left = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i, j: j - 1)\n    // Record and return the minimum path cost from the top-left corner to (i, j)\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* Minimum path sum: memoized search */\nfunction minPathSumDFSMem(grid, mem, i, j) {\n    // Stop searching at the top-left cell\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // If a row or column index is out of bounds, return a cost of +∞\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // If a result has already been recorded, return it directly\n    if (mem[i][j] !== -1) {\n        return mem[i][j];\n    }\n    // Minimum path costs to the cells on the left and above\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // Record and return the minimum path cost from the top-left corner to (i, j)\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* Minimum path sum: memoized search */\nfunction minPathSumDFSMem(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // Stop searching at the top-left cell\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // If a row or column index is out of bounds, return a cost of +∞\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // If a result has already been recorded, return it directly\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // Minimum path costs to the cells on the left and above\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // Record and return the minimum path cost from the top-left corner to (i, j)\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* Minimum path sum: memoized search */\nint minPathSumDFSMem(List&lt;List&lt;int&gt;&gt; grid, List&lt;List&lt;int&gt;&gt; mem, int i, int j) {\n  // Stop searching at the top-left cell\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // If a row or column index is out of bounds, return a large cost representing +∞\n  if (i &lt; 0 || j &lt; 0) {\n    // Dart int has no infinity value; use 2^31 as a sentinel for this example\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // If a result has already been recorded, return it directly\n  if (mem[i][j] != -1) {\n    return mem[i][j];\n  }\n  // Minimum path costs to the cells on the left and above\n  int up = minPathSumDFSMem(grid, mem, i - 1, j);\n  int left = minPathSumDFSMem(grid, mem, i, j - 1);\n  // Record and return the minimum path cost from the top-left corner to (i, j)\n  mem[i][j] = min(left, up) + grid[i][j];\n  return mem[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* Minimum path sum: memoized search */\nfn min_path_sum_dfs_mem(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // Stop searching at the top-left cell\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // If a row or column index is out of bounds, return i32::MAX to represent infinite cost\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // If a result has already been recorded, return it directly\n    if mem[i as usize][j as usize] != -1 {\n        return mem[i as usize][j as usize];\n    }\n    // Minimum path costs to the cells on the left and above\n    let up = min_path_sum_dfs_mem(grid, mem, i - 1, j);\n    let left = min_path_sum_dfs_mem(grid, mem, i, j - 1);\n    // Record and return the minimum path cost from the top-left corner to (i, j)\n    mem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];\n    mem[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* Minimum path sum: memoized search */\nint minPathSumDFSMem(int grid[MAX_SIZE][MAX_SIZE], int mem[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // Stop searching at the top-left cell\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // If a row or column index is out of bounds, return a cost of +∞\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // If a result has already been recorded, return it directly\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // Minimum path costs to the cells on the left and above\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // Record and return the minimum path cost from the top-left corner to (i, j)\n    mem[i][j] = myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.kt<pre><code>/* Minimum path sum: memoized search */\nfun minPathSumDFSMem(\n    grid: Array&lt;IntArray&gt;,\n    mem: Array&lt;IntArray&gt;,\n    i: Int,\n    j: Int\n): Int {\n    // Stop searching at the top-left cell\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // If a row or column index is out of bounds, return Int.MAX_VALUE as an infinite-cost sentinel\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // If a result has already been recorded, return it directly\n    if (mem[i][j] != -1) {\n        return mem[i][j]\n    }\n    // Minimum path costs to the cells on the left and above\n    val up = minPathSumDFSMem(grid, mem, i - 1, j)\n    val left = minPathSumDFSMem(grid, mem, i, j - 1)\n    // Record and return the minimum path cost from the top-left corner to (i, j)\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.rb<pre><code>### Minimum path sum: memoized search ###\ndef min_path_sum_dfs_mem(grid, mem, i, j)\n  # Stop searching at the top-left cell\n  return grid[0][0] if i == 0 &amp;&amp; j == 0\n  # If a row or column index is out of bounds, return a cost of +∞\n  return Float::INFINITY if i &lt; 0 || j &lt; 0\n  # If a result has already been recorded, return it directly\n  return mem[i][j] if mem[i][j] != -1\n  # Minimum path costs to the cells on the left and above\n  up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n  left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n  # Record and return the minimum path cost from the top-left corner to (i, j)\n  mem[i][j] = [left, up].min + grid[i][j]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>As shown in Figure 14-15, memoization ensures that every subproblem is solved only once. The time complexity is therefore determined by the number of states, the grid size \\(O(nm)\\).</p> <p></p> <p> Figure 14-15   Recursion tree for memoized search </p>","path":["Chapter 14.   Dynamic programming","14.3   Solving dynamic programming problems"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#3-method-3-dynamic-programming","level":3,"title":"3.   Method 3: Dynamic programming","text":"<p>The following code implements dynamic programming iteratively:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dp(grid: list[list[int]]) -&gt; int:\n    \"\"\"Minimum path sum: dynamic programming\"\"\"\n    n, m = len(grid), len(grid[0])\n    # Initialize the dp table\n    dp = [[0] * m for _ in range(n)]\n    dp[0][0] = grid[0][0]\n    # State transitions: first row\n    for j in range(1, m):\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    # State transitions: first column\n    for i in range(1, n):\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    # State transitions: remaining rows and columns\n    for i in range(1, n):\n        for j in range(1, m):\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n    return dp[n - 1][m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* Minimum path sum: dynamic programming */\nint minPathSumDP(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // Initialize the dp table\n    vector&lt;vector&lt;int&gt;&gt; dp(n, vector&lt;int&gt;(m));\n    dp[0][0] = grid[0][0];\n    // State transitions: first row\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // State transitions: first column\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // State transitions: remaining rows and columns\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* Minimum path sum: dynamic programming */\nint minPathSumDP(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // Initialize the dp table\n    int[][] dp = new int[n][m];\n    dp[0][0] = grid[0][0];\n    // State transitions: first row\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // State transitions: first column\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // State transitions: remaining rows and columns\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* Minimum path sum: dynamic programming */\nint MinPathSumDP(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // Initialize the dp table\n    int[,] dp = new int[n, m];\n    dp[0, 0] = grid[0][0];\n    // State transitions: first row\n    for (int j = 1; j &lt; m; j++) {\n        dp[0, j] = dp[0, j - 1] + grid[0][j];\n    }\n    // State transitions: first column\n    for (int i = 1; i &lt; n; i++) {\n        dp[i, 0] = dp[i - 1, 0] + grid[i][0];\n    }\n    // State transitions: remaining rows and columns\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i, j] = Math.Min(dp[i, j - 1], dp[i - 1, j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1, m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* Minimum path sum: dynamic programming */\nfunc minPathSumDP(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // Initialize the dp table\n    dp := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        dp[i] = make([]int, m)\n    }\n    dp[0][0] = grid[0][0]\n    // State transitions: first row\n    for j := 1; j &lt; m; j++ {\n        dp[0][j] = dp[0][j-1] + grid[0][j]\n    }\n    // State transitions: first column\n    for i := 1; i &lt; n; i++ {\n        dp[i][0] = dp[i-1][0] + grid[i][0]\n    }\n    // State transitions: remaining rows and columns\n    for i := 1; i &lt; n; i++ {\n        for j := 1; j &lt; m; j++ {\n            dp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(dp[i-1][j]))) + grid[i][j]\n        }\n    }\n    return dp[n-1][m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* Minimum path sum: dynamic programming */\nfunc minPathSumDP(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // Initialize the dp table\n    var dp = Array(repeating: Array(repeating: 0, count: m), count: n)\n    dp[0][0] = grid[0][0]\n    // State transitions: first row\n    for j in 1 ..&lt; m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // State transitions: first column\n    for i in 1 ..&lt; n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // State transitions: remaining rows and columns\n    for i in 1 ..&lt; n {\n        for j in 1 ..&lt; m {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* Minimum path sum: dynamic programming */\nfunction minPathSumDP(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // Initialize the dp table\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // State transitions: first row\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // State transitions: first column\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // State transitions: remaining rows and columns\n    for (let i = 1; i &lt; n; i++) {\n        for (let j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* Minimum path sum: dynamic programming */\nfunction minPathSumDP(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // Initialize the dp table\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // State transitions: first row\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // State transitions: first column\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // State transitions: remaining rows and columns\n    for (let i = 1; i &lt; n; i++) {\n        for (let j: number = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* Minimum path sum: dynamic programming */\nint minPathSumDP(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // Initialize the dp table\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n, (i) =&gt; List.filled(m, 0));\n  dp[0][0] = grid[0][0];\n  // State transitions: first row\n  for (int j = 1; j &lt; m; j++) {\n    dp[0][j] = dp[0][j - 1] + grid[0][j];\n  }\n  // State transitions: first column\n  for (int i = 1; i &lt; n; i++) {\n    dp[i][0] = dp[i - 1][0] + grid[i][0];\n  }\n  // State transitions: remaining rows and columns\n  for (int i = 1; i &lt; n; i++) {\n    for (int j = 1; j &lt; m; j++) {\n      dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n    }\n  }\n  return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* Minimum path sum: dynamic programming */\nfn min_path_sum_dp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // Initialize the dp table\n    let mut dp = vec![vec![0; m]; n];\n    dp[0][0] = grid[0][0];\n    // State transitions: first row\n    for j in 1..m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // State transitions: first column\n    for i in 1..n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // State transitions: remaining rows and columns\n    for i in 1..n {\n        for j in 1..m {\n            dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* Minimum path sum: dynamic programming */\nint minPathSumDP(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // Initialize the dp table\n    int **dp = malloc(n * sizeof(int *));\n    for (int i = 0; i &lt; n; i++) {\n        dp[i] = calloc(m, sizeof(int));\n    }\n    dp[0][0] = grid[0][0];\n    // State transitions: first row\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // State transitions: first column\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // State transitions: remaining rows and columns\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = myMin(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    int res = dp[n - 1][m - 1];\n    // Free memory\n    for (int i = 0; i &lt; n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* Minimum path sum: dynamic programming */\nfun minPathSumDP(grid: Array&lt;IntArray&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // Initialize the dp table\n    val dp = Array(n) { IntArray(m) }\n    dp[0][0] = grid[0][0]\n    // State transitions: first row\n    for (j in 1..&lt;m) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // State transitions: first column\n    for (i in 1..&lt;n) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // State transitions: remaining rows and columns\n    for (i in 1..&lt;n) {\n        for (j in 1..&lt;m) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>### Minimum path sum: dynamic programming ###\ndef min_path_sum_dp(grid)\n  n, m = grid.length, grid.first.length\n  # Initialize the dp table\n  dp = Array.new(n) { Array.new(m, 0) }\n  dp[0][0] = grid[0][0]\n  # State transitions: first row\n  (1...m).each { |j| dp[0][j] = dp[0][j - 1] + grid[0][j] }\n  # State transitions: first column\n  (1...n).each { |i| dp[i][0] = dp[i - 1][0] + grid[i][0] }\n  # State transitions: remaining rows and columns\n  for i in 1...n\n    for j in 1...m\n      dp[i][j] = [dp[i][j - 1], dp[i - 1][j]].min + grid[i][j]\n    end\n  end\n  dp[n -1][m -1]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 14-16 shows the state transitions for minimum path sum. They visit the entire grid, so the time complexity is \\(O(nm)\\).</p> <p>The array <code>dp</code> has dimensions \\(n \\times m\\), so the space complexity is \\(O(nm)\\).</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 14-16   Dynamic programming for minimum path sum </p>","path":["Chapter 14.   Dynamic programming","14.3   Solving dynamic programming problems"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#4-space-optimization","level":3,"title":"4.   Space optimization","text":"<p>Each cell depends only on the cells to its left and above, so a single-row array is enough to represent the \\(dp\\) table.</p> <p>Because <code>dp</code> stores only one row of states, we cannot initialize the entire first column in advance. Instead, update its entry when processing each row:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dp_comp(grid: list[list[int]]) -&gt; int:\n    \"\"\"Minimum path sum: space-optimized dynamic programming\"\"\"\n    n, m = len(grid), len(grid[0])\n    # Initialize the dp table\n    dp = [0] * m\n    # State transitions: first row\n    dp[0] = grid[0][0]\n    for j in range(1, m):\n        dp[j] = dp[j - 1] + grid[0][j]\n    # State transitions: remaining rows\n    for i in range(1, n):\n        # State transitions: first column\n        dp[0] = dp[0] + grid[i][0]\n        # State transitions: remaining columns\n        for j in range(1, m):\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n    return dp[m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* Minimum path sum: space-optimized dynamic programming */\nint minPathSumDPComp(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // Initialize the dp table\n    vector&lt;int&gt; dp(m);\n    // State transitions: first row\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // State transitions: remaining rows\n    for (int i = 1; i &lt; n; i++) {\n        // State transitions: first column\n        dp[0] = dp[0] + grid[i][0];\n        // State transitions: remaining columns\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* Minimum path sum: space-optimized dynamic programming */\nint minPathSumDPComp(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // Initialize the dp table\n    int[] dp = new int[m];\n    // State transitions: first row\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // State transitions: remaining rows\n    for (int i = 1; i &lt; n; i++) {\n        // State transitions: first column\n        dp[0] = dp[0] + grid[i][0];\n        // State transitions: remaining columns\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* Minimum path sum: space-optimized dynamic programming */\nint MinPathSumDPComp(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // Initialize the dp table\n    int[] dp = new int[m];\n    dp[0] = grid[0][0];\n    // State transitions: first row\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // State transitions: remaining rows\n    for (int i = 1; i &lt; n; i++) {\n        // State transitions: first column\n        dp[0] = dp[0] + grid[i][0];\n        // State transitions: remaining columns\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.Min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* Minimum path sum: space-optimized dynamic programming */\nfunc minPathSumDPComp(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // Initialize the dp table\n    dp := make([]int, m)\n    // State transitions: first row\n    dp[0] = grid[0][0]\n    for j := 1; j &lt; m; j++ {\n        dp[j] = dp[j-1] + grid[0][j]\n    }\n    // State transitions: remaining rows and columns\n    for i := 1; i &lt; n; i++ {\n        // State transitions: first column\n        dp[0] = dp[0] + grid[i][0]\n        // State transitions: remaining columns\n        for j := 1; j &lt; m; j++ {\n            dp[j] = int(math.Min(float64(dp[j-1]), float64(dp[j]))) + grid[i][j]\n        }\n    }\n    return dp[m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* Minimum path sum: space-optimized dynamic programming */\nfunc minPathSumDPComp(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // Initialize the dp table\n    var dp = Array(repeating: 0, count: m)\n    // State transitions: first row\n    dp[0] = grid[0][0]\n    for j in 1 ..&lt; m {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // State transitions: remaining rows\n    for i in 1 ..&lt; n {\n        // State transitions: first column\n        dp[0] = dp[0] + grid[i][0]\n        // State transitions: remaining columns\n        for j in 1 ..&lt; m {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* Minimum path sum: space-optimized dynamic programming */\nfunction minPathSumDPComp(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // Initialize the dp table\n    const dp = new Array(m);\n    // State transitions: first row\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // State transitions: remaining rows\n    for (let i = 1; i &lt; n; i++) {\n        // State transitions: first column\n        dp[0] = dp[0] + grid[i][0];\n        // State transitions: remaining columns\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* Minimum path sum: space-optimized dynamic programming */\nfunction minPathSumDPComp(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // Initialize the dp table\n    const dp = new Array(m);\n    // State transitions: first row\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // State transitions: remaining rows\n    for (let i = 1; i &lt; n; i++) {\n        // State transitions: first column\n        dp[0] = dp[0] + grid[i][0];\n        // State transitions: remaining columns\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* Minimum path sum: space-optimized dynamic programming */\nint minPathSumDPComp(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // Initialize the dp table\n  List&lt;int&gt; dp = List.filled(m, 0);\n  dp[0] = grid[0][0];\n  for (int j = 1; j &lt; m; j++) {\n    dp[j] = dp[j - 1] + grid[0][j];\n  }\n  // State transitions: remaining rows\n  for (int i = 1; i &lt; n; i++) {\n    // State transitions: first column\n    dp[0] = dp[0] + grid[i][0];\n    // State transitions: remaining columns\n    for (int j = 1; j &lt; m; j++) {\n      dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n    }\n  }\n  return dp[m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* Minimum path sum: space-optimized dynamic programming */\nfn min_path_sum_dp_comp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // Initialize the dp table\n    let mut dp = vec![0; m];\n    // State transitions: first row\n    dp[0] = grid[0][0];\n    for j in 1..m {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // State transitions: remaining rows\n    for i in 1..n {\n        // State transitions: first column\n        dp[0] = dp[0] + grid[i][0];\n        // State transitions: remaining columns\n        for j in 1..m {\n            dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    dp[m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* Minimum path sum: space-optimized dynamic programming */\nint minPathSumDPComp(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // Initialize the dp table\n    int *dp = calloc(m, sizeof(int));\n    // State transitions: first row\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // State transitions: remaining rows\n    for (int i = 1; i &lt; n; i++) {\n        // State transitions: first column\n        dp[0] = dp[0] + grid[i][0];\n        // State transitions: remaining columns\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = myMin(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    int res = dp[m - 1];\n    // Free memory\n    free(dp);\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* Minimum path sum: space-optimized dynamic programming */\nfun minPathSumDPComp(grid: Array&lt;IntArray&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // Initialize the dp table\n    val dp = IntArray(m)\n    // State transitions: first row\n    dp[0] = grid[0][0]\n    for (j in 1..&lt;m) {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // State transitions: remaining rows\n    for (i in 1..&lt;n) {\n        // State transitions: first column\n        dp[0] = dp[0] + grid[i][0]\n        // State transitions: remaining columns\n        for (j in 1..&lt;m) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>### Minimum path sum: space-optimized dynamic programming ###\ndef min_path_sum_dp_comp(grid)\n  n, m = grid.length, grid.first.length\n  # Initialize the dp table\n  dp = Array.new(m, 0)\n  # State transitions: first row\n  dp[0] = grid[0][0]\n  (1...m).each { |j| dp[j] = dp[j - 1] + grid[0][j] }\n  # State transitions: remaining rows\n  for i in 1...n\n    # State transitions: first column\n    dp[0] = dp[0] + grid[i][0]\n    # State transitions: remaining columns\n    (1...m).each { |j| dp[j] = [dp[j - 1], dp[j]].min + grid[i][j] }\n  end\n  dp[m - 1]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 14.   Dynamic programming","14.3   Solving dynamic programming problems"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/","level":1,"title":"14.6   Edit distance problem","text":"<p>Edit distance, also called Levenshtein distance, is the minimum number of edits needed to transform one string into another. It is commonly used in information retrieval and natural language processing to measure similarity between sequences.</p> <p>Question</p> <p>Given two strings \\(s\\) and \\(t\\), return the minimum number of edits needed to transform \\(s\\) into \\(t\\).</p> <p>Three edits are allowed: insert a character, delete a character, or replace a character with any other character.</p> <p>As shown in Figure 14-27, transforming <code>kitten</code> into <code>sitting</code> takes 3 edits: 2 replacements and 1 insertion. Transforming <code>hello</code> into <code>algo</code> also takes 3 edits: 2 replacements and 1 deletion.</p> <p></p> <p> Figure 14-27   Examples of edit distance </p> <p>Edit distance fits naturally into a decision tree model. Strings are nodes, and each decision (an edit) is an edge.</p> <p>As shown in Figure 14-28, without restrictions on the edits, each node can have many outgoing edges, one for each possible operation. Thus, many paths can transform <code>hello</code> into <code>algo</code>.</p> <p>From the decision tree perspective, the goal is to find the shortest path between nodes <code>hello</code> and <code>algo</code>.</p> <p></p> <p> Figure 14-28   Edit distance as a decision tree </p>","path":["Chapter 14.   Dynamic programming","14.6   Edit distance problem"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/#1-dynamic-programming-approach","level":3,"title":"1.   Dynamic programming approach","text":"<p>Step 1: Identify each decision, define the state, and establish the \\(dp\\) table</p> <p>Each decision applies one edit to string \\(s\\).</p> <p>We want each edit to reduce the problem's size so that we can form subproblems. Let the lengths of \\(s\\) and \\(t\\) be \\(n\\) and \\(m\\). First consider their last characters, \\(s[n-1]\\) and \\(t[m-1]\\).</p> <ul> <li>If \\(s[n-1]\\) and \\(t[m-1]\\) match, skip them and consider \\(s[n-2]\\) and \\(t[m-2]\\).</li> <li>If \\(s[n-1]\\) and \\(t[m-1]\\) differ, apply an edit to \\(s\\) (insertion, deletion, or replacement) to match the endings, allowing us to move on to a smaller problem.</li> </ul> <p>Each decision (edit) in \\(s\\) changes the characters still to be matched in \\(s\\) and \\(t\\). The state therefore specifies the \\(i\\)-th and \\(j\\)-th characters currently under consideration in \\(s\\) and \\(t\\), denoted by \\([i, j]\\).</p> <p>State \\([i, j]\\) represents the subproblem of finding the minimum number of edits needed to transform the first \\(i\\) characters of \\(s\\) into the first \\(j\\) characters of \\(t\\).</p> <p>This gives a two-dimensional \\(dp\\) table of size \\((n+1) \\times (m+1)\\).</p> <p>Step 2: Identify optimal substructure and derive the state transition equation</p> <p>For subproblem \\(dp[i, j]\\), the final characters are \\(s[i-1]\\) and \\(t[j-1]\\). The three possible edits give the cases shown in Figure 14-29.</p> <ol> <li>Insert \\(t[j-1]\\) after \\(s[i-1]\\), leaving subproblem \\(dp[i, j-1]\\).</li> <li>Delete \\(s[i-1]\\), leaving subproblem \\(dp[i-1, j]\\).</li> <li>Replace \\(s[i-1]\\) with \\(t[j-1]\\), leaving subproblem \\(dp[i-1, j-1]\\).</li> </ol> <p></p> <p> Figure 14-29   State transitions for edit distance </p> <p>The optimal substructure follows: the minimum edit count \\(dp[i, j]\\) is the smallest of \\(dp[i, j-1]\\), \\(dp[i-1, j]\\), and \\(dp[i-1, j-1]\\), plus \\(1\\) for the current edit. The state transition equation is:</p> \\[ dp[i, j] = \\min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1 \\] <p>When \\(s[i-1]\\) and \\(t[j-1]\\) match, no edit is needed for the current characters, and the state transition equation becomes:</p> \\[ dp[i, j] = dp[i-1, j-1] \\] <p>Step 3: Determine the boundary conditions and evaluation order</p> <p>If both strings are empty, \\(0\\) edits are needed, so \\(dp[0, 0] = 0\\). If \\(s\\) is empty but \\(t\\) is not, the minimum edit count is the length of \\(t\\), giving the first row \\(dp[0, j] = j\\). If \\(s\\) is nonempty but \\(t\\) is empty, the count is the length of \\(s\\), giving the first column \\(dp[i, 0] = i\\).</p> <p>The solution \\(dp[i, j]\\) depends on entries to its left, above, and upper left, so two nested loops can traverse the \\(dp\\) table in increasing index order.</p>","path":["Chapter 14.   Dynamic programming","14.6   Edit distance problem"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/#2-code-implementation","level":3,"title":"2.   Code implementation","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby edit_distance.py<pre><code>def edit_distance_dp(s: str, t: str) -&gt; int:\n    \"\"\"Edit distance: dynamic programming\"\"\"\n    n, m = len(s), len(t)\n    dp = [[0] * (m + 1) for _ in range(n + 1)]\n    # State transitions: first row and first column\n    for i in range(1, n + 1):\n        dp[i][0] = i\n    for j in range(1, m + 1):\n        dp[0][j] = j\n    # State transitions: remaining rows and columns\n    for i in range(1, n + 1):\n        for j in range(1, m + 1):\n            if s[i - 1] == t[j - 1]:\n                # If the two characters are equal, skip both of them\n                dp[i][j] = dp[i - 1][j - 1]\n            else:\n                # Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[i][j] = min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1\n    return dp[n][m]\n</code></pre> edit_distance.cpp<pre><code>/* Edit distance: dynamic programming */\nint editDistanceDP(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(m + 1, 0));\n    // State transitions: first row and first column\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // State transitions: remaining rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // If the two characters are equal, skip both of them\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.java<pre><code>/* Edit distance: dynamic programming */\nint editDistanceDP(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[][] dp = new int[n + 1][m + 1];\n    // State transitions: first row and first column\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // State transitions: remaining rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // If the two characters are equal, skip both of them\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[i][j] = Math.min(Math.min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.cs<pre><code>/* Edit distance: dynamic programming */\nint EditDistanceDP(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[,] dp = new int[n + 1, m + 1];\n    // State transitions: first row and first column\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i, 0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0, j] = j;\n    }\n    // State transitions: remaining rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // If the two characters are equal, skip both of them\n                dp[i, j] = dp[i - 1, j - 1];\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[i, j] = Math.Min(Math.Min(dp[i, j - 1], dp[i - 1, j]), dp[i - 1, j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n, m];\n}\n</code></pre> edit_distance.go<pre><code>/* Edit distance: dynamic programming */\nfunc editDistanceDP(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, m+1)\n    }\n    // State transitions: first row and first column\n    for i := 1; i &lt;= n; i++ {\n        dp[i][0] = i\n    }\n    for j := 1; j &lt;= m; j++ {\n        dp[0][j] = j\n    }\n    // State transitions: remaining rows and columns\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= m; j++ {\n            if s[i-1] == t[j-1] {\n                // If the two bytes are equal, skip both (each byte is one character for ASCII input)\n                dp[i][j] = dp[i-1][j-1]\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[i][j] = MinInt(MinInt(dp[i][j-1], dp[i-1][j]), dp[i-1][j-1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.swift<pre><code>/* Edit distance: dynamic programming */\nfunc editDistanceDP(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: Array(repeating: 0, count: m + 1), count: n + 1)\n    // State transitions: first row and first column\n    for i in 1 ... n {\n        dp[i][0] = i\n    }\n    for j in 1 ... m {\n        dp[0][j] = j\n    }\n    // State transitions: remaining rows and columns\n    for i in 1 ... n {\n        for j in 1 ... m {\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // If the two current UTF-8 bytes match, skip both\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.js<pre><code>/* Edit distance: dynamic programming */\nfunction editDistanceDP(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(m + 1).fill(0));\n    // State transitions: first row and first column\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // State transitions: remaining rows and columns\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // If the two characters are equal, skip both of them\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.ts<pre><code>/* Edit distance: dynamic programming */\nfunction editDistanceDP(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: m + 1 }, () =&gt; 0)\n    );\n    // State transitions: first row and first column\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // State transitions: remaining rows and columns\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // If the two characters are equal, skip both of them\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.dart<pre><code>/* Edit distance: dynamic programming */\nint editDistanceDP(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (_) =&gt; List.filled(m + 1, 0));\n  // State transitions: first row and first column\n  for (int i = 1; i &lt;= n; i++) {\n    dp[i][0] = i;\n  }\n  for (int j = 1; j &lt;= m; j++) {\n    dp[0][j] = j;\n  }\n  // State transitions: remaining rows and columns\n  for (int i = 1; i &lt;= n; i++) {\n    for (int j = 1; j &lt;= m; j++) {\n      if (s[i - 1] == t[j - 1]) {\n        // If the two characters are equal, skip both of them\n        dp[i][j] = dp[i - 1][j - 1];\n      } else {\n        // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n        dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n      }\n    }\n  }\n  return dp[n][m];\n}\n</code></pre> edit_distance.rs<pre><code>/* Edit distance: dynamic programming */\nfn edit_distance_dp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![vec![0; m + 1]; n + 1];\n    // State transitions: first row and first column\n    for i in 1..=n {\n        dp[i][0] = i as i32;\n    }\n    for j in 1..m {\n        dp[0][j] = j as i32;\n    }\n    // State transitions: remaining rows and columns\n    for i in 1..=n {\n        for j in 1..=m {\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // If the two characters are equal, skip both of them\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[i][j] =\n                    std::cmp::min(std::cmp::min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    dp[n][m]\n}\n</code></pre> edit_distance.c<pre><code>/* Edit distance: dynamic programming */\nint editDistanceDP(char *s, char *t, int n, int m) {\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(m + 1, sizeof(int));\n    }\n    // State transitions: first row and first column\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // State transitions: remaining rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // If the two characters are equal, skip both of them\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[i][j] = myMin(myMin(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    int res = dp[n][m];\n    // Free memory\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* Edit distance: dynamic programming */\nfun editDistanceDP(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = Array(n + 1) { IntArray(m + 1) }\n    // State transitions: first row and first column\n    for (i in 1..n) {\n        dp[i][0] = i\n    }\n    for (j in 1..m) {\n        dp[0][j] = j\n    }\n    // State transitions: remaining rows and columns\n    for (i in 1..n) {\n        for (j in 1..m) {\n            if (s[i - 1] == t[j - 1]) {\n                // If the two characters are equal, skip both of them\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.rb<pre><code>### Edit distance: dynamic programming ###\ndef edit_distance_dp(s, t)\n  n, m = s.length, t.length\n  dp = Array.new(n + 1) { Array.new(m + 1, 0) }\n  # State transitions: first row and first column\n  (1...(n + 1)).each { |i| dp[i][0] = i }\n  (1...(m + 1)).each { |j| dp[0][j] = j }\n  # State transitions: remaining rows and columns\n  for i in 1...(n + 1)\n    for j in 1...(m +1)\n      if s[i - 1] == t[j - 1]\n        # If the two characters are equal, skip both of them\n        dp[i][j] = dp[i - 1][j - 1]\n      else\n        # Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n        dp[i][j] = [dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]].min + 1\n      end\n    end\n  end\n  dp[n][m]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>As shown in Figure 14-30, the state transition process resembles that of the knapsack problem: both fill a two-dimensional grid.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 14-30   Dynamic programming for edit distance </p>","path":["Chapter 14.   Dynamic programming","14.6   Edit distance problem"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/#3-space-optimization","level":3,"title":"3.   Space optimization","text":"<p>The state \\(dp[i,j]\\) depends on \\(dp[i-1, j]\\) above, \\(dp[i, j-1]\\) to the left, and \\(dp[i-1, j-1]\\) at the upper left. Forward iteration loses \\(dp[i-1, j-1]\\), while backward iteration cannot compute \\(dp[i, j-1]\\) in advance. Neither order alone is sufficient.</p> <p>We can use a variable <code>leftup</code> to preserve the upper-left solution \\(dp[i-1, j-1]\\), leaving only the left and upper entries to consider. The situation then matches unbounded knapsack, and forward iteration works. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby edit_distance.py<pre><code>def edit_distance_dp_comp(s: str, t: str) -&gt; int:\n    \"\"\"Edit distance: space-optimized dynamic programming\"\"\"\n    n, m = len(s), len(t)\n    dp = [0] * (m + 1)\n    # State transitions: first row\n    for j in range(1, m + 1):\n        dp[j] = j\n    # State transitions: remaining rows\n    for i in range(1, n + 1):\n        # State transitions: first column\n        leftup = dp[0]  # Temporarily store dp[i-1, j-1]\n        dp[0] += 1\n        # State transitions: remaining columns\n        for j in range(1, m + 1):\n            temp = dp[j]\n            if s[i - 1] == t[j - 1]:\n                # If the two characters are equal, skip both of them\n                dp[j] = leftup\n            else:\n                # Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[j] = min(dp[j - 1], dp[j], leftup) + 1\n            leftup = temp  # Update to dp[i-1, j-1] for the next iteration\n    return dp[m]\n</code></pre> edit_distance.cpp<pre><code>/* Edit distance: space-optimized dynamic programming */\nint editDistanceDPComp(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;int&gt; dp(m + 1, 0);\n    // State transitions: first row\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // State transitions: remaining rows\n    for (int i = 1; i &lt;= n; i++) {\n        // State transitions: first column\n        int leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n        dp[0] = i;\n        // State transitions: remaining columns\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // If the two characters are equal, skip both of them\n                dp[j] = leftup;\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // Update to dp[i-1, j-1] for the next iteration\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.java<pre><code>/* Edit distance: space-optimized dynamic programming */\nint editDistanceDPComp(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[] dp = new int[m + 1];\n    // State transitions: first row\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // State transitions: remaining rows\n    for (int i = 1; i &lt;= n; i++) {\n        // State transitions: first column\n        int leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n        dp[0] = i;\n        // State transitions: remaining columns\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // If the two characters are equal, skip both of them\n                dp[j] = leftup;\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[j] = Math.min(Math.min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // Update to dp[i-1, j-1] for the next iteration\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.cs<pre><code>/* Edit distance: space-optimized dynamic programming */\nint EditDistanceDPComp(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[] dp = new int[m + 1];\n    // State transitions: first row\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // State transitions: remaining rows\n    for (int i = 1; i &lt;= n; i++) {\n        // State transitions: first column\n        int leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n        dp[0] = i;\n        // State transitions: remaining columns\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // If the two characters are equal, skip both of them\n                dp[j] = leftup;\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[j] = Math.Min(Math.Min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // Update to dp[i-1, j-1] for the next iteration\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.go<pre><code>/* Edit distance: space-optimized dynamic programming */\nfunc editDistanceDPComp(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([]int, m+1)\n    // State transitions: first row\n    for j := 1; j &lt;= m; j++ {\n        dp[j] = j\n    }\n    // State transitions: remaining rows\n    for i := 1; i &lt;= n; i++ {\n        // State transitions: first column\n        leftUp := dp[0] // Temporarily store dp[i-1, j-1]\n        dp[0] = i\n        // State transitions: remaining columns\n        for j := 1; j &lt;= m; j++ {\n            temp := dp[j]\n            if s[i-1] == t[j-1] {\n                // If the two bytes are equal, skip both (each byte is one character for ASCII input)\n                dp[j] = leftUp\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[j] = MinInt(MinInt(dp[j-1], dp[j]), leftUp) + 1\n            }\n            leftUp = temp // Update to dp[i-1, j-1] for the next iteration\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.swift<pre><code>/* Edit distance: space-optimized dynamic programming */\nfunc editDistanceDPComp(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: 0, count: m + 1)\n    // State transitions: first row\n    for j in 1 ... m {\n        dp[j] = j\n    }\n    // State transitions: remaining rows\n    for i in 1 ... n {\n        // State transitions: first column\n        var leftup = dp[0] // Temporarily store dp[i-1, j-1]\n        dp[0] = i\n        // State transitions: remaining columns\n        for j in 1 ... m {\n            let temp = dp[j]\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // If the two current UTF-8 bytes match, skip both\n                dp[j] = leftup\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n            }\n            leftup = temp // Update to dp[i-1, j-1] for the next iteration\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.js<pre><code>/* Edit distance: space-optimized dynamic programming */\nfunction editDistanceDPComp(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // State transitions: first row\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // State transitions: remaining rows\n    for (let i = 1; i &lt;= n; i++) {\n        // State transitions: first column\n        let leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n        dp[0] = i;\n        // State transitions: remaining columns\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // If the two characters are equal, skip both of them\n                dp[j] = leftup;\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // Update to dp[i-1, j-1] for the next iteration\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.ts<pre><code>/* Edit distance: space-optimized dynamic programming */\nfunction editDistanceDPComp(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // State transitions: first row\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // State transitions: remaining rows\n    for (let i = 1; i &lt;= n; i++) {\n        // State transitions: first column\n        let leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n        dp[0] = i;\n        // State transitions: remaining columns\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // If the two characters are equal, skip both of them\n                dp[j] = leftup;\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // Update to dp[i-1, j-1] for the next iteration\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.dart<pre><code>/* Edit distance: space-optimized dynamic programming */\nint editDistanceDPComp(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;int&gt; dp = List.filled(m + 1, 0);\n  // State transitions: first row\n  for (int j = 1; j &lt;= m; j++) {\n    dp[j] = j;\n  }\n  // State transitions: remaining rows\n  for (int i = 1; i &lt;= n; i++) {\n    // State transitions: first column\n    int leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n    dp[0] = i;\n    // State transitions: remaining columns\n    for (int j = 1; j &lt;= m; j++) {\n      int temp = dp[j];\n      if (s[i - 1] == t[j - 1]) {\n        // If the two characters are equal, skip both of them\n        dp[j] = leftup;\n      } else {\n        // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n        dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n      }\n      leftup = temp; // Update to dp[i-1, j-1] for the next iteration\n    }\n  }\n  return dp[m];\n}\n</code></pre> edit_distance.rs<pre><code>/* Edit distance: space-optimized dynamic programming */\nfn edit_distance_dp_comp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![0; m + 1];\n    // State transitions: first row\n    for j in 1..m {\n        dp[j] = j as i32;\n    }\n    // State transitions: remaining rows\n    for i in 1..=n {\n        // State transitions: first column\n        let mut leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n        dp[0] = i as i32;\n        // State transitions: remaining columns\n        for j in 1..=m {\n            let temp = dp[j];\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // If the two characters are equal, skip both of them\n                dp[j] = leftup;\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[j] = std::cmp::min(std::cmp::min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // Update to dp[i-1, j-1] for the next iteration\n        }\n    }\n    dp[m]\n}\n</code></pre> edit_distance.c<pre><code>/* Edit distance: space-optimized dynamic programming */\nint editDistanceDPComp(char *s, char *t, int n, int m) {\n    int *dp = calloc(m + 1, sizeof(int));\n    // State transitions: first row\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // State transitions: remaining rows\n    for (int i = 1; i &lt;= n; i++) {\n        // State transitions: first column\n        int leftup = dp[0]; // Temporarily store dp[i-1, j-1]\n        dp[0] = i;\n        // State transitions: remaining columns\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // If the two characters are equal, skip both of them\n                dp[j] = leftup;\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[j] = myMin(myMin(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // Update to dp[i-1, j-1] for the next iteration\n        }\n    }\n    int res = dp[m];\n    // Free memory\n    free(dp);\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* Edit distance: space-optimized dynamic programming */\nfun editDistanceDPComp(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = IntArray(m + 1)\n    // State transitions: first row\n    for (j in 1..m) {\n        dp[j] = j\n    }\n    // State transitions: remaining rows\n    for (i in 1..n) {\n        // State transitions: first column\n        var leftup = dp[0] // Temporarily store dp[i-1, j-1]\n        dp[0] = i\n        // State transitions: remaining columns\n        for (j in 1..m) {\n            val temp = dp[j]\n            if (s[i - 1] == t[j - 1]) {\n                // If the two characters are equal, skip both of them\n                dp[j] = leftup\n            } else {\n                // Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n            }\n            leftup = temp // Update to dp[i-1, j-1] for the next iteration\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.rb<pre><code>### Edit distance: space-optimized dynamic programming ###\ndef edit_distance_dp_comp(s, t)\n  n, m = s.length, t.length\n  dp = Array.new(m + 1, 0)\n  # State transitions: first row\n  (1...(m + 1)).each { |j| dp[j] = j }\n  # State transitions: remaining rows\n  for i in 1...(n + 1)\n    # State transitions: first column\n    leftup = dp.first # Temporarily store dp[i-1, j-1]\n    dp[0] += 1\n    # State transitions: remaining columns\n    for j in 1...(m + 1)\n      temp = dp[j]\n      if s[i - 1] == t[j - 1]\n        # If the two characters are equal, skip both of them\n        dp[j] = leftup\n      else\n        # Minimum edits = the minimum among the insertion, deletion, and replacement subproblems + 1\n        dp[j] = [dp[j - 1], dp[j], leftup].min + 1\n      end\n      leftup = temp # Update to dp[i-1, j-1] for the next iteration\n    end\n  end\n  dp[m]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 14.   Dynamic programming","14.6   Edit distance problem"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/","level":1,"title":"14.8   Exercises","text":"","path":["Chapter 14.   Dynamic programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1481-check-your-understanding","level":2,"title":"14.8.1   Check your understanding","text":"","path":["Chapter 14.   Dynamic programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1-when-is-dynamic-programming-a-good-fit","level":3,"title":"1.   When is dynamic programming a good fit?","text":"<p>A student says, “Whenever you can write a recurrence relation, you should use dynamic programming.” For each task below, decide whether dynamic programming, backtracking, or a loop or mathematical formula without a <code>dp</code> table is the best fit. Give one key reason for each choice.</p> <ol> <li>Find the minimum number of coins needed to make an amount of 6 using denominations <code>[1, 3, 4]</code>. Each denomination can be used any number of times.</li> <li>Output all permutations of <code>[1, 2, 3]</code>.</li> <li> <p>Calculate \\(1 + 2 + \\dots + n\\).</p> <p>For the task you consider suitable for dynamic programming, also explain what <code>dp[i]</code> represents.</p> </li> </ol> Reference answer <ol> <li> <p>Dynamic programming is a good fit. Let <code>dp[i]</code> be the minimum number of coins needed to make an amount of <code>i</code>.     For each coin <code>c</code> whose denomination does not exceed <code>i</code>, consider <code>dp[i-c] + 1</code> as a candidate answer,     then take the minimum of these candidates. Different choices repeatedly lead to the same amounts, and an optimal solution for a larger amount can be built from optimal solutions for smaller amounts.     The answer for an amount of 6 is 2: <code>3 + 3</code>.</p> </li> <li> <p>Use backtracking. The task requires generating all 6 permutations individually. Backtracking systematically makes a choice, continues the search,     then undoes the choice and explores other branches. Whatever method you use, outputting all the permutations requires enumerating them.</p> </li> <li> <p>A loop or the formula for an arithmetic series is sufficient. Although you can write <code>S(i) = S(i-1) + i</code>, calculating <code>S(i)</code> depends on just one smaller value, <code>S(i-1)</code>.     Each partial sum needs to be calculated only once, so there are no overlapping subproblems and no need for a <code>dp</code> table. Being able to write a recurrence does not mean you need dynamic programming.</p> </li> </ol>","path":["Chapter 14.   Dynamic programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#2-how-do-you-calculate-a-cell-in-the-knapsack-table","level":3,"title":"2.   How do you calculate a cell in the knapsack table?","text":"<p>Consider a 0-1 knapsack problem with item weights <code>wgt = [1, 2, 3]</code>, values <code>val = [5, 11, 15]</code>, and a knapsack capacity of 4. Here, <code>dp[i][c]</code> is the maximum value obtainable using only the first \\(i\\) items with a capacity limit of \\(c\\). The knapsack does not have to be filled exactly.</p> <p>Calculate just the state <code>dp[3][4]</code>. You are given <code>dp[2][4] = 16</code> and <code>dp[2][1] = 5</code>.</p> <ol> <li>What is the candidate value if you do not take item 3?</li> <li>If you take item 3, how much capacity remains, and what is the candidate value?</li> <li>What should <code>dp[3][4]</code> be? Which items does this solution take?</li> </ol> Reference answer <ol> <li> <p>If you do not take item 3, use the result for the first two items. The candidate value is <code>dp[2][4] = 16</code>.</p> </li> <li> <p>Item 3 weighs 3, so taking it leaves a capacity of \\(4-3=1\\). The candidate value is     <code>dp[2][1] + 15 = 5 + 15 = 20</code>.</p> </li> <li> <p>Comparing 16 and 20 gives <code>dp[3][4] = 20</code>. This solution takes items 1 and 3,     with a total weight of \\(1+3=4\\) and a total value of \\(5+15=20\\).</p> <p>Calculating this state illustrates a single “take it or leave it” comparison in the 0-1 knapsack problem.</p> </li> </ol>","path":["Chapter 14.   Dynamic programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#3-in-what-order-should-knapsack-capacities-be-updated","level":3,"title":"3.   In what order should knapsack capacities be updated?","text":"<p>A 0-1 knapsack problem has just one item, with weight 2 and value 5. The knapsack capacity is 4. Each item can be taken at most once, and the initial one-dimensional array is <code>dp = [0, 0, 0, 0, 0]</code>.</p> <p>A student processes this item by updating capacities from 2 to 4:</p> <ul> <li>Updating <code>dp[2]</code> gives 5.</li> <li>Updating <code>dp[3]</code> also gives 5.</li> <li>When updating <code>dp[4]</code>, the student uses the newly computed <code>dp[2]</code>, obtaining <code>dp[4] = 10</code>.</li> </ul> <ol> <li>Is <code>dp[4] = 10</code> correct? Why or why not?</li> <li>Given that each item can be taken at most once, what should <code>dp[4]</code> be?</li> <li>When processing each item, should capacities be updated from largest to smallest or from smallest to largest? What problem does this prevent?</li> </ol> Reference answer <ol> <li> <p>The result is incorrect. A value of 10 means taking the item worth 5 twice,     which violates the condition that each item can be taken at most once.</p> </li> <li> <p>The knapsack can contain only this one item, so the correct value of <code>dp[4]</code> is 5.</p> </li> <li> <p>Update capacities from largest to smallest: 4, then 3, then 2.     This ensures that when calculating <code>dp[c]</code>, the value read from <code>dp[c-2]</code> is still the result from before the current item was processed.     The current item therefore cannot be used more than once in the same round of updates.</p> </li> </ol>","path":["Chapter 14.   Dynamic programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1482-programming-exercises","level":2,"title":"14.8.2   Programming exercises","text":"","path":["Chapter 14.   Dynamic programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1-counting-ways-to-climb-stairs","level":3,"title":"1.   Counting ways to climb stairs","text":"<p>A staircase has <code>n</code> steps. You can climb either 1 or 2 steps at a time and must land exactly on step <code>n</code>. Calculate the number of different ways to climb the staircase. Assume <code>n &gt;= 1</code>; two ways differ only in their sequence of 1-step and 2-step moves. Use a one-dimensional dynamic programming array. For now, do not use the space optimization that retains only two states.</p> Hints <ol> <li>The final move to step i must be either a 1-step or a 2-step move.</li> <li>This gives dp[i] = dp[i-1] + dp[i-2].</li> <li>Handle n = 1 and n = 2 first, then fill the table starting at step 3.</li> </ol> <p>LeetCode</p>","path":["Chapter 14.   Dynamic programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#2-0-1-knapsack","level":3,"title":"2.   0-1 knapsack","text":"<p>You are given arrays <code>wgt</code> and <code>val</code> of equal length. Item <code>i</code> has a positive integer weight <code>wgt[i]</code> and a nonnegative integer value <code>val[i]</code>. The knapsack capacity <code>cap</code> is a nonnegative integer. Each item can be taken at most once. Find the maximum total value that fits in the knapsack without exceeding <code>cap</code> in total weight. Use one-dimensional dynamic programming.</p> Hints <ol> <li>Initialize an array dp of length cap + 1, where dp[c] is the maximum value obtainable with a capacity limit of c.</li> <li>When processing item i, compare dp[c] for leaving it out with dp[c-wgt[i]] + val[i] for taking it.</li> <li>Update capacities from largest to smallest to avoid taking the current item more than once in the same round.</li> </ol>","path":["Chapter 14.   Dynamic programming","14.8   Exercises"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/","level":1,"title":"14.1   Introduction to dynamic programming","text":"<p>Dynamic programming is an important algorithmic paradigm. It breaks a problem into smaller subproblems and stores their solutions to avoid repeated computation, greatly improving time efficiency.</p> <p>We will start with a classic example, give a brute-force backtracking solution, identify its overlapping subproblems, and gradually derive a more efficient dynamic programming solution.</p> <p>Climbing stairs</p> <p>Given a staircase with \\(n\\) steps, you may climb \\(1\\) or \\(2\\) steps at a time. How many ways are there to reach the top?</p> <p>As shown in Figure 14-1, a staircase with \\(3\\) steps has \\(3\\) possible ways to reach the top.</p> <p></p> <p> Figure 14-1   Number of ways to reach step 3 </p> <p>The goal is to count solutions, so we can use backtracking to enumerate every possibility. View climbing as a sequence of choices: starting from the ground, choose to climb \\(1\\) or \\(2\\) steps each time. Add \\(1\\) to the count whenever we reach the top, and prune any branch that goes beyond it. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_backtrack.py<pre><code>def backtrack(choices: list[int], state: int, n: int, res: list[int]) -&gt; int:\n    \"\"\"Backtracking\"\"\"\n    # When step n is reached, increase the number of ways by 1\n    if state == n:\n        res[0] += 1\n    # Iterate over all choices\n    for choice in choices:\n        # Prune: do not go beyond step n\n        if state + choice &gt; n:\n            continue\n        # Try: make a choice and update the state\n        backtrack(choices, state + choice, n, res)\n        # Backtrack\n\ndef climbing_stairs_backtrack(n: int) -&gt; int:\n    \"\"\"Climbing stairs: backtracking\"\"\"\n    choices = [1, 2]  # Choose to climb 1 or 2 steps\n    state = 0  # Start climbing from step 0\n    res = [0]  # Use res[0] to record the number of ways\n    backtrack(choices, state, n, res)\n    return res[0]\n</code></pre> climbing_stairs_backtrack.cpp<pre><code>/* Backtracking */\nvoid backtrack(vector&lt;int&gt; &amp;choices, int state, int n, vector&lt;int&gt; &amp;res) {\n    // When step n is reached, increase the number of ways by 1\n    if (state == n)\n        res[0]++;\n    // Iterate over all choices\n    for (auto &amp;choice : choices) {\n        // Prune: do not go beyond step n\n        if (state + choice &gt; n)\n            continue;\n        // Try: make a choice and update the state\n        backtrack(choices, state + choice, n, res);\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: backtracking */\nint climbingStairsBacktrack(int n) {\n    vector&lt;int&gt; choices = {1, 2}; // Choose to climb 1 or 2 steps\n    int state = 0;                // Start climbing from step 0\n    vector&lt;int&gt; res = {0};        // Use res[0] to record the number of ways\n    backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.java<pre><code>/* Backtracking */\nvoid backtrack(List&lt;Integer&gt; choices, int state, int n, List&lt;Integer&gt; res) {\n    // When step n is reached, increase the number of ways by 1\n    if (state == n)\n        res.set(0, res.get(0) + 1);\n    // Iterate over all choices\n    for (Integer choice : choices) {\n        // Prune: do not go beyond step n\n        if (state + choice &gt; n)\n            continue;\n        // Try: make a choice and update the state\n        backtrack(choices, state + choice, n, res);\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: backtracking */\nint climbingStairsBacktrack(int n) {\n    List&lt;Integer&gt; choices = Arrays.asList(1, 2); // Choose to climb 1 or 2 steps\n    int state = 0; // Start climbing from step 0\n    List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n    res.add(0); // Use res[0] to record the number of ways\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.cs<pre><code>/* Backtracking */\nvoid Backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n    // When step n is reached, increase the number of ways by 1\n    if (state == n)\n        res[0]++;\n    // Iterate over all choices\n    foreach (int choice in choices) {\n        // Prune: do not go beyond step n\n        if (state + choice &gt; n)\n            continue;\n        // Try: make a choice and update the state\n        Backtrack(choices, state + choice, n, res);\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: backtracking */\nint ClimbingStairsBacktrack(int n) {\n    List&lt;int&gt; choices = [1, 2]; // Choose to climb 1 or 2 steps\n    int state = 0; // Start climbing from step 0\n    List&lt;int&gt; res = [0]; // Use res[0] to record the number of ways\n    Backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.go<pre><code>/* Backtracking */\nfunc backtrack(choices []int, state, n int, res []int) {\n    // When step n is reached, increase the number of ways by 1\n    if state == n {\n        res[0] = res[0] + 1\n    }\n    // Iterate over all choices\n    for _, choice := range choices {\n        // Prune: do not go beyond step n\n        if state+choice &gt; n {\n            continue\n        }\n        // Try: make a choice and update the state\n        backtrack(choices, state+choice, n, res)\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: backtracking */\nfunc climbingStairsBacktrack(n int) int {\n    // Choose to climb 1 or 2 steps\n    choices := []int{1, 2}\n    // Start climbing from step 0\n    state := 0\n    res := make([]int, 1)\n    // Use res[0] to record the number of ways\n    res[0] = 0\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.swift<pre><code>/* Backtracking */\nfunc backtrack(choices: [Int], state: Int, n: Int, res: inout [Int]) {\n    // When step n is reached, increase the number of ways by 1\n    if state == n {\n        res[0] += 1\n    }\n    // Iterate over all choices\n    for choice in choices {\n        // Prune: do not go beyond step n\n        if state + choice &gt; n {\n            continue\n        }\n        // Try: make a choice and update the state\n        backtrack(choices: choices, state: state + choice, n: n, res: &amp;res)\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: backtracking */\nfunc climbingStairsBacktrack(n: Int) -&gt; Int {\n    let choices = [1, 2] // Choose to climb 1 or 2 steps\n    let state = 0 // Start climbing from step 0\n    var res: [Int] = []\n    res.append(0) // Use res[0] to record the number of ways\n    backtrack(choices: choices, state: state, n: n, res: &amp;res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.js<pre><code>/* Backtracking */\nfunction backtrack(choices, state, n, res) {\n    // When step n is reached, increase the number of ways by 1\n    if (state === n) res.set(0, res.get(0) + 1);\n    // Iterate over all choices\n    for (const choice of choices) {\n        // Prune: do not go beyond step n\n        if (state + choice &gt; n) continue;\n        // Try: make a choice and update the state\n        backtrack(choices, state + choice, n, res);\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: backtracking */\nfunction climbingStairsBacktrack(n) {\n    const choices = [1, 2]; // Choose to climb 1 or 2 steps\n    const state = 0; // Start climbing from step 0\n    const res = new Map();\n    res.set(0, 0); // Store the number of ways at key 0 in res\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.ts<pre><code>/* Backtracking */\nfunction backtrack(\n    choices: number[],\n    state: number,\n    n: number,\n    res: Map&lt;0, any&gt;\n): void {\n    // When step n is reached, increase the number of ways by 1\n    if (state === n) res.set(0, res.get(0) + 1);\n    // Iterate over all choices\n    for (const choice of choices) {\n        // Prune: do not go beyond step n\n        if (state + choice &gt; n) continue;\n        // Try: make a choice and update the state\n        backtrack(choices, state + choice, n, res);\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: backtracking */\nfunction climbingStairsBacktrack(n: number): number {\n    const choices = [1, 2]; // Choose to climb 1 or 2 steps\n    const state = 0; // Start climbing from step 0\n    const res = new Map();\n    res.set(0, 0); // Store the number of ways at key 0 in res\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.dart<pre><code>/* Backtracking */\nvoid backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n  // When step n is reached, increase the number of ways by 1\n  if (state == n) {\n    res[0]++;\n  }\n  // Iterate over all choices\n  for (int choice in choices) {\n    // Prune: do not go beyond step n\n    if (state + choice &gt; n) continue;\n    // Try: make a choice and update the state\n    backtrack(choices, state + choice, n, res);\n    // Backtrack\n  }\n}\n\n/* Climbing stairs: backtracking */\nint climbingStairsBacktrack(int n) {\n  List&lt;int&gt; choices = [1, 2]; // Choose to climb 1 or 2 steps\n  int state = 0; // Start climbing from step 0\n  List&lt;int&gt; res = [];\n  res.add(0); // Use res[0] to record the number of ways\n  backtrack(choices, state, n, res);\n  return res[0];\n}\n</code></pre> climbing_stairs_backtrack.rs<pre><code>/* Backtracking */\nfn backtrack(choices: &amp;[i32], state: i32, n: i32, res: &amp;mut [i32]) {\n    // When step n is reached, increase the number of ways by 1\n    if state == n {\n        res[0] = res[0] + 1;\n    }\n    // Iterate over all choices\n    for &amp;choice in choices {\n        // Prune: do not go beyond step n\n        if state + choice &gt; n {\n            continue;\n        }\n        // Try: make a choice and update the state\n        backtrack(choices, state + choice, n, res);\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: backtracking */\nfn climbing_stairs_backtrack(n: usize) -&gt; i32 {\n    let choices = vec![1, 2]; // Choose to climb 1 or 2 steps\n    let state = 0; // Start climbing from step 0\n    let mut res = Vec::new();\n    res.push(0); // Use res[0] to record the number of ways\n    backtrack(&amp;choices, state, n as i32, &amp;mut res);\n    res[0]\n}\n</code></pre> climbing_stairs_backtrack.c<pre><code>/* Backtracking */\nvoid backtrack(int *choices, int state, int n, int *res, int len) {\n    // When step n is reached, increase the number of ways by 1\n    if (state == n)\n        res[0]++;\n    // Iterate over all choices\n    for (int i = 0; i &lt; len; i++) {\n        int choice = choices[i];\n        // Prune: do not go beyond step n\n        if (state + choice &gt; n)\n            continue;\n        // Try: make a choice and update the state\n        backtrack(choices, state + choice, n, res, len);\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: backtracking */\nint climbingStairsBacktrack(int n) {\n    int choices[2] = {1, 2}; // Choose to climb 1 or 2 steps\n    int state = 0;           // Start climbing from step 0\n    int *res = (int *)malloc(sizeof(int));\n    *res = 0; // Use res[0] to record the number of ways\n    int len = sizeof(choices) / sizeof(int);\n    backtrack(choices, state, n, res, len);\n    int result = *res;\n    free(res);\n    return result;\n}\n</code></pre> climbing_stairs_backtrack.kt<pre><code>/* Backtracking */\nfun backtrack(\n    choices: MutableList&lt;Int&gt;,\n    state: Int,\n    n: Int,\n    res: MutableList&lt;Int&gt;\n) {\n    // When step n is reached, increase the number of ways by 1\n    if (state == n)\n        res[0] = res[0] + 1\n    // Iterate over all choices\n    for (choice in choices) {\n        // Prune: do not go beyond step n\n        if (state + choice &gt; n) continue\n        // Try: make a choice and update the state\n        backtrack(choices, state + choice, n, res)\n        // Backtrack\n    }\n}\n\n/* Climbing stairs: backtracking */\nfun climbingStairsBacktrack(n: Int): Int {\n    val choices = mutableListOf(1, 2) // Choose to climb 1 or 2 steps\n    val state = 0 // Start climbing from step 0\n    val res = mutableListOf&lt;Int&gt;()\n    res.add(0) // Use res[0] to record the number of ways\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.rb<pre><code>### Backtracking ###\ndef backtrack(choices, state, n, res)\n  # When step n is reached, increase the number of ways by 1\n  res[0] += 1 if state == n\n  # Iterate over all choices\n  for choice in choices\n    # Prune: do not go beyond step n\n    next if state + choice &gt; n\n\n    # Try: make a choice and update the state\n    backtrack(choices, state + choice, n, res)\n  end\n  # Backtrack\nend\n\n### Climbing stairs: backtracking ###\ndef climbing_stairs_backtrack(n)\n  choices = [1, 2] # Choose to climb 1 or 2 steps\n  state = 0 # Start climbing from step 0\n  res = [0] # Use res[0] to record the number of ways\n  backtrack(choices, state, n, res)\n  res.first\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 14.   Dynamic programming","14.1   Introduction to dynamic programming"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1411-method-1-brute-force-search","level":2,"title":"14.1.1   Method 1: Brute-force search","text":"<p>Backtracking usually does not explicitly decompose the problem. Instead, it treats a solution as a sequence of decisions and searches all possible solutions through trial and pruning.</p> <p>Let's analyze this problem through decomposition. Let \\(dp[i]\\) be the number of ways to reach step \\(i\\). The original problem is \\(dp[i]\\), and its subproblems include:</p> \\[ dp[i-1], dp[i-2], \\dots, dp[2], dp[1] \\] <p>Since each move climbs \\(1\\) or \\(2\\) steps, when we stand on step \\(i\\), we must previously have been on step \\(i - 1\\) or \\(i - 2\\). In other words, step \\(i\\) can only be reached from step \\(i -1\\) or \\(i - 2\\).</p> <p>This gives an important result: the number of ways to reach step \\(i - 1\\) plus the number of ways to reach step \\(i - 2\\) equals the number of ways to reach step \\(i\\). In symbols:</p> \\[ dp[i] = dp[i-1] + dp[i-2] \\] <p>Thus, the subproblems are linked by a recurrence: the solution to the original problem can be constructed from subproblem solutions. Figure 14-2 illustrates this relationship.</p> <p></p> <p> Figure 14-2   Recurrence for the number of ways </p> <p>The recurrence gives a brute-force search solution. Starting at \\(dp[n]\\), recursively express a larger problem as the sum of two smaller ones until reaching \\(dp[1]\\) or \\(dp[2]\\). Their solutions are known: \\(dp[1] = 1\\) and \\(dp[2] = 2\\), meaning steps \\(1\\) and \\(2\\) can be reached in \\(1\\) and \\(2\\) ways, respectively.</p> <p>Like the standard backtracking code, the following code performs depth-first search, but is more concise:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dfs.py<pre><code>def dfs(i: int) -&gt; int:\n    \"\"\"Search\"\"\"\n    # The values of dp[1] and dp[2] are known; return the corresponding value\n    if i == 1 or i == 2:\n        return i\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1) + dfs(i - 2)\n    return count\n\ndef climbing_stairs_dfs(n: int) -&gt; int:\n    \"\"\"Climbing stairs: search\"\"\"\n    return dfs(n)\n</code></pre> climbing_stairs_dfs.cpp<pre><code>/* Search */\nint dfs(int i) {\n    // The values of dp[1] and dp[2] are known; return them\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* Climbing stairs: search */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.java<pre><code>/* Search */\nint dfs(int i) {\n    // The values of dp[1] and dp[2] are known; return them\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* Climbing stairs: search */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.cs<pre><code>/* Search */\nint DFS(int i) {\n    // The values of dp[1] and dp[2] are known; return them\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1) + DFS(i - 2);\n    return count;\n}\n\n/* Climbing stairs: search */\nint ClimbingStairsDFS(int n) {\n    return DFS(n);\n}\n</code></pre> climbing_stairs_dfs.go<pre><code>/* Search */\nfunc dfs(i int) int {\n    // The values of dp[1] and dp[2] are known; return them\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfs(i-1) + dfs(i-2)\n    return count\n}\n\n/* Climbing stairs: search */\nfunc climbingStairsDFS(n int) int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.swift<pre><code>/* Search */\nfunc dfs(i: Int) -&gt; Int {\n    // The values of dp[1] and dp[2] are known; return them\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1) + dfs(i: i - 2)\n    return count\n}\n\n/* Climbing stairs: search */\nfunc climbingStairsDFS(n: Int) -&gt; Int {\n    dfs(i: n)\n}\n</code></pre> climbing_stairs_dfs.js<pre><code>/* Search */\nfunction dfs(i) {\n    // The values of dp[1] and dp[2] are known; return them\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* Climbing stairs: search */\nfunction climbingStairsDFS(n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.ts<pre><code>/* Search */\nfunction dfs(i: number): number {\n    // The values of dp[1] and dp[2] are known; return them\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* Climbing stairs: search */\nfunction climbingStairsDFS(n: number): number {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.dart<pre><code>/* Search */\nint dfs(int i) {\n  // The values of dp[1] and dp[2] are known; return them\n  if (i == 1 || i == 2) return i;\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1) + dfs(i - 2);\n  return count;\n}\n\n/* Climbing stairs: search */\nint climbingStairsDFS(int n) {\n  return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.rs<pre><code>/* Search */\nfn dfs(i: usize) -&gt; i32 {\n    // The values of dp[1] and dp[2] are known; return them\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1) + dfs(i - 2);\n    count\n}\n\n/* Climbing stairs: search */\nfn climbing_stairs_dfs(n: usize) -&gt; i32 {\n    dfs(n)\n}\n</code></pre> climbing_stairs_dfs.c<pre><code>/* Search */\nint dfs(int i) {\n    // The values of dp[1] and dp[2] are known; return them\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* Climbing stairs: search */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.kt<pre><code>/* Search */\nfun dfs(i: Int): Int {\n    // The values of dp[1] and dp[2] are known; return them\n    if (i == 1 || i == 2) return i\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1) + dfs(i - 2)\n    return count\n}\n\n/* Climbing stairs: search */\nfun climbingStairsDFS(n: Int): Int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.rb<pre><code>### Search ###\ndef dfs(i)\n  # The values of dp[1] and dp[2] are known; return them\n  return i if i == 1 || i == 2\n  # dp[i] = dp[i-1] + dp[i-2]\n  dfs(i - 1) + dfs(i - 2)\nend\n\n### Climbing stairs: search ###\ndef climbing_stairs_dfs(n)\n  dfs(n)\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 14-3 shows the recursion tree for brute-force search. For \\(dp[n]\\), its depth is \\(n\\), and the time complexity is \\(O(2^n)\\). Exponential growth is explosive, so a relatively large \\(n\\) leads to a very long wait.</p> <p></p> <p> Figure 14-3   Recursion tree for climbing stairs </p> <p>As Figure 14-3 shows, overlapping subproblems cause the exponential time complexity. For example, \\(dp[9]\\) splits into \\(dp[8]\\) and \\(dp[7]\\), while \\(dp[8]\\) splits into \\(dp[7]\\) and \\(dp[6]\\). Both contain \\(dp[7]\\).</p> <p>These subproblems in turn contain smaller overlapping subproblems, and the pattern repeats at deeper levels of the recursion tree. Most of the computation is wasted on solving the same subproblems repeatedly.</p>","path":["Chapter 14.   Dynamic programming","14.1   Introduction to dynamic programming"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1412-method-2-memoized-search","level":2,"title":"14.1.2   Method 2: Memoized search","text":"<p>To improve efficiency, we want every overlapping subproblem to be solved only once. Introduce an array <code>mem</code> to record each solution and prune repeated subproblems during the search.</p> <ol> <li>The first time we calculate \\(dp[i]\\), store it in <code>mem[i]</code> for later use.</li> <li>When \\(dp[i]\\) is needed again, retrieve it directly from <code>mem[i]</code> instead of recomputing it.</li> </ol> <p>The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dfs_mem.py<pre><code>def dfs(i: int, mem: list[int]) -&gt; int:\n    \"\"\"Memoized search\"\"\"\n    # The values of dp[1] and dp[2] are known; return the corresponding value\n    if i == 1 or i == 2:\n        return i\n    # If dp[i] has been recorded, return it directly\n    if mem[i] != -1:\n        return mem[i]\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    # Record dp[i]\n    mem[i] = count\n    return count\n\ndef climbing_stairs_dfs_mem(n: int) -&gt; int:\n    \"\"\"Climbing stairs: memoized search\"\"\"\n    # mem[i] stores the number of ways to reach step i; -1 means no result has been recorded\n    mem = [-1] * (n + 1)\n    return dfs(n, mem)\n</code></pre> climbing_stairs_dfs_mem.cpp<pre><code>/* Memoized search */\nint dfs(int i, vector&lt;int&gt; &amp;mem) {\n    // The values of dp[1] and dp[2] are known; return them\n    if (i == 1 || i == 2)\n        return i;\n    // If dp[i] has been recorded, return it directly\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // Record dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* Climbing stairs: memoized search */\nint climbingStairsDFSMem(int n) {\n    // mem[i] stores the number of ways to reach step i; -1 means no result has been recorded\n    vector&lt;int&gt; mem(n + 1, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.java<pre><code>/* Memoized search */\nint dfs(int i, int[] mem) {\n    // The values of dp[1] and dp[2] are known; return them\n    if (i == 1 || i == 2)\n        return i;\n    // If dp[i] has been recorded, return it directly\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // Record dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* Climbing stairs: memoized search */\nint climbingStairsDFSMem(int n) {\n    // mem[i] stores the number of ways to reach step i; -1 means no result has been recorded\n    int[] mem = new int[n + 1];\n    Arrays.fill(mem, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.cs<pre><code>/* Memoized search */\nint DFS(int i, int[] mem) {\n    // The values of dp[1] and dp[2] are known; return them\n    if (i == 1 || i == 2)\n        return i;\n    // If dp[i] has been recorded, return it directly\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1, mem) + DFS(i - 2, mem);\n    // Record dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* Climbing stairs: memoized search */\nint ClimbingStairsDFSMem(int n) {\n    // mem[i] stores the number of ways to reach step i; -1 means no result has been recorded\n    int[] mem = new int[n + 1];\n    Array.Fill(mem, -1);\n    return DFS(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.go<pre><code>/* Memoized search */\nfunc dfsMem(i int, mem []int) int {\n    // The values of dp[1] and dp[2] are known; return them\n    if i == 1 || i == 2 {\n        return i\n    }\n    // If dp[i] has been recorded, return it directly\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfsMem(i-1, mem) + dfsMem(i-2, mem)\n    // Record dp[i]\n    mem[i] = count\n    return count\n}\n\n/* Climbing stairs: memoized search */\nfunc climbingStairsDFSMem(n int) int {\n    // mem[i] stores the number of ways to reach step i; -1 means no result has been recorded\n    mem := make([]int, n+1)\n    for i := range mem {\n        mem[i] = -1\n    }\n    return dfsMem(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.swift<pre><code>/* Memoized search */\nfunc dfs(i: Int, mem: inout [Int]) -&gt; Int {\n    // The values of dp[1] and dp[2] are known; return them\n    if i == 1 || i == 2 {\n        return i\n    }\n    // If dp[i] has been recorded, return it directly\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1, mem: &amp;mem) + dfs(i: i - 2, mem: &amp;mem)\n    // Record dp[i]\n    mem[i] = count\n    return count\n}\n\n/* Climbing stairs: memoized search */\nfunc climbingStairsDFSMem(n: Int) -&gt; Int {\n    // mem[i] stores the number of ways to reach step i; -1 means no result has been recorded\n    var mem = Array(repeating: -1, count: n + 1)\n    return dfs(i: n, mem: &amp;mem)\n}\n</code></pre> climbing_stairs_dfs_mem.js<pre><code>/* Memoized search */\nfunction dfs(i, mem) {\n    // The values of dp[1] and dp[2] are known; return them\n    if (i === 1 || i === 2) return i;\n    // If dp[i] has been recorded, return it directly\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // Record dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* Climbing stairs: memoized search */\nfunction climbingStairsDFSMem(n) {\n    // mem[i] stores the number of ways to reach step i; -1 means no result has been recorded\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.ts<pre><code>/* Memoized search */\nfunction dfs(i: number, mem: number[]): number {\n    // The values of dp[1] and dp[2] are known; return them\n    if (i === 1 || i === 2) return i;\n    // If dp[i] has been recorded, return it directly\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // Record dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* Climbing stairs: memoized search */\nfunction climbingStairsDFSMem(n: number): number {\n    // mem[i] stores the number of ways to reach step i; -1 means no result has been recorded\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.dart<pre><code>/* Memoized search */\nint dfs(int i, List&lt;int&gt; mem) {\n  // The values of dp[1] and dp[2] are known; return them\n  if (i == 1 || i == 2) return i;\n  // If dp[i] has been recorded, return it directly\n  if (mem[i] != -1) return mem[i];\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n  // Record dp[i]\n  mem[i] = count;\n  return count;\n}\n\n/* Climbing stairs: memoized search */\nint climbingStairsDFSMem(int n) {\n  // mem[i] stores the number of ways to reach step i; -1 means no result has been recorded\n  List&lt;int&gt; mem = List.filled(n + 1, -1);\n  return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.rs<pre><code>/* Memoized search */\nfn dfs(i: usize, mem: &amp;mut [i32]) -&gt; i32 {\n    // The values of dp[1] and dp[2] are known; return them\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // If dp[i] has been recorded, return it directly\n    if mem[i] != -1 {\n        return mem[i];\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // Record dp[i]\n    mem[i] = count;\n    count\n}\n\n/* Climbing stairs: memoized search */\nfn climbing_stairs_dfs_mem(n: usize) -&gt; i32 {\n    // mem[i] stores the number of ways to reach step i; -1 means no result has been recorded\n    let mut mem = vec![-1; n + 1];\n    dfs(n, &amp;mut mem)\n}\n</code></pre> climbing_stairs_dfs_mem.c<pre><code>/* Memoized search */\nint dfs(int i, int *mem) {\n    // The values of dp[1] and dp[2] are known; return them\n    if (i == 1 || i == 2)\n        return i;\n    // If dp[i] has been recorded, return it directly\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // Record dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* Climbing stairs: memoized search */\nint climbingStairsDFSMem(int n) {\n    // mem[i] stores the number of ways to reach step i; -1 means no result has been recorded\n    int *mem = (int *)malloc((n + 1) * sizeof(int));\n    for (int i = 0; i &lt;= n; i++) {\n        mem[i] = -1;\n    }\n    int result = dfs(n, mem);\n    free(mem);\n    return result;\n}\n</code></pre> climbing_stairs_dfs_mem.kt<pre><code>/* Memoized search */\nfun dfs(i: Int, mem: IntArray): Int {\n    // The values of dp[1] and dp[2] are known; return them\n    if (i == 1 || i == 2) return i\n    // If dp[i] has been recorded, return it directly\n    if (mem[i] != -1) return mem[i]\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    // Record dp[i]\n    mem[i] = count\n    return count\n}\n\n/* Climbing stairs: memoized search */\nfun climbingStairsDFSMem(n: Int): Int {\n    // mem[i] stores the number of ways to reach step i; -1 means no result has been recorded\n    val mem = IntArray(n + 1)\n    mem.fill(-1)\n    return dfs(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.rb<pre><code>### Memoized search ###\ndef dfs(i, mem)\n  # The values of dp[1] and dp[2] are known; return them\n  return i if i == 1 || i == 2\n  # If dp[i] has been recorded, return it directly\n  return mem[i] if mem[i] != -1\n\n  # dp[i] = dp[i-1] + dp[i-2]\n  count = dfs(i - 1, mem) + dfs(i - 2, mem)\n  # Record dp[i]\n  mem[i] = count\nend\n\n### Climbing stairs: memoized search ###\ndef climbing_stairs_dfs_mem(n)\n  # mem[i] stores the number of ways to reach step i; -1 means no result has been recorded\n  mem = Array.new(n + 1, -1)\n  dfs(n, mem)\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>As shown in Figure 14-4, memoization makes every overlapping subproblem require only one computation, reducing the time complexity to \\(O(n)\\): a major improvement.</p> <p></p> <p> Figure 14-4   Recursion tree for memoized search </p>","path":["Chapter 14.   Dynamic programming","14.1   Introduction to dynamic programming"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1413-method-3-dynamic-programming","level":2,"title":"14.1.3   Method 3: Dynamic programming","text":"<p>Memoized search is a top-down approach: start with the original problem (the root), recursively divide larger subproblems into smaller ones until reaching the smallest subproblems with known solutions (the leaves), then collect their solutions as recursion returns to construct the solution to the original problem.</p> <p>In contrast, dynamic programming is a bottom-up approach: start with the smallest subproblem solutions and iteratively build solutions to larger subproblems until the original problem is solved.</p> <p>Because this dynamic programming process does not backtrack, it needs only iterative loops, not recursion. In the following code, the array <code>dp</code> stores subproblem solutions, serving the same role as <code>mem</code> in memoized search:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dp.py<pre><code>def climbing_stairs_dp(n: int) -&gt; int:\n    \"\"\"Climbing stairs: dynamic programming\"\"\"\n    if n == 1 or n == 2:\n        return n\n    # Initialize the dp table to store subproblem solutions\n    dp = [0] * (n + 1)\n    # Initial states: set the solutions to the smallest subproblems\n    dp[1], dp[2] = 1, 2\n    # State transitions: solve progressively larger subproblems using smaller ones\n    for i in range(3, n + 1):\n        dp[i] = dp[i - 1] + dp[i - 2]\n    return dp[n]\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* Climbing stairs: dynamic programming */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // Initialize the dp table to store subproblem solutions\n    vector&lt;int&gt; dp(n + 1);\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = 1;\n    dp[2] = 2;\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* Climbing stairs: dynamic programming */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // Initialize the dp table to store subproblem solutions\n    int[] dp = new int[n + 1];\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = 1;\n    dp[2] = 2;\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* Climbing stairs: dynamic programming */\nint ClimbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // Initialize the dp table to store subproblem solutions\n    int[] dp = new int[n + 1];\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = 1;\n    dp[2] = 2;\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* Climbing stairs: dynamic programming */\nfunc climbingStairsDP(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // Initialize the dp table to store subproblem solutions\n    dp := make([]int, n+1)\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = 1\n    dp[2] = 2\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = dp[i-1] + dp[i-2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* Climbing stairs: dynamic programming */\nfunc climbingStairsDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // Initialize the dp table to store subproblem solutions\n    var dp = Array(repeating: 0, count: n + 1)\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = 1\n    dp[2] = 2\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for i in 3 ... n {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* Climbing stairs: dynamic programming */\nfunction climbingStairsDP(n) {\n    if (n === 1 || n === 2) return n;\n    // Initialize the dp table to store subproblem solutions\n    const dp = new Array(n + 1).fill(-1);\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = 1;\n    dp[2] = 2;\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* Climbing stairs: dynamic programming */\nfunction climbingStairsDP(n: number): number {\n    if (n === 1 || n === 2) return n;\n    // Initialize the dp table to store subproblem solutions\n    const dp = new Array(n + 1).fill(-1);\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = 1;\n    dp[2] = 2;\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* Climbing stairs: dynamic programming */\nint climbingStairsDP(int n) {\n  if (n == 1 || n == 2) return n;\n  // Initialize the dp table to store subproblem solutions\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // Initial states: set the solutions to the smallest subproblems\n  dp[1] = 1;\n  dp[2] = 2;\n  // State transitions: solve progressively larger subproblems using smaller ones\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = dp[i - 1] + dp[i - 2];\n  }\n  return dp[n];\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* Climbing stairs: dynamic programming */\nfn climbing_stairs_dp(n: usize) -&gt; i32 {\n    // The values of dp[1] and dp[2] are known; return them\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    // Initialize the dp table to store subproblem solutions\n    let mut dp = vec![-1; n + 1];\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = 1;\n    dp[2] = 2;\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for i in 3..=n {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    dp[n]\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* Climbing stairs: dynamic programming */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // Initialize the dp table to store subproblem solutions\n    int *dp = (int *)malloc((n + 1) * sizeof(int));\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = 1;\n    dp[2] = 2;\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    int result = dp[n];\n    free(dp);\n    return result;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* Climbing stairs: dynamic programming */\nfun climbingStairsDP(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    // Initialize the dp table to store subproblem solutions\n    val dp = IntArray(n + 1)\n    // Initial states: set the solutions to the smallest subproblems\n    dp[1] = 1\n    dp[2] = 2\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for (i in 3..n) {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>### Climbing stairs: dynamic programming ###\ndef climbing_stairs_dp(n)\n  return n  if n == 1 || n == 2\n\n  # Initialize the dp table to store subproblem solutions\n  dp = Array.new(n + 1, 0)\n  # Initial states: set the solutions to the smallest subproblems\n  dp[1], dp[2] = 1, 2\n  # State transitions: solve progressively larger subproblems using smaller ones\n  (3...(n + 1)).each { |i| dp[i] = dp[i - 1] + dp[i - 2] }\n\n  dp[n]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 14-5 traces the code's execution.</p> <p></p> <p> Figure 14-5   Dynamic programming for climbing stairs </p> <p>Like backtracking, dynamic programming uses states to represent stages of solving a problem. Each state corresponds to a subproblem and its locally optimal solution. For climbing stairs, the state is the current step number \\(i\\).</p> <p>We can now summarize some common dynamic programming terminology.</p> <ul> <li>The array <code>dp</code> is called the dp table, and \\(dp[i]\\) stores the solution to the subproblem for state \\(i\\).</li> <li>The states for the smallest subproblems (steps \\(1\\) and \\(2\\)) are called the initial states.</li> <li>The recurrence \\(dp[i] = dp[i-1] + dp[i-2]\\) is called the state transition equation.</li> </ul>","path":["Chapter 14.   Dynamic programming","14.1   Introduction to dynamic programming"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1414-space-optimization","level":2,"title":"14.1.4   Space optimization","text":"<p>You may have noticed that \\(dp[i]\\) depends only on \\(dp[i-1]\\) and \\(dp[i-2]\\), so we do not need an entire <code>dp</code> array storing every subproblem solution. Two variables updated as we proceed are enough. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dp.py<pre><code>def climbing_stairs_dp_comp(n: int) -&gt; int:\n    \"\"\"Climbing stairs: space-optimized dynamic programming\"\"\"\n    if n == 1 or n == 2:\n        return n\n    a, b = 1, 2\n    for _ in range(3, n + 1):\n        a, b = b, a + b\n    return b\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* Climbing stairs: space-optimized dynamic programming */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* Climbing stairs: space-optimized dynamic programming */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* Climbing stairs: space-optimized dynamic programming */\nint ClimbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* Climbing stairs: space-optimized dynamic programming */\nfunc climbingStairsDPComp(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    a, b := 1, 2\n    // State transitions: solve progressively larger subproblems using smaller ones\n    for i := 3; i &lt;= n; i++ {\n        a, b = b, a+b\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* Climbing stairs: space-optimized dynamic programming */\nfunc climbingStairsDPComp(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    var a = 1\n    var b = 2\n    for _ in 3 ... n {\n        (a, b) = (b, a + b)\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* Climbing stairs: space-optimized dynamic programming */\nfunction climbingStairsDPComp(n) {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* Climbing stairs: space-optimized dynamic programming */\nfunction climbingStairsDPComp(n: number): number {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* Climbing stairs: space-optimized dynamic programming */\nint climbingStairsDPComp(int n) {\n  if (n == 1 || n == 2) return n;\n  int a = 1, b = 2;\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = a + b;\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* Climbing stairs: space-optimized dynamic programming */\nfn climbing_stairs_dp_comp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    let (mut a, mut b) = (1, 2);\n    for _ in 3..=n {\n        let tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    b\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* Climbing stairs: space-optimized dynamic programming */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* Climbing stairs: space-optimized dynamic programming */\nfun climbingStairsDPComp(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    var a = 1\n    var b = 2\n    for (i in 3..n) {\n        val temp = b\n        b += a\n        a = temp\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>### Climbing stairs: space-optimized dynamic programming ###\ndef climbing_stairs_dp_comp(n)\n  return n if n == 1 || n == 2\n\n  a, b = 1, 2\n  (3...(n + 1)).each { a, b = b, a + b }\n\n  b\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Removing the <code>dp</code> array reduces space complexity from \\(O(n)\\) to \\(O(1)\\).</p> <p>In dynamic programming, a state often depends on only a limited number of preceding states. We can keep just those needed, reducing the storage dimensions to save memory. This technique is called using rolling variables or a rolling array.</p>","path":["Chapter 14.   Dynamic programming","14.1   Introduction to dynamic programming"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/","level":1,"title":"14.4   0-1 knapsack problem","text":"<p>The knapsack problem is an excellent introduction to dynamic programming and one of its most common problem families. Variants include 0-1 knapsack, unbounded knapsack, and bounded knapsack.</p> <p>We will begin with the most common variant, 0-1 knapsack.</p> <p>Question</p> <p>Given \\(n\\) items, where item \\(i\\) has weight \\(wgt[i-1]\\) and value \\(val[i-1]\\), and a knapsack of capacity \\(cap\\), each item may be chosen only once. What is the maximum total value that fits within the capacity?</p> <p>As shown in Figure 14-17, item numbers \\(i\\) start at \\(1\\), while array indices start at \\(0\\). Item \\(i\\) therefore has weight \\(wgt[i-1]\\) and value \\(val[i-1]\\).</p> <p></p> <p> Figure 14-17   Example input for 0-1 knapsack </p> <p>We can view 0-1 knapsack as \\(n\\) rounds of decisions, with two choices for each item: leave it out or put it in. It therefore fits the decision tree model.</p> <p>The objective is to maximize value within a capacity limit, making dynamic programming a promising approach.</p> <p>Step 1: Identify each decision, define the state, and establish the \\(dp\\) table</p> <p>Leaving an item out keeps the available capacity unchanged; putting it in reduces the capacity. The state therefore consists of the current item number \\(i\\) and capacity \\(c\\), denoted by \\([i, c]\\).</p> <p>State \\([i, c]\\) represents the subproblem of finding the maximum value obtainable from the first \\(i\\) items with capacity \\(c\\), denoted by \\(dp[i, c]\\).</p> <p>We seek \\(dp[n, cap]\\), so we need a two-dimensional \\(dp\\) table of size \\((n+1) \\times (cap+1)\\).</p> <p>Step 2: Identify optimal substructure and derive the state transition equation</p> <p>After deciding what to do with item \\(i\\), the remaining subproblem concerns the first \\(i-1\\) items. There are two cases.</p> <ul> <li>Leave item \\(i\\) out: Capacity stays unchanged, and the remaining state is \\([i-1, c]\\).</li> <li>Put item \\(i\\) in: Capacity decreases by \\(wgt[i-1]\\), value increases by \\(val[i-1]\\), and the remaining state is \\([i-1, c-wgt[i-1]]\\).</li> </ul> <p>This reveals the optimal substructure: the maximum value \\(dp[i, c]\\) is the larger value obtained by leaving item \\(i\\) out or putting item \\(i\\) in. The state transition equation is:</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1]) \\] <p>If the current item's weight \\(wgt[i - 1]\\) exceeds the remaining capacity \\(c\\), the only option is to leave it out.</p> <p>Step 3: Determine the boundary conditions and evaluation order</p> <p>With no items or capacity \\(0\\), the maximum value is \\(0\\). Thus, the first column \\(dp[i, 0]\\) and first row \\(dp[0, c]\\) are both \\(0\\).</p> <p>The current state \\([i, c]\\) depends on \\([i-1, c]\\) above and \\([i-1, c-wgt[i-1]]\\) to the upper left, so two nested loops can traverse the \\(dp\\) table in increasing index order.</p> <p>We will now implement brute-force search, memoized search, and dynamic programming in that order.</p>","path":["Chapter 14.   Dynamic programming","14.4   0-1 knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#1-method-1-brute-force-search","level":3,"title":"1.   Method 1: Brute-force search","text":"<p>The search has the following elements.</p> <ul> <li>Parameters: The state \\([i, c]\\).</li> <li>Return value: The subproblem solution \\(dp[i, c]\\).</li> <li>Base case: If there are no items left, \\(i = 0\\), or the remaining capacity is \\(0\\), stop recursion and return value \\(0\\).</li> <li>Pruning: If the current item is heavier than the remaining capacity, only the choice to leave it out is allowed.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dfs(wgt: list[int], val: list[int], i: int, c: int) -&gt; int:\n    \"\"\"0-1 knapsack: brute-force search\"\"\"\n    # If all items have been considered or no capacity remains, return value 0\n    if i == 0 or c == 0:\n        return 0\n    # If the item exceeds the remaining capacity, it must be excluded\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs(wgt, val, i - 1, c)\n    # Calculate the maximum values obtained by excluding and including item i\n    no = knapsack_dfs(wgt, val, i - 1, c)\n    yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # Return the larger value from the two choices\n    return max(no, yes)\n</code></pre> knapsack.cpp<pre><code>/* 0-1 knapsack: brute-force search */\nint knapsackDFS(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int i, int c) {\n    // If all items have been considered or no capacity remains, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Return the larger value from the two choices\n    return max(no, yes);\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 knapsack: brute-force search */\nint knapsackDFS(int[] wgt, int[] val, int i, int c) {\n    // If all items have been considered or no capacity remains, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Return the larger value from the two choices\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 knapsack: brute-force search */\nint KnapsackDFS(int[] weight, int[] val, int i, int c) {\n    // If all items have been considered or no capacity remains, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFS(weight, val, i - 1, c);\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    int no = KnapsackDFS(weight, val, i - 1, c);\n    int yes = KnapsackDFS(weight, val, i - 1, c - weight[i - 1]) + val[i - 1];\n    // Return the larger value from the two choices\n    return Math.Max(no, yes);\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 knapsack: brute-force search */\nfunc knapsackDFS(wgt, val []int, i, c int) int {\n    // If all items have been considered or no capacity remains, return value 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if wgt[i-1] &gt; c {\n        return knapsackDFS(wgt, val, i-1, c)\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    no := knapsackDFS(wgt, val, i-1, c)\n    yes := knapsackDFS(wgt, val, i-1, c-wgt[i-1]) + val[i-1]\n    // Return the larger value from the two choices\n    return int(math.Max(float64(no), float64(yes)))\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 knapsack: brute-force search */\nfunc knapsackDFS(wgt: [Int], val: [Int], i: Int, c: Int) -&gt; Int {\n    // If all items have been considered or no capacity remains, return value 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if wgt[i - 1] &gt; c {\n        return knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    let no = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    let yes = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // Return the larger value from the two choices\n    return max(no, yes)\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 knapsack: brute-force search */\nfunction knapsackDFS(wgt, val, i, c) {\n    // If all items have been considered or no capacity remains, return value 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Return the larger value from the two choices\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 knapsack: brute-force search */\nfunction knapsackDFS(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    i: number,\n    c: number\n): number {\n    // If all items have been considered or no capacity remains, return value 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Return the larger value from the two choices\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 knapsack: brute-force search */\nint knapsackDFS(List&lt;int&gt; wgt, List&lt;int&gt; val, int i, int c) {\n  // If all items have been considered or no capacity remains, return value 0\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // If the item exceeds the remaining capacity, it must be excluded\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFS(wgt, val, i - 1, c);\n  }\n  // Calculate the maximum values obtained by excluding and including item i\n  int no = knapsackDFS(wgt, val, i - 1, c);\n  int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // Return the larger value from the two choices\n  return max(no, yes);\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 knapsack: brute-force search */\nfn knapsack_dfs(wgt: &amp;[i32], val: &amp;[i32], i: usize, c: usize) -&gt; i32 {\n    // If all items have been considered or no capacity remains, return value 0\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs(wgt, val, i - 1, c);\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    let no = knapsack_dfs(wgt, val, i - 1, c);\n    let yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // Return the larger value from the two choices\n    std::cmp::max(no, yes)\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 knapsack: brute-force search */\nint knapsackDFS(int wgt[], int val[], int i, int c) {\n    // If all items have been considered or no capacity remains, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Return the larger value from the two choices\n    return myMax(no, yes);\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 knapsack: brute-force search */\nfun knapsackDFS(\n    wgt: IntArray,\n    _val: IntArray,\n    i: Int,\n    c: Int\n): Int {\n    // If all items have been considered or no capacity remains, return value 0\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, _val, i - 1, c)\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    val no = knapsackDFS(wgt, _val, i - 1, c)\n    val yes = knapsackDFS(wgt, _val, i - 1, c - wgt[i - 1]) + _val[i - 1]\n    // Return the larger value from the two choices\n    return max(no, yes)\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 knapsack: brute-force search ###\ndef knapsack_dfs(wgt, val, i, c)\n  # If all items have been considered or no capacity remains, return value 0\n  return 0 if i == 0 || c == 0\n  # If the item exceeds the remaining capacity, it must be excluded\n  return knapsack_dfs(wgt, val, i - 1, c) if wgt[i - 1] &gt; c\n  # Calculate the maximum values obtained by excluding and including item i\n  no = knapsack_dfs(wgt, val, i - 1, c)\n  yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n  # Return the larger value from the two choices\n  [no, yes].max\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>As shown in Figure 14-18, each item creates two branches, excluding it or including it, so the time complexity is \\(O(2^n)\\).</p> <p>The recursion tree contains overlapping subproblems, such as \\(dp[1, 10]\\). Their number grows substantially with more items and larger capacity, especially when many items share the same weight.</p> <p></p> <p> Figure 14-18   Brute-force recursion tree for 0-1 knapsack </p>","path":["Chapter 14.   Dynamic programming","14.4   0-1 knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#2-method-2-memoized-search","level":3,"title":"2.   Method 2: Memoized search","text":"<p>To solve each overlapping subproblem only once, use a memoization table <code>mem</code>, where <code>mem[i][c]</code> stores \\(dp[i, c]\\).</p> <p>With memoization, the time complexity depends on the number of subproblems, giving \\(O(n \\times cap)\\). The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dfs_mem(\n    wgt: list[int], val: list[int], mem: list[list[int]], i: int, c: int\n) -&gt; int:\n    \"\"\"0-1 knapsack: memoized search\"\"\"\n    # If all items have been considered or no capacity remains, return value 0\n    if i == 0 or c == 0:\n        return 0\n    # If a result has already been recorded, return it directly\n    if mem[i][c] != -1:\n        return mem[i][c]\n    # If the item exceeds the remaining capacity, it must be excluded\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    # Calculate the maximum values obtained by excluding and including item i\n    no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # Record and return the larger value from the two choices\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 knapsack: memoized search */\nint knapsackDFSMem(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int c) {\n    // If all items have been considered or no capacity remains, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If a result has already been recorded, return it directly\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Record and return the larger value from the two choices\n    mem[i][c] = max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 knapsack: memoized search */\nint knapsackDFSMem(int[] wgt, int[] val, int[][] mem, int i, int c) {\n    // If all items have been considered or no capacity remains, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If a result has already been recorded, return it directly\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Record and return the larger value from the two choices\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 knapsack: memoized search */\nint KnapsackDFSMem(int[] weight, int[] val, int[][] mem, int i, int c) {\n    // If all items have been considered or no capacity remains, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If a result has already been recorded, return it directly\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFSMem(weight, val, mem, i - 1, c);\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    int no = KnapsackDFSMem(weight, val, mem, i - 1, c);\n    int yes = KnapsackDFSMem(weight, val, mem, i - 1, c - weight[i - 1]) + val[i - 1];\n    // Record and return the larger value from the two choices\n    mem[i][c] = Math.Max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 knapsack: memoized search */\nfunc knapsackDFSMem(wgt, val []int, mem [][]int, i, c int) int {\n    // If all items have been considered or no capacity remains, return value 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // If a result has already been recorded, return it directly\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if wgt[i-1] &gt; c {\n        return knapsackDFSMem(wgt, val, mem, i-1, c)\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    no := knapsackDFSMem(wgt, val, mem, i-1, c)\n    yes := knapsackDFSMem(wgt, val, mem, i-1, c-wgt[i-1]) + val[i-1]\n    // Return the larger value from the two choices\n    mem[i][c] = int(math.Max(float64(no), float64(yes)))\n    return mem[i][c]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 knapsack: memoized search */\nfunc knapsackDFSMem(wgt: [Int], val: [Int], mem: inout [[Int]], i: Int, c: Int) -&gt; Int {\n    // If all items have been considered or no capacity remains, return value 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // If a result has already been recorded, return it directly\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if wgt[i - 1] &gt; c {\n        return knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    let no = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    let yes = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // Record and return the larger value from the two choices\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 knapsack: memoized search */\nfunction knapsackDFSMem(wgt, val, mem, i, c) {\n    // If all items have been considered or no capacity remains, return value 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // If a result has already been recorded, return it directly\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Record and return the larger value from the two choices\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 knapsack: memoized search */\nfunction knapsackDFSMem(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    c: number\n): number {\n    // If all items have been considered or no capacity remains, return value 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // If a result has already been recorded, return it directly\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Record and return the larger value from the two choices\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 knapsack: memoized search */\nint knapsackDFSMem(\n  List&lt;int&gt; wgt,\n  List&lt;int&gt; val,\n  List&lt;List&lt;int&gt;&gt; mem,\n  int i,\n  int c,\n) {\n  // If all items have been considered or no capacity remains, return value 0\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // If a result has already been recorded, return it directly\n  if (mem[i][c] != -1) {\n    return mem[i][c];\n  }\n  // If the item exceeds the remaining capacity, it must be excluded\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFSMem(wgt, val, mem, i - 1, c);\n  }\n  // Calculate the maximum values obtained by excluding and including item i\n  int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n  int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // Record and return the larger value from the two choices\n  mem[i][c] = max(no, yes);\n  return mem[i][c];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 knapsack: memoized search */\nfn knapsack_dfs_mem(wgt: &amp;[i32], val: &amp;[i32], mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: usize, c: usize) -&gt; i32 {\n    // If all items have been considered or no capacity remains, return value 0\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // If a result has already been recorded, return it directly\n    if mem[i][c] != -1 {\n        return mem[i][c];\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    let no = knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    let yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // Record and return the larger value from the two choices\n    mem[i][c] = std::cmp::max(no, yes);\n    mem[i][c]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 knapsack: memoized search */\nint knapsackDFSMem(int wgt[], int val[], int memCols, int **mem, int i, int c) {\n    // If all items have been considered or no capacity remains, return value 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // If a result has already been recorded, return it directly\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    int no = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // Record and return the larger value from the two choices\n    mem[i][c] = myMax(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 knapsack: memoized search */\nfun knapsackDFSMem(\n    wgt: IntArray,\n    _val: IntArray,\n    mem: Array&lt;IntArray&gt;,\n    i: Int,\n    c: Int\n): Int {\n    // If all items have been considered or no capacity remains, return value 0\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // If a result has already been recorded, return it directly\n    if (mem[i][c] != -1) {\n        return mem[i][c]\n    }\n    // If the item exceeds the remaining capacity, it must be excluded\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, _val, mem, i - 1, c)\n    }\n    // Calculate the maximum values obtained by excluding and including item i\n    val no = knapsackDFSMem(wgt, _val, mem, i - 1, c)\n    val yes = knapsackDFSMem(wgt, _val, mem, i - 1, c - wgt[i - 1]) + _val[i - 1]\n    // Record and return the larger value from the two choices\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 knapsack: memoized search ###\ndef knapsack_dfs_mem(wgt, val, mem, i, c)\n  # If all items have been considered or no capacity remains, return value 0\n  return 0 if i == 0 || c == 0\n  # If a result has already been recorded, return it directly\n  return mem[i][c] if mem[i][c] != -1\n  # If the item exceeds the remaining capacity, it must be excluded\n  return knapsack_dfs_mem(wgt, val, mem, i - 1, c) if wgt[i - 1] &gt; c\n  # Calculate the maximum values obtained by excluding and including item i\n  no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n  yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n  # Record and return the larger value from the two choices\n  mem[i][c] = [no, yes].max\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 14-19 shows the branches pruned by memoized search.</p> <p></p> <p> Figure 14-19   Memoized recursion tree for 0-1 knapsack </p>","path":["Chapter 14.   Dynamic programming","14.4   0-1 knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#3-method-3-dynamic-programming","level":3,"title":"3.   Method 3: Dynamic programming","text":"<p>Dynamic programming essentially fills the \\(dp\\) table using state transitions, as shown in the code below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 knapsack: dynamic programming\"\"\"\n    n = len(wgt)\n    # Initialize the dp table\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # State transitions\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # Take the larger value obtained by excluding and including item i\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 knapsack: dynamic programming */\nint knapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // Initialize the dp table\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 knapsack: dynamic programming */\nint knapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // Initialize the dp table\n    int[][] dp = new int[n + 1][cap + 1];\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 knapsack: dynamic programming */\nint KnapsackDP(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // Initialize the dp table\n    int[,] dp = new int[n + 1, cap + 1];\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (weight[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i, c] = Math.Max(dp[i - 1, c - weight[i - 1]] + val[i - 1], dp[i - 1, c]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 knapsack: dynamic programming */\nfunc knapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // Initialize the dp table\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // State transitions\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i-1][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 knapsack: dynamic programming */\nfunc knapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // Initialize the dp table\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // State transitions\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 knapsack: dynamic programming */\nfunction knapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // Initialize the dp table\n    const dp = Array(n + 1)\n        .fill(0)\n        .map(() =&gt; Array(cap + 1).fill(0));\n    // State transitions\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 knapsack: dynamic programming */\nfunction knapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // Initialize the dp table\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // State transitions\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 knapsack: dynamic programming */\nint knapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // Initialize the dp table\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // State transitions\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // If item i exceeds the remaining capacity, do not choose it\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // Take the larger value obtained by excluding and including item i\n        dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 knapsack: dynamic programming */\nfn knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // Initialize the dp table\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // State transitions\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = std::cmp::max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1] as usize] + val[i - 1],\n                );\n            }\n        }\n    }\n    dp[n][cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 knapsack: dynamic programming */\nint knapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // Initialize the dp table\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = myMax(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // Free memory\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 knapsack: dynamic programming */\nfun knapsackDP(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // Initialize the dp table\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // State transitions\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 knapsack: dynamic programming ###\ndef knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # Initialize the dp table\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # State transitions\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # If item i exceeds the remaining capacity, do not choose it\n        dp[i][c] = dp[i - 1][c]\n      else\n        # Take the larger value obtained by excluding and including item i\n        dp[i][c] = [dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>As shown in Figure 14-20, both time and space complexity are determined by the size of <code>dp</code>: \\(O(n \\times cap)\\).</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 14-20   Dynamic programming for 0-1 knapsack </p>","path":["Chapter 14.   Dynamic programming","14.4   0-1 knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#4-space-optimization","level":3,"title":"4.   Space optimization","text":"<p>Each state depends only on the previous row, so two rolling arrays reduce space complexity from \\(O(n \\times cap)\\) to \\(O(cap)\\).</p> <p>Can we go further and use just one array? Each state depends on entries directly above or to the upper left. With one array, when we begin processing row \\(i\\), it still holds the states from row \\(i-1\\).</p> <ul> <li>With forward iteration, by the time we reach \\(dp[i, j]\\), the upper-left values \\(dp[i-1, 1]\\) ~ \\(dp[i-1, j-1]\\) may have been overwritten, making the transition incorrect.</li> <li>With backward iteration, these values are not overwritten before they are needed, so transitions remain correct.</li> </ul> <p>Figure 14-21 shows one array changing from row \\(i = 1\\) to row \\(i = 2\\). Consider how forward and backward iteration differ.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 14-21   Space-optimized dynamic programming for 0-1 knapsack </p> <p>In the implementation, simply remove the first dimension \\(i\\) from <code>dp</code> and reverse the inner loop:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 knapsack: space-optimized dynamic programming\"\"\"\n    n = len(wgt)\n    # Initialize the dp table\n    dp = [0] * (cap + 1)\n    # State transitions\n    for i in range(1, n + 1):\n        # Traverse in decreasing order\n        for c in range(cap, 0, -1):\n            if wgt[i - 1] &gt; c:\n                # If item i exceeds the remaining capacity, do not choose it\n                dp[c] = dp[c]\n            else:\n                # Take the larger value obtained by excluding and including item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 knapsack: space-optimized dynamic programming */\nint knapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // Initialize the dp table\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        // Traverse in decreasing order\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 knapsack: space-optimized dynamic programming */\nint knapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // Initialize the dp table\n    int[] dp = new int[cap + 1];\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        // Traverse in decreasing order\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 knapsack: space-optimized dynamic programming */\nint KnapsackDPComp(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // Initialize the dp table\n    int[] dp = new int[cap + 1];\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        // Traverse in decreasing order\n        for (int c = cap; c &gt; 0; c--) {\n            if (weight[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[c] = dp[c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = Math.Max(dp[c], dp[c - weight[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 knapsack: space-optimized dynamic programming */\nfunc knapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // Initialize the dp table\n    dp := make([]int, cap+1)\n    // State transitions\n    for i := 1; i &lt;= n; i++ {\n        // Traverse in decreasing order\n        for c := cap; c &gt;= 1; c-- {\n            if wgt[i-1] &lt;= c {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 knapsack: space-optimized dynamic programming */\nfunc knapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // Initialize the dp table\n    var dp = Array(repeating: 0, count: cap + 1)\n    // State transitions\n    for i in 1 ... n {\n        // Traverse in decreasing order\n        for c in (1 ... cap).reversed() {\n            if wgt[i - 1] &lt;= c {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 knapsack: space-optimized dynamic programming */\nfunction knapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // Initialize the dp table\n    const dp = Array(cap + 1).fill(0);\n    // State transitions\n    for (let i = 1; i &lt;= n; i++) {\n        // Traverse in decreasing order\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 knapsack: space-optimized dynamic programming */\nfunction knapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // Initialize the dp table\n    const dp = Array(cap + 1).fill(0);\n    // State transitions\n    for (let i = 1; i &lt;= n; i++) {\n        // Traverse in decreasing order\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 knapsack: space-optimized dynamic programming */\nint knapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // Initialize the dp table\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // State transitions\n  for (int i = 1; i &lt;= n; i++) {\n    // Traverse in decreasing order\n    for (int c = cap; c &gt;= 1; c--) {\n      if (wgt[i - 1] &lt;= c) {\n        // Take the larger value obtained by excluding and including item i\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 knapsack: space-optimized dynamic programming */\nfn knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // Initialize the dp table\n    let mut dp = vec![0; cap + 1];\n    // State transitions\n    for i in 1..=n {\n        // Traverse in decreasing order\n        for c in (1..=cap).rev() {\n            if wgt[i - 1] &lt;= c as i32 {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 knapsack: space-optimized dynamic programming */\nint knapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // Initialize the dp table\n    int *dp = calloc(cap + 1, sizeof(int));\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        // Traverse in decreasing order\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // Free memory\n    free(dp);\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 knapsack: space-optimized dynamic programming */\nfun knapsackDPComp(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // Initialize the dp table\n    val dp = IntArray(cap + 1)\n    // State transitions\n    for (i in 1..n) {\n        // Traverse in decreasing order\n        for (c in cap downTo 1) {\n            if (wgt[i - 1] &lt;= c) {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 knapsack: space-optimized dynamic programming ###\ndef knapsack_dp_comp(wgt, val, cap)\n  n = wgt.length\n  # Initialize the dp table\n  dp = Array.new(cap + 1, 0)\n  # State transitions\n  for i in 1...(n + 1)\n    # Traverse in decreasing order\n    for c in cap.downto(1)\n      if wgt[i - 1] &gt; c\n        # If item i exceeds the remaining capacity, do not choose it\n        dp[c] = dp[c]\n      else\n        # Take the larger value obtained by excluding and including item i\n        dp[c] = [dp[c], dp[c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[cap]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 14.   Dynamic programming","14.4   0-1 knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/summary/","level":1,"title":"14.7   Summary","text":"","path":["Chapter 14.   Dynamic programming","14.7   Summary"],"tags":[]},{"location":"chapter_dynamic_programming/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<ul> <li>Dynamic programming decomposes problems and stores subproblem solutions to avoid repeated computation and improve efficiency.</li> <li>Without a time limit, every dynamic programming problem can be solved by backtracking (brute-force search). However, the recursion tree contains many overlapping subproblems, making it very inefficient. Memoization stores computed solutions so that each overlapping subproblem is solved only once.</li> <li>Memoized search is a top-down recursive approach, while the corresponding dynamic programming solution works bottom up, like filling a table. Since the current state depends only on certain nearby states, removing a dimension from the \\(dp\\) table can reduce space complexity.</li> <li>Decomposing a problem into subproblems is a general algorithmic idea with different characteristics in divide and conquer, dynamic programming, and backtracking.</li> <li>Dynamic programming problems have three key properties: overlapping subproblems, optimal substructure, and the no-aftereffect property.</li> <li>A problem has optimal substructure if its optimal solution can be constructed from optimal solutions to subproblems.</li> <li>The no-aftereffect property means a state's future depends only on that state, not on the history leading to it. Many combinatorial optimization problems lack this property and cannot be solved quickly with dynamic programming.</li> </ul> <p>Knapsack problems</p> <ul> <li>Knapsack is one of the most typical dynamic programming problem families, with variants including 0-1, unbounded, and bounded knapsack.</li> <li>In 0-1 knapsack, the state value is the maximum value obtainable from the first \\(i\\) items within capacity \\(c\\). The choices to exclude or include an item reveal optimal substructure and give the state transition equation. After space optimization, backward iteration is necessary because each state depends on entries above and to the upper left, which must not be overwritten prematurely.</li> <li>Unbounded knapsack allows unlimited copies of each item, changing the transition for including an item. Because a state depends on entries directly above and to the left, space optimization requires forward iteration.</li> <li>Coin change is a variant of unbounded knapsack. Minimizing the number of coins instead of maximizing value changes \\(\\max()\\) to \\(\\min()\\) in the transition. Requiring the target amount exactly, rather than staying within a capacity, calls for using \\(amt + 1\\) to represent an infeasible solution.</li> <li>Coin change II counts combinations instead of minimizing coin count, so the transition replaces \\(\\min()\\) with addition.</li> </ul> <p>Edit distance</p> <ul> <li>Edit distance (Levenshtein distance) measures similarity between strings. It is the minimum number of edits needed to turn one string into another, allowing insertion, deletion, and replacement.</li> <li>The edit distance state is the minimum number of edits needed to transform the first \\(i\\) characters of \\(s\\) into the first \\(j\\) characters of \\(t\\). When \\(s[i-1] \\ne t[j-1]\\), insertion, deletion, and replacement each leave a corresponding subproblem, revealing optimal substructure and the state transition equation. When \\(s[i-1] = t[j-1]\\), no edit is needed for those characters.</li> <li>Edit distance depends on states above, to the left, and to the upper left, so neither forward nor backward iteration alone works after space optimization. A temporary variable preserves the upper-left state, reducing the situation to one like unbounded knapsack and allowing forward iteration.</li> </ul>","path":["Chapter 14.   Dynamic programming","14.7   Summary"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/","level":1,"title":"14.5   Unbounded knapsack problem","text":"<p>We will now solve another common knapsack variant, unbounded knapsack, and then examine a special case: coin change.</p>","path":["Chapter 14.   Dynamic programming","14.5   Unbounded knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1451-unbounded-knapsack","level":2,"title":"14.5.1   Unbounded knapsack","text":"<p>Question</p> <p>Given \\(n\\) items, where item \\(i\\) has weight \\(wgt[i-1]\\) and value \\(val[i-1]\\), and a knapsack of capacity \\(cap\\), each item may be chosen repeatedly. What is the maximum value that fits within the capacity? An example is shown in Figure 14-22.</p> <p></p> <p> Figure 14-22   Example input for unbounded knapsack </p>","path":["Chapter 14.   Dynamic programming","14.5   Unbounded knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1-dynamic-programming-approach","level":3,"title":"1.   Dynamic programming approach","text":"<p>Unbounded knapsack is very similar to 0-1 knapsack. The only difference is that there is no limit on how many times an item may be chosen.</p> <ul> <li>In 0-1 knapsack, there is only one of each item, so after including item \\(i\\), we can choose only from the first \\(i-1\\) items.</li> <li>In unbounded knapsack, each item has an unlimited supply, so after including item \\(i\\), we can still choose from the first \\(i\\) items.</li> </ul> <p>For unbounded knapsack, state \\([i, c]\\) has two cases.</p> <ul> <li>Leave item \\(i\\) out: As in 0-1 knapsack, the remaining state is \\([i-1, c]\\).</li> <li>Put item \\(i\\) in: Unlike 0-1 knapsack, the remaining state is \\([i, c-wgt[i-1]]\\).</li> </ul> <p>The state transition equation becomes:</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1]) \\]","path":["Chapter 14.   Dynamic programming","14.5   Unbounded knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2-code-implementation","level":3,"title":"2.   Code implementation","text":"<p>Compared with 0-1 knapsack, only one \\(i-1\\) in the transition changes to \\(i\\); the rest of the code is identical:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"Unbounded knapsack: dynamic programming\"\"\"\n    n = len(wgt)\n    # Initialize the dp table\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # State transitions\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # Take the larger value obtained by excluding and including item i\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* Unbounded knapsack: dynamic programming */\nint unboundedKnapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // Initialize the dp table\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* Unbounded knapsack: dynamic programming */\nint unboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // Initialize the dp table\n    int[][] dp = new int[n + 1][cap + 1];\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* Unbounded knapsack: dynamic programming */\nint UnboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // Initialize the dp table\n    int[,] dp = new int[n + 1, cap + 1];\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i, c] = Math.Max(dp[i - 1, c], dp[i, c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* Unbounded knapsack: dynamic programming */\nfunc unboundedKnapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // Initialize the dp table\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // State transitions\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* Unbounded knapsack: dynamic programming */\nfunc unboundedKnapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // Initialize the dp table\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // State transitions\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* Unbounded knapsack: dynamic programming */\nfunction unboundedKnapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // Initialize the dp table\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // State transitions\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* Unbounded knapsack: dynamic programming */\nfunction unboundedKnapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // Initialize the dp table\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // State transitions\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* Unbounded knapsack: dynamic programming */\nint unboundedKnapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // Initialize the dp table\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // State transitions\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // If item i exceeds the remaining capacity, do not choose it\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // Take the larger value obtained by excluding and including item i\n        dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* Unbounded knapsack: dynamic programming */\nfn unbounded_knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // Initialize the dp table\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // State transitions\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i][c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* Unbounded knapsack: dynamic programming */\nint unboundedKnapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // Initialize the dp table\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = myMax(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // Free memory\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* Unbounded knapsack: dynamic programming */\nfun unboundedKnapsackDP(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // Initialize the dp table\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // State transitions\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>### Unbounded knapsack: dynamic programming ###\ndef unbounded_knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # Initialize the dp table\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # State transitions\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # If item i exceeds the remaining capacity, do not choose it\n        dp[i][c] = dp[i - 1][c]\n      else\n        # Take the larger value obtained by excluding and including item i\n        dp[i][c] = [dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 14.   Dynamic programming","14.5   Unbounded knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3-space-optimization","level":3,"title":"3.   Space optimization","text":"<p>The current state depends on states to its left and above, so after space optimization, each row of the \\(dp\\) table must be processed in forward order.</p> <p>This is the opposite of 0-1 knapsack. Use Figure 14-23 to understand the difference.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 14-23   Space-optimized dynamic programming for unbounded knapsack </p> <p>The implementation simply removes the first dimension of <code>dp</code>:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"Unbounded knapsack: space-optimized dynamic programming\"\"\"\n    n = len(wgt)\n    # Initialize the dp table\n    dp = [0] * (cap + 1)\n    # State transitions\n    for i in range(1, n + 1):\n        # Traverse in increasing order\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # If item i exceeds the remaining capacity, do not choose it\n                dp[c] = dp[c]\n            else:\n                # Take the larger value obtained by excluding and including item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* Unbounded knapsack: space-optimized dynamic programming */\nint unboundedKnapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // Initialize the dp table\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[c] = dp[c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* Unbounded knapsack: space-optimized dynamic programming */\nint unboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // Initialize the dp table\n    int[] dp = new int[cap + 1];\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[c] = dp[c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* Unbounded knapsack: space-optimized dynamic programming */\nint UnboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // Initialize the dp table\n    int[] dp = new int[cap + 1];\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[c] = dp[c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = Math.Max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* Unbounded knapsack: space-optimized dynamic programming */\nfunc unboundedKnapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // Initialize the dp table\n    dp := make([]int, cap+1)\n    // State transitions\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[c] = dp[c]\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* Unbounded knapsack: space-optimized dynamic programming */\nfunc unboundedKnapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // Initialize the dp table\n    var dp = Array(repeating: 0, count: cap + 1)\n    // State transitions\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[c] = dp[c]\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* Unbounded knapsack: space-optimized dynamic programming */\nfunction unboundedKnapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // Initialize the dp table\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // State transitions\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[c] = dp[c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* Unbounded knapsack: space-optimized dynamic programming */\nfunction unboundedKnapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // Initialize the dp table\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // State transitions\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[c] = dp[c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* Unbounded knapsack: space-optimized dynamic programming */\nint unboundedKnapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // Initialize the dp table\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // State transitions\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // If item i exceeds the remaining capacity, do not choose it\n        dp[c] = dp[c];\n      } else {\n        // Take the larger value obtained by excluding and including item i\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* Unbounded knapsack: space-optimized dynamic programming */\nfn unbounded_knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // Initialize the dp table\n    let mut dp = vec![0; cap + 1];\n    // State transitions\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[c] = dp[c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* Unbounded knapsack: space-optimized dynamic programming */\nint unboundedKnapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // Initialize the dp table\n    int *dp = calloc(cap + 1, sizeof(int));\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[c] = dp[c];\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // Free memory\n    free(dp);\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* Unbounded knapsack: space-optimized dynamic programming */\nfun unboundedKnapsackDPComp(\n    wgt: IntArray,\n    _val: IntArray,\n    cap: Int\n): Int {\n    val n = wgt.size\n    // Initialize the dp table\n    val dp = IntArray(cap + 1)\n    // State transitions\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // If item i exceeds the remaining capacity, do not choose it\n                dp[c] = dp[c]\n            } else {\n                // Take the larger value obtained by excluding and including item i\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>### Unbounded knapsack: space-optimized dynamic programming ###\ndef unbounded_knapsack_dp_comp(wgt, val, cap)\n  n = wgt.length\n  # Initialize the dp table\n  dp = Array.new(cap + 1, 0)\n  # State transitions\n  for i in 1...(n + 1)\n    # Traverse in increasing order\n    for c in 1...(cap + 1)\n      if wgt[i -1] &gt; c\n        # If item i exceeds the remaining capacity, do not choose it\n        dp[c] = dp[c]\n      else\n        # Take the larger value obtained by excluding and including item i\n        dp[c] = [dp[c], dp[c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[cap]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 14.   Dynamic programming","14.5   Unbounded knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1452-coin-change-problem","level":2,"title":"14.5.2   Coin change problem","text":"<p>Knapsack represents a broad family of dynamic programming problems, with many variants such as coin change.</p> <p>Question</p> <p>Given \\(n\\) coin denominations, where denomination \\(i\\) has value \\(coins[i - 1]\\), and a target amount \\(amt\\), each denomination may be used repeatedly. What is the minimum number of coins needed to make the target? Return \\(-1\\) if it cannot be made. An example is shown in Figure 14-24.</p> <p></p> <p> Figure 14-24   Example input for coin change </p>","path":["Chapter 14.   Dynamic programming","14.5   Unbounded knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1-dynamic-programming-approach_1","level":3,"title":"1.   Dynamic programming approach","text":"<p>Coin change can be viewed as a special case of unbounded knapsack, with the following connections and differences.</p> <ul> <li>The two problems can be translated into one another: items correspond to coins, item weights to coin denominations, and knapsack capacity to the target amount.</li> <li>The optimization objectives are opposite: unbounded knapsack maximizes item value, while coin change minimizes the number of coins.</li> <li>Unbounded knapsack must stay within a capacity limit, whereas coin change must reach the target amount exactly.</li> </ul> <p>Step 1: Identify each decision, define the state, and establish the \\(dp\\) table</p> <p>State \\([i, a]\\) represents the subproblem of finding the minimum number of coins needed to make amount \\(a\\) using the first \\(i\\) denominations, denoted by \\(dp[i, a]\\).</p> <p>The two-dimensional \\(dp\\) table has size \\((n+1) \\times (amt+1)\\).</p> <p>Step 2: Identify optimal substructure and derive the state transition equation</p> <p>The state transition differs from unbounded knapsack in two ways.</p> <ul> <li>We want a minimum, so replace \\(\\max()\\) with \\(\\min()\\).</li> <li>We optimize the coin count rather than item value, so choosing a coin adds \\(+1\\).</li> </ul> \\[ dp[i, a] = \\min(dp[i-1, a], dp[i, a - coins[i-1]] + 1) \\] <p>Step 3: Determine the boundary conditions and evaluation order</p> <p>A target amount of \\(0\\) requires \\(0\\) coins, so every entry \\(dp[i, 0]\\) in the first column is \\(0\\).</p> <p>With no coins, any target amount \\(&gt; 0\\) is impossible. To let \\(\\min()\\) recognize and discard these infeasible solutions, represent them by \\(+ \\infty\\): set the first-row entries \\(dp[0, a]\\) to \\(+ \\infty\\).</p>","path":["Chapter 14.   Dynamic programming","14.5   Unbounded knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2-code-implementation_1","level":3,"title":"2.   Code implementation","text":"<p>Most programming languages do not provide an integer value for \\(+ \\infty\\), so the maximum <code>int</code> value is used instead. However, the \\(+ 1\\) in the transition may then overflow.</p> <p>Instead, use \\(amt + 1\\) to represent an infeasible solution, since making \\(amt\\) requires at most \\(amt\\) coins. Before returning, check whether \\(dp[n, amt]\\) equals \\(amt + 1\\). If so, return \\(-1\\) to indicate that the target cannot be made. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change.py<pre><code>def coin_change_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"Coin change: dynamic programming\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # Initialize the dp table\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # State transitions: first row and first column\n    for a in range(1, amt + 1):\n        dp[0][a] = MAX\n    # State transitions: remaining rows and columns\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # If the coin value exceeds the current amount a, do not choose coin i\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # Take the smaller result of not choosing and choosing coin i\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n    return dp[n][amt] if dp[n][amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* Coin change: dynamic programming */\nint coinChangeDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // Initialize the dp table\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // State transitions: first row and first column\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // State transitions: remaining rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* Coin change: dynamic programming */\nint coinChangeDP(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // Initialize the dp table\n    int[][] dp = new int[n + 1][amt + 1];\n    // State transitions: first row and first column\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // State transitions: remaining rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* Coin change: dynamic programming */\nint CoinChangeDP(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // Initialize the dp table\n    int[,] dp = new int[n + 1, amt + 1];\n    // State transitions: first row and first column\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0, a] = MAX;\n    }\n    // State transitions: remaining rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[i, a] = Math.Min(dp[i - 1, a], dp[i, a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n, amt] != MAX ? dp[n, amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* Coin change: dynamic programming */\nfunc coinChangeDP(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // Initialize the dp table\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // State transitions: first row and first column\n    for a := 1; a &lt;= amt; a++ {\n        dp[0][a] = max\n    }\n    // State transitions: remaining rows and columns\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // If the coin exceeds the current target amount a, do not choose coin i\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[i][a] = int(math.Min(float64(dp[i-1][a]), float64(dp[i][a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* Coin change: dynamic programming */\nfunc coinChangeDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // Initialize the dp table\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // State transitions: first row and first column\n    for a in 1 ... amt {\n        dp[0][a] = MAX\n    }\n    // State transitions: remaining rows and columns\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // If the coin exceeds the current target amount a, do not choose coin i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* Coin change: dynamic programming */\nfunction coinChangeDP(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // Initialize the dp table\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // State transitions: first row and first column\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // State transitions: remaining rows and columns\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* Coin change: dynamic programming */\nfunction coinChangeDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // Initialize the dp table\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // State transitions: first row and first column\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // State transitions: remaining rows and columns\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* Coin change: dynamic programming */\nint coinChangeDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // Initialize the dp table\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // State transitions: first row and first column\n  for (int a = 1; a &lt;= amt; a++) {\n    dp[0][a] = MAX;\n  }\n  // State transitions: remaining rows and columns\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // If the coin exceeds the current amount a, do not choose coin i\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // Take the smaller result of not choosing and choosing coin i\n        dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* Coin change: dynamic programming */\nfn coin_change_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // Initialize the dp table\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // State transitions: first row and first column\n    for a in 1..=amt {\n        dp[0][a] = max;\n    }\n    // State transitions: remaining rows and columns\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[i][a] = std::cmp::min(dp[i - 1][a], dp[i][a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* Coin change: dynamic programming */\nint coinChangeDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // Initialize the dp table\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // State transitions: first row and first column\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // State transitions: remaining rows and columns\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[i][a] = myMin(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[n][amt] != MAX ? dp[n][amt] : -1;\n    // Free memory\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* Coin change: dynamic programming */\nfun coinChangeDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // Initialize the dp table\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // State transitions: first row and first column\n    for (a in 1..amt) {\n        dp[0][a] = MAX\n    }\n    // State transitions: remaining rows and columns\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin value exceeds the current amount a, do not choose coin i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return if (dp[n][amt] != MAX) dp[n][amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>### Coin change: dynamic programming ###\ndef coin_change_dp(coins, amt)\n  n = coins.length\n  _MAX = amt + 1\n  # Initialize the dp table\n  dp = Array.new(n + 1) { Array.new(amt + 1, 0) }\n  # State transitions: first row and first column\n  (1...(amt + 1)).each { |a| dp[0][a] = _MAX }\n  # State transitions: remaining rows and columns\n  for i in 1...(n + 1)\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # If the coin exceeds the current amount a, do not choose coin i\n        dp[i][a] = dp[i - 1][a]\n      else\n        # Take the smaller result of not choosing and choosing coin i\n        dp[i][a] = [dp[i - 1][a], dp[i][a - coins[i - 1]] + 1].min\n      end\n    end\n  end\n  dp[n][amt] != _MAX ? dp[n][amt] : -1\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 14-25 shows dynamic programming for coin change, which closely resembles unbounded knapsack.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 14-25   Dynamic programming for coin change </p>","path":["Chapter 14.   Dynamic programming","14.5   Unbounded knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3-space-optimization_1","level":3,"title":"3.   Space optimization","text":"<p>Space optimization is the same as for unbounded knapsack:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change.py<pre><code>def coin_change_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"Coin change: space-optimized dynamic programming\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # Initialize the dp table\n    dp = [MAX] * (amt + 1)\n    dp[0] = 0\n    # State transitions\n    for i in range(1, n + 1):\n        # Traverse in increasing order\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # If the coin value exceeds the current amount a, do not choose coin i\n                dp[a] = dp[a]\n            else:\n                # Take the smaller result of not choosing and choosing coin i\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n    return dp[amt] if dp[amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* Coin change: space-optimized dynamic programming */\nint coinChangeDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // Initialize the dp table\n    vector&lt;int&gt; dp(amt + 1, MAX);\n    dp[0] = 0;\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[a] = dp[a];\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* Coin change: space-optimized dynamic programming */\nint coinChangeDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // Initialize the dp table\n    int[] dp = new int[amt + 1];\n    Arrays.fill(dp, MAX);\n    dp[0] = 0;\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[a] = dp[a];\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* Coin change: space-optimized dynamic programming */\nint CoinChangeDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // Initialize the dp table\n    int[] dp = new int[amt + 1];\n    Array.Fill(dp, MAX);\n    dp[0] = 0;\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[a] = dp[a];\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[a] = Math.Min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* Coin change: dynamic programming */\nfunc coinChangeDPComp(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // Initialize the dp table\n    dp := make([]int, amt+1)\n    for i := 1; i &lt;= amt; i++ {\n        dp[i] = max\n    }\n    // State transitions\n    for i := 1; i &lt;= n; i++ {\n        // Traverse in increasing order\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // If the coin exceeds the current target amount a, do not choose coin i\n                dp[a] = dp[a]\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[a] = int(math.Min(float64(dp[a]), float64(dp[a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* Coin change: space-optimized dynamic programming */\nfunc coinChangeDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // Initialize the dp table\n    var dp = Array(repeating: MAX, count: amt + 1)\n    dp[0] = 0\n    // State transitions\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // If the coin exceeds the current target amount a, do not choose coin i\n                dp[a] = dp[a]\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* Coin change: space-optimized dynamic programming */\nfunction coinChangeDPComp(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // Initialize the dp table\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // State transitions\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[a] = dp[a];\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* Coin change: space-optimized dynamic programming */\nfunction coinChangeDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // Initialize the dp table\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // State transitions\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[a] = dp[a];\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* Coin change: space-optimized dynamic programming */\nint coinChangeDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // Initialize the dp table\n  List&lt;int&gt; dp = List.filled(amt + 1, MAX);\n  dp[0] = 0;\n  // State transitions\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // If the coin exceeds the current amount a, do not choose coin i\n        dp[a] = dp[a];\n      } else {\n        // Take the smaller result of not choosing and choosing coin i\n        dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* Coin change: space-optimized dynamic programming */\nfn coin_change_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // Initialize the dp table\n    let mut dp = vec![0; amt + 1];\n    dp.fill(max);\n    dp[0] = 0;\n    // State transitions\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[a] = dp[a];\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[a] = std::cmp::min(dp[a], dp[a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* Coin change: space-optimized dynamic programming */\nint coinChangeDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // Initialize the dp table\n    int *dp = malloc((amt + 1) * sizeof(int));\n    for (int j = 1; j &lt;= amt; j++) {\n        dp[j] = MAX;\n    } \n    dp[0] = 0;\n\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[a] = dp[a];\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[a] = myMin(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[amt] != MAX ? dp[amt] : -1;\n    // Free memory\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* Coin change: space-optimized dynamic programming */\nfun coinChangeDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // Initialize the dp table\n    val dp = IntArray(amt + 1)\n    dp.fill(MAX)\n    dp[0] = 0\n    // State transitions\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin value exceeds the current amount a, do not choose coin i\n                dp[a] = dp[a]\n            } else {\n                // Take the smaller result of not choosing and choosing coin i\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return if (dp[amt] != MAX) dp[amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>### Coin change: space-optimized dynamic programming ###\ndef coin_change_dp_comp(coins, amt)\n  n = coins.length\n  _MAX = amt + 1\n  # Initialize the dp table\n  dp = Array.new(amt + 1, _MAX)\n  dp[0] = 0\n  # State transitions\n  for i in 1...(n + 1)\n    # Traverse in increasing order\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # If the coin exceeds the current amount a, do not choose coin i\n        dp[a] = dp[a]\n      else\n        # Take the smaller result of not choosing and choosing coin i\n        dp[a] = [dp[a], dp[a - coins[i - 1]] + 1].min\n      end\n    end\n  end\n  dp[amt] != _MAX ? dp[amt] : -1\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 14.   Dynamic programming","14.5   Unbounded knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1453-coin-change-problem-ii","level":2,"title":"14.5.3   Coin change problem II","text":"<p>Question</p> <p>Given \\(n\\) coin denominations, where denomination \\(i\\) has value \\(coins[i - 1]\\), and a target amount \\(amt\\), each denomination may be used repeatedly. How many combinations of coins make the target amount? An example is shown in Figure 14-26.</p> <p></p> <p> Figure 14-26   Example input for coin change II </p>","path":["Chapter 14.   Dynamic programming","14.5   Unbounded knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1-dynamic-programming-approach_2","level":3,"title":"1.   Dynamic programming approach","text":"<p>This time, we count combinations, so the subproblem becomes the number of combinations that make amount \\(a\\) using the first \\(i\\) denominations. The \\(dp\\) table remains a two-dimensional matrix of size \\((n+1) \\times (amt + 1)\\).</p> <p>The number of combinations for the current state is the sum of those that exclude the current coin and those that include it. The state transition equation is:</p> \\[ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]] \\] <p>A target of \\(0\\) can be made by choosing no coins, so initialize every first-column entry \\(dp[i, 0]\\) to \\(1\\). With no coins, any target \\(&gt;0\\) is impossible, so the first-row entries \\(dp[0, a]\\) are \\(0\\).</p>","path":["Chapter 14.   Dynamic programming","14.5   Unbounded knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2-code-implementation_2","level":3,"title":"2.   Code implementation","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change_ii.py<pre><code>def coin_change_ii_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"Coin change II: dynamic programming\"\"\"\n    n = len(coins)\n    # Initialize the dp table\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # Initialize the first column\n    for i in range(n + 1):\n        dp[i][0] = 1\n    # State transitions\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # If the coin value exceeds the current amount a, do not choose coin i\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # Add the numbers of ways without and with coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n    return dp[n][amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* Coin change II: dynamic programming */\nint coinChangeIIDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // Initialize the dp table\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // Initialize the first column\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* Coin change II: dynamic programming */\nint coinChangeIIDP(int[] coins, int amt) {\n    int n = coins.length;\n    // Initialize the dp table\n    int[][] dp = new int[n + 1][amt + 1];\n    // Initialize the first column\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* Coin change II: dynamic programming */\nint CoinChangeIIDP(int[] coins, int amt) {\n    int n = coins.Length;\n    // Initialize the dp table\n    int[,] dp = new int[n + 1, amt + 1];\n    // Initialize the first column\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i, 0] = 1;\n    }\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[i, a] = dp[i - 1, a] + dp[i, a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n, amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* Coin change II: dynamic programming */\nfunc coinChangeIIDP(coins []int, amt int) int {\n    n := len(coins)\n    // Initialize the dp table\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // Initialize the first column\n    for i := 0; i &lt;= n; i++ {\n        dp[i][0] = 1\n    }\n    // State transitions: remaining rows and columns\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // If the coin exceeds the current target amount a, do not choose coin i\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[i][a] = dp[i-1][a] + dp[i][a-coins[i-1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* Coin change II: dynamic programming */\nfunc coinChangeIIDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // Initialize the dp table\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // Initialize the first column\n    for i in 0 ... n {\n        dp[i][0] = 1\n    }\n    // State transitions\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // If the coin exceeds the current target amount a, do not choose coin i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* Coin change II: dynamic programming */\nfunction coinChangeIIDP(coins, amt) {\n    const n = coins.length;\n    // Initialize the dp table\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // Initialize the first column\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // State transitions\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* Coin change II: dynamic programming */\nfunction coinChangeIIDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // Initialize the dp table\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // Initialize the first column\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // State transitions\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* Coin change II: dynamic programming */\nint coinChangeIIDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // Initialize the dp table\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // Initialize the first column\n  for (int i = 0; i &lt;= n; i++) {\n    dp[i][0] = 1;\n  }\n  // State transitions\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // If the coin exceeds the current amount a, do not choose coin i\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // Add the numbers of ways without and with coin i\n        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[n][amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* Coin change II: dynamic programming */\nfn coin_change_ii_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // Initialize the dp table\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // Initialize the first column\n    for i in 0..=n {\n        dp[i][0] = 1;\n    }\n    // State transitions\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[n][amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* Coin change II: dynamic programming */\nint coinChangeIIDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // Initialize the dp table\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // Initialize the first column\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[n][amt];\n    // Free memory\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* Coin change II: dynamic programming */\nfun coinChangeIIDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // Initialize the dp table\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // Initialize the first column\n    for (i in 0..n) {\n        dp[i][0] = 1\n    }\n    // State transitions\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin value exceeds the current amount a, do not choose coin i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>### Coin change II: dynamic programming ###\ndef coin_change_ii_dp(coins, amt)\n  n = coins.length\n  # Initialize the dp table\n  dp = Array.new(n + 1) { Array.new(amt + 1, 0) }\n  # Initialize the first column\n  (0...(n + 1)).each { |i| dp[i][0] = 1 }\n  # State transitions\n  for i in 1...(n + 1)\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # If the coin exceeds the current amount a, do not choose coin i\n        dp[i][a] = dp[i - 1][a]\n      else\n        # Add the numbers of ways without and with coin i\n        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n      end\n    end\n  end\n  dp[n][amt]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 14.   Dynamic programming","14.5   Unbounded knapsack problem"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3-space-optimization_2","level":3,"title":"3.   Space optimization","text":"<p>The space optimization is the same: remove the coin dimension:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change_ii.py<pre><code>def coin_change_ii_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"Coin change II: space-optimized dynamic programming\"\"\"\n    n = len(coins)\n    # Initialize the dp table\n    dp = [0] * (amt + 1)\n    dp[0] = 1\n    # State transitions\n    for i in range(1, n + 1):\n        # Traverse in increasing order\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # If the coin value exceeds the current amount a, do not choose coin i\n                dp[a] = dp[a]\n            else:\n                # Add the numbers of ways without and with coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n    return dp[amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* Coin change II: space-optimized dynamic programming */\nint coinChangeIIDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // Initialize the dp table\n    vector&lt;int&gt; dp(amt + 1, 0);\n    dp[0] = 1;\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[a] = dp[a];\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* Coin change II: space-optimized dynamic programming */\nint coinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    // Initialize the dp table\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[a] = dp[a];\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* Coin change II: space-optimized dynamic programming */\nint CoinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    // Initialize the dp table\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[a] = dp[a];\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* Coin change II: space-optimized dynamic programming */\nfunc coinChangeIIDPComp(coins []int, amt int) int {\n    n := len(coins)\n    // Initialize the dp table\n    dp := make([]int, amt+1)\n    dp[0] = 1\n    // State transitions\n    for i := 1; i &lt;= n; i++ {\n        // Traverse in increasing order\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // If the coin exceeds the current target amount a, do not choose coin i\n                dp[a] = dp[a]\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[a] = dp[a] + dp[a-coins[i-1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* Coin change II: space-optimized dynamic programming */\nfunc coinChangeIIDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // Initialize the dp table\n    var dp = Array(repeating: 0, count: amt + 1)\n    dp[0] = 1\n    // State transitions\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // If the coin exceeds the current target amount a, do not choose coin i\n                dp[a] = dp[a]\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* Coin change II: space-optimized dynamic programming */\nfunction coinChangeIIDPComp(coins, amt) {\n    const n = coins.length;\n    // Initialize the dp table\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // State transitions\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[a] = dp[a];\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* Coin change II: space-optimized dynamic programming */\nfunction coinChangeIIDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // Initialize the dp table\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // State transitions\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[a] = dp[a];\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* Coin change II: space-optimized dynamic programming */\nint coinChangeIIDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // Initialize the dp table\n  List&lt;int&gt; dp = List.filled(amt + 1, 0);\n  dp[0] = 1;\n  // State transitions\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // If the coin exceeds the current amount a, do not choose coin i\n        dp[a] = dp[a];\n      } else {\n        // Add the numbers of ways without and with coin i\n        dp[a] = dp[a] + dp[a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* Coin change II: space-optimized dynamic programming */\nfn coin_change_ii_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // Initialize the dp table\n    let mut dp = vec![0; amt + 1];\n    dp[0] = 1;\n    // State transitions\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[a] = dp[a];\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* Coin change II: space-optimized dynamic programming */\nint coinChangeIIDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // Initialize the dp table\n    int *dp = calloc(amt + 1, sizeof(int));\n    dp[0] = 1;\n    // State transitions\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin exceeds the target amount, do not choose coin i\n                dp[a] = dp[a];\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[amt];\n    // Free memory\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* Coin change II: space-optimized dynamic programming */\nfun coinChangeIIDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // Initialize the dp table\n    val dp = IntArray(amt + 1)\n    dp[0] = 1\n    // State transitions\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // If the coin value exceeds the current amount a, do not choose coin i\n                dp[a] = dp[a]\n            } else {\n                // Add the numbers of ways without and with coin i\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>### Coin change II: space-optimized dynamic programming ###\ndef coin_change_ii_dp_comp(coins, amt)\n  n = coins.length\n  # Initialize the dp table\n  dp = Array.new(amt + 1, 0)\n  dp[0] = 1\n  # State transitions\n  for i in 1...(n + 1)\n    # Traverse in increasing order\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # If the coin exceeds the current amount a, do not choose coin i\n        dp[a] = dp[a]\n      else\n        # Add the numbers of ways without and with coin i\n        dp[a] = dp[a] + dp[a - coins[i - 1]]\n      end\n    end\n  end\n  dp[amt]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 14.   Dynamic programming","14.5   Unbounded knapsack problem"],"tags":[]},{"location":"chapter_graph/","level":1,"title":"Chapter 9.   Graphs","text":"<p>Abstract</p> <p>On the journey of life, we are like nodes joined by countless invisible edges.</p> <p>Every meeting and parting leaves a unique mark on this vast network.</p>","path":["Chapter 9.   Graphs"],"tags":[]},{"location":"chapter_graph/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>9.1   Graphs</li> <li>9.2   Basic graph operations</li> <li>9.3   Graph traversal</li> <li>9.4   Summary</li> <li>9.5   Exercises</li> </ul>","path":["Chapter 9.   Graphs"],"tags":[]},{"location":"chapter_graph/exercises/","level":1,"title":"9.5   Exercises","text":"","path":["Chapter 9.   Graphs","9.5   Exercises"],"tags":[]},{"location":"chapter_graph/exercises/#951-check-your-understanding","level":2,"title":"9.5.1   Check your understanding","text":"","path":["Chapter 9.   Graphs","9.5   Exercises"],"tags":[]},{"location":"chapter_graph/exercises/#1-representing-the-same-graph-in-two-ways","level":3,"title":"1.   Representing the same graph in two ways","text":"<p>An undirected graph has four vertices <code>A, B, C, D</code> and edges <code>A-B, A-C, B-C, C-D</code>.</p> <ol> <li>Write its adjacency list.</li> <li>Fill in its adjacency matrix using only 0 and 1.</li> <li>Which representation lets us determine whether <code>A</code> and <code>D</code> are directly connected by inspecting just one stored entry?</li> <li>If a graph has many vertices but few edges, which representation usually uses less space?</li> </ol> Reference answer <ol> <li> <p>The adjacency list is:</p> <pre><code>A: B, C\nB: A, C\nC: A, B, D\nD: C\n</code></pre> </li> <li> <p>The adjacency matrix is:</p> A B C D A 0 1 1 0 B 1 0 1 0 C 1 1 0 1 D 0 0 1 0 </li> <li> <p>In the adjacency matrix, we can inspect row <code>A</code>, column <code>D</code> directly. This makes it well suited to checking whether any two vertices are directly connected.</p> </li> <li> <p>When there are many vertices but few edges, an adjacency list records only existing edges. It usually uses less space than an adjacency matrix, which reserves an entry for every pair of vertices.</p> </li> </ol>","path":["Chapter 9.   Graphs","9.5   Exercises"],"tags":[]},{"location":"chapter_graph/exercises/#2-breadth-first-and-depth-first-traversal-orders","level":3,"title":"2.   Breadth-first and depth-first traversal orders","text":"<p>An undirected graph has vertices <code>A, B, C, D, E</code> and edges <code>A-B, A-C, B-D, C-D, D-E</code>.</p> <p>Start at A. Whenever several unvisited neighbors are available, choose them in alphabetical order:</p> <ol> <li>Write the breadth-first traversal (BFS) order.</li> <li>Write the recursive depth-first traversal (DFS) order.</li> <li>Why must both traversals keep track of vertices already visited?</li> </ol> Reference answer <ol> <li> <p>The BFS order is <code>A, B, C, D, E</code>. It first visits B and C, which are one edge away from A,     and then the more distant D and E.</p> </li> <li> <p>The DFS order is <code>A, B, D, C, E</code>. It proceeds into an unvisited neighbor of the current vertex,     first following <code>A → B → D → C</code>. When C has no new neighbors to visit, it returns to D and then visits E.</p> </li> <li> <p>The graph contains cycles, such as <code>A-B-D-C-A</code>. Without recording visited vertices,     traversal could repeatedly follow a cycle through the same vertices and fail to terminate.</p> </li> </ol>","path":["Chapter 9.   Graphs","9.5   Exercises"],"tags":[]},{"location":"chapter_graph/exercises/#3-can-one-bfs-visit-the-entire-graph","level":3,"title":"3.   Can one BFS visit the entire graph?","text":"<p>An undirected graph has vertices <code>A, B, C, D, E, F</code> and only the edges <code>A-B, B-C, D-E</code>.</p> <ol> <li>Which vertices can a single BFS starting from A visit?</li> <li>Based on your answer, has this BFS visited every vertex in the graph? Why?</li> <li>Suppose we scan all vertices alphabetically and start a new BFS whenever we encounter an unvisited vertex.     What are the starting vertices of these searches? How many separate connected parts, or connected components, does the graph have?</li> </ol> Reference answer <ol> <li> <p>Starting from A reaches only <code>A, B, C</code>.</p> </li> <li> <p>Not every vertex is visited. <code>D, E</code> form another connected part, and F is an isolated vertex.     None has a path to A, so they cannot be reached from A.</p> </li> <li> <p>The three BFS starting vertices are <code>A, D, F</code>, in order, visiting     <code>{A, B, C}</code>, <code>{D, E}</code>, and <code>{F}</code>, respectively. Thus, the graph has three connected components.</p> </li> </ol>","path":["Chapter 9.   Graphs","9.5   Exercises"],"tags":[]},{"location":"chapter_graph/exercises/#952-programming-exercises","level":2,"title":"9.5.2   Programming exercises","text":"","path":["Chapter 9.   Graphs","9.5   Exercises"],"tags":[]},{"location":"chapter_graph/exercises/#1-checking-for-a-path-in-an-undirected-graph","level":3,"title":"1.   Checking for a path in an undirected graph","text":"<p>You are given an undirected graph with \\(n\\) vertices numbered from \\(0\\) to \\(n-1\\). Each entry <code>[u, v]</code> in the array <code>edges</code> represents an undirected edge between vertices <code>u</code> and <code>v</code>.</p> <p>You are also given a starting vertex <code>source</code> and an ending vertex <code>destination</code>. First build an adjacency list from <code>edges</code>, then use BFS or DFS to determine whether a path exists from <code>source</code> to <code>destination</code>. Return <code>true</code> if it does, or <code>false</code> otherwise. The graph may contain cycles and may be disconnected.</p> Hints <ol> <li>Add each undirected edge in both directions.</li> <li>The graph may contain cycles, so keep track of vertices already visited.</li> <li>Starting from source, return true if you reach destination. Return false if traversal ends without reaching it.</li> </ol> <p>LeetCode</p>","path":["Chapter 9.   Graphs","9.5   Exercises"],"tags":[]},{"location":"chapter_graph/graph/","level":1,"title":"9.1   Graphs","text":"<p>A graph is a nonlinear data structure made up of vertices and edges. We can represent a graph \\(G\\) abstractly using a set of vertices \\(V\\) and a set of edges \\(E\\). The following example describes a graph with 5 vertices and 7 edges.</p> \\[ \\begin{aligned} V &amp; = \\{ 1, 2, 3, 4, 5 \\} \\newline E &amp; = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G &amp; = \\{ V, E \\} \\newline \\end{aligned} \\] <p>If we think of vertices as nodes and edges as references (pointers) connecting those nodes, a graph can be viewed as an extension of a linked list. As Figure 9-1 shows, network relationships (graphs) allow more freedom than linear relationships (linked lists) or divide-and-conquer relationships (trees), making them more complex.</p> <p></p> <p> Figure 9-1   Relationships between linked lists, trees, and graphs </p>","path":["Chapter 9.   Graphs","9.1   Graphs"],"tags":[]},{"location":"chapter_graph/graph/#911-common-graph-types-and-terminology","level":2,"title":"9.1.1   Common graph types and terminology","text":"<p>Depending on whether their edges have a direction, graphs are classified as undirected graphs or directed graphs, as shown in Figure 9-2.</p> <ul> <li>In an undirected graph, an edge represents a two-way connection between two vertices, such as a friendship on WeChat or QQ.</li> <li>In a directed graph, edges have a direction: the edges \\(A \\rightarrow B\\) and \\(A \\leftarrow B\\) are independent of each other. Examples include following and being followed on Weibo or Douyin.</li> </ul> <p></p> <p> Figure 9-2   Directed and undirected graphs </p> <p>Depending on whether all vertices are connected, graphs are classified as connected graphs or disconnected graphs, as shown in Figure 9-3.</p> <ul> <li>In a connected graph, every other vertex can be reached from any starting vertex.</li> <li>In a disconnected graph, at least one vertex cannot be reached from a given starting vertex.</li> </ul> <p></p> <p> Figure 9-3   Connected and disconnected graphs </p> <p>We can also assign a weight to each edge to obtain a weighted graph, as shown in Figure 9-4. For example, mobile games such as Honor of Kings calculate an affinity score between players based on the time they spend playing together. A weighted graph can represent this network of affinities.</p> <p></p> <p> Figure 9-4   Weighted and unweighted graphs </p> <p>The following terms are commonly used when discussing graphs.</p> <ul> <li>Adjacency: Two vertices are adjacent if an edge connects them. In Figure 9-4, the vertices adjacent to vertex 1 are 2, 3, and 5.</li> <li>Path: A sequence of edges followed from vertex A to vertex B is called a path from A to B. In Figure 9-4, the edge sequence 1-5-2-4 forms a path from vertex 1 to vertex 4.</li> <li>Degree: The number of edges incident to a vertex. In a directed graph, the in-degree is the number of edges pointing to the vertex, and the out-degree is the number of edges pointing away from it.</li> </ul>","path":["Chapter 9.   Graphs","9.1   Graphs"],"tags":[]},{"location":"chapter_graph/graph/#912-representing-a-graph","level":2,"title":"9.1.2   Representing a graph","text":"<p>Two common ways to represent a graph are an adjacency matrix and an adjacency list. We will use undirected graphs in the following examples.</p>","path":["Chapter 9.   Graphs","9.1   Graphs"],"tags":[]},{"location":"chapter_graph/graph/#1-adjacency-matrix","level":3,"title":"1.   Adjacency matrix","text":"<p>For a graph with \\(n\\) vertices, an adjacency matrix represents the graph using an \\(n \\times n\\) matrix. Each row (or column) represents a vertex, and the matrix entries represent edges: \\(1\\) or \\(0\\) indicates whether an edge exists between two vertices.</p> <p>As shown in Figure 9-5, let \\(M\\) be the adjacency matrix and \\(V\\) the vertex list. The entry \\(M[i, j] = 1\\) means that an edge connects vertices \\(V[i]\\) and \\(V[j]\\), whereas \\(M[i, j] = 0\\) means that no edge connects them.</p> <p></p> <p> Figure 9-5   Representing a graph with an adjacency matrix </p> <p>An adjacency matrix has the following properties.</p> <ul> <li>In a simple graph, a vertex cannot connect to itself, so the entries on the main diagonal do not represent meaningful connections.</li> <li>In an undirected graph, edges in the two directions are equivalent, so the adjacency matrix is symmetric about its main diagonal.</li> <li>Replacing the \\(1\\) and \\(0\\) entries with weights allows the matrix to represent a weighted graph.</li> </ul> <p>With an adjacency matrix, we can access a matrix entry directly to retrieve an edge. Adding, removing, looking up, and modifying edges are therefore efficient, each taking \\(O(1)\\) time. However, the matrix requires \\(O(n^2)\\) space and uses a substantial amount of memory.</p>","path":["Chapter 9.   Graphs","9.1   Graphs"],"tags":[]},{"location":"chapter_graph/graph/#2-adjacency-list","level":3,"title":"2.   Adjacency list","text":"<p>An adjacency list represents a graph using \\(n\\) linked lists whose nodes represent vertices. The \\(i\\)-th linked list corresponds to vertex \\(i\\) and stores all of its adjacent vertices (the vertices connected to it). Figure 9-6 shows an example of a graph stored as an adjacency list.</p> <p></p> <p> Figure 9-6   Representing a graph with an adjacency list </p> <p>An adjacency list stores only edges that actually exist. Since the total number of edges is usually much smaller than \\(n^2\\), it saves space. However, finding an edge requires traversing a linked list, so lookups are slower than with an adjacency matrix.</p> <p>As Figure 9-6 shows, an adjacency list closely resembles separate chaining in a hash table, so similar techniques can improve its efficiency. For example, a long linked list can be replaced with an AVL tree or a red-black tree to improve lookup time from \\(O(n)\\) to \\(O(\\log n)\\). Replacing it with a hash table can further reduce the time complexity to \\(O(1)\\).</p>","path":["Chapter 9.   Graphs","9.1   Graphs"],"tags":[]},{"location":"chapter_graph/graph/#913-common-applications-of-graphs","level":2,"title":"9.1.3   Common applications of graphs","text":"<p>As shown in Table 9-1, many real-world systems can be modeled as graphs, and their associated problems can be reduced to graph computation problems.</p> <p> Table 9-1   Common graphs in everyday life </p> Vertices Edges Graph computation problem Social network Users Friendships Recommending potential friends Subway network Stations Connections between stations Recommending the shortest route Solar system Celestial bodies Gravitational interactions between bodies Calculating planetary orbits","path":["Chapter 9.   Graphs","9.1   Graphs"],"tags":[]},{"location":"chapter_graph/graph_operations/","level":1,"title":"9.2   Basic graph operations","text":"<p>Basic graph operations fall into two categories: operations on edges and operations on vertices. Their implementations differ depending on whether the graph is represented by an adjacency matrix or an adjacency list.</p>","path":["Chapter 9.   Graphs","9.2   Basic graph operations"],"tags":[]},{"location":"chapter_graph/graph_operations/#921-adjacency-matrix-implementation","level":2,"title":"9.2.1   Adjacency matrix implementation","text":"<p>For an undirected graph with \\(n\\) vertices, the operations are implemented as shown in Figure 9-7.</p> <ul> <li>Add or remove an edge: Modify the corresponding entry in the adjacency matrix directly, taking \\(O(1)\\) time. Since the graph is undirected, update the edge in both directions.</li> <li>Add a vertex: Append one row and one column to the adjacency matrix and fill them with \\(0\\), taking \\(O(n)\\) time.</li> <li>Remove a vertex: Delete the corresponding row and column from the adjacency matrix. The worst case occurs when deleting the first row and column: \\((n-1)^2\\) entries must be shifted up and to the left, taking \\(O(n^2)\\) time.</li> <li>Initialize the graph: Given \\(n\\) vertices, initialize the vertex list <code>vertices</code> of length \\(n\\) in \\(O(n)\\) time, then initialize the \\(n \\times n\\) adjacency matrix <code>adjMat</code> in \\(O(n^2)\\) time.</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-7   Initializing an adjacency matrix and adding or removing edges and vertices </p> <p>The following code implements a graph using an adjacency matrix:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_adjacency_matrix.py<pre><code>class GraphAdjMat:\n    \"\"\"Undirected graph class implemented with an adjacency matrix\"\"\"\n\n    def __init__(self, vertices: list[int], edges: list[list[int]]):\n        \"\"\"Constructor\"\"\"\n        # Vertex list: elements are vertex values, and indices are vertex indices\n        self.vertices: list[int] = []\n        # Adjacency matrix: row and column indices correspond to vertex indices\n        self.adj_mat: list[list[int]] = []\n        # Add a vertex\n        for val in vertices:\n            self.add_vertex(val)\n        # Add an edge\n        # Note that the entries in edges are vertex indices, corresponding to element indices in vertices\n        for e in edges:\n            self.add_edge(e[0], e[1])\n\n    def size(self) -&gt; int:\n        \"\"\"Get the number of vertices\"\"\"\n        return len(self.vertices)\n\n    def add_vertex(self, val: int):\n        \"\"\"Add a vertex\"\"\"\n        n = self.size()\n        # Append the new vertex value to the vertex list\n        self.vertices.append(val)\n        # Add a row to the adjacency matrix\n        new_row = [0] * n\n        self.adj_mat.append(new_row)\n        # Add a column to the adjacency matrix\n        for row in self.adj_mat:\n            row.append(0)\n\n    def remove_vertex(self, index: int):\n        \"\"\"Remove a vertex\"\"\"\n        if index &gt;= self.size():\n            raise IndexError()\n        # Remove the vertex at index `index` from the vertex list\n        self.vertices.pop(index)\n        # Remove row `index` from the adjacency matrix\n        self.adj_mat.pop(index)\n        # Remove column `index` from the adjacency matrix\n        for row in self.adj_mat:\n            row.pop(index)\n\n    def add_edge(self, i: int, j: int):\n        \"\"\"Add an edge\"\"\"\n        # Parameters i and j are element indices in vertices\n        # Handle out-of-bounds or equal indices\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        # For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal\n        self.adj_mat[i][j] = 1\n        self.adj_mat[j][i] = 1\n\n    def remove_edge(self, i: int, j: int):\n        \"\"\"Remove an edge\"\"\"\n        # Parameters i and j are element indices in vertices\n        # Handle out-of-bounds or equal indices\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        self.adj_mat[i][j] = 0\n        self.adj_mat[j][i] = 0\n\n    def print(self):\n        \"\"\"Print the adjacency matrix\"\"\"\n        print(\"Vertex list =\", self.vertices)\n        print(\"Adjacency matrix =\")\n        print_matrix(self.adj_mat)\n</code></pre> graph_adjacency_matrix.cpp<pre><code>/* Undirected graph class implemented with an adjacency matrix */\nclass GraphAdjMat {\n    vector&lt;int&gt; vertices;       // Vertex list: elements are vertex values, and indices are vertex indices\n    vector&lt;vector&lt;int&gt;&gt; adjMat; // Adjacency matrix: row and column indices correspond to vertex indices\n\n  public:\n    /* Constructor */\n    GraphAdjMat(const vector&lt;int&gt; &amp;vertices, const vector&lt;vector&lt;int&gt;&gt; &amp;edges) {\n        // Add a vertex\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // Add an edge\n        // Note that the entries in edges are vertex indices, corresponding to element indices in vertices\n        for (const vector&lt;int&gt; &amp;edge : edges) {\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    int size() const {\n        return vertices.size();\n    }\n\n    /* Add a vertex */\n    void addVertex(int val) {\n        int n = size();\n        // Append the new vertex value to the vertex list\n        vertices.push_back(val);\n        // Add a row to the adjacency matrix\n        adjMat.emplace_back(vector&lt;int&gt;(n, 0));\n        // Add a column to the adjacency matrix\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.push_back(0);\n        }\n    }\n\n    /* Remove a vertex */\n    void removeVertex(int index) {\n        if (index &gt;= size()) {\n            throw out_of_range(\"Invalid vertex\");\n        }\n        // Remove the vertex at index `index` from the vertex list\n        vertices.erase(vertices.begin() + index);\n        // Remove row `index` from the adjacency matrix\n        adjMat.erase(adjMat.begin() + index);\n        // Remove column `index` from the adjacency matrix\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.erase(row.begin() + index);\n        }\n    }\n\n    /* Add an edge */\n    // Parameters i and j are element indices in vertices\n    void addEdge(int i, int j) {\n        // Handle out-of-bounds or equal indices\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"Invalid vertex\");\n        }\n        // For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* Remove an edge */\n    // Parameters i and j are element indices in vertices\n    void removeEdge(int i, int j) {\n        // Handle out-of-bounds or equal indices\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"Invalid vertex\");\n        }\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* Print the adjacency matrix */\n    void print() {\n        cout &lt;&lt; \"Vertex list = \";\n        printVector(vertices);\n        cout &lt;&lt; \"Adjacency matrix =\" &lt;&lt; endl;\n        printVectorMatrix(adjMat);\n    }\n};\n</code></pre> graph_adjacency_matrix.java<pre><code>/* Undirected graph class implemented with an adjacency matrix */\nclass GraphAdjMat {\n    List&lt;Integer&gt; vertices; // Vertex list: elements are vertex values, and indices are vertex indices\n    List&lt;List&lt;Integer&gt;&gt; adjMat; // Adjacency matrix: row and column indices correspond to vertex indices\n\n    /* Constructor */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = new ArrayList&lt;&gt;();\n        this.adjMat = new ArrayList&lt;&gt;();\n        // Add a vertex\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // Add an edge\n        // Note that the entries in edges are vertex indices, corresponding to element indices in vertices\n        for (int[] e : edges) {\n            addEdge(e[0], e[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    public int size() {\n        return vertices.size();\n    }\n\n    /* Add a vertex */\n    public void addVertex(int val) {\n        int n = size();\n        // Append the new vertex value to the vertex list\n        vertices.add(val);\n        // Add a row to the adjacency matrix\n        List&lt;Integer&gt; newRow = new ArrayList&lt;&gt;(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.add(0);\n        }\n        adjMat.add(newRow);\n        // Add a column to the adjacency matrix\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.add(0);\n        }\n    }\n\n    /* Remove a vertex */\n    public void removeVertex(int index) {\n        if (index &gt;= size())\n            throw new IndexOutOfBoundsException();\n        // Remove the vertex at index `index` from the vertex list\n        vertices.remove(index);\n        // Remove row `index` from the adjacency matrix\n        adjMat.remove(index);\n        // Remove column `index` from the adjacency matrix\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.remove(index);\n        }\n    }\n\n    /* Add an edge */\n    // Parameters i and j are element indices in vertices\n    public void addEdge(int i, int j) {\n        // Handle out-of-bounds or equal indices\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        // For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal\n        adjMat.get(i).set(j, 1);\n        adjMat.get(j).set(i, 1);\n    }\n\n    /* Remove an edge */\n    // Parameters i and j are element indices in vertices\n    public void removeEdge(int i, int j) {\n        // Handle out-of-bounds or equal indices\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        adjMat.get(i).set(j, 0);\n        adjMat.get(j).set(i, 0);\n    }\n\n    /* Print the adjacency matrix */\n    public void print() {\n        System.out.print(\"Vertex list = \");\n        System.out.println(vertices);\n        System.out.println(\"Adjacency matrix =\");\n        PrintUtil.printMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.cs<pre><code>/* Undirected graph class implemented with an adjacency matrix */\nclass GraphAdjMat {\n    List&lt;int&gt; vertices;     // Vertex list: elements are vertex values, and indices are vertex indices\n    List&lt;List&lt;int&gt;&gt; adjMat; // Adjacency matrix: row and column indices correspond to vertex indices\n\n    /* Constructor */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // Add a vertex\n        foreach (int val in vertices) {\n            AddVertex(val);\n        }\n        // Add an edge\n        // Note that the entries in edges are vertex indices, corresponding to element indices in vertices\n        foreach (int[] e in edges) {\n            AddEdge(e[0], e[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    int Size() {\n        return vertices.Count;\n    }\n\n    /* Add a vertex */\n    public void AddVertex(int val) {\n        int n = Size();\n        // Append the new vertex value to the vertex list\n        vertices.Add(val);\n        // Add a row to the adjacency matrix\n        List&lt;int&gt; newRow = new(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.Add(0);\n        }\n        adjMat.Add(newRow);\n        // Add a column to the adjacency matrix\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.Add(0);\n        }\n    }\n\n    /* Remove a vertex */\n    public void RemoveVertex(int index) {\n        if (index &gt;= Size())\n            throw new IndexOutOfRangeException();\n        // Remove the vertex at index `index` from the vertex list\n        vertices.RemoveAt(index);\n        // Remove row `index` from the adjacency matrix\n        adjMat.RemoveAt(index);\n        // Remove column `index` from the adjacency matrix\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.RemoveAt(index);\n        }\n    }\n\n    /* Add an edge */\n    // Parameters i and j are element indices in vertices\n    public void AddEdge(int i, int j) {\n        // Handle out-of-bounds or equal indices\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        // For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* Remove an edge */\n    // Parameters i and j are element indices in vertices\n    public void RemoveEdge(int i, int j) {\n        // Handle out-of-bounds or equal indices\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* Print the adjacency matrix */\n    public void Print() {\n        Console.Write(\"Vertex list = \");\n        PrintUtil.PrintList(vertices);\n        Console.WriteLine(\"Adjacency matrix =\");\n        PrintUtil.PrintMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.go<pre><code>/* Undirected graph type implemented with an adjacency matrix */\ntype graphAdjMat struct {\n    // Vertex list: elements are vertex values, and indices are vertex indices\n    vertices []int\n    // Adjacency matrix: row and column indices correspond to vertex indices\n    adjMat [][]int\n}\n\n/* Constructor */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n    // Allocate the adjacency matrix for all vertices\n    n := len(vertices)\n    adjMat := make([][]int, n)\n    for i := range adjMat {\n        adjMat[i] = make([]int, n)\n    }\n    // Initialize the graph\n    g := &amp;graphAdjMat{\n        vertices: vertices,\n        adjMat:   adjMat,\n    }\n    // Add all edges\n    // Note that the entries in edges are vertex indices, corresponding to element indices in vertices\n    for i := range edges {\n        g.addEdge(edges[i][0], edges[i][1])\n    }\n    return g\n}\n\n/* Get the number of vertices */\nfunc (g *graphAdjMat) size() int {\n    return len(g.vertices)\n}\n\n/* Add a vertex */\nfunc (g *graphAdjMat) addVertex(val int) {\n    n := g.size()\n    // Append the new vertex value to the vertex list\n    g.vertices = append(g.vertices, val)\n    // Add a row to the adjacency matrix\n    newRow := make([]int, n)\n    g.adjMat = append(g.adjMat, newRow)\n    // Add a column to the adjacency matrix\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i], 0)\n    }\n}\n\n/* Remove a vertex */\nfunc (g *graphAdjMat) removeVertex(index int) {\n    if index &gt;= g.size() {\n        return\n    }\n    // Remove the vertex at index `index` from the vertex list\n    g.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n    // Remove row `index` from the adjacency matrix\n    g.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n    // Remove column `index` from the adjacency matrix\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n    }\n}\n\n/* Add an edge */\n// Parameters i and j are element indices in vertices\nfunc (g *graphAdjMat) addEdge(i, j int) {\n    // Check for out-of-bounds or equal indices (the constructed error is not returned)\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index out of bounds\")\n    }\n    // For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal\n    g.adjMat[i][j] = 1\n    g.adjMat[j][i] = 1\n}\n\n/* Remove an edge */\n// Parameters i and j are element indices in vertices\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n    // Check for out-of-bounds or equal indices (the constructed error is not returned)\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index out of bounds\")\n    }\n    g.adjMat[i][j] = 0\n    g.adjMat[j][i] = 0\n}\n\n/* Print the adjacency matrix */\nfunc (g *graphAdjMat) print() {\n    fmt.Printf(\"\\tVertex list = %v\\n\", g.vertices)\n    fmt.Printf(\"\\tAdjacency matrix = \\n\")\n    for i := range g.adjMat {\n        fmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n    }\n}\n</code></pre> graph_adjacency_matrix.swift<pre><code>/* Undirected graph class implemented with an adjacency matrix */\nclass GraphAdjMat {\n    private var vertices: [Int] // Vertex list: elements are vertex values, and indices are vertex indices\n    private var adjMat: [[Int]] // Adjacency matrix: row and column indices correspond to vertex indices\n\n    /* Constructor */\n    init(vertices: [Int], edges: [[Int]]) {\n        self.vertices = []\n        adjMat = []\n        // Add all vertices\n        for val in vertices {\n            addVertex(val: val)\n        }\n        // Add all edges\n        // Note that the entries in edges are vertex indices, corresponding to element indices in vertices\n        for e in edges {\n            addEdge(i: e[0], j: e[1])\n        }\n    }\n\n    /* Get the number of vertices */\n    func size() -&gt; Int {\n        vertices.count\n    }\n\n    /* Add a vertex */\n    func addVertex(val: Int) {\n        let n = size()\n        // Append the new vertex value to the vertex list\n        vertices.append(val)\n        // Add a row to the adjacency matrix\n        let newRow = Array(repeating: 0, count: n)\n        adjMat.append(newRow)\n        // Add a column to the adjacency matrix\n        for i in adjMat.indices {\n            adjMat[i].append(0)\n        }\n    }\n\n    /* Remove a vertex */\n    func removeVertex(index: Int) {\n        if index &gt;= size() {\n            fatalError(\"Invalid index\")\n        }\n        // Remove the vertex at index `index` from the vertex list\n        vertices.remove(at: index)\n        // Remove row `index` from the adjacency matrix\n        adjMat.remove(at: index)\n        // Remove column `index` from the adjacency matrix\n        for i in adjMat.indices {\n            adjMat[i].remove(at: index)\n        }\n    }\n\n    /* Add an edge */\n    // Parameters i and j are element indices in vertices\n    func addEdge(i: Int, j: Int) {\n        // Handle out-of-bounds or equal indices\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"Invalid index\")\n        }\n        // For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* Remove an edge */\n    // Parameters i and j are element indices in vertices\n    func removeEdge(i: Int, j: Int) {\n        // Handle out-of-bounds or equal indices\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"Invalid index\")\n        }\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* Print the adjacency matrix */\n    func print() {\n        Swift.print(\"Vertex list = \", terminator: \"\")\n        Swift.print(vertices)\n        Swift.print(\"Adjacency matrix =\")\n        PrintUtil.printMatrix(matrix: adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.js<pre><code>/* Undirected graph class implemented with an adjacency matrix */\nclass GraphAdjMat {\n    vertices; // Vertex list: elements are vertex values, and indices are vertex indices\n    adjMat; // Adjacency matrix: row and column indices correspond to vertex indices\n\n    /* Constructor */\n    constructor(vertices, edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // Add a vertex\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // Add an edge\n        // Note that the entries in edges are vertex indices, corresponding to element indices in vertices\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    size() {\n        return this.vertices.length;\n    }\n\n    /* Add a vertex */\n    addVertex(val) {\n        const n = this.size();\n        // Append the new vertex value to the vertex list\n        this.vertices.push(val);\n        // Add a row to the adjacency matrix\n        const newRow = [];\n        for (let j = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // Add a column to the adjacency matrix\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* Remove a vertex */\n    removeVertex(index) {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index out of bounds');\n        }\n        // Remove the vertex at index `index` from the vertex list\n        this.vertices.splice(index, 1);\n\n        // Remove row `index` from the adjacency matrix\n        this.adjMat.splice(index, 1);\n        // Remove column `index` from the adjacency matrix\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* Add an edge */\n    // Parameters i and j are element indices in vertices\n    addEdge(i, j) {\n        // Handle out-of-bounds or equal indices\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index out of bounds');\n        }\n        // In an undirected graph, the adjacency matrix is symmetric about its main diagonal: (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* Remove an edge */\n    // Parameters i and j are element indices in vertices\n    removeEdge(i, j) {\n        // Handle out-of-bounds or equal indices\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index out of bounds');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* Print the adjacency matrix */\n    print() {\n        console.log('Vertex list = ', this.vertices);\n        console.log('Adjacency matrix =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.ts<pre><code>/* Undirected graph class implemented with an adjacency matrix */\nclass GraphAdjMat {\n    vertices: number[]; // Vertex list: elements are vertex values, and indices are vertex indices\n    adjMat: number[][]; // Adjacency matrix: row and column indices correspond to vertex indices\n\n    /* Constructor */\n    constructor(vertices: number[], edges: number[][]) {\n        this.vertices = [];\n        this.adjMat = [];\n        // Add a vertex\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // Add an edge\n        // Note that the entries in edges are vertex indices, corresponding to element indices in vertices\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    size(): number {\n        return this.vertices.length;\n    }\n\n    /* Add a vertex */\n    addVertex(val: number): void {\n        const n: number = this.size();\n        // Append the new vertex value to the vertex list\n        this.vertices.push(val);\n        // Add a row to the adjacency matrix\n        const newRow: number[] = [];\n        for (let j: number = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // Add a column to the adjacency matrix\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* Remove a vertex */\n    removeVertex(index: number): void {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index out of bounds');\n        }\n        // Remove the vertex at index `index` from the vertex list\n        this.vertices.splice(index, 1);\n\n        // Remove row `index` from the adjacency matrix\n        this.adjMat.splice(index, 1);\n        // Remove column `index` from the adjacency matrix\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* Add an edge */\n    // Parameters i and j are element indices in vertices\n    addEdge(i: number, j: number): void {\n        // Handle out-of-bounds or equal indices\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index out of bounds');\n        }\n        // In an undirected graph, the adjacency matrix is symmetric about its main diagonal: (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* Remove an edge */\n    // Parameters i and j are element indices in vertices\n    removeEdge(i: number, j: number): void {\n        // Handle out-of-bounds or equal indices\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index out of bounds');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* Print the adjacency matrix */\n    print(): void {\n        console.log('Vertex list = ', this.vertices);\n        console.log('Adjacency matrix =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.dart<pre><code>/* Undirected graph class implemented with an adjacency matrix */\nclass GraphAdjMat {\n  List&lt;int&gt; vertices = []; // Vertex entries: each element is a vertex value, and its index is the vertex index\n  List&lt;List&lt;int&gt;&gt; adjMat = []; //Adjacency matrix: row and column indices correspond to vertex indices\n\n  /* Constructor */\n  GraphAdjMat(List&lt;int&gt; vertices, List&lt;List&lt;int&gt;&gt; edges) {\n    this.vertices = [];\n    this.adjMat = [];\n    // Add a vertex\n    for (int val in vertices) {\n      addVertex(val);\n    }\n    // Add an edge\n    // Note that the entries in edges are vertex indices, corresponding to element indices in vertices\n    for (List&lt;int&gt; e in edges) {\n      addEdge(e[0], e[1]);\n    }\n  }\n\n  /* Get the number of vertices */\n  int size() {\n    return vertices.length;\n  }\n\n  /* Add a vertex */\n  void addVertex(int val) {\n    int n = size();\n    // Append the new vertex value to the vertex list\n    vertices.add(val);\n    // Add a row to the adjacency matrix\n    List&lt;int&gt; newRow = List.filled(n, 0, growable: true);\n    adjMat.add(newRow);\n    // Add a column to the adjacency matrix\n    for (List&lt;int&gt; row in adjMat) {\n      row.add(0);\n    }\n  }\n\n  /* Remove a vertex */\n  void removeVertex(int index) {\n    if (index &gt;= size()) {\n      throw IndexError;\n    }\n    // Remove the vertex at index `index` from the vertex list\n    vertices.removeAt(index);\n    // Remove row `index` from the adjacency matrix\n    adjMat.removeAt(index);\n    // Remove column `index` from the adjacency matrix\n    for (List&lt;int&gt; row in adjMat) {\n      row.removeAt(index);\n    }\n  }\n\n  /* Add an edge */\n  // Parameters i and j are element indices in vertices\n  void addEdge(int i, int j) {\n    // Handle out-of-bounds or equal indices\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    // For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal\n    adjMat[i][j] = 1;\n    adjMat[j][i] = 1;\n  }\n\n  /* Remove an edge */\n  // Parameters i and j are element indices in vertices\n  void removeEdge(int i, int j) {\n    // Handle out-of-bounds or equal indices\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    adjMat[i][j] = 0;\n    adjMat[j][i] = 0;\n  }\n\n  /* Print the adjacency matrix */\n  void printAdjMat() {\n    print(\"Vertex list = $vertices\");\n    print(\"Adjacency matrix = \");\n    printMatrix(adjMat);\n  }\n}\n</code></pre> graph_adjacency_matrix.rs<pre><code>/* Undirected graph type implemented with an adjacency matrix */\npub struct GraphAdjMat {\n    // Vertex list: elements are vertex values, and indices are vertex indices\n    pub vertices: Vec&lt;i32&gt;,\n    // Adjacency matrix: row and column indices correspond to vertex indices\n    pub adj_mat: Vec&lt;Vec&lt;i32&gt;&gt;,\n}\n\nimpl GraphAdjMat {\n    /* Constructor */\n    pub fn new(vertices: Vec&lt;i32&gt;, edges: Vec&lt;[usize; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjMat {\n            vertices: vec![],\n            adj_mat: vec![],\n        };\n        // Add a vertex\n        for val in vertices {\n            graph.add_vertex(val);\n        }\n        // Add an edge\n        // Note that the entries in edges are vertex indices, corresponding to element indices in vertices\n        for edge in edges {\n            graph.add_edge(edge[0], edge[1])\n        }\n\n        graph\n    }\n\n    /* Get the number of vertices */\n    pub fn size(&amp;self) -&gt; usize {\n        self.vertices.len()\n    }\n\n    /* Add a vertex */\n    pub fn add_vertex(&amp;mut self, val: i32) {\n        let n = self.size();\n        // Append the new vertex value to the vertex list\n        self.vertices.push(val);\n        // Add a row to the adjacency matrix\n        self.adj_mat.push(vec![0; n]);\n        // Add a column to the adjacency matrix\n        for row in self.adj_mat.iter_mut() {\n            row.push(0);\n        }\n    }\n\n    /* Remove a vertex */\n    pub fn remove_vertex(&amp;mut self, index: usize) {\n        if index &gt;= self.size() {\n            panic!(\"Invalid vertex index\")\n        }\n        // Remove the vertex at index `index` from the vertex list\n        self.vertices.remove(index);\n        // Remove row `index` from the adjacency matrix\n        self.adj_mat.remove(index);\n        // Remove column `index` from the adjacency matrix\n        for row in self.adj_mat.iter_mut() {\n            row.remove(index);\n        }\n    }\n\n    /* Add an edge */\n    pub fn add_edge(&amp;mut self, i: usize, j: usize) {\n        // Parameters i and j are element indices in vertices\n        // Handle out-of-bounds or equal indices\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"Invalid vertex index\")\n        }\n        // For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal\n        self.adj_mat[i][j] = 1;\n        self.adj_mat[j][i] = 1;\n    }\n\n    /* Remove an edge */\n    // Parameters i and j are element indices in vertices\n    pub fn remove_edge(&amp;mut self, i: usize, j: usize) {\n        // Parameters i and j are element indices in vertices\n        // Handle out-of-bounds or equal indices\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"Invalid vertex index\")\n        }\n        self.adj_mat[i][j] = 0;\n        self.adj_mat[j][i] = 0;\n    }\n\n    /* Print the adjacency matrix */\n    pub fn print(&amp;self) {\n        println!(\"Vertex list = {:?}\", self.vertices);\n        println!(\"Adjacency matrix =\");\n        println!(\"[\");\n        for row in &amp;self.adj_mat {\n            println!(\"  {:?},\", row);\n        }\n        println!(\"]\")\n    }\n}\n</code></pre> graph_adjacency_matrix.c<pre><code>/* Undirected graph structure implemented with an adjacency matrix */\ntypedef struct {\n    int vertices[MAX_SIZE];\n    int adjMat[MAX_SIZE][MAX_SIZE];\n    int size;\n} GraphAdjMat;\n\n/* Constructor */\nGraphAdjMat *newGraphAdjMat() {\n    GraphAdjMat *graph = (GraphAdjMat *)malloc(sizeof(GraphAdjMat));\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        for (int j = 0; j &lt; MAX_SIZE; j++) {\n            graph-&gt;adjMat[i][j] = 0;\n        }\n    }\n    return graph;\n}\n\n/* Destructor */\nvoid delGraphAdjMat(GraphAdjMat *graph) {\n    free(graph);\n}\n\n/* Add a vertex */\nvoid addVertex(GraphAdjMat *graph, int val) {\n    if (graph-&gt;size == MAX_SIZE) {\n        fprintf(stderr, \"The graph has reached its maximum number of vertices\\n\");\n        return;\n    }\n    // Add the vertex at index n and set row n and column n to zero\n    int n = graph-&gt;size;\n    graph-&gt;vertices[n] = val;\n    for (int i = 0; i &lt;= n; i++) {\n        graph-&gt;adjMat[n][i] = graph-&gt;adjMat[i][n] = 0;\n    }\n    graph-&gt;size++;\n}\n\n/* Remove a vertex */\nvoid removeVertex(GraphAdjMat *graph, int index) {\n    if (index &lt; 0 || index &gt;= graph-&gt;size) {\n        fprintf(stderr, \"Vertex index out of bounds\\n\");\n        return;\n    }\n    // Remove the vertex at index `index` from the vertex list\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        graph-&gt;vertices[i] = graph-&gt;vertices[i + 1];\n    }\n    // Remove row `index` from the adjacency matrix\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        for (int j = 0; j &lt; graph-&gt;size; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i + 1][j];\n        }\n    }\n    // Remove column `index` from the adjacency matrix\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        for (int j = index; j &lt; graph-&gt;size - 1; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i][j + 1];\n        }\n    }\n    graph-&gt;size--;\n}\n\n/* Add an edge */\n// Parameters i and j are element indices in vertices\nvoid addEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"Edge endpoint indices are out of bounds or equal\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 1;\n    graph-&gt;adjMat[j][i] = 1;\n}\n\n/* Remove an edge */\n// Parameters i and j are element indices in vertices\nvoid removeEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"Edge endpoint indices are out of bounds or equal\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 0;\n    graph-&gt;adjMat[j][i] = 0;\n}\n\n/* Print the adjacency matrix */\nvoid printGraphAdjMat(GraphAdjMat *graph) {\n    printf(\"Vertex list = \");\n    printArray(graph-&gt;vertices, graph-&gt;size);\n    printf(\"Adjacency matrix =\\n\");\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        printArray(graph-&gt;adjMat[i], graph-&gt;size);\n    }\n}\n</code></pre> graph_adjacency_matrix.kt<pre><code>/* Undirected graph class implemented with an adjacency matrix */\nclass GraphAdjMat(vertices: IntArray, edges: Array&lt;IntArray&gt;) {\n    val vertices = mutableListOf&lt;Int&gt;() // Vertex list: elements are vertex values, and indices are vertex indices\n    val adjMat = mutableListOf&lt;MutableList&lt;Int&gt;&gt;() // Adjacency matrix: row and column indices correspond to vertex indices\n\n    /* Constructor */\n    init {\n        // Add a vertex\n        for (vertex in vertices) {\n            addVertex(vertex)\n        }\n        // Add an edge\n        // Note that the entries in edges are vertex indices, corresponding to element indices in vertices\n        for (edge in edges) {\n            addEdge(edge[0], edge[1])\n        }\n    }\n\n    /* Get the number of vertices */\n    fun size(): Int {\n        return vertices.size\n    }\n\n    /* Add a vertex */\n    fun addVertex(_val: Int) {\n        val n = size()\n        // Append the new vertex value to the vertex list\n        vertices.add(_val)\n        // Add a row to the adjacency matrix\n        val newRow = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            newRow.add(0)\n        }\n        adjMat.add(newRow)\n        // Add a column to the adjacency matrix\n        for (row in adjMat) {\n            row.add(0)\n        }\n    }\n\n    /* Remove a vertex */\n    fun removeVertex(index: Int) {\n        if (index &gt;= size())\n            throw IndexOutOfBoundsException()\n        // Remove the vertex at index `index` from the vertex list\n        vertices.removeAt(index)\n        // Remove row `index` from the adjacency matrix\n        adjMat.removeAt(index)\n        // Remove column `index` from the adjacency matrix\n        for (row in adjMat) {\n            row.removeAt(index)\n        }\n    }\n\n    /* Add an edge */\n    // Parameters i and j are element indices in vertices\n    fun addEdge(i: Int, j: Int) {\n        // Handle out-of-bounds or equal indices\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw IndexOutOfBoundsException()\n        // For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* Remove an edge */\n    // Parameters i and j are element indices in vertices\n    fun removeEdge(i: Int, j: Int) {\n        // Handle out-of-bounds or equal indices\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw IndexOutOfBoundsException()\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* Print the adjacency matrix */\n    fun print() {\n        print(\"Vertex list = \")\n        println(vertices)\n        println(\"Adjacency matrix =\")\n        printMatrix(adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.rb<pre><code>### Undirected graph class implemented with an adjacency matrix ###\nclass GraphAdjMat\n  def initialize(vertices, edges)\n    ### Constructor ###\n    # Vertex list: elements are vertex values, and indices are vertex indices\n    @vertices = []\n    # Adjacency matrix: row and column indices correspond to vertex indices\n    @adj_mat = []\n    # Add a vertex\n    vertices.each { |val| add_vertex(val) }\n    # Add an edge\n    # Note that the entries in edges are vertex indices, corresponding to element indices in vertices\n    edges.each { |e| add_edge(e[0], e[1]) }\n  end\n\n  ### Get the number of vertices ###\n  def size\n    @vertices.length\n  end\n\n  ### Add a vertex ###\n  def add_vertex(val)\n    n = size\n    # Append the new vertex value to the vertex list\n    @vertices &lt;&lt; val\n    # Add a row to the adjacency matrix\n    new_row = Array.new(n, 0)\n    @adj_mat &lt;&lt; new_row\n    # Add a column to the adjacency matrix\n    @adj_mat.each { |row| row &lt;&lt; 0 }\n  end\n\n  ### Remove a vertex ###\n  def remove_vertex(index)\n    raise IndexError if index &gt;= size\n\n    # Remove the vertex at index `index` from the vertex list\n    @vertices.delete_at(index)\n    # Remove row `index` from the adjacency matrix\n    @adj_mat.delete_at(index)\n    # Remove column `index` from the adjacency matrix\n    @adj_mat.each { |row| row.delete_at(index) }\n  end\n\n  ### Add an edge ###\n  def add_edge(i, j)\n    # Parameters i and j are element indices in vertices\n    # Handle out-of-bounds or equal indices\n    if i &lt; 0 || j &lt; 0 || i &gt;= size || j &gt;= size || i == j\n      raise IndexError\n    end\n    # For an undirected graph, the adjacency matrix is symmetric about its main diagonal: entries (i, j) and (j, i) are equal\n    @adj_mat[i][j] = 1\n    @adj_mat[j][i] = 1\n  end\n\n  ### Remove an edge ###\n  def remove_edge(i, j)\n    # Parameters i and j are element indices in vertices\n    # Handle out-of-bounds or equal indices\n    if i &lt; 0 || j &lt; 0 || i &gt;= size || j &gt;= size || i == j\n      raise IndexError\n    end\n    @adj_mat[i][j] = 0\n    @adj_mat[j][i] = 0\n  end\n\n  ### Print the adjacency matrix ###\n  def __print__\n    puts \"Vertex list = #{@vertices}\"\n    puts 'Adjacency matrix ='\n    print_matrix(@adj_mat)\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 9.   Graphs","9.2   Basic graph operations"],"tags":[]},{"location":"chapter_graph/graph_operations/#922-adjacency-list-implementation","level":2,"title":"9.2.2   Adjacency list implementation","text":"<p>For an undirected graph with \\(n\\) vertices and \\(m\\) edges, the operations can be implemented as shown in Figure 9-8.</p> <ul> <li>Add an edge: Append the edge to the linked list of the corresponding vertex, taking \\(O(1)\\) time. Since the graph is undirected, add the edge in both directions.</li> <li>Remove an edge: Find and remove the specified edge in the corresponding vertex's linked list, taking \\(O(m)\\) time. For an undirected graph, remove the edge in both directions.</li> <li>Add a vertex: Add a linked list to the adjacency list with the new vertex as its head node, taking \\(O(1)\\) time.</li> <li>Remove a vertex: Traverse the entire adjacency list and remove every edge involving the specified vertex, taking \\(O(n + m)\\) time.</li> <li>Initialize the graph: Create \\(n\\) vertices and \\(2m\\) edge entries in the adjacency list, taking \\(O(n + m)\\) time.</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-8   Initializing an adjacency list and adding or removing edges and vertices </p> <p>The following code implements an adjacency list, with these differences from Figure 9-8.</p> <ul> <li>To make adding and removing vertices easier and simplify the code, we use lists (dynamic arrays) in place of linked lists.</li> <li>A hash table stores the adjacency list. Each <code>key</code> is a vertex instance, and its <code>value</code> is the list (linked list) of adjacent vertices.</li> </ul> <p>We also use the <code>Vertex</code> class to represent vertices in the adjacency list. If we identified vertices by list indices, as in the adjacency matrix, deleting the vertex at index \\(i\\) would require traversing the entire adjacency list and subtracting \\(1\\) from every index greater than \\(i\\). This would be inefficient. When each vertex is a unique <code>Vertex</code> instance, deleting one vertex requires no changes to the other vertices.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_adjacency_list.py<pre><code>class GraphAdjList:\n    \"\"\"Undirected graph class implemented with an adjacency list\"\"\"\n\n    def __init__(self, edges: list[list[Vertex]]):\n        \"\"\"Constructor\"\"\"\n        # Adjacency list: key is a vertex, value is the list of all its neighbors\n        self.adj_list = dict[Vertex, list[Vertex]]()\n        # Add all vertices and edges\n        for edge in edges:\n            self.add_vertex(edge[0])\n            self.add_vertex(edge[1])\n            self.add_edge(edge[0], edge[1])\n\n    def size(self) -&gt; int:\n        \"\"\"Get the number of vertices\"\"\"\n        return len(self.adj_list)\n\n    def add_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"Add an edge\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # Add edge vet1 - vet2\n        self.adj_list[vet1].append(vet2)\n        self.adj_list[vet2].append(vet1)\n\n    def remove_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"Remove an edge\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # Remove edge vet1 - vet2\n        self.adj_list[vet1].remove(vet2)\n        self.adj_list[vet2].remove(vet1)\n\n    def add_vertex(self, vet: Vertex):\n        \"\"\"Add a vertex\"\"\"\n        if vet in self.adj_list:\n            return\n        # Add a new list to the adjacency list\n        self.adj_list[vet] = []\n\n    def remove_vertex(self, vet: Vertex):\n        \"\"\"Remove a vertex\"\"\"\n        if vet not in self.adj_list:\n            raise ValueError()\n        # Remove the list corresponding to vertex vet from the adjacency list\n        self.adj_list.pop(vet)\n        # Traverse the lists of the other vertices and remove all edges incident to vet\n        for vertex in self.adj_list:\n            if vet in self.adj_list[vertex]:\n                self.adj_list[vertex].remove(vet)\n\n    def print(self):\n        \"\"\"Print the adjacency list\"\"\"\n        print(\"Adjacency list =\")\n        for vertex in self.adj_list:\n            tmp = [v.val for v in self.adj_list[vertex]]\n            print(f\"{vertex.val}: {tmp},\")\n</code></pre> graph_adjacency_list.cpp<pre><code>/* Undirected graph class implemented with an adjacency list */\nclass GraphAdjList {\n  public:\n    // Adjacency list: key is a vertex, value is the list of all its neighbors\n    unordered_map&lt;Vertex *, vector&lt;Vertex *&gt;&gt; adjList;\n\n    /* Remove the specified vertex from the vector */\n    void remove(vector&lt;Vertex *&gt; &amp;vec, Vertex *vet) {\n        for (int i = 0; i &lt; vec.size(); i++) {\n            if (vec[i] == vet) {\n                vec.erase(vec.begin() + i);\n                break;\n            }\n        }\n    }\n\n    /* Constructor */\n    GraphAdjList(const vector&lt;vector&lt;Vertex *&gt;&gt; &amp;edges) {\n        // Add all vertices and edges\n        for (const vector&lt;Vertex *&gt; &amp;edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    int size() {\n        return adjList.size();\n    }\n\n    /* Add an edge */\n    void addEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"Invalid vertex\");\n        // Add edge vet1 - vet2\n        adjList[vet1].push_back(vet2);\n        adjList[vet2].push_back(vet1);\n    }\n\n    /* Remove an edge */\n    void removeEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"Invalid vertex\");\n        // Remove edge vet1 - vet2\n        remove(adjList[vet1], vet2);\n        remove(adjList[vet2], vet1);\n    }\n\n    /* Add a vertex */\n    void addVertex(Vertex *vet) {\n        if (adjList.count(vet))\n            return;\n        // Add a new list to the adjacency list\n        adjList[vet] = vector&lt;Vertex *&gt;();\n    }\n\n    /* Remove a vertex */\n    void removeVertex(Vertex *vet) {\n        if (!adjList.count(vet))\n            throw invalid_argument(\"Invalid vertex\");\n        // Remove the list corresponding to vertex vet from the adjacency list\n        adjList.erase(vet);\n        // Traverse the lists of the other vertices and remove all edges incident to vet\n        for (auto &amp;adj : adjList) {\n            remove(adj.second, vet);\n        }\n    }\n\n    /* Print the adjacency list */\n    void print() {\n        cout &lt;&lt; \"Adjacency list =\" &lt;&lt; endl;\n        for (auto &amp;adj : adjList) {\n            const auto &amp;key = adj.first;\n            const auto &amp;vec = adj.second;\n            cout &lt;&lt; key-&gt;val &lt;&lt; \": \";\n            printVector(vetsToVals(vec));\n        }\n    }\n};\n</code></pre> graph_adjacency_list.java<pre><code>/* Undirected graph class implemented with an adjacency list */\nclass GraphAdjList {\n    // Adjacency list: key is a vertex, value is the list of all its neighbors\n    Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* Constructor */\n    public GraphAdjList(Vertex[][] edges) {\n        this.adjList = new HashMap&lt;&gt;();\n        // Add all vertices and edges\n        for (Vertex[] edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    public int size() {\n        return adjList.size();\n    }\n\n    /* Add an edge */\n    public void addEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // Add edge vet1 - vet2\n        adjList.get(vet1).add(vet2);\n        adjList.get(vet2).add(vet1);\n    }\n\n    /* Remove an edge */\n    public void removeEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // Remove edge vet1 - vet2\n        adjList.get(vet1).remove(vet2);\n        adjList.get(vet2).remove(vet1);\n    }\n\n    /* Add a vertex */\n    public void addVertex(Vertex vet) {\n        if (adjList.containsKey(vet))\n            return;\n        // Add a new list to the adjacency list\n        adjList.put(vet, new ArrayList&lt;&gt;());\n    }\n\n    /* Remove a vertex */\n    public void removeVertex(Vertex vet) {\n        if (!adjList.containsKey(vet))\n            throw new IllegalArgumentException();\n        // Remove the list corresponding to vertex vet from the adjacency list\n        adjList.remove(vet);\n        // Traverse the lists of the other vertices and remove all edges incident to vet\n        for (List&lt;Vertex&gt; list : adjList.values()) {\n            list.remove(vet);\n        }\n    }\n\n    /* Print the adjacency list */\n    public void print() {\n        System.out.println(\"Adjacency list =\");\n        for (Map.Entry&lt;Vertex, List&lt;Vertex&gt;&gt; pair : adjList.entrySet()) {\n            List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n            for (Vertex vertex : pair.getValue())\n                tmp.add(vertex.val);\n            System.out.println(pair.getKey().val + \": \" + tmp + \",\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.cs<pre><code>/* Undirected graph class implemented with an adjacency list */\nclass GraphAdjList {\n    // Adjacency list: key is a vertex, value is the list of all its neighbors\n    public Dictionary&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* Constructor */\n    public GraphAdjList(Vertex[][] edges) {\n        adjList = [];\n        // Add all vertices and edges\n        foreach (Vertex[] edge in edges) {\n            AddVertex(edge[0]);\n            AddVertex(edge[1]);\n            AddEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    int Size() {\n        return adjList.Count;\n    }\n\n    /* Add an edge */\n    public void AddEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // Add edge vet1 - vet2\n        adjList[vet1].Add(vet2);\n        adjList[vet2].Add(vet1);\n    }\n\n    /* Remove an edge */\n    public void RemoveEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // Remove edge vet1 - vet2\n        adjList[vet1].Remove(vet2);\n        adjList[vet2].Remove(vet1);\n    }\n\n    /* Add a vertex */\n    public void AddVertex(Vertex vet) {\n        if (adjList.ContainsKey(vet))\n            return;\n        // Add a new list to the adjacency list\n        adjList.Add(vet, []);\n    }\n\n    /* Remove a vertex */\n    public void RemoveVertex(Vertex vet) {\n        if (!adjList.ContainsKey(vet))\n            throw new InvalidOperationException();\n        // Remove the list corresponding to vertex vet from the adjacency list\n        adjList.Remove(vet);\n        // Traverse the lists of the other vertices and remove all edges incident to vet\n        foreach (List&lt;Vertex&gt; list in adjList.Values) {\n            list.Remove(vet);\n        }\n    }\n\n    /* Print the adjacency list */\n    public void Print() {\n        Console.WriteLine(\"Adjacency list =\");\n        foreach (KeyValuePair&lt;Vertex, List&lt;Vertex&gt;&gt; pair in adjList) {\n            List&lt;int&gt; tmp = [];\n            foreach (Vertex vertex in pair.Value)\n                tmp.Add(vertex.val);\n            Console.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.go<pre><code>/* Undirected graph type implemented with an adjacency list */\ntype graphAdjList struct {\n    // Adjacency list: key is a vertex, value is the list of all its neighbors\n    adjList map[Vertex][]Vertex\n}\n\n/* Constructor */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\n    g := &amp;graphAdjList{\n        adjList: make(map[Vertex][]Vertex),\n    }\n    // Add all vertices and edges\n    for _, edge := range edges {\n        g.addVertex(edge[0])\n        g.addVertex(edge[1])\n        g.addEdge(edge[0], edge[1])\n    }\n    return g\n}\n\n/* Get the number of vertices */\nfunc (g *graphAdjList) size() int {\n    return len(g.adjList)\n}\n\n/* Add an edge */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // Add edge vet1 - vet2 by appending each vertex to the other vertex's neighbor list\n    g.adjList[vet1] = append(g.adjList[vet1], vet2)\n    g.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n\n/* Remove an edge */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // Remove edge vet1 - vet2\n    g.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2)\n    g.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1)\n}\n\n/* Add a vertex */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if ok {\n        return\n    }\n    // Add a new list to the adjacency list\n    g.adjList[vet] = make([]Vertex, 0)\n}\n\n/* Remove a vertex */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if !ok {\n        panic(\"error\")\n    }\n    // Remove the list corresponding to vertex vet from the adjacency list\n    delete(g.adjList, vet)\n    // Traverse the lists of the other vertices and remove all edges incident to vet\n    for v, list := range g.adjList {\n        g.adjList[v] = DeleteSliceElms(list, vet)\n    }\n}\n\n/* Print the adjacency list */\nfunc (g *graphAdjList) print() {\n    var builder strings.Builder\n    fmt.Printf(\"Adjacency list = \\n\")\n    for k, v := range g.adjList {\n        builder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\n        for _, vet := range v {\n            builder.WriteString(strconv.Itoa(vet.Val) + \" \")\n        }\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> graph_adjacency_list.swift<pre><code>/* Undirected graph class implemented with an adjacency list */\nclass GraphAdjList {\n    // Adjacency list: key is a vertex, value is the list of all its neighbors\n    public private(set) var adjList: [Vertex: [Vertex]]\n\n    /* Constructor */\n    public init(edges: [[Vertex]]) {\n        adjList = [:]\n        // Add all vertices and edges\n        for edge in edges {\n            addVertex(vet: edge[0])\n            addVertex(vet: edge[1])\n            addEdge(vet1: edge[0], vet2: edge[1])\n        }\n    }\n\n    /* Get the number of vertices */\n    public func size() -&gt; Int {\n        adjList.count\n    }\n\n    /* Add an edge */\n    public func addEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"Invalid arguments\")\n        }\n        // Add edge vet1 - vet2\n        adjList[vet1]?.append(vet2)\n        adjList[vet2]?.append(vet1)\n    }\n\n    /* Remove an edge */\n    public func removeEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"Invalid arguments\")\n        }\n        // Remove edge vet1 - vet2\n        adjList[vet1]?.removeAll { $0 == vet2 }\n        adjList[vet2]?.removeAll { $0 == vet1 }\n    }\n\n    /* Add a vertex */\n    public func addVertex(vet: Vertex) {\n        if adjList[vet] != nil {\n            return\n        }\n        // Add a new list to the adjacency list\n        adjList[vet] = []\n    }\n\n    /* Remove a vertex */\n    public func removeVertex(vet: Vertex) {\n        if adjList[vet] == nil {\n            fatalError(\"Invalid arguments\")\n        }\n        // Remove the list corresponding to vertex vet from the adjacency list\n        adjList.removeValue(forKey: vet)\n        // Traverse the lists of the other vertices and remove all edges incident to vet\n        for key in adjList.keys {\n            adjList[key]?.removeAll { $0 == vet }\n        }\n    }\n\n    /* Print the adjacency list */\n    public func print() {\n        Swift.print(\"Adjacency list =\")\n        for (vertex, list) in adjList {\n            let list = list.map { $0.val }\n            Swift.print(\"\\(vertex.val): \\(list),\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.js<pre><code>/* Undirected graph class implemented with an adjacency list */\nclass GraphAdjList {\n    // Adjacency list: key is a vertex, value is the list of all its neighbors\n    adjList;\n\n    /* Constructor */\n    constructor(edges) {\n        this.adjList = new Map();\n        // Add all vertices and edges\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    size() {\n        return this.adjList.size;\n    }\n\n    /* Add an edge */\n    addEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Invalid argument');\n        }\n        // Add edge vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* Remove an edge */\n    removeEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2 ||\n            this.adjList.get(vet1).indexOf(vet2) === -1\n        ) {\n            throw new Error('Invalid argument');\n        }\n        // Remove edge vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* Add a vertex */\n    addVertex(vet) {\n        if (this.adjList.has(vet)) return;\n        // Add a new list to the adjacency list\n        this.adjList.set(vet, []);\n    }\n\n    /* Remove a vertex */\n    removeVertex(vet) {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Invalid argument');\n        }\n        // Remove the list corresponding to vertex vet from the adjacency list\n        this.adjList.delete(vet);\n        // Traverse the lists of the other vertices and remove all edges incident to vet\n        for (const set of this.adjList.values()) {\n            const index = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* Print the adjacency list */\n    print() {\n        console.log('Adjacency list =');\n        for (const [key, value] of this.adjList) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.ts<pre><code>/* Undirected graph class implemented with an adjacency list */\nclass GraphAdjList {\n    // Adjacency list: key is a vertex, value is the list of all its neighbors\n    adjList: Map&lt;Vertex, Vertex[]&gt;;\n\n    /* Constructor */\n    constructor(edges: Vertex[][]) {\n        this.adjList = new Map();\n        // Add all vertices and edges\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* Get the number of vertices */\n    size(): number {\n        return this.adjList.size;\n    }\n\n    /* Add an edge */\n    addEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Invalid argument');\n        }\n        // Add edge vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* Remove an edge */\n    removeEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2 ||\n            this.adjList.get(vet1).indexOf(vet2) === -1\n        ) {\n            throw new Error('Invalid argument');\n        }\n        // Remove edge vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* Add a vertex */\n    addVertex(vet: Vertex): void {\n        if (this.adjList.has(vet)) return;\n        // Add a new list to the adjacency list\n        this.adjList.set(vet, []);\n    }\n\n    /* Remove a vertex */\n    removeVertex(vet: Vertex): void {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Invalid argument');\n        }\n        // Remove the list corresponding to vertex vet from the adjacency list\n        this.adjList.delete(vet);\n        // Traverse the lists of the other vertices and remove all edges incident to vet\n        for (const set of this.adjList.values()) {\n            const index: number = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* Print the adjacency list */\n    print(): void {\n        console.log('Adjacency list =');\n        for (const [key, value] of this.adjList.entries()) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.dart<pre><code>/* Undirected graph class implemented with an adjacency list */\nclass GraphAdjList {\n  // Adjacency list: key is a vertex, value is the list of all its neighbors\n  Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList = {};\n\n  /* Constructor */\n  GraphAdjList(List&lt;List&lt;Vertex&gt;&gt; edges) {\n    for (List&lt;Vertex&gt; edge in edges) {\n      addVertex(edge[0]);\n      addVertex(edge[1]);\n      addEdge(edge[0], edge[1]);\n    }\n  }\n\n  /* Get the number of vertices */\n  int size() {\n    return adjList.length;\n  }\n\n  /* Add an edge */\n  void addEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // Add edge vet1 - vet2\n    adjList[vet1]!.add(vet2);\n    adjList[vet2]!.add(vet1);\n  }\n\n  /* Remove an edge */\n  void removeEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // Remove edge vet1 - vet2\n    adjList[vet1]!.remove(vet2);\n    adjList[vet2]!.remove(vet1);\n  }\n\n  /* Add a vertex */\n  void addVertex(Vertex vet) {\n    if (adjList.containsKey(vet)) return;\n    // Add a new list to the adjacency list\n    adjList[vet] = [];\n  }\n\n  /* Remove a vertex */\n  void removeVertex(Vertex vet) {\n    if (!adjList.containsKey(vet)) {\n      throw ArgumentError;\n    }\n    // Remove the list corresponding to vertex vet from the adjacency list\n    adjList.remove(vet);\n    // Traverse the lists of the other vertices and remove all edges incident to vet\n    adjList.forEach((key, value) {\n      value.remove(vet);\n    });\n  }\n\n  /* Print the adjacency list */\n  void printAdjList() {\n    print(\"Adjacency list =\");\n    adjList.forEach((key, value) {\n      List&lt;int&gt; tmp = [];\n      for (Vertex vertex in value) {\n        tmp.add(vertex.val);\n      }\n      print(\"${key.val}: $tmp,\");\n    });\n  }\n}\n</code></pre> graph_adjacency_list.rs<pre><code>/* Undirected graph type implemented with an adjacency list */\npub struct GraphAdjList {\n    // Adjacency list: key is a vertex, value is the list of all its neighbors\n    pub adj_list: HashMap&lt;Vertex, Vec&lt;Vertex&gt;&gt;, // A HashSet&lt;Vertex&gt; may be a better choice for the neighbor collection\n}\n\nimpl GraphAdjList {\n    /* Constructor */\n    pub fn new(edges: Vec&lt;[Vertex; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjList {\n            adj_list: HashMap::new(),\n        };\n        // Add all vertices and edges\n        for edge in edges {\n            graph.add_vertex(edge[0]);\n            graph.add_vertex(edge[1]);\n            graph.add_edge(edge[0], edge[1]);\n        }\n\n        graph\n    }\n\n    /* Get the number of vertices */\n    #[allow(unused)]\n    pub fn size(&amp;self) -&gt; usize {\n        self.adj_list.len()\n    }\n\n    /* Add an edge */\n    pub fn add_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if vet1 == vet2 {\n            panic!(\"Vertices must be distinct\");\n        }\n        // Add edge vet1 - vet2\n        self.adj_list.entry(vet1).or_default().push(vet2);\n        self.adj_list.entry(vet2).or_default().push(vet1);\n    }\n\n    /* Remove an edge */\n    #[allow(unused)]\n    pub fn remove_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if vet1 == vet2 {\n            panic!(\"Vertices must be distinct\");\n        }\n        // Remove edge vet1 - vet2\n        self.adj_list\n            .entry(vet1)\n            .and_modify(|v| v.retain(|&amp;e| e != vet2));\n        self.adj_list\n            .entry(vet2)\n            .and_modify(|v| v.retain(|&amp;e| e != vet1));\n    }\n\n    /* Add a vertex */\n    pub fn add_vertex(&amp;mut self, vet: Vertex) {\n        if self.adj_list.contains_key(&amp;vet) {\n            return;\n        }\n        // Add a new neighbor list to the adjacency list\n        self.adj_list.insert(vet, vec![]);\n    }\n\n    /* Remove a vertex */\n    #[allow(unused)]\n    pub fn remove_vertex(&amp;mut self, vet: Vertex) {\n        // Remove the neighbor list for vertex vet from the adjacency list\n        self.adj_list.remove(&amp;vet);\n        // Traverse the other neighbor lists and remove every edge incident to vet\n        for list in self.adj_list.values_mut() {\n            list.retain(|&amp;v| v != vet);\n        }\n    }\n\n    /* Print the adjacency list */\n    pub fn print(&amp;self) {\n        println!(\"Adjacency list =\");\n        for (vertex, list) in &amp;self.adj_list {\n            let list = list.iter().map(|vertex| vertex.val).collect::&lt;Vec&lt;i32&gt;&gt;();\n            println!(\"{}: {:?},\", vertex.val, list);\n        }\n    }\n}\n</code></pre> graph_adjacency_list.c<pre><code>/* Adjacency-list node structure */\ntypedef struct AdjListNode {\n    Vertex *vertex;           // Vertex\n    struct AdjListNode *next; // Next node\n} AdjListNode;\n\n/* Find the list node corresponding to a vertex */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* Helper for adding an edge */\nvoid addEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *node = (AdjListNode *)malloc(sizeof(AdjListNode));\n    node-&gt;vertex = vet;\n    // Insert at the front of the neighbor list\n    node-&gt;next = head-&gt;next;\n    head-&gt;next = node;\n}\n\n/* Helper for removing an edge */\nvoid removeEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *pre = head;\n    AdjListNode *cur = head-&gt;next;\n    // Search the linked list for the node corresponding to vet\n    while (cur != NULL &amp;&amp; cur-&gt;vertex != vet) {\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n    if (cur == NULL)\n        return;\n    // Remove the node corresponding to vet from the linked list\n    pre-&gt;next = cur-&gt;next;\n    // Free memory\n    free(cur);\n}\n\n/* Undirected graph structure implemented with an adjacency list */\ntypedef struct {\n    AdjListNode *heads[MAX_SIZE]; // Array of list heads\n    int size;                     // Number of vertices\n} GraphAdjList;\n\n/* Constructor */\nGraphAdjList *newGraphAdjList() {\n    GraphAdjList *graph = (GraphAdjList *)malloc(sizeof(GraphAdjList));\n    if (!graph) {\n        return NULL;\n    }\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        graph-&gt;heads[i] = NULL;\n    }\n    return graph;\n}\n\n/* Destructor */\nvoid delGraphAdjList(GraphAdjList *graph) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        AdjListNode *cur = graph-&gt;heads[i];\n        while (cur != NULL) {\n            AdjListNode *next = cur-&gt;next;\n            if (cur != graph-&gt;heads[i]) {\n                free(cur);\n            }\n            cur = next;\n        }\n        free(graph-&gt;heads[i]-&gt;vertex);\n        free(graph-&gt;heads[i]);\n    }\n    free(graph);\n}\n\n/* Find the list node corresponding to a vertex */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* Add an edge */\nvoid addEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL &amp;&amp; head1 != head2);\n    // Add edge vet1 - vet2\n    addEdgeHelper(head1, vet2);\n    addEdgeHelper(head2, vet1);\n}\n\n/* Remove an edge */\nvoid removeEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL);\n    // Remove edge vet1 - vet2\n    removeEdgeHelper(head1, head2-&gt;vertex);\n    removeEdgeHelper(head2, head1-&gt;vertex);\n}\n\n/* Add a vertex */\nvoid addVertex(GraphAdjList *graph, Vertex *vet) {\n    assert(graph != NULL &amp;&amp; graph-&gt;size &lt; MAX_SIZE);\n    AdjListNode *head = (AdjListNode *)malloc(sizeof(AdjListNode));\n    head-&gt;vertex = vet;\n    head-&gt;next = NULL;\n    // Add a new list to the adjacency list\n    graph-&gt;heads[graph-&gt;size++] = head;\n}\n\n/* Remove a vertex */\nvoid removeVertex(GraphAdjList *graph, Vertex *vet) {\n    AdjListNode *node = findNode(graph, vet);\n    assert(node != NULL);\n    // Remove the list corresponding to vertex vet from the adjacency list\n    AdjListNode *cur = node, *pre = NULL;\n    while (cur) {\n        pre = cur;\n        cur = cur-&gt;next;\n        free(pre);\n    }\n    // Traverse the lists of the other vertices and remove all edges incident to vet\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        cur = graph-&gt;heads[i];\n        pre = NULL;\n        while (cur) {\n            pre = cur;\n            cur = cur-&gt;next;\n            if (cur &amp;&amp; cur-&gt;vertex == vet) {\n                pre-&gt;next = cur-&gt;next;\n                free(cur);\n                break;\n            }\n        }\n    }\n    // Shift subsequent vertices forward to fill the gap\n    int i;\n    for (i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i] == node)\n            break;\n    }\n    for (int j = i; j &lt; graph-&gt;size - 1; j++) {\n        graph-&gt;heads[j] = graph-&gt;heads[j + 1];\n    }\n    graph-&gt;size--;\n    free(vet);\n}\n</code></pre> graph_adjacency_list.kt<pre><code>/* Undirected graph class implemented with an adjacency list */\nclass GraphAdjList(edges: Array&lt;Array&lt;Vertex?&gt;&gt;) {\n    // Adjacency list: key is a vertex, value is the list of all its neighbors\n    val adjList = HashMap&lt;Vertex, MutableList&lt;Vertex&gt;&gt;()\n\n    /* Constructor */\n    init {\n        // Add all vertices and edges\n        for (edge in edges) {\n            addVertex(edge[0]!!)\n            addVertex(edge[1]!!)\n            addEdge(edge[0]!!, edge[1]!!)\n        }\n    }\n\n    /* Get the number of vertices */\n    fun size(): Int {\n        return adjList.size\n    }\n\n    /* Add an edge */\n    fun addEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // Add edge vet1 - vet2\n        adjList[vet1]?.add(vet2)\n        adjList[vet2]?.add(vet1)\n    }\n\n    /* Remove an edge */\n    fun removeEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // Remove edge vet1 - vet2\n        adjList[vet1]?.remove(vet2)\n        adjList[vet2]?.remove(vet1)\n    }\n\n    /* Add a vertex */\n    fun addVertex(vet: Vertex) {\n        if (adjList.containsKey(vet))\n            return\n        // Add a new list to the adjacency list\n        adjList[vet] = mutableListOf()\n    }\n\n    /* Remove a vertex */\n    fun removeVertex(vet: Vertex) {\n        if (!adjList.containsKey(vet))\n            throw IllegalArgumentException()\n        // Remove the list corresponding to vertex vet from the adjacency list\n        adjList.remove(vet)\n        // Traverse the lists of the other vertices and remove all edges incident to vet\n        for (list in adjList.values) {\n            list.remove(vet)\n        }\n    }\n\n    /* Print the adjacency list */\n    fun print() {\n        println(\"Adjacency list =\")\n        for (pair in adjList.entries) {\n            val tmp = mutableListOf&lt;Int&gt;()\n            for (vertex in pair.value) {\n                tmp.add(vertex._val)\n            }\n            println(\"${pair.key._val}: $tmp,\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.rb<pre><code>### Undirected graph class implemented with an adjacency list ###\nclass GraphAdjList\n  attr_reader :adj_list\n\n  ### Constructor ###\n  def initialize(edges)\n    # Adjacency list: key is a vertex, value is the list of all its neighbors\n    @adj_list = {}\n    # Add all vertices and edges\n    for edge in edges\n      add_vertex(edge[0])\n      add_vertex(edge[1])\n      add_edge(edge[0], edge[1])\n    end\n  end\n\n  ### Get the number of vertices ###\n  def size\n    @adj_list.length\n  end\n\n  ### Add an edge ###\n  def add_edge(vet1, vet2)\n    raise ArgumentError if !@adj_list.include?(vet1) || !@adj_list.include?(vet2)\n\n    @adj_list[vet1] &lt;&lt; vet2\n    @adj_list[vet2] &lt;&lt; vet1\n  end\n\n  ### Remove an edge ###\n  def remove_edge(vet1, vet2)\n    raise ArgumentError if !@adj_list.include?(vet1) || !@adj_list.include?(vet2)\n\n    # Remove edge vet1 - vet2\n    @adj_list[vet1].delete(vet2)\n    @adj_list[vet2].delete(vet1)\n  end\n\n  ### Add a vertex ###\n  def add_vertex(vet)\n    return if @adj_list.include?(vet)\n\n    # Add a new neighbor list to the adjacency list\n    @adj_list[vet] = []\n  end\n\n  ### Remove a vertex ###\n  def remove_vertex(vet)\n    raise ArgumentError unless @adj_list.include?(vet)\n\n    # Remove the neighbor list for vertex vet from the adjacency list\n    @adj_list.delete(vet)\n    # Traverse the other neighbor lists and remove all edges incident to vet\n    for vertex in @adj_list\n      @adj_list[vertex.first].delete(vet) if @adj_list[vertex.first].include?(vet)\n    end\n  end\n\n  ### Print the adjacency list ###\n  def __print__\n    puts 'Adjacency list ='\n    for vertex in @adj_list\n      tmp = @adj_list[vertex.first].map { |v| v.val }\n      puts \"#{vertex.first.val}: #{tmp},\"\n    end\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 9.   Graphs","9.2   Basic graph operations"],"tags":[]},{"location":"chapter_graph/graph_operations/#923-comparing-efficiency","level":2,"title":"9.2.3   Comparing efficiency","text":"<p>For a graph with \\(n\\) vertices and \\(m\\) edges, Table 9-2 compares the time and space efficiency of adjacency matrices and adjacency lists. Here, \"adjacency list (linked lists)\" refers to the implementation in this section, while \"adjacency list (hash tables)\" specifically refers to an implementation in which every linked list is replaced by a hash table.</p> <p> Table 9-2   Comparing adjacency matrices and adjacency lists </p> Adjacency matrix Adjacency list (linked lists) Adjacency list (hash tables) Check adjacency \\(O(1)\\) \\(O(n)\\) \\(O(1)\\) Add an edge \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) Remove an edge \\(O(1)\\) \\(O(n)\\) \\(O(1)\\) Add a vertex \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) Remove a vertex \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) Memory usage \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\) <p>Table 9-2 suggests that an adjacency list using hash tables is the most efficient in both time and space. In practice, however, edge operations in an adjacency matrix are faster: each requires only a single array access or assignment. Overall, an adjacency matrix trades space for time, while an adjacency list trades time for space.</p>","path":["Chapter 9.   Graphs","9.2   Basic graph operations"],"tags":[]},{"location":"chapter_graph/graph_traversal/","level":1,"title":"9.3   Graph traversal","text":"<p>A tree represents one-to-many relationships, whereas a graph allows more freedom and can represent arbitrary many-to-many relationships. We can therefore view a tree as a special case of a graph. Accordingly, tree traversal is also a special case of graph traversal.</p> <p>Both graphs and trees use search algorithms for traversal. Graph traversal can likewise be divided into two types: breadth-first traversal and depth-first traversal.</p>","path":["Chapter 9.   Graphs","9.3   Graph traversal"],"tags":[]},{"location":"chapter_graph/graph_traversal/#931-breadth-first-traversal","level":2,"title":"9.3.1   Breadth-first traversal","text":"<p>Breadth-first traversal visits vertices from near to far. Starting from a vertex, it always visits the nearest vertices first and expands outward layer by layer. In Figure 9-9, we start at the top-left vertex and visit all its adjacent vertices, then visit all the adjacent vertices of the next vertex, and so on until every vertex has been visited.</p> <p></p> <p> Figure 9-9   Breadth-first traversal of a graph </p>","path":["Chapter 9.   Graphs","9.3   Graph traversal"],"tags":[]},{"location":"chapter_graph/graph_traversal/#1-algorithm-implementation","level":3,"title":"1.   Algorithm implementation","text":"<p>BFS is usually implemented with a queue, as shown in the code below. A queue's first-in, first-out behavior fits the near-to-far order of BFS.</p> <ol> <li>Enqueue the starting vertex <code>startVet</code> and begin the loop.</li> <li>In each iteration, dequeue the vertex at the front and record the visit, then enqueue all its adjacent vertices at the rear.</li> <li>Repeat step <code>2.</code> until every vertex has been visited.</li> </ol> <p>To avoid visiting a vertex more than once, we use a hash set <code>visited</code> to track which vertices have already been visited.</p> <p>Tip</p> <p>A hash set can be viewed as a hash table that stores only <code>key</code> entries, with no <code>value</code> entries. Adding, removing, looking up, and modifying a <code>key</code> take \\(O(1)\\) time. Because every <code>key</code> is unique, hash sets are commonly used to remove duplicate data.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_bfs.py<pre><code>def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"Breadth-first traversal\"\"\"\n    # Represent the graph with an adjacency list to access all neighbors of a given vertex\n    # Vertex traversal sequence\n    res = []\n    # Hash set recording visited vertices\n    visited = set[Vertex]([start_vet])\n    # Use a queue to implement BFS\n    que = deque[Vertex]([start_vet])\n    # Starting from start_vet, loop until all reachable vertices have been visited\n    while len(que) &gt; 0:\n        vet = que.popleft()  # Dequeue the front vertex\n        res.append(vet)  # Record the visited vertex\n        # Traverse all neighbors of this vertex\n        for adj_vet in graph.adj_list[vet]:\n            if adj_vet in visited:\n                continue  # Skip vertices that have already been visited\n            que.append(adj_vet)  # Enqueue only unvisited vertices\n            visited.add(adj_vet)  # Mark this vertex as visited\n    # Return the vertex traversal sequence\n    return res\n</code></pre> graph_bfs.cpp<pre><code>/* Breadth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nvector&lt;Vertex *&gt; graphBFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // Vertex traversal sequence\n    vector&lt;Vertex *&gt; res;\n    // Hash set recording visited vertices\n    unordered_set&lt;Vertex *&gt; visited = {startVet};\n    // Use a queue to implement BFS\n    queue&lt;Vertex *&gt; que;\n    que.push(startVet);\n    // Starting from startVet, continue until all reachable vertices have been visited\n    while (!que.empty()) {\n        Vertex *vet = que.front();\n        que.pop();          // Dequeue the front vertex\n        res.push_back(vet); // Record the visited vertex\n        // Traverse all neighbors of this vertex\n        for (auto adjVet : graph.adjList[vet]) {\n            if (visited.count(adjVet))\n                continue;            // Skip vertices that have already been visited\n            que.push(adjVet);        // Enqueue only unvisited vertices\n            visited.emplace(adjVet); // Mark this vertex as visited\n        }\n    }\n    // Return the vertex traversal sequence\n    return res;\n}\n</code></pre> graph_bfs.java<pre><code>/* Breadth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n    // Vertex traversal sequence\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // Hash set recording visited vertices\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    visited.add(startVet);\n    // Use a queue to implement BFS\n    Queue&lt;Vertex&gt; que = new LinkedList&lt;&gt;();\n    que.offer(startVet);\n    // Starting from startVet, loop until all reachable vertices have been visited\n    while (!que.isEmpty()) {\n        Vertex vet = que.poll(); // Dequeue the front vertex\n        res.add(vet);            // Record the visited vertex\n        // Traverse all neighbors of this vertex\n        for (Vertex adjVet : graph.adjList.get(vet)) {\n            if (visited.contains(adjVet))\n                continue;        // Skip vertices that have already been visited\n            que.offer(adjVet);   // Enqueue only unvisited vertices\n            visited.add(adjVet); // Mark this vertex as visited\n        }\n    }\n    // Return the vertex traversal sequence\n    return res;\n}\n</code></pre> graph_bfs.cs<pre><code>/* Breadth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nList&lt;Vertex&gt; GraphBFS(GraphAdjList graph, Vertex startVet) {\n    // Vertex traversal sequence\n    List&lt;Vertex&gt; res = [];\n    // Hash set recording visited vertices\n    HashSet&lt;Vertex&gt; visited = [startVet];\n    // Use a queue to implement BFS\n    Queue&lt;Vertex&gt; que = new();\n    que.Enqueue(startVet);\n    // Starting from startVet, loop until all reachable vertices have been visited\n    while (que.Count &gt; 0) {\n        Vertex vet = que.Dequeue(); // Dequeue the front vertex\n        res.Add(vet);               // Record the visited vertex\n        foreach (Vertex adjVet in graph.adjList[vet]) {\n            if (visited.Contains(adjVet)) {\n                continue;          // Skip vertices that have already been visited\n            }\n            que.Enqueue(adjVet);   // Enqueue only unvisited vertices\n            visited.Add(adjVet);   // Mark this vertex as visited\n        }\n    }\n\n    // Return the vertex traversal sequence\n    return res;\n}\n</code></pre> graph_bfs.go<pre><code>/* Breadth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // Vertex traversal sequence\n    res := make([]Vertex, 0)\n    // Hash set recording visited vertices\n    visited := make(map[Vertex]struct{})\n    visited[startVet] = struct{}{}\n    // Use a queue for BFS, represented here by a slice\n    queue := make([]Vertex, 0)\n    queue = append(queue, startVet)\n    // Starting from startVet, loop until all reachable vertices have been visited\n    for len(queue) &gt; 0 {\n        // Dequeue the front vertex\n        vet := queue[0]\n        queue = queue[1:]\n        // Record the visited vertex\n        res = append(res, vet)\n        // Traverse all neighbors of this vertex\n        for _, adjVet := range g.adjList[vet] {\n            _, isExist := visited[adjVet]\n            // Enqueue only unvisited vertices\n            if !isExist {\n                queue = append(queue, adjVet)\n                visited[adjVet] = struct{}{}\n            }\n        }\n    }\n    // Return the vertex traversal sequence\n    return res\n}\n</code></pre> graph_bfs.swift<pre><code>/* Breadth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // Vertex traversal sequence\n    var res: [Vertex] = []\n    // Hash set recording visited vertices\n    var visited: Set&lt;Vertex&gt; = [startVet]\n    // Use a queue to implement BFS\n    var que: [Vertex] = [startVet]\n    // Starting from startVet, loop until all reachable vertices have been visited\n    while !que.isEmpty {\n        let vet = que.removeFirst() // Dequeue the front vertex\n        res.append(vet) // Record the visited vertex\n        // Traverse all neighbors of this vertex\n        for adjVet in graph.adjList[vet] ?? [] {\n            if visited.contains(adjVet) {\n                continue // Skip vertices that have already been visited\n            }\n            que.append(adjVet) // Enqueue only unvisited vertices\n            visited.insert(adjVet) // Mark this vertex as visited\n        }\n    }\n    // Return the vertex traversal sequence\n    return res\n}\n</code></pre> graph_bfs.js<pre><code>/* Breadth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nfunction graphBFS(graph, startVet) {\n    // Vertex traversal sequence\n    const res = [];\n    // Hash set recording visited vertices\n    const visited = new Set();\n    visited.add(startVet);\n    // Use a queue to implement BFS\n    const que = [startVet];\n    // Starting from startVet, loop until all reachable vertices have been visited\n    while (que.length) {\n        const vet = que.shift(); // Dequeue the front vertex\n        res.push(vet); // Record the visited vertex\n        // Traverse all neighbors of this vertex\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // Skip vertices that have already been visited\n            }\n            que.push(adjVet); // Enqueue only unvisited vertices\n            visited.add(adjVet); // Mark this vertex as visited\n        }\n    }\n    // Return the vertex traversal sequence\n    return res;\n}\n</code></pre> graph_bfs.ts<pre><code>/* Breadth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // Vertex traversal sequence\n    const res: Vertex[] = [];\n    // Hash set recording visited vertices\n    const visited: Set&lt;Vertex&gt; = new Set();\n    visited.add(startVet);\n    // Use a queue to implement BFS\n    const que = [startVet];\n    // Starting from startVet, loop until all reachable vertices have been visited\n    while (que.length) {\n        const vet = que.shift(); // Dequeue the front vertex\n        res.push(vet); // Record the visited vertex\n        // Traverse all neighbors of this vertex\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // Skip vertices that have already been visited\n            }\n            que.push(adjVet); // Enqueue only unvisited vertices\n            visited.add(adjVet); // Mark this vertex as visited\n        }\n    }\n    // Return the vertex traversal sequence\n    return res;\n}\n</code></pre> graph_bfs.dart<pre><code>/* Breadth-first traversal */\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n  // Represent the graph with an adjacency list to access all neighbors of a given vertex\n  // Vertex traversal sequence\n  List&lt;Vertex&gt; res = [];\n  // Hash set recording visited vertices\n  Set&lt;Vertex&gt; visited = {};\n  visited.add(startVet);\n  // Use a queue to implement BFS\n  Queue&lt;Vertex&gt; que = Queue();\n  que.add(startVet);\n  // Starting from startVet, loop until all reachable vertices have been visited\n  while (que.isNotEmpty) {\n    Vertex vet = que.removeFirst(); // Dequeue the front vertex\n    res.add(vet); // Record the visited vertex\n    // Traverse all neighbors of this vertex\n    for (Vertex adjVet in graph.adjList[vet]!) {\n      if (visited.contains(adjVet)) {\n        continue; // Skip vertices that have already been visited\n      }\n      que.add(adjVet); // Enqueue only unvisited vertices\n      visited.add(adjVet); // Mark this vertex as visited\n    }\n  }\n  // Return the vertex traversal sequence\n  return res;\n}\n</code></pre> graph_bfs.rs<pre><code>/* Breadth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nfn graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // Vertex traversal sequence\n    let mut res = vec![];\n    // Hash set recording visited vertices\n    let mut visited = HashSet::new();\n    visited.insert(start_vet);\n    // Use a queue to implement BFS\n    let mut que = VecDeque::new();\n    que.push_back(start_vet);\n    // Start at start_vet and loop until all reachable vertices have been visited\n    while let Some(vet) = que.pop_front() {\n        res.push(vet); // Record the visited vertex\n\n        // Traverse all neighbors of this vertex\n        if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n            for &amp;adj_vet in adj_vets {\n                if visited.contains(&amp;adj_vet) {\n                    continue; // Skip vertices that have already been visited\n                }\n                que.push_back(adj_vet); // Enqueue only unvisited vertices\n                visited.insert(adj_vet); // Mark this vertex as visited\n            }\n        }\n    }\n    // Return the vertex traversal sequence\n    res\n}\n</code></pre> graph_bfs.c<pre><code>/* Vertex queue structure */\ntypedef struct {\n    Vertex *vertices[MAX_SIZE];\n    int front, rear, size;\n} Queue;\n\n/* Constructor */\nQueue *newQueue() {\n    Queue *q = (Queue *)malloc(sizeof(Queue));\n    q-&gt;front = q-&gt;rear = q-&gt;size = 0;\n    return q;\n}\n\n/* Check whether the queue is empty */\nint isEmpty(Queue *q) {\n    return q-&gt;size == 0;\n}\n\n/* Enqueue operation */\nvoid enqueue(Queue *q, Vertex *vet) {\n    q-&gt;vertices[q-&gt;rear] = vet;\n    q-&gt;rear = (q-&gt;rear + 1) % MAX_SIZE;\n    q-&gt;size++;\n}\n\n/* Dequeue operation */\nVertex *dequeue(Queue *q) {\n    Vertex *vet = q-&gt;vertices[q-&gt;front];\n    q-&gt;front = (q-&gt;front + 1) % MAX_SIZE;\n    q-&gt;size--;\n    return vet;\n}\n\n/* Check whether a vertex has been visited */\nint isVisited(Vertex **visited, int size, Vertex *vet) {\n    // Search for the vertex with a linear scan in O(n) time\n    for (int i = 0; i &lt; size; i++) {\n        if (visited[i] == vet)\n            return 1;\n    }\n    return 0;\n}\n\n/* Breadth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nvoid graphBFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize, Vertex **visited, int *visitedSize) {\n    // Use a queue to implement BFS\n    Queue *queue = newQueue();\n    enqueue(queue, startVet);\n    visited[(*visitedSize)++] = startVet;\n    // Starting from startVet, continue until all reachable vertices have been visited\n    while (!isEmpty(queue)) {\n        Vertex *vet = dequeue(queue); // Dequeue the front vertex\n        res[(*resSize)++] = vet;      // Record the visited vertex\n        // Traverse all neighbors of this vertex\n        AdjListNode *node = findNode(graph, vet);\n        while (node != NULL) {\n            // Skip vertices that have already been visited\n            if (!isVisited(visited, *visitedSize, node-&gt;vertex)) {\n                enqueue(queue, node-&gt;vertex);             // Enqueue only unvisited vertices\n                visited[(*visitedSize)++] = node-&gt;vertex; // Mark this vertex as visited\n            }\n            node = node-&gt;next;\n        }\n    }\n    // Free memory\n    free(queue);\n}\n</code></pre> graph_bfs.kt<pre><code>/* Breadth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nfun graphBFS(graph: GraphAdjList, startVet: Vertex): MutableList&lt;Vertex?&gt; {\n    // Vertex traversal sequence\n    val res = mutableListOf&lt;Vertex?&gt;()\n    // Hash set recording visited vertices\n    val visited = HashSet&lt;Vertex&gt;()\n    visited.add(startVet)\n    // Use a queue to implement BFS\n    val que = LinkedList&lt;Vertex&gt;()\n    que.offer(startVet)\n    // Starting from startVet, loop until all reachable vertices have been visited\n    while (!que.isEmpty()) {\n        val vet = que.poll() // Dequeue the front vertex\n        res.add(vet)         // Record the visited vertex\n        // Traverse all neighbors of this vertex\n        for (adjVet in graph.adjList[vet]!!) {\n            if (visited.contains(adjVet))\n                continue        // Skip vertices that have already been visited\n            que.offer(adjVet)   // Enqueue only unvisited vertices\n            visited.add(adjVet) // Mark this vertex as visited\n        }\n    }\n    // Return the vertex traversal sequence\n    return res\n}\n</code></pre> graph_bfs.rb<pre><code>### Breadth-first traversal ###\ndef graph_bfs(graph, start_vet)\n  # Represent the graph with an adjacency list to access all neighbors of a given vertex\n  # Vertex traversal sequence\n  res = []\n  # Hash set recording visited vertices\n  visited = Set.new([start_vet])\n  # Use a queue to implement BFS\n  que = [start_vet]\n  # Start from start_vet and loop until all reachable vertices have been visited\n  while que.length &gt; 0\n    vet = que.shift # Dequeue the front vertex\n    res &lt;&lt; vet # Record the visited vertex\n    # Traverse all neighbors of this vertex\n    for adj_vet in graph.adj_list[vet]\n      next if visited.include?(adj_vet) # Skip vertices that have already been visited\n      que &lt;&lt; adj_vet # Enqueue only unvisited vertices\n      visited.add(adj_vet) # Mark this vertex as visited\n    end\n  end\n  # Return the vertex traversal sequence\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>The code is fairly abstract; compare it with Figure 9-10 to build your understanding.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-10   Steps in breadth-first traversal of a graph </p> <p>Is the breadth-first traversal order unique?</p> <p>No. Breadth-first traversal only requires vertices to be visited from near to far; vertices at the same distance may be visited in any order. In Figure 9-10, vertices \\(1\\) and \\(3\\) can be visited in either order, and vertices \\(2\\), \\(4\\), and \\(6\\) can also be visited in any order.</p>","path":["Chapter 9.   Graphs","9.3   Graph traversal"],"tags":[]},{"location":"chapter_graph/graph_traversal/#2-complexity-analysis","level":3,"title":"2.   Complexity analysis","text":"<p>Time complexity: Every vertex is enqueued and dequeued once, taking \\(O(|V|)\\) time. While traversing adjacent vertices, every edge is visited \\(2\\) times because the graph is undirected, taking \\(O(2|E|)\\) time. The total time is \\(O(|V| + |E|)\\).</p> <p>Space complexity: The list <code>res</code>, hash set <code>visited</code>, and queue <code>que</code> each contain at most \\(|V|\\) vertices, requiring \\(O(|V|)\\) space.</p>","path":["Chapter 9.   Graphs","9.3   Graph traversal"],"tags":[]},{"location":"chapter_graph/graph_traversal/#932-depth-first-traversal","level":2,"title":"9.3.2   Depth-first traversal","text":"<p>Depth-first traversal follows a path as far as possible and returns when it can go no farther. In Figure 9-11, we start at the top-left vertex and move to an adjacent vertex of the current vertex, continuing until we reach a dead end. We then return and explore another path as far as possible, repeating this process until every vertex has been visited.</p> <p></p> <p> Figure 9-11   Depth-first traversal of a graph </p>","path":["Chapter 9.   Graphs","9.3   Graph traversal"],"tags":[]},{"location":"chapter_graph/graph_traversal/#1-algorithm-implementation_1","level":3,"title":"1.   Algorithm implementation","text":"<p>This pattern of going as far as possible and then returning is usually implemented with recursion. As with breadth-first traversal, we use a hash set <code>visited</code> to track visited vertices and avoid visiting them again.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_dfs.py<pre><code>def dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n    \"\"\"Depth-first traversal helper function\"\"\"\n    res.append(vet)  # Record the visited vertex\n    visited.add(vet)  # Mark this vertex as visited\n    # Traverse all neighbors of this vertex\n    for adjVet in graph.adj_list[vet]:\n        if adjVet in visited:\n            continue  # Skip vertices that have already been visited\n        # Recursively visit the neighboring vertex\n        dfs(graph, visited, res, adjVet)\n\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"Depth-first traversal\"\"\"\n    # Represent the graph with an adjacency list to access all neighbors of a given vertex\n    # Vertex traversal sequence\n    res = []\n    # Hash set recording visited vertices\n    visited = set[Vertex]()\n    dfs(graph, visited, res, start_vet)\n    return res\n</code></pre> graph_dfs.cpp<pre><code>/* Depth-first traversal helper function */\nvoid dfs(GraphAdjList &amp;graph, unordered_set&lt;Vertex *&gt; &amp;visited, vector&lt;Vertex *&gt; &amp;res, Vertex *vet) {\n    res.push_back(vet);   // Record the visited vertex\n    visited.emplace(vet); // Mark this vertex as visited\n    // Traverse all neighbors of this vertex\n    for (Vertex *adjVet : graph.adjList[vet]) {\n        if (visited.count(adjVet))\n            continue; // Skip vertices that have already been visited\n        // Recursively visit the neighboring vertex\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* Depth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nvector&lt;Vertex *&gt; graphDFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // Vertex traversal sequence\n    vector&lt;Vertex *&gt; res;\n    // Hash set recording visited vertices\n    unordered_set&lt;Vertex *&gt; visited;\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.java<pre><code>/* Depth-first traversal helper function */\nvoid dfs(GraphAdjList graph, Set&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.add(vet);     // Record the visited vertex\n    visited.add(vet); // Mark this vertex as visited\n    // Traverse all neighbors of this vertex\n    for (Vertex adjVet : graph.adjList.get(vet)) {\n        if (visited.contains(adjVet))\n            continue; // Skip vertices that have already been visited\n        // Recursively visit the neighboring vertex\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* Depth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n    // Vertex traversal sequence\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // Hash set recording visited vertices\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.cs<pre><code>/* Depth-first traversal helper function */\nvoid DFS(GraphAdjList graph, HashSet&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.Add(vet);     // Record the visited vertex\n    visited.Add(vet); // Mark this vertex as visited\n    // Traverse all neighbors of this vertex\n    foreach (Vertex adjVet in graph.adjList[vet]) {\n        if (visited.Contains(adjVet)) {\n            continue; // Skip vertices that have already been visited                             \n        }\n        // Recursively visit the neighboring vertex\n        DFS(graph, visited, res, adjVet);\n    }\n}\n\n/* Depth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nList&lt;Vertex&gt; GraphDFS(GraphAdjList graph, Vertex startVet) {\n    // Vertex traversal sequence\n    List&lt;Vertex&gt; res = [];\n    // Hash set recording visited vertices\n    HashSet&lt;Vertex&gt; visited = [];\n    DFS(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.go<pre><code>/* Depth-first traversal helper function */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n    // append returns an updated slice, which must be assigned back through the original slice pointer\n    *res = append(*res, vet)\n    visited[vet] = struct{}{}\n    // Traverse all neighbors of this vertex\n    for _, adjVet := range g.adjList[vet] {\n        _, isExist := visited[adjVet]\n        // Recursively visit the neighboring vertex\n        if !isExist {\n            dfs(g, visited, res, adjVet)\n        }\n    }\n}\n\n/* Depth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // Vertex traversal sequence\n    res := make([]Vertex, 0)\n    // Hash set recording visited vertices\n    visited := make(map[Vertex]struct{})\n    dfs(g, visited, &amp;res, startVet)\n    // Return the vertex traversal sequence\n    return res\n}\n</code></pre> graph_dfs.swift<pre><code>/* Depth-first traversal helper function */\nfunc dfs(graph: GraphAdjList, visited: inout Set&lt;Vertex&gt;, res: inout [Vertex], vet: Vertex) {\n    res.append(vet) // Record the visited vertex\n    visited.insert(vet) // Mark this vertex as visited\n    // Traverse all neighbors of this vertex\n    for adjVet in graph.adjList[vet] ?? [] {\n        if visited.contains(adjVet) {\n            continue // Skip vertices that have already been visited\n        }\n        // Recursively visit the neighboring vertex\n        dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: adjVet)\n    }\n}\n\n/* Depth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // Vertex traversal sequence\n    var res: [Vertex] = []\n    // Hash set recording visited vertices\n    var visited: Set&lt;Vertex&gt; = []\n    dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: startVet)\n    return res\n}\n</code></pre> graph_dfs.js<pre><code>/* Depth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nfunction dfs(graph, visited, res, vet) {\n    res.push(vet); // Record the visited vertex\n    visited.add(vet); // Mark this vertex as visited\n    // Traverse all neighbors of this vertex\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // Skip vertices that have already been visited\n        }\n        // Recursively visit the neighboring vertex\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* Depth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nfunction graphDFS(graph, startVet) {\n    // Vertex traversal sequence\n    const res = [];\n    // Hash set recording visited vertices\n    const visited = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.ts<pre><code>/* Depth-first traversal helper function */\nfunction dfs(\n    graph: GraphAdjList,\n    visited: Set&lt;Vertex&gt;,\n    res: Vertex[],\n    vet: Vertex\n): void {\n    res.push(vet); // Record the visited vertex\n    visited.add(vet); // Mark this vertex as visited\n    // Traverse all neighbors of this vertex\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // Skip vertices that have already been visited\n        }\n        // Recursively visit the neighboring vertex\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* Depth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // Vertex traversal sequence\n    const res: Vertex[] = [];\n    // Hash set recording visited vertices\n    const visited: Set&lt;Vertex&gt; = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.dart<pre><code>/* Depth-first traversal helper function */\nvoid dfs(\n  GraphAdjList graph,\n  Set&lt;Vertex&gt; visited,\n  List&lt;Vertex&gt; res,\n  Vertex vet,\n) {\n  res.add(vet); // Record the visited vertex\n  visited.add(vet); // Mark this vertex as visited\n  // Traverse all neighbors of this vertex\n  for (Vertex adjVet in graph.adjList[vet]!) {\n    if (visited.contains(adjVet)) {\n      continue; // Skip vertices that have already been visited\n    }\n    // Recursively visit the neighboring vertex\n    dfs(graph, visited, res, adjVet);\n  }\n}\n\n/* Depth-first traversal */\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n  // Vertex traversal sequence\n  List&lt;Vertex&gt; res = [];\n  // Hash set recording visited vertices\n  Set&lt;Vertex&gt; visited = {};\n  dfs(graph, visited, res, startVet);\n  return res;\n}\n</code></pre> graph_dfs.rs<pre><code>/* Depth-first traversal helper function */\nfn dfs(graph: &amp;GraphAdjList, visited: &amp;mut HashSet&lt;Vertex&gt;, res: &amp;mut Vec&lt;Vertex&gt;, vet: Vertex) {\n    res.push(vet); // Record the visited vertex\n    visited.insert(vet); // Mark this vertex as visited\n                         // Traverse all neighbors of this vertex\n    if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n        for &amp;adj_vet in adj_vets {\n            if visited.contains(&amp;adj_vet) {\n                continue; // Skip vertices that have already been visited\n            }\n            // Recursively visit the neighboring vertex\n            dfs(graph, visited, res, adj_vet);\n        }\n    }\n}\n\n/* Depth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nfn graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // Vertex traversal sequence\n    let mut res = vec![];\n    // Hash set recording visited vertices\n    let mut visited = HashSet::new();\n    dfs(&amp;graph, &amp;mut visited, &amp;mut res, start_vet);\n\n    res\n}\n</code></pre> graph_dfs.c<pre><code>/* Check whether a vertex has been visited */\nint isVisited(Vertex **res, int size, Vertex *vet) {\n    // Search for the vertex with a linear scan in O(n) time\n    for (int i = 0; i &lt; size; i++) {\n        if (res[i] == vet) {\n            return 1;\n        }\n    }\n    return 0;\n}\n\n/* Depth-first traversal helper function */\nvoid dfs(GraphAdjList *graph, Vertex **res, int *resSize, Vertex *vet) {\n    // Record the visited vertex\n    res[(*resSize)++] = vet;\n    // Traverse all neighbors of this vertex\n    AdjListNode *node = findNode(graph, vet);\n    while (node != NULL) {\n        // Skip vertices that have already been visited\n        if (!isVisited(res, *resSize, node-&gt;vertex)) {\n            // Recursively visit the neighboring vertex\n            dfs(graph, res, resSize, node-&gt;vertex);\n        }\n        node = node-&gt;next;\n    }\n}\n\n/* Depth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nvoid graphDFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize) {\n    dfs(graph, res, resSize, startVet);\n}\n</code></pre> graph_dfs.kt<pre><code>/* Depth-first traversal helper function */\nfun dfs(\n    graph: GraphAdjList,\n    visited: MutableSet&lt;Vertex?&gt;,\n    res: MutableList&lt;Vertex?&gt;,\n    vet: Vertex?\n) {\n    res.add(vet)     // Record the visited vertex\n    visited.add(vet) // Mark this vertex as visited\n    // Traverse all neighbors of this vertex\n    for (adjVet in graph.adjList[vet]!!) {\n        if (visited.contains(adjVet))\n            continue  // Skip vertices that have already been visited\n        // Recursively visit the neighboring vertex\n        dfs(graph, visited, res, adjVet)\n    }\n}\n\n/* Depth-first traversal */\n// Represent the graph with an adjacency list to access all neighbors of a given vertex\nfun graphDFS(graph: GraphAdjList, startVet: Vertex?): MutableList&lt;Vertex?&gt; {\n    // Vertex traversal sequence\n    val res = mutableListOf&lt;Vertex?&gt;()\n    // Hash set recording visited vertices\n    val visited = HashSet&lt;Vertex?&gt;()\n    dfs(graph, visited, res, startVet)\n    return res\n}\n</code></pre> graph_dfs.rb<pre><code>### Depth-first traversal helper function ###\ndef dfs(graph, visited, res, vet)\n  res &lt;&lt; vet # Record the visited vertex\n  visited.add(vet) # Mark this vertex as visited\n  # Traverse all neighbors of this vertex\n  for adj_vet in graph.adj_list[vet]\n    next if visited.include?(adj_vet) # Skip vertices that have already been visited\n    # Recursively visit the neighboring vertex\n    dfs(graph, visited, res, adj_vet)\n  end\nend\n\n### Depth-first traversal ###\ndef graph_dfs(graph, start_vet)\n  # Represent the graph with an adjacency list to access all neighbors of a given vertex\n  # Vertex traversal sequence\n  res = []\n  # Hash set recording visited vertices\n  visited = Set.new\n  dfs(graph, visited, res, start_vet)\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Figure 9-12 shows the steps of depth-first traversal.</p> <ul> <li>Straight dashed lines show recursive descent, where a new recursive call begins to visit another vertex.</li> <li>Curved dashed lines show backtracking, where a recursive call has returned to the point from which it was made.</li> </ul> <p>To deepen your understanding, follow Figure 9-12 alongside the code and mentally simulate (or draw on paper) the entire DFS process, including when each recursive call begins and returns.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 9-12   Steps in depth-first traversal of a graph </p> <p>Is the depth-first traversal order unique?</p> <p>Like breadth-first traversal, depth-first traversal does not have a unique order. From any vertex, we can explore in any direction first: rearranging the order of adjacent vertices still produces a depth-first traversal.</p> <p>For example, in tree traversal, \"root \\(\\rightarrow\\) left \\(\\rightarrow\\) right\", \"left \\(\\rightarrow\\) root \\(\\rightarrow\\) right\", and \"left \\(\\rightarrow\\) right \\(\\rightarrow\\) root\" correspond to preorder, inorder, and postorder traversal. They use three different visiting priorities, yet all are depth-first traversals.</p>","path":["Chapter 9.   Graphs","9.3   Graph traversal"],"tags":[]},{"location":"chapter_graph/graph_traversal/#2-complexity-analysis_1","level":3,"title":"2.   Complexity analysis","text":"<p>Time complexity: Every vertex is visited \\(1\\) time, taking \\(O(|V|)\\) time, and every edge is visited \\(2\\) times, taking \\(O(2|E|)\\) time. The total time is \\(O(|V| + |E|)\\).</p> <p>Space complexity: The list <code>res</code> and hash set <code>visited</code> each contain at most \\(|V|\\) vertices, and the maximum recursion depth is \\(|V|\\), so the space complexity is \\(O(|V|)\\).</p>","path":["Chapter 9.   Graphs","9.3   Graph traversal"],"tags":[]},{"location":"chapter_graph/summary/","level":1,"title":"9.4   Summary","text":"","path":["Chapter 9.   Graphs","9.4   Summary"],"tags":[]},{"location":"chapter_graph/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<ul> <li>A graph consists of vertices and edges and can be represented using a vertex set and an edge set.</li> <li>Network relationships (graphs) allow more freedom than linear relationships (linked lists) or divide-and-conquer relationships (trees), making them more complex.</li> <li>Edges in a directed graph have a direction; all vertices in a connected graph are reachable; and each edge in a weighted graph has a weight.</li> <li>An adjacency matrix represents a graph with a matrix. Each row (or column) represents a vertex, and entries represent edges, with \\(1\\) or \\(0\\) indicating whether an edge exists between two vertices. Adding, removing, looking up, and modifying edges are efficient, but the matrix uses substantial space.</li> <li>An adjacency list represents a graph with multiple linked lists. The \\(i\\)-th linked list corresponds to vertex \\(i\\) and stores all of its adjacent vertices. It uses less space than an adjacency matrix, but finding an edge requires traversing a linked list, making lookups slower.</li> <li>A long linked list in an adjacency list can be replaced with a red-black tree or a hash table to speed up lookups.</li> <li>In terms of algorithm design, an adjacency matrix trades space for time, while an adjacency list trades time for space.</li> <li>Graphs can model many real-world systems, such as social networks and subway networks.</li> <li>A tree is a special case of a graph, and tree traversal is a special case of graph traversal.</li> <li>Breadth-first graph traversal searches from near to far, expanding layer by layer. It is usually implemented with a queue.</li> <li>Depth-first graph traversal follows a path as far as possible and backtracks when it can go no farther. It is often implemented with recursion.</li> </ul>","path":["Chapter 9.   Graphs","9.4   Summary"],"tags":[]},{"location":"chapter_graph/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: Is a path defined as a sequence of vertices or a sequence of edges?</p> <p>Wikipedia's language editions give different definitions: the English edition defines a path as a sequence of edges, while the Chinese edition defines it as a sequence of vertices. The English wording is: In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.</p> <p>In this book, a path is treated as a sequence of edges rather than vertices. This is because multiple edges can connect the same two vertices, in which case each edge corresponds to a path.</p> <p>Q: Can some vertices in a disconnected graph remain unvisited?</p> <p>In a disconnected graph, at least one vertex cannot be reached from a given starting vertex. Traversing the entire graph requires multiple starting points so that all connected components are visited.</p> <p>Q: Must the adjacent vertices in an adjacency list appear in a particular order?</p> <p>They may appear in any order. In practice, however, it can be useful to sort them by a specific rule, such as the order in which vertices were added or their values. This helps us quickly find a vertex with a particular extreme value.</p>","path":["Chapter 9.   Graphs","9.4   Summary"],"tags":[]},{"location":"chapter_greedy/","level":1,"title":"Chapter 15.   Greedy algorithms","text":"<p>Abstract</p> <p>A sunflower turns toward the sun, always seeking the greatest opportunity to grow.</p> <p>Through a series of simple choices, a greedy strategy leads us step by step toward the best answer.</p>","path":["Chapter 15.   Greedy algorithms"],"tags":[]},{"location":"chapter_greedy/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>15.1   Greedy algorithms</li> <li>15.2   Fractional knapsack problem</li> <li>15.3   Maximum capacity problem</li> <li>15.4   Maximum product cutting problem</li> <li>15.5   Summary</li> <li>15.6   Exercises</li> </ul>","path":["Chapter 15.   Greedy algorithms"],"tags":[]},{"location":"chapter_greedy/exercises/","level":1,"title":"15.6   Exercises","text":"","path":["Chapter 15.   Greedy algorithms","15.6   Exercises"],"tags":[]},{"location":"chapter_greedy/exercises/#1561-check-your-understanding","level":2,"title":"15.6.1   Check your understanding","text":"","path":["Chapter 15.   Greedy algorithms","15.6   Exercises"],"tags":[]},{"location":"chapter_greedy/exercises/#1-is-choosing-the-largest-coin-always-best","level":3,"title":"1.   Is choosing the largest coin always best?","text":"<p>The coin denominations are <code>[1, 7, 10]</code>, and the target amount is 14.</p> <ol> <li>Follow the rule “always take the largest denomination that does not exceed the remaining amount,” and list the coins chosen.</li> <li>Is there a way to use fewer coins? If so, give one; if not, explain why.</li> <li>Can you conclude that this greedy strategy is correct for any set of coin denominations?</li> </ol> Reference answer <ol> <li> <p>The greedy strategy chooses <code>10 + 1 + 1 + 1 + 1</code>, using 5 coins in total.</p> </li> <li> <p>There is a solution with fewer coins: <code>7 + 7</code>, which uses only 2.</p> </li> <li> <p>No. This counterexample shows that choosing the largest available denomination at each step does not always minimize the number of coins for arbitrary denominations.     The largest immediate choice can rule out a better combination later.</p> </li> </ol>","path":["Chapter 15.   Greedy algorithms","15.6   Exercises"],"tags":[]},{"location":"chapter_greedy/exercises/#2-which-item-should-go-into-the-knapsack-first","level":3,"title":"2.   Which item should go into the knapsack first?","text":"<p>A knapsack with a capacity of 4 kg can hold the following items. You may take part of an item, and the value obtained is proportional to the weight taken.</p> <ul> <li>Item A weighs 4 kg and has a value of 20.</li> <li>Item B weighs 3 kg and has a value of 18.</li> </ul> <ol> <li>What is each item's value per kilogram? Which item should you take first?</li> <li>If you fill the knapsack using the greedy strategy for fractional knapsack, what is the final value?</li> <li>When items can be divided and the knapsack limits total weight, should you compare total value or value per kilogram? Why?</li> </ol> Reference answer <ol> <li> <p>A's value per kilogram is <code>20 ÷ 4 = 5</code>, and B's is <code>18 ÷ 3 = 6</code>.     Take B first because it has the higher value per unit of weight.</p> </li> <li> <p>Take all of B first, using 3 kg of capacity and gaining a value of 18. With 1 kg of capacity left,     take 1 kg of A for a value of 5. The final value is <code>18 + 5 = 23</code>.</p> </li> <li> <p>The knapsack limits total weight, and items can be divided, so compare value per unit of weight.     Although A has a higher total value, its value per kilogram is lower than B's. Taking all of A first would give a value of only 20.</p> </li> </ol>","path":["Chapter 15.   Greedy algorithms","15.6   Exercises"],"tags":[]},{"location":"chapter_greedy/exercises/#3-which-pointer-should-move-next","level":3,"title":"3.   Which pointer should move next?","text":"<p>The wall heights are <code>[1, 8, 6, 2, 5]</code>. Use two pointers, initially at opposite ends, to find the maximum container capacity. Capacity equals the height of the shorter of the two walls multiplied by the difference between their indices.</p> <ol> <li>Initially, the left pointer is at index 0 and the right pointer is at index 4. What is the current capacity? Which pointer should move next?</li> <li>After making the move you chose in question 1, what are the two pointer indices? What is the capacity now, and which pointer should move next?</li> <li>For the current pair of walls, you could move the pointer at the shorter wall or the pointer at the taller wall. Which move still has a chance of producing a larger capacity? Why?</li> </ol> Reference answer <ol> <li> <p>The current capacity is <code>min(1, 5) × (4 - 0) = 4</code>. The left wall is shorter, so move the left pointer.</p> </li> <li> <p>After moving the left pointer, the pointers are at indices 1 and 4. The current capacity is     <code>min(8, 5) × (4 - 1) = 15</code>. The right wall is shorter, so move the right pointer next.</p> </li> <li> <p>Only moving the pointer at the shorter wall can still produce a larger capacity. Moving the pointer at the taller wall reduces the distance between the walls, while the container's height remains limited by the shorter wall that has not moved.     That height can only stay the same or decrease, so the capacity cannot exceed its previous value. Moving the pointer at the shorter wall is the only way to have a chance of finding a taller wall.</p> </li> </ol>","path":["Chapter 15.   Greedy algorithms","15.6   Exercises"],"tags":[]},{"location":"chapter_greedy/exercises/#1562-programming-exercises","level":2,"title":"15.6.2   Programming exercises","text":"","path":["Chapter 15.   Greedy algorithms","15.6   Exercises"],"tags":[]},{"location":"chapter_greedy/exercises/#1-fractional-knapsack","level":3,"title":"1.   Fractional knapsack","text":"<p>You are given arrays <code>wgt</code> and <code>val</code> of equal length, where <code>wgt[i] &gt; 0</code> and <code>val[i] &gt;= 0</code>, and a knapsack capacity <code>cap &gt;= 0</code>. There is one of each item, but you may take just part of it. The value obtained is proportional to the fraction of the item's total weight that you take. Use a greedy algorithm to return the maximum total value the knapsack can hold, as a real number.</p> Hints <ol> <li>First calculate each item's value per unit of weight, val[i] / wgt[i]. Preserve the fractional part of the division result.</li> <li>Take items with a higher value per unit of weight first.</li> <li>If the remaining capacity is less than the current item's weight, take just enough of the item to fill the knapsack, then stop.</li> </ol>","path":["Chapter 15.   Greedy algorithms","15.6   Exercises"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/","level":1,"title":"15.2   Fractional knapsack problem","text":"<p>Question</p> <p>Given \\(n\\) items, where item \\(i\\) has weight \\(wgt[i-1]\\) and value \\(val[i-1]\\), and a knapsack with capacity \\(cap\\), each item may be selected only once. However, you may take a fraction of an item, with its value proportional to the weight taken. What is the maximum total value that fits within the knapsack's capacity? An example is shown in Figure 15-3.</p> <p></p> <p> Figure 15-3   Example input for the fractional knapsack problem </p> <p>The fractional knapsack problem is very similar to the 0-1 knapsack problem. The state consists of the current item \\(i\\) and capacity \\(c\\), and the goal is to maximize value within the knapsack's capacity.</p> <p>The difference is that we may take only part of an item. As shown in Figure 15-4, we can divide items into arbitrary fractions and calculate their values in proportion to their weights.</p> <ol> <li>For item \\(i\\), the value per unit weight is \\(val[i-1] / wgt[i-1]\\), which we call its unit value.</li> <li>If we take a portion of item \\(i\\) weighing \\(w\\), the value added to the knapsack is \\(w \\times val[i-1] / wgt[i-1]\\).</li> </ol> <p></p> <p> Figure 15-4   Value per unit weight of each item </p>","path":["Chapter 15.   Greedy algorithms","15.2   Fractional knapsack problem"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/#1-choosing-a-greedy-strategy","level":3,"title":"1.   Choosing a greedy strategy","text":"<p>Maximizing the total value in the knapsack essentially means maximizing the value obtained per unit weight. This leads to the greedy strategy shown in Figure 15-5.</p> <ol> <li>Sort the items by unit value in descending order.</li> <li>Iterate through the items, greedily choosing the item with the highest unit value at each step.</li> <li>If the remaining capacity is insufficient, take just enough of the current item to fill the knapsack.</li> </ol> <p></p> <p> Figure 15-5   Greedy strategy for the fractional knapsack problem </p>","path":["Chapter 15.   Greedy algorithms","15.2   Fractional knapsack problem"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/#2-code-implementation","level":3,"title":"2.   Code implementation","text":"<p>We define an <code>Item</code> class to make it easier to sort items by unit value. We repeatedly make greedy choices, stopping and returning the result when the knapsack is full:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby fractional_knapsack.py<pre><code>class Item:\n    \"\"\"Item\"\"\"\n\n    def __init__(self, w: int, v: int):\n        self.w = w  # Item weight\n        self.v = v  # Item value\n\ndef fractional_knapsack(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"Fractional knapsack: greedy algorithm\"\"\"\n    # Create a list of items, each with two attributes: weight and value\n    items = [Item(w, v) for w, v in zip(wgt, val)]\n    # Sort by value per unit weight, item.v / item.w, in descending order\n    items.sort(key=lambda item: item.v / item.w, reverse=True)\n    # Repeatedly make greedy choices\n    res = 0\n    for item in items:\n        if item.w &lt;= cap:\n            # If enough capacity remains, put the entire current item in the knapsack\n            res += item.v\n            cap -= item.w\n        else:\n            # If the remaining capacity is insufficient, put part of the current item in the knapsack\n            res += (item.v / item.w) * cap\n            # No capacity remains, so exit the loop\n            break\n    return res\n</code></pre> fractional_knapsack.cpp<pre><code>/* Item */\nclass Item {\n  public:\n    int w; // Item weight\n    int v; // Item value\n\n    Item(int w, int v) : w(w), v(v) {\n    }\n};\n\n/* Fractional knapsack: greedy algorithm */\ndouble fractionalKnapsack(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    // Create a list of items, each with two attributes: weight and value\n    vector&lt;Item&gt; items;\n    for (int i = 0; i &lt; wgt.size(); i++) {\n        items.push_back(Item(wgt[i], val[i]));\n    }\n    // Sort by value per unit weight, item.v / item.w, in descending order\n    sort(items.begin(), items.end(), [](Item &amp;a, Item &amp;b) { return (double)a.v / a.w &gt; (double)b.v / b.w; });\n    // Repeatedly make greedy choices\n    double res = 0;\n    for (auto &amp;item : items) {\n        if (item.w &lt;= cap) {\n            // If enough capacity remains, put the entire current item in the knapsack\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // If the remaining capacity is insufficient, put part of the current item in the knapsack\n            res += (double)item.v / item.w * cap;\n            // No capacity remains, so exit the loop\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.java<pre><code>/* Item */\nclass Item {\n    int w; // Item weight\n    int v; // Item value\n\n    public Item(int w, int v) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* Fractional knapsack: greedy algorithm */\ndouble fractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // Create a list of items, each with two attributes: weight and value\n    Item[] items = new Item[wgt.length];\n    for (int i = 0; i &lt; wgt.length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // Sort by value per unit weight, item.v / item.w, in descending order\n    Arrays.sort(items, Comparator.comparingDouble(item -&gt; -((double) item.v / item.w)));\n    // Repeatedly make greedy choices\n    double res = 0;\n    for (Item item : items) {\n        if (item.w &lt;= cap) {\n            // If enough capacity remains, put the entire current item in the knapsack\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // If the remaining capacity is insufficient, put part of the current item in the knapsack\n            res += (double) item.v / item.w * cap;\n            // No capacity remains, so exit the loop\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.cs<pre><code>/* Item */\nclass Item(int w, int v) {\n    public int w = w; // Item weight\n    public int v = v; // Item value\n}\n\n/* Fractional knapsack: greedy algorithm */\ndouble FractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // Create a list of items, each with two attributes: weight and value\n    Item[] items = new Item[wgt.Length];\n    for (int i = 0; i &lt; wgt.Length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // Sort by value per unit weight, item.v / item.w, in descending order\n    Array.Sort(items, (x, y) =&gt; (y.v / y.w).CompareTo(x.v / x.w));\n    // Repeatedly make greedy choices\n    double res = 0;\n    foreach (Item item in items) {\n        if (item.w &lt;= cap) {\n            // If enough capacity remains, put the entire current item in the knapsack\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // If the remaining capacity is insufficient, put part of the current item in the knapsack\n            res += (double)item.v / item.w * cap;\n            // No capacity remains, so exit the loop\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.go<pre><code>/* Item */\ntype Item struct {\n    w int // Item weight\n    v int // Item value\n}\n\n/* Fractional knapsack: greedy algorithm */\nfunc fractionalKnapsack(wgt []int, val []int, cap int) float64 {\n    // Create a list of items, each with two attributes: weight and value\n    items := make([]Item, len(wgt))\n    for i := 0; i &lt; len(wgt); i++ {\n        items[i] = Item{wgt[i], val[i]}\n    }\n    // Sort by value per unit weight, item.v / item.w, in descending order\n    sort.Slice(items, func(i, j int) bool {\n        return float64(items[i].v)/float64(items[i].w) &gt; float64(items[j].v)/float64(items[j].w)\n    })\n    // Repeatedly make greedy choices\n    res := 0.0\n    for _, item := range items {\n        if item.w &lt;= cap {\n            // If enough capacity remains, put the entire current item in the knapsack\n            res += float64(item.v)\n            cap -= item.w\n        } else {\n            // If the remaining capacity is insufficient, put part of the current item in the knapsack\n            res += float64(item.v) / float64(item.w) * float64(cap)\n            // No capacity remains, so exit the loop\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.swift<pre><code>/* Item */\nclass Item {\n    var w: Int // Item weight\n    var v: Int // Item value\n\n    init(w: Int, v: Int) {\n        self.w = w\n        self.v = v\n    }\n}\n\n/* Fractional knapsack: greedy algorithm */\nfunc fractionalKnapsack(wgt: [Int], val: [Int], cap: Int) -&gt; Double {\n    // Create a list of items, each with two attributes: weight and value\n    var items = zip(wgt, val).map { Item(w: $0, v: $1) }\n    // Sort by value per unit weight, item.v / item.w, in descending order\n    items.sort { -(Double($0.v) / Double($0.w)) &lt; -(Double($1.v) / Double($1.w)) }\n    // Repeatedly make greedy choices\n    var res = 0.0\n    var cap = cap\n    for item in items {\n        if item.w &lt;= cap {\n            // If enough capacity remains, put the entire current item in the knapsack\n            res += Double(item.v)\n            cap -= item.w\n        } else {\n            // If the remaining capacity is insufficient, put part of the current item in the knapsack\n            res += Double(item.v) / Double(item.w) * Double(cap)\n            // No capacity remains, so exit the loop\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.js<pre><code>/* Item */\nclass Item {\n    constructor(w, v) {\n        this.w = w; // Item weight\n        this.v = v; // Item value\n    }\n}\n\n/* Fractional knapsack: greedy algorithm */\nfunction fractionalKnapsack(wgt, val, cap) {\n    // Create a list of items, each with two attributes: weight and value\n    const items = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // Sort by value per unit weight, item.v / item.w, in descending order\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // Repeatedly make greedy choices\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // If enough capacity remains, put the entire current item in the knapsack\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // If the remaining capacity is insufficient, put part of the current item in the knapsack\n            res += (item.v / item.w) * cap;\n            // No capacity remains, so exit the loop\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.ts<pre><code>/* Item */\nclass Item {\n    w: number; // Item weight\n    v: number; // Item value\n\n    constructor(w: number, v: number) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* Fractional knapsack: greedy algorithm */\nfunction fractionalKnapsack(wgt: number[], val: number[], cap: number): number {\n    // Create a list of items, each with two attributes: weight and value\n    const items: Item[] = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // Sort by value per unit weight, item.v / item.w, in descending order\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // Repeatedly make greedy choices\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // If enough capacity remains, put the entire current item in the knapsack\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // If the remaining capacity is insufficient, put part of the current item in the knapsack\n            res += (item.v / item.w) * cap;\n            // No capacity remains, so exit the loop\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.dart<pre><code>/* Item */\nclass Item {\n  int w; // Item weight\n  int v; // Item value\n\n  Item(this.w, this.v);\n}\n\n/* Fractional knapsack: greedy algorithm */\ndouble fractionalKnapsack(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  // Create a list of items, each with two attributes: weight and value\n  List&lt;Item&gt; items = List.generate(wgt.length, (i) =&gt; Item(wgt[i], val[i]));\n  // Sort by value per unit weight, item.v / item.w, in descending order\n  items.sort((a, b) =&gt; (b.v / b.w).compareTo(a.v / a.w));\n  // Repeatedly make greedy choices\n  double res = 0;\n  for (Item item in items) {\n    if (item.w &lt;= cap) {\n      // If enough capacity remains, put the entire current item in the knapsack\n      res += item.v;\n      cap -= item.w;\n    } else {\n      // If the remaining capacity is insufficient, put part of the current item in the knapsack\n      res += item.v / item.w * cap;\n      // No capacity remains, so exit the loop\n      break;\n    }\n  }\n  return res;\n}\n</code></pre> fractional_knapsack.rs<pre><code>/* Item */\nstruct Item {\n    w: i32, // Item weight\n    v: i32, // Item value\n}\n\nimpl Item {\n    fn new(w: i32, v: i32) -&gt; Self {\n        Self { w, v }\n    }\n}\n\n/* Fractional knapsack: greedy algorithm */\nfn fractional_knapsack(wgt: &amp;[i32], val: &amp;[i32], mut cap: i32) -&gt; f64 {\n    // Create a list of items, each with two attributes: weight and value\n    let mut items = wgt\n        .iter()\n        .zip(val.iter())\n        .map(|(&amp;w, &amp;v)| Item::new(w, v))\n        .collect::&lt;Vec&lt;Item&gt;&gt;();\n    // Sort by value per unit weight, item.v / item.w, in descending order\n    items.sort_by(|a, b| {\n        (b.v as f64 / b.w as f64)\n            .partial_cmp(&amp;(a.v as f64 / a.w as f64))\n            .unwrap()\n    });\n    // Repeatedly make greedy choices\n    let mut res = 0.0;\n    for item in &amp;items {\n        if item.w &lt;= cap {\n            // If enough capacity remains, put the entire current item in the knapsack\n            res += item.v as f64;\n            cap -= item.w;\n        } else {\n            // If the remaining capacity is insufficient, put part of the current item in the knapsack\n            res += item.v as f64 / item.w as f64 * cap as f64;\n            // No capacity remains, so exit the loop\n            break;\n        }\n    }\n    res\n}\n</code></pre> fractional_knapsack.c<pre><code>/* Item */\ntypedef struct {\n    int w; // Item weight\n    int v; // Item value\n} Item;\n\n/* Fractional knapsack: greedy algorithm */\nfloat fractionalKnapsack(int wgt[], int val[], int itemCount, int cap) {\n    // Create a list of items, each with two attributes: weight and value\n    Item *items = malloc(sizeof(Item) * itemCount);\n    for (int i = 0; i &lt; itemCount; i++) {\n        items[i] = (Item){.w = wgt[i], .v = val[i]};\n    }\n    // Sort by value per unit weight, item.v / item.w, in descending order\n    qsort(items, (size_t)itemCount, sizeof(Item), sortByValueDensity);\n    // Repeatedly make greedy choices\n    float res = 0.0;\n    for (int i = 0; i &lt; itemCount; i++) {\n        if (items[i].w &lt;= cap) {\n            // If enough capacity remains, put the entire current item in the knapsack\n            res += items[i].v;\n            cap -= items[i].w;\n        } else {\n            // If the remaining capacity is insufficient, put part of the current item in the knapsack\n            res += (float)cap / items[i].w * items[i].v;\n            cap = 0;\n            break;\n        }\n    }\n    free(items);\n    return res;\n}\n</code></pre> fractional_knapsack.kt<pre><code>/* Item */\nclass Item(\n    val w: Int, // Item weight\n    val v: Int  // Item value\n)\n\n/* Fractional knapsack: greedy algorithm */\nfun fractionalKnapsack(wgt: IntArray, _val: IntArray, c: Int): Double {\n    // Create a list of items, each with two attributes: weight and value\n    var cap = c\n    val items = arrayOfNulls&lt;Item&gt;(wgt.size)\n    for (i in wgt.indices) {\n        items[i] = Item(wgt[i], _val[i])\n    }\n    // Sort by value per unit weight, item.v / item.w, in descending order\n    items.sortBy { item: Item? -&gt; -(item!!.v.toDouble() / item.w) }\n    // Repeatedly make greedy choices\n    var res = 0.0\n    for (item in items) {\n        if (item!!.w &lt;= cap) {\n            // If enough capacity remains, put the entire current item in the knapsack\n            res += item.v\n            cap -= item.w\n        } else {\n            // If the remaining capacity is insufficient, put part of the current item in the knapsack\n            res += item.v.toDouble() / item.w * cap\n            // No capacity remains, so exit the loop\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.rb<pre><code>### Item ###\nclass Item\n  attr_accessor :w # Item weight\n  attr_accessor :v # Item value\n\n  def initialize(w, v)\n    @w = w\n    @v = v\n  end\nend\n\n### Fractional knapsack: greedy algorithm ###\ndef fractional_knapsack(wgt, val, cap)\n  # Create a list of items, each with two attributes: weight and value\n  items = wgt.each_with_index.map { |w, i| Item.new(w, val[i]) }\n  # Sort by value per unit weight, item.v / item.w, in descending order\n  items.sort! { |a, b| (b.v.to_f / b.w) &lt;=&gt; (a.v.to_f / a.w) }\n  # Repeatedly make greedy choices\n  res = 0\n  for item in items\n    if item.w &lt;= cap\n      # If enough capacity remains, put the entire current item in the knapsack\n      res += item.v\n      cap -= item.w\n    else\n      # If the remaining capacity is insufficient, put part of the current item in the knapsack\n      res += (item.v.to_f / item.w) * cap\n      # No capacity remains, so exit the loop\n      break\n    end\n  end\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Built-in sorting algorithms typically take \\(O(n \\log n)\\) time and \\(O(\\log n)\\) or \\(O(n)\\) space, depending on the programming language's implementation.</p> <p>Apart from sorting, the worst case requires traversing the entire item list, so the time complexity is \\(O(n)\\), where \\(n\\) is the number of items.</p> <p>Initializing a list of <code>Item</code> objects requires \\(O(n)\\) space.</p>","path":["Chapter 15.   Greedy algorithms","15.2   Fractional knapsack problem"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/#3-proof-of-correctness","level":3,"title":"3.   Proof of correctness","text":"<p>We use proof by contradiction. Suppose item \\(x\\) has the highest unit value, and an algorithm produces a maximum value of <code>res</code>, but its solution does not include item \\(x\\).</p> <p>Now remove one unit of weight of any item from the knapsack and replace it with one unit of weight of item \\(x\\). Since \\(x\\) has the highest unit value, the total value after replacement must exceed <code>res</code>. This contradicts the assumption that <code>res</code> is optimal, showing that an optimal solution must include item \\(x\\).</p> <p>We can construct the same contradiction for other items in the solution. Thus, an item with a higher unit value is always a better choice, which establishes the greedy strategy.</p> <p>As shown in Figure 15-6, if we plot item weight on the horizontal axis and unit value on the vertical axis, the fractional knapsack problem becomes a problem of maximizing the area within a limited horizontal interval. This analogy provides a geometric view of why the greedy strategy works.</p> <p></p> <p> Figure 15-6   Geometric representation of the fractional knapsack problem </p>","path":["Chapter 15.   Greedy algorithms","15.2   Fractional knapsack problem"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/","level":1,"title":"15.1   Greedy algorithms","text":"<p>A greedy algorithm is a common approach to optimization problems. At each decision point, it chooses the option that looks best at that moment: it makes a locally optimal choice in the hope of reaching a globally optimal solution. Greedy algorithms are simple and efficient, with many practical applications.</p> <p>Both greedy algorithms and dynamic programming are commonly used for optimization. They share some features, such as relying on optimal substructure, but work differently.</p> <ul> <li>Dynamic programming considers the current decision in light of all decisions from earlier stages, using solutions to earlier subproblems to build the solution to the current subproblem.</li> <li>A greedy algorithm does not reconsider past decisions. It keeps moving forward, making greedy choices and shrinking the problem until it is solved.</li> </ul> <p>Let's explore how greedy algorithms work through the coin change problem, which you have already encountered in the section on the unbounded knapsack problem.</p> <p>Question</p> <p>Given \\(n\\) coin denominations, where denomination \\(i\\) has value \\(coins[i - 1]\\), and a target amount \\(amt\\), you may use each denomination any number of times. What is the minimum number of coins needed to make the target amount? Return \\(-1\\) if it cannot be made.</p> <p>The greedy strategy is shown in Figure 15-1. Given the remaining target amount, we greedily choose the largest coin that does not exceed it, repeating this step until we have made the target amount.</p> <p></p> <p> Figure 15-1   Greedy strategy for coin change </p> <p>The implementation is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change_greedy.py<pre><code>def coin_change_greedy(coins: list[int], amt: int) -&gt; int:\n    \"\"\"Coin change: greedy algorithm\"\"\"\n    # Assume the coins list is sorted\n    i = len(coins) - 1\n    count = 0\n    # Repeatedly make greedy choices until no amount remains\n    while amt &gt; 0:\n        # Find the largest coin no greater than the remaining amount\n        while i &gt; 0 and coins[i] &gt; amt:\n            i -= 1\n        # Choose coins[i]\n        amt -= coins[i]\n        count += 1\n    # Return -1 if no feasible solution was found\n    return count if amt == 0 else -1\n</code></pre> coin_change_greedy.cpp<pre><code>/* Coin change: greedy algorithm */\nint coinChangeGreedy(vector&lt;int&gt; &amp;coins, int amt) {\n    // Assume the coins list is sorted\n    int i = coins.size() - 1;\n    int count = 0;\n    // Repeatedly make greedy choices until no amount remains\n    while (amt &gt; 0) {\n        // Find the largest coin that does not exceed the remaining amount\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // Choose coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // Return -1 if no feasible solution was found\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.java<pre><code>/* Coin change: greedy algorithm */\nint coinChangeGreedy(int[] coins, int amt) {\n    // Assume the coins list is sorted\n    int i = coins.length - 1;\n    int count = 0;\n    // Repeatedly make greedy choices until no amount remains\n    while (amt &gt; 0) {\n        // Find the largest coin no greater than the remaining amount\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // Choose coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // Return -1 if no feasible solution was found\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.cs<pre><code>/* Coin change: greedy algorithm */\nint CoinChangeGreedy(int[] coins, int amt) {\n    // Assume the coins list is sorted\n    int i = coins.Length - 1;\n    int count = 0;\n    // Repeatedly make greedy choices until no amount remains\n    while (amt &gt; 0) {\n        // Find the largest coin no greater than the remaining amount\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // Choose coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // Return -1 if no feasible solution was found\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.go<pre><code>/* Coin change: greedy algorithm */\nfunc coinChangeGreedy(coins []int, amt int) int {\n    // Assume the coins list is sorted in ascending order\n    i := len(coins) - 1\n    count := 0\n    // Repeatedly make greedy choices until no amount remains\n    for amt &gt; 0 {\n        // Find the largest coin no greater than the remaining amount\n        for i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i--\n        }\n        // Choose coins[i]\n        amt -= coins[i]\n        count++\n    }\n    // Return -1 if no feasible solution was found\n    if amt != 0 {\n        return -1\n    }\n    return count\n}\n</code></pre> coin_change_greedy.swift<pre><code>/* Coin change: greedy algorithm */\nfunc coinChangeGreedy(coins: [Int], amt: Int) -&gt; Int {\n    // Assume the coins list is sorted in ascending order\n    var i = coins.count - 1\n    var count = 0\n    var amt = amt\n    // Repeatedly make greedy choices until no amount remains\n    while amt &gt; 0 {\n        // Find the largest coin no greater than the remaining amount\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1\n        }\n        // Choose coins[i]\n        amt -= coins[i]\n        count += 1\n    }\n    // Return -1 if no feasible solution was found\n    return amt == 0 ? count : -1\n}\n</code></pre> coin_change_greedy.js<pre><code>/* Coin change: greedy algorithm */\nfunction coinChangeGreedy(coins, amt) {\n    // Assume coins is sorted in ascending order\n    let i = coins.length - 1;\n    let count = 0;\n    // Repeatedly make greedy choices until no amount remains\n    while (amt &gt; 0) {\n        // Find the largest coin no greater than the remaining amount\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // Choose coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // Return -1 if no feasible solution was found\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.ts<pre><code>/* Coin change: greedy algorithm */\nfunction coinChangeGreedy(coins: number[], amt: number): number {\n    // Assume coins is sorted in ascending order\n    let i = coins.length - 1;\n    let count = 0;\n    // Repeatedly make greedy choices until no amount remains\n    while (amt &gt; 0) {\n        // Find the largest coin no greater than the remaining amount\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // Choose coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // Return -1 if no feasible solution was found\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.dart<pre><code>/* Coin change: greedy algorithm */\nint coinChangeGreedy(List&lt;int&gt; coins, int amt) {\n  // Assume the coins list is sorted in ascending order\n  int i = coins.length - 1;\n  int count = 0;\n  // Repeatedly make greedy choices until no amount remains\n  while (amt &gt; 0) {\n    // Find the largest coin no greater than the remaining amount\n    while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n      i--;\n    }\n    // Choose coins[i]\n    amt -= coins[i];\n    count++;\n  }\n  // Return -1 if no feasible solution was found\n  return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.rs<pre><code>/* Coin change: greedy algorithm */\nfn coin_change_greedy(coins: &amp;[i32], mut amt: i32) -&gt; i32 {\n    // Assume the coins list is sorted\n    let mut i = coins.len() - 1;\n    let mut count = 0;\n    // Repeatedly make greedy choices until no amount remains\n    while amt &gt; 0 {\n        // Find the largest coin that does not exceed the remaining amount\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1;\n        }\n        // Choose coins[i]\n        amt -= coins[i];\n        count += 1;\n    }\n    // Return -1 if no feasible solution was found\n    if amt == 0 {\n        count\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change_greedy.c<pre><code>/* Coin change: greedy algorithm */\nint coinChangeGreedy(int *coins, int size, int amt) {\n    // Assume the coins list is sorted\n    int i = size - 1;\n    int count = 0;\n    // Repeatedly make greedy choices until no amount remains\n    while (amt &gt; 0) {\n        // Find the largest coin no greater than the remaining amount\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // Choose coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // Return -1 if no feasible solution was found\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.kt<pre><code>/* Coin change: greedy algorithm */\nfun coinChangeGreedy(coins: IntArray, amt: Int): Int {\n    // Assume coins is sorted in ascending order\n    var am = amt\n    var i = coins.size - 1\n    var count = 0\n    // Repeatedly make greedy choices until no amount remains\n    while (am &gt; 0) {\n        // Find the largest coin no greater than the remaining amount\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; am) {\n            i--\n        }\n        // Choose coins[i]\n        am -= coins[i]\n        count++\n    }\n    // Return -1 if no feasible solution was found\n    return if (am == 0) count else -1\n}\n</code></pre> coin_change_greedy.rb<pre><code>### Coin change: greedy algorithm ###\ndef coin_change_greedy(coins, amt)\n  # Assume the coins list is sorted in ascending order\n  i = coins.length - 1\n  count = 0\n  # Repeatedly make greedy choices until no amount remains\n  while amt &gt; 0\n    # Find the largest coin no greater than the remaining amount\n    while i &gt; 0 &amp;&amp; coins[i] &gt; amt\n      i -= 1\n    end\n    # Choose coins[i]\n    amt -= coins[i]\n    count += 1\n  end\n  # Return -1 if no feasible solution was found\n  amt == 0 ? count : -1\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>You might find yourself thinking, \"So clean!\" The greedy algorithm solves coin change in only about ten lines of code.</p>","path":["Chapter 15.   Greedy algorithms","15.1   Greedy algorithms"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1511-strengths-and-limitations-of-greedy-algorithms","level":2,"title":"15.1.1   Strengths and limitations of greedy algorithms","text":"<p>Greedy algorithms are straightforward, easy to implement, and usually efficient. In the code above, let \\(\\min(coins)\\) be the smallest coin denomination. At most \\(amt / \\min(coins)\\) greedy choices are made, giving a time complexity of \\(O(amt / \\min(coins))\\). This is an order of magnitude lower than the dynamic programming solution's \\(O(n \\times amt)\\) time complexity.</p> <p>However, for some sets of coin denominations, a greedy algorithm cannot find the optimal solution. Figure 15-2 illustrates two examples.</p> <ul> <li>Successful case: \\(coins = [1, 5, 10, 20, 50, 100]\\). For these denominations, the greedy algorithm finds an optimal solution for any \\(amt\\).</li> <li>Counterexample: \\(coins = [1, 20, 50]\\). For \\(amt = 60\\), the greedy algorithm produces \\(50 + 1 \\times 10\\), using \\(11\\) coins. Dynamic programming finds the optimal solution \\(20 + 20 + 20\\), using only \\(3\\) coins.</li> <li>Counterexample: \\(coins = [1, 49, 50]\\). For \\(amt = 98\\), the greedy algorithm produces \\(50 + 1 \\times 48\\), using \\(49\\) coins. Dynamic programming finds the optimal solution \\(49 + 49\\), using only \\(2\\) coins.</li> </ul> <p></p> <p> Figure 15-2   Examples where the greedy algorithm fails to find the optimal solution </p> <p>In other words, a greedy algorithm cannot guarantee a globally optimal solution to coin change and may produce a very poor solution. Dynamic programming is better suited to this problem.</p> <p>In general, greedy algorithms are useful in two situations.</p> <ol> <li>An optimal solution is guaranteed: A greedy algorithm is often the best choice because it is usually more efficient than backtracking or dynamic programming.</li> <li>A near-optimal solution can be found: A greedy algorithm can still be useful. For many complex problems, finding a globally optimal solution is very difficult, so finding a good suboptimal solution efficiently is valuable.</li> </ol>","path":["Chapter 15.   Greedy algorithms","15.1   Greedy algorithms"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1512-properties-required-by-greedy-algorithms","level":2,"title":"15.1.2   Properties required by greedy algorithms","text":"<p>What kinds of problems are suitable for greedy algorithms? More specifically, when can a greedy algorithm guarantee an optimal solution?</p> <p>Greedy algorithms have stricter requirements than dynamic programming. Two properties are particularly important.</p> <ul> <li>Greedy-choice property: A greedy algorithm can guarantee an optimal solution only when locally optimal choices always lead to a globally optimal solution.</li> <li>Optimal substructure: An optimal solution to the original problem contains optimal solutions to its subproblems.</li> </ul> <p>Optimal substructure was covered in the dynamic programming chapter, so we will not repeat that discussion here. Some problems have optimal substructure that is not immediately apparent, yet can still be solved greedily.</p> <p>We will focus on how to determine whether the greedy-choice property holds. Its description may sound simple, but proving it is difficult for many problems.</p> <p>For coin change, for example, it is easy to give counterexamples that disprove the greedy-choice property, but proving it is harder. If asked which conditions on the coin denominations allow a greedy solution, we may be able to give only a vague answer based on intuition or examples, rather than a rigorous mathematical proof.</p> <p>Quote</p> <p>One paper gives an \\(O(n^3)\\)-time algorithm for determining whether a set of coin denominations allows a greedy algorithm to find an optimal solution for every amount.</p> <p>Pearson, D. A polynomial-time algorithm for the change-making problem[J]. Operations Research Letters, 2005, 33(3): 231-234.</p>","path":["Chapter 15.   Greedy algorithms","15.1   Greedy algorithms"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1513-steps-for-solving-a-problem-greedily","level":2,"title":"15.1.3   Steps for solving a problem greedily","text":"<p>Solving a greedy problem generally involves three steps.</p> <ol> <li>Analyze the problem: Understand its properties, including the state definition, optimization objective, and constraints. This step also appears in backtracking and dynamic programming.</li> <li>Choose a greedy strategy: Decide how to make a greedy choice at each step. Each choice should reduce the problem's size, eventually solving the whole problem.</li> <li>Prove correctness: Usually, we must establish the greedy-choice property and optimal substructure. This may require a mathematical proof, such as induction or proof by contradiction.</li> </ol> <p>Choosing the greedy strategy is the central step, but it can be difficult for the following reasons.</p> <ul> <li>Greedy strategies vary greatly from problem to problem. For many problems, the strategy is straightforward and can be found with a little thought and experimentation. For more complex problems, it may be well hidden, testing our problem-solving experience and algorithmic skills.</li> <li>Some greedy strategies are misleading. We may confidently devise a strategy, implement it, and submit the code, only to find that it fails some test cases. The strategy is only partially correct; the coin change problem above is a typical example.</li> </ul> <p>To ensure correctness, we should give the greedy strategy a rigorous mathematical proof, usually using contradiction or mathematical induction.</p> <p>However, such a proof can itself be difficult. When we do not know where to start, we often debug against test cases, gradually refining and checking the greedy strategy.</p>","path":["Chapter 15.   Greedy algorithms","15.1   Greedy algorithms"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1514-typical-greedy-problems","level":2,"title":"15.1.4   Typical greedy problems","text":"<p>Greedy algorithms often solve optimization problems with the greedy-choice property and optimal substructure. Some typical examples follow.</p> <ul> <li>Coin change: For certain sets of coin denominations, a greedy algorithm always finds an optimal solution.</li> <li>Interval scheduling: Given tasks that each occupy a time interval, the goal is to complete as many tasks as possible. Always choosing the available task that finishes earliest yields an optimal solution.</li> <li>Fractional knapsack: Given items and a weight limit, choose items to maximize total value without exceeding the limit. Choosing the item with the highest value-to-weight ratio at each step can yield an optimal solution in some settings.</li> <li>Stock trading: Given historical stock prices, you may buy and sell multiple times, but cannot buy again while holding stock until you sell it. The goal is to maximize profit.</li> <li>Huffman coding: This greedy algorithm is used for lossless data compression. It builds a Huffman tree by repeatedly merging the two nodes with the lowest frequencies. The resulting tree has the minimum weighted path length (encoding length).</li> <li>Dijkstra's algorithm: This greedy algorithm finds shortest paths from a given source vertex to every other vertex in a graph whose edge weights are all nonnegative.</li> </ul>","path":["Chapter 15.   Greedy algorithms","15.1   Greedy algorithms"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/","level":1,"title":"15.3   Maximum capacity problem","text":"<p>Question</p> <p>Given an array \\(ht\\), each element represents the height of a vertical wall. Any two walls, together with the space between them, form a container.</p> <p>The container's capacity is its height multiplied by its width (an area). Its height is determined by the shorter wall, and its width is the difference between the two walls' array indices.</p> <p>Choose two walls from the array to maximize the container's capacity, and return that maximum capacity. An example is shown in Figure 15-7.</p> <p></p> <p> Figure 15-7   Example input for the maximum capacity problem </p> <p>A container is formed by two walls, so the state of this problem is the pair of wall indices, denoted by \\([i, j]\\).</p> <p>Capacity is height times width, with the height determined by the shorter wall and the width equal to the difference between the wall indices. Writing the capacity as \\(cap[i, j]\\), we have:</p> \\[ cap[i, j] = \\min(ht[i], ht[j]) \\times (j - i) \\] <p>For an array of length \\(n\\), there are \\(C_n^2 = \\frac{n(n - 1)}{2}\\) pairs of walls (possible states). The most direct approach is to enumerate all states and find the maximum capacity, taking \\(O(n^2)\\) time.</p>","path":["Chapter 15.   Greedy algorithms","15.3   Maximum capacity problem"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/#1-choosing-a-greedy-strategy","level":3,"title":"1.   Choosing a greedy strategy","text":"<p>A more efficient solution is possible. Consider the state \\([i, j]\\) in Figure 15-8, where \\(i &lt; j\\) and \\(ht[i] &lt; ht[j]\\): wall \\(i\\) is shorter and wall \\(j\\) is taller.</p> <p></p> <p> Figure 15-8   Initial state </p> <p>As shown in Figure 15-9, moving the taller wall \\(j\\) inward toward the shorter wall \\(i\\) must reduce the capacity.</p> <p>Moving wall \\(j\\) inward always reduces the width \\(j-i\\). Since the shorter wall determines the height, the height can only stay the same (if \\(i\\) remains shorter) or decrease (if the new wall \\(j\\) is shorter).</p> <p></p> <p> Figure 15-9   State after moving the taller wall inward </p> <p>Conversely, only moving the shorter wall \\(i\\) inward can potentially increase the capacity. Although the width must decrease, the height may increase if the new wall \\(i\\) is taller. Figure 15-10 shows an example where moving the shorter wall increases the area.</p> <p></p> <p> Figure 15-10   State after moving the shorter wall inward </p> <p>This gives us a greedy strategy: initialize two pointers at opposite ends of the container. At each step, move the pointer at the shorter wall inward until the pointers meet.</p> <p>Figure 15-11 illustrates the greedy process.</p> <ol> <li>Initially, pointers \\(i\\) and \\(j\\) are at opposite ends of the array.</li> <li>Calculate the current capacity \\(cap[i, j]\\) and update the maximum capacity.</li> <li>Compare the heights of walls \\(i\\) and \\(j\\), and move the shorter wall inward by one position.</li> <li>Repeat steps <code>2.</code> and <code>3.</code> until \\(i\\) and \\(j\\) meet.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 15-11   Greedy steps for the maximum capacity problem </p>","path":["Chapter 15.   Greedy algorithms","15.3   Maximum capacity problem"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/#2-code-implementation","level":3,"title":"2.   Code implementation","text":"<p>The loop runs at most \\(n\\) times, so the time complexity is \\(O(n)\\).</p> <p>The variables \\(i\\), \\(j\\), and \\(res\\) require constant extra space, so the space complexity is \\(O(1)\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby max_capacity.py<pre><code>def max_capacity(ht: list[int]) -&gt; int:\n    \"\"\"Maximum container capacity: greedy algorithm\"\"\"\n    # Initialize i and j at opposite ends of the array\n    i, j = 0, len(ht) - 1\n    # The initial maximum capacity is 0\n    res = 0\n    # Repeatedly make greedy choices until the two sides meet\n    while i &lt; j:\n        # Update the maximum capacity\n        cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        # Move the shorter side inward\n        if ht[i] &lt; ht[j]:\n            i += 1\n        else:\n            j -= 1\n    return res\n</code></pre> max_capacity.cpp<pre><code>/* Maximum container capacity: greedy algorithm */\nint maxCapacity(vector&lt;int&gt; &amp;ht) {\n    // Initialize i and j at opposite ends of the array\n    int i = 0, j = ht.size() - 1;\n    // The initial maximum capacity is 0\n    int res = 0;\n    // Repeatedly make greedy choices until the two sides meet\n    while (i &lt; j) {\n        // Update the maximum capacity\n        int cap = min(ht[i], ht[j]) * (j - i);\n        res = max(res, cap);\n        // Move the shorter side inward\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.java<pre><code>/* Maximum container capacity: greedy algorithm */\nint maxCapacity(int[] ht) {\n    // Initialize i and j at opposite ends of the array\n    int i = 0, j = ht.length - 1;\n    // The initial maximum capacity is 0\n    int res = 0;\n    // Repeatedly make greedy choices until the two sides meet\n    while (i &lt; j) {\n        // Update the maximum capacity\n        int cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // Move the shorter side inward\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.cs<pre><code>/* Maximum container capacity: greedy algorithm */\nint MaxCapacity(int[] ht) {\n    // Initialize i and j at opposite ends of the array\n    int i = 0, j = ht.Length - 1;\n    // The initial maximum capacity is 0\n    int res = 0;\n    // Repeatedly make greedy choices until the two sides meet\n    while (i &lt; j) {\n        // Update the maximum capacity\n        int cap = Math.Min(ht[i], ht[j]) * (j - i);\n        res = Math.Max(res, cap);\n        // Move the shorter side inward\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.go<pre><code>/* Maximum container capacity: greedy algorithm */\nfunc maxCapacity(ht []int) int {\n    // Initialize i and j at opposite ends of the array\n    i, j := 0, len(ht)-1\n    // The initial maximum capacity is 0\n    res := 0\n    // Repeatedly make greedy choices until the two sides meet\n    for i &lt; j {\n        // Update the maximum capacity\n        capacity := int(math.Min(float64(ht[i]), float64(ht[j]))) * (j - i)\n        res = int(math.Max(float64(res), float64(capacity)))\n        // Move the shorter side inward\n        if ht[i] &lt; ht[j] {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.swift<pre><code>/* Maximum container capacity: greedy algorithm */\nfunc maxCapacity(ht: [Int]) -&gt; Int {\n    // Initialize i and j at opposite ends of the array\n    var i = ht.startIndex, j = ht.endIndex - 1\n    // The initial maximum capacity is 0\n    var res = 0\n    // Repeatedly make greedy choices until the two sides meet\n    while i &lt; j {\n        // Update the maximum capacity\n        let cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        // Move the shorter side inward\n        if ht[i] &lt; ht[j] {\n            i += 1\n        } else {\n            j -= 1\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.js<pre><code>/* Maximum container capacity: greedy algorithm */\nfunction maxCapacity(ht) {\n    // Initialize i and j at opposite ends of the array\n    let i = 0,\n        j = ht.length - 1;\n    // The initial maximum capacity is 0\n    let res = 0;\n    // Repeatedly make greedy choices until the two sides meet\n    while (i &lt; j) {\n        // Update the maximum capacity\n        const cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // Move the shorter side inward\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.ts<pre><code>/* Maximum container capacity: greedy algorithm */\nfunction maxCapacity(ht: number[]): number {\n    // Initialize i and j at opposite ends of the array\n    let i = 0,\n        j = ht.length - 1;\n    // The initial maximum capacity is 0\n    let res = 0;\n    // Repeatedly make greedy choices until the two sides meet\n    while (i &lt; j) {\n        // Update the maximum capacity\n        const cap: number = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // Move the shorter side inward\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.dart<pre><code>/* Maximum container capacity: greedy algorithm */\nint maxCapacity(List&lt;int&gt; ht) {\n  // Initialize i and j at opposite ends of the array\n  int i = 0, j = ht.length - 1;\n  // The initial maximum capacity is 0\n  int res = 0;\n  // Repeatedly make greedy choices until the two sides meet\n  while (i &lt; j) {\n    // Update the maximum capacity\n    int cap = min(ht[i], ht[j]) * (j - i);\n    res = max(res, cap);\n    // Move the shorter side inward\n    if (ht[i] &lt; ht[j]) {\n      i++;\n    } else {\n      j--;\n    }\n  }\n  return res;\n}\n</code></pre> max_capacity.rs<pre><code>/* Maximum container capacity: greedy algorithm */\nfn max_capacity(ht: &amp;[i32]) -&gt; i32 {\n    // Initialize i and j at opposite ends of the array\n    let mut i = 0;\n    let mut j = ht.len() - 1;\n    // The initial maximum capacity is 0\n    let mut res = 0;\n    // Repeatedly make greedy choices until the two sides meet\n    while i &lt; j {\n        // Update the maximum capacity\n        let cap = std::cmp::min(ht[i], ht[j]) * (j - i) as i32;\n        res = std::cmp::max(res, cap);\n        // Move the shorter side inward\n        if ht[i] &lt; ht[j] {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    res\n}\n</code></pre> max_capacity.c<pre><code>/* Maximum container capacity: greedy algorithm */\nint maxCapacity(int ht[], int htLength) {\n    // Initialize i and j at opposite ends of the array\n    int i = 0;\n    int j = htLength - 1;\n    // The initial maximum capacity is 0\n    int res = 0;\n    // Repeatedly make greedy choices until the two sides meet\n    while (i &lt; j) {\n        // Update the maximum capacity\n        int capacity = myMin(ht[i], ht[j]) * (j - i);\n        res = myMax(res, capacity);\n        // Move the shorter side inward\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.kt<pre><code>/* Maximum container capacity: greedy algorithm */\nfun maxCapacity(ht: IntArray): Int {\n    // Initialize i and j at opposite ends of the array\n    var i = 0\n    var j = ht.size - 1\n    // The initial maximum capacity is 0\n    var res = 0\n    // Repeatedly make greedy choices until the two sides meet\n    while (i &lt; j) {\n        // Update the maximum capacity\n        val cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        // Move the shorter side inward\n        if (ht[i] &lt; ht[j]) {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.rb<pre><code>### Maximum container capacity: greedy algorithm ###\ndef max_capacity(ht)\n  # Initialize i and j at opposite ends of the array\n  i, j = 0, ht.length - 1\n  # The initial maximum capacity is 0\n  res = 0\n\n  # Repeatedly make greedy choices until the two sides meet\n  while i &lt; j\n    # Update the maximum capacity\n    cap = [ht[i], ht[j]].min * (j - i)\n    res = [res, cap].max\n    # Move the shorter side inward\n    if ht[i] &lt; ht[j]\n      i += 1\n    else\n      j -= 1\n    end\n  end\n\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 15.   Greedy algorithms","15.3   Maximum capacity problem"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/#3-proof-of-correctness","level":3,"title":"3.   Proof of correctness","text":"<p>The greedy algorithm is faster than exhaustive search because each greedy choice skips some states.</p> <p>For example, in state \\(cap[i, j]\\), suppose \\(i\\) is the shorter wall and \\(j\\) the taller one. Moving the shorter wall \\(i\\) inward by one position skips the states shown in Figure 15-12. Their capacities will never be checked later.</p> \\[ cap[i, i+1], cap[i, i+2], \\dots, cap[i, j-2], cap[i, j-1] \\] <p></p> <p> Figure 15-12   States skipped by moving the shorter wall </p> <p>Observe that the skipped states are precisely those obtained by moving the taller wall \\(j\\) inward. We have already shown that moving the taller wall inward must reduce capacity. None of these skipped states can therefore be optimal, and skipping them cannot cause us to miss the optimal solution.</p> <p>This shows that moving the shorter wall is safe, and that the greedy strategy is valid.</p>","path":["Chapter 15.   Greedy algorithms","15.3   Maximum capacity problem"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/","level":1,"title":"15.4   Maximum product cutting problem","text":"<p>Question</p> <p>Given a positive integer \\(n\\), split it into a sum of at least two positive integers. What is the largest possible product of those integers? The problem is illustrated in Figure 15-13.</p> <p></p> <p> Figure 15-13   Definition of the maximum product cutting problem </p> <p>Suppose we split \\(n\\) into \\(m\\) integers that will be factors in the product. Denote the \\(i\\)-th factor by \\(n_i\\), so that</p> \\[ n = \\sum_{i=1}^{m}n_i \\] <p>Our goal is to maximize the product of all these integer factors:</p> \\[ \\max(\\prod_{i=1}^{m}n_i) \\] <p>We need to determine how many parts \\(m\\) to create and what each \\(n_i\\) should be.</p>","path":["Chapter 15.   Greedy algorithms","15.4   Maximum product cutting problem"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/#1-choosing-a-greedy-strategy","level":3,"title":"1.   Choosing a greedy strategy","text":"<p>The product of two integers is often greater than their sum. Suppose we split off a factor of \\(2\\) from \\(n\\), giving the product \\(2(n-2)\\). Comparing this product with \\(n\\) gives:</p> \\[ \\begin{aligned} 2(n-2) &amp; \\geq n \\newline 2n - n - 4 &amp; \\geq 0 \\newline n &amp; \\geq 4 \\end{aligned} \\] <p>As shown in Figure 15-14, when \\(n \\geq 4\\), splitting off a \\(2\\) increases the product. This suggests that every integer greater than or equal to \\(4\\) should be split further.</p> <p>Greedy strategy 1: If a factor \\(\\geq 4\\) appears in a split, divide it further. The final split should contain only the factors \\(1\\), \\(2\\), and \\(3\\).</p> <p></p> <p> Figure 15-14   Splitting increases the product </p> <p>Next, consider which factor is best. Of \\(1\\), \\(2\\), and \\(3\\), clearly \\(1\\) is the worst: \\(1 \\times (n-1) &lt; n\\) always holds, so splitting off a \\(1\\) reduces the product.</p> <p>As shown in Figure 15-15, when \\(n = 6\\), we have \\(3 \\times 3 &gt; 2 \\times 2 \\times 2\\). This means that splitting into factors of \\(3\\) is better than splitting into factors of \\(2\\).</p> <p>Greedy strategy 2: A split should contain at most two factors of \\(2\\), because three \\(2\\)s can always be replaced with two \\(3\\)s to obtain a larger product.</p> <p></p> <p> Figure 15-15   Optimal factors </p> <p>These observations lead to the following greedy strategy.</p> <ol> <li>Starting with \\(n\\), keep splitting off factors of \\(3\\) until the remainder is \\(0\\), \\(1\\), or \\(2\\).</li> <li>If the remainder is \\(0\\), then \\(n\\) is a multiple of \\(3\\), and no further action is needed.</li> <li>If the remainder is \\(2\\), keep it without splitting it further.</li> <li>If the remainder is \\(1\\), replace the last \\(3\\) and the remaining \\(1\\) with two \\(2\\)s, since \\(2 \\times 2 &gt; 1 \\times 3\\).</li> </ol>","path":["Chapter 15.   Greedy algorithms","15.4   Maximum product cutting problem"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/#2-code-implementation","level":3,"title":"2.   Code implementation","text":"<p>As shown in Figure 15-16, we do not need a loop to split the integer. Floor division gives the number of \\(3\\)s, denoted by \\(a\\), and the modulo operation gives the remainder \\(b\\). Thus:</p> \\[ n = 3 a + b \\] <p>For the boundary cases \\(n \\leq 3\\), we must split off a \\(1\\), giving a product of \\(1 \\times (n - 1)\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby max_product_cutting.py<pre><code>def max_product_cutting(n: int) -&gt; int:\n    \"\"\"Maximum product after splitting: greedy algorithm\"\"\"\n    # When n &lt;= 3, one part must be 1\n    if n &lt;= 3:\n        return 1 * (n - 1)\n    # Greedily split off parts of size 3; a is the number of 3s and b is the remainder\n    a, b = n // 3, n % 3\n    if b == 1:\n        # When the remainder is 1, replace one pair 1 * 3 with 2 * 2\n        return int(math.pow(3, a - 1)) * 2 * 2\n    if b == 2:\n        # When the remainder is 2, no adjustment is needed\n        return int(math.pow(3, a)) * 2\n    # When the remainder is 0, no adjustment is needed\n    return int(math.pow(3, a))\n</code></pre> max_product_cutting.cpp<pre><code>/* Maximum product after splitting: greedy algorithm */\nint maxProductCutting(int n) {\n    // When n &lt;= 3, one part must be 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // Greedily split off parts of size 3; a is the number of 3s and b is the remainder\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // When the remainder is 1, replace one pair 1 * 3 with 2 * 2\n        return (int)pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // When the remainder is 2, no adjustment is needed\n        return (int)pow(3, a) * 2;\n    }\n    // When the remainder is 0, no adjustment is needed\n    return (int)pow(3, a);\n}\n</code></pre> max_product_cutting.java<pre><code>/* Maximum product after splitting: greedy algorithm */\nint maxProductCutting(int n) {\n    // When n &lt;= 3, one part must be 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // Greedily split off parts of size 3; a is the number of 3s and b is the remainder\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // When the remainder is 1, replace one pair 1 * 3 with 2 * 2\n        return (int) Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // When the remainder is 2, no adjustment is needed\n        return (int) Math.pow(3, a) * 2;\n    }\n    // When the remainder is 0, no adjustment is needed\n    return (int) Math.pow(3, a);\n}\n</code></pre> max_product_cutting.cs<pre><code>/* Maximum product after splitting: greedy algorithm */\nint MaxProductCutting(int n) {\n    // When n &lt;= 3, one part must be 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // Greedily split off parts of size 3; a is the number of 3s and b is the remainder\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // When the remainder is 1, replace one pair 1 * 3 with 2 * 2\n        return (int)Math.Pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // When the remainder is 2, no adjustment is needed\n        return (int)Math.Pow(3, a) * 2;\n    }\n    // When the remainder is 0, no adjustment is needed\n    return (int)Math.Pow(3, a);\n}\n</code></pre> max_product_cutting.go<pre><code>/* Maximum product after splitting: greedy algorithm */\nfunc maxProductCutting(n int) int {\n    // When n &lt;= 3, one part must be 1\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // Greedily split off parts of size 3; a is the number of 3s and b is the remainder\n    a := n / 3\n    b := n % 3\n    if b == 1 {\n        // When the remainder is 1, replace one pair 1 * 3 with 2 * 2\n        return int(math.Pow(3, float64(a-1))) * 2 * 2\n    }\n    if b == 2 {\n        // When the remainder is 2, no adjustment is needed\n        return int(math.Pow(3, float64(a))) * 2\n    }\n    // When the remainder is 0, no adjustment is needed\n    return int(math.Pow(3, float64(a)))\n}\n</code></pre> max_product_cutting.swift<pre><code>/* Maximum product after splitting: greedy algorithm */\nfunc maxProductCutting(n: Int) -&gt; Int {\n    // When n &lt;= 3, one part must be 1\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // Greedily split off parts of size 3; a is the number of 3s and b is the remainder\n    let a = n / 3\n    let b = n % 3\n    if b == 1 {\n        // When the remainder is 1, replace one pair 1 * 3 with 2 * 2\n        return pow(3, a - 1) * 2 * 2\n    }\n    if b == 2 {\n        // When the remainder is 2, no adjustment is needed\n        return pow(3, a) * 2\n    }\n    // When the remainder is 0, no adjustment is needed\n    return pow(3, a)\n}\n</code></pre> max_product_cutting.js<pre><code>/* Maximum product after splitting: greedy algorithm */\nfunction maxProductCutting(n) {\n    // When n &lt;= 3, one part must be 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // Greedily split off parts of size 3; a is the number of 3s and b is the remainder\n    let a = Math.floor(n / 3);\n    let b = n % 3;\n    if (b === 1) {\n        // When the remainder is 1, replace one pair 1 * 3 with 2 * 2\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // When the remainder is 2, no adjustment is needed\n        return Math.pow(3, a) * 2;\n    }\n    // When the remainder is 0, no adjustment is needed\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.ts<pre><code>/* Maximum product after splitting: greedy algorithm */\nfunction maxProductCutting(n: number): number {\n    // When n &lt;= 3, one part must be 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // Greedily split off parts of size 3; a is the number of 3s and b is the remainder\n    let a: number = Math.floor(n / 3);\n    let b: number = n % 3;\n    if (b === 1) {\n        // When the remainder is 1, replace one pair 1 * 3 with 2 * 2\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // When the remainder is 2, no adjustment is needed\n        return Math.pow(3, a) * 2;\n    }\n    // When the remainder is 0, no adjustment is needed\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.dart<pre><code>/* Maximum product after splitting: greedy algorithm */\nint maxProductCutting(int n) {\n  // When n &lt;= 3, one part must be 1\n  if (n &lt;= 3) {\n    return 1 * (n - 1);\n  }\n  // Greedily split off parts of size 3; a is the number of 3s and b is the remainder\n  int a = n ~/ 3;\n  int b = n % 3;\n  if (b == 1) {\n    // When the remainder is 1, replace one pair 1 * 3 with 2 * 2\n    return (pow(3, a - 1) * 2 * 2).toInt();\n  }\n  if (b == 2) {\n    // When the remainder is 2, no adjustment is needed\n    return (pow(3, a) * 2).toInt();\n  }\n  // When the remainder is 0, no adjustment is needed\n  return pow(3, a).toInt();\n}\n</code></pre> max_product_cutting.rs<pre><code>/* Maximum product after splitting: greedy algorithm */\nfn max_product_cutting(n: i32) -&gt; i32 {\n    // When n &lt;= 3, one part must be 1\n    if n &lt;= 3 {\n        return 1 * (n - 1);\n    }\n    // Greedily split off parts of size 3; a is the number of 3s and b is the remainder\n    let a = n / 3;\n    let b = n % 3;\n    if b == 1 {\n        // When the remainder is 1, replace one pair 1 * 3 with 2 * 2\n        3_i32.pow(a as u32 - 1) * 2 * 2\n    } else if b == 2 {\n        // When the remainder is 2, no adjustment is needed\n        3_i32.pow(a as u32) * 2\n    } else {\n        // When the remainder is 0, no adjustment is needed\n        3_i32.pow(a as u32)\n    }\n}\n</code></pre> max_product_cutting.c<pre><code>/* Maximum product after splitting: greedy algorithm */\nint maxProductCutting(int n) {\n    // When n &lt;= 3, one part must be 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // Greedily split off parts of size 3; a is the number of 3s and b is the remainder\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // When the remainder is 1, replace one pair 1 * 3 with 2 * 2\n        return pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // When the remainder is 2, no adjustment is needed\n        return pow(3, a) * 2;\n    }\n    // When the remainder is 0, no adjustment is needed\n    return pow(3, a);\n}\n</code></pre> max_product_cutting.kt<pre><code>/* Maximum product after splitting: greedy algorithm */\nfun maxProductCutting(n: Int): Int {\n    // When n &lt;= 3, one part must be 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1)\n    }\n    // Greedily split off parts of size 3; a is the number of 3s and b is the remainder\n    val a = n / 3\n    val b = n % 3\n    if (b == 1) {\n        // When the remainder is 1, replace one pair 1 * 3 with 2 * 2\n        return 3.0.pow((a - 1)).toInt() * 2 * 2\n    }\n    if (b == 2) {\n        // When the remainder is 2, no adjustment is needed\n        return 3.0.pow(a).toInt() * 2 * 2\n    }\n    // When the remainder is 0, no adjustment is needed\n    return 3.0.pow(a).toInt()\n}\n</code></pre> max_product_cutting.rb<pre><code>### Maximum product after splitting: greedy algorithm ###\ndef max_product_cutting(n)\n  # When n &lt;= 3, one part must be 1\n  return 1 * (n - 1) if n &lt;= 3\n  # Greedily split off parts of size 3; a is the number of 3s and b is the remainder\n  a, b = n / 3, n % 3\n  # When the remainder is 1, replace one pair 1 * 3 with 2 * 2\n  return (3.pow(a - 1) * 2 * 2).to_i if b == 1\n  # When the remainder is 2, no adjustment is needed\n  return (3.pow(a) * 2).to_i if b == 2\n  # When the remainder is 0, no adjustment is needed\n  3.pow(a).to_i\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p></p> <p> Figure 15-16   Calculating the maximum product </p> <p>The time complexity depends on how the programming language implements exponentiation. Python, for example, offers three common ways to compute powers.</p> <ul> <li>The <code>**</code> operator and the <code>pow()</code> function both have a time complexity of \\(O(\\log⁡ a)\\).</li> <li>The <code>math.pow()</code> function calls the C library's <code>pow()</code> function internally, performing floating-point exponentiation in \\(O(1)\\) time.</li> </ul> <p>The variables \\(a\\) and \\(b\\) use constant extra space, so the space complexity is \\(O(1)\\).</p>","path":["Chapter 15.   Greedy algorithms","15.4   Maximum product cutting problem"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/#3-proof-of-correctness","level":3,"title":"3.   Proof of correctness","text":"<p>We use proof by contradiction, considering only \\(n \\geq 4\\).</p> <ol> <li>All factors are \\(\\leq 3\\): Suppose an optimal split contains a factor \\(x\\) that is \\(\\geq 4\\). We can split it further to obtain \\(2(x-2)\\), giving a greater (or equal) product. This contradicts the assumption.</li> <li>The split contains no \\(1\\): Suppose an optimal split contains a factor of \\(1\\). We can merge it into another factor to obtain a larger product, contradicting the assumption.</li> <li>The split contains at most two \\(2\\)s: Suppose an optimal split contains three \\(2\\)s. Replacing them with two \\(3\\)s gives a larger product, contradicting the assumption.</li> </ol>","path":["Chapter 15.   Greedy algorithms","15.4   Maximum product cutting problem"],"tags":[]},{"location":"chapter_greedy/summary/","level":1,"title":"15.5   Summary","text":"","path":["Chapter 15.   Greedy algorithms","15.5   Summary"],"tags":[]},{"location":"chapter_greedy/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<ul> <li>Greedy algorithms are commonly used for optimization. At each decision point, they make a locally optimal choice in the hope of reaching a globally optimal solution.</li> <li>A greedy algorithm repeatedly makes greedy choices, turning the problem into a smaller subproblem at each step until it is solved.</li> <li>Greedy algorithms are easy to implement and efficient. Their time complexity is usually lower than that of dynamic programming.</li> <li>For some coin denominations, a greedy algorithm is guaranteed to find an optimal coin change solution. For others, it may produce a very poor solution.</li> <li>Problems suited to greedy algorithms have two properties: the greedy-choice property and optimal substructure. The greedy-choice property establishes that the strategy works.</li> <li>Proving the greedy-choice property can be difficult for complex problems. Disproving it is often easier, as coin change illustrates.</li> <li>Solving a greedy problem involves three main steps: analyze the problem, choose a greedy strategy, and prove correctness. Choosing the strategy is the central step, while proving correctness is often the hardest.</li> <li>The fractional knapsack problem extends 0-1 knapsack by allowing fractions of items, making a greedy solution possible. The strategy can be proved correct by contradiction.</li> <li>The maximum capacity problem can be solved by exhaustive search in \\(O(n^2)\\) time. A greedy strategy that moves the shorter wall inward at each step reduces this to \\(O(n)\\).</li> <li>For the maximum product cutting problem, we derived two greedy strategies: continue splitting integers \\(\\geq 4\\), and favor factors of \\(3\\). The implementation uses exponentiation, so its time complexity depends on the exponentiation method, usually \\(O(1)\\) or \\(O(\\log n)\\).</li> </ul>","path":["Chapter 15.   Greedy algorithms","15.5   Summary"],"tags":[]},{"location":"chapter_hashing/","level":1,"title":"Chapter 6.   Hash tables","text":"<p>Abstract</p> <p>In the world of computers, a hash table is like a knowledgeable librarian.</p> <p>Knowing how to calculate a call number lets the librarian find the desired book quickly.</p>","path":["Chapter 6.   Hash tables"],"tags":[]},{"location":"chapter_hashing/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>6.1   Hash tables</li> <li>6.2   Hash collisions</li> <li>6.3   Hash algorithms</li> <li>6.4   Summary</li> <li>6.5   Exercises</li> </ul>","path":["Chapter 6.   Hash tables"],"tags":[]},{"location":"chapter_hashing/exercises/","level":1,"title":"6.5   Exercises","text":"","path":["Chapter 6.   Hash tables","6.5   Exercises"],"tags":[]},{"location":"chapter_hashing/exercises/#651-check-your-understanding","level":2,"title":"6.5.1   Check your understanding","text":"","path":["Chapter 6.   Hash tables","6.5   Exercises"],"tags":[]},{"location":"chapter_hashing/exercises/#1-searching-after-a-hash-collision","level":3,"title":"1.   Searching after a hash collision","text":"<p>A hash table has five buckets and uses the hash function \\(h(x)=x \\bmod 5\\). When elements collide, they are added in order to the list in that bucket. Insert <code>[1, 6, 11, 7]</code> in sequence:</p> <ol> <li>Write out the contents of buckets 0 through 4.</li> <li>When searching for 6, which bucket is visited first, and which elements are checked in order?</li> <li>Based on the bucket contents from question 1, did later elements overwrite earlier ones? Explain using this collision-handling method.</li> </ol> Reference answer <ol> <li> <p>Since \\(1\\bmod5=6\\bmod5=11\\bmod5=1\\) and \\(7\\bmod5=2\\), the buckets contain:</p> <pre><code>0: []\n1: [1, 6, 11]\n2: [7]\n3: []\n4: []\n</code></pre> </li> <li> <p>Searching for 6 first visits bucket 1 and compares against 1 and then 6, finding the target on the second comparison.</p> </li> <li> <p>Equal hash values mean only that the elements land in the same bucket, not that the elements themselves are equal. Separate chaining keeps all colliding elements in the bucket     and compares them one by one during a search, so 1, 6, and 11 do not overwrite one another.</p> </li> </ol>","path":["Chapter 6.   Hash tables","6.5   Exercises"],"tags":[]},{"location":"chapter_hashing/exercises/#2-where-do-elements-go-when-a-hash-table-grows","level":3,"title":"2.   Where do elements go when a hash table grows?","text":"<p>A hash table using separate chaining initially has five buckets and the hash function \\(h(x)=x\\bmod5\\). The keys <code>[1, 6, 11]</code> are all in bucket 1.</p> <p>The table is now expanded to seven buckets, and the hash function becomes \\(h(x)=x\\bmod7\\):</p> <ol> <li>Calculate the new bucket number for each of 1, 6, and 11.</li> <li>Which buckets contain elements after the expansion?</li> <li>Can the list in the old bucket 1 simply be copied unchanged into the new bucket 1? Explain using your results from questions 1 and 2.</li> </ol> Reference answer <ol> <li> <p>The new bucket numbers are:</p> <ul> <li>\\(1\\bmod7=1\\);</li> <li>\\(6\\bmod7=6\\);</li> <li>\\(11\\bmod7=4\\).</li> </ul> </li> <li> <p>Bucket 1 holds 1, bucket 4 holds 11, and bucket 6 holds 6. The three keys no longer share one bucket.</p> </li> <li> <p>The list cannot simply be copied unchanged. A bucket number is the remainder of the key divided by the number of buckets. When that number changes from 5 to 7, a key's bucket may change,     so every key's position must be recalculated. If we copied the old bucket 1 unchanged, subsequent searches for 6 and 11 using the new formula     would visit buckets 6 and 4, respectively, and fail to find them.</p> </li> </ol>","path":["Chapter 6.   Hash tables","6.5   Exercises"],"tags":[]},{"location":"chapter_hashing/exercises/#3-can-we-still-find-11-after-deleting-6","level":3,"title":"3.   Can we still find 11 after deleting 6?","text":"<p>A hash table has five positions with indices <code>0-4</code> and uses the hash function \\(h(x)=x\\bmod5\\). When a collision occurs, it starts at the computed index and searches to the right for the first empty position.</p> <p>Insert <code>[1, 6, 11]</code> in order:</p> <ol> <li>At which index is each number ultimately stored?</li> <li>Which indices are checked, in order, when searching for 11?</li> <li>Suppose deleting 6 marks its position as an empty slot that has never been used, and a search stops on reaching an empty slot.     What happens when we search for 11 again? Is the result correct? If not, how can we avoid the problem?</li> </ol> Reference answer <ol> <li> <p>The number 1 is stored at index 1. The number 6 also maps to index 1, so after the collision it is stored at index 2.     The number 11 likewise starts at index 1, skips occupied indices 1 and 2, and is finally stored at index 3.</p> </li> <li> <p>Searching for 11 checks indices <code>1, 2, 3</code> in order and finds it at index 3.</p> </li> <li> <p>If index 2 is marked as an empty slot that has never been used, a search for 11 checks index 1 and then stops at index 2,     incorrectly concluding that 11 is absent. Deletion should leave a deleted marker instead:     a search continues to the next index when it encounters this marker, wrapping from index 4 to index 0. Later insertions can still reuse the position.</p> </li> </ol>","path":["Chapter 6.   Hash tables","6.5   Exercises"],"tags":[]},{"location":"chapter_hashing/exercises/#652-programming-exercises","level":2,"title":"6.5.2   Programming exercises","text":"","path":["Chapter 6.   Hash tables","6.5   Exercises"],"tags":[]},{"location":"chapter_hashing/exercises/#1-comparing-the-characters-in-two-strings","level":3,"title":"1.   Comparing the characters in two strings","text":"<p>Given two strings <code>s</code> and <code>t</code> containing only lowercase English letters, you may rearrange the characters in <code>s</code> freely, but may not add, delete, or replace any characters.</p> <p>Determine whether the rearrangement can produce <code>t</code>. Return <code>true</code> if it can, or <code>false</code> otherwise. Use a hash table to record the frequency of each letter; do not sort the characters in either string.</p> Hints <ol> <li>Strings of different lengths cannot contain the same characters with the same frequencies.</li> <li>Use a hash table to record each letter's count; increment the appropriate count by 1 while scanning s.</li> <li>Decrement the appropriate count by 1 while scanning t. The character frequencies match only if all counts end at 0.</li> </ol> <p>LeetCode</p>","path":["Chapter 6.   Hash tables","6.5   Exercises"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/","level":1,"title":"6.3   Hash algorithms","text":"<p>The previous two sections explained how hash tables work and how collisions are resolved. Whether we use open addressing or separate chaining, these techniques keep the table working when collisions occur, but do not reduce how often collisions happen.</p> <p>Frequent collisions can severely degrade performance. In the separate-chaining table shown below, the ideal case distributes pairs evenly among buckets for the best lookup efficiency. In the worst case, all pairs occupy the same bucket, degrading the time complexity to \\(O(n)\\).</p> <p></p> <p> Figure 6-8   Best and worst cases for hash collisions </p> <p>The hash function determines how key-value pairs are distributed. Recall its two steps: calculate a hash value, then take it modulo the array length:</p> <pre><code>index = hash(key) % capacity\n</code></pre> <p>For a fixed <code>capacity</code>, the hash algorithm <code>hash()</code> determines the output, and therefore the distribution of pairs in the table.</p> <p>To lower the likelihood of collisions, we should therefore focus on designing <code>hash()</code>.</p>","path":["Chapter 6.   Hash tables","6.3   Hash algorithms"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#631-goals-of-a-hash-algorithm","level":2,"title":"6.3.1   Goals of a hash algorithm","text":"<p>To build a hash table that is both fast and reliable, its hash algorithm should have these properties.</p> <ul> <li>Determinism: The same input must always produce the same output, ensuring reliable lookup.</li> <li>Efficiency: Calculating a hash should be fast. Lower computational overhead makes the table more practical.</li> <li>Uniform distribution: The algorithm should distribute pairs evenly across the table. More uniform distribution means fewer collisions.</li> </ul> <p>Hash algorithms also have many applications beyond hash tables.</p> <ul> <li>Password storage: To protect passwords, systems usually store password hashes rather than plaintext passwords. When a user enters a password, the system hashes it and compares the result with the stored hash. A match means the password is accepted.</li> <li>Data-integrity checking: A sender can calculate a hash of the data and send it alongside the data. The receiver recomputes the hash and compares it with the received hash. A match indicates that the data is intact.</li> </ul> <p>Cryptographic applications require stronger security properties to prevent reverse engineering, such as recovering an original password from its hash.</p> <ul> <li>One-wayness: The hash should not reveal any information about the input data.</li> <li>Collision resistance: Finding two different inputs with the same hash should be extremely difficult.</li> <li>Avalanche effect: A small input change should produce a large, unpredictable output change.</li> </ul> <p>Uniform distribution and collision resistance are independent concepts. Uniform output does not guarantee collision resistance. For example, with random keys, <code>key % 100</code> can produce uniform outputs. But this algorithm is too simple: all keys with the same last two digits give the same output. It is therefore easy to infer a usable key from a hash and crack a password.</p>","path":["Chapter 6.   Hash tables","6.3   Hash algorithms"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#632-designing-a-hash-algorithm","level":2,"title":"6.3.2   Designing a hash algorithm","text":"<p>Designing a hash algorithm is complex and involves many factors. For less demanding uses, however, we can devise simple algorithms.</p> <ul> <li>Additive hashing: Add the ASCII codes of all input characters and use the sum as the hash.</li> <li>Multiplicative hashing: Use multiplication to mix values, multiplying by a constant each round and accumulating character ASCII codes into the hash.</li> <li>XOR hashing: Accumulate each input element into a hash using exclusive OR.</li> <li>Rotational hashing: Accumulate each character's ASCII code into a hash, rotating the hash before each accumulation.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby simple_hash.py<pre><code>def add_hash(key: str) -&gt; int:\n    \"\"\"Additive hash\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash += ord(c)\n    return hash % modulus\n\ndef mul_hash(key: str) -&gt; int:\n    \"\"\"Multiplicative hash\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = 31 * hash + ord(c)\n    return hash % modulus\n\ndef xor_hash(key: str) -&gt; int:\n    \"\"\"XOR hash\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash ^= ord(c)\n    return hash % modulus\n\ndef rot_hash(key: str) -&gt; int:\n    \"\"\"Rotating hash\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ ord(c)\n    return hash % modulus\n</code></pre> simple_hash.cpp<pre><code>/* Additive hash */\nint addHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* Multiplicative hash */\nint mulHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (31 * hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* XOR hash */\nint xorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash ^= (int)c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* Rotating hash */\nint rotHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.java<pre><code>/* Additive hash */\nint addHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* Multiplicative hash */\nint mulHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (31 * hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* XOR hash */\nint xorHash(String key) {\n    int hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash ^= (int) c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* Rotating hash */\nint rotHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n</code></pre> simple_hash.cs<pre><code>/* Additive hash */\nint AddHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* Multiplicative hash */\nint MulHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (31 * hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* XOR hash */\nint XorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash ^= c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* Rotating hash */\nint RotHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.go<pre><code>/* Additive hash */\nfunc addHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* Multiplicative hash */\nfunc mulHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (31*hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* XOR hash */\nfunc xorHash(key string) int {\n    hash := 0\n    modulus := 1000000007\n    for _, b := range []byte(key) {\n        fmt.Println(int(b))\n        hash ^= int(b)\n        hash = (31*hash + int(b)) % modulus\n    }\n    return hash &amp; modulus\n}\n\n/* Rotating hash */\nfunc rotHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ int64(b)) % modulus\n    }\n    return int(hash)\n}\n</code></pre> simple_hash.swift<pre><code>/* Additive hash */\nfunc addHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* Multiplicative hash */\nfunc mulHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (31 * hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* XOR hash */\nfunc xorHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash ^= Int(scalar.value)\n        }\n    }\n    return hash &amp; MODULUS\n}\n\n/* Rotating hash */\nfunc rotHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n</code></pre> simple_hash.js<pre><code>/* Additive hash */\nfunction addHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* Multiplicative hash */\nfunction mulHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* XOR hash */\nfunction xorHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash % MODULUS;\n}\n\n/* Rotating hash */\nfunction rotHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.ts<pre><code>/* Additive hash */\nfunction addHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* Multiplicative hash */\nfunction mulHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* XOR hash */\nfunction xorHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash % MODULUS;\n}\n\n/* Rotating hash */\nfunction rotHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.dart<pre><code>/* Additive hash */\nint addHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* Multiplicative hash */\nint mulHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (31 * hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* XOR hash */\nint xorHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash ^= key.codeUnitAt(i);\n  }\n  return hash &amp; MODULUS;\n}\n\n/* Rotating hash */\nint rotHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n</code></pre> simple_hash.rs<pre><code>/* Additive hash */\nfn add_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* Multiplicative hash */\nfn mul_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (31 * hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* XOR hash */\nfn xor_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash ^= c as i64;\n    }\n\n    (hash &amp; MODULUS) as i32\n}\n\n/* Rotating hash */\nfn rot_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n</code></pre> simple_hash.c<pre><code>/* Additive hash */\nint addHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* Multiplicative hash */\nint mulHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (31 * hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* XOR hash */\nint xorHash(char *key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash ^= (unsigned char)key[i];\n    }\n    return hash &amp; MODULUS;\n}\n\n/* Rotating hash */\nint rotHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (unsigned char)key[i]) % MODULUS;\n    }\n\n    return (int)hash;\n}\n</code></pre> simple_hash.kt<pre><code>/* Additive hash */\nfun addHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = (hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* Multiplicative hash */\nfun mulHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = (31 * hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* XOR hash */\nfun xorHash(key: String): Int {\n    var hash = 0\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = hash xor c.code\n    }\n    return hash and MODULUS\n}\n\n/* Rotating hash */\nfun rotHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = ((hash shl 4) xor (hash shr 28) xor c.code.toLong()) % MODULUS\n    }\n    return hash.toInt()\n}\n</code></pre> simple_hash.rb<pre><code>### Additive hash ###\ndef add_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash += c.ord }\n\n  hash % modulus\nend\n\n### Multiplicative hash ###\ndef mul_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash = 31 * hash + c.ord }\n\n  hash % modulus\nend\n\n### XOR hash ###\ndef xor_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash ^= c.ord }\n\n  hash % modulus\nend\n\n### Rotating hash ###\ndef rot_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.ord }\n\n  hash % modulus\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Each algorithm ends by taking the result modulo the large prime \\(1000000007\\) to keep it within a suitable range. Why emphasize a prime modulus? What goes wrong with a composite modulus? This is an interesting question.</p> <p>The conclusion is that using a large prime modulus helps make hash values as uniformly distributed as possible. Because a prime shares no nontrivial factors with other numbers, it reduces periodic patterns introduced by the modulo operation and thus avoids collisions.</p> <p>For example, choose the composite modulus \\(9\\), which is divisible by \\(3\\). Every key divisible by \\(3\\) then maps to one of just three hash values: \\(0\\), \\(3\\), or \\(6\\).</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 9 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\dots \\} \\end{aligned} \\] <p>If the input keys follow this arithmetic progression, the hash values cluster, increasing collisions. Now replace the modulus with the prime \\(13\\). With no common factor between the key pattern and the modulus, the output becomes noticeably more uniform.</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 13 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\dots \\} \\end{aligned} \\] <p>If the keys are already uniformly random, either a prime or a composite modulus can produce uniformly distributed hashes. When the keys have a periodic pattern, however, a composite modulus is more likely to cause clustering.</p> <p>We therefore usually choose a prime modulus, preferably a large one, to suppress periodic patterns as much as possible and make the algorithm more robust.</p>","path":["Chapter 6.   Hash tables","6.3   Hash algorithms"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#633-common-hash-algorithms","level":2,"title":"6.3.3   Common hash algorithms","text":"<p>The simple algorithms above are fragile and fall well short of our design goals. Addition and XOR are commutative, for example, so additive and XOR hashing cannot distinguish strings containing the same characters in different orders. This can increase collisions and create security problems.</p> <p>In practice, standard algorithms such as MD5, SHA-1, SHA-2, and SHA-3 are commonly used. They map inputs of arbitrary length to fixed-length hash values.</p> <p>Over nearly a century, hash algorithms have continually evolved. Some researchers work to improve their performance, while others, along with hackers, search for security weaknesses. Table 6-2 lists commonly used algorithms.</p> <ul> <li>MD5 and SHA-1 have been successfully attacked repeatedly and have been retired from security applications.</li> <li>SHA-256, part of the SHA-2 family, is among the most secure hash algorithms. No successful attack has been reported, so it is widely used in security applications and protocols.</li> <li>SHA-3 has lower implementation overhead and higher computational efficiency than SHA-2, but is not yet as widely used.</li> </ul> <p> Table 6-2   Common hash algorithms </p> MD5 SHA-1 SHA-2 SHA-3 Introduced 1992 1995 2002 2008 Output length 128 bit 160 bit 256/512 bit 224/256/384/512 bit Hash collisions More frequent More frequent Rare Rare Security Low; successfully attacked Low; successfully attacked High High Applications Deprecated; still used for integrity checks Deprecated Cryptocurrency transaction verification, digital signatures, and more Can replace SHA-2","path":["Chapter 6.   Hash tables","6.3   Hash algorithms"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#634-hash-values-of-data-structures","level":2,"title":"6.3.4   Hash values of data structures","text":"<p>Hash-table keys can be integers, floating-point numbers, strings, and other types. Programming languages usually provide built-in hashing for these types to calculate bucket indices. In Python, for example, <code>hash()</code> calculates hash values for various types.</p> <ul> <li>Integers and Booleans have hash values equal to their own values.</li> <li>Hashing floating-point numbers and strings is more complex; interested readers can explore the details.</li> <li>A tuple's hash combines the hashes of its individual elements into one value.</li> <li>An object's hash is based on its memory address. Overriding its hash method allows hashing by content instead.</li> </ul> <p>Tip</p> <p>Built-in hash functions differ in their definitions and implementations across programming languages.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby built_in_hash.py<pre><code>num = 3\nhash_num = hash(num)\n# The hash of the integer 3 is 3\n\nbol = True\nhash_bol = hash(bol)\n# The hash of the Boolean True is 1\n\ndec = 3.14159\nhash_dec = hash(dec)\n# The hash of the floating-point number 3.14159 is 326484311674566659\n\nstr = \"Hello 算法\"\nhash_str = hash(str)\n# The hash of the string “Hello 算法” is 4617003410720528961\n\ntup = (12836, \"小哈\")\nhash_tup = hash(tup)\n# The hash of the tuple (12836, '小哈') is 1029005403108185979\n\nobj = ListNode(0)\nhash_obj = hash(obj)\n# The hash of the node object &lt;ListNode object at 0x1058fd810&gt; is 274267521\n</code></pre> built_in_hash.cpp<pre><code>int num = 3;\nsize_t hashNum = hash&lt;int&gt;()(num);\n// The hash of the integer 3 is 3\n\nbool bol = true;\nsize_t hashBol = hash&lt;bool&gt;()(bol);\n// The hash of the Boolean 1 is 1\n\ndouble dec = 3.14159;\nsize_t hashDec = hash&lt;double&gt;()(dec);\n// The hash of the floating-point number 3.14159 is 4614256650576692846\n\nstring str = \"Hello 算法\";\nsize_t hashStr = hash&lt;string&gt;()(str);\n// The hash of the string “Hello 算法” is 15466937326284535026\n\n// In C++, built-in std::hash() provides hashing only for basic data types\n// Hashing for arrays and objects must be implemented separately\n</code></pre> built_in_hash.java<pre><code>int num = 3;\nint hashNum = Integer.hashCode(num);\n// The hash of the integer 3 is 3\n\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// The hash of the Boolean true is 1231\n\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// The hash of the floating-point number 3.14159 is -1340954729\n\nString str = \"Hello 算法\";\nint hashStr = str.hashCode();\n// The hash of the string “Hello 算法” is -727081396\n\nObject[] arr = { 12836, \"小哈\" };\nint hashTup = Arrays.hashCode(arr);\n// The hash of the array [12836, 小哈] is 1151158\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// The hash of the node object utils.ListNode@7dc5e7b4 is 2110121908\n</code></pre> built_in_hash.cs<pre><code>int num = 3;\nint hashNum = num.GetHashCode();\n// The hash of the integer 3 is 3;\n\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// The hash of the Boolean true is 1;\n\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// The hash of the floating-point number 3.14159 is -1340954729;\n\nstring str = \"Hello 算法\";\nint hashStr = str.GetHashCode();\n// The hash of the string “Hello 算法” is -586107568;\n\nobject[] arr = [12836, \"小哈\"];\nint hashTup = arr.GetHashCode();\n// The hash of the array [12836, 小哈] is 42931033;\n\nListNode obj = new(0);\nint hashObj = obj.GetHashCode();\n// The hash of the node object 0 is 39053774;\n</code></pre> built_in_hash.go<pre><code>// Go provides no built-in hash-code function\n</code></pre> built_in_hash.swift<pre><code>let num = 3\nlet hashNum = num.hashValue\n// The hash of the integer 3 is 9047044699613009734\n\nlet bol = true\nlet hashBol = bol.hashValue\n// The hash of the Boolean true is -4431640247352757451\n\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// The hash of the floating-point number 3.14159 is -2465384235396674631\n\nlet str = \"Hello 算法\"\nlet hashStr = str.hashValue\n// The hash of the string “Hello 算法” is -7850626797806988787\n\nlet arr = [AnyHashable(12836), AnyHashable(\"小哈\")]\nlet hashTup = arr.hashValue\n// The hash of the array [AnyHashable(12836), AnyHashable(\"小哈\")] is -2308633508154532996\n\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// The hash of the node object utils.ListNode is -2434780518035996159\n</code></pre> built_in_hash.js<pre><code>// JavaScript provides no built-in hash-code function\n</code></pre> built_in_hash.ts<pre><code>// TypeScript provides no built-in hash-code function\n</code></pre> built_in_hash.dart<pre><code>int num = 3;\nint hashNum = num.hashCode;\n// The hash of the integer 3 is 34803\n\nbool bol = true;\nint hashBol = bol.hashCode;\n// The hash of the Boolean true is 1231\n\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// The hash of the floating-point number 3.14159 is 2570631074981783\n\nString str = \"Hello 算法\";\nint hashStr = str.hashCode;\n// The hash of the string “Hello 算法” is 468167534\n\nList arr = [12836, \"小哈\"];\nint hashArr = arr.hashCode;\n// The hash of the array [12836, 小哈] is 976512528\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// The hash of the node object Instance of 'ListNode' is 1033450432\n</code></pre> built_in_hash.rs<pre><code>use std::collections::hash_map::DefaultHasher;\nuse std::hash::{Hash, Hasher};\n\nlet num = 3;\nlet mut num_hasher = DefaultHasher::new();\nnum.hash(&amp;mut num_hasher);\nlet hash_num = num_hasher.finish();\n// The hash of the integer 3 is 568126464209439262\n\nlet bol = true;\nlet mut bol_hasher = DefaultHasher::new();\nbol.hash(&amp;mut bol_hasher);\nlet hash_bol = bol_hasher.finish();\n// The hash of the Boolean true is 4952851536318644461\n\nlet dec: f32 = 3.14159;\nlet mut dec_hasher = DefaultHasher::new();\ndec.to_bits().hash(&amp;mut dec_hasher);\nlet hash_dec = dec_hasher.finish();\n// The hash of the floating-point number 3.14159 is 2566941990314602357\n\nlet str = \"Hello 算法\";\nlet mut str_hasher = DefaultHasher::new();\nstr.hash(&amp;mut str_hasher);\nlet hash_str = str_hasher.finish();\n// The hash of the string “Hello 算法” is 16092673739211250988\n\nlet arr = (&amp;12836, &amp;\"小哈\");\nlet mut tup_hasher = DefaultHasher::new();\narr.hash(&amp;mut tup_hasher);\nlet hash_tup = tup_hasher.finish();\n// The hash of the tuple (12836, \"小哈\") is 1885128010422702749\n\nlet node = ListNode::new(42);\nlet mut hasher = DefaultHasher::new();\nnode.borrow().val.hash(&amp;mut hasher);\nlet hash = hasher.finish();\n// The hash of the node object RefCell { value: ListNode { val: 42, next: None } } is 15387811073369036852\n</code></pre> built_in_hash.c<pre><code>// C provides no built-in hash-code function\n</code></pre> built_in_hash.kt<pre><code>val num = 3\nval hashNum = num.hashCode()\n// The hash of the integer 3 is 3\n\nval bol = true\nval hashBol = bol.hashCode()\n// The hash of the Boolean true is 1231\n\nval dec = 3.14159\nval hashDec = dec.hashCode()\n// The hash of the floating-point number 3.14159 is -1340954729\n\nval str = \"Hello 算法\"\nval hashStr = str.hashCode()\n// The hash of the string “Hello 算法” is -727081396\n\nval arr = arrayOf&lt;Any&gt;(12836, \"小哈\")\nval hashTup = arr.hashCode()\n// The hash of the array [12836, 小哈] is 189568618\n\nval obj = ListNode(0)\nval hashObj = obj.hashCode()\n// The hash of the node object utils.ListNode@1d81eb93 is 495053715\n</code></pre> built_in_hash.rb<pre><code>num = 3\nhash_num = num.hash\n# The hash of the integer 3 is -4385856518450339636\n\nbol = true\nhash_bol = bol.hash\n# The hash of the Boolean true is -1617938112149317027\n\ndec = 3.14159\nhash_dec = dec.hash\n# The hash of the floating-point number 3.14159 is -1479186995943067893\n\nstr = \"Hello 算法\"\nhash_str = str.hash\n# The hash of the string “Hello 算法” is -4075943250025831763\n\ntup = [12836, '小哈']\nhash_tup = tup.hash\n# The hash of the tuple (12836, '小哈') is 1999544809202288822\n\nobj = ListNode.new(0)\nhash_obj = obj.hash\n# The hash of the node object #&lt;ListNode:0x000078133140ab70&gt; is 4302940560806366381\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>In many programming languages, only immutable objects can serve as hash-table keys. If a list, a dynamic array, were used as a key, changing its contents would change its hash, making the original value impossible to retrieve from the table.</p> <p>Custom objects, such as linked-list nodes, can be hashable even though their member variables are mutable. Their hash is usually based on their memory address, which stays the same even when their contents change.</p> <p>You may notice different hash values when running the program in different consoles. Each time the Python interpreter starts, it adds a random salt to its string hash function. This helps prevent HashDoS attacks and improves security.</p>","path":["Chapter 6.   Hash tables","6.3   Hash algorithms"],"tags":[]},{"location":"chapter_hashing/hash_collision/","level":1,"title":"6.2   Hash collisions","text":"<p>As discussed in the previous section, a hash function's input space is usually much larger than its output space, so collisions are theoretically unavoidable. For example, if the input space is all integers and the output space is limited to the array's capacity, multiple integers must map to the same bucket index.</p> <p>Hash collisions can produce incorrect lookup results and make a hash table unusable. One solution is to expand the table whenever a collision occurs until the collision disappears. This simple, brute-force approach works, but is inefficient because expansion requires substantial data movement and hash computation. We can improve efficiency with the following strategies.</p> <ol> <li>Improve the data structure so that it works correctly even when collisions occur.</li> <li>Expand only when necessary, when collisions become sufficiently frequent.</li> </ol> <p>The main structural approaches are “separate chaining” and “open addressing.”</p>","path":["Chapter 6.   Hash tables","6.2   Hash collisions"],"tags":[]},{"location":"chapter_hashing/hash_collision/#621-separate-chaining","level":2,"title":"6.2.1   Separate chaining","text":"<p>In the original hash table, each bucket holds only one key-value pair. Separate chaining replaces this single element with a linked list. Each pair becomes a node, and all colliding pairs are stored in the same list. An example is shown below.</p> <p></p> <p> Figure 6-5   A hash table using separate chaining </p> <p>The operations change as follows.</p> <ul> <li>Lookup: Hash the input <code>key</code> to obtain a bucket index, access the list's head, and traverse the list comparing keys to find the target pair.</li> <li>Insertion: Use the hash function to access the list's head, then add the node containing the key-value pair to the list.</li> <li>Deletion: Use the hash function to locate the list, traverse it to find the target node, and remove that node.</li> </ul> <p>Separate chaining has the following limitations.</p> <ul> <li>More space: Linked lists store node pointers and consume more memory than arrays.</li> <li>Slower lookup: Finding an element requires a linear scan of the list.</li> </ul> <p>The code below gives a simple implementation. Keep two points in mind.</p> <ul> <li>Lists, implemented as dynamic arrays, replace linked lists to simplify the code. The hash table is an array of buckets, each of which is a list.</li> <li>The implementation includes expansion: when the load factor exceeds \\(\\frac{2}{3}\\), the table's capacity is doubled to \\(2\\) times its original size.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map_chaining.py<pre><code>class HashMapChaining:\n    \"\"\"Hash table with separate chaining\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        self.size = 0  # Number of key-value pairs\n        self.capacity = 4  # Hash table capacity\n        self.load_thres = 2.0 / 3.0  # Load factor threshold that triggers capacity growth\n        self.extend_ratio = 2  # Capacity growth factor\n        self.buckets = [[] for _ in range(self.capacity)]  # Bucket array\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"Hash function\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"Load factor\"\"\"\n        return self.size / self.capacity\n\n    def get(self, key: int) -&gt; str | None:\n        \"\"\"Lookup operation\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # Traverse the bucket; if key is found, return its corresponding val\n        for pair in bucket:\n            if pair.key == key:\n                return pair.val\n        # Return None if key is not found\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"Insert operation\"\"\"\n        # Grow the capacity when the load factor exceeds the threshold\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # Traverse the bucket; if the specified key is found, update its val and return\n        for pair in bucket:\n            if pair.key == key:\n                pair.val = val\n                return\n        # If key is absent, append the key-value pair at the end\n        pair = Pair(key, val)\n        bucket.append(pair)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"Remove operation\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # Traverse the bucket and remove the key-value pair\n        for pair in bucket:\n            if pair.key == key:\n                bucket.remove(pair)\n                self.size -= 1\n                break\n\n    def extend(self):\n        \"\"\"Grow the hash table capacity\"\"\"\n        # Keep a reference to the original hash table\n        buckets = self.buckets\n        # Initialize a new hash table with the expanded capacity\n        self.capacity *= self.extend_ratio\n        self.buckets = [[] for _ in range(self.capacity)]\n        self.size = 0\n        # Move the key-value pairs from the original hash table to the new one\n        for bucket in buckets:\n            for pair in bucket:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"Print the hash table\"\"\"\n        for bucket in self.buckets:\n            res = []\n            for pair in bucket:\n                res.append(str(pair.key) + \" -&gt; \" + pair.val)\n            print(res)\n</code></pre> hash_map_chaining.cpp<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n  private:\n    int size;                       // Number of key-value pairs\n    int capacity;                   // Hash table capacity\n    double loadThres;               // Load factor threshold that triggers capacity growth\n    int extendRatio;                // Capacity growth factor\n    vector&lt;vector&lt;Pair *&gt;&gt; buckets; // Bucket array\n\n  public:\n    /* Constructor */\n    HashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3.0), extendRatio(2) {\n        buckets.resize(capacity);\n    }\n\n    /* Destructor */\n    ~HashMapChaining() {\n        for (auto &amp;bucket : buckets) {\n            for (Pair *pair : bucket) {\n                // Free memory\n                delete pair;\n            }\n        }\n    }\n\n    /* Hash function */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* Load factor */\n    double loadFactor() {\n        return (double)size / (double)capacity;\n    }\n\n    /* Lookup operation */\n    string get(int key) {\n        int index = hashFunc(key);\n        // Traverse the bucket; if key is found, return its corresponding val\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                return pair-&gt;val;\n            }\n        }\n        // Return an empty string if key is not found\n        return \"\";\n    }\n\n    /* Insert operation */\n    void put(int key, string val) {\n        // Grow the capacity when the load factor exceeds the threshold\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        // Traverse the bucket; if the specified key is found, update its val and return\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                pair-&gt;val = val;\n                return;\n            }\n        }\n        // If key is absent, append the key-value pair at the end\n        buckets[index].push_back(new Pair(key, val));\n        size++;\n    }\n\n    /* Remove operation */\n    void remove(int key) {\n        int index = hashFunc(key);\n        auto &amp;bucket = buckets[index];\n        // Traverse the bucket and remove the key-value pair\n        for (int i = 0; i &lt; bucket.size(); i++) {\n            if (bucket[i]-&gt;key == key) {\n                Pair *tmp = bucket[i];\n                bucket.erase(bucket.begin() + i); // Remove the key-value pair from the list\n                delete tmp;                       // Free memory\n                size--;\n                return;\n            }\n        }\n    }\n\n    /* Grow the hash table capacity */\n    void extend() {\n        // Keep a copy of the original bucket array\n        vector&lt;vector&lt;Pair *&gt;&gt; bucketsTmp = buckets;\n        // Initialize a new hash table with the expanded capacity\n        capacity *= extendRatio;\n        buckets.clear();\n        buckets.resize(capacity);\n        size = 0;\n        // Move the key-value pairs from the original hash table to the new one\n        for (auto &amp;bucket : bucketsTmp) {\n            for (Pair *pair : bucket) {\n                put(pair-&gt;key, pair-&gt;val);\n                // Free memory\n                delete pair;\n            }\n        }\n    }\n\n    /* Print the hash table */\n    void print() {\n        for (auto &amp;bucket : buckets) {\n            cout &lt;&lt; \"[\";\n            for (Pair *pair : bucket) {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; \", \";\n            }\n            cout &lt;&lt; \"]\\n\";\n        }\n    }\n};\n</code></pre> hash_map_chaining.java<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n    int size; // Number of key-value pairs\n    int capacity; // Hash table capacity\n    double loadThres; // Load factor threshold that triggers capacity growth\n    int extendRatio; // Capacity growth factor\n    List&lt;List&lt;Pair&gt;&gt; buckets; // Bucket array\n\n    /* Constructor */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n    }\n\n    /* Hash function */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* Load factor */\n    double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* Lookup operation */\n    String get(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // Traverse the bucket; if key is found, return its corresponding val\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // Return null if key is not found\n        return null;\n    }\n\n    /* Insert operation */\n    void put(int key, String val) {\n        // Grow the capacity when the load factor exceeds the threshold\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // Traverse the bucket; if the specified key is found, update its val and return\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // If key is absent, append the key-value pair at the end\n        Pair pair = new Pair(key, val);\n        bucket.add(pair);\n        size++;\n    }\n\n    /* Remove operation */\n    void remove(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // Traverse the bucket and remove the key-value pair\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* Grow the hash table capacity */\n    void extend() {\n        // Keep a reference to the original hash table\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // Initialize a new hash table with the expanded capacity\n        capacity *= extendRatio;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n        size = 0;\n        // Move the key-value pairs from the original hash table to the new one\n        for (List&lt;Pair&gt; bucket : bucketsTmp) {\n            for (Pair pair : bucket) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print the hash table */\n    void print() {\n        for (List&lt;Pair&gt; bucket : buckets) {\n            List&lt;String&gt; res = new ArrayList&lt;&gt;();\n            for (Pair pair : bucket) {\n                res.add(pair.key + \" -&gt; \" + pair.val);\n            }\n            System.out.println(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.cs<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n    int size; // Number of key-value pairs\n    int capacity; // Hash table capacity\n    double loadThres; // Load factor threshold that triggers capacity growth\n    int extendRatio; // Capacity growth factor\n    List&lt;List&lt;Pair&gt;&gt; buckets; // Bucket array\n\n    /* Constructor */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n    }\n\n    /* Hash function */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* Load factor */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* Lookup operation */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        // Traverse the bucket; if key is found, return its corresponding val\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // Return null if key is not found\n        return null;\n    }\n\n    /* Insert operation */\n    public void Put(int key, string val) {\n        // Grow the capacity when the load factor exceeds the threshold\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        int index = HashFunc(key);\n        // Traverse the bucket; if the specified key is found, update its val and return\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // If key is absent, append the key-value pair at the end\n        buckets[index].Add(new Pair(key, val));\n        size++;\n    }\n\n    /* Remove operation */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // Traverse the bucket and remove the key-value pair\n        foreach (Pair pair in buckets[index].ToList()) {\n            if (pair.key == key) {\n                buckets[index].Remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* Grow the hash table capacity */\n    void Extend() {\n        // Keep a reference to the original hash table\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // Initialize a new hash table with the expanded capacity\n        capacity *= extendRatio;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n        size = 0;\n        // Move the key-value pairs from the original hash table to the new one\n        foreach (List&lt;Pair&gt; bucket in bucketsTmp) {\n            foreach (Pair pair in bucket) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print the hash table */\n    public void Print() {\n        foreach (List&lt;Pair&gt; bucket in buckets) {\n            List&lt;string&gt; res = [];\n            foreach (Pair pair in bucket) {\n                res.Add(pair.key + \" -&gt; \" + pair.val);\n            }\n            foreach (string kv in res) {\n                Console.WriteLine(kv);\n            }\n        }\n    }\n}\n</code></pre> hash_map_chaining.go<pre><code>/* Hash table with separate chaining */\ntype hashMapChaining struct {\n    size        int      // Number of key-value pairs\n    capacity    int      // Hash table capacity\n    loadThres   float64  // Load factor threshold that triggers capacity growth\n    extendRatio int      // Capacity growth factor\n    buckets     [][]pair // Bucket array\n}\n\n/* Constructor */\nfunc newHashMapChaining() *hashMapChaining {\n    buckets := make([][]pair, 4)\n    for i := 0; i &lt; 4; i++ {\n        buckets[i] = make([]pair, 0)\n    }\n    return &amp;hashMapChaining{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     buckets,\n    }\n}\n\n/* Hash function */\nfunc (m *hashMapChaining) hashFunc(key int) int {\n    return key % m.capacity\n}\n\n/* Load factor */\nfunc (m *hashMapChaining) loadFactor() float64 {\n    return float64(m.size) / float64(m.capacity)\n}\n\n/* Lookup operation */\nfunc (m *hashMapChaining) get(key int) string {\n    idx := m.hashFunc(key)\n    bucket := m.buckets[idx]\n    // Traverse the bucket; if key is found, return its corresponding val\n    for _, p := range bucket {\n        if p.key == key {\n            return p.val\n        }\n    }\n    // If key is not found, return an empty string\n    return \"\"\n}\n\n/* Insert operation */\nfunc (m *hashMapChaining) put(key int, val string) {\n    // Grow the capacity when the load factor exceeds the threshold\n    if m.loadFactor() &gt; m.loadThres {\n        m.extend()\n    }\n    idx := m.hashFunc(key)\n    // Traverse the bucket; if the specified key is found, update its val and return\n    for i := range m.buckets[idx] {\n        if m.buckets[idx][i].key == key {\n            m.buckets[idx][i].val = val\n            return\n        }\n    }\n    // If key is absent, append the key-value pair at the end\n    p := pair{\n        key: key,\n        val: val,\n    }\n    m.buckets[idx] = append(m.buckets[idx], p)\n    m.size += 1\n}\n\n/* Remove operation */\nfunc (m *hashMapChaining) remove(key int) {\n    idx := m.hashFunc(key)\n    // Traverse the bucket and remove the key-value pair\n    for i, p := range m.buckets[idx] {\n        if p.key == key {\n            // Remove the element from the slice\n            m.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\n            m.size -= 1\n            break\n        }\n    }\n}\n\n/* Grow the hash table capacity */\nfunc (m *hashMapChaining) extend() {\n    // Save a copy of the original hash table\n    tmpBuckets := make([][]pair, len(m.buckets))\n    for i := 0; i &lt; len(m.buckets); i++ {\n        tmpBuckets[i] = make([]pair, len(m.buckets[i]))\n        copy(tmpBuckets[i], m.buckets[i])\n    }\n    // Initialize a new hash table with the expanded capacity\n    m.capacity *= m.extendRatio\n    m.buckets = make([][]pair, m.capacity)\n    for i := 0; i &lt; m.capacity; i++ {\n        m.buckets[i] = make([]pair, 0)\n    }\n    m.size = 0\n    // Move the key-value pairs from the original hash table to the new one\n    for _, bucket := range tmpBuckets {\n        for _, p := range bucket {\n            m.put(p.key, p.val)\n        }\n    }\n}\n\n/* Print the hash table */\nfunc (m *hashMapChaining) print() {\n    var builder strings.Builder\n\n    for _, bucket := range m.buckets {\n        builder.WriteString(\"[\")\n        for _, p := range bucket {\n            builder.WriteString(strconv.Itoa(p.key) + \" -&gt; \" + p.val + \" \")\n        }\n        builder.WriteString(\"]\")\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> hash_map_chaining.swift<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n    var size: Int // Number of key-value pairs\n    var capacity: Int // Hash table capacity\n    var loadThres: Double // Load factor threshold that triggers capacity growth\n    var extendRatio: Int // Capacity growth factor\n    var buckets: [[Pair]] // Bucket array\n\n    /* Constructor */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: [], count: capacity)\n    }\n\n    /* Hash function */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* Load factor */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* Lookup operation */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // Traverse the bucket; if key is found, return its corresponding val\n        for pair in bucket {\n            if pair.key == key {\n                return pair.val\n            }\n        }\n        // Return nil if key is not found\n        return nil\n    }\n\n    /* Insert operation */\n    func put(key: Int, val: String) {\n        // Grow the capacity when the load factor exceeds the threshold\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // Traverse the bucket; if the specified key is found, update its val and return\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val\n                return\n            }\n        }\n        // If key is absent, append the key-value pair at the end\n        let pair = Pair(key: key, val: val)\n        buckets[index].append(pair)\n        size += 1\n    }\n\n    /* Remove operation */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // Traverse the bucket and remove the key-value pair\n        for (pairIndex, pair) in bucket.enumerated() {\n            if pair.key == key {\n                buckets[index].remove(at: pairIndex)\n                size -= 1\n                break\n            }\n        }\n    }\n\n    /* Grow the hash table capacity */\n    func extend() {\n        // Save the original bucket array\n        let bucketsTmp = buckets\n        // Initialize a new hash table with the expanded capacity\n        capacity *= extendRatio\n        buckets = Array(repeating: [], count: capacity)\n        size = 0\n        // Move the key-value pairs from the original hash table to the new one\n        for bucket in bucketsTmp {\n            for pair in bucket {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* Print the hash table */\n    func print() {\n        for bucket in buckets {\n            let res = bucket.map { \"\\($0.key) -&gt; \\($0.val)\" }\n            Swift.print(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.js<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n    #size; // Number of key-value pairs\n    #capacity; // Hash table capacity\n    #loadThres; // Load factor threshold that triggers capacity growth\n    #extendRatio; // Capacity growth factor\n    #buckets; // Bucket array\n\n    /* Constructor */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* Hash function */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* Load factor */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* Lookup operation */\n    get(key) {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // Traverse the bucket; if key is found, return its corresponding val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // Return null if key was not found\n        return null;\n    }\n\n    /* Insert operation */\n    put(key, val) {\n        // Grow the capacity when the load factor exceeds the threshold\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // Traverse the bucket; if the specified key is found, update its val and return\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // If key is absent, append the key-value pair at the end\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* Remove operation */\n    remove(key) {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // Traverse the bucket and remove the key-value pair\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* Grow the hash table capacity */\n    #extend() {\n        // Keep a reference to the original hash table\n        const bucketsTmp = this.#buckets;\n        // Initialize a new hash table with the expanded capacity\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // Move the key-value pairs from the original hash table to the new one\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print the hash table */\n    print() {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.ts<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n    #size: number; // Number of key-value pairs\n    #capacity: number; // Hash table capacity\n    #loadThres: number; // Load factor threshold that triggers capacity growth\n    #extendRatio: number; // Capacity growth factor\n    #buckets: Pair[][]; // Bucket array\n\n    /* Constructor */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* Hash function */\n    #hashFunc(key: number): number {\n        return key % this.#capacity;\n    }\n\n    /* Load factor */\n    #loadFactor(): number {\n        return this.#size / this.#capacity;\n    }\n\n    /* Lookup operation */\n    get(key: number): string | null {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // Traverse the bucket; if key is found, return its corresponding val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // Return null if key was not found\n        return null;\n    }\n\n    /* Insert operation */\n    put(key: number, val: string): void {\n        // Grow the capacity when the load factor exceeds the threshold\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // Traverse the bucket; if the specified key is found, update its val and return\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // If key is absent, append the key-value pair at the end\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* Remove operation */\n    remove(key: number): void {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // Traverse the bucket and remove the key-value pair\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* Grow the hash table capacity */\n    #extend(): void {\n        // Keep a reference to the original hash table\n        const bucketsTmp = this.#buckets;\n        // Initialize a new hash table with the expanded capacity\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // Move the key-value pairs from the original hash table to the new one\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print the hash table */\n    print(): void {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.dart<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n  late int size; // Number of key-value pairs\n  late int capacity; // Hash table capacity\n  late double loadThres; // Load factor threshold that triggers capacity growth\n  late int extendRatio; // Capacity growth factor\n  late List&lt;List&lt;Pair&gt;&gt; buckets; // Bucket array\n\n  /* Constructor */\n  HashMapChaining() {\n    size = 0;\n    capacity = 4;\n    loadThres = 2.0 / 3.0;\n    extendRatio = 2;\n    buckets = List.generate(capacity, (_) =&gt; []);\n  }\n\n  /* Hash function */\n  int hashFunc(int key) {\n    return key % capacity;\n  }\n\n  /* Load factor */\n  double loadFactor() {\n    return size / capacity;\n  }\n\n  /* Lookup operation */\n  String? get(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // Traverse the bucket; if key is found, return its corresponding val\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        return pair.val;\n      }\n    }\n    // Return null if key was not found\n    return null;\n  }\n\n  /* Insert operation */\n  void put(int key, String val) {\n    // Grow the capacity when the load factor exceeds the threshold\n    if (loadFactor() &gt; loadThres) {\n      extend();\n    }\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // Traverse the bucket; if the specified key is found, update its val and return\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        pair.val = val;\n        return;\n      }\n    }\n    // If key is absent, append the key-value pair at the end\n    Pair pair = Pair(key, val);\n    bucket.add(pair);\n    size++;\n  }\n\n  /* Remove operation */\n  void remove(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // Traverse the bucket and remove the key-value pair\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        bucket.remove(pair);\n        size--;\n        break;\n      }\n    }\n  }\n\n  /* Grow the hash table capacity */\n  void extend() {\n    // Keep a reference to the original hash table\n    List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n    // Initialize a new hash table with the expanded capacity\n    capacity *= extendRatio;\n    buckets = List.generate(capacity, (_) =&gt; []);\n    size = 0;\n    // Move the key-value pairs from the original hash table to the new one\n    for (List&lt;Pair&gt; bucket in bucketsTmp) {\n      for (Pair pair in bucket) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* Print the hash table */\n  void printHashMap() {\n    for (List&lt;Pair&gt; bucket in buckets) {\n      List&lt;String&gt; res = [];\n      for (Pair pair in bucket) {\n        res.add(\"${pair.key} -&gt; ${pair.val}\");\n      }\n      print(res);\n    }\n  }\n}\n</code></pre> hash_map_chaining.rs<pre><code>/* Hash table with separate chaining */\nstruct HashMapChaining {\n    size: usize,\n    capacity: usize,\n    load_thres: f32,\n    extend_ratio: usize,\n    buckets: Vec&lt;Vec&lt;Pair&gt;&gt;,\n}\n\nimpl HashMapChaining {\n    /* Constructor */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![vec![]; 4],\n        }\n    }\n\n    /* Hash function */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % self.capacity\n    }\n\n    /* Load factor */\n    fn load_factor(&amp;self) -&gt; f32 {\n        self.size as f32 / self.capacity as f32\n    }\n\n    /* Remove operation */\n    fn remove(&amp;mut self, key: i32) -&gt; Option&lt;String&gt; {\n        let index = self.hash_func(key);\n\n        // Traverse the bucket and remove the key-value pair\n        for (i, p) in self.buckets[index].iter_mut().enumerate() {\n            if p.key == key {\n                let pair = self.buckets[index].remove(i);\n                self.size -= 1;\n                return Some(pair.val);\n            }\n        }\n\n        // Return None if key is not found\n        None\n    }\n\n    /* Grow the hash table capacity */\n    fn extend(&amp;mut self) {\n        // Temporarily take ownership of the original buckets\n        let buckets_tmp = std::mem::take(&amp;mut self.buckets);\n\n        // Initialize a new hash table with the expanded capacity\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![Vec::new(); self.capacity as usize];\n        self.size = 0;\n\n        // Move the key-value pairs from the original hash table to the new one\n        for bucket in buckets_tmp {\n            for pair in bucket {\n                self.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print the hash table */\n    fn print(&amp;self) {\n        for bucket in &amp;self.buckets {\n            let mut res = Vec::new();\n            for pair in bucket {\n                res.push(format!(\"{} -&gt; {}\", pair.key, pair.val));\n            }\n            println!(\"{:?}\", res);\n        }\n    }\n\n    /* Insert operation */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // Grow the capacity when the load factor exceeds the threshold\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n\n        let index = self.hash_func(key);\n\n        // Traverse the bucket; if the specified key is found, update its val and return\n        for pair in self.buckets[index].iter_mut() {\n            if pair.key == key {\n                pair.val = val;\n                return;\n            }\n        }\n\n        // If key is absent, append the key-value pair at the end\n        let pair = Pair { key, val };\n        self.buckets[index].push(pair);\n        self.size += 1;\n    }\n\n    /* Lookup operation */\n    fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        let index = self.hash_func(key);\n\n        // Traverse the bucket; if key is found, return its corresponding val\n        for pair in self.buckets[index].iter() {\n            if pair.key == key {\n                return Some(&amp;pair.val);\n            }\n        }\n\n        // Return None if key is not found\n        None\n    }\n}\n</code></pre> hash_map_chaining.c<pre><code>/* Linked list node */\ntypedef struct Node {\n    Pair *pair;\n    struct Node *next;\n} Node;\n\n/* Hash table with separate chaining */\ntypedef struct {\n    int size;         // Number of key-value pairs\n    int capacity;     // Hash table capacity\n    double loadThres; // Load factor threshold that triggers capacity growth\n    int extendRatio;  // Capacity growth factor\n    Node **buckets;   // Bucket array\n} HashMapChaining;\n\n/* Constructor */\nHashMapChaining *newHashMapChaining() {\n    HashMapChaining *hashMap = (HashMapChaining *)malloc(sizeof(HashMapChaining));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    return hashMap;\n}\n\n/* Destructor */\nvoid delHashMapChaining(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        while (cur) {\n            Node *tmp = cur;\n            cur = cur-&gt;next;\n            free(tmp-&gt;pair);\n            free(tmp);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap);\n}\n\n/* Hash function */\nint hashFunc(HashMapChaining *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* Load factor */\ndouble loadFactor(HashMapChaining *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* Lookup operation */\nchar *get(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    // Traverse the bucket; if key is found, return its corresponding val\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            return cur-&gt;pair-&gt;val;\n        }\n        cur = cur-&gt;next;\n    }\n    return \"\"; // Return an empty string if key is not found\n}\n\n/* Insert operation */\nvoid put(HashMapChaining *hashMap, int key, const char *val) {\n    // Grow the capacity when the load factor exceeds the threshold\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    int index = hashFunc(hashMap, key);\n    // Traverse the bucket; if the specified key is found, update its val and return\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            strcpy(cur-&gt;pair-&gt;val, val); // If the specified key is found, update its val and return\n            return;\n        }\n        cur = cur-&gt;next;\n    }\n    // If key is absent, add the key-value pair at the head of the linked list\n    Pair *newPair = (Pair *)malloc(sizeof(Pair));\n    newPair-&gt;key = key;\n    strcpy(newPair-&gt;val, val);\n    Node *newNode = (Node *)malloc(sizeof(Node));\n    newNode-&gt;pair = newPair;\n    newNode-&gt;next = hashMap-&gt;buckets[index];\n    hashMap-&gt;buckets[index] = newNode;\n    hashMap-&gt;size++;\n}\n\n/* Grow the hash table capacity */\nvoid extend(HashMapChaining *hashMap) {\n    // Keep a reference to the original hash table\n    int oldCapacity = hashMap-&gt;capacity;\n    Node **oldBuckets = hashMap-&gt;buckets;\n    // Initialize a new hash table with the expanded capacity\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    hashMap-&gt;size = 0;\n    // Move the key-value pairs from the original hash table to the new one\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Node *cur = oldBuckets[i];\n        while (cur) {\n            put(hashMap, cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            Node *temp = cur;\n            cur = cur-&gt;next;\n            // Free memory\n            free(temp-&gt;pair);\n            free(temp);\n        }\n    }\n\n    free(oldBuckets);\n}\n\n/* Remove operation */\nvoid removeItem(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    Node *cur = hashMap-&gt;buckets[index];\n    Node *pre = NULL;\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            // Remove the key-value pair from the list\n            if (pre) {\n                pre-&gt;next = cur-&gt;next;\n            } else {\n                hashMap-&gt;buckets[index] = cur-&gt;next;\n            }\n            // Free memory\n            free(cur-&gt;pair);\n            free(cur);\n            hashMap-&gt;size--;\n            return;\n        }\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n}\n\n/* Print the hash table */\nvoid print(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        printf(\"[\");\n        while (cur) {\n            printf(\"%d -&gt; %s, \", cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            cur = cur-&gt;next;\n        }\n        printf(\"]\\n\");\n    }\n}\n</code></pre> hash_map_chaining.kt<pre><code>/* Hash table with separate chaining */\nclass HashMapChaining {\n    var size: Int // Number of key-value pairs\n    var capacity: Int // Hash table capacity\n    val loadThres: Double // Load factor threshold that triggers capacity growth\n    val extendRatio: Int // Capacity growth factor\n    var buckets: MutableList&lt;MutableList&lt;Pair&gt;&gt; // Bucket array\n\n    /* Constructor */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n    }\n\n    /* Hash function */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* Load factor */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* Lookup operation */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // Traverse the bucket; if key is found, return its corresponding val\n        for (pair in bucket) {\n            if (pair.key == key) return pair._val\n        }\n        // Return null if key is not found\n        return null\n    }\n\n    /* Insert operation */\n    fun put(key: Int, _val: String) {\n        // Grow the capacity when the load factor exceeds the threshold\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // Traverse the bucket; if the specified key is found, update its val and return\n        for (pair in bucket) {\n            if (pair.key == key) {\n                pair._val = _val\n                return\n            }\n        }\n        // If key is absent, append the key-value pair at the end\n        val pair = Pair(key, _val)\n        bucket.add(pair)\n        size++\n    }\n\n    /* Remove operation */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // Traverse the bucket and remove the key-value pair\n        for (pair in bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair)\n                size--\n                break\n            }\n        }\n    }\n\n    /* Grow the hash table capacity */\n    fun extend() {\n        // Keep a reference to the original hash table\n        val bucketsTmp = buckets\n        // Initialize a new hash table with the expanded capacity\n        capacity *= extendRatio\n        // A mutable list has no fixed size\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n        size = 0\n        // Move the key-value pairs from the original hash table to the new one\n        for (bucket in bucketsTmp) {\n            for (pair in bucket) {\n                put(pair.key, pair._val)\n            }\n        }\n    }\n\n    /* Print the hash table */\n    fun print() {\n        for (bucket in buckets) {\n            val res = mutableListOf&lt;String&gt;()\n            for (pair in bucket) {\n                val k = pair.key\n                val v = pair._val\n                res.add(\"$k -&gt; $v\")\n            }\n            println(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.rb<pre><code>### Hash table using chaining ###\nclass HashMapChaining\n  ### Constructor ###\n  def initialize\n    @size = 0 # Number of key-value pairs\n    @capacity = 4 # Hash table capacity\n    @load_thres = 2.0 / 3.0 # Load factor threshold that triggers capacity growth\n    @extend_ratio = 2 # Capacity growth factor\n    @buckets = Array.new(@capacity) { [] } # Bucket array\n  end\n\n  ### Hash function ###\n  def hash_func(key)\n    key % @capacity\n  end\n\n  ### Load factor ###\n  def load_factor\n    @size / @capacity\n  end\n\n  ### Lookup operation ###\n  def get(key)\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # Traverse the bucket; if key is found, return its corresponding val\n    for pair in bucket\n      return pair.val if pair.key == key\n    end\n    # Return nil if key is not found\n    nil\n  end\n\n  ### Insert operation ###\n  def put(key, val)\n    # Grow the capacity when the load factor exceeds the threshold\n    extend if load_factor &gt; @load_thres\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # Traverse the bucket; if the specified key is found, update its val and return\n    for pair in bucket\n      if pair.key == key\n        pair.val = val\n        return\n      end\n    end\n    # If key is absent, append the key-value pair at the end\n    pair = Pair.new(key, val)\n    bucket &lt;&lt; pair\n    @size += 1\n  end\n\n  ### Remove operation ###\n  def remove(key)\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # Traverse the bucket and remove the key-value pair\n    for pair in bucket\n      if pair.key == key\n        bucket.delete(pair)\n        @size -= 1\n        break\n      end\n    end\n  end\n\n  ### Grow the hash table capacity ###\n  def extend\n    # Temporarily store the original hash table\n    buckets = @buckets\n    # Initialize a new hash table with the expanded capacity\n    @capacity *= @extend_ratio\n    @buckets = Array.new(@capacity) { [] }\n    @size = 0\n    # Move the key-value pairs from the original hash table to the new one\n    for bucket in buckets\n      for pair in bucket\n        put(pair.key, pair.val)\n      end\n    end\n  end\n\n  ### Print the hash table ###\n  def print\n    for bucket in @buckets\n      res = []\n      for pair in bucket\n        res &lt;&lt; \"#{pair.key} -&gt; #{pair.val}\"\n      end\n      pp res\n    end\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>When a list becomes long, its \\(O(n)\\) lookup time is poor. Replacing it with an AVL tree or red-black tree improves lookup time to \\(O(\\log n)\\).</p>","path":["Chapter 6.   Hash tables","6.2   Hash collisions"],"tags":[]},{"location":"chapter_hashing/hash_collision/#622-open-addressing","level":2,"title":"6.2.2   Open addressing","text":"<p>Open addressing handles collisions by probing multiple locations without introducing another data structure. Common strategies include linear probing, quadratic probing, and multiple hashing.</p> <p>We use linear probing to explain how an open-addressing hash table works.</p>","path":["Chapter 6.   Hash tables","6.2   Hash collisions"],"tags":[]},{"location":"chapter_hashing/hash_collision/#1-linear-probing","level":3,"title":"1.   Linear probing","text":"<p>Linear probing searches with a fixed step size. Its operations differ from those of a basic hash table.</p> <ul> <li>Insertion: Calculate the bucket index using the hash function. If the bucket is occupied, scan forward from that position with a fixed step size, usually \\(1\\), until an empty bucket is found, then insert the element.</li> <li>Lookup: If a collision is encountered, scan forward with the same step size until the target is found, then return its <code>value</code>. An empty bucket means the target is absent, so return <code>None</code>.</li> </ul> <p>Figure 6-6 shows key-value pairs in an open-addressing table using linear probing. With this hash function, keys sharing the same last two digits map to the same bucket. Linear probing stores them successively in that bucket and the following buckets.</p> <p></p> <p> Figure 6-6   Distribution of key-value pairs with linear probing </p> <p>However, linear probing is prone to clustering. The longer a run of occupied positions becomes, the more likely a collision is to occur within it, causing the cluster to grow further. This feedback loop eventually degrades insertion, deletion, lookup, and update performance.</p> <p>We cannot simply delete an element from an open-addressing table. Deletion would leave an empty bucket, <code>None</code>. A lookup stops when linear probing reaches such a bucket, making later elements in the probe sequence inaccessible and potentially reporting them as absent, as shown below.</p> <p></p> <p> Figure 6-7   Lookup problems caused by deletion in open addressing </p> <p>To solve this, use lazy deletion: rather than simply removing the entry, mark its bucket with the constant <code>TOMBSTONE</code>. Both <code>None</code> and <code>TOMBSTONE</code> represent available buckets that can hold a new pair. Unlike <code>None</code>, however, a <code>TOMBSTONE</code> does not stop a search, because further pairs may lie beyond it.</p> <p>However, lazy deletion can accelerate performance degradation. Each deletion leaves a marker. As <code>TOMBSTONE</code> entries accumulate, searches may need to skip many of them before reaching the target, increasing search time.</p> <p>To address this, record the first <code>TOMBSTONE</code> encountered during probing and swap the target entry with it when found. On each lookup or insertion, this moves the entry closer to its ideal location—the start of the probe sequence—and improves lookup efficiency.</p> <p>The code below implements linear probing with lazy deletion. To use the table's space more fully, treat the array as circular: after reaching the end, wrap around to the beginning.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map_open_addressing.py<pre><code>class HashMapOpenAddressing:\n    \"\"\"Hash table with open addressing\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        self.size = 0  # Number of key-value pairs\n        self.capacity = 4  # Hash table capacity\n        self.load_thres = 2.0 / 3.0  # Load factor threshold that triggers capacity growth\n        self.extend_ratio = 2  # Capacity growth factor\n        self.buckets: list[Pair | None] = [None] * self.capacity  # Bucket array\n        self.TOMBSTONE = Pair(-1, \"-1\")  # Tombstone marker\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"Hash function\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"Load factor\"\"\"\n        return self.size / self.capacity\n\n    def find_bucket(self, key: int) -&gt; int:\n        \"\"\"Find the bucket index for key\"\"\"\n        index = self.hash_func(key)\n        first_tombstone = -1\n        # Probe linearly until an empty bucket is reached\n        while self.buckets[index] is not None:\n            # If key is found, return its bucket index\n            if self.buckets[index].key == key:\n                # If a tombstone was encountered earlier, move the key-value pair to that index\n                if first_tombstone != -1:\n                    self.buckets[first_tombstone] = self.buckets[index]\n                    self.buckets[index] = self.TOMBSTONE\n                    return first_tombstone  # Return the bucket index after the move\n                return index  # Return the bucket index\n            # Record the first tombstone encountered\n            if first_tombstone == -1 and self.buckets[index] is self.TOMBSTONE:\n                first_tombstone = index\n            # Calculate the bucket index, wrapping to the beginning after the end\n            index = (index + 1) % self.capacity\n        # If key is absent, return the index for insertion\n        return index if first_tombstone == -1 else first_tombstone\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"Lookup operation\"\"\"\n        # Find the bucket index for key\n        index = self.find_bucket(key)\n        # If the key-value pair is found, return its val\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            return self.buckets[index].val\n        # Return None if the key-value pair is absent\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"Insert operation\"\"\"\n        # Grow the capacity when the load factor exceeds the threshold\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        # Find the bucket index for key\n        index = self.find_bucket(key)\n        # If the key-value pair is found, overwrite val and return\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index].val = val\n            return\n        # If the key-value pair is absent, insert it\n        self.buckets[index] = Pair(key, val)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"Remove operation\"\"\"\n        # Find the bucket index for key\n        index = self.find_bucket(key)\n        # If the key-value pair is found, replace it with a tombstone\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index] = self.TOMBSTONE\n            self.size -= 1\n\n    def extend(self):\n        \"\"\"Grow the hash table capacity\"\"\"\n        # Keep a reference to the original hash table\n        buckets_tmp = self.buckets\n        # Initialize a new hash table with the expanded capacity\n        self.capacity *= self.extend_ratio\n        self.buckets = [None] * self.capacity\n        self.size = 0\n        # Move the key-value pairs from the original hash table to the new one\n        for pair in buckets_tmp:\n            if pair not in [None, self.TOMBSTONE]:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"Print the hash table\"\"\"\n        for pair in self.buckets:\n            if pair is None:\n                print(\"None\")\n            elif pair is self.TOMBSTONE:\n                print(\"TOMBSTONE\")\n            else:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> hash_map_open_addressing.cpp<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n  private:\n    int size;                             // Number of key-value pairs\n    int capacity = 4;                     // Hash table capacity\n    const double loadThres = 2.0 / 3.0;     // Load factor threshold that triggers capacity growth\n    const int extendRatio = 2;            // Capacity growth factor\n    vector&lt;Pair *&gt; buckets;               // Bucket array\n    Pair *TOMBSTONE = new Pair(-1, \"-1\"); // Tombstone marker\n\n  public:\n    /* Constructor */\n    HashMapOpenAddressing() : size(0), buckets(capacity, nullptr) {\n    }\n\n    /* Destructor */\n    ~HashMapOpenAddressing() {\n        for (Pair *pair : buckets) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                delete pair;\n            }\n        }\n        delete TOMBSTONE;\n    }\n\n    /* Hash function */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* Load factor */\n    double loadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* Find the bucket index for key */\n    int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // Probe linearly until an empty bucket is reached\n        while (buckets[index] != nullptr) {\n            // If key is found, return its bucket index\n            if (buckets[index]-&gt;key == key) {\n                // If a tombstone was encountered earlier, move the key-value pair to that index\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // Return the bucket index after the move\n                }\n                return index; // Return the bucket index\n            }\n            // Record the first tombstone encountered\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // Calculate the bucket index, wrapping to the beginning after the end\n            index = (index + 1) % capacity;\n        }\n        // If key is absent, return the index for insertion\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* Lookup operation */\n    string get(int key) {\n        // Find the bucket index for key\n        int index = findBucket(key);\n        // If the key-value pair is found, return its val\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]-&gt;val;\n        }\n        // Return an empty string if the key-value pair does not exist\n        return \"\";\n    }\n\n    /* Insert operation */\n    void put(int key, string val) {\n        // Grow the capacity when the load factor exceeds the threshold\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // Find the bucket index for key\n        int index = findBucket(key);\n        // If the key-value pair is found, overwrite val and return\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]-&gt;val = val;\n            return;\n        }\n        // If the key-value pair is absent, insert it\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* Remove operation */\n    void remove(int key) {\n        // Find the bucket index for key\n        int index = findBucket(key);\n        // If the key-value pair is found, replace it with a tombstone\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            delete buckets[index];\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* Grow the hash table capacity */\n    void extend() {\n        // Keep a copy of the original bucket array\n        vector&lt;Pair *&gt; bucketsTmp = buckets;\n        // Initialize a new hash table with the expanded capacity\n        capacity *= extendRatio;\n        buckets = vector&lt;Pair *&gt;(capacity, nullptr);\n        size = 0;\n        // Move the key-value pairs from the original hash table to the new one\n        for (Pair *pair : bucketsTmp) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                put(pair-&gt;key, pair-&gt;val);\n                delete pair;\n            }\n        }\n    }\n\n    /* Print the hash table */\n    void print() {\n        for (Pair *pair : buckets) {\n            if (pair == nullptr) {\n                cout &lt;&lt; \"nullptr\" &lt;&lt; endl;\n            } else if (pair == TOMBSTONE) {\n                cout &lt;&lt; \"TOMBSTONE\" &lt;&lt; endl;\n            } else {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; endl;\n            }\n        }\n    }\n};\n</code></pre> hash_map_open_addressing.java<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n    private int size; // Number of key-value pairs\n    private int capacity = 4; // Hash table capacity\n    private final double loadThres = 2.0 / 3.0; // Load factor threshold that triggers capacity growth\n    private final int extendRatio = 2; // Capacity growth factor\n    private Pair[] buckets; // Bucket array\n    private final Pair TOMBSTONE = new Pair(-1, \"-1\"); // Tombstone marker\n\n    /* Constructor */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* Hash function */\n    private int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* Load factor */\n    private double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* Find the bucket index for key */\n    private int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // Probe linearly until an empty bucket is reached\n        while (buckets[index] != null) {\n            // If key is found, return its bucket index\n            if (buckets[index].key == key) {\n                // If a tombstone was encountered earlier, move the key-value pair to that index\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // Return the bucket index after the move\n                }\n                return index; // Return the bucket index\n            }\n            // Record the first tombstone encountered\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // Calculate the bucket index, wrapping to the beginning after the end\n            index = (index + 1) % capacity;\n        }\n        // If key is absent, return the index for insertion\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* Lookup operation */\n    public String get(int key) {\n        // Find the bucket index for key\n        int index = findBucket(key);\n        // If the key-value pair is found, return its val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // Return null if the key-value pair is absent\n        return null;\n    }\n\n    /* Insert operation */\n    public void put(int key, String val) {\n        // Grow the capacity when the load factor exceeds the threshold\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // Find the bucket index for key\n        int index = findBucket(key);\n        // If the key-value pair is found, overwrite val and return\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // If the key-value pair is absent, insert it\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* Remove operation */\n    public void remove(int key) {\n        // Find the bucket index for key\n        int index = findBucket(key);\n        // If the key-value pair is found, replace it with a tombstone\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* Grow the hash table capacity */\n    private void extend() {\n        // Keep a reference to the original hash table\n        Pair[] bucketsTmp = buckets;\n        // Initialize a new hash table with the expanded capacity\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // Move the key-value pairs from the original hash table to the new one\n        for (Pair pair : bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print the hash table */\n    public void print() {\n        for (Pair pair : buckets) {\n            if (pair == null) {\n                System.out.println(\"null\");\n            } else if (pair == TOMBSTONE) {\n                System.out.println(\"TOMBSTONE\");\n            } else {\n                System.out.println(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.cs<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n    int size; // Number of key-value pairs\n    int capacity = 4; // Hash table capacity\n    double loadThres = 2.0 / 3.0; // Load factor threshold that triggers capacity growth\n    int extendRatio = 2; // Capacity growth factor\n    Pair[] buckets; // Bucket array\n    Pair TOMBSTONE = new(-1, \"-1\"); // Tombstone marker\n\n    /* Constructor */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* Hash function */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* Load factor */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* Find the bucket index for key */\n    int FindBucket(int key) {\n        int index = HashFunc(key);\n        int firstTombstone = -1;\n        // Probe linearly until an empty bucket is reached\n        while (buckets[index] != null) {\n            // If key is found, return its bucket index\n            if (buckets[index].key == key) {\n                // If a tombstone was encountered earlier, move the key-value pair to that index\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // Return the bucket index after the move\n                }\n                return index; // Return the bucket index\n            }\n            // Record the first tombstone encountered\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // Calculate the bucket index, wrapping to the beginning after the end\n            index = (index + 1) % capacity;\n        }\n        // If key is absent, return the index for insertion\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* Lookup operation */\n    public string? Get(int key) {\n        // Find the bucket index for key\n        int index = FindBucket(key);\n        // If the key-value pair is found, return its val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // Return null if the key-value pair is absent\n        return null;\n    }\n\n    /* Insert operation */\n    public void Put(int key, string val) {\n        // Grow the capacity when the load factor exceeds the threshold\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        // Find the bucket index for key\n        int index = FindBucket(key);\n        // If the key-value pair is found, overwrite val and return\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // If the key-value pair is absent, insert it\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* Remove operation */\n    public void Remove(int key) {\n        // Find the bucket index for key\n        int index = FindBucket(key);\n        // If the key-value pair is found, replace it with a tombstone\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* Grow the hash table capacity */\n    void Extend() {\n        // Keep a reference to the original hash table\n        Pair[] bucketsTmp = buckets;\n        // Initialize a new hash table with the expanded capacity\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // Move the key-value pairs from the original hash table to the new one\n        foreach (Pair pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print the hash table */\n    public void Print() {\n        foreach (Pair pair in buckets) {\n            if (pair == null) {\n                Console.WriteLine(\"null\");\n            } else if (pair == TOMBSTONE) {\n                Console.WriteLine(\"TOMBSTONE\");\n            } else {\n                Console.WriteLine(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.go<pre><code>/* Hash table with open addressing */\ntype hashMapOpenAddressing struct {\n    size        int     // Number of key-value pairs\n    capacity    int     // Hash table capacity\n    loadThres   float64 // Load factor threshold that triggers capacity growth\n    extendRatio int     // Capacity growth factor\n    buckets     []*pair // Bucket array\n    TOMBSTONE   *pair   // Tombstone marker\n}\n\n/* Constructor */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\n    return &amp;hashMapOpenAddressing{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     make([]*pair, 4),\n        TOMBSTONE:   &amp;pair{-1, \"-1\"},\n    }\n}\n\n/* Hash function */\nfunc (h *hashMapOpenAddressing) hashFunc(key int) int {\n    return key % h.capacity // Calculate the hash value from the key\n}\n\n/* Load factor */\nfunc (h *hashMapOpenAddressing) loadFactor() float64 {\n    return float64(h.size) / float64(h.capacity) // Calculate the current load factor\n}\n\n/* Find the bucket index for key */\nfunc (h *hashMapOpenAddressing) findBucket(key int) int {\n    index := h.hashFunc(key) // Get the initial index\n    firstTombstone := -1     // Record the position of the first TOMBSTONE encountered\n    for h.buckets[index] != nil {\n        if h.buckets[index].key == key {\n            if firstTombstone != -1 {\n                // If a tombstone was encountered earlier, move the key-value pair to that index\n                h.buckets[firstTombstone] = h.buckets[index]\n                h.buckets[index] = h.TOMBSTONE\n                return firstTombstone // Return the bucket index after the move\n            }\n            return index // Return the index found\n        }\n        if firstTombstone == -1 &amp;&amp; h.buckets[index] == h.TOMBSTONE {\n            firstTombstone = index // Record the position of the first tombstone encountered\n        }\n        index = (index + 1) % h.capacity // Probe linearly, wrapping to the beginning after the end\n    }\n    // If key is absent, return the index for insertion\n    if firstTombstone != -1 {\n        return firstTombstone\n    }\n    return index\n}\n\n/* Lookup operation */\nfunc (h *hashMapOpenAddressing) get(key int) string {\n    index := h.findBucket(key) // Find the bucket index for key\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        return h.buckets[index].val // If the key-value pair is found, return its val\n    }\n    return \"\" // If the key-value pair is absent, return \"\"\n}\n\n/* Insert operation */\nfunc (h *hashMapOpenAddressing) put(key int, val string) {\n    if h.loadFactor() &gt; h.loadThres {\n        h.extend() // Grow the capacity when the load factor exceeds the threshold\n    }\n    index := h.findBucket(key) // Find the bucket index for key\n    if h.buckets[index] == nil || h.buckets[index] == h.TOMBSTONE {\n        h.buckets[index] = &amp;pair{key, val} // If the key-value pair is absent, insert it\n        h.size++\n    } else {\n        h.buckets[index].val = val // If the key-value pair is found, overwrite val\n    }\n}\n\n/* Remove operation */\nfunc (h *hashMapOpenAddressing) remove(key int) {\n    index := h.findBucket(key) // Find the bucket index for key\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        h.buckets[index] = h.TOMBSTONE // If the key-value pair is found, replace it with a tombstone\n        h.size--\n    }\n}\n\n/* Grow the hash table capacity */\nfunc (h *hashMapOpenAddressing) extend() {\n    oldBuckets := h.buckets               // Keep a reference to the original hash table\n    h.capacity *= h.extendRatio           // Update the capacity\n    h.buckets = make([]*pair, h.capacity) // Initialize a new hash table with the expanded capacity\n    h.size = 0                            // Reset the size\n    // Move the key-value pairs from the original hash table to the new one\n    for _, pair := range oldBuckets {\n        if pair != nil &amp;&amp; pair != h.TOMBSTONE {\n            h.put(pair.key, pair.val)\n        }\n    }\n}\n\n/* Print the hash table */\nfunc (h *hashMapOpenAddressing) print() {\n    for _, pair := range h.buckets {\n        if pair == nil {\n            fmt.Println(\"nil\")\n        } else if pair == h.TOMBSTONE {\n            fmt.Println(\"TOMBSTONE\")\n        } else {\n            fmt.Printf(\"%d -&gt; %s\\n\", pair.key, pair.val)\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.swift<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n    var size: Int // Number of key-value pairs\n    var capacity: Int // Hash table capacity\n    var loadThres: Double // Load factor threshold that triggers capacity growth\n    var extendRatio: Int // Capacity growth factor\n    var buckets: [Pair?] // Bucket array\n    var TOMBSTONE: Pair // Tombstone marker\n\n    /* Constructor */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: nil, count: capacity)\n        TOMBSTONE = Pair(key: -1, val: \"-1\")\n    }\n\n    /* Hash function */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* Load factor */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* Find the bucket index for key */\n    func findBucket(key: Int) -&gt; Int {\n        var index = hashFunc(key: key)\n        var firstTombstone = -1\n        // Probe linearly until an empty bucket is reached\n        while buckets[index] != nil {\n            // If key is found, return its bucket index\n            if buckets[index]!.key == key {\n                // If a tombstone was encountered earlier, move the key-value pair to that index\n                if firstTombstone != -1 {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // Return the bucket index after the move\n                }\n                return index // Return the bucket index\n            }\n            // Record the first tombstone encountered\n            if firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE {\n                firstTombstone = index\n            }\n            // Calculate the bucket index, wrapping to the beginning after the end\n            index = (index + 1) % capacity\n        }\n        // If key is absent, return the index for insertion\n        return firstTombstone == -1 ? index : firstTombstone\n    }\n\n    /* Lookup operation */\n    func get(key: Int) -&gt; String? {\n        // Find the bucket index for key\n        let index = findBucket(key: key)\n        // If the key-value pair is found, return its val\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            return buckets[index]!.val\n        }\n        // Return nil if the key-value pair is absent\n        return nil\n    }\n\n    /* Insert operation */\n    func put(key: Int, val: String) {\n        // Grow the capacity when the load factor exceeds the threshold\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        // Find the bucket index for key\n        let index = findBucket(key: key)\n        // If the key-value pair is found, overwrite val and return\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index]!.val = val\n            return\n        }\n        // If the key-value pair is absent, insert it\n        buckets[index] = Pair(key: key, val: val)\n        size += 1\n    }\n\n    /* Remove operation */\n    func remove(key: Int) {\n        // Find the bucket index for key\n        let index = findBucket(key: key)\n        // If the key-value pair is found, replace it with a tombstone\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index] = TOMBSTONE\n            size -= 1\n        }\n    }\n\n    /* Grow the hash table capacity */\n    func extend() {\n        // Save the original bucket array\n        let bucketsTmp = buckets\n        // Initialize a new hash table with the expanded capacity\n        capacity *= extendRatio\n        buckets = Array(repeating: nil, count: capacity)\n        size = 0\n        // Move the key-value pairs from the original hash table to the new one\n        for pair in bucketsTmp {\n            if let pair, pair != TOMBSTONE {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* Print the hash table */\n    func print() {\n        for pair in buckets {\n            if pair == nil {\n                Swift.print(\"null\")\n            } else if pair == TOMBSTONE {\n                Swift.print(\"TOMBSTONE\")\n            } else {\n                Swift.print(\"\\(pair!.key) -&gt; \\(pair!.val)\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.js<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n    #size; // Number of key-value pairs\n    #capacity; // Hash table capacity\n    #loadThres; // Load factor threshold that triggers capacity growth\n    #extendRatio; // Capacity growth factor\n    #buckets; // Bucket array\n    #TOMBSTONE; // Tombstone marker\n\n    /* Constructor */\n    constructor() {\n        this.#size = 0; // Number of key-value pairs\n        this.#capacity = 4; // Hash table capacity\n        this.#loadThres = 2.0 / 3.0; // Load factor threshold that triggers capacity growth\n        this.#extendRatio = 2; // Capacity growth factor\n        this.#buckets = Array(this.#capacity).fill(null); // Bucket array\n        this.#TOMBSTONE = new Pair(-1, '-1'); // Tombstone marker\n    }\n\n    /* Hash function */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* Load factor */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* Find the bucket index for key */\n    #findBucket(key) {\n        let index = this.#hashFunc(key);\n        let firstTombstone = -1;\n        // Probe linearly until an empty bucket is reached\n        while (this.#buckets[index] !== null) {\n            // If key is found, return its bucket index\n            if (this.#buckets[index].key === key) {\n                // If a tombstone was encountered earlier, move the key-value pair to that index\n                if (firstTombstone !== -1) {\n                    this.#buckets[firstTombstone] = this.#buckets[index];\n                    this.#buckets[index] = this.#TOMBSTONE;\n                    return firstTombstone; // Return the bucket index after the move\n                }\n                return index; // Return the bucket index\n            }\n            // Record the first tombstone encountered\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.#buckets[index] === this.#TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // Calculate the bucket index, wrapping to the beginning after the end\n            index = (index + 1) % this.#capacity;\n        }\n        // If key is absent, return the index for insertion\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* Lookup operation */\n    get(key) {\n        // Find the bucket index for key\n        const index = this.#findBucket(key);\n        // If the key-value pair is found, return its val\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            return this.#buckets[index].val;\n        }\n        // Return null if the key-value pair does not exist\n        return null;\n    }\n\n    /* Insert operation */\n    put(key, val) {\n        // Grow the capacity when the load factor exceeds the threshold\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        // Find the bucket index for key\n        const index = this.#findBucket(key);\n        // If the key-value pair is found, overwrite val and return\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index].val = val;\n            return;\n        }\n        // If the key-value pair is absent, insert it\n        this.#buckets[index] = new Pair(key, val);\n        this.#size++;\n    }\n\n    /* Remove operation */\n    remove(key) {\n        // Find the bucket index for key\n        const index = this.#findBucket(key);\n        // If the key-value pair is found, replace it with a tombstone\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index] = this.#TOMBSTONE;\n            this.#size--;\n        }\n    }\n\n    /* Grow the hash table capacity */\n    #extend() {\n        // Keep a reference to the original hash table\n        const bucketsTmp = this.#buckets;\n        // Initialize a new hash table with the expanded capacity\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = Array(this.#capacity).fill(null);\n        this.#size = 0;\n        // Move the key-value pairs from the original hash table to the new one\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.#TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print the hash table */\n    print() {\n        for (const pair of this.#buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.#TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.ts<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n    private size: number; // Number of key-value pairs\n    private capacity: number; // Hash table capacity\n    private loadThres: number; // Load factor threshold that triggers capacity growth\n    private extendRatio: number; // Capacity growth factor\n    private buckets: Array&lt;Pair | null&gt;; // Bucket array\n    private TOMBSTONE: Pair; // Tombstone marker\n\n    /* Constructor */\n    constructor() {\n        this.size = 0; // Number of key-value pairs\n        this.capacity = 4; // Hash table capacity\n        this.loadThres = 2.0 / 3.0; // Load factor threshold that triggers capacity growth\n        this.extendRatio = 2; // Capacity growth factor\n        this.buckets = Array(this.capacity).fill(null); // Bucket array\n        this.TOMBSTONE = new Pair(-1, '-1'); // Tombstone marker\n    }\n\n    /* Hash function */\n    private hashFunc(key: number): number {\n        return key % this.capacity;\n    }\n\n    /* Load factor */\n    private loadFactor(): number {\n        return this.size / this.capacity;\n    }\n\n    /* Find the bucket index for key */\n    private findBucket(key: number): number {\n        let index = this.hashFunc(key);\n        let firstTombstone = -1;\n        // Probe linearly until an empty bucket is reached\n        while (this.buckets[index] !== null) {\n            // If key is found, return its bucket index\n            if (this.buckets[index]!.key === key) {\n                // If a tombstone was encountered earlier, move the key-value pair to that index\n                if (firstTombstone !== -1) {\n                    this.buckets[firstTombstone] = this.buckets[index];\n                    this.buckets[index] = this.TOMBSTONE;\n                    return firstTombstone; // Return the bucket index after the move\n                }\n                return index; // Return the bucket index\n            }\n            // Record the first tombstone encountered\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.buckets[index] === this.TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // Calculate the bucket index, wrapping to the beginning after the end\n            index = (index + 1) % this.capacity;\n        }\n        // If key is absent, return the index for insertion\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* Lookup operation */\n    get(key: number): string | null {\n        // Find the bucket index for key\n        const index = this.findBucket(key);\n        // If the key-value pair is found, return its val\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            return this.buckets[index]!.val;\n        }\n        // Return null if the key-value pair does not exist\n        return null;\n    }\n\n    /* Insert operation */\n    put(key: number, val: string): void {\n        // Grow the capacity when the load factor exceeds the threshold\n        if (this.loadFactor() &gt; this.loadThres) {\n            this.extend();\n        }\n        // Find the bucket index for key\n        const index = this.findBucket(key);\n        // If the key-value pair is found, overwrite val and return\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index]!.val = val;\n            return;\n        }\n        // If the key-value pair is absent, insert it\n        this.buckets[index] = new Pair(key, val);\n        this.size++;\n    }\n\n    /* Remove operation */\n    remove(key: number): void {\n        // Find the bucket index for key\n        const index = this.findBucket(key);\n        // If the key-value pair is found, replace it with a tombstone\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index] = this.TOMBSTONE;\n            this.size--;\n        }\n    }\n\n    /* Grow the hash table capacity */\n    private extend(): void {\n        // Keep a reference to the original hash table\n        const bucketsTmp = this.buckets;\n        // Initialize a new hash table with the expanded capacity\n        this.capacity *= this.extendRatio;\n        this.buckets = Array(this.capacity).fill(null);\n        this.size = 0;\n        // Move the key-value pairs from the original hash table to the new one\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* Print the hash table */\n    print(): void {\n        for (const pair of this.buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.dart<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n  late int _size; // Number of key-value pairs\n  int _capacity = 4; // Hash table capacity\n  double _loadThres = 2.0 / 3.0; // Load factor threshold that triggers capacity growth\n  int _extendRatio = 2; // Capacity growth factor\n  late List&lt;Pair?&gt; _buckets; // Bucket array\n  Pair _TOMBSTONE = Pair(-1, \"-1\"); // Tombstone marker\n\n  /* Constructor */\n  HashMapOpenAddressing() {\n    _size = 0;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n  }\n\n  /* Hash function */\n  int hashFunc(int key) {\n    return key % _capacity;\n  }\n\n  /* Load factor */\n  double loadFactor() {\n    return _size / _capacity;\n  }\n\n  /* Find the bucket index for key */\n  int findBucket(int key) {\n    int index = hashFunc(key);\n    int firstTombstone = -1;\n    // Probe linearly until an empty bucket is reached\n    while (_buckets[index] != null) {\n      // If key is found, return its bucket index\n      if (_buckets[index]!.key == key) {\n        // If a tombstone was encountered earlier, move the key-value pair to that index\n        if (firstTombstone != -1) {\n          _buckets[firstTombstone] = _buckets[index];\n          _buckets[index] = _TOMBSTONE;\n          return firstTombstone; // Return the bucket index after the move\n        }\n        return index; // Return the bucket index\n      }\n      // Record the first tombstone encountered\n      if (firstTombstone == -1 &amp;&amp; _buckets[index] == _TOMBSTONE) {\n        firstTombstone = index;\n      }\n      // Calculate the bucket index, wrapping to the beginning after the end\n      index = (index + 1) % _capacity;\n    }\n    // If key is absent, return the index for insertion\n    return firstTombstone == -1 ? index : firstTombstone;\n  }\n\n  /* Lookup operation */\n  String? get(int key) {\n    // Find the bucket index for key\n    int index = findBucket(key);\n    // If the key-value pair is found, return its val\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      return _buckets[index]!.val;\n    }\n    // Return null if the key-value pair does not exist\n    return null;\n  }\n\n  /* Insert operation */\n  void put(int key, String val) {\n    // Grow the capacity when the load factor exceeds the threshold\n    if (loadFactor() &gt; _loadThres) {\n      extend();\n    }\n    // Find the bucket index for key\n    int index = findBucket(key);\n    // If the key-value pair is found, overwrite val and return\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index]!.val = val;\n      return;\n    }\n    // If the key-value pair is absent, insert it\n    _buckets[index] = new Pair(key, val);\n    _size++;\n  }\n\n  /* Remove operation */\n  void remove(int key) {\n    // Find the bucket index for key\n    int index = findBucket(key);\n    // If the key-value pair is found, replace it with a tombstone\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index] = _TOMBSTONE;\n      _size--;\n    }\n  }\n\n  /* Grow the hash table capacity */\n  void extend() {\n    // Keep a reference to the original hash table\n    List&lt;Pair?&gt; bucketsTmp = _buckets;\n    // Initialize a new hash table with the expanded capacity\n    _capacity *= _extendRatio;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n    _size = 0;\n    // Move the key-value pairs from the original hash table to the new one\n    for (Pair? pair in bucketsTmp) {\n      if (pair != null &amp;&amp; pair != _TOMBSTONE) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* Print the hash table */\n  void printHashMap() {\n    for (Pair? pair in _buckets) {\n      if (pair == null) {\n        print(\"null\");\n      } else if (pair == _TOMBSTONE) {\n        print(\"TOMBSTONE\");\n      } else {\n        print(\"${pair.key} -&gt; ${pair.val}\");\n      }\n    }\n  }\n}\n</code></pre> hash_map_open_addressing.rs<pre><code>/* Hash table with open addressing */\nstruct HashMapOpenAddressing {\n    size: usize,                // Number of key-value pairs\n    capacity: usize,            // Hash table capacity\n    load_thres: f64,            // Load factor threshold that triggers capacity growth\n    extend_ratio: usize,        // Capacity growth factor\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;, // Bucket array\n    TOMBSTONE: Option&lt;Pair&gt;,    // Tombstone marker\n}\n\nimpl HashMapOpenAddressing {\n    /* Constructor */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![None; 4],\n            TOMBSTONE: Some(Pair {\n                key: -1,\n                val: \"-1\".to_string(),\n            }),\n        }\n    }\n\n    /* Hash function */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        (key % self.capacity as i32) as usize\n    }\n\n    /* Load factor */\n    fn load_factor(&amp;self) -&gt; f64 {\n        self.size as f64 / self.capacity as f64\n    }\n\n    /* Find the bucket index for key */\n    fn find_bucket(&amp;mut self, key: i32) -&gt; usize {\n        let mut index = self.hash_func(key);\n        let mut first_tombstone = -1;\n        // Probe linearly until an empty bucket is reached\n        while self.buckets[index].is_some() {\n            // If key is found, return the corresponding bucket index\n            if self.buckets[index].as_ref().unwrap().key == key {\n                // If a tombstone was encountered earlier, move the key-value pair to that index\n                if first_tombstone != -1 {\n                    self.buckets[first_tombstone as usize] = self.buckets[index].take();\n                    self.buckets[index] = self.TOMBSTONE.clone();\n                    return first_tombstone as usize; // Return the bucket index after the move\n                }\n                return index; // Return the bucket index\n            }\n            // Record the first tombstone encountered\n            if first_tombstone == -1 &amp;&amp; self.buckets[index] == self.TOMBSTONE {\n                first_tombstone = index as i32;\n            }\n            // Calculate the bucket index, wrapping to the beginning after the end\n            index = (index + 1) % self.capacity;\n        }\n        // If key is absent, return the index for insertion\n        if first_tombstone == -1 {\n            index\n        } else {\n            first_tombstone as usize\n        }\n    }\n\n    /* Lookup operation */\n    fn get(&amp;mut self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        // Find the bucket index for key\n        let index = self.find_bucket(key);\n        // If the key-value pair is found, return its val\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            return self.buckets[index].as_ref().map(|pair| &amp;pair.val as &amp;str);\n        }\n        // If the key-value pair does not exist, return None\n        None\n    }\n\n    /* Insert operation */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // Grow the capacity when the load factor exceeds the threshold\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n        // Find the bucket index for key\n        let index = self.find_bucket(key);\n        // If the key-value pair is found, overwrite val and return\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index].as_mut().unwrap().val = val;\n            return;\n        }\n        // If the key-value pair is absent, insert it\n        self.buckets[index] = Some(Pair { key, val });\n        self.size += 1;\n    }\n\n    /* Remove operation */\n    fn remove(&amp;mut self, key: i32) {\n        // Find the bucket index for key\n        let index = self.find_bucket(key);\n        // If the key-value pair is found, replace it with a tombstone\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index] = self.TOMBSTONE.clone();\n            self.size -= 1;\n        }\n    }\n\n    /* Grow the hash table capacity */\n    fn extend(&amp;mut self) {\n        // Temporarily save a copy of the original buckets\n        let buckets_tmp = self.buckets.clone();\n        // Initialize a new hash table with the expanded capacity\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![None; self.capacity];\n        self.size = 0;\n\n        // Move the key-value pairs from the original hash table to the new one\n        for pair in buckets_tmp {\n            if pair.is_none() || pair == self.TOMBSTONE {\n                continue;\n            }\n            let pair = pair.unwrap();\n\n            self.put(pair.key, pair.val);\n        }\n    }\n    /* Print the hash table */\n    fn print(&amp;self) {\n        for pair in &amp;self.buckets {\n            if pair.is_none() {\n                println!(\"null\");\n            } else if pair == &amp;self.TOMBSTONE {\n                println!(\"TOMBSTONE\");\n            } else {\n                let pair = pair.as_ref().unwrap();\n                println!(\"{} -&gt; {}\", pair.key, pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.c<pre><code>/* Hash table with open addressing */\ntypedef struct {\n    int size;         // Number of key-value pairs\n    int capacity;     // Hash table capacity\n    double loadThres; // Load factor threshold that triggers capacity growth\n    int extendRatio;  // Capacity growth factor\n    Pair **buckets;   // Bucket array\n    Pair *TOMBSTONE;  // Tombstone marker\n} HashMapOpenAddressing;\n\n/* Constructor */\nHashMapOpenAddressing *newHashMapOpenAddressing() {\n    HashMapOpenAddressing *hashMap = (HashMapOpenAddressing *)malloc(sizeof(HashMapOpenAddressing));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Pair **)calloc(hashMap-&gt;capacity, sizeof(Pair *));\n    hashMap-&gt;TOMBSTONE = (Pair *)malloc(sizeof(Pair));\n    hashMap-&gt;TOMBSTONE-&gt;key = -1;\n    hashMap-&gt;TOMBSTONE-&gt;val = \"-1\";\n\n    return hashMap;\n}\n\n/* Destructor */\nvoid delHashMapOpenAddressing(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap-&gt;TOMBSTONE);\n    free(hashMap);\n}\n\n/* Hash function */\nint hashFunc(HashMapOpenAddressing *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* Load factor */\ndouble loadFactor(HashMapOpenAddressing *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* Find the bucket index for key */\nint findBucket(HashMapOpenAddressing *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    int firstTombstone = -1;\n    // Probe linearly until an empty bucket is reached\n    while (hashMap-&gt;buckets[index] != NULL) {\n        // If key is found, return its bucket index\n        if (hashMap-&gt;buckets[index]-&gt;key == key) {\n            // If a tombstone was encountered earlier, move the key-value pair to that index\n            if (firstTombstone != -1) {\n                hashMap-&gt;buckets[firstTombstone] = hashMap-&gt;buckets[index];\n                hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n                return firstTombstone; // Return the bucket index after the move\n            }\n            return index; // Return the bucket index\n        }\n        // Record the first tombstone encountered\n        if (firstTombstone == -1 &amp;&amp; hashMap-&gt;buckets[index] == hashMap-&gt;TOMBSTONE) {\n            firstTombstone = index;\n        }\n        // Calculate the bucket index, wrapping to the beginning after the end\n        index = (index + 1) % hashMap-&gt;capacity;\n    }\n    // If key is absent, return the index for insertion\n    return firstTombstone == -1 ? index : firstTombstone;\n}\n\n/* Lookup operation */\nchar *get(HashMapOpenAddressing *hashMap, int key) {\n    // Find the bucket index for key\n    int index = findBucket(hashMap, key);\n    // If the key-value pair is found, return its val\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        return hashMap-&gt;buckets[index]-&gt;val;\n    }\n    // Return an empty string if the key-value pair does not exist\n    return \"\";\n}\n\n/* Insert operation */\nvoid put(HashMapOpenAddressing *hashMap, int key, char *val) {\n    // Grow the capacity when the load factor exceeds the threshold\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    // Find the bucket index for key\n    int index = findBucket(hashMap, key);\n    // If the key-value pair is found, overwrite val and return\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        free(hashMap-&gt;buckets[index]-&gt;val);\n        hashMap-&gt;buckets[index]-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n        strcpy(hashMap-&gt;buckets[index]-&gt;val, val);\n        hashMap-&gt;buckets[index]-&gt;val[strlen(val)] = '\\0';\n        return;\n    }\n    // If the key-value pair is absent, insert it\n    Pair *pair = (Pair *)malloc(sizeof(Pair));\n    pair-&gt;key = key;\n    pair-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n    strcpy(pair-&gt;val, val);\n    pair-&gt;val[strlen(val)] = '\\0';\n\n    hashMap-&gt;buckets[index] = pair;\n    hashMap-&gt;size++;\n}\n\n/* Remove operation */\nvoid removeItem(HashMapOpenAddressing *hashMap, int key) {\n    // Find the bucket index for key\n    int index = findBucket(hashMap, key);\n    // If the key-value pair is found, replace it with a tombstone\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        Pair *pair = hashMap-&gt;buckets[index];\n        free(pair-&gt;val);\n        free(pair);\n        hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n        hashMap-&gt;size--;\n    }\n}\n\n/* Grow the hash table capacity */\nvoid extend(HashMapOpenAddressing *hashMap) {\n    // Keep a reference to the original hash table\n    Pair **bucketsTmp = hashMap-&gt;buckets;\n    int oldCapacity = hashMap-&gt;capacity;\n    // Initialize a new hash table with the expanded capacity\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Pair **)calloc(hashMap-&gt;capacity, sizeof(Pair *));\n    hashMap-&gt;size = 0;\n    // Move the key-value pairs from the original hash table to the new one\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Pair *pair = bucketsTmp[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            put(hashMap, pair-&gt;key, pair-&gt;val);\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(bucketsTmp);\n}\n\n/* Print the hash table */\nvoid print(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair == NULL) {\n            printf(\"NULL\\n\");\n        } else if (pair == hashMap-&gt;TOMBSTONE) {\n            printf(\"TOMBSTONE\\n\");\n        } else {\n            printf(\"%d -&gt; %s\\n\", pair-&gt;key, pair-&gt;val);\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.kt<pre><code>/* Hash table with open addressing */\nclass HashMapOpenAddressing {\n    private var size: Int               // Number of key-value pairs\n    private var capacity: Int           // Hash table capacity\n    private val loadThres: Double       // Load factor threshold that triggers capacity growth\n    private val extendRatio: Int        // Capacity growth factor\n    private var buckets: Array&lt;Pair?&gt;   // Bucket array\n    private val TOMBSTONE: Pair         // Tombstone marker\n\n    /* Constructor */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = arrayOfNulls(capacity)\n        TOMBSTONE = Pair(-1, \"-1\")\n    }\n\n    /* Hash function */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* Load factor */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* Find the bucket index for key */\n    fun findBucket(key: Int): Int {\n        var index = hashFunc(key)\n        var firstTombstone = -1\n        // Probe linearly until an empty bucket is reached\n        while (buckets[index] != null) {\n            // If key is found, return its bucket index\n            if (buckets[index]?.key == key) {\n                // If a tombstone was encountered earlier, move the key-value pair to that index\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // Return the bucket index after the move\n                }\n                return index // Return the bucket index\n            }\n            // Record the first tombstone encountered\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index\n            }\n            // Calculate the bucket index, wrapping to the beginning after the end\n            index = (index + 1) % capacity\n        }\n        // If key is absent, return the index for insertion\n        return if (firstTombstone == -1) index else firstTombstone\n    }\n\n    /* Lookup operation */\n    fun get(key: Int): String? {\n        // Find the bucket index for key\n        val index = findBucket(key)\n        // If the key-value pair is found, return its val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]?._val\n        }\n        // Return null if the key-value pair is absent\n        return null\n    }\n\n    /* Insert operation */\n    fun put(key: Int, _val: String) {\n        // Grow the capacity when the load factor exceeds the threshold\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        // Find the bucket index for key\n        val index = findBucket(key)\n        // If the key-value pair is found, overwrite val and return\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]!!._val = _val\n            return\n        }\n        // If the key-value pair is absent, insert it\n        buckets[index] = Pair(key, _val)\n        size++\n    }\n\n    /* Remove operation */\n    fun remove(key: Int) {\n        // Find the bucket index for key\n        val index = findBucket(key)\n        // If the key-value pair is found, replace it with a tombstone\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE\n            size--\n        }\n    }\n\n    /* Grow the hash table capacity */\n    fun extend() {\n        // Keep a reference to the original hash table\n        val bucketsTmp = buckets\n        // Initialize a new hash table with the expanded capacity\n        capacity *= extendRatio\n        buckets = arrayOfNulls(capacity)\n        size = 0\n        // Move the key-value pairs from the original hash table to the new one\n        for (pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair._val)\n            }\n        }\n    }\n\n    /* Print the hash table */\n    fun print() {\n        for (pair in buckets) {\n            if (pair == null) {\n                println(\"null\")\n            } else if (pair == TOMBSTONE) {\n                println(\"TOMESTOME\")\n            } else {\n                println(\"${pair.key} -&gt; ${pair._val}\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.rb<pre><code>### Hash table with open addressing ###\nclass HashMapOpenAddressing\n  TOMBSTONE = Pair.new(-1, '-1') # Tombstone marker\n\n  ### Constructor ###\n  def initialize\n    @size = 0 # Number of key-value pairs\n    @capacity = 4 # Hash table capacity\n    @load_thres = 2.0 / 3.0 # Load factor threshold that triggers capacity growth\n    @extend_ratio = 2 # Capacity growth factor\n    @buckets = Array.new(@capacity) # Bucket array\n  end\n\n  ### Hash function ###\n  def hash_func(key)\n    key % @capacity\n  end\n\n  ### Load factor ###\n  def load_factor\n    @size / @capacity\n  end\n\n  ### Find the bucket index for key ###\n  def find_bucket(key)\n    index = hash_func(key)\n    first_tombstone = -1\n    # Probe linearly until an empty bucket is reached\n    while !@buckets[index].nil?\n      # If key is found, return its bucket index\n      if @buckets[index].key == key\n        # If a tombstone was encountered earlier, move the key-value pair to that index\n        if first_tombstone != -1\n          @buckets[first_tombstone] = @buckets[index]\n          @buckets[index] = TOMBSTONE\n          return first_tombstone # Return the bucket index after the move\n        end\n        return index # Return the bucket index\n      end\n      # Record the first tombstone encountered\n      first_tombstone = index if first_tombstone == -1 &amp;&amp; @buckets[index] == TOMBSTONE\n      # Calculate the bucket index, wrapping to the beginning after the end\n      index = (index + 1) % @capacity\n    end\n    # If key is absent, return the index for insertion\n    first_tombstone == -1 ? index : first_tombstone\n  end\n\n  ### Lookup operation ###\n  def get(key)\n    # Find the bucket index for key\n    index = find_bucket(key)\n    # If the key-value pair is found, return its val\n    return @buckets[index].val unless [nil, TOMBSTONE].include?(@buckets[index])\n    # Return nil if the key-value pair does not exist\n    nil\n  end\n\n  ### Insert operation ###\n  def put(key, val)\n    # Grow the capacity when the load factor exceeds the threshold\n    extend if load_factor &gt; @load_thres\n    # Find the bucket index for key\n    index = find_bucket(key)\n    # If the key-value pair is found, overwrite val and return\n    unless [nil, TOMBSTONE].include?(@buckets[index])\n      @buckets[index].val = val\n      return\n    end\n    # If the key-value pair is absent, insert it\n    @buckets[index] = Pair.new(key, val)\n    @size += 1\n  end\n\n  ### Remove operation ###\n  def remove(key)\n    # Find the bucket index for key\n    index = find_bucket(key)\n    # If the key-value pair is found, replace it with a tombstone\n    unless [nil, TOMBSTONE].include?(@buckets[index])\n      @buckets[index] = TOMBSTONE\n      @size -= 1\n    end\n  end\n\n  ### Grow the hash table capacity ###\n  def extend\n    # Keep a reference to the original hash table\n    buckets_tmp = @buckets\n    # Initialize a new hash table with the expanded capacity\n    @capacity *= @extend_ratio\n    @buckets = Array.new(@capacity)\n    @size = 0\n    # Move the key-value pairs from the original hash table to the new one\n    for pair in buckets_tmp\n      put(pair.key, pair.val) unless [nil, TOMBSTONE].include?(pair)\n    end\n  end\n\n  ### Print the hash table ###\n  def print\n    for pair in @buckets\n      if pair.nil?\n        puts \"Nil\"\n      elsif pair == TOMBSTONE\n        puts \"TOMBSTONE\"\n      else\n        puts \"#{pair.key} -&gt; #{pair.val}\"\n      end\n    end\n  end\nend\n</code></pre>","path":["Chapter 6.   Hash tables","6.2   Hash collisions"],"tags":[]},{"location":"chapter_hashing/hash_collision/#2-quadratic-probing","level":3,"title":"2.   Quadratic probing","text":"<p>Like linear probing, quadratic probing is a common open-addressing strategy. On a collision, it probes at distances determined by the square of the probe number—\\(1, 4, 9, \\dots\\)—rather than a fixed step size.</p> <p>Its main advantages are as follows.</p> <ul> <li>Squared probe distances aim to reduce the clustering caused by linear probing.</li> <li>Larger jumps help distribute entries more evenly while searching for empty positions.</li> </ul> <p>However, quadratic probing is not perfect.</p> <ul> <li>Clustering still occurs: some positions are more likely to be occupied than others.</li> <li>The growing squared distances may prevent probing the entire table, so an empty bucket may remain unreachable.</li> </ul>","path":["Chapter 6.   Hash tables","6.2   Hash collisions"],"tags":[]},{"location":"chapter_hashing/hash_collision/#3-multiple-hashing","level":3,"title":"3.   Multiple hashing","text":"<p>As its name suggests, multiple hashing probes using several hash functions, \\(f_1(x)\\), \\(f_2(x)\\), \\(f_3(x)\\), \\(\\dots\\).</p> <ul> <li>Insertion: If \\(f_1(x)\\) produces a collision, try \\(f_2(x)\\), and so on, inserting the element when an empty position is found.</li> <li>Lookup: Try the hash functions in the same order and return when the target is found. If an empty position is reached or all functions have been tried, the target is absent; return <code>None</code>.</li> </ul> <p>Multiple hashing is less prone to clustering than linear probing, but computing several hash functions adds overhead.</p> <p>Tip</p> <p>All open-addressing strategies—linear probing, quadratic probing, and multiple hashing—share the restriction that elements cannot simply be deleted.</p>","path":["Chapter 6.   Hash tables","6.2   Hash collisions"],"tags":[]},{"location":"chapter_hashing/hash_collision/#623-choices-made-by-programming-languages","level":2,"title":"6.2.3   Choices made by programming languages","text":"<p>Programming languages use different hash-table implementations. Here are a few examples.</p> <ul> <li>Python uses open addressing. Its <code>dict</code> probes using pseudorandom values.</li> <li>Java uses separate chaining. Since JDK 1.8, when a <code>HashMap</code>'s underlying array has length at least 64 and a chain reaches length 8, the chain is converted to a red-black tree to improve lookup performance.</li> <li>Since Go 1.24, the built-in <code>map</code> uses open addressing based on Swiss Tables by default. Go 1.23 and earlier use separate chaining, with up to 8 key-value pairs per bucket and linked overflow buckets when that capacity is exceeded.</li> </ul>","path":["Chapter 6.   Hash tables","6.2   Hash collisions"],"tags":[]},{"location":"chapter_hashing/hash_map/","level":1,"title":"6.1   Hash tables","text":"<p>A hash table, also called a hash map, supports efficient lookup by mapping a <code>key</code> to a <code>value</code>. Given a key, it can retrieve the corresponding value in \\(O(1)\\) time.</p> <p>Suppose we have \\(n\\) students, each with a name and a student ID. To support the query “given a student ID, return the corresponding name,” use the hash table shown below.</p> <p></p> <p> Figure 6-1   An abstract view of hash-table lookup </p> <p>Arrays and linked lists also support lookup. Table 6-1 compares their efficiency.</p> <ul> <li>Insertion: append an element to the end of the array or linked list in \\(O(1)\\) time.</li> <li>Lookup: because the array or list is unsorted, searching may require visiting every element, taking \\(O(n)\\) time.</li> <li>Deletion: first find the element, then remove it from the array or list, taking \\(O(n)\\) time.</li> </ul> <p> Table 6-1   Comparison of element-lookup efficiency </p> Array Linked list Hash table Find an element \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) Insert an element \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) Delete an element \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) <p>Hash-table insertion, deletion, lookup, and updating all take \\(O(1)\\) time, making them highly efficient.</p>","path":["Chapter 6.   Hash tables","6.1   Hash tables"],"tags":[]},{"location":"chapter_hashing/hash_map/#611-common-hash-table-operations","level":2,"title":"6.1.1   Common hash-table operations","text":"<p>Common operations include initialization, lookup, adding key-value pairs, and deleting pairs. Examples follow:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map.py<pre><code># Initialize a hash table\nhmap: dict = {}\n\n# Insertion\n# Add key-value pairs (key, value) to the hash table\nhmap[12836] = \"小哈\"\nhmap[15937] = \"小啰\"\nhmap[16750] = \"小算\"\nhmap[13276] = \"小法\"\nhmap[10583] = \"小鸭\"\n\n# Lookup\n# Look up key in the hash table to obtain value\nname: str = hmap[15937]\n\n# Deletion\n# Remove a key-value pair (key, value) from the hash table\nhmap.pop(10583)\n</code></pre> hash_map.cpp<pre><code>/* Initialize a hash table */\nunordered_map&lt;int, string&gt; map;\n\n/* Insertion */\n// Add key-value pairs (key, value) to the hash table\nmap[12836] = \"小哈\";\nmap[15937] = \"小啰\";\nmap[16750] = \"小算\";\nmap[13276] = \"小法\";\nmap[10583] = \"小鸭\";\n\n/* Lookup */\n// Look up key in the hash table to obtain value\nstring name = map[15937];\n\n/* Deletion */\n// Remove a key-value pair (key, value) from the hash table\nmap.erase(10583);\n</code></pre> hash_map.java<pre><code>/* Initialize a hash table */\nMap&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n\n/* Insertion */\n// Add key-value pairs (key, value) to the hash table\nmap.put(12836, \"小哈\");\nmap.put(15937, \"小啰\");\nmap.put(16750, \"小算\");\nmap.put(13276, \"小法\");\nmap.put(10583, \"小鸭\");\n\n/* Lookup */\n// Look up key in the hash table to obtain value\nString name = map.get(15937);\n\n/* Deletion */\n// Remove a key-value pair (key, value) from the hash table\nmap.remove(10583);\n</code></pre> hash_map.cs<pre><code>/* Initialize a hash table */\nDictionary&lt;int, string&gt; map = new() {\n    /* Insertion */\n    // Add key-value pairs (key, value) to the hash table\n    { 12836, \"小哈\" },\n    { 15937, \"小啰\" },\n    { 16750, \"小算\" },\n    { 13276, \"小法\" },\n    { 10583, \"小鸭\" }\n};\n\n/* Lookup */\n// Look up key in the hash table to obtain value\nstring name = map[15937];\n\n/* Deletion */\n// Remove a key-value pair (key, value) from the hash table\nmap.Remove(10583);\n</code></pre> hash_map_test.go<pre><code>/* Initialize a hash table */\nhmap := make(map[int]string)\n\n/* Insertion */\n// Add key-value pairs (key, value) to the hash table\nhmap[12836] = \"小哈\"\nhmap[15937] = \"小啰\"\nhmap[16750] = \"小算\"\nhmap[13276] = \"小法\"\nhmap[10583] = \"小鸭\"\n\n/* Lookup */\n// Look up key in the hash table to obtain value\nname := hmap[15937]\n\n/* Deletion */\n// Remove a key-value pair (key, value) from the hash table\ndelete(hmap, 10583)\n</code></pre> hash_map.swift<pre><code>/* Initialize a hash table */\nvar map: [Int: String] = [:]\n\n/* Insertion */\n// Add key-value pairs (key, value) to the hash table\nmap[12836] = \"小哈\"\nmap[15937] = \"小啰\"\nmap[16750] = \"小算\"\nmap[13276] = \"小法\"\nmap[10583] = \"小鸭\"\n\n/* Lookup */\n// Look up key in the hash table to obtain value\nlet name = map[15937]!\n\n/* Deletion */\n// Remove a key-value pair (key, value) from the hash table\nmap.removeValue(forKey: 10583)\n</code></pre> hash_map.js<pre><code>/* Initialize a hash table */\nconst map = new Map();\n/* Insertion */\n// Add key-value pairs (key, value) to the hash table\nmap.set(12836, '小哈');\nmap.set(15937, '小啰');\nmap.set(16750, '小算');\nmap.set(13276, '小法');\nmap.set(10583, '小鸭');\n\n/* Lookup */\n// Look up key in the hash table to obtain value\nlet name = map.get(15937);\n\n/* Deletion */\n// Remove a key-value pair (key, value) from the hash table\nmap.delete(10583);\n</code></pre> hash_map.ts<pre><code>/* Initialize a hash table */\nconst map = new Map&lt;number, string&gt;();\n/* Insertion */\n// Add key-value pairs (key, value) to the hash table\nmap.set(12836, '小哈');\nmap.set(15937, '小啰');\nmap.set(16750, '小算');\nmap.set(13276, '小法');\nmap.set(10583, '小鸭');\nconsole.info('\\nAfter insertion, the hash table is\\nKey -&gt; Value');\nconsole.info(map);\n\n/* Lookup */\n// Look up key in the hash table to obtain value\nlet name = map.get(15937);\nconsole.info('\\nLooking up student ID 15937 gives the name ' + name);\n\n/* Deletion */\n// Remove a key-value pair (key, value) from the hash table\nmap.delete(10583);\nconsole.info('\\nAfter deleting 10583, the hash table is\\nKey -&gt; Value');\nconsole.info(map);\n</code></pre> hash_map.dart<pre><code>/* Initialize a hash table */\nMap&lt;int, String&gt; map = {};\n\n/* Insertion */\n// Add key-value pairs (key, value) to the hash table\nmap[12836] = \"小哈\";\nmap[15937] = \"小啰\";\nmap[16750] = \"小算\";\nmap[13276] = \"小法\";\nmap[10583] = \"小鸭\";\n\n/* Lookup */\n// Look up key in the hash table to obtain value\nString name = map[15937];\n\n/* Deletion */\n// Remove a key-value pair (key, value) from the hash table\nmap.remove(10583);\n</code></pre> hash_map.rs<pre><code>use std::collections::HashMap;\n\n/* Initialize a hash table */\nlet mut map: HashMap&lt;i32, String&gt; = HashMap::new();\n\n/* Insertion */\n// Add key-value pairs (key, value) to the hash table\nmap.insert(12836, \"小哈\".to_string());\nmap.insert(15937, \"小啰\".to_string());\nmap.insert(16750, \"小算\".to_string());\nmap.insert(13279, \"小法\".to_string());\nmap.insert(10583, \"小鸭\".to_string());\n\n/* Lookup */\n// Look up key in the hash table to obtain value\nlet _name: Option&lt;&amp;String&gt; = map.get(&amp;15937);\n\n/* Deletion */\n// Remove a key-value pair (key, value) from the hash table\nlet _removed_value: Option&lt;String&gt; = map.remove(&amp;10583);\n</code></pre> hash_map.c<pre><code>// C provides no built-in hash table\n</code></pre> hash_map.kt<pre><code>/* Initialize a hash table */\nval map = HashMap&lt;Int,String&gt;()\n\n/* Insertion */\n// Add key-value pairs (key, value) to the hash table\nmap[12836] = \"小哈\"\nmap[15937] = \"小啰\"\nmap[16750] = \"小算\"\nmap[13276] = \"小法\"\nmap[10583] = \"小鸭\"\n\n/* Lookup */\n// Look up key in the hash table to obtain value\nval name = map[15937]\n\n/* Deletion */\n// Remove a key-value pair (key, value) from the hash table\nmap.remove(10583)\n</code></pre> hash_map.rb<pre><code># Initialize a hash table\nhmap = {}\n\n# Insertion\n# Add key-value pairs (key, value) to the hash table\nhmap[12836] = \"小哈\"\nhmap[15937] = \"小啰\"\nhmap[16750] = \"小算\"\nhmap[13276] = \"小法\"\nhmap[10583] = \"小鸭\"\n\n# Lookup\n# Look up key in the hash table to obtain value\nname = hmap[15937]\n\n# Deletion\n# Remove a key-value pair (key, value) from the hash table\nhmap.delete(10583)\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>There are three common ways to traverse a hash table: visit key-value pairs, keys only, or values only. Examples follow:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map.py<pre><code># Traverse the hash table\n# Traverse key-value pairs: key-&gt;value\nfor key, value in hmap.items():\n    print(key, \"-&gt;\", value)\n# Traverse keys only: key\nfor key in hmap.keys():\n    print(key)\n# Traverse values only: value\nfor value in hmap.values():\n    print(value)\n</code></pre> hash_map.cpp<pre><code>/* Traverse the hash table */\n// Traverse key-value pairs: key-&gt;value\nfor (auto kv: map) {\n    cout &lt;&lt; kv.first &lt;&lt; \" -&gt; \" &lt;&lt; kv.second &lt;&lt; endl;\n}\n// Use an iterator to traverse key-&gt;value pairs\nfor (auto iter = map.begin(); iter != map.end(); iter++) {\n    cout &lt;&lt; iter-&gt;first &lt;&lt; \"-&gt;\" &lt;&lt; iter-&gt;second &lt;&lt; endl;\n}\n</code></pre> hash_map.java<pre><code>/* Traverse the hash table */\n// Traverse key-value pairs: key-&gt;value\nfor (Map.Entry &lt;Integer, String&gt; kv: map.entrySet()) {\n    System.out.println(kv.getKey() + \" -&gt; \" + kv.getValue());\n}\n// Traverse keys only: key\nfor (int key: map.keySet()) {\n    System.out.println(key);\n}\n// Traverse values only: value\nfor (String val: map.values()) {\n    System.out.println(val);\n}\n</code></pre> hash_map.cs<pre><code>/* Traverse the hash table */\n// Traverse key-value pairs: Key-&gt;Value\nforeach (var kv in map) {\n    Console.WriteLine(kv.Key + \" -&gt; \" + kv.Value);\n}\n// Traverse keys only: key\nforeach (int key in map.Keys) {\n    Console.WriteLine(key);\n}\n// Traverse values only: value\nforeach (string val in map.Values) {\n    Console.WriteLine(val);\n}\n</code></pre> hash_map_test.go<pre><code>/* Traverse the hash table */\n// Traverse key-value pairs: key-&gt;value\nfor key, value := range hmap {\n    fmt.Println(key, \"-&gt;\", value)\n}\n// Traverse keys only: key\nfor key := range hmap {\n    fmt.Println(key)\n}\n// Traverse values only: value\nfor _, value := range hmap {\n    fmt.Println(value)\n}\n</code></pre> hash_map.swift<pre><code>/* Traverse the hash table */\n// Traverse key-value pairs: Key-&gt;Value\nfor (key, value) in map {\n    print(\"\\(key) -&gt; \\(value)\")\n}\n// Traverse keys only: Key\nfor key in map.keys {\n    print(key)\n}\n// Traverse values only: Value\nfor value in map.values {\n    print(value)\n}\n</code></pre> hash_map.js<pre><code>/* Traverse the hash table */\nconsole.info('\\nTraverse key-value pairs: Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\nTraverse keys only: Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\nTraverse values only: Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.ts<pre><code>/* Traverse the hash table */\nconsole.info('\\nTraverse key-value pairs: Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\nTraverse keys only: Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\nTraverse values only: Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.dart<pre><code>/* Traverse the hash table */\n// Traverse key-value pairs: Key-&gt;Value\nmap.forEach((key, value) {\n  print('$key -&gt; $value');\n});\n\n// Traverse keys only: Key\nmap.keys.forEach((key) {\n  print(key);\n});\n\n// Traverse values only: Value\nmap.values.forEach((value) {\n  print(value);\n});\n</code></pre> hash_map.rs<pre><code>/* Traverse the hash table */\n// Traverse key-value pairs: Key-&gt;Value\nfor (key, value) in &amp;map {\n    println!(\"{key} -&gt; {value}\");\n}\n\n// Traverse keys only: Key\nfor key in map.keys() {\n    println!(\"{key}\");\n}\n\n// Traverse values only: Value\nfor value in map.values() {\n    println!(\"{value}\");\n}\n</code></pre> hash_map.c<pre><code>// C provides no built-in hash table\n</code></pre> hash_map.kt<pre><code>/* Traverse the hash table */\n// Traverse key-value pairs: key-&gt;value\nfor ((key, value) in map) {\n    println(\"$key -&gt; $value\")\n}\n// Traverse keys only: key\nfor (key in map.keys) {\n    println(key)\n}\n// Traverse values only: value\nfor (_val in map.values) {\n    println(_val)\n}\n</code></pre> hash_map.rb<pre><code># Traverse the hash table\n# Traverse key-value pairs: key-&gt;value\nhmap.entries.each { |key, value| puts \"#{key} -&gt; #{value}\" }\n\n# Traverse keys only: key\nhmap.keys.each { |key| puts key }\n\n# Traverse values only: value\nhmap.values.each { |val| puts val }\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 6.   Hash tables","6.1   Hash tables"],"tags":[]},{"location":"chapter_hashing/hash_map/#612-a-simple-implementation","level":2,"title":"6.1.2   A simple implementation","text":"<p>Start with the simplest case: implement a hash table using just one array. Each slot is called a bucket and holds one key-value pair. A lookup finds the bucket corresponding to <code>key</code> and retrieves its <code>value</code>.</p> <p>How do we locate the bucket for a key? A hash function maps a larger input space into a smaller output space. Here, the inputs are all possible keys, and the outputs are bucket indices. Given a key, the hash function tells us where its key-value pair is stored in the array.</p> <p>For an input <code>key</code>, the calculation has two steps.</p> <ol> <li>Use a hash algorithm, <code>hash()</code>, to calculate a hash value.</li> <li>Take that value modulo the number of buckets, the array length <code>capacity</code>, to obtain the bucket index <code>index</code>.</li> </ol> <pre><code>index = hash(key) % capacity\n</code></pre> <p>Use <code>index</code> to access the bucket and retrieve <code>value</code>.</p> <p>If <code>capacity = 100</code> and <code>hash(key) = key</code>, the resulting hash function is <code>key % 100</code>. Figure 6-2 illustrates it with student IDs as keys and names as values.</p> <p></p> <p> Figure 6-2   How a hash function works </p> <p>The code below implements a simple hash table. The <code>Pair</code> class groups a <code>key</code> and <code>value</code> into a key-value pair.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_hash_map.py<pre><code>class Pair:\n    \"\"\"Key-value pair\"\"\"\n\n    def __init__(self, key: int, val: str):\n        self.key = key\n        self.val = val\n\nclass ArrayHashMap:\n    \"\"\"Hash table implemented with an array\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        # Initialize an array containing 100 buckets\n        self.buckets: list[Pair | None] = [None] * 100\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"Hash function\"\"\"\n        index = key % 100\n        return index\n\n    def get(self, key: int) -&gt; str | None:\n        \"\"\"Lookup operation\"\"\"\n        index: int = self.hash_func(key)\n        pair: Pair = self.buckets[index]\n        if pair is None:\n            return None\n        return pair.val\n\n    def put(self, key: int, val: str):\n        \"\"\"Insert and update operations\"\"\"\n        pair = Pair(key, val)\n        index: int = self.hash_func(key)\n        self.buckets[index] = pair\n\n    def remove(self, key: int):\n        \"\"\"Remove operation\"\"\"\n        index: int = self.hash_func(key)\n        # Set to None to mark removal\n        self.buckets[index] = None\n\n    def entry_set(self) -&gt; list[Pair]:\n        \"\"\"Get all key-value pairs\"\"\"\n        result: list[Pair] = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair)\n        return result\n\n    def key_set(self) -&gt; list[int]:\n        \"\"\"Get all keys\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.key)\n        return result\n\n    def value_set(self) -&gt; list[str]:\n        \"\"\"Get all values\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.val)\n        return result\n\n    def print(self):\n        \"\"\"Print the hash table\"\"\"\n        for pair in self.buckets:\n            if pair is not None:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> array_hash_map.cpp<pre><code>/* Key-value pair */\nstruct Pair {\n  public:\n    int key;\n    string val;\n    Pair(int key, string val) {\n        this-&gt;key = key;\n        this-&gt;val = val;\n    }\n};\n\n/* Hash table implemented with an array */\nclass ArrayHashMap {\n  private:\n    vector&lt;Pair *&gt; buckets;\n\n  public:\n    ArrayHashMap() {\n        // Initialize an array containing 100 buckets\n        buckets = vector&lt;Pair *&gt;(100);\n    }\n\n    ~ArrayHashMap() {\n        // Free memory\n        for (const auto &amp;bucket : buckets) {\n            delete bucket;\n        }\n        buckets.clear();\n    }\n\n    /* Hash function */\n    int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* Lookup operation */\n    string get(int key) {\n        int index = hashFunc(key);\n        Pair *pair = buckets[index];\n        if (pair == nullptr)\n            return \"\";\n        return pair-&gt;val;\n    }\n\n    /* Insert operation */\n    void put(int key, string val) {\n        Pair *pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* Remove operation */\n    void remove(int key) {\n        int index = hashFunc(key);\n        // Free the memory and set the pointer to nullptr\n        delete buckets[index];\n        buckets[index] = nullptr;\n    }\n\n    /* Get all key-value pairs */\n    vector&lt;Pair *&gt; pairSet() {\n        vector&lt;Pair *&gt; pairSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                pairSet.push_back(pair);\n            }\n        }\n        return pairSet;\n    }\n\n    /* Get all keys */\n    vector&lt;int&gt; keySet() {\n        vector&lt;int&gt; keySet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                keySet.push_back(pair-&gt;key);\n            }\n        }\n        return keySet;\n    }\n\n    /* Get all values */\n    vector&lt;string&gt; valueSet() {\n        vector&lt;string&gt; valueSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                valueSet.push_back(pair-&gt;val);\n            }\n        }\n        return valueSet;\n    }\n\n    /* Print the hash table */\n    void print() {\n        for (Pair *kv : pairSet()) {\n            cout &lt;&lt; kv-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; kv-&gt;val &lt;&lt; endl;\n        }\n    }\n};\n</code></pre> array_hash_map.java<pre><code>/* Key-value pair */\nclass Pair {\n    public int key;\n    public String val;\n\n    public Pair(int key, String val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* Hash table implemented with an array */\nclass ArrayHashMap {\n    private List&lt;Pair&gt; buckets;\n\n    public ArrayHashMap() {\n        // Initialize an array containing 100 buckets\n        buckets = new ArrayList&lt;&gt;();\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.add(null);\n        }\n    }\n\n    /* Hash function */\n    private int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* Lookup operation */\n    public String get(int key) {\n        int index = hashFunc(key);\n        Pair pair = buckets.get(index);\n        if (pair == null)\n            return null;\n        return pair.val;\n    }\n\n    /* Insert operation */\n    public void put(int key, String val) {\n        Pair pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets.set(index, pair);\n    }\n\n    /* Remove operation */\n    public void remove(int key) {\n        int index = hashFunc(key);\n        // Set the bucket to null to mark it as removed\n        buckets.set(index, null);\n    }\n\n    /* Get all key-value pairs */\n    public List&lt;Pair&gt; pairSet() {\n        List&lt;Pair&gt; pairSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                pairSet.add(pair);\n        }\n        return pairSet;\n    }\n\n    /* Get all keys */\n    public List&lt;Integer&gt; keySet() {\n        List&lt;Integer&gt; keySet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                keySet.add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* Get all values */\n    public List&lt;String&gt; valueSet() {\n        List&lt;String&gt; valueSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                valueSet.add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* Print the hash table */\n    public void print() {\n        for (Pair kv : pairSet()) {\n            System.out.println(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.cs<pre><code>/* Key-value pair: int-&gt;string */\nclass Pair(int key, string val) {\n    public int key = key;\n    public string val = val;\n}\n\n/* Hash table implemented with an array */\nclass ArrayHashMap {\n    List&lt;Pair?&gt; buckets;\n    public ArrayHashMap() {\n        // Initialize an array containing 100 buckets\n        buckets = [];\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.Add(null);\n        }\n    }\n\n    /* Hash function */\n    int HashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* Lookup operation */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        Pair? pair = buckets[index];\n        if (pair == null) return null;\n        return pair.val;\n    }\n\n    /* Insert operation */\n    public void Put(int key, string val) {\n        Pair pair = new(key, val);\n        int index = HashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* Remove operation */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // Set the bucket to null to mark it as removed\n        buckets[index] = null;\n    }\n\n    /* Get all key-value pairs */\n    public List&lt;Pair&gt; PairSet() {\n        List&lt;Pair&gt; pairSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                pairSet.Add(pair);\n        }\n        return pairSet;\n    }\n\n    /* Get all keys */\n    public List&lt;int&gt; KeySet() {\n        List&lt;int&gt; keySet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                keySet.Add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* Get all values */\n    public List&lt;string&gt; ValueSet() {\n        List&lt;string&gt; valueSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                valueSet.Add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* Print the hash table */\n    public void Print() {\n        foreach (Pair kv in PairSet()) {\n            Console.WriteLine(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.go<pre><code>/* Key-value pair */\ntype pair struct {\n    key int\n    val string\n}\n\n/* Hash table implemented with an array */\ntype arrayHashMap struct {\n    buckets []*pair\n}\n\n/* Initialize the hash table */\nfunc newArrayHashMap() *arrayHashMap {\n    // Initialize an array containing 100 buckets\n    buckets := make([]*pair, 100)\n    return &amp;arrayHashMap{buckets: buckets}\n}\n\n/* Hash function */\nfunc (a *arrayHashMap) hashFunc(key int) int {\n    index := key % 100\n    return index\n}\n\n/* Lookup operation */\nfunc (a *arrayHashMap) get(key int) string {\n    index := a.hashFunc(key)\n    pair := a.buckets[index]\n    if pair == nil {\n        return \"Not Found\"\n    }\n    return pair.val\n}\n\n/* Insert operation */\nfunc (a *arrayHashMap) put(key int, val string) {\n    pair := &amp;pair{key: key, val: val}\n    index := a.hashFunc(key)\n    a.buckets[index] = pair\n}\n\n/* Remove operation */\nfunc (a *arrayHashMap) remove(key int) {\n    index := a.hashFunc(key)\n    // Set to nil to mark the entry as removed\n    a.buckets[index] = nil\n}\n\n/* Get all key-value pairs */\nfunc (a *arrayHashMap) pairSet() []*pair {\n    var pairs []*pair\n    for _, pair := range a.buckets {\n        if pair != nil {\n            pairs = append(pairs, pair)\n        }\n    }\n    return pairs\n}\n\n/* Get all keys */\nfunc (a *arrayHashMap) keySet() []int {\n    var keys []int\n    for _, pair := range a.buckets {\n        if pair != nil {\n            keys = append(keys, pair.key)\n        }\n    }\n    return keys\n}\n\n/* Get all values */\nfunc (a *arrayHashMap) valueSet() []string {\n    var values []string\n    for _, pair := range a.buckets {\n        if pair != nil {\n            values = append(values, pair.val)\n        }\n    }\n    return values\n}\n\n/* Print the hash table */\nfunc (a *arrayHashMap) print() {\n    for _, pair := range a.buckets {\n        if pair != nil {\n            fmt.Println(pair.key, \"-&gt;\", pair.val)\n        }\n    }\n}\n</code></pre> array_hash_map.swift<pre><code>/* Key-value pair */\nclass Pair: Equatable {\n    public var key: Int\n    public var val: String\n\n    public init(key: Int, val: String) {\n        self.key = key\n        self.val = val\n    }\n\n    public static func == (lhs: Pair, rhs: Pair) -&gt; Bool {\n        lhs.key == rhs.key &amp;&amp; lhs.val == rhs.val\n    }\n}\n\n/* Hash table implemented with an array */\nclass ArrayHashMap {\n    private var buckets: [Pair?]\n\n    init() {\n        // Initialize an array containing 100 buckets\n        buckets = Array(repeating: nil, count: 100)\n    }\n\n    /* Hash function */\n    private func hashFunc(key: Int) -&gt; Int {\n        let index = key % 100\n        return index\n    }\n\n    /* Lookup operation */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let pair = buckets[index]\n        return pair?.val\n    }\n\n    /* Insert operation */\n    func put(key: Int, val: String) {\n        let pair = Pair(key: key, val: val)\n        let index = hashFunc(key: key)\n        buckets[index] = pair\n    }\n\n    /* Remove operation */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        // Set to nil to mark the entry as removed\n        buckets[index] = nil\n    }\n\n    /* Get all key-value pairs */\n    func pairSet() -&gt; [Pair] {\n        buckets.compactMap { $0 }\n    }\n\n    /* Get all keys */\n    func keySet() -&gt; [Int] {\n        buckets.compactMap { $0?.key }\n    }\n\n    /* Get all values */\n    func valueSet() -&gt; [String] {\n        buckets.compactMap { $0?.val }\n    }\n\n    /* Print the hash table */\n    func print() {\n        for pair in pairSet() {\n            Swift.print(\"\\(pair.key) -&gt; \\(pair.val)\")\n        }\n    }\n}\n</code></pre> array_hash_map.js<pre><code>/* Key-value pair: Number -&gt; String */\nclass Pair {\n    constructor(key, val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* Hash table implemented with an array */\nclass ArrayHashMap {\n    #buckets;\n    constructor() {\n        // Initialize an array containing 100 buckets\n        this.#buckets = new Array(100).fill(null);\n    }\n\n    /* Hash function */\n    #hashFunc(key) {\n        return key % 100;\n    }\n\n    /* Lookup operation */\n    get(key) {\n        let index = this.#hashFunc(key);\n        let pair = this.#buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* Insert operation */\n    set(key, val) {\n        let index = this.#hashFunc(key);\n        this.#buckets[index] = new Pair(key, val);\n    }\n\n    /* Remove operation */\n    delete(key) {\n        let index = this.#hashFunc(key);\n        // Set the bucket to null to mark it as deleted\n        this.#buckets[index] = null;\n    }\n\n    /* Get all key-value pairs */\n    entries() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* Get all keys */\n    keys() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* Get all values */\n    values() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* Print the hash table */\n    print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.ts<pre><code>/* Key-value pair: Number -&gt; String */\nclass Pair {\n    public key: number;\n    public val: string;\n\n    constructor(key: number, val: string) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* Hash table implemented with an array */\nclass ArrayHashMap {\n    private readonly buckets: (Pair | null)[];\n\n    constructor() {\n        // Initialize an array containing 100 buckets\n        this.buckets = new Array(100).fill(null);\n    }\n\n    /* Hash function */\n    private hashFunc(key: number): number {\n        return key % 100;\n    }\n\n    /* Lookup operation */\n    public get(key: number): string | null {\n        let index = this.hashFunc(key);\n        let pair = this.buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* Insert operation */\n    public set(key: number, val: string) {\n        let index = this.hashFunc(key);\n        this.buckets[index] = new Pair(key, val);\n    }\n\n    /* Remove operation */\n    public delete(key: number) {\n        let index = this.hashFunc(key);\n        // Set the bucket to null to mark it as deleted\n        this.buckets[index] = null;\n    }\n\n    /* Get all key-value pairs */\n    public entries(): (Pair | null)[] {\n        let arr: (Pair | null)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* Get all keys */\n    public keys(): (number | undefined)[] {\n        let arr: (number | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* Get all values */\n    public values(): (string | undefined)[] {\n        let arr: (string | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* Print the hash table */\n    public print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.dart<pre><code>/* Key-value pair */\nclass Pair {\n  int key;\n  String val;\n  Pair(this.key, this.val);\n}\n\n/* Hash table implemented with an array */\nclass ArrayHashMap {\n  late List&lt;Pair?&gt; _buckets;\n\n  ArrayHashMap() {\n    // Initialize an array containing 100 buckets\n    _buckets = List.filled(100, null);\n  }\n\n  /* Hash function */\n  int _hashFunc(int key) {\n    final int index = key % 100;\n    return index;\n  }\n\n  /* Lookup operation */\n  String? get(int key) {\n    final int index = _hashFunc(key);\n    final Pair? pair = _buckets[index];\n    if (pair == null) {\n      return null;\n    }\n    return pair.val;\n  }\n\n  /* Insert operation */\n  void put(int key, String val) {\n    final Pair pair = Pair(key, val);\n    final int index = _hashFunc(key);\n    _buckets[index] = pair;\n  }\n\n  /* Remove operation */\n  void remove(int key) {\n    final int index = _hashFunc(key);\n    _buckets[index] = null;\n  }\n\n  /* Get all key-value pairs */\n  List&lt;Pair&gt; pairSet() {\n    List&lt;Pair&gt; pairSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        pairSet.add(pair);\n      }\n    }\n    return pairSet;\n  }\n\n  /* Get all keys */\n  List&lt;int&gt; keySet() {\n    List&lt;int&gt; keySet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        keySet.add(pair.key);\n      }\n    }\n    return keySet;\n  }\n\n  /* Get all values */\n  List&lt;String&gt; values() {\n    List&lt;String&gt; valueSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        valueSet.add(pair.val);\n      }\n    }\n    return valueSet;\n  }\n\n  /* Print the hash table */\n  void printHashMap() {\n    for (final Pair kv in pairSet()) {\n      print(\"${kv.key} -&gt; ${kv.val}\");\n    }\n  }\n}\n</code></pre> array_hash_map.rs<pre><code>/* Key-value pair */\n#[derive(Debug, Clone, PartialEq)]\npub struct Pair {\n    pub key: i32,\n    pub val: String,\n}\n\n/* Hash table implemented with an array */\npub struct ArrayHashMap {\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;,\n}\n\nimpl ArrayHashMap {\n    pub fn new() -&gt; ArrayHashMap {\n        // Initialize an array containing 100 buckets\n        Self {\n            buckets: vec![None; 100],\n        }\n    }\n\n    /* Hash function */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % 100\n    }\n\n    /* Lookup operation */\n    pub fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;String&gt; {\n        let index = self.hash_func(key);\n        self.buckets[index].as_ref().map(|pair| &amp;pair.val)\n    }\n\n    /* Insert operation */\n    pub fn put(&amp;mut self, key: i32, val: &amp;str) {\n        let index = self.hash_func(key);\n        self.buckets[index] = Some(Pair {\n            key,\n            val: val.to_string(),\n        });\n    }\n\n    /* Remove operation */\n    pub fn remove(&amp;mut self, key: i32) {\n        let index = self.hash_func(key);\n        // Set to None to mark removal\n        self.buckets[index] = None;\n    }\n\n    /* Get all key-value pairs */\n    pub fn entry_set(&amp;self) -&gt; Vec&lt;&amp;Pair&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref())\n            .collect()\n    }\n\n    /* Get all keys */\n    pub fn key_set(&amp;self) -&gt; Vec&lt;&amp;i32&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.key))\n            .collect()\n    }\n\n    /* Get all values */\n    pub fn value_set(&amp;self) -&gt; Vec&lt;&amp;String&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.val))\n            .collect()\n    }\n\n    /* Print the hash table */\n    pub fn print(&amp;self) {\n        for pair in self.entry_set() {\n            println!(\"{} -&gt; {}\", pair.key, pair.val);\n        }\n    }\n}\n</code></pre> array_hash_map.c<pre><code>/* Key-value pair: int -&gt; string */\ntypedef struct {\n    int key;\n    char *val;\n} Pair;\n\n/* Hash table implemented with an array */\ntypedef struct {\n    Pair *buckets[MAX_SIZE];\n} ArrayHashMap;\n\n/* Constructor */\nArrayHashMap *newArrayHashMap() {\n    ArrayHashMap *hmap = malloc(sizeof(ArrayHashMap));\n    for (int i=0; i &lt; MAX_SIZE; i++) {\n        hmap-&gt;buckets[i] = NULL;\n    }\n    return hmap;\n}\n\n/* Destructor */\nvoid delArrayHashMap(ArrayHashMap *hmap) {\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            free(hmap-&gt;buckets[i]-&gt;val);\n            free(hmap-&gt;buckets[i]);\n        }\n    }\n    free(hmap);\n}\n\n/* Insert operation */\nvoid put(ArrayHashMap *hmap, const int key, const char *val) {\n    Pair *Pair = malloc(sizeof(Pair));\n    Pair-&gt;key = key;\n    Pair-&gt;val = malloc(strlen(val) + 1);\n    strcpy(Pair-&gt;val, val);\n\n    int index = hashFunc(key);\n    hmap-&gt;buckets[index] = Pair;\n}\n\n/* Remove operation */\nvoid removeItem(ArrayHashMap *hmap, const int key) {\n    int index = hashFunc(key);\n    free(hmap-&gt;buckets[index]-&gt;val);\n    free(hmap-&gt;buckets[index]);\n    hmap-&gt;buckets[index] = NULL;\n}\n\n/* Get all key-value pairs */\nvoid pairSet(ArrayHashMap *hmap, MapSet *set) {\n    Pair *entries;\n    int i = 0, index = 0;\n    int total = 0;\n    /* Count stored key-value pairs */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    entries = malloc(sizeof(Pair) * total);\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            entries[index].key = hmap-&gt;buckets[i]-&gt;key;\n            entries[index].val = malloc(strlen(hmap-&gt;buckets[i]-&gt;val) + 1);\n            strcpy(entries[index].val, hmap-&gt;buckets[i]-&gt;val);\n            index++;\n        }\n    }\n    set-&gt;set = entries;\n    set-&gt;len = total;\n}\n\n/* Get all keys */\nvoid keySet(ArrayHashMap *hmap, MapSet *set) {\n    int *keys;\n    int i = 0, index = 0;\n    int total = 0;\n    /* Count stored key-value pairs */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    keys = malloc(total * sizeof(int));\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            keys[index] = hmap-&gt;buckets[i]-&gt;key;\n            index++;\n        }\n    }\n    set-&gt;set = keys;\n    set-&gt;len = total;\n}\n\n/* Get all values */\nvoid valueSet(ArrayHashMap *hmap, MapSet *set) {\n    char **vals;\n    int i = 0, index = 0;\n    int total = 0;\n    /* Count stored key-value pairs */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    vals = malloc(total * sizeof(char *));\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            vals[index] = hmap-&gt;buckets[i]-&gt;val;\n            index++;\n        }\n    }\n    set-&gt;set = vals;\n    set-&gt;len = total;\n}\n\n/* Print the hash table */\nvoid print(ArrayHashMap *hmap) {\n    int i;\n    MapSet set;\n    pairSet(hmap, &amp;set);\n    Pair *entries = (Pair *)set.set;\n    for (i = 0; i &lt; set.len; i++) {\n        printf(\"%d -&gt; %s\\n\", entries[i].key, entries[i].val);\n    }\n    free(set.set);\n}\n</code></pre> array_hash_map.kt<pre><code>/* Key-value pair */\nclass Pair(\n    var key: Int,\n    var _val: String\n)\n\n/* Hash table implemented with an array */\nclass ArrayHashMap {\n    // Initialize an array containing 100 buckets\n    private val buckets = arrayOfNulls&lt;Pair&gt;(100)\n\n    /* Hash function */\n    fun hashFunc(key: Int): Int {\n        val index = key % 100\n        return index\n    }\n\n    /* Lookup operation */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val pair = buckets[index] ?: return null\n        return pair._val\n    }\n\n    /* Insert operation */\n    fun put(key: Int, _val: String) {\n        val pair = Pair(key, _val)\n        val index = hashFunc(key)\n        buckets[index] = pair\n    }\n\n    /* Remove operation */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        // Set the bucket to null to mark it as removed\n        buckets[index] = null\n    }\n\n    /* Get all key-value pairs */\n    fun pairSet(): MutableList&lt;Pair&gt; {\n        val pairSet = mutableListOf&lt;Pair&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                pairSet.add(pair)\n        }\n        return pairSet\n    }\n\n    /* Get all keys */\n    fun keySet(): MutableList&lt;Int&gt; {\n        val keySet = mutableListOf&lt;Int&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                keySet.add(pair.key)\n        }\n        return keySet\n    }\n\n    /* Get all values */\n    fun valueSet(): MutableList&lt;String&gt; {\n        val valueSet = mutableListOf&lt;String&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                valueSet.add(pair._val)\n        }\n        return valueSet\n    }\n\n    /* Print the hash table */\n    fun print() {\n        for (kv in pairSet()) {\n            val key = kv.key\n            val _val = kv._val\n            println(\"$key -&gt; $_val\")\n        }\n    }\n}\n</code></pre> array_hash_map.rb<pre><code>### Key-value pair ###\nclass Pair\n  attr_accessor :key, :val\n\n  def initialize(key, val)\n    @key = key\n    @val = val\n  end\nend\n\n### Hash table implemented with an array ###\nclass ArrayHashMap\n  ### Constructor ###\n  def initialize\n    # Initialize an array containing 100 buckets\n    @buckets = Array.new(100)\n  end\n\n  ### Hash function ###\n  def hash_func(key)\n    index = key % 100\n  end\n\n  ### Lookup operation ###\n  def get(key)\n    index = hash_func(key)\n    pair = @buckets[index]\n\n    return if pair.nil?\n    pair.val\n  end\n\n  ### Insert operation ###\n  def put(key, val)\n    pair = Pair.new(key, val)\n    index = hash_func(key)\n    @buckets[index] = pair\n  end\n\n  ### Remove operation ###\n  def remove(key)\n    index = hash_func(key)\n    # Set the bucket to nil to indicate removal\n    @buckets[index] = nil\n  end\n\n  ### Get all key-value pairs ###\n  def entry_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair unless pair.nil? }\n    result\n  end\n\n  ### Get all keys ###\n  def key_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair.key unless pair.nil? }\n    result\n  end\n\n  ### Get all values ###\n  def value_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair.val unless pair.nil? }\n    result\n  end\n\n  ### Print the hash table ###\n  def print\n    @buckets.each { |pair| puts \"#{pair.key} -&gt; #{pair.val}\" unless pair.nil? }\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 6.   Hash tables","6.1   Hash tables"],"tags":[]},{"location":"chapter_hashing/hash_map/#613-collisions-and-expansion","level":2,"title":"6.1.3   Collisions and expansion","text":"<p>A hash function maps the input space of all possible keys into the output space of array indices. The input space is usually far larger, so multiple inputs must theoretically map to the same output.</p> <p>For the function above, keys with the same last two digits produce the same output. Looking up student IDs 12836 and 20336 gives:</p> <pre><code>12836 % 100 = 36\n20336 % 100 = 36\n</code></pre> <p>As shown below, both IDs point to the same name, which is clearly incorrect. This situation, where different inputs produce the same output, is called a hash collision.</p> <p></p> <p> Figure 6-3   An example of a hash collision </p> <p>A larger table capacity \\(n\\) makes it less likely that several keys share a bucket, reducing collisions. Expanding the hash table can therefore reduce hash collisions.</p> <p>In the example below, <code>(136, A)</code> and <code>(236, D)</code> collide before expansion but no longer collide afterward.</p> <p></p> <p> Figure 6-4   Expanding a hash table </p> <p>As with array expansion, expanding a hash table requires moving every pair into a new table, which is time-consuming. Because <code>capacity</code> changes, the hash function must also recalculate every pair's position, adding further computation. Programming languages therefore usually reserve enough capacity to avoid frequent expansion.</p> <p>The load factor is the number of elements divided by the number of buckets. It measures how crowded the table is and often serves as the trigger for expansion. For example, in Java, a load factor above \\(0.75\\) causes the capacity to grow to \\(2\\) times its previous size.</p>","path":["Chapter 6.   Hash tables","6.1   Hash tables"],"tags":[]},{"location":"chapter_hashing/summary/","level":1,"title":"6.4   Summary","text":"","path":["Chapter 6.   Hash tables","6.4   Summary"],"tags":[]},{"location":"chapter_hashing/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<ul> <li>Given a <code>key</code>, a hash table can retrieve its <code>value</code> in \\(O(1)\\) time, making lookup highly efficient.</li> <li>Common operations include lookup, adding and deleting key-value pairs, and traversing the table.</li> <li>A hash function maps a <code>key</code> to an array index, allowing access to the corresponding bucket and its <code>value</code>.</li> <li>Two different keys may map to the same array index and cause incorrect lookup results. This is called a hash collision.</li> <li>A larger capacity lowers the likelihood of collisions, so expanding a hash table can alleviate them. Like array expansion, hash-table expansion is expensive.</li> <li>The load factor is the number of elements divided by the number of buckets. It indicates how crowded the table is and is commonly used to trigger expansion.</li> <li>Separate chaining replaces a single entry with a linked list and stores all colliding entries in that list. Long lists slow lookup; converting them to red-black trees can improve efficiency.</li> <li>Open addressing resolves collisions through repeated probing. Linear probing uses a fixed step size, is prone to clustering, and does not allow straightforward deletion. Multiple hashing uses several hash functions, reducing clustering but adding computational cost.</li> <li>Programming languages use different implementations: Java's <code>HashMap</code> uses separate chaining, while Python's <code>dict</code> uses open addressing.</li> <li>For hash tables, we want deterministic, efficient hashing with uniformly distributed outputs. Cryptographic hashing also requires collision resistance and an avalanche effect.</li> <li>Hash algorithms often use a large prime modulus to promote uniform hash values and reduce collisions.</li> <li>Common hash algorithms include MD5, SHA-1, SHA-2, and SHA-3. MD5 is often used for file-integrity checks, while SHA-2 is widely used in security applications and protocols.</li> <li>Programming languages usually provide built-in hashing for data types to calculate bucket indices. In general, only immutable objects are hashable.</li> </ul>","path":["Chapter 6.   Hash tables","6.4   Summary"],"tags":[]},{"location":"chapter_hashing/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: When does a hash table have \\(O(n)\\) time complexity?</p> <p>When collisions are severe, its time complexity degrades to \\(O(n)\\). With a well-designed hash function, suitable capacity, and evenly distributed collisions, it is \\(O(1)\\). We usually assume \\(O(1)\\) time when using a language's built-in hash table.</p> <p>Q: Why not use the hash function \\(f(x) = x\\)? Wouldn't that avoid collisions?</p> <p>With \\(f(x) = x\\), each element has a unique bucket index, making the structure equivalent to an array. But the input space is usually far larger than the output space, the array's length, so the final hashing step usually takes a remainder modulo that length. A hash table maps a larger state space into a smaller one while providing \\(O(1)\\) lookup.</p> <p>Q: Hash tables are implemented with arrays, linked lists, and binary trees. How can they be faster than those structures?</p> <p>First, greater time efficiency comes at the cost of space efficiency: a significant portion of a hash table's memory is unused.</p> <p>Second, the improvement applies only to particular operations. If an array or linked list can perform an operation with the same time complexity, it is usually faster because computing a hash adds overhead and increases the constant factor.</p> <p>Finally, hash-table performance can degrade. For example, separate chaining searches within linked lists or red-black trees, and there is still a risk of degeneration to \\(O(n)\\) time.</p> <p>Q: Does multiple hashing also prevent direct deletion? Can a deleted slot be reused?</p> <p>Multiple hashing is a form of open addressing, and all open-addressing methods require deletion markers rather than simply emptying a slot. A marked slot can be reused. When inserting a new element, if the hash functions locate such a slot, the new element can occupy it. This preserves the probe sequence while making effective use of space.</p> <p>Q: Why can a collision occur during a lookup with linear probing?</p> <p>The hash function leads to a bucket and a key-value pair, but the stored <code>key</code> may not match the requested one. That indicates a collision. Linear probing then continues with its predefined step size until it finds the correct pair or determines that none exists.</p> <p>Q: Why does expanding a hash table reduce collisions?</p> <p>The final hashing step often takes the remainder modulo the array length \\(n\\) to produce a valid index. After expansion, \\(n\\) changes, so the index of a given <code>key</code> may change too. Keys that previously shared a bucket may be redistributed among several buckets, reducing collisions.</p> <p>Q: If fast access is the goal, why not just use an array?</p> <p>When keys are consecutive integers in a small range, an array is simple and efficient. But for other key types, such as strings, we need a hash function to map keys to array indices and a bucket array to store entries. That structure is a hash table.</p>","path":["Chapter 6.   Hash tables","6.4   Summary"],"tags":[]},{"location":"chapter_heap/","level":1,"title":"Chapter 8.   Heaps","text":"<p>Abstract</p> <p>Heaps resemble mountain ranges, rising in layers of varied shapes.</p> <p>The peaks stand at different heights, yet the highest always catches the eye first.</p>","path":["Chapter 8.   Heaps"],"tags":[]},{"location":"chapter_heap/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>8.1   Heaps</li> <li>8.2   Building a heap</li> <li>8.3   The top-k problem</li> <li>8.4   Summary</li> <li>8.5   Exercises</li> </ul>","path":["Chapter 8.   Heaps"],"tags":[]},{"location":"chapter_heap/build_heap/","level":1,"title":"8.2   Building a heap","text":"<p>Sometimes we want to construct a heap from all the elements of a list. This process is called “building a heap.”</p>","path":["Chapter 8.   Heaps","8.2   Building a heap"],"tags":[]},{"location":"chapter_heap/build_heap/#821-building-through-insertion","level":2,"title":"8.2.1   Building through insertion","text":"<p>First create an empty heap, then traverse the list and insert each element: append it to the end of the heap, then sift it upward.</p> <p>Each insertion increases the heap's length by one. Since nodes are added to the binary tree from the top down, this constructs the heap “top down.”</p> <p>For \\(n\\) elements, each insertion takes \\(O(\\log{n})\\) time, so this method takes \\(O(n \\log n)\\) overall.</p>","path":["Chapter 8.   Heaps","8.2   Building a heap"],"tags":[]},{"location":"chapter_heap/build_heap/#822-building-by-sifting-nodes-down","level":2,"title":"8.2.2   Building by sifting nodes down","text":"<p>We can build a heap more efficiently in two steps.</p> <ol> <li>Copy all list elements into the heap without changing their order. The heap property may not yet hold.</li> <li>Traverse the heap in reverse level order, sifting down each non-leaf node.</li> </ol> <p>After a node is sifted down, the subtree rooted at that node is a valid heap. Since the traversal is reversed, the heap is built “bottom up.”</p> <p>Reverse traversal ensures that the subtrees below the current node are already valid heaps, which is necessary for sifting down the current node to work correctly.</p> <p>Leaf nodes have no children and are already valid heaps, so they need no sifting. As the code below shows, the last non-leaf node is the parent of the last node. Start there and traverse backward, sifting each node down:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def __init__(self, nums: list[int]):\n    \"\"\"Constructor: build a heap from the input list\"\"\"\n    # Place the list elements in the heap without changing their order\n    self.max_heap = nums\n    # Sift down from every non-leaf node\n    for i in range(self.parent(self.size() - 1), -1, -1):\n        self.sift_down(i)\n</code></pre> my_heap.cpp<pre><code>/* Constructor: build a heap from the input list */\nMaxHeap(vector&lt;int&gt; nums) {\n    // Place the list elements in the heap without changing their order\n    maxHeap = nums;\n    // Sift down from every non-leaf node\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.java<pre><code>/* Constructor: build a heap from the input list */\nMaxHeap(List&lt;Integer&gt; nums) {\n    // Place the list elements in the heap without changing their order\n    maxHeap = new ArrayList&lt;&gt;(nums);\n    // Sift down from every non-leaf node\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* Constructor: build a heap from an input list */\nMaxHeap(IEnumerable&lt;int&gt; nums) {\n    // Place the list elements in the heap without changing their order\n    maxHeap = new List&lt;int&gt;(nums);\n    // Sift down from every non-leaf node\n    var size = Parent(this.Size() - 1);\n    for (int i = size; i &gt;= 0; i--) {\n        SiftDown(i);\n    }\n}\n</code></pre> my_heap.go<pre><code>/* Constructor: build a heap from a slice */\nfunc newMaxHeap(nums []any) *maxHeap {\n    // Place the list elements in the heap without changing their order\n    h := &amp;maxHeap{data: nums}\n    for i := h.parent(len(h.data) - 1); i &gt;= 0; i-- {\n        // Sift down from every non-leaf node\n        h.siftDown(i)\n    }\n    return h\n}\n</code></pre> my_heap.swift<pre><code>/* Constructor: build a heap from the input list */\ninit(nums: [Int]) {\n    // Place the list elements in the heap without changing their order\n    maxHeap = nums\n    // Sift down from every non-leaf node\n    for i in (0 ... parent(i: size() - 1)).reversed() {\n        siftDown(i: i)\n    }\n}\n</code></pre> my_heap.js<pre><code>/* Constructor: create an empty heap or build a heap from the input list */\nconstructor(nums) {\n    // Place the list elements in the heap without changing their order\n    this.#maxHeap = nums === undefined ? [] : [...nums];\n    // Sift down from every non-leaf node\n    for (let i = this.#parent(this.size() - 1); i &gt;= 0; i--) {\n        this.#siftDown(i);\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* Constructor: create an empty heap or build a heap from the input list */\nconstructor(nums?: number[]) {\n    // Place the list elements in the heap without changing their order\n    this.maxHeap = nums === undefined ? [] : [...nums];\n    // Sift down from every non-leaf node\n    for (let i = this.parent(this.size() - 1); i &gt;= 0; i--) {\n        this.siftDown(i);\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* Constructor: build a heap from the input list */\nMaxHeap(List&lt;int&gt; nums) {\n  // Place the list elements in the heap without changing their order\n  _maxHeap = nums;\n  // Sift down from every non-leaf node\n  for (int i = _parent(size() - 1); i &gt;= 0; i--) {\n    siftDown(i);\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* Constructor: build a heap from the input list */\nfn new(nums: Vec&lt;i32&gt;) -&gt; Self {\n    // Place the list elements in the heap without changing their order\n    let mut heap = MaxHeap { max_heap: nums };\n    // Sift down from every non-leaf node\n    for i in (0..=Self::parent(heap.size() - 1)).rev() {\n        heap.sift_down(i);\n    }\n    heap\n}\n</code></pre> my_heap.c<pre><code>/* Constructor: build a heap from the input array */\nMaxHeap *newMaxHeap(int nums[], int size) {\n    // Place all elements in the heap\n    MaxHeap *maxHeap = (MaxHeap *)malloc(sizeof(MaxHeap));\n    maxHeap-&gt;size = size;\n    memcpy(maxHeap-&gt;data, nums, size * sizeof(int));\n    for (int i = parent(maxHeap, size - 1); i &gt;= 0; i--) {\n        // Sift down from every non-leaf node\n        siftDown(maxHeap, i);\n    }\n    return maxHeap;\n}\n</code></pre> my_heap.kt<pre><code>/* Max heap */\nclass MaxHeap(nums: MutableList&lt;Int&gt;?) {\n    // Use a resizable list instead of an array to avoid managing capacity growth\n    private val maxHeap = mutableListOf&lt;Int&gt;()\n\n    /* Constructor: build a heap from the input list */\n    init {\n        // Place the list elements in the heap without changing their order\n        maxHeap.addAll(nums!!)\n        // Sift down from every non-leaf node\n        for (i in parent(size() - 1) downTo 0) {\n            siftDown(i)\n        }\n    }\n\n    /* Get the left child index */\n    private fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* Get the right child index */\n    private fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* Get the parent index */\n    private fun parent(i: Int): Int {\n        return (i - 1) / 2 // Integer division (truncates toward zero in Kotlin)\n    }\n\n    /* Swap elements */\n    private fun swap(i: Int, j: Int) {\n        val temp = maxHeap[i]\n        maxHeap[i] = maxHeap[j]\n        maxHeap[j] = temp\n    }\n\n    /* Get the heap size */\n    fun size(): Int {\n        return maxHeap.size\n    }\n\n    /* Check whether the heap is empty */\n    fun isEmpty(): Boolean {\n        /* Check whether the heap is empty */\n        return size() == 0\n    }\n\n    /* Access the top element of the heap */\n    fun peek(): Int {\n        return maxHeap[0]\n    }\n\n    /* Push an element onto the heap */\n    fun push(_val: Int) {\n        // Add a node\n        maxHeap.add(_val)\n        // Sift up\n        siftUp(size() - 1)\n    }\n\n    /* Sift up starting from node i */\n    private fun siftUp(it: Int) {\n        // Kotlin function parameters are immutable, so create a temporary variable\n        var i = it\n        while (true) {\n            // Get the parent of node i\n            val p = parent(i)\n            // Stop sifting when the root has been passed or the node needs no repair\n            if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n            // Swap the two nodes\n            swap(i, p)\n            // Continue sifting up\n            i = p\n        }\n    }\n\n    /* Pop an element from the heap */\n    fun pop(): Int {\n        // Handle an empty heap\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // Swap the root with the rightmost leaf on the last level (swap the first and last elements)\n        swap(0, size() - 1)\n        // Remove a node\n        val _val = maxHeap.removeAt(size() - 1)\n        // Sift down\n        siftDown(0)\n        // Return the top element of the heap\n        return _val\n    }\n\n    /* Sift down starting from node i */\n    private fun siftDown(it: Int) {\n        // Kotlin function parameters are immutable, so create a temporary variable\n        var i = it\n        while (true) {\n            // Find the node with the largest value among i, l, and r, and store its index in ma\n            val l = left(i)\n            val r = right(i)\n            var ma = i\n            if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n            if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n            // Stop if node i is largest or both child indices l and r are out of bounds\n            if (ma == i) break\n            // Swap the two nodes\n            swap(i, ma)\n            // Continue sifting down\n            i = ma\n        }\n    }\n\n    /* Print the heap (binary tree) */\n    fun print() {\n        val queue = PriorityQueue { a: Int, b: Int -&gt; b - a }\n        queue.addAll(maxHeap)\n        printHeap(queue)\n    }\n}\n</code></pre> my_heap.rb<pre><code>### Constructor: build a heap from the input list ###\ndef initialize(nums)\n  # Place the list elements in the heap without changing their order\n  @max_heap = nums\n  # Sift down from every non-leaf node\n  parent(size - 1).downto(0) do |i|\n    sift_down(i)\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 8.   Heaps","8.2   Building a heap"],"tags":[]},{"location":"chapter_heap/build_heap/#823-complexity-analysis","level":2,"title":"8.2.3   Complexity analysis","text":"<p>Let us derive the time complexity of the second method.</p> <ul> <li>A complete binary tree with \\(n\\) nodes has \\((n + 1) / 2\\) leaves, where \\(/\\) denotes integer division rounded down. Therefore, \\(n / 2\\) nodes need sifting.</li> <li>A node can move at most as far as a leaf during sifting, so the maximum number of iterations is the tree height, \\(\\log n\\).</li> </ul> <p>Multiplying these gives a time complexity of \\(O(n \\log n)\\). However, this estimate is not tight, because it ignores the fact that there are far more nodes near the bottom of the tree than near the top.</p> <p>For a more accurate calculation, assume a perfect binary tree with \\(n\\) nodes and height \\(h\\). This simplifies the calculation without affecting the validity of the result.</p> <p></p> <p> Figure 8-5   Number of nodes at each level of a perfect binary tree </p> <p>As shown above, the maximum number of sift-down iterations for a node equals its distance to a leaf, which is its height. Summing “number of nodes \\(\\times\\) node height” over all levels therefore gives the total number of sifting iterations across all nodes.</p> \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{(h-1)}\\times1 \\] <p>To simplify this expression, use a standard series technique. First multiply \\(T(h)\\) by \\(2\\):</p> \\[ \\begin{aligned} T(h) &amp; = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{h-1}\\times1 \\newline 2 T(h) &amp; = 2^1h + 2^2(h-1) + 2^3(h-2) + \\dots + 2^{h}\\times1 \\newline \\end{aligned} \\] <p>Subtract the first expression, \\(T(h)\\), from the shifted expression, \\(2 T(h)\\):</p> \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\dots + 2^{h-1} + 2^h \\] <p>The powers of two in \\(T(h)\\) form a geometric series. Applying the sum formula gives:</p> \\[ \\begin{aligned} T(h) &amp; = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline &amp; = 2^{h+1} - h - 2 \\newline &amp; = O(2^h) \\end{aligned} \\] <p>A perfect binary tree of height \\(h\\) has \\(n = 2^{h+1} - 1\\) nodes, so \\(O(2^h) = O(n)\\). This shows that building a heap from an input list takes \\(O(n)\\) time and is highly efficient.</p>","path":["Chapter 8.   Heaps","8.2   Building a heap"],"tags":[]},{"location":"chapter_heap/exercises/","level":1,"title":"8.5   Exercises","text":"","path":["Chapter 8.   Heaps","8.5   Exercises"],"tags":[]},{"location":"chapter_heap/exercises/#851-check-your-understanding","level":2,"title":"8.5.1   Check your understanding","text":"","path":["Chapter 8.   Heaps","8.5   Exercises"],"tags":[]},{"location":"chapter_heap/exercises/#1-restoring-the-heap-after-inserting-10","level":3,"title":"1.   Restoring the heap after inserting 10","text":"<p>The array <code>[9, 7, 8, 3, 5]</code> represents a max heap. Insert the number 10.</p> <ol> <li>After appending 10 to the array, what is its parent's value?</li> <li>Sift up from the new node. Write the array after each swap.</li> <li>What is the final root element? How many swaps were made?</li> </ol> Reference answer <ol> <li> <p>The appended 10 has index 5, so its parent's index is     \\(\\lfloor(5-1)/2\\rfloor=2\\), and its parent's value is 8.</p> </li> <li> <p>Since 10 is greater than 8, the first swap produces <code>[9, 7, 10, 3, 5, 8]</code>.     Since 10 is also greater than its new parent, 9, the second swap produces <code>[10, 7, 9, 3, 5, 8]</code>.     The value 10 has now reached the root, so sifting stops.</p> </li> <li> <p>The final root is 10, and there were two swaps.</p> </li> </ol>","path":["Chapter 8.   Heaps","8.5   Exercises"],"tags":[]},{"location":"chapter_heap/exercises/#2-checking-parent-child-relationships-in-a-min-heap","level":3,"title":"2.   Checking parent-child relationships in a min heap","text":"<p>The array <code>[1, 4, 3, 7, 6, 2]</code> represents a complete binary tree. A min heap requires each parent to be no greater than its children. For index \\(i\\), the left and right child indices are \\(2i+1\\) and \\(2i+2\\), respectively.</p> <ol> <li>What are the indices and values of the children of index 2?</li> <li>The node at index 2 has value 3. Does its relationship with its children violate the min-heap property? If so, which two elements should be swapped?</li> <li>Based on your answer to question 2, write the array after the swap if one is needed, or explain why no swap is necessary. Finally, check the other parent-child relationships.</li> </ol> Reference answer <ol> <li> <p>The left child of index 2 is at index 5 and has value 2. Its right child would be at index 6, but the array has length 6, so that child does not exist.</p> </li> <li> <p>The parent value 3 exceeds the child value 2, violating the min-heap property. Swap the elements at indices 2 and 5.</p> </li> <li> <p>The resulting array is <code>[1, 4, 2, 7, 6, 3]</code>. Checking each relationship gives:     <code>1 ≤ 4</code>, <code>1 ≤ 2</code>; <code>4 ≤ 7</code>, <code>4 ≤ 6</code>; <code>2 ≤ 3</code>.     Every parent is no greater than its children, so the min-heap property now holds.</p> </li> </ol>","path":["Chapter 8.   Heaps","8.5   Exercises"],"tags":[]},{"location":"chapter_heap/exercises/#3-keeping-the-three-largest-numbers-with-a-min-heap","level":3,"title":"3.   Keeping the three largest numbers with a min heap","text":"<p>To keep the three largest numbers from the stream <code>[4, 1, 7, 3, 8]</code>, maintain a min heap of at most three elements.</p> <p>Insert the first three numbers in order. Once the heap is full, for each new number: if it is greater than the root, remove the root and insert the new number; otherwise, leave the heap unchanged.</p> <p>After reading each number, write the numbers retained in the heap and the root value. You need only give the retained numbers as a set, not their arrangement in the heap array.</p> Reference answer <p>The results after each number are:</p> Number read Retained numbers Root 4 <code>{4}</code> 4 1 <code>{1, 4}</code> 1 7 <code>{1, 4, 7}</code> 1 3 <code>{3, 4, 7}</code> 3 8 <code>{4, 7, 8}</code> 4 <p>Once the heap is full, its root is the smallest retained number. Replace the root with the new number only when the new number is larger. The final set <code>{4, 7, 8}</code> contains exactly the three largest numbers.</p>","path":["Chapter 8.   Heaps","8.5   Exercises"],"tags":[]},{"location":"chapter_heap/exercises/#852-programming-exercises","level":2,"title":"8.5.2   Programming exercises","text":"","path":["Chapter 8.   Heaps","8.5   Exercises"],"tags":[]},{"location":"chapter_heap/exercises/#1-finding-the-kth-largest-element-in-an-array","level":3,"title":"1.   Finding the kth largest element in an array","text":"<p>Given an integer array <code>nums</code> and an integer \\(k\\), where \\(1 \\le k \\le n\\) and \\(n\\) is the array length, return the element in position \\(k\\) when the array is ordered from largest to smallest.</p> <p>Count duplicate elements separately. For example, the second largest element of <code>[5, 5, 2]</code> is still 5. Use a min heap of at most \\(k\\) elements.</p> Hints <ol> <li>The kth largest element is the smallest of the k largest numbers.</li> <li>Insert each number into the min heap; if its size exceeds k, remove the minimum.</li> <li>When the traversal ends, the heap retains the k largest numbers, and its root is the answer.</li> </ol> <p>LeetCode</p>","path":["Chapter 8.   Heaps","8.5   Exercises"],"tags":[]},{"location":"chapter_heap/heap/","level":1,"title":"8.1   Heaps","text":"<p>A heap is a complete binary tree that satisfies a particular ordering condition. There are two main types, shown below.</p> <ul> <li>Min heap: every node's value is \\(\\leq\\) the values of its children.</li> <li>Max heap: every node's value is \\(\\geq\\) the values of its children.</li> </ul> <p></p> <p> Figure 8-1   A min heap and a max heap </p> <p>As a special kind of complete binary tree, a heap has the following properties.</p> <ul> <li>The lowest level is filled from left to right, and all other levels are full.</li> <li>The root is the “top” of the heap, while the rightmost node on the lowest level is its “last” node.</li> <li>In a max heap or min heap, the root holds the largest or smallest value, respectively.</li> </ul>","path":["Chapter 8.   Heaps","8.1   Heaps"],"tags":[]},{"location":"chapter_heap/heap/#811-common-heap-operations","level":2,"title":"8.1.1   Common heap operations","text":"<p>Many programming languages provide a priority queue, an abstract data structure whose elements are removed according to priority.</p> <p>Heaps are commonly used to implement priority queues. A max heap corresponds to a priority queue that removes elements from largest to smallest. From the user's perspective, the two can be treated as equivalent. This book therefore does not distinguish them strictly and uses the term “heap” for both.</p> <p>Table 8-1 lists common operations. Method names depend on the programming language.</p> <p> Table 8-1   Efficiency of heap operations </p> Method Description Time complexity <code>push()</code> Insert an element \\(O(\\log n)\\) <code>pop()</code> Remove the root element \\(O(\\log n)\\) <code>peek()</code> Access the root: the maximum in a max heap or minimum in a min heap \\(O(1)\\) <code>size()</code> Get the number of elements \\(O(1)\\) <code>isEmpty()</code> Check whether the heap is empty \\(O(1)\\) <p>In practice, use the heap or priority-queue class provided by the language.</p> <p>Just as a sorting algorithm can order values in ascending or descending order, we can switch between min heaps and max heaps by setting a <code>flag</code> or changing the <code>Comparator</code>:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby heap.py<pre><code># Initialize a min heap\nmin_heap, flag = [], 1\n# Initialize a max heap\nmax_heap, flag = [], -1\n\n# Python's heapq module implements a min heap by default\n# Negate elements before insertion to reverse their order and implement a max heap\n# Here, flag = 1 gives a min heap and flag = -1 gives a max heap\n\n# Insert elements into the heap\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n\n# Access the root element\npeek: int = flag * max_heap[0] # 5\n\n# Remove root elements\n# Removed elements form a descending sequence\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n\n# Get the heap size\nsize: int = len(max_heap)\n\n# Check whether the heap is empty\nis_empty: bool = not max_heap\n\n# Build a heap from a list\nmin_heap: list[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n</code></pre> heap.cpp<pre><code>/* Initialize a heap */\n// Initialize a min heap\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap;\n// Initialize a max heap\npriority_queue&lt;int, vector&lt;int&gt;, less&lt;int&gt;&gt; maxHeap;\n\n/* Insert elements into the heap */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n\n/* Access the root element */\nint peek = maxHeap.top(); // 5\n\n/* Remove root elements */\n// Removed elements form a descending sequence\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n\n/* Get the heap size */\nint size = maxHeap.size();\n\n/* Check whether the heap is empty */\nbool isEmpty = maxHeap.empty();\n\n/* Build a heap from a list */\nvector&lt;int&gt; input{1, 3, 2, 5, 4};\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap(input.begin(), input.end());\n</code></pre> heap.java<pre><code>/* Initialize a heap */\n// Initialize a min heap\nQueue&lt;Integer&gt; minHeap = new PriorityQueue&lt;&gt;();\n// Initialize a max heap by changing the Comparator with a lambda expression\nQueue&lt;Integer&gt; maxHeap = new PriorityQueue&lt;&gt;((a, b) -&gt; b - a);\n\n/* Insert elements into the heap */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n\n/* Access the root element */\nint peek = maxHeap.peek(); // 5\n\n/* Remove root elements */\n// Removed elements form a descending sequence\npeek = maxHeap.poll(); // 5\npeek = maxHeap.poll(); // 4\npeek = maxHeap.poll(); // 3\npeek = maxHeap.poll(); // 2\npeek = maxHeap.poll(); // 1\n\n/* Get the heap size */\nint size = maxHeap.size();\n\n/* Check whether the heap is empty */\nboolean isEmpty = maxHeap.isEmpty();\n\n/* Build a heap from a list */\nminHeap = new PriorityQueue&lt;&gt;(Arrays.asList(1, 3, 2, 5, 4));\n</code></pre> heap.cs<pre><code>/* Initialize a heap */\n// Initialize a min heap\nPriorityQueue&lt;int, int&gt; minHeap = new();\n// Initialize a max heap by changing the Comparer with a lambda expression\nPriorityQueue&lt;int, int&gt; maxHeap = new(Comparer&lt;int&gt;.Create((x, y) =&gt; y.CompareTo(x)));\n\n/* Insert elements into the heap */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n\n/* Access the root element */\nint peek = maxHeap.Peek();//5\n\n/* Remove root elements */\n// Removed elements form a descending sequence\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n\n/* Get the heap size */\nint size = maxHeap.Count;\n\n/* Check whether the heap is empty */\nbool isEmpty = maxHeap.Count == 0;\n\n/* Build a heap from a list */\nminHeap = new PriorityQueue&lt;int, int&gt;([(1, 1), (3, 3), (2, 2), (5, 5), (4, 4)]);\n</code></pre> heap.go<pre><code>// In Go, implement heap.Interface to build an integer max heap\n// Implementing heap.Interface also requires implementing sort.Interface\ntype intHeap []any\n\n// Push implements the heap.Interface method for inserting an element\nfunc (h *intHeap) Push(x any) {\n    // Push and Pop use pointer receivers\n    // because they change the slice length as well as its contents.\n    *h = append(*h, x.(int))\n}\n\n// Pop implements the heap.Interface method used to remove the root element\nfunc (h *intHeap) Pop() any {\n    // The element to remove has been placed at the end\n    last := (*h)[len(*h)-1]\n    *h = (*h)[:len(*h)-1]\n    return last\n}\n\n// Len implements a sort.Interface method\nfunc (h *intHeap) Len() int {\n    return len(*h)\n}\n\n// Less implements a sort.Interface method\nfunc (h *intHeap) Less(i, j int) bool {\n    // For a min heap, change this to a less-than comparison\n    return (*h)[i].(int) &gt; (*h)[j].(int)\n}\n\n// Swap implements a sort.Interface method\nfunc (h *intHeap) Swap(i, j int) {\n    (*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n\n// Top returns the root element\nfunc (h *intHeap) Top() any {\n    return (*h)[0]\n}\n\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n    /* Initialize a heap */\n    // Initialize a max heap\n    maxHeap := &amp;intHeap{}\n    heap.Init(maxHeap)\n    /* Insert elements into the heap */\n    // Call the heap.Interface methods to add elements\n    heap.Push(maxHeap, 1)\n    heap.Push(maxHeap, 3)\n    heap.Push(maxHeap, 2)\n    heap.Push(maxHeap, 4)\n    heap.Push(maxHeap, 5)\n\n    /* Access the root element */\n    top := maxHeap.Top()\n    fmt.Printf(\"The root element is %d\\n\", top)\n\n    /* Remove root elements */\n    // Call the heap.Interface methods to remove elements\n    heap.Pop(maxHeap) // 5\n    heap.Pop(maxHeap) // 4\n    heap.Pop(maxHeap) // 3\n    heap.Pop(maxHeap) // 2\n    heap.Pop(maxHeap) // 1\n\n    /* Get the heap size */\n    size := len(*maxHeap)\n    fmt.Printf(\"The heap contains %d elements\\n\", size)\n\n    /* Check whether the heap is empty */\n    isEmpty := len(*maxHeap) == 0\n    fmt.Printf(\"Is the heap empty: %t\\n\", isEmpty)\n}\n</code></pre> heap.swift<pre><code>/* Initialize a heap */\n// Swift's Heap type supports both minimum and maximum operations and requires swift-collections\nvar heap = Heap&lt;Int&gt;()\n\n/* Insert elements into the heap */\nheap.insert(1)\nheap.insert(3)\nheap.insert(2)\nheap.insert(5)\nheap.insert(4)\n\n/* Access the root element */\nvar peek = heap.max()!\n\n/* Remove root elements */\npeek = heap.removeMax() // 5\npeek = heap.removeMax() // 4\npeek = heap.removeMax() // 3\npeek = heap.removeMax() // 2\npeek = heap.removeMax() // 1\n\n/* Get the heap size */\nlet size = heap.count\n\n/* Check whether the heap is empty */\nlet isEmpty = heap.isEmpty\n\n/* Build a heap from a list */\nlet heap2 = Heap([1, 3, 2, 5, 4])\n</code></pre> heap.js<pre><code>// JavaScript provides no built-in Heap class\n</code></pre> heap.ts<pre><code>// TypeScript provides no built-in Heap class\n</code></pre> heap.dart<pre><code>// Dart provides no built-in Heap class\n</code></pre> heap.rs<pre><code>use std::collections::BinaryHeap;\nuse std::cmp::Reverse;\n\n/* Initialize a heap */\n// Initialize a min heap\nlet mut min_heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n// Initialize a max heap\nlet mut max_heap = BinaryHeap::new();\n\n/* Insert elements into the heap */\nmax_heap.push(1);\nmax_heap.push(3);\nmax_heap.push(2);\nmax_heap.push(5);\nmax_heap.push(4);\n\n/* Access the root element */\nlet peek = max_heap.peek().unwrap();  // 5\n\n/* Remove root elements */\n// Removed elements form a descending sequence\nlet peek = max_heap.pop().unwrap();   // 5\nlet peek = max_heap.pop().unwrap();   // 4\nlet peek = max_heap.pop().unwrap();   // 3\nlet peek = max_heap.pop().unwrap();   // 2\nlet peek = max_heap.pop().unwrap();   // 1\n\n/* Get the heap size */\nlet size = max_heap.len();\n\n/* Check whether the heap is empty */\nlet is_empty = max_heap.is_empty();\n\n/* Build a heap from a list */\nlet min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]);\n</code></pre> heap.c<pre><code>// C provides no built-in Heap class\n</code></pre> heap.kt<pre><code>/* Initialize a heap */\n// Initialize a min heap\nvar minHeap = PriorityQueue&lt;Int&gt;()\n// Initialize a max heap by changing the Comparator with a lambda expression\nval maxHeap = PriorityQueue { a: Int, b: Int -&gt; b - a }\n\n/* Insert elements into the heap */\nmaxHeap.offer(1)\nmaxHeap.offer(3)\nmaxHeap.offer(2)\nmaxHeap.offer(5)\nmaxHeap.offer(4)\n\n/* Access the root element */\nvar peek = maxHeap.peek() // 5\n\n/* Remove root elements */\n// Removed elements form a descending sequence\npeek = maxHeap.poll() // 5\npeek = maxHeap.poll() // 4\npeek = maxHeap.poll() // 3\npeek = maxHeap.poll() // 2\npeek = maxHeap.poll() // 1\n\n/* Get the heap size */\nval size = maxHeap.size\n\n/* Check whether the heap is empty */\nval isEmpty = maxHeap.isEmpty()\n\n/* Build a heap from a list */\nminHeap = PriorityQueue(mutableListOf(1, 3, 2, 5, 4))\n</code></pre> heap.rb<pre><code># Ruby provides no built-in Heap class\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 8.   Heaps","8.1   Heaps"],"tags":[]},{"location":"chapter_heap/heap/#812-implementing-a-heap","level":2,"title":"8.1.2   Implementing a heap","text":"<p>The following implementation is a max heap. To turn it into a min heap, reverse every ordering comparison, for example replacing \\(\\geq\\) with \\(\\leq\\). Interested readers can try this themselves.</p>","path":["Chapter 8.   Heaps","8.1   Heaps"],"tags":[]},{"location":"chapter_heap/heap/#1-storage-and-representation","level":3,"title":"1.   Storage and representation","text":"<p>The binary-tree chapter showed that complete binary trees are well suited to array representation. Since a heap is a complete binary tree, we will store it in an array.</p> <p>In an array representation, elements store node values and indices identify positions in the tree. Index-mapping formulas take the place of node pointers.</p> <p>As shown below, a node at index \\(i\\) has its left child at \\(2i + 1\\), its right child at \\(2i + 2\\), and its parent at \\((i - 1) / 2\\), rounded down. An out-of-bounds index represents a missing node.</p> <p></p> <p> Figure 8-2   Representing and storing a heap </p> <p>Wrap the index formulas in functions for convenient use:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def left(self, i: int) -&gt; int:\n    \"\"\"Get the left child index\"\"\"\n    return 2 * i + 1\n\ndef right(self, i: int) -&gt; int:\n    \"\"\"Get the right child index\"\"\"\n    return 2 * i + 2\n\ndef parent(self, i: int) -&gt; int:\n    \"\"\"Get the parent index\"\"\"\n    return (i - 1) // 2  # Floor division\n</code></pre> my_heap.cpp<pre><code>/* Get the left child index */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* Get the right child index */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* Get the parent index */\nint parent(int i) {\n    return (i - 1) / 2; // Integer division truncates toward zero\n}\n</code></pre> my_heap.java<pre><code>/* Get the left child index */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* Get the right child index */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* Get the parent index */\nint parent(int i) {\n    return (i - 1) / 2; // Integer division (truncates toward zero in Java)\n}\n</code></pre> my_heap.cs<pre><code>/* Get the left child index */\nint Left(int i) {\n    return 2 * i + 1;\n}\n\n/* Get the right child index */\nint Right(int i) {\n    return 2 * i + 2;\n}\n\n/* Get the parent index */\nint Parent(int i) {\n    return (i - 1) / 2; // Integer division (truncates toward zero in C#)\n}\n</code></pre> my_heap.go<pre><code>/* Get the left child index */\nfunc (h *maxHeap) left(i int) int {\n    return 2*i + 1\n}\n\n/* Get the right child index */\nfunc (h *maxHeap) right(i int) int {\n    return 2*i + 2\n}\n\n/* Get the parent index */\nfunc (h *maxHeap) parent(i int) int {\n    // Integer division (truncates toward zero in Go)\n    return (i - 1) / 2\n}\n</code></pre> my_heap.swift<pre><code>/* Get the left child index */\nfunc left(i: Int) -&gt; Int {\n    2 * i + 1\n}\n\n/* Get the right child index */\nfunc right(i: Int) -&gt; Int {\n    2 * i + 2\n}\n\n/* Get the parent index */\nfunc parent(i: Int) -&gt; Int {\n    (i - 1) / 2 // Integer division truncates toward zero\n}\n</code></pre> my_heap.js<pre><code>/* Get the left child index */\n#left(i) {\n    return 2 * i + 1;\n}\n\n/* Get the right child index */\n#right(i) {\n    return 2 * i + 2;\n}\n\n/* Get the parent index */\n#parent(i) {\n    return Math.floor((i - 1) / 2); // Floor division\n}\n</code></pre> my_heap.ts<pre><code>/* Get the left child index */\nleft(i: number): number {\n    return 2 * i + 1;\n}\n\n/* Get the right child index */\nright(i: number): number {\n    return 2 * i + 2;\n}\n\n/* Get the parent index */\nparent(i: number): number {\n    return Math.floor((i - 1) / 2); // Floor division\n}\n</code></pre> my_heap.dart<pre><code>/* Get the left child index */\nint _left(int i) {\n  return 2 * i + 1;\n}\n\n/* Get the right child index */\nint _right(int i) {\n  return 2 * i + 2;\n}\n\n/* Get the parent index */\nint _parent(int i) {\n  return (i - 1) ~/ 2; // Integer division\n}\n</code></pre> my_heap.rs<pre><code>/* Get the left child index */\nfn left(i: usize) -&gt; usize {\n    2 * i + 1\n}\n\n/* Get the right child index */\nfn right(i: usize) -&gt; usize {\n    2 * i + 2\n}\n\n/* Get the parent index */\nfn parent(i: usize) -&gt; usize {\n    (i - 1) / 2 // Floor division\n}\n</code></pre> my_heap.c<pre><code>/* Get the left child index */\nint left(MaxHeap *maxHeap, int i) {\n    return 2 * i + 1;\n}\n\n/* Get the right child index */\nint right(MaxHeap *maxHeap, int i) {\n    return 2 * i + 2;\n}\n\n/* Get the parent index */\nint parent(MaxHeap *maxHeap, int i) {\n    return (i - 1) / 2; // Integer division truncates toward zero\n}\n</code></pre> my_heap.kt<pre><code>/* Get the left child index */\nfun left(i: Int): Int {\n    return 2 * i + 1\n}\n\n/* Get the right child index */\nfun right(i: Int): Int {\n    return 2 * i + 2\n}\n\n/* Get the parent index */\nfun parent(i: Int): Int {\n    return (i - 1) / 2 // Integer division (truncates toward zero in Kotlin)\n}\n</code></pre> my_heap.rb<pre><code>### Get the left child index ###\ndef left(i)\n  2 * i + 1\nend\n\n### Get the right child index ###\ndef right(i)\n  2 * i + 2\nend\n\n### Get the parent index ###\ndef parent(i)\n  (i - 1) / 2     # Floor division\nend\n</code></pre>","path":["Chapter 8.   Heaps","8.1   Heaps"],"tags":[]},{"location":"chapter_heap/heap/#2-accessing-the-root","level":3,"title":"2.   Accessing the root","text":"<p>The heap's root is the first element of the list:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def peek(self) -&gt; int:\n    \"\"\"Access the top element of the heap\"\"\"\n    return self.max_heap[0]\n</code></pre> my_heap.cpp<pre><code>/* Access the top element of the heap */\nint peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.java<pre><code>/* Access the top element of the heap */\nint peek() {\n    return maxHeap.get(0);\n}\n</code></pre> my_heap.cs<pre><code>/* Access the top element of the heap */\nint Peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.go<pre><code>/* Access the top element of the heap */\nfunc (h *maxHeap) peek() any {\n    return h.data[0]\n}\n</code></pre> my_heap.swift<pre><code>/* Access the top element of the heap */\nfunc peek() -&gt; Int {\n    maxHeap[0]\n}\n</code></pre> my_heap.js<pre><code>/* Access the top element of the heap */\npeek() {\n    return this.#maxHeap[0];\n}\n</code></pre> my_heap.ts<pre><code>/* Access the top element of the heap */\npeek(): number {\n    return this.maxHeap[0];\n}\n</code></pre> my_heap.dart<pre><code>/* Access the top element of the heap */\nint peek() {\n  return _maxHeap[0];\n}\n</code></pre> my_heap.rs<pre><code>/* Access the top element of the heap */\nfn peek(&amp;self) -&gt; Option&lt;i32&gt; {\n    self.max_heap.first().copied()\n}\n</code></pre> my_heap.c<pre><code>/* Access the top element of the heap */\nint peek(MaxHeap *maxHeap) {\n    return maxHeap-&gt;data[0];\n}\n</code></pre> my_heap.kt<pre><code>/* Access the top element of the heap */\nfun peek(): Int {\n    return maxHeap[0]\n}\n</code></pre> my_heap.rb<pre><code>### Access the top element of the heap ###\ndef peek\n  @max_heap[0]\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 8.   Heaps","8.1   Heaps"],"tags":[]},{"location":"chapter_heap/heap/#3-inserting-an-element","level":3,"title":"3.   Inserting an element","text":"<p>To insert <code>val</code>, first append it at the end of the heap. Since it may be larger than other elements, this can violate the heap property. We must restore the property along the path from the inserted node to the root, an operation called heapifying.</p> <p>Start at the inserted node and sift upward. As shown below, compare it with its parent and swap them if the inserted value is larger. Repeat upward until the root is reached or no swap is needed.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 8-3   Steps for inserting into a heap </p> <p>With \\(n\\) nodes, the tree height is \\(O(\\log n)\\), so sifting takes at most \\(O(\\log n)\\) iterations. Insertion therefore takes \\(O(\\log n)\\) time. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def push(self, val: int):\n    \"\"\"Push an element onto the heap\"\"\"\n    # Add a node\n    self.max_heap.append(val)\n    # Sift up\n    self.sift_up(self.size() - 1)\n\ndef sift_up(self, i: int):\n    \"\"\"Sift up starting from node i\"\"\"\n    while True:\n        # Get the parent of node i\n        p = self.parent(i)\n        # Stop sifting when the root has been passed or the node needs no repair\n        if p &lt; 0 or self.max_heap[i] &lt;= self.max_heap[p]:\n            break\n        # Swap the two nodes\n        self.swap(i, p)\n        # Continue sifting up\n        i = p\n</code></pre> my_heap.cpp<pre><code>/* Push an element onto the heap */\nvoid push(int val) {\n    // Add a node\n    maxHeap.push_back(val);\n    // Sift up\n    siftUp(size() - 1);\n}\n\n/* Sift up starting from node i */\nvoid siftUp(int i) {\n    while (true) {\n        // Get the parent of node i\n        int p = parent(i);\n        // Stop sifting when the root has been passed or the node needs no repair\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // Swap the two nodes\n        swap(maxHeap[i], maxHeap[p]);\n        // Continue sifting up\n        i = p;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* Push an element onto the heap */\nvoid push(int val) {\n    // Add a node\n    maxHeap.add(val);\n    // Sift up\n    siftUp(size() - 1);\n}\n\n/* Sift up starting from node i */\nvoid siftUp(int i) {\n    while (true) {\n        // Get the parent of node i\n        int p = parent(i);\n        // Stop sifting when the root has been passed or the node needs no repair\n        if (p &lt; 0 || maxHeap.get(i) &lt;= maxHeap.get(p))\n            break;\n        // Swap the two nodes\n        swap(i, p);\n        // Continue sifting up\n        i = p;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* Push an element onto the heap */\nvoid Push(int val) {\n    // Add a node\n    maxHeap.Add(val);\n    // Sift up\n    SiftUp(Size() - 1);\n}\n\n/* Sift up starting from node i */\nvoid SiftUp(int i) {\n    while (true) {\n        // Get the parent of node i\n        int p = Parent(i);\n        // Stop sifting if the root has been passed or the node needs no adjustment\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // Swap the two nodes\n        Swap(i, p);\n        // Continue sifting up\n        i = p;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* Push an element onto the heap */\nfunc (h *maxHeap) push(val any) {\n    // Add a node\n    h.data = append(h.data, val)\n    // Sift up\n    h.siftUp(len(h.data) - 1)\n}\n\n/* Sift up starting from node i */\nfunc (h *maxHeap) siftUp(i int) {\n    for true {\n        // Get the parent of node i\n        p := h.parent(i)\n        // Stop sifting when the root has been passed or the node needs no repair\n        if p &lt; 0 || h.data[i].(int) &lt;= h.data[p].(int) {\n            break\n        }\n        // Swap the two nodes\n        h.swap(i, p)\n        // Continue sifting up\n        i = p\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* Push an element onto the heap */\nfunc push(val: Int) {\n    // Add a node\n    maxHeap.append(val)\n    // Sift up\n    siftUp(i: size() - 1)\n}\n\n/* Sift up starting from node i */\nfunc siftUp(i: Int) {\n    var i = i\n    while true {\n        // Get the parent of node i\n        let p = parent(i: i)\n        // Stop when the parent index is invalid or the node already satisfies the heap order\n        if p &lt; 0 || maxHeap[i] &lt;= maxHeap[p] {\n            break\n        }\n        // Swap the two nodes\n        swap(i: i, j: p)\n        // Continue sifting up\n        i = p\n    }\n}\n</code></pre> my_heap.js<pre><code>/* Push an element onto the heap */\npush(val) {\n    // Add a node\n    this.#maxHeap.push(val);\n    // Sift up\n    this.#siftUp(this.size() - 1);\n}\n\n/* Sift up starting from node i */\n#siftUp(i) {\n    while (true) {\n        // Get the parent of node i\n        const p = this.#parent(i);\n        // Stop sifting when the root has been passed or the node needs no repair\n        if (p &lt; 0 || this.#maxHeap[i] &lt;= this.#maxHeap[p]) break;\n        // Swap the two nodes\n        this.#swap(i, p);\n        // Continue sifting up\n        i = p;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* Push an element onto the heap */\npush(val: number): void {\n    // Add a node\n    this.maxHeap.push(val);\n    // Sift up\n    this.siftUp(this.size() - 1);\n}\n\n/* Sift up starting from node i */\nsiftUp(i: number): void {\n    while (true) {\n        // Get the parent of node i\n        const p = this.parent(i);\n        // Stop sifting when the root has been passed or the node needs no repair\n        if (p &lt; 0 || this.maxHeap[i] &lt;= this.maxHeap[p]) break;\n        // Swap the two nodes\n        this.swap(i, p);\n        // Continue sifting up\n        i = p;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* Push an element onto the heap */\nvoid push(int val) {\n  // Add a node\n  _maxHeap.add(val);\n  // Sift up\n  siftUp(size() - 1);\n}\n\n/* Sift up starting from node i */\nvoid siftUp(int i) {\n  while (true) {\n    // Get the parent of node i\n    int p = _parent(i);\n    // Stop sifting when the root has been passed or the node needs no repair\n    if (p &lt; 0 || _maxHeap[i] &lt;= _maxHeap[p]) {\n      break;\n    }\n    // Swap the two nodes\n    _swap(i, p);\n    // Continue sifting up\n    i = p;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* Push an element onto the heap */\nfn push(&amp;mut self, val: i32) {\n    // Add a node\n    self.max_heap.push(val);\n    // Sift up\n    self.sift_up(self.size() - 1);\n}\n\n/* Sift up starting from node i */\nfn sift_up(&amp;mut self, mut i: usize) {\n    loop {\n        // Stop sifting up if node i is already the root\n        if i == 0 {\n            break;\n        }\n        // Get the parent of node i\n        let p = Self::parent(i);\n        // Stop sifting up when the node already satisfies the heap property\n        if self.max_heap[i] &lt;= self.max_heap[p] {\n            break;\n        }\n        // Swap the two nodes\n        self.swap(i, p);\n        // Continue sifting up\n        i = p;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* Push an element onto the heap */\nvoid push(MaxHeap *maxHeap, int val) {\n    // Normally, the number of nodes should remain below this limit\n    if (maxHeap-&gt;size == MAX_SIZE) {\n        printf(\"Heap is full!\");\n        return;\n    }\n    // Add a node\n    maxHeap-&gt;data[maxHeap-&gt;size] = val;\n    maxHeap-&gt;size++;\n\n    // Sift up\n    siftUp(maxHeap, maxHeap-&gt;size - 1);\n}\n\n/* Sift up starting from node i */\nvoid siftUp(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // Get the parent of node i\n        int p = parent(maxHeap, i);\n        // Stop sifting when the root has been passed or the node needs no repair\n        if (p &lt; 0 || maxHeap-&gt;data[i] &lt;= maxHeap-&gt;data[p]) {\n            break;\n        }\n        // Swap the two nodes\n        swap(maxHeap, i, p);\n        // Continue sifting up\n        i = p;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* Push an element onto the heap */\nfun push(_val: Int) {\n    // Add a node\n    maxHeap.add(_val)\n    // Sift up\n    siftUp(size() - 1)\n}\n\n/* Sift up starting from node i */\nfun siftUp(it: Int) {\n    // Kotlin function parameters are immutable, so create a temporary variable\n    var i = it\n    while (true) {\n        // Get the parent of node i\n        val p = parent(i)\n        // Stop sifting when the root has been passed or the node needs no repair\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n        // Swap the two nodes\n        swap(i, p)\n        // Continue sifting up\n        i = p\n    }\n}\n</code></pre> my_heap.rb<pre><code>### Push an element onto the heap ###\ndef push(val)\n  # Add a node\n  @max_heap &lt;&lt; val\n  # Sift up\n  sift_up(size - 1)\nend\n\n### Sift up starting from node i ###\ndef sift_up(i)\n  loop do\n    # Get the parent of node i\n    p = parent(i)\n    # Stop sifting when the root has been passed or the node needs no repair\n    break if p &lt; 0 || @max_heap[i] &lt;= @max_heap[p]\n    # Swap the two nodes\n    swap(i, p)\n    # Continue sifting up\n    i = p\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 8.   Heaps","8.1   Heaps"],"tags":[]},{"location":"chapter_heap/heap/#4-removing-the-root","level":3,"title":"4.   Removing the root","text":"<p>The root is the first element of the list. Removing it directly would change every remaining index, making it difficult to restore the heap property. To minimize index changes, use the following steps.</p> <ol> <li>Swap the root with the last element, the rightmost leaf.</li> <li>Remove the last element from the list. Because of the swap, this is the original root.</li> <li>Starting at the root, sift downward.</li> </ol> <p>As shown below, sifting down proceeds in the opposite direction from sifting up. Compare the root with its two children and swap it with the larger child when needed. Continue downward until a leaf is reached or no swap is necessary.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 8-4   Steps for removing the root of a heap </p> <p>Like insertion, removing the root takes \\(O(\\log n)\\) time. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def pop(self) -&gt; int:\n    \"\"\"Pop an element from the heap\"\"\"\n    # Handle an empty heap\n    if self.is_empty():\n        raise IndexError(\"Heap is empty\")\n    # Swap the root with the rightmost leaf on the last level (swap the first and last elements)\n    self.swap(0, self.size() - 1)\n    # Remove a node\n    val = self.max_heap.pop()\n    # Sift down\n    self.sift_down(0)\n    # Return the top element of the heap\n    return val\n\ndef sift_down(self, i: int):\n    \"\"\"Sift down starting from node i\"\"\"\n    while True:\n        # Find the node with the largest value among i, l, and r, and store its index in ma\n        l, r, ma = self.left(i), self.right(i), i\n        if l &lt; self.size() and self.max_heap[l] &gt; self.max_heap[ma]:\n            ma = l\n        if r &lt; self.size() and self.max_heap[r] &gt; self.max_heap[ma]:\n            ma = r\n        # Stop if node i is largest or both child indices l and r are out of bounds\n        if ma == i:\n            break\n        # Swap the two nodes\n        self.swap(i, ma)\n        # Continue sifting down\n        i = ma\n</code></pre> my_heap.cpp<pre><code>/* Pop an element from the heap */\nvoid pop() {\n    // Handle an empty heap\n    if (isEmpty()) {\n        throw out_of_range(\"Heap is empty\");\n    }\n    // Swap the root with the rightmost leaf on the last level (swap the first and last elements)\n    swap(maxHeap[0], maxHeap[size() - 1]);\n    // Remove a node\n    maxHeap.pop_back();\n    // Sift down\n    siftDown(0);\n}\n\n/* Sift down starting from node i */\nvoid siftDown(int i) {\n    while (true) {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if (ma == i)\n            break;\n        swap(maxHeap[i], maxHeap[ma]);\n        // Continue sifting down\n        i = ma;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* Pop an element from the heap */\nint pop() {\n    // Handle an empty heap\n    if (isEmpty())\n        throw new IndexOutOfBoundsException();\n    // Swap the root with the rightmost leaf on the last level (swap the first and last elements)\n    swap(0, size() - 1);\n    // Remove a node\n    int val = maxHeap.remove(size() - 1);\n    // Sift down\n    siftDown(0);\n    // Return the top element of the heap\n    return val;\n}\n\n/* Sift down starting from node i */\nvoid siftDown(int i) {\n    while (true) {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap.get(l) &gt; maxHeap.get(ma))\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap.get(r) &gt; maxHeap.get(ma))\n            ma = r;\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if (ma == i)\n            break;\n        // Swap the two nodes\n        swap(i, ma);\n        // Continue sifting down\n        i = ma;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* Pop an element from the heap */\nint Pop() {\n    // Handle an empty heap\n    if (IsEmpty())\n        throw new IndexOutOfRangeException();\n    // Swap the root with the rightmost leaf on the last level (swap the first and last elements)\n    Swap(0, Size() - 1);\n    // Remove a node\n    int val = maxHeap.Last();\n    maxHeap.RemoveAt(Size() - 1);\n    // Sift down\n    SiftDown(0);\n    // Return the top element of the heap\n    return val;\n}\n\n/* Sift down starting from node i */\nvoid SiftDown(int i) {\n    while (true) {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        int l = Left(i), r = Right(i), ma = i;\n        if (l &lt; Size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; Size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // Stop sifting if node i has the largest value or has no children\n        if (ma == i) break;\n        // Swap the two nodes\n        Swap(i, ma);\n        // Continue sifting down\n        i = ma;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* Pop an element from the heap */\nfunc (h *maxHeap) pop() any {\n    // Handle an empty heap\n    if h.isEmpty() {\n        fmt.Println(\"error\")\n        return nil\n    }\n    // Swap the root with the rightmost leaf on the last level (swap the first and last elements)\n    h.swap(0, h.size()-1)\n    // Remove a node\n    val := h.data[len(h.data)-1]\n    h.data = h.data[:len(h.data)-1]\n    // Sift down\n    h.siftDown(0)\n\n    // Return the top element of the heap\n    return val\n}\n\n/* Sift down starting from node i */\nfunc (h *maxHeap) siftDown(i int) {\n    for true {\n        // Find the node with the largest value among i, l, and r, and store its index in max\n        l, r, max := h.left(i), h.right(i), i\n        if l &lt; h.size() &amp;&amp; h.data[l].(int) &gt; h.data[max].(int) {\n            max = l\n        }\n        if r &lt; h.size() &amp;&amp; h.data[r].(int) &gt; h.data[max].(int) {\n            max = r\n        }\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if max == i {\n            break\n        }\n        // Swap the two nodes\n        h.swap(i, max)\n        // Continue sifting down\n        i = max\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* Pop an element from the heap */\nfunc pop() -&gt; Int {\n    // Handle an empty heap\n    if isEmpty() {\n        fatalError(\"Heap is empty\")\n    }\n    // Swap the root with the rightmost leaf on the last level (swap the first and last elements)\n    swap(i: 0, j: size() - 1)\n    // Remove a node\n    let val = maxHeap.remove(at: size() - 1)\n    // Sift down\n    siftDown(i: 0)\n    // Return the top element of the heap\n    return val\n}\n\n/* Sift down starting from node i */\nfunc siftDown(i: Int) {\n    var i = i\n    while true {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        let l = left(i: i)\n        let r = right(i: i)\n        var ma = i\n        if l &lt; size(), maxHeap[l] &gt; maxHeap[ma] {\n            ma = l\n        }\n        if r &lt; size(), maxHeap[r] &gt; maxHeap[ma] {\n            ma = r\n        }\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if ma == i {\n            break\n        }\n        // Swap the two nodes\n        swap(i: i, j: ma)\n        // Continue sifting down\n        i = ma\n    }\n}\n</code></pre> my_heap.js<pre><code>/* Pop an element from the heap */\npop() {\n    // Handle an empty heap\n    if (this.isEmpty()) throw new Error('Heap is empty');\n    // Swap the root with the rightmost leaf on the last level (swap the first and last elements)\n    this.#swap(0, this.size() - 1);\n    // Remove a node\n    const val = this.#maxHeap.pop();\n    // Sift down\n    this.#siftDown(0);\n    // Return the top element of the heap\n    return val;\n}\n\n/* Sift down starting from node i */\n#siftDown(i) {\n    while (true) {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        const l = this.#left(i),\n            r = this.#right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.#maxHeap[l] &gt; this.#maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.#maxHeap[r] &gt; this.#maxHeap[ma]) ma = r;\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if (ma === i) break;\n        // Swap the two nodes\n        this.#swap(i, ma);\n        // Continue sifting down\n        i = ma;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* Pop an element from the heap */\npop(): number {\n    // Handle an empty heap\n    if (this.isEmpty()) throw new RangeError('Heap is empty');\n    // Swap the root with the rightmost leaf on the last level (swap the first and last elements)\n    this.swap(0, this.size() - 1);\n    // Remove a node\n    const val = this.maxHeap.pop();\n    // Sift down\n    this.siftDown(0);\n    // Return the top element of the heap\n    return val;\n}\n\n/* Sift down starting from node i */\nsiftDown(i: number): void {\n    while (true) {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        const l = this.left(i),\n            r = this.right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.maxHeap[l] &gt; this.maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.maxHeap[r] &gt; this.maxHeap[ma]) ma = r;\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if (ma === i) break;\n        // Swap the two nodes\n        this.swap(i, ma);\n        // Continue sifting down\n        i = ma;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* Pop an element from the heap */\nint pop() {\n  // Handle an empty heap\n  if (isEmpty()) throw Exception('Heap is empty');\n  // Swap the root with the rightmost leaf on the last level (swap the first and last elements)\n  _swap(0, size() - 1);\n  // Remove a node\n  int val = _maxHeap.removeLast();\n  // Sift down\n  siftDown(0);\n  // Return the top element of the heap\n  return val;\n}\n\n/* Sift down starting from node i */\nvoid siftDown(int i) {\n  while (true) {\n    // Find the node with the largest value among i, l, and r, and store its index in ma\n    int l = _left(i);\n    int r = _right(i);\n    int ma = i;\n    if (l &lt; size() &amp;&amp; _maxHeap[l] &gt; _maxHeap[ma]) ma = l;\n    if (r &lt; size() &amp;&amp; _maxHeap[r] &gt; _maxHeap[ma]) ma = r;\n    // Stop if node i is largest or both child indices l and r are out of bounds\n    if (ma == i) break;\n    // Swap the two nodes\n    _swap(i, ma);\n    // Continue sifting down\n    i = ma;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* Pop an element from the heap */\nfn pop(&amp;mut self) -&gt; i32 {\n    // Handle an empty heap\n    if self.is_empty() {\n        panic!(\"Index out of bounds\");\n    }\n    // Swap the root with the rightmost leaf on the last level (swap the first and last elements)\n    self.swap(0, self.size() - 1);\n    // Remove a node\n    let val = self.max_heap.pop().unwrap();\n    // Sift down\n    self.sift_down(0);\n    // Return the top element of the heap\n    val\n}\n\n/* Sift down starting from node i */\nfn sift_down(&amp;mut self, mut i: usize) {\n    loop {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        let (l, r, mut ma) = (Self::left(i), Self::right(i), i);\n        if l &lt; self.size() &amp;&amp; self.max_heap[l] &gt; self.max_heap[ma] {\n            ma = l;\n        }\n        if r &lt; self.size() &amp;&amp; self.max_heap[r] &gt; self.max_heap[ma] {\n            ma = r;\n        }\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if ma == i {\n            break;\n        }\n        // Swap the two nodes\n        self.swap(i, ma);\n        // Continue sifting down\n        i = ma;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* Pop an element from the heap */\nint pop(MaxHeap *maxHeap) {\n    // Handle an empty heap\n    if (isEmpty(maxHeap)) {\n        printf(\"Heap is empty!\");\n        return INT_MAX;\n    }\n    // Swap the root with the rightmost leaf on the last level (swap the first and last elements)\n    swap(maxHeap, 0, size(maxHeap) - 1);\n    // Remove a node\n    int val = maxHeap-&gt;data[maxHeap-&gt;size - 1];\n    maxHeap-&gt;size--;\n    // Sift down\n    siftDown(maxHeap, 0);\n\n    // Return the top element of the heap\n    return val;\n}\n\n/* Sift down starting from node i */\nvoid siftDown(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // Find the node with the largest value among i, l, and r, and store its index in max\n        int l = left(maxHeap, i);\n        int r = right(maxHeap, i);\n        int max = i;\n        if (l &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[l] &gt; maxHeap-&gt;data[max]) {\n            max = l;\n        }\n        if (r &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[r] &gt; maxHeap-&gt;data[max]) {\n            max = r;\n        }\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if (max == i) {\n            break;\n        }\n        // Swap the two nodes\n        swap(maxHeap, i, max);\n        // Continue sifting down\n        i = max;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* Pop an element from the heap */\nfun pop(): Int {\n    // Handle an empty heap\n    if (isEmpty()) throw IndexOutOfBoundsException()\n    // Swap the root with the rightmost leaf on the last level (swap the first and last elements)\n    swap(0, size() - 1)\n    // Remove a node\n    val _val = maxHeap.removeAt(size() - 1)\n    // Sift down\n    siftDown(0)\n    // Return the top element of the heap\n    return _val\n}\n\n/* Sift down starting from node i */\nfun siftDown(it: Int) {\n    // Kotlin function parameters are immutable, so create a temporary variable\n    var i = it\n    while (true) {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        val l = left(i)\n        val r = right(i)\n        var ma = i\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if (ma == i) break\n        // Swap the two nodes\n        swap(i, ma)\n        // Continue sifting down\n        i = ma\n    }\n}\n</code></pre> my_heap.rb<pre><code>### Pop an element from the heap ###\ndef pop\n  # Handle an empty heap\n  raise IndexError, \"Heap is empty\" if is_empty?\n  # Swap the root with the rightmost leaf on the last level (swap the first and last elements)\n  swap(0, size - 1)\n  # Remove a node\n  val = @max_heap.pop\n  # Sift down\n  sift_down(0)\n  # Return the top element of the heap\n  val\nend\n\n### Sift down starting from node i ###\ndef sift_down(i)\n  loop do\n    # Find the node with the largest value among i, l, and r, and store its index in ma\n    l, r, ma = left(i), right(i), i\n    ma = l if l &lt; size &amp;&amp; @max_heap[l] &gt; @max_heap[ma]\n    ma = r if r &lt; size &amp;&amp; @max_heap[r] &gt; @max_heap[ma]\n\n    # Stop if node i is largest or both child indices l and r are out of bounds\n    break if ma == i\n\n    # Swap the two nodes\n    swap(i, ma)\n    # Continue sifting down\n    i = ma\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 8.   Heaps","8.1   Heaps"],"tags":[]},{"location":"chapter_heap/heap/#813-common-applications","level":2,"title":"8.1.3   Common applications","text":"<ul> <li>Priority queues: Heaps are usually the first choice for implementing priority queues. Enqueueing and dequeueing both take \\(O(\\log n)\\) time, and building a heap takes \\(O(n)\\).</li> <li>Heap sort: Build a heap from a dataset, then repeatedly remove its root to obtain sorted data. In practice, a more elegant implementation is usually used; see the heap-sort section.</li> <li>Finding the largest \\(k\\) elements: This is both a classic algorithm problem and a common application, such as selecting the 10 hottest news stories for Weibo's trending list or the 10 best-selling products.</li> </ul>","path":["Chapter 8.   Heaps","8.1   Heaps"],"tags":[]},{"location":"chapter_heap/summary/","level":1,"title":"8.4   Summary","text":"","path":["Chapter 8.   Heaps","8.4   Summary"],"tags":[]},{"location":"chapter_heap/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<ul> <li>A heap is a complete binary tree. Depending on its ordering condition, it is either a max heap or a min heap. The root holds the largest or smallest element, respectively.</li> <li>A priority queue removes elements according to their priority and is usually implemented with a heap.</li> <li>Common heap operations and their time complexities include insertion in \\(O(\\log n)\\), removing the root in \\(O(\\log n)\\), and accessing the root in \\(O(1)\\).</li> <li>Complete binary trees are well suited to array representation, so heaps are usually stored in arrays.</li> <li>Sifting restores the heap property and is used during both insertion and removal.</li> <li>Building a heap from \\(n\\) input elements can be optimized to \\(O(n)\\) time.</li> <li>Top-k is a classic algorithm problem that a heap can solve efficiently in \\(O(n \\log k)\\) time.</li> </ul>","path":["Chapter 8.   Heaps","8.4   Summary"],"tags":[]},{"location":"chapter_heap/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: Is the heap data structure the same concept as the heap in memory management?</p> <p>No; they simply share a name. In computer memory, the heap is used for dynamic memory allocation, allowing a running program to store data. A program can request heap memory for structures such as objects and arrays. When that data is no longer needed, the memory must be released to prevent leaks. Compared with stack memory, heap memory requires more careful management; misuse can lead to memory leaks, dangling pointers, and other problems.</p>","path":["Chapter 8.   Heaps","8.4   Summary"],"tags":[]},{"location":"chapter_heap/top_k/","level":1,"title":"8.3   The top-k problem","text":"<p>Question</p> <p>Given an unsorted array <code>nums</code> of length \\(n\\), return its largest \\(k\\) elements.</p> <p>We first introduce two straightforward solutions, then a more efficient solution using a heap.</p>","path":["Chapter 8.   Heaps","8.3   The top-k problem"],"tags":[]},{"location":"chapter_heap/top_k/#831-method-1-repeated-selection","level":2,"title":"8.3.1   Method 1: Repeated selection","text":"<p>As shown below, scan the array \\(k\\) times, selecting the \\(1\\)st, \\(2\\)nd, \\(\\dots\\), and \\(k\\)th largest elements in successive rounds. This takes \\(O(nk)\\) time.</p> <p>This method is suitable only when \\(k \\ll n\\). As \\(k\\) approaches \\(n\\), its time complexity approaches \\(O(n^2)\\), making it very slow.</p> <p></p> <p> Figure 8-6   Finding the largest k elements by repeated scans </p> <p>Tip</p> <p>When \\(k = n\\), this produces a fully sorted sequence and is equivalent to selection sort.</p>","path":["Chapter 8.   Heaps","8.3   The top-k problem"],"tags":[]},{"location":"chapter_heap/top_k/#832-method-2-sorting","level":2,"title":"8.3.2   Method 2: Sorting","text":"<p>As shown below, sort <code>nums</code> and return its rightmost \\(k\\) elements. This takes \\(O(n \\log n)\\) time.</p> <p>This method does more work than necessary: we need only the largest \\(k\\) elements, not a sorted order for all the others.</p> <p></p> <p> Figure 8-7   Finding the largest k elements by sorting </p>","path":["Chapter 8.   Heaps","8.3   The top-k problem"],"tags":[]},{"location":"chapter_heap/top_k/#833-method-3-a-heap","level":2,"title":"8.3.3   Method 3: A heap","text":"<p>A heap solves the top-k problem more efficiently, following the steps below.</p> <ol> <li>Initialize a min heap, whose root holds its smallest element.</li> <li>Insert the first \\(k\\) array elements into the heap one by one.</li> <li>Starting with element \\(k + 1\\), if the current element is larger than the root, remove the root and insert the current element.</li> <li>After the traversal, the heap contains the largest \\(k\\) elements.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 8-8   Finding the largest k elements with a heap </p> <p>An implementation follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby top_k.py<pre><code>def top_k_heap(nums: list[int], k: int) -&gt; list[int]:\n    \"\"\"Use a heap to find the k largest elements in an array\"\"\"\n    # Initialize a min heap\n    heap = []\n    # Push the first k array elements onto the heap\n    for i in range(k):\n        heapq.heappush(heap, nums[i])\n    # Starting with element k+1, keep the heap size at k\n    for i in range(k, len(nums)):\n        # If the current element is larger than the heap top, pop the top element and push the current element\n        if nums[i] &gt; heap[0]:\n            heapq.heappop(heap)\n            heapq.heappush(heap, nums[i])\n    return heap\n</code></pre> top_k.cpp<pre><code>/* Use a heap to find the k largest elements in an array */\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; topKHeap(vector&lt;int&gt; &amp;nums, int k) {\n    // Initialize a min heap\n    priority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; heap;\n    // Push the first k array elements onto the heap\n    for (int i = 0; i &lt; k; i++) {\n        heap.push(nums[i]);\n    }\n    // Starting with element k+1, keep the heap size at k\n    for (int i = k; i &lt; nums.size(); i++) {\n        // If the current element is larger than the heap top, pop the top element and push the current element\n        if (nums[i] &gt; heap.top()) {\n            heap.pop();\n            heap.push(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.java<pre><code>/* Use a heap to find the k largest elements in an array */\nQueue&lt;Integer&gt; topKHeap(int[] nums, int k) {\n    // Initialize a min heap\n    Queue&lt;Integer&gt; heap = new PriorityQueue&lt;Integer&gt;();\n    // Push the first k array elements onto the heap\n    for (int i = 0; i &lt; k; i++) {\n        heap.offer(nums[i]);\n    }\n    // Starting with element k+1, keep the heap size at k\n    for (int i = k; i &lt; nums.length; i++) {\n        // If the current element is larger than the heap top, pop the top element and push the current element\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll();\n            heap.offer(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.cs<pre><code>/* Use a heap to find the k largest elements in an array */\nPriorityQueue&lt;int, int&gt; TopKHeap(int[] nums, int k) {\n    // Initialize a min heap\n    PriorityQueue&lt;int, int&gt; heap = new();\n    // Push the first k array elements onto the heap\n    for (int i = 0; i &lt; k; i++) {\n        heap.Enqueue(nums[i], nums[i]);\n    }\n    // Starting with element k+1, keep the heap size at k\n    for (int i = k; i &lt; nums.Length; i++) {\n        // If the current element is larger than the heap top, pop the top element and push the current element\n        if (nums[i] &gt; heap.Peek()) {\n            heap.Dequeue();\n            heap.Enqueue(nums[i], nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.go<pre><code>/* Use a heap to find the k largest elements in an array */\nfunc topKHeap(nums []int, k int) *minHeap {\n    // Initialize a min heap\n    h := &amp;minHeap{}\n    heap.Init(h)\n    // Push the first k array elements onto the heap\n    for i := 0; i &lt; k; i++ {\n        heap.Push(h, nums[i])\n    }\n    // Starting with element k+1, keep the heap size at k\n    for i := k; i &lt; len(nums); i++ {\n        // If the current element is larger than the heap top, pop the top element and push the current element\n        if nums[i] &gt; h.Top().(int) {\n            heap.Pop(h)\n            heap.Push(h, nums[i])\n        }\n    }\n    return h\n}\n</code></pre> top_k.swift<pre><code>/* Use a heap to find the k largest elements in an array */\nfunc topKHeap(nums: [Int], k: Int) -&gt; [Int] {\n    // Build a heap from the first k elements and use it as a min heap\n    var heap = Heap(nums.prefix(k))\n    // Starting with element k+1, keep the heap size at k\n    for i in nums.indices.dropFirst(k) {\n        // If the current element exceeds the heap minimum, remove the minimum and insert the current element\n        if nums[i] &gt; heap.min()! {\n            _ = heap.removeMin()\n            heap.insert(nums[i])\n        }\n    }\n    return heap.unordered\n}\n</code></pre> top_k.js<pre><code>/* Push an element onto the heap */\nfunction pushMinHeap(maxHeap, val) {\n    // Negate the value\n    maxHeap.push(-val);\n}\n\n/* Pop an element from the heap */\nfunction popMinHeap(maxHeap) {\n    // Negate the value\n    return -maxHeap.pop();\n}\n\n/* Access the top element of the heap */\nfunction peekMinHeap(maxHeap) {\n    // Negate the value\n    return -maxHeap.peek();\n}\n\n/* Get the elements in the heap */\nfunction getMinHeap(maxHeap) {\n    // Negate the value\n    return maxHeap.getMaxHeap().map((num) =&gt; -num);\n}\n\n/* Use a heap to find the k largest elements in an array */\nfunction topKHeap(nums, k) {\n    // Initialize a min heap\n    // Note: negate all values in the heap to simulate a min heap with a max heap\n    const maxHeap = new MaxHeap([]);\n    // Push the first k array elements onto the heap\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // Starting with element k+1, keep the heap size at k\n    for (let i = k; i &lt; nums.length; i++) {\n        // If the current element is larger than the heap top, pop the top element and push the current element\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // Return the elements in the heap\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.ts<pre><code>/* Push an element onto the heap */\nfunction pushMinHeap(maxHeap: MaxHeap, val: number): void {\n    // Negate the value\n    maxHeap.push(-val);\n}\n\n/* Pop an element from the heap */\nfunction popMinHeap(maxHeap: MaxHeap): number {\n    // Negate the value\n    return -maxHeap.pop();\n}\n\n/* Access the top element of the heap */\nfunction peekMinHeap(maxHeap: MaxHeap): number {\n    // Negate the value\n    return -maxHeap.peek();\n}\n\n/* Get the elements in the heap */\nfunction getMinHeap(maxHeap: MaxHeap): number[] {\n    // Negate the value\n    return maxHeap.getMaxHeap().map((num: number) =&gt; -num);\n}\n\n/* Use a heap to find the k largest elements in an array */\nfunction topKHeap(nums: number[], k: number): number[] {\n    // Initialize a min heap\n    // Note: negate all values in the heap to simulate a min heap with a max heap\n    const maxHeap = new MaxHeap([]);\n    // Push the first k array elements onto the heap\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // Starting with element k+1, keep the heap size at k\n    for (let i = k; i &lt; nums.length; i++) {\n        // If the current element is larger than the heap top, pop the top element and push the current element\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // Return the elements in the heap\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.dart<pre><code>/* Use a heap to find the k largest elements in an array */\nMinHeap topKHeap(List&lt;int&gt; nums, int k) {\n  // Initialize a min heap with the first k elements of the array\n  MinHeap heap = MinHeap(nums.sublist(0, k));\n  // Starting with element k+1, keep the heap size at k\n  for (int i = k; i &lt; nums.length; i++) {\n    // If the current element is larger than the heap top, pop the top element and push the current element\n    if (nums[i] &gt; heap.peek()) {\n      heap.pop();\n      heap.push(nums[i]);\n    }\n  }\n  return heap;\n}\n</code></pre> top_k.rs<pre><code>/* Use a heap to find the k largest elements in an array */\nfn top_k_heap(nums: Vec&lt;i32&gt;, k: usize) -&gt; BinaryHeap&lt;Reverse&lt;i32&gt;&gt; {\n    // BinaryHeap is a max heap; Reverse reverses the comparison order to implement a min heap\n    let mut heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n    // Push the first k array elements onto the heap\n    for &amp;num in nums.iter().take(k) {\n        heap.push(Reverse(num));\n    }\n    // Starting with element k+1, keep the heap size at k\n    for &amp;num in nums.iter().skip(k) {\n        // If the current element is larger than the heap top, pop the top element and push the current element\n        if num &gt; heap.peek().unwrap().0 {\n            heap.pop();\n            heap.push(Reverse(num));\n        }\n    }\n    heap\n}\n</code></pre> top_k.c<pre><code>/* Push an element onto the heap */\nvoid pushMinHeap(MaxHeap *maxHeap, int val) {\n    // Negate the element\n    push(maxHeap, -val);\n}\n\n/* Pop an element from the heap */\nint popMinHeap(MaxHeap *maxHeap) {\n    // Negate the element\n    return -pop(maxHeap);\n}\n\n/* Access the top element of the heap */\nint peekMinHeap(MaxHeap *maxHeap) {\n    // Negate the element\n    return -peek(maxHeap);\n}\n\n/* Get the elements in the heap */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // Negate every element in the heap and store the results in array res\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n/* Get the elements in the heap */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // Negate every element in the heap and store the results in array res\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n// Use a heap to find the k largest elements in an array\nint *topKHeap(int *nums, int sizeNums, int k) {\n    // Initialize a min heap\n    // Note: negate every element to simulate a min heap with a max heap\n    int *empty = (int *)malloc(0);\n    MaxHeap *maxHeap = newMaxHeap(empty, 0);\n    // Push the first k array elements onto the heap\n    for (int i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // Starting with element k+1, keep the heap size at k\n    for (int i = k; i &lt; sizeNums; i++) {\n        // If the current element is larger than the heap top, pop the top element and push the current element\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    int *res = getMinHeap(maxHeap);\n    // Free memory\n    delMaxHeap(maxHeap);\n    return res;\n}\n</code></pre> top_k.kt<pre><code>/* Use a heap to find the k largest elements in an array */\nfun topKHeap(nums: IntArray, k: Int): Queue&lt;Int&gt; {\n    // Initialize a min heap\n    val heap = PriorityQueue&lt;Int&gt;()\n    // Push the first k array elements onto the heap\n    for (i in 0..&lt;k) {\n        heap.offer(nums[i])\n    }\n    // Starting with element k+1, keep the heap size at k\n    for (i in k..&lt;nums.size) {\n        // If the current element is larger than the heap top, pop the top element and push the current element\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll()\n            heap.offer(nums[i])\n        }\n    }\n    return heap\n}\n</code></pre> top_k.rb<pre><code>### Use a heap to find the k largest elements in an array ###\ndef top_k_heap(nums, k)\n  # Initialize a min heap\n  # Note: negate all values in the heap to simulate a min heap with a max heap\n  max_heap = MaxHeap.new([])\n\n  # Push the first k array elements onto the heap\n  for i in 0...k\n    push_min_heap(max_heap, nums[i])\n  end\n\n  # Starting with element k+1, keep the heap size at k\n  for i in k...nums.length\n    # If the current element is larger than the heap top, pop the top element and push the current element\n    if nums[i] &gt; peek_min_heap(max_heap)\n      pop_min_heap(max_heap)\n      push_min_heap(max_heap, nums[i])\n    end\n  end\n\n  get_min_heap(max_heap)\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>There are \\(n\\) rounds of heap insertion and removal to consider, and the heap has at most \\(k\\) elements, so the time complexity is \\(O(n \\log k)\\). This is highly efficient: when \\(k\\) is small, the complexity approaches \\(O(n)\\); when \\(k\\) is large, it does not exceed \\(O(n \\log n)\\).</p> <p>This approach also suits dynamic data streams. As new data arrives, we can maintain the heap to update the largest \\(k\\) elements continuously.</p>","path":["Chapter 8.   Heaps","8.3   The top-k problem"],"tags":[]},{"location":"chapter_hello_algo/","level":1,"title":"Foreword","text":"<p>A few years ago, I shared a series of solutions to the “Coding Interviews” problems on LeetCode and received encouragement and support from many readers. In our conversations, one question came up more than any other: “How do I get started with algorithms?” Over time, I became deeply interested in this question.</p> <p>Jumping into practice problems with little preparation seems to be the most popular approach: simple, direct, and effective. Yet solving problems is rather like playing Minesweeper. Strong independent learners can clear the mines one by one, while those with a weak foundation may run into one setback after another and gradually retreat in frustration. Reading a textbook from cover to cover is another common approach. For job seekers, however, theses, applications, written tests, and interview preparation already consume most of their energy, making a thick textbook a formidable challenge.</p> <p>If you face similar difficulties, I am glad this book has found you. It is my answer to that question—perhaps not the optimal solution, but at least a constructive attempt. Although this book alone will not secure a job offer, it will guide you through the “knowledge map” of data structures and algorithms. It will show you the shapes, sizes, and locations of different “mines” and teach you ways to clear them. With these skills, I believe you can approach practice problems and technical reading more confidently and gradually build a complete body of knowledge.</p> <p>I wholeheartedly agree with Professor Feynman's words: “Knowledge isn't free. You have to pay attention.” In this sense, the book is not entirely “free.” To honor the precious attention you invest in it, I will devote as much of my own attention as I can to writing it.</p> <p>I am aware of the limits of my knowledge. Although the content has undergone considerable refinement, many errors surely remain. I sincerely welcome corrections and suggestions from teachers and fellow learners.</p> <p></p> Hello, algorithms! <p>Computers have transformed the world. Their speed and remarkable programmability make them an ideal medium for executing algorithms and processing data. From the lifelike graphics of video games and the intelligent decisions of self-driving cars to AlphaGo's brilliant games and ChatGPT's natural interactions, these applications are all compelling expressions of algorithms running on computers.</p> <p>In fact, algorithms and data structures existed throughout the world long before computers. Early algorithms were relatively simple, such as ancient counting methods and steps for making tools. As civilization advanced, algorithms became more refined and complex. From exquisite craftsmanship and industrial products that boosted productivity to the scientific laws governing the universe, ingenious algorithmic ideas lie behind almost everything, whether ordinary or astonishing.</p> <p>Data structures are equally widespread. Systems ranging from social networks to subway routes can be modeled as “graphs.” Social organizations, from nations to families, often have the characteristics of “trees.” Winter clothing behaves like a “stack”: what goes on first comes off last. A tube of shuttlecocks is like a “queue,” with items inserted at one end and removed at the other. A dictionary resembles a “hash table,” allowing quick lookup of a desired entry.</p> <p>Through clear animated illustrations and runnable code examples, this book aims to help readers understand the core concepts of algorithms and data structures and implement them through programming. Building on that foundation, it seeks to reveal how algorithms take shape in our complex world and to show their beauty. I hope you find it helpful!</p>","path":["Foreword"],"tags":[]},{"location":"chapter_introduction/","level":1,"title":"Chapter 1.   Getting to know algorithms","text":"<p>Abstract</p> <p>A young woman dances gracefully among the data, the melody of algorithms rippling through her skirt.</p> <p>She invites you to join her. Follow her steps into a world of algorithms filled with logic and beauty.</p>","path":["Chapter 1.   Getting to know algorithms"],"tags":[]},{"location":"chapter_introduction/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>1.1   Algorithms are everywhere</li> <li>1.2   What is an algorithm?</li> <li>1.3   Summary</li> </ul>","path":["Chapter 1.   Getting to know algorithms"],"tags":[]},{"location":"chapter_introduction/algorithms_are_everywhere/","level":1,"title":"1.1   Algorithms are everywhere","text":"<p>When we hear the word “algorithm,” we naturally think of mathematics. In fact, many algorithms do not involve complex mathematics. Instead, they rely on basic logic that we encounter throughout everyday life.</p> <p>Before we formally explore algorithms, here is an interesting fact: you have already learned many algorithms without realizing it, and you routinely use them in everyday life. The following examples illustrate this point.</p> <p>Example 1: Looking up a word in a dictionary. In a Chinese dictionary, each character has a corresponding pinyin pronunciation, and entries are arranged alphabetically by pinyin. Suppose we want to find a character whose pinyin begins with \\(r\\). We would usually follow the process shown in Figure 1-1.</p> <ol> <li>Open the dictionary roughly halfway through and check the initial letter on that page. Suppose it is \\(m\\).</li> <li>Since \\(r\\) comes after \\(m\\) in the pinyin alphabet, rule out the first half of the dictionary and narrow the search to the second half.</li> <li>Repeat steps <code>1.</code> and <code>2.</code> until you find the pages whose pinyin entries begin with \\(r\\).</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 1-1   Steps for looking up a word in a dictionary </p> <p>This dictionary skill, learned in elementary school, is actually the well-known “binary search” algorithm. From a data structure perspective, we can view the dictionary as a sorted “array”; from an algorithm perspective, the sequence of lookup operations above is “binary search.”</p> <p>Example 2: Arranging playing cards. When playing cards, we need to arrange our hand from smallest to largest each round. The process is shown in Figure 1-2.</p> <ol> <li>Divide the cards into “sorted” and “unsorted” sections, initially treating the single leftmost card as sorted.</li> <li>Take a card from the unsorted section and insert it into the correct position in the sorted section. The two leftmost cards are now sorted.</li> <li>Repeat step <code>2.</code>, inserting one card from the unsorted section into the sorted section each round, until all the cards are sorted.</li> </ol> <p></p> <p> Figure 1-2   Steps for sorting playing cards </p> <p>This method of arranging playing cards is essentially “insertion sort,” an algorithm that is very efficient for small datasets. Insertion sort appears in the library sorting functions of many programming languages.</p> <p>Example 3: Making change. Suppose we buy goods costing \\(69\\) yuan at a supermarket and pay the cashier \\(100\\) yuan. The cashier needs to give us \\(31\\) yuan in change and naturally reasons through the process shown in Figure 1-3.</p> <ol> <li>The available denominations smaller than \\(31\\) yuan are \\(1\\), \\(5\\), \\(10\\), and \\(20\\) yuan.</li> <li>Take the largest available denomination, \\(20\\) yuan, leaving \\(31 - 20 = 11\\) yuan.</li> <li>Take the largest denomination that fits the remaining amount, \\(10\\) yuan, leaving \\(11 - 10 = 1\\) yuan.</li> <li>Take the largest denomination that fits the remaining amount, \\(1\\) yuan, leaving \\(1 - 1 = 0\\) yuan.</li> <li>The change is complete: \\(20 + 10 + 1 = 31\\) yuan.</li> </ol> <p></p> <p> Figure 1-3   The process of making change </p> <p>At each step above, we make the choice that looks best at the moment—using the largest possible denomination—and eventually obtain a valid way to make change. In terms of data structures and algorithms, this is a “greedy” algorithm.</p> <p>From cooking a meal to traveling between stars, solving almost any problem involves algorithms. Computers allow us to store data structures in memory through programming and write code that calls on CPUs and GPUs to execute algorithms. We can therefore bring real-world problems to computers and solve complex problems more efficiently.</p> <p>Tip</p> <p>If concepts such as data structures, algorithms, arrays, and binary search still seem unclear, keep reading. This book will guide you into the world of data structures and algorithms.</p>","path":["Chapter 1.   Getting to know algorithms","1.1   Algorithms are everywhere"],"tags":[]},{"location":"chapter_introduction/summary/","level":1,"title":"1.3   Summary","text":"","path":["Chapter 1.   Getting to know algorithms","1.3   Summary"],"tags":[]},{"location":"chapter_introduction/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<ul> <li>Algorithms are everywhere in daily life; they are not remote, inaccessible knowledge. In fact, we have already learned many algorithms without realizing it and use them to solve problems large and small.</li> <li>Looking up a word in a dictionary follows the same principle as binary search. Binary search illustrates the important algorithmic idea of divide and conquer.</li> <li>Arranging playing cards closely resembles insertion sort. Insertion sort is well suited to small datasets.</li> <li>Making change is essentially a greedy algorithm: at each step, make the choice that looks best at the moment.</li> <li>An algorithm is a set of instructions or steps that solves a specific problem within a finite time. A data structure is a way of organizing and storing data in a computer.</li> <li>Data structures and algorithms are closely connected. Data structures provide the foundation for algorithms, while algorithms bring data structures to life.</li> <li>We can compare data structures and algorithms to assembling building blocks: the blocks represent data; their shapes and connections represent data structures; and the assembly steps represent algorithms.</li> </ul>","path":["Chapter 1.   Getting to know algorithms","1.3   Summary"],"tags":[]},{"location":"chapter_introduction/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: As a programmer, I have never used algorithms to solve problems in my daily work. Common algorithms are already built into programming languages, so I can simply use them. Does this mean the problems we encounter at work are not difficult enough to require algorithms?</p> <p>If specific job skills are like the techniques of a martial art, foundational subjects are more like the underlying strength that makes those techniques effective.</p> <p>I believe the purpose of learning algorithms—and other foundational subjects—is not to implement them from scratch at work. It is to use what we have learned to respond and make judgments professionally when solving problems, improving the overall quality of our work. For a simple example, every programming language has a built-in sorting function:</p> <ul> <li>Without studying data structures and algorithms, we might pass any dataset to this sorting function. It runs smoothly and performs reasonably well, so nothing seems wrong.</li> <li>After studying algorithms, however, we know that the built-in sorting function has a time complexity of \\(O(n \\log n)\\). If the data consists of integers with a fixed number of digits, such as student IDs, we can use the more efficient “radix sort” to reduce the time complexity to \\(O(nk)\\), where \\(k\\) is the number of digits. For large datasets, the time saved can create considerable value through lower costs, a better user experience, and so on.</li> </ul> <p>In engineering, optimal solutions are difficult to reach for many problems, and many others are solved only “well enough.” How difficult a problem appears depends both on its nature and on the knowledge of the person examining it. The more knowledge and experience we have, the more deeply we can analyze a problem and the more elegantly we can solve it.</p> <p>Q: Now that AI agents write almost all our code, do we still need to learn algorithms?</p> <p>To quote Mu Li: “An agent is an ability multiplier: it multiplies your ability by 10. It is not an adder that adds 10 to your ability.”</p> <p>If AI multiplies our ability, then that starting ability matters too. I think two kinds of ability are especially worth developing. The first is communication, particularly collaboration with AI: state the goal clearly, supply the necessary context, and correct the course promptly. Explaining a problem to AI also helps us clarify it for ourselves. The second is foundational knowledge. AI can fill in the details, but we need to understand the basic concepts ourselves to ask relevant questions, judge what is correct, and turn its output into useful results.</p> <p>One distinctive feature of AI is that, unlike earlier tools, it has no clear boundaries to its functionality. It is a little like water, taking different shapes in different people's hands. The stronger a person's abilities and the better their taste, the more pronounced AI's multiplying effect becomes.</p>","path":["Chapter 1.   Getting to know algorithms","1.3   Summary"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/","level":1,"title":"1.2   What is an algorithm?","text":"","path":["Chapter 1.   Getting to know algorithms","1.2   What is an algorithm?"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/#121-definition-of-an-algorithm","level":2,"title":"1.2.1   Definition of an algorithm","text":"<p>An algorithm is a set of instructions or steps that solves a specific problem within a finite time. It has the following properties.</p> <ul> <li>The problem is well defined, with clear definitions of its inputs and outputs.</li> <li>The algorithm is feasible: it can finish within a finite number of steps, using finite time and memory.</li> <li>Every step has a definite meaning. Given the same inputs and execution conditions, the output is always the same.</li> </ul>","path":["Chapter 1.   Getting to know algorithms","1.2   What is an algorithm?"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/#122-definition-of-a-data-structure","level":2,"title":"1.2.2   Definition of a data structure","text":"<p>A data structure is a way of organizing and storing data. It encompasses the data itself, the relationships between data items, and the methods used to operate on them. Its design aims to achieve the following goals.</p> <ul> <li>Use as little space as possible to conserve computer memory.</li> <li>Make data operations as fast as possible, including access, insertion, deletion, and updating.</li> <li>Provide a concise representation of data and its logical relationships so that algorithms can run efficiently.</li> </ul> <p>Designing a data structure involves tradeoffs. Improving one aspect often requires compromising another. Here are two examples.</p> <ul> <li>Compared with arrays, linked lists make insertion and deletion easier, at the expense of data access speed.</li> <li>Compared with linked lists, graphs provide richer information about logical relationships but require more memory.</li> </ul>","path":["Chapter 1.   Getting to know algorithms","1.2   What is an algorithm?"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/#123-the-relationship-between-data-structures-and-algorithms","level":2,"title":"1.2.3   The relationship between data structures and algorithms","text":"<p>As Figure 1-4 shows, data structures and algorithms are closely intertwined in three ways.</p> <ul> <li>Data structures are the foundation of algorithms. They provide algorithms with data in a structured form and methods for operating on that data.</li> <li>Algorithms bring data structures to life. A data structure by itself only stores information; it must be combined with an algorithm to solve a specific problem.</li> <li>An algorithm can often be implemented using different data structures, but its efficiency may vary greatly. Choosing the right data structure is crucial.</li> </ul> <p></p> <p> Figure 1-4   The relationship between data structures and algorithms </p> <p>Data structures and algorithms are like assembling the building blocks shown in Figure 1-5. A set of building blocks includes not only many pieces but also detailed assembly instructions. By following those instructions step by step, we can build an intricate model.</p> <p></p> <p> Figure 1-5   Assembling building blocks </p> <p>Table 1-1 describes this analogy in detail.</p> <p> Table 1-1   Comparing data structures and algorithms to building blocks </p> Data structures and algorithms Building blocks Input data Unassembled blocks Data structure How the blocks are organized, including their shapes, sizes, and connections Algorithm A sequence of steps for assembling the blocks into the desired form Output data The finished model <p>Data structures and algorithms are independent of programming languages. This is why the book can provide implementations in multiple languages.</p> <p>A common shorthand</p> <p>In everyday discussions, we often shorten “data structures and algorithms” to “algorithms.” For example, the familiar algorithm problems on LeetCode actually test knowledge of both data structures and algorithms.</p>","path":["Chapter 1.   Getting to know algorithms","1.2   What is an algorithm?"],"tags":[]},{"location":"chapter_paperbook/","level":1,"title":"Print edition (Chinese)","text":"<p>After a long period of refinement, the Simplified Chinese print edition of Hello Algo is finally here! A line of poetry captures how I feel:</p> <p>Chase the wind and moon without pause; beyond the open plains lie the mountains of spring.</p> <p></p> <p>The following video introduces the print edition and shares some of my thoughts on:</p> <ul> <li>The importance of learning data structures and algorithms.</li> <li>Why I chose Python for the print edition.</li> <li>What sharing knowledge means to me.</li> </ul> <p>I am new to making videos. Please show your support with a like, a coin, and a favorite—thank you!</p> <p>Here are some snapshots of the print edition:</p> <p></p> <p></p>","path":["Print edition (Chinese)"],"tags":[]},{"location":"chapter_paperbook/#strengths-and-limitations","level":2,"title":"Strengths and limitations","text":"<p>Here are a few things about the print edition that may pleasantly surprise you:</p> <ul> <li>Full-color printing makes the most of the book's animation-style illustrations.</li> <li>Carefully selected paper reproduces colors faithfully while preserving the distinctive feel of a printed book.</li> <li>Formatting is more standardized than in the web edition; for example, formulas in figures are italicized.</li> <li>A fold-out mind map and bookmarks are included at no extra cost.</li> <li>The print, web, and PDF editions have synchronized content, so you can switch freely between them.</li> </ul> <p>Tip</p> <p>Synchronizing the print and web editions is difficult, so there may be minor differences. Thank you for your understanding!</p> <p>Of course, there are also a few points to consider before buying:</p> <ul> <li>The book uses Python, which may not be your main language. You can treat Python as pseudocode and focus on understanding the ideas.</li> <li>Full-color printing greatly improves the reading experience for illustrations and code, but costs somewhat more than black-and-white printing.</li> </ul> <p>Tip</p> <p>“Print quality” and “price” are like “time efficiency” and “space efficiency” in algorithms: it is difficult to optimize both. In my view, print quality corresponds to time efficiency and deserves greater priority.</p>","path":["Print edition (Chinese)"],"tags":[]},{"location":"chapter_paperbook/#where-to-buy","level":2,"title":"Where to buy","text":"<p>If you are interested in the Simplified Chinese print edition, consider getting a copy. We have arranged a 50% discount on the new book. Follow this link or scan the QR code below:</p> <p></p>","path":["Print edition (Chinese)"],"tags":[]},{"location":"chapter_paperbook/#afterword","level":2,"title":"Afterword","text":"<p>At first, I underestimated the work involved in publishing a print edition. I assumed that as long as the open-source project was well maintained, some form of automation could generate the printed book. Experience showed that producing a printed book differs greatly from updating an open-source project, and moving between the two requires considerable extra work.</p> <p>There is still a long road from a first draft to a manuscript ready for publication. It takes close collaboration and sustained refinement by the author and the publisher's planning, editorial, design, and marketing teams. My thanks go to Turing's commissioning editor Junhua Wang and everyone at Posts &amp; Telecom Press and Turing Community who took part in publishing this book!</p> <p>I hope this book will help you!</p>","path":["Print edition (Chinese)"],"tags":[]},{"location":"chapter_preface/","level":1,"title":"Chapter 0.   Preface","text":"<p>Abstract</p> <p>Algorithms are like a beautiful symphony, with each line of code flowing to its rhythm.</p> <p>May this book play softly in your mind, leaving a melody that is both distinctive and lasting.</p>","path":["Chapter 0.   Preface"],"tags":[]},{"location":"chapter_preface/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>0.1   About this book</li> <li>0.2   How to use this book</li> <li>0.3   Summary</li> </ul>","path":["Chapter 0.   Preface"],"tags":[]},{"location":"chapter_preface/about_the_book/","level":1,"title":"0.1   About this book","text":"<p>This project aims to create an open-source, free, beginner-friendly introduction to data structures and algorithms.</p> <ul> <li>Animated illustrations throughout the book make concepts clear and accessible, with a gentle learning curve that guides beginners through the landscape of data structures and algorithms.</li> <li>Source code runs with one click, helping readers improve their programming skills through practice and understand how algorithms work and how data structures are implemented.</li> <li>We encourage readers to learn together. Ask questions and share insights in the comments, and make progress through discussion.</li> </ul>","path":["Chapter 0.   Preface","0.1   About this book"],"tags":[]},{"location":"chapter_preface/about_the_book/#011-who-this-book-is-for","level":2,"title":"0.1.1   Who this book is for","text":"<p>If you are new to algorithms, have never studied them, or have some experience with practice problems but only a hazy understanding of data structures and algorithms—sometimes feeling you understand, sometimes not—this book is written for you!</p> <p>If you have already solved a substantial number of problems and know most common problem types, this book can help you review and organize your knowledge. The repository's source code can serve as a “problem-solving toolkit” or an “algorithm dictionary.”</p> <p>If you are an expert in algorithms, we welcome your valuable suggestions and invite you to contribute to the book.</p> <p>Prerequisites</p> <p>You should have basic programming skills in at least one language and be able to read and write simple code.</p>","path":["Chapter 0.   Preface","0.1   About this book"],"tags":[]},{"location":"chapter_preface/about_the_book/#012-organization","level":2,"title":"0.1.2   Organization","text":"<p>Figure 0-1 shows the book's main topics.</p> <ul> <li>Complexity analysis: Criteria and methods for evaluating data structures and algorithms, including how to derive time and space complexity, common complexity classes, and examples.</li> <li>Data structures: Basic data types and ways to classify data structures; definitions, strengths, weaknesses, common operations, variants, typical applications, and implementations of arrays, linked lists, stacks, queues, hash tables, trees, heaps, graphs, and more.</li> <li>Algorithms: Definitions, strengths, weaknesses, efficiency, applications, solution steps, and example problems for searching, sorting, divide and conquer, backtracking, dynamic programming, greedy algorithms, and more.</li> </ul> <p></p> <p> Figure 0-1   The book's main topics </p>","path":["Chapter 0.   Preface","0.1   About this book"],"tags":[]},{"location":"chapter_preface/about_the_book/#013-acknowledgments","level":2,"title":"0.1.3   Acknowledgments","text":"<p>This book continues to improve through the collective efforts of many contributors in the open-source community. Thank you to everyone who has invested time and energy in writing it. In the order automatically generated by GitHub, they are: krahets, coderonion, Gonglja, nuomi1, Reanon, justin-tse, hpstory, danielsss, curtishd, night-cruise, S-N-O-R-L-A-X, rongyi, msk397, gvenusleo, khoaxuantu, rivertwilight, K3v123, gyt95, zhuoqinyue, yuelinxin, Zuoxun, mingXta, Phoenix0415, FangYuan33, GN-Yu, longsizhuo, pengchzn, QiLOL, Cathay-Chen, guowei-gong, xBLACKICEx, IsChristina, JoseHung, qualifier1024, hello-ikun, magentaqin, Guanngxu, thomasq0, sunshinesDL, L-Super, Transmigration-zhou, WSL0809, Slone123c, lhxsm, yuan0221, what-is-me, theNefelibatas, Shyam-Chen, sangxiaai, longranger2, codeberg-user, xiongsp, JeffersonHuang, prinpal, seven1240, Wonderdch, malone6, xiaomiusa87, gaofer, bluebean-cloud, a16su, SamJin98, hongyun-robot, nanlei, XiaChuerwu, yd-j, iron-irax, mgisr, steventimes, junminhong, heshuyue, danny900714, Nigh, Dr-XYZ, MolDuM, XC-Zero, reeswell, PXG-XPG, NI-SW, Horbin-Magician, Enlightenus, YangXuanyi, xjr7670, beatrix-chan, DullSword, qq909244296, iStig, boloboloda, hts0000, gledfish, fbigm, echo1937, jiaxianhua, wenjianmin, keshida, kilikilikid, lclc6, lwbaptx, linyejoe2, liuxjerry, szu17dmy, dshlstarr, Yucao-cy, coderlef, czruby, bongbongbakudan, beintentional, ZongYangL, ZhongYuuu, ZhongGuanbin, hezhizhen, linzeyan, ZJKung, JTCPOWI, KawaiiAsh, luluxia, xb534, ztkuaikuai, yw-1021, ElaBosak233, baagod, zhouLion, yishangzhang, yi427, yanedie, yabo083, weibk, wangwang105, th1nk3r-ing, tao363, 4yDX3906, syd168, sslmj2020, smilelsb, siqyka, selear, sdshaoda, Xi-Row, popozhu, nuquist19, noobcodemaker, XiaoK29, chadyi, lyl625760, lucaswangdev, llql1211, 0130w, shanghai-Jerry, EJackYang, Javesun99, eltociear, lipusheng, KNChiu, BlindTerran, ShiMaRing, lovelock, FreddieLi, FloranceYeh, fanchenggang, gltianwen, goerll, nedchu, curly210102, CuB3y0nd, KraHsu, CarrotDLaw, youshaoXG, bubble9um, Asashishi, Asa0oo0o0o, fanenr, eagleanurag, akshiterate, 52coder, foursevenlove, KorsChen, hopkings2008, yang-le, realwujing, Evilrabbit520, Umer-Jahangir, Turing-1024-Lee, Suremotoo, paoxiaomooo, Chieko-Seren, Senrian, Allen-Scai, 19santosh99, ymmmas, Risuntsy, Richard-Zhang1019, RafaelCaso, qingpeng9802, primexiao, Urbaner3, codetypess, nidhoggfgg, MwumLi, CreatorMetaSky, martinx, ZnYang2018, hugtyftg, logan-qiu, psychelzh, Kunchen-Luo, Keynman and KeiichiKasai.</p> <p>The code was reviewed by coderonion, curtishd, Gonglja, gvenusleo, hpstory, justin-tse, khoaxuantu, krahets, night-cruise, nuomi1, Reanon, and rongyi, listed alphabetically. Thank you for the time and effort you devoted to ensuring consistent, well-written code across languages.</p> <p>The English edition was reviewed by yuelinxin, K3v123, magentaqin, QiLOL, Phoenix0415, SamJin98, yanedie, RafaelCaso, pengchzn, and thomasq0; the Japanese edition by eltociear; the Russian edition by И. А. Шевкун and Yuyan Huang; and the Traditional Chinese edition by Shyam-Chen and Dr-XYZ. Their contributions have made the book accessible to a wider readership. Thank you all.</p> <p>The tool for generating the book's ePub edition was developed by zhongfq. Thank you for giving readers a more flexible way to read.</p> <p>Many people helped me while I was writing this book.</p> <ul> <li>Thank you to my mentor at work, Dr. Xi Li. During a long conversation, your encouragement to “get started” strengthened my resolve to write this book;</li> <li>Thank you to my girlfriend Paopao, the book's first reader. Your many suggestions from a beginner's perspective have made it more accessible to newcomers;</li> <li>Thank you to Tengbao, Qibao, and Feibao for giving the book a creative name that brings back fond memories of writing our first line of code, “Hello World!”;</li> <li>Thank you to Xiaoquan for your professional help with intellectual property, which played an important role in improving this open-source book;</li> <li>Thank you to Sutong for designing the beautiful cover and logo, and for patiently revising them many times to satisfy my perfectionism;</li> <li>Thank you to @squidfunk for your typography advice and for developing the open-source documentation theme Material-for-MkDocs.</li> </ul> <p>While writing, I read many textbooks and articles on data structures and algorithms. These works provided excellent models and helped ensure the accuracy and quality of this book. My thanks go to all the teachers and predecessors whose outstanding contributions made this possible!</p> <p>The book advocates learning through both thought and hands-on practice, an approach deeply inspired by Dive into Deep Learning. I warmly recommend this excellent book to every reader.</p> <p>My heartfelt thanks to my parents. Your constant support and encouragement have given me the opportunity to do this wonderfully engaging work.</p>","path":["Chapter 0.   Preface","0.1   About this book"],"tags":[]},{"location":"chapter_preface/suggestions/","level":1,"title":"0.2   How to use this book","text":"<p>Tip</p> <p>For the best reading experience, we recommend reading this section in full.</p>","path":["Chapter 0.   Preface","0.2   How to use this book"],"tags":[]},{"location":"chapter_preface/suggestions/#021-writing-conventions","level":2,"title":"0.2.1   Writing conventions","text":"<ul> <li>Sections with <code>*</code> after the title are optional and cover more difficult material. If your time is limited, you can skip them at first.</li> <li>Technical terms appear in bold in the print and PDF editions or are underlined in the web edition, such as array. Remembering these terms will help you read technical literature.</li> <li>Key points and summary statements appear in bold and deserve particular attention.</li> <li>Words and phrases used in a specific sense appear in “quotation marks” to avoid ambiguity.</li> <li>When terminology differs between programming languages, this book follows Python; for example, it uses <code>None</code> to represent a null value.</li> <li>For a more compact layout, this book sometimes departs from a language's usual commenting conventions. Comments fall into three main categories: heading comments, explanatory comments, and multiline comments.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>\"\"\"Heading comment identifying a function, class, test case, and so on\"\"\"\n\n# Explanatory comment describing the code in detail\n\n\"\"\"\nMultiline\ncomment\n\"\"\"\n</code></pre> <pre><code>/* Heading comment identifying a function, class, test case, and so on */\n\n// Explanatory comment describing the code in detail\n\n/**\n * Multiline\n * comment\n */\n</code></pre> <pre><code>/* Heading comment identifying a function, class, test case, and so on */\n\n// Explanatory comment describing the code in detail\n\n/**\n * Multiline\n * comment\n */\n</code></pre> <pre><code>/* Heading comment identifying a function, class, test case, and so on */\n\n// Explanatory comment describing the code in detail\n\n/**\n * Multiline\n * comment\n */\n</code></pre> <pre><code>/* Heading comment identifying a function, class, test case, and so on */\n\n// Explanatory comment describing the code in detail\n\n/**\n * Multiline\n * comment\n */\n</code></pre> <pre><code>/* Heading comment identifying a function, class, test case, and so on */\n\n// Explanatory comment describing the code in detail\n\n/**\n * Multiline\n * comment\n */\n</code></pre> <pre><code>/* Heading comment identifying a function, class, test case, and so on */\n\n// Explanatory comment describing the code in detail\n\n/**\n * Multiline\n * comment\n */\n</code></pre> <pre><code>/* Heading comment identifying a function, class, test case, and so on */\n\n// Explanatory comment describing the code in detail\n\n/**\n * Multiline\n * comment\n */\n</code></pre> <pre><code>/* Heading comment identifying a function, class, test case, and so on */\n\n// Explanatory comment describing the code in detail\n\n/**\n * Multiline\n * comment\n */\n</code></pre> <pre><code>/* Heading comment identifying a function, class, test case, and so on */\n\n// Explanatory comment describing the code in detail\n\n// Multiline\n// comment\n</code></pre> <pre><code>/* Heading comment identifying a function, class, test case, and so on */\n\n// Explanatory comment describing the code in detail\n\n/**\n * Multiline\n * comment\n */\n</code></pre> <pre><code>/* Heading comment identifying a function, class, test case, and so on */\n\n// Explanatory comment describing the code in detail\n\n/**\n * Multiline\n * comment\n */\n</code></pre> <pre><code>### Heading comment identifying a function, class, test case, and so on ###\n\n# Explanatory comment describing the code in detail\n\n# Multiline\n# comment\n</code></pre>","path":["Chapter 0.   Preface","0.2   How to use this book"],"tags":[]},{"location":"chapter_preface/suggestions/#022-learn-efficiently-with-animated-illustrations","level":2,"title":"0.2.2   Learn efficiently with animated illustrations","text":"<p>Compared with text, videos and images convey information in a denser, more structured form that is easier to understand. In this book, key and difficult concepts are presented mainly through animated illustrations, with text providing explanations and additional detail.</p> <p>When a passage includes an animated illustration like the one below, focus on the illustration and use the text to support it, drawing on both to understand the material.</p> <p></p> <p> Figure 0-2   An example of an animated illustration </p>","path":["Chapter 0.   Preface","0.2   How to use this book"],"tags":[]},{"location":"chapter_preface/suggestions/#023-deepen-your-understanding-through-coding","level":2,"title":"0.2.3   Deepen your understanding through coding","text":"<p>The accompanying code is hosted in the GitHub repository. As Figure 0-3 shows, the source code includes test cases and can be run with one click.</p> <p>If time permits, we recommend typing out the code yourself while referring to the examples. If your study time is limited, at least read and run all the code.</p> <p>Writing code often teaches us more than reading it. Hands-on practice is where real learning happens.</p> <p></p> <p> Figure 0-3   An example of running code </p> <p>Getting ready to run the code involves three main steps.</p> <p>Step 1: Set up a local programming environment. Follow the tutorial in the appendix. If your environment is already set up, skip this step.</p> <p>Step 2: Clone or download the repository. Visit the GitHub repository. If you have Git installed, clone the repository with the following command:</p> <pre><code>git clone https://github.com/krahets/hello-algo.git\n</code></pre> <p>Alternatively, click “Download ZIP” in the location shown below to download the code as an archive, then extract it on your computer.</p> <p></p> <p> Figure 0-4   Cloning the repository and downloading the code </p> <p>Step 3: Run the source code. As shown below, when a code block has a filename at the top, you can find the corresponding source file in the repository's <code>en/codes</code> folder. These files can be run with one click, saving unnecessary debugging time so you can focus on learning.</p> <p></p> <p> Figure 0-5   A code block labeled with its source filename </p> <p>In addition to running code locally, the web edition supports visual execution of Python code through pythontutor. Click “Visualize execution” beneath a code block to expand the view and watch the algorithm run. You can also click “View full screen” for a larger display, as shown in Figure 0-6.</p> <p></p> <p> Figure 0-6   Python code visualized in PythonTutor </p>","path":["Chapter 0.   Preface","0.2   How to use this book"],"tags":[]},{"location":"chapter_preface/suggestions/#024-learn-together-through-questions-and-discussion","level":2,"title":"0.2.4   Learn together through questions and discussion","text":"<p>As you read, do not simply skip concepts you have not understood. You are welcome to ask questions in the comments. My fellow contributors and I will do our best to answer, usually within two days.</p> <p>As shown below, every section in the web edition has a comments area at the bottom. I hope you will read it regularly. Other readers' questions can reveal gaps in your own understanding and encourage deeper thought. I also hope you will generously answer their questions, share your insights, and help them make progress.</p> <p></p> <p> Figure 0-7   An example of the comments area </p>","path":["Chapter 0.   Preface","0.2   How to use this book"],"tags":[]},{"location":"chapter_preface/suggestions/#025-a-learning-path-for-algorithms","level":2,"title":"0.2.5   A learning path for algorithms","text":"<p>Broadly speaking, learning data structures and algorithms can be divided into three stages.</p> <ol> <li>Stage 1: Learn the fundamentals. Become familiar with the properties and uses of data structures, and study how different algorithms work, their steps, applications, and efficiency.</li> <li>Stage 2: Solve practice problems. We recommend starting with popular problems and solving at least 100 to become familiar with common algorithmic problems. When you first begin, forgetting what you have learned may be a challenge, but this is normal. Review problems according to the “Ebbinghaus forgetting curve”; after 3–5 rounds of repetition, they will usually be firmly in memory. See this GitHub repository for recommended problem lists and practice plans.</li> <li>Stage 3: Build a systematic understanding. Read algorithm articles, problem-solving frameworks, and textbooks to keep expanding your knowledge. In your practice, try more advanced strategies: group problems by topic, solve one problem in several ways, or apply one solution to several problems. You can find advice on these strategies in various communities.</li> </ol> <p>As Figure 0-8 shows, this book mainly covers “Stage 1,” helping you approach Stages 2 and 3 more efficiently.</p> <p></p> <p> Figure 0-8   A learning path for algorithms </p>","path":["Chapter 0.   Preface","0.2   How to use this book"],"tags":[]},{"location":"chapter_preface/summary/","level":1,"title":"0.3   Summary","text":"","path":["Chapter 0.   Preface","0.3   Summary"],"tags":[]},{"location":"chapter_preface/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<ul> <li>This book is primarily for beginners in algorithms. If you already have some background, it can help you review your knowledge systematically, and its source code can serve as a toolkit for solving practice problems.</li> <li>The book has three main parts—complexity analysis, data structures, and algorithms—covering most topics in the field.</li> <li>For beginners, reading an introductory book early on is essential and can help avoid many detours.</li> <li>The animated illustrations usually explain key or challenging concepts. Pay particular attention to them as you read.</li> <li>Practice is the best way to learn programming. We strongly recommend running the source code and writing it out yourself.</li> <li>Every section in the web edition has a comments area. You are welcome to share your questions and insights at any time.</li> </ul>","path":["Chapter 0.   Preface","0.3   Summary"],"tags":[]},{"location":"chapter_reference/","level":1,"title":"References","text":"<p>[1] Thomas H. Cormen, et al. Introduction to Algorithms (3<sup>rd</sup> Edition).</p> <p>[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1<sup>st</sup> Edition).</p> <p>[3] Robert Sedgewick, et al. Algorithms (4<sup>th</sup> Edition).</p> <p>[4] Weimin Yan. Data Structures (C Edition).</p> <p>[5] Junhui Deng. Data Structures (C++ Edition, 3<sup>rd</sup> Edition).</p> <p>[6] Mark Allen Weiss; translated by Yue Chen. Data Structures and Algorithm Analysis in Java (3<sup>rd</sup> Edition).</p> <p>[7] Jie Cheng. A Friendly Guide to Data Structures.</p> <p>[8] Zheng Wang. The Beauty of Data Structures and Algorithms.</p> <p>[9] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6<sup>th</sup> Edition).</p> <p>[10] Aston Zhang, et al. Dive into Deep Learning.</p>","path":["References"],"tags":[]},{"location":"chapter_searching/","level":1,"title":"Chapter 10.   Searching","text":"<p>Abstract</p> <p>Searching is an adventure into the unknown. We may need to explore every corner of a mysterious space, or we may be able to pinpoint our target quickly.</p> <p>On this journey, every exploration may reveal an unexpected answer.</p>","path":["Chapter 10.   Searching"],"tags":[]},{"location":"chapter_searching/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>10.1   Binary search</li> <li>10.2   Finding an insertion point with binary search</li> <li>10.3   Finding boundaries with binary search</li> <li>10.4   Optimizing with hashing</li> <li>10.5   Search algorithms revisited</li> <li>10.6   Summary</li> <li>10.7   Exercises</li> </ul>","path":["Chapter 10.   Searching"],"tags":[]},{"location":"chapter_searching/binary_search/","level":1,"title":"10.1   Binary search","text":"<p>Binary search is an efficient search algorithm based on divide and conquer. It uses the sorted order of the data to halve the search range each round, continuing until it finds the target element or the search interval becomes empty.</p> <p>Question</p> <p>Given an array <code>nums</code> of length \\(n\\), with distinct elements arranged in ascending order, find and return the index of <code>target</code>. If the array does not contain this element, return \\(-1\\). An example is shown below.</p> <p></p> <p> Figure 10-1   Example data for binary search </p> <p>As shown below, initialize pointers \\(i = 0\\) and \\(j = n - 1\\) to the first and last elements, respectively. They represent the search interval \\([0, n - 1]\\). Square brackets denote a closed interval, which includes its endpoints.</p> <p>Next, repeat the following two steps.</p> <ol> <li>Calculate the midpoint index \\(m = \\lfloor {(i + j) / 2} \\rfloor\\), where \\(\\lfloor \\: \\rfloor\\) denotes rounding down.</li> <li>Compare <code>nums[m]</code> with <code>target</code>. There are three cases.<ol> <li>If <code>nums[m] &lt; target</code>, then <code>target</code> lies in \\([m + 1, j]\\), so set \\(i = m + 1\\).</li> <li>If <code>nums[m] &gt; target</code>, then <code>target</code> lies in \\([i, m - 1]\\), so set \\(j = m - 1\\).</li> <li>If <code>nums[m] = target</code>, we have found <code>target</code>, so return index \\(m\\).</li> </ol> </li> </ol> <p>If the array does not contain the target, the search interval eventually becomes empty. Return \\(-1\\) in this case.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 10-2   The binary search process </p> <p>Because \\(i\\) and \\(j\\) are both of type <code>int</code>, \\(i + j\\) may exceed the range of <code>int</code>. To avoid integer overflow, we usually calculate the midpoint using \\(m = \\lfloor {i + (j - i) / 2} \\rfloor\\).</p> <p>The implementation is shown below:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search.py<pre><code>def binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"Binary search (closed interval)\"\"\"\n    # Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements\n    i, j = 0, len(nums) - 1\n    # Loop until the search interval is empty (when i &gt; j)\n    while i &lt;= j:\n        # Python integers can grow arbitrarily large, subject to available memory, so integer overflow is not a concern\n        m = (i + j) // 2  # Calculate the midpoint index m\n        if nums[m] &lt; target:\n            i = m + 1  # In this case, target must be in [m+1, j] if it exists\n        elif nums[m] &gt; target:\n            j = m - 1  # In this case, target must be in [i, m-1] if it exists\n        else:\n            return m  # The target element was found; return its index\n    return -1  # The target element was not found; return -1\n</code></pre> binary_search.cpp<pre><code>/* Binary search (closed interval) */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    // Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements\n    int i = 0, j = nums.size() - 1;\n    // Loop until the search interval is empty (when i &gt; j)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target)    // In this case, target must be in [m+1, j] if it exists\n            i = m + 1;\n        else if (nums[m] &gt; target) // In this case, target must be in [i, m-1] if it exists\n            j = m - 1;\n        else // The target element was found; return its index\n            return m;\n    }\n    // The target element was not found; return -1\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* Binary search (closed interval) */\nint binarySearch(int[] nums, int target) {\n    // Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements\n    int i = 0, j = nums.length - 1;\n    // Loop until the search interval is empty (when i &gt; j)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) // In this case, target must be in [m+1, j] if it exists\n            i = m + 1;\n        else if (nums[m] &gt; target) // In this case, target must be in [i, m-1] if it exists\n            j = m - 1;\n        else // The target element was found; return its index\n            return m;\n    }\n    // The target element was not found; return -1\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* Binary search (closed interval) */\nint BinarySearch(int[] nums, int target) {\n    // Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements\n    int i = 0, j = nums.Length - 1;\n    // Loop until the search interval is empty (when i &gt; j)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2;   // Calculate the midpoint index m\n        if (nums[m] &lt; target)      // In this case, target must be in [m+1, j] if it exists\n            i = m + 1;\n        else if (nums[m] &gt; target) // In this case, target must be in [i, m-1] if it exists\n            j = m - 1;\n        else                       // The target element was found; return its index\n            return m;\n    }\n    // The target element was not found; return -1\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* Binary search (closed interval) */\nfunc binarySearch(nums []int, target int) int {\n    // Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements\n    i, j := 0, len(nums)-1\n    // Loop until the search interval is empty (when i &gt; j)\n    for i &lt;= j {\n        m := i + (j-i)/2      // Calculate the midpoint index m\n        if nums[m] &lt; target { // In this case, target must be in [m+1, j] if it exists\n            i = m + 1\n        } else if nums[m] &gt; target { // In this case, target must be in [i, m-1] if it exists\n            j = m - 1\n        } else { // The target element was found; return its index\n            return m\n        }\n    }\n    // The target element was not found; return -1\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* Binary search (closed interval) */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    // Loop until the search interval is empty (when i &gt; j)\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // Calculate the midpoint index m\n        if nums[m] &lt; target { // In this case, target must be in [m+1, j] if it exists\n            i = m + 1\n        } else if nums[m] &gt; target { // In this case, target must be in [i, m-1] if it exists\n            j = m - 1\n        } else { // The target element was found; return its index\n            return m\n        }\n    }\n    // The target element was not found; return -1\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* Binary search (closed interval) */\nfunction binarySearch(nums, target) {\n    // Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements\n    let i = 0,\n        j = nums.length - 1;\n    // Loop until the search interval is empty (when i &gt; j)\n    while (i &lt;= j) {\n        // Calculate midpoint index m, using parseInt() to round down\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // In this case, target must be in [m+1, j] if it exists\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // In this case, target must be in [i, m-1] if it exists\n            j = m - 1;\n        else return m; // The target element was found; return its index\n    }\n    // The target element was not found; return -1\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* Binary search (closed interval) */\nfunction binarySearch(nums: number[], target: number): number {\n    // Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements\n    let i = 0,\n        j = nums.length - 1;\n    // Loop until the search interval is empty (when i &gt; j)\n    while (i &lt;= j) {\n        // Calculate the midpoint index m\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // In this case, target must be in [m+1, j] if it exists\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // In this case, target must be in [i, m-1] if it exists\n            j = m - 1;\n        } else {\n            // The target element was found; return its index\n            return m;\n        }\n    }\n    return -1; // The target element was not found; return -1\n}\n</code></pre> binary_search.dart<pre><code>/* Binary search (closed interval) */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  // Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements\n  int i = 0, j = nums.length - 1;\n  // Loop until the search interval is empty (when i &gt; j)\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // Calculate the midpoint index m\n    if (nums[m] &lt; target) {\n      // In this case, target must be in [m+1, j] if it exists\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // In this case, target must be in [i, m-1] if it exists\n      j = m - 1;\n    } else {\n      // The target element was found; return its index\n      return m;\n    }\n  }\n  // The target element was not found; return -1\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* Binary search (closed interval) */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements\n    let mut i = 0;\n    let mut j = nums.len() as i32 - 1;\n    // Loop until the search interval is empty (when i &gt; j)\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // Calculate the midpoint index m\n        if nums[m as usize] &lt; target {\n            // In this case, target must be in [m+1, j] if it exists\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // In this case, target must be in [i, m-1] if it exists\n            j = m - 1;\n        } else {\n            // The target element was found; return its index\n            return m;\n        }\n    }\n    // The target element was not found; return -1\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* Binary search (closed interval) */\nint binarySearch(int *nums, int len, int target) {\n    // Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements\n    int i = 0, j = len - 1;\n    // Loop until the search interval is empty (when i &gt; j)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target)    // In this case, target must be in [m+1, j] if it exists\n            i = m + 1;\n        else if (nums[m] &gt; target) // In this case, target must be in [i, m-1] if it exists\n            j = m - 1;\n        else // The target element was found; return its index\n            return m;\n    }\n    // The target element was not found; return -1\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* Binary search (closed interval) */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    // Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements\n    var i = 0\n    var j = nums.size - 1\n    // Loop until the search interval is empty (when i &gt; j)\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // Calculate the midpoint index m\n        if (nums[m] &lt; target) // In this case, target must be in [m+1, j] if it exists\n            i = m + 1\n        else if (nums[m] &gt; target) // In this case, target must be in [i, m-1] if it exists\n            j = m - 1\n        else  // The target element was found; return its index\n            return m\n    }\n    // The target element was not found; return -1\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>### Binary search (closed interval) ###\ndef binary_search(nums, target)\n  # Initialize the closed interval [0, n-1], with i and j pointing to the first and last elements\n  i, j = 0, nums.length - 1\n\n  # Loop until the search interval is empty (when i &gt; j)\n  while i &lt;= j\n    # Ruby integers can grow arbitrarily large, limited by available memory, so integer overflow is not a concern here\n    m = (i + j) / 2   # Calculate the midpoint index m\n\n    if nums[m] &lt; target\n      i = m + 1 # In this case, target must be in [m+1, j] if it exists\n    elsif nums[m] &gt; target\n      j = m - 1 # In this case, target must be in [i, m-1] if it exists\n    else\n      return m  # The target element was found; return its index\n    end\n  end\n\n  -1  # The target element was not found; return -1\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Time complexity is \\(O(\\log n)\\): the search interval is halved each round, so the loop runs \\(\\log_2 n\\) times.</p> <p>Space complexity is \\(O(1)\\): pointers \\(i\\) and \\(j\\) use constant space.</p>","path":["Chapter 10.   Searching","10.1   Binary search"],"tags":[]},{"location":"chapter_searching/binary_search/#1011-ways-to-represent-the-interval","level":2,"title":"10.1.1   Ways to represent the interval","text":"<p>Besides the closed interval above, another common representation is a “left-closed, right-open” interval, written \\([0, n)\\). It includes the left endpoint but excludes the right endpoint. With this representation, \\([i, j)\\) is empty when \\(i = j\\).</p> <p>We can implement an equivalent binary search algorithm using this representation:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search.py<pre><code>def binary_search_lcro(nums: list[int], target: int) -&gt; int:\n    \"\"\"Binary search (left-closed, right-open interval)\"\"\"\n    # Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last\n    i, j = 0, len(nums)\n    # Loop until the search interval is empty (when i = j)\n    while i &lt; j:\n        m = (i + j) // 2  # Calculate the midpoint index m\n        if nums[m] &lt; target:\n            i = m + 1  # In this case, target must be in [m+1, j) if it exists\n        elif nums[m] &gt; target:\n            j = m  # In this case, target must be in [i, m) if it exists\n        else:\n            return m  # The target element was found; return its index\n    return -1  # The target element was not found; return -1\n</code></pre> binary_search.cpp<pre><code>/* Binary search (left-closed, right-open interval) */\nint binarySearchLCRO(vector&lt;int&gt; &amp;nums, int target) {\n    // Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last\n    int i = 0, j = nums.size();\n    // Loop until the search interval is empty (when i = j)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target)    // In this case, target must be in [m+1, j) if it exists\n            i = m + 1;\n        else if (nums[m] &gt; target) // In this case, target must be in [i, m) if it exists\n            j = m;\n        else // The target element was found; return its index\n            return m;\n    }\n    // The target element was not found; return -1\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* Binary search (left-closed, right-open interval) */\nint binarySearchLCRO(int[] nums, int target) {\n    // Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last\n    int i = 0, j = nums.length;\n    // Loop until the search interval is empty (when i = j)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) // In this case, target must be in [m+1, j) if it exists\n            i = m + 1;\n        else if (nums[m] &gt; target) // In this case, target must be in [i, m) if it exists\n            j = m;\n        else // The target element was found; return its index\n            return m;\n    }\n    // The target element was not found; return -1\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* Binary search (left-closed, right-open interval) */\nint BinarySearchLCRO(int[] nums, int target) {\n    // Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last\n    int i = 0, j = nums.Length;\n    // Loop until the search interval is empty (when i = j)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2;   // Calculate the midpoint index m\n        if (nums[m] &lt; target)      // In this case, target must be in [m+1, j) if it exists\n            i = m + 1;\n        else if (nums[m] &gt; target) // In this case, target must be in [i, m) if it exists\n            j = m;\n        else                       // The target element was found; return its index\n            return m;\n    }\n    // The target element was not found; return -1\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* Binary search (left-closed, right-open interval) */\nfunc binarySearchLCRO(nums []int, target int) int {\n    // Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last\n    i, j := 0, len(nums)\n    // Loop until the search interval is empty (when i = j)\n    for i &lt; j {\n        m := i + (j-i)/2      // Calculate the midpoint index m\n        if nums[m] &lt; target { // In this case, target must be in [m+1, j) if it exists\n            i = m + 1\n        } else if nums[m] &gt; target { // In this case, target must be in [i, m) if it exists\n            j = m\n        } else { // The target element was found; return its index\n            return m\n        }\n    }\n    // The target element was not found; return -1\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* Binary search (left-closed, right-open interval) */\nfunc binarySearchLCRO(nums: [Int], target: Int) -&gt; Int {\n    // Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last\n    var i = nums.startIndex\n    var j = nums.endIndex\n    // Loop until the search interval is empty (when i = j)\n    while i &lt; j {\n        let m = i + (j - i) / 2 // Calculate the midpoint index m\n        if nums[m] &lt; target { // In this case, target must be in [m+1, j) if it exists\n            i = m + 1\n        } else if nums[m] &gt; target { // In this case, target must be in [i, m) if it exists\n            j = m\n        } else { // The target element was found; return its index\n            return m\n        }\n    }\n    // The target element was not found; return -1\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* Binary search (left-closed, right-open interval) */\nfunction binarySearchLCRO(nums, target) {\n    // Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last\n    let i = 0,\n        j = nums.length;\n    // Loop until the search interval is empty (when i = j)\n    while (i &lt; j) {\n        // Calculate midpoint index m, using parseInt() to round down\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // In this case, target must be in [m+1, j) if it exists\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // In this case, target must be in [i, m) if it exists\n            j = m;\n        // The target element was found; return its index\n        else return m;\n    }\n    // The target element was not found; return -1\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* Binary search (left-closed, right-open interval) */\nfunction binarySearchLCRO(nums: number[], target: number): number {\n    // Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last\n    let i = 0,\n        j = nums.length;\n    // Loop until the search interval is empty (when i = j)\n    while (i &lt; j) {\n        // Calculate the midpoint index m\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // In this case, target must be in [m+1, j) if it exists\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // In this case, target must be in [i, m) if it exists\n            j = m;\n        } else {\n            // The target element was found; return its index\n            return m;\n        }\n    }\n    return -1; // The target element was not found; return -1\n}\n</code></pre> binary_search.dart<pre><code>/* Binary search (left-closed, right-open interval) */\nint binarySearchLCRO(List&lt;int&gt; nums, int target) {\n  // Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last\n  int i = 0, j = nums.length;\n  // Loop until the search interval is empty (when i = j)\n  while (i &lt; j) {\n    int m = i + (j - i) ~/ 2; // Calculate the midpoint index m\n    if (nums[m] &lt; target) {\n      // In this case, target must be in [m+1, j) if it exists\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // In this case, target must be in [i, m) if it exists\n      j = m;\n    } else {\n      // The target element was found; return its index\n      return m;\n    }\n  }\n  // The target element was not found; return -1\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* Binary search (left-closed, right-open interval) */\nfn binary_search_lcro(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last\n    let mut i = 0;\n    let mut j = nums.len() as i32;\n    // Loop until the search interval is empty (when i = j)\n    while i &lt; j {\n        let m = i + (j - i) / 2; // Calculate the midpoint index m\n        if nums[m as usize] &lt; target {\n            // In this case, target must be in [m+1, j) if it exists\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // In this case, target must be in [i, m) if it exists\n            j = m;\n        } else {\n            // The target element was found; return its index\n            return m;\n        }\n    }\n    // The target element was not found; return -1\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* Binary search (left-closed, right-open interval) */\nint binarySearchLCRO(int *nums, int len, int target) {\n    // Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last\n    int i = 0, j = len;\n    // Loop until the search interval is empty (when i = j)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target)    // In this case, target must be in [m+1, j) if it exists\n            i = m + 1;\n        else if (nums[m] &gt; target) // In this case, target must be in [i, m) if it exists\n            j = m;\n        else // The target element was found; return its index\n            return m;\n    }\n    // The target element was not found; return -1\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* Binary search (left-closed, right-open interval) */\nfun binarySearchLCRO(nums: IntArray, target: Int): Int {\n    // Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last\n    var i = 0\n    var j = nums.size\n    // Loop until the search interval is empty (when i = j)\n    while (i &lt; j) {\n        val m = i + (j - i) / 2 // Calculate the midpoint index m\n        if (nums[m] &lt; target) // In this case, target must be in [m+1, j) if it exists\n            i = m + 1\n        else if (nums[m] &gt; target) // In this case, target must be in [i, m) if it exists\n            j = m\n        else  // The target element was found; return its index\n            return m\n    }\n    // The target element was not found; return -1\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>### Binary search (left-closed, right-open interval) ###\ndef binary_search_lcro(nums, target)\n  # Initialize the left-closed, right-open interval [0, n), with i at the first element and j one past the last\n  i, j = 0, nums.length\n\n  # Loop until the search interval is empty (when i = j)\n  while i &lt; j\n    # Calculate the midpoint index m\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # In this case, target must be in [m+1, j) if it exists\n    elsif nums[m] &gt; target\n      j = m - 1 # In this case, target must be in [i, m) if it exists\n    else\n      return m  # The target element was found; return its index\n    end\n  end\n\n  -1  # The target element was not found; return -1\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>As shown below, the two interval representations require different initialization, loop conditions, and operations for narrowing the interval.</p> <p>Since a closed interval includes both endpoints, the operations that narrow the interval through pointers \\(i\\) and \\(j\\) are symmetric. This makes mistakes less likely, so we generally recommend the closed-interval version.</p> <p></p> <p> Figure 10-3   Two ways to define the interval </p>","path":["Chapter 10.   Searching","10.1   Binary search"],"tags":[]},{"location":"chapter_searching/binary_search/#1012-strengths-and-limitations","level":2,"title":"10.1.2   Strengths and limitations","text":"<p>Binary search performs well in both time and space.</p> <ul> <li>Binary search is time-efficient. Its logarithmic time complexity offers a significant advantage for large datasets. For example, when \\(n = 2^{20}\\), linear search requires \\(2^{20} = 1048576\\) iterations, whereas binary search needs only \\(\\log_2 2^{20} = 20\\).</li> <li>Binary search requires no extra storage. It uses less space than search algorithms that rely on additional storage, such as hash lookup.</li> </ul> <p>However, binary search is not suitable in every situation, mainly for the following reasons.</p> <ul> <li>Binary search requires sorted data. Sorting unsorted input solely to perform a binary search is not worthwhile, because sorting usually takes \\(O(n \\log n)\\) time, more than either linear or binary search. When insertions are frequent, preserving the array's order requires inserting each element at a specific position, an expensive \\(O(n)\\) operation.</li> <li>Binary search is suitable only for arrays. It needs to jump between nonadjacent elements, which is inefficient in a linked list. It is therefore unsuitable for linked lists and data structures based on them.</li> <li>Linear search performs better on small datasets. Each iteration of linear search needs only one comparison. Binary search needs one addition, one division, 1–3 comparisons, and one more addition or subtraction, totaling 4–6 elementary operations. When \\(n\\) is small, linear search can therefore be faster.</li> </ul>","path":["Chapter 10.   Searching","10.1   Binary search"],"tags":[]},{"location":"chapter_searching/binary_search_edge/","level":1,"title":"10.3   Finding boundaries with binary search","text":"","path":["Chapter 10.   Searching","10.3   Finding boundaries with binary search"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#1031-finding-the-left-boundary","level":2,"title":"10.3.1   Finding the left boundary","text":"<p>Question</p> <p>Given a sorted array <code>nums</code> of length \\(n\\) that may contain duplicate elements, return the index of the leftmost occurrence of <code>target</code>. If the array does not contain this element, return \\(-1\\).</p> <p>Recall the method for finding an insertion point with binary search: when the search finishes, \\(i\\) points to the leftmost <code>target</code>. Finding the insertion point therefore amounts to finding the index of the leftmost <code>target</code>.</p> <p>We can use the insertion-point function to find the left boundary. However, <code>target</code> may be absent from the array, leading to either of the following outcomes.</p> <ul> <li>The insertion index \\(i\\) is out of bounds.</li> <li><code>nums[i]</code> is not equal to <code>target</code>.</li> </ul> <p>In either case, simply return \\(-1\\). The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_edge.py<pre><code>def binary_search_left_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"Find the leftmost occurrence of target using binary search\"\"\"\n    # This is equivalent to finding the insertion point of target\n    i = binary_search_insertion(nums, target)\n    # If target was not found, return -1\n    if i == len(nums) or nums[i] != target:\n        return -1\n    # Target was found; return index i\n    return i\n</code></pre> binary_search_edge.cpp<pre><code>/* Find the leftmost occurrence of target using binary search */\nint binarySearchLeftEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // This is equivalent to finding the insertion point of target\n    int i = binarySearchInsertion(nums, target);\n    // If target was not found, return -1\n    if (i == nums.size() || nums[i] != target) {\n        return -1;\n    }\n    // Target was found; return index i\n    return i;\n}\n</code></pre> binary_search_edge.java<pre><code>/* Find the leftmost occurrence of target using binary search */\nint binarySearchLeftEdge(int[] nums, int target) {\n    // This is equivalent to finding the insertion point of target\n    int i = binary_search_insertion.binarySearchInsertion(nums, target);\n    // If target was not found, return -1\n    if (i == nums.length || nums[i] != target) {\n        return -1;\n    }\n    // Target was found; return index i\n    return i;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* Find the leftmost occurrence of target using binary search */\nint BinarySearchLeftEdge(int[] nums, int target) {\n    // This is equivalent to finding the insertion point of target\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target);\n    // If target was not found, return -1\n    if (i == nums.Length || nums[i] != target) {\n        return -1;\n    }\n    // Target was found; return index i\n    return i;\n}\n</code></pre> binary_search_edge.go<pre><code>/* Find the leftmost occurrence of target using binary search */\nfunc binarySearchLeftEdge(nums []int, target int) int {\n    // This is equivalent to finding the insertion point of target\n    i := binarySearchInsertion(nums, target)\n    // If target was not found, return -1\n    if i == len(nums) || nums[i] != target {\n        return -1\n    }\n    // Target was found; return index i\n    return i\n}\n</code></pre> binary_search_edge.swift<pre><code>/* Find the leftmost occurrence of target using binary search */\nfunc binarySearchLeftEdge(nums: [Int], target: Int) -&gt; Int {\n    // This is equivalent to finding the insertion point of target\n    let i = binarySearchInsertion(nums: nums, target: target)\n    // If target was not found, return -1\n    if i == nums.endIndex || nums[i] != target {\n        return -1\n    }\n    // Target was found; return index i\n    return i\n}\n</code></pre> binary_search_edge.js<pre><code>/* Find the leftmost occurrence of target using binary search */\nfunction binarySearchLeftEdge(nums, target) {\n    // This is equivalent to finding the insertion point of target\n    const i = binarySearchInsertion(nums, target);\n    // If target was not found, return -1\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // Target was found; return index i\n    return i;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* Find the leftmost occurrence of target using binary search */\nfunction binarySearchLeftEdge(nums: Array&lt;number&gt;, target: number): number {\n    // This is equivalent to finding the insertion point of target\n    const i = binarySearchInsertion(nums, target);\n    // If target was not found, return -1\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // Target was found; return index i\n    return i;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* Find the leftmost occurrence of target using binary search */\nint binarySearchLeftEdge(List&lt;int&gt; nums, int target) {\n  // This is equivalent to finding the insertion point of target\n  int i = binarySearchInsertion(nums, target);\n  // If target was not found, return -1\n  if (i == nums.length || nums[i] != target) {\n    return -1;\n  }\n  // Target was found; return index i\n  return i;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* Find the leftmost occurrence of target using binary search */\nfn binary_search_left_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // This is equivalent to finding the insertion point of target\n    let i = binary_search_insertion(nums, target);\n    // If target was not found, return -1\n    if i == nums.len() as i32 || nums[i as usize] != target {\n        return -1;\n    }\n    // Target was found; return index i\n    i\n}\n</code></pre> binary_search_edge.c<pre><code>/* Find the leftmost occurrence of target using binary search */\nint binarySearchLeftEdge(int *nums, int numSize, int target) {\n    // This is equivalent to finding the insertion point of target\n    int i = binarySearchInsertion(nums, numSize, target);\n    // If target was not found, return -1\n    if (i == numSize || nums[i] != target) {\n        return -1;\n    }\n    // Target was found; return index i\n    return i;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* Find the leftmost occurrence of target using binary search */\nfun binarySearchLeftEdge(nums: IntArray, target: Int): Int {\n    // This is equivalent to finding the insertion point of target\n    val i = binarySearchInsertion(nums, target)\n    // If target was not found, return -1\n    if (i == nums.size || nums[i] != target) {\n        return -1\n    }\n    // Target was found; return index i\n    return i\n}\n</code></pre> binary_search_edge.rb<pre><code>### Find the leftmost occurrence of target using binary search ###\ndef binary_search_left_edge(nums, target)\n  # This is equivalent to finding the insertion point of target\n  i = binary_search_insertion(nums, target)\n\n  # If target was not found, return -1\n  return -1 if i == nums.length || nums[i] != target\n\n  i # Target was found; return index i\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 10.   Searching","10.3   Finding boundaries with binary search"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#1032-finding-the-right-boundary","level":2,"title":"10.3.2   Finding the right boundary","text":"<p>How can we find the rightmost <code>target</code>? The most direct approach is to modify the pointer update for <code>nums[m] == target</code>. We omit the code here; interested readers can implement it themselves.</p> <p>Here are two shortcuts.</p>","path":["Chapter 10.   Searching","10.3   Finding boundaries with binary search"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#1-reusing-the-left-boundary-search","level":3,"title":"1.   Reusing the left-boundary search","text":"<p>We can use the function for finding the leftmost element to find the rightmost one: turn the search for the rightmost <code>target</code> into a search for the leftmost <code>target + 1</code>.</p> <p>As shown below, when the search finishes, pointer \\(i\\) points to the leftmost <code>target + 1</code>, if it exists, while \\(j\\) points to the rightmost <code>target</code>. We therefore return \\(j\\).</p> <p></p> <p> Figure 10-7   Turning a right-boundary search into a left-boundary search </p> <p>The returned insertion point is \\(i\\), so subtract \\(1\\) to obtain \\(j\\):</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_edge.py<pre><code>def binary_search_right_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"Find the rightmost occurrence of target using binary search\"\"\"\n    # Reduce the problem to finding the leftmost insertion point of target + 1\n    i = binary_search_insertion(nums, target + 1)\n    # j is the candidate rightmost target index; i is the index of the first element greater than target, or len(nums)\n    j = i - 1\n    # If target was not found, return -1\n    if j == -1 or nums[j] != target:\n        return -1\n    # Target was found; return index j\n    return j\n</code></pre> binary_search_edge.cpp<pre><code>/* Find the rightmost occurrence of target using binary search */\nint binarySearchRightEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // Reduce the problem to finding the leftmost insertion point of target + 1\n    int i = binarySearchInsertion(nums, target + 1);\n    // j points to the rightmost target, while i points to the first element greater than target\n    int j = i - 1;\n    // If target was not found, return -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // Target was found; return index j\n    return j;\n}\n</code></pre> binary_search_edge.java<pre><code>/* Find the rightmost occurrence of target using binary search */\nint binarySearchRightEdge(int[] nums, int target) {\n    // Reduce the problem to finding the leftmost insertion point of target + 1\n    int i = binary_search_insertion.binarySearchInsertion(nums, target + 1);\n    // j points to the rightmost target, while i points to the first element greater than target\n    int j = i - 1;\n    // If target was not found, return -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // Target was found; return index j\n    return j;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* Find the rightmost occurrence of target using binary search */\nint BinarySearchRightEdge(int[] nums, int target) {\n    // Reduce the problem to finding the leftmost insertion point of target + 1\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target + 1);\n    // j is the candidate index of the rightmost target; i is the insertion point after all elements &lt;= target\n    int j = i - 1;\n    // If target was not found, return -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // Target was found; return index j\n    return j;\n}\n</code></pre> binary_search_edge.go<pre><code>/* Find the rightmost occurrence of target using binary search */\nfunc binarySearchRightEdge(nums []int, target int) int {\n    // For integer values, find the leftmost insertion point of target + 1 (assuming no overflow)\n    i := binarySearchInsertion(nums, target+1)\n    // i is the index of the first element greater than target (or len(nums)); j is the candidate index of the rightmost target\n    j := i - 1\n    // If target was not found, return -1\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // Target was found; return index j\n    return j\n}\n</code></pre> binary_search_edge.swift<pre><code>/* Find the rightmost occurrence of target using binary search */\nfunc binarySearchRightEdge(nums: [Int], target: Int) -&gt; Int {\n    // Reduce the problem to finding the leftmost insertion point of target + 1\n    let i = binarySearchInsertion(nums: nums, target: target + 1)\n    // i is the first index with a value greater than target (or nums.endIndex); j is the candidate index of the rightmost target\n    let j = i - 1\n    // If target was not found, return -1\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // Target was found; return index j\n    return j\n}\n</code></pre> binary_search_edge.js<pre><code>/* Find the rightmost occurrence of target using binary search */\nfunction binarySearchRightEdge(nums, target) {\n    // Reduce the problem to finding the leftmost insertion point of target + 1\n    const i = binarySearchInsertion(nums, target + 1);\n    // j points to the rightmost target, while i points to the first element greater than target\n    const j = i - 1;\n    // If target was not found, return -1\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // Target was found; return index j\n    return j;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* Find the rightmost occurrence of target using binary search */\nfunction binarySearchRightEdge(nums: Array&lt;number&gt;, target: number): number {\n    // Reduce the problem to finding the leftmost insertion point of target + 1\n    const i = binarySearchInsertion(nums, target + 1);\n    // For integer elements, i is the first index greater than target (or the array length); j is the candidate rightmost target index\n    const j = i - 1;\n    // If target was not found, return -1\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // Target was found; return index j\n    return j;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* Find the rightmost occurrence of target using binary search */\nint binarySearchRightEdge(List&lt;int&gt; nums, int target) {\n  // Reduce the problem to finding the leftmost insertion point of target + 1\n  int i = binarySearchInsertion(nums, target + 1);\n  // i is the first index greater than target (or the array length); j is the candidate rightmost target index\n  int j = i - 1;\n  // If target was not found, return -1\n  if (j == -1 || nums[j] != target) {\n    return -1;\n  }\n  // Target was found; return index j\n  return j;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* Find the rightmost occurrence of target using binary search */\nfn binary_search_right_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // Reduce the problem to finding the leftmost insertion point of target + 1\n    let i = binary_search_insertion(nums, target + 1);\n    // j points to the rightmost target, while i points to the first element greater than target\n    let j = i - 1;\n    // If target was not found, return -1\n    if j == -1 || nums[j as usize] != target {\n        return -1;\n    }\n    // Target was found; return index j\n    j\n}\n</code></pre> binary_search_edge.c<pre><code>/* Find the rightmost occurrence of target using binary search */\nint binarySearchRightEdge(int *nums, int numSize, int target) {\n    // Reduce the problem to finding the leftmost insertion point of target + 1\n    int i = binarySearchInsertion(nums, numSize, target + 1);\n    // j points to the rightmost target, while i points to the first element greater than target\n    int j = i - 1;\n    // If target was not found, return -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // Target was found; return index j\n    return j;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* Find the rightmost occurrence of target using binary search */\nfun binarySearchRightEdge(nums: IntArray, target: Int): Int {\n    // For integer values, find the leftmost insertion point of target + 1 (assuming no overflow)\n    val i = binarySearchInsertion(nums, target + 1)\n    // i is the index of the first element greater than target (or the array length); j is the candidate rightmost target index\n    val j = i - 1\n    // If target was not found, return -1\n    if (j == -1 || nums[j] != target) {\n        return -1\n    }\n    // Target was found; return index j\n    return j\n}\n</code></pre> binary_search_edge.rb<pre><code>### Find the rightmost occurrence of target using binary search ###\ndef binary_search_right_edge(nums, target)\n  # For integer elements, find the leftmost insertion point of target + 1\n  i = binary_search_insertion(nums, target + 1)\n\n  # i is the index of the first element greater than target (or the array length); j is the candidate rightmost target index\n  j = i - 1\n\n  # If target was not found, return -1\n  return -1 if j == -1 || nums[j] != target\n\n  j # Target was found; return index j\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 10.   Searching","10.3   Finding boundaries with binary search"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#2-turning-a-boundary-search-into-an-element-search","level":3,"title":"2.   Turning a boundary search into an element search","text":"<p>When the array does not contain <code>target</code>, \\(i\\) ultimately points to the first element greater than <code>target</code>, while \\(j\\) points to the rightmost element smaller than <code>target</code>.</p> <p>As shown below, we can therefore construct a value absent from the array and use it to find the left and right boundaries.</p> <ul> <li>To find the leftmost <code>target</code>, search for <code>target - 0.5</code> and return pointer \\(i\\).</li> <li>To find the rightmost <code>target</code>, search for <code>target + 0.5</code> and return pointer \\(j\\).</li> </ul> <p></p> <p> Figure 10-8   Turning a boundary search into an element search </p> <p>The code is omitted here, but keep two points in mind.</p> <ul> <li>The given array contains no fractional values, so we do not need to handle equality.</li> <li>Since this method introduces fractional values, change the function's <code>target</code> variable to a floating-point type. No change is needed in Python.</li> </ul>","path":["Chapter 10.   Searching","10.3   Finding boundaries with binary search"],"tags":[]},{"location":"chapter_searching/binary_search_insertion/","level":1,"title":"10.2   Finding an insertion point with binary search","text":"<p>Binary search can do more than find a target element. It also solves many variations of the problem, such as finding where to insert a target.</p>","path":["Chapter 10.   Searching","10.2   Finding an insertion point with binary search"],"tags":[]},{"location":"chapter_searching/binary_search_insertion/#1021-when-there-are-no-duplicates","level":2,"title":"10.2.1   When there are no duplicates","text":"<p>Question</p> <p>Given a sorted array <code>nums</code> of length \\(n\\) with no duplicate elements and a value <code>target</code>, insert <code>target</code> into <code>nums</code> while preserving its order. If <code>target</code> is already present, insert the new value to its left. Return the index of the newly inserted <code>target</code>, as illustrated below.</p> <p></p> <p> Figure 10-4   Example data for finding an insertion point </p> <p>To reuse the binary search code from the previous section, we need to answer two questions.</p> <p>Question 1: If the array contains <code>target</code>, is the insertion index the index of that element?</p> <p>The problem asks us to insert <code>target</code> to the left of the equal element, so the new <code>target</code> takes the original one's position. Thus, if the array contains <code>target</code>, the insertion index is the index of that <code>target</code>.</p> <p>Question 2: If the array does not contain <code>target</code>, which index is the insertion point?</p> <p>Consider the search process more closely. Pointer \\(i\\) moves when <code>nums[m] &lt; target</code>, so \\(i\\) approaches elements greater than or equal to <code>target</code>. Similarly, pointer \\(j\\) keeps approaching elements less than or equal to <code>target</code>.</p> <p>When the search ends, \\(i\\) points to the first element greater than <code>target</code>, and \\(j\\) points to the rightmost element smaller than <code>target</code>. Thus, when <code>target</code> is absent, the insertion index is \\(i\\). The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_insertion.py<pre><code>def binary_search_insertion_simple(nums: list[int], target: int) -&gt; int:\n    \"\"\"Find an insertion point using binary search (no duplicates)\"\"\"\n    i, j = 0, len(nums) - 1  # Initialize the closed interval [0, n-1]\n    while i &lt;= j:\n        m = (i + j) // 2  # Calculate the midpoint index m\n        if nums[m] &lt; target:\n            i = m + 1  # Target must be in [m+1, j] if it exists\n        elif nums[m] &gt; target:\n            j = m - 1  # Target must be in [i, m-1] if it exists\n        else:\n            return m  # Target was found; return insertion point m\n    # Target was not found; return insertion point i\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* Find an insertion point using binary search (no duplicates) */\nint binarySearchInsertionSimple(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // Initialize the closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // Target must be in [m+1, j] if it exists\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // Target must be in [i, m-1] if it exists\n        } else {\n            return m; // Target was found; return insertion point m\n        }\n    }\n    // Target was not found; return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* Find an insertion point using binary search (no duplicates) */\nint binarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // Initialize the closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // Target must be in [m+1, j] if it exists\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // Target must be in [i, m-1] if it exists\n        } else {\n            return m; // Target was found; return insertion point m\n        }\n    }\n    // Target was not found; return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* Find an insertion point using binary search (no duplicates) */\nint BinarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // Initialize the closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // Target must be in [m+1, j] if it exists\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // Target must be in [i, m-1] if it exists\n        } else {\n            return m; // Target was found; return insertion point m\n        }\n    }\n    // Target was not found; return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* Find an insertion point using binary search (no duplicates) */\nfunc binarySearchInsertionSimple(nums []int, target int) int {\n    // Initialize the closed interval [0, n-1]\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // Calculate the midpoint index m\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // Target must be in [m+1, j] if it exists\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // Target must be in [i, m-1] if it exists\n            j = m - 1\n        } else {\n            // Target was found; return insertion point m\n            return m\n        }\n    }\n    // Target was not found; return insertion point i\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* Find an insertion point using binary search (no duplicates) */\nfunc binarySearchInsertionSimple(nums: [Int], target: Int) -&gt; Int {\n    // Initialize the closed interval [0, n-1]\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // Calculate the midpoint index m\n        if nums[m] &lt; target {\n            i = m + 1 // Target must be in [m+1, j] if it exists\n        } else if nums[m] &gt; target {\n            j = m - 1 // Target must be in [i, m-1] if it exists\n        } else {\n            return m // Target was found; return insertion point m\n        }\n    }\n    // Target was not found; return insertion point i\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* Find an insertion point using binary search (no duplicates) */\nfunction binarySearchInsertionSimple(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // Initialize the closed interval [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // Calculate midpoint index m, using Math.floor() to round down\n        if (nums[m] &lt; target) {\n            i = m + 1; // Target must be in [m+1, j] if it exists\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // Target must be in [i, m-1] if it exists\n        } else {\n            return m; // Target was found; return insertion point m\n        }\n    }\n    // Target was not found; return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* Find an insertion point using binary search (no duplicates) */\nfunction binarySearchInsertionSimple(\n    nums: Array&lt;number&gt;,\n    target: number\n): number {\n    let i = 0,\n        j = nums.length - 1; // Initialize the closed interval [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // Calculate midpoint index m, using Math.floor() to round down\n        if (nums[m] &lt; target) {\n            i = m + 1; // Target must be in [m+1, j] if it exists\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // Target must be in [i, m-1] if it exists\n        } else {\n            return m; // Target was found; return insertion point m\n        }\n    }\n    // Target was not found; return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* Find an insertion point using binary search (no duplicates) */\nint binarySearchInsertionSimple(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // Initialize the closed interval [0, n-1]\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // Calculate the midpoint index m\n    if (nums[m] &lt; target) {\n      i = m + 1; // Target must be in [m+1, j] if it exists\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // Target must be in [i, m-1] if it exists\n    } else {\n      return m; // Target was found; return insertion point m\n    }\n  }\n  // Target was not found; return insertion point i\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* Find an insertion point using binary search (no duplicates) */\nfn binary_search_insertion_simple(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // Initialize the closed interval [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // Calculate the midpoint index m\n        if nums[m as usize] &lt; target {\n            i = m + 1; // Target must be in [m+1, j] if it exists\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // Target must be in [i, m-1] if it exists\n        } else {\n            return m;\n        }\n    }\n    // Target was not found; return insertion point i\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* Find an insertion point using binary search (no duplicates) */\nint binarySearchInsertionSimple(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // Initialize the closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // Target must be in [m+1, j] if it exists\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // Target must be in [i, m-1] if it exists\n        } else {\n            return m; // Target was found; return insertion point m\n        }\n    }\n    // Target was not found; return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* Find an insertion point using binary search (no duplicates) */\nfun binarySearchInsertionSimple(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // Initialize the closed interval [0, n-1]\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1 // Target must be in [m+1, j] if it exists\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // Target must be in [i, m-1] if it exists\n        } else {\n            return m // Target was found; return insertion point m\n        }\n    }\n    // Target was not found; return insertion point i\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>### Find an insertion point using binary search (no duplicates) ###\ndef binary_search_insertion_simple(nums, target)\n  # Initialize the closed interval [0, n-1]\n  i, j = 0, nums.length - 1\n\n  while i &lt;= j\n    # Calculate the midpoint index m\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # Target must be in [m+1, j] if it exists\n    elsif nums[m] &gt; target\n      j = m - 1 # Target must be in [i, m-1] if it exists\n    else\n      return m  # Target was found; return insertion point m\n    end\n  end\n\n  i # Target was not found; return insertion point i\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 10.   Searching","10.2   Finding an insertion point with binary search"],"tags":[]},{"location":"chapter_searching/binary_search_insertion/#1022-when-duplicates-are-present","level":2,"title":"10.2.2   When duplicates are present","text":"<p>Question</p> <p>Modify the previous problem to allow duplicate elements in the array. All other requirements remain the same.</p> <p>If the array contains multiple occurrences of <code>target</code>, ordinary binary search can return the index of one occurrence, but cannot determine how many more occurrences lie to its left or right.</p> <p>The target must be inserted as far left as possible, so we need the index of the leftmost <code>target</code>. One initial approach follows the steps below.</p> <ol> <li>Run binary search to find the index of any <code>target</code>, denoted by \\(k\\).</li> <li>Starting at index \\(k\\), scan leftward and return when the leftmost <code>target</code> is reached.</li> </ol> <p></p> <p> Figure 10-5   Finding the insertion point among duplicates with a linear scan </p> <p>Although this works, it includes a linear search and therefore takes \\(O(n)\\) time. It is inefficient when the array contains many occurrences of <code>target</code>.</p> <p>Now consider extending binary search itself. As shown below, the overall process remains the same: each round calculates midpoint index \\(m\\) and compares <code>target</code> with <code>nums[m]</code>. The cases are as follows.</p> <ul> <li>If <code>nums[m] &lt; target</code> or <code>nums[m] &gt; target</code>, we have not found <code>target</code>. Narrow the interval as in ordinary binary search, bringing pointers \\(i\\) and \\(j\\) closer to <code>target</code>.</li> <li>If <code>nums[m] == target</code>, elements smaller than <code>target</code> lie in \\([i, m - 1]\\). Narrow the interval by setting \\(j = m - 1\\), bringing pointer \\(j\\) closer to elements smaller than <code>target</code>.</li> </ul> <p>When the loop finishes, \\(i\\) points to the leftmost <code>target</code>, and \\(j\\) points to the rightmost element smaller than <code>target</code>. Index \\(i\\) is therefore the insertion point.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 10-6   Steps for finding an insertion point among duplicates </p> <p>In the code below, the branches for <code>nums[m] &gt; target</code> and <code>nums[m] == target</code> perform the same operation, so they can be combined.</p> <p>Even so, we can leave them separate to make the logic clearer and easier to read.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_insertion.py<pre><code>def binary_search_insertion(nums: list[int], target: int) -&gt; int:\n    \"\"\"Find an insertion point using binary search (with duplicates)\"\"\"\n    i, j = 0, len(nums) - 1  # Initialize the closed interval [0, n-1]\n    while i &lt;= j:\n        m = (i + j) // 2  # Calculate the midpoint index m\n        if nums[m] &lt; target:\n            i = m + 1  # Target must be in [m+1, j] if it exists\n        elif nums[m] &gt; target:\n            j = m - 1  # Target must be in [i, m-1] if it exists\n        else:\n            j = m - 1  # Continue in [i, m-1] to find the rightmost element smaller than target\n    # Return insertion point i\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* Find an insertion point using binary search (with duplicates) */\nint binarySearchInsertion(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // Initialize the closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // Target must be in [m+1, j] if it exists\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // Target must be in [i, m-1] if it exists\n        } else {\n            j = m - 1; // Continue searching [i, m-1] for the insertion point before any equal elements\n        }\n    }\n    // Return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* Find an insertion point using binary search (with duplicates) */\nint binarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // Initialize the closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // Target must be in [m+1, j] if it exists\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // Target must be in [i, m-1] if it exists\n        } else {\n            j = m - 1; // Continue in [i, m-1] to find the rightmost element smaller than target\n        }\n    }\n    // Return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* Find an insertion point using binary search (with duplicates) */\nint BinarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // Initialize the closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // Target must be in [m+1, j] if it exists\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // Target must be in [i, m-1] if it exists\n        } else {\n            j = m - 1; // Continue in [i, m-1] to find the rightmost element smaller than target\n        }\n    }\n    // Return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* Find an insertion point using binary search (with duplicates) */\nfunc binarySearchInsertion(nums []int, target int) int {\n    // Initialize the closed interval [0, n-1]\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // Calculate the midpoint index m\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // Target must be in [m+1, j] if it exists\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // Target must be in [i, m-1] if it exists\n            j = m - 1\n        } else {\n            // Continue in [i, m-1] to find the rightmost element smaller than target\n            j = m - 1\n        }\n    }\n    // Return insertion point i\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* Find an insertion point using binary search (with duplicates) */\nfunc binarySearchInsertion(nums: [Int], target: Int) -&gt; Int {\n    // Initialize the closed interval [0, n-1]\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // Calculate the midpoint index m\n        if nums[m] &lt; target {\n            i = m + 1 // Target must be in [m+1, j] if it exists\n        } else if nums[m] &gt; target {\n            j = m - 1 // Target must be in [i, m-1] if it exists\n        } else {\n            j = m - 1 // Continue in [i, m-1] to find the rightmost element smaller than target\n        }\n    }\n    // Return insertion point i\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* Find an insertion point using binary search (with duplicates) */\nfunction binarySearchInsertion(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // Initialize the closed interval [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // Calculate midpoint index m, using Math.floor() to round down\n        if (nums[m] &lt; target) {\n            i = m + 1; // Target must be in [m+1, j] if it exists\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // Target must be in [i, m-1] if it exists\n        } else {\n            j = m - 1; // Continue in [i, m-1] to find the rightmost element smaller than target\n        }\n    }\n    // Return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* Find an insertion point using binary search (with duplicates) */\nfunction binarySearchInsertion(nums: Array&lt;number&gt;, target: number): number {\n    let i = 0,\n        j = nums.length - 1; // Initialize the closed interval [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // Calculate midpoint index m, using Math.floor() to round down\n        if (nums[m] &lt; target) {\n            i = m + 1; // Target must be in [m+1, j] if it exists\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // Target must be in [i, m-1] if it exists\n        } else {\n            j = m - 1; // Continue in [i, m-1] to find the rightmost element smaller than target\n        }\n    }\n    // Return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* Find an insertion point using binary search (with duplicates) */\nint binarySearchInsertion(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // Initialize the closed interval [0, n-1]\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // Calculate the midpoint index m\n    if (nums[m] &lt; target) {\n      i = m + 1; // Target must be in [m+1, j] if it exists\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // Target must be in [i, m-1] if it exists\n    } else {\n      j = m - 1; // Continue in [i, m-1] to find the rightmost element smaller than target\n    }\n  }\n  // Return insertion point i\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* Find an insertion point using binary search (with duplicates) */\npub fn binary_search_insertion(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // Initialize the closed interval [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // Calculate the midpoint index m\n        if nums[m as usize] &lt; target {\n            i = m + 1; // Target must be in [m+1, j] if it exists\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // Target must be in [i, m-1] if it exists\n        } else {\n            j = m - 1; // Continue searching [i, m-1] for the insertion point before any equal elements\n        }\n    }\n    // Return insertion point i\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* Find an insertion point using binary search (with duplicates) */\nint binarySearchInsertion(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // Initialize the closed interval [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1; // Target must be in [m+1, j] if it exists\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // Target must be in [i, m-1] if it exists\n        } else {\n            j = m - 1; // Continue in [i, m-1] to find the rightmost element smaller than target\n        }\n    }\n    // Return insertion point i\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* Find an insertion point using binary search (with duplicates) */\nfun binarySearchInsertion(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // Initialize the closed interval [0, n-1]\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // Calculate the midpoint index m\n        if (nums[m] &lt; target) {\n            i = m + 1 // Target must be in [m+1, j] if it exists\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // Target must be in [i, m-1] if it exists\n        } else {\n            j = m - 1 // Continue in [i, m-1] to find the rightmost element smaller than target\n        }\n    }\n    // Return insertion point i\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>### Find an insertion point using binary search (with duplicates) ###\ndef binary_search_insertion(nums, target)\n  # Initialize the closed interval [0, n-1]\n  i, j = 0, nums.length - 1\n\n  while i &lt;= j\n    # Calculate the midpoint index m\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # Target must be in [m+1, j] if it exists\n    elsif nums[m] &gt; target\n      j = m - 1 # Target must be in [i, m-1] if it exists\n    else\n      j = m - 1 # Continue in [i, m-1] to find the rightmost element smaller than target\n    end\n  end\n\n  i # Return insertion point i\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Tip</p> <p>All the code in this section uses closed intervals. Interested readers can implement versions using left-closed, right-open intervals.</p> <p>Viewed more broadly, binary search simply assigns a search goal to each of pointers \\(i\\) and \\(j\\). The goal may be a particular element, such as <code>target</code>, or a range of elements, such as those smaller than <code>target</code>.</p> <p>As the interval is repeatedly halved, pointers \\(i\\) and \\(j\\) approach their assigned goals. Eventually they either find the answer or stop after crossing a boundary.</p>","path":["Chapter 10.   Searching","10.2   Finding an insertion point with binary search"],"tags":[]},{"location":"chapter_searching/exercises/","level":1,"title":"10.7   Exercises","text":"","path":["Chapter 10.   Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/exercises/#1071-check-your-understanding","level":2,"title":"10.7.1   Check your understanding","text":"","path":["Chapter 10.   Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/exercises/#1-how-binary-search-narrows-the-search-interval","level":3,"title":"1.   How binary search narrows the search interval","text":"<p>Search for 16 in the sorted array <code>[2, 5, 8, 12, 16, 23, 38]</code>. Use a closed interval <code>[i, j]</code> and the midpoint \\(m=i+(j-i)/2\\), rounded down.</p> <p>For each iteration, write <code>(i, j, m)</code>, the midpoint element, and how the interval is narrowed next, until the target is found.</p> Reference answer <p>The search proceeds as follows:</p> Iteration <code>(i, j, m)</code> Midpoint element Next step 1 <code>(0, 6, 3)</code> 12 <code>12 &lt; 16</code>, so set <code>i = 4</code> 2 <code>(4, 6, 5)</code> 23 <code>23 &gt; 16</code>, so set <code>j = 4</code> 3 <code>(4, 4, 4)</code> 16 Target found; return index 4 <p>Since the array is sorted, when the midpoint value is below the target, we can exclude the midpoint and everything to its left. When it is above the target, we can exclude the midpoint and everything to its right.</p>","path":["Chapter 10.   Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/exercises/#2-left-and-right-boundaries-of-repeated-elements","level":3,"title":"2.   Left and right boundaries of repeated elements","text":"<p>Search for 2 in the array <code>[1, 2, 2, 2, 4, 6]</code>. A student uses binary search, finds the target at index 2, and immediately returns, saying: “Index 2 is the left boundary of the number 2.”</p> <ol> <li>Is the student correct? What are the left and right boundaries of 2? Explain.</li> <li>When finding the left boundary, which side should we continue searching if the midpoint element equals the target?</li> <li>Which side should we continue searching when finding the right boundary? State only the direction; you do not need to give the full search process.</li> </ol> Reference answer <ol> <li> <p>The student is incorrect. Returning immediately upon finding a 2 guarantees only that some occurrence has been found, not the leftmost or rightmost one.     Here, the left boundary is index 1 and the right boundary is index 3.</p> </li> <li> <p>When finding the left boundary, continue searching to the left even if the midpoint element equals 2.     For example, with a closed interval, set <code>j = m - 1</code>.</p> </li> <li> <p>When finding the right boundary, continue searching to the right after the midpoint element equals 2.     For example, set <code>i = m + 1</code>.</p> </li> </ol>","path":["Chapter 10.   Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/exercises/#3-choosing-a-search-method-for-different-data","level":3,"title":"3.   Choosing a search method for different data","text":"<p>Choose a suitable method from linear search, binary search, and a hash table for each situation, and explain:</p> <ol> <li>Repeatedly searching \\(10^7\\) integers that are already sorted and will not change, without building an additional data structure.</li> <li>Repeatedly checking whether a key exists in a data set with frequent insertions and deletions, with no need to preserve order or perform range queries.</li> <li>Searching for a value only once in an unsorted array.</li> </ol> Reference answer <ol> <li>Binary search: the data is sorted and static, giving \\(O(\\log n)\\) time without an additional data structure.</li> <li>A hash table: when the hash function distributes keys fairly evenly among buckets, insertion, deletion, and key lookup all have average time complexity \\(O(1)\\).</li> <li> <p>Scan directly from beginning to end. For a single search, sorting or building a hash table would first require processing the entire array,     without reducing the total work for this task.</p> <p>The choice depends on whether the data is sorted, whether additional structures are allowed, how many queries there are, and which operations are needed.</p> </li> </ol>","path":["Chapter 10.   Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/exercises/#1072-programming-exercises","level":2,"title":"10.7.2   Programming exercises","text":"","path":["Chapter 10.   Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/exercises/#1-binary-search-in-a-sorted-array","level":3,"title":"1.   Binary search in a sorted array","text":"<p>Given an integer array <code>nums</code> in strictly increasing order and a target value <code>target</code>, find <code>target</code> using binary search. Return its array index if it exists, or -1 otherwise.</p> Hints <ol> <li>The initial interval has left = 0 and right = n - 1; it is nonempty while left &lt;= right.</li> <li>Calculate the midpoint with mid = left + (right - left) // 2.</li> <li>If <code>nums[mid]</code> is less than <code>target</code>, move the left boundary to <code>mid + 1</code>; if <code>nums[mid]</code> is greater than <code>target</code>, move the right boundary to <code>mid - 1</code>; return immediately if they are equal.</li> </ol> <p>LeetCode</p>","path":["Chapter 10.   Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/exercises/#2-finding-an-insertion-position-in-a-sorted-array","level":3,"title":"2.   Finding an insertion position in a sorted array","text":"<p>You are given an integer array <code>nums</code> in strictly increasing order and a target value <code>target</code>.</p> <p>If <code>target</code> is already in the array, return its index. Otherwise, return the position at which <code>target</code> can be inserted while preserving strictly increasing order. The answer may be 0 or the array length. Use binary search.</p> Hints <ol> <li>The answer may be 0 or the array length n.</li> <li>With a closed interval, if <code>nums[mid]</code> is greater than or equal to <code>target</code>, set <code>right = mid - 1</code> to keep checking farther left; otherwise, set <code>left = mid + 1</code>.</li> <li>When the loop ends, left is the insertion position.</li> </ol> <p>LeetCode</p>","path":["Chapter 10.   Searching","10.7   Exercises"],"tags":[]},{"location":"chapter_searching/replace_linear_by_hashing/","level":1,"title":"10.4   Optimizing with hashing","text":"<p>In algorithm problems, we often reduce time complexity by replacing linear search with hash lookup. Let us explore this strategy through a problem.</p> <p>Question</p> <p>Given an integer array <code>nums</code> and a target value <code>target</code>, find two elements whose sum equals <code>target</code> and return their array indices. Any valid solution is acceptable.</p>","path":["Chapter 10.   Searching","10.4   Optimizing with hashing"],"tags":[]},{"location":"chapter_searching/replace_linear_by_hashing/#1041-linear-search-trading-time-for-space","level":2,"title":"10.4.1   Linear search: trading time for space","text":"<p>Consider examining every possible pair directly. As shown below, use two nested loops and check whether the two integers sum to <code>target</code> each time. If they do, return their indices.</p> <p></p> <p> Figure 10-9   Solving two sum with linear search </p> <p>The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby two_sum.py<pre><code>def two_sum_brute_force(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"Method 1: brute-force enumeration\"\"\"\n    # Two nested loops give a time complexity of O(n^2)\n    for i in range(len(nums) - 1):\n        for j in range(i + 1, len(nums)):\n            if nums[i] + nums[j] == target:\n                return [i, j]\n    return []\n</code></pre> two_sum.cpp<pre><code>/* Method 1: brute-force enumeration */\nvector&lt;int&gt; twoSumBruteForce(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // Two nested loops give a time complexity of O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return {i, j};\n        }\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* Method 1: brute-force enumeration */\nint[] twoSumBruteForce(int[] nums, int target) {\n    int size = nums.length;\n    // Two nested loops give a time complexity of O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return new int[] { i, j };\n        }\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* Method 1: brute-force enumeration */\nint[] TwoSumBruteForce(int[] nums, int target) {\n    int size = nums.Length;\n    // Two nested loops give a time complexity of O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return [i, j];\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* Method 1: brute-force enumeration */\nfunc twoSumBruteForce(nums []int, target int) []int {\n    size := len(nums)\n    // Two nested loops give a time complexity of O(n^2)\n    for i := 0; i &lt; size-1; i++ {\n        for j := i + 1; j &lt; size; j++ {\n            if nums[i]+nums[j] == target {\n                return []int{i, j}\n            }\n        }\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* Method 1: brute-force enumeration */\nfunc twoSumBruteForce(nums: [Int], target: Int) -&gt; [Int] {\n    // Two nested loops give a time complexity of O(n^2)\n    for i in nums.indices.dropLast() {\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[i] + nums[j] == target {\n                return [i, j]\n            }\n        }\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* Method 1: brute-force enumeration */\nfunction twoSumBruteForce(nums, target) {\n    const n = nums.length;\n    // Two nested loops give a time complexity of O(n^2)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* Method 1: brute-force enumeration */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\n    const n = nums.length;\n    // Two nested loops give a time complexity of O(n^2)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* Method 1: brute-force enumeration */\nList&lt;int&gt; twoSumBruteForce(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // Two nested loops give a time complexity of O(n^2)\n  for (var i = 0; i &lt; size - 1; i++) {\n    for (var j = i + 1; j &lt; size; j++) {\n      if (nums[i] + nums[j] == target) return [i, j];\n    }\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* Method 1: brute-force enumeration */\npub fn two_sum_brute_force(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    let size = nums.len();\n    // Two nested loops give a time complexity of O(n^2)\n    for i in 0..size - 1 {\n        for j in i + 1..size {\n            if nums[i] + nums[j] == target {\n                return Some(vec![i as i32, j as i32]);\n            }\n        }\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* Method 1: brute-force enumeration */\nint *twoSumBruteForce(int *nums, int numsSize, int target, int *returnSize) {\n    for (int i = 0; i &lt; numsSize; ++i) {\n        for (int j = i + 1; j &lt; numsSize; ++j) {\n            if (nums[i] + nums[j] == target) {\n                int *res = malloc(sizeof(int) * 2);\n                res[0] = i, res[1] = j;\n                *returnSize = 2;\n                return res;\n            }\n        }\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* Method 1: brute-force enumeration */\nfun twoSumBruteForce(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // Two nested loops give a time complexity of O(n^2)\n    for (i in 0..&lt;size - 1) {\n        for (j in i + 1..&lt;size) {\n            if (nums[i] + nums[j] == target) return intArrayOf(i, j)\n        }\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>### Method 1: brute-force enumeration ###\ndef two_sum_brute_force(nums, target)\n  # Two nested loops give a time complexity of O(n^2)\n  for i in 0...(nums.length - 1)\n    for j in (i + 1)...nums.length\n      return [i, j] if nums[i] + nums[j] == target\n    end\n  end\n\n  []\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>This approach takes \\(O(n^2)\\) time and \\(O(1)\\) space, making it very slow for large datasets.</p>","path":["Chapter 10.   Searching","10.4   Optimizing with hashing"],"tags":[]},{"location":"chapter_searching/replace_linear_by_hashing/#1042-hash-lookup-trading-space-for-time","level":2,"title":"10.4.2   Hash lookup: trading space for time","text":"<p>Now use a hash table whose keys are array elements and whose values are their indices. Iterate through the array, performing the steps below each round.</p> <ol> <li>Check whether <code>target - nums[i]</code> is in the hash table. If it is, return the indices of the two elements.</li> <li>Add the key-value pair consisting of <code>nums[i]</code> and index <code>i</code> to the hash table.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> Figure 10-10   Solving two sum with an auxiliary hash table </p> <p>The implementation below requires only one loop:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby two_sum.py<pre><code>def two_sum_hash_table(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"Method 2: auxiliary hash table\"\"\"\n    # The auxiliary hash table uses O(n) space\n    dic = {}\n    # A single loop with average O(1) hash operations gives an average time complexity of O(n)\n    for i in range(len(nums)):\n        if target - nums[i] in dic:\n            return [dic[target - nums[i]], i]\n        dic[nums[i]] = i\n    return []\n</code></pre> two_sum.cpp<pre><code>/* Method 2: auxiliary hash table */\nvector&lt;int&gt; twoSumHashTable(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // The auxiliary hash table uses O(n) space\n    unordered_map&lt;int, int&gt; dic;\n    // A single loop gives a time complexity of O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.find(target - nums[i]) != dic.end()) {\n            return {dic[target - nums[i]], i};\n        }\n        dic.emplace(nums[i], i);\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* Method 2: auxiliary hash table */\nint[] twoSumHashTable(int[] nums, int target) {\n    int size = nums.length;\n    // The auxiliary hash table uses O(n) space\n    Map&lt;Integer, Integer&gt; dic = new HashMap&lt;&gt;();\n    // A single loop gives a time complexity of O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.containsKey(target - nums[i])) {\n            return new int[] { dic.get(target - nums[i]), i };\n        }\n        dic.put(nums[i], i);\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* Method 2: auxiliary hash table */\nint[] TwoSumHashTable(int[] nums, int target) {\n    int size = nums.Length;\n    // The auxiliary hash table uses O(n) space\n    Dictionary&lt;int, int&gt; dic = [];\n    // A single loop gives a time complexity of O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.ContainsKey(target - nums[i])) {\n            return [dic[target - nums[i]], i];\n        }\n        dic.Add(nums[i], i);\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* Method 2: auxiliary hash table */\nfunc twoSumHashTable(nums []int, target int) []int {\n    // The auxiliary hash table uses O(n) space\n    hashTable := map[int]int{}\n    // A single loop gives a time complexity of O(n)\n    for idx, val := range nums {\n        if preIdx, ok := hashTable[target-val]; ok {\n            return []int{preIdx, idx}\n        }\n        hashTable[val] = idx\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* Method 2: auxiliary hash table */\nfunc twoSumHashTable(nums: [Int], target: Int) -&gt; [Int] {\n    // The auxiliary hash table uses O(n) space\n    var dic: [Int: Int] = [:]\n    // A single loop gives a time complexity of O(n)\n    for i in nums.indices {\n        if let j = dic[target - nums[i]] {\n            return [j, i]\n        }\n        dic[nums[i]] = i\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* Method 2: auxiliary hash table */\nfunction twoSumHashTable(nums, target) {\n    // The auxiliary hash table uses O(n) space\n    let m = {};\n    // A single loop gives a time complexity of O(n)\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (m[target - nums[i]] !== undefined) {\n            return [m[target - nums[i]], i];\n        } else {\n            m[nums[i]] = i;\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* Method 2: auxiliary hash table */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n    // The auxiliary hash table uses O(n) space\n    let m: Map&lt;number, number&gt; = new Map();\n    // A single loop gives a time complexity of O(n)\n    for (let i = 0; i &lt; nums.length; i++) {\n        let index = m.get(target - nums[i]);\n        if (index !== undefined) {\n            return [index, i];\n        } else {\n            m.set(nums[i], i);\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* Method 2: auxiliary hash table */\nList&lt;int&gt; twoSumHashTable(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // The auxiliary hash table uses O(n) space\n  Map&lt;int, int&gt; dic = HashMap();\n  // A single loop gives a time complexity of O(n)\n  for (var i = 0; i &lt; size; i++) {\n    if (dic.containsKey(target - nums[i])) {\n      return [dic[target - nums[i]]!, i];\n    }\n    dic.putIfAbsent(nums[i], () =&gt; i);\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* Method 2: auxiliary hash table */\npub fn two_sum_hash_table(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    // The auxiliary hash table uses O(n) space\n    let mut dic = HashMap::new();\n    // A single loop gives a time complexity of O(n)\n    for (i, num) in nums.iter().enumerate() {\n        match dic.get(&amp;(target - num)) {\n            Some(v) =&gt; return Some(vec![*v as i32, i as i32]),\n            None =&gt; dic.insert(num, i as i32),\n        };\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* Hash table */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // Implemented using uthash.h\n} HashTable;\n\n/* Look up a key in the hash table */\nHashTable *find(HashTable *h, int key) {\n    HashTable *tmp;\n    HASH_FIND_INT(h, &amp;key, tmp);\n    return tmp;\n}\n\n/* Insert an entry into the hash table */\nvoid insert(HashTable **h, int key, int val) {\n    HashTable *t = find(*h, key);\n    if (t == NULL) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = key, tmp-&gt;val = val;\n        HASH_ADD_INT(*h, key, tmp);\n    } else {\n        t-&gt;val = val;\n    }\n}\n\n/* Method 2: auxiliary hash table */\nint *twoSumHashTable(int *nums, int numsSize, int target, int *returnSize) {\n    HashTable *hashtable = NULL;\n    for (int i = 0; i &lt; numsSize; i++) {\n        HashTable *t = find(hashtable, target - nums[i]);\n        if (t != NULL) {\n            int *res = malloc(sizeof(int) * 2);\n            res[0] = t-&gt;val, res[1] = i;\n            *returnSize = 2;\n            return res;\n        }\n        insert(&amp;hashtable, nums[i], i);\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* Method 2: auxiliary hash table */\nfun twoSumHashTable(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // The auxiliary hash table uses O(n) space\n    val dic = HashMap&lt;Int, Int&gt;()\n    // A single loop gives a time complexity of O(n)\n    for (i in 0..&lt;size) {\n        if (dic.containsKey(target - nums[i])) {\n            return intArrayOf(dic[target - nums[i]]!!, i)\n        }\n        dic[nums[i]] = i\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>### Method 2: auxiliary hash table ###\ndef two_sum_hash_table(nums, target)\n  # The auxiliary hash table uses O(n) space\n  dic = {}\n  # A single loop gives a time complexity of O(n)\n  for i in 0...nums.length\n    return [dic[target - nums[i]], i] if dic.has_key?(target - nums[i])\n\n    dic[nums[i]] = i\n  end\n\n  []\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Hash lookup reduces the time complexity from \\(O(n^2)\\) to \\(O(n)\\), greatly improving efficiency.</p> <p>Maintaining an additional hash table requires \\(O(n)\\) space. Even so, this approach offers a better balance of time and space efficiency, making it the optimal solution to this problem.</p>","path":["Chapter 10.   Searching","10.4   Optimizing with hashing"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/","level":1,"title":"10.5   Search algorithms revisited","text":"<p>A search algorithm finds one or more elements that satisfy specified conditions in a data structure, such as an array, linked list, tree, or graph.</p> <p>Search algorithms fall into two categories according to their approach.</p> <ul> <li>Locate the target by traversing the data structure, as in traversals of arrays, linked lists, trees, and graphs.</li> <li>Find elements efficiently by using the organization of the data or prior information about it, as in binary search, hash lookup, and binary search tree lookup.</li> </ul> <p>These ideas have all appeared in earlier chapters, so search algorithms are already familiar to us. In this section, we revisit them from a more systematic perspective.</p>","path":["Chapter 10.   Searching","10.5   Search algorithms revisited"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/#1051-brute-force-search","level":2,"title":"10.5.1   Brute-force search","text":"<p>Brute-force search locates a target by visiting the elements of a data structure one by one.</p> <ul> <li>“Linear search” applies to linear data structures such as arrays and linked lists. Starting at one end, it visits elements in sequence until it finds the target or reaches the other end without finding it.</li> <li>“Breadth-first search” and “depth-first search” are two strategies for traversing graphs and trees. Breadth-first search starts at an initial node and searches level by level, visiting nearby nodes before more distant ones. Depth-first search follows a path as far as possible, then backtracks and tries other paths until the whole structure has been traversed.</li> </ul> <p>Brute-force search is simple and broadly applicable. It requires neither preprocessing the data nor building an additional data structure.</p> <p>However, these algorithms take \\(O(n)\\) time, where \\(n\\) is the number of elements, so they perform poorly on large datasets.</p>","path":["Chapter 10.   Searching","10.5   Search algorithms revisited"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/#1052-adaptive-search","level":2,"title":"10.5.2   Adaptive search","text":"<p>Adaptive search uses properties of the data, such as its ordering, to optimize the search process and locate the target more efficiently.</p> <ul> <li>“Binary search” uses sorted data to search efficiently and applies only to arrays.</li> <li>“Hash lookup” uses a hash table to map lookup keys to target data as key-value pairs.</li> <li>“Tree search” uses comparisons of node values in a particular tree structure, such as a binary search tree, to rule out nodes quickly and locate the target.</li> </ul> <p>These algorithms are efficient, achieving time complexities of \\(O(\\log n)\\) or even \\(O(1)\\).</p> <p>However, they often require preprocessing the data. Binary search requires sorting the array in advance; hash lookup and tree search require additional data structures, whose maintenance also costs time and space.</p> <p>Tip</p> <p>Adaptive search algorithms are often called lookup algorithms. Their main purpose is to retrieve target elements quickly from specific data structures.</p>","path":["Chapter 10.   Searching","10.5   Search algorithms revisited"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/#1053-choosing-a-search-method","level":2,"title":"10.5.3   Choosing a search method","text":"<p>Given a dataset of size \\(n\\), we can find a target using linear search, binary search, tree search, hash lookup, and other methods. Figure 10-11 illustrates how these methods work.</p> <p></p> <p> Figure 10-11   Different search strategies </p> <p>Table 10-1 compares their efficiency and properties.</p> <p> Table 10-1   Efficiency of lookup algorithms </p> Linear search Binary search Tree search Hash lookup Find an element \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) Insert an element \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) Delete an element \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) Extra space \\(O(1)\\) \\(O(1)\\) \\(O(n)\\) \\(O(n)\\) Preprocessing / Sort \\(O(n \\log n)\\) Build a tree \\(O(n \\log n)\\) Build a hash table \\(O(n)\\) Data ordering Unsorted Sorted Sorted Unsorted <p>The choice also depends on dataset size, required search performance, and how frequently the data is queried and updated.</p> <p>Linear search</p> <ul> <li>It is broadly applicable and requires no preprocessing. If we need only one lookup, preprocessing for the other three methods takes longer than a linear search itself.</li> <li>It suits small datasets, where time complexity has less influence on actual performance.</li> <li>It suits frequently updated data because no additional data structure needs to be maintained.</li> </ul> <p>Binary search</p> <ul> <li>It suits large datasets, with predictable performance and a worst-case time complexity of \\(O(\\log n)\\).</li> <li>The dataset cannot be excessively large, because an array requires contiguous memory.</li> <li>It is unsuitable for frequent insertions and deletions, since maintaining a sorted array is expensive.</li> </ul> <p>Hash lookup</p> <ul> <li>It suits situations demanding very fast queries, with an average time complexity of \\(O(1)\\).</li> <li>It is unsuitable when sorted data or range queries are needed, because a hash table does not preserve order.</li> <li>It depends heavily on the hash function and collision-resolution strategy, creating a greater risk of performance degradation.</li> <li>It is unsuitable for excessively large datasets, because a hash table needs extra space to minimize collisions and maintain good query performance.</li> </ul> <p>Tree search</p> <ul> <li>It suits very large datasets, because tree nodes are stored at separate locations in memory.</li> <li>It suits situations requiring sorted data or range queries.</li> <li>Repeated insertions and deletions can make a binary search tree skewed, degrading its time complexity to \\(O(n)\\).</li> <li>An AVL tree or red-black tree keeps operations at \\(O(\\log n)\\), but maintaining balance adds overhead.</li> </ul>","path":["Chapter 10.   Searching","10.5   Search algorithms revisited"],"tags":[]},{"location":"chapter_searching/summary/","level":1,"title":"10.6   Summary","text":"","path":["Chapter 10.   Searching","10.6   Summary"],"tags":[]},{"location":"chapter_searching/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<ul> <li>Binary search relies on sorted data and repeatedly halves the search interval. It requires ordered input and applies only to arrays or data structures based on arrays.</li> <li>Brute-force search locates data by traversing a data structure. Linear search suits arrays and linked lists; breadth-first and depth-first search suit graphs and trees. These algorithms are broadly applicable and require no preprocessing, but their \\(O(n)\\) time complexity is relatively high.</li> <li>Hash lookup, tree search, and binary search efficiently locate targets in particular data structures. Their time complexities can be \\(O(\\log n)\\) or even \\(O(1)\\), but they usually rely on additional data structures.</li> <li>In practice, choose a search method by considering dataset size, required search performance, query frequency, update frequency, and related factors.</li> <li>Linear search suits small or frequently updated datasets; binary search suits large, sorted datasets; hash lookup suits fast queries without range-query requirements; tree search suits large, dynamic datasets that need ordering and range queries.</li> <li>Replacing linear search with hash lookup is a common optimization that reduces lookup time from \\(O(n)\\) to \\(O(1)\\).</li> </ul>","path":["Chapter 10.   Searching","10.6   Summary"],"tags":[]},{"location":"chapter_sorting/","level":1,"title":"Chapter 11.   Sorting","text":"<p>Abstract</p> <p>Sorting is like a magic key that turns chaos into order, helping us understand and process data more efficiently.</p> <p>From a simple ascending sequence to a complex arrangement by category, sorting reveals the harmonious beauty of data.</p>","path":["Chapter 11.   Sorting"],"tags":[]},{"location":"chapter_sorting/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>11.1   Sorting algorithms</li> <li>11.2   Selection sort</li> <li>11.3   Bubble sort</li> <li>11.4   Insertion sort</li> <li>11.5   Quick sort</li> <li>11.6   Merge sort</li> <li>11.7   Heap sort</li> <li>11.8   Bucket sort</li> <li>11.9   Counting sort</li> <li>11.10   Radix sort</li> <li>11.11   Summary</li> <li>11.12   Exercises</li> </ul>","path":["Chapter 11.   Sorting"],"tags":[]},{"location":"chapter_sorting/bubble_sort/","level":1,"title":"11.3   Bubble sort","text":"<p>Bubble sort sorts by repeatedly comparing and swapping adjacent elements. The process resembles bubbles rising from the bottom to the top, which gives the algorithm its name.</p> <p>As shown below, we can simulate bubbling through swaps. Scan from the left end of the array to the right, comparing each pair of adjacent elements. If the left element is greater than the right, swap them. After the scan, the largest element has moved to the far right.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 11-4   Simulating bubbling through element swaps </p>","path":["Chapter 11.   Sorting","11.3   Bubble sort"],"tags":[]},{"location":"chapter_sorting/bubble_sort/#1131-algorithm-steps","level":2,"title":"11.3.1   Algorithm steps","text":"<p>For an array of length \\(n\\), bubble sort proceeds as shown below.</p> <ol> <li>First, perform a bubbling pass over all \\(n\\) elements, moving the largest element to its correct position.</li> <li>Next, perform a pass over the remaining \\(n - 1\\) elements, moving the second-largest element to its correct position.</li> <li>Continue in this way. After \\(n - 1\\) passes, the largest \\(n - 1\\) elements are all in their correct positions.</li> <li>The one remaining element must be the smallest and needs no further sorting. The array is now sorted.</li> </ol> <p></p> <p> Figure 11-5   The bubble sort process </p> <p>An implementation follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby bubble_sort.py<pre><code>def bubble_sort(nums: list[int]):\n    \"\"\"Bubble sort\"\"\"\n    n = len(nums)\n    # Outer loop: the unsorted range is [0, i]\n    for i in range(n - 1, 0, -1):\n        # Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # Swap nums[j] and nums[j + 1]\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n</code></pre> bubble_sort.cpp<pre><code>/* Bubble sort */\nvoid bubbleSort(vector&lt;int&gt; &amp;nums) {\n    // Outer loop: the unsorted range is [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                // Use the std::swap() function here\n                swap(nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* Bubble sort */\nvoid bubbleSort(int[] nums) {\n    // Outer loop: the unsorted range is [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* Bubble sort */\nvoid BubbleSort(int[] nums) {\n    // Outer loop: the unsorted range is [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* Bubble sort */\nfunc bubbleSort(nums []int) {\n    // Outer loop: the unsorted range is [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // Swap nums[j] and nums[j + 1]\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* Bubble sort */\nfunc bubbleSort(nums: inout [Int]) {\n    // Outer loop: the unsorted range is [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // Swap nums[j] and nums[j + 1]\n                nums.swapAt(j, j + 1)\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* Bubble sort */\nfunction bubbleSort(nums) {\n    // Outer loop: the unsorted range is [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* Bubble sort */\nfunction bubbleSort(nums: number[]): void {\n    // Outer loop: the unsorted range is [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* Bubble sort */\nvoid bubbleSort(List&lt;int&gt; nums) {\n  // Outer loop: the unsorted range is [0, i]\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // Swap nums[j] and nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n      }\n    }\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* Bubble sort */\nfn bubble_sort(nums: &amp;mut [i32]) {\n    // Outer loop: the unsorted range is [0, i]\n    for i in (1..nums.len()).rev() {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // Swap nums[j] and nums[j + 1]\n                nums.swap(j, j + 1);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* Bubble sort */\nvoid bubbleSort(int nums[], int size) {\n    // Outer loop: the unsorted range is [0, i]\n    for (int i = size - 1; i &gt; 0; i--) {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* Bubble sort */\nfun bubbleSort(nums: IntArray) {\n    // Outer loop: the unsorted range is [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>### Bubble sort ###\ndef bubble_sort(nums)\n  n = nums.length\n  # Outer loop: the unsorted range is [0, i]\n  for i in (n - 1).downto(1)\n    # Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # Swap nums[j] and nums[j + 1]\n        nums[j], nums[j + 1] = nums[j + 1], nums[j]\n      end\n    end\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 11.   Sorting","11.3   Bubble sort"],"tags":[]},{"location":"chapter_sorting/bubble_sort/#1132-improving-efficiency","level":2,"title":"11.3.2   Improving efficiency","text":"<p>If a bubbling pass performs no swaps, the array is already sorted and we can return immediately. We can therefore add a variable <code>flag</code> to detect this situation and stop as soon as it occurs.</p> <p>With this optimization, the worst-case and average time complexities remain \\(O(n^2)\\). However, a fully sorted input achieves the best-case time complexity of \\(O(n)\\).</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby bubble_sort.py<pre><code>def bubble_sort_with_flag(nums: list[int]):\n    \"\"\"Bubble sort (flag optimization)\"\"\"\n    n = len(nums)\n    # Outer loop: the unsorted range is [0, i]\n    for i in range(n - 1, 0, -1):\n        flag = False  # Initialize the flag\n        # Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # Swap nums[j] and nums[j + 1]\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n                flag = True  # Record that a swap occurred\n        if not flag:\n            break  # Exit immediately if this pass made no swaps\n</code></pre> bubble_sort.cpp<pre><code>/* Bubble sort (flag optimization)*/\nvoid bubbleSortWithFlag(vector&lt;int&gt; &amp;nums) {\n    // Outer loop: the unsorted range is [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        bool flag = false; // Initialize the flag\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                // Use the std::swap() function here\n                swap(nums[j], nums[j + 1]);\n                flag = true; // Record that a swap occurred\n            }\n        }\n        if (!flag)\n            break; // Exit immediately if this pass made no swaps\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* Bubble sort (flag optimization) */\nvoid bubbleSortWithFlag(int[] nums) {\n    // Outer loop: the unsorted range is [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        boolean flag = false; // Initialize the flag\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // Record that a swap occurred\n            }\n        }\n        if (!flag)\n            break; // Exit immediately if this pass made no swaps\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* Bubble sort (flag optimization)*/\nvoid BubbleSortWithFlag(int[] nums) {\n    // Outer loop: the unsorted range is [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        bool flag = false; // Initialize the flag\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                flag = true;  // Record that a swap occurred\n            }\n        }\n        if (!flag) break;     // Exit immediately if this pass made no swaps\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* Bubble sort (flag optimization)*/\nfunc bubbleSortWithFlag(nums []int) {\n    // Outer loop: the unsorted range is [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        flag := false // Initialize the flag\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // Swap nums[j] and nums[j + 1]\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n                flag = true // Record that a swap occurred\n            }\n        }\n        if flag == false { // Exit immediately if this pass made no swaps\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* Bubble sort (flag optimization)*/\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n    // Outer loop: the unsorted range is [0, i]\n    for i in nums.indices.dropFirst().reversed() {\n        var flag = false // Initialize the flag\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // Swap nums[j] and nums[j + 1]\n                nums.swapAt(j, j + 1)\n                flag = true // Record that a swap occurred\n            }\n        }\n        if !flag { // Exit immediately if this pass made no swaps\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* Bubble sort (flag optimization)*/\nfunction bubbleSortWithFlag(nums) {\n    // Outer loop: the unsorted range is [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // Initialize the flag\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // Record that a swap occurred\n            }\n        }\n        if (!flag) break; // Exit immediately if this pass made no swaps\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* Bubble sort (flag optimization)*/\nfunction bubbleSortWithFlag(nums: number[]): void {\n    // Outer loop: the unsorted range is [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // Initialize the flag\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // Record that a swap occurred\n            }\n        }\n        if (!flag) break; // Exit immediately if this pass made no swaps\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* Bubble sort (flag optimization)*/\nvoid bubbleSortWithFlag(List&lt;int&gt; nums) {\n  // Outer loop: the unsorted range is [0, i]\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    bool flag = false; // Initialize the flag\n    // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // Swap nums[j] and nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        flag = true; // Record that a swap occurred\n      }\n    }\n    if (!flag) break; // Exit immediately if this pass made no swaps\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* Bubble sort (flag optimization) */\nfn bubble_sort_with_flag(nums: &amp;mut [i32]) {\n    // Outer loop: the unsorted range is [0, i]\n    for i in (1..nums.len()).rev() {\n        let mut flag = false; // Initialize the flag\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // Swap nums[j] and nums[j + 1]\n                nums.swap(j, j + 1);\n                flag = true; // Record that a swap occurred\n            }\n        }\n        if !flag {\n            break; // Exit immediately if this pass made no swaps\n        };\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* Bubble sort (flag optimization)*/\nvoid bubbleSortWithFlag(int nums[], int size) {\n    // Outer loop: the unsorted range is [0, i]\n    for (int i = size - 1; i &gt; 0; i--) {\n        bool flag = false;\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n                flag = true;\n            }\n        }\n        if (!flag)\n            break;\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* Bubble sort (flag optimization) */\nfun bubbleSortWithFlag(nums: IntArray) {\n    // Outer loop: the unsorted range is [0, i]\n    for (i in nums.size - 1 downTo 1) {\n        var flag = false // Initialize the flag\n        // Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // Swap nums[j] and nums[j + 1]\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n                flag = true // Record that a swap occurred\n            }\n        }\n        if (!flag) break // Exit immediately if this pass made no swaps\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>### Bubble sort (flag optimization)###\ndef bubble_sort_with_flag(nums)\n  n = nums.length\n  # Outer loop: the unsorted range is [0, i]\n  for i in (n - 1).downto(1)\n    flag = false # Initialize the flag\n\n    # Inner loop: move the largest element in the unsorted range [0, i] to its right end through swaps\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # Swap nums[j] and nums[j + 1]\n        nums[j], nums[j + 1] = nums[j + 1], nums[j]\n        flag = true # Record that a swap occurred\n      end\n    end\n\n    break unless flag # Exit immediately if this pass made no swaps\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 11.   Sorting","11.3   Bubble sort"],"tags":[]},{"location":"chapter_sorting/bubble_sort/#1133-algorithm-properties","level":2,"title":"11.3.3   Algorithm properties","text":"<ul> <li>Time complexity: \\(O(n^2)\\); adaptive. Successive passes traverse \\(n - 1\\), \\(n - 2\\), \\(\\dots\\), \\(2\\), and \\(1\\) positions, totaling \\((n - 1) n / 2\\). With the <code>flag</code> optimization, the best-case time complexity is \\(O(n)\\).</li> <li>Space complexity: \\(O(1)\\); in-place. Pointers \\(i\\) and \\(j\\) use constant extra space.</li> <li>Stable. Equal elements are not swapped during a bubbling pass.</li> </ul>","path":["Chapter 11.   Sorting","11.3   Bubble sort"],"tags":[]},{"location":"chapter_sorting/bucket_sort/","level":1,"title":"11.8   Bucket sort","text":"<p>The sorting algorithms discussed so far are “comparison-based”: they sort by comparing elements. Their worst-case time complexity has a lower bound of \\(\\Omega(n \\log n)\\). We now turn to several “non-comparison” sorting algorithms, which can achieve linear time complexity.</p> <p>Bucket sort is a classic application of divide and conquer. It creates ordered buckets, each corresponding to a range of values, and distributes the data evenly among them. It then sorts each bucket separately and combines the results in bucket order.</p>","path":["Chapter 11.   Sorting","11.8   Bucket sort"],"tags":[]},{"location":"chapter_sorting/bucket_sort/#1181-algorithm-steps","level":2,"title":"11.8.1   Algorithm steps","text":"<p>Consider an array of length \\(n\\) containing floating-point values in \\([0, 1)\\). Bucket sort proceeds as shown below.</p> <ol> <li>Initialize \\(k\\) buckets and distribute the \\(n\\) elements among the \\(k\\) buckets.</li> <li>Sort each bucket separately, using the programming language's built-in sorting function here.</li> <li>Combine the results in ascending bucket order.</li> </ol> <p></p> <p> Figure 11-13   The bucket sort process </p> <p>The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby bucket_sort.py<pre><code>def bucket_sort(nums: list[float]):\n    \"\"\"Bucket sort\"\"\"\n    # Initialize k = n/2 buckets, expecting 2 elements per bucket\n    k = len(nums) // 2\n    buckets = [[] for _ in range(k)]\n    # 1. Distribute the array elements among the buckets\n    for num in nums:\n        # The input range is [0, 1); use num * k to map each value to an index in [0, k-1]\n        i = int(num * k)\n        # Add num to bucket i\n        buckets[i].append(num)\n    # 2. Sort each bucket\n    for bucket in buckets:\n        # Use the built-in sorting function, or replace it with another sorting algorithm\n        bucket.sort()\n    # 3. Traverse the buckets to combine the results\n    i = 0\n    for bucket in buckets:\n        for num in bucket:\n            nums[i] = num\n            i += 1\n</code></pre> bucket_sort.cpp<pre><code>/* Bucket sort */\nvoid bucketSort(vector&lt;float&gt; &amp;nums) {\n    // Initialize k = n/2 buckets, expecting 2 elements per bucket\n    int k = nums.size() / 2;\n    vector&lt;vector&lt;float&gt;&gt; buckets(k);\n    // 1. Distribute the array elements among the buckets\n    for (float num : nums) {\n        // The input range is [0, 1); use num * k to map each value to an index in [0, k-1]\n        int i = num * k;\n        // Add num to bucket i\n        buckets[i].push_back(num);\n    }\n    // 2. Sort each bucket\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        // Use the built-in sorting function, or replace it with another sorting algorithm\n        sort(bucket.begin(), bucket.end());\n    }\n    // 3. Traverse the buckets to combine the results\n    int i = 0;\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.java<pre><code>/* Bucket sort */\nvoid bucketSort(float[] nums) {\n    // Initialize k = n/2 buckets, expecting 2 elements per bucket\n    int k = nums.length / 2;\n    List&lt;List&lt;Float&gt;&gt; buckets = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; k; i++) {\n        buckets.add(new ArrayList&lt;&gt;());\n    }\n    // 1. Distribute the array elements among the buckets\n    for (float num : nums) {\n        // The input range is [0, 1); use num * k to map each value to an index in [0, k-1]\n        int i = (int) (num * k);\n        // Add num to bucket i\n        buckets.get(i).add(num);\n    }\n    // 2. Sort each bucket\n    for (List&lt;Float&gt; bucket : buckets) {\n        // Use the built-in sorting function, or replace it with another sorting algorithm\n        Collections.sort(bucket);\n    }\n    // 3. Traverse the buckets to combine the results\n    int i = 0;\n    for (List&lt;Float&gt; bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.cs<pre><code>/* Bucket sort */\nvoid BucketSort(float[] nums) {\n    // Initialize k = n/2 buckets, expecting 2 elements per bucket\n    int k = nums.Length / 2;\n    List&lt;List&lt;float&gt;&gt; buckets = [];\n    for (int i = 0; i &lt; k; i++) {\n        buckets.Add([]);\n    }\n    // 1. Distribute the array elements among the buckets\n    foreach (float num in nums) {\n        // The input range is [0, 1); use num * k to map each value to an index in [0, k-1]\n        int i = (int)(num * k);\n        // Add num to bucket i\n        buckets[i].Add(num);\n    }\n    // 2. Sort each bucket\n    foreach (List&lt;float&gt; bucket in buckets) {\n        // Use the built-in sorting function, or replace it with another sorting algorithm\n        bucket.Sort();\n    }\n    // 3. Traverse the buckets to combine the results\n    int j = 0;\n    foreach (List&lt;float&gt; bucket in buckets) {\n        foreach (float num in bucket) {\n            nums[j++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.go<pre><code>/* Bucket sort */\nfunc bucketSort(nums []float64) {\n    // Initialize k = floor(n/2) buckets, expecting about 2 elements per bucket for uniformly distributed input\n    k := len(nums) / 2\n    buckets := make([][]float64, k)\n    for i := 0; i &lt; k; i++ {\n        buckets[i] = make([]float64, 0)\n    }\n    // 1. Distribute the array elements among the buckets\n    for _, num := range nums {\n        // The input range is [0, 1); use num * k to map each value to an index in [0, k-1]\n        i := int(num * float64(k))\n        // Add num to bucket i\n        buckets[i] = append(buckets[i], num)\n    }\n    // 2. Sort each bucket\n    for i := 0; i &lt; k; i++ {\n        // Use the standard slice-sorting function, or replace it with another sorting algorithm\n        sort.Float64s(buckets[i])\n    }\n    // 3. Traverse the buckets to combine the results\n    i := 0\n    for _, bucket := range buckets {\n        for _, num := range bucket {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> bucket_sort.swift<pre><code>/* Bucket sort */\nfunc bucketSort(nums: inout [Double]) {\n    // Initialize k = n/2 buckets, expecting 2 elements per bucket\n    let k = nums.count / 2\n    var buckets = (0 ..&lt; k).map { _ in [Double]() }\n    // 1. Distribute the array elements among the buckets\n    for num in nums {\n        // The input range is [0, 1); use num * k to map each value to an index in [0, k-1]\n        let i = Int(num * Double(k))\n        // Add num to bucket i\n        buckets[i].append(num)\n    }\n    // 2. Sort each bucket\n    for i in buckets.indices {\n        // Use the built-in sorting function, or replace it with another sorting algorithm\n        buckets[i].sort()\n    }\n    // 3. Traverse the buckets to combine the results\n    var i = nums.startIndex\n    for bucket in buckets {\n        for num in bucket {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> bucket_sort.js<pre><code>/* Bucket sort */\nfunction bucketSort(nums) {\n    // Initialize ceil(n/2) buckets, aiming for about 2 elements per bucket\n    const k = nums.length / 2;\n    const buckets = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. Distribute the array elements among the buckets\n    for (const num of nums) {\n        // The input range is [0, 1); floor(num * k) maps values to indices from 0 to ceil(k)-1\n        const i = Math.floor(num * k);\n        // Add num to bucket i\n        buckets[i].push(num);\n    }\n    // 2. Sort each bucket\n    for (const bucket of buckets) {\n        // Use the built-in sorting function, or replace it with another sorting algorithm\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. Traverse the buckets to combine the results\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.ts<pre><code>/* Bucket sort */\nfunction bucketSort(nums: number[]): void {\n    // Set k = n/2 and create ceil(k) buckets, aiming for about 2 elements per bucket\n    const k = nums.length / 2;\n    const buckets: number[][] = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. Distribute the array elements among the buckets\n    for (const num of nums) {\n        // The input range is [0, 1); floor(num * k) maps each value to an index in [0, ceil(k)-1]\n        const i = Math.floor(num * k);\n        // Add num to bucket i\n        buckets[i].push(num);\n    }\n    // 2. Sort each bucket\n    for (const bucket of buckets) {\n        // Use the built-in sorting function, or replace it with another sorting algorithm\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. Traverse the buckets to combine the results\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.dart<pre><code>/* Bucket sort */\nvoid bucketSort(List&lt;double&gt; nums) {\n  // Initialize k = floor(n/2) buckets, aiming for about 2 elements per bucket\n  int k = nums.length ~/ 2;\n  List&lt;List&lt;double&gt;&gt; buckets = List.generate(k, (index) =&gt; []);\n\n  // 1. Distribute the array elements among the buckets\n  for (double _num in nums) {\n    // The input range is [0, 1); use _num * k to map each value to an index in [0, k-1]\n    int i = (_num * k).toInt();\n    // Add _num to bucket i\n    buckets[i].add(_num);\n  }\n  // 2. Sort each bucket\n  for (List&lt;double&gt; bucket in buckets) {\n    bucket.sort();\n  }\n  // 3. Traverse the buckets to combine the results\n  int i = 0;\n  for (List&lt;double&gt; bucket in buckets) {\n    for (double _num in bucket) {\n      nums[i++] = _num;\n    }\n  }\n}\n</code></pre> bucket_sort.rs<pre><code>/* Bucket sort */\nfn bucket_sort(nums: &amp;mut [f64]) {\n    // Initialize k = n/2 buckets, expecting 2 elements per bucket\n    let k = nums.len() / 2;\n    let mut buckets = vec![vec![]; k];\n    // 1. Distribute the array elements among the buckets\n    for &amp;num in nums.iter() {\n        // The input range is [0, 1); use num * k to map each value to an index in [0, k-1]\n        let i = (num * k as f64) as usize;\n        // Add num to bucket i\n        buckets[i].push(num);\n    }\n    // 2. Sort each bucket\n    for bucket in &amp;mut buckets {\n        // Use the built-in sorting function, or replace it with another sorting algorithm\n        bucket.sort_by(|a, b| a.partial_cmp(b).unwrap());\n    }\n    // 3. Traverse the buckets to combine the results\n    let mut i = 0;\n    for bucket in buckets.iter() {\n        for &amp;num in bucket.iter() {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> bucket_sort.c<pre><code>/* Bucket sort */\nvoid bucketSort(float nums[], int n) {\n    int k = n / 2;                                 // Initialize k = n/2 buckets\n    int *sizes = malloc(k * sizeof(int));          // Record the size of each bucket\n    float **buckets = malloc(k * sizeof(float *)); // Array of dynamic arrays (buckets)\n    // Preallocate enough space for each bucket\n    for (int i = 0; i &lt; k; ++i) {\n        buckets[i] = (float *)malloc(n * sizeof(float));\n        sizes[i] = 0;\n    }\n    // 1. Distribute the array elements among the buckets\n    for (int i = 0; i &lt; n; ++i) {\n        int idx = (int)(nums[i] * k);\n        buckets[idx][sizes[idx]++] = nums[i];\n    }\n    // 2. Sort each bucket\n    for (int i = 0; i &lt; k; ++i) {\n        qsort(buckets[i], sizes[i], sizeof(float), compare);\n    }\n    // 3. Merge the sorted buckets\n    int idx = 0;\n    for (int i = 0; i &lt; k; ++i) {\n        for (int j = 0; j &lt; sizes[i]; ++j) {\n            nums[idx++] = buckets[i][j];\n        }\n        // Free memory\n        free(buckets[i]);\n    }\n}\n</code></pre> bucket_sort.kt<pre><code>/* Bucket sort */\nfun bucketSort(nums: FloatArray) {\n    // Initialize k = floor(n/2) buckets, expecting about 2 elements per bucket\n    val k = nums.size / 2\n    val buckets = mutableListOf&lt;MutableList&lt;Float&gt;&gt;()\n    for (i in 0..&lt;k) {\n        buckets.add(mutableListOf())\n    }\n    // 1. Distribute the array elements among the buckets\n    for (num in nums) {\n        // The input range is [0, 1); use num * k to map each value to an index in [0, k-1]\n        val i = (num * k).toInt()\n        // Add num to bucket i\n        buckets[i].add(num)\n    }\n    // 2. Sort each bucket\n    for (bucket in buckets) {\n        // Use the built-in sorting function, or replace it with another sorting algorithm\n        bucket.sort()\n    }\n    // 3. Traverse the buckets to combine the results\n    var i = 0\n    for (bucket in buckets) {\n        for (num in bucket) {\n            nums[i++] = num\n        }\n    }\n}\n</code></pre> bucket_sort.rb<pre><code>### Bucket sort ###\ndef bucket_sort(nums)\n  # Initialize k = floor(n/2) buckets, aiming for about 2 elements per bucket\n  k = nums.length / 2\n  buckets = Array.new(k) { [] }\n\n  # 1. Distribute the array elements among the buckets\n  nums.each do |num|\n    # The input range is [0, 1); use num * k to map each value to an index in [0, k-1]\n    i = (num * k).to_i\n    # Add num to bucket i\n    buckets[i] &lt;&lt; num\n  end\n\n  # 2. Sort each bucket\n  buckets.each do |bucket|\n    # Use the built-in sorting function, or replace it with another sorting algorithm\n    bucket.sort!\n  end\n\n  # 3. Traverse the buckets to combine the results\n  i = 0\n  buckets.each do |bucket|\n    bucket.each do |num|\n      nums[i] = num\n      i += 1\n    end\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 11.   Sorting","11.8   Bucket sort"],"tags":[]},{"location":"chapter_sorting/bucket_sort/#1182-algorithm-properties","level":2,"title":"11.8.2   Algorithm properties","text":"<p>Bucket sort is useful for very large datasets. For example, suppose the input contains one million elements and memory limitations prevent loading all of them at once. We can divide them into 1,000 buckets, sort the buckets separately, and combine the results.</p> <ul> <li>Time complexity: \\(O(n + k)\\). If the elements are evenly distributed, each bucket contains \\(\\frac{n}{k}\\) elements. If sorting one bucket takes \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) time, sorting all buckets takes \\(O(n \\log\\frac{n}{k})\\). When the number of buckets \\(k\\) is relatively large, this approaches \\(O(n)\\). Combining the results visits all buckets and elements, taking \\(O(n + k)\\) time. In the worst case, all data falls into one bucket, and sorting it takes \\(O(n^2)\\) time.</li> <li>Space complexity: \\(O(n + k)\\); not in-place. Extra space is needed for \\(k\\) buckets and a total of \\(n\\) elements.</li> <li>Whether bucket sort is stable depends on whether the algorithm used to sort within each bucket is stable.</li> </ul>","path":["Chapter 11.   Sorting","11.8   Bucket sort"],"tags":[]},{"location":"chapter_sorting/bucket_sort/#1183-distributing-elements-evenly","level":2,"title":"11.8.3   Distributing elements evenly","text":"<p>Bucket sort can theoretically achieve \\(O(n)\\) time. The key is distributing elements evenly among the buckets, since real data is often unevenly distributed. Suppose we want to divide all products on Taobao evenly into 10 buckets by price. There are many products below 100 yuan but few above 1,000 yuan. Splitting the price range into 10 equal intervals would leave very different numbers of products in the buckets.</p> <p>To obtain an even distribution, we can first choose rough boundaries and divide the data into 3 buckets. Then split the more crowded buckets into 3 smaller buckets, repeating until all buckets contain roughly equal numbers of elements.</p> <p>As shown below, this effectively creates a recursion tree whose leaf values should be as balanced as possible. We need not use 3 buckets at every step; the subdivision can be chosen to suit the data.</p> <p></p> <p> Figure 11-14   Recursively subdividing buckets </p> <p>If we know the probability distribution of product prices in advance, we can choose bucket boundaries according to that distribution. We do not necessarily need to measure the distribution explicitly; we can approximate it with a probability model suited to the data.</p> <p>In Figure 11-15, we assume prices follow a normal distribution. This lets us choose suitable price intervals and distribute products evenly among the buckets.</p> <p></p> <p> Figure 11-15   Defining buckets from a probability distribution </p>","path":["Chapter 11.   Sorting","11.8   Bucket sort"],"tags":[]},{"location":"chapter_sorting/counting_sort/","level":1,"title":"11.9   Counting sort","text":"<p>Counting sort sorts by counting how often each element occurs. It is typically used for integer arrays.</p>","path":["Chapter 11.   Sorting","11.9   Counting sort"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1191-a-simple-implementation","level":2,"title":"11.9.1   A simple implementation","text":"<p>Start with a simple example: an array <code>nums</code> of length \\(n\\) containing nonnegative integers. The overall process is shown below.</p> <ol> <li>Traverse the array to find its maximum value, denoted by \\(m\\), then create an auxiliary array <code>counter</code> of length \\(m + 1\\).</li> <li>Use <code>counter</code> to count occurrences of each number in <code>nums</code>. Here, <code>counter[num]</code> is the number of occurrences of <code>num</code>. Simply traverse <code>nums</code> and increment <code>counter[num]</code> by \\(1\\) for each current value <code>num</code>.</li> <li>Because the indices of <code>counter</code> are inherently ordered, the numbers are effectively already sorted. Traverse <code>counter</code> and write each number into <code>nums</code> in ascending order, repeating it according to its count.</li> </ol> <p></p> <p> Figure 11-16   The counting sort process </p> <p>The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby counting_sort.py<pre><code>def counting_sort_naive(nums: list[int]):\n    \"\"\"Counting sort\"\"\"\n    # Simple implementation; cannot be used to sort objects\n    # 1. Find the maximum array element m\n    m = max(nums)\n    # 2. Count the occurrences of each number\n    # counter[num] is the number of occurrences of num\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. Traverse counter and fill the original array nums with the elements\n    i = 0\n    for num in range(m + 1):\n        for _ in range(counter[num]):\n            nums[i] = num\n            i += 1\n</code></pre> counting_sort.cpp<pre><code>/* Counting sort */\n// Simple implementation; cannot be used to sort objects\nvoid countingSortNaive(vector&lt;int&gt; &amp;nums) {\n    // 1. Find the maximum array element m\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. Traverse counter and fill the original array nums with the elements\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.java<pre><code>/* Counting sort */\n// Simple implementation; cannot be used to sort objects\nvoid countingSortNaive(int[] nums) {\n    // 1. Find the maximum array element m\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. Traverse counter and fill the original array nums with the elements\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* Counting sort */\n// Simple implementation; cannot be used to sort objects\nvoid CountingSortNaive(int[] nums) {\n    // 1. Find the maximum array element m\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. Traverse counter and fill the original array nums with the elements\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* Counting sort */\n// Simple implementation; cannot be used to sort objects\nfunc countingSortNaive(nums []int) {\n    // 1. Find the maximum array element m\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. Traverse counter and fill the original array nums with the elements\n    for i, num := 0, 0; num &lt; m+1; num++ {\n        for j := 0; j &lt; counter[num]; j++ {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.swift<pre><code>/* Counting sort */\n// Simple implementation; cannot be used to sort objects\nfunc countingSortNaive(nums: inout [Int]) {\n    // 1. Find the maximum array element m\n    let m = nums.max()!\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. Traverse counter and fill the original array nums with the elements\n    var i = 0\n    for num in 0 ..&lt; m + 1 {\n        for _ in 0 ..&lt; counter[num] {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* Counting sort */\n// Simple implementation; cannot be used to sort objects\nfunction countingSortNaive(nums) {\n    // 1. Find the maximum array element m\n    let m = Math.max(...nums);\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. Traverse counter and fill the original array nums with the elements\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* Counting sort */\n// Simple implementation; cannot be used to sort objects\nfunction countingSortNaive(nums: number[]): void {\n    // 1. Find the maximum array element m\n    let m: number = Math.max(...nums);\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. Traverse counter and fill the original array nums with the elements\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* Counting sort */\n// Simple implementation; cannot be used to sort objects\nvoid countingSortNaive(List&lt;int&gt; nums) {\n  // 1. Find the maximum array element m\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. Count the occurrences of each number\n  // counter[_num] is the number of occurrences of _num\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. Traverse counter and fill the original array nums with the elements\n  int i = 0;\n  for (int _num = 0; _num &lt; m + 1; _num++) {\n    for (int j = 0; j &lt; counter[_num]; j++, i++) {\n      nums[i] = _num;\n    }\n  }\n}\n</code></pre> counting_sort.rs<pre><code>/* Counting sort */\n// Simple implementation; cannot be used to sort objects\nfn counting_sort_naive(nums: &amp;mut [i32]) {\n    // 1. Find the maximum array element m\n    let m = *nums.iter().max().unwrap();\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    let mut counter = vec![0; m as usize + 1];\n    for &amp;num in nums.iter() {\n        counter[num as usize] += 1;\n    }\n    // 3. Traverse counter and fill the original array nums with the elements\n    let mut i = 0;\n    for num in 0..m + 1 {\n        for _ in 0..counter[num as usize] {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> counting_sort.c<pre><code>/* Counting sort */\n// Simple implementation; cannot be used to sort objects\nvoid countingSortNaive(int nums[], int size) {\n    // 1. Find the maximum array element m\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    int *counter = calloc(m + 1, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. Traverse counter and fill the original array nums with the elements\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n    // 4. Free memory\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* Counting sort */\n// Simple implementation; cannot be used to sort objects\nfun countingSortNaive(nums: IntArray) {\n    // 1. Find the maximum array element m\n    var m = 0\n    for (num in nums) {\n        m = max(m, num)\n    }\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. Traverse counter and fill the original array nums with the elements\n    var i = 0\n    for (num in 0..&lt;m + 1) {\n        var j = 0\n        while (j &lt; counter[num]) {\n            nums[i] = num\n            j++\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.rb<pre><code>### Counting sort ###\ndef counting_sort_naive(nums)\n  # Simple implementation; cannot be used to sort objects\n  # 1. Find the maximum array element m\n  m = 0\n  nums.each { |num| m = [m, num].max }\n  # 2. Count the occurrences of each number\n  # counter[num] is the number of occurrences of num\n  counter = Array.new(m + 1, 0)\n  nums.each { |num| counter[num] += 1 }\n  # 3. Traverse counter and fill the original array nums with the elements\n  i = 0\n  for num in 0...(m + 1)\n    (0...counter[num]).each do\n      nums[i] = num\n      i += 1\n    end\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>How counting sort relates to bucket sort</p> <p>From a bucket-sort perspective, each index of <code>counter</code> is a bucket, and counting occurrences is equivalent to distributing elements into their corresponding buckets. Counting sort is essentially a special case of bucket sort for integer data.</p>","path":["Chapter 11.   Sorting","11.9   Counting sort"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1192-a-complete-implementation","level":2,"title":"11.9.2   A complete implementation","text":"<p>You may have noticed that step <code>3.</code> above fails when the input consists of objects. Suppose the inputs are product objects and we want to sort them by price, a member variable of the class. The algorithm above can produce only the sorted prices.</p> <p>How can we obtain the sorted original data? First, calculate the “prefix sums” of <code>counter</code>. The prefix sum <code>prefix[i]</code> is the sum of the array entries up to index <code>i</code>:</p> \\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\] <p>The prefix sum has a precise meaning: <code>prefix[num] - 1</code> is the index of the last occurrence of <code>num</code> in the result array <code>res</code>. This tells us where each element belongs. Next, traverse the original array <code>nums</code> in reverse and perform two steps for each element <code>num</code>.</p> <ol> <li>Place <code>num</code> at index <code>prefix[num] - 1</code> in <code>res</code>.</li> <li>Decrease <code>prefix[num]</code> by \\(1\\) to obtain the index for placing the next occurrence of <code>num</code>.</li> </ol> <p>After the traversal, <code>res</code> contains the sorted result. Finally, copy <code>res</code> over the original array <code>nums</code>. Figure 11-17 shows the complete process.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 11-17   Steps of counting sort </p> <p>The implementation is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby counting_sort.py<pre><code>def counting_sort(nums: list[int]):\n    \"\"\"Counting sort\"\"\"\n    # Full implementation; stable and adaptable to sorting objects\n    # 1. Find the maximum array element m\n    m = max(nums)\n    # 2. Count the occurrences of each number\n    # counter[num] is the number of occurrences of num\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. Compute prefix sums of counter to convert occurrence counts into end positions\n    # That is, counter[num]-1 is the index of the last occurrence of num in res\n    for i in range(m):\n        counter[i + 1] += counter[i]\n    # 4. Traverse nums in reverse and place each element in the result array res\n    # Initialize array res to store the result\n    n = len(nums)\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        num = nums[i]\n        res[counter[num] - 1] = num  # Place num at the corresponding index\n        counter[num] -= 1  # Decrement the prefix sum by 1 to determine where to place the next occurrence of num\n    # Overwrite the original array nums with the result array res\n    for i in range(n):\n        nums[i] = res[i]\n</code></pre> counting_sort.cpp<pre><code>/* Counting sort */\n// Stable implementation that can be adapted to sort objects\nvoid countingSort(vector&lt;int&gt; &amp;nums) {\n    // 1. Find the maximum array element m\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. Compute prefix sums of counter to convert occurrence counts into end positions\n    // That is, counter[num]-1 is the index of the last occurrence of num in res\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. Traverse nums in reverse and place each element in the result array res\n    // Initialize array res to store the result\n    int n = nums.size();\n    vector&lt;int&gt; res(n);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // Place num at the corresponding index\n        counter[num]--;              // Decrement the prefix sum by 1 to determine where to place the next occurrence of num\n    }\n    // Overwrite the original array nums with the result array res\n    nums = res;\n}\n</code></pre> counting_sort.java<pre><code>/* Counting sort */\n// Full implementation; stable and adaptable to sorting objects\nvoid countingSort(int[] nums) {\n    // 1. Find the maximum array element m\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. Compute prefix sums of counter to convert occurrence counts into end positions\n    // That is, counter[num]-1 is the index of the last occurrence of num in res\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. Traverse nums in reverse and place each element in the result array res\n    // Initialize array res to store the result\n    int n = nums.length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // Place num at the corresponding index\n        counter[num]--; // Decrement the prefix sum by 1 to determine where to place the next occurrence of num\n    }\n    // Overwrite the original array nums with the result array res\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* Counting sort */\n// Full implementation; stable and adaptable to sorting objects\nvoid CountingSort(int[] nums) {\n    // 1. Find the maximum array element m\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. Compute prefix sums of counter to convert occurrence counts into end positions\n    // That is, counter[num]-1 is the index of the last occurrence of num in res\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. Traverse nums in reverse and place each element in the result array res\n    // Initialize array res to store the result\n    int n = nums.Length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // Place num at the corresponding index\n        counter[num]--; // Decrement the prefix sum by 1 to determine where to place the next occurrence of num\n    }\n    // Overwrite the original array nums with the result array res\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* Counting sort */\n// Full implementation; stable and adaptable to sorting objects\nfunc countingSort(nums []int) {\n    // 1. Find the maximum array element m\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. Compute prefix sums of counter to convert occurrence counts into end positions\n    // That is, counter[num]-1 is the index of the last occurrence of num in res\n    for i := 0; i &lt; m; i++ {\n        counter[i+1] += counter[i]\n    }\n    // 4. Traverse nums in reverse and place each element in the result array res\n    // Initialize array res to store the result\n    n := len(nums)\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        num := nums[i]\n        // Place num at the corresponding index\n        res[counter[num]-1] = num\n        // Decrement the prefix sum by 1 to determine where to place the next occurrence of num\n        counter[num]--\n    }\n    // Overwrite the original array nums with the result array res\n    copy(nums, res)\n}\n</code></pre> counting_sort.swift<pre><code>/* Counting sort */\n// Full implementation; stable and adaptable to sorting objects\nfunc countingSort(nums: inout [Int]) {\n    // 1. Find the maximum array element m\n    let m = nums.max()!\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. Compute prefix sums of counter to convert occurrence counts into end positions\n    // That is, counter[num]-1 is the index of the last occurrence of num in res\n    for i in 0 ..&lt; m {\n        counter[i + 1] += counter[i]\n    }\n    // 4. Traverse nums in reverse and place each element in the result array res\n    // Initialize array res to store the result\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let num = nums[i]\n        res[counter[num] - 1] = num // Place num at the corresponding index\n        counter[num] -= 1 // Decrement the prefix sum by 1 to determine where to place the next occurrence of num\n    }\n    // Overwrite the original array nums with the result array res\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* Counting sort */\n// Full implementation; stable and adaptable to sorting objects\nfunction countingSort(nums) {\n    // 1. Find the maximum array element m\n    let m = Math.max(...nums);\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. Compute prefix sums of counter to convert occurrence counts into end positions\n    // That is, counter[num]-1 is the index of the last occurrence of num in res\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. Traverse nums in reverse and place each element in the result array res\n    // Initialize array res to store the result\n    const n = nums.length;\n    const res = new Array(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // Place num at the corresponding index\n        counter[num]--; // Decrement the prefix sum by 1 to determine where to place the next occurrence of num\n    }\n    // Overwrite the original array nums with the result array res\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* Counting sort */\n// Full stable implementation; this approach can be adapted to sort objects\nfunction countingSort(nums: number[]): void {\n    // 1. Find the maximum array element m\n    let m: number = Math.max(...nums);\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. Compute prefix sums of counter to convert occurrence counts into end positions\n    // That is, counter[num]-1 is the index of the last occurrence of num in res\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. Traverse nums in reverse and place each element in the result array res\n    // Initialize array res to store the result\n    const n = nums.length;\n    const res: number[] = new Array&lt;number&gt;(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // Place num at the corresponding index\n        counter[num]--; // Decrement the prefix sum by 1 to determine where to place the next occurrence of num\n    }\n    // Overwrite the original array nums with the result array res\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* Counting sort */\n// Full stable implementation; this approach can be adapted to sort objects\nvoid countingSort(List&lt;int&gt; nums) {\n  // 1. Find the maximum array element m\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. Count the occurrences of each number\n  // counter[_num] is the number of occurrences of _num\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. Compute prefix sums of counter to convert occurrence counts into end positions\n  // That is, counter[_num]-1 is the index of the last occurrence of _num in res\n  for (int i = 0; i &lt; m; i++) {\n    counter[i + 1] += counter[i];\n  }\n  // 4. Traverse nums in reverse and place each element in the result array res\n  // Initialize array res to store the result\n  int n = nums.length;\n  List&lt;int&gt; res = List.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int _num = nums[i];\n    res[counter[_num] - 1] = _num; // Place _num at the corresponding index\n    counter[_num]--; // Decrement the prefix sum by 1 to determine where to place the next occurrence of _num\n  }\n  // Overwrite the original array nums with the result array res\n  nums.setAll(0, res);\n}\n</code></pre> counting_sort.rs<pre><code>/* Counting sort */\n// Stable implementation that can be adapted to sort objects\nfn counting_sort(nums: &amp;mut [i32]) {\n    // 1. Find the maximum array element m\n    let m = *nums.iter().max().unwrap() as usize;\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    let mut counter = vec![0; m + 1];\n    for &amp;num in nums.iter() {\n        counter[num as usize] += 1;\n    }\n    // 3. Compute prefix sums of counter to convert occurrence counts into end positions\n    // That is, counter[num]-1 is the index of the last occurrence of num in res\n    for i in 0..m {\n        counter[i + 1] += counter[i];\n    }\n    // 4. Traverse nums in reverse and place each element in the result array res\n    // Initialize array res to store the result\n    let n = nums.len();\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let num = nums[i];\n        res[counter[num as usize] - 1] = num; // Place num at the corresponding index\n        counter[num as usize] -= 1; // Decrement the prefix sum by 1 to determine where to place the next occurrence of num\n    }\n    // Overwrite the original array nums with the result array res\n    nums.copy_from_slice(&amp;res)\n}\n</code></pre> counting_sort.c<pre><code>/* Counting sort */\n// Full, stable implementation; can be adapted to sort objects\nvoid countingSort(int nums[], int size) {\n    // 1. Find the maximum array element m\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    int *counter = calloc(m, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. Compute prefix sums of counter to convert occurrence counts into end positions\n    // That is, counter[num]-1 is the index of the last occurrence of num in res\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. Traverse nums in reverse and place each element in the result array res\n    // Initialize array res to store the result\n    int *res = malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // Place num at the corresponding index\n        counter[num]--;              // Decrement the prefix sum by 1 to determine where to place the next occurrence of num\n    }\n    // Overwrite the original array nums with the result array res\n    memcpy(nums, res, size * sizeof(int));\n    // 5. Free memory\n    free(res);\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* Counting sort */\n// Full implementation; stable and adaptable to sorting objects\nfun countingSort(nums: IntArray) {\n    // 1. Find the maximum array element m\n    var m = 0\n    for (num in nums) {\n        m = max(m, num)\n    }\n    // 2. Count the occurrences of each number\n    // counter[num] is the number of occurrences of num\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. Compute prefix sums of counter to convert occurrence counts into end positions\n    // That is, counter[num]-1 is the index of the last occurrence of num in res\n    for (i in 0..&lt;m) {\n        counter[i + 1] += counter[i]\n    }\n    // 4. Traverse nums in reverse and place each element in the result array res\n    // Initialize array res to store the result\n    val n = nums.size\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val num = nums[i]\n        res[counter[num] - 1] = num // Place num at the corresponding index\n        counter[num]-- // Decrement the prefix sum by 1 to determine where to place the next occurrence of num\n    }\n    // Overwrite the original array nums with the result array res\n    for (i in 0..&lt;n) {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.rb<pre><code>### Counting sort ###\ndef counting_sort(nums)\n  # Full, stable implementation; the approach can be adapted to sort objects\n  # 1. Find the maximum array element m\n  m = nums.max\n  # 2. Count the occurrences of each number\n  # counter[num] is the number of occurrences of num\n  counter = Array.new(m + 1, 0)\n  nums.each { |num| counter[num] += 1 }\n  # 3. Compute prefix sums of counter to convert occurrence counts into end positions\n  # That is, counter[num]-1 is the index of the last occurrence of num in res\n  (0...m).each { |i| counter[i + 1] += counter[i] }\n  # 4. Traverse nums in reverse and place each element in the result array res\n  # Initialize array res to store the result\n  n = nums.length\n  res = Array.new(n, 0)\n  (n - 1).downto(0).each do |i|\n    num = nums[i]\n    res[counter[num] - 1] = num # Place num at the corresponding index\n    counter[num] -= 1 # Decrement the prefix sum by 1 to determine where to place the next occurrence of num\n  end\n  # Overwrite the original array nums with the result array res\n  (0...n).each { |i| nums[i] = res[i] }\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 11.   Sorting","11.9   Counting sort"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1193-algorithm-properties","level":2,"title":"11.9.3   Algorithm properties","text":"<ul> <li>Time complexity: \\(O(n + m)\\); not adaptive. Traversing <code>nums</code> and <code>counter</code> each takes linear time. Usually \\(n \\gg m\\), so the complexity approaches \\(O(n)\\).</li> <li>Space complexity: \\(O(n + m)\\); not in-place. The algorithm uses arrays <code>res</code> and <code>counter</code> of lengths \\(n\\) and \\(m\\), respectively.</li> <li>Stable. Since elements are placed into <code>res</code> from right to left, traversing <code>nums</code> in reverse preserves the relative order of equal elements. A forward traversal also produces a correctly sorted result, but it is not stable.</li> </ul>","path":["Chapter 11.   Sorting","11.9   Counting sort"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1194-limitations","level":2,"title":"11.9.4   Limitations","text":"<p>Counting sort may seem ingenious: it sorts efficiently simply by counting occurrences. However, its prerequisites are relatively restrictive.</p> <p>Counting sort applies only to nonnegative integers. For other data types, the values must be convertible to nonnegative integers without changing their relative order. For example, if an integer array contains negative numbers, add a constant to every element to make them all positive, sort, and then reverse the transformation.</p> <p>Counting sort suits large datasets with a relatively small value range. In the example above, \\(m\\) must not be too large, or the algorithm will consume excessive space. When \\(n \\ll m\\), counting sort takes \\(O(m)\\) time and may be slower than an \\(O(n \\log n)\\) sorting algorithm.</p>","path":["Chapter 11.   Sorting","11.9   Counting sort"],"tags":[]},{"location":"chapter_sorting/exercises/","level":1,"title":"11.12   Exercises","text":"","path":["Chapter 11.   Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/exercises/#11121-check-your-understanding","level":2,"title":"11.12.1   Check your understanding","text":"","path":["Chapter 11.   Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/exercises/#1-the-first-passes-of-selection-sort-and-bubble-sort","level":3,"title":"1.   The first passes of selection sort and bubble sort","text":"<p>Given the array <code>[4, 2, 5, 1, 3]</code>, sort in ascending order throughout this exercise.</p> <ol> <li>Simulate the first two passes of selection sort. Write the array after each pass and identify the positions whose final values are now fixed.</li> <li>Simulate the first pass of bubble sort. Write the array and the number of swaps, and identify the position whose final value is now fixed.</li> </ol> Reference answer <ol> <li> <p>The first two passes are:</p> Pass Array Explanation 1 <code>[1, 2, 5, 4, 3]</code> Swap the minimum element, 1, with the first element 2 <code>[1, 2, 5, 4, 3]</code> The value 2 is already at index 1, so no swap is needed <p>The first two positions are now fixed. Subsequent passes need only continue selecting the minimum from <code>[5, 4, 3]</code>.</p> </li> <li> <p>Compare adjacent elements in order: swap 4 and 2, leave 4 and 5 unchanged, swap 5 and 1, then swap 5 and 3.     The result is <code>[2, 4, 1, 3, 5]</code>, with three swaps. The maximum element, 5, has moved to the end, fixing the last position.</p> </li> </ol>","path":["Chapter 11.   Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/exercises/#2-does-the-order-of-equal-elements-change","level":3,"title":"2.   Does the order of equal elements change?","text":"<p>In the array \\([2_a, 2_b, 1]\\), \\(2_a\\) and \\(2_b\\) have equal values; the subscripts mark their original order.</p> <ol> <li>Write the array after the first pass of selection sort. Has the relative order of \\(2_a\\) and \\(2_b\\) changed?</li> <li>Write the array after the first pass of bubble sort. Has the relative order of \\(2_a\\) and \\(2_b\\) changed?</li> <li>Based on your answers, explain how the two algorithms differ in preserving the original order of equal elements.</li> </ol> Reference answer <ol> <li> <p>The first pass of selection sort selects the minimum element, 1, and swaps it with the first element, \\(2_a\\), producing     \\([1, 2_b, 2_a]\\). The swap moves \\(2_a\\) after \\(2_b\\), changing their relative order.</p> </li> <li> <p>Bubble sort first compares \\(2_a\\) with \\(2_b\\) and does not swap them because they are equal. It then compares and swaps \\(2_b\\) and 1.     After the first pass, the array is \\([2_a, 1, 2_b]\\). Since \\(2_a\\) still precedes \\(2_b\\), their relative order is unchanged.</p> </li> <li> <p>In this example, selection sort changes the original order of equal elements. Bubble sort swaps adjacent elements only when the left element     is greater than the right one. Equal elements are never swapped with each other, so their original order is preserved.</p> </li> </ol>","path":["Chapter 11.   Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/exercises/#3-comparing-counting-sort-and-radix-sort","level":3,"title":"3.   Comparing counting sort and radix sort","text":"<p>A school needs to sort many student IDs, each exactly eight digits long. Answer the following:</p> <ol> <li>How many passes does radix sort need when starting with the least significant digit?</li> <li>If counting sort treats each student ID directly as an integer, why would many counting entries go unused?</li> <li>Based on your answers, which would you choose for sorting many eight-digit student IDs: counting sort or radix sort? Why?</li> </ol> Reference answer <ol> <li> <p>Each ID has eight digits, so there are eight passes, from the least to the most significant digit. Each pass groups digits only by values 0 through 9.</p> </li> <li> <p>Direct counting reserves an entry for every possible eight-digit value, but actual students use only a small fraction of them.     Most counts would remain 0.</p> </li> <li> <p>Choose radix sort. It uses the fixed number of digits and the ten possible values per digit, requiring only eight stable grouping passes.     Counting sort using each entire eight-digit ID as an integer index would reserve entries for many values that never occur.</p> </li> </ol>","path":["Chapter 11.   Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/exercises/#11122-programming-exercises","level":2,"title":"11.12.2   Programming exercises","text":"","path":["Chapter 11.   Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/exercises/#1-sorting-an-array-with-merge-sort","level":3,"title":"1.   Sorting an array with merge sort","text":"<p>Given an integer array <code>nums</code>, implement merge sort yourself to arrange its elements in nondecreasing order and return the result. Do not call the language's built-in sorting function.</p> Hints <ol> <li>An interval of length at most 1 is already sorted.</li> <li>Split the interval in half at the midpoint and recursively sort each half.</li> <li>Use two pointers to merge the sorted halves, then write the result back to the original array.</li> </ol> <p>LeetCode</p>","path":["Chapter 11.   Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/exercises/#2-sorting-an-integer-array-with-counting-sort","level":3,"title":"2.   Sorting an integer array with counting sort","text":"<p>You are given an integer array <code>nums</code> and a nonnegative integer \\(K\\). Every element lies between \\(0\\) and \\(K\\).</p> <p>Implement counting sort, write the elements back into <code>nums</code> in nondecreasing order, and return <code>nums</code>. Do not determine the order by comparing elements with one another, and do not use the language's built-in sorting function.</p> Hints <ol> <li>Since every element is between 0 and K, its value can directly index the counting array.</li> <li>First scan nums, incrementing the count at the corresponding position by 1.</li> <li>Then scan the counting array from 0 through K. If x occurs a given number of times, write that many consecutive copies of x into nums.</li> </ol>","path":["Chapter 11.   Sorting","11.12   Exercises"],"tags":[]},{"location":"chapter_sorting/heap_sort/","level":1,"title":"11.7   Heap sort","text":"<p>Tip</p> <p>Before reading this section, make sure you have completed the chapter on heaps.</p> <p>Heap sort is an efficient sorting algorithm built on the heap data structure. We can implement it with the heap-building and removal operations we have already learned.</p> <ol> <li>Build a min heap from the input array. The smallest element is now at the root.</li> <li>Repeatedly remove the root and record the removed elements to obtain an ascending sequence.</li> </ol> <p>Although this works, it requires an extra array to store the removed elements, wasting space. In practice, we usually use a more elegant implementation.</p>","path":["Chapter 11.   Sorting","11.7   Heap sort"],"tags":[]},{"location":"chapter_sorting/heap_sort/#1171-algorithm-steps","level":2,"title":"11.7.1   Algorithm steps","text":"<p>For an array of length \\(n\\), heap sort proceeds as shown below.</p> <ol> <li>Build a max heap from the input array. The largest element is now at the root.</li> <li>Swap the root, the first element, with the last element of the heap. Reduce the heap's length by \\(1\\) and increase the number of sorted elements by \\(1\\).</li> <li>Starting at the root, sift down to restore the heap property.</li> <li>Repeat steps <code>2.</code> and <code>3.</code>. After \\(n - 1\\) rounds, the array is sorted.</li> </ol> <p>Tip</p> <p>In fact, removing the root of a heap also includes steps <code>2.</code> and <code>3.</code>, with the additional step of removing an element from the array.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 11-12   Steps of heap sort </p> <p>The implementation uses the same <code>sift_down()</code> function introduced in the heap chapter. Because the heap shrinks as maximum elements are extracted, add a length parameter \\(n\\) to specify its current valid length. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby heap_sort.py<pre><code>def sift_down(nums: list[int], n: int, i: int):\n    \"\"\"Sift down from node i in a heap of size n\"\"\"\n    while True:\n        # Find the node with the largest value among i, l, and r, and store its index in ma\n        l = 2 * i + 1\n        r = 2 * i + 2\n        ma = i\n        if l &lt; n and nums[l] &gt; nums[ma]:\n            ma = l\n        if r &lt; n and nums[r] &gt; nums[ma]:\n            ma = r\n        # Stop if node i is largest or both child indices l and r are out of bounds\n        if ma == i:\n            break\n        # Swap the two nodes\n        nums[i], nums[ma] = nums[ma], nums[i]\n        # Continue sifting down\n        i = ma\n\ndef heap_sort(nums: list[int]):\n    \"\"\"Heap sort\"\"\"\n    # Build the heap by sifting down from every non-leaf node\n    for i in range(len(nums) // 2 - 1, -1, -1):\n        sift_down(nums, len(nums), i)\n    # Extract the largest element from the heap, repeating for n-1 rounds\n    for i in range(len(nums) - 1, 0, -1):\n        # Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)\n        nums[0], nums[i] = nums[i], nums[0]\n        # Sift down starting at the root\n        sift_down(nums, i, 0)\n</code></pre> heap_sort.cpp<pre><code>/* Sift down from node i in a heap of size n */\nvoid siftDown(vector&lt;int&gt; &amp;nums, int n, int i) {\n    while (true) {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if (ma == i) {\n            break;\n        }\n        // Swap the two nodes\n        swap(nums[i], nums[ma]);\n        // Continue sifting down\n        i = ma;\n    }\n}\n\n/* Heap sort */\nvoid heapSort(vector&lt;int&gt; &amp;nums) {\n    // Build the heap by sifting down from every non-leaf node\n    for (int i = nums.size() / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, nums.size(), i);\n    }\n    // Extract the largest element from the heap, repeating for n-1 rounds\n    for (int i = nums.size() - 1; i &gt; 0; --i) {\n        // Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)\n        swap(nums[0], nums[i]);\n        // Sift down starting at the root\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.java<pre><code>/* Sift down from node i in a heap of size n */\nvoid siftDown(int[] nums, int n, int i) {\n    while (true) {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if (ma == i)\n            break;\n        // Swap the two nodes\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // Continue sifting down\n        i = ma;\n    }\n}\n\n/* Heap sort */\nvoid heapSort(int[] nums) {\n    // Build the heap by sifting down from every non-leaf node\n    for (int i = nums.length / 2 - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // Extract the largest element from the heap, repeating for n-1 rounds\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // Sift down starting at the root\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.cs<pre><code>/* Sift down from node i in a heap of size n */\nvoid SiftDown(int[] nums, int n, int i) {\n    while (true) {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if (ma == i)\n            break;\n        // Swap the two nodes\n        (nums[ma], nums[i]) = (nums[i], nums[ma]);\n        // Continue sifting down\n        i = ma;\n    }\n}\n\n/* Heap sort */\nvoid HeapSort(int[] nums) {\n    // Build the heap by sifting down from every non-leaf node\n    for (int i = nums.Length / 2 - 1; i &gt;= 0; i--) {\n        SiftDown(nums, nums.Length, i);\n    }\n    // Extract the largest element from the heap, repeating for n-1 rounds\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)\n        (nums[i], nums[0]) = (nums[0], nums[i]);\n        // Sift down starting at the root\n        SiftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.go<pre><code>/* Sift down from node i in a heap of size n */\nfunc siftDown(nums *[]int, n, i int) {\n    for true {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        l := 2*i + 1\n        r := 2*i + 2\n        ma := i\n        if l &lt; n &amp;&amp; (*nums)[l] &gt; (*nums)[ma] {\n            ma = l\n        }\n        if r &lt; n &amp;&amp; (*nums)[r] &gt; (*nums)[ma] {\n            ma = r\n        }\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if ma == i {\n            break\n        }\n        // Swap the two nodes\n        (*nums)[i], (*nums)[ma] = (*nums)[ma], (*nums)[i]\n        // Continue sifting down\n        i = ma\n    }\n}\n\n/* Heap sort */\nfunc heapSort(nums *[]int) {\n    // Build the heap by sifting down from every non-leaf node\n    for i := len(*nums)/2 - 1; i &gt;= 0; i-- {\n        siftDown(nums, len(*nums), i)\n    }\n    // Extract the largest element from the heap, repeating for n-1 rounds\n    for i := len(*nums) - 1; i &gt; 0; i-- {\n        // Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)\n        (*nums)[0], (*nums)[i] = (*nums)[i], (*nums)[0]\n        // Sift down starting at the root\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.swift<pre><code>/* Sift down from node i in a heap of size n */\nfunc siftDown(nums: inout [Int], n: Int, i: Int) {\n    var i = i\n    while true {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        let l = 2 * i + 1\n        let r = 2 * i + 2\n        var ma = i\n        if l &lt; n, nums[l] &gt; nums[ma] {\n            ma = l\n        }\n        if r &lt; n, nums[r] &gt; nums[ma] {\n            ma = r\n        }\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if ma == i {\n            break\n        }\n        // Swap the two nodes\n        nums.swapAt(i, ma)\n        // Continue sifting down\n        i = ma\n    }\n}\n\n/* Heap sort */\nfunc heapSort(nums: inout [Int]) {\n    // Build the heap by sifting down from every non-leaf node\n    for i in stride(from: nums.count / 2 - 1, through: 0, by: -1) {\n        siftDown(nums: &amp;nums, n: nums.count, i: i)\n    }\n    // Extract the largest element from the heap, repeating for n-1 rounds\n    for i in nums.indices.dropFirst().reversed() {\n        // Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)\n        nums.swapAt(0, i)\n        // Sift down starting at the root\n        siftDown(nums: &amp;nums, n: i, i: 0)\n    }\n}\n</code></pre> heap_sort.js<pre><code>/* Sift down from node i in a heap of size n */\nfunction siftDown(nums, n, i) {\n    while (true) {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if (ma === i) {\n            break;\n        }\n        // Swap the two nodes\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // Continue sifting down\n        i = ma;\n    }\n}\n\n/* Heap sort */\nfunction heapSort(nums) {\n    // Build the heap by sifting down from every non-leaf node\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // Extract the largest element from the heap, repeating for n-1 rounds\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // Sift down starting at the root\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.ts<pre><code>/* Sift down from node i in a heap of size n */\nfunction siftDown(nums: number[], n: number, i: number): void {\n    while (true) {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if (ma === i) {\n            break;\n        }\n        // Swap the two nodes\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // Continue sifting down\n        i = ma;\n    }\n}\n\n/* Heap sort */\nfunction heapSort(nums: number[]): void {\n    // Build the heap by sifting down from every non-leaf node\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // Extract the largest element from the heap, repeating for n-1 rounds\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // Sift down starting at the root\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.dart<pre><code>/* Sift down from node i in a heap of size n */\nvoid siftDown(List&lt;int&gt; nums, int n, int i) {\n  while (true) {\n    // Find the node with the largest value among i, l, and r, and store its index in ma\n    int l = 2 * i + 1;\n    int r = 2 * i + 2;\n    int ma = i;\n    if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) ma = l;\n    if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) ma = r;\n    // Stop if node i is largest or both child indices l and r are out of bounds\n    if (ma == i) break;\n    // Swap the two nodes\n    int temp = nums[i];\n    nums[i] = nums[ma];\n    nums[ma] = temp;\n    // Continue sifting down\n    i = ma;\n  }\n}\n\n/* Heap sort */\nvoid heapSort(List&lt;int&gt; nums) {\n  // Build the heap by sifting down from every non-leaf node\n  for (int i = nums.length ~/ 2 - 1; i &gt;= 0; i--) {\n    siftDown(nums, nums.length, i);\n  }\n  // Extract the largest element from the heap, repeating for n-1 rounds\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)\n    int tmp = nums[0];\n    nums[0] = nums[i];\n    nums[i] = tmp;\n    // Sift down starting at the root\n    siftDown(nums, i, 0);\n  }\n}\n</code></pre> heap_sort.rs<pre><code>/* Sift down from node i in a heap of size n */\nfn sift_down(nums: &amp;mut [i32], n: usize, mut i: usize) {\n    loop {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let mut ma = i;\n        if l &lt; n &amp;&amp; nums[l] &gt; nums[ma] {\n            ma = l;\n        }\n        if r &lt; n &amp;&amp; nums[r] &gt; nums[ma] {\n            ma = r;\n        }\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if ma == i {\n            break;\n        }\n        // Swap the two nodes\n        nums.swap(i, ma);\n        // Continue sifting down\n        i = ma;\n    }\n}\n\n/* Heap sort */\nfn heap_sort(nums: &amp;mut [i32]) {\n    // Build the heap by sifting down from every non-leaf node\n    for i in (0..nums.len() / 2).rev() {\n        sift_down(nums, nums.len(), i);\n    }\n    // Extract the largest element from the heap, repeating for n-1 rounds\n    for i in (1..nums.len()).rev() {\n        // Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)\n        nums.swap(0, i);\n        // Sift down starting at the root\n        sift_down(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.c<pre><code>/* Sift down from node i in a heap of size n */\nvoid siftDown(int nums[], int n, int i) {\n    while (1) {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if (ma == i) {\n            break;\n        }\n        // Swap the two nodes\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // Continue sifting down\n        i = ma;\n    }\n}\n\n/* Heap sort */\nvoid heapSort(int nums[], int n) {\n    // Build the heap by sifting down from every non-leaf node\n    for (int i = n / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, n, i);\n    }\n    // Extract the largest element from the heap, repeating for n-1 rounds\n    for (int i = n - 1; i &gt; 0; --i) {\n        // Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // Sift down starting at the root\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.kt<pre><code>/* Sift down from node i in a heap of size n */\nfun siftDown(nums: IntArray, n: Int, li: Int) {\n    var i = li\n    while (true) {\n        // Find the node with the largest value among i, l, and r, and store its index in ma\n        val l = 2 * i + 1\n        val r = 2 * i + 2\n        var ma = i\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) \n            ma = l\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) \n            ma = r\n        // Stop if node i is largest or both child indices l and r are out of bounds\n        if (ma == i) \n            break\n        // Swap the two nodes\n        val temp = nums[i]\n        nums[i] = nums[ma]\n        nums[ma] = temp\n        // Continue sifting down\n        i = ma\n    }\n}\n\n/* Heap sort */\nfun heapSort(nums: IntArray) {\n    // Build the heap by sifting down from every non-leaf node\n    for (i in nums.size / 2 - 1 downTo 0) {\n        siftDown(nums, nums.size, i)\n    }\n    // Extract the largest element from the heap, repeating for n-1 rounds\n    for (i in nums.size - 1 downTo 1) {\n        // Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)\n        val temp = nums[0]\n        nums[0] = nums[i]\n        nums[i] = temp\n        // Sift down starting at the root\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.rb<pre><code>### Sift down from node i in a heap of size n ###\ndef sift_down(nums, n, i)\n  while true\n    # Find the node with the largest value among i, l, and r, and store its index in ma\n    l = 2 * i + 1\n    r = 2 * i + 2\n    ma = i\n    ma = l if l &lt; n &amp;&amp; nums[l] &gt; nums[ma]\n    ma = r if r &lt; n &amp;&amp; nums[r] &gt; nums[ma]\n    # Stop if node i is largest or both child indices l and r are out of bounds\n    break if ma == i\n    # Swap the two nodes\n    nums[i], nums[ma] = nums[ma], nums[i]\n    # Continue sifting down\n    i = ma\n  end\nend\n\n### Heap sort ###\ndef heap_sort(nums)\n  # Build the heap by sifting down from every non-leaf node\n  (nums.length / 2 - 1).downto(0) do |i|\n    sift_down(nums, nums.length, i)\n  end\n  # Extract the largest element from the heap, repeating for n-1 rounds\n  (nums.length - 1).downto(1) do |i|\n    # Swap the root with the rightmost leaf on the last level (the first and last elements of the current heap)\n    nums[0], nums[i] = nums[i], nums[0]\n    # Sift down starting at the root\n    sift_down(nums, i, 0)\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 11.   Sorting","11.7   Heap sort"],"tags":[]},{"location":"chapter_sorting/heap_sort/#1172-algorithm-properties","level":2,"title":"11.7.2   Algorithm properties","text":"<ul> <li>Time complexity: \\(O(n \\log n)\\); not adaptive. Building the heap takes \\(O(n)\\) time. Extracting its maximum takes \\(O(\\log n)\\) time and is repeated \\(n - 1\\) times.</li> <li>Space complexity: \\(O(1)\\); in-place. A few pointer variables use \\(O(1)\\) space. Swapping and sifting take place within the original array.</li> <li>Not stable. Swapping the root with the last element of the heap may change the relative order of equal elements.</li> </ul>","path":["Chapter 11.   Sorting","11.7   Heap sort"],"tags":[]},{"location":"chapter_sorting/insertion_sort/","level":1,"title":"11.4   Insertion sort","text":"<p>Insertion sort is a simple sorting algorithm that works much like arranging a hand of playing cards.</p> <p>Choose a base element from the unsorted section, compare it with the elements in the sorted section to its left, and insert it into the correct position.</p> <p>Figure 11-6 shows how to insert an element into an array. Let the base element be <code>base</code>. Shift all elements between the target index and <code>base</code> one position to the right, then assign <code>base</code> to the target index.</p> <p></p> <p> Figure 11-6   A single insertion operation </p>","path":["Chapter 11.   Sorting","11.4   Insertion sort"],"tags":[]},{"location":"chapter_sorting/insertion_sort/#1141-algorithm-steps","level":2,"title":"11.4.1   Algorithm steps","text":"<p>The overall insertion sort process is shown below.</p> <ol> <li>Initially, the first element of the array is treated as sorted.</li> <li>Select the second element as <code>base</code> and insert it into the correct position. The first two elements are now sorted.</li> <li>Select the third element as <code>base</code> and insert it into the correct position. The first three elements are now sorted.</li> <li>Continue in this way. In the final round, select the last element as <code>base</code> and insert it into the correct position. All elements are now sorted.</li> </ol> <p></p> <p> Figure 11-7   The insertion sort process </p> <p>An implementation follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby insertion_sort.py<pre><code>def insertion_sort(nums: list[int]):\n    \"\"\"Insertion sort\"\"\"\n    # Outer loop: the sorted range is [0, i-1]\n    for i in range(1, len(nums)):\n        base = nums[i]\n        j = i - 1\n        # Inner loop: insert base at the correct position in the sorted range [0, i-1]\n        while j &gt;= 0 and nums[j] &gt; base:\n            nums[j + 1] = nums[j]  # Shift nums[j] one position to the right\n            j -= 1\n        nums[j + 1] = base  # Place base at the correct position\n</code></pre> insertion_sort.cpp<pre><code>/* Insertion sort */\nvoid insertionSort(vector&lt;int&gt; &amp;nums) {\n    // Outer loop: the sorted range is [0, i-1]\n    for (int i = 1; i &lt; nums.size(); i++) {\n        int base = nums[i], j = i - 1;\n        // Inner loop: insert base at the correct position in the sorted range [0, i-1]\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // Shift nums[j] one position to the right\n            j--;\n        }\n        nums[j + 1] = base; // Place base at the correct position\n    }\n}\n</code></pre> insertion_sort.java<pre><code>/* Insertion sort */\nvoid insertionSort(int[] nums) {\n    // Outer loop: the sorted range is [0, i-1]\n    for (int i = 1; i &lt; nums.length; i++) {\n        int base = nums[i], j = i - 1;\n        // Inner loop: insert base at the correct position in the sorted range [0, i-1]\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // Shift nums[j] one position to the right\n            j--;\n        }\n        nums[j + 1] = base;        // Place base at the correct position\n    }\n}\n</code></pre> insertion_sort.cs<pre><code>/* Insertion sort */\nvoid InsertionSort(int[] nums) {\n    // Outer loop: the sorted range is [0, i-1]\n    for (int i = 1; i &lt; nums.Length; i++) {\n        int bas = nums[i], j = i - 1;\n        // Inner loop: insert bas at the correct position in the sorted range [0, i-1]\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; bas) {\n            nums[j + 1] = nums[j]; // Shift nums[j] one position to the right\n            j--;\n        }\n        nums[j + 1] = bas;         // Place bas at the correct position\n    }\n}\n</code></pre> insertion_sort.go<pre><code>/* Insertion sort */\nfunc insertionSort(nums []int) {\n    // Outer loop: the sorted range is [0, i-1]\n    for i := 1; i &lt; len(nums); i++ {\n        base := nums[i]\n        j := i - 1\n        // Inner loop: insert base at the correct position in the sorted range [0, i-1]\n        for j &gt;= 0 &amp;&amp; nums[j] &gt; base {\n            nums[j+1] = nums[j] // Shift nums[j] one position to the right\n            j--\n        }\n        nums[j+1] = base // Place base at the correct position\n    }\n}\n</code></pre> insertion_sort.swift<pre><code>/* Insertion sort */\nfunc insertionSort(nums: inout [Int]) {\n    // Outer loop: the sorted range is [0, i-1]\n    for i in nums.indices.dropFirst() {\n        let base = nums[i]\n        var j = i - 1\n        // Inner loop: insert base at the correct position in the sorted range [0, i-1]\n        while j &gt;= 0, nums[j] &gt; base {\n            nums[j + 1] = nums[j] // Shift nums[j] one position to the right\n            j -= 1\n        }\n        nums[j + 1] = base // Place base at the correct position\n    }\n}\n</code></pre> insertion_sort.js<pre><code>/* Insertion sort */\nfunction insertionSort(nums) {\n    // Outer loop: the sorted range is [0, i-1]\n    for (let i = 1; i &lt; nums.length; i++) {\n        let base = nums[i],\n            j = i - 1;\n        // Inner loop: insert base at the correct position in the sorted range [0, i-1]\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // Shift nums[j] one position to the right\n            j--;\n        }\n        nums[j + 1] = base; // Place base at the correct position\n    }\n}\n</code></pre> insertion_sort.ts<pre><code>/* Insertion sort */\nfunction insertionSort(nums: number[]): void {\n    // Outer loop: the sorted range is [0, i-1]\n    for (let i = 1; i &lt; nums.length; i++) {\n        const base = nums[i];\n        let j = i - 1;\n        // Inner loop: insert base at the correct position in the sorted range [0, i-1]\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // Shift nums[j] one position to the right\n            j--;\n        }\n        nums[j + 1] = base; // Place base at the correct position\n    }\n}\n</code></pre> insertion_sort.dart<pre><code>/* Insertion sort */\nvoid insertionSort(List&lt;int&gt; nums) {\n  // Outer loop: the sorted range is [0, i-1]\n  for (int i = 1; i &lt; nums.length; i++) {\n    int base = nums[i], j = i - 1;\n    // Inner loop: insert base at the correct position in the sorted range [0, i-1]\n    while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n      nums[j + 1] = nums[j]; // Shift nums[j] one position to the right\n      j--;\n    }\n    nums[j + 1] = base; // Place base at the correct position\n  }\n}\n</code></pre> insertion_sort.rs<pre><code>/* Insertion sort */\nfn insertion_sort(nums: &amp;mut [i32]) {\n    // Outer loop: the sorted range is [0, i-1]\n    for i in 1..nums.len() {\n        let (base, mut j) = (nums[i], (i - 1) as i32);\n        // Inner loop: insert base at the correct position in the sorted range [0, i-1]\n        while j &gt;= 0 &amp;&amp; nums[j as usize] &gt; base {\n            nums[(j + 1) as usize] = nums[j as usize]; // Shift nums[j] one position to the right\n            j -= 1;\n        }\n        nums[(j + 1) as usize] = base; // Place base at the correct position\n    }\n}\n</code></pre> insertion_sort.c<pre><code>/* Insertion sort */\nvoid insertionSort(int nums[], int size) {\n    // Outer loop: the sorted range is [0, i-1]\n    for (int i = 1; i &lt; size; i++) {\n        int base = nums[i], j = i - 1;\n        // Inner loop: insert base at the correct position in the sorted range [0, i-1]\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            // Shift nums[j] one position to the right\n            nums[j + 1] = nums[j];\n            j--;\n        }\n        // Place base at the correct position\n        nums[j + 1] = base;\n    }\n}\n</code></pre> insertion_sort.kt<pre><code>/* Insertion sort */\nfun insertionSort(nums: IntArray) {\n    //Outer loop: the sorted prefix grows to 1, 2, ..., n elements\n    for (i in nums.indices) {\n        val base = nums[i]\n        var j = i - 1\n        // Inner loop: insert base at the correct position in the sorted range [0, i-1]\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j] // Shift nums[j] one position to the right\n            j--\n        }\n        nums[j + 1] = base        // Place base at the correct position\n    }\n}\n</code></pre> insertion_sort.rb<pre><code>### Insertion sort ###\ndef insertion_sort(nums)\n  n = nums.length\n  # Outer loop: the sorted range is [0, i-1]\n  for i in 1...n\n    base = nums[i]\n    j = i - 1\n    # Inner loop: insert base at the correct position in the sorted range [0, i-1]\n    while j &gt;= 0 &amp;&amp; nums[j] &gt; base\n      nums[j + 1] = nums[j] # Shift nums[j] one position to the right\n      j -= 1\n    end\n    nums[j + 1] = base # Place base at the correct position\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 11.   Sorting","11.4   Insertion sort"],"tags":[]},{"location":"chapter_sorting/insertion_sort/#1142-algorithm-properties","level":2,"title":"11.4.2   Algorithm properties","text":"<ul> <li>Time complexity: \\(O(n^2)\\); adaptive. In the worst case, the insertion operations require \\(n - 1\\), \\(n-2\\), \\(\\dots\\), \\(2\\), and \\(1\\) iterations, totaling \\((n - 1) n / 2\\), so the time complexity is \\(O(n^2)\\). For ordered data, an insertion can stop early. A fully sorted input achieves the best-case time complexity of \\(O(n)\\).</li> <li>Space complexity: \\(O(1)\\); in-place. Pointers \\(i\\) and \\(j\\) use constant extra space.</li> <li>Stable. Each element is inserted to the right of equal elements, preserving their order.</li> </ul>","path":["Chapter 11.   Sorting","11.4   Insertion sort"],"tags":[]},{"location":"chapter_sorting/insertion_sort/#1143-why-use-insertion-sort","level":2,"title":"11.4.3   Why use insertion sort?","text":"<p>Insertion sort takes \\(O(n^2)\\) time, while quick sort, which we will study shortly, takes \\(O(n \\log n)\\). Despite its higher time complexity, insertion sort is usually faster for small datasets.</p> <p>This resembles the comparison between linear and binary search. Algorithms such as quick sort, with \\(O(n \\log n)\\) time complexity, use divide and conquer and tend to involve more elementary operations. When the dataset is small, \\(n^2\\) and \\(n \\log n\\) are relatively close, so asymptotic complexity is not the dominant factor; the number of elementary operations per round matters more.</p> <p>In fact, the built-in sorting functions of many programming languages, such as Java, use insertion sort. The general approach is to use a divide-and-conquer algorithm such as quick sort for long arrays and insertion sort directly for short ones.</p> <p>Although bubble sort, selection sort, and insertion sort all take \\(O(n^2)\\) time, insertion sort is used much more often in practice, mainly for the following reasons.</p> <ul> <li>Bubble sort swaps elements, using a temporary variable and three elementary operations. Insertion sort shifts elements by assignment, requiring just one operation. Bubble sort therefore usually has higher computational overhead.</li> <li>Selection sort takes \\(O(n^2)\\) time in every case. For partially sorted data, insertion sort is usually more efficient than selection sort.</li> <li>Selection sort is not stable and therefore cannot be used for multilevel sorting.</li> </ul>","path":["Chapter 11.   Sorting","11.4   Insertion sort"],"tags":[]},{"location":"chapter_sorting/merge_sort/","level":1,"title":"11.6   Merge sort","text":"<p>Merge sort is a divide-and-conquer sorting algorithm with the “division” and “merging” phases shown below.</p> <ol> <li>Division: recursively split the array at its midpoint, turning the problem of sorting a long array into problems of sorting shorter arrays.</li> <li>Merging: stop dividing when a subarray has length 1, then repeatedly merge two shorter sorted arrays into one longer sorted array until the process is complete.</li> </ol> <p></p> <p> Figure 11-10   The division and merging phases of merge sort </p>","path":["Chapter 11.   Sorting","11.6   Merge sort"],"tags":[]},{"location":"chapter_sorting/merge_sort/#1161-algorithm-steps","level":2,"title":"11.6.1   Algorithm steps","text":"<p>As shown below, the division phase works from the top down, recursively splitting each array into two subarrays at its midpoint.</p> <ol> <li>Calculate the midpoint <code>mid</code>, then recursively divide the left subarray, <code>[left, mid]</code>, and the right subarray, <code>[mid + 1, right]</code>.</li> <li>Repeat step <code>1.</code> recursively until each subarray has length 1.</li> </ol> <p>The merging phase works from the bottom up, combining the left and right subarrays into a sorted array. Merging begins with subarrays of length 1, so every subarray involved in this phase is already sorted.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 11-11   Steps of merge sort </p> <p>Merge sort follows the same recursive order as a postorder traversal of a binary tree.</p> <ul> <li>Postorder traversal: recurse into the left subtree, then the right subtree, and finally process the root.</li> <li>Merge sort: recurse into the left subarray, then the right subarray, and finally merge them.</li> </ul> <p>The implementation follows. The interval to merge in <code>nums</code> is <code>[left, right]</code>, while its corresponding interval in <code>tmp</code> is <code>[0, right - left]</code>.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby merge_sort.py<pre><code>def merge(nums: list[int], left: int, mid: int, right: int):\n    \"\"\"Merge the left and right subarrays\"\"\"\n    # The left subarray spans [left, mid], and the right subarray spans [mid+1, right]\n    # Create a temporary array tmp to store the merged result\n    tmp = [0] * (right - left + 1)\n    # Initialize the starting indices of the left and right subarrays\n    i, j, k = left, mid + 1, 0\n    # While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array\n    while i &lt;= mid and j &lt;= right:\n        if nums[i] &lt;= nums[j]:\n            tmp[k] = nums[i]\n            i += 1\n        else:\n            tmp[k] = nums[j]\n            j += 1\n        k += 1\n    # Copy the remaining elements of the left and right subarrays into the temporary array\n    while i &lt;= mid:\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    while j &lt;= right:\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    # Copy the elements of temporary array tmp back into the corresponding range of the original array nums\n    for k in range(0, len(tmp)):\n        nums[left + k] = tmp[k]\n\ndef merge_sort(nums: list[int], left: int, right: int):\n    \"\"\"Merge sort\"\"\"\n    # Base case\n    if left &gt;= right:\n        return  # Stop recursing when the subarray has at most one element\n    # Divide phase\n    mid = (left + right) // 2 # Calculate the midpoint\n    merge_sort(nums, left, mid)  # Recurse on the left subarray\n    merge_sort(nums, mid + 1, right)  # Recurse on the right subarray\n    # Merge phase\n    merge(nums, left, mid, right)\n</code></pre> merge_sort.cpp<pre><code>/* Merge the left and right subarrays */\nvoid merge(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    // The left subarray spans [left, mid], and the right subarray spans [mid+1, right]\n    // Create a temporary array tmp to store the merged result\n    vector&lt;int&gt; tmp(right - left + 1);\n    // Initialize the starting indices of the left and right subarrays\n    int i = left, j = mid + 1, k = 0;\n    // While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // Copy the elements of temporary array tmp back into the corresponding range of the original array nums\n    for (k = 0; k &lt; tmp.size(); k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* Merge sort */\nvoid mergeSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // Base case\n    if (left &gt;= right)\n        return; // Stop recursing when the subarray has at most one element\n    // Divide phase\n    int mid = left + (right - left) / 2;    // Calculate the midpoint\n    mergeSort(nums, left, mid);      // Recurse on the left subarray\n    mergeSort(nums, mid + 1, right); // Recurse on the right subarray\n    // Merge phase\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.java<pre><code>/* Merge the left and right subarrays */\nvoid merge(int[] nums, int left, int mid, int right) {\n    // The left subarray spans [left, mid], and the right subarray spans [mid+1, right]\n    // Create a temporary array tmp to store the merged result\n    int[] tmp = new int[right - left + 1];\n    // Initialize the starting indices of the left and right subarrays\n    int i = left, j = mid + 1, k = 0;\n    // While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // Copy the elements of temporary array tmp back into the corresponding range of the original array nums\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* Merge sort */\nvoid mergeSort(int[] nums, int left, int right) {\n    // Base case\n    if (left &gt;= right)\n        return; // Stop recursing when the subarray has at most one element\n    // Divide phase\n    int mid = left + (right - left) / 2; // Calculate the midpoint\n    mergeSort(nums, left, mid); // Recurse on the left subarray\n    mergeSort(nums, mid + 1, right); // Recurse on the right subarray\n    // Merge phase\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.cs<pre><code>/* Merge the left and right subarrays */\nvoid Merge(int[] nums, int left, int mid, int right) {\n    // The left subarray spans [left, mid], and the right subarray spans [mid+1, right]\n    // Create a temporary array tmp to store the merged result\n    int[] tmp = new int[right - left + 1];\n    // Initialize the starting indices of the left and right subarrays\n    int i = left, j = mid + 1, k = 0;\n    // While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // Copy the elements of temporary array tmp back into the corresponding range of the original array nums\n    for (k = 0; k &lt; tmp.Length; ++k) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* Merge sort */\nvoid MergeSort(int[] nums, int left, int right) {\n    // Base case\n    if (left &gt;= right) return;       // Stop recursing when the subarray has at most one element\n    // Divide phase\n    int mid = left + (right - left) / 2;    // Calculate the midpoint\n    MergeSort(nums, left, mid);      // Recurse on the left subarray\n    MergeSort(nums, mid + 1, right); // Recurse on the right subarray\n    // Merge phase\n    Merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.go<pre><code>/* Merge the left and right subarrays */\nfunc merge(nums []int, left, mid, right int) {\n    // The left subarray spans [left, mid], and the right subarray spans [mid+1, right]\n    // Create a temporary array tmp to store the merged result\n    tmp := make([]int, right-left+1)\n    // Initialize the starting indices of the left and right subarrays\n    i, j, k := left, mid+1, 0\n    // While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array\n    for i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i++\n        } else {\n            tmp[k] = nums[j]\n            j++\n        }\n        k++\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    for i &lt;= mid {\n        tmp[k] = nums[i]\n        i++\n        k++\n    }\n    for j &lt;= right {\n        tmp[k] = nums[j]\n        j++\n        k++\n    }\n    // Copy the elements of temporary array tmp back into the corresponding range of the original array nums\n    for k := 0; k &lt; len(tmp); k++ {\n        nums[left+k] = tmp[k]\n    }\n}\n\n/* Merge sort */\nfunc mergeSort(nums []int, left, right int) {\n    // Base case\n    if left &gt;= right {\n        return\n    }\n    // Divide phase\n    mid := left + (right - left) / 2\n    mergeSort(nums, left, mid)\n    mergeSort(nums, mid+1, right)\n    // Merge phase\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.swift<pre><code>/* Merge the left and right subarrays */\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n    // The left subarray spans [left, mid], and the right subarray spans [mid+1, right]\n    // Create a temporary array tmp to store the merged result\n    var tmp = Array(repeating: 0, count: right - left + 1)\n    // Initialize the starting indices of the left and right subarrays\n    var i = left, j = mid + 1, k = 0\n    // While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array\n    while i &lt;= mid, j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i += 1\n        } else {\n            tmp[k] = nums[j]\n            j += 1\n        }\n        k += 1\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while i &lt;= mid {\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    }\n    // Copy the elements of temporary array tmp back into the corresponding range of the original array nums\n    for k in tmp.indices {\n        nums[left + k] = tmp[k]\n    }\n}\n\n/* Merge sort */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n    // Base case\n    if left &gt;= right { // Stop recursing when the subarray has at most one element\n        return\n    }\n    // Divide phase\n    let mid = left + (right - left) / 2 // Calculate the midpoint\n    mergeSort(nums: &amp;nums, left: left, right: mid) // Recurse on the left subarray\n    mergeSort(nums: &amp;nums, left: mid + 1, right: right) // Recurse on the right subarray\n    // Merge phase\n    merge(nums: &amp;nums, left: left, mid: mid, right: right)\n}\n</code></pre> merge_sort.js<pre><code>/* Merge the left and right subarrays */\nfunction merge(nums, left, mid, right) {\n    // The left subarray spans [left, mid], and the right subarray spans [mid+1, right]\n    // Create a temporary array tmp to store the merged result\n    const tmp = new Array(right - left + 1);\n    // Initialize the starting indices of the left and right subarrays\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // Copy the elements of temporary array tmp back into the corresponding range of the original array nums\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* Merge sort */\nfunction mergeSort(nums, left, right) {\n    // Base case\n    if (left &gt;= right) return; // Stop recursing when the subarray has at most one element\n    // Divide phase\n    let mid = Math.floor(left + (right - left) / 2); // Calculate the midpoint\n    mergeSort(nums, left, mid); // Recurse on the left subarray\n    mergeSort(nums, mid + 1, right); // Recurse on the right subarray\n    // Merge phase\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.ts<pre><code>/* Merge the left and right subarrays */\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n    // The left subarray spans [left, mid], and the right subarray spans [mid+1, right]\n    // Create a temporary array tmp to store the merged result\n    const tmp = new Array(right - left + 1);\n    // Initialize the starting indices of the left and right subarrays\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // Copy the elements of temporary array tmp back into the corresponding range of the original array nums\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* Merge sort */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n    // Base case\n    if (left &gt;= right) return; // Stop recursing when the subarray has zero or one element\n    // Divide phase\n    let mid = Math.floor(left + (right - left) / 2); // Calculate the midpoint\n    mergeSort(nums, left, mid); // Recurse on the left subarray\n    mergeSort(nums, mid + 1, right); // Recurse on the right subarray\n    // Merge phase\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.dart<pre><code>/* Merge the left and right subarrays */\nvoid merge(List&lt;int&gt; nums, int left, int mid, int right) {\n  // The left subarray spans [left, mid], and the right subarray spans [mid+1, right]\n  // Create a temporary array tmp to store the merged result\n  List&lt;int&gt; tmp = List.filled(right - left + 1, 0);\n  // Initialize the starting indices of the left and right subarrays\n  int i = left, j = mid + 1, k = 0;\n  // While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array\n  while (i &lt;= mid &amp;&amp; j &lt;= right) {\n    if (nums[i] &lt;= nums[j])\n      tmp[k++] = nums[i++];\n    else\n      tmp[k++] = nums[j++];\n  }\n  // Copy the remaining elements of the left and right subarrays into the temporary array\n  while (i &lt;= mid) {\n    tmp[k++] = nums[i++];\n  }\n  while (j &lt;= right) {\n    tmp[k++] = nums[j++];\n  }\n  // Copy the elements of temporary array tmp back into the corresponding range of the original array nums\n  for (k = 0; k &lt; tmp.length; k++) {\n    nums[left + k] = tmp[k];\n  }\n}\n\n/* Merge sort */\nvoid mergeSort(List&lt;int&gt; nums, int left, int right) {\n  // Base case\n  if (left &gt;= right) return; // Stop recursing when the subarray has at most one element\n  // Divide phase\n  int mid = left + (right - left) ~/ 2; // Calculate the midpoint\n  mergeSort(nums, left, mid); // Recurse on the left subarray\n  mergeSort(nums, mid + 1, right); // Recurse on the right subarray\n  // Merge phase\n  merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.rs<pre><code>/* Merge the left and right subarrays */\nfn merge(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) {\n    // The left subarray spans [left, mid], and the right subarray spans [mid+1, right]\n    // Create a temporary array tmp to store the merged result\n    let tmp_size = right - left + 1;\n    let mut tmp = vec![0; tmp_size];\n    // Initialize the starting indices of the left and right subarrays\n    let (mut i, mut j, mut k) = (left, mid + 1, 0);\n    // While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array\n    while i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i];\n            i += 1;\n        } else {\n            tmp[k] = nums[j];\n            j += 1;\n        }\n        k += 1;\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while i &lt;= mid {\n        tmp[k] = nums[i];\n        k += 1;\n        i += 1;\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j];\n        k += 1;\n        j += 1;\n    }\n    // Copy the elements of temporary array tmp back into the corresponding range of the original array nums\n    for k in 0..tmp_size {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* Merge sort */\nfn merge_sort(nums: &amp;mut [i32], left: usize, right: usize) {\n    // Base case\n    if left &gt;= right {\n        return; // Stop recursing when the subarray has at most one element\n    }\n\n    // Divide phase\n    let mid = left + (right - left) / 2; // Calculate the midpoint\n    merge_sort(nums, left, mid); // Recurse on the left subarray\n    merge_sort(nums, mid + 1, right); // Recurse on the right subarray\n\n    // Merge phase\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.c<pre><code>/* Merge the left and right subarrays */\nvoid merge(int *nums, int left, int mid, int right) {\n    // The left subarray spans [left, mid], and the right subarray spans [mid+1, right]\n    // Create a temporary array tmp to store the merged result\n    int tmpSize = right - left + 1;\n    int *tmp = (int *)malloc(tmpSize * sizeof(int));\n    // Initialize the starting indices of the left and right subarrays\n    int i = left, j = mid + 1, k = 0;\n    // While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // Copy the elements of temporary array tmp back into the corresponding range of the original array nums\n    for (k = 0; k &lt; tmpSize; ++k) {\n        nums[left + k] = tmp[k];\n    }\n    // Free memory\n    free(tmp);\n}\n\n/* Merge sort */\nvoid mergeSort(int *nums, int left, int right) {\n    // Base case\n    if (left &gt;= right)\n        return; // Stop recursing when the subarray has at most one element\n    // Divide phase\n    int mid = left + (right - left) / 2;    // Calculate the midpoint\n    mergeSort(nums, left, mid);      // Recurse on the left subarray\n    mergeSort(nums, mid + 1, right); // Recurse on the right subarray\n    // Merge phase\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.kt<pre><code>/* Merge the left and right subarrays */\nfun merge(nums: IntArray, left: Int, mid: Int, right: Int) {\n    // The left subarray spans [left, mid], and the right subarray spans [mid+1, right]\n    // Create a temporary array tmp to store the merged result\n    val tmp = IntArray(right - left + 1)\n    // Initialize the starting indices of the left and right subarrays\n    var i = left\n    var j = mid + 1\n    var k = 0\n    // While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++]\n        else\n            tmp[k++] = nums[j++]\n    }\n    // Copy the remaining elements of the left and right subarrays into the temporary array\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++]\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++]\n    }\n    // Copy the elements of temporary array tmp back into the corresponding range of the original array nums\n    for (l in tmp.indices) {\n        nums[left + l] = tmp[l]\n    }\n}\n\n/* Merge sort */\nfun mergeSort(nums: IntArray, left: Int, right: Int) {\n    // Base case\n    if (left &gt;= right) return  // Stop recursing when the subarray has at most one element\n    // Divide phase\n    val mid = left + (right - left) / 2 // Calculate the midpoint\n    mergeSort(nums, left, mid) // Recurse on the left subarray\n    mergeSort(nums, mid + 1, right) // Recurse on the right subarray\n    // Merge phase\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.rb<pre><code>### Merge the left and right subarrays ###\ndef merge(nums, left, mid, right)\n  # The left subarray spans [left, mid], and the right subarray spans [mid+1, right]\n  # Create a temporary array tmp to store the merged result\n  tmp = Array.new(right - left + 1, 0)\n  # Initialize the starting indices of the left and right subarrays\n  i, j, k = left, mid + 1, 0\n  # While both subarrays have elements remaining, compare them and copy the smaller element into the temporary array\n  while i &lt;= mid &amp;&amp; j &lt;= right\n    if nums[i] &lt;= nums[j]\n      tmp[k] = nums[i]\n      i += 1\n    else\n      tmp[k] = nums[j]\n      j += 1\n    end\n    k += 1\n  end\n  # Copy the remaining elements of the left and right subarrays into the temporary array\n  while i &lt;= mid\n    tmp[k] = nums[i]\n    i += 1\n    k += 1\n  end\n  while j &lt;= right\n    tmp[k] = nums[j]\n    j += 1\n    k += 1\n  end\n  # Copy the elements of temporary array tmp back into the corresponding range of the original array nums\n  (0...tmp.length).each do |k|\n    nums[left + k] = tmp[k]\n  end\nend\n\n### Merge sort ###\ndef merge_sort(nums, left, right)\n  # Base case\n  # Stop recursing when the subarray contains at most one element\n  return if left &gt;= right\n  # Divide phase\n  mid = left + (right - left) / 2 # Calculate the midpoint\n  merge_sort(nums, left, mid) # Recurse on the left subarray\n  merge_sort(nums, mid + 1, right) # Recurse on the right subarray\n  # Merge phase\n  merge(nums, left, mid, right)\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 11.   Sorting","11.6   Merge sort"],"tags":[]},{"location":"chapter_sorting/merge_sort/#1162-algorithm-properties","level":2,"title":"11.6.2   Algorithm properties","text":"<ul> <li>Time complexity: \\(O(n \\log n)\\); not adaptive. Division produces a recursion tree of height \\(\\log n\\). Merging performs a total of \\(n\\) operations at each level, giving \\(O(n \\log n)\\) overall.</li> <li>Space complexity: \\(O(n)\\); not in-place. The recursion depth is \\(\\log n\\), requiring \\(O(\\log n)\\) space for stack frames. Merging uses an auxiliary array requiring \\(O(n)\\) extra space.</li> <li>Stable. Merging preserves the order of equal elements.</li> </ul>","path":["Chapter 11.   Sorting","11.6   Merge sort"],"tags":[]},{"location":"chapter_sorting/merge_sort/#1163-sorting-linked-lists","level":2,"title":"11.6.3   Sorting linked lists","text":"<p>Merge sort has a significant advantage over other sorting algorithms for linked lists: it can reduce the space complexity of sorting a linked list to \\(O(1)\\).</p> <ul> <li>Division: use iteration instead of recursion to divide the list, eliminating the space needed for recursive stack frames.</li> <li>Merging: inserting and deleting linked-list nodes requires only changing references or pointers. Merging two short sorted lists into a longer one therefore needs no additional list.</li> </ul> <p>The implementation is fairly involved. Interested readers can consult further resources for details.</p>","path":["Chapter 11.   Sorting","11.6   Merge sort"],"tags":[]},{"location":"chapter_sorting/quick_sort/","level":1,"title":"11.5   Quick sort","text":"<p>Quick sort is an efficient, widely used sorting algorithm based on divide and conquer.</p> <p>Its core operation is “partitioning”: choose an array element as the “pivot,” move all smaller elements to its left, and move all larger elements to its right. The partitioning process is shown below.</p> <ol> <li>Choose the leftmost element as the pivot and initialize pointers <code>i</code> and <code>j</code> at the two ends of the array.</li> <li>In a loop, use <code>i</code> to find the first element greater than the pivot and <code>j</code> to find the first element smaller than it, then swap those elements.</li> <li>Repeat step <code>2.</code> until <code>i</code> and <code>j</code> meet. Finally, swap the pivot into the boundary between the two subarrays.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 11-8   Steps of partitioning </p> <p>After partitioning, the original array has three parts: the left subarray, the pivot, and the right subarray. Every element in the left subarray is \\(\\leq\\) the pivot, which is \\(\\leq\\) every element in the right subarray. We now need only sort the two subarrays.</p> <p>Quick sort's divide-and-conquer strategy</p> <p>Partitioning reduces the problem of sorting one long array to sorting two shorter arrays.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"Two-pointer partitioning\"\"\"\n    # Use nums[left] as the pivot\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # Scan from right to left for the first element smaller than the pivot\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # Scan from left to right for the first element larger than the pivot\n        # Swap the elements\n        nums[i], nums[j] = nums[j], nums[i]\n    # Swap the pivot into the boundary position between the two subarrays\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # Return the pivot index\n</code></pre> quick_sort.cpp<pre><code>/* Two-pointer partitioning */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;                // Scan from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;                // Scan from left to right for the first element larger than the pivot\n        swap(nums[i], nums[j]); // Swap these two elements\n    }\n    swap(nums[i], nums[left]);  // Swap the pivot into the boundary position between the two subarrays\n    return i;                   // Return the pivot index\n}\n</code></pre> quick_sort.java<pre><code>/* Swap the elements */\nvoid swap(int[] nums, int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* Two-pointer partitioning */\nint partition(int[] nums, int left, int right) {\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // Scan from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // Scan from left to right for the first element larger than the pivot\n        swap(nums, i, j); // Swap these two elements\n    }\n    swap(nums, i, left);  // Swap the pivot into the boundary position between the two subarrays\n    return i;             // Return the pivot index\n}\n</code></pre> quick_sort.cs<pre><code>/* Swap the elements */\nvoid Swap(int[] nums, int i, int j) {\n    (nums[j], nums[i]) = (nums[i], nums[j]);\n}\n\n/* Two-pointer partitioning */\nint Partition(int[] nums, int left, int right) {\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // Scan from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // Scan from left to right for the first element larger than the pivot\n        Swap(nums, i, j); // Swap these two elements\n    }\n    Swap(nums, i, left);  // Swap the pivot into the boundary position between the two subarrays\n    return i;             // Return the pivot index\n}\n</code></pre> quick_sort.go<pre><code>/* Two-pointer partitioning */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n    // Use nums[left] as the pivot\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- // Scan from right to left for the first element smaller than the pivot\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ // Scan from left to right for the first element larger than the pivot\n        }\n        // Swap the elements\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    // Swap the pivot into the boundary position between the two subarrays\n    nums[i], nums[left] = nums[left], nums[i]\n    return i // Return the pivot index\n}\n</code></pre> quick_sort.swift<pre><code>/* Two-pointer partitioning */\nfunc partition(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // Use nums[left] as the pivot\n    var i = left\n    var j = right\n    while i &lt; j {\n        while i &lt; j, nums[j] &gt;= nums[left] {\n            j -= 1 // Scan from right to left for the first element smaller than the pivot\n        }\n        while i &lt; j, nums[i] &lt;= nums[left] {\n            i += 1 // Scan from left to right for the first element larger than the pivot\n        }\n        nums.swapAt(i, j) // Swap these two elements\n    }\n    nums.swapAt(i, left) // Swap the pivot into the boundary position between the two subarrays\n    return i // Return the pivot index\n}\n</code></pre> quick_sort.js<pre><code>/* Swap the elements */\nswap(nums, i, j) {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* Two-pointer partitioning */\npartition(nums, left, right) {\n    // Use nums[left] as the pivot\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // Scan from right to left for the first element smaller than the pivot\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // Scan from left to right for the first element larger than the pivot\n        }\n        // Swap the elements\n        this.swap(nums, i, j); // Swap these two elements\n    }\n    this.swap(nums, i, left); // Swap the pivot into the boundary position between the two subarrays\n    return i; // Return the pivot index\n}\n</code></pre> quick_sort.ts<pre><code>/* Swap the elements */\nswap(nums: number[], i: number, j: number): void {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* Two-pointer partitioning */\npartition(nums: number[], left: number, right: number): number {\n    // Use nums[left] as the pivot\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // Scan from right to left for the first element smaller than the pivot\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // Scan from left to right for the first element larger than the pivot\n        }\n        // Swap the elements\n        this.swap(nums, i, j); // Swap these two elements\n    }\n    this.swap(nums, i, left); // Swap the pivot into the boundary position between the two subarrays\n    return i; // Return the pivot index\n}\n</code></pre> quick_sort.dart<pre><code>/* Swap the elements */\nvoid _swap(List&lt;int&gt; nums, int i, int j) {\n  int tmp = nums[i];\n  nums[i] = nums[j];\n  nums[j] = tmp;\n}\n\n/* Two-pointer partitioning */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // Use nums[left] as the pivot\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // Scan from right to left for the first element smaller than the pivot\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // Scan from left to right for the first element larger than the pivot\n    _swap(nums, i, j); // Swap these two elements\n  }\n  _swap(nums, i, left); // Swap the pivot into the boundary position between the two subarrays\n  return i; // Return the pivot index\n}\n</code></pre> quick_sort.rs<pre><code>/* Two-pointer partitioning */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // Use nums[left] as the pivot\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // Scan from right to left for the first element smaller than the pivot\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // Scan from left to right for the first element larger than the pivot\n        }\n        nums.swap(i, j); // Swap these two elements\n    }\n    nums.swap(i, left); // Swap the pivot into the boundary position between the two subarrays\n    i // Return the pivot index\n}\n</code></pre> quick_sort.c<pre><code>/* Swap the elements */\nvoid swap(int nums[], int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* Two-pointer partitioning */\nint partition(int nums[], int left, int right) {\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j--; // Scan from right to left for the first element smaller than the pivot\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i++; // Scan from left to right for the first element larger than the pivot\n        }\n        // Swap these two elements\n        swap(nums, i, j);\n    }\n    // Swap the pivot into the boundary position between the two subarrays\n    swap(nums, i, left);\n    // Return the pivot index\n    return i;\n}\n</code></pre> quick_sort.kt<pre><code>/* Swap the elements */\nfun swap(nums: IntArray, i: Int, j: Int) {\n    val temp = nums[i]\n    nums[i] = nums[j]\n    nums[j] = temp\n}\n\n/* Two-pointer partitioning */\nfun partition(nums: IntArray, left: Int, right: Int): Int {\n    // Use nums[left] as the pivot\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--           // Scan from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++           // Scan from left to right for the first element larger than the pivot\n        swap(nums, i, j)  // Swap these two elements\n    }\n    swap(nums, i, left)   // Swap the pivot into the boundary position between the two subarrays\n    return i              // Return the pivot index\n}\n</code></pre> quick_sort.rb<pre><code>### Two-pointer partitioning ###\ndef partition(nums, left, right)\n  # Use nums[left] as the pivot\n  i, j = left, right\n  while i &lt; j\n    while i &lt; j &amp;&amp; nums[j] &gt;= nums[left]\n      j -= 1 # Scan from right to left for the first element smaller than the pivot\n    end\n    while i &lt; j &amp;&amp; nums[i] &lt;= nums[left]\n      i += 1 # Scan from left to right for the first element larger than the pivot\n    end\n    # Swap the elements\n    nums[i], nums[j] = nums[j], nums[i]\n  end\n  # Swap the pivot into the boundary position between the two subarrays\n  nums[i], nums[left] = nums[left], nums[i]\n  i # Return the pivot index\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 11.   Sorting","11.5   Quick sort"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1151-algorithm-steps","level":2,"title":"11.5.1   Algorithm steps","text":"<p>The overall quick sort process is shown below.</p> <ol> <li>Partition the original array to obtain unsorted left and right subarrays.</li> <li>Recursively partition the left and right subarrays.</li> <li>Continue until each subarray has length 1, completing the sort.</li> </ol> <p></p> <p> Figure 11-9   The quick sort process </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"Quick sort\"\"\"\n    # Stop recursing when the subarray has at most one element\n    if left &gt;= right:\n        return\n    # Two-pointer partitioning\n    pivot = self.partition(nums, left, right)\n    # Recurse on the left and right subarrays\n    self.quick_sort(nums, left, pivot - 1)\n    self.quick_sort(nums, pivot + 1, right)\n</code></pre> quick_sort.cpp<pre><code>/* Quick sort */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // Stop recursing when the subarray has at most one element\n    if (left &gt;= right)\n        return;\n    // Two-pointer partitioning\n    int pivot = partition(nums, left, right);\n    // Recurse on the left and right subarrays\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.java<pre><code>/* Quick sort */\nvoid quickSort(int[] nums, int left, int right) {\n    // Stop recursing when the subarray has at most one element\n    if (left &gt;= right)\n        return;\n    // Two-pointer partitioning\n    int pivot = partition(nums, left, right);\n    // Recurse on the left and right subarrays\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.cs<pre><code>/* Quick sort */\nvoid QuickSort(int[] nums, int left, int right) {\n    // Stop recursing when the subarray has at most one element\n    if (left &gt;= right)\n        return;\n    // Two-pointer partitioning\n    int pivot = Partition(nums, left, right);\n    // Recurse on the left and right subarrays\n    QuickSort(nums, left, pivot - 1);\n    QuickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.go<pre><code>/* Quick sort */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n    // Stop recursing when the subarray has at most one element\n    if left &gt;= right {\n        return\n    }\n    // Two-pointer partitioning\n    pivot := q.partition(nums, left, right)\n    // Recurse on the left and right subarrays\n    q.quickSort(nums, left, pivot-1)\n    q.quickSort(nums, pivot+1, right)\n}\n</code></pre> quick_sort.swift<pre><code>/* Quick sort */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n    // Stop recursing when the subarray has at most one element\n    if left &gt;= right {\n        return\n    }\n    // Two-pointer partitioning\n    let pivot = partition(nums: &amp;nums, left: left, right: right)\n    // Recurse on the left and right subarrays\n    quickSort(nums: &amp;nums, left: left, right: pivot - 1)\n    quickSort(nums: &amp;nums, left: pivot + 1, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* Quick sort */\nquickSort(nums, left, right) {\n    // Stop recursing when the subarray has at most one element\n    if (left &gt;= right) return;\n    // Two-pointer partitioning\n    const pivot = this.partition(nums, left, right);\n    // Recurse on the left and right subarrays\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.ts<pre><code>/* Quick sort */\nquickSort(nums: number[], left: number, right: number): void {\n    // Stop recursing when the subarray has zero or one element\n    if (left &gt;= right) {\n        return;\n    }\n    // Two-pointer partitioning\n    const pivot = this.partition(nums, left, right);\n    // Recurse on the left and right subarrays\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.dart<pre><code>/* Quick sort */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // Stop recursing when the subarray has at most one element\n  if (left &gt;= right) return;\n  // Two-pointer partitioning\n  int pivot = _partition(nums, left, right);\n  // Recurse on the left and right subarrays\n  quickSort(nums, left, pivot - 1);\n  quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.rs<pre><code>/* Quick sort */\npub fn quick_sort(left: i32, right: i32, nums: &amp;mut [i32]) {\n    // Stop recursing when the subarray has at most one element\n    if left &gt;= right {\n        return;\n    }\n    // Two-pointer partitioning\n    let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n    // Recurse on the left and right subarrays\n    Self::quick_sort(left, pivot - 1, nums);\n    Self::quick_sort(pivot + 1, right, nums);\n}\n</code></pre> quick_sort.c<pre><code>/* Quick sort */\nvoid quickSort(int nums[], int left, int right) {\n    // Stop recursing when the subarray has at most one element\n    if (left &gt;= right) {\n        return;\n    }\n    // Two-pointer partitioning\n    int pivot = partition(nums, left, right);\n    // Recurse on the left and right subarrays\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.kt<pre><code>/* Quick sort */\nfun quickSort(nums: IntArray, left: Int, right: Int) {\n    // Stop recursing when the subarray has at most one element\n    if (left &gt;= right) return\n    // Two-pointer partitioning\n    val pivot = partition(nums, left, right)\n    // Recurse on the left and right subarrays\n    quickSort(nums, left, pivot - 1)\n    quickSort(nums, pivot + 1, right)\n}\n</code></pre> quick_sort.rb<pre><code>### Quick sort ###\ndef quick_sort(nums, left, right)\n  # Continue while the subarray contains more than one element\n  if left &lt; right\n    # Two-pointer partitioning\n    pivot = partition(nums, left, right)\n    # Recurse on the left and right subarrays\n    quick_sort(nums, left, pivot - 1)\n    quick_sort(nums, pivot + 1, right)\n  end\n  nums\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 11.   Sorting","11.5   Quick sort"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1152-algorithm-properties","level":2,"title":"11.5.2   Algorithm properties","text":"<ul> <li>Time complexity: \\(O(n \\log n)\\); not adaptive. On average, partitioning produces \\(\\log n\\) recursive levels, with \\(n\\) loop iterations in total at each level, giving \\(O(n \\log n)\\) time. In the worst case, each partition splits an array of length \\(n\\) into subarrays of lengths \\(0\\) and \\(n - 1\\). The recursion reaches \\(n\\) levels, each involving up to \\(n\\) iterations, for \\(O(n^2)\\) time overall.</li> <li>Space complexity: \\(O(n)\\); in-place. A fully reverse-sorted input reaches the worst-case recursion depth of \\(n\\), requiring \\(O(n)\\) space for stack frames. Sorting takes place in the original array without an auxiliary array.</li> <li>Not stable. In the final partitioning step, the pivot may be swapped to the right of an equal element.</li> </ul>","path":["Chapter 11.   Sorting","11.5   Quick sort"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1153-why-is-quick-sort-quick","level":2,"title":"11.5.3   Why is quick sort quick?","text":"<p>As its name suggests, quick sort has an efficiency advantage. Although its average time complexity matches that of merge sort and heap sort, it is usually faster, mainly for the following reasons.</p> <ul> <li>The worst case is unlikely. Quick sort's worst-case time complexity is \\(O(n^2)\\), making its performance less predictable than merge sort's, but it runs in \\(O(n \\log n)\\) time in the vast majority of cases.</li> <li>Efficient cache use. During partitioning, the system can load the whole subarray into cache, making element access efficient. Algorithms such as heap sort jump between elements and do not share this advantage.</li> <li>Small constant factors. Of these three algorithms, quick sort performs the fewest comparisons, assignments, swaps, and similar operations overall. This resembles the reason insertion sort is faster than bubble sort.</li> </ul>","path":["Chapter 11.   Sorting","11.5   Quick sort"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1154-improving-pivot-selection","level":2,"title":"11.5.4   Improving pivot selection","text":"<p>Quick sort can become slower on certain inputs. Consider a fully reverse-sorted array. Because the leftmost element is chosen as the pivot, partitioning moves it to the far right, leaving subarrays of lengths \\(n - 1\\) and \\(0\\). Continuing recursively produces a subarray of length \\(0\\) after every partition. Divide and conquer becomes ineffective, and quick sort degenerates into a process resembling bubble sort.</p> <p>To reduce this risk, we can improve how the pivot is selected. For example, choose a random element as the pivot. With bad luck, however, every pivot may still be poor, leading to disappointing performance.</p> <p>Programming languages generally produce pseudorandom numbers. A test case constructed specifically for a pseudorandom sequence can still degrade quick sort's performance.</p> <p>For further improvement, choose three candidate elements, usually the first, last, and middle elements, and use their median as the pivot. This greatly increases the chance that the pivot is neither too small nor too large. We can use more candidates to improve robustness further. This approach greatly reduces the likelihood of degradation to \\(O(n^2)\\).</p> <p>An implementation follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def median_three(self, nums: list[int], left: int, mid: int, right: int) -&gt; int:\n    \"\"\"Select the median of three candidate elements\"\"\"\n    l, m, r = nums[left], nums[mid], nums[right]\n    if (l &lt;= m &lt;= r) or (r &lt;= m &lt;= l):\n        return mid  # m lies between l and r\n    if (m &lt;= l &lt;= r) or (r &lt;= l &lt;= m):\n        return left  # l lies between m and r\n    return right\n\ndef partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"Two-pointer partitioning (median of three)\"\"\"\n    # Select the median of the leftmost, middle, and rightmost elements as the pivot\n    med = self.median_three(nums, left, (left + right) // 2, right)\n    # Swap the median into the leftmost position of the current subarray\n    nums[left], nums[med] = nums[med], nums[left]\n    # Use nums[left] as the pivot\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # Scan from right to left for the first element smaller than the pivot\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # Scan from left to right for the first element larger than the pivot\n        # Swap the elements\n        nums[i], nums[j] = nums[j], nums[i]\n    # Swap the pivot into the boundary position between the two subarrays\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # Return the pivot index\n</code></pre> quick_sort.cpp<pre><code>/* Select the median of three candidate elements */\nint medianThree(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m lies between l and r\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l lies between m and r\n    return right;\n}\n\n/* Two-pointer partitioning (median of three) */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // Select the median of three candidate elements\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // Swap the median into the leftmost position of the current subarray\n    swap(nums[left], nums[med]);\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;                // Scan from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;                // Scan from left to right for the first element larger than the pivot\n        swap(nums[i], nums[j]); // Swap these two elements\n    }\n    swap(nums[i], nums[left]);  // Swap the pivot into the boundary position between the two subarrays\n    return i;                   // Return the pivot index\n}\n</code></pre> quick_sort.java<pre><code>/* Select the median of three candidate elements */\nint medianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m lies between l and r\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l lies between m and r\n    return right;\n}\n\n/* Two-pointer partitioning (median of three) */\nint partition(int[] nums, int left, int right) {\n    // Select the median of three candidate elements\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // Swap the median into the leftmost position of the current subarray\n    swap(nums, left, med);\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // Scan from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // Scan from left to right for the first element larger than the pivot\n        swap(nums, i, j); // Swap these two elements\n    }\n    swap(nums, i, left);  // Swap the pivot into the boundary position between the two subarrays\n    return i;             // Return the pivot index\n}\n</code></pre> quick_sort.cs<pre><code>/* Select the median of three candidate elements */\nint MedianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m lies between l and r\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l lies between m and r\n    return right;\n}\n\n/* Two-pointer partitioning (median of three) */\nint Partition(int[] nums, int left, int right) {\n    // Select the median of three candidate elements\n    int med = MedianThree(nums, left, (left + right) / 2, right);\n    // Swap the median into the leftmost position of the current subarray\n    Swap(nums, left, med);\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // Scan from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // Scan from left to right for the first element larger than the pivot\n        Swap(nums, i, j); // Swap these two elements\n    }\n    Swap(nums, i, left);  // Swap the pivot into the boundary position between the two subarrays\n    return i;             // Return the pivot index\n}\n</code></pre> quick_sort.go<pre><code>/* Select the median of three candidate elements */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n    l, m, r := nums[left], nums[mid], nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m lies between l and r\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l lies between m and r\n    }\n    return right\n}\n\n/* Two-pointer partitioning (median of three)*/\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n    // Select the median of the left, middle, and right elements as the pivot\n    med := q.medianThree(nums, left, (left+right)/2, right)\n    // Swap the median into the leftmost position of the current subarray\n    nums[left], nums[med] = nums[med], nums[left]\n    // Use nums[left] as the pivot\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- //Scan from right to left for the first element smaller than the pivot\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ //Scan from left to right for the first element larger than the pivot\n        }\n        //Swap the elements\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    //Swap the pivot into the boundary position between the two subarrays\n    nums[i], nums[left] = nums[left], nums[i]\n    return i //Return the pivot index\n}\n</code></pre> quick_sort.swift<pre><code>/* Select the median of three candidate elements */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -&gt; Int {\n    let l = nums[left]\n    let m = nums[mid]\n    let r = nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m lies between l and r\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l lies between m and r\n    }\n    return right\n}\n\n/* Two-pointer partitioning (median of three) */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // Select the median of three candidate elements\n    let med = medianThree(nums: nums, left: left, mid: left + (right - left) / 2, right: right)\n    // Swap the median into the leftmost position of the current subarray\n    nums.swapAt(left, med)\n    return partition(nums: &amp;nums, left: left, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* Select the median of three candidate elements */\nmedianThree(nums, left, mid, right) {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m lies between l and r\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l lies between m and r\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* Two-pointer partitioning (median of three) */\npartition(nums, left, right) {\n    // Select the median of three candidate elements\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // Swap the median into the leftmost position of the current subarray\n    this.swap(nums, left, med);\n    // Use nums[left] as the pivot\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // Scan from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // Scan from left to right for the first element larger than the pivot\n        this.swap(nums, i, j); // Swap these two elements\n    }\n    this.swap(nums, i, left); // Swap the pivot into the boundary position between the two subarrays\n    return i; // Return the pivot index\n}\n</code></pre> quick_sort.ts<pre><code>/* Select the median of three candidate elements */\nmedianThree(\n    nums: number[],\n    left: number,\n    mid: number,\n    right: number\n): number {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m lies between l and r\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l lies between m and r\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* Two-pointer partitioning (median of three) */\npartition(nums: number[], left: number, right: number): number {\n    // Select the median of three candidate elements\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // Swap the median into the leftmost position of the current subarray\n    this.swap(nums, left, med);\n    // Use nums[left] as the pivot\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j--; // Scan from right to left for the first element smaller than the pivot\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i++; // Scan from left to right for the first element larger than the pivot\n        }\n        this.swap(nums, i, j); // Swap these two elements\n    }\n    this.swap(nums, i, left); // Swap the pivot into the boundary position between the two subarrays\n    return i; // Return the pivot index\n}\n</code></pre> quick_sort.dart<pre><code>/* Select the median of three candidate elements */\nint _medianThree(List&lt;int&gt; nums, int left, int mid, int right) {\n  int l = nums[left], m = nums[mid], r = nums[right];\n  if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n    return mid; // m lies between l and r\n  if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n    return left; // l lies between m and r\n  return right;\n}\n\n/* Two-pointer partitioning (median of three) */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // Select the median of three candidate elements\n  int med = _medianThree(nums, left, (left + right) ~/ 2, right);\n  // Swap the median into the leftmost position of the current subarray\n  _swap(nums, left, med);\n  // Use nums[left] as the pivot\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // Scan from right to left for the first element smaller than the pivot\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // Scan from left to right for the first element larger than the pivot\n    _swap(nums, i, j); // Swap these two elements\n  }\n  _swap(nums, i, left); // Swap the pivot into the boundary position between the two subarrays\n  return i; // Return the pivot index\n}\n</code></pre> quick_sort.rs<pre><code>/* Select the median of three candidate elements */\nfn median_three(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) -&gt; usize {\n    let (l, m, r) = (nums[left], nums[mid], nums[right]);\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid; // m lies between l and r\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left; // l lies between m and r\n    }\n    right\n}\n\n/* Two-pointer partitioning (median of three) */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // Select the median of three candidate elements\n    let med = Self::median_three(nums, left, (left + right) / 2, right);\n    // Swap the median into the leftmost position of the current subarray\n    nums.swap(left, med);\n    // Use nums[left] as the pivot\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // Scan from right to left for the first element smaller than the pivot\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // Scan from left to right for the first element larger than the pivot\n        }\n        nums.swap(i, j); // Swap these two elements\n    }\n    nums.swap(i, left); // Swap the pivot into the boundary position between the two subarrays\n    i // Return the pivot index\n}\n</code></pre> quick_sort.c<pre><code>/* Select the median of three candidate elements */\nint medianThree(int nums[], int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m lies between l and r\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l lies between m and r\n    return right;\n}\n\n/* Two-pointer partitioning (median of three) */\nint partitionMedian(int nums[], int left, int right) {\n    // Select the median of three candidate elements\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // Swap the median into the leftmost position of the array\n    swap(nums, left, med);\n    // Use nums[left] as the pivot\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--; // Scan from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // Scan from left to right for the first element larger than the pivot\n        swap(nums, i, j); // Swap these two elements\n    }\n    swap(nums, i, left); // Swap the pivot into the boundary position between the two subarrays\n    return i;            // Return the pivot index\n}\n</code></pre> quick_sort.kt<pre><code>/* Select the median of three candidate elements */\nfun medianThree(nums: IntArray, left: Int, mid: Int, right: Int): Int {\n    val l = nums[left]\n    val m = nums[mid]\n    val r = nums[right]\n    if ((m in l..r) || (m in r..l))\n        return mid  // m lies between l and r\n    if ((l in m..r) || (l in r..m))\n        return left // l lies between m and r\n    return right\n}\n\n/* Two-pointer partitioning (median of three) */\nfun partitionMedian(nums: IntArray, left: Int, right: Int): Int {\n    // Select the median of three candidate elements\n    val med = medianThree(nums, left, (left + right) / 2, right)\n    // Swap the median into the leftmost position of the current subarray\n    swap(nums, left, med)\n    // Use nums[left] as the pivot\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--                      // Scan from right to left for the first element smaller than the pivot\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++                      // Scan from left to right for the first element larger than the pivot\n        swap(nums, i, j)             // Swap these two elements\n    }\n    swap(nums, i, left)              // Swap the pivot into the boundary position between the two subarrays\n    return i                         // Return the pivot index\n}\n</code></pre> quick_sort.rb<pre><code>### Select the median of three candidate elements ###\ndef median_three(nums, left, mid, right)\n  # Select the median of three candidate elements\n  _l, _m, _r = nums[left], nums[mid], nums[right]\n  # _m lies between _l and _r\n  return mid if (_l &lt;= _m &amp;&amp; _m &lt;= _r) || (_r &lt;= _m &amp;&amp; _m &lt;= _l)\n  # _l lies between _m and _r\n  return left if (_m &lt;= _l &amp;&amp; _l &lt;= _r) || (_r &lt;= _l &amp;&amp; _l &lt;= _m)\n  return right\nend\n\n### Two-pointer partitioning (median of three)###\ndef partition(nums, left, right)\n  ### Use nums[left] as the pivot\n  med = median_three(nums, left, (left + right) / 2, right)\n  # Swap the median into the leftmost position of the current subarray\n  nums[left], nums[med] = nums[med], nums[left]\n  i, j = left, right\n  while i &lt; j\n    while i &lt; j &amp;&amp; nums[j] &gt;= nums[left]\n      j -= 1 # Scan from right to left for the first element smaller than the pivot\n    end\n    while i &lt; j &amp;&amp; nums[i] &lt;= nums[left]\n      i += 1 # Scan from left to right for the first element larger than the pivot\n    end\n    # Swap the elements\n    nums[i], nums[j] = nums[j], nums[i]\n  end\n  # Swap the pivot into the boundary position between the two subarrays\n  nums[i], nums[left] = nums[left], nums[i]\n  i # Return the pivot index\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 11.   Sorting","11.5   Quick sort"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1155-reducing-recursion-depth","level":2,"title":"11.5.5   Reducing recursion depth","text":"<p>Quick sort may use considerable space on certain inputs. For a fully sorted input, let the current subarray have length \\(m\\). Each partition produces a left subarray of length \\(0\\) and a right subarray of length \\(m - 1\\). Each recursive call therefore reduces the problem size by only one element, producing a recursion tree of height \\(n - 1\\) and requiring \\(O(n)\\) space for stack frames.</p> <p>To prevent stack frames from accumulating, compare the two subarray lengths after each partition and recurse only on the shorter one. Since that subarray has length at most \\(n / 2\\), the recursion depth is at most \\(\\log n\\), reducing the worst-case space complexity to \\(O(\\log n)\\). The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"Quick sort (recursion depth optimization)\"\"\"\n    # Stop when the subarray has at most one element\n    while left &lt; right:\n        # Perform two-pointer partitioning\n        pivot = self.partition(nums, left, right)\n        # Apply quick sort to the shorter of the two subarrays\n        if pivot - left &lt; right - pivot:\n            self.quick_sort(nums, left, pivot - 1)  # Recursively sort the left subarray\n            left = pivot + 1  # The remaining unsorted range is [pivot + 1, right]\n        else:\n            self.quick_sort(nums, pivot + 1, right)  # Recursively sort the right subarray\n            right = pivot - 1  # The remaining unsorted range is [left, pivot - 1]\n</code></pre> quick_sort.cpp<pre><code>/* Quick sort (recursion depth optimization) */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // Stop when the subarray has at most one element\n    while (left &lt; right) {\n        // Perform two-pointer partitioning\n        int pivot = partition(nums, left, right);\n        // Apply quick sort to the shorter of the two subarrays\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // Recursively sort the left subarray\n            left = pivot + 1;                 // The remaining unsorted range is [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right); // Recursively sort the right subarray\n            right = pivot - 1;                 // The remaining unsorted range is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.java<pre><code>/* Quick sort (recursion depth optimization) */\nvoid quickSort(int[] nums, int left, int right) {\n    // Stop when the subarray has at most one element\n    while (left &lt; right) {\n        // Perform two-pointer partitioning\n        int pivot = partition(nums, left, right);\n        // Apply quick sort to the shorter of the two subarrays\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // Recursively sort the left subarray\n            left = pivot + 1; // The remaining unsorted range is [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right); // Recursively sort the right subarray\n            right = pivot - 1; // The remaining unsorted range is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.cs<pre><code>/* Quick sort (recursion depth optimization) */\nvoid QuickSort(int[] nums, int left, int right) {\n    // Stop when the subarray has at most one element\n    while (left &lt; right) {\n        // Perform two-pointer partitioning\n        int pivot = Partition(nums, left, right);\n        // Apply quick sort to the shorter of the two subarrays\n        if (pivot - left &lt; right - pivot) {\n            QuickSort(nums, left, pivot - 1);  // Recursively sort the left subarray\n            left = pivot + 1;  // The remaining unsorted range is [pivot + 1, right]\n        } else {\n            QuickSort(nums, pivot + 1, right); // Recursively sort the right subarray\n            right = pivot - 1; // The remaining unsorted range is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.go<pre><code>/* Quick sort (recursion depth optimization)*/\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n    // Stop when the subarray has at most one element\n    for left &lt; right {\n        // Perform two-pointer partitioning\n        pivot := q.partition(nums, left, right)\n        // Apply quick sort to the shorter of the two subarrays\n        if pivot-left &lt; right-pivot {\n            q.quickSort(nums, left, pivot-1) // Recursively sort the left subarray\n            left = pivot + 1                 // The remaining unsorted range is [pivot + 1, right]\n        } else {\n            q.quickSort(nums, pivot+1, right) // Recursively sort the right subarray\n            right = pivot - 1                 // The remaining unsorted range is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.swift<pre><code>/* Quick sort (recursion depth optimization) */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\n    var left = left\n    var right = right\n    // Stop when the subarray has at most one element\n    while left &lt; right {\n        // Perform two-pointer partitioning\n        let pivot = partition(nums: &amp;nums, left: left, right: right)\n        // Apply quick sort to the shorter of the two subarrays\n        if (pivot - left) &lt; (right - pivot) {\n            quickSortTailCall(nums: &amp;nums, left: left, right: pivot - 1) // Recursively sort the left subarray\n            left = pivot + 1 // The remaining unsorted range is [pivot + 1, right]\n        } else {\n            quickSortTailCall(nums: &amp;nums, left: pivot + 1, right: right) // Recursively sort the right subarray\n            right = pivot - 1 // The remaining unsorted range is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.js<pre><code>/* Quick sort (recursion depth optimization) */\nquickSort(nums, left, right) {\n    // Stop when the subarray has at most one element\n    while (left &lt; right) {\n        // Perform two-pointer partitioning\n        let pivot = this.partition(nums, left, right);\n        // Apply quick sort to the shorter of the two subarrays\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // Recursively sort the left subarray\n            left = pivot + 1; // The remaining unsorted range is [pivot + 1, right]\n        } else {\n            this.quickSort(nums, pivot + 1, right); // Recursively sort the right subarray\n            right = pivot - 1; // The remaining unsorted range is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.ts<pre><code>/* Quick sort (recursion depth optimization) */\nquickSort(nums: number[], left: number, right: number): void {\n    // Stop when the subarray has zero or one element\n    while (left &lt; right) {\n        // Perform two-pointer partitioning\n        let pivot = this.partition(nums, left, right);\n        // Apply quick sort to the shorter of the two subarrays\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // Recursively sort the left subarray\n            left = pivot + 1; // The remaining unsorted range is [pivot + 1, right]\n        } else {\n            this.quickSort(nums, pivot + 1, right); // Recursively sort the right subarray\n            right = pivot - 1; // The remaining unsorted range is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.dart<pre><code>/* Quick sort (recursion depth optimization) */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // Stop when the subarray has at most one element\n  while (left &lt; right) {\n    // Perform two-pointer partitioning\n    int pivot = _partition(nums, left, right);\n    // Apply quick sort to the shorter of the two subarrays\n    if (pivot - left &lt; right - pivot) {\n      quickSort(nums, left, pivot - 1); // Recursively sort the left subarray\n      left = pivot + 1; // The remaining unsorted range is [pivot + 1, right]\n    } else {\n      quickSort(nums, pivot + 1, right); // Recursively sort the right subarray\n      right = pivot - 1; // The remaining unsorted range is [left, pivot - 1]\n    }\n  }\n}\n</code></pre> quick_sort.rs<pre><code>/* Quick sort (recursion depth optimization) */\npub fn quick_sort(mut left: i32, mut right: i32, nums: &amp;mut [i32]) {\n    // Stop when the subarray has at most one element\n    while left &lt; right {\n        // Perform two-pointer partitioning\n        let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n        // Apply quick sort to the shorter of the two subarrays\n        if pivot - left &lt; right - pivot {\n            Self::quick_sort(left, pivot - 1, nums); // Recursively sort the left subarray\n            left = pivot + 1; // The remaining unsorted range is [pivot + 1, right]\n        } else {\n            Self::quick_sort(pivot + 1, right, nums); // Recursively sort the right subarray\n            right = pivot - 1; // The remaining unsorted range is [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.c<pre><code>/* Quick sort (recursion depth optimization) */\nvoid quickSortTailCall(int nums[], int left, int right) {\n    // Stop when the subarray has at most one element\n    while (left &lt; right) {\n        // Perform two-pointer partitioning\n        int pivot = partition(nums, left, right);\n        // Apply quick sort to the shorter of the two subarrays\n        if (pivot - left &lt; right - pivot) {\n            // Recursively sort the left subarray\n            quickSortTailCall(nums, left, pivot - 1);\n            // The remaining unsorted range is [pivot + 1, right]\n            left = pivot + 1;\n        } else {\n            // Recursively sort the right subarray\n            quickSortTailCall(nums, pivot + 1, right);\n            // The remaining unsorted range is [left, pivot - 1]\n            right = pivot - 1;\n        }\n    }\n}\n</code></pre> quick_sort.kt<pre><code>/* Quick sort (recursion depth optimization) */\nfun quickSortTailCall(nums: IntArray, left: Int, right: Int) {\n    // Stop when the subarray has at most one element\n    var l = left\n    var r = right\n    while (l &lt; r) {\n        // Perform two-pointer partitioning\n        val pivot = partition(nums, l, r)\n        // Apply quick sort to the shorter of the two subarrays\n        if (pivot - l &lt; r - pivot) {\n            quickSort(nums, l, pivot - 1) // Recursively sort the left subarray\n            l = pivot + 1 // The remaining unsorted range is [pivot + 1, r]\n        } else {\n            quickSort(nums, pivot + 1, r) // Recursively sort the right subarray\n            r = pivot - 1 // The remaining unsorted range is [l, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.rb<pre><code>### Quick sort (recursion depth optimization)###\ndef quick_sort(nums, left, right)\n  # Continue while the subarray contains more than one element\n  while left &lt; right\n    # Two-pointer partitioning\n    pivot = partition(nums, left, right)\n    # Apply quick sort to the shorter of the two subarrays\n    if pivot - left &lt; right - pivot\n      quick_sort(nums, left, pivot - 1)\n      left = pivot + 1 # The remaining unsorted range is [pivot + 1, right]\n    else\n      quick_sort(nums, pivot + 1, right)\n      right = pivot - 1 # The remaining unsorted range is [left, pivot - 1]\n    end\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 11.   Sorting","11.5   Quick sort"],"tags":[]},{"location":"chapter_sorting/radix_sort/","level":1,"title":"11.10   Radix sort","text":"<p>The previous section introduced counting sort, which suits a large number of elements \\(n\\) with a small value range \\(m\\). Suppose we need to sort \\(n = 10^6\\) student IDs, each an \\(8\\)-digit number. The value range \\(m = 10^8\\) is very large, so counting sort would require substantial memory. Radix sort avoids this problem.</p> <p>Radix sort shares counting sort's core idea of sorting by counting occurrences. It also uses the relative significance of digit positions, sorting one digit at a time to obtain the final order.</p>","path":["Chapter 11.   Sorting","11.10   Radix sort"],"tags":[]},{"location":"chapter_sorting/radix_sort/#11101-algorithm-steps","level":2,"title":"11.10.1   Algorithm steps","text":"<p>For the student IDs, number the least significant digit as \\(1\\) and the most significant as \\(8\\). Radix sort proceeds as shown below.</p> <ol> <li>Initialize the digit position \\(k = 1\\).</li> <li>Apply counting sort to the \\(k\\)th digit of each student ID. The data is now sorted in ascending order by its \\(k\\)th digit.</li> <li>Increase the digit position \\(k\\) by \\(1\\) and return to step <code>2.</code>. Continue until all digit positions have been processed.</li> </ol> <p></p> <p> Figure 11-18   The radix sort process </p> <p>Now consider the implementation. For a number \\(x\\) in base \\(d\\), its \\(k\\)th digit \\(x_k\\) can be obtained with the following formula:</p> \\[ x_k = \\lfloor\\frac{x}{d^{k-1}}\\rfloor \\bmod d \\] <p>Here, \\(\\lfloor a \\rfloor\\) rounds the floating-point value \\(a\\) down, and \\(\\bmod \\: d\\) takes the remainder modulo \\(d\\). For the student IDs, \\(d = 10\\) and \\(k \\in [1, 8]\\).</p> <p>We also need to modify counting sort slightly so that it sorts by the \\(k\\)th digit:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby radix_sort.py<pre><code>def digit(num: int, exp: int) -&gt; int:\n    \"\"\"Get the k-th digit of num, where exp = 10^(k-1)\"\"\"\n    # Passing exp instead of k avoids repeatedly performing expensive exponentiation here\n    return (num // exp) % 10\n\ndef counting_sort_digit(nums: list[int], exp: int):\n    \"\"\"Counting sort (sort nums by the k-th digit)\"\"\"\n    # Decimal digits range from 0 to 9, so the bucket array needs length 10\n    counter = [0] * 10\n    n = len(nums)\n    # Count the occurrences of each digit from 0 to 9\n    for i in range(n):\n        d = digit(nums[i], exp)  # Get the k-th digit of nums[i] and store it in d\n        counter[d] += 1  # Count the occurrences of digit d\n    # Compute prefix sums to convert occurrence counts into array positions\n    for i in range(1, 10):\n        counter[i] += counter[i - 1]\n    # Traverse in reverse and place each element into res according to the bucket counts\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        d = digit(nums[i], exp)\n        j = counter[d] - 1  # Get the destination index j for digit d in the array\n        res[j] = nums[i]  # Place the current element at index j\n        counter[d] -= 1  # Decrease the count for d by 1\n    # Overwrite the original array nums with the result\n    for i in range(n):\n        nums[i] = res[i]\n\ndef radix_sort(nums: list[int]):\n    \"\"\"Radix sort\"\"\"\n    # Get the largest array element to determine the maximum number of digits\n    m = max(nums)\n    # Process digits from least significant to most significant\n    exp = 1\n    while exp &lt;= m:\n        # Apply counting sort to the k-th digit of the array elements\n        # k = 1 -&gt; exp = 1\n        # k = 2 -&gt; exp = 10\n        # That is, exp = 10^(k-1)\n        counting_sort_digit(nums, exp)\n        exp *= 10\n</code></pre> radix_sort.cpp<pre><code>/* Get the k-th digit of num, where exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // Passing exp instead of k avoids repeatedly performing expensive exponentiation here\n    return (num / exp) % 10;\n}\n\n/* Counting sort (sort nums by the k-th digit) */\nvoid countingSortDigit(vector&lt;int&gt; &amp;nums, int exp) {\n    // Decimal digits range from 0 to 9, so the bucket array needs length 10\n    vector&lt;int&gt; counter(10, 0);\n    int n = nums.size();\n    // Count the occurrences of each digit from 0 to 9\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // Get the k-th digit of nums[i] and store it in d\n        counter[d]++;                // Count the occurrences of digit d\n    }\n    // Compute prefix sums to convert occurrence counts into array positions\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // Traverse in reverse and place each element into res according to the bucket counts\n    vector&lt;int&gt; res(n, 0);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // Get the destination index j for digit d in the array\n        res[j] = nums[i];       // Place the current element at index j\n        counter[d]--;           // Decrease the count for d by 1\n    }\n    // Overwrite the original array nums with the result\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* Radix sort */\nvoid radixSort(vector&lt;int&gt; &amp;nums) {\n    // Get the largest array element to determine the maximum number of digits\n    int m = *max_element(nums.begin(), nums.end());\n    // Process digits from least significant to most significant\n    for (int exp = 1; exp &lt;= m; exp *= 10)\n        // Apply counting sort to the k-th digit of the array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // That is, exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.java<pre><code>/* Get the k-th digit of num, where exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // Passing exp instead of k avoids repeatedly performing expensive exponentiation here\n    return (num / exp) % 10;\n}\n\n/* Counting sort (sort nums by the k-th digit) */\nvoid countingSortDigit(int[] nums, int exp) {\n    // Decimal digits range from 0 to 9, so the bucket array needs length 10\n    int[] counter = new int[10];\n    int n = nums.length;\n    // Count the occurrences of each digit from 0 to 9\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // Get the k-th digit of nums[i] and store it in d\n        counter[d]++;                // Count the occurrences of digit d\n    }\n    // Compute prefix sums to convert occurrence counts into array positions\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // Traverse in reverse and place each element into res according to the bucket counts\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // Get the destination index j for digit d in the array\n        res[j] = nums[i];       // Place the current element at index j\n        counter[d]--;           // Decrease the count for d by 1\n    }\n    // Overwrite the original array nums with the result\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* Radix sort */\nvoid radixSort(int[] nums) {\n    // Get the largest array element to determine the maximum number of digits\n    int m = Integer.MIN_VALUE;\n    for (int num : nums)\n        if (num &gt; m)\n            m = num;\n    // Process digits from least significant to most significant\n    for (int exp = 1; exp &lt;= m; exp *= 10) {\n        // Apply counting sort to the k-th digit of the array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // That is, exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.cs<pre><code>/* Get the k-th digit of num, where exp = 10^(k-1) */\nint Digit(int num, int exp) {\n    // Passing exp instead of k avoids repeatedly performing expensive exponentiation here\n    return (num / exp) % 10;\n}\n\n/* Counting sort (sort nums by the k-th digit) */\nvoid CountingSortDigit(int[] nums, int exp) {\n    // Decimal digits range from 0 to 9, so the bucket array needs length 10\n    int[] counter = new int[10];\n    int n = nums.Length;\n    // Count the occurrences of each digit from 0 to 9\n    for (int i = 0; i &lt; n; i++) {\n        int d = Digit(nums[i], exp); // Get the k-th digit of nums[i] and store it in d\n        counter[d]++;                // Count the occurrences of digit d\n    }\n    // Compute prefix sums to convert occurrence counts into array positions\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // Traverse in reverse and place each element into res according to the bucket counts\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = Digit(nums[i], exp);\n        int j = counter[d] - 1; // Get the destination index j for digit d in the array\n        res[j] = nums[i];       // Place the current element at index j\n        counter[d]--;           // Decrease the count for d by 1\n    }\n    // Overwrite the original array nums with the result\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* Radix sort */\nvoid RadixSort(int[] nums) {\n    // Get the largest array element to determine the maximum number of digits\n    int m = int.MinValue;\n    foreach (int num in nums) {\n        if (num &gt; m) m = num;\n    }\n    // Process digits from least significant to most significant\n    for (int exp = 1; exp &lt;= m; exp *= 10) {\n        // Apply counting sort to the k-th digit of the array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // That is, exp = 10^(k-1)\n        CountingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.go<pre><code>/* Get the k-th digit of num, where exp = 10^(k-1) */\nfunc digit(num, exp int) int {\n    // Passing exp instead of k avoids repeatedly performing expensive exponentiation here\n    return (num / exp) % 10\n}\n\n/* Counting sort (sort nums by the k-th digit) */\nfunc countingSortDigit(nums []int, exp int) {\n    // Decimal digits range from 0 to 9, so the bucket array needs length 10\n    counter := make([]int, 10)\n    n := len(nums)\n    // Count the occurrences of each digit from 0 to 9\n    for i := 0; i &lt; n; i++ {\n        d := digit(nums[i], exp) // Get the k-th digit of nums[i] and store it in d\n        counter[d]++             // Count the occurrences of digit d\n    }\n    // Compute prefix sums to convert occurrence counts into array positions\n    for i := 1; i &lt; 10; i++ {\n        counter[i] += counter[i-1]\n    }\n    // Traverse in reverse and place each element into res according to the bucket counts\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        d := digit(nums[i], exp)\n        j := counter[d] - 1 // Get the destination index j for digit d in the array\n        res[j] = nums[i]    // Place the current element at index j\n        counter[d]--        // Decrease the count for d by 1\n    }\n    // Overwrite the original array nums with the result\n    for i := 0; i &lt; n; i++ {\n        nums[i] = res[i]\n    }\n}\n\n/* Radix sort */\nfunc radixSort(nums []int) {\n    // Get the largest array element to determine the maximum number of digits\n    max := math.MinInt\n    for _, num := range nums {\n        if num &gt; max {\n            max = num\n        }\n    }\n    // Process digits from least significant to most significant\n    for exp := 1; max &gt;= exp; exp *= 10 {\n        // Apply counting sort to the k-th digit of the array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // That is, exp = 10^(k-1)\n        countingSortDigit(nums, exp)\n    }\n}\n</code></pre> radix_sort.swift<pre><code>/* Get the k-th digit of num, where exp = 10^(k-1) */\nfunc digit(num: Int, exp: Int) -&gt; Int {\n    // Passing exp instead of k avoids repeatedly performing expensive exponentiation here\n    (num / exp) % 10\n}\n\n/* Counting sort (sort nums by the k-th digit) */\nfunc countingSortDigit(nums: inout [Int], exp: Int) {\n    // Decimal digits range from 0 to 9, so the bucket array needs length 10\n    var counter = Array(repeating: 0, count: 10)\n    // Count the occurrences of each digit from 0 to 9\n    for i in nums.indices {\n        let d = digit(num: nums[i], exp: exp) // Get the k-th digit of nums[i] and store it in d\n        counter[d] += 1 // Count the occurrences of digit d\n    }\n    // Compute prefix sums to convert occurrence counts into array positions\n    for i in 1 ..&lt; 10 {\n        counter[i] += counter[i - 1]\n    }\n    // Traverse in reverse and place each element into res according to the bucket counts\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let d = digit(num: nums[i], exp: exp)\n        let j = counter[d] - 1 // Get the destination index j for digit d in the array\n        res[j] = nums[i] // Place the current element at index j\n        counter[d] -= 1 // Decrease the count for d by 1\n    }\n    // Overwrite the original array nums with the result\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n\n/* Radix sort */\nfunc radixSort(nums: inout [Int]) {\n    // Get the largest array element to determine the maximum number of digits\n    var m = Int.min\n    for num in nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // Process digits from least significant to most significant\n    for exp in sequence(first: 1, next: { m &gt;= ($0 * 10) ? $0 * 10 : nil }) {\n        // Apply counting sort to the k-th digit of the array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // That is, exp = 10^(k-1)\n        countingSortDigit(nums: &amp;nums, exp: exp)\n    }\n}\n</code></pre> radix_sort.js<pre><code>/* Get the k-th digit of num, where exp = 10^(k-1) */\nfunction digit(num, exp) {\n    // Passing exp instead of k avoids repeatedly performing expensive exponentiation here\n    return Math.floor(num / exp) % 10;\n}\n\n/* Counting sort (sort nums by the k-th digit) */\nfunction countingSortDigit(nums, exp) {\n    // Decimal digits range from 0 to 9, so the bucket array needs length 10\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // Count the occurrences of each digit from 0 to 9\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // Get the k-th digit of nums[i] and store it in d\n        counter[d]++; // Count the occurrences of digit d\n    }\n    // Compute prefix sums to convert occurrence counts into array positions\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // Traverse in reverse and place each element into res according to the bucket counts\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // Get the destination index j for digit d in the array\n        res[j] = nums[i]; // Place the current element at index j\n        counter[d]--; // Decrease the count for d by 1\n    }\n    // Overwrite the original array nums with the result\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* Radix sort */\nfunction radixSort(nums) {\n    // Get the largest array element to determine the maximum number of digits\n    let m = Math.max(... nums);\n    // Process digits from least significant to most significant\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // Apply counting sort to the k-th digit of the array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // That is, exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.ts<pre><code>/* Get the k-th digit of num, where exp = 10^(k-1) */\nfunction digit(num: number, exp: number): number {\n    // Passing exp instead of k avoids repeatedly performing expensive exponentiation here\n    return Math.floor(num / exp) % 10;\n}\n\n/* Counting sort (sort nums by the k-th digit) */\nfunction countingSortDigit(nums: number[], exp: number): void {\n    // Decimal digits range from 0 to 9, so the bucket array needs length 10\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // Count the occurrences of each digit from 0 to 9\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // Get the k-th digit of nums[i] and store it in d\n        counter[d]++; // Count the occurrences of digit d\n    }\n    // Compute prefix sums to convert occurrence counts into array positions\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // Traverse in reverse and place each element into res according to the bucket counts\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // Get the destination index j for digit d in the array\n        res[j] = nums[i]; // Place the current element at index j\n        counter[d]--; // Decrease the count for d by 1\n    }\n    // Overwrite the original array nums with the result\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* Radix sort */\nfunction radixSort(nums: number[]): void {\n    // Get the largest array element to determine the maximum number of digits\n    let m: number = Math.max(... nums);\n    // Process digits from least significant to most significant\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // Apply counting sort to the k-th digit of the array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // That is, exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.dart<pre><code>/* Get the k-th digit of _num, where exp = 10^(k-1) */\nint digit(int _num, int exp) {\n  // Passing exp instead of k avoids repeatedly performing expensive exponentiation here\n  return (_num ~/ exp) % 10;\n}\n\n/* Counting sort (sort nums by the k-th digit) */\nvoid countingSortDigit(List&lt;int&gt; nums, int exp) {\n  // Decimal digits range from 0 to 9, so the bucket array needs length 10\n  List&lt;int&gt; counter = List&lt;int&gt;.filled(10, 0);\n  int n = nums.length;\n  // Count the occurrences of each digit from 0 to 9\n  for (int i = 0; i &lt; n; i++) {\n    int d = digit(nums[i], exp); // Get the k-th digit of nums[i] and store it in d\n    counter[d]++; // Count the occurrences of digit d\n  }\n  // Compute prefix sums to convert occurrence counts into array positions\n  for (int i = 1; i &lt; 10; i++) {\n    counter[i] += counter[i - 1];\n  }\n  // Traverse in reverse and place each element into res according to the bucket counts\n  List&lt;int&gt; res = List&lt;int&gt;.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int d = digit(nums[i], exp);\n    int j = counter[d] - 1; // Get the destination index j for digit d in the array\n    res[j] = nums[i]; // Place the current element at index j\n    counter[d]--; // Decrease the count for d by 1\n  }\n  // Overwrite the original array nums with the result\n  for (int i = 0; i &lt; n; i++) nums[i] = res[i];\n}\n\n/* Radix sort */\nvoid radixSort(List&lt;int&gt; nums) {\n  // Get the largest array element to determine the maximum number of digits\n  // On native platforms, Dart int values are 64-bit integers\n  int m = -1 &lt;&lt; 63;\n  for (int _num in nums) if (_num &gt; m) m = _num;\n  // Process digits from least significant to most significant\n  for (int exp = 1; exp &lt;= m; exp *= 10)\n    // Apply counting sort to the k-th digit of the array elements\n    // k = 1 -&gt; exp = 1\n    // k = 2 -&gt; exp = 10\n    // That is, exp = 10^(k-1)\n    countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.rs<pre><code>/* Get the k-th digit of num, where exp = 10^(k-1) */\nfn digit(num: i32, exp: i32) -&gt; usize {\n    // Passing exp instead of k avoids repeatedly performing expensive exponentiation here\n    return ((num / exp) % 10) as usize;\n}\n\n/* Counting sort (sort nums by the k-th digit) */\nfn counting_sort_digit(nums: &amp;mut [i32], exp: i32) {\n    // Decimal digits range from 0 to 9, so the bucket array needs length 10\n    let mut counter = [0; 10];\n    let n = nums.len();\n    // Count the occurrences of each digit from 0 to 9\n    for i in 0..n {\n        let d = digit(nums[i], exp); // Get the k-th digit of nums[i] and store it in d\n        counter[d] += 1; // Count the occurrences of digit d\n    }\n    // Compute prefix sums to convert occurrence counts into array positions\n    for i in 1..10 {\n        counter[i] += counter[i - 1];\n    }\n    // Traverse in reverse and place each element into res according to the bucket counts\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let d = digit(nums[i], exp);\n        let j = counter[d] - 1; // Get the destination index j for digit d in the array\n        res[j] = nums[i]; // Place the current element at index j\n        counter[d] -= 1; // Decrease the count for d by 1\n    }\n    // Overwrite the original array nums with the result\n    nums.copy_from_slice(&amp;res);\n}\n\n/* Radix sort */\nfn radix_sort(nums: &amp;mut [i32]) {\n    // Get the largest array element to determine the maximum number of digits\n    let m = *nums.into_iter().max().unwrap();\n    // Process digits from least significant to most significant\n    let mut exp = 1;\n    while exp &lt;= m {\n        counting_sort_digit(nums, exp);\n        exp *= 10;\n    }\n}\n</code></pre> radix_sort.c<pre><code>/* Get the k-th digit of num, where exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // Passing exp instead of k avoids repeatedly performing expensive exponentiation here\n    return (num / exp) % 10;\n}\n\n/* Counting sort (sort nums by the k-th digit) */\nvoid countingSortDigit(int nums[], int size, int exp) {\n    // Decimal digits range from 0 to 9, so the bucket array needs length 10\n    int *counter = (int *)malloc((sizeof(int) * 10));\n    memset(counter, 0, sizeof(int) * 10); // Initialize the counts to 0\n    // Count the occurrences of each digit from 0 to 9\n    for (int i = 0; i &lt; size; i++) {\n        // Get the k-th digit of nums[i] and store it in d\n        int d = digit(nums[i], exp);\n        // Count the occurrences of digit d\n        counter[d]++;\n    }\n    // Compute prefix sums to convert occurrence counts into array positions\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // Traverse in reverse and place each element into res according to the bucket counts\n    int *res = (int *)malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // Get the destination index j for digit d in the array\n        res[j] = nums[i];       // Place the current element at index j\n        counter[d]--;           // Decrease the count for d by 1\n    }\n    // Overwrite the original array nums with the result\n    for (int i = 0; i &lt; size; i++) {\n        nums[i] = res[i];\n    }\n    // Free memory\n    free(res);\n    free(counter);\n}\n\n/* Radix sort */\nvoid radixSort(int nums[], int size) {\n    // Get the largest array element to determine the maximum number of digits\n    int max = INT32_MIN;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; max) {\n            max = nums[i];\n        }\n    }\n    // Process digits from least significant to most significant\n    for (int exp = 1; max &gt;= exp; exp *= 10)\n        // Apply counting sort to the k-th digit of the array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // That is, exp = 10^(k-1)\n        countingSortDigit(nums, size, exp);\n}\n</code></pre> radix_sort.kt<pre><code>/* Get the k-th digit of num, where exp = 10^(k-1) */\nfun digit(num: Int, exp: Int): Int {\n    // Passing exp instead of k avoids repeatedly performing expensive exponentiation here\n    return (num / exp) % 10\n}\n\n/* Counting sort (sort nums by the k-th digit) */\nfun countingSortDigit(nums: IntArray, exp: Int) {\n    // Decimal digits range from 0 to 9, so the bucket array needs length 10\n    val counter = IntArray(10)\n    val n = nums.size\n    // Count the occurrences of each digit from 0 to 9\n    for (i in 0..&lt;n) {\n        val d = digit(nums[i], exp) // Get the k-th digit of nums[i] and store it in d\n        counter[d]++                // Count the occurrences of digit d\n    }\n    // Compute prefix sums to convert occurrence counts into array positions\n    for (i in 1..9) {\n        counter[i] += counter[i - 1]\n    }\n    // Traverse in reverse and place each element into res according to the bucket counts\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val d = digit(nums[i], exp)\n        val j = counter[d] - 1 // Get the destination index j for digit d in the array\n        res[j] = nums[i]       // Place the current element at index j\n        counter[d]--           // Decrease the count for d by 1\n    }\n    // Overwrite the original array nums with the result\n    for (i in 0..&lt;n)\n        nums[i] = res[i]\n}\n\n/* Radix sort */\nfun radixSort(nums: IntArray) {\n    // Get the largest array element to determine the maximum number of digits\n    var m = Int.MIN_VALUE\n    for (num in nums) if (num &gt; m) m = num\n    var exp = 1\n    // Process digits from least significant to most significant\n    while (exp &lt;= m) {\n        // Apply counting sort to the k-th digit of the array elements\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // That is, exp = 10^(k-1)\n        countingSortDigit(nums, exp)\n        exp *= 10\n    }\n}\n</code></pre> radix_sort.rb<pre><code>### Get the k-th digit of num, where exp = 10^(k-1) ###\ndef digit(num, exp)\n  # Passing exp instead of k avoids repeatedly performing expensive exponentiation here\n  (num / exp) % 10\nend\n\n### Counting sort (sort nums by the k-th digit)###\ndef counting_sort_digit(nums, exp)\n  # Decimal digits range from 0 to 9, so the bucket array needs length 10\n  counter = Array.new(10, 0)\n  n = nums.length\n  # Count the occurrences of each digit from 0 to 9\n  for i in 0...n\n    d = digit(nums[i], exp) # Get the k-th digit of nums[i] and store it in d\n    counter[d] += 1 # Count the occurrences of digit d\n  end\n  # Compute prefix sums to convert occurrence counts into array positions\n  (1...10).each { |i| counter[i] += counter[i - 1] }\n  # Traverse in reverse and place each element into res according to the bucket counts\n  res = Array.new(n, 0)\n  for i in (n - 1).downto(0)\n    d = digit(nums[i], exp)\n    j = counter[d] - 1 # Get the destination index j for digit d in the array\n    res[j] = nums[i] # Place the current element at index j\n    counter[d] -= 1 # Decrease the count for d by 1\n  end\n  # Overwrite the original array nums with the result\n  (0...n).each { |i| nums[i] = res[i] }\nend\n\n### Radix sort ###\ndef radix_sort(nums)\n  # Get the largest array element to determine the maximum number of digits\n  m = nums.max\n  # Process digits from least significant to most significant\n  exp = 1\n  while exp &lt;= m\n    # Apply counting sort to the k-th digit of the array elements\n    # k = 1 -&gt; exp = 1\n    # k = 2 -&gt; exp = 10\n    # That is, exp = 10^(k-1)\n    counting_sort_digit(nums, exp)\n    exp *= 10\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Why start with the least significant digit?</p> <p>In successive rounds, a later sort takes precedence over an earlier one. For example, if the first round orders two elements as \\(a &lt; b\\) but the second orders them as \\(a &gt; b\\), the second result replaces the first. Higher digits have greater priority, so lower digits must be sorted first.</p>","path":["Chapter 11.   Sorting","11.10   Radix sort"],"tags":[]},{"location":"chapter_sorting/radix_sort/#11102-algorithm-properties","level":2,"title":"11.10.2   Algorithm properties","text":"<p>Compared with counting sort, radix sort handles a larger range of values, provided the data can be represented with a fixed number of digits and that number is not too large. For example, floating-point numbers are unsuitable because their digit count \\(k\\) may be so large that \\(O(nk) \\gg O(n^2)\\).</p> <ul> <li>Time complexity: \\(O(nk)\\); not adaptive. For \\(n\\) elements represented in base \\(d\\) with at most \\(k\\) digits, counting sort on one digit takes \\(O(n + d)\\) time. Sorting all \\(k\\) digits takes \\(O((n + d)k)\\). Usually \\(d\\) and \\(k\\) are relatively small, so the complexity approaches \\(O(n)\\).</li> <li>Space complexity: \\(O(n + d)\\); not in-place. Like counting sort, radix sort requires arrays <code>res</code> and <code>counter</code> of lengths \\(n\\) and \\(d\\).</li> <li>Stable. Radix sort is stable when the counting sort used within it is stable. If counting sort is not stable, radix sort cannot guarantee a correctly sorted result.</li> </ul>","path":["Chapter 11.   Sorting","11.10   Radix sort"],"tags":[]},{"location":"chapter_sorting/selection_sort/","level":1,"title":"11.2   Selection sort","text":"<p>Selection sort works very simply: in each round, select the smallest element from the unsorted section and place it at the end of the sorted section.</p> <p>For an array of length \\(n\\), the algorithm proceeds as shown below.</p> <ol> <li>Initially, all elements are unsorted, so the unsorted index interval is \\([0, n-1]\\).</li> <li>Select the smallest element in \\([0, n-1]\\) and swap it with the element at index \\(0\\). The first element is now sorted.</li> <li>Select the smallest element in \\([1, n-1]\\) and swap it with the element at index \\(1\\). The first two elements are now sorted.</li> <li>Continue in this way. After \\(n - 1\\) rounds of selection and swapping, the first \\(n - 1\\) elements are sorted.</li> <li>The remaining element must be the largest and needs no sorting. The array is now sorted.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 11-2   Steps of selection sort </p> <p>In the code, \\(k\\) tracks the smallest element in the unsorted section:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby selection_sort.py<pre><code>def selection_sort(nums: list[int]):\n    \"\"\"Selection sort\"\"\"\n    n = len(nums)\n    # Outer loop: the unsorted range is [i, n-1]\n    for i in range(n - 1):\n        # Inner loop: find the smallest element in the unsorted range\n        k = i\n        for j in range(i + 1, n):\n            if nums[j] &lt; nums[k]:\n                k = j  # Record the index of the smallest element\n        # Swap the smallest element with the first element of the unsorted range\n        nums[i], nums[k] = nums[k], nums[i]\n</code></pre> selection_sort.cpp<pre><code>/* Selection sort */\nvoid selectionSort(vector&lt;int&gt; &amp;nums) {\n    int n = nums.size();\n    // Outer loop: the unsorted range is [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // Inner loop: find the smallest element in the unsorted range\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // Record the index of the smallest element\n        }\n        // Swap the smallest element with the first element of the unsorted range\n        swap(nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.java<pre><code>/* Selection sort */\nvoid selectionSort(int[] nums) {\n    int n = nums.length;\n    // Outer loop: the unsorted range is [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // Inner loop: find the smallest element in the unsorted range\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // Record the index of the smallest element\n        }\n        // Swap the smallest element with the first element of the unsorted range\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.cs<pre><code>/* Selection sort */\nvoid SelectionSort(int[] nums) {\n    int n = nums.Length;\n    // Outer loop: the unsorted range is [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // Inner loop: find the smallest element in the unsorted range\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // Record the index of the smallest element\n        }\n        // Swap the smallest element with the first element of the unsorted range\n        (nums[k], nums[i]) = (nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.go<pre><code>/* Selection sort */\nfunc selectionSort(nums []int) {\n    n := len(nums)\n    // Outer loop: the unsorted range is [i, n-1]\n    for i := 0; i &lt; n-1; i++ {\n        // Inner loop: find the smallest element in the unsorted range\n        k := i\n        for j := i + 1; j &lt; n; j++ {\n            if nums[j] &lt; nums[k] {\n                // Record the index of the smallest element\n                k = j\n            }\n        }\n        // Swap the smallest element with the first element of the unsorted range\n        nums[i], nums[k] = nums[k], nums[i]\n\n    }\n}\n</code></pre> selection_sort.swift<pre><code>/* Selection sort */\nfunc selectionSort(nums: inout [Int]) {\n    // Outer loop: the unsorted range is [i, n-1]\n    for i in nums.indices.dropLast() {\n        // Inner loop: find the smallest element in the unsorted range\n        var k = i\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[j] &lt; nums[k] {\n                k = j // Record the index of the smallest element\n            }\n        }\n        // Swap the smallest element with the first element of the unsorted range\n        nums.swapAt(i, k)\n    }\n}\n</code></pre> selection_sort.js<pre><code>/* Selection sort */\nfunction selectionSort(nums) {\n    let n = nums.length;\n    // Outer loop: the unsorted range is [i, n-1]\n    for (let i = 0; i &lt; n - 1; i++) {\n        // Inner loop: find the smallest element in the unsorted range\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // Record the index of the smallest element\n            }\n        }\n        // Swap the smallest element with the first element of the unsorted range\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.ts<pre><code>/* Selection sort */\nfunction selectionSort(nums: number[]): void {\n    let n = nums.length;\n    // Outer loop: the unsorted range is [i, n-1]\n    for (let i = 0; i &lt; n - 1; i++) {\n        // Inner loop: find the smallest element in the unsorted range\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // Record the index of the smallest element\n            }\n        }\n        // Swap the smallest element with the first element of the unsorted range\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.dart<pre><code>/* Selection sort */\nvoid selectionSort(List&lt;int&gt; nums) {\n  int n = nums.length;\n  // Outer loop: the unsorted range is [i, n-1]\n  for (int i = 0; i &lt; n - 1; i++) {\n    // Inner loop: find the smallest element in the unsorted range\n    int k = i;\n    for (int j = i + 1; j &lt; n; j++) {\n      if (nums[j] &lt; nums[k]) k = j; // Record the index of the smallest element\n    }\n    // Swap the smallest element with the first element of the unsorted range\n    int temp = nums[i];\n    nums[i] = nums[k];\n    nums[k] = temp;\n  }\n}\n</code></pre> selection_sort.rs<pre><code>/* Selection sort */\nfn selection_sort(nums: &amp;mut [i32]) {\n    if nums.is_empty() {\n        return;\n    }\n    let n = nums.len();\n    // Outer loop: the unsorted range is [i, n-1]\n    for i in 0..n - 1 {\n        // Inner loop: find the smallest element in the unsorted range\n        let mut k = i;\n        for j in i + 1..n {\n            if nums[j] &lt; nums[k] {\n                k = j; // Record the index of the smallest element\n            }\n        }\n        // Swap the smallest element with the first element of the unsorted range\n        nums.swap(i, k);\n    }\n}\n</code></pre> selection_sort.c<pre><code>/* Selection sort */\nvoid selectionSort(int nums[], int n) {\n    // Outer loop: the unsorted range is [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // Inner loop: find the smallest element in the unsorted range\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // Record the index of the smallest element\n        }\n        // Swap the smallest element with the first element of the unsorted range\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.kt<pre><code>/* Selection sort */\nfun selectionSort(nums: IntArray) {\n    val n = nums.size\n    // Outer loop: the unsorted range is [i, n-1]\n    for (i in 0..&lt;n - 1) {\n        var k = i\n        // Inner loop: find the smallest element in the unsorted range\n        for (j in i + 1..&lt;n) {\n            if (nums[j] &lt; nums[k])\n                k = j // Record the index of the smallest element\n        }\n        // Swap the smallest element with the first element of the unsorted range\n        val temp = nums[i]\n        nums[i] = nums[k]\n        nums[k] = temp\n    }\n}\n</code></pre> selection_sort.rb<pre><code>### Selection sort ###\ndef selection_sort(nums)\n  n = nums.length\n  # Outer loop: the unsorted range is [i, n-1]\n  for i in 0...(n - 1)\n    # Inner loop: find the smallest element in the unsorted range\n    k = i\n    for j in (i + 1)...n\n      if nums[j] &lt; nums[k]\n        k = j # Record the index of the smallest element\n      end\n    end\n    # Swap the smallest element with the first element of the unsorted range\n    nums[i], nums[k] = nums[k], nums[i]\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 11.   Sorting","11.2   Selection sort"],"tags":[]},{"location":"chapter_sorting/selection_sort/#1121-algorithm-properties","level":2,"title":"11.2.1   Algorithm properties","text":"<ul> <li>Time complexity: \\(O(n^2)\\); not adaptive. The outer loop runs \\(n - 1\\) times. The inner loop runs \\(n - 1\\) times in the first round and \\(1\\) time in the last, giving successive counts of \\(n - 1\\), \\(n - 2\\), \\(\\dots\\), \\(2\\), and \\(1\\), which sum to \\(\\frac{n(n - 1)}{2}\\).</li> <li>Space complexity: \\(O(1)\\); in-place. Pointers \\(i\\) and \\(j\\) use constant extra space.</li> <li>Not stable. As shown below, <code>nums[i]</code> may be swapped to the right of an equal element, changing their relative order.</li> </ul> <p> Figure 11-3   An example showing that selection sort is not stable </p>","path":["Chapter 11.   Sorting","11.2   Selection sort"],"tags":[]},{"location":"chapter_sorting/sorting_algorithm/","level":1,"title":"11.1   Sorting algorithms","text":"<p>A sorting algorithm arranges a dataset in a specified order. Sorting is widely used because ordered data is often easier to search, analyze, and process efficiently.</p> <p>As shown below, the data may consist of integers, floating-point numbers, characters, strings, and other types. The ordering rule can be chosen as needed: numerical magnitude, ASCII character order, or a custom rule.</p> <p></p> <p> Figure 11-1   Examples of data types and ordering rules </p>","path":["Chapter 11.   Sorting","11.1   Sorting algorithms"],"tags":[]},{"location":"chapter_sorting/sorting_algorithm/#1111-evaluation-criteria","level":2,"title":"11.1.1   Evaluation criteria","text":"<p>Efficiency: We want low time complexity and few operations overall, corresponding to smaller constant factors in the complexity. Efficiency is especially important for large datasets.</p> <p>In-place operation: An in-place sort works directly on the original array without an auxiliary array, saving memory. It usually moves data fewer times and runs faster as well.</p> <p>Stability: A stable sort preserves the relative order of equal elements in the array.</p> <p>Stable sorting is necessary for multilevel sorting. Suppose a table stores student names in its first column and ages in its second. An unstable sort may destroy the input's existing order:</p> <pre><code># The input is already sorted by name\n# (name, age)\n  ('A', 19)\n  ('B', 18)\n  ('C', 21)\n  ('D', 19)\n  ('E', 23)\n\n# Suppose an unstable algorithm sorts the list by age.\n# The relative positions of ('D', 19) and ('A', 19) change,\n# so the original ordering by name is lost\n  ('B', 18)\n  ('D', 19)\n  ('A', 19)\n  ('C', 21)\n  ('E', 23)\n</code></pre> <p>Adaptiveness: An adaptive sort uses ordering already present in the input to reduce computation and improve efficiency. Its best-case time complexity is usually better than its average complexity.</p> <p>Comparison-based or not: A comparison-based sort uses comparison operators (\\(&lt;\\), \\(=\\), \\(&gt;\\)) to determine the relative order of elements and sort the array. Its worst-case time complexity has a lower bound of \\(\\Omega(n \\log n)\\). A non-comparison sort does not use comparison operators and can achieve \\(O(n)\\) time, but is less broadly applicable.</p>","path":["Chapter 11.   Sorting","11.1   Sorting algorithms"],"tags":[]},{"location":"chapter_sorting/sorting_algorithm/#1112-the-ideal-sorting-algorithm","level":2,"title":"11.1.2   The ideal sorting algorithm","text":"<p>Fast, in-place, stable, adaptive, and broadly applicable. No sorting algorithm with all these properties has yet been found. We must therefore choose according to the characteristics of the data and the requirements of the problem.</p> <p>Next, we will study different sorting algorithms and analyze their strengths and weaknesses using these criteria.</p>","path":["Chapter 11.   Sorting","11.1   Sorting algorithms"],"tags":[]},{"location":"chapter_sorting/summary/","level":1,"title":"11.11   Summary","text":"","path":["Chapter 11.   Sorting","11.11   Summary"],"tags":[]},{"location":"chapter_sorting/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<ul> <li>Bubble sort sorts by swapping adjacent elements. Adding a flag to stop early improves its best-case time complexity to \\(O(n)\\).</li> <li>Insertion sort inserts one element from the unsorted section into its correct position in the sorted section each round. Although its time complexity is \\(O(n^2)\\), its low operation count makes it popular for small datasets.</li> <li>Quick sort sorts through partitioning. Repeatedly choosing the worst pivot can degrade its time complexity to \\(O(n^2)\\). Median or random pivot selection reduces this risk. Recursing on the shorter subarray reduces recursion depth and improves space complexity to \\(O(\\log n)\\).</li> <li>Merge sort has division and merging phases, exemplifying divide and conquer. Sorting an array requires an auxiliary array and \\(O(n)\\) space, while sorting a linked list can be optimized to use \\(O(1)\\) space.</li> <li>Bucket sort has three steps: distribute data into buckets, sort within each bucket, and combine the results. It also uses divide and conquer and suits very large datasets. The key is distributing data evenly.</li> <li>Counting sort is a special case of bucket sort that sorts by counting occurrences. It suits large datasets with a limited value range and requires data that can be converted to positive integers.</li> <li>Radix sort sorts one digit at a time and requires data that can be represented as numbers with a fixed number of digits.</li> <li>Ideally, a sorting algorithm would be efficient, stable, in-place, and adaptive. As with other data structures and algorithms, however, no single sorting algorithm meets all these requirements. In practice, choose one according to the data's properties.</li> <li>Figure 11-19 compares common sorting algorithms in efficiency, stability, in-place operation, and adaptiveness.</li> </ul> <p> Figure 11-19   Comparison of sorting algorithms </p>","path":["Chapter 11.   Sorting","11.11   Summary"],"tags":[]},{"location":"chapter_sorting/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: When is stability necessary in a sorting algorithm?</p> <p>We often sort objects by one of their attributes. For example, students have names and heights, and we want a multilevel sort. First sort by name to obtain <code>(A, 180) (B, 185) (C, 170) (D, 170)</code>, then sort by height. An unstable algorithm may produce <code>(D, 170) (C, 170) (A, 180) (B, 185)</code>.</p> <p>Students D and C have switched positions, breaking the ordering by name. This is not what we want.</p> <p>Q: Can the right-to-left and left-to-right scans in partitioning be reversed?</p> <p>No. When the leftmost element is the pivot, the right-to-left scan must come first. This may seem counterintuitive, so let us examine why.</p> <p>The final step of <code>partition()</code> swaps <code>nums[left]</code> and <code>nums[i]</code>. Afterward, every element to the left of the pivot must be <code>&lt;=</code> the pivot, so <code>nums[left] &gt;= nums[i]</code> must hold before that swap. If we scan left to right first and fail to find a larger element, the loop stops at <code>i == j</code>, where it is possible that <code>nums[j] == nums[i] &gt; nums[left]</code>. The final swap would then move an element larger than the pivot to the far left, causing partitioning to fail.</p> <p>For example, given <code>[0, 0, 0, 0, 1]</code>, scanning left to right first produces <code>[1, 0, 0, 0, 0]</code> after partitioning, which is incorrect.</p> <p>Conversely, if we choose <code>nums[right]</code> as the pivot, we must scan left to right first.</p> <p>Q: In the recursion-depth optimization for quick sort, why does choosing the shorter subarray keep the depth at most \\(\\log n\\)?</p> <p>Recursion depth is the number of recursive calls that have not yet returned. Each partition divides the array into two subarrays. After the optimization, the subarray chosen for recursion is at most half the original length. Even in the worst case, where it is always exactly half, the depth is \\(\\log n\\).</p> <p>In the original quick sort, we may repeatedly recurse on the larger subarray, with lengths \\(n\\), \\(n - 1\\), \\(\\dots\\), \\(2\\), and \\(1\\) in the worst case, giving depth \\(n\\). The optimization prevents this.</p> <p>Q: If all array elements are equal, does quick sort take \\(O(n^2)\\) time? How can we handle this degeneration?</p> <p>Yes. In this case, consider partitioning into three sections: elements smaller than, equal to, and greater than the pivot. Recurse only on the smaller and greater sections. If all elements are equal, a single partition then completes the sort.</p> <p>Q: Why is bucket sort's worst-case time complexity \\(O(n^2)\\)?</p> <p>In the worst case, all elements fall into one bucket. If an \\(O(n^2)\\) algorithm is used to sort that bucket, the overall time complexity is \\(O(n^2)\\).</p>","path":["Chapter 11.   Sorting","11.11   Summary"],"tags":[]},{"location":"chapter_stack_and_queue/","level":1,"title":"Chapter 5.   Stacks and queues","text":"<p>Abstract</p> <p>A stack is like cats piled on top of one another; a queue is like cats standing in line.</p> <p>They represent last-in, first-out and first-in, first-out order, respectively.</p>","path":["Chapter 5.   Stacks and queues"],"tags":[]},{"location":"chapter_stack_and_queue/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>5.1   Stacks</li> <li>5.2   Queues</li> <li>5.3   Deques</li> <li>5.4   Summary</li> <li>5.5   Exercises</li> </ul>","path":["Chapter 5.   Stacks and queues"],"tags":[]},{"location":"chapter_stack_and_queue/deque/","level":1,"title":"5.3   Deques","text":"<p>A queue allows removal only at the front and insertion only at the rear. As shown below, a double-ended queue, or deque, is more flexible: elements can be added or removed at either end.</p> <p></p> <p> Figure 5-7   Deque operations </p>","path":["Chapter 5.   Stacks and queues","5.3   Deques"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#531-common-deque-operations","level":2,"title":"5.3.1   Common deque operations","text":"<p>Table 5-3 lists common operations. Actual method names depend on the programming language.</p> <p> Table 5-3   Efficiency of deque operations </p> Method Description Time complexity <code>push_first()</code> Add an element at the front \\(O(1)\\) <code>push_last()</code> Add an element at the rear \\(O(1)\\) <code>pop_first()</code> Remove the front element \\(O(1)\\) <code>pop_last()</code> Remove the rear element \\(O(1)\\) <code>peek_first()</code> Access the front element \\(O(1)\\) <code>peek_last()</code> Access the rear element \\(O(1)\\) <p>Again, we can use a deque class already provided by the programming language:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby deque.py<pre><code>from collections import deque\n\n# Initialize a deque\ndeq: deque[int] = deque()\n\n# Enqueue elements\ndeq.append(2)      # Add at the rear\ndeq.append(5)\ndeq.append(4)\ndeq.appendleft(3)  # Add at the front\ndeq.appendleft(1)\n\n# Access elements\nfront: int = deq[0]  # Front element\nrear: int = deq[-1]  # Rear element\n\n# Dequeue elements\npop_front: int = deq.popleft()  # Remove the front element\npop_rear: int = deq.pop()       # Remove the rear element\n\n# Get the deque size\nsize: int = len(deq)\n\n# Check whether the deque is empty\nis_empty: bool = len(deq) == 0\n</code></pre> deque.cpp<pre><code>/* Initialize a deque */\ndeque&lt;int&gt; deque;\n\n/* Enqueue elements */\ndeque.push_back(2);   // Add at the rear\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // Add at the front\ndeque.push_front(1);\n\n/* Access elements */\nint front = deque.front(); // Front element\nint back = deque.back();   // Rear element\n\n/* Dequeue elements */\ndeque.pop_front();  // Remove the front element\ndeque.pop_back();   // Remove the rear element\n\n/* Get the deque size */\nint size = deque.size();\n\n/* Check whether the deque is empty */\nbool empty = deque.empty();\n</code></pre> deque.java<pre><code>/* Initialize a deque */\nDeque&lt;Integer&gt; deque = new LinkedList&lt;&gt;();\n\n/* Enqueue elements */\ndeque.offerLast(2);   // Add at the rear\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // Add at the front\ndeque.offerFirst(1);\n\n/* Access elements */\nint peekFirst = deque.peekFirst();  // Front element\nint peekLast = deque.peekLast();    // Rear element\n\n/* Dequeue elements */\nint popFirst = deque.pollFirst();  // Remove the front element\nint popLast = deque.pollLast();    // Remove the rear element\n\n/* Get the deque size */\nint size = deque.size();\n\n/* Check whether the deque is empty */\nboolean isEmpty = deque.isEmpty();\n</code></pre> deque.cs<pre><code>/* Initialize a deque */\n// In C#, use a LinkedList as a deque\nLinkedList&lt;int&gt; deque = new();\n\n/* Enqueue elements */\ndeque.AddLast(2);   // Add at the rear\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // Add at the front\ndeque.AddFirst(1);\n\n/* Access elements */\nint peekFirst = deque.First.Value;  // Front element\nint peekLast = deque.Last.Value;    // Rear element\n\n/* Dequeue elements */\ndeque.RemoveFirst();  // Remove the front element\ndeque.RemoveLast();   // Remove the rear element\n\n/* Get the deque size */\nint size = deque.Count;\n\n/* Check whether the deque is empty */\nbool isEmpty = deque.Count == 0;\n</code></pre> deque_test.go<pre><code>/* Initialize a deque */\n// In Go, use a list as a deque\ndeque := list.New()\n\n/* Enqueue elements */\ndeque.PushBack(2)      // Add at the rear\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // Add at the front\ndeque.PushFront(1)\n\n/* Access elements */\nfront := deque.Front() // Front element\nrear := deque.Back()   // Rear element\n\n/* Dequeue elements */\ndeque.Remove(front)    // Remove the front element\ndeque.Remove(rear)     // Remove the rear element\n\n/* Get the deque size */\nsize := deque.Len()\n\n/* Check whether the deque is empty */\nisEmpty := deque.Len() == 0\n</code></pre> deque.swift<pre><code>/* Initialize a deque */\n// Swift has no built-in deque class; use an Array as a deque\nvar deque: [Int] = []\n\n/* Enqueue elements */\ndeque.append(2) // Add at the rear\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // Add at the front\ndeque.insert(1, at: 0)\n\n/* Access elements */\nlet peekFirst = deque.first! // Front element\nlet peekLast = deque.last! // Rear element\n\n/* Dequeue elements */\n// When using an Array, popFirst takes O(n) time\nlet popFirst = deque.removeFirst() // Remove the front element\nlet popLast = deque.removeLast() // Remove the rear element\n\n/* Get the deque size */\nlet size = deque.count\n\n/* Check whether the deque is empty */\nlet isEmpty = deque.isEmpty\n</code></pre> deque.js<pre><code>/* Initialize a deque */\n// JavaScript has no built-in deque; use an Array instead\nconst deque = [];\n\n/* Enqueue elements */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// Note that this is an array, so unshift() takes O(n) time\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* Access elements */\nconst peekFirst = deque[0];\nconst peekLast = deque[deque.length - 1];\n\n/* Dequeue elements */\n// Note that this is an array, so shift() takes O(n) time\nconst popFront = deque.shift();\nconst popBack = deque.pop();\n\n/* Get the deque size */\nconst size = deque.length;\n\n/* Check whether the deque is empty */\nconst isEmpty = size === 0;\n</code></pre> deque.ts<pre><code>/* Initialize a deque */\n// TypeScript has no built-in deque; use an Array instead\nconst deque: number[] = [];\n\n/* Enqueue elements */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// Note that this is an array, so unshift() takes O(n) time\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* Access elements */\nconst peekFirst: number = deque[0];\nconst peekLast: number = deque[deque.length - 1];\n\n/* Dequeue elements */\n// Note that this is an array, so shift() takes O(n) time\nconst popFront: number = deque.shift() as number;\nconst popBack: number = deque.pop() as number;\n\n/* Get the deque size */\nconst size: number = deque.length;\n\n/* Check whether the deque is empty */\nconst isEmpty: boolean = size === 0;\n</code></pre> deque.dart<pre><code>/* Initialize a deque */\n// In Dart, Queue is defined as a deque\nQueue&lt;int&gt; deque = Queue&lt;int&gt;();\n\n/* Enqueue elements */\ndeque.addLast(2);  // Add at the rear\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // Add at the front\ndeque.addFirst(1);\n\n/* Access elements */\nint peekFirst = deque.first; // Front element\nint peekLast = deque.last;   // Rear element\n\n/* Dequeue elements */\nint popFirst = deque.removeFirst(); // Remove the front element\nint popLast = deque.removeLast();   // Remove the rear element\n\n/* Get the deque size */\nint size = deque.length;\n\n/* Check whether the deque is empty */\nbool isEmpty = deque.isEmpty;\n</code></pre> deque.rs<pre><code>/* Initialize a deque */\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* Enqueue elements */\ndeque.push_back(2);  // Add at the rear\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // Add at the front\ndeque.push_front(1);\n\n/* Access elements */\nif let Some(front) = deque.front() { // Front element\n}\nif let Some(rear) = deque.back() {   // Rear element\n}\n\n/* Dequeue elements */\nif let Some(pop_front) = deque.pop_front() { // Remove the front element\n}\nif let Some(pop_rear) = deque.pop_back() {   // Remove the rear element\n}\n\n/* Get the deque size */\nlet size = deque.len();\n\n/* Check whether the deque is empty */\nlet is_empty = deque.is_empty();\n</code></pre> deque.c<pre><code>// C provides no built-in deque\n</code></pre> deque.kt<pre><code>/* Initialize a deque */\nval deque = LinkedList&lt;Int&gt;()\n\n/* Enqueue elements */\ndeque.offerLast(2)  // Add at the rear\ndeque.offerLast(5)\ndeque.offerLast(4)\ndeque.offerFirst(3) // Add at the front\ndeque.offerFirst(1)\n\n/* Access elements */\nval peekFirst = deque.peekFirst() // Front element\nval peekLast = deque.peekLast()   // Rear element\n\n/* Dequeue elements */\nval popFirst = deque.pollFirst() // Remove the front element\nval popLast = deque.pollLast()   // Remove the rear element\n\n/* Get the deque size */\nval size = deque.size\n\n/* Check whether the deque is empty */\nval isEmpty = deque.isEmpty()\n</code></pre> deque.rb<pre><code># Initialize a deque\n# Ruby has no built-in deque; use an Array instead\ndeque = []\n\n# Enqueue elements\ndeque &lt;&lt; 2\ndeque &lt;&lt; 5\ndeque &lt;&lt; 4\n# Note that this is an array, so Array#unshift takes O(n) time\ndeque.unshift(3)\ndeque.unshift(1)\n\n# Access elements\npeek_first = deque.first\npeek_last = deque.last\n\n# Dequeue elements\n# Note that this is an array, so Array#shift takes O(n) time\npop_front = deque.shift\npop_back = deque.pop\n\n# Get the deque size\nsize = deque.length\n\n# Check whether the deque is empty\nis_empty = size.zero?\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 5.   Stacks and queues","5.3   Deques"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#532-implementing-a-deque","level":2,"title":"5.3.2   Implementing a deque *","text":"<p>A deque is implemented much like a queue, using either a linked list or an array as the underlying structure.</p>","path":["Chapter 5.   Stacks and queues","5.3   Deques"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#1-implementation-with-a-doubly-linked-list","level":3,"title":"1.   Implementation with a doubly linked list","text":"<p>In the previous section, we used a singly linked list for a queue because it easily supports removing the head for dequeueing and adding after the tail for enqueueing.</p> <p>A deque supports both insertion and removal at both ends. It therefore needs the symmetric operations in the other direction as well. For this reason, we use a doubly linked list.</p> <p>As shown below, treat the list's head and tail as the deque's front and rear, and support adding and removing nodes at both ends.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 5-8   Insertion and removal in a linked-list deque </p> <p>The implementation is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linkedlist_deque.py<pre><code>class ListNode:\n    \"\"\"Doubly linked list node\"\"\"\n\n    def __init__(self, val: int):\n        \"\"\"Constructor\"\"\"\n        self.val: int = val\n        self.next: ListNode | None = None  # Reference to the next node\n        self.prev: ListNode | None = None  # Reference to the previous node\n\nclass LinkedListDeque:\n    \"\"\"Deque implemented with a doubly linked list\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        self._front: ListNode | None = None  # Head node front\n        self._rear: ListNode | None = None  # Tail node rear\n        self._size: int = 0  # Deque size\n\n    def size(self) -&gt; int:\n        \"\"\"Get the deque size\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"Check whether the deque is empty\"\"\"\n        return self._size == 0\n\n    def push(self, num: int, is_front: bool):\n        \"\"\"Enqueue operation\"\"\"\n        node = ListNode(num)\n        # If the linked list is empty, make both front and rear point to node\n        if self.is_empty():\n            self._front = self._rear = node\n        # Enqueue at the front\n        elif is_front:\n            # Add node to the head of the linked list\n            self._front.prev = node\n            node.next = self._front\n            self._front = node  # Update the head node\n        # Enqueue at the rear\n        else:\n            # Add node to the tail of the linked list\n            self._rear.next = node\n            node.prev = self._rear\n            self._rear = node  # Update the tail node\n        self._size += 1  # Update the deque size\n\n    def push_first(self, num: int):\n        \"\"\"Enqueue at the front\"\"\"\n        self.push(num, True)\n\n    def push_last(self, num: int):\n        \"\"\"Enqueue at the rear\"\"\"\n        self.push(num, False)\n\n    def pop(self, is_front: bool) -&gt; int:\n        \"\"\"Dequeue operation\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Deque is empty\")\n        # When the deque size is 1, the head and tail point to the same node\n        if self._size == 1:\n            val: int = self._front.val\n            self._front = None\n            self._rear = None\n        # Dequeue from the front\n        elif is_front:\n            val: int = self._front.val  # Save the head node value\n            # Remove the head node\n            fnext: ListNode | None = self._front.next\n            fnext.prev = None\n            self._front.next = None\n            self._front = fnext  # Update the head node\n        # Dequeue from the rear\n        else:\n            val: int = self._rear.val  # Save the tail node value\n            # Remove the tail node\n            rprev: ListNode | None = self._rear.prev\n            rprev.next = None\n            self._rear.prev = None\n            self._rear = rprev  # Update the tail node\n        self._size -= 1  # Update the deque size\n        return val\n\n    def pop_first(self) -&gt; int:\n        \"\"\"Dequeue from the front\"\"\"\n        return self.pop(True)\n\n    def pop_last(self) -&gt; int:\n        \"\"\"Dequeue from the rear\"\"\"\n        return self.pop(False)\n\n    def peek_first(self) -&gt; int:\n        \"\"\"Access the front element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Deque is empty\")\n        return self._front.val\n\n    def peek_last(self) -&gt; int:\n        \"\"\"Access the rear element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Deque is empty\")\n        return self._rear.val\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"Return an array for printing\"\"\"\n        node = self._front\n        res = [0] * self.size()\n        for i in range(self.size()):\n            res[i] = node.val\n            node = node.next\n        return res\n</code></pre> linkedlist_deque.cpp<pre><code>/* Doubly linked list node */\nstruct DoublyListNode {\n    int val;              // Node value\n    DoublyListNode *next; // Pointer to the next node\n    DoublyListNode *prev; // Pointer to the previous node\n    DoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n    }\n};\n\n/* Deque implemented with a doubly linked list */\nclass LinkedListDeque {\n  private:\n    DoublyListNode *front, *rear; // Head node front and tail node rear\n    int queSize = 0;              // Deque size\n\n  public:\n    /* Constructor */\n    LinkedListDeque() : front(nullptr), rear(nullptr) {\n    }\n\n    /* Destructor */\n    ~LinkedListDeque() {\n        // Traverse the linked list and delete each node to free its memory\n        DoublyListNode *pre, *cur = front;\n        while (cur != nullptr) {\n            pre = cur;\n            cur = cur-&gt;next;\n            delete pre;\n        }\n    }\n\n    /* Get the deque size */\n    int size() {\n        return queSize;\n    }\n\n    /* Check whether the deque is empty */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* Enqueue operation */\n    void push(int num, bool isFront) {\n        DoublyListNode *node = new DoublyListNode(num);\n        // If the linked list is empty, make both front and rear point to node\n        if (isEmpty())\n            front = rear = node;\n        // Enqueue at the front\n        else if (isFront) {\n            // Add node to the head of the linked list\n            front-&gt;prev = node;\n            node-&gt;next = front;\n            front = node; // Update the head node\n        // Enqueue at the rear\n        } else {\n            // Add node to the tail of the linked list\n            rear-&gt;next = node;\n            node-&gt;prev = rear;\n            rear = node; // Update the tail node\n        }\n        queSize++; // Update the deque size\n    }\n\n    /* Enqueue at the front */\n    void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* Enqueue at the rear */\n    void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* Dequeue operation */\n    int pop(bool isFront) {\n        if (isEmpty())\n            throw out_of_range(\"Deque is empty\");\n        int val;\n        // When the deque size is 1, the head and tail point to the same node\n        if (queSize == 1) {\n            val = front-&gt;val;\n            delete front; // Both point to the same node, so free it only once\n            front = nullptr;\n            rear = nullptr;\n        }\n        // Dequeue from the front\n        else if (isFront) {\n            val = front-&gt;val; // Save the head node value\n            // Remove the head node\n            DoublyListNode *fNext = front-&gt;next;\n            fNext-&gt;prev = nullptr;\n            front-&gt;next = nullptr;\n            delete front;\n            front = fNext; // Update the head node\n        // Dequeue from the rear\n        } else {\n            val = rear-&gt;val; // Save the tail node value\n            // Remove the tail node\n            DoublyListNode *rPrev = rear-&gt;prev;\n            rPrev-&gt;next = nullptr;\n            rear-&gt;prev = nullptr;\n            delete rear;\n            rear = rPrev; // Update the tail node\n        }\n        queSize--; // Update the deque size\n        return val;\n    }\n\n    /* Dequeue from the front */\n    int popFirst() {\n        return pop(true);\n    }\n\n    /* Dequeue from the rear */\n    int popLast() {\n        return pop(false);\n    }\n\n    /* Access the front element */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"Deque is empty\");\n        return front-&gt;val;\n    }\n\n    /* Access the rear element */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"Deque is empty\");\n        return rear-&gt;val;\n    }\n\n    /* Return a vector for printing */\n    vector&lt;int&gt; toVector() {\n        DoublyListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_deque.java<pre><code>/* Doubly linked list node */\nclass ListNode {\n    int val; // Node value\n    ListNode next; // Reference to the next node\n    ListNode prev; // Reference to the previous node\n\n    ListNode(int val) {\n        this.val = val;\n        prev = next = null;\n    }\n}\n\n/* Deque implemented with a doubly linked list */\nclass LinkedListDeque {\n    private ListNode front, rear; // Head node front and tail node rear\n    private int queSize = 0; // Deque size\n\n    public LinkedListDeque() {\n        front = rear = null;\n    }\n\n    /* Get the deque size */\n    public int size() {\n        return queSize;\n    }\n\n    /* Check whether the deque is empty */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* Enqueue operation */\n    private void push(int num, boolean isFront) {\n        ListNode node = new ListNode(num);\n        // If the linked list is empty, make both front and rear point to node\n        if (isEmpty())\n            front = rear = node;\n        // Enqueue at the front\n        else if (isFront) {\n            // Add node to the head of the linked list\n            front.prev = node;\n            node.next = front;\n            front = node; // Update the head node\n        // Enqueue at the rear\n        } else {\n            // Add node to the tail of the linked list\n            rear.next = node;\n            node.prev = rear;\n            rear = node; // Update the tail node\n        }\n        queSize++; // Update the queue size\n    }\n\n    /* Enqueue at the front */\n    public void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* Enqueue at the rear */\n    public void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* Dequeue operation */\n    private int pop(boolean isFront) {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        int val;\n        // When the queue size is 1, the head and tail point to the same node\n        if (queSize == 1) {\n            val = front.val;\n            front = null;\n            rear = null;\n        }\n        // Dequeue from the front\n        else if (isFront) {\n            val = front.val; // Save the head node value\n            // Remove the head node\n            ListNode fNext = front.next;\n            fNext.prev = null;\n            front.next = null;\n            front = fNext; // Update the head node\n        // Dequeue from the rear\n        } else {\n            val = rear.val; // Save the tail node value\n            // Remove the tail node\n            ListNode rPrev = rear.prev;\n            rPrev.next = null;\n            rear.prev = null;\n            rear = rPrev; // Update the tail node\n        }\n        queSize--; // Update the queue size\n        return val;\n    }\n\n    /* Dequeue from the front */\n    public int popFirst() {\n        return pop(true);\n    }\n\n    /* Dequeue from the rear */\n    public int popLast() {\n        return pop(false);\n    }\n\n    /* Access the front element */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* Access the rear element */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return rear.val;\n    }\n\n    /* Return an array for printing */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.cs<pre><code>/* Doubly linked list node */\nclass ListNode(int val) {\n    public int val = val;       // Node value\n    public ListNode? next = null; // Reference to the next node\n    public ListNode? prev = null; // Reference to the previous node\n}\n\n/* Deque implemented with a doubly linked list */\nclass LinkedListDeque {\n    ListNode? front, rear; // Head node front and tail node rear\n    int queSize = 0;      // Deque size\n\n    public LinkedListDeque() {\n        front = null;\n        rear = null;\n    }\n\n    /* Get the deque size */\n    public int Size() {\n        return queSize;\n    }\n\n    /* Check whether the deque is empty */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* Enqueue operation */\n    void Push(int num, bool isFront) {\n        ListNode node = new(num);\n        // If the linked list is empty, make both front and rear point to node\n        if (IsEmpty()) {\n            front = node;\n            rear = node;\n        }\n        // Enqueue at the front\n        else if (isFront) {\n            // Add node to the head of the linked list\n            front!.prev = node;\n            node.next = front;\n            front = node; // Update the head node                           \n        }\n        // Enqueue at the rear\n        else {\n            // Add node to the tail of the linked list\n            rear!.next = node;\n            node.prev = rear;\n            rear = node;  // Update the tail node\n        }\n\n        queSize++; // Update the queue size\n    }\n\n    /* Enqueue at the front */\n    public void PushFirst(int num) {\n        Push(num, true);\n    }\n\n    /* Enqueue at the rear */\n    public void PushLast(int num) {\n        Push(num, false);\n    }\n\n    /* Dequeue operation */\n    int? Pop(bool isFront) {\n        if (IsEmpty())\n            throw new Exception();\n        int? val;\n        // When the queue size is 1, the head and tail point to the same node\n        if (queSize == 1) {\n            val = front!.val;\n            front = null;\n            rear = null;\n        }\n        // Dequeue from the front\n        else if (isFront) {\n            val = front!.val; // Save the head node value\n            // Remove the head node\n            ListNode? fNext = front?.next;\n            fNext!.prev = null;\n            front.next = null;\n            front = fNext;   // Update the head node\n        }\n        // Dequeue from the rear\n        else {\n            val = rear!.val;  // Save the tail node value\n            // Remove the tail node\n            ListNode? rPrev = rear?.prev;\n            rPrev!.next = null;\n            rear.prev = null;\n            rear = rPrev;    // Update the tail node\n        }\n\n        queSize--; // Update the queue size\n        return val;\n    }\n\n    /* Dequeue from the front */\n    public int? PopFirst() {\n        return Pop(true);\n    }\n\n    /* Dequeue from the rear */\n    public int? PopLast() {\n        return Pop(false);\n    }\n\n    /* Access the front element */\n    public int? PeekFirst() {\n        if (IsEmpty())\n            throw new Exception();\n        return front?.val;\n    }\n\n    /* Access the rear element */\n    public int? PeekLast() {\n        if (IsEmpty())\n            throw new Exception();\n        return rear?.val;\n    }\n\n    /* Return an array for printing */\n    public int?[] ToArray() {\n        ListNode? node = front;\n        int?[] res = new int?[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node?.val;\n            node = node?.next;\n        }\n\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.go<pre><code>/* Deque implemented with a doubly linked list */\ntype linkedListDeque struct {\n    // Use the standard library list package\n    data *list.List\n}\n\n/* Initialize the deque */\nfunc newLinkedListDeque() *linkedListDeque {\n    return &amp;linkedListDeque{\n        data: list.New(),\n    }\n}\n\n/* Enqueue an element at the front */\nfunc (s *linkedListDeque) pushFirst(value any) {\n    s.data.PushFront(value)\n}\n\n/* Enqueue an element at the rear */\nfunc (s *linkedListDeque) pushLast(value any) {\n    s.data.PushBack(value)\n}\n\n/* Dequeue the front element */\nfunc (s *linkedListDeque) popFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* Dequeue the rear element */\nfunc (s *linkedListDeque) popLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* Access the front element */\nfunc (s *linkedListDeque) peekFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* Access the rear element */\nfunc (s *linkedListDeque) peekLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* Get the queue size */\nfunc (s *linkedListDeque) size() int {\n    return s.data.Len()\n}\n\n/* Check whether the queue is empty */\nfunc (s *linkedListDeque) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* Get the list for printing */\nfunc (s *linkedListDeque) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_deque.swift<pre><code>/* Doubly linked list node */\nclass ListNode {\n    var val: Int // Node value\n    var next: ListNode? // Reference to the next node\n    weak var prev: ListNode? // Reference to the previous node\n\n    init(val: Int) {\n        self.val = val\n    }\n}\n\n/* Deque implemented with a doubly linked list */\nclass LinkedListDeque {\n    private var front: ListNode? // Head node front\n    private var rear: ListNode? // Tail node rear\n    private var _size: Int // Deque size\n\n    init() {\n        _size = 0\n    }\n\n    /* Get the deque size */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* Check whether the deque is empty */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* Enqueue operation */\n    private func push(num: Int, isFront: Bool) {\n        let node = ListNode(val: num)\n        // If the linked list is empty, make both front and rear point to node\n        if isEmpty() {\n            front = node\n            rear = node\n        }\n        // Enqueue at the front\n        else if isFront {\n            // Add node to the head of the linked list\n            front?.prev = node\n            node.next = front\n            front = node // Update the head node\n        }\n        // Enqueue at the rear\n        else {\n            // Add node to the tail of the linked list\n            rear?.next = node\n            node.prev = rear\n            rear = node // Update the tail node\n        }\n        _size += 1 // Update the deque size\n    }\n\n    /* Enqueue at the front */\n    func pushFirst(num: Int) {\n        push(num: num, isFront: true)\n    }\n\n    /* Enqueue at the rear */\n    func pushLast(num: Int) {\n        push(num: num, isFront: false)\n    }\n\n    /* Dequeue operation */\n    private func pop(isFront: Bool) -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Deque is empty\")\n        }\n        let val: Int\n        // When the deque size is 1, the head and tail point to the same node\n        if _size == 1 {\n            val = front!.val\n            front = nil\n            rear = nil\n        }\n        // Dequeue from the front\n        else if isFront {\n            val = front!.val // Save the head node value\n            // Remove the head node\n            let fNext = front?.next\n            fNext?.prev = nil\n            front?.next = nil\n            front = fNext // Update the head node\n        }\n        // Dequeue from the rear\n        else {\n            val = rear!.val // Save the tail node value\n            // Remove the tail node\n            let rPrev = rear?.prev\n            rPrev?.next = nil\n            rear?.prev = nil\n            rear = rPrev // Update the tail node\n        }\n        _size -= 1 // Update the deque size\n        return val\n    }\n\n    /* Dequeue from the front */\n    func popFirst() -&gt; Int {\n        pop(isFront: true)\n    }\n\n    /* Dequeue from the rear */\n    func popLast() -&gt; Int {\n        pop(isFront: false)\n    }\n\n    /* Access the front element */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Deque is empty\")\n        }\n        return front!.val\n    }\n\n    /* Access the rear element */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Deque is empty\")\n        }\n        return rear!.val\n    }\n\n    /* Return an array for printing */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.js<pre><code>/* Doubly linked list node */\nclass ListNode {\n    prev; // Reference (pointer) to the previous node\n    next; // Reference (pointer) to the next node\n    val; // Node value\n\n    constructor(val) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* Deque implemented with a doubly linked list */\nclass LinkedListDeque {\n    #front; // Head node front\n    #rear; // Tail node rear\n    #queSize; // Deque size\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n        this.#queSize = 0;\n    }\n\n    /* Enqueue at the rear */\n    pushLast(val) {\n        const node = new ListNode(val);\n        // If the linked list is empty, make both front and rear point to node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // Add node to the tail of the linked list\n            this.#rear.next = node;\n            node.prev = this.#rear;\n            this.#rear = node; // Update the tail node\n        }\n        this.#queSize++;\n    }\n\n    /* Enqueue at the front */\n    pushFirst(val) {\n        const node = new ListNode(val);\n        // If the linked list is empty, make both front and rear point to node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // Add node to the head of the linked list\n            this.#front.prev = node;\n            node.next = this.#front;\n            this.#front = node; // Update the head node\n        }\n        this.#queSize++;\n    }\n\n    /* Dequeue from the rear */\n    popLast() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#rear.val; // Store the tail node value\n        // When the queue size is 1, the head and tail point to the same node\n        if (this.#queSize === 1) {\n            this.#front = null;\n            this.#rear = null;\n        }\n        // Remove the tail node\n        else {\n            let temp = this.#rear.prev;\n            temp.next = null;\n            this.#rear.prev = null;\n            this.#rear = temp; // Update the tail node\n        }\n        this.#queSize--;\n        return value;\n    }\n\n    /* Dequeue from the front */\n    popFirst() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#front.val; // Store the head node value\n        // When the queue size is 1, the head and tail point to the same node\n        if (this.#queSize === 1) {\n            this.#front = null;\n            this.#rear = null;\n        }\n        // Remove the head node\n        else {\n            let temp = this.#front.next;\n            temp.prev = null;\n            this.#front.next = null;\n            this.#front = temp; // Update the head node\n        }\n        this.#queSize--;\n        return value;\n    }\n\n    /* Access the rear element */\n    peekLast() {\n        return this.#queSize === 0 ? null : this.#rear.val;\n    }\n\n    /* Access the front element */\n    peekFirst() {\n        return this.#queSize === 0 ? null : this.#front.val;\n    }\n\n    /* Get the deque size */\n    size() {\n        return this.#queSize;\n    }\n\n    /* Check whether the deque is empty */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* Print the deque */\n    print() {\n        const arr = [];\n        let temp = this.#front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.ts<pre><code>/* Doubly linked list node */\nclass ListNode {\n    prev: ListNode; // Reference (pointer) to the previous node\n    next: ListNode; // Reference (pointer) to the next node\n    val: number; // Node value\n\n    constructor(val: number) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* Deque implemented with a doubly linked list */\nclass LinkedListDeque {\n    private front: ListNode; // Head node front\n    private rear: ListNode; // Tail node rear\n    private queSize: number; // Deque size\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n        this.queSize = 0;\n    }\n\n    /* Enqueue at the rear */\n    pushLast(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // If the linked list is empty, make both front and rear point to node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // Add node to the tail of the linked list\n            this.rear.next = node;\n            node.prev = this.rear;\n            this.rear = node; // Update the tail node\n        }\n        this.queSize++;\n    }\n\n    /* Enqueue at the front */\n    pushFirst(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // If the linked list is empty, make both front and rear point to node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // Add node to the head of the linked list\n            this.front.prev = node;\n            node.next = this.front;\n            this.front = node; // Update the head node\n        }\n        this.queSize++;\n    }\n\n    /* Dequeue from the rear */\n    popLast(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.rear!.val; // Store the tail node value\n        // When the queue size is 1, the head and tail point to the same node\n        if (this.queSize === 1) {\n            this.front = null;\n            this.rear = null;\n        }\n        // Remove the tail node\n        else {\n            let temp: ListNode = this.rear!.prev;\n            temp!.next = null;\n            this.rear!.prev = null;\n            this.rear = temp; // Update the tail node\n        }\n        this.queSize--;\n        return value;\n    }\n\n    /* Dequeue from the front */\n    popFirst(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.front.val; // Store the head node value\n        // When the queue size is 1, the head and tail point to the same node\n        if (this.queSize === 1) {\n            this.front = null;\n            this.rear = null;\n        }\n        // Remove the head node\n        else {\n            let temp: ListNode = this.front!.next;\n            temp!.prev = null;\n            this.front!.next = null;\n            this.front = temp; // Update the head node\n        }\n        this.queSize--;\n        return value;\n    }\n\n    /* Access the rear element */\n    peekLast(): number {\n        return this.queSize === 0 ? null : this.rear.val;\n    }\n\n    /* Access the front element */\n    peekFirst(): number {\n        return this.queSize === 0 ? null : this.front.val;\n    }\n\n    /* Get the deque size */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* Check whether the deque is empty */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* Print the deque */\n    print(): void {\n        const arr: number[] = [];\n        let temp: ListNode = this.front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.dart<pre><code>/* Doubly linked list node */\nclass ListNode {\n  int val; // Node value\n  ListNode? next; // Reference to the next node\n  ListNode? prev; // Reference to the previous node\n\n  ListNode(this.val, {this.next, this.prev});\n}\n\n/* Deque implemented with a doubly linked list */\nclass LinkedListDeque {\n  late ListNode? _front; // Head node _front\n  late ListNode? _rear; // Tail node _rear\n  int _queSize = 0; // Deque size\n\n  LinkedListDeque() {\n    this._front = null;\n    this._rear = null;\n  }\n\n  /* Get the deque size */\n  int size() {\n    return this._queSize;\n  }\n\n  /* Check whether the deque is empty */\n  bool isEmpty() {\n    return size() == 0;\n  }\n\n  /* Enqueue operation */\n  void push(int _num, bool isFront) {\n    final ListNode node = ListNode(_num);\n    if (isEmpty()) {\n      // If the linked list is empty, make both _front and _rear point to node\n      _front = _rear = node;\n    } else if (isFront) {\n      // Enqueue at the front\n      // Add node to the head of the linked list\n      _front!.prev = node;\n      node.next = _front;\n      _front = node; // Update the head node\n    } else {\n      // Enqueue at the rear\n      // Add node to the tail of the linked list\n      _rear!.next = node;\n      node.prev = _rear;\n      _rear = node; // Update the tail node\n    }\n    _queSize++; // Update the queue size\n  }\n\n  /* Enqueue at the front */\n  void pushFirst(int _num) {\n    push(_num, true);\n  }\n\n  /* Enqueue at the rear */\n  void pushLast(int _num) {\n    push(_num, false);\n  }\n\n  /* Dequeue operation */\n  int? pop(bool isFront) {\n    // If the queue is empty, return null immediately\n    if (isEmpty()) {\n      return null;\n    }\n    final int val;\n    // When the queue size is 1, the head and tail point to the same node\n    if (_queSize == 1) {\n      val = _front!.val;\n      _front = null;\n      _rear = null;\n    }\n    // Dequeue from the front\n    else if (isFront) {\n      val = _front!.val; // Save the head node value\n      // Remove the head node\n      ListNode? fNext = _front!.next;\n      fNext!.prev = null;\n      _front!.next = null;\n      _front = fNext; // Update the head node\n    // Dequeue from the rear\n    } else {\n      val = _rear!.val; // Save the tail node value\n      // Remove the tail node\n      ListNode? rPrev = _rear!.prev;\n      rPrev!.next = null;\n      _rear!.prev = null;\n      _rear = rPrev; // Update the tail node\n    }\n    _queSize--; // Update the queue size\n    return val;\n  }\n\n  /* Dequeue from the front */\n  int? popFirst() {\n    return pop(true);\n  }\n\n  /* Dequeue from the rear */\n  int? popLast() {\n    return pop(false);\n  }\n\n  /* Access the front element */\n  int? peekFirst() {\n    return _front?.val;\n  }\n\n  /* Access the rear element */\n  int? peekLast() {\n    return _rear?.val;\n  }\n\n  /* Return an array for printing */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; _queSize; i++) {\n      res.add(node!.val);\n      node = node.next;\n    }\n    return res;\n  }\n}\n</code></pre> linkedlist_deque.rs<pre><code>/* Doubly linked list node */\npub struct ListNode&lt;T&gt; {\n    pub val: T,                                 // Node value\n    pub next: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // Pointer to the next node\n    pub prev: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // Pointer to the previous node\n}\n\nimpl&lt;T&gt; ListNode&lt;T&gt; {\n    pub fn new(val: T) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n        Rc::new(RefCell::new(ListNode {\n            val,\n            next: None,\n            prev: None,\n        }))\n    }\n}\n\n/* Deque implemented with a doubly linked list */\n#[allow(dead_code)]\npub struct LinkedListDeque&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // Head node front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // Tail node rear\n    que_size: usize,                         // Deque size\n}\n\nimpl&lt;T: Copy&gt; LinkedListDeque&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* Get the deque size */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* Check whether the deque is empty */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.que_size == 0;\n    }\n\n    /* Enqueue operation */\n    fn push(&amp;mut self, num: T, is_front: bool) {\n        let node = ListNode::new(num);\n        // Enqueue at the front\n        if is_front {\n            match self.front.take() {\n                // If the linked list is empty, make both front and rear point to node\n                None =&gt; {\n                    self.rear = Some(node.clone());\n                    self.front = Some(node);\n                }\n                // Add node to the head of the linked list\n                Some(old_front) =&gt; {\n                    old_front.borrow_mut().prev = Some(node.clone());\n                    node.borrow_mut().next = Some(old_front);\n                    self.front = Some(node); // Update the head node\n                }\n            }\n        }\n        // Enqueue at the rear\n        else {\n            match self.rear.take() {\n                // If the linked list is empty, make both front and rear point to node\n                None =&gt; {\n                    self.front = Some(node.clone());\n                    self.rear = Some(node);\n                }\n                // Add node to the tail of the linked list\n                Some(old_rear) =&gt; {\n                    old_rear.borrow_mut().next = Some(node.clone());\n                    node.borrow_mut().prev = Some(old_rear);\n                    self.rear = Some(node); // Update the tail node\n                }\n            }\n        }\n        self.que_size += 1; // Update the deque size\n    }\n\n    /* Enqueue at the front */\n    pub fn push_first(&amp;mut self, num: T) {\n        self.push(num, true);\n    }\n\n    /* Enqueue at the rear */\n    pub fn push_last(&amp;mut self, num: T) {\n        self.push(num, false);\n    }\n\n    /* Dequeue operation */\n    fn pop(&amp;mut self, is_front: bool) -&gt; Option&lt;T&gt; {\n        // If the deque is empty, return None immediately\n        if self.is_empty() {\n            return None;\n        };\n        // Dequeue from the front\n        if is_front {\n            self.front.take().map(|old_front| {\n                match old_front.borrow_mut().next.take() {\n                    Some(new_front) =&gt; {\n                        new_front.borrow_mut().prev.take();\n                        self.front = Some(new_front); // Update the head node\n                    }\n                    None =&gt; {\n                        self.rear.take();\n                    }\n                }\n                self.que_size -= 1; // Update the deque size\n                old_front.borrow().val\n            })\n        }\n        // Dequeue from the rear\n        else {\n            self.rear.take().map(|old_rear| {\n                match old_rear.borrow_mut().prev.take() {\n                    Some(new_rear) =&gt; {\n                        new_rear.borrow_mut().next.take();\n                        self.rear = Some(new_rear); // Update the tail node\n                    }\n                    None =&gt; {\n                        self.front.take();\n                    }\n                }\n                self.que_size -= 1; // Update the deque size\n                old_rear.borrow().val\n            })\n        }\n    }\n\n    /* Dequeue from the front */\n    pub fn pop_first(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(true);\n    }\n\n    /* Dequeue from the rear */\n    pub fn pop_last(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(false);\n    }\n\n    /* Access the front element */\n    pub fn peek_first(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* Access the rear element */\n    pub fn peek_last(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.rear.as_ref()\n    }\n\n    /* Return a vector for printing */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        let mut res: Vec&lt;T&gt; = Vec::new();\n        fn recur&lt;T: Copy&gt;(cur: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, res: &amp;mut Vec&lt;T&gt;) {\n            if let Some(cur) = cur {\n                res.push(cur.borrow().val);\n                recur(cur.borrow().next.as_ref(), res);\n            }\n        }\n\n        recur(head, &amp;mut res);\n        res\n    }\n}\n</code></pre> linkedlist_deque.c<pre><code>/* Doubly linked list node */\ntypedef struct DoublyListNode {\n    int val;                     // Node value\n    struct DoublyListNode *next; // Next node\n    struct DoublyListNode *prev; // Previous node\n} DoublyListNode;\n\n/* Constructor */\nDoublyListNode *newDoublyListNode(int num) {\n    DoublyListNode *new = (DoublyListNode *)malloc(sizeof(DoublyListNode));\n    new-&gt;val = num;\n    new-&gt;next = NULL;\n    new-&gt;prev = NULL;\n    return new;\n}\n\n/* Destructor */\nvoid delDoublyListNode(DoublyListNode *node) {\n    free(node);\n}\n\n/* Deque implemented with a doubly linked list */\ntypedef struct {\n    DoublyListNode *front, *rear; // Head node front and tail node rear\n    int queSize;                  // Deque size\n} LinkedListDeque;\n\n/* Constructor */\nLinkedListDeque *newLinkedListDeque() {\n    LinkedListDeque *deque = (LinkedListDeque *)malloc(sizeof(LinkedListDeque));\n    deque-&gt;front = NULL;\n    deque-&gt;rear = NULL;\n    deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* Destructor */\nvoid delLinkedListdeque(LinkedListDeque *deque) {\n    // Free all nodes\n    for (int i = 0; i &lt; deque-&gt;queSize &amp;&amp; deque-&gt;front != NULL; i++) {\n        DoublyListNode *tmp = deque-&gt;front;\n        deque-&gt;front = deque-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // Free the deque structure\n    free(deque);\n}\n\n/* Get the queue size */\nint size(LinkedListDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* Check whether the queue is empty */\nbool empty(LinkedListDeque *deque) {\n    return (size(deque) == 0);\n}\n\n/* Enqueue */\nvoid push(LinkedListDeque *deque, int num, bool isFront) {\n    DoublyListNode *node = newDoublyListNode(num);\n    // If the list is empty, make both front and rear point to node\n    if (empty(deque)) {\n        deque-&gt;front = deque-&gt;rear = node;\n    }\n    // Enqueue at the front\n    else if (isFront) {\n        // Add node to the head of the linked list\n        deque-&gt;front-&gt;prev = node;\n        node-&gt;next = deque-&gt;front;\n        deque-&gt;front = node; // Update the head node\n    }\n    // Enqueue at the rear\n    else {\n        // Add node to the tail of the linked list\n        deque-&gt;rear-&gt;next = node;\n        node-&gt;prev = deque-&gt;rear;\n        deque-&gt;rear = node;\n    }\n    deque-&gt;queSize++; // Update the queue size\n}\n\n/* Enqueue at the front */\nvoid pushFirst(LinkedListDeque *deque, int num) {\n    push(deque, num, true);\n}\n\n/* Enqueue at the rear */\nvoid pushLast(LinkedListDeque *deque, int num) {\n    push(deque, num, false);\n}\n\n/* Access the front element */\nint peekFirst(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;front);\n    return deque-&gt;front-&gt;val;\n}\n\n/* Access the rear element */\nint peekLast(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;rear);\n    return deque-&gt;rear-&gt;val;\n}\n\n/* Dequeue */\nint pop(LinkedListDeque *deque, bool isFront) {\n    if (empty(deque))\n        return -1;\n    int val;\n    // When the queue size is 1, the head and tail point to the same node\n    if (deque-&gt;queSize == 1) {\n        val = deque-&gt;front-&gt;val;\n        delDoublyListNode(deque-&gt;front); // Both point to the same node, so free it only once\n        deque-&gt;front = NULL;\n        deque-&gt;rear = NULL;\n    }\n    // Dequeue from the front\n    else if (isFront) {\n        val = peekFirst(deque); // Save the head node value\n        DoublyListNode *fNext = deque-&gt;front-&gt;next;\n        fNext-&gt;prev = NULL;\n        deque-&gt;front-&gt;next = NULL;\n        delDoublyListNode(deque-&gt;front);\n        deque-&gt;front = fNext; // Update the head node\n    }\n    // Dequeue from the rear\n    else {\n        val = peekLast(deque); // Save the tail node value\n        DoublyListNode *rPrev = deque-&gt;rear-&gt;prev;\n        rPrev-&gt;next = NULL;\n        deque-&gt;rear-&gt;prev = NULL;\n        delDoublyListNode(deque-&gt;rear);\n        deque-&gt;rear = rPrev; // Update the tail node\n    }\n    deque-&gt;queSize--; // Update the queue size\n    return val;\n}\n\n/* Dequeue from the front */\nint popFirst(LinkedListDeque *deque) {\n    return pop(deque, true);\n}\n\n/* Dequeue from the rear */\nint popLast(LinkedListDeque *deque) {\n    return pop(deque, false);\n}\n\n/* Print the queue */\nvoid printLinkedListDeque(LinkedListDeque *deque) {\n    int *arr = malloc(sizeof(int) * deque-&gt;queSize);\n    // Copy the linked list data into an array\n    int i;\n    DoublyListNode *node;\n    for (i = 0, node = deque-&gt;front; i &lt; deque-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, deque-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_deque.kt<pre><code>/* Doubly linked list node */\nclass ListNode(var _val: Int) {\n    // The node value is stored in _val\n    var next: ListNode? = null // Reference to the next node\n    var prev: ListNode? = null // Reference to the previous node\n}\n\n/* Deque implemented with a doubly linked list */\nclass LinkedListDeque {\n    private var front: ListNode? = null // Head node front\n    private var rear: ListNode? = null // Tail node rear\n    private var queSize: Int = 0 // Deque size\n\n    /* Get the deque size */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* Check whether the deque is empty */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* Enqueue operation */\n    fun push(num: Int, isFront: Boolean) {\n        val node = ListNode(num)\n        // If the linked list is empty, make both front and rear point to node\n        if (isEmpty()) {\n            rear = node\n            front = rear\n            // Enqueue at the front\n        } else if (isFront) {\n            // Add node to the head of the linked list\n            front?.prev = node\n            node.next = front\n            front = node // Update the head node\n            // Enqueue at the rear\n        } else {\n            // Add node to the tail of the linked list\n            rear?.next = node\n            node.prev = rear\n            rear = node // Update the tail node\n        }\n        queSize++ // Update the queue size\n    }\n\n    /* Enqueue at the front */\n    fun pushFirst(num: Int) {\n        push(num, true)\n    }\n\n    /* Enqueue at the rear */\n    fun pushLast(num: Int) {\n        push(num, false)\n    }\n\n    /* Dequeue operation */\n    fun pop(isFront: Boolean): Int {\n        if (isEmpty()) \n            throw IndexOutOfBoundsException()\n        val _val: Int\n        // When the queue size is 1, the head and tail point to the same node\n        if (queSize == 1) {\n            _val = front!!._val\n            front = null\n            rear = null\n        }\n        // Dequeue from the front\n        else if (isFront) {\n            _val = front!!._val // Save the head node value\n            // Remove the head node\n            val fNext = front!!.next\n            fNext!!.prev = null\n            front!!.next = null\n            front = fNext // Update the head node\n            // Dequeue from the rear\n        } else {\n            _val = rear!!._val // Save the tail node value\n            // Remove the tail node\n            val rPrev = rear!!.prev\n            rPrev!!.next = null\n            rear!!.prev = null\n            rear = rPrev // Update the tail node\n        }\n        queSize-- // Update the queue size\n        return _val\n    }\n\n    /* Dequeue from the front */\n    fun popFirst(): Int {\n        return pop(true)\n    }\n\n    /* Dequeue from the rear */\n    fun popLast(): Int {\n        return pop(false)\n    }\n\n    /* Access the front element */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!._val\n    }\n\n    /* Access the rear element */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return rear!!._val\n    }\n\n    /* Return an array for printing */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!._val\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.rb<pre><code>=begin\nFile: linkedlist_deque.rb\nCreated Time: 2024-04-06\nAuthor: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)\n=end\n\n### Doubly linked list node\nclass ListNode\n  attr_accessor :val\n  attr_accessor :next # Reference to the next node\n  attr_accessor :prev # Reference to the previous node\n\n  ### Constructor ###\n  def initialize(val)\n    @val = val\n  end\nend\n\n### Deque implemented with a doubly linked list ###\nclass LinkedListDeque\n  ### Get the deque size ###\n  attr_reader :size\n\n  ### Constructor ###\n  def initialize\n    @front = nil  # Head node front\n    @rear = nil   # Tail node rear\n    @size = 0     # Deque size\n  end\n\n  ### Check whether the deque is empty ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### Enqueue operation ###\n  def push(num, is_front)\n    node = ListNode.new(num)\n    # If the list is empty, point both front and rear to node\n    if is_empty?\n      @front = @rear = node\n    # Enqueue at the front\n    elsif is_front\n      # Add node to the head of the linked list\n      @front.prev = node\n      node.next = @front\n      @front = node # Update the head node\n    # Enqueue at the rear\n    else\n      # Add node to the tail of the linked list\n      @rear.next = node\n      node.prev = @rear\n      @rear = node # Update the tail node\n    end\n    @size += 1 # Update the queue size\n  end\n\n  ### Enqueue at the front ###\n  def push_first(num)\n    push(num, true)\n  end\n\n  ### Enqueue at the rear ###\n  def push_last(num)\n    push(num, false)\n  end\n\n  ### Dequeue operation ###\n  def pop(is_front)\n    raise IndexError, 'Deque is empty' if is_empty?\n\n    # When the queue size is 1, the head and tail point to the same node\n    if @size == 1\n      val = @front.val\n      @front = nil\n      @rear = nil\n    # Dequeue from the front\n    elsif is_front\n      val = @front.val # Save the head node value\n      # Remove the head node\n      fnext = @front.next\n      fnext.prev = nil\n      @front.next = nil\n      @front = fnext # Update the head node\n    # Dequeue from the rear\n    else\n      val = @rear.val # Save the tail node value\n      # Remove the tail node\n      rprev = @rear.prev\n      rprev.next = nil\n      @rear.prev = nil\n      @rear = rprev # Update the tail node\n    end\n    @size -= 1 # Update the queue size\n\n    val\n  end\n\n  ### Dequeue from the front ###\n  def pop_first\n    pop(true)\n  end\n\n  ### Dequeue from the rear ###\n  def pop_last\n    pop(false)\n  end\n\n  ### Access the front element ###\n  def peek_first\n    raise IndexError, 'Deque is empty' if is_empty?\n\n    @front.val\n  end\n\n  ### Access the rear element ###\n  def peek_last\n    raise IndexError, 'Deque is empty' if is_empty?\n\n    @rear.val\n  end\n\n  ### Return an array for printing ###\n  def to_array\n    node = @front\n    res = Array.new(size, 0)\n    for i in 0...size\n      res[i] = node.val\n      node = node.next\n    end\n    res\n  end\nend\n</code></pre>","path":["Chapter 5.   Stacks and queues","5.3   Deques"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#2-implementation-with-an-array","level":3,"title":"2.   Implementation with an array","text":"<p>As with an array-based queue, we can use a circular array to implement a deque, as shown below.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 5-9   Insertion and removal in an array-based deque </p> <p>Starting with the queue implementation, we need only add methods for insertion at the front and removal at the rear:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_deque.py<pre><code>class ArrayDeque:\n    \"\"\"Deque implemented with a circular array\"\"\"\n\n    def __init__(self, capacity: int):\n        \"\"\"Constructor\"\"\"\n        self._nums: list[int] = [0] * capacity\n        self._front: int = 0\n        self._size: int = 0\n\n    def capacity(self) -&gt; int:\n        \"\"\"Get the deque capacity\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"Get the deque size\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"Check whether the deque is empty\"\"\"\n        return self._size == 0\n\n    def index(self, i: int) -&gt; int:\n        \"\"\"Calculate an index in the circular array\"\"\"\n        # Use modulo to connect the end of the array to its beginning\n        # When i passes the end of the array, wrap around to the beginning\n        # When i passes the beginning of the array, wrap around to the end\n        return (i + self.capacity()) % self.capacity()\n\n    def push_first(self, num: int):\n        \"\"\"Enqueue at the front\"\"\"\n        if self._size == self.capacity():\n            print(\"Deque is full\")\n            return\n        # Move the front pointer one position to the left\n        # Use modulo to wrap front to the end when it passes the beginning of the array\n        self._front = self.index(self._front - 1)\n        # Add num to the front\n        self._nums[self._front] = num\n        self._size += 1\n\n    def push_last(self, num: int):\n        \"\"\"Enqueue at the rear\"\"\"\n        if self._size == self.capacity():\n            print(\"Deque is full\")\n            return\n        # Calculate the rear pointer, which points one position past the last element\n        rear = self.index(self._front + self._size)\n        # Add num to the rear\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop_first(self) -&gt; int:\n        \"\"\"Dequeue from the front\"\"\"\n        num = self.peek_first()\n        # Advance the front pointer by one position\n        self._front = self.index(self._front + 1)\n        self._size -= 1\n        return num\n\n    def pop_last(self) -&gt; int:\n        \"\"\"Dequeue from the rear\"\"\"\n        num = self.peek_last()\n        self._size -= 1\n        return num\n\n    def peek_first(self) -&gt; int:\n        \"\"\"Access the front element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Deque is empty\")\n        return self._nums[self._front]\n\n    def peek_last(self) -&gt; int:\n        \"\"\"Access the rear element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Deque is empty\")\n        # Calculate the index of the last element\n        last = self.index(self._front + self._size - 1)\n        return self._nums[last]\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"Return an array for printing\"\"\"\n        # Convert only the list elements within the occupied range\n        res = []\n        for i in range(self._size):\n            res.append(self._nums[self.index(self._front + i)])\n        return res\n</code></pre> array_deque.cpp<pre><code>/* Deque implemented with a circular array */\nclass ArrayDeque {\n  private:\n    vector&lt;int&gt; nums; // Array for storing deque elements\n    int front;        // Front index, identifying the first element\n    int queSize;      // Deque size\n\n  public:\n    /* Constructor */\n    ArrayDeque(int capacity) {\n        nums.resize(capacity);\n        front = queSize = 0;\n    }\n\n    /* Get the deque capacity */\n    int capacity() {\n        return nums.size();\n    }\n\n    /* Get the deque size */\n    int size() {\n        return queSize;\n    }\n\n    /* Check whether the deque is empty */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* Calculate an index in the circular array */\n    int index(int i) {\n        // Use modulo to connect the end of the array to its beginning\n        // When i passes the end of the array, wrap around to the beginning\n        // When i passes the beginning of the array, wrap around to the end\n        return (i + capacity()) % capacity();\n    }\n\n    /* Enqueue at the front */\n    void pushFirst(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"Deque is full\" &lt;&lt; endl;\n            return;\n        }\n        // Move the front index one position to the left\n        // Use modulo to wrap front to the end when it passes the beginning of the array\n        front = index(front - 1);\n        // Add num to the front\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* Enqueue at the rear */\n    void pushLast(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"Deque is full\" &lt;&lt; endl;\n            return;\n        }\n        // Calculate the rear index, one position past the last element\n        int rear = index(front + queSize);\n        // Add num to the rear\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* Dequeue from the front */\n    int popFirst() {\n        int num = peekFirst();\n        // Advance the front index by one position\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* Dequeue from the rear */\n    int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* Access the front element */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"Deque is empty\");\n        return nums[front];\n    }\n\n    /* Access the rear element */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"Deque is empty\");\n        // Calculate the index of the last element\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* Return a vector for printing */\n    vector&lt;int&gt; toVector() {\n        // Copy only the elements in the occupied range\n        vector&lt;int&gt; res(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n};\n</code></pre> array_deque.java<pre><code>/* Deque implemented with a circular array */\nclass ArrayDeque {\n    private int[] nums; // Array that stores the deque elements\n    private int front; // Front pointer, pointing to the first element\n    private int queSize; // Deque size\n\n    /* Constructor */\n    public ArrayDeque(int capacity) {\n        this.nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* Get the deque capacity */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* Get the deque size */\n    public int size() {\n        return queSize;\n    }\n\n    /* Check whether the deque is empty */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* Calculate an index in the circular array */\n    private int index(int i) {\n        // Use modulo to connect the end of the array to its beginning\n        // When i passes the end of the array, wrap around to the beginning\n        // When i passes the beginning of the array, wrap around to the end\n        return (i + capacity()) % capacity();\n    }\n\n    /* Enqueue at the front */\n    public void pushFirst(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"Deque is full\");\n            return;\n        }\n        // Move the front pointer one position to the left\n        // Use modulo to wrap front to the end when it passes the beginning of the array\n        front = index(front - 1);\n        // Add num to the front\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* Enqueue at the rear */\n    public void pushLast(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"Deque is full\");\n            return;\n        }\n        // Calculate the rear pointer, which points one position past the last element\n        int rear = index(front + queSize);\n        // Add num to the rear\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* Dequeue from the front */\n    public int popFirst() {\n        int num = peekFirst();\n        // Advance the front pointer by one position\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* Dequeue from the rear */\n    public int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* Access the front element */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* Access the rear element */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        // Calculate the index of the last element\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* Return an array for printing */\n    public int[] toArray() {\n        // Convert only the list elements within the occupied range\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.cs<pre><code>/* Deque implemented with a circular array */\nclass ArrayDeque {\n    int[] nums;  // Array that stores the deque elements\n    int front;   // Front pointer, pointing to the first element\n    int queSize; // Deque size\n\n    /* Constructor */\n    public ArrayDeque(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* Get the deque capacity */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* Get the deque size */\n    public int Size() {\n        return queSize;\n    }\n\n    /* Check whether the deque is empty */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* Calculate an index in the circular array */\n    int Index(int i) {\n        // Use modulo to connect the end of the array to its beginning\n        // When i passes the end of the array, wrap around to the beginning\n        // When i passes the beginning of the array, wrap around to the end\n        return (i + Capacity()) % Capacity();\n    }\n\n    /* Enqueue at the front */\n    public void PushFirst(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"Deque is full\");\n            return;\n        }\n        // Move the front pointer one position to the left\n        // Use modulo to wrap front to the end when it passes the beginning of the array\n        front = Index(front - 1);\n        // Add num to the front\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* Enqueue at the rear */\n    public void PushLast(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"Deque is full\");\n            return;\n        }\n        // Calculate the rear pointer, which points one position past the last element\n        int rear = Index(front + queSize);\n        // Add num to the rear\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* Dequeue from the front */\n    public int PopFirst() {\n        int num = PeekFirst();\n        // Advance the front pointer by one position\n        front = Index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* Dequeue from the rear */\n    public int PopLast() {\n        int num = PeekLast();\n        queSize--;\n        return num;\n    }\n\n    /* Access the front element */\n    public int PeekFirst() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        return nums[front];\n    }\n\n    /* Access the rear element */\n    public int PeekLast() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        // Calculate the index of the last element\n        int last = Index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* Return an array for printing */\n    public int[] ToArray() {\n        // Copy only the elements within the occupied range\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[Index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.go<pre><code>/* Deque implemented with a circular array */\ntype arrayDeque struct {\n    nums        []int // Array that stores the deque elements\n    front       int   // Front pointer, pointing to the first element\n    queSize     int   // Deque size\n    queCapacity int   // Queue capacity (the maximum number of elements)\n}\n\n/* Initialize the queue */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\n    return &amp;arrayDeque{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* Get the deque size */\nfunc (q *arrayDeque) size() int {\n    return q.queSize\n}\n\n/* Check whether the deque is empty */\nfunc (q *arrayDeque) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* Calculate an index in the circular array */\nfunc (q *arrayDeque) index(i int) int {\n    // Use modulo to connect the end of the array to its beginning\n    // When i passes the end of the array, wrap around to the beginning\n    // When i passes the beginning of the array, wrap around to the end\n    return (i + q.queCapacity) % q.queCapacity\n}\n\n/* Enqueue at the front */\nfunc (q *arrayDeque) pushFirst(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"Deque is full\")\n        return\n    }\n    // Move the front pointer one position to the left\n    // Use modulo to wrap front to the end when it passes the beginning of the array\n    q.front = q.index(q.front - 1)\n    // Add num to the front\n    q.nums[q.front] = num\n    q.queSize++\n}\n\n/* Enqueue at the rear */\nfunc (q *arrayDeque) pushLast(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"Deque is full\")\n        return\n    }\n    // Calculate the rear pointer, which points one position past the last element\n    rear := q.index(q.front + q.queSize)\n    // Add num to the rear\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* Dequeue from the front */\nfunc (q *arrayDeque) popFirst() any {\n    num := q.peekFirst()\n    if num == nil {\n        return nil\n    }\n    // Advance the front pointer by one position\n    q.front = q.index(q.front + 1)\n    q.queSize--\n    return num\n}\n\n/* Dequeue from the rear */\nfunc (q *arrayDeque) popLast() any {\n    num := q.peekLast()\n    if num == nil {\n        return nil\n    }\n    q.queSize--\n    return num\n}\n\n/* Access the front element */\nfunc (q *arrayDeque) peekFirst() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* Access the rear element */\nfunc (q *arrayDeque) peekLast() any {\n    if q.isEmpty() {\n        return nil\n    }\n    // Calculate the index of the last element\n    last := q.index(q.front + q.queSize - 1)\n    return q.nums[last]\n}\n\n/* Get a slice for printing */\nfunc (q *arrayDeque) toSlice() []int {\n    // Convert only the list elements within the occupied range\n    res := make([]int, q.queSize)\n    for i, j := 0, q.front; i &lt; q.queSize; i++ {\n        res[i] = q.nums[q.index(j)]\n        j++\n    }\n    return res\n}\n</code></pre> array_deque.swift<pre><code>/* Deque implemented with a circular array */\nclass ArrayDeque {\n    private var nums: [Int] // Array that stores the deque elements\n    private var front: Int // Front index, identifying the first element\n    private var _size: Int // Deque size\n\n    /* Constructor */\n    init(capacity: Int) {\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* Get the deque capacity */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* Get the deque size */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* Check whether the deque is empty */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* Calculate an index in the circular array */\n    private func index(i: Int) -&gt; Int {\n        // Use modulo to connect the end of the array to its beginning\n        // When i passes the end of the array, wrap around to the beginning\n        // When i passes the beginning of the array, wrap around to the end\n        (i + capacity()) % capacity()\n    }\n\n    /* Enqueue at the front */\n    func pushFirst(num: Int) {\n        if size() == capacity() {\n            print(\"Deque is full\")\n            return\n        }\n        // Move the front index one position to the left\n        // Use modulo to wrap front to the end when it passes the beginning of the array\n        front = index(i: front - 1)\n        // Add num to the front\n        nums[front] = num\n        _size += 1\n    }\n\n    /* Enqueue at the rear */\n    func pushLast(num: Int) {\n        if size() == capacity() {\n            print(\"Deque is full\")\n            return\n        }\n        // Calculate the rear index, one position past the last element\n        let rear = index(i: front + size())\n        // Add num to the rear\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* Dequeue from the front */\n    func popFirst() -&gt; Int {\n        let num = peekFirst()\n        // Advance the front index by one position\n        front = index(i: front + 1)\n        _size -= 1\n        return num\n    }\n\n    /* Dequeue from the rear */\n    func popLast() -&gt; Int {\n        let num = peekLast()\n        _size -= 1\n        return num\n    }\n\n    /* Access the front element */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Deque is empty\")\n        }\n        return nums[front]\n    }\n\n    /* Access the rear element */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Deque is empty\")\n        }\n        // Calculate the index of the last element\n        let last = index(i: front + size() - 1)\n        return nums[last]\n    }\n\n    /* Return an array for printing */\n    func toArray() -&gt; [Int] {\n        // Include only the elements in the occupied range\n        (front ..&lt; front + size()).map { nums[index(i: $0)] }\n    }\n}\n</code></pre> array_deque.js<pre><code>/* Deque implemented with a circular array */\nclass ArrayDeque {\n    #nums; // Array for storing deque elements\n    #front; // Front pointer, pointing to the first element\n    #queSize; // Deque size\n\n    /* Constructor */\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n        this.#front = 0;\n        this.#queSize = 0;\n    }\n\n    /* Get the deque capacity */\n    capacity() {\n        return this.#nums.length;\n    }\n\n    /* Get the deque size */\n    size() {\n        return this.#queSize;\n    }\n\n    /* Check whether the deque is empty */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* Calculate an index in the circular array */\n    index(i) {\n        // Use modulo to connect the end of the array to its beginning\n        // When i passes the end of the array, wrap around to the beginning\n        // When i passes the beginning of the array, wrap around to the end\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* Enqueue at the front */\n    pushFirst(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('Deque is full');\n            return;\n        }\n        // Move the front pointer one position to the left\n        // Use modulo to wrap front to the end when it passes the beginning of the array\n        this.#front = this.index(this.#front - 1);\n        // Add num to the front\n        this.#nums[this.#front] = num;\n        this.#queSize++;\n    }\n\n    /* Enqueue at the rear */\n    pushLast(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('Deque is full');\n            return;\n        }\n        // Calculate the rear pointer, which points one position past the last element\n        const rear = this.index(this.#front + this.#queSize);\n        // Add num to the rear\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* Dequeue from the front */\n    popFirst() {\n        const num = this.peekFirst();\n        // Advance the front pointer by one position\n        this.#front = this.index(this.#front + 1);\n        this.#queSize--;\n        return num;\n    }\n\n    /* Dequeue from the rear */\n    popLast() {\n        const num = this.peekLast();\n        this.#queSize--;\n        return num;\n    }\n\n    /* Access the front element */\n    peekFirst() {\n        if (this.isEmpty()) throw new Error('Deque is empty');\n        return this.#nums[this.#front];\n    }\n\n    /* Access the rear element */\n    peekLast() {\n        if (this.isEmpty()) throw new Error('Deque is empty');\n        // Calculate the index of the last element\n        const last = this.index(this.#front + this.#queSize - 1);\n        return this.#nums[last];\n    }\n\n    /* Return an array for printing */\n    toArray() {\n        // Convert only the list elements within the occupied range\n        const res = [];\n        for (let i = 0, j = this.#front; i &lt; this.#queSize; i++, j++) {\n            res[i] = this.#nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.ts<pre><code>/* Deque implemented with a circular array */\nclass ArrayDeque {\n    private nums: number[]; // Array for storing deque elements\n    private front: number; // Front pointer, pointing to the first element\n    private queSize: number; // Deque size\n\n    /* Constructor */\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = 0;\n        this.queSize = 0;\n    }\n\n    /* Get the deque capacity */\n    capacity(): number {\n        return this.nums.length;\n    }\n\n    /* Get the deque size */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* Check whether the deque is empty */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* Calculate an index in the circular array */\n    index(i: number): number {\n        // Use modulo to connect the end of the array to its beginning\n        // When i passes the end of the array, wrap around to the beginning\n        // When i passes the beginning of the array, wrap around to the end\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* Enqueue at the front */\n    pushFirst(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('Deque is full');\n            return;\n        }\n        // Move the front pointer one position to the left\n        // Use modulo to wrap front to the end when it passes the beginning of the array\n        this.front = this.index(this.front - 1);\n        // Add num to the front\n        this.nums[this.front] = num;\n        this.queSize++;\n    }\n\n    /* Enqueue at the rear */\n    pushLast(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('Deque is full');\n            return;\n        }\n        // Calculate the rear pointer, which points one position past the last element\n        const rear: number = this.index(this.front + this.queSize);\n        // Add num to the rear\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* Dequeue from the front */\n    popFirst(): number {\n        const num: number = this.peekFirst();\n        // Advance the front pointer by one position\n        this.front = this.index(this.front + 1);\n        this.queSize--;\n        return num;\n    }\n\n    /* Dequeue from the rear */\n    popLast(): number {\n        const num: number = this.peekLast();\n        this.queSize--;\n        return num;\n    }\n\n    /* Access the front element */\n    peekFirst(): number {\n        if (this.isEmpty()) throw new Error('Deque is empty');\n        return this.nums[this.front];\n    }\n\n    /* Access the rear element */\n    peekLast(): number {\n        if (this.isEmpty()) throw new Error('Deque is empty');\n        // Calculate the index of the last element\n        const last = this.index(this.front + this.queSize - 1);\n        return this.nums[last];\n    }\n\n    /* Return an array for printing */\n    toArray(): number[] {\n        // Convert only the list elements within the occupied range\n        const res: number[] = [];\n        for (let i = 0, j = this.front; i &lt; this.queSize; i++, j++) {\n            res[i] = this.nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.dart<pre><code>/* Deque implemented with a circular array */\nclass ArrayDeque {\n  late List&lt;int&gt; _nums; // Array for storing deque elements\n  late int _front; // Front pointer, pointing to the first element\n  late int _queSize; // Deque size\n\n  /* Constructor */\n  ArrayDeque(int capacity) {\n    this._nums = List.filled(capacity, 0);\n    this._front = this._queSize = 0;\n  }\n\n  /* Get the deque capacity */\n  int capacity() {\n    return _nums.length;\n  }\n\n  /* Get the deque size */\n  int size() {\n    return _queSize;\n  }\n\n  /* Check whether the deque is empty */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* Calculate an index in the circular array */\n  int index(int i) {\n    // Use modulo to connect the end of the array to its beginning\n    // When i passes the end of the array, wrap around to the beginning\n    // When i passes the beginning of the array, wrap around to the end\n    return (i + capacity()) % capacity();\n  }\n\n  /* Enqueue at the front */\n  void pushFirst(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"Deque is full\");\n    }\n    // Move the front pointer one position to the left\n    // Use modulo to wrap _front to the end when it passes the beginning of the array\n    _front = index(_front - 1);\n    // Add _num to the front\n    _nums[_front] = _num;\n    _queSize++;\n  }\n\n  /* Enqueue at the rear */\n  void pushLast(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"Deque is full\");\n    }\n    // Calculate the rear pointer, which points one position past the last element\n    int rear = index(_front + _queSize);\n    // Add _num to the rear\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* Dequeue from the front */\n  int popFirst() {\n    int _num = peekFirst();\n    // Move the front pointer one position to the right\n    _front = index(_front + 1);\n    _queSize--;\n    return _num;\n  }\n\n  /* Dequeue from the rear */\n  int popLast() {\n    int _num = peekLast();\n    _queSize--;\n    return _num;\n  }\n\n  /* Access the front element */\n  int peekFirst() {\n    if (isEmpty()) {\n      throw Exception(\"Deque is empty\");\n    }\n    return _nums[_front];\n  }\n\n  /* Access the rear element */\n  int peekLast() {\n    if (isEmpty()) {\n      throw Exception(\"Deque is empty\");\n    }\n    // Calculate the index of the last element\n    int last = index(_front + _queSize - 1);\n    return _nums[last];\n  }\n\n  /* Return an array for printing */\n  List&lt;int&gt; toArray() {\n    // Convert only the list elements within the occupied range\n    List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[index(j)];\n    }\n    return res;\n  }\n}\n</code></pre> array_deque.rs<pre><code>/* Deque implemented with a circular array */\nstruct ArrayDeque&lt;T&gt; {\n    nums: Vec&lt;T&gt;,    // Array for storing deque elements\n    front: usize,    // Front index, identifying the first element\n    que_size: usize, // Deque size\n}\n\nimpl&lt;T: Copy + Default&gt; ArrayDeque&lt;T&gt; {\n    /* Constructor */\n    pub fn new(capacity: usize) -&gt; Self {\n        Self {\n            nums: vec![T::default(); capacity],\n            front: 0,\n            que_size: 0,\n        }\n    }\n\n    /* Get the deque capacity */\n    pub fn capacity(&amp;self) -&gt; usize {\n        self.nums.len()\n    }\n\n    /* Get the deque size */\n    pub fn size(&amp;self) -&gt; usize {\n        self.que_size\n    }\n\n    /* Check whether the deque is empty */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* Calculate an index in the circular array */\n    fn index(&amp;self, i: i32) -&gt; usize {\n        // Use modulo to connect the end of the array to its beginning\n        // When i passes the end of the array, wrap around to the beginning\n        // When i passes the beginning of the array, wrap around to the end\n        ((i + self.capacity() as i32) % self.capacity() as i32) as usize\n    }\n\n    /* Enqueue at the front */\n    pub fn push_first(&amp;mut self, num: T) {\n        if self.que_size == self.capacity() {\n            println!(\"Deque is full\");\n            return;\n        }\n        // Move the front index one position to the left\n        // Use modulo to wrap front to the end when it passes the beginning of the array\n        self.front = self.index(self.front as i32 - 1);\n        // Add num to the front\n        self.nums[self.front] = num;\n        self.que_size += 1;\n    }\n\n    /* Enqueue at the rear */\n    pub fn push_last(&amp;mut self, num: T) {\n        if self.que_size == self.capacity() {\n            println!(\"Deque is full\");\n            return;\n        }\n        // Calculate the rear index, one position past the last element\n        let rear = self.index(self.front as i32 + self.que_size as i32);\n        // Add num to the rear\n        self.nums[rear] = num;\n        self.que_size += 1;\n    }\n\n    /* Dequeue from the front */\n    fn pop_first(&amp;mut self) -&gt; T {\n        let num = self.peek_first();\n        // Advance the front index by one position\n        self.front = self.index(self.front as i32 + 1);\n        self.que_size -= 1;\n        num\n    }\n\n    /* Dequeue from the rear */\n    fn pop_last(&amp;mut self) -&gt; T {\n        let num = self.peek_last();\n        self.que_size -= 1;\n        num\n    }\n\n    /* Access the front element */\n    fn peek_first(&amp;self) -&gt; T {\n        if self.is_empty() {\n            panic!(\"Deque is empty\")\n        };\n        self.nums[self.front]\n    }\n\n    /* Access the rear element */\n    fn peek_last(&amp;self) -&gt; T {\n        if self.is_empty() {\n            panic!(\"Deque is empty\")\n        };\n        // Calculate the index of the last element\n        let last = self.index(self.front as i32 + self.que_size as i32 - 1);\n        self.nums[last]\n    }\n\n    /* Return a vector for printing */\n    fn to_array(&amp;self) -&gt; Vec&lt;T&gt; {\n        // Convert only the list elements within the occupied range\n        let mut res = vec![T::default(); self.que_size];\n        let mut j = self.front;\n        for i in 0..self.que_size {\n            res[i] = self.nums[self.index(j as i32)];\n            j += 1;\n        }\n        res\n    }\n}\n</code></pre> array_deque.c<pre><code>/* Deque implemented with a circular array */\ntypedef struct {\n    int *nums;       // Array that stores the queue elements\n    int front;       // Front pointer, pointing to the first element\n    int queSize;     // Current number of elements in the deque\n    int queCapacity; // Queue capacity\n} ArrayDeque;\n\n/* Constructor */\nArrayDeque *newArrayDeque(int capacity) {\n    ArrayDeque *deque = (ArrayDeque *)malloc(sizeof(ArrayDeque));\n    // Initialize the array\n    deque-&gt;queCapacity = capacity;\n    deque-&gt;nums = (int *)malloc(sizeof(int) * deque-&gt;queCapacity);\n    deque-&gt;front = deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* Destructor */\nvoid delArrayDeque(ArrayDeque *deque) {\n    free(deque-&gt;nums);\n    free(deque);\n}\n\n/* Get the deque capacity */\nint capacity(ArrayDeque *deque) {\n    return deque-&gt;queCapacity;\n}\n\n/* Get the deque size */\nint size(ArrayDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* Check whether the deque is empty */\nbool empty(ArrayDeque *deque) {\n    return deque-&gt;queSize == 0;\n}\n\n/* Calculate an index in the circular array */\nint dequeIndex(ArrayDeque *deque, int i) {\n    // Use modulo to connect the end of the array to its beginning\n    // When i passes the end of the array, wrap to the beginning\n    // When i passes the beginning of the array, wrap around to the end\n    return ((i + capacity(deque)) % capacity(deque));\n}\n\n/* Enqueue at the front */\nvoid pushFirst(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"Deque is full\\r\\n\");\n        return;\n    }\n    // Move the front pointer one position to the left\n    // Use modulo to wrap front to the end when it passes the beginning\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front - 1);\n    // Add num to the front\n    deque-&gt;nums[deque-&gt;front] = num;\n    deque-&gt;queSize++;\n}\n\n/* Enqueue at the rear */\nvoid pushLast(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"Deque is full\\r\\n\");\n        return;\n    }\n    // Calculate the rear pointer, which points one position past the last element\n    int rear = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize);\n    // Add num to the rear\n    deque-&gt;nums[rear] = num;\n    deque-&gt;queSize++;\n}\n\n/* Access the front element */\nint peekFirst(ArrayDeque *deque) {\n    // Access error: the deque is empty\n    assert(empty(deque) == 0);\n    return deque-&gt;nums[deque-&gt;front];\n}\n\n/* Access the rear element */\nint peekLast(ArrayDeque *deque) {\n    // Access error: the deque is empty\n    assert(empty(deque) == 0);\n    int last = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize - 1);\n    return deque-&gt;nums[last];\n}\n\n/* Dequeue from the front */\nint popFirst(ArrayDeque *deque) {\n    int num = peekFirst(deque);\n    // Advance the front pointer by one position\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front + 1);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* Dequeue from the rear */\nint popLast(ArrayDeque *deque) {\n    int num = peekLast(deque);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* Return an array for printing */\nint *toArray(ArrayDeque *deque, int *queSize) {\n    *queSize = deque-&gt;queSize;\n    int *res = (int *)calloc(deque-&gt;queSize, sizeof(int));\n    int j = deque-&gt;front;\n    for (int i = 0; i &lt; deque-&gt;queSize; i++) {\n        res[i] = deque-&gt;nums[j % deque-&gt;queCapacity];\n        j++;\n    }\n    return res;\n}\n</code></pre> array_deque.kt<pre><code>/* Constructor */\nclass ArrayDeque(capacity: Int) {\n    private var nums: IntArray = IntArray(capacity) // Array that stores the deque elements\n    private var front: Int = 0 // Front pointer, pointing to the first element\n    private var queSize: Int = 0 // Deque size\n\n    /* Get the deque capacity */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* Get the deque size */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* Check whether the deque is empty */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* Calculate an index in the circular array */\n    private fun index(i: Int): Int {\n        // Use modulo to connect the end of the array to its beginning\n        // When i passes the end of the array, wrap around to the beginning\n        // When i passes the beginning of the array, wrap around to the end\n        return (i + capacity()) % capacity()\n    }\n\n    /* Enqueue at the front */\n    fun pushFirst(num: Int) {\n        if (queSize == capacity()) {\n            println(\"Deque is full\")\n            return\n        }\n        // Move the front pointer one position to the left\n        // Use modulo to wrap front to the end when it passes the beginning of the array\n        front = index(front - 1)\n        // Add num to the front\n        nums[front] = num\n        queSize++\n    }\n\n    /* Enqueue at the rear */\n    fun pushLast(num: Int) {\n        if (queSize == capacity()) {\n            println(\"Deque is full\")\n            return\n        }\n        // Calculate the rear pointer, which points one position past the last element\n        val rear = index(front + queSize)\n        // Add num to the rear\n        nums[rear] = num\n        queSize++\n    }\n\n    /* Dequeue from the front */\n    fun popFirst(): Int {\n        val num = peekFirst()\n        // Advance the front pointer by one position\n        front = index(front + 1)\n        queSize--\n        return num\n    }\n\n    /* Dequeue from the rear */\n    fun popLast(): Int {\n        val num = peekLast()\n        queSize--\n        return num\n    }\n\n    /* Access the front element */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* Access the rear element */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // Calculate the index of the last element\n        val last = index(front + queSize - 1)\n        return nums[last]\n    }\n\n    /* Return an array for printing */\n    fun toArray(): IntArray {\n        // Copy only the elements within the occupied range\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[index(j)]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_deque.rb<pre><code>### Deque implemented with a circular array ###\nclass ArrayDeque\n  ### Get the deque size ###\n  attr_reader :size\n\n  ### Constructor ###\n  def initialize(capacity)\n    @nums = Array.new(capacity, 0)\n    @front = 0\n    @size = 0\n  end\n\n  ### Get the deque capacity ###\n  def capacity\n    @nums.length\n  end\n\n  ### Check whether the deque is empty ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### Enqueue at the front ###\n  def push_first(num)\n    if size == capacity\n      puts 'Deque is full'\n      return\n    end\n\n    # Move the front pointer one position to the left\n    # Use modulo to wrap front to the end when it passes the beginning of the array\n    @front = index(@front - 1)\n    # Add num to the front\n    @nums[@front] = num\n    @size += 1\n  end\n\n  ### Enqueue at the rear ###\n  def push_last(num)\n    if size == capacity\n      puts 'Deque is full'\n      return\n    end\n\n    # Calculate the rear pointer, which points one position past the last element\n    rear = index(@front + size)\n    # Add num to the rear\n    @nums[rear] = num\n    @size += 1\n  end\n\n  ### Dequeue from the front ###\n  def pop_first\n    num = peek_first\n    # Advance the front pointer by one position\n    @front = index(@front + 1)\n    @size -= 1\n    num\n  end\n\n  ### Dequeue from the rear ###\n  def pop_last\n    num = peek_last\n    @size -= 1\n    num\n  end\n\n  ### Access the front element ###\n  def peek_first\n    raise IndexError, 'Deque is empty' if is_empty?\n\n    @nums[@front]\n  end\n\n  ### Access the rear element ###\n  def peek_last\n    raise IndexError, 'Deque is empty' if is_empty?\n\n    # Calculate the index of the last element\n    last = index(@front + size - 1)\n    @nums[last]\n  end\n\n  ### Return an array for printing ###\n  def to_array\n    # Convert only the list elements within the occupied range\n    res = []\n    for i in 0...size\n      res &lt;&lt; @nums[index(@front + i)]\n    end\n    res\n  end\n\n  private\n\n  ### Calculate an index in the circular array ###\n  def index(i)\n    # Use modulo to connect the end of the array to its beginning\n    # When i passes the end of the array, wrap around to the beginning\n    # When i passes the beginning of the array, wrap around to the end\n    (i + capacity) % capacity\n  end\nend\n</code></pre>","path":["Chapter 5.   Stacks and queues","5.3   Deques"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#533-applications","level":2,"title":"5.3.3   Applications","text":"<p>A deque combines stack and queue behavior, so it supports all their applications while offering greater flexibility.</p> <p>Software undo is usually implemented with a stack: the system uses <code>push</code> to record each change and <code>pop</code> to undo it. Because resources are limited, software often restricts the history to a fixed number of steps, such as \\(50\\). When the stack grows beyond \\(50\\), the oldest entry must be removed from the bottom, the front of the deque. A stack cannot provide this operation, so a deque is needed instead. The core undo behavior still follows last-in, first-out order; the deque simply provides flexibility for the additional logic.</p>","path":["Chapter 5.   Stacks and queues","5.3   Deques"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/","level":1,"title":"5.5   Exercises","text":"","path":["Chapter 5.   Stacks and queues","5.5   Exercises"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#551-check-your-understanding","level":2,"title":"5.5.1   Check your understanding","text":"","path":["Chapter 5.   Stacks and queues","5.5   Exercises"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#1-which-element-comes-out-first-in-a-stack-or-queue","level":3,"title":"1.   Which element comes out first in a stack or queue?","text":"<p>Start with an empty stack <code>S</code> and an empty queue <code>Q</code>. Perform the same operations on each:</p> <p>Step 1: Add <code>A</code>. Step 2: Add <code>B</code>. Step 3: Remove and record one element. Step 4: Add <code>C</code>. Step 5: Keep removing and recording elements until the container is empty.</p> <p>Write the order in which elements are removed from <code>S</code> and from <code>Q</code>. Explain the difference using last in, first out or first in, first out.</p> Reference answer <p>The removal order for stack <code>S</code> is <code>B, C, A</code>. After adding <code>A, B</code>, we first pop <code>B</code>, the most recently added element, and then add <code>C</code>. We then pop <code>C, A</code> in order. This is last in, first out.</p> <p>The removal order for queue <code>Q</code> is <code>A, B, C</code>. After adding <code>A, B</code>, we first remove <code>A</code>, the earliest element added. We then add <code>C</code> and remove <code>B, C</code> in order. This is first in, first out.</p>","path":["Chapter 5.   Stacks and queues","5.5   Exercises"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#2-what-happens-when-the-rear-passes-the-end-of-the-array","level":3,"title":"2.   What happens when the rear passes the end of the array?","text":"<p>A queue uses a circular array of length 5, with indices <code>0-4</code>. Currently, <code>front = 3</code> and <code>size = 2</code>, with queue elements <code>A, B</code> at indices 3 and 4, respectively.</p> <ol> <li>When enqueueing <code>C</code>, at which index should <code>C</code> be placed? What is <code>size</code> afterward?</li> <li>If we then dequeue once, which element is removed? What are the new values of <code>front</code> and <code>size</code>?</li> <li>What is the logical order from front to rear now? Does dequeuing require moving the other array elements? Why?</li> </ol> Reference answer <ol> <li> <p>The new element's position is     <code>(front + size) % 5 = (3 + 2) % 5 = 0</code>,     so <code>C</code> goes at index 0. After enqueueing, <code>size = 3</code>.</p> </li> <li> <p>Dequeueing removes the current front element, <code>A</code>. The new front index is     <code>(3 + 1) % 5 = 4</code>, so <code>front = 4</code> and <code>size = 2</code>.</p> </li> <li> <p>The valid elements are logically ordered as <code>B, C</code>, with <code>B</code> at index 4 and <code>C</code> at index 0.     Dequeueing only advances <code>front</code> and updates <code>size</code>. The circular array uses the remainder operation to wrap indices back to the beginning,     so the other elements do not need to shift forward together.</p> </li> </ol>","path":["Chapter 5.   Stacks and queues","5.5   Exercises"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#3-operations-at-both-ends-of-a-deque","level":3,"title":"3.   Operations at both ends of a deque","text":"<p>Here, <code>push_first</code> adds at the front, <code>push_last</code> adds at the rear, <code>pop_first</code> removes from the front, and <code>pop_last</code> removes from the rear.</p> <p>Starting with an empty deque <code>deq</code>, perform:</p> <ol> <li><code>push_last(A)</code></li> <li><code>push_last(B)</code></li> <li><code>push_first(C)</code></li> <li><code>pop_last()</code></li> <li><code>push_last(D)</code></li> <li><code>pop_first()</code></li> </ol> <ol> <li>Which elements are removed by the two pop operations?</li> <li>Which elements remain, from front to rear, after all the operations?</li> <li>Examine these six steps. Can a queue that allows additions only at the rear and removals only at the front perform them all? If not, identify the unsupported operations. Then explain whether a deque can perform them all, and why.</li> </ol> Reference answer <p>After the first three steps, the deque contains <code>[C, A, B]</code> from front to rear.</p> <ol> <li> <p><code>pop_last()</code> removes <code>B</code>. After adding <code>D</code>, the deque is <code>[C, A, D]</code>.     Then <code>pop_first()</code> removes <code>C</code>.</p> </li> <li> <p>The remaining elements are <code>[A, D]</code>.</p> </li> <li> <p>A queue that allows additions only at the rear and removals only at the front cannot perform every operation:     step 3, <code>push_first(C)</code>, adds at the front, and step 4, <code>pop_last()</code>, removes from the rear. Neither operation is supported by that queue.     A deque supports additions and removals at both ends, so it can perform all six steps.</p> </li> </ol>","path":["Chapter 5.   Stacks and queues","5.5   Exercises"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#552-programming-exercises","level":2,"title":"5.5.2   Programming exercises","text":"","path":["Chapter 5.   Stacks and queues","5.5   Exercises"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#1-checking-a-bracket-sequence","level":3,"title":"1.   Checking a bracket sequence","text":"<p>Given a string <code>s</code> containing only the three types of brackets <code>()</code>, <code>[]</code>, and <code>{}</code>, use a stack to determine whether it is valid.</p> <p>A valid sequence must satisfy both conditions: each closing bracket matches the type of the most recent unmatched opening bracket, and no unmatched opening brackets remain at the end. Return a Boolean result.</p> Hints <ol> <li>You can map each closing bracket to its matching opening bracket.</li> <li>When you encounter a closing bracket, first check whether the stack is empty, then check whether its top matches.</li> <li>The stack must also be empty when the scan finishes.</li> </ol> <p>LeetCode</p>","path":["Chapter 5.   Stacks and queues","5.5   Exercises"],"tags":[]},{"location":"chapter_stack_and_queue/queue/","level":1,"title":"5.2   Queues","text":"<p>A queue is a linear data structure that follows a first-in, first-out rule. It models a line of people: newcomers join at the back, while those at the front leave one by one.</p> <p>As shown below, the two ends are called the “front” and the “rear.” Adding an element at the rear is called “enqueueing,” and removing an element from the front is called “dequeueing.”</p> <p></p> <p> Figure 5-4   The first-in, first-out rule of a queue </p>","path":["Chapter 5.   Stacks and queues","5.2   Queues"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#521-common-queue-operations","level":2,"title":"5.2.1   Common queue operations","text":"<p>Table 5-2 lists common operations. Method names vary between programming languages; here we use the same names as for stacks.</p> <p> Table 5-2   Efficiency of queue operations </p> Method Description Time complexity <code>push()</code> Enqueue an element at the rear \\(O(1)\\) <code>pop()</code> Dequeue the front element \\(O(1)\\) <code>peek()</code> Access the front element \\(O(1)\\) <p>We can use a queue class already provided by the programming language:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby queue.py<pre><code>from collections import deque\n\n# Initialize a queue\n# In Python, we usually use the deque class as a queue\n# Although queue.Queue() is a dedicated queue class, it is less convenient and is not recommended here\nque: deque[int] = deque()\n\n# Enqueue elements\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n\n# Access the front element\nfront: int = que[0]\n\n# Dequeue an element\npop: int = que.popleft()\n\n# Get the queue size\nsize: int = len(que)\n\n# Check whether the queue is empty\nis_empty: bool = len(que) == 0\n</code></pre> queue.cpp<pre><code>/* Initialize a queue */\nqueue&lt;int&gt; queue;\n\n/* Enqueue elements */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* Access the front element */\nint front = queue.front();\n\n/* Dequeue an element */\nqueue.pop();\n\n/* Get the queue size */\nint size = queue.size();\n\n/* Check whether the queue is empty */\nbool empty = queue.empty();\n</code></pre> queue.java<pre><code>/* Initialize a queue */\nQueue&lt;Integer&gt; queue = new LinkedList&lt;&gt;();\n\n/* Enqueue elements */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n\n/* Access the front element */\nint peek = queue.peek();\n\n/* Dequeue an element */\nint pop = queue.poll();\n\n/* Get the queue size */\nint size = queue.size();\n\n/* Check whether the queue is empty */\nboolean isEmpty = queue.isEmpty();\n</code></pre> queue.cs<pre><code>/* Initialize a queue */\nQueue&lt;int&gt; queue = new();\n\n/* Enqueue elements */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n\n/* Access the front element */\nint peek = queue.Peek();\n\n/* Dequeue an element */\nint pop = queue.Dequeue();\n\n/* Get the queue size */\nint size = queue.Count;\n\n/* Check whether the queue is empty */\nbool isEmpty = queue.Count == 0;\n</code></pre> queue_test.go<pre><code>/* Initialize a queue */\n// In Go, use a list as a queue\nqueue := list.New()\n\n/* Enqueue elements */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n\n/* Access the front element */\npeek := queue.Front()\n\n/* Dequeue an element */\npop := queue.Front()\nqueue.Remove(pop)\n\n/* Get the queue size */\nsize := queue.Len()\n\n/* Check whether the queue is empty */\nisEmpty := queue.Len() == 0\n</code></pre> queue.swift<pre><code>/* Initialize a queue */\n// Swift has no built-in queue class; use an Array as a queue\nvar queue: [Int] = []\n\n/* Enqueue elements */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n\n/* Access the front element */\nlet peek = queue.first!\n\n/* Dequeue an element */\n// Because this is an array, removeFirst takes O(n) time\nlet pop = queue.removeFirst()\n\n/* Get the queue size */\nlet size = queue.count\n\n/* Check whether the queue is empty */\nlet isEmpty = queue.isEmpty\n</code></pre> queue.js<pre><code>/* Initialize a queue */\n// JavaScript has no built-in queue; use an Array as a queue\nconst queue = [];\n\n/* Enqueue elements */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* Access the front element */\nconst peek = queue[0];\n\n/* Dequeue an element */\n// The underlying structure is an array, so shift() takes O(n) time\nconst pop = queue.shift();\n\n/* Get the queue size */\nconst size = queue.length;\n\n/* Check whether the queue is empty */\nconst empty = queue.length === 0;\n</code></pre> queue.ts<pre><code>/* Initialize a queue */\n// TypeScript has no built-in queue; use an Array as a queue\nconst queue: number[] = [];\n\n/* Enqueue elements */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* Access the front element */\nconst peek = queue[0];\n\n/* Dequeue an element */\n// The underlying structure is an array, so shift() takes O(n) time\nconst pop = queue.shift();\n\n/* Get the queue size */\nconst size = queue.length;\n\n/* Check whether the queue is empty */\nconst empty = queue.length === 0;\n</code></pre> queue.dart<pre><code>/* Initialize a queue */\n// In Dart, Queue is a deque and can also be used as a queue\nQueue&lt;int&gt; queue = Queue();\n\n/* Enqueue elements */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n\n/* Access the front element */\nint peek = queue.first;\n\n/* Dequeue an element */\nint pop = queue.removeFirst();\n\n/* Get the queue size */\nint size = queue.length;\n\n/* Check whether the queue is empty */\nbool isEmpty = queue.isEmpty;\n</code></pre> queue.rs<pre><code>/* Initialize a deque */\n// In Rust, use a deque as an ordinary queue\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* Enqueue elements */\ndeque.push_back(1);\ndeque.push_back(3);\ndeque.push_back(2);\ndeque.push_back(5);\ndeque.push_back(4);\n\n/* Access the front element */\nif let Some(front) = deque.front() {\n}\n\n/* Dequeue an element */\nif let Some(pop) = deque.pop_front() {\n}\n\n/* Get the queue size */\nlet size = deque.len();\n\n/* Check whether the queue is empty */\nlet is_empty = deque.is_empty();\n</code></pre> queue.c<pre><code>// C provides no built-in queue\n</code></pre> queue.kt<pre><code>/* Initialize a queue */\nval queue = LinkedList&lt;Int&gt;()\n\n/* Enqueue elements */\nqueue.offer(1)\nqueue.offer(3)\nqueue.offer(2)\nqueue.offer(5)\nqueue.offer(4)\n\n/* Access the front element */\nval peek = queue.peek()\n\n/* Dequeue an element */\nval pop = queue.poll()\n\n/* Get the queue size */\nval size = queue.size\n\n/* Check whether the queue is empty */\nval isEmpty = queue.isEmpty()\n</code></pre> queue.rb<pre><code># Initialize a queue\n# Ruby's built-in Thread::Queue has no peek or traversal methods; use an Array as a queue\nqueue = []\n\n# Enqueue elements\nqueue.push(1)\nqueue.push(3)\nqueue.push(2)\nqueue.push(5)\nqueue.push(4)\n\n# Access the front element of the queue\npeek = queue.first\n\n# Dequeue an element\n# Note that this is an array, so Array#shift takes O(n) time\npop = queue.shift\n\n# Get the queue size\nsize = queue.length\n\n# Check whether the queue is empty\nis_empty = queue.empty?\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 5.   Stacks and queues","5.2   Queues"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#522-implementing-a-queue","level":2,"title":"5.2.2   Implementing a queue","text":"<p>A queue needs a data structure that allows insertion at one end and deletion at the other. Both linked lists and arrays can meet this requirement.</p>","path":["Chapter 5.   Stacks and queues","5.2   Queues"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#1-implementation-with-a-linked-list","level":3,"title":"1.   Implementation with a linked list","text":"<p>As shown below, treat the linked list's head and tail as the queue's front and rear. Nodes may be added only at the rear and removed only from the front.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> Figure 5-5   Enqueue and dequeue in a linked-list queue </p> <p>Here is an implementation using a linked list:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linkedlist_queue.py<pre><code>class LinkedListQueue:\n    \"\"\"Queue implemented with a linked list\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        self._front: ListNode | None = None  # Head node front\n        self._rear: ListNode | None = None  # Tail node rear\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"Get the queue size\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"Check whether the queue is empty\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"Enqueue\"\"\"\n        # Add num after the tail node\n        node = ListNode(num)\n        # If the queue is empty, make both the head and tail point to this node\n        if self._front is None:\n            self._front = node\n            self._rear = node\n        # If the queue is not empty, add this node after the tail node\n        else:\n            self._rear.next = node\n            self._rear = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"Dequeue\"\"\"\n        num = self.peek()\n        # Remove the head node\n        self._front = self._front.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"Access the front element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Queue is empty\")\n        return self._front.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"Convert to a list for printing\"\"\"\n        queue = []\n        temp = self._front\n        while temp:\n            queue.append(temp.val)\n            temp = temp.next\n        return queue\n</code></pre> linkedlist_queue.cpp<pre><code>/* Queue implemented with a linked list */\nclass LinkedListQueue {\n  private:\n    ListNode *front, *rear; // Head node front and tail node rear\n    int queSize;\n\n  public:\n    LinkedListQueue() {\n        front = nullptr;\n        rear = nullptr;\n        queSize = 0;\n    }\n\n    ~LinkedListQueue() {\n        // Traverse the linked list and delete each node to free its memory\n        freeMemoryLinkedList(front);\n    }\n\n    /* Get the queue size */\n    int size() {\n        return queSize;\n    }\n\n    /* Check whether the queue is empty */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* Enqueue */\n    void push(int num) {\n        // Add num after the tail node\n        ListNode *node = new ListNode(num);\n        // If the queue is empty, make both the head and tail point to this node\n        if (front == nullptr) {\n            front = node;\n            rear = node;\n        }\n        // If the queue is not empty, add this node after the tail node\n        else {\n            rear-&gt;next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* Dequeue */\n    int pop() {\n        int num = peek();\n        // Remove the head node\n        ListNode *tmp = front;\n        front = front-&gt;next;\n        // Free memory\n        delete tmp;\n        queSize--;\n        return num;\n    }\n\n    /* Access the front element */\n    int peek() {\n        if (size() == 0)\n            throw out_of_range(\"Queue is empty\");\n        return front-&gt;val;\n    }\n\n    /* Convert the linked list to a vector and return it */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_queue.java<pre><code>/* Queue implemented with a linked list */\nclass LinkedListQueue {\n    private ListNode front, rear; // Head node front and tail node rear\n    private int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* Get the queue size */\n    public int size() {\n        return queSize;\n    }\n\n    /* Check whether the queue is empty */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* Enqueue */\n    public void push(int num) {\n        // Add num after the tail node\n        ListNode node = new ListNode(num);\n        // If the queue is empty, make both the head and tail point to this node\n        if (front == null) {\n            front = node;\n            rear = node;\n        // If the queue is not empty, add this node after the tail node\n        } else {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* Dequeue */\n    public int pop() {\n        int num = peek();\n        // Remove the head node\n        front = front.next;\n        queSize--;\n        return num;\n    }\n\n    /* Access the front element */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* Convert the linked list to an array and return it */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.cs<pre><code>/* Queue implemented with a linked list */\nclass LinkedListQueue {\n    ListNode? front, rear;  // Head node front and tail node rear \n    int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* Get the queue size */\n    public int Size() {\n        return queSize;\n    }\n\n    /* Check whether the queue is empty */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* Enqueue */\n    public void Push(int num) {\n        // Add num after the tail node\n        ListNode node = new(num);\n        // If the queue is empty, make both the head and tail point to this node\n        if (front == null) {\n            front = node;\n            rear = node;\n            // If the queue is not empty, add this node after the tail node\n        } else if (rear != null) {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* Dequeue */\n    public int Pop() {\n        int num = Peek();\n        // Remove the head node\n        front = front?.next;\n        queSize--;\n        return num;\n    }\n\n    /* Access the front element */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return front!.val;\n    }\n\n    /* Convert the linked list to an array and return it */\n    public int[] ToArray() {\n        if (front == null)\n            return [];\n\n        ListNode? node = front;\n        int[] res = new int[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.go<pre><code>/* Queue implemented with a linked list */\ntype linkedListQueue struct {\n    // Use the standard library list package to implement the queue\n    data *list.List\n}\n\n/* Initialize the queue */\nfunc newLinkedListQueue() *linkedListQueue {\n    return &amp;linkedListQueue{\n        data: list.New(),\n    }\n}\n\n/* Enqueue */\nfunc (s *linkedListQueue) push(value any) {\n    s.data.PushBack(value)\n}\n\n/* Dequeue */\nfunc (s *linkedListQueue) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* Access the front element */\nfunc (s *linkedListQueue) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* Get the queue size */\nfunc (s *linkedListQueue) size() int {\n    return s.data.Len()\n}\n\n/* Check whether the queue is empty */\nfunc (s *linkedListQueue) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* Get the list for printing */\nfunc (s *linkedListQueue) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_queue.swift<pre><code>/* Queue implemented with a linked list */\nclass LinkedListQueue {\n    private var front: ListNode? // Head node\n    private var rear: ListNode? // Tail node\n    private var _size: Int\n\n    init() {\n        _size = 0\n    }\n\n    /* Get the queue size */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* Check whether the queue is empty */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* Enqueue */\n    func push(num: Int) {\n        // Add num after the tail node\n        let node = ListNode(x: num)\n        // If the queue is empty, make both the head and tail point to this node\n        if front == nil {\n            front = node\n            rear = node\n        }\n        // If the queue is not empty, add this node after the tail node\n        else {\n            rear?.next = node\n            rear = node\n        }\n        _size += 1\n    }\n\n    /* Dequeue */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // Remove the head node\n        front = front?.next\n        _size -= 1\n        return num\n    }\n\n    /* Access the front element */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Queue is empty\")\n        }\n        return front!.val\n    }\n\n    /* Convert the linked list to an array and return it */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.js<pre><code>/* Queue implemented with a linked list */\nclass LinkedListQueue {\n    #front; // Head node #front\n    #rear; // Tail node #rear\n    #queSize = 0;\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n    }\n\n    /* Get the queue size */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* Check whether the queue is empty */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* Enqueue */\n    push(num) {\n        // Add num after the tail node\n        const node = new ListNode(num);\n        // If the queue is empty, make both the head and tail point to this node\n        if (!this.#front) {\n            this.#front = node;\n            this.#rear = node;\n            // If the queue is not empty, add this node after the tail node\n        } else {\n            this.#rear.next = node;\n            this.#rear = node;\n        }\n        this.#queSize++;\n    }\n\n    /* Dequeue */\n    pop() {\n        const num = this.peek();\n        // Remove the head node\n        this.#front = this.#front.next;\n        this.#queSize--;\n        return num;\n    }\n\n    /* Access the front element */\n    peek() {\n        if (this.size === 0) throw new Error('Queue is empty');\n        return this.#front.val;\n    }\n\n    /* Convert the linked list to an Array and return it */\n    toArray() {\n        let node = this.#front;\n        const res = new Array(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.ts<pre><code>/* Queue implemented with a linked list */\nclass LinkedListQueue {\n    private front: ListNode | null; // Head node front\n    private rear: ListNode | null; // Tail node rear\n    private queSize: number = 0;\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n    }\n\n    /* Get the queue size */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* Check whether the queue is empty */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* Enqueue */\n    push(num: number): void {\n        // Add num after the tail node\n        const node = new ListNode(num);\n        // If the queue is empty, make both the head and tail point to this node\n        if (!this.front) {\n            this.front = node;\n            this.rear = node;\n            // If the queue is not empty, add this node after the tail node\n        } else {\n            this.rear!.next = node;\n            this.rear = node;\n        }\n        this.queSize++;\n    }\n\n    /* Dequeue */\n    pop(): number {\n        const num = this.peek();\n        if (!this.front) throw new Error('Queue is empty');\n        // Remove the head node\n        this.front = this.front.next;\n        this.queSize--;\n        return num;\n    }\n\n    /* Access the front element */\n    peek(): number {\n        if (this.size === 0) throw new Error('Queue is empty');\n        return this.front!.val;\n    }\n\n    /* Convert the linked list to an Array and return it */\n    toArray(): number[] {\n        let node = this.front;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.dart<pre><code>/* Queue implemented with a linked list */\nclass LinkedListQueue {\n  ListNode? _front; // Head node _front\n  ListNode? _rear; // Tail node _rear\n  int _queSize = 0; // Queue size\n\n  LinkedListQueue() {\n    _front = null;\n    _rear = null;\n  }\n\n  /* Get the queue size */\n  int size() {\n    return _queSize;\n  }\n\n  /* Check whether the queue is empty */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* Enqueue */\n  void push(int _num) {\n    // Add _num after the tail node\n    final node = ListNode(_num);\n    // If the queue is empty, make both the head and tail point to this node\n    if (_front == null) {\n      _front = node;\n      _rear = node;\n    } else {\n      // If the queue is not empty, add this node after the tail node\n      _rear!.next = node;\n      _rear = node;\n    }\n    _queSize++;\n  }\n\n  /* Dequeue */\n  int pop() {\n    final int _num = peek();\n    // Remove the head node\n    _front = _front!.next;\n    _queSize--;\n    return _num;\n  }\n\n  /* Access the front element */\n  int peek() {\n    if (_queSize == 0) {\n      throw Exception('Queue is empty');\n    }\n    return _front!.val;\n  }\n\n  /* Convert the linked list to an array and return it */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; queue = [];\n    while (node != null) {\n      queue.add(node.val);\n      node = node.next;\n    }\n    return queue;\n  }\n}\n</code></pre> linkedlist_queue.rs<pre><code>/* Queue implemented with a linked list */\n#[allow(dead_code)]\npub struct LinkedListQueue&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // Head node front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // Tail node rear\n    que_size: usize,                         // Queue size\n}\n\nimpl&lt;T: Copy&gt; LinkedListQueue&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* Get the queue size */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* Check whether the queue is empty */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.que_size == 0;\n    }\n\n    /* Enqueue */\n    pub fn push(&amp;mut self, num: T) {\n        // Add num after the tail node\n        let new_rear = ListNode::new(num);\n        match self.rear.take() {\n            // If the queue is not empty, add this node after the tail node\n            Some(old_rear) =&gt; {\n                old_rear.borrow_mut().next = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n            // If the queue is empty, make both the head and tail point to this node\n            None =&gt; {\n                self.front = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n        }\n        self.que_size += 1;\n    }\n\n    /* Dequeue */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.front.take().map(|old_front| {\n            match old_front.borrow_mut().next.take() {\n                Some(new_front) =&gt; {\n                    self.front = Some(new_front);\n                }\n                None =&gt; {\n                    self.rear.take();\n                }\n            }\n            self.que_size -= 1;\n            old_front.borrow().val\n        })\n    }\n\n    /* Access the front element */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* Convert the linked list to a vector and return it */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        let mut res: Vec&lt;T&gt; = Vec::new();\n\n        fn recur&lt;T: Copy&gt;(cur: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, res: &amp;mut Vec&lt;T&gt;) {\n            if let Some(cur) = cur {\n                res.push(cur.borrow().val);\n                recur(cur.borrow().next.as_ref(), res);\n            }\n        }\n\n        recur(head, &amp;mut res);\n\n        res\n    }\n}\n</code></pre> linkedlist_queue.c<pre><code>/* Queue implemented with a linked list */\ntypedef struct {\n    ListNode *front, *rear;\n    int queSize;\n} LinkedListQueue;\n\n/* Constructor */\nLinkedListQueue *newLinkedListQueue() {\n    LinkedListQueue *queue = (LinkedListQueue *)malloc(sizeof(LinkedListQueue));\n    queue-&gt;front = NULL;\n    queue-&gt;rear = NULL;\n    queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* Destructor */\nvoid delLinkedListQueue(LinkedListQueue *queue) {\n    // Free all nodes\n    while (queue-&gt;front != NULL) {\n        ListNode *tmp = queue-&gt;front;\n        queue-&gt;front = queue-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // Free the queue structure\n    free(queue);\n}\n\n/* Get the queue size */\nint size(LinkedListQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* Check whether the queue is empty */\nbool empty(LinkedListQueue *queue) {\n    return (size(queue) == 0);\n}\n\n/* Enqueue */\nvoid push(LinkedListQueue *queue, int num) {\n    // Add node at the tail\n    ListNode *node = newListNode(num);\n    // If the queue is empty, make both the head and tail point to this node\n    if (queue-&gt;front == NULL) {\n        queue-&gt;front = node;\n        queue-&gt;rear = node;\n    }\n    // If the queue is not empty, add this node after the tail node\n    else {\n        queue-&gt;rear-&gt;next = node;\n        queue-&gt;rear = node;\n    }\n    queue-&gt;queSize++;\n}\n\n/* Access the front element */\nint peek(LinkedListQueue *queue) {\n    assert(size(queue) &amp;&amp; queue-&gt;front);\n    return queue-&gt;front-&gt;val;\n}\n\n/* Dequeue */\nint pop(LinkedListQueue *queue) {\n    int num = peek(queue);\n    ListNode *tmp = queue-&gt;front;\n    queue-&gt;front = queue-&gt;front-&gt;next;\n    free(tmp);\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* Print the queue */\nvoid printLinkedListQueue(LinkedListQueue *queue) {\n    int *arr = malloc(sizeof(int) * queue-&gt;queSize);\n    // Copy the linked list data into an array\n    int i;\n    ListNode *node;\n    for (i = 0, node = queue-&gt;front; i &lt; queue-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, queue-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_queue.kt<pre><code>/* Queue implemented with a linked list */\nclass LinkedListQueue(\n    // Head node front and tail node rear\n    private var front: ListNode? = null,\n    private var rear: ListNode? = null,\n    private var queSize: Int = 0\n) {\n\n    /* Get the queue size */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* Check whether the queue is empty */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* Enqueue */\n    fun push(num: Int) {\n        // Add num after the tail node\n        val node = ListNode(num)\n        // If the queue is empty, make both the head and tail point to this node\n        if (front == null) {\n            front = node\n            rear = node\n            // If the queue is not empty, add this node after the tail node\n        } else {\n            rear?.next = node\n            rear = node\n        }\n        queSize++\n    }\n\n    /* Dequeue */\n    fun pop(): Int {\n        val num = peek()\n        // Remove the head node\n        front = front?.next\n        queSize--\n        return num\n    }\n\n    /* Access the front element */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!._val\n    }\n\n    /* Convert the linked list to an array and return it */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!._val\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.rb<pre><code>### Queue implemented with a linked list ###\nclass LinkedListQueue\n  ### Get the queue size ###\n  attr_reader :size\n\n  ### Constructor ###\n  def initialize\n    @front = nil  # Head node front\n    @rear = nil   # Tail node rear\n    @size = 0\n  end\n\n  ### Check whether the queue is empty ###\n  def is_empty?\n    @front.nil?\n  end\n\n  ### Enqueue ###\n  def push(num)\n    # Add num after the tail node\n    node = ListNode.new(num)\n\n    # If the queue is empty, point both the head and tail to this node\n    if @front.nil?\n      @front = node\n      @rear = node\n    # If the queue is not empty, append this node after the tail\n    else\n      @rear.next = node\n      @rear = node\n    end\n\n    @size += 1\n  end\n\n  ### Dequeue ###\n  def pop\n    num = peek\n    # Remove the head node\n    @front = @front.next\n    @size -= 1\n    num\n  end\n\n  ### Access the front element ###\n  def peek\n    raise IndexError, 'Queue is empty' if is_empty?\n\n    @front.val\n  end\n\n  ### Convert the linked list to an Array and return it ###\n  def to_array\n    queue = []\n    temp = @front\n    while temp\n      queue &lt;&lt; temp.val\n      temp = temp.next\n    end\n    queue\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 5.   Stacks and queues","5.2   Queues"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#2-implementation-with-an-array","level":3,"title":"2.   Implementation with an array","text":"<p>Removing the first element of an array takes \\(O(n)\\) time, which would make dequeueing inefficient. A useful technique avoids this problem.</p> <p>Use a variable <code>front</code> for the index of the front element and a variable <code>size</code> for the queue's length. Define <code>rear = front + size</code>, so <code>rear</code> points to the position immediately after the last element.</p> <p>With this design, the occupied interval is <code>[front, rear - 1]</code>. Operations work as shown below.</p> <ul> <li>Enqueue: assign the new element to index <code>rear</code> and increase <code>size</code> by 1.</li> <li>Dequeue: increase <code>front</code> by 1 and decrease <code>size</code> by 1.</li> </ul> <p>Both operations require only a constant amount of work and take \\(O(1)\\) time.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> Figure 5-6   Enqueue and dequeue in an array-based queue </p> <p>There is a problem: repeated enqueueing and dequeueing move both <code>front</code> and <code>rear</code> to the right, and they cannot continue once they reach the end of the array. To solve this, treat the array as circular, with its end connected to its beginning.</p> <p>When <code>front</code> or <code>rear</code> passes the array's end, wrap it back to the beginning. The remainder operation implements this cyclic behavior:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_queue.py<pre><code>class ArrayQueue:\n    \"\"\"Queue implemented with a circular array\"\"\"\n\n    def __init__(self, size: int):\n        \"\"\"Constructor\"\"\"\n        self._nums: list[int] = [0] * size  # Array that stores the queue elements\n        self._front: int = 0  # Front pointer, pointing to the first element\n        self._size: int = 0  # Queue size\n\n    def capacity(self) -&gt; int:\n        \"\"\"Get the queue capacity\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"Get the queue size\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"Check whether the queue is empty\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"Enqueue\"\"\"\n        if self._size == self.capacity():\n            raise IndexError(\"Queue is full\")\n        # Calculate the rear pointer, which points one position past the last element\n        # Use modulo to wrap rear to the beginning when it passes the end of the array\n        rear: int = (self._front + self._size) % self.capacity()\n        # Add num to the rear\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"Dequeue\"\"\"\n        num: int = self.peek()\n        # Advance the front pointer by one position, wrapping to the beginning if it passes the end\n        self._front = (self._front + 1) % self.capacity()\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"Access the front element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Queue is empty\")\n        return self._nums[self._front]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"Return a list for printing\"\"\"\n        res = [0] * self.size()\n        j: int = self._front\n        for i in range(self.size()):\n            res[i] = self._nums[(j % self.capacity())]\n            j += 1\n        return res\n</code></pre> array_queue.cpp<pre><code>/* Queue implemented with a circular array */\nclass ArrayQueue {\n  private:\n    int *nums;       // Array that stores the queue elements\n    int front;       // Front index, identifying the first element\n    int queSize;     // Queue size\n    int queCapacity; // Queue capacity\n\n  public:\n    ArrayQueue(int capacity) {\n        // Initialize the array\n        nums = new int[capacity];\n        queCapacity = capacity;\n        front = queSize = 0;\n    }\n\n    ~ArrayQueue() {\n        delete[] nums;\n    }\n\n    /* Get the queue capacity */\n    int capacity() {\n        return queCapacity;\n    }\n\n    /* Get the queue size */\n    int size() {\n        return queSize;\n    }\n\n    /* Check whether the queue is empty */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* Enqueue */\n    void push(int num) {\n        if (queSize == queCapacity) {\n            cout &lt;&lt; \"Queue is full\" &lt;&lt; endl;\n            return;\n        }\n        // Calculate the rear index, one position past the last element\n        // Use modulo to wrap rear to the beginning when it passes the end of the array\n        int rear = (front + queSize) % queCapacity;\n        // Add num to the rear\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* Dequeue */\n    int pop() {\n        int num = peek();\n        // Advance the front index by one position, wrapping to the beginning if it passes the end\n        front = (front + 1) % queCapacity;\n        queSize--;\n        return num;\n    }\n\n    /* Access the front element */\n    int peek() {\n        if (isEmpty())\n            throw out_of_range(\"Queue is empty\");\n        return nums[front];\n    }\n\n    /* Convert the array to a vector and return it */\n    vector&lt;int&gt; toVector() {\n        // Copy only the elements in the occupied range\n        vector&lt;int&gt; arr(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            arr[i] = nums[j % queCapacity];\n        }\n        return arr;\n    }\n};\n</code></pre> array_queue.java<pre><code>/* Queue implemented with a circular array */\nclass ArrayQueue {\n    private int[] nums; // Array that stores the queue elements\n    private int front; // Front pointer, pointing to the first element\n    private int queSize; // Queue size\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* Get the queue capacity */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* Get the queue size */\n    public int size() {\n        return queSize;\n    }\n\n    /* Check whether the queue is empty */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* Enqueue */\n    public void push(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"Queue is full\");\n            return;\n        }\n        // Calculate the rear pointer, which points one position past the last element\n        // Use modulo to wrap rear to the beginning when it passes the end of the array\n        int rear = (front + queSize) % capacity();\n        // Add num to the rear\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* Dequeue */\n    public int pop() {\n        int num = peek();\n        // Advance the front pointer by one position, wrapping to the beginning if it passes the end\n        front = (front + 1) % capacity();\n        queSize--;\n        return num;\n    }\n\n    /* Access the front element */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* Return an array */\n    public int[] toArray() {\n        // Convert only the list elements within the occupied range\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.cs<pre><code>/* Queue implemented with a circular array */\nclass ArrayQueue {\n    int[] nums;  // Array that stores the queue elements\n    int front;   // Front pointer, pointing to the first element\n    int queSize; // Queue size\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* Get the queue capacity */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* Get the queue size */\n    public int Size() {\n        return queSize;\n    }\n\n    /* Check whether the queue is empty */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* Enqueue */\n    public void Push(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"Queue is full\");\n            return;\n        }\n        // Calculate the rear pointer, which points one position past the last element\n        // Use modulo to wrap rear to the beginning when it passes the end of the array\n        int rear = (front + queSize) % Capacity();\n        // Add num to the rear\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* Dequeue */\n    public int Pop() {\n        int num = Peek();\n        // Advance the front pointer by one position, wrapping to the beginning if it passes the end\n        front = (front + 1) % Capacity();\n        queSize--;\n        return num;\n    }\n\n    /* Access the front element */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return nums[front];\n    }\n\n    /* Return an array */\n    public int[] ToArray() {\n        // Copy only the elements within the occupied range\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % this.Capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.go<pre><code>/* Queue implemented with a circular array */\ntype arrayQueue struct {\n    nums        []int // Array that stores the queue elements\n    front       int   // Front pointer, pointing to the first element\n    queSize     int   // Queue size\n    queCapacity int   // Queue capacity (the maximum number of elements)\n}\n\n/* Initialize the queue */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\n    return &amp;arrayQueue{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* Get the queue size */\nfunc (q *arrayQueue) size() int {\n    return q.queSize\n}\n\n/* Check whether the queue is empty */\nfunc (q *arrayQueue) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* Enqueue */\nfunc (q *arrayQueue) push(num int) {\n    // The queue is full when queSize == queCapacity\n    if q.queSize == q.queCapacity {\n        return\n    }\n    // Calculate the rear pointer, which points one position past the last element\n    // Use modulo to wrap rear to the beginning when it passes the end of the array\n    rear := (q.front + q.queSize) % q.queCapacity\n    // Add num to the rear\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* Dequeue */\nfunc (q *arrayQueue) pop() any {\n    num := q.peek()\n    if num == nil {\n        return nil\n    }\n\n    // Advance the front pointer by one position, wrapping to the beginning if it passes the end\n    q.front = (q.front + 1) % q.queCapacity\n    q.queSize--\n    return num\n}\n\n/* Access the front element */\nfunc (q *arrayQueue) peek() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* Get a slice for printing */\nfunc (q *arrayQueue) toSlice() []int {\n    rear := (q.front + q.queSize)\n    if rear &gt;= q.queCapacity {\n        rear %= q.queCapacity\n        return append(q.nums[q.front:], q.nums[:rear]...)\n    }\n    return q.nums[q.front:rear]\n}\n</code></pre> array_queue.swift<pre><code>/* Queue implemented with a circular array */\nclass ArrayQueue {\n    private var nums: [Int] // Array that stores the queue elements\n    private var front: Int // Front index, identifying the first element\n    private var _size: Int // Queue size\n\n    init(capacity: Int) {\n        // Initialize the array\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* Get the queue capacity */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* Get the queue size */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* Check whether the queue is empty */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* Enqueue */\n    func push(num: Int) {\n        if size() == capacity() {\n            print(\"Queue is full\")\n            return\n        }\n        // Calculate the rear index, one position past the last element\n        // Use modulo to wrap rear to the beginning when it passes the end of the array\n        let rear = (front + size()) % capacity()\n        // Add num to the rear\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* Dequeue */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // Advance the front index by one position, wrapping to the beginning if it passes the end\n        front = (front + 1) % capacity()\n        _size -= 1\n        return num\n    }\n\n    /* Access the front element */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Queue is empty\")\n        }\n        return nums[front]\n    }\n\n    /* Return an array */\n    func toArray() -&gt; [Int] {\n        // Include only the elements in the occupied range\n        (front ..&lt; front + size()).map { nums[$0 % capacity()] }\n    }\n}\n</code></pre> array_queue.js<pre><code>/* Queue implemented with a circular array */\nclass ArrayQueue {\n    #nums; // Array that stores the queue elements\n    #front = 0; // Front pointer, pointing to the first element\n    #queSize = 0; // Queue size\n\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n    }\n\n    /* Get the queue capacity */\n    get capacity() {\n        return this.#nums.length;\n    }\n\n    /* Get the queue size */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* Check whether the queue is empty */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* Enqueue */\n    push(num) {\n        if (this.size === this.capacity) {\n            console.log('Queue is full');\n            return;\n        }\n        // Calculate the rear pointer, which points one position past the last element\n        // Use modulo to wrap rear to the beginning when it passes the end of the array\n        const rear = (this.#front + this.size) % this.capacity;\n        // Add num to the rear\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* Dequeue */\n    pop() {\n        const num = this.peek();\n        // Advance the front pointer by one position, wrapping to the beginning if it passes the end\n        this.#front = (this.#front + 1) % this.capacity;\n        this.#queSize--;\n        return num;\n    }\n\n    /* Access the front element */\n    peek() {\n        if (this.isEmpty()) throw new Error('Queue is empty');\n        return this.#nums[this.#front];\n    }\n\n    /* Return an Array */\n    toArray() {\n        // Convert only the list elements within the occupied range\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.#front; i &lt; this.size; i++, j++) {\n            arr[i] = this.#nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.ts<pre><code>/* Queue implemented with a circular array */\nclass ArrayQueue {\n    private nums: number[]; // Array that stores the queue elements\n    private front: number; // Front pointer, pointing to the first element\n    private queSize: number; // Queue size\n\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = this.queSize = 0;\n    }\n\n    /* Get the queue capacity */\n    get capacity(): number {\n        return this.nums.length;\n    }\n\n    /* Get the queue size */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* Check whether the queue is empty */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* Enqueue */\n    push(num: number): void {\n        if (this.size === this.capacity) {\n            console.log('Queue is full');\n            return;\n        }\n        // Calculate the rear pointer, which points one position past the last element\n        // Use modulo to wrap rear to the beginning when it passes the end of the array\n        const rear = (this.front + this.queSize) % this.capacity;\n        // Add num to the rear\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* Dequeue */\n    pop(): number {\n        const num = this.peek();\n        // Advance the front pointer by one position, wrapping to the beginning if it passes the end\n        this.front = (this.front + 1) % this.capacity;\n        this.queSize--;\n        return num;\n    }\n\n    /* Access the front element */\n    peek(): number {\n        if (this.isEmpty()) throw new Error('Queue is empty');\n        return this.nums[this.front];\n    }\n\n    /* Return an Array */\n    toArray(): number[] {\n        // Convert only the list elements within the occupied range\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.front; i &lt; this.size; i++, j++) {\n            arr[i] = this.nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.dart<pre><code>/* Queue implemented with a circular array */\nclass ArrayQueue {\n  late List&lt;int&gt; _nums; // Array that stores the queue elements\n  late int _front; // Front pointer, pointing to the first element\n  late int _queSize; // Queue size\n\n  ArrayQueue(int capacity) {\n    _nums = List.filled(capacity, 0);\n    _front = _queSize = 0;\n  }\n\n  /* Get the queue capacity */\n  int capaCity() {\n    return _nums.length;\n  }\n\n  /* Get the queue size */\n  int size() {\n    return _queSize;\n  }\n\n  /* Check whether the queue is empty */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* Enqueue */\n  void push(int _num) {\n    if (_queSize == capaCity()) {\n      throw Exception(\"Queue is full\");\n    }\n    // Calculate the rear pointer, which points one position past the last element\n    // Use modulo to wrap rear to the beginning when it passes the end of the array\n    int rear = (_front + _queSize) % capaCity();\n    // Add _num to the rear\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* Dequeue */\n  int pop() {\n    int _num = peek();\n    // Advance the front pointer by one position, wrapping to the beginning if it passes the end\n    _front = (_front + 1) % capaCity();\n    _queSize--;\n    return _num;\n  }\n\n  /* Access the front element */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"Queue is empty\");\n    }\n    return _nums[_front];\n  }\n\n  /* Return an array */\n  List&lt;int&gt; toArray() {\n    // Convert only the list elements within the occupied range\n    final List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[j % capaCity()];\n    }\n    return res;\n  }\n}\n</code></pre> array_queue.rs<pre><code>/* Queue implemented with a circular array */\nstruct ArrayQueue&lt;T&gt; {\n    nums: Vec&lt;T&gt;,      // Array that stores the queue elements\n    front: i32,        // Front index, identifying the first element\n    que_size: i32,     // Queue size\n    que_capacity: i32, // Queue capacity\n}\n\nimpl&lt;T: Copy + Default&gt; ArrayQueue&lt;T&gt; {\n    /* Constructor */\n    fn new(capacity: i32) -&gt; ArrayQueue&lt;T&gt; {\n        ArrayQueue {\n            nums: vec![T::default(); capacity as usize],\n            front: 0,\n            que_size: 0,\n            que_capacity: capacity,\n        }\n    }\n\n    /* Get the queue capacity */\n    fn capacity(&amp;self) -&gt; i32 {\n        self.que_capacity\n    }\n\n    /* Get the queue size */\n    fn size(&amp;self) -&gt; i32 {\n        self.que_size\n    }\n\n    /* Check whether the queue is empty */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* Enqueue */\n    fn push(&amp;mut self, num: T) {\n        if self.que_size == self.capacity() {\n            println!(\"Queue is full\");\n            return;\n        }\n        // Calculate the rear index, one position past the last element\n        // Use modulo to wrap rear to the beginning when it passes the end of the array\n        let rear = (self.front + self.que_size) % self.que_capacity;\n        // Add num to the rear\n        self.nums[rear as usize] = num;\n        self.que_size += 1;\n    }\n\n    /* Dequeue */\n    fn pop(&amp;mut self) -&gt; T {\n        let num = self.peek();\n        // Advance the front index by one position, wrapping to the beginning if it passes the end\n        self.front = (self.front + 1) % self.que_capacity;\n        self.que_size -= 1;\n        num\n    }\n\n    /* Access the front element */\n    fn peek(&amp;self) -&gt; T {\n        if self.is_empty() {\n            panic!(\"Index out of bounds\");\n        }\n        self.nums[self.front as usize]\n    }\n\n    /* Return a vector */\n    fn to_vector(&amp;self) -&gt; Vec&lt;T&gt; {\n        let cap = self.que_capacity;\n        let mut j = self.front;\n        let mut arr = vec![T::default(); cap as usize];\n        for i in 0..self.que_size {\n            arr[i as usize] = self.nums[(j % cap) as usize];\n            j += 1;\n        }\n        arr\n    }\n}\n</code></pre> array_queue.c<pre><code>/* Queue implemented with a circular array */\ntypedef struct {\n    int *nums;       // Array that stores the queue elements\n    int front;       // Front pointer, pointing to the first element\n    int queSize;     // Current number of elements in the queue\n    int queCapacity; // Queue capacity\n} ArrayQueue;\n\n/* Constructor */\nArrayQueue *newArrayQueue(int capacity) {\n    ArrayQueue *queue = (ArrayQueue *)malloc(sizeof(ArrayQueue));\n    // Initialize the array\n    queue-&gt;queCapacity = capacity;\n    queue-&gt;nums = (int *)malloc(sizeof(int) * queue-&gt;queCapacity);\n    queue-&gt;front = queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* Destructor */\nvoid delArrayQueue(ArrayQueue *queue) {\n    free(queue-&gt;nums);\n    free(queue);\n}\n\n/* Get the queue capacity */\nint capacity(ArrayQueue *queue) {\n    return queue-&gt;queCapacity;\n}\n\n/* Get the queue size */\nint size(ArrayQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* Check whether the queue is empty */\nbool empty(ArrayQueue *queue) {\n    return queue-&gt;queSize == 0;\n}\n\n/* Access the front element */\nint peek(ArrayQueue *queue) {\n    assert(size(queue) != 0);\n    return queue-&gt;nums[queue-&gt;front];\n}\n\n/* Enqueue */\nvoid push(ArrayQueue *queue, int num) {\n    if (size(queue) == capacity(queue)) {\n        printf(\"Queue is full\\r\\n\");\n        return;\n    }\n    // Calculate the rear pointer, which points one position past the last element\n    // Use modulo to wrap rear to the beginning when it passes the end of the array\n    int rear = (queue-&gt;front + queue-&gt;queSize) % queue-&gt;queCapacity;\n    // Add num to the rear\n    queue-&gt;nums[rear] = num;\n    queue-&gt;queSize++;\n}\n\n/* Dequeue */\nint pop(ArrayQueue *queue) {\n    int num = peek(queue);\n    // Advance the front pointer by one position, wrapping to the beginning if it passes the end\n    queue-&gt;front = (queue-&gt;front + 1) % queue-&gt;queCapacity;\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* Return an array for printing */\nint *toArray(ArrayQueue *queue, int *queSize) {\n    *queSize = queue-&gt;queSize;\n    int *res = (int *)calloc(queue-&gt;queSize, sizeof(int));\n    int j = queue-&gt;front;\n    for (int i = 0; i &lt; queue-&gt;queSize; i++) {\n        res[i] = queue-&gt;nums[j % queue-&gt;queCapacity];\n        j++;\n    }\n    return res;\n}\n</code></pre> array_queue.kt<pre><code>/* Queue implemented with a circular array */\nclass ArrayQueue(capacity: Int) {\n    private val nums: IntArray = IntArray(capacity) // Array that stores the queue elements\n    private var front: Int = 0 // Front pointer, pointing to the first element\n    private var queSize: Int = 0 // Queue size\n\n    /* Get the queue capacity */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* Get the queue size */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* Check whether the queue is empty */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* Enqueue */\n    fun push(num: Int) {\n        if (queSize == capacity()) {\n            println(\"Queue is full\")\n            return\n        }\n        // Calculate the rear pointer, which points one position past the last element\n        // Use modulo to wrap rear to the beginning when it passes the end of the array\n        val rear = (front + queSize) % capacity()\n        // Add num to the rear\n        nums[rear] = num\n        queSize++\n    }\n\n    /* Dequeue */\n    fun pop(): Int {\n        val num = peek()\n        // Advance the front pointer by one position, wrapping to the beginning if it passes the end\n        front = (front + 1) % capacity()\n        queSize--\n        return num\n    }\n\n    /* Access the front element */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* Return an array */\n    fun toArray(): IntArray {\n        // Copy only the elements within the occupied range\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[j % capacity()]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_queue.rb<pre><code>### Queue implemented with a circular array ###\nclass ArrayQueue\n  ### Get the queue size ###\n  attr_reader :size\n\n  ### Constructor ###\n  def initialize(size)\n    @nums = Array.new(size, 0) # Array that stores the queue elements\n    @front = 0 # Front pointer, pointing to the first element\n    @size = 0 # Queue size\n  end\n\n  ### Get the queue capacity ###\n  def capacity\n    @nums.length\n  end\n\n  ### Check whether the queue is empty ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### Enqueue ###\n  def push(num)\n    raise IndexError, 'Queue is full' if size == capacity\n\n    # Calculate the rear pointer, which points one position past the last element\n    # Use modulo to wrap rear to the beginning when it passes the end of the array\n    rear = (@front + size) % capacity\n    # Add num to the rear\n    @nums[rear] = num\n    @size += 1\n  end\n\n  ### Dequeue ###\n  def pop\n    num = peek\n    # Advance the front pointer by one position, wrapping to the beginning if it passes the end\n    @front = (@front + 1) % capacity\n    @size -= 1\n    num\n  end\n\n  ### Access the front element ###\n  def peek\n    raise IndexError, 'Queue is empty' if is_empty?\n\n    @nums[@front]\n  end\n\n  ### Return a list for printing ###\n  def to_array\n    res = Array.new(size, 0)\n    j = @front\n\n    for i in 0...size\n      res[i] = @nums[j % capacity]\n      j += 1\n    end\n\n    res\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>This queue still has a limitation: its capacity is fixed. We can address it by replacing the array with a dynamic array and adding resizing. Interested readers can try implementing this themselves.</p> <p>The comparison between these two implementations is the same as for stacks, so we will not repeat it here.</p>","path":["Chapter 5.   Stacks and queues","5.2   Queues"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#523-typical-applications","level":2,"title":"5.2.3   Typical applications","text":"<ul> <li>Taobao orders. When a shopper places an order, it enters a queue, and the system processes orders in sequence. During the Singles' Day shopping festival, enormous numbers of orders arrive in a short time, making high concurrency a major engineering challenge.</li> <li>Tasks awaiting processing. Queues preserve processing order wherever first-come, first-served behavior is needed, such as printer jobs or restaurant orders.</li> </ul>","path":["Chapter 5.   Stacks and queues","5.2   Queues"],"tags":[]},{"location":"chapter_stack_and_queue/stack/","level":1,"title":"5.1   Stacks","text":"<p>A stack is a linear data structure that follows a last-in, first-out rule.</p> <p>Think of a stack of plates on a table. If we can move only one plate at a time, reaching the bottom plate requires removing all the plates above it first. Replace the plates with elements such as integers, characters, or objects, and we have a stack.</p> <p>As shown below, the upper end is the “top” of the stack and the lower end is the “bottom.” Adding an element to the top is called “pushing,” and removing the top element is called “popping.”</p> <p></p> <p> Figure 5-1   The last-in, first-out rule of a stack </p>","path":["Chapter 5.   Stacks and queues","5.1   Stacks"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#511-common-stack-operations","level":2,"title":"5.1.1   Common stack operations","text":"<p>Table 5-1 lists common operations. Actual method names depend on the programming language; here we use the familiar names <code>push()</code>, <code>pop()</code>, and <code>peek()</code>.</p> <p> Table 5-1   Efficiency of stack operations </p> Method Description Time complexity <code>push()</code> Push an element onto the top \\(O(1)\\) <code>pop()</code> Remove the top element \\(O(1)\\) <code>peek()</code> Access the top element \\(O(1)\\) <p>We can usually use a language's built-in stack class. If no dedicated class is provided, use an array or linked list as a stack and avoid operations unrelated to stack behavior.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby stack.py<pre><code># Initialize a stack\n# Python has no built-in stack class; use a list as a stack\nstack: list[int] = []\n\n# Push elements\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n# Access the top element\npeek: int = stack[-1]\n\n# Pop an element\npop: int = stack.pop()\n\n# Get the stack size\nsize: int = len(stack)\n\n# Check whether the stack is empty\nis_empty: bool = len(stack) == 0\n</code></pre> stack.cpp<pre><code>/* Initialize a stack */\nstack&lt;int&gt; stack;\n\n/* Push elements */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access the top element */\nint top = stack.top();\n\n/* Pop an element */\nstack.pop(); // No return value\n\n/* Get the stack size */\nint size = stack.size();\n\n/* Check whether the stack is empty */\nbool empty = stack.empty();\n</code></pre> stack.java<pre><code>/* Initialize a stack */\nStack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n\n/* Push elements */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access the top element */\nint peek = stack.peek();\n\n/* Pop an element */\nint pop = stack.pop();\n\n/* Get the stack size */\nint size = stack.size();\n\n/* Check whether the stack is empty */\nboolean isEmpty = stack.isEmpty();\n</code></pre> stack.cs<pre><code>/* Initialize a stack */\nStack&lt;int&gt; stack = new();\n\n/* Push elements */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n\n/* Access the top element */\nint peek = stack.Peek();\n\n/* Pop an element */\nint pop = stack.Pop();\n\n/* Get the stack size */\nint size = stack.Count;\n\n/* Check whether the stack is empty */\nbool isEmpty = stack.Count == 0;\n</code></pre> stack_test.go<pre><code>/* Initialize a stack */\n// In Go, a slice is recommended for use as a stack\nvar stack []int\n\n/* Push elements */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n\n/* Access the top element */\npeek := stack[len(stack)-1]\n\n/* Pop an element */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n\n/* Get the stack size */\nsize := len(stack)\n\n/* Check whether the stack is empty */\nisEmpty := len(stack) == 0\n</code></pre> stack.swift<pre><code>/* Initialize a stack */\n// Swift has no built-in stack class; use an Array as a stack\nvar stack: [Int] = []\n\n/* Push elements */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n/* Access the top element */\nlet peek = stack.last!\n\n/* Pop an element */\nlet pop = stack.removeLast()\n\n/* Get the stack size */\nlet size = stack.count\n\n/* Check whether the stack is empty */\nlet isEmpty = stack.isEmpty\n</code></pre> stack.js<pre><code>/* Initialize a stack */\n// JavaScript has no built-in stack class; use an Array as a stack\nconst stack = [];\n\n/* Push elements */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access the top element */\nconst peek = stack[stack.length-1];\n\n/* Pop an element */\nconst pop = stack.pop();\n\n/* Get the stack size */\nconst size = stack.length;\n\n/* Check whether the stack is empty */\nconst is_empty = stack.length === 0;\n</code></pre> stack.ts<pre><code>/* Initialize a stack */\n// TypeScript has no built-in stack class; use an Array as a stack\nconst stack: number[] = [];\n\n/* Push elements */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access the top element */\nconst peek = stack[stack.length - 1];\n\n/* Pop an element */\nconst pop = stack.pop();\n\n/* Get the stack size */\nconst size = stack.length;\n\n/* Check whether the stack is empty */\nconst is_empty = stack.length === 0;\n</code></pre> stack.dart<pre><code>/* Initialize a stack */\n// Dart has no built-in stack class; use a List as a stack\nList&lt;int&gt; stack = [];\n\n/* Push elements */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n\n/* Access the top element */\nint peek = stack.last;\n\n/* Pop an element */\nint pop = stack.removeLast();\n\n/* Get the stack size */\nint size = stack.length;\n\n/* Check whether the stack is empty */\nbool isEmpty = stack.isEmpty;\n</code></pre> stack.rs<pre><code>/* Initialize a stack */\n// Use a Vec as a stack\nlet mut stack: Vec&lt;i32&gt; = Vec::new();\n\n/* Push elements */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* Access the top element */\nlet top = stack.last().unwrap();\n\n/* Pop an element */\nlet pop = stack.pop().unwrap();\n\n/* Get the stack size */\nlet size = stack.len();\n\n/* Check whether the stack is empty */\nlet is_empty = stack.is_empty();\n</code></pre> stack.c<pre><code>// C provides no built-in stack\n</code></pre> stack.kt<pre><code>/* Initialize a stack */\nval stack = Stack&lt;Int&gt;()\n\n/* Push elements */\nstack.push(1)\nstack.push(3)\nstack.push(2)\nstack.push(5)\nstack.push(4)\n\n/* Access the top element */\nval peek = stack.peek()\n\n/* Pop an element */\nval pop = stack.pop()\n\n/* Get the stack size */\nval size = stack.size\n\n/* Check whether the stack is empty */\nval isEmpty = stack.isEmpty()\n</code></pre> stack.rb<pre><code># Initialize a stack\n# Ruby has no built-in stack class; use an Array as a stack\nstack = []\n\n# Push elements\nstack &lt;&lt; 1\nstack &lt;&lt; 3\nstack &lt;&lt; 2\nstack &lt;&lt; 5\nstack &lt;&lt; 4\n\n# Access the top element\npeek = stack.last\n\n# Pop an element\npop = stack.pop\n\n# Get the stack size\nsize = stack.length\n\n# Check whether the stack is empty\nis_empty = stack.empty?\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 5.   Stacks and queues","5.1   Stacks"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#512-implementing-a-stack","level":2,"title":"5.1.2   Implementing a stack","text":"<p>To understand how a stack works more deeply, let us implement a stack class ourselves.</p> <p>A stack follows last-in, first-out order, so elements can be added or removed only at the top. Arrays and linked lists allow insertion and deletion at arbitrary positions, so a stack can be viewed as a restricted array or linked list. By hiding irrelevant operations, we make its public behavior conform to the stack's rules.</p>","path":["Chapter 5.   Stacks and queues","5.1   Stacks"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#1-implementation-with-a-linked-list","level":3,"title":"1.   Implementation with a linked list","text":"<p>When implementing a stack with a linked list, treat the head as the top and the tail as the bottom.</p> <p>As shown below, pushing simply inserts a node at the head, an operation called “head insertion.” Popping removes the head node.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> Figure 5-2   Push and pop in a linked-list stack </p> <p>Here is an implementation using a linked list:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linkedlist_stack.py<pre><code>class LinkedListStack:\n    \"\"\"Stack implemented with a linked list\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        self._peek: ListNode | None = None\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"Get the stack size\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"Check whether the stack is empty\"\"\"\n        return self._size == 0\n\n    def push(self, val: int):\n        \"\"\"Push\"\"\"\n        node = ListNode(val)\n        node.next = self._peek\n        self._peek = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"Pop\"\"\"\n        num = self.peek()\n        self._peek = self._peek.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"Access the top element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Stack is empty\")\n        return self._peek.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"Convert to a list for printing\"\"\"\n        arr = []\n        node = self._peek\n        while node:\n            arr.append(node.val)\n            node = node.next\n        arr.reverse()\n        return arr\n</code></pre> linkedlist_stack.cpp<pre><code>/* Stack implemented with a linked list */\nclass LinkedListStack {\n  private:\n    ListNode *stackTop; // Use the head node as the stack top\n    int stkSize;        // Stack size\n\n  public:\n    LinkedListStack() {\n        stackTop = nullptr;\n        stkSize = 0;\n    }\n\n    ~LinkedListStack() {\n        // Traverse the linked list and delete each node to free its memory\n        freeMemoryLinkedList(stackTop);\n    }\n\n    /* Get the stack size */\n    int size() {\n        return stkSize;\n    }\n\n    /* Check whether the stack is empty */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* Push */\n    void push(int num) {\n        ListNode *node = new ListNode(num);\n        node-&gt;next = stackTop;\n        stackTop = node;\n        stkSize++;\n    }\n\n    /* Pop */\n    int pop() {\n        int num = top();\n        ListNode *tmp = stackTop;\n        stackTop = stackTop-&gt;next;\n        // Free memory\n        delete tmp;\n        stkSize--;\n        return num;\n    }\n\n    /* Access the top element */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"Stack is empty\");\n        return stackTop-&gt;val;\n    }\n\n    /* Convert the linked list to a vector and return it */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = stackTop;\n        vector&lt;int&gt; res(size());\n        for (int i = res.size() - 1; i &gt;= 0; i--) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_stack.java<pre><code>/* Stack implemented with a linked list */\nclass LinkedListStack {\n    private ListNode stackPeek; // Use the head node as the stack top\n    private int stkSize = 0; // Stack size\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* Get the stack size */\n    public int size() {\n        return stkSize;\n    }\n\n    /* Check whether the stack is empty */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* Push */\n    public void push(int num) {\n        ListNode node = new ListNode(num);\n        node.next = stackPeek;\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* Pop */\n    public int pop() {\n        int num = peek();\n        stackPeek = stackPeek.next;\n        stkSize--;\n        return num;\n    }\n\n    /* Access the top element */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stackPeek.val;\n    }\n\n    /* Convert the list to an array and return it */\n    public int[] toArray() {\n        ListNode node = stackPeek;\n        int[] res = new int[size()];\n        for (int i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.cs<pre><code>/* Stack implemented with a linked list */\nclass LinkedListStack {\n    ListNode? stackPeek;  // Use the head node as the stack top\n    int stkSize = 0;   // Stack size\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* Get the stack size */\n    public int Size() {\n        return stkSize;\n    }\n\n    /* Check whether the stack is empty */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* Push */\n    public void Push(int num) {\n        ListNode node = new(num) {\n            next = stackPeek\n        };\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* Pop */\n    public int Pop() {\n        int num = Peek();\n        stackPeek = stackPeek!.next;\n        stkSize--;\n        return num;\n    }\n\n    /* Access the top element */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stackPeek!.val;\n    }\n\n    /* Convert the linked list to an array and return it */\n    public int[] ToArray() {\n        if (stackPeek == null)\n            return [];\n\n        ListNode? node = stackPeek;\n        int[] res = new int[Size()];\n        for (int i = res.Length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.go<pre><code>/* Stack implemented with a linked list */\ntype linkedListStack struct {\n    // Use the standard library list package to implement the stack\n    data *list.List\n}\n\n/* Initialize the stack */\nfunc newLinkedListStack() *linkedListStack {\n    return &amp;linkedListStack{\n        data: list.New(),\n    }\n}\n\n/* Push */\nfunc (s *linkedListStack) push(value int) {\n    s.data.PushBack(value)\n}\n\n/* Pop */\nfunc (s *linkedListStack) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* Access the top element */\nfunc (s *linkedListStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* Get the stack size */\nfunc (s *linkedListStack) size() int {\n    return s.data.Len()\n}\n\n/* Check whether the stack is empty */\nfunc (s *linkedListStack) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* Get the list for printing */\nfunc (s *linkedListStack) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_stack.swift<pre><code>/* Stack implemented with a linked list */\nclass LinkedListStack {\n    private var _peek: ListNode? // Use the head node as the stack top\n    private var _size: Int // Stack size\n\n    init() {\n        _size = 0\n    }\n\n    /* Get the stack size */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* Check whether the stack is empty */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* Push */\n    func push(num: Int) {\n        let node = ListNode(x: num)\n        node.next = _peek\n        _peek = node\n        _size += 1\n    }\n\n    /* Pop */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        _peek = _peek?.next\n        _size -= 1\n        return num\n    }\n\n    /* Access the top element */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Stack is empty\")\n        }\n        return _peek!.val\n    }\n\n    /* Convert the linked list to an array and return it */\n    func toArray() -&gt; [Int] {\n        var node = _peek\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices.reversed() {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.js<pre><code>/* Stack implemented with a linked list */\nclass LinkedListStack {\n    #stackPeek; // Use the head node as the stack top\n    #stkSize = 0; // Stack size\n\n    constructor() {\n        this.#stackPeek = null;\n    }\n\n    /* Get the stack size */\n    get size() {\n        return this.#stkSize;\n    }\n\n    /* Check whether the stack is empty */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* Push */\n    push(num) {\n        const node = new ListNode(num);\n        node.next = this.#stackPeek;\n        this.#stackPeek = node;\n        this.#stkSize++;\n    }\n\n    /* Pop */\n    pop() {\n        const num = this.peek();\n        this.#stackPeek = this.#stackPeek.next;\n        this.#stkSize--;\n        return num;\n    }\n\n    /* Access the top element */\n    peek() {\n        if (!this.#stackPeek) throw new Error('Stack is empty');\n        return this.#stackPeek.val;\n    }\n\n    /* Convert the linked list to an Array and return it */\n    toArray() {\n        let node = this.#stackPeek;\n        const res = new Array(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.ts<pre><code>/* Stack implemented with a linked list */\nclass LinkedListStack {\n    private stackPeek: ListNode | null; // Use the head node as the stack top\n    private stkSize: number = 0; // Stack size\n\n    constructor() {\n        this.stackPeek = null;\n    }\n\n    /* Get the stack size */\n    get size(): number {\n        return this.stkSize;\n    }\n\n    /* Check whether the stack is empty */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* Push */\n    push(num: number): void {\n        const node = new ListNode(num);\n        node.next = this.stackPeek;\n        this.stackPeek = node;\n        this.stkSize++;\n    }\n\n    /* Pop */\n    pop(): number {\n        const num = this.peek();\n        if (!this.stackPeek) throw new Error('Stack is empty');\n        this.stackPeek = this.stackPeek.next;\n        this.stkSize--;\n        return num;\n    }\n\n    /* Access the top element */\n    peek(): number {\n        if (!this.stackPeek) throw new Error('Stack is empty');\n        return this.stackPeek.val;\n    }\n\n    /* Convert the linked list to an Array and return it */\n    toArray(): number[] {\n        let node = this.stackPeek;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.dart<pre><code>/* Stack implemented with a linked list */\nclass LinkedListStack {\n  ListNode? _stackPeek; // Use the head node as the stack top\n  int _stkSize = 0; // Stack size\n\n  LinkedListStack() {\n    _stackPeek = null;\n  }\n\n  /* Get the stack size */\n  int size() {\n    return _stkSize;\n  }\n\n  /* Check whether the stack is empty */\n  bool isEmpty() {\n    return _stkSize == 0;\n  }\n\n  /* Push */\n  void push(int _num) {\n    final ListNode node = ListNode(_num);\n    node.next = _stackPeek;\n    _stackPeek = node;\n    _stkSize++;\n  }\n\n  /* Pop */\n  int pop() {\n    final int _num = peek();\n    _stackPeek = _stackPeek!.next;\n    _stkSize--;\n    return _num;\n  }\n\n  /* Access the top element */\n  int peek() {\n    if (_stackPeek == null) {\n      throw Exception(\"Stack is empty\");\n    }\n    return _stackPeek!.val;\n  }\n\n  /* Convert the linked list to a List and return it */\n  List&lt;int&gt; toList() {\n    ListNode? node = _stackPeek;\n    List&lt;int&gt; list = [];\n    while (node != null) {\n      list.add(node.val);\n      node = node.next;\n    }\n    list = list.reversed.toList();\n    return list;\n  }\n}\n</code></pre> linkedlist_stack.rs<pre><code>/* Stack implemented with a linked list */\n#[allow(dead_code)]\npub struct LinkedListStack&lt;T&gt; {\n    stack_peek: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // Use the head node as the stack top\n    stk_size: usize,                              // Stack size\n}\n\nimpl&lt;T: Copy&gt; LinkedListStack&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            stack_peek: None,\n            stk_size: 0,\n        }\n    }\n\n    /* Get the stack size */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.stk_size;\n    }\n\n    /* Check whether the stack is empty */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* Push */\n    pub fn push(&amp;mut self, num: T) {\n        let node = ListNode::new(num);\n        node.borrow_mut().next = self.stack_peek.take();\n        self.stack_peek = Some(node);\n        self.stk_size += 1;\n    }\n\n    /* Pop */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack_peek.take().map(|old_head| {\n            self.stack_peek = old_head.borrow_mut().next.take();\n            self.stk_size -= 1;\n\n            old_head.borrow().val\n        })\n    }\n\n    /* Access the top element */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.stack_peek.as_ref()\n    }\n\n    /* Convert the linked list to a vector and return it */\n    pub fn to_array(&amp;self) -&gt; Vec&lt;T&gt; {\n        fn _to_array&lt;T: Sized + Copy&gt;(head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n            if let Some(node) = head {\n                let mut nums = _to_array(node.borrow().next.as_ref());\n                nums.push(node.borrow().val);\n                return nums;\n            }\n            return Vec::new();\n        }\n\n        _to_array(self.peek())\n    }\n}\n</code></pre> linkedlist_stack.c<pre><code>/* Stack implemented with a linked list */\ntypedef struct {\n    ListNode *top; // Use the head node as the stack top\n    int size;      // Stack size\n} LinkedListStack;\n\n/* Constructor */\nLinkedListStack *newLinkedListStack() {\n    LinkedListStack *s = malloc(sizeof(LinkedListStack));\n    s-&gt;top = NULL;\n    s-&gt;size = 0;\n    return s;\n}\n\n/* Destructor */\nvoid delLinkedListStack(LinkedListStack *s) {\n    while (s-&gt;top) {\n        ListNode *n = s-&gt;top-&gt;next;\n        free(s-&gt;top);\n        s-&gt;top = n;\n    }\n    free(s);\n}\n\n/* Get the stack size */\nint size(LinkedListStack *s) {\n    return s-&gt;size;\n}\n\n/* Check whether the stack is empty */\nbool isEmpty(LinkedListStack *s) {\n    return size(s) == 0;\n}\n\n/* Push */\nvoid push(LinkedListStack *s, int num) {\n    ListNode *node = (ListNode *)malloc(sizeof(ListNode));\n    node-&gt;next = s-&gt;top; // Set the new node's pointer field\n    node-&gt;val = num;     // Set the new node's data field\n    s-&gt;top = node;       // Update the stack top\n    s-&gt;size++;           // Update the stack size\n}\n\n/* Access the top element */\nint peek(LinkedListStack *s) {\n    if (s-&gt;size == 0) {\n        printf(\"Stack is empty\\n\");\n        return INT_MAX;\n    }\n    return s-&gt;top-&gt;val;\n}\n\n/* Pop */\nint pop(LinkedListStack *s) {\n    int val = peek(s);\n    ListNode *tmp = s-&gt;top;\n    s-&gt;top = s-&gt;top-&gt;next;\n    // Free memory\n    free(tmp);\n    s-&gt;size--;\n    return val;\n}\n</code></pre> linkedlist_stack.kt<pre><code>/* Stack implemented with a linked list */\nclass LinkedListStack(\n    private var stackPeek: ListNode? = null, // Use the head node as the stack top\n    private var stkSize: Int = 0 // Stack size\n) {\n\n    /* Get the stack size */\n    fun size(): Int {\n        return stkSize\n    }\n\n    /* Check whether the stack is empty */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* Push */\n    fun push(num: Int) {\n        val node = ListNode(num)\n        node.next = stackPeek\n        stackPeek = node\n        stkSize++\n    }\n\n    /* Pop */\n    fun pop(): Int? {\n        val num = peek()\n        stackPeek = stackPeek?.next\n        stkSize--\n        return num\n    }\n\n    /* Access the top element */\n    fun peek(): Int? {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stackPeek?._val\n    }\n\n    /* Convert the list to an array and return it */\n    fun toArray(): IntArray {\n        var node = stackPeek\n        val res = IntArray(size())\n        for (i in res.size - 1 downTo 0) {\n            res[i] = node?._val!!\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.rb<pre><code>### Stack implemented with a linked list ###\nclass LinkedListStack\n  attr_reader :size\n\n  ### Constructor ###\n  def initialize\n    @size = 0\n  end\n\n  ### Check whether the stack is empty ###\n  def is_empty?\n    @peek.nil?\n  end\n\n  ### Push ###\n  def push(val)\n    node = ListNode.new(val)\n    node.next = @peek\n    @peek = node\n    @size += 1\n  end\n\n  ### Pop ###\n  def pop\n    num = peek\n    @peek = @peek.next\n    @size -= 1\n    num\n  end\n\n  ### Access the top element ###\n  def peek\n    raise IndexError, 'Stack is empty' if is_empty?\n\n    @peek.val\n  end\n\n  ### Convert the linked list to an Array and return it ###\n  def to_array\n    arr = []\n    node = @peek\n    while node\n      arr &lt;&lt; node.val\n      node = node.next\n    end\n    arr.reverse\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 5.   Stacks and queues","5.1   Stacks"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#2-implementation-with-an-array","level":3,"title":"2.   Implementation with an array","text":"<p>For an array-based stack, use the end of the array as the top. As shown below, pushing and popping correspond to adding and removing the last element, both taking \\(O(1)\\) time.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> Figure 5-3   Push and pop in an array-based stack </p> <p>Since elements may keep arriving, use a dynamic array so that resizing is handled for us. An implementation follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_stack.py<pre><code>class ArrayStack:\n    \"\"\"Stack implemented with an array\"\"\"\n\n    def __init__(self):\n        \"\"\"Constructor\"\"\"\n        self._stack: list[int] = []\n\n    def size(self) -&gt; int:\n        \"\"\"Get the stack size\"\"\"\n        return len(self._stack)\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"Check whether the stack is empty\"\"\"\n        return self.size() == 0\n\n    def push(self, item: int):\n        \"\"\"Push\"\"\"\n        self._stack.append(item)\n\n    def pop(self) -&gt; int:\n        \"\"\"Pop\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Stack is empty\")\n        return self._stack.pop()\n\n    def peek(self) -&gt; int:\n        \"\"\"Access the top element\"\"\"\n        if self.is_empty():\n            raise IndexError(\"Stack is empty\")\n        return self._stack[-1]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"Return a list for printing\"\"\"\n        return self._stack\n</code></pre> array_stack.cpp<pre><code>/* Stack implemented with an array */\nclass ArrayStack {\n  private:\n    vector&lt;int&gt; stack;\n\n  public:\n    /* Get the stack size */\n    int size() {\n        return stack.size();\n    }\n\n    /* Check whether the stack is empty */\n    bool isEmpty() {\n        return stack.size() == 0;\n    }\n\n    /* Push */\n    void push(int num) {\n        stack.push_back(num);\n    }\n\n    /* Pop */\n    int pop() {\n        int num = top();\n        stack.pop_back();\n        return num;\n    }\n\n    /* Access the top element */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"Stack is empty\");\n        return stack.back();\n    }\n\n    /* Return a vector */\n    vector&lt;int&gt; toVector() {\n        return stack;\n    }\n};\n</code></pre> array_stack.java<pre><code>/* Stack implemented with an array */\nclass ArrayStack {\n    private ArrayList&lt;Integer&gt; stack;\n\n    public ArrayStack() {\n        // Initialize the list (dynamic array)\n        stack = new ArrayList&lt;&gt;();\n    }\n\n    /* Get the stack size */\n    public int size() {\n        return stack.size();\n    }\n\n    /* Check whether the stack is empty */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* Push */\n    public void push(int num) {\n        stack.add(num);\n    }\n\n    /* Pop */\n    public int pop() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.remove(size() - 1);\n    }\n\n    /* Access the top element */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.get(size() - 1);\n    }\n\n    /* Convert the list to an array and return it */\n    public Object[] toArray() {\n        return stack.toArray();\n    }\n}\n</code></pre> array_stack.cs<pre><code>/* Stack implemented with an array */\nclass ArrayStack {\n    List&lt;int&gt; stack;\n    public ArrayStack() {\n        // Initialize the list (dynamic array)\n        stack = [];\n    }\n\n    /* Get the stack size */\n    public int Size() {\n        return stack.Count;\n    }\n\n    /* Check whether the stack is empty */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* Push */\n    public void Push(int num) {\n        stack.Add(num);\n    }\n\n    /* Pop */\n    public int Pop() {\n        if (IsEmpty())\n            throw new Exception();\n        var val = Peek();\n        stack.RemoveAt(Size() - 1);\n        return val;\n    }\n\n    /* Access the top element */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stack[Size() - 1];\n    }\n\n    /* Convert the list to an array and return it */\n    public int[] ToArray() {\n        return [.. stack];\n    }\n}\n</code></pre> array_stack.go<pre><code>/* Stack implemented with an array */\ntype arrayStack struct {\n    data []int // Data\n}\n\n/* Initialize the stack */\nfunc newArrayStack() *arrayStack {\n    return &amp;arrayStack{\n        // Set the stack length to 0 and its capacity to 16\n        data: make([]int, 0, 16),\n    }\n}\n\n/* Stack size */\nfunc (s *arrayStack) size() int {\n    return len(s.data)\n}\n\n/* Check whether the stack is empty */\nfunc (s *arrayStack) isEmpty() bool {\n    return s.size() == 0\n}\n\n/* Push */\nfunc (s *arrayStack) push(v int) {\n    // Appending grows the slice capacity automatically when needed\n    s.data = append(s.data, v)\n}\n\n/* Pop */\nfunc (s *arrayStack) pop() any {\n    val := s.peek()\n    s.data = s.data[:len(s.data)-1]\n    return val\n}\n\n/* Get the top element */\nfunc (s *arrayStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    val := s.data[len(s.data)-1]\n    return val\n}\n\n/* Get a slice for printing */\nfunc (s *arrayStack) toSlice() []int {\n    return s.data\n}\n</code></pre> array_stack.swift<pre><code>/* Stack implemented with an array */\nclass ArrayStack {\n    private var stack: [Int]\n\n    init() {\n        // Initialize the list (dynamic array)\n        stack = []\n    }\n\n    /* Get the stack size */\n    func size() -&gt; Int {\n        stack.count\n    }\n\n    /* Check whether the stack is empty */\n    func isEmpty() -&gt; Bool {\n        stack.isEmpty\n    }\n\n    /* Push */\n    func push(num: Int) {\n        stack.append(num)\n    }\n\n    /* Pop */\n    @discardableResult\n    func pop() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Stack is empty\")\n        }\n        return stack.removeLast()\n    }\n\n    /* Access the top element */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"Stack is empty\")\n        }\n        return stack.last!\n    }\n\n    /* Return the stack elements as an array */\n    func toArray() -&gt; [Int] {\n        stack\n    }\n}\n</code></pre> array_stack.js<pre><code>/* Stack implemented with an array */\nclass ArrayStack {\n    #stack;\n    constructor() {\n        this.#stack = [];\n    }\n\n    /* Get the stack size */\n    get size() {\n        return this.#stack.length;\n    }\n\n    /* Check whether the stack is empty */\n    isEmpty() {\n        return this.#stack.length === 0;\n    }\n\n    /* Push */\n    push(num) {\n        this.#stack.push(num);\n    }\n\n    /* Pop */\n    pop() {\n        if (this.isEmpty()) throw new Error('Stack is empty');\n        return this.#stack.pop();\n    }\n\n    /* Access the top element */\n    top() {\n        if (this.isEmpty()) throw new Error('Stack is empty');\n        return this.#stack[this.#stack.length - 1];\n    }\n\n    /* Return an Array */\n    toArray() {\n        return this.#stack;\n    }\n}\n</code></pre> array_stack.ts<pre><code>/* Stack implemented with an array */\nclass ArrayStack {\n    private stack: number[];\n    constructor() {\n        this.stack = [];\n    }\n\n    /* Get the stack size */\n    get size(): number {\n        return this.stack.length;\n    }\n\n    /* Check whether the stack is empty */\n    isEmpty(): boolean {\n        return this.stack.length === 0;\n    }\n\n    /* Push */\n    push(num: number): void {\n        this.stack.push(num);\n    }\n\n    /* Pop */\n    pop(): number | undefined {\n        if (this.isEmpty()) throw new Error('Stack is empty');\n        return this.stack.pop();\n    }\n\n    /* Access the top element */\n    top(): number | undefined {\n        if (this.isEmpty()) throw new Error('Stack is empty');\n        return this.stack[this.stack.length - 1];\n    }\n\n    /* Return an Array */\n    toArray() {\n        return this.stack;\n    }\n}\n</code></pre> array_stack.dart<pre><code>/* Stack implemented with an array */\nclass ArrayStack {\n  late List&lt;int&gt; _stack;\n  ArrayStack() {\n    _stack = [];\n  }\n\n  /* Get the stack size */\n  int size() {\n    return _stack.length;\n  }\n\n  /* Check whether the stack is empty */\n  bool isEmpty() {\n    return _stack.isEmpty;\n  }\n\n  /* Push */\n  void push(int _num) {\n    _stack.add(_num);\n  }\n\n  /* Pop */\n  int pop() {\n    if (isEmpty()) {\n      throw Exception(\"Stack is empty\");\n    }\n    return _stack.removeLast();\n  }\n\n  /* Access the top element */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"Stack is empty\");\n    }\n    return _stack.last;\n  }\n\n  /* Return the underlying stack array */\n  List&lt;int&gt; toArray() =&gt; _stack;\n}\n</code></pre> array_stack.rs<pre><code>/* Stack implemented with an array */\nstruct ArrayStack&lt;T&gt; {\n    stack: Vec&lt;T&gt;,\n}\n\nimpl&lt;T&gt; ArrayStack&lt;T&gt; {\n    /* Initialize the stack */\n    fn new() -&gt; ArrayStack&lt;T&gt; {\n        ArrayStack::&lt;T&gt; {\n            stack: Vec::&lt;T&gt;::new(),\n        }\n    }\n\n    /* Get the stack size */\n    fn size(&amp;self) -&gt; usize {\n        self.stack.len()\n    }\n\n    /* Check whether the stack is empty */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.size() == 0\n    }\n\n    /* Push */\n    fn push(&amp;mut self, num: T) {\n        self.stack.push(num);\n    }\n\n    /* Pop */\n    fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack.pop()\n    }\n\n    /* Access the top element */\n    fn peek(&amp;self) -&gt; Option&lt;&amp;T&gt; {\n        if self.is_empty() {\n            panic!(\"Stack is empty\")\n        };\n        self.stack.last()\n    }\n\n    /* Return a reference to the vector */\n    fn to_array(&amp;self) -&gt; &amp;Vec&lt;T&gt; {\n        &amp;self.stack\n    }\n}\n</code></pre> array_stack.c<pre><code>/* Stack implemented with an array */\ntypedef struct {\n    int *data;\n    int size;\n} ArrayStack;\n\n/* Constructor */\nArrayStack *newArrayStack() {\n    ArrayStack *stack = malloc(sizeof(ArrayStack));\n    // Initialize with a large capacity to avoid resizing\n    stack-&gt;data = malloc(sizeof(int) * MAX_SIZE);\n    stack-&gt;size = 0;\n    return stack;\n}\n\n/* Destructor */\nvoid delArrayStack(ArrayStack *stack) {\n    free(stack-&gt;data);\n    free(stack);\n}\n\n/* Get the stack size */\nint size(ArrayStack *stack) {\n    return stack-&gt;size;\n}\n\n/* Check whether the stack is empty */\nbool isEmpty(ArrayStack *stack) {\n    return stack-&gt;size == 0;\n}\n\n/* Push */\nvoid push(ArrayStack *stack, int num) {\n    if (stack-&gt;size == MAX_SIZE) {\n        printf(\"Stack is full\\n\");\n        return;\n    }\n    stack-&gt;data[stack-&gt;size] = num;\n    stack-&gt;size++;\n}\n\n/* Access the top element */\nint peek(ArrayStack *stack) {\n    if (stack-&gt;size == 0) {\n        printf(\"Stack is empty\\n\");\n        return INT_MAX;\n    }\n    return stack-&gt;data[stack-&gt;size - 1];\n}\n\n/* Pop */\nint pop(ArrayStack *stack) {\n    int val = peek(stack);\n    stack-&gt;size--;\n    return val;\n}\n</code></pre> array_stack.kt<pre><code>/* Stack implemented with an array */\nclass ArrayStack {\n    // Initialize the list (dynamic array)\n    private val stack = mutableListOf&lt;Int&gt;()\n\n    /* Get the stack size */\n    fun size(): Int {\n        return stack.size\n    }\n\n    /* Check whether the stack is empty */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* Push */\n    fun push(num: Int) {\n        stack.add(num)\n    }\n\n    /* Pop */\n    fun pop(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack.removeAt(size() - 1)\n    }\n\n    /* Access the top element */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack[size() - 1]\n    }\n\n    /* Convert the list to an array and return it */\n    fun toArray(): Array&lt;Any&gt; {\n        return stack.toTypedArray()\n    }\n}\n</code></pre> array_stack.rb<pre><code>### Stack implemented with an array ###\nclass ArrayStack\n  ### Constructor ###\n  def initialize\n    @stack = []\n  end\n\n  ### Get the stack size ###\n  def size\n    @stack.length\n  end\n\n  ### Check whether the stack is empty ###\n  def is_empty?\n    @stack.empty?\n  end\n\n  ### Push ###\n  def push(item)\n    @stack &lt;&lt; item\n  end\n\n  ### Pop ###\n  def pop\n    raise IndexError, 'Stack is empty' if is_empty?\n\n    @stack.pop\n  end\n\n  ### Access the top element ###\n  def peek\n    raise IndexError, 'Stack is empty' if is_empty?\n\n    @stack.last\n  end\n\n  ### Return a list for printing ###\n  def to_array\n    @stack\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 5.   Stacks and queues","5.1   Stacks"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#513-comparing-the-implementations","level":2,"title":"5.1.3   Comparing the implementations","text":"<p>Supported operations</p> <p>Both implementations support all stack operations. An array also allows random access, but that is outside the stack abstraction and is generally not used.</p> <p>Time efficiency</p> <p>In an array-based stack, pushes and pops operate in preallocated contiguous memory, giving good cache locality and high efficiency. However, a push that exceeds the array's capacity triggers expansion, making that particular push an \\(O(n)\\) operation.</p> <p>A linked list grows flexibly and avoids the slowdown associated with array expansion. However, pushing requires creating a node object and updating pointers, so it is relatively slower. If the element is already a node object, the creation step can be skipped to improve efficiency.</p> <p>For elements of primitive types such as <code>int</code> or <code>double</code>, we can therefore draw the following conclusions.</p> <ul> <li>An array-based stack slows down when expansion is triggered, but expansion is infrequent, so average efficiency is higher.</li> <li>A linked-list stack provides more consistent performance.</li> </ul> <p>Space efficiency</p> <p>When a dynamic array is initialized, its initial capacity may exceed actual needs. Expansion typically multiplies the capacity by a fixed factor, such as 2, which may again leave excess space. An array-based stack can therefore waste some space.</p> <p>However, linked-list nodes also store pointers, so each node occupies relatively more space.</p> <p>We cannot say universally which implementation uses less memory; the answer depends on the situation.</p>","path":["Chapter 5.   Stacks and queues","5.1   Stacks"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#514-typical-applications","level":2,"title":"5.1.4   Typical applications","text":"<ul> <li>Browser back and forward navigation, and software undo and redo. When a new web page is opened, the browser pushes the previous page onto a stack. Going back pops that page. Supporting both back and forward requires two stacks working together.</li> <li>Program memory management. Each function call adds a stack frame containing the function's context. During recursion, frames are pushed as calls descend and popped as they return.</li> </ul>","path":["Chapter 5.   Stacks and queues","5.1   Stacks"],"tags":[]},{"location":"chapter_stack_and_queue/summary/","level":1,"title":"5.4   Summary","text":"","path":["Chapter 5.   Stacks and queues","5.4   Summary"],"tags":[]},{"location":"chapter_stack_and_queue/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<ul> <li>A stack follows last-in, first-out order and can be implemented with an array or linked list.</li> <li>An array-based stack offers higher average time efficiency, but a push that triggers expansion takes \\(O(n)\\) time. A linked-list stack provides more consistent performance.</li> <li>An array-based stack may waste some space. However, linked-list nodes require more memory than individual array elements.</li> <li>A queue follows first-in, first-out order and can also be implemented with an array or linked list. Their time and space tradeoffs resemble those of stack implementations.</li> <li>A deque is a more flexible queue that allows insertion and removal at both ends.</li> </ul>","path":["Chapter 5.   Stacks and queues","5.4   Summary"],"tags":[]},{"location":"chapter_stack_and_queue/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: Are browser back and forward navigation implemented with a doubly linked list?</p> <p>Browser navigation fundamentally reflects stack behavior. Visiting a new page adds it to the top of a stack; clicking Back removes it. A deque conveniently supports additional operations, as discussed in the deque section.</p> <p>Q: After popping a node, do we need to free its memory?</p> <p>If the node will still be used, do not free it. If it is no longer needed, languages such as <code>Java</code> and <code>Python</code> use automatic garbage collection, so manual deallocation is unnecessary. In <code>C</code> and <code>C++</code>, memory must be released manually.</p> <p>Q: A deque looks like two stacks joined together. What is it useful for?</p> <p>A deque resembles a combination of a stack and a queue, or two stacks joined together. It combines stack and queue behavior, supporting all their applications with greater flexibility.</p> <p>Q: How exactly do undo and redo work?</p> <p>Use two stacks: <code>A</code> for undo and <code>B</code> for redo.</p> <ol> <li>Whenever the user performs an action, push it onto <code>A</code> and clear <code>B</code>.</li> <li>To undo, pop the most recent action from <code>A</code> and push it onto <code>B</code>.</li> <li>To redo, pop the most recent action from <code>B</code> and push it onto <code>A</code>.</li> </ol>","path":["Chapter 5.   Stacks and queues","5.4   Summary"],"tags":[]},{"location":"chapter_tree/","level":1,"title":"Chapter 7.   Trees","text":"<p>Abstract</p> <p>A towering tree brims with life, its roots deep, its leaves lush, and its branches spreading wide.</p> <p>It gives vivid form to the idea of dividing data to conquer a problem.</p>","path":["Chapter 7.   Trees"],"tags":[]},{"location":"chapter_tree/#chapter-contents","level":2,"title":"Chapter contents","text":"<ul> <li>7.1   Binary trees</li> <li>7.2   Traversing a binary tree</li> <li>7.3   Array representation of a binary tree</li> <li>7.4   Binary search trees</li> <li>7.5   AVL trees *</li> <li>7.6   Summary</li> <li>7.7   Exercises</li> </ul>","path":["Chapter 7.   Trees"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/","level":1,"title":"7.3   Array representation of a binary tree","text":"<p>In a linked representation, a binary tree stores data in <code>TreeNode</code> nodes connected by pointers. The previous section introduced basic operations on binary trees in this form.</p> <p>Can we represent a binary tree with an array instead? Yes.</p>","path":["Chapter 7.   Trees","7.3   Array representation of a binary tree"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/#731-representing-a-perfect-binary-tree","level":2,"title":"7.3.1   Representing a perfect binary tree","text":"<p>Start with a simple case. Store every node of a perfect binary tree in an array in level order. Each node then corresponds to a unique array index.</p> <p>Level-order traversal gives us a mapping between parent and child indices: if a node has index \\(i\\), its left child has index \\(2i + 1\\) and its right child has index \\(2i + 2\\). Figure 7-12 illustrates this relationship.</p> <p></p> <p> Figure 7-12   Array representation of a perfect binary tree </p> <p>The mapping formulas serve the same role as node references or pointers in a linked representation. Given any node in the array, we can use them to access its left or right child.</p>","path":["Chapter 7.   Trees","7.3   Array representation of a binary tree"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/#732-representing-an-arbitrary-binary-tree","level":2,"title":"7.3.2   Representing an arbitrary binary tree","text":"<p>A perfect binary tree is a special case. Other binary trees often contain many missing nodes, <code>None</code>, at intermediate levels. Ordinary level-order traversal omits these positions, so its sequence alone cannot tell us their number or locations. Multiple tree structures can therefore produce the same level-order sequence.</p> <p>As shown below, the array representation above no longer works for a non-perfect binary tree.</p> <p></p> <p> Figure 7-13   Multiple binary trees can share a level-order sequence </p> <p>To solve this, include every <code>None</code> position explicitly in the level-order sequence. The sequence then represents the tree uniquely, as shown below. Example code follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># Array representation of a binary tree\n# Use None for empty positions\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Use INT_MAX, the maximum int value, for empty positions\nvector&lt;int&gt; tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Use the wrapper class Integer so that null can mark empty positions\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Use the nullable type int? so that null can mark empty positions\nint?[] tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Use a slice of any so that nil can mark empty positions\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Use the optional type Int? so that nil can mark empty positions\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Use null for empty positions\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Use null for empty positions\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Use the nullable type int? so that null can mark empty positions\nList&lt;int?&gt; tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Use None for empty positions\nlet tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)];\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Use the maximum int value for empty positions, so node values must not equal INT_MAX\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* Array representation of a binary tree */\n// Use null for empty positions\nval tree = arrayOf( 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 )\n</code></pre> <pre><code>### Array representation of a binary tree ###\n# Use nil for empty positions\ntree = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <p></p> <p> Figure 7-14   Array representation of an arbitrary binary tree </p> <p>Complete binary trees are particularly well suited to array representation. By definition, missing positions occur only at the right end of the lowest level, so all <code>None</code> entries appear at the end of the level-order sequence.</p> <p>We can therefore omit all <code>None</code> entries when storing a complete binary tree in an array. An example is shown below.</p> <p></p> <p> Figure 7-15   Array representation of a complete binary tree </p> <p>The following code implements an array-based binary tree with these operations.</p> <ul> <li>Given a node, get its value, left or right child, and parent.</li> <li>Obtain preorder, inorder, postorder, and level-order traversal sequences.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_binary_tree.py<pre><code>class ArrayBinaryTree:\n    \"\"\"Binary tree class using an array representation\"\"\"\n\n    def __init__(self, arr: list[int | None]):\n        \"\"\"Constructor\"\"\"\n        self._tree = list(arr)\n\n    def size(self):\n        \"\"\"Length of the backing list\"\"\"\n        return len(self._tree)\n\n    def val(self, i: int) -&gt; int | None:\n        \"\"\"Get the value of the node at index i\"\"\"\n        # If the index is out of bounds, return None to represent an empty position\n        if i &lt; 0 or i &gt;= self.size():\n            return None\n        return self._tree[i]\n\n    def left(self, i: int) -&gt; int | None:\n        \"\"\"Get the left child index of the node at index i\"\"\"\n        return 2 * i + 1\n\n    def right(self, i: int) -&gt; int | None:\n        \"\"\"Get the right child index of the node at index i\"\"\"\n        return 2 * i + 2\n\n    def parent(self, i: int) -&gt; int | None:\n        \"\"\"Get the parent index of the node at index i\"\"\"\n        return (i - 1) // 2\n\n    def level_order(self) -&gt; list[int]:\n        \"\"\"Level-order traversal\"\"\"\n        self.res = []\n        # Traverse the array directly\n        for i in range(self.size()):\n            if self.val(i) is not None:\n                self.res.append(self.val(i))\n        return self.res\n\n    def dfs(self, i: int, order: str):\n        \"\"\"Depth-first traversal\"\"\"\n        if self.val(i) is None:\n            return\n        # Preorder traversal\n        if order == \"pre\":\n            self.res.append(self.val(i))\n        self.dfs(self.left(i), order)\n        # Inorder traversal\n        if order == \"in\":\n            self.res.append(self.val(i))\n        self.dfs(self.right(i), order)\n        # Postorder traversal\n        if order == \"post\":\n            self.res.append(self.val(i))\n\n    def pre_order(self) -&gt; list[int]:\n        \"\"\"Preorder traversal\"\"\"\n        self.res = []\n        self.dfs(0, order=\"pre\")\n        return self.res\n\n    def in_order(self) -&gt; list[int]:\n        \"\"\"Inorder traversal\"\"\"\n        self.res = []\n        self.dfs(0, order=\"in\")\n        return self.res\n\n    def post_order(self) -&gt; list[int]:\n        \"\"\"Postorder traversal\"\"\"\n        self.res = []\n        self.dfs(0, order=\"post\")\n        return self.res\n</code></pre> array_binary_tree.cpp<pre><code>/* Binary tree class using an array representation */\nclass ArrayBinaryTree {\n  public:\n    /* Constructor */\n    ArrayBinaryTree(vector&lt;int&gt; arr) {\n        tree = arr;\n    }\n\n    /* Number of positions in the array */\n    int size() {\n        return tree.size();\n    }\n\n    /* Get the value of the node at index i */\n    int val(int i) {\n        // If the index is out of bounds, return INT_MAX to represent an empty position\n        if (i &lt; 0 || i &gt;= size())\n            return INT_MAX;\n        return tree[i];\n    }\n\n    /* Get the left child index of the node at index i */\n    int left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* Get the right child index of the node at index i */\n    int right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* Get the parent index of the node at index i */\n    int parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* Level-order traversal */\n    vector&lt;int&gt; levelOrder() {\n        vector&lt;int&gt; res;\n        // Traverse the array directly\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != INT_MAX)\n                res.push_back(val(i));\n        }\n        return res;\n    }\n\n    /* Preorder traversal */\n    vector&lt;int&gt; preOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* Inorder traversal */\n    vector&lt;int&gt; inOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* Postorder traversal */\n    vector&lt;int&gt; postOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"post\", res);\n        return res;\n    }\n\n  private:\n    vector&lt;int&gt; tree;\n\n    /* Depth-first traversal */\n    void dfs(int i, string order, vector&lt;int&gt; &amp;res) {\n        // Return if this position is empty\n        if (val(i) == INT_MAX)\n            return;\n        // Preorder traversal\n        if (order == \"pre\")\n            res.push_back(val(i));\n        dfs(left(i), order, res);\n        // Inorder traversal\n        if (order == \"in\")\n            res.push_back(val(i));\n        dfs(right(i), order, res);\n        // Postorder traversal\n        if (order == \"post\")\n            res.push_back(val(i));\n    }\n};\n</code></pre> array_binary_tree.java<pre><code>/* Binary tree class using an array representation */\nclass ArrayBinaryTree {\n    private List&lt;Integer&gt; tree;\n\n    /* Constructor */\n    public ArrayBinaryTree(List&lt;Integer&gt; arr) {\n        tree = new ArrayList&lt;&gt;(arr);\n    }\n\n    /* List size */\n    public int size() {\n        return tree.size();\n    }\n\n    /* Get the value of the node at index i */\n    public Integer val(int i) {\n        // If the index is out of bounds, return null to represent an empty position\n        if (i &lt; 0 || i &gt;= size())\n            return null;\n        return tree.get(i);\n    }\n\n    /* Get the left child index of the node at index i */\n    public Integer left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* Get the right child index of the node at index i */\n    public Integer right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* Get the parent index of the node at index i */\n    public Integer parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* Level-order traversal */\n    public List&lt;Integer&gt; levelOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        // Traverse the array directly\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != null)\n                res.add(val(i));\n        }\n        return res;\n    }\n\n    /* Depth-first traversal */\n    private void dfs(Integer i, String order, List&lt;Integer&gt; res) {\n        // Return if this position is empty\n        if (val(i) == null)\n            return;\n        // Preorder traversal\n        if (\"pre\".equals(order))\n            res.add(val(i));\n        dfs(left(i), order, res);\n        // Inorder traversal\n        if (\"in\".equals(order))\n            res.add(val(i));\n        dfs(right(i), order, res);\n        // Postorder traversal\n        if (\"post\".equals(order))\n            res.add(val(i));\n    }\n\n    /* Preorder traversal */\n    public List&lt;Integer&gt; preOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* Inorder traversal */\n    public List&lt;Integer&gt; inOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* Postorder traversal */\n    public List&lt;Integer&gt; postOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.cs<pre><code>/* Binary tree class using an array representation */\nclass ArrayBinaryTree(List&lt;int?&gt; arr) {\n    List&lt;int?&gt; tree = new(arr);\n\n    /* List size */\n    public int Size() {\n        return tree.Count;\n    }\n\n    /* Get the value of the node at index i */\n    public int? Val(int i) {\n        // If the index is out of bounds, return null to represent an empty position\n        if (i &lt; 0 || i &gt;= Size())\n            return null;\n        return tree[i];\n    }\n\n    /* Get the left child index of the node at index i */\n    public int Left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* Get the right child index of the node at index i */\n    public int Right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* Get the parent index of the node at index i */\n    public int Parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* Level-order traversal */\n    public List&lt;int&gt; LevelOrder() {\n        List&lt;int&gt; res = [];\n        // Traverse the array directly\n        for (int i = 0; i &lt; Size(); i++) {\n            if (Val(i).HasValue)\n                res.Add(Val(i)!.Value);\n        }\n        return res;\n    }\n\n    /* Depth-first traversal */\n    void DFS(int i, string order, List&lt;int&gt; res) {\n        // Return if this position is empty\n        if (!Val(i).HasValue)\n            return;\n        // Preorder traversal\n        if (order == \"pre\")\n            res.Add(Val(i)!.Value);\n        DFS(Left(i), order, res);\n        // Inorder traversal\n        if (order == \"in\")\n            res.Add(Val(i)!.Value);\n        DFS(Right(i), order, res);\n        // Postorder traversal\n        if (order == \"post\")\n            res.Add(Val(i)!.Value);\n    }\n\n    /* Preorder traversal */\n    public List&lt;int&gt; PreOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"pre\", res);\n        return res;\n    }\n\n    /* Inorder traversal */\n    public List&lt;int&gt; InOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"in\", res);\n        return res;\n    }\n\n    /* Postorder traversal */\n    public List&lt;int&gt; PostOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.go<pre><code>/* Binary tree type using an array representation */\ntype arrayBinaryTree struct {\n    tree []any\n}\n\n/* Constructor */\nfunc newArrayBinaryTree(arr []any) *arrayBinaryTree {\n    return &amp;arrayBinaryTree{\n        tree: arr,\n    }\n}\n\n/* Length of the backing slice */\nfunc (abt *arrayBinaryTree) size() int {\n    return len(abt.tree)\n}\n\n/* Get the value of the node at index i */\nfunc (abt *arrayBinaryTree) val(i int) any {\n    // If the index is out of bounds, return nil to represent an empty position\n    if i &lt; 0 || i &gt;= abt.size() {\n        return nil\n    }\n    return abt.tree[i]\n}\n\n/* Get the left child index of the node at index i */\nfunc (abt *arrayBinaryTree) left(i int) int {\n    return 2*i + 1\n}\n\n/* Get the right child index of the node at index i */\nfunc (abt *arrayBinaryTree) right(i int) int {\n    return 2*i + 2\n}\n\n/* Get the parent index of the node at index i */\nfunc (abt *arrayBinaryTree) parent(i int) int {\n    return (i - 1) / 2\n}\n\n/* Level-order traversal */\nfunc (abt *arrayBinaryTree) levelOrder() []any {\n    var res []any\n    // Traverse the array directly\n    for i := 0; i &lt; abt.size(); i++ {\n        if abt.val(i) != nil {\n            res = append(res, abt.val(i))\n        }\n    }\n    return res\n}\n\n/* Depth-first traversal */\nfunc (abt *arrayBinaryTree) dfs(i int, order string, res *[]any) {\n    // Return if this position is empty\n    if abt.val(i) == nil {\n        return\n    }\n    // Preorder traversal\n    if order == \"pre\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.left(i), order, res)\n    // Inorder traversal\n    if order == \"in\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.right(i), order, res)\n    // Postorder traversal\n    if order == \"post\" {\n        *res = append(*res, abt.val(i))\n    }\n}\n\n/* Preorder traversal */\nfunc (abt *arrayBinaryTree) preOrder() []any {\n    var res []any\n    abt.dfs(0, \"pre\", &amp;res)\n    return res\n}\n\n/* Inorder traversal */\nfunc (abt *arrayBinaryTree) inOrder() []any {\n    var res []any\n    abt.dfs(0, \"in\", &amp;res)\n    return res\n}\n\n/* Postorder traversal */\nfunc (abt *arrayBinaryTree) postOrder() []any {\n    var res []any\n    abt.dfs(0, \"post\", &amp;res)\n    return res\n}\n</code></pre> array_binary_tree.swift<pre><code>/* Binary tree class using an array representation */\nclass ArrayBinaryTree {\n    private var tree: [Int?]\n\n    /* Constructor */\n    init(arr: [Int?]) {\n        tree = arr\n    }\n\n    /* Length of the backing array */\n    func size() -&gt; Int {\n        tree.count\n    }\n\n    /* Get the value of the node at index i */\n    func val(i: Int) -&gt; Int? {\n        // If the index is out of bounds, return nil to represent an empty position\n        if i &lt; 0 || i &gt;= size() {\n            return nil\n        }\n        return tree[i]\n    }\n\n    /* Get the left child index of the node at index i */\n    func left(i: Int) -&gt; Int {\n        2 * i + 1\n    }\n\n    /* Get the right child index of the node at index i */\n    func right(i: Int) -&gt; Int {\n        2 * i + 2\n    }\n\n    /* Get the parent index of the node at index i */\n    func parent(i: Int) -&gt; Int {\n        (i - 1) / 2\n    }\n\n    /* Level-order traversal */\n    func levelOrder() -&gt; [Int] {\n        var res: [Int] = []\n        // Traverse the array directly\n        for i in 0 ..&lt; size() {\n            if let val = val(i: i) {\n                res.append(val)\n            }\n        }\n        return res\n    }\n\n    /* Depth-first traversal */\n    private func dfs(i: Int, order: String, res: inout [Int]) {\n        // Return if this position is empty\n        guard let val = val(i: i) else {\n            return\n        }\n        // Preorder traversal\n        if order == \"pre\" {\n            res.append(val)\n        }\n        dfs(i: left(i: i), order: order, res: &amp;res)\n        // Inorder traversal\n        if order == \"in\" {\n            res.append(val)\n        }\n        dfs(i: right(i: i), order: order, res: &amp;res)\n        // Postorder traversal\n        if order == \"post\" {\n            res.append(val)\n        }\n    }\n\n    /* Preorder traversal */\n    func preOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"pre\", res: &amp;res)\n        return res\n    }\n\n    /* Inorder traversal */\n    func inOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"in\", res: &amp;res)\n        return res\n    }\n\n    /* Postorder traversal */\n    func postOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"post\", res: &amp;res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.js<pre><code>/* Binary tree class using an array representation */\nclass ArrayBinaryTree {\n    #tree;\n\n    /* Constructor */\n    constructor(arr) {\n        this.#tree = arr;\n    }\n\n    /* Array length */\n    size() {\n        return this.#tree.length;\n    }\n\n    /* Get the value of the node at index i */\n    val(i) {\n        // If the index is out of bounds, return null to indicate an empty position\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* Get the left child index of the node at index i */\n    left(i) {\n        return 2 * i + 1;\n    }\n\n    /* Get the right child index of the node at index i */\n    right(i) {\n        return 2 * i + 2;\n    }\n\n    /* Get the parent index of the node at index i */\n    parent(i) {\n        return Math.floor((i - 1) / 2); // Floor division\n    }\n\n    /* Level-order traversal */\n    levelOrder() {\n        let res = [];\n        // Traverse the array directly\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* Depth-first traversal */\n    #dfs(i, order, res) {\n        // Return if this position is empty\n        if (this.val(i) === null) return;\n        // Preorder traversal\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // Inorder traversal\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // Postorder traversal\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* Preorder traversal */\n    preOrder() {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* Inorder traversal */\n    inOrder() {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* Postorder traversal */\n    postOrder() {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.ts<pre><code>/* Binary tree class using an array representation */\nclass ArrayBinaryTree {\n    #tree: (number | null)[];\n\n    /* Constructor */\n    constructor(arr: (number | null)[]) {\n        this.#tree = arr;\n    }\n\n    /* Array length, including empty positions */\n    size(): number {\n        return this.#tree.length;\n    }\n\n    /* Get the value of the node at index i */\n    val(i: number): number | null {\n        // If the index is out of bounds, return null to indicate an empty position\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* Get the left child index of the node at index i */\n    left(i: number): number {\n        return 2 * i + 1;\n    }\n\n    /* Get the right child index of the node at index i */\n    right(i: number): number {\n        return 2 * i + 2;\n    }\n\n    /* Get the parent index of the node at index i */\n    parent(i: number): number {\n        return Math.floor((i - 1) / 2); // Floor division\n    }\n\n    /* Level-order traversal */\n    levelOrder(): number[] {\n        let res = [];\n        // Traverse the array directly\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* Depth-first traversal */\n    #dfs(i: number, order: Order, res: (number | null)[]): void {\n        // Return if this position is empty\n        if (this.val(i) === null) return;\n        // Preorder traversal\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // Inorder traversal\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // Postorder traversal\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* Preorder traversal */\n    preOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* Inorder traversal */\n    inOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* Postorder traversal */\n    postOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.dart<pre><code>/* Binary tree class using an array representation */\nclass ArrayBinaryTree {\n  late List&lt;int?&gt; _tree;\n\n  /* Constructor */\n  ArrayBinaryTree(this._tree);\n\n  /* Array length, including empty positions */\n  int size() {\n    return _tree.length;\n  }\n\n  /* Get the value of the node at index i */\n  int? val(int i) {\n    // If the index is out of bounds, return null to indicate an empty position\n    if (i &lt; 0 || i &gt;= size()) {\n      return null;\n    }\n    return _tree[i];\n  }\n\n  /* Get the left child index of the node at index i */\n  int? left(int i) {\n    return 2 * i + 1;\n  }\n\n  /* Get the right child index of the node at index i */\n  int? right(int i) {\n    return 2 * i + 2;\n  }\n\n  /* Get the parent index of the node at index i */\n  int? parent(int i) {\n    return (i - 1) ~/ 2;\n  }\n\n  /* Level-order traversal */\n  List&lt;int&gt; levelOrder() {\n    List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; size(); i++) {\n      if (val(i) != null) {\n        res.add(val(i)!);\n      }\n    }\n    return res;\n  }\n\n  /* Depth-first traversal */\n  void dfs(int i, String order, List&lt;int?&gt; res) {\n    // Return if this position is empty\n    if (val(i) == null) {\n      return;\n    }\n    // Preorder traversal\n    if (order == 'pre') {\n      res.add(val(i));\n    }\n    dfs(left(i)!, order, res);\n    // Inorder traversal\n    if (order == 'in') {\n      res.add(val(i));\n    }\n    dfs(right(i)!, order, res);\n    // Postorder traversal\n    if (order == 'post') {\n      res.add(val(i));\n    }\n  }\n\n  /* Preorder traversal */\n  List&lt;int?&gt; preOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'pre', res);\n    return res;\n  }\n\n  /* Inorder traversal */\n  List&lt;int?&gt; inOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'in', res);\n    return res;\n  }\n\n  /* Postorder traversal */\n  List&lt;int?&gt; postOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'post', res);\n    return res;\n  }\n}\n</code></pre> array_binary_tree.rs<pre><code>/* Binary tree type using an array representation */\nstruct ArrayBinaryTree {\n    tree: Vec&lt;Option&lt;i32&gt;&gt;,\n}\n\nimpl ArrayBinaryTree {\n    /* Constructor */\n    fn new(arr: Vec&lt;Option&lt;i32&gt;&gt;) -&gt; Self {\n        Self { tree: arr }\n    }\n\n    /* Number of positions in the array */\n    fn size(&amp;self) -&gt; i32 {\n        self.tree.len() as i32\n    }\n\n    /* Get the value of the node at index i */\n    fn val(&amp;self, i: i32) -&gt; Option&lt;i32&gt; {\n        // If the index is out of bounds, return None to represent an empty position\n        if i &lt; 0 || i &gt;= self.size() {\n            None\n        } else {\n            self.tree[i as usize]\n        }\n    }\n\n    /* Get the left child index of the node at index i */\n    fn left(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 1\n    }\n\n    /* Get the right child index of the node at index i */\n    fn right(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 2\n    }\n\n    /* Get the parent index of the node at index i */\n    fn parent(&amp;self, i: i32) -&gt; i32 {\n        (i - 1) / 2\n    }\n\n    /* Level-order traversal */\n    fn level_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        self.tree.iter().filter_map(|&amp;x| x).collect()\n    }\n\n    /* Depth-first traversal */\n    fn dfs(&amp;self, i: i32, order: &amp;'static str, res: &amp;mut Vec&lt;i32&gt;) {\n        if self.val(i).is_none() {\n            return;\n        }\n        let val = self.val(i).unwrap();\n        // Preorder traversal\n        if order == \"pre\" {\n            res.push(val);\n        }\n        self.dfs(self.left(i), order, res);\n        // Inorder traversal\n        if order == \"in\" {\n            res.push(val);\n        }\n        self.dfs(self.right(i), order, res);\n        // Postorder traversal\n        if order == \"post\" {\n            res.push(val);\n        }\n    }\n\n    /* Preorder traversal */\n    fn pre_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"pre\", &amp;mut res);\n        res\n    }\n\n    /* Inorder traversal */\n    fn in_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"in\", &amp;mut res);\n        res\n    }\n\n    /* Postorder traversal */\n    fn post_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"post\", &amp;mut res);\n        res\n    }\n}\n</code></pre> array_binary_tree.c<pre><code>/* Binary tree structure represented by an array */\ntypedef struct {\n    int *tree;\n    int size;\n} ArrayBinaryTree;\n\n/* Constructor */\nArrayBinaryTree *newArrayBinaryTree(int *arr, int arrSize) {\n    ArrayBinaryTree *abt = (ArrayBinaryTree *)malloc(sizeof(ArrayBinaryTree));\n    abt-&gt;tree = malloc(sizeof(int) * arrSize);\n    memcpy(abt-&gt;tree, arr, sizeof(int) * arrSize);\n    abt-&gt;size = arrSize;\n    return abt;\n}\n\n/* Destructor */\nvoid delArrayBinaryTree(ArrayBinaryTree *abt) {\n    free(abt-&gt;tree);\n    free(abt);\n}\n\n/* Array length */\nint size(ArrayBinaryTree *abt) {\n    return abt-&gt;size;\n}\n\n/* Get the value of the node at index i */\nint val(ArrayBinaryTree *abt, int i) {\n    // If the index is out of bounds, return INT_MAX to represent an empty position\n    if (i &lt; 0 || i &gt;= size(abt))\n        return INT_MAX;\n    return abt-&gt;tree[i];\n}\n\n/* Level-order traversal */\nint *levelOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    // Traverse the array directly\n    for (int i = 0; i &lt; size(abt); i++) {\n        if (val(abt, i) != INT_MAX)\n            res[index++] = val(abt, i);\n    }\n    *returnSize = index;\n    return res;\n}\n\n/* Depth-first traversal */\nvoid dfs(ArrayBinaryTree *abt, int i, char *order, int *res, int *index) {\n    // Return if this position is empty\n    if (val(abt, i) == INT_MAX)\n        return;\n    // Preorder traversal\n    if (strcmp(order, \"pre\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, left(i), order, res, index);\n    // Inorder traversal\n    if (strcmp(order, \"in\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, right(i), order, res, index);\n    // Postorder traversal\n    if (strcmp(order, \"post\") == 0)\n        res[(*index)++] = val(abt, i);\n}\n\n/* Preorder traversal */\nint *preOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"pre\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* Inorder traversal */\nint *inOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"in\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* Postorder traversal */\nint *postOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"post\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n</code></pre> array_binary_tree.kt<pre><code>/* Binary tree class using an array representation */\nclass ArrayBinaryTree(val tree: MutableList&lt;Int?&gt;) {\n    /* List size */\n    fun size(): Int {\n        return tree.size\n    }\n\n    /* Get the value of the node at index i */\n    fun _val(i: Int): Int? {\n        // If the index is out of bounds, return null to represent an empty position\n        if (i &lt; 0 || i &gt;= size()) return null\n        return tree[i]\n    }\n\n    /* Get the left child index of the node at index i */\n    fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* Get the right child index of the node at index i */\n    fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* Get the parent index of the node at index i */\n    fun parent(i: Int): Int {\n        return (i - 1) / 2\n    }\n\n    /* Level-order traversal */\n    fun levelOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        // Traverse the array directly\n        for (i in 0..&lt;size()) {\n            if (_val(i) != null)\n                res.add(_val(i))\n        }\n        return res\n    }\n\n    /* Depth-first traversal */\n    fun dfs(i: Int, order: String, res: MutableList&lt;Int?&gt;) {\n        // Return if this position is empty\n        if (_val(i) == null)\n            return\n        // Preorder traversal\n        if (\"pre\" == order)\n            res.add(_val(i))\n        dfs(left(i), order, res)\n        // Inorder traversal\n        if (\"in\" == order)\n            res.add(_val(i))\n        dfs(right(i), order, res)\n        // Postorder traversal\n        if (\"post\" == order)\n            res.add(_val(i))\n    }\n\n    /* Preorder traversal */\n    fun preOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"pre\", res)\n        return res\n    }\n\n    /* Inorder traversal */\n    fun inOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"in\", res)\n        return res\n    }\n\n    /* Postorder traversal */\n    fun postOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"post\", res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.rb<pre><code>### Binary tree class using an array representation ###\nclass ArrayBinaryTree\n  ### Constructor ###\n  def initialize(arr)\n    @tree = arr.to_a\n  end\n\n  ### Length of the array representation ###\n  def size\n    @tree.length\n  end\n\n  ### Get the value of the node at index i ###\n  def val(i)\n    # If the index is out of bounds, return nil to indicate an empty position\n    return if i &lt; 0 || i &gt;= size\n\n    @tree[i]\n  end\n\n  ### Get the left child index of the node at index i ###\n  def left(i)\n    2 * i + 1\n  end\n\n  ### Get the right child index of the node at index i ###\n  def right(i)\n    2 * i + 2\n  end\n\n  ### Get the parent index of the node at index i ###\n  def parent(i)\n    (i - 1) / 2\n  end\n\n  ### Level-order traversal ###\n  def level_order\n    @res = []\n\n    # Traverse the array directly\n    for i in 0...size\n      @res &lt;&lt; val(i) unless val(i).nil?\n    end\n\n    @res\n  end\n\n  ### Depth-first traversal ###\n  def dfs(i, order)\n    return if val(i).nil?\n    # Preorder traversal\n    @res &lt;&lt; val(i) if order == :pre\n    dfs(left(i), order)\n    # Inorder traversal\n    @res &lt;&lt; val(i) if order == :in\n    dfs(right(i), order)\n    # Postorder traversal\n    @res &lt;&lt; val(i) if order == :post\n  end\n\n  ### Preorder traversal ###\n  def pre_order\n    @res = []\n    dfs(0, :pre)\n    @res\n  end\n\n  ### Inorder traversal ###\n  def in_order\n    @res = []\n    dfs(0, :in)\n    @res\n  end\n\n  ### Postorder traversal ###\n  def post_order\n    @res = []\n    dfs(0, :post)\n    @res\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 7.   Trees","7.3   Array representation of a binary tree"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/#733-strengths-and-limitations","level":2,"title":"7.3.3   Strengths and limitations","text":"<p>Array representation offers the following advantages.</p> <ul> <li>Contiguous memory is cache-friendly, making access and traversal fast.</li> <li>No pointers need to be stored, saving space.</li> <li>Nodes can be accessed randomly by index.</li> </ul> <p>However, it also has limitations.</p> <ul> <li>Arrays require contiguous memory, making them unsuitable for extremely large trees.</li> <li>Adding and removing nodes requires array insertion and deletion operations, which are inefficient.</li> <li>If the tree has many missing nodes, actual node data occupies only a small fraction of the array, wasting space.</li> </ul>","path":["Chapter 7.   Trees","7.3   Array representation of a binary tree"],"tags":[]},{"location":"chapter_tree/avl_tree/","level":1,"title":"7.5   AVL trees *","text":"<p>The binary-search-tree section explained that repeated insertions and deletions can make a tree degenerate into a linked list. All operations then degrade from \\(O(\\log n)\\) to \\(O(n)\\) time.</p> <p>In the example below, just two deletions turn the binary search tree into a linked list.</p> <p></p> <p> Figure 7-24   An AVL-tree example degenerating after deletions </p> <p>Likewise, inserting two nodes into the perfect binary tree below makes it heavily skewed to the left, degrading lookup performance.</p> <p></p> <p> Figure 7-25   An AVL-tree example degenerating after insertions </p> <p>In 1962, G. M. Adelson-Velsky and E. M. Landis introduced the AVL tree in their paper “An algorithm for the organization of information.” They described operations that prevent degeneration through repeated insertions and deletions, keeping the time complexity of operations at \\(O(\\log n)\\). AVL trees therefore maintain efficient data operations when insertions, deletions, lookups, and updates are frequent, making them valuable in practice.</p>","path":["Chapter 7.   Trees","7.5   AVL trees *"],"tags":[]},{"location":"chapter_tree/avl_tree/#751-avl-tree-terminology","level":2,"title":"7.5.1   AVL tree terminology","text":"<p>An AVL tree is both a binary search tree and a balanced binary tree, satisfying the properties of both. It is therefore a balanced binary search tree.</p>","path":["Chapter 7.   Trees","7.5   AVL trees *"],"tags":[]},{"location":"chapter_tree/avl_tree/#1-node-height","level":3,"title":"1.   Node height","text":"<p>AVL operations need node heights, so add a <code>height</code> variable to the node class:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class TreeNode:\n    \"\"\"AVL tree node class\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                 # Node value\n        self.height: int = 0                # Node height\n        self.left: TreeNode | None = None   # Left child reference\n        self.right: TreeNode | None = None  # Right child reference\n</code></pre> <pre><code>/* AVL tree node class */\nstruct TreeNode {\n    int val{};          // Node value\n    int height = 0;     // Node height\n    TreeNode *left{};   // Left child\n    TreeNode *right{};  // Right child\n    TreeNode() = default;\n    explicit TreeNode(int x) : val(x){}\n};\n</code></pre> <pre><code>/* AVL tree node class */\nclass TreeNode {\n    public int val;        // Node value\n    public int height;     // Node height\n    public TreeNode left;  // Left child\n    public TreeNode right; // Right child\n    public TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* AVL tree node class */\nclass TreeNode(int? x) {\n    public int? val = x;    // Node value\n    public int height;      // Node height\n    public TreeNode? left;  // Left child reference\n    public TreeNode? right; // Right child reference\n}\n</code></pre> <pre><code>/* AVL tree node struct */\ntype TreeNode struct {\n    Val    int       // Node value\n    Height int       // Node height\n    Left   *TreeNode // Left child reference\n    Right  *TreeNode // Right child reference\n}\n</code></pre> <pre><code>/* AVL tree node class */\nclass TreeNode {\n    var val: Int // Node value\n    var height: Int // Node height\n    var left: TreeNode? // Left child\n    var right: TreeNode? // Right child\n\n    init(x: Int) {\n        val = x\n        height = 0\n    }\n}\n</code></pre> <pre><code>/* AVL tree node class */\nclass TreeNode {\n    val; // Node value\n    height; //Node height\n    left; // Left child pointer\n    right; // Right child pointer\n    constructor(val, left, right, height) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL tree node class */\nclass TreeNode {\n    val: number;            // Node value\n    height: number;         // Node height\n    left: TreeNode | null;  // Left child pointer\n    right: TreeNode | null; // Right child pointer\n    constructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL tree node class */\nclass TreeNode {\n  int val;         // Node value\n  int height;      // Node height\n  TreeNode? left;  // Left child\n  TreeNode? right; // Right child\n  TreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* AVL tree node struct */\nstruct TreeNode {\n    val: i32,                               // Node value\n    height: i32,                            // Node height\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // Left child\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // Right child\n}\n\nimpl TreeNode {\n    /* Constructor */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            height: 0,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* AVL tree node struct */\ntypedef struct TreeNode {\n    int val;\n    int height;\n    struct TreeNode *left;\n    struct TreeNode *right;\n} TreeNode;\n\n/* Constructor */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* AVL tree node class */\nclass TreeNode(val _val: Int) {  // Node value\n    var height: Int = 0          // Node height\n    var left: TreeNode? = null   // Left child\n    var right: TreeNode? = null  // Right child\n}\n</code></pre> <pre><code>### AVL tree node class ###\nclass TreeNode\n  attr_accessor :val    # Node value\n  attr_accessor :height # Node height\n  attr_accessor :left   # Left child reference\n  attr_accessor :right  # Right child reference\n\n  def initialize(val)\n    @val = val\n    @height = 0\n  end\nend\n</code></pre> <p>A node's height is the distance to its farthest leaf, measured in edges. A leaf has height \\(0\\), while a missing node has height \\(-1\\). We create two helper functions to retrieve and update a node's height:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def height(self, node: TreeNode | None) -&gt; int:\n    \"\"\"Get the node height\"\"\"\n    # An empty subtree has height -1; a leaf has height 0\n    if node is not None:\n        return node.height\n    return -1\n\ndef update_height(self, node: TreeNode | None):\n    \"\"\"Update the node height\"\"\"\n    # The node height equals the height of its taller subtree + 1\n    node.height = max([self.height(node.left), self.height(node.right)]) + 1\n</code></pre> avl_tree.cpp<pre><code>/* Get the node height */\nint height(TreeNode *node) {\n    // A null node has height -1; a leaf has height 0\n    return node == nullptr ? -1 : node-&gt;height;\n}\n\n/* Update the node height */\nvoid updateHeight(TreeNode *node) {\n    // The node height equals the height of its taller subtree + 1\n    node-&gt;height = max(height(node-&gt;left), height(node-&gt;right)) + 1;\n}\n</code></pre> avl_tree.java<pre><code>/* Get the node height */\nint height(TreeNode node) {\n    // A null node has height -1; a leaf has height 0\n    return node == null ? -1 : node.height;\n}\n\n/* Update the node height */\nvoid updateHeight(TreeNode node) {\n    // The node height equals the height of its taller subtree + 1\n    node.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.cs<pre><code>/* Get the node height */\nint Height(TreeNode? node) {\n    // A null node has height -1; a leaf has height 0\n    return node == null ? -1 : node.height;\n}\n\n/* Update the node height */\nvoid UpdateHeight(TreeNode node) {\n    // The node height equals the height of its taller subtree + 1\n    node.height = Math.Max(Height(node.left), Height(node.right)) + 1;\n}\n</code></pre> avl_tree.go<pre><code>/* Get the node height */\nfunc (t *aVLTree) height(node *TreeNode) int {\n    // A nil node has height -1; a leaf has height 0\n    if node != nil {\n        return node.Height\n    }\n    return -1\n}\n\n/* Update the node height */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\n    lh := t.height(node.Left)\n    rh := t.height(node.Right)\n    // The node height equals the height of its taller subtree + 1\n    if lh &gt; rh {\n        node.Height = lh + 1\n    } else {\n        node.Height = rh + 1\n    }\n}\n</code></pre> avl_tree.swift<pre><code>/* Get the node height */\nfunc height(node: TreeNode?) -&gt; Int {\n    // A nil node has height -1; a leaf has height 0\n    node?.height ?? -1\n}\n\n/* Update the node height */\nfunc updateHeight(node: TreeNode?) {\n    // The node height equals the height of its taller subtree + 1\n    node?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n</code></pre> avl_tree.js<pre><code>/* Get the node height */\nheight(node) {\n    // A null node has height -1; a leaf has height 0\n    return node === null ? -1 : node.height;\n}\n\n/* Update the node height */\n#updateHeight(node) {\n    // The node height equals the height of its taller subtree + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.ts<pre><code>/* Get the node height */\nheight(node: TreeNode): number {\n    // A null node has height -1; a leaf has height 0\n    return node === null ? -1 : node.height;\n}\n\n/* Update the node height */\nupdateHeight(node: TreeNode): void {\n    // The node height equals the height of its taller subtree + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.dart<pre><code>/* Get the node height */\nint height(TreeNode? node) {\n  // A null node has height -1; a leaf has height 0\n  return node == null ? -1 : node.height;\n}\n\n/* Update the node height */\nvoid updateHeight(TreeNode? node) {\n  // The node height equals the height of its taller subtree + 1\n  node!.height = max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.rs<pre><code>/* Get the node height */\nfn height(node: OptionTreeNodeRc) -&gt; i32 {\n    // A null node has height -1; a leaf has height 0\n    match node {\n        Some(node) =&gt; node.borrow().height,\n        None =&gt; -1,\n    }\n}\n\n/* Update the node height */\nfn update_height(node: OptionTreeNodeRc) {\n    if let Some(node) = node {\n        let left = node.borrow().left.clone();\n        let right = node.borrow().right.clone();\n        // The node height equals the height of its taller subtree + 1\n        node.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* Get the node height */\nint height(TreeNode *node) {\n    // A null node has height -1; a leaf has height 0\n    if (node != NULL) {\n        return node-&gt;height;\n    }\n    return -1;\n}\n\n/* Update the node height */\nvoid updateHeight(TreeNode *node) {\n    int lh = height(node-&gt;left);\n    int rh = height(node-&gt;right);\n    // The node height equals the height of its taller subtree + 1\n    if (lh &gt; rh) {\n        node-&gt;height = lh + 1;\n    } else {\n        node-&gt;height = rh + 1;\n    }\n}\n</code></pre> avl_tree.kt<pre><code>/* Get the node height */\nfun height(node: TreeNode?): Int {\n    // A null node has height -1; a leaf has height 0\n    return node?.height ?: -1\n}\n\n/* Update the node height */\nfun updateHeight(node: TreeNode?) {\n    // The node height equals the height of its taller subtree + 1\n    node?.height = max(height(node?.left), height(node?.right)) + 1\n}\n</code></pre> avl_tree.rb<pre><code>### Get the node height ###\ndef height(node)\n  # A nil node has height -1; a leaf has height 0\n  return node.height unless node.nil?\n\n  -1\nend\n\n### Update the node height ###\ndef update_height(node)\n  # The node height equals the height of its taller subtree + 1\n  node.height = [height(node.left), height(node.right)].max + 1\nend\n</code></pre>","path":["Chapter 7.   Trees","7.5   AVL trees *"],"tags":[]},{"location":"chapter_tree/avl_tree/#2-balance-factor","level":3,"title":"2.   Balance factor","text":"<p>A node's balance factor is its left subtree's height minus its right subtree's height. A missing node has balance factor \\(0\\). Wrap this calculation in a function for convenient use:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def balance_factor(self, node: TreeNode | None) -&gt; int:\n    \"\"\"Get the balance factor\"\"\"\n    # An empty subtree has balance factor 0\n    if node is None:\n        return 0\n    # Node balance factor = left subtree height - right subtree height\n    return self.height(node.left) - self.height(node.right)\n</code></pre> avl_tree.cpp<pre><code>/* Get the balance factor */\nint balanceFactor(TreeNode *node) {\n    // A null node has balance factor 0\n    if (node == nullptr)\n        return 0;\n    // Node balance factor = left subtree height - right subtree height\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.java<pre><code>/* Get the balance factor */\nint balanceFactor(TreeNode node) {\n    // A null node has balance factor 0\n    if (node == null)\n        return 0;\n    // Node balance factor = left subtree height - right subtree height\n    return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.cs<pre><code>/* Get the balance factor */\nint BalanceFactor(TreeNode? node) {\n    // A null node has balance factor 0\n    if (node == null) return 0;\n    // Node balance factor = left subtree height - right subtree height\n    return Height(node.left) - Height(node.right);\n}\n</code></pre> avl_tree.go<pre><code>/* Get the balance factor */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n    // A nil node has balance factor 0\n    if node == nil {\n        return 0\n    }\n    // Node balance factor = left subtree height - right subtree height\n    return t.height(node.Left) - t.height(node.Right)\n}\n</code></pre> avl_tree.swift<pre><code>/* Get the balance factor */\nfunc balanceFactor(node: TreeNode?) -&gt; Int {\n    // A nil node has balance factor 0\n    guard let node = node else { return 0 }\n    // Node balance factor = left subtree height - right subtree height\n    return height(node: node.left) - height(node: node.right)\n}\n</code></pre> avl_tree.js<pre><code>/* Get the balance factor */\nbalanceFactor(node) {\n    // A null node has balance factor 0\n    if (node === null) return 0;\n    // Node balance factor = left subtree height - right subtree height\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.ts<pre><code>/* Get the balance factor */\nbalanceFactor(node: TreeNode): number {\n    // A null node has balance factor 0\n    if (node === null) return 0;\n    // Node balance factor = left subtree height - right subtree height\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.dart<pre><code>/* Get the balance factor */\nint balanceFactor(TreeNode? node) {\n  // A null node has balance factor 0\n  if (node == null) return 0;\n  // Node balance factor = left subtree height - right subtree height\n  return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.rs<pre><code>/* Get the balance factor */\nfn balance_factor(node: OptionTreeNodeRc) -&gt; i32 {\n    match node {\n        // A null node has balance factor 0\n        None =&gt; 0,\n        // Node balance factor = left subtree height - right subtree height\n        Some(node) =&gt; {\n            Self::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())\n        }\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* Get the balance factor */\nint balanceFactor(TreeNode *node) {\n    // A null node has balance factor 0\n    if (node == NULL) {\n        return 0;\n    }\n    // Node balance factor = left subtree height - right subtree height\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.kt<pre><code>/* Get the balance factor */\nfun balanceFactor(node: TreeNode?): Int {\n    // A null node has balance factor 0\n    if (node == null) return 0\n    // Node balance factor = left subtree height - right subtree height\n    return height(node.left) - height(node.right)\n}\n</code></pre> avl_tree.rb<pre><code>### Get the balance factor ###\ndef balance_factor(node)\n  # A nil node has balance factor 0\n  return 0 if node.nil?\n\n  # Node balance factor = left subtree height - right subtree height\n  height(node.left) - height(node.right)\nend\n</code></pre> <p>Tip</p> <p>If a node's balance factor is \\(f\\), every node in an AVL tree satisfies \\(-1 \\le f \\le 1\\).</p>","path":["Chapter 7.   Trees","7.5   AVL trees *"],"tags":[]},{"location":"chapter_tree/avl_tree/#752-avl-tree-rotations","level":2,"title":"7.5.2   AVL tree rotations","text":"<p>The defining operation of an AVL tree is rotation, which restores balance without changing the inorder traversal sequence. A rotation preserves the binary-search-tree property while restoring the balanced-tree property.</p> <p>A node whose balance factor has absolute value \\(&gt; 1\\) is “unbalanced.” Depending on the imbalance, we use one of four rotations: right, left, right-then-left, or left-then-right. Let us examine them in detail.</p>","path":["Chapter 7.   Trees","7.5   AVL trees *"],"tags":[]},{"location":"chapter_tree/avl_tree/#1-right-rotation","level":3,"title":"1.   Right rotation","text":"<p>In Figure 7-26, balance factors appear beneath the nodes. Viewed from the bottom up, node 3 is the first unbalanced node. Focus on the subtree rooted there, naming its root <code>node</code> and its left child <code>child</code>. A right rotation restores balance while preserving the binary-search-tree property.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 7-26   Steps of a right rotation </p> <p>If <code>child</code> has a right child, denoted by <code>grand_child</code>, add one step to the rotation: make <code>grand_child</code> the left child of <code>node</code>, as shown below.</p> <p></p> <p> Figure 7-27   A right rotation with grand_child </p> <p>“Rotating right” is a visual description; the implementation updates node pointers:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def right_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"Right rotation\"\"\"\n    child = node.left\n    grand_child = child.right\n    # Rotate node to the right around child\n    child.right = node\n    node.left = grand_child\n    # Update the node height\n    self.update_height(node)\n    self.update_height(child)\n    # Return the root of the rotated subtree\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* Right rotation */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;left;\n    TreeNode *grandChild = child-&gt;right;\n    // Rotate node to the right around child\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // Update the node height\n    updateHeight(node);\n    updateHeight(child);\n    // Return the root of the rotated subtree\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* Right rotation */\nTreeNode rightRotate(TreeNode node) {\n    TreeNode child = node.left;\n    TreeNode grandChild = child.right;\n    // Rotate node to the right around child\n    child.right = node;\n    node.left = grandChild;\n    // Update the node height\n    updateHeight(node);\n    updateHeight(child);\n    // Return the root of the rotated subtree\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* Right rotation */\nTreeNode? RightRotate(TreeNode? node) {\n    TreeNode? child = node?.left;\n    TreeNode? grandChild = child?.right;\n    // Rotate node to the right around child\n    child.right = node;\n    node.left = grandChild;\n    // Update the node height\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // Return the root of the rotated subtree\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* Right rotation */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\n    child := node.Left\n    grandChild := child.Right\n    // Rotate node to the right around child\n    child.Right = node\n    node.Left = grandChild\n    // Update the node height\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // Return the root of the rotated subtree\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* Right rotation */\nfunc rightRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.left\n    let grandChild = child?.right\n    // Rotate node to the right around child\n    child?.right = node\n    node?.left = grandChild\n    // Update the node height\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // Return the root of the rotated subtree\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* Right rotation */\n#rightRotate(node) {\n    const child = node.left;\n    const grandChild = child.right;\n    // Rotate node to the right around child\n    child.right = node;\n    node.left = grandChild;\n    // Update the node height\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // Return the root of the rotated subtree\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* Right rotation */\nrightRotate(node: TreeNode): TreeNode {\n    const child = node.left;\n    const grandChild = child.right;\n    // Rotate node to the right around child\n    child.right = node;\n    node.left = grandChild;\n    // Update the node height\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // Return the root of the rotated subtree\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* Right rotation */\nTreeNode? rightRotate(TreeNode? node) {\n  TreeNode? child = node!.left;\n  TreeNode? grandChild = child!.right;\n  // Rotate node to the right around child\n  child.right = node;\n  node.left = grandChild;\n  // Update the node height\n  updateHeight(node);\n  updateHeight(child);\n  // Return the root of the rotated subtree\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* Right rotation */\nfn right_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().left.clone().unwrap();\n            let grand_child = child.borrow().right.clone();\n            // Rotate node to the right around child\n            child.borrow_mut().right = Some(node.clone());\n            node.borrow_mut().left = grand_child;\n            // Update the node height\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // Return the root of the rotated subtree\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* Right rotation */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;left;\n    grandChild = child-&gt;right;\n    // Rotate node to the right around child\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // Update the node height\n    updateHeight(node);\n    updateHeight(child);\n    // Return the root of the rotated subtree\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* Right rotation */\nfun rightRotate(node: TreeNode?): TreeNode {\n    val child = node!!.left\n    val grandChild = child!!.right\n    // Rotate node to the right around child\n    child.right = node\n    node.left = grandChild\n    // Update the node height\n    updateHeight(node)\n    updateHeight(child)\n    // Return the root of the rotated subtree\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>### Right rotation ###\ndef right_rotate(node)\n  child = node.left\n  grand_child = child.right\n  # Rotate node to the right around child\n  child.right = node\n  node.left = grand_child\n  # Update the node height\n  update_height(node)\n  update_height(child)\n  # Return the root of the rotated subtree\n  child\nend\n</code></pre>","path":["Chapter 7.   Trees","7.5   AVL trees *"],"tags":[]},{"location":"chapter_tree/avl_tree/#2-left-rotation","level":3,"title":"2.   Left rotation","text":"<p>For the mirror image of the unbalanced tree above, perform a left rotation, as shown below.</p> <p></p> <p> Figure 7-28   A left rotation </p> <p>Similarly, if <code>child</code> has a left child, denoted by <code>grand_child</code>, make it the right child of <code>node</code> during the rotation.</p> <p></p> <p> Figure 7-29   A left rotation with grand_child </p> <p>Right and left rotations are mirror images, as are the imbalances they resolve. To obtain left-rotation code, exchange every <code>left</code> and <code>right</code> in the right-rotation implementation:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def left_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"Left rotation\"\"\"\n    child = node.right\n    grand_child = child.left\n    # Rotate node to the left around child\n    child.left = node\n    node.right = grand_child\n    # Update the node height\n    self.update_height(node)\n    self.update_height(child)\n    # Return the root of the rotated subtree\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* Left rotation */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;right;\n    TreeNode *grandChild = child-&gt;left;\n    // Rotate node to the left around child\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // Update the node height\n    updateHeight(node);\n    updateHeight(child);\n    // Return the root of the rotated subtree\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* Left rotation */\nTreeNode leftRotate(TreeNode node) {\n    TreeNode child = node.right;\n    TreeNode grandChild = child.left;\n    // Rotate node to the left around child\n    child.left = node;\n    node.right = grandChild;\n    // Update the node height\n    updateHeight(node);\n    updateHeight(child);\n    // Return the root of the rotated subtree\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* Left rotation */\nTreeNode? LeftRotate(TreeNode? node) {\n    TreeNode? child = node?.right;\n    TreeNode? grandChild = child?.left;\n    // Rotate node to the left around child\n    child.left = node;\n    node.right = grandChild;\n    // Update the node height\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // Return the root of the rotated subtree\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* Left rotation */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\n    child := node.Right\n    grandChild := child.Left\n    // Rotate node to the left around child\n    child.Left = node\n    node.Right = grandChild\n    // Update the node height\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // Return the root of the rotated subtree\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* Left rotation */\nfunc leftRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.right\n    let grandChild = child?.left\n    // Rotate node to the left around child\n    child?.left = node\n    node?.right = grandChild\n    // Update the node height\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // Return the root of the rotated subtree\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* Left rotation */\n#leftRotate(node) {\n    const child = node.right;\n    const grandChild = child.left;\n    // Rotate node to the left around child\n    child.left = node;\n    node.right = grandChild;\n    // Update the node height\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // Return the root of the rotated subtree\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* Left rotation */\nleftRotate(node: TreeNode): TreeNode {\n    const child = node.right;\n    const grandChild = child.left;\n    // Rotate node to the left around child\n    child.left = node;\n    node.right = grandChild;\n    // Update the node height\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // Return the root of the rotated subtree\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* Left rotation */\nTreeNode? leftRotate(TreeNode? node) {\n  TreeNode? child = node!.right;\n  TreeNode? grandChild = child!.left;\n  // Rotate node to the left around child\n  child.left = node;\n  node.right = grandChild;\n  // Update the node height\n  updateHeight(node);\n  updateHeight(child);\n  // Return the root of the rotated subtree\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* Left rotation */\nfn left_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().right.clone().unwrap();\n            let grand_child = child.borrow().left.clone();\n            // Rotate node to the left around child\n            child.borrow_mut().left = Some(node.clone());\n            node.borrow_mut().right = grand_child;\n            // Update the node height\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // Return the root of the rotated subtree\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* Left rotation */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;right;\n    grandChild = child-&gt;left;\n    // Rotate node to the left around child\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // Update the node height\n    updateHeight(node);\n    updateHeight(child);\n    // Return the root of the rotated subtree\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* Left rotation */\nfun leftRotate(node: TreeNode?): TreeNode {\n    val child = node!!.right\n    val grandChild = child!!.left\n    // Rotate node to the left around child\n    child.left = node\n    node.right = grandChild\n    // Update the node height\n    updateHeight(node)\n    updateHeight(child)\n    // Return the root of the rotated subtree\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>### Left rotation ###\ndef left_rotate(node)\n  child = node.right\n  grand_child = child.left\n  # Rotate node to the left around child\n  child.left = node\n  node.right = grand_child\n  # Update the node height\n  update_height(node)\n  update_height(child)\n  # Return the root of the rotated subtree\n  child\nend\n</code></pre>","path":["Chapter 7.   Trees","7.5   AVL trees *"],"tags":[]},{"location":"chapter_tree/avl_tree/#3-left-then-right-rotation","level":3,"title":"3.   Left-then-right rotation","text":"<p>For unbalanced node 3 below, neither a left nor a right rotation alone restores balance. First rotate <code>child</code> to the left, then rotate <code>node</code> to the right.</p> <p></p> <p> Figure 7-30   A left-then-right rotation </p>","path":["Chapter 7.   Trees","7.5   AVL trees *"],"tags":[]},{"location":"chapter_tree/avl_tree/#4-right-then-left-rotation","level":3,"title":"4.   Right-then-left rotation","text":"<p>For the mirror-image case below, first rotate <code>child</code> to the right, then rotate <code>node</code> to the left.</p> <p></p> <p> Figure 7-31   A right-then-left rotation </p>","path":["Chapter 7.   Trees","7.5   AVL trees *"],"tags":[]},{"location":"chapter_tree/avl_tree/#5-choosing-a-rotation","level":3,"title":"5.   Choosing a rotation","text":"<p>The four cases below correspond to the examples above and require, respectively, right, left-then-right, right-then-left, and left rotations.</p> <p></p> <p> Figure 7-32   The four AVL rotation cases </p> <p>As Table 7-3 shows, the signs of the unbalanced node's balance factor and that of the child on its taller side determine the case.</p> <p> Table 7-3   Conditions for choosing a rotation </p> Unbalanced node's balance factor Child's balance factor Rotation \\(&gt; 1\\) (left-heavy) \\(\\geq 0\\) Right \\(&gt; 1\\) (left-heavy) \\(&lt;0\\) Left, then right \\(&lt; -1\\) (right-heavy) \\(\\leq 0\\) Left \\(&lt; -1\\) (right-heavy) \\(&gt;0\\) Right, then left <p>Wrap the rotation logic in one function. It can then handle every imbalance and restore the affected node's balance. The code is as follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"Perform rotations to restore balance in this subtree\"\"\"\n    # Get the balance factor of node\n    balance_factor = self.balance_factor(node)\n    # Left-heavy tree\n    if balance_factor &gt; 1:\n        if self.balance_factor(node.left) &gt;= 0:\n            # Rotate right\n            return self.right_rotate(node)\n        else:\n            # Rotate left, then right\n            node.left = self.left_rotate(node.left)\n            return self.right_rotate(node)\n    # Right-heavy tree\n    elif balance_factor &lt; -1:\n        if self.balance_factor(node.right) &lt;= 0:\n            # Rotate left\n            return self.left_rotate(node)\n        else:\n            # Rotate right, then left\n            node.right = self.right_rotate(node.right)\n            return self.left_rotate(node)\n    # The tree is balanced; return without rotating\n    return node\n</code></pre> avl_tree.cpp<pre><code>/* Perform rotations to restore balance in this subtree */\nTreeNode *rotate(TreeNode *node) {\n    // Get the balance factor of node\n    int _balanceFactor = balanceFactor(node);\n    // Left-heavy tree\n    if (_balanceFactor &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // Rotate right\n            return rightRotate(node);\n        } else {\n            // Rotate left, then right\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // Right-heavy tree\n    if (_balanceFactor &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // Rotate left\n            return leftRotate(node);\n        } else {\n            // Rotate right, then left\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // The tree is balanced; return without rotating\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* Perform rotations to restore balance in this subtree */\nTreeNode rotate(TreeNode node) {\n    // Get the balance factor of node\n    int balanceFactor = balanceFactor(node);\n    // Left-heavy tree\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // Rotate right\n            return rightRotate(node);\n        } else {\n            // Rotate left, then right\n            node.left = leftRotate(node.left);\n            return rightRotate(node);\n        }\n    }\n    // Right-heavy tree\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // Rotate left\n            return leftRotate(node);\n        } else {\n            // Rotate right, then left\n            node.right = rightRotate(node.right);\n            return leftRotate(node);\n        }\n    }\n    // The tree is balanced; return without rotating\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* Perform rotations to restore balance in this subtree */\nTreeNode? Rotate(TreeNode? node) {\n    // Get the balance factor of node\n    int balanceFactorInt = BalanceFactor(node);\n    // Left-heavy tree\n    if (balanceFactorInt &gt; 1) {\n        if (BalanceFactor(node?.left) &gt;= 0) {\n            // Rotate right\n            return RightRotate(node);\n        } else {\n            // Rotate left, then right\n            node!.left = LeftRotate(node!.left);\n            return RightRotate(node);\n        }\n    }\n    // Right-heavy tree\n    if (balanceFactorInt &lt; -1) {\n        if (BalanceFactor(node?.right) &lt;= 0) {\n            // Rotate left\n            return LeftRotate(node);\n        } else {\n            // Rotate right, then left\n            node!.right = RightRotate(node!.right);\n            return LeftRotate(node);\n        }\n    }\n    // The tree is balanced; return without rotating\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* Perform rotations to restore balance in this subtree */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n    // Get the balance factor of node\n    // Go favors short local variable names; bf holds the result of t.balanceFactor(node)\n    bf := t.balanceFactor(node)\n    // Left-heavy tree\n    if bf &gt; 1 {\n        if t.balanceFactor(node.Left) &gt;= 0 {\n            // Rotate right\n            return t.rightRotate(node)\n        } else {\n            // Rotate left, then right\n            node.Left = t.leftRotate(node.Left)\n            return t.rightRotate(node)\n        }\n    }\n    // Right-heavy tree\n    if bf &lt; -1 {\n        if t.balanceFactor(node.Right) &lt;= 0 {\n            // Rotate left\n            return t.leftRotate(node)\n        } else {\n            // Rotate right, then left\n            node.Right = t.rightRotate(node.Right)\n            return t.leftRotate(node)\n        }\n    }\n    // The tree is balanced; return without rotating\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* Perform rotations to restore balance in this subtree */\nfunc rotate(node: TreeNode?) -&gt; TreeNode? {\n    // Get the balance factor of node\n    let balanceFactor = balanceFactor(node: node)\n    // Left-heavy tree\n    if balanceFactor &gt; 1 {\n        if self.balanceFactor(node: node?.left) &gt;= 0 {\n            // Rotate right\n            return rightRotate(node: node)\n        } else {\n            // Rotate left, then right\n            node?.left = leftRotate(node: node?.left)\n            return rightRotate(node: node)\n        }\n    }\n    // Right-heavy tree\n    if balanceFactor &lt; -1 {\n        if self.balanceFactor(node: node?.right) &lt;= 0 {\n            // Rotate left\n            return leftRotate(node: node)\n        } else {\n            // Rotate right, then left\n            node?.right = rightRotate(node: node?.right)\n            return leftRotate(node: node)\n        }\n    }\n    // The tree is balanced; return without rotating\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* Perform rotations to restore balance in this subtree */\n#rotate(node) {\n    // Get the balance factor of node\n    const balanceFactor = this.balanceFactor(node);\n    // Left-heavy tree\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // Rotate right\n            return this.#rightRotate(node);\n        } else {\n            // Rotate left, then right\n            node.left = this.#leftRotate(node.left);\n            return this.#rightRotate(node);\n        }\n    }\n    // Right-heavy tree\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // Rotate left\n            return this.#leftRotate(node);\n        } else {\n            // Rotate right, then left\n            node.right = this.#rightRotate(node.right);\n            return this.#leftRotate(node);\n        }\n    }\n    // The tree is balanced; return without rotating\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* Perform rotations to restore balance in this subtree */\nrotate(node: TreeNode): TreeNode {\n    // Get the balance factor of node\n    const balanceFactor = this.balanceFactor(node);\n    // Left-heavy tree\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // Rotate right\n            return this.rightRotate(node);\n        } else {\n            // Rotate left, then right\n            node.left = this.leftRotate(node.left);\n            return this.rightRotate(node);\n        }\n    }\n    // Right-heavy tree\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // Rotate left\n            return this.leftRotate(node);\n        } else {\n            // Rotate right, then left\n            node.right = this.rightRotate(node.right);\n            return this.leftRotate(node);\n        }\n    }\n    // The tree is balanced; return without rotating\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* Perform rotations to restore balance in this subtree */\nTreeNode? rotate(TreeNode? node) {\n  // Get the balance factor of node\n  int factor = balanceFactor(node);\n  // Left-heavy tree\n  if (factor &gt; 1) {\n    if (balanceFactor(node!.left) &gt;= 0) {\n      // Rotate right\n      return rightRotate(node);\n    } else {\n      // Rotate left, then right\n      node.left = leftRotate(node.left);\n      return rightRotate(node);\n    }\n  }\n  // Right-heavy tree\n  if (factor &lt; -1) {\n    if (balanceFactor(node!.right) &lt;= 0) {\n      // Rotate left\n      return leftRotate(node);\n    } else {\n      // Rotate right, then left\n      node.right = rightRotate(node.right);\n      return leftRotate(node);\n    }\n  }\n  // The tree is balanced; return without rotating\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* Perform rotations to restore balance in this subtree */\nfn rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    // Get the balance factor of node\n    let balance_factor = Self::balance_factor(node.clone());\n    // Left-heavy tree\n    if balance_factor &gt; 1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().left.clone()) &gt;= 0 {\n            // Rotate right\n            Self::right_rotate(Some(node))\n        } else {\n            // Rotate left, then right\n            let left = node.borrow().left.clone();\n            node.borrow_mut().left = Self::left_rotate(left);\n            Self::right_rotate(Some(node))\n        }\n    }\n    // Right-heavy tree\n    else if balance_factor &lt; -1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().right.clone()) &lt;= 0 {\n            // Rotate left\n            Self::left_rotate(Some(node))\n        } else {\n            // Rotate right, then left\n            let right = node.borrow().right.clone();\n            node.borrow_mut().right = Self::right_rotate(right);\n            Self::left_rotate(Some(node))\n        }\n    } else {\n        // The tree is balanced; return without rotating\n        node\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* Perform rotations to restore balance in this subtree */\nTreeNode *rotate(TreeNode *node) {\n    // Get the balance factor of node\n    int bf = balanceFactor(node);\n    // Left-heavy tree\n    if (bf &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // Rotate right\n            return rightRotate(node);\n        } else {\n            // Rotate left, then right\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // Right-heavy tree\n    if (bf &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // Rotate left\n            return leftRotate(node);\n        } else {\n            // Rotate right, then left\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // The tree is balanced; return without rotating\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* Perform rotations to restore balance in this subtree */\nfun rotate(node: TreeNode): TreeNode {\n    // Get the balance factor of node\n    val balanceFactor = balanceFactor(node)\n    // Left-heavy tree\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // Rotate right\n            return rightRotate(node)\n        } else {\n            // Rotate left, then right\n            node.left = leftRotate(node.left)\n            return rightRotate(node)\n        }\n    }\n    // Right-heavy tree\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // Rotate left\n            return leftRotate(node)\n        } else {\n            // Rotate right, then left\n            node.right = rightRotate(node.right)\n            return leftRotate(node)\n        }\n    }\n    // The tree is balanced; return without rotating\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### Perform rotations to restore balance in this subtree ###\ndef rotate(node)\n  # Get the balance factor of node\n  balance_factor = balance_factor(node)\n  # Left-heavy tree\n  if balance_factor &gt; 1\n    if balance_factor(node.left) &gt;= 0\n      # Rotate right\n      return right_rotate(node)\n    else\n      # Rotate left, then right\n      node.left = left_rotate(node.left)\n      return right_rotate(node)\n    end\n  # Right-heavy tree\n  elsif balance_factor &lt; -1\n    if balance_factor(node.right) &lt;= 0\n      # Rotate left\n      return left_rotate(node)\n    else\n      # Rotate right, then left\n      node.right = right_rotate(node.right)\n      return left_rotate(node)\n    end\n  end\n  # The tree is balanced; return without rotating\n  node\nend\n</code></pre>","path":["Chapter 7.   Trees","7.5   AVL trees *"],"tags":[]},{"location":"chapter_tree/avl_tree/#753-common-avl-tree-operations","level":2,"title":"7.5.3   Common AVL tree operations","text":"","path":["Chapter 7.   Trees","7.5   AVL trees *"],"tags":[]},{"location":"chapter_tree/avl_tree/#1-inserting-a-node","level":3,"title":"1.   Inserting a node","text":"<p>Insertion largely follows the binary-search-tree procedure. The difference is that inserting a node may unbalance nodes along the path to the root. Starting at the inserted node, work upward and perform rotations to restore balance at every affected node:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def insert(self, val):\n    \"\"\"Insert a node\"\"\"\n    self._root = self.insert_helper(self._root, val)\n\ndef insert_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode:\n    \"\"\"Insert a node recursively (helper method)\"\"\"\n    if node is None:\n        return TreeNode(val)\n    # 1. Find the insertion position and insert the node\n    if val &lt; node.val:\n        node.left = self.insert_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.insert_helper(node.right, val)\n    else:\n        # Return without inserting a duplicate node\n        return node\n    # Update the node height\n    self.update_height(node)\n    # 2. Perform rotations to restore balance in this subtree\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* Insert a node */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* Insert a node recursively (helper method) */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return new TreeNode(val);\n    /* 1. Find the insertion position and insert the node */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    else\n        return node;    // Return without inserting a duplicate node\n    updateHeight(node); // Update the node height\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = rotate(node);\n    // Return the root of the subtree\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* Insert a node */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* Insert a node recursively (helper method) */\nTreeNode insertHelper(TreeNode node, int val) {\n    if (node == null)\n        return new TreeNode(val);\n    /* 1. Find the insertion position and insert the node */\n    if (val &lt; node.val)\n        node.left = insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = insertHelper(node.right, val);\n    else\n        return node; // Return without inserting a duplicate node\n    updateHeight(node); // Update the node height\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = rotate(node);\n    // Return the subtree root\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* Insert a node */\nvoid Insert(int val) {\n    root = InsertHelper(root, val);\n}\n\n/* Insert a node recursively (helper method) */\nTreeNode? InsertHelper(TreeNode? node, int val) {\n    if (node == null) return new TreeNode(val);\n    /* 1. Find the insertion position and insert the node */\n    if (val &lt; node.val)\n        node.left = InsertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = InsertHelper(node.right, val);\n    else\n        return node;     // Return without inserting a duplicate node\n    UpdateHeight(node);  // Update the node height\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = Rotate(node);\n    // Return the subtree root\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* Insert a node */\nfunc (t *aVLTree) insert(val int) {\n    t.root = t.insertHelper(t.root, val)\n}\n\n/* Insert a node recursively (helper function) */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return NewTreeNode(val)\n    }\n    /* 1. Find the insertion position and insert the node */\n    if val &lt; node.Val.(int) {\n        node.Left = t.insertHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.insertHelper(node.Right, val)\n    } else {\n        // Return without inserting a duplicate node\n        return node\n    }\n    // Update the node height\n    t.updateHeight(node)\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = t.rotate(node)\n    // Return the subtree root\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* Insert a node */\nfunc insert(val: Int) {\n    root = insertHelper(node: root, val: val)\n}\n\n/* Insert a node recursively (helper method) */\nfunc insertHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return TreeNode(x: val)\n    }\n    /* 1. Find the insertion position and insert the node */\n    if val &lt; node!.val {\n        node?.left = insertHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = insertHelper(node: node?.right, val: val)\n    } else {\n        return node // Return without inserting a duplicate node\n    }\n    updateHeight(node: node) // Update the node height\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = rotate(node: node)\n    // Return the subtree root\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* Insert a node */\ninsert(val) {\n    this.root = this.#insertHelper(this.root, val);\n}\n\n/* Insert a node recursively (helper method) */\n#insertHelper(node, val) {\n    if (node === null) return new TreeNode(val);\n    /* 1. Find the insertion position and insert the node */\n    if (val &lt; node.val) node.left = this.#insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#insertHelper(node.right, val);\n    else return node; // Return without inserting a duplicate node\n    this.#updateHeight(node); // Update the node height\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = this.#rotate(node);\n    // Return the root of the subtree\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* Insert a node */\ninsert(val: number): void {\n    this.root = this.insertHelper(this.root, val);\n}\n\n/* Insert a node recursively (helper method) */\ninsertHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return new TreeNode(val);\n    /* 1. Find the insertion position and insert the node */\n    if (val &lt; node.val) {\n        node.left = this.insertHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.insertHelper(node.right, val);\n    } else {\n        return node; // Return without inserting a duplicate node\n    }\n    this.updateHeight(node); // Update the node height\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = this.rotate(node);\n    // Return the root of the subtree\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* Insert a node */\nvoid insert(int val) {\n  root = insertHelper(root, val);\n}\n\n/* Insert a node recursively (helper method) */\nTreeNode? insertHelper(TreeNode? node, int val) {\n  if (node == null) return TreeNode(val);\n  /* 1. Find the insertion position and insert the node */\n  if (val &lt; node.val)\n    node.left = insertHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = insertHelper(node.right, val);\n  else\n    return node; // Return without inserting a duplicate node\n  updateHeight(node); // Update the node height\n  /* 2. Perform rotations to restore balance in this subtree */\n  node = rotate(node);\n  // Return the root of the subtree\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* Insert a node */\nfn insert(&amp;mut self, val: i32) {\n    self.root = Self::insert_helper(self.root.clone(), val);\n}\n\n/* Insert a node recursively (helper method) */\nfn insert_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. Find the insertion position and insert the node */\n            match {\n                let node_val = node.borrow().val;\n                node_val\n            }\n            .cmp(&amp;val)\n            {\n                Ordering::Greater =&gt; {\n                    let left = node.borrow().left.clone();\n                    node.borrow_mut().left = Self::insert_helper(left, val);\n                }\n                Ordering::Less =&gt; {\n                    let right = node.borrow().right.clone();\n                    node.borrow_mut().right = Self::insert_helper(right, val);\n                }\n                Ordering::Equal =&gt; {\n                    return Some(node); // Return without inserting a duplicate node\n                }\n            }\n            Self::update_height(Some(node.clone())); // Update the node height\n\n            /* 2. Perform rotations to restore balance in this subtree */\n            node = Self::rotate(Some(node)).unwrap();\n            // Return the root of the subtree\n            Some(node)\n        }\n        None =&gt; Some(TreeNode::new(val)),\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* Insert a node */\nvoid insert(AVLTree *tree, int val) {\n    tree-&gt;root = insertHelper(tree-&gt;root, val);\n}\n\n/* Recursively insert a node (helper function) */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == NULL) {\n        return newTreeNode(val);\n    }\n    /* 1. Find the insertion position and insert the node */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    } else {\n        // Return without inserting a duplicate node\n        return node;\n    }\n    // Update the node height\n    updateHeight(node);\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = rotate(node);\n    // Return the root of the subtree\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* Insert a node */\nfun insert(_val: Int) {\n    root = insertHelper(root, _val)\n}\n\n/* Insert a node recursively (helper method) */\nfun insertHelper(n: TreeNode?, _val: Int): TreeNode {\n    if (n == null)\n        return TreeNode(_val)\n    var node = n\n    /* 1. Find the insertion position and insert the node */\n    if (_val &lt; node._val)\n        node.left = insertHelper(node.left, _val)\n    else if (_val &gt; node._val)\n        node.right = insertHelper(node.right, _val)\n    else\n        return node // Return without inserting a duplicate node\n    updateHeight(node) // Update the node height\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = rotate(node)\n    // Return the subtree root\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### Insert a node ###\ndef insert(val)\n  @root = insert_helper(@root, val)\nend\n\n### Insert a node recursively (helper method)###\ndef insert_helper(node, val)\n  return TreeNode.new(val) if node.nil?\n  # 1. Find the insertion position and insert the node\n  if val &lt; node.val\n    node.left = insert_helper(node.left, val)\n  elsif val &gt; node.val\n    node.right = insert_helper(node.right, val)\n  else\n    # Return without inserting a duplicate node\n    return node\n  end\n  # Update the node height\n  update_height(node)\n  # 2. Perform rotations to restore balance in this subtree\n  rotate(node)\nend\n</code></pre>","path":["Chapter 7.   Trees","7.5   AVL trees *"],"tags":[]},{"location":"chapter_tree/avl_tree/#2-deleting-a-node","level":3,"title":"2.   Deleting a node","text":"<p>Similarly, extend binary-search-tree deletion with rotations from the bottom up to restore balance at every affected node:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def remove(self, val: int):\n    \"\"\"Remove a node\"\"\"\n    self._root = self.remove_helper(self._root, val)\n\ndef remove_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode | None:\n    \"\"\"Remove a node recursively (helper method)\"\"\"\n    if node is None:\n        return None\n    # 1. Find and remove the node\n    if val &lt; node.val:\n        node.left = self.remove_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.remove_helper(node.right, val)\n    else:\n        if node.left is None or node.right is None:\n            child = node.left or node.right\n            # Number of children = 0: remove node and return\n            if child is None:\n                return None\n            # Number of children = 1: remove node directly\n            else:\n                node = child\n        else:\n            # Number of children = 2: remove the inorder successor and use its value to replace the current node value\n            temp = node.right\n            while temp.left is not None:\n                temp = temp.left\n            node.right = self.remove_helper(node.right, temp.val)\n            node.val = temp.val\n    # Update the node height\n    self.update_height(node)\n    # 2. Perform rotations to restore balance in this subtree\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* Remove a node */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* Remove a node recursively (helper method) */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return nullptr;\n    /* 1. Find and remove the node */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    else {\n        if (node-&gt;left == nullptr || node-&gt;right == nullptr) {\n            TreeNode *child = node-&gt;left != nullptr ? node-&gt;left : node-&gt;right;\n            // Number of children = 0: remove node and return\n            if (child == nullptr) {\n                delete node;\n                return nullptr;\n            }\n            // Number of children = 1: remove node directly\n            else {\n                delete node;\n                node = child;\n            }\n        } else {\n            // Number of children = 2: remove the inorder successor and use its value to replace the current node value\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != nullptr) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    updateHeight(node); // Update the node height\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = rotate(node);\n    // Return the root of the subtree\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* Remove a node */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* Remove a node recursively (helper method) */\nTreeNode removeHelper(TreeNode node, int val) {\n    if (node == null)\n        return null;\n    /* 1. Find and remove the node */\n    if (val &lt; node.val)\n        node.left = removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = removeHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode child = node.left != null ? node.left : node.right;\n            // Number of children = 0: remove node and return\n            if (child == null)\n                return null;\n            // Number of children = 1: remove node directly\n            else\n                node = child;\n        } else {\n            // Number of children = 2: remove the inorder successor and use its value to replace the current node value\n            TreeNode temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    updateHeight(node); // Update the node height\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = rotate(node);\n    // Return the subtree root\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* Remove a node */\nvoid Remove(int val) {\n    root = RemoveHelper(root, val);\n}\n\n/* Remove a node recursively (helper method) */\nTreeNode? RemoveHelper(TreeNode? node, int val) {\n    if (node == null) return null;\n    /* 1. Find and remove the node */\n    if (val &lt; node.val)\n        node.left = RemoveHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = RemoveHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode? child = node.left ?? node.right;\n            // Number of children = 0: remove node and return\n            if (child == null)\n                return null;\n            // Number of children = 1: remove node directly\n            else\n                node = child;\n        } else {\n            // Number of children = 2: remove the inorder successor and use its value to replace the current node value\n            TreeNode? temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = RemoveHelper(node.right, temp.val!.Value);\n            node.val = temp.val;\n        }\n    }\n    UpdateHeight(node);  // Update the node height\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = Rotate(node);\n    // Return the subtree root\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* Remove a node */\nfunc (t *aVLTree) remove(val int) {\n    t.root = t.removeHelper(t.root, val)\n}\n\n/* Remove a node recursively (helper function) */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return nil\n    }\n    /* 1. Find and remove the node */\n    if val &lt; node.Val.(int) {\n        node.Left = t.removeHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.removeHelper(node.Right, val)\n    } else {\n        if node.Left == nil || node.Right == nil {\n            child := node.Left\n            if node.Right != nil {\n                child = node.Right\n            }\n            if child == nil {\n                // Number of children = 0: remove node and return\n                return nil\n            } else {\n                // Number of children = 1: remove node directly\n                node = child\n            }\n        } else {\n            // Number of children = 2: remove the inorder successor and use its value to replace the current node value\n            temp := node.Right\n            for temp.Left != nil {\n                temp = temp.Left\n            }\n            node.Right = t.removeHelper(node.Right, temp.Val.(int))\n            node.Val = temp.Val\n        }\n    }\n    // Update the node height\n    t.updateHeight(node)\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = t.rotate(node)\n    // Return the subtree root\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* Remove a node */\nfunc remove(val: Int) {\n    root = removeHelper(node: root, val: val)\n}\n\n/* Remove a node recursively (helper method) */\nfunc removeHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return nil\n    }\n    /* 1. Find and remove the node */\n    if val &lt; node!.val {\n        node?.left = removeHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = removeHelper(node: node?.right, val: val)\n    } else {\n        if node?.left == nil || node?.right == nil {\n            let child = node?.left ?? node?.right\n            // Number of children = 0: remove node and return\n            if child == nil {\n                return nil\n            }\n            // Number of children = 1: remove node directly\n            else {\n                node = child\n            }\n        } else {\n            // Number of children = 2: remove the inorder successor and use its value to replace the current node value\n            var temp = node?.right\n            while temp?.left != nil {\n                temp = temp?.left\n            }\n            node?.right = removeHelper(node: node?.right, val: temp!.val)\n            node?.val = temp!.val\n        }\n    }\n    updateHeight(node: node) // Update the node height\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = rotate(node: node)\n    // Return the subtree root\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* Remove a node */\nremove(val) {\n    this.root = this.#removeHelper(this.root, val);\n}\n\n/* Remove a node recursively (helper method) */\n#removeHelper(node, val) {\n    if (node === null) return null;\n    /* 1. Find and remove the node */\n    if (val &lt; node.val) node.left = this.#removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#removeHelper(node.right, val);\n    else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // Number of children = 0: remove node and return\n            if (child === null) return null;\n            // Number of children = 1: remove node directly\n            else node = child;\n        } else {\n            // Number of children = 2: remove the inorder successor and use its value to replace the current node value\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.#removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.#updateHeight(node); // Update the node height\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = this.#rotate(node);\n    // Return the root of the subtree\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* Remove a node */\nremove(val: number): void {\n    this.root = this.removeHelper(this.root, val);\n}\n\n/* Remove a node recursively (helper method) */\nremoveHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return null;\n    /* 1. Find and remove the node */\n    if (val &lt; node.val) {\n        node.left = this.removeHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.removeHelper(node.right, val);\n    } else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // Number of children = 0: remove node and return\n            if (child === null) {\n                return null;\n            } else {\n                // Number of children = 1: remove node directly\n                node = child;\n            }\n        } else {\n            // Number of children = 2: remove the inorder successor and use its value to replace the current node value\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.updateHeight(node); // Update the node height\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = this.rotate(node);\n    // Return the root of the subtree\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* Remove a node */\nvoid remove(int val) {\n  root = removeHelper(root, val);\n}\n\n/* Remove a node recursively (helper method) */\nTreeNode? removeHelper(TreeNode? node, int val) {\n  if (node == null) return null;\n  /* 1. Find and remove the node */\n  if (val &lt; node.val)\n    node.left = removeHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = removeHelper(node.right, val);\n  else {\n    if (node.left == null || node.right == null) {\n      TreeNode? child = node.left ?? node.right;\n      // Number of children = 0: remove node and return\n      if (child == null)\n        return null;\n      // Number of children = 1: remove node directly\n      else\n        node = child;\n    } else {\n      // Number of children = 2: remove the inorder successor and use its value to replace the current node value\n      TreeNode? temp = node.right;\n      while (temp!.left != null) {\n        temp = temp.left;\n      }\n      node.right = removeHelper(node.right, temp.val);\n      node.val = temp.val;\n    }\n  }\n  updateHeight(node); // Update the node height\n  /* 2. Perform rotations to restore balance in this subtree */\n  node = rotate(node);\n  // Return the root of the subtree\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* Remove a node */\nfn remove(&amp;self, val: i32) {\n    Self::remove_helper(self.root.clone(), val);\n}\n\n/* Remove a node recursively (helper method) */\nfn remove_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. Find and remove the node */\n            if val &lt; node.borrow().val {\n                let left = node.borrow().left.clone();\n                node.borrow_mut().left = Self::remove_helper(left, val);\n            } else if val &gt; node.borrow().val {\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, val);\n            } else if node.borrow().left.is_none() || node.borrow().right.is_none() {\n                let child = if node.borrow().left.is_some() {\n                    node.borrow().left.clone()\n                } else {\n                    node.borrow().right.clone()\n                };\n                match child {\n                    // Number of children = 0: remove node and return\n                    None =&gt; {\n                        return None;\n                    }\n                    // Number of children = 1: remove node directly\n                    Some(child) =&gt; node = child,\n                }\n            } else {\n                // Number of children = 2: remove the inorder successor and use its value to replace the current node value\n                let mut temp = node.borrow().right.clone().unwrap();\n                loop {\n                    let temp_left = temp.borrow().left.clone();\n                    if temp_left.is_none() {\n                        break;\n                    }\n                    temp = temp_left.unwrap();\n                }\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);\n                node.borrow_mut().val = temp.borrow().val;\n            }\n            Self::update_height(Some(node.clone())); // Update the node height\n\n            /* 2. Perform rotations to restore balance in this subtree */\n            node = Self::rotate(Some(node)).unwrap();\n            // Return the root of the subtree\n            Some(node)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* Remove a node */\n// The name remove conflicts with the function declared in stdio.h\nvoid removeItem(AVLTree *tree, int val) {\n    TreeNode *root = removeHelper(tree-&gt;root, val);\n}\n\n/* Recursively remove a node (helper function) */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    TreeNode *child, *grandChild;\n    if (node == NULL) {\n        return NULL;\n    }\n    /* 1. Find and remove the node */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    } else {\n        if (node-&gt;left == NULL || node-&gt;right == NULL) {\n            child = node-&gt;left;\n            if (node-&gt;right != NULL) {\n                child = node-&gt;right;\n            }\n            // Number of children = 0: remove node and return\n            if (child == NULL) {\n                return NULL;\n            } else {\n                // Number of children = 1: remove node directly\n                node = child;\n            }\n        } else {\n            // Number of children = 2: remove the inorder successor and use its value to replace the current node value\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != NULL) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    // Update the node height\n    updateHeight(node);\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = rotate(node);\n    // Return the root of the subtree\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* Remove a node */\nfun remove(_val: Int) {\n    root = removeHelper(root, _val)\n}\n\n/* Remove a node recursively (helper method) */\nfun removeHelper(n: TreeNode?, _val: Int): TreeNode? {\n    var node = n ?: return null\n    /* 1. Find and remove the node */\n    if (_val &lt; node._val)\n        node.left = removeHelper(node.left, _val)\n    else if (_val &gt; node._val)\n        node.right = removeHelper(node.right, _val)\n    else {\n        if (node.left == null || node.right == null) {\n            val child = if (node.left != null)\n                node.left\n            else\n                node.right\n            // Number of children = 0: remove node and return\n            if (child == null)\n                return null\n            // Number of children = 1: remove node directly\n            else\n                node = child\n        } else {\n            // Number of children = 2: remove the inorder successor and use its value to replace the current node value\n            var temp = node.right\n            while (temp!!.left != null) {\n                temp = temp.left\n            }\n            node.right = removeHelper(node.right, temp._val)\n            node._val = temp._val\n        }\n    }\n    updateHeight(node) // Update the node height\n    /* 2. Perform rotations to restore balance in this subtree */\n    node = rotate(node)\n    // Return the subtree root\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### Remove a node ###\ndef remove(val)\n  @root = remove_helper(@root, val)\nend\n\n### Remove a node recursively (helper method)###\ndef remove_helper(node, val)\n  return if node.nil?\n  # 1. Find and remove the node\n  if val &lt; node.val\n    node.left = remove_helper(node.left, val)\n  elsif val &gt; node.val\n    node.right = remove_helper(node.right, val)\n  else\n    if node.left.nil? || node.right.nil?\n      child = node.left || node.right\n      # Number of children = 0: remove node and return\n      return if child.nil?\n      # Number of children = 1: remove node directly\n      node = child\n    else\n      # Number of children = 2: remove the inorder successor and use its value to replace the current node value\n      temp = node.right\n      while !temp.left.nil?\n        temp = temp.left\n      end\n      node.right = remove_helper(node.right, temp.val)\n      node.val = temp.val\n    end\n  end\n  # Update the node height\n  update_height(node)\n  # 2. Perform rotations to restore balance in this subtree\n  rotate(node)\nend\n</code></pre>","path":["Chapter 7.   Trees","7.5   AVL trees *"],"tags":[]},{"location":"chapter_tree/avl_tree/#3-finding-a-node","level":3,"title":"3.   Finding a node","text":"<p>Lookup is identical to binary-search-tree lookup, so we will not repeat it here.</p>","path":["Chapter 7.   Trees","7.5   AVL trees *"],"tags":[]},{"location":"chapter_tree/avl_tree/#754-typical-applications","level":2,"title":"7.5.4   Typical applications","text":"<ul> <li>Organizing and storing large datasets, particularly when lookups are frequent and insertions and deletions are less frequent.</li> <li>Building database indexing systems.</li> <li>Red-black trees are another common kind of balanced binary search tree. Their balance requirements are looser than those of AVL trees, so insertion and deletion need fewer rotations and are more efficient on average.</li> </ul>","path":["Chapter 7.   Trees","7.5   AVL trees *"],"tags":[]},{"location":"chapter_tree/binary_search_tree/","level":1,"title":"7.4   Binary search trees","text":"<p>A binary search tree satisfies the following conditions, as shown below.</p> <ol> <li>Every value in the left subtree is \\(&lt;\\) the root value, which is \\(&lt;\\) every value in the right subtree.</li> <li>The left and right subtrees of every node are also binary search trees and satisfy condition <code>1.</code>.</li> </ol> <p></p> <p> Figure 7-16   A binary search tree </p>","path":["Chapter 7.   Trees","7.4   Binary search trees"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#741-binary-search-tree-operations","level":2,"title":"7.4.1   Binary search tree operations","text":"<p>We encapsulate the tree in a <code>BinarySearchTree</code> class with a member variable <code>root</code> pointing to its root node.</p>","path":["Chapter 7.   Trees","7.4   Binary search trees"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#1-finding-a-node","level":3,"title":"1.   Finding a node","text":"<p>To find a target value <code>num</code>, use the tree's ordering property. As shown below, initialize a node reference <code>cur</code> at <code>root</code> and repeatedly compare <code>cur.val</code> with <code>num</code>.</p> <ul> <li>If <code>cur.val &lt; num</code>, the target lies in the right subtree, so set <code>cur = cur.right</code>.</li> <li>If <code>cur.val &gt; num</code>, the target lies in the left subtree, so set <code>cur = cur.left</code>.</li> <li>If <code>cur.val = num</code>, the target has been found; exit the loop and return the node.</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 7-17   Finding a node in a binary search tree </p> <p>Searching a binary search tree follows the same principle as binary search: each round eliminates half the possibilities. The number of iterations is at most the tree's height, giving \\(O(\\log n)\\) time when the tree is balanced. An implementation follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_tree.py<pre><code>def search(self, num: int) -&gt; TreeNode | None:\n    \"\"\"Find a node\"\"\"\n    cur = self._root\n    # Search in a loop, stopping if cur becomes None\n    while cur is not None:\n        # If present, the target node is in the right subtree of cur\n        if cur.val &lt; num:\n            cur = cur.right\n        # If present, the target node is in the left subtree of cur\n        elif cur.val &gt; num:\n            cur = cur.left\n        # The target node was found; exit the loop\n        else:\n            break\n    return cur\n</code></pre> binary_search_tree.cpp<pre><code>/* Find a node */\nTreeNode *search(int num) {\n    TreeNode *cur = root;\n    // Continue searching while the current node pointer is not null\n    while (cur != nullptr) {\n        // The target node lies in the right subtree of cur\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // The target node lies in the left subtree of cur\n        else if (cur-&gt;val &gt; num)\n            cur = cur-&gt;left;\n        // The target node was found; exit the loop\n        else\n            break;\n    }\n    // Return the target node, or nullptr if it was not found\n    return cur;\n}\n</code></pre> binary_search_tree.java<pre><code>/* Find a node */\nTreeNode search(int num) {\n    TreeNode cur = root;\n    // Search in a loop until reaching a null reference beyond a leaf\n    while (cur != null) {\n        // The target node lies in the right subtree of cur\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // The target node lies in the left subtree of cur\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // The target node was found; exit the loop\n        else\n            break;\n    }\n    // Return the target node\n    return cur;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* Find a node */\nTreeNode? Search(int num) {\n    TreeNode? cur = root;\n    // Search in a loop, stopping if a null reference is reached\n    while (cur != null) {\n        // The target node lies in the right subtree of cur\n        if (cur.val &lt; num) cur =\n            cur.right;\n        // The target node lies in the left subtree of cur\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // The target node was found; exit the loop\n        else\n            break;\n    }\n    // Return the target node\n    return cur;\n}\n</code></pre> binary_search_tree.go<pre><code>/* Find a node */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\n    node := bst.root\n    // Continue searching while the current node is not nil\n    for node != nil {\n        if node.Val.(int) &lt; num {\n            // If the target node exists, it lies in the right subtree of node\n            node = node.Right\n        } else if node.Val.(int) &gt; num {\n            // If the target node exists, it lies in the left subtree of node\n            node = node.Left\n        } else {\n            // The target node was found; exit the loop\n            break\n        }\n    }\n    // Return the target node, or nil if it was not found\n    return node\n}\n</code></pre> binary_search_tree.swift<pre><code>/* Find a node */\nfunc search(num: Int) -&gt; TreeNode? {\n    var cur = root\n    // Search in a loop, stopping if cur becomes nil\n    while cur != nil {\n        // If present, the target node lies in the right subtree of cur\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // If present, the target node lies in the left subtree of cur\n        else if cur!.val &gt; num {\n            cur = cur?.left\n        }\n        // The target node was found; exit the loop\n        else {\n            break\n        }\n    }\n    // Return the target node, or nil if it was not found\n    return cur\n}\n</code></pre> binary_search_tree.js<pre><code>/* Find a node */\nsearch(num) {\n    let cur = this.root;\n    // Search in a loop until reaching a null reference beyond a leaf\n    while (cur !== null) {\n        // The target node lies in the right subtree of cur\n        if (cur.val &lt; num) cur = cur.right;\n        // The target node lies in the left subtree of cur\n        else if (cur.val &gt; num) cur = cur.left;\n        // The target node was found; exit the loop\n        else break;\n    }\n    // Return the target node\n    return cur;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* Find a node */\nsearch(num: number): TreeNode | null {\n    let cur = this.root;\n    // Search in a loop, stopping if a null reference is reached\n    while (cur !== null) {\n        // The target node lies in the right subtree of cur\n        if (cur.val &lt; num) cur = cur.right;\n        // The target node lies in the left subtree of cur\n        else if (cur.val &gt; num) cur = cur.left;\n        // The target node was found; exit the loop\n        else break;\n    }\n    // Return the target node\n    return cur;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* Find a node */\nTreeNode? search(int _num) {\n  TreeNode? cur = _root;\n  // Search in a loop, stopping if a null reference is reached\n  while (cur != null) {\n    // The target node lies in the right subtree of cur\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // The target node lies in the left subtree of cur\n    else if (cur.val &gt; _num)\n      cur = cur.left;\n    // The target node was found; exit the loop\n    else\n      break;\n  }\n  // Return the target node\n  return cur;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* Find a node */\npub fn search(&amp;self, num: i32) -&gt; OptionTreeNodeRc {\n    let mut cur = self.root.clone();\n    // Search while cur refers to a node\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // The target node lies in the right subtree of cur\n            Ordering::Greater =&gt; cur = node.borrow().right.clone(),\n            // The target node lies in the left subtree of cur\n            Ordering::Less =&gt; cur = node.borrow().left.clone(),\n            // The target node was found; exit the loop\n            Ordering::Equal =&gt; break,\n        }\n    }\n\n    // Return the matching node, or None if no match was found\n    cur\n}\n</code></pre> binary_search_tree.c<pre><code>/* Find a node */\nTreeNode *search(BinarySearchTree *bst, int num) {\n    TreeNode *cur = bst-&gt;root;\n    // Search in a loop until reaching a null reference beyond a leaf\n    while (cur != NULL) {\n        if (cur-&gt;val &lt; num) {\n            // The target node lies in the right subtree of cur\n            cur = cur-&gt;right;\n        } else if (cur-&gt;val &gt; num) {\n            // The target node lies in the left subtree of cur\n            cur = cur-&gt;left;\n        } else {\n            // The target node was found; exit the loop\n            break;\n        }\n    }\n    // Return the target node\n    return cur;\n}\n</code></pre> binary_search_tree.kt<pre><code>/* Find a node */\nfun search(num: Int): TreeNode? {\n    var cur = root\n    // Continue searching while cur is not null\n    while (cur != null) {\n        // The target node lies in the right subtree of cur\n        cur = if (cur._val &lt; num)\n            cur.right\n        // The target node lies in the left subtree of cur\n        else if (cur._val &gt; num)\n            cur.left\n        // The target node was found; exit the loop\n        else\n            break\n    }\n    // Return the target node\n    return cur\n}\n</code></pre> binary_search_tree.rb<pre><code>### Find a node ###\ndef search(num)\n  cur = @root\n\n  # Search in a loop, stopping if cur becomes nil\n  while !cur.nil?\n    # The target node lies in the right subtree of cur\n    if cur.val &lt; num\n      cur = cur.right\n    # The target node lies in the left subtree of cur\n    elsif cur.val &gt; num\n      cur = cur.left\n    # The target node was found; exit the loop\n    else\n      break\n    end\n  end\n\n  cur\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 7.   Trees","7.4   Binary search trees"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#2-inserting-a-node","level":3,"title":"2.   Inserting a node","text":"<p>To insert <code>num</code> while preserving the ordering “left subtree &lt; root &lt; right subtree,” follow the process below.</p> <ol> <li>Find the insertion position: as in a search, start at the root and move down according to comparisons with <code>num</code>. Stop when the traversal passes a leaf and reaches <code>None</code>.</li> <li>Insert at that position: create a node containing <code>num</code> and place it at the <code>None</code> position.</li> </ol> <p></p> <p> Figure 7-18   Inserting a node into a binary search tree </p> <p>Keep two details in mind when implementing this.</p> <ul> <li>Duplicate values are not allowed under this definition. If the value already exists, return without inserting it.</li> <li>Keep a reference <code>pre</code> to the node from the previous iteration. When the traversal reaches <code>None</code>, <code>pre</code> gives its parent so that the new node can be attached.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_tree.py<pre><code>def insert(self, num: int):\n    \"\"\"Insert a node\"\"\"\n    # If the tree is empty, initialize the root node\n    if self._root is None:\n        self._root = TreeNode(num)\n        return\n    # Search in a loop, stopping if cur becomes None\n    cur, pre = self._root, None\n    while cur is not None:\n        # Return immediately if a duplicate node is found\n        if cur.val == num:\n            return\n        pre = cur\n        # The insertion position lies in the right subtree of cur\n        if cur.val &lt; num:\n            cur = cur.right\n        # The insertion position lies in the left subtree of cur\n        else:\n            cur = cur.left\n    # Insert a node\n    node = TreeNode(num)\n    if pre.val &lt; num:\n        pre.right = node\n    else:\n        pre.left = node\n</code></pre> binary_search_tree.cpp<pre><code>/* Insert a node */\nvoid insert(int num) {\n    // If the tree is empty, initialize the root node\n    if (root == nullptr) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode *cur = root, *pre = nullptr;\n    // Continue searching while the current node pointer is not null\n    while (cur != nullptr) {\n        // Return immediately if a duplicate node is found\n        if (cur-&gt;val == num)\n            return;\n        pre = cur;\n        // The insertion position lies in the right subtree of cur\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // The insertion position lies in the left subtree of cur\n        else\n            cur = cur-&gt;left;\n    }\n    // Insert a node\n    TreeNode *node = new TreeNode(num);\n    if (pre-&gt;val &lt; num)\n        pre-&gt;right = node;\n    else\n        pre-&gt;left = node;\n}\n</code></pre> binary_search_tree.java<pre><code>/* Insert a node */\nvoid insert(int num) {\n    // If the tree is empty, initialize the root node\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode cur = root, pre = null;\n    // Search in a loop until reaching a null reference beyond a leaf\n    while (cur != null) {\n        // Return immediately if a duplicate node is found\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // The insertion position lies in the right subtree of cur\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // The insertion position lies in the left subtree of cur\n        else\n            cur = cur.left;\n    }\n    // Insert a node\n    TreeNode node = new TreeNode(num);\n    if (pre.val &lt; num)\n        pre.right = node;\n    else\n        pre.left = node;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* Insert a node */\nvoid Insert(int num) {\n    // If the tree is empty, initialize the root node\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode? cur = root, pre = null;\n    // Search in a loop, stopping if a null reference is reached\n    while (cur != null) {\n        // Return immediately if a duplicate node is found\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // The insertion position lies in the right subtree of cur\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // The insertion position lies in the left subtree of cur\n        else\n            cur = cur.left;\n    }\n\n    // Insert a node\n    TreeNode node = new(num);\n    if (pre != null) {\n        if (pre.val &lt; num)\n            pre.right = node;\n        else\n            pre.left = node;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* Insert a node */\nfunc (bst *binarySearchTree) insert(num int) {\n    cur := bst.root\n    // If the tree is empty, initialize the root node\n    if cur == nil {\n        bst.root = NewTreeNode(num)\n        return\n    }\n    // Track the parent of the node to insert\n    var pre *TreeNode = nil\n    // Continue searching while the current node is not nil\n    for cur != nil {\n        if cur.Val == num {\n            return\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            cur = cur.Right\n        } else {\n            cur = cur.Left\n        }\n    }\n    // Insert a node\n    node := NewTreeNode(num)\n    if pre.Val.(int) &lt; num {\n        pre.Right = node\n    } else {\n        pre.Left = node\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* Insert a node */\nfunc insert(num: Int) {\n    // If the tree is empty, initialize the root node\n    if root == nil {\n        root = TreeNode(x: num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // Search in a loop, stopping if cur becomes nil\n    while cur != nil {\n        // Return immediately if a duplicate node is found\n        if cur!.val == num {\n            return\n        }\n        pre = cur\n        // The insertion position lies in the right subtree of cur\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // The insertion position lies in the left subtree of cur\n        else {\n            cur = cur?.left\n        }\n    }\n    // Insert a node\n    let node = TreeNode(x: num)\n    if pre!.val &lt; num {\n        pre?.right = node\n    } else {\n        pre?.left = node\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* Insert a node */\ninsert(num) {\n    // If the tree is empty, initialize the root node\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur = this.root,\n        pre = null;\n    // Search in a loop until reaching a null reference beyond a leaf\n    while (cur !== null) {\n        // Return immediately if a duplicate node is found\n        if (cur.val === num) return;\n        pre = cur;\n        // The insertion position lies in the right subtree of cur\n        if (cur.val &lt; num) cur = cur.right;\n        // The insertion position lies in the left subtree of cur\n        else cur = cur.left;\n    }\n    // Insert a node\n    const node = new TreeNode(num);\n    if (pre.val &lt; num) pre.right = node;\n    else pre.left = node;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* Insert a node */\ninsert(num: number): void {\n    // If the tree is empty, initialize the root node\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // Search in a loop, stopping if a null reference is reached\n    while (cur !== null) {\n        // Return immediately if a duplicate node is found\n        if (cur.val === num) return;\n        pre = cur;\n        // The insertion position lies in the right subtree of cur\n        if (cur.val &lt; num) cur = cur.right;\n        // The insertion position lies in the left subtree of cur\n        else cur = cur.left;\n    }\n    // Insert a node\n    const node = new TreeNode(num);\n    if (pre!.val &lt; num) pre!.right = node;\n    else pre!.left = node;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* Insert a node */\nvoid insert(int _num) {\n  // If the tree is empty, initialize the root node\n  if (_root == null) {\n    _root = TreeNode(_num);\n    return;\n  }\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // Search in a loop, stopping if a null reference is reached\n  while (cur != null) {\n    // Return immediately if a duplicate node is found\n    if (cur.val == _num) return;\n    pre = cur;\n    // The insertion position lies in the right subtree of cur\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // The insertion position lies in the left subtree of cur\n    else\n      cur = cur.left;\n  }\n  // Insert a node\n  TreeNode? node = TreeNode(_num);\n  if (pre!.val &lt; _num)\n    pre.right = node;\n  else\n    pre.left = node;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* Insert a node */\npub fn insert(&amp;mut self, num: i32) {\n    // If the tree is empty, initialize the root node\n    if self.root.is_none() {\n        self.root = Some(TreeNode::new(num));\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // Search while cur refers to a node\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // Return immediately if a duplicate node is found\n            Ordering::Equal =&gt; return,\n            // The insertion position lies in the right subtree of cur\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // The insertion position lies in the left subtree of cur\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // Insert a node\n    let pre = pre.unwrap();\n    let node = Some(TreeNode::new(num));\n    if num &gt; pre.borrow().val {\n        pre.borrow_mut().right = node;\n    } else {\n        pre.borrow_mut().left = node;\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* Insert a node */\nvoid insert(BinarySearchTree *bst, int num) {\n    // If the tree is empty, initialize the root node\n    if (bst-&gt;root == NULL) {\n        bst-&gt;root = newTreeNode(num);\n        return;\n    }\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // Search in a loop until reaching a null reference beyond a leaf\n    while (cur != NULL) {\n        // Return immediately if a duplicate node is found\n        if (cur-&gt;val == num) {\n            return;\n        }\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // The insertion position lies in the right subtree of cur\n            cur = cur-&gt;right;\n        } else {\n            // The insertion position lies in the left subtree of cur\n            cur = cur-&gt;left;\n        }\n    }\n    // Insert a node\n    TreeNode *node = newTreeNode(num);\n    if (pre-&gt;val &lt; num) {\n        pre-&gt;right = node;\n    } else {\n        pre-&gt;left = node;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* Insert a node */\nfun insert(num: Int) {\n    // If the tree is empty, initialize the root node\n    if (root == null) {\n        root = TreeNode(num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode? = null\n    // Continue searching while cur is not null\n    while (cur != null) {\n        // Return immediately if a duplicate node is found\n        if (cur._val == num)\n            return\n        pre = cur\n        // The insertion position lies in the right subtree of cur\n        cur = if (cur._val &lt; num)\n            cur.right\n        // The insertion position lies in the left subtree of cur\n        else\n            cur.left\n    }\n    // Insert a node\n    val node = TreeNode(num)\n    if (pre?._val!! &lt; num)\n        pre.right = node\n    else\n        pre.left = node\n}\n</code></pre> binary_search_tree.rb<pre><code>### Insert a node ###\ndef insert(num)\n  # If the tree is empty, initialize the root node\n  if @root.nil?\n    @root = TreeNode.new(num)\n    return\n  end\n\n  # Search in a loop, stopping if cur becomes nil\n  cur, pre = @root, nil\n  while !cur.nil?\n    # Return immediately if a duplicate node is found\n    return if cur.val == num\n\n    pre = cur\n    # The insertion position lies in the right subtree of cur\n    if cur.val &lt; num\n      cur = cur.right\n    # The insertion position lies in the left subtree of cur\n    else\n      cur = cur.left\n    end\n  end\n\n  # Insert a node\n  node = TreeNode.new(num)\n  if pre.val &lt; num\n    pre.right = node\n  else\n    pre.left = node\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Like lookup, insertion takes \\(O(\\log n)\\) time.</p>","path":["Chapter 7.   Trees","7.4   Binary search trees"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#3-deleting-a-node","level":3,"title":"3.   Deleting a node","text":"<p>First locate the target node, then remove it while preserving “left subtree &lt; root &lt; right subtree.” The procedure depends on whether the target has 0, 1, or 2 children.</p> <p>As shown below, a node of degree \\(0\\) is a leaf and can be removed directly.</p> <p></p> <p> Figure 7-19   Deleting a node of degree 0 </p> <p>As shown below, replace a node of degree \\(1\\) with its only child.</p> <p></p> <p> Figure 7-20   Deleting a node of degree 1 </p> <p>A node of degree \\(2\\) cannot simply be removed; another node must replace it. To preserve “left subtree \\(&lt;\\) root \\(&lt;\\) right subtree,” use either the smallest node in the right subtree or the largest node in the left subtree.</p> <p>Suppose we choose the smallest node in the right subtree, the next node in inorder traversal. The process is shown below.</p> <ol> <li>Find the target node's successor in inorder traversal and call it <code>tmp</code>.</li> <li>Replace the target node's value with <code>tmp</code>'s value, then recursively delete <code>tmp</code> from the tree.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> Figure 7-21   Deleting a node of degree 2 </p> <p>Deletion also takes \\(O(\\log n)\\) time: locating the target takes \\(O(\\log n)\\), and finding its inorder successor takes \\(O(\\log n)\\). An implementation follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_tree.py<pre><code>def remove(self, num: int):\n    \"\"\"Remove a node\"\"\"\n    # Return immediately if the tree is empty\n    if self._root is None:\n        return\n    # Search in a loop, stopping if cur becomes None\n    cur, pre = self._root, None\n    while cur is not None:\n        # The node to remove was found; exit the loop\n        if cur.val == num:\n            break\n        pre = cur\n        # If present, the node to remove is in the right subtree of cur\n        if cur.val &lt; num:\n            cur = cur.right\n        # If present, the node to remove is in the left subtree of cur\n        else:\n            cur = cur.left\n    # Return immediately if the node to remove does not exist\n    if cur is None:\n        return\n\n    # Number of children = 0 or 1\n    if cur.left is None or cur.right is None:\n        # With 0 or 1 child, child is None or that child, respectively\n        child = cur.left or cur.right\n        # Remove node cur\n        if cur != self._root:\n            if pre.left == cur:\n                pre.left = child\n            else:\n                pre.right = child\n        else:\n            # If the removed node is the root, assign the new root\n            self._root = child\n    # Number of children = 2\n    else:\n        # Get the inorder successor of cur\n        tmp: TreeNode = cur.right\n        while tmp.left is not None:\n            tmp = tmp.left\n        # Recursively remove node tmp\n        self.remove(tmp.val)\n        # Replace the value of cur with the value of tmp\n        cur.val = tmp.val\n</code></pre> binary_search_tree.cpp<pre><code>/* Remove a node */\nvoid remove(int num) {\n    // Return immediately if the tree is empty\n    if (root == nullptr)\n        return;\n    TreeNode *cur = root, *pre = nullptr;\n    // Continue searching while the current node pointer is not null\n    while (cur != nullptr) {\n        // The node to remove was found; exit the loop\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        // The node to remove lies in the right subtree of cur\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // The node to remove lies in the left subtree of cur\n        else\n            cur = cur-&gt;left;\n    }\n    // Return immediately if the node to remove does not exist\n    if (cur == nullptr)\n        return;\n    // Number of children = 0 or 1\n    if (cur-&gt;left == nullptr || cur-&gt;right == nullptr) {\n        // With 0 or 1 child, child is nullptr or that child, respectively\n        TreeNode *child = cur-&gt;left != nullptr ? cur-&gt;left : cur-&gt;right;\n        // Remove node cur\n        if (cur != root) {\n            if (pre-&gt;left == cur)\n                pre-&gt;left = child;\n            else\n                pre-&gt;right = child;\n        } else {\n            // If the removed node is the root, assign the new root\n            root = child;\n        }\n        // Free memory\n        delete cur;\n    }\n    // Number of children = 2\n    else {\n        // Get the inorder successor of cur\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != nullptr) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // Recursively remove node tmp\n        remove(tmp-&gt;val);\n        // Replace the value of cur with the value of tmp\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.java<pre><code>/* Remove a node */\nvoid remove(int num) {\n    // Return immediately if the tree is empty\n    if (root == null)\n        return;\n    TreeNode cur = root, pre = null;\n    // Search in a loop until reaching a null reference beyond a leaf\n    while (cur != null) {\n        // The node to remove was found; exit the loop\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // The node to remove lies in the right subtree of cur\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // The node to remove lies in the left subtree of cur\n        else\n            cur = cur.left;\n    }\n    // Return immediately if the node to remove does not exist\n    if (cur == null)\n        return;\n    // Number of children = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // With 0 / 1 children, child is null / the child node, respectively\n        TreeNode child = cur.left != null ? cur.left : cur.right;\n        // Remove node cur\n        if (cur != root) {\n            if (pre.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // If the removed node is the root, assign the new root\n            root = child;\n        }\n    }\n    // Number of children = 2\n    else {\n        // Get the inorder successor of cur\n        TreeNode tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // Recursively remove node tmp\n        remove(tmp.val);\n        // Replace the value of cur with the value of tmp\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.cs<pre><code>/* Remove a node */\nvoid Remove(int num) {\n    // Return immediately if the tree is empty\n    if (root == null)\n        return;\n    TreeNode? cur = root, pre = null;\n    // Search in a loop, stopping if a null reference is reached\n    while (cur != null) {\n        // The node to remove was found; exit the loop\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // The node to remove lies in the right subtree of cur\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // The node to remove lies in the left subtree of cur\n        else\n            cur = cur.left;\n    }\n    // Return immediately if the node to remove does not exist\n    if (cur == null)\n        return;\n    // Number of children = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // With 0 / 1 children, child is null / the child node, respectively\n        TreeNode? child = cur.left ?? cur.right;\n        // Remove node cur\n        if (cur != root) {\n            if (pre!.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // If the removed node is the root, assign the new root\n            root = child;\n        }\n    }\n    // Number of children = 2\n    else {\n        // Get the inorder successor of cur\n        TreeNode? tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // Recursively remove node tmp\n        Remove(tmp.val!.Value);\n        // Replace the value of cur with the value of tmp\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* Remove a node */\nfunc (bst *binarySearchTree) remove(num int) {\n    cur := bst.root\n    // Return immediately if the tree is empty\n    if cur == nil {\n        return\n    }\n    // Track the parent of the node to remove\n    var pre *TreeNode = nil\n    // Continue searching while the current node is not nil\n    for cur != nil {\n        if cur.Val == num {\n            break\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            // The node to remove is in the right subtree\n            cur = cur.Right\n        } else {\n            // The node to remove is in the left subtree\n            cur = cur.Left\n        }\n    }\n    // Return immediately if the node to remove does not exist\n    if cur == nil {\n        return\n    }\n    // The node has 0 or 1 child\n    if cur.Left == nil || cur.Right == nil {\n        var child *TreeNode = nil\n        // Get the child of the node to remove\n        if cur.Left != nil {\n            child = cur.Left\n        } else {\n            child = cur.Right\n        }\n        // Remove node cur\n        if cur != bst.root {\n            if pre.Left == cur {\n                pre.Left = child\n            } else {\n                pre.Right = child\n            }\n        } else {\n            // If the removed node is the root, assign the new root\n            bst.root = child\n        }\n        // The node has 2 children\n    } else {\n        // Get the inorder successor of cur, the node to remove\n        tmp := cur.Right\n        for tmp.Left != nil {\n            tmp = tmp.Left\n        }\n        // Recursively remove node tmp\n        bst.remove(tmp.Val.(int))\n        // Replace the value of cur with the value of tmp\n        cur.Val = tmp.Val\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* Remove a node */\nfunc remove(num: Int) {\n    // Return immediately if the tree is empty\n    if root == nil {\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // Search in a loop, stopping if cur becomes nil\n    while cur != nil {\n        // The node to remove was found; exit the loop\n        if cur!.val == num {\n            break\n        }\n        pre = cur\n        // If present, the node to remove lies in the right subtree of cur\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // If present, the node to remove lies in the left subtree of cur\n        else {\n            cur = cur?.left\n        }\n    }\n    // Return immediately if the node to remove does not exist\n    if cur == nil {\n        return\n    }\n    // Number of children = 0 or 1\n    if cur?.left == nil || cur?.right == nil {\n        // With 0 / 1 children, child is nil / the child node, respectively\n        let child = cur?.left ?? cur?.right\n        // Remove node cur\n        if cur !== root {\n            if pre?.left === cur {\n                pre?.left = child\n            } else {\n                pre?.right = child\n            }\n        } else {\n            // If the removed node is the root, assign the new root\n            root = child\n        }\n    }\n    // Number of children = 2\n    else {\n        // Get the inorder successor of cur\n        var tmp = cur?.right\n        while tmp?.left != nil {\n            tmp = tmp?.left\n        }\n        // Recursively remove node tmp\n        remove(num: tmp!.val)\n        // Replace the value of cur with the value of tmp\n        cur?.val = tmp!.val\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* Remove a node */\nremove(num) {\n    // Return immediately if the tree is empty\n    if (this.root === null) return;\n    let cur = this.root,\n        pre = null;\n    // Search in a loop until reaching a null reference beyond a leaf\n    while (cur !== null) {\n        // The node to remove was found; exit the loop\n        if (cur.val === num) break;\n        pre = cur;\n        // The node to remove lies in the right subtree of cur\n        if (cur.val &lt; num) cur = cur.right;\n        // The node to remove lies in the left subtree of cur\n        else cur = cur.left;\n    }\n    // Return immediately if the node to remove does not exist\n    if (cur === null) return;\n    // Number of children = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // With 0 / 1 children, child is null / the child node, respectively\n        const child = cur.left !== null ? cur.left : cur.right;\n        // Remove node cur\n        if (cur !== this.root) {\n            if (pre.left === cur) pre.left = child;\n            else pre.right = child;\n        } else {\n            // If the removed node is the root, assign the new root\n            this.root = child;\n        }\n    }\n    // Number of children = 2\n    else {\n        // Get the inorder successor of cur\n        let tmp = cur.right;\n        while (tmp.left !== null) {\n            tmp = tmp.left;\n        }\n        // Recursively remove node tmp\n        this.remove(tmp.val);\n        // Replace the value of cur with the value of tmp\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.ts<pre><code>/* Remove a node */\nremove(num: number): void {\n    // Return immediately if the tree is empty\n    if (this.root === null) return;\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // Search in a loop, stopping if a null reference is reached\n    while (cur !== null) {\n        // The node to remove was found; exit the loop\n        if (cur.val === num) break;\n        pre = cur;\n        // The node to remove lies in the right subtree of cur\n        if (cur.val &lt; num) cur = cur.right;\n        // The node to remove lies in the left subtree of cur\n        else cur = cur.left;\n    }\n    // Return immediately if the node to remove does not exist\n    if (cur === null) return;\n    // Number of children = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // With 0 / 1 children, child is null / the child node, respectively\n        const child: TreeNode | null =\n            cur.left !== null ? cur.left : cur.right;\n        // Remove node cur\n        if (cur !== this.root) {\n            if (pre!.left === cur) pre!.left = child;\n            else pre!.right = child;\n        } else {\n            // If the removed node is the root, assign the new root\n            this.root = child;\n        }\n    }\n    // Number of children = 2\n    else {\n        // Get the inorder successor of cur\n        let tmp: TreeNode | null = cur.right;\n        while (tmp!.left !== null) {\n            tmp = tmp!.left;\n        }\n        // Recursively remove node tmp\n        this.remove(tmp!.val);\n        // Replace the value of cur with the value of tmp\n        cur.val = tmp!.val;\n    }\n}\n</code></pre> binary_search_tree.dart<pre><code>/* Remove a node */\nvoid remove(int _num) {\n  // Return immediately if the tree is empty\n  if (_root == null) return;\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // Search in a loop, stopping if a null reference is reached\n  while (cur != null) {\n    // The node to remove was found; exit the loop\n    if (cur.val == _num) break;\n    pre = cur;\n    // The node to remove lies in the right subtree of cur\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // The node to remove lies in the left subtree of cur\n    else\n      cur = cur.left;\n  }\n  // Return if the node to remove does not exist\n  if (cur == null) return;\n  // Number of children = 0 or 1\n  if (cur.left == null || cur.right == null) {\n    // With 0 / 1 children, child is null / the child node, respectively\n    TreeNode? child = cur.left ?? cur.right;\n    // Remove node cur\n    if (cur != _root) {\n      if (pre!.left == cur)\n        pre.left = child;\n      else\n        pre.right = child;\n    } else {\n      // If the removed node is the root, assign the new root\n      _root = child;\n    }\n  } else {\n    // Number of children = 2\n    // Get the inorder successor of cur\n    TreeNode? tmp = cur.right;\n    while (tmp!.left != null) {\n      tmp = tmp.left;\n    }\n    // Recursively remove node tmp\n    remove(tmp.val);\n    // Replace the value of cur with the value of tmp\n    cur.val = tmp.val;\n  }\n}\n</code></pre> binary_search_tree.rs<pre><code>/* Remove a node */\npub fn remove(&amp;mut self, num: i32) {\n    // Return immediately if the tree is empty\n    if self.root.is_none() {\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // Search while cur refers to a node\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // The node to remove was found; exit the loop\n            Ordering::Equal =&gt; break,\n            // The node to remove lies in the right subtree of cur\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // The node to remove lies in the left subtree of cur\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // Return immediately if the node to remove does not exist\n    if cur.is_none() {\n        return;\n    }\n    let cur = cur.unwrap();\n    let (left_child, right_child) = (cur.borrow().left.clone(), cur.borrow().right.clone());\n    match (left_child.clone(), right_child.clone()) {\n        // Number of children = 0 or 1\n        (None, None) | (Some(_), None) | (None, Some(_)) =&gt; {\n            // With 0 or 1 child, child is None or that child, respectively\n            let child = left_child.or(right_child);\n            let pre = pre.unwrap();\n            // Remove node cur\n            if !Rc::ptr_eq(&amp;cur, self.root.as_ref().unwrap()) {\n                let left = pre.borrow().left.clone();\n                if left.is_some() &amp;&amp; Rc::ptr_eq(left.as_ref().unwrap(), &amp;cur) {\n                    pre.borrow_mut().left = child;\n                } else {\n                    pre.borrow_mut().right = child;\n                }\n            } else {\n                // If the removed node is the root, assign the new root\n                self.root = child;\n            }\n        }\n        // Number of children = 2\n        (Some(_), Some(_)) =&gt; {\n            // Get the inorder successor of cur\n            let mut tmp = cur.borrow().right.clone();\n            while let Some(node) = tmp.clone() {\n                if node.borrow().left.is_some() {\n                    tmp = node.borrow().left.clone();\n                } else {\n                    break;\n                }\n            }\n            let tmp_val = tmp.unwrap().borrow().val;\n            // Recursively remove node tmp\n            self.remove(tmp_val);\n            // Replace the value of cur with the value of tmp\n            cur.borrow_mut().val = tmp_val;\n        }\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* Remove a node */\n// The name remove conflicts with the function declared in stdio.h\nvoid removeItem(BinarySearchTree *bst, int num) {\n    // Return immediately if the tree is empty\n    if (bst-&gt;root == NULL)\n        return;\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // Search in a loop until reaching a null reference beyond a leaf\n    while (cur != NULL) {\n        // The node to remove was found; exit the loop\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // The node to remove is in cur's right subtree\n            cur = cur-&gt;right;\n        } else {\n            // The node to remove is in cur's left subtree\n            cur = cur-&gt;left;\n        }\n    }\n    // Return immediately if the node to remove does not exist\n    if (cur == NULL)\n        return;\n    // Check whether the node to remove has children\n    if (cur-&gt;left == NULL || cur-&gt;right == NULL) {\n        /* Number of children = 0 or 1 */\n        // With 0 or 1 child, child is NULL or that child, respectively\n        TreeNode *child = cur-&gt;left != NULL ? cur-&gt;left : cur-&gt;right;\n        // Remove node cur\n        if (pre-&gt;left == cur) {\n            pre-&gt;left = child;\n        } else {\n            pre-&gt;right = child;\n        }\n        // Free memory\n        free(cur);\n    } else {\n        /* Number of children = 2 */\n        // Get the inorder successor of cur\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != NULL) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // Recursively remove node tmp\n        removeItem(bst, tmp-&gt;val);\n        // Replace the value of cur with the value of tmp\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* Remove a node */\nfun remove(num: Int) {\n    // Return immediately if the tree is empty\n    if (root == null)\n        return\n    var cur = root\n    var pre: TreeNode? = null\n    // Continue searching while cur is not null\n    while (cur != null) {\n        // The node to remove was found; exit the loop\n        if (cur._val == num)\n            break\n        pre = cur\n        // The node to remove lies in the right subtree of cur\n        cur = if (cur._val &lt; num)\n            cur.right\n        // The node to remove lies in the left subtree of cur\n        else\n            cur.left\n    }\n    // Return immediately if the node to remove does not exist\n    if (cur == null)\n        return\n    // Number of children = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // With 0 / 1 children, child is null / the child node, respectively\n        val child = if (cur.left != null)\n            cur.left\n        else\n            cur.right\n        // Remove node cur\n        if (cur != root) {\n            if (pre!!.left == cur)\n                pre.left = child\n            else\n                pre.right = child\n        } else {\n            // If the removed node is the root, assign the new root\n            root = child\n        }\n        // Number of children = 2\n    } else {\n        // Get the inorder successor of cur\n        var tmp = cur.right\n        while (tmp!!.left != null) {\n            tmp = tmp.left\n        }\n        // Recursively remove node tmp\n        remove(tmp._val)\n        // Replace the value of cur with the value of tmp\n        cur._val = tmp._val\n    }\n}\n</code></pre> binary_search_tree.rb<pre><code>### Remove a node ###\ndef remove(num)\n  # Return immediately if the tree is empty\n  return if @root.nil?\n\n  # Search in a loop, stopping if cur becomes nil\n  cur, pre = @root, nil\n  while !cur.nil?\n    # The node to remove was found; exit the loop\n    break if cur.val == num\n\n    pre = cur\n    # The node to remove lies in the right subtree of cur\n    if cur.val &lt; num\n      cur = cur.right\n    # The node to remove lies in the left subtree of cur\n    else\n      cur = cur.left\n    end\n  end\n  # Return immediately if the node to remove does not exist\n  return if cur.nil?\n\n  # Number of children = 0 or 1\n  if cur.left.nil? || cur.right.nil?\n    # With 0 / 1 children, child is nil / the child node, respectively\n    child = cur.left || cur.right\n    # Remove node cur\n    if cur != @root\n      if pre.left == cur\n        pre.left = child\n      else\n        pre.right = child\n      end\n    else\n      # If the removed node is the root, assign the new root\n      @root = child\n    end\n  # Number of children = 2\n  else\n    # Get the inorder successor of cur\n    tmp = cur.right\n    while !tmp.left.nil?\n      tmp = tmp.left\n    end\n    # Recursively remove node tmp\n    remove(tmp.val)\n    # Replace the value of cur with the value of tmp\n    cur.val = tmp.val\n  end\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 7.   Trees","7.4   Binary search trees"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#4-inorder-traversal-produces-sorted-data","level":3,"title":"4.   Inorder traversal produces sorted data","text":"<p>As shown below, inorder traversal visits “left \\(\\rightarrow\\) root \\(\\rightarrow\\) right,” while a binary search tree obeys “left child \\(&lt;\\) root \\(&lt;\\) right child.”</p> <p>Thus, inorder traversal always visits the next smallest node, giving an important property: the inorder traversal of a binary search tree is in ascending order.</p> <p>This lets us retrieve sorted data from the tree in just \\(O(n)\\) time, with no additional sorting.</p> <p></p> <p> Figure 7-22   Inorder traversal of a binary search tree </p>","path":["Chapter 7.   Trees","7.4   Binary search trees"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#742-efficiency","level":2,"title":"7.4.2   Efficiency","text":"<p>Suppose we want to store a dataset in an array or a binary search tree. As Table 7-2 shows, the tree's operations take logarithmic time, offering consistent and efficient performance. An array is faster only when insertions are frequent but lookups and deletions are infrequent.</p> <p> Table 7-2   Efficiency of arrays and search trees </p> Unsorted array Binary search tree Find an element \\(O(n)\\) \\(O(\\log n)\\) Insert an element \\(O(1)\\) \\(O(\\log n)\\) Delete an element \\(O(n)\\) \\(O(\\log n)\\) <p>Ideally, the tree is balanced, allowing any node to be found within \\(\\log n\\) iterations.</p> <p>However, repeated insertions and deletions may cause it to degenerate into a linked list, as shown below. The time complexity of its operations then degrades to \\(O(n)\\).</p> <p></p> <p> Figure 7-23   Degeneration of a binary search tree </p>","path":["Chapter 7.   Trees","7.4   Binary search trees"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#743-common-applications","level":2,"title":"7.4.3   Common applications","text":"<ul> <li>Multilevel indexes in systems, supporting efficient lookup, insertion, and deletion.</li> <li>The underlying data structure for certain search algorithms.</li> <li>Storing a data stream while maintaining its order.</li> </ul>","path":["Chapter 7.   Trees","7.4   Binary search trees"],"tags":[]},{"location":"chapter_tree/binary_tree/","level":1,"title":"7.1   Binary trees","text":"<p>A binary tree is a nonlinear data structure representing relationships between ancestors and descendants. It embodies the divide-and-conquer idea of splitting in two. Like a linked list, it is built from nodes, each containing a value and references to its left and right children.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class TreeNode:\n    \"\"\"Binary tree node class\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # Node value\n        self.left: TreeNode | None = None  # Left child reference\n        self.right: TreeNode | None = None # Right child reference\n</code></pre> <pre><code>/* Binary tree node struct */\nstruct TreeNode {\n    int val;          // Node value\n    TreeNode *left;   // Left child pointer\n    TreeNode *right;  // Right child pointer\n    TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n</code></pre> <pre><code>/* Binary tree node class */\nclass TreeNode {\n    int val;         // Node value\n    TreeNode left;   // Left child reference\n    TreeNode right;  // Right child reference\n    TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* Binary tree node class */\nclass TreeNode(int? x) {\n    public int? val = x;    // Node value\n    public TreeNode? left;  // Left child reference\n    public TreeNode? right; // Right child reference\n}\n</code></pre> <pre><code>/* Binary tree node struct */\ntype TreeNode struct {\n    Val   int\n    Left  *TreeNode\n    Right *TreeNode\n}\n/* Constructor */\nfunc NewTreeNode(v int) *TreeNode {\n    return &amp;TreeNode{\n        Left:  nil, // Left child pointer\n        Right: nil, // Right child pointer\n        Val:   v,   // Node value\n    }\n}\n</code></pre> <pre><code>/* Binary tree node class */\nclass TreeNode {\n    var val: Int // Node value\n    var left: TreeNode? // Left child reference\n    var right: TreeNode? // Right child reference\n\n    init(x: Int) {\n        val = x\n    }\n}\n</code></pre> <pre><code>/* Binary tree node class */\nclass TreeNode {\n    val; // Node value\n    left; // Left child pointer\n    right; // Right child pointer\n    constructor(val, left, right) {\n        this.val = val === undefined ? 0 : val;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* Binary tree node class */\nclass TreeNode {\n    val: number;\n    left: TreeNode | null;\n    right: TreeNode | null;\n\n    constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val; // Node value\n        this.left = left === undefined ? null : left; // Left child reference\n        this.right = right === undefined ? null : right; // Right child reference\n    }\n}\n</code></pre> <pre><code>/* Binary tree node class */\nclass TreeNode {\n  int val;         // Node value\n  TreeNode? left;  // Left child reference\n  TreeNode? right; // Right child reference\n  TreeNode(this.val, [this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* Binary tree node struct */\nstruct TreeNode {\n    val: i32,                               // Node value\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // Left child reference\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // Right child reference\n}\n\nimpl TreeNode {\n    /* Constructor */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* Binary tree node struct */\ntypedef struct TreeNode {\n    int val;                // Node value\n    int height;             // Node height\n    struct TreeNode *left;  // Left child pointer\n    struct TreeNode *right; // Right child pointer\n} TreeNode;\n\n/* Constructor */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* Binary tree node class */\nclass TreeNode(val _val: Int) {  // Node value\n    var left: TreeNode? = null   // Left child reference\n    var right: TreeNode? = null  // Right child reference\n}\n</code></pre> <pre><code>### Binary tree node class ###\nclass TreeNode\n  attr_accessor :val    # Node value\n  attr_accessor :left   # Left child reference\n  attr_accessor :right  # Right child reference\n\n  def initialize(val)\n    @val = val\n  end\nend\n</code></pre> <p>Each node has two references or pointers, one to its left child and one to its right child. The node is their parent. Its left child and all nodes below that child form its left subtree; the right subtree is defined similarly.</p> <p>Every non-leaf node has children and at least one nonempty subtree. In Figure 7-1, if node 2 is the parent, its left and right children are nodes 4 and 5. Its left subtree consists of node 4 and everything below it, and its right subtree consists of node 5 and everything below it.</p> <p></p> <p> Figure 7-1   Parents, children, and subtrees </p>","path":["Chapter 7.   Trees","7.1   Binary trees"],"tags":[]},{"location":"chapter_tree/binary_tree/#711-common-terminology","level":2,"title":"7.1.1   Common terminology","text":"<p>Figure 7-2 illustrates common binary-tree terminology.</p> <ul> <li>Root: the topmost node, which has no parent.</li> <li>Leaf: a node with no children; both of its child references point to <code>None</code>.</li> <li>Edge: a connection between two nodes, represented by a reference or pointer.</li> <li>A node's level: levels increase from top to bottom, with the root at level 1.</li> <li>A node's degree: its number of children, which is 0, 1, or 2 in a binary tree.</li> <li>A tree's height: the number of edges from the root to its farthest leaf.</li> <li>A node's depth: the number of edges from the root to that node.</li> <li>A node's height: the number of edges from that node to its farthest leaf.</li> </ul> <p></p> <p> Figure 7-2   Common binary-tree terminology </p> <p>Tip</p> <p>Height and depth are usually measured in edges, but some problems or textbooks count nodes instead. Under that convention, add 1 to both height and depth.</p>","path":["Chapter 7.   Trees","7.1   Binary trees"],"tags":[]},{"location":"chapter_tree/binary_tree/#712-basic-operations","level":2,"title":"7.1.2   Basic operations","text":"","path":["Chapter 7.   Trees","7.1   Binary trees"],"tags":[]},{"location":"chapter_tree/binary_tree/#1-initializing-a-binary-tree","level":3,"title":"1.   Initializing a binary tree","text":"<p>As with a linked list, first create the nodes, then connect them with references or pointers.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree.py<pre><code># Initialize a binary tree\n# Initialize nodes\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# Connect nodes with references (pointers)\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.cpp<pre><code>/* Initialize a binary tree */\n// Initialize nodes\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// Connect nodes with references (pointers)\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.java<pre><code>// Initialize nodes\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// Connect nodes with references (pointers)\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.cs<pre><code>/* Initialize a binary tree */\n// Initialize nodes\nTreeNode n1 = new(1);\nTreeNode n2 = new(2);\nTreeNode n3 = new(3);\nTreeNode n4 = new(4);\nTreeNode n5 = new(5);\n// Connect nodes with references (pointers)\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.go<pre><code>/* Initialize a binary tree */\n// Initialize nodes\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// Connect nodes with references (pointers)\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n</code></pre> binary_tree.swift<pre><code>// Initialize nodes\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// Connect nodes with references (pointers)\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.js<pre><code>/* Initialize a binary tree */\n// Initialize nodes\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// Connect nodes with references (pointers)\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.ts<pre><code>/* Initialize a binary tree */\n// Initialize nodes\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// Connect nodes with references (pointers)\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.dart<pre><code>/* Initialize a binary tree */\n// Initialize nodes\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// Connect nodes with references (pointers)\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.rs<pre><code>// Initialize nodes\nlet n1 = TreeNode::new(1);\nlet n2 = TreeNode::new(2);\nlet n3 = TreeNode::new(3);\nlet n4 = TreeNode::new(4);\nlet n5 = TreeNode::new(5);\n// Connect nodes with references (pointers)\nn1.borrow_mut().left = Some(n2.clone());\nn1.borrow_mut().right = Some(n3);\nn2.borrow_mut().left = Some(n4);\nn2.borrow_mut().right = Some(n5);\n</code></pre> binary_tree.c<pre><code>/* Initialize a binary tree */\n// Initialize nodes\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// Connect nodes with references (pointers)\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.kt<pre><code>// Initialize nodes\nval n1 = TreeNode(1)\nval n2 = TreeNode(2)\nval n3 = TreeNode(3)\nval n4 = TreeNode(4)\nval n5 = TreeNode(5)\n// Connect nodes with references (pointers)\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.rb<pre><code># Initialize a binary tree\n# Initialize nodes\nn1 = TreeNode.new(1)\nn2 = TreeNode.new(2)\nn3 = TreeNode.new(3)\nn4 = TreeNode.new(4)\nn5 = TreeNode.new(5)\n# Connect nodes with references (pointers)\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 7.   Trees","7.1   Binary trees"],"tags":[]},{"location":"chapter_tree/binary_tree/#2-inserting-and-deleting-nodes","level":3,"title":"2.   Inserting and deleting nodes","text":"<p>As with linked lists, insertion and deletion can be implemented by changing pointers. An example is shown below.</p> <p></p> <p> Figure 7-3   Inserting and deleting a binary-tree node </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree.py<pre><code># Insert and delete a node\np = TreeNode(0)\n# Insert node P between n1 and n2\nn1.left = p\np.left = n2\n# Delete node P\nn1.left = n2\n</code></pre> binary_tree.cpp<pre><code>/* Insert and delete a node */\nTreeNode* P = new TreeNode(0);\n// Insert node P between n1 and n2\nn1-&gt;left = P;\nP-&gt;left = n2;\n// Delete node P\nn1-&gt;left = n2;\n// Free the memory\ndelete P;\n</code></pre> binary_tree.java<pre><code>TreeNode P = new TreeNode(0);\n// Insert node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Delete node P\nn1.left = n2;\n</code></pre> binary_tree.cs<pre><code>/* Insert and delete a node */\nTreeNode P = new(0);\n// Insert node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Delete node P\nn1.left = n2;\n</code></pre> binary_tree.go<pre><code>/* Insert and delete a node */\n// Insert node P between n1 and n2\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// Delete node P\nn1.Left = n2\n</code></pre> binary_tree.swift<pre><code>let P = TreeNode(x: 0)\n// Insert node P between n1 and n2\nn1.left = P\nP.left = n2\n// Delete node P\nn1.left = n2\n</code></pre> binary_tree.js<pre><code>/* Insert and delete a node */\nlet P = new TreeNode(0);\n// Insert node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Delete node P\nn1.left = n2;\n</code></pre> binary_tree.ts<pre><code>/* Insert and delete a node */\nconst P = new TreeNode(0);\n// Insert node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Delete node P\nn1.left = n2;\n</code></pre> binary_tree.dart<pre><code>/* Insert and delete a node */\nTreeNode P = new TreeNode(0);\n// Insert node P between n1 and n2\nn1.left = P;\nP.left = n2;\n// Delete node P\nn1.left = n2;\n</code></pre> binary_tree.rs<pre><code>let p = TreeNode::new(0);\n// Insert node P between n1 and n2\nn1.borrow_mut().left = Some(p.clone());\np.borrow_mut().left = Some(n2.clone());\n// Delete node p\nn1.borrow_mut().left = Some(n2);\n</code></pre> binary_tree.c<pre><code>/* Insert and delete a node */\nTreeNode *P = newTreeNode(0);\n// Insert node P between n1 and n2\nn1-&gt;left = P;\nP-&gt;left = n2;\n// Delete node P\nn1-&gt;left = n2;\n// Free the memory\nfree(P);\n</code></pre> binary_tree.kt<pre><code>val P = TreeNode(0)\n// Insert node P between n1 and n2\nn1.left = P\nP.left = n2\n// Delete node P\nn1.left = n2\n</code></pre> binary_tree.rb<pre><code># Insert and delete a node\n_p = TreeNode.new(0)\n# Insert node _p between n1 and n2\nn1.left = _p\n_p.left = n2\n# Delete the node\nn1.left = n2\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Tip</p> <p>Inserting a node may change the tree's logical structure, while deleting a node generally means deleting it and all its subtrees. Meaningful insertion and deletion operations therefore usually require several coordinated steps.</p>","path":["Chapter 7.   Trees","7.1   Binary trees"],"tags":[]},{"location":"chapter_tree/binary_tree/#713-common-types-of-binary-trees","level":2,"title":"7.1.3   Common types of binary trees","text":"","path":["Chapter 7.   Trees","7.1   Binary trees"],"tags":[]},{"location":"chapter_tree/binary_tree/#1-perfect-binary-trees","level":3,"title":"1.   Perfect binary trees","text":"<p>In a perfect binary tree, every level is completely filled, as shown below. Leaves have degree \\(0\\), and all other nodes have degree \\(2\\). A tree of height \\(h\\) has \\(2^{h+1} - 1\\) nodes, an exponential relationship reminiscent of cell division in nature.</p> <p>Tip</p> <p>In Chinese-language resources, a perfect binary tree is often called 满二叉树.</p> <p></p> <p> Figure 7-4   A perfect binary tree </p>","path":["Chapter 7.   Trees","7.1   Binary trees"],"tags":[]},{"location":"chapter_tree/binary_tree/#2-complete-binary-trees","level":3,"title":"2.   Complete binary trees","text":"<p>Only the lowest level of a complete binary tree may be incomplete, and its nodes must fill that level consecutively from left to right, as shown below. Every perfect binary tree is also complete.</p> <p></p> <p> Figure 7-5   A complete binary tree </p>","path":["Chapter 7.   Trees","7.1   Binary trees"],"tags":[]},{"location":"chapter_tree/binary_tree/#3-full-binary-trees","level":3,"title":"3.   Full binary trees","text":"<p>In a full binary tree, every node other than a leaf has exactly two children, as shown below.</p> <p></p> <p> Figure 7-6   A full binary tree </p>","path":["Chapter 7.   Trees","7.1   Binary trees"],"tags":[]},{"location":"chapter_tree/binary_tree/#4-balanced-binary-trees","level":3,"title":"4.   Balanced binary trees","text":"<p>In a balanced binary tree, the heights of the left and right subtrees of every node differ by at most 1, as shown below.</p> <p></p> <p> Figure 7-7   A balanced binary tree </p>","path":["Chapter 7.   Trees","7.1   Binary trees"],"tags":[]},{"location":"chapter_tree/binary_tree/#714-degeneration-of-a-binary-tree","level":2,"title":"7.1.4   Degeneration of a binary tree","text":"<p>Figure 7-8 shows ideal and degenerate tree structures. When every level is full, the tree is perfect. When every node lies along one side, the tree degenerates into a linked list.</p> <ul> <li>A perfect binary tree is the ideal case, making full use of divide and conquer.</li> <li>A linked list is the opposite extreme: operations become linear, with time complexity degrading to \\(O(n)\\).</li> </ul> <p></p> <p> Figure 7-8   Best and worst binary-tree structures </p> <p>As Table 7-1 shows, quantities such as leaf count, total node count, and height reach their maximum or minimum values in these best and worst structures.</p> <p> Table 7-1   Best and worst binary-tree structures </p> Perfect binary tree Linked list Nodes at level \\(i\\) \\(2^{i-1}\\) \\(1\\) Leaves in a tree of height \\(h\\) \\(2^h\\) \\(1\\) Total nodes in a tree of height \\(h\\) \\(2^{h+1} - 1\\) \\(h + 1\\) Height of a tree with \\(n\\) nodes \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)","path":["Chapter 7.   Trees","7.1   Binary trees"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/","level":1,"title":"7.2   Traversing a binary tree","text":"<p>In its physical representation, a tree is a linked data structure, so traversal follows pointers from node to node. Unlike a linked list, however, a tree is nonlinear. Traversing it is more complex and requires a search algorithm.</p> <p>Common binary-tree traversals include level-order, preorder, inorder, and postorder traversal.</p>","path":["Chapter 7.   Trees","7.2   Traversing a binary tree"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#721-level-order-traversal","level":2,"title":"7.2.1   Level-order traversal","text":"<p>As shown below, level-order traversal visits the tree level by level from top to bottom, moving from left to right within each level.</p> <p>Level-order traversal is a form of breadth-first traversal, also called breadth-first search (BFS). It explores outward one layer at a time.</p> <p></p> <p> Figure 7-9   Level-order traversal of a binary tree </p>","path":["Chapter 7.   Trees","7.2   Traversing a binary tree"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#1-implementation","level":3,"title":"1.   Implementation","text":"<p>Breadth-first traversal usually uses a queue. A queue's first-in, first-out rule matches the traversal's level-by-level progression. The implementation follows:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree_bfs.py<pre><code>def level_order(root: TreeNode | None) -&gt; list[int]:\n    \"\"\"Level-order traversal\"\"\"\n    # Initialize a queue and enqueue the root\n    queue: deque[TreeNode] = deque()\n    queue.append(root)\n    # Initialize a list to store the traversal sequence\n    res = []\n    while queue:\n        node: TreeNode = queue.popleft()  # Dequeue from the queue\n        res.append(node.val)  # Store the node value\n        if node.left is not None:\n            queue.append(node.left)  # Enqueue the left child\n        if node.right is not None:\n            queue.append(node.right)  # Enqueue the right child\n    return res\n</code></pre> binary_tree_bfs.cpp<pre><code>/* Level-order traversal */\nvector&lt;int&gt; levelOrder(TreeNode *root) {\n    // Initialize a queue and enqueue the root\n    queue&lt;TreeNode *&gt; queue;\n    queue.push(root);\n    // Initialize a list to store the traversal sequence\n    vector&lt;int&gt; vec;\n    while (!queue.empty()) {\n        TreeNode *node = queue.front();\n        queue.pop();              // Dequeue from the queue\n        vec.push_back(node-&gt;val); // Store the node value\n        if (node-&gt;left != nullptr)\n            queue.push(node-&gt;left); // Enqueue the left child\n        if (node-&gt;right != nullptr)\n            queue.push(node-&gt;right); // Enqueue the right child\n    }\n    return vec;\n}\n</code></pre> binary_tree_bfs.java<pre><code>/* Level-order traversal */\nList&lt;Integer&gt; levelOrder(TreeNode root) {\n    // Initialize a queue and enqueue the root\n    Queue&lt;TreeNode&gt; queue = new LinkedList&lt;&gt;();\n    queue.add(root);\n    // Initialize a list to store the traversal sequence\n    List&lt;Integer&gt; list = new ArrayList&lt;&gt;();\n    while (!queue.isEmpty()) {\n        TreeNode node = queue.poll(); // Dequeue from the queue\n        list.add(node.val);           // Store the node value\n        if (node.left != null)\n            queue.offer(node.left);   // Enqueue the left child\n        if (node.right != null)\n            queue.offer(node.right);  // Enqueue the right child\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.cs<pre><code>/* Level-order traversal */\nList&lt;int&gt; LevelOrder(TreeNode root) {\n    // Initialize a queue and enqueue the root\n    Queue&lt;TreeNode&gt; queue = new();\n    queue.Enqueue(root);\n    // Initialize a list to store the traversal sequence\n    List&lt;int&gt; list = [];\n    while (queue.Count != 0) {\n        TreeNode node = queue.Dequeue(); // Dequeue from the queue\n        list.Add(node.val!.Value);       // Store the node value\n        if (node.left != null)\n            queue.Enqueue(node.left);    // Enqueue the left child\n        if (node.right != null)\n            queue.Enqueue(node.right);   // Enqueue the right child\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.go<pre><code>/* Level-order traversal */\nfunc levelOrder(root *TreeNode) []any {\n    // Initialize a queue and enqueue the root\n    queue := list.New()\n    queue.PushBack(root)\n    // Initialize a slice to store the traversal sequence\n    nums := make([]any, 0)\n    for queue.Len() &gt; 0 {\n        // Dequeue from the queue\n        node := queue.Remove(queue.Front()).(*TreeNode)\n        // Store the node value\n        nums = append(nums, node.Val)\n        if node.Left != nil {\n            // Enqueue the left child\n            queue.PushBack(node.Left)\n        }\n        if node.Right != nil {\n            // Enqueue the right child\n            queue.PushBack(node.Right)\n        }\n    }\n    return nums\n}\n</code></pre> binary_tree_bfs.swift<pre><code>/* Level-order traversal */\nfunc levelOrder(root: TreeNode) -&gt; [Int] {\n    // Initialize a queue and enqueue the root\n    var queue: [TreeNode] = [root]\n    // Initialize a list to store the traversal sequence\n    var list: [Int] = []\n    while !queue.isEmpty {\n        let node = queue.removeFirst() // Dequeue from the queue\n        list.append(node.val) // Store the node value\n        if let left = node.left {\n            queue.append(left) // Enqueue the left child\n        }\n        if let right = node.right {\n            queue.append(right) // Enqueue the right child\n        }\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.js<pre><code>/* Level-order traversal */\nfunction levelOrder(root) {\n    // Initialize a queue and enqueue the root\n    const queue = [root];\n    // Initialize a list to store the traversal sequence\n    const list = [];\n    while (queue.length) {\n        let node = queue.shift(); // Dequeue from the queue\n        list.push(node.val); // Store the node value\n        if (node.left) queue.push(node.left); // Enqueue the left child\n        if (node.right) queue.push(node.right); // Enqueue the right child\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.ts<pre><code>/* Level-order traversal */\nfunction levelOrder(root: TreeNode | null): number[] {\n    // Initialize a queue and enqueue the root\n    const queue = [root];\n    // Initialize a list to store the traversal sequence\n    const list: number[] = [];\n    while (queue.length) {\n        let node = queue.shift() as TreeNode; // Dequeue from the queue\n        list.push(node.val); // Store the node value\n        if (node.left) {\n            queue.push(node.left); // Enqueue the left child\n        }\n        if (node.right) {\n            queue.push(node.right); // Enqueue the right child\n        }\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.dart<pre><code>/* Level-order traversal */\nList&lt;int&gt; levelOrder(TreeNode? root) {\n  // Initialize a queue and enqueue the root\n  Queue&lt;TreeNode?&gt; queue = Queue();\n  queue.add(root);\n  // Initialize a list to store the traversal sequence\n  List&lt;int&gt; res = [];\n  while (queue.isNotEmpty) {\n    TreeNode? node = queue.removeFirst(); // Dequeue from the queue\n    res.add(node!.val); // Store the node value\n    if (node.left != null) queue.add(node.left); // Enqueue the left child\n    if (node.right != null) queue.add(node.right); // Enqueue the right child\n  }\n  return res;\n}\n</code></pre> binary_tree_bfs.rs<pre><code>/* Level-order traversal */\nfn level_order(root: &amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    // Initialize a queue and enqueue the root\n    let mut que = VecDeque::new();\n    que.push_back(root.clone());\n    // Initialize a list to store the traversal sequence\n    let mut vec = Vec::new();\n\n    while let Some(node) = que.pop_front() {\n        // Dequeue from the queue\n        vec.push(node.borrow().val); // Store the node value\n        if let Some(left) = node.borrow().left.as_ref() {\n            que.push_back(left.clone()); // Enqueue the left child\n        }\n        if let Some(right) = node.borrow().right.as_ref() {\n            que.push_back(right.clone()); // Enqueue the right child\n        };\n    }\n    vec\n}\n</code></pre> binary_tree_bfs.c<pre><code>/* Level-order traversal */\nint *levelOrder(TreeNode *root, int *size) {\n    /* Auxiliary queue */\n    int front, rear;\n    int index, *arr;\n    TreeNode *node;\n    TreeNode **queue;\n\n    /* Auxiliary queue */\n    queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);\n    // Queue indices\n    front = 0, rear = 0;\n    // Enqueue the root node\n    queue[rear++] = root;\n    // Initialize a list to store the traversal sequence\n    /* Auxiliary array */\n    arr = (int *)malloc(sizeof(int) * MAX_SIZE);\n    // Array index\n    index = 0;\n    while (front &lt; rear) {\n        // Dequeue from the queue\n        node = queue[front++];\n        // Store the node value\n        arr[index++] = node-&gt;val;\n        if (node-&gt;left != NULL) {\n            // Enqueue the left child\n            queue[rear++] = node-&gt;left;\n        }\n        if (node-&gt;right != NULL) {\n            // Enqueue the right child\n            queue[rear++] = node-&gt;right;\n        }\n    }\n    // Update the array length\n    *size = index;\n    arr = realloc(arr, sizeof(int) * (*size));\n\n    // Free the auxiliary queue array\n    free(queue);\n    return arr;\n}\n</code></pre> binary_tree_bfs.kt<pre><code>/* Level-order traversal */\nfun levelOrder(root: TreeNode?): MutableList&lt;Int&gt; {\n    // Initialize a queue and enqueue the root\n    val queue = LinkedList&lt;TreeNode?&gt;()\n    queue.add(root)\n    // Initialize a list to store the traversal sequence\n    val list = mutableListOf&lt;Int&gt;()\n    while (queue.isNotEmpty()) {\n        val node = queue.poll()      // Dequeue from the queue\n        list.add(node?._val!!)       // Store the node value\n        if (node.left != null)\n            queue.offer(node.left)   // Enqueue the left child\n        if (node.right != null)\n            queue.offer(node.right)  // Enqueue the right child\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.rb<pre><code>### Level-order traversal ###\ndef level_order(root)\n  # Initialize a queue and enqueue the root\n  queue = [root]\n  # Initialize a list to store the traversal sequence\n  res = []\n  while !queue.empty?\n    node = queue.shift # Dequeue from the queue\n    res &lt;&lt; node.val # Store the node value\n    queue &lt;&lt; node.left unless node.left.nil? # Enqueue the left child\n    queue &lt;&lt; node.right unless node.right.nil? # Enqueue the right child\n  end\n  res\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p>","path":["Chapter 7.   Trees","7.2   Traversing a binary tree"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#2-complexity-analysis","level":3,"title":"2.   Complexity analysis","text":"<ul> <li>Time complexity: \\(O(n)\\). Each of the \\(n\\) nodes is visited once, taking \\(O(n)\\) time.</li> <li>Space complexity: \\(O(n)\\). In the worst case, a perfect binary tree, the queue holds up to \\((n + 1) / 2\\) nodes just before the lowest level is visited, requiring \\(O(n)\\) space.</li> </ul>","path":["Chapter 7.   Trees","7.2   Traversing a binary tree"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#722-preorder-inorder-and-postorder-traversal","level":2,"title":"7.2.2   Preorder, inorder, and postorder traversal","text":"<p>Preorder, inorder, and postorder traversal are forms of depth-first traversal, also called depth-first search (DFS). They follow a path to its end, then backtrack and continue.</p> <p>Figure 7-10 illustrates depth-first traversal. Imagine walking once around the outside of the entire binary tree. At each node, you encounter three positions corresponding to preorder, inorder, and postorder visits.</p> <p></p> <p> Figure 7-10   Preorder, inorder, and postorder traversal of a binary search tree </p>","path":["Chapter 7.   Trees","7.2   Traversing a binary tree"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#1-implementation_1","level":3,"title":"1.   Implementation","text":"<p>Depth-first search is usually implemented recursively:</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree_dfs.py<pre><code>def pre_order(root: TreeNode | None):\n    \"\"\"Preorder traversal\"\"\"\n    if root is None:\n        return\n    # Visit order: root -&gt; left subtree -&gt; right subtree\n    res.append(root.val)\n    pre_order(root=root.left)\n    pre_order(root=root.right)\n\ndef in_order(root: TreeNode | None):\n    \"\"\"Inorder traversal\"\"\"\n    if root is None:\n        return\n    # Visit order: left subtree -&gt; root -&gt; right subtree\n    in_order(root=root.left)\n    res.append(root.val)\n    in_order(root=root.right)\n\ndef post_order(root: TreeNode | None):\n    \"\"\"Postorder traversal\"\"\"\n    if root is None:\n        return\n    # Visit order: left subtree -&gt; right subtree -&gt; root\n    post_order(root=root.left)\n    post_order(root=root.right)\n    res.append(root.val)\n</code></pre> binary_tree_dfs.cpp<pre><code>/* Preorder traversal */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // Visit order: root -&gt; left subtree -&gt; right subtree\n    vec.push_back(root-&gt;val);\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n\n/* Inorder traversal */\nvoid inOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // Visit order: left subtree -&gt; root -&gt; right subtree\n    inOrder(root-&gt;left);\n    vec.push_back(root-&gt;val);\n    inOrder(root-&gt;right);\n}\n\n/* Postorder traversal */\nvoid postOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // Visit order: left subtree -&gt; right subtree -&gt; root\n    postOrder(root-&gt;left);\n    postOrder(root-&gt;right);\n    vec.push_back(root-&gt;val);\n}\n</code></pre> binary_tree_dfs.java<pre><code>/* Preorder traversal */\nvoid preOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // Visit order: root -&gt; left subtree -&gt; right subtree\n    list.add(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* Inorder traversal */\nvoid inOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // Visit order: left subtree -&gt; root -&gt; right subtree\n    inOrder(root.left);\n    list.add(root.val);\n    inOrder(root.right);\n}\n\n/* Postorder traversal */\nvoid postOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // Visit order: left subtree -&gt; right subtree -&gt; root\n    postOrder(root.left);\n    postOrder(root.right);\n    list.add(root.val);\n}\n</code></pre> binary_tree_dfs.cs<pre><code>/* Preorder traversal */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) return;\n    // Visit order: root -&gt; left subtree -&gt; right subtree\n    list.Add(root.val!.Value);\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n\n/* Inorder traversal */\nvoid InOrder(TreeNode? root) {\n    if (root == null) return;\n    // Visit order: left subtree -&gt; root -&gt; right subtree\n    InOrder(root.left);\n    list.Add(root.val!.Value);\n    InOrder(root.right);\n}\n\n/* Postorder traversal */\nvoid PostOrder(TreeNode? root) {\n    if (root == null) return;\n    // Visit order: left subtree -&gt; right subtree -&gt; root\n    PostOrder(root.left);\n    PostOrder(root.right);\n    list.Add(root.val!.Value);\n}\n</code></pre> binary_tree_dfs.go<pre><code>/* Preorder traversal */\nfunc preOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // Visit order: root -&gt; left subtree -&gt; right subtree\n    nums = append(nums, node.Val)\n    preOrder(node.Left)\n    preOrder(node.Right)\n}\n\n/* Inorder traversal */\nfunc inOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // Visit order: left subtree -&gt; root -&gt; right subtree\n    inOrder(node.Left)\n    nums = append(nums, node.Val)\n    inOrder(node.Right)\n}\n\n/* Postorder traversal */\nfunc postOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // Visit order: left subtree -&gt; right subtree -&gt; root\n    postOrder(node.Left)\n    postOrder(node.Right)\n    nums = append(nums, node.Val)\n}\n</code></pre> binary_tree_dfs.swift<pre><code>/* Preorder traversal */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // Visit order: root -&gt; left subtree -&gt; right subtree\n    list.append(root.val)\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n\n/* Inorder traversal */\nfunc inOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // Visit order: left subtree -&gt; root -&gt; right subtree\n    inOrder(root: root.left)\n    list.append(root.val)\n    inOrder(root: root.right)\n}\n\n/* Postorder traversal */\nfunc postOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // Visit order: left subtree -&gt; right subtree -&gt; root\n    postOrder(root: root.left)\n    postOrder(root: root.right)\n    list.append(root.val)\n}\n</code></pre> binary_tree_dfs.js<pre><code>/* Preorder traversal */\nfunction preOrder(root) {\n    if (root === null) return;\n    // Visit order: root -&gt; left subtree -&gt; right subtree\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* Inorder traversal */\nfunction inOrder(root) {\n    if (root === null) return;\n    // Visit order: left subtree -&gt; root -&gt; right subtree\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* Postorder traversal */\nfunction postOrder(root) {\n    if (root === null) return;\n    // Visit order: left subtree -&gt; right subtree -&gt; root\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.ts<pre><code>/* Preorder traversal */\nfunction preOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // Visit order: root -&gt; left subtree -&gt; right subtree\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* Inorder traversal */\nfunction inOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // Visit order: left subtree -&gt; root -&gt; right subtree\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* Postorder traversal */\nfunction postOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // Visit order: left subtree -&gt; right subtree -&gt; root\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.dart<pre><code>/* Preorder traversal */\nvoid preOrder(TreeNode? node) {\n  if (node == null) return;\n  // Visit order: root -&gt; left subtree -&gt; right subtree\n  list.add(node.val);\n  preOrder(node.left);\n  preOrder(node.right);\n}\n\n/* Inorder traversal */\nvoid inOrder(TreeNode? node) {\n  if (node == null) return;\n  // Visit order: left subtree -&gt; root -&gt; right subtree\n  inOrder(node.left);\n  list.add(node.val);\n  inOrder(node.right);\n}\n\n/* Postorder traversal */\nvoid postOrder(TreeNode? node) {\n  if (node == null) return;\n  // Visit order: left subtree -&gt; right subtree -&gt; root\n  postOrder(node.left);\n  postOrder(node.right);\n  list.add(node.val);\n}\n</code></pre> binary_tree_dfs.rs<pre><code>/* Preorder traversal */\nfn pre_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // Visit order: root -&gt; left subtree -&gt; right subtree\n            let node = node.borrow();\n            res.push(node.val);\n            dfs(node.left.as_ref(), res);\n            dfs(node.right.as_ref(), res);\n        }\n    }\n    dfs(root, &amp;mut result);\n\n    result\n}\n\n/* Inorder traversal */\nfn in_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // Visit order: left subtree -&gt; root -&gt; right subtree\n            let node = node.borrow();\n            dfs(node.left.as_ref(), res);\n            res.push(node.val);\n            dfs(node.right.as_ref(), res);\n        }\n    }\n    dfs(root, &amp;mut result);\n\n    result\n}\n\n/* Postorder traversal */\nfn post_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // Visit order: left subtree -&gt; right subtree -&gt; root\n            let node = node.borrow();\n            dfs(node.left.as_ref(), res);\n            dfs(node.right.as_ref(), res);\n            res.push(node.val);\n        }\n    }\n\n    dfs(root, &amp;mut result);\n\n    result\n}\n</code></pre> binary_tree_dfs.c<pre><code>/* Preorder traversal */\nvoid preOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // Visit order: root -&gt; left subtree -&gt; right subtree\n    arr[(*size)++] = root-&gt;val;\n    preOrder(root-&gt;left, size);\n    preOrder(root-&gt;right, size);\n}\n\n/* Inorder traversal */\nvoid inOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // Visit order: left subtree -&gt; root -&gt; right subtree\n    inOrder(root-&gt;left, size);\n    arr[(*size)++] = root-&gt;val;\n    inOrder(root-&gt;right, size);\n}\n\n/* Postorder traversal */\nvoid postOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // Visit order: left subtree -&gt; right subtree -&gt; root\n    postOrder(root-&gt;left, size);\n    postOrder(root-&gt;right, size);\n    arr[(*size)++] = root-&gt;val;\n}\n</code></pre> binary_tree_dfs.kt<pre><code>/* Preorder traversal */\nfun preOrder(root: TreeNode?) {\n    if (root == null) return\n    // Visit order: root -&gt; left subtree -&gt; right subtree\n    list.add(root._val)\n    preOrder(root.left)\n    preOrder(root.right)\n}\n\n/* Inorder traversal */\nfun inOrder(root: TreeNode?) {\n    if (root == null) return\n    // Visit order: left subtree -&gt; root -&gt; right subtree\n    inOrder(root.left)\n    list.add(root._val)\n    inOrder(root.right)\n}\n\n/* Postorder traversal */\nfun postOrder(root: TreeNode?) {\n    if (root == null) return\n    // Visit order: left subtree -&gt; right subtree -&gt; root\n    postOrder(root.left)\n    postOrder(root.right)\n    list.add(root._val)\n}\n</code></pre> binary_tree_dfs.rb<pre><code>### Preorder traversal ###\ndef pre_order(root)\n  return if root.nil?\n\n  # Visit order: root -&gt; left subtree -&gt; right subtree\n  $res &lt;&lt; root.val\n  pre_order(root.left)\n  pre_order(root.right)\nend\n\n### Inorder traversal ###\ndef in_order(root)\n  return if root.nil?\n\n  # Visit order: left subtree -&gt; root -&gt; right subtree\n  in_order(root.left)\n  $res &lt;&lt; root.val\n  in_order(root.right)\nend\n\n### Postorder traversal ###\ndef post_order(root)\n  return if root.nil?\n\n  # Visit order: left subtree -&gt; right subtree -&gt; root\n  post_order(root.left)\n  post_order(root.right)\n  $res &lt;&lt; root.val\nend\n</code></pre> Visualize execution <p> View full screen &gt;</p> <p>Tip</p> <p>Depth-first search can also be implemented iteratively. Interested readers can explore this approach themselves.</p> <p>Figure 7-11 shows the recursive process for preorder traversal, with two opposing phases: descending into calls and returning from them.</p> <ol> <li>Descending starts a new function call and visits the next node.</li> <li>Returning ends a function call, indicating that the current node has been fully processed.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> Figure 7-11   The recursive process of preorder traversal </p>","path":["Chapter 7.   Trees","7.2   Traversing a binary tree"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#2-complexity-analysis_1","level":3,"title":"2.   Complexity analysis","text":"<ul> <li>Time complexity: \\(O(n)\\). Every node is visited once, taking \\(O(n)\\) time.</li> <li>Space complexity: \\(O(n)\\). In the worst case, the tree degenerates into a linked list, recursion reaches depth \\(n\\), and stack frames require \\(O(n)\\) space.</li> </ul>","path":["Chapter 7.   Trees","7.2   Traversing a binary tree"],"tags":[]},{"location":"chapter_tree/exercises/","level":1,"title":"7.7   Exercises","text":"","path":["Chapter 7.   Trees","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#771-check-your-understanding","level":2,"title":"7.7.1   Check your understanding","text":"","path":["Chapter 7.   Trees","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#1-complete-full-and-perfect-binary-trees","level":3,"title":"1.   Complete, full, and perfect binary trees","text":"<p>The following arrays represent binary trees in level order, with <code>None</code> marking an empty position:</p> <ul> <li>Tree A: <code>[1, 2, 3, 4, 5, 6]</code></li> <li>Tree B: <code>[1, 2, 3, None, None, 6, 7]</code></li> </ul> <ol> <li>Which tree is a complete binary tree?</li> <li>Which is a full binary tree, meaning that every non-leaf node has two children?</li> <li>Is either tree a perfect binary tree? Explain for each tree.</li> </ol> Reference answer <ol> <li> <p>Tree A is complete. Only its bottom level is not full, and the nodes occupy consecutive positions from left to right.     Tree B is not complete: its bottom level has empty positions on the left but still has nodes farther to the right.</p> </li> <li> <p>Tree B is full: nodes 1 and 3 each have two children, and all the other nodes are leaves.     Tree A is not full because node 3 has only its left child, 6.</p> </li> <li> <p>Neither tree is perfect, because neither has a completely filled bottom level.</p> </li> </ol>","path":["Chapter 7.   Trees","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#2-three-traversal-orders-for-the-same-tree","level":3,"title":"2.   Three traversal orders for the same tree","text":"<p>Store the array <code>[1, 2, 3, 4, 5, 6, 7]</code> in a complete binary tree in level order.</p> <ol> <li>Draw the tree.</li> <li>Write its preorder, inorder, and postorder traversal sequences.</li> <li>In the inorder sequence, which parts of the tree do the sequences to the left and right of root node 1 correspond to?</li> </ol> Reference answer <ol> <li> <p>The tree is:</p> <pre><code>      1\n    /   \\\n   2     3\n  / \\   / \\\n 4   5 6   7\n</code></pre> </li> <li> <p>Preorder: <code>1, 2, 4, 5, 3, 6, 7</code>;     inorder: <code>4, 2, 5, 1, 6, 3, 7</code>;     postorder: <code>4, 5, 2, 6, 7, 3, 1</code>.</p> </li> <li> <p>The sequence <code>4, 2, 5</code> to the left of root node 1 is the inorder traversal of the left subtree.     The sequence <code>6, 3, 7</code> to its right is the inorder traversal of the right subtree.</p> </li> </ol>","path":["Chapter 7.   Trees","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#3-comparing-two-binary-search-trees","level":3,"title":"3.   Comparing two binary search trees","text":"<p>Insert each of the following sequences from left to right into an empty binary search tree:</p> <ul> <li>Sequence A: <code>[4, 2, 6, 1, 3, 5, 7]</code></li> <li>Sequence B: <code>[1, 2, 3, 4, 5, 6, 7]</code></li> </ul> <ol> <li>Write the nodes visited when searching for 7 in each tree.</li> <li>If height is measured by the number of edges from the root to the farthest leaf, what is each tree's height?</li> <li>Based on your answers, do the two trees search for 7 equally efficiently? Explain using their shapes and search paths.</li> </ol> Reference answer <ol> <li> <p>In the tree built from sequence A, the search path is <code>4 → 6 → 7</code>.     In the tree built from sequence B, it is <code>1 → 2 → 3 → 4 → 5 → 6 → 7</code>.</p> </li> <li> <p>The first tree has every level filled and a height of 2. The second tree has only right children and a height of 6.</p> </li> <li> <p>The searches for 7 are not equally efficient. Insertion order changes the shape and height of a binary search tree. Searching for 7 visits only three nodes in the first tree,     but must visit seven in sequence in the second. The taller the tree, the more nodes may need to be compared along a path in the worst case.</p> </li> </ol>","path":["Chapter 7.   Trees","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#772-programming-exercises","level":2,"title":"7.7.2   Programming exercises","text":"","path":["Chapter 7.   Trees","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#1-maximum-depth-of-a-binary-tree","level":3,"title":"1.   Maximum depth of a binary tree","text":"<p>You are given the root node <code>root</code> of a binary tree. Each node contains an integer value and references to its left and right children.</p> <p>The maximum depth is the number of nodes on the path from the root to the farthest leaf. Return the maximum depth of the tree; an empty tree has maximum depth 0. Use recursion.</p> Hints <ol> <li>Depth is counted in nodes here. A tree containing only a root has maximum depth 1.</li> <li>Have the recursive function return the maximum depth of the subtree rooted at the current node.</li> <li>Return 0 for an empty node; otherwise, return max(depth(left), depth(right)) + 1.</li> </ol> <p>LeetCode</p>","path":["Chapter 7.   Trees","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#2-traversing-a-binary-tree-level-by-level","level":3,"title":"2.   Traversing a binary tree level by level","text":"<p>Given the root node <code>root</code> of a binary tree, use a queue to visit all nodes level by level from top to bottom, visiting each level from left to right.</p> <p>Return a two-dimensional array: the first subarray contains the values on the root's level, the second contains those on the next level, and so on. Return an empty array if the tree is empty.</p> Hints <ol> <li>Level-order traversal visits earlier enqueued nodes first, so use a queue.</li> <li>At the start of each round, the nodes in the queue belong to exactly one level.</li> <li>Record the queue's length first, then dequeue that many nodes and enqueue their children.</li> </ol> <p>LeetCode</p>","path":["Chapter 7.   Trees","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/exercises/#3-the-kth-smallest-element-in-a-binary-search-tree","level":3,"title":"3.   The kth smallest element in a binary search tree","text":"<p>A binary search tree contains <code>n</code> nodes with distinct values. If all node values are sorted in ascending order, their positions are numbered starting from 1.</p> <p>Given the root node <code>root</code> and an integer <code>k</code> satisfying <code>1 &lt;= k &lt;= n</code>, return the node value in position <code>k</code>. Find the answer directly during inorder traversal, without first collecting all node values.</p> Hints <ol> <li>An inorder traversal of a binary search tree visits node values in ascending order.</li> <li>Inorder traversal processes the left subtree, the current node, and then the right subtree. Increment a counter by 1 when visiting the current node.</li> <li>When the counter first reaches k, the current node's value is the answer; there is no need to continue traversing.</li> </ol> <p>LeetCode</p>","path":["Chapter 7.   Trees","7.7   Exercises"],"tags":[]},{"location":"chapter_tree/summary/","level":1,"title":"7.6   Summary","text":"","path":["Chapter 7.   Trees","7.6   Summary"],"tags":[]},{"location":"chapter_tree/summary/#1-key-takeaways","level":3,"title":"1.   Key takeaways","text":"<ul> <li>A binary tree is a nonlinear data structure that embodies the divide-and-conquer idea of splitting in two. Each node contains a value and two pointers, one to each child.</li> <li>A node's left or right child, together with everything below it, forms its left or right subtree.</li> <li>Tree terminology includes root, leaf, level, degree, edge, height, and depth.</li> <li>Initializing a binary tree and inserting or deleting nodes resemble linked-list operations.</li> <li>Common types include perfect, complete, full, and balanced binary trees. A perfect tree is the ideal form; a linked list is the worst degenerate form.</li> <li>A binary tree can be represented by an array of node values and empty positions in level order. Index mappings between parents and children replace pointers.</li> <li>Level-order traversal is breadth-first search: it expands outward one layer at a time and usually uses a queue.</li> <li>Preorder, inorder, and postorder traversal are depth-first search: they follow paths to their ends before backtracking and usually use recursion.</li> <li>A binary search tree supports efficient lookup, insertion, and deletion in \\(O(\\log n)\\) time. If it degenerates into a linked list, all these operations degrade to \\(O(n)\\).</li> <li>An AVL tree is a balanced binary search tree that uses rotations to maintain balance through insertions and deletions.</li> <li>AVL rotations include right rotation, left rotation, right-then-left rotation, and left-then-right rotation. After an insertion or deletion, rotations are performed from the bottom up to restore balance.</li> </ul>","path":["Chapter 7.   Trees","7.6   Summary"],"tags":[]},{"location":"chapter_tree/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: For a binary tree with only one node, are both the tree's height and the root's depth \\(0\\)?</p> <p>Yes. Height and depth are usually defined by the number of edges traversed.</p> <p>Q: The text says binary-tree insertion and deletion usually involve a sequence of coordinated operations. What does that mean? Does it refer to releasing resources held by child nodes?</p> <p>For example, deleting a node from a binary search tree has three cases, and each case requires several steps involving nodes.</p> <p>Q: Why does depth-first traversal have preorder, inorder, and postorder variants? What is each used for?</p> <p>Just as an array can be traversed forward or backward, these are three ways to visit a binary tree and obtain a particular ordering. In a binary search tree, values satisfy <code>left child value &lt; root value &lt; right child value</code>, so visiting “left \\(\\rightarrow\\) root \\(\\rightarrow\\) right” produces a sorted node sequence.</p> <p>Q: A right rotation adjusts <code>node</code>, <code>child</code>, and <code>grand_child</code>. What happens to the link between <code>node</code> and its parent? Doesn't the rotation break it?</p> <p>View the operation recursively. <code>right_rotate(root)</code> receives a subtree's root and ends with <code>return child</code>, returning the new root of that subtree. The connection to its parent is updated after the function returns; it is outside the rotation function's responsibility.</p> <p>Q: In C++, how do you decide which methods are <code>private</code> and which are <code>public</code>? Why is <code>height()</code> public but <code>updateHeight()</code> private?</p> <p>The main consideration is where the method is used. A method used only within the class should be <code>private</code>. Calling <code>updateHeight()</code> alone is not meaningful to a user; it is just one step of insertion or deletion. In contrast, <code>height()</code> retrieves a node's height, much like <code>vector.size()</code>, so it is <code>public</code> for convenient access.</p> <p>Q: How do you build a binary search tree from input data? Is choosing the root important?</p> <p>Yes. The construction is provided by the <code>build_tree()</code> method in the binary-search-tree code. Usually, we sort the input, choose its middle element as the root, and recursively construct the left and right subtrees. This keeps the tree as balanced as possible.</p> <p>Q: Must strings in Java always be compared with <code>equals()</code>?</p> <p>For primitive types in Java, <code>==</code> compares variable values. For reference types, the two forms work differently.</p> <ul> <li><code>==</code> compares whether two variables refer to the same object, meaning the same location in memory.</li> <li><code>equals()</code> compares whether two objects have equal values.</li> </ul> <p>Use <code>equals()</code> when comparing values. However, strings initialized with <code>String a = \"hi\"; String b = \"hi\";</code> are stored in the string pool and refer to the same object, so <code>a == b</code> also works for these strings.</p> <p>Q: Just before breadth-first traversal reaches the lowest level, does the queue contain \\(2^h\\) nodes?</p> <p>Yes. For example, a perfect binary tree of height \\(h = 2\\) has \\(n = 7\\) nodes, with \\(4 = 2^h = (n + 1) / 2\\) nodes on its lowest level.</p>","path":["Chapter 7.   Trees","7.6   Summary"],"tags":[]}]}
\ No newline at end of file
diff --git a/en/sitemap.xml b/en/sitemap.xml
index 06eda04b9..7075b27de 100644
--- a/en/sitemap.xml
+++ b/en/sitemap.xml
@@ -354,4 +354,7 @@
       <url>
         <loc>https://www.hello-algo.com/en/chapter_reference/</loc>
       </url>
+      <url>
+        <loc>https://www.hello-algo.com/en/chapter_paperbook/</loc>
+      </url>
 </urlset>
\ No newline at end of file
diff --git a/en/stylesheets/animation_player.css b/en/stylesheets/animation_player.css
index 9c1d18f06..c7e789263 100644
--- a/en/stylesheets/animation_player.css
+++ b/en/stylesheets/animation_player.css
@@ -176,4 +176,4 @@
     font-size: 0.7rem;
   }
 }
-/*! update cache: 20261029171308 */
+/*! update cache: 20261030193205 */
diff --git a/en/stylesheets/extra.css b/en/stylesheets/extra.css
index d5c179c5a..a1b5d751b 100644
--- a/en/stylesheets/extra.css
+++ b/en/stylesheets/extra.css
@@ -477,10 +477,75 @@ html:has(body[data-md-color-scheme="default"]) {
   font-weight: normal;
 }
 
+/* Keep long English chapter labels inside the hero at tablet/desktop widths. */
 .hero-div--en {
+  width: min(112vh, 76vw);
+  height: min(84vh, 57vw);
   font-size: min(1.65vh, 2.3vw);
 }
 
+.hero-div--en > a > span {
+  z-index: 1;
+}
+
+/* The illustrated chapter map becomes readable touch targets on small screens. */
+@media screen and (max-width: 600px) {
+  .home-div:has(> .hero-div--en) {
+    display: block;
+    height: auto !important;
+    margin-top: 0 !important;
+    padding: 0.5rem 0 1rem !important;
+  }
+
+  .home-div > .hero-div--en {
+    display: grid;
+    grid-template-columns: repeat(2, minmax(0, 1fr));
+    gap: 0.35rem 0.5rem;
+    width: calc(100% - 1.5rem);
+    height: auto;
+    margin: 0.5rem auto 1rem;
+  }
+
+  .home-div > .hero-div--en > img {
+    display: none;
+  }
+
+  .home-div > .hero-div--en > a {
+    display: flex;
+    align-items: center;
+    gap: 0.4rem;
+    min-width: 0;
+    min-height: 48px;
+    padding: 0.3rem 0.15rem;
+    text-align: left;
+  }
+
+  .home-div > .hero-div--en > a > img {
+    position: static;
+    flex-shrink: 0;
+    width: 32px !important;
+    height: 32px !important;
+    object-fit: contain;
+  }
+
+  .home-div > .hero-div--en > a > span {
+    position: static;
+    min-width: 0;
+    transform: none;
+    white-space: normal;
+    font-size: 13px;
+    line-height: 1.3;
+  }
+
+  .home-div > .hero-div--en + .heading-div {
+    position: relative;
+    left: auto;
+    bottom: auto;
+    transform: none;
+    padding: 0 0.75rem;
+  }
+}
+
 .hero-div--ru {
   font-size: min(1.45vh, 2vw);
 }
@@ -931,4 +996,4 @@ a:hover .device-on-hover {
     max-width: 100%;
   }
 }
-/*! update cache: 20261029171308 */
+/*! update cache: 20261030193205 */
diff --git a/en/stylesheets/giscus-dark.css b/en/stylesheets/giscus-dark.css
index 9f9bd00ac..81590aebf 100644
--- a/en/stylesheets/giscus-dark.css
+++ b/en/stylesheets/giscus-dark.css
@@ -122,4 +122,4 @@ main .gsc-loading-image {
 .gsc-reply-content::-webkit-scrollbar-track {
   background: transparent;
 }
-/*! update cache: 20261029171308 */
+/*! update cache: 20261030193205 */
diff --git a/en/stylesheets/giscus-light.css b/en/stylesheets/giscus-light.css
index c31eca3ca..8966f19a2 100644
--- a/en/stylesheets/giscus-light.css
+++ b/en/stylesheets/giscus-light.css
@@ -153,4 +153,4 @@ main {
 .gsc-reply-content::-webkit-scrollbar-track {
   background: transparent;
 }
-/*! update cache: 20261029171308 */
+/*! update cache: 20261030193205 */
diff --git a/ja/javascripts/animation_player.js b/ja/javascripts/animation_player.js
index 8930fb753..588983256 100644
--- a/ja/javascripts/animation_player.js
+++ b/ja/javascripts/animation_player.js
@@ -301,4 +301,4 @@
     initAutoSlide();
   }
 })();
-/*! update cache: 20261029171338 */
+/*! update cache: 20261030193234 */
diff --git a/ja/javascripts/katex.js b/ja/javascripts/katex.js
index 1b22cce08..9c2faef50 100644
--- a/ja/javascripts/katex.js
+++ b/ja/javascripts/katex.js
@@ -8,4 +8,4 @@ document$.subscribe(({ body }) => {
     ],
   });
 });
-/*! update cache: 20261029171338 */
+/*! update cache: 20261030193234 */
diff --git a/ja/javascripts/mathjax.js b/ja/javascripts/mathjax.js
index aae8477a3..9e6c50ea0 100644
--- a/ja/javascripts/mathjax.js
+++ b/ja/javascripts/mathjax.js
@@ -15,4 +15,4 @@ window.MathJax = {
 document$.subscribe(() => {
   MathJax.typesetPromise();
 });
-/*! update cache: 20261029171338 */
+/*! update cache: 20261030193234 */
diff --git a/ja/javascripts/starfield.js b/ja/javascripts/starfield.js
index 5edc1dd09..63c30eff6 100644
--- a/ja/javascripts/starfield.js
+++ b/ja/javascripts/starfield.js
@@ -469,4 +469,4 @@
 
   return Starfield;
 });
-/*! update cache: 20261029171338 */
+/*! update cache: 20261030193234 */
diff --git a/ja/stylesheets/animation_player.css b/ja/stylesheets/animation_player.css
index 9361f7020..c107f3bce 100644
--- a/ja/stylesheets/animation_player.css
+++ b/ja/stylesheets/animation_player.css
@@ -176,4 +176,4 @@
     font-size: 0.7rem;
   }
 }
-/*! update cache: 20261029171338 */
+/*! update cache: 20261030193234 */
diff --git a/ja/stylesheets/extra.css b/ja/stylesheets/extra.css
index b39f9c814..6c9458382 100644
--- a/ja/stylesheets/extra.css
+++ b/ja/stylesheets/extra.css
@@ -477,10 +477,75 @@ html:has(body[data-md-color-scheme="default"]) {
   font-weight: normal;
 }
 
+/* Keep long English chapter labels inside the hero at tablet/desktop widths. */
 .hero-div--en {
+  width: min(112vh, 76vw);
+  height: min(84vh, 57vw);
   font-size: min(1.65vh, 2.3vw);
 }
 
+.hero-div--en > a > span {
+  z-index: 1;
+}
+
+/* The illustrated chapter map becomes readable touch targets on small screens. */
+@media screen and (max-width: 600px) {
+  .home-div:has(> .hero-div--en) {
+    display: block;
+    height: auto !important;
+    margin-top: 0 !important;
+    padding: 0.5rem 0 1rem !important;
+  }
+
+  .home-div > .hero-div--en {
+    display: grid;
+    grid-template-columns: repeat(2, minmax(0, 1fr));
+    gap: 0.35rem 0.5rem;
+    width: calc(100% - 1.5rem);
+    height: auto;
+    margin: 0.5rem auto 1rem;
+  }
+
+  .home-div > .hero-div--en > img {
+    display: none;
+  }
+
+  .home-div > .hero-div--en > a {
+    display: flex;
+    align-items: center;
+    gap: 0.4rem;
+    min-width: 0;
+    min-height: 48px;
+    padding: 0.3rem 0.15rem;
+    text-align: left;
+  }
+
+  .home-div > .hero-div--en > a > img {
+    position: static;
+    flex-shrink: 0;
+    width: 32px !important;
+    height: 32px !important;
+    object-fit: contain;
+  }
+
+  .home-div > .hero-div--en > a > span {
+    position: static;
+    min-width: 0;
+    transform: none;
+    white-space: normal;
+    font-size: 13px;
+    line-height: 1.3;
+  }
+
+  .home-div > .hero-div--en + .heading-div {
+    position: relative;
+    left: auto;
+    bottom: auto;
+    transform: none;
+    padding: 0 0.75rem;
+  }
+}
+
 .hero-div--ru {
   font-size: min(1.45vh, 2vw);
 }
@@ -931,4 +996,4 @@ a:hover .device-on-hover {
     max-width: 100%;
   }
 }
-/*! update cache: 20261029171338 */
+/*! update cache: 20261030193234 */
diff --git a/ja/stylesheets/giscus-dark.css b/ja/stylesheets/giscus-dark.css
index 38794d0db..001572ff2 100644
--- a/ja/stylesheets/giscus-dark.css
+++ b/ja/stylesheets/giscus-dark.css
@@ -122,4 +122,4 @@ main .gsc-loading-image {
 .gsc-reply-content::-webkit-scrollbar-track {
   background: transparent;
 }
-/*! update cache: 20261029171338 */
+/*! update cache: 20261030193234 */
diff --git a/ja/stylesheets/giscus-light.css b/ja/stylesheets/giscus-light.css
index 0541d77fb..ab90b5f08 100644
--- a/ja/stylesheets/giscus-light.css
+++ b/ja/stylesheets/giscus-light.css
@@ -153,4 +153,4 @@ main {
 .gsc-reply-content::-webkit-scrollbar-track {
   background: transparent;
 }
-/*! update cache: 20261029171338 */
+/*! update cache: 20261030193234 */
diff --git a/javascripts/animation_player.js b/javascripts/animation_player.js
index c049ba8f8..9dfd45096 100644
--- a/javascripts/animation_player.js
+++ b/javascripts/animation_player.js
@@ -301,4 +301,4 @@
     initAutoSlide();
   }
 })();
-/*! update cache: 20261029171211 */
+/*! update cache: 20261030193105 */
diff --git a/javascripts/katex.js b/javascripts/katex.js
index 909025b83..7b317d3c8 100644
--- a/javascripts/katex.js
+++ b/javascripts/katex.js
@@ -8,4 +8,4 @@ document$.subscribe(({ body }) => {
     ],
   });
 });
-/*! update cache: 20261029171211 */
+/*! update cache: 20261030193105 */
diff --git a/javascripts/mathjax.js b/javascripts/mathjax.js
index 3f7cbfda1..47aeb5d02 100644
--- a/javascripts/mathjax.js
+++ b/javascripts/mathjax.js
@@ -15,4 +15,4 @@ window.MathJax = {
 document$.subscribe(() => {
   MathJax.typesetPromise();
 });
-/*! update cache: 20261029171211 */
+/*! update cache: 20261030193105 */
diff --git a/javascripts/starfield.js b/javascripts/starfield.js
index 5f7e7f40a..4c45f9187 100644
--- a/javascripts/starfield.js
+++ b/javascripts/starfield.js
@@ -469,4 +469,4 @@
 
   return Starfield;
 });
-/*! update cache: 20261029171211 */
+/*! update cache: 20261030193105 */
diff --git a/ko/chapter_hello_algo/index.html b/ko/chapter_hello_algo/index.html
index fb0efda57..b91f2e269 100644
--- a/ko/chapter_hello_algo/index.html
+++ b/ko/chapter_hello_algo/index.html
@@ -4543,9 +4543,9 @@
 
 <!-- Page content -->
 <h1 id="_1">머리말<a class="headerlink" href="#_1" title="고유 링크">&para;</a></h1>
-<p>몇 년 전, 나는 리트코드에 『剑指 Offer』의 문제 풀이 시리즈를 공유했고 많은 독자의 격려와 지지를 받았다. 독자들과 이야기하면서 가장 자주 받은 질문은 “알고리즘 공부를 어떻게 시작하면 좋을까요?”였다. 점차 나는 이 질문에 깊은 관심을 갖게 되었다.</p>
-<p>아무것도 모르는 상태에서 무작정 문제를 푸는 방식이 가장 인기 있는 방법인 듯하다. 간단하고 직접적이며 효과도 있다. 하지만 문제 풀이는 ‘지뢰 찾기’ 게임과 비슷하다. 스스로 공부하는 능력이 뛰어난 사람은 지뢰를 하나씩 무사히 제거하지만, 기초가 부족한 사람은 지뢰에 거듭 부딪혀 좌절하며 점점 물러서기 쉽다. 교재를 처음부터 끝까지 읽는 것도 흔한 방법이다. 그러나 취업을 준비하는 사람에게는 졸업 논문 작성, 이력서 제출, 필기시험과 면접 준비만으로도 대부분의 에너지가 소모되므로, 두꺼운 책을 파고드는 일이 큰 도전이 되곤 한다.</p>
-<p>비슷한 어려움을 겪고 있다면, 이 책이 여러분을 ‘찾아온’ 것은 다행스러운 일이다. 이 책은 그 질문에 대한 나의 답이다. 최적의 답은 아닐지라도, 적어도 적극적인 시도는 될 것이다. 이 책만으로 곧바로 취업에 성공하기는 어렵겠지만, 자료 구조와 알고리즘의 ‘지식 지도’를 탐험하도록 안내하고 여러 ‘지뢰’의 모양, 크기, 분포를 알려 주며 다양한 ‘지뢰 제거 방법’을 익히도록 도울 것이다. 이러한 능력을 갖추면 문제를 풀고 문헌을 읽는 일이 한결 수월해지고, 차츰 완전한 지식 체계를 세울 수 있을 것이다.</p>
+<p>몇 년 전, 나는 리트코드에 『剑指 Offer』(IT 대기업 코딩 면접 대비서)의 문제 풀이 시리즈를 공유했고 많은 독자의 격려와 지지를 받았다. 독자들과 이야기하면서 가장 자주 받은 질문은 “알고리즘 공부를 어떻게 시작하면 좋을까요?”였다. 점차 나는 이 질문에 깊은 관심을 갖게 되었다.</p>
+<p>아무것도 모르는 상태에서 무작정 문제를 푸는 방식이 가장 인기 있는 방법인 듯하다. 간단하고 직접적이며 효과도 있다. 하지만 문제 풀이는 ‘지뢰 찾기’ 게임과 비슷하다. 스스로 학습하는 능력이 뛰어난 사람은 지뢰를 하나씩 찾아 제거하며 앞으로 나아가지만, 기초가 부족한 사람은 지뢰를 거듭 밟으며 좌절하고, 결국 앞으로 나아가지 못한 채 주저앉기 쉽다. 알고리즘 교재를 처음부터 끝까지 읽는 것도 흔한 방법이다. 그러나 취업을 준비하는 사람에게는 졸업 논문 작성, 이력서 제출, 필기시험과 면접 준비만으로도 대부분의 에너지가 소모되므로, 두꺼운 책을 파고드는 일이 큰 도전이 되곤 한다.</p>
+<p>비슷한 어려움을 겪고 있다면, 다행히도 이 책이 여러분을 '찾아왔다'고 말할 수 있을 것이다. 이 책은 앞서 던진 질문에 대한 나의 답이다. 최선의 해답은 아닐지라도, 적어도 앞으로 나아가는 데 도움이 되는 시도라고 생각한다. 물론 이 책 한 권만으로 곧바로 취업에 성공할 수는 없을 것이다. 하지만 자료 구조와 알고리즘의 '지식 지도'를 탐험하도록 안내하고, 여러 ‘지뢰’의 모양과 크기, 분포를 이해하며 다양한 ‘지뢰 제거 방법’을 익힐 수 있도록 도울 것이다. 이를 바탕으로 알고리즘 문제를 풀거나 기술 문헌을 읽는 일이 한층 수월해지고, 차근차근 체계적인 지식을 쌓아 나갈 수 있으리라 믿는다.</p>
 <p>나는 파인만 교수의 말에 깊이 공감한다. “지식은 공짜가 아니다. 주의를 기울여야 한다.” 이런 의미에서 이 책은 완전히 ‘무료’인 것은 아니다. 여러분이 이 책에 쏟는 소중한 ‘주의와 관심’을 헛되게 하지 않도록, 나 역시 최대한의 ‘주의와 관심’을 기울여 집필에 최선을 다하고자 한다.</p>
 <p>내 학식과 재능이 부족함을 잘 알고 있다. 책의 내용을 한동안 다듬어 왔지만 여전히 많은 오류가 있을 것이다. 선생님과 학생 여러분의 비판과 지적을 부탁드린다.</p>
 <p><img alt="Hello 알고리즘" class="cover-image" src="../assets/covers/chapter_hello_algo.jpg" /></p>
@@ -4554,7 +4554,7 @@
 </div>
 
 <p>컴퓨터의 등장은 세상을 크게 바꾸었다. 빠른 계산 능력과 뛰어난 프로그래밍 가능성을 갖춘 컴퓨터는 알고리즘을 실행하고 데이터를 처리하는 이상적인 수단이 되었다. 비디오 게임의 사실적인 화면, 자율 주행의 지능적인 판단, AlphaGo의 멋진 대국, ChatGPT의 자연스러운 상호작용까지, 이 모든 응용은 컴퓨터 위에서 알고리즘이 정교하게 구현된 결과다.</p>
-<p>사실 컴퓨터가 등장하기 전부터 알고리즘과 자료 구조는 세상 곳곳에 존재했다. 초기의 알고리즘은 고대의 셈법이나 도구 제작 절차처럼 비교적 단순했다. 문명이 발전하면서 알고리즘도 점점 더 정교하고 복잡해졌다. 장인의 놀라운 솜씨에서 생산력을 높여 준 산업 제품에 이르기까지, 나아가 우주의 운행을 설명하는 과학 법칙에 이르기까지, 평범하거나 경이로운 거의 모든 것의 이면에는 정교한 알고리즘적 사고가 숨어 있다.</p>
+<p>사실 컴퓨터가 등장하기 전부터 알고리즘과 자료 구조는 세상 곳곳에 존재했다. 초기의 알고리즘은 고대의 셈법이나 도구 제작 절차처럼 비교적 단순했다. 문명이 발전하면서 알고리즘도 점점 더 정교하고 복잡해졌다. 장인의 놀라운 솜씨에서 생산력을 높여 준 산업 제품에 이르기까지, 나아가 우주의 다양한 현상을 설명하는 과학 법칙에 이르기까지, 평범하거나 경이로운 거의 모든 것의 이면에는 정교한 알고리즘적 사고가 숨어 있다.</p>
 <p>자료 구조 역시 어디에나 있다. 크게는 사회 관계망부터 작게는 지하철 노선까지 많은 시스템을 ‘그래프’로 모델링할 수 있다. 국가부터 가정에 이르기까지 사회의 주요 조직 형태는 ‘트리’의 특성을 보인다. 겨울옷은 ‘스택’과 같아서 가장 먼저 입은 옷을 가장 나중에 벗는다. 셔틀콕 통은 한쪽으로 넣고 다른 쪽으로 꺼낸다는 점에서 ‘큐’와 닮았다. 사전은 원하는 표제어를 빠르게 찾을 수 있다는 점에서 ‘해시 테이블’과 비슷하다.</p>
 <p>이 책은 명확하고 이해하기 쉬운 애니메이션 그림과 실행 가능한 코드 예제를 통해 독자가 알고리즘과 자료 구조의 핵심 개념을 이해하고 직접 프로그래밍으로 구현할 수 있도록 돕는 것을 목표로 한다. 이를 바탕으로 복잡한 세상 속에서 알고리즘이 생생하게 드러나는 모습을 보여 주며 알고리즘의 아름다움을 전하고자 한다. 이 책이 여러분에게 도움이 되기를 바란다!</p>
 
diff --git a/ko/javascripts/animation_player.js b/ko/javascripts/animation_player.js
index ed65c012d..4c395737e 100644
--- a/ko/javascripts/animation_player.js
+++ b/ko/javascripts/animation_player.js
@@ -301,4 +301,4 @@
     initAutoSlide();
   }
 })();
-/*! update cache: 20261029171440 */
+/*! update cache: 20261030193337 */
diff --git a/ko/javascripts/katex.js b/ko/javascripts/katex.js
index 0bb69e2a1..8537df67e 100644
--- a/ko/javascripts/katex.js
+++ b/ko/javascripts/katex.js
@@ -8,4 +8,4 @@ document$.subscribe(({ body }) => {
     ],
   });
 });
-/*! update cache: 20261029171440 */
+/*! update cache: 20261030193337 */
diff --git a/ko/javascripts/mathjax.js b/ko/javascripts/mathjax.js
index 68c0b2167..59a33e0dd 100644
--- a/ko/javascripts/mathjax.js
+++ b/ko/javascripts/mathjax.js
@@ -15,4 +15,4 @@ window.MathJax = {
 document$.subscribe(() => {
   MathJax.typesetPromise();
 });
-/*! update cache: 20261029171440 */
+/*! update cache: 20261030193337 */
diff --git a/ko/javascripts/starfield.js b/ko/javascripts/starfield.js
index 9befcd956..2d09c0fd9 100644
--- a/ko/javascripts/starfield.js
+++ b/ko/javascripts/starfield.js
@@ -469,4 +469,4 @@
 
   return Starfield;
 });
-/*! update cache: 20261029171440 */
+/*! update cache: 20261030193337 */
diff --git a/ko/search.json b/ko/search.json
index 920c85e99..3f6a53562 100644
--- a/ko/search.json
+++ b/ko/search.json
@@ -1 +1 @@
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VS Code 공식 웹사이트에서 운영체제에 맞는 버전을 내려받아 설치한다.</p> <p></p> <p> 그림 16-1   공식 웹사이트에서 VS Code 내려받기 </p> <p>VS Code는 풍부한 확장 생태계를 갖추고 있어 대부분의 프로그래밍 언어를 실행하고 디버깅할 수 있다. Python을 예로 들면, ‘Python Extension Pack’을 설치한 뒤 Python 코드를 디버깅할 수 있다. 설치 과정은 아래 그림(16-2)과 같다.</p> <p></p> <p> 그림 16-2   VS Code 확장 설치 </p>","path":["제 16 장   부록","16.1   프로그래밍 환경 설치"],"tags":[]},{"location":"chapter_appendix/installation/#1612","level":2,"title":"16.1.2   언어 실행 환경 설치","text":"","path":["제 16 장   부록","16.1   프로그래밍 환경 설치"],"tags":[]},{"location":"chapter_appendix/installation/#1-python","level":3,"title":"1.   Python 환경","text":"<ol> <li>Miniconda3를 내려받아 설치한다. 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설치"],"tags":[]},{"location":"chapter_appendix/installation/#7-javascript","level":3,"title":"7.   JavaScript 환경","text":"<ol> <li>Node.js를 내려받아 설치한다.</li> <li>(선택 사항) VS Code 확장 마켓플레이스에서 <code>Prettier</code>를 검색하여 코드 포매터를 설치한다.</li> </ol>","path":["제 16 장   부록","16.1   프로그래밍 환경 설치"],"tags":[]},{"location":"chapter_appendix/installation/#8-typescript","level":3,"title":"8.   TypeScript 환경","text":"<ol> <li>JavaScript 환경과 같은 순서로 설치한다.</li> <li>TypeScript Execute (tsx)를 설치한다.</li> <li>VS Code 확장 마켓플레이스에서 <code>typescript</code>를 검색하여 Pretty TypeScript Errors를 설치한다.</li> </ol>","path":["제 16 장   부록","16.1   프로그래밍 환경 설치"],"tags":[]},{"location":"chapter_appendix/installation/#9-dart","level":3,"title":"9.   Dart 환경","text":"<ol> <li>Dart를 내려받아 설치한다.</li> <li>VS Code 확장 마켓플레이스에서 <code>dart</code>를 검색하여 Dart를 설치한다.</li> </ol>","path":["제 16 장   부록","16.1   프로그래밍 환경 설치"],"tags":[]},{"location":"chapter_appendix/installation/#10-rust","level":3,"title":"10.   Rust 환경","text":"<ol> <li>Rust를 내려받아 설치한다.</li> <li>VS Code 확장 마켓플레이스에서 <code>rust</code>를 검색하여 rust-analyzer를 설치한다.</li> </ol>","path":["제 16 장   부록","16.1   프로그래밍 환경 설치"],"tags":[]},{"location":"chapter_appendix/terminology/","level":1,"title":"16.3   용어집","text":"<p>아래 표(16-1)에는 이 책에 나오는 주요 용어를 정리했다. 영어 문헌을 읽을 때 도움이 되므로 각 용어의 영어 명칭을 기억해 두기를 권장한다.</p> <p> 표 16-1   자료 구조와 알고리즘의 주요 용어 </p> English 한국어 algorithm 알고리즘 data structure 자료 구조 code 코드 file 파일 function 함수 method 메서드 variable 변수 asymptotic complexity analysis 점근적 복잡도 분석 time complexity 시간 복잡도 space complexity 공간 복잡도 loop 반복문 iteration 반복 recursion 재귀 tail recursion 꼬리 재귀 recursion tree 재귀 트리 big-\\(O\\) notation 빅 \\(O\\) 표기법 asymptotic upper bound 점근적 상한 sign-magnitude 부호와 크기 표현 1’s complement 1의 보수 2’s complement 2의 보수 array 배열 index 인덱스 linked list 연결 리스트 linked list node, list node 연결 리스트 노드 head node 머리 노드 tail node 꼬리 노드 list 리스트 dynamic array 동적 배열 hard disk 하드 디스크 random-access memory (RAM) 메모리 cache memory 캐시 메모리 cache miss 캐시 미스 cache hit rate 캐시 적중률 stack 스택 top of the stack 스택의 맨 위 bottom of the stack 스택의 맨 아래 queue 큐 double-ended queue 덱 front of the queue 큐의 앞 rear of the queue 큐의 뒤 hash table 해시 테이블 hash set 해시 집합 bucket 버킷 hash function 해시 함수 hash collision 해시 충돌 load factor 적재율 separate chaining 체이닝 open addressing 개방 주소법 linear probing 선형 탐사 lazy deletion 지연 삭제 binary tree 이진 트리 tree node 트리 노드 left-child node 왼쪽 자식 노드 right-child node 오른쪽 자식 노드 parent node 부모 노드 left subtree 왼쪽 서브트리 right subtree 오른쪽 서브트리 root node 루트 노드 leaf node 리프 노드 edge 간선 level 레벨 degree 차수 height 높이 depth 깊이 perfect binary tree 포화 이진 트리 complete binary tree 완전 이진 트리 full binary tree 정 이진 트리 balanced binary tree 균형 이진 트리 binary search tree 이진 탐색 트리 AVL tree AVL 트리 red-black tree 레드-블랙 트리 level-order traversal 레벨 순서 순회 breadth-first traversal 너비 우선 순회 depth-first traversal 깊이 우선 순회 pre-order traversal 전위 순회 in-order traversal 중위 순회 post-order traversal 후위 순회 balanced binary search tree 균형 이진 탐색 트리 balance factor 균형 인수 heap 힙 max heap 최대 힙 min heap 최소 힙 priority queue 우선순위 큐 heapify 힙 재정렬 top-\\(k\\) problem Top-\\(k\\) 문제 graph 그래프 vertex 정점 undirected graph 무방향 그래프 directed graph 방향 그래프 connected graph 연결 그래프 disconnected graph 비연결 그래프 weighted graph 가중 그래프 adjacency 인접 path 경로 in-degree 진입 차수 out-degree 진출 차수 adjacency matrix 인접 행렬 adjacency list 인접 리스트 breadth-first search 너비 우선 탐색 depth-first search 깊이 우선 탐색 binary search 이진 탐색 searching algorithm 탐색 알고리즘 sorting algorithm 정렬 알고리즘 selection sort 선택 정렬 bubble sort 버블 정렬 insertion sort 삽입 정렬 quick sort 퀵 정렬 merge sort 병합 정렬 heap sort 힙 정렬 bucket sort 버킷 정렬 counting sort 계수 정렬 radix sort 기수 정렬 divide and conquer 분할 정복 hanota problem 하노이의 탑 문제 backtracking algorithm 백트래킹 알고리즘 constraint 제약 조건 solution 해 state 상태 pruning 가지치기 permutations problem 모든 순열 구하기 subset-sum problem 부분 집합의 합 문제 \\(n\\)-queens problem \\(n\\)-퀸 문제 dynamic programming 동적 계획법 initial state 초기 상태 state-transition equation 상태 전이 식 knapsack problem 배낭 문제 edit distance problem 편집 거리 문제 greedy algorithm 탐욕 알고리즘","path":["제 16 장   부록","16.3   용어집"],"tags":[]},{"location":"chapter_array_and_linkedlist/","level":1,"title":"제 4 장   배열과 연결 리스트","text":"<p>이 장을 시작하며</p> <p>자료 구조의 세계는 두꺼운 벽돌벽과 같다.</p> <p>배열의 벽돌은 가지런히 빈틈없이 붙어 있다. 연결 리스트의 벽돌은 곳곳에 흩어져 있으며, 이들을 잇는 덩굴이 벽돌 틈 사이를 자유롭게 누빈다.</p>","path":["제 4 장   배열과 연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>4.1   배열</li> <li>4.2   연결 리스트</li> <li>4.3   리스트</li> <li>4.4   메모리와 캐시 *</li> <li>4.5   요약</li> <li>4.6   연습 문제</li> </ul>","path":["제 4 장   배열과 연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/","level":1,"title":"4.1   배열","text":"<p>배열(array)은 같은 자료형의 원소를 연속된 메모리 공간에 저장하는 선형 자료 구조다. 배열에서 원소가 있는 위치를 해당 원소의 인덱스(index)라고 한다. 아래 그림(4-1)은 배열의 주요 개념과 저장 방식을 보여 준다.</p> <p></p> <p> 그림 4-1   배열의 정의와 저장 방식 </p>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#411","level":2,"title":"4.1.1   배열의 기본 연산","text":"","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#1","level":3,"title":"1.   배열 초기화","text":"<p>필요에 따라 초깃값을 지정하지 않거나 지정하는 두 가지 초기화 방식을 사용할 수 있다. 초깃값을 지정하지 않으면 대부분의 프로그래밍 언어는 배열 원소를 \\(0\\)으로 초기화한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code># 배열 초기화\narr: list[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> array.cpp<pre><code>/* 배열 초기화 */\n// 스택에 저장\nint arr[5] = {};\nint nums[5] = { 1, 3, 2, 5, 4 };\n// 힙에 저장(공간을 직접 해제해야 함)\nint* arr1 = new int[5] {};\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n</code></pre> array.java<pre><code>/* 배열 초기화 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n</code></pre> array.cs<pre><code>/* 배열 초기화 */\nint[] arr = new int[5]; // [ 0, 0, 0, 0, 0 ]\nint[] nums = [1, 3, 2, 5, 4];\n</code></pre> array.go<pre><code>/* 배열 초기화 */\nvar arr [5]int\n// Go에서 길이를 지정하면([5]int) 배열이고, 지정하지 않으면([]int) 슬라이스다\n// Go 배열의 길이는 컴파일 시 결정되도록 설계되었으므로 상수로만 길이를 지정할 수 있다\n// 용량 확장 메서드 extend()를 쉽게 구현하기 위해 아래에서는 슬라이스(Slice)를 배열(Array)로 간주한다\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> array.swift<pre><code>/* 배열 초기화 */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n</code></pre> array.js<pre><code>/* 배열 초기화 */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n</code></pre> array.ts<pre><code>/* 배열 초기화 */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> array.dart<pre><code>/* 배열 초기화 */\nList&lt;int&gt; arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> array.rs<pre><code>/* 배열 초기화 */\nlet arr: [i32; 5] = [0; 5]; // [0, 0, 0, 0, 0]\nlet slice: &amp;[i32] = &amp;[0; 5];\n// Rust에서 길이를 지정하면([i32; 5]) 배열이고, 지정하지 않으면(&amp;[i32]) 슬라이스다\n// Rust 배열의 길이는 컴파일 시 결정되도록 설계되었으므로 상수로만 길이를 지정할 수 있다\n// Vector는 Rust에서 일반적으로 동적 배열로 사용하는 자료형이다\n// 용량 확장 메서드 extend()를 쉽게 구현하기 위해 아래에서는 vector를 배열(array)로 간주한다\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> array.c<pre><code>/* 배열 초기화 */\nint arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n</code></pre> array.kt<pre><code>/* 배열 초기화 */\nvar arr = IntArray(5) // { 0, 0, 0, 0, 0 }\nvar nums = intArrayOf(1, 3, 2, 5, 4)\n</code></pre> array.rb<pre><code># 배열 초기화\narr = Array.new(5, 0)\nnums = [1, 3, 2, 5, 4]\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#2","level":3,"title":"2.   원소 접근","text":"<p>배열 원소는 연속된 메모리 공간에 저장되므로 메모리 주소를 매우 쉽게 계산할 수 있다. 배열의 메모리 주소(첫 원소의 메모리 주소)와 원소의 인덱스가 주어지면, 아래 그림(4-2)의 식으로 해당 원소의 메모리 주소를 구하여 직접 접근할 수 있다.</p> <p></p> <p> 그림 4-2   배열 원소의 메모리 주소 계산 </p> <p>위 그림(4-2)에서 배열의 첫 원소 인덱스는 \\(0\\)이다. \\(1\\)부터 세는 편이 더 자연스러워 보이므로 직관에 어긋날 수 있다. 그러나 주소 계산식의 관점에서 인덱스는 본질적으로 메모리 주소의 오프셋이다. 첫 원소의 주소 오프셋은 \\(0\\)이므로 인덱스가 \\(0\\)인 것은 합리적이다.</p> <p>배열의 원소 접근은 매우 효율적이며, 어떤 원소든 \\(O(1)\\) 시간에 임의 접근할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def random_access(nums: list[int]) -&gt; int:\n    \"\"\"원소에 임의 접근\"\"\"\n    # 구간 [0, len(nums)-1]에서 숫자 하나를 무작위로 뽑는다\n    random_index = random.randint(0, len(nums) - 1)\n    # 무작위 원소를 가져와 반환한다\n    random_num = nums[random_index]\n    return random_num\n</code></pre> array.cpp<pre><code>/* 원소에 임의 접근 */\nint randomAccess(int *nums, int size) {\n    // 구간 [0, size)에서 무작위로 수 하나 선택\n    int randomIndex = rand() % size;\n    // 무작위 원소를 가져와 반환한다\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.java<pre><code>/* 원소에 임의 접근 */\nint randomAccess(int[] nums) {\n    // 구간 [0, nums.length)에서 숫자 하나를 무작위로 뽑는다\n    int randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n    // 무작위 원소를 가져와 반환한다\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.cs<pre><code>/* 원소에 임의 접근 */\nint RandomAccess(int[] nums) {\n    Random random = new();\n    // 구간 [0, nums.Length)에서 무작위로 수 하나 선택\n    int randomIndex = random.Next(nums.Length);\n    // 무작위 원소를 가져와 반환한다\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.go<pre><code>/* 원소에 임의 접근 */\nfunc randomAccess(nums []int) (randomNum int) {\n    // 구간 [0, nums.length)에서 숫자 하나를 무작위로 뽑는다\n    randomIndex := rand.Intn(len(nums))\n    // 무작위 원소를 가져와 반환한다\n    randomNum = nums[randomIndex]\n    return\n}\n</code></pre> array.swift<pre><code>/* 원소에 임의 접근 */\nfunc randomAccess(nums: [Int]) -&gt; Int {\n    // 구간 [0, nums.count)에서 숫자 하나를 무작위로 뽑는다\n    let randomIndex = nums.indices.randomElement()!\n    // 무작위 원소를 가져와 반환한다\n    let randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.js<pre><code>/* 원소에 임의 접근 */\nfunction randomAccess(nums) {\n    // 구간 [0, nums.length)에서 숫자 하나를 무작위로 뽑는다\n    const random_index = Math.floor(Math.random() * nums.length);\n    // 무작위 원소를 가져와 반환한다\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.ts<pre><code>/* 원소에 임의 접근 */\nfunction randomAccess(nums: number[]): number {\n    // 구간 [0, nums.length)에서 숫자 하나를 무작위로 뽑는다\n    const random_index = Math.floor(Math.random() * nums.length);\n    // 무작위 원소를 가져와 반환한다\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.dart<pre><code>/* 원소에 임의 접근 */\nint randomAccess(List&lt;int&gt; nums) {\n  // 구간 [0, nums.length)에서 숫자 하나를 무작위로 뽑는다\n  int randomIndex = Random().nextInt(nums.length);\n  // 무작위 원소를 가져와 반환한다\n  int randomNum = nums[randomIndex];\n  return randomNum;\n}\n</code></pre> array.rs<pre><code>/* 원소에 임의 접근 */\nfn random_access(nums: &amp;[i32]) -&gt; i32 {\n    // 구간 [0, nums.len())에서 숫자 하나를 무작위로 뽑는다\n    let random_index = rand::thread_rng().gen_range(0..nums.len());\n    // 무작위 원소를 가져와 반환한다\n    let random_num = nums[random_index];\n    random_num\n}\n</code></pre> array.c<pre><code>/* 원소에 임의 접근 */\nint randomAccess(int *nums, int size) {\n    // 구간 [0, size)에서 무작위로 수 하나 선택\n    int randomIndex = rand() % size;\n    // 무작위 원소를 가져와 반환한다\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.kt<pre><code>/* 원소에 임의 접근 */\nfun randomAccess(nums: IntArray): Int {\n    // 구간 [0, nums.size)에서 숫자 하나를 무작위로 뽑는다\n    val randomIndex = ThreadLocalRandom.current().nextInt(0, nums.size)\n    // 무작위 원소를 가져와 반환한다\n    val randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.rb<pre><code>### 원소에 임의 접근 ###\ndef random_access(nums)\n  # 구간 [0, nums.length)에서 숫자 하나를 무작위로 뽑는다\n  random_index = Random.rand(0...nums.length)\n\n  # 무작위 원소를 가져와 반환한다\n  nums[random_index]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#3","level":3,"title":"3.   원소 삽입","text":"<p>배열 원소는 메모리에서 ‘빈틈없이 붙어 있어’ 사이에 다른 데이터를 넣을 공간이 없다. 아래 그림(4-3)과 같이 배열 중간에 원소를 삽입하려면 해당 위치 이후의 모든 원소를 한 칸씩 뒤로 옮긴 다음, 그 인덱스에 새 원소를 대입해야 한다.</p> <p></p> <p> 그림 4-3   배열에 원소를 삽입하는 예 </p> <p>배열의 길이는 고정되어 있으므로 원소를 하나 삽입하면 반드시 마지막 원소가 ‘사라진다’는 점에 주의하자. 이 문제의 해결책은 ‘리스트’ 절에서 다룬다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def insert(nums: list[int], num: int, index: int):\n    \"\"\"배열의 인덱스 index에 원소 num 삽입\"\"\"\n    # 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for i in range(len(nums) - 1, index, -1):\n        nums[i] = nums[i - 1]\n    # index 위치의 원소에 num을 대입한다\n    nums[index] = num\n</code></pre> array.cpp<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nvoid insert(int *nums, int size, int num, int index) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num;\n}\n</code></pre> array.java<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nvoid insert(int[] nums, int num, int index) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for (int i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num;\n}\n</code></pre> array.cs<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nvoid Insert(int[] nums, int num, int index) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for (int i = nums.Length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num;\n}\n</code></pre> array.go<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nfunc insert(nums []int, num int, index int) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for i := len(nums) - 1; i &gt; index; i-- {\n        nums[i] = nums[i-1]\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num\n}\n</code></pre> array.swift<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for i in stride(from: nums.count - 1, to: index, by: -1) {\n        nums[i] = nums[i - 1]\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num\n}\n</code></pre> array.js<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nfunction insert(nums, num, index) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num;\n}\n</code></pre> array.ts<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nfunction insert(nums: number[], num: number, index: number): void {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num;\n}\n</code></pre> array.dart<pre><code>/* 배열의 인덱스 index에 원소 _num 삽입 */\nvoid insert(List&lt;int&gt; nums, int _num, int index) {\n  // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n  for (var i = nums.length - 1; i &gt; index; i--) {\n    nums[i] = nums[i - 1];\n  }\n  // 인덱스 index의 원소에 _num을 대입한다\n  nums[index] = _num;\n}\n</code></pre> array.rs<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nfn insert(nums: &amp;mut [i32], num: i32, index: usize) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for i in (index + 1..nums.len()).rev() {\n        nums[i] = nums[i - 1];\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num;\n}\n</code></pre> array.c<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nvoid insert(int *nums, int size, int num, int index) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num;\n}\n</code></pre> array.kt<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nfun insert(nums: IntArray, num: Int, index: Int) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for (i in nums.size - 1 downTo index + 1) {\n        nums[i] = nums[i - 1]\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num\n}\n</code></pre> array.rb<pre><code>### 배열의 인덱스 index에 원소 num 삽입 ###\ndef insert(nums, num, index)\n  # 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n  for i in (nums.length - 1).downto(index + 1)\n    nums[i] = nums[i - 1]\n  end\n\n  # index 위치의 원소에 num을 대입한다\n  nums[index] = num\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#4","level":3,"title":"4.   원소 삭제","text":"<p>마찬가지로 아래 그림(4-4)과 같이 인덱스 \\(i\\)의 원소를 삭제하려면 인덱스 \\(i\\) 뒤의 원소를 모두 한 칸씩 앞으로 옮겨야 한다.</p> <p></p> <p> 그림 4-4   배열에서 원소를 삭제하는 예 </p> <p>삭제가 끝난 뒤 원래의 마지막 원소는 ‘의미 없는’ 값이 되므로 따로 수정할 필요가 없다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def remove(nums: list[int], index: int):\n    \"\"\"인덱스 index에 있는 원소 삭제\"\"\"\n    # 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for i in range(index, len(nums) - 1):\n        nums[i] = nums[i + 1]\n</code></pre> array.cpp<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nvoid remove(int *nums, int size, int index) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.java<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nvoid remove(int[] nums, int index) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for (int i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.cs<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nvoid Remove(int[] nums, int index) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for (int i = index; i &lt; nums.Length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.go<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nfunc remove(nums []int, index int) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for i := index; i &lt; len(nums)-1; i++ {\n        nums[i] = nums[i+1]\n    }\n}\n</code></pre> array.swift<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nfunc remove(nums: inout [Int], index: Int) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for i in nums.indices.dropFirst(index).dropLast() {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.js<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nfunction remove(nums, index) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.ts<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nfunction remove(nums: number[], index: number): void {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.dart<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nvoid remove(List&lt;int&gt; nums, int index) {\n  // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n  for (var i = index; i &lt; nums.length - 1; i++) {\n    nums[i] = nums[i + 1];\n  }\n}\n</code></pre> array.rs<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nfn remove(nums: &amp;mut [i32], index: usize) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for i in index..nums.len() - 1 {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.c<pre><code>/* 인덱스 index에 있는 원소 삭제 */\n// 주의: stdio.h가 remove라는 이름을 사용한다\nvoid removeItem(int *nums, int size, int index) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.kt<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nfun remove(nums: IntArray, index: Int) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for (i in index..&lt;nums.size - 1) {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.rb<pre><code>### 인덱스 index에 있는 원소 삭제 ###\ndef remove(nums, index)\n  # 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n  for i in index...(nums.length - 1)\n    nums[i] = nums[i + 1]\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>배열의 삽입과 삭제 연산에는 다음과 같은 단점이 있다.</p> <ul> <li>높은 시간 복잡도: 삽입과 삭제의 평균 시간 복잡도는 모두 \\(O(n)\\)이며, \\(n\\)은 배열 길이다.</li> <li>원소 손실: 배열의 길이는 바꿀 수 없으므로 삽입 후 길이 범위를 벗어나는 원소가 사라진다.</li> <li>메모리 낭비: 긴 배열을 초기화하고 앞부분만 사용하면 삽입 시 사라지는 마지막 원소는 모두 ‘의미 없는’ 값이 된다. 하지만 이 방식은 메모리 공간 일부를 낭비한다.</li> </ul>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#5","level":3,"title":"5.   배열 순회","text":"<p>대부분의 프로그래밍 언어에서는 인덱스로 배열을 순회할 수도 있고, 배열의 각 원소를 직접 얻으며 순회할 수도 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def traverse(nums: list[int]):\n    \"\"\"배열 순회\"\"\"\n    count = 0\n    # 인덱스를 통해 배열을 순회한다\n    for i in range(len(nums)):\n        count += nums[i]\n    # 배열의 원소를 직접 순회한다\n    for num in nums:\n        count += num\n    # 데이터의 인덱스와 원소를 함께 순회한다\n    for i, num in enumerate(nums):\n        count += nums[i]\n        count += num\n</code></pre> array.cpp<pre><code>/* 배열 순회 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // 인덱스를 통해 배열을 순회한다\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.java<pre><code>/* 배열 순회 */\nvoid traverse(int[] nums) {\n    int count = 0;\n    // 인덱스를 통해 배열을 순회한다\n    for (int i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // 배열의 원소를 직접 순회한다\n    for (int num : nums) {\n        count += num;\n    }\n}\n</code></pre> array.cs<pre><code>/* 배열 순회 */\nvoid Traverse(int[] nums) {\n    int count = 0;\n    // 인덱스를 통해 배열을 순회한다\n    for (int i = 0; i &lt; nums.Length; i++) {\n        count += nums[i];\n    }\n    // 배열의 원소를 직접 순회한다\n    foreach (int num in nums) {\n        count += num;\n    }\n}\n</code></pre> array.go<pre><code>/* 배열 순회 */\nfunc traverse(nums []int) {\n    count := 0\n    // 인덱스를 통해 배열을 순회한다\n    for i := 0; i &lt; len(nums); i++ {\n        count += nums[i]\n    }\n    count = 0\n    // 배열의 원소를 직접 순회한다\n    for _, num := range nums {\n        count += num\n    }\n    // 데이터의 인덱스와 원소를 함께 순회한다\n    for i, num := range nums {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.swift<pre><code>/* 배열 순회 */\nfunc traverse(nums: [Int]) {\n    var count = 0\n    // 인덱스를 통해 배열을 순회한다\n    for i in nums.indices {\n        count += nums[i]\n    }\n    // 배열의 원소를 직접 순회한다\n    for num in nums {\n        count += num\n    }\n    // 데이터의 인덱스와 원소를 함께 순회한다\n    for (i, num) in nums.enumerated() {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.js<pre><code>/* 배열 순회 */\nfunction traverse(nums) {\n    let count = 0;\n    // 인덱스를 통해 배열을 순회한다\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // 배열의 원소를 직접 순회한다\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.ts<pre><code>/* 배열 순회 */\nfunction traverse(nums: number[]): void {\n    let count = 0;\n    // 인덱스를 통해 배열을 순회한다\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // 배열의 원소를 직접 순회한다\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.dart<pre><code>/* 배열 원소 순회 */\nvoid traverse(List&lt;int&gt; nums) {\n  int count = 0;\n  // 인덱스를 통해 배열을 순회한다\n  for (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n  }\n  // 배열의 원소를 직접 순회한다\n  for (int _num in nums) {\n    count += _num;\n  }\n  // forEach 메서드로 배열을 순회한다\n  nums.forEach((_num) {\n    count += _num;\n  });\n}\n</code></pre> array.rs<pre><code>/* 배열 순회 */\nfn traverse(nums: &amp;[i32]) {\n    let mut _count = 0;\n    // 인덱스를 통해 배열을 순회한다\n    for i in 0..nums.len() {\n        _count += nums[i];\n    }\n    // 배열의 원소를 직접 순회한다\n    _count = 0;\n    for &amp;num in nums {\n        _count += num;\n    }\n}\n</code></pre> array.c<pre><code>/* 배열 순회 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // 인덱스를 통해 배열을 순회한다\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.kt<pre><code>/* 배열 순회 */\nfun traverse(nums: IntArray) {\n    var count = 0\n    // 인덱스를 통해 배열을 순회한다\n    for (i in nums.indices) {\n        count += nums[i]\n    }\n    // 배열의 원소를 직접 순회한다\n    for (j in nums) {\n        count += j\n    }\n}\n</code></pre> array.rb<pre><code>### 배열 순회 ###\ndef traverse(nums)\n  count = 0\n\n  # 인덱스를 통해 배열을 순회한다\n  for i in 0...nums.length\n    count += nums[i]\n  end\n\n  # 배열의 원소를 직접 순회한다\n  for num in nums\n    count += num\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#6","level":3,"title":"6.   원소 탐색","text":"<p>배열에서 특정 원소를 찾으려면 배열을 순회하면서 매번 값이 일치하는지 확인하고, 일치하면 해당 인덱스를 출력한다.</p> <p>배열은 선형 자료 구조이므로 이 탐색 연산을 ‘선형 탐색’이라고 한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def find(nums: list[int], target: int) -&gt; int:\n    \"\"\"배열에서 지정한 원소 탐색\"\"\"\n    for i in range(len(nums)):\n        if nums[i] == target:\n            return i\n    return -1\n</code></pre> array.cpp<pre><code>/* 배열에서 지정한 원소 탐색 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.java<pre><code>/* 배열에서 지정한 원소 탐색 */\nint find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.cs<pre><code>/* 배열에서 지정한 원소 탐색 */\nint Find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.go<pre><code>/* 배열에서 지정한 원소 탐색 */\nfunc find(nums []int, target int) (index int) {\n    index = -1\n    for i := 0; i &lt; len(nums); i++ {\n        if nums[i] == target {\n            index = i\n            break\n        }\n    }\n    return\n}\n</code></pre> array.swift<pre><code>/* 배열에서 지정한 원소 탐색 */\nfunc find(nums: [Int], target: Int) -&gt; Int {\n    for i in nums.indices {\n        if nums[i] == target {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> array.js<pre><code>/* 배열에서 지정한 원소 탐색 */\nfunction find(nums, target) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) return i;\n    }\n    return -1;\n}\n</code></pre> array.ts<pre><code>/* 배열에서 지정한 원소 탐색 */\nfunction find(nums: number[], target: number): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> array.dart<pre><code>/* 배열에서 지정한 원소 탐색 */\nint find(List&lt;int&gt; nums, int target) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    if (nums[i] == target) return i;\n  }\n  return -1;\n}\n</code></pre> array.rs<pre><code>/* 배열에서 지정한 원소 탐색 */\nfn find(nums: &amp;[i32], target: i32) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        if nums[i] == target {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> array.c<pre><code>/* 배열에서 지정한 원소 탐색 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.kt<pre><code>/* 배열에서 지정한 원소 탐색 */\nfun find(nums: IntArray, target: Int): Int {\n    for (i in nums.indices) {\n        if (nums[i] == target)\n            return i\n    }\n    return -1\n}\n</code></pre> array.rb<pre><code>### 배열에서 지정한 원소 탐색 ###\ndef find(nums, target)\n  for i in 0...nums.length\n    return i if nums[i] == target\n  end\n\n  -1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#7","level":3,"title":"7.   배열 용량 확장","text":"<p>복잡한 시스템 환경에서는 배열 바로 뒤의 메모리 공간을 사용할 수 있다고 보장하기 어려우므로, 배열 용량을 안전하게 늘릴 수 없다. 따라서 대부분의 프로그래밍 언어에서 배열 길이는 바꿀 수 없다.</p> <p>배열 용량을 늘리려면 더 큰 배열을 새로 만들고 기존 원소를 차례로 복사해야 한다. 이는 \\(O(n)\\) 연산으로, 배열이 매우 크면 시간이 많이 든다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def extend(nums: list[int], enlarge: int) -&gt; list[int]:\n    \"\"\"배열 길이 확장\"\"\"\n    # 길이를 확장한 배열을 초기화한다\n    res = [0] * (len(nums) + enlarge)\n    # 원래 배열의 모든 원소를 새 배열에 복사한다\n    for i in range(len(nums)):\n        res[i] = nums[i]\n    # 확장한 새 배열을 반환한다\n    return res\n</code></pre> array.cpp<pre><code>/* 배열 길이 확장 */\nint *extend(int *nums, int size, int enlarge) {\n    // 길이를 확장한 배열을 초기화한다\n    int *res = new int[size + enlarge];\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // 메모리 해제\n    delete[] nums;\n    // 확장한 새 배열을 반환한다\n    return res;\n}\n</code></pre> array.java<pre><code>/* 배열 길이 확장 */\nint[] extend(int[] nums, int enlarge) {\n    // 길이를 확장한 배열을 초기화한다\n    int[] res = new int[nums.length + enlarge];\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for (int i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // 확장한 새 배열을 반환한다\n    return res;\n}\n</code></pre> array.cs<pre><code>/* 배열 길이 확장 */\nint[] Extend(int[] nums, int enlarge) {\n    // 길이를 확장한 배열을 초기화한다\n    int[] res = new int[nums.Length + enlarge];\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for (int i = 0; i &lt; nums.Length; i++) {\n        res[i] = nums[i];\n    }\n    // 확장한 새 배열을 반환한다\n    return res;\n}\n</code></pre> array.go<pre><code>/* 배열 길이 확장 */\nfunc extend(nums []int, enlarge int) []int {\n    // 길이를 확장한 배열을 초기화한다\n    res := make([]int, len(nums)+enlarge)\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for i, num := range nums {\n        res[i] = num\n    }\n    // 확장한 새 배열을 반환한다\n    return res\n}\n</code></pre> array.swift<pre><code>/* 배열 길이 확장 */\nfunc extend(nums: [Int], enlarge: Int) -&gt; [Int] {\n    // 길이를 확장한 배열을 초기화한다\n    var res = Array(repeating: 0, count: nums.count + enlarge)\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for i in nums.indices {\n        res[i] = nums[i]\n    }\n    // 확장한 새 배열을 반환한다\n    return res\n}\n</code></pre> array.js<pre><code>/* 배열 길이 확장 */\n// JavaScript의 Array는 동적 배열이므로 직접 확장할 수 있다\n// 학습의 편의를 위해 이 함수에서는 Array를 길이가 고정된 배열로 취급한다\nfunction extend(nums, enlarge) {\n    // 길이를 확장한 배열을 초기화한다\n    const res = new Array(nums.length + enlarge).fill(0);\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // 확장한 새 배열을 반환한다\n    return res;\n}\n</code></pre> array.ts<pre><code>/* 배열 길이 확장 */\n// TypeScript의 Array는 동적 배열이므로 직접 확장할 수 있다\n// 학습의 편의를 위해 이 함수에서는 Array를 길이가 고정된 배열로 취급한다\nfunction extend(nums: number[], enlarge: number): number[] {\n    // 길이를 확장한 배열을 초기화한다\n    const res = new Array(nums.length + enlarge).fill(0);\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // 확장한 새 배열을 반환한다\n    return res;\n}\n</code></pre> array.dart<pre><code>/* 배열 길이 확장 */\nList&lt;int&gt; extend(List&lt;int&gt; nums, int enlarge) {\n  // 길이를 확장한 배열을 초기화한다\n  List&lt;int&gt; res = List.filled(nums.length + enlarge, 0);\n  // 원래 배열의 모든 원소를 새 배열에 복사한다\n  for (var i = 0; i &lt; nums.length; i++) {\n    res[i] = nums[i];\n  }\n  // 확장한 새 배열을 반환한다\n  return res;\n}\n</code></pre> array.rs<pre><code>/* 배열 길이 확장 */\nfn extend(nums: &amp;[i32], enlarge: usize) -&gt; Vec&lt;i32&gt; {\n    // 길이를 확장한 배열을 초기화한다\n    let mut res: Vec&lt;i32&gt; = vec![0; nums.len() + enlarge];\n    // 원래 배열의 모든 원소를 새 배열로 복사한다\n    res[0..nums.len()].copy_from_slice(nums);\n\n    // 확장한 새 배열을 반환한다\n    res\n}\n</code></pre> array.c<pre><code>/* 배열 길이 확장 */\nint *extend(int *nums, int size, int enlarge) {\n    // 길이를 확장한 배열을 초기화한다\n    int *res = (int *)malloc(sizeof(int) * (size + enlarge));\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // 확장한 공간 초기화\n    for (int i = size; i &lt; size + enlarge; i++) {\n        res[i] = 0;\n    }\n    // 확장한 새 배열을 반환한다\n    return res;\n}\n</code></pre> array.kt<pre><code>/* 배열 길이 확장 */\nfun extend(nums: IntArray, enlarge: Int): IntArray {\n    // 길이를 확장한 배열을 초기화한다\n    val res = IntArray(nums.size + enlarge)\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for (i in nums.indices) {\n        res[i] = nums[i]\n    }\n    // 확장한 새 배열을 반환한다\n    return res\n}\n</code></pre> array.rb<pre><code>### 배열 길이 확장 ###\n# Ruby의 Array는 동적 배열이므로 직접 확장할 수 있다\n# 학습의 편의를 위해 이 함수에서는 Array를 길이가 고정된 배열로 취급한다\ndef extend(nums, enlarge)\n  # 길이를 확장한 배열을 초기화한다\n  res = Array.new(nums.length + enlarge, 0)\n\n  # 원래 배열의 모든 원소를 새 배열에 복사한다\n  for i in 0...nums.length\n    res[i] = nums[i]\n  end\n\n  # 확장한 새 배열을 반환한다\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#412","level":2,"title":"4.1.2   배열의 장점과 한계","text":"<p>배열은 연속된 메모리 공간에 저장되고 원소의 자료형이 모두 같다. 이러한 풍부한 사전 정보를 활용하면 시스템은 자료 구조의 연산 효율을 최적화할 수 있다.</p> <ul> <li>높은 공간 효율: 연속된 메모리 블록을 데이터에 할당하므로 추가적인 구조 비용이 없다.</li> <li>임의 접근 지원: 어떤 원소든 \\(O(1)\\) 시간에 접근할 수 있다.</li> <li>캐시 지역성: 배열 원소에 접근할 때 컴퓨터는 그 원소뿐 아니라 주변 데이터도 캐시에 저장한다. 이를 통해 이후 연산의 실행 속도를 높일 수 있다.</li> </ul> <p>연속 공간 저장은 양날의 검으로, 다음과 같은 한계가 있다.</p> <ul> <li>낮은 삽입 및 삭제 효율: 원소가 많으면 삽입이나 삭제 시 많은 원소를 이동해야 한다.</li> <li>고정된 길이: 초기화한 뒤에는 길이가 고정된다. 용량을 늘리려면 모든 데이터를 새 배열로 복사해야 하므로 비용이 크다.</li> <li>공간 낭비: 실제 필요한 크기보다 크게 할당하면 남는 공간이 낭비된다.</li> </ul>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#413","level":2,"title":"4.1.3   배열의 대표적인 활용","text":"<p>배열은 기본적이면서도 흔히 쓰이는 자료 구조다. 다양한 알고리즘에 자주 사용되며 복잡한 자료 구조를 구현하는 데도 쓰인다.</p> <ul> <li>임의 접근: 일부 표본을 무작위로 추출하려면 데이터를 배열에 저장하고 무작위 수열을 생성하여 인덱스에 따라 표본을 뽑을 수 있다.</li> <li>정렬과 탐색: 배열은 정렬 및 탐색 알고리즘에서 가장 많이 쓰이는 자료 구조다. 퀵 정렬, 병합 정렬, 이진 탐색 등은 주로 배열에서 수행된다.</li> <li>조회 테이블: 원소나 대응 관계를 빠르게 찾고 싶을 때 배열을 조회 테이블로 사용할 수 있다. 예를 들어 문자와 ASCII 코드의 대응을 구현하려면 문자의 ASCII 코드 값을 인덱스로 삼고, 해당 원소를 배열의 대응 위치에 저장하면 된다.</li> <li>머신 러닝: 신경망에서는 벡터, 행렬, 텐서 사이의 선형 대수 연산을 많이 사용하며, 이 데이터들은 모두 배열 형태로 구성된다. 배열은 신경망 프로그래밍에서 가장 많이 쓰는 자료 구조다.</li> <li>자료 구조 구현: 배열로 스택, 큐, 해시 테이블, 힙, 그래프 등을 구현할 수 있다. 예를 들어 그래프의 인접 행렬 표현은 실제로 이차원 배열이다.</li> </ul>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/","level":1,"title":"4.6   연습 문제","text":"","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#461","level":2,"title":"4.6.1   개념 확인","text":"","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#1","level":3,"title":"1.   배열과 연결 리스트에서 원소 찾기","text":"<p>배열과 단일 연결 리스트에 모두 <code>[A, B, C, D, E]</code>가 순서대로 저장되어 있다. 네 번째 원소 <code>D</code>를 읽으려고 한다.</p> <ol> <li>배열에서는 어떤 인덱스를 바로 사용할 수 있는가?</li> <li>단일 연결 리스트에서는 머리 노드 <code>A</code>부터 <code>next</code>를 따라 어떤 노드를 차례대로 거쳐야 하는가?</li> <li>읽으려는 원소가 뒤쪽에 있을수록 두 구조에 필요한 단계 수는 어떻게 변하는가? 위치를 기준으로 반복해서 읽을 때는 어느 구조가 더 적합하며, 그 이유는 무엇인가?</li> </ol> 참고 답안 <ol> <li> <p>인덱스가 0부터 시작한다면 네 번째 원소의 인덱스는 3이므로, 배열에서는 <code>arr[3]</code>을 바로 읽을 수 있다.</p> </li> <li> <p>단일 연결 리스트는 맨 앞부터 시작해야 한다. 접근 경로는 <code>A → B → C → D</code>이며, <code>next</code>를 따라 3번 이동해야 한다.</p> </li> <li> <p>배열은 시작 주소와 인덱스로 원소의 위치를 바로 계산할 수 있으므로 위치를 기준으로 접근하는 시간 복잡도가 \\(O(1)\\)이다.     단일 연결 리스트에서 \\(k\\)번째 노드에 접근하려면 머리 노드부터 <code>next</code>를 따라 \\(k-1\\)번 이동해야 하므로,     최악의 경우 \\(O(n)\\) 시간이 필요하다.</p> <p>여기서는 위치를 기준으로 읽는 연산만 비교한 것이며, 연결 리스트가 모든 연산에서 더 느리다는 뜻은 아니다.</p> </li> </ol>","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#2","level":3,"title":"2.   배열과 연결 리스트에 원소 삽입하기","text":"<p>배열과 단일 연결 리스트에 모두 <code>A, B, C, D</code>가 저장되어 있다. <code>B</code> 뒤에 <code>X</code>를 삽입하려고 한다.</p> <ul> <li>배열의 용량은 5이고 현재 상태는 <code>[A, B, C, D, _]</code>다.</li> <li>연결 리스트는 <code>A → B → C → D</code>이며, 노드 <code>B</code>를 가리키는 참조를 이미 가지고 있다.</li> </ul> <ol> <li>배열에서는 어떤 원소를 이동해야 하는가? 삽입 후 배열을 적어 보자.</li> <li>연결 리스트에서는 <code>X.next</code>와 <code>B.next</code>를 어떤 순서로 수정해야 하는가? 삽입 후 연결 리스트를 적어 보자.</li> <li>삽입 효율성을 비교할 때 ‘노드 B의 참조를 이미 가지고 있다’는 조건을 강조하는 이유는 무엇인가?</li> </ol> 참고 답안 <ol> <li> <p>배열에서는 먼저 <code>D</code>를 오른쪽으로 한 칸 옮기고, 다음으로 <code>C</code>를 오른쪽으로 한 칸 옮긴 뒤, <code>X</code>를 인덱스 2에 넣는다.     결과는 <code>[A, B, X, C, D]</code>다.</p> </li> <li> <p><code>B.next</code>는 원래 <code>C</code>를 가리킨다. 먼저 <code>X.next = B.next</code>로 설정하여 <code>X</code>가 <code>C</code>를 가리키게 하고,     다음으로 <code>B.next = X</code>로 설정한다. 결과는 <code>A → B → X → C → D</code>다.     기존 연결을 저장하지 않은 채 <code>B.next</code>를 먼저 덮어쓰면 <code>C</code>를 찾지 못할 수 있다.</p> </li> <li> <p><code>B</code>의 위치를 알고 있다면 연결 리스트의 삽입은 연결 두 개만 수정하면 되므로 \\(O(1)\\) 시간에 완료할 수 있다.     맨 앞부터 <code>B</code>를 찾아야 한다면 탐색 자체에 \\(O(n)\\) 시간이 걸릴 수 있다.</p> </li> </ol>","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#3","level":3,"title":"3.   리스트의 용량은 어떻게 늘어나는가","text":"<p>배열로 구현한 리스트의 현재 내용은 <code>[A, B, C]</code>이고, 길이는 <code>size = 3</code>, 용량은 <code>capacity = 4</code>다. 용량이 부족해지면 새 배열의 용량을 기존의 2배로 늘리기로 한다.</p> <ol> <li><code>D</code>를 추가한 뒤 리스트의 길이와 용량은 각각 얼마인가? 용량을 늘려야 하는가?</li> <li>이어서 <code>E</code>를 추가하면 용량은 얼마가 되는가? 기존 원소를 몇 개 복사해야 하는가?</li> <li>내부 배열의 길이는 바꿀 수 없는데, 리스트의 용량은 어떻게 늘어나는 것처럼 보이는가?</li> </ol> 참고 답안 <ol> <li> <p><code>D</code>는 마지막 빈자리에 넣을 수 있다. 내용은 <code>[A, B, C, D]</code>가 되고,     <code>size = 4</code>, <code>capacity = 4</code>이며 용량 확장은 필요 없다.</p> </li> <li> <p>이어서 <code>E</code>를 추가할 때는 빈자리가 없으므로 용량이 8인 새 배열을 만들어야 한다.     기존 원소 <code>A, B, C, D</code> 네 개를 복사한 뒤 <code>E</code>를 넣는다.     이때 <code>size = 5</code>, <code>capacity = 8</code>이다.</p> </li> <li> <p>원래 배열 자체의 길이가 늘어난 것은 아니다. 더 큰 새 배열을 만들고 기존 원소를 복사한 뒤,     새 배열을 내부 저장 공간으로 사용하기 때문에 사용자에게는 용량이 늘어난 것으로 보인다.</p> </li> </ol>","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#462","level":2,"title":"4.6.2   프로그래밍 연습","text":"","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#1-1","level":3,"title":"1.   배열로 표현한 큰 정수에 1 더하기","text":"<p>배열 <code>digits</code>에는 음이 아닌 정수의 각 자릿수가 왼쪽부터 저장되어 있다. 예를 들어 <code>[3, 0, 8]</code>은 308을 나타낸다. 숫자 0은 <code>[0]</code>으로 나타내며, 그 밖의 입력은 첫 번째 자릿수가 0이 아니다.</p> <p>십진수의 세로셈 덧셈 과정을 따라 이 정수에 1을 더한 뒤, 결과를 같은 배열 형식으로 반환하라. <code>digits</code>를 직접 수정해도 된다. 맨 앞에 새로운 자리 올림이 생기면 더 긴 배열을 반환할 수 있다.</p> 풀이 힌트 <ol> <li>세로셈 덧셈처럼 배열의 마지막 자릿수부터 시작한다.</li> <li>현재 자릿수가 9보다 작으면 1을 더한 뒤 바로 반환할 수 있다.</li> <li>현재 자릿수가 9라면 0으로 바꾼다. 모든 자릿수가 9였다면 맨 앞에 1을 추가해야 한다.</li> </ol> <p>LeetCode</p>","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#2_1","level":3,"title":"2.   단일 연결 리스트 뒤집기","text":"<p>단일 연결 리스트의 머리 노드 <code>head</code>가 주어진다. 각 노드에는 값과 다음 노드를 가리키는 <code>next</code>가 있다.</p> <p>반복 방식으로 모든 노드 사이의 연결 방향을 뒤집고, 뒤집힌 리스트의 머리 노드를 반환하라. 새로운 연결 리스트 노드를 만들어서는 안 된다.</p> 풀이 힌트 <ol> <li>먼저 종이에 연결된 노드 세 개와 prev, cur 두 포인터를 그려 보자.</li> <li>cur.next를 바꾸기 전에 nxt에 원래 다음 노드를 반드시 저장해야 한다.</li> <li>cur.next의 방향을 뒤집은 뒤 prev = cur, cur = nxt로 설정하고 원래 리스트의 다음 노드를 계속 처리한다.</li> </ol> <p>LeetCode</p>","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/","level":1,"title":"4.2   연결 리스트","text":"<p>메모리 공간은 모든 프로그램이 함께 사용하는 자원이다. 복잡한 시스템 실행 환경에서는 빈 메모리 공간이 곳곳에 흩어져 있을 수 있다. 배열을 저장할 메모리는 연속적이어야 하므로, 배열이 매우 크면 그만큼 큰 연속 공간을 확보하지 못할 수도 있다. 이때 연결 리스트의 유연성이 장점으로 드러난다.</p> <p>연결 리스트(linked list)는 각 원소가 노드 객체이며, 노드들이 ‘참조’로 연결된 선형 자료 구조다. 참조는 다음 노드의 메모리 주소를 기록하므로, 이를 통해 현재 노드에서 다음 노드에 접근할 수 있다.</p> <p>이러한 설계 덕분에 각 노드는 메모리 곳곳에 분산되어 저장될 수 있으며, 메모리 주소가 연속적일 필요가 없다.</p> <p></p> <p> 그림 4-5   연결 리스트의 정의와 저장 방식 </p> <p>위 그림(4-5)에서 연결 리스트를 구성하는 단위는 노드(node) 객체다. 각 노드는 노드의 ‘값’과 다음 노드를 가리키는 ‘참조’라는 두 데이터를 포함한다.</p> <ul> <li>연결 리스트의 첫 노드를 ‘머리 노드’, 마지막 노드를 ‘꼬리 노드’라고 한다.</li> <li>꼬리 노드는 ‘없음’을 가리키며, Java, C++, Python에서는 이를 각각 <code>null</code>, <code>nullptr</code>, <code>None</code>으로 표기한다.</li> <li>C, C++, Go, Rust처럼 포인터를 지원하는 언어에서는 위의 ‘참조’를 ‘포인터’로 바꾸어 이해하면 된다.</li> </ul> <p>다음 코드와 같이 연결 리스트 노드 <code>ListNode</code>는 값 외에도 참조(포인터)를 하나 더 저장해야 한다. 따라서 같은 양의 데이터를 저장할 때 연결 리스트는 배열보다 메모리 공간을 더 많이 차지한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class ListNode:\n    \"\"\"연결 리스트 노드 클래스\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val               # 노드 값\n        self.next: ListNode | None = None # 다음 노드를 가리키는 참조\n</code></pre> <pre><code>/* 연결 리스트 노드 구조체 */\nstruct ListNode {\n    int val;         // 노드 값\n    ListNode *next;  // 다음 노드를 가리키는 포인터\n    ListNode(int x) : val(x), next(nullptr) {}  // 생성자\n};\n</code></pre> <pre><code>/* 연결 리스트 노드 클래스 */\nclass ListNode {\n    int val;        // 노드 값\n    ListNode next;  // 다음 노드를 가리키는 참조\n    ListNode(int x) { val = x; }  // 생성자\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 클래스 */\nclass ListNode(int x) {  //생성자\n    int val = x;         // 노드 값\n    ListNode? next;      // 다음 노드를 가리키는 참조\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 구조체 */\ntype ListNode struct {\n    Val  int       // 노드 값\n    Next *ListNode // 다음 노드를 가리키는 포인터\n}\n\n// NewListNode 생성자: 새 연결 리스트 생성\nfunc NewListNode(val int) *ListNode {\n    return &amp;ListNode{\n        Val:  val,\n        Next: nil,\n    }\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 클래스 */\nclass ListNode {\n    var val: Int // 노드 값\n    var next: ListNode? // 다음 노드를 가리키는 참조\n\n    init(x: Int) { // 생성자\n        val = x\n    }\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 클래스 */\nclass ListNode {\n    constructor(val, next) {\n        this.val = (val === undefined ? 0 : val);       // 노드 값\n        this.next = (next === undefined ? null : next); // 다음 노드를 가리키는 참조\n    }\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 클래스 */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    constructor(val?: number, next?: ListNode | null) {\n        this.val = val === undefined ? 0 : val;        // 노드 값\n        this.next = next === undefined ? null : next;  // 다음 노드를 가리키는 참조\n    }\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 클래스 */\nclass ListNode {\n  int val; // 노드 값\n  ListNode? next; // 다음 노드를 가리키는 참조\n  ListNode(this.val, [this.next]); // 생성자\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* 연결 리스트 노드 클래스 */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // 노드 값\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // 다음 노드를 가리키는 포인터\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 구조체 */\ntypedef struct ListNode {\n    int val;               // 노드 값\n    struct ListNode *next; // 다음 노드를 가리키는 포인터\n} ListNode;\n\n/* 생성자 */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 클래스 */\n// 생성자\nclass ListNode(x: Int) {\n    val _val: Int = x          // 노드 값\n    var next: ListNode? = null // 다음 노드를 가리키는 참조\n}\n</code></pre> <pre><code># 연결 리스트 노드 클래스\nclass ListNode\n  attr_accessor :val  # 노드 값\n  attr_accessor :next # 다음 노드를 가리키는 참조\n\n  def initialize(val=0, next_node=nil)\n    @val = val\n    @next = next_node\n  end\nend\n</code></pre>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#421","level":2,"title":"4.2.1   연결 리스트의 기본 연산","text":"","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#1","level":3,"title":"1.   연결 리스트 초기화","text":"<p>연결 리스트를 만드는 과정은 두 단계다. 먼저 각 노드 객체를 초기화한 다음, 노드 사이의 참조 관계를 설정한다. 초기화가 끝나면 머리 노드부터 시작하여 <code>next</code> 참조를 따라 모든 노드에 차례로 접근할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code># 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화\n# 각 노드 초기화\nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# 노드 사이의 참조 설정\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.cpp<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// 노드 사이의 참조 설정\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.java<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// 노드 사이의 참조 설정\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.cs<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nListNode n0 = new(1);\nListNode n1 = new(3);\nListNode n2 = new(2);\nListNode n3 = new(5);\nListNode n4 = new(4);\n// 노드 사이의 참조 설정\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.go<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// 노드 사이의 참조 설정\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n</code></pre> linked_list.swift<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// 노드 사이의 참조 설정\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.js<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// 노드 사이의 참조 설정\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.ts<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// 노드 사이의 참조 설정\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.dart<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\\\n// 각 노드 초기화\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// 노드 사이의 참조 설정\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rs<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nlet n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None }));\nlet n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None }));\nlet n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None }));\nlet n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None }));\nlet n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None }));\n\n// 노드 사이의 참조 설정\nn0.borrow_mut().next = Some(n1.clone());\nn1.borrow_mut().next = Some(n2.clone());\nn2.borrow_mut().next = Some(n3.clone());\nn3.borrow_mut().next = Some(n4.clone());\n</code></pre> linked_list.c<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// 노드 사이의 참조 설정\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.kt<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nval n0 = ListNode(1)\nval n1 = ListNode(3)\nval n2 = ListNode(2)\nval n3 = ListNode(5)\nval n4 = ListNode(4)\n// 노드 사이의 참조 설정\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rb<pre><code># 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화\n# 각 노드 초기화\nn0 = ListNode.new(1)\nn1 = ListNode.new(3)\nn2 = ListNode.new(2)\nn3 = ListNode.new(5)\nn4 = ListNode.new(4)\n# 노드 사이의 참조 설정\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>배열은 전체가 하나의 변수다. 예를 들어 배열 <code>nums</code>에는 <code>nums[0]</code>, <code>nums[1]</code> 등의 원소가 들어 있다. 반면 연결 리스트는 독립된 여러 노드 객체로 이루어진다. 보통 머리 노드로 연결 리스트 전체를 지칭한다. 예를 들어 위 코드의 연결 리스트는 연결 리스트 <code>n0</code>라고 부를 수 있다.</p>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#2","level":3,"title":"2.   노드 삽입","text":"<p>연결 리스트에 노드를 삽입하기는 매우 쉽다. 아래 그림(4-6)과 같이 인접한 노드 <code>n0</code>와 <code>n1</code> 사이에 새 노드 <code>P</code>를 삽입하려면 노드 참조(포인터) 두 개만 바꾸면 된다. 시간 복잡도는 \\(O(1)\\)이다.</p> <p>반면 배열의 원소 삽입은 시간 복잡도가 \\(O(n)\\)이므로 데이터가 많을 때 효율이 낮다.</p> <p></p> <p> 그림 4-6   연결 리스트에 노드를 삽입하는 예 </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def insert(n0: ListNode, P: ListNode):\n    \"\"\"연결 리스트의 노드 n0 뒤에 노드 P 삽입\"\"\"\n    n1 = n0.next\n    P.next = n1\n    n0.next = P\n</code></pre> linked_list.cpp<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.java<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nvoid insert(ListNode n0, ListNode P) {\n    ListNode n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.cs<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nvoid Insert(ListNode n0, ListNode P) {\n    ListNode? n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.go<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nfunc insertNode(n0 *ListNode, P *ListNode) {\n    n1 := n0.Next\n    P.Next = n1\n    n0.Next = P\n}\n</code></pre> linked_list.swift<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nfunc insert(n0: ListNode, P: ListNode) {\n    let n1 = n0.next\n    P.next = n1\n    n0.next = P\n}\n</code></pre> linked_list.js<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nfunction insert(n0, P) {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.ts<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nfunction insert(n0: ListNode, P: ListNode): void {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.dart<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nvoid insert(ListNode n0, ListNode P) {\n  ListNode? n1 = n0.next;\n  P.next = n1;\n  n0.next = P;\n}\n</code></pre> linked_list.rs<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\n#[allow(non_snake_case)]\npub fn insert&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, P: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    let n1 = n0.borrow_mut().next.take();\n    P.borrow_mut().next = n1;\n    n0.borrow_mut().next = Some(P);\n}\n</code></pre> linked_list.c<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.kt<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nfun insert(n0: ListNode?, p: ListNode?) {\n    val n1 = n0?.next\n    p?.next = n1\n    n0?.next = p\n}\n</code></pre> linked_list.rb<pre><code>### 연결 리스트의 노드 n0 뒤에 노드 _p 삽입 ###\n# Ruby에서 `p`는 내장 함수이고 `P`는 상수이므로, 대신 `_p`를 사용할 수 있다\ndef insert(n0, _p)\n  n1 = n0.next\n  _p.next = n1\n  n0.next = _p\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#3","level":3,"title":"3.   노드 삭제","text":"<p>아래 그림(4-7)과 같이 연결 리스트의 노드 삭제도 매우 간단하다. 노드 하나의 참조(포인터)만 바꾸면 된다.</p> <p>삭제 후에도 노드 <code>P</code>는 여전히 <code>n1</code>을 가리키지만, 연결 리스트를 순회해서는 더 이상 <code>P</code>에 접근할 수 없다. 따라서 <code>P</code>는 이미 이 연결 리스트에 속하지 않는다.</p> <p></p> <p> 그림 4-7   연결 리스트의 노드 삭제 </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def remove(n0: ListNode):\n    \"\"\"연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제\"\"\"\n    if not n0.next:\n        return\n    # n0 -&gt; P -&gt; n1\n    P = n0.next\n    n1 = P.next\n    n0.next = n1\n</code></pre> linked_list.cpp<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nvoid remove(ListNode *n0) {\n    if (n0-&gt;next == nullptr)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // 메모리 해제\n    delete P;\n}\n</code></pre> linked_list.java<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nvoid remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.cs<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nvoid Remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode? n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.go<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nfunc removeItem(n0 *ListNode) {\n    if n0.Next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    P := n0.Next\n    n1 := P.Next\n    n0.Next = n1\n}\n</code></pre> linked_list.swift<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nfunc remove(n0: ListNode) {\n    if n0.next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    let P = n0.next\n    let n1 = P?.next\n    n0.next = n1\n}\n</code></pre> linked_list.js<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nfunction remove(n0) {\n    if (!n0.next) return;\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.ts<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nfunction remove(n0: ListNode): void {\n    if (!n0.next) {\n        return;\n    }\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.dart<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nvoid remove(ListNode n0) {\n  if (n0.next == null) return;\n  // n0 -&gt; P -&gt; n1\n  ListNode P = n0.next!;\n  ListNode? n1 = P.next;\n  n0.next = n1;\n}\n</code></pre> linked_list.rs<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\n#[allow(non_snake_case)]\npub fn remove&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    // n0 -&gt; P -&gt; n1\n    let P = n0.borrow_mut().next.take();\n    if let Some(node) = P {\n        let n1 = node.borrow_mut().next.take();\n        n0.borrow_mut().next = n1;\n    }\n}\n</code></pre> linked_list.c<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\n// 주의: stdio.h가 remove라는 이름을 사용한다\nvoid removeItem(ListNode *n0) {\n    if (!n0-&gt;next)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // 메모리 해제\n    free(P);\n}\n</code></pre> linked_list.kt<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nfun remove(n0: ListNode?) {\n    if (n0?.next == null)\n        return\n    // n0 -&gt; P -&gt; n1\n    val p = n0.next\n    val n1 = p?.next\n    n0.next = n1\n}\n</code></pre> linked_list.rb<pre><code>### 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 ###\ndef remove(n0)\n  return if n0.next.nil?\n\n  # n0 -&gt; remove_node -&gt; n1\n  remove_node = n0.next\n  n1 = remove_node.next\n  n0.next = n1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#4","level":3,"title":"4.   노드 접근","text":"<p>연결 리스트의 노드 접근 효율은 낮다. 앞 절에서 설명했듯 배열은 어떤 원소든 \\(O(1)\\) 시간에 접근할 수 있다. 그러나 연결 리스트에서는 머리 노드부터 하나씩 뒤로 순회하여 목표 노드를 찾아야 한다. 즉, \\(i\\)번째 노드에 접근하려면 \\(i - 1\\)번 반복해야 하므로 시간 복잡도는 \\(O(n)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def access(head: ListNode, index: int) -&gt; ListNode | None:\n    \"\"\"연결 리스트에서 인덱스가 index인 노드에 접근\"\"\"\n    for _ in range(index):\n        if not head:\n            return None\n        head = head.next\n    return head\n</code></pre> linked_list.cpp<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == nullptr)\n            return nullptr;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.java<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nListNode access(ListNode head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.cs<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nListNode? Access(ListNode? head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.go<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nfunc access(head *ListNode, index int) *ListNode {\n    for i := 0; i &lt; index; i++ {\n        if head == nil {\n            return nil\n        }\n        head = head.Next\n    }\n    return head\n}\n</code></pre> linked_list.swift<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nfunc access(head: ListNode, index: Int) -&gt; ListNode? {\n    var head: ListNode? = head\n    for _ in 0 ..&lt; index {\n        if head == nil {\n            return nil\n        }\n        head = head?.next\n    }\n    return head\n}\n</code></pre> linked_list.js<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nfunction access(head, index) {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.ts<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nfunction access(head: ListNode | null, index: number): ListNode | null {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.dart<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nListNode? access(ListNode? head, int index) {\n  for (var i = 0; i &lt; index; i++) {\n    if (head == null) return null;\n    head = head.next;\n  }\n  return head;\n}\n</code></pre> linked_list.rs<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\npub fn access&lt;T&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, index: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n    fn dfs&lt;T&gt;(\n        head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,\n        index: i32,\n    ) -&gt; Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        if index &lt;= 0 {\n            return head.cloned();\n        }\n\n        if let Some(node) = head {\n            dfs(node.borrow().next.as_ref(), index - 1)\n        } else {\n            None\n        }\n    }\n\n    dfs(Some(head).as_ref(), index)\n}\n</code></pre> linked_list.c<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == NULL)\n            return NULL;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.kt<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nfun access(head: ListNode?, index: Int): ListNode? {\n    var h = head\n    for (i in 0..&lt;index) {\n        if (h == null)\n            return null\n        h = h.next\n    }\n    return h\n}\n</code></pre> linked_list.rb<pre><code>### 연결 리스트에서 인덱스가 index인 노드에 접근 ###\ndef access(head, index)\n  for i in 0...index\n    return nil if head.nil?\n    head = head.next\n  end\n\n  head\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#5","level":3,"title":"5.   노드 탐색","text":"<p>연결 리스트를 순회하여 값이 <code>target</code>인 노드를 찾고, 그 노드의 인덱스를 출력한다. 이 과정도 선형 탐색에 해당한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def find(head: ListNode, target: int) -&gt; int:\n    \"\"\"연결 리스트에서 값이 target인 첫 번째 노드 탐색\"\"\"\n    index = 0\n    while head:\n        if head.val == target:\n            return index\n        head = head.next\n        index += 1\n    return -1\n</code></pre> linked_list.cpp<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head != nullptr) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.java<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nint find(ListNode head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.cs<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nint Find(ListNode? head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.go<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nfunc findNode(head *ListNode, target int) int {\n    index := 0\n    for head != nil {\n        if head.Val == target {\n            return index\n        }\n        head = head.Next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.swift<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nfunc find(head: ListNode, target: Int) -&gt; Int {\n    var head: ListNode? = head\n    var index = 0\n    while head != nil {\n        if head?.val == target {\n            return index\n        }\n        head = head?.next\n        index += 1\n    }\n    return -1\n}\n</code></pre> linked_list.js<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nfunction find(head, target) {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.ts<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nfunction find(head: ListNode | null, target: number): number {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.dart<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nint find(ListNode? head, int target) {\n  int index = 0;\n  while (head != null) {\n    if (head.val == target) {\n      return index;\n    }\n    head = head.next;\n    index++;\n  }\n  return -1;\n}\n</code></pre> linked_list.rs<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\npub fn find&lt;T: PartialEq&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, target: T) -&gt; i32 {\n    fn find&lt;T: PartialEq&gt;(head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, target: T, idx: i32) -&gt; i32 {\n        if let Some(node) = head {\n            if node.borrow().val == target {\n                return idx;\n            }\n            return find(node.borrow().next.as_ref(), target, idx + 1);\n        } else {\n            -1\n        }\n    }\n\n    find(Some(head).as_ref(), target, 0)\n}\n</code></pre> linked_list.c<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.kt<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nfun find(head: ListNode?, target: Int): Int {\n    var index = 0\n    var h = head\n    while (h != null) {\n        if (h._val == target)\n            return index\n        h = h.next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.rb<pre><code>### 연결 리스트에서 값이 target인 첫 번째 노드 탐색 ###\ndef find(head, target)\n  index = 0\n  while head\n    return index if head.val == target\n    head = head.next\n    index += 1\n  end\n\n  -1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#422-vs","level":2,"title":"4.2.2   배열 vs. 연결 리스트","text":"<p>아래 표(4-1)는 배열과 연결 리스트의 특징과 연산 효율을 비교한다. 둘은 서로 반대되는 저장 전략을 사용하므로 성질과 연산 효율도 대조적이다.</p> <p> 표 4-1   배열과 연결 리스트의 효율 비교 </p> 배열 연결 리스트 저장 방식 연속된 메모리 공간 분산된 메모리 공간 용량 확장 길이를 바꿀 수 없음 유연하게 확장 가능 메모리 효율 원소의 메모리 사용량은 적지만 공간을 낭비할 수 있음 원소의 메모리 사용량이 많음 원소 접근 \\(O(1)\\) \\(O(n)\\) 원소 추가 \\(O(n)\\) \\(O(1)\\) 원소 삭제 \\(O(n)\\) \\(O(1)\\)","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#423","level":2,"title":"4.2.3   대표적인 연결 리스트 종류","text":"<p>아래 그림(4-8)과 같이 대표적인 연결 리스트에는 세 종류가 있다.</p> <ul> <li>단일 연결 리스트: 앞에서 소개한 일반적인 연결 리스트다. 각 노드는 값과 다음 노드를 가리키는 참조라는 두 데이터를 포함한다. 첫 노드는 머리 노드, 마지막 노드는 꼬리 노드이며, 꼬리 노드는 <code>None</code>을 가리킨다.</li> <li>원형 연결 리스트: 단일 연결 리스트의 꼬리 노드가 머리 노드를 가리키도록 앞뒤를 이으면 원형 연결 리스트가 된다. 원형 연결 리스트에서는 어느 노드든 머리 노드로 볼 수 있다.</li> <li>이중 연결 리스트: 단일 연결 리스트와 달리 양방향 참조를 기록한다. 각 노드는 후속 노드(다음 노드)와 선행 노드(이전 노드)를 가리키는 참조(포인터)를 모두 포함한다. 양방향으로 순회할 수 있어 단일 연결 리스트보다 유연하지만, 그만큼 메모리 공간도 더 많이 차지한다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class ListNode:\n    \"\"\"이중 연결 리스트 노드 클래스\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # 노드 값\n        self.next: ListNode | None = None  # 후속 노드를 가리키는 참조\n        self.prev: ListNode | None = None  # 선행 노드를 가리키는 참조\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 구조체 */\nstruct ListNode {\n    int val;         // 노드 값\n    ListNode *next;  // 후속 노드를 가리키는 포인터\n    ListNode *prev;  // 선행 노드를 가리키는 포인터\n    ListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // 생성자\n};\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 클래스 */\nclass ListNode {\n    int val;        // 노드 값\n    ListNode next;  // 후속 노드를 가리키는 참조\n    ListNode prev;  // 선행 노드를 가리키는 참조\n    ListNode(int x) { val = x; }  // 생성자\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 클래스 */\nclass ListNode(int x) {  // 생성자\n    int val = x;    // 노드 값\n    ListNode next;  // 후속 노드를 가리키는 참조\n    ListNode prev;  // 선행 노드를 가리키는 참조\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 구조체 */\ntype DoublyListNode struct {\n    Val  int             // 노드 값\n    Next *DoublyListNode // 후속 노드를 가리키는 포인터\n    Prev *DoublyListNode // 선행 노드를 가리키는 포인터\n}\n\n// NewDoublyListNode 초기화\nfunc NewDoublyListNode(val int) *DoublyListNode {\n    return &amp;DoublyListNode{\n        Val:  val,\n        Next: nil,\n        Prev: nil,\n    }\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 클래스 */\nclass ListNode {\n    var val: Int // 노드 값\n    var next: ListNode? // 후속 노드를 가리키는 참조\n    var prev: ListNode? // 선행 노드를 가리키는 참조\n\n    init(x: Int) { // 생성자\n        val = x\n    }\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 클래스 */\nclass ListNode {\n    constructor(val, next, prev) {\n        this.val = val  ===  undefined ? 0 : val;        // 노드 값\n        this.next = next  ===  undefined ? null : next;  // 후속 노드를 가리키는 참조\n        this.prev = prev  ===  undefined ? null : prev;  // 선행 노드를 가리키는 참조\n    }\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 클래스 */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    prev: ListNode | null;\n    constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\n        this.val = val  ===  undefined ? 0 : val;        // 노드 값\n        this.next = next  ===  undefined ? null : next;  // 후속 노드를 가리키는 참조\n        this.prev = prev  ===  undefined ? null : prev;  // 선행 노드를 가리키는 참조\n    }\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 클래스 */\nclass ListNode {\n    int val;        // 노드 값\n    ListNode? next;  // 후속 노드를 가리키는 참조\n    ListNode? prev;  // 선행 노드를 가리키는 참조\n    ListNode(this.val, [this.next, this.prev]);  // 생성자\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* 이중 연결 리스트 노드 자료형 */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // 노드 값\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // 후속 노드를 가리키는 포인터\n    prev: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // 선행 노드를 가리키는 포인터\n}\n\n/* 생성자 */\nimpl ListNode {\n    fn new(val: i32) -&gt; Self {\n        ListNode {\n            val,\n            next: None,\n            prev: None,\n        }\n    }\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 구조체 */\ntypedef struct ListNode {\n    int val;               // 노드 값\n    struct ListNode *next; // 후속 노드를 가리키는 포인터\n    struct ListNode *prev; // 선행 노드를 가리키는 포인터\n} ListNode;\n\n/* 생성자 */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    node-&gt;prev = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 클래스 */\n// 생성자\nclass ListNode(x: Int) {\n    val _val: Int = x           // 노드 값\n    var next: ListNode? = null  // 후속 노드를 가리키는 참조\n    var prev: ListNode? = null  // 선행 노드를 가리키는 참조\n}\n</code></pre> <pre><code># 이중 연결 리스트 노드 클래스\nclass ListNode\n  attr_accessor :val    # 노드 값\n  attr_accessor :next   # 후속 노드를 가리키는 참조\n  attr_accessor :prev   # 선행 노드를 가리키는 참조\n\n  def initialize(val=0, next_node=nil, prev_node=nil)\n    @val = val\n    @next = next_node\n    @prev = prev_node\n  end\nend\n</code></pre> <p></p> <p> 그림 4-8   대표적인 연결 리스트 종류 </p>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#424","level":2,"title":"4.2.4   연결 리스트의 대표적인 활용","text":"<p>단일 연결 리스트는 보통 스택, 큐, 해시 테이블, 그래프 등의 자료 구조를 구현하는 데 사용된다.</p> <ul> <li>스택과 큐: 삽입과 삭제를 모두 연결 리스트의 같은 끝에서 수행하면 후입선출 특성이 나타나 스택에 해당한다. 한쪽 끝에서 삽입하고 반대쪽 끝에서 삭제하면 선입선출 특성이 나타나 큐에 해당한다.</li> <li>해시 테이블: 체이닝은 해시 충돌을 해결하는 대표적인 방법 중 하나로, 충돌하는 모든 원소를 하나의 연결 리스트에 넣는다.</li> <li>그래프: 인접 리스트는 그래프를 표현하는 흔한 방법이다. 각 정점에 연결 리스트 하나를 연결하고, 그 리스트의 각 원소로 해당 정점과 연결된 다른 정점을 나타낸다.</li> </ul> <p>이중 연결 리스트는 이전 원소와 다음 원소를 빠르게 찾아야 하는 상황에 자주 사용된다.</p> <ul> <li>고급 자료 구조: 레드-블랙 트리나 B 트리에서는 노드의 부모 노드에 접근해야 한다. 이중 연결 리스트처럼 노드에 부모를 가리키는 참조를 저장하여 구현할 수 있다.</li> <li>브라우저 방문 기록: 사용자가 웹 브라우저의 앞으로 또는 뒤로 버튼을 누르면 이전이나 다음에 방문했던 페이지를 알아야 한다. 이중 연결 리스트를 사용하면 이러한 연산이 간단해진다.</li> <li>LRU 알고리즘: 캐시 교체 알고리즘인 LRU에서는 가장 오랫동안 사용하지 않은 데이터를 빠르게 찾고 노드도 빠르게 추가하고 삭제해야 한다. 이때 이중 연결 리스트가 매우 적합하다.</li> </ul> <p>원형 연결 리스트는 운영체제의 자원 스케줄링처럼 주기적인 연산이 필요한 상황에 자주 사용된다.</p> <ul> <li>라운드 로빈 스케줄링 알고리즘: 운영체제의 대표적인 CPU 스케줄링 알고리즘으로, 여러 프로세스를 순환한다. 각 프로세스에 타임 슬라이스를 할당하고, 시간이 다 되면 CPU가 다음 프로세스로 전환한다. 이러한 순환은 원형 연결 리스트로 구현할 수 있다.</li> <li>데이터 버퍼: 일부 데이터 버퍼도 원형 연결 리스트로 구현할 수 있다. 예를 들어 오디오나 동영상 플레이어에서 데이터 스트림을 여러 버퍼 블록으로 나누어 원형 연결 리스트에 넣으면 끊김 없이 재생할 수 있다.</li> </ul>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/","level":1,"title":"4.3   리스트","text":"<p>리스트(list)는 순서가 있는 원소의 모음을 나타내는 추상적인 자료 구조 개념이다. 원소 접근, 수정, 추가, 삭제, 순회 등의 연산을 지원하며, 사용자가 용량 제한을 신경 쓰지 않아도 된다. 리스트는 연결 리스트나 배열로 구현할 수 있다.</p> <ul> <li>연결 리스트는 원소의 삽입, 삭제, 탐색, 갱신을 지원하고 유연하게 용량을 늘릴 수 있으므로 자연스럽게 리스트로 볼 수 있다.</li> <li>배열도 원소의 삽입, 삭제, 탐색, 갱신을 지원하지만 길이를 바꿀 수 없으므로 길이가 제한된 리스트로 볼 수 있다.</li> </ul> <p>배열로 리스트를 구현하면 길이를 바꿀 수 없다는 특성 때문에 실용성이 낮아진다. 보통 저장할 데이터 양을 미리 알 수 없어 적절한 길이를 정하기 어렵기 때문이다. 길이가 너무 짧으면 필요를 충족하지 못하고, 너무 길면 메모리 공간을 낭비한다.</p> <p>이 문제는 동적 배열(dynamic array)로 리스트를 구현하여 해결할 수 있다. 동적 배열은 배열의 장점을 이어받으면서도 프로그램 실행 중에 용량을 동적으로 늘릴 수 있다.</p> <p>실제로 많은 프로그래밍 언어의 표준 라이브러리가 제공하는 리스트는 동적 배열로 구현된다. Python의 <code>list</code>, Java의 <code>ArrayList</code>, C++의 <code>vector</code>, C#의 <code>List</code> 등이 그 예다. 이후 설명에서는 ‘리스트’와 ‘동적 배열’을 같은 개념으로 다룬다.</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#431","level":2,"title":"4.3.1   리스트의 기본 연산","text":"","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#1","level":3,"title":"1.   리스트 초기화","text":"<p>보통 ‘초깃값 없음’과 ‘초깃값 지정’의 두 가지 초기화 방식을 사용한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># 리스트 초기화\n# 초깃값 없음\nnums1: list[int] = []\n# 초깃값 지정\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> list.cpp<pre><code>/* 리스트 초기화 */\n// C++의 vector가 이 글에서 설명하는 nums에 해당한다\n// 초깃값 없음\nvector&lt;int&gt; nums1;\n// 초깃값 지정\nvector&lt;int&gt; nums = { 1, 3, 2, 5, 4 };\n</code></pre> list.java<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;();\n// 초깃값 지정(배열은 int[]의 래퍼형인 Integer[]여야 함)\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList&lt;Integer&gt; nums = new ArrayList&lt;&gt;(Arrays.asList(numbers));\n</code></pre> list.cs<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nList&lt;int&gt; nums1 = [];\n// 초깃값 지정\nint[] numbers = [1, 3, 2, 5, 4];\nList&lt;int&gt; nums = [.. numbers];\n</code></pre> list_test.go<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nnums1 := []int{}\n// 초깃값 지정\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> list.swift<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nlet nums1: [Int] = []\n// 초깃값 지정\nvar nums = [1, 3, 2, 5, 4]\n</code></pre> list.js<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nconst nums1 = [];\n// 초깃값 지정\nconst nums = [1, 3, 2, 5, 4];\n</code></pre> list.ts<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nconst nums1: number[] = [];\n// 초깃값 지정\nconst nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> list.dart<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nList&lt;int&gt; nums1 = [];\n// 초깃값 지정\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> list.rs<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nlet nums1: Vec&lt;i32&gt; = Vec::new();\n// 초깃값 지정\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> list.c<pre><code>// C는 내장 동적 배열을 제공하지 않는다\n</code></pre> list.kt<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nvar nums1 = listOf&lt;Int&gt;()\n// 초깃값 지정\nvar numbers = arrayOf(1, 3, 2, 5, 4)\nvar nums = numbers.toMutableList()\n</code></pre> list.rb<pre><code># 리스트 초기화\n# 초깃값 없음\nnums1 = []\n# 초깃값 지정\nnums = [1, 3, 2, 5, 4]\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#2","level":3,"title":"2.   원소 접근","text":"<p>리스트는 본질적으로 배열이므로 원소에 접근하고 갱신하는 데 \\(O(1)\\) 시간이 걸려 매우 효율적이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># 원소 접근\nnum: int = nums[1]  # 인덱스 1의 원소에 접근\n\n# 원소 갱신\nnums[1] = 0    # 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.cpp<pre><code>/* 원소 접근 */\nint num = nums[1];  // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums[1] = 0;  // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.java<pre><code>/* 원소 접근 */\nint num = nums.get(1);  // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums.set(1, 0);  // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.cs<pre><code>/* 원소 접근 */\nint num = nums[1];  // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums[1] = 0;  // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list_test.go<pre><code>/* 원소 접근 */\nnum := nums[1]  // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums[1] = 0     // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.swift<pre><code>/* 원소 접근 */\nlet num = nums[1] // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums[1] = 0 // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.js<pre><code>/* 원소 접근 */\nconst num = nums[1];  // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums[1] = 0;  // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.ts<pre><code>/* 원소 접근 */\nconst num: number = nums[1];  // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums[1] = 0;  // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.dart<pre><code>/* 원소 접근 */\nint num = nums[1];  // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums[1] = 0;  // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.rs<pre><code>/* 원소 접근 */\nlet num: i32 = nums[1];  // 인덱스 1의 원소에 접근\n/* 원소 갱신 */\nnums[1] = 0;             // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.c<pre><code>// C는 내장 동적 배열을 제공하지 않는다\n</code></pre> list.kt<pre><code>/* 원소 접근 */\nval num = nums[1]       // 인덱스 1의 원소에 접근\n/* 원소 갱신 */\nnums[1] = 0             // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.rb<pre><code># 원소 접근\nnum = nums[1] # 인덱스 1의 원소에 접근\n# 원소 갱신\nnums[1] = 0 # 인덱스 1의 원소를 0으로 갱신\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#3","level":3,"title":"3.   원소 삽입과 삭제","text":"<p>배열과 달리 리스트는 원소를 자유롭게 추가하고 삭제할 수 있다. 끝에 원소를 추가하는 시간 복잡도는 \\(O(1)\\)이지만, 삽입과 삭제의 효율은 배열과 같아 시간 복잡도가 \\(O(n)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># 리스트 비우기\nnums.clear()\n\n# 끝에 원소 추가\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n# 중간에 원소 삽입\nnums.insert(3, 6)  # 인덱스 3에 숫자 6 삽입\n\n# 원소 삭제\nnums.pop(3)        # 인덱스 3의 원소 삭제\n</code></pre> list.cpp<pre><code>/* 리스트 비우기 */\nnums.clear();\n\n/* 끝에 원소 추가 */\nnums.push_back(1);\nnums.push_back(3);\nnums.push_back(2);\nnums.push_back(5);\nnums.push_back(4);\n\n/* 중간에 원소 삽입 */\nnums.insert(nums.begin() + 3, 6);  // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.erase(nums.begin() + 3);      // 인덱스 3의 원소 삭제\n</code></pre> list.java<pre><code>/* 리스트 비우기 */\nnums.clear();\n\n/* 끝에 원소 추가 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* 중간에 원소 삽입 */\nnums.add(3, 6);  // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.remove(3);  // 인덱스 3의 원소 삭제\n</code></pre> list.cs<pre><code>/* 리스트 비우기 */\nnums.Clear();\n\n/* 끝에 원소 추가 */\nnums.Add(1);\nnums.Add(3);\nnums.Add(2);\nnums.Add(5);\nnums.Add(4);\n\n/* 중간에 원소 삽입 */\nnums.Insert(3, 6);  // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.RemoveAt(3);  // 인덱스 3의 원소 삭제\n</code></pre> list_test.go<pre><code>/* 리스트 비우기 */\nnums = nil\n\n/* 끝에 원소 추가 */\nnums = append(nums, 1)\nnums = append(nums, 3)\nnums = append(nums, 2)\nnums = append(nums, 5)\nnums = append(nums, 4)\n\n/* 중간에 원소 삽입 */\nnums = append(nums[:3], append([]int{6}, nums[3:]...)...) // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums = append(nums[:3], nums[4:]...) // 인덱스 3의 원소 삭제\n</code></pre> list.swift<pre><code>/* 리스트 비우기 */\nnums.removeAll()\n\n/* 끝에 원소 추가 */\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n/* 중간에 원소 삽입 */\nnums.insert(6, at: 3) // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.remove(at: 3) // 인덱스 3의 원소 삭제\n</code></pre> list.js<pre><code>/* 리스트 비우기 */\nnums.length = 0;\n\n/* 끝에 원소 추가 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* 중간에 원소 삽입 */\nnums.splice(3, 0, 6); // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.splice(3, 1);  // 인덱스 3의 원소 삭제\n</code></pre> list.ts<pre><code>/* 리스트 비우기 */\nnums.length = 0;\n\n/* 끝에 원소 추가 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* 중간에 원소 삽입 */\nnums.splice(3, 0, 6); // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.splice(3, 1);  // 인덱스 3의 원소 삭제\n</code></pre> list.dart<pre><code>/* 리스트 비우기 */\nnums.clear();\n\n/* 끝에 원소 추가 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* 중간에 원소 삽입 */\nnums.insert(3, 6); // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.removeAt(3); // 인덱스 3의 원소 삭제\n</code></pre> list.rs<pre><code>/* 리스트 비우기 */\nnums.clear();\n\n/* 끝에 원소 추가 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* 중간에 원소 삽입 */\nnums.insert(3, 6);  // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.remove(3);    // 인덱스 3의 원소 삭제\n</code></pre> list.c<pre><code>// C는 내장 동적 배열을 제공하지 않는다\n</code></pre> list.kt<pre><code>/* 리스트 비우기 */\nnums.clear();\n\n/* 끝에 원소 추가 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* 중간에 원소 삽입 */\nnums.add(3, 6);  // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.remove(3);  // 인덱스 3의 원소 삭제\n</code></pre> list.rb<pre><code># 리스트 비우기\nnums.clear\n\n# 끝에 원소 추가\nnums &lt;&lt; 1\nnums &lt;&lt; 3\nnums &lt;&lt; 2\nnums &lt;&lt; 5\nnums &lt;&lt; 4\n\n# 중간에 원소 삽입\nnums.insert(3, 6) # 인덱스 3에 숫자 6 삽입\n\n# 원소 삭제\nnums.delete_at(3) # 인덱스 3의 원소 삭제\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#4","level":3,"title":"4.   리스트 순회","text":"<p>배열처럼 리스트도 인덱스로 순회하거나 각 원소를 직접 순회할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># 인덱스로 리스트 순회\ncount = 0\nfor i in range(len(nums)):\n    count += nums[i]\n\n# 리스트 원소를 직접 순회\nfor num in nums:\n    count += num\n</code></pre> list.cpp<pre><code>/* 인덱스로 리스트 순회 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums[i];\n}\n\n/* 리스트 원소를 직접 순회 */\ncount = 0;\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.java<pre><code>/* 인덱스로 리스트 순회 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums.get(i);\n}\n\n/* 리스트 원소를 직접 순회 */\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.cs<pre><code>/* 인덱스로 리스트 순회 */\nint count = 0;\nfor (int i = 0; i &lt; nums.Count; i++) {\n    count += nums[i];\n}\n\n/* 리스트 원소를 직접 순회 */\ncount = 0;\nforeach (int num in nums) {\n    count += num;\n}\n</code></pre> list_test.go<pre><code>/* 인덱스로 리스트 순회 */\ncount := 0\nfor i := 0; i &lt; len(nums); i++ {\n    count += nums[i]\n}\n\n/* 리스트 원소를 직접 순회 */\ncount = 0\nfor _, num := range nums {\n    count += num\n}\n</code></pre> list.swift<pre><code>/* 인덱스로 리스트 순회 */\nvar count = 0\nfor i in nums.indices {\n    count += nums[i]\n}\n\n/* 리스트 원소를 직접 순회 */\ncount = 0\nfor num in nums {\n    count += num\n}\n</code></pre> list.js<pre><code>/* 인덱스로 리스트 순회 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* 리스트 원소를 직접 순회 */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.ts<pre><code>/* 인덱스로 리스트 순회 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* 리스트 원소를 직접 순회 */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.dart<pre><code>/* 인덱스로 리스트 순회 */\nint count = 0;\nfor (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* 리스트 원소를 직접 순회 */\ncount = 0;\nfor (var num in nums) {\n    count += num;\n}\n</code></pre> list.rs<pre><code>// 인덱스로 리스트 순회\nlet mut _count = 0;\nfor i in 0..nums.len() {\n    _count += nums[i];\n}\n\n// 리스트 원소를 직접 순회\n_count = 0;\nfor num in &amp;nums {\n    _count += num;\n}\n</code></pre> list.c<pre><code>// C는 내장 동적 배열을 제공하지 않는다\n</code></pre> list.kt<pre><code>/* 인덱스로 리스트 순회 */\nvar count = 0\nfor (i in nums.indices) {\n    count += nums[i]\n}\n\n/* 리스트 원소를 직접 순회 */\nfor (num in nums) {\n    count += num\n}\n</code></pre> list.rb<pre><code># 인덱스로 리스트 순회\ncount = 0\nfor i in 0...nums.length\n    count += nums[i]\nend\n\n# 리스트 원소를 직접 순회\ncount = 0\nfor num in nums\n    count += num\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#5","level":3,"title":"5.   리스트 연결","text":"<p>새 리스트 <code>nums1</code>이 주어지면 기존 리스트의 끝에 이어 붙일 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># 두 리스트 연결\nnums1: list[int] = [6, 8, 7, 10, 9]\nnums += nums1  # 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.cpp<pre><code>/* 두 리스트 연결 */\nvector&lt;int&gt; nums1 = { 6, 8, 7, 10, 9 };\n// 리스트 nums1을 nums 뒤에 연결\nnums.insert(nums.end(), nums1.begin(), nums1.end());\n</code></pre> list.java<pre><code>/* 두 리스트 연결 */\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nnums.addAll(nums1);  // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.cs<pre><code>/* 두 리스트 연결 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.AddRange(nums1);  // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list_test.go<pre><code>/* 두 리스트 연결 */\nnums1 := []int{6, 8, 7, 10, 9}\nnums = append(nums, nums1...)  // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.swift<pre><code>/* 두 리스트 연결 */\nlet nums1 = [6, 8, 7, 10, 9]\nnums.append(contentsOf: nums1) // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.js<pre><code>/* 두 리스트 연결 */\nconst nums1 = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.ts<pre><code>/* 두 리스트 연결 */\nconst nums1: number[] = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.dart<pre><code>/* 두 리스트 연결 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.addAll(nums1);  // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.rs<pre><code>/* 두 리스트 연결 */\nlet nums1: Vec&lt;i32&gt; = vec![6, 8, 7, 10, 9];\nnums.extend(nums1);\n</code></pre> list.c<pre><code>// C는 내장 동적 배열을 제공하지 않는다\n</code></pre> list.kt<pre><code>/* 두 리스트 연결 */\nval nums1 = intArrayOf(6, 8, 7, 10, 9).toMutableList()\nnums.addAll(nums1)  // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.rb<pre><code># 두 리스트 연결\nnums1 = [6, 8, 7, 10, 9]\nnums += nums1\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#6","level":3,"title":"6.   리스트 정렬","text":"<p>리스트를 정렬하면 배열 관련 알고리즘 문제에 자주 등장하는 ‘이진 탐색’과 ‘투 포인터’ 알고리즘을 사용할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># 리스트 정렬\nnums.sort()  # 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.cpp<pre><code>/* 리스트 정렬 */\nsort(nums.begin(), nums.end());  // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.java<pre><code>/* 리스트 정렬 */\nCollections.sort(nums);  // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.cs<pre><code>/* 리스트 정렬 */\nnums.Sort(); // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list_test.go<pre><code>/* 리스트 정렬 */\nsort.Ints(nums)  // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.swift<pre><code>/* 리스트 정렬 */\nnums.sort() // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.js<pre><code>/* 리스트 정렬 */\nnums.sort((a, b) =&gt; a - b);  // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.ts<pre><code>/* 리스트 정렬 */\nnums.sort((a, b) =&gt; a - b);  // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.dart<pre><code>/* 리스트 정렬 */\nnums.sort(); // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.rs<pre><code>/* 리스트 정렬 */\nnums.sort(); // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.c<pre><code>// C는 내장 동적 배열을 제공하지 않는다\n</code></pre> list.kt<pre><code>/* 리스트 정렬 */\nnums.sort() // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.rb<pre><code># 리스트 정렬\nnums = nums.sort { |a, b| a &lt;=&gt; b } # 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#432","level":2,"title":"4.3.2   리스트 구현","text":"<p>Java, C++, Python 등 많은 프로그래밍 언어는 리스트를 내장하고 있다. 이들의 구현은 비교적 복잡하며 초기 용량, 확장 배수 등의 매개변수도 정교하게 설정되어 있다. 관심 있는 독자는 소스 코드를 살펴보며 학습할 수 있다.</p> <p>리스트의 작동 원리를 더 깊이 이해하기 위해 간단한 리스트를 구현해 보자. 핵심 설계는 다음 세 가지다.</p> <ul> <li>초기 용량: 배열의 초기 용량을 적절히 선택한다. 이 예에서는 10을 초기 용량으로 정한다.</li> <li>개수 기록: 현재 원소 수를 기록하는 변수 <code>size</code>를 선언하고, 삽입과 삭제에 따라 즉시 갱신한다. 이 변수를 통해 리스트의 끝을 찾고 용량 확장이 필요한지 판단할 수 있다.</li> <li>용량 확장 방식: 원소를 삽입할 때 리스트가 가득 찼으면 용량을 늘려야 한다. 확장 배수에 따라 더 큰 배열을 만든 뒤 기존 원소를 모두 차례로 옮긴다. 이 예에서는 매번 배열 용량을 이전의 2배로 늘린다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_list.py<pre><code>class MyList:\n    \"\"\"리스트 클래스\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        self._capacity: int = 10  # 리스트 용량\n        self._arr: list[int] = [0] * self._capacity  # 배열(리스트 원소 저장)\n        self._size: int = 0  # 리스트 길이(현재 원소 개수)\n        self._extend_ratio: int = 2  # 리스트의 용량을 확장할 때마다 적용하는 배수\n\n    def size(self) -&gt; int:\n        \"\"\"리스트 길이(현재 원소 개수) 반환\"\"\"\n        return self._size\n\n    def capacity(self) -&gt; int:\n        \"\"\"리스트 용량 반환\"\"\"\n        return self._capacity\n\n    def get(self, index: int) -&gt; int:\n        \"\"\"원소 접근\"\"\"\n        # 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"인덱스가 범위를 벗어났다\")\n        return self._arr[index]\n\n    def set(self, num: int, index: int):\n        \"\"\"원소 갱신\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"인덱스가 범위를 벗어났다\")\n        self._arr[index] = num\n\n    def add(self, num: int):\n        \"\"\"맨 뒤에 원소 추가\"\"\"\n        # 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if self.size() == self.capacity():\n            self.extend_capacity()\n        self._arr[self._size] = num\n        self._size += 1\n\n    def insert(self, num: int, index: int):\n        \"\"\"중간에 원소 삽입\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"인덱스가 범위를 벗어났다\")\n        # 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if self._size == self.capacity():\n            self.extend_capacity()\n        # 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for j in range(self._size - 1, index - 1, -1):\n            self._arr[j + 1] = self._arr[j]\n        self._arr[index] = num\n        # 원소 개수를 갱신한다\n        self._size += 1\n\n    def remove(self, index: int) -&gt; int:\n        \"\"\"원소 삭제\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"인덱스가 범위를 벗어났다\")\n        num = self._arr[index]\n        # 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for j in range(index, self._size - 1):\n            self._arr[j] = self._arr[j + 1]\n        # 원소 개수를 갱신한다\n        self._size -= 1\n        # 삭제한 원소를 반환한다\n        return num\n\n    def extend_capacity(self):\n        \"\"\"리스트 용량 확장\"\"\"\n        # 원래 배열 길이의 _extend_ratio 배인 새 배열을 만들고 원래 배열을 새 배열에 복사한다\n        self._arr = self._arr + [0] * self.capacity() * (self._extend_ratio - 1)\n        # 리스트 용량을 갱신한다\n        self._capacity = len(self._arr)\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"유효한 길이만큼의 리스트 반환\"\"\"\n        return self._arr[: self._size]\n</code></pre> my_list.cpp<pre><code>/* 리스트 클래스 */\nclass MyList {\n  private:\n    int *arr;             // 배열(리스트 원소 저장)\n    int arrCapacity = 10; // 리스트 용량\n    int arrSize = 0;      // 리스트 길이(현재 원소 개수)\n    int extendRatio = 2;   // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n  public:\n    /* 생성자 */\n    MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* 소멸자 */\n    ~MyList() {\n        delete[] arr;\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환*/\n    int size() {\n        return arrSize;\n    }\n\n    /* 리스트 용량 반환 */\n    int capacity() {\n        return arrCapacity;\n    }\n\n    /* 원소 접근 */\n    int get(int index) {\n        // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"인덱스가 범위를 벗어났다\");\n        return arr[index];\n    }\n\n    /* 원소 갱신 */\n    void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"인덱스가 범위를 벗어났다\");\n        arr[index] = num;\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    void add(int num) {\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (size() == capacity())\n            extendCapacity();\n        arr[size()] = num;\n        // 원소 개수를 갱신한다\n        arrSize++;\n    }\n\n    /* 중간에 원소 삽입 */\n    void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"인덱스가 범위를 벗어났다\");\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (size() == capacity())\n            extendCapacity();\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for (int j = size() - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // 원소 개수를 갱신한다\n        arrSize++;\n    }\n\n    /* 원소 삭제 */\n    int remove(int index) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"인덱스가 범위를 벗어났다\");\n        int num = arr[index];\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for (int j = index; j &lt; size() - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // 원소 개수를 갱신한다\n        arrSize--;\n        // 삭제한 원소를 반환한다\n        return num;\n    }\n\n    /* 리스트 용량 확장 */\n    void extendCapacity() {\n        // 길이가 기존 배열의 extendRatio배인 새 배열 생성\n        int newCapacity = capacity() * extendRatio;\n        int *tmp = arr;\n        arr = new int[newCapacity];\n        // 원래 배열의 모든 원소를 새 배열에 복사한다\n        for (int i = 0; i &lt; size(); i++) {\n            arr[i] = tmp[i];\n        }\n        // 메모리 해제\n        delete[] tmp;\n        arrCapacity = newCapacity;\n    }\n\n    /* 출력을 위해 리스트를 Vector로 변환 */\n    vector&lt;int&gt; toVector() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        vector&lt;int&gt; vec(size());\n        for (int i = 0; i &lt; size(); i++) {\n            vec[i] = arr[i];\n        }\n        return vec;\n    }\n};\n</code></pre> my_list.java<pre><code>/* 리스트 클래스 */\nclass MyList {\n    private int[] arr; // 배열(리스트 원소 저장)\n    private int capacity = 10; // 리스트 용량\n    private int size = 0; // 리스트 길이(현재 원소 개수)\n    private int extendRatio = 2; // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n    /* 생성자 */\n    public MyList() {\n        arr = new int[capacity];\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환 */\n    public int size() {\n        return size;\n    }\n\n    /* 리스트 용량 반환 */\n    public int capacity() {\n        return capacity;\n    }\n\n    /* 원소 접근 */\n    public int get(int index) {\n        // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\");\n        return arr[index];\n    }\n\n    /* 원소 갱신 */\n    public void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\");\n        arr[index] = num;\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    public void add(int num) {\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (size == capacity())\n            extendCapacity();\n        arr[size] = num;\n        // 원소 개수를 갱신한다\n        size++;\n    }\n\n    /* 중간에 원소 삽입 */\n    public void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\");\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (size == capacity())\n            extendCapacity();\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for (int j = size - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // 원소 개수를 갱신한다\n        size++;\n    }\n\n    /* 원소 삭제 */\n    public int remove(int index) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\");\n        int num = arr[index];\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for (int j = index; j &lt; size - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // 원소 개수를 갱신한다\n        size--;\n        // 삭제한 원소를 반환한다\n        return num;\n    }\n\n    /* 리스트 용량 확장 */\n    public void extendCapacity() {\n        // 길이가 원래 배열의 extendRatio배인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n        arr = Arrays.copyOf(arr, capacity() * extendRatio);\n        // 리스트 용량을 갱신한다\n        capacity = arr.length;\n    }\n\n    /* 리스트를 배열로 변환 */\n    public int[] toArray() {\n        int size = size();\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        int[] arr = new int[size];\n        for (int i = 0; i &lt; size; i++) {\n            arr[i] = get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.cs<pre><code>/* 리스트 클래스 */\nclass MyList {\n    private int[] arr;           // 배열(리스트 원소 저장)\n    private int arrCapacity = 10;    // 리스트 용량\n    private int arrSize = 0;         // 리스트 길이(현재 원소 개수)\n    private readonly int extendRatio = 2;  // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n    /* 생성자 */\n    public MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환*/\n    public int Size() {\n        return arrSize;\n    }\n\n    /* 리스트 용량 반환 */\n    public int Capacity() {\n        return arrCapacity;\n    }\n\n    /* 원소 접근 */\n    public int Get(int index) {\n        // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"인덱스가 범위를 벗어났다\");\n        return arr[index];\n    }\n\n    /* 원소 갱신 */\n    public void Set(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"인덱스가 범위를 벗어났다\");\n        arr[index] = num;\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    public void Add(int num) {\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        arr[arrSize] = num;\n        // 원소 개수를 갱신한다\n        arrSize++;\n    }\n\n    /* 중간에 원소 삽입 */\n    public void Insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"인덱스가 범위를 벗어났다\");\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for (int j = arrSize - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // 원소 개수를 갱신한다\n        arrSize++;\n    }\n\n    /* 원소 삭제 */\n    public int Remove(int index) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"인덱스가 범위를 벗어났다\");\n        int num = arr[index];\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for (int j = index; j &lt; arrSize - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // 원소 개수를 갱신한다\n        arrSize--;\n        // 삭제한 원소를 반환한다\n        return num;\n    }\n\n    /* 리스트 용량 확장 */\n    public void ExtendCapacity() {\n        // 길이가 arrCapacity * extendRatio인 배열을 만들고 기존 배열을 새 배열에 복사\n        Array.Resize(ref arr, arrCapacity * extendRatio);\n        // 리스트 용량을 갱신한다\n        arrCapacity = arr.Length;\n    }\n\n    /* 리스트를 배열로 변환 */\n    public int[] ToArray() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        int[] arr = new int[arrSize];\n        for (int i = 0; i &lt; arrSize; i++) {\n            arr[i] = Get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.go<pre><code>/* 리스트 클래스 */\ntype myList struct {\n    arrCapacity int\n    arr         []int\n    arrSize     int\n    extendRatio int\n}\n\n/* 생성자 */\nfunc newMyList() *myList {\n    return &amp;myList{\n        arrCapacity: 10,              // 리스트 용량\n        arr:         make([]int, 10), // 배열(리스트 원소 저장)\n        arrSize:     0,               // 리스트 길이(현재 원소 개수)\n        extendRatio: 2,               // 리스트의 용량을 확장할 때마다 적용하는 배수\n    }\n}\n\n/* 리스트 길이(현재 원소 개수) 반환 */\nfunc (l *myList) size() int {\n    return l.arrSize\n}\n\n/*  리스트 용량 반환 */\nfunc (l *myList) capacity() int {\n    return l.arrCapacity\n}\n\n/* 원소 접근 */\nfunc (l *myList) get(index int) int {\n    // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"인덱스가 범위를 벗어났다\")\n    }\n    return l.arr[index]\n}\n\n/* 원소 갱신 */\nfunc (l *myList) set(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"인덱스가 범위를 벗어났다\")\n    }\n    l.arr[index] = num\n}\n\n/* 맨 뒤에 원소 추가 */\nfunc (l *myList) add(num int) {\n    // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    l.arr[l.arrSize] = num\n    // 원소 개수를 갱신한다\n    l.arrSize++\n}\n\n/* 중간에 원소 삽입 */\nfunc (l *myList) insert(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"인덱스가 범위를 벗어났다\")\n    }\n    // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n    for j := l.arrSize - 1; j &gt;= index; j-- {\n        l.arr[j+1] = l.arr[j]\n    }\n    l.arr[index] = num\n    // 원소 개수를 갱신한다\n    l.arrSize++\n}\n\n/* 원소 삭제 */\nfunc (l *myList) remove(index int) int {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"인덱스가 범위를 벗어났다\")\n    }\n    num := l.arr[index]\n    // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n    for j := index; j &lt; l.arrSize-1; j++ {\n        l.arr[j] = l.arr[j+1]\n    }\n    // 원소 개수를 갱신한다\n    l.arrSize--\n    // 삭제한 원소를 반환한다\n    return num\n}\n\n/* 리스트 용량 확장 */\nfunc (l *myList) extendCapacity() {\n    // 길이가 원래 배열의 extendRatio배인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n    l.arr = append(l.arr, make([]int, l.arrCapacity*(l.extendRatio-1))...)\n    // 리스트 용량을 갱신한다\n    l.arrCapacity = len(l.arr)\n}\n\n/* 유효한 길이만큼의 리스트 반환 */\nfunc (l *myList) toArray() []int {\n    // 유효한 길이 범위 안의 리스트 원소만 변환한다\n    return l.arr[:l.arrSize]\n}\n</code></pre> my_list.swift<pre><code>/* 리스트 클래스 */\nclass MyList {\n    private var arr: [Int] // 배열(리스트 원소 저장)\n    private var _capacity: Int // 리스트 용량\n    private var _size: Int // 리스트 길이(현재 원소 개수)\n    private let extendRatio: Int // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n    /* 생성자 */\n    init() {\n        _capacity = 10\n        _size = 0\n        extendRatio = 2\n        arr = Array(repeating: 0, count: _capacity)\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환*/\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* 리스트 용량 반환 */\n    func capacity() -&gt; Int {\n        _capacity\n    }\n\n    /* 원소 접근 */\n    func get(index: Int) -&gt; Int {\n        // 인덱스가 범위를 벗어나면 오류를 발생시킨다. 아래에서도 동일하다\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"인덱스가 범위를 벗어났다\")\n        }\n        return arr[index]\n    }\n\n    /* 원소 갱신 */\n    func set(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"인덱스가 범위를 벗어났다\")\n        }\n        arr[index] = num\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    func add(num: Int) {\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if size() == capacity() {\n            extendCapacity()\n        }\n        arr[size()] = num\n        // 원소 개수를 갱신한다\n        _size += 1\n    }\n\n    /* 중간에 원소 삽입 */\n    func insert(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"인덱스가 범위를 벗어났다\")\n        }\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if size() == capacity() {\n            extendCapacity()\n        }\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for j in (index ..&lt; size()).reversed() {\n            arr[j + 1] = arr[j]\n        }\n        arr[index] = num\n        // 원소 개수를 갱신한다\n        _size += 1\n    }\n\n    /* 원소 삭제 */\n    @discardableResult\n    func remove(index: Int) -&gt; Int {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"인덱스가 범위를 벗어났다\")\n        }\n        let num = arr[index]\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for j in index ..&lt; (size() - 1) {\n            arr[j] = arr[j + 1]\n        }\n        // 원소 개수를 갱신한다\n        _size -= 1\n        // 삭제한 원소를 반환한다\n        return num\n    }\n\n    /* 리스트 용량 확장 */\n    func extendCapacity() {\n        // 길이가 원래 배열의 extendRatio배인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n        arr = arr + Array(repeating: 0, count: capacity() * (extendRatio - 1))\n        // 리스트 용량을 갱신한다\n        _capacity = arr.count\n    }\n\n    /* 리스트를 배열로 변환 */\n    func toArray() -&gt; [Int] {\n        Array(arr.prefix(size()))\n    }\n}\n</code></pre> my_list.js<pre><code>/* 리스트 클래스 */\nclass MyList {\n    #arr = new Array(); // 배열(리스트 원소 저장)\n    #capacity = 10; // 리스트 용량\n    #size = 0; // 리스트 길이(현재 원소 개수)\n    #extendRatio = 2; // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n    /* 생성자 */\n    constructor() {\n        this.#arr = new Array(this.#capacity);\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환*/\n    size() {\n        return this.#size;\n    }\n\n    /* 리스트 용량 반환 */\n    capacity() {\n        return this.#capacity;\n    }\n\n    /* 원소 접근 */\n    get(index) {\n        // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('인덱스가 범위를 벗어났다');\n        return this.#arr[index];\n    }\n\n    /* 원소 갱신 */\n    set(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('인덱스가 범위를 벗어났다');\n        this.#arr[index] = num;\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    add(num) {\n        // 길이가 용량과 같으면 용량을 확장해야 한다\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // 새 원소를 리스트 맨 뒤에 추가한다\n        this.#arr[this.#size] = num;\n        this.#size++;\n    }\n\n    /* 중간에 원소 삽입 */\n    insert(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('인덱스가 범위를 벗어났다');\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for (let j = this.#size - 1; j &gt;= index; j--) {\n            this.#arr[j + 1] = this.#arr[j];\n        }\n        // 원소 개수를 갱신한다\n        this.#arr[index] = num;\n        this.#size++;\n    }\n\n    /* 원소 삭제 */\n    remove(index) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('인덱스가 범위를 벗어났다');\n        let num = this.#arr[index];\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for (let j = index; j &lt; this.#size - 1; j++) {\n            this.#arr[j] = this.#arr[j + 1];\n        }\n        // 원소 개수를 갱신한다\n        this.#size--;\n        // 삭제한 원소를 반환한다\n        return num;\n    }\n\n    /* 리스트 용량 확장 */\n    extendCapacity() {\n        // 길이가 원래 배열의 extendRatio배인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n        this.#arr = this.#arr.concat(\n            new Array(this.capacity() * (this.#extendRatio - 1))\n        );\n        // 리스트 용량을 갱신한다\n        this.#capacity = this.#arr.length;\n    }\n\n    /* 리스트를 배열로 변환 */\n    toArray() {\n        let size = this.size();\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.ts<pre><code>/* 리스트 클래스 */\nclass MyList {\n    private arr: Array&lt;number&gt;; // 배열(리스트 원소 저장)\n    private _capacity: number = 10; // 리스트 용량\n    private _size: number = 0; // 리스트 길이(현재 원소 개수)\n    private extendRatio: number = 2; // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n    /* 생성자 */\n    constructor() {\n        this.arr = new Array(this._capacity);\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환*/\n    public size(): number {\n        return this._size;\n    }\n\n    /* 리스트 용량 반환 */\n    public capacity(): number {\n        return this._capacity;\n    }\n\n    /* 원소 접근 */\n    public get(index: number): number {\n        // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('인덱스가 범위를 벗어났다');\n        return this.arr[index];\n    }\n\n    /* 원소 갱신 */\n    public set(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('인덱스가 범위를 벗어났다');\n        this.arr[index] = num;\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    public add(num: number): void {\n        // 길이가 용량과 같으면 용량을 확장해야 한다\n        if (this._size === this._capacity) this.extendCapacity();\n        // 새 원소를 리스트 맨 뒤에 추가한다\n        this.arr[this._size] = num;\n        this._size++;\n    }\n\n    /* 중간에 원소 삽입 */\n    public insert(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('인덱스가 범위를 벗어났다');\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (this._size === this._capacity) {\n            this.extendCapacity();\n        }\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for (let j = this._size - 1; j &gt;= index; j--) {\n            this.arr[j + 1] = this.arr[j];\n        }\n        // 원소 개수를 갱신한다\n        this.arr[index] = num;\n        this._size++;\n    }\n\n    /* 원소 삭제 */\n    public remove(index: number): number {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('인덱스가 범위를 벗어났다');\n        let num = this.arr[index];\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for (let j = index; j &lt; this._size - 1; j++) {\n            this.arr[j] = this.arr[j + 1];\n        }\n        // 원소 개수를 갱신한다\n        this._size--;\n        // 삭제한 원소를 반환한다\n        return num;\n    }\n\n    /* 리스트 용량 확장 */\n    public extendCapacity(): void {\n        // 길이가 size인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n        this.arr = this.arr.concat(\n            new Array(this.capacity() * (this.extendRatio - 1))\n        );\n        // 리스트 용량을 갱신한다\n        this._capacity = this.arr.length;\n    }\n\n    /* 리스트를 배열로 변환 */\n    public toArray(): number[] {\n        let size = this.size();\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.dart<pre><code>/* 리스트 클래스 */\nclass MyList {\n  late List&lt;int&gt; _arr; // 배열(리스트 원소 저장)\n  int _capacity = 10; // 리스트 용량\n  int _size = 0; // 리스트 길이(현재 원소 개수)\n  int _extendRatio = 2; // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n  /* 생성자 */\n  MyList() {\n    _arr = List.filled(_capacity, 0);\n  }\n\n  /* 리스트 길이(현재 원소 개수) 반환*/\n  int size() =&gt; _size;\n\n  /* 리스트 용량 반환 */\n  int capacity() =&gt; _capacity;\n\n  /* 원소 접근 */\n  int get(int index) {\n    if (index &gt;= _size) throw RangeError('인덱스가 범위를 벗어났다');\n    return _arr[index];\n  }\n\n  /* 원소 갱신 */\n  void set(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('인덱스가 범위를 벗어났다');\n    _arr[index] = _num;\n  }\n\n  /* 맨 뒤에 원소 추가 */\n  void add(int _num) {\n    // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n    if (_size == _capacity) extendCapacity();\n    _arr[_size] = _num;\n    // 원소 개수를 갱신한다\n    _size++;\n  }\n\n  /* 중간에 원소 삽입 */\n  void insert(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('인덱스가 범위를 벗어났다');\n    // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n    if (_size == _capacity) extendCapacity();\n    // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n    for (var j = _size - 1; j &gt;= index; j--) {\n      _arr[j + 1] = _arr[j];\n    }\n    _arr[index] = _num;\n    // 원소 개수를 갱신한다\n    _size++;\n  }\n\n  /* 원소 삭제 */\n  int remove(int index) {\n    if (index &gt;= _size) throw RangeError('인덱스가 범위를 벗어났다');\n    int _num = _arr[index];\n    // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n    for (var j = index; j &lt; _size - 1; j++) {\n      _arr[j] = _arr[j + 1];\n    }\n    // 원소 개수를 갱신한다\n    _size--;\n    // 삭제한 원소를 반환한다\n    return _num;\n  }\n\n  /* 리스트 용량 확장 */\n  void extendCapacity() {\n    // 길이가 원래 배열의 _extendRatio배인 새 배열을 생성한다\n    final _newNums = List.filled(_capacity * _extendRatio, 0);\n    // 원래 배열을 새 배열에 복사한다\n    List.copyRange(_newNums, 0, _arr);\n    // _arr의 참조를 갱신한다\n    _arr = _newNums;\n    // 리스트 용량을 갱신한다\n    _capacity = _arr.length;\n  }\n\n  /* 리스트를 배열로 변환 */\n  List&lt;int&gt; toArray() {\n    List&lt;int&gt; arr = [];\n    for (var i = 0; i &lt; _size; i++) {\n      arr.add(get(i));\n    }\n    return arr;\n  }\n}\n</code></pre> my_list.rs<pre><code>/* 리스트 클래스 */\n#[allow(dead_code)]\nstruct MyList {\n    arr: Vec&lt;i32&gt;,       // 배열(리스트 원소 저장)\n    capacity: usize,     // 리스트 용량\n    size: usize,         // 리스트 길이(현재 원소 개수)\n    extend_ratio: usize, // 리스트의 용량을 확장할 때마다 적용하는 배수\n}\n\n#[allow(unused, unused_comparisons)]\nimpl MyList {\n    /* 생성자 */\n    pub fn new(capacity: usize) -&gt; Self {\n        let mut vec = vec![0; capacity];\n        Self {\n            arr: vec,\n            capacity,\n            size: 0,\n            extend_ratio: 2,\n        }\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환*/\n    pub fn size(&amp;self) -&gt; usize {\n        return self.size;\n    }\n\n    /* 리스트 용량 반환 */\n    pub fn capacity(&amp;self) -&gt; usize {\n        return self.capacity;\n    }\n\n    /* 원소 접근 */\n    pub fn get(&amp;self, index: usize) -&gt; i32 {\n        // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if index &gt;= self.size {\n            panic!(\"인덱스가 범위를 벗어났다\")\n        };\n        return self.arr[index];\n    }\n\n    /* 원소 갱신 */\n    pub fn set(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size {\n            panic!(\"인덱스가 범위를 벗어났다\")\n        };\n        self.arr[index] = num;\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    pub fn add(&amp;mut self, num: i32) {\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        self.arr[self.size] = num;\n        // 원소 개수를 갱신한다\n        self.size += 1;\n    }\n\n    /* 중간에 원소 삽입 */\n    pub fn insert(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size() {\n            panic!(\"인덱스가 범위를 벗어났다\")\n        };\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for j in (index..self.size).rev() {\n            self.arr[j + 1] = self.arr[j];\n        }\n        self.arr[index] = num;\n        // 원소 개수를 갱신한다\n        self.size += 1;\n    }\n\n    /* 원소 삭제 */\n    pub fn remove(&amp;mut self, index: usize) -&gt; i32 {\n        if index &gt;= self.size() {\n            panic!(\"인덱스가 범위를 벗어났다\")\n        };\n        let num = self.arr[index];\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for j in index..self.size - 1 {\n            self.arr[j] = self.arr[j + 1];\n        }\n        // 원소 개수를 갱신한다\n        self.size -= 1;\n        // 삭제한 원소를 반환한다\n        return num;\n    }\n\n    /* 리스트 용량 확장 */\n    pub fn extend_capacity(&amp;mut self) {\n        // 길이가 원래 배열의 extend_ratio배인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n        let new_capacity = self.capacity * self.extend_ratio;\n        self.arr.resize(new_capacity, 0);\n        // 리스트 용량을 갱신한다\n        self.capacity = new_capacity;\n    }\n\n    /* 리스트를 배열로 변환 */\n    pub fn to_array(&amp;self) -&gt; Vec&lt;i32&gt; {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        let mut arr = Vec::new();\n        for i in 0..self.size {\n            arr.push(self.get(i));\n        }\n        arr\n    }\n}\n</code></pre> my_list.c<pre><code>/* 리스트 클래스 */\ntypedef struct {\n    int *arr;        // 배열(리스트 원소 저장)\n    int capacity;    // 리스트 용량\n    int size;        // 리스트 크기\n    int extendRatio; // 리스트 용량을 확장할 때마다 적용하는 배수\n} MyList;\n\n/* 생성자 */\nMyList *newMyList() {\n    MyList *nums = malloc(sizeof(MyList));\n    nums-&gt;capacity = 10;\n    nums-&gt;arr = malloc(sizeof(int) * nums-&gt;capacity);\n    nums-&gt;size = 0;\n    nums-&gt;extendRatio = 2;\n    return nums;\n}\n\n/* 소멸자 */\nvoid delMyList(MyList *nums) {\n    free(nums-&gt;arr);\n    free(nums);\n}\n\n/* 리스트 길이 구하기 */\nint size(MyList *nums) {\n    return nums-&gt;size;\n}\n\n/* 리스트 용량 반환 */\nint capacity(MyList *nums) {\n    return nums-&gt;capacity;\n}\n\n/* 원소 접근 */\nint get(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    return nums-&gt;arr[index];\n}\n\n/* 원소 갱신 */\nvoid set(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    nums-&gt;arr[index] = num;\n}\n\n/* 맨 뒤에 원소 추가 */\nvoid add(MyList *nums, int num) {\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // 용량 확장\n    }\n    nums-&gt;arr[size(nums)] = num;\n    nums-&gt;size++;\n}\n\n/* 중간에 원소 삽입 */\nvoid insert(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // 용량 확장\n    }\n    for (int i = size(nums); i &gt; index; --i) {\n        nums-&gt;arr[i] = nums-&gt;arr[i - 1];\n    }\n    nums-&gt;arr[index] = num;\n    nums-&gt;size++;\n}\n\n/* 원소 삭제 */\n// 주의: stdio.h가 remove라는 이름을 사용한다\nint removeItem(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    int num = nums-&gt;arr[index];\n    for (int i = index; i &lt; size(nums) - 1; i++) {\n        nums-&gt;arr[i] = nums-&gt;arr[i + 1];\n    }\n    nums-&gt;size--;\n    return num;\n}\n\n/* 리스트 용량 확장 */\nvoid extendCapacity(MyList *nums) {\n    // 먼저 공간 할당\n    int newCapacity = capacity(nums) * nums-&gt;extendRatio;\n    int *extend = (int *)malloc(sizeof(int) * newCapacity);\n    int *temp = nums-&gt;arr;\n\n    // 기존 데이터를 새 공간에 복사\n    for (int i = 0; i &lt; size(nums); i++)\n        extend[i] = nums-&gt;arr[i];\n\n    // 기존 데이터 공간 해제\n    free(temp);\n\n    // 새 데이터로 갱신\n    nums-&gt;arr = extend;\n    nums-&gt;capacity = newCapacity;\n}\n\n/* 출력을 위해 리스트를 Array로 변환 */\nint *toArray(MyList *nums) {\n    return nums-&gt;arr;\n}\n</code></pre> my_list.kt<pre><code>/* 리스트 클래스 */\nclass MyList {\n    private var arr: IntArray = intArrayOf() // 배열(리스트 원소 저장)\n    private var capacity: Int = 10 // 리스트 용량\n    private var size: Int = 0 // 리스트 길이(현재 원소 개수)\n    private var extendRatio: Int = 2 // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n    /* 생성자 */\n    init {\n        arr = IntArray(capacity)\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환 */\n    fun size(): Int {\n        return size\n    }\n\n    /* 리스트 용량 반환 */\n    fun capacity(): Int {\n        return capacity\n    }\n\n    /* 원소 접근 */\n    fun get(index: Int): Int {\n        // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\")\n        return arr[index]\n    }\n\n    /* 원소 갱신 */\n    fun set(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\")\n        arr[index] = num\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    fun add(num: Int) {\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (size == capacity())\n            extendCapacity()\n        arr[size] = num\n        // 원소 개수를 갱신한다\n        size++\n    }\n\n    /* 중간에 원소 삽입 */\n    fun insert(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\")\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (size == capacity())\n            extendCapacity()\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for (j in size - 1 downTo index)\n            arr[j + 1] = arr[j]\n        arr[index] = num\n        // 원소 개수를 갱신한다\n        size++\n    }\n\n    /* 원소 삭제 */\n    fun remove(index: Int): Int {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\")\n        val num = arr[index]\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for (j in index..&lt;size - 1)\n            arr[j] = arr[j + 1]\n        // 원소 개수를 갱신한다\n        size--\n        // 삭제한 원소를 반환한다\n        return num\n    }\n\n    /* 리스트 용량 확장 */\n    fun extendCapacity() {\n        // 길이가 원래 배열의 extendRatio배인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n        arr = arr.copyOf(capacity() * extendRatio)\n        // 리스트 용량을 갱신한다\n        capacity = arr.size\n    }\n\n    /* 리스트를 배열로 변환 */\n    fun toArray(): IntArray {\n        val size = size()\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        val arr = IntArray(size)\n        for (i in 0..&lt;size) {\n            arr[i] = get(i)\n        }\n        return arr\n    }\n}\n</code></pre> my_list.rb<pre><code>### 리스트 클래스 ###\nclass MyList\n  attr_reader :size       # 리스트 길이(현재 원소 개수) 반환\n  attr_reader :capacity   # 리스트 용량 반환\n\n  ### 생성자 ###\n  def initialize\n    @capacity = 10\n    @size = 0\n    @extend_ratio = 2\n    @arr = Array.new(capacity)\n  end\n\n  ### 원소 접근 ###\n  def get(index)\n    # 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n    raise IndexError, \"인덱스가 범위를 벗어났다\" if index &lt; 0 || index &gt;= size\n    @arr[index]\n  end\n\n  ### 원소 접근 ###\n  def set(index, num)\n    raise IndexError, \"인덱스가 범위를 벗어났다\" if index &lt; 0 || index &gt;= size\n    @arr[index] = num\n  end\n\n  ### 맨 뒤에 원소 추가 ###\n  def add(num)\n    # 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n    extend_capacity if size == capacity\n    @arr[size] = num\n\n    # 원소 개수를 갱신한다\n    @size += 1\n  end\n\n  ### 중간에 원소 삽입 ###\n  def insert(index, num)\n    raise IndexError, \"인덱스가 범위를 벗어났다\" if index &lt; 0 || index &gt;= size\n\n    # 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n    extend_capacity if size == capacity\n\n    # 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n    for j in (size - 1).downto(index)\n      @arr[j + 1] = @arr[j]\n    end\n    @arr[index] = num\n\n    # 원소 개수를 갱신한다\n    @size += 1\n  end\n\n  ### 원소 삭제 ###\n  def remove(index)\n    raise IndexError, \"인덱스가 범위를 벗어났다\" if index &lt; 0 || index &gt;= size\n    num = @arr[index]\n\n    # 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n    for j in index...size\n      @arr[j] = @arr[j + 1]\n    end\n\n    # 원소 개수를 갱신한다\n    @size -= 1\n\n    # 삭제한 원소를 반환한다\n    num\n  end\n\n  ### 리스트 용량 확장 ###\n  def extend_capacity\n    # 길이가 원래 배열의 extend_ratio배인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n    arr = @arr.dup + Array.new(capacity * (@extend_ratio - 1))\n    # 리스트 용량을 갱신한다\n    @capacity = arr.length\n  end\n\n  ### 리스트를 배열로 변환 ###\n  def to_array\n    sz = size\n    # 유효한 길이 범위 안의 리스트 원소만 변환한다\n    arr = Array.new(sz)\n    for i in 0...sz\n      arr[i] = get(i)\n    end\n    arr\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/","level":1,"title":"4.4   메모리와 캐시 *","text":"<p>이 장의 앞선 두 절에서는 기본적이고 중요한 두 자료 구조인 배열과 연결 리스트를 살펴보았다. 이들은 각각 ‘연속 저장’과 ‘분산 저장’이라는 물리적 구조를 대표한다.</p> <p>실제로 물리적 구조는 프로그램의 메모리 및 캐시 사용 효율을 크게 좌우하며, 나아가 알고리즘 프로그램의 전체 성능에 영향을 준다.</p>","path":["제 4 장   배열과 연결 리스트","4.4   메모리와 캐시 *"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/#441","level":2,"title":"4.4.1   컴퓨터의 저장 장치","text":"<p>컴퓨터에는 하드 디스크(hard disk), 메모리(random-access memory, RAM), 캐시(cache memory)라는 세 종류의 저장 장치가 있다. 아래 표(4-2)는 컴퓨터 시스템에서 각 장치가 맡는 역할과 성능 특징을 보여 준다.</p> <p> 표 4-2   컴퓨터의 저장 장치 </p> 하드 디스크 메모리 캐시 용도 운영체제, 프로그램, 파일 등의 데이터를 장기간 저장 실행 중인 프로그램과 처리 중인 데이터를 임시 저장 자주 접근하는 데이터와 명령어를 저장하여 CPU의 메모리 접근 횟수를 줄임 휘발성 전원이 꺼져도 데이터가 사라지지 않음 전원이 꺼지면 데이터가 사라짐 전원이 꺼지면 데이터가 사라짐 용량 큼, TB 단위 작음, GB 단위 매우 작음, MB 단위 속도 느림, 수백~수천 MB/s 빠름, 수십 GB/s 매우 빠름, 수십~수백 GB/s 가격(중국 위안) 저렴함, GB당 수십 펀~수 위안 비쌈, GB당 수십~수백 위안 매우 비쌈, CPU 가격에 포함 <p>컴퓨터의 저장 시스템은 아래 그림(4-9)과 같은 피라미드 구조로 생각할 수 있다. 꼭대기에 가까울수록 속도는 빠르고, 용량은 작으며, 비용은 높다. 이러한 계층적 설계는 우연이 아니라 컴퓨터 과학자와 엔지니어들이 깊이 고민한 결과다.</p> <ul> <li>메모리로 하드 디스크를 대체하기는 어렵다. 우선 메모리의 데이터는 전원이 꺼지면 사라지므로 장기 저장에 적합하지 않다. 또한 메모리 가격은 하드 디스크의 수십 배여서 소비자 시장에 보편적으로 적용하기 어렵다.</li> <li>캐시에서 큰 용량과 빠른 속도를 동시에 얻기는 어렵다. L1, L2, L3 캐시의 용량이 커질수록 물리적 크기도 커지고 CPU 코어와의 거리도 멀어진다. 그 결과 데이터 전송 시간이 늘고 원소 접근 지연도 커진다. 현재 기술에서 다단계 캐시 구조는 용량, 속도, 비용 사이의 최적 절충점이다.</li> </ul> <p></p> <p> 그림 4-9   컴퓨터 저장 시스템 </p> <p>참고</p> <p>컴퓨터의 저장 계층 구조는 속도, 용량, 비용 사이의 정교한 균형을 보여 준다. 이러한 절충은 모든 산업 분야에 널리 존재하며, 서로 다른 장점과 제약 사이에서 최적의 균형을 찾아야 한다.</p> <p>전체적으로 하드 디스크는 많은 데이터를 장기간 저장하고, 메모리는 실행 중인 프로그램이 처리하는 데이터를 임시 저장하며, 캐시는 자주 접근하는 데이터와 명령어를 저장하여 실행 효율을 높인다. 세 장치는 서로 협력하여 컴퓨터 시스템이 효율적으로 작동하게 한다.</p> <p>아래 그림(4-10)과 같이 프로그램이 실행되면 하드 디스크의 데이터를 메모리로 읽어 와 CPU가 계산에 사용한다. 캐시는 CPU의 일부로 볼 수 있으며, 메모리에서 데이터를 지능적으로 불러와 CPU에 빠른 데이터 읽기를 제공한다. 이를 통해 실행 효율을 크게 높이고 더 느린 메모리에 대한 의존을 줄인다.</p> <p></p> <p> 그림 4-10   하드 디스크, 메모리, 캐시 사이의 데이터 흐름 </p>","path":["제 4 장   배열과 연결 리스트","4.4   메모리와 캐시 *"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/#442","level":2,"title":"4.4.2   자료 구조의 메모리 효율","text":"<p>메모리 공간 활용 측면에서 배열과 연결 리스트에는 각각 장점과 한계가 있다.</p> <p>한편으로 메모리는 한정되어 있으며 같은 메모리 영역을 여러 프로그램이 공유할 수 없으므로, 자료 구조가 공간을 최대한 효율적으로 사용하기를 바란다. 배열 원소는 빈틈없이 붙어 있어 연결 리스트 노드 사이의 참조(포인터)를 저장할 추가 공간이 필요 없으므로 공간 효율이 더 높다. 하지만 배열은 충분한 연속 메모리를 한꺼번에 할당해야 하므로 공간 낭비가 발생할 수 있으며, 용량 확장에도 시간과 공간이 추가로 든다. 반면 연결 리스트는 ‘노드’ 단위로 메모리를 동적으로 할당하고 회수하므로 더 유연하다.</p> <p>다른 한편으로 프로그램이 실행되면서 메모리 할당과 해제가 반복되면 빈 메모리의 단편화가 점점 심해져 메모리 활용 효율이 낮아진다. 배열은 연속 저장 방식을 사용하므로 상대적으로 단편화를 덜 일으킨다. 반면 연결 리스트의 원소는 분산되어 있으므로 삽입과 삭제가 잦으면 단편화가 더 쉽게 발생한다.</p>","path":["제 4 장   배열과 연결 리스트","4.4   메모리와 캐시 *"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/#443","level":2,"title":"4.4.3   자료 구조의 캐시 효율","text":"<p>캐시는 메모리보다 용량은 훨씬 작지만 속도는 훨씬 빨라 프로그램 실행 속도에 매우 중요한 역할을 한다. 용량이 제한되어 있어 자주 사용하는 데이터 일부만 저장할 수 있으므로, CPU가 접근하려는 데이터가 캐시에 없으면 캐시 미스(cache miss)가 발생한다. 이때 CPU는 더 느린 메모리에서 필요한 데이터를 불러와야 한다.</p> <p>따라서 ‘캐시 미스’가 적을수록 CPU의 데이터 읽기 및 쓰기 효율이 높아지고, 프로그램 성능도 좋아진다. CPU가 캐시에서 데이터를 성공적으로 가져오는 비율을 캐시 적중률(cache hit rate)이라고 하며, 캐시 효율을 측정하는 지표로 자주 사용한다.</p> <p>캐시는 효율을 최대한 높이기 위해 다음과 같은 데이터 로딩 방식을 사용한다.</p> <ul> <li>캐시 라인: 캐시는 데이터를 한 바이트씩 저장하고 불러오는 대신 캐시 라인 단위로 처리한다. 캐시 라인 단위 전송은 바이트 단위 전송보다 효율적이다.</li> <li>프리페치: 프로세서는 순차 접근이나 고정 간격 접근 같은 데이터 접근 패턴을 예측하고, 그 패턴에 따라 데이터를 미리 캐시에 불러와 적중률을 높인다.</li> <li>공간 지역성: 어떤 데이터에 접근하면 가까운 시일 안에 주변 데이터에도 접근할 가능성이 있다. 따라서 캐시는 데이터를 불러올 때 주변 데이터도 함께 불러와 적중률을 높인다.</li> <li>시간 지역성: 어떤 데이터에 접근하면 머지않아 다시 접근할 가능성이 높다. 캐시는 이 원리를 이용해 최근 접근한 데이터를 유지함으로써 적중률을 높인다.</li> </ul> <p>실제로 배열과 연결 리스트의 캐시 활용 효율은 서로 다르며, 주로 다음과 같은 차이가 있다.</p> <ul> <li>공간 사용량: 연결 리스트 원소는 배열 원소보다 더 많은 공간을 차지하므로 캐시에 담을 수 있는 유효 데이터의 양이 적다.</li> <li>캐시 라인: 연결 리스트의 데이터는 메모리 곳곳에 흩어져 있지만 캐시는 ‘라인 단위’로 불러오므로, 불필요한 데이터를 가져오는 비율이 높다.</li> <li>프리페치: 배열의 데이터 접근 패턴은 연결 리스트보다 ‘예측 가능성’이 높다. 즉, 시스템이 다음에 불러올 데이터를 더 쉽게 예상할 수 있다.</li> <li>공간 지역성: 배열은 한곳에 모인 메모리 공간에 저장되므로, 불러온 데이터 주변의 데이터에도 곧 접근할 가능성이 더 높다.</li> </ul> <p>대체로 배열은 캐시 적중률이 더 높아 연산 효율에서도 연결 리스트보다 유리한 경우가 많다. 이 때문에 알고리즘 문제를 풀 때는 배열 기반 자료 구조를 더 선호하는 경우가 많다.</p> <p>다만 캐시 효율이 높다고 해서 모든 상황에서 배열이 연결 리스트보다 좋은 것은 아니다. 실제로 어떤 자료 구조를 쓸지는 구체적인 요구에 따라 결정해야 한다. 예를 들어 배열과 연결 리스트 모두 ‘스택’을 구현할 수 있지만(다음 장에서 자세히 설명한다), 적합한 상황은 다르다.</p> <ul> <li>알고리즘 문제를 풀 때는 배열 기반 스택을 선호한다. 배열에 일정한 메모리 공간을 미리 할당하기만 하면, 높은 연산 효율과 임의 접근 기능을 얻을 수 있기 때문이다.</li> <li>데이터가 매우 많고 변화가 잦으며 스택의 예상 크기를 가늠하기 어렵다면 연결 리스트 기반 스택이 더 적합하다. 연결 리스트는 많은 데이터를 메모리 여러 곳에 분산 저장할 수 있으며, 배열 용량 확장에 따른 추가 비용도 피할 수 있다.</li> </ul>","path":["제 4 장   배열과 연결 리스트","4.4   메모리와 캐시 *"],"tags":[]},{"location":"chapter_array_and_linkedlist/summary/","level":1,"title":"4.5   요약","text":"","path":["제 4 장   배열과 연결 리스트","4.5   요약"],"tags":[]},{"location":"chapter_array_and_linkedlist/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>배열과 연결 리스트는 두 가지 기본 자료 구조로, 컴퓨터 메모리에 데이터를 저장하는 연속 공간 저장과 분산 공간 저장을 각각 대표한다. 둘의 특징은 서로 보완적이다.</li> <li>배열은 임의 접근을 지원하고 메모리 사용량이 적다. 그러나 원소 삽입과 삭제의 효율이 낮으며 초기화한 뒤 길이를 바꿀 수 없다.</li> <li>연결 리스트는 참조(포인터)를 바꾸어 노드를 효율적으로 삽입하고 삭제하며, 길이도 유연하게 조절할 수 있다. 하지만 노드 접근 효율이 낮고 메모리를 더 많이 사용한다. 대표적인 종류에는 단일 연결 리스트, 원형 연결 리스트, 이중 연결 리스트가 있다.</li> <li>리스트는 삽입, 삭제, 탐색, 갱신을 지원하는 순서 있는 원소의 모음으로, 보통 동적 배열로 구현한다. 배열의 장점을 유지하면서 길이를 유연하게 조절할 수 있다.</li> <li>리스트의 등장으로 배열의 실용성이 크게 높아졌지만, 메모리 공간 일부가 낭비될 수 있다.</li> <li>프로그램이 실행될 때 데이터는 주로 메모리에 저장된다. 배열은 메모리 공간 효율이 더 높고, 연결 리스트는 메모리를 더 유연하게 사용한다.</li> <li>캐시는 캐시 라인, 프리페치, 공간 지역성, 시간 지역성 등의 로딩 방식을 통해 CPU에 빠른 데이터 접근을 제공하여 프로그램 실행 효율을 크게 높인다.</li> <li>배열은 캐시 적중률이 더 높으므로 보통 연결 리스트보다 효율적이다. 자료 구조를 선택할 때는 구체적인 요구와 상황에 맞춰야 한다.</li> </ul>","path":["제 4 장   배열과 연결 리스트","4.5   요약"],"tags":[]},{"location":"chapter_array_and_linkedlist/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 배열을 스택에 저장하는지 힙에 저장하는지에 따라 시간 효율과 공간 효율이 달라지는가?</p> <p>스택에 저장한 배열도 힙에 저장한 배열도 연속된 메모리 공간에 있으므로, 데이터 연산 효율은 대체로 같다. 다만 스택과 힙의 특성 때문에 다음과 같은 차이가 생긴다.</p> <ol> <li>할당 및 해제 효율: 스택은 비교적 작은 메모리 영역으로 컴파일러가 자동으로 할당한다. 힙은 더 크며 코드에서 동적으로 할당할 수 있고 단편화도 더 쉽게 발생한다. 따라서 힙의 할당 및 해제 연산은 보통 스택보다 느리다.</li> <li>크기 제한: 스택 메모리는 비교적 작고 힙의 크기는 대개 사용 가능한 메모리에 의해 제한된다. 따라서 큰 배열을 저장하기에는 힙이 더 적합하다.</li> <li>유연성: 스택에 저장하는 배열의 크기는 컴파일 시 정해야 하지만, 힙에 저장하는 배열의 크기는 실행 중에 동적으로 정할 수 있다.</li> </ol> <p>Q: 배열은 원소의 자료형이 같아야 한다고 하는데, 연결 리스트에서는 왜 이를 강조하지 않는가?</p> <p>연결 리스트는 참조(포인터)로 연결된 노드들로 이루어진다. 각 노드에는 <code>int</code>, <code>double</code>, <code>string</code>, <code>object</code> 등 서로 다른 자료형의 데이터를 저장할 수 있다.</p> <p>반면 배열 원소는 자료형이 같아야 오프셋을 계산하여 해당 원소의 위치를 구할 수 있다. 예를 들어 배열에 <code>int</code>와 <code>long</code>이 함께 있으면 원소 하나가 각각 4바이트와 8바이트를 차지한다. ‘원소 길이’가 두 가지이므로 다음 식으로 오프셋을 계산할 수 없다.</p> <pre><code># 원소 메모리 주소 = 배열 메모리 주소(첫 원소의 메모리 주소) + 원소 길이 * 원소 인덱스\n</code></pre> <p>Q: 노드 <code>P</code>를 삭제한 뒤 <code>P.next</code>를 <code>None</code>으로 설정해야 하는가?</p> <p><code>P.next</code>를 바꾸지 않아도 된다. 해당 연결 리스트에서는 머리 노드부터 꼬리 노드까지 순회해도 더 이상 <code>P</code>를 만나지 않는다. 즉, <code>P</code>는 이미 연결 리스트에서 삭제되었으므로 어디를 가리키든 이 연결 리스트에 영향을 주지 않는다.</p> <p>자료 구조와 알고리즘 문제를 푸는 관점에서는 프로그램 논리가 올바르기만 하면 연결을 끊지 않아도 된다. 표준 라이브러리 관점에서는 연결을 끊는 편이 더 안전하고 논리도 명확하다. 연결을 끊지 않은 상태에서 삭제한 노드가 정상적으로 회수되지 않으면, 뒤따르는 노드의 메모리 회수에도 영향을 줄 수 있다.</p> <p>Q: 연결 리스트의 삽입과 삭제는 시간 복잡도가 \\(O(1)\\)이라고 한다. 하지만 먼저 원소를 찾는 데 \\(O(n)\\)이 필요한데, 전체 시간 복잡도는 왜 \\(O(n)\\)이 아닌가?</p> <p>원소를 먼저 찾고 삭제한다면 시간 복잡도는 실제로 \\(O(n)\\)이다. 다만 연결 리스트의 \\(O(1)\\) 삽입 및 삭제는 다른 응용에서 장점을 발휘한다. 예를 들어 덱은 연결 리스트로 구현하기에 적합하다. 머리 노드와 꼬리 노드를 항상 가리키는 포인터 변수를 유지하면 매번 삽입과 삭제를 \\(O(1)\\)에 수행할 수 있다.</p> <p>Q: ‘연결 리스트의 정의와 저장 방식’ 그림에서 노드 포인터를 저장하는 연한 파란색 부분은 하나의 메모리 주소를 차지하는가? 아니면 노드 값과 반씩 차지하는가?</p> <p>그 그림은 정성적인 설명이다. 정확한 크기는 구체적인 상황에 따라 분석해야 한다.</p> <ul> <li>노드 값이 차지하는 공간은 자료형에 따라 다르다. 예를 들어 <code>int</code>, <code>long</code>, <code>double</code>, 인스턴스 객체 등이 있다.</li> <li>포인터 변수가 차지하는 메모리 크기는 운영체제와 컴파일 환경에 따라 정해지며, 대개 8바이트 또는 4바이트다.</li> </ul> <p>Q: 리스트 끝에 원소를 추가하는 연산은 항상 \\(O(1)\\)인가?</p> <p>원소를 추가할 때 리스트의 길이를 넘으면 먼저 용량을 늘려야 한다. 시스템이 새 메모리 영역을 요청하고 기존 원소 전체를 옮기므로, 이때의 시간 복잡도는 \\(O(n)\\)이다.</p> <p>Q: ‘리스트는 배열의 실용성을 크게 높였지만 메모리 공간 일부를 낭비할 수 있다’는 설명에서, 공간 낭비는 용량, 길이, 확장 배수 같은 추가 변수의 메모리를 뜻하는가?</p> <p>여기서 공간 낭비는 주로 두 가지를 뜻한다. 하나는 리스트에 초기 길이를 설정하지만 그만큼 필요하지 않을 수 있다는 것이다. 다른 하나는 잦은 확장을 피하려고 보통 \\(\\times 1.5\\)처럼 일정한 계수를 곱해 용량을 늘린다는 것이다. 이때도 빈자리가 많이 생기며, 보통 이를 전부 채우지는 못한다.</p> <p>Q: Python에서 <code>n = [1, 2, 3]</code>을 초기화하면 세 원소의 주소가 이어져 있다. 하지만 <code>m = [2, 1, 3]</code>을 초기화하면 원소들의 id가 연속적이지 않고 <code>n</code>의 같은 값과 각각 일치한다. 주소가 연속적이지 않아도 <code>m</code>은 배열인가?</p> <p>리스트 원소를 연결 리스트 노드로 바꾸어 <code>n = [n1, n2, n3, n4, n5]</code>라고 하면, 이 다섯 노드 객체도 보통 메모리 곳곳에 흩어져 있다. 그래도 리스트 인덱스가 주어지면 \\(O(1)\\) 시간에 노드의 메모리 주소를 얻어 해당 노드에 접근할 수 있다. 배열에 저장된 것은 노드 자체가 아니라 노드의 참조이기 때문이다.</p> <p>많은 언어와 달리 Python에서는 숫자도 객체로 감싸져 있으며, 리스트에는 숫자 자체가 아닌 숫자의 참조가 저장된다. 따라서 두 배열의 같은 숫자가 같은 id를 가지며, 숫자들의 메모리 주소는 연속적일 필요가 없다.</p> <p>Q: C++ STL의 <code>std::list</code>는 이중 연결 리스트를 구현하는데, 일부 알고리즘 책에서는 거의 직접 쓰지 않는 것 같다. 어떤 한계가 있기 때문인가?</p> <p>한편으로는 보통 배열로 알고리즘을 구현하는 것을 선호하며, 꼭 필요할 때만 연결 리스트를 사용한다. 주된 이유는 두 가지다.</p> <ul> <li>공간 비용: 원소마다 이전 원소와 다음 원소를 위한 포인터가 두 개 더 필요하므로 <code>std::list</code>는 보통 <code>std::vector</code>보다 공간을 많이 차지한다.</li> <li>캐시에 불리함: 데이터가 연속적으로 저장되지 않아 <code>std::list</code>의 캐시 활용률이 낮다. 일반적으로는 <code>std::vector</code>의 성능이 더 좋다.</li> </ul> <p>다른 한편으로 연결 리스트가 필요한 경우는 주로 이진 트리와 그래프다. 스택과 큐는 연결 리스트 대신 프로그래밍 언어가 제공하는 <code>stack</code>과 <code>queue</code>를 쓰는 경우가 많다.</p> <p>Q: <code>res = [[0]] * n</code>으로 만든 이차원 리스트에서 각 <code>[0]</code>은 서로 독립적인가?</p> <p>독립적이지 않다. 이 이차원 리스트의 모든 <code>[0]</code>은 실제로 같은 객체를 참조한다. 한 원소를 수정하면 대응하는 원소가 모두 함께 바뀐다.</p> <p>각 <code>[0]</code>을 독립적으로 만들려면 <code>res = [[0] for _ in range(n)]</code>을 사용하면 된다. 이 방식은 서로 독립적인 <code>[0]</code> 리스트 객체 \\(n\\)개를 초기화한다.</p> <p>Q: <code>res = [0] * n</code>으로 만든 리스트에서 각 정수 0은 서로 독립적인가?</p> <p>이 리스트의 모든 정수 0은 같은 객체를 참조한다. Python은 객체 재사용을 극대화하고 성능을 높이기 위해 작은 정수(보통 -5부터 256까지)를 캐시 풀로 관리하기 때문이다.</p> <p>모두 같은 객체를 가리켜도 리스트의 각 원소는 독립적으로 바꿀 수 있다. Python의 정수는 ‘불변 객체’이기 때문이다. 원소를 바꾸면 기존 객체 자체가 바뀌는 것이 아니라 다른 객체를 참조하게 된다.</p> <p>그러나 리스트 원소가 리스트, 딕셔너리, 클래스 인스턴스 같은 ‘가변 객체’라면, 원소의 내용을 수정할 때 객체 자체가 바뀌므로 이를 참조하는 모든 원소에서 같은 변화가 나타난다.</p>","path":["제 4 장   배열과 연결 리스트","4.5   요약"],"tags":[]},{"location":"chapter_backtracking/","level":1,"title":"제 13 장   백트래킹","text":"<p>이 장을 시작하며</p> <p>우리는 미로 속 탐험가처럼 앞으로 나아가는 길에서 어려움을 만날 수 있다.</p> <p>백트래킹의 힘은 다시 출발하고 끊임없이 시도하여, 마침내 빛으로 향하는 출구를 찾게 해 준다.</p>","path":["제 13 장   백트래킹"],"tags":[]},{"location":"chapter_backtracking/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>13.1   백트래킹 알고리즘</li> <li>13.2   모든 순열 문제</li> <li>13.3   부분 집합 합 문제</li> <li>13.4   n-퀸 문제</li> <li>13.5   요약</li> <li>13.6   연습 문제</li> </ul>","path":["제 13 장   백트래킹"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/","level":1,"title":"13.1   백트래킹 알고리즘","text":"<p>백트래킹 알고리즘(backtracking algorithm)은 모든 경우를 열거하여 문제를 푸는 방법이다. 초기 상태에서 출발하여 가능한 모든 해결 방법을 완전 탐색하고, 올바른 해를 만나면 기록한다. 해를 찾거나 가능한 선택을 모두 시도해도 찾을 수 없을 때까지 진행하는 것이 핵심이다.</p> <p>백트래킹은 보통 '깊이 우선 탐색'으로 해 공간을 순회한다. '이진 트리' 장에서 전위, 중위, 후위 순회가 모두 깊이 우선 탐색이라고 설명했다. 이제 전위 순회를 이용한 백트래킹 문제를 만들어 작동 원리를 차근차근 알아보자.</p> <p>예제 1</p> <p>이진 트리가 주어졌을 때 값이 \\(7\\)인 모든 노드를 찾아 기록하고, 노드 리스트를 반환하라.</p> <p>이 문제에서는 트리를 전위 순회하며 현재 노드의 값이 \\(7\\)인지 확인하고, 맞으면 해당 노드의 값을 결과 리스트 <code>res</code>에 추가한다. 과정은 아래 그림(13-1)과 코드에 나타나 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_i_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"전위 순회: 예제 1\"\"\"\n    if root is None:\n        return\n    if root.val == 7:\n        # 해를 기록한다\n        res.append(root)\n    pre_order(root.left)\n    pre_order(root.right)\n</code></pre> preorder_traversal_i_compact.cpp<pre><code>/* 전위 순회: 예제 1 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // 해를 기록한다\n        res.push_back(root);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.java<pre><code>/* 전위 순회: 예제 1 */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // 해를 기록한다\n        res.add(root);\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.cs<pre><code>/* 전위 순회: 예제 1 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // 해를 기록한다\n        res.Add(root);\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.go<pre><code>/* 전위 순회: 예제 1 */\nfunc preOrderI(root *TreeNode, res *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    if (root.Val).(int) == 7 {\n        // 해를 기록한다\n        *res = append(*res, root)\n    }\n    preOrderI(root.Left, res)\n    preOrderI(root.Right, res)\n}\n</code></pre> preorder_traversal_i_compact.swift<pre><code>/* 전위 순회: 예제 1 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    if root.val == 7 {\n        // 해를 기록한다\n        res.append(root)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n</code></pre> preorder_traversal_i_compact.js<pre><code>/* 전위 순회: 예제 1 */\nfunction preOrder(root, res) {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // 해를 기록한다\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.ts<pre><code>/* 전위 순회: 예제 1 */\nfunction preOrder(root: TreeNode | null, res: TreeNode[]): void {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // 해를 기록한다\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.dart<pre><code>/* 전위 순회: 예제 1 */\nvoid preOrder(TreeNode? root, List&lt;TreeNode&gt; res) {\n  if (root == null) {\n    return;\n  }\n  if (root.val == 7) {\n    // 해를 기록한다\n    res.add(root);\n  }\n  preOrder(root.left, res);\n  preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.rs<pre><code>/* 전위 순회: 예제 1 */\nfn pre_order(res: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        if node.borrow().val == 7 {\n            // 해를 기록한다\n            res.push(node.clone());\n        }\n        pre_order(res, node.borrow().left.as_ref());\n        pre_order(res, node.borrow().right.as_ref());\n    }\n}\n</code></pre> preorder_traversal_i_compact.c<pre><code>/* 전위 순회: 예제 1 */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // 해를 기록한다\n        res[resSize++] = root;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.kt<pre><code>/* 전위 순회: 예제 1 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    if (root._val == 7) {\n        // 해를 기록한다\n        res!!.add(root)\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n}\n</code></pre> preorder_traversal_i_compact.rb<pre><code>### 전위 순회: 예제 1 ###\ndef pre_order(root)\n  return unless root\n\n  # 해를 기록한다\n  $res &lt;&lt; root if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 13-1   전위 순회로 노드 탐색 </p>","path":["제 13 장   백트래킹","13.1   백트래킹 알고리즘"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1311","level":2,"title":"13.1.1   시도와 되돌리기","text":"<p>백트래킹이라고 부르는 이유는 해 공간을 탐색할 때 '시도'와 '되돌리기' 전략을 사용하기 때문이다. 탐색 중 더 진행할 수 없거나 조건을 만족하는 해를 얻을 수 없는 상태를 만나면, 직전 선택을 취소하고 이전 상태로 돌아가 다른 선택을 시도한다.</p> <p>예제 1에서는 각 노드의 방문이 한 번의 '시도'이며, 리프 노드를 지나거나 부모 노드로 돌아가는 <code>return</code>이 '되돌리기'에 해당한다.</p> <p>되돌리기에는 함수 반환만 포함되는 것이 아니다. 이를 설명하기 위해 예제 1을 조금 확장해 보자.</p> <p>예제 2</p> <p>이진 트리에서 값이 \\(7\\)인 모든 노드를 찾아 루트 노드에서 각 노드까지의 경로를 반환하라.</p> <p>예제 1의 코드에 리스트 <code>path</code>를 추가하여 방문한 노드의 경로를 기록해야 한다. 값이 \\(7\\)인 노드를 방문하면 <code>path</code>를 복사하여 결과 리스트 <code>res</code>에 추가한다. 순회를 마치면 <code>res</code>에 모든 해가 저장된다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_ii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"전위 순회: 예제 2\"\"\"\n    if root is None:\n        return\n    # 시도\n    path.append(root)\n    if root.val == 7:\n        # 해를 기록한다\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # 되돌리기\n    path.pop()\n</code></pre> preorder_traversal_ii_compact.cpp<pre><code>/* 전위 순회: 예제 2 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    // 시도\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // 해를 기록한다\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // 되돌리기\n    path.pop_back();\n}\n</code></pre> preorder_traversal_ii_compact.java<pre><code>/* 전위 순회: 예제 2 */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    // 시도\n    path.add(root);\n    if (root.val == 7) {\n        // 해를 기록한다\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // 되돌리기\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_ii_compact.cs<pre><code>/* 전위 순회: 예제 2 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    // 시도\n    path.Add(root);\n    if (root.val == 7) {\n        // 해를 기록한다\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // 되돌리기\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_ii_compact.go<pre><code>/* 전위 순회: 예제 2 */\nfunc preOrderII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    // 시도\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // 해를 기록한다\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderII(root.Left, res, path)\n    preOrderII(root.Right, res, path)\n    // 되돌리기\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_ii_compact.swift<pre><code>/* 전위 순회: 예제 2 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // 시도\n    path.append(root)\n    if root.val == 7 {\n        // 해를 기록한다\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // 되돌리기\n    path.removeLast()\n}\n</code></pre> preorder_traversal_ii_compact.js<pre><code>/* 전위 순회: 예제 2 */\nfunction preOrder(root, path, res) {\n    if (root === null) {\n        return;\n    }\n    // 시도\n    path.push(root);\n    if (root.val === 7) {\n        // 해를 기록한다\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // 되돌리기\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.ts<pre><code>/* 전위 순회: 예제 2 */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    if (root === null) {\n        return;\n    }\n    // 시도\n    path.push(root);\n    if (root.val === 7) {\n        // 해를 기록한다\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // 되돌리기\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.dart<pre><code>/* 전위 순회: 예제 2 */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null) {\n    return;\n  }\n\n  // 시도\n  path.add(root);\n  if (root.val == 7) {\n    // 해를 기록한다\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // 되돌리기\n  path.removeLast();\n}\n</code></pre> preorder_traversal_ii_compact.rs<pre><code>/* 전위 순회: 예제 2 */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        // 시도\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // 해를 기록한다\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.as_ref());\n        pre_order(res, path, node.borrow().right.as_ref());\n        // 되돌리기\n        path.pop();\n    }\n}\n</code></pre> preorder_traversal_ii_compact.c<pre><code>/* 전위 순회: 예제 2 */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    // 시도\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // 해를 기록한다\n        for (int i = 0; i &lt; pathSize; ++i) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // 되돌리기\n    pathSize--;\n}\n</code></pre> preorder_traversal_ii_compact.kt<pre><code>/* 전위 순회: 예제 2 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    // 시도\n    path!!.add(root)\n    if (root._val == 7) {\n        // 해를 기록한다\n        res!!.add(path!!.toMutableList())\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // 되돌리기\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_ii_compact.rb<pre><code>### 전위 순회: 예제 2 ###\ndef pre_order(root)\n  return unless root\n\n  # 시도\n  $path &lt;&lt; root\n\n  # 해를 기록한다\n  $res &lt;&lt; $path.dup if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\n\n  # 되돌리기\n  $path.pop\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>매번 '시도'할 때는 현재 노드를 <code>path</code>에 추가하여 경로를 기록한다. '되돌리기' 전에는 이 노드를 <code>path</code>에서 꺼내어 이번 시도 이전의 상태를 복원해야 한다.</p> <p>아래 그림(13-2)의 과정을 보면 시도와 되돌리기를 각각 '전진'과 '취소'로 이해할 수 있으며, 두 연산은 서로 반대 방향이다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 13-2   시도와 되돌리기 </p>","path":["제 13 장   백트래킹","13.1   백트래킹 알고리즘"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1312","level":2,"title":"13.1.2   가지치기","text":"<p>복잡한 백트래킹 문제에는 보통 하나 이상의 제약 조건이 있으며, 이 제약 조건을 '가지치기'에 활용할 수 있다.</p> <p>예제 3</p> <p>이진 트리에서 값이 \\(7\\)인 모든 노드를 찾아 루트 노드에서 각 노드까지의 경로를 반환하라. 단, 경로에 값이 \\(3\\)인 노드가 포함되면 안 된다.</p> <p>이 제약을 만족하려면 가지치기 연산을 추가해야 한다. 탐색 중 값이 \\(3\\)인 노드를 만나면 더 탐색하지 않고 조기에 반환한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_iii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"전위 순회: 예제 3\"\"\"\n    # 가지치기\n    if root is None or root.val == 3:\n        return\n    # 시도\n    path.append(root)\n    if root.val == 7:\n        # 해를 기록한다\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # 되돌리기\n    path.pop()\n</code></pre> preorder_traversal_iii_compact.cpp<pre><code>/* 전위 순회: 예제 3 */\nvoid preOrder(TreeNode *root) {\n    // 가지치기\n    if (root == nullptr || root-&gt;val == 3) {\n        return;\n    }\n    // 시도\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // 해를 기록한다\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // 되돌리기\n    path.pop_back();\n}\n</code></pre> preorder_traversal_iii_compact.java<pre><code>/* 전위 순회: 예제 3 */\nvoid preOrder(TreeNode root) {\n    // 가지치기\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // 시도\n    path.add(root);\n    if (root.val == 7) {\n        // 해를 기록한다\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // 되돌리기\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_iii_compact.cs<pre><code>/* 전위 순회: 예제 3 */\nvoid PreOrder(TreeNode? root) {\n    // 가지치기\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // 시도\n    path.Add(root);\n    if (root.val == 7) {\n        // 해를 기록한다\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // 되돌리기\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_iii_compact.go<pre><code>/* 전위 순회: 예제 3 */\nfunc preOrderIII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    // 가지치기\n    if root == nil || root.Val == 3 {\n        return\n    }\n    // 시도\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // 해를 기록한다\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderIII(root.Left, res, path)\n    preOrderIII(root.Right, res, path)\n    // 되돌리기\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_iii_compact.swift<pre><code>/* 전위 순회: 예제 3 */\nfunc preOrder(root: TreeNode?) {\n    // 가지치기\n    guard let root = root, root.val != 3 else {\n        return\n    }\n    // 시도\n    path.append(root)\n    if root.val == 7 {\n        // 해를 기록한다\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // 되돌리기\n    path.removeLast()\n}\n</code></pre> preorder_traversal_iii_compact.js<pre><code>/* 전위 순회: 예제 3 */\nfunction preOrder(root, path, res) {\n    // 가지치기\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // 시도\n    path.push(root);\n    if (root.val === 7) {\n        // 해를 기록한다\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // 되돌리기\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.ts<pre><code>/* 전위 순회: 예제 3 */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // 가지치기\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // 시도\n    path.push(root);\n    if (root.val === 7) {\n        // 해를 기록한다\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // 되돌리기\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.dart<pre><code>/* 전위 순회: 예제 3 */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null || root.val == 3) {\n    return;\n  }\n\n  // 시도\n  path.add(root);\n  if (root.val == 7) {\n    // 해를 기록한다\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // 되돌리기\n  path.removeLast();\n}\n</code></pre> preorder_traversal_iii_compact.rs<pre><code>/* 전위 순회: 예제 3 */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    // 가지치기\n    if root.is_none() || root.as_ref().unwrap().borrow().val == 3 {\n        return;\n    }\n    if let Some(node) = root {\n        // 시도\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // 해를 기록한다\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.as_ref());\n        pre_order(res, path, node.borrow().right.as_ref());\n        // 되돌리기\n        path.pop();\n    }\n}\n</code></pre> preorder_traversal_iii_compact.c<pre><code>/* 전위 순회: 예제 3 */\nvoid preOrder(TreeNode *root) {\n    // 가지치기\n    if (root == NULL || root-&gt;val == 3) {\n        return;\n    }\n    // 시도\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // 해를 기록한다\n        for (int i = 0; i &lt; pathSize; i++) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // 되돌리기\n    pathSize--;\n}\n</code></pre> preorder_traversal_iii_compact.kt<pre><code>/* 전위 순회: 예제 3 */\nfun preOrder(root: TreeNode?) {\n    // 가지치기\n    if (root == null || root._val == 3) {\n        return\n    }\n    // 시도\n    path!!.add(root)\n    if (root._val == 7) {\n        // 해를 기록한다\n        res!!.add(path!!.toMutableList())\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // 되돌리기\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_iii_compact.rb<pre><code>### 전위 순회: 예제 3 ###\ndef pre_order(root)\n  # 가지치기\n  return if !root || root.val == 3\n\n  # 시도\n  $path.append(root)\n\n  # 해를 기록한다\n  $res &lt;&lt; $path.dup if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\n\n  # 되돌리기\n  $path.pop\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>'가지치기'는 매우 직관적인 이름이다. 아래 그림(13-3)과 같이 탐색 중 제약 조건을 만족하지 않는 탐색 가지를 '잘라 내어' 의미 없는 시도를 많이 피하고 탐색 효율을 높인다.</p> <p></p> <p> 그림 13-3   제약 조건에 따른 가지치기 </p>","path":["제 13 장   백트래킹","13.1   백트래킹 알고리즘"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1313","level":2,"title":"13.1.3   기본 틀 코드","text":"<p>이제 백트래킹의 '시도, 되돌리기, 가지치기'라는 기본 틀을 추출하여 코드의 범용성을 높여 보자.</p> <p>아래 기본 틀 코드에서 <code>state</code>는 문제의 현재 상태를, <code>choices</code>는 그 상태에서 가능한 선택들을 나타낸다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def backtrack(state: State, choices: list[choice], res: list[state]):\n    \"\"\"백트래킹 알고리즘의 기본 틀\"\"\"\n    # 해인지 확인\n    if is_solution(state):\n        # 해 기록\n        record_solution(state, res)\n        # 더 이상 탐색하지 않는다\n        return\n    # 모든 선택지 순회\n    for choice in choices:\n        # 가지치기: 선택이 유효한지 확인\n        if is_valid(state, choice):\n            # 시도: 선택하고 상태 갱신\n            make_choice(state, choice)\n            backtrack(state, choices, res)\n            # 되돌리기: 선택을 취소하고 이전 상태 복원\n            undo_choice(state, choice)\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nvoid backtrack(State *state, vector&lt;Choice *&gt; &amp;choices, vector&lt;State *&gt; &amp;res) {\n    // 해인지 확인\n    if (isSolution(state)) {\n        // 해 기록\n        recordSolution(state, res);\n        // 더 이상 탐색하지 않는다\n        return;\n    }\n    // 모든 선택지 순회\n    for (Choice choice : choices) {\n        // 가지치기: 선택이 유효한지 확인\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nvoid backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // 해인지 확인\n    if (isSolution(state)) {\n        // 해 기록\n        recordSolution(state, res);\n        // 더 이상 탐색하지 않는다\n        return;\n    }\n    // 모든 선택지 순회\n    for (Choice choice : choices) {\n        // 가지치기: 선택이 유효한지 확인\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nvoid Backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // 해인지 확인\n    if (IsSolution(state)) {\n        // 해 기록\n        RecordSolution(state, res);\n        // 더 이상 탐색하지 않는다\n        return;\n    }\n    // 모든 선택지 순회\n    foreach (Choice choice in choices) {\n        // 가지치기: 선택이 유효한지 확인\n        if (IsValid(state, choice)) {\n            // 시도: 선택하고 상태 갱신\n            MakeChoice(state, choice);\n            Backtrack(state, choices, res);\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nfunc backtrack(state *State, choices []Choice, res *[]State) {\n    // 해인지 확인\n    if isSolution(state) {\n        // 해 기록\n        recordSolution(state, res)\n        // 더 이상 탐색하지 않는다\n        return\n    }\n    // 모든 선택지 순회\n    for _, choice := range choices {\n        // 가지치기: 선택이 유효한지 확인\n        if isValid(state, choice) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nfunc backtrack(state: inout State, choices: [Choice], res: inout [State]) {\n    // 해인지 확인\n    if isSolution(state: state) {\n        // 해 기록\n        recordSolution(state: state, res: &amp;res)\n        // 더 이상 탐색하지 않는다\n        return\n    }\n    // 모든 선택지 순회\n    for choice in choices {\n        // 가지치기: 선택이 유효한지 확인\n        if isValid(state: state, choice: choice) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state: &amp;state, choice: choice)\n            backtrack(state: &amp;state, choices: choices, res: &amp;res)\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nfunction backtrack(state, choices, res) {\n    // 해인지 확인\n    if (isSolution(state)) {\n        // 해 기록\n        recordSolution(state, res);\n        // 더 이상 탐색하지 않는다\n        return;\n    }\n    // 모든 선택지 순회\n    for (let choice of choices) {\n        // 가지치기: 선택이 유효한지 확인\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nfunction backtrack(state: State, choices: Choice[], res: State[]): void {\n    // 해인지 확인\n    if (isSolution(state)) {\n        // 해 기록\n        recordSolution(state, res);\n        // 더 이상 탐색하지 않는다\n        return;\n    }\n    // 모든 선택지 순회\n    for (let choice of choices) {\n        // 가지치기: 선택이 유효한지 확인\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nvoid backtrack(State state, List&lt;Choice&gt;, List&lt;State&gt; res) {\n  // 해인지 확인\n  if (isSolution(state)) {\n    // 해 기록\n    recordSolution(state, res);\n    // 더 이상 탐색하지 않는다\n    return;\n  }\n  // 모든 선택지 순회\n  for (Choice choice in choices) {\n    // 가지치기: 선택이 유효한지 확인\n    if (isValid(state, choice)) {\n      // 시도: 선택하고 상태 갱신\n      makeChoice(state, choice);\n      backtrack(state, choices, res);\n      // 되돌리기: 선택을 취소하고 이전 상태 복원\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nfn backtrack(state: &amp;mut State, choices: &amp;Vec&lt;Choice&gt;, res: &amp;mut Vec&lt;State&gt;) {\n    // 해인지 확인\n    if is_solution(state) {\n        // 해 기록\n        record_solution(state, res);\n        // 더 이상 탐색하지 않는다\n        return;\n    }\n    // 모든 선택지 순회\n    for choice in choices {\n        // 가지치기: 선택이 유효한지 확인\n        if is_valid(state, choice) {\n            // 시도: 선택하고 상태 갱신\n            make_choice(state, choice);\n            backtrack(state, choices, res);\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undo_choice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nvoid backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) {\n    // 해인지 확인\n    if (isSolution(state)) {\n        // 해 기록\n        recordSolution(state, res, numRes);\n        // 더 이상 탐색하지 않는다\n        return;\n    }\n    // 모든 선택지 순회\n    for (int i = 0; i &lt; numChoices; i++) {\n        // 가지치기: 선택이 유효한지 확인\n        if (isValid(state, &amp;choices[i])) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state, &amp;choices[i]);\n            backtrack(state, choices, numChoices, res, numRes);\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state, &amp;choices[i]);\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nfun backtrack(state: State?, choices: List&lt;Choice?&gt;, res: List&lt;State?&gt;?) {\n    // 해인지 확인\n    if (isSolution(state)) {\n        // 해 기록\n        recordSolution(state, res)\n        // 더 이상 탐색하지 않는다\n        return\n    }\n    // 모든 선택지 순회\n    for (choice in choices) {\n        // 가지치기: 선택이 유효한지 확인\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>### 백트래킹 알고리즘의 기본 틀 ###\ndef backtrack(state, choices, res)\n    # 해인지 확인\n    if is_solution?(state)\n        # 해 기록\n        record_solution(state, res)\n        return\n    end\n\n    # 모든 선택지 순회\n    for choice in choices\n        # 가지치기: 선택이 유효한지 확인\n        if is_valid?(state, choice)\n            # 시도: 선택하고 상태 갱신\n            make_choice(state, choice)\n            backtrack(state, choices, res)\n            # 되돌리기: 선택을 취소하고 이전 상태 복원\n            undo_choice(state, choice)\n        end\n    end\nend\n</code></pre> <p>이제 기본 틀 코드로 예제 3을 해결해 보자. 상태 <code>state</code>는 노드 순회 경로, 선택지 <code>choices</code>는 현재 노드의 왼쪽과 오른쪽 자식 노드, 결과 <code>res</code>는 경로들의 리스트다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_iii_template.py<pre><code>def is_solution(state: list[TreeNode]) -&gt; bool:\n    \"\"\"현재 상태가 해인지 판단\"\"\"\n    return state and state[-1].val == 7\n\ndef record_solution(state: list[TreeNode], res: list[list[TreeNode]]):\n    \"\"\"해를 기록한다\"\"\"\n    res.append(list(state))\n\ndef is_valid(state: list[TreeNode], choice: TreeNode) -&gt; bool:\n    \"\"\"현재 상태에서 해당 선택이 유효한지 판단\"\"\"\n    return choice is not None and choice.val != 3\n\ndef make_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"상태 갱신\"\"\"\n    state.append(choice)\n\ndef undo_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"상태 복원\"\"\"\n    state.pop()\n\ndef backtrack(\n    state: list[TreeNode], choices: list[TreeNode], res: list[list[TreeNode]]\n):\n    \"\"\"백트래킹 알고리즘: 예제 3\"\"\"\n    # 해인지 확인한다\n    if is_solution(state):\n        # 해를 기록한다\n        record_solution(state, res)\n    # 모든 선택지를 순회한다\n    for choice in choices:\n        # 가지치기: 선택이 유효한지 확인한다\n        if is_valid(state, choice):\n            # 시도: 선택하고 상태를 갱신한다\n            make_choice(state, choice)\n            # 다음 선택을 진행한다\n            backtrack(state, [choice.left, choice.right], res)\n            # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undo_choice(state, choice)\n</code></pre> preorder_traversal_iii_template.cpp<pre><code>/* 현재 상태가 해인지 판단 */\nbool isSolution(vector&lt;TreeNode *&gt; &amp;state) {\n    return !state.empty() &amp;&amp; state.back()-&gt;val == 7;\n}\n\n/* 해를 기록한다 */\nvoid recordSolution(vector&lt;TreeNode *&gt; &amp;state, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    res.push_back(state);\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nbool isValid(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    return choice != nullptr &amp;&amp; choice-&gt;val != 3;\n}\n\n/* 상태 갱신 */\nvoid makeChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.push_back(choice);\n}\n\n/* 상태 복원 */\nvoid undoChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.pop_back();\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nvoid backtrack(vector&lt;TreeNode *&gt; &amp;state, vector&lt;TreeNode *&gt; &amp;choices, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    // 해인지 확인한다\n    if (isSolution(state)) {\n        // 해를 기록한다\n        recordSolution(state, res);\n    }\n    // 모든 선택지를 순회한다\n    for (TreeNode *choice : choices) {\n        // 가지치기: 선택이 유효한지 확인한다\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(state, choice);\n            // 다음 선택을 진행한다\n            vector&lt;TreeNode *&gt; nextChoices{choice-&gt;left, choice-&gt;right};\n            backtrack(state, nextChoices, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.java<pre><code>/* 현재 상태가 해인지 판단 */\nboolean isSolution(List&lt;TreeNode&gt; state) {\n    return !state.isEmpty() &amp;&amp; state.get(state.size() - 1).val == 7;\n}\n\n/* 해를 기록한다 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.add(new ArrayList&lt;&gt;(state));\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nboolean isValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* 상태 갱신 */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.add(choice);\n}\n\n/* 상태 복원 */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.remove(state.size() - 1);\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nvoid backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // 해인지 확인한다\n    if (isSolution(state)) {\n        // 해를 기록한다\n        recordSolution(state, res);\n    }\n    // 모든 선택지를 순회한다\n    for (TreeNode choice : choices) {\n        // 가지치기: 선택이 유효한지 확인한다\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(state, choice);\n            // 다음 선택을 진행한다\n            backtrack(state, Arrays.asList(choice.left, choice.right), res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.cs<pre><code>/* 현재 상태가 해인지 판단 */\nbool IsSolution(List&lt;TreeNode&gt; state) {\n    return state.Count != 0 &amp;&amp; state[^1].val == 7;\n}\n\n/* 해를 기록한다 */\nvoid RecordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.Add(new List&lt;TreeNode&gt;(state));\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nbool IsValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* 상태 갱신 */\nvoid MakeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.Add(choice);\n}\n\n/* 상태 복원 */\nvoid UndoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.RemoveAt(state.Count - 1);\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nvoid Backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // 해인지 확인한다\n    if (IsSolution(state)) {\n        // 해를 기록한다\n        RecordSolution(state, res);\n    }\n    // 모든 선택지를 순회한다\n    foreach (TreeNode choice in choices) {\n        // 가지치기: 선택이 유효한지 확인한다\n        if (IsValid(state, choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            MakeChoice(state, choice);\n            // 다음 선택을 진행한다\n            Backtrack(state, [choice.left!, choice.right!], res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.go<pre><code>/* 현재 상태가 해인지 판단 */\nfunc isSolution(state *[]*TreeNode) bool {\n    return len(*state) != 0 &amp;&amp; (*state)[len(*state)-1].Val == 7\n}\n\n/* 해를 기록한다 */\nfunc recordSolution(state *[]*TreeNode, res *[][]*TreeNode) {\n    *res = append(*res, append([]*TreeNode{}, *state...))\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nfunc isValid(state *[]*TreeNode, choice *TreeNode) bool {\n    return choice != nil &amp;&amp; choice.Val != 3\n}\n\n/* 상태 갱신 */\nfunc makeChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = append(*state, choice)\n}\n\n/* 상태 복원 */\nfunc undoChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = (*state)[:len(*state)-1]\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nfunc backtrackIII(state *[]*TreeNode, choices *[]*TreeNode, res *[][]*TreeNode) {\n    // 해인지 확인한다\n    if isSolution(state) {\n        // 해를 기록한다\n        recordSolution(state, res)\n    }\n    // 모든 선택지를 순회한다\n    for _, choice := range *choices {\n        // 가지치기: 선택이 유효한지 확인한다\n        if isValid(state, choice) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(state, choice)\n            // 다음 선택을 진행한다\n            temp := make([]*TreeNode, 0)\n            temp = append(temp, choice.Left, choice.Right)\n            backtrackIII(state, &amp;temp, res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.swift<pre><code>/* 현재 상태가 해인지 판단 */\nfunc isSolution(state: [TreeNode]) -&gt; Bool {\n    !state.isEmpty &amp;&amp; state.last!.val == 7\n}\n\n/* 해를 기록한다 */\nfunc recordSolution(state: [TreeNode], res: inout [[TreeNode]]) {\n    res.append(state)\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nfunc isValid(state: [TreeNode], choice: TreeNode?) -&gt; Bool {\n    choice != nil &amp;&amp; choice!.val != 3\n}\n\n/* 상태 갱신 */\nfunc makeChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.append(choice)\n}\n\n/* 상태 복원 */\nfunc undoChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.removeLast()\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nfunc backtrack(state: inout [TreeNode], choices: [TreeNode], res: inout [[TreeNode]]) {\n    // 해인지 확인한다\n    if isSolution(state: state) {\n        recordSolution(state: state, res: &amp;res)\n    }\n    // 모든 선택지를 순회한다\n    for choice in choices {\n        // 가지치기: 선택이 유효한지 확인한다\n        if isValid(state: state, choice: choice) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(state: &amp;state, choice: choice)\n            // 다음 선택을 진행한다\n            backtrack(state: &amp;state, choices: [choice.left, choice.right].compactMap { $0 }, res: &amp;res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.js<pre><code>/* 현재 상태가 해인지 판단 */\nfunction isSolution(state) {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* 해를 기록한다 */\nfunction recordSolution(state, res) {\n    res.push([...state]);\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nfunction isValid(state, choice) {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* 상태 갱신 */\nfunction makeChoice(state, choice) {\n    state.push(choice);\n}\n\n/* 상태 복원 */\nfunction undoChoice(state) {\n    state.pop();\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nfunction backtrack(state, choices, res) {\n    // 해인지 확인한다\n    if (isSolution(state)) {\n        // 해를 기록한다\n        recordSolution(state, res);\n    }\n    // 모든 선택지를 순회한다\n    for (const choice of choices) {\n        // 가지치기: 선택이 유효한지 확인한다\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(state, choice);\n            // 다음 선택을 진행한다\n            backtrack(state, [choice.left, choice.right], res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.ts<pre><code>/* 현재 상태가 해인지 판단 */\nfunction isSolution(state: TreeNode[]): boolean {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* 해를 기록한다 */\nfunction recordSolution(state: TreeNode[], res: TreeNode[][]): void {\n    res.push([...state]);\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nfunction isValid(state: TreeNode[], choice: TreeNode): boolean {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* 상태 갱신 */\nfunction makeChoice(state: TreeNode[], choice: TreeNode): void {\n    state.push(choice);\n}\n\n/* 상태 복원 */\nfunction undoChoice(state: TreeNode[]): void {\n    state.pop();\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nfunction backtrack(\n    state: TreeNode[],\n    choices: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // 해인지 확인한다\n    if (isSolution(state)) {\n        // 해를 기록한다\n        recordSolution(state, res);\n    }\n    // 모든 선택지를 순회한다\n    for (const choice of choices) {\n        // 가지치기: 선택이 유효한지 확인한다\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(state, choice);\n            // 다음 선택을 진행한다\n            backtrack(state, [choice.left, choice.right], res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.dart<pre><code>/* 현재 상태가 해인지 판단 */\nbool isSolution(List&lt;TreeNode&gt; state) {\n  return state.isNotEmpty &amp;&amp; state.last.val == 7;\n}\n\n/* 해를 기록한다 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n  res.add(List.from(state));\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nbool isValid(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* 상태 갱신 */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.add(choice!);\n}\n\n/* 상태 복원 */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.removeLast();\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nvoid backtrack(\n  List&lt;TreeNode&gt; state,\n  List&lt;TreeNode?&gt; choices,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  // 해인지 확인한다\n  if (isSolution(state)) {\n    // 해를 기록한다\n    recordSolution(state, res);\n  }\n  // 모든 선택지를 순회한다\n  for (TreeNode? choice in choices) {\n    // 가지치기: 선택이 유효한지 확인한다\n    if (isValid(state, choice)) {\n      // 시도: 선택하고 상태를 갱신한다\n      makeChoice(state, choice);\n      // 다음 선택을 진행한다\n      backtrack(state, [choice!.left, choice.right], res);\n      // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> preorder_traversal_iii_template.rs<pre><code>/* 현재 상태가 해인지 판단 */\nfn is_solution(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\n    return !state.is_empty() &amp;&amp; state.last().unwrap().borrow().val == 7;\n}\n\n/* 해를 기록한다 */\nfn record_solution(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    res.push(state.clone());\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nfn is_valid(_: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\n    return choice.is_some() &amp;&amp; choice.unwrap().borrow().val != 3;\n}\n\n/* 상태 갱신 */\nfn make_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.push(choice);\n}\n\n/* 상태 복원 */\nfn undo_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, _: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.pop();\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nfn backtrack(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    choices: &amp;Vec&lt;Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    // 해인지 확인한다\n    if is_solution(state) {\n        // 해를 기록한다\n        record_solution(state, res);\n    }\n    // 모든 선택지를 순회한다\n    for &amp;choice in choices.iter() {\n        // 가지치기: 선택이 유효한지 확인한다\n        if is_valid(state, choice) {\n            // 시도: 선택하고 상태를 갱신한다\n            make_choice(state, choice.unwrap().clone());\n            // 다음 선택을 진행한다\n            backtrack(\n                state,\n                &amp;vec![\n                    choice.unwrap().borrow().left.as_ref(),\n                    choice.unwrap().borrow().right.as_ref(),\n                ],\n                res,\n            );\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undo_choice(state, choice.unwrap().clone());\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.c<pre><code>/* 현재 상태가 해인지 판단 */\nbool isSolution(void) {\n    return pathSize &gt; 0 &amp;&amp; path[pathSize - 1]-&gt;val == 7;\n}\n\n/* 해를 기록한다 */\nvoid recordSolution(void) {\n    for (int i = 0; i &lt; pathSize; i++) {\n        res[resSize][i] = path[i];\n    }\n    resSize++;\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nbool isValid(TreeNode *choice) {\n    return choice != NULL &amp;&amp; choice-&gt;val != 3;\n}\n\n/* 상태 갱신 */\nvoid makeChoice(TreeNode *choice) {\n    path[pathSize++] = choice;\n}\n\n/* 상태 복원 */\nvoid undoChoice(void) {\n    pathSize--;\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nvoid backtrack(TreeNode *choices[2]) {\n    // 해인지 확인한다\n    if (isSolution()) {\n        // 해를 기록한다\n        recordSolution();\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; 2; i++) {\n        TreeNode *choice = choices[i];\n        // 가지치기: 선택이 유효한지 확인한다\n        if (isValid(choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(choice);\n            // 다음 선택을 진행한다\n            TreeNode *nextChoices[2] = {choice-&gt;left, choice-&gt;right};\n            backtrack(nextChoices);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice();\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.kt<pre><code>/* 현재 상태가 해인지 판단 */\nfun isSolution(state: MutableList&lt;TreeNode?&gt;): Boolean {\n    return state.isNotEmpty() &amp;&amp; state[state.size - 1]?._val == 7\n}\n\n/* 해를 기록한다 */\nfun recordSolution(state: MutableList&lt;TreeNode?&gt;?, res: MutableList&lt;MutableList&lt;TreeNode?&gt;?&gt;) {\n    res.add(state!!.toMutableList())\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nfun isValid(state: MutableList&lt;TreeNode?&gt;?, choice: TreeNode?): Boolean {\n    return choice != null &amp;&amp; choice._val != 3\n}\n\n/* 상태 갱신 */\nfun makeChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.add(choice)\n}\n\n/* 상태 복원 */\nfun undoChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.removeLast()\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nfun backtrack(\n    state: MutableList&lt;TreeNode?&gt;,\n    choices: MutableList&lt;TreeNode?&gt;,\n    res: MutableList&lt;MutableList&lt;TreeNode?&gt;?&gt;\n) {\n    // 해인지 확인한다\n    if (isSolution(state)) {\n        // 해를 기록한다\n        recordSolution(state, res)\n    }\n    // 모든 선택지를 순회한다\n    for (choice in choices) {\n        // 가지치기: 선택이 유효한지 확인한다\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(state, choice)\n            // 다음 선택을 진행한다\n            backtrack(state, mutableListOf(choice!!.left, choice.right), res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.rb<pre><code>### 현재 상태가 해인지 판단 ###\ndef is_solution?(state)\n  !state.empty? &amp;&amp; state.last.val == 7\nend\n\n### 해를 기록한다 ###\ndef record_solution(state, res)\n  res &lt;&lt; state.dup\nend\n\n### 현재 상태에서 해당 선택이 유효한지 판단 ###\ndef is_valid?(state, choice)\n  choice &amp;&amp; choice.val != 3\nend\n\n### 상태 갱신 ###\ndef make_choice(state, choice)\n  state &lt;&lt; choice\nend\n\n### 상태 복원 ###\ndef undo_choice(state, choice)\n  state.pop\nend\n\n### 백트래킹 알고리즘: 예제 3 ###\ndef backtrack(state, choices, res)\n  # 해인지 확인한다\n  record_solution(state, res) if is_solution?(state)\n\n  # 모든 선택지를 순회한다\n  for choice in choices\n    # 가지치기: 선택이 유효한지 확인한다\n    if is_valid?(state, choice)\n      # 시도: 선택하고 상태를 갱신한다\n      make_choice(state, choice)\n      # 다음 선택을 진행한다\n      backtrack(state, [choice.left, choice.right], res)\n      # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n      undo_choice(state, choice)\n    end\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>문제의 요구에 따르면 값이 \\(7\\)인 노드를 찾은 뒤에도 탐색을 계속해야 한다. 따라서 해를 기록한 뒤의 <code>return</code> 문을 삭제해야 한다. 아래 그림(13-4)은 <code>return</code> 문을 유지하거나 삭제했을 때의 탐색 과정을 비교한다.</p> <p></p> <p> 그림 13-4   return 유지 여부에 따른 탐색 과정 비교 </p> <p>전위 순회 기반 코드에 비해 백트래킹 기본 틀을 사용한 코드는 다소 장황하지만 범용성이 더 좋다. 실제로 많은 백트래킹 문제를 이 틀 안에서 해결할 수 있다. 구체적인 문제에 맞게 <code>state</code>와 <code>choices</code>를 정의하고, 틀 안의 각 메서드만 구현하면 된다.</p>","path":["제 13 장   백트래킹","13.1   백트래킹 알고리즘"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1314","level":2,"title":"13.1.4   주요 용어","text":"<p>알고리즘 문제를 더 명확히 분석하기 위해 백트래킹에서 자주 쓰는 용어를 정리하고, 예제 3에 대응하는 사례를 아래 표(13-1)에 나타냈다.</p> <p> 표 13-1   백트래킹 알고리즘의 주요 용어 </p> 용어 정의 예제 3 해(solution) 문제의 특정 조건을 만족하는 답으로, 하나 또는 여러 개일 수 있다 루트 노드에서 노드 \\(7\\)까지 제약 조건을 만족하는 모든 경로 제약 조건(constraint) 해의 유효성을 제한하는 조건으로, 보통 가지치기에 사용한다 경로에 노드 \\(3\\)이 포함되지 않는다 상태(state) 이미 내린 선택을 포함하여 특정 시점의 문제 상황을 나타낸다 현재까지 방문한 노드의 경로, 즉 노드 리스트 <code>path</code> 시도(attempt) 가능한 선택으로 해 공간을 탐색하는 과정으로, 선택, 상태 갱신, 해인지 확인하는 작업을 포함한다 왼쪽(오른쪽) 자식 노드를 재귀적으로 방문하고 <code>path</code>에 추가한 뒤 값이 \\(7\\)인지 확인한다 되돌리기(backtracking) 제약 조건을 만족하지 않는 상태를 만나면 이전 선택을 취소하고 직전 상태로 돌아간다 리프 노드를 지나거나 노드 방문을 마치거나 값이 \\(3\\)인 노드를 만나면 탐색을 멈추고 함수가 반환된다 가지치기(pruning) 문제의 특성과 제약 조건에 따라 의미 없는 탐색 경로를 피하여 효율을 높이는 방법이다 값이 \\(3\\)인 노드를 만나면 탐색을 더 진행하지 않는다 <p>참고</p> <p>문제, 해, 상태 등의 개념은 분할 정복, 백트래킹, 동적 계획법, 탐욕 알고리즘 등에서 공통으로 사용된다.</p>","path":["제 13 장   백트래킹","13.1   백트래킹 알고리즘"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1315","level":2,"title":"13.1.5   장점과 한계","text":"<p>백트래킹은 본질적으로 깊이 우선 탐색 알고리즘으로, 조건을 만족하는 해를 찾을 때까지 가능한 모든 해결 방법을 시도한다. 모든 가능한 해를 찾을 수 있으며, 적절히 가지치기하면 매우 효율적이라는 장점이 있다.</p> <p>하지만 규모가 크거나 복잡한 문제에서는 백트래킹의 실행 효율을 받아들이기 어려울 수 있다.</p> <ul> <li>시간: 상태 공간의 모든 가능성을 순회해야 하는 경우가 많아 시간 복잡도가 지수 또는 팩토리얼 수준에 이를 수 있다.</li> <li>공간: 재귀 호출 중 현재 상태(경로, 가지치기용 보조 변수 등)를 저장해야 하므로 깊이가 크면 공간 요구량도 매우 커질 수 있다.</li> </ul> <p>그렇더라도 일부 탐색 문제와 제약 만족 문제에서는 백트래킹이 여전히 최선의 해결 방법이다. 어떤 선택이 유효한 해를 만들지 예측할 수 없으므로 가능한 모든 선택을 순회해야 하기 때문이다. 이때는 효율을 어떻게 최적화하느냐가 핵심이며, 대표적인 방법은 두 가지다.</p> <ul> <li>가지치기: 해가 나올 수 없는 경로를 탐색하지 않아 시간과 공간을 절약한다.</li> <li>휴리스틱 탐색: 탐색에 전략이나 추정값을 도입하여 유효한 해가 나올 가능성이 높은 경로를 먼저 탐색한다.</li> </ul>","path":["제 13 장   백트래킹","13.1   백트래킹 알고리즘"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1316","level":2,"title":"13.1.6   대표적인 백트래킹 문제","text":"<p>백트래킹은 많은 탐색 문제, 제약 만족 문제, 조합 최적화 문제를 해결하는 데 사용할 수 있다.</p> <p>탐색 문제: 특정 조건을 만족하는 해결 방법을 찾는 것이 목표다.</p> <ul> <li>모든 순열 문제: 집합이 주어졌을 때 가능한 모든 배열 순서를 구한다.</li> <li>부분 집합 합 문제: 집합과 목표 합이 주어졌을 때 원소의 합이 목표 합인 모든 부분 집합을 찾는다.</li> <li>하노이의 탑 문제: 기둥 세 개와 크기가 다른 원판들이 주어졌을 때 모든 원판을 한 기둥에서 다른 기둥으로 옮긴다. 한 번에 하나만 옮길 수 있고 큰 원판을 작은 원판 위에 놓으면 안 된다.</li> </ul> <p>제약 만족 문제: 모든 제약 조건을 만족하는 해를 찾는 것이 목표다.</p> <ul> <li>\\(n\\)-퀸: \\(n \\times n\\) 체스판에 퀸 \\(n\\) 개를 서로 공격하지 않도록 놓는다.</li> <li>스도쿠: \\(9 \\times 9\\) 격자에 \\(1\\) ~ \\(9\\)를 넣되, 각 행과 열 및 각 \\(3 \\times 3\\) 부분 격자에서 숫자가 중복되지 않게 한다.</li> <li>그래프 색칠 문제: 무방향 그래프의 각 정점을 가능한 한 적은 색으로 칠하되, 인접한 정점은 서로 다른 색을 갖게 한다.</li> </ul> <p>조합 최적화 문제: 조합 공간에서 특정 조건을 만족하는 최적해를 찾는 것이 목표다.</p> <ul> <li>0-1 배낭 문제: 가치와 무게가 있는 물건들과 배낭이 주어졌을 때, 용량 제한 안에서 총가치가 최대가 되도록 물건을 고른다.</li> <li>외판원 문제: 그래프의 한 점에서 출발하여 다른 모든 점을 정확히 한 번 방문한 뒤 출발점으로 돌아오는 최단 경로를 구한다.</li> <li>최대 클리크 문제: 무방향 그래프에서 가장 큰 완전 부분 그래프, 즉 임의의 두 정점 사이에 간선이 있는 부분 그래프를 찾는다.</li> </ul> <p>많은 조합 최적화 문제에서 백트래킹이 최적의 해결 방법은 아니라는 점에 주의하자.</p> <ul> <li>0-1 배낭 문제는 더 높은 시간 효율을 위해 보통 동적 계획법으로 해결한다.</li> <li>외판원 문제는 유명한 NP-Hard 문제이며, 유전 알고리즘과 개미 군집 알고리즘 등이 자주 쓰인다.</li> <li>최대 클리크 문제는 그래프 이론의 대표적인 문제로, 탐욕 알고리즘 같은 휴리스틱 알고리즘을 사용할 수 있다.</li> </ul>","path":["제 13 장   백트래킹","13.1   백트래킹 알고리즘"],"tags":[]},{"location":"chapter_backtracking/exercises/","level":1,"title":"13.6   연습 문제","text":"","path":["제 13 장   백트래킹","13.6   연습 문제"],"tags":[]},{"location":"chapter_backtracking/exercises/#1361","level":2,"title":"13.6.1   개념 확인","text":"","path":["제 13 장   백트래킹","13.6   연습 문제"],"tags":[]},{"location":"chapter_backtracking/exercises/#1","level":3,"title":"1.   이 순열 알고리즘은 결과를 빠뜨리는가","text":"<p>한 백트래킹 알고리즘이 <code>1, 2, 3</code> 순서로 숫자를 시도하며 모든 순열을 생성한다. 숫자 <code>x</code>를 선택할 때마다 다음을 수행한다.</p> <ol> <li>현재 경로의 끝에 <code>x</code>를 추가한다.</li> <li><code>x</code>를 ‘사용함’으로 표시한다.</li> <li>재귀 호출로 다음 위치를 채운다.</li> </ol> <p>재귀 호출이 반환된 뒤, 학생은 경로 끝의 <code>x</code>만 삭제하고 다음 숫자를 계속 시도했다.</p> <ol> <li>알고리즘이 처음 얻는 순열은 무엇인가? 순열 6개를 모두 얻을 수 있는가?</li> <li>재귀 호출에서 이전 단계로 돌아가기 전에 경로 끝의 숫자를 삭제하는 것만으로 충분한가? 충분하지 않다면 무엇을 더 해야 하는지 이유와 함께 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>처음 얻는 순열은 <code>[1, 2, 3]</code>이지만, 모든 순열을 얻을 수는 없다. 반환하면서 경로는 짧아져도     숫자 1, 2, 3은 여전히 모두 ‘사용함’으로 표시되어 있어 이후 분기에서 선택할 숫자가 없기 때문이다.</p> </li> <li> <p>충분하지 않다. 경로 끝의 <code>x</code>를 삭제한 뒤에는 <code>x</code>를 다시 ‘사용하지 않음’으로 표시해야 한다.     현재 경로와 사용 여부 표시가 함께 탐색 상태를 나타낸다. 선택할 때 두 곳을 수정했으므로 되돌아갈 때도 두 곳을 모두 복원해야     다른 분기에서 <code>x</code>를 다시 선택할 수 있다.</p> </li> </ol>","path":["제 13 장   백트래킹","13.6   연습 문제"],"tags":[]},{"location":"chapter_backtracking/exercises/#2","level":3,"title":"2.   숫자를 선택하는 순서가 중요한가","text":"<p>정렬된 배열 <code>[2, 3, 5]</code>와 목표 값 5가 주어지며, 각 숫자는 반복해서 선택할 수 있다. 알고리즘에서는 각 탐색 경로의 숫자가 작은 것부터 큰 것 순으로만 나타나도록 제한한다.</p> <ol> <li>서로 다른 어떤 조합을 얻을 수 있는가?</li> <li>같은 숫자들을 다른 순서로 반복해서 탐색할 필요가 없는 이유는 무엇인가? ‘작은 것부터 큰 것 순’이라는 제한은 어떤 역할을 하는가?</li> <li>현재 경로가 <code>[3]</code>이고 2가 더 필요한 상황에서 다음 후보 숫자가 3이다. 이때 이 단계의 나머지 후보를 모두 확인하지 않고 멈출 수 있는 이유는 무엇인가?</li> </ol> 참고 답안 <ol> <li> <p>서로 다른 조합은 <code>[2, 3]</code>과 <code>[5]</code>다.</p> </li> <li> <p>이 문제에서는 <code>[2, 3]</code>과 <code>[3, 2]</code>를 같은 조합으로 취급하므로 숫자를 선택하는 순서는 답에 반영하지 않는다.     경로의 숫자가 작은 것부터 큰 것 순으로 나타나도록 정하면 탐색 중에 <code>[3, 2]</code> 같은 중복 조합을 바로 건너뛸 수 있다.</p> </li> <li> <p>현재 2가 더 필요한데 후보 숫자 3은 이미 2보다 크다. 배열이 정렬되어 있으므로     3 뒤의 후보는 더 크기만 할 뿐이고, 모두 현재 조합에 추가할 수 없다. 따라서 이 단계의 확인을 바로 끝낼 수 있다.</p> </li> </ol>","path":["제 13 장   백트래킹","13.6   연습 문제"],"tags":[]},{"location":"chapter_backtracking/exercises/#3","level":3,"title":"3.   다음 퀸은 어디에 놓을 수 있는가","text":"<p><code>4 × 4</code> 체스판에 행별로 퀸을 놓으며, 행과 열의 인덱스는 모두 0부터 시작한다. 현재 <code>(0, 1)</code>과 <code>(1, 3)</code>에 퀸을 놓았고, 이제 인덱스 2인 행에 다음 퀸을 놓으려 한다.</p> <ol> <li>‘같은 열’이라는 이유로 제외되는 열은 어디인가?</li> <li>남은 열 가운데 ‘같은 대각선’이라는 이유로 제외되는 위치는 어디인가?</li> <li>인덱스 2인 행에서 시도해 볼 수 있는 위치는 어디인가?</li> </ol> 참고 답안 <ol> <li> <p>인덱스 1과 3인 열에 이미 퀸이 있으므로 <code>(2, 1)</code>과 <code>(2, 3)</code>은 제외된다.</p> </li> <li> <p>남은 위치 중 <code>(2, 2)</code>는 <code>(1, 3)</code>과 같은 대각선에 있으므로 역시 제외된다.     <code>(2, 0)</code>은 이미 놓인 두 퀸 중 어느 것과도 같은 열이나 같은 대각선에 있지 않다.</p> </li> <li> <p>인덱스 2인 행에서 시도할 수 있는 위치는 <code>(2, 0)</code>뿐이다.</p> <p>이 단계에서는 현재 배치가 조건에 맞는다는 것만 확인한 셈이다. 이후 체스판을 완성할 수 없다면 여전히 되돌아가 더 앞 단계의 다른 선택을 시도해야 한다.</p> </li> </ol>","path":["제 13 장   백트래킹","13.6   연습 문제"],"tags":[]},{"location":"chapter_backtracking/exercises/#1362","level":2,"title":"13.6.2   프로그래밍 연습","text":"","path":["제 13 장   백트래킹","13.6   연습 문제"],"tags":[]},{"location":"chapter_backtracking/exercises/#1_1","level":3,"title":"1.   중복 원소가 없는 모든 순열","text":"<p>정수 배열 <code>nums</code>에는 원소가 적어도 하나 있으며, 모든 원소가 서로 다르다. 각 원소를 한 번씩 사용하여 만들 수 있는 모든 순서를 나열하고, 각 순서를 배열로 반환하라. 결과에서 순열들이 나오는 순서는 상관없다. 백트래킹을 사용하고, 각 위치의 원소가 현재 순열에 이미 선택되었는지 불리언 배열로 기록하라.</p> 풀이 힌트 <ol> <li>재귀 깊이는 순열의 몇 번째 위치를 채우는 중인지를 나타낸다.</li> <li>각 단계에서는 아직 사용하지 않은 원소만 시도한다.</li> <li>경로 길이가 <code>nums</code>의 길이에 도달하면 경로의 복사본을 답에 추가한다.</li> </ol> <p>LeetCode</p>","path":["제 13 장   백트래킹","13.6   연습 문제"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/","level":1,"title":"13.4   n-퀸 문제","text":"<p>문제</p> <p>체스 규칙에서 퀸은 같은 행, 열 또는 대각선에 있는 말을 공격할 수 있다. 퀸 \\(n\\) 개와 \\(n \\times n\\) 크기의 체스판이 주어졌을 때, 퀸들이 서로 공격할 수 없도록 배치하는 방법을 찾아라.</p> <p>아래 그림(13-15)과 같이 \\(n = 4\\)이면 해는 두 개다. 백트래킹 관점에서 \\(n \\times n\\) 체스판의 \\(n^2\\) 개 칸은 모든 선택지 <code>choices</code>를 제공한다. 퀸을 하나씩 놓는 동안 체스판은 계속 바뀌며, 각 시점의 체스판이 상태 <code>state</code>다.</p> <p></p> <p> 그림 13-15   4-퀸 문제의 해 </p> <p>아래 그림(13-16)은 세 가지 제약 조건을 보여 준다. 여러 퀸이 같은 행, 같은 열, 같은 대각선에 있으면 안 된다. 대각선은 주대각선 방향 <code>\\</code>과 부대각선 방향 <code>/</code>의 두 종류로 나뉜다.</p> <p></p> <p> 그림 13-16   n-퀸 문제의 제약 조건 </p>","path":["제 13 장   백트래킹","13.4   n-퀸 문제"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/#1","level":3,"title":"1.   한 행씩 배치하는 전략","text":"<p>퀸 수와 체스판의 행 수가 모두 \\(n\\)이므로 각 행에는 퀸을 정확히 하나씩 놓아야 한다는 사실을 쉽게 도출할 수 있다.</p> <p>따라서 첫 행부터 마지막 행까지 매 행에 퀸 하나씩 놓는 전략을 사용할 수 있다.</p> <p>아래 그림(13-17)은 4-퀸 문제에서 한 행씩 배치하는 과정이다. 공간 제약 때문에 첫 행의 탐색 가지 중 하나만 펼쳤으며, 열과 대각선 제약을 만족하지 않는 배치는 모두 가지치기했다.</p> <p></p> <p> 그림 13-17   한 행씩 배치하는 전략 </p> <p>본질적으로 행별 배치 전략 자체가 가지치기 역할을 한다. 같은 행에 여러 퀸이 놓이는 모든 탐색 가지를 피하기 때문이다.</p>","path":["제 13 장   백트래킹","13.4   n-퀸 문제"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/#2","level":3,"title":"2.   열과 대각선 가지치기","text":"<p>열 제약을 만족하기 위해 길이 \\(n\\)의 불리언 배열 <code>cols</code>에 각 열의 퀸 존재 여부를 기록할 수 있다. 배치를 결정하기 전에 <code>cols</code>로 이미 퀸이 있는 열을 가지치기하고, 백트래킹 중 <code>cols</code>의 상태를 동적으로 갱신한다.</p> <p>참고</p> <p>행렬의 시작점은 왼쪽 위이며, 행 인덱스는 아래로 갈수록, 열 인덱스는 오른쪽으로 갈수록 커진다.</p> <p>대각선 제약은 어떻게 처리할까? 한 칸의 행과 열 인덱스를 \\((row, col)\\)이라고 하자. 주대각선 방향의 대각선 하나를 고르면, 그 위의 모든 칸에서 행 인덱스와 열 인덱스의 차가 같다. 즉, 같은 주대각선 방향 대각선에서는 \\(row - col\\)이 일정하다.</p> <p>따라서 두 칸이 \\(row_1 - col_1 = row_2 - col_2\\)를 만족하면 반드시 같은 주대각선 방향 대각선에 있다. 이 규칙을 이용하여 아래 그림(13-18)의 배열 <code>diags1</code>에 각 대각선의 퀸 존재 여부를 기록할 수 있다.</p> <p>마찬가지로 같은 부대각선 방향 대각선의 모든 칸에서는 \\(row + col\\)이 일정하다. 배열 <code>diags2</code>를 이용하여 이 방향의 제약도 처리할 수 있다.</p> <p></p> <p> 그림 13-18   열 제약과 대각선 제약 처리 </p>","path":["제 13 장   백트래킹","13.4   n-퀸 문제"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/#3","level":3,"title":"3.   코드 구현","text":"<p>\\(n\\) 차 정방행렬에서 \\(row - col\\)의 범위는 \\([-n + 1, n - 1]\\), \\(row + col\\)의 범위는 \\([0, 2n - 2]\\)이다. 따라서 두 방향의 대각선 수는 각각 \\(2n - 1\\) 개이며, 배열 <code>diags1</code>과 <code>diags2</code>의 길이도 모두 \\(2n - 1\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby n_queens.py<pre><code>def backtrack(\n    row: int,\n    n: int,\n    state: list[list[str]],\n    res: list[list[list[str]]],\n    cols: list[bool],\n    diags1: list[bool],\n    diags2: list[bool],\n):\n    \"\"\"백트래킹 알고리즘: n-퀸\"\"\"\n    # 모든 행에 배치했으면 해를 기록한다\n    if row == n:\n        res.append([list(row) for row in state])\n        return\n    # 모든 열을 순회한다\n    for col in range(n):\n        # 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        diag1 = row - col + n - 1\n        diag2 = row + col\n        # 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if not cols[col] and not diags1[diag1] and not diags2[diag2]:\n            # 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = \"Q\"\n            cols[col] = diags1[diag1] = diags2[diag2] = True\n            # 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            # 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = \"#\"\n            cols[col] = diags1[diag1] = diags2[diag2] = False\n\ndef n_queens(n: int) -&gt; list[list[list[str]]]:\n    \"\"\"n-퀸 문제 풀기\"\"\"\n    # n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    state = [[\"#\" for _ in range(n)] for _ in range(n)]\n    cols = [False] * n  # 열에 퀸이 있는지 기록한다\n    diags1 = [False] * (2 * n - 1)  # 주대각선에 퀸이 있는지 기록한다\n    diags2 = [False] * (2 * n - 1)  # 부대각선에 퀸이 있는지 기록한다\n    res = []\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n</code></pre> n_queens.cpp<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nvoid backtrack(int row, int n, vector&lt;vector&lt;string&gt;&gt; &amp;state, vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; &amp;res, vector&lt;bool&gt; &amp;cols,\n               vector&lt;bool&gt; &amp;diags1, vector&lt;bool&gt; &amp;diags2) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if (row == n) {\n        res.push_back(state);\n        return;\n    }\n    // 모든 열을 순회한다\n    for (int col = 0; col &lt; n; col++) {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nvector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; nQueens(int n) {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    vector&lt;vector&lt;string&gt;&gt; state(n, vector&lt;string&gt;(n, \"#\"));\n    vector&lt;bool&gt; cols(n, false);           // 열에 퀸이 있는지 기록한다\n    vector&lt;bool&gt; diags1(2 * n - 1, false); // 주대각선에 퀸이 있는지 기록한다\n    vector&lt;bool&gt; diags2(2 * n - 1, false); // 부대각선에 퀸이 있는지 기록한다\n    vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; res;\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.java<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nvoid backtrack(int row, int n, List&lt;List&lt;String&gt;&gt; state, List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n        boolean[] cols, boolean[] diags1, boolean[] diags2) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if (row == n) {\n        List&lt;List&lt;String&gt;&gt; copyState = new ArrayList&lt;&gt;();\n        for (List&lt;String&gt; sRow : state) {\n            copyState.add(new ArrayList&lt;&gt;(sRow));\n        }\n        res.add(copyState);\n        return;\n    }\n    // 모든 열을 순회한다\n    for (int col = 0; col &lt; n; col++) {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state.get(row).set(col, \"Q\");\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state.get(row).set(col, \"#\");\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    List&lt;List&lt;String&gt;&gt; state = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;String&gt; row = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            row.add(\"#\");\n        }\n        state.add(row);\n    }\n    boolean[] cols = new boolean[n]; // 열에 퀸이 있는지 기록한다\n    boolean[] diags1 = new boolean[2 * n - 1]; // 주대각선에 퀸이 있는지 기록한다\n    boolean[] diags2 = new boolean[2 * n - 1]; // 부대각선에 퀸이 있는지 기록한다\n    List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = new ArrayList&lt;&gt;();\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.cs<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nvoid Backtrack(int row, int n, List&lt;List&lt;string&gt;&gt; state, List&lt;List&lt;List&lt;string&gt;&gt;&gt; res,\n        bool[] cols, bool[] diags1, bool[] diags2) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if (row == n) {\n        List&lt;List&lt;string&gt;&gt; copyState = [];\n        foreach (List&lt;string&gt; sRow in state) {\n            copyState.Add(new List&lt;string&gt;(sRow));\n        }\n        res.Add(copyState);\n        return;\n    }\n    // 모든 열을 순회한다\n    for (int col = 0; col &lt; n; col++) {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // 다음 행에 배치한다\n            Backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nList&lt;List&lt;List&lt;string&gt;&gt;&gt; NQueens(int n) {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    List&lt;List&lt;string&gt;&gt; state = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;string&gt; row = [];\n        for (int j = 0; j &lt; n; j++) {\n            row.Add(\"#\");\n        }\n        state.Add(row);\n    }\n    bool[] cols = new bool[n]; // 열에 퀸이 있는지 기록한다\n    bool[] diags1 = new bool[2 * n - 1]; // 주대각선에 퀸이 있는지 기록한다\n    bool[] diags2 = new bool[2 * n - 1]; // 부대각선에 퀸이 있는지 기록한다\n    List&lt;List&lt;List&lt;string&gt;&gt;&gt; res = [];\n\n    Backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.go<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if row == n {\n        newState := make([][]string, len(*state))\n        for i, _ := range newState {\n            newState[i] = make([]string, len((*state)[0]))\n            copy(newState[i], (*state)[i])\n\n        }\n        *res = append(*res, newState)\n        return\n    }\n    // 모든 열을 순회한다\n    for col := 0; col &lt; n; col++ {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        diag1 := row - col + n - 1\n        diag2 := row + col\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if !(*cols)[col] &amp;&amp; !(*diags1)[diag1] &amp;&amp; !(*diags2)[diag2] {\n            // 시도: 해당 칸에 퀸을 놓는다\n            (*state)[row][col] = \"Q\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n            // 다음 행에 배치한다\n            backtrack(row+1, n, state, res, cols, diags1, diags2)\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            (*state)[row][col] = \"#\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nfunc nQueens(n int) [][][]string {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    state := make([][]string, n)\n    for i := 0; i &lt; n; i++ {\n        row := make([]string, n)\n        for i := 0; i &lt; n; i++ {\n            row[i] = \"#\"\n        }\n        state[i] = row\n    }\n    // 열에 퀸이 있는지 기록한다\n    cols := make([]bool, n)\n    diags1 := make([]bool, 2*n-1)\n    diags2 := make([]bool, 2*n-1)\n    res := make([][][]string, 0)\n    backtrack(0, n, &amp;state, &amp;res, &amp;cols, &amp;diags1, &amp;diags2)\n    return res\n}\n</code></pre> n_queens.swift<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nfunc backtrack(row: Int, n: Int, state: inout [[String]], res: inout [[[String]]], cols: inout [Bool], diags1: inout [Bool], diags2: inout [Bool]) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if row == n {\n        res.append(state)\n        return\n    }\n    // 모든 열을 순회한다\n    for col in 0 ..&lt; n {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        let diag1 = row - col + n - 1\n        let diag2 = row + col\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = \"Q\"\n            cols[col] = true\n            diags1[diag1] = true\n            diags2[diag2] = true\n            // 다음 행에 배치한다\n            backtrack(row: row + 1, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = \"#\"\n            cols[col] = false\n            diags1[diag1] = false\n            diags2[diag2] = false\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nfunc nQueens(n: Int) -&gt; [[[String]]] {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    var state = Array(repeating: Array(repeating: \"#\", count: n), count: n)\n    var cols = Array(repeating: false, count: n) // 열에 퀸이 있는지 기록한다\n    var diags1 = Array(repeating: false, count: 2 * n - 1) // 주대각선에 퀸이 있는지 기록한다\n    var diags2 = Array(repeating: false, count: 2 * n - 1) // 부대각선에 퀸이 있는지 기록한다\n    var res: [[[String]]] = []\n\n    backtrack(row: 0, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n\n    return res\n}\n</code></pre> n_queens.js<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nfunction backtrack(row, n, state, res, cols, diags1, diags2) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // 모든 열을 순회한다\n    for (let col = 0; col &lt; n; col++) {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nfunction nQueens(n) {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // 열에 퀸이 있는지 기록한다\n    const diags1 = Array(2 * n - 1).fill(false); // 주대각선에 퀸이 있는지 기록한다\n    const diags2 = Array(2 * n - 1).fill(false); // 부대각선에 퀸이 있는지 기록한다\n    const res = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.ts<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nfunction backtrack(\n    row: number,\n    n: number,\n    state: string[][],\n    res: string[][][],\n    cols: boolean[],\n    diags1: boolean[],\n    diags2: boolean[]\n): void {\n    // 모든 행에 배치했으면 해를 기록한다\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // 모든 열을 순회한다\n    for (let col = 0; col &lt; n; col++) {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nfunction nQueens(n: number): string[][][] {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // 열에 퀸이 있는지 기록한다\n    const diags1 = Array(2 * n - 1).fill(false); // 주대각선에 퀸이 있는지 기록한다\n    const diags2 = Array(2 * n - 1).fill(false); // 부대각선에 퀸이 있는지 기록한다\n    const res: string[][][] = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.dart<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nvoid backtrack(\n  int row,\n  int n,\n  List&lt;List&lt;String&gt;&gt; state,\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n  List&lt;bool&gt; cols,\n  List&lt;bool&gt; diags1,\n  List&lt;bool&gt; diags2,\n) {\n  // 모든 행에 배치했으면 해를 기록한다\n  if (row == n) {\n    List&lt;List&lt;String&gt;&gt; copyState = [];\n    for (List&lt;String&gt; sRow in state) {\n      copyState.add(List.from(sRow));\n    }\n    res.add(copyState);\n    return;\n  }\n  // 모든 열을 순회한다\n  for (int col = 0; col &lt; n; col++) {\n    // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n    int diag1 = row - col + n - 1;\n    int diag2 = row + col;\n    // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n    if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n      // 시도: 해당 칸에 퀸을 놓는다\n      state[row][col] = \"Q\";\n      cols[col] = true;\n      diags1[diag1] = true;\n      diags2[diag2] = true;\n      // 다음 행에 배치한다\n      backtrack(row + 1, n, state, res, cols, diags1, diags2);\n      // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n      state[row][col] = \"#\";\n      cols[col] = false;\n      diags1[diag1] = false;\n      diags2[diag2] = false;\n    }\n  }\n}\n\n/* n-퀸 문제 풀기 */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n  // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n  List&lt;List&lt;String&gt;&gt; state = List.generate(n, (index) =&gt; List.filled(n, \"#\"));\n  List&lt;bool&gt; cols = List.filled(n, false); // 열에 퀸이 있는지 기록한다\n  List&lt;bool&gt; diags1 = List.filled(2 * n - 1, false); // 주대각선에 퀸이 있는지 기록한다\n  List&lt;bool&gt; diags2 = List.filled(2 * n - 1, false); // 부대각선에 퀸이 있는지 기록한다\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = [];\n\n  backtrack(0, n, state, res, cols, diags1, diags2);\n\n  return res;\n}\n</code></pre> n_queens.rs<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nfn backtrack(\n    row: usize,\n    n: usize,\n    state: &amp;mut Vec&lt;Vec&lt;String&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt;,\n    cols: &amp;mut [bool],\n    diags1: &amp;mut [bool],\n    diags2: &amp;mut [bool],\n) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if row == n {\n        res.push(state.clone());\n        return;\n    }\n    // 모든 열을 순회한다\n    for col in 0..n {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        let diag1 = row + n - 1 - col;\n        let diag2 = row + col;\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = \"Q\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (true, true, true);\n            // 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = \"#\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (false, false, false);\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nfn n_queens(n: usize) -&gt; Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    let mut state: Vec&lt;Vec&lt;String&gt;&gt; = vec![vec![\"#\".to_string(); n]; n];\n    let mut cols = vec![false; n]; // 열에 퀸이 있는지 기록한다\n    let mut diags1 = vec![false; 2 * n - 1]; // 주대각선에 퀸이 있는지 기록한다\n    let mut diags2 = vec![false; 2 * n - 1]; // 부대각선에 퀸이 있는지 기록한다\n    let mut res: Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; = Vec::new();\n\n    backtrack(\n        0,\n        n,\n        &amp;mut state,\n        &amp;mut res,\n        &amp;mut cols,\n        &amp;mut diags1,\n        &amp;mut diags2,\n    );\n\n    res\n}\n</code></pre> n_queens.c<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nvoid backtrack(int row, int n, char state[MAX_SIZE][MAX_SIZE], char ***res, int *resSize, bool cols[MAX_SIZE],\n               bool diags1[2 * MAX_SIZE - 1], bool diags2[2 * MAX_SIZE - 1]) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if (row == n) {\n        res[*resSize] = (char **)malloc(sizeof(char *) * n);\n        for (int i = 0; i &lt; n; ++i) {\n            res[*resSize][i] = (char *)malloc(sizeof(char) * (n + 1));\n            strcpy(res[*resSize][i], state[i]);\n        }\n        (*resSize)++;\n        return;\n    }\n    // 모든 열을 순회한다\n    for (int col = 0; col &lt; n; col++) {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, resSize, cols, diags1, diags2);\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nchar ***nQueens(int n, int *returnSize) {\n    char state[MAX_SIZE][MAX_SIZE];\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    for (int i = 0; i &lt; n; ++i) {\n        for (int j = 0; j &lt; n; ++j) {\n            state[i][j] = '#';\n        }\n        state[i][n] = '\\0';\n    }\n    bool cols[MAX_SIZE] = {false};           // 열에 퀸이 있는지 기록한다\n    bool diags1[2 * MAX_SIZE - 1] = {false}; // 주대각선에 퀸이 있는지 기록한다\n    bool diags2[2 * MAX_SIZE - 1] = {false}; // 부대각선에 퀸이 있는지 기록한다\n\n    char ***res = (char ***)malloc(sizeof(char **) * MAX_SIZE);\n    *returnSize = 0;\n    backtrack(0, n, state, res, returnSize, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.kt<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nfun backtrack(\n    row: Int,\n    n: Int,\n    state: MutableList&lt;MutableList&lt;String&gt;&gt;,\n    res: MutableList&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt;,\n    cols: BooleanArray,\n    diags1: BooleanArray,\n    diags2: BooleanArray\n) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if (row == n) {\n        val copyState = mutableListOf&lt;MutableList&lt;String&gt;&gt;()\n        for (sRow in state) {\n            copyState.add(sRow.toMutableList())\n        }\n        res.add(copyState)\n        return\n    }\n    // 모든 열을 순회한다\n    for (col in 0..&lt;n) {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        val diag1 = row - col + n - 1\n        val diag2 = row + col\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = \"Q\"\n            diags2[diag2] = true\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n            // 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = \"#\"\n            diags2[diag2] = false\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nfun nQueens(n: Int): MutableList&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt; {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    val state = mutableListOf&lt;MutableList&lt;String&gt;&gt;()\n    for (i in 0..&lt;n) {\n        val row = mutableListOf&lt;String&gt;()\n        for (j in 0..&lt;n) {\n            row.add(\"#\")\n        }\n        state.add(row)\n    }\n    val cols = BooleanArray(n) // 열에 퀸이 있는지 기록한다\n    val diags1 = BooleanArray(2 * n - 1) // 주대각선에 퀸이 있는지 기록한다\n    val diags2 = BooleanArray(2 * n - 1) // 부대각선에 퀸이 있는지 기록한다\n    val res = mutableListOf&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt;()\n\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n}\n</code></pre> n_queens.rb<pre><code>### 백트래킹 알고리즘: n-퀸 ###\ndef backtrack(row, n, state, res, cols, diags1, diags2)\n  # 모든 행에 배치했으면 해를 기록한다\n  if row == n\n    res &lt;&lt; state.map { |row| row.dup }\n    return\n  end\n\n  # 모든 열을 순회한다\n  for col in 0...n\n    # 해당 칸이 속한 주대각선과 부대각선을 계산한다\n    diag1 = row - col + n - 1\n    diag2 = row + col\n    # 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n    if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]\n      # 시도: 해당 칸에 퀸을 놓는다\n      state[row][col] = \"Q\"\n      cols[col] = diags1[diag1] = diags2[diag2] = true\n      # 다음 행에 배치한다\n      backtrack(row + 1, n, state, res, cols, diags1, diags2)\n      # 되돌리기: 해당 칸을 빈칸으로 되돌린다\n      state[row][col] = \"#\"\n      cols[col] = diags1[diag1] = diags2[diag2] = false\n    end\n  end\nend\n\n### n-퀸 문제 풀기 ###\ndef n_queens(n)\n  # n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n  state = Array.new(n) { Array.new(n, \"#\") }\n  cols = Array.new(n, false) # 열에 퀸이 있는지 기록한다\n  diags1 = Array.new(2 * n - 1, false) # 주대각선에 퀸이 있는지 기록한다\n  diags2 = Array.new(2 * n - 1, false) # 부대각선에 퀸이 있는지 기록한다\n  res = []\n  backtrack(0, n, state, res, cols, diags1, diags2)\n\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>행별로 \\(n\\) 번 배치할 때 열 제약을 고려하면 첫 행부터 마지막 행까지 선택지 수가 \\(n\\), \\(n-1\\), \\(\\dots\\), \\(2\\), \\(1\\)이므로 \\(O(n!)\\) 시간이 걸린다. 해를 기록할 때 행렬 <code>state</code>를 복사하여 <code>res</code>에 추가해야 하며, 복사에는 \\(O(n^2)\\) 시간이 든다. 따라서 전체 시간 복잡도는 \\(O(n! \\cdot n^2)\\)이다. 실제로는 대각선 제약에 따른 가지치기도 탐색 공간을 크게 줄이므로 위 복잡도보다 탐색 효율이 좋은 경우가 많다.</p> <p>배열 <code>state</code>는 \\(O(n^2)\\) 공간을, <code>cols</code>, <code>diags1</code>, <code>diags2</code>는 각각 \\(O(n)\\) 공간을 사용한다. 최대 재귀 깊이가 \\(n\\)이므로 스택 프레임 공간은 \\(O(n)\\)이다. 따라서 공간 복잡도는 \\(O(n^2)\\)이다.</p>","path":["제 13 장   백트래킹","13.4   n-퀸 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/","level":1,"title":"13.2   모든 순열 문제","text":"<p>모든 순열 문제는 백트래킹 알고리즘의 대표적인 응용이다. 배열이나 문자열 같은 집합이 주어졌을 때 원소들의 가능한 모든 순열을 찾는 문제로 정의된다.</p> <p>아래 표(13-2)에는 입력 배열과 그에 대응하는 모든 순열의 예시가 나와 있다.</p> <p> 표 13-2   모든 순열의 예시 </p> 입력 배열 모든 순열 \\([1]\\) \\([1]\\) \\([1, 2]\\) \\([1, 2], [2, 1]\\) \\([1, 2, 3]\\) \\([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\\)","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1321","level":2,"title":"13.2.1   값이 같은 원소가 없는 경우","text":"<p>문제</p> <p>중복 원소가 없는 정수 배열을 입력받아 가능한 모든 순열을 반환하라.</p> <p>백트래킹 관점에서는 순열을 만드는 과정을 연속된 선택의 결과로 볼 수 있다. 입력 배열이 \\([1, 2, 3]\\)이고 \\(1\\), \\(3\\), \\(2\\) 순서로 선택하면 순열 \\([1, 3, 2]\\)를 얻는다. 되돌리기는 선택 하나를 취소한 뒤 다른 선택을 계속 시도한다는 뜻이다.</p> <p>백트래킹 코드에서 후보 집합 <code>choices</code>는 입력 배열의 모든 원소이고, 상태 <code>state</code>는 지금까지 선택한 원소들이다. 각 원소는 한 번만 선택할 수 있으므로 <code>state</code> 안의 원소는 모두 중복되지 않아야 한다.</p> <p>아래 그림(13-5)과 같이 탐색 과정을 재귀 트리로 펼칠 수 있다. 각 노드는 현재 상태 <code>state</code>를 나타낸다. 루트에서 시작해 세 차례 선택하면 리프에 도달하며, 각 리프 노드는 순열 하나에 대응한다.</p> <p></p> <p> 그림 13-5   모든 순열의 재귀 트리 </p>","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1","level":3,"title":"1.   중복 선택 가지치기","text":"<p>각 원소를 한 번만 선택하도록 불리언 배열 <code>selected</code>를 도입하자. <code>selected[i]</code>는 <code>choices[i]</code>를 이미 선택했는지 나타내며, 이를 바탕으로 다음과 같이 가지치기한다.</p> <ul> <li><code>choices[i]</code>를 선택하면 <code>selected[i]</code>를 \\(\\text{True}\\)로 설정하여 이미 선택했음을 표시한다.</li> <li>선택지 리스트 <code>choices</code>를 순회할 때 이미 선택한 모든 노드를 건너뛰어 가지치기한다.</li> </ul> <p>아래 그림(13-6)처럼 첫 번째에 1, 두 번째에 3, 세 번째에 2를 선택한다면, 두 번째에서는 원소 1의 가지를, 세 번째에서는 원소 1과 3의 가지를 잘라 내야 한다.</p> <p></p> <p> 그림 13-6   모든 순열의 가지치기 예시 </p> <p>위 그림(13-6)에서 이 가지치기가 탐색 공간의 크기를 \\(O(n^n)\\)에서 \\(O(n!)\\)로 줄이는 것을 볼 수 있다.</p>","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#2","level":3,"title":"2.   코드 구현","text":"<p>위 내용을 정리했으니 기본 틀 코드의 '빈칸을 채울' 수 있다. 전체 코드를 짧게 만들기 위해 각 함수를 따로 구현하지 않고 <code>backtrack()</code> 함수 안에 직접 작성한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby permutations_i.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"백트래킹 알고리즘: 모든 순열 I\"\"\"\n    # 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # 모든 선택지를 순회한다\n    for i, choice in enumerate(choices):\n        # 가지치기: 원소를 중복하여 선택할 수 없다\n        if not selected[i]:\n            # 시도: 선택하고 상태를 갱신한다\n            selected[i] = True\n            state.append(choice)\n            # 다음 선택을 진행한다\n            backtrack(state, choices, selected, res)\n            # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = False\n            state.pop()\n\ndef permutations_i(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"모든 순열 I\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_i.cpp<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if (!selected[i]) {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true;\n            state.push_back(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* 모든 순열 I */\nvector&lt;vector&lt;int&gt;&gt; permutationsI(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_i.java<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if (!selected[i]) {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true;\n            state.add(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* 모든 순열 I */\nList&lt;List&lt;Integer&gt;&gt; permutationsI(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_i.cs<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if (!selected[i]) {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true;\n            state.Add(choice);\n            // 다음 선택을 진행한다\n            Backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* 모든 순열 I */\nList&lt;List&lt;int&gt;&gt; PermutationsI(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_i.go<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nfunc backtrackI(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // 모든 선택지를 순회한다\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if !(*selected)[i] {\n            // 시도: 선택하고 상태를 갱신한다\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // 다음 선택을 진행한다\n            backtrackI(state, choices, selected, res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* 모든 순열 I */\nfunc permutationsI(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackI(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_i.swift<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // 모든 선택지를 순회한다\n    for (i, choice) in choices.enumerated() {\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if !selected[i] {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true\n            state.append(choice)\n            // 다음 선택을 진행한다\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* 모든 순열 I */\nfunc permutationsI(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_i.js<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nfunction backtrack(state, choices, selected, res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    choices.forEach((choice, i) =&gt; {\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if (!selected[i]) {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true;\n            state.push(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* 모든 순열 I */\nfunction permutationsI(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.ts<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    choices.forEach((choice, i) =&gt; {\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if (!selected[i]) {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true;\n            state.push(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* 모든 순열 I */\nfunction permutationsI(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.dart<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // 모든 선택지를 순회한다\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // 가지치기: 원소를 중복하여 선택할 수 없다\n    if (!selected[i]) {\n      // 시도: 선택하고 상태를 갱신한다\n      selected[i] = true;\n      state.add(choice);\n      // 다음 선택을 진행한다\n      backtrack(state, choices, selected, res);\n      // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* 모든 순열 I */\nList&lt;List&lt;int&gt;&gt; permutationsI(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_i.rs<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if !selected[i] {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true;\n            state.push(choice);\n            // 다음 선택을 진행한다\n            backtrack(state.clone(), choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop();\n        }\n    }\n}\n\n/* 모든 순열 I */\nfn permutations_i(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new(); // 상태(부분 집합)\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_i.c<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if (!selected[i]) {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true;\n            state[stateSize] = choice;\n            // 다음 선택을 진행한다\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n        }\n    }\n}\n\n/* 모든 순열 I */\nint **permutationsI(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_i.kt<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.size == choices.size) {\n        res.add(state.toMutableList())\n        return\n    }\n    // 모든 선택지를 순회한다\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if (!selected[i]) {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true\n            state.add(choice)\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* 모든 순열 I */\nfun permutationsI(nums: IntArray): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;()\n    backtrack(mutableListOf(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_i.rb<pre><code>### 백트래킹 알고리즘: 모든 순열 I ###\ndef backtrack(state, choices, selected, res)\n  # 상태의 길이가 원소 개수와 같으면 해를 기록한다\n  if state.length == choices.length\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # 모든 선택지를 순회한다\n  choices.each_with_index do |choice, i|\n    # 가지치기: 원소를 중복하여 선택할 수 없다\n    unless selected[i]\n      # 시도: 선택하고 상태를 갱신한다\n      selected[i] = true\n      state &lt;&lt; choice\n      # 다음 선택을 진행한다\n      backtrack(state, choices, selected, res)\n      # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n      selected[i] = false\n      state.pop\n    end\n  end\nend\n\n### 모든 순열 I ###\ndef permutations_i(nums)\n  res = []\n  backtrack([], nums, Array.new(nums.length, false), res)\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1322","level":2,"title":"13.2.2   값이 같은 원소가 있는 경우","text":"<p>문제</p> <p>중복 원소가 있을 수 있는 정수 배열을 입력받아 중복되지 않는 모든 순열을 반환하라.</p> <p>입력 배열이 \\([1, 1, 2]\\)라고 하자. 두 원소 \\(1\\)을 쉽게 구분하기 위해 두 번째 \\(1\\)을 \\(\\hat{1}\\)로 표시한다.</p> <p>아래 그림(13-7)과 같이 앞의 방법으로 만든 순열의 절반은 중복된다.</p> <p></p> <p> 그림 13-7   중복된 순열 </p> <p>중복된 순열은 어떻게 제거할까? 가장 직접적으로는 해시 집합을 이용해 결과에서 중복을 제거할 수 있다. 하지만 이 방법은 충분히 간결하지 않다. 중복된 순열을 만드는 탐색 가지는 불필요하므로 미리 알아내어 잘라 내야 하며, 그러면 알고리즘의 효율을 더 높일 수 있다.</p>","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1_1","level":3,"title":"1.   같은 값의 원소 가지치기","text":"<p>아래 그림(13-8)에서 첫 번째 선택으로 \\(1\\)을 고르든 \\(\\hat{1}\\)을 고르든 같다. 두 선택 아래에서 만들어지는 모든 순열이 중복되므로 \\(\\hat{1}\\)을 가지치기해야 한다.</p> <p>마찬가지로 첫 번째에 \\(2\\)를 선택한 뒤에도 두 번째의 \\(1\\)과 \\(\\hat{1}\\)이 중복 가지를 만들므로 두 번째의 \\(\\hat{1}\\)도 잘라 내야 한다.</p> <p>본질적으로 한 차례의 선택에서 값이 같은 여러 원소를 한 번만 선택하도록 보장하는 것이 목표다.</p> <p></p> <p> 그림 13-8   중복된 순열 가지치기 </p>","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#2_1","level":3,"title":"2.   코드 구현","text":"<p>앞 문제의 코드를 바탕으로 매 차례의 선택에서 해시 집합 <code>duplicated</code>를 만들어 이미 시도한 원소를 기록하고 중복 원소를 가지치기한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby permutations_ii.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"백트래킹 알고리즘: 모든 순열 II\"\"\"\n    # 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # 모든 선택지를 순회한다\n    duplicated = set[int]()\n    for i, choice in enumerate(choices):\n        # 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if not selected[i] and choice not in duplicated:\n            # 시도: 선택하고 상태를 갱신한다\n            duplicated.add(choice)  # 선택한 원소의 값을 기록한다\n            selected[i] = True\n            state.append(choice)\n            # 다음 선택을 진행한다\n            backtrack(state, choices, selected, res)\n            # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = False\n            state.pop()\n\ndef permutations_ii(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"모든 순열 II\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_ii.cpp<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    unordered_set&lt;int&gt; duplicated;\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if (!selected[i] &amp;&amp; duplicated.find(choice) == duplicated.end()) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.emplace(choice); // 선택한 원소의 값을 기록한다\n            selected[i] = true;\n            state.push_back(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* 모든 순열 II */\nvector&lt;vector&lt;int&gt;&gt; permutationsII(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_ii.java<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    Set&lt;Integer&gt; duplicated = new HashSet&lt;Integer&gt;();\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.add(choice); // 선택한 원소의 값을 기록한다\n            selected[i] = true;\n            state.add(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* 모든 순열 II */\nList&lt;List&lt;Integer&gt;&gt; permutationsII(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_ii.cs<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    HashSet&lt;int&gt; duplicated = [];\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if (!selected[i] &amp;&amp; !duplicated.Contains(choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.Add(choice); // 선택한 원소의 값을 기록한다\n            selected[i] = true;\n            state.Add(choice);\n            // 다음 선택을 진행한다\n            Backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* 모든 순열 II */\nList&lt;List&lt;int&gt;&gt; PermutationsII(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_ii.go<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nfunc backtrackII(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // 모든 선택지를 순회한다\n    duplicated := make(map[int]struct{}, 0)\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if _, ok := duplicated[choice]; !ok &amp;&amp; !(*selected)[i] {\n            // 시도: 선택하고 상태를 갱신한다\n            // 선택한 원소의 값을 기록한다\n            duplicated[choice] = struct{}{}\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // 다음 선택을 진행한다\n            backtrackII(state, choices, selected, res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* 모든 순열 II */\nfunc permutationsII(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackII(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_ii.swift<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // 모든 선택지를 순회한다\n    var duplicated: Set&lt;Int&gt; = []\n    for (i, choice) in choices.enumerated() {\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if !selected[i], !duplicated.contains(choice) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.insert(choice) // 선택한 원소의 값을 기록한다\n            selected[i] = true\n            state.append(choice)\n            // 다음 선택을 진행한다\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* 모든 순열 II */\nfunc permutationsII(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_ii.js<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nfunction backtrack(state, choices, selected, res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.add(choice); // 선택한 원소의 값을 기록한다\n            selected[i] = true;\n            state.push(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* 모든 순열 II */\nfunction permutationsII(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.ts<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.add(choice); // 선택한 원소의 값을 기록한다\n            selected[i] = true;\n            state.push(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* 모든 순열 II */\nfunction permutationsII(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.dart<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // 모든 선택지를 순회한다\n  Set&lt;int&gt; duplicated = {};\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n    if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n      // 시도: 선택하고 상태를 갱신한다\n      duplicated.add(choice); // 선택한 원소의 값을 기록한다\n      selected[i] = true;\n      state.add(choice);\n      // 다음 선택을 진행한다\n      backtrack(state, choices, selected, res);\n      // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* 모든 순열 II */\nList&lt;List&lt;int&gt;&gt; permutationsII(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_ii.rs<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    let mut duplicated = HashSet::&lt;i32&gt;::new();\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if !selected[i] &amp;&amp; !duplicated.contains(&amp;choice) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.insert(choice); // 선택한 원소의 값을 기록한다\n            selected[i] = true;\n            state.push(choice);\n            // 다음 선택을 진행한다\n            backtrack(state.clone(), choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop();\n        }\n    }\n}\n\n/* 모든 순열 II */\nfn permutations_ii(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new();\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_ii.c<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // 모든 선택지를 순회한다\n    bool duplicated[MAX_SIZE] = {false};\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if (!selected[i] &amp;&amp; !duplicated[choice]) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated[choice] = true; // 선택한 원소의 값을 기록한다\n            selected[i] = true;\n            state[stateSize] = choice;\n            // 다음 선택을 진행한다\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n        }\n    }\n}\n\n/* 모든 순열 II */\nint **permutationsII(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_ii.kt<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.size == choices.size) {\n        res.add(state.toMutableList())\n        return\n    }\n    // 모든 선택지를 순회한다\n    val duplicated = HashSet&lt;Int&gt;()\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.add(choice) // 선택한 원소의 값을 기록한다\n            selected[i] = true\n            state.add(choice)\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* 모든 순열 II */\nfun permutationsII(nums: IntArray): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;()\n    backtrack(mutableListOf(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_ii.rb<pre><code>### 백트래킹 알고리즘: 모든 순열 II ###\ndef backtrack(state, choices, selected, res)\n  # 상태의 길이가 원소 개수와 같으면 해를 기록한다\n  if state.length == choices.length\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # 모든 선택지를 순회한다\n  duplicated = Set.new\n  choices.each_with_index do |choice, i|\n    # 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n    if !selected[i] &amp;&amp; !duplicated.include?(choice)\n      # 시도: 선택하고 상태를 갱신한다\n      duplicated.add(choice)\n      selected[i] = true\n      state &lt;&lt; choice\n      # 다음 선택을 진행한다\n      backtrack(state, choices, selected, res)\n      # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n      selected[i] = false\n      state.pop\n    end\n  end\nend\n\n### 모든 순열 II ###\ndef permutations_ii(nums)\n  res = []\n  backtrack([], nums, Array.new(nums.length, false), res)\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>모든 원소가 서로 다르다면 \\(n\\) 개 원소의 순열은 \\(n!\\) 가지(팩토리얼)다. 결과를 기록할 때마다 길이 \\(n\\)의 리스트를 복사하는 데 \\(O(n)\\) 시간이 걸린다. 따라서 시간 복잡도는 \\(O(n!n)\\)이다.</p> <p>최대 재귀 깊이는 \\(n\\)이므로 \\(O(n)\\)의 스택 프레임 공간을 사용한다. <code>selected</code>는 \\(O(n)\\) 공간을 사용한다. 동시에 최대 \\(n\\) 개의 <code>duplicated</code>가 존재하여 \\(O(n^2)\\) 공간을 사용한다. 따라서 공간 복잡도는 \\(O(n^2)\\)이다.</p>","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#3","level":3,"title":"3.   두 가지 가지치기 비교","text":"<p><code>selected</code>와 <code>duplicated</code>는 모두 가지치기에 쓰이지만 목적이 다르다.</p> <ul> <li>중복 선택 가지치기: 전체 탐색 과정에 <code>selected</code>는 하나만 존재한다. 현재 상태에 어떤 원소가 포함되어 있는지를 기록하여 같은 원소가 <code>state</code>에 중복 등장하지 않게 한다.</li> <li>같은 값의 원소 가지치기: 매 차례의 선택(호출된 각 <code>backtrack</code> 함수)마다 <code>duplicated</code>가 하나씩 있다. 이번 순회(<code>for</code> 반복문)에서 이미 선택한 원소를 기록하여 같은 값의 원소를 한 번만 선택하게 한다.</li> </ul> <p>아래 그림(13-9)은 두 가지 가지치기 조건이 적용되는 범위를 보여 준다. 각 트리 노드는 선택 하나를 나타내며, 루트에서 리프까지의 경로에 있는 노드들이 하나의 순열을 이룬다.</p> <p></p> <p> 그림 13-9   두 가지 가지치기 조건의 적용 범위 </p>","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/","level":1,"title":"13.3   부분 집합 합 문제","text":"","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1331","level":2,"title":"13.3.1   중복 원소가 없는 경우","text":"<p>문제</p> <p>양의 정수 배열 <code>nums</code>와 목표 양의 정수 <code>target</code>이 주어진다. 원소의 합이 <code>target</code>인 모든 조합을 찾아라. 배열에는 중복 원소가 없고, 각 원소는 여러 번 선택할 수 있다. 조합들을 리스트로 반환하되 중복 조합이 없어야 한다.</p> <p>예를 들어 입력 집합이 \\(\\{3, 4, 5\\}\\)이고 목표 정수가 \\(9\\)이면 해는 \\(\\{3, 3, 3\\}, \\{4, 5\\}\\)다. 다음 두 가지에 주의하자.</p> <ul> <li>입력 집합의 원소는 횟수 제한 없이 반복 선택할 수 있다.</li> <li>부분 집합에서는 원소의 순서를 구분하지 않는다. 예를 들어 \\(\\{4, 5\\}\\)와 \\(\\{5, 4\\}\\)는 같은 부분 집합이다.</li> </ul>","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1","level":3,"title":"1.   모든 순열 풀이 참고","text":"<p>모든 순열 문제처럼 부분 집합을 만드는 과정을 연속된 선택의 결과로 볼 수 있다. 선택할 때마다 '원소의 합'을 갱신하고, 합이 <code>target</code>이 되면 해당 부분 집합을 결과 리스트에 기록한다.</p> <p>모든 순열 문제와 달리 이 문제에서는 집합의 원소를 횟수 제한 없이 선택할 수 있다. 따라서 선택 여부를 기록하는 불리언 리스트 <code>selected</code>가 필요하지 않다. 모든 순열 코드를 조금 수정하면 다음과 같은 첫 풀이를 얻는다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby subset_sum_i_naive.py<pre><code>def backtrack(\n    state: list[int],\n    target: int,\n    total: int,\n    choices: list[int],\n    res: list[list[int]],\n):\n    \"\"\"백트래킹 알고리즘: 부분 집합 합 I\"\"\"\n    # 부분 집합의 합이 target과 같으면 해를 기록한다\n    if total == target:\n        res.append(list(state))\n        return\n    # 모든 선택지를 순회한다\n    for i in range(len(choices)):\n        # 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if total + choices[i] &gt; target:\n            continue\n        # 시도: 선택하고 원소의 합 total을 갱신한다\n        state.append(choices[i])\n        # 다음 선택을 진행한다\n        backtrack(state, target, total + choices[i], choices, res)\n        # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop()\n\ndef subset_sum_i_naive(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"부분 집합 합 I 풀기(중복 부분 집합 포함)\"\"\"\n    state = []  # 상태(부분 집합)\n    total = 0  # 부분 집합의 합\n    res = []  # 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, total, nums, res)\n    return res\n</code></pre> subset_sum_i_naive.cpp<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, int total, vector&lt;int&gt; &amp;choices, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (total == target) {\n        res.push_back(state);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (size_t i = 0; i &lt; choices.size(); i++) {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.push_back(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target, total + choices[i], choices, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop_back();\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nvector&lt;vector&lt;int&gt;&gt; subsetSumINaive(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;       // 상태(부분 집합)\n    int total = 0;           // 부분 집합의 합\n    vector&lt;vector&lt;int&gt;&gt; res; // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.java<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int total, int[] choices, List&lt;List&lt;Integer&gt;&gt; res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (total == target) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; choices.length; i++) {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.add(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target, total + choices[i], choices, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.remove(state.size() - 1);\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nList&lt;List&lt;Integer&gt;&gt; subsetSumINaive(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // 상태(부분 집합)\n    int total = 0; // 부분 집합의 합\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.cs<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid Backtrack(List&lt;int&gt; state, int target, int total, int[] choices, List&lt;List&lt;int&gt;&gt; res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (total == target) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; choices.Length; i++) {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.Add(choices[i]);\n        // 다음 선택을 진행한다\n        Backtrack(state, target, total + choices[i], choices, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nList&lt;List&lt;int&gt;&gt; SubsetSumINaive(int[] nums, int target) {\n    List&lt;int&gt; state = []; // 상태(부분 집합)\n    int total = 0; // 부분 집합의 합\n    List&lt;List&lt;int&gt;&gt; res = []; // 결과 리스트(부분 집합 리스트)\n    Backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.go<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunc backtrackSubsetSumINaive(total, target int, state, choices *[]int, res *[][]int) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == total {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // 모든 선택지를 순회한다\n    for i := 0; i &lt; len(*choices); i++ {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if total+(*choices)[i] &gt; target {\n            continue\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        *state = append(*state, (*choices)[i])\n        // 다음 선택을 진행한다\n        backtrackSubsetSumINaive(total+(*choices)[i], target, state, choices, res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nfunc subsetSumINaive(nums []int, target int) [][]int {\n    state := make([]int, 0) // 상태(부분 집합)\n    total := 0              // 부분 집합의 합\n    res := make([][]int, 0) // 결과 리스트(부분 집합 리스트)\n    backtrackSubsetSumINaive(total, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.swift<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunc backtrack(state: inout [Int], target: Int, total: Int, choices: [Int], res: inout [[Int]]) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if total == target {\n        res.append(state)\n        return\n    }\n    // 모든 선택지를 순회한다\n    for i in choices.indices {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if total + choices[i] &gt; target {\n            continue\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.append(choices[i])\n        // 다음 선택을 진행한다\n        backtrack(state: &amp;state, target: target, total: total + choices[i], choices: choices, res: &amp;res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.removeLast()\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nfunc subsetSumINaive(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // 상태(부분 집합)\n    let total = 0 // 부분 집합의 합\n    var res: [[Int]] = [] // 결과 리스트(부분 집합 리스트)\n    backtrack(state: &amp;state, target: target, total: total, choices: nums, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.js<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunction backtrack(state, target, total, choices, res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (let i = 0; i &lt; choices.length; i++) {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target, total + choices[i], choices, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nfunction subsetSumINaive(nums, target) {\n    const state = []; // 상태(부분 집합)\n    const total = 0; // 부분 집합의 합\n    const res = []; // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.ts<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    total: number,\n    choices: number[],\n    res: number[][]\n): void {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (let i = 0; i &lt; choices.length; i++) {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target, total + choices[i], choices, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nfunction subsetSumINaive(nums: number[], target: number): number[][] {\n    const state = []; // 상태(부분 집합)\n    const total = 0; // 부분 집합의 합\n    const res = []; // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.dart<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  int total,\n  List&lt;int&gt; choices,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // 부분 집합의 합이 target과 같으면 해를 기록한다\n  if (total == target) {\n    res.add(List.from(state));\n    return;\n  }\n  // 모든 선택지를 순회한다\n  for (int i = 0; i &lt; choices.length; i++) {\n    // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n    if (total + choices[i] &gt; target) {\n      continue;\n    }\n    // 시도: 선택하고 원소의 합 total을 갱신한다\n    state.add(choices[i]);\n    // 다음 선택을 진행한다\n    backtrack(state, target, total + choices[i], choices, res);\n    // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n    state.removeLast();\n  }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nList&lt;List&lt;int&gt;&gt; subsetSumINaive(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // 상태(부분 집합)\n  int total = 0; // 원소의 합\n  List&lt;List&lt;int&gt;&gt; res = []; // 결과 리스트(부분 집합 리스트)\n  backtrack(state, target, total, nums, res);\n  return res;\n}\n</code></pre> subset_sum_i_naive.rs<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    total: i32,\n    choices: &amp;[i32],\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if total == target {\n        res.push(state.clone());\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for i in 0..choices.len() {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if total + choices[i] &gt; target {\n            continue;\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target, total + choices[i], choices, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nfn subset_sum_i_naive(nums: &amp;[i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // 상태(부분 집합)\n    let total = 0; // 부분 집합의 합\n    let mut res = Vec::new(); // 결과 리스트(부분 집합 리스트)\n    backtrack(&amp;mut state, target, total, nums, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i_naive.c<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(int target, int total, int *choices, int choicesSize) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (total == target) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; choicesSize; i++) {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state[stateSize++] = choices[i];\n        // 다음 선택을 진행한다\n        backtrack(target, total + choices[i], choices, choicesSize);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        stateSize--;\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nvoid subsetSumINaive(int *nums, int numsSize, int target) {\n    resSize = 0; // 해의 개수를 0으로 초기화\n    backtrack(target, 0, nums, numsSize);\n}\n</code></pre> subset_sum_i_naive.kt<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    total: Int,\n    choices: IntArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (total == target) {\n        res.add(state.toMutableList())\n        return\n    }\n    // 모든 선택지를 순회한다\n    for (i in choices.indices) {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if (total + choices[i] &gt; target) {\n            continue\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.add(choices[i])\n        // 다음 선택을 진행한다\n        backtrack(state, target, total + choices[i], choices, res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nfun subsetSumINaive(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // 상태(부분 집합)\n    val total = 0 // 부분 집합의 합\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, total, nums, res)\n    return res\n}\n</code></pre> subset_sum_i_naive.rb<pre><code>### 백트래킹 알고리즘: 부분 집합 합 I ###\ndef backtrack(state, target, total, choices, res)\n  # 부분 집합의 합이 target과 같으면 해를 기록한다\n  if total == target\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # 모든 선택지를 순회한다\n  for i in 0...choices.length\n    # 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n    next if total + choices[i] &gt; target\n    # 시도: 선택하고 원소의 합 total을 갱신한다\n    state &lt;&lt; choices[i]\n    # 다음 선택을 진행한다\n    backtrack(state, target, total + choices[i], choices, res)\n    # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n    state.pop\n  end\nend\n\n### 부분 집합 합 I 풀기(중복 부분 집합 포함)###\ndef subset_sum_i_naive(nums, target)\n  state = [] # 상태(부분 집합)\n  total = 0 # 부분 집합의 합\n  res = [] # 결과 리스트(부분 집합 리스트)\n  backtrack(state, target, total, nums, res)\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>위 코드에 배열 \\([3, 4, 5]\\)와 목표 원소 \\(9\\)를 넣으면 \\([3, 3, 3], [4, 5], [5, 4]\\)가 나온다. 합이 \\(9\\)인 모든 부분 집합을 찾았지만 \\([4, 5]\\)와 \\([5, 4]\\)는 중복이다.</p> <p>탐색 과정에서는 선택 순서를 구분하지만 부분 집합에서는 구분하지 않기 때문이다. 아래 그림(13-10)에서 \\(4\\) 다음 \\(5\\)를 고르는 것과 \\(5\\) 다음 \\(4\\)를 고르는 것은 서로 다른 가지이지만 같은 부분 집합에 대응한다.</p> <p></p> <p> 그림 13-10   부분 집합 탐색과 범위 초과 가지치기 </p> <p>중복 부분 집합을 제거하는 직접적인 방법은 결과 리스트에서 중복을 제거하는 것이다. 하지만 다음 두 이유로 매우 비효율적이다.</p> <ul> <li>배열 원소가 많고 특히 <code>target</code>이 크면 탐색 과정에서 중복 부분 집합이 매우 많이 생긴다.</li> <li>부분 집합(배열)이 같은지 비교하는 데 시간이 많이 든다. 먼저 배열을 정렬한 뒤 원소를 하나씩 비교해야 한다.</li> </ul>","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#2","level":3,"title":"2.   중복 부분 집합 가지치기","text":"<p>탐색 중 가지치기로 중복을 제거해 보자. 아래 그림(13-11)에서 중복 부분 집합은 배열 원소를 서로 다른 순서로 선택할 때 생긴다. 예를 들면 다음과 같다.</p> <ol> <li>첫 번째와 두 번째에 각각 \\(3\\)과 \\(4\\)를 선택하면 이 두 원소를 포함하는 모든 부분 집합 \\([3, 4, \\dots]\\)가 만들어진다.</li> <li>이후 첫 번째에 \\(4\\)를 선택하면 두 번째에서는 \\(3\\)을 건너뛰어야 한다. 이 선택으로 만드는 \\([4, 3, \\dots]\\)는 <code>1.</code> 단계에서 만든 부분 집합과 완전히 중복되기 때문이다.</li> </ol> <p>탐색에서 각 레벨의 선택은 왼쪽부터 오른쪽 순서로 시도하므로, 오른쪽으로 갈수록 잘라 내는 가지가 많아진다.</p> <ol> <li>첫 두 차례에 \\(3\\)과 \\(5\\)를 선택하여 부분 집합 \\([3, 5, \\dots]\\)를 만든다.</li> <li>첫 두 차례에 \\(4\\)와 \\(5\\)를 선택하여 부분 집합 \\([4, 5, \\dots]\\)를 만든다.</li> <li>첫 번째에 \\(5\\)를 선택한다면 두 번째에서는 \\(3\\)과 \\(4\\)를 건너뛰어야 한다. \\([5, 3, \\dots]\\)와 \\([5, 4, \\dots]\\)는 <code>1.</code> 단계와 <code>2.</code> 단계의 부분 집합과 완전히 중복되기 때문이다.</li> </ol> <p></p> <p> 그림 13-11   선택 순서의 차이로 생기는 중복 부분 집합 </p> <p>정리하면 입력 배열이 \\([x_1, x_2, \\dots, x_n]\\)일 때 선택 순서 \\([x_{i_1}, x_{i_2}, \\dots, x_{i_m}]\\)는 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\)을 만족해야 한다. 이 조건을 만족하지 않는 선택 순서는 중복을 만들므로 가지치기해야 한다.</p>","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#3","level":3,"title":"3.   코드 구현","text":"<p>이 가지치기를 위해 순회 시작점을 나타내는 변수 <code>start</code>를 초기화한다. \\(x_{i}\\)를 선택한 뒤 다음 차례는 인덱스 \\(i\\)부터 순회하도록 설정한다. 그러면 선택 순서가 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\)을 만족하여 부분 집합의 유일성을 보장한다.</p> <p>이 밖에도 다음 두 가지를 최적화했다.</p> <ul> <li>탐색을 시작하기 전에 배열 <code>nums</code>를 정렬한다. 선택지를 순회하다가 부분 집합의 합이 <code>target</code>을 넘으면 반복문을 바로 종료한다. 뒤의 원소들은 더 크므로 합도 반드시 <code>target</code>을 넘기 때문이다.</li> <li>원소 합 변수 <code>total</code>을 없애고 <code>target</code>에서 값을 빼는 방식으로 합을 추적한다. <code>target</code>이 \\(0\\)이 되면 해를 기록한다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby subset_sum_i.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"백트래킹 알고리즘: 부분 집합 합 I\"\"\"\n    # 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0:\n        res.append(list(state))\n        return\n    # 모든 선택지를 순회한다\n    # 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for i in range(start, len(choices)):\n        # 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        # 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target - choices[i] &lt; 0:\n            break\n        # 시도: 선택하고 target, start를 갱신한다\n        state.append(choices[i])\n        # 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i, res)\n        # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop()\n\ndef subset_sum_i(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"부분 집합 합 I 풀기\"\"\"\n    state = []  # 상태(부분 집합)\n    nums.sort()  # nums를 정렬한다\n    start = 0  # 순회 시작 위치\n    res = []  # 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_i.cpp<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for (int i = start; i &lt; choices.size(); i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push_back(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop_back();\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nvector&lt;vector&lt;int&gt;&gt; subsetSumI(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // 상태(부분 집합)\n    sort(nums.begin(), nums.end()); // nums를 정렬한다\n    int start = 0;                  // 순회 시작 위치\n    vector&lt;vector&lt;int&gt;&gt; res;        // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.java<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for (int i = start; i &lt; choices.length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.add(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.remove(state.size() - 1);\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nList&lt;List&lt;Integer&gt;&gt; subsetSumI(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // 상태(부분 집합)\n    Arrays.sort(nums); // nums를 정렬한다\n    int start = 0; // 순회 시작 위치\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.cs<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for (int i = start; i &lt; choices.Length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.Add(choices[i]);\n        // 다음 선택을 진행한다\n        Backtrack(state, target - choices[i], choices, i, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nList&lt;List&lt;int&gt;&gt; SubsetSumI(int[] nums, int target) {\n    List&lt;int&gt; state = []; // 상태(부분 집합)\n    Array.Sort(nums); // nums를 정렬한다\n    int start = 0; // 순회 시작 위치\n    List&lt;List&lt;int&gt;&gt; res = []; // 결과 리스트(부분 집합 리스트)\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.go<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunc backtrackSubsetSumI(start, target int, state, choices *[]int, res *[][]int) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for i := start; i &lt; len(*choices); i++ {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        *state = append(*state, (*choices)[i])\n        // 다음 선택을 진행한다\n        backtrackSubsetSumI(i, target-(*choices)[i], state, choices, res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nfunc subsetSumI(nums []int, target int) [][]int {\n    state := make([]int, 0) // 상태(부분 집합)\n    sort.Ints(nums)         // nums를 정렬한다\n    start := 0              // 순회 시작 위치\n    res := make([][]int, 0) // 결과 리스트(부분 집합 리스트)\n    backtrackSubsetSumI(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.swift<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for i in choices.indices.dropFirst(start) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.append(choices[i])\n        // 다음 선택을 진행한다\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i, res: &amp;res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.removeLast()\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nfunc subsetSumI(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // 상태(부분 집합)\n    let nums = nums.sorted() // nums를 정렬한다\n    let start = 0 // 순회 시작 위치\n    var res: [[Int]] = [] // 결과 리스트(부분 집합 리스트)\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.js<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunction backtrack(state, target, choices, start, res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for (let i = start; i &lt; choices.length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nfunction subsetSumI(nums, target) {\n    const state = []; // 상태(부분 집합)\n    nums.sort((a, b) =&gt; a - b); // nums를 정렬한다\n    const start = 0; // 순회 시작 위치\n    const res = []; // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.ts<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for (let i = start; i &lt; choices.length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nfunction subsetSumI(nums: number[], target: number): number[][] {\n    const state = []; // 상태(부분 집합)\n    nums.sort((a, b) =&gt; a - b); // nums를 정렬한다\n    const start = 0; // 순회 시작 위치\n    const res = []; // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.dart<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // 부분 집합의 합이 target과 같으면 해를 기록한다\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // 모든 선택지를 순회한다\n  // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n  for (int i = start; i &lt; choices.length; i++) {\n    // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n    // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // 시도: 선택하고 target, start를 갱신한다\n    state.add(choices[i]);\n    // 다음 선택을 진행한다\n    backtrack(state, target - choices[i], choices, i, res);\n    // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n    state.removeLast();\n  }\n}\n\n/* 부분 집합 합 I 풀기 */\nList&lt;List&lt;int&gt;&gt; subsetSumI(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // 상태(부분 집합)\n  nums.sort(); // nums를 정렬한다\n  int start = 0; // 순회 시작 위치\n  List&lt;List&lt;int&gt;&gt; res = []; // 결과 리스트(부분 집합 리스트)\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_i.rs<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0 {\n        res.push(state.clone());\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for i in start..choices.len() {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nfn subset_sum_i(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // 상태(부분 집합)\n    nums.sort(); // nums를 정렬한다\n    let start = 0; // 순회 시작 위치\n    let mut res = Vec::new(); // 결과 리스트(부분 집합 리스트)\n    backtrack(&amp;mut state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i.c<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; ++i) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for (int i = start; i &lt; choicesSize; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state[stateSize] = choices[i];\n        stateSize++;\n        // 다음 선택을 진행한다\n        backtrack(target - choices[i], choices, choicesSize, i);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        stateSize--;\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nvoid subsetSumI(int *nums, int numsSize, int target) {\n    qsort(nums, numsSize, sizeof(int), cmp); // nums를 정렬한다\n    int start = 0;                           // 순회 시작 위치\n    backtrack(target, nums, numsSize, start);\n}\n</code></pre> subset_sum_i.kt<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.add(state.toMutableList())\n        return\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for (i in start..&lt;choices.size) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.add(choices[i])\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i, res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nfun subsetSumI(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // 상태(부분 집합)\n    nums.sort() // nums를 정렬한다\n    val start = 0 // 순회 시작 위치\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_i.rb<pre><code>### 백트래킹 알고리즘: 부분 집합 합 I ###\ndef backtrack(state, target, choices, start, res)\n  # 부분 집합의 합이 target과 같으면 해를 기록한다\n  if target.zero?\n    res &lt;&lt; state.dup\n    return\n  end\n  # 모든 선택지를 순회한다\n  # 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n  for i in start...choices.length\n    # 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n    # 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n    break if target - choices[i] &lt; 0\n    # 시도: 선택하고 target, start를 갱신한다\n    state &lt;&lt; choices[i]\n    # 다음 선택을 진행한다\n    backtrack(state, target - choices[i], choices, i, res)\n    # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n    state.pop\n  end\nend\n\n### 부분 집합 합 I 풀기 ###\ndef subset_sum_i(nums, target)\n  state = [] # 상태(부분 집합)\n  nums.sort! # nums를 정렬한다\n  start = 0 # 순회 시작 위치\n  res = [] # 결과 리스트(부분 집합 리스트)\n  backtrack(state, target, nums, start, res)\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(13-12)은 배열 \\([3, 4, 5]\\)와 목표 원소 \\(9\\)를 위 코드에 넣었을 때의 전체 백트래킹 과정이다.</p> <p></p> <p> 그림 13-12   부분 집합 합 I의 백트래킹 과정 </p>","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1332","level":2,"title":"13.3.2   중복 원소가 있는 경우","text":"<p>문제</p> <p>양의 정수 배열 <code>nums</code>와 목표 양의 정수 <code>target</code>이 주어진다. 원소의 합이 <code>target</code>인 모든 조합을 찾아라. 배열에는 중복 원소가 있을 수 있고, 각 원소는 한 번만 선택할 수 있다. 조합들을 리스트로 반환하되 중복 조합이 없어야 한다.</p> <p>앞 문제와 달리 입력 배열에 중복 원소가 있을 수 있어 새로운 문제가 생긴다. 예를 들어 배열 \\([4, \\hat{4}, 5]\\)와 목표 원소 \\(9\\)를 기존 코드에 넣으면 \\([4, 5], [\\hat{4}, 5]\\)가 나와 부분 집합이 중복된다.</p> <p>중복의 원인은 값이 같은 원소를 한 차례의 선택에서 여러 번 고르기 때문이다. 아래 그림(13-13)에서 첫 차례의 세 선택지 중 두 개가 \\(4\\)여서 중복된 두 탐색 가지가 만들어지고, 같은 부분 집합을 출력한다. 마찬가지로 두 번째 차례의 두 \\(4\\)도 중복을 만든다.</p> <p></p> <p> 그림 13-13   같은 값의 원소로 생기는 중복 부분 집합 </p>","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1_1","level":3,"title":"1.   같은 값의 원소 가지치기","text":"<p>이를 해결하려면 각 차례에서 같은 값의 원소를 한 번만 선택하도록 제한해야 한다. 구현 방법은 간단하면서도 기발하다. 배열이 정렬되어 있으므로 같은 원소끼리는 인접한다. 따라서 한 차례의 선택에서 현재 원소가 왼쪽 원소와 같다면 이미 선택한 값이라는 뜻이므로 바로 건너뛴다.</p> <p>또한 이 문제에서는 각 배열 원소를 한 번만 선택할 수 있다. 다행히 <code>start</code>로 이 제약도 만족시킬 수 있다. \\(x_{i}\\)를 선택한 뒤 다음 차례는 인덱스 \\(i + 1\\)부터 배열의 끝을 향해 순회하도록 설정한다. 그러면 중복 부분 집합과 같은 원소의 반복 선택을 모두 막을 수 있다.</p>","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#2_1","level":3,"title":"2.   코드 구현","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby subset_sum_ii.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"백트래킹 알고리즘: 부분 집합 합 II\"\"\"\n    # 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0:\n        res.append(list(state))\n        return\n    # 모든 선택지를 순회한다\n    # 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    # 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for i in range(start, len(choices)):\n        # 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        # 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target - choices[i] &lt; 0:\n            break\n        # 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if i &gt; start and choices[i] == choices[i - 1]:\n            continue\n        # 시도: 선택하고 target, start를 갱신한다\n        state.append(choices[i])\n        # 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop()\n\ndef subset_sum_ii(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"부분 집합 합 II 풀기\"\"\"\n    state = []  # 상태(부분 집합)\n    nums.sort()  # nums를 정렬한다\n    start = 0  # 순회 시작 위치\n    res = []  # 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_ii.cpp<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for (int i = start; i &lt; choices.size(); i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push_back(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop_back();\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nvector&lt;vector&lt;int&gt;&gt; subsetSumII(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // 상태(부분 집합)\n    sort(nums.begin(), nums.end()); // nums를 정렬한다\n    int start = 0;                  // 순회 시작 위치\n    vector&lt;vector&lt;int&gt;&gt; res;        // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.java<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for (int i = start; i &lt; choices.length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.add(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.remove(state.size() - 1);\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nList&lt;List&lt;Integer&gt;&gt; subsetSumII(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // 상태(부분 집합)\n    Arrays.sort(nums); // nums를 정렬한다\n    int start = 0; // 순회 시작 위치\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.cs<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for (int i = start; i &lt; choices.Length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.Add(choices[i]);\n        // 다음 선택을 진행한다\n        Backtrack(state, target - choices[i], choices, i + 1, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nList&lt;List&lt;int&gt;&gt; SubsetSumII(int[] nums, int target) {\n    List&lt;int&gt; state = []; // 상태(부분 집합)\n    Array.Sort(nums); // nums를 정렬한다\n    int start = 0; // 순회 시작 위치\n    List&lt;List&lt;int&gt;&gt; res = []; // 결과 리스트(부분 집합 리스트)\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.go<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nfunc backtrackSubsetSumII(start, target int, state, choices *[]int, res *[][]int) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for i := start; i &lt; len(*choices); i++ {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if i &gt; start &amp;&amp; (*choices)[i] == (*choices)[i-1] {\n            continue\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        *state = append(*state, (*choices)[i])\n        // 다음 선택을 진행한다\n        backtrackSubsetSumII(i+1, target-(*choices)[i], state, choices, res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nfunc subsetSumII(nums []int, target int) [][]int {\n    state := make([]int, 0) // 상태(부분 집합)\n    sort.Ints(nums)         // nums를 정렬한다\n    start := 0              // 순회 시작 위치\n    res := make([][]int, 0) // 결과 리스트(부분 집합 리스트)\n    backtrackSubsetSumII(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.swift<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for i in choices.indices.dropFirst(start) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if i &gt; start, choices[i] == choices[i - 1] {\n            continue\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.append(choices[i])\n        // 다음 선택을 진행한다\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i + 1, res: &amp;res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.removeLast()\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nfunc subsetSumII(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // 상태(부분 집합)\n    let nums = nums.sorted() // nums를 정렬한다\n    let start = 0 // 순회 시작 위치\n    var res: [[Int]] = [] // 결과 리스트(부분 집합 리스트)\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.js<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nfunction backtrack(state, target, choices, start, res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for (let i = start; i &lt; choices.length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nfunction subsetSumII(nums, target) {\n    const state = []; // 상태(부분 집합)\n    nums.sort((a, b) =&gt; a - b); // nums를 정렬한다\n    const start = 0; // 순회 시작 위치\n    const res = []; // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.ts<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for (let i = start; i &lt; choices.length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nfunction subsetSumII(nums: number[], target: number): number[][] {\n    const state = []; // 상태(부분 집합)\n    nums.sort((a, b) =&gt; a - b); // nums를 정렬한다\n    const start = 0; // 순회 시작 위치\n    const res = []; // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.dart<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // 부분 집합의 합이 target과 같으면 해를 기록한다\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // 모든 선택지를 순회한다\n  // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n  // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n  for (int i = start; i &lt; choices.length; i++) {\n    // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n    // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n    if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n      continue;\n    }\n    // 시도: 선택하고 target, start를 갱신한다\n    state.add(choices[i]);\n    // 다음 선택을 진행한다\n    backtrack(state, target - choices[i], choices, i + 1, res);\n    // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n    state.removeLast();\n  }\n}\n\n/* 부분 집합 합 II 풀기 */\nList&lt;List&lt;int&gt;&gt; subsetSumII(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // 상태(부분 집합)\n  nums.sort(); // nums를 정렬한다\n  int start = 0; // 순회 시작 위치\n  List&lt;List&lt;int&gt;&gt; res = []; // 결과 리스트(부분 집합 리스트)\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_ii.rs<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0 {\n        res.push(state.clone());\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for i in start..choices.len() {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if i &gt; start &amp;&amp; choices[i] == choices[i - 1] {\n            continue;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nfn subset_sum_ii(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // 상태(부분 집합)\n    nums.sort(); // nums를 정렬한다\n    let start = 0; // 순회 시작 위치\n    let mut res = Vec::new(); // 결과 리스트(부분 집합 리스트)\n    backtrack(&amp;mut state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_ii.c<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for (int i = start; i &lt; choicesSize; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 넘으면 바로 건너뛴다\n        if (target - choices[i] &lt; 0) {\n            continue;\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state[stateSize] = choices[i];\n        stateSize++;\n        // 다음 선택을 진행한다\n        backtrack(target - choices[i], choices, choicesSize, i + 1);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        stateSize--;\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nvoid subsetSumII(int *nums, int numsSize, int target) {\n    // nums를 정렬한다\n    qsort(nums, numsSize, sizeof(int), cmp);\n    // 백트래킹 시작\n    backtrack(target, nums, numsSize, 0);\n}\n</code></pre> subset_sum_ii.kt<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.add(state.toMutableList())\n        return\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for (i in start..&lt;choices.size) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.add(choices[i])\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nfun subsetSumII(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // 상태(부분 집합)\n    nums.sort() // nums를 정렬한다\n    val start = 0 // 순회 시작 위치\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_ii.rb<pre><code>### 백트래킹 알고리즘: 부분 집합 합 II ###\ndef backtrack(state, target, choices, start, res)\n  # 부분 집합의 합이 target과 같으면 해를 기록한다\n  if target.zero?\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # 모든 선택지를 순회한다\n  # 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n  # 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n  for i in start...choices.length\n    # 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n    # 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n    break if target - choices[i] &lt; 0\n    # 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n    next if i &gt; start &amp;&amp; choices[i] == choices[i - 1]\n    # 시도: 선택하고 target, start를 갱신한다\n    state &lt;&lt; choices[i]\n    # 다음 선택을 진행한다\n    backtrack(state, target - choices[i], choices, i + 1, res)\n    # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n    state.pop\n  end\nend\n\n### 부분 집합 합 II 풀기 ###\ndef subset_sum_ii(nums, target)\n  state = [] # 상태(부분 집합)\n  nums.sort! # nums를 정렬한다\n  start = 0 # 순회 시작 위치\n  res = [] # 결과 리스트(부분 집합 리스트)\n  backtrack(state, target, nums, start, res)\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(13-14)은 배열 \\([4, 4, 5]\\)와 목표 원소 \\(9\\)의 백트래킹 과정으로, 네 가지 가지치기를 포함한다. 그림과 코드 주석을 함께 보면서 전체 탐색 과정과 각 가지치기가 어떻게 작동하는지 이해해 보자.</p> <p></p> <p> 그림 13-14   부분 집합 합 II의 백트래킹 과정 </p>","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/summary/","level":1,"title":"13.5   요약","text":"","path":["제 13 장   백트래킹","13.5   요약"],"tags":[]},{"location":"chapter_backtracking/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>백트래킹은 본질적으로 모든 경우를 열거하는 방법이다. 해 공간을 깊이 우선으로 순회하여 조건에 맞는 해를 찾고 기록하며, 모든 해를 찾거나 순회가 끝나면 종료한다.</li> <li>백트래킹의 탐색 과정은 시도와 되돌리기로 이루어진다. 깊이 우선 탐색으로 여러 선택을 시도하고, 제약 조건을 만족하지 않으면 직전 선택을 취소하여 이전 상태로 돌아간 뒤 다른 선택을 계속 시도한다. 시도와 되돌리기는 서로 반대 방향의 연산이다.</li> <li>백트래킹 문제에는 보통 여러 제약 조건이 있고, 이를 가지치기에 사용할 수 있다. 가지치기는 불필요한 탐색 가지를 일찍 종료하여 효율을 크게 높인다.</li> <li>백트래킹은 주로 탐색 문제와 제약 만족 문제에 사용한다. 조합 최적화 문제도 해결할 수 있지만, 더 효율적이거나 더 좋은 결과를 내는 풀이가 있는 경우가 많다.</li> <li>모든 순열 문제는 주어진 집합 원소들의 가능한 모든 순열을 찾는 것이 목표다. 배열로 각 원소의 선택 여부를 기록하고 동일 원소를 중복 선택하는 가지를 잘라 내어 각 원소를 한 번만 선택하도록 한다.</li> <li>모든 순열 문제에서 집합에 중복 원소가 있으면 결과에 중복된 순열이 생긴다. 각 차례에서 같은 값의 원소를 한 번만 선택하도록 제한해야 하며, 보통 해시 집합으로 구현한다.</li> <li>부분 집합 합 문제는 주어진 집합에서 합이 목표 값인 모든 부분 집합을 찾는다. 집합은 원소 순서를 구분하지 않지만 탐색은 모든 순서의 결과를 출력하므로 중복이 생긴다. 백트래킹 전에 데이터를 정렬하고 각 차례의 순회 시작점을 변수로 지정하여 중복 부분 집합을 만드는 가지를 잘라 낸다.</li> <li>부분 집합 합 문제에서는 배열의 같은 값 원소가 중복 집합을 만들 수 있다. 배열이 정렬되어 있다는 전제를 이용해 인접 원소의 값이 같은지 확인하여 가지치기하고, 같은 값을 매 차례 한 번만 선택하도록 한다.</li> <li>\\(n\\)-퀸 문제는 \\(n \\times n\\) 체스판에 퀸 \\(n\\) 개를 서로 공격하지 않게 배치하는 문제다. 행, 열, 주대각선 방향, 부대각선 방향의 제약이 있다. 행 제약을 만족하려고 행별 배치 전략을 사용하여 각 행에 퀸 하나를 놓는다.</li> <li>열 제약과 대각선 제약은 비슷하게 처리한다. 열 제약은 배열에 각 열의 퀸 존재 여부를 기록하여 선택한 칸이 유효한지 판단한다. 대각선 제약은 두 배열로 각 방향 대각선의 퀸 존재 여부를 기록한다. 핵심은 같은 주대각선(부대각선) 방향 대각선의 칸들이 만족하는 행과 열 인덱스의 규칙을 찾는 것이다.</li> </ul>","path":["제 13 장   백트래킹","13.5   요약"],"tags":[]},{"location":"chapter_backtracking/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 백트래킹과 재귀의 관계는 어떻게 이해하면 되는가?</p> <p>백트래킹은 '알고리즘 전략'이고, 재귀는 '도구'에 더 가깝다.</p> <ul> <li>백트래킹은 보통 재귀로 구현한다. 백트래킹은 재귀의 활용 사례 중 하나이며, 탐색 문제에 재귀를 적용한 것이다.</li> <li>재귀의 구조는 '부분 문제로 분해'하는 풀이 패턴을 보여 주며, 분할 정복, 백트래킹, 동적 계획법(메모이제이션을 이용한 재귀) 등의 문제에 자주 사용한다.</li> </ul>","path":["제 13 장   백트래킹","13.5   요약"],"tags":[]},{"location":"chapter_computational_complexity/","level":1,"title":"제 2 장   복잡도 분석","text":"<p>이 장을 시작하며</p> <p>복잡도 분석은 광활한 알고리즘 우주에서 시간과 공간을 안내하는 길잡이와 같다.</p> <p>시간과 공간이라는 두 차원을 깊이 탐구하며 더 우아한 해법을 찾도록 이끌어 준다.</p>","path":["제 2 장   복잡도 분석"],"tags":[]},{"location":"chapter_computational_complexity/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>2.1   알고리즘 효율 평가</li> <li>2.2   반복과 재귀</li> <li>2.3   시간 복잡도</li> <li>2.4   공간 복잡도</li> <li>2.5   요약</li> <li>2.6   연습 문제</li> </ul>","path":["제 2 장   복잡도 분석"],"tags":[]},{"location":"chapter_computational_complexity/exercises/","level":1,"title":"2.6   연습 문제","text":"","path":["제 2 장   복잡도 분석","2.6   연습 문제"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#261","level":2,"title":"2.6.1   개념 확인","text":"","path":["제 2 장   복잡도 분석","2.6   연습 문제"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#1","level":3,"title":"1.   반복과 재귀의 시간과 공간","text":"<p>다음 두 코드는 모두 \\(1 + 2 + \\dots + n\\)을 계산한다(\\(n \\ge 1\\)로 가정한다). <code>n</code>을 4로 설정하고, 프로그램이 실제로 실행되는 순서에 따라 질문에 답한 뒤 두 방식의 효율성을 비교해 보자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby complexity_exercises.py<pre><code>def sum_iter(n: int) -&gt; int:\n    \"\"\"반복으로 합 구하기\"\"\"\n    res = 0\n    for i in range(1, n + 1):\n        res += i\n    return res\n\ndef sum_recur(n: int) -&gt; int:\n    \"\"\"재귀로 합 구하기\"\"\"\n    if n == 1:\n        return 1\n    return n + sum_recur(n - 1)\n</code></pre> complexity_exercises.cpp<pre><code>/* 반복으로 합 구하기 */\nint sumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; ++i) {\n        res += i;\n    }\n    return res;\n}\n\n/* 재귀로 합 구하기 */\nint sumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.java<pre><code>/* 반복으로 합 구하기 */\nint sumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 재귀로 합 구하기 */\nint sumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.cs<pre><code>/* 반복으로 합 구하기 */\nint SumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 재귀로 합 구하기 */\nint SumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + SumRecur(n - 1);\n}\n</code></pre> complexity_exercises.go<pre><code>/* 반복으로 합 구하기 */\nfunc sumIter(n int) int {\n    res := 0\n    for i := 1; i &lt;= n; i++ {\n        res += i\n    }\n    return res\n}\n\n/* 재귀로 합 구하기 */\nfunc sumRecur(n int) int {\n    if n == 1 {\n        return 1\n    }\n    return n + sumRecur(n-1)\n}\n</code></pre> complexity_exercises.swift<pre><code>/* 반복으로 합 구하기 */\nfunc sumIter(n: Int) -&gt; Int {\n    var res = 0\n    for i in 1 ... n {\n        res += i\n    }\n    return res\n}\n\n/* 재귀로 합 구하기 */\nfunc sumRecur(n: Int) -&gt; Int {\n    if n == 1 {\n        return 1\n    }\n    return n + sumRecur(n: n - 1)\n}\n</code></pre> complexity_exercises.js<pre><code>/* 반복으로 합 구하기 */\nfunction sumIter(n) {\n    let res = 0;\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 재귀로 합 구하기 */\nfunction sumRecur(n) {\n    if (n === 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.ts<pre><code>/* 반복으로 합 구하기 */\nfunction sumIter(n: number): number {\n    let res = 0;\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 재귀로 합 구하기 */\nfunction sumRecur(n: number): number {\n    if (n === 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.dart<pre><code>/* 반복으로 합 구하기 */\nint sumIter(int n) {\n  int res = 0;\n  for (int i = 1; i &lt;= n; i++) {\n    res += i;\n  }\n  return res;\n}\n\n/* 재귀로 합 구하기 */\nint sumRecur(int n) {\n  if (n == 1) {\n    return 1;\n  }\n  return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.rs<pre><code>/* 반복으로 합 구하기 */\nfn sum_iter(n: i32) -&gt; i32 {\n    let mut res = 0;\n    for i in 1..=n {\n        res += i;\n    }\n    res\n}\n\n/* 재귀로 합 구하기 */\nfn sum_recur(n: i32) -&gt; i32 {\n    if n == 1 {\n        return 1;\n    }\n    n + sum_recur(n - 1)\n}\n</code></pre> complexity_exercises.c<pre><code>/* 반복으로 합 구하기 */\nint sumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 재귀로 합 구하기 */\nint sumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.kt<pre><code>/* 반복으로 합 구하기 */\nfun sumIter(n: Int): Int {\n    var res = 0\n    for (i in 1..n) {\n        res += i\n    }\n    return res\n}\n\n/* 재귀로 합 구하기 */\nfun sumRecur(n: Int): Int {\n    if (n == 1) {\n        return 1\n    }\n    return n + sumRecur(n - 1)\n}\n</code></pre> complexity_exercises.rb<pre><code>### 반복으로 합 구하기 ###\ndef sum_iter(n)\n  res = 0\n  for i in 1..n\n    res += i\n  end\n  res\nend\n\n### 재귀로 합 구하기 ###\ndef sum_recur(n)\n  return 1 if n == 1\n\n  n + sum_recur(n - 1)\nend\n</code></pre> <ol> <li><code>n = 4</code>를 입력하여 반복 함수를 실행할 때, 각 반복이 끝난 뒤 누적 변수 <code>res</code>의 값은 각각 얼마인가?</li> <li><code>n = 4</code>를 입력하여 재귀 함수를 실행할 때, 매개변수 <code>n</code>은 어떤 값을 순서대로 갖는가? 가장 깊은 호출부터 반환할 때 결과는 어떻게 구해지는가?</li> <li>두 방식의 시간 복잡도와 공간 복잡도는 각각 얼마인가? 1번과 2번의 실행 과정을 바탕으로 이유를 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>반복 변수 <code>i</code>는 차례대로 <code>1, 2, 3, 4</code>가 되고, 각 반복이 끝날 때 <code>res</code>는     <code>1, 3, 6, 10</code>으로 변한다. 따라서 반복 함수는 10을 반환한다.</p> </li> <li> <p>매개변수 <code>n</code>은 <code>4 → 3 → 2 → 1</code> 순서로 변한다.     가장 깊은 호출이 1을 반환한 뒤, 각 호출은 차례대로 <code>2 + 1 = 3</code>, <code>3 + 3 = 6</code>, <code>4 + 6 = 10</code>을 얻는다.     가장 깊은 지점에서는 4번의 함수 호출이 모두 아직 끝나지 않은 상태다.</p> </li> <li> <p>두 코드 모두 \\(n\\)에 비례하는 횟수만큼 반복하거나 호출하므로 시간 복잡도는 모두 \\(O(n)\\)이다.     공간 복잡도는 다르다. 반복 방식은 상수 개의 변수만 사용하므로 \\(O(1)\\)이다.     재귀 방식은 종료 조건에 도달할 때까지 앞선 호출들이 반환값을 기다려야 하므로 호출 스택에 최대 \\(n\\)번의 호출이 동시에 저장된다.     따라서 공간 복잡도는 \\(O(n)\\)이다.</p> <p>공간 복잡도를 분석할 때는 코드에 선언된 변수뿐 아니라 재귀 호출이 사용하는 공간도 고려해야 한다.</p> </li> </ol>","path":["제 2 장   복잡도 분석","2.6   연습 문제"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#2","level":3,"title":"2.   세 코드의 시간 복잡도","text":"<p>다음 세 코드 조각은 모두 양의 정수 \\(n\\)을 입력받는다. 시간 복잡도가 낮은 순서대로 나열하고 각각의 복잡도를 적어 보자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby complexity_exercises.py<pre><code>def linear_loop(n: int) -&gt; int:\n    \"\"\"선형 차수 반복문\"\"\"\n    res = 0\n    for i in range(n):\n        res += i\n    return res\n\ndef quadratic_loop(n: int) -&gt; int:\n    \"\"\"제곱 차수 반복문\"\"\"\n    res = 0\n    for i in range(n):\n        for j in range(i, n):\n            res += j\n    return res\n\ndef logarithmic_loop(n: int) -&gt; int:\n    \"\"\"로그 차수 반복문\"\"\"\n    while n &gt; 1:\n        n //= 2\n    return n\n</code></pre> complexity_exercises.cpp<pre><code>/* 선형 차수 반복문 */\nint linearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; ++i) {\n        res += i;\n    }\n    return res;\n}\n\n/* 제곱 차수 반복문 */\nint quadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; ++i) {\n        for (int j = i; j &lt; n; ++j) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* 로그 차수 반복문 */\nint logarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.java<pre><code>/* 선형 차수 반복문 */\nint linearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 제곱 차수 반복문 */\nint quadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* 로그 차수 반복문 */\nint logarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.cs<pre><code>/* 선형 차수 반복문 */\nint LinearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 제곱 차수 반복문 */\nint QuadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* 로그 차수 반복문 */\nint LogarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.go<pre><code>/* 선형 차수 반복문 */\nfunc linearLoop(n int) int {\n    res := 0\n    for i := 0; i &lt; n; i++ {\n        res += i\n    }\n    return res\n}\n\n/* 제곱 차수 반복문 */\nfunc quadraticLoop(n int) int {\n    res := 0\n    for i := 0; i &lt; n; i++ {\n        for j := i; j &lt; n; j++ {\n            res += j\n        }\n    }\n    return res\n}\n\n/* 로그 차수 반복문 */\nfunc logarithmicLoop(n int) int {\n    for n &gt; 1 {\n        n /= 2\n    }\n    return n\n}\n</code></pre> complexity_exercises.swift<pre><code>/* 선형 차수 반복문 */\nfunc linearLoop(n: Int) -&gt; Int {\n    var res = 0\n    for i in 0 ..&lt; n {\n        res += i\n    }\n    return res\n}\n\n/* 제곱 차수 반복문 */\nfunc quadraticLoop(n: Int) -&gt; Int {\n    var res = 0\n    for i in 0 ..&lt; n {\n        for j in i ..&lt; n {\n            res += j\n        }\n    }\n    return res\n}\n\n/* 로그 차수 반복문 */\nfunc logarithmicLoop(n: Int) -&gt; Int {\n    var n = n\n    while n &gt; 1 {\n        n /= 2\n    }\n    return n\n}\n</code></pre> complexity_exercises.js<pre><code>/* 선형 차수 반복문 */\nfunction linearLoop(n) {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 제곱 차수 반복문 */\nfunction quadraticLoop(n) {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* 로그 차수 반복문 */\nfunction logarithmicLoop(n) {\n    while (n &gt; 1) {\n        n = Math.floor(n / 2);\n    }\n    return n;\n}\n</code></pre> complexity_exercises.ts<pre><code>/* 선형 차수 반복문 */\nfunction linearLoop(n: number): number {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 제곱 차수 반복문 */\nfunction quadraticLoop(n: number): number {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* 로그 차수 반복문 */\nfunction logarithmicLoop(n: number): number {\n    while (n &gt; 1) {\n        n = Math.floor(n / 2);\n    }\n    return n;\n}\n</code></pre> complexity_exercises.dart<pre><code>/* 선형 차수 반복문 */\nint linearLoop(int n) {\n  int res = 0;\n  for (int i = 0; i &lt; n; i++) {\n    res += i;\n  }\n  return res;\n}\n\n/* 제곱 차수 반복문 */\nint quadraticLoop(int n) {\n  int res = 0;\n  for (int i = 0; i &lt; n; i++) {\n    for (int j = i; j &lt; n; j++) {\n      res += j;\n    }\n  }\n  return res;\n}\n\n/* 로그 차수 반복문 */\nint logarithmicLoop(int n) {\n  while (n &gt; 1) {\n    n ~/= 2;\n  }\n  return n;\n}\n</code></pre> complexity_exercises.rs<pre><code>/* 선형 차수 반복문 */\nfn linear_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    for i in 0..n {\n        res += i;\n    }\n    res\n}\n\n/* 제곱 차수 반복문 */\nfn quadratic_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    for i in 0..n {\n        for j in i..n {\n            res += j;\n        }\n    }\n    res\n}\n\n/* 로그 차수 반복문 */\nfn logarithmic_loop(mut n: i32) -&gt; i32 {\n    while n &gt; 1 {\n        n /= 2;\n    }\n    n\n}\n</code></pre> complexity_exercises.c<pre><code>/* 선형 차수 반복문 */\nint linearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 제곱 차수 반복문 */\nint quadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* 로그 차수 반복문 */\nint logarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.kt<pre><code>/* 선형 차수 반복문 */\nfun linearLoop(n: Int): Int {\n    var res = 0\n    for (i in 0 until n) {\n        res += i\n    }\n    return res\n}\n\n/* 제곱 차수 반복문 */\nfun quadraticLoop(n: Int): Int {\n    var res = 0\n    for (i in 0 until n) {\n        for (j in i until n) {\n            res += j\n        }\n    }\n    return res\n}\n\n/* 로그 차수 반복문 */\nfun logarithmicLoop(n: Int): Int {\n    var value = n\n    while (value &gt; 1) {\n        value /= 2\n    }\n    return value\n}\n</code></pre> complexity_exercises.rb<pre><code>### 선형 차수 반복문 ###\ndef linear_loop(n)\n  res = 0\n  for i in 0...n\n    res += i\n  end\n  res\nend\n\n### 제곱 차수 반복문 ###\ndef quadratic_loop(n)\n  res = 0\n  for i in 0...n\n    for j in i...n\n      res += j\n    end\n  end\n  res\nend\n\n### 로그 차수 반복문 ###\ndef logarithmic_loop(n)\n  n /= 2 while n &gt; 1\n  n\nend\n</code></pre> 참고 답안 <p>낮은 순서대로 코드 3은 \\(O(\\log n)\\), 코드 1은 \\(O(n)\\), 코드 2는 \\(O(n^2)\\)이다. 코드 3은 매번 \\(n\\)을 절반으로 줄이므로 반복 횟수는 약 \\(\\log_2 n\\)번이다. 코드 1의 반복문은 정확히 \\(n\\)번 실행된다. 코드 2의 안쪽 반복 횟수는 차례대로 \\(n,n-1,\\dots,1\\)이므로 총 \\(n(n+1)/2\\)번이며, 제곱 차수에 해당한다.</p>","path":["제 2 장   복잡도 분석","2.6   연습 문제"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#3","level":3,"title":"3.   어느 뒤집기 방식이 공간을 더 절약하는가","text":"<p>배열 <code>nums</code>의 모든 원소를 역순으로 뒤집는 두 가지 방법이 있다.</p> <ol> <li>길이가 같은 새 배열 <code>res</code>를 만들고 원소를 역순으로 복사한 뒤 반환한다.</li> <li> <p>두 인덱스 <code>i</code>와 <code>j</code>를 각각 처음과 끝에서 가운데로 이동시키며 <code>nums[i]</code>와 <code>nums[j]</code>를 쌍으로 교환한다.</p> <p>두 방법의 공간 복잡도는 각각 얼마인가? 어느 방법이 ‘제자리’ 연산에 해당하는가?</p> </li> </ol> 참고 답안 <ol> <li> <p>입력과 길이가 같은 보조 배열이 필요하므로 공간 복잡도는 \\(O(n)\\)이다.</p> </li> <li> <p>인덱스 변수 두 개만 사용하므로     공간 복잡도는 \\(O(1)\\)이며, 제자리 연산이다.</p> <p>주의할 점은 제자리 뒤집기가 입력 배열을 변경한다는 것이다. 입력을 변경해도 되는 경우에만 우선적으로 선택해야 한다. 원본 배열을 보존해야 한다면 첫 번째 방법의 복사 비용은 피할 수 없다.</p> </li> </ol>","path":["제 2 장   복잡도 분석","2.6   연습 문제"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#262","level":2,"title":"2.6.2   프로그래밍 연습","text":"","path":["제 2 장   복잡도 분석","2.6   연습 문제"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#1_1","level":3,"title":"1.   피보나치 수","text":"<p>피보나치 수열은 \\(F(0)=0\\), \\(F(1)=1\\)이며, \\(n\\ge2\\)일 때 \\(F(n)=F(n-1)+F(n-2)\\)를 만족한다.</p> <p>음이 아닌 정수 <code>n</code>이 주어질 때, 재귀를 사용하지 않고 반복문으로 \\(F(n)\\)을 계산하여 반환하라.</p> 풀이 힌트 <ol> <li>n이 0 또는 1인 경우를 먼저 따로 처리한다.</li> <li>다음 항을 계산할 때는 앞선 두 항만 필요하므로 수열 전체를 저장할 필요가 없다.</li> <li>두 변수를 갱신할 때 아직 사용해야 하는 이전 값을 너무 일찍 덮어쓰지 않도록 주의한다.</li> </ol> <p>LeetCode</p>","path":["제 2 장   복잡도 분석","2.6   연습 문제"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/","level":1,"title":"2.2   반복과 재귀","text":"<p>알고리즘에서는 특정 작업을 반복해서 실행하는 일이 흔하며, 이는 복잡도 분석과 밀접하게 연결되어 있다. 따라서 시간 복잡도와 공간 복잡도를 소개하기 전에, 프로그램에서 작업을 반복 실행하는 두 가지 기본 제어 구조인 반복과 재귀부터 알아보자.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#221","level":2,"title":"2.2.1   반복","text":"<p>반복(iteration)은 특정 작업을 거듭 실행하는 제어 구조다. 반복에서는 일정한 조건을 만족하는 동안 코드 일부를 계속 실행하다가, 조건을 더 이상 만족하지 않으면 멈춘다.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for","level":3,"title":"1.   for 반복문","text":"<p><code>for</code> 반복문은 가장 흔한 반복 형식 중 하나로, 반복 횟수를 미리 알고 있을 때 적합하다.</p> <p>다음 함수는 <code>for</code> 반복문으로 \\(1 + 2 + \\dots + n\\)을 계산하고, 그 합을 변수 <code>res</code>에 기록한다. Python의 <code>range(a, b)</code>는 왼쪽 끝을 포함하고 오른쪽 끝은 포함하지 않는 구간이므로, 순회 범위는 \\(a, a + 1, \\dots, b-1\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def for_loop(n: int) -&gt; int:\n    \"\"\"for 반복문\"\"\"\n    res = 0\n    # 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for i in range(1, n + 1):\n        res += i\n    return res\n</code></pre> iteration.cpp<pre><code>/* for 반복문 */\nint forLoop(int n) {\n    int res = 0;\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for (int i = 1; i &lt;= n; ++i) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* for 반복문 */\nint forLoop(int n) {\n    int res = 0;\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* for 반복문 */\nint ForLoop(int n) {\n    int res = 0;\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* for 반복문 */\nfunc forLoop(n int) int {\n    res := 0\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for i := 1; i &lt;= n; i++ {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* for 반복문 */\nfunc forLoop(n: Int) -&gt; Int {\n    var res = 0\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for i in 1 ... n {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* for 반복문 */\nfunction forLoop(n) {\n    let res = 0;\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* for 반복문 */\nfunction forLoop(n: number): number {\n    let res = 0;\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* for 반복문 */\nint forLoop(int n) {\n  int res = 0;\n  // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n  for (int i = 1; i &lt;= n; i++) {\n    res += i;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* for 반복문 */\nfn for_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for i in 1..=n {\n        res += i;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* for 반복문 */\nint forLoop(int n) {\n    int res = 0;\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* for 반복문 */\nfun forLoop(n: Int): Int {\n    var res = 0\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for (i in 1..n) {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### for 반복문 ###\ndef for_loop(n)\n  res = 0\n\n  # 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n  for i in 1..n\n    res += i\n  end\n\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(2-1)은 이 합계 함수의 순서도다.</p> <p></p> <p> 그림 2-1   합계 함수의 순서도 </p> <p>이 합계 함수의 연산 횟수는 입력 데이터 크기 \\(n\\)에 비례한다. 이를 ‘선형 관계’라고도 한다. 실제로 시간 복잡도가 설명하는 것이 바로 이 ‘선형 관계’다. 관련 내용은 다음 절에서 자세히 소개한다.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while","level":3,"title":"2.   while 반복문","text":"<p><code>for</code> 반복문과 마찬가지로 <code>while</code> 반복문도 반복을 구현하는 방법이다. <code>while</code> 반복문은 매번 먼저 조건을 검사하여 참이면 실행을 계속하고, 그렇지 않으면 반복을 끝낸다.</p> <p>다음은 <code>while</code> 반복문으로 \\(1 + 2 + \\dots + n\\)을 계산하는 코드다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def while_loop(n: int) -&gt; int:\n    \"\"\"while 반복문\"\"\"\n    res = 0\n    i = 1  # 조건 변수를 초기화한다\n    # 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while i &lt;= n:\n        res += i\n        i += 1  # 조건 변수를 갱신한다\n    return res\n</code></pre> iteration.cpp<pre><code>/* while 반복문 */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        i++; // 조건 변수를 갱신한다\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while 반복문 */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        i++; // 조건 변수를 갱신한다\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while 반복문 */\nint WhileLoop(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        i += 1; // 조건 변수를 갱신한다\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while 반복문 */\nfunc whileLoop(n int) int {\n    res := 0\n    // 조건 변수를 초기화한다\n    i := 1\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for i &lt;= n {\n        res += i\n        // 조건 변수를 갱신한다\n        i++\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while 반복문 */\nfunc whileLoop(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while i &lt;= n {\n        res += i\n        i += 1 // 조건 변수를 갱신한다\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while 반복문 */\nfunction whileLoop(n) {\n    let res = 0;\n    let i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        i++; // 조건 변수를 갱신한다\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while 반복문 */\nfunction whileLoop(n: number): number {\n    let res = 0;\n    let i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        i++; // 조건 변수를 갱신한다\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while 반복문 */\nint whileLoop(int n) {\n  int res = 0;\n  int i = 1; // 조건 변수를 초기화한다\n  // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n  while (i &lt;= n) {\n    res += i;\n    i++; // 조건 변수를 갱신한다\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while 반복문 */\nfn while_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // 조건 변수를 초기화한다\n\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while i &lt;= n {\n        res += i;\n        i += 1; // 조건 변수를 갱신한다\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while 반복문 */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        i++; // 조건 변수를 갱신한다\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while 반복문 */\nfun whileLoop(n: Int): Int {\n    var res = 0\n    var i = 1 // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while (i &lt;= n) {\n        res += i\n        i++ // 조건 변수를 갱신한다\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### while 반복문 ###\ndef while_loop(n)\n  res = 0\n  i = 1 # 조건 변수를 초기화한다\n\n  # 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n  while i &lt;= n\n    res += i\n    i += 1 # 조건 변수를 갱신한다\n  end\n\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p><code>while</code> 반복문은 <code>for</code> 반복문보다 자유도가 높다. <code>while</code> 반복문에서는 조건 변수의 초기화와 갱신 단계를 자유롭게 설계할 수 있다.</p> <p>예를 들어 다음 코드에서는 조건 변수 \\(i\\)를 매번 두 차례 갱신한다. 이런 경우에는 <code>for</code> 반복문으로 구현하기가 다소 불편하다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def while_loop_ii(n: int) -&gt; int:\n    \"\"\"while 반복문(두 번 갱신)\"\"\"\n    res = 0\n    i = 1  # 조건 변수를 초기화한다\n    # 반복문으로 1, 4, 10, ...의 합을 구한다\n    while i &lt;= n:\n        res += i\n        # 조건 변수를 갱신한다\n        i += 1\n        i *= 2\n    return res\n</code></pre> iteration.cpp<pre><code>/* while 반복문(두 번 갱신) */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        // 조건 변수를 갱신한다\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while 반복문(두 번 갱신) */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        // 조건 변수를 갱신한다\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while 반복문(두 번 갱신) */\nint WhileLoopII(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        // 조건 변수를 갱신한다\n        i += 1; \n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while 반복문(두 번 갱신) */\nfunc whileLoopII(n int) int {\n    res := 0\n    // 조건 변수를 초기화한다\n    i := 1\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    for i &lt;= n {\n        res += i\n        // 조건 변수를 갱신한다\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while 반복문(두 번 갱신) */\nfunc whileLoopII(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while i &lt;= n {\n        res += i\n        // 조건 변수를 갱신한다\n        i += 1\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while 반복문(두 번 갱신) */\nfunction whileLoopII(n) {\n    let res = 0;\n    let i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        // 조건 변수를 갱신한다\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while 반복문(두 번 갱신) */\nfunction whileLoopII(n: number): number {\n    let res = 0;\n    let i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        // 조건 변수를 갱신한다\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while 반복문(두 번 갱신) */\nint whileLoopII(int n) {\n  int res = 0;\n  int i = 1; // 조건 변수를 초기화한다\n  // 반복문으로 1, 4, 10, ...의 합을 구한다\n  while (i &lt;= n) {\n    res += i;\n    // 조건 변수를 갱신한다\n    i++;\n    i *= 2;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while 반복문(두 번 갱신) */\nfn while_loop_ii(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // 조건 변수를 초기화한다\n\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while i &lt;= n {\n        res += i;\n        // 조건 변수를 갱신한다\n        i += 1;\n        i *= 2;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while 반복문(두 번 갱신) */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        // 조건 변수를 갱신한다\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while 반복문(두 번 갱신) */\nfun whileLoopII(n: Int): Int {\n    var res = 0\n    var i = 1 // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while (i &lt;= n) {\n        res += i\n        // 조건 변수를 갱신한다\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### while 반복문(두 번 갱신)###\ndef while_loop_ii(n)\n  res = 0\n  i = 1 # 조건 변수를 초기화한다\n\n  # 반복문으로 1, 4, 10, ...의 합을 구한다\n  while i &lt;= n\n    res += i\n    # 조건 변수를 갱신한다\n    i += 1\n    i *= 2\n  end\n\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>전반적으로 <code>for</code> 반복문은 코드가 더 간결하고, <code>while</code> 반복문은 더 유연하다. 둘 다 반복 구조를 구현할 수 있으며, 어떤 것을 쓸지는 문제의 구체적인 요구에 따라 결정해야 한다.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#3","level":3,"title":"3.   중첩 반복문","text":"<p>하나의 반복 구조 안에 다른 반복 구조를 중첩할 수 있다. 다음은 <code>for</code> 반복문을 사용한 예다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def nested_for_loop(n: int) -&gt; str:\n    \"\"\"이중 for 반복문\"\"\"\n    res = \"\"\n    # i = 1, 2, ..., n-1, n을 반복한다\n    for i in range(1, n + 1):\n        # j = 1, 2, ..., n-1, n을 반복한다\n        for j in range(1, n + 1):\n            res += f\"({i}, {j}), \"\n    return res\n</code></pre> iteration.cpp<pre><code>/* 이중 for 반복문 */\nstring nestedForLoop(int n) {\n    ostringstream res;\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for (int i = 1; i &lt;= n; ++i) {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for (int j = 1; j &lt;= n; ++j) {\n            res &lt;&lt; \"(\" &lt;&lt; i &lt;&lt; \", \" &lt;&lt; j &lt;&lt; \"), \";\n        }\n    }\n    return res.str();\n}\n</code></pre> iteration.java<pre><code>/* 이중 for 반복문 */\nString nestedForLoop(int n) {\n    StringBuilder res = new StringBuilder();\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for (int i = 1; i &lt;= n; i++) {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for (int j = 1; j &lt;= n; j++) {\n            res.append(\"(\" + i + \", \" + j + \"), \");\n        }\n    }\n    return res.toString();\n}\n</code></pre> iteration.cs<pre><code>/* 이중 for 반복문 */\nstring NestedForLoop(int n) {\n    StringBuilder res = new();\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for (int i = 1; i &lt;= n; i++) {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for (int j = 1; j &lt;= n; j++) {\n            res.Append($\"({i}, {j}), \");\n        }\n    }\n    return res.ToString();\n}\n</code></pre> iteration.go<pre><code>/* 이중 for 반복문 */\nfunc nestedForLoop(n int) string {\n    res := \"\"\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= n; j++ {\n            // j = 1, 2, ..., n-1, n을 반복한다\n            res += fmt.Sprintf(\"(%d, %d), \", i, j)\n        }\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* 이중 for 반복문 */\nfunc nestedForLoop(n: Int) -&gt; String {\n    var res = \"\"\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for i in 1 ... n {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for j in 1 ... n {\n            res.append(\"(\\(i), \\(j)), \")\n        }\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* 이중 for 반복문 */\nfunction nestedForLoop(n) {\n    let res = '';\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for (let i = 1; i &lt;= n; i++) {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* 이중 for 반복문 */\nfunction nestedForLoop(n: number): string {\n    let res = '';\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for (let i = 1; i &lt;= n; i++) {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* 이중 for 반복문 */\nString nestedForLoop(int n) {\n  String res = \"\";\n  // i = 1, 2, ..., n-1, n을 반복한다\n  for (int i = 1; i &lt;= n; i++) {\n    // j = 1, 2, ..., n-1, n을 반복한다\n    for (int j = 1; j &lt;= n; j++) {\n      res += \"($i, $j), \";\n    }\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* 이중 for 반복문 */\nfn nested_for_loop(n: i32) -&gt; String {\n    let mut res = vec![];\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for i in 1..=n {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for j in 1..=n {\n            res.push(format!(\"({}, {}), \", i, j));\n        }\n    }\n    res.join(\"\")\n}\n</code></pre> iteration.c<pre><code>/* 이중 for 반복문 */\nchar *nestedForLoop(int n) {\n    // n * n은 해당 점의 개수이며, \"(i, j), \"의 최대 문자열 길이는 6+10*2이다. 마지막 널 문자 \\0을 위한 공간도 추가한다\n    int size = n * n * 26 + 1;\n    char *res = malloc(size * sizeof(char));\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for (int i = 1; i &lt;= n; i++) {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for (int j = 1; j &lt;= n; j++) {\n            char tmp[26];\n            snprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\n            strncat(res, tmp, size - strlen(res) - 1);\n        }\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* 이중 for 반복문 */\nfun nestedForLoop(n: Int): String {\n    val res = StringBuilder()\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for (i in 1..n) {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for (j in 1..n) {\n            res.append(\" ($i, $j), \")\n        }\n    }\n    return res.toString()\n}\n</code></pre> iteration.rb<pre><code>### 이중 for 반복문 ###\ndef nested_for_loop(n)\n  res = \"\"\n\n  # i = 1, 2, ..., n-1, n을 반복한다\n  for i in 1..n\n    # j = 1, 2, ..., n-1, n을 반복한다\n    for j in 1..n\n      res += \"(#{i}, #{j}), \"\n    end\n  end\n\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(2-2)은 이 중첩 반복문의 순서도다.</p> <p></p> <p> 그림 2-2   중첩 반복문의 순서도 </p> <p>이 경우 함수의 연산 횟수는 \\(n^2\\)에 비례한다. 즉, 알고리즘 실행 시간과 입력 데이터 크기 \\(n\\) 사이에는 ‘제곱 관계’가 있다.</p> <p>반복문을 계속 중첩할 수도 있다. 한 번 중첩할 때마다 ‘차원이 하나씩 늘어나’ 시간 복잡도가 ‘세제곱 관계’, ‘네제곱 관계’ 등으로 높아진다.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#222","level":2,"title":"2.2.2   재귀","text":"<p>재귀(recursion)는 함수가 자기 자신을 호출하여 문제를 해결하는 알고리즘 전략이다. 크게 두 단계로 이루어진다.</p> <ol> <li>내려가기: 프로그램이 자기 자신을 계속 더 깊이 호출한다. 보통 더 작거나 단순한 인수를 전달하며, ‘종료 조건’에 도달할 때까지 이어진다.</li> <li>돌아오기: ‘종료 조건’에 도달하면 가장 깊은 재귀 함수부터 한 단계씩 반환하며, 각 단계의 결과를 모은다.</li> </ol> <p>구현 관점에서 보면 재귀 코드는 크게 세 요소로 이루어진다.</p> <ol> <li>종료 조건: ‘내려가기’에서 ‘돌아오기’로 전환할 시점을 결정한다.</li> <li>재귀 호출: ‘내려가기’에 해당한다. 함수가 자기 자신을 호출하며, 보통 더 작거나 단순한 인수를 입력한다.</li> <li>결과 반환: ‘돌아오기’에 해당한다. 현재 재귀 단계의 결과를 이전 단계로 반환한다.</li> </ol> <p>다음 코드를 보면, 함수 <code>recur(n)</code>을 호출하기만 하면 \\(1 + 2 + \\dots + n\\)을 계산할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def recur(n: int) -&gt; int:\n    \"\"\"재귀\"\"\"\n    # 종료 조건\n    if n == 1:\n        return 1\n    # 내려가기: 재귀 호출\n    res = recur(n - 1)\n    # 돌아오기: 결과 반환\n    return n + res\n</code></pre> recursion.cpp<pre><code>/* 재귀 */\nint recur(int n) {\n    // 종료 조건\n    if (n == 1)\n        return 1;\n    // 내려가기: 재귀 호출\n    int res = recur(n - 1);\n    // 돌아오기: 결과 반환\n    return n + res;\n}\n</code></pre> recursion.java<pre><code>/* 재귀 */\nint recur(int n) {\n    // 종료 조건\n    if (n == 1)\n        return 1;\n    // 내려가기: 재귀 호출\n    int res = recur(n - 1);\n    // 돌아오기: 결과 반환\n    return n + res;\n}\n</code></pre> recursion.cs<pre><code>/* 재귀 */\nint Recur(int n) {\n    // 종료 조건\n    if (n == 1)\n        return 1;\n    // 내려가기: 재귀 호출\n    int res = Recur(n - 1);\n    // 돌아오기: 결과 반환\n    return n + res;\n}\n</code></pre> recursion.go<pre><code>/* 재귀 */\nfunc recur(n int) int {\n    // 종료 조건\n    if n == 1 {\n        return 1\n    }\n    // 내려가기: 재귀 호출\n    res := recur(n - 1)\n    // 돌아오기: 결과 반환\n    return n + res\n}\n</code></pre> recursion.swift<pre><code>/* 재귀 */\nfunc recur(n: Int) -&gt; Int {\n    // 종료 조건\n    if n == 1 {\n        return 1\n    }\n    // 내려가기: 재귀 호출\n    let res = recur(n: n - 1)\n    // 돌아오기: 결과 반환\n    return n + res\n}\n</code></pre> recursion.js<pre><code>/* 재귀 */\nfunction recur(n) {\n    // 종료 조건\n    if (n === 1) return 1;\n    // 내려가기: 재귀 호출\n    const res = recur(n - 1);\n    // 돌아오기: 결과 반환\n    return n + res;\n}\n</code></pre> recursion.ts<pre><code>/* 재귀 */\nfunction recur(n: number): number {\n    // 종료 조건\n    if (n === 1) return 1;\n    // 내려가기: 재귀 호출\n    const res = recur(n - 1);\n    // 돌아오기: 결과 반환\n    return n + res;\n}\n</code></pre> recursion.dart<pre><code>/* 재귀 */\nint recur(int n) {\n  // 종료 조건\n  if (n == 1) return 1;\n  // 내려가기: 재귀 호출\n  int res = recur(n - 1);\n  // 돌아오기: 결과 반환\n  return n + res;\n}\n</code></pre> recursion.rs<pre><code>/* 재귀 */\nfn recur(n: i32) -&gt; i32 {\n    // 종료 조건\n    if n == 1 {\n        return 1;\n    }\n    // 내려가기: 재귀 호출\n    let res = recur(n - 1);\n    // 돌아오기: 결과 반환\n    n + res\n}\n</code></pre> recursion.c<pre><code>/* 재귀 */\nint recur(int n) {\n    // 종료 조건\n    if (n == 1)\n        return 1;\n    // 내려가기: 재귀 호출\n    int res = recur(n - 1);\n    // 돌아오기: 결과 반환\n    return n + res;\n}\n</code></pre> recursion.kt<pre><code>/* 재귀 */\nfun recur(n: Int): Int {\n    // 종료 조건\n    if (n == 1)\n        return 1\n    // 내려가기: 재귀 호출\n    val res = recur(n - 1)\n    // 돌아오기: 결과 반환\n    return n + res\n}\n</code></pre> recursion.rb<pre><code>### 재귀 ###\ndef recur(n)\n  # 종료 조건\n  return 1 if n == 1\n  # 내려가기: 재귀 호출\n  res = recur(n - 1)\n  # 돌아오기: 결과 반환\n  n + res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(2-3)은 이 함수의 재귀 과정을 보여 준다.</p> <p></p> <p> 그림 2-3   합계 함수의 재귀 과정 </p> <p>반복과 재귀는 계산 결과가 같을 수 있지만, 문제를 생각하고 해결하는 방식은 완전히 다르다.</p> <ul> <li>반복: ‘상향식’으로 문제를 해결한다. 가장 기본적인 단계에서 시작해 그 단계를 계속 반복하거나 누적하여 작업을 완료한다.</li> <li>재귀: ‘하향식’으로 문제를 해결한다. 원래 문제를 같은 형태의 더 작은 부분 문제로 나눈다. 그런 다음 부분 문제를 더 작은 부분 문제로 계속 나누다가 기저 사례에 도달하면 멈춘다(기저 사례의 해는 이미 알고 있다).</li> </ul> <p>앞의 합계 함수를 예로 들어, 문제를 \\(f(n) = 1 + 2 + \\dots + n\\)이라고 하자.</p> <ul> <li>반복: 반복문에서 합을 구하는 과정을 재현한다. \\(1\\)부터 \\(n\\)까지 순회하면서 매번 더하면 \\(f(n)\\)을 구할 수 있다.</li> <li>재귀: 문제를 부분 문제 \\(f(n) = n + f(n-1)\\)로 나누고, 계속 재귀적으로 나누다가 기저 사례 \\(f(1) = 1\\)에 도달하면 멈춘다.</li> </ul>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#1","level":3,"title":"1.   호출 스택","text":"<p>재귀 함수가 자기 자신을 호출할 때마다 시스템은 새로 실행되는 함수를 위해 메모리를 할당하고 지역 변수, 호출 주소 등의 정보를 저장한다. 이로 인해 두 가지 결과가 나타난다.</p> <ul> <li>함수의 문맥 데이터는 ‘스택 프레임 공간’이라는 메모리 영역에 저장되며, 함수가 반환된 뒤에야 해제된다. 따라서 재귀는 보통 반복보다 더 많은 메모리 공간을 사용한다.</li> <li>재귀 함수 호출에는 추가 비용이 발생한다. 따라서 재귀는 보통 반복문보다 시간 효율이 낮다.</li> </ul> <p>아래 그림(2-4)과 같이 종료 조건에 도달하기 전에는 아직 반환되지 않은 재귀 함수가 \\(n\\)개 동시에 존재하므로, 재귀 깊이는 \\(n\\)이다.</p> <p></p> <p> 그림 2-4   재귀 호출 깊이 </p> <p>실제로 프로그래밍 언어가 허용하는 재귀 깊이는 보통 제한되어 있으며, 재귀가 너무 깊어지면 스택 오버플로 오류가 발생할 수 있다.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#2","level":3,"title":"2.   꼬리 재귀","text":"<p>흥미롭게도 함수가 반환하기 직전의 마지막 단계에서 재귀 호출을 수행하면, 컴파일러나 인터프리터가 이를 최적화하여 반복과 비슷한 공간 효율을 얻을 수 있다. 이런 경우를 꼬리 재귀(tail recursion)라고 한다.</p> <ul> <li>일반 재귀: 함수가 이전 단계의 함수로 반환된 뒤에도 실행할 코드가 남아 있으므로, 시스템은 이전 호출의 문맥을 저장해야 한다.</li> <li>꼬리 재귀: 재귀 호출이 함수 반환 전의 마지막 연산이다. 따라서 이전 단계로 반환된 뒤에 다른 연산을 수행할 필요가 없으므로, 시스템은 이전 함수의 문맥을 저장하지 않아도 된다.</li> </ul> <p>\\(1 + 2 + \\dots + n\\)을 계산하는 예에서 결과 변수 <code>res</code>를 함수 매개변수로 두면 꼬리 재귀를 구현할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def tail_recur(n, res):\n    \"\"\"꼬리 재귀\"\"\"\n    # 종료 조건\n    if n == 0:\n        return res\n    # 꼬리 재귀 호출\n    return tail_recur(n - 1, res + n)\n</code></pre> recursion.cpp<pre><code>/* 꼬리 재귀 */\nint tailRecur(int n, int res) {\n    // 종료 조건\n    if (n == 0)\n        return res;\n    // 꼬리 재귀 호출\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.java<pre><code>/* 꼬리 재귀 */\nint tailRecur(int n, int res) {\n    // 종료 조건\n    if (n == 0)\n        return res;\n    // 꼬리 재귀 호출\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.cs<pre><code>/* 꼬리 재귀 */\nint TailRecur(int n, int res) {\n    // 종료 조건\n    if (n == 0)\n        return res;\n    // 꼬리 재귀 호출\n    return TailRecur(n - 1, res + n);\n}\n</code></pre> recursion.go<pre><code>/* 꼬리 재귀 */\nfunc tailRecur(n int, res int) int {\n    // 종료 조건\n    if n == 0 {\n        return res\n    }\n    // 꼬리 재귀 호출\n    return tailRecur(n-1, res+n)\n}\n</code></pre> recursion.swift<pre><code>/* 꼬리 재귀 */\nfunc tailRecur(n: Int, res: Int) -&gt; Int {\n    // 종료 조건\n    if n == 0 {\n        return res\n    }\n    // 꼬리 재귀 호출\n    return tailRecur(n: n - 1, res: res + n)\n}\n</code></pre> recursion.js<pre><code>/* 꼬리 재귀 */\nfunction tailRecur(n, res) {\n    // 종료 조건\n    if (n === 0) return res;\n    // 꼬리 재귀 호출\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.ts<pre><code>/* 꼬리 재귀 */\nfunction tailRecur(n: number, res: number): number {\n    // 종료 조건\n    if (n === 0) return res;\n    // 꼬리 재귀 호출\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.dart<pre><code>/* 꼬리 재귀 */\nint tailRecur(int n, int res) {\n  // 종료 조건\n  if (n == 0) return res;\n  // 꼬리 재귀 호출\n  return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.rs<pre><code>/* 꼬리 재귀 */\nfn tail_recur(n: i32, res: i32) -&gt; i32 {\n    // 종료 조건\n    if n == 0 {\n        return res;\n    }\n    // 꼬리 재귀 호출\n    tail_recur(n - 1, res + n)\n}\n</code></pre> recursion.c<pre><code>/* 꼬리 재귀 */\nint tailRecur(int n, int res) {\n    // 종료 조건\n    if (n == 0)\n        return res;\n    // 꼬리 재귀 호출\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.kt<pre><code>/* 꼬리 재귀 */\ntailrec fun tailRecur(n: Int, res: Int): Int {\n    // tailrec 키워드를 추가하여 꼬리 재귀 최적화를 활성화한다\n    // 종료 조건\n    if (n == 0)\n        return res\n    // 꼬리 재귀 호출\n    return tailRecur(n - 1, res + n)\n}\n</code></pre> recursion.rb<pre><code>### 꼬리 재귀 ###\ndef tail_recur(n, res)\n  # 종료 조건\n  return res if n == 0\n  # 꼬리 재귀 호출\n  tail_recur(n - 1, res + n)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>꼬리 재귀의 실행 과정은 아래 그림(2-5)과 같다. 일반 재귀와 꼬리 재귀를 비교하면 합을 구하는 시점이 다르다.</p> <ul> <li>일반 재귀: ‘돌아오기’ 과정에서 합을 구한다. 각 단계가 반환된 뒤에 덧셈을 한 번씩 수행한다.</li> <li>꼬리 재귀: ‘내려가기’ 과정에서 합을 구한다. ‘돌아오기’ 과정에서는 단계별로 반환만 하면 된다.</li> </ul> <p></p> <p> 그림 2-5   꼬리 재귀 과정 </p> <p>참고</p> <p>많은 컴파일러와 인터프리터가 꼬리 재귀 최적화를 지원하지 않는다는 점에 주의하자. 예를 들어 Python은 기본적으로 꼬리 재귀 최적화를 지원하지 않으므로, 함수가 꼬리 재귀 형태여도 스택 오버플로 문제가 발생할 수 있다.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#3_1","level":3,"title":"3.   재귀 트리","text":"<p>‘분할 정복’과 관련된 알고리즘 문제에서는 재귀가 반복보다 생각하기 쉽고 코드도 읽기 쉬운 경우가 많다. ‘피보나치 수열’을 예로 살펴보자.</p> <p>문제</p> <p>피보나치 수열 \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\)가 주어졌을 때, 이 수열의 \\(n\\)번째 수를 구하라.</p> <p>피보나치 수열의 \\(n\\)번째 수를 \\(f(n)\\)이라고 하면 다음 두 가지 사실을 쉽게 알 수 있다.</p> <ul> <li>수열의 첫 두 수는 \\(f(1) = 0\\)과 \\(f(2) = 1\\)이다.</li> <li>수열의 각 수는 앞의 두 수의 합이다. 즉, \\(f(n) = f(n - 1) + f(n - 2)\\)이다.</li> </ul> <p>점화 관계에 따라 재귀 호출을 하고 첫 두 수를 종료 조건으로 삼으면 재귀 코드를 작성할 수 있다. <code>fib(n)</code>을 호출하면 피보나치 수열의 \\(n\\)번째 수를 얻는다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def fib(n: int) -&gt; int:\n    \"\"\"피보나치 수열: 재귀\"\"\"\n    # 종료 조건 f(1) = 0, f(2) = 1\n    if n == 1 or n == 2:\n        return n - 1\n    # 재귀 호출 f(n) = f(n-1) + f(n-2)\n    res = fib(n - 1) + fib(n - 2)\n    # 결과 f(n)을 반환한다\n    return res\n</code></pre> recursion.cpp<pre><code>/* 피보나치 수열: 재귀 */\nint fib(int n) {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // 결과 f(n)을 반환한다\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* 피보나치 수열: 재귀 */\nint fib(int n) {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // 결과 f(n)을 반환한다\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* 피보나치 수열: 재귀 */\nint Fib(int n) {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    int res = Fib(n - 1) + Fib(n - 2);\n    // 결과 f(n)을 반환한다\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* 피보나치 수열: 재귀 */\nfunc fib(n int) int {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    res := fib(n-1) + fib(n-2)\n    // 결과 f(n)을 반환한다\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* 피보나치 수열: 재귀 */\nfunc fib(n: Int) -&gt; Int {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    let res = fib(n: n - 1) + fib(n: n - 2)\n    // 결과 f(n)을 반환한다\n    return res\n}\n</code></pre> recursion.js<pre><code>/* 피보나치 수열: 재귀 */\nfunction fib(n) {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // 결과 f(n)을 반환한다\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* 피보나치 수열: 재귀 */\nfunction fib(n: number): number {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // 결과 f(n)을 반환한다\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* 피보나치 수열: 재귀 */\nint fib(int n) {\n  // 종료 조건 f(1) = 0, f(2) = 1\n  if (n == 1 || n == 2) return n - 1;\n  // 재귀 호출 f(n) = f(n-1) + f(n-2)\n  int res = fib(n - 1) + fib(n - 2);\n  // 결과 f(n)을 반환한다\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* 피보나치 수열: 재귀 */\nfn fib(n: i32) -&gt; i32 {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1;\n    }\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    let res = fib(n - 1) + fib(n - 2);\n    // 결과를 반환한다\n    res\n}\n</code></pre> recursion.c<pre><code>/* 피보나치 수열: 재귀 */\nint fib(int n) {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // 결과 f(n)을 반환한다\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* 피보나치 수열: 재귀 */\nfun fib(n: Int): Int {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    val res = fib(n - 1) + fib(n - 2)\n    // 결과 f(n)을 반환한다\n    return res\n}\n</code></pre> recursion.rb<pre><code>### 피보나치 수열: 재귀 ###\ndef fib(n)\n  # 종료 조건 f(1) = 0, f(2) = 1\n  return n - 1 if n == 1 || n == 2\n  # 재귀 호출 f(n) = f(n-1) + f(n-2)\n  res = fib(n - 1) + fib(n - 2)\n  # 결과 f(n)을 반환한다\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>위 코드는 함수 안에서 두 번의 재귀 호출을 수행한다. 이는 하나의 호출에서 두 개의 호출 분기가 생긴다는 뜻이다. 아래 그림(2-6)과 같이 재귀 호출을 계속하면 최종적으로 \\(n\\)개 층으로 이루어진 재귀 트리(recursion tree)가 만들어진다.</p> <p></p> <p> 그림 2-6   피보나치 수열의 재귀 트리 </p> <p>본질적으로 재귀는 ‘문제를 더 작은 부분 문제로 나누는’ 사고방식을 보여 준다. 이러한 분할 정복 전략은 매우 중요하다.</p> <ul> <li>알고리즘 관점에서 탐색, 정렬, 백트래킹, 분할 정복, 동적 계획법 등 중요한 알고리즘 전략 다수가 이 사고방식을 직간접적으로 활용한다.</li> <li>자료 구조 관점에서 재귀는 연결 리스트, 트리, 그래프와 관련된 문제를 다루는 데 자연스럽게 어울린다. 이들은 분할 정복 관점으로 분석하기에 매우 적합하기 때문이다.</li> </ul>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#223","level":2,"title":"2.2.3   두 방식의 비교","text":"<p>앞의 내용을 정리하면, 아래 표(2-1)와 같이 반복과 재귀는 구현, 성능, 적용 범위에서 차이가 있다.</p> <p> 표 2-1   반복과 재귀의 특징 비교 </p> 반복 재귀 구현 방식 반복 구조 함수가 자기 자신을 호출 시간 효율 보통 효율이 높고 함수 호출 비용이 없음 함수 호출마다 비용이 발생함 메모리 사용 보통 고정 크기의 메모리 공간을 사용함 함수 호출이 누적되면 스택 프레임 공간을 많이 사용할 수 있음 적합한 문제 간단한 반복 작업에 적합하며 코드가 직관적이고 읽기 쉬움 트리, 그래프, 분할 정복, 백트래킹 등 부분 문제로 나누는 문제에 적합하며 코드 구조가 간결하고 명확함 <p>참고</p> <p>다음 내용이 이해하기 어렵다면 ‘스택’ 장을 읽은 뒤에 다시 복습해도 좋다.</p> <p>그렇다면 반복과 재귀는 내부적으로 어떤 관계가 있을까? 앞의 재귀 함수를 예로 들면, 합을 구하는 연산은 재귀의 ‘돌아오기’ 단계에서 수행된다. 즉, 가장 먼저 호출된 함수가 실제로는 가장 나중에 자신의 덧셈을 마친다. 이 작동 방식은 스택의 ‘후입선출’ 원리와 통한다.</p> <p>실제로 ‘호출 스택’, ‘스택 프레임 공간’ 같은 재귀 관련 용어에도 재귀와 스택의 밀접한 관계가 드러난다.</p> <ol> <li>내려가기: 함수가 호출되면 시스템은 ‘호출 스택’에 새로운 스택 프레임을 할당하여 함수의 지역 변수, 매개변수, 반환 주소 등의 데이터를 저장한다.</li> <li>돌아오기: 함수가 실행을 마치고 반환되면 해당 스택 프레임이 ‘호출 스택’에서 제거되고, 이전 함수의 실행 환경이 복원된다.</li> </ol> <p>따라서 명시적인 스택으로 호출 스택의 동작을 재현하면, 재귀를 반복 형태로 바꿀 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def for_loop_recur(n: int) -&gt; int:\n    \"\"\"반복으로 재귀 구현\"\"\"\n    # 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    stack = []\n    res = 0\n    # 내려가기: 재귀 호출\n    for i in range(n, 0, -1):\n        # 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.append(i)\n    # 돌아오기: 결과 반환\n    while stack:\n        # 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.pop()\n    # res = 1+2+3+...+n\n    return res\n</code></pre> recursion.cpp<pre><code>/* 반복으로 재귀 구현 */\nint forLoopRecur(int n) {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    stack&lt;int&gt; stack;\n    int res = 0;\n    // 내려가기: 재귀 호출\n    for (int i = n; i &gt; 0; i--) {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.push(i);\n    }\n    // 돌아오기: 결과 반환\n    while (!stack.empty()) {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.top();\n        stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* 반복으로 재귀 구현 */\nint forLoopRecur(int n) {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    Stack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n    int res = 0;\n    // 내려가기: 재귀 호출\n    for (int i = n; i &gt; 0; i--) {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.push(i);\n    }\n    // 돌아오기: 결과 반환\n    while (!stack.isEmpty()) {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* 반복으로 재귀 구현 */\nint ForLoopRecur(int n) {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    Stack&lt;int&gt; stack = new();\n    int res = 0;\n    // 내려가기: 재귀 호출\n    for (int i = n; i &gt; 0; i--) {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.Push(i);\n    }\n    // 돌아오기: 결과 반환\n    while (stack.Count &gt; 0) {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.Pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* 반복으로 재귀 구현 */\nfunc forLoopRecur(n int) int {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    stack := list.New()\n    res := 0\n    // 내려가기: 재귀 호출\n    for i := n; i &gt; 0; i-- {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.PushBack(i)\n    }\n    // 돌아오기: 결과 반환\n    for stack.Len() != 0 {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.Back().Value.(int)\n        stack.Remove(stack.Back())\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* 반복으로 재귀 구현 */\nfunc forLoopRecur(n: Int) -&gt; Int {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    var stack: [Int] = []\n    var res = 0\n    // 내려가기: 재귀 호출\n    for i in (1 ... n).reversed() {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.append(i)\n    }\n    // 돌아오기: 결과 반환\n    while !stack.isEmpty {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.removeLast()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.js<pre><code>/* 반복으로 재귀 구현 */\nfunction forLoopRecur(n) {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    const stack = [];\n    let res = 0;\n    // 내려가기: 재귀 호출\n    for (let i = n; i &gt; 0; i--) {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.push(i);\n    }\n    // 돌아오기: 결과 반환\n    while (stack.length) {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* 반복으로 재귀 구현 */\nfunction forLoopRecur(n: number): number {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다 \n    const stack: number[] = [];\n    let res: number = 0;\n    // 내려가기: 재귀 호출\n    for (let i = n; i &gt; 0; i--) {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.push(i);\n    }\n    // 돌아오기: 결과 반환\n    while (stack.length) {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* 반복으로 재귀 구현 */\nint forLoopRecur(int n) {\n  // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n  List&lt;int&gt; stack = [];\n  int res = 0;\n  // 내려가기: 재귀 호출\n  for (int i = n; i &gt; 0; i--) {\n    // 스택에 삽입하여 내려가는 과정을 모방한다\n    stack.add(i);\n  }\n  // 돌아오기: 결과 반환\n  while (!stack.isEmpty) {\n    // 스택에서 꺼내어 돌아오는 과정을 모방한다\n    res += stack.removeLast();\n  }\n  // res = 1+2+3+...+n\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* 반복으로 재귀 구현 */\nfn for_loop_recur(n: i32) -&gt; i32 {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    let mut stack = Vec::new();\n    let mut res = 0;\n    // 내려가기: 재귀 호출\n    for i in (1..=n).rev() {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.push(i);\n    }\n    // 돌아오기: 결과 반환\n    while !stack.is_empty() {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.pop().unwrap();\n    }\n    // res = 1+2+3+...+n\n    res\n}\n</code></pre> recursion.c<pre><code>/* 반복으로 재귀 구현 */\nint forLoopRecur(int n) {\n    int stack[1000]; // 큰 배열을 사용하여 스택을 구현한다\n    int top = -1;    // 스택의 맨 위 인덱스\n    int res = 0;\n    // 내려가기: 재귀 호출\n    for (int i = n; i &gt; 0; i--) {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack[1 + top++] = i;\n    }\n    // 돌아오기: 결과 반환\n    while (top &gt;= 0) {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack[top--];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* 반복으로 재귀 구현 */\nfun forLoopRecur(n: Int): Int {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    val stack = Stack&lt;Int&gt;()\n    var res = 0\n    // 내려가기: 재귀 호출\n    for (i in n downTo 0) {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.push(i)\n    }\n    // 돌아오기: 결과 반환\n    while (stack.isNotEmpty()) {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.pop()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.rb<pre><code>### 반복으로 재귀 구현 ###\ndef for_loop_recur(n)\n  # 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n  stack = []\n  res = 0\n\n  # 내려가기: 재귀 호출\n  for i in n.downto(0)\n    # 스택에 삽입하여 내려가는 과정을 모방한다\n    stack &lt;&lt; i\n  end\n  # 돌아오기: 결과 반환\n  while !stack.empty?\n    res += stack.pop\n  end\n\n  # res = 1+2+3+...+n\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>위 코드를 보면 재귀를 반복으로 바꾸자 코드가 더 복잡해졌다. 반복과 재귀는 많은 경우 서로 변환할 수 있지만, 다음 두 가지 이유로 항상 그럴 가치가 있는 것은 아니다.</p> <ul> <li>변환한 코드가 더 이해하기 어렵고 가독성이 떨어질 수 있다.</li> <li>복잡한 문제에서는 시스템 호출 스택의 동작을 재현하기가 매우 어려울 수 있다.</li> </ul> <p>결국 반복과 재귀 중 어떤 것을 선택할지는 문제의 구체적인 성격에 달려 있다. 실제 프로그래밍에서는 둘의 장단점을 비교하고 상황에 맞는 방법을 선택하는 것이 중요하다.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/performance_evaluation/","level":1,"title":"2.1   알고리즘 효율 평가","text":"<p>알고리즘을 설계할 때는 다음 두 단계의 목표를 차례로 추구한다.</p> <ol> <li>문제의 해법 찾기: 알고리즘은 정해진 입력 범위 안에서 문제의 올바른 해를 안정적으로 구해야 한다.</li> <li>최적의 해법 찾기: 같은 문제에도 여러 해법이 있을 수 있으므로, 가능한 한 효율적인 알고리즘을 찾고자 한다.</li> </ol> <p>즉, 문제를 해결할 수 있다는 전제하에 알고리즘 효율은 알고리즘의 우수성을 평가하는 주요 지표가 된다. 여기에는 다음 두 가지 측면이 있다.</p> <ul> <li>시간 효율: 알고리즘의 실행 시간이 얼마나 긴가.</li> <li>공간 효율: 알고리즘이 차지하는 메모리 공간이 얼마나 큰가.</li> </ul> <p>간단히 말해, 우리의 목표는 ‘빠르면서도 메모리를 적게 사용하는’ 자료 구조와 알고리즘을 설계하는 것이다. 알고리즘 효율을 효과적으로 평가하는 일도 매우 중요하다. 그래야 여러 알고리즘을 비교하고, 이를 바탕으로 알고리즘을 설계하고 최적화할 수 있기 때문이다.</p> <p>효율 평가 방법은 크게 실제 측정과 이론적 추정으로 나뉜다.</p>","path":["제 2 장   복잡도 분석","2.1   알고리즘 효율 평가"],"tags":[]},{"location":"chapter_computational_complexity/performance_evaluation/#211","level":2,"title":"2.1.1   실제 측정","text":"<p>같은 문제를 해결할 수 있는 알고리즘 <code>A</code>와 알고리즘 <code>B</code>가 있으며, 두 알고리즘의 효율을 비교해야 한다고 하자. 가장 직접적인 방법은 컴퓨터에서 두 알고리즘을 실행하고 실행 시간과 메모리 사용량을 관찰해 기록하는 것이다. 이러한 평가 방식은 실제 상황을 반영할 수 있지만 한계도 크다.</p> <p>한편으로는 측정 환경에 따른 영향을 배제하기 어렵다. 하드웨어 구성은 알고리즘 성능에 영향을 준다. 예를 들어 병렬성이 높은 알고리즘은 멀티코어 CPU에서 실행하기에 더 적합하고, 메모리 연산이 집중된 알고리즘은 고성능 메모리에서 더 좋은 성능을 보인다. 즉, 같은 알고리즘도 컴퓨터에 따라 측정 결과가 다를 수 있다. 따라서 다양한 컴퓨터에서 측정해 평균 효율을 구해야 하지만, 이는 현실적이지 않다.</p> <p>다른 한편으로는 충분한 측정을 수행하는 데 매우 많은 자원이 든다. 입력 데이터의 양이 달라지면 알고리즘의 효율도 달라진다. 예를 들어 입력 데이터가 적을 때는 알고리즘 <code>A</code>의 실행 시간이 알고리즘 <code>B</code>보다 짧지만, 입력 데이터가 많을 때는 결과가 정반대일 수 있다. 따라서 설득력 있는 결론을 얻으려면 다양한 크기의 입력 데이터를 측정해야 하며, 여기에는 많은 컴퓨팅 자원이 필요하다.</p>","path":["제 2 장   복잡도 분석","2.1   알고리즘 효율 평가"],"tags":[]},{"location":"chapter_computational_complexity/performance_evaluation/#212","level":2,"title":"2.1.2   이론적 추정","text":"<p>실제 측정에는 큰 한계가 있으므로, 계산만으로 알고리즘 효율을 평가하는 방법을 생각해 볼 수 있다. 이러한 추정 방법을 점근적 복잡도 분석(asymptotic complexity analysis), 줄여서 복잡도 분석이라고 한다.</p> <p>복잡도 분석은 알고리즘 실행에 필요한 시간 및 공간 자원과 입력 데이터 크기의 관계를 나타낸다. 입력 데이터 크기가 증가할 때 알고리즘 실행에 필요한 시간과 공간이 증가하는 추세를 설명한다. 이 정의는 다소 이해하기 어려우므로 세 가지 핵심으로 나누어 살펴보자.</p> <ul> <li>‘시간 및 공간 자원’은 각각 시간 복잡도(time complexity)와 공간 복잡도(space complexity)에 해당한다.</li> <li>‘입력 데이터 크기가 증가할 때’는 복잡도가 알고리즘의 실행 효율과 입력 데이터 크기 사이의 관계를 반영한다는 뜻이다.</li> <li>‘시간과 공간이 증가하는 추세’는 복잡도 분석이 실행 시간이나 사용 공간의 구체적인 값보다 시간 또는 공간이 얼마나 빠르게 증가하는지에 주목한다는 뜻이다.</li> </ul> <p>복잡도 분석은 실제 측정 방법의 단점을 극복한다. 구체적으로 다음과 같은 장점이 있다.</p> <ul> <li>실제로 코드를 실행하지 않아도 되므로 에너지를 절약하고 환경 부담을 줄인다.</li> <li>측정 환경에 의존하지 않아 분석 결과를 모든 실행 플랫폼에 적용할 수 있다.</li> <li>데이터 양에 따른 알고리즘 효율, 특히 데이터가 많을 때의 알고리즘 성능을 보여 준다.</li> </ul> <p>참고</p> <p>복잡도 개념이 아직 혼란스러워도 걱정할 필요 없다. 이후 절에서 자세히 소개한다.</p> <p>복잡도 분석은 알고리즘 효율을 평가하는 ‘자’와 같다. 이를 통해 알고리즘 실행에 필요한 시간과 공간 자원을 가늠하고 여러 알고리즘의 효율을 비교할 수 있다.</p> <p>복잡도는 수학적 개념이므로 초보자에게는 추상적으로 느껴지고 배우기도 다소 어려울 수 있다. 이런 관점에서는 복잡도 분석이 맨 처음 소개하기에 적절하지 않을 수도 있다. 그러나 어떤 자료 구조나 알고리즘의 특징을 논할 때는 실행 속도와 공간 사용량에 대한 분석을 피하기 어렵다.</p> <p>따라서 자료 구조와 알고리즘을 깊이 학습하기 전에, 복잡도 분석을 기초적으로 이해하여 간단한 알고리즘의 복잡도를 분석할 수 있도록 하는 것을 권한다.</p>","path":["제 2 장   복잡도 분석","2.1   알고리즘 효율 평가"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/","level":1,"title":"2.4   공간 복잡도","text":"<p>공간 복잡도(space complexity)는 데이터 양이 많아질 때 알고리즘이 차지하는 메모리 공간이 증가하는 추세를 측정한다. 시간 복잡도와 매우 비슷한 개념으로, ‘실행 시간’을 ‘차지하는 메모리 공간’으로 바꾸면 된다.</p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#241","level":2,"title":"2.4.1   알고리즘과 관련된 공간","text":"<p>알고리즘이 실행 중 사용하는 메모리 공간에는 주로 다음과 같은 종류가 있다.</p> <ul> <li>입력 공간: 알고리즘의 입력 데이터를 저장한다.</li> <li>임시 공간: 실행 중의 변수, 객체, 함수 문맥 등의 데이터를 저장한다.</li> <li>출력 공간: 알고리즘의 출력 데이터를 저장한다.</li> </ul> <p>일반적으로 공간 복잡도에 포함하는 범위는 ‘임시 공간’과 ‘출력 공간’의 합이다.</p> <p>임시 공간은 다시 세 부분으로 나눌 수 있다.</p> <ul> <li>임시 데이터: 실행 중의 각종 상수, 변수, 객체 등을 저장한다.</li> <li>스택 프레임 공간: 호출한 함수의 문맥 데이터를 저장한다. 시스템은 함수가 호출될 때마다 스택 맨 위에 스택 프레임을 만들며, 함수가 반환되면 이 공간을 해제한다.</li> <li>명령어 공간: 컴파일된 프로그램 명령어를 저장하며, 실제 계산에서는 보통 무시한다.</li> </ul> <p>프로그램의 공간 복잡도를 분석할 때는 아래 그림(2-15)과 같이 보통 임시 데이터, 스택 프레임 공간, 출력 데이터의 세 부분을 계산한다.</p> <p></p> <p> 그림 2-15   알고리즘이 사용하는 공간 </p> <p>관련 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class Node:\n    \"\"\"클래스\"\"\"\n    def __init__(self, x: int):\n        self.val: int = x              # 노드 값\n        self.next: Node | None = None  # 다음 노드를 가리키는 참조\n\ndef function() -&gt; int:\n    \"\"\"함수\"\"\"\n    # 몇 가지 연산 수행...\n    return 0\n\ndef algorithm(n) -&gt; int:  # 입력 데이터\n    A = 0                 # 임시 데이터(상수, 보통 대문자로 표기)\n    b = 0                 # 임시 데이터(변수)\n    node = Node(0)        # 임시 데이터(객체)\n    c = function()        # 스택 프레임 공간(함수 호출)\n    return A + b + c      # 출력 데이터\n</code></pre> <pre><code>/* 구조체 */\nstruct Node {\n    int val;\n    Node *next;\n    Node(int x) : val(x), next(nullptr) {}\n};\n\n/* 함수 */\nint func() {\n    // 몇 가지 연산 수행...\n    return 0;\n}\n\nint algorithm(int n) {        // 입력 데이터\n    const int a = 0;          // 임시 데이터(상수)\n    int b = 0;                // 임시 데이터(변수)\n    Node* node = new Node(0); // 임시 데이터(객체)\n    int c = func();           // 스택 프레임 공간(함수 호출)\n    return a + b + c;         // 출력 데이터\n}\n</code></pre> <pre><code>/* 클래스 */\nclass Node {\n    int val;\n    Node next;\n    Node(int x) { val = x; }\n}\n\n/* 함수 */\nint function() {\n    // 몇 가지 연산 수행...\n    return 0;\n}\n\nint algorithm(int n) {        // 입력 데이터\n    final int a = 0;          // 임시 데이터(상수)\n    int b = 0;                // 임시 데이터(변수)\n    Node node = new Node(0);  // 임시 데이터(객체)\n    int c = function();       // 스택 프레임 공간(함수 호출)\n    return a + b + c;         // 출력 데이터\n}\n</code></pre> <pre><code>/* 클래스 */\nclass Node(int x) {\n    int val = x;\n    Node next;\n}\n\n/* 함수 */\nint Function() {\n    // 몇 가지 연산 수행...\n    return 0;\n}\n\nint Algorithm(int n) {        // 입력 데이터\n    const int a = 0;          // 임시 데이터(상수)\n    int b = 0;                // 임시 데이터(변수)\n    Node node = new(0);       // 임시 데이터(객체)\n    int c = Function();       // 스택 프레임 공간(함수 호출)\n    return a + b + c;         // 출력 데이터\n}\n</code></pre> <pre><code>/* 구조체 */\ntype node struct {\n    val  int\n    next *node\n}\n\n/* node 구조체 생성  */\nfunc newNode(val int) *node {\n    return &amp;node{val: val}\n}\n\n/* 함수 */\nfunc function() int {\n    // 몇 가지 연산 수행...\n    return 0\n}\n\nfunc algorithm(n int) int { // 입력 데이터\n    const a = 0             // 임시 데이터(상수)\n    b := 0                  // 임시 데이터(변수)\n    newNode(0)              // 임시 데이터(객체)\n    c := function()         // 스택 프레임 공간(함수 호출)\n    return a + b + c        // 출력 데이터\n}\n</code></pre> <pre><code>/* 클래스 */\nclass Node {\n    var val: Int\n    var next: Node?\n\n    init(x: Int) {\n        val = x\n    }\n}\n\n/* 함수 */\nfunc function() -&gt; Int {\n    // 몇 가지 연산 수행...\n    return 0\n}\n\nfunc algorithm(n: Int) -&gt; Int { // 입력 데이터\n    let a = 0             // 임시 데이터(상수)\n    var b = 0             // 임시 데이터(변수)\n    let node = Node(x: 0) // 임시 데이터(객체)\n    let c = function()    // 스택 프레임 공간(함수 호출)\n    return a + b + c      // 출력 데이터\n}\n</code></pre> <pre><code>/* 클래스 */\nclass Node {\n    val;\n    next;\n    constructor(val) {\n        this.val = val === undefined ? 0 : val; // 노드 값\n        this.next = null;                       // 다음 노드를 가리키는 참조\n    }\n}\n\n/* 함수 */\nfunction constFunc() {\n    // 몇 가지 연산 수행\n    return 0;\n}\n\nfunction algorithm(n) {       // 입력 데이터\n    const a = 0;              // 임시 데이터(상수)\n    let b = 0;                // 임시 데이터(변수)\n    const node = new Node(0); // 임시 데이터(객체)\n    const c = constFunc();    // 스택 프레임 공간(함수 호출)\n    return a + b + c;         // 출력 데이터\n}\n</code></pre> <pre><code>/* 클래스 */\nclass Node {\n    val: number;\n    next: Node | null;\n    constructor(val?: number) {\n        this.val = val === undefined ? 0 : val; // 노드 값\n        this.next = null;                       // 다음 노드를 가리키는 참조\n    }\n}\n\n/* 함수 */\nfunction constFunc(): number {\n    // 몇 가지 연산 수행\n    return 0;\n}\n\nfunction algorithm(n: number): number { // 입력 데이터\n    const a = 0;                        // 임시 데이터(상수)\n    let b = 0;                          // 임시 데이터(변수)\n    const node = new Node(0);           // 임시 데이터(객체)\n    const c = constFunc();              // 스택 프레임 공간(함수 호출)\n    return a + b + c;                   // 출력 데이터\n}\n</code></pre> <pre><code>/* 클래스 */\nclass Node {\n  int val;\n  Node next;\n  Node(this.val, [this.next]);\n}\n\n/* 함수 */\nint function() {\n  // 몇 가지 연산 수행...\n  return 0;\n}\n\nint algorithm(int n) {  // 입력 데이터\n  const int a = 0;      // 임시 데이터(상수)\n  int b = 0;            // 임시 데이터(변수)\n  Node node = Node(0);  // 임시 데이터(객체)\n  int c = function();   // 스택 프레임 공간(함수 호출)\n  return a + b + c;     // 출력 데이터\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* 구조체 */\nstruct Node {\n    val: i32,\n    next: Option&lt;Rc&lt;RefCell&lt;Node&gt;&gt;&gt;,\n}\n\n/* Node 구조체 생성 */\nimpl Node {\n    fn new(val: i32) -&gt; Self {\n        Self { val: val, next: None }\n    }\n}\n\n/* 함수 */\nfn function() -&gt; i32 {      \n    // 몇 가지 연산 수행...\n    return 0;\n}\n\nfn algorithm(n: i32) -&gt; i32 {       // 입력 데이터\n    const a: i32 = 0;               // 임시 데이터(상수)\n    let mut b = 0;                  // 임시 데이터(변수)\n    let node = Node::new(0);        // 임시 데이터(객체)\n    let c = function();             // 스택 프레임 공간(함수 호출)\n    return a + b + c;               // 출력 데이터\n}\n</code></pre> <pre><code>/* 함수 */\nint func() {\n    // 몇 가지 연산 수행...\n    return 0;\n}\n\nint algorithm(int n) { // 입력 데이터\n    const int a = 0;   // 임시 데이터(상수)\n    int b = 0;         // 임시 데이터(변수)\n    int c = func();    // 스택 프레임 공간(함수 호출)\n    return a + b + c;  // 출력 데이터\n}\n</code></pre> <pre><code>/* 클래스 */\nclass Node(var _val: Int) {\n    var next: Node? = null\n}\n\n/* 함수 */\nfun function(): Int {\n    // 몇 가지 연산 수행...\n    return 0\n}\n\nfun algorithm(n: Int): Int { // 입력 데이터\n    val a = 0                // 임시 데이터(상수)\n    var b = 0                // 임시 데이터(변수)\n    val node = Node(0)       // 임시 데이터(객체)\n    val c = function()       // 스택 프레임 공간(함수 호출)\n    return a + b + c         // 출력 데이터\n}\n</code></pre> <pre><code>### 클래스 ###\nclass Node\n    attr_accessor :val      # 노드 값\n    attr_accessor :next     # 다음 노드를 가리키는 참조\n\n    def initialize(x)\n        @val = x\n    end\nend\n\n### 함수 ###\ndef function\n    # 몇 가지 연산 수행...\n    0\nend\n\n### 알고리즘 ###\ndef algorithm(n)        # 입력 데이터\n    a = 0               # 임시 데이터(상수)\n    b = 0               # 임시 데이터(변수)\n    node = Node.new(0)  # 임시 데이터(객체)\n    c = function        # 스택 프레임 공간(함수 호출)\n    a + b + c           # 출력 데이터\nend\n</code></pre>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#242","level":2,"title":"2.4.2   계산 방법","text":"<p>공간 복잡도를 구하는 방법은 시간 복잡도와 대체로 같다. 계산 대상을 ‘연산 횟수’에서 ‘사용 공간의 크기’로 바꾸면 된다.</p> <p>시간 복잡도와 달리 보통 최악 공간 복잡도에만 관심을 둔다. 메모리 공간은 반드시 충족해야 하는 요구 사항이므로, 어떤 입력 데이터에서도 충분한 메모리 공간이 확보되어 있어야 하기 때문이다.</p> <p>다음 코드를 보면 최악 공간 복잡도에서 ‘최악’은 두 가지 의미를 지닌다.</p> <ol> <li>가장 불리한 입력 데이터를 기준으로 한다: \\(n &lt; 10\\)이면 공간 복잡도는 \\(O(1)\\)이다. 그러나 \\(n &gt; 10\\)이면 초기화한 배열 <code>nums</code>가 \\(O(n)\\) 공간을 차지하므로 최악 공간 복잡도는 \\(O(n)\\)이다.</li> <li>알고리즘 실행 중 최대 메모리 사용량을 기준으로 한다: 예를 들어 프로그램이 마지막 줄을 실행하기 전에는 \\(O(1)\\) 공간을 차지하지만, 배열 <code>nums</code>를 초기화하면 \\(O(n)\\) 공간을 차지한다. 따라서 최악 공간 복잡도는 \\(O(n)\\)이다.</li> </ol> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def algorithm(n: int):\n    a = 0               # O(1)\n    b = [0] * 10000     # O(1)\n    if n &gt; 10:\n        nums = [0] * n  # O(n)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    vector&lt;int&gt; b(10000);    // O(1)\n    if (n &gt; 10)\n        vector&lt;int&gt; nums(n); // O(n)\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10)\n        int[] nums = new int[n]; // O(n)\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10) {\n        int[] nums = new int[n]; // O(n)\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 0                      // O(1)\n    b := make([]int, 10000)     // O(1)\n    var nums []int\n    if n &gt; 10 {\n        nums := make([]int, n)  // O(n)\n    }\n    fmt.Println(a, b, nums)\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    let a = 0 // O(1)\n    let b = Array(repeating: 0, count: 10000) // O(1)\n    if n &gt; 10 {\n        let nums = Array(repeating: 0, count: n) // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 0;                            // O(1)\n  List&lt;int&gt; b = List.filled(10000, 0);  // O(1)\n  if (n &gt; 10) {\n    List&lt;int&gt; nums = List.filled(n, 0); // O(n)\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let a = 0;                              // O(1)\n    let b = [0; 10000];                     // O(1)\n    if n &gt; 10 {\n        let nums = vec![0; n as usize];     // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    int b[10000];            // O(1)\n    if (n &gt; 10)\n        int nums[n] = {0};   // O(n)\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    val a = 0                    // O(1)\n    val b = IntArray(10000)      // O(1)\n    if (n &gt; 10) {\n        val nums = IntArray(n)   // O(n)\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 0                           # O(1)\n    b = Array.new(10000)            # O(1)\n    nums = Array.new(n) if n &gt; 10   # O(n)\nend\n</code></pre> <p>재귀 함수에서는 스택 프레임 공간을 계산하는 데 주의해야 한다. 다음 코드를 살펴보자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def function() -&gt; int:\n    # 몇 가지 연산 수행\n    return 0\n\ndef loop(n: int):\n    \"\"\"반복문의 공간 복잡도는 O(1)\"\"\"\n    for _ in range(n):\n        function()\n\ndef recur(n: int):\n    \"\"\"재귀의 공간 복잡도는 O(n)\"\"\"\n    if n == 1:\n        return\n    return recur(n - 1)\n</code></pre> <pre><code>int func() {\n    // 몇 가지 연산 수행\n    return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n    // 몇 가지 연산 수행\n    return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    recur(n - 1);\n}\n</code></pre> <pre><code>int Function() {\n    // 몇 가지 연산 수행\n    return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nvoid Loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nint Recur(int n) {\n    if (n == 1) return 1;\n    return Recur(n - 1);\n}\n</code></pre> <pre><code>func function() int {\n    // 몇 가지 연산 수행\n    return 0\n}\n\n/* 반복문의 공간 복잡도는 O(1) */\nfunc loop(n int) {\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n}\n\n/* 재귀의 공간 복잡도는 O(n) */\nfunc recur(n int) {\n    if n == 1 {\n        return\n    }\n    recur(n - 1)\n}\n</code></pre> <pre><code>@discardableResult\nfunc function() -&gt; Int {\n    // 몇 가지 연산 수행\n    return 0\n}\n\n/* 반복문의 공간 복잡도는 O(1) */\nfunc loop(n: Int) {\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n\n/* 재귀의 공간 복잡도는 O(n) */\nfunc recur(n: Int) {\n    if n == 1 {\n        return\n    }\n    recur(n: n - 1)\n}\n</code></pre> <pre><code>function constFunc() {\n    // 몇 가지 연산 수행\n    return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nfunction loop(n) {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nfunction recur(n) {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>function constFunc(): number {\n    // 몇 가지 연산 수행\n    return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nfunction loop(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nfunction recur(n: number): void {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n  // 몇 가지 연산 수행\n  return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nvoid loop(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nvoid recur(int n) {\n  if (n == 1) return;\n  recur(n - 1);\n}\n</code></pre> <pre><code>fn function() -&gt; i32 {\n    // 몇 가지 연산 수행\n    return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nfn loop(n: i32) {\n    for i in 0..n {\n        function();\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nfn recur(n: i32) {\n    if n == 1 {\n        return;\n    }\n    recur(n - 1);\n}\n</code></pre> <pre><code>int func() {\n    // 몇 가지 연산 수행\n    return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    recur(n - 1);\n}\n</code></pre> <pre><code>fun function(): Int {\n    // 몇 가지 연산 수행\n    return 0\n}\n/* 반복문의 공간 복잡도는 O(1) */\nfun loop(n: Int) {\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nfun recur(n: Int) {\n    if (n == 1) return\n    return recur(n - 1)\n}\n</code></pre> <pre><code>def function\n    # 몇 가지 연산 수행\n    0\nend\n\n### 반복문의 공간 복잡도는 O(1) ###\ndef loop(n)\n    (0...n).each { function }\nend\n\n### 재귀의 공간 복잡도는 O(n) ###\ndef recur(n)\n    return if n == 1\n    recur(n - 1)\nend\n</code></pre> <p>함수 <code>loop()</code>와 <code>recur()</code>의 시간 복잡도는 모두 \\(O(n)\\)이지만 공간 복잡도는 다르다.</p> <ul> <li>함수 <code>loop()</code>는 반복문에서 <code>function()</code>을 \\(n\\)번 호출한다. 매번 <code>function()</code>이 반환되어 스택 프레임 공간이 해제되므로, 공간 복잡도는 여전히 \\(O(1)\\)이다.</li> <li>재귀 함수 <code>recur()</code>의 실행 중에는 아직 반환되지 않은 <code>recur()</code>가 \\(n\\)개 동시에 존재하므로, \\(O(n)\\)의 스택 프레임 공간을 차지한다.</li> </ul>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#243","level":2,"title":"2.4.3   대표적인 유형","text":"<p>입력 데이터 크기를 \\(n\\)이라고 할 때, 아래 그림(2-16)은 대표적인 공간 복잡도 유형을 낮은 순서대로 보여 준다.</p> \\[ \\begin{aligned} &amp; O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \\newline &amp; \\text{상수} &lt; \\text{로그} &lt; \\text{선형} &lt; \\text{제곱} &lt; \\text{지수} \\end{aligned} \\] <p></p> <p> 그림 2-16   대표적인 공간 복잡도 유형 </p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#1-o1","level":3,"title":"1.   상수 공간 \\(O(1)\\)","text":"<p>상수 공간은 개수가 입력 데이터 크기 \\(n\\)과 무관한 상수, 변수, 객체에서 흔히 나타난다.</p> <p>반복문에서 변수를 초기화하거나 함수를 호출할 때 사용하는 메모리는 다음 반복에 들어가면 해제되므로 공간이 누적되지 않는다. 따라서 공간 복잡도는 여전히 \\(O(1)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def function() -&gt; int:\n    \"\"\"함수\"\"\"\n    # 몇 가지 연산을 수행한다\n    return 0\n\ndef constant(n: int):\n    \"\"\"상수 차수\"\"\"\n    # 상수, 변수, 객체는 O(1) 공간을 차지한다\n    a = 0\n    nums = [0] * 10000\n    node = ListNode(0)\n    # 반복문 안의 변수는 O(1) 공간을 차지한다\n    for _ in range(n):\n        c = 0\n    # 반복문 안의 함수는 O(1) 공간을 차지한다\n    for _ in range(n):\n        function()\n</code></pre> space_complexity.cpp<pre><code>/* 함수 */\nint func() {\n    // 몇 가지 연산을 수행한다\n    return 0;\n}\n\n/* 상수 차수 */\nvoid constant(int n) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    const int a = 0;\n    int b = 0;\n    vector&lt;int&gt; nums(10000);\n    ListNode node(0);\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* 함수 */\nint function() {\n    // 몇 가지 연산을 수행한다\n    return 0;\n}\n\n/* 상수 차수 */\nvoid constant(int n) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    final int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new ListNode(0);\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* 함수 */\nint Function() {\n    // 몇 가지 연산을 수행한다\n    return 0;\n}\n\n/* 상수 차수 */\nvoid Constant(int n) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new(0);\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* 함수 */\nfunc function() int {\n    // 몇 가지 연산을 수행한다...\n    return 0\n}\n\n/* 상수 차수 */\nfunc spaceConstant(n int) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    const a = 0\n    b := 0\n    nums := make([]int, 10000)\n    node := newNode(0)\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    var c int\n    for i := 0; i &lt; n; i++ {\n        c = 0\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n    b += 0\n    c += 0\n    nums[0] = 0\n    node.val = 0\n}\n</code></pre> space_complexity.swift<pre><code>/* 함수 */\n@discardableResult\nfunc function() -&gt; Int {\n    // 몇 가지 연산을 수행한다\n    return 0\n}\n\n/* 상수 차수 */\nfunc constant(n: Int) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    let a = 0\n    var b = 0\n    let nums = Array(repeating: 0, count: 10000)\n    let node = ListNode(x: 0)\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for _ in 0 ..&lt; n {\n        let c = 0\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n</code></pre> space_complexity.js<pre><code>/* 함수 */\nfunction constFunc() {\n    // 몇 가지 연산을 수행한다\n    return 0;\n}\n\n/* 상수 차수 */\nfunction constant(n) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* 함수 */\nfunction constFunc(): number {\n    // 몇 가지 연산을 수행한다\n    return 0;\n}\n\n/* 상수 차수 */\nfunction constant(n: number): void {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* 함수 */\nint function() {\n  // 몇 가지 연산을 수행한다\n  return 0;\n}\n\n/* 상수 차수 */\nvoid constant(int n) {\n  // 상수, 변수, 객체는 O(1) 공간을 차지한다\n  final int a = 0;\n  int b = 0;\n  List&lt;int&gt; nums = List.filled(10000, 0);\n  ListNode node = ListNode(0);\n  // 반복문 안의 변수는 O(1) 공간을 차지한다\n  for (var i = 0; i &lt; n; i++) {\n    int c = 0;\n  }\n  // 반복문 안의 함수는 O(1) 공간을 차지한다\n  for (var i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* 함수 */\nfn function() -&gt; i32 {\n    // 몇 가지 연산을 수행한다\n    return 0;\n}\n\n/* 상수 차수 */\n#[allow(unused)]\nfn constant(n: i32) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    const A: i32 = 0;\n    let b = 0;\n    let nums = vec![0; 10000];\n    let node = ListNode::new(0);\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for i in 0..n {\n        let c = 0;\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for i in 0..n {\n        function();\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* 함수 */\nint func() {\n    // 몇 가지 연산을 수행한다\n    return 0;\n}\n\n/* 상수 차수 */\nvoid constant(int n) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    const int a = 0;\n    int b = 0;\n    int nums[1000];\n    ListNode *node = newListNode(0);\n    free(node);\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* 함수 */\nfun function(): Int {\n    // 몇 가지 연산을 수행한다\n    return 0\n}\n\n/* 상수 차수 */\nfun constant(n: Int) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    val a = 0\n    var b = 0\n    val nums = Array(10000) { 0 }\n    val node = ListNode(0)\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for (i in 0..&lt;n) {\n        val c = 0\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### 함수 ###\ndef function\n  # 몇 가지 연산을 수행한다\n  0\nend\n\n### 상수 차수 ###\ndef constant(n)\n  # 상수, 변수, 객체는 O(1) 공간을 차지한다\n  a = 0\n  nums = [0] * 10000\n  node = ListNode.new\n\n  # 반복문 안의 변수는 O(1) 공간을 차지한다\n  (0...n).each { c = 0 }\n  # 반복문 안의 함수는 O(1) 공간을 차지한다\n  (0...n).each { function }\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#2-on","level":3,"title":"2.   선형 공간 \\(O(n)\\)","text":"<p>선형 공간은 원소 수가 \\(n\\)에 비례하는 배열, 연결 리스트, 스택, 큐 등에 흔히 나타난다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def linear(n: int):\n    \"\"\"선형 차수\"\"\"\n    # 길이가 n인 리스트는 O(n) 공간을 차지한다\n    nums = [0] * n\n    # 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    hmap = dict[int, str]()\n    for i in range(n):\n        hmap[i] = str(i)\n</code></pre> space_complexity.cpp<pre><code>/* 선형 차수 */\nvoid linear(int n) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    vector&lt;int&gt; nums(n);\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    vector&lt;ListNode&gt; nodes;\n    for (int i = 0; i &lt; n; i++) {\n        nodes.push_back(ListNode(i));\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    unordered_map&lt;int, string&gt; map;\n    for (int i = 0; i &lt; n; i++) {\n        map[i] = to_string(i);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* 선형 차수 */\nvoid linear(int n) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    int[] nums = new int[n];\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    List&lt;ListNode&gt; nodes = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        nodes.add(new ListNode(i));\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    Map&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        map.put(i, String.valueOf(i));\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* 선형 차수 */\nvoid Linear(int n) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    int[] nums = new int[n];\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    List&lt;ListNode&gt; nodes = [];\n    for (int i = 0; i &lt; n; i++) {\n        nodes.Add(new ListNode(i));\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    Dictionary&lt;int, string&gt; map = [];\n    for (int i = 0; i &lt; n; i++) {\n        map.Add(i, i.ToString());\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* 선형 차수 */\nfunc spaceLinear(n int) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    _ = make([]int, n)\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    var nodes []*node\n    for i := 0; i &lt; n; i++ {\n        nodes = append(nodes, newNode(i))\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    m := make(map[int]string, n)\n    for i := 0; i &lt; n; i++ {\n        m[i] = strconv.Itoa(i)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* 선형 차수 */\nfunc linear(n: Int) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    let nums = Array(repeating: 0, count: n)\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    let nodes = (0 ..&lt; n).map { ListNode(x: $0) }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    let map = Dictionary(uniqueKeysWithValues: (0 ..&lt; n).map { ($0, \"\\($0)\") })\n}\n</code></pre> space_complexity.js<pre><code>/* 선형 차수 */\nfunction linear(n) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    const nums = new Array(n);\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    const nodes = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* 선형 차수 */\nfunction linear(n: number): void {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    const nums = new Array(n);\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    const nodes: ListNode[] = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* 선형 차수 */\nvoid linear(int n) {\n  // 길이가 n인 배열은 O(n) 공간을 차지한다\n  List&lt;int&gt; nums = List.filled(n, 0);\n  // 길이가 n인 리스트는 O(n) 공간을 차지한다\n  List&lt;ListNode&gt; nodes = [];\n  for (var i = 0; i &lt; n; i++) {\n    nodes.add(ListNode(i));\n  }\n  // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n  Map&lt;int, String&gt; map = HashMap();\n  for (var i = 0; i &lt; n; i++) {\n    map.putIfAbsent(i, () =&gt; i.toString());\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* 선형 차수 */\n#[allow(unused)]\nfn linear(n: i32) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    let mut nums = vec![0; n as usize];\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    let mut nodes = Vec::new();\n    for i in 0..n {\n        nodes.push(ListNode::new(i))\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    let mut map = HashMap::new();\n    for i in 0..n {\n        map.insert(i, i.to_string());\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* 해시 테이블 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // uthash.h를 기반으로 구현한다\n} HashTable;\n\n/* 선형 차수 */\nvoid linear(int n) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    int *nums = malloc(sizeof(int) * n);\n    free(nums);\n\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    ListNode **nodes = malloc(sizeof(ListNode *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        nodes[i] = newListNode(i);\n    }\n    // 메모리 해제\n    for (int i = 0; i &lt; n; i++) {\n        free(nodes[i]);\n    }\n    free(nodes);\n\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    HashTable *h = NULL;\n    for (int i = 0; i &lt; n; i++) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = i;\n        tmp-&gt;val = i;\n        HASH_ADD_INT(h, key, tmp);\n    }\n\n    // 메모리 해제\n    HashTable *curr, *tmp;\n    HASH_ITER(hh, h, curr, tmp) {\n        HASH_DEL(h, curr);\n        free(curr);\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* 선형 차수 */\nfun linear(n: Int) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    val nums = Array(n) { 0 }\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    val nodes = mutableListOf&lt;ListNode&gt;()\n    for (i in 0..&lt;n) {\n        nodes.add(ListNode(i))\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    val map = mutableMapOf&lt;Int, String&gt;()\n    for (i in 0..&lt;n) {\n        map[i] = i.toString()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### 선형 차수 ###\ndef linear(n)\n  # 길이가 n인 리스트는 O(n) 공간을 차지한다\n  nums = Array.new(n, 0)\n\n  # 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n  hmap = {}\n  for i in 0...n\n    hmap[i] = i.to_s\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(2-17)과 같이 이 함수의 재귀 깊이는 \\(n\\)이다. 즉, 아직 반환되지 않은 <code>linear_recur()</code> 함수가 \\(n\\)개 동시에 존재하며 \\(O(n)\\) 크기의 스택 프레임 공간을 사용한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def linear_recur(n: int):\n    \"\"\"선형 차수(재귀 구현)\"\"\"\n    print(\"재귀 n =\", n)\n    if n == 1:\n        return\n    linear_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* 선형 차수(재귀 구현) */\nvoid linearRecur(int n) {\n    cout &lt;&lt; \"재귀 n = \" &lt;&lt; n &lt;&lt; endl;\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* 선형 차수(재귀 구현) */\nvoid linearRecur(int n) {\n    System.out.println(\"재귀 n = \" + n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* 선형 차수(재귀 구현) */\nvoid LinearRecur(int n) {\n    Console.WriteLine(\"재귀 n = \" + n);\n    if (n == 1) return;\n    LinearRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* 선형 차수(재귀 구현) */\nfunc spaceLinearRecur(n int) {\n    fmt.Println(\"재귀 n =\", n)\n    if n == 1 {\n        return\n    }\n    spaceLinearRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* 선형 차수(재귀 구현) */\nfunc linearRecur(n: Int) {\n    print(\"재귀 n = \\(n)\")\n    if n == 1 {\n        return\n    }\n    linearRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* 선형 차수(재귀 구현) */\nfunction linearRecur(n) {\n    console.log(`재귀 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* 선형 차수(재귀 구현) */\nfunction linearRecur(n: number): void {\n    console.log(`재귀 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* 선형 차수(재귀 구현) */\nvoid linearRecur(int n) {\n  print('재귀 n = $n');\n  if (n == 1) return;\n  linearRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* 선형 차수(재귀 구현) */\nfn linear_recur(n: i32) {\n    println!(\"재귀 n = {}\", n);\n    if n == 1 {\n        return;\n    };\n    linear_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* 선형 차수(재귀 구현) */\nvoid linearRecur(int n) {\n    printf(\"재귀 n = %d\\r\\n\", n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.kt<pre><code>/* 선형 차수(재귀 구현) */\nfun linearRecur(n: Int) {\n    println(\"재귀 n = $n\")\n    if (n == 1)\n        return\n    linearRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>### 선형 차수(재귀 구현)###\ndef linear_recur(n)\n  puts \"재귀 n = #{n}\"\n  return if n == 1\n  linear_recur(n - 1)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 2-17   재귀 함수에서 발생하는 선형 공간 복잡도 </p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#3-on2","level":3,"title":"3.   제곱 공간 \\(O(n^2)\\)","text":"<p>제곱 공간은 원소 수가 \\(n\\)의 제곱에 비례하는 행렬과 그래프에 흔히 나타난다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def quadratic(n: int):\n    \"\"\"제곱 차수\"\"\"\n    # 이차원 리스트는 O(n^2) 공간을 차지한다\n    num_matrix = [[0] * n for _ in range(n)]\n</code></pre> space_complexity.cpp<pre><code>/* 제곱 차수 */\nvoid quadratic(int n) {\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    vector&lt;vector&lt;int&gt;&gt; numMatrix;\n    for (int i = 0; i &lt; n; i++) {\n        vector&lt;int&gt; tmp;\n        for (int j = 0; j &lt; n; j++) {\n            tmp.push_back(0);\n        }\n        numMatrix.push_back(tmp);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* 제곱 차수 */\nvoid quadratic(int n) {\n    // 행렬은 O(n^2) 공간을 차지한다\n    int[][] numMatrix = new int[n][n];\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    List&lt;List&lt;Integer&gt;&gt; numList = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            tmp.add(0);\n        }\n        numList.add(tmp);\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* 제곱 차수 */\nvoid Quadratic(int n) {\n    // 행렬은 O(n^2) 공간을 차지한다\n    int[,] numMatrix = new int[n, n];\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    List&lt;List&lt;int&gt;&gt; numList = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;int&gt; tmp = [];\n        for (int j = 0; j &lt; n; j++) {\n            tmp.Add(0);\n        }\n        numList.Add(tmp);\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* 제곱 차수 */\nfunc spaceQuadratic(n int) {\n    // 행렬은 O(n^2) 공간을 차지한다\n    numMatrix := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        numMatrix[i] = make([]int, n)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* 제곱 차수 */\nfunc quadratic(n: Int) {\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    let numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n</code></pre> space_complexity.js<pre><code>/* 제곱 차수 */\nfunction quadratic(n) {\n    // 행렬은 O(n^2) 공간을 차지한다\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* 제곱 차수 */\nfunction quadratic(n: number): void {\n    // 행렬은 O(n^2) 공간을 차지한다\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* 제곱 차수 */\nvoid quadratic(int n) {\n  // 행렬은 O(n^2) 공간을 차지한다\n  List&lt;List&lt;int&gt;&gt; numMatrix = List.generate(n, (_) =&gt; List.filled(n, 0));\n  // 이차원 리스트는 O(n^2) 공간을 차지한다\n  List&lt;List&lt;int&gt;&gt; numList = [];\n  for (var i = 0; i &lt; n; i++) {\n    List&lt;int&gt; tmp = [];\n    for (int j = 0; j &lt; n; j++) {\n      tmp.add(0);\n    }\n    numList.add(tmp);\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* 제곱 차수 */\n#[allow(unused)]\nfn quadratic(n: i32) {\n    // 행렬은 O(n^2) 공간을 차지한다\n    let num_matrix = vec![vec![0; n as usize]; n as usize];\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    let mut num_list = Vec::new();\n    for i in 0..n {\n        let mut tmp = Vec::new();\n        for j in 0..n {\n            tmp.push(0);\n        }\n        num_list.push(tmp);\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* 제곱 차수 */\nvoid quadratic(int n) {\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    int **numMatrix = malloc(sizeof(int *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        int *tmp = malloc(sizeof(int) * n);\n        for (int j = 0; j &lt; n; j++) {\n            tmp[j] = 0;\n        }\n        numMatrix[i] = tmp;\n    }\n\n    // 메모리 해제\n    for (int i = 0; i &lt; n; i++) {\n        free(numMatrix[i]);\n    }\n    free(numMatrix);\n}\n</code></pre> space_complexity.kt<pre><code>/* 제곱 차수 */\nfun quadratic(n: Int) {\n    // 행렬은 O(n^2) 공간을 차지한다\n    val numMatrix = arrayOfNulls&lt;Array&lt;Int&gt;?&gt;(n)\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    val numList = mutableListOf&lt;MutableList&lt;Int&gt;&gt;()\n    for (i in 0..&lt;n) {\n        val tmp = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            tmp.add(0)\n        }\n        numList.add(tmp)\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### 제곱 차수 ###\ndef quadratic(n)\n  # 이차원 리스트는 O(n^2) 공간을 차지한다\n  Array.new(n) { Array.new(n, 0) }\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(2-18)과 같이 이 함수의 재귀 깊이는 \\(n\\)이고, 각 재귀 함수에서 길이가 각각 \\(n\\), \\(n-1\\), \\(\\dots\\), \\(2\\), \\(1\\)인 배열을 초기화한다. 평균 길이는 \\(n / 2\\)이므로 전체적으로 \\(O(n^2)\\) 공간을 차지한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def quadratic_recur(n: int) -&gt; int:\n    \"\"\"제곱 차수(재귀 구현)\"\"\"\n    if n &lt;= 0:\n        return 0\n    # 배열 nums의 길이는 n, n-1, ..., 2, 1이다\n    nums = [0] * n\n    return quadratic_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* 제곱 차수(재귀 구현) */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    vector&lt;int&gt; nums(n);\n    cout &lt;&lt; \"재귀 n = \" &lt;&lt; n &lt;&lt; \" 일 때 nums의 길이 = \" &lt;&lt; nums.size() &lt;&lt; endl;\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* 제곱 차수(재귀 구현) */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    // 배열 nums의 길이는 n, n-1, ..., 2, 1이다\n    int[] nums = new int[n];\n    System.out.println(\"재귀 n = \" + n + \" 일 때 nums의 길이 = \" + nums.length);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* 제곱 차수(재귀 구현) */\nint QuadraticRecur(int n) {\n    if (n &lt;= 0) return 0;\n    int[] nums = new int[n];\n    Console.WriteLine(\"재귀 n = \" + n + \" 일 때 nums의 길이 = \" + nums.Length);\n    return QuadraticRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* 제곱 차수(재귀 구현) */\nfunc spaceQuadraticRecur(n int) int {\n    if n &lt;= 0 {\n        return 0\n    }\n    nums := make([]int, n)\n    fmt.Printf(\"재귀 n = %d에서 nums의 길이 = %d \\n\", n, len(nums))\n    return spaceQuadraticRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* 제곱 차수(재귀 구현) */\n@discardableResult\nfunc quadraticRecur(n: Int) -&gt; Int {\n    if n &lt;= 0 {\n        return 0\n    }\n    // 배열 nums의 길이는 n, n-1, ..., 2, 1이다\n    let nums = Array(repeating: 0, count: n)\n    print(\"재귀(n = \\(n))의 nums 길이 = \\(nums.count)\")\n    return quadraticRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* 제곱 차수(재귀 구현) */\nfunction quadraticRecur(n) {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`재귀 n = ${n}일 때 nums의 길이 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* 제곱 차수(재귀 구현) */\nfunction quadraticRecur(n: number): number {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`재귀 n = ${n}일 때 nums의 길이 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* 제곱 차수(재귀 구현) */\nint quadraticRecur(int n) {\n  if (n &lt;= 0) return 0;\n  List&lt;int&gt; nums = List.filled(n, 0);\n  print('재귀(n = $n)의 nums 길이 = ${nums.length}');\n  return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* 제곱 차수(재귀 구현) */\nfn quadratic_recur(n: i32) -&gt; i32 {\n    if n &lt;= 0 {\n        return 0;\n    };\n    // 배열 nums의 길이는 n, n-1, ..., 2, 1이다\n    let nums = vec![0; n as usize];\n    println!(\"재귀 n = {}에서 nums의 길이 = {}\", n, nums.len());\n    return quadratic_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* 제곱 차수(재귀 구현) */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    int *nums = malloc(sizeof(int) * n);\n    printf(\"재귀 n = %d에서 nums의 길이 = %d\\r\\n\", n, n);\n    int res = quadraticRecur(n - 1);\n    free(nums);\n    return res;\n}\n</code></pre> space_complexity.kt<pre><code>/* 제곱 차수(재귀 구현) */\ntailrec fun quadraticRecur(n: Int): Int {\n    if (n &lt;= 0)\n        return 0\n    // 배열 nums의 길이는 n, n-1, ..., 2, 1이다\n    val nums = Array(n) { 0 }\n    println(\"재귀(n = $n)의 nums 길이 = ${nums.size}\")\n    return quadraticRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>### 제곱 차수(재귀 구현)###\ndef quadratic_recur(n)\n  return 0 unless n &gt; 0\n\n  # 배열 nums의 길이는 n, n-1, ..., 2, 1이다\n  nums = Array.new(n, 0)\n  quadratic_recur(n - 1)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 2-18   재귀 함수에서 발생하는 제곱 공간 복잡도 </p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#4-o2n","level":3,"title":"4.   지수 공간 \\(O(2^n)\\)","text":"<p>지수 공간은 이진 트리에 흔히 나타난다. 아래 그림(2-19)에서 층수가 \\(n\\)인 ‘포화 이진 트리’의 노드 수는 \\(2^n - 1\\)이므로 \\(O(2^n)\\) 공간을 차지한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def build_tree(n: int) -&gt; TreeNode | None:\n    \"\"\"지수 차수(포화 이진 트리 구성)\"\"\"\n    if n == 0:\n        return None\n    root = TreeNode(0)\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n    return root\n</code></pre> space_complexity.cpp<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return nullptr;\n    TreeNode *root = new TreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.java<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nTreeNode buildTree(int n) {\n    if (n == 0)\n        return null;\n    TreeNode root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.cs<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nTreeNode? BuildTree(int n) {\n    if (n == 0) return null;\n    TreeNode root = new(0) {\n        left = BuildTree(n - 1),\n        right = BuildTree(n - 1)\n    };\n    return root;\n}\n</code></pre> space_complexity.go<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nfunc buildTree(n int) *TreeNode {\n    if n == 0 {\n        return nil\n    }\n    root := NewTreeNode(0)\n    root.Left = buildTree(n - 1)\n    root.Right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.swift<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nfunc buildTree(n: Int) -&gt; TreeNode? {\n    if n == 0 {\n        return nil\n    }\n    let root = TreeNode(x: 0)\n    root.left = buildTree(n: n - 1)\n    root.right = buildTree(n: n - 1)\n    return root\n}\n</code></pre> space_complexity.js<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nfunction buildTree(n) {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.ts<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nfunction buildTree(n: number): TreeNode | null {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.dart<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nTreeNode? buildTree(int n) {\n  if (n == 0) return null;\n  TreeNode root = TreeNode(0);\n  root.left = buildTree(n - 1);\n  root.right = buildTree(n - 1);\n  return root;\n}\n</code></pre> space_complexity.rs<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nfn build_tree(n: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    if n == 0 {\n        return None;\n    };\n    let root = TreeNode::new(0);\n    root.borrow_mut().left = build_tree(n - 1);\n    root.borrow_mut().right = build_tree(n - 1);\n    return Some(root);\n}\n</code></pre> space_complexity.c<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return NULL;\n    TreeNode *root = newTreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.kt<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nfun buildTree(n: Int): TreeNode? {\n    if (n == 0)\n        return null\n    val root = TreeNode(0)\n    root.left = buildTree(n - 1)\n    root.right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.rb<pre><code>### 지수 차수(포화 이진 트리 구성)###\ndef build_tree(n)\n  return if n == 0\n\n  TreeNode.new.tap do |root|\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 2-19   포화 이진 트리에서 발생하는 지수 공간 복잡도 </p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#5-olog-n","level":3,"title":"5.   로그 공간 \\(O(\\log n)\\)","text":"<p>로그 공간은 분할 정복 알고리즘에 흔히 나타난다. 예를 들어 병합 정렬은 길이가 \\(n\\)인 배열을 입력받아 재귀 호출마다 중간 지점에서 반으로 나눈다. 높이가 \\(\\log n\\)인 재귀 트리가 만들어지므로 \\(O(\\log n)\\)의 스택 프레임 공간을 사용한다.</p> <p>또 다른 예로 수를 문자열로 변환하는 경우를 보자. 양의 정수 \\(n\\)을 입력받으면 자릿수는 \\(\\lfloor \\log_{10} n \\rfloor + 1\\)이며, 대응하는 문자열 길이도 \\(\\lfloor \\log_{10} n \\rfloor + 1\\)이다. 따라서 공간 복잡도는 \\(O(\\log_{10} n + 1) = O(\\log n)\\)이다.</p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#244","level":2,"title":"2.4.4   시간과 공간의 절충","text":"<p>이상적으로는 알고리즘의 시간 복잡도와 공간 복잡도가 모두 최적이기를 바란다. 그러나 실제로 두 복잡도를 동시에 최적화하기는 대개 매우 어렵다.</p> <p>시간 복잡도를 낮추려면 보통 공간 복잡도가 높아지는 대가를 치러야 하며, 그 반대도 마찬가지다. 메모리 공간을 더 사용하여 실행 속도를 높이는 방식을 ‘공간으로 시간을 절약하기’, 반대의 방식을 ‘시간으로 공간을 절약하기’라고 한다.</p> <p>어떤 방식을 선택할지는 어느 쪽을 더 중시하는지에 달려 있다. 대부분의 경우 시간이 공간보다 귀하므로 ‘공간으로 시간을 절약하기’가 더 널리 쓰인다. 물론 데이터 양이 매우 많다면 공간 복잡도를 관리하는 것 역시 중요하다.</p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/summary/","level":1,"title":"2.5   요약","text":"","path":["제 2 장   복잡도 분석","2.5   요약"],"tags":[]},{"location":"chapter_computational_complexity/summary/#1","level":3,"title":"1.   핵심 정리","text":"<p>알고리즘 효율 평가</p> <ul> <li>시간 효율과 공간 효율은 알고리즘의 우수성을 평가하는 두 가지 주요 지표다.</li> <li>실제 측정으로 알고리즘 효율을 평가할 수 있지만, 측정 환경의 영향을 없애기 어렵고 많은 컴퓨팅 자원이 든다.</li> <li>복잡도 분석은 실제 측정의 단점을 극복한다. 분석 결과는 모든 실행 플랫폼에 적용할 수 있으며, 데이터 크기에 따른 알고리즘 효율을 보여 준다.</li> </ul> <p>시간 복잡도</p> <ul> <li>시간 복잡도는 데이터 양이 증가할 때 알고리즘 실행 시간이 증가하는 추세를 측정하여 효율을 효과적으로 평가한다. 다만 입력 데이터가 적거나 시간 복잡도가 같은 경우처럼, 알고리즘 효율의 우열을 정확히 비교할 수 없는 상황도 있다.</li> <li>최악 시간 복잡도는 빅 \\(O\\) 표기법으로 나타내며 함수의 점근적 상한에 해당한다. \\(n\\)이 양의 무한대로 향할 때 연산 횟수 \\(T(n)\\)의 증가 차수를 반영한다.</li> <li>시간 복잡도를 구하는 과정은 두 단계다. 먼저 연산 횟수를 세고, 그다음 점근적 상한을 판단한다.</li> <li>대표적인 시간 복잡도를 낮은 순서대로 나열하면 \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n \\log n)\\), \\(O(n^2)\\), \\(O(2^n)\\), \\(O(n!)\\) 등이 있다.</li> <li>일부 알고리즘의 시간 복잡도는 고정되어 있지 않고 입력 데이터의 분포에 따라 달라진다. 시간 복잡도는 최악, 최선, 평균 시간 복잡도로 나뉜다. 최선의 경우에 도달하려면 대개 입력 데이터가 엄격한 조건을 만족해야 하므로 최선 시간 복잡도는 거의 쓰지 않는다.</li> <li>평균 시간 복잡도는 무작위 입력 데이터에서의 실행 효율을 반영하므로 실제 응용에서의 알고리즘 성능에 가장 가깝다. 이를 계산하려면 입력 데이터의 분포와 이를 종합한 기댓값을 구해야 한다.</li> </ul> <p>공간 복잡도</p> <ul> <li>공간 복잡도는 시간 복잡도와 비슷하게, 데이터 양이 증가할 때 알고리즘이 차지하는 메모리 공간이 증가하는 추세를 측정한다.</li> <li>알고리즘 실행에 쓰이는 메모리 공간은 입력 공간, 임시 공간, 출력 공간으로 나뉜다. 보통 입력 공간은 공간 복잡도 계산에 포함하지 않는다. 임시 공간은 임시 데이터, 스택 프레임 공간, 명령어 공간으로 나뉘며, 스택 프레임 공간은 보통 재귀 함수에서만 공간 복잡도에 영향을 준다.</li> <li>보통 최악 공간 복잡도에만 관심을 둔다. 즉, 가장 불리한 입력 데이터와 실행 시점을 기준으로 공간 복잡도를 계산한다.</li> <li>대표적인 공간 복잡도를 낮은 순서대로 나열하면 \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n^2)\\), \\(O(2^n)\\) 등이 있다.</li> </ul>","path":["제 2 장   복잡도 분석","2.5   요약"],"tags":[]},{"location":"chapter_computational_complexity/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 꼬리 재귀의 공간 복잡도는 \\(O(1)\\)인가?</p> <p>이론적으로 꼬리 재귀 함수의 공간 복잡도는 \\(O(1)\\)로 최적화할 수 있다. 그러나 Java, Python, C++, Go, C# 등 대다수 프로그래밍 언어는 자동 꼬리 재귀 최적화를 지원하지 않으므로, 보통 공간 복잡도를 \\(O(n)\\)으로 본다.</p> <p>Q: 함수와 메서드라는 두 용어는 어떻게 다른가?</p> <p>함수(function)는 독립적으로 실행할 수 있으며, 모든 인수가 명시적으로 전달된다. 메서드(method)는 객체와 연결되어 있고, 자신을 호출하는 객체에 암묵적으로 전달되며, 클래스 인스턴스에 포함된 데이터를 조작할 수 있다.</p> <p>몇 가지 흔히 쓰는 프로그래밍 언어를 예로 살펴보자.</p> <ul> <li>C는 절차적 프로그래밍 언어로 객체 지향 개념이 없으므로 함수만 있다. 다만 구조체(struct)를 만들어 객체 지향 프로그래밍을 흉내 낼 수 있으며, 구조체에 연결된 함수는 다른 언어의 메서드에 해당한다.</li> <li>Java와 C#은 객체 지향 프로그래밍 언어로, 코드 블록(메서드)은 보통 클래스의 일부다. 정적 메서드는 클래스에 묶여 있고 특정 인스턴스 변수에 접근할 수 없으므로 함수와 비슷하게 동작한다.</li> <li>C++와 Python은 절차적 프로그래밍(함수)과 객체 지향 프로그래밍(메서드)을 모두 지원한다.</li> </ul> <p>Q: ‘대표적인 공간 복잡도 유형’ 그림은 사용 공간의 절대적인 크기를 나타내는가?</p> <p>아니다. 이 그림은 공간 복잡도를 보여 주므로, 사용 공간의 절대적인 크기가 아니라 증가 추세를 나타낸다.</p> <p>\\(n = 8\\)이라고 놓으면 각 곡선의 값이 해당 함수와 일치하지 않을 수 있다. 이는 각 곡선에 상수항을 넣어 값의 범위를 보기 편한 크기로 줄였기 때문이다.</p> <p>실제로는 각 방법의 ‘상수항’에 해당하는 비용을 보통 알 수 없으므로, \\(n = 8\\)일 때는 복잡도만으로 최적의 해법을 고르기 어렵다. 반면 \\(n = 8^5\\)일 때는 증가 추세가 지배적인 영향을 주므로 훨씬 쉽게 선택할 수 있다.</p> <p>Q 실제 사용 상황에 따라 시간이나 공간을 희생하도록 알고리즘을 설계하기도 하는가?</p> <p>실제 응용에서는 대부분 공간을 더 사용하여 시간을 줄이는 쪽을 선택한다. 예를 들어 데이터베이스 인덱스에서는 보통 B+ 트리나 해시 인덱스를 구축한다. 메모리 공간을 많이 사용하는 대신 \\(O(\\log n)\\), 나아가 \\(O(1)\\)의 효율적인 조회가 가능하다.</p> <p>공간 자원이 귀한 상황에서는 시간을 더 들여 공간을 절약하기도 한다. 예를 들어 임베디드 개발에서는 장치 메모리가 귀하므로, 해시 테이블 대신 배열에서 순차 탐색을 수행할 수 있다. 메모리 사용량을 줄이는 대가로 탐색 속도가 느려진다.</p>","path":["제 2 장   복잡도 분석","2.5   요약"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/","level":1,"title":"2.3   시간 복잡도","text":"<p>실행 시간은 알고리즘 효율을 직관적이고 정확하게 보여 준다. 코드의 실행 시간을 정확히 추정하려면 어떻게 해야 할까?</p> <ol> <li>실행 플랫폼을 정한다. 하드웨어 구성, 프로그래밍 언어, 시스템 환경 등이 포함되며, 모두 코드의 실행 효율에 영향을 준다.</li> <li>각 연산에 필요한 실행 시간을 평가한다. 예를 들어 덧셈 연산 <code>+</code>는 1 ns, 곱셈 연산 <code>*</code>는 10 ns, 출력 연산 <code>print()</code>는 5 ns가 걸린다고 하자.</li> <li>코드에 있는 모든 연산을 센다. 그런 다음 각 연산의 실행 시간을 모두 더해 전체 실행 시간을 구한다.</li> </ol> <p>예를 들어 다음 코드에서 입력 데이터 크기는 \\(n\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># 어떤 실행 플랫폼에서\ndef algorithm(n: int):\n    a = 2      # 1 ns\n    a = a + 1  # 1 ns\n    a = a * 2  # 10 ns\n    # n번 반복\n    for _ in range(n):  # 1 ns\n        print(0)        # 5 ns\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        cout &lt;&lt; 0 &lt;&lt; endl;         // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        System.out.println(0);     // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nvoid Algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        Console.WriteLine(0);      // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nfunc algorithm(n int) {\n    a := 2     // 1 ns\n    a = a + 1  // 1 ns\n    a = a * 2  // 10 ns\n    // n번 반복\n    for i := 0; i &lt; n; i++ {  // 1 ns\n        fmt.Println(a)        // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nfunc algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // n번 반복\n    for _ in 0 ..&lt; n { // 1 ns\n        print(0) // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nfunction algorithm(n) {\n    var a = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // n번 반복\n    for(let i = 0; i &lt; n; i++) { // 1 ns\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nfunction algorithm(n: number): void {\n    var a: number = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // n번 반복\n    for(let i = 0; i &lt; n; i++) { // 1 ns\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nvoid algorithm(int n) {\n  int a = 2; // 1 ns\n  a = a + 1; // 1 ns\n  a = a * 2; // 10 ns\n  // n번 반복\n  for (int i = 0; i &lt; n; i++) { // 1 ns\n    print(0); // 5 ns\n  }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nfn algorithm(n: i32) {\n    let mut a = 2;      // 1 ns\n    a = a + 1;          // 1 ns\n    a = a * 2;          // 10 ns\n    // n번 반복\n    for _ in 0..n {     // 1 ns\n        println!(\"{}\", 0);  // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) {   // 1 ns\n        printf(\"%d\", 0);            // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nfun algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // n번 반복\n    for (i in 0..&lt;n) {  // 1 ns\n        println(0)      // 5 ns\n    }\n}\n</code></pre> <pre><code># 어떤 실행 플랫폼에서\ndef algorithm(n)\n    a = 2       # 1 ns\n    a = a + 1   # 1 ns\n    a = a * 2   # 10 ns\n    # n번 반복\n    (0...n).each do # 1 ns\n        puts 0      # 5 ns\n    end\nend\n</code></pre> <p>위 방법으로 구한 알고리즘의 실행 시간은 \\((6n + 12)\\) ns다.</p> \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] <p>그러나 실제로 알고리즘의 실행 시간을 이렇게 계산하는 것은 합리적이지도 현실적이지도 않다. 우선 알고리즘은 여러 플랫폼에서 실행되어야 하므로 추정 시간을 특정 실행 플랫폼에 묶고 싶지 않다. 또한 각 연산의 실행 시간을 알아내기 어려워 추정 과정이 매우 복잡해진다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#231","level":2,"title":"2.3.1   시간의 증가 추세 계산","text":"<p>시간 복잡도 분석이 계산하는 것은 알고리즘 실행 시간 자체가 아니라, 데이터 양이 많아질 때 알고리즘 실행 시간이 증가하는 추세다.</p> <p>‘시간의 증가 추세’는 추상적인 개념이므로 예를 통해 이해해 보자. 입력 데이터 크기가 \\(n\\)이고, 세 알고리즘 <code>A</code>, <code>B</code>, <code>C</code>가 주어졌다고 하자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># 알고리즘 A의 시간 복잡도: 상수 시간\ndef algorithm_A(n: int):\n    print(0)\n# 알고리즘 B의 시간 복잡도: 선형 시간\ndef algorithm_B(n: int):\n    for _ in range(n):\n        print(0)\n# 알고리즘 C의 시간 복잡도: 상수 시간\ndef algorithm_C(n: int):\n    for _ in range(1000000):\n        print(0)\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nvoid algorithm_A(int n) {\n    cout &lt;&lt; 0 &lt;&lt; endl;\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nvoid algorithm_A(int n) {\n    System.out.println(0);\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        System.out.println(0);\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        System.out.println(0);\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nvoid AlgorithmA(int n) {\n    Console.WriteLine(0);\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nvoid AlgorithmB(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Console.WriteLine(0);\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nvoid AlgorithmC(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        Console.WriteLine(0);\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nfunc algorithm_A(n int) {\n    fmt.Println(0)\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nfunc algorithm_B(n int) {\n    for i := 0; i &lt; n; i++ {\n        fmt.Println(0)\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nfunc algorithm_C(n int) {\n    for i := 0; i &lt; 1000000; i++ {\n        fmt.Println(0)\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nfunc algorithmA(n: Int) {\n    print(0)\n}\n\n// 알고리즘 B의 시간 복잡도: 선형 시간\nfunc algorithmB(n: Int) {\n    for _ in 0 ..&lt; n {\n        print(0)\n    }\n}\n\n// 알고리즘 C의 시간 복잡도: 상수 시간\nfunc algorithmC(n: Int) {\n    for _ in 0 ..&lt; 1_000_000 {\n        print(0)\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nfunction algorithm_A(n) {\n    console.log(0);\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nfunction algorithm_B(n) {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nfunction algorithm_C(n) {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nfunction algorithm_A(n: number): void {\n    console.log(0);\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nfunction algorithm_B(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nfunction algorithm_C(n: number): void {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nvoid algorithmA(int n) {\n  print(0);\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nvoid algorithmB(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    print(0);\n  }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nvoid algorithmC(int n) {\n  for (int i = 0; i &lt; 1000000; i++) {\n    print(0);\n  }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nfn algorithm_A(n: i32) {\n    println!(\"{}\", 0);\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nfn algorithm_B(n: i32) {\n    for _ in 0..n {\n        println!(\"{}\", 0);\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nfn algorithm_C(n: i32) {\n    for _ in 0..1000000 {\n        println!(\"{}\", 0);\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nvoid algorithm_A(int n) {\n    printf(\"%d\", 0);\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        printf(\"%d\", 0);\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        printf(\"%d\", 0);\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nfun algoritm_A(n: Int) {\n    println(0)\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nfun algorithm_B(n: Int) {\n    for (i in 0..&lt;n){\n        println(0)\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nfun algorithm_C(n: Int) {\n    for (i in 0..&lt;1000000) {\n        println(0)\n    }\n}\n</code></pre> <pre><code># 알고리즘 A의 시간 복잡도: 상수 시간\ndef algorithm_A(n)\n    puts 0\nend\n\n# 알고리즘 B의 시간 복잡도: 선형 시간\ndef algorithm_B(n)\n    (0...n).each { puts 0 }\nend\n\n# 알고리즘 C의 시간 복잡도: 상수 시간\ndef algorithm_C(n)\n    (0...1_000_000).each { puts 0 }\nend\n</code></pre> <p>아래 그림(2-7)은 이 세 알고리즘 함수의 시간 복잡도를 보여 준다.</p> <ul> <li>알고리즘 <code>A</code>에는 출력 연산이 \\(1\\)번뿐이므로, \\(n\\)이 커져도 실행 시간이 늘어나지 않는다. 이런 알고리즘의 시간 복잡도를 ‘상수 시간’이라고 한다.</li> <li>알고리즘 <code>B</code>의 출력 연산은 \\(n\\)번 반복되므로, \\(n\\)이 커짐에 따라 실행 시간이 선형으로 증가한다. 이런 알고리즘의 시간 복잡도를 ‘선형 시간’이라고 한다.</li> <li>알고리즘 <code>C</code>의 출력 연산은 \\(1000000\\)번 반복된다. 실행 시간은 길지만 입력 데이터 크기 \\(n\\)과는 무관하다. 따라서 <code>C</code>의 시간 복잡도는 <code>A</code>와 같은 ‘상수 시간’이다.</li> </ul> <p></p> <p> 그림 2-7   알고리즘 A, B, C의 시간 증가 추세 </p> <p>알고리즘의 실행 시간을 직접 계산하는 방식과 비교하면 시간 복잡도 분석에는 어떤 특징이 있을까?</p> <ul> <li>시간 복잡도는 알고리즘 효율을 효과적으로 평가할 수 있다. 예를 들어 알고리즘 <code>B</code>의 실행 시간은 선형으로 증가하므로 \\(n &gt; 1\\)이면 알고리즘 <code>A</code>보다 느리고, \\(n &gt; 1000000\\)이면 알고리즘 <code>C</code>보다 느리다. 실제로 입력 데이터 크기 \\(n\\)이 충분히 크면 ‘상수 시간’ 알고리즘은 반드시 ‘선형 시간’ 알고리즘보다 유리하다. 이것이 바로 시간의 증가 추세가 뜻하는 바다.</li> <li>시간 복잡도를 구하는 방법은 더 간단하다. 실행 플랫폼과 연산 종류는 알고리즘 실행 시간의 증가 추세와 무관하다. 따라서 시간 복잡도 분석에서는 모든 연산의 실행 시간을 같은 ‘단위 시간’으로 볼 수 있다. 그러면 ‘연산 실행 시간 계산’을 ‘연산 횟수 계산’으로 단순화할 수 있어 추정이 훨씬 쉬워진다.</li> <li>시간 복잡도에도 한계는 있다. 예를 들어 알고리즘 <code>A</code>와 <code>C</code>의 시간 복잡도는 같지만 실제 실행 시간에는 큰 차이가 있다. 마찬가지로 알고리즘 <code>B</code>의 시간 복잡도는 <code>C</code>보다 높지만, 입력 데이터 크기 \\(n\\)이 작으면 <code>B</code>가 <code>C</code>보다 확실히 유리하다. 이런 경우에는 시간 복잡도만으로 효율의 우열을 판단하기 어렵다. 물론 이러한 문제에도 불구하고 복잡도 분석은 여전히 알고리즘 효율을 평가하는 가장 효과적이고 널리 쓰이는 방법이다.</li> </ul>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#232","level":2,"title":"2.3.2   함수의 점근적 상한","text":"<p>입력 크기가 \\(n\\)인 다음 함수를 살펴보자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def algorithm(n: int):\n    a = 1      # +1\n    a = a + 1  # +1\n    a = a * 2  # +1\n    # n번 반복\n    for i in range(n):  # +1\n        print(0)        # +1\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) { // +1(매번 i ++ 실행)\n        cout &lt;&lt; 0 &lt;&lt; endl;    // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) { // +1(매번 i ++ 실행)\n        System.out.println(0);    // +1\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) {   // +1(매번 i ++ 실행)\n        Console.WriteLine(0);   // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1      // +1\n    a = a + 1   // +1\n    a = a * 2   // +1\n    // n번 반복\n    for i := 0; i &lt; n; i++ {   // +1\n        fmt.Println(a)         // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // n번 반복\n    for _ in 0 ..&lt; n { // +1\n        print(0) // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    var a = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // n번 반복\n    for(let i = 0; i &lt; n; i++){ // +1(매번 i ++ 실행)\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void{\n    var a: number = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // n번 반복\n    for(let i = 0; i &lt; n; i++){ // +1(매번 i ++ 실행)\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +1\n  a = a + 1; // +1\n  a = a * 2; // +1\n  // n번 반복\n  for (int i = 0; i &lt; n; i++) { // +1(매번 i ++ 실행)\n    print(0); // +1\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;   // +1\n    a = a + 1;      // +1\n    a = a * 2;      // +1\n\n    // n번 반복\n    for _ in 0..n { // +1(매번 i ++ 실행)\n        println!(\"{}\", 0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) {   // +1(매번 i ++ 실행)\n        printf(\"%d\", 0);            // +1\n    }\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // n번 반복\n    for (i in 0..&lt;n) { // +1(매번 i ++ 실행)\n        println(0) // +1\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 1       # +1\n    a = a + 1   # +1\n    a = a * 2   # +1\n    # n번 반복\n    (0...n).each do # +1\n        puts 0      # +1\n    end\nend\n</code></pre> <p>알고리즘의 연산 횟수를 입력 데이터 크기 \\(n\\)에 대한 함수 \\(T(n)\\)으로 나타내면, 위 함수의 연산 횟수는 다음과 같다.</p> \\[ T(n) = 3 + 2n \\] <p>\\(T(n)\\)은 일차 함수이므로 실행 시간이 선형으로 증가한다. 따라서 시간 복잡도는 선형 시간이다.</p> <p>선형 시간 복잡도를 \\(O(n)\\)으로 표기한다. 이 수학 기호를 빅 \\(O\\) 표기법(big-\\(O\\) notation)이라고 하며, 함수 \\(T(n)\\)의 점근적 상한(asymptotic upper bound)을 나타낸다.</p> <p>시간 복잡도 분석은 본질적으로 ‘연산 횟수 \\(T(n)\\)’의 점근적 상한을 계산하는 것으로, 명확한 수학적 정의가 있다.</p> <p>함수의 점근적 상한</p> <p>양의 실수 \\(c\\)와 실수 \\(n_0\\)가 존재하여 모든 \\(n &gt; n_0\\)에 대해 \\(T(n) \\leq c \\cdot f(n)\\)이면, \\(f(n)\\)은 \\(T(n)\\)의 점근적 상한을 제공한다고 하며 \\(T(n) = O(f(n))\\)으로 표기한다.</p> <p>아래 그림(2-8)과 같이 점근적 상한을 계산한다는 것은 함수 \\(f(n)\\)을 찾는 것이다. 이 함수는 \\(n\\)이 무한대로 향할 때 \\(T(n)\\)과 \\(f(n)\\)의 증가 차수가 같고 상수 계수 \\(c\\)만큼만 차이가 나도록 해야 한다.</p> <p></p> <p> 그림 2-8   함수의 점근적 상한 </p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#233","level":2,"title":"2.3.3   계산 방법","text":"<p>점근적 상한은 수학적인 느낌이 강하므로, 완전히 이해되지 않아도 걱정할 필요 없다. 우선 계산 방법을 익힌 뒤 반복해서 연습하면 그 수학적 의미도 점차 이해할 수 있다.</p> <p>정의에 따르면 \\(f(n)\\)을 정하면 시간 복잡도 \\(O(f(n))\\)을 얻는다. 그렇다면 점근적 상한 \\(f(n)\\)은 어떻게 정할까? 크게 두 단계로 나뉜다. 먼저 연산 횟수를 세고, 그다음 점근적 상한을 판단한다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#1-1","level":3,"title":"1.   1단계: 연산 횟수 세기","text":"<p>코드를 위에서 아래로 한 줄씩 살펴보며 계산하면 된다. 다만 앞서 본 \\(c \\cdot f(n)\\)의 상수 계수 \\(c\\)는 어떤 크기든 가능하므로, 연산 횟수 \\(T(n)\\)의 각종 계수와 상수항은 무시해도 된다. 이 원리를 바탕으로 계산을 단순화하는 다음 요령을 얻을 수 있다.</p> <ol> <li>\\(T(n)\\)의 상수를 무시한다. 상수는 \\(n\\)과 무관하므로 시간 복잡도에 영향을 주지 않는다.</li> <li>모든 계수를 생략한다. 예를 들어 \\(2n\\)번, \\(5n + 1\\)번 등의 반복은 모두 \\(n\\)번으로 단순화할 수 있다. \\(n\\) 앞의 계수는 시간 복잡도에 영향을 주지 않기 때문이다.</li> <li>중첩 반복문에는 곱셈을 사용한다. 전체 연산 횟수는 바깥쪽 반복문과 안쪽 반복문의 연산 횟수의 곱이며, 각 반복문에는 여전히 <code>1.</code>과 <code>2.</code>의 요령을 따로 적용할 수 있다.</li> </ol> <p>다음 함수에 위 요령을 적용하여 연산 횟수를 세어 보자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def algorithm(n: int):\n    a = 1      # +0(요령 1)\n    a = a + n  # +0(요령 1)\n    # +n(요령 2)\n    for i in range(5 * n + 1):\n        print(0)\n    # +n*n(요령 3)\n    for i in range(2 * n):\n        for j in range(n + 1):\n            print(0)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0(요령 1)\n    a = a + n;  // +0(요령 1)\n    // +n(요령 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n    // +n*n(요령 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            cout &lt;&lt; 0 &lt;&lt; endl;\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0(요령 1)\n    a = a + n;  // +0(요령 1)\n    // +n(요령 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        System.out.println(0);\n    }\n    // +n*n(요령 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            System.out.println(0);\n        }\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +0(요령 1)\n    a = a + n;  // +0(요령 1)\n    // +n(요령 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        Console.WriteLine(0);\n    }\n    // +n*n(요령 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            Console.WriteLine(0);\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1     // +0(요령 1)\n    a = a + n  // +0(요령 1)\n    // +n(요령 2)\n    for i := 0; i &lt; 5 * n + 1; i++ {\n        fmt.Println(0)\n    }\n    // +n*n(요령 3)\n    for i := 0; i &lt; 2 * n; i++ {\n        for j := 0; j &lt; n + 1; j++ {\n            fmt.Println(0)\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +0(요령 1)\n    a = a + n // +0(요령 1)\n    // +n(요령 2)\n    for _ in 0 ..&lt; (5 * n + 1) {\n        print(0)\n    }\n    // +n*n(요령 3)\n    for _ in 0 ..&lt; (2 * n) {\n        for _ in 0 ..&lt; (n + 1) {\n            print(0)\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    let a = 1;  // +0(요령 1)\n    a = a + n;  // +0(요령 1)\n    // +n(요령 2)\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n(요령 3)\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    let a = 1;  // +0(요령 1)\n    a = a + n;  // +0(요령 1)\n    // +n(요령 2)\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n(요령 3)\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +0(요령 1)\n  a = a + n; // +0(요령 1)\n  // +n(요령 2)\n  for (int i = 0; i &lt; 5 * n + 1; i++) {\n    print(0);\n  }\n  // +n*n(요령 3)\n  for (int i = 0; i &lt; 2 * n; i++) {\n    for (int j = 0; j &lt; n + 1; j++) {\n      print(0);\n    }\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;     // +0(요령 1)\n    a = a + n;        // +0(요령 1)\n\n    // +n(요령 2)\n    for i in 0..(5 * n + 1) {\n        println!(\"{}\", 0);\n    }\n\n    // +n*n(요령 3)\n    for i in 0..(2 * n) {\n        for j in 0..(n + 1) {\n            println!(\"{}\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0(요령 1)\n    a = a + n;  // +0(요령 1)\n    // +n(요령 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        printf(\"%d\", 0);\n    }\n    // +n*n(요령 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            printf(\"%d\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1   // +0(요령 1)\n    a = a + n   // +0(요령 1)\n    // +n(요령 2)\n    for (i in 0..&lt;5 * n + 1) {\n        println(0)\n    }\n    // +n*n(요령 3)\n    for (i in 0..&lt;2 * n) {\n        for (j in 0..&lt;n + 1) {\n            println(0)\n        }\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 1       # +0(요령 1)\n    a = a + n   # +0(요령 1)\n    # +n(요령 2)\n    (0...(5 * n + 1)).each do { puts 0 }\n    # +n*n(요령 3)\n    (0...(2 * n)).each do\n        (0...(n + 1)).each do { puts 0 }\n    end\nend\n</code></pre> <p>아래 식은 위 요령을 적용하기 전과 후의 계산 결과를 보여 준다. 두 결과에서 구한 시간 복잡도는 모두 \\(O(n^2)\\)이다.</p> \\[ \\begin{aligned} T(n) &amp; = 2n(n + 1) + (5n + 1) + 2 &amp; \\text{빠짐없이 계산 (-.-|||)} \\newline &amp; = 2n^2 + 7n + 3 \\newline T(n) &amp; = n^2 + n &amp; \\text{간단히 계산 (o.O)} \\end{aligned} \\]","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#2-2","level":3,"title":"2.   2단계: 점근적 상한 판단","text":"<p>시간 복잡도는 \\(T(n)\\)에서 가장 높은 차수의 항으로 결정된다. \\(n\\)이 무한대로 향하면 가장 높은 차수의 항이 지배적인 영향을 주므로 나머지 항의 영향은 무시할 수 있기 때문이다.</p> <p>아래 표(2-2)는 몇 가지 예를 보여 준다. 일부 값이 과장되어 있는 이유는 ‘계수로는 차수를 뒤집을 수 없다’는 결론을 강조하기 위해서다. \\(n\\)이 무한대로 향하면 이러한 상수는 중요하지 않게 된다.</p> <p> 표 2-2   연산 횟수에 따른 시간 복잡도 </p> 연산 횟수 \\(T(n)\\) 시간 복잡도 \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#234","level":2,"title":"2.3.4   대표적인 유형","text":"<p>입력 데이터 크기를 \\(n\\)이라고 할 때, 대표적인 시간 복잡도 유형은 아래 그림(2-9)과 같다(낮은 순서대로 나열).</p> \\[ \\begin{aligned} &amp; O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n \\log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!) \\newline &amp; \\text{상수} &lt; \\text{로그} &lt; \\text{선형} &lt; \\text{선형 로그} &lt; \\text{제곱} &lt; \\text{지수} &lt; \\text{팩토리얼} \\end{aligned} \\] <p></p> <p> 그림 2-9   대표적인 시간 복잡도 유형 </p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#1-o1","level":3,"title":"1.   상수 시간 \\(O(1)\\)","text":"<p>상수 시간의 연산 횟수는 입력 데이터 크기 \\(n\\)과 무관하며, \\(n\\)이 변해도 달라지지 않는다.</p> <p>다음 함수에서 연산 횟수 <code>size</code>는 매우 클 수 있지만 입력 데이터 크기 \\(n\\)과 무관하므로, 시간 복잡도는 여전히 \\(O(1)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def constant(n: int) -&gt; int:\n    \"\"\"상수 차수\"\"\"\n    count = 0\n    size = 100000\n    for _ in range(size):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 상수 차수 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 상수 차수 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 상수 차수 */\nint Constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 상수 차수 */\nfunc constant(n int) int {\n    count := 0\n    size := 100000\n    for i := 0; i &lt; size; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 상수 차수 */\nfunc constant(n: Int) -&gt; Int {\n    var count = 0\n    let size = 100_000\n    for _ in 0 ..&lt; size {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 상수 차수 */\nfunction constant(n) {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 상수 차수 */\nfunction constant(n: number): number {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 상수 차수 */\nint constant(int n) {\n  int count = 0;\n  int size = 100000;\n  for (var i = 0; i &lt; size; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 상수 차수 */\nfn constant(n: i32) -&gt; i32 {\n    _ = n;\n    let mut count = 0;\n    let size = 100_000;\n    for _ in 0..size {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 상수 차수 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 상수 차수 */\nfun constant(n: Int): Int {\n    var count = 0\n    val size = 100000\n    for (i in 0..&lt;size)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 상수 차수 ###\ndef constant(n)\n  count = 0\n  size = 100000\n\n  (0...size).each { count += 1 }\n\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#2-on","level":3,"title":"2.   선형 시간 \\(O(n)\\)","text":"<p>선형 시간의 연산 횟수는 입력 데이터 크기 \\(n\\)에 대해 선형으로 증가한다. 선형 시간은 보통 단일 반복문에서 나타난다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def linear(n: int) -&gt; int:\n    \"\"\"선형 차수\"\"\"\n    count = 0\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 선형 차수 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 선형 차수 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 선형 차수 */\nint Linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 선형 차수 */\nfunc linear(n int) int {\n    count := 0\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 선형 차수 */\nfunc linear(n: Int) -&gt; Int {\n    var count = 0\n    for _ in 0 ..&lt; n {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 선형 차수 */\nfunction linear(n) {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 선형 차수 */\nfunction linear(n: number): number {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 선형 차수 */\nint linear(int n) {\n  int count = 0;\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 선형 차수 */\nfn linear(n: i32) -&gt; i32 {\n    let mut count = 0;\n    for _ in 0..n {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 선형 차수 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 선형 차수 */\nfun linear(n: Int): Int {\n    var count = 0\n    for (i in 0..&lt;n)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 선형 차수 ###\ndef linear(n)\n  count = 0\n  (0...n).each { count += 1 }\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>배열 순회와 연결 리스트 순회 등의 연산은 시간 복잡도가 모두 \\(O(n)\\)이다. 여기서 \\(n\\)은 배열이나 연결 리스트의 길이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def array_traversal(nums: list[int]) -&gt; int:\n    \"\"\"선형 차수(배열 순회)\"\"\"\n    count = 0\n    # 반복 횟수는 배열 길이에 비례한다\n    for num in nums:\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 선형 차수(배열 순회) */\nint arrayTraversal(vector&lt;int&gt; &amp;nums) {\n    int count = 0;\n    // 반복 횟수는 배열 길이에 비례한다\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 선형 차수(배열 순회) */\nint arrayTraversal(int[] nums) {\n    int count = 0;\n    // 반복 횟수는 배열 길이에 비례한다\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 선형 차수(배열 순회) */\nint ArrayTraversal(int[] nums) {\n    int count = 0;\n    // 반복 횟수는 배열 길이에 비례한다\n    foreach (int num in nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 선형 차수(배열 순회) */\nfunc arrayTraversal(nums []int) int {\n    count := 0\n    // 반복 횟수는 배열 길이에 비례한다\n    for range nums {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 선형 차수(배열 순회) */\nfunc arrayTraversal(nums: [Int]) -&gt; Int {\n    var count = 0\n    // 반복 횟수는 배열 길이에 비례한다\n    for _ in nums {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 선형 차수(배열 순회) */\nfunction arrayTraversal(nums) {\n    let count = 0;\n    // 반복 횟수는 배열 길이에 비례한다\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 선형 차수(배열 순회) */\nfunction arrayTraversal(nums: number[]): number {\n    let count = 0;\n    // 반복 횟수는 배열 길이에 비례한다\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 선형 차수(배열 순회) */\nint arrayTraversal(List&lt;int&gt; nums) {\n  int count = 0;\n  // 반복 횟수는 배열 길이에 비례한다\n  for (var _num in nums) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 선형 차수(배열 순회) */\nfn array_traversal(nums: &amp;[i32]) -&gt; i32 {\n    let mut count = 0;\n    // 반복 횟수는 배열 길이에 비례한다\n    for _ in nums {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 선형 차수(배열 순회) */\nint arrayTraversal(int *nums, int n) {\n    int count = 0;\n    // 반복 횟수는 배열 길이에 비례한다\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 선형 차수(배열 순회) */\nfun arrayTraversal(nums: IntArray): Int {\n    var count = 0\n    // 반복 횟수는 배열 길이에 비례한다\n    for (num in nums) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 선형 차수(배열 순회)###\ndef array_traversal(nums)\n  count = 0\n\n  # 반복 횟수는 배열 길이에 비례한다\n  for num in nums\n    count += 1\n  end\n\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>입력 데이터 크기 \\(n\\)은 입력 데이터의 종류에 따라 구체적으로 정해야 한다는 점에 주의하자. 첫 번째 예에서는 변수 \\(n\\)이 입력 데이터 크기이고, 두 번째 예에서는 배열의 길이 \\(n\\)이 데이터 크기다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#3-on2","level":3,"title":"3.   제곱 시간 \\(O(n^2)\\)","text":"<p>제곱 시간의 연산 횟수는 입력 데이터 크기 \\(n\\)의 제곱에 비례하여 증가한다. 보통 중첩 반복문에서 나타나며, 바깥쪽과 안쪽 반복문의 시간 복잡도가 각각 \\(O(n)\\)이므로 전체 시간 복잡도는 \\(O(n^2)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def quadratic(n: int) -&gt; int:\n    \"\"\"제곱 차수\"\"\"\n    count = 0\n    # 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for i in range(n):\n        for j in range(n):\n            count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 제곱 차수 */\nint quadratic(int n) {\n    int count = 0;\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 제곱 차수 */\nint quadratic(int n) {\n    int count = 0;\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 제곱 차수 */\nint Quadratic(int n) {\n    int count = 0;\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 제곱 차수 */\nfunc quadratic(n int) int {\n    count := 0\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; n; j++ {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 제곱 차수 */\nfunc quadratic(n: Int) -&gt; Int {\n    var count = 0\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; n {\n            count += 1\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 제곱 차수 */\nfunction quadratic(n) {\n    let count = 0;\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 제곱 차수 */\nfunction quadratic(n: number): number {\n    let count = 0;\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 제곱 차수 */\nint quadratic(int n) {\n  int count = 0;\n  // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n  for (int i = 0; i &lt; n; i++) {\n    for (int j = 0; j &lt; n; j++) {\n      count++;\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 제곱 차수 */\nfn quadratic(n: i32) -&gt; i32 {\n    let mut count = 0;\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for _ in 0..n {\n        for _ in 0..n {\n            count += 1;\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 제곱 차수 */\nint quadratic(int n) {\n    int count = 0;\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 제곱 차수 */\nfun quadratic(n: Int): Int {\n    var count = 0\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;n) {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 제곱 차수 ###\ndef quadratic(n)\n  count = 0\n\n  # 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n  for i in 0...n\n    for j in 0...n\n      count += 1\n    end\n  end\n\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(2-10)은 상수 시간, 선형 시간, 제곱 시간의 세 가지 시간 복잡도를 비교한다.</p> <p></p> <p> 그림 2-10   상수 시간, 선형 시간, 제곱 시간의 시간 복잡도 </p> <p>버블 정렬을 예로 들면, 바깥쪽 반복문은 \\(n - 1\\)번 실행되고 안쪽 반복문은 \\(n-1\\), \\(n-2\\), \\(\\dots\\), \\(2\\), \\(1\\)번, 평균 \\(n / 2\\)번 실행된다. 따라서 시간 복잡도는 \\(O((n - 1) n / 2) = O(n^2)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def bubble_sort(nums: list[int]) -&gt; int:\n    \"\"\"제곱 차수(버블 정렬)\"\"\"\n    count = 0  # 카운터\n    # 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in range(len(nums) - 1, 0, -1):\n        # 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # nums[j]와 nums[j + 1]을 교환한다\n                tmp: int = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3  # 원소 교환에는 단위 연산 3회가 포함된다\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 제곱 차수(버블 정렬) */\nint bubbleSort(vector&lt;int&gt; &amp;nums) {\n    int count = 0; // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 제곱 차수(버블 정렬) */\nint bubbleSort(int[] nums) {\n    int count = 0; // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 제곱 차수(버블 정렬) */\nint BubbleSort(int[] nums) {\n    int count = 0;  // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                count += 3;  // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 제곱 차수(버블 정렬) */\nfunc bubbleSort(nums []int) int {\n    count := 0 // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                tmp := nums[j]\n                nums[j] = nums[j+1]\n                nums[j+1] = tmp\n                count += 3 // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 제곱 차수(버블 정렬) */\nfunc bubbleSort(nums: inout [Int]) -&gt; Int {\n    var count = 0 // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in nums.indices.dropFirst().reversed() {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3 // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 제곱 차수(버블 정렬) */\nfunction bubbleSort(nums) {\n    let count = 0; // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 제곱 차수(버블 정렬) */\nfunction bubbleSort(nums: number[]): number {\n    let count = 0; // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 제곱 차수(버블 정렬) */\nint bubbleSort(List&lt;int&gt; nums) {\n  int count = 0; // 카운터\n  // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n  for (var i = nums.length - 1; i &gt; 0; i--) {\n    // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n    for (var j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // nums[j]와 nums[j + 1]을 교환한다\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        count += 3; // 원소 교환에는 단위 연산 3회가 포함된다\n      }\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 제곱 차수(버블 정렬) */\nfn bubble_sort(nums: &amp;mut [i32]) -&gt; i32 {\n    let mut count = 0; // 카운터\n\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in (1..nums.len()).rev() {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 제곱 차수(버블 정렬) */\nint bubbleSort(int *nums, int n) {\n    int count = 0; // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = n - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 제곱 차수(버블 정렬) */\nfun bubbleSort(nums: IntArray): Int {\n    var count = 0 // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (i in nums.size - 1 downTo 1) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n                count += 3 // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 제곱 차수(버블 정렬)###\ndef bubble_sort(nums)\n  count = 0  # 카운터\n\n  # 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n  for i in (nums.length - 1).downto(0)\n    # 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # nums[j]와 nums[j + 1]을 교환한다\n        tmp = nums[j]\n        nums[j] = nums[j + 1]\n        nums[j + 1] = tmp\n        count += 3 # 원소 교환에는 단위 연산 3회가 포함된다\n      end\n    end\n  end\n\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#4-o2n","level":3,"title":"4.   지수 시간 \\(O(2^n)\\)","text":"<p>생물학의 ‘세포 분열’은 지수적 증가의 대표적인 예다. 처음에는 세포가 \\(1\\)개이고, 한 차례 분열하면 \\(2\\)개, 두 차례 분열하면 \\(4\\)개가 된다. 이처럼 \\(n\\)차례 분열하면 세포는 \\(2^n\\)개가 된다.</p> <p>아래 그림(2-11)과 코드는 세포 분열 과정을 재현하며, 시간 복잡도는 \\(O(2^n)\\)이다. 입력 \\(n\\)은 분열이 진행되는 라운드 수이고, 반환값 <code>count</code>는 전체 분열 횟수라는 점에 주의하자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def exponential(n: int) -&gt; int:\n    \"\"\"지수 차수(반복문 구현)\"\"\"\n    count = 0\n    base = 1\n    # 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for _ in range(n):\n        for _ in range(base):\n            count += 1\n        base *= 2\n    # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 지수 차수(반복문 구현) */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 지수 차수(반복문 구현) */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 지수 차수(반복문 구현) */\nint Exponential(int n) {\n    int count = 0, bas = 1;\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 지수 차수(반복문 구현)*/\nfunc exponential(n int) int {\n    count, base := 0, 1\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; base; j++ {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 지수 차수(반복문 구현) */\nfunc exponential(n: Int) -&gt; Int {\n    var count = 0\n    var base = 1\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; base {\n            count += 1\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 지수 차수(반복문 구현) */\nfunction exponential(n) {\n    let count = 0,\n        base = 1;\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 지수 차수(반복문 구현) */\nfunction exponential(n: number): number {\n    let count = 0,\n        base = 1;\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 지수 차수(반복문 구현) */\nint exponential(int n) {\n  int count = 0, base = 1;\n  // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n  for (var i = 0; i &lt; n; i++) {\n    for (var j = 0; j &lt; base; j++) {\n      count++;\n    }\n    base *= 2;\n  }\n  // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 지수 차수(반복문 구현) */\nfn exponential(n: i32) -&gt; i32 {\n    let mut count = 0;\n    let mut base = 1;\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for _ in 0..n {\n        for _ in 0..base {\n            count += 1\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 지수 차수(반복문 구현) */\nint exponential(int n) {\n    int count = 0;\n    int bas = 1;\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 지수 차수(반복문 구현) */\nfun exponential(n: Int): Int {\n    var count = 0\n    var base = 1\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;base) {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 지수 차수(반복문 구현)###\ndef exponential(n)\n  count, base = 0, 1\n\n  # 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n  (0...n).each do\n    (0...base).each { count += 1 }\n    base *= 2\n  end\n\n  # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 2-11   지수 시간 복잡도 </p> <p>실제 알고리즘에서 지수 시간은 재귀 함수에 자주 나타난다. 예를 들어 다음 코드는 재귀적으로 둘로 나뉘며, \\(n\\)번 나뉜 뒤 멈춘다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def exp_recur(n: int) -&gt; int:\n    \"\"\"지수 차수(재귀 구현)\"\"\"\n    if n == 1:\n        return 1\n    return exp_recur(n - 1) + exp_recur(n - 1) + 1\n</code></pre> time_complexity.cpp<pre><code>/* 지수 차수(재귀 구현) */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* 지수 차수(재귀 구현) */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* 지수 차수(재귀 구현) */\nint ExpRecur(int n) {\n    if (n == 1) return 1;\n    return ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* 지수 차수(재귀 구현)*/\nfunc expRecur(n int) int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n-1) + expRecur(n-1) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* 지수 차수(재귀 구현) */\nfunc expRecur(n: Int) -&gt; Int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* 지수 차수(재귀 구현) */\nfunction expRecur(n) {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* 지수 차수(재귀 구현) */\nfunction expRecur(n: number): number {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* 지수 차수(재귀 구현) */\nint expRecur(int n) {\n  if (n == 1) return 1;\n  return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* 지수 차수(재귀 구현) */\nfn exp_recur(n: i32) -&gt; i32 {\n    if n == 1 {\n        return 1;\n    }\n    exp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* 지수 차수(재귀 구현) */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* 지수 차수(재귀 구현) */\nfun expRecur(n: Int): Int {\n    if (n == 1) {\n        return 1\n    }\n    return expRecur(n - 1) + expRecur(n - 1) + 1\n}\n</code></pre> time_complexity.rb<pre><code>### 지수 차수(재귀 구현)###\ndef exp_recur(n)\n  return 1 if n == 1\n  exp_recur(n - 1) + exp_recur(n - 1) + 1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>지수 시간은 매우 빠르게 증가하며 완전 열거(완전 탐색, 백트래킹 등)에 흔히 나타난다. 데이터가 많은 문제에서는 지수 시간을 감당하기 어려우므로, 보통 동적 계획법이나 탐욕 알고리즘 등으로 해결해야 한다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#5-olog-n","level":3,"title":"5.   로그 시간 \\(O(\\log n)\\)","text":"<p>지수 시간과 반대로 로그 시간은 ‘매번 절반으로 줄어드는’ 상황을 나타낸다. 입력 데이터 크기가 \\(n\\)이고 매번 절반으로 줄어들므로, 반복 횟수는 \\(2^n\\)의 역함수인 \\(\\log_2 n\\)이다.</p> <p>아래 그림(2-12)과 코드는 ‘매번 절반으로 줄어드는’ 과정을 재현한다. 시간 복잡도는 \\(O(\\log_2 n)\\)이며, 줄여서 \\(O(\\log n)\\)으로 쓴다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def logarithmic(n: int) -&gt; int:\n    \"\"\"로그 차수(반복문 구현)\"\"\"\n    count = 0\n    while n &gt; 1:\n        n = n / 2\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 로그 차수(반복문 구현) */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 로그 차수(반복문 구현) */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 로그 차수(반복문 구현) */\nint Logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n /= 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 로그 차수(반복문 구현)*/\nfunc logarithmic(n int) int {\n    count := 0\n    for n &gt; 1 {\n        n = n / 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 로그 차수(반복문 구현) */\nfunc logarithmic(n: Int) -&gt; Int {\n    var count = 0\n    var n = n\n    while n &gt; 1 {\n        n = n / 2\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 로그 차수(반복문 구현) */\nfunction logarithmic(n) {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 로그 차수(반복문 구현) */\nfunction logarithmic(n: number): number {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 로그 차수(반복문 구현) */\nint logarithmic(int n) {\n  int count = 0;\n  while (n &gt; 1) {\n    n = n ~/ 2;\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 로그 차수(반복문 구현) */\nfn logarithmic(mut n: i32) -&gt; i32 {\n    let mut count = 0;\n    while n &gt; 1 {\n        n = n / 2;\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 로그 차수(반복문 구현) */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 로그 차수(반복문 구현) */\nfun logarithmic(n: Int): Int {\n    var n1 = n\n    var count = 0\n    while (n1 &gt; 1) {\n        n1 /= 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 로그 차수(반복문 구현)###\ndef logarithmic(n)\n  count = 0\n\n  while n &gt; 1\n    n /= 2\n    count += 1\n  end\n\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 2-12   로그 시간 복잡도 </p> <p>지수 시간처럼 로그 시간도 재귀 함수에 자주 나타난다. 다음 코드는 높이가 \\(\\log_2 n\\)인 재귀 트리를 만든다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def log_recur(n: int) -&gt; int:\n    \"\"\"로그 차수(재귀 구현)\"\"\"\n    if n &lt;= 1:\n        return 0\n    return log_recur(n / 2) + 1\n</code></pre> time_complexity.cpp<pre><code>/* 로그 차수(재귀 구현) */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* 로그 차수(재귀 구현) */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* 로그 차수(재귀 구현) */\nint LogRecur(int n) {\n    if (n &lt;= 1) return 0;\n    return LogRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* 로그 차수(재귀 구현)*/\nfunc logRecur(n int) int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n/2) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* 로그 차수(재귀 구현) */\nfunc logRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n: n / 2) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* 로그 차수(재귀 구현) */\nfunction logRecur(n) {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* 로그 차수(재귀 구현) */\nfunction logRecur(n: number): number {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* 로그 차수(재귀 구현) */\nint logRecur(int n) {\n  if (n &lt;= 1) return 0;\n  return logRecur(n ~/ 2) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* 로그 차수(재귀 구현) */\nfn log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 0;\n    }\n    log_recur(n / 2) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* 로그 차수(재귀 구현) */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* 로그 차수(재귀 구현) */\nfun logRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 0\n    return logRecur(n / 2) + 1\n}\n</code></pre> time_complexity.rb<pre><code>### 로그 차수(재귀 구현)###\ndef log_recur(n)\n  return 0 unless n &gt; 1\n  log_recur(n / 2) + 1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>로그 시간은 분할 정복 전략에 기반한 알고리즘에 자주 나타나며, ‘하나를 여러 개로 나누기’, ‘복잡한 것을 단순하게 만들기’라는 알고리즘 사고방식을 보여 준다. 증가 속도가 느려 상수 시간 다음으로 이상적인 시간 복잡도다.</p> <p>\\(O(\\log n)\\)의 밑은 무엇인가?</p> <p>정확히 말하면, ‘하나를 \\(m\\)개로 나누기’에 해당하는 시간 복잡도는 \\(O(\\log_m n)\\)이다. 로그의 밑 변환 공식을 적용하면 밑이 달라도 같은 시간 복잡도를 얻는다.</p> \\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\] <p>즉, 밑 \\(m\\)은 복잡도에 영향을 주지 않고 바꿀 수 있다. 따라서 보통 밑 \\(m\\)을 생략하고 로그 시간을 \\(O(\\log n)\\)으로 표기한다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#6-on-log-n","level":3,"title":"6.   선형 로그 시간 \\(O(n \\log n)\\)","text":"<p>선형 로그 시간은 중첩 반복문에 자주 나타나며, 두 반복문의 시간 복잡도는 각각 \\(O(\\log n)\\)과 \\(O(n)\\)이다. 관련 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def linear_log_recur(n: int) -&gt; int:\n    \"\"\"선형 로그 차수\"\"\"\n    if n &lt;= 1:\n        return 1\n    # 둘로 나누어 부분 문제의 크기를 절반으로 줄인다\n    count = linear_log_recur(n // 2) + linear_log_recur(n // 2)\n    # 현재 부분 문제에는 n회의 연산이 포함된다\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 선형 로그 차수 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 선형 로그 차수 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 선형 로그 차수 */\nint LinearLogRecur(int n) {\n    if (n &lt;= 1) return 1;\n    int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 선형 로그 차수 */\nfunc linearLogRecur(n int) int {\n    if n &lt;= 1 {\n        return 1\n    }\n    count := linearLogRecur(n/2) + linearLogRecur(n/2)\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 선형 로그 차수 */\nfunc linearLogRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 1\n    }\n    var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\n    for _ in stride(from: 0, to: n, by: 1) {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 선형 로그 차수 */\nfunction linearLogRecur(n) {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 선형 로그 차수 */\nfunction linearLogRecur(n: number): number {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 선형 로그 차수 */\nint linearLogRecur(int n) {\n  if (n &lt;= 1) return 1;\n  int count = linearLogRecur(n ~/ 2) + linearLogRecur(n ~/ 2);\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 선형 로그 차수 */\nfn linear_log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 1;\n    }\n    let mut count = linear_log_recur(n / 2) + linear_log_recur(n / 2);\n    for _ in 0..n {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> time_complexity.c<pre><code>/* 선형 로그 차수 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 선형 로그 차수 */\nfun linearLogRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 1\n    var count = linearLogRecur(n / 2) + linearLogRecur(n / 2)\n    for (i in 0..&lt;n) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 선형 로그 차수 ###\ndef linear_log_recur(n)\n  return 1 unless n &gt; 1\n\n  count = linear_log_recur(n / 2) + linear_log_recur(n / 2)\n  (0...n).each { count += 1 }\n\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(2-13)은 선형 로그 시간이 만들어지는 방식을 보여 준다. 이진 트리의 각 층에서 수행하는 전체 연산 횟수는 \\(n\\)이고, 층수는 \\(\\log_2 n + 1\\)이므로 시간 복잡도는 \\(O(n \\log n)\\)이다.</p> <p></p> <p> 그림 2-13   선형 로그 시간 복잡도 </p> <p>퀵 정렬, 병합 정렬, 힙 정렬 등 널리 쓰이는 정렬 알고리즘의 시간 복잡도는 보통 \\(O(n \\log n)\\)이다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#7-on","level":3,"title":"7.   팩토리얼 시간 \\(O(n!)\\)","text":"<p>팩토리얼 시간은 수학의 ‘모든 순열 구하기’ 문제에 해당한다. 서로 다른 원소 \\(n\\)개가 주어졌을 때 가능한 모든 순열을 구하면, 그 개수는 다음과 같다.</p> \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\] <p>팩토리얼은 보통 재귀로 구현한다. 아래 그림(2-14)과 코드에서 첫 번째 층은 \\(n\\)개로, 두 번째 층은 \\(n - 1\\)개로 분기하며, 이를 반복하다가 \\(n\\)번째 층에 도달하면 분기를 멈춘다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def factorial_recur(n: int) -&gt; int:\n    \"\"\"팩토리얼 차수(재귀 구현)\"\"\"\n    if n == 0:\n        return 1\n    count = 0\n    # 하나에서 n개로 분열한다\n    for _ in range(n):\n        count += factorial_recur(n - 1)\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 팩토리얼 차수(재귀 구현) */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // 하나에서 n개로 분열한다\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 팩토리얼 차수(재귀 구현) */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // 하나에서 n개로 분열한다\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 팩토리얼 차수(재귀 구현) */\nint FactorialRecur(int n) {\n    if (n == 0) return 1;\n    int count = 0;\n    // 하나에서 n개로 분열한다\n    for (int i = 0; i &lt; n; i++) {\n        count += FactorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 팩토리얼 차수(재귀 구현) */\nfunc factorialRecur(n int) int {\n    if n == 0 {\n        return 1\n    }\n    count := 0\n    // 하나에서 n개로 분열한다\n    for i := 0; i &lt; n; i++ {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 팩토리얼 차수(재귀 구현) */\nfunc factorialRecur(n: Int) -&gt; Int {\n    if n == 0 {\n        return 1\n    }\n    var count = 0\n    // 하나에서 n개로 분열한다\n    for _ in 0 ..&lt; n {\n        count += factorialRecur(n: n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 팩토리얼 차수(재귀 구현) */\nfunction factorialRecur(n) {\n    if (n === 0) return 1;\n    let count = 0;\n    // 하나에서 n개로 분열한다\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 팩토리얼 차수(재귀 구현) */\nfunction factorialRecur(n: number): number {\n    if (n === 0) return 1;\n    let count = 0;\n    // 하나에서 n개로 분열한다\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 팩토리얼 차수(재귀 구현) */\nint factorialRecur(int n) {\n  if (n == 0) return 1;\n  int count = 0;\n  // 하나에서 n개로 분열한다\n  for (var i = 0; i &lt; n; i++) {\n    count += factorialRecur(n - 1);\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 팩토리얼 차수(재귀 구현) */\nfn factorial_recur(n: i32) -&gt; i32 {\n    if n == 0 {\n        return 1;\n    }\n    let mut count = 0;\n    // 하나에서 n개로 분열한다\n    for _ in 0..n {\n        count += factorial_recur(n - 1);\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 팩토리얼 차수(재귀 구현) */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 팩토리얼 차수(재귀 구현) */\nfun factorialRecur(n: Int): Int {\n    if (n == 0)\n        return 1\n    var count = 0\n    // 하나에서 n개로 분열한다\n    for (i in 0..&lt;n) {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 팩토리얼 차수(재귀 구현)###\ndef factorial_recur(n)\n  return 1 if n == 0\n\n  count = 0\n  # 하나에서 n개로 분열한다\n  (0...n).each { count += factorial_recur(n - 1) }\n\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 2-14   팩토리얼 시간 복잡도 </p> <p>\\(n \\geq 4\\)이면 항상 \\(n! &gt; 2^n\\)이므로 팩토리얼 시간은 지수 시간보다 더 빠르게 증가한다. 따라서 \\(n\\)이 클 때는 이 역시 감당하기 어렵다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#235","level":2,"title":"2.3.5   최악, 최선, 평균 시간 복잡도","text":"<p>알고리즘의 시간 효율은 고정되어 있지 않고 입력 데이터의 분포에 따라 달라지는 경우가 많다. 길이가 \\(n\\)인 배열 <code>nums</code>를 입력받는다고 하자. <code>nums</code>에는 \\(1\\)부터 \\(n\\)까지의 수가 한 번씩 들어 있으며, 원소 순서는 무작위로 섞여 있다. 목표는 원소 \\(1\\)의 인덱스를 반환하는 것이다. 여기서 다음과 같은 결론을 얻는다.</p> <ul> <li><code>nums = [?, ?, ..., 1]</code>, 즉 마지막 원소가 \\(1\\)이면 배열 전체를 순회해야 하므로 최악 시간 복잡도 \\(O(n)\\)에 도달한다.</li> <li><code>nums = [1, ?, ?, ...]</code>, 즉 첫 번째 원소가 \\(1\\)이면 배열 길이에 관계없이 더 순회할 필요가 없으므로 최선 시간 복잡도 \\(\\Omega(1)\\)에 도달한다.</li> </ul> <p>‘최악 시간 복잡도’는 함수의 점근적 상한에 해당하며 빅 \\(O\\) 표기법으로 나타낸다. 이에 대응하여 ‘최선 시간 복잡도’는 함수의 점근적 하한에 해당하며 \\(\\Omega\\) 표기법으로 나타낸다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby worst_best_time_complexity.py<pre><code>def random_numbers(n: int) -&gt; list[int]:\n    \"\"\"원소가 1, 2, ..., n이며 순서가 섞인 배열 생성\"\"\"\n    # 배열 nums =: 1, 2, 3, ..., n을 생성한다\n    nums = [i for i in range(1, n + 1)]\n    # 배열 원소를 무작위로 섞는다\n    random.shuffle(nums)\n    return nums\n\ndef find_one(nums: list[int]) -&gt; int:\n    \"\"\"배열 nums에서 숫자 1이 있는 인덱스 탐색\"\"\"\n    for i in range(len(nums)):\n        # 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        # 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if nums[i] == 1:\n            return i\n    return -1\n</code></pre> worst_best_time_complexity.cpp<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nvector&lt;int&gt; randomNumbers(int n) {\n    vector&lt;int&gt; nums(n);\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // 시스템 시간으로 난수 시드를 생성한다\n    unsigned seed = chrono::system_clock::now().time_since_epoch().count();\n    // 배열 원소를 무작위로 섞는다\n    shuffle(nums.begin(), nums.end(), default_random_engine(seed));\n    return nums;\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nint findOne(vector&lt;int&gt; &amp;nums) {\n    for (int i = 0; i &lt; nums.size(); i++) {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.java<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nint[] randomNumbers(int n) {\n    Integer[] nums = new Integer[n];\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // 배열 원소를 무작위로 섞는다\n    Collections.shuffle(Arrays.asList(nums));\n    // Integer[] -&gt; int[]\n    int[] res = new int[n];\n    for (int i = 0; i &lt; n; i++) {\n        res[i] = nums[i];\n    }\n    return res;\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nint findOne(int[] nums) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.cs<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nint[] RandomNumbers(int n) {\n    int[] nums = new int[n];\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n\n    // 배열 원소를 무작위로 섞는다\n    for (int i = 0; i &lt; nums.Length; i++) {\n        int index = new Random().Next(i, nums.Length);\n        (nums[i], nums[index]) = (nums[index], nums[i]);\n    }\n    return nums;\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nint FindOne(int[] nums) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.go<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nfunc randomNumbers(n int) []int {\n    nums := make([]int, n)\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for i := 0; i &lt; n; i++ {\n        nums[i] = i + 1\n    }\n    // 배열 원소를 무작위로 섞는다\n    rand.Shuffle(len(nums), func(i, j int) {\n        nums[i], nums[j] = nums[j], nums[i]\n    })\n    return nums\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nfunc findOne(nums []int) int {\n    for i := 0; i &lt; len(nums); i++ {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.swift<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nfunc randomNumbers(n: Int) -&gt; [Int] {\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    var nums = Array(1 ... n)\n    // 배열 원소를 무작위로 섞는다\n    nums.shuffle()\n    return nums\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nfunc findOne(nums: [Int]) -&gt; Int {\n    for i in nums.indices {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.js<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nfunction randomNumbers(n) {\n    const nums = Array(n);\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // 배열 원소를 무작위로 섞는다\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nfunction findOne(nums) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.ts<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nfunction randomNumbers(n: number): number[] {\n    const nums = Array(n);\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // 배열 원소를 무작위로 섞는다\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nfunction findOne(nums: number[]): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.dart<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nList&lt;int&gt; randomNumbers(int n) {\n  final nums = List.filled(n, 0);\n  // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n  for (var i = 0; i &lt; n; i++) {\n    nums[i] = i + 1;\n  }\n  // 배열 원소를 무작위로 섞는다\n  nums.shuffle();\n\n  return nums;\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nint findOne(List&lt;int&gt; nums) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n    // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n    if (nums[i] == 1) return i;\n  }\n\n  return -1;\n}\n</code></pre> worst_best_time_complexity.rs<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nfn random_numbers(n: i32) -&gt; Vec&lt;i32&gt; {\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    let mut nums = (1..=n).collect::&lt;Vec&lt;i32&gt;&gt;();\n    // 배열 원소를 무작위로 섞는다\n    nums.shuffle(&amp;mut thread_rng());\n    nums\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nfn find_one(nums: &amp;[i32]) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if nums[i] == 1 {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> worst_best_time_complexity.c<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nint *randomNumbers(int n) {\n    // 힙 메모리를 할당한다(원소 개수가 n이고 원소 타입이 int인 1차원 가변 길이 배열을 생성한다)\n    int *nums = (int *)malloc(n * sizeof(int));\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // 배열 원소를 무작위로 섞는다\n    for (int i = n - 1; i &gt; 0; i--) {\n        int j = rand() % (i + 1);\n        int temp = nums[i];\n        nums[i] = nums[j];\n        nums[j] = temp;\n    }\n    return nums;\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nint findOne(int *nums, int n) {\n    for (int i = 0; i &lt; n; i++) {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.kt<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nfun randomNumbers(n: Int): Array&lt;Int?&gt; {\n    val nums = IntArray(n)\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for (i in 0..&lt;n) {\n        nums[i] = i + 1\n    }\n    // 배열 원소를 무작위로 섞는다\n    nums.shuffle()\n    val res = arrayOfNulls&lt;Int&gt;(n)\n    for (i in 0..&lt;n) {\n        res[i] = nums[i]\n    }\n    return res\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nfun findOne(nums: Array&lt;Int?&gt;): Int {\n    for (i in nums.indices) {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if (nums[i] == 1)\n            return i\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.rb<pre><code>### 원소가 1, 2, ..., n이며 순서가 섞인 배열 생성 ###\ndef random_numbers(n)\n  # 배열 nums =: 1, 2, 3, ..., n을 생성한다\n  nums = Array.new(n) { |i| i + 1 }\n  # 배열 원소를 무작위로 섞는다\n  nums.shuffle!\nend\n\n### 배열 nums에서 숫자 1이 있는 인덱스 탐색 ###\ndef find_one(nums)\n  for i in 0...nums.length\n    # 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n    # 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n    return i if nums[i] == 1\n  end\n\n  -1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>실제로 최선 시간 복잡도는 거의 사용하지 않는다. 보통 최선의 경우가 발생할 확률이 매우 낮아 오해를 줄 수 있기 때문이다. 최악 시간 복잡도는 효율의 안전한 기준을 제시하므로 더 실용적이며, 이를 통해 안심하고 알고리즘을 사용할 수 있다.</p> <p>앞의 예에서 최악 시간 복잡도와 최선 시간 복잡도는 ‘특수한 데이터 분포’에서만 나타난다. 이러한 경우는 발생 확률이 낮을 수 있어 실제 실행 효율을 제대로 반영하지 못한다. 반면 평균 시간 복잡도는 무작위 입력 데이터에서의 알고리즘 실행 효율을 나타내며, \\(\\Theta\\) 표기법으로 표시한다.</p> <p>일부 알고리즘은 무작위 데이터 분포에서의 평균 상황을 간단히 계산할 수 있다. 앞의 예에서는 입력 배열이 섞여 있으므로 원소 \\(1\\)이 어느 인덱스에 있을 확률도 같다. 따라서 평균 반복 횟수는 배열 길이의 절반인 \\(n / 2\\)이며, 평균 시간 복잡도는 \\(\\Theta(n / 2) = \\Theta(n)\\)이다.</p> <p>하지만 더 복잡한 알고리즘에서는 데이터 분포에 따른 전체 기댓값을 분석하기 어려워 평균 시간 복잡도를 구하기 힘든 경우가 많다. 이럴 때는 보통 최악 시간 복잡도를 알고리즘 효율의 평가 기준으로 삼는다.</p> <p>\\(\\Theta\\) 기호는 왜 보기 드문가?</p> <p>\\(O\\) 기호가 워낙 익숙해서인지 평균 시간 복잡도에도 이를 자주 사용한다. 그러나 엄밀하게는 올바른 표기 방식이 아니다. 이 책이나 다른 자료에서 ‘평균 시간 복잡도 \\(O(n)\\)’ 같은 표현을 만나면 \\(\\Theta(n)\\)으로 이해하면 된다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_data_structure/","level":1,"title":"제 3 장   자료 구조","text":"<p>이 장을 시작하며</p> <p>자료 구조는 튼튼하면서도 다양한 형태를 지닌 뼈대와 같다.</p> <p>데이터를 질서 있게 조직하는 설계도를 제공하며, 알고리즘은 그 토대 위에서 생동감을 얻는다.</p>","path":["제 3 장   자료 구조"],"tags":[]},{"location":"chapter_data_structure/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>3.1   자료 구조의 분류</li> <li>3.2   기본 자료형</li> <li>3.3   수의 인코딩 *</li> <li>3.4   문자 인코딩 *</li> <li>3.5   요약</li> <li>3.6   연습 문제</li> </ul>","path":["제 3 장   자료 구조"],"tags":[]},{"location":"chapter_data_structure/basic_data_types/","level":1,"title":"3.2   기본 자료형","text":"<p>컴퓨터의 데이터라고 하면 텍스트, 이미지, 동영상, 음성, 3D 모델 등 다양한 형태가 떠오른다. 이 데이터들은 조직된 형태는 서로 다르지만, 모두 여러 기본 자료형으로 이루어져 있다.</p> <p>기본 자료형은 CPU가 직접 연산할 수 있는 자료형으로, 알고리즘에서 직접 사용된다. 주로 다음과 같은 종류가 있다.</p> <ul> <li>정수형 <code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>.</li> <li>소수를 표현하는 부동 소수점형 <code>float</code>, <code>double</code>.</li> <li>여러 언어의 글자, 문장 부호, 나아가 이모지 등을 표현하는 문자형 <code>char</code>.</li> <li>‘예’와 ‘아니요’를 판단하는 불리언형 <code>bool</code>.</li> </ul> <p>기본 자료형은 컴퓨터에 이진수 형태로 저장된다. 이진수 한 자리가 \\(1\\)비트다. 현대 운영체제의 대다수에서 \\(1\\)바이트(byte)는 \\(8\\)비트(bit)로 이루어진다.</p> <p>기본 자료형의 값 범위는 차지하는 공간의 크기에 따라 결정된다. Java를 예로 살펴보자.</p> <ul> <li>정수형 <code>byte</code>는 \\(1\\)바이트 = \\(8\\)비트를 차지하며, \\(2^{8}\\)개의 수를 표현할 수 있다.</li> <li>정수형 <code>int</code>는 \\(4\\)바이트 = \\(32\\)비트를 차지하며, \\(2^{32}\\)개의 수를 표현할 수 있다.</li> </ul> <p>아래 표(3-1)는 Java의 여러 기본 자료형이 차지하는 공간, 값 범위, 기본값을 나열한다. 표를 외울 필요는 없으며, 대략 이해한 뒤 필요할 때 찾아보면 된다.</p> <p> 표 3-1   기본 자료형의 공간 사용량과 값 범위 </p> 자료형 기호 공간 사용량 최솟값 최댓값 기본값 정수 <code>byte</code> 1바이트 \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) \\(0\\) <code>short</code> 2바이트 \\(-2^{15}\\) \\(2^{15} - 1\\) \\(0\\) <code>int</code> 4바이트 \\(-2^{31}\\) \\(2^{31} - 1\\) \\(0\\) <code>long</code> 8바이트 \\(-2^{63}\\) \\(2^{63} - 1\\) \\(0\\) 부동 소수점 <code>float</code> 4바이트 \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0\\text{f}\\) <code>double</code> 8바이트 \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) \\(0.0\\) 문자 <code>char</code> 2바이트 \\(0\\) \\(2^{16} - 1\\) \\(0\\) 불리언 <code>bool</code> 1바이트 \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\) <p>위 표(3-1)는 Java의 기본 자료형을 기준으로 한다. 프로그래밍 언어마다 자료형을 정의하는 방식이 있으므로, 공간 사용량과 값 범위, 기본값은 다를 수 있다.</p> <ul> <li>Python의 정수형 <code>int</code>는 사용 가능한 메모리만 충분하면 크기에 제한이 없다. 부동 소수점형 <code>float</code>는 64비트 배정밀도다. <code>char</code> 자료형은 없으며, 문자 하나는 실제로 길이가 1인 문자열 <code>str</code>이다.</li> <li>C와 C++는 기본 자료형의 크기를 명확히 규정하지 않으므로 구현과 플랫폼에 따라 다르다. 위 표(3-1)는 Linux, macOS 등의 64비트 Unix 운영체제에서 사용하는 LP64 데이터 모델을 따른다.</li> <li>문자형 <code>char</code>의 크기는 C와 C++에서 1바이트이며, 대부분의 프로그래밍 언어에서는 사용하는 문자 인코딩 방식에 따라 정해진다. 자세한 내용은 ‘문자 인코딩’ 절에서 설명한다.</li> <li>불리언 값을 표현하는 데는 1비트(\\(0\\) 또는 \\(1\\))면 충분하지만, 메모리에서는 보통 1바이트로 저장한다. 현대 컴퓨터 CPU가 대개 1바이트를 주소 지정이 가능한 최소 메모리 단위로 사용하기 때문이다.</li> </ul> <p>그렇다면 기본 자료형과 자료 구조는 어떤 관계가 있을까? 자료 구조란 컴퓨터에서 데이터를 조직하고 저장하는 방식이다. 이 문장의 핵심은 ‘데이터’보다 ‘구조’에 있다.</p> <p>‘한 줄로 늘어선 수’를 표현하려면 자연스럽게 배열을 떠올린다. 배열의 선형 구조가 수 사이의 인접 관계와 순서 관계를 표현할 수 있기 때문이다. 그러나 저장할 내용이 정수 <code>int</code>인지, 소수 <code>float</code>인지, 문자 <code>char</code>인지는 ‘자료 구조’와 무관하다.</p> <p>즉, 기본 자료형은 데이터의 ‘내용 유형’을 제공하고, 자료 구조는 데이터의 ‘조직 방식’을 제공한다. 다음 코드는 같은 자료 구조인 배열을 사용하여 <code>int</code>, <code>float</code>, <code>char</code>, <code>bool</code> 등 서로 다른 기본 자료형을 저장하고 표현한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># 여러 기본 자료형으로 배열 초기화\nnumbers: list[int] = [0] * 5\ndecimals: list[float] = [0.0] * 5\n# Python의 문자는 실제로 길이가 1인 문자열이다\ncharacters: list[str] = ['0'] * 5\nbools: list[bool] = [False] * 5\n# Python의 리스트는 여러 기본 자료형과 객체 참조를 자유롭게 저장할 수 있다\ndata = [0, 0.0, 'a', False, ListNode(0)]\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool bools[5];\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] bools = new boolean[5];\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] bools = new bool[5];\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar bools = [5]bool{}\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nlet numbers = Array(repeating: 0, count: 5)\nlet decimals = Array(repeating: 0.0, count: 5)\nlet characters: [Character] = Array(repeating: \"a\", count: 5)\nlet bools = Array(repeating: false, count: 5)\n</code></pre> <pre><code>// JavaScript의 배열은 여러 기본 자료형과 객체를 자유롭게 저장할 수 있다\nconst array = [0, 0.0, 'a', false];\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst bools: boolean[] = [];\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nList&lt;int&gt; numbers = List.filled(5, 0);\nList&lt;double&gt; decimals = List.filled(5, 0.0);\nList&lt;String&gt; characters = List.filled(5, 'a');\nList&lt;bool&gt; bools = List.filled(5, false);\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nlet numbers: Vec&lt;i32&gt; = vec![0; 5];\nlet decimals: Vec&lt;f32&gt; = vec![0.0; 5];\nlet characters: Vec&lt;char&gt; = vec!['0'; 5];\nlet bools: Vec&lt;bool&gt; = vec![false; 5];\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool bools[10];\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nval numbers = IntArray(5)\nval decinals = FloatArray(5)\nval characters = CharArray(5)\nval bools = BooleanArray(5)\n</code></pre> <pre><code># Ruby의 리스트는 여러 기본 자료형과 객체 참조를 자유롭게 저장할 수 있다\ndata = [0, 0.0, 'a', false, ListNode(0)]\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 3 장   자료 구조","3.2   기본 자료형"],"tags":[]},{"location":"chapter_data_structure/character_encoding/","level":1,"title":"3.4   문자 인코딩 *","text":"<p>컴퓨터에서는 모든 데이터를 이진수 형태로 저장하며, 문자 <code>char</code>도 예외가 아니다. 문자를 표현하려면 각 문자와 이진수 사이의 일대일 대응 관계를 정한 ‘문자 집합’이 필요하다. 문자 집합이 있으면 컴퓨터는 대응표를 찾아 이진수를 문자로 바꿀 수 있다.</p>","path":["제 3 장   자료 구조","3.4   문자 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#341-ascii","level":2,"title":"3.4.1   ASCII 문자 집합","text":"<p>ASCII 코드는 가장 초기에 등장한 문자 집합으로, 정식 명칭은 American Standard Code for Information Interchange(미국 정보 교환 표준 부호)다. 7비트 이진수(한 바이트의 하위 7비트)로 문자 하나를 표현하여 최대 128개의 서로 다른 문자를 나타낸다. 아래 그림(3-6)과 같이 영문 대소문자, 숫자 0 ~ 9, 일부 문장 부호와 제어 문자(줄바꿈, 탭 등)를 포함한다.</p> <p></p> <p> 그림 3-6   ASCII 코드 </p> <p>그러나 ASCII 코드는 영어만 표현할 수 있다. 컴퓨터가 전 세계로 보급되면서 더 많은 언어를 표현할 수 있는 EASCII 문자 집합이 등장했다. ASCII의 7비트를 8비트로 확장하여 서로 다른 문자 256개를 표현한다.</p> <p>이후 세계 여러 지역에 맞는 EASCII 문자 집합이 잇따라 등장했다. 이 문자 집합들은 앞의 128개 문자는 모두 ASCII 코드로 통일하고, 뒤의 128개 문자는 각 언어의 필요에 맞게 서로 다르게 정의했다.</p>","path":["제 3 장   자료 구조","3.4   문자 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#342-gbk","level":2,"title":"3.4.2   GBK 문자 집합","text":"<p>하지만 EASCII 코드도 여러 언어에 필요한 문자 수를 충족하지 못했다. 예를 들어 한자는 거의 10만 자에 이르며, 일상에서 사용하는 것만 해도 수천 자다. 중국 국가표준총국은 1980년에 한자 6763자를 수록한 GB2312 문자 집합을 발표하여 컴퓨터의 한자 처리에 필요한 기본적인 요구를 충족했다.</p> <p>그러나 GB2312는 일부 드문 한자와 번체자를 처리할 수 없었다. GBK 문자 집합은 GB2312를 확장한 것으로 총 21886자의 한자를 수록한다. GBK 인코딩에서는 ASCII 문자를 1바이트로, 한자를 2바이트로 표현한다.</p>","path":["제 3 장   자료 구조","3.4   문자 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#343-unicode","level":2,"title":"3.4.3   Unicode 문자 집합","text":"<p>컴퓨터 기술이 빠르게 발전하면서 문자 집합과 인코딩 표준이 다양하게 등장했지만, 이는 여러 문제를 낳았다. 한편으로는 문자 집합이 대개 특정 언어의 문자만 정의하여 다국어 환경에서 제대로 작동하지 못했다. 다른 한편으로는 같은 언어에도 여러 문자 집합 표준이 있어, 두 컴퓨터의 인코딩 표준이 다르면 정보를 주고받을 때 글자가 깨졌다.</p> <p>당시 연구자들은 이렇게 생각했다. 세계의 모든 언어와 기호를 담을 만큼 완전한 문자 집합을 만들면 다국어 환경과 글자 깨짐 문제를 해결할 수 있지 않을까? 이러한 생각에서 크고 포괄적인 문자 집합 Unicode가 탄생했다.</p> <p>Unicode는 중국어로 ‘통일 부호’라는 뜻의 이름으로 불리며, 이론적으로 100만 개 이상의 문자를 수용할 수 있다. 전 세계 문자를 하나의 문자 집합에 통합하고, 여러 언어의 문자를 처리하고 표시하는 공통 문자 집합을 제공하여 인코딩 표준의 차이로 발생하는 글자 깨짐을 줄이는 것이 목표다. 1991년 발표된 뒤로 새로운 언어와 문자가 계속 추가되었다. 2022년 9월 기준으로 Unicode에는 여러 언어의 문자, 기호, 이모지 등을 포함하여 149186개의 문자가 들어 있다.</p> <p>Unicode는 범용 문자 집합으로서 각 문자에 고유한 ‘코드 포인트’(문자 번호)를 부여한다. 범위는 U+0000부터 U+10FFFF까지이며, 하나로 통일된 문자 번호 공간을 이룬다. 그러나 Unicode는 이러한 문자 코드 포인트를 컴퓨터에 어떻게 저장할지는 정하지 않는다. 길이가 다른 Unicode 코드 포인트들이 한 텍스트에 함께 나타나면 시스템은 문자를 어떻게 해석할까? 예를 들어 길이가 2바이트인 인코딩이 주어졌을 때, 2바이트 문자 하나인지 1바이트 문자 두 개인지 어떻게 판단할까?</p> <p>이 문제의 직접적인 해결책은 모든 문자를 같은 길이로 인코딩하여 저장하는 것이다. 아래 그림(3-7)에서 ‘Hello’의 각 문자는 1바이트, ‘算法’의 각 문자는 2바이트를 차지한다. 상위 비트에 0을 채우면 ‘Hello 算法’의 모든 문자를 2바이트 길이로 인코딩할 수 있다. 그러면 시스템이 2바이트마다 문자 하나를 해석하여 원래 문구를 복원할 수 있다.</p> <p></p> <p> 그림 3-7   Unicode 인코딩 예 </p> <p>하지만 ASCII 코드에서 보았듯이 영어 문자를 인코딩하는 데는 1바이트면 충분하다. 위 방식을 사용하면 영어 텍스트가 ASCII 인코딩의 두 배 공간을 차지하여 메모리를 크게 낭비한다. 따라서 더 효율적인 Unicode 인코딩 방법이 필요하다.</p>","path":["제 3 장   자료 구조","3.4   문자 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#344-utf-8","level":2,"title":"3.4.4   UTF-8 인코딩","text":"<p>현재 UTF-8은 세계에서 가장 널리 쓰이는 Unicode 인코딩 방식이다. 가변 길이 인코딩으로, 문자의 복잡성에 따라 1~4바이트로 문자 하나를 표현한다. ASCII 문자는 1바이트, 라틴 문자와 그리스 문자는 2바이트, 자주 쓰는 중국어 문자는 3바이트, 일부 드문 문자는 4바이트가 필요하다.</p> <p>UTF-8 인코딩 규칙은 복잡하지 않으며, 다음 두 경우로 나뉜다.</p> <ul> <li>길이가 1바이트인 문자는 최상위 비트를 \\(0\\)으로 두고 나머지 7비트에 Unicode 코드 포인트를 넣는다. ASCII 문자는 Unicode 문자 집합의 처음 128개 코드 포인트를 차지한다. 즉, UTF-8 인코딩은 ASCII 코드와 하위 호환된다. 따라서 오래된 ASCII 텍스트도 UTF-8로 해석할 수 있다.</li> <li>길이가 \\(n\\)바이트인 문자(\\(n &gt; 1\\))는 첫 바이트의 상위 \\(n\\)비트를 모두 \\(1\\)로, \\(n + 1\\)번째 비트를 \\(0\\)으로 둔다. 두 번째 바이트부터는 각 바이트의 상위 2비트를 \\(10\\)으로 두고, 나머지 모든 비트에 문자의 Unicode 코드 포인트를 채운다.</li> </ul> <p>아래 그림(3-8)은 ‘Hello算法’의 UTF-8 인코딩을 보여 준다. 상위 \\(n\\)비트가 모두 \\(1\\)이므로 시스템은 맨 앞에서 연속된 \\(1\\)의 개수를 읽어 문자 길이가 \\(n\\)임을 알 수 있다.</p> <p>그렇다면 나머지 바이트의 상위 2비트를 모두 \\(10\\)으로 두는 이유는 무엇일까? 이 \\(10\\)은 검사 표식 역할을 한다. 시스템이 잘못된 바이트에서 텍스트를 해석하기 시작했을 때, 바이트 앞부분의 \\(10\\)을 통해 이상을 빠르게 감지할 수 있다.</p> <p>\\(10\\)을 검사 표식으로 쓰는 이유는 UTF-8 규칙상 문자 인코딩의 맨 앞 두 비트가 \\(10\\)일 수 없기 때문이다. 이는 귀류법으로 증명할 수 있다. 문자 인코딩의 맨 앞 두 비트가 \\(10\\)이라고 가정하면, 길이가 \\(1\\)바이트인 ASCII 문자라는 뜻이다. 하지만 ASCII 코드의 최상위 비트는 \\(0\\)이어야 하므로 가정과 모순된다.</p> <p></p> <p> 그림 3-8   UTF-8 인코딩 예 </p> <p>UTF-8 외에도 다음 두 인코딩 방식이 널리 쓰인다.</p> <ul> <li>UTF-16 인코딩: 2바이트 또는 4바이트로 문자 하나를 표현한다. 모든 ASCII 문자와 자주 쓰는 비영어권 문자는 2바이트로 표현하고, 일부 문자는 4바이트가 필요하다. 2바이트 문자는 UTF-16 인코딩 값이 Unicode 코드 포인트와 같다.</li> <li>UTF-32 인코딩: 모든 문자에 4바이트를 사용한다. 따라서 UTF-8이나 UTF-16보다 공간을 더 많이 차지하며, 특히 ASCII 문자의 비중이 큰 텍스트에서 차이가 크다.</li> </ul> <p>저장 공간 관점에서 UTF-8은 영어 문자를 1바이트만으로 표현하므로 매우 효율적이다. 반면 중국어 같은 일부 비영어권 문자는 UTF-16으로 인코딩하는 것이 더 효율적이다. UTF-16은 2바이트면 되지만 UTF-8은 3바이트가 필요할 수 있기 때문이다.</p> <p>호환성 관점에서는 UTF-8의 범용성이 가장 뛰어나며, 많은 도구와 라이브러리가 UTF-8을 우선 지원한다.</p>","path":["제 3 장   자료 구조","3.4   문자 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#345","level":2,"title":"3.4.5   프로그래밍 언어의 문자 인코딩","text":"<p>과거의 대다수 프로그래밍 언어에서는 실행 중인 프로그램의 문자열에 UTF-16이나 UTF-32 같은 고정 길이 인코딩을 사용했다. 고정 길이 인코딩에서는 문자열을 배열처럼 다룰 수 있어 다음과 같은 장점이 있다.</p> <ul> <li>임의 접근: UTF-16 문자열은 임의 접근이 쉽다. UTF-8은 가변 길이 인코딩이므로 \\(i\\)번째 문자를 찾으려면 문자열의 처음부터 \\(i\\)번째 문자까지 순회해야 하며, \\(O(n)\\) 시간이 필요하다.</li> <li>문자 수 계산: 임의 접근과 마찬가지로 UTF-16 문자열의 길이 계산도 \\(O(1)\\) 연산이다. 반면 UTF-8 문자열의 길이를 계산하려면 문자열 전체를 순회해야 한다.</li> <li>문자열 연산: UTF-16 문자열에서는 분할, 연결, 삽입, 삭제 등 여러 문자열 연산을 더 쉽게 수행할 수 있다. UTF-8 문자열에서는 잘못된 UTF-8 인코딩이 생기지 않도록 보장하기 위해 보통 추가 계산이 필요하다.</li> </ul> <p>프로그래밍 언어의 문자 인코딩 설계는 여러 요소가 관련된 흥미로운 주제다.</p> <ul> <li>Java의 <code>String</code>은 UTF-16 인코딩을 사용하며 각 문자는 2바이트를 차지한다. Java를 처음 설계할 당시에는 16비트로 가능한 모든 문자를 표현할 수 있다고 생각했기 때문이다. 그러나 이는 잘못된 판단이었다. 이후 Unicode 규격이 16비트를 넘어 확장되면서, 현재 Java의 문자는 16비트 값 두 개로 이루어진 ‘서로게이트 쌍’으로 표현될 수 있다.</li> <li>JavaScript와 TypeScript의 문자열이 UTF-16을 사용하는 이유도 Java와 비슷하다. Netscape가 1995년에 JavaScript를 처음 발표했을 때 Unicode는 초기 발전 단계였으며, 당시에는 16비트 인코딩으로 모든 Unicode 문자를 표현할 수 있었다.</li> <li>C#이 UTF-16을 사용하는 주된 이유는 .NET 플랫폼을 Microsoft가 설계했으며, Windows 운영체제를 비롯한 Microsoft의 여러 기술에서 UTF-16을 널리 사용하기 때문이다.</li> </ul> <p>이 언어들은 문자 수를 과소평가했기 때문에 16비트를 넘는 Unicode 문자를 표현하려고 ‘서로게이트 쌍’을 도입할 수밖에 없었다. 이는 불가피한 선택이었다. 한편으로는 서로게이트 쌍이 포함된 문자열에서 문자 하나가 2바이트 또는 4바이트를 차지하여 고정 길이 인코딩의 장점을 잃는다. 다른 한편으로는 서로게이트 쌍을 처리하는 코드가 더 필요해 프로그래밍이 복잡해지고 디버깅도 어려워진다.</p> <p>이러한 이유로 일부 프로그래밍 언어는 다른 인코딩 방식을 채택했다.</p> <ul> <li>Python의 <code>str</code>은 Unicode 인코딩과 유연한 문자열 표현을 사용한다. 저장되는 문자 길이는 문자열에서 가장 큰 Unicode 코드 포인트에 따라 정해진다. 모든 문자가 ASCII이면 문자마다 1바이트를 사용한다. ASCII 범위를 넘는 문자가 있지만 모두 기본 다국어 평면(BMP)에 속하면 문자마다 2바이트를 사용한다. BMP를 벗어나는 문자가 있으면 문자마다 4바이트를 사용한다.</li> <li>Go의 <code>string</code>은 내부적으로 UTF-8 인코딩을 사용한다. Go는 Unicode 코드 포인트 하나를 나타내는 <code>rune</code> 자료형도 제공한다.</li> <li>Rust의 <code>str</code>과 <code>String</code>은 내부적으로 UTF-8 인코딩을 사용한다. Rust도 Unicode 코드 포인트 하나를 나타내는 <code>char</code> 자료형을 제공한다.</li> </ul> <p>위 내용은 프로그래밍 언어 안에서 문자열을 저장하는 방식에 관한 것이다. 이는 문자열을 파일에 저장하거나 네트워크로 전송하는 방식과는 별개의 문제다. 파일 저장이나 네트워크 전송에서는 보통 호환성과 공간 효율을 높이기 위해 문자열을 UTF-8 형식으로 인코딩한다.</p>","path":["제 3 장   자료 구조","3.4   문자 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/classification_of_data_structure/","level":1,"title":"3.1   자료 구조의 분류","text":"<p>대표적인 자료 구조로는 배열, 연결 리스트, 스택, 큐, 해시 테이블, 트리, 힙, 그래프가 있다. 이들은 ‘논리적 구조’와 ‘물리적 구조’라는 두 관점에서 분류할 수 있다.</p>","path":["제 3 장   자료 구조","3.1   자료 구조의 분류"],"tags":[]},{"location":"chapter_data_structure/classification_of_data_structure/#311","level":2,"title":"3.1.1   논리적 구조: 선형과 비선형","text":"<p>논리적 구조는 데이터 원소 사이의 논리적 관계를 드러낸다. 배열과 연결 리스트에서는 데이터가 일정한 순서로 배열되어 데이터 사이의 선형 관계를 나타낸다. 트리에서는 데이터가 위에서 아래로 계층을 이루어 ‘조상’과 ‘자손’ 사이의 파생 관계를 나타낸다. 그래프는 노드와 간선으로 이루어져 복잡한 네트워크 관계를 표현한다.</p> <p>아래 그림(3-1)과 같이 논리적 구조는 크게 ‘선형’과 ‘비선형’으로 나뉜다. 선형 구조는 데이터가 논리적으로 일렬로 배열되어 있어 비교적 직관적이다. 비선형 구조는 이와 달리 일렬로 배열되지 않는다.</p> <ul> <li>선형 자료 구조: 배열, 연결 리스트, 스택, 큐, 해시 테이블이 해당하며, 원소 사이에 일대일의 순서 관계가 있다.</li> <li>비선형 자료 구조: 트리, 힙, 그래프, 해시 테이블이 해당한다.</li> </ul> <p>비선형 자료 구조는 다시 트리 구조와 네트워크 구조로 나뉜다.</p> <ul> <li>트리 구조: 트리, 힙, 해시 테이블이 해당하며, 원소 사이에 일대다 관계가 있다.</li> <li>네트워크 구조: 그래프가 해당하며, 원소 사이에 다대다 관계가 있다.</li> </ul> <p></p> <p> 그림 3-1   선형 자료 구조와 비선형 자료 구조 </p>","path":["제 3 장   자료 구조","3.1   자료 구조의 분류"],"tags":[]},{"location":"chapter_data_structure/classification_of_data_structure/#312","level":2,"title":"3.1.2   물리적 구조: 연속과 분산","text":"<p>알고리즘 프로그램이 실행될 때 처리 중인 데이터는 주로 메모리에 저장된다. 아래 그림(3-2)은 컴퓨터 메모리 모듈을 보여 주며, 검은색 칩마다 메모리 공간이 들어 있다. 메모리는 각 셀에 일정한 크기의 데이터를 저장할 수 있는 거대한 Excel 표라고 생각할 수 있다.</p> <p>시스템은 메모리 주소를 통해 원하는 위치의 데이터에 접근한다. 아래 그림(3-2)과 같이 컴퓨터는 특정 규칙에 따라 표의 각 셀에 번호를 부여하여 모든 메모리 공간이 고유한 메모리 주소를 갖게 한다. 프로그램은 이 주소를 이용해 메모리의 데이터에 접근한다.</p> <p></p> <p> 그림 3-2   메모리 모듈, 메모리 공간, 메모리 주소 </p> <p>참고</p> <p>메모리를 Excel 표에 빗댄 것은 단순화한 비유다. 실제 메모리의 작동 방식은 더 복잡하며 주소 공간, 메모리 관리, 캐시, 가상 메모리, 물리 메모리 등의 개념이 관련된다.</p> <p>메모리는 모든 프로그램이 공유하는 자원이다. 어떤 메모리 영역을 한 프로그램이 사용하면 보통 다른 프로그램은 동시에 사용할 수 없다. 따라서 자료 구조와 알고리즘을 설계할 때 메모리 자원은 중요한 고려 사항이다. 예를 들어 알고리즘의 최대 메모리 사용량은 시스템의 남은 메모리를 넘지 않아야 한다. 연속된 큰 메모리 공간이 부족하다면 분산된 메모리 공간에 저장할 수 있는 자료 구조를 선택해야 한다.</p> <p>아래 그림(3-3)과 같이 물리적 구조는 데이터가 컴퓨터 메모리에 저장되는 방식을 나타내며, 연속 공간 저장(배열)과 분산 공간 저장(연결 리스트)으로 나뉜다. 물리적 구조는 데이터 접근, 갱신, 삽입, 삭제 등의 연산 방식을 근본적으로 결정한다. 두 물리적 구조는 시간 효율과 공간 효율에서 서로 보완적인 특징을 지닌다.</p> <p></p> <p> 그림 3-3   연속 공간 저장과 분산 공간 저장 </p> <p>모든 자료 구조는 배열, 연결 리스트 또는 둘의 조합으로 구현된다는 점도 알아 두자. 예를 들어 스택과 큐는 배열이나 연결 리스트로 구현할 수 있으며, 해시 테이블 구현에는 배열과 연결 리스트가 함께 포함될 수 있다.</p> <ul> <li>배열로 구현할 수 있는 것: 스택, 큐, 해시 테이블, 트리, 힙, 그래프, 행렬, 텐서(차원이 \\(\\geq 3\\)인 배열) 등.</li> <li>연결 리스트로 구현할 수 있는 것: 스택, 큐, 해시 테이블, 트리, 힙, 그래프 등.</li> </ul> <p>연결 리스트는 초기화한 뒤에도 프로그램 실행 중에 길이를 조정할 수 있으므로 ‘동적 자료 구조’라고도 한다. 배열은 초기화한 뒤 길이를 바꿀 수 없으므로 ‘정적 자료 구조’라고도 한다. 다만 배열도 메모리를 다시 할당하면 길이를 바꿀 수 있어 어느 정도 ‘동적 특성’을 가질 수 있다.</p> <p>참고</p> <p>물리적 구조를 이해하기 어렵다면 다음 장을 먼저 읽은 뒤 이 절을 복습하는 것을 권한다.</p>","path":["제 3 장   자료 구조","3.1   자료 구조의 분류"],"tags":[]},{"location":"chapter_data_structure/exercises/","level":1,"title":"3.6   연습 문제","text":"","path":["제 3 장   자료 구조","3.6   연습 문제"],"tags":[]},{"location":"chapter_data_structure/exercises/#361","level":2,"title":"3.6.1   개념 확인","text":"","path":["제 3 장   자료 구조","3.6   연습 문제"],"tags":[]},{"location":"chapter_data_structure/exercises/#1","level":3,"title":"1.   일상 속 데이터의 관계","text":"<p>데이터 사이의 관계를 바탕으로 다음 세 상황에 대해 ‘선형 구조, 트리 구조, 네트워크 구조’ 중 하나를 고르고 이유를 설명하라.</p> <ol> <li>학생들이 한 줄로 서 있고, 각자 바로 앞과 뒤에 있는 사람만 신경 쓴다.</li> <li>학교를 ‘학교 → 학년 → 학급’의 계층으로 나누어 관리한다.</li> <li>도시의 도로가 여러 교차로를 연결한다. 한 교차로에서 여러 다른 교차로로 갈 수 있으며 순환하는 경로도 생길 수 있다.</li> </ol> 참고 답안 <ol> <li> <p>선형 구조다. 맨 앞과 맨 뒤에 선 사람을 제외하면 각 사람은 앞뒤로 한 사람씩만 이웃하며, 관계가 하나의 선을 따라 이어진다.</p> </li> <li> <p>트리 구조다. 각 학급은 한 학년에 속하고 각 학년은 학교에 속하므로, 관계가 위에서 아래로 계층적으로 펼쳐진다.</p> </li> <li> <p>네트워크 구조다. 한 교차로가 여러 교차로와 연결될 수 있고 경로들이 순환을 이룰 수도 있으므로, 하나의 순서나 엄격한 계층으로 나열할 수 없다.</p> <p>구조를 판단할 때는 메모리에서 차지하는 공간보다 원소 사이의 관계를 먼저 살펴보아야 한다.</p> </li> </ol>","path":["제 3 장   자료 구조","3.6   연습 문제"],"tags":[]},{"location":"chapter_data_structure/exercises/#2","level":3,"title":"2.   논리적 순서를 메모리에 저장하는 방법","text":"<p>논리적 순서 <code>A → B → C</code>를 저장하기 위해 다음과 같이 단순화한 두 메모리 배치 방식을 생각해 보자.</p> <ul> <li>방법 가: <code>A, B, C</code>를 각각 번호가 <code>20, 21, 22</code>인 메모리 칸에 저장한다.</li> <li>방법 나: <code>A, B, C</code>를 각각 번호가 <code>20, 7, 31</code>인 메모리 칸에 저장하고, <code>A</code>에 <code>B</code>의 위치를, <code>B</code>에 <code>C</code>의 위치를 기록한다.</li> </ul> <ol> <li>어느 방법이 연속 공간 저장이고, 어느 방법이 분산 공간 저장인가?</li> <li>두 방법은 각각 배열과 연결 리스트 중 어느 쪽에 더 가까운가?</li> <li>방법 나는 메모리 칸의 번호가 크기순으로 정렬되어 있지 않은데, 어떻게 <code>A → B → C</code>의 논리적 순서를 나타낼 수 있는가?</li> </ol> 참고 답안 <ol> <li> <p>방법 가는 연속된 메모리 칸을 사용하므로 연속 공간 저장이다. 방법 나는 노드가 여러 위치에 흩어져 있으므로 분산 공간 저장이다.</p> </li> <li> <p>방법 가는 배열에, 방법 나는 연결 리스트에 더 가깝다.</p> </li> <li> <p>논리적 순서는 메모리 칸 번호의 크기가 아니라 노드 사이에 기록된 연결 관계로 결정된다.     <code>A</code>에 기록된 위치에서 <code>B</code>를 찾고, 다시 <code>B</code>에 기록된 위치에서 <code>C</code>를 찾을 수 있으므로 <code>A, B, C</code>에 차례대로 접근할 수 있다.</p> <p>이는 논리적 구조와 물리적 구조가 같은 데이터 집합을 바라보는 서로 다른 두 관점임을 보여 준다.</p> </li> </ol>","path":["제 3 장   자료 구조","3.6   연습 문제"],"tags":[]},{"location":"chapter_data_structure/exercises/#3","level":3,"title":"3.   과제 기록의 자료형과 구조","text":"<p>한 학습 모임에서 학생 4명의 과제 제출 여부를 좌석 순서대로 기록한 결과가 다음과 같다.</p> <p><code>[true, false, true, true]</code></p> <ol> <li>각 원소에는 어떤 기본 자료형이 적합한가?</li> <li>이 4개 원소를 좌석 순서대로 한 줄에 나열한 것은 어떤 논리적 구조인가?</li> <li>나중에 각 학생의 과제 점수 <code>[90, 0, 85, 100]</code>를 기록하도록 바꾼다면, 달라지는 것은 데이터의 ‘내용 유형’인가, ‘조직 방식’인가? 이유를 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>각 원소는 ‘예’ 또는 ‘아니요’만 나타내므로 불리언 자료형 <code>bool</code>이 적합하다.</p> </li> <li> <p>원소들이 좌석 순서대로 나열되어 선형 구조를 이루며, 배열에 저장할 수 있다.</p> </li> <li> <p>달라지는 것은 내용 유형이다. 원소가 불리언 값에서 정수로 바뀐다. 조직 방식은 바뀌지 않는다.     데이터는 여전히 좌석 순서대로 한 줄에 나열되어 있으므로 배열이라는 선형 구조를 사용할 수 있다.</p> <p>기본 자료형은 ‘무엇을 저장하는가’를, 자료 구조는 ‘데이터를 어떻게 구성하는가’를 설명한다.</p> </li> </ol>","path":["제 3 장   자료 구조","3.6   연습 문제"],"tags":[]},{"location":"chapter_data_structure/exercises/#362","level":2,"title":"3.6.2   프로그래밍 연습","text":"","path":["제 3 장   자료 구조","3.6   연습 문제"],"tags":[]},{"location":"chapter_data_structure/exercises/#1-1","level":3,"title":"1.   이진수 표현에서 1의 개수 세기","text":"<p>음이 아닌 정수 <code>n</code>이 주어질 때, 이진수 표현에 포함된 1의 개수를 구하라.</p> <p>비트 연산을 사용해야 하며, 이진수 표현을 문자열로 변환하거나 1의 개수를 바로 세어 주는 내장 함수를 사용해서는 안 된다.</p> 풀이 힌트 <ol> <li>n &amp; 1로 n의 가장 오른쪽 비트를 얻어 그 비트가 1인지 확인할 수 있다.</li> <li>오른쪽으로 한 비트 이동하면 현재 가장 오른쪽 비트가 제거된다. 대부분의 언어에서는 &gt;&gt; 연산자를 사용한다.</li> <li>비트를 하나씩 확인하고 오른쪽으로 이동하는 방법을 완성한 뒤, n &amp; (n - 1)이 n의 가장 오른쪽 1을 0으로 바꾼다는 점도 살펴보자.</li> </ol> <p>LeetCode</p>","path":["제 3 장   자료 구조","3.6   연습 문제"],"tags":[]},{"location":"chapter_data_structure/number_encoding/","level":1,"title":"3.3   수의 인코딩 *","text":"<p>참고</p> <p>이 책에서 제목에 * 기호가 있는 절은 선택 학습 내용이다. 시간이 부족하거나 이해하기 어렵다면 먼저 건너뛰고, 필수 내용을 학습한 뒤 따로 도전해도 좋다.</p>","path":["제 3 장   자료 구조","3.3   수의 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/number_encoding/#331-1-2","level":2,"title":"3.3.1   부호와 크기, 1의 보수, 2의 보수","text":"<p>앞 절의 표에서 모든 정수형은 표현할 수 있는 음수가 양수보다 하나 더 많았다. 예를 들어 <code>byte</code>의 값 범위는 \\([-128, 127]\\)이다. 직관과 어긋나는 이 현상의 이유를 이해하려면 부호와 크기, 1의 보수, 2의 보수에 대해 알아야 한다.</p> <p>먼저 수는 컴퓨터에 ‘2의 보수’ 형태로 저장된다는 점을 알아 두자. 그 이유를 분석하기 전에 세 가지 표현을 정의하겠다.</p> <ul> <li>부호와 크기: 이진수 표현의 최상위 비트를 부호 비트로 사용한다. \\(0\\)은 양수, \\(1\\)은 음수를 나타내며 나머지 비트는 수의 크기를 나타낸다.</li> <li>1의 보수: 양수의 1의 보수는 부호와 크기 표현과 같다. 음수의 1의 보수는 부호와 크기 표현에서 부호 비트를 제외한 모든 비트를 반전한 것이다.</li> <li>2의 보수: 양수의 2의 보수는 부호와 크기 표현과 같다. 음수의 2의 보수는 1의 보수에 \\(1\\)을 더한 것이다.</li> </ul> <p>아래 그림(3-4)은 부호와 크기, 1의 보수, 2의 보수를 서로 변환하는 방법을 보여 준다.</p> <p></p> <p> 그림 3-4   부호와 크기, 1의 보수, 2의 보수의 상호 변환 </p> <p>부호와 크기(sign-magnitude) 표현은 가장 직관적이지만 한계가 있다. 우선 음수의 부호와 크기 표현은 연산에 그대로 사용할 수 없다. 예를 들어 이 표현으로 \\(1 + (-2)\\)를 계산하면 결과가 \\(-3\\)이 되어 명백히 틀린다.</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 + 1000 \\; 0010 \\newline &amp; = 1000 \\; 0011 \\newline &amp; \\rightarrow -3 \\end{aligned} \\] <p>이 문제를 해결하기 위해 1의 보수(1's complement)가 도입되었다. 부호와 크기 표현을 먼저 1의 보수로 바꾸고 \\(1 + (-2)\\)를 계산한 뒤, 결과를 다시 부호와 크기 표현으로 바꾸면 올바른 결과인 \\(-1\\)을 얻는다.</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 \\; \\text{(부호와 크기)} + 1000 \\; 0010 \\; \\text{(부호와 크기)} \\newline &amp; = 0000 \\; 0001 \\; \\text{(1의 보수)} + 1111  \\; 1101 \\; \\text{(1의 보수)} \\newline &amp; = 1111 \\; 1110 \\; \\text{(1의 보수)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(부호와 크기)} \\newline &amp; \\rightarrow -1 \\end{aligned} \\] <p>또한 부호와 크기 표현에서는 0을 \\(+0\\)과 \\(-0\\)의 두 가지 방식으로 나타낸다. 즉, 0에 서로 다른 두 이진 인코딩이 대응하므로 모호함이 생길 수 있다. 예를 들어 조건을 판단할 때 양의 0과 음의 0을 구분하지 않으면 결과가 틀릴 수 있다. 이러한 모호함을 처리하려면 추가 판단 연산이 필요하므로 컴퓨터의 연산 효율이 낮아질 수 있다.</p> \\[ \\begin{aligned} +0 &amp; \\rightarrow 0000 \\; 0000 \\newline -0 &amp; \\rightarrow 1000 \\; 0000 \\end{aligned} \\] <p>1의 보수에도 부호와 크기 표현처럼 양의 0과 음의 0의 모호함이 있다. 그래서 2의 보수(2's complement)가 도입되었다. 먼저 음의 0을 부호와 크기, 1의 보수, 2의 보수로 변환하는 과정을 살펴보자.</p> \\[ \\begin{aligned} -0 \\rightarrow \\; &amp; 1000 \\; 0000 \\; \\text{(부호와 크기)} \\newline = \\; &amp; 1111 \\; 1111 \\; \\text{(1의 보수)} \\newline = 1 \\; &amp; 0000 \\; 0000 \\; \\text{(2의 보수)} \\newline \\end{aligned} \\] <p>음의 0의 1의 보수에 \\(1\\)을 더하면 자리 올림이 발생한다. 그러나 <code>byte</code>는 길이가 8비트뿐이므로 9번째 비트로 넘친 \\(1\\)은 버려진다. 즉, 음의 0의 2의 보수는 \\(0000 \\; 0000\\)으로, 양의 0의 2의 보수와 같다. 따라서 2의 보수 표현에는 0이 하나만 존재하며, 양의 0과 음의 0의 모호함이 해결된다.</p> <p>마지막 의문이 남는다. <code>byte</code>의 값 범위는 \\([-128, 127]\\)인데, 하나 더 있는 음수 \\(-128\\)은 어떻게 얻을까? 구간 \\([-127, +127]\\)의 모든 정수에는 대응하는 부호와 크기, 1의 보수, 2의 보수 표현이 있으며, 부호와 크기 표현과 2의 보수는 서로 변환할 수 있다.</p> <p>그러나 2의 보수 \\(1000 \\; 0000\\)은 예외로, 대응하는 부호와 크기 표현이 없다. 변환 방법을 적용하면 부호와 크기 표현으로 \\(0000 \\; 0000\\)을 얻는다. 하지만 이 표현은 수 \\(0\\)을 나타내며 2의 보수도 자기 자신이어야 하므로 모순이다. 컴퓨터에서는 이 특수한 2의 보수 \\(1000 \\; 0000\\)이 \\(-128\\)을 나타내도록 정한다. 실제로 \\((-1) + (-127)\\)을 2의 보수로 계산한 결과는 \\(-128\\)이다.</p> \\[ \\begin{aligned} &amp; (-127) + (-1) \\newline &amp; \\rightarrow 1111 \\; 1111 \\; \\text{(부호와 크기)} + 1000 \\; 0001 \\; \\text{(부호와 크기)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(1의 보수)} + 1111  \\; 1110 \\; \\text{(1의 보수)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(2의 보수)} + 1111  \\; 1111 \\; \\text{(2의 보수)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(2의 보수)} \\newline &amp; \\rightarrow -128 \\end{aligned} \\] <p>위의 계산이 모두 덧셈이라는 사실을 발견했을 것이다. 여기에는 중요한 사실이 담겨 있다. 컴퓨터 내부의 하드웨어 회로는 주로 덧셈 연산을 기반으로 설계된다. 덧셈은 곱셈, 나눗셈, 뺄셈 같은 다른 연산보다 하드웨어로 구현하기 쉽고, 병렬 처리도 쉬우며, 연산 속도도 빠르기 때문이다.</p> <p>그렇다고 컴퓨터가 덧셈만 할 수 있다는 뜻은 아니다. 덧셈과 몇 가지 기본 논리 연산을 결합하면 다양한 수학 연산을 구현할 수 있다. 예를 들어 뺄셈 \\(a - b\\)는 덧셈 \\(a + (-b)\\)로 바꿀 수 있고, 곱셈과 나눗셈도 여러 번의 덧셈이나 뺄셈으로 바꿀 수 있다.</p> <p>이제 컴퓨터가 2의 보수를 사용하는 이유를 정리할 수 있다. 2의 보수 표현을 쓰면 같은 회로와 연산으로 양수와 음수의 덧셈을 처리할 수 있다. 뺄셈을 위한 별도 하드웨어 회로를 설계할 필요가 없으며, 양의 0과 음의 0의 모호함도 따로 처리하지 않아도 된다. 이는 하드웨어 설계를 크게 단순화하고 연산 효율을 높인다.</p> <p>2의 보수는 매우 정교하게 설계된 방식이지만, 분량상 여기까지만 소개한다. 관심 있는 독자는 더 깊이 알아보기를 권한다.</p>","path":["제 3 장   자료 구조","3.3   수의 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/number_encoding/#332","level":2,"title":"3.3.2   부동 소수점 인코딩","text":"<p>자세히 살펴보면 이런 의문이 생긴다. <code>int</code>와 <code>float</code>는 모두 4바이트로 길이가 같은데, 왜 <code>float</code>의 값 범위가 <code>int</code>보다 훨씬 넓을까? <code>float</code>는 소수도 표현해야 하므로 오히려 값 범위가 줄어들 것 같아 직관과 어긋난다.</p> <p>실제로 이는 부동 소수점형 <code>float</code>가 다른 표현 방식을 사용하기 때문이다. 길이가 32비트인 이진수를 다음과 같이 쓰자.</p> \\[ b_{31} b_{30} b_{29} \\ldots b_2 b_1 b_0 \\] <p>IEEE 754 표준에 따르면 32비트 <code>float</code>는 다음 세 부분으로 이루어진다.</p> <ul> <li>부호 비트 \\(\\mathrm{S}\\): 1비트를 차지하며 \\(b_{31}\\)에 해당한다.</li> <li>지수 비트 \\(\\mathrm{E}\\): 8비트를 차지하며 \\(b_{30} b_{29} \\ldots b_{23}\\)에 해당한다.</li> <li>가수의 소수부 비트 \\(\\mathrm{N}\\): 23비트를 차지하며 \\(b_{22} b_{21} \\ldots b_0\\)에 해당한다.</li> </ul> <p>이진수 <code>float</code>가 나타내는 값을 계산하는 방법은 다음과 같다.</p> \\[ \\text {val} = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\] <p>이를 십진수로 바꾸면 다음 식을 얻는다.</p> \\[ \\text {val}=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\] <p>각 항의 값 범위는 다음과 같다.</p> \\[ \\begin{aligned} \\mathrm{S} \\in &amp; \\{ 0, 1\\}, \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = &amp; (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\] <p></p> <p> 그림 3-5   IEEE 754 표준에 따른 float 계산 예 </p> <p>위 그림(3-5)에서 예로 든 데이터가 \\(\\mathrm{S} = 0\\), \\(\\mathrm{E} = 124\\), \\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\)이면 다음과 같다.</p> \\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\] <p>이제 처음의 질문에 답할 수 있다. <code>float</code>의 표현 방식에는 지수 비트가 있으므로 값 범위가 <code>int</code>보다 훨씬 넓다. 위 계산에 따르면 <code>float</code>로 표현할 수 있는 가장 큰 양수는 \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\)이며, 부호 비트를 바꾸면 가장 작은 음수를 얻는다.</p> <p>부동 소수점형 <code>float</code>는 값 범위를 넓히는 대신 정밀도를 희생한다. 정수형 <code>int</code>는 32비트 전체로 수를 표현하므로 수들이 균일하게 분포한다. 반면 부동 소수점형 <code>float</code>는 지수 비트 때문에 값이 커질수록 인접한 두 수 사이의 간격도 커지는 경향이 있다.</p> <p>아래 표(3-2)와 같이 지수 비트 \\(\\mathrm{E} = 0\\)과 \\(\\mathrm{E} = 255\\)는 특별한 의미를 가지며, 0, 무한대, \\(\\mathrm{NaN}\\) 등을 표현하는 데 사용된다.</p> <p> 표 3-2   지수 비트의 의미 </p> 지수 비트 E 소수부 \\(\\mathrm{N} = 0\\) 소수부 \\(\\mathrm{N} \\ne 0\\) 계산식 \\(0\\) \\(\\pm 0\\) 비정규화 수 \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) 정규화 수 정규화 수 \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\) <p>비정규화 수는 부동 소수점의 정밀도를 크게 높인다는 점도 알아 두자. 가장 작은 양의 정규화 수는 \\(2^{-126}\\)이고, 가장 작은 양의 비정규화 수는 \\(2^{-126} \\times 2^{-23}\\)이다.</p> <p>배정밀도 <code>double</code>도 <code>float</code>와 비슷한 표현 방식을 사용하므로 여기서는 자세한 설명을 생략한다.</p>","path":["제 3 장   자료 구조","3.3   수의 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/summary/","level":1,"title":"3.5   요약","text":"","path":["제 3 장   자료 구조","3.5   요약"],"tags":[]},{"location":"chapter_data_structure/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>자료 구조는 논리적 구조와 물리적 구조라는 두 관점에서 분류할 수 있다. 논리적 구조는 데이터 원소 사이의 논리적 관계를 설명하고, 물리적 구조는 컴퓨터 메모리에 데이터를 저장하는 방식을 설명한다.</li> <li>대표적인 논리적 구조에는 선형, 트리형, 네트워크형 등이 있다. 보통 논리적 구조에 따라 자료 구조를 선형(배열, 연결 리스트, 스택, 큐)과 비선형(트리, 그래프, 힙)으로 나눈다. 해시 테이블 구현에는 선형과 비선형 자료 구조가 함께 포함될 수 있다.</li> <li>프로그램이 실행되면 데이터는 컴퓨터 메모리에 저장된다. 각 메모리 공간에는 대응하는 메모리 주소가 있으며, 프로그램은 이 주소로 데이터에 접근한다.</li> <li>물리적 구조는 주로 연속 공간 저장(배열)과 분산 공간 저장(연결 리스트)으로 나뉜다. 모든 자료 구조는 배열, 연결 리스트 또는 둘의 조합으로 구현된다.</li> <li>컴퓨터의 기본 자료형에는 정수형 <code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>, 부동 소수점형 <code>float</code>, <code>double</code>, 문자형 <code>char</code>, 불리언형 <code>bool</code>이 있다. 값 범위는 차지하는 공간의 크기와 표현 방식에 따라 결정된다.</li> <li>부호와 크기, 1의 보수, 2의 보수는 컴퓨터에서 수를 인코딩하는 세 가지 방식으로, 서로 변환할 수 있다. 정수의 부호와 크기 표현에서 최상위 비트는 부호 비트이고, 나머지 비트는 수의 크기를 나타낸다.</li> <li>정수는 컴퓨터에 2의 보수 형태로 저장된다. 이 표현에서는 양수와 음수의 덧셈을 똑같이 처리할 수 있어 뺄셈만을 위한 별도 하드웨어 회로가 필요 없으며, 양의 0과 음의 0의 모호함도 없다.</li> <li>부동 소수점 인코딩은 부호 1비트, 지수 8비트, 가수의 소수부 23비트로 이루어진다. 지수 비트 덕분에 정수보다 훨씬 넓은 값 범위를 갖는 대신 정밀도를 희생한다.</li> <li>ASCII 코드는 가장 초기에 등장한 영어 문자 집합으로, 길이가 1바이트이며 128개의 문자를 수록한다. GBK는 널리 쓰이는 중국어 문자 집합으로 2만 자 이상의 한자를 수록한다. Unicode는 전 세계 여러 언어의 문자를 담은 완전한 문자 집합 표준을 제공하여 인코딩 방식의 차이로 발생하는 글자 깨짐을 해결하고자 한다.</li> <li>UTF-8은 가장 널리 쓰이는 Unicode 인코딩 방식으로 범용성이 매우 좋다. 확장성이 뛰어난 가변 길이 인코딩이며 저장 공간을 효율적으로 활용한다. UTF-16과 UTF-32는 고정 길이 인코딩 방식이다. 중국어를 인코딩할 때 UTF-16은 UTF-8보다 공간을 적게 차지한다. Java, C# 등의 언어는 기본적으로 UTF-16을 사용한다.</li> </ul>","path":["제 3 장   자료 구조","3.5   요약"],"tags":[]},{"location":"chapter_data_structure/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 해시 테이블에는 왜 선형 자료 구조와 비선형 자료 구조가 함께 포함되는가?</p> <p>해시 테이블의 기반은 배열이며, 해시 충돌을 해결하기 위해 ‘체이닝’을 사용할 수 있다(뒤의 ‘해시 충돌’ 절에서 다룬다). 배열의 각 버킷은 연결 리스트를 가리키며, 연결 리스트 길이가 일정 기준을 넘으면 트리(보통 레드-블랙 트리)로 바뀔 수 있다.</p> <p>저장 관점에서 해시 테이블의 기반은 배열이고, 각 버킷의 슬롯에는 값 하나, 연결 리스트 또는 트리가 들어갈 수 있다. 따라서 해시 테이블에는 선형 자료 구조(배열, 연결 리스트)와 비선형 자료 구조(트리)가 함께 포함될 수 있다.</p> <p>Q: <code>char</code> 자료형의 길이는 1바이트인가?</p> <p><code>char</code>의 길이는 프로그래밍 언어가 사용하는 인코딩 방식에 따라 정해진다. 예를 들어 Java, JavaScript, TypeScript, C#은 모두 UTF-16 인코딩으로 Unicode 코드 포인트를 저장하므로 <code>char</code>의 길이는 2바이트다.</p> <p>Q: 배열로 구현한 자료 구조를 ‘정적 자료 구조’라고 부르면 모호하지 않은가? 스택에서도 원소를 꺼내거나 넣을 수 있는데, 이런 연산은 모두 ‘동적’이다.</p> <p>스택은 동적으로 데이터를 조작할 수 있지만 자료 구조는 여전히 ‘정적’이다. 즉, 길이를 바꿀 수 없다. 배열 기반 자료 구조는 원소를 동적으로 추가하거나 삭제할 수 있지만 용량은 고정되어 있다. 데이터 양이 미리 할당한 크기를 넘으면 더 큰 배열을 새로 만들고 기존 내용을 복사해야 한다.</p> <p>Q: 스택이나 큐를 만들 때 크기를 지정하지 않았는데, 왜 ‘정적 자료 구조’인가?</p> <p>고급 프로그래밍 언어에서는 스택이나 큐의 초기 용량을 직접 지정하지 않아도 클래스 내부에서 자동으로 처리한다. 예를 들어 Java <code>ArrayList</code>의 초기 용량은 보통 10이다. 용량 확장도 자동으로 이루어진다. 자세한 내용은 뒤의 ‘리스트’ 절에서 설명한다.</p> <p>Q: 부호와 크기 표현을 2의 보수로 바꾸려면 ‘반전한 뒤 1을 더한다’. 그렇다면 반대 변환은 역연산인 ‘1을 뺀 뒤 반전하기’여야 할 텐데, ‘반전한 뒤 1을 더하기’로도 가능한 이유는 무엇인가?</p> <p>두 표현의 상호 변환은 실제로 ‘보수’를 구하는 과정이기 때문이다. 먼저 보수를 정의하자. \\(a + b = c\\)라면 \\(a\\)를 \\(c\\)에 대한 \\(b\\)의 보수라 하고, 반대로 \\(b\\)도 \\(c\\)에 대한 \\(a\\)의 보수라고 한다.</p> <p>\\(n = 4\\)비트 이진수 \\(0010\\)이 주어졌다고 하자. 이를 부호와 크기 표현으로 보면(부호 비트는 고려하지 않는다), 2의 보수는 ‘반전한 뒤 1을 더해’ 얻는다.</p> \\[ 0010 \\rightarrow 1101 \\rightarrow 1110 \\] <p>원래 표현과 2의 보수의 합은 \\(0010 + 1110 = 10000\\)이다. 즉, 2의 보수 \\(1110\\)은 \\(10000\\)에 대한 원래 수 \\(0010\\)의 ‘보수’다. 따라서 ‘반전한 뒤 1을 더하기’는 실제로 \\(10000\\)에 대한 보수를 계산하는 과정이다.</p> <p>그렇다면 \\(10000\\)에 대한 2의 보수 \\(1110\\)의 ‘보수’는 무엇일까? 똑같이 ‘반전한 뒤 1을 더해서’ 얻을 수 있다.</p> \\[ 1110 \\rightarrow 0001 \\rightarrow 0010 \\] <p>즉, 원래 표현과 2의 보수는 \\(10000\\)에 대해 서로의 ‘보수’다. 따라서 두 방향의 변환에 같은 연산인 ‘반전한 뒤 1을 더하기’를 사용할 수 있다.</p> <p>물론 역연산인 ‘1을 뺀 뒤 반전하기’로도 2의 보수 \\(1110\\)에서 원래 표현을 구할 수 있다.</p> \\[ 1110 \\rightarrow 1101 \\rightarrow 0010 \\] <p>정리하면 ‘반전한 뒤 1을 더하기’와 ‘1을 뺀 뒤 반전하기’는 모두 \\(10000\\)에 대한 보수를 구하는 연산이므로 서로 동등하다.</p> <p>본질적으로 ‘반전’은 \\(1111\\)에 대한 보수를 구하는 연산이다(항상 <code>부호와 크기 표현 + 1의 보수 = 1111</code>이 성립하기 때문이다). 여기에 1을 더해서 얻은 2의 보수는 \\(10000\\)에 대한 보수다.</p> <p>위 설명은 \\(n = 4\\)를 예로 들었지만, 자릿수가 임의의 값인 이진수에도 일반화할 수 있다.</p>","path":["제 3 장   자료 구조","3.5   요약"],"tags":[]},{"location":"chapter_divide_and_conquer/","level":1,"title":"제 12 장   분할 정복","text":"<p>이 장을 시작하며</p> <p>어려운 문제도 한 단계씩 나눌 때마다 더욱 단순해진다.</p> <p>분할 정복은 중요한 사실을 알려 준다. 쉬운 것부터 시작하면 모든 것이 더 이상 복잡하지 않다.</p>","path":["제 12 장   분할 정복"],"tags":[]},{"location":"chapter_divide_and_conquer/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>12.1   분할 정복 알고리즘</li> <li>12.2   분할 정복 탐색 전략</li> <li>12.3   이진 트리 구성 문제</li> <li>12.4   하노이의 탑 문제</li> <li>12.5   요약</li> <li>12.6   연습 문제</li> </ul>","path":["제 12 장   분할 정복"],"tags":[]},{"location":"chapter_divide_and_conquer/binary_search_recur/","level":1,"title":"12.2   분할 정복 탐색 전략","text":"<p>앞에서 배웠듯이 탐색 알고리즘은 크게 두 종류로 나뉜다.</p> <ul> <li>완전 탐색: 자료 구조를 순회하며, 시간 복잡도는 \\(O(n)\\)이다.</li> <li>적응형 탐색: 데이터 고유의 구성 방식이나 사전 정보를 이용하며, 시간 복잡도가 \\(O(\\log n)\\) 또는 \\(O(1)\\)에 이를 수 있다.</li> </ul> <p>실제로 시간 복잡도가 \\(O(\\log n)\\)인 탐색 알고리즘은 보통 분할 정복 전략을 기반으로 구현된다. 이진 탐색과 트리가 그 예다.</p> <ul> <li>이진 탐색은 각 단계에서 원래 문제(배열에서 목표 원소 찾기)를 더 작은 문제(배열의 절반에서 목표 원소 찾기)로 나눈다. 배열이 비거나 목표 원소를 찾을 때까지 계속한다.</li> <li>트리는 분할 정복 사고의 대표적인 예다. 이진 탐색 트리, AVL 트리, 힙 등의 자료 구조에서 여러 연산의 시간 복잡도는 \\(O(\\log n)\\)이다.</li> </ul> <p>이진 탐색의 분할 정복 전략은 다음과 같다.</p> <ul> <li>문제를 나눌 수 있다: 중간 원소와 목표 원소를 비교하여 원래 문제(배열에서 찾기)를 부분 문제(배열의 절반에서 찾기)로 재귀적으로 나눈다.</li> <li>부분 문제가 독립적이다: 이진 탐색은 매번 부분 문제 하나만 처리하며 다른 부분 문제의 영향을 받지 않는다.</li> <li>부분 문제의 해를 합칠 필요가 없다: 특정 원소 하나를 찾는 것이 목적이므로 부분 문제의 해를 합칠 필요가 없다. 부분 문제를 해결하면 원래 문제도 함께 해결된다.</li> </ul> <p>분할 정복이 탐색 효율을 높이는 근본적인 이유는 완전 탐색이 매번 선택지 하나만 제외하는 데 비해 분할 정복 탐색은 매번 선택지의 절반을 제외할 수 있기 때문이다.</p>","path":["제 12 장   분할 정복","12.2   분할 정복 탐색 전략"],"tags":[]},{"location":"chapter_divide_and_conquer/binary_search_recur/#1","level":3,"title":"1.   분할 정복으로 이진 탐색 구현","text":"<p>앞 장에서는 이진 탐색을 반복 방식으로 구현했다. 이제 분할 정복(재귀)으로 구현해 보자.</p> <p>문제</p> <p>길이가 \\(n\\)이고 모든 원소가 서로 다른 정렬된 배열 <code>nums</code>가 주어진다. 원소 <code>target</code>을 찾아라.</p> <p>분할 정복의 관점에서 탐색 구간 \\([i, j]\\)에 해당하는 부분 문제를 \\(f(i, j)\\)라고 하자.</p> <p>원래 문제 \\(f(0, n-1)\\)에서 시작하여 다음 단계로 이진 탐색을 수행한다.</p> <ol> <li>탐색 구간 \\([i, j]\\)의 중간 인덱스 \\(m\\)을 계산하고, 이를 기준으로 탐색 구간의 절반을 제외한다.</li> <li>크기가 절반으로 줄어든 부분 문제 \\(f(i, m-1)\\) 또는 \\(f(m+1, j)\\)를 재귀적으로 푼다.</li> <li><code>target</code>을 찾거나 구간이 빌 때까지 <code>1.</code> 단계와 <code>2.</code> 단계를 반복한 뒤 반환한다.</li> </ol> <p>아래 그림(12-4)은 배열에서 원소 \\(6\\)을 이진 탐색하는 분할 정복 과정을 보여 준다.</p> <p></p> <p> 그림 12-4   이진 탐색의 분할 정복 과정 </p> <p>구현 코드에서는 문제 \\(f(i, j)\\)를 푸는 재귀 함수 <code>dfs()</code>를 정의한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_recur.py<pre><code>def dfs(nums: list[int], target: int, i: int, j: int) -&gt; int:\n    \"\"\"이진 탐색: 문제 f(i, j)\"\"\"\n    # 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if i &gt; j:\n        return -1\n    # 중간 인덱스 m을 계산한다\n    m = (i + j) // 2\n    if nums[m] &lt; target:\n        # 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m + 1, j)\n    elif nums[m] &gt; target:\n        # 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m - 1)\n    else:\n        # 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m\n\ndef binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"이진 탐색\"\"\"\n    n = len(nums)\n    # 문제 f(0, n-1)을 푼다\n    return dfs(nums, target, 0, n - 1)\n</code></pre> binary_search_recur.cpp<pre><code>/* 이진 탐색: 문제 f(i, j) */\nint dfs(vector&lt;int&gt; &amp;nums, int target, int i, int j) {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if (i &gt; j) {\n        return -1;\n    }\n    // 중간 인덱스 m을 계산한다\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m;\n    }\n}\n\n/* 이진 탐색 */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    int n = nums.size();\n    // 문제 f(0, n-1)을 푼다\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.java<pre><code>/* 이진 탐색: 문제 f(i, j) */\nint dfs(int[] nums, int target, int i, int j) {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if (i &gt; j) {\n        return -1;\n    }\n    // 중간 인덱스 m을 계산한다\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m;\n    }\n}\n\n/* 이진 탐색 */\nint binarySearch(int[] nums, int target) {\n    int n = nums.length;\n    // 문제 f(0, n-1)을 푼다\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.cs<pre><code>/* 이진 탐색: 문제 f(i, j) */\nint DFS(int[] nums, int target, int i, int j) {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if (i &gt; j) {\n        return -1;\n    }\n    // 중간 인덱스 m을 계산한다\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return DFS(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return DFS(nums, target, i, m - 1);\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m;\n    }\n}\n\n/* 이진 탐색 */\nint BinarySearch(int[] nums, int target) {\n    int n = nums.Length;\n    // 문제 f(0, n-1)을 푼다\n    return DFS(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.go<pre><code>/* 이진 탐색: 문제 f(i, j) */\nfunc dfs(nums []int, target, i, j int) int {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if i &gt; j {\n        return -1\n    }\n    //    중간 인덱스를 계산한다\n    m := i + ((j - i) &gt;&gt; 1)\n    //중간 원소와 목표 원소의 크기를 비교한다\n    if nums[m] &lt; target {\n        // 중간 원소가 더 작으면 배열의 오른쪽 절반을 재귀적으로 탐색한다\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m+1, j)\n    } else if nums[m] &gt; target {\n        // 중간 원소가 더 크면 배열의 왼쪽 절반을 재귀적으로 탐색한다\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m-1)\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m\n    }\n}\n\n/* 이진 탐색 */\nfunc binarySearch(nums []int, target int) int {\n    n := len(nums)\n    return dfs(nums, target, 0, n-1)\n}\n</code></pre> binary_search_recur.swift<pre><code>/* 이진 탐색: 문제 f(i, j) */\nfunc dfs(nums: [Int], target: Int, i: Int, j: Int) -&gt; Int {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if i &gt; j {\n        return -1\n    }\n    // 중간 인덱스 m을 계산한다\n    let m = (i + j) / 2\n    if nums[m] &lt; target {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums: nums, target: target, i: m + 1, j: j)\n    } else if nums[m] &gt; target {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums: nums, target: target, i: i, j: m - 1)\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m\n    }\n}\n\n/* 이진 탐색 */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // 문제 f(0, n-1)을 푼다\n    dfs(nums: nums, target: target, i: nums.startIndex, j: nums.endIndex - 1)\n}\n</code></pre> binary_search_recur.js<pre><code>/* 이진 탐색: 문제 f(i, j) */\nfunction dfs(nums, target, i, j) {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if (i &gt; j) {\n        return -1;\n    }\n    // 중간 인덱스 m을 계산한다\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m;\n    }\n}\n\n/* 이진 탐색 */\nfunction binarySearch(nums, target) {\n    const n = nums.length;\n    // 문제 f(0, n-1)을 푼다\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.ts<pre><code>/* 이진 탐색: 문제 f(i, j) */\nfunction dfs(nums: number[], target: number, i: number, j: number): number {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if (i &gt; j) {\n        return -1;\n    }\n    // 중간 인덱스 m을 계산한다\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m;\n    }\n}\n\n/* 이진 탐색 */\nfunction binarySearch(nums: number[], target: number): number {\n    const n = nums.length;\n    // 문제 f(0, n-1)을 푼다\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.dart<pre><code>/* 이진 탐색: 문제 f(i, j) */\nint dfs(List&lt;int&gt; nums, int target, int i, int j) {\n  // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n  if (i &gt; j) {\n    return -1;\n  }\n  // 중간 인덱스 m을 계산한다\n  int m = (i + j) ~/ 2;\n  if (nums[m] &lt; target) {\n    // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n    return dfs(nums, target, m + 1, j);\n  } else if (nums[m] &gt; target) {\n    // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n    return dfs(nums, target, i, m - 1);\n  } else {\n    // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n    return m;\n  }\n}\n\n/* 이진 탐색 */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  int n = nums.length;\n  // 문제 f(0, n-1)을 푼다\n  return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.rs<pre><code>/* 이진 탐색: 문제 f(i, j) */\nfn dfs(nums: &amp;[i32], target: i32, i: i32, j: i32) -&gt; i32 {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if i &gt; j {\n        return -1;\n    }\n    let m: i32 = i + (j - i) / 2;\n    if nums[m as usize] &lt; target {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m + 1, j);\n    } else if nums[m as usize] &gt; target {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m;\n    }\n}\n\n/* 이진 탐색 */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let n = nums.len() as i32;\n    // 문제 f(0, n-1)을 푼다\n    dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.c<pre><code>/* 이진 탐색: 문제 f(i, j) */\nint dfs(int nums[], int target, int i, int j) {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if (i &gt; j) {\n        return -1;\n    }\n    // 중간 인덱스 m을 계산한다\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m;\n    }\n}\n\n/* 이진 탐색 */\nint binarySearch(int nums[], int target, int numsSize) {\n    int n = numsSize;\n    // 문제 f(0, n-1)을 푼다\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.kt<pre><code>/* 이진 탐색: 문제 f(i, j) */\nfun dfs(\n    nums: IntArray,\n    target: Int,\n    i: Int,\n    j: Int\n): Int {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if (i &gt; j) {\n        return -1\n    }\n    // 중간 인덱스 m을 계산한다\n    val m = (i + j) / 2\n    return if (nums[m] &lt; target) {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        dfs(nums, target, m + 1, j)\n    } else if (nums[m] &gt; target) {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        dfs(nums, target, i, m - 1)\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        m\n    }\n}\n\n/* 이진 탐색 */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    val n = nums.size\n    // 문제 f(0, n-1)을 푼다\n    return dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.rb<pre><code>### 이진 탐색: 문제 f(i, j) ###\ndef dfs(nums, target, i, j)\n  # 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n  return -1 if i &gt; j\n\n  # 중간 인덱스 m을 계산한다\n  m = (i + j) / 2\n\n  if nums[m] &lt; target\n    # 부분 문제 f(m+1, j)를 재귀적으로 푼다\n    return dfs(nums, target, m + 1, j)\n  elsif nums[m] &gt; target\n    # 부분 문제 f(i, m-1)를 재귀적으로 푼다\n    return dfs(nums, target, i, m - 1)\n  else\n    # 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n    return m\n  end\nend\n\n### 이진 탐색 ###\ndef binary_search(nums, target)\n  n = nums.length\n  # 문제 f(0, n-1)을 푼다\n  dfs(nums, target, 0, n - 1)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 12 장   분할 정복","12.2   분할 정복 탐색 전략"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/","level":1,"title":"12.3   이진 트리 구성 문제","text":"<p>문제</p> <p>이진 트리의 전위 순회 <code>preorder</code>와 중위 순회 <code>inorder</code>가 주어진다. 이를 이용해 이진 트리를 구성하고 루트 노드를 반환하라. 트리에는 값이 중복되는 노드가 없다고 가정한다(아래 그림(12-5) 참고).</p> <p></p> <p> 그림 12-5   이진 트리 구성 예시 데이터 </p>","path":["제 12 장   분할 정복","12.3   이진 트리 구성 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#1","level":3,"title":"1.   분할 정복 문제인지 판단","text":"<p>원래 문제는 <code>preorder</code>와 <code>inorder</code>로 이진 트리를 구성하는 것으로, 전형적인 분할 정복 문제다.</p> <ul> <li>문제를 나눌 수 있다: 원래 문제를 왼쪽 서브트리 구성과 오른쪽 서브트리 구성이라는 두 부분 문제로 나누고, 루트 노드 초기화 연산을 하나 더할 수 있다. 각 서브트리(부분 문제)에도 같은 방법을 적용해 더 작은 서브트리(부분 문제)로 나눌 수 있으며, 가장 작은 부분 문제인 빈 서브트리에 도달하면 종료한다.</li> <li>부분 문제가 독립적이다: 왼쪽 서브트리와 오른쪽 서브트리는 겹치는 부분 없이 서로 독립적이다. 왼쪽 서브트리를 구성할 때는 중위 순회와 전위 순회에서 왼쪽 서브트리에 해당하는 부분만 보면 된다. 오른쪽도 마찬가지다.</li> <li>부분 문제의 해를 합칠 수 있다: 왼쪽 서브트리와 오른쪽 서브트리(부분 문제의 해)를 얻으면 루트 노드에 연결하여 원래 문제의 해를 만들 수 있다.</li> </ul>","path":["제 12 장   분할 정복","12.3   이진 트리 구성 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#2","level":3,"title":"2.   서브트리를 나누는 방법","text":"<p>위 분석에 따르면 분할 정복으로 해결할 수 있다. 그런데 전위 순회 <code>preorder</code>와 중위 순회 <code>inorder</code>로 왼쪽과 오른쪽 서브트리를 어떻게 나눌까?</p> <p>정의에 따르면 <code>preorder</code>와 <code>inorder</code>는 모두 세 부분으로 나뉜다.</p> <ul> <li>전위 순회: <code>[ 루트 노드 | 왼쪽 서브트리 | 오른쪽 서브트리 ]</code>. 예를 들어 위 그림(12-5)의 트리는 <code>[ 3 | 9 | 2 1 7 ]</code>에 해당한다.</li> <li>중위 순회: <code>[ 왼쪽 서브트리 | 루트 노드 | 오른쪽 서브트리 ]</code>. 예를 들어 위 그림(12-5)의 트리는 <code>[ 9 | 3 | 1 2 7 ]</code>에 해당한다.</li> </ul> <p>위 그림(12-5)의 데이터를 예로 들면 다음 그림(12-6)의 단계로 나눌 수 있다.</p> <ol> <li>전위 순회의 첫 원소 3은 루트 노드의 값이다.</li> <li><code>inorder</code>에서 루트 노드 3의 인덱스를 찾아 이를 기준으로 <code>inorder</code>를 <code>[ 9 | 3 | 1 2 7 ]</code>로 나눈다.</li> <li><code>inorder</code>의 분할 결과로 왼쪽 서브트리의 노드 수는 1, 오른쪽은 3임을 알 수 있다. 따라서 <code>preorder</code>를 <code>[ 3 | 9 | 2 1 7 ]</code>로 나눌 수 있다.</li> </ol> <p></p> <p> 그림 12-6   전위 순회와 중위 순회에서 서브트리 나누기 </p>","path":["제 12 장   분할 정복","12.3   이진 트리 구성 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#3","level":3,"title":"3.   변수로 서브트리 구간 표현","text":"<p>위 방법으로 <code>preorder</code>와 <code>inorder</code>에서 루트 노드, 왼쪽 서브트리, 오른쪽 서브트리의 인덱스 구간을 구했다. 이 구간들을 표현하려면 몇 개의 포인터 변수가 필요하다.</p> <ul> <li>현재 트리의 루트 노드가 <code>preorder</code>에서 갖는 인덱스를 \\(i\\)라고 한다.</li> <li>현재 트리의 루트 노드가 <code>inorder</code>에서 갖는 인덱스를 \\(m\\)이라고 한다.</li> <li>현재 트리가 <code>inorder</code>에서 차지하는 인덱스 구간을 \\([l, r]\\)이라고 한다.</li> </ul> <p>아래 표(12-1)와 같이 이 변수들로 <code>preorder</code>에서 루트 노드의 인덱스와 <code>inorder</code>에서 서브트리의 인덱스 구간을 나타낼 수 있다.</p> <p> 표 12-1   전위 순회와 중위 순회에서 루트 노드와 서브트리의 인덱스 </p> <code>preorder</code>에서 루트 노드의 인덱스 <code>inorder</code>에서 서브트리의 인덱스 구간 현재 트리 \\(i\\) \\([l, r]\\) 왼쪽 서브트리 \\(i + 1\\) \\([l, m-1]\\) 오른쪽 서브트리 \\(i + 1 + (m - l)\\) \\([m+1, r]\\) <p>오른쪽 서브트리의 루트 인덱스에 있는 \\((m-l)\\)은 '왼쪽 서브트리의 노드 수'를 뜻한다. 아래 그림(12-7)과 함께 이해해 보자.</p> <p></p> <p> 그림 12-7   루트 노드와 양쪽 서브트리의 인덱스 구간 표현 </p>","path":["제 12 장   분할 정복","12.3   이진 트리 구성 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#4","level":3,"title":"4.   코드 구현","text":"<p>\\(m\\)을 더 효율적으로 조회하기 위해 해시 테이블 <code>hmap</code>에 배열 <code>inorder</code>의 원소와 인덱스 사이의 매핑을 저장한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby build_tree.py<pre><code>def dfs(\n    preorder: list[int],\n    inorder_map: dict[int, int],\n    i: int,\n    l: int,\n    r: int,\n) -&gt; TreeNode | None:\n    \"\"\"이진 트리 구성: 분할 정복\"\"\"\n    # 서브트리 구간이 비어 있으면 종료한다\n    if r - l &lt; 0:\n        return None\n    # 루트 노드를 초기화한다\n    root = TreeNode(preorder[i])\n    # m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    m = inorder_map[preorder[i]]\n    # 부분 문제: 왼쪽 서브트리 구성\n    root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n    # 부분 문제: 오른쪽 서브트리 구성\n    root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n    # 루트 노드를 반환한다\n    return root\n\ndef build_tree(preorder: list[int], inorder: list[int]) -&gt; TreeNode | None:\n    \"\"\"이진 트리 구성\"\"\"\n    # inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    inorder_map = {val: i for i, val in enumerate(inorder)}\n    root = dfs(preorder, inorder_map, 0, 0, len(inorder) - 1)\n    return root\n</code></pre> build_tree.cpp<pre><code>/* 이진 트리 구성: 분할 정복 */\nTreeNode *dfs(vector&lt;int&gt; &amp;preorder, unordered_map&lt;int, int&gt; &amp;inorderMap, int i, int l, int r) {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if (r - l &lt; 0)\n        return NULL;\n    // 루트 노드를 초기화한다\n    TreeNode *root = new TreeNode(preorder[i]);\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    int m = inorderMap[preorder[i]];\n    // 부분 문제: 왼쪽 서브트리 구성\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // 부분 문제: 오른쪽 서브트리 구성\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // 루트 노드를 반환한다\n    return root;\n}\n\n/* 이진 트리 구성 */\nTreeNode *buildTree(vector&lt;int&gt; &amp;preorder, vector&lt;int&gt; &amp;inorder) {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    unordered_map&lt;int, int&gt; inorderMap;\n    for (int i = 0; i &lt; inorder.size(); i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorder.size() - 1);\n    return root;\n}\n</code></pre> build_tree.java<pre><code>/* 이진 트리 구성: 분할 정복 */\nTreeNode dfs(int[] preorder, Map&lt;Integer, Integer&gt; inorderMap, int i, int l, int r) {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if (r - l &lt; 0)\n        return null;\n    // 루트 노드를 초기화한다\n    TreeNode root = new TreeNode(preorder[i]);\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    int m = inorderMap.get(preorder[i]);\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // 루트 노드를 반환한다\n    return root;\n}\n\n/* 이진 트리 구성 */\nTreeNode buildTree(int[] preorder, int[] inorder) {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    Map&lt;Integer, Integer&gt; inorderMap = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; inorder.length; i++) {\n        inorderMap.put(inorder[i], i);\n    }\n    TreeNode root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.cs<pre><code>/* 이진 트리 구성: 분할 정복 */\nTreeNode? DFS(int[] preorder, Dictionary&lt;int, int&gt; inorderMap, int i, int l, int r) {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if (r - l &lt; 0)\n        return null;\n    // 루트 노드를 초기화한다\n    TreeNode root = new(preorder[i]);\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    int m = inorderMap[preorder[i]];\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.left = DFS(preorder, inorderMap, i + 1, l, m - 1);\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.right = DFS(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // 루트 노드를 반환한다\n    return root;\n}\n\n/* 이진 트리 구성 */\nTreeNode? BuildTree(int[] preorder, int[] inorder) {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    Dictionary&lt;int, int&gt; inorderMap = [];\n    for (int i = 0; i &lt; inorder.Length; i++) {\n        inorderMap.TryAdd(inorder[i], i);\n    }\n    TreeNode? root = DFS(preorder, inorderMap, 0, 0, inorder.Length - 1);\n    return root;\n}\n</code></pre> build_tree.go<pre><code>/* 이진 트리 구성: 분할 정복 */\nfunc dfsBuildTree(preorder []int, inorderMap map[int]int, i, l, r int) *TreeNode {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if r-l &lt; 0 {\n        return nil\n    }\n    // 루트 노드를 초기화한다\n    root := NewTreeNode(preorder[i])\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    m := inorderMap[preorder[i]]\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.Left = dfsBuildTree(preorder, inorderMap, i+1, l, m-1)\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.Right = dfsBuildTree(preorder, inorderMap, i+1+m-l, m+1, r)\n    // 루트 노드를 반환한다\n    return root\n}\n\n/* 이진 트리 구성 */\nfunc buildTree(preorder, inorder []int) *TreeNode {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    inorderMap := make(map[int]int, len(inorder))\n    for i := 0; i &lt; len(inorder); i++ {\n        inorderMap[inorder[i]] = i\n    }\n\n    root := dfsBuildTree(preorder, inorderMap, 0, 0, len(inorder)-1)\n    return root\n}\n</code></pre> build_tree.swift<pre><code>/* 이진 트리 구성: 분할 정복 */\nfunc dfs(preorder: [Int], inorderMap: [Int: Int], i: Int, l: Int, r: Int) -&gt; TreeNode? {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if r - l &lt; 0 {\n        return nil\n    }\n    // 루트 노드를 초기화한다\n    let root = TreeNode(x: preorder[i])\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    let m = inorderMap[preorder[i]]!\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.left = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1, l: l, r: m - 1)\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.right = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1 + m - l, l: m + 1, r: r)\n    // 루트 노드를 반환한다\n    return root\n}\n\n/* 이진 트리 구성 */\nfunc buildTree(preorder: [Int], inorder: [Int]) -&gt; TreeNode? {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    let inorderMap = inorder.enumerated().reduce(into: [:]) { $0[$1.element] = $1.offset }\n    return dfs(preorder: preorder, inorderMap: inorderMap, i: inorder.startIndex, l: inorder.startIndex, r: inorder.endIndex - 1)\n}\n</code></pre> build_tree.js<pre><code>/* 이진 트리 구성: 분할 정복 */\nfunction dfs(preorder, inorderMap, i, l, r) {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if (r - l &lt; 0) return null;\n    // 루트 노드를 초기화한다\n    const root = new TreeNode(preorder[i]);\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    const m = inorderMap.get(preorder[i]);\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // 루트 노드를 반환한다\n    return root;\n}\n\n/* 이진 트리 구성 */\nfunction buildTree(preorder, inorder) {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    let inorderMap = new Map();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.ts<pre><code>/* 이진 트리 구성: 분할 정복 */\nfunction dfs(\n    preorder: number[],\n    inorderMap: Map&lt;number, number&gt;,\n    i: number,\n    l: number,\n    r: number\n): TreeNode | null {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if (r - l &lt; 0) return null;\n    // 루트 노드를 초기화한다\n    const root: TreeNode = new TreeNode(preorder[i]);\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    const m = inorderMap.get(preorder[i]);\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // 루트 노드를 반환한다\n    return root;\n}\n\n/* 이진 트리 구성 */\nfunction buildTree(preorder: number[], inorder: number[]): TreeNode | null {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    let inorderMap = new Map&lt;number, number&gt;();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.dart<pre><code>/* 이진 트리 구성: 분할 정복 */\nTreeNode? dfs(\n  List&lt;int&gt; preorder,\n  Map&lt;int, int&gt; inorderMap,\n  int i,\n  int l,\n  int r,\n) {\n  // 서브트리 구간이 비어 있으면 종료한다\n  if (r - l &lt; 0) {\n    return null;\n  }\n  // 루트 노드를 초기화한다\n  TreeNode? root = TreeNode(preorder[i]);\n  // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n  int m = inorderMap[preorder[i]]!;\n  // 부분 문제: 왼쪽 서브트리 구성\n  root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n  // 부분 문제: 오른쪽 서브트리 구성\n  root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n  // 루트 노드를 반환한다\n  return root;\n}\n\n/* 이진 트리 구성 */\nTreeNode? buildTree(List&lt;int&gt; preorder, List&lt;int&gt; inorder) {\n  // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n  Map&lt;int, int&gt; inorderMap = {};\n  for (int i = 0; i &lt; inorder.length; i++) {\n    inorderMap[inorder[i]] = i;\n  }\n  TreeNode? root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n  return root;\n}\n</code></pre> build_tree.rs<pre><code>/* 이진 트리 구성: 분할 정복 */\nfn dfs(\n    preorder: &amp;[i32],\n    inorder_map: &amp;HashMap&lt;i32, i32&gt;,\n    i: i32,\n    l: i32,\n    r: i32,\n) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if r - l &lt; 0 {\n        return None;\n    }\n    // 루트 노드를 초기화한다\n    let root = TreeNode::new(preorder[i as usize]);\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    let m = inorder_map.get(&amp;preorder[i as usize]).unwrap();\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.borrow_mut().left = dfs(preorder, inorder_map, i + 1, l, m - 1);\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.borrow_mut().right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r);\n    // 루트 노드를 반환한다\n    Some(root)\n}\n\n/* 이진 트리 구성 */\nfn build_tree(preorder: &amp;[i32], inorder: &amp;[i32]) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    let mut inorder_map: HashMap&lt;i32, i32&gt; = HashMap::new();\n    for i in 0..inorder.len() {\n        inorder_map.insert(inorder[i], i as i32);\n    }\n    let root = dfs(preorder, &amp;inorder_map, 0, 0, inorder.len() as i32 - 1);\n    root\n}\n</code></pre> build_tree.c<pre><code>/* 이진 트리 구성: 분할 정복 */\nTreeNode *dfs(int *preorder, int *inorderMap, int i, int l, int r, int size) {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if (r - l &lt; 0)\n        return NULL;\n    // 루트 노드를 초기화한다\n    TreeNode *root = (TreeNode *)malloc(sizeof(TreeNode));\n    root-&gt;val = preorder[i];\n    root-&gt;left = NULL;\n    root-&gt;right = NULL;\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    int m = inorderMap[preorder[i]];\n    // 부분 문제: 왼쪽 서브트리 구성\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1, size);\n    // 부분 문제: 오른쪽 서브트리 구성\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r, size);\n    // 루트 노드를 반환한다\n    return root;\n}\n\n/* 이진 트리 구성 */\nTreeNode *buildTree(int *preorder, int preorderSize, int *inorder, int inorderSize) {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    int *inorderMap = (int *)malloc(sizeof(int) * MAX_SIZE);\n    for (int i = 0; i &lt; inorderSize; i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorderSize - 1, inorderSize);\n    free(inorderMap);\n    return root;\n}\n</code></pre> build_tree.kt<pre><code>/* 이진 트리 구성: 분할 정복 */\nfun dfs(\n    preorder: IntArray,\n    inorderMap: Map&lt;Int?, Int?&gt;,\n    i: Int,\n    l: Int,\n    r: Int\n): TreeNode? {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if (r - l &lt; 0) return null\n    // 루트 노드를 초기화한다\n    val root = TreeNode(preorder[i])\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    val m = inorderMap[preorder[i]]!!\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1)\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r)\n    // 루트 노드를 반환한다\n    return root\n}\n\n/* 이진 트리 구성 */\nfun buildTree(preorder: IntArray, inorder: IntArray): TreeNode? {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    val inorderMap = HashMap&lt;Int?, Int?&gt;()\n    for (i in inorder.indices) {\n        inorderMap[inorder[i]] = i\n    }\n    val root = dfs(preorder, inorderMap, 0, 0, inorder.size - 1)\n    return root\n}\n</code></pre> build_tree.rb<pre><code>### 이진 트리 구성: 분할 정복 ###\ndef dfs(preorder, inorder_map, i, l, r)\n  # 서브트리 구간이 비어 있으면 종료한다\n  return if r - l &lt; 0\n\n  # 루트 노드를 초기화한다\n  root = TreeNode.new(preorder[i])\n  # m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n  m = inorder_map[preorder[i]]\n  # 부분 문제: 왼쪽 서브트리 구성\n  root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n  # 부분 문제: 오른쪽 서브트리 구성\n  root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n\n  # 루트 노드를 반환한다\n  root\nend\n\n### 이진 트리 구성 ###\ndef build_tree(preorder, inorder)\n  # inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n  inorder_map = {}\n  inorder.each_with_index { |val, i| inorder_map[val] = i }\n  dfs(preorder, inorder_map, 0, 0, inorder.length - 1)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(12-8)은 이진 트리 구성의 재귀 과정을 보여 준다. 각 노드는 아래로 '호출'하는 과정에서 만들어지고, 간선(참조)은 위로 '반환'하는 과정에서 만들어진다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 12-8   이진 트리 구성의 재귀 과정 </p> <p>각 재귀 함수에서 전위 순회 <code>preorder</code>와 중위 순회 <code>inorder</code>를 나눈 결과는 아래 그림(12-9)과 같다.</p> <p></p> <p> 그림 12-9   각 재귀 함수의 분할 결과 </p> <p>트리의 노드 수가 \\(n\\)일 때 노드 하나를 초기화하는 데(재귀 함수 <code>dfs()</code> 한 번 실행) \\(O(1)\\) 시간이 걸린다. 따라서 전체 시간 복잡도는 \\(O(n)\\)이다.</p> <p>해시 테이블은 <code>inorder</code> 원소와 인덱스의 매핑을 저장하므로 공간 복잡도가 \\(O(n)\\)이다. 최악의 경우인 이진 트리가 연결 리스트로 퇴화한 경우에는 재귀 깊이가 \\(n\\)에 이르러 \\(O(n)\\)의 스택 프레임 공간을 사용한다. 따라서 전체 공간 복잡도는 \\(O(n)\\)이다.</p>","path":["제 12 장   분할 정복","12.3   이진 트리 구성 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/","level":1,"title":"12.1   분할 정복 알고리즘","text":"<p>분할 정복(divide and conquer)은 '나누어 해결한다'는 뜻으로, 매우 중요하고 흔히 쓰이는 알고리즘 전략이다. 보통 재귀로 구현하며 '분할'과 '정복'의 두 단계로 이루어진다.</p> <ol> <li>분할(나누는 단계): 원래 문제를 둘 이상의 부분 문제로 재귀적으로 나누고, 가장 작은 부분 문제에 도달하면 멈춘다.</li> <li>정복(합치는 단계): 해를 이미 아는 가장 작은 부분 문제부터 시작하여 아래에서 위로 부분 문제의 해를 합침으로써 원래 문제의 해를 구성한다.</li> </ol> <p>아래 그림(12-1)의 '병합 정렬'은 분할 정복 전략의 대표적인 응용이다.</p> <ol> <li>분할: 원래 배열(원래 문제)을 두 부분 배열(부분 문제)로 재귀적으로 나누고, 부분 배열에 원소가 하나만 남으면(가장 작은 부분 문제) 멈춘다.</li> <li>정복: 정렬된 부분 배열(부분 문제의 해)을 아래에서 위로 합쳐 정렬된 원래 배열(원래 문제의 해)을 얻는다.</li> </ol> <p></p> <p> 그림 12-1   병합 정렬의 분할 정복 전략 </p>","path":["제 12 장   분할 정복","12.1   분할 정복 알고리즘"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1211","level":2,"title":"12.1.1   분할 정복 문제를 판단하는 방법","text":"<p>문제가 분할 정복에 적합한지는 보통 다음 기준으로 판단할 수 있다.</p> <ol> <li>문제를 나눌 수 있다: 원래 문제를 크기가 더 작고 형태가 비슷한 부분 문제로 나눌 수 있으며, 같은 방식으로 재귀적으로 나눌 수 있다.</li> <li>부분 문제가 독립적이다: 부분 문제들이 겹치지 않고 서로 의존하지 않으며, 독립적으로 해결할 수 있다.</li> <li>부분 문제의 해를 합칠 수 있다: 부분 문제의 해를 합쳐 원래 문제의 해를 얻는다.</li> </ol> <p>병합 정렬은 위 세 기준을 모두 만족한다.</p> <ol> <li>문제를 나눌 수 있다: 배열(원래 문제)을 두 부분 배열(부분 문제)로 재귀적으로 나눈다.</li> <li>부분 문제가 독립적이다: 각 부분 배열을 독립적으로 정렬할 수 있다(부분 문제를 독립적으로 해결할 수 있다).</li> <li>부분 문제의 해를 합칠 수 있다: 정렬된 두 부분 배열(부분 문제의 해)을 하나의 정렬된 배열(원래 문제의 해)로 합칠 수 있다.</li> </ol>","path":["제 12 장   분할 정복","12.1   분할 정복 알고리즘"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1212","level":2,"title":"12.1.2   분할 정복으로 효율 높이기","text":"<p>분할 정복은 알고리즘 문제를 효과적으로 해결할 뿐 아니라 효율을 높이는 경우도 많다. 퀵 정렬, 병합 정렬, 힙 정렬이 선택 정렬, 버블 정렬, 삽입 정렬보다 빠른 이유도 분할 정복 전략을 사용하기 때문이다.</p> <p>여기서 궁금해진다. 분할 정복이 알고리즘의 효율을 높이는 근본적인 원리는 무엇일까? 큰 문제를 여러 부분 문제로 나누고, 부분 문제를 해결하고, 그 해를 합쳐 원래 문제의 해를 구하는 과정이 왜 원래 문제를 직접 푸는 것보다 효율적일까? 연산 수와 병렬 계산이라는 두 관점에서 살펴볼 수 있다.</p>","path":["제 12 장   분할 정복","12.1   분할 정복 알고리즘"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1","level":3,"title":"1.   연산 수 최적화","text":"<p>'버블 정렬'은 길이 \\(n\\)의 배열을 처리하는 데 \\(O(n^2)\\) 시간이 걸린다. 아래 그림(12-2)처럼 중간에서 두 부분 배열로 나눈다고 하자. 분할에는 \\(O(n)\\) 시간, 각 부분 배열의 정렬에는 \\(O((n / 2)^2)\\) 시간, 두 부분 배열의 병합에는 \\(O(n)\\) 시간이 걸리므로 전체 시간 복잡도는 다음과 같다.</p> \\[ O(n + (\\frac{n}{2})^2 \\times 2 + n) = O(\\frac{n^2}{2} + 2n) \\] <p></p> <p> 그림 12-2   배열 분할 전후의 버블 정렬 </p> <p>이어서 왼쪽은 분할 전, 오른쪽은 분할 후의 총연산 수를 나타내는 다음 부등식을 계산해 보자.</p> \\[ \\begin{aligned} n^2 &amp; &gt; \\frac{n^2}{2} + 2n \\newline n^2 - \\frac{n^2}{2} - 2n &amp; &gt; 0 \\newline n(n - 4) &amp; &gt; 0 \\end{aligned} \\] <p>이는 \\(n &gt; 4\\)일 때 분할 후의 연산 수가 더 적어 정렬 효율이 높아질 것임을 뜻한다. 분할 후에도 시간 복잡도는 여전히 이차인 \\(O(n^2)\\)이며, 상수항만 작아졌다는 점에 주의하자.</p> <p>여기서 더 나아가 부분 배열을 계속 중간에서 두 개로 나누고, 원소가 하나만 남을 때까지 반복하면 어떨까? 이 발상은 바로 '병합 정렬'이며 시간 복잡도는 \\(O(n \\log n)\\)이다.</p> <p>다시 생각해 보자. 분할 지점을 더 많이 두어 원래 배열을 \\(k\\) 개의 부분 배열로 고르게 나누면 어떨까? 이는 '버킷 정렬'과 매우 비슷하다. 대규모 데이터 정렬에 적합하며 이론적으로 시간 복잡도가 \\(O(n + k)\\)에 이를 수 있다.</p>","path":["제 12 장   분할 정복","12.1   분할 정복 알고리즘"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#2","level":3,"title":"2.   병렬 계산 최적화","text":"<p>분할 정복으로 만들어진 부분 문제는 서로 독립적이므로 보통 병렬로 해결할 수 있다. 즉, 분할 정복은 알고리즘의 시간 복잡도를 줄일 뿐 아니라 운영체제의 병렬 최적화에도 유리하다.</p> <p>병렬 최적화는 멀티코어나 다중 프로세서 환경에서 특히 효과적이다. 여러 부분 문제를 동시에 처리하여 계산 자원을 더 충분히 활용하고 전체 실행 시간을 크게 줄일 수 있기 때문이다.</p> <p>예를 들어 아래 그림(12-3)의 '버킷 정렬'에서 대규모 데이터를 각 버킷에 고르게 분배하면, 버킷별 정렬 작업을 여러 계산 장치에 나누어 맡긴 뒤 결과를 합칠 수 있다.</p> <p></p> <p> 그림 12-3   버킷 정렬의 병렬 계산 </p>","path":["제 12 장   분할 정복","12.1   분할 정복 알고리즘"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1213","level":2,"title":"12.1.3   분할 정복의 주요 응용","text":"<p>분할 정복은 여러 대표적인 알고리즘 문제를 해결할 수 있다.</p> <ul> <li>가장 가까운 점 쌍 찾기: 먼저 점 집합을 두 부분으로 나누고 각 부분에서 가장 가까운 점 쌍을 찾은 뒤, 마지막으로 두 부분에 걸쳐 있는 가장 가까운 점 쌍을 찾는다.</li> <li>큰 정수의 곱셈: Karatsuba 알고리즘은 큰 정수의 곱셈을 더 작은 정수들의 곱셈과 덧셈으로 나눈다.</li> <li>행렬 곱셈: Strassen 알고리즘은 큰 행렬의 곱셈을 작은 행렬들의 곱셈과 덧셈으로 나눈다.</li> <li>하노이의 탑 문제: 재귀로 해결할 수 있는 하노이의 탑은 분할 정복 전략의 대표적인 응용이다.</li> <li>역전 쌍 구하기: 수열에서 앞의 수가 뒤의 수보다 크면 두 수가 역전 쌍을 이룬다. 역전 쌍 문제는 분할 정복의 아이디어로 병합 정렬을 이용하여 해결할 수 있다.</li> </ul> <p>또한 분할 정복은 알고리즘과 자료 구조의 설계에도 매우 널리 쓰인다.</p> <ul> <li>이진 탐색: 정렬된 배열을 중간 인덱스에서 두 부분으로 나누고, 목표 값과 중간 원소를 비교한 결과에 따라 제외할 절반을 정한다. 남은 구간에서 같은 이진 탐색 연산을 수행한다.</li> <li>병합 정렬: 이 절의 앞부분에서 소개했으므로 다시 설명하지 않는다.</li> <li>퀵 정렬: 피벗을 하나 고르고 피벗보다 작은 원소와 큰 원소를 각각 두 부분 배열로 나눈다. 부분 배열에 원소가 하나만 남을 때까지 같은 분할을 반복한다.</li> <li>버킷 정렬: 데이터를 여러 버킷에 분산하고 각 버킷의 원소를 정렬한 뒤, 버킷별 원소를 차례로 꺼내 정렬된 배열을 얻는다.</li> <li>트리: 이진 탐색 트리, AVL 트리, 레드-블랙 트리, B 트리, B+ 트리 등의 탐색, 삽입, 삭제 연산은 모두 분할 정복 전략의 응용으로 볼 수 있다.</li> <li>힙: 특수한 완전 이진 트리인 힙의 삽입, 삭제, 재정렬 등에도 분할 정복의 아이디어가 숨어 있다.</li> <li>해시 테이블: 분할 정복을 직접 적용하지는 않지만 일부 해시 충돌 해결 방법에는 간접적으로 사용된다. 예를 들어 체이닝의 긴 연결 리스트를 레드-블랙 트리로 바꾸어 조회 효율을 높인다.</li> </ul> <p>이처럼 분할 정복은 조용히 스며들어 작용하는 알고리즘 사고방식이며, 다양한 알고리즘과 자료 구조 속에 숨어 있다.</p>","path":["제 12 장   분할 정복","12.1   분할 정복 알고리즘"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/","level":1,"title":"12.6   연습 문제","text":"","path":["제 12 장   분할 정복","12.6   연습 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1261","level":2,"title":"12.6.1   개념 확인","text":"","path":["제 12 장   분할 정복","12.6   연습 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1","level":3,"title":"1.   어떤 작업에 분할 정복이 적합한가","text":"<p>한 학생이 ‘먼저 둘로 나누고, 각각 해결한 뒤, 결과를 합치는’ 방식으로 다음 작업을 수행하려 한다. 각 작업이 ‘분할 정복에 적합함’, ‘분할 정복이 가능하지만 전체 작업량은 줄어들지 않음’, ‘두 부분을 독립적으로 해결할 수 없음’ 중 어디에 해당하는지 판단하고 이유를 설명하라.</p> <ol> <li>정렬되지 않은 배열을 정렬한다.</li> <li>배열의 최댓값을 구한다.</li> <li>일련의 스택 연산 <code>push(x)</code>, <code>pop()</code>을 순서대로 실행하고 각 <code>pop()</code>에서 얻은 원소를 출력한다.</li> </ol> 참고 답안 <ol> <li>적합함: 반으로 나눈 뒤 두 부분을 독립적으로 정렬하고, \\(O(n)\\) 시간에 병합한다. 이것이 바로 병합 정렬이다.</li> <li>분할 정복이 가능하지만 전체 작업량은 줄어들지 않음: 왼쪽과 오른쪽을 합치면 여전히 \\(n\\)개 원소를 모두 확인해야 하므로,     직접 순회하는 것과 마찬가지로 \\(O(n)\\) 시간이 든다.</li> <li>두 부분을 독립적으로 해결할 수 없음: 후반부를 시작할 때의 스택 내용은 전반부의 실행 결과에 따라 달라진다.     따라서 서로의 결과를 모르는 상태에서 두 부분을 독립적으로 완료할 수 없다.</li> </ol>","path":["제 12 장   분할 정복","12.6   연습 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#2","level":3,"title":"2.   빠른 거듭제곱은 계산을 어떻게 줄이는가","text":"<p>다음 재귀 함수는 분할 정복으로 \\(x^n\\)을 계산한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby fast_power.py<pre><code>def fast_pow(x: int, n: int) -&gt; int:\n    \"\"\"빠른 거듭제곱\"\"\"\n    if n == 0:\n        return 1\n    half = fast_pow(x, n // 2)\n    if n % 2 == 0:\n        return half * half\n    return half * half * x\n</code></pre> fast_power.cpp<pre><code>/* 빠른 거듭제곱 */\nint fastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = fastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.java<pre><code>/* 빠른 거듭제곱 */\nint fastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = fastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.cs<pre><code>/* 빠른 거듭제곱 */\nint FastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = FastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.go<pre><code>/* 빠른 거듭제곱 */\nfunc fastPow(x, n int) int {\n    if n == 0 {\n        return 1\n    }\n    half := fastPow(x, n/2)\n    if n%2 == 0 {\n        return half * half\n    }\n    return half * half * x\n}\n</code></pre> fast_power.swift<pre><code>/* 빠른 거듭제곱 */\nfunc fastPow(x: Int, n: Int) -&gt; Int {\n    if n == 0 {\n        return 1\n    }\n    let half = fastPow(x: x, n: n / 2)\n    if n % 2 == 0 {\n        return half * half\n    }\n    return half * half * x\n}\n</code></pre> fast_power.js<pre><code>/* 빠른 거듭제곱 */\nfunction fastPow(x, n) {\n    if (n === 0) {\n        return 1;\n    }\n    const half = fastPow(x, Math.floor(n / 2));\n    if (n % 2 === 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.ts<pre><code>/* 빠른 거듭제곱 */\nfunction fastPow(x: number, n: number): number {\n    if (n === 0) {\n        return 1;\n    }\n    const half = fastPow(x, Math.floor(n / 2));\n    if (n % 2 === 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.dart<pre><code>/* 빠른 거듭제곱 */\nint fastPow(int x, int n) {\n  if (n == 0) {\n    return 1;\n  }\n  int half = fastPow(x, n ~/ 2);\n  if (n % 2 == 0) {\n    return half * half;\n  }\n  return half * half * x;\n}\n</code></pre> fast_power.rs<pre><code>/* 빠른 거듭제곱 */\nfn fast_pow(x: i32, n: i32) -&gt; i32 {\n    if n == 0 {\n        return 1;\n    }\n    let half = fast_pow(x, n / 2);\n    if n % 2 == 0 {\n        return half * half;\n    }\n    half * half * x\n}\n</code></pre> fast_power.c<pre><code>/* 빠른 거듭제곱 */\nint fastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = fastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.kt<pre><code>/* 빠른 거듭제곱 */\nfun fastPow(x: Int, n: Int): Int {\n    if (n == 0) {\n        return 1\n    }\n    val half = fastPow(x, n / 2)\n    if (n % 2 == 0) {\n        return half * half\n    }\n    return half * half * x\n}\n</code></pre> fast_power.rb<pre><code>### 빠른 거듭제곱 ###\ndef fast_pow(x, n)\n  return 1 if n == 0\n\n  half = fast_pow(x, n / 2)\n  return half * half if n.even?\n\n  half * half * x\nend\n</code></pre> <p><code>x = 3</code>, <code>n = 5</code>로 설정하고 이 함수를 실행한다.</p> <ol> <li>재귀 호출에서 매개변수 <code>n</code>은 차례대로 어떤 값이 되는가?</li> <li>가장 깊은 호출부터 반환할 때 각 호출은 차례대로 어떤 값을 반환하는가?</li> <li>곱셈 양쪽에서 같은 부분 문제를 한 번씩 호출하지 않고, 먼저 재귀 호출의 결과를 <code>half</code>에 저장하는 이유는 무엇인가?</li> </ol> 참고 답안 <ol> <li> <p>매개변수는 <code>5 → 2 → 1 → 0</code> 순으로 바뀐다. 종료 조건에 도달할 때까지 매번 지수를 절반으로 줄인다.</p> </li> <li> <p><code>n = 0</code>이면 1을 반환한다. <code>n = 1</code>이면 \\(1×1×3=3\\)을 반환한다.     <code>n = 2</code>이면 \\(3×3=9\\)를, <code>n = 5</code>이면 \\(9×9×3=243\\)을 반환한다.</p> </li> <li> <p>곱셈 양쪽에서 같은 부분 문제를 한 번씩 호출하면 두 재귀 호출은 완전히 같은 계산을 수행한다.     먼저 결과를 <code>half</code>에 저장하면 각 단계에서 재귀 호출을 한 번만 하면 되고, 재귀 깊이는 대략 \\(\\log n\\)이다.     두 번 호출하면 같은 계산이 대량으로 반복된다.</p> </li> </ol>","path":["제 12 장   분할 정복","12.6   연습 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#3","level":3,"title":"3.   순회 결과에서 왼쪽 서브트리와 오른쪽 서브트리 나누기","text":"<p>중복된 노드가 없는 이진 트리의 전위 순회와 중위 순회 결과는 다음과 같다.</p> <ul> <li>전위 순회: <code>[A, B, D, E, C]</code></li> <li>중위 순회: <code>[D, B, E, A, C]</code></li> </ul> <p>루트 노드 수준에서만 나누고, 더 깊이 재귀적으로 나누거나 트리 전체를 그릴 필요는 없다.</p> <ol> <li>루트 노드는 무엇인가?</li> <li>중위 순회에서 왼쪽 서브트리와 오른쪽 서브트리에 해당하는 구간은 각각 어디인가?</li> <li>전위 순회에서 왼쪽 서브트리와 오른쪽 서브트리에 해당하는 구간은 각각 어디인가? 루트 노드의 직접적인 자식은 누구인가?</li> </ol> 참고 답안 <ol> <li> <p>전위 순회의 첫 번째 노드가 루트 노드이므로 루트는 <code>A</code>다.</p> </li> <li> <p><code>A</code>를 기준으로 중위 순회가 두 부분으로 나뉜다. 왼쪽 서브트리는 <code>[D, B, E]</code>, 오른쪽 서브트리는 <code>[C]</code>다.</p> </li> <li> <p>왼쪽 서브트리에 노드가 3개 있으므로 전위 순회에서 루트 노드 <code>A</code> 다음의 원소 3개가 왼쪽 서브트리에 속한다.     즉 <code>[B, D, E]</code>가 왼쪽 서브트리이고, 나머지 <code>[C]</code>가 오른쪽 서브트리다.     따라서 루트 노드의 왼쪽 자식은 <code>B</code>, 오른쪽 자식은 <code>C</code>다.</p> </li> </ol>","path":["제 12 장   분할 정복","12.6   연습 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1262","level":2,"title":"12.6.2   프로그래밍 연습","text":"","path":["제 12 장   분할 정복","12.6   연습 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1_1","level":3,"title":"1.   빠른 거듭제곱","text":"<p>실수 <code>x</code>와 정수 <code>n</code>이 주어진다. 언어에서 제공하는 거듭제곱 함수를 호출하지 않고 \\(x^n\\)을 계산하라. 재귀적인 분할 정복을 사용하여 매번 지수를 절반으로 줄이고, 이미 계산한 부분 문제의 결과를 재사용해야 한다. 이 문제에서는 <code>x = 0</code>인 경우를 포함하여 \\(x^0=1\\)로 정한다. <code>n &lt; 0</code>일 때는 <code>x != 0</code>임이 보장되며, 답을 \\((1/x)^{-n}\\)으로 바꾸어 구할 수 있다.</p> 풀이 힌트 <ol> <li>n이 0이면 답은 1이다.</li> <li>x의 n // 2제곱을 계산하여 half에 저장한다. 같은 재귀 호출을 두 번 하지 마라.</li> <li>n &lt; 0이면 먼저 x를 1 / x로 바꾸고 n을 -n으로 바꾼다. C++이나 Java에서는 n을 먼저 64비트 정수로 변환하면 가장 작은 32비트 정수의 부호를 반대로 바꿀 때 발생하는 오버플로를 피할 수 있다.</li> </ol> <p>LeetCode</p>","path":["제 12 장   분할 정복","12.6   연습 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/","level":1,"title":"12.4   하노이의 탑 문제","text":"<p>병합 정렬과 이진 트리 구성에서는 원래 문제를 크기가 절반인 두 부분 문제로 나누었다. 하지만 하노이의 탑 문제에서는 다른 분할 전략을 사용한다.</p> <p>문제</p> <p>기둥 세 개를 <code>A</code>, <code>B</code>, <code>C</code>라고 하자. 처음에는 기둥 <code>A</code>에 원판 \\(n\\) 개가 위에서 아래로 작은 것부터 큰 것 순서로 꽂혀 있다. 목표는 원래 순서를 유지하면서 이 원판 \\(n\\) 개를 기둥 <code>C</code>로 옮기는 것이다(아래 그림(12-10) 참고). 원판을 옮길 때는 다음 규칙을 지켜야 한다.</p> <ol> <li>원판은 한 기둥의 맨 위에서만 꺼내어 다른 기둥의 맨 위에 놓을 수 있다.</li> <li>한 번에 원판 하나만 옮길 수 있다.</li> <li>작은 원판은 항상 큰 원판 위에 있어야 한다.</li> </ol> <p></p> <p> 그림 12-10   하노이의 탑 문제 예시 </p> <p>크기가 \\(i\\)인 하노이의 탑 문제를 \\(f(i)\\)라고 하자. 예를 들어 \\(f(3)\\)은 원판 \\(3\\) 개를 <code>A</code>에서 <code>C</code>로 옮기는 문제다.</p>","path":["제 12 장   분할 정복","12.4   하노이의 탑 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/#1","level":3,"title":"1.   기본 경우 살펴보기","text":"<p>아래 그림(12-11)과 같이 \\(f(1)\\), 즉 원판이 하나뿐이면 <code>A</code>에서 <code>C</code>로 바로 옮기면 된다.</p> &lt;1&gt;&lt;2&gt; <p></p> <p></p> <p> 그림 12-11   크기 1인 문제의 해 </p> <p>아래 그림(12-12)과 같이 \\(f(2)\\), 즉 원판이 두 개이면 작은 원판이 항상 큰 원판 위에 있어야 하므로 <code>B</code>를 이용해야 한다.</p> <ol> <li>위의 작은 원판을 <code>A</code>에서 <code>B</code>로 옮긴다.</li> <li>큰 원판을 <code>A</code>에서 <code>C</code>로 옮긴다.</li> <li>작은 원판을 <code>B</code>에서 <code>C</code>로 옮긴다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> 그림 12-12   크기 2인 문제의 해 </p> <p>\\(f(2)\\)의 해결 과정은 <code>B</code>를 이용하여 원판 두 개를 <code>A</code>에서 <code>C</code>로 옮기는 것으로 요약할 수 있다. 여기서 <code>C</code>를 목표 기둥, <code>B</code>를 보조 기둥이라고 한다.</p>","path":["제 12 장   분할 정복","12.4   하노이의 탑 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/#2","level":3,"title":"2.   부분 문제로 나누기","text":"<p>\\(f(3)\\), 즉 원판이 세 개이면 상황이 조금 더 복잡해진다.</p> <p>\\(f(1)\\)과 \\(f(2)\\)의 해를 이미 알고 있으므로 분할 정복의 관점에서 생각해 보자. <code>A</code> 위쪽의 원판 두 개를 하나의 묶음으로 보고 아래 그림(12-13)의 단계를 수행하면 세 원판을 <code>A</code>에서 <code>C</code>로 옮길 수 있다.</p> <ol> <li><code>B</code>를 목표 기둥, <code>C</code>를 보조 기둥으로 삼아 원판 두 개를 <code>A</code>에서 <code>B</code>로 옮긴다.</li> <li><code>A</code>에 남은 원판 하나를 <code>C</code>로 바로 옮긴다.</li> <li><code>C</code>를 목표 기둥, <code>A</code>를 보조 기둥으로 삼아 원판 두 개를 <code>B</code>에서 <code>C</code>로 옮긴다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> 그림 12-13   크기 3인 문제의 해 </p> <p>본질적으로 문제 \\(f(3)\\)을 부분 문제 \\(f(2)\\) 두 개와 \\(f(1)\\) 하나로 나눈 것이다. 세 부분 문제를 순서대로 해결하면 원래 문제도 해결된다. 이는 부분 문제들이 독립적이고 해를 합칠 수 있음을 보여 준다.</p> <p>이제 아래 그림(12-14)과 같이 하노이의 탑을 푸는 분할 정복 전략을 정리할 수 있다. 원래 문제 \\(f(n)\\)을 부분 문제 \\(f(n-1)\\) 두 개와 \\(f(1)\\) 하나로 나누고, 다음 순서로 해결한다.</p> <ol> <li><code>C</code>를 이용하여 원판 \\(n-1\\) 개를 <code>A</code>에서 <code>B</code>로 옮긴다.</li> <li>남은 원판 \\(1\\) 개를 <code>A</code>에서 <code>C</code>로 바로 옮긴다.</li> <li><code>A</code>를 이용하여 원판 \\(n-1\\) 개를 <code>B</code>에서 <code>C</code>로 옮긴다.</li> </ol> <p>두 부분 문제 \\(f(n-1)\\)도 같은 방식으로 재귀적으로 나눌 수 있으며, 가장 작은 부분 문제 \\(f(1)\\)에 도달하면 멈춘다. \\(f(1)\\)의 해는 이미 알고 있고, 한 번만 옮기면 된다.</p> <p></p> <p> 그림 12-14   하노이의 탑을 푸는 분할 정복 전략 </p>","path":["제 12 장   분할 정복","12.4   하노이의 탑 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/#3","level":3,"title":"3.   코드 구현","text":"<p>코드에서는 재귀 함수 <code>dfs(i, src, buf, tar)</code>를 정의한다. 이 함수는 기둥 <code>src</code> 위쪽의 원판 \\(i\\) 개를 보조 기둥 <code>buf</code>를 이용하여 목표 기둥 <code>tar</code>로 옮긴다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hanota.py<pre><code>def move(src: list[int], tar: list[int]):\n    \"\"\"원판 하나 이동\"\"\"\n    # src의 맨 위에서 원판 하나를 꺼낸다\n    pan = src.pop()\n    # 원판을 tar의 맨 위에 놓는다\n    tar.append(pan)\n\ndef dfs(i: int, src: list[int], buf: list[int], tar: list[int]):\n    \"\"\"하노이의 탑 문제 f(i) 풀기\"\"\"\n    # src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if i == 1:\n        move(src, tar)\n        return\n    # 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, tar, buf)\n    # 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, tar)\n    # 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, src, tar)\n\ndef solve_hanota(A: list[int], B: list[int], C: list[int]):\n    \"\"\"하노이의 탑 문제 풀기\"\"\"\n    n = len(A)\n    # B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(n, A, B, C)\n</code></pre> hanota.cpp<pre><code>/* 원판 하나 이동 */\nvoid move(vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;tar) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    int pan = src.back();\n    src.pop_back();\n    // 원판을 tar의 맨 위에 놓는다\n    tar.push_back(pan);\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nvoid dfs(int i, vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;buf, vector&lt;int&gt; &amp;tar) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, tar, buf);\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, tar);\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, src, tar);\n}\n\n/* 하노이의 탑 문제 풀기 */\nvoid solveHanota(vector&lt;int&gt; &amp;A, vector&lt;int&gt; &amp;B, vector&lt;int&gt; &amp;C) {\n    int n = A.size();\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.java<pre><code>/* 원판 하나 이동 */\nvoid move(List&lt;Integer&gt; src, List&lt;Integer&gt; tar) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    Integer pan = src.remove(src.size() - 1);\n    // 원판을 tar의 맨 위에 놓는다\n    tar.add(pan);\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nvoid dfs(int i, List&lt;Integer&gt; src, List&lt;Integer&gt; buf, List&lt;Integer&gt; tar) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, tar, buf);\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, tar);\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, src, tar);\n}\n\n/* 하노이의 탑 문제 풀기 */\nvoid solveHanota(List&lt;Integer&gt; A, List&lt;Integer&gt; B, List&lt;Integer&gt; C) {\n    int n = A.size();\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.cs<pre><code>/* 원판 하나 이동 */\nvoid Move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    int pan = src[^1];\n    src.RemoveAt(src.Count - 1);\n    // 원판을 tar의 맨 위에 놓는다\n    tar.Add(pan);\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nvoid DFS(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if (i == 1) {\n        Move(src, tar);\n        return;\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    DFS(i - 1, src, tar, buf);\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    Move(src, tar);\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    DFS(i - 1, buf, src, tar);\n}\n\n/* 하노이의 탑 문제 풀기 */\nvoid SolveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n    int n = A.Count;\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    DFS(n, A, B, C);\n}\n</code></pre> hanota.go<pre><code>/* 원판 하나 이동 */\nfunc move(src, tar *list.List) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    pan := src.Back()\n    // 원판을 tar의 맨 위에 놓는다\n    tar.PushBack(pan.Value)\n    // src의 맨 위 원판을 제거한다\n    src.Remove(pan)\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nfunc dfsHanota(i int, src, buf, tar *list.List) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if i == 1 {\n        move(src, tar)\n        return\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfsHanota(i-1, src, tar, buf)\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, tar)\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfsHanota(i-1, buf, src, tar)\n}\n\n/* 하노이의 탑 문제 풀기 */\nfunc solveHanota(A, B, C *list.List) {\n    n := A.Len()\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfsHanota(n, A, B, C)\n}\n</code></pre> hanota.swift<pre><code>/* 원판 하나 이동 */\nfunc move(src: inout [Int], tar: inout [Int]) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    let pan = src.popLast()!\n    // 원판을 tar의 맨 위에 놓는다\n    tar.append(pan)\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nfunc dfs(i: Int, src: inout [Int], buf: inout [Int], tar: inout [Int]) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if i == 1 {\n        move(src: &amp;src, tar: &amp;tar)\n        return\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i: i - 1, src: &amp;src, buf: &amp;tar, tar: &amp;buf)\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src: &amp;src, tar: &amp;tar)\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i: i - 1, src: &amp;buf, buf: &amp;src, tar: &amp;tar)\n}\n\n/* 하노이의 탑 문제 풀기 */\nfunc solveHanota(A: inout [Int], B: inout [Int], C: inout [Int]) {\n    let n = A.count\n    // 리스트의 맨 뒤가 기둥의 맨 위이다\n    // B를 이용하여 src의 맨 위 원판 n개를 C로 옮긴다\n    dfs(i: n, src: &amp;A, buf: &amp;B, tar: &amp;C)\n}\n</code></pre> hanota.js<pre><code>/* 원판 하나 이동 */\nfunction move(src, tar) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    const pan = src.pop();\n    // 원판을 tar의 맨 위에 놓는다\n    tar.push(pan);\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nfunction dfs(i, src, buf, tar) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, tar, buf);\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, tar);\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, src, tar);\n}\n\n/* 하노이의 탑 문제 풀기 */\nfunction solveHanota(A, B, C) {\n    const n = A.length;\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.ts<pre><code>/* 원판 하나 이동 */\nfunction move(src: number[], tar: number[]): void {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    const pan = src.pop();\n    // 원판을 tar의 맨 위에 놓는다\n    tar.push(pan);\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nfunction dfs(i: number, src: number[], buf: number[], tar: number[]): void {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, tar, buf);\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, tar);\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, src, tar);\n}\n\n/* 하노이의 탑 문제 풀기 */\nfunction solveHanota(A: number[], B: number[], C: number[]): void {\n    const n = A.length;\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.dart<pre><code>/* 원판 하나 이동 */\nvoid move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n  // src의 맨 위에서 원판 하나를 꺼낸다\n  int pan = src.removeLast();\n  // 원판을 tar의 맨 위에 놓는다\n  tar.add(pan);\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nvoid dfs(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n  // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n  if (i == 1) {\n    move(src, tar);\n    return;\n  }\n  // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n  dfs(i - 1, src, tar, buf);\n  // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n  move(src, tar);\n  // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n  dfs(i - 1, buf, src, tar);\n}\n\n/* 하노이의 탑 문제 풀기 */\nvoid solveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n  int n = A.length;\n  // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n  dfs(n, A, B, C);\n}\n</code></pre> hanota.rs<pre><code>/* 원판 하나 이동 */\nfn move_pan(src: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    let pan = src.pop().unwrap();\n    // 원판을 tar의 맨 위에 놓는다\n    tar.push(pan);\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nfn dfs(i: i32, src: &amp;mut Vec&lt;i32&gt;, buf: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if i == 1 {\n        move_pan(src, tar);\n        return;\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, tar, buf);\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move_pan(src, tar);\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, src, tar);\n}\n\n/* 하노이의 탑 문제 풀기 */\nfn solve_hanota(A: &amp;mut Vec&lt;i32&gt;, B: &amp;mut Vec&lt;i32&gt;, C: &amp;mut Vec&lt;i32&gt;) {\n    let n = A.len() as i32;\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.c<pre><code>/* 원판 하나 이동 */\nvoid move(int *src, int *srcSize, int *tar, int *tarSize) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    int pan = src[*srcSize - 1];\n    src[*srcSize - 1] = 0;\n    (*srcSize)--;\n    // 원판을 tar의 맨 위에 놓는다\n    tar[*tarSize] = pan;\n    (*tarSize)++;\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nvoid dfs(int i, int *src, int *srcSize, int *buf, int *bufSize, int *tar, int *tarSize) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if (i == 1) {\n        move(src, srcSize, tar, tarSize);\n        return;\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, srcSize, tar, tarSize, buf, bufSize);\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, srcSize, tar, tarSize);\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, bufSize, src, srcSize, tar, tarSize);\n}\n\n/* 하노이의 탑 문제 풀기 */\nvoid solveHanota(int *A, int *ASize, int *B, int *BSize, int *C, int *CSize) {\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(*ASize, A, ASize, B, BSize, C, CSize);\n}\n</code></pre> hanota.kt<pre><code>/* 원판 하나 이동 */\nfun move(src: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    val pan = src.removeAt(src.size - 1)\n    // 원판을 tar의 맨 위에 놓는다\n    tar.add(pan)\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nfun dfs(i: Int, src: MutableList&lt;Int&gt;, buf: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if (i == 1) {\n        move(src, tar)\n        return\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, tar, buf)\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, tar)\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, src, tar)\n}\n\n/* 하노이의 탑 문제 풀기 */\nfun solveHanota(A: MutableList&lt;Int&gt;, B: MutableList&lt;Int&gt;, C: MutableList&lt;Int&gt;) {\n    val n = A.size\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(n, A, B, C)\n}\n</code></pre> hanota.rb<pre><code>### 원판 하나 이동 ###\ndef move(src, tar)\n  # src의 맨 위에서 원판 하나를 꺼낸다\n  pan = src.pop\n  # 원판을 tar의 맨 위에 놓는다\n  tar &lt;&lt; pan\nend\n\n### 하노이의 탑 문제 f(i) 풀기 ###\ndef dfs(i, src, buf, tar)\n  # src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n  if i == 1\n    move(src, tar)\n    return\n  end\n\n  # 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n  dfs(i - 1, src, tar, buf)\n  # 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n  move(src, tar)\n  # 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n  dfs(i - 1, buf, src, tar)\nend\n\n### 하노이의 탑 문제 풀기 ###\ndef solve_hanota(_A, _B, _C)\n  n = _A.length\n  # B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n  dfs(n, _A, _B, _C)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(12-15)과 같이 하노이의 탑 문제는 높이가 \\(n\\)인 재귀 트리를 이룬다. 각 노드는 부분 문제 하나를 나타내며 호출된 <code>dfs()</code> 함수 하나에 대응한다. 따라서 시간 복잡도는 \\(O(2^n)\\), 공간 복잡도는 \\(O(n)\\)이다.</p> <p></p> <p> 그림 12-15   하노이의 탑 문제의 재귀 트리 </p> <p>하노이의 탑 전설</p> <p>하노이의 탑 문제에는 오래된 전설이 전해진다. 고대 인도의 한 사원에서 승려들이 세 개의 높은 다이아몬드 기둥과 크기가 서로 다른 금 원판 \\(64\\) 개를 가지고 있었다. 승려들은 원판을 계속 옮기며 마지막 원판이 올바른 위치에 놓이는 순간 세상이 끝난다고 믿었다.</p> <p>하지만 매초 원판을 하나씩 옮겨도 총 \\(2^{64} \\approx 1.84×10^{19}\\) 초, 약 \\(5850\\) 억 년이 필요하다. 이는 현재 추정하는 우주의 나이보다 훨씬 길다. 따라서 전설이 사실이라고 해도 세상의 종말을 걱정할 필요는 없을 것이다.</p>","path":["제 12 장   분할 정복","12.4   하노이의 탑 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/summary/","level":1,"title":"12.5   요약","text":"","path":["제 12 장   분할 정복","12.5   요약"],"tags":[]},{"location":"chapter_divide_and_conquer/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>분할 정복은 흔히 쓰이는 알고리즘 설계 전략이다. 분할(나누기)과 정복(합치기)의 두 단계로 이루어지며 보통 재귀로 구현한다.</li> <li>분할 정복 문제를 판단하는 기준은 문제를 나눌 수 있는지, 부분 문제들이 독립적인지, 부분 문제의 해를 합칠 수 있는지다.</li> <li>병합 정렬은 분할 정복 전략의 대표적인 응용이다. 배열을 같은 길이의 두 부분 배열로 재귀적으로 나누고, 원소가 하나만 남으면 단계적으로 병합하여 정렬한다.</li> <li>분할 정복 전략을 도입하면 알고리즘 효율이 높아지는 경우가 많다. 연산 수를 줄일 수 있고, 시스템의 병렬 최적화에도 유리하기 때문이다.</li> <li>분할 정복은 여러 알고리즘 문제를 해결할 뿐 아니라 자료 구조와 알고리즘의 설계에도 널리 사용되어 곳곳에서 찾아볼 수 있다.</li> <li>적응형 탐색은 완전 탐색보다 효율적이다. 시간 복잡도가 \\(O(\\log n)\\)인 탐색 알고리즘은 보통 분할 정복 전략을 기반으로 구현된다.</li> <li>이진 탐색은 분할 정복 전략의 또 다른 대표적인 응용으로, 부분 문제의 해를 합치는 단계가 없다. 재귀적 분할 정복으로 이진 탐색을 구현할 수 있다.</li> <li>이진 트리 구성 문제에서는 트리를 구성하는 원래 문제를 왼쪽 서브트리와 오른쪽 서브트리를 구성하는 부분 문제로 나눌 수 있다. 전위 순회와 중위 순회의 인덱스 구간을 나누어 구현한다.</li> <li>하노이의 탑에서는 크기 \\(n\\)인 문제를 크기 \\(n-1\\)인 부분 문제 두 개와 크기 \\(1\\)인 부분 문제 하나로 나눌 수 있다. 세 부분 문제를 순서대로 해결하면 원래 문제도 해결된다.</li> </ul>","path":["제 12 장   분할 정복","12.5   요약"],"tags":[]},{"location":"chapter_dynamic_programming/","level":1,"title":"제 14 장   동적 계획법","text":"<p>이 장을 시작하며</p> <p>시냇물은 강으로 모이고, 강물은 바다로 흘러든다.</p> <p>동적 계획법은 작은 문제의 해를 모아 큰 문제의 답을 만들며, 한 걸음씩 문제 해결의 건너편으로 우리를 이끈다.</p>","path":["제 14 장   동적 계획법"],"tags":[]},{"location":"chapter_dynamic_programming/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>14.1   동적 계획법 첫걸음</li> <li>14.2   동적 계획법 문제의 특성</li> <li>14.3   동적 계획법 문제 풀이 방법</li> <li>14.4   0-1 배낭 문제</li> <li>14.5   무한 배낭 문제</li> <li>14.6   편집 거리 문제</li> <li>14.7   요약</li> <li>14.8   연습 문제</li> </ul>","path":["제 14 장   동적 계획법"],"tags":[]},{"location":"chapter_dynamic_programming/dp_problem_features/","level":1,"title":"14.2   동적 계획법 문제의 특성","text":"<p>앞 절에서는 부분 문제로 나누어 원래 문제를 푸는 동적 계획법을 배웠다. 부분 문제로 나누는 것은 일반적인 알고리즘 아이디어지만 분할 정복, 동적 계획법, 백트래킹에서는 강조점이 다르다.</p> <ul> <li>분할 정복은 원래 문제를 독립적인 여러 부분 문제로 재귀적으로 나누어 가장 작은 부분 문제까지 내려간다. 이후 되돌아오면서 해를 합쳐 원래 문제의 해를 얻는다.</li> <li>동적 계획법도 문제를 재귀적으로 나누지만, 분할 정복과 달리 부분 문제들이 서로 의존하며 분해 과정에서 중복 부분 문제가 많이 생긴다.</li> <li>백트래킹은 시도와 되돌리기로 가능한 모든 해를 열거하고 가지치기로 불필요한 탐색 가지를 피한다. 원래 문제의 해는 일련의 의사 결정 단계로 이루어지며, 각 결정 단계에 앞선 부분 수열을 부분 문제로 볼 수 있다.</li> </ul> <p>동적 계획법은 최적화 문제에 자주 쓰인다. 이러한 문제에는 중복 부분 문제뿐 아니라 최적 부분 구조와 무후효성이라는 두 가지 특성이 더 있다.</p>","path":["제 14 장   동적 계획법","14.2   동적 계획법 문제의 특성"],"tags":[]},{"location":"chapter_dynamic_programming/dp_problem_features/#1421","level":2,"title":"14.2.1   최적 부분 구조","text":"<p>최적 부분 구조의 개념을 더 잘 보여 주도록 계단 오르기 문제를 조금 바꿔 보자.</p> <p>계단 오르기의 최소 비용</p> <p>계단을 한 번에 \\(1\\) 칸 또는 \\(2\\) 칸 오를 수 있다. 각 칸에는 그곳에서 지불할 비용인 음이 아닌 정수가 적혀 있다. 음이 아닌 정수 배열 \\(cost\\)가 주어지며, \\(cost[i]\\)는 \\(i\\) 번째 칸의 비용이고 \\(cost[0]\\)은 바닥(출발점)에 해당한다. 맨 위에 도달하는 데 필요한 최소 비용을 구하라.</p> <p>아래 그림(14-6)과 같이 \\(1\\), \\(2\\), \\(3\\) 번째 칸의 비용이 각각 \\(1\\), \\(10\\), \\(1\\)이면 바닥에서 \\(3\\) 번째 칸까지 오르는 최소 비용은 \\(2\\)다.</p> <p></p> <p> 그림 14-6   3번째 칸까지 오르는 최소 비용 </p> <p>\\(i\\) 번째 칸까지 오르는 데 지불한 누적 비용을 \\(dp[i]\\)라고 하자. \\(i\\) 번째 칸에는 \\(i - 1\\) 번째 또는 \\(i - 2\\) 번째 칸에서만 올 수 있으므로 \\(dp[i]\\)는 \\(dp[i - 1] + cost[i]\\) 또는 \\(dp[i - 2] + cost[i]\\)다. 비용을 최소화하려면 둘 중 작은 값을 선택해야 한다.</p> \\[ dp[i] = \\min(dp[i-1], dp[i-2]) + cost[i] \\] <p>여기서 최적 부분 구조의 의미를 알 수 있다. 원래 문제의 최적해는 부분 문제의 최적해로 구성된다.</p> <p>이 문제는 분명히 최적 부분 구조를 가진다. 두 부분 문제의 최적해 \\(dp[i-1]\\)과 \\(dp[i-2]\\) 중 더 좋은 것을 골라 원래 문제 \\(dp[i]\\)의 최적해를 구성하기 때문이다.</p> <p>그렇다면 앞 절의 계단 오르기 문제도 최적 부분 구조를 가질까? 방법 수를 구하는 것이 목표이므로 단순한 계수 문제처럼 보인다. 하지만 '최대 방법 수를 구하라'고 바꿔 보면 문제의 의미는 같은데 최적 부분 구조가 드러난다. \\(n\\) 번째 칸의 최대 방법 수는 \\(n-1\\) 번째와 \\(n-2\\) 번째 칸의 최대 방법 수를 더한 값이다. 이처럼 최적 부분 구조의 해석은 유연하며 문제마다 다른 의미를 가질 수 있다.</p> <p>상태 전이 식과 초기 상태 \\(dp[1] = cost[1]\\), \\(dp[2] = cost[2]\\)를 이용하면 다음 동적 계획법 코드를 얻는다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp(cost: list[int]) -&gt; int:\n    \"\"\"계단 오르기의 최소 비용: 동적 계획법\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    # 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    dp = [0] * (n + 1)\n    # 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1], dp[2] = cost[1], cost[2]\n    # 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in range(3, n + 1):\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    return dp[n]\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nint minCostClimbingStairsDP(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    vector&lt;int&gt; dp(n + 1);\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nint minCostClimbingStairsDP(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int[] dp = new int[n + 1];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nint MinCostClimbingStairsDP(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int[] dp = new int[n + 1];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.Min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nfunc minCostClimbingStairsDP(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    dp := make([]int, n+1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = min(dp[i-1], dp[i-2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nfunc minCostClimbingStairsDP(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    var dp = Array(repeating: 0, count: n + 1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in 3 ... n {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nfunction minCostClimbingStairsDP(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    const dp = new Array(n + 1);\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nfunction minCostClimbingStairsDP(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    const dp = new Array(n + 1);\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nint minCostClimbingStairsDP(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n  dp[1] = cost[1];\n  dp[2] = cost[2];\n  // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n  }\n  return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nfn min_cost_climbing_stairs_dp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    let mut dp = vec![-1; n + 1];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in 3..=n {\n        dp[i] = cmp::min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nint minCostClimbingStairsDP(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int *dp = calloc(n + 1, sizeof(int));\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = myMin(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    int res = dp[n];\n    // 메모리 해제\n    free(dp);\n    return res;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nfun minCostClimbingStairsDP(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    val dp = IntArray(n + 1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (i in 3..n) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>### 계단 오르기의 최소 비용: 동적 계획법 ###\ndef min_cost_climbing_stairs_dp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  # 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n  dp = Array.new(n + 1, 0)\n  # 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n  dp[1], dp[2] = cost[1], cost[2]\n  # 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n  (3...(n + 1)).each { |i| dp[i] = [dp[i - 1], dp[i - 2]].min + cost[i] }\n  dp[n]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-7)은 위 코드의 동적 계획법 과정이다.</p> <p></p> <p> 그림 14-7   계단 오르기 최소 비용의 동적 계획법 과정 </p> <p>이 문제도 공간을 최적화하여 1차원을 0차원으로 압축하고, 공간 복잡도를 \\(O(n)\\)에서 \\(O(1)\\)로 줄일 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp_comp(cost: list[int]) -&gt; int:\n    \"\"\"계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    a, b = cost[1], cost[2]\n    for i in range(3, n + 1):\n        a, b = b, min(a, b) + cost[i]\n    return b\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nint minCostClimbingStairsDPComp(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nint minCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nint MinCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.Min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nfunc minCostClimbingStairsDPComp(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    a, b := cost[1], cost[2]\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i := 3; i &lt;= n; i++ {\n        tmp := b\n        b = min(a, tmp) + cost[i]\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nfunc minCostClimbingStairsDPComp(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    var (a, b) = (cost[1], cost[2])\n    for i in 3 ... n {\n        (a, b) = (b, min(a, b) + cost[i])\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nfunction minCostClimbingStairsDPComp(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nfunction minCostClimbingStairsDPComp(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nint minCostClimbingStairsDPComp(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  int a = cost[1], b = cost[2];\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = min(a, tmp) + cost[i];\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nfn min_cost_climbing_stairs_dp_comp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    };\n    let (mut a, mut b) = (cost[1], cost[2]);\n    for i in 3..=n {\n        let tmp = b;\n        b = cmp::min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    b\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nint minCostClimbingStairsDPComp(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = myMin(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nfun minCostClimbingStairsDPComp(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    var a = cost[1]\n    var b = cost[2]\n    for (i in 3..n) {\n        val tmp = b\n        b = min(a, tmp) + cost[i]\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>### 계단 오르기의 최소 비용: 동적 계획법 ###\ndef min_cost_climbing_stairs_dp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  # 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n  dp = Array.new(n + 1, 0)\n  # 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n  dp[1], dp[2] = cost[1], cost[2]\n  # 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n  (3...(n + 1)).each { |i| dp[i] = [dp[i - 1], dp[i - 2]].min + cost[i] }\n  dp[n]\nend\n\n# 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법\ndef min_cost_climbing_stairs_dp_comp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  a, b = cost[1], cost[2]\n  (3...(n + 1)).each { |i| a, b = b, [a, b].min + cost[i] }\n  b\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.2   동적 계획법 문제의 특성"],"tags":[]},{"location":"chapter_dynamic_programming/dp_problem_features/#1422","level":2,"title":"14.2.2   무후효성","text":"<p>무후효성은 동적 계획법이 문제를 효과적으로 해결할 수 있게 하는 중요한 특성이다. 어떤 상태가 주어졌을 때 이후의 전개가 현재 상태에만 의존하고, 과거에 거쳐 온 모든 상태와는 무관하다는 뜻이다.</p> <p>계단 오르기에서 상태 \\(i\\)는 \\(1\\) 칸 또는 \\(2\\) 칸 오르는 선택에 따라 상태 \\(i+1\\) 또는 \\(i+2\\)로 이어진다. 두 선택을 할 때 상태 \\(i\\) 이전의 상태를 고려할 필요는 없다. 과거 상태는 \\(i\\) 이후의 전개에 영향을 주지 않기 때문이다.</p> <p>하지만 계단 오르기에 제약 하나를 추가하면 상황이 달라진다.</p> <p>제약이 있는 계단 오르기</p> <p>총 \\(n\\) 칸의 계단을 한 번에 \\(1\\) 칸 또는 \\(2\\) 칸 오를 수 있다. 단, 연속 두 번 \\(1\\) 칸씩 오를 수는 없다. 맨 위까지 오르는 방법은 몇 가지인가?</p> <p>아래 그림(14-8)에서는 \\(3\\) 번째 칸까지 오르는 유효한 방법이 \\(2\\) 가지만 남는다. \\(1\\) 칸씩 세 번 연속 오르는 방법은 제약 조건을 만족하지 않아 제외된다.</p> <p></p> <p> 그림 14-8   제약이 있을 때 3번째 칸까지 오르는 방법 수 </p> <p>이 문제에서 직전에 \\(1\\) 칸 올라왔다면 다음에는 반드시 \\(2\\) 칸 올라야 한다. 즉, 다음 선택은 현재 상태(현재 칸 번호)만으로 정해지지 않고 이전 상태(직전에 있던 칸 번호)에도 의존한다.</p> <p>따라서 기존 상태 정의로는 무후효성을 만족하지 않으며, 상태 전이 식 \\(dp[i] = dp[i-1] + dp[i-2]\\)도 더 이상 유효하지 않다. \\(dp[i-1]\\)은 이번에 \\(1\\) 칸 오르는 경우인데, 그 안에는 '직전에도 \\(1\\) 칸 올라온' 방법이 많이 들어 있다. 제약을 만족하려면 \\(dp[i-1]\\)을 그대로 \\(dp[i]\\)에 더할 수 없다.</p> <p>이를 위해 상태 정의를 확장해야 한다. 상태 \\([i, j]\\)는 현재 \\(i\\) 번째 칸에 있고 직전에 \\(j\\) 칸 올라왔음을 나타내며, \\(j \\in \\{1, 2\\}\\)이다. 이렇게 정의하면 직전에 \\(1\\) 칸 왔는지 \\(2\\) 칸 왔는지를 구분하여 현재 상태의 이전 상태를 판단할 수 있다.</p> <ul> <li>직전에 \\(1\\) 칸 왔다면 그 앞에서는 \\(2\\) 칸만 올 수 있다. 즉, \\(dp[i, 1]\\)은 \\(dp[i-1, 2]\\)에서만 전이된다.</li> <li>직전에 \\(2\\) 칸 왔다면 그 앞에서는 \\(1\\) 칸이나 \\(2\\) 칸 모두 가능하다. 즉, \\(dp[i, 2]\\)는 \\(dp[i-2, 1]\\) 또는 \\(dp[i-2, 2]\\)에서 전이된다.</li> </ul> <p>아래 그림(14-9)과 같이 이 정의에서 \\(dp[i, j]\\)는 상태 \\([i, j]\\)의 방법 수를 나타낸다. 상태 전이 식은 다음과 같다.</p> \\[ \\begin{cases} dp[i, 1] = dp[i-1, 2] \\\\ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] \\end{cases} \\] <p></p> <p> 그림 14-9   제약을 고려한 점화 관계 </p> <p>마지막으로 \\(dp[n, 1] + dp[n, 2]\\)를 반환하면 된다. 두 값의 합은 \\(n\\) 번째 칸까지 오르는 전체 방법 수다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_constraint_dp.py<pre><code>def climbing_stairs_constraint_dp(n: int) -&gt; int:\n    \"\"\"제약 조건이 있는 계단 오르기: 동적 계획법\"\"\"\n    if n == 1 or n == 2:\n        return 1\n    # 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    dp = [[0] * 3 for _ in range(n + 1)]\n    # 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1], dp[1][2] = 1, 0\n    dp[2][1], dp[2][2] = 0, 1\n    # 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in range(3, n + 1):\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    return dp[n][1] + dp[n][2]\n</code></pre> climbing_stairs_constraint_dp.cpp<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(3, 0));\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.java<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int[][] dp = new int[n + 1][3];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.cs<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nint ClimbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int[,] dp = new int[n + 1, 3];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1, 1] = 1;\n    dp[1, 2] = 0;\n    dp[2, 1] = 0;\n    dp[2, 2] = 1;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i, 1] = dp[i - 1, 2];\n        dp[i, 2] = dp[i - 2, 1] + dp[i - 2, 2];\n    }\n    return dp[n, 1] + dp[n, 2];\n}\n</code></pre> climbing_stairs_constraint_dp.go<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nfunc climbingStairsConstraintDP(n int) int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    dp := make([][3]int, n+1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i := 3; i &lt;= n; i++ {\n        dp[i][1] = dp[i-1][2]\n        dp[i][2] = dp[i-2][1] + dp[i-2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.swift<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nfunc climbingStairsConstraintDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    var dp = Array(repeating: Array(repeating: 0, count: 3), count: n + 1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in 3 ... n {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.js<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nfunction climbingStairsConstraintDP(n) {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    const dp = Array.from(new Array(n + 1), () =&gt; new Array(3));\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.ts<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nfunction climbingStairsConstraintDP(n: number): number {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(3));\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.dart<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nint climbingStairsConstraintDP(int n) {\n  if (n == 1 || n == 2) {\n    return 1;\n  }\n  // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(3, 0));\n  // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n  dp[1][1] = 1;\n  dp[1][2] = 0;\n  dp[2][1] = 0;\n  dp[2][2] = 1;\n  // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i][1] = dp[i - 1][2];\n    dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n  }\n  return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.rs<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nfn climbing_stairs_constraint_dp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return 1;\n    };\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    let mut dp = vec![vec![-1; 3]; n + 1];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in 3..=n {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.c<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(3, sizeof(int));\n    }\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    int res = dp[n][1] + dp[n][2];\n    // 메모리 해제\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> climbing_stairs_constraint_dp.kt<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nfun climbingStairsConstraintDP(n: Int): Int {\n    if (n == 1 || n == 2) {\n        return 1\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    val dp = Array(n + 1) { IntArray(3) }\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (i in 3..n) {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.rb<pre><code>### 제약 조건이 있는 계단 오르기: 동적 계획법 ###\ndef climbing_stairs_constraint_dp(n)\n  return 1 if n == 1 || n == 2\n\n  # 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n  dp = Array.new(n + 1) { Array.new(3, 0) }\n  # 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n  dp[1][1], dp[1][2] = 1, 0\n  dp[2][1], dp[2][2] = 0, 1\n  # 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n  for i in 3...(n + 1)\n    dp[i][1] = dp[i - 1][2]\n    dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n  end\n\n  dp[n][1] + dp[n][2]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>위 예시는 이전 상태 하나만 추가로 고려하면 되므로 상태 정의를 확장하여 무후효성을 다시 만족시킬 수 있었다. 하지만 일부 문제는 '과거의 영향'이 매우 강하다.</p> <p>계단 오르기와 장애물 생성</p> <p>총 \\(n\\) 칸의 계단을 한 번에 \\(1\\) 칸 또는 \\(2\\) 칸 오를 수 있다. \\(i\\) 번째 칸에 도착하면 시스템이 자동으로 \\(2i\\) 번째 칸에 장애물을 놓으며, 이후의 모든 이동에서는 \\(2i\\) 번째 칸으로 갈 수 없다. 예를 들어 처음 두 번에 \\(2\\), \\(3\\) 번째 칸에 도착했다면 이후에는 \\(4\\), \\(6\\) 번째 칸에 갈 수 없다. 맨 위까지 오르는 방법은 몇 가지인가?</p> <p>이 문제에서 다음 이동은 과거의 모든 상태에 의존한다. 매번 이동할 때마다 더 높은 칸에 장애물이 생겨 이후 이동에 영향을 주기 때문이다. 이런 문제는 동적 계획법으로 해결하기 어려운 경우가 많다.</p> <p>외판원 문제 같은 복잡한 조합 최적화 문제 중에는 무후효성을 만족하지 않는 경우가 많다. 이런 문제에는 보통 휴리스틱 탐색, 유전 알고리즘, 강화 학습 등의 다른 방법을 사용하여 제한된 시간 안에 활용 가능한 국소 최적해를 구한다.</p>","path":["제 14 장   동적 계획법","14.2   동적 계획법 문제의 특성"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/","level":1,"title":"14.3   동적 계획법 문제 풀이 방법","text":"<p>앞의 두 절에서는 동적 계획법 문제의 주요 특성을 살펴보았다. 이제 더 실용적인 두 질문을 알아보자.</p> <ol> <li>어떤 문제가 동적 계획법 문제인지 어떻게 판단할까?</li> <li>동적 계획법 문제는 어디서 시작하며, 전체 풀이 단계는 어떻게 될까?</li> </ol>","path":["제 14 장   동적 계획법","14.3   동적 계획법 문제 풀이 방법"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1431","level":2,"title":"14.3.1   문제 판단","text":"<p>대체로 중복 부분 문제와 최적 부분 구조가 있고 무후효성을 만족하면 동적 계획법에 적합하다. 하지만 문제 설명에서 이러한 특성을 바로 끌어내기는 어렵다. 따라서 보통 조건을 완화하여 먼저 백트래킹(모든 경우 열거)으로 풀기에 적합한지 살펴본다.</p> <p>백트래킹에 적합한 문제는 보통 '의사 결정 트리 모형'을 따른다. 각 노드가 결정 하나를, 각 경로가 결정의 순서를 나타내는 트리 구조로 표현할 수 있는 문제다.</p> <p>즉, 명확한 결정 개념이 있고 연속된 결정을 통해 해가 만들어진다면 의사 결정 트리 모형을 만족하며, 보통 백트래킹으로 해결할 수 있다.</p> <p>여기에 더해 동적 계획법 문제로 판단할 때 '가산점'이 되는 요소도 있다.</p> <ul> <li>최댓값, 최솟값, 가장 많은 수, 가장 적은 수 등의 최적화 표현이 포함되어 있다.</li> <li>상태를 리스트, 다차원 행렬, 트리로 표현할 수 있고, 한 상태와 주변 상태들 사이에 점화 관계가 있다.</li> </ul> <p>반대로 '감점' 요소도 있다.</p> <ul> <li>최적해 하나가 아니라 가능한 모든 해결 방법을 찾는 것이 목표다.</li> <li>문제 설명에 순열이나 조합의 특성이 뚜렷하며, 구체적인 여러 방법을 반환해야 한다.</li> </ul> <p>의사 결정 트리 모형을 만족하면서 '가산점' 요소가 뚜렷하다면, 우선 동적 계획법 문제라고 가정하고 풀이 과정에서 확인해 볼 수 있다.</p>","path":["제 14 장   동적 계획법","14.3   동적 계획법 문제 풀이 방법"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1432","level":2,"title":"14.3.2   문제 해결 단계","text":"<p>동적 계획법의 풀이 흐름은 문제의 성격과 난이도에 따라 다르지만, 보통 결정 설명, 상태 정의, \\(dp\\) 테이블 구성, 상태 전이 식 도출, 경계 조건 결정 등의 단계를 따른다.</p> <p>단계를 더 구체적으로 보여 주기 위해 대표적인 '최소 경로 합' 문제를 예로 든다.</p> <p>문제</p> <p>\\(n \\times m\\)의 2차원 격자 <code>grid</code>가 주어진다. 각 칸에는 그 칸의 비용인 음이 아닌 정수가 들어 있다. 로봇은 왼쪽 위에서 출발하여 매번 아래 또는 오른쪽으로 한 칸씩만 이동하고, 오른쪽 아래에 도착한다. 왼쪽 위에서 오른쪽 아래까지의 최소 경로 합을 반환하라.</p> <p>아래 그림(14-10)의 격자에서는 최소 경로 합이 \\(13\\)이다.</p> <p></p> <p> 그림 14-10   최소 경로 합 예시 데이터 </p> <p>1단계: 매 차례의 결정을 생각하고 상태를 정의하여 \\(dp\\) 테이블을 얻는다</p> <p>이 문제에서 매 차례의 결정은 현재 칸에서 아래 또는 오른쪽으로 한 칸 이동하는 것이다. 현재 칸의 행과 열 인덱스가 \\([i, j]\\)라면 이동 후에는 \\([i+1, j]\\) 또는 \\([i, j+1]\\)이 된다. 따라서 상태는 행 인덱스와 열 인덱스라는 두 변수를 포함해야 하며 \\([i, j]\\)로 표시한다.</p> <p>상태 \\([i, j]\\)의 부분 문제는 출발점 \\([0, 0]\\)에서 \\([i, j]\\)까지의 최소 경로 합이고, 해를 \\(dp[i, j]\\)라고 한다.</p> <p>이로써 아래 그림(14-11)처럼 입력 격자 \\(grid\\)와 크기가 같은 2차원 \\(dp\\) 행렬을 얻는다.</p> <p></p> <p> 그림 14-11   상태 정의와 dp 테이블 </p> <p>참고</p> <p>동적 계획법과 백트래킹 과정은 결정의 순서로 표현할 수 있으며, 상태는 모든 결정 변수로 이루어진다. 상태에는 풀이의 진행 정도를 나타내는 모든 변수가 포함되어야 한다. 즉, 다음 상태를 도출하기에 충분한 정보가 있어야 한다.</p> <p>각 상태는 부분 문제 하나에 대응하며, 모든 부분 문제의 해를 저장하기 위해 \\(dp\\) 테이블을 정의한다. 상태의 독립 변수 각각이 \\(dp\\) 테이블의 한 차원이 된다. 본질적으로 \\(dp\\) 테이블은 상태와 부분 문제의 해 사이의 매핑이다.</p> <p>2단계: 최적 부분 구조를 찾아 상태 전이 식을 도출한다</p> <p>상태 \\([i, j]\\)는 위쪽 칸 \\([i-1, j]\\) 또는 왼쪽 칸 \\([i, j-1]\\)에서만 올 수 있다. 따라서 최적 부분 구조는 다음과 같다. \\([i, j]\\)까지의 최소 경로 합은 \\([i, j-1]\\)까지의 최소 경로 합과 \\([i-1, j]\\)까지의 최소 경로 합 중 작은 쪽에 의해 결정된다.</p> <p>위 분석으로 아래 그림(14-12)의 상태 전이 식을 얻는다.</p> \\[ dp[i, j] = \\min(dp[i-1, j], dp[i, j-1]) + grid[i, j] \\] <p></p> <p> 그림 14-12   최적 부분 구조와 상태 전이 식 </p> <p>참고</p> <p>정의한 \\(dp\\) 테이블을 바탕으로 원래 문제와 부분 문제의 관계를 생각하고, 부분 문제의 최적해로 원래 문제의 최적해를 만드는 방법, 즉 최적 부분 구조를 찾는다.</p> <p>최적 부분 구조를 찾으면 이를 이용해 상태 전이 식을 세울 수 있다.</p> <p>3단계: 경계 조건과 상태 전이 순서를 정한다</p> <p>이 문제에서 첫 행의 상태는 왼쪽 상태에서만, 첫 열의 상태는 위쪽 상태에서만 올 수 있다. 따라서 첫 행 \\(i = 0\\)과 첫 열 \\(j = 0\\)이 경계 조건이다.</p> <p>아래 그림(14-13)과 같이 각 칸은 왼쪽 칸과 위쪽 칸에서 전이되므로, 바깥 반복문은 행을, 안쪽 반복문은 열을 순회한다.</p> <p></p> <p> 그림 14-13   경계 조건과 상태 전이 순서 </p> <p>참고</p> <p>경계 조건은 동적 계획법에서 \\(dp\\) 테이블을 초기화하는 데, 탐색에서는 가지치기에 사용한다.</p> <p>상태 전이 순서의 핵심은 현재 문제의 해를 계산하기 전에 그 해가 의존하는 모든 작은 부분 문제의 해가 올바르게 계산되어 있도록 하는 것이다.</p> <p>이제 동적 계획법 코드를 바로 작성할 수 있다. 하지만 부분 문제로 나누는 사고는 위에서 아래로 진행하므로, '완전 탐색 \\(\\rightarrow\\) 메모이제이션을 이용한 탐색 \\(\\rightarrow\\) 동적 계획법' 순서로 구현하는 편이 사고 흐름에 더 자연스럽다.</p>","path":["제 14 장   동적 계획법","14.3   동적 계획법 문제 풀이 방법"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1-1","level":3,"title":"1.   방법 1: 완전 탐색","text":"<p>상태 \\([i, j]\\)에서 탐색을 시작하여 더 작은 상태 \\([i-1, j]\\)와 \\([i, j-1]\\)로 계속 나눈다. 재귀 함수에는 다음 요소가 포함된다.</p> <ul> <li>재귀 매개변수: 상태 \\([i, j]\\).</li> <li>반환값: \\([0, 0]\\)에서 \\([i, j]\\)까지의 최소 경로 합 \\(dp[i, j]\\).</li> <li>종료 조건: \\(i = 0\\)이고 \\(j = 0\\)이면 비용 \\(grid[0, 0]\\)을 반환한다.</li> <li>가지치기: \\(i &lt; 0\\) 또는 \\(j &lt; 0\\)이면 인덱스가 범위를 벗어나므로 불가능함을 나타내는 비용 \\(+\\infty\\)를 반환한다.</li> </ul> <p>구현 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dfs(grid: list[list[int]], i: int, j: int) -&gt; int:\n    \"\"\"최소 경로 합: 완전 탐색\"\"\"\n    # 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    up = min_path_sum_dfs(grid, i - 1, j)\n    left = min_path_sum_dfs(grid, i, j - 1)\n    # 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return min(left, up) + grid[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* 최소 경로 합: 완전 탐색 */\nint minPathSumDFS(vector&lt;vector&lt;int&gt;&gt; &amp;grid, int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.java<pre><code>/* 최소 경로 합: 완전 탐색 */\nint minPathSumDFS(int[][] grid, int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* 최소 경로 합: 완전 탐색 */\nint MinPathSumDFS(int[][] grid, int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    int up = MinPathSumDFS(grid, i - 1, j);\n    int left = MinPathSumDFS(grid, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return Math.Min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* 최소 경로 합: 완전 탐색 */\nfunc minPathSumDFS(grid [][]int, i, j int) int {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    up := minPathSumDFS(grid, i-1, j)\n    left := minPathSumDFS(grid, i, j-1)\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return int(math.Min(float64(left), float64(up))) + grid[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* 최소 경로 합: 완전 탐색 */\nfunc minPathSumDFS(grid: [[Int]], i: Int, j: Int) -&gt; Int {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    let up = minPathSumDFS(grid: grid, i: i - 1, j: j)\n    let left = minPathSumDFS(grid: grid, i: i, j: j - 1)\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* 최소 경로 합: 완전 탐색 */\nfunction minPathSumDFS(grid, i, j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* 최소 경로 합: 완전 탐색 */\nfunction minPathSumDFS(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i === 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* 최소 경로 합: 완전 탐색 */\nint minPathSumDFS(List&lt;List&lt;int&gt;&gt; grid, int i, int j) {\n  // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n  if (i &lt; 0 || j &lt; 0) {\n    // Dart의 int 타입은 범위가 고정된 정수이며, 무한대를 나타내는 값이 없다\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n  int up = minPathSumDFS(grid, i - 1, j);\n  int left = minPathSumDFS(grid, i, j - 1);\n  // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n  return min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* 최소 경로 합: 완전 탐색 */\nfn min_path_sum_dfs(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    let up = min_path_sum_dfs(grid, i - 1, j);\n    let left = min_path_sum_dfs(grid, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    std::cmp::min(left, up) + grid[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* 최소 경로 합: 완전 탐색 */\nint minPathSumDFS(int grid[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.kt<pre><code>/* 최소 경로 합: 완전 탐색 */\nfun minPathSumDFS(grid: Array&lt;IntArray&gt;, i: Int, j: Int): Int {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    val up = minPathSumDFS(grid, i - 1, j)\n    val left = minPathSumDFS(grid, i, j - 1)\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.rb<pre><code>### 최소 경로 합: 완전 탐색 ###\ndef min_path_sum_dfs(grid, i, j)\n  # 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n  return grid[i][j] if i == 0 &amp;&amp; j == 0\n  # 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n  return Float::INFINITY if i &lt; 0 || j &lt; 0\n  # 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n  up = min_path_sum_dfs(grid, i - 1, j)\n  left = min_path_sum_dfs(grid, i, j - 1)\n  # 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n  [left, up].min + grid[i][j]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-14)은 \\(dp[2, 1]\\)을 루트로 하는 재귀 트리다. 중복 부분 문제가 포함되어 있으며, 그 수는 격자 <code>grid</code>의 크기가 커질수록 급격히 늘어난다.</p> <p>중복 부분 문제가 생기는 근본적인 이유는 왼쪽 위에서 어떤 칸에 도달하는 경로가 여러 개 있기 때문이다.</p> <p></p> <p> 그림 14-14   완전 탐색의 재귀 트리 </p> <p>각 상태에는 아래와 오른쪽이라는 두 선택이 있고, 왼쪽 위에서 오른쪽 아래까지는 총 \\(m + n - 2\\) 번 이동해야 한다. 따라서 최악의 시간 복잡도는 \\(O(2^{m + n})\\)이며, \\(n\\)과 \\(m\\)은 각각 격자의 행 수와 열 수다. 이 계산은 경계 근처의 상황을 고려하지 않았다. 격자 경계에서는 선택이 하나뿐이므로 실제 경로 수는 이보다 적다.</p>","path":["제 14 장   동적 계획법","14.3   동적 계획법 문제 풀이 방법"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#2-2","level":3,"title":"2.   방법 2: 메모이제이션을 이용한 탐색","text":"<p>격자 <code>grid</code>와 같은 크기의 메모이제이션 리스트 <code>mem</code>을 도입하여 각 부분 문제의 해를 기록하고 중복 부분 문제를 가지치기한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dfs_mem(\n    grid: list[list[int]], mem: list[list[int]], i: int, j: int\n) -&gt; int:\n    \"\"\"최소 경로 합: 메모이제이션을 이용한 탐색\"\"\"\n    # 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # 기록이 이미 있으면 바로 반환한다\n    if mem[i][j] != -1:\n        return mem[i][j]\n    # 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n    left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n    # 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nint minPathSumDFSMem(vector&lt;vector&lt;int&gt;&gt; &amp;grid, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.java<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nint minPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nint MinPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    int up = MinPathSumDFSMem(grid, mem, i - 1, j);\n    int left = MinPathSumDFSMem(grid, mem, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = Math.Min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nfunc minPathSumDFSMem(grid, mem [][]int, i, j int) int {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    up := minPathSumDFSMem(grid, mem, i-1, j)\n    left := minPathSumDFSMem(grid, mem, i, j-1)\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = int(math.Min(float64(left), float64(up))) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nfunc minPathSumDFSMem(grid: [[Int]], mem: inout [[Int]], i: Int, j: Int) -&gt; Int {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    let up = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i - 1, j: j)\n    let left = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i, j: j - 1)\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nfunction minPathSumDFSMem(grid, mem, i, j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][j] !== -1) {\n        return mem[i][j];\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nfunction minPathSumDFSMem(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nint minPathSumDFSMem(List&lt;List&lt;int&gt;&gt; grid, List&lt;List&lt;int&gt;&gt; mem, int i, int j) {\n  // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n  if (i &lt; 0 || j &lt; 0) {\n    // Dart의 int 타입은 범위가 고정된 정수이며, 무한대를 나타내는 값이 없다\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // 기록이 이미 있으면 바로 반환한다\n  if (mem[i][j] != -1) {\n    return mem[i][j];\n  }\n  // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n  int up = minPathSumDFSMem(grid, mem, i - 1, j);\n  int left = minPathSumDFSMem(grid, mem, i, j - 1);\n  // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n  mem[i][j] = min(left, up) + grid[i][j];\n  return mem[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nfn min_path_sum_dfs_mem(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if mem[i as usize][j as usize] != -1 {\n        return mem[i as usize][j as usize];\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    let up = min_path_sum_dfs_mem(grid, mem, i - 1, j);\n    let left = min_path_sum_dfs_mem(grid, mem, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];\n    mem[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nint minPathSumDFSMem(int grid[MAX_SIZE][MAX_SIZE], int mem[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.kt<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nfun minPathSumDFSMem(\n    grid: Array&lt;IntArray&gt;,\n    mem: Array&lt;IntArray&gt;,\n    i: Int,\n    j: Int\n): Int {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][j] != -1) {\n        return mem[i][j]\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    val up = minPathSumDFSMem(grid, mem, i - 1, j)\n    val left = minPathSumDFSMem(grid, mem, i, j - 1)\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.rb<pre><code>### 최소 경로 합: 메모이제이션을 이용한 탐색 ###\ndef min_path_sum_dfs_mem(grid, mem, i, j)\n  # 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n  return grid[0][0] if i == 0 &amp;&amp; j == 0\n  # 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n  return Float::INFINITY if i &lt; 0 || j &lt; 0\n  # 기록이 이미 있으면 바로 반환한다\n  return mem[i][j] if mem[i][j] != -1\n  # 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n  up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n  left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n  # 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n  mem[i][j] = [left, up].min + grid[i][j]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-15)과 같이 메모이제이션을 도입하면 모든 부분 문제의 해를 한 번만 계산한다. 따라서 시간 복잡도는 전체 상태 수, 즉 격자 크기인 \\(O(nm)\\)에 의해 결정된다.</p> <p></p> <p> 그림 14-15   메모이제이션을 이용한 탐색의 재귀 트리 </p>","path":["제 14 장   동적 계획법","14.3   동적 계획법 문제 풀이 방법"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#3-3","level":3,"title":"3.   방법 3: 동적 계획법","text":"<p>반복을 이용한 동적 계획법 구현은 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dp(grid: list[list[int]]) -&gt; int:\n    \"\"\"최소 경로 합: 동적 계획법\"\"\"\n    n, m = len(grid), len(grid[0])\n    # dp 테이블을 초기화한다\n    dp = [[0] * m for _ in range(n)]\n    dp[0][0] = grid[0][0]\n    # 상태 전이: 첫 번째 행\n    for j in range(1, m):\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    # 상태 전이: 첫 번째 열\n    for i in range(1, n):\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    # 상태 전이: 나머지 행과 열\n    for i in range(1, n):\n        for j in range(1, m):\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n    return dp[n - 1][m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* 최소 경로 합: 동적 계획법 */\nint minPathSumDP(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // dp 테이블을 초기화한다\n    vector&lt;vector&lt;int&gt;&gt; dp(n, vector&lt;int&gt;(m));\n    dp[0][0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // 상태 전이: 첫 번째 열\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* 최소 경로 합: 동적 계획법 */\nint minPathSumDP(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // dp 테이블을 초기화한다\n    int[][] dp = new int[n][m];\n    dp[0][0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // 상태 전이: 첫 번째 열\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* 최소 경로 합: 동적 계획법 */\nint MinPathSumDP(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // dp 테이블을 초기화한다\n    int[,] dp = new int[n, m];\n    dp[0, 0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt; m; j++) {\n        dp[0, j] = dp[0, j - 1] + grid[0][j];\n    }\n    // 상태 전이: 첫 번째 열\n    for (int i = 1; i &lt; n; i++) {\n        dp[i, 0] = dp[i - 1, 0] + grid[i][0];\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i, j] = Math.Min(dp[i, j - 1], dp[i - 1, j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1, m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* 최소 경로 합: 동적 계획법 */\nfunc minPathSumDP(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // dp 테이블을 초기화한다\n    dp := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        dp[i] = make([]int, m)\n    }\n    dp[0][0] = grid[0][0]\n    // 상태 전이: 첫 번째 행\n    for j := 1; j &lt; m; j++ {\n        dp[0][j] = dp[0][j-1] + grid[0][j]\n    }\n    // 상태 전이: 첫 번째 열\n    for i := 1; i &lt; n; i++ {\n        dp[i][0] = dp[i-1][0] + grid[i][0]\n    }\n    // 상태 전이: 나머지 행과 열\n    for i := 1; i &lt; n; i++ {\n        for j := 1; j &lt; m; j++ {\n            dp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(dp[i-1][j]))) + grid[i][j]\n        }\n    }\n    return dp[n-1][m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* 최소 경로 합: 동적 계획법 */\nfunc minPathSumDP(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: Array(repeating: 0, count: m), count: n)\n    dp[0][0] = grid[0][0]\n    // 상태 전이: 첫 번째 행\n    for j in 1 ..&lt; m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // 상태 전이: 첫 번째 열\n    for i in 1 ..&lt; n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // 상태 전이: 나머지 행과 열\n    for i in 1 ..&lt; n {\n        for j in 1 ..&lt; m {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* 최소 경로 합: 동적 계획법 */\nfunction minPathSumDP(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // 상태 전이: 첫 번째 열\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // 상태 전이: 나머지 행과 열\n    for (let i = 1; i &lt; n; i++) {\n        for (let j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* 최소 경로 합: 동적 계획법 */\nfunction minPathSumDP(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // 상태 전이: 첫 번째 열\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // 상태 전이: 나머지 행과 열\n    for (let i = 1; i &lt; n; i++) {\n        for (let j: number = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* 최소 경로 합: 동적 계획법 */\nint minPathSumDP(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // dp 테이블을 초기화한다\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n, (i) =&gt; List.filled(m, 0));\n  dp[0][0] = grid[0][0];\n  // 상태 전이: 첫 번째 행\n  for (int j = 1; j &lt; m; j++) {\n    dp[0][j] = dp[0][j - 1] + grid[0][j];\n  }\n  // 상태 전이: 첫 번째 열\n  for (int i = 1; i &lt; n; i++) {\n    dp[i][0] = dp[i - 1][0] + grid[i][0];\n  }\n  // 상태 전이: 나머지 행과 열\n  for (int i = 1; i &lt; n; i++) {\n    for (int j = 1; j &lt; m; j++) {\n      dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n    }\n  }\n  return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* 최소 경로 합: 동적 계획법 */\nfn min_path_sum_dp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // dp 테이블을 초기화한다\n    let mut dp = vec![vec![0; m]; n];\n    dp[0][0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for j in 1..m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // 상태 전이: 첫 번째 열\n    for i in 1..n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // 상태 전이: 나머지 행과 열\n    for i in 1..n {\n        for j in 1..m {\n            dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* 최소 경로 합: 동적 계획법 */\nint minPathSumDP(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // dp 테이블을 초기화한다\n    int **dp = malloc(n * sizeof(int *));\n    for (int i = 0; i &lt; n; i++) {\n        dp[i] = calloc(m, sizeof(int));\n    }\n    dp[0][0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // 상태 전이: 첫 번째 열\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = myMin(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    int res = dp[n - 1][m - 1];\n    // 메모리 해제\n    for (int i = 0; i &lt; n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* 최소 경로 합: 동적 계획법 */\nfun minPathSumDP(grid: Array&lt;IntArray&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // dp 테이블을 초기화한다\n    val dp = Array(n) { IntArray(m) }\n    dp[0][0] = grid[0][0]\n    // 상태 전이: 첫 번째 행\n    for (j in 1..&lt;m) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // 상태 전이: 첫 번째 열\n    for (i in 1..&lt;n) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // 상태 전이: 나머지 행과 열\n    for (i in 1..&lt;n) {\n        for (j in 1..&lt;m) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>### 최소 경로 합: 동적 계획법 ###\ndef min_path_sum_dp(grid)\n  n, m = grid.length, grid.first.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(n) { Array.new(m, 0) }\n  dp[0][0] = grid[0][0]\n  # 상태 전이: 첫 번째 행\n  (1...m).each { |j| dp[0][j] = dp[0][j - 1] + grid[0][j] }\n  # 상태 전이: 첫 번째 열\n  (1...n).each { |i| dp[i][0] = dp[i - 1][0] + grid[i][0] }\n  # 상태 전이: 나머지 행과 열\n  for i in 1...n\n    for j in 1...m\n      dp[i][j] = [dp[i][j - 1], dp[i - 1][j]].min + grid[i][j]\n    end\n  end\n  dp[n -1][m -1]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-16)의 최소 경로 합 상태 전이 과정에서는 전체 격자를 순회하므로 시간 복잡도는 \\(O(nm)\\)이다.</p> <p>배열 <code>dp</code>의 크기는 \\(n \\times m\\)이므로 공간 복잡도는 \\(O(nm)\\)이다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 14-16   최소 경로 합의 동적 계획법 과정 </p>","path":["제 14 장   동적 계획법","14.3   동적 계획법 문제 풀이 방법"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#4","level":3,"title":"4.   공간 최적화","text":"<p>각 칸은 왼쪽 칸과 위쪽 칸에만 의존하므로 한 행짜리 배열만으로 \\(dp\\) 테이블을 구현할 수 있다.</p> <p>배열 <code>dp</code>가 한 행의 상태만 나타내므로 첫 열의 상태를 미리 초기화할 수는 없다. 대신 각 행을 순회할 때 갱신해야 한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dp_comp(grid: list[list[int]]) -&gt; int:\n    \"\"\"최소 경로 합: 공간을 최적화한 동적 계획법\"\"\"\n    n, m = len(grid), len(grid[0])\n    # dp 테이블을 초기화한다\n    dp = [0] * m\n    # 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0]\n    for j in range(1, m):\n        dp[j] = dp[j - 1] + grid[0][j]\n    # 상태 전이: 나머지 행\n    for i in range(1, n):\n        # 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0]\n        # 상태 전이: 나머지 열\n        for j in range(1, m):\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n    return dp[m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nint minPathSumDPComp(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // dp 테이블을 초기화한다\n    vector&lt;int&gt; dp(m);\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt; n; i++) {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0];\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nint minPathSumDPComp(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[m];\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt; n; i++) {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0];\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nint MinPathSumDPComp(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[m];\n    dp[0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt; n; i++) {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0];\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.Min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nfunc minPathSumDPComp(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // dp 테이블을 초기화한다\n    dp := make([]int, m)\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0]\n    for j := 1; j &lt; m; j++ {\n        dp[j] = dp[j-1] + grid[0][j]\n    }\n    // 상태 전이: 나머지 행과 열\n    for i := 1; i &lt; n; i++ {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0]\n        // 상태 전이: 나머지 열\n        for j := 1; j &lt; m; j++ {\n            dp[j] = int(math.Min(float64(dp[j-1]), float64(dp[j]))) + grid[i][j]\n        }\n    }\n    return dp[m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nfunc minPathSumDPComp(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: 0, count: m)\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0]\n    for j in 1 ..&lt; m {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // 상태 전이: 나머지 행\n    for i in 1 ..&lt; n {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0]\n        // 상태 전이: 나머지 열\n        for j in 1 ..&lt; m {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nfunction minPathSumDPComp(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // dp 테이블을 초기화한다\n    const dp = new Array(m);\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // 상태 전이: 나머지 행\n    for (let i = 1; i &lt; n; i++) {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0];\n        // 상태 전이: 나머지 열\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nfunction minPathSumDPComp(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // dp 테이블을 초기화한다\n    const dp = new Array(m);\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // 상태 전이: 나머지 행\n    for (let i = 1; i &lt; n; i++) {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0];\n        // 상태 전이: 나머지 열\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nint minPathSumDPComp(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // dp 테이블을 초기화한다\n  List&lt;int&gt; dp = List.filled(m, 0);\n  dp[0] = grid[0][0];\n  for (int j = 1; j &lt; m; j++) {\n    dp[j] = dp[j - 1] + grid[0][j];\n  }\n  // 상태 전이: 나머지 행\n  for (int i = 1; i &lt; n; i++) {\n    // 상태 전이: 첫 번째 열\n    dp[0] = dp[0] + grid[i][0];\n    // 상태 전이: 나머지 열\n    for (int j = 1; j &lt; m; j++) {\n      dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n    }\n  }\n  return dp[m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nfn min_path_sum_dp_comp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // dp 테이블을 초기화한다\n    let mut dp = vec![0; m];\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0];\n    for j in 1..m {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // 상태 전이: 나머지 행\n    for i in 1..n {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0];\n        // 상태 전이: 나머지 열\n        for j in 1..m {\n            dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    dp[m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nint minPathSumDPComp(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // dp 테이블을 초기화한다\n    int *dp = calloc(m, sizeof(int));\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt; n; i++) {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0];\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = myMin(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    int res = dp[m - 1];\n    // 메모리 해제\n    free(dp);\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nfun minPathSumDPComp(grid: Array&lt;IntArray&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // dp 테이블을 초기화한다\n    val dp = IntArray(m)\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0]\n    for (j in 1..&lt;m) {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // 상태 전이: 나머지 행\n    for (i in 1..&lt;n) {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0]\n        // 상태 전이: 나머지 열\n        for (j in 1..&lt;m) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>### 최소 경로 합: 공간을 최적화한 동적 계획법 ###\ndef min_path_sum_dp_comp(grid)\n  n, m = grid.length, grid.first.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(m, 0)\n  # 상태 전이: 첫 번째 행\n  dp[0] = grid[0][0]\n  (1...m).each { |j| dp[j] = dp[j - 1] + grid[0][j] }\n  # 상태 전이: 나머지 행\n  for i in 1...n\n    # 상태 전이: 첫 번째 열\n    dp[0] = dp[0] + grid[i][0]\n    # 상태 전이: 나머지 열\n    (1...m).each { |j| dp[j] = [dp[j - 1], dp[j]].min + grid[i][j] }\n  end\n  dp[m - 1]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.3   동적 계획법 문제 풀이 방법"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/","level":1,"title":"14.6   편집 거리 문제","text":"<p>편집 거리 또는 Levenshtein 거리는 두 문자열을 서로 바꾸는 데 필요한 최소 수정 횟수다. 정보 검색과 자연어 처리에서 두 시퀀스의 유사도를 측정할 때 자주 사용한다.</p> <p>문제</p> <p>두 문자열 \\(s\\)와 \\(t\\)를 입력받아 \\(s\\)를 \\(t\\)로 바꾸는 데 필요한 최소 편집 횟수를 반환하라.</p> <p>문자열에 문자를 하나 삽입하거나, 하나 삭제하거나, 한 문자를 임의의 다른 문자로 바꾸는 세 연산을 할 수 있다.</p> <p>아래 그림(14-27)과 같이 <code>kitten</code>을 <code>sitting</code>으로 바꾸려면 교체 2회와 추가 1회로 총 3회 편집해야 한다. <code>hello</code>를 <code>algo</code>로 바꾸려면 교체 2회와 삭제 1회로 총 3회가 필요하다.</p> <p></p> <p> 그림 14-27   편집 거리 예시 데이터 </p> <p>편집 거리 문제는 의사 결정 트리 모형으로 자연스럽게 설명할 수 있다. 문자열은 트리 노드에, 한 번의 결정(편집 연산)은 간선 하나에 대응한다.</p> <p>아래 그림(14-28)과 같이 연산을 제한하지 않으면 각 노드에서 다양한 연산에 대응하는 많은 간선이 뻗어 나온다. 즉, <code>hello</code>에서 <code>algo</code>로 가는 경로는 매우 많다.</p> <p>의사 결정 트리의 관점에서 이 문제의 목표는 노드 <code>hello</code>와 노드 <code>algo</code> 사이의 최단 경로를 구하는 것이다.</p> <p></p> <p> 그림 14-28   의사 결정 트리로 표현한 편집 거리 문제 </p>","path":["제 14 장   동적 계획법","14.6   편집 거리 문제"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/#1","level":3,"title":"1.   동적 계획법 접근","text":"<p>1단계: 매 차례의 결정을 생각하고 상태를 정의하여 \\(dp\\) 테이블을 얻는다</p> <p>매 차례의 결정은 문자열 \\(s\\)에 편집 연산을 한 번 적용하는 것이다.</p> <p>부분 문제를 만들려면 편집하는 동안 문제의 크기가 점차 줄어들어야 한다. 문자열 \\(s\\)와 \\(t\\)의 길이를 각각 \\(n\\)과 \\(m\\)이라고 하고, 두 문자열의 마지막 문자 \\(s[n-1]\\)과 \\(t[m-1]\\)을 먼저 살펴보자.</p> <ul> <li>\\(s[n-1]\\)과 \\(t[m-1]\\)이 같으면 두 문자를 건너뛰고 \\(s[n-2]\\)와 \\(t[m-2]\\)를 바로 살펴볼 수 있다.</li> <li>\\(s[n-1]\\)과 \\(t[m-1]\\)이 다르면 \\(s\\)에 편집(삽입, 삭제, 교체)을 한 번 적용하여 끝 문자가 같아지게 하고, 이를 건너뛰어 더 작은 문제를 살펴본다.</li> </ul> <p>즉, 문자열 \\(s\\)에서 결정(편집 연산)을 할 때마다 \\(s\\)와 \\(t\\)에서 아직 맞추어야 하는 문자들이 달라진다. 따라서 현재 살펴보는 \\(s\\)의 \\(i\\) 번째 문자와 \\(t\\)의 \\(j\\) 번째 문자를 상태로 정의하고 \\([i, j]\\)라고 한다.</p> <p>상태 \\([i, j]\\)의 부분 문제는 \\(s\\)의 앞 \\(i\\) 개 문자를 \\(t\\)의 앞 \\(j\\) 개 문자로 바꾸는 최소 편집 횟수다.</p> <p>이로써 \\((n+1) \\times (m+1)\\) 크기의 2차원 \\(dp\\) 테이블을 얻는다.</p> <p>2단계: 최적 부분 구조를 찾아 상태 전이 식을 도출한다</p> <p>부분 문제 \\(dp[i, j]\\)에 해당하는 두 문자열의 끝 문자는 \\(s[i-1]\\)과 \\(t[j-1]\\)이다. 편집 연산에 따라 아래 그림(14-29)의 세 경우로 나뉜다.</p> <ol> <li>\\(s[i-1]\\) 뒤에 \\(t[j-1]\\)을 추가하면 부분 문제 \\(dp[i, j-1]\\)이 남는다.</li> <li>\\(s[i-1]\\)을 삭제하면 부분 문제 \\(dp[i-1, j]\\)가 남는다.</li> <li>\\(s[i-1]\\)을 \\(t[j-1]\\)로 바꾸면 부분 문제 \\(dp[i-1, j-1]\\)이 남는다.</li> </ol> <p></p> <p> 그림 14-29   편집 거리의 상태 전이 </p> <p>위 분석으로 최적 부분 구조를 얻는다. \\(dp[i, j]\\)의 최소 편집 횟수는 \\(dp[i, j-1]\\), \\(dp[i-1, j]\\), \\(dp[i-1, j-1]\\) 중 가장 작은 편집 횟수에 이번 편집 횟수 \\(1\\)을 더한 값이다. 상태 전이 식은 다음과 같다.</p> \\[ dp[i, j] = \\min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1 \\] <p>\\(s[i-1]\\)과 \\(t[j-1]\\)이 같으면 현재 문자를 편집할 필요가 없으며, 이때 상태 전이 식은 다음과 같다.</p> \\[ dp[i, j] = dp[i-1, j-1] \\] <p>3단계: 경계 조건과 상태 전이 순서를 정한다</p> <p>두 문자열이 모두 비어 있으면 편집 횟수는 \\(0\\)이므로 \\(dp[0, 0] = 0\\)이다. \\(s\\)만 비어 있고 \\(t\\)는 비어 있지 않으면 최소 편집 횟수는 \\(t\\)의 길이이므로 첫 행은 \\(dp[0, j] = j\\)다. 반대로 \\(s\\)는 비어 있지 않고 \\(t\\)가 비어 있으면 최소 편집 횟수는 \\(s\\)의 길이이므로 첫 열은 \\(dp[i, 0] = i\\)다.</p> <p>상태 전이 식에서 \\(dp[i, j]\\)는 왼쪽, 위쪽, 왼쪽 위의 해에 의존하므로 이중 반복문으로 전체 \\(dp\\) 테이블을 정방향으로 순회하면 된다.</p>","path":["제 14 장   동적 계획법","14.6   편집 거리 문제"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/#2","level":3,"title":"2.   코드 구현","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby edit_distance.py<pre><code>def edit_distance_dp(s: str, t: str) -&gt; int:\n    \"\"\"편집 거리: 동적 계획법\"\"\"\n    n, m = len(s), len(t)\n    dp = [[0] * (m + 1) for _ in range(n + 1)]\n    # 상태 전이: 첫 번째 행과 첫 번째 열\n    for i in range(1, n + 1):\n        dp[i][0] = i\n    for j in range(1, m + 1):\n        dp[0][j] = j\n    # 상태 전이: 나머지 행과 열\n    for i in range(1, n + 1):\n        for j in range(1, m + 1):\n            if s[i - 1] == t[j - 1]:\n                # 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1]\n            else:\n                # 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] = min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1\n    return dp[n][m]\n</code></pre> edit_distance.cpp<pre><code>/* 편집 거리: 동적 계획법 */\nint editDistanceDP(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(m + 1, 0));\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.java<pre><code>/* 편집 거리: 동적 계획법 */\nint editDistanceDP(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[][] dp = new int[n + 1][m + 1];\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] = Math.min(Math.min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.cs<pre><code>/* 편집 거리: 동적 계획법 */\nint EditDistanceDP(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[,] dp = new int[n + 1, m + 1];\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i, 0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0, j] = j;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i, j] = dp[i - 1, j - 1];\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i, j] = Math.Min(Math.Min(dp[i, j - 1], dp[i - 1, j]), dp[i - 1, j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n, m];\n}\n</code></pre> edit_distance.go<pre><code>/* 편집 거리: 동적 계획법 */\nfunc editDistanceDP(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, m+1)\n    }\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for i := 1; i &lt;= n; i++ {\n        dp[i][0] = i\n    }\n    for j := 1; j &lt;= m; j++ {\n        dp[0][j] = j\n    }\n    // 상태 전이: 나머지 행과 열\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= m; j++ {\n            if s[i-1] == t[j-1] {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i-1][j-1]\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] = MinInt(MinInt(dp[i][j-1], dp[i-1][j]), dp[i-1][j-1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.swift<pre><code>/* 편집 거리: 동적 계획법 */\nfunc editDistanceDP(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: Array(repeating: 0, count: m + 1), count: n + 1)\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for i in 1 ... n {\n        dp[i][0] = i\n    }\n    for j in 1 ... m {\n        dp[0][j] = j\n    }\n    // 상태 전이: 나머지 행과 열\n    for i in 1 ... n {\n        for j in 1 ... m {\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.js<pre><code>/* 편집 거리: 동적 계획법 */\nfunction editDistanceDP(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(m + 1).fill(0));\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.ts<pre><code>/* 편집 거리: 동적 계획법 */\nfunction editDistanceDP(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: m + 1 }, () =&gt; 0)\n    );\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.dart<pre><code>/* 편집 거리: 동적 계획법 */\nint editDistanceDP(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (_) =&gt; List.filled(m + 1, 0));\n  // 상태 전이: 첫 번째 행과 첫 번째 열\n  for (int i = 1; i &lt;= n; i++) {\n    dp[i][0] = i;\n  }\n  for (int j = 1; j &lt;= m; j++) {\n    dp[0][j] = j;\n  }\n  // 상태 전이: 나머지 행과 열\n  for (int i = 1; i &lt;= n; i++) {\n    for (int j = 1; j &lt;= m; j++) {\n      if (s[i - 1] == t[j - 1]) {\n        // 두 문자가 같으면 두 문자를 바로 건너뛴다\n        dp[i][j] = dp[i - 1][j - 1];\n      } else {\n        // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n        dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n      }\n    }\n  }\n  return dp[n][m];\n}\n</code></pre> edit_distance.rs<pre><code>/* 편집 거리: 동적 계획법 */\nfn edit_distance_dp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![vec![0; m + 1]; n + 1];\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for i in 1..=n {\n        dp[i][0] = i as i32;\n    }\n    for j in 1..m {\n        dp[0][j] = j as i32;\n    }\n    // 상태 전이: 나머지 행과 열\n    for i in 1..=n {\n        for j in 1..=m {\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] =\n                    std::cmp::min(std::cmp::min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    dp[n][m]\n}\n</code></pre> edit_distance.c<pre><code>/* 편집 거리: 동적 계획법 */\nint editDistanceDP(char *s, char *t, int n, int m) {\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(m + 1, sizeof(int));\n    }\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] = myMin(myMin(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    int res = dp[n][m];\n    // 메모리 해제\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* 편집 거리: 동적 계획법 */\nfun editDistanceDP(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = Array(n + 1) { IntArray(m + 1) }\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (i in 1..n) {\n        dp[i][0] = i\n    }\n    for (j in 1..m) {\n        dp[0][j] = j\n    }\n    // 상태 전이: 나머지 행과 열\n    for (i in 1..n) {\n        for (j in 1..m) {\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.rb<pre><code>### 편집 거리: 동적 계획법 ###\ndef edit_distance_dp(s, t)\n  n, m = s.length, t.length\n  dp = Array.new(n + 1) { Array.new(m + 1, 0) }\n  # 상태 전이: 첫 번째 행과 첫 번째 열\n  (1...(n + 1)).each { |i| dp[i][0] = i }\n  (1...(m + 1)).each { |j| dp[0][j] = j }\n  # 상태 전이: 나머지 행과 열\n  for i in 1...(n + 1)\n    for j in 1...(m +1)\n      if s[i - 1] == t[j - 1]\n        # 두 문자가 같으면 두 문자를 바로 건너뛴다\n        dp[i][j] = dp[i - 1][j - 1]\n      else\n        # 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n        dp[i][j] = [dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]].min + 1\n      end\n    end\n  end\n  dp[n][m]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-30)과 같이 편집 거리의 상태 전이는 배낭 문제와 매우 비슷하며, 모두 2차원 격자를 채우는 과정으로 볼 수 있다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 14-30   편집 거리의 동적 계획법 과정 </p>","path":["제 14 장   동적 계획법","14.6   편집 거리 문제"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/#3","level":3,"title":"3.   공간 최적화","text":"<p>\\(dp[i,j]\\)는 위쪽 \\(dp[i-1, j]\\), 왼쪽 \\(dp[i, j-1]\\), 왼쪽 위 \\(dp[i-1, j-1]\\)에서 전이된다. 그런데 정방향 순회는 왼쪽 위 값 \\(dp[i-1, j-1]\\)을 잃고, 역방향 순회는 \\(dp[i, j-1]\\)을 미리 계산하지 못하므로 두 순서 모두 그대로는 사용할 수 없다.</p> <p>이를 위해 변수 <code>leftup</code>에 왼쪽 위의 해 \\(dp[i-1, j-1]\\)을 잠시 저장하면 왼쪽과 위쪽의 해만 고려하면 된다. 이때는 무한 배낭 문제와 같은 상황이므로 정방향 순회를 사용할 수 있다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby edit_distance.py<pre><code>def edit_distance_dp_comp(s: str, t: str) -&gt; int:\n    \"\"\"편집 거리: 공간을 최적화한 동적 계획법\"\"\"\n    n, m = len(s), len(t)\n    dp = [0] * (m + 1)\n    # 상태 전이: 첫 번째 행\n    for j in range(1, m + 1):\n        dp[j] = j\n    # 상태 전이: 나머지 행\n    for i in range(1, n + 1):\n        # 상태 전이: 첫 번째 열\n        leftup = dp[0]  # dp[i-1, j-1]을 임시로 저장한다\n        dp[0] += 1\n        # 상태 전이: 나머지 열\n        for j in range(1, m + 1):\n            temp = dp[j]\n            if s[i - 1] == t[j - 1]:\n                # 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup\n            else:\n                # 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = min(dp[j - 1], dp[j], leftup) + 1\n            leftup = temp  # 다음 반복의 dp[i-1, j-1]로 갱신한다\n    return dp[m]\n</code></pre> edit_distance.cpp<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nint editDistanceDPComp(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;int&gt; dp(m + 1, 0);\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt;= n; i++) {\n        // 상태 전이: 첫 번째 열\n        int leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i;\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup;\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.java<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nint editDistanceDPComp(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[] dp = new int[m + 1];\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt;= n; i++) {\n        // 상태 전이: 첫 번째 열\n        int leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i;\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup;\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = Math.min(Math.min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.cs<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nint EditDistanceDPComp(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[] dp = new int[m + 1];\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt;= n; i++) {\n        // 상태 전이: 첫 번째 열\n        int leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i;\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup;\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = Math.Min(Math.Min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.go<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nfunc editDistanceDPComp(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([]int, m+1)\n    // 상태 전이: 첫 번째 행\n    for j := 1; j &lt;= m; j++ {\n        dp[j] = j\n    }\n    // 상태 전이: 나머지 행\n    for i := 1; i &lt;= n; i++ {\n        // 상태 전이: 첫 번째 열\n        leftUp := dp[0] // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i\n        // 상태 전이: 나머지 열\n        for j := 1; j &lt;= m; j++ {\n            temp := dp[j]\n            if s[i-1] == t[j-1] {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftUp\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = MinInt(MinInt(dp[j-1], dp[j]), leftUp) + 1\n            }\n            leftUp = temp // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.swift<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nfunc editDistanceDPComp(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: 0, count: m + 1)\n    // 상태 전이: 첫 번째 행\n    for j in 1 ... m {\n        dp[j] = j\n    }\n    // 상태 전이: 나머지 행\n    for i in 1 ... n {\n        // 상태 전이: 첫 번째 열\n        var leftup = dp[0] // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i\n        // 상태 전이: 나머지 열\n        for j in 1 ... m {\n            let temp = dp[j]\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n            }\n            leftup = temp // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.js<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nfunction editDistanceDPComp(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // 상태 전이: 첫 번째 행\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // 상태 전이: 나머지 행\n    for (let i = 1; i &lt;= n; i++) {\n        // 상태 전이: 첫 번째 열\n        let leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i;\n        // 상태 전이: 나머지 열\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup;\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.ts<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nfunction editDistanceDPComp(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // 상태 전이: 첫 번째 행\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // 상태 전이: 나머지 행\n    for (let i = 1; i &lt;= n; i++) {\n        // 상태 전이: 첫 번째 열\n        let leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i;\n        // 상태 전이: 나머지 열\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup;\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.dart<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nint editDistanceDPComp(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;int&gt; dp = List.filled(m + 1, 0);\n  // 상태 전이: 첫 번째 행\n  for (int j = 1; j &lt;= m; j++) {\n    dp[j] = j;\n  }\n  // 상태 전이: 나머지 행\n  for (int i = 1; i &lt;= n; i++) {\n    // 상태 전이: 첫 번째 열\n    int leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n    dp[0] = i;\n    // 상태 전이: 나머지 열\n    for (int j = 1; j &lt;= m; j++) {\n      int temp = dp[j];\n      if (s[i - 1] == t[j - 1]) {\n        // 두 문자가 같으면 두 문자를 바로 건너뛴다\n        dp[j] = leftup;\n      } else {\n        // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n        dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n      }\n      leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n    }\n  }\n  return dp[m];\n}\n</code></pre> edit_distance.rs<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nfn edit_distance_dp_comp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![0; m + 1];\n    // 상태 전이: 첫 번째 행\n    for j in 1..m {\n        dp[j] = j as i32;\n    }\n    // 상태 전이: 나머지 행\n    for i in 1..=n {\n        // 상태 전이: 첫 번째 열\n        let mut leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i as i32;\n        // 상태 전이: 나머지 열\n        for j in 1..=m {\n            let temp = dp[j];\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup;\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = std::cmp::min(std::cmp::min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    dp[m]\n}\n</code></pre> edit_distance.c<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nint editDistanceDPComp(char *s, char *t, int n, int m) {\n    int *dp = calloc(m + 1, sizeof(int));\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt;= n; i++) {\n        // 상태 전이: 첫 번째 열\n        int leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i;\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup;\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = myMin(myMin(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    int res = dp[m];\n    // 메모리 해제\n    free(dp);\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nfun editDistanceDPComp(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = IntArray(m + 1)\n    // 상태 전이: 첫 번째 행\n    for (j in 1..m) {\n        dp[j] = j\n    }\n    // 상태 전이: 나머지 행\n    for (i in 1..n) {\n        // 상태 전이: 첫 번째 열\n        var leftup = dp[0] // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i\n        // 상태 전이: 나머지 열\n        for (j in 1..m) {\n            val temp = dp[j]\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n            }\n            leftup = temp // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.rb<pre><code>### 편집 거리: 공간을 최적화한 동적 계획법 ###\ndef edit_distance_dp_comp(s, t)\n  n, m = s.length, t.length\n  dp = Array.new(m + 1, 0)\n  # 상태 전이: 첫 번째 행\n  (1...(m + 1)).each { |j| dp[j] = j }\n  # 상태 전이: 나머지 행\n  for i in 1...(n + 1)\n    # 상태 전이: 첫 번째 열\n    leftup = dp.first # dp[i-1, j-1]을 임시로 저장한다\n    dp[0] += 1\n    # 상태 전이: 나머지 열\n    for j in 1...(m + 1)\n      temp = dp[j]\n      if s[i - 1] == t[j - 1]\n        # 두 문자가 같으면 두 문자를 바로 건너뛴다\n        dp[j] = leftup\n      else\n        # 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n        dp[j] = [dp[j - 1], dp[j], leftup].min + 1\n      end\n      leftup = temp # 다음 반복의 dp[i-1, j-1]로 갱신한다\n    end\n  end\n  dp[m]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.6   편집 거리 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/","level":1,"title":"14.8   연습 문제","text":"","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1481","level":2,"title":"14.8.1   개념 확인","text":"","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1","level":3,"title":"1.   언제 동적 계획법을 사용하는 것이 적합한가","text":"<p>한 학생이 “점화식을 쓸 수만 있으면 동적 계획법을 사용해야 한다.”라고 말했다. 다음 세 작업에 동적 계획법, 백트래킹, 또는 <code>dp</code> 테이블을 만들지 않는 반복문이나 수학 공식 중 어느 방법이 더 적합한지 판단하고, 핵심 이유를 하나씩 적어 보자.</p> <ol> <li>액면가가 <code>[1, 3, 4]</code>인 동전으로 금액 6을 만들 때 필요한 동전의 최소 개수를 구한다. 각 동전은 반복해서 사용할 수 있다.</li> <li><code>[1, 2, 3]</code>의 모든 순열을 출력한다.</li> <li> <p>\\(1 + 2 + \\dots + n\\)을 계산한다.</p> <p>동적 계획법이 적합하다고 판단한 작업에서는 <code>dp[i]</code>가 무엇을 나타내는지도 설명하라.</p> </li> </ol> 참고 답안 <ol> <li> <p>동적 계획법이 적합하다. <code>dp[i]</code>를 금액 <code>i</code>를 만드는 데 필요한 동전의 최소 개수로 정의한다.     <code>i</code> 이하인 각 동전 <code>c</code>에 대해 <code>dp[i-c] + 1</code>을 후보 답으로 삼고,     이 후보들 중 최솟값을 선택하면 된다. 서로 다른 선택에서 같은 금액을 반복해서 만나며, 더 큰 금액의 최적해도 더 작은 금액의 최적해로 구성할 수 있다.     금액 6의 답은 2이며, <code>3 + 3</code>으로 만들 수 있다.</p> </li> <li> <p>백트래킹을 사용해야 한다. 문제에서는 순열 6개를 모두 하나씩 생성해야 한다. 백트래킹은 하나를 선택하고 탐색을 이어 간 뒤,     선택을 취소하고 다른 분기를 시도하는 과정을 체계적으로 수행할 수 있다. 어떤 방법을 쓰더라도 이 순열들을 실제로 출력하려면 열거 과정을 생략할 수 없다.</p> </li> <li> <p>반복문이나 등차수열의 합 공식을 사용하면 된다. <code>S(i) = S(i-1) + i</code>라는 점화식을 쓸 수는 있지만, <code>S(i)</code>를 계산할 때는 더 작은 값 <code>S(i-1)</code> 하나에만 의존한다.     각 부분합을 한 번만 계산하면 되고 중복되는 부분 문제가 없으므로 <code>dp</code> 테이블을 만들 필요가 없다. ‘점화식을 쓸 수 있다’고 해서 ‘동적 계획법이 필요하다’는 뜻은 아니다.</p> </li> </ol>","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#2","level":3,"title":"2.   배낭 테이블의 한 칸은 어떻게 계산하는가","text":"<p>0-1 배낭 문제에서 물건의 무게는 <code>wgt = [1, 2, 3]</code>, 가치는 <code>val = [5, 11, 15]</code>이며, 배낭 용량은 4다. <code>dp[i][c]</code>는 앞의 물건 \\(i\\)개만 고려하고 배낭 용량의 상한이 \\(c\\)일 때 얻을 수 있는 최대 가치를 나타낸다. 배낭을 반드시 가득 채울 필요는 없다.</p> <p>지금은 상태 <code>dp[3][4]</code>만 계산한다. <code>dp[2][4] = 16</code>, <code>dp[2][1] = 5</code>임을 알고 있다.</p> <ol> <li>세 번째 물건을 선택하지 않을 때 후보 가치는 얼마인가?</li> <li>세 번째 물건을 선택하면 용량이 얼마나 남으며, 후보 가치는 얼마인가?</li> <li><code>dp[3][4]</code>는 얼마로 정해야 하는가? 이는 어떤 물건들을 선택한 결과인가?</li> </ol> 참고 답안 <ol> <li> <p>세 번째 물건을 선택하지 않으면 앞의 두 물건에 대한 결과를 그대로 사용하므로 후보 가치는 <code>dp[2][4] = 16</code>이다.</p> </li> <li> <p>세 번째 물건의 무게는 3이므로 넣은 뒤 남는 용량은 \\(4-3=1\\)이다. 후보 가치는     <code>dp[2][1] + 15 = 5 + 15 = 20</code>이다.</p> </li> <li> <p>16과 20을 비교하면 <code>dp[3][4] = 20</code>을 얻는다. 이는 첫 번째와 세 번째 물건을 선택한 결과로,     총무게는 \\(1+3=4\\), 총가치는 \\(5+15=20\\)이다.</p> <p>이 상태의 계산은 0-1 배낭 문제에서 한 번의 ‘선택하거나 선택하지 않거나’ 비교를 보여 준다.</p> </li> </ol>","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#3","level":3,"title":"3.   배낭 용량은 어떤 순서로 갱신해야 하는가","text":"<p>0-1 배낭 문제에 물건이 하나뿐이며, 무게는 2, 가치는 5다. 배낭 용량은 4다. 각 물건은 최대 한 번만 선택할 수 있고, 초기 일차원 배열은 <code>dp = [0, 0, 0, 0, 0]</code>이다.</p> <p>한 학생이 이 물건을 처리하면서 용량을 2부터 4까지 갱신했다.</p> <ul> <li><code>dp[2]</code>를 갱신한 뒤 5를 얻었다.</li> <li><code>dp[3]</code>을 갱신한 뒤에도 5를 얻었다.</li> <li><code>dp[4]</code>를 갱신할 때 방금 구한 <code>dp[2]</code>를 다시 사용하여 <code>dp[4] = 10</code>을 얻었다.</li> </ul> <ol> <li><code>dp[4] = 10</code>이라는 결과가 올바른가? 이유는 무엇인가?</li> <li>‘각 물건은 최대 한 번만 선택할 수 있다’는 조건에 따르면 <code>dp[4]</code>는 얼마여야 하는가?</li> <li>각 물건을 처리할 때 용량을 큰 것부터 작은 것 순으로 갱신해야 하는가, 그 반대인가? 그렇게 하면 어떤 문제를 피할 수 있는가?</li> </ol> 참고 답안 <ol> <li> <p>올바르지 않다. 가치 10은 가치가 5인 이 물건을 두 번 넣은 것과 같으므로,     ‘각 물건은 최대 한 번만 선택할 수 있다’는 조건에 어긋난다.</p> </li> <li> <p>배낭에는 이 물건 하나만 넣을 수 있으므로 올바른 <code>dp[4]</code> 값은 5다.</p> </li> <li> <p>용량을 큰 것부터 작은 것 순으로, 즉 4, 3, 2 순으로 갱신해야 한다.     그러면 <code>dp[c]</code>를 계산할 때 읽는 <code>dp[c-2]</code>는 아직 현재 물건을 처리하기 전의 결과이므로,     같은 회전에서 현재 물건을 반복해서 사용하지 않게 된다.</p> </li> </ol>","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1482","level":2,"title":"14.8.2   프로그래밍 연습","text":"","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1_1","level":3,"title":"1.   계단을 오르는 방법의 수","text":"<p>계단은 총 <code>n</code>칸이다. 한 번에 1칸 또는 2칸만 올라갈 수 있으며, 정확히 <code>n</code>번째 칸에 도착해야 한다. 서로 다른 방법이 모두 몇 가지인지 계산하라. <code>n &gt;= 1</code>이며, 각 걸음에서 1칸을 오르는지 2칸을 오르는지에 따라서만 방법을 구분한다. 일차원 동적 계획법 배열을 사용하라. 상태 두 개만 유지하는 공간 최적화 방식은 아직 사용하지 마라.</p> 풀이 힌트 <ol> <li>i번째 칸에 도착하는 마지막 걸음은 1칸 또는 2칸을 오르는 것뿐이다.</li> <li>따라서 dp[i] = dp[i-1] + dp[i-2]다.</li> <li>n이 1과 2인 경우를 먼저 처리한 뒤, 세 번째 칸부터 테이블을 채운다.</li> </ol> <p>LeetCode</p>","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#2-0-1","level":3,"title":"2.   0-1 배낭 문제","text":"<p>길이가 같은 배열 <code>wgt</code>와 <code>val</code>이 주어진다. <code>i</code>번째 물건의 무게 <code>wgt[i]</code>는 양의 정수이고, 가치 <code>val[i]</code>는 음이 아닌 정수다. 배낭 용량 <code>cap</code>은 음이 아닌 정수다. 각 물건은 최대 한 번만 선택할 수 있다. 총무게가 <code>cap</code>을 넘지 않는 조건에서 배낭에 넣을 수 있는 최대 총가치를 구하라. 일차원 동적 계획법으로 구현하라.</p> 풀이 힌트 <ol> <li>길이가 cap + 1인 배열 dp를 초기화한다. dp[c]는 용량 상한이 c일 때의 최대 가치를 나타낸다.</li> <li>물건 i를 처리할 때 ‘선택하지 않는 경우’의 dp[c]와 ‘선택하는 경우’의 dp[c-wgt[i]] + val[i]를 비교한다.</li> <li>같은 회전에서 현재 물건을 반복해서 선택하지 않도록 용량을 반드시 큰 것부터 작은 것 순으로 갱신해야 한다.</li> </ol>","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/","level":1,"title":"14.1   동적 계획법 첫걸음","text":"<p>동적 계획법(dynamic programming)은 중요한 알고리즘 패러다임이다. 문제를 더 작은 부분 문제들로 나누고 그 해를 저장하여 중복 계산을 피함으로써 시간 효율을 크게 높인다.</p> <p>이 절에서는 대표적인 예제에서 출발하여 먼저 완전 탐색을 수행하는 백트래킹 풀이를 제시한다. 그 안의 중복 부분 문제를 살펴보고, 더 효율적인 동적 계획법 풀이를 단계적으로 도출한다.</p> <p>계단 오르기</p> <p>총 \\(n\\) 칸인 계단이 있고, 한 번에 \\(1\\) 칸 또는 \\(2\\) 칸을 오를 수 있다. 맨 위까지 오르는 방법은 몇 가지인가?</p> <p>아래 그림(14-1)과 같이 \\(3\\) 칸짜리 계단의 맨 위에 오르는 방법은 총 \\(3\\) 가지다.</p> <p></p> <p> 그림 14-1   3번째 칸까지 오르는 방법 수 </p> <p>이 문제는 방법의 수를 구하는 것이므로 백트래킹으로 모든 가능성을 열거할 수 있다. 계단 오르기를 여러 차례의 선택 과정으로 생각해 보자. 바닥에서 출발하여 매번 \\(1\\) 칸 또는 \\(2\\) 칸을 고른다. 맨 위에 도착할 때마다 방법 수를 \\(1\\) 늘리고, 맨 위를 넘어가면 가지치기한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_backtrack.py<pre><code>def backtrack(choices: list[int], state: int, n: int, res: list[int]) -&gt; int:\n    \"\"\"백트래킹\"\"\"\n    # n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if state == n:\n        res[0] += 1\n    # 모든 선택지를 순회한다\n    for choice in choices:\n        # 가지치기: n번째 계단을 넘어갈 수 없다\n        if state + choice &gt; n:\n            continue\n        # 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res)\n        # 되돌리기\n\ndef climbing_stairs_backtrack(n: int) -&gt; int:\n    \"\"\"계단 오르기: 백트래킹\"\"\"\n    choices = [1, 2]  # 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    state = 0  # 0번째 계단에서 오르기 시작한다\n    res = [0]  # res[0]에 방법의 수를 기록한다\n    backtrack(choices, state, n, res)\n    return res[0]\n</code></pre> climbing_stairs_backtrack.cpp<pre><code>/* 백트래킹 */\nvoid backtrack(vector&lt;int&gt; &amp;choices, int state, int n, vector&lt;int&gt; &amp;res) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if (state == n)\n        res[0]++;\n    // 모든 선택지를 순회한다\n    for (auto &amp;choice : choices) {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if (state + choice &gt; n)\n            continue;\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res);\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nint climbingStairsBacktrack(int n) {\n    vector&lt;int&gt; choices = {1, 2}; // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    int state = 0;                // 0번째 계단에서 오르기 시작한다\n    vector&lt;int&gt; res = {0};        // res[0]에 방법의 수를 기록한다\n    backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.java<pre><code>/* 백트래킹 */\nvoid backtrack(List&lt;Integer&gt; choices, int state, int n, List&lt;Integer&gt; res) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if (state == n)\n        res.set(0, res.get(0) + 1);\n    // 모든 선택지를 순회한다\n    for (Integer choice : choices) {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if (state + choice &gt; n)\n            continue;\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res);\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nint climbingStairsBacktrack(int n) {\n    List&lt;Integer&gt; choices = Arrays.asList(1, 2); // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    int state = 0; // 0번째 계단에서 오르기 시작한다\n    List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n    res.add(0); // res[0]에 방법의 수를 기록한다\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.cs<pre><code>/* 백트래킹 */\nvoid Backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if (state == n)\n        res[0]++;\n    // 모든 선택지를 순회한다\n    foreach (int choice in choices) {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if (state + choice &gt; n)\n            continue;\n        // 시도: 선택하고 상태를 갱신한다\n        Backtrack(choices, state + choice, n, res);\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nint ClimbingStairsBacktrack(int n) {\n    List&lt;int&gt; choices = [1, 2]; // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    int state = 0; // 0번째 계단에서 오르기 시작한다\n    List&lt;int&gt; res = [0]; // res[0]에 방법의 수를 기록한다\n    Backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.go<pre><code>/* 백트래킹 */\nfunc backtrack(choices []int, state, n int, res []int) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if state == n {\n        res[0] = res[0] + 1\n    }\n    // 모든 선택지를 순회한다\n    for _, choice := range choices {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if state+choice &gt; n {\n            continue\n        }\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state+choice, n, res)\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nfunc climbingStairsBacktrack(n int) int {\n    // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    choices := []int{1, 2}\n    // 0번째 계단에서 오르기 시작한다\n    state := 0\n    res := make([]int, 1)\n    // res[0]에 방법의 수를 기록한다\n    res[0] = 0\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.swift<pre><code>/* 백트래킹 */\nfunc backtrack(choices: [Int], state: Int, n: Int, res: inout [Int]) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if state == n {\n        res[0] += 1\n    }\n    // 모든 선택지를 순회한다\n    for choice in choices {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if state + choice &gt; n {\n            continue\n        }\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices: choices, state: state + choice, n: n, res: &amp;res)\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nfunc climbingStairsBacktrack(n: Int) -&gt; Int {\n    let choices = [1, 2] // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    let state = 0 // 0번째 계단에서 오르기 시작한다\n    var res: [Int] = []\n    res.append(0) // res[0]에 방법의 수를 기록한다\n    backtrack(choices: choices, state: state, n: n, res: &amp;res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.js<pre><code>/* 백트래킹 */\nfunction backtrack(choices, state, n, res) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if (state === n) res.set(0, res.get(0) + 1);\n    // 모든 선택지를 순회한다\n    for (const choice of choices) {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if (state + choice &gt; n) continue;\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res);\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nfunction climbingStairsBacktrack(n) {\n    const choices = [1, 2]; // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    const state = 0; // 0번째 계단에서 오르기 시작한다\n    const res = new Map();\n    res.set(0, 0); // res[0]에 방법의 수를 기록한다\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.ts<pre><code>/* 백트래킹 */\nfunction backtrack(\n    choices: number[],\n    state: number,\n    n: number,\n    res: Map&lt;0, any&gt;\n): void {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if (state === n) res.set(0, res.get(0) + 1);\n    // 모든 선택지를 순회한다\n    for (const choice of choices) {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if (state + choice &gt; n) continue;\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res);\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nfunction climbingStairsBacktrack(n: number): number {\n    const choices = [1, 2]; // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    const state = 0; // 0번째 계단에서 오르기 시작한다\n    const res = new Map();\n    res.set(0, 0); // res[0]에 방법의 수를 기록한다\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.dart<pre><code>/* 백트래킹 */\nvoid backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n  // n번째 계단에 도달하면 방법의 수에 1을 더한다\n  if (state == n) {\n    res[0]++;\n  }\n  // 모든 선택지를 순회한다\n  for (int choice in choices) {\n    // 가지치기: n번째 계단을 넘어갈 수 없다\n    if (state + choice &gt; n) continue;\n    // 시도: 선택하고 상태를 갱신한다\n    backtrack(choices, state + choice, n, res);\n    // 되돌리기\n  }\n}\n\n/* 계단 오르기: 백트래킹 */\nint climbingStairsBacktrack(int n) {\n  List&lt;int&gt; choices = [1, 2]; // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n  int state = 0; // 0번째 계단에서 오르기 시작한다\n  List&lt;int&gt; res = [];\n  res.add(0); // res[0]에 방법의 수를 기록한다\n  backtrack(choices, state, n, res);\n  return res[0];\n}\n</code></pre> climbing_stairs_backtrack.rs<pre><code>/* 백트래킹 */\nfn backtrack(choices: &amp;[i32], state: i32, n: i32, res: &amp;mut [i32]) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if state == n {\n        res[0] = res[0] + 1;\n    }\n    // 모든 선택지를 순회한다\n    for &amp;choice in choices {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if state + choice &gt; n {\n            continue;\n        }\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res);\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nfn climbing_stairs_backtrack(n: usize) -&gt; i32 {\n    let choices = vec![1, 2]; // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    let state = 0; // 0번째 계단에서 오르기 시작한다\n    let mut res = Vec::new();\n    res.push(0); // res[0]에 방법의 수를 기록한다\n    backtrack(&amp;choices, state, n as i32, &amp;mut res);\n    res[0]\n}\n</code></pre> climbing_stairs_backtrack.c<pre><code>/* 백트래킹 */\nvoid backtrack(int *choices, int state, int n, int *res, int len) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if (state == n)\n        res[0]++;\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; len; i++) {\n        int choice = choices[i];\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if (state + choice &gt; n)\n            continue;\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res, len);\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nint climbingStairsBacktrack(int n) {\n    int choices[2] = {1, 2}; // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    int state = 0;           // 0번째 계단에서 오르기 시작한다\n    int *res = (int *)malloc(sizeof(int));\n    *res = 0; // res[0]에 방법의 수를 기록한다\n    int len = sizeof(choices) / sizeof(int);\n    backtrack(choices, state, n, res, len);\n    int result = *res;\n    free(res);\n    return result;\n}\n</code></pre> climbing_stairs_backtrack.kt<pre><code>/* 백트래킹 */\nfun backtrack(\n    choices: MutableList&lt;Int&gt;,\n    state: Int,\n    n: Int,\n    res: MutableList&lt;Int&gt;\n) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if (state == n)\n        res[0] = res[0] + 1\n    // 모든 선택지를 순회한다\n    for (choice in choices) {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if (state + choice &gt; n) continue\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res)\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nfun climbingStairsBacktrack(n: Int): Int {\n    val choices = mutableListOf(1, 2) // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    val state = 0 // 0번째 계단에서 오르기 시작한다\n    val res = mutableListOf&lt;Int&gt;()\n    res.add(0) // res[0]에 방법의 수를 기록한다\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.rb<pre><code>### 백트래킹 ###\ndef backtrack(choices, state, n, res)\n  # n번째 계단에 도달하면 방법의 수에 1을 더한다\n  res[0] += 1 if state == n\n  # 모든 선택지를 순회한다\n  for choice in choices\n    # 가지치기: n번째 계단을 넘어갈 수 없다\n    next if state + choice &gt; n\n\n    # 시도: 선택하고 상태를 갱신한다\n    backtrack(choices, state + choice, n, res)\n  end\n  # 되돌리기\nend\n\n### 계단 오르기: 백트래킹 ###\ndef climbing_stairs_backtrack(n)\n  choices = [1, 2] # 한 계단 또는 두 계단 오르기를 선택할 수 있다\n  state = 0 # 0번째 계단에서 오르기 시작한다\n  res = [0] # res[0]에 방법의 수를 기록한다\n  backtrack(choices, state, n, res)\n  res.first\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.1   동적 계획법 첫걸음"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1411-1","level":2,"title":"14.1.1   방법 1: 완전 탐색","text":"<p>백트래킹은 보통 문제를 명시적으로 나누지 않고 일련의 의사 결정 단계로 바라보며, 시도와 가지치기를 통해 가능한 모든 해를 찾는다.</p> <p>이 문제를 부분 문제로 나누는 관점에서 분석해 보자. \\(i\\) 번째 칸에 오르는 방법 수를 \\(dp[i]\\)라고 하면 \\(dp[i]\\)가 원래 문제이고, 다음은 그 부분 문제들이다.</p> \\[ dp[i-1], dp[i-2], \\dots, dp[2], dp[1] \\] <p>한 번에 \\(1\\) 칸 또는 \\(2\\) 칸만 오를 수 있으므로, 현재 \\(i\\) 번째 칸에 있다면 직전에는 \\(i - 1\\) 번째 또는 \\(i - 2\\) 번째 칸에 있었어야 한다. 즉, \\(i -1\\) 번째 또는 \\(i - 2\\) 번째 칸에서만 \\(i\\) 번째 칸으로 올라올 수 있다.</p> <p>여기서 중요한 결론을 얻는다. \\(i - 1\\) 번째 칸에 오르는 방법 수와 \\(i - 2\\) 번째 칸에 오르는 방법 수를 더하면 \\(i\\) 번째 칸에 오르는 방법 수가 된다. 식으로 나타내면 다음과 같다.</p> \\[ dp[i] = dp[i-1] + dp[i-2] \\] <p>이는 계단 오르기 문제의 부분 문제들 사이에 점화 관계가 있으며, 부분 문제의 해로 원래 문제의 해를 구성할 수 있다는 뜻이다. 아래 그림(14-2)이 이 점화 관계를 보여 준다.</p> <p></p> <p> 그림 14-2   방법 수의 점화 관계 </p> <p>점화식으로 완전 탐색 풀이를 얻을 수 있다. \\(dp[n]\\)에서 출발해 큰 문제 하나를 더 작은 두 문제의 합으로 재귀적으로 나누고, 가장 작은 부분 문제 \\(dp[1]\\)과 \\(dp[2]\\)에 도달하면 반환한다. 이들의 해는 \\(dp[1] = 1\\), \\(dp[2] = 2\\)로 이미 알려져 있다. 즉, \\(1\\), \\(2\\) 번째 칸에 오르는 방법은 각각 \\(1\\), \\(2\\) 가지다.</p> <p>아래 코드는 일반적인 백트래킹 코드와 마찬가지로 깊이 우선 탐색이지만 더 간결하다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dfs.py<pre><code>def dfs(i: int) -&gt; int:\n    \"\"\"탐색\"\"\"\n    # dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 or i == 2:\n        return i\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1) + dfs(i - 2)\n    return count\n\ndef climbing_stairs_dfs(n: int) -&gt; int:\n    \"\"\"계단 오르기: 탐색\"\"\"\n    return dfs(n)\n</code></pre> climbing_stairs_dfs.cpp<pre><code>/* 탐색 */\nint dfs(int i) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* 계단 오르기: 탐색 */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.java<pre><code>/* 탐색 */\nint dfs(int i) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* 계단 오르기: 탐색 */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.cs<pre><code>/* 탐색 */\nint DFS(int i) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1) + DFS(i - 2);\n    return count;\n}\n\n/* 계단 오르기: 탐색 */\nint ClimbingStairsDFS(int n) {\n    return DFS(n);\n}\n</code></pre> climbing_stairs_dfs.go<pre><code>/* 탐색 */\nfunc dfs(i int) int {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfs(i-1) + dfs(i-2)\n    return count\n}\n\n/* 계단 오르기: 탐색 */\nfunc climbingStairsDFS(n int) int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.swift<pre><code>/* 탐색 */\nfunc dfs(i: Int) -&gt; Int {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1) + dfs(i: i - 2)\n    return count\n}\n\n/* 계단 오르기: 탐색 */\nfunc climbingStairsDFS(n: Int) -&gt; Int {\n    dfs(i: n)\n}\n</code></pre> climbing_stairs_dfs.js<pre><code>/* 탐색 */\nfunction dfs(i) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* 계단 오르기: 탐색 */\nfunction climbingStairsDFS(n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.ts<pre><code>/* 탐색 */\nfunction dfs(i: number): number {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* 계단 오르기: 탐색 */\nfunction climbingStairsDFS(n: number): number {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.dart<pre><code>/* 탐색 */\nint dfs(int i) {\n  // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n  if (i == 1 || i == 2) return i;\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1) + dfs(i - 2);\n  return count;\n}\n\n/* 계단 오르기: 탐색 */\nint climbingStairsDFS(int n) {\n  return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.rs<pre><code>/* 탐색 */\nfn dfs(i: usize) -&gt; i32 {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1) + dfs(i - 2);\n    count\n}\n\n/* 계단 오르기: 탐색 */\nfn climbing_stairs_dfs(n: usize) -&gt; i32 {\n    dfs(n)\n}\n</code></pre> climbing_stairs_dfs.c<pre><code>/* 탐색 */\nint dfs(int i) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* 계단 오르기: 탐색 */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.kt<pre><code>/* 탐색 */\nfun dfs(i: Int): Int {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2) return i\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1) + dfs(i - 2)\n    return count\n}\n\n/* 계단 오르기: 탐색 */\nfun climbingStairsDFS(n: Int): Int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.rb<pre><code>### 탐색 ###\ndef dfs(i)\n  # dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n  return i if i == 1 || i == 2\n  # dp[i] = dp[i-1] + dp[i-2]\n  dfs(i - 1) + dfs(i - 2)\nend\n\n### 계단 오르기: 탐색 ###\ndef climbing_stairs_dfs(n)\n  dfs(n)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-3)은 완전 탐색으로 만들어지는 재귀 트리다. 문제 \\(dp[n]\\)의 재귀 트리 깊이는 \\(n\\)이고 시간 복잡도는 \\(O(2^n)\\)이다. 지수적 증가는 폭발적이므로 비교적 큰 \\(n\\)을 입력하면 매우 오래 기다려야 한다.</p> <p></p> <p> 그림 14-3   계단 오르기의 재귀 트리 </p> <p>위 그림(14-3)에서 지수 시간 복잡도는 '중복 부분 문제' 때문에 생긴다. 예를 들어 \\(dp[9]\\)는 \\(dp[8]\\)과 \\(dp[7]\\)로, \\(dp[8]\\)은 \\(dp[7]\\)과 \\(dp[6]\\)으로 나뉘므로 두 경우 모두 부분 문제 \\(dp[7]\\)을 포함한다.</p> <p>그 아래의 부분 문제에도 더 작은 중복 부분 문제가 계속 생겨난다. 계산 자원의 대부분은 이러한 중복 부분 문제에 낭비된다.</p>","path":["제 14 장   동적 계획법","14.1   동적 계획법 첫걸음"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1412-2","level":2,"title":"14.1.2   방법 2: 메모이제이션을 이용한 탐색","text":"<p>효율을 높이려면 모든 중복 부분 문제를 한 번씩만 계산해야 한다. 이를 위해 각 부분 문제의 해를 기록하는 배열 <code>mem</code>을 만들고, 탐색 중 중복 부분 문제를 가지치기한다.</p> <ol> <li>\\(dp[i]\\)를 처음 계산하면 나중에 사용할 수 있도록 <code>mem[i]</code>에 기록한다.</li> <li>\\(dp[i]\\)가 다시 필요하면 <code>mem[i]</code>에서 결과를 바로 가져와 중복 계산을 피한다.</li> </ol> <p>코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dfs_mem.py<pre><code>def dfs(i: int, mem: list[int]) -&gt; int:\n    \"\"\"메모이제이션을 이용한 탐색\"\"\"\n    # dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 or i == 2:\n        return i\n    # dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if mem[i] != -1:\n        return mem[i]\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    # dp[i]를 기록한다\n    mem[i] = count\n    return count\n\ndef climbing_stairs_dfs_mem(n: int) -&gt; int:\n    \"\"\"계단 오르기: 메모이제이션을 이용한 탐색\"\"\"\n    # mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    mem = [-1] * (n + 1)\n    return dfs(n, mem)\n</code></pre> climbing_stairs_dfs_mem.cpp<pre><code>/* 메모이제이션을 이용한 탐색 */\nint dfs(int i, vector&lt;int&gt; &amp;mem) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // dp[i]를 기록한다\n    mem[i] = count;\n    return count;\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nint climbingStairsDFSMem(int n) {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    vector&lt;int&gt; mem(n + 1, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.java<pre><code>/* 메모이제이션을 이용한 탐색 */\nint dfs(int i, int[] mem) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // dp[i]를 기록한다\n    mem[i] = count;\n    return count;\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nint climbingStairsDFSMem(int n) {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    int[] mem = new int[n + 1];\n    Arrays.fill(mem, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.cs<pre><code>/* 메모이제이션을 이용한 탐색 */\nint DFS(int i, int[] mem) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1, mem) + DFS(i - 2, mem);\n    // dp[i]를 기록한다\n    mem[i] = count;\n    return count;\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nint ClimbingStairsDFSMem(int n) {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    int[] mem = new int[n + 1];\n    Array.Fill(mem, -1);\n    return DFS(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.go<pre><code>/* 메모이제이션을 이용한 탐색 */\nfunc dfsMem(i int, mem []int) int {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfsMem(i-1, mem) + dfsMem(i-2, mem)\n    // dp[i]를 기록한다\n    mem[i] = count\n    return count\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nfunc climbingStairsDFSMem(n int) int {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    mem := make([]int, n+1)\n    for i := range mem {\n        mem[i] = -1\n    }\n    return dfsMem(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.swift<pre><code>/* 메모이제이션을 이용한 탐색 */\nfunc dfs(i: Int, mem: inout [Int]) -&gt; Int {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1, mem: &amp;mem) + dfs(i: i - 2, mem: &amp;mem)\n    // dp[i]를 기록한다\n    mem[i] = count\n    return count\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nfunc climbingStairsDFSMem(n: Int) -&gt; Int {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    var mem = Array(repeating: -1, count: n + 1)\n    return dfs(i: n, mem: &amp;mem)\n}\n</code></pre> climbing_stairs_dfs_mem.js<pre><code>/* 메모이제이션을 이용한 탐색 */\nfunction dfs(i, mem) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i === 1 || i === 2) return i;\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // dp[i]를 기록한다\n    mem[i] = count;\n    return count;\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nfunction climbingStairsDFSMem(n) {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.ts<pre><code>/* 메모이제이션을 이용한 탐색 */\nfunction dfs(i: number, mem: number[]): number {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i === 1 || i === 2) return i;\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // dp[i]를 기록한다\n    mem[i] = count;\n    return count;\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nfunction climbingStairsDFSMem(n: number): number {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.dart<pre><code>/* 메모이제이션을 이용한 탐색 */\nint dfs(int i, List&lt;int&gt; mem) {\n  // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n  if (i == 1 || i == 2) return i;\n  // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n  if (mem[i] != -1) return mem[i];\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n  // dp[i]를 기록한다\n  mem[i] = count;\n  return count;\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nint climbingStairsDFSMem(int n) {\n  // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n  List&lt;int&gt; mem = List.filled(n + 1, -1);\n  return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.rs<pre><code>/* 메모이제이션을 이용한 탐색 */\nfn dfs(i: usize, mem: &amp;mut [i32]) -&gt; i32 {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if mem[i] != -1 {\n        return mem[i];\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // dp[i]를 기록한다\n    mem[i] = count;\n    count\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nfn climbing_stairs_dfs_mem(n: usize) -&gt; i32 {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    let mut mem = vec![-1; n + 1];\n    dfs(n, &amp;mut mem)\n}\n</code></pre> climbing_stairs_dfs_mem.c<pre><code>/* 메모이제이션을 이용한 탐색 */\nint dfs(int i, int *mem) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // dp[i]를 기록한다\n    mem[i] = count;\n    return count;\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nint climbingStairsDFSMem(int n) {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    int *mem = (int *)malloc((n + 1) * sizeof(int));\n    for (int i = 0; i &lt;= n; i++) {\n        mem[i] = -1;\n    }\n    int result = dfs(n, mem);\n    free(mem);\n    return result;\n}\n</code></pre> climbing_stairs_dfs_mem.kt<pre><code>/* 메모이제이션을 이용한 탐색 */\nfun dfs(i: Int, mem: IntArray): Int {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2) return i\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if (mem[i] != -1) return mem[i]\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    // dp[i]를 기록한다\n    mem[i] = count\n    return count\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nfun climbingStairsDFSMem(n: Int): Int {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    val mem = IntArray(n + 1)\n    mem.fill(-1)\n    return dfs(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.rb<pre><code>### 메모이제이션을 이용한 탐색 ###\ndef dfs(i, mem)\n  # dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n  return i if i == 1 || i == 2\n  # dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n  return mem[i] if mem[i] != -1\n\n  # dp[i] = dp[i-1] + dp[i-2]\n  count = dfs(i - 1, mem) + dfs(i - 2, mem)\n  # dp[i]를 기록한다\n  mem[i] = count\nend\n\n### 계단 오르기: 메모이제이션을 이용한 탐색 ###\ndef climbing_stairs_dfs_mem(n)\n  # mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n  mem = Array.new(n + 1, -1)\n  dfs(n, mem)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-4)과 같이 메모이제이션을 적용하면 모든 중복 부분 문제를 한 번만 계산하므로 시간 복잡도가 \\(O(n)\\)으로 개선된다. 이는 매우 큰 발전이다.</p> <p></p> <p> 그림 14-4   메모이제이션을 이용한 탐색의 재귀 트리 </p>","path":["제 14 장   동적 계획법","14.1   동적 계획법 첫걸음"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1413-3","level":2,"title":"14.1.3   방법 3: 동적 계획법","text":"<p>메모이제이션을 이용한 탐색은 '위에서 아래로' 진행하는 방법이다. 원래 문제(루트 노드)에서 시작하여 큰 부분 문제를 작은 부분 문제로 재귀적으로 나누고, 해를 이미 아는 가장 작은 부분 문제(리프 노드)에 도달한다. 이후 되돌아오면서 각 단계의 해를 모아 원래 문제의 해를 구성한다.</p> <p>반대로 동적 계획법은 '아래에서 위로' 진행하는 방법이다. 가장 작은 부분 문제의 해에서 시작하여 반복적으로 더 큰 부분 문제의 해를 구성하고, 마침내 원래 문제의 해를 얻는다.</p> <p>동적 계획법에는 되돌아오는 과정이 없으므로 재귀 없이 반복문만으로 구현할 수 있다. 아래 코드에서는 부분 문제의 해를 저장하는 배열 <code>dp</code>를 초기화한다. 이는 메모이제이션을 이용한 탐색의 배열 <code>mem</code>과 같은 기록 역할을 한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dp.py<pre><code>def climbing_stairs_dp(n: int) -&gt; int:\n    \"\"\"계단 오르기: 동적 계획법\"\"\"\n    if n == 1 or n == 2:\n        return n\n    # 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    dp = [0] * (n + 1)\n    # 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1], dp[2] = 1, 2\n    # 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in range(3, n + 1):\n        dp[i] = dp[i - 1] + dp[i - 2]\n    return dp[n]\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* 계단 오르기: 동적 계획법 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    vector&lt;int&gt; dp(n + 1);\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1;\n    dp[2] = 2;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* 계단 오르기: 동적 계획법 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int[] dp = new int[n + 1];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1;\n    dp[2] = 2;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* 계단 오르기: 동적 계획법 */\nint ClimbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int[] dp = new int[n + 1];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1;\n    dp[2] = 2;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* 계단 오르기: 동적 계획법 */\nfunc climbingStairsDP(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    dp := make([]int, n+1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1\n    dp[2] = 2\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = dp[i-1] + dp[i-2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* 계단 오르기: 동적 계획법 */\nfunc climbingStairsDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    var dp = Array(repeating: 0, count: n + 1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1\n    dp[2] = 2\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in 3 ... n {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* 계단 오르기: 동적 계획법 */\nfunction climbingStairsDP(n) {\n    if (n === 1 || n === 2) return n;\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    const dp = new Array(n + 1).fill(-1);\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1;\n    dp[2] = 2;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* 계단 오르기: 동적 계획법 */\nfunction climbingStairsDP(n: number): number {\n    if (n === 1 || n === 2) return n;\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    const dp = new Array(n + 1).fill(-1);\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1;\n    dp[2] = 2;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* 계단 오르기: 동적 계획법 */\nint climbingStairsDP(int n) {\n  if (n == 1 || n == 2) return n;\n  // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n  dp[1] = 1;\n  dp[2] = 2;\n  // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = dp[i - 1] + dp[i - 2];\n  }\n  return dp[n];\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* 계단 오르기: 동적 계획법 */\nfn climbing_stairs_dp(n: usize) -&gt; i32 {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    let mut dp = vec![-1; n + 1];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1;\n    dp[2] = 2;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in 3..=n {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    dp[n]\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* 계단 오르기: 동적 계획법 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int *dp = (int *)malloc((n + 1) * sizeof(int));\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1;\n    dp[2] = 2;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    int result = dp[n];\n    free(dp);\n    return result;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* 계단 오르기: 동적 계획법 */\nfun climbingStairsDP(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    val dp = IntArray(n + 1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1\n    dp[2] = 2\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (i in 3..n) {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>### 계단 오르기: 동적 계획법 ###\ndef climbing_stairs_dp(n)\n  return n  if n == 1 || n == 2\n\n  # 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n  dp = Array.new(n + 1, 0)\n  # 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n  dp[1], dp[2] = 1, 2\n  # 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n  (3...(n + 1)).each { |i| dp[i] = dp[i - 1] + dp[i - 2] }\n\n  dp[n]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-5)은 위 코드의 실행 과정을 보여 준다.</p> <p></p> <p> 그림 14-5   계단 오르기의 동적 계획법 과정 </p> <p>백트래킹과 마찬가지로 동적 계획법도 '상태'로 문제 해결의 특정 단계를 나타낸다. 각 상태는 부분 문제 하나와 그에 해당하는 국소 최적해에 대응한다. 예를 들어 계단 오르기 문제의 상태는 현재 위치한 계단의 칸 번호 \\(i\\)로 정의한다.</p> <p>위 내용을 바탕으로 동적 계획법의 주요 용어를 정리할 수 있다.</p> <ul> <li>배열 <code>dp</code>를 dp 테이블이라고 하며, \\(dp[i]\\)는 상태 \\(i\\)에 대응하는 부분 문제의 해다.</li> <li>가장 작은 부분 문제에 해당하는 상태(계단의 \\(1\\), \\(2\\) 번째 칸)를 초기 상태라고 한다.</li> <li>점화식 \\(dp[i] = dp[i-1] + dp[i-2]\\)를 상태 전이 식이라고 한다.</li> </ul>","path":["제 14 장   동적 계획법","14.1   동적 계획법 첫걸음"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1414","level":2,"title":"14.1.4   공간 최적화","text":"<p>주의 깊게 읽었다면 \\(dp[i]\\)는 \\(dp[i-1]\\)과 \\(dp[i-2]\\)에만 의존하므로 배열 <code>dp</code>에 모든 부분 문제의 해를 저장할 필요가 없음을 알아차렸을 것이다. 변수 두 개를 차례로 갱신하며 진행하면 된다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dp.py<pre><code>def climbing_stairs_dp_comp(n: int) -&gt; int:\n    \"\"\"계단 오르기: 공간을 최적화한 동적 계획법\"\"\"\n    if n == 1 or n == 2:\n        return n\n    a, b = 1, 2\n    for _ in range(3, n + 1):\n        a, b = b, a + b\n    return b\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nint ClimbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nfunc climbingStairsDPComp(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    a, b := 1, 2\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i := 3; i &lt;= n; i++ {\n        a, b = b, a+b\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nfunc climbingStairsDPComp(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    var a = 1\n    var b = 2\n    for _ in 3 ... n {\n        (a, b) = (b, a + b)\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nfunction climbingStairsDPComp(n) {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nfunction climbingStairsDPComp(n: number): number {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nint climbingStairsDPComp(int n) {\n  if (n == 1 || n == 2) return n;\n  int a = 1, b = 2;\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = a + b;\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nfn climbing_stairs_dp_comp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    let (mut a, mut b) = (1, 2);\n    for _ in 3..=n {\n        let tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    b\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nfun climbingStairsDPComp(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    var a = 1\n    var b = 2\n    for (i in 3..n) {\n        val temp = b\n        b += a\n        a = temp\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>### 계단 오르기: 공간을 최적화한 동적 계획법 ###\ndef climbing_stairs_dp_comp(n)\n  return n if n == 1 || n == 2\n\n  a, b = 1, 2\n  (3...(n + 1)).each { a, b = b, a + b }\n\n  b\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>위 코드에서는 배열 <code>dp</code>의 공간을 없앴으므로 공간 복잡도가 \\(O(n)\\)에서 \\(O(1)\\)로 줄어든다.</p> <p>동적 계획법의 현재 상태는 앞선 몇 개의 상태에만 의존하는 경우가 많다. 이때 필요한 상태만 남겨 '차원을 줄이면' 메모리를 절약할 수 있다. 이 공간 최적화 기법을 '롤링 변수' 또는 '롤링 배열'이라고 한다.</p>","path":["제 14 장   동적 계획법","14.1   동적 계획법 첫걸음"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/","level":1,"title":"14.4   0-1 배낭 문제","text":"<p>배낭 문제는 동적 계획법 입문에 매우 좋은 예제이며 가장 흔한 문제 유형이다. 0-1 배낭, 무한 배낭, 개수 제한 배낭 등 여러 변형이 있다.</p> <p>이 절에서는 가장 흔한 0-1 배낭 문제부터 해결한다.</p> <p>문제</p> <p>물건 \\(n\\) 개와 용량이 \\(cap\\)인 배낭이 주어진다. \\(i\\) 번째 물건의 무게는 \\(wgt[i-1]\\), 가치는 \\(val[i-1]\\)이다. 각 물건을 한 번만 선택할 수 있을 때, 배낭 용량 제한 안에 담을 수 있는 물건의 최대 가치를 구하라.</p> <p>아래 그림(14-17)에서 물건 번호 \\(i\\)는 \\(1\\)부터, 배열 인덱스는 \\(0\\)부터 시작하므로 물건 \\(i\\)의 무게와 가치는 각각 \\(wgt[i-1]\\), \\(val[i-1]\\)이다.</p> <p></p> <p> 그림 14-17   0-1 배낭 문제 예시 데이터 </p> <p>0-1 배낭 문제는 \\(n\\) 차례의 결정으로 이루어진 과정으로 볼 수 있다. 각 물건에는 넣지 않기와 넣기라는 두 결정이 있으므로 의사 결정 트리 모형을 만족한다.</p> <p>목표가 '배낭 용량 제한 안에 담을 수 있는 최대 가치'이므로 동적 계획법 문제일 가능성이 높다.</p> <p>1단계: 매 차례의 결정을 생각하고 상태를 정의하여 \\(dp\\) 테이블을 얻는다</p> <p>물건을 넣지 않으면 배낭 용량은 그대로이고, 넣으면 용량이 줄어든다. 따라서 현재 물건 번호 \\(i\\)와 배낭 용량 \\(c\\)로 상태를 정의하고 \\([i, c]\\)라고 한다.</p> <p>상태 \\([i, c]\\)의 부분 문제는 앞의 \\(i\\) 개 물건을 용량 \\(c\\)인 배낭에 담았을 때의 최대 가치이며, 이를 \\(dp[i, c]\\)라고 한다.</p> <p>구할 값은 \\(dp[n, cap]\\)이므로 \\((n+1) \\times (cap+1)\\) 크기의 2차원 \\(dp\\) 테이블이 필요하다.</p> <p>2단계: 최적 부분 구조를 찾아 상태 전이 식을 도출한다</p> <p>물건 \\(i\\)에 대한 결정을 내리면 앞의 \\(i-1\\) 개 물건을 결정하는 부분 문제가 남는다. 다음 두 경우로 나뉜다.</p> <ul> <li>물건 \\(i\\)를 넣지 않음: 용량은 그대로이며 상태는 \\([i-1, c]\\)로 바뀐다.</li> <li>물건 \\(i\\)를 넣음: 용량이 \\(wgt[i-1]\\) 줄고 가치가 \\(val[i-1]\\) 늘며, 상태는 \\([i-1, c-wgt[i-1]]\\)로 바뀐다.</li> </ul> <p>이 분석에서 최적 부분 구조가 드러난다. 최대 가치 \\(dp[i, c]\\)는 물건 \\(i\\)를 넣지 않는 경우와 물건 \\(i\\)를 넣는 경우 중 가치가 더 큰 쪽이다. 상태 전이 식은 다음과 같다.</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1]) \\] <p>현재 물건의 무게 \\(wgt[i - 1]\\)가 남은 용량 \\(c\\)를 넘으면 넣지 않는 선택만 가능하다.</p> <p>3단계: 경계 조건과 상태 전이 순서를 정한다</p> <p>물건이 없거나 배낭 용량이 \\(0\\)이면 최대 가치는 \\(0\\)이다. 즉, 첫 열 \\(dp[i, 0]\\)과 첫 행 \\(dp[0, c]\\)는 모두 \\(0\\)이다.</p> <p>현재 상태 \\([i, c]\\)는 바로 위의 \\([i-1, c]\\)와 왼쪽 위의 \\([i-1, c-wgt[i-1]]\\)에서 전이되므로, 이중 반복문으로 전체 \\(dp\\) 테이블을 정방향으로 순회하면 된다.</p> <p>이 분석을 바탕으로 완전 탐색, 메모이제이션을 이용한 탐색, 동적 계획법 순서로 구현한다.</p>","path":["제 14 장   동적 계획법","14.4   0-1 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#1-1","level":3,"title":"1.   방법 1: 완전 탐색","text":"<p>탐색 코드에는 다음 요소가 포함된다.</p> <ul> <li>재귀 매개변수: 상태 \\([i, c]\\).</li> <li>반환값: 부분 문제의 해 \\(dp[i, c]\\).</li> <li>종료 조건: 물건 번호가 범위를 벗어나 \\(i = 0\\)이 되거나 남은 용량이 \\(0\\)이면 재귀를 종료하고 가치 \\(0\\)을 반환한다.</li> <li>가지치기: 현재 물건의 무게가 남은 배낭 용량보다 크면 넣지 않는 선택만 할 수 있다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dfs(wgt: list[int], val: list[int], i: int, c: int) -&gt; int:\n    \"\"\"0-1 배낭: 완전 탐색\"\"\"\n    # 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 or c == 0:\n        return 0\n    # 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs(wgt, val, i - 1, c)\n    # 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    no = knapsack_dfs(wgt, val, i - 1, c)\n    yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # 두 경우 중 더 큰 가치를 반환한다\n    return max(no, yes)\n</code></pre> knapsack.cpp<pre><code>/* 0-1 배낭: 완전 탐색 */\nint knapsackDFS(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 반환한다\n    return max(no, yes);\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 배낭: 완전 탐색 */\nint knapsackDFS(int[] wgt, int[] val, int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 반환한다\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 배낭: 완전 탐색 */\nint KnapsackDFS(int[] weight, int[] val, int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFS(weight, val, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = KnapsackDFS(weight, val, i - 1, c);\n    int yes = KnapsackDFS(weight, val, i - 1, c - weight[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 반환한다\n    return Math.Max(no, yes);\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 배낭: 완전 탐색 */\nfunc knapsackDFS(wgt, val []int, i, c int) int {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i-1] &gt; c {\n        return knapsackDFS(wgt, val, i-1, c)\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    no := knapsackDFS(wgt, val, i-1, c)\n    yes := knapsackDFS(wgt, val, i-1, c-wgt[i-1]) + val[i-1]\n    // 두 경우 중 더 큰 가치를 반환한다\n    return int(math.Max(float64(no), float64(yes)))\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 배낭: 완전 탐색 */\nfunc knapsackDFS(wgt: [Int], val: [Int], i: Int, c: Int) -&gt; Int {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i - 1] &gt; c {\n        return knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    let no = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    let yes = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // 두 경우 중 더 큰 가치를 반환한다\n    return max(no, yes)\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 배낭: 완전 탐색 */\nfunction knapsackDFS(wgt, val, i, c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 반환한다\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 배낭: 완전 탐색 */\nfunction knapsackDFS(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    i: number,\n    c: number\n): number {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 반환한다\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 배낭: 완전 탐색 */\nint knapsackDFS(List&lt;int&gt; wgt, List&lt;int&gt; val, int i, int c) {\n  // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFS(wgt, val, i - 1, c);\n  }\n  // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n  int no = knapsackDFS(wgt, val, i - 1, c);\n  int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // 두 경우 중 더 큰 가치를 반환한다\n  return max(no, yes);\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 배낭: 완전 탐색 */\nfn knapsack_dfs(wgt: &amp;[i32], val: &amp;[i32], i: usize, c: usize) -&gt; i32 {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs(wgt, val, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    let no = knapsack_dfs(wgt, val, i - 1, c);\n    let yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 반환한다\n    std::cmp::max(no, yes)\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 배낭: 완전 탐색 */\nint knapsackDFS(int wgt[], int val[], int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 반환한다\n    return myMax(no, yes);\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 배낭: 완전 탐색 */\nfun knapsackDFS(\n    wgt: IntArray,\n    _val: IntArray,\n    i: Int,\n    c: Int\n): Int {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, _val, i - 1, c)\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    val no = knapsackDFS(wgt, _val, i - 1, c)\n    val yes = knapsackDFS(wgt, _val, i - 1, c - wgt[i - 1]) + _val[i - 1]\n    // 두 경우 중 더 큰 가치를 반환한다\n    return max(no, yes)\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 배낭: 완전 탐색 ###\ndef knapsack_dfs(wgt, val, i, c)\n  # 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n  return 0 if i == 0 || c == 0\n  # 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n  return knapsack_dfs(wgt, val, i - 1, c) if wgt[i - 1] &gt; c\n  # 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n  no = knapsack_dfs(wgt, val, i - 1, c)\n  yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n  # 두 경우 중 더 큰 가치를 반환한다\n  [no, yes].max\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-18)과 같이 물건마다 선택하지 않기와 선택하기라는 두 탐색 가지가 생기므로 시간 복잡도는 \\(O(2^n)\\)이다.</p> <p>재귀 트리에는 \\(dp[1, 10]\\) 같은 중복 부분 문제가 있음을 쉽게 알 수 있다. 물건이 많고 용량이 크며, 특히 무게가 같은 물건이 많을수록 중복 부분 문제의 수가 크게 늘어난다.</p> <p></p> <p> 그림 14-18   0-1 배낭 문제의 완전 탐색 재귀 트리 </p>","path":["제 14 장   동적 계획법","14.4   0-1 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#2-2","level":3,"title":"2.   방법 2: 메모이제이션을 이용한 탐색","text":"<p>중복 부분 문제를 한 번만 계산하도록 메모이제이션 리스트 <code>mem</code>에 부분 문제의 해를 저장한다. <code>mem[i][c]</code>는 \\(dp[i, c]\\)에 대응한다.</p> <p>메모이제이션을 도입하면 시간 복잡도는 부분 문제의 수에 의해 결정되므로 \\(O(n \\times cap)\\)이다. 구현 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dfs_mem(\n    wgt: list[int], val: list[int], mem: list[list[int]], i: int, c: int\n) -&gt; int:\n    \"\"\"0-1 배낭: 메모이제이션을 이용한 탐색\"\"\"\n    # 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 or c == 0:\n        return 0\n    # 기록이 이미 있으면 바로 반환한다\n    if mem[i][c] != -1:\n        return mem[i][c]\n    # 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    # 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nint knapsackDFSMem(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nint knapsackDFSMem(int[] wgt, int[] val, int[][] mem, int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nint KnapsackDFSMem(int[] weight, int[] val, int[][] mem, int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFSMem(weight, val, mem, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = KnapsackDFSMem(weight, val, mem, i - 1, c);\n    int yes = KnapsackDFSMem(weight, val, mem, i - 1, c - weight[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = Math.Max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nfunc knapsackDFSMem(wgt, val []int, mem [][]int, i, c int) int {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i-1] &gt; c {\n        return knapsackDFSMem(wgt, val, mem, i-1, c)\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    no := knapsackDFSMem(wgt, val, mem, i-1, c)\n    yes := knapsackDFSMem(wgt, val, mem, i-1, c-wgt[i-1]) + val[i-1]\n    // 두 경우 중 더 큰 가치를 반환한다\n    mem[i][c] = int(math.Max(float64(no), float64(yes)))\n    return mem[i][c]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nfunc knapsackDFSMem(wgt: [Int], val: [Int], mem: inout [[Int]], i: Int, c: Int) -&gt; Int {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i - 1] &gt; c {\n        return knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    let no = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    let yes = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nfunction knapsackDFSMem(wgt, val, mem, i, c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nfunction knapsackDFSMem(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    c: number\n): number {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nint knapsackDFSMem(\n  List&lt;int&gt; wgt,\n  List&lt;int&gt; val,\n  List&lt;List&lt;int&gt;&gt; mem,\n  int i,\n  int c,\n) {\n  // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // 기록이 이미 있으면 바로 반환한다\n  if (mem[i][c] != -1) {\n    return mem[i][c];\n  }\n  // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFSMem(wgt, val, mem, i - 1, c);\n  }\n  // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n  int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n  int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // 두 경우 중 더 큰 가치를 기록하고 반환한다\n  mem[i][c] = max(no, yes);\n  return mem[i][c];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nfn knapsack_dfs_mem(wgt: &amp;[i32], val: &amp;[i32], mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: usize, c: usize) -&gt; i32 {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if mem[i][c] != -1 {\n        return mem[i][c];\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    let no = knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    let yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = std::cmp::max(no, yes);\n    mem[i][c]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nint knapsackDFSMem(int wgt[], int val[], int memCols, int **mem, int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = myMax(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nfun knapsackDFSMem(\n    wgt: IntArray,\n    _val: IntArray,\n    mem: Array&lt;IntArray&gt;,\n    i: Int,\n    c: Int\n): Int {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][c] != -1) {\n        return mem[i][c]\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, _val, mem, i - 1, c)\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    val no = knapsackDFSMem(wgt, _val, mem, i - 1, c)\n    val yes = knapsackDFSMem(wgt, _val, mem, i - 1, c - wgt[i - 1]) + _val[i - 1]\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 배낭: 메모이제이션을 이용한 탐색 ###\ndef knapsack_dfs_mem(wgt, val, mem, i, c)\n  # 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n  return 0 if i == 0 || c == 0\n  # 기록이 이미 있으면 바로 반환한다\n  return mem[i][c] if mem[i][c] != -1\n  # 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n  return knapsack_dfs_mem(wgt, val, mem, i - 1, c) if wgt[i - 1] &gt; c\n  # 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n  no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n  yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n  # 두 경우 중 더 큰 가치를 기록하고 반환한다\n  mem[i][c] = [no, yes].max\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-19)은 메모이제이션을 이용한 탐색에서 잘라 낸 가지들을 보여 준다.</p> <p></p> <p> 그림 14-19   0-1 배낭 문제의 메모이제이션 탐색 재귀 트리 </p>","path":["제 14 장   동적 계획법","14.4   0-1 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#3-3","level":3,"title":"3.   방법 3: 동적 계획법","text":"<p>동적 계획법은 본질적으로 상태 전이를 통해 \\(dp\\) 테이블을 채우는 과정이다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 배낭: 동적 계획법\"\"\"\n    n = len(wgt)\n    # dp 테이블을 초기화한다\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # 상태 전이\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 배낭: 동적 계획법 */\nint knapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // dp 테이블을 초기화한다\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 배낭: 동적 계획법 */\nint knapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // dp 테이블을 초기화한다\n    int[][] dp = new int[n + 1][cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 배낭: 동적 계획법 */\nint KnapsackDP(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // dp 테이블을 초기화한다\n    int[,] dp = new int[n + 1, cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (weight[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i, c] = Math.Max(dp[i - 1, c - weight[i - 1]] + val[i - 1], dp[i - 1, c]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 배낭: 동적 계획법 */\nfunc knapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // dp 테이블을 초기화한다\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // 상태 전이\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i-1][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 배낭: 동적 계획법 */\nfunc knapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // 상태 전이\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 배낭: 동적 계획법 */\nfunction knapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array(n + 1)\n        .fill(0)\n        .map(() =&gt; Array(cap + 1).fill(0));\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 배낭: 동적 계획법 */\nfunction knapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 배낭: 동적 계획법 */\nint knapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // dp 테이블을 초기화한다\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // 상태 전이\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 배낭: 동적 계획법 */\nfn knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // dp 테이블을 초기화한다\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // 상태 전이\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = std::cmp::max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1] as usize] + val[i - 1],\n                );\n            }\n        }\n    }\n    dp[n][cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 배낭: 동적 계획법 */\nint knapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // dp 테이블을 초기화한다\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = myMax(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // 메모리 해제\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 배낭: 동적 계획법 */\nfun knapsackDP(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // dp 테이블을 초기화한다\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // 상태 전이\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 배낭: 동적 계획법 ###\ndef knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # 상태 전이\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[i][c] = dp[i - 1][c]\n      else\n        # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[i][c] = [dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-20)과 같이 시간 복잡도와 공간 복잡도는 모두 배열 <code>dp</code>의 크기인 \\(O(n \\times cap)\\)이다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 14-20   0-1 배낭 문제의 동적 계획법 과정 </p>","path":["제 14 장   동적 계획법","14.4   0-1 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#4","level":3,"title":"4.   공간 최적화","text":"<p>각 상태는 바로 윗행의 상태에만 의존하므로 두 배열을 번갈아 갱신하면 공간 복잡도를 \\(O(n \\times cap)\\)에서 \\(O(cap)\\)으로 줄일 수 있다.</p> <p>배열 하나만으로도 가능할까? 각 상태는 바로 위 또는 왼쪽 위 칸에서 전이된다. 배열 하나를 쓴다면 \\(i\\) 번째 행의 순회를 시작할 때 그 배열에는 아직 \\(i-1\\) 번째 행의 상태가 저장되어 있다.</p> <ul> <li>정방향으로 순회하면 \\(dp[i, j]\\)에 도달했을 때 왼쪽 위 \\(dp[i-1, 1]\\) ~ \\(dp[i-1, j-1]\\)의 값이 이미 덮어써졌을 수 있어 올바른 상태 전이를 할 수 없다.</li> <li>역방향으로 순회하면 값이 덮어써지는 문제가 없어 상태 전이를 올바르게 수행할 수 있다.</li> </ul> <p>아래 그림(14-21)은 배열 하나로 \\(i = 1\\) 행에서 \\(i = 2\\) 행으로 바꾸는 과정이다. 정방향과 역방향 순회의 차이를 생각해 보자.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 14-21   공간 최적화 후 0-1 배낭의 동적 계획법 과정 </p> <p>구현에서는 배열 <code>dp</code>의 첫 번째 차원 \\(i\\)를 없애고 안쪽 반복문을 역방향 순회로 바꾸기만 하면 된다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 배낭: 공간을 최적화한 동적 계획법\"\"\"\n    n = len(wgt)\n    # dp 테이블을 초기화한다\n    dp = [0] * (cap + 1)\n    # 상태 전이\n    for i in range(1, n + 1):\n        # 역방향 순회\n        for c in range(cap, 0, -1):\n            if wgt[i - 1] &gt; c:\n                # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c]\n            else:\n                # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nint knapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // dp 테이블을 초기화한다\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        // 역방향 순회\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nint knapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        // 역방향 순회\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nint KnapsackDPComp(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        // 역방향 순회\n        for (int c = cap; c &gt; 0; c--) {\n            if (weight[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.Max(dp[c], dp[c - weight[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nfunc knapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // dp 테이블을 초기화한다\n    dp := make([]int, cap+1)\n    // 상태 전이\n    for i := 1; i &lt;= n; i++ {\n        // 역방향 순회\n        for c := cap; c &gt;= 1; c-- {\n            if wgt[i-1] &lt;= c {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nfunc knapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: 0, count: cap + 1)\n    // 상태 전이\n    for i in 1 ... n {\n        // 역방향 순회\n        for c in (1 ... cap).reversed() {\n            if wgt[i - 1] &lt;= c {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nfunction knapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array(cap + 1).fill(0);\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        // 역방향 순회\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nfunction knapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array(cap + 1).fill(0);\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        // 역방향 순회\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nint knapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // dp 테이블을 초기화한다\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // 상태 전이\n  for (int i = 1; i &lt;= n; i++) {\n    // 역방향 순회\n    for (int c = cap; c &gt;= 1; c--) {\n      if (wgt[i - 1] &lt;= c) {\n        // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nfn knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // dp 테이블을 초기화한다\n    let mut dp = vec![0; cap + 1];\n    // 상태 전이\n    for i in 1..=n {\n        // 역방향 순회\n        for c in (1..=cap).rev() {\n            if wgt[i - 1] &lt;= c as i32 {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nint knapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // dp 테이블을 초기화한다\n    int *dp = calloc(cap + 1, sizeof(int));\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        // 역방향 순회\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // 메모리 해제\n    free(dp);\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nfun knapsackDPComp(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // dp 테이블을 초기화한다\n    val dp = IntArray(cap + 1)\n    // 상태 전이\n    for (i in 1..n) {\n        // 역방향 순회\n        for (c in cap downTo 1) {\n            if (wgt[i - 1] &lt;= c) {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 배낭: 공간을 최적화한 동적 계획법 ###\ndef knapsack_dp_comp(wgt, val, cap)\n  n = wgt.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(cap + 1, 0)\n  # 상태 전이\n  for i in 1...(n + 1)\n    # 역방향 순회\n    for c in cap.downto(1)\n      if wgt[i - 1] &gt; c\n        # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[c] = dp[c]\n      else\n        # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[c] = [dp[c], dp[c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[cap]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.4   0-1 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/summary/","level":1,"title":"14.7   요약","text":"","path":["제 14 장   동적 계획법","14.7   요약"],"tags":[]},{"location":"chapter_dynamic_programming/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>동적 계획법은 문제를 나누고 부분 문제의 해를 저장하여 중복 계산을 피함으로써 계산 효율을 높인다.</li> <li>시간을 고려하지 않는다면 모든 동적 계획법 문제는 백트래킹(완전 탐색)으로 풀 수 있다. 하지만 재귀 트리에 중복 부분 문제가 매우 많아 비효율적이다. 메모이제이션 리스트에 계산한 해를 저장하면 중복 부분 문제를 한 번만 계산하도록 보장할 수 있다.</li> <li>메모이제이션을 이용한 탐색은 위에서 아래로 진행하는 재귀 풀이이며, 이에 대응하는 동적 계획법은 '표를 채우듯' 아래에서 위로 진행하는 반복 풀이이다. 현재 상태는 일부 주변 상태에만 의존하므로 \\(dp\\) 테이블의 한 차원을 없애 공간 복잡도를 줄일 수 있다.</li> <li>부분 문제로 나누는 것은 일반적인 알고리즘 아이디어이며, 분할 정복, 동적 계획법, 백트래킹에서 서로 다른 특성을 가진다.</li> <li>동적 계획법 문제의 세 가지 주요 특성은 중복 부분 문제, 최적 부분 구조, 무후효성이다.</li> <li>원래 문제의 최적해를 부분 문제의 최적해로 구성할 수 있다면 최적 부분 구조를 가진다.</li> <li>무후효성이란 한 상태의 이후 전개가 현재 상태에만 의존하고 과거의 모든 상태와는 무관하다는 뜻이다. 많은 조합 최적화 문제는 무후효성을 갖지 않아 동적 계획법으로 빠르게 풀 수 없다.</li> </ul> <p>배낭 문제</p> <ul> <li>배낭 문제는 가장 대표적인 동적 계획법 문제 중 하나로, 0-1 배낭, 무한 배낭, 개수 제한 배낭 등의 변형이 있다.</li> <li>0-1 배낭의 상태는 앞 \\(i\\) 개 물건을 용량 \\(c\\)인 배낭에 담을 때의 최대 가치로 정의한다. 넣지 않기와 넣기라는 두 결정으로 최적 부분 구조와 상태 전이 식을 얻는다. 공간 최적화에서는 바로 위와 왼쪽 위 상태에 의존하므로, 왼쪽 위 값이 덮어써지지 않도록 리스트를 역순으로 순회해야 한다.</li> <li>무한 배낭은 각 물건의 선택 횟수에 제한이 없어 물건을 넣는 상태 전이가 0-1 배낭과 다르다. 바로 위와 바로 왼쪽 상태에 의존하므로 공간 최적화에서는 정방향 순회를 해야 한다.</li> <li>동전 교환은 무한 배낭의 변형이다. '최대' 가치 대신 '최소' 동전 수를 구하므로 상태 전이 식의 \\(\\max()\\)를 \\(\\min()\\)으로 바꾼다. 용량을 '넘지 않는' 것이 아니라 목표 금액을 '정확히' 만들어야 하므로, 만들 수 없는 유효하지 않은 해를 \\(amt + 1\\)로 나타낸다.</li> <li>동전 교환 II는 '최소 동전 수' 대신 '동전 조합 수'를 구하므로 상태 전이 식의 \\(\\min()\\)을 합 연산으로 바꾼다.</li> </ul> <p>편집 거리 문제</p> <ul> <li>편집 거리(Levenshtein 거리)는 두 문자열의 유사도를 측정하며, 한 문자열을 다른 문자열로 바꾸는 최소 편집 횟수로 정의된다. 편집 연산에는 추가, 삭제, 교체가 있다.</li> <li>편집 거리의 상태는 \\(s\\)의 앞 \\(i\\) 개 문자를 \\(t\\)의 앞 \\(j\\) 개 문자로 바꾸는 최소 편집 횟수로 정의한다. \\(s[i-1] \\ne t[j-1]\\)이면 추가, 삭제, 교체의 세 결정이 있고 각각 남는 부분 문제가 있다. 이를 바탕으로 최적 부분 구조와 상태 전이 식을 구한다. \\(s[i-1] = t[j-1]\\)이면 현재 문자를 편집할 필요가 없다.</li> <li>편집 거리의 상태는 바로 위, 바로 왼쪽, 왼쪽 위 상태에 의존하므로 공간 최적화 후에는 정방향이나 역방향 순회만으로 올바른 전이를 할 수 없다. 변수를 하나 두어 왼쪽 위 상태를 임시 저장하면 무한 배낭과 같은 상황이 되어 정방향 순회가 가능하다.</li> </ul>","path":["제 14 장   동적 계획법","14.7   요약"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/","level":1,"title":"14.5   무한 배낭 문제","text":"<p>이 절에서는 또 다른 대표적인 배낭 문제인 무한 배낭을 먼저 풀고, 그 특수한 경우인 동전 교환을 알아본다.</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1451","level":2,"title":"14.5.1   무한 배낭 문제","text":"<p>문제</p> <p>물건 \\(n\\) 개와 용량이 \\(cap\\)인 배낭이 주어진다. \\(i\\) 번째 물건의 무게는 \\(wgt[i-1]\\), 가치는 \\(val[i-1]\\)이다. 각 물건을 반복해서 선택할 수 있을 때, 용량 제한 안에 담을 수 있는 최대 가치를 구하라. 예시는 아래 그림(14-22)과 같다.</p> <p></p> <p> 그림 14-22   무한 배낭 문제 예시 데이터 </p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1","level":3,"title":"1.   동적 계획법 접근","text":"<p>무한 배낭 문제는 0-1 배낭 문제와 매우 비슷하며, 물건을 선택하는 횟수에 제한이 없다는 점만 다르다.</p> <ul> <li>0-1 배낭에서는 종류별 물건이 하나뿐이므로 물건 \\(i\\)를 넣은 뒤에는 앞의 \\(i-1\\) 개 물건에서만 선택할 수 있다.</li> <li>무한 배낭에서는 각 종류의 물건이 무한히 많으므로 물건 \\(i\\)를 넣은 뒤에도 앞의 \\(i\\) 개 물건에서 계속 선택할 수 있다.</li> </ul> <p>무한 배낭의 규칙에 따라 상태 \\([i, c]\\)의 변화는 두 경우로 나뉜다.</p> <ul> <li>물건 \\(i\\)를 넣지 않음: 0-1 배낭과 동일하게 \\([i-1, c]\\)로 전이한다.</li> <li>물건 \\(i\\)를 넣음: 0-1 배낭과 달리 \\([i, c-wgt[i-1]]\\)로 전이한다.</li> </ul> <p>따라서 상태 전이 식은 다음과 같이 바뀐다.</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1]) \\]","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2","level":3,"title":"2.   코드 구현","text":"<p>두 문제의 코드를 비교하면 상태 전이의 한 곳이 \\(i-1\\)에서 \\(i\\)로 바뀌었을 뿐 나머지는 완전히 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"무한 배낭: 동적 계획법\"\"\"\n    n = len(wgt)\n    # dp 테이블을 초기화한다\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # 상태 전이\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* 무한 배낭: 동적 계획법 */\nint unboundedKnapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // dp 테이블을 초기화한다\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* 무한 배낭: 동적 계획법 */\nint unboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // dp 테이블을 초기화한다\n    int[][] dp = new int[n + 1][cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* 무한 배낭: 동적 계획법 */\nint UnboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // dp 테이블을 초기화한다\n    int[,] dp = new int[n + 1, cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i, c] = Math.Max(dp[i - 1, c], dp[i, c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* 무한 배낭: 동적 계획법 */\nfunc unboundedKnapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // dp 테이블을 초기화한다\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // 상태 전이\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* 무한 배낭: 동적 계획법 */\nfunc unboundedKnapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // 상태 전이\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* 무한 배낭: 동적 계획법 */\nfunction unboundedKnapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* 무한 배낭: 동적 계획법 */\nfunction unboundedKnapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* 무한 배낭: 동적 계획법 */\nint unboundedKnapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // dp 테이블을 초기화한다\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // 상태 전이\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* 무한 배낭: 동적 계획법 */\nfn unbounded_knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // dp 테이블을 초기화한다\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // 상태 전이\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i][c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* 무한 배낭: 동적 계획법 */\nint unboundedKnapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // dp 테이블을 초기화한다\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = myMax(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // 메모리 해제\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* 무한 배낭: 동적 계획법 */\nfun unboundedKnapsackDP(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // dp 테이블을 초기화한다\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // 상태 전이\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>### 무한 배낭: 동적 계획법 ###\ndef unbounded_knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # 상태 전이\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[i][c] = dp[i - 1][c]\n      else\n        # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[i][c] = [dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3","level":3,"title":"3.   공간 최적화","text":"<p>현재 상태는 왼쪽과 위쪽 상태에서 전이되므로 공간 최적화 후에는 \\(dp\\) 테이블의 각 행을 정방향으로 순회해야 한다.</p> <p>이는 0-1 배낭의 순회 순서와 정반대다. 아래 그림(14-23)으로 차이를 이해해 보자.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 14-23   공간 최적화 후 무한 배낭의 동적 계획법 과정 </p> <p>구현은 간단하다. 배열 <code>dp</code>의 첫 번째 차원을 없애면 된다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"무한 배낭: 공간을 최적화한 동적 계획법\"\"\"\n    n = len(wgt)\n    # dp 테이블을 초기화한다\n    dp = [0] * (cap + 1)\n    # 상태 전이\n    for i in range(1, n + 1):\n        # 정방향 순회\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c]\n            else:\n                # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nint unboundedKnapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // dp 테이블을 초기화한다\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nint unboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nint UnboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.Max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nfunc unboundedKnapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // dp 테이블을 초기화한다\n    dp := make([]int, cap+1)\n    // 상태 전이\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nfunc unboundedKnapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: 0, count: cap + 1)\n    // 상태 전이\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nfunction unboundedKnapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nfunction unboundedKnapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nint unboundedKnapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // dp 테이블을 초기화한다\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // 상태 전이\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[c] = dp[c];\n      } else {\n        // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nfn unbounded_knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // dp 테이블을 초기화한다\n    let mut dp = vec![0; cap + 1];\n    // 상태 전이\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nint unboundedKnapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // dp 테이블을 초기화한다\n    int *dp = calloc(cap + 1, sizeof(int));\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // 메모리 해제\n    free(dp);\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nfun unboundedKnapsackDPComp(\n    wgt: IntArray,\n    _val: IntArray,\n    cap: Int\n): Int {\n    val n = wgt.size\n    // dp 테이블을 초기화한다\n    val dp = IntArray(cap + 1)\n    // 상태 전이\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>### 무한 배낭: 동적 계획법 ###\ndef unbounded_knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # 상태 전이\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[i][c] = dp[i - 1][c]\n      else\n        # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[i][c] = [dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n\n### 무한 배낭: 공간을 최적화한 동적 계획법 ##3\ndef unbounded_knapsack_dp_comp(wgt, val, cap)\n  n = wgt.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(cap + 1, 0)\n  # 상태 전이\n  for i in 1...(n + 1)\n    # 정방향 순회\n    for c in 1...(cap + 1)\n      if wgt[i -1] &gt; c\n        # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[c] = dp[c]\n      else\n        # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[c] = [dp[c], dp[c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[cap]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1452","level":2,"title":"14.5.2   동전 교환 문제","text":"<p>배낭 문제는 동적 계획법의 큰 문제군을 대표하며, 동전 교환 같은 여러 변형이 있다.</p> <p>문제</p> <p>동전 \\(n\\) 종류가 주어지며 \\(i\\) 번째 동전의 액면가는 \\(coins[i - 1]\\)이고 목표 금액은 \\(amt\\)다. 각 종류의 동전을 반복해서 선택할 수 있을 때, 목표 금액을 만드는 최소 동전 수를 구하라. 목표 금액을 만들 수 없으면 \\(-1\\)을 반환하라. 예시는 아래 그림(14-24)과 같다.</p> <p></p> <p> 그림 14-24   동전 교환 문제 예시 데이터 </p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_1","level":3,"title":"1.   동적 계획법 접근","text":"<p>동전 교환은 무한 배낭 문제의 특수한 경우로 볼 수 있으며, 다음과 같은 공통점과 차이점이 있다.</p> <ul> <li>두 문제는 서로 변환할 수 있다. '물건'은 '동전', '물건 무게'는 '동전 액면가', '배낭 용량'은 '목표 금액'에 대응한다.</li> <li>최적화 목표가 반대다. 무한 배낭은 물건 가치를 최대화하고, 동전 교환은 동전 수를 최소화한다.</li> <li>무한 배낭은 용량을 '넘지 않는' 해를 구하지만, 동전 교환은 목표 금액을 '정확히' 만드는 해를 구한다.</li> </ul> <p>1단계: 매 차례의 결정을 생각하고 상태를 정의하여 \\(dp\\) 테이블을 얻는다</p> <p>상태 \\([i, a]\\)의 부분 문제는 앞의 \\(i\\) 종류 동전으로 금액 \\(a\\)를 만드는 최소 동전 수이며, 이를 \\(dp[i, a]\\)라고 한다.</p> <p>2차원 \\(dp\\) 테이블의 크기는 \\((n+1) \\times (amt+1)\\)이다.</p> <p>2단계: 최적 부분 구조를 찾아 상태 전이 식을 도출한다</p> <p>이 문제의 상태 전이 식은 무한 배낭과 다음 두 가지가 다르다.</p> <ul> <li>최솟값을 구하므로 연산자 \\(\\max()\\)를 \\(\\min()\\)으로 바꾼다.</li> <li>최적화 대상은 상품 가치가 아니라 동전 수이므로, 동전을 선택하면 \\(+1\\)만 수행한다.</li> </ul> \\[ dp[i, a] = \\min(dp[i-1, a], dp[i, a - coins[i-1]] + 1) \\] <p>3단계: 경계 조건과 상태 전이 순서를 정한다</p> <p>목표 금액이 \\(0\\)이면 최소 동전 수는 \\(0\\)이다. 즉, 첫 열의 모든 \\(dp[i, 0]\\)은 \\(0\\)이다.</p> <p>동전이 없으면 \\(&gt; 0\\)인 어떤 목표 금액도 만들 수 없으므로 유효하지 않은 해다. 상태 전이 식의 \\(\\min()\\)이 이를 알아보고 제외하도록 \\(+ \\infty\\)로 나타낸다. 즉, 첫 행의 모든 \\(dp[0, a]\\)를 \\(+ \\infty\\)로 둔다.</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_1","level":3,"title":"2.   코드 구현","text":"<p>대부분의 프로그래밍 언어는 \\(+ \\infty\\) 변수를 제공하지 않아 정수 타입 <code>int</code>의 최댓값으로 대신해야 한다. 하지만 그러면 상태 전이 식의 \\(+ 1\\) 연산에서 정수 오버플로가 발생할 수 있다.</p> <p>이를 피하기 위해 유효하지 않은 해를 \\(amt + 1\\)로 나타낸다. \\(amt\\)를 만드는 동전 수는 많아도 \\(amt\\) 개이기 때문이다. 마지막에 \\(dp[n, amt]\\)가 \\(amt + 1\\)인지 확인하고, 같으면 목표 금액을 만들 수 없다는 뜻으로 \\(-1\\)을 반환한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change.py<pre><code>def coin_change_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"동전 교환: 동적 계획법\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # dp 테이블을 초기화한다\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # 상태 전이: 첫 번째 행과 첫 번째 열\n    for a in range(1, amt + 1):\n        dp[0][a] = MAX\n    # 상태 전이: 나머지 행과 열\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n    return dp[n][amt] if dp[n][amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* 동전 교환: 동적 계획법 */\nint coinChangeDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* 동전 교환: 동적 계획법 */\nint coinChangeDP(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    int[][] dp = new int[n + 1][amt + 1];\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* 동전 교환: 동적 계획법 */\nint CoinChangeDP(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    int[,] dp = new int[n + 1, amt + 1];\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0, a] = MAX;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i, a] = Math.Min(dp[i - 1, a], dp[i, a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n, amt] != MAX ? dp[n, amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* 동전 교환: 동적 계획법 */\nfunc coinChangeDP(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // dp 테이블을 초기화한다\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for a := 1; a &lt;= amt; a++ {\n        dp[0][a] = max\n    }\n    // 상태 전이: 나머지 행과 열\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = int(math.Min(float64(dp[i-1][a]), float64(dp[i][a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* 동전 교환: 동적 계획법 */\nfunc coinChangeDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for a in 1 ... amt {\n        dp[0][a] = MAX\n    }\n    // 상태 전이: 나머지 행과 열\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* 동전 교환: 동적 계획법 */\nfunction coinChangeDP(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* 동전 교환: 동적 계획법 */\nfunction coinChangeDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* 동전 교환: 동적 계획법 */\nint coinChangeDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // dp 테이블을 초기화한다\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // 상태 전이: 첫 번째 행과 첫 번째 열\n  for (int a = 1; a &lt;= amt; a++) {\n    dp[0][a] = MAX;\n  }\n  // 상태 전이: 나머지 행과 열\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n        dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* 동전 교환: 동적 계획법 */\nfn coin_change_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // dp 테이블을 초기화한다\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for a in 1..=amt {\n        dp[0][a] = max;\n    }\n    // 상태 전이: 나머지 행과 열\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = std::cmp::min(dp[i - 1][a], dp[i][a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* 동전 교환: 동적 계획법 */\nint coinChangeDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = myMin(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[n][amt] != MAX ? dp[n][amt] : -1;\n    // 메모리 해제\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* 동전 교환: 동적 계획법 */\nfun coinChangeDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // dp 테이블을 초기화한다\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (a in 1..amt) {\n        dp[0][a] = MAX\n    }\n    // 상태 전이: 나머지 행과 열\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return if (dp[n][amt] != MAX) dp[n][amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>### 동전 교환: 동적 계획법 ###\ndef coin_change_dp(coins, amt)\n  n = coins.length\n  _MAX = amt + 1\n  # dp 테이블을 초기화한다\n  dp = Array.new(n + 1) { Array.new(amt + 1, 0) }\n  # 상태 전이: 첫 번째 행과 첫 번째 열\n  (1...(amt + 1)).each { |a| dp[0][a] = _MAX }\n  # 상태 전이: 나머지 행과 열\n  for i in 1...(n + 1)\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[i][a] = dp[i - 1][a]\n      else\n        # 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n        dp[i][a] = [dp[i - 1][a], dp[i][a - coins[i - 1]] + 1].min\n      end\n    end\n  end\n  dp[n][amt] != _MAX ? dp[n][amt] : -1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-25)의 동전 교환 동적 계획법 과정은 무한 배낭 문제와 매우 비슷하다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 14-25   동전 교환 문제의 동적 계획법 과정 </p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_1","level":3,"title":"3.   공간 최적화","text":"<p>동전 교환의 공간 최적화는 무한 배낭과 같은 방식으로 처리한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change.py<pre><code>def coin_change_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"동전 교환: 공간을 최적화한 동적 계획법\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # dp 테이블을 초기화한다\n    dp = [MAX] * (amt + 1)\n    dp[0] = 0\n    # 상태 전이\n    for i in range(1, n + 1):\n        # 정방향 순회\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            else:\n                # 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n    return dp[amt] if dp[amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nint coinChangeDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    vector&lt;int&gt; dp(amt + 1, MAX);\n    dp[0] = 0;\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nint coinChangeDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[amt + 1];\n    Arrays.fill(dp, MAX);\n    dp[0] = 0;\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nint CoinChangeDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[amt + 1];\n    Array.Fill(dp, MAX);\n    dp[0] = 0;\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = Math.Min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* 동전 교환: 동적 계획법 */\nfunc coinChangeDPComp(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // dp 테이블을 초기화한다\n    dp := make([]int, amt+1)\n    for i := 1; i &lt;= amt; i++ {\n        dp[i] = max\n    }\n    // 상태 전이\n    for i := 1; i &lt;= n; i++ {\n        // 정방향 순회\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = int(math.Min(float64(dp[a]), float64(dp[a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nfunc coinChangeDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: MAX, count: amt + 1)\n    dp[0] = 0\n    // 상태 전이\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nfunction coinChangeDPComp(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nfunction coinChangeDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nint coinChangeDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // dp 테이블을 초기화한다\n  List&lt;int&gt; dp = List.filled(amt + 1, MAX);\n  dp[0] = 0;\n  // 상태 전이\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[a] = dp[a];\n      } else {\n        // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n        dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nfn coin_change_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // dp 테이블을 초기화한다\n    let mut dp = vec![0; amt + 1];\n    dp.fill(max);\n    dp[0] = 0;\n    // 상태 전이\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = std::cmp::min(dp[a], dp[a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nint coinChangeDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    int *dp = malloc((amt + 1) * sizeof(int));\n    for (int j = 1; j &lt;= amt; j++) {\n        dp[j] = MAX;\n    } \n    dp[0] = 0;\n\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = myMin(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[amt] != MAX ? dp[amt] : -1;\n    // 메모리 해제\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nfun coinChangeDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // dp 테이블을 초기화한다\n    val dp = IntArray(amt + 1)\n    dp.fill(MAX)\n    dp[0] = 0\n    // 상태 전이\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return if (dp[amt] != MAX) dp[amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>### 동전 교환: 공간을 최적화한 동적 계획법 ###\ndef coin_change_dp_comp(coins, amt)\n  n = coins.length\n  _MAX = amt + 1\n  # dp 테이블을 초기화한다\n  dp = Array.new(amt + 1, _MAX)\n  dp[0] = 0\n  # 상태 전이\n  for i in 1...(n + 1)\n    # 정방향 순회\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[a] = dp[a]\n      else\n        # 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n        dp[a] = [dp[a], dp[a - coins[i - 1]] + 1].min\n      end\n    end\n  end\n  dp[amt] != _MAX ? dp[amt] : -1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1453-ii","level":2,"title":"14.5.3   동전 교환 문제 II","text":"<p>문제</p> <p>동전 \\(n\\) 종류가 주어지며 \\(i\\) 번째 동전의 액면가는 \\(coins[i - 1]\\)이고 목표 금액은 \\(amt\\)다. 각 종류의 동전을 반복해서 선택할 수 있을 때, 목표 금액을 만드는 동전 조합 수를 구하라. 예시는 아래 그림(14-26)과 같다.</p> <p></p> <p> 그림 14-26   동전 교환 문제 II 예시 데이터 </p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_2","level":3,"title":"1.   동적 계획법 접근","text":"<p>앞 문제와 달리 이번에는 조합 수를 구하므로 부분 문제는 앞의 \\(i\\) 종류 동전으로 금액 \\(a\\)를 만드는 조합 수가 된다. \\(dp\\) 테이블은 여전히 \\((n+1) \\times (amt + 1)\\) 크기의 2차원 행렬이다.</p> <p>현재 상태의 조합 수는 현재 동전을 선택하지 않는 경우와 선택하는 경우의 조합 수를 더한 값이다. 상태 전이 식은 다음과 같다.</p> \\[ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]] \\] <p>목표 금액이 \\(0\\)이면 동전을 하나도 선택하지 않고도 만들 수 있으므로 첫 열의 모든 \\(dp[i, 0]\\)을 \\(1\\)로 초기화한다. 동전이 없으면 \\(&gt;0\\)인 목표 금액을 만들 수 없으므로 첫 행의 모든 \\(dp[0, a]\\)는 \\(0\\)이다.</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_2","level":3,"title":"2.   코드 구현","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change_ii.py<pre><code>def coin_change_ii_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"동전 교환 II: 동적 계획법\"\"\"\n    n = len(coins)\n    # dp 테이블을 초기화한다\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # 첫 번째 열을 초기화한다\n    for i in range(n + 1):\n        dp[i][0] = 1\n    # 상태 전이\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n    return dp[n][amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* 동전 교환 II: 동적 계획법 */\nint coinChangeIIDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // dp 테이블을 초기화한다\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // 첫 번째 열을 초기화한다\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* 동전 교환 II: 동적 계획법 */\nint coinChangeIIDP(int[] coins, int amt) {\n    int n = coins.length;\n    // dp 테이블을 초기화한다\n    int[][] dp = new int[n + 1][amt + 1];\n    // 첫 번째 열을 초기화한다\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* 동전 교환 II: 동적 계획법 */\nint CoinChangeIIDP(int[] coins, int amt) {\n    int n = coins.Length;\n    // dp 테이블을 초기화한다\n    int[,] dp = new int[n + 1, amt + 1];\n    // 첫 번째 열을 초기화한다\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i, 0] = 1;\n    }\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i, a] = dp[i - 1, a] + dp[i, a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n, amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* 동전 교환 II: 동적 계획법 */\nfunc coinChangeIIDP(coins []int, amt int) int {\n    n := len(coins)\n    // dp 테이블을 초기화한다\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // 첫 번째 열을 초기화한다\n    for i := 0; i &lt;= n; i++ {\n        dp[i][0] = 1\n    }\n    // 상태 전이: 나머지 행과 열\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i-1][a] + dp[i][a-coins[i-1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* 동전 교환 II: 동적 계획법 */\nfunc coinChangeIIDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // 첫 번째 열을 초기화한다\n    for i in 0 ... n {\n        dp[i][0] = 1\n    }\n    // 상태 전이\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* 동전 교환 II: 동적 계획법 */\nfunction coinChangeIIDP(coins, amt) {\n    const n = coins.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // 첫 번째 열을 초기화한다\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* 동전 교환 II: 동적 계획법 */\nfunction coinChangeIIDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // 첫 번째 열을 초기화한다\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* 동전 교환 II: 동적 계획법 */\nint coinChangeIIDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // dp 테이블을 초기화한다\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // 첫 번째 열을 초기화한다\n  for (int i = 0; i &lt;= n; i++) {\n    dp[i][0] = 1;\n  }\n  // 상태 전이\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[n][amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* 동전 교환 II: 동적 계획법 */\nfn coin_change_ii_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // dp 테이블을 초기화한다\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // 첫 번째 열을 초기화한다\n    for i in 0..=n {\n        dp[i][0] = 1;\n    }\n    // 상태 전이\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[n][amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* 동전 교환 II: 동적 계획법 */\nint coinChangeIIDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // dp 테이블을 초기화한다\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // 첫 번째 열을 초기화한다\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[n][amt];\n    // 메모리 해제\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* 동전 교환 II: 동적 계획법 */\nfun coinChangeIIDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // dp 테이블을 초기화한다\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // 첫 번째 열을 초기화한다\n    for (i in 0..n) {\n        dp[i][0] = 1\n    }\n    // 상태 전이\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>### 동전 교환 II: 동적 계획법 ###\ndef coin_change_ii_dp(coins, amt)\n  n = coins.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(n + 1) { Array.new(amt + 1, 0) }\n  # 첫 번째 열을 초기화한다\n  (0...(n + 1)).each { |i| dp[i][0] = 1 }\n  # 상태 전이\n  for i in 1...(n + 1)\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[i][a] = dp[i - 1][a]\n      else\n        # 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n      end\n    end\n  end\n  dp[n][amt]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_2","level":3,"title":"3.   공간 최적화","text":"<p>공간 최적화 방법도 같으며, 동전 차원을 없애면 된다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change_ii.py<pre><code>def coin_change_ii_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"동전 교환 II: 공간을 최적화한 동적 계획법\"\"\"\n    n = len(coins)\n    # dp 테이블을 초기화한다\n    dp = [0] * (amt + 1)\n    dp[0] = 1\n    # 상태 전이\n    for i in range(1, n + 1):\n        # 정방향 순회\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            else:\n                # 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n    return dp[amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nint coinChangeIIDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // dp 테이블을 초기화한다\n    vector&lt;int&gt; dp(amt + 1, 0);\n    dp[0] = 1;\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nint coinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nint CoinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nfunc coinChangeIIDPComp(coins []int, amt int) int {\n    n := len(coins)\n    // dp 테이블을 초기화한다\n    dp := make([]int, amt+1)\n    dp[0] = 1\n    // 상태 전이\n    for i := 1; i &lt;= n; i++ {\n        // 정방향 순회\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a-coins[i-1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nfunc coinChangeIIDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: 0, count: amt + 1)\n    dp[0] = 1\n    // 상태 전이\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nfunction coinChangeIIDPComp(coins, amt) {\n    const n = coins.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nfunction coinChangeIIDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nint coinChangeIIDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // dp 테이블을 초기화한다\n  List&lt;int&gt; dp = List.filled(amt + 1, 0);\n  dp[0] = 1;\n  // 상태 전이\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[a] = dp[a];\n      } else {\n        // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n        dp[a] = dp[a] + dp[a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nfn coin_change_ii_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // dp 테이블을 초기화한다\n    let mut dp = vec![0; amt + 1];\n    dp[0] = 1;\n    // 상태 전이\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nint coinChangeIIDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // dp 테이블을 초기화한다\n    int *dp = calloc(amt + 1, sizeof(int));\n    dp[0] = 1;\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[amt];\n    // 메모리 해제\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nfun coinChangeIIDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // dp 테이블을 초기화한다\n    val dp = IntArray(amt + 1)\n    dp[0] = 1\n    // 상태 전이\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>### 동전 교환 II: 공간을 최적화한 동적 계획법 ###\ndef coin_change_ii_dp_comp(coins, amt)\n  n = coins.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(amt + 1, 0)\n  dp[0] = 1\n  # 상태 전이\n  for i in 1...(n + 1)\n    # 정방향 순회\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[a] = dp[a]\n      else\n        # 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n        dp[a] = dp[a] + dp[a - coins[i - 1]]\n      end\n    end\n  end\n  dp[amt]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_graph/","level":1,"title":"제 9 장   그래프","text":"<p>이 장을 시작하며</p> <p>삶이라는 여정에서 우리는 하나하나의 노드처럼, 보이지 않는 수많은 간선으로 연결되어 있다.</p> <p>만남과 헤어짐은 매번 이 거대한 네트워크 그래프에 저마다의 흔적을 남긴다.</p>","path":["제 9 장   그래프"],"tags":[]},{"location":"chapter_graph/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>9.1   그래프</li> <li>9.2   그래프의 기본 연산</li> <li>9.3   그래프 순회</li> <li>9.4   요약</li> <li>9.5   연습 문제</li> </ul>","path":["제 9 장   그래프"],"tags":[]},{"location":"chapter_graph/exercises/","level":1,"title":"9.5   연습 문제","text":"","path":["제 9 장   그래프","9.5   연습 문제"],"tags":[]},{"location":"chapter_graph/exercises/#951","level":2,"title":"9.5.1   개념 확인","text":"","path":["제 9 장   그래프","9.5   연습 문제"],"tags":[]},{"location":"chapter_graph/exercises/#1","level":3,"title":"1.   같은 그래프를 두 가지 방식으로 표현하기","text":"<p>무방향 그래프의 정점은 <code>A, B, C, D</code> 4개이며, 간선은 <code>A-B, A-C, B-C, C-D</code>다.</p> <ol> <li>이 그래프의 인접 리스트를 적어 보자.</li> <li>0과 1만 들어가는 인접 행렬을 완성해 보자.</li> <li><code>A</code>와 <code>D</code>가 직접 연결되어 있는지 판단할 때, 저장된 항목 하나만 확인하면 되는 표현 방식은 무엇인가?</li> <li>정점이 많고 간선이 적다면 일반적으로 어느 표현 방식이 공간을 더 절약하는가?</li> </ol> 참고 답안 <ol> <li> <p>인접 리스트는 다음과 같다.</p> <pre><code>A: B, C\nB: A, C\nC: A, B, D\nD: C\n</code></pre> </li> <li> <p>인접 행렬은 다음과 같다.</p> A B C D A 0 1 1 0 B 1 0 1 0 C 1 1 0 1 D 0 0 1 0 </li> <li> <p>인접 행렬에서는 <code>A</code>행과 <code>D</code>열이 만나는 항목을 바로 확인할 수 있다. 따라서 임의의 두 정점이 직접 연결되어 있는지 판단하기에 적합하다.</p> </li> <li> <p>정점이 많고 간선이 적을 때 인접 리스트는 실제로 존재하는 간선만 기록한다. 따라서 모든 정점 쌍에 대해 자리를 마련하는 인접 행렬보다 일반적으로 공간을 더 절약한다.</p> </li> </ol>","path":["제 9 장   그래프","9.5   연습 문제"],"tags":[]},{"location":"chapter_graph/exercises/#2","level":3,"title":"2.   너비 우선 탐색과 깊이 우선 탐색의 방문 순서","text":"<p>무방향 그래프의 정점은 <code>A, B, C, D, E</code>이며, 간선은 <code>A-B, A-C, B-D, C-D, D-E</code>다.</p> <p>A에서 출발하고, 방문하지 않은 인접 정점이 여러 개면 알파벳순으로 선택한다고 하자.</p> <ol> <li>너비 우선 탐색(BFS)의 방문 순서를 적어 보자.</li> <li>재귀 깊이 우선 탐색(DFS)의 방문 순서를 적어 보자.</li> <li>두 탐색 모두 이미 방문한 정점을 기록해야 하는 이유는 무엇인가?</li> </ol> 참고 답안 <ol> <li> <p>BFS의 방문 순서는 <code>A, B, C, D, E</code>다. A에서 간선 하나로 도달하는 B와 C를 먼저 방문하고,     그다음 더 멀리 있는 D와 E를 방문한다.</p> </li> <li> <p>DFS의 방문 순서는 <code>A, B, D, C, E</code>다. 현재 정점에서 방문하지 않은 인접 정점으로 차례대로 들어가므로     먼저 <code>A → B → D → C</code>를 따라간다. C에서 새로 방문할 인접 정점이 없으면 D로 돌아와 E를 방문한다.</p> </li> <li> <p>그래프에는 <code>A-B-D-C-A</code>와 같은 사이클이 있다. 이미 방문한 정점을 기록하지 않으면     같은 정점들을 사이클을 따라 반복해서 방문하여 탐색이 정상적으로 끝나지 않을 수 있다.</p> </li> </ol>","path":["제 9 장   그래프","9.5   연습 문제"],"tags":[]},{"location":"chapter_graph/exercises/#3-bfs","level":3,"title":"3.   BFS 한 번으로 그래프 전체를 방문할 수 있는가","text":"<p>무방향 그래프의 정점은 <code>A, B, C, D, E, F</code>이며, 간선은 <code>A-B, B-C, D-E</code>뿐이다.</p> <ol> <li>A에서 BFS를 한 번 시작하면 어떤 정점을 방문할 수 있는가?</li> <li>1번의 결과를 바탕으로, 이 BFS가 그래프의 모든 정점을 방문했는지 판단하고 이유를 설명하라.</li> <li>모든 정점을 알파벳순으로 확인하면서 방문하지 않은 정점을 만날 때마다 BFS를 새로 시작한다면,     각 BFS의 시작 정점은 무엇인가? 이 그래프는 서로 연결되지 않은 부분인 연결 요소 몇 개로 이루어져 있는가?</li> </ol> 참고 답안 <ol> <li> <p>A에서 출발하면 <code>A, B, C</code>만 방문할 수 있다.</p> </li> <li> <p>모든 정점을 방문한 것은 아니다. <code>D, E</code>는 서로 연결된 별개의 부분을 이루고, F는 홀로 떨어진 정점이다.     이 정점들과 A 사이에는 경로가 없으므로 A에서 출발해서는 도달할 수 없다.</p> </li> <li> <p>세 번의 BFS는 차례대로 <code>A, D, F</code>에서 시작하여 각각     <code>{A, B, C}</code>, <code>{D, E}</code>, <code>{F}</code>를 방문한다. 따라서 이 그래프의 연결 요소는 3개다.</p> </li> </ol>","path":["제 9 장   그래프","9.5   연습 문제"],"tags":[]},{"location":"chapter_graph/exercises/#952","level":2,"title":"9.5.2   프로그래밍 연습","text":"","path":["제 9 장   그래프","9.5   연습 문제"],"tags":[]},{"location":"chapter_graph/exercises/#1_1","level":3,"title":"1.   무방향 그래프에 경로가 존재하는지 판단하기","text":"<p>정점이 \\(n\\)개인 무방향 그래프가 주어지며, 정점 번호는 \\(0\\)부터 \\(n-1\\)까지다. 배열 <code>edges</code>의 각 항목 <code>[u, v]</code>는 정점 <code>u</code>와 <code>v</code> 사이에 무방향 간선이 있음을 나타낸다.</p> <p>시작 정점 <code>source</code>와 도착 정점 <code>destination</code>도 주어진다. 먼저 <code>edges</code>로 인접 리스트를 만든 뒤 BFS 또는 DFS를 사용하여 <code>source</code>에서 <code>destination</code>까지의 경로가 있는지 판단하라. 있으면 <code>true</code>를, 없으면 <code>false</code>를 반환하라. 그래프에는 사이클이 있을 수 있으며, 모든 정점이 서로 연결되어 있지 않을 수도 있다.</p> 풀이 힌트 <ol> <li>무방향 간선은 양쪽 방향을 모두 추가해야 한다.</li> <li>그래프에는 사이클이 있을 수 있으므로 이미 방문한 노드를 반드시 기록해야 한다.</li> <li>source에서 출발하여 destination을 만나면 true를 반환한다. 탐색을 마칠 때까지 만나지 못하면 false를 반환한다.</li> </ol> <p>LeetCode</p>","path":["제 9 장   그래프","9.5   연습 문제"],"tags":[]},{"location":"chapter_graph/graph/","level":1,"title":"9.1   그래프","text":"<p>그래프(graph)는 정점(vertex)과 간선(edge)으로 이루어진 비선형 자료 구조다. 그래프 \\(G\\)는 정점 집합 \\(V\\)와 간선 집합 \\(E\\)로 추상적으로 표현할 수 있다. 다음 예시는 정점 5개와 간선 7개로 이루어진 그래프를 나타낸다.</p> \\[ \\begin{aligned} V &amp; = \\{ 1, 2, 3, 4, 5 \\} \\newline E &amp; = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G &amp; = \\{ V, E \\} \\newline \\end{aligned} \\] <p>정점을 노드로, 간선을 노드들을 연결하는 참조(포인터)로 보면 그래프는 연결 리스트를 확장한 자료 구조로 생각할 수 있다. 아래 그림(9-1)과 같이 선형 관계(연결 리스트)와 분할 정복 관계(트리)에 비해 네트워크 관계(그래프)는 자유도가 높아 더 복잡하다.</p> <p></p> <p> 그림 9-1   연결 리스트, 트리, 그래프 사이의 관계 </p>","path":["제 9 장   그래프","9.1   그래프"],"tags":[]},{"location":"chapter_graph/graph/#911","level":2,"title":"9.1.1   그래프의 주요 종류와 용어","text":"<p>간선에 방향이 있는지에 따라 아래 그림(9-2)과 같이 무방향 그래프(undirected graph)와 방향 그래프(directed graph)로 나뉜다.</p> <ul> <li>무방향 그래프에서 간선은 두 정점 사이의 '양방향' 연결 관계를 나타낸다. 위챗이나 QQ의 '친구 관계'가 그 예다.</li> <li>방향 그래프에서 간선은 방향을 가진다. 즉, \\(A \\rightarrow B\\)와 \\(A \\leftarrow B\\) 방향의 간선은 서로 독립적이다. 웨이보나 더우인의 '팔로우'와 '팔로워' 관계가 그 예다.</li> </ul> <p></p> <p> 그림 9-2   방향 그래프와 무방향 그래프 </p> <p>모든 정점이 연결되어 있는지에 따라 아래 그림(9-3)과 같이 연결 그래프(connected graph)와 비연결 그래프(disconnected graph)로 나뉜다.</p> <ul> <li>연결 그래프에서는 한 정점에서 출발하여 다른 모든 정점에 도달할 수 있다.</li> <li>비연결 그래프에서는 한 정점에서 출발했을 때 도달할 수 없는 정점이 하나 이상 존재한다.</li> </ul> <p></p> <p> 그림 9-3   연결 그래프와 비연결 그래프 </p> <p>간선에 '가중치' 변수를 추가하면 아래 그림(9-4)과 같은 가중 그래프(weighted graph)를 얻는다. 예를 들어 『왕자영요』 같은 모바일 게임에서는 함께 게임한 시간을 바탕으로 플레이어 간의 '친밀도'를 계산한다. 이러한 친밀도 네트워크는 가중 그래프로 표현할 수 있다.</p> <p></p> <p> 그림 9-4   가중 그래프와 비가중 그래프 </p> <p>그래프 자료 구조에서 자주 사용하는 용어는 다음과 같다.</p> <ul> <li>인접(adjacency): 두 정점 사이에 간선이 있으면 두 정점이 '인접한다'고 한다. 위 그림(9-4)에서 정점 1에 인접한 정점은 2, 3, 5다.</li> <li>경로(path): 정점 A에서 정점 B로 갈 때 거치는 간선의 순서를 A에서 B로 가는 '경로'라고 한다. 위 그림(9-4)에서 간선 순서 1-5-2-4는 정점 1에서 정점 4로 가는 경로다.</li> <li>차수(degree): 한 정점에 연결된 간선의 수다. 방향 그래프에서는 진입 차수(in-degree)가 해당 정점으로 들어오는 간선 수를, 진출 차수(out-degree)가 해당 정점에서 나가는 간선 수를 나타낸다.</li> </ul>","path":["제 9 장   그래프","9.1   그래프"],"tags":[]},{"location":"chapter_graph/graph/#912","level":2,"title":"9.1.2   그래프의 표현","text":"<p>그래프를 표현하는 대표적인 방법은 '인접 행렬'과 '인접 리스트'다. 아래에서는 무방향 그래프를 예로 든다.</p>","path":["제 9 장   그래프","9.1   그래프"],"tags":[]},{"location":"chapter_graph/graph/#1","level":3,"title":"1.   인접 행렬","text":"<p>그래프의 정점 수를 \\(n\\)이라고 하자. 인접 행렬(adjacency matrix)은 \\(n \\times n\\) 크기의 행렬로 그래프를 표현한다. 각 행(열)은 정점 하나를, 행렬의 원소는 간선을 나타내며, \\(1\\) 또는 \\(0\\)으로 두 정점 사이에 간선이 있는지 표시한다.</p> <p>아래 그림(9-5)과 같이 인접 행렬을 \\(M\\), 정점 리스트를 \\(V\\)라고 하면, 행렬 원소 \\(M[i, j] = 1\\)은 정점 \\(V[i]\\)와 정점 \\(V[j]\\) 사이에 간선이 있음을 뜻한다. 반대로 \\(M[i, j] = 0\\)은 두 정점 사이에 간선이 없음을 뜻한다.</p> <p></p> <p> 그림 9-5   그래프의 인접 행렬 표현 </p> <p>인접 행렬에는 다음 성질이 있다.</p> <ul> <li>단순 그래프에서는 정점이 자기 자신과 연결될 수 없으므로, 인접 행렬의 주대각선 원소는 의미가 없다.</li> <li>무방향 그래프에서는 두 방향의 간선이 같으므로, 인접 행렬은 주대각선에 대해 대칭이다.</li> <li>인접 행렬의 원소 \\(1\\)과 \\(0\\)을 가중치로 바꾸면 가중 그래프를 표현할 수 있다.</li> </ul> <p>인접 행렬로 그래프를 표현하면 행렬 원소에 직접 접근하여 간선을 얻을 수 있다. 따라서 추가, 삭제, 조회, 수정 연산이 매우 효율적이며 시간 복잡도는 모두 \\(O(1)\\)이다. 하지만 행렬의 공간 복잡도는 \\(O(n^2)\\)이므로 메모리를 많이 차지한다.</p>","path":["제 9 장   그래프","9.1   그래프"],"tags":[]},{"location":"chapter_graph/graph/#2","level":3,"title":"2.   인접 리스트","text":"<p>인접 리스트(adjacency list)는 \\(n\\) 개의 연결 리스트로 그래프를 표현하며, 연결 리스트의 노드는 정점을 나타낸다. \\(i\\) 번째 연결 리스트는 정점 \\(i\\)에 대응하고, 해당 정점에 인접한 모든 정점(간선으로 연결된 정점)을 저장한다. 아래 그림(9-6)은 인접 리스트로 그래프를 저장한 예다.</p> <p></p> <p> 그림 9-6   그래프의 인접 리스트 표현 </p> <p>인접 리스트는 실제로 존재하는 간선만 저장한다. 전체 간선 수는 보통 \\(n^2\\)보다 훨씬 작으므로 공간을 더 절약할 수 있다. 하지만 간선을 찾으려면 연결 리스트를 순회해야 하므로 시간 효율은 인접 행렬보다 낮다.</p> <p>위 그림(9-6)을 보면 인접 리스트의 구조는 해시 테이블의 '체이닝'과 매우 비슷하므로, 비슷한 방법으로 효율을 높일 수 있다. 예를 들어 연결 리스트가 길면 AVL 트리나 레드-블랙 트리로 바꾸어 시간 복잡도를 \\(O(n)\\)에서 \\(O(\\log n)\\)으로 개선할 수 있다. 연결 리스트를 해시 테이블로 바꾸면 시간 복잡도를 \\(O(1)\\)로 줄일 수도 있다.</p>","path":["제 9 장   그래프","9.1   그래프"],"tags":[]},{"location":"chapter_graph/graph/#913","level":2,"title":"9.1.3   그래프의 주요 응용","text":"<p>아래 표(9-1)와 같이 현실의 많은 시스템을 그래프로 모델링할 수 있으며, 관련 문제를 그래프 계산 문제로 환원할 수 있다.</p> <p> 표 9-1   일상에서 볼 수 있는 그래프 </p> 정점 간선 그래프 계산 문제 소셜 네트워크 사용자 친구 관계 잠재적 친구 추천 지하철 노선 역 역 사이의 연결 관계 최단 경로 추천 태양계 천체 천체 사이의 만유인력 행성 궤도 계산","path":["제 9 장   그래프","9.1   그래프"],"tags":[]},{"location":"chapter_graph/graph_operations/","level":1,"title":"9.2   그래프의 기본 연산","text":"<p>그래프의 기본 연산은 '간선'에 대한 연산과 '정점'에 대한 연산으로 나뉜다. '인접 행렬'과 '인접 리스트' 중 어느 표현을 사용하는지에 따라 구현 방법이 달라진다.</p>","path":["제 9 장   그래프","9.2   그래프의 기본 연산"],"tags":[]},{"location":"chapter_graph/graph_operations/#921","level":2,"title":"9.2.1   인접 행렬 기반 구현","text":"<p>정점 수가 \\(n\\)인 무방향 그래프가 주어졌을 때 각 연산은 아래 그림(9-7)과 같이 구현한다.</p> <ul> <li>간선 추가 또는 삭제: 인접 행렬에서 해당 간선을 직접 수정하면 되며, \\(O(1)\\) 시간이 걸린다. 무방향 그래프이므로 두 방향의 간선을 함께 갱신해야 한다.</li> <li>정점 추가: 인접 행렬의 끝에 행과 열을 하나씩 추가하고 모두 \\(0\\)으로 채운다. \\(O(n)\\) 시간이 걸린다.</li> <li>정점 삭제: 인접 행렬에서 행과 열을 하나씩 삭제한다. 첫 번째 행과 열을 삭제할 때가 최악의 경우로, \\((n-1)^2\\) 개 원소를 '왼쪽 위로 이동'해야 하므로 \\(O(n^2)\\) 시간이 걸린다.</li> <li>초기화: 정점 \\(n\\) 개를 전달받아 길이가 \\(n\\)인 정점 리스트 <code>vertices</code>를 초기화하는 데 \\(O(n)\\) 시간이 걸린다. \\(n \\times n\\) 크기의 인접 행렬 <code>adjMat</code>을 초기화하는 데는 \\(O(n^2)\\) 시간이 걸린다.</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 9-7   인접 행렬의 초기화, 간선 추가와 삭제, 정점 추가와 삭제 </p> <p>다음은 인접 행렬로 그래프를 표현한 구현 코드다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_adjacency_matrix.py<pre><code>class GraphAdjMat:\n    \"\"\"인접 행렬로 구현한 무방향 그래프 클래스\"\"\"\n\n    def __init__(self, vertices: list[int], edges: list[list[int]]):\n        \"\"\"생성자\"\"\"\n        # 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n        self.vertices: list[int] = []\n        # 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n        self.adj_mat: list[list[int]] = []\n        # 정점 추가\n        for val in vertices:\n            self.add_vertex(val)\n        # 간선 추가\n        # edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for e in edges:\n            self.add_edge(e[0], e[1])\n\n    def size(self) -&gt; int:\n        \"\"\"정점 개수 반환\"\"\"\n        return len(self.vertices)\n\n    def add_vertex(self, val: int):\n        \"\"\"정점 추가\"\"\"\n        n = self.size()\n        # 정점 리스트에 새 정점의 값을 추가한다\n        self.vertices.append(val)\n        # 인접 행렬에 행 하나를 추가한다\n        new_row = [0] * n\n        self.adj_mat.append(new_row)\n        # 인접 행렬에 열 하나를 추가한다\n        for row in self.adj_mat:\n            row.append(0)\n\n    def remove_vertex(self, index: int):\n        \"\"\"정점 삭제\"\"\"\n        if index &gt;= self.size():\n            raise IndexError()\n        # 정점 리스트에서 인덱스 index의 정점을 제거한다\n        self.vertices.pop(index)\n        # 인접 행렬에서 인덱스 index의 행을 삭제한다\n        self.adj_mat.pop(index)\n        # 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for row in self.adj_mat:\n            row.pop(index)\n\n    def add_edge(self, i: int, j: int):\n        \"\"\"간선 추가\"\"\"\n        # 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n        # 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        # 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n        self.adj_mat[i][j] = 1\n        self.adj_mat[j][i] = 1\n\n    def remove_edge(self, i: int, j: int):\n        \"\"\"간선 삭제\"\"\"\n        # 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n        # 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        self.adj_mat[i][j] = 0\n        self.adj_mat[j][i] = 0\n\n    def print(self):\n        \"\"\"인접 행렬 출력\"\"\"\n        print(\"정점 리스트 =\", self.vertices)\n        print(\"인접 행렬 =\")\n        print_matrix(self.adj_mat)\n</code></pre> graph_adjacency_matrix.cpp<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat {\n    vector&lt;int&gt; vertices;       // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    vector&lt;vector&lt;int&gt;&gt; adjMat; // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n  public:\n    /* 생성자 */\n    GraphAdjMat(const vector&lt;int&gt; &amp;vertices, const vector&lt;vector&lt;int&gt;&gt; &amp;edges) {\n        // 정점 추가\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for (const vector&lt;int&gt; &amp;edge : edges) {\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    int size() const {\n        return vertices.size();\n    }\n\n    /* 정점 추가 */\n    void addVertex(int val) {\n        int n = size();\n        // 정점 리스트에 새 정점의 값을 추가한다\n        vertices.push_back(val);\n        // 인접 행렬에 행 하나를 추가한다\n        adjMat.emplace_back(vector&lt;int&gt;(n, 0));\n        // 인접 행렬에 열 하나를 추가한다\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.push_back(0);\n        }\n    }\n\n    /* 정점 삭제 */\n    void removeVertex(int index) {\n        if (index &gt;= size()) {\n            throw out_of_range(\"정점이 존재하지 않는다\");\n        }\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        vertices.erase(vertices.begin() + index);\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        adjMat.erase(adjMat.begin() + index);\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.erase(row.begin() + index);\n        }\n    }\n\n    /* 간선 추가 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    void addEdge(int i, int j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"정점이 존재하지 않는다\");\n        }\n        // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    void removeEdge(int i, int j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"정점이 존재하지 않는다\");\n        }\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* 인접 행렬 출력 */\n    void print() {\n        cout &lt;&lt; \"정점 리스트 = \";\n        printVector(vertices);\n        cout &lt;&lt; \"인접 행렬 =\" &lt;&lt; endl;\n        printVectorMatrix(adjMat);\n    }\n};\n</code></pre> graph_adjacency_matrix.java<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat {\n    List&lt;Integer&gt; vertices; // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    List&lt;List&lt;Integer&gt;&gt; adjMat; // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n    /* 생성자 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = new ArrayList&lt;&gt;();\n        this.adjMat = new ArrayList&lt;&gt;();\n        // 정점 추가\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for (int[] e : edges) {\n            addEdge(e[0], e[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    public int size() {\n        return vertices.size();\n    }\n\n    /* 정점 추가 */\n    public void addVertex(int val) {\n        int n = size();\n        // 정점 리스트에 새 정점의 값을 추가한다\n        vertices.add(val);\n        // 인접 행렬에 행 하나를 추가한다\n        List&lt;Integer&gt; newRow = new ArrayList&lt;&gt;(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.add(0);\n        }\n        adjMat.add(newRow);\n        // 인접 행렬에 열 하나를 추가한다\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.add(0);\n        }\n    }\n\n    /* 정점 삭제 */\n    public void removeVertex(int index) {\n        if (index &gt;= size())\n            throw new IndexOutOfBoundsException();\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        vertices.remove(index);\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        adjMat.remove(index);\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.remove(index);\n        }\n    }\n\n    /* 간선 추가 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    public void addEdge(int i, int j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n        adjMat.get(i).set(j, 1);\n        adjMat.get(j).set(i, 1);\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    public void removeEdge(int i, int j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        adjMat.get(i).set(j, 0);\n        adjMat.get(j).set(i, 0);\n    }\n\n    /* 인접 행렬 출력 */\n    public void print() {\n        System.out.print(\"정점 리스트 = \");\n        System.out.println(vertices);\n        System.out.println(\"인접 행렬 =\");\n        PrintUtil.printMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.cs<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat {\n    List&lt;int&gt; vertices;     // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    List&lt;List&lt;int&gt;&gt; adjMat; // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n    /* 생성자 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // 정점 추가\n        foreach (int val in vertices) {\n            AddVertex(val);\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        foreach (int[] e in edges) {\n            AddEdge(e[0], e[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    int Size() {\n        return vertices.Count;\n    }\n\n    /* 정점 추가 */\n    public void AddVertex(int val) {\n        int n = Size();\n        // 정점 리스트에 새 정점의 값을 추가한다\n        vertices.Add(val);\n        // 인접 행렬에 행 하나를 추가한다\n        List&lt;int&gt; newRow = new(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.Add(0);\n        }\n        adjMat.Add(newRow);\n        // 인접 행렬에 열 하나를 추가한다\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.Add(0);\n        }\n    }\n\n    /* 정점 삭제 */\n    public void RemoveVertex(int index) {\n        if (index &gt;= Size())\n            throw new IndexOutOfRangeException();\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        vertices.RemoveAt(index);\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        adjMat.RemoveAt(index);\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.RemoveAt(index);\n        }\n    }\n\n    /* 간선 추가 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    public void AddEdge(int i, int j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    public void RemoveEdge(int i, int j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* 인접 행렬 출력 */\n    public void Print() {\n        Console.Write(\"정점 리스트 = \");\n        PrintUtil.PrintList(vertices);\n        Console.WriteLine(\"인접 행렬 =\");\n        PrintUtil.PrintMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.go<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\ntype graphAdjMat struct {\n    // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    vertices []int\n    // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n    adjMat [][]int\n}\n\n/* 생성자 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n    // 정점 추가\n    n := len(vertices)\n    adjMat := make([][]int, n)\n    for i := range adjMat {\n        adjMat[i] = make([]int, n)\n    }\n    // 그래프 초기화\n    g := &amp;graphAdjMat{\n        vertices: vertices,\n        adjMat:   adjMat,\n    }\n    // 간선 추가\n    // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n    for i := range edges {\n        g.addEdge(edges[i][0], edges[i][1])\n    }\n    return g\n}\n\n/* 정점 개수 반환 */\nfunc (g *graphAdjMat) size() int {\n    return len(g.vertices)\n}\n\n/* 정점 추가 */\nfunc (g *graphAdjMat) addVertex(val int) {\n    n := g.size()\n    // 정점 리스트에 새 정점의 값을 추가한다\n    g.vertices = append(g.vertices, val)\n    // 인접 행렬에 행 하나를 추가한다\n    newRow := make([]int, n)\n    g.adjMat = append(g.adjMat, newRow)\n    // 인접 행렬에 열 하나를 추가한다\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i], 0)\n    }\n}\n\n/* 정점 삭제 */\nfunc (g *graphAdjMat) removeVertex(index int) {\n    if index &gt;= g.size() {\n        return\n    }\n    // 정점 리스트에서 인덱스 index의 정점을 제거한다\n    g.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n    // 인접 행렬에서 인덱스 index의 행을 삭제한다\n    g.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n    // 인접 행렬에서 인덱스 index의 열을 삭제한다\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n    }\n}\n\n/* 간선 추가 */\n// 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\nfunc (g *graphAdjMat) addEdge(i, j int) {\n    // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"인덱스가 범위를 벗어났다\")\n    }\n    // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n    g.adjMat[i][j] = 1\n    g.adjMat[j][i] = 1\n}\n\n/* 간선 삭제 */\n// 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n    // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"인덱스가 범위를 벗어났다\")\n    }\n    g.adjMat[i][j] = 0\n    g.adjMat[j][i] = 0\n}\n\n/* 인접 행렬 출력 */\nfunc (g *graphAdjMat) print() {\n    fmt.Printf(\"\\t정점 리스트 = %v\\n\", g.vertices)\n    fmt.Printf(\"\\t인접 행렬 = \\n\")\n    for i := range g.adjMat {\n        fmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n    }\n}\n</code></pre> graph_adjacency_matrix.swift<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat {\n    private var vertices: [Int] // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    private var adjMat: [[Int]] // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n    /* 생성자 */\n    init(vertices: [Int], edges: [[Int]]) {\n        self.vertices = []\n        adjMat = []\n        // 정점 추가\n        for val in vertices {\n            addVertex(val: val)\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for e in edges {\n            addEdge(i: e[0], j: e[1])\n        }\n    }\n\n    /* 정점 개수 반환 */\n    func size() -&gt; Int {\n        vertices.count\n    }\n\n    /* 정점 추가 */\n    func addVertex(val: Int) {\n        let n = size()\n        // 정점 리스트에 새 정점의 값을 추가한다\n        vertices.append(val)\n        // 인접 행렬에 행 하나를 추가한다\n        let newRow = Array(repeating: 0, count: n)\n        adjMat.append(newRow)\n        // 인접 행렬에 열 하나를 추가한다\n        for i in adjMat.indices {\n            adjMat[i].append(0)\n        }\n    }\n\n    /* 정점 삭제 */\n    func removeVertex(index: Int) {\n        if index &gt;= size() {\n            fatalError(\"범위를 벗어남\")\n        }\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        vertices.remove(at: index)\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        adjMat.remove(at: index)\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for i in adjMat.indices {\n            adjMat[i].remove(at: index)\n        }\n    }\n\n    /* 간선 추가 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    func addEdge(i: Int, j: Int) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"범위를 벗어남\")\n        }\n        // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    func removeEdge(i: Int, j: Int) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"범위를 벗어남\")\n        }\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* 인접 행렬 출력 */\n    func print() {\n        Swift.print(\"정점 리스트 = \", terminator: \"\")\n        Swift.print(vertices)\n        Swift.print(\"인접 행렬 =\")\n        PrintUtil.printMatrix(matrix: adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.js<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat {\n    vertices; // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    adjMat; // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n    /* 생성자 */\n    constructor(vertices, edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // 정점 추가\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    size() {\n        return this.vertices.length;\n    }\n\n    /* 정점 추가 */\n    addVertex(val) {\n        const n = this.size();\n        // 정점 리스트에 새 정점의 값을 추가한다\n        this.vertices.push(val);\n        // 인접 행렬에 행 하나를 추가한다\n        const newRow = [];\n        for (let j = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // 인접 행렬에 열 하나를 추가한다\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* 정점 삭제 */\n    removeVertex(index) {\n        if (index &gt;= this.size()) {\n            throw new RangeError('인덱스가 범위를 벗어났다');\n        }\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        this.vertices.splice(index, 1);\n\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        this.adjMat.splice(index, 1);\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* 간선 추가 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    addEdge(i, j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('인덱스가 범위를 벗어났다');\n        }\n        // 무방향 그래프의 인접 행렬은 주대각선을 기준으로 대칭이므로 (i, j) === (j, i)를 만족한다\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    removeEdge(i, j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('인덱스가 범위를 벗어났다');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* 인접 행렬 출력 */\n    print() {\n        console.log('정점 리스트 = ', this.vertices);\n        console.log('인접 행렬 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.ts<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat {\n    vertices: number[]; // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    adjMat: number[][]; // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n    /* 생성자 */\n    constructor(vertices: number[], edges: number[][]) {\n        this.vertices = [];\n        this.adjMat = [];\n        // 정점 추가\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    size(): number {\n        return this.vertices.length;\n    }\n\n    /* 정점 추가 */\n    addVertex(val: number): void {\n        const n: number = this.size();\n        // 정점 리스트에 새 정점의 값을 추가한다\n        this.vertices.push(val);\n        // 인접 행렬에 행 하나를 추가한다\n        const newRow: number[] = [];\n        for (let j: number = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // 인접 행렬에 열 하나를 추가한다\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* 정점 삭제 */\n    removeVertex(index: number): void {\n        if (index &gt;= this.size()) {\n            throw new RangeError('인덱스가 범위를 벗어났다');\n        }\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        this.vertices.splice(index, 1);\n\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        this.adjMat.splice(index, 1);\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* 간선 추가 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    addEdge(i: number, j: number): void {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('인덱스가 범위를 벗어났다');\n        }\n        // 무방향 그래프의 인접 행렬은 주대각선을 기준으로 대칭이므로 (i, j) === (j, i)를 만족한다\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    removeEdge(i: number, j: number): void {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('인덱스가 범위를 벗어났다');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* 인접 행렬 출력 */\n    print(): void {\n        console.log('정점 리스트 = ', this.vertices);\n        console.log('인접 행렬 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.dart<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat {\n  List&lt;int&gt; vertices = []; // 정점 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n  List&lt;List&lt;int&gt;&gt; adjMat = []; //인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n  /* 생성자 */\n  GraphAdjMat(List&lt;int&gt; vertices, List&lt;List&lt;int&gt;&gt; edges) {\n    this.vertices = [];\n    this.adjMat = [];\n    // 정점 추가\n    for (int val in vertices) {\n      addVertex(val);\n    }\n    // 간선 추가\n    // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n    for (List&lt;int&gt; e in edges) {\n      addEdge(e[0], e[1]);\n    }\n  }\n\n  /* 정점 개수 반환 */\n  int size() {\n    return vertices.length;\n  }\n\n  /* 정점 추가 */\n  void addVertex(int val) {\n    int n = size();\n    // 정점 리스트에 새 정점의 값을 추가한다\n    vertices.add(val);\n    // 인접 행렬에 행 하나를 추가한다\n    List&lt;int&gt; newRow = List.filled(n, 0, growable: true);\n    adjMat.add(newRow);\n    // 인접 행렬에 열 하나를 추가한다\n    for (List&lt;int&gt; row in adjMat) {\n      row.add(0);\n    }\n  }\n\n  /* 정점 삭제 */\n  void removeVertex(int index) {\n    if (index &gt;= size()) {\n      throw IndexError;\n    }\n    // 정점 리스트에서 인덱스 index의 정점을 제거한다\n    vertices.removeAt(index);\n    // 인접 행렬에서 인덱스 index의 행을 삭제한다\n    adjMat.removeAt(index);\n    // 인접 행렬에서 인덱스 index의 열을 삭제한다\n    for (List&lt;int&gt; row in adjMat) {\n      row.removeAt(index);\n    }\n  }\n\n  /* 간선 추가 */\n  // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n  void addEdge(int i, int j) {\n    // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n    adjMat[i][j] = 1;\n    adjMat[j][i] = 1;\n  }\n\n  /* 간선 삭제 */\n  // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n  void removeEdge(int i, int j) {\n    // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    adjMat[i][j] = 0;\n    adjMat[j][i] = 0;\n  }\n\n  /* 인접 행렬 출력 */\n  void printAdjMat() {\n    print(\"정점 리스트 = $vertices\");\n    print(\"인접 행렬 = \");\n    printMatrix(adjMat);\n  }\n}\n</code></pre> graph_adjacency_matrix.rs<pre><code>/* 인접 행렬로 구현한 무방향 그래프 자료형 */\npub struct GraphAdjMat {\n    // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    pub vertices: Vec&lt;i32&gt;,\n    // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n    pub adj_mat: Vec&lt;Vec&lt;i32&gt;&gt;,\n}\n\nimpl GraphAdjMat {\n    /* 생성자 */\n    pub fn new(vertices: Vec&lt;i32&gt;, edges: Vec&lt;[usize; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjMat {\n            vertices: vec![],\n            adj_mat: vec![],\n        };\n        // 정점 추가\n        for val in vertices {\n            graph.add_vertex(val);\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for edge in edges {\n            graph.add_edge(edge[0], edge[1])\n        }\n\n        graph\n    }\n\n    /* 정점 개수 반환 */\n    pub fn size(&amp;self) -&gt; usize {\n        self.vertices.len()\n    }\n\n    /* 정점 추가 */\n    pub fn add_vertex(&amp;mut self, val: i32) {\n        let n = self.size();\n        // 정점 리스트에 새 정점의 값을 추가한다\n        self.vertices.push(val);\n        // 인접 행렬에 행 하나를 추가한다\n        self.adj_mat.push(vec![0; n]);\n        // 인접 행렬에 열 하나를 추가한다\n        for row in self.adj_mat.iter_mut() {\n            row.push(0);\n        }\n    }\n\n    /* 정점 삭제 */\n    pub fn remove_vertex(&amp;mut self, index: usize) {\n        if index &gt;= self.size() {\n            panic!(\"인덱스 오류\")\n        }\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        self.vertices.remove(index);\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        self.adj_mat.remove(index);\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for row in self.adj_mat.iter_mut() {\n            row.remove(index);\n        }\n    }\n\n    /* 간선 추가 */\n    pub fn add_edge(&amp;mut self, i: usize, j: usize) {\n        // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"인덱스 오류\")\n        }\n        // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n        self.adj_mat[i][j] = 1;\n        self.adj_mat[j][i] = 1;\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    pub fn remove_edge(&amp;mut self, i: usize, j: usize) {\n        // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"인덱스 오류\")\n        }\n        self.adj_mat[i][j] = 0;\n        self.adj_mat[j][i] = 0;\n    }\n\n    /* 인접 행렬 출력 */\n    pub fn print(&amp;self) {\n        println!(\"정점 리스트 = {:?}\", self.vertices);\n        println!(\"인접 행렬 =\");\n        println!(\"[\");\n        for row in &amp;self.adj_mat {\n            println!(\"  {:?},\", row);\n        }\n        println!(\"]\")\n    }\n}\n</code></pre> graph_adjacency_matrix.c<pre><code>/* 인접 행렬로 구현한 무방향 그래프 구조체 */\ntypedef struct {\n    int vertices[MAX_SIZE];\n    int adjMat[MAX_SIZE][MAX_SIZE];\n    int size;\n} GraphAdjMat;\n\n/* 생성자 */\nGraphAdjMat *newGraphAdjMat() {\n    GraphAdjMat *graph = (GraphAdjMat *)malloc(sizeof(GraphAdjMat));\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        for (int j = 0; j &lt; MAX_SIZE; j++) {\n            graph-&gt;adjMat[i][j] = 0;\n        }\n    }\n    return graph;\n}\n\n/* 소멸자 */\nvoid delGraphAdjMat(GraphAdjMat *graph) {\n    free(graph);\n}\n\n/* 정점 추가 */\nvoid addVertex(GraphAdjMat *graph, int val) {\n    if (graph-&gt;size == MAX_SIZE) {\n        fprintf(stderr, \"그래프의 정점 개수가 최댓값에 도달했다\\n\");\n        return;\n    }\n    // n번째 정점을 추가하고 n번째 행과 열을 0으로 설정한다\n    int n = graph-&gt;size;\n    graph-&gt;vertices[n] = val;\n    for (int i = 0; i &lt;= n; i++) {\n        graph-&gt;adjMat[n][i] = graph-&gt;adjMat[i][n] = 0;\n    }\n    graph-&gt;size++;\n}\n\n/* 정점 삭제 */\nvoid removeVertex(GraphAdjMat *graph, int index) {\n    if (index &lt; 0 || index &gt;= graph-&gt;size) {\n        fprintf(stderr, \"정점 인덱스가 범위를 벗어났다\\n\");\n        return;\n    }\n    // 정점 리스트에서 인덱스 index의 정점을 제거한다\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        graph-&gt;vertices[i] = graph-&gt;vertices[i + 1];\n    }\n    // 인접 행렬에서 인덱스 index의 행을 삭제한다\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        for (int j = 0; j &lt; graph-&gt;size; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i + 1][j];\n        }\n    }\n    // 인접 행렬에서 인덱스 index의 열을 삭제한다\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        for (int j = index; j &lt; graph-&gt;size - 1; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i][j + 1];\n        }\n    }\n    graph-&gt;size--;\n}\n\n/* 간선 추가 */\n// 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\nvoid addEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"간선 인덱스가 범위를 벗어나거나 서로 같다\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 1;\n    graph-&gt;adjMat[j][i] = 1;\n}\n\n/* 간선 삭제 */\n// 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\nvoid removeEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"간선 인덱스가 범위를 벗어나거나 서로 같다\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 0;\n    graph-&gt;adjMat[j][i] = 0;\n}\n\n/* 인접 행렬 출력 */\nvoid printGraphAdjMat(GraphAdjMat *graph) {\n    printf(\"정점 리스트 = \");\n    printArray(graph-&gt;vertices, graph-&gt;size);\n    printf(\"인접 행렬 =\\n\");\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        printArray(graph-&gt;adjMat[i], graph-&gt;size);\n    }\n}\n</code></pre> graph_adjacency_matrix.kt<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat(vertices: IntArray, edges: Array&lt;IntArray&gt;) {\n    val vertices = mutableListOf&lt;Int&gt;() // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    val adjMat = mutableListOf&lt;MutableList&lt;Int&gt;&gt;() // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n    /* 생성자 */\n    init {\n        // 정점 추가\n        for (vertex in vertices) {\n            addVertex(vertex)\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for (edge in edges) {\n            addEdge(edge[0], edge[1])\n        }\n    }\n\n    /* 정점 개수 반환 */\n    fun size(): Int {\n        return vertices.size\n    }\n\n    /* 정점 추가 */\n    fun addVertex(_val: Int) {\n        val n = size()\n        // 정점 리스트에 새 정점의 값을 추가한다\n        vertices.add(_val)\n        // 인접 행렬에 행 하나를 추가한다\n        val newRow = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            newRow.add(0)\n        }\n        adjMat.add(newRow)\n        // 인접 행렬에 열 하나를 추가한다\n        for (row in adjMat) {\n            row.add(0)\n        }\n    }\n\n    /* 정점 삭제 */\n    fun removeVertex(index: Int) {\n        if (index &gt;= size())\n            throw IndexOutOfBoundsException()\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        vertices.removeAt(index)\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        adjMat.removeAt(index)\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for (row in adjMat) {\n            row.removeAt(index)\n        }\n    }\n\n    /* 간선 추가 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    fun addEdge(i: Int, j: Int) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw IndexOutOfBoundsException()\n        // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    fun removeEdge(i: Int, j: Int) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw IndexOutOfBoundsException()\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* 인접 행렬 출력 */\n    fun print() {\n        print(\"정점 리스트 = \")\n        println(vertices)\n        println(\"인접 행렬 =\")\n        printMatrix(adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.rb<pre><code>### 인접 행렬로 구현한 무방향 그래프 클래스 ###\nclass GraphAdjMat\n  def initialize(vertices, edges)\n    ### 생성자 ###\n    # 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    @vertices = []\n    # 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n    @adj_mat = []\n    # 정점 추가\n    vertices.each { |val| add_vertex(val) }\n    # 간선 추가\n    # edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n    edges.each { |e| add_edge(e[0], e[1]) }\n  end\n\n  ### 정점 개수 반환 ###\n  def size\n    @vertices.length\n  end\n\n  ### 정점 추가 ###\n  def add_vertex(val)\n    n = size\n    # 정점 리스트에 새 정점의 값을 추가한다\n    @vertices &lt;&lt; val\n    # 인접 행렬에 행 하나를 추가한다\n    new_row = Array.new(n, 0)\n    @adj_mat &lt;&lt; new_row\n    # 인접 행렬에 열 하나를 추가한다\n    @adj_mat.each { |row| row &lt;&lt; 0 }\n  end\n\n  ### 정점 삭제 ###\n  def remove_vertex(index)\n    raise IndexError if index &gt;= size\n\n    # 정점 리스트에서 인덱스 index의 정점을 제거한다\n    @vertices.delete_at(index)\n    # 인접 행렬에서 인덱스 index의 행을 삭제한다\n    @adj_mat.delete_at(index)\n    # 인접 행렬에서 인덱스 index의 열을 삭제한다\n    @adj_mat.each { |row| row.delete_at(index) }\n  end\n\n  ### 간선 추가 ###\n  def add_edge(i, j)\n    # 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    # 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n    if i &lt; 0 || j &lt; 0 || i &gt;= size || j &gt;= size || i == j\n      raise IndexError\n    end\n    # 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n    @adj_mat[i][j] = 1\n    @adj_mat[j][i] = 1\n  end\n\n  ### 간선 삭제 ###\n  def remove_edge(i, j)\n    # 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    # 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n    if i &lt; 0 || j &lt; 0 || i &gt;= size || j &gt;= size || i == j\n      raise IndexError\n    end\n    @adj_mat[i][j] = 0\n    @adj_mat[j][i] = 0\n  end\n\n  ### 인접 행렬 출력 ###\n  def __print__\n    puts \"정점 리스트 = #{@vertices}\"\n    puts '인접 행렬 ='\n    print_matrix(@adj_mat)\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 9 장   그래프","9.2   그래프의 기본 연산"],"tags":[]},{"location":"chapter_graph/graph_operations/#922","level":2,"title":"9.2.2   인접 리스트 기반 구현","text":"<p>무방향 그래프의 전체 정점 수가 \\(n\\), 전체 간선 수가 \\(m\\)이라면 아래 그림(9-8)의 방법으로 각 연산을 구현할 수 있다.</p> <ul> <li>간선 추가: 해당 정점에 대응하는 연결 리스트 끝에 간선을 추가한다. \\(O(1)\\) 시간이 걸린다. 무방향 그래프이므로 두 방향의 간선을 함께 추가해야 한다.</li> <li>간선 삭제: 해당 정점에 대응하는 연결 리스트에서 지정한 간선을 찾아 삭제한다. \\(O(m)\\) 시간이 걸린다. 무방향 그래프에서는 두 방향의 간선을 함께 삭제해야 한다.</li> <li>정점 추가: 인접 리스트에 연결 리스트 하나를 추가하고 새 정점을 그 연결 리스트의 머리 노드로 둔다. \\(O(1)\\) 시간이 걸린다.</li> <li>정점 삭제: 인접 리스트 전체를 순회하며 지정한 정점이 포함된 모든 간선을 삭제한다. \\(O(n + m)\\) 시간이 걸린다.</li> <li>초기화: 인접 리스트에 정점 \\(n\\) 개와 간선 \\(2m\\) 개를 만든다. \\(O(n + m)\\) 시간이 걸린다.</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 9-8   인접 리스트의 초기화, 간선 추가와 삭제, 정점 추가와 삭제 </p> <p>다음은 인접 리스트의 구현 코드다. 위 그림(9-8)과 비교하면 실제 코드에는 다음 차이가 있다.</p> <ul> <li>정점을 편리하게 추가하고 삭제하며 코드를 간단하게 만들기 위해 연결 리스트 대신 리스트(동적 배열)를 사용한다.</li> <li>해시 테이블로 인접 리스트를 저장한다. <code>key</code>는 정점 인스턴스이고, <code>value</code>는 그 정점의 인접 정점 리스트(연결 리스트)다.</li> </ul> <p>또한 인접 리스트에서 정점을 나타내기 위해 <code>Vertex</code> 클래스를 사용한다. 인접 행렬처럼 리스트 인덱스로 정점을 구분한다면 인덱스가 \\(i\\)인 정점을 삭제할 때 인접 리스트 전체를 순회하면서 \\(i\\)보다 큰 모든 인덱스를 \\(1\\)씩 줄여야 하므로 매우 비효율적이기 때문이다. 반면 정점마다 고유한 <code>Vertex</code> 인스턴스를 사용하면 어느 정점을 삭제해도 다른 정점을 수정할 필요가 없다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_adjacency_list.py<pre><code>class GraphAdjList:\n    \"\"\"인접 리스트로 구현한 무방향 그래프 클래스\"\"\"\n\n    def __init__(self, edges: list[list[Vertex]]):\n        \"\"\"생성자\"\"\"\n        # 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n        self.adj_list = dict[Vertex, list[Vertex]]()\n        # 모든 정점과 간선을 추가한다\n        for edge in edges:\n            self.add_vertex(edge[0])\n            self.add_vertex(edge[1])\n            self.add_edge(edge[0], edge[1])\n\n    def size(self) -&gt; int:\n        \"\"\"정점 개수 반환\"\"\"\n        return len(self.adj_list)\n\n    def add_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"간선 추가\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # 간선 vet1 - vet2를 추가한다\n        self.adj_list[vet1].append(vet2)\n        self.adj_list[vet2].append(vet1)\n\n    def remove_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"간선 삭제\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # 간선 vet1 - vet2를 삭제한다\n        self.adj_list[vet1].remove(vet2)\n        self.adj_list[vet2].remove(vet1)\n\n    def add_vertex(self, vet: Vertex):\n        \"\"\"정점 추가\"\"\"\n        if vet in self.adj_list:\n            return\n        # 인접 리스트에 새 연결 리스트를 추가한다\n        self.adj_list[vet] = []\n\n    def remove_vertex(self, vet: Vertex):\n        \"\"\"정점 삭제\"\"\"\n        if vet not in self.adj_list:\n            raise ValueError()\n        # 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        self.adj_list.pop(vet)\n        # 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for vertex in self.adj_list:\n            if vet in self.adj_list[vertex]:\n                self.adj_list[vertex].remove(vet)\n\n    def print(self):\n        \"\"\"인접 리스트 출력\"\"\"\n        print(\"인접 리스트 =\")\n        for vertex in self.adj_list:\n            tmp = [v.val for v in self.adj_list[vertex]]\n            print(f\"{vertex.val}: {tmp},\")\n</code></pre> graph_adjacency_list.cpp<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList {\n  public:\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    unordered_map&lt;Vertex *, vector&lt;Vertex *&gt;&gt; adjList;\n\n    /* vector에서 지정한 노드 삭제 */\n    void remove(vector&lt;Vertex *&gt; &amp;vec, Vertex *vet) {\n        for (int i = 0; i &lt; vec.size(); i++) {\n            if (vec[i] == vet) {\n                vec.erase(vec.begin() + i);\n                break;\n            }\n        }\n    }\n\n    /* 생성자 */\n    GraphAdjList(const vector&lt;vector&lt;Vertex *&gt;&gt; &amp;edges) {\n        // 모든 정점과 간선을 추가한다\n        for (const vector&lt;Vertex *&gt; &amp;edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    int size() {\n        return adjList.size();\n    }\n\n    /* 간선 추가 */\n    void addEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"정점이 존재하지 않는다\");\n        // 간선 vet1 - vet2를 추가한다\n        adjList[vet1].push_back(vet2);\n        adjList[vet2].push_back(vet1);\n    }\n\n    /* 간선 삭제 */\n    void removeEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"정점이 존재하지 않는다\");\n        // 간선 vet1 - vet2를 삭제한다\n        remove(adjList[vet1], vet2);\n        remove(adjList[vet2], vet1);\n    }\n\n    /* 정점 추가 */\n    void addVertex(Vertex *vet) {\n        if (adjList.count(vet))\n            return;\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        adjList[vet] = vector&lt;Vertex *&gt;();\n    }\n\n    /* 정점 삭제 */\n    void removeVertex(Vertex *vet) {\n        if (!adjList.count(vet))\n            throw invalid_argument(\"정점이 존재하지 않는다\");\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        adjList.erase(vet);\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for (auto &amp;adj : adjList) {\n            remove(adj.second, vet);\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    void print() {\n        cout &lt;&lt; \"인접 리스트 =\" &lt;&lt; endl;\n        for (auto &amp;adj : adjList) {\n            const auto &amp;key = adj.first;\n            const auto &amp;vec = adj.second;\n            cout &lt;&lt; key-&gt;val &lt;&lt; \": \";\n            printVector(vetsToVals(vec));\n        }\n    }\n};\n</code></pre> graph_adjacency_list.java<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* 생성자 */\n    public GraphAdjList(Vertex[][] edges) {\n        this.adjList = new HashMap&lt;&gt;();\n        // 모든 정점과 간선을 추가한다\n        for (Vertex[] edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    public int size() {\n        return adjList.size();\n    }\n\n    /* 간선 추가 */\n    public void addEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // 간선 vet1 - vet2를 추가한다\n        adjList.get(vet1).add(vet2);\n        adjList.get(vet2).add(vet1);\n    }\n\n    /* 간선 삭제 */\n    public void removeEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // 간선 vet1 - vet2를 삭제한다\n        adjList.get(vet1).remove(vet2);\n        adjList.get(vet2).remove(vet1);\n    }\n\n    /* 정점 추가 */\n    public void addVertex(Vertex vet) {\n        if (adjList.containsKey(vet))\n            return;\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        adjList.put(vet, new ArrayList&lt;&gt;());\n    }\n\n    /* 정점 삭제 */\n    public void removeVertex(Vertex vet) {\n        if (!adjList.containsKey(vet))\n            throw new IllegalArgumentException();\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        adjList.remove(vet);\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for (List&lt;Vertex&gt; list : adjList.values()) {\n            list.remove(vet);\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    public void print() {\n        System.out.println(\"인접 리스트 =\");\n        for (Map.Entry&lt;Vertex, List&lt;Vertex&gt;&gt; pair : adjList.entrySet()) {\n            List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n            for (Vertex vertex : pair.getValue())\n                tmp.add(vertex.val);\n            System.out.println(pair.getKey().val + \": \" + tmp + \",\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.cs<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    public Dictionary&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* 생성자 */\n    public GraphAdjList(Vertex[][] edges) {\n        adjList = [];\n        // 모든 정점과 간선을 추가한다\n        foreach (Vertex[] edge in edges) {\n            AddVertex(edge[0]);\n            AddVertex(edge[1]);\n            AddEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    int Size() {\n        return adjList.Count;\n    }\n\n    /* 간선 추가 */\n    public void AddEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // 간선 vet1 - vet2를 추가한다\n        adjList[vet1].Add(vet2);\n        adjList[vet2].Add(vet1);\n    }\n\n    /* 간선 삭제 */\n    public void RemoveEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // 간선 vet1 - vet2를 삭제한다\n        adjList[vet1].Remove(vet2);\n        adjList[vet2].Remove(vet1);\n    }\n\n    /* 정점 추가 */\n    public void AddVertex(Vertex vet) {\n        if (adjList.ContainsKey(vet))\n            return;\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        adjList.Add(vet, []);\n    }\n\n    /* 정점 삭제 */\n    public void RemoveVertex(Vertex vet) {\n        if (!adjList.ContainsKey(vet))\n            throw new InvalidOperationException();\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        adjList.Remove(vet);\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        foreach (List&lt;Vertex&gt; list in adjList.Values) {\n            list.Remove(vet);\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    public void Print() {\n        Console.WriteLine(\"인접 리스트 =\");\n        foreach (KeyValuePair&lt;Vertex, List&lt;Vertex&gt;&gt; pair in adjList) {\n            List&lt;int&gt; tmp = [];\n            foreach (Vertex vertex in pair.Value)\n                tmp.Add(vertex.val);\n            Console.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.go<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\ntype graphAdjList struct {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    adjList map[Vertex][]Vertex\n}\n\n/* 생성자 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\n    g := &amp;graphAdjList{\n        adjList: make(map[Vertex][]Vertex),\n    }\n    // 모든 정점과 간선을 추가한다\n    for _, edge := range edges {\n        g.addVertex(edge[0])\n        g.addVertex(edge[1])\n        g.addEdge(edge[0], edge[1])\n    }\n    return g\n}\n\n/* 정점 개수 반환 */\nfunc (g *graphAdjList) size() int {\n    return len(g.adjList)\n}\n\n/* 간선 추가 */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"오류\")\n    }\n    // 간선 vet1 - vet2를 추가하고 익명 struct{}를 추가한다\n    g.adjList[vet1] = append(g.adjList[vet1], vet2)\n    g.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n\n/* 간선 삭제 */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"오류\")\n    }\n    // 간선 vet1 - vet2를 삭제한다\n    g.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2)\n    g.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1)\n}\n\n/* 정점 추가 */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if ok {\n        return\n    }\n    // 인접 리스트에 새 연결 리스트를 추가한다\n    g.adjList[vet] = make([]Vertex, 0)\n}\n\n/* 정점 삭제 */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if !ok {\n        panic(\"오류\")\n    }\n    // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n    delete(g.adjList, vet)\n    // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n    for v, list := range g.adjList {\n        g.adjList[v] = DeleteSliceElms(list, vet)\n    }\n}\n\n/* 인접 리스트 출력 */\nfunc (g *graphAdjList) print() {\n    var builder strings.Builder\n    fmt.Printf(\"인접 리스트 = \\n\")\n    for k, v := range g.adjList {\n        builder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\n        for _, vet := range v {\n            builder.WriteString(strconv.Itoa(vet.Val) + \" \")\n        }\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> graph_adjacency_list.swift<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    public private(set) var adjList: [Vertex: [Vertex]]\n\n    /* 생성자 */\n    public init(edges: [[Vertex]]) {\n        adjList = [:]\n        // 모든 정점과 간선을 추가한다\n        for edge in edges {\n            addVertex(vet: edge[0])\n            addVertex(vet: edge[1])\n            addEdge(vet1: edge[0], vet2: edge[1])\n        }\n    }\n\n    /* 정점 개수 반환 */\n    public func size() -&gt; Int {\n        adjList.count\n    }\n\n    /* 간선 추가 */\n    public func addEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"인수 오류\")\n        }\n        // 간선 vet1 - vet2를 추가한다\n        adjList[vet1]?.append(vet2)\n        adjList[vet2]?.append(vet1)\n    }\n\n    /* 간선 삭제 */\n    public func removeEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"인수 오류\")\n        }\n        // 간선 vet1 - vet2를 삭제한다\n        adjList[vet1]?.removeAll { $0 == vet2 }\n        adjList[vet2]?.removeAll { $0 == vet1 }\n    }\n\n    /* 정점 추가 */\n    public func addVertex(vet: Vertex) {\n        if adjList[vet] != nil {\n            return\n        }\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        adjList[vet] = []\n    }\n\n    /* 정점 삭제 */\n    public func removeVertex(vet: Vertex) {\n        if adjList[vet] == nil {\n            fatalError(\"인수 오류\")\n        }\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        adjList.removeValue(forKey: vet)\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for key in adjList.keys {\n            adjList[key]?.removeAll { $0 == vet }\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    public func print() {\n        Swift.print(\"인접 리스트 =\")\n        for (vertex, list) in adjList {\n            let list = list.map { $0.val }\n            Swift.print(\"\\(vertex.val): \\(list),\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.js<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    adjList;\n\n    /* 생성자 */\n    constructor(edges) {\n        this.adjList = new Map();\n        // 모든 정점과 간선을 추가한다\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    size() {\n        return this.adjList.size;\n    }\n\n    /* 간선 추가 */\n    addEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('잘못된 인수');\n        }\n        // 간선 vet1 - vet2를 추가한다\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* 간선 삭제 */\n    removeEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2 ||\n            this.adjList.get(vet1).indexOf(vet2) === -1\n        ) {\n            throw new Error('잘못된 인수');\n        }\n        // 간선 vet1 - vet2를 삭제한다\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* 정점 추가 */\n    addVertex(vet) {\n        if (this.adjList.has(vet)) return;\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        this.adjList.set(vet, []);\n    }\n\n    /* 정점 삭제 */\n    removeVertex(vet) {\n        if (!this.adjList.has(vet)) {\n            throw new Error('잘못된 인수');\n        }\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        this.adjList.delete(vet);\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for (const set of this.adjList.values()) {\n            const index = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    print() {\n        console.log('인접 리스트 =');\n        for (const [key, value] of this.adjList) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.ts<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    adjList: Map&lt;Vertex, Vertex[]&gt;;\n\n    /* 생성자 */\n    constructor(edges: Vertex[][]) {\n        this.adjList = new Map();\n        // 모든 정점과 간선을 추가한다\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    size(): number {\n        return this.adjList.size;\n    }\n\n    /* 간선 추가 */\n    addEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('잘못된 인수');\n        }\n        // 간선 vet1 - vet2를 추가한다\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* 간선 삭제 */\n    removeEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2 ||\n            this.adjList.get(vet1).indexOf(vet2) === -1\n        ) {\n            throw new Error('잘못된 인수');\n        }\n        // 간선 vet1 - vet2를 삭제한다\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* 정점 추가 */\n    addVertex(vet: Vertex): void {\n        if (this.adjList.has(vet)) return;\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        this.adjList.set(vet, []);\n    }\n\n    /* 정점 삭제 */\n    removeVertex(vet: Vertex): void {\n        if (!this.adjList.has(vet)) {\n            throw new Error('잘못된 인수');\n        }\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        this.adjList.delete(vet);\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for (const set of this.adjList.values()) {\n            const index: number = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    print(): void {\n        console.log('인접 리스트 =');\n        for (const [key, value] of this.adjList.entries()) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.dart<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList {\n  // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n  Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList = {};\n\n  /* 생성자 */\n  GraphAdjList(List&lt;List&lt;Vertex&gt;&gt; edges) {\n    for (List&lt;Vertex&gt; edge in edges) {\n      addVertex(edge[0]);\n      addVertex(edge[1]);\n      addEdge(edge[0], edge[1]);\n    }\n  }\n\n  /* 정점 개수 반환 */\n  int size() {\n    return adjList.length;\n  }\n\n  /* 간선 추가 */\n  void addEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // 간선 vet1 - vet2를 추가한다\n    adjList[vet1]!.add(vet2);\n    adjList[vet2]!.add(vet1);\n  }\n\n  /* 간선 삭제 */\n  void removeEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // 간선 vet1 - vet2를 삭제한다\n    adjList[vet1]!.remove(vet2);\n    adjList[vet2]!.remove(vet1);\n  }\n\n  /* 정점 추가 */\n  void addVertex(Vertex vet) {\n    if (adjList.containsKey(vet)) return;\n    // 인접 리스트에 새 연결 리스트를 추가한다\n    adjList[vet] = [];\n  }\n\n  /* 정점 삭제 */\n  void removeVertex(Vertex vet) {\n    if (!adjList.containsKey(vet)) {\n      throw ArgumentError;\n    }\n    // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n    adjList.remove(vet);\n    // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n    adjList.forEach((key, value) {\n      value.remove(vet);\n    });\n  }\n\n  /* 인접 리스트 출력 */\n  void printAdjList() {\n    print(\"인접 리스트 =\");\n    adjList.forEach((key, value) {\n      List&lt;int&gt; tmp = [];\n      for (Vertex vertex in value) {\n        tmp.add(vertex.val);\n      }\n      print(\"${key.val}: $tmp,\");\n    });\n  }\n}\n</code></pre> graph_adjacency_list.rs<pre><code>/* 인접 리스트로 구현한 무방향 그래프 자료형 */\npub struct GraphAdjList {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    pub adj_list: HashMap&lt;Vertex, Vec&lt;Vertex&gt;&gt;, // 값 부분에는 HashSet&lt;Vertex&gt;가 더 적합할까?\n}\n\nimpl GraphAdjList {\n    /* 생성자 */\n    pub fn new(edges: Vec&lt;[Vertex; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjList {\n            adj_list: HashMap::new(),\n        };\n        // 모든 정점과 간선을 추가한다\n        for edge in edges {\n            graph.add_vertex(edge[0]);\n            graph.add_vertex(edge[1]);\n            graph.add_edge(edge[0], edge[1]);\n        }\n\n        graph\n    }\n\n    /* 정점 개수 반환 */\n    #[allow(unused)]\n    pub fn size(&amp;self) -&gt; usize {\n        self.adj_list.len()\n    }\n\n    /* 간선 추가 */\n    pub fn add_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if vet1 == vet2 {\n            panic!(\"값 오류\");\n        }\n        // 간선 vet1 - vet2를 추가한다\n        self.adj_list.entry(vet1).or_default().push(vet2);\n        self.adj_list.entry(vet2).or_default().push(vet1);\n    }\n\n    /* 간선 삭제 */\n    #[allow(unused)]\n    pub fn remove_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if vet1 == vet2 {\n            panic!(\"값 오류\");\n        }\n        // 간선 vet1 - vet2를 삭제한다\n        self.adj_list\n            .entry(vet1)\n            .and_modify(|v| v.retain(|&amp;e| e != vet2));\n        self.adj_list\n            .entry(vet2)\n            .and_modify(|v| v.retain(|&amp;e| e != vet1));\n    }\n\n    /* 정점 추가 */\n    pub fn add_vertex(&amp;mut self, vet: Vertex) {\n        if self.adj_list.contains_key(&amp;vet) {\n            return;\n        }\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        self.adj_list.insert(vet, vec![]);\n    }\n\n    /* 정점 삭제 */\n    #[allow(unused)]\n    pub fn remove_vertex(&amp;mut self, vet: Vertex) {\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        self.adj_list.remove(&amp;vet);\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for list in self.adj_list.values_mut() {\n            list.retain(|&amp;v| v != vet);\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    pub fn print(&amp;self) {\n        println!(\"인접 리스트 =\");\n        for (vertex, list) in &amp;self.adj_list {\n            let list = list.iter().map(|vertex| vertex.val).collect::&lt;Vec&lt;i32&gt;&gt;();\n            println!(\"{}: {:?},\", vertex.val, list);\n        }\n    }\n}\n</code></pre> graph_adjacency_list.c<pre><code>/* 노드 구조체 */\ntypedef struct AdjListNode {\n    Vertex *vertex;           // 정점\n    struct AdjListNode *next; // 다음 노드\n} AdjListNode;\n\n/* 정점에 해당하는 노드 찾기 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* 간선 추가 보조 함수 */\nvoid addEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *node = (AdjListNode *)malloc(sizeof(AdjListNode));\n    node-&gt;vertex = vet;\n    // 맨 앞에 삽입한다\n    node-&gt;next = head-&gt;next;\n    head-&gt;next = node;\n}\n\n/* 간선 삭제 보조 함수 */\nvoid removeEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *pre = head;\n    AdjListNode *cur = head-&gt;next;\n    // 연결 리스트에서 vet에 해당하는 노드를 찾는다\n    while (cur != NULL &amp;&amp; cur-&gt;vertex != vet) {\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n    if (cur == NULL)\n        return;\n    // 연결 리스트에서 vet에 해당하는 노드를 삭제한다\n    pre-&gt;next = cur-&gt;next;\n    // 메모리 해제\n    free(cur);\n}\n\n/* 인접 리스트로 구현한 무방향 그래프 클래스 */\ntypedef struct {\n    AdjListNode *heads[MAX_SIZE]; // 노드 배열\n    int size;                     // 노드 개수\n} GraphAdjList;\n\n/* 생성자 */\nGraphAdjList *newGraphAdjList() {\n    GraphAdjList *graph = (GraphAdjList *)malloc(sizeof(GraphAdjList));\n    if (!graph) {\n        return NULL;\n    }\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        graph-&gt;heads[i] = NULL;\n    }\n    return graph;\n}\n\n/* 소멸자 */\nvoid delGraphAdjList(GraphAdjList *graph) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        AdjListNode *cur = graph-&gt;heads[i];\n        while (cur != NULL) {\n            AdjListNode *next = cur-&gt;next;\n            if (cur != graph-&gt;heads[i]) {\n                free(cur);\n            }\n            cur = next;\n        }\n        free(graph-&gt;heads[i]-&gt;vertex);\n        free(graph-&gt;heads[i]);\n    }\n    free(graph);\n}\n\n/* 정점에 해당하는 노드 찾기 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* 간선 추가 */\nvoid addEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL &amp;&amp; head1 != head2);\n    // 간선 vet1 - vet2를 추가한다\n    addEdgeHelper(head1, vet2);\n    addEdgeHelper(head2, vet1);\n}\n\n/* 간선 삭제 */\nvoid removeEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL);\n    // 간선 vet1 - vet2를 삭제한다\n    removeEdgeHelper(head1, head2-&gt;vertex);\n    removeEdgeHelper(head2, head1-&gt;vertex);\n}\n\n/* 정점 추가 */\nvoid addVertex(GraphAdjList *graph, Vertex *vet) {\n    assert(graph != NULL &amp;&amp; graph-&gt;size &lt; MAX_SIZE);\n    AdjListNode *head = (AdjListNode *)malloc(sizeof(AdjListNode));\n    head-&gt;vertex = vet;\n    head-&gt;next = NULL;\n    // 인접 리스트에 새 연결 리스트를 추가한다\n    graph-&gt;heads[graph-&gt;size++] = head;\n}\n\n/* 정점 삭제 */\nvoid removeVertex(GraphAdjList *graph, Vertex *vet) {\n    AdjListNode *node = findNode(graph, vet);\n    assert(node != NULL);\n    // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n    AdjListNode *cur = node, *pre = NULL;\n    while (cur) {\n        pre = cur;\n        cur = cur-&gt;next;\n        free(pre);\n    }\n    // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        cur = graph-&gt;heads[i];\n        pre = NULL;\n        while (cur) {\n            pre = cur;\n            cur = cur-&gt;next;\n            if (cur &amp;&amp; cur-&gt;vertex == vet) {\n                pre-&gt;next = cur-&gt;next;\n                free(cur);\n                break;\n            }\n        }\n    }\n    // 해당 정점 뒤의 정점을 앞으로 옮겨 빈자리를 채운다\n    int i;\n    for (i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i] == node)\n            break;\n    }\n    for (int j = i; j &lt; graph-&gt;size - 1; j++) {\n        graph-&gt;heads[j] = graph-&gt;heads[j + 1];\n    }\n    graph-&gt;size--;\n    free(vet);\n}\n</code></pre> graph_adjacency_list.kt<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList(edges: Array&lt;Array&lt;Vertex?&gt;&gt;) {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    val adjList = HashMap&lt;Vertex, MutableList&lt;Vertex&gt;&gt;()\n\n    /* 생성자 */\n    init {\n        // 모든 정점과 간선을 추가한다\n        for (edge in edges) {\n            addVertex(edge[0]!!)\n            addVertex(edge[1]!!)\n            addEdge(edge[0]!!, edge[1]!!)\n        }\n    }\n\n    /* 정점 개수 반환 */\n    fun size(): Int {\n        return adjList.size\n    }\n\n    /* 간선 추가 */\n    fun addEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // 간선 vet1 - vet2를 추가한다\n        adjList[vet1]?.add(vet2)\n        adjList[vet2]?.add(vet1)\n    }\n\n    /* 간선 삭제 */\n    fun removeEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // 간선 vet1 - vet2를 삭제한다\n        adjList[vet1]?.remove(vet2)\n        adjList[vet2]?.remove(vet1)\n    }\n\n    /* 정점 추가 */\n    fun addVertex(vet: Vertex) {\n        if (adjList.containsKey(vet))\n            return\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        adjList[vet] = mutableListOf()\n    }\n\n    /* 정점 삭제 */\n    fun removeVertex(vet: Vertex) {\n        if (!adjList.containsKey(vet))\n            throw IllegalArgumentException()\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        adjList.remove(vet)\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for (list in adjList.values) {\n            list.remove(vet)\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    fun print() {\n        println(\"인접 리스트 =\")\n        for (pair in adjList.entries) {\n            val tmp = mutableListOf&lt;Int&gt;()\n            for (vertex in pair.value) {\n                tmp.add(vertex._val)\n            }\n            println(\"${pair.key._val}: $tmp,\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.rb<pre><code>### 인접 리스트로 구현한 무방향 그래프 클래스 ###\nclass GraphAdjList\n  attr_reader :adj_list\n\n  ### 생성자 ###\n  def initialize(edges)\n    # 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    @adj_list = {}\n    # 모든 정점과 간선을 추가한다\n    for edge in edges\n      add_vertex(edge[0])\n      add_vertex(edge[1])\n      add_edge(edge[0], edge[1])\n    end\n  end\n\n  ### 정점 개수 반환 ###\n  def size\n    @adj_list.length\n  end\n\n  ### 간선 추가 ###\n  def add_edge(vet1, vet2)\n    raise ArgumentError if !@adj_list.include?(vet1) || !@adj_list.include?(vet2)\n\n    @adj_list[vet1] &lt;&lt; vet2\n    @adj_list[vet2] &lt;&lt; vet1\n  end\n\n  ### 간선 삭제 ###\n  def remove_edge(vet1, vet2)\n    raise ArgumentError if !@adj_list.include?(vet1) || !@adj_list.include?(vet2)\n\n    # 간선 vet1 - vet2를 삭제한다\n    @adj_list[vet1].delete(vet2)\n    @adj_list[vet2].delete(vet1)\n  end\n\n  ### 정점 추가 ###\n  def add_vertex(vet)\n    return if @adj_list.include?(vet)\n\n    # 인접 리스트에 새 연결 리스트를 추가한다\n    @adj_list[vet] = []\n  end\n\n  ### 정점 삭제 ###\n  def remove_vertex(vet)\n    raise ArgumentError unless @adj_list.include?(vet)\n\n    # 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n    @adj_list.delete(vet)\n    # 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n    for vertex in @adj_list\n      @adj_list[vertex.first].delete(vet) if @adj_list[vertex.first].include?(vet)\n    end\n  end\n\n  ### 인접 리스트 출력 ###\n  def __print__\n    puts '인접 리스트 ='\n    for vertex in @adj_list\n      tmp = @adj_list[vertex.first].map { |v| v.val }\n      puts \"#{vertex.first.val}: #{tmp},\"\n    end\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 9 장   그래프","9.2   그래프의 기본 연산"],"tags":[]},{"location":"chapter_graph/graph_operations/#923","level":2,"title":"9.2.3   효율 비교","text":"<p>그래프에 정점 \\(n\\) 개와 간선 \\(m\\) 개가 있다고 하자. 아래 표(9-2)는 인접 행렬과 인접 리스트의 시간 효율과 공간 효율을 비교한다. 여기서 인접 리스트(연결 리스트)는 이 절의 구현에 해당하며, 인접 리스트(해시 테이블)는 모든 연결 리스트를 해시 테이블로 바꾼 구현을 뜻한다.</p> <p> 표 9-2   인접 행렬과 인접 리스트 비교 </p> 인접 행렬 인접 리스트(연결 리스트) 인접 리스트(해시 테이블) 인접 여부 확인 \\(O(1)\\) \\(O(n)\\) \\(O(1)\\) 간선 추가 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) 간선 삭제 \\(O(1)\\) \\(O(n)\\) \\(O(1)\\) 정점 추가 \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) 정점 삭제 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) 메모리 사용량 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\) <p>위 표(9-2)만 보면 인접 리스트(해시 테이블)의 시간 효율과 공간 효율이 가장 좋아 보인다. 하지만 실제로는 배열 접근이나 대입 한 번이면 되는 인접 행렬의 간선 연산이 더 효율적이다. 종합하면 인접 행렬은 '공간을 더 써서 시간을 절약하는' 원리를, 인접 리스트는 '시간을 더 써서 공간을 절약하는' 원리를 보여 준다.</p>","path":["제 9 장   그래프","9.2   그래프의 기본 연산"],"tags":[]},{"location":"chapter_graph/graph_traversal/","level":1,"title":"9.3   그래프 순회","text":"<p>트리는 '일대다' 관계를 나타내지만, 그래프는 자유도가 더 높아 임의의 '다대다' 관계를 표현할 수 있다. 따라서 트리는 그래프의 특수한 경우로 볼 수 있다. 트리 순회 역시 그래프 순회의 특수한 경우다.</p> <p>그래프와 트리 모두 탐색 알고리즘을 이용하여 순회한다. 그래프 순회도 너비 우선 순회와 깊이 우선 순회의 두 가지로 나뉜다.</p>","path":["제 9 장   그래프","9.3   그래프 순회"],"tags":[]},{"location":"chapter_graph/graph_traversal/#931","level":2,"title":"9.3.1   너비 우선 순회","text":"<p>너비 우선 순회는 가까운 곳에서 먼 곳으로 진행하는 순회 방식이다. 한 노드에서 출발하여 항상 거리가 가장 가까운 정점을 먼저 방문하고, 한 층씩 바깥으로 확장한다. 아래 그림(9-9)에서는 왼쪽 위 정점에서 출발하여 해당 정점의 모든 인접 정점을 먼저 순회한 다음, 다음 정점의 모든 인접 정점을 순회한다. 이를 모든 정점을 방문할 때까지 반복한다.</p> <p></p> <p> 그림 9-9   그래프의 너비 우선 순회 </p>","path":["제 9 장   그래프","9.3   그래프 순회"],"tags":[]},{"location":"chapter_graph/graph_traversal/#1","level":3,"title":"1.   알고리즘 구현","text":"<p>BFS는 보통 큐로 구현하며, 코드는 다음과 같다. 큐의 '선입선출' 성질은 BFS의 '가까운 곳부터 먼 곳으로'라는 발상과 잘 맞는다.</p> <ol> <li>순회의 시작 정점 <code>startVet</code>를 큐에 넣고 반복문을 시작한다.</li> <li>매 반복에서 큐의 맨 앞 정점을 꺼내 방문을 기록한 다음, 해당 정점의 모든 인접 정점을 큐의 끝에 넣는다.</li> <li>모든 정점을 방문할 때까지 <code>2.</code> 단계를 반복한 뒤 종료한다.</li> </ol> <p>같은 정점을 여러 번 순회하지 않도록 해시 집합 <code>visited</code>를 사용하여 방문한 노드를 기록한다.</p> <p>참고</p> <p>해시 집합은 <code>value</code> 없이 <code>key</code>만 저장하는 해시 테이블로 볼 수 있다. <code>key</code>의 추가, 삭제, 조회, 수정 연산을 \\(O(1)\\) 시간에 수행한다. <code>key</code>는 중복될 수 없으므로, 해시 집합은 데이터 중복 제거 등에 자주 사용된다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_bfs.py<pre><code>def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"너비 우선 순회\"\"\"\n    # 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\n    # 정점 순회 순서\n    res = []\n    # 방문한 정점을 기록하는 해시 집합\n    visited = set[Vertex]([start_vet])\n    # 큐를 사용하여 BFS를 구현한다\n    que = deque[Vertex]([start_vet])\n    # 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while len(que) &gt; 0:\n        vet = que.popleft()  # 큐 맨 앞의 정점을 꺼낸다\n        res.append(vet)  # 방문한 정점을 기록한다\n        # 해당 정점의 모든 인접 정점을 순회한다\n        for adj_vet in graph.adj_list[vet]:\n            if adj_vet in visited:\n                continue  # 이미 방문한 정점은 건너뛴다\n            que.append(adj_vet)  # 방문하지 않은 정점만 큐에 삽입한다\n            visited.add(adj_vet)  # 해당 정점을 방문했다고 표시한다\n    # 정점 순회 순서를 반환한다\n    return res\n</code></pre> graph_bfs.cpp<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nvector&lt;Vertex *&gt; graphBFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // 정점 순회 순서\n    vector&lt;Vertex *&gt; res;\n    // 방문한 정점을 기록하는 해시 집합\n    unordered_set&lt;Vertex *&gt; visited = {startVet};\n    // 큐를 사용하여 BFS를 구현한다\n    queue&lt;Vertex *&gt; que;\n    que.push(startVet);\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while (!que.empty()) {\n        Vertex *vet = que.front();\n        que.pop();          // 큐 맨 앞의 정점을 꺼낸다\n        res.push_back(vet); // 방문한 정점을 기록한다\n        // 해당 정점의 모든 인접 정점을 순회한다\n        for (auto adjVet : graph.adjList[vet]) {\n            if (visited.count(adjVet))\n                continue;            // 이미 방문한 정점은 건너뛴다\n            que.push(adjVet);        // 방문하지 않은 정점만 큐에 삽입한다\n            visited.emplace(adjVet); // 해당 정점을 방문했다고 표시한다\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    return res;\n}\n</code></pre> graph_bfs.java<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n    // 정점 순회 순서\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // 방문한 정점을 기록하는 해시 집합\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    visited.add(startVet);\n    // 큐를 사용하여 BFS를 구현한다\n    Queue&lt;Vertex&gt; que = new LinkedList&lt;&gt;();\n    que.offer(startVet);\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while (!que.isEmpty()) {\n        Vertex vet = que.poll(); // 큐 맨 앞의 정점을 꺼낸다\n        res.add(vet);            // 방문한 정점을 기록한다\n        // 해당 정점의 모든 인접 정점을 순회한다\n        for (Vertex adjVet : graph.adjList.get(vet)) {\n            if (visited.contains(adjVet))\n                continue;        // 이미 방문한 정점은 건너뛴다\n            que.offer(adjVet);   // 방문하지 않은 정점만 큐에 삽입한다\n            visited.add(adjVet); // 해당 정점을 방문했다고 표시한다\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    return res;\n}\n</code></pre> graph_bfs.cs<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nList&lt;Vertex&gt; GraphBFS(GraphAdjList graph, Vertex startVet) {\n    // 정점 순회 순서\n    List&lt;Vertex&gt; res = [];\n    // 방문한 정점을 기록하는 해시 집합\n    HashSet&lt;Vertex&gt; visited = [startVet];\n    // 큐를 사용하여 BFS를 구현한다\n    Queue&lt;Vertex&gt; que = new();\n    que.Enqueue(startVet);\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while (que.Count &gt; 0) {\n        Vertex vet = que.Dequeue(); // 큐 맨 앞의 정점을 꺼낸다\n        res.Add(vet);               // 방문한 정점을 기록한다\n        foreach (Vertex adjVet in graph.adjList[vet]) {\n            if (visited.Contains(adjVet)) {\n                continue;          // 이미 방문한 정점은 건너뛴다\n            }\n            que.Enqueue(adjVet);   // 방문하지 않은 정점만 큐에 삽입한다\n            visited.Add(adjVet);   // 해당 정점을 방문했다고 표시한다\n        }\n    }\n\n    // 정점 순회 순서를 반환한다\n    return res;\n}\n</code></pre> graph_bfs.go<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // 정점 순회 순서\n    res := make([]Vertex, 0)\n    // 방문한 정점을 기록하는 해시 집합\n    visited := make(map[Vertex]struct{})\n    visited[startVet] = struct{}{}\n    // 큐로 BFS를 구현하며, 슬라이스로 큐를 구현한다\n    queue := make([]Vertex, 0)\n    queue = append(queue, startVet)\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    for len(queue) &gt; 0 {\n        // 큐 맨 앞의 정점을 꺼낸다\n        vet := queue[0]\n        queue = queue[1:]\n        // 방문한 정점을 기록한다\n        res = append(res, vet)\n        // 해당 정점의 모든 인접 정점을 순회한다\n        for _, adjVet := range g.adjList[vet] {\n            _, isExist := visited[adjVet]\n            // 방문하지 않은 정점만 큐에 삽입한다\n            if !isExist {\n                queue = append(queue, adjVet)\n                visited[adjVet] = struct{}{}\n            }\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    return res\n}\n</code></pre> graph_bfs.swift<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // 정점 순회 순서\n    var res: [Vertex] = []\n    // 방문한 정점을 기록하는 해시 집합\n    var visited: Set&lt;Vertex&gt; = [startVet]\n    // 큐를 사용하여 BFS를 구현한다\n    var que: [Vertex] = [startVet]\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while !que.isEmpty {\n        let vet = que.removeFirst() // 큐 맨 앞의 정점을 꺼낸다\n        res.append(vet) // 방문한 정점을 기록한다\n        // 해당 정점의 모든 인접 정점을 순회한다\n        for adjVet in graph.adjList[vet] ?? [] {\n            if visited.contains(adjVet) {\n                continue // 이미 방문한 정점은 건너뛴다\n            }\n            que.append(adjVet) // 방문하지 않은 정점만 큐에 삽입한다\n            visited.insert(adjVet) // 해당 정점을 방문했다고 표시한다\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    return res\n}\n</code></pre> graph_bfs.js<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunction graphBFS(graph, startVet) {\n    // 정점 순회 순서\n    const res = [];\n    // 방문한 정점을 기록하는 해시 집합\n    const visited = new Set();\n    visited.add(startVet);\n    // 큐를 사용하여 BFS를 구현한다\n    const que = [startVet];\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while (que.length) {\n        const vet = que.shift(); // 큐 맨 앞의 정점을 꺼낸다\n        res.push(vet); // 방문한 정점을 기록한다\n        // 해당 정점의 모든 인접 정점을 순회한다\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // 이미 방문한 정점은 건너뛴다\n            }\n            que.push(adjVet); // 방문하지 않은 정점만 큐에 삽입한다\n            visited.add(adjVet); // 해당 정점을 방문했다고 표시한다\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    return res;\n}\n</code></pre> graph_bfs.ts<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // 정점 순회 순서\n    const res: Vertex[] = [];\n    // 방문한 정점을 기록하는 해시 집합\n    const visited: Set&lt;Vertex&gt; = new Set();\n    visited.add(startVet);\n    // 큐를 사용하여 BFS를 구현한다\n    const que = [startVet];\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while (que.length) {\n        const vet = que.shift(); // 큐 맨 앞의 정점을 꺼낸다\n        res.push(vet); // 방문한 정점을 기록한다\n        // 해당 정점의 모든 인접 정점을 순회한다\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // 이미 방문한 정점은 건너뛴다\n            }\n            que.push(adjVet); // 방문하지 않은 정점만 큐에 삽입한다\n            visited.add(adjVet); // 해당 정점을 방문했다고 표시한다\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    return res;\n}\n</code></pre> graph_bfs.dart<pre><code>/* 너비 우선 순회 */\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n  // 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\n  // 정점 순회 순서\n  List&lt;Vertex&gt; res = [];\n  // 방문한 정점을 기록하는 해시 집합\n  Set&lt;Vertex&gt; visited = {};\n  visited.add(startVet);\n  // 큐를 사용하여 BFS를 구현한다\n  Queue&lt;Vertex&gt; que = Queue();\n  que.add(startVet);\n  // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n  while (que.isNotEmpty) {\n    Vertex vet = que.removeFirst(); // 큐 맨 앞의 정점을 꺼낸다\n    res.add(vet); // 방문한 정점을 기록한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for (Vertex adjVet in graph.adjList[vet]!) {\n      if (visited.contains(adjVet)) {\n        continue; // 이미 방문한 정점은 건너뛴다\n      }\n      que.add(adjVet); // 방문하지 않은 정점만 큐에 삽입한다\n      visited.add(adjVet); // 해당 정점을 방문했다고 표시한다\n    }\n  }\n  // 정점 순회 순서를 반환한다\n  return res;\n}\n</code></pre> graph_bfs.rs<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfn graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // 정점 순회 순서\n    let mut res = vec![];\n    // 방문한 정점을 기록하는 해시 집합\n    let mut visited = HashSet::new();\n    visited.insert(start_vet);\n    // 큐를 사용하여 BFS를 구현한다\n    let mut que = VecDeque::new();\n    que.push_back(start_vet);\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while let Some(vet) = que.pop_front() {\n        res.push(vet); // 방문한 정점을 기록한다\n\n        // 해당 정점의 모든 인접 정점을 순회한다\n        if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n            for &amp;adj_vet in adj_vets {\n                if visited.contains(&amp;adj_vet) {\n                    continue; // 이미 방문한 정점은 건너뛴다\n                }\n                que.push_back(adj_vet); // 방문하지 않은 정점만 큐에 삽입한다\n                visited.insert(adj_vet); // 해당 정점을 방문했다고 표시한다\n            }\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    res\n}\n</code></pre> graph_bfs.c<pre><code>/* 노드 큐 구조체 */\ntypedef struct {\n    Vertex *vertices[MAX_SIZE];\n    int front, rear, size;\n} Queue;\n\n/* 생성자 */\nQueue *newQueue() {\n    Queue *q = (Queue *)malloc(sizeof(Queue));\n    q-&gt;front = q-&gt;rear = q-&gt;size = 0;\n    return q;\n}\n\n/* 큐가 비어 있는지 판단 */\nint isEmpty(Queue *q) {\n    return q-&gt;size == 0;\n}\n\n/* 큐 삽입 연산 */\nvoid enqueue(Queue *q, Vertex *vet) {\n    q-&gt;vertices[q-&gt;rear] = vet;\n    q-&gt;rear = (q-&gt;rear + 1) % MAX_SIZE;\n    q-&gt;size++;\n}\n\n/* 큐에서 꺼내는 연산 */\nVertex *dequeue(Queue *q) {\n    Vertex *vet = q-&gt;vertices[q-&gt;front];\n    q-&gt;front = (q-&gt;front + 1) % MAX_SIZE;\n    q-&gt;size--;\n    return vet;\n}\n\n/* 정점을 이미 방문했는지 확인 */\nint isVisited(Vertex **visited, int size, Vertex *vet) {\n    // O(n) 시간에 순회하여 노드를 찾는다\n    for (int i = 0; i &lt; size; i++) {\n        if (visited[i] == vet)\n            return 1;\n    }\n    return 0;\n}\n\n/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nvoid graphBFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize, Vertex **visited, int *visitedSize) {\n    // 큐를 사용하여 BFS를 구현한다\n    Queue *queue = newQueue();\n    enqueue(queue, startVet);\n    visited[(*visitedSize)++] = startVet;\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while (!isEmpty(queue)) {\n        Vertex *vet = dequeue(queue); // 큐 맨 앞의 정점을 꺼낸다\n        res[(*resSize)++] = vet;      // 방문한 정점을 기록한다\n        // 해당 정점의 모든 인접 정점을 순회한다\n        AdjListNode *node = findNode(graph, vet);\n        while (node != NULL) {\n            // 이미 방문한 정점은 건너뛴다\n            if (!isVisited(visited, *visitedSize, node-&gt;vertex)) {\n                enqueue(queue, node-&gt;vertex);             // 방문하지 않은 정점만 큐에 삽입한다\n                visited[(*visitedSize)++] = node-&gt;vertex; // 해당 정점을 방문했다고 표시한다\n            }\n            node = node-&gt;next;\n        }\n    }\n    // 메모리 해제\n    free(queue);\n}\n</code></pre> graph_bfs.kt<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfun graphBFS(graph: GraphAdjList, startVet: Vertex): MutableList&lt;Vertex?&gt; {\n    // 정점 순회 순서\n    val res = mutableListOf&lt;Vertex?&gt;()\n    // 방문한 정점을 기록하는 해시 집합\n    val visited = HashSet&lt;Vertex&gt;()\n    visited.add(startVet)\n    // 큐를 사용하여 BFS를 구현한다\n    val que = LinkedList&lt;Vertex&gt;()\n    que.offer(startVet)\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while (!que.isEmpty()) {\n        val vet = que.poll() // 큐 맨 앞의 정점을 꺼낸다\n        res.add(vet)         // 방문한 정점을 기록한다\n        // 해당 정점의 모든 인접 정점을 순회한다\n        for (adjVet in graph.adjList[vet]!!) {\n            if (visited.contains(adjVet))\n                continue        // 이미 방문한 정점은 건너뛴다\n            que.offer(adjVet)   // 방문하지 않은 정점만 큐에 삽입한다\n            visited.add(adjVet) // 해당 정점을 방문했다고 표시한다\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    return res\n}\n</code></pre> graph_bfs.rb<pre><code>### 너비 우선 순회 ###\ndef graph_bfs(graph, start_vet)\n  # 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\n  # 정점 순회 순서\n  res = []\n  # 방문한 정점을 기록하는 해시 집합\n  visited = Set.new([start_vet])\n  # 큐를 사용하여 BFS를 구현한다\n  que = [start_vet]\n  # 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n  while que.length &gt; 0\n    vet = que.shift # 큐 맨 앞의 정점을 꺼낸다\n    res &lt;&lt; vet # 방문한 정점을 기록한다\n    # 해당 정점의 모든 인접 정점을 순회한다\n    for adj_vet in graph.adj_list[vet]\n      next if visited.include?(adj_vet) # 이미 방문한 정점은 건너뛴다\n      que &lt;&lt; adj_vet # 방문하지 않은 정점만 큐에 삽입한다\n      visited.add(adj_vet) # 해당 정점을 방문했다고 표시한다\n    end\n  end\n  # 정점 순회 순서를 반환한다\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>코드만으로는 추상적일 수 있으므로 아래 그림(9-10)과 함께 살펴보며 이해를 깊이 다져 보자.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 9-10   그래프의 너비 우선 순회 단계 </p> <p>너비 우선 순회 순서는 유일한가?</p> <p>유일하지 않다. 너비 우선 순회는 '가까운 곳부터 먼 곳으로' 순회하기만 하면 되므로, 거리가 같은 여러 정점의 방문 순서는 자유롭게 바꿀 수 있다. 위 그림(9-10)에서는 정점 \\(1\\), \\(3\\)의 방문 순서를 바꿀 수 있고, 정점 \\(2\\), \\(4\\), \\(6\\)의 순서도 자유롭게 바꿀 수 있다.</p>","path":["제 9 장   그래프","9.3   그래프 순회"],"tags":[]},{"location":"chapter_graph/graph_traversal/#2","level":3,"title":"2.   복잡도 분석","text":"<p>시간 복잡도: 모든 정점은 큐에 한 번 들어갔다가 한 번 나오므로 \\(O(|V|)\\) 시간이 걸린다. 인접 정점을 순회할 때는 무방향 그래프이므로 모든 간선을 \\(2\\) 번씩 방문하여 \\(O(2|E|)\\) 시간이 걸린다. 전체 시간은 \\(O(|V| + |E|)\\)이다.</p> <p>공간 복잡도: 리스트 <code>res</code>, 해시 집합 <code>visited</code>, 큐 <code>que</code>에 저장되는 정점 수는 각각 최대 \\(|V|\\)이므로 \\(O(|V|)\\) 공간을 사용한다.</p>","path":["제 9 장   그래프","9.3   그래프 순회"],"tags":[]},{"location":"chapter_graph/graph_traversal/#932","level":2,"title":"9.3.2   깊이 우선 순회","text":"<p>깊이 우선 순회는 먼저 끝까지 나아가고, 더 갈 곳이 없으면 되돌아오는 순회 방식이다. 아래 그림(9-11)에서는 왼쪽 위 정점에서 출발하여 현재 정점의 인접 정점 하나를 방문한다. 끝에 도달하면 되돌아오고, 다시 끝까지 나아갔다가 돌아오는 과정을 모든 정점을 순회할 때까지 반복한다.</p> <p></p> <p> 그림 9-11   그래프의 깊이 우선 순회 </p>","path":["제 9 장   그래프","9.3   그래프 순회"],"tags":[]},{"location":"chapter_graph/graph_traversal/#1_1","level":3,"title":"1.   알고리즘 구현","text":"<p>'끝까지 갔다가 돌아오는' 알고리즘 패턴은 보통 재귀로 구현한다. 너비 우선 순회와 마찬가지로 깊이 우선 순회에서도 해시 집합 <code>visited</code>로 이미 방문한 정점을 기록하여 중복 방문을 방지한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_dfs.py<pre><code>def dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n    \"\"\"깊이 우선 순회 보조 함수\"\"\"\n    res.append(vet)  # 방문한 정점을 기록한다\n    visited.add(vet)  # 해당 정점을 방문했다고 표시한다\n    # 해당 정점의 모든 인접 정점을 순회한다\n    for adjVet in graph.adj_list[vet]:\n        if adjVet in visited:\n            continue  # 이미 방문한 정점은 건너뛴다\n        # 인접 정점을 재귀적으로 방문한다\n        dfs(graph, visited, res, adjVet)\n\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"깊이 우선 순회\"\"\"\n    # 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\n    # 정점 순회 순서\n    res = []\n    # 방문한 정점을 기록하는 해시 집합\n    visited = set[Vertex]()\n    dfs(graph, visited, res, start_vet)\n    return res\n</code></pre> graph_dfs.cpp<pre><code>/* 깊이 우선 순회 보조 함수 */\nvoid dfs(GraphAdjList &amp;graph, unordered_set&lt;Vertex *&gt; &amp;visited, vector&lt;Vertex *&gt; &amp;res, Vertex *vet) {\n    res.push_back(vet);   // 방문한 정점을 기록한다\n    visited.emplace(vet); // 해당 정점을 방문했다고 표시한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for (Vertex *adjVet : graph.adjList[vet]) {\n        if (visited.count(adjVet))\n            continue; // 이미 방문한 정점은 건너뛴다\n        // 인접 정점을 재귀적으로 방문한다\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nvector&lt;Vertex *&gt; graphDFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // 정점 순회 순서\n    vector&lt;Vertex *&gt; res;\n    // 방문한 정점을 기록하는 해시 집합\n    unordered_set&lt;Vertex *&gt; visited;\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.java<pre><code>/* 깊이 우선 순회 보조 함수 */\nvoid dfs(GraphAdjList graph, Set&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.add(vet);     // 방문한 정점을 기록한다\n    visited.add(vet); // 해당 정점을 방문했다고 표시한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for (Vertex adjVet : graph.adjList.get(vet)) {\n        if (visited.contains(adjVet))\n            continue; // 이미 방문한 정점은 건너뛴다\n        // 인접 정점을 재귀적으로 방문한다\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n    // 정점 순회 순서\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // 방문한 정점을 기록하는 해시 집합\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.cs<pre><code>/* 깊이 우선 순회 보조 함수 */\nvoid DFS(GraphAdjList graph, HashSet&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.Add(vet);     // 방문한 정점을 기록한다\n    visited.Add(vet); // 해당 정점을 방문했다고 표시한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    foreach (Vertex adjVet in graph.adjList[vet]) {\n        if (visited.Contains(adjVet)) {\n            continue; // 이미 방문한 정점은 건너뛴다                             \n        }\n        // 인접 정점을 재귀적으로 방문한다\n        DFS(graph, visited, res, adjVet);\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nList&lt;Vertex&gt; GraphDFS(GraphAdjList graph, Vertex startVet) {\n    // 정점 순회 순서\n    List&lt;Vertex&gt; res = [];\n    // 방문한 정점을 기록하는 해시 집합\n    HashSet&lt;Vertex&gt; visited = [];\n    DFS(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.go<pre><code>/* 깊이 우선 순회 보조 함수 */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n    // append 연산은 새 참조를 반환하므로 원래 참조에 새 슬라이스의 참조를 다시 대입해야 한다\n    *res = append(*res, vet)\n    visited[vet] = struct{}{}\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for _, adjVet := range g.adjList[vet] {\n        _, isExist := visited[adjVet]\n        // 인접 정점을 재귀적으로 방문한다\n        if !isExist {\n            dfs(g, visited, res, adjVet)\n        }\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // 정점 순회 순서\n    res := make([]Vertex, 0)\n    // 방문한 정점을 기록하는 해시 집합\n    visited := make(map[Vertex]struct{})\n    dfs(g, visited, &amp;res, startVet)\n    // 정점 순회 순서를 반환한다\n    return res\n}\n</code></pre> graph_dfs.swift<pre><code>/* 깊이 우선 순회 보조 함수 */\nfunc dfs(graph: GraphAdjList, visited: inout Set&lt;Vertex&gt;, res: inout [Vertex], vet: Vertex) {\n    res.append(vet) // 방문한 정점을 기록한다\n    visited.insert(vet) // 해당 정점을 방문했다고 표시한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for adjVet in graph.adjList[vet] ?? [] {\n        if visited.contains(adjVet) {\n            continue // 이미 방문한 정점은 건너뛴다\n        }\n        // 인접 정점을 재귀적으로 방문한다\n        dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: adjVet)\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // 정점 순회 순서\n    var res: [Vertex] = []\n    // 방문한 정점을 기록하는 해시 집합\n    var visited: Set&lt;Vertex&gt; = []\n    dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: startVet)\n    return res\n}\n</code></pre> graph_dfs.js<pre><code>/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunction dfs(graph, visited, res, vet) {\n    res.push(vet); // 방문한 정점을 기록한다\n    visited.add(vet); // 해당 정점을 방문했다고 표시한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // 이미 방문한 정점은 건너뛴다\n        }\n        // 인접 정점을 재귀적으로 방문한다\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunction graphDFS(graph, startVet) {\n    // 정점 순회 순서\n    const res = [];\n    // 방문한 정점을 기록하는 해시 집합\n    const visited = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.ts<pre><code>/* 깊이 우선 순회 보조 함수 */\nfunction dfs(\n    graph: GraphAdjList,\n    visited: Set&lt;Vertex&gt;,\n    res: Vertex[],\n    vet: Vertex\n): void {\n    res.push(vet); // 방문한 정점을 기록한다\n    visited.add(vet); // 해당 정점을 방문했다고 표시한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // 이미 방문한 정점은 건너뛴다\n        }\n        // 인접 정점을 재귀적으로 방문한다\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // 정점 순회 순서\n    const res: Vertex[] = [];\n    // 방문한 정점을 기록하는 해시 집합\n    const visited: Set&lt;Vertex&gt; = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.dart<pre><code>/* 깊이 우선 순회 보조 함수 */\nvoid dfs(\n  GraphAdjList graph,\n  Set&lt;Vertex&gt; visited,\n  List&lt;Vertex&gt; res,\n  Vertex vet,\n) {\n  res.add(vet); // 방문한 정점을 기록한다\n  visited.add(vet); // 해당 정점을 방문했다고 표시한다\n  // 해당 정점의 모든 인접 정점을 순회한다\n  for (Vertex adjVet in graph.adjList[vet]!) {\n    if (visited.contains(adjVet)) {\n      continue; // 이미 방문한 정점은 건너뛴다\n    }\n    // 인접 정점을 재귀적으로 방문한다\n    dfs(graph, visited, res, adjVet);\n  }\n}\n\n/* 깊이 우선 순회 */\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n  // 정점 순회 순서\n  List&lt;Vertex&gt; res = [];\n  // 방문한 정점을 기록하는 해시 집합\n  Set&lt;Vertex&gt; visited = {};\n  dfs(graph, visited, res, startVet);\n  return res;\n}\n</code></pre> graph_dfs.rs<pre><code>/* 깊이 우선 순회 보조 함수 */\nfn dfs(graph: &amp;GraphAdjList, visited: &amp;mut HashSet&lt;Vertex&gt;, res: &amp;mut Vec&lt;Vertex&gt;, vet: Vertex) {\n    res.push(vet); // 방문한 정점을 기록한다\n    visited.insert(vet); // 해당 정점을 방문했다고 표시한다\n                         // 해당 정점의 모든 인접 정점을 순회한다\n    if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n        for &amp;adj_vet in adj_vets {\n            if visited.contains(&amp;adj_vet) {\n                continue; // 이미 방문한 정점은 건너뛴다\n            }\n            // 인접 정점을 재귀적으로 방문한다\n            dfs(graph, visited, res, adj_vet);\n        }\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfn graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // 정점 순회 순서\n    let mut res = vec![];\n    // 방문한 정점을 기록하는 해시 집합\n    let mut visited = HashSet::new();\n    dfs(&amp;graph, &amp;mut visited, &amp;mut res, start_vet);\n\n    res\n}\n</code></pre> graph_dfs.c<pre><code>/* 정점을 이미 방문했는지 확인 */\nint isVisited(Vertex **res, int size, Vertex *vet) {\n    // O(n) 시간에 순회하여 노드를 찾는다\n    for (int i = 0; i &lt; size; i++) {\n        if (res[i] == vet) {\n            return 1;\n        }\n    }\n    return 0;\n}\n\n/* 깊이 우선 순회 보조 함수 */\nvoid dfs(GraphAdjList *graph, Vertex **res, int *resSize, Vertex *vet) {\n    // 방문한 정점을 기록한다\n    res[(*resSize)++] = vet;\n    // 해당 정점의 모든 인접 정점을 순회한다\n    AdjListNode *node = findNode(graph, vet);\n    while (node != NULL) {\n        // 이미 방문한 정점은 건너뛴다\n        if (!isVisited(res, *resSize, node-&gt;vertex)) {\n            // 인접 정점을 재귀적으로 방문한다\n            dfs(graph, res, resSize, node-&gt;vertex);\n        }\n        node = node-&gt;next;\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nvoid graphDFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize) {\n    dfs(graph, res, resSize, startVet);\n}\n</code></pre> graph_dfs.kt<pre><code>/* 깊이 우선 순회 보조 함수 */\nfun dfs(\n    graph: GraphAdjList,\n    visited: MutableSet&lt;Vertex?&gt;,\n    res: MutableList&lt;Vertex?&gt;,\n    vet: Vertex?\n) {\n    res.add(vet)     // 방문한 정점을 기록한다\n    visited.add(vet) // 해당 정점을 방문했다고 표시한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for (adjVet in graph.adjList[vet]!!) {\n        if (visited.contains(adjVet))\n            continue  // 이미 방문한 정점은 건너뛴다\n        // 인접 정점을 재귀적으로 방문한다\n        dfs(graph, visited, res, adjVet)\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfun graphDFS(graph: GraphAdjList, startVet: Vertex?): MutableList&lt;Vertex?&gt; {\n    // 정점 순회 순서\n    val res = mutableListOf&lt;Vertex?&gt;()\n    // 방문한 정점을 기록하는 해시 집합\n    val visited = HashSet&lt;Vertex?&gt;()\n    dfs(graph, visited, res, startVet)\n    return res\n}\n</code></pre> graph_dfs.rb<pre><code>### 깊이 우선 순회 보조 함수 ###\ndef dfs(graph, visited, res, vet)\n  res &lt;&lt; vet # 방문한 정점을 기록한다\n  visited.add(vet) # 해당 정점을 방문했다고 표시한다\n  # 해당 정점의 모든 인접 정점을 순회한다\n  for adj_vet in graph.adj_list[vet]\n    next if visited.include?(adj_vet) # 이미 방문한 정점은 건너뛴다\n    # 인접 정점을 재귀적으로 방문한다\n    dfs(graph, visited, res, adj_vet)\n  end\nend\n\n### 깊이 우선 순회 ###\ndef graph_dfs(graph, start_vet)\n  # 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\n  # 정점 순회 순서\n  res = []\n  # 방문한 정점을 기록하는 해시 집합\n  visited = Set.new\n  dfs(graph, visited, res, start_vet)\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>깊이 우선 순회의 진행 과정은 아래 그림(9-12)과 같다.</p> <ul> <li>직선 점선은 아래로 내려가는 재귀 호출을 나타내며, 새 정점을 방문하기 위해 새로운 재귀 메서드를 호출했다는 뜻이다.</li> <li>곡선 점선은 위로 되돌아가는 백트래킹을 나타내며, 재귀 메서드가 반환되어 이를 호출한 위치로 돌아왔다는 뜻이다.</li> </ul> <p>이해를 깊이 다지기 위해 아래 그림(9-12)과 코드를 함께 보면서, 각 재귀 메서드가 언제 호출되고 언제 반환되는지까지 포함하여 전체 DFS 과정을 머릿속으로 따라가 보자(또는 직접 그려 보자).</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 9-12   그래프의 깊이 우선 순회 단계 </p> <p>깊이 우선 순회 순서는 유일한가?</p> <p>너비 우선 순회와 마찬가지로 깊이 우선 순회 순서도 유일하지 않다. 한 정점에서 어느 방향을 먼저 탐색해도 되므로, 인접 정점의 순서를 자유롭게 바꾸어도 모두 깊이 우선 순회에 해당한다.</p> <p>트리 순회를 예로 들면 '루트 \\(\\rightarrow\\) 왼쪽 \\(\\rightarrow\\) 오른쪽', '왼쪽 \\(\\rightarrow\\) 루트 \\(\\rightarrow\\) 오른쪽', '왼쪽 \\(\\rightarrow\\) 오른쪽 \\(\\rightarrow\\) 루트'는 각각 전위, 중위, 후위 순회에 해당한다. 세 방식은 방문 우선순위가 다르지만 모두 깊이 우선 순회다.</p>","path":["제 9 장   그래프","9.3   그래프 순회"],"tags":[]},{"location":"chapter_graph/graph_traversal/#2_1","level":3,"title":"2.   복잡도 분석","text":"<p>시간 복잡도: 모든 정점을 \\(1\\) 번씩 방문하는 데 \\(O(|V|)\\) 시간이 걸리고, 모든 간선을 \\(2\\) 번씩 방문하는 데 \\(O(2|E|)\\) 시간이 걸린다. 전체 시간은 \\(O(|V| + |E|)\\)이다.</p> <p>공간 복잡도: 리스트 <code>res</code>와 해시 집합 <code>visited</code>의 정점 수는 각각 최대 \\(|V|\\)이고, 재귀 깊이도 최대 \\(|V|\\)이므로 \\(O(|V|)\\) 공간을 사용한다.</p>","path":["제 9 장   그래프","9.3   그래프 순회"],"tags":[]},{"location":"chapter_graph/summary/","level":1,"title":"9.4   요약","text":"","path":["제 9 장   그래프","9.4   요약"],"tags":[]},{"location":"chapter_graph/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>그래프는 정점과 간선으로 이루어지며, 정점 집합과 간선 집합으로 표현할 수 있다.</li> <li>선형 관계(연결 리스트)와 분할 정복 관계(트리)에 비해 네트워크 관계(그래프)는 자유도가 높아 더 복잡하다.</li> <li>방향 그래프의 간선에는 방향이 있고, 연결 그래프에서는 모든 정점에 도달할 수 있으며, 가중 그래프의 각 간선에는 가중치 변수가 있다.</li> <li>인접 행렬은 행렬로 그래프를 표현한다. 각 행(열)은 정점 하나를, 행렬 원소는 간선을 나타내며, \\(1\\) 또는 \\(0\\)으로 두 정점 사이에 간선이 있는지 표시한다. 인접 행렬은 추가, 삭제, 조회, 수정 연산이 매우 효율적이지만 공간을 많이 차지한다.</li> <li>인접 리스트는 여러 연결 리스트로 그래프를 표현한다. \\(i\\) 번째 연결 리스트는 정점 \\(i\\)에 대응하고, 해당 정점의 모든 인접 정점을 저장한다. 인접 행렬보다 공간을 절약하지만, 간선을 찾으려면 연결 리스트를 순회해야 하므로 시간 효율이 낮다.</li> <li>인접 리스트 안의 연결 리스트가 너무 길면 레드-블랙 트리나 해시 테이블로 바꾸어 조회 효율을 높일 수 있다.</li> <li>알고리즘의 관점에서 보면 인접 행렬은 '공간을 더 써서 시간을 절약'하고, 인접 리스트는 '시간을 더 써서 공간을 절약'한다.</li> <li>그래프는 소셜 네트워크나 지하철 노선 등 다양한 현실 시스템을 모델링하는 데 사용할 수 있다.</li> <li>트리는 그래프의 특수한 경우이며, 트리 순회도 그래프 순회의 특수한 경우다.</li> <li>그래프의 너비 우선 순회는 가까운 곳에서 먼 곳으로 한 층씩 확장하는 탐색 방식이며, 보통 큐로 구현한다.</li> <li>그래프의 깊이 우선 순회는 먼저 끝까지 나아가고 더 갈 곳이 없으면 되돌아오는 탐색 방식이며, 보통 재귀로 구현한다.</li> </ul>","path":["제 9 장   그래프","9.4   요약"],"tags":[]},{"location":"chapter_graph/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 경로는 정점의 순서로 정의하는가, 간선의 순서로 정의하는가?</p> <p>위키백과의 언어별 정의는 서로 다르다. 영어판에서는 '경로는 간선의 순서'라고 하고, 중국어판에서는 '경로는 정점의 순서'라고 한다. 영어판 원문을 옮기면 다음과 같다. '그래프 이론에서 그래프의 경로는 정점의 나열을 연결하는 간선의 유한하거나 무한한 나열이다.'</p> <p>이 절에서는 경로를 정점의 순서가 아니라 간선의 순서로 본다. 두 정점 사이에 여러 간선이 연결될 수 있으며, 이때 각 간선은 각각 하나의 경로에 해당하기 때문이다.</p> <p>Q: 비연결 그래프에는 순회할 수 없는 정점이 있는가?</p> <p>비연결 그래프에서는 한 정점에서 출발했을 때 도달할 수 없는 정점이 하나 이상 존재한다. 그래프의 모든 연결 요소를 순회하려면 시작점을 여러 개 설정해야 한다.</p> <p>Q: 인접 리스트에서 '해당 정점과 연결된 모든 정점'의 순서에 제약이 있는가?</p> <p>순서는 자유롭다. 하지만 실제 응용에서는 지정한 규칙에 따라 정렬해야 할 수도 있다. 예를 들어 정점이 추가된 순서나 정점 값의 크기순으로 정렬하면, '특정 기준에서 최댓값이나 최솟값을 가진' 정점을 빠르게 찾는 데 도움이 된다.</p>","path":["제 9 장   그래프","9.4   요약"],"tags":[]},{"location":"chapter_greedy/","level":1,"title":"제 15 장   탐욕","text":"<p>이 장을 시작하며</p> <p>해바라기는 태양을 따라 돌며 매 순간 자신의 성장 가능성을 최대한 추구한다.</p> <p>탐욕 전략은 단순한 선택을 거듭하면서 차츰 최선의 답으로 나아간다.</p>","path":["제 15 장   탐욕"],"tags":[]},{"location":"chapter_greedy/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>15.1   탐욕 알고리즘</li> <li>15.2   분할 가능한 배낭 문제</li> <li>15.3   최대 용량 문제</li> <li>15.4   정수 분할의 최대 곱 문제</li> <li>15.5   요약</li> <li>15.6   연습 문제</li> </ul>","path":["제 15 장   탐욕"],"tags":[]},{"location":"chapter_greedy/exercises/","level":1,"title":"15.6   연습 문제","text":"","path":["제 15 장   탐욕","15.6   연습 문제"],"tags":[]},{"location":"chapter_greedy/exercises/#1561","level":2,"title":"15.6.1   개념 확인","text":"","path":["제 15 장   탐욕","15.6   연습 문제"],"tags":[]},{"location":"chapter_greedy/exercises/#1","level":3,"title":"1.   매번 가장 큰 동전을 고르면 항상 최선인가","text":"<p>동전의 액면가는 <code>[1, 7, 10]</code>이며, 목표 금액은 14다.</p> <ol> <li>‘남은 금액을 넘지 않는 가장 큰 액면가의 동전을 매번 고른다’는 규칙을 실행하고 선택한 동전을 적어 보자.</li> <li>동전을 더 적게 사용하는 방법이 있는가? 있으면 하나를 적고, 없으면 이유를 설명하라.</li> <li>이 결과로 해당 탐욕 전략이 어떤 동전 액면가에서든 올바르다고 할 수 있는가?</li> </ol> 참고 답안 <ol> <li> <p>탐욕 방식은 <code>10 + 1 + 1 + 1 + 1</code>을 차례대로 선택하여 총 5개의 동전을 사용한다.</p> </li> <li> <p>더 적게 사용하는 방법이 있다. <code>7 + 7</code>이면 동전 2개만 필요하다.</p> </li> <li> <p>그럴 수 없다. 이 반례는 임의의 동전 액면가에서 현재 고를 수 있는 가장 큰 액면가를 매번 선택한다고 해서 반드시 최소 동전 개수를 얻는 것은 아님을 보여 준다.     당장 가장 큰 것을 선택하면 이후에 더 좋은 조합을 만들 기회를 잃을 수 있다.</p> </li> </ol>","path":["제 15 장   탐욕","15.6   연습 문제"],"tags":[]},{"location":"chapter_greedy/exercises/#2","level":3,"title":"2.   배낭에 어떤 물건을 먼저 넣어야 하는가","text":"<p>용량이 4킬로그램인 배낭에 다음 물건을 넣을 수 있다. 물건의 일부만 넣어도 되며, 얻는 가치는 넣은 무게에 비례한다.</p> <ul> <li>물건 A: 무게 4킬로그램, 가치 20</li> <li>물건 B: 무게 3킬로그램, 가치 18</li> </ul> <ol> <li>두 물건의 킬로그램당 가치는 각각 얼마인가? 어느 물건을 먼저 넣어야 하는가?</li> <li>분할 가능한 배낭 문제의 탐욕 전략으로 배낭을 가득 채우면 최종 가치는 얼마인가?</li> <li>물건을 나눌 수 있고 배낭이 총무게를 제한하는 상황에서는 물건을 고를 때 총가치와 킬로그램당 가치 중 무엇을 비교해야 하는가? 이유도 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>A의 킬로그램당 가치는 <code>20 ÷ 4 = 5</code>, B의 킬로그램당 가치는 <code>18 ÷ 3 = 6</code>이다.     따라서 단위 무게당 가치가 더 높은 B를 먼저 넣어야 한다.</p> </li> <li> <p>먼저 B를 전부 넣으면 3킬로그램을 차지하고 가치 18을 얻는다. 남은 용량 1킬로그램에는     A를 1킬로그램 넣어 가치 5를 얻는다. 최종 가치는 <code>18 + 5 = 23</code>이다.</p> </li> <li> <p>배낭은 총무게를 제한하고 물건을 나눌 수 있으므로 단위 무게당 가치를 비교해야 한다.     A는 총가치가 더 높지만 킬로그램당 가치는 B보다 낮다. A부터 넣어 배낭을 채우면 가치 20밖에 얻지 못한다.</p> </li> </ol>","path":["제 15 장   탐욕","15.6   연습 문제"],"tags":[]},{"location":"chapter_greedy/exercises/#3","level":3,"title":"3.   두 포인터를 다음에 어떻게 움직여야 하는가","text":"<p>칸막이 높이가 <code>[1, 8, 6, 2, 5]</code>일 때 양 끝에서 시작하는 두 포인터로 최대 용량을 찾는다. 용량은 ‘두 칸막이 중 더 낮은 높이 × 두 칸막이의 인덱스 차이’다.</p> <ol> <li>처음에는 왼쪽 포인터가 인덱스 0, 오른쪽 포인터가 인덱스 4에 있다. 현재 용량은 얼마인가? 다음에는 어느 포인터를 움직여야 하는가?</li> <li>1번에서 선택한 대로 한 번 움직이면 두 포인터는 각각 어느 인덱스에 있는가? 이때 용량은 얼마이며, 다음에는 어느 포인터를 움직여야 하는가?</li> <li>현재 두 칸막이에서 더 낮은 쪽의 포인터를 움직일 수도 있고, 더 높은 쪽의 포인터를 움직일 수도 있다. 어느 쪽을 움직여야 더 큰 용량을 얻을 가능성이 남는가? 이유는 무엇인가?</li> </ol> 참고 답안 <ol> <li> <p>현재 용량은 <code>min(1, 5) × (4 - 0) = 4</code>다. 왼쪽 칸막이가 더 낮으므로 왼쪽 포인터를 움직인다.</p> </li> <li> <p>왼쪽 포인터를 움직이면 두 포인터는 각각 인덱스 1과 4에 있다. 현재 용량은     <code>min(8, 5) × (4 - 1) = 15</code>다. 오른쪽 칸막이가 더 낮으므로 다음에는 오른쪽 포인터를 움직인다.</p> </li> <li> <p>더 낮은 칸막이 쪽의 포인터를 움직여야 더 큰 용량을 얻을 가능성이 남는다. 더 높은 칸막이를 움직이면 두 칸막이 사이의 거리는 반드시 줄어드는 반면, 용기의 높이는 여전히 움직이지 않은 낮은 칸막이에 의해 제한되므로     그대로이거나 더 낮아질 뿐이다. 따라서 용량은 움직이기 전보다 커질 수 없다. 낮은 칸막이를 움직여야만 더 높은 칸막이를 만날 기회가 생긴다.</p> </li> </ol>","path":["제 15 장   탐욕","15.6   연습 문제"],"tags":[]},{"location":"chapter_greedy/exercises/#1562","level":2,"title":"15.6.2   프로그래밍 연습","text":"","path":["제 15 장   탐욕","15.6   연습 문제"],"tags":[]},{"location":"chapter_greedy/exercises/#1_1","level":3,"title":"1.   분할 가능한 배낭 문제","text":"<p>길이가 같은 배열 <code>wgt</code>와 <code>val</code>이 주어진다. <code>wgt[i] &gt; 0</code>, <code>val[i] &gt;= 0</code>이며 배낭 용량은 <code>cap &gt;= 0</code>이다. 각 종류의 물건은 하나씩만 있지만, 그중 일부만 넣어도 된다. 얻는 가치는 넣은 무게가 해당 물건의 총무게에서 차지하는 비율로 계산한다. 탐욕 알고리즘을 사용하여 배낭에서 얻을 수 있는 최대 총가치를 실수로 반환하라.</p> 풀이 힌트 <ol> <li>먼저 각 물건의 단위 무게당 가치 val[i] / wgt[i]를 계산한다. 나눗셈 결과의 소수 부분을 유지해야 한다.</li> <li>단위 무게당 가치가 높은 물건부터 배낭에 넣는다.</li> <li>남은 용량이 현재 물건의 무게보다 작으면 배낭을 정확히 채우는 만큼만 넣고 끝낸다.</li> </ol>","path":["제 15 장   탐욕","15.6   연습 문제"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/","level":1,"title":"15.2   분할 가능한 배낭 문제","text":"<p>문제</p> <p>물건 \\(n\\) 개와 용량이 \\(cap\\)인 배낭이 주어진다. \\(i\\) 번째 물건의 무게는 \\(wgt[i-1]\\), 가치는 \\(val[i-1]\\)이다. 각 물건은 한 번만 선택할 수 있지만, 일부만 선택할 수 있으며 가치는 선택한 무게의 비율에 따라 계산한다. 용량 제한 안에서 배낭에 담을 수 있는 최대 가치를 구하라. 예시는 아래 그림(15-3)과 같다.</p> <p></p> <p> 그림 15-3   분할 가능한 배낭 문제 예시 데이터 </p> <p>분할 가능한 배낭 문제는 전체적으로 0-1 배낭 문제와 매우 비슷하다. 상태는 현재 물건 \\(i\\)와 용량 \\(c\\)를 포함하며, 목표는 용량 제한 안에서 최대 가치를 구하는 것이다.</p> <p>차이점은 물건의 일부만 선택할 수 있다는 점이다. 아래 그림(15-4)과 같이 물건을 원하는 대로 나누고 무게 비율에 따라 그 가치를 계산할 수 있다.</p> <ol> <li>물건 \\(i\\)의 단위 무게당 가치는 \\(val[i-1] / wgt[i-1]\\)이며, 줄여서 단위 가치라고 한다.</li> <li>물건 \\(i\\)의 일부를 무게 \\(w\\)만큼 넣으면 배낭의 가치는 \\(w \\times val[i-1] / wgt[i-1]\\)만큼 늘어난다.</li> </ol> <p></p> <p> 그림 15-4   물건의 단위 무게당 가치 </p>","path":["제 15 장   탐욕","15.2   분할 가능한 배낭 문제"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/#1","level":3,"title":"1.   탐욕 전략 결정","text":"<p>배낭 안의 총가치를 최대화하는 것은 본질적으로 단위 무게당 물건 가치를 최대화하는 것이다. 여기서 아래 그림(15-5)의 탐욕 전략을 도출할 수 있다.</p> <ol> <li>물건을 단위 가치가 높은 순서로 정렬한다.</li> <li>모든 물건을 순회하며 매번 단위 가치가 가장 높은 물건을 탐욕적으로 선택한다.</li> <li>남은 용량이 부족하면 현재 물건의 일부로 배낭을 가득 채운다.</li> </ol> <p></p> <p> 그림 15-5   분할 가능한 배낭 문제의 탐욕 전략 </p>","path":["제 15 장   탐욕","15.2   분할 가능한 배낭 문제"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/#2","level":3,"title":"2.   코드 구현","text":"<p>단위 가치순으로 정렬하기 쉽도록 물건 클래스 <code>Item</code>을 만든다. 탐욕적 선택을 반복하다가 배낭이 가득 차면 반복문을 빠져나와 해를 반환한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby fractional_knapsack.py<pre><code>class Item:\n    \"\"\"물건\"\"\"\n\n    def __init__(self, w: int, v: int):\n        self.w = w  # 물건의 무게\n        self.v = v  # 물건의 가치\n\ndef fractional_knapsack(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"분할 가능한 배낭: 탐욕\"\"\"\n    # 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    items = [Item(w, v) for w, v in zip(wgt, val)]\n    # 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    items.sort(key=lambda item: item.v / item.w, reverse=True)\n    # 탐욕적 선택을 반복한다\n    res = 0\n    for item in items:\n        if item.w &lt;= cap:\n            # 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v\n            cap -= item.w\n        else:\n            # 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += (item.v / item.w) * cap\n            # 남은 용량이 없으므로 반복문을 종료한다\n            break\n    return res\n</code></pre> fractional_knapsack.cpp<pre><code>/* 물건 */\nclass Item {\n  public:\n    int w; // 물건의 무게\n    int v; // 물건의 가치\n\n    Item(int w, int v) : w(w), v(v) {\n    }\n};\n\n/* 분할 가능한 배낭: 탐욕 */\ndouble fractionalKnapsack(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    vector&lt;Item&gt; items;\n    for (int i = 0; i &lt; wgt.size(); i++) {\n        items.push_back(Item(wgt[i], val[i]));\n    }\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    sort(items.begin(), items.end(), [](Item &amp;a, Item &amp;b) { return (double)a.v / a.w &gt; (double)b.v / b.w; });\n    // 탐욕적 선택을 반복한다\n    double res = 0;\n    for (auto &amp;item : items) {\n        if (item.w &lt;= cap) {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += (double)item.v / item.w * cap;\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.java<pre><code>/* 물건 */\nclass Item {\n    int w; // 물건의 무게\n    int v; // 물건의 가치\n\n    public Item(int w, int v) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* 분할 가능한 배낭: 탐욕 */\ndouble fractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    Item[] items = new Item[wgt.length];\n    for (int i = 0; i &lt; wgt.length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    Arrays.sort(items, Comparator.comparingDouble(item -&gt; -((double) item.v / item.w)));\n    // 탐욕적 선택을 반복한다\n    double res = 0;\n    for (Item item : items) {\n        if (item.w &lt;= cap) {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += (double) item.v / item.w * cap;\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.cs<pre><code>/* 물건 */\nclass Item(int w, int v) {\n    public int w = w; // 물건의 무게\n    public int v = v; // 물건의 가치\n}\n\n/* 분할 가능한 배낭: 탐욕 */\ndouble FractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    Item[] items = new Item[wgt.Length];\n    for (int i = 0; i &lt; wgt.Length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    Array.Sort(items, (x, y) =&gt; (y.v / y.w).CompareTo(x.v / x.w));\n    // 탐욕적 선택을 반복한다\n    double res = 0;\n    foreach (Item item in items) {\n        if (item.w &lt;= cap) {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += (double)item.v / item.w * cap;\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.go<pre><code>/* 물건 */\ntype Item struct {\n    w int // 물건의 무게\n    v int // 물건의 가치\n}\n\n/* 분할 가능한 배낭: 탐욕 */\nfunc fractionalKnapsack(wgt []int, val []int, cap int) float64 {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    items := make([]Item, len(wgt))\n    for i := 0; i &lt; len(wgt); i++ {\n        items[i] = Item{wgt[i], val[i]}\n    }\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    sort.Slice(items, func(i, j int) bool {\n        return float64(items[i].v)/float64(items[i].w) &gt; float64(items[j].v)/float64(items[j].w)\n    })\n    // 탐욕적 선택을 반복한다\n    res := 0.0\n    for _, item := range items {\n        if item.w &lt;= cap {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += float64(item.v)\n            cap -= item.w\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += float64(item.v) / float64(item.w) * float64(cap)\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.swift<pre><code>/* 물건 */\nclass Item {\n    var w: Int // 물건의 무게\n    var v: Int // 물건의 가치\n\n    init(w: Int, v: Int) {\n        self.w = w\n        self.v = v\n    }\n}\n\n/* 분할 가능한 배낭: 탐욕 */\nfunc fractionalKnapsack(wgt: [Int], val: [Int], cap: Int) -&gt; Double {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    var items = zip(wgt, val).map { Item(w: $0, v: $1) }\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    items.sort { -(Double($0.v) / Double($0.w)) &lt; -(Double($1.v) / Double($1.w)) }\n    // 탐욕적 선택을 반복한다\n    var res = 0.0\n    var cap = cap\n    for item in items {\n        if item.w &lt;= cap {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += Double(item.v)\n            cap -= item.w\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += Double(item.v) / Double(item.w) * Double(cap)\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.js<pre><code>/* 물건 */\nclass Item {\n    constructor(w, v) {\n        this.w = w; // 물건의 무게\n        this.v = v; // 물건의 가치\n    }\n}\n\n/* 분할 가능한 배낭: 탐욕 */\nfunction fractionalKnapsack(wgt, val, cap) {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    const items = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // 탐욕적 선택을 반복한다\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += (item.v / item.w) * cap;\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.ts<pre><code>/* 물건 */\nclass Item {\n    w: number; // 물건의 무게\n    v: number; // 물건의 가치\n\n    constructor(w: number, v: number) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* 분할 가능한 배낭: 탐욕 */\nfunction fractionalKnapsack(wgt: number[], val: number[], cap: number): number {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    const items: Item[] = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // 탐욕적 선택을 반복한다\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += (item.v / item.w) * cap;\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.dart<pre><code>/* 물건 */\nclass Item {\n  int w; // 물건의 무게\n  int v; // 물건의 가치\n\n  Item(this.w, this.v);\n}\n\n/* 분할 가능한 배낭: 탐욕 */\ndouble fractionalKnapsack(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n  List&lt;Item&gt; items = List.generate(wgt.length, (i) =&gt; Item(wgt[i], val[i]));\n  // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n  items.sort((a, b) =&gt; (b.v / b.w).compareTo(a.v / a.w));\n  // 탐욕적 선택을 반복한다\n  double res = 0;\n  for (Item item in items) {\n    if (item.w &lt;= cap) {\n      // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n      res += item.v;\n      cap -= item.w;\n    } else {\n      // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n      res += item.v / item.w * cap;\n      // 남은 용량이 없으므로 반복문을 종료한다\n      break;\n    }\n  }\n  return res;\n}\n</code></pre> fractional_knapsack.rs<pre><code>/* 물건 */\nstruct Item {\n    w: i32, // 물건의 무게\n    v: i32, // 물건의 가치\n}\n\nimpl Item {\n    fn new(w: i32, v: i32) -&gt; Self {\n        Self { w, v }\n    }\n}\n\n/* 분할 가능한 배낭: 탐욕 */\nfn fractional_knapsack(wgt: &amp;[i32], val: &amp;[i32], mut cap: i32) -&gt; f64 {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    let mut items = wgt\n        .iter()\n        .zip(val.iter())\n        .map(|(&amp;w, &amp;v)| Item::new(w, v))\n        .collect::&lt;Vec&lt;Item&gt;&gt;();\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    items.sort_by(|a, b| {\n        (b.v as f64 / b.w as f64)\n            .partial_cmp(&amp;(a.v as f64 / a.w as f64))\n            .unwrap()\n    });\n    // 탐욕적 선택을 반복한다\n    let mut res = 0.0;\n    for item in &amp;items {\n        if item.w &lt;= cap {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v as f64;\n            cap -= item.w;\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += item.v as f64 / item.w as f64 * cap as f64;\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break;\n        }\n    }\n    res\n}\n</code></pre> fractional_knapsack.c<pre><code>/* 물건 */\ntypedef struct {\n    int w; // 물건의 무게\n    int v; // 물건의 가치\n} Item;\n\n/* 분할 가능한 배낭: 탐욕 */\nfloat fractionalKnapsack(int wgt[], int val[], int itemCount, int cap) {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    Item *items = malloc(sizeof(Item) * itemCount);\n    for (int i = 0; i &lt; itemCount; i++) {\n        items[i] = (Item){.w = wgt[i], .v = val[i]};\n    }\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    qsort(items, (size_t)itemCount, sizeof(Item), sortByValueDensity);\n    // 탐욕적 선택을 반복한다\n    float res = 0.0;\n    for (int i = 0; i &lt; itemCount; i++) {\n        if (items[i].w &lt;= cap) {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += items[i].v;\n            cap -= items[i].w;\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += (float)cap / items[i].w * items[i].v;\n            cap = 0;\n            break;\n        }\n    }\n    free(items);\n    return res;\n}\n</code></pre> fractional_knapsack.kt<pre><code>/* 물건 */\nclass Item(\n    val w: Int, // 물건\n    val v: Int  // 물건의 가치\n)\n\n/* 분할 가능한 배낭: 탐욕 */\nfun fractionalKnapsack(wgt: IntArray, _val: IntArray, c: Int): Double {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    var cap = c\n    val items = arrayOfNulls&lt;Item&gt;(wgt.size)\n    for (i in wgt.indices) {\n        items[i] = Item(wgt[i], _val[i])\n    }\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    items.sortBy { item: Item? -&gt; -(item!!.v.toDouble() / item.w) }\n    // 탐욕적 선택을 반복한다\n    var res = 0.0\n    for (item in items) {\n        if (item!!.w &lt;= cap) {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v\n            cap -= item.w\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += item.v.toDouble() / item.w * cap\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.rb<pre><code>### 물건 ###\nclass Item\n  attr_accessor :w # 물건의 무게\n  attr_accessor :v # 물건의 가치\n\n  def initialize(w, v)\n    @w = w\n    @v = v\n  end\nend\n\n### 분할 가능한 배낭: 탐욕 ###\ndef fractional_knapsack(wgt, val, cap)\n  # 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n  items = wgt.each_with_index.map { |w, i| Item.new(w, val[i]) }\n  # 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n  items.sort! { |a, b| (b.v.to_f / b.w) &lt;=&gt; (a.v.to_f / a.w) }\n  # 탐욕적 선택을 반복한다\n  res = 0\n  for item in items\n    if item.w &lt;= cap\n      # 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n      res += item.v\n      cap -= item.w\n    else\n      # 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n      res += (item.v.to_f / item.w) * cap\n      # 남은 용량이 없으므로 반복문을 종료한다\n      break\n    end\n  end\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>내장 정렬 알고리즘의 시간 복잡도는 보통 \\(O(n \\log n)\\)이고, 공간 복잡도는 언어의 구체적인 구현에 따라 대체로 \\(O(\\log n)\\) 또는 \\(O(n)\\)이다.</p> <p>정렬을 제외하면 최악의 경우 물건 리스트 전체를 순회해야 하므로 시간 복잡도는 \\(O(n)\\)이다. 여기서 \\(n\\)은 물건 수다.</p> <p><code>Item</code> 객체 리스트를 초기화하므로 공간 복잡도는 \\(O(n)\\)이다.</p>","path":["제 15 장   탐욕","15.2   분할 가능한 배낭 문제"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/#3","level":3,"title":"3.   정확성 증명","text":"<p>귀류법을 사용하자. 물건 \\(x\\)의 단위 가치가 가장 높고, 어떤 알고리즘으로 구한 최대 가치가 <code>res</code>인데 그 해에 \\(x\\)가 없다고 가정한다.</p> <p>배낭에서 임의의 물건을 단위 무게만큼 꺼내고 같은 무게의 물건 \\(x\\)로 바꾼다. \\(x\\)의 단위 가치가 가장 높으므로 바꾼 뒤의 총가치는 반드시 <code>res</code>보다 커진다. 이는 <code>res</code>가 최적해라는 가정에 모순이므로 최적해에는 물건 \\(x\\)가 포함되어야 한다.</p> <p>그 해의 다른 물건에도 같은 모순을 구성할 수 있다. 즉, 단위 가치가 더 높은 물건이 항상 더 좋은 선택이며, 탐욕 전략이 유효함을 알 수 있다.</p> <p>아래 그림(15-6)과 같이 물건 무게를 가로축, 단위 가치를 세로축으로 보면, 분할 가능한 배낭 문제는 '제한된 가로축 구간에서 최대 넓이를 구하는' 문제로 바뀐다. 이 비유는 탐욕 전략의 유효성을 기하학적으로 이해하는 데 도움이 된다.</p> <p></p> <p> 그림 15-6   분할 가능한 배낭 문제의 기하학적 표현 </p>","path":["제 15 장   탐욕","15.2   분할 가능한 배낭 문제"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/","level":1,"title":"15.1   탐욕 알고리즘","text":"<p>탐욕 알고리즘(greedy algorithm)은 최적화 문제를 푸는 대표적인 알고리즘이다. 매 결정 단계에서 현재 가장 좋아 보이는 선택, 즉 국소적으로 최적인 결정을 탐욕적으로 내려 전역 최적해를 얻으려는 것이 기본 아이디어다. 간결하고 효율적이어서 많은 실제 문제에 널리 쓰인다.</p> <p>탐욕 알고리즘과 동적 계획법은 모두 최적화 문제에 자주 사용한다. 최적 부분 구조에 의존한다는 공통점 등이 있지만 작동 원리는 다르다.</p> <ul> <li>동적 계획법은 이전 단계의 모든 결정을 바탕으로 현재 결정을 고려하고, 과거 부분 문제의 해로 현재 부분 문제의 해를 구성한다.</li> <li>탐욕 알고리즘은 과거 결정을 고려하지 않고 계속 앞으로 나아가며 탐욕적으로 선택한다. 문제의 범위를 점차 줄여 해결될 때까지 진행한다.</li> </ul> <p>먼저 '동전 교환' 예제로 탐욕 알고리즘의 원리를 알아보자. '무한 배낭 문제' 절에서 소개한 문제라 익숙할 것이다.</p> <p>문제</p> <p>동전 \\(n\\) 종류가 주어지며 \\(i\\) 번째 동전의 액면가는 \\(coins[i - 1]\\)이고 목표 금액은 \\(amt\\)다. 각 종류의 동전을 반복해서 선택할 수 있을 때 목표 금액을 만드는 최소 동전 수를 구하라. 만들 수 없다면 \\(-1\\)을 반환하라.</p> <p>이 문제에서 사용하는 탐욕 전략은 아래 그림(15-1)과 같다. 주어진 목표 금액에 대해 그 금액을 넘지 않으면서 가장 가까운 액면가의 동전을 탐욕적으로 고르고, 목표 금액을 만들 때까지 반복한다.</p> <p></p> <p> 그림 15-1   동전 교환의 탐욕 전략 </p> <p>구현 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change_greedy.py<pre><code>def coin_change_greedy(coins: list[int], amt: int) -&gt; int:\n    \"\"\"동전 교환: 탐욕\"\"\"\n    # coins 리스트가 정렬되어 있다고 가정한다\n    i = len(coins) - 1\n    count = 0\n    # 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while amt &gt; 0:\n        # 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while i &gt; 0 and coins[i] &gt; amt:\n            i -= 1\n        # coins[i]를 선택한다\n        amt -= coins[i]\n        count += 1\n    # 가능한 방법을 찾지 못하면 -1을 반환한다\n    return count if amt == 0 else -1\n</code></pre> coin_change_greedy.cpp<pre><code>/* 동전 교환: 탐욕 */\nint coinChangeGreedy(vector&lt;int&gt; &amp;coins, int amt) {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    int i = coins.size() - 1;\n    int count = 0;\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while (amt &gt; 0) {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i];\n        count++;\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.java<pre><code>/* 동전 교환: 탐욕 */\nint coinChangeGreedy(int[] coins, int amt) {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    int i = coins.length - 1;\n    int count = 0;\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while (amt &gt; 0) {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i];\n        count++;\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.cs<pre><code>/* 동전 교환: 탐욕 */\nint CoinChangeGreedy(int[] coins, int amt) {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    int i = coins.Length - 1;\n    int count = 0;\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while (amt &gt; 0) {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i];\n        count++;\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.go<pre><code>/* 동전 교환: 탐욕 */\nfunc coinChangeGreedy(coins []int, amt int) int {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    i := len(coins) - 1\n    count := 0\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    for amt &gt; 0 {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        for i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i--\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i]\n        count++\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    if amt != 0 {\n        return -1\n    }\n    return count\n}\n</code></pre> coin_change_greedy.swift<pre><code>/* 동전 교환: 탐욕 */\nfunc coinChangeGreedy(coins: [Int], amt: Int) -&gt; Int {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    var i = coins.count - 1\n    var count = 0\n    var amt = amt\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while amt &gt; 0 {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i]\n        count += 1\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return amt == 0 ? count : -1\n}\n</code></pre> coin_change_greedy.js<pre><code>/* 동전 교환: 탐욕 */\nfunction coinChangeGreedy(coins, amt) {\n    // coins 배열이 정렬되어 있다고 가정한다\n    let i = coins.length - 1;\n    let count = 0;\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while (amt &gt; 0) {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i];\n        count++;\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.ts<pre><code>/* 동전 교환: 탐욕 */\nfunction coinChangeGreedy(coins: number[], amt: number): number {\n    // coins 배열이 정렬되어 있다고 가정한다\n    let i = coins.length - 1;\n    let count = 0;\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while (amt &gt; 0) {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i];\n        count++;\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.dart<pre><code>/* 동전 교환: 탐욕 */\nint coinChangeGreedy(List&lt;int&gt; coins, int amt) {\n  // coins 리스트가 정렬되어 있다고 가정한다\n  int i = coins.length - 1;\n  int count = 0;\n  // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n  while (amt &gt; 0) {\n    // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n    while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n      i--;\n    }\n    // coins[i]를 선택한다\n    amt -= coins[i];\n    count++;\n  }\n  // 가능한 방법을 찾지 못하면 -1을 반환한다\n  return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.rs<pre><code>/* 동전 교환: 탐욕 */\nfn coin_change_greedy(coins: &amp;[i32], mut amt: i32) -&gt; i32 {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    let mut i = coins.len() - 1;\n    let mut count = 0;\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while amt &gt; 0 {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1;\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i];\n        count += 1;\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    if amt == 0 {\n        count\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change_greedy.c<pre><code>/* 동전 교환: 탐욕 */\nint coinChangeGreedy(int *coins, int size, int amt) {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    int i = size - 1;\n    int count = 0;\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while (amt &gt; 0) {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i];\n        count++;\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.kt<pre><code>/* 동전 교환: 탐욕 */\nfun coinChangeGreedy(coins: IntArray, amt: Int): Int {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    var am = amt\n    var i = coins.size - 1\n    var count = 0\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while (am &gt; 0) {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; am) {\n            i--\n        }\n        // coins[i]를 선택한다\n        am -= coins[i]\n        count++\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return if (am == 0) count else -1\n}\n</code></pre> coin_change_greedy.rb<pre><code>### 동전 교환: 탐욕 ###\ndef coin_change_greedy(coins, amt)\n  # coins 리스트가 정렬되어 있다고 가정한다\n  i = coins.length - 1\n  count = 0\n  # 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n  while amt &gt; 0\n    # 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n    while i &gt; 0 &amp;&amp; coins[i] &gt; amt\n      i -= 1\n    end\n    # coins[i]를 선택한다\n    amt -= coins[i]\n    count += 1\n  end\n  # 가능한 방법을 찾지 못하면 -1을 반환한다\n  amt == 0 ? count : -1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>'정말 간결하다!'라는 감탄이 나올 만하다. 탐욕 알고리즘은 약 열 줄의 코드만으로 동전 교환 문제를 해결했다.</p>","path":["제 15 장   탐욕","15.1   탐욕 알고리즘"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1511","level":2,"title":"15.1.1   탐욕 알고리즘의 장점과 한계","text":"<p>탐욕 알고리즘은 연산이 직접적이고 구현이 간단할 뿐 아니라 보통 효율도 매우 높다. 위 코드에서 최소 액면가를 \\(\\min(coins)\\)라고 하면 탐욕적 선택은 최대 \\(amt / \\min(coins)\\) 번 반복하므로 시간 복잡도는 \\(O(amt / \\min(coins))\\)이다. 동적 계획법의 \\(O(n \\times amt)\\)보다 한 차수 낮다.</p> <p>하지만 일부 동전 액면가 조합에서는 최적해를 찾지 못한다. 아래 그림(15-2)은 두 예를 보여 준다.</p> <ul> <li>성립 예 \\(coins = [1, 5, 10, 20, 50, 100]\\): 이 동전 조합에서는 어떤 \\(amt\\)에 대해서도 탐욕 알고리즘이 최적해를 찾는다.</li> <li>반례 \\(coins = [1, 20, 50]\\): \\(amt = 60\\)이면 탐욕 알고리즘은 \\(50 + 1 \\times 10\\)으로 동전 \\(11\\) 개를 고른다. 하지만 동적 계획법은 \\(20 + 20 + 20\\)이라는 최적해를 찾아 \\(3\\) 개만 사용한다.</li> <li>반례 \\(coins = [1, 49, 50]\\): \\(amt = 98\\)이면 탐욕 알고리즘은 \\(50 + 1 \\times 48\\)로 동전 \\(49\\) 개를 고른다. 하지만 동적 계획법은 \\(49 + 49\\)라는 최적해를 찾아 \\(2\\) 개만 사용한다.</li> </ul> <p></p> <p> 그림 15-2   탐욕 알고리즘이 최적해를 찾지 못하는 예 </p> <p>즉, 동전 교환에서 탐욕 알고리즘은 전역 최적해를 보장하지 못하고 매우 나쁜 해를 찾을 수도 있다. 이 문제는 동적 계획법으로 푸는 편이 더 적합하다.</p> <p>일반적으로 탐욕 알고리즘을 적용하는 상황은 다음 두 가지다.</p> <ol> <li>최적해를 보장할 수 있다: 보통 백트래킹이나 동적 계획법보다 효율적이므로 이런 경우에는 탐욕 알고리즘이 가장 좋은 선택인 경우가 많다.</li> <li>최적해에 가까운 해를 찾을 수 있다: 이 경우에도 탐욕 알고리즘을 쓸 수 있다. 복잡한 문제에서는 전역 최적해를 찾기 매우 어려우므로, 효율적으로 차선의 해를 찾는 것만으로도 유용하다.</li> </ol>","path":["제 15 장   탐욕","15.1   탐욕 알고리즘"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1512","level":2,"title":"15.1.2   탐욕 알고리즘의 특성","text":"<p>그렇다면 어떤 문제가 탐욕 알고리즘에 적합할까? 다시 말해 어떤 경우에 최적해를 보장할 수 있을까?</p> <p>탐욕 알고리즘은 동적 계획법보다 적용 조건이 엄격하며, 주로 다음 두 성질에 주목한다.</p> <ul> <li>탐욕적 선택 속성: 국소 최적 선택이 항상 전역 최적해로 이어져야만 탐욕 알고리즘이 최적해를 보장할 수 있다.</li> <li>최적 부분 구조: 원래 문제의 최적해가 부분 문제의 최적해를 포함한다.</li> </ul> <p>최적 부분 구조는 '동적 계획법' 장에서 설명했으므로 반복하지 않는다. 일부 문제는 최적 부분 구조가 뚜렷하지 않아도 탐욕 알고리즘으로 해결할 수 있다는 점을 알아 두자.</p> <p>여기서는 탐욕적 선택 속성을 판단하는 방법을 주로 살펴본다. 설명은 간단해 보이지만 많은 문제에서 이 속성을 증명하기는 쉽지 않다.</p> <p>예를 들어 동전 교환에서는 반례로 탐욕적 선택 속성을 부정하기는 쉽지만 증명하기는 어렵다. 어떤 조건을 만족하는 동전 조합을 탐욕 알고리즘으로 풀 수 있는가라는 질문에는 직관이나 사례를 바탕으로 애매한 답을 내놓기 쉽고, 엄밀한 수학적 증명을 제시하기는 어렵다.</p> <p>참고 문헌</p> <p>어떤 동전 조합에 대해 탐욕 알고리즘이 모든 금액의 최적해를 찾을 수 있는지 판단하는 \\(O(n^3)\\) 알고리즘을 제시한 논문이 있다.</p> <p>Pearson, D. A polynomial-time algorithm for the change-making problem[J]. Operations Research Letters, 2005, 33(3): 231-234.</p>","path":["제 15 장   탐욕","15.1   탐욕 알고리즘"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1513","level":2,"title":"15.1.3   탐욕 알고리즘의 문제 해결 단계","text":"<p>탐욕 문제의 풀이 과정은 대체로 다음 세 단계다.</p> <ol> <li>문제 분석: 상태 정의, 최적화 목표, 제약 조건 등 문제의 특성을 정리하고 이해한다. 백트래킹과 동적 계획법에서도 필요한 단계다.</li> <li>탐욕 전략 결정: 매 단계에서 어떻게 탐욕적으로 선택할지 정한다. 각 단계에서 문제 크기를 줄여 최종적으로 전체 문제를 해결할 수 있어야 한다.</li> <li>정확성 증명: 보통 탐욕적 선택 속성과 최적 부분 구조를 증명해야 한다. 귀납법이나 귀류법 같은 수학적 증명이 필요할 수 있다.</li> </ol> <p>탐욕 전략을 정하는 것이 핵심 단계지만, 다음 이유로 쉽지 않을 수 있다.</p> <ul> <li>문제마다 탐욕 전략이 크게 다르다. 많은 문제는 전략이 비교적 명확하여 대략적인 생각과 시도만으로 찾을 수 있다. 반면 복잡한 문제에서는 전략이 잘 드러나지 않아 문제 풀이 경험과 알고리즘 역량이 크게 요구된다.</li> <li>일부 탐욕 전략은 그럴듯해 보이지만 함정이 있다. 자신 있게 전략을 설계하고 코드를 작성해 제출해도 일부 테스트를 통과하지 못할 수 있다. 전략이 '부분적으로만 옳기' 때문이다. 앞에서 본 동전 교환이 대표적인 사례다.</li> </ul> <p>정확성을 보장하려면 탐욕 전략을 엄밀히 증명해야 하며, 보통 귀류법이나 수학적 귀납법을 사용한다.</p> <p>하지만 정확성 증명 자체도 쉽지 않을 수 있다. 실마리가 없으면 보통 테스트 사례로 코드를 디버깅하면서 탐욕 전략을 조금씩 수정하고 검증한다.</p>","path":["제 15 장   탐욕","15.1   탐욕 알고리즘"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1514","level":2,"title":"15.1.4   대표적인 탐욕 알고리즘 문제","text":"<p>탐욕 알고리즘은 탐욕적 선택 속성과 최적 부분 구조를 만족하는 최적화 문제에 자주 사용한다. 대표적인 예는 다음과 같다.</p> <ul> <li>동전 거스름돈 문제: 일부 동전 조합에서는 탐욕 알고리즘이 항상 최적해를 얻는다.</li> <li>구간 스케줄링 문제: 각 작업이 일정 시간 구간에 걸쳐 진행될 때 가능한 한 많은 작업을 완료하는 것이 목표다. 매번 가장 일찍 끝나는 작업을 고르면 최적해를 얻을 수 있다.</li> <li>분할 가능한 배낭 문제: 물건들과 적재 용량이 주어졌을 때 총무게가 용량을 넘지 않으면서 총가치가 최대가 되도록 물건을 고른다. 매번 무게 대비 가치의 비율(가치 / 무게)이 가장 높은 물건을 고르면 일부 경우에 최적해를 얻을 수 있다.</li> <li>주식 매매 문제: 주가의 과거 기록이 주어졌을 때 여러 번 매매하여 최대 이익을 구한다. 단, 이미 주식을 보유하고 있으면 팔기 전에는 다시 살 수 없다.</li> <li>허프먼 코딩: 무손실 데이터 압축에 쓰이는 탐욕 알고리즘이다. 매번 빈도가 가장 낮은 두 노드를 합쳐 허프먼 트리를 만들면, 최종 트리의 가중 경로 길이(부호 길이)가 최소가 된다.</li> <li>Dijkstra 알고리즘: 모든 간선 가중치가 음이 아닌 그래프에서 주어진 시작 정점부터 나머지 정점까지의 최단 경로를 구하는 탐욕 알고리즘이다.</li> </ul>","path":["제 15 장   탐욕","15.1   탐욕 알고리즘"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/","level":1,"title":"15.3   최대 용량 문제","text":"<p>문제</p> <p>배열 \\(ht\\)의 각 원소는 수직 칸막이의 높이를 나타낸다. 배열에서 임의의 두 칸막이와 그 사이의 공간으로 용기를 만들 수 있다.</p> <p>용량은 높이와 너비의 곱(넓이)이다. 높이는 더 낮은 칸막이로 결정되고, 너비는 두 칸막이의 배열 인덱스 차이다.</p> <p>용량이 최대가 되도록 두 칸막이를 골라 최대 용량을 반환하라. 예시는 아래 그림(15-7)과 같다.</p> <p></p> <p> 그림 15-7   최대 용량 문제 예시 데이터 </p> <p>임의의 두 칸막이로 용기를 만들므로 이 문제의 상태는 두 칸막이의 인덱스 \\([i, j]\\)다.</p> <p>문제에 따르면 용량은 높이와 너비의 곱이고, 높이는 낮은 칸막이로 결정되며 너비는 두 인덱스의 차다. 용량을 \\(cap[i, j]\\)라고 하면 다음 공식을 얻는다.</p> \\[ cap[i, j] = \\min(ht[i], ht[j]) \\times (j - i) \\] <p>배열 길이가 \\(n\\)이면 두 칸막이의 조합 수(전체 상태 수)는 \\(C_n^2 = \\frac{n(n - 1)}{2}\\)이다. 가장 직접적으로는 모든 상태를 열거하여 최대 용량을 구할 수 있으며, 시간 복잡도는 \\(O(n^2)\\)이다.</p>","path":["제 15 장   탐욕","15.3   최대 용량 문제"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/#1","level":3,"title":"1.   탐욕 전략 결정","text":"<p>더 효율적인 풀이도 있다. 아래 그림(15-8)처럼 인덱스 \\(i &lt; j\\)이고 높이 \\(ht[i] &lt; ht[j]\\)인 상태 \\([i, j]\\)를 고르자. 즉, \\(i\\)는 낮은 칸막이, \\(j\\)는 높은 칸막이다.</p> <p></p> <p> 그림 15-8   초기 상태 </p> <p>아래 그림(15-9)과 같이 높은 칸막이 \\(j\\)를 낮은 칸막이 \\(i\\) 쪽으로 옮기면 용량은 반드시 작아진다.</p> <p>\\(j\\)를 옮기면 너비 \\(j-i\\)는 확실히 줄어든다. 높이는 낮은 칸막이로 결정되므로 그대로이거나(\\(i\\)가 계속 더 낮음), 더 낮아질 수만 있다(이동한 \\(j\\)가 더 낮아짐).</p> <p></p> <p> 그림 15-9   높은 칸막이를 안쪽으로 옮긴 상태 </p> <p>반대로 생각하면 낮은 칸막이 \\(i\\)를 안쪽으로 옮겨야만 용량이 커질 가능성이 있다. 너비는 반드시 줄지만 높이는 커질 수도 있기 때문이다(이동한 \\(i\\)가 더 높을 수 있다). 아래 그림(15-10)에서는 낮은 칸막이를 옮긴 뒤 넓이가 커진다.</p> <p></p> <p> 그림 15-10   낮은 칸막이를 안쪽으로 옮긴 상태 </p> <p>따라서 두 포인터를 용기의 양 끝에 놓고, 매번 낮은 칸막이 쪽 포인터를 안쪽으로 옮겨 두 포인터가 만날 때까지 진행하는 탐욕 전략을 도출할 수 있다.</p> <p>아래 그림(15-11)은 탐욕 전략의 실행 과정이다.</p> <ol> <li>처음에는 포인터 \\(i\\)와 \\(j\\)를 배열의 양 끝에 둔다.</li> <li>현재 상태의 용량 \\(cap[i, j]\\)를 계산하고 최대 용량을 갱신한다.</li> <li>칸막이 \\(i\\)와 \\(j\\)의 높이를 비교하여 낮은 쪽을 안쪽으로 한 칸 옮긴다.</li> <li>\\(i\\)와 \\(j\\)가 만날 때까지 <code>2.</code> 단계와 <code>3.</code> 단계를 반복한다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 15-11   최대 용량 문제의 탐욕 과정 </p>","path":["제 15 장   탐욕","15.3   최대 용량 문제"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/#2","level":3,"title":"2.   코드 구현","text":"<p>반복문은 최대 \\(n\\) 회 실행되므로 시간 복잡도는 \\(O(n)\\)이다.</p> <p>변수 \\(i\\), \\(j\\), \\(res\\)는 상수 크기의 추가 공간을 사용하므로 공간 복잡도는 \\(O(1)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby max_capacity.py<pre><code>def max_capacity(ht: list[int]) -&gt; int:\n    \"\"\"최대 용량: 탐욕\"\"\"\n    # i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    i, j = 0, len(ht) - 1\n    # 최대 용량의 초깃값은 0이다\n    res = 0\n    # 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while i &lt; j:\n        # 최대 용량을 갱신한다\n        cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        # 낮은 칸막이를 안쪽으로 이동한다\n        if ht[i] &lt; ht[j]:\n            i += 1\n        else:\n            j -= 1\n    return res\n</code></pre> max_capacity.cpp<pre><code>/* 최대 용량: 탐욕 */\nint maxCapacity(vector&lt;int&gt; &amp;ht) {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    int i = 0, j = ht.size() - 1;\n    // 최대 용량의 초깃값은 0이다\n    int res = 0;\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while (i &lt; j) {\n        // 최대 용량을 갱신한다\n        int cap = min(ht[i], ht[j]) * (j - i);\n        res = max(res, cap);\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.java<pre><code>/* 최대 용량: 탐욕 */\nint maxCapacity(int[] ht) {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    int i = 0, j = ht.length - 1;\n    // 최대 용량의 초깃값은 0이다\n    int res = 0;\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while (i &lt; j) {\n        // 최대 용량을 갱신한다\n        int cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.cs<pre><code>/* 최대 용량: 탐욕 */\nint MaxCapacity(int[] ht) {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    int i = 0, j = ht.Length - 1;\n    // 최대 용량의 초깃값은 0이다\n    int res = 0;\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while (i &lt; j) {\n        // 최대 용량을 갱신한다\n        int cap = Math.Min(ht[i], ht[j]) * (j - i);\n        res = Math.Max(res, cap);\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.go<pre><code>/* 최대 용량: 탐욕 */\nfunc maxCapacity(ht []int) int {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    i, j := 0, len(ht)-1\n    // 최대 용량의 초깃값은 0이다\n    res := 0\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    for i &lt; j {\n        // 최대 용량을 갱신한다\n        capacity := int(math.Min(float64(ht[i]), float64(ht[j]))) * (j - i)\n        res = int(math.Max(float64(res), float64(capacity)))\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if ht[i] &lt; ht[j] {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.swift<pre><code>/* 최대 용량: 탐욕 */\nfunc maxCapacity(ht: [Int]) -&gt; Int {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    var i = ht.startIndex, j = ht.endIndex - 1\n    // 최대 용량의 초깃값은 0이다\n    var res = 0\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while i &lt; j {\n        // 최대 용량을 갱신한다\n        let cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if ht[i] &lt; ht[j] {\n            i += 1\n        } else {\n            j -= 1\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.js<pre><code>/* 최대 용량: 탐욕 */\nfunction maxCapacity(ht) {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    let i = 0,\n        j = ht.length - 1;\n    // 최대 용량의 초깃값은 0이다\n    let res = 0;\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while (i &lt; j) {\n        // 최대 용량을 갱신한다\n        const cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.ts<pre><code>/* 최대 용량: 탐욕 */\nfunction maxCapacity(ht: number[]): number {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    let i = 0,\n        j = ht.length - 1;\n    // 최대 용량의 초깃값은 0이다\n    let res = 0;\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while (i &lt; j) {\n        // 최대 용량을 갱신한다\n        const cap: number = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.dart<pre><code>/* 최대 용량: 탐욕 */\nint maxCapacity(List&lt;int&gt; ht) {\n  // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n  int i = 0, j = ht.length - 1;\n  // 최대 용량의 초깃값은 0이다\n  int res = 0;\n  // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n  while (i &lt; j) {\n    // 최대 용량을 갱신한다\n    int cap = min(ht[i], ht[j]) * (j - i);\n    res = max(res, cap);\n    // 낮은 칸막이를 안쪽으로 이동한다\n    if (ht[i] &lt; ht[j]) {\n      i++;\n    } else {\n      j--;\n    }\n  }\n  return res;\n}\n</code></pre> max_capacity.rs<pre><code>/* 최대 용량: 탐욕 */\nfn max_capacity(ht: &amp;[i32]) -&gt; i32 {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    let mut i = 0;\n    let mut j = ht.len() - 1;\n    // 최대 용량의 초깃값은 0이다\n    let mut res = 0;\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while i &lt; j {\n        // 최대 용량을 갱신한다\n        let cap = std::cmp::min(ht[i], ht[j]) * (j - i) as i32;\n        res = std::cmp::max(res, cap);\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if ht[i] &lt; ht[j] {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    res\n}\n</code></pre> max_capacity.c<pre><code>/* 최대 용량: 탐욕 */\nint maxCapacity(int ht[], int htLength) {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    int i = 0;\n    int j = htLength - 1;\n    // 최대 용량의 초깃값은 0이다\n    int res = 0;\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while (i &lt; j) {\n        // 최대 용량을 갱신한다\n        int capacity = myMin(ht[i], ht[j]) * (j - i);\n        res = myMax(res, capacity);\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.kt<pre><code>/* 최대 용량: 탐욕 */\nfun maxCapacity(ht: IntArray): Int {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    var i = 0\n    var j = ht.size - 1\n    // 최대 용량의 초깃값은 0이다\n    var res = 0\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while (i &lt; j) {\n        // 최대 용량을 갱신한다\n        val cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if (ht[i] &lt; ht[j]) {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.rb<pre><code>### 최대 용량: 탐욕 ###\ndef max_capacity(ht)\n  # i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n  i, j = 0, ht.length - 1\n  # 최대 용량의 초깃값은 0이다\n  res = 0\n\n  # 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n  while i &lt; j\n    # 최대 용량을 갱신한다\n    cap = [ht[i], ht[j]].min * (j - i)\n    res = [res, cap].max\n    # 낮은 칸막이를 안쪽으로 이동한다\n    if ht[i] &lt; ht[j]\n      i += 1\n    else\n      j -= 1\n    end\n  end\n\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 15 장   탐욕","15.3   최대 용량 문제"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/#3","level":3,"title":"3.   정확성 증명","text":"<p>탐욕 알고리즘이 모든 경우 열거보다 빠른 이유는 매번 일부 상태를 '건너뛰기' 때문이다.</p> <p>예를 들어 상태 \\(cap[i, j]\\)에서 \\(i\\)가 낮은 칸막이, \\(j\\)가 높은 칸막이라고 하자. 낮은 \\(i\\)를 탐욕적으로 한 칸 안쪽으로 옮기면 아래 그림(15-12)의 상태들을 건너뛴다. 이는 이후에 그 상태들의 용량을 확인할 수 없다는 뜻이다.</p> \\[ cap[i, i+1], cap[i, i+2], \\dots, cap[i, j-2], cap[i, j-1] \\] <p></p> <p> 그림 15-12   낮은 칸막이를 옮겨 건너뛴 상태들 </p> <p>건너뛴 상태들은 사실 높은 칸막이 \\(j\\)를 안쪽으로 옮긴 모든 상태다. 앞에서 높은 칸막이를 안쪽으로 옮기면 용량이 반드시 줄어듦을 보였다. 따라서 건너뛴 상태는 최적해가 될 수 없으며, 이를 건너뛰어도 최적해를 놓치지 않는다.</p> <p>이 분석은 낮은 칸막이를 옮기는 연산이 '안전'하며 탐욕 전략이 유효함을 보여 준다.</p>","path":["제 15 장   탐욕","15.3   최대 용량 문제"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/","level":1,"title":"15.4   정수 분할의 최대 곱 문제","text":"<p>문제</p> <p>양의 정수 \\(n\\)을 두 개 이상의 양의 정수의 합으로 나누었을 때, 나눈 정수들의 곱이 최대 얼마가 되는지 구하라. 아래 그림(15-13)을 참고하자.</p> <p></p> <p> 그림 15-13   정수 분할의 최대 곱 문제 정의 </p> <p>\\(n\\)을 \\(m\\) 개의 정수 인수로 나누고 \\(i\\) 번째 인수를 \\(n_i\\)라고 하면 다음과 같다.</p> \\[ n = \\sum_{i=1}^{m}n_i \\] <p>목표는 모든 정수 인수의 곱을 최대화하는 것이다.</p> \\[ \\max(\\prod_{i=1}^{m}n_i) \\] <p>생각해야 할 것은 나누는 개수 \\(m\\)과 각 \\(n_i\\)의 값이다.</p>","path":["제 15 장   탐욕","15.4   정수 분할의 최대 곱 문제"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/#1","level":3,"title":"1.   탐욕 전략 결정","text":"<p>경험상 두 정수의 곱은 합보다 큰 경우가 많다. \\(n\\)에서 인수 \\(2\\)를 나누어 내면 곱은 \\(2(n-2)\\)다. 이를 \\(n\\)과 비교해 보자.</p> \\[ \\begin{aligned} 2(n-2) &amp; \\geq n \\newline 2n - n - 4 &amp; \\geq 0 \\newline n &amp; \\geq 4 \\end{aligned} \\] <p>아래 그림(15-14)과 같이 \\(n \\geq 4\\)이면 \\(2\\)를 나누어 낸 뒤 곱이 커진다. 이는 \\(4\\) 이상의 정수는 모두 나누어야 한다는 뜻이다.</p> <p>탐욕 전략 1: 분할에 \\(\\geq 4\\)인 인수가 있으면 계속 나누어야 한다. 최종 분할에는 \\(1\\), \\(2\\), \\(3\\)만 있어야 한다.</p> <p></p> <p> 그림 15-14   분할로 커지는 곱 </p> <p>이제 어느 인수가 가장 좋은지 생각해 보자. \\(1\\), \\(2\\), \\(3\\) 중 \\(1\\)이 가장 나쁘다는 것은 분명하다. \\(1 \\times (n-1) &lt; n\\)이 항상 성립하므로 \\(1\\)을 나누어 내면 오히려 곱이 줄어들기 때문이다.</p> <p>아래 그림(15-15)에서 \\(n = 6\\)일 때 \\(3 \\times 3 &gt; 2 \\times 2 \\times 2\\)다. 따라서 \\(2\\)보다 \\(3\\)으로 나누는 편이 더 좋다.</p> <p>탐욕 전략 2: 분할에는 \\(2\\)가 많아도 두 개만 있어야 한다. \\(2\\) 세 개는 언제나 \\(3\\) 두 개로 바꾸어 더 큰 곱을 얻을 수 있기 때문이다.</p> <p></p> <p> 그림 15-15   최적인 분할 인수 </p> <p>위 내용을 종합하면 다음 탐욕 전략을 얻는다.</p> <ol> <li>정수 \\(n\\)에서 인수 \\(3\\)을 계속 나누어 내고, 나머지가 \\(0\\), \\(1\\), \\(2\\)가 되면 멈춘다.</li> <li>나머지가 \\(0\\)이면 \\(n\\)이 \\(3\\)의 배수이므로 추가로 처리하지 않는다.</li> <li>나머지가 \\(2\\)이면 더 나누지 않고 그대로 둔다.</li> <li>나머지가 \\(1\\)이면 \\(2 \\times 2 &gt; 1 \\times 3\\)이므로 마지막 \\(3\\)과 나머지 \\(1\\)을 \\(2\\) 두 개로 바꾼다.</li> </ol>","path":["제 15 장   탐욕","15.4   정수 분할의 최대 곱 문제"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/#2","level":3,"title":"2.   코드 구현","text":"<p>아래 그림(15-16)과 같이 반복문으로 정수를 나눌 필요 없이 내림 정수 나눗셈으로 \\(3\\)의 개수 \\(a\\)를, 나머지 연산으로 나머지 \\(b\\)를 구할 수 있다. 이때 다음이 성립한다.</p> \\[ n = 3 a + b \\] <p>경계 경우인 \\(n \\leq 3\\)에서는 반드시 \\(1\\)을 하나 나누어 내야 하므로 곱은 \\(1 \\times (n - 1)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby max_product_cutting.py<pre><code>def max_product_cutting(n: int) -&gt; int:\n    \"\"\"분할 후의 최대 곱: 탐욕\"\"\"\n    # n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if n &lt;= 3:\n        return 1 * (n - 1)\n    # 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    a, b = n // 3, n % 3\n    if b == 1:\n        # 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return int(math.pow(3, a - 1)) * 2 * 2\n    if b == 2:\n        # 나머지가 2이면 별도로 처리하지 않는다\n        return int(math.pow(3, a)) * 2\n    # 나머지가 0이면 별도로 처리하지 않는다\n    return int(math.pow(3, a))\n</code></pre> max_product_cutting.cpp<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nint maxProductCutting(int n) {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return (int)pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return (int)pow(3, a) * 2;\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return (int)pow(3, a);\n}\n</code></pre> max_product_cutting.java<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nint maxProductCutting(int n) {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return (int) Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return (int) Math.pow(3, a) * 2;\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return (int) Math.pow(3, a);\n}\n</code></pre> max_product_cutting.cs<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nint MaxProductCutting(int n) {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return (int)Math.Pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return (int)Math.Pow(3, a) * 2;\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return (int)Math.Pow(3, a);\n}\n</code></pre> max_product_cutting.go<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nfunc maxProductCutting(n int) int {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    a := n / 3\n    b := n % 3\n    if b == 1 {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return int(math.Pow(3, float64(a-1))) * 2 * 2\n    }\n    if b == 2 {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return int(math.Pow(3, float64(a))) * 2\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return int(math.Pow(3, float64(a)))\n}\n</code></pre> max_product_cutting.swift<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nfunc maxProductCutting(n: Int) -&gt; Int {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    let a = n / 3\n    let b = n % 3\n    if b == 1 {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return pow(3, a - 1) * 2 * 2\n    }\n    if b == 2 {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return pow(3, a) * 2\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return pow(3, a)\n}\n</code></pre> max_product_cutting.js<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nfunction maxProductCutting(n) {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    let a = Math.floor(n / 3);\n    let b = n % 3;\n    if (b === 1) {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return Math.pow(3, a) * 2;\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.ts<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nfunction maxProductCutting(n: number): number {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    let a: number = Math.floor(n / 3);\n    let b: number = n % 3;\n    if (b === 1) {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return Math.pow(3, a) * 2;\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.dart<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nint maxProductCutting(int n) {\n  // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n  if (n &lt;= 3) {\n    return 1 * (n - 1);\n  }\n  // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n  int a = n ~/ 3;\n  int b = n % 3;\n  if (b == 1) {\n    // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n    return (pow(3, a - 1) * 2 * 2).toInt();\n  }\n  if (b == 2) {\n    // 나머지가 2이면 별도로 처리하지 않는다\n    return (pow(3, a) * 2).toInt();\n  }\n  // 나머지가 0이면 별도로 처리하지 않는다\n  return pow(3, a).toInt();\n}\n</code></pre> max_product_cutting.rs<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nfn max_product_cutting(n: i32) -&gt; i32 {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if n &lt;= 3 {\n        return 1 * (n - 1);\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    let a = n / 3;\n    let b = n % 3;\n    if b == 1 {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        3_i32.pow(a as u32 - 1) * 2 * 2\n    } else if b == 2 {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        3_i32.pow(a as u32) * 2\n    } else {\n        // 나머지가 0이면 별도로 처리하지 않는다\n        3_i32.pow(a as u32)\n    }\n}\n</code></pre> max_product_cutting.c<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nint maxProductCutting(int n) {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return pow(3, a) * 2;\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return pow(3, a);\n}\n</code></pre> max_product_cutting.kt<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nfun maxProductCutting(n: Int): Int {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if (n &lt;= 3) {\n        return 1 * (n - 1)\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    val a = n / 3\n    val b = n % 3\n    if (b == 1) {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return 3.0.pow((a - 1)).toInt() * 2 * 2\n    }\n    if (b == 2) {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return 3.0.pow(a).toInt() * 2 * 2\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return 3.0.pow(a).toInt()\n}\n</code></pre> max_product_cutting.rb<pre><code>### 분할 후의 최대 곱: 탐욕 ###\ndef max_product_cutting(n)\n  # n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n  return 1 * (n - 1) if n &lt;= 3\n  # 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n  a, b = n / 3, n % 3\n  # 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n  return (3.pow(a - 1) * 2 * 2).to_i if b == 1\n  # 나머지가 2이면 별도로 처리하지 않는다\n  return (3.pow(a) * 2).to_i if b == 2\n  # 나머지가 0이면 별도로 처리하지 않는다\n  3.pow(a).to_i\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 15-16   정수 분할 최대 곱의 계산 방법 </p> <p>시간 복잡도는 프로그래밍 언어의 거듭제곱 구현 방식에 달려 있다. Python에서 자주 쓰는 거듭제곱 계산 방법은 세 가지다.</p> <ul> <li>연산자 <code>**</code>와 함수 <code>pow()</code>의 시간 복잡도는 모두 \\(O(\\log⁡ a)\\)이다.</li> <li>함수 <code>math.pow()</code>는 내부적으로 C 라이브러리의 <code>pow()</code>를 호출하여 부동 소수점 거듭제곱을 계산하며, 시간 복잡도는 \\(O(1)\\)이다.</li> </ul> <p>변수 \\(a\\)와 \\(b\\)는 상수 크기의 추가 공간을 사용하므로 공간 복잡도는 \\(O(1)\\)이다.</p>","path":["제 15 장   탐욕","15.4   정수 분할의 최대 곱 문제"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/#3","level":3,"title":"3.   정확성 증명","text":"<p>귀류법을 사용하며 \\(n \\geq 4\\)인 경우만 분석한다.</p> <ol> <li>모든 인수는 \\(\\leq 3\\)이다: 최적 분할에 \\(\\geq 4\\)인 인수 \\(x\\)가 있다고 가정하자. 이를 \\(2(x-2)\\)로 더 나누면 곱이 커지거나 같아진다. 이는 가정과 모순이다.</li> <li>분할에 \\(1\\)이 없다: 최적 분할에 인수 \\(1\\)이 있다고 가정하자. 이를 다른 인수에 합치면 더 큰 곱을 얻으므로 가정과 모순이다.</li> <li>분할에 \\(2\\)는 최대 두 개다: 최적 분할에 \\(2\\)가 세 개 있다고 가정하자. \\(3\\) 두 개로 바꾸면 곱이 더 커지므로 가정과 모순이다.</li> </ol>","path":["제 15 장   탐욕","15.4   정수 분할의 최대 곱 문제"],"tags":[]},{"location":"chapter_greedy/summary/","level":1,"title":"15.5   요약","text":"","path":["제 15 장   탐욕","15.5   요약"],"tags":[]},{"location":"chapter_greedy/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>탐욕 알고리즘은 주로 최적화 문제에 사용하며, 매 결정 단계에서 국소 최적 선택을 내려 전역 최적해를 얻으려는 원리다.</li> <li>탐욕 알고리즘은 선택을 하나씩 반복하며 매번 문제를 더 작은 부분 문제로 바꾸고, 해결될 때까지 진행한다.</li> <li>탐욕 알고리즘은 구현이 간단하고 풀이 효율이 높다. 시간 복잡도는 보통 동적 계획법보다 낮다.</li> <li>동전 교환에서는 일부 동전 조합에 대해 최적해를 보장하지만, 다른 조합에서는 매우 나쁜 해를 찾을 수도 있다.</li> <li>탐욕 알고리즘에 적합한 문제는 탐욕적 선택 속성과 최적 부분 구조를 가진다. 탐욕적 선택 속성은 전략의 유효성을 나타낸다.</li> <li>복잡한 문제에서는 탐욕적 선택 속성의 증명이 쉽지 않다. 동전 교환처럼 반례로 부정하는 편이 상대적으로 쉬운 경우가 있다.</li> <li>탐욕 문제의 풀이는 문제 분석, 탐욕 전략 결정, 정확성 증명의 세 단계로 나뉜다. 전략 결정이 핵심이며, 정확성 증명이 어려운 경우가 많다.</li> <li>분할 가능한 배낭은 0-1 배낭에서 물건의 일부를 고를 수 있도록 바꾼 문제이므로 탐욕 알고리즘으로 풀 수 있다. 전략의 정확성은 귀류법으로 증명할 수 있다.</li> <li>최대 용량 문제는 모든 경우를 열거하면 시간 복잡도가 \\(O(n^2)\\)이다. 매번 낮은 칸막이를 안쪽으로 옮기는 탐욕 전략으로 \\(O(n)\\)까지 최적화할 수 있다.</li> <li>정수 분할의 최대 곱 문제에서는 \\(\\geq 4\\)인 정수를 계속 나누어야 하고 최적인 인수는 \\(3\\)이라는 두 탐욕 전략을 차례로 도출했다. 코드에는 거듭제곱이 포함되므로 시간 복잡도는 구현 방식에 따라 보통 \\(O(1)\\) 또는 \\(O(\\log n)\\)이다.</li> </ul>","path":["제 15 장   탐욕","15.5   요약"],"tags":[]},{"location":"chapter_hashing/","level":1,"title":"제 6 장   해시 테이블","text":"<p>이 장을 시작하며</p> <p>컴퓨터 세계에서 해시 테이블은 영리한 사서와 같다.</p> <p>청구 기호를 계산하는 방법을 알기에 원하는 책을 빠르게 찾아낸다.</p>","path":["제 6 장   해시 테이블"],"tags":[]},{"location":"chapter_hashing/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>6.1   해시 테이블</li> <li>6.2   해시 충돌</li> <li>6.3   해시 알고리즘</li> <li>6.4   요약</li> <li>6.5   연습 문제</li> </ul>","path":["제 6 장   해시 테이블"],"tags":[]},{"location":"chapter_hashing/exercises/","level":1,"title":"6.5   연습 문제","text":"","path":["제 6 장   해시 테이블","6.5   연습 문제"],"tags":[]},{"location":"chapter_hashing/exercises/#651","level":2,"title":"6.5.1   개념 확인","text":"","path":["제 6 장   해시 테이블","6.5   연습 문제"],"tags":[]},{"location":"chapter_hashing/exercises/#1","level":3,"title":"1.   해시 충돌이 일어난 뒤 탐색하는 방법","text":"<p>버킷이 5개인 해시 테이블의 해시 함수는 \\(h(x)=x \\bmod 5\\)다. 충돌이 나면 해당 버킷의 리스트에 원소를 순서대로 넣는다. <code>[1, 6, 11, 7]</code>을 차례대로 삽입한다.</p> <ol> <li>0~4번 버킷의 내용을 적어 보자.</li> <li>6을 찾을 때 어느 버킷으로 들어가고, 어떤 원소를 차례대로 확인하는가?</li> <li>1번에서 적은 버킷의 내용을 보면 나중에 삽입한 원소가 먼저 삽입한 원소를 덮어썼는가? 이 충돌 처리 방식과 연결하여 이유를 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>\\(1\\bmod5=6\\bmod5=11\\bmod5=1\\)이고 \\(7\\bmod5=2\\)이므로 각 버킷의 내용은 다음과 같다.</p> <pre><code>0: []\n1: [1, 6, 11]\n2: [7]\n3: []\n4: []\n</code></pre> </li> <li> <p>6을 찾을 때는 먼저 1번 버킷으로 들어가 1, 6을 차례대로 비교한다. 두 번째 비교에서 목표를 찾는다.</p> </li> <li> <p>해시값이 같다는 것은 같은 버킷에 들어간다는 뜻일 뿐, 원소들이 같다는 뜻은 아니다. 체이닝은 충돌한 원소를 모두 버킷에 저장하고,     탐색할 때 하나씩 비교한다. 따라서 1, 6, 11은 서로를 덮어쓰지 않는다.</p> </li> </ol>","path":["제 6 장   해시 테이블","6.5   연습 문제"],"tags":[]},{"location":"chapter_hashing/exercises/#2","level":3,"title":"2.   해시 테이블의 용량을 늘리면 원소는 어디로 가는가","text":"<p>체이닝을 사용하는 해시 테이블에 원래 5개의 버킷이 있고, 해시 함수는 \\(h(x)=x\\bmod5\\)다. 키 <code>[1, 6, 11]</code>은 모두 1번 버킷에 있다.</p> <p>이제 버킷 수를 7개로 늘리고 해시 함수도 \\(h(x)=x\\bmod7\\)로 바꾼다.</p> <ol> <li>1, 6, 11의 새 버킷 번호를 각각 계산하라.</li> <li>용량을 늘린 뒤 어느 버킷에 원소가 들어 있는가?</li> <li>기존 1번 버킷의 리스트를 새 1번 버킷에 그대로 복사해도 되는가? 1번과 2번의 결과를 바탕으로 이유를 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>새 버킷 번호는 각각 다음과 같다.</p> <ul> <li>\\(1\\bmod7=1\\)</li> <li>\\(6\\bmod7=6\\)</li> <li>\\(11\\bmod7=4\\)</li> </ul> </li> <li> <p>1번 버킷에 1, 4번 버킷에 11, 6번 버킷에 6이 저장된다. 세 키가 더 이상 하나의 버킷에 모여 있지 않다.</p> </li> <li> <p>그대로 복사해서는 안 된다. 버킷 번호는 키를 버킷 수로 나눈 나머지로 구한다. 버킷 수가 5에서 7로 바뀌면 같은 키의 버킷 번호도 달라질 수 있으므로,     모든 키의 위치를 다시 계산해야 한다. 이전 1번 버킷을 그대로 복사하면 새 식으로 6과 11을 찾을 때     각각 6번 버킷과 4번 버킷으로 가게 되어 찾지 못한다.</p> </li> </ol>","path":["제 6 장   해시 테이블","6.5   연습 문제"],"tags":[]},{"location":"chapter_hashing/exercises/#3-6-11","level":3,"title":"3.   6을 삭제한 뒤에도 11을 찾을 수 있는가","text":"<p>해시 테이블에 5개의 자리가 있고 인덱스는 <code>0～4</code>이며, 해시 함수는 \\(h(x)=x\\bmod5\\)다. 충돌이 나면 해시 함수로 계산한 인덱스부터 오른쪽으로 이동하며 처음 만나는 빈자리를 찾는다.</p> <p><code>[1, 6, 11]</code>을 차례대로 삽입한다.</p> <ol> <li>세 수는 최종적으로 각각 어느 인덱스에 저장되는가?</li> <li>11을 찾을 때 어떤 인덱스를 차례대로 확인하는가?</li> <li>6을 삭제하면서 그 자리를 ‘한 번도 사용하지 않은 빈자리’로 바꾸고, 탐색은 빈자리를 만나면 멈춘다고 하자.     다시 11을 찾으면 어떤 일이 일어나는가? 그 결과는 올바른가? 문제가 있다면 어떻게 피할 수 있는가?</li> </ol> 참고 답안 <ol> <li> <p>1은 인덱스 1에 저장된다. 6도 인덱스 1로 매핑되지만 충돌하므로 인덱스 2에 저장된다.     11 역시 인덱스 1부터 시작해 이미 사용 중인 인덱스 1과 2를 지나 인덱스 3에 저장된다.</p> </li> <li> <p>11을 찾을 때는 인덱스 <code>1, 2, 3</code>을 차례대로 확인하고 인덱스 3에서 찾는다.</p> </li> <li> <p>인덱스 2를 ‘한 번도 사용하지 않음’을 뜻하는 빈자리로 바꾸면, 11을 찾다가 인덱스 1을 확인한 뒤 인덱스 2에서 멈춘다.     따라서 11이 없다고 잘못 판단한다. 삭제할 때는 ‘삭제됨’ 표시를 남겨야 한다.     탐색 중 이 표시를 만나면 다음 인덱스를 계속 확인하고, 인덱스 4를 지나면 0으로 돌아간다. 이후 삽입할 때는 이 자리를 다시 사용할 수 있다.</p> </li> </ol>","path":["제 6 장   해시 테이블","6.5   연습 문제"],"tags":[]},{"location":"chapter_hashing/exercises/#652","level":2,"title":"6.5.2   프로그래밍 연습","text":"","path":["제 6 장   해시 테이블","6.5   연습 문제"],"tags":[]},{"location":"chapter_hashing/exercises/#1_1","level":3,"title":"1.   두 문자열의 문자 구성 비교하기","text":"<p>영어 소문자만 포함하는 두 문자열 <code>s</code>와 <code>t</code>가 주어진다. <code>s</code>의 문자 위치는 자유롭게 바꿀 수 있지만 문자를 추가하거나 삭제하거나 다른 문자로 바꿀 수는 없다.</p> <p>순서를 바꿔 <code>t</code>를 만들 수 있는지 판단하고, 가능하면 <code>true</code>, 그렇지 않으면 <code>false</code>를 반환하라. 문자열의 문자를 정렬하지 말고 해시 테이블로 각 문자의 등장 횟수를 기록하라.</p> 풀이 힌트 <ol> <li>두 문자열의 길이가 다르면 문자 구성도 반드시 다르다.</li> <li>해시 테이블에 각 문자의 개수를 기록한다. s를 순회할 때 해당 개수를 1씩 늘린다.</li> <li>t를 순회할 때 해당 개수를 1씩 줄인다. 마지막에 모든 개수가 0이어야 문자 구성이 같다.</li> </ol> <p>LeetCode</p>","path":["제 6 장   해시 테이블","6.5   연습 문제"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/","level":1,"title":"6.3   해시 알고리즘","text":"<p>앞의 두 절에서는 해시 테이블의 작동 원리와 해시 충돌 처리 방법을 알아보았다. 하지만 개방 주소법이든 체이닝이든 충돌이 발생해도 해시 테이블이 정상적으로 작동하게 할 뿐, 충돌 발생 자체를 줄이지는 못한다.</p> <p>충돌이 너무 자주 발생하면 성능이 급격히 나빠진다. 아래 그림(6-8)의 체이닝 해시 테이블에서 이상적인 경우는 키-값 쌍이 각 버킷에 고르게 분포하여 최고의 조회 효율을 얻는 것이다. 최악의 경우에는 모든 키-값 쌍이 같은 버킷에 들어가 시간 복잡도가 \\(O(n)\\)으로 나빠진다.</p> <p></p> <p> 그림 6-8   해시 충돌의 최선과 최악의 경우 </p> <p>키-값 쌍의 분포는 해시 함수가 결정한다. 해시 함수는 먼저 해시값을 계산하고 배열 길이로 나눈 나머지를 구한다.</p> <pre><code>index = hash(key) % capacity\n</code></pre> <p>위 식에서 해시 테이블 용량 <code>capacity</code>가 고정되면 해시 알고리즘 <code>hash()</code>가 출력값을 결정하며, 나아가 키-값 쌍의 분포를 결정한다.</p> <p>따라서 해시 충돌 확률을 낮추려면 해시 알고리즘 <code>hash()</code>의 설계에 집중해야 한다.</p>","path":["제 6 장   해시 테이블","6.3   해시 알고리즘"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#631","level":2,"title":"6.3.1   해시 알고리즘의 목표","text":"<p>‘빠르고 안정적인’ 해시 테이블을 구현하려면 해시 알고리즘이 다음 특징을 갖추어야 한다.</p> <ul> <li>결정성: 같은 입력에는 항상 같은 출력을 생성해야 한다. 그래야 해시 테이블을 신뢰할 수 있다.</li> <li>높은 효율: 해시값 계산이 충분히 빨라야 한다. 계산 비용이 작을수록 해시 테이블의 실용성이 높아진다.</li> <li>균등 분포: 키-값 쌍이 해시 테이블에 고르게 분포하도록 해야 한다. 분포가 고를수록 충돌 확률은 낮아진다.</li> </ul> <p>해시 알고리즘은 해시 테이블 외에도 여러 분야에서 널리 사용된다.</p> <ul> <li>비밀번호 저장: 사용자의 비밀번호를 보호하기 위해 시스템은 보통 평문 대신 해시값을 저장한다. 사용자가 비밀번호를 입력하면 해시값을 계산해 저장된 값과 비교하고, 두 값이 같으면 올바른 비밀번호로 판단한다.</li> <li>데이터 무결성 검사: 송신자는 데이터의 해시값을 계산해 함께 보낼 수 있다. 수신자는 받은 데이터의 해시값을 다시 계산하여 전달받은 해시값과 비교한다. 둘이 같으면 데이터가 온전하다고 판단한다.</li> </ul> <p>암호학 관련 응용에서는 해시값으로 원래 비밀번호를 추론하는 등의 역공학을 막기 위해 더 높은 수준의 보안 특성이 필요하다.</p> <ul> <li>일방향성: 해시값으로 입력 데이터에 관한 정보를 역으로 추론할 수 없어야 한다.</li> <li>충돌 저항성: 해시값이 같은 서로 다른 두 입력을 찾기가 극히 어려워야 한다.</li> <li>눈사태 효과: 입력의 작은 변화가 출력의 크고 예측할 수 없는 변화로 이어져야 한다.</li> </ul> <p>‘균등 분포’와 ‘충돌 저항성’은 서로 독립된 개념이다. 균등 분포를 만족한다고 충돌 저항성도 만족하는 것은 아니다. 예를 들어 <code>key</code>가 무작위로 입력되면 해시 함수 <code>key % 100</code>은 고르게 분포하는 출력을 만들 수 있다. 하지만 이 알고리즘은 너무 단순하여 마지막 두 자리가 같은 모든 <code>key</code>의 출력이 같다. 따라서 해시값에서 유효한 <code>key</code>를 쉽게 역추론하여 비밀번호를 깰 수 있다.</p>","path":["제 6 장   해시 테이블","6.3   해시 알고리즘"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#632","level":2,"title":"6.3.2   해시 알고리즘의 설계","text":"<p>해시 알고리즘 설계는 여러 요소를 고려해야 하는 복잡한 문제다. 다만 요구 수준이 높지 않은 상황에서는 간단한 알고리즘을 설계할 수 있다.</p> <ul> <li>덧셈 해싱: 입력의 각 문자에 해당하는 ASCII 코드를 모두 더한 값을 해시값으로 삼는다.</li> <li>곱셈 해싱: 곱셈의 비상관성을 이용하여 매번 상수를 곱하고, 각 문자의 ASCII 코드를 해시값에 누적한다.</li> <li>XOR 해싱: 입력 데이터의 각 원소를 XOR 연산으로 하나의 해시값에 누적한다.</li> <li>회전 해싱: 각 문자의 ASCII 코드를 해시값에 누적하되, 누적하기 전에 해시값에 회전 연산을 수행한다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby simple_hash.py<pre><code>def add_hash(key: str) -&gt; int:\n    \"\"\"덧셈 해시\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash += ord(c)\n    return hash % modulus\n\ndef mul_hash(key: str) -&gt; int:\n    \"\"\"곱셈 해시\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = 31 * hash + ord(c)\n    return hash % modulus\n\ndef xor_hash(key: str) -&gt; int:\n    \"\"\"XOR 해시\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash ^= ord(c)\n    return hash % modulus\n\ndef rot_hash(key: str) -&gt; int:\n    \"\"\"회전 해시\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ ord(c)\n    return hash % modulus\n</code></pre> simple_hash.cpp<pre><code>/* 덧셈 해시 */\nint addHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* 곱셈 해시 */\nint mulHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (31 * hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* XOR 해시 */\nint xorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash ^= (int)c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* 회전 해시 */\nint rotHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.java<pre><code>/* 덧셈 해시 */\nint addHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* 곱셈 해시 */\nint mulHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (31 * hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* XOR 해시 */\nint xorHash(String key) {\n    int hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash ^= (int) c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* 회전 해시 */\nint rotHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n</code></pre> simple_hash.cs<pre><code>/* 덧셈 해시 */\nint AddHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* 곱셈 해시 */\nint MulHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (31 * hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* XOR 해시 */\nint XorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash ^= c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* 회전 해시 */\nint RotHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.go<pre><code>/* 덧셈 해시 */\nfunc addHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* 곱셈 해시 */\nfunc mulHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (31*hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* XOR 해시 */\nfunc xorHash(key string) int {\n    hash := 0\n    modulus := 1000000007\n    for _, b := range []byte(key) {\n        fmt.Println(int(b))\n        hash ^= int(b)\n        hash = (31*hash + int(b)) % modulus\n    }\n    return hash &amp; modulus\n}\n\n/* 회전 해시 */\nfunc rotHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ int64(b)) % modulus\n    }\n    return int(hash)\n}\n</code></pre> simple_hash.swift<pre><code>/* 덧셈 해시 */\nfunc addHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* 곱셈 해시 */\nfunc mulHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (31 * hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* XOR 해시 */\nfunc xorHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash ^= Int(scalar.value)\n        }\n    }\n    return hash &amp; MODULUS\n}\n\n/* 회전 해시 */\nfunc rotHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n</code></pre> simple_hash.js<pre><code>/* 덧셈 해시 */\nfunction addHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* 곱셈 해시 */\nfunction mulHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* XOR 해시 */\nfunction xorHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash % MODULUS;\n}\n\n/* 회전 해시 */\nfunction rotHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.ts<pre><code>/* 덧셈 해시 */\nfunction addHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* 곱셈 해시 */\nfunction mulHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* XOR 해시 */\nfunction xorHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash % MODULUS;\n}\n\n/* 회전 해시 */\nfunction rotHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.dart<pre><code>/* 덧셈 해시 */\nint addHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* 곱셈 해시 */\nint mulHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (31 * hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* XOR 해시 */\nint xorHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash ^= key.codeUnitAt(i);\n  }\n  return hash &amp; MODULUS;\n}\n\n/* 회전 해시 */\nint rotHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n</code></pre> simple_hash.rs<pre><code>/* 덧셈 해시 */\nfn add_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* 곱셈 해시 */\nfn mul_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (31 * hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* XOR 해시 */\nfn xor_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash ^= c as i64;\n    }\n\n    (hash &amp; MODULUS) as i32\n}\n\n/* 회전 해시 */\nfn rot_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n</code></pre> simple_hash.c<pre><code>/* 덧셈 해시 */\nint addHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* 곱셈 해시 */\nint mulHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (31 * hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* XOR 해시 */\nint xorHash(char *key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash ^= (unsigned char)key[i];\n    }\n    return hash &amp; MODULUS;\n}\n\n/* 회전 해시 */\nint rotHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (unsigned char)key[i]) % MODULUS;\n    }\n\n    return (int)hash;\n}\n</code></pre> simple_hash.kt<pre><code>/* 덧셈 해시 */\nfun addHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = (hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* 곱셈 해시 */\nfun mulHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = (31 * hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* XOR 해시 */\nfun xorHash(key: String): Int {\n    var hash = 0\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = hash xor c.code\n    }\n    return hash and MODULUS\n}\n\n/* 회전 해시 */\nfun rotHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = ((hash shl 4) xor (hash shr 28) xor c.code.toLong()) % MODULUS\n    }\n    return hash.toInt()\n}\n</code></pre> simple_hash.rb<pre><code>### 덧셈 해시 ###\ndef add_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash += c.ord }\n\n  hash % modulus\nend\n\n### 곱셈 해시 ###\ndef mul_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash = 31 * hash + c.ord }\n\n  hash % modulus\nend\n\n### XOR 해시 ###\ndef xor_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash ^= c.ord }\n\n  hash % modulus\nend\n\n### 회전 해시 ###\ndef rot_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.ord }\n\n  hash % modulus\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>각 알고리즘의 마지막 단계에서는 큰 소수 \\(1000000007\\)로 나눈 나머지를 구하여 해시값이 적절한 범위에 있도록 한다. 왜 소수로 나누어야 할까? 합성수로 나누면 어떤 단점이 있을까? 생각해 볼 만한 흥미로운 문제다.</p> <p>결론부터 말하면 큰 소수를 모듈러 연산의 법으로 사용하면 해시값의 균등 분포를 최대한 보장할 수 있다. 소수는 다른 수와 공약수를 갖지 않으므로 나머지 연산에서 생기는 주기적 패턴을 줄여 충돌을 피할 수 있기 때문이다.</p> <p>예를 들어 합성수 \\(9\\)를 법으로 선택했다고 하자. 이 수는 \\(3\\)으로 나누어떨어지므로, \\(3\\)으로 나누어떨어지는 모든 <code>key</code>는 \\(0\\), \\(3\\), \\(6\\)의 세 해시값에 매핑된다.</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 9 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\dots \\} \\end{aligned} \\] <p>입력 <code>key</code>가 마침 이런 등차수열 분포를 따르면 해시값에 군집이 생겨 충돌이 심해진다. 이제 <code>modulus</code>를 소수 \\(13\\)으로 바꾸면 <code>key</code>와 <code>modulus</code> 사이에 공약수가 없으므로 출력 해시값의 균등성이 뚜렷하게 좋아진다.</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 13 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\dots \\} \\end{aligned} \\] <p><code>key</code>가 무작위로 균등하게 분포함을 보장할 수 있다면 소수와 합성수 중 어느 것을 법으로 써도 균등한 해시값을 얻을 수 있다. 반면 <code>key</code>의 분포에 주기성이 있으면 합성수로 나눌 때 군집이 생기기 쉽다.</p> <p>따라서 보통 소수를 법으로 선택하며, 가능한 한 큰 소수를 사용해 주기적 패턴을 줄이고 해시 알고리즘의 안정성을 높인다.</p>","path":["제 6 장   해시 테이블","6.3   해시 알고리즘"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#633","level":2,"title":"6.3.3   대표적인 해시 알고리즘","text":"<p>앞에서 소개한 간단한 해시 알고리즘은 비교적 ‘취약하여’ 설계 목표에 크게 못 미친다. 예를 들어 덧셈과 XOR는 교환법칙을 만족하므로, 이 두 해싱 방식은 내용은 같고 순서만 다른 문자열을 구분하지 못한다. 이는 해시 충돌을 늘리고 보안 문제를 일으킬 수 있다.</p> <p>실제로는 MD5, SHA-1, SHA-2, SHA-3 같은 표준 해시 알고리즘을 주로 사용한다. 이들은 길이가 임의인 입력 데이터를 고정 길이의 해시값으로 매핑한다.</p> <p>지난 한 세기 가까이 해시 알고리즘은 계속 개선되고 최적화되어 왔다. 일부 연구자는 성능을 높이는 데 힘쓰고, 다른 연구자와 해커는 보안 문제를 찾는 데 집중한다. 아래 표(6-2)는 실제 응용에서 흔히 쓰이는 해시 알고리즘을 보여 준다.</p> <ul> <li>MD5와 SHA-1은 여러 차례 공격에 성공한 사례가 있어 각종 보안 응용에서 사용이 중단되었다.</li> <li>SHA-2 계열의 SHA-256은 가장 안전한 해시 알고리즘 중 하나로, 아직 성공한 공격 사례가 없어 각종 보안 응용과 프로토콜에서 자주 사용된다.</li> <li>SHA-3는 SHA-2보다 구현 비용이 낮고 계산 효율이 높지만, 현재 보급 범위는 SHA-2 계열에 못 미친다.</li> </ul> <p> 표 6-2   대표적인 해시 알고리즘 </p> MD5 SHA-1 SHA-2 SHA-3 발표 시기 1992 1995 2002 2008 출력 길이 128 bit 160 bit 256/512 bit 224/256/384/512 bit 해시 충돌 비교적 많음 비교적 많음 매우 적음 매우 적음 보안 수준 낮음, 공격 성공 사례가 있음 낮음, 공격 성공 사례가 있음 높음 높음 활용 사용 중단, 데이터 무결성 검사에는 여전히 사용 사용 중단 암호화폐 거래 검증, 전자 서명 등 SHA-2를 대체할 수 있음","path":["제 6 장   해시 테이블","6.3   해시 알고리즘"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#634","level":2,"title":"6.3.4   자료 구조의 해시값","text":"<p>해시 테이블의 <code>key</code>는 정수, 소수, 문자열 등의 자료형일 수 있다. 프로그래밍 언어는 보통 이러한 자료형에 내장 해시 알고리즘을 제공하여 버킷 인덱스를 계산한다. 예를 들어 Python에서는 <code>hash()</code> 함수로 여러 자료형의 해시값을 구할 수 있다.</p> <ul> <li>정수와 불리언의 해시값은 그 값 자체다.</li> <li>부동 소수점 수와 문자열의 해시값 계산은 더 복잡하다. 관심 있는 독자는 따로 알아보자.</li> <li>튜플은 각 원소의 해시값을 계산한 뒤 이를 결합하여 하나의 해시값을 얻는다.</li> <li>객체의 해시값은 메모리 주소를 기반으로 생성된다. 객체의 해시 메서드를 재정의하면 내용을 기반으로 해시값을 만들 수도 있다.</li> </ul> <p>참고</p> <p>프로그래밍 언어마다 내장 해시값 계산 함수의 정의와 방식이 다르다는 점에 주의하자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby built_in_hash.py<pre><code>num = 3\nhash_num = hash(num)\n# 정수 3의 해시값은 3\n\nbol = True\nhash_bol = hash(bol)\n# 불리언 True의 해시값은 1\n\ndec = 3.14159\nhash_dec = hash(dec)\n# 소수 3.14159의 해시값은 326484311674566659\n\nstr = \"Hello 算法\"\nhash_str = hash(str)\n# 문자열 “Hello 算法”의 해시값은 4617003410720528961\n\ntup = (12836, \"小哈\")\nhash_tup = hash(tup)\n# 튜플 (12836, '小哈')의 해시값은 1029005403108185979\n\nobj = ListNode(0)\nhash_obj = hash(obj)\n# 노드 객체 &lt;ListNode object at 0x1058fd810&gt;의 해시값은 274267521\n</code></pre> built_in_hash.cpp<pre><code>int num = 3;\nsize_t hashNum = hash&lt;int&gt;()(num);\n// 정수 3의 해시값은 3\n\nbool bol = true;\nsize_t hashBol = hash&lt;bool&gt;()(bol);\n// 불리언 1의 해시값은 1\n\ndouble dec = 3.14159;\nsize_t hashDec = hash&lt;double&gt;()(dec);\n// 소수 3.14159의 해시값은 4614256650576692846\n\nstring str = \"Hello 算法\";\nsize_t hashStr = hash&lt;string&gt;()(str);\n// 문자열 “Hello 算法”의 해시값은 15466937326284535026\n\n// C++의 내장 std::hash()는 기본 자료형의 해시값 계산만 제공한다\n// 배열과 객체의 해시값 계산은 직접 구현해야 한다\n</code></pre> built_in_hash.java<pre><code>int num = 3;\nint hashNum = Integer.hashCode(num);\n// 정수 3의 해시값은 3\n\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// 불리언 true의 해시값은 1231\n\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// 소수 3.14159의 해시값은 -1340954729\n\nString str = \"Hello 算法\";\nint hashStr = str.hashCode();\n// 문자열 “Hello 算法”의 해시값은 -727081396\n\nObject[] arr = { 12836, \"小哈\" };\nint hashTup = Arrays.hashCode(arr);\n// 배열 [12836, 小哈]의 해시값은 1151158\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// 노드 객체 utils.ListNode@7dc5e7b4의 해시값은 2110121908\n</code></pre> built_in_hash.cs<pre><code>int num = 3;\nint hashNum = num.GetHashCode();\n// 정수 3의 해시값은 3;\n\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// 불리언 true의 해시값은 1;\n\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// 소수 3.14159의 해시값은 -1340954729;\n\nstring str = \"Hello 算法\";\nint hashStr = str.GetHashCode();\n// 문자열 “Hello 算法”의 해시값은 -586107568;\n\nobject[] arr = [12836, \"小哈\"];\nint hashTup = arr.GetHashCode();\n// 배열 [12836, 小哈]의 해시값은 42931033;\n\nListNode obj = new(0);\nint hashObj = obj.GetHashCode();\n// 노드 객체 0의 해시값은 39053774;\n</code></pre> built_in_hash.go<pre><code>// Go는 내장 hash code 함수를 제공하지 않는다\n</code></pre> built_in_hash.swift<pre><code>let num = 3\nlet hashNum = num.hashValue\n// 정수 3의 해시값은 9047044699613009734\n\nlet bol = true\nlet hashBol = bol.hashValue\n// 불리언 true의 해시값은 -4431640247352757451\n\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// 소수 3.14159의 해시값은 -2465384235396674631\n\nlet str = \"Hello 算法\"\nlet hashStr = str.hashValue\n// 문자열 “Hello 算法”의 해시값은 -7850626797806988787\n\nlet arr = [AnyHashable(12836), AnyHashable(\"小哈\")]\nlet hashTup = arr.hashValue\n// 배열 [AnyHashable(12836), AnyHashable(\"小哈\")]의 해시값은 -2308633508154532996\n\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// 노드 객체 utils.ListNode의 해시값은 -2434780518035996159\n</code></pre> built_in_hash.js<pre><code>// JavaScript는 내장 hash code 함수를 제공하지 않는다\n</code></pre> built_in_hash.ts<pre><code>// TypeScript는 내장 hash code 함수를 제공하지 않는다\n</code></pre> built_in_hash.dart<pre><code>int num = 3;\nint hashNum = num.hashCode;\n// 정수 3의 해시값은 34803\n\nbool bol = true;\nint hashBol = bol.hashCode;\n// 불리언 true의 해시값은 1231\n\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// 소수 3.14159의 해시값은 2570631074981783\n\nString str = \"Hello 算法\";\nint hashStr = str.hashCode;\n// 문자열 “Hello 算法”의 해시값은 468167534\n\nList arr = [12836, \"小哈\"];\nint hashArr = arr.hashCode;\n// 배열 [12836, 小哈]의 해시값은 976512528\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// 노드 객체 Instance of 'ListNode'의 해시값은 1033450432\n</code></pre> built_in_hash.rs<pre><code>use std::collections::hash_map::DefaultHasher;\nuse std::hash::{Hash, Hasher};\n\nlet num = 3;\nlet mut num_hasher = DefaultHasher::new();\nnum.hash(&amp;mut num_hasher);\nlet hash_num = num_hasher.finish();\n// 정수 3의 해시값은 568126464209439262\n\nlet bol = true;\nlet mut bol_hasher = DefaultHasher::new();\nbol.hash(&amp;mut bol_hasher);\nlet hash_bol = bol_hasher.finish();\n// 불리언 true의 해시값은 4952851536318644461\n\nlet dec: f32 = 3.14159;\nlet mut dec_hasher = DefaultHasher::new();\ndec.to_bits().hash(&amp;mut dec_hasher);\nlet hash_dec = dec_hasher.finish();\n// 소수 3.14159의 해시값은 2566941990314602357\n\nlet str = \"Hello 算法\";\nlet mut str_hasher = DefaultHasher::new();\nstr.hash(&amp;mut str_hasher);\nlet hash_str = str_hasher.finish();\n// 문자열 “Hello 算法”의 해시값은 16092673739211250988\n\nlet arr = (&amp;12836, &amp;\"小哈\");\nlet mut tup_hasher = DefaultHasher::new();\narr.hash(&amp;mut tup_hasher);\nlet hash_tup = tup_hasher.finish();\n// 튜플 (12836, \"小哈\")의 해시값은 1885128010422702749\n\nlet node = ListNode::new(42);\nlet mut hasher = DefaultHasher::new();\nnode.borrow().val.hash(&amp;mut hasher);\nlet hash = hasher.finish();\n// 노드 객체 RefCell { value: ListNode { val: 42, next: None } }의 해시값은 15387811073369036852\n</code></pre> built_in_hash.c<pre><code>// C는 내장 hash code 함수를 제공하지 않는다\n</code></pre> built_in_hash.kt<pre><code>val num = 3\nval hashNum = num.hashCode()\n// 정수 3의 해시값은 3\n\nval bol = true\nval hashBol = bol.hashCode()\n// 불리언 true의 해시값은 1231\n\nval dec = 3.14159\nval hashDec = dec.hashCode()\n// 소수 3.14159의 해시값은 -1340954729\n\nval str = \"Hello 算法\"\nval hashStr = str.hashCode()\n// 문자열 “Hello 算法”의 해시값은 -727081396\n\nval arr = arrayOf&lt;Any&gt;(12836, \"小哈\")\nval hashTup = arr.hashCode()\n// 배열 [12836, 小哈]의 해시값은 189568618\n\nval obj = ListNode(0)\nval hashObj = obj.hashCode()\n// 노드 객체 utils.ListNode@1d81eb93의 해시값은 495053715\n</code></pre> built_in_hash.rb<pre><code>num = 3\nhash_num = num.hash\n# 정수 3의 해시값은 -4385856518450339636\n\nbol = true\nhash_bol = bol.hash\n# 불리언 true의 해시값은 -1617938112149317027\n\ndec = 3.14159\nhash_dec = dec.hash\n# 소수 3.14159의 해시값은 -1479186995943067893\n\nstr = \"Hello 算法\"\nhash_str = str.hash\n# 문자열 “Hello 算法”의 해시값은 -4075943250025831763\n\ntup = [12836, '小哈']\nhash_tup = tup.hash\n# 튜플 (12836, '小哈')의 해시값은 1999544809202288822\n\nobj = ListNode.new(0)\nhash_obj = obj.hash\n# 노드 객체 #&lt;ListNode:0x000078133140ab70&gt;의 해시값은 4302940560806366381\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>많은 프로그래밍 언어에서는 불변 객체만 해시 테이블의 <code>key</code>로 사용할 수 있다. 리스트(동적 배열)를 <code>key</code>로 사용하면 내용이 바뀔 때 해시값도 달라져, 기존의 <code>value</code>를 조회할 수 없게 된다.</p> <p>연결 리스트 노드 같은 사용자 정의 객체는 멤버 변수가 바뀔 수 있어도 해시 가능하다. 객체의 해시값은 보통 메모리 주소를 기반으로 생성되기 때문이다. 내용이 바뀌어도 메모리 주소가 그대로면 해시값도 바뀌지 않는다.</p> <p>서로 다른 콘솔에서 실행하면 출력되는 해시값이 다르다는 점을 발견했을 것이다. Python 인터프리터가 시작될 때마다 문자열 해시 함수에 무작위 솔트(salt) 값을 추가하기 때문이다. 이 방식은 HashDoS 공격을 효과적으로 방지하여 해시 알고리즘의 보안성을 높인다.</p>","path":["제 6 장   해시 테이블","6.3   해시 알고리즘"],"tags":[]},{"location":"chapter_hashing/hash_collision/","level":1,"title":"6.2   해시 충돌","text":"<p>앞 절에서 설명했듯이 해시 함수의 입력 공간은 보통 출력 공간보다 훨씬 크므로, 이론적으로 해시 충돌은 피할 수 없다. 예를 들어 입력 공간이 모든 정수이고 출력 공간이 배열 용량만큼이라면, 여러 정수가 같은 버킷 인덱스에 매핑될 수밖에 없다.</p> <p>해시 충돌은 잘못된 조회 결과를 일으켜 해시 테이블의 사용 가능성을 크게 해친다. 이를 해결하려면 충돌이 생길 때마다 해시 테이블을 확장하여 충돌이 없어질 때까지 반복할 수 있다. 단순하고 효과적인 방법이지만, 확장할 때 많은 데이터를 옮기고 해시값을 계산해야 하므로 효율은 매우 낮다. 효율을 높이려면 다음 전략을 사용할 수 있다.</p> <ol> <li>해시 테이블의 자료 구조를 개선하여 해시 충돌이 발생해도 정상적으로 작동하게 한다.</li> <li>필요한 경우, 즉 해시 충돌이 심할 때만 용량을 확장한다.</li> </ol> <p>해시 테이블의 구조를 개선하는 대표적인 방법은 ‘체이닝’과 ‘개방 주소법’이다.</p>","path":["제 6 장   해시 테이블","6.2   해시 충돌"],"tags":[]},{"location":"chapter_hashing/hash_collision/#621","level":2,"title":"6.2.1   체이닝","text":"<p>기본 해시 테이블에서는 버킷 하나에 키-값 쌍 하나만 저장할 수 있다. 체이닝(separate chaining)은 단일 원소를 연결 리스트로 바꾸고, 키-값 쌍을 노드로 삼아 충돌한 키-값 쌍을 모두 같은 연결 리스트에 저장한다. 아래 그림(6-5)은 체이닝 해시 테이블의 예다.</p> <p></p> <p> 그림 6-5   체이닝 해시 테이블 </p> <p>체이닝 해시 테이블에서는 연산 방식이 다음과 같이 바뀐다.</p> <ul> <li>원소 조회: <code>key</code>를 입력하여 해시 함수로 버킷 인덱스를 구한 뒤 연결 리스트의 머리 노드에 접근한다. 이후 리스트를 순회하며 <code>key</code>를 비교하여 목표 키-값 쌍을 찾는다.</li> <li>원소 추가: 해시 함수로 연결 리스트의 머리 노드에 접근한 뒤, 노드(키-값 쌍)를 리스트에 추가한다.</li> <li>원소 삭제: 해시 함수의 결과로 연결 리스트의 맨 앞에 접근한 뒤, 순회하여 목표 노드를 찾아 삭제한다.</li> </ul> <p>체이닝에는 다음과 같은 한계가 있다.</p> <ul> <li>공간 사용량 증가: 연결 리스트에는 노드 포인터가 있으므로 배열보다 메모리를 더 많이 사용한다.</li> <li>조회 효율 감소: 해당 원소를 찾으려면 연결 리스트를 선형으로 순회해야 한다.</li> </ul> <p>다음 코드는 간단한 체이닝 해시 테이블을 구현한다. 두 가지에 주의하자.</p> <ul> <li>코드를 단순화하기 위해 연결 리스트 대신 리스트(동적 배열)를 사용한다. 따라서 해시 테이블(배열)은 여러 버킷을 포함하며, 각 버킷은 리스트다.</li> <li>아래 구현에는 용량 확장 메서드가 포함된다. 적재율이 \\(\\frac{2}{3}\\)를 넘으면 해시 테이블을 기존 크기의 \\(2\\)배로 늘린다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map_chaining.py<pre><code>class HashMapChaining:\n    \"\"\"체이닝 해시 테이블\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        self.size = 0  # 키-값 쌍 개수\n        self.capacity = 4  # 해시 테이블 용량\n        self.load_thres = 2.0 / 3.0  # 용량 확장을 수행하는 적재율 임계값\n        self.extend_ratio = 2  # 용량 확장 배수\n        self.buckets = [[] for _ in range(self.capacity)]  # 버킷 배열\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"해시 함수\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"적재율\"\"\"\n        return self.size / self.capacity\n\n    def get(self, key: int) -&gt; str | None:\n        \"\"\"조회 연산\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for pair in bucket:\n            if pair.key == key:\n                return pair.val\n        # key를 찾지 못하면 None을 반환한다\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"추가 연산\"\"\"\n        # 적재율이 임계값을 초과하면 용량을 확장한다\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for pair in bucket:\n            if pair.key == key:\n                pair.val = val\n                return\n        # 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        pair = Pair(key, val)\n        bucket.append(pair)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"삭제 연산\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # 버킷을 순회하며 키-값 쌍을 삭제한다\n        for pair in bucket:\n            if pair.key == key:\n                bucket.remove(pair)\n                self.size -= 1\n                break\n\n    def extend(self):\n        \"\"\"해시 테이블 용량 확장\"\"\"\n        # 원래 해시 테이블을 임시로 저장한다\n        buckets = self.buckets\n        # 용량을 확장한 새 해시 테이블을 초기화한다\n        self.capacity *= self.extend_ratio\n        self.buckets = [[] for _ in range(self.capacity)]\n        self.size = 0\n        # 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for bucket in buckets:\n            for pair in bucket:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"해시 테이블 출력\"\"\"\n        for bucket in self.buckets:\n            res = []\n            for pair in bucket:\n                res.append(str(pair.key) + \" -&gt; \" + pair.val)\n            print(res)\n</code></pre> hash_map_chaining.cpp<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n  private:\n    int size;                       // 키-값 쌍 개수\n    int capacity;                   // 해시 테이블 용량\n    double loadThres;               // 용량 확장을 수행하는 적재율 임계값\n    int extendRatio;                // 용량 확장 배수\n    vector&lt;vector&lt;Pair *&gt;&gt; buckets; // 버킷 배열\n\n  public:\n    /* 생성자 */\n    HashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3.0), extendRatio(2) {\n        buckets.resize(capacity);\n    }\n\n    /* 소멸자 */\n    ~HashMapChaining() {\n        for (auto &amp;bucket : buckets) {\n            for (Pair *pair : bucket) {\n                // 메모리 해제\n                delete pair;\n            }\n        }\n    }\n\n    /* 해시 함수 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* 적재율 */\n    double loadFactor() {\n        return (double)size / (double)capacity;\n    }\n\n    /* 조회 연산 */\n    string get(int key) {\n        int index = hashFunc(key);\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                return pair-&gt;val;\n            }\n        }\n        // key를 찾지 못하면 빈 문자열을 반환한다\n        return \"\";\n    }\n\n    /* 추가 연산 */\n    void put(int key, string val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                pair-&gt;val = val;\n                return;\n            }\n        }\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        buckets[index].push_back(new Pair(key, val));\n        size++;\n    }\n\n    /* 삭제 연산 */\n    void remove(int key) {\n        int index = hashFunc(key);\n        auto &amp;bucket = buckets[index];\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        for (int i = 0; i &lt; bucket.size(); i++) {\n            if (bucket[i]-&gt;key == key) {\n                Pair *tmp = bucket[i];\n                bucket.erase(bucket.begin() + i); // 해당 키-값 쌍을 삭제한다\n                delete tmp;                       // 메모리 해제\n                size--;\n                return;\n            }\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    void extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        vector&lt;vector&lt;Pair *&gt;&gt; bucketsTmp = buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio;\n        buckets.clear();\n        buckets.resize(capacity);\n        size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (auto &amp;bucket : bucketsTmp) {\n            for (Pair *pair : bucket) {\n                put(pair-&gt;key, pair-&gt;val);\n                // 메모리 해제\n                delete pair;\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    void print() {\n        for (auto &amp;bucket : buckets) {\n            cout &lt;&lt; \"[\";\n            for (Pair *pair : bucket) {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; \", \";\n            }\n            cout &lt;&lt; \"]\\n\";\n        }\n    }\n};\n</code></pre> hash_map_chaining.java<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n    int size; // 키-값 쌍 개수\n    int capacity; // 해시 테이블 용량\n    double loadThres; // 용량 확장을 수행하는 적재율 임계값\n    int extendRatio; // 용량 확장 배수\n    List&lt;List&lt;Pair&gt;&gt; buckets; // 버킷 배열\n\n    /* 생성자 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n    }\n\n    /* 해시 함수 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* 적재율 */\n    double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* 조회 연산 */\n    String get(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // key를 찾지 못하면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    void put(int key, String val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        Pair pair = new Pair(key, val);\n        bucket.add(pair);\n        size++;\n    }\n\n    /* 삭제 연산 */\n    void remove(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    void extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n        size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (List&lt;Pair&gt; bucket : bucketsTmp) {\n            for (Pair pair : bucket) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    void print() {\n        for (List&lt;Pair&gt; bucket : buckets) {\n            List&lt;String&gt; res = new ArrayList&lt;&gt;();\n            for (Pair pair : bucket) {\n                res.add(pair.key + \" -&gt; \" + pair.val);\n            }\n            System.out.println(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.cs<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n    int size; // 키-값 쌍 개수\n    int capacity; // 해시 테이블 용량\n    double loadThres; // 용량 확장을 수행하는 적재율 임계값\n    int extendRatio; // 용량 확장 배수\n    List&lt;List&lt;Pair&gt;&gt; buckets; // 버킷 배열\n\n    /* 생성자 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n    }\n\n    /* 해시 함수 */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* 적재율 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* 조회 연산 */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // key를 찾지 못하면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    public void Put(int key, string val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        int index = HashFunc(key);\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        buckets[index].Add(new Pair(key, val));\n        size++;\n    }\n\n    /* 삭제 연산 */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        foreach (Pair pair in buckets[index].ToList()) {\n            if (pair.key == key) {\n                buckets[index].Remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    void Extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n        size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        foreach (List&lt;Pair&gt; bucket in bucketsTmp) {\n            foreach (Pair pair in bucket) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    public void Print() {\n        foreach (List&lt;Pair&gt; bucket in buckets) {\n            List&lt;string&gt; res = [];\n            foreach (Pair pair in bucket) {\n                res.Add(pair.key + \" -&gt; \" + pair.val);\n            }\n            foreach (string kv in res) {\n                Console.WriteLine(kv);\n            }\n        }\n    }\n}\n</code></pre> hash_map_chaining.go<pre><code>/* 체이닝 해시 테이블 */\ntype hashMapChaining struct {\n    size        int      // 키-값 쌍 개수\n    capacity    int      // 해시 테이블 용량\n    loadThres   float64  // 용량 확장을 수행하는 적재율 임계값\n    extendRatio int      // 용량 확장 배수\n    buckets     [][]pair // 버킷 배열\n}\n\n/* 생성자 */\nfunc newHashMapChaining() *hashMapChaining {\n    buckets := make([][]pair, 4)\n    for i := 0; i &lt; 4; i++ {\n        buckets[i] = make([]pair, 0)\n    }\n    return &amp;hashMapChaining{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     buckets,\n    }\n}\n\n/* 해시 함수 */\nfunc (m *hashMapChaining) hashFunc(key int) int {\n    return key % m.capacity\n}\n\n/* 적재율 */\nfunc (m *hashMapChaining) loadFactor() float64 {\n    return float64(m.size) / float64(m.capacity)\n}\n\n/* 조회 연산 */\nfunc (m *hashMapChaining) get(key int) string {\n    idx := m.hashFunc(key)\n    bucket := m.buckets[idx]\n    // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n    for _, p := range bucket {\n        if p.key == key {\n            return p.val\n        }\n    }\n    // key를 찾지 못하면 빈 문자열을 반환한다\n    return \"\"\n}\n\n/* 추가 연산 */\nfunc (m *hashMapChaining) put(key int, val string) {\n    // 적재율이 임계값을 초과하면 용량을 확장한다\n    if m.loadFactor() &gt; m.loadThres {\n        m.extend()\n    }\n    idx := m.hashFunc(key)\n    // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n    for i := range m.buckets[idx] {\n        if m.buckets[idx][i].key == key {\n            m.buckets[idx][i].val = val\n            return\n        }\n    }\n    // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n    p := pair{\n        key: key,\n        val: val,\n    }\n    m.buckets[idx] = append(m.buckets[idx], p)\n    m.size += 1\n}\n\n/* 삭제 연산 */\nfunc (m *hashMapChaining) remove(key int) {\n    idx := m.hashFunc(key)\n    // 버킷을 순회하며 키-값 쌍을 삭제한다\n    for i, p := range m.buckets[idx] {\n        if p.key == key {\n            // 슬라이스에서 삭제한다\n            m.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\n            m.size -= 1\n            break\n        }\n    }\n}\n\n/* 해시 테이블 용량 확장 */\nfunc (m *hashMapChaining) extend() {\n    // 원래 해시 테이블을 임시로 저장한다\n    tmpBuckets := make([][]pair, len(m.buckets))\n    for i := 0; i &lt; len(m.buckets); i++ {\n        tmpBuckets[i] = make([]pair, len(m.buckets[i]))\n        copy(tmpBuckets[i], m.buckets[i])\n    }\n    // 용량을 확장한 새 해시 테이블을 초기화한다\n    m.capacity *= m.extendRatio\n    m.buckets = make([][]pair, m.capacity)\n    for i := 0; i &lt; m.capacity; i++ {\n        m.buckets[i] = make([]pair, 0)\n    }\n    m.size = 0\n    // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for _, bucket := range tmpBuckets {\n        for _, p := range bucket {\n            m.put(p.key, p.val)\n        }\n    }\n}\n\n/* 해시 테이블 출력 */\nfunc (m *hashMapChaining) print() {\n    var builder strings.Builder\n\n    for _, bucket := range m.buckets {\n        builder.WriteString(\"[\")\n        for _, p := range bucket {\n            builder.WriteString(strconv.Itoa(p.key) + \" -&gt; \" + p.val + \" \")\n        }\n        builder.WriteString(\"]\")\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> hash_map_chaining.swift<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n    var size: Int // 키-값 쌍 개수\n    var capacity: Int // 해시 테이블 용량\n    var loadThres: Double // 용량 확장을 수행하는 적재율 임계값\n    var extendRatio: Int // 용량 확장 배수\n    var buckets: [[Pair]] // 버킷 배열\n\n    /* 생성자 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: [], count: capacity)\n    }\n\n    /* 해시 함수 */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* 적재율 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* 조회 연산 */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for pair in bucket {\n            if pair.key == key {\n                return pair.val\n            }\n        }\n        // key를 찾지 못하면 nil을 반환한다\n        return nil\n    }\n\n    /* 추가 연산 */\n    func put(key: Int, val: String) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val\n                return\n            }\n        }\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        let pair = Pair(key: key, val: val)\n        buckets[index].append(pair)\n        size += 1\n    }\n\n    /* 삭제 연산 */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        for (pairIndex, pair) in bucket.enumerated() {\n            if pair.key == key {\n                buckets[index].remove(at: pairIndex)\n                size -= 1\n                break\n            }\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    func extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        let bucketsTmp = buckets\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio\n        buckets = Array(repeating: [], count: capacity)\n        size = 0\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for bucket in bucketsTmp {\n            for pair in bucket {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    func print() {\n        for bucket in buckets {\n            let res = bucket.map { \"\\($0.key) -&gt; \\($0.val)\" }\n            Swift.print(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.js<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n    #size; // 키-값 쌍 개수\n    #capacity; // 해시 테이블 용량\n    #loadThres; // 용량 확장을 수행하는 적재율 임계값\n    #extendRatio; // 용량 확장 배수\n    #buckets; // 버킷 배열\n\n    /* 생성자 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* 해시 함수 */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* 적재율 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* 조회 연산 */\n    get(key) {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // key를 찾지 못하면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    put(key, val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* 삭제 연산 */\n    remove(key) {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    #extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        const bucketsTmp = this.#buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    print() {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.ts<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n    #size: number; // 키-값 쌍 개수\n    #capacity: number; // 해시 테이블 용량\n    #loadThres: number; // 용량 확장을 수행하는 적재율 임계값\n    #extendRatio: number; // 용량 확장 배수\n    #buckets: Pair[][]; // 버킷 배열\n\n    /* 생성자 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* 해시 함수 */\n    #hashFunc(key: number): number {\n        return key % this.#capacity;\n    }\n\n    /* 적재율 */\n    #loadFactor(): number {\n        return this.#size / this.#capacity;\n    }\n\n    /* 조회 연산 */\n    get(key: number): string | null {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // key를 찾지 못하면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    put(key: number, val: string): void {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* 삭제 연산 */\n    remove(key: number): void {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    #extend(): void {\n        // 원래 해시 테이블을 임시로 저장한다\n        const bucketsTmp = this.#buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    print(): void {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.dart<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n  late int size; // 키-값 쌍 개수\n  late int capacity; // 해시 테이블 용량\n  late double loadThres; // 용량 확장을 수행하는 적재율 임계값\n  late int extendRatio; // 용량 확장 배수\n  late List&lt;List&lt;Pair&gt;&gt; buckets; // 버킷 배열\n\n  /* 생성자 */\n  HashMapChaining() {\n    size = 0;\n    capacity = 4;\n    loadThres = 2.0 / 3.0;\n    extendRatio = 2;\n    buckets = List.generate(capacity, (_) =&gt; []);\n  }\n\n  /* 해시 함수 */\n  int hashFunc(int key) {\n    return key % capacity;\n  }\n\n  /* 적재율 */\n  double loadFactor() {\n    return size / capacity;\n  }\n\n  /* 조회 연산 */\n  String? get(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        return pair.val;\n      }\n    }\n    // key를 찾지 못하면 null을 반환한다\n    return null;\n  }\n\n  /* 추가 연산 */\n  void put(int key, String val) {\n    // 적재율이 임계값을 초과하면 용량을 확장한다\n    if (loadFactor() &gt; loadThres) {\n      extend();\n    }\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        pair.val = val;\n        return;\n      }\n    }\n    // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n    Pair pair = Pair(key, val);\n    bucket.add(pair);\n    size++;\n  }\n\n  /* 삭제 연산 */\n  void remove(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // 버킷을 순회하며 키-값 쌍을 삭제한다\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        bucket.remove(pair);\n        size--;\n        break;\n      }\n    }\n  }\n\n  /* 해시 테이블 용량 확장 */\n  void extend() {\n    // 원래 해시 테이블을 임시로 저장한다\n    List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n    // 용량을 확장한 새 해시 테이블을 초기화한다\n    capacity *= extendRatio;\n    buckets = List.generate(capacity, (_) =&gt; []);\n    size = 0;\n    // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for (List&lt;Pair&gt; bucket in bucketsTmp) {\n      for (Pair pair in bucket) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* 해시 테이블 출력 */\n  void printHashMap() {\n    for (List&lt;Pair&gt; bucket in buckets) {\n      List&lt;String&gt; res = [];\n      for (Pair pair in bucket) {\n        res.add(\"${pair.key} -&gt; ${pair.val}\");\n      }\n      print(res);\n    }\n  }\n}\n</code></pre> hash_map_chaining.rs<pre><code>/* 체이닝 해시 테이블 */\nstruct HashMapChaining {\n    size: usize,\n    capacity: usize,\n    load_thres: f32,\n    extend_ratio: usize,\n    buckets: Vec&lt;Vec&lt;Pair&gt;&gt;,\n}\n\nimpl HashMapChaining {\n    /* 생성자 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![vec![]; 4],\n        }\n    }\n\n    /* 해시 함수 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % self.capacity\n    }\n\n    /* 적재율 */\n    fn load_factor(&amp;self) -&gt; f32 {\n        self.size as f32 / self.capacity as f32\n    }\n\n    /* 삭제 연산 */\n    fn remove(&amp;mut self, key: i32) -&gt; Option&lt;String&gt; {\n        let index = self.hash_func(key);\n\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        for (i, p) in self.buckets[index].iter_mut().enumerate() {\n            if p.key == key {\n                let pair = self.buckets[index].remove(i);\n                self.size -= 1;\n                return Some(pair.val);\n            }\n        }\n\n        // key를 찾지 못하면 None을 반환한다\n        None\n    }\n\n    /* 해시 테이블 용량 확장 */\n    fn extend(&amp;mut self) {\n        // 원래 해시 테이블을 임시로 저장한다\n        let buckets_tmp = std::mem::take(&amp;mut self.buckets);\n\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![Vec::new(); self.capacity as usize];\n        self.size = 0;\n\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for bucket in buckets_tmp {\n            for pair in bucket {\n                self.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    fn print(&amp;self) {\n        for bucket in &amp;self.buckets {\n            let mut res = Vec::new();\n            for pair in bucket {\n                res.push(format!(\"{} -&gt; {}\", pair.key, pair.val));\n            }\n            println!(\"{:?}\", res);\n        }\n    }\n\n    /* 추가 연산 */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n\n        let index = self.hash_func(key);\n\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for pair in self.buckets[index].iter_mut() {\n            if pair.key == key {\n                pair.val = val;\n                return;\n            }\n        }\n\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        let pair = Pair { key, val };\n        self.buckets[index].push(pair);\n        self.size += 1;\n    }\n\n    /* 조회 연산 */\n    fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        let index = self.hash_func(key);\n\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for pair in self.buckets[index].iter() {\n            if pair.key == key {\n                return Some(&amp;pair.val);\n            }\n        }\n\n        // key를 찾지 못하면 None을 반환한다\n        None\n    }\n}\n</code></pre> hash_map_chaining.c<pre><code>/* 연결 리스트 노드 */\ntypedef struct Node {\n    Pair *pair;\n    struct Node *next;\n} Node;\n\n/* 체이닝 해시 테이블 */\ntypedef struct {\n    int size;         // 키-값 쌍 개수\n    int capacity;     // 해시 테이블 용량\n    double loadThres; // 용량 확장을 수행하는 적재율 임계값\n    int extendRatio;  // 용량 확장 배수\n    Node **buckets;   // 버킷 배열\n} HashMapChaining;\n\n/* 생성자 */\nHashMapChaining *newHashMapChaining() {\n    HashMapChaining *hashMap = (HashMapChaining *)malloc(sizeof(HashMapChaining));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    return hashMap;\n}\n\n/* 소멸자 */\nvoid delHashMapChaining(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        while (cur) {\n            Node *tmp = cur;\n            cur = cur-&gt;next;\n            free(tmp-&gt;pair);\n            free(tmp);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap);\n}\n\n/* 해시 함수 */\nint hashFunc(HashMapChaining *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* 적재율 */\ndouble loadFactor(HashMapChaining *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* 조회 연산 */\nchar *get(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            return cur-&gt;pair-&gt;val;\n        }\n        cur = cur-&gt;next;\n    }\n    return \"\"; // key를 찾지 못하면 빈 문자열을 반환한다\n}\n\n/* 추가 연산 */\nvoid put(HashMapChaining *hashMap, int key, const char *val) {\n    // 적재율이 임계값을 초과하면 용량을 확장한다\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    int index = hashFunc(hashMap, key);\n    // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            strcpy(cur-&gt;pair-&gt;val, val); // 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n            return;\n        }\n        cur = cur-&gt;next;\n    }\n    // 해당 key가 없으면 키-값 쌍을 연결 리스트의 맨 앞에 추가한다\n    Pair *newPair = (Pair *)malloc(sizeof(Pair));\n    newPair-&gt;key = key;\n    strcpy(newPair-&gt;val, val);\n    Node *newNode = (Node *)malloc(sizeof(Node));\n    newNode-&gt;pair = newPair;\n    newNode-&gt;next = hashMap-&gt;buckets[index];\n    hashMap-&gt;buckets[index] = newNode;\n    hashMap-&gt;size++;\n}\n\n/* 해시 테이블 용량 확장 */\nvoid extend(HashMapChaining *hashMap) {\n    // 원래 해시 테이블을 임시로 저장한다\n    int oldCapacity = hashMap-&gt;capacity;\n    Node **oldBuckets = hashMap-&gt;buckets;\n    // 용량을 확장한 새 해시 테이블을 초기화한다\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    hashMap-&gt;size = 0;\n    // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Node *cur = oldBuckets[i];\n        while (cur) {\n            put(hashMap, cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            Node *temp = cur;\n            cur = cur-&gt;next;\n            // 메모리 해제\n            free(temp-&gt;pair);\n            free(temp);\n        }\n    }\n\n    free(oldBuckets);\n}\n\n/* 삭제 연산 */\nvoid removeItem(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    Node *cur = hashMap-&gt;buckets[index];\n    Node *pre = NULL;\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            // 해당 키-값 쌍을 삭제한다\n            if (pre) {\n                pre-&gt;next = cur-&gt;next;\n            } else {\n                hashMap-&gt;buckets[index] = cur-&gt;next;\n            }\n            // 메모리 해제\n            free(cur-&gt;pair);\n            free(cur);\n            hashMap-&gt;size--;\n            return;\n        }\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n}\n\n/* 해시 테이블 출력 */\nvoid print(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        printf(\"[\");\n        while (cur) {\n            printf(\"%d -&gt; %s, \", cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            cur = cur-&gt;next;\n        }\n        printf(\"]\\n\");\n    }\n}\n</code></pre> hash_map_chaining.kt<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n    var size: Int // 키-값 쌍 개수\n    var capacity: Int // 해시 테이블 용량\n    val loadThres: Double // 용량 확장을 수행하는 적재율 임계값\n    val extendRatio: Int // 용량 확장 배수\n    var buckets: MutableList&lt;MutableList&lt;Pair&gt;&gt; // 버킷 배열\n\n    /* 생성자 */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n    }\n\n    /* 해시 함수 */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* 적재율 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* 조회 연산 */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for (pair in bucket) {\n            if (pair.key == key) return pair._val\n        }\n        // key를 찾지 못하면 null을 반환한다\n        return null\n    }\n\n    /* 추가 연산 */\n    fun put(key: Int, _val: String) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for (pair in bucket) {\n            if (pair.key == key) {\n                pair._val = _val\n                return\n            }\n        }\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        val pair = Pair(key, _val)\n        bucket.add(pair)\n        size++\n    }\n\n    /* 삭제 연산 */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        for (pair in bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair)\n                size--\n                break\n            }\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    fun extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        val bucketsTmp = buckets\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio\n        // mutablelist는 크기가 고정되어 있지 않다\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n        size = 0\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (bucket in bucketsTmp) {\n            for (pair in bucket) {\n                put(pair.key, pair._val)\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    fun print() {\n        for (bucket in buckets) {\n            val res = mutableListOf&lt;String&gt;()\n            for (pair in bucket) {\n                val k = pair.key\n                val v = pair._val\n                res.add(\"$k -&gt; $v\")\n            }\n            println(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.rb<pre><code>### 체이닝 해시 테이블 ###\nclass HashMapChaining\n  ### 생성자 ###\n  def initialize\n    @size = 0 # 키-값 쌍 개수\n    @capacity = 4 # 해시 테이블 용량\n    @load_thres = 2.0 / 3.0 # 용량 확장을 수행하는 적재율 임계값\n    @extend_ratio = 2 # 용량 확장 배수\n    @buckets = Array.new(@capacity) { [] } # 버킷 배열\n  end\n\n  ### 해시 함수 ###\n  def hash_func(key)\n    key % @capacity\n  end\n\n  ### 적재율 ###\n  def load_factor\n    @size / @capacity\n  end\n\n  ### 조회 연산 ###\n  def get(key)\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n    for pair in bucket\n      return pair.val if pair.key == key\n    end\n    # key를 찾지 못하면 nil을 반환한다\n    nil\n  end\n\n  ### 추가 연산 ###\n  def put(key, val)\n    # 적재율이 임계값을 초과하면 용량을 확장한다\n    extend if load_factor &gt; @load_thres\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n    for pair in bucket\n      if pair.key == key\n        pair.val = val\n        return\n      end\n    end\n    # 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n    pair = Pair.new(key, val)\n    bucket &lt;&lt; pair\n    @size += 1\n  end\n\n  ### 삭제 연산 ###\n  def remove(key)\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # 버킷을 순회하며 키-값 쌍을 삭제한다\n    for pair in bucket\n      if pair.key == key\n        bucket.delete(pair)\n        @size -= 1\n        break\n      end\n    end\n  end\n\n  ### 해시 테이블 용량 확장 ###\n  def extend\n    # 원래 해시 테이블을 임시로 저장한다\n    buckets = @buckets\n    # 용량을 확장한 새 해시 테이블을 초기화한다\n    @capacity *= @extend_ratio\n    @buckets = Array.new(@capacity) { [] }\n    @size = 0\n    # 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for bucket in buckets\n      for pair in bucket\n        put(pair.key, pair.val)\n      end\n    end\n  end\n\n  ### 해시 테이블 출력 ###\n  def print\n    for bucket in @buckets\n      res = []\n      for pair in bucket\n        res &lt;&lt; \"#{pair.key} -&gt; #{pair.val}\"\n      end\n      pp res\n    end\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>연결 리스트가 매우 길어지면 \\(O(n)\\)의 조회 효율은 나빠진다. 이때 연결 리스트를 ‘AVL 트리’나 ‘레드-블랙 트리’로 바꾸면 조회의 시간 복잡도를 \\(O(\\log n)\\)으로 개선할 수 있다.</p>","path":["제 6 장   해시 테이블","6.2   해시 충돌"],"tags":[]},{"location":"chapter_hashing/hash_collision/#622","level":2,"title":"6.2.2   개방 주소법","text":"<p>개방 주소법(open addressing)은 추가 자료 구조를 도입하지 않고 ‘여러 차례 탐사’하여 해시 충돌을 처리한다. 대표적인 탐사 방식에는 선형 탐사, 이차 탐사, 다중 해싱 등이 있다.</p> <p>선형 탐사를 예로 개방 주소법 해시 테이블의 작동 원리를 알아보자.</p>","path":["제 6 장   해시 테이블","6.2   해시 충돌"],"tags":[]},{"location":"chapter_hashing/hash_collision/#1","level":3,"title":"1.   선형 탐사","text":"<p>선형 탐사는 고정 간격의 선형 탐색으로 탐사하며, 일반 해시 테이블과 연산 방식이 다르다.</p> <ul> <li>원소 삽입: 해시 함수로 버킷 인덱스를 구한다. 이미 원소가 있으면 충돌 위치부터 뒤로 일정한 간격(보통 \\(1\\))으로 순회하여 빈 버킷을 찾고 원소를 넣는다.</li> <li>원소 탐색: 해시 충돌이 있으면 같은 간격으로 뒤를 순회하다가 해당 원소를 찾으면 <code>value</code>를 반환한다. 빈 버킷을 만나면 목표 원소가 없다는 뜻이므로 <code>None</code>을 반환한다.</li> </ul> <p>아래 그림(6-6)은 개방 주소법(선형 탐사) 해시 테이블의 키-값 쌍 분포를 보여 준다. 이 해시 함수에서는 마지막 두 자리가 같은 <code>key</code>가 모두 같은 버킷에 매핑된다. 선형 탐사를 통해 이들은 해당 버킷과 그 뒤의 버킷에 차례로 저장된다.</p> <p></p> <p> 그림 6-6   개방 주소법(선형 탐사) 해시 테이블의 키-값 쌍 분포 </p> <p>하지만 선형 탐사에서는 ‘군집 현상’이 쉽게 발생한다. 배열에서 연속으로 채워진 구간이 길수록 그 구간에서 충돌할 가능성도 커진다. 그 결과 군집이 더욱 커지는 악순환이 생기고, 결국 삽입, 삭제, 탐색, 갱신의 효율이 낮아진다.</p> <p>개방 주소법 해시 테이블에서는 원소를 곧바로 삭제하면 안 된다는 점에 주의하자. 삭제하면 배열에 빈 버킷 <code>None</code>이 생긴다. 조회 중 선형 탐사가 이 버킷에 도달하면 바로 반환하므로 그 뒤의 원소에 접근할 수 없게 된다. 아래 그림(6-7)과 같이 프로그램이 그 원소들을 존재하지 않는다고 잘못 판단할 수 있다.</p> <p></p> <p> 그림 6-7   개방 주소법에서 원소 삭제로 발생하는 조회 문제 </p> <p>이 문제는 지연 삭제(lazy deletion)로 해결할 수 있다. 원소를 해시 테이블에서 곧바로 제거하지 않고 상수 <code>TOMBSTONE</code>으로 해당 버킷을 표시한다. <code>None</code>과 <code>TOMBSTONE</code>은 모두 빈 버킷을 나타내며 새 키-값 쌍을 넣을 수 있다. 하지만 <code>TOMBSTONE</code>을 만나면 그 뒤에도 키-값 쌍이 있을 수 있으므로 탐사를 계속해야 한다.</p> <p>그러나 지연 삭제는 해시 테이블의 성능 저하를 가속할 수 있다. 삭제할 때마다 삭제 표시가 생기므로 <code>TOMBSTONE</code>이 늘어날수록 탐색 시간도 길어진다. 목표 원소를 찾기 위해 여러 <code>TOMBSTONE</code>을 건너뛰어야 할 수 있기 때문이다.</p> <p>이를 위해 선형 탐사 중 처음 만난 <code>TOMBSTONE</code>의 인덱스를 기록하고, 찾은 목표 원소와 그 <code>TOMBSTONE</code>의 위치를 교환하는 방법을 생각할 수 있다. 조회하거나 추가할 때마다 원소가 이상적인 위치(탐사 시작점)에 더 가까운 버킷으로 옮겨지므로 조회 효율이 좋아진다.</p> <p>다음 코드는 지연 삭제를 포함한 개방 주소법(선형 탐사) 해시 테이블을 구현한다. 공간을 더 충분히 활용하기 위해 해시 테이블을 ‘원형 배열’로 보고, 끝을 넘으면 처음으로 돌아와 순회를 계속한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map_open_addressing.py<pre><code>class HashMapOpenAddressing:\n    \"\"\"개방 주소법 해시 테이블\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        self.size = 0  # 키-값 쌍 개수\n        self.capacity = 4  # 해시 테이블 용량\n        self.load_thres = 2.0 / 3.0  # 용량 확장을 수행하는 적재율 임계값\n        self.extend_ratio = 2  # 용량 확장 배수\n        self.buckets: list[Pair | None] = [None] * self.capacity  # 버킷 배열\n        self.TOMBSTONE = Pair(-1, \"-1\")  # 삭제 표시\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"해시 함수\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"적재율\"\"\"\n        return self.size / self.capacity\n\n    def find_bucket(self, key: int) -&gt; int:\n        \"\"\"key에 해당하는 버킷 인덱스 탐색\"\"\"\n        index = self.hash_func(key)\n        first_tombstone = -1\n        # 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while self.buckets[index] is not None:\n            # key를 만나면 해당 버킷 인덱스를 반환한다\n            if self.buckets[index].key == key:\n                # 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if first_tombstone != -1:\n                    self.buckets[first_tombstone] = self.buckets[index]\n                    self.buckets[index] = self.TOMBSTONE\n                    return first_tombstone  # 옮긴 후의 버킷 인덱스를 반환한다\n                return index  # 버킷 인덱스를 반환한다\n            # 처음 만난 삭제 표시를 기록한다\n            if first_tombstone == -1 and self.buckets[index] is self.TOMBSTONE:\n                first_tombstone = index\n            # 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % self.capacity\n        # key가 없으면 추가할 위치의 인덱스를 반환한다\n        return index if first_tombstone == -1 else first_tombstone\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"조회 연산\"\"\"\n        # key에 해당하는 버킷 인덱스 탐색\n        index = self.find_bucket(key)\n        # 키-값 쌍을 찾으면 해당 val을 반환한다\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            return self.buckets[index].val\n        # 키-값 쌍이 없으면 None을 반환한다\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"추가 연산\"\"\"\n        # 적재율이 임계값을 초과하면 용량을 확장한다\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        # key에 해당하는 버킷 인덱스 탐색\n        index = self.find_bucket(key)\n        # 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index].val = val\n            return\n        # 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        self.buckets[index] = Pair(key, val)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"삭제 연산\"\"\"\n        # key에 해당하는 버킷 인덱스 탐색\n        index = self.find_bucket(key)\n        # 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index] = self.TOMBSTONE\n            self.size -= 1\n\n    def extend(self):\n        \"\"\"해시 테이블 용량 확장\"\"\"\n        # 원래 해시 테이블을 임시로 저장한다\n        buckets_tmp = self.buckets\n        # 용량을 확장한 새 해시 테이블을 초기화한다\n        self.capacity *= self.extend_ratio\n        self.buckets = [None] * self.capacity\n        self.size = 0\n        # 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for pair in buckets_tmp:\n            if pair not in [None, self.TOMBSTONE]:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"해시 테이블 출력\"\"\"\n        for pair in self.buckets:\n            if pair is None:\n                print(\"None\")\n            elif pair is self.TOMBSTONE:\n                print(\"TOMBSTONE\")\n            else:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> hash_map_open_addressing.cpp<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n  private:\n    int size;                             // 키-값 쌍 개수\n    int capacity = 4;                     // 해시 테이블 용량\n    const double loadThres = 2.0 / 3.0;     // 용량 확장을 수행하는 적재율 임계값\n    const int extendRatio = 2;            // 용량 확장 배수\n    vector&lt;Pair *&gt; buckets;               // 버킷 배열\n    Pair *TOMBSTONE = new Pair(-1, \"-1\"); // 삭제 표시\n\n  public:\n    /* 생성자 */\n    HashMapOpenAddressing() : size(0), buckets(capacity, nullptr) {\n    }\n\n    /* 소멸자 */\n    ~HashMapOpenAddressing() {\n        for (Pair *pair : buckets) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                delete pair;\n            }\n        }\n        delete TOMBSTONE;\n    }\n\n    /* 해시 함수 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* 적재율 */\n    double loadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while (buckets[index] != nullptr) {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if (buckets[index]-&gt;key == key) {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index; // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % capacity;\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* 조회 연산 */\n    string get(int key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = findBucket(key);\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]-&gt;val;\n        }\n        // 키-값 쌍이 없으면 빈 문자열을 반환한다\n        return \"\";\n    }\n\n    /* 추가 연산 */\n    void put(int key, string val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = findBucket(key);\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]-&gt;val = val;\n            return;\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* 삭제 연산 */\n    void remove(int key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = findBucket(key);\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            delete buckets[index];\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    void extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        vector&lt;Pair *&gt; bucketsTmp = buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio;\n        buckets = vector&lt;Pair *&gt;(capacity, nullptr);\n        size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (Pair *pair : bucketsTmp) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                put(pair-&gt;key, pair-&gt;val);\n                delete pair;\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    void print() {\n        for (Pair *pair : buckets) {\n            if (pair == nullptr) {\n                cout &lt;&lt; \"nullptr\" &lt;&lt; endl;\n            } else if (pair == TOMBSTONE) {\n                cout &lt;&lt; \"TOMBSTONE\" &lt;&lt; endl;\n            } else {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; endl;\n            }\n        }\n    }\n};\n</code></pre> hash_map_open_addressing.java<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n    private int size; // 키-값 쌍 개수\n    private int capacity = 4; // 해시 테이블 용량\n    private final double loadThres = 2.0 / 3.0; // 용량 확장을 수행하는 적재율 임계값\n    private final int extendRatio = 2; // 용량 확장 배수\n    private Pair[] buckets; // 버킷 배열\n    private final Pair TOMBSTONE = new Pair(-1, \"-1\"); // 삭제 표시\n\n    /* 생성자 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* 해시 함수 */\n    private int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* 적재율 */\n    private double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    private int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while (buckets[index] != null) {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if (buckets[index].key == key) {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index; // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % capacity;\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* 조회 연산 */\n    public String get(int key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = findBucket(key);\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // 키-값 쌍이 없으면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    public void put(int key, String val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = findBucket(key);\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* 삭제 연산 */\n    public void remove(int key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = findBucket(key);\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    private void extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        Pair[] bucketsTmp = buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (Pair pair : bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    public void print() {\n        for (Pair pair : buckets) {\n            if (pair == null) {\n                System.out.println(\"null\");\n            } else if (pair == TOMBSTONE) {\n                System.out.println(\"TOMBSTONE\");\n            } else {\n                System.out.println(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.cs<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n    int size; // 키-값 쌍 개수\n    int capacity = 4; // 해시 테이블 용량\n    double loadThres = 2.0 / 3.0; // 용량 확장을 수행하는 적재율 임계값\n    int extendRatio = 2; // 용량 확장 배수\n    Pair[] buckets; // 버킷 배열\n    Pair TOMBSTONE = new(-1, \"-1\"); // 삭제 표시\n\n    /* 생성자 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* 해시 함수 */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* 적재율 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    int FindBucket(int key) {\n        int index = HashFunc(key);\n        int firstTombstone = -1;\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while (buckets[index] != null) {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if (buckets[index].key == key) {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index; // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % capacity;\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* 조회 연산 */\n    public string? Get(int key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = FindBucket(key);\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // 키-값 쌍이 없으면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    public void Put(int key, string val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = FindBucket(key);\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* 삭제 연산 */\n    public void Remove(int key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = FindBucket(key);\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    void Extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        Pair[] bucketsTmp = buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        foreach (Pair pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    public void Print() {\n        foreach (Pair pair in buckets) {\n            if (pair == null) {\n                Console.WriteLine(\"null\");\n            } else if (pair == TOMBSTONE) {\n                Console.WriteLine(\"TOMBSTONE\");\n            } else {\n                Console.WriteLine(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.go<pre><code>/* 개방 주소법 해시 테이블 */\ntype hashMapOpenAddressing struct {\n    size        int     // 키-값 쌍 개수\n    capacity    int     // 해시 테이블 용량\n    loadThres   float64 // 용량 확장을 수행하는 적재율 임계값\n    extendRatio int     // 용량 확장 배수\n    buckets     []*pair // 버킷 배열\n    TOMBSTONE   *pair   // 삭제 표시\n}\n\n/* 생성자 */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\n    return &amp;hashMapOpenAddressing{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     make([]*pair, 4),\n        TOMBSTONE:   &amp;pair{-1, \"-1\"},\n    }\n}\n\n/* 해시 함수 */\nfunc (h *hashMapOpenAddressing) hashFunc(key int) int {\n    return key % h.capacity // 키를 바탕으로 해시값을 계산한다\n}\n\n/* 적재율 */\nfunc (h *hashMapOpenAddressing) loadFactor() float64 {\n    return float64(h.size) / float64(h.capacity) // 현재 적재율을 계산한다\n}\n\n/* key에 해당하는 버킷 인덱스 탐색 */\nfunc (h *hashMapOpenAddressing) findBucket(key int) int {\n    index := h.hashFunc(key) // 초기 인덱스를 구한다\n    firstTombstone := -1     // 처음 만난 TOMBSTONE의 위치를 기록한다\n    for h.buckets[index] != nil {\n        if h.buckets[index].key == key {\n            if firstTombstone != -1 {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                h.buckets[firstTombstone] = h.buckets[index]\n                h.buckets[index] = h.TOMBSTONE\n                return firstTombstone // 옮긴 후의 버킷 인덱스를 반환한다\n            }\n            return index // 찾은 인덱스를 반환한다\n        }\n        if firstTombstone == -1 &amp;&amp; h.buckets[index] == h.TOMBSTONE {\n            firstTombstone = index // 처음 만난 삭제 표시의 위치를 기록한다\n        }\n        index = (index + 1) % h.capacity // 선형 탐사를 수행하며, 끝을 넘어가면 처음으로 돌아온다\n    }\n    // key가 없으면 추가할 위치의 인덱스를 반환한다\n    if firstTombstone != -1 {\n        return firstTombstone\n    }\n    return index\n}\n\n/* 조회 연산 */\nfunc (h *hashMapOpenAddressing) get(key int) string {\n    index := h.findBucket(key) // key에 해당하는 버킷 인덱스 탐색\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        return h.buckets[index].val // 키-값 쌍을 찾으면 해당 val을 반환한다\n    }\n    return \"\" // 키-값 쌍이 없으면 \"\"를 반환한다\n}\n\n/* 추가 연산 */\nfunc (h *hashMapOpenAddressing) put(key int, val string) {\n    if h.loadFactor() &gt; h.loadThres {\n        h.extend() // 적재율이 임계값을 초과하면 용량을 확장한다\n    }\n    index := h.findBucket(key) // key에 해당하는 버킷 인덱스 탐색\n    if h.buckets[index] == nil || h.buckets[index] == h.TOMBSTONE {\n        h.buckets[index] = &amp;pair{key, val} // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        h.size++\n    } else {\n        h.buckets[index].val = val // 키-값 쌍을 찾으면 val을 덮어쓴다\n    }\n}\n\n/* 삭제 연산 */\nfunc (h *hashMapOpenAddressing) remove(key int) {\n    index := h.findBucket(key) // key에 해당하는 버킷 인덱스 탐색\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        h.buckets[index] = h.TOMBSTONE // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        h.size--\n    }\n}\n\n/* 해시 테이블 용량 확장 */\nfunc (h *hashMapOpenAddressing) extend() {\n    oldBuckets := h.buckets               // 원래 해시 테이블을 임시로 저장한다\n    h.capacity *= h.extendRatio           // 용량을 갱신한다\n    h.buckets = make([]*pair, h.capacity) // 용량을 확장한 새 해시 테이블을 초기화한다\n    h.size = 0                            // 크기를 초기화한다\n    // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for _, pair := range oldBuckets {\n        if pair != nil &amp;&amp; pair != h.TOMBSTONE {\n            h.put(pair.key, pair.val)\n        }\n    }\n}\n\n/* 해시 테이블 출력 */\nfunc (h *hashMapOpenAddressing) print() {\n    for _, pair := range h.buckets {\n        if pair == nil {\n            fmt.Println(\"nil\")\n        } else if pair == h.TOMBSTONE {\n            fmt.Println(\"TOMBSTONE\")\n        } else {\n            fmt.Printf(\"%d -&gt; %s\\n\", pair.key, pair.val)\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.swift<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n    var size: Int // 키-값 쌍 개수\n    var capacity: Int // 해시 테이블 용량\n    var loadThres: Double // 용량 확장을 수행하는 적재율 임계값\n    var extendRatio: Int // 용량 확장 배수\n    var buckets: [Pair?] // 버킷 배열\n    var TOMBSTONE: Pair // 삭제 표시\n\n    /* 생성자 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: nil, count: capacity)\n        TOMBSTONE = Pair(key: -1, val: \"-1\")\n    }\n\n    /* 해시 함수 */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* 적재율 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    func findBucket(key: Int) -&gt; Int {\n        var index = hashFunc(key: key)\n        var firstTombstone = -1\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while buckets[index] != nil {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if buckets[index]!.key == key {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if firstTombstone != -1 {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE {\n                firstTombstone = index\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % capacity\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        return firstTombstone == -1 ? index : firstTombstone\n    }\n\n    /* 조회 연산 */\n    func get(key: Int) -&gt; String? {\n        // key에 해당하는 버킷 인덱스 탐색\n        let index = findBucket(key: key)\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            return buckets[index]!.val\n        }\n        // 키-값 쌍이 없으면 null을 반환한다\n        return nil\n    }\n\n    /* 추가 연산 */\n    func put(key: Int, val: String) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        let index = findBucket(key: key)\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index]!.val = val\n            return\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        buckets[index] = Pair(key: key, val: val)\n        size += 1\n    }\n\n    /* 삭제 연산 */\n    func remove(key: Int) {\n        // key에 해당하는 버킷 인덱스 탐색\n        let index = findBucket(key: key)\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index] = TOMBSTONE\n            size -= 1\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    func extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        let bucketsTmp = buckets\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio\n        buckets = Array(repeating: nil, count: capacity)\n        size = 0\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for pair in bucketsTmp {\n            if let pair, pair != TOMBSTONE {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    func print() {\n        for pair in buckets {\n            if pair == nil {\n                Swift.print(\"null\")\n            } else if pair == TOMBSTONE {\n                Swift.print(\"TOMBSTONE\")\n            } else {\n                Swift.print(\"\\(pair!.key) -&gt; \\(pair!.val)\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.js<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n    #size; // 키-값 쌍 개수\n    #capacity; // 해시 테이블 용량\n    #loadThres; // 용량 확장을 수행하는 적재율 임계값\n    #extendRatio; // 용량 확장 배수\n    #buckets; // 버킷 배열\n    #TOMBSTONE; // 삭제 표시\n\n    /* 생성자 */\n    constructor() {\n        this.#size = 0; // 키-값 쌍 개수\n        this.#capacity = 4; // 해시 테이블 용량\n        this.#loadThres = 2.0 / 3.0; // 용량 확장을 수행하는 적재율 임계값\n        this.#extendRatio = 2; // 용량 확장 배수\n        this.#buckets = Array(this.#capacity).fill(null); // 버킷 배열\n        this.#TOMBSTONE = new Pair(-1, '-1'); // 삭제 표시\n    }\n\n    /* 해시 함수 */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* 적재율 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    #findBucket(key) {\n        let index = this.#hashFunc(key);\n        let firstTombstone = -1;\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while (this.#buckets[index] !== null) {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if (this.#buckets[index].key === key) {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if (firstTombstone !== -1) {\n                    this.#buckets[firstTombstone] = this.#buckets[index];\n                    this.#buckets[index] = this.#TOMBSTONE;\n                    return firstTombstone; // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index; // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.#buckets[index] === this.#TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % this.#capacity;\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* 조회 연산 */\n    get(key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        const index = this.#findBucket(key);\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            return this.#buckets[index].val;\n        }\n        // 키-값 쌍이 없으면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    put(key, val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        const index = this.#findBucket(key);\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index].val = val;\n            return;\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        this.#buckets[index] = new Pair(key, val);\n        this.#size++;\n    }\n\n    /* 삭제 연산 */\n    remove(key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        const index = this.#findBucket(key);\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index] = this.#TOMBSTONE;\n            this.#size--;\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    #extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        const bucketsTmp = this.#buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = Array(this.#capacity).fill(null);\n        this.#size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.#TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    print() {\n        for (const pair of this.#buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.#TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.ts<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n    private size: number; // 키-값 쌍 개수\n    private capacity: number; // 해시 테이블 용량\n    private loadThres: number; // 용량 확장을 수행하는 적재율 임계값\n    private extendRatio: number; // 용량 확장 배수\n    private buckets: Array&lt;Pair | null&gt;; // 버킷 배열\n    private TOMBSTONE: Pair; // 삭제 표시\n\n    /* 생성자 */\n    constructor() {\n        this.size = 0; // 키-값 쌍 개수\n        this.capacity = 4; // 해시 테이블 용량\n        this.loadThres = 2.0 / 3.0; // 용량 확장을 수행하는 적재율 임계값\n        this.extendRatio = 2; // 용량 확장 배수\n        this.buckets = Array(this.capacity).fill(null); // 버킷 배열\n        this.TOMBSTONE = new Pair(-1, '-1'); // 삭제 표시\n    }\n\n    /* 해시 함수 */\n    private hashFunc(key: number): number {\n        return key % this.capacity;\n    }\n\n    /* 적재율 */\n    private loadFactor(): number {\n        return this.size / this.capacity;\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    private findBucket(key: number): number {\n        let index = this.hashFunc(key);\n        let firstTombstone = -1;\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while (this.buckets[index] !== null) {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if (this.buckets[index]!.key === key) {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if (firstTombstone !== -1) {\n                    this.buckets[firstTombstone] = this.buckets[index];\n                    this.buckets[index] = this.TOMBSTONE;\n                    return firstTombstone; // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index; // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.buckets[index] === this.TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % this.capacity;\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* 조회 연산 */\n    get(key: number): string | null {\n        // key에 해당하는 버킷 인덱스 탐색\n        const index = this.findBucket(key);\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            return this.buckets[index]!.val;\n        }\n        // 키-값 쌍이 없으면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    put(key: number, val: string): void {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (this.loadFactor() &gt; this.loadThres) {\n            this.extend();\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        const index = this.findBucket(key);\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index]!.val = val;\n            return;\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        this.buckets[index] = new Pair(key, val);\n        this.size++;\n    }\n\n    /* 삭제 연산 */\n    remove(key: number): void {\n        // key에 해당하는 버킷 인덱스 탐색\n        const index = this.findBucket(key);\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index] = this.TOMBSTONE;\n            this.size--;\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    private extend(): void {\n        // 원래 해시 테이블을 임시로 저장한다\n        const bucketsTmp = this.buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        this.capacity *= this.extendRatio;\n        this.buckets = Array(this.capacity).fill(null);\n        this.size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    print(): void {\n        for (const pair of this.buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.dart<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n  late int _size; // 키-값 쌍 개수\n  int _capacity = 4; // 해시 테이블 용량\n  double _loadThres = 2.0 / 3.0; // 용량 확장을 수행하는 적재율 임계값\n  int _extendRatio = 2; // 용량 확장 배수\n  late List&lt;Pair?&gt; _buckets; // 버킷 배열\n  Pair _TOMBSTONE = Pair(-1, \"-1\"); // 삭제 표시\n\n  /* 생성자 */\n  HashMapOpenAddressing() {\n    _size = 0;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n  }\n\n  /* 해시 함수 */\n  int hashFunc(int key) {\n    return key % _capacity;\n  }\n\n  /* 적재율 */\n  double loadFactor() {\n    return _size / _capacity;\n  }\n\n  /* key에 해당하는 버킷 인덱스 탐색 */\n  int findBucket(int key) {\n    int index = hashFunc(key);\n    int firstTombstone = -1;\n    // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n    while (_buckets[index] != null) {\n      // key를 만나면 해당 버킷 인덱스를 반환한다\n      if (_buckets[index]!.key == key) {\n        // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n        if (firstTombstone != -1) {\n          _buckets[firstTombstone] = _buckets[index];\n          _buckets[index] = _TOMBSTONE;\n          return firstTombstone; // 옮긴 후의 버킷 인덱스를 반환한다\n        }\n        return index; // 버킷 인덱스를 반환한다\n      }\n      // 처음 만난 삭제 표시를 기록한다\n      if (firstTombstone == -1 &amp;&amp; _buckets[index] == _TOMBSTONE) {\n        firstTombstone = index;\n      }\n      // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n      index = (index + 1) % _capacity;\n    }\n    // key가 없으면 추가할 위치의 인덱스를 반환한다\n    return firstTombstone == -1 ? index : firstTombstone;\n  }\n\n  /* 조회 연산 */\n  String? get(int key) {\n    // key에 해당하는 버킷 인덱스 탐색\n    int index = findBucket(key);\n    // 키-값 쌍을 찾으면 해당 val을 반환한다\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      return _buckets[index]!.val;\n    }\n    // 키-값 쌍이 없으면 null을 반환한다\n    return null;\n  }\n\n  /* 추가 연산 */\n  void put(int key, String val) {\n    // 적재율이 임계값을 초과하면 용량을 확장한다\n    if (loadFactor() &gt; _loadThres) {\n      extend();\n    }\n    // key에 해당하는 버킷 인덱스 탐색\n    int index = findBucket(key);\n    // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index]!.val = val;\n      return;\n    }\n    // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n    _buckets[index] = new Pair(key, val);\n    _size++;\n  }\n\n  /* 삭제 연산 */\n  void remove(int key) {\n    // key에 해당하는 버킷 인덱스 탐색\n    int index = findBucket(key);\n    // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index] = _TOMBSTONE;\n      _size--;\n    }\n  }\n\n  /* 해시 테이블 용량 확장 */\n  void extend() {\n    // 원래 해시 테이블을 임시로 저장한다\n    List&lt;Pair?&gt; bucketsTmp = _buckets;\n    // 용량을 확장한 새 해시 테이블을 초기화한다\n    _capacity *= _extendRatio;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n    _size = 0;\n    // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for (Pair? pair in bucketsTmp) {\n      if (pair != null &amp;&amp; pair != _TOMBSTONE) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* 해시 테이블 출력 */\n  void printHashMap() {\n    for (Pair? pair in _buckets) {\n      if (pair == null) {\n        print(\"null\");\n      } else if (pair == _TOMBSTONE) {\n        print(\"TOMBSTONE\");\n      } else {\n        print(\"${pair.key} -&gt; ${pair.val}\");\n      }\n    }\n  }\n}\n</code></pre> hash_map_open_addressing.rs<pre><code>/* 개방 주소법 해시 테이블 */\nstruct HashMapOpenAddressing {\n    size: usize,                // 키-값 쌍 개수\n    capacity: usize,            // 해시 테이블 용량\n    load_thres: f64,            // 용량 확장을 수행하는 적재율 임계값\n    extend_ratio: usize,        // 용량 확장 배수\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;, // 버킷 배열\n    TOMBSTONE: Option&lt;Pair&gt;,    // 삭제 표시\n}\n\nimpl HashMapOpenAddressing {\n    /* 생성자 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![None; 4],\n            TOMBSTONE: Some(Pair {\n                key: -1,\n                val: \"-1\".to_string(),\n            }),\n        }\n    }\n\n    /* 해시 함수 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        (key % self.capacity as i32) as usize\n    }\n\n    /* 적재율 */\n    fn load_factor(&amp;self) -&gt; f64 {\n        self.size as f64 / self.capacity as f64\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    fn find_bucket(&amp;mut self, key: i32) -&gt; usize {\n        let mut index = self.hash_func(key);\n        let mut first_tombstone = -1;\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while self.buckets[index].is_some() {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if self.buckets[index].as_ref().unwrap().key == key {\n                // 앞서 삭제 표시를 만났다면 키-값 쌍을 해당 인덱스로 옮긴다\n                if first_tombstone != -1 {\n                    self.buckets[first_tombstone as usize] = self.buckets[index].take();\n                    self.buckets[index] = self.TOMBSTONE.clone();\n                    return first_tombstone as usize; // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index; // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if first_tombstone == -1 &amp;&amp; self.buckets[index] == self.TOMBSTONE {\n                first_tombstone = index as i32;\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % self.capacity;\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        if first_tombstone == -1 {\n            index\n        } else {\n            first_tombstone as usize\n        }\n    }\n\n    /* 조회 연산 */\n    fn get(&amp;mut self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        // key에 해당하는 버킷 인덱스 탐색\n        let index = self.find_bucket(key);\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            return self.buckets[index].as_ref().map(|pair| &amp;pair.val as &amp;str);\n        }\n        // 키-값 쌍이 없으면 null을 반환한다\n        None\n    }\n\n    /* 추가 연산 */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        let index = self.find_bucket(key);\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index].as_mut().unwrap().val = val;\n            return;\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        self.buckets[index] = Some(Pair { key, val });\n        self.size += 1;\n    }\n\n    /* 삭제 연산 */\n    fn remove(&amp;mut self, key: i32) {\n        // key에 해당하는 버킷 인덱스 탐색\n        let index = self.find_bucket(key);\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index] = self.TOMBSTONE.clone();\n            self.size -= 1;\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    fn extend(&amp;mut self) {\n        // 원래 해시 테이블을 임시로 저장한다\n        let buckets_tmp = self.buckets.clone();\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![None; self.capacity];\n        self.size = 0;\n\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for pair in buckets_tmp {\n            if pair.is_none() || pair == self.TOMBSTONE {\n                continue;\n            }\n            let pair = pair.unwrap();\n\n            self.put(pair.key, pair.val);\n        }\n    }\n    /* 해시 테이블 출력 */\n    fn print(&amp;self) {\n        for pair in &amp;self.buckets {\n            if pair.is_none() {\n                println!(\"null\");\n            } else if pair == &amp;self.TOMBSTONE {\n                println!(\"TOMBSTONE\");\n            } else {\n                let pair = pair.as_ref().unwrap();\n                println!(\"{} -&gt; {}\", pair.key, pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.c<pre><code>/* 개방 주소법 해시 테이블 */\ntypedef struct {\n    int size;         // 키-값 쌍 개수\n    int capacity;     // 해시 테이블 용량\n    double loadThres; // 용량 확장을 수행하는 적재율 임계값\n    int extendRatio;  // 용량 확장 배수\n    Pair **buckets;   // 버킷 배열\n    Pair *TOMBSTONE;  // 삭제 표시\n} HashMapOpenAddressing;\n\n/* 생성자 */\nHashMapOpenAddressing *newHashMapOpenAddressing() {\n    HashMapOpenAddressing *hashMap = (HashMapOpenAddressing *)malloc(sizeof(HashMapOpenAddressing));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Pair **)calloc(hashMap-&gt;capacity, sizeof(Pair *));\n    hashMap-&gt;TOMBSTONE = (Pair *)malloc(sizeof(Pair));\n    hashMap-&gt;TOMBSTONE-&gt;key = -1;\n    hashMap-&gt;TOMBSTONE-&gt;val = \"-1\";\n\n    return hashMap;\n}\n\n/* 소멸자 */\nvoid delHashMapOpenAddressing(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap-&gt;TOMBSTONE);\n    free(hashMap);\n}\n\n/* 해시 함수 */\nint hashFunc(HashMapOpenAddressing *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* 적재율 */\ndouble loadFactor(HashMapOpenAddressing *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* key에 해당하는 버킷 인덱스 탐색 */\nint findBucket(HashMapOpenAddressing *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    int firstTombstone = -1;\n    // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n    while (hashMap-&gt;buckets[index] != NULL) {\n        // key를 만나면 해당 버킷 인덱스를 반환한다\n        if (hashMap-&gt;buckets[index]-&gt;key == key) {\n            // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n            if (firstTombstone != -1) {\n                hashMap-&gt;buckets[firstTombstone] = hashMap-&gt;buckets[index];\n                hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n                return firstTombstone; // 옮긴 후의 버킷 인덱스를 반환한다\n            }\n            return index; // 버킷 인덱스를 반환한다\n        }\n        // 처음 만난 삭제 표시를 기록한다\n        if (firstTombstone == -1 &amp;&amp; hashMap-&gt;buckets[index] == hashMap-&gt;TOMBSTONE) {\n            firstTombstone = index;\n        }\n        // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n        index = (index + 1) % hashMap-&gt;capacity;\n    }\n    // key가 없으면 추가할 위치의 인덱스를 반환한다\n    return firstTombstone == -1 ? index : firstTombstone;\n}\n\n/* 조회 연산 */\nchar *get(HashMapOpenAddressing *hashMap, int key) {\n    // key에 해당하는 버킷 인덱스 탐색\n    int index = findBucket(hashMap, key);\n    // 키-값 쌍을 찾으면 해당 val을 반환한다\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        return hashMap-&gt;buckets[index]-&gt;val;\n    }\n    // 키-값 쌍이 없으면 빈 문자열을 반환한다\n    return \"\";\n}\n\n/* 추가 연산 */\nvoid put(HashMapOpenAddressing *hashMap, int key, char *val) {\n    // 적재율이 임계값을 초과하면 용량을 확장한다\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    // key에 해당하는 버킷 인덱스 탐색\n    int index = findBucket(hashMap, key);\n    // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        free(hashMap-&gt;buckets[index]-&gt;val);\n        hashMap-&gt;buckets[index]-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n        strcpy(hashMap-&gt;buckets[index]-&gt;val, val);\n        hashMap-&gt;buckets[index]-&gt;val[strlen(val)] = '\\0';\n        return;\n    }\n    // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n    Pair *pair = (Pair *)malloc(sizeof(Pair));\n    pair-&gt;key = key;\n    pair-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n    strcpy(pair-&gt;val, val);\n    pair-&gt;val[strlen(val)] = '\\0';\n\n    hashMap-&gt;buckets[index] = pair;\n    hashMap-&gt;size++;\n}\n\n/* 삭제 연산 */\nvoid removeItem(HashMapOpenAddressing *hashMap, int key) {\n    // key에 해당하는 버킷 인덱스 탐색\n    int index = findBucket(hashMap, key);\n    // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        Pair *pair = hashMap-&gt;buckets[index];\n        free(pair-&gt;val);\n        free(pair);\n        hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n        hashMap-&gt;size--;\n    }\n}\n\n/* 해시 테이블 용량 확장 */\nvoid extend(HashMapOpenAddressing *hashMap) {\n    // 원래 해시 테이블을 임시로 저장한다\n    Pair **bucketsTmp = hashMap-&gt;buckets;\n    int oldCapacity = hashMap-&gt;capacity;\n    // 용량을 확장한 새 해시 테이블을 초기화한다\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Pair **)calloc(hashMap-&gt;capacity, sizeof(Pair *));\n    hashMap-&gt;size = 0;\n    // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Pair *pair = bucketsTmp[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            put(hashMap, pair-&gt;key, pair-&gt;val);\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(bucketsTmp);\n}\n\n/* 해시 테이블 출력 */\nvoid print(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair == NULL) {\n            printf(\"NULL\\n\");\n        } else if (pair == hashMap-&gt;TOMBSTONE) {\n            printf(\"TOMBSTONE\\n\");\n        } else {\n            printf(\"%d -&gt; %s\\n\", pair-&gt;key, pair-&gt;val);\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.kt<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n    private var size: Int               // 키-값 쌍 개수\n    private var capacity: Int           // 해시 테이블 용량\n    private val loadThres: Double       // 용량 확장을 수행하는 적재율 임계값\n    private val extendRatio: Int        // 용량 확장 배수\n    private var buckets: Array&lt;Pair?&gt;   // 버킷 배열\n    private val TOMBSTONE: Pair         // 삭제 표시\n\n    /* 생성자 */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = arrayOfNulls(capacity)\n        TOMBSTONE = Pair(-1, \"-1\")\n    }\n\n    /* 해시 함수 */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* 적재율 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    fun findBucket(key: Int): Int {\n        var index = hashFunc(key)\n        var firstTombstone = -1\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while (buckets[index] != null) {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if (buckets[index]?.key == key) {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % capacity\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        return if (firstTombstone == -1) index else firstTombstone\n    }\n\n    /* 조회 연산 */\n    fun get(key: Int): String? {\n        // key에 해당하는 버킷 인덱스 탐색\n        val index = findBucket(key)\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]?._val\n        }\n        // 키-값 쌍이 없으면 null을 반환한다\n        return null\n    }\n\n    /* 추가 연산 */\n    fun put(key: Int, _val: String) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        val index = findBucket(key)\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]!!._val = _val\n            return\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        buckets[index] = Pair(key, _val)\n        size++\n    }\n\n    /* 삭제 연산 */\n    fun remove(key: Int) {\n        // key에 해당하는 버킷 인덱스 탐색\n        val index = findBucket(key)\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE\n            size--\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    fun extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        val bucketsTmp = buckets\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio\n        buckets = arrayOfNulls(capacity)\n        size = 0\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair._val)\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    fun print() {\n        for (pair in buckets) {\n            if (pair == null) {\n                println(\"null\")\n            } else if (pair == TOMBSTONE) {\n                println(\"TOMESTOME\")\n            } else {\n                println(\"${pair.key} -&gt; ${pair._val}\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.rb<pre><code>### 개방 주소법 해시 테이블 ###\nclass HashMapOpenAddressing\n  TOMBSTONE = Pair.new(-1, '-1') # 삭제 표시\n\n  ### 생성자 ###\n  def initialize\n    @size = 0 # 키-값 쌍 개수\n    @capacity = 4 # 해시 테이블 용량\n    @load_thres = 2.0 / 3.0 # 용량 확장을 수행하는 적재율 임계값\n    @extend_ratio = 2 # 용량 확장 배수\n    @buckets = Array.new(@capacity) # 버킷 배열\n  end\n\n  ### 해시 함수 ###\n  def hash_func(key)\n    key % @capacity\n  end\n\n  ### 적재율 ###\n  def load_factor\n    @size / @capacity\n  end\n\n  ### key에 해당하는 버킷 인덱스 탐색 ###\n  def find_bucket(key)\n    index = hash_func(key)\n    first_tombstone = -1\n    # 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n    while !@buckets[index].nil?\n      # key를 만나면 해당 버킷 인덱스를 반환한다\n      if @buckets[index].key == key\n        # 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n        if first_tombstone != -1\n          @buckets[first_tombstone] = @buckets[index]\n          @buckets[index] = TOMBSTONE\n          return first_tombstone # 옮긴 후의 버킷 인덱스를 반환한다\n        end\n        return index # 버킷 인덱스를 반환한다\n      end\n      # 처음 만난 삭제 표시를 기록한다\n      first_tombstone = index if first_tombstone == -1 &amp;&amp; @buckets[index] == TOMBSTONE\n      # 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n      index = (index + 1) % @capacity\n    end\n    # key가 없으면 추가할 위치의 인덱스를 반환한다\n    first_tombstone == -1 ? index : first_tombstone\n  end\n\n  ### 조회 연산 ###\n  def get(key)\n    # key에 해당하는 버킷 인덱스 탐색\n    index = find_bucket(key)\n    # 키-값 쌍을 찾으면 해당 val을 반환한다\n    return @buckets[index].val unless [nil, TOMBSTONE].include?(@buckets[index])\n    # 키-값 쌍이 없으면 nil을 반환한다\n    nil\n  end\n\n  ### 추가 연산 ###\n  def put(key, val)\n    # 적재율이 임계값을 초과하면 용량을 확장한다\n    extend if load_factor &gt; @load_thres\n    # key에 해당하는 버킷 인덱스 탐색\n    index = find_bucket(key)\n    # 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n    unless [nil, TOMBSTONE].include?(@buckets[index])\n      @buckets[index].val = val\n      return\n    end\n    # 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n    @buckets[index] = Pair.new(key, val)\n    @size += 1\n  end\n\n  ### 삭제 연산 ###\n  def remove(key)\n    # key에 해당하는 버킷 인덱스 탐색\n    index = find_bucket(key)\n    # 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n    unless [nil, TOMBSTONE].include?(@buckets[index])\n      @buckets[index] = TOMBSTONE\n      @size -= 1\n    end\n  end\n\n  ### 해시 테이블 용량 확장 ###\n  def extend\n    # 원래 해시 테이블을 임시로 저장한다\n    buckets_tmp = @buckets\n    # 용량을 확장한 새 해시 테이블을 초기화한다\n    @capacity *= @extend_ratio\n    @buckets = Array.new(@capacity)\n    @size = 0\n    # 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for pair in buckets_tmp\n      put(pair.key, pair.val) unless [nil, TOMBSTONE].include?(pair)\n    end\n  end\n\n  ### 해시 테이블 출력 ###\n  def print\n    for pair in @buckets\n      if pair.nil?\n        puts \"Nil\"\n      elsif pair == TOMBSTONE\n        puts \"TOMBSTONE\"\n      else\n        puts \"#{pair.key} -&gt; #{pair.val}\"\n      end\n    end\n  end\nend\n</code></pre>","path":["제 6 장   해시 테이블","6.2   해시 충돌"],"tags":[]},{"location":"chapter_hashing/hash_collision/#2","level":3,"title":"2.   이차 탐사","text":"<p>이차 탐사는 선형 탐사처럼 개방 주소법의 대표적인 전략이다. 충돌이 발생하면 고정 간격으로 이동하는 대신 ‘탐사 횟수의 제곱’만큼, 즉 \\(1, 4, 9, \\dots\\)칸을 이동한다.</p> <p>이차 탐사의 주요 장점은 다음과 같다.</p> <ul> <li>탐사 횟수의 제곱만큼 떨어진 위치로 이동하여 선형 탐사의 군집 현상을 완화하려 한다.</li> <li>더 먼 거리로 건너뛰며 빈 위치를 찾으므로 데이터를 더 고르게 분포시키는 데 도움이 된다.</li> </ul> <p>하지만 이차 탐사도 완벽하지는 않다.</p> <ul> <li>일부 위치가 다른 위치보다 더 쉽게 채워지는 군집 현상이 여전히 존재한다.</li> <li>제곱의 증가 때문에 해시 테이블 전체를 탐사하지 못할 수 있다. 즉, 빈 버킷이 있어도 그곳에 도달하지 못할 수 있다.</li> </ul>","path":["제 6 장   해시 테이블","6.2   해시 충돌"],"tags":[]},{"location":"chapter_hashing/hash_collision/#3","level":3,"title":"3.   다중 해싱","text":"<p>이름 그대로 여러 해시 함수 \\(f_1(x)\\), \\(f_2(x)\\), \\(f_3(x)\\), \\(\\dots\\)를 사용하여 탐사하는 방법이다.</p> <ul> <li>원소 삽입: 해시 함수 \\(f_1(x)\\)에서 충돌하면 \\(f_2(x)\\)를 시도한다. 이런 식으로 빈 위치를 찾을 때까지 계속한 뒤 원소를 넣는다.</li> <li>원소 탐색: 같은 해시 함수 순서로 탐색하여 목표 원소를 찾으면 반환한다. 빈 위치를 만나거나 모든 함수를 시도했다면 원소가 없다는 뜻이므로 <code>None</code>을 반환한다.</li> </ul> <p>선형 탐사보다 군집 현상이 덜 발생하지만, 여러 해시 함수로 인해 추가 계산이 필요하다.</p> <p>참고</p> <p>개방 주소법의 선형 탐사, 이차 탐사, 다중 해싱 모두 ‘원소를 곧바로 삭제할 수 없다’는 문제가 있다는 점에 주의하자.</p>","path":["제 6 장   해시 테이블","6.2   해시 충돌"],"tags":[]},{"location":"chapter_hashing/hash_collision/#623","level":2,"title":"6.2.3   프로그래밍 언어의 선택","text":"<p>프로그래밍 언어마다 해시 테이블 구현 전략이 다르다. 몇 가지 예를 살펴보자.</p> <ul> <li>Python은 개방 주소법을 사용한다. 딕셔너리 <code>dict</code>는 의사 난수를 이용해 탐사한다.</li> <li>Java는 체이닝을 사용한다. JDK 1.8부터는 <code>HashMap</code> 내부 배열 길이가 64 이상이고 연결 리스트 길이가 8 이상이면, 탐색 성능을 높이기 위해 연결 리스트를 레드-블랙 트리로 바꾼다.</li> <li>Go 1.24부터 내장 <code>map</code>은 기본적으로 Swiss Table 기반의 개방 주소법을 사용한다. Go 1.23 이하에서는 체이닝을 사용하며, 버킷 하나에 키-값 쌍을 최대 8개 저장하고 용량을 넘으면 오버플로 버킷을 연결한다.</li> </ul>","path":["제 6 장   해시 테이블","6.2   해시 충돌"],"tags":[]},{"location":"chapter_hashing/hash_map/","level":1,"title":"6.1   해시 테이블","text":"<p>해시 테이블(hash table)은 해시 표라고도 하며, 키 <code>key</code>와 값 <code>value</code> 사이의 대응 관계를 만들어 원소를 효율적으로 조회한다. 구체적으로 키 <code>key</code>를 입력하면 해당 값 <code>value</code>를 \\(O(1)\\) 시간에 얻을 수 있다.</p> <p>아래 그림(6-1)과 같이 학생 \\(n\\)명에게 각각 ‘이름’과 ‘학번’이라는 두 데이터가 있다고 하자. ‘학번을 입력하면 해당 이름을 반환하는’ 조회 기능을 구현하려면 그림의 해시 테이블을 사용할 수 있다.</p> <p></p> <p> 그림 6-1   해시 테이블의 추상적 표현 </p> <p>해시 테이블 외에 배열과 연결 리스트로도 조회 기능을 구현할 수 있다. 효율 비교는 아래 표(6-1)와 같다.</p> <ul> <li>원소 추가: 배열(연결 리스트)의 끝에 원소를 추가하면 되므로 \\(O(1)\\) 시간이 걸린다.</li> <li>원소 조회: 배열(연결 리스트)이 정렬되어 있지 않으므로 모든 원소를 순회해야 하며 \\(O(n)\\) 시간이 걸린다.</li> <li>원소 삭제: 먼저 원소를 찾은 뒤 배열(연결 리스트)에서 삭제해야 하므로 \\(O(n)\\) 시간이 걸린다.</li> </ul> <p> 표 6-1   원소 조회 효율 비교 </p> 배열 연결 리스트 해시 테이블 원소 탐색 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) 원소 추가 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) 원소 삭제 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) <p>이처럼 해시 테이블에서는 삽입, 삭제, 탐색, 갱신의 시간 복잡도가 모두 \\(O(1)\\)로 매우 효율적이다.</p>","path":["제 6 장   해시 테이블","6.1   해시 테이블"],"tags":[]},{"location":"chapter_hashing/hash_map/#611","level":2,"title":"6.1.1   해시 테이블의 기본 연산","text":"<p>대표적인 연산에는 초기화, 조회, 키-값 쌍 추가와 삭제 등이 있다. 예제 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map.py<pre><code># 해시 테이블 초기화\nhmap: dict = {}\n\n# 추가 연산\n# 해시 테이블에 키-값 쌍 (key, value) 추가\nhmap[12836] = \"小哈\"\nhmap[15937] = \"小啰\"\nhmap[16750] = \"小算\"\nhmap[13276] = \"小法\"\nhmap[10583] = \"小鸭\"\n\n# 조회 연산\n# 해시 테이블에 키 key를 입력하여 값 value를 얻음\nname: str = hmap[15937]\n\n# 삭제 연산\n# 해시 테이블에서 키-값 쌍 (key, value) 삭제\nhmap.pop(10583)\n</code></pre> hash_map.cpp<pre><code>/* 해시 테이블 초기화 */\nunordered_map&lt;int, string&gt; map;\n\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap[12836] = \"小哈\";\nmap[15937] = \"小啰\";\nmap[16750] = \"小算\";\nmap[13276] = \"小法\";\nmap[10583] = \"小鸭\";\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nstring name = map[15937];\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.erase(10583);\n</code></pre> hash_map.java<pre><code>/* 해시 테이블 초기화 */\nMap&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap.put(12836, \"小哈\");\nmap.put(15937, \"小啰\");\nmap.put(16750, \"小算\");\nmap.put(13276, \"小法\");\nmap.put(10583, \"小鸭\");\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nString name = map.get(15937);\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.remove(10583);\n</code></pre> hash_map.cs<pre><code>/* 해시 테이블 초기화 */\nDictionary&lt;int, string&gt; map = new() {\n    /* 추가 연산 */\n    // 해시 테이블에 키-값 쌍 (key, value) 추가\n    { 12836, \"小哈\" },\n    { 15937, \"小啰\" },\n    { 16750, \"小算\" },\n    { 13276, \"小法\" },\n    { 10583, \"小鸭\" }\n};\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nstring name = map[15937];\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.Remove(10583);\n</code></pre> hash_map_test.go<pre><code>/* 해시 테이블 초기화 */\nhmap := make(map[int]string)\n\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nhmap[12836] = \"小哈\"\nhmap[15937] = \"小啰\"\nhmap[16750] = \"小算\"\nhmap[13276] = \"小法\"\nhmap[10583] = \"小鸭\"\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nname := hmap[15937]\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\ndelete(hmap, 10583)\n</code></pre> hash_map.swift<pre><code>/* 해시 테이블 초기화 */\nvar map: [Int: String] = [:]\n\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap[12836] = \"小哈\"\nmap[15937] = \"小啰\"\nmap[16750] = \"小算\"\nmap[13276] = \"小法\"\nmap[10583] = \"小鸭\"\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nlet name = map[15937]!\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.removeValue(forKey: 10583)\n</code></pre> hash_map.js<pre><code>/* 해시 테이블 초기화 */\nconst map = new Map();\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap.set(12836, '小哈');\nmap.set(15937, '小啰');\nmap.set(16750, '小算');\nmap.set(13276, '小法');\nmap.set(10583, '小鸭');\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nlet name = map.get(15937);\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.delete(10583);\n</code></pre> hash_map.ts<pre><code>/* 해시 테이블 초기화 */\nconst map = new Map&lt;number, string&gt;();\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap.set(12836, '小哈');\nmap.set(15937, '小啰');\nmap.set(16750, '小算');\nmap.set(13276, '小法');\nmap.set(10583, '小鸭');\nconsole.info('\\n추가한 뒤의 해시 테이블\\nKey -&gt; Value');\nconsole.info(map);\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nlet name = map.get(15937);\nconsole.info('\\n학번 15937을 입력하여 찾은 이름 ' + name);\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.delete(10583);\nconsole.info('\\n10583을 삭제한 뒤의 해시 테이블\\nKey -&gt; Value');\nconsole.info(map);\n</code></pre> hash_map.dart<pre><code>/* 해시 테이블 초기화 */\nMap&lt;int, String&gt; map = {};\n\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap[12836] = \"小哈\";\nmap[15937] = \"小啰\";\nmap[16750] = \"小算\";\nmap[13276] = \"小法\";\nmap[10583] = \"小鸭\";\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nString name = map[15937];\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.remove(10583);\n</code></pre> hash_map.rs<pre><code>use std::collections::HashMap;\n\n/* 해시 테이블 초기화 */\nlet mut map: HashMap&lt;i32, String&gt; = HashMap::new();\n\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap.insert(12836, \"小哈\".to_string());\nmap.insert(15937, \"小啰\".to_string());\nmap.insert(16750, \"小算\".to_string());\nmap.insert(13279, \"小法\".to_string());\nmap.insert(10583, \"小鸭\".to_string());\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nlet _name: Option&lt;&amp;String&gt; = map.get(&amp;15937);\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nlet _removed_value: Option&lt;String&gt; = map.remove(&amp;10583);\n</code></pre> hash_map.c<pre><code>// C는 내장 해시 테이블을 제공하지 않는다\n</code></pre> hash_map.kt<pre><code>/* 해시 테이블 초기화 */\nval map = HashMap&lt;Int,String&gt;()\n\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap[12836] = \"小哈\"\nmap[15937] = \"小啰\"\nmap[16750] = \"小算\"\nmap[13276] = \"小法\"\nmap[10583] = \"小鸭\"\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nval name = map[15937]\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.remove(10583)\n</code></pre> hash_map.rb<pre><code># 해시 테이블 초기화\nhmap = {}\n\n# 추가 연산\n# 해시 테이블에 키-값 쌍 (key, value) 추가\nhmap[12836] = \"小哈\"\nhmap[15937] = \"小啰\"\nhmap[16750] = \"小算\"\nhmap[13276] = \"小法\"\nhmap[10583] = \"小鸭\"\n\n# 조회 연산\n# 해시 테이블에 키 key를 입력하여 값 value를 얻음\nname = hmap[15937]\n\n# 삭제 연산\n# 해시 테이블에서 키-값 쌍 (key, value) 삭제\nhmap.delete(10583)\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>해시 테이블을 순회하는 대표적인 방식은 키-값 쌍 순회, 키 순회, 값 순회의 세 가지다. 예제 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map.py<pre><code># 해시 테이블 순회\n# 키-값 쌍 key-&gt;value 순회\nfor key, value in hmap.items():\n    print(key, \"-&gt;\", value)\n# 키 key만 순회\nfor key in hmap.keys():\n    print(key)\n# 값 value만 순회\nfor value in hmap.values():\n    print(value)\n</code></pre> hash_map.cpp<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 key-&gt;value 순회\nfor (auto kv: map) {\n    cout &lt;&lt; kv.first &lt;&lt; \" -&gt; \" &lt;&lt; kv.second &lt;&lt; endl;\n}\n// 이터레이터로 key-&gt;value 순회\nfor (auto iter = map.begin(); iter != map.end(); iter++) {\n    cout &lt;&lt; iter-&gt;first &lt;&lt; \"-&gt;\" &lt;&lt; iter-&gt;second &lt;&lt; endl;\n}\n</code></pre> hash_map.java<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 key-&gt;value 순회\nfor (Map.Entry &lt;Integer, String&gt; kv: map.entrySet()) {\n    System.out.println(kv.getKey() + \" -&gt; \" + kv.getValue());\n}\n// 키 key만 순회\nfor (int key: map.keySet()) {\n    System.out.println(key);\n}\n// 값 value만 순회\nfor (String val: map.values()) {\n    System.out.println(val);\n}\n</code></pre> hash_map.cs<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 Key-&gt;Value 순회\nforeach (var kv in map) {\n    Console.WriteLine(kv.Key + \" -&gt; \" + kv.Value);\n}\n// 키 key만 순회\nforeach (int key in map.Keys) {\n    Console.WriteLine(key);\n}\n// 값 value만 순회\nforeach (string val in map.Values) {\n    Console.WriteLine(val);\n}\n</code></pre> hash_map_test.go<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 key-&gt;value 순회\nfor key, value := range hmap {\n    fmt.Println(key, \"-&gt;\", value)\n}\n// 키 key만 순회\nfor key := range hmap {\n    fmt.Println(key)\n}\n// 값 value만 순회\nfor _, value := range hmap {\n    fmt.Println(value)\n}\n</code></pre> hash_map.swift<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 Key-&gt;Value 순회\nfor (key, value) in map {\n    print(\"\\(key) -&gt; \\(value)\")\n}\n// 키 Key만 순회\nfor key in map.keys {\n    print(key)\n}\n// 값 Value만 순회\nfor value in map.values {\n    print(value)\n}\n</code></pre> hash_map.js<pre><code>/* 해시 테이블 순회 */\nconsole.info('\\n키-값 쌍 Key-&gt;Value 순회');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n키 Key만 순회');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\n값 Value만 순회');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.ts<pre><code>/* 해시 테이블 순회 */\nconsole.info('\\n키-값 쌍 Key-&gt;Value 순회');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n키 Key만 순회');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\n값 Value만 순회');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.dart<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 Key-&gt;Value 순회\nmap.forEach((key, value) {\n  print('$key -&gt; $value');\n});\n\n// 키 Key만 순회\nmap.keys.forEach((key) {\n  print(key);\n});\n\n// 값 Value만 순회\nmap.values.forEach((value) {\n  print(value);\n});\n</code></pre> hash_map.rs<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 Key-&gt;Value 순회\nfor (key, value) in &amp;map {\n    println!(\"{key} -&gt; {value}\");\n}\n\n// 키 Key만 순회\nfor key in map.keys() {\n    println!(\"{key}\");\n}\n\n// 값 Value만 순회\nfor value in map.values() {\n    println!(\"{value}\");\n}\n</code></pre> hash_map.c<pre><code>// C는 내장 해시 테이블을 제공하지 않는다\n</code></pre> hash_map.kt<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 key-&gt;value 순회\nfor ((key, value) in map) {\n    println(\"$key -&gt; $value\")\n}\n// 키 key만 순회\nfor (key in map.keys) {\n    println(key)\n}\n// 값 value만 순회\nfor (_val in map.values) {\n    println(_val)\n}\n</code></pre> hash_map.rb<pre><code># 해시 테이블 순회\n# 키-값 쌍 key-&gt;value 순회\nhmap.entries.each { |key, value| puts \"#{key} -&gt; #{value}\" }\n\n# 키 key만 순회\nhmap.keys.each { |key| puts key }\n\n# 값 value만 순회\nhmap.values.each { |val| puts val }\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 6 장   해시 테이블","6.1   해시 테이블"],"tags":[]},{"location":"chapter_hashing/hash_map/#612","level":2,"title":"6.1.2   간단한 해시 테이블 구현","text":"<p>먼저 가장 간단한 경우로 배열 하나만 사용하여 해시 테이블을 구현해 보자. 해시 테이블에서는 배열의 각 자리를 버킷(bucket)이라고 하며, 버킷마다 키-값 쌍 하나를 저장할 수 있다. 따라서 조회 연산은 <code>key</code>에 해당하는 버킷을 찾고 그 안에서 <code>value</code>를 얻는 것이다.</p> <p>그렇다면 <code>key</code>로 해당 버킷을 어떻게 찾을까? 이를 위해 해시 함수(hash function)를 사용한다. 해시 함수는 큰 입력 공간을 작은 출력 공간에 매핑한다. 해시 테이블에서 입력 공간은 모든 <code>key</code>이고, 출력 공간은 모든 버킷(배열 인덱스)이다. 즉, <code>key</code>를 입력하면 해시 함수를 통해 그 키-값 쌍이 배열에 저장되는 위치를 얻을 수 있다.</p> <p><code>key</code>를 입력했을 때 해시 함수의 계산 과정은 다음 두 단계로 나뉜다.</p> <ol> <li>어떤 해시 알고리즘 <code>hash()</code>로 해시값을 계산한다.</li> <li>해시값을 버킷 수(배열 길이) <code>capacity</code>로 나눈 나머지를 구하여 <code>key</code>에 해당하는 버킷(배열 인덱스) <code>index</code>를 얻는다.</li> </ol> <pre><code>index = hash(key) % capacity\n</code></pre> <p>그런 다음 <code>index</code>로 해시 테이블의 해당 버킷에 접근하여 <code>value</code>를 얻는다.</p> <p>배열 길이가 <code>capacity = 100</code>이고 해시 알고리즘이 <code>hash(key) = key</code>이면, 해시 함수는 <code>key % 100</code>이다. 아래 그림(6-2)은 학번을 <code>key</code>, 이름을 <code>value</code>로 삼아 해시 함수의 작동 원리를 보여 준다.</p> <p></p> <p> 그림 6-2   해시 함수의 작동 원리 </p> <p>다음 코드는 간단한 해시 테이블을 구현한다. <code>key</code>와 <code>value</code>를 <code>Pair</code> 클래스로 묶어 키-값 쌍을 표현한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_hash_map.py<pre><code>class Pair:\n    \"\"\"키-값 쌍\"\"\"\n\n    def __init__(self, key: int, val: str):\n        self.key = key\n        self.val = val\n\nclass ArrayHashMap:\n    \"\"\"배열로 구현한 해시 테이블\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        # 버킷 100개를 포함하는 배열을 초기화한다\n        self.buckets: list[Pair | None] = [None] * 100\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"해시 함수\"\"\"\n        index = key % 100\n        return index\n\n    def get(self, key: int) -&gt; str | None:\n        \"\"\"조회 연산\"\"\"\n        index: int = self.hash_func(key)\n        pair: Pair = self.buckets[index]\n        if pair is None:\n            return None\n        return pair.val\n\n    def put(self, key: int, val: str):\n        \"\"\"추가 및 갱신 연산\"\"\"\n        pair = Pair(key, val)\n        index: int = self.hash_func(key)\n        self.buckets[index] = pair\n\n    def remove(self, key: int):\n        \"\"\"삭제 연산\"\"\"\n        index: int = self.hash_func(key)\n        # None으로 설정하여 삭제를 나타낸다\n        self.buckets[index] = None\n\n    def entry_set(self) -&gt; list[Pair]:\n        \"\"\"모든 키-값 쌍 반환\"\"\"\n        result: list[Pair] = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair)\n        return result\n\n    def key_set(self) -&gt; list[int]:\n        \"\"\"모든 키 반환\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.key)\n        return result\n\n    def value_set(self) -&gt; list[str]:\n        \"\"\"모든 값 반환\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.val)\n        return result\n\n    def print(self):\n        \"\"\"해시 테이블 출력\"\"\"\n        for pair in self.buckets:\n            if pair is not None:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> array_hash_map.cpp<pre><code>/* 키-값 쌍 */\nstruct Pair {\n  public:\n    int key;\n    string val;\n    Pair(int key, string val) {\n        this-&gt;key = key;\n        this-&gt;val = val;\n    }\n};\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n  private:\n    vector&lt;Pair *&gt; buckets;\n\n  public:\n    ArrayHashMap() {\n        // 버킷 100개를 포함하는 배열을 초기화한다\n        buckets = vector&lt;Pair *&gt;(100);\n    }\n\n    ~ArrayHashMap() {\n        // 메모리 해제\n        for (const auto &amp;bucket : buckets) {\n            delete bucket;\n        }\n        buckets.clear();\n    }\n\n    /* 해시 함수 */\n    int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* 조회 연산 */\n    string get(int key) {\n        int index = hashFunc(key);\n        Pair *pair = buckets[index];\n        if (pair == nullptr)\n            return \"\";\n        return pair-&gt;val;\n    }\n\n    /* 추가 연산 */\n    void put(int key, string val) {\n        Pair *pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* 삭제 연산 */\n    void remove(int key) {\n        int index = hashFunc(key);\n        // 메모리를 해제하고 nullptr로 설정한다\n        delete buckets[index];\n        buckets[index] = nullptr;\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    vector&lt;Pair *&gt; pairSet() {\n        vector&lt;Pair *&gt; pairSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                pairSet.push_back(pair);\n            }\n        }\n        return pairSet;\n    }\n\n    /* 모든 키 반환 */\n    vector&lt;int&gt; keySet() {\n        vector&lt;int&gt; keySet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                keySet.push_back(pair-&gt;key);\n            }\n        }\n        return keySet;\n    }\n\n    /* 모든 값 반환 */\n    vector&lt;string&gt; valueSet() {\n        vector&lt;string&gt; valueSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                valueSet.push_back(pair-&gt;val);\n            }\n        }\n        return valueSet;\n    }\n\n    /* 해시 테이블 출력 */\n    void print() {\n        for (Pair *kv : pairSet()) {\n            cout &lt;&lt; kv-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; kv-&gt;val &lt;&lt; endl;\n        }\n    }\n};\n</code></pre> array_hash_map.java<pre><code>/* 키-값 쌍 */\nclass Pair {\n    public int key;\n    public String val;\n\n    public Pair(int key, String val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n    private List&lt;Pair&gt; buckets;\n\n    public ArrayHashMap() {\n        // 버킷 100개를 포함하는 배열을 초기화한다\n        buckets = new ArrayList&lt;&gt;();\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.add(null);\n        }\n    }\n\n    /* 해시 함수 */\n    private int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* 조회 연산 */\n    public String get(int key) {\n        int index = hashFunc(key);\n        Pair pair = buckets.get(index);\n        if (pair == null)\n            return null;\n        return pair.val;\n    }\n\n    /* 추가 연산 */\n    public void put(int key, String val) {\n        Pair pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets.set(index, pair);\n    }\n\n    /* 삭제 연산 */\n    public void remove(int key) {\n        int index = hashFunc(key);\n        // null로 설정하여 삭제를 나타낸다\n        buckets.set(index, null);\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    public List&lt;Pair&gt; pairSet() {\n        List&lt;Pair&gt; pairSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                pairSet.add(pair);\n        }\n        return pairSet;\n    }\n\n    /* 모든 키 반환 */\n    public List&lt;Integer&gt; keySet() {\n        List&lt;Integer&gt; keySet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                keySet.add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* 모든 값 반환 */\n    public List&lt;String&gt; valueSet() {\n        List&lt;String&gt; valueSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                valueSet.add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* 해시 테이블 출력 */\n    public void print() {\n        for (Pair kv : pairSet()) {\n            System.out.println(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.cs<pre><code>/* 키-값 쌍 int-&gt;string */\nclass Pair(int key, string val) {\n    public int key = key;\n    public string val = val;\n}\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n    List&lt;Pair?&gt; buckets;\n    public ArrayHashMap() {\n        // 버킷 100개를 포함하는 배열을 초기화한다\n        buckets = [];\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.Add(null);\n        }\n    }\n\n    /* 해시 함수 */\n    int HashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* 조회 연산 */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        Pair? pair = buckets[index];\n        if (pair == null) return null;\n        return pair.val;\n    }\n\n    /* 추가 연산 */\n    public void Put(int key, string val) {\n        Pair pair = new(key, val);\n        int index = HashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* 삭제 연산 */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // null로 설정하여 삭제를 나타낸다\n        buckets[index] = null;\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    public List&lt;Pair&gt; PairSet() {\n        List&lt;Pair&gt; pairSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                pairSet.Add(pair);\n        }\n        return pairSet;\n    }\n\n    /* 모든 키 반환 */\n    public List&lt;int&gt; KeySet() {\n        List&lt;int&gt; keySet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                keySet.Add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* 모든 값 반환 */\n    public List&lt;string&gt; ValueSet() {\n        List&lt;string&gt; valueSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                valueSet.Add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* 해시 테이블 출력 */\n    public void Print() {\n        foreach (Pair kv in PairSet()) {\n            Console.WriteLine(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.go<pre><code>/* 키-값 쌍 */\ntype pair struct {\n    key int\n    val string\n}\n\n/* 배열로 구현한 해시 테이블 */\ntype arrayHashMap struct {\n    buckets []*pair\n}\n\n/* 해시 테이블 초기화 */\nfunc newArrayHashMap() *arrayHashMap {\n    // 버킷 100개를 포함하는 배열을 초기화한다\n    buckets := make([]*pair, 100)\n    return &amp;arrayHashMap{buckets: buckets}\n}\n\n/* 해시 함수 */\nfunc (a *arrayHashMap) hashFunc(key int) int {\n    index := key % 100\n    return index\n}\n\n/* 조회 연산 */\nfunc (a *arrayHashMap) get(key int) string {\n    index := a.hashFunc(key)\n    pair := a.buckets[index]\n    if pair == nil {\n        return \"찾을 수 없음\"\n    }\n    return pair.val\n}\n\n/* 추가 연산 */\nfunc (a *arrayHashMap) put(key int, val string) {\n    pair := &amp;pair{key: key, val: val}\n    index := a.hashFunc(key)\n    a.buckets[index] = pair\n}\n\n/* 삭제 연산 */\nfunc (a *arrayHashMap) remove(key int) {\n    index := a.hashFunc(key)\n    // nil로 설정하여 삭제를 나타낸다\n    a.buckets[index] = nil\n}\n\n/* 모든 키-값 쌍 반환 */\nfunc (a *arrayHashMap) pairSet() []*pair {\n    var pairs []*pair\n    for _, pair := range a.buckets {\n        if pair != nil {\n            pairs = append(pairs, pair)\n        }\n    }\n    return pairs\n}\n\n/* 모든 키 반환 */\nfunc (a *arrayHashMap) keySet() []int {\n    var keys []int\n    for _, pair := range a.buckets {\n        if pair != nil {\n            keys = append(keys, pair.key)\n        }\n    }\n    return keys\n}\n\n/* 모든 값 반환 */\nfunc (a *arrayHashMap) valueSet() []string {\n    var values []string\n    for _, pair := range a.buckets {\n        if pair != nil {\n            values = append(values, pair.val)\n        }\n    }\n    return values\n}\n\n/* 해시 테이블 출력 */\nfunc (a *arrayHashMap) print() {\n    for _, pair := range a.buckets {\n        if pair != nil {\n            fmt.Println(pair.key, \"-&gt;\", pair.val)\n        }\n    }\n}\n</code></pre> array_hash_map.swift<pre><code>/* 키-값 쌍 */\nclass Pair: Equatable {\n    public var key: Int\n    public var val: String\n\n    public init(key: Int, val: String) {\n        self.key = key\n        self.val = val\n    }\n\n    public static func == (lhs: Pair, rhs: Pair) -&gt; Bool {\n        lhs.key == rhs.key &amp;&amp; lhs.val == rhs.val\n    }\n}\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n    private var buckets: [Pair?]\n\n    init() {\n        // 버킷 100개를 포함하는 배열을 초기화한다\n        buckets = Array(repeating: nil, count: 100)\n    }\n\n    /* 해시 함수 */\n    private func hashFunc(key: Int) -&gt; Int {\n        let index = key % 100\n        return index\n    }\n\n    /* 조회 연산 */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let pair = buckets[index]\n        return pair?.val\n    }\n\n    /* 추가 연산 */\n    func put(key: Int, val: String) {\n        let pair = Pair(key: key, val: val)\n        let index = hashFunc(key: key)\n        buckets[index] = pair\n    }\n\n    /* 삭제 연산 */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        // nil로 설정하여 삭제를 나타낸다\n        buckets[index] = nil\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    func pairSet() -&gt; [Pair] {\n        buckets.compactMap { $0 }\n    }\n\n    /* 모든 키 반환 */\n    func keySet() -&gt; [Int] {\n        buckets.compactMap { $0?.key }\n    }\n\n    /* 모든 값 반환 */\n    func valueSet() -&gt; [String] {\n        buckets.compactMap { $0?.val }\n    }\n\n    /* 해시 테이블 출력 */\n    func print() {\n        for pair in pairSet() {\n            Swift.print(\"\\(pair.key) -&gt; \\(pair.val)\")\n        }\n    }\n}\n</code></pre> array_hash_map.js<pre><code>/* 키-값 쌍 Number -&gt; String */\nclass Pair {\n    constructor(key, val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n    #buckets;\n    constructor() {\n        // 버킷 100개를 포함하는 배열을 초기화한다\n        this.#buckets = new Array(100).fill(null);\n    }\n\n    /* 해시 함수 */\n    #hashFunc(key) {\n        return key % 100;\n    }\n\n    /* 조회 연산 */\n    get(key) {\n        let index = this.#hashFunc(key);\n        let pair = this.#buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* 추가 연산 */\n    set(key, val) {\n        let index = this.#hashFunc(key);\n        this.#buckets[index] = new Pair(key, val);\n    }\n\n    /* 삭제 연산 */\n    delete(key) {\n        let index = this.#hashFunc(key);\n        // null로 설정하여 삭제를 나타낸다\n        this.#buckets[index] = null;\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    entries() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* 모든 키 반환 */\n    keys() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* 모든 값 반환 */\n    values() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* 해시 테이블 출력 */\n    print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.ts<pre><code>/* 키-값 쌍 Number -&gt; String */\nclass Pair {\n    public key: number;\n    public val: string;\n\n    constructor(key: number, val: string) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n    private readonly buckets: (Pair | null)[];\n\n    constructor() {\n        // 버킷 100개를 포함하는 배열을 초기화한다\n        this.buckets = new Array(100).fill(null);\n    }\n\n    /* 해시 함수 */\n    private hashFunc(key: number): number {\n        return key % 100;\n    }\n\n    /* 조회 연산 */\n    public get(key: number): string | null {\n        let index = this.hashFunc(key);\n        let pair = this.buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* 추가 연산 */\n    public set(key: number, val: string) {\n        let index = this.hashFunc(key);\n        this.buckets[index] = new Pair(key, val);\n    }\n\n    /* 삭제 연산 */\n    public delete(key: number) {\n        let index = this.hashFunc(key);\n        // null로 설정하여 삭제를 나타낸다\n        this.buckets[index] = null;\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    public entries(): (Pair | null)[] {\n        let arr: (Pair | null)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* 모든 키 반환 */\n    public keys(): (number | undefined)[] {\n        let arr: (number | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* 모든 값 반환 */\n    public values(): (string | undefined)[] {\n        let arr: (string | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* 해시 테이블 출력 */\n    public print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.dart<pre><code>/* 키-값 쌍 */\nclass Pair {\n  int key;\n  String val;\n  Pair(this.key, this.val);\n}\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n  late List&lt;Pair?&gt; _buckets;\n\n  ArrayHashMap() {\n    // 버킷 100개를 포함하는 배열을 초기화한다\n    _buckets = List.filled(100, null);\n  }\n\n  /* 해시 함수 */\n  int _hashFunc(int key) {\n    final int index = key % 100;\n    return index;\n  }\n\n  /* 조회 연산 */\n  String? get(int key) {\n    final int index = _hashFunc(key);\n    final Pair? pair = _buckets[index];\n    if (pair == null) {\n      return null;\n    }\n    return pair.val;\n  }\n\n  /* 추가 연산 */\n  void put(int key, String val) {\n    final Pair pair = Pair(key, val);\n    final int index = _hashFunc(key);\n    _buckets[index] = pair;\n  }\n\n  /* 삭제 연산 */\n  void remove(int key) {\n    final int index = _hashFunc(key);\n    _buckets[index] = null;\n  }\n\n  /* 모든 키-값 쌍 반환 */\n  List&lt;Pair&gt; pairSet() {\n    List&lt;Pair&gt; pairSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        pairSet.add(pair);\n      }\n    }\n    return pairSet;\n  }\n\n  /* 모든 키 반환 */\n  List&lt;int&gt; keySet() {\n    List&lt;int&gt; keySet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        keySet.add(pair.key);\n      }\n    }\n    return keySet;\n  }\n\n  /* 모든 값 반환 */\n  List&lt;String&gt; values() {\n    List&lt;String&gt; valueSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        valueSet.add(pair.val);\n      }\n    }\n    return valueSet;\n  }\n\n  /* 해시 테이블 출력 */\n  void printHashMap() {\n    for (final Pair kv in pairSet()) {\n      print(\"${kv.key} -&gt; ${kv.val}\");\n    }\n  }\n}\n</code></pre> array_hash_map.rs<pre><code>/* 키-값 쌍 */\n#[derive(Debug, Clone, PartialEq)]\npub struct Pair {\n    pub key: i32,\n    pub val: String,\n}\n\n/* 배열로 구현한 해시 테이블 */\npub struct ArrayHashMap {\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;,\n}\n\nimpl ArrayHashMap {\n    pub fn new() -&gt; ArrayHashMap {\n        // 버킷 100개를 포함하는 배열을 초기화한다\n        Self {\n            buckets: vec![None; 100],\n        }\n    }\n\n    /* 해시 함수 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % 100\n    }\n\n    /* 조회 연산 */\n    pub fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;String&gt; {\n        let index = self.hash_func(key);\n        self.buckets[index].as_ref().map(|pair| &amp;pair.val)\n    }\n\n    /* 추가 연산 */\n    pub fn put(&amp;mut self, key: i32, val: &amp;str) {\n        let index = self.hash_func(key);\n        self.buckets[index] = Some(Pair {\n            key,\n            val: val.to_string(),\n        });\n    }\n\n    /* 삭제 연산 */\n    pub fn remove(&amp;mut self, key: i32) {\n        let index = self.hash_func(key);\n        // None으로 설정하여 삭제를 나타낸다\n        self.buckets[index] = None;\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    pub fn entry_set(&amp;self) -&gt; Vec&lt;&amp;Pair&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref())\n            .collect()\n    }\n\n    /* 모든 키 반환 */\n    pub fn key_set(&amp;self) -&gt; Vec&lt;&amp;i32&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.key))\n            .collect()\n    }\n\n    /* 모든 값 반환 */\n    pub fn value_set(&amp;self) -&gt; Vec&lt;&amp;String&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.val))\n            .collect()\n    }\n\n    /* 해시 테이블 출력 */\n    pub fn print(&amp;self) {\n        for pair in self.entry_set() {\n            println!(\"{} -&gt; {}\", pair.key, pair.val);\n        }\n    }\n}\n</code></pre> array_hash_map.c<pre><code>/* 키-값 쌍 int-&gt;string */\ntypedef struct {\n    int key;\n    char *val;\n} Pair;\n\n/* 배열로 구현한 해시 테이블 */\ntypedef struct {\n    Pair *buckets[MAX_SIZE];\n} ArrayHashMap;\n\n/* 생성자 */\nArrayHashMap *newArrayHashMap() {\n    ArrayHashMap *hmap = malloc(sizeof(ArrayHashMap));\n    for (int i=0; i &lt; MAX_SIZE; i++) {\n        hmap-&gt;buckets[i] = NULL;\n    }\n    return hmap;\n}\n\n/* 소멸자 */\nvoid delArrayHashMap(ArrayHashMap *hmap) {\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            free(hmap-&gt;buckets[i]-&gt;val);\n            free(hmap-&gt;buckets[i]);\n        }\n    }\n    free(hmap);\n}\n\n/* 추가 연산 */\nvoid put(ArrayHashMap *hmap, const int key, const char *val) {\n    Pair *Pair = malloc(sizeof(Pair));\n    Pair-&gt;key = key;\n    Pair-&gt;val = malloc(strlen(val) + 1);\n    strcpy(Pair-&gt;val, val);\n\n    int index = hashFunc(key);\n    hmap-&gt;buckets[index] = Pair;\n}\n\n/* 삭제 연산 */\nvoid removeItem(ArrayHashMap *hmap, const int key) {\n    int index = hashFunc(key);\n    free(hmap-&gt;buckets[index]-&gt;val);\n    free(hmap-&gt;buckets[index]);\n    hmap-&gt;buckets[index] = NULL;\n}\n\n/* 모든 키-값 쌍 반환 */\nvoid pairSet(ArrayHashMap *hmap, MapSet *set) {\n    Pair *entries;\n    int i = 0, index = 0;\n    int total = 0;\n    /* 유효한 키-값 쌍의 개수를 센다 */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    entries = malloc(sizeof(Pair) * total);\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            entries[index].key = hmap-&gt;buckets[i]-&gt;key;\n            entries[index].val = malloc(strlen(hmap-&gt;buckets[i]-&gt;val) + 1);\n            strcpy(entries[index].val, hmap-&gt;buckets[i]-&gt;val);\n            index++;\n        }\n    }\n    set-&gt;set = entries;\n    set-&gt;len = total;\n}\n\n/* 모든 키 반환 */\nvoid keySet(ArrayHashMap *hmap, MapSet *set) {\n    int *keys;\n    int i = 0, index = 0;\n    int total = 0;\n    /* 유효한 키-값 쌍의 개수를 센다 */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    keys = malloc(total * sizeof(int));\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            keys[index] = hmap-&gt;buckets[i]-&gt;key;\n            index++;\n        }\n    }\n    set-&gt;set = keys;\n    set-&gt;len = total;\n}\n\n/* 모든 값 반환 */\nvoid valueSet(ArrayHashMap *hmap, MapSet *set) {\n    char **vals;\n    int i = 0, index = 0;\n    int total = 0;\n    /* 유효한 키-값 쌍의 개수를 센다 */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    vals = malloc(total * sizeof(char *));\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            vals[index] = hmap-&gt;buckets[i]-&gt;val;\n            index++;\n        }\n    }\n    set-&gt;set = vals;\n    set-&gt;len = total;\n}\n\n/* 해시 테이블 출력 */\nvoid print(ArrayHashMap *hmap) {\n    int i;\n    MapSet set;\n    pairSet(hmap, &amp;set);\n    Pair *entries = (Pair *)set.set;\n    for (i = 0; i &lt; set.len; i++) {\n        printf(\"%d -&gt; %s\\n\", entries[i].key, entries[i].val);\n    }\n    free(set.set);\n}\n</code></pre> array_hash_map.kt<pre><code>/* 키-값 쌍 */\nclass Pair(\n    var key: Int,\n    var _val: String\n)\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n    // 버킷 100개를 포함하는 배열을 초기화한다\n    private val buckets = arrayOfNulls&lt;Pair&gt;(100)\n\n    /* 해시 함수 */\n    fun hashFunc(key: Int): Int {\n        val index = key % 100\n        return index\n    }\n\n    /* 조회 연산 */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val pair = buckets[index] ?: return null\n        return pair._val\n    }\n\n    /* 추가 연산 */\n    fun put(key: Int, _val: String) {\n        val pair = Pair(key, _val)\n        val index = hashFunc(key)\n        buckets[index] = pair\n    }\n\n    /* 삭제 연산 */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        // null로 설정하여 삭제를 나타낸다\n        buckets[index] = null\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    fun pairSet(): MutableList&lt;Pair&gt; {\n        val pairSet = mutableListOf&lt;Pair&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                pairSet.add(pair)\n        }\n        return pairSet\n    }\n\n    /* 모든 키 반환 */\n    fun keySet(): MutableList&lt;Int&gt; {\n        val keySet = mutableListOf&lt;Int&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                keySet.add(pair.key)\n        }\n        return keySet\n    }\n\n    /* 모든 값 반환 */\n    fun valueSet(): MutableList&lt;String&gt; {\n        val valueSet = mutableListOf&lt;String&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                valueSet.add(pair._val)\n        }\n        return valueSet\n    }\n\n    /* 해시 테이블 출력 */\n    fun print() {\n        for (kv in pairSet()) {\n            val key = kv.key\n            val _val = kv._val\n            println(\"$key -&gt; $_val\")\n        }\n    }\n}\n</code></pre> array_hash_map.rb<pre><code>### 키-값 쌍 ###\nclass Pair\n  attr_accessor :key, :val\n\n  def initialize(key, val)\n    @key = key\n    @val = val\n  end\nend\n\n### 배열로 구현한 해시 테이블 ###\nclass ArrayHashMap\n  ### 생성자 ###\n  def initialize\n    # 버킷 100개를 포함하는 배열을 초기화한다\n    @buckets = Array.new(100)\n  end\n\n  ### 해시 함수 ###\n  def hash_func(key)\n    index = key % 100\n  end\n\n  ### 조회 연산 ###\n  def get(key)\n    index = hash_func(key)\n    pair = @buckets[index]\n\n    return if pair.nil?\n    pair.val\n  end\n\n  ### 추가 연산 ###\n  def put(key, val)\n    pair = Pair.new(key, val)\n    index = hash_func(key)\n    @buckets[index] = pair\n  end\n\n  ### 삭제 연산 ###\n  def remove(key)\n    index = hash_func(key)\n    # nil로 설정하여 삭제를 나타낸다\n    @buckets[index] = nil\n  end\n\n  ### 모든 키-값 쌍 반환 ###\n  def entry_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair unless pair.nil? }\n    result\n  end\n\n  ### 모든 키 반환 ###\n  def key_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair.key unless pair.nil? }\n    result\n  end\n\n  ### 모든 값 반환 ###\n  def value_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair.val unless pair.nil? }\n    result\n  end\n\n  ### 해시 테이블 출력 ###\n  def print\n    @buckets.each { |pair| puts \"#{pair.key} -&gt; #{pair.val}\" unless pair.nil? }\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 6 장   해시 테이블","6.1   해시 테이블"],"tags":[]},{"location":"chapter_hashing/hash_map/#613","level":2,"title":"6.1.3   해시 충돌과 용량 확장","text":"<p>해시 함수는 본질적으로 모든 <code>key</code>로 이루어진 입력 공간을 배열의 모든 인덱스로 이루어진 출력 공간에 매핑한다. 입력 공간은 대개 출력 공간보다 훨씬 크므로 이론적으로 ‘여러 입력이 같은 출력에 대응하는’ 경우가 반드시 존재한다.</p> <p>앞의 해시 함수에서는 입력 <code>key</code>의 마지막 두 자리가 같으면 출력도 같다. 예를 들어 학번이 12836과 20336인 두 학생을 조회하면 다음과 같다.</p> <pre><code>12836 % 100 = 36\n20336 % 100 = 36\n</code></pre> <p>아래 그림(6-3)에서 두 학번이 같은 이름을 가리키는데, 이는 명백히 잘못되었다. 이처럼 여러 입력이 같은 출력에 대응하는 현상을 해시 충돌(hash collision)이라고 한다.</p> <p></p> <p> 그림 6-3   해시 충돌의 예 </p> <p>해시 테이블의 용량 \\(n\\)이 커질수록 여러 <code>key</code>가 같은 버킷에 들어갈 확률은 낮아지고 충돌도 줄어든다. 따라서 해시 테이블의 용량을 늘려 해시 충돌을 줄일 수 있다.</p> <p>아래 그림(6-4)에서 확장 전에는 키-값 쌍 <code>(136, A)</code>와 <code>(236, D)</code>가 충돌하지만 확장 후에는 충돌이 사라진다.</p> <p></p> <p> 그림 6-4   해시 테이블의 용량 확장 </p> <p>배열 확장과 마찬가지로 해시 테이블을 확장하려면 모든 키-값 쌍을 기존 테이블에서 새 테이블로 옮겨야 하므로 시간이 많이 든다. 또한 용량 <code>capacity</code>가 바뀌므로 해시 함수로 모든 키-값 쌍의 저장 위치를 다시 계산해야 하며, 이는 확장 비용을 더욱 늘린다. 따라서 프로그래밍 언어는 보통 충분히 큰 용량을 미리 확보하여 잦은 확장을 방지한다.</p> <p>적재율(load factor)은 해시 테이블의 원소 수를 버킷 수로 나눈 값으로, 해시 충돌의 심각도를 측정하는 중요한 개념이다. 용량 확장을 시작하는 조건으로도 자주 사용된다. 예를 들어 Java에서는 적재율이 \\(0.75\\)를 넘으면 해시 테이블의 용량을 기존의 \\(2\\)배로 늘린다.</p>","path":["제 6 장   해시 테이블","6.1   해시 테이블"],"tags":[]},{"location":"chapter_hashing/summary/","level":1,"title":"6.4   요약","text":"","path":["제 6 장   해시 테이블","6.4   요약"],"tags":[]},{"location":"chapter_hashing/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>해시 테이블은 <code>key</code>를 입력받아 \\(O(1)\\) 시간에 <code>value</code>를 조회하므로 매우 효율적이다.</li> <li>대표적인 연산에는 조회, 키-값 쌍 추가와 삭제, 해시 테이블 순회 등이 있다.</li> <li>해시 함수는 <code>key</code>를 배열 인덱스로 매핑하여 해당 버킷에 접근하고 <code>value</code>를 얻도록 한다.</li> <li>서로 다른 두 <code>key</code>가 해시 함수를 거쳐 같은 배열 인덱스를 얻으면 잘못된 조회 결과가 발생할 수 있다. 이를 해시 충돌이라고 한다.</li> <li>해시 테이블 용량이 클수록 충돌 확률이 낮으므로 용량 확장으로 충돌을 완화할 수 있다. 배열과 마찬가지로 해시 테이블의 용량 확장 비용도 크다.</li> <li>적재율은 원소 수를 버킷 수로 나눈 값이며, 해시 충돌의 심각도를 반영한다. 용량 확장의 시작 조건으로 자주 사용된다.</li> <li>체이닝은 단일 원소를 연결 리스트로 바꾸어 충돌하는 모든 원소를 같은 리스트에 저장한다. 리스트가 너무 길면 조회 효율이 낮아지므로, 레드-블랙 트리로 바꾸어 효율을 높일 수 있다.</li> <li>개방 주소법은 여러 차례 탐사하여 해시 충돌을 처리한다. 선형 탐사는 고정 간격을 사용하며, 원소를 삭제할 수 없고 군집이 생기기 쉽다는 단점이 있다. 다중 해싱은 여러 해시 함수로 탐사하므로 군집은 덜 생기지만 계산량이 늘어난다.</li> <li>프로그래밍 언어마다 해시 테이블 구현이 다르다. 예를 들어 Java의 <code>HashMap</code>은 체이닝을, Python의 <code>dict</code>는 개방 주소법을 사용한다.</li> <li>해시 테이블의 해시 알고리즘에는 결정성, 높은 효율, 균등 분포가 필요하다. 암호학에서는 충돌 저항성과 눈사태 효과도 갖추어야 한다.</li> <li>해시 알고리즘은 해시값의 균등 분포를 최대한 보장하고 충돌을 줄이기 위해 보통 큰 소수를 모듈러 연산의 법으로 사용한다.</li> <li>대표적인 해시 알고리즘으로 MD5, SHA-1, SHA-2, SHA-3 등이 있다. MD5는 파일 무결성 검사에, SHA-2는 보안 응용과 프로토콜에 자주 쓰인다.</li> <li>프로그래밍 언어는 보통 자료형별 내장 해시 알고리즘을 제공하여 버킷 인덱스를 계산한다. 일반적으로 불변 객체만 해시 가능하다.</li> </ul>","path":["제 6 장   해시 테이블","6.4   요약"],"tags":[]},{"location":"chapter_hashing/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 해시 테이블의 시간 복잡도가 \\(O(n)\\)이 되는 경우는 언제인가?</p> <p>해시 충돌이 심하면 시간 복잡도가 \\(O(n)\\)으로 나빠진다. 해시 함수를 잘 설계하고 용량을 적절히 정하며 충돌이 비교적 고르게 분포하면 \\(O(1)\\)이다. 프로그래밍 언어의 내장 해시 테이블을 사용할 때는 보통 시간 복잡도를 \\(O(1)\\)로 본다.</p> <p>Q: 해시 함수 \\(f(x) = x\\)를 사용하면 충돌이 없을 텐데, 왜 그렇게 하지 않는가?</p> <p>\\(f(x) = x\\)에서는 각 원소에 고유한 버킷 인덱스가 대응하므로 배열과 같다. 그러나 입력 공간은 보통 출력 공간(배열 길이)보다 훨씬 크므로, 해시 함수의 마지막 단계에서 대개 배열 길이로 나눈 나머지를 구한다. 즉, 해시 테이블의 목적은 큰 상태 공간을 작은 공간에 매핑하면서 \\(O(1)\\)의 조회 효율을 제공하는 것이다.</p> <p>Q: 해시 테이블도 배열, 연결 리스트, 이진 트리로 구현되는데 어떻게 이들보다 효율이 높을 수 있는가?</p> <p>첫째, 시간 효율은 높아지지만 공간 효율은 낮아진다. 해시 테이블에는 사용하지 않는 메모리가 상당 부분 있다.</p> <p>둘째, 특정 활용 상황에서만 시간 효율이 높아진다. 같은 기능을 배열이나 연결 리스트로 같은 시간 복잡도에 구현할 수 있다면 보통 그쪽이 더 빠르다. 해시 함수 계산에 비용이 들어 시간 복잡도의 상수항이 더 크기 때문이다.</p> <p>마지막으로 해시 테이블의 시간 복잡도도 나빠질 수 있다. 예를 들어 체이닝에서는 연결 리스트나 레드-블랙 트리에서 탐색하므로, 여전히 \\(O(n)\\) 시간으로 나빠질 위험이 있다.</p> <p>Q: 다중 해싱도 원소를 곧바로 삭제할 수 없는가? 삭제 표시를 한 공간은 다시 사용할 수 있는가?</p> <p>다중 해싱은 개방 주소법의 한 종류다. 개방 주소법은 모두 원소를 곧바로 삭제할 수 없으므로 삭제 표시를 해야 한다. 표시된 공간은 다시 사용할 수 있다. 새 원소를 삽입할 때 해시 함수로 삭제 표시가 있는 위치를 찾으면 그곳에 새 원소를 넣을 수 있다. 이렇게 하면 탐사 순서를 유지하면서 공간 활용률도 보장한다.</p> <p>Q: 선형 탐사에서는 원소를 찾을 때도 왜 해시 충돌이 발생하는가?</p> <p>해시 함수로 해당 버킷과 키-값 쌍을 찾았는데 <code>key</code>가 일치하지 않으면 해시 충돌이 있다는 뜻이다. 따라서 미리 정한 간격으로 뒤를 차례로 탐색하다가 올바른 키-값 쌍을 찾거나, 찾을 수 없다고 판단하면 종료한다.</p> <p>Q: 해시 테이블의 용량 확장은 왜 해시 충돌을 완화하는가?</p> <p>해시 함수는 마지막에 대개 배열 길이 \\(n\\)으로 나눈 나머지를 구하여 출력이 인덱스 범위에 들어가도록 한다. 용량을 늘리면 \\(n\\)이 바뀌고 <code>key</code>에 대응하는 인덱스도 달라질 수 있다. 원래 같은 버킷에 있던 여러 <code>key</code>가 여러 버킷으로 분산될 수 있으므로 충돌이 완화된다.</p> <p>Q: 효율적으로 저장하고 접근하려는 목적이라면 배열을 바로 사용하면 되지 않는가?</p> <p><code>key</code>가 좁은 범위의 연속된 정수라면 배열을 바로 쓰는 것이 간단하고 효율적이다. 하지만 문자열처럼 다른 자료형이면 해시 함수로 <code>key</code>를 배열 인덱스에 매핑하고 버킷 배열에 원소를 저장해야 한다. 이것이 바로 해시 테이블이다.</p>","path":["제 6 장   해시 테이블","6.4   요약"],"tags":[]},{"location":"chapter_heap/","level":1,"title":"제 8 장   힙","text":"<p>이 장을 시작하며</p> <p>힙은 겹겹이 이어진 산봉우리처럼 높낮이도 모습도 제각각이다.</p> <p>봉우리마다 높이가 다르지만, 가장 높은 봉우리는 언제나 가장 먼저 눈에 들어온다.</p>","path":["제 8 장   힙"],"tags":[]},{"location":"chapter_heap/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>8.1   힙</li> <li>8.2   힙 구성 연산</li> <li>8.3   Top-k 문제</li> <li>8.4   요약</li> <li>8.5   연습 문제</li> </ul>","path":["제 8 장   힙"],"tags":[]},{"location":"chapter_heap/build_heap/","level":1,"title":"8.2   힙 구성 연산","text":"<p>어떤 상황에서는 리스트의 모든 원소를 사용해 힙을 구성하고 싶을 수 있다. 이 과정을 '힙 구성 연산'이라고 한다.</p>","path":["제 8 장   힙","8.2   힙 구성 연산"],"tags":[]},{"location":"chapter_heap/build_heap/#821","level":2,"title":"8.2.1   힙 삽입 연산을 이용한 구현","text":"<p>먼저 빈 힙을 만든 다음, 리스트를 순회하면서 각 원소에 대해 차례로 '힙 삽입 연산'을 수행한다. 즉, 원소를 힙의 끝에 추가한 뒤 해당 원소부터 '아래에서 위로' 힙을 재정렬한다.</p> <p>원소 하나를 힙에 삽입할 때마다 힙의 길이가 1씩 늘어난다. 이진 트리에 위에서 아래 순서로 노드를 추가하므로 힙은 '위에서 아래로' 구성된다.</p> <p>원소의 개수를 \\(n\\)이라고 하자. 각 원소의 힙 삽입 연산에는 \\(O(\\log{n})\\) 시간이 걸리므로, 이 힙 구성 방법의 시간 복잡도는 \\(O(n \\log n)\\)이다.</p>","path":["제 8 장   힙","8.2   힙 구성 연산"],"tags":[]},{"location":"chapter_heap/build_heap/#822","level":2,"title":"8.2.2   순회하며 힙을 재정렬하는 구현","text":"<p>실제로는 다음 두 단계로 더 효율적인 힙 구성 방법을 구현할 수 있다.</p> <ol> <li>리스트의 모든 원소를 그대로 힙에 추가한다. 이 시점에서는 아직 힙의 성질을 만족하지 않는다.</li> <li>힙을 역순으로 순회하면서(레벨 순서 순회의 역순), 각 내부 노드에 대해 차례로 '위에서 아래로 힙 재정렬'을 수행한다.</li> </ol> <p>노드 하나를 재정렬할 때마다 그 노드를 루트로 하는 서브트리가 올바른 부분 힙이 된다. 역순으로 순회하므로 힙은 '아래에서 위로' 구성된다.</p> <p>역순으로 순회하는 이유는 현재 노드 아래의 서브트리들이 이미 올바른 부분 힙임을 보장하기 위해서다. 그래야 현재 노드를 재정렬하는 작업이 유효해진다.</p> <p>리프 노드는 자식 노드가 없으므로 그 자체로 올바른 부분 힙이며, 재정렬할 필요가 없다. 아래 코드와 같이 마지막 내부 노드는 마지막 노드의 부모 노드다. 여기서부터 역순으로 순회하며 힙 재정렬을 수행한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def __init__(self, nums: list[int]):\n    \"\"\"생성자: 입력 리스트를 바탕으로 힙 구성\"\"\"\n    # 리스트 원소를 그대로 힙에 추가한다\n    self.max_heap = nums\n    # 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for i in range(self.parent(self.size() - 1), -1, -1):\n        self.sift_down(i)\n</code></pre> my_heap.cpp<pre><code>/* 생성자: 입력 리스트를 바탕으로 힙 구성 */\nMaxHeap(vector&lt;int&gt; nums) {\n    // 리스트 원소를 그대로 힙에 추가한다\n    maxHeap = nums;\n    // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.java<pre><code>/* 생성자: 입력 리스트를 바탕으로 힙 구성 */\nMaxHeap(List&lt;Integer&gt; nums) {\n    // 리스트 원소를 그대로 힙에 추가한다\n    maxHeap = new ArrayList&lt;&gt;(nums);\n    // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* 생성자: 입력 리스트로 힙 구성 */\nMaxHeap(IEnumerable&lt;int&gt; nums) {\n    // 리스트 원소를 그대로 힙에 추가한다\n    maxHeap = new List&lt;int&gt;(nums);\n    // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    var size = Parent(this.Size() - 1);\n    for (int i = size; i &gt;= 0; i--) {\n        SiftDown(i);\n    }\n}\n</code></pre> my_heap.go<pre><code>/* 생성자: 슬라이스를 바탕으로 힙 구성 */\nfunc newMaxHeap(nums []any) *maxHeap {\n    // 리스트 원소를 그대로 힙에 추가한다\n    h := &amp;maxHeap{data: nums}\n    for i := h.parent(len(h.data) - 1); i &gt;= 0; i-- {\n        // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n        h.siftDown(i)\n    }\n    return h\n}\n</code></pre> my_heap.swift<pre><code>/* 생성자: 입력 리스트를 바탕으로 힙 구성 */\ninit(nums: [Int]) {\n    // 리스트 원소를 그대로 힙에 추가한다\n    maxHeap = nums\n    // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for i in (0 ... parent(i: size() - 1)).reversed() {\n        siftDown(i: i)\n    }\n}\n</code></pre> my_heap.js<pre><code>/* 생성자: 빈 힙을 만들거나 입력 리스트로 힙을 구성한다 */\nconstructor(nums) {\n    // 리스트 원소를 그대로 힙에 추가한다\n    this.#maxHeap = nums === undefined ? [] : [...nums];\n    // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (let i = this.#parent(this.size() - 1); i &gt;= 0; i--) {\n        this.#siftDown(i);\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* 생성자: 빈 힙을 만들거나 입력 리스트로 힙을 구성한다 */\nconstructor(nums?: number[]) {\n    // 리스트 원소를 그대로 힙에 추가한다\n    this.maxHeap = nums === undefined ? [] : [...nums];\n    // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (let i = this.parent(this.size() - 1); i &gt;= 0; i--) {\n        this.siftDown(i);\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* 생성자: 입력 리스트를 바탕으로 힙 구성 */\nMaxHeap(List&lt;int&gt; nums) {\n  // 리스트 원소를 그대로 힙에 추가한다\n  _maxHeap = nums;\n  // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n  for (int i = _parent(size() - 1); i &gt;= 0; i--) {\n    siftDown(i);\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* 생성자: 입력 리스트를 바탕으로 힙 구성 */\nfn new(nums: Vec&lt;i32&gt;) -&gt; Self {\n    // 리스트 원소를 그대로 힙에 추가한다\n    let mut heap = MaxHeap { max_heap: nums };\n    // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for i in (0..=Self::parent(heap.size() - 1)).rev() {\n        heap.sift_down(i);\n    }\n    heap\n}\n</code></pre> my_heap.c<pre><code>/* 생성자: 슬라이스를 바탕으로 힙 구성 */\nMaxHeap *newMaxHeap(int nums[], int size) {\n    // 모든 원소를 힙에 삽입한다\n    MaxHeap *maxHeap = (MaxHeap *)malloc(sizeof(MaxHeap));\n    maxHeap-&gt;size = size;\n    memcpy(maxHeap-&gt;data, nums, size * sizeof(int));\n    for (int i = parent(maxHeap, size - 1); i &gt;= 0; i--) {\n        // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n        siftDown(maxHeap, i);\n    }\n    return maxHeap;\n}\n</code></pre> my_heap.kt<pre><code>/* 최대 힙 */\nclass MaxHeap(nums: MutableList&lt;Int&gt;?) {\n    // 배열 대신 리스트를 사용하므로 용량 확장을 고려할 필요가 없다\n    private val maxHeap = mutableListOf&lt;Int&gt;()\n\n    /* 생성자: 입력 리스트를 바탕으로 힙 구성 */\n    init {\n        // 리스트 원소를 그대로 힙에 추가한다\n        maxHeap.addAll(nums!!)\n        // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n        for (i in parent(size() - 1) downTo 0) {\n            siftDown(i)\n        }\n    }\n\n    /* 왼쪽 자식 노드의 인덱스 반환 */\n    private fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* 오른쪽 자식 노드의 인덱스 반환 */\n    private fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* 부모 노드의 인덱스 반환 */\n    private fun parent(i: Int): Int {\n        return (i - 1) / 2 // 나눗셈의 결과를 내림한다\n    }\n\n    /* 원소 교환 */\n    private fun swap(i: Int, j: Int) {\n        val temp = maxHeap[i]\n        maxHeap[i] = maxHeap[j]\n        maxHeap[j] = temp\n    }\n\n    /* 힙 크기 반환 */\n    fun size(): Int {\n        return maxHeap.size\n    }\n\n    /* 힙이 비어 있는지 판단 */\n    fun isEmpty(): Boolean {\n        /* 힙이 비어 있는지 판단 */\n        return size() == 0\n    }\n\n    /* 힙의 루트 원소 접근 */\n    fun peek(): Int {\n        return maxHeap[0]\n    }\n\n    /* 힙에 원소 삽입 */\n    fun push(_val: Int) {\n        // 노드 추가\n        maxHeap.add(_val)\n        // 아래에서 위로 힙을 재정렬한다\n        siftUp(size() - 1)\n    }\n\n    /* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\n    private fun siftUp(it: Int) {\n        // Kotlin의 함수 매개변수는 변경할 수 없으므로 임시 변수를 생성한다\n        var i = it\n        while (true) {\n            // 노드 i의 부모 노드를 구한다\n            val p = parent(i)\n            // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n            if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n            // 두 노드를 교환한다\n            swap(i, p)\n            // 위쪽으로 힙 재정렬을 반복한다\n            i = p\n        }\n    }\n\n    /* 힙에서 원소 꺼내기 */\n    fun pop(): Int {\n        // 비어 있는 경우를 처리한다\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        swap(0, size() - 1)\n        // 노드 삭제\n        val _val = maxHeap.removeAt(size() - 1)\n        // 위에서 아래로 힙을 재정렬한다\n        siftDown(0)\n        // 힙의 루트 원소를 반환한다\n        return _val\n    }\n\n    /* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\n    private fun siftDown(it: Int) {\n        // Kotlin의 함수 매개변수는 변경할 수 없으므로 임시 변수를 생성한다\n        var i = it\n        while (true) {\n            // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n            val l = left(i)\n            val r = right(i)\n            var ma = i\n            if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n            if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n            // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n            if (ma == i) break\n            // 두 노드를 교환한다\n            swap(i, ma)\n            // 아래쪽으로 힙 재정렬을 반복한다\n            i = ma\n        }\n    }\n\n    /* 힙 출력(이진 트리) */\n    fun print() {\n        val queue = PriorityQueue { a: Int, b: Int -&gt; b - a }\n        queue.addAll(maxHeap)\n        printHeap(queue)\n    }\n}\n</code></pre> my_heap.rb<pre><code>### 생성자: 입력 리스트를 바탕으로 힙 구성 ###\ndef initialize(nums)\n  # 리스트 원소를 그대로 힙에 추가한다\n  @max_heap = nums\n  # 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n  parent(size - 1).downto(0) do |i|\n    sift_down(i)\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 8 장   힙","8.2   힙 구성 연산"],"tags":[]},{"location":"chapter_heap/build_heap/#823","level":2,"title":"8.2.3   복잡도 분석","text":"<p>이제 두 번째 힙 구성 방법의 시간 복잡도를 계산해 보자.</p> <ul> <li>완전 이진 트리의 노드 수가 \\(n\\)이라면, 리프 노드의 수는 \\((n + 1) / 2\\)이다. 여기서 \\(/\\)는 내림 정수 나눗셈이다. 따라서 재정렬해야 하는 노드의 수는 \\(n / 2\\)이다.</li> <li>위에서 아래로 힙을 재정렬할 때 각 노드는 최대 리프 노드까지 내려갈 수 있으므로, 최대 반복 횟수는 이진 트리의 높이인 \\(\\log n\\)이다.</li> </ul> <p>위의 두 값을 곱하면 힙 구성 과정의 시간 복잡도가 \\(O(n \\log n)\\)이 된다. 하지만 이 추정은 정확하지 않다. 이진 트리에서는 아래쪽 노드의 수가 위쪽 노드보다 훨씬 많다는 성질을 고려하지 않았기 때문이다.</p> <p>이제 더 정확하게 계산해 보자. 계산을 간단히 하기 위해 노드 수가 \\(n\\)이고 높이가 \\(h\\)인 '포화 이진 트리'가 주어졌다고 가정한다. 이 가정은 계산 결과의 정확성에 영향을 주지 않는다.</p> <p></p> <p> 그림 8-5   포화 이진 트리의 레벨별 노드 수 </p> <p>위 그림(8-5)에서 노드를 '위에서 아래로 재정렬'하는 최대 반복 횟수는 그 노드에서 리프 노드까지의 거리와 같으며, 이 거리는 바로 '노드의 높이'다. 따라서 각 레벨의 '노드 수 \\(\\times\\) 노드 높이'를 더하면 모든 노드에 대한 힙 재정렬의 총반복 횟수를 구할 수 있다.</p> \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{(h-1)}\\times1 \\] <p>위 식을 정리하려면 중등 수학의 수열 지식이 필요하다. 먼저 \\(T(h)\\)에 \\(2\\)를 곱하면 다음과 같다.</p> \\[ \\begin{aligned} T(h) &amp; = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{h-1}\\times1 \\newline 2 T(h) &amp; = 2^1h + 2^2(h-1) + 2^3(h-2) + \\dots + 2^{h}\\times1 \\newline \\end{aligned} \\] <p>두 식의 항을 한 칸씩 어긋나게 놓고 아래 식 \\(2 T(h)\\)에서 위 식 \\(T(h)\\)를 빼면 다음을 얻는다.</p> \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\dots + 2^{h-1} + 2^h \\] <p>위 식에서 \\(T(h)\\)를 이루는 항들이 등비수열을 이루므로 합 공식을 바로 적용하여 시간 복잡도를 구할 수 있다.</p> \\[ \\begin{aligned} T(h) &amp; = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline &amp; = 2^{h+1} - h - 2 \\newline &amp; = O(2^h) \\end{aligned} \\] <p>또한 높이가 \\(h\\)인 포화 이진 트리의 노드 수는 \\(n = 2^{h+1} - 1\\)이므로, 복잡도가 \\(O(2^h) = O(n)\\)임을 쉽게 알 수 있다. 위 계산은 리스트를 입력받아 힙을 구성하는 시간 복잡도가 \\(O(n)\\)으로 매우 효율적임을 보여 준다.</p>","path":["제 8 장   힙","8.2   힙 구성 연산"],"tags":[]},{"location":"chapter_heap/exercises/","level":1,"title":"8.5   연습 문제","text":"","path":["제 8 장   힙","8.5   연습 문제"],"tags":[]},{"location":"chapter_heap/exercises/#851","level":2,"title":"8.5.1   개념 확인","text":"","path":["제 8 장   힙","8.5   연습 문제"],"tags":[]},{"location":"chapter_heap/exercises/#1-10","level":3,"title":"1.   10을 힙에 넣은 뒤 어떻게 조정하는가","text":"<p>배열 <code>[9, 7, 8, 3, 5]</code>는 최대 힙을 나타낸다. 여기에 숫자 10을 삽입한다.</p> <ol> <li>먼저 10을 배열 끝에 추가한다. 부모 노드의 값은 얼마인가?</li> <li>새 노드부터 위로 힙을 재정렬하며, 교환할 때마다 배열을 적어 보자.</li> <li>최종 루트 원소는 무엇이며, 총 몇 번 교환했는가?</li> </ol> 참고 답안 <ol> <li> <p>추가한 10의 인덱스는 5이고, 부모 인덱스는     \\(\\lfloor(5-1)/2\\rfloor=2\\)이므로 부모 노드의 값은 8이다.</p> </li> <li> <p>10이 8보다 크므로 첫 번째 교환 후에는 <code>[9, 7, 10, 3, 5, 8]</code>이 된다.     10은 부모 9보다도 크므로 두 번째 교환 후에는 <code>[10, 7, 9, 3, 5, 8]</code>이 된다.     10이 루트에 도달했으므로 힙 재정렬을 마친다.</p> </li> <li> <p>최종 루트 값은 10이며, 총 2번 교환했다.</p> </li> </ol>","path":["제 8 장   힙","8.5   연습 문제"],"tags":[]},{"location":"chapter_heap/exercises/#2","level":3,"title":"2.   최소 힙의 부모와 자식 관계 확인하기","text":"<p>배열 <code>[1, 4, 3, 7, 6, 2]</code>는 완전 이진 트리를 나타낸다. 최소 힙에서는 모든 부모의 값이 자식의 값보다 작거나 같아야 한다. 인덱스 \\(i\\)의 왼쪽과 오른쪽 자식 인덱스는 각각 \\(2i+1\\), \\(2i+2\\)다.</p> <ol> <li>인덱스 2의 자식 인덱스와 값은 각각 무엇인가?</li> <li>인덱스 2의 값은 3이다. 이 노드와 자식의 관계가 최소 힙 규칙에 어긋나는가? 그렇다면 어떤 두 원소를 교환해야 하는가?</li> <li>2번의 판단에 따라, 규칙에 어긋나면 교환 후 배열을 적고 그렇지 않으면 교환할 필요가 없음을 설명하라. 마지막으로 나머지 부모와 자식 관계도 확인하라.</li> </ol> 참고 답안 <ol> <li> <p>인덱스 2의 왼쪽 자식 인덱스는 5이고 값은 2다. 오른쪽 자식 인덱스는 6이지만 배열 길이가 6이므로 오른쪽 자식은 없다.</p> </li> <li> <p>부모의 값 3이 자식의 값 2보다 크므로 최소 힙 규칙에 어긋난다. 인덱스 2와 5의 원소를 교환해야 한다.</p> </li> <li> <p>교환 후 배열은 <code>[1, 4, 2, 7, 6, 3]</code>이다. 관계를 하나씩 확인하면     <code>1 ≤ 4</code>, <code>1 ≤ 2</code>, <code>4 ≤ 7</code>, <code>4 ≤ 6</code>, <code>2 ≤ 3</code>이다.     모든 부모의 값이 자식의 값보다 작거나 같으므로 최소 힙 규칙을 만족한다.</p> </li> </ol>","path":["제 8 장   힙","8.5   연습 문제"],"tags":[]},{"location":"chapter_heap/exercises/#3","level":3,"title":"3.   최소 힙으로 가장 큰 수 세 개 유지하기","text":"<p>데이터 스트림 <code>[4, 1, 7, 3, 8]</code>에서 가장 큰 수 3개를 유지하려면 크기가 3을 넘지 않는 최소 힙을 사용할 수 있다.</p> <p>먼저 앞의 3개 수를 차례대로 최소 힙에 넣는다. 힙이 가득 찬 뒤에는 새 수를 읽을 때마다 루트보다 크면 루트를 삭제하고 새 수를 넣는다. 그렇지 않으면 힙을 그대로 둔다.</p> <p>각 수를 읽은 뒤 힙에 남아 있는 수와 루트 값을 적어 보자. 남아 있는 수는 집합으로만 적으면 되며, 힙 배열에서의 순서는 적지 않아도 된다.</p> 참고 답안 <p>각 수를 읽은 뒤의 결과는 다음과 같다.</p> 읽은 수 남아 있는 수 루트 4 <code>{4}</code> 4 1 <code>{1, 4}</code> 1 7 <code>{1, 4, 7}</code> 1 3 <code>{3, 4, 7}</code> 3 8 <code>{4, 7, 8}</code> 4 <p>힙이 가득 차면 루트는 현재 남아 있는 수 중 가장 작은 수다. 새 수가 루트보다 클 때만 새 수로 루트를 대체한다. 최종적으로 남은 <code>{4, 7, 8}</code>이 바로 가장 큰 수 3개다.</p>","path":["제 8 장   힙","8.5   연습 문제"],"tags":[]},{"location":"chapter_heap/exercises/#852","level":2,"title":"8.5.2   프로그래밍 연습","text":"","path":["제 8 장   힙","8.5   연습 문제"],"tags":[]},{"location":"chapter_heap/exercises/#1-k","level":3,"title":"1.   배열에서 k번째로 큰 원소 찾기","text":"<p>정수 배열 <code>nums</code>와 정수 \\(k\\)가 주어진다. 여기서 \\(1 \\le k \\le n\\)이며 \\(n\\)은 배열 길이다. 배열을 큰 것부터 나열했을 때 \\(k\\)번째 위치의 원소를 반환하라.</p> <p>중복 원소도 각각 센다. 예를 들어 <code>[5, 5, 2]</code>에서 두 번째로 큰 원소 역시 5다. 크기가 \\(k\\)를 넘지 않는 최소 힙을 사용하여 해결하라.</p> 풀이 힌트 <ol> <li>k번째로 큰 원소는 가장 큰 k개 수 중 가장 작은 수다.</li> <li>각 수를 최소 힙에 넣고 크기가 k를 넘으면 최솟값을 꺼낸다.</li> <li>순회가 끝나면 힙에 가장 큰 k개 수가 남으며, 루트가 답이다.</li> </ol> <p>LeetCode</p>","path":["제 8 장   힙","8.5   연습 문제"],"tags":[]},{"location":"chapter_heap/heap/","level":1,"title":"8.1   힙","text":"<p>힙(heap)은 특정 조건을 만족하는 완전 이진 트리로, 아래 그림(8-1)과 같이 크게 두 종류로 나뉜다.</p> <ul> <li>최소 힙(min heap): 모든 노드의 값이 자식 노드의 값보다 작거나 같다(\\(\\leq\\)).</li> <li>최대 힙(max heap): 모든 노드의 값이 자식 노드의 값보다 크거나 같다(\\(\\geq\\)).</li> </ul> <p></p> <p> 그림 8-1   최소 힙과 최대 힙 </p> <p>힙은 완전 이진 트리의 특수한 형태이므로 다음 성질을 가진다.</p> <ul> <li>가장 아래 레벨은 왼쪽부터 노드가 채워지며, 나머지 레벨은 모두 가득 차 있다.</li> <li>이진 트리의 루트 노드를 '힙의 루트', 가장 아래 레벨에서 맨 오른쪽에 있는 노드를 '힙의 끝'이라고 한다.</li> <li>최대 힙(최소 힙)에서는 힙의 루트 원소(루트 노드)의 값이 가장 크다(작다).</li> </ul>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/heap/#811","level":2,"title":"8.1.1   힙의 주요 연산","text":"<p>많은 프로그래밍 언어에서 제공하는 것은 우선순위 큐(priority queue)라는 점을 알아 두자. 우선순위 큐는 우선순위에 따라 정렬되는 큐로 정의되는 추상 자료 구조다.</p> <p>실제로 힙은 보통 우선순위 큐를 구현하는 데 사용되며, 최대 힙은 원소를 큰 값부터 작은 값 순서로 꺼내는 우선순위 큐에 해당한다. 사용하는 입장에서는 '우선순위 큐'와 '힙'을 동등한 자료 구조로 볼 수 있다. 따라서 이 책에서는 둘을 특별히 구분하지 않고 모두 '힙'이라고 한다.</p> <p>힙의 주요 연산은 아래 표(8-1)와 같다. 메서드 이름은 프로그래밍 언어에 따라 달라진다.</p> <p> 표 8-1   힙 연산의 효율 </p> 메서드 이름 설명 시간 복잡도 <code>push()</code> 힙에 원소 삽입 \\(O(\\log n)\\) <code>pop()</code> 힙의 루트 원소 꺼내기 \\(O(\\log n)\\) <code>peek()</code> 힙의 루트 원소에 접근(최대 / 최소 힙에서는 각각 최댓값 / 최솟값) \\(O(1)\\) <code>size()</code> 힙의 원소 수 구하기 \\(O(1)\\) <code>isEmpty()</code> 힙이 비어 있는지 확인 \\(O(1)\\) <p>실제 응용에서는 프로그래밍 언어가 제공하는 힙 클래스(또는 우선순위 큐 클래스)를 바로 사용할 수 있다.</p> <p>정렬 알고리즘에서 '오름차순 정렬'과 '내림차순 정렬'을 선택하듯이, <code>flag</code>를 설정하거나 <code>Comparator</code>를 수정하여 '최소 힙'과 '최대 힙'을 전환할 수 있다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby heap.py<pre><code># 최소 힙 초기화\nmin_heap, flag = [], 1\n# 최대 힙 초기화\nmax_heap, flag = [], -1\n\n# Python의 heapq 모듈은 기본적으로 최소 힙을 구현한다\n# 원소의 부호를 바꾼 뒤 삽입하면 대소 관계가 뒤집히므로 최대 힙을 구현할 수 있다\n# 이 예제에서 flag = 1이면 최소 힙, flag = -1이면 최대 힙에 해당한다\n\n# 힙에 원소 삽입\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n\n# 힙의 루트 원소 구하기\npeek: int = flag * max_heap[0] # 5\n\n# 힙의 루트 원소 꺼내기\n# 꺼낸 원소들은 큰 값부터 작은 값 순서로 나열된다\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n\n# 힙의 크기 구하기\nsize: int = len(max_heap)\n\n# 힙이 비어 있는지 확인\nis_empty: bool = not max_heap\n\n# 리스트를 입력받아 힙 구성\nmin_heap: list[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n</code></pre> heap.cpp<pre><code>/* 힙 초기화 */\n// 최소 힙 초기화\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap;\n// 최대 힙 초기화\npriority_queue&lt;int, vector&lt;int&gt;, less&lt;int&gt;&gt; maxHeap;\n\n/* 힙에 원소 삽입 */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n\n/* 힙의 루트 원소 구하기 */\nint peek = maxHeap.top(); // 5\n\n/* 힙의 루트 원소 꺼내기 */\n// 꺼낸 원소들은 큰 값부터 작은 값 순서로 나열된다\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n\n/* 힙의 크기 구하기 */\nint size = maxHeap.size();\n\n/* 힙이 비어 있는지 확인 */\nbool isEmpty = maxHeap.empty();\n\n/* 리스트를 입력받아 힙 구성 */\nvector&lt;int&gt; input{1, 3, 2, 5, 4};\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap(input.begin(), input.end());\n</code></pre> heap.java<pre><code>/* 힙 초기화 */\n// 최소 힙 초기화\nQueue&lt;Integer&gt; minHeap = new PriorityQueue&lt;&gt;();\n// 최대 힙 초기화(lambda 식으로 Comparator를 수정하면 된다)\nQueue&lt;Integer&gt; maxHeap = new PriorityQueue&lt;&gt;((a, b) -&gt; b - a);\n\n/* 힙에 원소 삽입 */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n\n/* 힙의 루트 원소 구하기 */\nint peek = maxHeap.peek(); // 5\n\n/* 힙의 루트 원소 꺼내기 */\n// 꺼낸 원소들은 큰 값부터 작은 값 순서로 나열된다\npeek = maxHeap.poll(); // 5\npeek = maxHeap.poll(); // 4\npeek = maxHeap.poll(); // 3\npeek = maxHeap.poll(); // 2\npeek = maxHeap.poll(); // 1\n\n/* 힙의 크기 구하기 */\nint size = maxHeap.size();\n\n/* 힙이 비어 있는지 확인 */\nboolean isEmpty = maxHeap.isEmpty();\n\n/* 리스트를 입력받아 힙 구성 */\nminHeap = new PriorityQueue&lt;&gt;(Arrays.asList(1, 3, 2, 5, 4));\n</code></pre> heap.cs<pre><code>/* 힙 초기화 */\n// 최소 힙 초기화\nPriorityQueue&lt;int, int&gt; minHeap = new();\n// 최대 힙 초기화(lambda 식으로 Comparer를 수정하면 된다)\nPriorityQueue&lt;int, int&gt; maxHeap = new(Comparer&lt;int&gt;.Create((x, y) =&gt; y.CompareTo(x)));\n\n/* 힙에 원소 삽입 */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n\n/* 힙의 루트 원소 구하기 */\nint peek = maxHeap.Peek();//5\n\n/* 힙의 루트 원소 꺼내기 */\n// 꺼낸 원소들은 큰 값부터 작은 값 순서로 나열된다\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n\n/* 힙의 크기 구하기 */\nint size = maxHeap.Count;\n\n/* 힙이 비어 있는지 확인 */\nbool isEmpty = maxHeap.Count == 0;\n\n/* 리스트를 입력받아 힙 구성 */\nminHeap = new PriorityQueue&lt;int, int&gt;([(1, 1), (3, 3), (2, 2), (5, 5), (4, 4)]);\n</code></pre> heap.go<pre><code>// Go에서는 heap.Interface를 구현하여 정수 최대 힙을 만들 수 있다\n// heap.Interface를 구현하려면 sort.Interface도 구현해야 한다\ntype intHeap []any\n\n// Push는 heap.Interface의 메서드로, 힙에 원소를 삽입한다\nfunc (h *intHeap) Push(x any) {\n    // Push와 Pop은 포인터 리시버(pointer receiver)를 매개변수로 사용한다\n    // 슬라이스의 내용뿐 아니라 길이도 수정하기 때문이다.\n    *h = append(*h, x.(int))\n}\n\n// Pop은 heap.Interface의 메서드로, 힙의 루트 원소를 꺼낸다\nfunc (h *intHeap) Pop() any {\n    // 꺼낼 원소는 맨 끝에 저장되어 있다\n    last := (*h)[len(*h)-1]\n    *h = (*h)[:len(*h)-1]\n    return last\n}\n\n// Len은 sort.Interface의 메서드다\nfunc (h *intHeap) Len() int {\n    return len(*h)\n}\n\n// Less는 sort.Interface의 메서드다\nfunc (h *intHeap) Less(i, j int) bool {\n    // 최소 힙을 구현하려면 부등호를 &lt;로 바꿔야 한다\n    return (*h)[i].(int) &gt; (*h)[j].(int)\n}\n\n// Swap은 sort.Interface의 메서드다\nfunc (h *intHeap) Swap(i, j int) {\n    (*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n\n// Top은 힙의 루트 원소를 구한다\nfunc (h *intHeap) Top() any {\n    return (*h)[0]\n}\n\n/* 실행 코드 */\nfunc TestHeap(t *testing.T) {\n    /* 힙 초기화 */\n    // 최대 힙 초기화\n    maxHeap := &amp;intHeap{}\n    heap.Init(maxHeap)\n    /* 힙에 원소 삽입 */\n    // heap.Interface의 메서드를 호출하여 원소를 추가한다\n    heap.Push(maxHeap, 1)\n    heap.Push(maxHeap, 3)\n    heap.Push(maxHeap, 2)\n    heap.Push(maxHeap, 4)\n    heap.Push(maxHeap, 5)\n\n    /* 힙의 루트 원소 구하기 */\n    top := maxHeap.Top()\n    fmt.Printf(\"힙의 루트 원소: %d\\n\", top)\n\n    /* 힙의 루트 원소 꺼내기 */\n    // heap.Interface의 메서드를 호출하여 원소를 제거한다\n    heap.Pop(maxHeap) // 5\n    heap.Pop(maxHeap) // 4\n    heap.Pop(maxHeap) // 3\n    heap.Pop(maxHeap) // 2\n    heap.Pop(maxHeap) // 1\n\n    /* 힙의 크기 구하기 */\n    size := len(*maxHeap)\n    fmt.Printf(\"힙의 원소 수: %d\\n\", size)\n\n    /* 힙이 비어 있는지 확인 */\n    isEmpty := len(*maxHeap) == 0\n    fmt.Printf(\"힙이 비어 있는가: %t\\n\", isEmpty)\n}\n</code></pre> heap.swift<pre><code>/* 힙 초기화 */\n// Swift의 Heap 타입은 최대 힙과 최소 힙을 모두 지원하며, swift-collections를 가져와야 한다\nvar heap = Heap&lt;Int&gt;()\n\n/* 힙에 원소 삽입 */\nheap.insert(1)\nheap.insert(3)\nheap.insert(2)\nheap.insert(5)\nheap.insert(4)\n\n/* 힙의 루트 원소 구하기 */\nvar peek = heap.max()!\n\n/* 힙의 루트 원소 꺼내기 */\npeek = heap.removeMax() // 5\npeek = heap.removeMax() // 4\npeek = heap.removeMax() // 3\npeek = heap.removeMax() // 2\npeek = heap.removeMax() // 1\n\n/* 힙의 크기 구하기 */\nlet size = heap.count\n\n/* 힙이 비어 있는지 확인 */\nlet isEmpty = heap.isEmpty\n\n/* 리스트를 입력받아 힙 구성 */\nlet heap2 = Heap([1, 3, 2, 5, 4])\n</code></pre> heap.js<pre><code>// JavaScript는 내장 Heap 클래스를 제공하지 않는다\n</code></pre> heap.ts<pre><code>// TypeScript는 내장 Heap 클래스를 제공하지 않는다\n</code></pre> heap.dart<pre><code>// Dart는 내장 Heap 클래스를 제공하지 않는다\n</code></pre> heap.rs<pre><code>use std::collections::BinaryHeap;\nuse std::cmp::Reverse;\n\n/* 힙 초기화 */\n// 최소 힙 초기화\nlet mut min_heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n// 최대 힙 초기화\nlet mut max_heap = BinaryHeap::new();\n\n/* 힙에 원소 삽입 */\nmax_heap.push(1);\nmax_heap.push(3);\nmax_heap.push(2);\nmax_heap.push(5);\nmax_heap.push(4);\n\n/* 힙의 루트 원소 구하기 */\nlet peek = max_heap.peek().unwrap();  // 5\n\n/* 힙의 루트 원소 꺼내기 */\n// 꺼낸 원소들은 큰 값부터 작은 값 순서로 나열된다\nlet peek = max_heap.pop().unwrap();   // 5\nlet peek = max_heap.pop().unwrap();   // 4\nlet peek = max_heap.pop().unwrap();   // 3\nlet peek = max_heap.pop().unwrap();   // 2\nlet peek = max_heap.pop().unwrap();   // 1\n\n/* 힙의 크기 구하기 */\nlet size = max_heap.len();\n\n/* 힙이 비어 있는지 확인 */\nlet is_empty = max_heap.is_empty();\n\n/* 리스트를 입력받아 힙 구성 */\nlet min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]);\n</code></pre> heap.c<pre><code>// C는 내장 Heap 클래스를 제공하지 않는다\n</code></pre> heap.kt<pre><code>/* 힙 초기화 */\n// 최소 힙 초기화\nvar minHeap = PriorityQueue&lt;Int&gt;()\n// 최대 힙 초기화(lambda 식으로 Comparator를 수정하면 된다)\nval maxHeap = PriorityQueue { a: Int, b: Int -&gt; b - a }\n\n/* 힙에 원소 삽입 */\nmaxHeap.offer(1)\nmaxHeap.offer(3)\nmaxHeap.offer(2)\nmaxHeap.offer(5)\nmaxHeap.offer(4)\n\n/* 힙의 루트 원소 구하기 */\nvar peek = maxHeap.peek() // 5\n\n/* 힙의 루트 원소 꺼내기 */\n// 꺼낸 원소들은 큰 값부터 작은 값 순서로 나열된다\npeek = maxHeap.poll() // 5\npeek = maxHeap.poll() // 4\npeek = maxHeap.poll() // 3\npeek = maxHeap.poll() // 2\npeek = maxHeap.poll() // 1\n\n/* 힙의 크기 구하기 */\nval size = maxHeap.size\n\n/* 힙이 비어 있는지 확인 */\nval isEmpty = maxHeap.isEmpty()\n\n/* 리스트를 입력받아 힙 구성 */\nminHeap = PriorityQueue(mutableListOf(1, 3, 2, 5, 4))\n</code></pre> heap.rb<pre><code># Ruby는 내장 Heap 클래스를 제공하지 않는다\n</code></pre> 실행 시각화 <p> 전체 화면 &gt;</p>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/heap/#812","level":2,"title":"8.1.2   힙의 구현","text":"<p>아래에서는 최대 힙을 구현한다. 이를 최소 힙으로 바꾸려면 모든 대소 비교 조건을 반대로 바꾸면 된다(예를 들어 \\(\\geq\\)를 \\(\\leq\\)로 바꾼다). 관심 있는 독자는 직접 구현해 보자.</p>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/heap/#1","level":3,"title":"1.   힙의 저장과 표현","text":"<p>'이진 트리' 장에서 살펴보았듯이 완전 이진 트리는 배열로 표현하기에 매우 적합하다. 힙은 완전 이진 트리이므로 배열을 사용하여 힙을 저장한다.</p> <p>배열로 이진 트리를 표현할 때 원소는 노드의 값을, 인덱스는 이진 트리에서 노드의 위치를 나타낸다. 노드 포인터는 인덱스 매핑 공식으로 구현한다.</p> <p>아래 그림(8-2)과 같이 인덱스 \\(i\\)가 주어지면 왼쪽 자식 노드의 인덱스는 \\(2i + 1\\), 오른쪽 자식 노드의 인덱스는 \\(2i + 2\\), 부모 노드의 인덱스는 \\((i - 1) / 2\\)(내림 정수 나눗셈)이다. 인덱스가 범위를 벗어나면 빈 노드이거나 노드가 존재하지 않는다는 뜻이다.</p> <p></p> <p> 그림 8-2   힙의 표현과 저장 </p> <p>나중에 편리하게 사용할 수 있도록 인덱스 매핑 공식을 함수로 만들 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def left(self, i: int) -&gt; int:\n    \"\"\"왼쪽 자식 노드의 인덱스 반환\"\"\"\n    return 2 * i + 1\n\ndef right(self, i: int) -&gt; int:\n    \"\"\"오른쪽 자식 노드의 인덱스 반환\"\"\"\n    return 2 * i + 2\n\ndef parent(self, i: int) -&gt; int:\n    \"\"\"부모 노드의 인덱스 반환\"\"\"\n    return (i - 1) // 2  # 나눗셈의 결과를 내림한다\n</code></pre> my_heap.cpp<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* 부모 노드의 인덱스 반환 */\nint parent(int i) {\n    return (i - 1) / 2; // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.java<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* 부모 노드의 인덱스 반환 */\nint parent(int i) {\n    return (i - 1) / 2; // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.cs<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nint Left(int i) {\n    return 2 * i + 1;\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nint Right(int i) {\n    return 2 * i + 2;\n}\n\n/* 부모 노드의 인덱스 반환 */\nint Parent(int i) {\n    return (i - 1) / 2; // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.go<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nfunc (h *maxHeap) left(i int) int {\n    return 2*i + 1\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nfunc (h *maxHeap) right(i int) int {\n    return 2*i + 2\n}\n\n/* 부모 노드의 인덱스 반환 */\nfunc (h *maxHeap) parent(i int) int {\n    // 나눗셈의 결과를 내림한다\n    return (i - 1) / 2\n}\n</code></pre> my_heap.swift<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nfunc left(i: Int) -&gt; Int {\n    2 * i + 1\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nfunc right(i: Int) -&gt; Int {\n    2 * i + 2\n}\n\n/* 부모 노드의 인덱스 반환 */\nfunc parent(i: Int) -&gt; Int {\n    (i - 1) / 2 // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.js<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\n#left(i) {\n    return 2 * i + 1;\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\n#right(i) {\n    return 2 * i + 2;\n}\n\n/* 부모 노드의 인덱스 반환 */\n#parent(i) {\n    return Math.floor((i - 1) / 2); // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.ts<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nleft(i: number): number {\n    return 2 * i + 1;\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nright(i: number): number {\n    return 2 * i + 2;\n}\n\n/* 부모 노드의 인덱스 반환 */\nparent(i: number): number {\n    return Math.floor((i - 1) / 2); // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.dart<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nint _left(int i) {\n  return 2 * i + 1;\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nint _right(int i) {\n  return 2 * i + 2;\n}\n\n/* 부모 노드의 인덱스 반환 */\nint _parent(int i) {\n  return (i - 1) ~/ 2; // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.rs<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nfn left(i: usize) -&gt; usize {\n    2 * i + 1\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nfn right(i: usize) -&gt; usize {\n    2 * i + 2\n}\n\n/* 부모 노드의 인덱스 반환 */\nfn parent(i: usize) -&gt; usize {\n    (i - 1) / 2 // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.c<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nint left(MaxHeap *maxHeap, int i) {\n    return 2 * i + 1;\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nint right(MaxHeap *maxHeap, int i) {\n    return 2 * i + 2;\n}\n\n/* 부모 노드의 인덱스 반환 */\nint parent(MaxHeap *maxHeap, int i) {\n    return (i - 1) / 2; // 내림\n}\n</code></pre> my_heap.kt<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nfun left(i: Int): Int {\n    return 2 * i + 1\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nfun right(i: Int): Int {\n    return 2 * i + 2\n}\n\n/* 부모 노드의 인덱스 반환 */\nfun parent(i: Int): Int {\n    return (i - 1) / 2 // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.rb<pre><code>### 왼쪽 자식 노드의 인덱스 반환 ###\ndef left(i)\n  2 * i + 1\nend\n\n### 오른쪽 자식 노드의 인덱스 반환 ###\ndef right(i)\n  2 * i + 2\nend\n\n### 부모 노드의 인덱스 반환 ###\ndef parent(i)\n  (i - 1) / 2     # 나눗셈의 결과를 내림한다\nend\n</code></pre>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/heap/#2","level":3,"title":"2.   힙의 루트 원소에 접근","text":"<p>힙의 루트 원소는 이진 트리의 루트 노드, 즉 리스트의 첫 번째 원소다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def peek(self) -&gt; int:\n    \"\"\"힙의 루트 원소 접근\"\"\"\n    return self.max_heap[0]\n</code></pre> my_heap.cpp<pre><code>/* 힙의 루트 원소 접근 */\nint peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.java<pre><code>/* 힙의 루트 원소 접근 */\nint peek() {\n    return maxHeap.get(0);\n}\n</code></pre> my_heap.cs<pre><code>/* 힙의 루트 원소 접근 */\nint Peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.go<pre><code>/* 힙의 루트 원소 접근 */\nfunc (h *maxHeap) peek() any {\n    return h.data[0]\n}\n</code></pre> my_heap.swift<pre><code>/* 힙의 루트 원소 접근 */\nfunc peek() -&gt; Int {\n    maxHeap[0]\n}\n</code></pre> my_heap.js<pre><code>/* 힙의 루트 원소 접근 */\npeek() {\n    return this.#maxHeap[0];\n}\n</code></pre> my_heap.ts<pre><code>/* 힙의 루트 원소 접근 */\npeek(): number {\n    return this.maxHeap[0];\n}\n</code></pre> my_heap.dart<pre><code>/* 힙의 루트 원소 접근 */\nint peek() {\n  return _maxHeap[0];\n}\n</code></pre> my_heap.rs<pre><code>/* 힙의 루트 원소 접근 */\nfn peek(&amp;self) -&gt; Option&lt;i32&gt; {\n    self.max_heap.first().copied()\n}\n</code></pre> my_heap.c<pre><code>/* 힙의 루트 원소 접근 */\nint peek(MaxHeap *maxHeap) {\n    return maxHeap-&gt;data[0];\n}\n</code></pre> my_heap.kt<pre><code>/* 힙의 루트 원소 접근 */\nfun peek(): Int {\n    return maxHeap[0]\n}\n</code></pre> my_heap.rb<pre><code>### 힙의 루트 원소 접근 ###\ndef peek\n  @max_heap[0]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/heap/#3","level":3,"title":"3.   힙에 원소 삽입","text":"<p>원소 <code>val</code>이 주어지면 먼저 힙의 끝에 추가한다. 추가한 <code>val</code>이 힙의 다른 원소보다 클 수 있으므로, 힙의 성립 조건이 깨질 수 있다. 따라서 삽입한 노드에서 루트 노드까지의 경로에 있는 노드들을 조정해야 한다. 이 연산을 힙 재정렬(heapify)이라고 한다.</p> <p>삽입한 노드부터 아래에서 위로 힙을 재정렬해 보자. 아래 그림(8-3)과 같이 삽입한 노드의 값과 부모 노드의 값을 비교하여 삽입한 노드가 더 크면 서로 바꾼다. 이 작업을 반복하여 아래에서 위로 힙의 노드들을 조정하고, 루트 노드를 지나거나 바꿀 필요가 없는 노드를 만나면 종료한다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 8-3   힙에 원소를 삽입하는 단계 </p> <p>전체 노드 수가 \\(n\\)이면 트리의 높이는 \\(O(\\log n)\\)이다. 따라서 힙 재정렬의 반복 횟수는 최대 \\(O(\\log n)\\)이며, 힙에 원소를 삽입하는 연산의 시간 복잡도는 \\(O(\\log n)\\)이다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def push(self, val: int):\n    \"\"\"힙에 원소 삽입\"\"\"\n    # 노드 추가\n    self.max_heap.append(val)\n    # 아래에서 위로 힙을 재정렬한다\n    self.sift_up(self.size() - 1)\n\ndef sift_up(self, i: int):\n    \"\"\"노드 i에서 시작하여 아래에서 위로 힙 재정렬\"\"\"\n    while True:\n        # 노드 i의 부모 노드를 구한다\n        p = self.parent(i)\n        # 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if p &lt; 0 or self.max_heap[i] &lt;= self.max_heap[p]:\n            break\n        # 두 노드를 교환한다\n        self.swap(i, p)\n        # 위쪽으로 힙 재정렬을 반복한다\n        i = p\n</code></pre> my_heap.cpp<pre><code>/* 힙에 원소 삽입 */\nvoid push(int val) {\n    // 노드 추가\n    maxHeap.push_back(val);\n    // 아래에서 위로 힙을 재정렬한다\n    siftUp(size() - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nvoid siftUp(int i) {\n    while (true) {\n        // 노드 i의 부모 노드를 구한다\n        int p = parent(i);\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // 두 노드를 교환한다\n        swap(maxHeap[i], maxHeap[p]);\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* 힙에 원소 삽입 */\nvoid push(int val) {\n    // 노드 추가\n    maxHeap.add(val);\n    // 아래에서 위로 힙을 재정렬한다\n    siftUp(size() - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nvoid siftUp(int i) {\n    while (true) {\n        // 노드 i의 부모 노드를 구한다\n        int p = parent(i);\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if (p &lt; 0 || maxHeap.get(i) &lt;= maxHeap.get(p))\n            break;\n        // 두 노드를 교환한다\n        swap(i, p);\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* 힙에 원소 삽입 */\nvoid Push(int val) {\n    // 노드 추가\n    maxHeap.Add(val);\n    // 아래에서 위로 힙을 재정렬한다\n    SiftUp(Size() - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nvoid SiftUp(int i) {\n    while (true) {\n        // 노드 i의 부모 노드를 구한다\n        int p = Parent(i);\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 종료한다\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // 두 노드를 교환한다\n        Swap(i, p);\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* 힙에 원소 삽입 */\nfunc (h *maxHeap) push(val any) {\n    // 노드 추가\n    h.data = append(h.data, val)\n    // 아래에서 위로 힙을 재정렬한다\n    h.siftUp(len(h.data) - 1)\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nfunc (h *maxHeap) siftUp(i int) {\n    for true {\n        // 노드 i의 부모 노드를 구한다\n        p := h.parent(i)\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if p &lt; 0 || h.data[i].(int) &lt;= h.data[p].(int) {\n            break\n        }\n        // 두 노드를 교환한다\n        h.swap(i, p)\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* 힙에 원소 삽입 */\nfunc push(val: Int) {\n    // 노드 추가\n    maxHeap.append(val)\n    // 아래에서 위로 힙을 재정렬한다\n    siftUp(i: size() - 1)\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nfunc siftUp(i: Int) {\n    var i = i\n    while true {\n        // 노드 i의 부모 노드를 구한다\n        let p = parent(i: i)\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if p &lt; 0 || maxHeap[i] &lt;= maxHeap[p] {\n            break\n        }\n        // 두 노드를 교환한다\n        swap(i: i, j: p)\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p\n    }\n}\n</code></pre> my_heap.js<pre><code>/* 힙에 원소 삽입 */\npush(val) {\n    // 노드 추가\n    this.#maxHeap.push(val);\n    // 아래에서 위로 힙을 재정렬한다\n    this.#siftUp(this.size() - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\n#siftUp(i) {\n    while (true) {\n        // 노드 i의 부모 노드를 구한다\n        const p = this.#parent(i);\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if (p &lt; 0 || this.#maxHeap[i] &lt;= this.#maxHeap[p]) break;\n        // 두 노드를 교환한다\n        this.#swap(i, p);\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* 힙에 원소 삽입 */\npush(val: number): void {\n    // 노드 추가\n    this.maxHeap.push(val);\n    // 아래에서 위로 힙을 재정렬한다\n    this.siftUp(this.size() - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nsiftUp(i: number): void {\n    while (true) {\n        // 노드 i의 부모 노드를 구한다\n        const p = this.parent(i);\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if (p &lt; 0 || this.maxHeap[i] &lt;= this.maxHeap[p]) break;\n        // 두 노드를 교환한다\n        this.swap(i, p);\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* 힙에 원소 삽입 */\nvoid push(int val) {\n  // 노드 추가\n  _maxHeap.add(val);\n  // 아래에서 위로 힙을 재정렬한다\n  siftUp(size() - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nvoid siftUp(int i) {\n  while (true) {\n    // 노드 i의 부모 노드를 구한다\n    int p = _parent(i);\n    // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n    if (p &lt; 0 || _maxHeap[i] &lt;= _maxHeap[p]) {\n      break;\n    }\n    // 두 노드를 교환한다\n    _swap(i, p);\n    // 위쪽으로 힙 재정렬을 반복한다\n    i = p;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* 힙에 원소 삽입 */\nfn push(&amp;mut self, val: i32) {\n    // 노드 추가\n    self.max_heap.push(val);\n    // 아래에서 위로 힙을 재정렬한다\n    self.sift_up(self.size() - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nfn sift_up(&amp;mut self, mut i: usize) {\n    loop {\n        // 노드 i가 이미 힙의 루트이므로 힙 재정렬을 마친다\n        if i == 0 {\n            break;\n        }\n        // 노드 i의 부모 노드를 구한다\n        let p = Self::parent(i);\n        // 노드를 조정할 필요가 없으면 힙 재정렬을 마친다\n        if self.max_heap[i] &lt;= self.max_heap[p] {\n            break;\n        }\n        // 두 노드를 교환한다\n        self.swap(i, p);\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* 힙에 원소 삽입 */\nvoid push(MaxHeap *maxHeap, int val) {\n    // 기본적으로 이렇게 많은 노드를 추가해서는 안 된다\n    if (maxHeap-&gt;size == MAX_SIZE) {\n        printf(\"힙이 가득 찼다!\");\n        return;\n    }\n    // 노드 추가\n    maxHeap-&gt;data[maxHeap-&gt;size] = val;\n    maxHeap-&gt;size++;\n\n    // 아래에서 위로 힙을 재정렬한다\n    siftUp(maxHeap, maxHeap-&gt;size - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nvoid siftUp(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // 노드 i의 부모 노드를 구한다\n        int p = parent(maxHeap, i);\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if (p &lt; 0 || maxHeap-&gt;data[i] &lt;= maxHeap-&gt;data[p]) {\n            break;\n        }\n        // 두 노드를 교환한다\n        swap(maxHeap, i, p);\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* 힙에 원소 삽입 */\nfun push(_val: Int) {\n    // 노드 추가\n    maxHeap.add(_val)\n    // 아래에서 위로 힙을 재정렬한다\n    siftUp(size() - 1)\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nfun siftUp(it: Int) {\n    // Kotlin의 함수 매개변수는 변경할 수 없으므로 임시 변수를 생성한다\n    var i = it\n    while (true) {\n        // 노드 i의 부모 노드를 구한다\n        val p = parent(i)\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n        // 두 노드를 교환한다\n        swap(i, p)\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p\n    }\n}\n</code></pre> my_heap.rb<pre><code>### 힙에 원소 삽입 ###\ndef push(val)\n  # 노드 추가\n  @max_heap &lt;&lt; val\n  # 아래에서 위로 힙을 재정렬한다\n  sift_up(size - 1)\nend\n\n### 노드 i에서 시작하여 아래에서 위로 힙 재정렬 ###\ndef sift_up(i)\n  loop do\n    # 노드 i의 부모 노드를 구한다\n    p = parent(i)\n    # 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n    break if p &lt; 0 || @max_heap[i] &lt;= @max_heap[p]\n    # 두 노드를 교환한다\n    swap(i, p)\n    # 위쪽으로 힙 재정렬을 반복한다\n    i = p\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/heap/#4","level":3,"title":"4.   힙의 루트 원소 꺼내기","text":"<p>힙의 루트 원소는 이진 트리의 루트 노드, 즉 리스트의 첫 번째 원소다. 리스트에서 첫 번째 원소를 바로 삭제하면 이진 트리의 모든 노드 인덱스가 바뀌어, 이후 힙 재정렬로 복구하기가 어려워진다. 원소 인덱스의 변화를 최소화하기 위해 다음 절차를 사용한다.</p> <ol> <li>힙의 루트 원소와 끝 원소를 교환한다(루트 노드와 맨 오른쪽 리프 노드를 교환한다).</li> <li>교환 후 리스트에서 힙의 끝 원소를 삭제한다(이미 교환했으므로 실제로 삭제되는 것은 원래의 루트 원소다).</li> <li>루트 노드부터 위에서 아래로 힙을 재정렬한다.</li> </ol> <p>아래 그림(8-4)과 같이 '위에서 아래로 힙 재정렬'은 '아래에서 위로 힙 재정렬'과 진행 방향이 반대다. 루트 노드의 값을 두 자식 노드의 값과 비교하여, 더 큰 값을 가진 자식 노드와 루트 노드를 교환한다. 이 작업을 반복하며 리프 노드를 지나거나 교환할 필요가 없는 노드를 만나면 종료한다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 8-4   힙의 루트 원소를 꺼내는 단계 </p> <p>원소 삽입과 마찬가지로 힙의 루트 원소를 꺼내는 연산의 시간 복잡도도 \\(O(\\log n)\\)이다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def pop(self) -&gt; int:\n    \"\"\"힙에서 원소 꺼내기\"\"\"\n    # 비어 있는 경우를 처리한다\n    if self.is_empty():\n        raise IndexError(\"힙이 비어 있다\")\n    # 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    self.swap(0, self.size() - 1)\n    # 노드 삭제\n    val = self.max_heap.pop()\n    # 위에서 아래로 힙을 재정렬한다\n    self.sift_down(0)\n    # 힙의 루트 원소를 반환한다\n    return val\n\ndef sift_down(self, i: int):\n    \"\"\"노드 i에서 시작하여 위에서 아래로 힙 재정렬\"\"\"\n    while True:\n        # 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        l, r, ma = self.left(i), self.right(i), i\n        if l &lt; self.size() and self.max_heap[l] &gt; self.max_heap[ma]:\n            ma = l\n        if r &lt; self.size() and self.max_heap[r] &gt; self.max_heap[ma]:\n            ma = r\n        # 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if ma == i:\n            break\n        # 두 노드를 교환한다\n        self.swap(i, ma)\n        # 아래쪽으로 힙 재정렬을 반복한다\n        i = ma\n</code></pre> my_heap.cpp<pre><code>/* 힙에서 원소 꺼내기 */\nvoid pop() {\n    // 비어 있는 경우를 처리한다\n    if (isEmpty()) {\n        throw out_of_range(\"힙이 비어 있다\");\n    }\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    swap(maxHeap[0], maxHeap[size() - 1]);\n    // 노드 삭제\n    maxHeap.pop_back();\n    // 위에서 아래로 힙을 재정렬한다\n    siftDown(0);\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nvoid siftDown(int i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i)\n            break;\n        swap(maxHeap[i], maxHeap[ma]);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* 힙에서 원소 꺼내기 */\nint pop() {\n    // 비어 있는 경우를 처리한다\n    if (isEmpty())\n        throw new IndexOutOfBoundsException();\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    swap(0, size() - 1);\n    // 노드 삭제\n    int val = maxHeap.remove(size() - 1);\n    // 위에서 아래로 힙을 재정렬한다\n    siftDown(0);\n    // 힙의 루트 원소를 반환한다\n    return val;\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nvoid siftDown(int i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap.get(l) &gt; maxHeap.get(ma))\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap.get(r) &gt; maxHeap.get(ma))\n            ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i)\n            break;\n        // 두 노드를 교환한다\n        swap(i, ma);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* 힙에서 원소 꺼내기 */\nint Pop() {\n    // 비어 있는 경우를 처리한다\n    if (IsEmpty())\n        throw new IndexOutOfRangeException();\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    Swap(0, Size() - 1);\n    // 노드 삭제\n    int val = maxHeap.Last();\n    maxHeap.RemoveAt(Size() - 1);\n    // 위에서 아래로 힙을 재정렬한다\n    SiftDown(0);\n    // 힙의 루트 원소를 반환한다\n    return val;\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nvoid SiftDown(int i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        int l = Left(i), r = Right(i), ma = i;\n        if (l &lt; Size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; Size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // 노드 i의 값이 가장 크거나 리프 노드를 넘어가면 힙 재정렬을 종료한다\n        if (ma == i) break;\n        // 두 노드를 교환한다\n        Swap(i, ma);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* 힙에서 원소 꺼내기 */\nfunc (h *maxHeap) pop() any {\n    // 비어 있는 경우를 처리한다\n    if h.isEmpty() {\n        fmt.Println(\"오류\")\n        return nil\n    }\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    h.swap(0, h.size()-1)\n    // 노드 삭제\n    val := h.data[len(h.data)-1]\n    h.data = h.data[:len(h.data)-1]\n    // 위에서 아래로 힙을 재정렬한다\n    h.siftDown(0)\n\n    // 힙의 루트 원소를 반환한다\n    return val\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nfunc (h *maxHeap) siftDown(i int) {\n    for true {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 max로 표시한다\n        l, r, max := h.left(i), h.right(i), i\n        if l &lt; h.size() &amp;&amp; h.data[l].(int) &gt; h.data[max].(int) {\n            max = l\n        }\n        if r &lt; h.size() &amp;&amp; h.data[r].(int) &gt; h.data[max].(int) {\n            max = r\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if max == i {\n            break\n        }\n        // 두 노드를 교환한다\n        h.swap(i, max)\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = max\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* 힙에서 원소 꺼내기 */\nfunc pop() -&gt; Int {\n    // 비어 있는 경우를 처리한다\n    if isEmpty() {\n        fatalError(\"힙이 비어 있다\")\n    }\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    swap(i: 0, j: size() - 1)\n    // 노드 삭제\n    let val = maxHeap.remove(at: size() - 1)\n    // 위에서 아래로 힙을 재정렬한다\n    siftDown(i: 0)\n    // 힙의 루트 원소를 반환한다\n    return val\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nfunc siftDown(i: Int) {\n    var i = i\n    while true {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        let l = left(i: i)\n        let r = right(i: i)\n        var ma = i\n        if l &lt; size(), maxHeap[l] &gt; maxHeap[ma] {\n            ma = l\n        }\n        if r &lt; size(), maxHeap[r] &gt; maxHeap[ma] {\n            ma = r\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if ma == i {\n            break\n        }\n        // 두 노드를 교환한다\n        swap(i: i, j: ma)\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma\n    }\n}\n</code></pre> my_heap.js<pre><code>/* 힙에서 원소 꺼내기 */\npop() {\n    // 비어 있는 경우를 처리한다\n    if (this.isEmpty()) throw new Error('힙이 비어 있다');\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    this.#swap(0, this.size() - 1);\n    // 노드 삭제\n    const val = this.#maxHeap.pop();\n    // 위에서 아래로 힙을 재정렬한다\n    this.#siftDown(0);\n    // 힙의 루트 원소를 반환한다\n    return val;\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\n#siftDown(i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        const l = this.#left(i),\n            r = this.#right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.#maxHeap[l] &gt; this.#maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.#maxHeap[r] &gt; this.#maxHeap[ma]) ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma === i) break;\n        // 두 노드를 교환한다\n        this.#swap(i, ma);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* 힙에서 원소 꺼내기 */\npop(): number {\n    // 비어 있는 경우를 처리한다\n    if (this.isEmpty()) throw new RangeError('힙이 비어 있다.');\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    this.swap(0, this.size() - 1);\n    // 노드 삭제\n    const val = this.maxHeap.pop();\n    // 위에서 아래로 힙을 재정렬한다\n    this.siftDown(0);\n    // 힙의 루트 원소를 반환한다\n    return val;\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nsiftDown(i: number): void {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        const l = this.left(i),\n            r = this.right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.maxHeap[l] &gt; this.maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.maxHeap[r] &gt; this.maxHeap[ma]) ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma === i) break;\n        // 두 노드를 교환한다\n        this.swap(i, ma);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* 힙에서 원소 꺼내기 */\nint pop() {\n  // 비어 있는 경우를 처리한다\n  if (isEmpty()) throw Exception('힙이 비어 있다');\n  // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n  _swap(0, size() - 1);\n  // 노드 삭제\n  int val = _maxHeap.removeLast();\n  // 위에서 아래로 힙을 재정렬한다\n  siftDown(0);\n  // 힙의 루트 원소를 반환한다\n  return val;\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nvoid siftDown(int i) {\n  while (true) {\n    // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n    int l = _left(i);\n    int r = _right(i);\n    int ma = i;\n    if (l &lt; size() &amp;&amp; _maxHeap[l] &gt; _maxHeap[ma]) ma = l;\n    if (r &lt; size() &amp;&amp; _maxHeap[r] &gt; _maxHeap[ma]) ma = r;\n    // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n    if (ma == i) break;\n    // 두 노드를 교환한다\n    _swap(i, ma);\n    // 아래쪽으로 힙 재정렬을 반복한다\n    i = ma;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* 힙에서 원소 꺼내기 */\nfn pop(&amp;mut self) -&gt; i32 {\n    // 비어 있는 경우를 처리한다\n    if self.is_empty() {\n        panic!(\"인덱스가 범위를 벗어났다\");\n    }\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    self.swap(0, self.size() - 1);\n    // 노드 삭제\n    let val = self.max_heap.pop().unwrap();\n    // 위에서 아래로 힙을 재정렬한다\n    self.sift_down(0);\n    // 힙의 루트 원소를 반환한다\n    val\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nfn sift_down(&amp;mut self, mut i: usize) {\n    loop {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        let (l, r, mut ma) = (Self::left(i), Self::right(i), i);\n        if l &lt; self.size() &amp;&amp; self.max_heap[l] &gt; self.max_heap[ma] {\n            ma = l;\n        }\n        if r &lt; self.size() &amp;&amp; self.max_heap[r] &gt; self.max_heap[ma] {\n            ma = r;\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if ma == i {\n            break;\n        }\n        // 두 노드를 교환한다\n        self.swap(i, ma);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* 힙에서 원소 꺼내기 */\nint pop(MaxHeap *maxHeap) {\n    // 비어 있는 경우를 처리한다\n    if (isEmpty(maxHeap)) {\n        printf(\"힙이 비어 있다!\");\n        return INT_MAX;\n    }\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    swap(maxHeap, 0, size(maxHeap) - 1);\n    // 노드 삭제\n    int val = maxHeap-&gt;data[maxHeap-&gt;size - 1];\n    maxHeap-&gt;size--;\n    // 위에서 아래로 힙을 재정렬한다\n    siftDown(maxHeap, 0);\n\n    // 힙의 루트 원소를 반환한다\n    return val;\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nvoid siftDown(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 max로 표시한다\n        int l = left(maxHeap, i);\n        int r = right(maxHeap, i);\n        int max = i;\n        if (l &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[l] &gt; maxHeap-&gt;data[max]) {\n            max = l;\n        }\n        if (r &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[r] &gt; maxHeap-&gt;data[max]) {\n            max = r;\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (max == i) {\n            break;\n        }\n        // 두 노드를 교환한다\n        swap(maxHeap, i, max);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = max;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* 힙에서 원소 꺼내기 */\nfun pop(): Int {\n    // 비어 있는 경우를 처리한다\n    if (isEmpty()) throw IndexOutOfBoundsException()\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    swap(0, size() - 1)\n    // 노드 삭제\n    val _val = maxHeap.removeAt(size() - 1)\n    // 위에서 아래로 힙을 재정렬한다\n    siftDown(0)\n    // 힙의 루트 원소를 반환한다\n    return _val\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nfun siftDown(it: Int) {\n    // Kotlin의 함수 매개변수는 변경할 수 없으므로 임시 변수를 생성한다\n    var i = it\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        val l = left(i)\n        val r = right(i)\n        var ma = i\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i) break\n        // 두 노드를 교환한다\n        swap(i, ma)\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma\n    }\n}\n</code></pre> my_heap.rb<pre><code>### 힙에서 원소 꺼내기 ###\ndef pop\n  # 비어 있는 경우를 처리한다\n  raise IndexError, \"힙이 비어 있다\" if is_empty?\n  # 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n  swap(0, size - 1)\n  # 노드 삭제\n  val = @max_heap.pop\n  # 위에서 아래로 힙을 재정렬한다\n  sift_down(0)\n  # 힙의 루트 원소를 반환한다\n  val\nend\n\n### 노드 i에서 시작하여 위에서 아래로 힙 재정렬 ###\ndef sift_down(i)\n  loop do\n    # 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n    l, r, ma = left(i), right(i), i\n    ma = l if l &lt; size &amp;&amp; @max_heap[l] &gt; @max_heap[ma]\n    ma = r if r &lt; size &amp;&amp; @max_heap[r] &gt; @max_heap[ma]\n\n    # 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n    break if ma == i\n\n    # 두 노드를 교환한다\n    swap(i, ma)\n    # 아래쪽으로 힙 재정렬을 반복한다\n    i = ma\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/heap/#813","level":2,"title":"8.1.3   힙의 주요 응용","text":"<ul> <li>우선순위 큐: 힙은 우선순위 큐를 구현할 때 가장 먼저 고려하는 자료 구조다. 삽입과 꺼내기 연산의 시간 복잡도는 모두 \\(O(\\log n)\\)이고, 힙 구성 연산은 \\(O(n)\\)으로 모두 매우 효율적이다.</li> <li>힙 정렬: 데이터가 주어지면 이를 이용해 힙을 만든 뒤 원소를 계속 꺼내어 정렬된 데이터를 얻을 수 있다. 다만 힙 정렬은 보통 더 간결한 방법으로 구현한다. 자세한 내용은 '힙 정렬' 절을 참고하자.</li> <li>가장 큰 \\(k\\) 개 원소 구하기: 대표적인 알고리즘 문제이자 흔한 응용 사례다. 예를 들어 인기 상위 10개 뉴스를 웨이보의 인기 검색어로 선정하거나, 판매량 상위 10개 상품을 고를 때 사용할 수 있다.</li> </ul>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/summary/","level":1,"title":"8.4   요약","text":"","path":["제 8 장   힙","8.4   요약"],"tags":[]},{"location":"chapter_heap/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>힙은 완전 이진 트리이며, 만족하는 조건에 따라 최대 힙과 최소 힙으로 나뉜다. 최대(최소) 힙의 루트 원소는 가장 큰(작은) 값이다.</li> <li>우선순위 큐는 원소를 꺼내는 우선순위가 있는 큐로 정의되며, 보통 힙으로 구현한다.</li> <li>힙의 주요 연산과 시간 복잡도는 원소 삽입 \\(O(\\log n)\\), 루트 원소 꺼내기 \\(O(\\log n)\\), 루트 원소 접근 \\(O(1)\\) 등이다.</li> <li>완전 이진 트리는 배열로 표현하기에 매우 적합하므로, 힙은 보통 배열에 저장한다.</li> <li>힙 재정렬 연산은 힙의 성질을 유지하는 데 사용하며, 삽입과 꺼내기 연산 모두에서 필요하다.</li> <li>\\(n\\) 개의 원소를 입력받아 힙을 구성하는 시간 복잡도는 \\(O(n)\\)까지 최적화할 수 있어 매우 효율적이다.</li> <li>Top-k는 대표적인 알고리즘 문제다. 힙 자료 구조를 사용하면 \\(O(n \\log k)\\)의 시간 복잡도로 효율적으로 해결할 수 있다.</li> </ul>","path":["제 8 장   힙","8.4   요약"],"tags":[]},{"location":"chapter_heap/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 자료 구조의 '힙'과 메모리 관리의 '힙'은 같은 개념인가?</p> <p>두 개념은 서로 다르며, 우연히 이름만 '힙'으로 같을 뿐이다. 컴퓨터 시스템 메모리의 힙은 동적 메모리 할당에 사용되는 영역으로, 프로그램이 실행 중에 데이터를 저장하는 데 사용할 수 있다. 프로그램은 객체나 배열 같은 복잡한 구조를 저장하기 위해 일정량의 힙 메모리를 요청할 수 있다. 데이터가 더 이상 필요하지 않으면 메모리 누수를 막기 위해 해당 메모리를 해제해야 한다. 스택 메모리에 비해 힙 메모리는 관리와 사용에 더 주의해야 하며, 잘못 사용하면 메모리 누수나 유효하지 않은 포인터 등의 문제가 발생할 수 있다.</p>","path":["제 8 장   힙","8.4   요약"],"tags":[]},{"location":"chapter_heap/top_k/","level":1,"title":"8.3   Top-k 문제","text":"<p>문제</p> <p>길이가 \\(n\\)인 정렬되지 않은 배열 <code>nums</code>가 주어졌을 때, 배열에서 가장 큰 \\(k\\) 개의 원소를 반환하라.</p> <p>이 문제에 대해 먼저 발상이 비교적 직접적인 두 가지 풀이를 살펴보고, 이어서 더 효율적인 힙 기반 풀이를 소개한다.</p>","path":["제 8 장   힙","8.3   Top-k 문제"],"tags":[]},{"location":"chapter_heap/top_k/#831-1","level":2,"title":"8.3.1   방법 1: 순회하며 선택","text":"<p>아래 그림(8-6)과 같이 \\(k\\) 번 순회하면서 매 순회에서 각각 \\(1\\), \\(2\\), \\(\\dots\\), \\(k\\) 번째로 큰 원소를 뽑을 수 있다. 시간 복잡도는 \\(O(nk)\\)이다.</p> <p>이 방법은 \\(k \\ll n\\)인 경우에만 적합하다. \\(k\\)가 \\(n\\)에 가까워지면 시간 복잡도가 \\(O(n^2)\\)에 가까워져 매우 오래 걸리기 때문이다.</p> <p></p> <p> 그림 8-6   순회로 가장 큰 k개의 원소 찾기 </p> <p>참고</p> <p>\\(k = n\\)이면 전체가 정렬된 수열을 얻을 수 있으며, 이는 '선택 정렬' 알고리즘과 같다.</p>","path":["제 8 장   힙","8.3   Top-k 문제"],"tags":[]},{"location":"chapter_heap/top_k/#832-2","level":2,"title":"8.3.2   방법 2: 정렬","text":"<p>아래 그림(8-7)과 같이 배열 <code>nums</code>를 먼저 정렬한 뒤 맨 오른쪽의 \\(k\\) 개 원소를 반환할 수 있다. 시간 복잡도는 \\(O(n \\log n)\\)이다.</p> <p>이 방법은 필요한 것보다 더 많은 일을 한다. 가장 큰 \\(k\\) 개 원소만 찾으면 되며, 나머지 원소를 정렬할 필요는 없기 때문이다.</p> <p></p> <p> 그림 8-7   정렬로 가장 큰 k개의 원소 찾기 </p>","path":["제 8 장   힙","8.3   Top-k 문제"],"tags":[]},{"location":"chapter_heap/top_k/#833-3","level":2,"title":"8.3.3   방법 3: 힙","text":"<p>힙을 이용하면 Top-k 문제를 더 효율적으로 해결할 수 있다. 과정은 아래 그림(8-8)과 같다.</p> <ol> <li>루트 원소가 가장 작은 최소 힙을 초기화한다.</li> <li>먼저 배열의 앞 \\(k\\) 개 원소를 차례로 힙에 삽입한다.</li> <li>\\(k + 1\\) 번째 원소부터 시작하여, 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 삽입한다.</li> <li>순회를 마치면 힙에 남은 원소가 가장 큰 \\(k\\) 개 원소다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 8-8   힙으로 가장 큰 k개의 원소 찾기 </p> <p>예시 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby top_k.py<pre><code>def top_k_heap(nums: list[int], k: int) -&gt; list[int]:\n    \"\"\"힙을 이용하여 배열에서 가장 큰 k개 원소 탐색\"\"\"\n    # 최소 힙 초기화\n    heap = []\n    # 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for i in range(k):\n        heapq.heappush(heap, nums[i])\n    # k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for i in range(k, len(nums)):\n        # 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if nums[i] &gt; heap[0]:\n            heapq.heappop(heap)\n            heapq.heappush(heap, nums[i])\n    return heap\n</code></pre> top_k.cpp<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; topKHeap(vector&lt;int&gt; &amp;nums, int k) {\n    // 최소 힙 초기화\n    priority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; heap;\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for (int i = 0; i &lt; k; i++) {\n        heap.push(nums[i]);\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for (int i = k; i &lt; nums.size(); i++) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if (nums[i] &gt; heap.top()) {\n            heap.pop();\n            heap.push(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.java<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nQueue&lt;Integer&gt; topKHeap(int[] nums, int k) {\n    // 최소 힙 초기화\n    Queue&lt;Integer&gt; heap = new PriorityQueue&lt;Integer&gt;();\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for (int i = 0; i &lt; k; i++) {\n        heap.offer(nums[i]);\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for (int i = k; i &lt; nums.length; i++) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll();\n            heap.offer(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.cs<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nPriorityQueue&lt;int, int&gt; TopKHeap(int[] nums, int k) {\n    // 최소 힙 초기화\n    PriorityQueue&lt;int, int&gt; heap = new();\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for (int i = 0; i &lt; k; i++) {\n        heap.Enqueue(nums[i], nums[i]);\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for (int i = k; i &lt; nums.Length; i++) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if (nums[i] &gt; heap.Peek()) {\n            heap.Dequeue();\n            heap.Enqueue(nums[i], nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.go<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nfunc topKHeap(nums []int, k int) *minHeap {\n    // 최소 힙 초기화\n    h := &amp;minHeap{}\n    heap.Init(h)\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for i := 0; i &lt; k; i++ {\n        heap.Push(h, nums[i])\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for i := k; i &lt; len(nums); i++ {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if nums[i] &gt; h.Top().(int) {\n            heap.Pop(h)\n            heap.Push(h, nums[i])\n        }\n    }\n    return h\n}\n</code></pre> top_k.swift<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nfunc topKHeap(nums: [Int], k: Int) -&gt; [Int] {\n    // 최소 힙을 초기화하여 앞쪽 k개 원소로 힙을 구성한다\n    var heap = Heap(nums.prefix(k))\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for i in nums.indices.dropFirst(k) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if nums[i] &gt; heap.min()! {\n            _ = heap.removeMin()\n            heap.insert(nums[i])\n        }\n    }\n    return heap.unordered\n}\n</code></pre> top_k.js<pre><code>/* 힙에 원소 삽입 */\nfunction pushMinHeap(maxHeap, val) {\n    // 원소의 부호를 반전한다\n    maxHeap.push(-val);\n}\n\n/* 힙에서 원소 꺼내기 */\nfunction popMinHeap(maxHeap) {\n    // 원소의 부호를 반전한다\n    return -maxHeap.pop();\n}\n\n/* 힙의 루트 원소 접근 */\nfunction peekMinHeap(maxHeap) {\n    // 원소의 부호를 반전한다\n    return -maxHeap.peek();\n}\n\n/* 힙의 원소를 반환한다 */\nfunction getMinHeap(maxHeap) {\n    // 원소의 부호를 반전한다\n    return maxHeap.getMaxHeap().map((num) =&gt; -num);\n}\n\n/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nfunction topKHeap(nums, k) {\n    // 최소 힙 초기화\n    // 힙의 모든 원소의 부호를 반전하여 최대 힙으로 최소 힙을 구현한다는 점에 유의한다\n    const maxHeap = new MaxHeap([]);\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for (let i = k; i &lt; nums.length; i++) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // 힙의 원소를 반환한다\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.ts<pre><code>/* 힙에 원소 삽입 */\nfunction pushMinHeap(maxHeap: MaxHeap, val: number): void {\n    // 원소의 부호를 반전한다\n    maxHeap.push(-val);\n}\n\n/* 힙에서 원소 꺼내기 */\nfunction popMinHeap(maxHeap: MaxHeap): number {\n    // 원소의 부호를 반전한다\n    return -maxHeap.pop();\n}\n\n/* 힙의 루트 원소 접근 */\nfunction peekMinHeap(maxHeap: MaxHeap): number {\n    // 원소의 부호를 반전한다\n    return -maxHeap.peek();\n}\n\n/* 힙의 원소를 반환한다 */\nfunction getMinHeap(maxHeap: MaxHeap): number[] {\n    // 원소의 부호를 반전한다\n    return maxHeap.getMaxHeap().map((num: number) =&gt; -num);\n}\n\n/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nfunction topKHeap(nums: number[], k: number): number[] {\n    // 최소 힙 초기화\n    // 힙의 모든 원소의 부호를 반전하여 최대 힙으로 최소 힙을 구현한다는 점에 유의한다\n    const maxHeap = new MaxHeap([]);\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for (let i = k; i &lt; nums.length; i++) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // 힙의 원소를 반환한다\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.dart<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nMinHeap topKHeap(List&lt;int&gt; nums, int k) {\n  // 최소 힙을 초기화하고 배열의 앞쪽 k개 원소를 힙에 삽입한다\n  MinHeap heap = MinHeap(nums.sublist(0, k));\n  // k+1번째 원소부터 힙의 길이를 k로 유지한다\n  for (int i = k; i &lt; nums.length; i++) {\n    // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n    if (nums[i] &gt; heap.peek()) {\n      heap.pop();\n      heap.push(nums[i]);\n    }\n  }\n  return heap;\n}\n</code></pre> top_k.rs<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nfn top_k_heap(nums: Vec&lt;i32&gt;, k: usize) -&gt; BinaryHeap&lt;Reverse&lt;i32&gt;&gt; {\n    // BinaryHeap은 최대 힙이므로 Reverse로 원소의 순서를 뒤집어 최소 힙을 구현한다\n    let mut heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for &amp;num in nums.iter().take(k) {\n        heap.push(Reverse(num));\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for &amp;num in nums.iter().skip(k) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if num &gt; heap.peek().unwrap().0 {\n            heap.pop();\n            heap.push(Reverse(num));\n        }\n    }\n    heap\n}\n</code></pre> top_k.c<pre><code>/* 힙에 원소 삽입 */\nvoid pushMinHeap(MaxHeap *maxHeap, int val) {\n    // 원소의 부호를 반전한다\n    push(maxHeap, -val);\n}\n\n/* 힙에서 원소 꺼내기 */\nint popMinHeap(MaxHeap *maxHeap) {\n    // 원소의 부호를 반전한다\n    return -pop(maxHeap);\n}\n\n/* 힙의 루트 원소 접근 */\nint peekMinHeap(MaxHeap *maxHeap) {\n    // 원소의 부호를 반전한다\n    return -peek(maxHeap);\n}\n\n/* 힙의 원소를 반환한다 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // 힙의 모든 원소의 부호를 반전하여 res 배열에 저장한다\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n/* 힙의 원소를 반환한다 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // 힙의 모든 원소의 부호를 반전하여 res 배열에 저장한다\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n// 힙으로 배열에서 가장 큰 k개 원소를 찾는 함수\nint *topKHeap(int *nums, int sizeNums, int k) {\n    // 최소 힙 초기화\n    // 힙의 모든 원소의 부호를 반전하여 최대 힙으로 최소 힙을 구현한다는 점에 유의한다\n    int *empty = (int *)malloc(0);\n    MaxHeap *maxHeap = newMaxHeap(empty, 0);\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for (int i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for (int i = k; i &lt; sizeNums; i++) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    int *res = getMinHeap(maxHeap);\n    // 메모리 해제\n    delMaxHeap(maxHeap);\n    return res;\n}\n</code></pre> top_k.kt<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nfun topKHeap(nums: IntArray, k: Int): Queue&lt;Int&gt; {\n    // 최소 힙 초기화\n    val heap = PriorityQueue&lt;Int&gt;()\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for (i in 0..&lt;k) {\n        heap.offer(nums[i])\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for (i in k..&lt;nums.size) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll()\n            heap.offer(nums[i])\n        }\n    }\n    return heap\n}\n</code></pre> top_k.rb<pre><code>### 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 ###\ndef top_k_heap(nums, k)\n  # 최소 힙 초기화\n  # 힙의 모든 원소의 부호를 반전하여 최대 힙으로 최소 힙을 구현한다는 점에 유의한다\n  max_heap = MaxHeap.new([])\n\n  # 배열의 앞쪽 k개 원소를 힙에 삽입한다\n  for i in 0...k\n    push_min_heap(max_heap, nums[i])\n  end\n\n  # k+1번째 원소부터 힙의 길이를 k로 유지한다\n  for i in k...nums.length\n    # 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n    if nums[i] &gt; peek_min_heap(max_heap)\n      pop_min_heap(max_heap)\n      push_min_heap(max_heap, nums[i])\n    end\n  end\n\n  get_min_heap(max_heap)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>삽입과 꺼내기를 총 \\(n\\) 차례 수행하고 힙의 최대 길이는 \\(k\\)이므로, 시간 복잡도는 \\(O(n \\log k)\\)이다. 이 방법은 매우 효율적이다. \\(k\\)가 작으면 시간 복잡도가 \\(O(n)\\)에 가까워지고, \\(k\\)가 커도 \\(O(n \\log n)\\)을 넘지 않는다.</p> <p>또한 이 방법은 동적으로 들어오는 데이터 스트림에도 적합하다. 데이터가 계속 추가될 때 힙 안의 원소를 지속적으로 관리하면 가장 큰 \\(k\\) 개 원소를 동적으로 갱신할 수 있다.</p>","path":["제 8 장   힙","8.3   Top-k 문제"],"tags":[]},{"location":"chapter_hello_algo/","level":1,"title":"머리말","text":"<p>몇 년 전, 나는 리트코드에 『剑指 Offer』의 문제 풀이 시리즈를 공유했고 많은 독자의 격려와 지지를 받았다. 독자들과 이야기하면서 가장 자주 받은 질문은 “알고리즘 공부를 어떻게 시작하면 좋을까요?”였다. 점차 나는 이 질문에 깊은 관심을 갖게 되었다.</p> <p>아무것도 모르는 상태에서 무작정 문제를 푸는 방식이 가장 인기 있는 방법인 듯하다. 간단하고 직접적이며 효과도 있다. 하지만 문제 풀이는 ‘지뢰 찾기’ 게임과 비슷하다. 스스로 공부하는 능력이 뛰어난 사람은 지뢰를 하나씩 무사히 제거하지만, 기초가 부족한 사람은 지뢰에 거듭 부딪혀 좌절하며 점점 물러서기 쉽다. 교재를 처음부터 끝까지 읽는 것도 흔한 방법이다. 그러나 취업을 준비하는 사람에게는 졸업 논문 작성, 이력서 제출, 필기시험과 면접 준비만으로도 대부분의 에너지가 소모되므로, 두꺼운 책을 파고드는 일이 큰 도전이 되곤 한다.</p> <p>비슷한 어려움을 겪고 있다면, 이 책이 여러분을 ‘찾아온’ 것은 다행스러운 일이다. 이 책은 그 질문에 대한 나의 답이다. 최적의 답은 아닐지라도, 적어도 적극적인 시도는 될 것이다. 이 책만으로 곧바로 취업에 성공하기는 어렵겠지만, 자료 구조와 알고리즘의 ‘지식 지도’를 탐험하도록 안내하고 여러 ‘지뢰’의 모양, 크기, 분포를 알려 주며 다양한 ‘지뢰 제거 방법’을 익히도록 도울 것이다. 이러한 능력을 갖추면 문제를 풀고 문헌을 읽는 일이 한결 수월해지고, 차츰 완전한 지식 체계를 세울 수 있을 것이다.</p> <p>나는 파인만 교수의 말에 깊이 공감한다. “지식은 공짜가 아니다. 주의를 기울여야 한다.” 이런 의미에서 이 책은 완전히 ‘무료’인 것은 아니다. 여러분이 이 책에 쏟는 소중한 ‘주의와 관심’을 헛되게 하지 않도록, 나 역시 최대한의 ‘주의와 관심’을 기울여 집필에 최선을 다하고자 한다.</p> <p>내 학식과 재능이 부족함을 잘 알고 있다. 책의 내용을 한동안 다듬어 왔지만 여전히 많은 오류가 있을 것이다. 선생님과 학생 여러분의 비판과 지적을 부탁드린다.</p> <p></p> Hello, 알고리즘! <p>컴퓨터의 등장은 세상을 크게 바꾸었다. 빠른 계산 능력과 뛰어난 프로그래밍 가능성을 갖춘 컴퓨터는 알고리즘을 실행하고 데이터를 처리하는 이상적인 수단이 되었다. 비디오 게임의 사실적인 화면, 자율 주행의 지능적인 판단, AlphaGo의 멋진 대국, ChatGPT의 자연스러운 상호작용까지, 이 모든 응용은 컴퓨터 위에서 알고리즘이 정교하게 구현된 결과다.</p> <p>사실 컴퓨터가 등장하기 전부터 알고리즘과 자료 구조는 세상 곳곳에 존재했다. 초기의 알고리즘은 고대의 셈법이나 도구 제작 절차처럼 비교적 단순했다. 문명이 발전하면서 알고리즘도 점점 더 정교하고 복잡해졌다. 장인의 놀라운 솜씨에서 생산력을 높여 준 산업 제품에 이르기까지, 나아가 우주의 운행을 설명하는 과학 법칙에 이르기까지, 평범하거나 경이로운 거의 모든 것의 이면에는 정교한 알고리즘적 사고가 숨어 있다.</p> <p>자료 구조 역시 어디에나 있다. 크게는 사회 관계망부터 작게는 지하철 노선까지 많은 시스템을 ‘그래프’로 모델링할 수 있다. 국가부터 가정에 이르기까지 사회의 주요 조직 형태는 ‘트리’의 특성을 보인다. 겨울옷은 ‘스택’과 같아서 가장 먼저 입은 옷을 가장 나중에 벗는다. 셔틀콕 통은 한쪽으로 넣고 다른 쪽으로 꺼낸다는 점에서 ‘큐’와 닮았다. 사전은 원하는 표제어를 빠르게 찾을 수 있다는 점에서 ‘해시 테이블’과 비슷하다.</p> <p>이 책은 명확하고 이해하기 쉬운 애니메이션 그림과 실행 가능한 코드 예제를 통해 독자가 알고리즘과 자료 구조의 핵심 개념을 이해하고 직접 프로그래밍으로 구현할 수 있도록 돕는 것을 목표로 한다. 이를 바탕으로 복잡한 세상 속에서 알고리즘이 생생하게 드러나는 모습을 보여 주며 알고리즘의 아름다움을 전하고자 한다. 이 책이 여러분에게 도움이 되기를 바란다!</p>","path":["머리말"],"tags":[]},{"location":"chapter_introduction/","level":1,"title":"제 1 장   알고리즘과의 첫 만남","text":"<p>이 장을 시작하며</p> <p>한 소녀가 데이터와 어우러져 우아하게 춤을 춘다. 치맛자락에는 알고리즘의 선율이 나부낀다.</p> <p>소녀가 함께 춤추자고 손을 내민다. 그 발걸음을 따라 논리와 아름다움으로 가득한 알고리즘의 세계에 들어서 보자.</p>","path":["제 1 장   알고리즘과의 첫 만남"],"tags":[]},{"location":"chapter_introduction/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>1.1   어디에나 있는 알고리즘</li> <li>1.2   알고리즘이란 무엇인가</li> <li>1.3   요약</li> </ul>","path":["제 1 장   알고리즘과의 첫 만남"],"tags":[]},{"location":"chapter_introduction/algorithms_are_everywhere/","level":1,"title":"1.1   어디에나 있는 알고리즘","text":"<p>‘알고리즘’이라는 말을 들으면 자연스럽게 수학을 떠올리게 된다. 하지만 실제로 많은 알고리즘은 복잡한 수학보다는 기본적인 논리에 더 많이 의존하며, 이러한 논리는 일상생활 곳곳에서 찾아볼 수 있다.</p> <p>알고리즘을 본격적으로 살펴보기 전에 흥미로운 사실 하나를 소개하고자 한다. 여러분은 이미 자신도 모르게 많은 알고리즘을 익혔고, 일상생활에서 자연스럽게 사용하고 있다. 몇 가지 구체적인 예를 통해 이 사실을 확인해 보자.</p> <p>예 1: 사전에서 단어 찾기. 중국어 사전에서는 각 한자에 대응하는 병음이 있고, 표제어는 병음의 알파벳순으로 배열되어 있다. 병음의 첫 글자가 \\(r\\)인 한자를 찾는다고 가정하면, 보통 아래 그림(1-1)과 같은 방법을 사용한다.</p> <ol> <li>사전의 중간쯤을 펼쳐 그 페이지의 병음 첫 글자를 확인한다. 첫 글자가 \\(m\\)이라고 가정하자.</li> <li>병음의 알파벳순에서 \\(r\\)은 \\(m\\) 뒤에 있으므로 사전의 앞쪽 절반을 제외하고 탐색 범위를 뒤쪽 절반으로 좁힌다.</li> <li>병음의 첫 글자가 \\(r\\)인 페이지를 찾을 때까지 단계 <code>1.</code>과 단계 <code>2.</code>를 반복한다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 1-1   사전에서 단어를 찾는 단계 </p> <p>초등학생도 익히는 사전 찾기는 사실 유명한 ‘이진 탐색’ 알고리즘이다. 자료 구조의 관점에서 사전은 정렬된 ‘배열’로 볼 수 있고, 알고리즘의 관점에서 앞서 살펴본 사전 찾기의 일련의 동작은 ‘이진 탐색’으로 볼 수 있다.</p> <p>예 2: 카드 정리하기. 카드 게임을 할 때는 매번 손에 든 카드를 작은 수부터 큰 수 순서로 정리한다. 그 과정은 아래 그림(1-2)과 같다.</p> <ol> <li>카드를 ‘정렬된 부분’과 ‘정렬되지 않은 부분’으로 나누고, 처음에는 맨 왼쪽 카드 1장이 이미 정렬되어 있다고 가정한다.</li> <li>정렬되지 않은 부분에서 카드 한 장을 꺼내 정렬된 부분의 올바른 위치에 끼워 넣는다. 그러면 맨 왼쪽 카드 2장이 정렬된 상태가 된다.</li> <li>단계 <code>2.</code>를 반복하여 매번 정렬되지 않은 부분의 카드 한 장을 정렬된 부분에 끼워 넣는다. 모든 카드가 정렬될 때까지 계속한다.</li> </ol> <p></p> <p> 그림 1-2   카드를 정렬하는 단계 </p> <p>이러한 카드 정리 방법은 본질적으로 ‘삽입 정렬’ 알고리즘이며, 작은 데이터 집합을 처리할 때 매우 효율적이다. 많은 프로그래밍 언어의 정렬 라이브러리 함수에서도 삽입 정렬이 사용된다.</p> <p>예 3: 거스름돈 주기. 마트에서 \\(69\\)위안어치 물건을 사고 계산원에게 \\(100\\)위안을 냈다고 가정하자. 계산원은 \\(31\\)위안을 거슬러 주어야 하며, 자연스럽게 아래 그림(1-3)과 같은 과정을 떠올린다.</p> <ol> <li>선택할 수 있는 것은 액면가가 \\(31\\)위안보다 작은 화폐로, \\(1\\)위안, \\(5\\)위안, \\(10\\)위안, \\(20\\)위안이 있다.</li> <li>선택지 중 가장 큰 \\(20\\)위안을 꺼내면 \\(31 - 20 = 11\\)위안이 남는다.</li> <li>남은 선택지 중 가장 큰 \\(10\\)위안을 꺼내면 \\(11 - 10 = 1\\)위안이 남는다.</li> <li>남은 선택지 중 가장 큰 \\(1\\)위안을 꺼내면 \\(1 - 1 = 0\\)위안이 남는다.</li> <li>거스름돈을 모두 주었다. 그 구성은 \\(20 + 10 + 1 = 31\\)위안이다.</li> </ol> <p></p> <p> 그림 1-3   거스름돈을 주는 과정 </p> <p>위 과정에서는 각 단계마다 현재 가장 좋아 보이는 선택, 즉 가능한 한 액면가가 큰 화폐를 고른다. 그 결과 거스름돈을 줄 수 있는 방법을 얻는다. 자료 구조와 알고리즘의 관점에서 이 방법은 본질적으로 ‘탐욕’ 알고리즘이다.</p> <p>작게는 요리 한 가지를 만드는 일부터 크게는 성간 항해에 이르기까지, 거의 모든 문제의 해결에는 알고리즘이 필요하다. 컴퓨터의 등장으로 우리는 프로그래밍을 통해 자료 구조를 메모리에 저장하고, CPU와 GPU가 알고리즘을 실행하도록 코드를 작성할 수 있게 되었다. 이로써 현실의 문제를 컴퓨터로 옮겨 다양한 복잡한 문제를 더 효율적으로 해결할 수 있다.</p> <p>참고</p> <p>자료 구조, 알고리즘, 배열, 이진 탐색 등의 개념이 아직 막연하게 느껴져도 계속 읽어 나가면 된다. 이 책이 자료 구조와 알고리즘의 지식 세계로 안내할 것이다.</p>","path":["제 1 장   알고리즘과의 첫 만남","1.1   어디에나 있는 알고리즘"],"tags":[]},{"location":"chapter_introduction/summary/","level":1,"title":"1.3   요약","text":"","path":["제 1 장   알고리즘과의 첫 만남","1.3   요약"],"tags":[]},{"location":"chapter_introduction/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>알고리즘은 일상생활 어디에나 있으며, 손에 닿지 않는 어려운 지식이 아니다. 사실 우리는 이미 자신도 모르게 많은 알고리즘을 익혀 크고 작은 생활 속 문제를 해결하고 있다.</li> <li>사전에서 단어를 찾는 원리는 이진 탐색 알고리즘과 같다. 이진 탐색에는 분할 정복이라는 중요한 알고리즘적 사고가 담겨 있다.</li> <li>카드를 정리하는 과정은 삽입 정렬 알고리즘과 매우 비슷하다. 삽입 정렬은 작은 데이터 집합을 정렬하는 데 적합하다.</li> <li>거스름돈을 주는 과정은 본질적으로 탐욕 알고리즘이다. 각 단계에서 현재 가장 좋아 보이는 선택을 한다.</li> <li>알고리즘은 유한한 시간 안에 특정 문제를 해결하는 명령 또는 작업 단계의 집합이고, 자료 구조는 컴퓨터에서 데이터를 구성하고 저장하는 방식이다.</li> <li>자료 구조와 알고리즘은 긴밀하게 연결되어 있다. 자료 구조는 알고리즘의 토대이며, 알고리즘은 자료 구조에 생명력을 불어넣는다.</li> <li>자료 구조와 알고리즘은 블록 조립에 비유할 수 있다. 블록은 데이터를, 블록의 모양과 연결 방식 등은 자료 구조를, 블록을 조립하는 단계는 알고리즘을 나타낸다.</li> </ul>","path":["제 1 장   알고리즘과의 첫 만남","1.3   요약"],"tags":[]},{"location":"chapter_introduction/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 프로그래머로 일하면서 직접 알고리즘으로 문제를 해결해 본 적이 없다. 자주 쓰는 알고리즘은 프로그래밍 언어에서 이미 제공하므로 그대로 사용하면 된다. 그렇다면 업무에서 다루는 문제가 아직 알고리즘이 필요할 정도는 아니라는 뜻인가?</p> <p>구체적인 업무 기술을 무술의 ‘동작’에 비유한다면, 기초 과목은 ‘내공’에 더 가깝다.</p> <p>알고리즘을 비롯한 기초 과목을 공부하는 의미는 업무에서 이를 처음부터 직접 구현하는 데 있지 않다고 생각한다. 배운 지식을 바탕으로 문제에 전문적으로 대응하고 판단하여 업무의 전반적인 품질을 높이는 데 의미가 있다. 간단한 예로, 프로그래밍 언어마다 내장 정렬 함수를 제공한다.</p> <ul> <li>자료 구조와 알고리즘을 배우지 않았다면 어떤 데이터든 이 정렬 함수에 넘길 수 있다. 실행도 원활하고 성능도 괜찮으니 별문제가 없어 보인다.</li> <li>하지만 알고리즘을 배웠다면 내장 정렬 함수의 시간 복잡도가 \\(O(n \\log n)\\)임을 안다. 주어진 데이터가 학번처럼 자릿수가 고정된 정수라면 더 효율적인 ‘기수 정렬’을 사용하여 시간 복잡도를 \\(O(nk)\\)로 낮출 수 있다. 여기서 \\(k\\)는 자릿수다. 데이터의 양이 많을 때는 이렇게 절약한 실행 시간이 비용 절감이나 사용자 경험 개선 등 큰 가치를 만들어 낼 수 있다.</li> </ul> <p>공학 분야의 많은 문제는 최적해를 얻기 어렵고, 그저 ‘어느 정도’ 해결된 상태로 남아 있다. 문제의 난이도는 문제 자체의 성질뿐 아니라 그 문제를 바라보는 사람의 지식에도 달려 있다. 지식이 체계적이고 경험이 많을수록 문제를 더 깊이 분석하고 더 우아하게 해결할 수 있다.</p> <p>Q: 이제는 거의 모든 코드를 AI 에이전트가 작성하는데, 그래도 알고리즘을 배워야 하는가?</p> <p>리무(李沐)의 말을 인용하자면, “에이전트는 능력 증폭기다. 능력에 10을 곱하는 것이지, 10을 더하는 덧셈기가 아니다.”</p> <p>AI가 능력 증폭기라면 증폭의 바탕이 되는 ‘능력’도 중요하다. 특히 두 가지 능력을 쌓을 가치가 있다고 생각한다. 첫째는 표현 능력, 특히 AI와 협업하는 능력이다. 목표를 명확히 설명하고 배경을 충분히 제공하며 잘못된 방향은 제때 바로잡아야 한다. AI에게 문제를 설명하는 일은 스스로 문제를 명확히 이해하는 데도 도움이 된다. 둘째는 기초 지식이다. 세부 사항은 AI가 보완할 수 있지만, 핵심을 짚어 질문하고 옳고 그름을 판단하며 AI의 출력을 유용한 결과물로 만들려면 기본 개념은 스스로 이해해야 한다.</p> <p>AI의 특별한 점은 과거의 도구처럼 기능의 경계가 뚜렷하지 않다는 것이다. AI는 물과 비슷해서 누구의 손에 들어가느냐에 따라 다른 모습을 띤다. 사람의 능력이 뛰어나고 안목이 좋을수록 AI의 증폭 효과도 더욱 뚜렷해진다.</p>","path":["제 1 장   알고리즘과의 첫 만남","1.3   요약"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/","level":1,"title":"1.2   알고리즘이란 무엇인가","text":"","path":["제 1 장   알고리즘과의 첫 만남","1.2   알고리즘이란 무엇인가"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/#121","level":2,"title":"1.2.1   알고리즘의 정의","text":"<p>알고리즘(algorithm)은 유한한 시간 안에 특정 문제를 해결하는 명령 또는 작업 단계의 집합이며, 다음과 같은 특성을 갖는다.</p> <ul> <li>문제가 명확하고 입력과 출력이 분명하게 정의되어 있다.</li> <li>실행 가능하며, 유한한 단계와 시간, 메모리 공간 안에서 완료할 수 있다.</li> <li>각 단계의 의미가 명확하여 입력과 실행 조건이 같으면 항상 같은 출력을 얻는다.</li> </ul>","path":["제 1 장   알고리즘과의 첫 만남","1.2   알고리즘이란 무엇인가"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/#122","level":2,"title":"1.2.2   자료 구조의 정의","text":"<p>자료 구조(data structure)는 데이터를 구성하고 저장하는 방식으로, 데이터의 내용, 데이터 사이의 관계, 데이터를 조작하는 방법을 포함한다. 자료 구조는 다음과 같은 목표를 고려하여 설계한다.</p> <ul> <li>컴퓨터 메모리를 절약하도록 공간 사용량을 최대한 줄인다.</li> <li>데이터 접근, 추가, 삭제, 갱신 등의 연산을 최대한 빠르게 수행한다.</li> <li>알고리즘이 효율적으로 실행되도록 간결한 데이터 표현과 논리적 정보를 제공한다.</li> </ul> <p>자료 구조를 설계하는 과정에는 수많은 절충이 따른다. 어느 한 측면을 개선하려면 다른 측면을 양보해야 할 때가 많다. 두 가지 예를 살펴보자.</p> <ul> <li>연결 리스트는 배열보다 데이터를 추가하거나 삭제하기가 편리하지만, 데이터에 접근하는 속도는 느리다.</li> <li>그래프는 연결 리스트보다 더 풍부한 논리적 정보를 제공하지만, 더 많은 메모리 공간이 필요하다.</li> </ul>","path":["제 1 장   알고리즘과의 첫 만남","1.2   알고리즘이란 무엇인가"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/#123","level":2,"title":"1.2.3   자료 구조와 알고리즘의 관계","text":"<p>아래 그림(1-4)과 같이 자료 구조와 알고리즘은 밀접하게 연결되어 있다. 그 관계는 다음 세 가지 측면에서 살펴볼 수 있다.</p> <ul> <li>자료 구조는 알고리즘의 토대다. 자료 구조는 알고리즘에 체계적으로 저장된 데이터와 이를 조작하는 방법을 제공한다.</li> <li>알고리즘은 자료 구조에 생명력을 불어넣는다. 자료 구조 자체는 데이터 정보만 저장하며, 알고리즘과 결합해야 특정 문제를 해결할 수 있다.</li> <li>같은 알고리즘도 서로 다른 자료 구조를 바탕으로 구현할 수 있지만, 실행 효율은 크게 달라질 수 있다. 따라서 적절한 자료 구조를 선택하는 것이 중요하다.</li> </ul> <p></p> <p> 그림 1-4   자료 구조와 알고리즘의 관계 </p> <p>자료 구조와 알고리즘은 아래 그림(1-5)의 조립 블록에 비유할 수 있다. 블록 한 세트에는 여러 부품뿐 아니라 자세한 조립 설명서도 들어 있다. 설명서를 따라 차례대로 조립하면 멋진 블록 모형을 만들 수 있다.</p> <p></p> <p> 그림 1-5   블록 조립 </p> <p>구체적인 대응 관계는 아래 표(1-1)와 같다.</p> <p> 표 1-1   자료 구조와 알고리즘을 블록 조립에 비유하기 </p> 자료 구조와 알고리즘 블록 조립 입력 데이터 조립하기 전의 블록 자료 구조 모양, 크기, 연결 방식 등 블록의 구성 형태 알고리즘 블록을 원하는 형태로 조립하는 일련의 작업 단계 출력 데이터 완성된 블록 모형 <p>자료 구조와 알고리즘은 프로그래밍 언어에 종속되지 않는다. 이 때문에 이 책에서는 여러 프로그래밍 언어로 작성한 구현을 제공할 수 있다.</p> <p>관례적으로 사용하는 줄임말</p> <p>실제로 이야기할 때는 ‘자료 구조와 알고리즘’을 흔히 ‘알고리즘’이라고 줄여 부른다. 예를 들어 널리 알려진 리트코드의 알고리즘 문제는 실제로 자료 구조와 알고리즘 두 분야의 지식을 함께 평가한다.</p>","path":["제 1 장   알고리즘과의 첫 만남","1.2   알고리즘이란 무엇인가"],"tags":[]},{"location":"chapter_preface/","level":1,"title":"제 0 장   들어가며","text":"<p>이 장을 시작하며</p> <p>알고리즘은 아름다운 교향곡과 같다. 코드 한 줄 한 줄이 리듬처럼 흐른다.</p> <p>이 책이 여러분의 머릿속에 잔잔히 울려 퍼져, 특별하고 깊은 선율을 남기기를 바란다.</p>","path":["제 0 장   들어가며"],"tags":[]},{"location":"chapter_preface/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>0.1   이 책에 대하여</li> <li>0.2   이 책을 활용하는 방법</li> <li>0.3   요약</li> </ul>","path":["제 0 장   들어가며"],"tags":[]},{"location":"chapter_preface/about_the_book/","level":1,"title":"0.1   이 책에 대하여","text":"<p>이 프로젝트는 오픈 소스로 공개되는 무료 자료 구조와 알고리즘 입문서를 만드는 것을 목표로 한다. 처음 배우는 사람도 쉽게 접근할 수 있는 책을 지향한다.</p> <ul> <li>책 전체에 애니메이션 그림을 사용하여 내용을 명확하고 이해하기 쉽게 설명한다. 난이도를 완만하게 높여 가며 초보자가 자료 구조와 알고리즘의 지식 지도를 탐험하도록 안내한다.</li> <li>소스 코드를 클릭 한 번으로 실행할 수 있다. 독자가 실습을 통해 프로그래밍 능력을 키우고 알고리즘의 동작 원리와 자료 구조의 내부 구현을 이해하도록 돕는다.</li> <li>독자들이 서로 도우며 공부하기를 권장한다. 댓글에서 질문하고 의견을 나누며 함께 성장할 수 있다.</li> </ul>","path":["제 0 장   들어가며","0.1   이 책에 대하여"],"tags":[]},{"location":"chapter_preface/about_the_book/#011","level":2,"title":"0.1.1   대상 독자","text":"<p>알고리즘을 처음 접하는 입문자이거나, 문제를 풀어 본 경험은 조금 있지만 자료 구조와 알고리즘에 대한 이해가 막연해 알 것 같다가도 모르겠다는 생각이 든다면, 이 책은 바로 여러분을 위한 책이다!</p> <p>이미 상당한 수의 문제를 풀었고 대부분의 문제 유형에 익숙하다면, 이 책을 통해 알고리즘 지식을 되짚고 체계적으로 정리할 수 있다. 저장소의 소스 코드는 ‘문제 풀이 도구 모음’이나 ‘알고리즘 사전’처럼 활용할 수 있다.</p> <p>알고리즘의 ‘고수’라면 소중한 의견을 보내 주거나 집필에 함께 참여해 주기를 기대한다.</p> <p>선수 지식</p> <p>적어도 한 가지 언어의 프로그래밍 기초를 알고, 간단한 코드를 읽고 작성할 수 있어야 한다.</p>","path":["제 0 장   들어가며","0.1   이 책에 대하여"],"tags":[]},{"location":"chapter_preface/about_the_book/#012","level":2,"title":"0.1.2   내용 구성","text":"<p>이 책의 주요 내용은 아래 그림(0-1)과 같다.</p> <ul> <li>복잡도 분석: 자료 구조와 알고리즘을 평가하는 기준과 방법을 다룬다. 시간 복잡도와 공간 복잡도의 계산 방법, 대표적인 유형과 예제 등을 살펴본다.</li> <li>자료 구조: 기본 자료형과 자료 구조의 분류 방법을 다룬다. 배열, 연결 리스트, 스택, 큐, 해시 테이블, 트리, 힙, 그래프 등의 정의, 장단점, 자주 쓰는 연산, 대표적인 유형과 응용, 구현 방법 등을 살펴본다.</li> <li>알고리즘: 탐색, 정렬, 분할 정복, 백트래킹, 동적 계획법, 탐욕 등의 정의, 장단점, 효율성, 활용 상황, 문제 해결 단계와 예제 문제 등을 살펴본다.</li> </ul> <p></p> <p> 그림 0-1   이 책의 주요 내용 </p>","path":["제 0 장   들어가며","0.1   이 책에 대하여"],"tags":[]},{"location":"chapter_preface/about_the_book/#013","level":2,"title":"0.1.3   감사의 말","text":"<p>이 책은 오픈 소스 커뮤니티의 수많은 기여자가 함께 노력한 덕분에 계속 발전하고 있다. 시간과 노력을 들여 준 모든 집필 참여자에게 감사드린다. 참여자는 다음과 같다(GitHub가 자동으로 생성한 순서): krahets, coderonion, Gonglja, nuomi1, Reanon, justin-tse, hpstory, danielsss, curtishd, night-cruise, S-N-O-R-L-A-X, rongyi, msk397, gvenusleo, khoaxuantu, rivertwilight, K3v123, gyt95, zhuoqinyue, yuelinxin, Zuoxun, mingXta, Phoenix0415, FangYuan33, GN-Yu, longsizhuo, pengchzn, QiLOL, Cathay-Chen, guowei-gong, xBLACKICEx, IsChristina, JoseHung, qualifier1024, hello-ikun, magentaqin, Guanngxu, thomasq0, sunshinesDL, L-Super, Transmigration-zhou, WSL0809, Slone123c, lhxsm, yuan0221, what-is-me, theNefelibatas, Shyam-Chen, sangxiaai, longranger2, codeberg-user, xiongsp, JeffersonHuang, prinpal, seven1240, Wonderdch, malone6, xiaomiusa87, gaofer, bluebean-cloud, a16su, SamJin98, hongyun-robot, nanlei, XiaChuerwu, yd-j, iron-irax, mgisr, steventimes, junminhong, heshuyue, danny900714, Nigh, Dr-XYZ, MolDuM, XC-Zero, reeswell, PXG-XPG, NI-SW, Horbin-Magician, Enlightenus, YangXuanyi, xjr7670, beatrix-chan, DullSword, qq909244296, iStig, boloboloda, hts0000, gledfish, fbigm, echo1937, jiaxianhua, wenjianmin, keshida, kilikilikid, lclc6, lwbaptx, linyejoe2, liuxjerry, szu17dmy, dshlstarr, Yucao-cy, coderlef, czruby, bongbongbakudan, beintentional, ZongYangL, ZhongYuuu, ZhongGuanbin, hezhizhen, linzeyan, ZJKung, JTCPOWI, KawaiiAsh, luluxia, xb534, ztkuaikuai, yw-1021, ElaBosak233, baagod, zhouLion, yishangzhang, yi427, yanedie, yabo083, weibk, wangwang105, th1nk3r-ing, tao363, 4yDX3906, syd168, sslmj2020, smilelsb, siqyka, selear, sdshaoda, Xi-Row, popozhu, nuquist19, noobcodemaker, XiaoK29, chadyi, lyl625760, lucaswangdev, llql1211, 0130w, shanghai-Jerry, EJackYang, Javesun99, eltociear, lipusheng, KNChiu, BlindTerran, ShiMaRing, lovelock, FreddieLi, FloranceYeh, fanchenggang, gltianwen, goerll, nedchu, curly210102, CuB3y0nd, KraHsu, CarrotDLaw, youshaoXG, bubble9um, Asashishi, Asa0oo0o0o, fanenr, eagleanurag, akshiterate, 52coder, foursevenlove, KorsChen, hopkings2008, yang-le, realwujing, Evilrabbit520, Umer-Jahangir, Turing-1024-Lee, Suremotoo, paoxiaomooo, Chieko-Seren, Senrian, Allen-Scai, 19santosh99, ymmmas, Risuntsy, Richard-Zhang1019, RafaelCaso, qingpeng9802, primexiao, Urbaner3, codetypess, nidhoggfgg, MwumLi, CreatorMetaSky, martinx, ZnYang2018, hugtyftg, logan-qiu, psychelzh, Kunchen-Luo, Keynman 및 KeiichiKasai.</p> <p>이 책의 코드 검토는 coderonion, curtishd, Gonglja, gvenusleo, hpstory, justin-tse, khoaxuantu, krahets, night-cruise, nuomi1, Reanon 및 rongyi가 맡았다(알파벳순). 시간과 노력을 들여 준 이들에게 감사드린다. 이들 덕분에 각 언어의 코드가 일관된 규칙과 형식을 유지할 수 있었다.</p> <p>영문판은 yuelinxin, K3v123, magentaqin, QiLOL, Phoenix0415, SamJin98, yanedie, RafaelCaso, pengchzn 및 thomasq0가 검토했다. 일본어판은 eltociear가, 러시아어판은 И. А. Шевкун과 Yuyan Huang이, 번체 중국어판은 Shyam-Chen과 Dr-XYZ가 검토했다. 이들의 기여 덕분에 더 많은 독자에게 이 책을 전할 수 있게 되었다. 모두에게 감사드린다.</p> <p>이 책의 ePub 전자책 생성 도구는 zhongfq가 개발했다. 독자에게 더 유연한 독서 방식을 제공해 준 그의 기여에 감사드린다.</p> <p>이 책을 쓰는 동안 많은 분의 도움을 받았다.</p> <ul> <li>회사에서 나의 멘토였던 리시(李汐) 박사에게 감사드린다. 긴 대화 중 “어서 행동에 옮겨 보라”고 격려해 준 덕분에 이 책을 쓰겠다는 결심을 굳힐 수 있었다.</li> <li>이 책의 첫 독자가 되어 준 여자 친구 파오파오(泡泡)에게 감사드린다. 알고리즘 입문자의 시각에서 많은 소중한 의견을 주어 초보자가 읽기 더 좋은 책이 되었다.</li> <li>텅바오(腾宝), 치바오(琦宝), 페이바오(飞宝)에게 감사드린다. 이 책에 창의적인 이름을 지어 주어 모두가 첫 코드 “Hello World!”를 작성하던 아름다운 기억을 떠올리게 했다.</li> <li>지식재산권에 관한 전문적인 도움을 준 샤오취안(校铨)에게 감사드린다. 이 도움은 오픈 소스 도서인 이 책을 완성해 가는 데 중요한 역할을 했다.</li> <li>아름다운 표지와 로고를 디자인해 준 쑤퉁(苏潼)에게 감사드린다. 나의 지나친 꼼꼼함 때문에 여러 번 수정을 요청했는데도 인내심 있게 작업해 주었다.</li> <li>편집과 조판에 관한 조언을 해 주고 오픈 소스 문서 테마 Material-for-MkDocs를 개발한 @squidfunk에게 감사드린다.</li> </ul> <p>집필하면서 자료 구조와 알고리즘에 관한 많은 교재와 글을 읽었다. 이 작품들은 훌륭한 본보기가 되어 이 책의 정확성과 품질을 뒷받침해 주었다. 모든 선생님과 선배의 뛰어난 기여에 감사드린다!</p> <p>이 책은 손으로 실습하고 머리로 생각하며 배우는 방식을 권장한다. 이 점에서는 『실습으로 배우는 딥러닝(Dive into Deep Learning)』에서 깊은 영감을 받았다. 독자 여러분에게 이 훌륭한 책을 적극 추천한다.</p> <p>부모님께 진심으로 감사드린다. 늘 보내 주신 지지와 격려 덕분에 이처럼 흥미로운 일을 할 기회를 얻었다.</p>","path":["제 0 장   들어가며","0.1   이 책에 대하여"],"tags":[]},{"location":"chapter_preface/suggestions/","level":1,"title":"0.2   이 책을 활용하는 방법","text":"<p>참고</p> <p>가장 좋은 독서 경험을 위해 이 절을 끝까지 읽어 보기를 권장한다.</p>","path":["제 0 장   들어가며","0.2   이 책을 활용하는 방법"],"tags":[]},{"location":"chapter_preface/suggestions/#021","level":2,"title":"0.2.1   글의 표현과 서식에 관한 약속","text":"<ul> <li>제목 뒤에 <code>*</code>가 붙은 절은 선택해서 읽는 심화 내용이다. 시간이 부족하면 우선 건너뛰어도 된다.</li> <li>웹 버전에서는 전문 용어를 밑줄로 표시한다. 예를 들면 배열(array)과 같다. 문헌을 읽을 때 도움이 되므로 용어를 기억해 두기를 권장한다.</li> <li>핵심 내용과 요약 문장은 굵게 표시한다. 이러한 문장은 특히 주의 깊게 읽을 필요가 있다.</li> <li>특정한 의미로 사용하는 단어나 문구는 모호함을 피하기 위해 ‘따옴표’로 표시한다.</li> <li>프로그래밍 언어마다 명칭이 다를 때는 Python을 기준으로 삼는다. 예를 들어 ‘비어 있음’을 나타낼 때 <code>None</code>을 사용한다.</li> <li>이 책은 내용을 더 간결하게 배치하기 위해 프로그래밍 언어의 주석 작성 규칙을 일부 따르지 않는다. 주석은 주로 제목 주석, 설명 주석, 여러 줄 주석의 세 유형으로 나뉜다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>\"\"\"함수, 클래스, 테스트 사례 등을 표시하는 제목 주석\"\"\"\n\n# 코드를 자세히 설명하는 설명 주석\n\n\"\"\"\n여러 줄\n주석\n\"\"\"\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n// 여러 줄\n// 주석\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>### 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 ###\n\n# 코드를 자세히 설명하는 설명 주석\n\n# 여러 줄\n# 주석\n</code></pre>","path":["제 0 장   들어가며","0.2   이 책을 활용하는 방법"],"tags":[]},{"location":"chapter_preface/suggestions/#022","level":2,"title":"0.2.2   애니메이션 그림으로 효율적으로 배우기","text":"<p>영상과 그림은 글보다 정보 밀도가 높고 구조가 뚜렷하여 이해하기 쉽다. 이 책에서는 핵심 내용과 어려운 개념을 주로 애니메이션 그림으로 보여 주고, 글은 설명과 보충을 담당한다.</p> <p>책을 읽다가 아래 그림(0-2)과 같은 애니메이션 설명을 만나면 그림을 중심으로 보고 글을 보조 자료로 활용하여 두 가지를 함께 이해하기 바란다.</p> <p></p> <p> 그림 0-2   애니메이션 그림 예시 </p>","path":["제 0 장   들어가며","0.2   이 책을 활용하는 방법"],"tags":[]},{"location":"chapter_preface/suggestions/#023","level":2,"title":"0.2.3   코드 실습으로 이해를 깊게 하기","text":"<p>이 책의 예제 코드는 GitHub 저장소에 있다. 아래 그림(0-3)과 같이 소스 코드에는 테스트 사례가 포함되어 있어 클릭 한 번으로 실행할 수 있다.</p> <p>시간이 허락한다면 예제 코드를 참고하여 직접 한 번 작성해 보기를 권장한다. 공부할 시간이 부족하더라도 적어도 모든 코드를 읽고 실행해 보자.</p> <p>코드를 읽는 것보다 직접 작성하는 과정에서 더 많은 것을 얻는 경우가 많다. 직접 해 보며 배우는 것이 진짜 배움이다.</p> <p></p> <p> 그림 0-3   코드 실행 예시 </p> <p>코드를 실행하기 위한 준비는 크게 세 단계로 나뉜다.</p> <p>1단계: 로컬 프로그래밍 환경 설치. 부록의 안내에 따라 설치한다. 이미 설치했다면 이 단계는 건너뛰어도 된다.</p> <p>2단계: 코드 저장소 복제 또는 내려받기. GitHub 저장소로 이동한다. Git이 설치되어 있다면 다음 명령으로 저장소를 복제할 수 있다.</p> <pre><code>git clone https://github.com/krahets/hello-algo.git\n</code></pre> <p>물론 아래 그림(0-4)에 표시된 위치에서 ‘Download ZIP’ 버튼을 눌러 코드 압축 파일을 바로 내려받은 뒤 로컬에서 압축을 풀어도 된다.</p> <p></p> <p> 그림 0-4   저장소 복제와 코드 내려받기 </p> <p>3단계: 소스 코드 실행. 아래 그림(0-5)처럼 위쪽에 파일명이 표시된 코드 블록은 저장소의 <code>ko/codes</code> 폴더에서 해당 한국어판 소스 파일을 찾을 수 있다. 소스 파일은 클릭 한 번으로 실행할 수 있으므로 불필요한 디버깅 시간을 줄이고 학습 내용에 집중할 수 있다.</p> <p></p> <p> 그림 0-5   코드 블록과 이에 대응하는 소스 파일 </p> <p>로컬에서 실행하는 것 외에 웹 버전에서는 Python 코드의 실행 과정을 시각화할 수도 있다(pythontutor 기반). 아래 그림(0-6)처럼 코드 블록 아래의 ‘코드 시각화’를 누르면 화면이 펼쳐져 알고리즘 코드의 실행 과정을 관찰할 수 있다. ‘전체 화면’을 누르면 더 넓은 화면에서 볼 수 있다.</p> <p></p> <p> 그림 0-6   Python 코드 실행 시각화 </p>","path":["제 0 장   들어가며","0.2   이 책을 활용하는 방법"],"tags":[]},{"location":"chapter_preface/suggestions/#024","level":2,"title":"0.2.4   질문과 토론으로 함께 성장하기","text":"<p>책을 읽다가 이해하지 못한 내용을 만나도 쉽게 건너뛰지 않기를 바란다. 댓글란에 질문을 남기면 나와 동료들이 성심껏 답변하겠다. 보통 이틀 안에 답변을 받을 수 있다.</p> <p>아래 그림(0-7)처럼 웹 버전의 각 장 아래에는 댓글란이 있다. 댓글의 내용에도 관심을 가져 주기 바란다. 다른 사람들이 겪은 문제를 살펴보면서 놓친 내용을 보완하고 더 깊이 생각할 수 있다. 또한 다른 독자의 질문에 답하고 자신의 생각을 나누어 서로의 성장을 도울 수 있기를 기대한다.</p> <p></p> <p> 그림 0-7   댓글란 예시 </p>","path":["제 0 장   들어가며","0.2   이 책을 활용하는 방법"],"tags":[]},{"location":"chapter_preface/suggestions/#025","level":2,"title":"0.2.5   알고리즘 학습 경로","text":"<p>자료 구조와 알고리즘을 배우는 과정은 크게 세 단계로 나눌 수 있다.</p> <ol> <li>1단계: 알고리즘 입문. 다양한 자료 구조의 특성과 사용법에 익숙해지고, 각 알고리즘의 원리, 절차, 용도, 효율성 등을 배운다.</li> <li>2단계: 알고리즘 문제 풀기. 많이 다루는 문제부터 시작하여 최소 100문제 정도를 풀고 주요 알고리즘 문제에 익숙해지기를 권장한다. 처음 문제를 풀 때는 배운 내용을 잊는 일이 어려움으로 느껴질 수 있지만, 이는 매우 자연스러운 일이므로 걱정하지 않아도 된다. ‘에빙하우스의 망각 곡선’에 맞추어 문제를 복습하면 보통 3~5번 반복한 뒤에는 확실하게 기억할 수 있다. 추천 문제 목록과 학습 계획은 이 GitHub 저장소를 참고하기 바란다.</li> <li>3단계: 지식 체계 세우기. 학습 측면에서는 알고리즘 전문 글, 문제 해결 틀, 알고리즘 교재를 읽으며 지식 체계를 꾸준히 넓힐 수 있다. 문제 풀이 측면에서는 주제별 분류, 한 문제를 여러 방법으로 풀기, 한 해법을 여러 문제에 적용하기 등의 심화 전략을 시도할 수 있다. 관련 경험과 조언은 여러 커뮤니티에서 찾아볼 수 있다.</li> </ol> <p>아래 그림(0-8)처럼 이 책은 주로 ‘1단계’를 다루며, 이후 2단계와 3단계의 학습을 더 효율적으로 진행하도록 돕는 것을 목표로 한다.</p> <p></p> <p> 그림 0-8   알고리즘 학습 경로 </p>","path":["제 0 장   들어가며","0.2   이 책을 활용하는 방법"],"tags":[]},{"location":"chapter_preface/summary/","level":1,"title":"0.3   요약","text":"","path":["제 0 장   들어가며","0.3   요약"],"tags":[]},{"location":"chapter_preface/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>이 책은 주로 알고리즘 입문자를 대상으로 한다. 이미 기초를 갖추었다면 알고리즘 지식을 체계적으로 복습하는 데 도움이 되며, 소스 코드는 ‘문제 풀이 도구 모음’으로 활용할 수 있다.</li> <li>내용은 크게 복잡도 분석, 자료 구조, 알고리즘의 세 부분으로 구성되어 있으며, 이 분야의 주요 주제를 폭넓게 다룬다.</li> <li>알고리즘 초보자에게는 처음 공부할 때 입문서를 읽는 일이 매우 중요하다. 불필요하게 돌아가는 길을 많이 줄일 수 있다.</li> <li>애니메이션 그림은 주로 핵심 내용과 어려운 개념을 설명하는 데 쓰인다. 책을 읽을 때 이러한 내용을 특히 주의 깊게 살펴보자.</li> <li>실습은 프로그래밍을 배우는 가장 좋은 방법이다. 소스 코드를 실행하고 직접 코드를 작성해 볼 것을 적극 권장한다.</li> <li>웹 버전의 각 장에는 댓글란이 있다. 궁금한 점이나 생각을 언제든 나누기 바란다.</li> </ul>","path":["제 0 장   들어가며","0.3   요약"],"tags":[]},{"location":"chapter_reference/","level":1,"title":"참고 문헌","text":"<p>[1] Thomas H. Cormen, et al. Introduction to Algorithms (3<sup>rd</sup> Edition).</p> <p>[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1<sup>st</sup> Edition).</p> <p>[3] Robert Sedgewick, et al. Algorithms (4<sup>th</sup> Edition).</p> <p>[4] 옌웨이민(严蔚敏). 『자료 구조(C 언어판)』(数据结构（C 语言版）).</p> <p>[5] 덩쥔후이(邓俊辉). 『자료 구조(C++ 언어판, 제3판)』(数据结构（C++ 语言版，第三版）).</p> <p>[6] Mark Allen Weiss 지음, 천웨(陈越) 옮김. 『자료 구조와 알고리즘 분석: Java 언어로 설명하기(제3판)』(数据结构与算法分析：Java语言描述（第三版）).</p> <p>[7] 청제(程杰). 『쉽게 풀어 쓴 자료 구조』(大话数据结构).</p> <p>[8] 왕정(王争). 『자료 구조와 알고리즘의 아름다움』(数据结构与算法之美).</p> <p>[9] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6<sup>th</sup> Edition).</p> <p>[10] Aston Zhang, et al. Dive into Deep Learning.</p>","path":["참고 문헌"],"tags":[]},{"location":"chapter_searching/","level":1,"title":"제 10 장   탐색","text":"<p>이 장을 시작하며</p> <p>탐색은 미지의 세계로 떠나는 모험이다. 신비로운 공간 구석구석을 모두 돌아봐야 할 수도, 목표를 빠르게 찾아낼 수도 있다.</p> <p>무언가를 찾아 나서는 이 여정에서는 매번 새로운 곳을 살필 때마다 뜻밖의 답을 만날 수 있다.</p>","path":["제 10 장   탐색"],"tags":[]},{"location":"chapter_searching/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>10.1   이진 탐색</li> <li>10.2   이진 탐색으로 삽입 위치 찾기</li> <li>10.3   이진 탐색으로 경계 찾기</li> <li>10.4   해시를 이용한 최적화 전략</li> <li>10.5   탐색 알고리즘 다시 보기</li> <li>10.6   요약</li> <li>10.7   연습 문제</li> </ul>","path":["제 10 장   탐색"],"tags":[]},{"location":"chapter_searching/binary_search/","level":1,"title":"10.1   이진 탐색","text":"<p>이진 탐색(binary search)은 분할 정복 전략에 기반한 효율적인 탐색 알고리즘이다. 데이터가 정렬되어 있다는 성질을 이용하여 매번 탐색 범위를 절반으로 줄이며, 목표 원소를 찾거나 탐색 구간이 빌 때까지 반복한다.</p> <p>문제</p> <p>길이가 \\(n\\)인 배열 <code>nums</code>가 주어진다. 원소는 오름차순으로 정렬되어 있고 중복되지 않는다. 이 배열에서 원소 <code>target</code>을 찾아 그 인덱스를 반환하라. 배열에 해당 원소가 없으면 \\(-1\\)을 반환하라. 예시는 아래 그림(10-1)과 같다.</p> <p></p> <p> 그림 10-1   이진 탐색 예시 데이터 </p> <p>아래 그림(10-2)과 같이 포인터 \\(i = 0\\)과 \\(j = n - 1\\)을 초기화하여 각각 배열의 첫 원소와 마지막 원소를 가리키게 한다. 이는 탐색 구간 \\([0, n - 1]\\)을 나타낸다. 대괄호는 경곗값 자체를 포함하는 닫힌 구간을 뜻한다.</p> <p>이어서 다음 두 단계를 반복한다.</p> <ol> <li>중간 인덱스 \\(m = \\lfloor {(i + j) / 2} \\rfloor\\)를 계산한다. 여기서 \\(\\lfloor \\: \\rfloor\\)는 내림 연산이다.</li> <li><code>nums[m]</code>과 <code>target</code>의 대소 관계를 비교하며, 다음 세 경우로 나뉜다.<ol> <li><code>nums[m] &lt; target</code>이면 <code>target</code>은 구간 \\([m + 1, j]\\)에 있으므로 \\(i = m + 1\\)을 수행한다.</li> <li><code>nums[m] &gt; target</code>이면 <code>target</code>은 구간 \\([i, m - 1]\\)에 있으므로 \\(j = m - 1\\)을 수행한다.</li> <li><code>nums[m] = target</code>이면 <code>target</code>을 찾았으므로 인덱스 \\(m\\)을 반환한다.</li> </ol> </li> </ol> <p>배열에 목표 원소가 없으면 탐색 구간은 결국 비게 된다. 이때 \\(-1\\)을 반환한다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 10-2   이진 탐색 과정 </p> <p>\\(i\\)와 \\(j\\)가 모두 <code>int</code> 타입이므로 \\(i + j\\)가 <code>int</code> 타입의 표현 범위를 벗어날 수 있다. 정수 오버플로를 피하기 위해 보통 \\(m = \\lfloor {i + (j - i) / 2} \\rfloor\\) 공식으로 중간 인덱스를 계산한다.</p> <p>코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search.py<pre><code>def binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"이진 탐색(양 끝을 포함하는 닫힌 구간)\"\"\"\n    # 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    i, j = 0, len(nums) - 1\n    # 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while i &lt;= j:\n        # 이론적으로 Python의 수는 메모리 크기에 따라 무한히 커질 수 있으므로 큰 수의 범위 초과를 고려할 필요가 없다\n        m = (i + j) // 2  # 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target:\n            i = m + 1  # 이 경우 target은 구간 [m+1, j]에 있다\n        elif nums[m] &gt; target:\n            j = m - 1  # 이 경우 target은 구간 [i, m-1]에 있다\n        else:\n            return m  # 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n    return -1  # 목표 원소를 찾지 못했으므로 -1을 반환한다\n</code></pre> binary_search.cpp<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    int i = 0, j = nums.size() - 1;\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target)    // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1;\n        else // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nint binarySearch(int[] nums, int target) {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    int i = 0, j = nums.length - 1;\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1;\n        else // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nint BinarySearch(int[] nums, int target) {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    int i = 0, j = nums.Length - 1;\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2;   // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target)      // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1;\n        else                       // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nfunc binarySearch(nums []int, target int) int {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    i, j := 0, len(nums)-1\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    for i &lt;= j {\n        m := i + (j-i)/2      // 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target { // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1\n        } else if nums[m] &gt; target { // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1\n        } else { // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m\n        }\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target { // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1\n        } else if nums[m] &gt; target { // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1\n        } else { // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m\n        }\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nfunction binarySearch(nums, target) {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    let i = 0,\n        j = nums.length - 1;\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while (i &lt;= j) {\n        // 중간 인덱스 m을 계산하며, parseInt()로 소수 부분을 버린다\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1;\n        else return m; // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nfunction binarySearch(nums: number[], target: number): number {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    let i = 0,\n        j = nums.length - 1;\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while (i &lt;= j) {\n        // 중간 인덱스 m을 계산한다\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1;\n        } else {\n            // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n        }\n    }\n    return -1; // 목표 원소를 찾지 못했으므로 -1을 반환한다\n}\n</code></pre> binary_search.dart<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n  int i = 0, j = nums.length - 1;\n  // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // 중간 인덱스 m을 계산한다\n    if (nums[m] &lt; target) {\n      // 이 경우 target은 구간 [m+1, j]에 있다\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // 이 경우 target은 구간 [i, m-1]에 있다\n      j = m - 1;\n    } else {\n      // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n      return m;\n    }\n  }\n  // 목표 원소를 찾지 못했으므로 -1을 반환한다\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    let mut i = 0;\n    let mut j = nums.len() as i32 - 1;\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if nums[m as usize] &lt; target {\n            // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1;\n        } else {\n            // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n        }\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nint binarySearch(int *nums, int len, int target) {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    int i = 0, j = len - 1;\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target)    // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1;\n        else // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    var i = 0\n    var j = nums.size - 1\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1\n        else  // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>### 이진 탐색(양 끝을 포함하는 닫힌 구간) ###\ndef binary_search(nums, target)\n  # 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n  i, j = 0, nums.length - 1\n\n  # 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n  while i &lt;= j\n    # Ruby의 정수는 이론상 메모리가 허용하는 만큼 커질 수 있으므로 큰 수의 오버플로를 고려할 필요가 없다\n    m = (i + j) / 2   # 중간 인덱스 m을 계산한다\n\n    if nums[m] &lt; target\n      i = m + 1 # 이 경우 target은 구간 [m+1, j]에 있다\n    elsif nums[m] &gt; target\n      j = m - 1 # 이 경우 target은 구간 [i, m-1]에 있다\n    else\n      return m  # 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n    end\n  end\n\n  -1  # 목표 원소를 찾지 못했으므로 -1을 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>시간 복잡도는 \\(O(\\log n)\\)이다. 이진 탐색의 반복문에서 구간이 매번 절반으로 줄어들므로 반복 횟수는 \\(\\log_2 n\\)이다.</p> <p>공간 복잡도는 \\(O(1)\\)이다. 포인터 \\(i\\)와 \\(j\\)는 상수 크기의 공간을 사용한다.</p>","path":["제 10 장   탐색","10.1   이진 탐색"],"tags":[]},{"location":"chapter_searching/binary_search/#1011","level":2,"title":"10.1.1   구간 표현 방법","text":"<p>위의 양끝이 닫힌 구간 외에 '왼쪽은 닫히고 오른쪽은 열린' 구간도 자주 사용한다. \\([0, n)\\)으로 정의하며 왼쪽 경곗값은 포함하고 오른쪽 경곗값은 포함하지 않는다. 이 표현에서 구간 \\([i, j)\\)는 \\(i = j\\)일 때 비어 있다.</p> <p>이 표현을 사용하여 같은 기능의 이진 탐색 알고리즘을 구현할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search.py<pre><code>def binary_search_lcro(nums: list[int], target: int) -&gt; int:\n    \"\"\"이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간)\"\"\"\n    # 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    i, j = 0, len(nums)\n    # 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while i &lt; j:\n        m = (i + j) // 2  # 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target:\n            i = m + 1  # 이 경우 target은 구간 [m+1, j)에 있다\n        elif nums[m] &gt; target:\n            j = m  # 이 경우 target은 구간 [i, m)에 있다\n        else:\n            return m  # 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n    return -1  # 목표 원소를 찾지 못했으므로 -1을 반환한다\n</code></pre> binary_search.cpp<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nint binarySearchLCRO(vector&lt;int&gt; &amp;nums, int target) {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    int i = 0, j = nums.size();\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target)    // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m)에 있다\n            j = m;\n        else // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nint binarySearchLCRO(int[] nums, int target) {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    int i = 0, j = nums.length;\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m)에 있다\n            j = m;\n        else // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nint BinarySearchLCRO(int[] nums, int target) {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    int i = 0, j = nums.Length;\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2;   // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target)      // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m)에 있다\n            j = m;\n        else                       // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nfunc binarySearchLCRO(nums []int, target int) int {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    i, j := 0, len(nums)\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    for i &lt; j {\n        m := i + (j-i)/2      // 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target { // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1\n        } else if nums[m] &gt; target { // 이 경우 target은 구간 [i, m)에 있다\n            j = m\n        } else { // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m\n        }\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nfunc binarySearchLCRO(nums: [Int], target: Int) -&gt; Int {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    var i = nums.startIndex\n    var j = nums.endIndex\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while i &lt; j {\n        let m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target { // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1\n        } else if nums[m] &gt; target { // 이 경우 target은 구간 [i, m)에 있다\n            j = m\n        } else { // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m\n        }\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nfunction binarySearchLCRO(nums, target) {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    let i = 0,\n        j = nums.length;\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while (i &lt; j) {\n        // 중간 인덱스 m을 계산하며, parseInt()로 소수 부분을 버린다\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // 이 경우 target은 구간 [i, m)에 있다\n            j = m;\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        else return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nfunction binarySearchLCRO(nums: number[], target: number): number {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    let i = 0,\n        j = nums.length;\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while (i &lt; j) {\n        // 중간 인덱스 m을 계산한다\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // 이 경우 target은 구간 [i, m)에 있다\n            j = m;\n        } else {\n            // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n        }\n    }\n    return -1; // 목표 원소를 찾지 못했으므로 -1을 반환한다\n}\n</code></pre> binary_search.dart<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nint binarySearchLCRO(List&lt;int&gt; nums, int target) {\n  // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n  int i = 0, j = nums.length;\n  // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n  while (i &lt; j) {\n    int m = i + (j - i) ~/ 2; // 중간 인덱스 m을 계산한다\n    if (nums[m] &lt; target) {\n      // 이 경우 target은 구간 [m+1, j)에 있다\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // 이 경우 target은 구간 [i, m)에 있다\n      j = m;\n    } else {\n      // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n      return m;\n    }\n  }\n  // 목표 원소를 찾지 못했으므로 -1을 반환한다\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nfn binary_search_lcro(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    let mut i = 0;\n    let mut j = nums.len() as i32;\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while i &lt; j {\n        let m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if nums[m as usize] &lt; target {\n            // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // 이 경우 target은 구간 [i, m)에 있다\n            j = m;\n        } else {\n            // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n        }\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nint binarySearchLCRO(int *nums, int len, int target) {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    int i = 0, j = len;\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target)    // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m)에 있다\n            j = m;\n        else // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nfun binarySearchLCRO(nums: IntArray, target: Int): Int {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    var i = 0\n    var j = nums.size\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while (i &lt; j) {\n        val m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m)에 있다\n            j = m\n        else  // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>### 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) ###\ndef binary_search_lcro(nums, target)\n  # 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n  i, j = 0, nums.length\n\n  # 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n  while i &lt; j\n    # 중간 인덱스 m을 계산한다\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # 이 경우 target은 구간 [m+1, j)에 있다\n    elsif nums[m] &gt; target\n      j = m - 1 # 이 경우 target은 구간 [i, m)에 있다\n    else\n      return m  # 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n    end\n  end\n\n  -1  # 목표 원소를 찾지 못했으므로 -1을 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(10-3)과 같이 구간을 표현하는 방식에 따라 이진 탐색 알고리즘의 초기화, 반복 조건, 구간 축소 연산이 모두 달라진다.</p> <p>'양끝이 닫힌 구간'에서는 왼쪽과 오른쪽 경계를 모두 포함하므로, 포인터 \\(i\\)와 \\(j\\)로 구간을 줄이는 연산도 대칭적이다. 실수할 가능성이 더 작으므로 일반적으로 '양끝이 닫힌 구간'으로 작성하기를 권장한다.</p> <p></p> <p> 그림 10-3   두 가지 구간 정의 </p>","path":["제 10 장   탐색","10.1   이진 탐색"],"tags":[]},{"location":"chapter_searching/binary_search/#1012","level":2,"title":"10.1.2   장점과 한계","text":"<p>이진 탐색은 시간과 공간 양쪽에서 좋은 성능을 보인다.</p> <ul> <li>이진 탐색은 시간 효율이 높다. 데이터가 많을수록 로그 시간 복잡도의 이점이 뚜렷하다. 예를 들어 데이터 크기가 \\(n = 2^{20}\\)이면 선형 탐색은 \\(2^{20} = 1048576\\) 번 반복해야 하지만, 이진 탐색은 \\(\\log_2 2^{20} = 20\\) 번이면 된다.</li> <li>이진 탐색은 추가 공간이 필요하지 않다. 해시 탐색처럼 추가 공간을 사용하는 탐색 알고리즘보다 공간을 절약한다.</li> </ul> <p>하지만 이진 탐색이 모든 상황에 적합한 것은 아니다. 주요 이유는 다음과 같다.</p> <ul> <li>이진 탐색은 정렬된 데이터에만 적용할 수 있다. 입력이 정렬되지 않았다면 이진 탐색만을 위해 정렬하는 것은 오히려 손해다. 정렬 알고리즘의 시간 복잡도는 보통 \\(O(n \\log n)\\)으로 선형 탐색과 이진 탐색보다 높기 때문이다. 원소를 자주 삽입하는 상황에서도 정렬 상태를 유지하려면 특정 위치에 삽입해야 하므로, \\(O(n)\\)의 큰 시간 비용이 든다.</li> <li>이진 탐색은 배열에만 적합하다. 원소를 건너뛰며(연속적이지 않게) 접근해야 하는데, 연결 리스트에서는 이런 접근이 비효율적이다. 따라서 연결 리스트나 연결 리스트 기반 자료 구조에는 적합하지 않다.</li> <li>데이터가 적을 때는 선형 탐색의 성능이 더 좋다. 선형 탐색은 매번 비교 1회만 필요하지만, 이진 탐색은 덧셈 1회, 나눗셈 1회, 비교 1 ~ 3회, 덧셈(뺄셈) 1회로 총 4 ~ 6회의 기본 연산이 필요하다. 따라서 데이터 수 \\(n\\)이 작으면 선형 탐색이 오히려 더 빠르다.</li> </ul>","path":["제 10 장   탐색","10.1   이진 탐색"],"tags":[]},{"location":"chapter_searching/binary_search_edge/","level":1,"title":"10.3   이진 탐색으로 경계 찾기","text":"","path":["제 10 장   탐색","10.3   이진 탐색으로 경계 찾기"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#1031","level":2,"title":"10.3.1   왼쪽 경계 찾기","text":"<p>문제</p> <p>중복 원소를 포함할 수 있는 길이 \\(n\\)의 정렬된 배열 <code>nums</code>가 주어진다. 배열에서 가장 왼쪽에 있는 원소 <code>target</code>의 인덱스를 반환하라. 해당 원소가 없으면 \\(-1\\)을 반환하라.</p> <p>이진 탐색으로 삽입 위치를 찾는 방법을 떠올려 보자. 탐색이 끝나면 \\(i\\)는 가장 왼쪽의 <code>target</code>을 가리킨다. 따라서 삽입 위치 찾기는 본질적으로 가장 왼쪽 <code>target</code>의 인덱스를 찾는 과정이다.</p> <p>삽입 위치를 찾는 함수를 이용하여 왼쪽 경계를 찾아 보자. 배열에 <code>target</code>이 없을 수도 있으며, 이때는 다음 두 결과 중 하나가 나올 수 있다.</p> <ul> <li>삽입 위치의 인덱스 \\(i\\)가 범위를 벗어난다.</li> <li>원소 <code>nums[i]</code>가 <code>target</code>과 다르다.</li> </ul> <p>위 두 경우에는 \\(-1\\)을 바로 반환하면 된다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_edge.py<pre><code>def binary_search_left_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"가장 왼쪽에 있는 target을 이진 탐색\"\"\"\n    # target의 삽입 위치를 찾는 것과 같다\n    i = binary_search_insertion(nums, target)\n    # target을 찾지 못했으므로 -1을 반환한다\n    if i == len(nums) or nums[i] != target:\n        return -1\n    # target을 찾았으므로 인덱스 i를 반환한다\n    return i\n</code></pre> binary_search_edge.cpp<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nint binarySearchLeftEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // target의 삽입 위치를 찾는 것과 같다\n    int i = binarySearchInsertion(nums, target);\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (i == nums.size() || nums[i] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_edge.java<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nint binarySearchLeftEdge(int[] nums, int target) {\n    // target의 삽입 위치를 찾는 것과 같다\n    int i = binary_search_insertion.binarySearchInsertion(nums, target);\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (i == nums.length || nums[i] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nint BinarySearchLeftEdge(int[] nums, int target) {\n    // target의 삽입 위치를 찾는 것과 같다\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target);\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (i == nums.Length || nums[i] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_edge.go<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nfunc binarySearchLeftEdge(nums []int, target int) int {\n    // target의 삽입 위치를 찾는 것과 같다\n    i := binarySearchInsertion(nums, target)\n    // target을 찾지 못했으므로 -1을 반환한다\n    if i == len(nums) || nums[i] != target {\n        return -1\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i\n}\n</code></pre> binary_search_edge.swift<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nfunc binarySearchLeftEdge(nums: [Int], target: Int) -&gt; Int {\n    // target의 삽입 위치를 찾는 것과 같다\n    let i = binarySearchInsertion(nums: nums, target: target)\n    // target을 찾지 못했으므로 -1을 반환한다\n    if i == nums.endIndex || nums[i] != target {\n        return -1\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i\n}\n</code></pre> binary_search_edge.js<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nfunction binarySearchLeftEdge(nums, target) {\n    // target의 삽입 위치를 찾는 것과 같다\n    const i = binarySearchInsertion(nums, target);\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nfunction binarySearchLeftEdge(nums: Array&lt;number&gt;, target: number): number {\n    // target의 삽입 위치를 찾는 것과 같다\n    const i = binarySearchInsertion(nums, target);\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nint binarySearchLeftEdge(List&lt;int&gt; nums, int target) {\n  // target의 삽입 위치를 찾는 것과 같다\n  int i = binarySearchInsertion(nums, target);\n  // target을 찾지 못했으므로 -1을 반환한다\n  if (i == nums.length || nums[i] != target) {\n    return -1;\n  }\n  // target을 찾았으므로 인덱스 i를 반환한다\n  return i;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nfn binary_search_left_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // target의 삽입 위치를 찾는 것과 같다\n    let i = binary_search_insertion(nums, target);\n    // target을 찾지 못했으므로 -1을 반환한다\n    if i == nums.len() as i32 || nums[i as usize] != target {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    i\n}\n</code></pre> binary_search_edge.c<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nint binarySearchLeftEdge(int *nums, int numSize, int target) {\n    // target의 삽입 위치를 찾는 것과 같다\n    int i = binarySearchInsertion(nums, numSize, target);\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (i == numSize || nums[i] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nfun binarySearchLeftEdge(nums: IntArray, target: Int): Int {\n    // target의 삽입 위치를 찾는 것과 같다\n    val i = binarySearchInsertion(nums, target)\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (i == nums.size || nums[i] != target) {\n        return -1\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i\n}\n</code></pre> binary_search_edge.rb<pre><code>### 가장 왼쪽에 있는 target을 이진 탐색 ###\ndef binary_search_left_edge(nums, target)\n  # target의 삽입 위치를 찾는 것과 같다\n  i = binary_search_insertion(nums, target)\n\n  # target을 찾지 못했으므로 -1을 반환한다\n  return -1 if i == nums.length || nums[i] != target\n\n  i # target을 찾았으므로 인덱스 i를 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 10 장   탐색","10.3   이진 탐색으로 경계 찾기"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#1032","level":2,"title":"10.3.2   오른쪽 경계 찾기","text":"<p>가장 오른쪽에 있는 <code>target</code>은 어떻게 찾을까? 가장 직접적인 방법은 코드를 수정하여 <code>nums[m] == target</code>일 때 포인터로 구간을 줄이는 방식을 바꾸는 것이다. 코드는 생략하니, 관심 있는 독자는 직접 구현해 보자.</p> <p>아래에서는 더 기발한 두 가지 방법을 소개한다.</p>","path":["제 10 장   탐색","10.3   이진 탐색으로 경계 찾기"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#1","level":3,"title":"1.   왼쪽 경계 찾기 재활용","text":"<p>가장 왼쪽 원소를 찾는 함수로 가장 오른쪽 원소도 찾을 수 있다. 구체적으로는 가장 오른쪽 <code>target</code>을 찾는 문제를 가장 왼쪽 <code>target + 1</code>을 찾는 문제로 바꾼다.</p> <p>아래 그림(10-7)과 같이 탐색이 끝나면 포인터 \\(i\\)는 가장 왼쪽 <code>target + 1</code>(존재한다면)을 가리키고, \\(j\\)는 가장 오른쪽 <code>target</code>을 가리킨다. 따라서 \\(j\\)를 반환하면 된다.</p> <p></p> <p> 그림 10-7   오른쪽 경계 찾기를 왼쪽 경계 찾기로 변환 </p> <p>반환되는 삽입 위치는 \\(i\\)이므로, 여기서 \\(1\\)을 빼야 \\(j\\)를 얻을 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_edge.py<pre><code>def binary_search_right_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"가장 오른쪽에 있는 target을 이진 탐색\"\"\"\n    # 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    i = binary_search_insertion(nums, target + 1)\n    # j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    j = i - 1\n    # target을 찾지 못했으므로 -1을 반환한다\n    if j == -1 or nums[j] != target:\n        return -1\n    # target을 찾았으므로 인덱스 j를 반환한다\n    return j\n</code></pre> binary_search_edge.cpp<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nint binarySearchRightEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    int i = binarySearchInsertion(nums, target + 1);\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    int j = i - 1;\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j;\n}\n</code></pre> binary_search_edge.java<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nint binarySearchRightEdge(int[] nums, int target) {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    int i = binary_search_insertion.binarySearchInsertion(nums, target + 1);\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    int j = i - 1;\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nint BinarySearchRightEdge(int[] nums, int target) {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target + 1);\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    int j = i - 1;\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j;\n}\n</code></pre> binary_search_edge.go<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nfunc binarySearchRightEdge(nums []int, target int) int {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    i := binarySearchInsertion(nums, target+1)\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    j := i - 1\n    // target을 찾지 못했으므로 -1을 반환한다\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j\n}\n</code></pre> binary_search_edge.swift<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nfunc binarySearchRightEdge(nums: [Int], target: Int) -&gt; Int {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    let i = binarySearchInsertion(nums: nums, target: target + 1)\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    let j = i - 1\n    // target을 찾지 못했으므로 -1을 반환한다\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j\n}\n</code></pre> binary_search_edge.js<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nfunction binarySearchRightEdge(nums, target) {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    const i = binarySearchInsertion(nums, target + 1);\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    const j = i - 1;\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nfunction binarySearchRightEdge(nums: Array&lt;number&gt;, target: number): number {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    const i = binarySearchInsertion(nums, target + 1);\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    const j = i - 1;\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nint binarySearchRightEdge(List&lt;int&gt; nums, int target) {\n  // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n  int i = binarySearchInsertion(nums, target + 1);\n  // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n  int j = i - 1;\n  // target을 찾지 못했으므로 -1을 반환한다\n  if (j == -1 || nums[j] != target) {\n    return -1;\n  }\n  // target을 찾았으므로 인덱스 j를 반환한다\n  return j;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nfn binary_search_right_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    let i = binary_search_insertion(nums, target + 1);\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    let j = i - 1;\n    // target을 찾지 못했으므로 -1을 반환한다\n    if j == -1 || nums[j as usize] != target {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    j\n}\n</code></pre> binary_search_edge.c<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nint binarySearchRightEdge(int *nums, int numSize, int target) {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    int i = binarySearchInsertion(nums, numSize, target + 1);\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    int j = i - 1;\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nfun binarySearchRightEdge(nums: IntArray, target: Int): Int {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    val i = binarySearchInsertion(nums, target + 1)\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    val j = i - 1\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (j == -1 || nums[j] != target) {\n        return -1\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j\n}\n</code></pre> binary_search_edge.rb<pre><code>### 가장 오른쪽에 있는 target을 이진 탐색 ###\ndef binary_search_right_edge(nums, target)\n  # 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n  i = binary_search_insertion(nums, target + 1)\n\n  # j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n  j = i - 1\n\n  # target을 찾지 못했으므로 -1을 반환한다\n  return -1 if j == -1 || nums[j] != target\n\n  j # target을 찾았으므로 인덱스 j를 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 10 장   탐색","10.3   이진 탐색으로 경계 찾기"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#2","level":3,"title":"2.   원소 찾기로 변환","text":"<p>배열에 <code>target</code>이 없으면 최종적으로 \\(i\\)는 <code>target</code>보다 큰 첫 번째 원소를, \\(j\\)는 <code>target</code>보다 작은 가장 오른쪽 원소를 가리킨다.</p> <p>따라서 아래 그림(10-8)과 같이 배열에 없는 원소를 만들어 왼쪽 경계와 오른쪽 경계를 찾을 수 있다.</p> <ul> <li>가장 왼쪽 <code>target</code> 찾기: <code>target - 0.5</code>를 탐색하고 포인터 \\(i\\)를 반환하는 것으로 바꿀 수 있다.</li> <li>가장 오른쪽 <code>target</code> 찾기: <code>target + 0.5</code>를 탐색하고 포인터 \\(j\\)를 반환하는 것으로 바꿀 수 있다.</li> </ul> <p></p> <p> 그림 10-8   경계 찾기를 원소 찾기로 변환 </p> <p>코드는 생략하며, 다음 두 가지에 주의하자.</p> <ul> <li>주어진 배열에는 소수가 없으므로 값이 같은 경우의 처리는 고려하지 않아도 된다.</li> <li>이 방법은 소수를 사용하므로 함수의 변수 <code>target</code>을 부동 소수점 타입으로 바꿔야 한다(Python은 수정할 필요가 없다).</li> </ul>","path":["제 10 장   탐색","10.3   이진 탐색으로 경계 찾기"],"tags":[]},{"location":"chapter_searching/binary_search_insertion/","level":1,"title":"10.2   이진 탐색으로 삽입 위치 찾기","text":"<p>이진 탐색은 목표 원소를 찾는 데만 쓰이는 것이 아니다. 목표 원소를 삽입할 위치를 찾는 등 여러 변형 문제에도 사용할 수 있다.</p>","path":["제 10 장   탐색","10.2   이진 탐색으로 삽입 위치 찾기"],"tags":[]},{"location":"chapter_searching/binary_search_insertion/#1021","level":2,"title":"10.2.1   중복 원소가 없는 경우","text":"<p>문제</p> <p>중복 원소가 없는 길이 \\(n\\)의 정렬된 배열 <code>nums</code>와 원소 <code>target</code>이 주어진다. 정렬 상태를 유지하면서 <code>target</code>을 배열 <code>nums</code>에 삽입하려고 한다. 이미 <code>target</code>이 있으면 그 왼쪽에 삽입한다. 삽입 후 <code>target</code>의 배열 인덱스를 반환하라. 예시는 아래 그림(10-4)과 같다.</p> <p></p> <p> 그림 10-4   이진 탐색으로 삽입 위치를 찾는 예시 데이터 </p> <p>앞 절의 이진 탐색 코드를 재사용하려면 다음 두 질문에 답해야 한다.</p> <p>질문 1: 배열에 <code>target</code>이 있을 때 삽입 위치의 인덱스는 해당 원소의 인덱스인가?</p> <p>문제에서는 값이 같은 원소의 왼쪽에 <code>target</code>을 삽입하라고 한다. 따라서 새로 삽입하는 <code>target</code>이 기존 <code>target</code>의 자리를 차지한다. 즉, 배열에 <code>target</code>이 있으면 삽입 위치의 인덱스는 그 <code>target</code>의 인덱스다.</p> <p>질문 2: 배열에 <code>target</code>이 없을 때 삽입 위치는 어느 원소의 인덱스인가?</p> <p>이진 탐색 과정을 더 살펴보자. <code>nums[m] &lt; target</code>이면 \\(i\\)가 이동하므로 포인터 \\(i\\)는 <code>target</code> 이상인 원소에 가까워진다. 마찬가지로 포인터 \\(j\\)는 항상 <code>target</code> 이하인 원소에 가까워진다.</p> <p>따라서 이진 탐색이 끝나면 \\(i\\)는 <code>target</code>보다 큰 첫 번째 원소를, \\(j\\)는 <code>target</code>보다 작은 가장 오른쪽 원소를 가리킨다. 배열에 <code>target</code>이 없을 때 삽입 인덱스가 \\(i\\)임을 쉽게 알 수 있다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_insertion.py<pre><code>def binary_search_insertion_simple(nums: list[int], target: int) -&gt; int:\n    \"\"\"삽입 위치의 이진 탐색(중복 원소 없음)\"\"\"\n    i, j = 0, len(nums) - 1  # 닫힌 구간 [0, n-1]을 초기화한다\n    while i &lt;= j:\n        m = (i + j) // 2  # 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target:\n            i = m + 1  # target은 구간 [m+1, j]에 있다\n        elif nums[m] &gt; target:\n            j = m - 1  # target은 구간 [i, m-1]에 있다\n        else:\n            return m  # target을 찾았으므로 삽입 위치 m을 반환한다\n    # target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nint binarySearchInsertionSimple(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            return m; // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nint binarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            return m; // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nint BinarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            return m; // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nfunc binarySearchInsertionSimple(nums []int, target int) int {\n    // 닫힌 구간 [0, n-1]을 초기화한다\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // 중간 인덱스 m을 계산한다\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target은 구간 [m+1, j]에 있다\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target은 구간 [i, m-1]에 있다\n            j = m - 1\n        } else {\n            // target을 찾았으므로 삽입 위치 m을 반환한다\n            return m\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nfunc binarySearchInsertionSimple(nums: [Int], target: Int) -&gt; Int {\n    // 닫힌 구간 [0, n-1]을 초기화한다\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target {\n            i = m + 1 // target은 구간 [m+1, j]에 있다\n        } else if nums[m] &gt; target {\n            j = m - 1 // target은 구간 [i, m-1]에 있다\n        } else {\n            return m // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nfunction binarySearchInsertionSimple(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // 중간 인덱스 m을 계산하며, Math.floor()로 내림한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            return m; // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nfunction binarySearchInsertionSimple(\n    nums: Array&lt;number&gt;,\n    target: number\n): number {\n    let i = 0,\n        j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // 중간 인덱스 m을 계산하며, Math.floor()로 내림한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            return m; // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nint binarySearchInsertionSimple(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // 중간 인덱스 m을 계산한다\n    if (nums[m] &lt; target) {\n      i = m + 1; // target은 구간 [m+1, j]에 있다\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target은 구간 [i, m-1]에 있다\n    } else {\n      return m; // target을 찾았으므로 삽입 위치 m을 반환한다\n    }\n  }\n  // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nfn binary_search_insertion_simple(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // 닫힌 구간 [0, n-1]을 초기화한다\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if nums[m as usize] &lt; target {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            return m;\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nint binarySearchInsertionSimple(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            return m; // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nfun binarySearchInsertionSimple(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1 // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // target은 구간 [i, m-1]에 있다\n        } else {\n            return m // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>### 삽입 위치의 이진 탐색(중복 원소 없음) ###\ndef binary_search_insertion_simple(nums, target)\n  # 닫힌 구간 [0, n-1]을 초기화한다\n  i, j = 0, nums.length - 1\n\n  while i &lt;= j\n    # 중간 인덱스 m을 계산한다\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # target은 구간 [m+1, j]에 있다\n    elsif nums[m] &gt; target\n      j = m - 1 # target은 구간 [i, m-1]에 있다\n    else\n      return m  # target을 찾았으므로 삽입 위치 m을 반환한다\n    end\n  end\n\n  i # target을 찾지 못했으므로 삽입 위치 i를 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 10 장   탐색","10.2   이진 탐색으로 삽입 위치 찾기"],"tags":[]},{"location":"chapter_searching/binary_search_insertion/#1022","level":2,"title":"10.2.2   중복 원소가 있는 경우","text":"<p>문제</p> <p>앞 문제에서 배열에 중복 원소가 있을 수 있다는 조건만 추가하고, 나머지 조건은 그대로 둔다.</p> <p>배열에 <code>target</code>이 여러 개 있다면 일반적인 이진 탐색으로는 그중 하나의 인덱스만 구할 수 있다. 그 원소의 왼쪽과 오른쪽에 <code>target</code>이 몇 개 더 있는지는 알 수 없다.</p> <p>문제에서는 목표 원소를 가장 왼쪽에 삽입하라고 한다. 따라서 배열에서 가장 왼쪽 <code>target</code>의 인덱스를 찾아야 한다. 먼저 아래 그림(10-5)과 같은 방법을 생각해 볼 수 있다.</p> <ol> <li>이진 탐색으로 임의의 <code>target</code>의 인덱스를 구하고 이를 \\(k\\)라고 한다.</li> <li>인덱스 \\(k\\)부터 왼쪽으로 선형 순회하여 가장 왼쪽 <code>target</code>을 찾으면 반환한다.</li> </ol> <p></p> <p> 그림 10-5   선형 탐색으로 중복 원소의 삽입 위치 찾기 </p> <p>이 방법도 가능하지만 선형 탐색이 포함되어 시간 복잡도가 \\(O(n)\\)이다. 배열에 중복된 <code>target</code>이 많으면 매우 비효율적이다.</p> <p>이제 이진 탐색 코드를 확장해 보자. 아래 그림(10-6)과 같이 전체 흐름은 그대로 유지한다. 매 반복에서 중간 인덱스 \\(m\\)을 계산하고, <code>target</code>과 <code>nums[m]</code>의 대소 관계에 따라 다음과 같이 처리한다.</p> <ul> <li><code>nums[m] &lt; target</code> 또는 <code>nums[m] &gt; target</code>이면 아직 <code>target</code>을 찾지 못했으므로, 일반적인 이진 탐색처럼 구간을 줄여 포인터 \\(i\\)와 \\(j\\)가 <code>target</code>에 가까워지게 한다.</li> <li><code>nums[m] == target</code>이면 <code>target</code>보다 작은 원소는 구간 \\([i, m - 1]\\)에 있으므로, \\(j = m - 1\\)로 구간을 줄여 포인터 \\(j\\)가 <code>target</code>보다 작은 원소에 가까워지게 한다.</li> </ul> <p>반복이 끝나면 \\(i\\)는 가장 왼쪽 <code>target</code>을, \\(j\\)는 <code>target</code>보다 작은 가장 오른쪽 원소를 가리킨다. 따라서 인덱스 \\(i\\)가 삽입 위치다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 10-6   이진 탐색으로 중복 원소의 삽입 위치를 찾는 단계 </p> <p>아래 코드를 보면 <code>nums[m] &gt; target</code>과 <code>nums[m] == target</code> 분기에서 수행하는 연산이 같으므로 두 분기를 합칠 수 있다.</p> <p>그렇더라도 조건을 나누어 두면 논리가 더 명확하고 가독성이 좋으므로 그대로 둘 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_insertion.py<pre><code>def binary_search_insertion(nums: list[int], target: int) -&gt; int:\n    \"\"\"삽입 위치의 이진 탐색(중복 원소 있음)\"\"\"\n    i, j = 0, len(nums) - 1  # 닫힌 구간 [0, n-1]을 초기화한다\n    while i &lt;= j:\n        m = (i + j) // 2  # 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target:\n            i = m + 1  # target은 구간 [m+1, j]에 있다\n        elif nums[m] &gt; target:\n            j = m - 1  # target은 구간 [i, m-1]에 있다\n        else:\n            j = m - 1  # target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n    # 삽입 위치 i를 반환한다\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nint binarySearchInsertion(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nint binarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nint BinarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nfunc binarySearchInsertion(nums []int, target int) int {\n    // 닫힌 구간 [0, n-1]을 초기화한다\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // 중간 인덱스 m을 계산한다\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target은 구간 [m+1, j]에 있다\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target은 구간 [i, m-1]에 있다\n            j = m - 1\n        } else {\n            // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n            j = m - 1\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nfunc binarySearchInsertion(nums: [Int], target: Int) -&gt; Int {\n    // 닫힌 구간 [0, n-1]을 초기화한다\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target {\n            i = m + 1 // target은 구간 [m+1, j]에 있다\n        } else if nums[m] &gt; target {\n            j = m - 1 // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1 // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nfunction binarySearchInsertion(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // 중간 인덱스 m을 계산하며, Math.floor()로 내림한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nfunction binarySearchInsertion(nums: Array&lt;number&gt;, target: number): number {\n    let i = 0,\n        j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // 중간 인덱스 m을 계산하며, Math.floor()로 내림한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nint binarySearchInsertion(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // 중간 인덱스 m을 계산한다\n    if (nums[m] &lt; target) {\n      i = m + 1; // target은 구간 [m+1, j]에 있다\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target은 구간 [i, m-1]에 있다\n    } else {\n      j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n    }\n  }\n  // 삽입 위치 i를 반환한다\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\npub fn binary_search_insertion(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // 닫힌 구간 [0, n-1]을 초기화한다\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if nums[m as usize] &lt; target {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nint binarySearchInsertion(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nfun binarySearchInsertion(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1 // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1 // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>### 삽입 위치의 이진 탐색(중복 원소 있음) ###\ndef binary_search_insertion(nums, target)\n  # 닫힌 구간 [0, n-1]을 초기화한다\n  i, j = 0, nums.length - 1\n\n  while i &lt;= j\n    # 중간 인덱스 m을 계산한다\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # target은 구간 [m+1, j]에 있다\n    elsif nums[m] &gt; target\n      j = m - 1 # target은 구간 [i, m-1]에 있다\n    else\n      j = m - 1 # target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n    end\n  end\n\n  i # 삽입 위치 i를 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>참고</p> <p>이 절의 코드는 모두 '양끝이 닫힌 구간'을 사용한다. 관심 있는 독자는 '왼쪽은 닫히고 오른쪽은 열린 구간'으로 직접 구현해 보자.</p> <p>종합하면 이진 탐색은 포인터 \\(i\\)와 \\(j\\)에 각각 탐색 목표를 정해 주는 과정이다. 목표는 특정 원소(예: <code>target</code>)일 수도 있고, 원소의 범위(예: <code>target</code>보다 작은 원소)일 수도 있다.</p> <p>구간을 계속 반으로 나누는 동안 포인터 \\(i\\)와 \\(j\\)는 미리 정한 목표에 점차 가까워진다. 결국 답을 찾거나 경계를 넘어선 뒤 멈춘다.</p>","path":["제 10 장   탐색","10.2   이진 탐색으로 삽입 위치 찾기"],"tags":[]},{"location":"chapter_searching/exercises/","level":1,"title":"10.7   연습 문제","text":"","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/exercises/#1071","level":2,"title":"10.7.1   개념 확인","text":"","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/exercises/#1","level":3,"title":"1.   이진 탐색은 탐색 구간을 어떻게 줄이는가","text":"<p>정렬된 배열 <code>[2, 5, 8, 12, 16, 23, 38]</code>에서 16을 찾는다. 양 끝을 포함하는 닫힌 구간 <code>[i, j]</code>를 사용하며, 중간 인덱스는 \\(m=i+(j-i)/2\\)다. 나눗셈 결과의 소수 부분은 버린다.</p> <p>목표 값을 찾을 때까지 매 단계의 <code>(i, j, m)</code>과 중간 원소, 다음 단계에서 구간을 어떻게 줄이는지 적어 보자.</p> 참고 답안 <p>각 단계의 탐색 과정은 다음과 같다.</p> 단계 <code>(i, j, m)</code> 중간 원소 다음 동작 1 <code>(0, 6, 3)</code> 12 <code>12 &lt; 16</code>이므로 <code>i = 4</code>로 설정 2 <code>(4, 6, 5)</code> 23 <code>23 &gt; 16</code>이므로 <code>j = 4</code>로 설정 3 <code>(4, 4, 4)</code> 16 목표 값을 찾았으므로 인덱스 4를 반환 <p>배열이 정렬되어 있으므로 중간 값이 목표 값보다 작으면 중간 원소와 그 왼쪽을 제외할 수 있다. 중간 값이 목표 값보다 크면 중간 원소와 그 오른쪽을 제외할 수 있다.</p>","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/exercises/#2","level":3,"title":"2.   중복 원소의 왼쪽 경계와 오른쪽 경계","text":"<p>배열 <code>[1, 2, 2, 2, 4, 6]</code>에서 숫자 2를 찾는다. 한 학생이 이진 탐색으로 인덱스 2에서 목표 값을 찾자마자 반환한 뒤, “인덱스 2가 숫자 2의 왼쪽 경계다.”라고 말했다.</p> <ol> <li>이 학생의 말이 맞는가? 숫자 2의 왼쪽 경계와 오른쪽 경계는 각각 어디인지 이유와 함께 설명하라.</li> <li>왼쪽 경계를 찾을 때 중간 원소가 목표 값과 같다면, 다음에는 어느 쪽을 계속 탐색해야 하는가?</li> <li>오른쪽 경계를 찾을 때는 어느 쪽을 계속 탐색해야 하는가? 탐색 과정 전체를 적지 말고 방향만 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>이 학생의 말은 틀렸다. 2를 하나 찾자마자 반환하면 어떤 2를 찾았다는 것만 보장할 수 있고, 가장 왼쪽이나 가장 오른쪽의 2를 찾았다고 보장할 수는 없다.     이 문제에서 왼쪽 경계는 인덱스 1, 오른쪽 경계는 인덱스 3이다.</p> </li> <li> <p>왼쪽 경계를 찾을 때는 중간 원소가 2와 같더라도 왼쪽을 계속 탐색해야 한다.     예를 들어 양 끝을 포함하는 닫힌 구간을 사용하는 경우 <code>j = m - 1</code>로 설정한다.</p> </li> <li> <p>오른쪽 경계를 찾을 때는 중간 원소가 2와 같으면 오른쪽을 계속 탐색해야 한다.     예를 들어 <code>i = m + 1</code>로 설정한다.</p> </li> </ol>","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/exercises/#3","level":3,"title":"3.   데이터에 따라 어떤 탐색 방법을 선택해야 하는가","text":"<p>다음 세 상황에 알맞은 방법을 ‘선형 탐색, 이진 탐색, 해시 테이블’ 중에서 고르고 이유를 설명하라.</p> <ol> <li>이미 정렬되어 있고 더 이상 변하지 않는 정수 \\(10^7\\)개에서 반복해서 값을 찾으며, 다른 자료 구조를 추가로 만들지 않는다.</li> <li>삽입과 삭제가 빈번한 데이터 집합에서 특정 키가 있는지 반복해서 확인한다. 정렬 상태를 유지할 필요가 없고 범위 탐색도 하지 않는다.</li> <li>정렬되지 않은 배열에서 특정 값을 한 번만 찾는다.</li> </ol> 참고 답안 <ol> <li>이진 탐색: 데이터가 정렬되어 있고 변하지 않으므로 \\(O(\\log n)\\) 시간에 탐색할 수 있으며, 추가 공간도 필요하지 않다.</li> <li>해시 테이블: 해시 함수가 키를 각 버킷에 비교적 고르게 분산한다면 삽입, 삭제, 키를 이용한 탐색의 평균 시간 복잡도가 모두 \\(O(1)\\)이다.</li> <li> <p>처음부터 끝까지 직접 순회: 한 번만 찾는 경우에는 정렬하거나 해시 테이블을 만드는 데에도 먼저 배열 전체를 처리해야 하므로,     이번 작업의 전체 작업량이 줄어들지 않는다.</p> <p>어떤 방법을 선택할지는 데이터의 정렬 여부, 추가 자료 구조의 허용 여부, 조회 횟수, 수행해야 하는 연산에 따라 달라진다.</p> </li> </ol>","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/exercises/#1072","level":2,"title":"10.7.2   프로그래밍 연습","text":"","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/exercises/#1_1","level":3,"title":"1.   정렬된 배열의 이진 탐색","text":"<p>엄격한 오름차순으로 정렬된 정수 배열 <code>nums</code>와 목표 값 <code>target</code>이 주어진다. 이진 탐색으로 <code>target</code>을 찾아라. 존재하면 배열 인덱스를 반환하고, 존재하지 않으면 -1을 반환하라.</p> 풀이 힌트 <ol> <li>초기 구간은 left = 0, right = n - 1이며, 구간이 비어 있지 않을 조건은 left &lt;= right다.</li> <li>mid = left + (right - left) // 2로 중간 인덱스를 계산한다.</li> <li><code>nums[mid]</code>가 <code>target</code>보다 작으면 왼쪽 경계를 <code>mid + 1</code>로 옮긴다. <code>nums[mid]</code>가 <code>target</code>보다 크면 오른쪽 경계를 <code>mid - 1</code>로 옮긴다. 같으면 즉시 반환한다.</li> </ol> <p>LeetCode</p>","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/exercises/#2_1","level":3,"title":"2.   정렬된 배열의 삽입 위치","text":"<p>엄격한 오름차순으로 정렬된 정수 배열 <code>nums</code>와 목표 값 <code>target</code>이 주어진다.</p> <p><code>target</code>이 이미 배열에 있으면 그 인덱스를 반환하라. 없으면 <code>target</code>을 삽입해도 엄격한 오름차순을 유지할 수 있는 위치를 반환하라. 답은 0일 수도 있고 배열 길이와 같을 수도 있다. 이진 탐색을 사용하라.</p> 풀이 힌트 <ol> <li>답은 0일 수도 있고 배열 길이 n일 수도 있다.</li> <li>양 끝을 포함하는 닫힌 구간을 사용할 때 <code>nums[mid]</code>가 <code>target</code>보다 크거나 같으면 <code>right = mid - 1</code>로 설정하여 더 왼쪽의 위치를 계속 확인한다. 그렇지 않으면 <code>left = mid + 1</code>로 설정한다.</li> <li>반복문이 끝나면 left가 삽입 위치다.</li> </ol> <p>LeetCode</p>","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/replace_linear_by_hashing/","level":1,"title":"10.4   해시를 이용한 최적화 전략","text":"<p>알고리즘 문제에서는 선형 탐색을 해시 탐색으로 바꾸어 시간 복잡도를 줄이는 경우가 많다. 한 가지 문제를 통해 더 자세히 알아보자.</p> <p>문제</p> <p>정수 배열 <code>nums</code>와 목표 원소 <code>target</code>이 주어진다. 배열에서 '합'이 <code>target</code>인 두 원소를 찾아 그 인덱스를 반환하라. 답이 여러 개라면 아무거나 하나를 반환해도 된다.</p>","path":["제 10 장   탐색","10.4   해시를 이용한 최적화 전략"],"tags":[]},{"location":"chapter_searching/replace_linear_by_hashing/#1041","level":2,"title":"10.4.1   선형 탐색: 시간을 더 써서 공간 절약","text":"<p>가능한 모든 조합을 직접 순회해 보자. 아래 그림(10-9)과 같이 이중 반복문을 사용하여 매번 두 정수의 합이 <code>target</code>인지 확인하고, 맞으면 두 인덱스를 반환한다.</p> <p></p> <p> 그림 10-9   선형 탐색으로 두 수의 합 구하기 </p> <p>코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby two_sum.py<pre><code>def two_sum_brute_force(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"방법 1: 완전 열거\"\"\"\n    # 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for i in range(len(nums) - 1):\n        for j in range(i + 1, len(nums)):\n            if nums[i] + nums[j] == target:\n                return [i, j]\n    return []\n</code></pre> two_sum.cpp<pre><code>/* 방법 1: 완전 열거 */\nvector&lt;int&gt; twoSumBruteForce(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return {i, j};\n        }\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* 방법 1: 완전 열거 */\nint[] twoSumBruteForce(int[] nums, int target) {\n    int size = nums.length;\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return new int[] { i, j };\n        }\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* 방법 1: 완전 열거 */\nint[] TwoSumBruteForce(int[] nums, int target) {\n    int size = nums.Length;\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return [i, j];\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* 방법 1: 완전 열거 */\nfunc twoSumBruteForce(nums []int, target int) []int {\n    size := len(nums)\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for i := 0; i &lt; size-1; i++ {\n        for j := i + 1; j &lt; size; j++ {\n            if nums[i]+nums[j] == target {\n                return []int{i, j}\n            }\n        }\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* 방법 1: 완전 열거 */\nfunc twoSumBruteForce(nums: [Int], target: Int) -&gt; [Int] {\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for i in nums.indices.dropLast() {\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[i] + nums[j] == target {\n                return [i, j]\n            }\n        }\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* 방법 1: 완전 열거 */\nfunction twoSumBruteForce(nums, target) {\n    const n = nums.length;\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* 방법 1: 완전 열거 */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\n    const n = nums.length;\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* 방법 1: 완전 열거 */\nList&lt;int&gt; twoSumBruteForce(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n  for (var i = 0; i &lt; size - 1; i++) {\n    for (var j = i + 1; j &lt; size; j++) {\n      if (nums[i] + nums[j] == target) return [i, j];\n    }\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* 방법 1: 완전 열거 */\npub fn two_sum_brute_force(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    let size = nums.len();\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for i in 0..size - 1 {\n        for j in i + 1..size {\n            if nums[i] + nums[j] == target {\n                return Some(vec![i as i32, j as i32]);\n            }\n        }\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* 방법 1: 완전 열거 */\nint *twoSumBruteForce(int *nums, int numsSize, int target, int *returnSize) {\n    for (int i = 0; i &lt; numsSize; ++i) {\n        for (int j = i + 1; j &lt; numsSize; ++j) {\n            if (nums[i] + nums[j] == target) {\n                int *res = malloc(sizeof(int) * 2);\n                res[0] = i, res[1] = j;\n                *returnSize = 2;\n                return res;\n            }\n        }\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* 방법 1: 완전 열거 */\nfun twoSumBruteForce(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for (i in 0..&lt;size - 1) {\n        for (j in i + 1..&lt;size) {\n            if (nums[i] + nums[j] == target) return intArrayOf(i, j)\n        }\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>### 방법 1: 완전 열거 ###\ndef two_sum_brute_force(nums, target)\n  # 이중 반복문이며 시간 복잡도는 O(n^2)이다\n  for i in 0...(nums.length - 1)\n    for j in (i + 1)...nums.length\n      return [i, j] if nums[i] + nums[j] == target\n    end\n  end\n\n  []\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>이 방법의 시간 복잡도는 \\(O(n^2)\\), 공간 복잡도는 \\(O(1)\\)이다. 데이터가 많으면 시간이 매우 오래 걸린다.</p>","path":["제 10 장   탐색","10.4   해시를 이용한 최적화 전략"],"tags":[]},{"location":"chapter_searching/replace_linear_by_hashing/#1042","level":2,"title":"10.4.2   해시 탐색: 공간을 더 써서 시간 절약","text":"<p>키와 값으로 각각 배열 원소와 그 인덱스를 저장하는 해시 테이블을 사용해 보자. 배열을 순회하면서 매번 아래 그림(10-10)의 단계를 수행한다.</p> <ol> <li>수 <code>target - nums[i]</code>가 해시 테이블에 있는지 확인하고, 있으면 두 원소의 인덱스를 바로 반환한다.</li> <li><code>nums[i]</code>와 인덱스 <code>i</code>로 이루어진 키-값 쌍을 해시 테이블에 추가한다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> 그림 10-10   보조 해시 테이블로 두 수의 합 구하기 </p> <p>구현 코드는 다음과 같으며, 단일 반복문만 있으면 된다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby two_sum.py<pre><code>def two_sum_hash_table(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"방법 2: 보조 해시 테이블\"\"\"\n    # 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    dic = {}\n    # 단일 반복문이며 시간 복잡도는 O(n)이다\n    for i in range(len(nums)):\n        if target - nums[i] in dic:\n            return [dic[target - nums[i]], i]\n        dic[nums[i]] = i\n    return []\n</code></pre> two_sum.cpp<pre><code>/* 방법 2: 보조 해시 테이블 */\nvector&lt;int&gt; twoSumHashTable(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    unordered_map&lt;int, int&gt; dic;\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.find(target - nums[i]) != dic.end()) {\n            return {dic[target - nums[i]], i};\n        }\n        dic.emplace(nums[i], i);\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* 방법 2: 보조 해시 테이블 */\nint[] twoSumHashTable(int[] nums, int target) {\n    int size = nums.length;\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    Map&lt;Integer, Integer&gt; dic = new HashMap&lt;&gt;();\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.containsKey(target - nums[i])) {\n            return new int[] { dic.get(target - nums[i]), i };\n        }\n        dic.put(nums[i], i);\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* 방법 2: 보조 해시 테이블 */\nint[] TwoSumHashTable(int[] nums, int target) {\n    int size = nums.Length;\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    Dictionary&lt;int, int&gt; dic = [];\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.ContainsKey(target - nums[i])) {\n            return [dic[target - nums[i]], i];\n        }\n        dic.Add(nums[i], i);\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* 방법 2: 보조 해시 테이블 */\nfunc twoSumHashTable(nums []int, target int) []int {\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    hashTable := map[int]int{}\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for idx, val := range nums {\n        if preIdx, ok := hashTable[target-val]; ok {\n            return []int{preIdx, idx}\n        }\n        hashTable[val] = idx\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* 방법 2: 보조 해시 테이블 */\nfunc twoSumHashTable(nums: [Int], target: Int) -&gt; [Int] {\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    var dic: [Int: Int] = [:]\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for i in nums.indices {\n        if let j = dic[target - nums[i]] {\n            return [j, i]\n        }\n        dic[nums[i]] = i\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* 방법 2: 보조 해시 테이블 */\nfunction twoSumHashTable(nums, target) {\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    let m = {};\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (m[target - nums[i]] !== undefined) {\n            return [m[target - nums[i]], i];\n        } else {\n            m[nums[i]] = i;\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* 방법 2: 보조 해시 테이블 */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    let m: Map&lt;number, number&gt; = new Map();\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for (let i = 0; i &lt; nums.length; i++) {\n        let index = m.get(target - nums[i]);\n        if (index !== undefined) {\n            return [index, i];\n        } else {\n            m.set(nums[i], i);\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* 방법 2: 보조 해시 테이블 */\nList&lt;int&gt; twoSumHashTable(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n  Map&lt;int, int&gt; dic = HashMap();\n  // 단일 반복문이며 시간 복잡도는 O(n)이다\n  for (var i = 0; i &lt; size; i++) {\n    if (dic.containsKey(target - nums[i])) {\n      return [dic[target - nums[i]]!, i];\n    }\n    dic.putIfAbsent(nums[i], () =&gt; i);\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* 방법 2: 보조 해시 테이블 */\npub fn two_sum_hash_table(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    let mut dic = HashMap::new();\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for (i, num) in nums.iter().enumerate() {\n        match dic.get(&amp;(target - num)) {\n            Some(v) =&gt; return Some(vec![*v as i32, i as i32]),\n            None =&gt; dic.insert(num, i as i32),\n        };\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* 해시 테이블 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // uthash.h를 기반으로 구현한다\n} HashTable;\n\n/* 해시 테이블 조회 */\nHashTable *find(HashTable *h, int key) {\n    HashTable *tmp;\n    HASH_FIND_INT(h, &amp;key, tmp);\n    return tmp;\n}\n\n/* 해시 테이블에 원소 삽입 */\nvoid insert(HashTable **h, int key, int val) {\n    HashTable *t = find(*h, key);\n    if (t == NULL) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = key, tmp-&gt;val = val;\n        HASH_ADD_INT(*h, key, tmp);\n    } else {\n        t-&gt;val = val;\n    }\n}\n\n/* 방법 2: 보조 해시 테이블 */\nint *twoSumHashTable(int *nums, int numsSize, int target, int *returnSize) {\n    HashTable *hashtable = NULL;\n    for (int i = 0; i &lt; numsSize; i++) {\n        HashTable *t = find(hashtable, target - nums[i]);\n        if (t != NULL) {\n            int *res = malloc(sizeof(int) * 2);\n            res[0] = t-&gt;val, res[1] = i;\n            *returnSize = 2;\n            return res;\n        }\n        insert(&amp;hashtable, nums[i], i);\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* 방법 2: 보조 해시 테이블 */\nfun twoSumHashTable(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    val dic = HashMap&lt;Int, Int&gt;()\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for (i in 0..&lt;size) {\n        if (dic.containsKey(target - nums[i])) {\n            return intArrayOf(dic[target - nums[i]]!!, i)\n        }\n        dic[nums[i]] = i\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>### 방법 2: 보조 해시 테이블 ###\ndef two_sum_hash_table(nums, target)\n  # 보조 해시 테이블의 공간 복잡도는 O(n)이다\n  dic = {}\n  # 단일 반복문이며 시간 복잡도는 O(n)이다\n  for i in 0...nums.length\n    return [dic[target - nums[i]], i] if dic.has_key?(target - nums[i])\n\n    dic[nums[i]] = i\n  end\n\n  []\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>이 방법은 해시 탐색으로 시간 복잡도를 \\(O(n^2)\\)에서 \\(O(n)\\)으로 줄여 실행 효율을 크게 높인다.</p> <p>해시 테이블을 추가로 유지해야 하므로 공간 복잡도는 \\(O(n)\\)이다. 그렇더라도 시간과 공간 효율의 균형이 더 좋으므로 이 문제의 최적 풀이에 해당한다.</p>","path":["제 10 장   탐색","10.4   해시를 이용한 최적화 전략"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/","level":1,"title":"10.5   탐색 알고리즘 다시 보기","text":"<p>탐색 알고리즘(searching algorithm)은 배열, 연결 리스트, 트리, 그래프 등의 자료 구조에서 특정 조건을 만족하는 원소 하나 또는 여러 개를 찾는 데 사용한다.</p> <p>탐색 알고리즘은 구현 아이디어에 따라 다음 두 종류로 나뉜다.</p> <ul> <li>배열, 연결 리스트, 트리, 그래프의 순회처럼 자료 구조를 순회하여 목표 원소를 찾는다.</li> <li>이진 탐색, 해시 탐색, 이진 탐색 트리의 탐색처럼 데이터의 구성 방식이나 데이터에 대한 사전 정보를 이용하여 원소를 효율적으로 찾는다.</li> </ul> <p>이 내용은 모두 앞 장들에서 이미 소개했으므로 탐색 알고리즘은 낯설지 않을 것이다. 이 절에서는 더 체계적인 관점에서 탐색 알고리즘을 다시 살펴본다.</p>","path":["제 10 장   탐색","10.5   탐색 알고리즘 다시 보기"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/#1051","level":2,"title":"10.5.1   완전 탐색","text":"<p>완전 탐색은 자료 구조의 각 원소를 순회하여 목표 원소를 찾는다.</p> <ul> <li>'선형 탐색'은 배열이나 연결 리스트 같은 선형 자료 구조에 적합하다. 자료 구조의 한쪽 끝에서 원소를 하나씩 방문하며, 목표 원소를 찾거나 찾지 못한 채 반대쪽 끝에 도달하면 종료한다.</li> <li>'너비 우선 탐색'과 '깊이 우선 탐색'은 그래프와 트리의 두 가지 순회 전략이다. 너비 우선 탐색은 시작 노드에서 한 층씩 탐색하며 가까운 노드부터 먼 노드 순서로 방문한다. 깊이 우선 탐색은 시작 노드에서 한 경로를 따라 끝까지 간 뒤 되돌아와 다른 경로를 시도하며, 전체 자료 구조를 순회할 때까지 반복한다.</li> </ul> <p>완전 탐색은 간단하고 범용성이 좋으며, 데이터를 전처리하거나 추가 자료 구조를 사용할 필요가 없다.</p> <p>하지만 이러한 알고리즘의 시간 복잡도는 \\(O(n)\\)이다. 여기서 \\(n\\)은 원소 수이며, 데이터가 많으면 성능이 떨어진다.</p>","path":["제 10 장   탐색","10.5   탐색 알고리즘 다시 보기"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/#1052","level":2,"title":"10.5.2   적응형 탐색","text":"<p>적응형 탐색은 정렬 여부 같은 데이터 고유의 성질을 이용하여 탐색 과정을 최적화하고, 목표 원소를 더 효율적으로 찾는다.</p> <ul> <li>'이진 탐색'은 데이터의 정렬 상태를 이용하여 효율적으로 찾으며, 배열에만 적합하다.</li> <li>'해시 탐색'은 해시 테이블로 탐색 데이터와 목표 데이터 사이에 키-값 매핑을 만들어 조회한다.</li> <li>'트리 탐색'은 이진 탐색 트리 같은 특정 트리 구조에서 노드 값을 비교하여 탐색 대상에서 빠르게 제외하고 목표 원소를 찾는다.</li> </ul> <p>이러한 알고리즘은 효율이 높아 시간 복잡도가 \\(O(\\log n)\\), 나아가 \\(O(1)\\)에 이를 수 있다.</p> <p>하지만 이런 알고리즘을 사용하려면 데이터를 전처리해야 하는 경우가 많다. 예를 들어 이진 탐색은 배열을 미리 정렬해야 하며, 해시 탐색과 트리 탐색은 추가 자료 구조가 필요하다. 이러한 자료 구조를 유지하는 데도 시간과 공간이 추가로 든다.</p> <p>참고</p> <p>적응형 탐색 알고리즘은 조회 알고리즘이라고도 하며, 주로 특정 자료 구조에서 목표 원소를 빠르게 조회하는 데 사용한다.</p>","path":["제 10 장   탐색","10.5   탐색 알고리즘 다시 보기"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/#1053","level":2,"title":"10.5.3   탐색 방법 선택","text":"<p>크기가 \\(n\\)인 데이터 집합이 주어지면 선형 탐색, 이진 탐색, 트리 탐색, 해시 탐색 등 다양한 방법으로 목표 원소를 찾을 수 있다. 각 방법의 동작 원리는 아래 그림(10-11)과 같다.</p> <p></p> <p> 그림 10-11   다양한 탐색 전략 </p> <p>위 방법들의 연산 효율과 특성은 아래 표(10-1)와 같다.</p> <p> 표 10-1   탐색 알고리즘 효율 비교 </p> 선형 탐색 이진 탐색 트리 탐색 해시 탐색 원소 탐색 \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) 원소 삽입 \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) 원소 삭제 \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) 추가 공간 \\(O(1)\\) \\(O(1)\\) \\(O(n)\\) \\(O(n)\\) 데이터 전처리 / 정렬 \\(O(n \\log n)\\) 트리 구성 \\(O(n \\log n)\\) 해시 테이블 구성 \\(O(n)\\) 데이터 정렬 여부 정렬되지 않음 정렬됨 정렬됨 정렬되지 않음 <p>탐색 알고리즘을 선택할 때는 데이터 규모, 요구되는 탐색 성능, 데이터 조회와 갱신 빈도 등도 고려해야 한다.</p> <p>선형 탐색</p> <ul> <li>범용성이 좋으며 데이터 전처리가 전혀 필요하지 않다. 데이터를 한 번만 조회한다면 나머지 세 방법은 전처리 시간만으로도 선형 탐색보다 오래 걸린다.</li> <li>데이터 규모가 작을 때 적합하다. 이때는 시간 복잡도가 효율에 미치는 영향이 작다.</li> <li>데이터를 추가로 관리할 필요가 없으므로 데이터 갱신이 잦은 상황에 적합하다.</li> </ul> <p>이진 탐색</p> <ul> <li>데이터가 많을 때 적합하며, 최악의 시간 복잡도가 \\(O(\\log n)\\)으로 성능이 안정적이다.</li> <li>배열을 저장하려면 연속된 메모리 공간이 필요하므로 데이터가 지나치게 많으면 안 된다.</li> <li>정렬된 배열을 유지하는 비용이 크므로 데이터 추가와 삭제가 잦은 상황에는 적합하지 않다.</li> </ul> <p>해시 탐색</p> <ul> <li>평균 시간 복잡도가 \\(O(1)\\)이므로 매우 높은 조회 성능이 필요한 상황에 적합하다.</li> <li>해시 테이블은 데이터의 정렬 상태를 유지할 수 없으므로 정렬된 데이터나 범위 탐색이 필요한 상황에는 적합하지 않다.</li> <li>해시 함수와 해시 충돌 처리 전략에 크게 의존하므로 성능이 저하될 위험이 비교적 크다.</li> <li>충돌을 최대한 줄여 좋은 조회 성능을 내려면 추가 공간이 필요하므로 데이터가 지나치게 많은 경우에는 적합하지 않다.</li> </ul> <p>트리 탐색</p> <ul> <li>트리 노드는 메모리에 분산 저장되므로 대규모 데이터에 적합하다.</li> <li>데이터의 정렬 상태를 유지하거나 범위 탐색을 해야 하는 상황에 적합하다.</li> <li>노드를 계속 추가하고 삭제하다 보면 이진 탐색 트리가 한쪽으로 치우쳐 시간 복잡도가 \\(O(n)\\)으로 저하될 수 있다.</li> <li>AVL 트리나 레드-블랙 트리를 사용하면 각 연산을 안정적으로 \\(O(\\log n)\\)에 수행할 수 있지만, 트리의 균형을 유지하는 데 추가 비용이 든다.</li> </ul>","path":["제 10 장   탐색","10.5   탐색 알고리즘 다시 보기"],"tags":[]},{"location":"chapter_searching/summary/","level":1,"title":"10.6   요약","text":"","path":["제 10 장   탐색","10.6   요약"],"tags":[]},{"location":"chapter_searching/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>이진 탐색은 데이터가 정렬되어 있다는 성질을 이용하여 반복할 때마다 탐색 구간을 절반씩 줄인다. 입력 데이터가 정렬되어 있어야 하며, 배열이나 배열 기반 자료 구조에만 적합하다.</li> <li>완전 탐색은 자료 구조를 순회하여 데이터를 찾는다. 선형 탐색은 배열과 연결 리스트에, 너비 우선 탐색과 깊이 우선 탐색은 그래프와 트리에 적합하다. 범용성이 좋고 전처리가 필요하지 않지만, 시간 복잡도가 \\(O(n)\\)으로 비교적 높다.</li> <li>해시 탐색, 트리 탐색, 이진 탐색은 특정 자료 구조에서 목표 원소를 빠르게 찾는 효율적인 탐색 방법이다. 시간 복잡도가 \\(O(\\log n)\\) 또는 \\(O(1)\\)에 이를 만큼 효율적이지만, 보통 추가 자료 구조가 필요하다.</li> <li>실제로는 데이터 규모, 요구되는 탐색 성능, 데이터 조회와 갱신 빈도 등을 구체적으로 분석하여 적합한 탐색 방법을 골라야 한다.</li> <li>선형 탐색은 작거나 자주 갱신되는 데이터에, 이진 탐색은 크고 정렬된 데이터에 적합하다. 해시 탐색은 높은 조회 효율이 필요하면서 범위 조회는 필요 없는 데이터에, 트리 탐색은 순서 유지와 범위 조회가 필요한 대규모 동적 데이터에 적합하다.</li> <li>선형 탐색을 해시 탐색으로 바꾸는 것은 실행 시간을 최적화하는 흔한 전략으로, 시간 복잡도를 \\(O(n)\\)에서 \\(O(1)\\)로 줄일 수 있다.</li> </ul>","path":["제 10 장   탐색","10.6   요약"],"tags":[]},{"location":"chapter_sorting/","level":1,"title":"제 11 장   정렬","text":"<p>이 장을 시작하며</p> <p>정렬은 혼란을 질서로 바꾸는 마법의 열쇠와 같아서, 데이터를 더 효율적으로 이해하고 처리하게 해 준다.</p> <p>단순한 오름차순이든 복잡한 분류와 배열이든, 정렬은 데이터가 지닌 조화로운 아름다움을 보여 준다.</p>","path":["제 11 장   정렬"],"tags":[]},{"location":"chapter_sorting/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>11.1   정렬 알고리즘</li> <li>11.2   선택 정렬</li> <li>11.3   버블 정렬</li> <li>11.4   삽입 정렬</li> <li>11.5   퀵 정렬</li> <li>11.6   병합 정렬</li> <li>11.7   힙 정렬</li> <li>11.8   버킷 정렬</li> <li>11.9   계수 정렬</li> <li>11.10   기수 정렬</li> <li>11.11   요약</li> <li>11.12   연습 문제</li> </ul>","path":["제 11 장   정렬"],"tags":[]},{"location":"chapter_sorting/bubble_sort/","level":1,"title":"11.3   버블 정렬","text":"<p>버블 정렬(bubble sort)은 인접한 원소를 계속 비교하고 교환하여 정렬한다. 이 과정이 거품이 아래에서 위로 올라오는 모습과 비슷하여 버블 정렬이라는 이름이 붙었다.</p> <p>아래 그림(11-4)과 같이 원소 교환으로 거품이 올라오는 과정을 흉내 낼 수 있다. 배열의 맨 왼쪽부터 오른쪽으로 순회하며 인접한 두 원소를 비교하고, '왼쪽 원소 &gt; 오른쪽 원소'이면 둘을 교환한다. 순회를 마치면 가장 큰 원소가 배열의 맨 오른쪽으로 이동한다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 11-4   원소 교환으로 거품의 이동 구현 </p>","path":["제 11 장   정렬","11.3   버블 정렬"],"tags":[]},{"location":"chapter_sorting/bubble_sort/#1131","level":2,"title":"11.3.1   알고리즘 과정","text":"<p>배열의 길이가 \\(n\\)일 때 버블 정렬의 단계는 아래 그림(11-5)과 같다.</p> <ol> <li>먼저 \\(n\\) 개 원소에 '버블 연산'을 수행하여 배열에서 가장 큰 원소를 올바른 위치로 옮긴다.</li> <li>이어서 남은 \\(n - 1\\) 개 원소에 '버블 연산'을 수행하여 두 번째로 큰 원소를 올바른 위치로 옮긴다.</li> <li>같은 과정을 반복하여 \\(n - 1\\) 회의 '버블 연산'을 수행하면 큰 순서로 \\(n - 1\\) 개 원소가 모두 올바른 위치에 놓인다.</li> <li>남은 원소 하나는 반드시 가장 작은 원소이므로 정렬할 필요가 없으며, 배열 정렬이 완료된다.</li> </ol> <p></p> <p> 그림 11-5   버블 정렬 과정 </p> <p>예시 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby bubble_sort.py<pre><code>def bubble_sort(nums: list[int]):\n    \"\"\"버블 정렬\"\"\"\n    n = len(nums)\n    # 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in range(n - 1, 0, -1):\n        # 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # nums[j]와 nums[j + 1]을 교환한다\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n</code></pre> bubble_sort.cpp<pre><code>/* 버블 정렬 */\nvoid bubbleSort(vector&lt;int&gt; &amp;nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                // 여기서는 std::swap() 함수를 사용한다\n                swap(nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* 버블 정렬 */\nvoid bubbleSort(int[] nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* 버블 정렬 */\nvoid BubbleSort(int[] nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* 버블 정렬 */\nfunc bubbleSort(nums []int) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* 버블 정렬 */\nfunc bubbleSort(nums: inout [Int]) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in nums.indices.dropFirst().reversed() {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                nums.swapAt(j, j + 1)\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* 버블 정렬 */\nfunction bubbleSort(nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* 버블 정렬 */\nfunction bubbleSort(nums: number[]): void {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* 버블 정렬 */\nvoid bubbleSort(List&lt;int&gt; nums) {\n  // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // nums[j]와 nums[j + 1]을 교환한다\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n      }\n    }\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* 버블 정렬 */\nfn bubble_sort(nums: &amp;mut [i32]) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in (1..nums.len()).rev() {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                nums.swap(j, j + 1);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* 버블 정렬 */\nvoid bubbleSort(int nums[], int size) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = size - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* 버블 정렬 */\nfun bubbleSort(nums: IntArray) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (i in nums.size - 1 downTo 1) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>### 버블 정렬 ###\ndef bubble_sort(nums)\n  n = nums.length\n  # 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n  for i in (n - 1).downto(1)\n    # 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # nums[j]와 nums[j + 1]을 교환한다\n        nums[j], nums[j + 1] = nums[j + 1], nums[j]\n      end\n    end\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.3   버블 정렬"],"tags":[]},{"location":"chapter_sorting/bubble_sort/#1132","level":2,"title":"11.3.2   효율 최적화","text":"<p>어느 회의 '버블 연산'에서 교환이 한 번도 일어나지 않았다면 배열은 이미 정렬되어 있으므로 결과를 바로 반환할 수 있다. 따라서 이 상황을 감지하는 플래그 <code>flag</code>를 추가하고, 발생하는 즉시 반환할 수 있다.</p> <p>최적화 후에도 버블 정렬의 최악 및 평균 시간 복잡도는 \\(O(n^2)\\)이다. 하지만 입력 배열이 완전히 정렬되어 있으면 최선 시간 복잡도인 \\(O(n)\\)에 도달한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby bubble_sort.py<pre><code>def bubble_sort_with_flag(nums: list[int]):\n    \"\"\"버블 정렬(플래그 최적화)\"\"\"\n    n = len(nums)\n    # 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in range(n - 1, 0, -1):\n        flag = False  # 플래그를 초기화한다\n        # 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # nums[j]와 nums[j + 1]을 교환한다\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n                flag = True  # 원소가 교환되었음을 기록한다\n        if not flag:\n            break  # 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n</code></pre> bubble_sort.cpp<pre><code>/* 버블 정렬(플래그 최적화)*/\nvoid bubbleSortWithFlag(vector&lt;int&gt; &amp;nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        bool flag = false; // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                // 여기서는 std::swap() 함수를 사용한다\n                swap(nums[j], nums[j + 1]);\n                flag = true; // 원소가 교환되었음을 기록한다\n            }\n        }\n        if (!flag)\n            break; // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* 버블 정렬(플래그 최적화) */\nvoid bubbleSortWithFlag(int[] nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        boolean flag = false; // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // 원소가 교환되었음을 기록한다\n            }\n        }\n        if (!flag)\n            break; // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* 버블 정렬(플래그 최적화)*/\nvoid BubbleSortWithFlag(int[] nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        bool flag = false; // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                flag = true;  // 원소가 교환되었음을 기록한다\n            }\n        }\n        if (!flag) break;     // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* 버블 정렬(플래그 최적화)*/\nfunc bubbleSortWithFlag(nums []int) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        flag := false // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n                flag = true // 원소가 교환되었음을 기록한다\n            }\n        }\n        if flag == false { // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* 버블 정렬(플래그 최적화)*/\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in nums.indices.dropFirst().reversed() {\n        var flag = false // 플래그를 초기화한다\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                nums.swapAt(j, j + 1)\n                flag = true // 원소가 교환되었음을 기록한다\n            }\n        }\n        if !flag { // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* 버블 정렬(플래그 최적화)*/\nfunction bubbleSortWithFlag(nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // 원소가 교환되었음을 기록한다\n            }\n        }\n        if (!flag) break; // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* 버블 정렬(플래그 최적화)*/\nfunction bubbleSortWithFlag(nums: number[]): void {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // 원소가 교환되었음을 기록한다\n            }\n        }\n        if (!flag) break; // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* 버블 정렬(플래그 최적화)*/\nvoid bubbleSortWithFlag(List&lt;int&gt; nums) {\n  // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    bool flag = false; // 플래그를 초기화한다\n    // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // nums[j]와 nums[j + 1]을 교환한다\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        flag = true; // 원소가 교환되었음을 기록한다\n      }\n    }\n    if (!flag) break; // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* 버블 정렬(플래그 최적화) */\nfn bubble_sort_with_flag(nums: &amp;mut [i32]) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in (1..nums.len()).rev() {\n        let mut flag = false; // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                nums.swap(j, j + 1);\n                flag = true; // 원소가 교환되었음을 기록한다\n            }\n        }\n        if !flag {\n            break; // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n        };\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* 버블 정렬(플래그 최적화)*/\nvoid bubbleSortWithFlag(int nums[], int size) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = size - 1; i &gt; 0; i--) {\n        bool flag = false;\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n                flag = true;\n            }\n        }\n        if (!flag)\n            break;\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* 버블 정렬(플래그 최적화) */\nfun bubbleSortWithFlag(nums: IntArray) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (i in nums.size - 1 downTo 1) {\n        var flag = false // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n                flag = true // 원소가 교환되었음을 기록한다\n            }\n        }\n        if (!flag) break // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>### 버블 정렬(플래그 최적화)###\ndef bubble_sort_with_flag(nums)\n  n = nums.length\n  # 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n  for i in (n - 1).downto(1)\n    flag = false # 플래그를 초기화한다\n\n    # 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # nums[j]와 nums[j + 1]을 교환한다\n        nums[j], nums[j + 1] = nums[j + 1], nums[j]\n        flag = true # 원소가 교환되었음을 기록한다\n      end\n    end\n\n    break unless flag # 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.3   버블 정렬"],"tags":[]},{"location":"chapter_sorting/bubble_sort/#1133","level":2,"title":"11.3.3   알고리즘 특성","text":"<ul> <li>시간 복잡도 \\(O(n^2)\\), 적응형 정렬: 각 회의 '버블 연산'에서 순회하는 배열 길이는 차례로 \\(n - 1\\), \\(n - 2\\), \\(\\dots\\), \\(2\\), \\(1\\)이며, 합은 \\((n - 1) n / 2\\)이다. <code>flag</code>를 도입하여 최적화하면 최선 시간 복잡도가 \\(O(n)\\)이 된다.</li> <li>공간 복잡도 \\(O(1)\\), 제자리 정렬: 포인터 \\(i\\)와 \\(j\\)는 상수 크기의 추가 공간을 사용한다.</li> <li>안정 정렬: '버블 연산'에서 값이 같은 원소는 교환하지 않는다.</li> </ul>","path":["제 11 장   정렬","11.3   버블 정렬"],"tags":[]},{"location":"chapter_sorting/bucket_sort/","level":1,"title":"11.8   버킷 정렬","text":"<p>앞에서 살펴본 정렬 알고리즘은 모두 원소의 크기를 비교하여 정렬하는 '비교 기반 정렬 알고리즘'이다. 이 알고리즘들의 최악 시간 복잡도 하한은 \\(\\Omega(n \\log n)\\)이다. 이제 시간 복잡도가 선형에 이를 수 있는 몇 가지 '비교를 사용하지 않는 정렬 알고리즘'을 살펴보자.</p> <p>버킷 정렬(bucket sort)은 분할 정복 전략의 대표적인 응용이다. 크기순으로 배열된 여러 버킷을 만들고 각 버킷에 데이터 범위를 대응시킨 뒤, 데이터를 각 버킷에 고르게 분배한다. 이어서 버킷별로 내부 정렬을 수행하고, 마지막으로 버킷 순서대로 모든 데이터를 합친다.</p>","path":["제 11 장   정렬","11.8   버킷 정렬"],"tags":[]},{"location":"chapter_sorting/bucket_sort/#1181","level":2,"title":"11.8.1   알고리즘 과정","text":"<p>길이가 \\(n\\)이고 원소가 범위 \\([0, 1)\\)의 부동 소수점 수인 배열을 생각해 보자. 버킷 정렬의 과정은 아래 그림(11-13)과 같다.</p> <ol> <li>버킷 \\(k\\) 개를 초기화하고 원소 \\(n\\) 개를 이 \\(k\\) 개의 버킷에 분배한다.</li> <li>각 버킷을 따로 정렬한다(여기서는 프로그래밍 언어의 내장 정렬 함수를 사용한다).</li> <li>작은 버킷부터 큰 버킷 순서로 결과를 합친다.</li> </ol> <p></p> <p> 그림 11-13   버킷 정렬 알고리즘 과정 </p> <p>코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby bucket_sort.py<pre><code>def bucket_sort(nums: list[float]):\n    \"\"\"버킷 정렬\"\"\"\n    # 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    k = len(nums) // 2\n    buckets = [[] for _ in range(k)]\n    # 1. 배열 원소를 각 버킷에 배정한다\n    for num in nums:\n        # 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        i = int(num * k)\n        # num을 버킷 i에 추가한다\n        buckets[i].append(num)\n    # 2. 각 버킷을 정렬한다\n    for bucket in buckets:\n        # 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        bucket.sort()\n    # 3. 버킷을 순회하여 결과를 합친다\n    i = 0\n    for bucket in buckets:\n        for num in bucket:\n            nums[i] = num\n            i += 1\n</code></pre> bucket_sort.cpp<pre><code>/* 버킷 정렬 */\nvoid bucketSort(vector&lt;float&gt; &amp;nums) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    int k = nums.size() / 2;\n    vector&lt;vector&lt;float&gt;&gt; buckets(k);\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for (float num : nums) {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        int i = num * k;\n        // num을 버킷 bucket_idx에 추가한다\n        buckets[i].push_back(num);\n    }\n    // 2. 각 버킷을 정렬한다\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        sort(bucket.begin(), bucket.end());\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    int i = 0;\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.java<pre><code>/* 버킷 정렬 */\nvoid bucketSort(float[] nums) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    int k = nums.length / 2;\n    List&lt;List&lt;Float&gt;&gt; buckets = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; k; i++) {\n        buckets.add(new ArrayList&lt;&gt;());\n    }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for (float num : nums) {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        int i = (int) (num * k);\n        // num을 버킷 i에 추가한다\n        buckets.get(i).add(num);\n    }\n    // 2. 각 버킷을 정렬한다\n    for (List&lt;Float&gt; bucket : buckets) {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        Collections.sort(bucket);\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    int i = 0;\n    for (List&lt;Float&gt; bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.cs<pre><code>/* 버킷 정렬 */\nvoid BucketSort(float[] nums) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    int k = nums.Length / 2;\n    List&lt;List&lt;float&gt;&gt; buckets = [];\n    for (int i = 0; i &lt; k; i++) {\n        buckets.Add([]);\n    }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    foreach (float num in nums) {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        int i = (int)(num * k);\n        // num을 버킷 i에 추가한다\n        buckets[i].Add(num);\n    }\n    // 2. 각 버킷을 정렬한다\n    foreach (List&lt;float&gt; bucket in buckets) {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        bucket.Sort();\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    int j = 0;\n    foreach (List&lt;float&gt; bucket in buckets) {\n        foreach (float num in bucket) {\n            nums[j++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.go<pre><code>/* 버킷 정렬 */\nfunc bucketSort(nums []float64) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    k := len(nums) / 2\n    buckets := make([][]float64, k)\n    for i := 0; i &lt; k; i++ {\n        buckets[i] = make([]float64, 0)\n    }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for _, num := range nums {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        i := int(num * float64(k))\n        // num을 버킷 i에 추가한다\n        buckets[i] = append(buckets[i], num)\n    }\n    // 2. 각 버킷을 정렬한다\n    for i := 0; i &lt; k; i++ {\n        // 내장 슬라이스 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        sort.Float64s(buckets[i])\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    i := 0\n    for _, bucket := range buckets {\n        for _, num := range bucket {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> bucket_sort.swift<pre><code>/* 버킷 정렬 */\nfunc bucketSort(nums: inout [Double]) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    let k = nums.count / 2\n    var buckets = (0 ..&lt; k).map { _ in [Double]() }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for num in nums {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        let i = Int(num * Double(k))\n        // num을 버킷 i에 추가한다\n        buckets[i].append(num)\n    }\n    // 2. 각 버킷을 정렬한다\n    for i in buckets.indices {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        buckets[i].sort()\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    var i = nums.startIndex\n    for bucket in buckets {\n        for num in bucket {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> bucket_sort.js<pre><code>/* 버킷 정렬 */\nfunction bucketSort(nums) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    const k = nums.length / 2;\n    const buckets = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for (const num of nums) {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        const i = Math.floor(num * k);\n        // num을 버킷 i에 추가한다\n        buckets[i].push(num);\n    }\n    // 2. 각 버킷을 정렬한다\n    for (const bucket of buckets) {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.ts<pre><code>/* 버킷 정렬 */\nfunction bucketSort(nums: number[]): void {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    const k = nums.length / 2;\n    const buckets: number[][] = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for (const num of nums) {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        const i = Math.floor(num * k);\n        // num을 버킷 i에 추가한다\n        buckets[i].push(num);\n    }\n    // 2. 각 버킷을 정렬한다\n    for (const bucket of buckets) {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.dart<pre><code>/* 버킷 정렬 */\nvoid bucketSort(List&lt;double&gt; nums) {\n  // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n  int k = nums.length ~/ 2;\n  List&lt;List&lt;double&gt;&gt; buckets = List.generate(k, (index) =&gt; []);\n\n  // 1. 배열 원소를 각 버킷에 배정한다\n  for (double _num in nums) {\n    // 입력 데이터의 범위는 [0, 1)이며, _num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n    int i = (_num * k).toInt();\n    // _num을 버킷 bucket_idx에 추가한다\n    buckets[i].add(_num);\n  }\n  // 2. 각 버킷을 정렬한다\n  for (List&lt;double&gt; bucket in buckets) {\n    bucket.sort();\n  }\n  // 3. 버킷을 순회하여 결과를 합친다\n  int i = 0;\n  for (List&lt;double&gt; bucket in buckets) {\n    for (double _num in bucket) {\n      nums[i++] = _num;\n    }\n  }\n}\n</code></pre> bucket_sort.rs<pre><code>/* 버킷 정렬 */\nfn bucket_sort(nums: &amp;mut [f64]) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    let k = nums.len() / 2;\n    let mut buckets = vec![vec![]; k];\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for &amp;num in nums.iter() {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        let i = (num * k as f64) as usize;\n        // num을 버킷 i에 추가한다\n        buckets[i].push(num);\n    }\n    // 2. 각 버킷을 정렬한다\n    for bucket in &amp;mut buckets {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        bucket.sort_by(|a, b| a.partial_cmp(b).unwrap());\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    let mut i = 0;\n    for bucket in buckets.iter() {\n        for &amp;num in bucket.iter() {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> bucket_sort.c<pre><code>/* 버킷 정렬 */\nvoid bucketSort(float nums[], int n) {\n    int k = n / 2;                                 // k = n/2개의 버킷을 초기화한다\n    int *sizes = malloc(k * sizeof(int));          // 각 버킷의 크기를 기록한다\n    float **buckets = malloc(k * sizeof(float *)); // 동적 배열의 배열(버킷)\n    // 각 버킷에 충분한 공간을 미리 할당한다\n    for (int i = 0; i &lt; k; ++i) {\n        buckets[i] = (float *)malloc(n * sizeof(float));\n        sizes[i] = 0;\n    }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for (int i = 0; i &lt; n; ++i) {\n        int idx = (int)(nums[i] * k);\n        buckets[idx][sizes[idx]++] = nums[i];\n    }\n    // 2. 각 버킷을 정렬한다\n    for (int i = 0; i &lt; k; ++i) {\n        qsort(buckets[i], sizes[i], sizeof(float), compare);\n    }\n    // 3. 정렬한 버킷을 병합한다\n    int idx = 0;\n    for (int i = 0; i &lt; k; ++i) {\n        for (int j = 0; j &lt; sizes[i]; ++j) {\n            nums[idx++] = buckets[i][j];\n        }\n        // 메모리 해제\n        free(buckets[i]);\n    }\n}\n</code></pre> bucket_sort.kt<pre><code>/* 버킷 정렬 */\nfun bucketSort(nums: FloatArray) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    val k = nums.size / 2\n    val buckets = mutableListOf&lt;MutableList&lt;Float&gt;&gt;()\n    for (i in 0..&lt;k) {\n        buckets.add(mutableListOf())\n    }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for (num in nums) {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        val i = (num * k).toInt()\n        // num을 버킷 i에 추가한다\n        buckets[i].add(num)\n    }\n    // 2. 각 버킷을 정렬한다\n    for (bucket in buckets) {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        bucket.sort()\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    var i = 0\n    for (bucket in buckets) {\n        for (num in bucket) {\n            nums[i++] = num\n        }\n    }\n}\n</code></pre> bucket_sort.rb<pre><code>### 버킷 정렬 ###\ndef bucket_sort(nums)\n  # 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n  k = nums.length / 2\n  buckets = Array.new(k) { [] }\n\n  # 1. 배열 원소를 각 버킷에 배정한다\n  nums.each do |num|\n    # 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n    i = (num * k).to_i\n    # num을 버킷 i에 추가한다\n    buckets[i] &lt;&lt; num\n  end\n\n  # 2. 각 버킷을 정렬한다\n  buckets.each do |bucket|\n    # 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n    bucket.sort!\n  end\n\n  # 3. 버킷을 순회하여 결과를 합친다\n  i = 0\n  buckets.each do |bucket|\n    bucket.each do |num|\n      nums[i] = num\n      i += 1\n    end\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.8   버킷 정렬"],"tags":[]},{"location":"chapter_sorting/bucket_sort/#1182","level":2,"title":"11.8.2   알고리즘 특성","text":"<p>버킷 정렬은 매우 큰 데이터를 처리하는 데 적합하다. 예를 들어 입력에 원소가 100만 개 있고 공간 제약 때문에 시스템 메모리에 모든 데이터를 한 번에 올릴 수 없다고 하자. 이때 데이터를 1000개 버킷으로 나누어 각각 정렬한 다음 결과를 합칠 수 있다.</p> <ul> <li>시간 복잡도 \\(O(n + k)\\): 원소가 각 버킷에 고르게 분포한다면 버킷당 원소 수는 \\(\\frac{n}{k}\\)이다. 버킷 하나를 정렬하는 데 \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) 시간이 걸린다고 하면 모든 버킷의 정렬에는 \\(O(n \\log\\frac{n}{k})\\) 시간이 걸린다. 버킷 수 \\(k\\)가 충분히 크면 시간 복잡도는 \\(O(n)\\)에 가까워진다. 결과를 합칠 때는 모든 버킷과 원소를 순회하므로 \\(O(n + k)\\) 시간이 든다. 최악의 경우 모든 데이터가 하나의 버킷에 들어가며, 그 버킷을 정렬하는 데 \\(O(n^2)\\) 시간이 걸린다.</li> <li>공간 복잡도 \\(O(n + k)\\), 제자리 정렬이 아님: 버킷 \\(k\\) 개와 총 \\(n\\) 개의 원소를 위한 추가 공간이 필요하다.</li> <li>버킷 정렬의 안정성은 버킷 내부를 정렬하는 알고리즘의 안정성에 달려 있다.</li> </ul>","path":["제 11 장   정렬","11.8   버킷 정렬"],"tags":[]},{"location":"chapter_sorting/bucket_sort/#1183","level":2,"title":"11.8.3   고르게 분배하는 방법","text":"<p>버킷 정렬의 시간 복잡도는 이론적으로 \\(O(n)\\)에 이를 수 있다. 핵심은 원소를 각 버킷에 고르게 분배하는 것인데, 실제 데이터는 균일하게 분포하지 않는 경우가 많다. 예를 들어 타오바오의 모든 상품을 가격 범위에 따라 10개 버킷에 고르게 나누고 싶다고 하자. 상품 가격은 불균일하여 100위안 미만의 상품은 매우 많고 1000위안 초과 상품은 매우 적다. 가격 구간을 같은 폭으로 10등분하면 버킷별 상품 수에 큰 차이가 생긴다.</p> <p>고르게 분배하려면 먼저 대략적인 경계를 정해 데이터를 버킷 3개로 나눌 수 있다. 분배한 뒤 상품이 많은 버킷을 다시 3개로 나누고, 모든 버킷의 원소 수가 비슷해질 때까지 반복한다.</p> <p>아래 그림(11-14)과 같이 이 방법은 본질적으로 재귀 트리를 만드는 과정이며, 리프 노드의 값을 가능한 한 고르게 만드는 것이 목표다. 반드시 매번 버킷 3개로 나눌 필요는 없으며, 데이터 특성에 맞추어 분할 방식을 유연하게 선택할 수 있다.</p> <p></p> <p> 그림 11-14   버킷을 재귀적으로 분할 </p> <p>상품 가격의 확률 분포를 미리 알고 있다면 그 분포에 따라 각 버킷의 가격 경계를 설정할 수 있다. 데이터 분포를 반드시 따로 집계할 필요는 없으며, 데이터 특성에 따라 적절한 확률 모형으로 근사할 수도 있다.</p> <p>아래 그림(11-15)에서는 상품 가격이 정규 분포를 따른다고 가정한다. 이를 바탕으로 가격 구간을 적절히 정하면 상품을 각 버킷에 고르게 분배할 수 있다.</p> <p></p> <p> 그림 11-15   확률 분포에 따른 버킷 분할 </p>","path":["제 11 장   정렬","11.8   버킷 정렬"],"tags":[]},{"location":"chapter_sorting/counting_sort/","level":1,"title":"11.9   계수 정렬","text":"<p>계수 정렬(counting sort)은 원소의 개수를 세어 정렬하며, 주로 정수 배열에 사용한다.</p>","path":["제 11 장   정렬","11.9   계수 정렬"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1191","level":2,"title":"11.9.1   간단한 구현","text":"<p>먼저 간단한 예를 보자. 길이가 \\(n\\)인 배열 <code>nums</code>의 모든 원소가 '음이 아닌 정수'라고 하자. 계수 정렬의 전체 과정은 아래 그림(11-16)과 같다.</p> <ol> <li>배열을 순회하여 가장 큰 수 \\(m\\)을 찾고, 길이가 \\(m + 1\\)인 보조 배열 <code>counter</code>를 만든다.</li> <li><code>counter</code>를 이용하여 <code>nums</code>에서 각 수가 나타나는 횟수를 센다. <code>counter[num]</code>은 수 <code>num</code>의 등장 횟수다. 세는 방법은 간단하다. <code>nums</code>를 순회하면서 현재 수를 <code>num</code>이라고 할 때 <code>counter[num]</code>을 \\(1\\)씩 늘리면 된다.</li> <li><code>counter</code>의 인덱스는 본래 순서대로 배열되어 있으므로, 모든 수가 이미 정렬된 셈이다. 이어서 <code>counter</code>를 순회하며 각 수의 등장 횟수만큼 작은 수부터 큰 수 순서로 <code>nums</code>에 채우면 된다.</li> </ol> <p></p> <p> 그림 11-16   계수 정렬 과정 </p> <p>코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby counting_sort.py<pre><code>def counting_sort_naive(nums: list[int]):\n    \"\"\"계수 정렬\"\"\"\n    # 간단한 구현이며, 객체 정렬에는 사용할 수 없다\n    # 1. 배열의 최댓값 m을 구한다\n    m = max(nums)\n    # 2. 각 숫자의 등장 횟수를 센다\n    # counter[num]은 num의 등장 횟수를 나타낸다\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    i = 0\n    for num in range(m + 1):\n        for _ in range(counter[num]):\n            nums[i] = num\n            i += 1\n</code></pre> counting_sort.cpp<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nvoid countingSortNaive(vector&lt;int&gt; &amp;nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.java<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nvoid countingSortNaive(int[] nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nvoid CountingSortNaive(int[] nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nfunc countingSortNaive(nums []int) {\n    // 1. 배열의 최댓값 m을 구한다\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    for i, num := 0, 0; num &lt; m+1; num++ {\n        for j := 0; j &lt; counter[num]; j++ {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.swift<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nfunc countingSortNaive(nums: inout [Int]) {\n    // 1. 배열의 최댓값 m을 구한다\n    let m = nums.max()!\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    var i = 0\n    for num in 0 ..&lt; m + 1 {\n        for _ in 0 ..&lt; counter[num] {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nfunction countingSortNaive(nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    let m = Math.max(...nums);\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nfunction countingSortNaive(nums: number[]): void {\n    // 1. 배열의 최댓값 m을 구한다\n    let m: number = Math.max(...nums);\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nvoid countingSortNaive(List&lt;int&gt; nums) {\n  // 1. 배열의 최댓값 m을 구한다\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. 각 숫자의 등장 횟수를 센다\n  // counter[_num]은 _num의 등장 횟수를 나타낸다\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n  int i = 0;\n  for (int _num = 0; _num &lt; m + 1; _num++) {\n    for (int j = 0; j &lt; counter[_num]; j++, i++) {\n      nums[i] = _num;\n    }\n  }\n}\n</code></pre> counting_sort.rs<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nfn counting_sort_naive(nums: &amp;mut [i32]) {\n    // 1. 배열의 최댓값 m을 구한다\n    let m = *nums.iter().max().unwrap();\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    let mut counter = vec![0; m as usize + 1];\n    for &amp;num in nums.iter() {\n        counter[num as usize] += 1;\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    let mut i = 0;\n    for num in 0..m + 1 {\n        for _ in 0..counter[num as usize] {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> counting_sort.c<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nvoid countingSortNaive(int nums[], int size) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    int *counter = calloc(m + 1, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n    // 4. 메모리를 해제한다\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nfun countingSortNaive(nums: IntArray) {\n    // 1. 배열의 최댓값 m을 구한다\n    var m = 0\n    for (num in nums) {\n        m = max(m, num)\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    var i = 0\n    for (num in 0..&lt;m + 1) {\n        var j = 0\n        while (j &lt; counter[num]) {\n            nums[i] = num\n            j++\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.rb<pre><code>### 계수 정렬 ###\ndef counting_sort_naive(nums)\n  # 간단한 구현이며, 객체 정렬에는 사용할 수 없다\n  # 1. 배열의 최댓값 m을 구한다\n  m = 0\n  nums.each { |num| m = [m, num].max }\n  # 2. 각 숫자의 등장 횟수를 센다\n  # counter[num]은 num의 등장 횟수를 나타낸다\n  counter = Array.new(m + 1, 0)\n  nums.each { |num| counter[num] += 1 }\n  # 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n  i = 0\n  for num in 0...(m + 1)\n    (0...counter[num]).each do\n      nums[i] = num\n      i += 1\n    end\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>계수 정렬과 버킷 정렬의 관계</p> <p>버킷 정렬의 관점에서는 계수 배열 <code>counter</code>의 각 인덱스를 하나의 버킷으로, 개수를 세는 과정을 원소를 해당 버킷에 분배하는 과정으로 볼 수 있다. 본질적으로 계수 정렬은 정수 데이터에 적용되는 버킷 정렬의 특수한 경우다.</p>","path":["제 11 장   정렬","11.9   계수 정렬"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1192","level":2,"title":"11.9.2   완전한 구현","text":"<p>주의 깊게 읽었다면 입력 데이터가 객체일 때 위의 <code>3.</code> 단계가 작동하지 않는다는 것을 알아차렸을 것이다. 입력이 상품 객체이고 상품 가격(클래스의 멤버 변수)에 따라 정렬하려 한다고 하자. 위 알고리즘으로는 가격의 정렬 결과만 얻을 수 있다.</p> <p>그렇다면 원래 데이터의 정렬 결과는 어떻게 얻을까? 먼저 <code>counter</code>의 '접두 합'을 계산한다. 이름 그대로 인덱스 <code>i</code>의 접두 합 <code>prefix[i]</code>는 배열의 앞 <code>i</code> 개 원소를 더한 값이다.</p> \\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\] <p>접두 합은 명확한 의미를 가진다. <code>prefix[num] - 1</code>은 결과 배열 <code>res</code>에서 원소 <code>num</code>이 마지막으로 나타나는 인덱스다. 이는 각 원소가 결과 배열의 어디에 놓여야 하는지 알려 주는 핵심 정보다. 이어서 원래 배열 <code>nums</code>의 각 원소 <code>num</code>을 역순으로 순회하며 매번 다음 두 단계를 수행한다.</p> <ol> <li>배열 <code>res</code>의 인덱스 <code>prefix[num] - 1</code>에 <code>num</code>을 넣는다.</li> <li>접두 합 <code>prefix[num]</code>을 \\(1\\) 줄여 다음에 <code>num</code>을 넣을 인덱스를 얻는다.</li> </ol> <p>순회를 마치면 배열 <code>res</code>에 정렬된 결과가 들어 있다. 마지막으로 <code>res</code>를 원래 배열 <code>nums</code>에 덮어쓰면 된다. 아래 그림(11-17)은 계수 정렬의 전체 과정을 보여 준다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 11-17   계수 정렬 단계 </p> <p>계수 정렬의 구현 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby counting_sort.py<pre><code>def counting_sort(nums: list[int]):\n    \"\"\"계수 정렬\"\"\"\n    # 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\n    # 1. 배열의 최댓값 m을 구한다\n    m = max(nums)\n    # 2. 각 숫자의 등장 횟수를 센다\n    # counter[num]은 num의 등장 횟수를 나타낸다\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    # 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for i in range(m):\n        counter[i + 1] += counter[i]\n    # 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    # 결과를 기록할 배열 res를 초기화한다\n    n = len(nums)\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        num = nums[i]\n        res[counter[num] - 1] = num  # num을 해당 인덱스에 배치한다\n        counter[num] -= 1  # 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    # 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    for i in range(n):\n        nums[i] = res[i]\n</code></pre> counting_sort.cpp<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nvoid countingSort(vector&lt;int&gt; &amp;nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    int n = nums.size();\n    vector&lt;int&gt; res(n);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // num을 해당 인덱스에 배치한다\n        counter[num]--;              // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    nums = res;\n}\n</code></pre> counting_sort.java<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nvoid countingSort(int[] nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    int n = nums.length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // num을 해당 인덱스에 배치한다\n        counter[num]--; // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nvoid CountingSort(int[] nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    int n = nums.Length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // num을 해당 인덱스에 배치한다\n        counter[num]--; // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nfunc countingSort(nums []int) {\n    // 1. 배열의 최댓값 m을 구한다\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for i := 0; i &lt; m; i++ {\n        counter[i+1] += counter[i]\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    n := len(nums)\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        num := nums[i]\n        // num을 해당 인덱스에 배치한다\n        res[counter[num]-1] = num\n        // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n        counter[num]--\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    copy(nums, res)\n}\n</code></pre> counting_sort.swift<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nfunc countingSort(nums: inout [Int]) {\n    // 1. 배열의 최댓값 m을 구한다\n    let m = nums.max()!\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for i in 0 ..&lt; m {\n        counter[i + 1] += counter[i]\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let num = nums[i]\n        res[counter[num] - 1] = num // num을 해당 인덱스에 배치한다\n        counter[num] -= 1 // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nfunction countingSort(nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    let m = Math.max(...nums);\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    const n = nums.length;\n    const res = new Array(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // num을 해당 인덱스에 배치한다\n        counter[num]--; // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nfunction countingSort(nums: number[]): void {\n    // 1. 배열의 최댓값 m을 구한다\n    let m: number = Math.max(...nums);\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    const n = nums.length;\n    const res: number[] = new Array&lt;number&gt;(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // num을 해당 인덱스에 배치한다\n        counter[num]--; // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nvoid countingSort(List&lt;int&gt; nums) {\n  // 1. 배열의 최댓값 m을 구한다\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. 각 숫자의 등장 횟수를 센다\n  // counter[_num]은 _num의 등장 횟수를 나타낸다\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n  // 즉, counter[_num]-1은 res에서 _num이 마지막으로 등장하는 인덱스이다\n  for (int i = 0; i &lt; m; i++) {\n    counter[i + 1] += counter[i];\n  }\n  // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n  // 결과를 기록할 배열 res를 초기화한다\n  int n = nums.length;\n  List&lt;int&gt; res = List.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int _num = nums[i];\n    res[counter[_num] - 1] = _num; // _num을 해당 인덱스에 배치한다\n    counter[_num]--; // 접두 합을 1 감소시켜 다음에 _num을 배치할 인덱스를 구한다\n  }\n  // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n  nums.setAll(0, res);\n}\n</code></pre> counting_sort.rs<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nfn counting_sort(nums: &amp;mut [i32]) {\n    // 1. 배열의 최댓값 m을 구한다\n    let m = *nums.iter().max().unwrap() as usize;\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    let mut counter = vec![0; m + 1];\n    for &amp;num in nums.iter() {\n        counter[num as usize] += 1;\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for i in 0..m {\n        counter[i + 1] += counter[i];\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    let n = nums.len();\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let num = nums[i];\n        res[counter[num as usize] - 1] = num; // num을 해당 인덱스에 배치한다\n        counter[num as usize] -= 1; // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    nums.copy_from_slice(&amp;res)\n}\n</code></pre> counting_sort.c<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nvoid countingSort(int nums[], int size) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    int *counter = calloc(m, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    int *res = malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // num을 해당 인덱스에 배치한다\n        counter[num]--;              // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    memcpy(nums, res, size * sizeof(int));\n    // 5. 메모리를 해제한다\n    free(res);\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nfun countingSort(nums: IntArray) {\n    // 1. 배열의 최댓값 m을 구한다\n    var m = 0\n    for (num in nums) {\n        m = max(m, num)\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for (i in 0..&lt;m) {\n        counter[i + 1] += counter[i]\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    val n = nums.size\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val num = nums[i]\n        res[counter[num] - 1] = num // num을 해당 인덱스에 배치한다\n        counter[num]-- // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    for (i in 0..&lt;n) {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.rb<pre><code>### 계수 정렬 ###\ndef counting_sort(nums)\n  # 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\n  # 1. 배열의 최댓값 m을 구한다\n  m = nums.max\n  # 2. 각 숫자의 등장 횟수를 센다\n  # counter[num]은 num의 등장 횟수를 나타낸다\n  counter = Array.new(m + 1, 0)\n  nums.each { |num| counter[num] += 1 }\n  # 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n  # 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n  (0...m).each { |i| counter[i + 1] += counter[i] }\n  # 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n  # 결과를 기록할 배열 res를 초기화한다\n  n = nums.length\n  res = Array.new(n, 0)\n  (n - 1).downto(0).each do |i|\n    num = nums[i]\n    res[counter[num] - 1] = num # num을 해당 인덱스에 배치한다\n    counter[num] -= 1 # 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n  end\n  # 결과 배열 res로 원래 배열 nums를 덮어쓴다\n  (0...n).each { |i| nums[i] = res[i] }\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.9   계수 정렬"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1193","level":2,"title":"11.9.3   알고리즘 특성","text":"<ul> <li>시간 복잡도 \\(O(n + m)\\), 비적응형 정렬: <code>nums</code>와 <code>counter</code>를 순회하며, 각각 선형 시간이 걸린다. 보통 \\(n \\gg m\\)이므로 시간 복잡도는 \\(O(n)\\)에 가까워진다.</li> <li>공간 복잡도 \\(O(n + m)\\), 제자리 정렬이 아님: 길이가 각각 \\(n\\)과 \\(m\\)인 배열 <code>res</code>와 <code>counter</code>를 사용한다.</li> <li>안정 정렬: <code>res</code>에는 '오른쪽에서 왼쪽으로' 원소를 채우므로, <code>nums</code>를 역순으로 순회하면 값이 같은 원소의 상대적 위치가 바뀌지 않아 안정 정렬이 된다. 정방향으로 순회해도 정렬 결과 자체는 올바르지만 안정적이지 않다.</li> </ul>","path":["제 11 장   정렬","11.9   계수 정렬"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1194","level":2,"title":"11.9.4   한계","text":"<p>여기까지 보면 계수 정렬이 매우 기발하다고 느낄 수 있다. 개수만 세어도 효율적으로 정렬할 수 있기 때문이다. 하지만 계수 정렬을 사용하기 위한 전제 조건은 비교적 엄격하다.</p> <p>계수 정렬은 음이 아닌 정수에만 적합하다. 다른 타입에 적용하려면 해당 데이터를 음이 아닌 정수로 바꿀 수 있어야 하며, 변환 과정에서 원소 간 대소 관계가 바뀌면 안 된다. 예를 들어 음수가 포함된 정수 배열은 모든 수에 일정한 상수를 더해 양수로 바꾼 뒤, 정렬 후 다시 원래 값으로 돌릴 수 있다.</p> <p>계수 정렬은 데이터 수는 많지만 값의 범위는 작은 경우에 적합하다. 위 예시에서 \\(m\\)이 너무 크면 공간을 지나치게 많이 차지한다. 또한 \\(n \\ll m\\)이면 계수 정렬에 \\(O(m)\\) 시간이 걸리므로 \\(O(n \\log n)\\) 정렬 알고리즘보다 느릴 수도 있다.</p>","path":["제 11 장   정렬","11.9   계수 정렬"],"tags":[]},{"location":"chapter_sorting/exercises/","level":1,"title":"11.12   연습 문제","text":"","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/exercises/#11121","level":2,"title":"11.12.1   개념 확인","text":"","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/exercises/#1","level":3,"title":"1.   선택 정렬과 버블 정렬의 처음 몇 회전","text":"<p>배열 <code>[4, 2, 5, 1, 3]</code>이 주어진다. 다음은 모두 오름차순으로 정렬한다.</p> <ol> <li>선택 정렬의 첫 두 회전을 따라가며 각 회전이 끝난 뒤의 배열을 적고, 어느 위치가 확정되었는지 설명하라.</li> <li>버블 정렬의 첫 회전을 따라가며 배열 상태와 교환 횟수를 적고, 어느 위치가 확정되었는지 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>첫 두 회전은 다음과 같다.</p> 회전 배열 상태 설명 1 <code>[1, 2, 5, 4, 3]</code> 가장 작은 원소 1을 첫 번째 원소와 교환 2 <code>[1, 2, 5, 4, 3]</code> 값 2가 이미 인덱스 1에 있으므로 교환할 필요 없음 <p>이때 앞의 두 위치는 확정되었으므로, 이후에는 <code>[5, 4, 3]</code>에서 계속 최솟값을 선택하면 된다.</p> </li> <li> <p>이웃한 원소를 차례대로 비교한다. 4와 2는 교환하고, 4와 5는 교환하지 않으며, 5와 1을 교환한 뒤 5와 3을 교환한다.     결과는 <code>[2, 4, 1, 3, 5]</code>이고 총 3번 교환한다. 가장 큰 원소 5가 배열 끝으로 이동했으므로 마지막 위치가 확정되었다.</p> </li> </ol>","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/exercises/#2","level":3,"title":"2.   값이 같은 원소의 순서가 달라지는가","text":"<p>배열 \\([2_a, 2_b, 1]\\)에서 \\(2_a\\)와 \\(2_b\\)는 값이 같지만, 아래 첨자로 원래 순서를 표시했다.</p> <ol> <li>선택 정렬의 첫 회전이 끝난 뒤의 배열을 적어 보자. \\(2_a\\)와 \\(2_b\\)의 상대적인 순서가 바뀌었는가?</li> <li>버블 정렬의 첫 회전이 끝난 뒤의 배열을 적어 보자. \\(2_a\\)와 \\(2_b\\)의 상대적인 순서가 바뀌었는가?</li> <li>앞의 두 결과를 바탕으로, 값이 같은 원소의 원래 순서를 유지하는 측면에서 두 정렬이 어떻게 다른지 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>선택 정렬의 첫 회전에서는 가장 작은 원소 1을 골라 첫 원소 \\(2_a\\)와 교환한다. 결과는     \\([1, 2_b, 2_a]\\)다. \\(2_a\\)가 \\(2_b\\) 뒤로 옮겨졌으므로 두 원소의 상대적인 순서가 바뀌었다.</p> </li> <li> <p>버블 정렬은 먼저 \\(2_a\\)와 \\(2_b\\)를 비교한다. 두 값이 같으므로 교환하지 않는다. 이어서 \\(2_b\\)와 1을 비교하여 교환한다.     첫 회전이 끝나면 \\([2_a, 1, 2_b]\\)가 된다. \\(2_a\\)가 여전히 \\(2_b\\) 앞에 있으므로 상대적인 순서는 바뀌지 않았다.</p> </li> <li> <p>이 예에서 선택 정렬은 값이 같은 원소의 원래 순서를 바꾼다. 버블 정렬은 왼쪽 원소가 오른쪽 원소보다 클 때만     이웃한 원소를 교환한다. 값이 같은 원소끼리는 교환하지 않으므로 원래의 앞뒤 순서를 유지할 수 있다.</p> </li> </ol>","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/exercises/#3","level":3,"title":"3.   계수 정렬과 기수 정렬 비교하기","text":"<p>학교에서 자릿수가 8자리로 고정된 학번을 대량으로 정렬하려 한다. 다음에 답하라.</p> <ol> <li>가장 낮은 자리부터 처리하는 기수 정렬은 총 몇 회전이 필요한가?</li> <li>학번을 그대로 정수로 취급하여 계수 정렬을 하면, 사용하지 않는 계수 배열의 자리를 많이 마련하게 되는 이유는 무엇인가?</li> <li>앞의 두 결과를 바탕으로, 8자리로 고정된 학번을 대량으로 처리하려면 계수 정렬과 기수 정렬 중 어느 것을 선택하겠는가? 이유도 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>학번은 8자리이므로 가장 낮은 자리부터 가장 높은 자리까지 총 8회전이 필요하다. 매 회전에서는 0~9에 따라 그룹을 나누기만 하면 된다.</p> </li> <li> <p>학번 전체의 출현 횟수를 직접 세려면 가능한 모든 8자리 값에 대해 자리를 마련해야 한다. 하지만 실제 학생이 사용하는 학번은 그중 일부에 불과하므로,     계수 배열의 대부분은 계속 0으로 남는다.</p> </li> <li> <p>기수 정렬을 선택해야 한다. ‘자릿수가 고정되어 있고, 각 자리에 올 수 있는 값은 10가지뿐’이라는 특징을 이용하면 안정적인 그룹 나누기를 8회 반복하는 것으로 충분하다.     계수 정렬에서 8자리 학번 전체를 정수 인덱스로 사용하면 실제로는 나타나지 않을 수많은 값에 대해서도 계수 배열의 자리를 마련해야 한다.</p> </li> </ol>","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/exercises/#11122","level":2,"title":"11.12.2   프로그래밍 연습","text":"","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/exercises/#1_1","level":3,"title":"1.   병합 정렬로 배열 정렬하기","text":"<p>정수 배열 <code>nums</code>가 주어진다. 병합 정렬을 직접 구현하여 원소를 비내림차순으로 정렬한 뒤 반환하라. 언어에서 제공하는 정렬 함수를 호출하지 마라.</p> 풀이 힌트 <ol> <li>구간 길이가 1 이하면 이미 정렬되어 있다.</li> <li>중간 지점에서 구간을 둘로 나누고 각각 재귀적으로 정렬한다.</li> <li>두 포인터로 정렬된 두 구간을 병합한 뒤, 결과를 원래 배열에 다시 쓴다.</li> </ol> <p>LeetCode</p>","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/exercises/#2_1","level":3,"title":"2.   계수 정렬로 정수 배열 정렬하기","text":"<p>정수 배열 <code>nums</code>와 음이 아닌 정수 \\(K\\)가 주어진다. 배열의 모든 원소는 \\(0\\) 이상 \\(K\\) 이하이다.</p> <p>계수 정렬을 구현하여 비내림차순으로 정렬한 결과를 <code>nums</code>에 다시 쓰고 <code>nums</code>를 반환하라. 원소의 대소 비교로 순서를 결정하거나 언어에서 제공하는 정렬 함수를 호출하지 마라.</p> 풀이 힌트 <ol> <li>모든 원소가 0 이상 K 이하이므로 원소 값을 계수 배열의 인덱스로 바로 사용할 수 있다.</li> <li>먼저 nums를 한 번 순회하면서 해당 위치의 계수를 1씩 늘린다.</li> <li>이어서 계수 배열을 0부터 K까지 순회한다. 값 x가 나타난 횟수만큼 nums에 x를 연속해서 쓴다.</li> </ol>","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/heap_sort/","level":1,"title":"11.7   힙 정렬","text":"<p>참고</p> <p>이 절을 읽기 전에 '힙' 장을 먼저 학습하자.</p> <p>힙 정렬(heap sort)은 힙 자료 구조를 이용한 효율적인 정렬 알고리즘이다. 이미 배운 '힙 구성 연산'과 '힙에서 원소를 꺼내는 연산'으로 구현할 수 있다.</p> <ol> <li>배열을 입력받아 최소 힙을 구성한다. 이때 가장 작은 원소가 힙의 루트에 놓인다.</li> <li>힙에서 원소를 계속 꺼내며 순서대로 기록하면 오름차순으로 정렬된 수열을 얻는다.</li> </ol> <p>위 방법도 가능하지만 꺼낸 원소를 저장할 추가 배열이 필요하여 공간을 낭비한다. 실제로는 보통 더 간결한 구현 방식을 사용한다.</p>","path":["제 11 장   정렬","11.7   힙 정렬"],"tags":[]},{"location":"chapter_sorting/heap_sort/#1171","level":2,"title":"11.7.1   알고리즘 과정","text":"<p>배열의 길이가 \\(n\\)일 때 힙 정렬의 과정은 아래 그림(11-12)과 같다.</p> <ol> <li>배열을 입력받아 최대 힙을 구성한다. 완료하면 가장 큰 원소가 힙의 루트에 놓인다.</li> <li>힙의 루트 원소(첫 번째 원소)와 끝 원소(마지막 원소)를 교환한다. 교환 후 힙의 길이는 \\(1\\) 줄고, 정렬된 원소 수는 \\(1\\) 늘어난다.</li> <li>힙의 루트 원소부터 위에서 아래로 힙 재정렬(sift down)을 수행한다. 재정렬을 마치면 힙의 성질이 복구된다.</li> <li><code>2.</code> 단계와 <code>3.</code> 단계를 반복한다. \\(n - 1\\) 회 반복하면 배열 정렬이 완료된다.</li> </ol> <p>참고</p> <p>힙에서 원소를 꺼내는 연산에도 <code>2.</code> 단계와 <code>3.</code> 단계가 포함되며, 실제로 원소를 꺼내는 단계가 하나 더 있을 뿐이다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 11-12   힙 정렬 단계 </p> <p>구현에서는 '힙' 장과 동일한 하향 힙 재정렬 함수 <code>sift_down()</code>을 사용한다. 가장 큰 원소를 꺼낼 때마다 힙의 길이가 줄어들므로, 현재 유효한 힙 길이를 지정하는 매개변수 \\(n\\)을 <code>sift_down()</code>에 추가해야 한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby heap_sort.py<pre><code>def sift_down(nums: list[int], n: int, i: int):\n    \"\"\"길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬\"\"\"\n    while True:\n        # 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        l = 2 * i + 1\n        r = 2 * i + 2\n        ma = i\n        if l &lt; n and nums[l] &gt; nums[ma]:\n            ma = l\n        if r &lt; n and nums[r] &gt; nums[ma]:\n            ma = r\n        # 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if ma == i:\n            break\n        # 두 노드를 교환한다\n        nums[i], nums[ma] = nums[ma], nums[i]\n        # 아래쪽으로 힙 재정렬을 반복한다\n        i = ma\n\ndef heap_sort(nums: list[int]):\n    \"\"\"힙 정렬\"\"\"\n    # 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for i in range(len(nums) // 2 - 1, -1, -1):\n        sift_down(nums, len(nums), i)\n    # 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for i in range(len(nums) - 1, 0, -1):\n        # 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        nums[0], nums[i] = nums[i], nums[0]\n        # 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        sift_down(nums, i, 0)\n</code></pre> heap_sort.cpp<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nvoid siftDown(vector&lt;int&gt; &amp;nums, int n, int i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i) {\n            break;\n        }\n        // 두 노드를 교환한다\n        swap(nums[i], nums[ma]);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n\n/* 힙 정렬 */\nvoid heapSort(vector&lt;int&gt; &amp;nums) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (int i = nums.size() / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, nums.size(), i);\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for (int i = nums.size() - 1; i &gt; 0; --i) {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        swap(nums[0], nums[i]);\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.java<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nvoid siftDown(int[] nums, int n, int i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i)\n            break;\n        // 두 노드를 교환한다\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n\n/* 힙 정렬 */\nvoid heapSort(int[] nums) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (int i = nums.length / 2 - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.cs<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nvoid SiftDown(int[] nums, int n, int i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i)\n            break;\n        // 두 노드를 교환한다\n        (nums[ma], nums[i]) = (nums[i], nums[ma]);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n\n/* 힙 정렬 */\nvoid HeapSort(int[] nums) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (int i = nums.Length / 2 - 1; i &gt;= 0; i--) {\n        SiftDown(nums, nums.Length, i);\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        (nums[i], nums[0]) = (nums[0], nums[i]);\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        SiftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.go<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nfunc siftDown(nums *[]int, n, i int) {\n    for true {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        l := 2*i + 1\n        r := 2*i + 2\n        ma := i\n        if l &lt; n &amp;&amp; (*nums)[l] &gt; (*nums)[ma] {\n            ma = l\n        }\n        if r &lt; n &amp;&amp; (*nums)[r] &gt; (*nums)[ma] {\n            ma = r\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if ma == i {\n            break\n        }\n        // 두 노드를 교환한다\n        (*nums)[i], (*nums)[ma] = (*nums)[ma], (*nums)[i]\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma\n    }\n}\n\n/* 힙 정렬 */\nfunc heapSort(nums *[]int) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for i := len(*nums)/2 - 1; i &gt;= 0; i-- {\n        siftDown(nums, len(*nums), i)\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for i := len(*nums) - 1; i &gt; 0; i-- {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        (*nums)[0], (*nums)[i] = (*nums)[i], (*nums)[0]\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.swift<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nfunc siftDown(nums: inout [Int], n: Int, i: Int) {\n    var i = i\n    while true {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        let l = 2 * i + 1\n        let r = 2 * i + 2\n        var ma = i\n        if l &lt; n, nums[l] &gt; nums[ma] {\n            ma = l\n        }\n        if r &lt; n, nums[r] &gt; nums[ma] {\n            ma = r\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if ma == i {\n            break\n        }\n        // 두 노드를 교환한다\n        nums.swapAt(i, ma)\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma\n    }\n}\n\n/* 힙 정렬 */\nfunc heapSort(nums: inout [Int]) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for i in stride(from: nums.count / 2 - 1, through: 0, by: -1) {\n        siftDown(nums: &amp;nums, n: nums.count, i: i)\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for i in nums.indices.dropFirst().reversed() {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        nums.swapAt(0, i)\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums: &amp;nums, n: i, i: 0)\n    }\n}\n</code></pre> heap_sort.js<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nfunction siftDown(nums, n, i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma === i) {\n            break;\n        }\n        // 두 노드를 교환한다\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n\n/* 힙 정렬 */\nfunction heapSort(nums) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.ts<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nfunction siftDown(nums: number[], n: number, i: number): void {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma === i) {\n            break;\n        }\n        // 두 노드를 교환한다\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n\n/* 힙 정렬 */\nfunction heapSort(nums: number[]): void {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.dart<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nvoid siftDown(List&lt;int&gt; nums, int n, int i) {\n  while (true) {\n    // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n    int l = 2 * i + 1;\n    int r = 2 * i + 2;\n    int ma = i;\n    if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) ma = l;\n    if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) ma = r;\n    // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n    if (ma == i) break;\n    // 두 노드를 교환한다\n    int temp = nums[i];\n    nums[i] = nums[ma];\n    nums[ma] = temp;\n    // 아래쪽으로 힙 재정렬을 반복한다\n    i = ma;\n  }\n}\n\n/* 힙 정렬 */\nvoid heapSort(List&lt;int&gt; nums) {\n  // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n  for (int i = nums.length ~/ 2 - 1; i &gt;= 0; i--) {\n    siftDown(nums, nums.length, i);\n  }\n  // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    int tmp = nums[0];\n    nums[0] = nums[i];\n    nums[i] = tmp;\n    // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n    siftDown(nums, i, 0);\n  }\n}\n</code></pre> heap_sort.rs<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nfn sift_down(nums: &amp;mut [i32], n: usize, mut i: usize) {\n    loop {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let mut ma = i;\n        if l &lt; n &amp;&amp; nums[l] &gt; nums[ma] {\n            ma = l;\n        }\n        if r &lt; n &amp;&amp; nums[r] &gt; nums[ma] {\n            ma = r;\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if ma == i {\n            break;\n        }\n        // 두 노드를 교환한다\n        nums.swap(i, ma);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n\n/* 힙 정렬 */\nfn heap_sort(nums: &amp;mut [i32]) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for i in (0..nums.len() / 2).rev() {\n        sift_down(nums, nums.len(), i);\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for i in (1..nums.len()).rev() {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        nums.swap(0, i);\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        sift_down(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.c<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nvoid siftDown(int nums[], int n, int i) {\n    while (1) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i) {\n            break;\n        }\n        // 두 노드를 교환한다\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n\n/* 힙 정렬 */\nvoid heapSort(int nums[], int n) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (int i = n / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, n, i);\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for (int i = n - 1; i &gt; 0; --i) {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.kt<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nfun siftDown(nums: IntArray, n: Int, li: Int) {\n    var i = li\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        val l = 2 * i + 1\n        val r = 2 * i + 2\n        var ma = i\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) \n            ma = l\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) \n            ma = r\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i) \n            break\n        // 두 노드를 교환한다\n        val temp = nums[i]\n        nums[i] = nums[ma]\n        nums[ma] = temp\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma\n    }\n}\n\n/* 힙 정렬 */\nfun heapSort(nums: IntArray) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (i in nums.size / 2 - 1 downTo 0) {\n        siftDown(nums, nums.size, i)\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for (i in nums.size - 1 downTo 1) {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        val temp = nums[0]\n        nums[0] = nums[i]\n        nums[i] = temp\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.rb<pre><code>### 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 ###\ndef sift_down(nums, n, i)\n  while true\n    # 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n    l = 2 * i + 1\n    r = 2 * i + 2\n    ma = i\n    ma = l if l &lt; n &amp;&amp; nums[l] &gt; nums[ma]\n    ma = r if r &lt; n &amp;&amp; nums[r] &gt; nums[ma]\n    # 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n    break if ma == i\n    # 두 노드를 교환한다\n    nums[i], nums[ma] = nums[ma], nums[i]\n    # 아래쪽으로 힙 재정렬을 반복한다\n    i = ma\n  end\nend\n\n### 힙 정렬 ###\ndef heap_sort(nums)\n  # 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n  (nums.length / 2 - 1).downto(0) do |i|\n    sift_down(nums, nums.length, i)\n  end\n  # 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n  (nums.length - 1).downto(1) do |i|\n    # 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    nums[0], nums[i] = nums[i], nums[0]\n    # 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n    sift_down(nums, i, 0)\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.7   힙 정렬"],"tags":[]},{"location":"chapter_sorting/heap_sort/#1172","level":2,"title":"11.7.2   알고리즘 특성","text":"<ul> <li>시간 복잡도 \\(O(n \\log n)\\), 비적응형 정렬: 힙 구성에는 \\(O(n)\\) 시간이 걸린다. 힙에서 최댓값을 꺼내는 시간 복잡도는 \\(O(\\log n)\\)이며, 총 \\(n - 1\\) 회 반복한다.</li> <li>공간 복잡도 \\(O(1)\\), 제자리 정렬: 몇 개의 포인터 변수가 \\(O(1)\\) 공간을 사용한다. 원소 교환과 힙 재정렬은 모두 원래 배열에서 수행한다.</li> <li>불안정 정렬: 힙의 루트 원소와 끝 원소를 교환할 때 값이 같은 원소들의 상대적 위치가 바뀔 수 있다.</li> </ul>","path":["제 11 장   정렬","11.7   힙 정렬"],"tags":[]},{"location":"chapter_sorting/insertion_sort/","level":1,"title":"11.4   삽입 정렬","text":"<p>삽입 정렬(insertion sort)은 간단한 정렬 알고리즘으로, 손으로 카드 한 벌을 정리하는 과정과 원리가 매우 비슷하다.</p> <p>구체적으로는 미정렬 구간에서 기준 원소 하나를 골라, 왼쪽의 정렬된 구간에 있는 원소들과 하나씩 크기를 비교한 뒤 올바른 위치에 삽입한다.</p> <p>아래 그림(11-6)은 배열에 원소를 삽입하는 과정을 보여 준다. 기준 원소를 <code>base</code>라고 하면 목표 인덱스부터 <code>base</code> 사이의 모든 원소를 오른쪽으로 한 칸씩 이동한 뒤, 목표 인덱스에 <code>base</code>를 대입한다.</p> <p></p> <p> 그림 11-6   한 번의 삽입 연산 </p>","path":["제 11 장   정렬","11.4   삽입 정렬"],"tags":[]},{"location":"chapter_sorting/insertion_sort/#1141","level":2,"title":"11.4.1   알고리즘 과정","text":"<p>삽입 정렬의 전체 과정은 아래 그림(11-7)과 같다.</p> <ol> <li>처음에는 배열의 첫 번째 원소가 정렬되어 있다.</li> <li>배열의 두 번째 원소를 <code>base</code>로 골라 올바른 위치에 삽입하면 배열의 앞 2개 원소가 정렬된다.</li> <li>세 번째 원소를 <code>base</code>로 골라 올바른 위치에 삽입하면 배열의 앞 3개 원소가 정렬된다.</li> <li>같은 과정을 반복한다. 마지막 회에서 마지막 원소를 <code>base</code>로 골라 올바른 위치에 삽입하면 모든 원소가 정렬된다.</li> </ol> <p></p> <p> 그림 11-7   삽입 정렬 과정 </p> <p>예시 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby insertion_sort.py<pre><code>def insertion_sort(nums: list[int]):\n    \"\"\"삽입 정렬\"\"\"\n    # 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for i in range(1, len(nums)):\n        base = nums[i]\n        j = i - 1\n        # 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while j &gt;= 0 and nums[j] &gt; base:\n            nums[j + 1] = nums[j]  # nums[j]를 오른쪽으로 한 칸 이동한다\n            j -= 1\n        nums[j + 1] = base  # 올바른 위치에 base를 대입한다\n</code></pre> insertion_sort.cpp<pre><code>/* 삽입 정렬 */\nvoid insertionSort(vector&lt;int&gt; &amp;nums) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for (int i = 1; i &lt; nums.size(); i++) {\n        int base = nums[i], j = i - 1;\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // nums[j]를 오른쪽으로 한 칸 이동한다\n            j--;\n        }\n        nums[j + 1] = base; // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.java<pre><code>/* 삽입 정렬 */\nvoid insertionSort(int[] nums) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for (int i = 1; i &lt; nums.length; i++) {\n        int base = nums[i], j = i - 1;\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // nums[j]를 오른쪽으로 한 칸 이동한다\n            j--;\n        }\n        nums[j + 1] = base;        // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.cs<pre><code>/* 삽입 정렬 */\nvoid InsertionSort(int[] nums) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for (int i = 1; i &lt; nums.Length; i++) {\n        int bas = nums[i], j = i - 1;\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; bas) {\n            nums[j + 1] = nums[j]; // nums[j]를 오른쪽으로 한 칸 이동한다\n            j--;\n        }\n        nums[j + 1] = bas;         // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.go<pre><code>/* 삽입 정렬 */\nfunc insertionSort(nums []int) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for i := 1; i &lt; len(nums); i++ {\n        base := nums[i]\n        j := i - 1\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        for j &gt;= 0 &amp;&amp; nums[j] &gt; base {\n            nums[j+1] = nums[j] // nums[j]를 오른쪽으로 한 칸 이동한다\n            j--\n        }\n        nums[j+1] = base // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.swift<pre><code>/* 삽입 정렬 */\nfunc insertionSort(nums: inout [Int]) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for i in nums.indices.dropFirst() {\n        let base = nums[i]\n        var j = i - 1\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while j &gt;= 0, nums[j] &gt; base {\n            nums[j + 1] = nums[j] // nums[j]를 오른쪽으로 한 칸 이동한다\n            j -= 1\n        }\n        nums[j + 1] = base // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.js<pre><code>/* 삽입 정렬 */\nfunction insertionSort(nums) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for (let i = 1; i &lt; nums.length; i++) {\n        let base = nums[i],\n            j = i - 1;\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // nums[j]를 오른쪽으로 한 칸 이동한다\n            j--;\n        }\n        nums[j + 1] = base; // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.ts<pre><code>/* 삽입 정렬 */\nfunction insertionSort(nums: number[]): void {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for (let i = 1; i &lt; nums.length; i++) {\n        const base = nums[i];\n        let j = i - 1;\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // nums[j]를 오른쪽으로 한 칸 이동한다\n            j--;\n        }\n        nums[j + 1] = base; // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.dart<pre><code>/* 삽입 정렬 */\nvoid insertionSort(List&lt;int&gt; nums) {\n  // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n  for (int i = 1; i &lt; nums.length; i++) {\n    int base = nums[i], j = i - 1;\n    // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n    while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n      nums[j + 1] = nums[j]; // nums[j]를 오른쪽으로 한 칸 이동한다\n      j--;\n    }\n    nums[j + 1] = base; // 올바른 위치에 base를 대입한다\n  }\n}\n</code></pre> insertion_sort.rs<pre><code>/* 삽입 정렬 */\nfn insertion_sort(nums: &amp;mut [i32]) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for i in 1..nums.len() {\n        let (base, mut j) = (nums[i], (i - 1) as i32);\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while j &gt;= 0 &amp;&amp; nums[j as usize] &gt; base {\n            nums[(j + 1) as usize] = nums[j as usize]; // nums[j]를 오른쪽으로 한 칸 이동한다\n            j -= 1;\n        }\n        nums[(j + 1) as usize] = base; // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.c<pre><code>/* 삽입 정렬 */\nvoid insertionSort(int nums[], int size) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for (int i = 1; i &lt; size; i++) {\n        int base = nums[i], j = i - 1;\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            // nums[j]를 오른쪽으로 한 칸 이동한다\n            nums[j + 1] = nums[j];\n            j--;\n        }\n        // 올바른 위치에 base를 대입한다\n        nums[j + 1] = base;\n    }\n}\n</code></pre> insertion_sort.kt<pre><code>/* 삽입 정렬 */\nfun insertionSort(nums: IntArray) {\n    //바깥쪽 반복문: 정렬된 원소의 수는 1, 2, ..., n이 된다\n    for (i in nums.indices) {\n        val base = nums[i]\n        var j = i - 1\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j] // nums[j]를 오른쪽으로 한 칸 이동한다\n            j--\n        }\n        nums[j + 1] = base        // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.rb<pre><code>### 삽입 정렬 ###\ndef insertion_sort(nums)\n  n = nums.length\n  # 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n  for i in 1...n\n    base = nums[i]\n    j = i - 1\n    # 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n    while j &gt;= 0 &amp;&amp; nums[j] &gt; base\n      nums[j + 1] = nums[j] # nums[j]를 오른쪽으로 한 칸 이동한다\n      j -= 1\n    end\n    nums[j + 1] = base # 올바른 위치에 base를 대입한다\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.4   삽입 정렬"],"tags":[]},{"location":"chapter_sorting/insertion_sort/#1142","level":2,"title":"11.4.2   알고리즘 특성","text":"<ul> <li>시간 복잡도 \\(O(n^2)\\), 적응형 정렬: 최악의 경우 각 삽입 연산에서 \\(n - 1\\), \\(n-2\\), \\(\\dots\\), \\(2\\), \\(1\\) 번 반복해야 한다. 합은 \\((n - 1) n / 2\\)이므로 시간 복잡도는 \\(O(n^2)\\)이다. 정렬된 데이터를 만나면 삽입 연산이 일찍 끝난다. 입력 배열이 완전히 정렬되어 있으면 최선 시간 복잡도 \\(O(n)\\)에 도달한다.</li> <li>공간 복잡도 \\(O(1)\\), 제자리 정렬: 포인터 \\(i\\)와 \\(j\\)는 상수 크기의 추가 공간을 사용한다.</li> <li>안정 정렬: 삽입할 때 값이 같은 원소의 오른쪽에 삽입하므로 상대적 순서가 바뀌지 않는다.</li> </ul>","path":["제 11 장   정렬","11.4   삽입 정렬"],"tags":[]},{"location":"chapter_sorting/insertion_sort/#1143","level":2,"title":"11.4.3   삽입 정렬의 장점","text":"<p>삽입 정렬의 시간 복잡도는 \\(O(n^2)\\)이고, 곧 배울 퀵 정렬의 시간 복잡도는 \\(O(n \\log n)\\)이다. 삽입 정렬의 시간 복잡도가 더 높지만 데이터가 적으면 보통 삽입 정렬이 더 빠르다.</p> <p>이는 선형 탐색과 이진 탐색의 적합한 사용 상황에서 내린 결론과 비슷하다. 퀵 정렬 같은 \\(O(n \\log n)\\) 알고리즘은 분할 정복 전략에 기반한 정렬 알고리즘으로, 기본 연산이 더 많은 경우가 많다. 데이터가 적으면 \\(n^2\\)과 \\(n \\log n\\)의 값이 비슷하여 시간 복잡도가 결정적인 요인이 되지 못하고, 매 반복에서 수행하는 기본 연산 수가 결정적인 역할을 한다.</p> <p>실제로 Java 등 여러 프로그래밍 언어의 내장 정렬 함수는 삽입 정렬을 활용한다. 대체로 긴 배열에는 퀵 정렬 같은 분할 정복 기반 정렬을 사용하고, 짧은 배열에는 삽입 정렬을 바로 적용한다.</p> <p>버블 정렬, 선택 정렬, 삽입 정렬의 시간 복잡도는 모두 \\(O(n^2)\\)이지만, 실제로는 삽입 정렬이 버블 정렬이나 선택 정렬보다 훨씬 자주 사용된다. 주된 이유는 다음과 같다.</p> <ul> <li>버블 정렬은 원소 교환으로 구현하므로 임시 변수가 필요하고 총 3회의 기본 연산을 수행한다. 삽입 정렬은 원소 대입으로 구현하므로 기본 연산이 1회만 필요하다. 따라서 버블 정렬의 계산 비용은 보통 삽입 정렬보다 크다.</li> <li>선택 정렬의 시간 복잡도는 어떤 경우에도 \\(O(n^2)\\)이다. 데이터가 부분적으로 정렬되어 있다면 삽입 정렬이 보통 선택 정렬보다 효율적이다.</li> <li>선택 정렬은 불안정하므로 여러 기준으로 순차 정렬하는 데 사용할 수 없다.</li> </ul>","path":["제 11 장   정렬","11.4   삽입 정렬"],"tags":[]},{"location":"chapter_sorting/merge_sort/","level":1,"title":"11.6   병합 정렬","text":"<p>병합 정렬(merge sort)은 분할 정복 전략에 기반한 정렬 알고리즘으로, 아래 그림(11-10)의 '분할' 단계와 '병합' 단계로 이루어진다.</p> <ol> <li>분할 단계: 재귀적으로 배열을 계속 중간에서 나누어, 긴 배열을 정렬하는 문제를 짧은 배열을 정렬하는 문제로 바꾼다.</li> <li>병합 단계: 부분 배열의 길이가 1이 되면 분할을 멈추고 병합을 시작한다. 정렬된 짧은 왼쪽 배열과 오른쪽 배열을 하나의 더 긴 정렬된 배열로 계속 합치며, 끝날 때까지 반복한다.</li> </ol> <p></p> <p> 그림 11-10   병합 정렬의 분할 단계와 병합 단계 </p>","path":["제 11 장   정렬","11.6   병합 정렬"],"tags":[]},{"location":"chapter_sorting/merge_sort/#1161","level":2,"title":"11.6.1   알고리즘 과정","text":"<p>아래 그림(11-11)과 같이 '분할 단계'에서는 위에서 아래로 재귀하며 배열을 중간에서 두 부분 배열로 나눈다.</p> <ol> <li>배열의 중간 인덱스 <code>mid</code>를 계산하고 왼쪽 부분 배열(구간 <code>[left, mid]</code>)과 오른쪽 부분 배열(구간 <code>[mid + 1, right]</code>)을 재귀적으로 나눈다.</li> <li>부분 배열 구간의 길이가 1이 될 때까지 <code>1.</code> 단계를 재귀적으로 반복한다.</li> </ol> <p>'병합 단계'에서는 아래에서 위로 왼쪽 부분 배열과 오른쪽 부분 배열을 하나의 정렬된 배열로 합친다. 길이가 1인 부분 배열부터 병합하므로, 병합 단계의 각 부분 배열은 이미 정렬되어 있다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 11-11   병합 정렬 단계 </p> <p>병합 정렬의 재귀 순서는 이진 트리의 후위 순회와 같다는 것을 알 수 있다.</p> <ul> <li>후위 순회: 왼쪽 서브트리에 재귀한 다음 오른쪽 서브트리에 재귀하고, 마지막으로 루트 노드를 처리한다.</li> <li>병합 정렬: 왼쪽 부분 배열에 재귀한 다음 오른쪽 부분 배열에 재귀하고, 마지막으로 병합한다.</li> </ul> <p>병합 정렬의 구현은 다음 코드와 같다. <code>nums</code>에서 병합할 구간은 <code>[left, right]</code>이고, 이에 대응하는 <code>tmp</code>의 구간은 <code>[0, right - left]</code>라는 점에 주의하자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby merge_sort.py<pre><code>def merge(nums: list[int], left: int, mid: int, right: int):\n    \"\"\"왼쪽 부분 배열과 오른쪽 부분 배열 병합\"\"\"\n    # 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    # 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    tmp = [0] * (right - left + 1)\n    # 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    i, j, k = left, mid + 1, 0\n    # 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while i &lt;= mid and j &lt;= right:\n        if nums[i] &lt;= nums[j]:\n            tmp[k] = nums[i]\n            i += 1\n        else:\n            tmp[k] = nums[j]\n            j += 1\n        k += 1\n    # 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while i &lt;= mid:\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    while j &lt;= right:\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    # 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for k in range(0, len(tmp)):\n        nums[left + k] = tmp[k]\n\ndef merge_sort(nums: list[int], left: int, right: int):\n    \"\"\"병합 정렬\"\"\"\n    # 종료 조건\n    if left &gt;= right:\n        return  # 부분 배열의 길이가 1이면 재귀를 종료한다\n    # 분할 단계\n    mid = (left + right) // 2 # 중간 위치를 계산한다\n    merge_sort(nums, left, mid)  # 왼쪽 부분 배열에 재귀 호출한다\n    merge_sort(nums, mid + 1, right)  # 오른쪽 부분 배열에 재귀 호출한다\n    # 병합 단계\n    merge(nums, left, mid, right)\n</code></pre> merge_sort.cpp<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nvoid merge(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    vector&lt;int&gt; tmp(right - left + 1);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    int i = left, j = mid + 1, k = 0;\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for (k = 0; k &lt; tmp.size(); k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* 병합 정렬 */\nvoid mergeSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // 종료 조건\n    if (left &gt;= right)\n        return; // 부분 배열의 길이가 1이면 재귀를 종료한다\n    // 분할 단계\n    int mid = left + (right - left) / 2;    // 중간 위치를 계산한다\n    mergeSort(nums, left, mid);      // 왼쪽 부분 배열에 재귀 호출한다\n    mergeSort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.java<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nvoid merge(int[] nums, int left, int mid, int right) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    int[] tmp = new int[right - left + 1];\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    int i = left, j = mid + 1, k = 0;\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* 병합 정렬 */\nvoid mergeSort(int[] nums, int left, int right) {\n    // 종료 조건\n    if (left &gt;= right)\n        return; // 부분 배열의 길이가 1이면 재귀를 종료한다\n    // 분할 단계\n    int mid = left + (right - left) / 2; // 중간 위치를 계산한다\n    mergeSort(nums, left, mid); // 왼쪽 부분 배열에 재귀 호출한다\n    mergeSort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.cs<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nvoid Merge(int[] nums, int left, int mid, int right) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    int[] tmp = new int[right - left + 1];\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    int i = left, j = mid + 1, k = 0;\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for (k = 0; k &lt; tmp.Length; ++k) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* 병합 정렬 */\nvoid MergeSort(int[] nums, int left, int right) {\n    // 종료 조건\n    if (left &gt;= right) return;       // 부분 배열의 길이가 1이면 재귀를 종료한다\n    // 분할 단계\n    int mid = left + (right - left) / 2;    // 중간 위치를 계산한다\n    MergeSort(nums, left, mid);      // 왼쪽 부분 배열에 재귀 호출한다\n    MergeSort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    Merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.go<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nfunc merge(nums []int, left, mid, right int) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    tmp := make([]int, right-left+1)\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    i, j, k := left, mid+1, 0\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    for i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i++\n        } else {\n            tmp[k] = nums[j]\n            j++\n        }\n        k++\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    for i &lt;= mid {\n        tmp[k] = nums[i]\n        i++\n        k++\n    }\n    for j &lt;= right {\n        tmp[k] = nums[j]\n        j++\n        k++\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for k := 0; k &lt; len(tmp); k++ {\n        nums[left+k] = tmp[k]\n    }\n}\n\n/* 병합 정렬 */\nfunc mergeSort(nums []int, left, right int) {\n    // 종료 조건\n    if left &gt;= right {\n        return\n    }\n    // 분할 단계\n    mid := left + (right - left) / 2\n    mergeSort(nums, left, mid)\n    mergeSort(nums, mid+1, right)\n    // 병합 단계\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.swift<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    var tmp = Array(repeating: 0, count: right - left + 1)\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    var i = left, j = mid + 1, k = 0\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while i &lt;= mid, j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i += 1\n        } else {\n            tmp[k] = nums[j]\n            j += 1\n        }\n        k += 1\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while i &lt;= mid {\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for k in tmp.indices {\n        nums[left + k] = tmp[k]\n    }\n}\n\n/* 병합 정렬 */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n    // 종료 조건\n    if left &gt;= right { // 부분 배열의 길이가 1이면 재귀를 종료한다\n        return\n    }\n    // 분할 단계\n    let mid = left + (right - left) / 2 // 중간 위치를 계산한다\n    mergeSort(nums: &amp;nums, left: left, right: mid) // 왼쪽 부분 배열에 재귀 호출한다\n    mergeSort(nums: &amp;nums, left: mid + 1, right: right) // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    merge(nums: &amp;nums, left: left, mid: mid, right: right)\n}\n</code></pre> merge_sort.js<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nfunction merge(nums, left, mid, right) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    const tmp = new Array(right - left + 1);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* 병합 정렬 */\nfunction mergeSort(nums, left, right) {\n    // 종료 조건\n    if (left &gt;= right) return; // 부분 배열의 길이가 1이면 재귀를 종료한다\n    // 분할 단계\n    let mid = Math.floor(left + (right - left) / 2); // 중간 위치를 계산한다\n    mergeSort(nums, left, mid); // 왼쪽 부분 배열에 재귀 호출한다\n    mergeSort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.ts<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    const tmp = new Array(right - left + 1);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* 병합 정렬 */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n    // 종료 조건\n    if (left &gt;= right) return; // 부분 배열의 길이가 1이면 재귀를 종료한다\n    // 분할 단계\n    let mid = Math.floor(left + (right - left) / 2); // 중간 위치를 계산한다\n    mergeSort(nums, left, mid); // 왼쪽 부분 배열에 재귀 호출한다\n    mergeSort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.dart<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nvoid merge(List&lt;int&gt; nums, int left, int mid, int right) {\n  // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n  // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n  List&lt;int&gt; tmp = List.filled(right - left + 1, 0);\n  // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n  int i = left, j = mid + 1, k = 0;\n  // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n  while (i &lt;= mid &amp;&amp; j &lt;= right) {\n    if (nums[i] &lt;= nums[j])\n      tmp[k++] = nums[i++];\n    else\n      tmp[k++] = nums[j++];\n  }\n  // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n  while (i &lt;= mid) {\n    tmp[k++] = nums[i++];\n  }\n  while (j &lt;= right) {\n    tmp[k++] = nums[j++];\n  }\n  // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n  for (k = 0; k &lt; tmp.length; k++) {\n    nums[left + k] = tmp[k];\n  }\n}\n\n/* 병합 정렬 */\nvoid mergeSort(List&lt;int&gt; nums, int left, int right) {\n  // 종료 조건\n  if (left &gt;= right) return; // 부분 배열의 길이가 1이면 재귀를 종료한다\n  // 분할 단계\n  int mid = left + (right - left) ~/ 2; // 중간 위치를 계산한다\n  mergeSort(nums, left, mid); // 왼쪽 부분 배열에 재귀 호출한다\n  mergeSort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n  // 병합 단계\n  merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.rs<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nfn merge(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    let tmp_size = right - left + 1;\n    let mut tmp = vec![0; tmp_size];\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    let (mut i, mut j, mut k) = (left, mid + 1, 0);\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i];\n            i += 1;\n        } else {\n            tmp[k] = nums[j];\n            j += 1;\n        }\n        k += 1;\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while i &lt;= mid {\n        tmp[k] = nums[i];\n        k += 1;\n        i += 1;\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j];\n        k += 1;\n        j += 1;\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for k in 0..tmp_size {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* 병합 정렬 */\nfn merge_sort(nums: &amp;mut [i32], left: usize, right: usize) {\n    // 종료 조건\n    if left &gt;= right {\n        return; // 부분 배열의 길이가 1이면 재귀를 종료한다\n    }\n\n    // 분할 단계\n    let mid = left + (right - left) / 2; // 중간 위치를 계산한다\n    merge_sort(nums, left, mid); // 왼쪽 부분 배열에 재귀 호출한다\n    merge_sort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n\n    // 병합 단계\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.c<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nvoid merge(int *nums, int left, int mid, int right) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    int tmpSize = right - left + 1;\n    int *tmp = (int *)malloc(tmpSize * sizeof(int));\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    int i = left, j = mid + 1, k = 0;\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for (k = 0; k &lt; tmpSize; ++k) {\n        nums[left + k] = tmp[k];\n    }\n    // 메모리 해제\n    free(tmp);\n}\n\n/* 병합 정렬 */\nvoid mergeSort(int *nums, int left, int right) {\n    // 종료 조건\n    if (left &gt;= right)\n        return; // 부분 배열의 길이가 1이면 재귀를 종료한다\n    // 분할 단계\n    int mid = left + (right - left) / 2;    // 중간 위치를 계산한다\n    mergeSort(nums, left, mid);      // 왼쪽 부분 배열에 재귀 호출한다\n    mergeSort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.kt<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nfun merge(nums: IntArray, left: Int, mid: Int, right: Int) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    val tmp = IntArray(right - left + 1)\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    var i = left\n    var j = mid + 1\n    var k = 0\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++]\n        else\n            tmp[k++] = nums[j++]\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++]\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++]\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for (l in tmp.indices) {\n        nums[left + l] = tmp[l]\n    }\n}\n\n/* 병합 정렬 */\nfun mergeSort(nums: IntArray, left: Int, right: Int) {\n    // 종료 조건\n    if (left &gt;= right) return  // 부분 배열의 길이가 1이면 재귀를 종료한다\n    // 분할 단계\n    val mid = left + (right - left) / 2 // 중간 위치를 계산한다\n    mergeSort(nums, left, mid) // 왼쪽 부분 배열에 재귀 호출한다\n    mergeSort(nums, mid + 1, right) // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.rb<pre><code>### 왼쪽 부분 배열과 오른쪽 부분 배열 병합 ###\ndef merge(nums, left, mid, right)\n  # 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n  # 병합 결과를 저장할 임시 배열 tmp를 생성한다\n  tmp = Array.new(right - left + 1, 0)\n  # 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n  i, j, k = left, mid + 1, 0\n  # 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n  while i &lt;= mid &amp;&amp; j &lt;= right\n    if nums[i] &lt;= nums[j]\n      tmp[k] = nums[i]\n      i += 1\n    else\n      tmp[k] = nums[j]\n      j += 1\n    end\n    k += 1\n  end\n  # 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n  while i &lt;= mid\n    tmp[k] = nums[i]\n    i += 1\n    k += 1\n  end\n  while j &lt;= right\n    tmp[k] = nums[j]\n    j += 1\n    k += 1\n  end\n  # 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n  (0...tmp.length).each do |k|\n    nums[left + k] = tmp[k]\n  end\nend\n\n### 병합 정렬 ###\ndef merge_sort(nums, left, right)\n  # 종료 조건\n  # 부분 배열의 길이가 1이면 재귀를 종료한다\n  return if left &gt;= right\n  # 분할 단계\n  mid = left + (right - left) / 2 # 중간 위치를 계산한다\n  merge_sort(nums, left, mid) # 왼쪽 부분 배열에 재귀 호출한다\n  merge_sort(nums, mid + 1, right) # 오른쪽 부분 배열에 재귀 호출한다\n  # 병합 단계\n  merge(nums, left, mid, right)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.6   병합 정렬"],"tags":[]},{"location":"chapter_sorting/merge_sort/#1162","level":2,"title":"11.6.2   알고리즘 특성","text":"<ul> <li>시간 복잡도 \\(O(n \\log n)\\), 비적응형 정렬: 분할 과정에서 높이 \\(\\log n\\)의 재귀 트리가 만들어지고, 각 레벨에서 병합에 필요한 총연산 수는 \\(n\\)이므로 전체 시간 복잡도는 \\(O(n \\log n)\\)이다.</li> <li>공간 복잡도 \\(O(n)\\), 제자리 정렬이 아님: 재귀 깊이가 \\(\\log n\\)이므로 \\(O(\\log n)\\) 크기의 스택 프레임 공간을 사용한다. 병합에는 보조 배열이 필요하여 \\(O(n)\\)의 추가 공간을 사용한다.</li> <li>안정 정렬: 병합 과정에서 값이 같은 원소들의 순서가 유지된다.</li> </ul>","path":["제 11 장   정렬","11.6   병합 정렬"],"tags":[]},{"location":"chapter_sorting/merge_sort/#1163","level":2,"title":"11.6.3   연결 리스트 정렬","text":"<p>연결 리스트에서는 병합 정렬이 다른 정렬 알고리즘보다 뚜렷한 장점을 가진다. 연결 리스트 정렬의 공간 복잡도를 \\(O(1)\\)까지 최적화할 수 있다.</p> <ul> <li>분할 단계: 연결 리스트를 나눌 때 '재귀' 대신 '반복'을 사용하면 재귀에 필요한 스택 프레임 공간을 없앨 수 있다.</li> <li>병합 단계: 연결 리스트에서는 참조(포인터)만 바꾸어 노드를 추가하거나 삭제할 수 있다. 따라서 정렬된 짧은 연결 리스트 두 개를 하나의 긴 연결 리스트로 합칠 때 추가 연결 리스트를 만들 필요가 없다.</li> </ul> <p>구체적인 구현은 다소 복잡하므로, 관심 있는 독자는 관련 자료를 찾아 학습해 보자.</p>","path":["제 11 장   정렬","11.6   병합 정렬"],"tags":[]},{"location":"chapter_sorting/quick_sort/","level":1,"title":"11.5   퀵 정렬","text":"<p>퀵 정렬(quick sort)은 분할 정복 전략에 기반한 정렬 알고리즘으로, 실행 효율이 높고 널리 사용된다.</p> <p>퀵 정렬의 핵심 연산은 '센티널 분할'이다. 배열의 한 원소를 '피벗'으로 고른 다음, 피벗보다 작은 원소는 모두 왼쪽으로, 큰 원소는 모두 오른쪽으로 옮기는 것이 목표다. 구체적인 과정은 아래 그림(11-8)과 같다.</p> <ol> <li>배열의 맨 왼쪽 원소를 피벗으로 고르고, 포인터 <code>i</code>와 <code>j</code>가 배열의 양 끝을 가리키도록 초기화한다.</li> <li>반복문을 실행하여 매번 <code>i</code>(<code>j</code>)로 피벗보다 큰(작은) 첫 번째 원소를 각각 찾은 뒤 두 원소를 교환한다.</li> <li><code>i</code>와 <code>j</code>가 만날 때까지 <code>2.</code> 단계를 반복한다. 마지막으로 피벗을 두 부분 배열의 경계로 옮긴다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 11-8   센티널 분할 단계 </p> <p>분할을 마치면 원래 배열은 왼쪽 부분 배열, 피벗, 오른쪽 부분 배열의 세 부분으로 나뉘며, '왼쪽 부분 배열의 모든 원소 \\(\\leq\\) 피벗 \\(\\leq\\) 오른쪽 부분 배열의 모든 원소'를 만족한다. 따라서 이후에는 두 부분 배열만 정렬하면 된다.</p> <p>퀵 정렬의 분할 정복 전략</p> <p>센티널 분할은 본질적으로 긴 배열 하나를 정렬하는 문제를 짧은 배열 두 개를 정렬하는 문제로 단순화한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"센티널 분할\"\"\"\n    # nums[left]를 피벗으로 삼는다\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        # 원소 교환\n        nums[i], nums[j] = nums[j], nums[i]\n    # 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # 피벗의 인덱스를 반환한다\n</code></pre> quick_sort.cpp<pre><code>/* 센티널 분할 */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;                // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;                // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        swap(nums[i], nums[j]); // 두 원소를 교환한다\n    }\n    swap(nums[i], nums[left]);  // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i;                   // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.java<pre><code>/* 원소 교환 */\nvoid swap(int[] nums, int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* 센티널 분할 */\nint partition(int[] nums, int left, int right) {\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        swap(nums, i, j); // 두 원소를 교환한다\n    }\n    swap(nums, i, left);  // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i;             // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.cs<pre><code>/* 원소 교환 */\nvoid Swap(int[] nums, int i, int j) {\n    (nums[j], nums[i]) = (nums[i], nums[j]);\n}\n\n/* 센티널 분할 */\nint Partition(int[] nums, int left, int right) {\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        Swap(nums, i, j); // 두 원소를 교환한다\n    }\n    Swap(nums, i, left);  // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i;             // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.go<pre><code>/* 센티널 분할 */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n    // nums[left]를 피벗으로 삼는다\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        // 원소 교환\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    nums[i], nums[left] = nums[left], nums[i]\n    return i // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.swift<pre><code>/* 센티널 분할 */\nfunc partition(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // nums[left]를 피벗으로 삼는다\n    var i = left\n    var j = right\n    while i &lt; j {\n        while i &lt; j, nums[j] &gt;= nums[left] {\n            j -= 1 // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        while i &lt; j, nums[i] &lt;= nums[left] {\n            i += 1 // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        nums.swapAt(i, j) // 두 원소를 교환한다\n    }\n    nums.swapAt(i, left) // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.js<pre><code>/* 원소 교환 */\nswap(nums, i, j) {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* 센티널 분할 */\npartition(nums, left, right) {\n    // nums[left]를 피벗으로 삼는다\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        // 원소 교환\n        this.swap(nums, i, j); // 두 원소를 교환한다\n    }\n    this.swap(nums, i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i; // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.ts<pre><code>/* 원소 교환 */\nswap(nums: number[], i: number, j: number): void {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* 센티널 분할 */\npartition(nums: number[], left: number, right: number): number {\n    // nums[left]를 피벗으로 삼는다\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        // 원소 교환\n        this.swap(nums, i, j); // 두 원소를 교환한다\n    }\n    this.swap(nums, i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i; // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.dart<pre><code>/* 원소 교환 */\nvoid _swap(List&lt;int&gt; nums, int i, int j) {\n  int tmp = nums[i];\n  nums[i] = nums[j];\n  nums[j] = tmp;\n}\n\n/* 센티널 분할 */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // nums[left]를 피벗으로 삼는다\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n    _swap(nums, i, j); // 두 원소를 교환한다\n  }\n  _swap(nums, i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n  return i; // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.rs<pre><code>/* 센티널 분할 */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // nums[left]를 피벗으로 삼는다\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        nums.swap(i, j); // 두 원소를 교환한다\n    }\n    nums.swap(i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    i // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.c<pre><code>/* 원소 교환 */\nvoid swap(int nums[], int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* 센티널 분할 */\nint partition(int nums[], int left, int right) {\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j--; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i++; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        // 두 원소를 교환한다\n        swap(nums, i, j);\n    }\n    // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    swap(nums, i, left);\n    // 피벗의 인덱스를 반환한다\n    return i;\n}\n</code></pre> quick_sort.kt<pre><code>/* 원소 교환 */\nfun swap(nums: IntArray, i: Int, j: Int) {\n    val temp = nums[i]\n    nums[i] = nums[j]\n    nums[j] = temp\n}\n\n/* 센티널 분할 */\nfun partition(nums: IntArray, left: Int, right: Int): Int {\n    // nums[left]를 피벗으로 삼는다\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--           // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++           // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        swap(nums, i, j)  // 두 원소를 교환한다\n    }\n    swap(nums, i, left)   // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i              // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.rb<pre><code>### 센티널 분할 ###\ndef partition(nums, left, right)\n  # nums[left]를 피벗으로 삼는다\n  i, j = left, right\n  while i &lt; j\n    while i &lt; j &amp;&amp; nums[j] &gt;= nums[left]\n      j -= 1 # 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n    end\n    while i &lt; j &amp;&amp; nums[i] &lt;= nums[left]\n      i += 1 # 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n    end\n    # 원소 교환\n    nums[i], nums[j] = nums[j], nums[i]\n  end\n  # 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n  nums[i], nums[left] = nums[left], nums[i]\n  i # 피벗의 인덱스를 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.5   퀵 정렬"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1151","level":2,"title":"11.5.1   알고리즘 과정","text":"<p>퀵 정렬의 전체 과정은 아래 그림(11-9)과 같다.</p> <ol> <li>먼저 원래 배열에 '센티널 분할'을 한 번 수행하여 정렬되지 않은 왼쪽 부분 배열과 오른쪽 부분 배열을 얻는다.</li> <li>이어서 왼쪽 부분 배열과 오른쪽 부분 배열에 각각 '센티널 분할'을 재귀적으로 수행한다.</li> <li>부분 배열의 길이가 1이 될 때까지 재귀를 계속하면 배열 전체의 정렬이 완료된다.</li> </ol> <p></p> <p> 그림 11-9   퀵 정렬 과정 </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"퀵 정렬\"\"\"\n    # 부분 배열의 길이가 1이면 재귀를 종료한다\n    if left &gt;= right:\n        return\n    # 센티널 분할\n    pivot = self.partition(nums, left, right)\n    # 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    self.quick_sort(nums, left, pivot - 1)\n    self.quick_sort(nums, pivot + 1, right)\n</code></pre> quick_sort.cpp<pre><code>/* 퀵 정렬 */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if (left &gt;= right)\n        return;\n    // 센티널 분할\n    int pivot = partition(nums, left, right);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.java<pre><code>/* 퀵 정렬 */\nvoid quickSort(int[] nums, int left, int right) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if (left &gt;= right)\n        return;\n    // 센티널 분할\n    int pivot = partition(nums, left, right);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.cs<pre><code>/* 퀵 정렬 */\nvoid QuickSort(int[] nums, int left, int right) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if (left &gt;= right)\n        return;\n    // 센티널 분할\n    int pivot = Partition(nums, left, right);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    QuickSort(nums, left, pivot - 1);\n    QuickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.go<pre><code>/* 퀵 정렬 */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if left &gt;= right {\n        return\n    }\n    // 센티널 분할\n    pivot := q.partition(nums, left, right)\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    q.quickSort(nums, left, pivot-1)\n    q.quickSort(nums, pivot+1, right)\n}\n</code></pre> quick_sort.swift<pre><code>/* 퀵 정렬 */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if left &gt;= right {\n        return\n    }\n    // 센티널 분할\n    let pivot = partition(nums: &amp;nums, left: left, right: right)\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    quickSort(nums: &amp;nums, left: left, right: pivot - 1)\n    quickSort(nums: &amp;nums, left: pivot + 1, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* 퀵 정렬 */\nquickSort(nums, left, right) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if (left &gt;= right) return;\n    // 센티널 분할\n    const pivot = this.partition(nums, left, right);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.ts<pre><code>/* 퀵 정렬 */\nquickSort(nums: number[], left: number, right: number): void {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if (left &gt;= right) {\n        return;\n    }\n    // 센티널 분할\n    const pivot = this.partition(nums, left, right);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.dart<pre><code>/* 퀵 정렬 */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // 부분 배열의 길이가 1이면 재귀를 종료한다\n  if (left &gt;= right) return;\n  // 센티널 분할\n  int pivot = _partition(nums, left, right);\n  // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n  quickSort(nums, left, pivot - 1);\n  quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.rs<pre><code>/* 퀵 정렬 */\npub fn quick_sort(left: i32, right: i32, nums: &amp;mut [i32]) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if left &gt;= right {\n        return;\n    }\n    // 센티널 분할\n    let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    Self::quick_sort(left, pivot - 1, nums);\n    Self::quick_sort(pivot + 1, right, nums);\n}\n</code></pre> quick_sort.c<pre><code>/* 퀵 정렬 */\nvoid quickSort(int nums[], int left, int right) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if (left &gt;= right) {\n        return;\n    }\n    // 센티널 분할\n    int pivot = partition(nums, left, right);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.kt<pre><code>/* 퀵 정렬 */\nfun quickSort(nums: IntArray, left: Int, right: Int) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if (left &gt;= right) return\n    // 센티널 분할\n    val pivot = partition(nums, left, right)\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    quickSort(nums, left, pivot - 1)\n    quickSort(nums, pivot + 1, right)\n}\n</code></pre> quick_sort.rb<pre><code>### 퀵 정렬 클래스 ###\ndef quick_sort(nums, left, right)\n  # 부분 배열의 길이가 1이 아니면 재귀 호출한다\n  if left &lt; right\n    # 센티널 분할\n    pivot = partition(nums, left, right)\n    # 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    quick_sort(nums, left, pivot - 1)\n    quick_sort(nums, pivot + 1, right)\n  end\n  nums\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.5   퀵 정렬"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1152","level":2,"title":"11.5.2   알고리즘 특성","text":"<ul> <li>시간 복잡도 \\(O(n \\log n)\\), 비적응형 정렬: 평균적으로 분할의 재귀 레벨 수는 \\(\\log n\\)이고 각 레벨의 총반복 횟수는 \\(n\\)이므로 전체 시간은 \\(O(n \\log n)\\)이다. 최악의 경우 매 분할에서 길이 \\(n\\)의 배열이 길이 \\(0\\)과 \\(n - 1\\)인 두 부분 배열로 나뉜다. 이때 재귀 레벨 수는 \\(n\\)에 이르고 각 레벨의 반복 횟수는 \\(n\\)이므로 전체 시간은 \\(O(n^2)\\)이다.</li> <li>공간 복잡도 \\(O(n)\\), 제자리 정렬: 입력 배열이 완전히 역순이면 재귀 깊이가 최악의 값인 \\(n\\)에 이르므로 \\(O(n)\\)의 스택 프레임 공간을 사용한다. 정렬 자체는 추가 배열 없이 원래 배열에서 수행한다.</li> <li>불안정 정렬: 분할의 마지막 단계에서 피벗이 값이 같은 원소의 오른쪽으로 교환될 수 있다.</li> </ul>","path":["제 11 장   정렬","11.5   퀵 정렬"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1153","level":2,"title":"11.5.3   퀵 정렬이 빠른 이유","text":"<p>이름에서도 알 수 있듯이 퀵 정렬은 효율 측면에서 장점이 있다. 평균 시간 복잡도는 '병합 정렬', '힙 정렬'과 같지만 보통 퀵 정렬이 더 효율적이다. 주요 이유는 다음과 같다.</p> <ul> <li>최악의 경우가 발생할 확률이 낮다: 퀵 정렬의 최악 시간 복잡도는 \\(O(n^2)\\)으로 병합 정렬만큼 안정적이지 않지만, 대부분의 경우 \\(O(n \\log n)\\) 시간에 실행된다.</li> <li>캐시 활용 효율이 높다: 분할을 수행할 때 시스템이 부분 배열 전체를 캐시에 올릴 수 있어 원소 접근이 효율적이다. 반면 '힙 정렬' 같은 알고리즘은 원소를 건너뛰며 접근하므로 이런 특성이 없다.</li> <li>복잡도의 상수 계수가 작다: 위 세 알고리즘 중 퀵 정렬은 비교, 대입, 교환 등의 총연산 수가 가장 적다. '삽입 정렬'이 '버블 정렬'보다 빠른 이유와 비슷하다.</li> </ul>","path":["제 11 장   정렬","11.5   퀵 정렬"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1154","level":2,"title":"11.5.4   피벗 선택 최적화","text":"<p>일부 입력에서는 퀵 정렬의 시간 효율이 떨어질 수 있다. 극단적인 예로 입력 배열이 완전히 역순이라고 하자. 맨 왼쪽 원소를 피벗으로 고르면 분할 후 피벗이 맨 오른쪽으로 이동하여 왼쪽 부분 배열의 길이는 \\(n - 1\\), 오른쪽 부분 배열의 길이는 \\(0\\)이 된다. 이런 식으로 재귀하면 매번 한쪽 부분 배열의 길이가 \\(0\\)이 되어 분할 정복 전략이 제 기능을 하지 못하고, 퀵 정렬은 '버블 정렬'과 비슷한 형태로 퇴화한다.</p> <p>이 상황을 최대한 피하려면 분할에 사용할 피벗의 선택 전략을 최적화할 수 있다. 예를 들어 원소 하나를 무작위로 골라 피벗으로 삼을 수 있다. 하지만 운이 나빠 매번 좋지 않은 피벗을 고르면 여전히 만족스러운 효율을 얻지 못한다.</p> <p>프로그래밍 언어가 일반적으로 생성하는 것은 '의사 난수'라는 점에 주의하자. 의사 난수 수열에 맞추어 특정 테스트 사례를 만들면 퀵 정렬의 효율이 여전히 저하될 수 있다.</p> <p>더 개선하려면 배열에서 후보 원소 세 개(보통 첫 원소, 마지막 원소, 중간 원소)를 고르고, 이 세 후보의 중앙값을 피벗으로 사용한다. 그러면 피벗이 '너무 작지도 크지도 않을' 확률이 크게 높아진다. 후보를 더 많이 골라 알고리즘의 견고함을 높일 수도 있다. 이 방법을 사용하면 시간 복잡도가 \\(O(n^2)\\)으로 저하될 확률이 크게 줄어든다.</p> <p>예시 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def median_three(self, nums: list[int], left: int, mid: int, right: int) -&gt; int:\n    \"\"\"세 후보 원소의 중앙값 선택\"\"\"\n    l, m, r = nums[left], nums[mid], nums[right]\n    if (l &lt;= m &lt;= r) or (r &lt;= m &lt;= l):\n        return mid  # m은 l과 r 사이에 있다\n    if (m &lt;= l &lt;= r) or (r &lt;= l &lt;= m):\n        return left  # l은 m과 r 사이에 있다\n    return right\n\ndef partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"센티널 분할(세 수의 중앙값 선택)\"\"\"\n    # nums[left]를 피벗으로 삼는다\n    med = self.median_three(nums, left, (left + right) // 2, right)\n    # 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    nums[left], nums[med] = nums[med], nums[left]\n    # nums[left]를 피벗으로 삼는다\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        # 원소 교환\n        nums[i], nums[j] = nums[j], nums[i]\n    # 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # 피벗의 인덱스를 반환한다\n</code></pre> quick_sort.cpp<pre><code>/* 세 후보 원소의 중앙값 선택 */\nint medianThree(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m은 l과 r 사이에 있다\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l은 m과 r 사이에 있다\n    return right;\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // 세 후보 원소의 중앙값 선택\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    swap(nums[left], nums[med]);\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;                // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;                // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        swap(nums[i], nums[j]); // 두 원소를 교환한다\n    }\n    swap(nums[i], nums[left]);  // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i;                   // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.java<pre><code>/* 세 후보 원소의 중앙값 선택 */\nint medianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m은 l과 r 사이에 있다\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l은 m과 r 사이에 있다\n    return right;\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nint partition(int[] nums, int left, int right) {\n    // 세 후보 원소의 중앙값 선택\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    swap(nums, left, med);\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        swap(nums, i, j); // 두 원소를 교환한다\n    }\n    swap(nums, i, left);  // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i;             // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.cs<pre><code>/* 세 후보 원소의 중앙값 선택 */\nint MedianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m은 l과 r 사이에 있다\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l은 m과 r 사이에 있다\n    return right;\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nint Partition(int[] nums, int left, int right) {\n    // 세 후보 원소의 중앙값 선택\n    int med = MedianThree(nums, left, (left + right) / 2, right);\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    Swap(nums, left, med);\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        Swap(nums, i, j); // 두 원소를 교환한다\n    }\n    Swap(nums, i, left);  // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i;             // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.go<pre><code>/* 세 후보 원소의 중앙값 선택 */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n    l, m, r := nums[left], nums[mid], nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m은 l과 r 사이에 있다\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l은 m과 r 사이에 있다\n    }\n    return right\n}\n\n/* 센티널 분할(세 수의 중앙값 선택)*/\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n    // nums[left]를 피벗으로 삼는다\n    med := q.medianThree(nums, left, (left+right)/2, right)\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    nums[left], nums[med] = nums[med], nums[left]\n    // nums[left]를 피벗으로 삼는다\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- //오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ //왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        //원소 교환\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    //피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    nums[i], nums[left] = nums[left], nums[i]\n    return i //피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.swift<pre><code>/* 세 후보 원소의 중앙값 선택 */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -&gt; Int {\n    let l = nums[left]\n    let m = nums[mid]\n    let r = nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m은 l과 r 사이에 있다\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l은 m과 r 사이에 있다\n    }\n    return right\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // 세 후보 원소의 중앙값 선택\n    let med = medianThree(nums: nums, left: left, mid: left + (right - left) / 2, right: right)\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    nums.swapAt(left, med)\n    return partition(nums: &amp;nums, left: left, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* 세 후보 원소의 중앙값 선택 */\nmedianThree(nums, left, mid, right) {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m은 l과 r 사이에 있다\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l은 m과 r 사이에 있다\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\npartition(nums, left, right) {\n    // 세 후보 원소의 중앙값 선택\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    this.swap(nums, left, med);\n    // nums[left]를 피벗으로 삼는다\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        this.swap(nums, i, j); // 두 원소를 교환한다\n    }\n    this.swap(nums, i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i; // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.ts<pre><code>/* 세 후보 원소의 중앙값 선택 */\nmedianThree(\n    nums: number[],\n    left: number,\n    mid: number,\n    right: number\n): number {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m은 l과 r 사이에 있다\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l은 m과 r 사이에 있다\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\npartition(nums: number[], left: number, right: number): number {\n    // 세 후보 원소의 중앙값 선택\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    this.swap(nums, left, med);\n    // nums[left]를 피벗으로 삼는다\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j--; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i++; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        this.swap(nums, i, j); // 두 원소를 교환한다\n    }\n    this.swap(nums, i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i; // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.dart<pre><code>/* 세 후보 원소의 중앙값 선택 */\nint _medianThree(List&lt;int&gt; nums, int left, int mid, int right) {\n  int l = nums[left], m = nums[mid], r = nums[right];\n  if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n    return mid; // m은 l과 r 사이에 있다\n  if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n    return left; // l은 m과 r 사이에 있다\n  return right;\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // 세 후보 원소의 중앙값 선택\n  int med = _medianThree(nums, left, (left + right) ~/ 2, right);\n  // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n  _swap(nums, left, med);\n  // nums[left]를 피벗으로 삼는다\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n    _swap(nums, i, j); // 두 원소를 교환한다\n  }\n  _swap(nums, i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n  return i; // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.rs<pre><code>/* 세 후보 원소의 중앙값 선택 */\nfn median_three(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) -&gt; usize {\n    let (l, m, r) = (nums[left], nums[mid], nums[right]);\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid; // m은 l과 r 사이에 있다\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left; // l은 m과 r 사이에 있다\n    }\n    right\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // 세 후보 원소의 중앙값 선택\n    let med = Self::median_three(nums, left, (left + right) / 2, right);\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    nums.swap(left, med);\n    // nums[left]를 피벗으로 삼는다\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        nums.swap(i, j); // 두 원소를 교환한다\n    }\n    nums.swap(i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    i // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.c<pre><code>/* 세 후보 원소의 중앙값 선택 */\nint medianThree(int nums[], int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m은 l과 r 사이에 있다\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l은 m과 r 사이에 있다\n    return right;\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nint partitionMedian(int nums[], int left, int right) {\n    // 세 후보 원소의 중앙값 선택\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    swap(nums, left, med);\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        swap(nums, i, j); // 두 원소를 교환한다\n    }\n    swap(nums, i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i;            // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.kt<pre><code>/* 세 후보 원소의 중앙값 선택 */\nfun medianThree(nums: IntArray, left: Int, mid: Int, right: Int): Int {\n    val l = nums[left]\n    val m = nums[mid]\n    val r = nums[right]\n    if ((m in l..r) || (m in r..l))\n        return mid  // m은 l과 r 사이에 있다\n    if ((l in m..r) || (l in r..m))\n        return left // l은 m과 r 사이에 있다\n    return right\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nfun partitionMedian(nums: IntArray, left: Int, right: Int): Int {\n    // 세 후보 원소의 중앙값 선택\n    val med = medianThree(nums, left, (left + right) / 2, right)\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    swap(nums, left, med)\n    // nums[left]를 피벗으로 삼는다\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--                      // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++                      // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        swap(nums, i, j)             // 두 원소를 교환한다\n    }\n    swap(nums, i, left)              // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i                         // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.rb<pre><code>### 세 후보 원소의 중앙값 선택 ###\ndef median_three(nums, left, mid, right)\n  # 세 후보 원소의 중앙값 선택\n  _l, _m, _r = nums[left], nums[mid], nums[right]\n  # m은 l과 r 사이에 있다\n  return mid if (_l &lt;= _m &amp;&amp; _m &lt;= _r) || (_r &lt;= _m &amp;&amp; _m &lt;= _l)\n  # l은 m과 r 사이에 있다\n  return left if (_m &lt;= _l &amp;&amp; _l &lt;= _r) || (_r &lt;= _l &amp;&amp; _l &lt;= _m)\n  return right\nend\n\n### 센티널 분할(세 수의 중앙값 선택)###\ndef partition(nums, left, right)\n  ### nums[left]를 피벗으로 삼는다\n  med = median_three(nums, left, (left + right) / 2, right)\n  # 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n  nums[left], nums[med] = nums[med], nums[left]\n  i, j = left, right\n  while i &lt; j\n    while i &lt; j &amp;&amp; nums[j] &gt;= nums[left]\n      j -= 1 # 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n    end\n    while i &lt; j &amp;&amp; nums[i] &lt;= nums[left]\n      i += 1 # 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n    end\n    # 원소 교환\n    nums[i], nums[j] = nums[j], nums[i]\n  end\n  # 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n  nums[i], nums[left] = nums[left], nums[i]\n  i # 피벗의 인덱스를 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.5   퀵 정렬"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1155","level":2,"title":"11.5.5   재귀 깊이 최적화","text":"<p>일부 입력에서는 퀵 정렬이 많은 공간을 차지할 수 있다. 완전히 정렬된 입력 배열을 예로 들어 보자. 재귀 중인 부분 배열의 길이가 \\(m\\)이라면 매 분할에서 길이 \\(0\\)인 왼쪽 부분 배열과 길이 \\(m - 1\\)인 오른쪽 부분 배열이 생긴다. 이는 재귀 한 단계마다 줄어드는 문제 크기가 매우 작다는 뜻이다(원소 하나만 줄어든다). 재귀 트리의 높이는 \\(n - 1\\)에 이르며, \\(O(n)\\) 크기의 스택 프레임 공간이 필요하다.</p> <p>스택 프레임 공간이 쌓이지 않도록 매번 분할을 마친 뒤 두 부분 배열의 길이를 비교하여 더 짧은 부분 배열에만 재귀를 수행할 수 있다. 짧은 쪽의 길이는 \\(n / 2\\)를 넘지 않으므로 재귀 깊이를 \\(\\log n\\) 이하로 유지할 수 있다. 이로써 최악의 공간 복잡도를 \\(O(\\log n)\\)으로 개선한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"퀵 정렬(재귀 깊이 최적화)\"\"\"\n    # 부분 배열의 길이가 1이면 종료한다\n    while left &lt; right:\n        # 센티널 분할 연산\n        pivot = self.partition(nums, left, right)\n        # 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if pivot - left &lt; right - pivot:\n            self.quick_sort(nums, left, pivot - 1)  # 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1  # 남은 미정렬 구간은 [pivot + 1, right]이다\n        else:\n            self.quick_sort(nums, pivot + 1, right)  # 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1  # 남은 미정렬 구간은 [left, pivot - 1]이다\n</code></pre> quick_sort.cpp<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // 부분 배열의 길이가 1이면 종료한다\n    while (left &lt; right) {\n        // 센티널 분할 연산\n        int pivot = partition(nums, left, right);\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1;                 // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            quickSort(nums, pivot + 1, right); // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1;                 // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.java<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nvoid quickSort(int[] nums, int left, int right) {\n    // 부분 배열의 길이가 1이면 종료한다\n    while (left &lt; right) {\n        // 센티널 분할 연산\n        int pivot = partition(nums, left, right);\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1; // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            quickSort(nums, pivot + 1, right); // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1; // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.cs<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nvoid QuickSort(int[] nums, int left, int right) {\n    // 부분 배열의 길이가 1이면 종료한다\n    while (left &lt; right) {\n        // 센티널 분할 연산\n        int pivot = Partition(nums, left, right);\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - left &lt; right - pivot) {\n            QuickSort(nums, left, pivot - 1);  // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1;  // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            QuickSort(nums, pivot + 1, right); // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1; // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.go<pre><code>/* 퀵 정렬(재귀 깊이 최적화)*/\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n    // 부분 배열의 길이가 1이면 종료한다\n    for left &lt; right {\n        // 센티널 분할 연산\n        pivot := q.partition(nums, left, right)\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if pivot-left &lt; right-pivot {\n            q.quickSort(nums, left, pivot-1) // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1                 // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            q.quickSort(nums, pivot+1, right) // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1                 // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.swift<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\n    var left = left\n    var right = right\n    // 부분 배열의 길이가 1이면 종료한다\n    while left &lt; right {\n        // 센티널 분할 연산\n        let pivot = partition(nums: &amp;nums, left: left, right: right)\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - left) &lt; (right - pivot) {\n            quickSortTailCall(nums: &amp;nums, left: left, right: pivot - 1) // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1 // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            quickSortTailCall(nums: &amp;nums, left: pivot + 1, right: right) // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1 // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.js<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nquickSort(nums, left, right) {\n    // 부분 배열의 길이가 1이면 종료한다\n    while (left &lt; right) {\n        // 센티널 분할 연산\n        let pivot = this.partition(nums, left, right);\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1; // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            this.quickSort(nums, pivot + 1, right); // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1; // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.ts<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nquickSort(nums: number[], left: number, right: number): void {\n    // 부분 배열의 길이가 1이면 종료한다\n    while (left &lt; right) {\n        // 센티널 분할 연산\n        let pivot = this.partition(nums, left, right);\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1; // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            this.quickSort(nums, pivot + 1, right); // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1; // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.dart<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // 부분 배열의 길이가 1이면 종료한다\n  while (left &lt; right) {\n    // 센티널 분할 연산\n    int pivot = _partition(nums, left, right);\n    // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n    if (pivot - left &lt; right - pivot) {\n      quickSort(nums, left, pivot - 1); // 왼쪽 부분 배열을 재귀적으로 정렬한다\n      left = pivot + 1; // 남은 미정렬 구간은 [pivot + 1, right]이다\n    } else {\n      quickSort(nums, pivot + 1, right); // 오른쪽 부분 배열을 재귀적으로 정렬한다\n      right = pivot - 1; // 남은 미정렬 구간은 [left, pivot - 1]이다\n    }\n  }\n}\n</code></pre> quick_sort.rs<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\npub fn quick_sort(mut left: i32, mut right: i32, nums: &amp;mut [i32]) {\n    // 부분 배열의 길이가 1이면 종료한다\n    while left &lt; right {\n        // 센티널 분할 연산\n        let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if pivot - left &lt; right - pivot {\n            Self::quick_sort(left, pivot - 1, nums); // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1; // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            Self::quick_sort(pivot + 1, right, nums); // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1; // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.c<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nvoid quickSortTailCall(int nums[], int left, int right) {\n    // 부분 배열의 길이가 1이면 종료한다\n    while (left &lt; right) {\n        // 센티널 분할 연산\n        int pivot = partition(nums, left, right);\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - left &lt; right - pivot) {\n            // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            quickSortTailCall(nums, left, pivot - 1);\n            // 남은 미정렬 구간은 [pivot + 1, right]이다\n            left = pivot + 1;\n        } else {\n            // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            quickSortTailCall(nums, pivot + 1, right);\n            // 남은 미정렬 구간은 [left, pivot - 1]이다\n            right = pivot - 1;\n        }\n    }\n}\n</code></pre> quick_sort.kt<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nfun quickSortTailCall(nums: IntArray, left: Int, right: Int) {\n    // 부분 배열의 길이가 1이면 종료한다\n    var l = left\n    var r = right\n    while (l &lt; r) {\n        // 센티널 분할 연산\n        val pivot = partition(nums, l, r)\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - l &lt; r - pivot) {\n            quickSort(nums, l, pivot - 1) // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            l = pivot + 1 // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            quickSort(nums, pivot + 1, r) // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            r = pivot - 1 // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.rb<pre><code>### 퀵 정렬(재귀 깊이 최적화)###\ndef quick_sort(nums, left, right)\n  # 부분 배열의 길이가 1이 아니면 재귀 호출한다\n  while left &lt; right\n    # 센티널 분할\n    pivot = partition(nums, left, right)\n    # 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n    if pivot - left &lt; right - pivot\n      quick_sort(nums, left, pivot - 1)\n      left = pivot + 1 # 남은 미정렬 구간은 [pivot + 1, right]이다\n    else\n      quick_sort(nums, pivot + 1, right)\n      right = pivot - 1 # 남은 미정렬 구간은 [left, pivot - 1]이다\n    end\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.5   퀵 정렬"],"tags":[]},{"location":"chapter_sorting/radix_sort/","level":1,"title":"11.10   기수 정렬","text":"<p>앞 절에서 배운 계수 정렬은 데이터 수 \\(n\\)은 많고 값의 범위 \\(m\\)은 작은 경우에 적합하다. 학번 \\(n = 10^6\\) 개를 정렬해야 하며, 학번이 \\(8\\) 자리 수라고 하자. 이때 값의 범위는 \\(m = 10^8\\)로 매우 커서 계수 정렬에는 많은 메모리가 필요하다. 기수 정렬을 사용하면 이 문제를 피할 수 있다.</p> <p>기수 정렬(radix sort)의 핵심 아이디어는 계수 정렬과 같으며, 개수를 세어 정렬한다. 여기에 자릿수 사이의 순서 관계를 이용하여 각 자릿수를 차례로 정렬함으로써 최종 정렬 결과를 얻는다.</p>","path":["제 11 장   정렬","11.10   기수 정렬"],"tags":[]},{"location":"chapter_sorting/radix_sort/#11101","level":2,"title":"11.10.1   알고리즘 과정","text":"<p>학번을 예로 들어 가장 낮은 자리를 \\(1\\) 번째, 가장 높은 자리를 \\(8\\) 번째라고 하자. 기수 정렬의 과정은 아래 그림(11-18)과 같다.</p> <ol> <li>자릿수 \\(k = 1\\)로 초기화한다.</li> <li>학번의 \\(k\\) 번째 자리를 기준으로 '계수 정렬'을 수행한다. 완료하면 데이터는 \\(k\\) 번째 자리의 값에 따라 오름차순으로 정렬된다.</li> <li>\\(k\\)를 \\(1\\) 늘리고 <code>2.</code> 단계로 돌아가 반복한다. 모든 자리를 정렬하면 종료한다.</li> </ol> <p></p> <p> 그림 11-18   기수 정렬 알고리즘 과정 </p> <p>이제 코드 구현을 살펴보자. \\(d\\) 진법으로 표현한 수 \\(x\\)의 \\(k\\) 번째 자리 \\(x_k\\)는 다음 공식으로 구할 수 있다.</p> \\[ x_k = \\lfloor\\frac{x}{d^{k-1}}\\rfloor \\bmod d \\] <p>여기서 \\(\\lfloor a \\rfloor\\)는 부동 소수점 수 \\(a\\)를 내림하는 연산이고, \\(\\bmod \\: d\\)는 \\(d\\)로 나눈 나머지를 구하는 연산이다. 학번 데이터에서는 \\(d = 10\\)이고 \\(k \\in [1, 8]\\)이다.</p> <p>또한 수의 \\(k\\) 번째 자리를 기준으로 정렬할 수 있도록 계수 정렬 코드를 조금 수정해야 한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby radix_sort.py<pre><code>def digit(num: int, exp: int) -&gt; int:\n    \"\"\"원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다\"\"\"\n    # k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return (num // exp) % 10\n\ndef counting_sort_digit(nums: list[int], exp: int):\n    \"\"\"계수 정렬(nums의 k번째 자릿값을 기준으로 정렬)\"\"\"\n    # 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    counter = [0] * 10\n    n = len(nums)\n    # 0~9 각 숫자의 등장 횟수를 센다\n    for i in range(n):\n        d = digit(nums[i], exp)  # nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d] += 1  # 숫자 d의 등장 횟수를 센다\n    # 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for i in range(1, 10):\n        counter[i] += counter[i - 1]\n    # 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        d = digit(nums[i], exp)\n        j = counter[d] - 1  # 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i]  # 현재 원소를 인덱스 j에 채운다\n        counter[d] -= 1  # d의 개수를 1 줄인다\n    # 결과로 원래 배열 nums를 덮어쓴다\n    for i in range(n):\n        nums[i] = res[i]\n\ndef radix_sort(nums: list[int]):\n    \"\"\"기수 정렬\"\"\"\n    # 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    m = max(nums)\n    # 낮은 자리에서 높은 자리 순서로 순회한다\n    exp = 1\n    while exp &lt;= m:\n        # 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        # k = 1 -&gt; exp = 1\n        # k = 2 -&gt; exp = 10\n        # 즉, exp = 10^(k-1)이다\n        counting_sort_digit(nums, exp)\n        exp *= 10\n</code></pre> radix_sort.cpp<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nint digit(int num, int exp) {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return (num / exp) % 10;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nvoid countingSortDigit(vector&lt;int&gt; &amp;nums, int exp) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    vector&lt;int&gt; counter(10, 0);\n    int n = nums.size();\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d]++;                // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    vector&lt;int&gt; res(n, 0);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i];       // 현재 원소를 인덱스 j에 채운다\n        counter[d]--;           // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* 기수 정렬 */\nvoid radixSort(vector&lt;int&gt; &amp;nums) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    int m = *max_element(nums.begin(), nums.end());\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for (int exp = 1; exp &lt;= m; exp *= 10)\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.java<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nint digit(int num, int exp) {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return (num / exp) % 10;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nvoid countingSortDigit(int[] nums, int exp) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    int[] counter = new int[10];\n    int n = nums.length;\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d]++;                // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i];       // 현재 원소를 인덱스 j에 채운다\n        counter[d]--;           // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* 기수 정렬 */\nvoid radixSort(int[] nums) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    int m = Integer.MIN_VALUE;\n    for (int num : nums)\n        if (num &gt; m)\n            m = num;\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for (int exp = 1; exp &lt;= m; exp *= 10) {\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.cs<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nint Digit(int num, int exp) {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return (num / exp) % 10;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nvoid CountingSortDigit(int[] nums, int exp) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    int[] counter = new int[10];\n    int n = nums.Length;\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for (int i = 0; i &lt; n; i++) {\n        int d = Digit(nums[i], exp); // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d]++;                // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = Digit(nums[i], exp);\n        int j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i];       // 현재 원소를 인덱스 j에 채운다\n        counter[d]--;           // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* 기수 정렬 */\nvoid RadixSort(int[] nums) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    int m = int.MinValue;\n    foreach (int num in nums) {\n        if (num &gt; m) m = num;\n    }\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for (int exp = 1; exp &lt;= m; exp *= 10) {\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        CountingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.go<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nfunc digit(num, exp int) int {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return (num / exp) % 10\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nfunc countingSortDigit(nums []int, exp int) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    counter := make([]int, 10)\n    n := len(nums)\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for i := 0; i &lt; n; i++ {\n        d := digit(nums[i], exp) // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d]++             // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for i := 1; i &lt; 10; i++ {\n        counter[i] += counter[i-1]\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        d := digit(nums[i], exp)\n        j := counter[d] - 1 // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i]    // 현재 원소를 인덱스 j에 채운다\n        counter[d]--        // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for i := 0; i &lt; n; i++ {\n        nums[i] = res[i]\n    }\n}\n\n/* 기수 정렬 */\nfunc radixSort(nums []int) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    max := math.MinInt\n    for _, num := range nums {\n        if num &gt; max {\n            max = num\n        }\n    }\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for exp := 1; max &gt;= exp; exp *= 10 {\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums, exp)\n    }\n}\n</code></pre> radix_sort.swift<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nfunc digit(num: Int, exp: Int) -&gt; Int {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    (num / exp) % 10\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nfunc countingSortDigit(nums: inout [Int], exp: Int) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    var counter = Array(repeating: 0, count: 10)\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for i in nums.indices {\n        let d = digit(num: nums[i], exp: exp) // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d] += 1 // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for i in 1 ..&lt; 10 {\n        counter[i] += counter[i - 1]\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let d = digit(num: nums[i], exp: exp)\n        let j = counter[d] - 1 // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i] // 현재 원소를 인덱스 j에 채운다\n        counter[d] -= 1 // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n\n/* 기수 정렬 */\nfunc radixSort(nums: inout [Int]) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    var m = Int.min\n    for num in nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for exp in sequence(first: 1, next: { m &gt;= ($0 * 10) ? $0 * 10 : nil }) {\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums: &amp;nums, exp: exp)\n    }\n}\n</code></pre> radix_sort.js<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nfunction digit(num, exp) {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return Math.floor(num / exp) % 10;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nfunction countingSortDigit(nums, exp) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d]++; // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i]; // 현재 원소를 인덱스 j에 채운다\n        counter[d]--; // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* 기수 정렬 */\nfunction radixSort(nums) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    let m = Math.max(... nums);\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.ts<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nfunction digit(num: number, exp: number): number {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return Math.floor(num / exp) % 10;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nfunction countingSortDigit(nums: number[], exp: number): void {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d]++; // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i]; // 현재 원소를 인덱스 j에 채운다\n        counter[d]--; // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* 기수 정렬 */\nfunction radixSort(nums: number[]): void {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    let m: number = Math.max(... nums);\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.dart<pre><code>/* 원소 _num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nint digit(int _num, int exp) {\n  // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n  return (_num ~/ exp) % 10;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nvoid countingSortDigit(List&lt;int&gt; nums, int exp) {\n  // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n  List&lt;int&gt; counter = List&lt;int&gt;.filled(10, 0);\n  int n = nums.length;\n  // 0~9 각 숫자의 등장 횟수를 센다\n  for (int i = 0; i &lt; n; i++) {\n    int d = digit(nums[i], exp); // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n    counter[d]++; // 숫자 d의 등장 횟수를 센다\n  }\n  // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n  for (int i = 1; i &lt; 10; i++) {\n    counter[i] += counter[i - 1];\n  }\n  // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n  List&lt;int&gt; res = List&lt;int&gt;.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int d = digit(nums[i], exp);\n    int j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n    res[j] = nums[i]; // 현재 원소를 인덱스 j에 채운다\n    counter[d]--; // d의 개수를 1 줄인다\n  }\n  // 결과로 원래 배열 nums를 덮어쓴다\n  for (int i = 0; i &lt; n; i++) nums[i] = res[i];\n}\n\n/* 기수 정렬 */\nvoid radixSort(List&lt;int&gt; nums) {\n  // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n  // Dart에서 int의 길이는 64비트이다\n  int m = -1 &lt;&lt; 63;\n  for (int _num in nums) if (_num &gt; m) m = _num;\n  // 낮은 자리에서 높은 자리 순서로 순회한다\n  for (int exp = 1; exp &lt;= m; exp *= 10)\n    // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n    // k = 1 -&gt; exp = 1\n    // k = 2 -&gt; exp = 10\n    // 즉, exp = 10^(k-1)이다\n    countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.rs<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nfn digit(num: i32, exp: i32) -&gt; usize {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return ((num / exp) % 10) as usize;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nfn counting_sort_digit(nums: &amp;mut [i32], exp: i32) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    let mut counter = [0; 10];\n    let n = nums.len();\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for i in 0..n {\n        let d = digit(nums[i], exp); // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d] += 1; // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for i in 1..10 {\n        counter[i] += counter[i - 1];\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let d = digit(nums[i], exp);\n        let j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i]; // 현재 원소를 인덱스 j에 채운다\n        counter[d] -= 1; // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    nums.copy_from_slice(&amp;res);\n}\n\n/* 기수 정렬 */\nfn radix_sort(nums: &amp;mut [i32]) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    let m = *nums.into_iter().max().unwrap();\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    let mut exp = 1;\n    while exp &lt;= m {\n        counting_sort_digit(nums, exp);\n        exp *= 10;\n    }\n}\n</code></pre> radix_sort.c<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nint digit(int num, int exp) {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return (num / exp) % 10;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nvoid countingSortDigit(int nums[], int size, int exp) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    int *counter = (int *)malloc((sizeof(int) * 10));\n    memset(counter, 0, sizeof(int) * 10); // 이후 메모리 해제를 지원하도록 0으로 초기화한다\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for (int i = 0; i &lt; size; i++) {\n        // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        int d = digit(nums[i], exp);\n        // 숫자 d의 등장 횟수를 센다\n        counter[d]++;\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    int *res = (int *)malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i];       // 현재 원소를 인덱스 j에 채운다\n        counter[d]--;           // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for (int i = 0; i &lt; size; i++) {\n        nums[i] = res[i];\n    }\n    // 메모리 해제\n    free(res);\n    free(counter);\n}\n\n/* 기수 정렬 */\nvoid radixSort(int nums[], int size) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    int max = INT32_MIN;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; max) {\n            max = nums[i];\n        }\n    }\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for (int exp = 1; max &gt;= exp; exp *= 10)\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums, size, exp);\n}\n</code></pre> radix_sort.kt<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nfun digit(num: Int, exp: Int): Int {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return (num / exp) % 10\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nfun countingSortDigit(nums: IntArray, exp: Int) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    val counter = IntArray(10)\n    val n = nums.size\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for (i in 0..&lt;n) {\n        val d = digit(nums[i], exp) // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d]++                // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for (i in 1..9) {\n        counter[i] += counter[i - 1]\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val d = digit(nums[i], exp)\n        val j = counter[d] - 1 // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i]       // 현재 원소를 인덱스 j에 채운다\n        counter[d]--           // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for (i in 0..&lt;n)\n        nums[i] = res[i]\n}\n\n/* 기수 정렬 */\nfun radixSort(nums: IntArray) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    var m = Int.MIN_VALUE\n    for (num in nums) if (num &gt; m) m = num\n    var exp = 1\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    while (exp &lt;= m) {\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums, exp)\n        exp *= 10\n    }\n}\n</code></pre> radix_sort.rb<pre><code>### 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 ###\ndef digit(num, exp)\n  # k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n  (num / exp) % 10\nend\n\n### 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬)###\ndef counting_sort_digit(nums, exp)\n  # 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n  counter = Array.new(10, 0)\n  n = nums.length\n  # 0~9 각 숫자의 등장 횟수를 센다\n  for i in 0...n\n    d = digit(nums[i], exp) # nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n    counter[d] += 1 # 숫자 d의 등장 횟수를 센다\n  end\n  # 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n  (1...10).each { |i| counter[i] += counter[i - 1] }\n  # 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n  res = Array.new(n, 0)\n  for i in (n - 1).downto(0)\n    d = digit(nums[i], exp)\n    j = counter[d] - 1 # 배열에서 d의 인덱스 j를 구한다\n    res[j] = nums[i] # 현재 원소를 인덱스 j에 채운다\n    counter[d] -= 1 # d의 개수를 1 줄인다\n  end\n  # 결과로 원래 배열 nums를 덮어쓴다\n  (0...n).each { |i| nums[i] = res[i] }\nend\n\n### 기수 정렬 ###\ndef radix_sort(nums)\n  # 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n  m = nums.max\n  # 낮은 자리에서 높은 자리 순서로 순회한다\n  exp = 1\n  while exp &lt;= m\n    # 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n    # k = 1 -&gt; exp = 1\n    # k = 2 -&gt; exp = 10\n    # 즉, exp = 10^(k-1)이다\n    counting_sort_digit(nums, exp)\n    exp *= 10\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>왜 가장 낮은 자리부터 정렬하는가?</p> <p>여러 차례 연속으로 정렬하면 뒤의 정렬 결과가 앞의 결과를 덮어쓴다. 예를 들어 첫 번째 정렬에서는 \\(a &lt; b\\)였지만 두 번째 정렬에서는 \\(a &gt; b\\)라면, 두 번째 결과가 첫 번째 결과를 대신한다. 높은 자릿수가 낮은 자릿수보다 우선하므로 낮은 자리부터 정렬한 뒤 높은 자리를 정렬해야 한다.</p>","path":["제 11 장   정렬","11.10   기수 정렬"],"tags":[]},{"location":"chapter_sorting/radix_sort/#11102","level":2,"title":"11.10.2   알고리즘 특성","text":"<p>기수 정렬은 계수 정렬보다 값의 범위가 큰 경우에 적합하다. 단, 데이터를 고정된 자릿수의 형식으로 표현할 수 있어야 하며 자릿수가 지나치게 크면 안 된다. 예를 들어 부동 소수점 수는 자릿수 \\(k\\)가 너무 커서 시간 복잡도가 \\(O(nk) \\gg O(n^2)\\)이 될 수 있으므로 기수 정렬에 적합하지 않다.</p> <ul> <li>시간 복잡도 \\(O(nk)\\), 비적응형 정렬: 데이터 수가 \\(n\\), 진법이 \\(d\\), 최대 자릿수가 \\(k\\)라면 한 자리를 계수 정렬하는 데 \\(O(n + d)\\) 시간이 걸리고, \\(k\\) 자리 전체를 정렬하는 데 \\(O((n + d)k)\\) 시간이 걸린다. 일반적으로 \\(d\\)와 \\(k\\)는 모두 비교적 작으므로 시간 복잡도는 \\(O(n)\\)에 가까워진다.</li> <li>공간 복잡도 \\(O(n + d)\\), 제자리 정렬이 아님: 계수 정렬과 마찬가지로 길이가 \\(n\\)과 \\(d\\)인 배열 <code>res</code>와 <code>counter</code>가 필요하다.</li> <li>안정 정렬: 계수 정렬이 안정적이면 기수 정렬도 안정적이다. 계수 정렬이 불안정하면 기수 정렬은 올바른 정렬 결과를 보장할 수 없다.</li> </ul>","path":["제 11 장   정렬","11.10   기수 정렬"],"tags":[]},{"location":"chapter_sorting/selection_sort/","level":1,"title":"11.2   선택 정렬","text":"<p>선택 정렬(selection sort)의 원리는 매우 간단하다. 반복문을 실행하여 매번 정렬되지 않은 구간에서 가장 작은 원소를 고른 뒤 정렬된 구간의 끝에 놓는다.</p> <p>배열의 길이가 \\(n\\)일 때 선택 정렬의 과정은 아래 그림(11-2)과 같다.</p> <ol> <li>처음에는 모든 원소가 정렬되지 않았으므로 미정렬 구간(인덱스)은 \\([0, n-1]\\)이다.</li> <li>구간 \\([0, n-1]\\)에서 가장 작은 원소를 골라 인덱스 \\(0\\)의 원소와 교환한다. 그러면 배열의 앞 1개 원소가 정렬된다.</li> <li>구간 \\([1, n-1]\\)에서 가장 작은 원소를 골라 인덱스 \\(1\\)의 원소와 교환한다. 그러면 배열의 앞 2개 원소가 정렬된다.</li> <li>같은 과정을 반복한다. \\(n - 1\\) 번 선택하고 교환하면 배열의 앞 \\(n - 1\\) 개 원소가 정렬된다.</li> <li>남은 원소 하나는 반드시 가장 큰 원소이므로 정렬할 필요가 없으며, 배열 정렬이 완료된다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 11-2   선택 정렬 단계 </p> <p>코드에서는 \\(k\\)로 미정렬 구간의 최소 원소를 기록한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby selection_sort.py<pre><code>def selection_sort(nums: list[int]):\n    \"\"\"선택 정렬\"\"\"\n    n = len(nums)\n    # 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for i in range(n - 1):\n        # 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        k = i\n        for j in range(i + 1, n):\n            if nums[j] &lt; nums[k]:\n                k = j  # 최솟값의 인덱스를 기록한다\n        # 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        nums[i], nums[k] = nums[k], nums[i]\n</code></pre> selection_sort.cpp<pre><code>/* 선택 정렬 */\nvoid selectionSort(vector&lt;int&gt; &amp;nums) {\n    int n = nums.size();\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for (int i = 0; i &lt; n - 1; i++) {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // 최솟값의 인덱스를 기록한다\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        swap(nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.java<pre><code>/* 선택 정렬 */\nvoid selectionSort(int[] nums) {\n    int n = nums.length;\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for (int i = 0; i &lt; n - 1; i++) {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // 최솟값의 인덱스를 기록한다\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.cs<pre><code>/* 선택 정렬 */\nvoid SelectionSort(int[] nums) {\n    int n = nums.Length;\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for (int i = 0; i &lt; n - 1; i++) {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // 최솟값의 인덱스를 기록한다\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        (nums[k], nums[i]) = (nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.go<pre><code>/* 선택 정렬 */\nfunc selectionSort(nums []int) {\n    n := len(nums)\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for i := 0; i &lt; n-1; i++ {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        k := i\n        for j := i + 1; j &lt; n; j++ {\n            if nums[j] &lt; nums[k] {\n                // 최솟값의 인덱스를 기록한다\n                k = j\n            }\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        nums[i], nums[k] = nums[k], nums[i]\n\n    }\n}\n</code></pre> selection_sort.swift<pre><code>/* 선택 정렬 */\nfunc selectionSort(nums: inout [Int]) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for i in nums.indices.dropLast() {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        var k = i\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[j] &lt; nums[k] {\n                k = j // 최솟값의 인덱스를 기록한다\n            }\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        nums.swapAt(i, k)\n    }\n}\n</code></pre> selection_sort.js<pre><code>/* 선택 정렬 */\nfunction selectionSort(nums) {\n    let n = nums.length;\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for (let i = 0; i &lt; n - 1; i++) {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // 최솟값의 인덱스를 기록한다\n            }\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.ts<pre><code>/* 선택 정렬 */\nfunction selectionSort(nums: number[]): void {\n    let n = nums.length;\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for (let i = 0; i &lt; n - 1; i++) {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // 최솟값의 인덱스를 기록한다\n            }\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.dart<pre><code>/* 선택 정렬 */\nvoid selectionSort(List&lt;int&gt; nums) {\n  int n = nums.length;\n  // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n  for (int i = 0; i &lt; n - 1; i++) {\n    // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n    int k = i;\n    for (int j = i + 1; j &lt; n; j++) {\n      if (nums[j] &lt; nums[k]) k = j; // 최솟값의 인덱스를 기록한다\n    }\n    // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n    int temp = nums[i];\n    nums[i] = nums[k];\n    nums[k] = temp;\n  }\n}\n</code></pre> selection_sort.rs<pre><code>/* 선택 정렬 */\nfn selection_sort(nums: &amp;mut [i32]) {\n    if nums.is_empty() {\n        return;\n    }\n    let n = nums.len();\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for i in 0..n - 1 {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        let mut k = i;\n        for j in i + 1..n {\n            if nums[j] &lt; nums[k] {\n                k = j; // 최솟값의 인덱스를 기록한다\n            }\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        nums.swap(i, k);\n    }\n}\n</code></pre> selection_sort.c<pre><code>/* 선택 정렬 */\nvoid selectionSort(int nums[], int n) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for (int i = 0; i &lt; n - 1; i++) {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // 최솟값의 인덱스를 기록한다\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.kt<pre><code>/* 선택 정렬 */\nfun selectionSort(nums: IntArray) {\n    val n = nums.size\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for (i in 0..&lt;n - 1) {\n        var k = i\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        for (j in i + 1..&lt;n) {\n            if (nums[j] &lt; nums[k])\n                k = j // 최솟값의 인덱스를 기록한다\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        val temp = nums[i]\n        nums[i] = nums[k]\n        nums[k] = temp\n    }\n}\n</code></pre> selection_sort.rb<pre><code>### 선택 정렬 ###\ndef selection_sort(nums)\n  n = nums.length\n  # 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n  for i in 0...(n - 1)\n    # 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n    k = i\n    for j in (i + 1)...n\n      if nums[j] &lt; nums[k]\n        k = j # 최솟값의 인덱스를 기록한다\n      end\n    end\n    # 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n    nums[i], nums[k] = nums[k], nums[i]\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.2   선택 정렬"],"tags":[]},{"location":"chapter_sorting/selection_sort/#1121","level":2,"title":"11.2.1   알고리즘 특성","text":"<ul> <li>시간 복잡도 \\(O(n^2)\\), 비적응형 정렬: 바깥 반복문은 \\(n - 1\\) 번 실행된다. 안쪽 반복문은 첫 회에 \\(n - 1\\) 번, 마지막 회에 \\(1\\) 번 실행된다. 즉, 각 회의 실행 횟수는 \\(n - 1\\), \\(n - 2\\), \\(\\dots\\), \\(2\\), \\(1\\)이고 합은 \\(\\frac{n(n - 1)}{2}\\)이다.</li> <li>공간 복잡도 \\(O(1)\\), 제자리 정렬: 포인터 \\(i\\)와 \\(j\\)는 상수 크기의 추가 공간을 사용한다.</li> <li>불안정 정렬: 아래 그림(11-3)과 같이 원소 <code>nums[i]</code>가 자신과 값이 같은 원소의 오른쪽으로 교환되어 두 원소의 상대적 순서가 바뀔 수 있다.</li> </ul> <p> 그림 11-3   선택 정렬의 불안정성 예시 </p>","path":["제 11 장   정렬","11.2   선택 정렬"],"tags":[]},{"location":"chapter_sorting/sorting_algorithm/","level":1,"title":"11.1   정렬 알고리즘","text":"<p>정렬 알고리즘(sorting algorithm)은 데이터 집합을 특정 순서로 나열하는 데 사용한다. 정렬된 데이터는 대체로 더 효율적으로 탐색, 분석, 처리할 수 있으므로 정렬 알고리즘은 널리 활용된다.</p> <p>아래 그림(11-1)과 같이 정렬할 데이터의 타입은 정수, 부동 소수점 수, 문자, 문자열 등 다양하다. 정렬 기준도 수의 크기, 문자의 ASCII 코드 순서, 사용자 정의 규칙 등 필요에 따라 정할 수 있다.</p> <p></p> <p> 그림 11-1   데이터 타입과 정렬 기준의 예 </p>","path":["제 11 장   정렬","11.1   정렬 알고리즘"],"tags":[]},{"location":"chapter_sorting/sorting_algorithm/#1111","level":2,"title":"11.1.1   평가 기준","text":"<p>실행 효율: 정렬 알고리즘은 시간 복잡도가 가능한 한 낮고, 전체 연산 수도 적기를 기대한다(시간 복잡도의 상수항이 작아야 한다). 데이터가 많을 때는 실행 효율이 특히 중요하다.</p> <p>제자리 여부: 이름 그대로 제자리 정렬은 추가 보조 배열 없이 원래 배열을 직접 조작하여 정렬하므로 메모리를 절약한다. 보통 데이터 이동 연산이 적어 실행 속도도 더 빠르다.</p> <p>안정성: 안정 정렬은 정렬 후에도 값이 같은 원소들의 배열 내 상대적 순서가 바뀌지 않는다.</p> <p>안정 정렬은 여러 기준으로 순차 정렬할 때 필요한 조건이다. 학생 정보를 저장한 표의 첫 번째 열이 이름, 두 번째 열이 나이라고 하자. 이때 불안정 정렬을 사용하면 입력 데이터가 기존에 정렬되어 있던 순서가 깨질 수 있다.</p> <pre><code># 입력 데이터는 이름순으로 정렬되어 있다\n# (name, age)\n  ('A', 19)\n  ('B', 18)\n  ('C', 21)\n  ('D', 19)\n  ('E', 23)\n\n# 불안정 정렬 알고리즘으로 리스트를 나이순으로 정렬하면,\n# 결과에서 ('D', 19)와 ('A', 19)의 상대적 위치가 바뀌어,\n# 입력 데이터의 이름순 정렬 상태가 사라질 수 있다\n  ('B', 18)\n  ('D', 19)\n  ('A', 19)\n  ('C', 21)\n  ('E', 23)\n</code></pre> <p>적응성: 적응형 정렬은 입력 데이터에 이미 존재하는 순서 정보를 이용해 계산량을 줄이고 시간 효율을 높일 수 있다. 적응형 정렬 알고리즘의 최선 시간 복잡도는 보통 평균 시간 복잡도보다 낮다.</p> <p>비교 기반 여부: 비교 기반 정렬은 비교 연산자(\\(&lt;\\), \\(=\\), \\(&gt;\\))로 원소의 상대적 순서를 판단하여 배열 전체를 정렬한다. 최악의 시간 복잡도 하한은 \\(\\Omega(n \\log n)\\)이다. 반면 비교를 사용하지 않는 정렬은 비교 연산자를 사용하지 않으며 시간 복잡도가 \\(O(n)\\)에 이를 수 있지만, 범용성은 상대적으로 낮다.</p>","path":["제 11 장   정렬","11.1   정렬 알고리즘"],"tags":[]},{"location":"chapter_sorting/sorting_algorithm/#1112","level":2,"title":"11.1.2   이상적인 정렬 알고리즘","text":"<p>빠른 실행, 제자리 정렬, 안정성, 적응성, 높은 범용성을 모두 갖추면 이상적이다. 하지만 지금까지 이러한 특성을 모두 갖춘 정렬 알고리즘은 발견되지 않았다. 따라서 데이터의 구체적인 특성과 문제의 요구에 맞추어 정렬 알고리즘을 선택해야 한다.</p> <p>이제 다양한 정렬 알고리즘을 배우고, 위 평가 기준에 따라 각각의 장단점을 분석해 보자.</p>","path":["제 11 장   정렬","11.1   정렬 알고리즘"],"tags":[]},{"location":"chapter_sorting/summary/","level":1,"title":"11.11   요약","text":"","path":["제 11 장   정렬","11.11   요약"],"tags":[]},{"location":"chapter_sorting/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>버블 정렬은 인접한 원소를 교환하여 정렬한다. 플래그를 추가해 조기에 반환하도록 하면 최선 시간 복잡도를 \\(O(n)\\)으로 최적화할 수 있다.</li> <li>삽입 정렬은 매번 미정렬 구간의 원소를 정렬된 구간의 올바른 위치에 삽입한다. 시간 복잡도는 \\(O(n^2)\\)이지만 기본 연산 수가 비교적 적어, 데이터가 적은 정렬 작업에서 매우 많이 사용된다.</li> <li>퀵 정렬은 센티널 분할 연산으로 정렬한다. 분할할 때마다 최악의 피벗을 고르면 시간 복잡도가 \\(O(n^2)\\)으로 저하될 수 있다. 중앙값이나 무작위 원소를 피벗으로 사용하면 이런 확률을 줄일 수 있다. 더 짧은 부분 구간에 먼저 재귀하면 재귀 깊이를 효과적으로 줄여 공간 복잡도를 \\(O(\\log n)\\)으로 최적화할 수 있다.</li> <li>병합 정렬은 분할과 병합의 두 단계로 이루어지며 분할 정복 전략을 전형적으로 보여 준다. 배열을 정렬할 때는 보조 배열이 필요하여 공간 복잡도가 \\(O(n)\\)이지만, 연결 리스트 정렬은 \\(O(1)\\)까지 최적화할 수 있다.</li> <li>버킷 정렬은 데이터를 버킷에 분배하고, 각 버킷 안에서 정렬하고, 결과를 합치는 세 단계로 이루어진다. 역시 분할 정복 전략을 보여 주며 매우 큰 데이터에 적합하다. 핵심은 데이터를 고르게 분배하는 것이다.</li> <li>계수 정렬은 버킷 정렬의 특수한 경우로, 데이터의 등장 횟수를 세어 정렬한다. 데이터 수는 많지만 값의 범위가 제한된 경우에 적합하며, 데이터를 양의 정수로 변환할 수 있어야 한다.</li> <li>기수 정렬은 자릿수별로 정렬하여 전체 데이터를 정렬하며, 데이터를 고정된 자릿수의 수로 표현할 수 있어야 한다.</li> <li>우리는 효율, 안정성, 제자리 동작, 적응성 등의 장점을 갖춘 정렬 알고리즘을 원한다. 하지만 다른 자료 구조와 알고리즘처럼 이 모든 조건을 동시에 만족하는 정렬 알고리즘은 없다. 실제 응용에서는 데이터의 특성에 따라 적합한 알고리즘을 선택해야 한다.</li> <li>아래 그림(11-19)은 주요 정렬 알고리즘의 효율, 안정성, 제자리 여부, 적응성 등을 비교한다.</li> </ul> <p> 그림 11-19   정렬 알고리즘 비교 </p>","path":["제 11 장   정렬","11.11   요약"],"tags":[]},{"location":"chapter_sorting/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 정렬 알고리즘의 안정성은 어떤 상황에서 필요한가?</p> <p>실제로 객체의 특정 속성을 기준으로 정렬하는 경우가 있다. 예를 들어 학생에게 이름과 키라는 두 속성이 있고, 여러 기준으로 순차 정렬하고 싶다고 하자. 먼저 이름순으로 정렬하여 <code>(A, 180) (B, 185) (C, 170) (D, 170)</code>을 얻은 뒤 키순으로 정렬한다. 정렬 알고리즘이 불안정하면 <code>(D, 170) (C, 170) (A, 180) (B, 185)</code>가 나올 수 있다.</p> <p>학생 D와 C의 위치가 바뀌어 이름순 정렬이 깨졌음을 알 수 있으며, 이는 원하는 결과가 아니다.</p> <p>Q: 센티널 분할에서 '오른쪽에서 왼쪽으로 찾기'와 '왼쪽에서 오른쪽으로 찾기'의 순서를 바꿔도 되는가?</p> <p>안 된다. 맨 왼쪽 원소를 피벗으로 선택하면 반드시 '오른쪽에서 왼쪽으로 찾기'를 먼저 수행하고, 그다음 '왼쪽에서 오른쪽으로 찾기'를 수행해야 한다. 직관과 다르게 느껴질 수 있으니 이유를 분석해 보자.</p> <p>분할 함수 <code>partition()</code>의 마지막 단계는 <code>nums[left]</code>와 <code>nums[i]</code>를 교환하는 것이다. 교환 후에는 피벗 왼쪽의 모든 원소에 대해 '원소 <code>&lt;=</code> 피벗'이 성립해야 하므로, 마지막 교환 전에 반드시 <code>nums[left] &gt;= nums[i]</code>가 성립해야 한다. 먼저 '왼쪽에서 오른쪽으로 찾기'를 수행했는데 피벗보다 큰 원소를 찾지 못한다면, <code>i == j</code>일 때 반복문을 빠져나오며 이때 <code>nums[j] == nums[i] &gt; nums[left]</code>일 수 있다. 즉, 마지막 교환이 피벗보다 큰 원소를 맨 왼쪽으로 옮겨 분할을 실패하게 할 수 있다.</p> <p>예를 들어 배열이 <code>[0, 0, 0, 0, 1]</code>일 때 '왼쪽에서 오른쪽으로 찾기'를 먼저 하면, 분할 결과가 <code>[1, 0, 0, 0, 0]</code>으로 잘못 나온다.</p> <p>더 생각해 보면 <code>nums[right]</code>를 피벗으로 선택했을 때는 반대로 '왼쪽에서 오른쪽으로 찾기'를 먼저 해야 한다.</p> <p>Q: 퀵 정렬의 재귀 깊이를 최적화할 때 짧은 배열을 선택하면 왜 깊이가 \\(\\log n\\)을 넘지 않는가?</p> <p>재귀 깊이는 현재 아직 반환되지 않은 재귀 메서드의 수다. 매 분할에서 원래 배열을 두 부분 배열로 나눈다. 재귀 깊이를 최적화하면 아래로 재귀하는 부분 배열의 길이는 최대 원래 길이의 절반이다. 최악의 경우 매번 정확히 절반이라고 해도 최종 재귀 깊이는 \\(\\log n\\)이다.</p> <p>원래 퀵 정렬에서는 더 긴 배열에 계속 재귀할 수 있다. 최악의 경우 길이가 \\(n\\), \\(n - 1\\), \\(\\dots\\), \\(2\\), \\(1\\)로 줄어 재귀 깊이가 \\(n\\)이 된다. 재귀 깊이 최적화는 이 상황을 방지한다.</p> <p>Q: 배열의 모든 원소가 같으면 퀵 정렬의 시간 복잡도는 \\(O(n^2)\\)인가? 이런 퇴화는 어떻게 처리하는가?</p> <p>그렇다. 이 경우 배열을 피벗보다 작은 부분, 같은 부분, 큰 부분의 세 부분으로 나누고 작은 부분과 큰 부분에만 재귀하는 방법을 고려할 수 있다. 이렇게 하면 모든 원소가 같은 입력 배열은 분할 한 번으로 정렬된다.</p> <p>Q: 버킷 정렬의 최악 시간 복잡도는 왜 \\(O(n^2)\\)인가?</p> <p>최악의 경우 모든 원소가 같은 버킷에 들어간다. 이 원소들을 \\(O(n^2)\\) 알고리즘으로 정렬한다면 시간 복잡도는 \\(O(n^2)\\)이다.</p>","path":["제 11 장   정렬","11.11   요약"],"tags":[]},{"location":"chapter_stack_and_queue/","level":1,"title":"제 5 장   스택과 큐","text":"<p>이 장을 시작하며</p> <p>스택은 고양이들이 포개져 있는 모습과 같고, 큐는 고양이들이 줄을 서 있는 모습과 같다.</p> <p>둘은 각각 후입선출과 선입선출의 논리적 관계를 나타낸다.</p>","path":["제 5 장   스택과 큐"],"tags":[]},{"location":"chapter_stack_and_queue/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>5.1   스택</li> <li>5.2   큐</li> <li>5.3   덱</li> <li>5.4   요약</li> <li>5.5   연습 문제</li> </ul>","path":["제 5 장   스택과 큐"],"tags":[]},{"location":"chapter_stack_and_queue/deque/","level":1,"title":"5.3   덱","text":"<p>큐에서는 맨 앞 원소를 삭제하거나 맨 뒤에 원소를 추가하는 것만 가능하다. 아래 그림(5-7)의 덱(double-ended queue)은 앞뒤 양쪽에서 원소를 추가하고 삭제할 수 있어 더 유연하다.</p> <p></p> <p> 그림 5-7   덱의 연산 </p>","path":["제 5 장   스택과 큐","5.3   덱"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#531","level":2,"title":"5.3.1   덱의 기본 연산","text":"<p>덱의 기본 연산은 아래 표(5-3)와 같다. 구체적인 메서드 이름은 사용하는 프로그래밍 언어에 따라 정해진다.</p> <p> 표 5-3   덱의 연산 효율 </p> 메서드 설명 시간 복잡도 <code>push_first()</code> 맨 앞에 원소 추가 \\(O(1)\\) <code>push_last()</code> 맨 뒤에 원소 추가 \\(O(1)\\) <code>pop_first()</code> 맨 앞 원소 삭제 \\(O(1)\\) <code>pop_last()</code> 맨 뒤 원소 삭제 \\(O(1)\\) <code>peek_first()</code> 맨 앞 원소에 접근 \\(O(1)\\) <code>peek_last()</code> 맨 뒤 원소에 접근 \\(O(1)\\) <p>마찬가지로 프로그래밍 언어가 제공하는 덱 클래스를 바로 사용할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby deque.py<pre><code>from collections import deque\n\n# 덱 초기화\ndeq: deque[int] = deque()\n\n# 덱에 원소 삽입\ndeq.append(2)      # 맨 뒤에 추가\ndeq.append(5)\ndeq.append(4)\ndeq.appendleft(3)  # 맨 앞에 추가\ndeq.appendleft(1)\n\n# 원소 접근\nfront: int = deq[0]  # 맨 앞 원소\nrear: int = deq[-1]  # 맨 뒤 원소\n\n# 덱에서 원소 꺼내기\npop_front: int = deq.popleft()  # 맨 앞 원소 꺼내기\npop_rear: int = deq.pop()       # 맨 뒤 원소 꺼내기\n\n# 덱 길이 구하기\nsize: int = len(deq)\n\n# 덱이 비어 있는지 확인\nis_empty: bool = len(deq) == 0\n</code></pre> deque.cpp<pre><code>/* 덱 초기화 */\ndeque&lt;int&gt; deque;\n\n/* 덱에 원소 삽입 */\ndeque.push_back(2);   // 맨 뒤에 추가\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // 맨 앞에 추가\ndeque.push_front(1);\n\n/* 원소 접근 */\nint front = deque.front(); // 맨 앞 원소\nint back = deque.back();   // 맨 뒤 원소\n\n/* 덱에서 원소 꺼내기 */\ndeque.pop_front();  // 맨 앞 원소 꺼내기\ndeque.pop_back();   // 맨 뒤 원소 꺼내기\n\n/* 덱 길이 구하기 */\nint size = deque.size();\n\n/* 덱이 비어 있는지 확인 */\nbool empty = deque.empty();\n</code></pre> deque.java<pre><code>/* 덱 초기화 */\nDeque&lt;Integer&gt; deque = new LinkedList&lt;&gt;();\n\n/* 덱에 원소 삽입 */\ndeque.offerLast(2);   // 맨 뒤에 추가\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // 맨 앞에 추가\ndeque.offerFirst(1);\n\n/* 원소 접근 */\nint peekFirst = deque.peekFirst();  // 맨 앞 원소\nint peekLast = deque.peekLast();    // 맨 뒤 원소\n\n/* 덱에서 원소 꺼내기 */\nint popFirst = deque.pollFirst();  // 맨 앞 원소 꺼내기\nint popLast = deque.pollLast();    // 맨 뒤 원소 꺼내기\n\n/* 덱 길이 구하기 */\nint size = deque.size();\n\n/* 덱이 비어 있는지 확인 */\nboolean isEmpty = deque.isEmpty();\n</code></pre> deque.cs<pre><code>/* 덱 초기화 */\n// C#에서는 연결 리스트 LinkedList를 덱으로 사용한다\nLinkedList&lt;int&gt; deque = new();\n\n/* 덱에 원소 삽입 */\ndeque.AddLast(2);   // 맨 뒤에 추가\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // 맨 앞에 추가\ndeque.AddFirst(1);\n\n/* 원소 접근 */\nint peekFirst = deque.First.Value;  // 맨 앞 원소\nint peekLast = deque.Last.Value;    // 맨 뒤 원소\n\n/* 덱에서 원소 꺼내기 */\ndeque.RemoveFirst();  // 맨 앞 원소 꺼내기\ndeque.RemoveLast();   // 맨 뒤 원소 꺼내기\n\n/* 덱 길이 구하기 */\nint size = deque.Count;\n\n/* 덱이 비어 있는지 확인 */\nbool isEmpty = deque.Count == 0;\n</code></pre> deque_test.go<pre><code>/* 덱 초기화 */\n// Go에서는 list를 덱으로 사용한다\ndeque := list.New()\n\n/* 덱에 원소 삽입 */\ndeque.PushBack(2)      // 맨 뒤에 추가\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // 맨 앞에 추가\ndeque.PushFront(1)\n\n/* 원소 접근 */\nfront := deque.Front() // 맨 앞 원소\nrear := deque.Back()   // 맨 뒤 원소\n\n/* 덱에서 원소 꺼내기 */\ndeque.Remove(front)    // 맨 앞 원소 꺼내기\ndeque.Remove(rear)     // 맨 뒤 원소 꺼내기\n\n/* 덱 길이 구하기 */\nsize := deque.Len()\n\n/* 덱이 비어 있는지 확인 */\nisEmpty := deque.Len() == 0\n</code></pre> deque.swift<pre><code>/* 덱 초기화 */\n// Swift에는 내장 덱 클래스가 없으므로 Array를 덱으로 사용할 수 있다\nvar deque: [Int] = []\n\n/* 덱에 원소 삽입 */\ndeque.append(2) // 맨 뒤에 추가\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // 맨 앞에 추가\ndeque.insert(1, at: 0)\n\n/* 원소 접근 */\nlet peekFirst = deque.first! // 맨 앞 원소\nlet peekLast = deque.last! // 맨 뒤 원소\n\n/* 덱에서 원소 꺼내기 */\n// Array로 구현하면 popFirst의 복잡도는 O(n)이다\nlet popFirst = deque.removeFirst() // 맨 앞 원소 꺼내기\nlet popLast = deque.removeLast() // 맨 뒤 원소 꺼내기\n\n/* 덱 길이 구하기 */\nlet size = deque.count\n\n/* 덱이 비어 있는지 확인 */\nlet isEmpty = deque.isEmpty\n</code></pre> deque.js<pre><code>/* 덱 초기화 */\n// JavaScript에는 내장 덱이 없으므로 Array를 덱으로 사용해야 한다\nconst deque = [];\n\n/* 덱에 원소 삽입 */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// 배열이므로 unshift() 메서드의 시간 복잡도는 O(n)임에 주의\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* 원소 접근 */\nconst peekFirst = deque[0];\nconst peekLast = deque[deque.length - 1];\n\n/* 덱에서 원소 꺼내기 */\n// 배열이므로 shift() 메서드의 시간 복잡도는 O(n)임에 주의\nconst popFront = deque.shift();\nconst popBack = deque.pop();\n\n/* 덱 길이 구하기 */\nconst size = deque.length;\n\n/* 덱이 비어 있는지 확인 */\nconst isEmpty = size === 0;\n</code></pre> deque.ts<pre><code>/* 덱 초기화 */\n// TypeScript에는 내장 덱이 없으므로 Array를 덱으로 사용해야 한다\nconst deque: number[] = [];\n\n/* 덱에 원소 삽입 */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// 배열이므로 unshift() 메서드의 시간 복잡도는 O(n)임에 주의\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* 원소 접근 */\nconst peekFirst: number = deque[0];\nconst peekLast: number = deque[deque.length - 1];\n\n/* 덱에서 원소 꺼내기 */\n// 배열이므로 shift() 메서드의 시간 복잡도는 O(n)임에 주의\nconst popFront: number = deque.shift() as number;\nconst popBack: number = deque.pop() as number;\n\n/* 덱 길이 구하기 */\nconst size: number = deque.length;\n\n/* 덱이 비어 있는지 확인 */\nconst isEmpty: boolean = size === 0;\n</code></pre> deque.dart<pre><code>/* 덱 초기화 */\n// Dart의 Queue는 덱으로 정의되어 있다\nQueue&lt;int&gt; deque = Queue&lt;int&gt;();\n\n/* 덱에 원소 삽입 */\ndeque.addLast(2);  // 맨 뒤에 추가\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // 맨 앞에 추가\ndeque.addFirst(1);\n\n/* 원소 접근 */\nint peekFirst = deque.first; // 맨 앞 원소\nint peekLast = deque.last;   // 맨 뒤 원소\n\n/* 덱에서 원소 꺼내기 */\nint popFirst = deque.removeFirst(); // 맨 앞 원소 꺼내기\nint popLast = deque.removeLast();   // 맨 뒤 원소 꺼내기\n\n/* 덱 길이 구하기 */\nint size = deque.length;\n\n/* 덱이 비어 있는지 확인 */\nbool isEmpty = deque.isEmpty;\n</code></pre> deque.rs<pre><code>/* 덱 초기화 */\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* 덱에 원소 삽입 */\ndeque.push_back(2);  // 맨 뒤에 추가\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // 맨 앞에 추가\ndeque.push_front(1);\n\n/* 원소 접근 */\nif let Some(front) = deque.front() { // 맨 앞 원소\n}\nif let Some(rear) = deque.back() {   // 맨 뒤 원소\n}\n\n/* 덱에서 원소 꺼내기 */\nif let Some(pop_front) = deque.pop_front() { // 맨 앞 원소 꺼내기\n}\nif let Some(pop_rear) = deque.pop_back() {   // 맨 뒤 원소 꺼내기\n}\n\n/* 덱 길이 구하기 */\nlet size = deque.len();\n\n/* 덱이 비어 있는지 확인 */\nlet is_empty = deque.is_empty();\n</code></pre> deque.c<pre><code>// C는 내장 덱을 제공하지 않는다\n</code></pre> deque.kt<pre><code>/* 덱 초기화 */\nval deque = LinkedList&lt;Int&gt;()\n\n/* 덱에 원소 삽입 */\ndeque.offerLast(2)  // 맨 뒤에 추가\ndeque.offerLast(5)\ndeque.offerLast(4)\ndeque.offerFirst(3) // 맨 앞에 추가\ndeque.offerFirst(1)\n\n/* 원소 접근 */\nval peekFirst = deque.peekFirst() // 맨 앞 원소\nval peekLast = deque.peekLast()   // 맨 뒤 원소\n\n/* 덱에서 원소 꺼내기 */\nval popFirst = deque.pollFirst() // 맨 앞 원소 꺼내기\nval popLast = deque.pollLast()   // 맨 뒤 원소 꺼내기\n\n/* 덱 길이 구하기 */\nval size = deque.size\n\n/* 덱이 비어 있는지 확인 */\nval isEmpty = deque.isEmpty()\n</code></pre> deque.rb<pre><code># 덱 초기화\n# Ruby에는 내장 덱이 없으므로 Array를 덱으로 사용해야 한다\ndeque = []\n\n# 덱에 원소 삽입\ndeque &lt;&lt; 2\ndeque &lt;&lt; 5\ndeque &lt;&lt; 4\n# 배열이므로 Array#unshift 메서드의 시간 복잡도는 O(n)임에 주의\ndeque.unshift(3)\ndeque.unshift(1)\n\n# 원소 접근\npeek_first = deque.first\npeek_last = deque.last\n\n# 덱에서 원소 꺼내기\n# 배열이므로 Array#shift 메서드의 시간 복잡도는 O(n)임에 주의\npop_front = deque.shift\npop_back = deque.pop\n\n# 덱 길이 구하기\nsize = deque.length\n\n# 덱이 비어 있는지 확인\nis_empty = size.zero?\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 5 장   스택과 큐","5.3   덱"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#532","level":2,"title":"5.3.2   덱 구현 *","text":"<p>덱의 구현은 큐와 비슷하며, 기반 자료 구조로 연결 리스트나 배열을 선택할 수 있다.</p>","path":["제 5 장   스택과 큐","5.3   덱"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#1","level":3,"title":"1.   이중 연결 리스트 기반 구현","text":"<p>앞 절에서는 일반적인 단일 연결 리스트로 큐를 구현했다. 머리 노드를 삭제하는 연산(큐에서 꺼내기)과 꼬리 노드 뒤에 새 노드를 추가하는 연산(큐에 삽입하기)을 쉽게 수행할 수 있기 때문이다.</p> <p>덱에서는 앞과 뒤 모두 삽입과 꺼내기를 수행할 수 있다. 즉, 큐와 대칭되는 반대 방향의 연산도 구현해야 한다. 이를 위해 ‘이중 연결 리스트’를 덱의 기반 자료 구조로 사용한다.</p> <p>아래 그림(5-8)과 같이 이중 연결 리스트의 머리 노드와 꼬리 노드를 덱의 맨 앞과 맨 뒤로 보고, 양 끝에서 노드를 추가하고 삭제하는 기능을 구현한다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 5-8   연결 리스트 기반 덱의 삽입과 꺼내기 </p> <p>구현 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linkedlist_deque.py<pre><code>class ListNode:\n    \"\"\"이중 연결 리스트 노드\"\"\"\n\n    def __init__(self, val: int):\n        \"\"\"생성자\"\"\"\n        self.val: int = val\n        self.next: ListNode | None = None  # 다음 노드 참조\n        self.prev: ListNode | None = None  # 이전 노드 참조\n\nclass LinkedListDeque:\n    \"\"\"이중 연결 리스트로 구현한 덱\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        self._front: ListNode | None = None  # 머리 노드 front\n        self._rear: ListNode | None = None  # 꼬리 노드 rear\n        self._size: int = 0  # 덱의 길이\n\n    def size(self) -&gt; int:\n        \"\"\"덱의 길이 반환\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"덱이 비어 있는지 판단\"\"\"\n        return self._size == 0\n\n    def push(self, num: int, is_front: bool):\n        \"\"\"큐 삽입 연산\"\"\"\n        node = ListNode(num)\n        # 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if self.is_empty():\n            self._front = self._rear = node\n        # 맨 앞에 삽입하는 연산\n        elif is_front:\n            # node를 연결 리스트의 맨 앞에 추가한다\n            self._front.prev = node\n            node.next = self._front\n            self._front = node  # 머리 노드를 갱신한다\n        # 맨 뒤에 삽입하는 연산\n        else:\n            # node를 연결 리스트의 맨 뒤에 추가한다\n            self._rear.next = node\n            node.prev = self._rear\n            self._rear = node  # 꼬리 노드를 갱신한다\n        self._size += 1  # 큐 길이를 갱신한다\n\n    def push_first(self, num: int):\n        \"\"\"맨 앞에 삽입\"\"\"\n        self.push(num, True)\n\n    def push_last(self, num: int):\n        \"\"\"맨 뒤에 삽입\"\"\"\n        self.push(num, False)\n\n    def pop(self, is_front: bool) -&gt; int:\n        \"\"\"큐에서 꺼내는 연산\"\"\"\n        if self.is_empty():\n            raise IndexError(\"덱이 비어 있다\")\n        # 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if self._size == 1:\n            val: int = self._front.val\n            self._front = None\n            self._rear = None\n        # 맨 앞에서 꺼내는 연산\n        elif is_front:\n            val: int = self._front.val  # 머리 노드의 값을 임시로 저장한다\n            # 머리 노드를 삭제한다\n            fnext: ListNode | None = self._front.next\n            fnext.prev = None\n            self._front.next = None\n            self._front = fnext  # 머리 노드를 갱신한다\n        # 맨 뒤에서 꺼내는 연산\n        else:\n            val: int = self._rear.val  # 꼬리 노드의 값을 임시로 저장한다\n            # 꼬리 노드를 삭제한다\n            rprev: ListNode | None = self._rear.prev\n            rprev.next = None\n            self._rear.prev = None\n            self._rear = rprev  # 꼬리 노드를 갱신한다\n        self._size -= 1  # 큐 길이를 갱신한다\n        return val\n\n    def pop_first(self) -&gt; int:\n        \"\"\"맨 앞에서 꺼내기\"\"\"\n        return self.pop(True)\n\n    def pop_last(self) -&gt; int:\n        \"\"\"맨 뒤에서 꺼내기\"\"\"\n        return self.pop(False)\n\n    def peek_first(self) -&gt; int:\n        \"\"\"맨 앞 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"덱이 비어 있다\")\n        return self._front.val\n\n    def peek_last(self) -&gt; int:\n        \"\"\"맨 뒤 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"덱이 비어 있다\")\n        return self._rear.val\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"출력에 사용할 배열 반환\"\"\"\n        node = self._front\n        res = [0] * self.size()\n        for i in range(self.size()):\n            res[i] = node.val\n            node = node.next\n        return res\n</code></pre> linkedlist_deque.cpp<pre><code>/* 이중 연결 리스트 노드 */\nstruct DoublyListNode {\n    int val;              // 노드 값\n    DoublyListNode *next; // 다음 노드 포인터\n    DoublyListNode *prev; // 이전 노드 포인터\n    DoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n    }\n};\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n  private:\n    DoublyListNode *front, *rear; // 머리 노드 front, 꼬리 노드 rear\n    int queSize = 0;              // 덱의 길이\n\n  public:\n    /* 생성자 */\n    LinkedListDeque() : front(nullptr), rear(nullptr) {\n    }\n\n    /* 소멸자 */\n    ~LinkedListDeque() {\n        // 연결 리스트를 순회하며 노드를 삭제하여 메모리를 해제한다\n        DoublyListNode *pre, *cur = front;\n        while (cur != nullptr) {\n            pre = cur;\n            cur = cur-&gt;next;\n            delete pre;\n        }\n    }\n\n    /* 덱의 길이 반환 */\n    int size() {\n        return queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* 큐 삽입 연산 */\n    void push(int num, bool isFront) {\n        DoublyListNode *node = new DoublyListNode(num);\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (isEmpty())\n            front = rear = node;\n        // 맨 앞에 삽입하는 연산\n        else if (isFront) {\n            // node를 연결 리스트의 맨 앞에 추가한다\n            front-&gt;prev = node;\n            node-&gt;next = front;\n            front = node; // 머리 노드를 갱신한다\n        // 맨 뒤에 삽입하는 연산\n        } else {\n            // node를 연결 리스트의 맨 뒤에 추가한다\n            rear-&gt;next = node;\n            node-&gt;prev = rear;\n            rear = node; // 꼬리 노드를 갱신한다\n        }\n        queSize++; // 큐 길이를 갱신한다\n    }\n\n    /* 맨 앞에 삽입 */\n    void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* 맨 뒤에 삽입 */\n    void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* 큐에서 꺼내는 연산 */\n    int pop(bool isFront) {\n        if (isEmpty())\n            throw out_of_range(\"큐가 비어 있다\");\n        int val;\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (queSize == 1) {\n            val = front-&gt;val;\n            delete front; // 같은 노드를 가리키므로 하나만 해제하면 된다\n            front = nullptr;\n            rear = nullptr;\n        }\n        // 맨 앞에서 꺼내는 연산\n        else if (isFront) {\n            val = front-&gt;val; // 머리 노드의 값을 임시로 저장한다\n            // 머리 노드를 삭제한다\n            DoublyListNode *fNext = front-&gt;next;\n            fNext-&gt;prev = nullptr;\n            front-&gt;next = nullptr;\n            delete front;\n            front = fNext; // 머리 노드를 갱신한다\n        // 맨 뒤에서 꺼내는 연산\n        } else {\n            val = rear-&gt;val; // 꼬리 노드의 값을 임시로 저장한다\n            // 꼬리 노드를 삭제한다\n            DoublyListNode *rPrev = rear-&gt;prev;\n            rPrev-&gt;next = nullptr;\n            rear-&gt;prev = nullptr;\n            delete rear;\n            rear = rPrev; // 꼬리 노드를 갱신한다\n        }\n        queSize--; // 큐 길이를 갱신한다\n        return val;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    int popFirst() {\n        return pop(true);\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    int popLast() {\n        return pop(false);\n    }\n\n    /* 맨 앞 원소 접근 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"덱이 비어 있다\");\n        return front-&gt;val;\n    }\n\n    /* 맨 뒤 원소 접근 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"덱이 비어 있다\");\n        return rear-&gt;val;\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    vector&lt;int&gt; toVector() {\n        DoublyListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_deque.java<pre><code>/* 이중 연결 리스트 노드 */\nclass ListNode {\n    int val; // 노드 값\n    ListNode next; // 다음 노드 참조\n    ListNode prev; // 이전 노드 참조\n\n    ListNode(int val) {\n        this.val = val;\n        prev = next = null;\n    }\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n    private ListNode front, rear; // 머리 노드 front, 꼬리 노드 rear\n    private int queSize = 0; // 덱의 길이\n\n    public LinkedListDeque() {\n        front = rear = null;\n    }\n\n    /* 덱의 길이 반환 */\n    public int size() {\n        return queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* 큐 삽입 연산 */\n    private void push(int num, boolean isFront) {\n        ListNode node = new ListNode(num);\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (isEmpty())\n            front = rear = node;\n        // 맨 앞에 삽입하는 연산\n        else if (isFront) {\n            // node를 연결 리스트의 맨 앞에 추가한다\n            front.prev = node;\n            node.next = front;\n            front = node; // 머리 노드를 갱신한다\n        // 맨 뒤에 삽입하는 연산\n        } else {\n            // node를 연결 리스트의 맨 뒤에 추가한다\n            rear.next = node;\n            node.prev = rear;\n            rear = node; // 꼬리 노드를 갱신한다\n        }\n        queSize++; // 큐 길이를 갱신한다\n    }\n\n    /* 맨 앞에 삽입 */\n    public void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* 맨 뒤에 삽입 */\n    public void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* 큐에서 꺼내는 연산 */\n    private int pop(boolean isFront) {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        int val;\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (queSize == 1) {\n            val = front.val;\n            front = null;\n            rear = null;\n        }\n        // 맨 앞에서 꺼내는 연산\n        else if (isFront) {\n            val = front.val; // 머리 노드의 값을 임시로 저장한다\n            // 머리 노드를 삭제한다\n            ListNode fNext = front.next;\n            fNext.prev = null;\n            front.next = null;\n            front = fNext; // 머리 노드를 갱신한다\n        // 맨 뒤에서 꺼내는 연산\n        } else {\n            val = rear.val; // 꼬리 노드의 값을 임시로 저장한다\n            // 꼬리 노드를 삭제한다\n            ListNode rPrev = rear.prev;\n            rPrev.next = null;\n            rear.prev = null;\n            rear = rPrev; // 꼬리 노드를 갱신한다\n        }\n        queSize--; // 큐 길이를 갱신한다\n        return val;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    public int popFirst() {\n        return pop(true);\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    public int popLast() {\n        return pop(false);\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* 맨 뒤 원소 접근 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return rear.val;\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.cs<pre><code>/* 이중 연결 리스트 노드 */\nclass ListNode(int val) {\n    public int val = val;       // 노드 값\n    public ListNode? next = null; // 다음 노드 참조\n    public ListNode? prev = null; // 이전 노드 참조\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n    ListNode? front, rear; // 머리 노드 front, 꼬리 노드 rear\n    int queSize = 0;      // 덱의 길이\n\n    public LinkedListDeque() {\n        front = null;\n        rear = null;\n    }\n\n    /* 덱의 길이 반환 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* 큐 삽입 연산 */\n    void Push(int num, bool isFront) {\n        ListNode node = new(num);\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (IsEmpty()) {\n            front = node;\n            rear = node;\n        }\n        // 맨 앞에 삽입하는 연산\n        else if (isFront) {\n            // node를 연결 리스트의 맨 앞에 추가한다\n            front!.prev = node;\n            node.next = front;\n            front = node; // 머리 노드를 갱신한다                           \n        }\n        // 맨 뒤에 삽입하는 연산\n        else {\n            // node를 연결 리스트의 맨 뒤에 추가한다\n            rear!.next = node;\n            node.prev = rear;\n            rear = node;  // 꼬리 노드를 갱신한다\n        }\n\n        queSize++; // 큐 길이를 갱신한다\n    }\n\n    /* 맨 앞에 삽입 */\n    public void PushFirst(int num) {\n        Push(num, true);\n    }\n\n    /* 맨 뒤에 삽입 */\n    public void PushLast(int num) {\n        Push(num, false);\n    }\n\n    /* 큐에서 꺼내는 연산 */\n    int? Pop(bool isFront) {\n        if (IsEmpty())\n            throw new Exception();\n        int? val;\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (queSize == 1) {\n            val = front!.val;\n            front = null;\n            rear = null;\n        }\n        // 맨 앞에서 꺼내는 연산\n        else if (isFront) {\n            val = front!.val; // 머리 노드의 값을 임시로 저장한다\n            // 머리 노드를 삭제한다\n            ListNode? fNext = front?.next;\n            fNext!.prev = null;\n            front.next = null;\n            front = fNext;   // 머리 노드를 갱신한다\n        }\n        // 맨 뒤에서 꺼내는 연산\n        else {\n            val = rear!.val;  // 꼬리 노드의 값을 임시로 저장한다\n            // 꼬리 노드를 삭제한다\n            ListNode? rPrev = rear?.prev;\n            rPrev!.next = null;\n            rear.prev = null;\n            rear = rPrev;    // 꼬리 노드를 갱신한다\n        }\n\n        queSize--; // 큐 길이를 갱신한다\n        return val;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    public int? PopFirst() {\n        return Pop(true);\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    public int? PopLast() {\n        return Pop(false);\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int? PeekFirst() {\n        if (IsEmpty())\n            throw new Exception();\n        return front?.val;\n    }\n\n    /* 맨 뒤 원소 접근 */\n    public int? PeekLast() {\n        if (IsEmpty())\n            throw new Exception();\n        return rear?.val;\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    public int?[] ToArray() {\n        ListNode? node = front;\n        int?[] res = new int?[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node?.val;\n            node = node?.next;\n        }\n\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.go<pre><code>/* 이중 연결 리스트로 구현한 덱 */\ntype linkedListDeque struct {\n    // 내장 패키지 list를 사용한다\n    data *list.List\n}\n\n/* 덱 초기화 */\nfunc newLinkedListDeque() *linkedListDeque {\n    return &amp;linkedListDeque{\n        data: list.New(),\n    }\n}\n\n/* 맨 앞에 원소 삽입 */\nfunc (s *linkedListDeque) pushFirst(value any) {\n    s.data.PushFront(value)\n}\n\n/* 맨 뒤에 원소 삽입 */\nfunc (s *linkedListDeque) pushLast(value any) {\n    s.data.PushBack(value)\n}\n\n/* 맨 앞 원소를 꺼낸다 */\nfunc (s *linkedListDeque) popFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* 맨 뒤 원소를 꺼낸다 */\nfunc (s *linkedListDeque) popLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* 맨 앞 원소 접근 */\nfunc (s *linkedListDeque) peekFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* 맨 뒤 원소 접근 */\nfunc (s *linkedListDeque) peekLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* 큐의 길이 반환 */\nfunc (s *linkedListDeque) size() int {\n    return s.data.Len()\n}\n\n/* 큐가 비어 있는지 판단 */\nfunc (s *linkedListDeque) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* 출력에 사용할 List 반환 */\nfunc (s *linkedListDeque) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_deque.swift<pre><code>/* 이중 연결 리스트 노드 */\nclass ListNode {\n    var val: Int // 노드 값\n    var next: ListNode? // 다음 노드 참조\n    weak var prev: ListNode? // 이전 노드 참조\n\n    init(val: Int) {\n        self.val = val\n    }\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n    private var front: ListNode? // 머리 노드 front\n    private var rear: ListNode? // 꼬리 노드 rear\n    private var _size: Int // 덱의 길이\n\n    init() {\n        _size = 0\n    }\n\n    /* 덱의 길이 반환 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* 큐 삽입 연산 */\n    private func push(num: Int, isFront: Bool) {\n        let node = ListNode(val: num)\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if isEmpty() {\n            front = node\n            rear = node\n        }\n        // 맨 앞에 삽입하는 연산\n        else if isFront {\n            // node를 연결 리스트의 맨 앞에 추가한다\n            front?.prev = node\n            node.next = front\n            front = node // 머리 노드를 갱신한다\n        }\n        // 맨 뒤에 삽입하는 연산\n        else {\n            // node를 연결 리스트의 맨 뒤에 추가한다\n            rear?.next = node\n            node.prev = rear\n            rear = node // 꼬리 노드를 갱신한다\n        }\n        _size += 1 // 큐 길이를 갱신한다\n    }\n\n    /* 맨 앞에 삽입 */\n    func pushFirst(num: Int) {\n        push(num: num, isFront: true)\n    }\n\n    /* 맨 뒤에 삽입 */\n    func pushLast(num: Int) {\n        push(num: num, isFront: false)\n    }\n\n    /* 큐에서 꺼내는 연산 */\n    private func pop(isFront: Bool) -&gt; Int {\n        if isEmpty() {\n            fatalError(\"덱이 비어 있다\")\n        }\n        let val: Int\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if _size == 1 {\n            val = front!.val\n            front = nil\n            rear = nil\n        }\n        // 맨 앞에서 꺼내는 연산\n        else if isFront {\n            val = front!.val // 머리 노드의 값을 임시로 저장한다\n            // 머리 노드를 삭제한다\n            let fNext = front?.next\n            fNext?.prev = nil\n            front?.next = nil\n            front = fNext // 머리 노드를 갱신한다\n        }\n        // 맨 뒤에서 꺼내는 연산\n        else {\n            val = rear!.val // 꼬리 노드의 값을 임시로 저장한다\n            // 꼬리 노드를 삭제한다\n            let rPrev = rear?.prev\n            rPrev?.next = nil\n            rear?.prev = nil\n            rear = rPrev // 꼬리 노드를 갱신한다\n        }\n        _size -= 1 // 큐 길이를 갱신한다\n        return val\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    func popFirst() -&gt; Int {\n        pop(isFront: true)\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    func popLast() -&gt; Int {\n        pop(isFront: false)\n    }\n\n    /* 맨 앞 원소 접근 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"덱이 비어 있다\")\n        }\n        return front!.val\n    }\n\n    /* 맨 뒤 원소 접근 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"덱이 비어 있다\")\n        }\n        return rear!.val\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.js<pre><code>/* 이중 연결 리스트 노드 */\nclass ListNode {\n    prev; // 이전 노드 참조(포인터)\n    next; // 다음 노드 참조(포인터)\n    val; // 노드 값\n\n    constructor(val) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n    #front; // 머리 노드 front\n    #rear; // 꼬리 노드 rear\n    #queSize; // 덱의 길이\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n        this.#queSize = 0;\n    }\n\n    /* 맨 뒤에 삽입하는 연산 */\n    pushLast(val) {\n        const node = new ListNode(val);\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // node를 연결 리스트의 맨 뒤에 추가한다\n            this.#rear.next = node;\n            node.prev = this.#rear;\n            this.#rear = node; // 꼬리 노드를 갱신한다\n        }\n        this.#queSize++;\n    }\n\n    /* 맨 앞에 삽입하는 연산 */\n    pushFirst(val) {\n        const node = new ListNode(val);\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // node를 연결 리스트의 맨 앞에 추가한다\n            this.#front.prev = node;\n            node.next = this.#front;\n            this.#front = node; // 머리 노드를 갱신한다\n        }\n        this.#queSize++;\n    }\n\n    /* 맨 뒤에서 꺼내는 연산 */\n    popLast() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#rear.val; // 꼬리 노드의 값을 저장한다\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (this.#queSize === 1) {\n            this.#front = null;\n            this.#rear = null;\n        }\n        // 꼬리 노드를 삭제한다\n        else {\n            let temp = this.#rear.prev;\n            temp.next = null;\n            this.#rear.prev = null;\n            this.#rear = temp; // 꼬리 노드를 갱신한다\n        }\n        this.#queSize--;\n        return value;\n    }\n\n    /* 맨 앞에서 꺼내는 연산 */\n    popFirst() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#front.val; // 머리 노드의 값을 저장한다\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (this.#queSize === 1) {\n            this.#front = null;\n            this.#rear = null;\n        }\n        // 머리 노드를 삭제한다\n        else {\n            let temp = this.#front.next;\n            temp.prev = null;\n            this.#front.next = null;\n            this.#front = temp; // 머리 노드를 갱신한다\n        }\n        this.#queSize--;\n        return value;\n    }\n\n    /* 맨 뒤 원소 접근 */\n    peekLast() {\n        return this.#queSize === 0 ? null : this.#rear.val;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peekFirst() {\n        return this.#queSize === 0 ? null : this.#front.val;\n    }\n\n    /* 덱의 길이 반환 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* 덱을 출력한다 */\n    print() {\n        const arr = [];\n        let temp = this.#front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.ts<pre><code>/* 이중 연결 리스트 노드 */\nclass ListNode {\n    prev: ListNode; // 이전 노드 참조(포인터)\n    next: ListNode; // 다음 노드 참조(포인터)\n    val: number; // 노드 값\n\n    constructor(val: number) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n    private front: ListNode; // 머리 노드 front\n    private rear: ListNode; // 꼬리 노드 rear\n    private queSize: number; // 덱의 길이\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n        this.queSize = 0;\n    }\n\n    /* 맨 뒤에 삽입하는 연산 */\n    pushLast(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // node를 연결 리스트의 맨 뒤에 추가한다\n            this.rear.next = node;\n            node.prev = this.rear;\n            this.rear = node; // 꼬리 노드를 갱신한다\n        }\n        this.queSize++;\n    }\n\n    /* 맨 앞에 삽입하는 연산 */\n    pushFirst(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // node를 연결 리스트의 맨 앞에 추가한다\n            this.front.prev = node;\n            node.next = this.front;\n            this.front = node; // 머리 노드를 갱신한다\n        }\n        this.queSize++;\n    }\n\n    /* 맨 뒤에서 꺼내는 연산 */\n    popLast(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.rear!.val; // 꼬리 노드의 값을 저장한다\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (this.queSize === 1) {\n            this.front = null;\n            this.rear = null;\n        }\n        // 꼬리 노드를 삭제한다\n        else {\n            let temp: ListNode = this.rear!.prev;\n            temp!.next = null;\n            this.rear!.prev = null;\n            this.rear = temp; // 꼬리 노드를 갱신한다\n        }\n        this.queSize--;\n        return value;\n    }\n\n    /* 맨 앞에서 꺼내는 연산 */\n    popFirst(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.front.val; // 머리 노드의 값을 저장한다\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (this.queSize === 1) {\n            this.front = null;\n            this.rear = null;\n        }\n        // 머리 노드를 삭제한다\n        else {\n            let temp: ListNode = this.front!.next;\n            temp!.prev = null;\n            this.front!.next = null;\n            this.front = temp; // 머리 노드를 갱신한다\n        }\n        this.queSize--;\n        return value;\n    }\n\n    /* 맨 뒤 원소 접근 */\n    peekLast(): number {\n        return this.queSize === 0 ? null : this.rear.val;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peekFirst(): number {\n        return this.queSize === 0 ? null : this.front.val;\n    }\n\n    /* 덱의 길이 반환 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* 덱을 출력한다 */\n    print(): void {\n        const arr: number[] = [];\n        let temp: ListNode = this.front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.dart<pre><code>/* 이중 연결 리스트 노드 */\nclass ListNode {\n  int val; // 노드 값\n  ListNode? next; // 다음 노드 참조\n  ListNode? prev; // 이전 노드 참조\n\n  ListNode(this.val, {this.next, this.prev});\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n  late ListNode? _front; // 머리 노드 _front\n  late ListNode? _rear; // 꼬리 노드 _rear\n  int _queSize = 0; // 덱의 길이\n\n  LinkedListDeque() {\n    this._front = null;\n    this._rear = null;\n  }\n\n  /* 덱의 길이 반환 */\n  int size() {\n    return this._queSize;\n  }\n\n  /* 덱이 비어 있는지 판단 */\n  bool isEmpty() {\n    return size() == 0;\n  }\n\n  /* 큐 삽입 연산 */\n  void push(int _num, bool isFront) {\n    final ListNode node = ListNode(_num);\n    if (isEmpty()) {\n      // 연결 리스트가 비어 있으면 _front와 _rear가 모두 node를 가리키게 한다\n      _front = _rear = node;\n    } else if (isFront) {\n      // 맨 앞에 삽입하는 연산\n      // node를 연결 리스트의 맨 앞에 추가한다\n      _front!.prev = node;\n      node.next = _front;\n      _front = node; // 머리 노드를 갱신한다\n    } else {\n      // 맨 뒤에 삽입하는 연산\n      // node를 연결 리스트의 맨 뒤에 추가한다\n      _rear!.next = node;\n      node.prev = _rear;\n      _rear = node; // 꼬리 노드를 갱신한다\n    }\n    _queSize++; // 큐 길이를 갱신한다\n  }\n\n  /* 맨 앞에 삽입 */\n  void pushFirst(int _num) {\n    push(_num, true);\n  }\n\n  /* 맨 뒤에 삽입 */\n  void pushLast(int _num) {\n    push(_num, false);\n  }\n\n  /* 큐에서 꺼내는 연산 */\n  int? pop(bool isFront) {\n    // 큐가 비어 있으면 바로 null을 반환한다\n    if (isEmpty()) {\n      return null;\n    }\n    final int val;\n    // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n    if (_queSize == 1) {\n      val = _front!.val;\n      _front = null;\n      _rear = null;\n    }\n    // 맨 앞에서 꺼내는 연산\n    else if (isFront) {\n      val = _front!.val; // 머리 노드의 값을 임시로 저장한다\n      // 머리 노드를 삭제한다\n      ListNode? fNext = _front!.next;\n      fNext!.prev = null;\n      _front!.next = null;\n      _front = fNext; // 머리 노드를 갱신한다\n    // 맨 뒤에서 꺼내는 연산\n    } else {\n      val = _rear!.val; // 꼬리 노드의 값을 임시로 저장한다\n      // 꼬리 노드를 삭제한다\n      ListNode? rPrev = _rear!.prev;\n      rPrev!.next = null;\n      _rear!.prev = null;\n      _rear = rPrev; // 꼬리 노드를 갱신한다\n    }\n    _queSize--; // 큐 길이를 갱신한다\n    return val;\n  }\n\n  /* 맨 앞에서 꺼내기 */\n  int? popFirst() {\n    return pop(true);\n  }\n\n  /* 맨 뒤에서 꺼내기 */\n  int? popLast() {\n    return pop(false);\n  }\n\n  /* 맨 앞 원소 접근 */\n  int? peekFirst() {\n    return _front?.val;\n  }\n\n  /* 맨 뒤 원소 접근 */\n  int? peekLast() {\n    return _rear?.val;\n  }\n\n  /* 출력에 사용할 배열 반환 */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; _queSize; i++) {\n      res.add(node!.val);\n      node = node.next;\n    }\n    return res;\n  }\n}\n</code></pre> linkedlist_deque.rs<pre><code>/* 이중 연결 리스트 노드 */\npub struct ListNode&lt;T&gt; {\n    pub val: T,                                 // 노드 값\n    pub next: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // 다음 노드 포인터\n    pub prev: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // 이전 노드 포인터\n}\n\nimpl&lt;T&gt; ListNode&lt;T&gt; {\n    pub fn new(val: T) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n        Rc::new(RefCell::new(ListNode {\n            val,\n            next: None,\n            prev: None,\n        }))\n    }\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\n#[allow(dead_code)]\npub struct LinkedListDeque&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // 머리 노드 front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // 꼬리 노드 rear\n    que_size: usize,                         // 덱의 길이\n}\n\nimpl&lt;T: Copy&gt; LinkedListDeque&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* 덱의 길이 반환 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.que_size == 0;\n    }\n\n    /* 큐 삽입 연산 */\n    fn push(&amp;mut self, num: T, is_front: bool) {\n        let node = ListNode::new(num);\n        // 맨 앞에 삽입하는 연산\n        if is_front {\n            match self.front.take() {\n                // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n                None =&gt; {\n                    self.rear = Some(node.clone());\n                    self.front = Some(node);\n                }\n                // node를 연결 리스트의 맨 앞에 추가한다\n                Some(old_front) =&gt; {\n                    old_front.borrow_mut().prev = Some(node.clone());\n                    node.borrow_mut().next = Some(old_front);\n                    self.front = Some(node); // 머리 노드를 갱신한다\n                }\n            }\n        }\n        // 맨 뒤에 삽입하는 연산\n        else {\n            match self.rear.take() {\n                // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n                None =&gt; {\n                    self.front = Some(node.clone());\n                    self.rear = Some(node);\n                }\n                // node를 연결 리스트의 맨 뒤에 추가한다\n                Some(old_rear) =&gt; {\n                    old_rear.borrow_mut().next = Some(node.clone());\n                    node.borrow_mut().prev = Some(old_rear);\n                    self.rear = Some(node); // 꼬리 노드를 갱신한다\n                }\n            }\n        }\n        self.que_size += 1; // 큐 길이를 갱신한다\n    }\n\n    /* 맨 앞에 삽입 */\n    pub fn push_first(&amp;mut self, num: T) {\n        self.push(num, true);\n    }\n\n    /* 맨 뒤에 삽입 */\n    pub fn push_last(&amp;mut self, num: T) {\n        self.push(num, false);\n    }\n\n    /* 큐에서 꺼내는 연산 */\n    fn pop(&amp;mut self, is_front: bool) -&gt; Option&lt;T&gt; {\n        // 큐가 비어 있으면 바로 None을 반환한다\n        if self.is_empty() {\n            return None;\n        };\n        // 맨 앞에서 꺼내는 연산\n        if is_front {\n            self.front.take().map(|old_front| {\n                match old_front.borrow_mut().next.take() {\n                    Some(new_front) =&gt; {\n                        new_front.borrow_mut().prev.take();\n                        self.front = Some(new_front); // 머리 노드를 갱신한다\n                    }\n                    None =&gt; {\n                        self.rear.take();\n                    }\n                }\n                self.que_size -= 1; // 큐 길이를 갱신한다\n                old_front.borrow().val\n            })\n        }\n        // 맨 뒤에서 꺼내는 연산\n        else {\n            self.rear.take().map(|old_rear| {\n                match old_rear.borrow_mut().prev.take() {\n                    Some(new_rear) =&gt; {\n                        new_rear.borrow_mut().next.take();\n                        self.rear = Some(new_rear); // 꼬리 노드를 갱신한다\n                    }\n                    None =&gt; {\n                        self.front.take();\n                    }\n                }\n                self.que_size -= 1; // 큐 길이를 갱신한다\n                old_rear.borrow().val\n            })\n        }\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    pub fn pop_first(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(true);\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    pub fn pop_last(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(false);\n    }\n\n    /* 맨 앞 원소 접근 */\n    pub fn peek_first(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* 맨 뒤 원소 접근 */\n    pub fn peek_last(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.rear.as_ref()\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        let mut res: Vec&lt;T&gt; = Vec::new();\n        fn recur&lt;T: Copy&gt;(cur: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, res: &amp;mut Vec&lt;T&gt;) {\n            if let Some(cur) = cur {\n                res.push(cur.borrow().val);\n                recur(cur.borrow().next.as_ref(), res);\n            }\n        }\n\n        recur(head, &amp;mut res);\n        res\n    }\n}\n</code></pre> linkedlist_deque.c<pre><code>/* 이중 연결 리스트 노드 */\ntypedef struct DoublyListNode {\n    int val;                     // 노드 값\n    struct DoublyListNode *next; // 다음 노드\n    struct DoublyListNode *prev; // 이전 노드\n} DoublyListNode;\n\n/* 생성자 */\nDoublyListNode *newDoublyListNode(int num) {\n    DoublyListNode *new = (DoublyListNode *)malloc(sizeof(DoublyListNode));\n    new-&gt;val = num;\n    new-&gt;next = NULL;\n    new-&gt;prev = NULL;\n    return new;\n}\n\n/* 소멸자 */\nvoid delDoublyListNode(DoublyListNode *node) {\n    free(node);\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\ntypedef struct {\n    DoublyListNode *front, *rear; // 머리 노드 front, 꼬리 노드 rear\n    int queSize;                  // 덱의 길이\n} LinkedListDeque;\n\n/* 생성자 */\nLinkedListDeque *newLinkedListDeque() {\n    LinkedListDeque *deque = (LinkedListDeque *)malloc(sizeof(LinkedListDeque));\n    deque-&gt;front = NULL;\n    deque-&gt;rear = NULL;\n    deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* 소멸자 */\nvoid delLinkedListdeque(LinkedListDeque *deque) {\n    // 모든 노드를 해제한다\n    for (int i = 0; i &lt; deque-&gt;queSize &amp;&amp; deque-&gt;front != NULL; i++) {\n        DoublyListNode *tmp = deque-&gt;front;\n        deque-&gt;front = deque-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // deque 구조체를 해제한다\n    free(deque);\n}\n\n/* 큐의 길이 반환 */\nint size(LinkedListDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* 큐가 비어 있는지 판단 */\nbool empty(LinkedListDeque *deque) {\n    return (size(deque) == 0);\n}\n\n/* 큐에 삽입 */\nvoid push(LinkedListDeque *deque, int num, bool isFront) {\n    DoublyListNode *node = newDoublyListNode(num);\n    // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n    if (empty(deque)) {\n        deque-&gt;front = deque-&gt;rear = node;\n    }\n    // 맨 앞에 삽입하는 연산\n    else if (isFront) {\n        // node를 연결 리스트의 맨 앞에 추가한다\n        deque-&gt;front-&gt;prev = node;\n        node-&gt;next = deque-&gt;front;\n        deque-&gt;front = node; // 머리 노드를 갱신한다\n    }\n    // 맨 뒤에 삽입하는 연산\n    else {\n        // node를 연결 리스트의 맨 뒤에 추가한다\n        deque-&gt;rear-&gt;next = node;\n        node-&gt;prev = deque-&gt;rear;\n        deque-&gt;rear = node;\n    }\n    deque-&gt;queSize++; // 큐 길이를 갱신한다\n}\n\n/* 맨 앞에 삽입 */\nvoid pushFirst(LinkedListDeque *deque, int num) {\n    push(deque, num, true);\n}\n\n/* 맨 뒤에 삽입 */\nvoid pushLast(LinkedListDeque *deque, int num) {\n    push(deque, num, false);\n}\n\n/* 맨 앞 원소 접근 */\nint peekFirst(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;front);\n    return deque-&gt;front-&gt;val;\n}\n\n/* 맨 뒤 원소 접근 */\nint peekLast(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;rear);\n    return deque-&gt;rear-&gt;val;\n}\n\n/* 큐에서 꺼내기 */\nint pop(LinkedListDeque *deque, bool isFront) {\n    if (empty(deque))\n        return -1;\n    int val;\n    // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n    if (deque-&gt;queSize == 1) {\n        val = deque-&gt;front-&gt;val;\n        delDoublyListNode(deque-&gt;front); // 같은 노드를 가리키므로 하나만 해제하면 된다\n        deque-&gt;front = NULL;\n        deque-&gt;rear = NULL;\n    }\n    // 맨 앞에서 꺼내는 연산\n    else if (isFront) {\n        val = peekFirst(deque); // 머리 노드의 값을 임시로 저장한다\n        DoublyListNode *fNext = deque-&gt;front-&gt;next;\n        fNext-&gt;prev = NULL;\n        deque-&gt;front-&gt;next = NULL;\n        delDoublyListNode(deque-&gt;front);\n        deque-&gt;front = fNext; // 머리 노드를 갱신한다\n    }\n    // 맨 뒤에서 꺼내는 연산\n    else {\n        val = peekLast(deque); // 꼬리 노드의 값을 임시로 저장한다\n        DoublyListNode *rPrev = deque-&gt;rear-&gt;prev;\n        rPrev-&gt;next = NULL;\n        deque-&gt;rear-&gt;prev = NULL;\n        delDoublyListNode(deque-&gt;rear);\n        deque-&gt;rear = rPrev; // 꼬리 노드를 갱신한다\n    }\n    deque-&gt;queSize--; // 큐 길이를 갱신한다\n    return val;\n}\n\n/* 맨 앞에서 꺼내기 */\nint popFirst(LinkedListDeque *deque) {\n    return pop(deque, true);\n}\n\n/* 맨 뒤에서 꺼내기 */\nint popLast(LinkedListDeque *deque) {\n    return pop(deque, false);\n}\n\n/* 큐 출력 */\nvoid printLinkedListDeque(LinkedListDeque *deque) {\n    int *arr = malloc(sizeof(int) * deque-&gt;queSize);\n    // 연결 리스트의 데이터를 배열에 복사한다\n    int i;\n    DoublyListNode *node;\n    for (i = 0, node = deque-&gt;front; i &lt; deque-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, deque-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_deque.kt<pre><code>/* 이중 연결 리스트 노드 */\nclass ListNode(var _val: Int) {\n    // 노드 값\n    var next: ListNode? = null // 다음 노드 참조\n    var prev: ListNode? = null // 이전 노드 참조\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n    private var front: ListNode? = null // 머리 노드 front\n    private var rear: ListNode? = null // 꼬리 노드 rear\n    private var queSize: Int = 0 // 덱의 길이\n\n    /* 덱의 길이 반환 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* 큐 삽입 연산 */\n    fun push(num: Int, isFront: Boolean) {\n        val node = ListNode(num)\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (isEmpty()) {\n            rear = node\n            front = rear\n            // 맨 앞에 삽입하는 연산\n        } else if (isFront) {\n            // node를 연결 리스트의 맨 앞에 추가한다\n            front?.prev = node\n            node.next = front\n            front = node // 머리 노드를 갱신한다\n            // 맨 뒤에 삽입하는 연산\n        } else {\n            // node를 연결 리스트의 맨 뒤에 추가한다\n            rear?.next = node\n            node.prev = rear\n            rear = node // 꼬리 노드를 갱신한다\n        }\n        queSize++ // 큐 길이를 갱신한다\n    }\n\n    /* 맨 앞에 삽입 */\n    fun pushFirst(num: Int) {\n        push(num, true)\n    }\n\n    /* 맨 뒤에 삽입 */\n    fun pushLast(num: Int) {\n        push(num, false)\n    }\n\n    /* 큐에서 꺼내는 연산 */\n    fun pop(isFront: Boolean): Int {\n        if (isEmpty()) \n            throw IndexOutOfBoundsException()\n        val _val: Int\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (queSize == 1) {\n            _val = front!!._val\n            front = null\n            rear = null\n        }\n        // 맨 앞에서 꺼내는 연산\n        else if (isFront) {\n            _val = front!!._val // 머리 노드의 값을 임시로 저장한다\n            // 머리 노드를 삭제한다\n            val fNext = front!!.next\n            fNext!!.prev = null\n            front!!.next = null\n            front = fNext // 머리 노드를 갱신한다\n            // 맨 뒤에서 꺼내는 연산\n        } else {\n            _val = rear!!._val // 꼬리 노드의 값을 임시로 저장한다\n            // 꼬리 노드를 삭제한다\n            val rPrev = rear!!.prev\n            rPrev!!.next = null\n            rear!!.prev = null\n            rear = rPrev // 꼬리 노드를 갱신한다\n        }\n        queSize-- // 큐 길이를 갱신한다\n        return _val\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    fun popFirst(): Int {\n        return pop(true)\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    fun popLast(): Int {\n        return pop(false)\n    }\n\n    /* 맨 앞 원소 접근 */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!._val\n    }\n\n    /* 맨 뒤 원소 접근 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return rear!!._val\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!._val\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.rb<pre><code>=begin\nFile: linkedlist_deque.rb\nCreated Time: 2024-04-06\nAuthor: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)\n=end\n\n### 이중 연결 리스트 노드\nclass ListNode\n  attr_accessor :val\n  attr_accessor :next # 다음 노드 참조\n  attr_accessor :prev # 이전 노드 참조\n\n  ### 생성자 ###\n  def initialize(val)\n    @val = val\n  end\nend\n\n### 이중 연결 리스트로 구현한 덱 ###\nclass LinkedListDeque\n  ### 덱의 길이 반환 ###\n  attr_reader :size\n\n  ### 생성자 ###\n  def initialize\n    @front = nil  # 머리 노드 front\n    @rear = nil   # 꼬리 노드 rear\n    @size = 0     # 덱의 길이\n  end\n\n  ### 덱이 비어 있는지 판단 ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### 큐 삽입 연산 ###\n  def push(num, is_front)\n    node = ListNode.new(num)\n    # 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n    if is_empty?\n      @front = @rear = node\n    # 맨 앞에 삽입하는 연산\n    elsif is_front\n      # node를 연결 리스트의 맨 앞에 추가한다\n      @front.prev = node\n      node.next = @front\n      @front = node # 머리 노드를 갱신한다\n    # 맨 뒤에 삽입하는 연산\n    else\n      # node를 연결 리스트의 맨 뒤에 추가한다\n      @rear.next = node\n      node.prev = @rear\n      @rear = node # 꼬리 노드를 갱신한다\n    end\n    @size += 1 # 큐 길이를 갱신한다\n  end\n\n  ### 맨 앞에 삽입 ###\n  def push_first(num)\n    push(num, true)\n  end\n\n  ### 맨 뒤에 삽입 ###\n  def push_last(num)\n    push(num, false)\n  end\n\n  ### 큐에서 꺼내는 연산 ###\n  def pop(is_front)\n    raise IndexError, '덱이 비어 있다' if is_empty?\n\n    # 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n    if @size == 1\n      val = @front.val\n      @front = nil\n      @rear = nil\n    # 맨 앞에서 꺼내는 연산\n    elsif is_front\n      val = @front.val # 머리 노드의 값을 임시로 저장한다\n      # 머리 노드를 삭제한다\n      fnext = @front.next\n      fnext.prev = nil\n      @front.next = nil\n      @front = fnext # 머리 노드를 갱신한다\n    # 맨 뒤에서 꺼내는 연산\n    else\n      val = @rear.val # 꼬리 노드의 값을 임시로 저장한다\n      # 꼬리 노드를 삭제한다\n      rprev = @rear.prev\n      rprev.next = nil\n      @rear.prev = nil\n      @rear = rprev # 꼬리 노드를 갱신한다\n    end\n    @size -= 1 # 큐 길이를 갱신한다\n\n    val\n  end\n\n  ### 맨 앞에서 꺼내기 ###\n  def pop_first\n    pop(true)\n  end\n\n  ### 맨 앞에서 꺼내기 ###\n  def pop_last\n    pop(false)\n  end\n\n  ### 맨 앞 원소 접근 ###\n  def peek_first\n    raise IndexError, '덱이 비어 있다' if is_empty?\n\n    @front.val\n  end\n\n  ### 맨 뒤 원소 접근 ###\n  def peek_last\n    raise IndexError, '덱이 비어 있다' if is_empty?\n\n    @rear.val\n  end\n\n  ### 출력에 사용할 배열 반환 ###\n  def to_array\n    node = @front\n    res = Array.new(size, 0)\n    for i in 0...size\n      res[i] = node.val\n      node = node.next\n    end\n    res\n  end\nend\n</code></pre>","path":["제 5 장   스택과 큐","5.3   덱"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#2","level":3,"title":"2.   배열 기반 구현","text":"<p>아래 그림(5-9)과 같이 배열 기반 큐와 마찬가지로 원형 배열을 사용하여 덱을 구현할 수 있다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 5-9   배열 기반 덱의 삽입과 꺼내기 </p> <p>큐 구현에 ‘맨 앞에 삽입하기’와 ‘맨 뒤에서 꺼내기’ 메서드만 추가하면 된다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_deque.py<pre><code>class ArrayDeque:\n    \"\"\"원형 배열로 구현한 덱\"\"\"\n\n    def __init__(self, capacity: int):\n        \"\"\"생성자\"\"\"\n        self._nums: list[int] = [0] * capacity\n        self._front: int = 0\n        self._size: int = 0\n\n    def capacity(self) -&gt; int:\n        \"\"\"덱의 용량 반환\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"덱의 길이 반환\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"덱이 비어 있는지 판단\"\"\"\n        return self._size == 0\n\n    def index(self, i: int) -&gt; int:\n        \"\"\"원형 배열 인덱스 계산\"\"\"\n        # 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        # i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        # i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        return (i + self.capacity()) % self.capacity()\n\n    def push_first(self, num: int):\n        \"\"\"맨 앞에 삽입\"\"\"\n        if self._size == self.capacity():\n            print(\"덱이 가득 찼다\")\n            return\n        # 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        # 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        self._front = self.index(self._front - 1)\n        # num을 맨 앞에 추가한다\n        self._nums[self._front] = num\n        self._size += 1\n\n    def push_last(self, num: int):\n        \"\"\"맨 뒤에 삽입\"\"\"\n        if self._size == self.capacity():\n            print(\"덱이 가득 찼다\")\n            return\n        # 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        rear = self.index(self._front + self._size)\n        # num을 맨 뒤에 추가한다\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop_first(self) -&gt; int:\n        \"\"\"맨 앞에서 꺼내기\"\"\"\n        num = self.peek_first()\n        # 맨 앞 포인터를 뒤로 한 칸 이동한다\n        self._front = self.index(self._front + 1)\n        self._size -= 1\n        return num\n\n    def pop_last(self) -&gt; int:\n        \"\"\"맨 뒤에서 꺼내기\"\"\"\n        num = self.peek_last()\n        self._size -= 1\n        return num\n\n    def peek_first(self) -&gt; int:\n        \"\"\"맨 앞 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"덱이 비어 있다\")\n        return self._nums[self._front]\n\n    def peek_last(self) -&gt; int:\n        \"\"\"맨 뒤 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"덱이 비어 있다\")\n        # 마지막 원소의 인덱스를 계산한다\n        last = self.index(self._front + self._size - 1)\n        return self._nums[last]\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"출력에 사용할 배열 반환\"\"\"\n        # 유효한 길이 범위 안의 리스트 원소만 변환한다\n        res = []\n        for i in range(self._size):\n            res.append(self._nums[self.index(self._front + i)])\n        return res\n</code></pre> array_deque.cpp<pre><code>/* 원형 배열로 구현한 덱 */\nclass ArrayDeque {\n  private:\n    vector&lt;int&gt; nums; // 덱의 원소를 저장하는 배열\n    int front;        // 맨 앞 원소를 가리키는 맨 앞 포인터\n    int queSize;      // 덱 길이\n\n  public:\n    /* 생성자 */\n    ArrayDeque(int capacity) {\n        nums.resize(capacity);\n        front = queSize = 0;\n    }\n\n    /* 덱의 용량 반환 */\n    int capacity() {\n        return nums.size();\n    }\n\n    /* 덱의 길이 반환 */\n    int size() {\n        return queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    int index(int i) {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        return (i + capacity()) % capacity();\n    }\n\n    /* 맨 앞에 삽입 */\n    void pushFirst(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"덱이 가득 찼다\" &lt;&lt; endl;\n            return;\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        front = index(front - 1);\n        // num을 맨 앞에 추가한다\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* 맨 뒤에 삽입 */\n    void pushLast(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"덱이 가득 찼다\" &lt;&lt; endl;\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        int rear = index(front + queSize);\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    int popFirst() {\n        int num = peekFirst();\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"덱이 비어 있다\");\n        return nums[front];\n    }\n\n    /* 맨 뒤 원소 접근 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"덱이 비어 있다\");\n        // 마지막 원소의 인덱스를 계산한다\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    vector&lt;int&gt; toVector() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        vector&lt;int&gt; res(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n};\n</code></pre> array_deque.java<pre><code>/* 원형 배열로 구현한 덱 */\nclass ArrayDeque {\n    private int[] nums; // 덱의 원소를 저장하는 배열\n    private int front; // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private int queSize; // 덱 길이\n\n    /* 생성자 */\n    public ArrayDeque(int capacity) {\n        this.nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* 덱의 용량 반환 */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* 덱의 길이 반환 */\n    public int size() {\n        return queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    private int index(int i) {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        return (i + capacity()) % capacity();\n    }\n\n    /* 맨 앞에 삽입 */\n    public void pushFirst(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"덱이 가득 찼다\");\n            return;\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        front = index(front - 1);\n        // num을 맨 앞에 추가한다\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* 맨 뒤에 삽입 */\n    public void pushLast(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"덱이 가득 찼다\");\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        int rear = index(front + queSize);\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    public int popFirst() {\n        int num = peekFirst();\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    public int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* 맨 뒤 원소 접근 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        // 마지막 원소의 인덱스를 계산한다\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    public int[] toArray() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.cs<pre><code>/* 원형 배열로 구현한 덱 */\nclass ArrayDeque {\n    int[] nums;  // 덱의 원소를 저장하는 배열\n    int front;   // 맨 앞 원소를 가리키는 맨 앞 포인터\n    int queSize; // 덱 길이\n\n    /* 생성자 */\n    public ArrayDeque(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* 덱의 용량 반환 */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* 덱의 길이 반환 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    int Index(int i) {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        return (i + Capacity()) % Capacity();\n    }\n\n    /* 맨 앞에 삽입 */\n    public void PushFirst(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"덱이 가득 찼다\");\n            return;\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        front = Index(front - 1);\n        // num을 맨 앞에 추가한다\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* 맨 뒤에 삽입 */\n    public void PushLast(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"덱이 가득 찼다\");\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        int rear = Index(front + queSize);\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    public int PopFirst() {\n        int num = PeekFirst();\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        front = Index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    public int PopLast() {\n        int num = PeekLast();\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int PeekFirst() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        return nums[front];\n    }\n\n    /* 맨 뒤 원소 접근 */\n    public int PeekLast() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        // 마지막 원소의 인덱스를 계산한다\n        int last = Index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    public int[] ToArray() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[Index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.go<pre><code>/* 원형 배열로 구현한 덱 */\ntype arrayDeque struct {\n    nums        []int // 덱의 원소를 저장하는 배열\n    front       int   // 맨 앞 원소를 가리키는 맨 앞 포인터\n    queSize     int   // 덱 길이\n    queCapacity int   // 큐 용량(저장할 수 있는 최대 원소 개수)\n}\n\n/* 큐 초기화 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\n    return &amp;arrayDeque{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* 덱의 길이 반환 */\nfunc (q *arrayDeque) size() int {\n    return q.queSize\n}\n\n/* 덱이 비어 있는지 판단 */\nfunc (q *arrayDeque) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* 원형 배열 인덱스 계산 */\nfunc (q *arrayDeque) index(i int) int {\n    // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n    // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n    // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n    return (i + q.queCapacity) % q.queCapacity\n}\n\n/* 맨 앞에 삽입 */\nfunc (q *arrayDeque) pushFirst(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"덱이 가득 찼다\")\n        return\n    }\n    // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n    // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n    q.front = q.index(q.front - 1)\n    // num을 맨 앞에 추가한다\n    q.nums[q.front] = num\n    q.queSize++\n}\n\n/* 맨 뒤에 삽입 */\nfunc (q *arrayDeque) pushLast(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"덱이 가득 찼다\")\n        return\n    }\n    // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    rear := q.index(q.front + q.queSize)\n    // num을 맨 뒤에 추가한다\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* 맨 앞에서 꺼내기 */\nfunc (q *arrayDeque) popFirst() any {\n    num := q.peekFirst()\n    if num == nil {\n        return nil\n    }\n    // 맨 앞 포인터를 뒤로 한 칸 이동한다\n    q.front = q.index(q.front + 1)\n    q.queSize--\n    return num\n}\n\n/* 맨 뒤에서 꺼내기 */\nfunc (q *arrayDeque) popLast() any {\n    num := q.peekLast()\n    if num == nil {\n        return nil\n    }\n    q.queSize--\n    return num\n}\n\n/* 맨 앞 원소 접근 */\nfunc (q *arrayDeque) peekFirst() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* 맨 뒤 원소 접근 */\nfunc (q *arrayDeque) peekLast() any {\n    if q.isEmpty() {\n        return nil\n    }\n    // 마지막 원소의 인덱스를 계산한다\n    last := q.index(q.front + q.queSize - 1)\n    return q.nums[last]\n}\n\n/* 출력에 사용할 Slice 반환 */\nfunc (q *arrayDeque) toSlice() []int {\n    // 유효한 길이 범위 안의 리스트 원소만 변환한다\n    res := make([]int, q.queSize)\n    for i, j := 0, q.front; i &lt; q.queSize; i++ {\n        res[i] = q.nums[q.index(j)]\n        j++\n    }\n    return res\n}\n</code></pre> array_deque.swift<pre><code>/* 원형 배열로 구현한 덱 */\nclass ArrayDeque {\n    private var nums: [Int] // 덱의 원소를 저장하는 배열\n    private var front: Int // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private var _size: Int // 덱 길이\n\n    /* 생성자 */\n    init(capacity: Int) {\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* 덱의 용량 반환 */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* 덱의 길이 반환 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    private func index(i: Int) -&gt; Int {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        (i + capacity()) % capacity()\n    }\n\n    /* 맨 앞에 삽입 */\n    func pushFirst(num: Int) {\n        if size() == capacity() {\n            print(\"덱이 가득 찼다\")\n            return\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        front = index(i: front - 1)\n        // num을 맨 앞에 추가한다\n        nums[front] = num\n        _size += 1\n    }\n\n    /* 맨 뒤에 삽입 */\n    func pushLast(num: Int) {\n        if size() == capacity() {\n            print(\"덱이 가득 찼다\")\n            return\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        let rear = index(i: front + size())\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    func popFirst() -&gt; Int {\n        let num = peekFirst()\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        front = index(i: front + 1)\n        _size -= 1\n        return num\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    func popLast() -&gt; Int {\n        let num = peekLast()\n        _size -= 1\n        return num\n    }\n\n    /* 맨 앞 원소 접근 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"덱이 비어 있다\")\n        }\n        return nums[front]\n    }\n\n    /* 맨 뒤 원소 접근 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"덱이 비어 있다\")\n        }\n        // 마지막 원소의 인덱스를 계산한다\n        let last = index(i: front + size() - 1)\n        return nums[last]\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    func toArray() -&gt; [Int] {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        (front ..&lt; front + size()).map { nums[index(i: $0)] }\n    }\n}\n</code></pre> array_deque.js<pre><code>/* 원형 배열로 구현한 덱 */\nclass ArrayDeque {\n    #nums; // 덱의 원소를 저장하는 배열\n    #front; // 맨 앞 원소를 가리키는 맨 앞 포인터\n    #queSize; // 덱 길이\n\n    /* 생성자 */\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n        this.#front = 0;\n        this.#queSize = 0;\n    }\n\n    /* 덱의 용량 반환 */\n    capacity() {\n        return this.#nums.length;\n    }\n\n    /* 덱의 길이 반환 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    index(i) {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* 맨 앞에 삽입 */\n    pushFirst(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('덱이 가득 찼다');\n            return;\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        this.#front = this.index(this.#front - 1);\n        // num을 맨 앞에 추가한다\n        this.#nums[this.#front] = num;\n        this.#queSize++;\n    }\n\n    /* 맨 뒤에 삽입 */\n    pushLast(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('덱이 가득 찼다');\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        const rear = this.index(this.#front + this.#queSize);\n        // num을 맨 뒤에 추가한다\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    popFirst() {\n        const num = this.peekFirst();\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        this.#front = this.index(this.#front + 1);\n        this.#queSize--;\n        return num;\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    popLast() {\n        const num = this.peekLast();\n        this.#queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peekFirst() {\n        if (this.isEmpty()) throw new Error('덱이 비어 있다.');\n        return this.#nums[this.#front];\n    }\n\n    /* 맨 뒤 원소 접근 */\n    peekLast() {\n        if (this.isEmpty()) throw new Error('덱이 비어 있다.');\n        // 마지막 원소의 인덱스를 계산한다\n        const last = this.index(this.#front + this.#queSize - 1);\n        return this.#nums[last];\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    toArray() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        const res = [];\n        for (let i = 0, j = this.#front; i &lt; this.#queSize; i++, j++) {\n            res[i] = this.#nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.ts<pre><code>/* 원형 배열로 구현한 덱 */\nclass ArrayDeque {\n    private nums: number[]; // 덱의 원소를 저장하는 배열\n    private front: number; // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private queSize: number; // 덱 길이\n\n    /* 생성자 */\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = 0;\n        this.queSize = 0;\n    }\n\n    /* 덱의 용량 반환 */\n    capacity(): number {\n        return this.nums.length;\n    }\n\n    /* 덱의 길이 반환 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    index(i: number): number {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* 맨 앞에 삽입 */\n    pushFirst(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('덱이 가득 찼다');\n            return;\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        this.front = this.index(this.front - 1);\n        // num을 맨 앞에 추가한다\n        this.nums[this.front] = num;\n        this.queSize++;\n    }\n\n    /* 맨 뒤에 삽입 */\n    pushLast(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('덱이 가득 찼다');\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        const rear: number = this.index(this.front + this.queSize);\n        // num을 맨 뒤에 추가한다\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    popFirst(): number {\n        const num: number = this.peekFirst();\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        this.front = this.index(this.front + 1);\n        this.queSize--;\n        return num;\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    popLast(): number {\n        const num: number = this.peekLast();\n        this.queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peekFirst(): number {\n        if (this.isEmpty()) throw new Error('덱이 비어 있다.');\n        return this.nums[this.front];\n    }\n\n    /* 맨 뒤 원소 접근 */\n    peekLast(): number {\n        if (this.isEmpty()) throw new Error('덱이 비어 있다.');\n        // 마지막 원소의 인덱스를 계산한다\n        const last = this.index(this.front + this.queSize - 1);\n        return this.nums[last];\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    toArray(): number[] {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        const res: number[] = [];\n        for (let i = 0, j = this.front; i &lt; this.queSize; i++, j++) {\n            res[i] = this.nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.dart<pre><code>/* 원형 배열로 구현한 덱 */\nclass ArrayDeque {\n  late List&lt;int&gt; _nums; // 덱의 원소를 저장하는 배열\n  late int _front; // 맨 앞 원소를 가리키는 맨 앞 포인터\n  late int _queSize; // 덱 길이\n\n  /* 생성자 */\n  ArrayDeque(int capacity) {\n    this._nums = List.filled(capacity, 0);\n    this._front = this._queSize = 0;\n  }\n\n  /* 덱의 용량 반환 */\n  int capacity() {\n    return _nums.length;\n  }\n\n  /* 덱의 길이 반환 */\n  int size() {\n    return _queSize;\n  }\n\n  /* 덱이 비어 있는지 판단 */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* 원형 배열 인덱스 계산 */\n  int index(int i) {\n    // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n    // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n    // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n    return (i + capacity()) % capacity();\n  }\n\n  /* 맨 앞에 삽입 */\n  void pushFirst(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"덱이 가득 찼다\");\n    }\n    // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n    // 나머지 연산을 사용하여 _front가 배열의 시작을 넘어가면 끝으로 돌아가게 한다\n    _front = index(_front - 1);\n    // _num을 맨 앞에 추가한다\n    _nums[_front] = _num;\n    _queSize++;\n  }\n\n  /* 맨 뒤에 삽입 */\n  void pushLast(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"덱이 가득 찼다\");\n    }\n    // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    int rear = index(_front + _queSize);\n    // _num을 맨 뒤에 추가한다\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* 맨 앞에서 꺼내기 */\n  int popFirst() {\n    int _num = peekFirst();\n    // 맨 앞 포인터를 오른쪽으로 한 칸 이동한다\n    _front = index(_front + 1);\n    _queSize--;\n    return _num;\n  }\n\n  /* 맨 뒤에서 꺼내기 */\n  int popLast() {\n    int _num = peekLast();\n    _queSize--;\n    return _num;\n  }\n\n  /* 맨 앞 원소 접근 */\n  int peekFirst() {\n    if (isEmpty()) {\n      throw Exception(\"덱이 비어 있다\");\n    }\n    return _nums[_front];\n  }\n\n  /* 맨 뒤 원소 접근 */\n  int peekLast() {\n    if (isEmpty()) {\n      throw Exception(\"덱이 비어 있다\");\n    }\n    // 마지막 원소의 인덱스를 계산한다\n    int last = index(_front + _queSize - 1);\n    return _nums[last];\n  }\n\n  /* 출력에 사용할 배열 반환 */\n  List&lt;int&gt; toArray() {\n    // 유효한 길이 범위 안의 리스트 원소만 변환한다\n    List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[index(j)];\n    }\n    return res;\n  }\n}\n</code></pre> array_deque.rs<pre><code>/* 원형 배열로 구현한 덱 */\nstruct ArrayDeque&lt;T&gt; {\n    nums: Vec&lt;T&gt;,    // 덱의 원소를 저장하는 배열\n    front: usize,    // 맨 앞 원소를 가리키는 맨 앞 포인터\n    que_size: usize, // 덱 길이\n}\n\nimpl&lt;T: Copy + Default&gt; ArrayDeque&lt;T&gt; {\n    /* 생성자 */\n    pub fn new(capacity: usize) -&gt; Self {\n        Self {\n            nums: vec![T::default(); capacity],\n            front: 0,\n            que_size: 0,\n        }\n    }\n\n    /* 덱의 용량 반환 */\n    pub fn capacity(&amp;self) -&gt; usize {\n        self.nums.len()\n    }\n\n    /* 덱의 길이 반환 */\n    pub fn size(&amp;self) -&gt; usize {\n        self.que_size\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    fn index(&amp;self, i: i32) -&gt; usize {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        ((i + self.capacity() as i32) % self.capacity() as i32) as usize\n    }\n\n    /* 맨 앞에 삽입 */\n    pub fn push_first(&amp;mut self, num: T) {\n        if self.que_size == self.capacity() {\n            println!(\"덱이 가득 찼다\");\n            return;\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        self.front = self.index(self.front as i32 - 1);\n        // num을 맨 앞에 추가한다\n        self.nums[self.front] = num;\n        self.que_size += 1;\n    }\n\n    /* 맨 뒤에 삽입 */\n    pub fn push_last(&amp;mut self, num: T) {\n        if self.que_size == self.capacity() {\n            println!(\"덱이 가득 찼다\");\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        let rear = self.index(self.front as i32 + self.que_size as i32);\n        // num을 맨 뒤에 추가한다\n        self.nums[rear] = num;\n        self.que_size += 1;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    fn pop_first(&amp;mut self) -&gt; T {\n        let num = self.peek_first();\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        self.front = self.index(self.front as i32 + 1);\n        self.que_size -= 1;\n        num\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    fn pop_last(&amp;mut self) -&gt; T {\n        let num = self.peek_last();\n        self.que_size -= 1;\n        num\n    }\n\n    /* 맨 앞 원소 접근 */\n    fn peek_first(&amp;self) -&gt; T {\n        if self.is_empty() {\n            panic!(\"덱이 비어 있다\")\n        };\n        self.nums[self.front]\n    }\n\n    /* 맨 뒤 원소 접근 */\n    fn peek_last(&amp;self) -&gt; T {\n        if self.is_empty() {\n            panic!(\"덱이 비어 있다\")\n        };\n        // 마지막 원소의 인덱스를 계산한다\n        let last = self.index(self.front as i32 + self.que_size as i32 - 1);\n        self.nums[last]\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    fn to_array(&amp;self) -&gt; Vec&lt;T&gt; {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        let mut res = vec![T::default(); self.que_size];\n        let mut j = self.front;\n        for i in 0..self.que_size {\n            res[i] = self.nums[self.index(j as i32)];\n            j += 1;\n        }\n        res\n    }\n}\n</code></pre> array_deque.c<pre><code>/* 원형 배열로 구현한 덱 */\ntypedef struct {\n    int *nums;       // 큐의 원소를 저장하는 배열\n    int front;       // 맨 앞 원소를 가리키는 맨 앞 포인터\n    int queSize;     // 맨 뒤 포인터, 큐의 맨 뒤 + 1을 가리킨다\n    int queCapacity; // 큐 용량\n} ArrayDeque;\n\n/* 생성자 */\nArrayDeque *newArrayDeque(int capacity) {\n    ArrayDeque *deque = (ArrayDeque *)malloc(sizeof(ArrayDeque));\n    // 배열 초기화\n    deque-&gt;queCapacity = capacity;\n    deque-&gt;nums = (int *)malloc(sizeof(int) * deque-&gt;queCapacity);\n    deque-&gt;front = deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* 소멸자 */\nvoid delArrayDeque(ArrayDeque *deque) {\n    free(deque-&gt;nums);\n    free(deque);\n}\n\n/* 덱의 용량 반환 */\nint capacity(ArrayDeque *deque) {\n    return deque-&gt;queCapacity;\n}\n\n/* 덱의 길이 반환 */\nint size(ArrayDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* 덱이 비어 있는지 판단 */\nbool empty(ArrayDeque *deque) {\n    return deque-&gt;queSize == 0;\n}\n\n/* 원형 배열 인덱스 계산 */\nint dequeIndex(ArrayDeque *deque, int i) {\n    // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n    // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n    // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n    return ((i + capacity(deque)) % capacity(deque));\n}\n\n/* 맨 앞에 삽입 */\nvoid pushFirst(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"덱이 가득 찼다\\r\\n\");\n        return;\n    }\n    // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n    // 나머지 연산으로 front가 배열의 처음을 넘어가면 끝으로 돌아가게 한다\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front - 1);\n    // num을 큐의 맨 앞에 추가한다\n    deque-&gt;nums[deque-&gt;front] = num;\n    deque-&gt;queSize++;\n}\n\n/* 맨 뒤에 삽입 */\nvoid pushLast(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"덱이 가득 찼다\\r\\n\");\n        return;\n    }\n    // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    int rear = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize);\n    // num을 맨 뒤에 추가한다\n    deque-&gt;nums[rear] = num;\n    deque-&gt;queSize++;\n}\n\n/* 맨 앞 원소 접근 */\nint peekFirst(ArrayDeque *deque) {\n    // 접근 예외: 덱이 비어 있다\n    assert(empty(deque) == 0);\n    return deque-&gt;nums[deque-&gt;front];\n}\n\n/* 맨 뒤 원소 접근 */\nint peekLast(ArrayDeque *deque) {\n    // 접근 예외: 덱이 비어 있다\n    assert(empty(deque) == 0);\n    int last = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize - 1);\n    return deque-&gt;nums[last];\n}\n\n/* 맨 앞에서 꺼내기 */\nint popFirst(ArrayDeque *deque) {\n    int num = peekFirst(deque);\n    // 맨 앞 포인터를 뒤로 한 칸 이동한다\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front + 1);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* 맨 뒤에서 꺼내기 */\nint popLast(ArrayDeque *deque) {\n    int num = peekLast(deque);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* 출력에 사용할 배열 반환 */\nint *toArray(ArrayDeque *deque, int *queSize) {\n    *queSize = deque-&gt;queSize;\n    int *res = (int *)calloc(deque-&gt;queSize, sizeof(int));\n    int j = deque-&gt;front;\n    for (int i = 0; i &lt; deque-&gt;queSize; i++) {\n        res[i] = deque-&gt;nums[j % deque-&gt;queCapacity];\n        j++;\n    }\n    return res;\n}\n</code></pre> array_deque.kt<pre><code>/* 생성자 */\nclass ArrayDeque(capacity: Int) {\n    private var nums: IntArray = IntArray(capacity) // 덱의 원소를 저장하는 배열\n    private var front: Int = 0 // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private var queSize: Int = 0 // 덱 길이\n\n    /* 덱의 용량 반환 */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* 덱의 길이 반환 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    private fun index(i: Int): Int {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        return (i + capacity()) % capacity()\n    }\n\n    /* 맨 앞에 삽입 */\n    fun pushFirst(num: Int) {\n        if (queSize == capacity()) {\n            println(\"덱이 가득 찼다\")\n            return\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        front = index(front - 1)\n        // num을 맨 앞에 추가한다\n        nums[front] = num\n        queSize++\n    }\n\n    /* 맨 뒤에 삽입 */\n    fun pushLast(num: Int) {\n        if (queSize == capacity()) {\n            println(\"덱이 가득 찼다\")\n            return\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        val rear = index(front + queSize)\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num\n        queSize++\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    fun popFirst(): Int {\n        val num = peekFirst()\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        front = index(front + 1)\n        queSize--\n        return num\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    fun popLast(): Int {\n        val num = peekLast()\n        queSize--\n        return num\n    }\n\n    /* 맨 앞 원소 접근 */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* 맨 뒤 원소 접근 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // 마지막 원소의 인덱스를 계산한다\n        val last = index(front + queSize - 1)\n        return nums[last]\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    fun toArray(): IntArray {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[index(j)]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_deque.rb<pre><code>### 원형 배열로 구현한 덱 ###\nclass ArrayDeque\n  ### 덱의 길이 반환 ###\n  attr_reader :size\n\n  ### 생성자 ###\n  def initialize(capacity)\n    @nums = Array.new(capacity, 0)\n    @front = 0\n    @size = 0\n  end\n\n  ### 덱의 용량 반환 ###\n  def capacity\n    @nums.length\n  end\n\n  ### 덱이 비어 있는지 판단 ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### 맨 앞에 삽입 ###\n  def push_first(num)\n    if size == capacity\n      puts '덱이 가득 찼다'\n      return\n    end\n\n    # 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n    # 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n    @front = index(@front - 1)\n    # num을 맨 앞에 추가한다\n    @nums[@front] = num\n    @size += 1\n  end\n\n  ### 맨 뒤에 삽입 ###\n  def push_last(num)\n    if size == capacity\n      puts '덱이 가득 찼다'\n      return\n    end\n\n    # 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    rear = index(@front + size)\n    # num을 맨 뒤에 추가한다\n    @nums[rear] = num\n    @size += 1\n  end\n\n  ### 맨 앞에서 꺼내기 ###\n  def pop_first\n    num = peek_first\n    # 맨 앞 포인터를 뒤로 한 칸 이동한다\n    @front = index(@front + 1)\n    @size -= 1\n    num\n  end\n\n  ### 맨 뒤에서 꺼내기 ###\n  def pop_last\n    num = peek_last\n    @size -= 1\n    num\n  end\n\n  ### 맨 앞 원소 접근 ###\n  def peek_first\n    raise IndexError, '덱이 비어 있다' if is_empty?\n\n    @nums[@front]\n  end\n\n  ### 맨 뒤 원소 접근 ###\n  def peek_last\n    raise IndexError, '덱이 비어 있다' if is_empty?\n\n    # 마지막 원소의 인덱스를 계산한다\n    last = index(@front + size - 1)\n    @nums[last]\n  end\n\n  ### 출력에 사용할 배열 반환 ###\n  def to_array\n    # 유효한 길이 범위 안의 리스트 원소만 변환한다\n    res = []\n    for i in 0...size\n      res &lt;&lt; @nums[index(@front + i)]\n    end\n    res\n  end\n\n  private\n\n  ### 원형 배열 인덱스 계산 ###\n  def index(i)\n    # 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n    # i가 배열의 끝을 넘어가면 처음으로 돌아온다\n    # i가 배열의 처음을 넘어가면 끝으로 돌아온다\n    (i + capacity) % capacity\n  end\nend\n</code></pre>","path":["제 5 장   스택과 큐","5.3   덱"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#533","level":2,"title":"5.3.3   덱의 활용","text":"<p>덱은 스택과 큐의 논리를 모두 갖추므로 둘의 모든 활용 상황에 적용할 수 있으며, 더 높은 자유도도 제공한다.</p> <p>소프트웨어의 ‘실행 취소’는 보통 스택으로 구현한다. 시스템이 각 변경 작업을 스택에 <code>push</code>하고, <code>pop</code>으로 취소한다. 다만 자원 제약 때문에 보통 취소 가능한 단계 수를 제한한다(예를 들어 \\(50\\)단계만 저장). 스택 길이가 \\(50\\)을 넘으면 바닥(덱의 맨 앞)에서 삭제해야 한다. 스택은 이 기능을 지원하지 않으므로 덱으로 대체해야 한다. ‘실행 취소’의 핵심 논리는 여전히 스택의 후입선출 원리를 따르며, 덱은 여기에 필요한 추가 논리를 더 유연하게 구현하게 해 준다.</p>","path":["제 5 장   스택과 큐","5.3   덱"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/","level":1,"title":"5.5   연습 문제","text":"","path":["제 5 장   스택과 큐","5.5   연습 문제"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#551","level":2,"title":"5.5.1   개념 확인","text":"","path":["제 5 장   스택과 큐","5.5   연습 문제"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#1","level":3,"title":"1.   스택과 큐에서 무엇이 먼저 나오는가","text":"<p>빈 스택 <code>S</code>와 빈 큐 <code>Q</code>를 준비하고, 각각에 같은 연산을 수행한다.</p> <p>1단계: <code>A</code>를 넣는다. 2단계: <code>B</code>를 넣는다. 3단계: 원소 하나를 꺼내 기록한다. 4단계: <code>C</code>를 넣는다. 5단계: 비어 있을 때까지 원소를 계속 꺼내 기록한다.</p> <p><code>S</code>와 <code>Q</code>에서 원소가 나오는 순서를 각각 적고, ‘후입선출’ 또는 ‘선입선출’로 차이를 설명하라.</p> 참고 답안 <p>스택 <code>S</code>에서 나오는 순서는 <code>B, C, A</code>다. <code>A, B</code>를 넣은 뒤 가장 최근에 넣은 <code>B</code>를 먼저 꺼내고 <code>C</code>를 넣는다. 이후 <code>C, A</code>를 차례대로 꺼내므로 ‘후입선출’의 특성을 보인다.</p> <p>큐 <code>Q</code>에서 나오는 순서는 <code>A, B, C</code>다. <code>A, B</code>를 넣은 뒤 가장 먼저 넣은 <code>A</code>를 꺼내고, <code>C</code>를 넣는다. 이후 <code>B, C</code>를 차례대로 꺼내므로 ‘선입선출’의 특성을 보인다.</p>","path":["제 5 장   스택과 큐","5.5   연습 문제"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#2","level":3,"title":"2.   큐의 뒤가 배열의 끝을 넘어가면 어떻게 되는가","text":"<p>길이가 5인 원형 배열로 큐를 구현하며, 배열의 인덱스는 <code>0～4</code>다. 현재 <code>front = 3</code>, <code>size = 2</code>이고 큐의 <code>A, B</code>는 각각 인덱스 3과 4에 있다.</p> <ol> <li>‘<code>C</code>를 큐에 넣기’를 수행하면 <code>C</code>를 어느 인덱스에 저장해야 하는가? 삽입 후 <code>size</code>는 얼마인가?</li> <li>이어서 큐에서 한 번 꺼내면 어떤 원소가 나오는가? 새로운 <code>front</code>와 <code>size</code>는 각각 얼마인가?</li> <li>이때 큐의 앞에서 뒤로 이어지는 논리적 순서는 무엇인가? 원소를 꺼낼 때 배열의 다른 원소도 이동해야 하는가? 이유를 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>새 원소의 위치는     <code>(front + size) % 5 = (3 + 2) % 5 = 0</code>이므로,     <code>C</code>는 인덱스 0에 저장된다. 삽입 후 <code>size = 3</code>이다.</p> </li> <li> <p>현재 큐의 맨 앞 원소인 <code>A</code>가 나온다. 새로운 맨 앞 인덱스는     <code>(3 + 1) % 5 = 4</code>이므로 <code>front = 4</code>, <code>size = 2</code>다.</p> </li> <li> <p>유효한 원소의 논리적 순서는 <code>B, C</code>이며, <code>B</code>는 인덱스 4에, <code>C</code>는 인덱스 0에 있다.     원소를 꺼낼 때는 <code>front</code>를 이동하고 <code>size</code>만 수정하면 된다. 원형 배열에서는 나머지 연산으로 인덱스가 처음으로 돌아가므로,     다른 원소를 모두 앞으로 옮길 필요가 없다.</p> </li> </ol>","path":["제 5 장   스택과 큐","5.5   연습 문제"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#3","level":3,"title":"3.   덱의 양 끝 연산","text":"<p>여기서는 <code>push_first</code>를 앞에 넣기, <code>push_last</code>를 뒤에 넣기, <code>pop_first</code>를 앞에서 꺼내기, <code>pop_last</code>를 뒤에서 꺼내기로 정의한다.</p> <p>빈 덱 <code>deq</code>에 다음 연산을 차례대로 수행한다.</p> <ol> <li><code>push_last(A)</code></li> <li><code>push_last(B)</code></li> <li><code>push_first(C)</code></li> <li><code>pop_last()</code></li> <li><code>push_last(D)</code></li> <li><code>pop_first()</code></li> </ol> <ol> <li>두 번 꺼낸 원소는 각각 무엇인가?</li> <li>모든 연산을 마친 뒤 앞에서 뒤까지 어떤 원소가 남는가?</li> <li>뒤에서만 넣고 앞에서만 삭제할 수 있는 큐로 이 6단계를 모두 수행할 수 있는가? 불가능하다면 수행할 수 없는 연산을 찾아라. 이어서 덱은 모두 수행할 수 있는지와 그 이유를 설명하라.</li> </ol> 참고 답안 <p>처음 세 단계를 마치면 덱의 앞에서 뒤까지는 <code>[C, A, B]</code>다.</p> <ol> <li> <p><code>pop_last()</code>로 <code>B</code>를 꺼낸다. <code>D</code>를 넣으면 <code>[C, A, D]</code>가 되고,     <code>pop_first()</code>로 <code>C</code>를 꺼낸다.</p> </li> <li> <p>마지막에는 <code>[A, D]</code>가 남는다.</p> </li> <li> <p>뒤에서만 넣고 앞에서만 삭제하는 큐로는 모든 연산을 수행할 수 없다.     3단계의 <code>push_first(C)</code>는 앞에 넣어야 하고, 4단계의 <code>pop_last()</code>는 뒤에서 삭제해야 하므로 이 큐가 지원하는 연산 범위를 벗어난다.     덱은 양 끝에서 모두 삽입과 삭제가 가능하므로 6단계를 전부 수행할 수 있다.</p> </li> </ol>","path":["제 5 장   스택과 큐","5.5   연습 문제"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#552","level":2,"title":"5.5.2   프로그래밍 연습","text":"","path":["제 5 장   스택과 큐","5.5   연습 문제"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#1_1","level":3,"title":"1.   괄호 문자열 검사하기","text":"<p>세 종류의 괄호 <code>()</code>, <code>[]</code>, <code>{}</code>만 포함하는 문자열 <code>s</code>가 주어진다. 스택을 사용하여 올바른 괄호 문자열인지 판단하라.</p> <p>올바른 문자열은 다음 두 조건을 모두 만족해야 한다. 닫는 괄호는 가장 최근에 등장한, 아직 짝이 없는 여는 괄호와 종류가 같아야 하며, 순회를 마친 뒤 짝이 없는 여는 괄호가 남아 있어서는 안 된다. 판단 결과를 불리언 값으로 반환하라.</p> 풀이 힌트 <ol> <li>‘닫는 괄호에서 대응하는 여는 괄호로의 매핑’을 만들 수 있다.</li> <li>닫는 괄호를 만나면 먼저 스택이 비었는지 확인한 뒤 맨 위 괄호가 짝이 맞는지 확인한다.</li> <li>순회가 끝난 뒤에도 스택이 비어 있어야 한다.</li> </ol> <p>LeetCode</p>","path":["제 5 장   스택과 큐","5.5   연습 문제"],"tags":[]},{"location":"chapter_stack_and_queue/queue/","level":1,"title":"5.2   큐","text":"<p>큐(queue)는 선입선출 규칙을 따르는 선형 자료 구조다. 이름 그대로 줄 서는 현상을 재현한다. 새로 온 사람은 계속 줄의 뒤에 서고, 맨 앞에 있는 사람부터 하나씩 떠난다.</p> <p>아래 그림(5-4)과 같이 큐의 앞쪽을 ‘큐의 맨 앞’, 뒤쪽을 ‘큐의 맨 뒤’라고 한다. 맨 뒤에 원소를 추가하는 연산을 ‘큐에 삽입하기’, 맨 앞 원소를 삭제하는 연산을 ‘큐에서 꺼내기’라고 한다.</p> <p></p> <p> 그림 5-4   큐의 선입선출 규칙 </p>","path":["제 5 장   스택과 큐","5.2   큐"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#521","level":2,"title":"5.2.1   큐의 기본 연산","text":"<p>큐의 기본 연산은 아래 표(5-2)와 같다. 메서드 이름은 프로그래밍 언어에 따라 다를 수 있으며, 여기서는 스택과 같은 이름을 사용한다.</p> <p> 표 5-2   큐의 연산 효율 </p> 메서드 설명 시간 복잡도 <code>push()</code> 큐에 원소 삽입, 즉 맨 뒤에 추가 \\(O(1)\\) <code>pop()</code> 큐의 맨 앞 원소를 꺼냄 \\(O(1)\\) <code>peek()</code> 큐의 맨 앞 원소에 접근 \\(O(1)\\) <p>프로그래밍 언어가 제공하는 큐 클래스를 바로 사용할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby queue.py<pre><code>from collections import deque\n\n# 큐 초기화\n# Python에서는 보통 덱 클래스 deque를 큐로 사용한다\n# queue.Queue()는 전용 큐 클래스이지만 사용이 불편하므로 권하지 않는다\nque: deque[int] = deque()\n\n# 큐에 원소 삽입\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n\n# 큐의 맨 앞 원소에 접근\nfront: int = que[0]\n\n# 큐에서 원소 꺼내기\npop: int = que.popleft()\n\n# 큐 길이 구하기\nsize: int = len(que)\n\n# 큐가 비어 있는지 확인\nis_empty: bool = len(que) == 0\n</code></pre> queue.cpp<pre><code>/* 큐 초기화 */\nqueue&lt;int&gt; queue;\n\n/* 큐에 원소 삽입 */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* 큐의 맨 앞 원소에 접근 */\nint front = queue.front();\n\n/* 큐에서 원소 꺼내기 */\nqueue.pop();\n\n/* 큐 길이 구하기 */\nint size = queue.size();\n\n/* 큐가 비어 있는지 확인 */\nbool empty = queue.empty();\n</code></pre> queue.java<pre><code>/* 큐 초기화 */\nQueue&lt;Integer&gt; queue = new LinkedList&lt;&gt;();\n\n/* 큐에 원소 삽입 */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n\n/* 큐의 맨 앞 원소에 접근 */\nint peek = queue.peek();\n\n/* 큐에서 원소 꺼내기 */\nint pop = queue.poll();\n\n/* 큐 길이 구하기 */\nint size = queue.size();\n\n/* 큐가 비어 있는지 확인 */\nboolean isEmpty = queue.isEmpty();\n</code></pre> queue.cs<pre><code>/* 큐 초기화 */\nQueue&lt;int&gt; queue = new();\n\n/* 큐에 원소 삽입 */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n\n/* 큐의 맨 앞 원소에 접근 */\nint peek = queue.Peek();\n\n/* 큐에서 원소 꺼내기 */\nint pop = queue.Dequeue();\n\n/* 큐 길이 구하기 */\nint size = queue.Count;\n\n/* 큐가 비어 있는지 확인 */\nbool isEmpty = queue.Count == 0;\n</code></pre> queue_test.go<pre><code>/* 큐 초기화 */\n// Go에서는 list를 큐로 사용한다\nqueue := list.New()\n\n/* 큐에 원소 삽입 */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n\n/* 큐의 맨 앞 원소에 접근 */\npeek := queue.Front()\n\n/* 큐에서 원소 꺼내기 */\npop := queue.Front()\nqueue.Remove(pop)\n\n/* 큐 길이 구하기 */\nsize := queue.Len()\n\n/* 큐가 비어 있는지 확인 */\nisEmpty := queue.Len() == 0\n</code></pre> queue.swift<pre><code>/* 큐 초기화 */\n// Swift에는 내장 큐 클래스가 없으므로 Array를 큐로 사용할 수 있다\nvar queue: [Int] = []\n\n/* 큐에 원소 삽입 */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n\n/* 큐의 맨 앞 원소에 접근 */\nlet peek = queue.first!\n\n/* 큐에서 원소 꺼내기 */\n// 배열이므로 removeFirst의 복잡도는 O(n)이다\nlet pop = queue.removeFirst()\n\n/* 큐 길이 구하기 */\nlet size = queue.count\n\n/* 큐가 비어 있는지 확인 */\nlet isEmpty = queue.isEmpty\n</code></pre> queue.js<pre><code>/* 큐 초기화 */\n// JavaScript에는 내장 큐가 없으므로 Array를 큐로 사용할 수 있다\nconst queue = [];\n\n/* 큐에 원소 삽입 */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* 큐의 맨 앞 원소에 접근 */\nconst peek = queue[0];\n\n/* 큐에서 원소 꺼내기 */\n// 기반이 배열이므로 shift() 메서드의 시간 복잡도는 O(n)이다\nconst pop = queue.shift();\n\n/* 큐 길이 구하기 */\nconst size = queue.length;\n\n/* 큐가 비어 있는지 확인 */\nconst empty = queue.length === 0;\n</code></pre> queue.ts<pre><code>/* 큐 초기화 */\n// TypeScript에는 내장 큐가 없으므로 Array를 큐로 사용할 수 있다\nconst queue: number[] = [];\n\n/* 큐에 원소 삽입 */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* 큐의 맨 앞 원소에 접근 */\nconst peek = queue[0];\n\n/* 큐에서 원소 꺼내기 */\n// 기반이 배열이므로 shift() 메서드의 시간 복잡도는 O(n)이다\nconst pop = queue.shift();\n\n/* 큐 길이 구하기 */\nconst size = queue.length;\n\n/* 큐가 비어 있는지 확인 */\nconst empty = queue.length === 0;\n</code></pre> queue.dart<pre><code>/* 큐 초기화 */\n// Dart의 Queue 클래스는 덱이며 일반 큐로도 사용할 수 있다\nQueue&lt;int&gt; queue = Queue();\n\n/* 큐에 원소 삽입 */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n\n/* 큐의 맨 앞 원소에 접근 */\nint peek = queue.first;\n\n/* 큐에서 원소 꺼내기 */\nint pop = queue.removeFirst();\n\n/* 큐 길이 구하기 */\nint size = queue.length;\n\n/* 큐가 비어 있는지 확인 */\nbool isEmpty = queue.isEmpty;\n</code></pre> queue.rs<pre><code>/* 덱 초기화 */\n// Rust에서는 덱을 일반 큐로 사용한다\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* 큐에 원소 삽입 */\ndeque.push_back(1);\ndeque.push_back(3);\ndeque.push_back(2);\ndeque.push_back(5);\ndeque.push_back(4);\n\n/* 큐의 맨 앞 원소에 접근 */\nif let Some(front) = deque.front() {\n}\n\n/* 큐에서 원소 꺼내기 */\nif let Some(pop) = deque.pop_front() {\n}\n\n/* 큐 길이 구하기 */\nlet size = deque.len();\n\n/* 큐가 비어 있는지 확인 */\nlet is_empty = deque.is_empty();\n</code></pre> queue.c<pre><code>// C는 내장 큐를 제공하지 않는다\n</code></pre> queue.kt<pre><code>/* 큐 초기화 */\nval queue = LinkedList&lt;Int&gt;()\n\n/* 큐에 원소 삽입 */\nqueue.offer(1)\nqueue.offer(3)\nqueue.offer(2)\nqueue.offer(5)\nqueue.offer(4)\n\n/* 큐의 맨 앞 원소에 접근 */\nval peek = queue.peek()\n\n/* 큐에서 원소 꺼내기 */\nval pop = queue.poll()\n\n/* 큐 길이 구하기 */\nval size = queue.size\n\n/* 큐가 비어 있는지 확인 */\nval isEmpty = queue.isEmpty()\n</code></pre> queue.rb<pre><code># 큐 초기화\n# Ruby의 내장 큐(Thread::Queue)에는 peek와 순회 메서드가 없으므로 Array를 큐로 사용할 수 있다\nqueue = []\n\n# 큐에 원소 삽입\nqueue.push(1)\nqueue.push(3)\nqueue.push(2)\nqueue.push(5)\nqueue.push(4)\n\n# 큐 원소에 접근\npeek = queue.first\n\n# 큐에서 원소 꺼내기\n# 배열이므로 Array#shift 메서드의 시간 복잡도는 O(n)임에 주의\npop = queue.shift\n\n# 큐 길이 구하기\nsize = queue.length\n\n# 큐가 비어 있는지 확인\nis_empty = queue.empty?\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 5 장   스택과 큐","5.2   큐"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#522","level":2,"title":"5.2.2   큐 구현","text":"<p>큐를 구현하려면 한쪽 끝에서 원소를 추가하고 반대쪽 끝에서 삭제할 수 있는 자료 구조가 필요하다. 연결 리스트와 배열 모두 이 조건을 만족한다.</p>","path":["제 5 장   스택과 큐","5.2   큐"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#1","level":3,"title":"1.   연결 리스트 기반 구현","text":"<p>아래 그림(5-5)과 같이 연결 리스트의 ‘머리 노드’와 ‘꼬리 노드’를 각각 ‘큐의 맨 앞’과 ‘큐의 맨 뒤’로 볼 수 있다. 맨 뒤에서는 노드 추가만, 맨 앞에서는 노드 삭제만 허용한다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> 그림 5-5   연결 리스트 기반 큐의 삽입과 꺼내기 </p> <p>연결 리스트로 큐를 구현한 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linkedlist_queue.py<pre><code>class LinkedListQueue:\n    \"\"\"연결 리스트로 구현한 큐\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        self._front: ListNode | None = None  # 머리 노드 front\n        self._rear: ListNode | None = None  # 꼬리 노드 rear\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"큐의 길이 반환\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"큐가 비어 있는지 판단\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"큐에 삽입\"\"\"\n        # 꼬리 노드 뒤에 num을 추가한다\n        node = ListNode(num)\n        # 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if self._front is None:\n            self._front = node\n            self._rear = node\n        # 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        else:\n            self._rear.next = node\n            self._rear = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"큐에서 꺼내기\"\"\"\n        num = self.peek()\n        # 머리 노드를 삭제한다\n        self._front = self._front.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"맨 앞 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"큐가 비어 있다\")\n        return self._front.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"출력에 사용할 리스트로 변환\"\"\"\n        queue = []\n        temp = self._front\n        while temp:\n            queue.append(temp.val)\n            temp = temp.next\n        return queue\n</code></pre> linkedlist_queue.cpp<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue {\n  private:\n    ListNode *front, *rear; // 머리 노드 front, 꼬리 노드 rear\n    int queSize;\n\n  public:\n    LinkedListQueue() {\n        front = nullptr;\n        rear = nullptr;\n        queSize = 0;\n    }\n\n    ~LinkedListQueue() {\n        // 연결 리스트를 순회하며 노드를 삭제하여 메모리를 해제한다\n        freeMemoryLinkedList(front);\n    }\n\n    /* 큐의 길이 반환 */\n    int size() {\n        return queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* 큐에 삽입 */\n    void push(int num) {\n        // 꼬리 노드 뒤에 num을 추가한다\n        ListNode *node = new ListNode(num);\n        // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if (front == nullptr) {\n            front = node;\n            rear = node;\n        }\n        // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        else {\n            rear-&gt;next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    int pop() {\n        int num = peek();\n        // 머리 노드를 삭제한다\n        ListNode *tmp = front;\n        front = front-&gt;next;\n        // 메모리 해제\n        delete tmp;\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    int peek() {\n        if (size() == 0)\n            throw out_of_range(\"큐가 비어 있다\");\n        return front-&gt;val;\n    }\n\n    /* 연결 리스트를 Vector로 변환하여 반환 */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_queue.java<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue {\n    private ListNode front, rear; // 머리 노드 front, 꼬리 노드 rear\n    private int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* 큐의 길이 반환 */\n    public int size() {\n        return queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* 큐에 삽입 */\n    public void push(int num) {\n        // 꼬리 노드 뒤에 num을 추가한다\n        ListNode node = new ListNode(num);\n        // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if (front == null) {\n            front = node;\n            rear = node;\n        // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        } else {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    public int pop() {\n        int num = peek();\n        // 머리 노드를 삭제한다\n        front = front.next;\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.cs<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue {\n    ListNode? front, rear;  // 머리 노드 front, 꼬리 노드 rear \n    int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* 큐의 길이 반환 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* 큐에 삽입 */\n    public void Push(int num) {\n        // 꼬리 노드 뒤에 num을 추가한다\n        ListNode node = new(num);\n        // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if (front == null) {\n            front = node;\n            rear = node;\n            // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        } else if (rear != null) {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    public int Pop() {\n        int num = Peek();\n        // 머리 노드를 삭제한다\n        front = front?.next;\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return front!.val;\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    public int[] ToArray() {\n        if (front == null)\n            return [];\n\n        ListNode? node = front;\n        int[] res = new int[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.go<pre><code>/* 연결 리스트로 구현한 큐 */\ntype linkedListQueue struct {\n    // 내장 패키지 list로 큐를 구현한다\n    data *list.List\n}\n\n/* 큐 초기화 */\nfunc newLinkedListQueue() *linkedListQueue {\n    return &amp;linkedListQueue{\n        data: list.New(),\n    }\n}\n\n/* 큐에 삽입 */\nfunc (s *linkedListQueue) push(value any) {\n    s.data.PushBack(value)\n}\n\n/* 큐에서 꺼내기 */\nfunc (s *linkedListQueue) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* 맨 앞 원소 접근 */\nfunc (s *linkedListQueue) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* 큐의 길이 반환 */\nfunc (s *linkedListQueue) size() int {\n    return s.data.Len()\n}\n\n/* 큐가 비어 있는지 판단 */\nfunc (s *linkedListQueue) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* 출력에 사용할 List 반환 */\nfunc (s *linkedListQueue) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_queue.swift<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue {\n    private var front: ListNode? // 머리 노드\n    private var rear: ListNode? // 꼬리 노드\n    private var _size: Int\n\n    init() {\n        _size = 0\n    }\n\n    /* 큐의 길이 반환 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* 큐에 삽입 */\n    func push(num: Int) {\n        // 꼬리 노드 뒤에 num을 추가한다\n        let node = ListNode(x: num)\n        // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if front == nil {\n            front = node\n            rear = node\n        }\n        // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        else {\n            rear?.next = node\n            rear = node\n        }\n        _size += 1\n    }\n\n    /* 큐에서 꺼내기 */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // 머리 노드를 삭제한다\n        front = front?.next\n        _size -= 1\n        return num\n    }\n\n    /* 맨 앞 원소 접근 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"큐가 비어 있다\")\n        }\n        return front!.val\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.js<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue {\n    #front; // 머리 노드 #front\n    #rear; // 꼬리 노드 #rear\n    #queSize = 0;\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n    }\n\n    /* 큐의 길이 반환 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* 큐에 삽입 */\n    push(num) {\n        // 꼬리 노드 뒤에 num을 추가한다\n        const node = new ListNode(num);\n        // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if (!this.#front) {\n            this.#front = node;\n            this.#rear = node;\n            // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        } else {\n            this.#rear.next = node;\n            this.#rear = node;\n        }\n        this.#queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    pop() {\n        const num = this.peek();\n        // 머리 노드를 삭제한다\n        this.#front = this.#front.next;\n        this.#queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peek() {\n        if (this.size === 0) throw new Error('큐가 비어 있다');\n        return this.#front.val;\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    toArray() {\n        let node = this.#front;\n        const res = new Array(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.ts<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue {\n    private front: ListNode | null; // 머리 노드 front\n    private rear: ListNode | null; // 꼬리 노드 rear\n    private queSize: number = 0;\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n    }\n\n    /* 큐의 길이 반환 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* 큐에 삽입 */\n    push(num: number): void {\n        // 꼬리 노드 뒤에 num을 추가한다\n        const node = new ListNode(num);\n        // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if (!this.front) {\n            this.front = node;\n            this.rear = node;\n            // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        } else {\n            this.rear!.next = node;\n            this.rear = node;\n        }\n        this.queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    pop(): number {\n        const num = this.peek();\n        if (!this.front) throw new Error('큐가 비어 있다');\n        // 머리 노드를 삭제한다\n        this.front = this.front.next;\n        this.queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peek(): number {\n        if (this.size === 0) throw new Error('큐가 비어 있다');\n        return this.front!.val;\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    toArray(): number[] {\n        let node = this.front;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.dart<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue {\n  ListNode? _front; // 머리 노드 _front\n  ListNode? _rear; // 꼬리 노드 _rear\n  int _queSize = 0; // 큐 길이\n\n  LinkedListQueue() {\n    _front = null;\n    _rear = null;\n  }\n\n  /* 큐의 길이 반환 */\n  int size() {\n    return _queSize;\n  }\n\n  /* 큐가 비어 있는지 판단 */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* 큐에 삽입 */\n  void push(int _num) {\n    // 꼬리 노드 뒤에 _num을 추가한다\n    final node = ListNode(_num);\n    // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n    if (_front == null) {\n      _front = node;\n      _rear = node;\n    } else {\n      // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n      _rear!.next = node;\n      _rear = node;\n    }\n    _queSize++;\n  }\n\n  /* 큐에서 꺼내기 */\n  int pop() {\n    final int _num = peek();\n    // 머리 노드를 삭제한다\n    _front = _front!.next;\n    _queSize--;\n    return _num;\n  }\n\n  /* 맨 앞 원소 접근 */\n  int peek() {\n    if (_queSize == 0) {\n      throw Exception('큐가 비어 있다');\n    }\n    return _front!.val;\n  }\n\n  /* 연결 리스트를 Array로 변환하여 반환 */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; queue = [];\n    while (node != null) {\n      queue.add(node.val);\n      node = node.next;\n    }\n    return queue;\n  }\n}\n</code></pre> linkedlist_queue.rs<pre><code>/* 연결 리스트로 구현한 큐 */\n#[allow(dead_code)]\npub struct LinkedListQueue&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // 머리 노드 front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // 꼬리 노드 rear\n    que_size: usize,                         // 큐의 길이\n}\n\nimpl&lt;T: Copy&gt; LinkedListQueue&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* 큐의 길이 반환 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.que_size == 0;\n    }\n\n    /* 큐에 삽입 */\n    pub fn push(&amp;mut self, num: T) {\n        // 꼬리 노드 뒤에 num을 추가한다\n        let new_rear = ListNode::new(num);\n        match self.rear.take() {\n            // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n            Some(old_rear) =&gt; {\n                old_rear.borrow_mut().next = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n            // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n            None =&gt; {\n                self.front = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n        }\n        self.que_size += 1;\n    }\n\n    /* 큐에서 꺼내기 */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.front.take().map(|old_front| {\n            match old_front.borrow_mut().next.take() {\n                Some(new_front) =&gt; {\n                    self.front = Some(new_front);\n                }\n                None =&gt; {\n                    self.rear.take();\n                }\n            }\n            self.que_size -= 1;\n            old_front.borrow().val\n        })\n    }\n\n    /* 맨 앞 원소 접근 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        let mut res: Vec&lt;T&gt; = Vec::new();\n\n        fn recur&lt;T: Copy&gt;(cur: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, res: &amp;mut Vec&lt;T&gt;) {\n            if let Some(cur) = cur {\n                res.push(cur.borrow().val);\n                recur(cur.borrow().next.as_ref(), res);\n            }\n        }\n\n        recur(head, &amp;mut res);\n\n        res\n    }\n}\n</code></pre> linkedlist_queue.c<pre><code>/* 연결 리스트로 구현한 큐 */\ntypedef struct {\n    ListNode *front, *rear;\n    int queSize;\n} LinkedListQueue;\n\n/* 생성자 */\nLinkedListQueue *newLinkedListQueue() {\n    LinkedListQueue *queue = (LinkedListQueue *)malloc(sizeof(LinkedListQueue));\n    queue-&gt;front = NULL;\n    queue-&gt;rear = NULL;\n    queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* 소멸자 */\nvoid delLinkedListQueue(LinkedListQueue *queue) {\n    // 모든 노드를 해제한다\n    while (queue-&gt;front != NULL) {\n        ListNode *tmp = queue-&gt;front;\n        queue-&gt;front = queue-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // queue 구조체를 해제한다\n    free(queue);\n}\n\n/* 큐의 길이 반환 */\nint size(LinkedListQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* 큐가 비어 있는지 판단 */\nbool empty(LinkedListQueue *queue) {\n    return (size(queue) == 0);\n}\n\n/* 큐에 삽입 */\nvoid push(LinkedListQueue *queue, int num) {\n    // 꼬리 노드 위치에 node를 추가한다\n    ListNode *node = newListNode(num);\n    // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n    if (queue-&gt;front == NULL) {\n        queue-&gt;front = node;\n        queue-&gt;rear = node;\n    }\n    // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n    else {\n        queue-&gt;rear-&gt;next = node;\n        queue-&gt;rear = node;\n    }\n    queue-&gt;queSize++;\n}\n\n/* 맨 앞 원소 접근 */\nint peek(LinkedListQueue *queue) {\n    assert(size(queue) &amp;&amp; queue-&gt;front);\n    return queue-&gt;front-&gt;val;\n}\n\n/* 큐에서 꺼내기 */\nint pop(LinkedListQueue *queue) {\n    int num = peek(queue);\n    ListNode *tmp = queue-&gt;front;\n    queue-&gt;front = queue-&gt;front-&gt;next;\n    free(tmp);\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* 큐 출력 */\nvoid printLinkedListQueue(LinkedListQueue *queue) {\n    int *arr = malloc(sizeof(int) * queue-&gt;queSize);\n    // 연결 리스트의 데이터를 배열에 복사한다\n    int i;\n    ListNode *node;\n    for (i = 0, node = queue-&gt;front; i &lt; queue-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, queue-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_queue.kt<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue(\n    // 머리 노드 front, 꼬리 노드 rear\n    private var front: ListNode? = null,\n    private var rear: ListNode? = null,\n    private var queSize: Int = 0\n) {\n\n    /* 큐의 길이 반환 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* 큐에 삽입 */\n    fun push(num: Int) {\n        // 꼬리 노드 뒤에 num을 추가한다\n        val node = ListNode(num)\n        // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if (front == null) {\n            front = node\n            rear = node\n            // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        } else {\n            rear?.next = node\n            rear = node\n        }\n        queSize++\n    }\n\n    /* 큐에서 꺼내기 */\n    fun pop(): Int {\n        val num = peek()\n        // 머리 노드를 삭제한다\n        front = front?.next\n        queSize--\n        return num\n    }\n\n    /* 맨 앞 원소 접근 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!._val\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!._val\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.rb<pre><code>### 연결 리스트로 구현한 큐 ###\nclass LinkedListQueue\n  ### 큐의 길이 반환 ###\n  attr_reader :size\n\n  ### 생성자 ###\n  def initialize\n    @front = nil  # 머리 노드 front\n    @rear = nil   # 꼬리 노드 rear\n    @size = 0\n  end\n\n  ### 큐가 비어 있는지 판단 ###\n  def is_empty?\n    @front.nil?\n  end\n\n  ### 큐에 삽입 ###\n  def push(num)\n    # 꼬리 노드 뒤에 num을 추가한다\n    node = ListNode.new(num)\n\n    # 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n    if @front.nil?\n      @front = node\n      @rear = node\n    # 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n    else\n      @rear.next = node\n      @rear = node\n    end\n\n    @size += 1\n  end\n\n  ### 큐에서 꺼내기 ###\n  def pop\n    num = peek\n    # 머리 노드를 삭제한다\n    @front = @front.next\n    @size -= 1\n    num\n  end\n\n  ### 맨 앞 원소 접근 ###\n  def peek\n    raise IndexError, '큐가 비어 있다' if is_empty?\n\n    @front.val\n  end\n\n  ### 연결 리스트를 Array로 변환하여 반환 ###\n  def to_array\n    queue = []\n    temp = @front\n    while temp\n      queue &lt;&lt; temp.val\n      temp = temp.next\n    end\n    queue\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 5 장   스택과 큐","5.2   큐"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#2","level":3,"title":"2.   배열 기반 구현","text":"<p>배열에서 첫 원소를 삭제하는 시간 복잡도는 \\(O(n)\\)이므로, 큐에서 원소를 꺼내는 효율이 낮아진다. 그러나 다음과 같은 방법으로 이 문제를 피할 수 있다.</p> <p>변수 <code>front</code>로 큐의 맨 앞 원소 인덱스를 가리키고, 변수 <code>size</code>로 큐의 길이를 기록한다. <code>rear = front + size</code>로 정의하면 <code>rear</code>는 큐의 맨 뒤 원소 바로 다음 위치를 가리킨다.</p> <p>이 설계에서는 배열에서 원소가 들어 있는 유효 구간이 <code>[front, rear - 1]</code>이며, 각 연산의 구현 방법은 아래 그림(5-6)과 같다.</p> <ul> <li>삽입: 인덱스 <code>rear</code>에 입력 원소를 대입하고 <code>size</code>를 1 늘린다.</li> <li>꺼내기: <code>front</code>를 1 늘리고 <code>size</code>를 1 줄이면 된다.</li> </ul> <p>두 연산 모두 한 번의 처리만 필요하므로 시간 복잡도는 \\(O(1)\\)이다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> 그림 5-6   배열 기반 큐의 삽입과 꺼내기 </p> <p>여기에는 한 가지 문제가 있다. 삽입과 꺼내기를 반복하면 <code>front</code>와 <code>rear</code>가 계속 오른쪽으로 이동하다가 배열의 끝에 도달하면 더 이상 움직일 수 없다. 이를 해결하려면 배열을 앞뒤가 이어진 ‘원형 배열’로 볼 수 있다.</p> <p>원형 배열에서는 <code>front</code>나 <code>rear</code>가 배열의 끝을 넘으면 곧바로 처음으로 돌아와 순회를 계속하게 해야 한다. 이 주기적 규칙은 ‘나머지 연산’으로 구현할 수 있다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_queue.py<pre><code>class ArrayQueue:\n    \"\"\"원형 배열로 구현한 큐\"\"\"\n\n    def __init__(self, size: int):\n        \"\"\"생성자\"\"\"\n        self._nums: list[int] = [0] * size  # 큐의 원소를 저장하는 배열\n        self._front: int = 0  # 맨 앞 원소를 가리키는 맨 앞 포인터\n        self._size: int = 0  # 큐 길이\n\n    def capacity(self) -&gt; int:\n        \"\"\"큐의 용량 반환\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"큐의 길이 반환\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"큐가 비어 있는지 판단\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"큐에 삽입\"\"\"\n        if self._size == self.capacity():\n            raise IndexError(\"큐가 가득 찼다\")\n        # 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        # 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        rear: int = (self._front + self._size) % self.capacity()\n        # num을 맨 뒤에 추가한다\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"큐에서 꺼내기\"\"\"\n        num: int = self.peek()\n        # 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        self._front = (self._front + 1) % self.capacity()\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"맨 앞 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"큐가 비어 있다\")\n        return self._nums[self._front]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"출력에 사용할 리스트 반환\"\"\"\n        res = [0] * self.size()\n        j: int = self._front\n        for i in range(self.size()):\n            res[i] = self._nums[(j % self.capacity())]\n            j += 1\n        return res\n</code></pre> array_queue.cpp<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue {\n  private:\n    int *nums;       // 큐의 원소를 저장하는 배열\n    int front;       // 맨 앞 원소를 가리키는 맨 앞 포인터\n    int queSize;     // 큐 길이\n    int queCapacity; // 큐 용량\n\n  public:\n    ArrayQueue(int capacity) {\n        // 배열 초기화\n        nums = new int[capacity];\n        queCapacity = capacity;\n        front = queSize = 0;\n    }\n\n    ~ArrayQueue() {\n        delete[] nums;\n    }\n\n    /* 큐의 용량 반환 */\n    int capacity() {\n        return queCapacity;\n    }\n\n    /* 큐의 길이 반환 */\n    int size() {\n        return queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* 큐에 삽입 */\n    void push(int num) {\n        if (queSize == queCapacity) {\n            cout &lt;&lt; \"큐가 가득 찼다\" &lt;&lt; endl;\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        int rear = (front + queSize) % queCapacity;\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    int pop() {\n        int num = peek();\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        front = (front + 1) % queCapacity;\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    int peek() {\n        if (isEmpty())\n            throw out_of_range(\"큐가 비어 있다\");\n        return nums[front];\n    }\n\n    /* 배열을 Vector로 변환하여 반환 */\n    vector&lt;int&gt; toVector() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        vector&lt;int&gt; arr(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            arr[i] = nums[j % queCapacity];\n        }\n        return arr;\n    }\n};\n</code></pre> array_queue.java<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue {\n    private int[] nums; // 큐의 원소를 저장하는 배열\n    private int front; // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private int queSize; // 큐 길이\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* 큐의 용량 반환 */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* 큐의 길이 반환 */\n    public int size() {\n        return queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* 큐에 삽입 */\n    public void push(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"큐가 가득 찼다\");\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        int rear = (front + queSize) % capacity();\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    public int pop() {\n        int num = peek();\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        front = (front + 1) % capacity();\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* 배열 반환 */\n    public int[] toArray() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.cs<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue {\n    int[] nums;  // 큐의 원소를 저장하는 배열\n    int front;   // 맨 앞 원소를 가리키는 맨 앞 포인터\n    int queSize; // 큐 길이\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* 큐의 용량 반환 */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* 큐의 길이 반환 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* 큐에 삽입 */\n    public void Push(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"큐가 가득 찼다\");\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        int rear = (front + queSize) % Capacity();\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    public int Pop() {\n        int num = Peek();\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        front = (front + 1) % Capacity();\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return nums[front];\n    }\n\n    /* 배열 반환 */\n    public int[] ToArray() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % this.Capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.go<pre><code>/* 원형 배열로 구현한 큐 */\ntype arrayQueue struct {\n    nums        []int // 큐의 원소를 저장하는 배열\n    front       int   // 맨 앞 원소를 가리키는 맨 앞 포인터\n    queSize     int   // 큐 길이\n    queCapacity int   // 큐 용량(저장할 수 있는 최대 원소 개수)\n}\n\n/* 큐 초기화 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\n    return &amp;arrayQueue{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* 큐의 길이 반환 */\nfunc (q *arrayQueue) size() int {\n    return q.queSize\n}\n\n/* 큐가 비어 있는지 판단 */\nfunc (q *arrayQueue) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* 큐에 삽입 */\nfunc (q *arrayQueue) push(num int) {\n    // rear == queCapacity이면 큐가 가득 찼음을 나타낸다\n    if q.queSize == q.queCapacity {\n        return\n    }\n    // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n    rear := (q.front + q.queSize) % q.queCapacity\n    // num을 맨 뒤에 추가한다\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* 큐에서 꺼내기 */\nfunc (q *arrayQueue) pop() any {\n    num := q.peek()\n    if num == nil {\n        return nil\n    }\n\n    // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n    q.front = (q.front + 1) % q.queCapacity\n    q.queSize--\n    return num\n}\n\n/* 맨 앞 원소 접근 */\nfunc (q *arrayQueue) peek() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* 출력에 사용할 Slice 반환 */\nfunc (q *arrayQueue) toSlice() []int {\n    rear := (q.front + q.queSize)\n    if rear &gt;= q.queCapacity {\n        rear %= q.queCapacity\n        return append(q.nums[q.front:], q.nums[:rear]...)\n    }\n    return q.nums[q.front:rear]\n}\n</code></pre> array_queue.swift<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue {\n    private var nums: [Int] // 큐의 원소를 저장하는 배열\n    private var front: Int // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private var _size: Int // 큐 길이\n\n    init(capacity: Int) {\n        // 배열 초기화\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* 큐의 용량 반환 */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* 큐의 길이 반환 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* 큐에 삽입 */\n    func push(num: Int) {\n        if size() == capacity() {\n            print(\"큐가 가득 찼다\")\n            return\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        let rear = (front + size()) % capacity()\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* 큐에서 꺼내기 */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        front = (front + 1) % capacity()\n        _size -= 1\n        return num\n    }\n\n    /* 맨 앞 원소 접근 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"큐가 비어 있다\")\n        }\n        return nums[front]\n    }\n\n    /* 배열 반환 */\n    func toArray() -&gt; [Int] {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        (front ..&lt; front + size()).map { nums[$0 % capacity()] }\n    }\n}\n</code></pre> array_queue.js<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue {\n    #nums; // 큐의 원소를 저장하는 배열\n    #front = 0; // 맨 앞 원소를 가리키는 맨 앞 포인터\n    #queSize = 0; // 큐 길이\n\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n    }\n\n    /* 큐의 용량 반환 */\n    get capacity() {\n        return this.#nums.length;\n    }\n\n    /* 큐의 길이 반환 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* 큐에 삽입 */\n    push(num) {\n        if (this.size === this.capacity) {\n            console.log('큐가 가득 찼다');\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        const rear = (this.#front + this.size) % this.capacity;\n        // num을 맨 뒤에 추가한다\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    pop() {\n        const num = this.peek();\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        this.#front = (this.#front + 1) % this.capacity;\n        this.#queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peek() {\n        if (this.isEmpty()) throw new Error('큐가 비어 있다');\n        return this.#nums[this.#front];\n    }\n\n    /* Array를 반환한다 */\n    toArray() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.#front; i &lt; this.size; i++, j++) {\n            arr[i] = this.#nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.ts<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue {\n    private nums: number[]; // 큐의 원소를 저장하는 배열\n    private front: number; // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private queSize: number; // 큐 길이\n\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = this.queSize = 0;\n    }\n\n    /* 큐의 용량 반환 */\n    get capacity(): number {\n        return this.nums.length;\n    }\n\n    /* 큐의 길이 반환 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* 큐에 삽입 */\n    push(num: number): void {\n        if (this.size === this.capacity) {\n            console.log('큐가 가득 찼다');\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        const rear = (this.front + this.queSize) % this.capacity;\n        // num을 맨 뒤에 추가한다\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    pop(): number {\n        const num = this.peek();\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        this.front = (this.front + 1) % this.capacity;\n        this.queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peek(): number {\n        if (this.isEmpty()) throw new Error('큐가 비어 있다');\n        return this.nums[this.front];\n    }\n\n    /* Array를 반환한다 */\n    toArray(): number[] {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.front; i &lt; this.size; i++, j++) {\n            arr[i] = this.nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.dart<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue {\n  late List&lt;int&gt; _nums; // 큐의 원소를 저장하는 배열\n  late int _front; // 맨 앞 원소를 가리키는 맨 앞 포인터\n  late int _queSize; // 큐 길이\n\n  ArrayQueue(int capacity) {\n    _nums = List.filled(capacity, 0);\n    _front = _queSize = 0;\n  }\n\n  /* 큐의 용량 반환 */\n  int capaCity() {\n    return _nums.length;\n  }\n\n  /* 큐의 길이 반환 */\n  int size() {\n    return _queSize;\n  }\n\n  /* 큐가 비어 있는지 판단 */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* 큐에 삽입 */\n  void push(int _num) {\n    if (_queSize == capaCity()) {\n      throw Exception(\"큐가 가득 찼다\");\n    }\n    // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n    int rear = (_front + _queSize) % capaCity();\n    // _num을 맨 뒤에 추가한다\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* 큐에서 꺼내기 */\n  int pop() {\n    int _num = peek();\n    // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n    _front = (_front + 1) % capaCity();\n    _queSize--;\n    return _num;\n  }\n\n  /* 맨 앞 원소 접근 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"큐가 비어 있다\");\n    }\n    return _nums[_front];\n  }\n\n  /* Array를 반환한다 */\n  List&lt;int&gt; toArray() {\n    // 유효한 길이 범위 안의 리스트 원소만 변환한다\n    final List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[j % capaCity()];\n    }\n    return res;\n  }\n}\n</code></pre> array_queue.rs<pre><code>/* 원형 배열로 구현한 큐 */\nstruct ArrayQueue&lt;T&gt; {\n    nums: Vec&lt;T&gt;,      // 큐의 원소를 저장하는 배열\n    front: i32,        // 맨 앞 원소를 가리키는 맨 앞 포인터\n    que_size: i32,     // 큐 길이\n    que_capacity: i32, // 큐 용량\n}\n\nimpl&lt;T: Copy + Default&gt; ArrayQueue&lt;T&gt; {\n    /* 생성자 */\n    fn new(capacity: i32) -&gt; ArrayQueue&lt;T&gt; {\n        ArrayQueue {\n            nums: vec![T::default(); capacity as usize],\n            front: 0,\n            que_size: 0,\n            que_capacity: capacity,\n        }\n    }\n\n    /* 큐의 용량 반환 */\n    fn capacity(&amp;self) -&gt; i32 {\n        self.que_capacity\n    }\n\n    /* 큐의 길이 반환 */\n    fn size(&amp;self) -&gt; i32 {\n        self.que_size\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* 큐에 삽입 */\n    fn push(&amp;mut self, num: T) {\n        if self.que_size == self.capacity() {\n            println!(\"큐가 가득 찼다\");\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        let rear = (self.front + self.que_size) % self.que_capacity;\n        // num을 맨 뒤에 추가한다\n        self.nums[rear as usize] = num;\n        self.que_size += 1;\n    }\n\n    /* 큐에서 꺼내기 */\n    fn pop(&amp;mut self) -&gt; T {\n        let num = self.peek();\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        self.front = (self.front + 1) % self.que_capacity;\n        self.que_size -= 1;\n        num\n    }\n\n    /* 맨 앞 원소 접근 */\n    fn peek(&amp;self) -&gt; T {\n        if self.is_empty() {\n            panic!(\"인덱스가 범위를 벗어났다\");\n        }\n        self.nums[self.front as usize]\n    }\n\n    /* 배열 반환 */\n    fn to_vector(&amp;self) -&gt; Vec&lt;T&gt; {\n        let cap = self.que_capacity;\n        let mut j = self.front;\n        let mut arr = vec![T::default(); cap as usize];\n        for i in 0..self.que_size {\n            arr[i as usize] = self.nums[(j % cap) as usize];\n            j += 1;\n        }\n        arr\n    }\n}\n</code></pre> array_queue.c<pre><code>/* 원형 배열로 구현한 큐 */\ntypedef struct {\n    int *nums;       // 큐의 원소를 저장하는 배열\n    int front;       // 맨 앞 원소를 가리키는 맨 앞 포인터\n    int queSize;     // 현재 큐의 원소 개수\n    int queCapacity; // 큐 용량\n} ArrayQueue;\n\n/* 생성자 */\nArrayQueue *newArrayQueue(int capacity) {\n    ArrayQueue *queue = (ArrayQueue *)malloc(sizeof(ArrayQueue));\n    // 배열 초기화\n    queue-&gt;queCapacity = capacity;\n    queue-&gt;nums = (int *)malloc(sizeof(int) * queue-&gt;queCapacity);\n    queue-&gt;front = queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* 소멸자 */\nvoid delArrayQueue(ArrayQueue *queue) {\n    free(queue-&gt;nums);\n    free(queue);\n}\n\n/* 큐의 용량 반환 */\nint capacity(ArrayQueue *queue) {\n    return queue-&gt;queCapacity;\n}\n\n/* 큐의 길이 반환 */\nint size(ArrayQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* 큐가 비어 있는지 판단 */\nbool empty(ArrayQueue *queue) {\n    return queue-&gt;queSize == 0;\n}\n\n/* 맨 앞 원소 접근 */\nint peek(ArrayQueue *queue) {\n    assert(size(queue) != 0);\n    return queue-&gt;nums[queue-&gt;front];\n}\n\n/* 큐에 삽입 */\nvoid push(ArrayQueue *queue, int num) {\n    if (size(queue) == capacity(queue)) {\n        printf(\"큐가 가득 찼다\\r\\n\");\n        return;\n    }\n    // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n    int rear = (queue-&gt;front + queue-&gt;queSize) % queue-&gt;queCapacity;\n    // num을 맨 뒤에 추가한다\n    queue-&gt;nums[rear] = num;\n    queue-&gt;queSize++;\n}\n\n/* 큐에서 꺼내기 */\nint pop(ArrayQueue *queue) {\n    int num = peek(queue);\n    // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n    queue-&gt;front = (queue-&gt;front + 1) % queue-&gt;queCapacity;\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* 출력에 사용할 배열 반환 */\nint *toArray(ArrayQueue *queue, int *queSize) {\n    *queSize = queue-&gt;queSize;\n    int *res = (int *)calloc(queue-&gt;queSize, sizeof(int));\n    int j = queue-&gt;front;\n    for (int i = 0; i &lt; queue-&gt;queSize; i++) {\n        res[i] = queue-&gt;nums[j % queue-&gt;queCapacity];\n        j++;\n    }\n    return res;\n}\n</code></pre> array_queue.kt<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue(capacity: Int) {\n    private val nums: IntArray = IntArray(capacity) // 큐의 원소를 저장하는 배열\n    private var front: Int = 0 // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private var queSize: Int = 0 // 큐 길이\n\n    /* 큐의 용량 반환 */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* 큐의 길이 반환 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* 큐에 삽입 */\n    fun push(num: Int) {\n        if (queSize == capacity()) {\n            println(\"큐가 가득 찼다\")\n            return\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        val rear = (front + queSize) % capacity()\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num\n        queSize++\n    }\n\n    /* 큐에서 꺼내기 */\n    fun pop(): Int {\n        val num = peek()\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        front = (front + 1) % capacity()\n        queSize--\n        return num\n    }\n\n    /* 맨 앞 원소 접근 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* 배열 반환 */\n    fun toArray(): IntArray {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[j % capacity()]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_queue.rb<pre><code>### 원형 배열로 구현한 큐 ###\nclass ArrayQueue\n  ### 큐의 길이 반환 ###\n  attr_reader :size\n\n  ### 생성자 ###\n  def initialize(size)\n    @nums = Array.new(size, 0) # 큐의 원소를 저장하는 배열\n    @front = 0 # 맨 앞 원소를 가리키는 맨 앞 포인터\n    @size = 0 # 큐 길이\n  end\n\n  ### 큐의 용량 반환 ###\n  def capacity\n    @nums.length\n  end\n\n  ### 큐가 비어 있는지 판단 ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### 큐에 삽입 ###\n  def push(num)\n    raise IndexError, '큐가 가득 찼다' if size == capacity\n\n    # 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    # 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n    rear = (@front + size) % capacity\n    # num을 맨 뒤에 추가한다\n    @nums[rear] = num\n    @size += 1\n  end\n\n  ### 큐에서 꺼내기 ###\n  def pop\n    num = peek\n    # 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n    @front = (@front + 1) % capacity\n    @size -= 1\n    num\n  end\n\n  ### 맨 앞 원소 접근 ###\n  def peek\n    raise IndexError, '큐가 비어 있다' if is_empty?\n\n    @nums[@front]\n  end\n\n  ### 출력에 사용할 리스트 반환 ###\n  def to_array\n    res = Array.new(size, 0)\n    j = @front\n\n    for i in 0...size\n      res[i] = @nums[j % capacity]\n      j += 1\n    end\n\n    res\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>위 큐 구현에는 여전히 길이를 바꿀 수 없다는 한계가 있다. 하지만 배열을 동적 배열로 바꾸어 용량 확장 방식을 도입하면 쉽게 해결할 수 있다. 관심 있는 독자는 직접 구현해 보자.</p> <p>두 구현의 비교 결과는 스택과 같으므로 여기서는 반복해서 설명하지 않는다.</p>","path":["제 5 장   스택과 큐","5.2   큐"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#523","level":2,"title":"5.2.3   큐의 대표적인 활용","text":"<ul> <li>타오바오 주문. 구매자가 주문하면 큐에 들어가고, 시스템은 순서대로 처리한다. 광군제 기간에는 짧은 시간에 엄청난 주문이 몰리므로 높은 동시성을 처리하는 일이 엔지니어들의 핵심 과제가 된다.</li> <li>여러 종류의 대기 작업. 프린터 작업 대기열, 식당 음식 제공 대기열 등 ‘먼저 온 순서’를 지켜야 하는 상황에서 큐는 처리 순서를 효과적으로 유지한다.</li> </ul>","path":["제 5 장   스택과 큐","5.2   큐"],"tags":[]},{"location":"chapter_stack_and_queue/stack/","level":1,"title":"5.1   스택","text":"<p>스택(stack)은 후입선출 원리를 따르는 선형 자료 구조다.</p> <p>스택은 탁자 위에 쌓인 접시에 비유할 수 있다. 한 번에 접시 하나만 옮길 수 있다면 맨 아래 접시를 꺼내려면 위의 접시들을 차례로 치워야 한다. 접시를 정수, 문자, 객체 등 여러 자료형의 원소로 바꾸면 스택이라는 자료 구조가 된다.</p> <p>아래 그림(5-1)과 같이 원소가 쌓인 맨 위를 ‘스택의 꼭대기’, 맨 아래를 ‘스택의 바닥’이라고 한다. 맨 위에 원소를 추가하는 연산을 ‘스택에 삽입하기’, 맨 위 원소를 삭제하는 연산을 ‘스택에서 꺼내기’라고 한다.</p> <p></p> <p> 그림 5-1   스택의 후입선출 규칙 </p>","path":["제 5 장   스택과 큐","5.1   스택"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#511","level":2,"title":"5.1.1   스택의 기본 연산","text":"<p>스택의 기본 연산은 아래 표(5-1)와 같다. 구체적인 메서드 이름은 프로그래밍 언어에 따라 다르며, 여기서는 흔히 쓰이는 <code>push()</code>, <code>pop()</code>, <code>peek()</code>를 예로 든다.</p> <p> 표 5-1   스택의 연산 효율 </p> 메서드 설명 시간 복잡도 <code>push()</code> 원소를 스택에 삽입(맨 위에 추가) \\(O(1)\\) <code>pop()</code> 스택 맨 위 원소를 꺼냄 \\(O(1)\\) <code>peek()</code> 스택 맨 위 원소에 접근 \\(O(1)\\) <p>보통 프로그래밍 언어에 내장된 스택 클래스를 바로 사용할 수 있다. 전용 스택 클래스가 없는 언어에서는 ‘배열’이나 ‘연결 리스트’를 스택으로 사용하면서, 프로그램 논리상 스택과 무관한 연산은 사용하지 않으면 된다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby stack.py<pre><code># 스택 초기화\n# Python에는 내장 스택 클래스가 없으므로 list를 스택으로 사용할 수 있다\nstack: list[int] = []\n\n# 스택에 원소 삽입\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n# 스택 맨 위 원소에 접근\npeek: int = stack[-1]\n\n# 스택에서 원소 꺼내기\npop: int = stack.pop()\n\n# 스택 길이 구하기\nsize: int = len(stack)\n\n# 비어 있는지 확인\nis_empty: bool = len(stack) == 0\n</code></pre> stack.cpp<pre><code>/* 스택 초기화 */\nstack&lt;int&gt; stack;\n\n/* 스택에 원소 삽입 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* 스택 맨 위 원소에 접근 */\nint top = stack.top();\n\n/* 스택에서 원소 꺼내기 */\nstack.pop(); // 반환값 없음\n\n/* 스택 길이 구하기 */\nint size = stack.size();\n\n/* 비어 있는지 확인 */\nbool empty = stack.empty();\n</code></pre> stack.java<pre><code>/* 스택 초기화 */\nStack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n\n/* 스택에 원소 삽입 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* 스택 맨 위 원소에 접근 */\nint peek = stack.peek();\n\n/* 스택에서 원소 꺼내기 */\nint pop = stack.pop();\n\n/* 스택 길이 구하기 */\nint size = stack.size();\n\n/* 비어 있는지 확인 */\nboolean isEmpty = stack.isEmpty();\n</code></pre> stack.cs<pre><code>/* 스택 초기화 */\nStack&lt;int&gt; stack = new();\n\n/* 스택에 원소 삽입 */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n\n/* 스택 맨 위 원소에 접근 */\nint peek = stack.Peek();\n\n/* 스택에서 원소 꺼내기 */\nint pop = stack.Pop();\n\n/* 스택 길이 구하기 */\nint size = stack.Count;\n\n/* 비어 있는지 확인 */\nbool isEmpty = stack.Count == 0;\n</code></pre> stack_test.go<pre><code>/* 스택 초기화 */\n// Go에서는 Slice를 스택으로 사용하는 것을 권한다\nvar stack []int\n\n/* 스택에 원소 삽입 */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n\n/* 스택 맨 위 원소에 접근 */\npeek := stack[len(stack)-1]\n\n/* 스택에서 원소 꺼내기 */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n\n/* 스택 길이 구하기 */\nsize := len(stack)\n\n/* 비어 있는지 확인 */\nisEmpty := len(stack) == 0\n</code></pre> stack.swift<pre><code>/* 스택 초기화 */\n// Swift에는 내장 스택 클래스가 없으므로 Array를 스택으로 사용할 수 있다\nvar stack: [Int] = []\n\n/* 스택에 원소 삽입 */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n/* 스택 맨 위 원소에 접근 */\nlet peek = stack.last!\n\n/* 스택에서 원소 꺼내기 */\nlet pop = stack.removeLast()\n\n/* 스택 길이 구하기 */\nlet size = stack.count\n\n/* 비어 있는지 확인 */\nlet isEmpty = stack.isEmpty\n</code></pre> stack.js<pre><code>/* 스택 초기화 */\n// JavaScript에는 내장 스택 클래스가 없으므로 Array를 스택으로 사용할 수 있다\nconst stack = [];\n\n/* 스택에 원소 삽입 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* 스택 맨 위 원소에 접근 */\nconst peek = stack[stack.length-1];\n\n/* 스택에서 원소 꺼내기 */\nconst pop = stack.pop();\n\n/* 스택 길이 구하기 */\nconst size = stack.length;\n\n/* 비어 있는지 확인 */\nconst is_empty = stack.length === 0;\n</code></pre> stack.ts<pre><code>/* 스택 초기화 */\n// TypeScript에는 내장 스택 클래스가 없으므로 Array를 스택으로 사용할 수 있다\nconst stack: number[] = [];\n\n/* 스택에 원소 삽입 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* 스택 맨 위 원소에 접근 */\nconst peek = stack[stack.length - 1];\n\n/* 스택에서 원소 꺼내기 */\nconst pop = stack.pop();\n\n/* 스택 길이 구하기 */\nconst size = stack.length;\n\n/* 비어 있는지 확인 */\nconst is_empty = stack.length === 0;\n</code></pre> stack.dart<pre><code>/* 스택 초기화 */\n// Dart에는 내장 스택 클래스가 없으므로 List를 스택으로 사용할 수 있다\nList&lt;int&gt; stack = [];\n\n/* 스택에 원소 삽입 */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n\n/* 스택 맨 위 원소에 접근 */\nint peek = stack.last;\n\n/* 스택에서 원소 꺼내기 */\nint pop = stack.removeLast();\n\n/* 스택 길이 구하기 */\nint size = stack.length;\n\n/* 비어 있는지 확인 */\nbool isEmpty = stack.isEmpty;\n</code></pre> stack.rs<pre><code>/* 스택 초기화 */\n// Vec를 스택으로 사용\nlet mut stack: Vec&lt;i32&gt; = Vec::new();\n\n/* 스택에 원소 삽입 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* 스택 맨 위 원소에 접근 */\nlet top = stack.last().unwrap();\n\n/* 스택에서 원소 꺼내기 */\nlet pop = stack.pop().unwrap();\n\n/* 스택 길이 구하기 */\nlet size = stack.len();\n\n/* 비어 있는지 확인 */\nlet is_empty = stack.is_empty();\n</code></pre> stack.c<pre><code>// C는 내장 스택을 제공하지 않는다\n</code></pre> stack.kt<pre><code>/* 스택 초기화 */\nval stack = Stack&lt;Int&gt;()\n\n/* 스택에 원소 삽입 */\nstack.push(1)\nstack.push(3)\nstack.push(2)\nstack.push(5)\nstack.push(4)\n\n/* 스택 맨 위 원소에 접근 */\nval peek = stack.peek()\n\n/* 스택에서 원소 꺼내기 */\nval pop = stack.pop()\n\n/* 스택 길이 구하기 */\nval size = stack.size\n\n/* 비어 있는지 확인 */\nval isEmpty = stack.isEmpty()\n</code></pre> stack.rb<pre><code># 스택 초기화\n# Ruby에는 내장 스택 클래스가 없으므로 Array를 스택으로 사용할 수 있다\nstack = []\n\n# 스택에 원소 삽입\nstack &lt;&lt; 1\nstack &lt;&lt; 3\nstack &lt;&lt; 2\nstack &lt;&lt; 5\nstack &lt;&lt; 4\n\n# 스택 맨 위 원소에 접근\npeek = stack.last\n\n# 스택에서 원소 꺼내기\npop = stack.pop\n\n# 스택 길이 구하기\nsize = stack.length\n\n# 비어 있는지 확인\nis_empty = stack.empty?\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 5 장   스택과 큐","5.1   스택"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#512","level":2,"title":"5.1.2   스택 구현","text":"<p>스택의 작동 원리를 깊이 이해하기 위해 스택 클래스를 직접 구현해 보자.</p> <p>스택은 후입선출 원리를 따르므로 맨 위에서만 원소를 추가하거나 삭제할 수 있다. 반면 배열과 연결 리스트는 어느 위치에서든 추가와 삭제가 가능하므로, 스택은 제약을 가한 배열이나 연결 리스트로 볼 수 있다. 즉, 일부 무관한 연산을 ‘가려서’ 외부에 보이는 논리가 스택의 특성에 맞도록 만들 수 있다.</p>","path":["제 5 장   스택과 큐","5.1   스택"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#1","level":3,"title":"1.   연결 리스트 기반 구현","text":"<p>연결 리스트로 스택을 구현할 때는 머리 노드를 스택의 꼭대기로, 꼬리 노드를 바닥으로 볼 수 있다.</p> <p>아래 그림(5-2)과 같이 스택에 삽입할 때는 연결 리스트의 맨 앞에 원소를 넣으면 된다. 이 방식을 ‘머리 삽입’이라고 한다. 스택에서 꺼낼 때는 머리 노드를 삭제하면 된다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> 그림 5-2   연결 리스트 기반 스택의 삽입과 꺼내기 </p> <p>연결 리스트로 스택을 구현한 예제 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linkedlist_stack.py<pre><code>class LinkedListStack:\n    \"\"\"연결 리스트로 구현한 스택\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        self._peek: ListNode | None = None\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"스택의 길이 반환\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"스택이 비어 있는지 판단\"\"\"\n        return self._size == 0\n\n    def push(self, val: int):\n        \"\"\"스택에 삽입\"\"\"\n        node = ListNode(val)\n        node.next = self._peek\n        self._peek = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"스택에서 꺼내기\"\"\"\n        num = self.peek()\n        self._peek = self._peek.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"스택 맨 위 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"스택이 비어 있다\")\n        return self._peek.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"출력에 사용할 리스트로 변환\"\"\"\n        arr = []\n        node = self._peek\n        while node:\n            arr.append(node.val)\n            node = node.next\n        arr.reverse()\n        return arr\n</code></pre> linkedlist_stack.cpp<pre><code>/* 연결 리스트로 구현한 스택 */\nclass LinkedListStack {\n  private:\n    ListNode *stackTop; // 머리 노드를 스택의 맨 위로 사용한다\n    int stkSize;        // 스택의 길이\n\n  public:\n    LinkedListStack() {\n        stackTop = nullptr;\n        stkSize = 0;\n    }\n\n    ~LinkedListStack() {\n        // 연결 리스트를 순회하며 노드를 삭제하여 메모리를 해제한다\n        freeMemoryLinkedList(stackTop);\n    }\n\n    /* 스택의 길이 반환 */\n    int size() {\n        return stkSize;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* 스택에 삽입 */\n    void push(int num) {\n        ListNode *node = new ListNode(num);\n        node-&gt;next = stackTop;\n        stackTop = node;\n        stkSize++;\n    }\n\n    /* 스택에서 꺼내기 */\n    int pop() {\n        int num = top();\n        ListNode *tmp = stackTop;\n        stackTop = stackTop-&gt;next;\n        // 메모리 해제\n        delete tmp;\n        stkSize--;\n        return num;\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"스택이 비어 있다\");\n        return stackTop-&gt;val;\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = stackTop;\n        vector&lt;int&gt; res(size());\n        for (int i = res.size() - 1; i &gt;= 0; i--) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_stack.java<pre><code>/* 연결 리스트로 구현한 스택 */\nclass LinkedListStack {\n    private ListNode stackPeek; // 머리 노드를 스택의 맨 위로 사용한다\n    private int stkSize = 0; // 스택의 길이\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* 스택의 길이 반환 */\n    public int size() {\n        return stkSize;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* 스택에 삽입 */\n    public void push(int num) {\n        ListNode node = new ListNode(num);\n        node.next = stackPeek;\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* 스택에서 꺼내기 */\n    public int pop() {\n        int num = peek();\n        stackPeek = stackPeek.next;\n        stkSize--;\n        return num;\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stackPeek.val;\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    public int[] toArray() {\n        ListNode node = stackPeek;\n        int[] res = new int[size()];\n        for (int i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.cs<pre><code>/* 연결 리스트로 구현한 스택 */\nclass LinkedListStack {\n    ListNode? stackPeek;  // 머리 노드를 스택의 맨 위로 사용한다\n    int stkSize = 0;   // 스택의 길이\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* 스택의 길이 반환 */\n    public int Size() {\n        return stkSize;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* 스택에 삽입 */\n    public void Push(int num) {\n        ListNode node = new(num) {\n            next = stackPeek\n        };\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* 스택에서 꺼내기 */\n    public int Pop() {\n        int num = Peek();\n        stackPeek = stackPeek!.next;\n        stkSize--;\n        return num;\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stackPeek!.val;\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    public int[] ToArray() {\n        if (stackPeek == null)\n            return [];\n\n        ListNode? node = stackPeek;\n        int[] res = new int[Size()];\n        for (int i = res.Length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.go<pre><code>/* 연결 리스트로 구현한 스택 */\ntype linkedListStack struct {\n    // 내장 패키지 list로 스택을 구현한다\n    data *list.List\n}\n\n/* 스택 초기화 */\nfunc newLinkedListStack() *linkedListStack {\n    return &amp;linkedListStack{\n        data: list.New(),\n    }\n}\n\n/* 스택에 삽입 */\nfunc (s *linkedListStack) push(value int) {\n    s.data.PushBack(value)\n}\n\n/* 스택에서 꺼내기 */\nfunc (s *linkedListStack) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* 스택 맨 위 원소 접근 */\nfunc (s *linkedListStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* 스택의 길이 반환 */\nfunc (s *linkedListStack) size() int {\n    return s.data.Len()\n}\n\n/* 스택이 비어 있는지 판단 */\nfunc (s *linkedListStack) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* 출력에 사용할 List 반환 */\nfunc (s *linkedListStack) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_stack.swift<pre><code>/* 연결 리스트로 구현한 스택 */\nclass LinkedListStack {\n    private var _peek: ListNode? // 머리 노드를 스택의 맨 위로 사용한다\n    private var _size: Int // 스택의 길이\n\n    init() {\n        _size = 0\n    }\n\n    /* 스택의 길이 반환 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* 스택에 삽입 */\n    func push(num: Int) {\n        let node = ListNode(x: num)\n        node.next = _peek\n        _peek = node\n        _size += 1\n    }\n\n    /* 스택에서 꺼내기 */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        _peek = _peek?.next\n        _size -= 1\n        return num\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"스택이 비어 있다\")\n        }\n        return _peek!.val\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    func toArray() -&gt; [Int] {\n        var node = _peek\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices.reversed() {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.js<pre><code>/* 연결 리스트로 구현한 스택 */\nclass LinkedListStack {\n    #stackPeek; // 머리 노드를 스택의 맨 위로 사용한다\n    #stkSize = 0; // 스택의 길이\n\n    constructor() {\n        this.#stackPeek = null;\n    }\n\n    /* 스택의 길이 반환 */\n    get size() {\n        return this.#stkSize;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* 스택에 삽입 */\n    push(num) {\n        const node = new ListNode(num);\n        node.next = this.#stackPeek;\n        this.#stackPeek = node;\n        this.#stkSize++;\n    }\n\n    /* 스택에서 꺼내기 */\n    pop() {\n        const num = this.peek();\n        this.#stackPeek = this.#stackPeek.next;\n        this.#stkSize--;\n        return num;\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    peek() {\n        if (!this.#stackPeek) throw new Error('스택이 비어 있다');\n        return this.#stackPeek.val;\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    toArray() {\n        let node = this.#stackPeek;\n        const res = new Array(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.ts<pre><code>/* 연결 리스트로 구현한 스택 */\nclass LinkedListStack {\n    private stackPeek: ListNode | null; // 머리 노드를 스택의 맨 위로 사용한다\n    private stkSize: number = 0; // 스택의 길이\n\n    constructor() {\n        this.stackPeek = null;\n    }\n\n    /* 스택의 길이 반환 */\n    get size(): number {\n        return this.stkSize;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* 스택에 삽입 */\n    push(num: number): void {\n        const node = new ListNode(num);\n        node.next = this.stackPeek;\n        this.stackPeek = node;\n        this.stkSize++;\n    }\n\n    /* 스택에서 꺼내기 */\n    pop(): number {\n        const num = this.peek();\n        if (!this.stackPeek) throw new Error('스택이 비어 있다');\n        this.stackPeek = this.stackPeek.next;\n        this.stkSize--;\n        return num;\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    peek(): number {\n        if (!this.stackPeek) throw new Error('스택이 비어 있다');\n        return this.stackPeek.val;\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    toArray(): number[] {\n        let node = this.stackPeek;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.dart<pre><code>/* 연결 리스트 클래스로 구현한 스택 */\nclass LinkedListStack {\n  ListNode? _stackPeek; // 머리 노드를 스택의 맨 위로 사용한다\n  int _stkSize = 0; // 스택의 길이\n\n  LinkedListStack() {\n    _stackPeek = null;\n  }\n\n  /* 스택의 길이 반환 */\n  int size() {\n    return _stkSize;\n  }\n\n  /* 스택이 비어 있는지 판단 */\n  bool isEmpty() {\n    return _stkSize == 0;\n  }\n\n  /* 스택에 삽입 */\n  void push(int _num) {\n    final ListNode node = ListNode(_num);\n    node.next = _stackPeek;\n    _stackPeek = node;\n    _stkSize++;\n  }\n\n  /* 스택에서 꺼내기 */\n  int pop() {\n    final int _num = peek();\n    _stackPeek = _stackPeek!.next;\n    _stkSize--;\n    return _num;\n  }\n\n  /* 스택 맨 위 원소 접근 */\n  int peek() {\n    if (_stackPeek == null) {\n      throw Exception(\"스택이 비어 있다\");\n    }\n    return _stackPeek!.val;\n  }\n\n  /* 연결 리스트를 List로 변환하여 반환 */\n  List&lt;int&gt; toList() {\n    ListNode? node = _stackPeek;\n    List&lt;int&gt; list = [];\n    while (node != null) {\n      list.add(node.val);\n      node = node.next;\n    }\n    list = list.reversed.toList();\n    return list;\n  }\n}\n</code></pre> linkedlist_stack.rs<pre><code>/* 연결 리스트로 구현한 스택 */\n#[allow(dead_code)]\npub struct LinkedListStack&lt;T&gt; {\n    stack_peek: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // 머리 노드를 스택의 맨 위로 사용한다\n    stk_size: usize,                              // 스택의 길이\n}\n\nimpl&lt;T: Copy&gt; LinkedListStack&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            stack_peek: None,\n            stk_size: 0,\n        }\n    }\n\n    /* 스택의 길이 반환 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.stk_size;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* 스택에 삽입 */\n    pub fn push(&amp;mut self, num: T) {\n        let node = ListNode::new(num);\n        node.borrow_mut().next = self.stack_peek.take();\n        self.stack_peek = Some(node);\n        self.stk_size += 1;\n    }\n\n    /* 스택에서 꺼내기 */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack_peek.take().map(|old_head| {\n            self.stack_peek = old_head.borrow_mut().next.take();\n            self.stk_size -= 1;\n\n            old_head.borrow().val\n        })\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.stack_peek.as_ref()\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    pub fn to_array(&amp;self) -&gt; Vec&lt;T&gt; {\n        fn _to_array&lt;T: Sized + Copy&gt;(head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n            if let Some(node) = head {\n                let mut nums = _to_array(node.borrow().next.as_ref());\n                nums.push(node.borrow().val);\n                return nums;\n            }\n            return Vec::new();\n        }\n\n        _to_array(self.peek())\n    }\n}\n</code></pre> linkedlist_stack.c<pre><code>/* 연결 리스트로 구현한 스택 */\ntypedef struct {\n    ListNode *top; // 머리 노드를 스택의 맨 위로 사용한다\n    int size;      // 스택의 길이\n} LinkedListStack;\n\n/* 생성자 */\nLinkedListStack *newLinkedListStack() {\n    LinkedListStack *s = malloc(sizeof(LinkedListStack));\n    s-&gt;top = NULL;\n    s-&gt;size = 0;\n    return s;\n}\n\n/* 소멸자 */\nvoid delLinkedListStack(LinkedListStack *s) {\n    while (s-&gt;top) {\n        ListNode *n = s-&gt;top-&gt;next;\n        free(s-&gt;top);\n        s-&gt;top = n;\n    }\n    free(s);\n}\n\n/* 스택의 길이 반환 */\nint size(LinkedListStack *s) {\n    return s-&gt;size;\n}\n\n/* 스택이 비어 있는지 판단 */\nbool isEmpty(LinkedListStack *s) {\n    return size(s) == 0;\n}\n\n/* 스택에 삽입 */\nvoid push(LinkedListStack *s, int num) {\n    ListNode *node = (ListNode *)malloc(sizeof(ListNode));\n    node-&gt;next = s-&gt;top; // 새로 추가한 노드의 포인터 필드를 갱신한다\n    node-&gt;val = num;     // 새로 추가한 노드의 데이터 필드를 갱신한다\n    s-&gt;top = node;       // 스택의 맨 위를 갱신한다\n    s-&gt;size++;           // 스택 크기를 갱신한다\n}\n\n/* 스택 맨 위 원소 접근 */\nint peek(LinkedListStack *s) {\n    if (s-&gt;size == 0) {\n        printf(\"스택이 비어 있다\\n\");\n        return INT_MAX;\n    }\n    return s-&gt;top-&gt;val;\n}\n\n/* 스택에서 꺼내기 */\nint pop(LinkedListStack *s) {\n    int val = peek(s);\n    ListNode *tmp = s-&gt;top;\n    s-&gt;top = s-&gt;top-&gt;next;\n    // 메모리 해제\n    free(tmp);\n    s-&gt;size--;\n    return val;\n}\n</code></pre> linkedlist_stack.kt<pre><code>/* 연결 리스트로 구현한 스택 */\nclass LinkedListStack(\n    private var stackPeek: ListNode? = null, // 머리 노드를 스택의 맨 위로 사용한다\n    private var stkSize: Int = 0 // 스택의 길이\n) {\n\n    /* 스택의 길이 반환 */\n    fun size(): Int {\n        return stkSize\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* 스택에 삽입 */\n    fun push(num: Int) {\n        val node = ListNode(num)\n        node.next = stackPeek\n        stackPeek = node\n        stkSize++\n    }\n\n    /* 스택에서 꺼내기 */\n    fun pop(): Int? {\n        val num = peek()\n        stackPeek = stackPeek?.next\n        stkSize--\n        return num\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    fun peek(): Int? {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stackPeek?._val\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    fun toArray(): IntArray {\n        var node = stackPeek\n        val res = IntArray(size())\n        for (i in res.size - 1 downTo 0) {\n            res[i] = node?._val!!\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.rb<pre><code>### 연결 리스트로 구현한 스택 ###\nclass LinkedListStack\n  attr_reader :size\n\n  ### 생성자 ###\n  def initialize\n    @size = 0\n  end\n\n  ### 스택이 비어 있는지 판단 ###\n  def is_empty?\n    @peek.nil?\n  end\n\n  ### 스택에 삽입 ###\n  def push(val)\n    node = ListNode.new(val)\n    node.next = @peek\n    @peek = node\n    @size += 1\n  end\n\n  ### 스택에서 꺼내기 ###\n  def pop\n    num = peek\n    @peek = @peek.next\n    @size -= 1\n    num\n  end\n\n  ### 스택 맨 위 원소 접근 ###\n  def peek\n    raise IndexError, '스택이 비어 있다' if is_empty?\n\n    @peek.val\n  end\n\n  ### 연결 리스트를 Array로 변환하여 반환 ###\n  def to_array\n    arr = []\n    node = @peek\n    while node\n      arr &lt;&lt; node.val\n      node = node.next\n    end\n    arr.reverse\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 5 장   스택과 큐","5.1   스택"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#2","level":3,"title":"2.   배열 기반 구현","text":"<p>배열로 스택을 구현할 때는 배열의 끝을 스택의 꼭대기로 볼 수 있다. 아래 그림(5-3)과 같이 삽입과 꺼내기는 각각 배열 끝에 원소를 추가하고 삭제하는 연산에 해당하며, 시간 복잡도는 모두 \\(O(1)\\)이다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> 그림 5-3   배열 기반 스택의 삽입과 꺼내기 </p> <p>스택에 삽입되는 원소가 계속 늘어날 수 있으므로 동적 배열을 사용하면 배열의 용량 확장을 직접 처리하지 않아도 된다. 예제 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_stack.py<pre><code>class ArrayStack:\n    \"\"\"배열로 구현한 스택\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        self._stack: list[int] = []\n\n    def size(self) -&gt; int:\n        \"\"\"스택의 길이 반환\"\"\"\n        return len(self._stack)\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"스택이 비어 있는지 판단\"\"\"\n        return self.size() == 0\n\n    def push(self, item: int):\n        \"\"\"스택에 삽입\"\"\"\n        self._stack.append(item)\n\n    def pop(self) -&gt; int:\n        \"\"\"스택에서 꺼내기\"\"\"\n        if self.is_empty():\n            raise IndexError(\"스택이 비어 있다\")\n        return self._stack.pop()\n\n    def peek(self) -&gt; int:\n        \"\"\"스택 맨 위 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"스택이 비어 있다\")\n        return self._stack[-1]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"출력에 사용할 리스트 반환\"\"\"\n        return self._stack\n</code></pre> array_stack.cpp<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n  private:\n    vector&lt;int&gt; stack;\n\n  public:\n    /* 스택의 길이 반환 */\n    int size() {\n        return stack.size();\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    bool isEmpty() {\n        return stack.size() == 0;\n    }\n\n    /* 스택에 삽입 */\n    void push(int num) {\n        stack.push_back(num);\n    }\n\n    /* 스택에서 꺼내기 */\n    int pop() {\n        int num = top();\n        stack.pop_back();\n        return num;\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"스택이 비어 있다\");\n        return stack.back();\n    }\n\n    /* Vector 반환 */\n    vector&lt;int&gt; toVector() {\n        return stack;\n    }\n};\n</code></pre> array_stack.java<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n    private ArrayList&lt;Integer&gt; stack;\n\n    public ArrayStack() {\n        // 리스트(동적 배열)를 초기화한다\n        stack = new ArrayList&lt;&gt;();\n    }\n\n    /* 스택의 길이 반환 */\n    public int size() {\n        return stack.size();\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* 스택에 삽입 */\n    public void push(int num) {\n        stack.add(num);\n    }\n\n    /* 스택에서 꺼내기 */\n    public int pop() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.remove(size() - 1);\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.get(size() - 1);\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    public Object[] toArray() {\n        return stack.toArray();\n    }\n}\n</code></pre> array_stack.cs<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n    List&lt;int&gt; stack;\n    public ArrayStack() {\n        // 리스트(동적 배열)를 초기화한다\n        stack = [];\n    }\n\n    /* 스택의 길이 반환 */\n    public int Size() {\n        return stack.Count;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* 스택에 삽입 */\n    public void Push(int num) {\n        stack.Add(num);\n    }\n\n    /* 스택에서 꺼내기 */\n    public int Pop() {\n        if (IsEmpty())\n            throw new Exception();\n        var val = Peek();\n        stack.RemoveAt(Size() - 1);\n        return val;\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stack[Size() - 1];\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    public int[] ToArray() {\n        return [.. stack];\n    }\n}\n</code></pre> array_stack.go<pre><code>/* 배열로 구현한 스택 */\ntype arrayStack struct {\n    data []int // 데이터\n}\n\n/* 스택 초기화 */\nfunc newArrayStack() *arrayStack {\n    return &amp;arrayStack{\n        // 스택의 길이를 0, 용량을 16으로 설정한다\n        data: make([]int, 0, 16),\n    }\n}\n\n/* 스택의 길이 */\nfunc (s *arrayStack) size() int {\n    return len(s.data)\n}\n\n/* 스택이 비어 있는지 판단 */\nfunc (s *arrayStack) isEmpty() bool {\n    return s.size() == 0\n}\n\n/* 스택에 삽입 */\nfunc (s *arrayStack) push(v int) {\n    // 슬라이스는 용량이 자동으로 확장된다\n    s.data = append(s.data, v)\n}\n\n/* 스택에서 꺼내기 */\nfunc (s *arrayStack) pop() any {\n    val := s.peek()\n    s.data = s.data[:len(s.data)-1]\n    return val\n}\n\n/* 스택 맨 위 원소 반환 */\nfunc (s *arrayStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    val := s.data[len(s.data)-1]\n    return val\n}\n\n/* 출력에 사용할 Slice 반환 */\nfunc (s *arrayStack) toSlice() []int {\n    return s.data\n}\n</code></pre> array_stack.swift<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n    private var stack: [Int]\n\n    init() {\n        // 리스트(동적 배열)를 초기화한다\n        stack = []\n    }\n\n    /* 스택의 길이 반환 */\n    func size() -&gt; Int {\n        stack.count\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    func isEmpty() -&gt; Bool {\n        stack.isEmpty\n    }\n\n    /* 스택에 삽입 */\n    func push(num: Int) {\n        stack.append(num)\n    }\n\n    /* 스택에서 꺼내기 */\n    @discardableResult\n    func pop() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"스택이 비어 있다\")\n        }\n        return stack.removeLast()\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"스택이 비어 있다\")\n        }\n        return stack.last!\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    func toArray() -&gt; [Int] {\n        stack\n    }\n}\n</code></pre> array_stack.js<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n    #stack;\n    constructor() {\n        this.#stack = [];\n    }\n\n    /* 스택의 길이 반환 */\n    get size() {\n        return this.#stack.length;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    isEmpty() {\n        return this.#stack.length === 0;\n    }\n\n    /* 스택에 삽입 */\n    push(num) {\n        this.#stack.push(num);\n    }\n\n    /* 스택에서 꺼내기 */\n    pop() {\n        if (this.isEmpty()) throw new Error('스택이 비어 있다');\n        return this.#stack.pop();\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    top() {\n        if (this.isEmpty()) throw new Error('스택이 비어 있다');\n        return this.#stack[this.#stack.length - 1];\n    }\n\n    /* Array를 반환한다 */\n    toArray() {\n        return this.#stack;\n    }\n}\n</code></pre> array_stack.ts<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n    private stack: number[];\n    constructor() {\n        this.stack = [];\n    }\n\n    /* 스택의 길이 반환 */\n    get size(): number {\n        return this.stack.length;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    isEmpty(): boolean {\n        return this.stack.length === 0;\n    }\n\n    /* 스택에 삽입 */\n    push(num: number): void {\n        this.stack.push(num);\n    }\n\n    /* 스택에서 꺼내기 */\n    pop(): number | undefined {\n        if (this.isEmpty()) throw new Error('스택이 비어 있다');\n        return this.stack.pop();\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    top(): number | undefined {\n        if (this.isEmpty()) throw new Error('스택이 비어 있다');\n        return this.stack[this.stack.length - 1];\n    }\n\n    /* Array를 반환한다 */\n    toArray() {\n        return this.stack;\n    }\n}\n</code></pre> array_stack.dart<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n  late List&lt;int&gt; _stack;\n  ArrayStack() {\n    _stack = [];\n  }\n\n  /* 스택의 길이 반환 */\n  int size() {\n    return _stack.length;\n  }\n\n  /* 스택이 비어 있는지 판단 */\n  bool isEmpty() {\n    return _stack.isEmpty;\n  }\n\n  /* 스택에 삽입 */\n  void push(int _num) {\n    _stack.add(_num);\n  }\n\n  /* 스택에서 꺼내기 */\n  int pop() {\n    if (isEmpty()) {\n      throw Exception(\"스택이 비어 있다\");\n    }\n    return _stack.removeLast();\n  }\n\n  /* 스택 맨 위 원소 접근 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"스택이 비어 있다\");\n    }\n    return _stack.last;\n  }\n\n  /* 스택을 Array로 변환하여 반환 */\n  List&lt;int&gt; toArray() =&gt; _stack;\n}\n</code></pre> array_stack.rs<pre><code>/* 배열로 구현한 스택 */\nstruct ArrayStack&lt;T&gt; {\n    stack: Vec&lt;T&gt;,\n}\n\nimpl&lt;T&gt; ArrayStack&lt;T&gt; {\n    /* 스택 초기화 */\n    fn new() -&gt; ArrayStack&lt;T&gt; {\n        ArrayStack::&lt;T&gt; {\n            stack: Vec::&lt;T&gt;::new(),\n        }\n    }\n\n    /* 스택의 길이 반환 */\n    fn size(&amp;self) -&gt; usize {\n        self.stack.len()\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.size() == 0\n    }\n\n    /* 스택에 삽입 */\n    fn push(&amp;mut self, num: T) {\n        self.stack.push(num);\n    }\n\n    /* 스택에서 꺼내기 */\n    fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack.pop()\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    fn peek(&amp;self) -&gt; Option&lt;&amp;T&gt; {\n        if self.is_empty() {\n            panic!(\"스택이 비어 있다\")\n        };\n        self.stack.last()\n    }\n\n    /* &amp;Vec를 반환한다 */\n    fn to_array(&amp;self) -&gt; &amp;Vec&lt;T&gt; {\n        &amp;self.stack\n    }\n}\n</code></pre> array_stack.c<pre><code>/* 배열로 구현한 스택 */\ntypedef struct {\n    int *data;\n    int size;\n} ArrayStack;\n\n/* 생성자 */\nArrayStack *newArrayStack() {\n    ArrayStack *stack = malloc(sizeof(ArrayStack));\n    // 용량 확장을 피하도록 큰 용량으로 초기화한다\n    stack-&gt;data = malloc(sizeof(int) * MAX_SIZE);\n    stack-&gt;size = 0;\n    return stack;\n}\n\n/* 소멸자 */\nvoid delArrayStack(ArrayStack *stack) {\n    free(stack-&gt;data);\n    free(stack);\n}\n\n/* 스택의 길이 반환 */\nint size(ArrayStack *stack) {\n    return stack-&gt;size;\n}\n\n/* 스택이 비어 있는지 판단 */\nbool isEmpty(ArrayStack *stack) {\n    return stack-&gt;size == 0;\n}\n\n/* 스택에 삽입 */\nvoid push(ArrayStack *stack, int num) {\n    if (stack-&gt;size == MAX_SIZE) {\n        printf(\"스택이 가득 찼다\\n\");\n        return;\n    }\n    stack-&gt;data[stack-&gt;size] = num;\n    stack-&gt;size++;\n}\n\n/* 스택 맨 위 원소 접근 */\nint peek(ArrayStack *stack) {\n    if (stack-&gt;size == 0) {\n        printf(\"스택이 비어 있다\\n\");\n        return INT_MAX;\n    }\n    return stack-&gt;data[stack-&gt;size - 1];\n}\n\n/* 스택에서 꺼내기 */\nint pop(ArrayStack *stack) {\n    int val = peek(stack);\n    stack-&gt;size--;\n    return val;\n}\n</code></pre> array_stack.kt<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n    // 리스트(동적 배열)를 초기화한다\n    private val stack = mutableListOf&lt;Int&gt;()\n\n    /* 스택의 길이 반환 */\n    fun size(): Int {\n        return stack.size\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* 스택에 삽입 */\n    fun push(num: Int) {\n        stack.add(num)\n    }\n\n    /* 스택에서 꺼내기 */\n    fun pop(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack.removeAt(size() - 1)\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack[size() - 1]\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    fun toArray(): Array&lt;Any&gt; {\n        return stack.toTypedArray()\n    }\n}\n</code></pre> array_stack.rb<pre><code>### 배열로 구현한 스택 ###\nclass ArrayStack\n  ### 생성자 ###\n  def initialize\n    @stack = []\n  end\n\n  ### 스택의 길이 반환 ###\n  def size\n    @stack.length\n  end\n\n  ### 스택이 비어 있는지 판단 ###\n  def is_empty?\n    @stack.empty?\n  end\n\n  ### 스택에 삽입 ###\n  def push(item)\n    @stack &lt;&lt; item\n  end\n\n  ### 스택에서 꺼내기 ###\n  def pop\n    raise IndexError, '스택이 비어 있다' if is_empty?\n\n    @stack.pop\n  end\n\n  ### 스택 맨 위 원소 접근 ###\n  def peek\n    raise IndexError, '스택이 비어 있다' if is_empty?\n\n    @stack.last\n  end\n\n  ### 출력에 사용할 리스트 반환 ###\n  def to_array\n    @stack\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 5 장   스택과 큐","5.1   스택"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#513","level":2,"title":"5.1.3   두 구현의 비교","text":"<p>지원 연산</p> <p>두 구현 모두 스택 정의에 포함된 모든 연산을 지원한다. 배열 기반 구현은 임의 접근도 지원하지만, 이는 스택의 정의를 벗어나므로 일반적으로 사용하지 않는다.</p> <p>시간 효율</p> <p>배열 기반 구현에서는 삽입과 꺼내기를 미리 할당한 연속 메모리에서 수행한다. 캐시 지역성이 좋아 효율이 높다. 다만 삽입 시 배열 용량을 넘으면 확장이 발생하여 그 삽입 연산의 시간 복잡도는 \\(O(n)\\)이 된다.</p> <p>연결 리스트 기반 구현은 크기를 유연하게 늘릴 수 있어 배열 확장에 따른 효율 저하가 없다. 하지만 삽입 시 노드 객체를 초기화하고 포인터를 수정해야 하므로 상대적으로 효율이 낮다. 삽입할 원소 자체가 노드 객체라면 초기화 단계를 생략하여 효율을 높일 수 있다.</p> <p>따라서 삽입하고 꺼내는 원소가 <code>int</code>나 <code>double</code> 같은 기본 자료형이라면 다음과 같이 정리할 수 있다.</p> <ul> <li>배열 기반 스택은 용량 확장이 발생할 때 효율이 낮아지지만, 확장은 드물게 일어나므로 평균 효율은 더 높다.</li> <li>연결 리스트 기반 스택은 더 안정적인 효율을 제공한다.</li> </ul> <p>공간 효율</p> <p>리스트를 초기화할 때 시스템은 ‘초기 용량’을 할당하며, 이 용량은 실제 필요보다 클 수 있다. 또한 확장 시 보통 2배처럼 일정한 비율로 늘리므로 확장 후 용량도 실제 필요보다 클 수 있다. 따라서 배열 기반 스택은 어느 정도 공간을 낭비할 수 있다.</p> <p>반면 연결 리스트 노드는 포인터를 추가로 저장해야 하므로 노드 하나가 차지하는 공간이 상대적으로 크다.</p> <p>따라서 어느 구현이 메모리를 더 절약하는지는 단정할 수 없으며 구체적인 상황에 따라 분석해야 한다.</p>","path":["제 5 장   스택과 큐","5.1   스택"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#514","level":2,"title":"5.1.4   스택의 대표적인 활용","text":"<ul> <li>브라우저의 뒤로 및 앞으로, 소프트웨어의 실행 취소 및 다시 실행. 새 웹 페이지를 열 때마다 브라우저는 이전 페이지를 스택에 넣어 뒤로 가기로 돌아갈 수 있게 한다. 뒤로 가기는 실제로 스택에서 원소를 꺼내는 연산이다. 뒤로 가기와 앞으로 가기를 함께 지원하려면 스택 두 개를 사용해야 한다.</li> <li>프로그램 메모리 관리. 함수가 호출될 때마다 시스템은 스택 맨 위에 스택 프레임을 추가해 함수의 문맥 정보를 기록한다. 재귀 함수에서는 아래로 호출을 이어 가는 단계에서 스택에 계속 넣고, 위로 되돌아오는 단계에서 계속 꺼낸다.</li> </ul>","path":["제 5 장   스택과 큐","5.1   스택"],"tags":[]},{"location":"chapter_stack_and_queue/summary/","level":1,"title":"5.4   요약","text":"","path":["제 5 장   스택과 큐","5.4   요약"],"tags":[]},{"location":"chapter_stack_and_queue/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>스택은 후입선출 원리를 따르는 자료 구조로, 배열이나 연결 리스트로 구현할 수 있다.</li> <li>시간 효율 측면에서 배열 기반 스택의 평균 효율은 더 높지만, 용량 확장이 발생하면 한 번의 삽입 연산의 시간 복잡도가 \\(O(n)\\)으로 나빠진다. 반면 연결 리스트 기반 스택은 더 안정적인 효율을 보인다.</li> <li>공간 효율 측면에서 배열 기반 스택은 어느 정도 공간을 낭비할 수 있다. 다만 연결 리스트의 노드 하나는 배열 원소보다 더 많은 메모리를 차지한다.</li> <li>큐는 선입선출 원리를 따르는 자료 구조로, 역시 배열이나 연결 리스트로 구현할 수 있다. 시간 및 공간 효율의 비교 결과는 앞의 스택과 비슷하다.</li> <li>덱은 자유도가 더 높은 큐로, 양 끝에서 원소를 추가하고 삭제할 수 있다.</li> </ul>","path":["제 5 장   스택과 큐","5.4   요약"],"tags":[]},{"location":"chapter_stack_and_queue/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 브라우저의 앞으로 가기와 뒤로 가기는 이중 연결 리스트로 구현되는가?</p> <p>브라우저의 앞으로 가기와 뒤로 가기는 본질적으로 ‘스택’을 보여 준다. 사용자가 새 페이지를 방문하면 맨 위에 추가되고, 뒤로 버튼을 누르면 그 페이지가 맨 위에서 꺼내진다. ‘덱’ 절에서 설명했듯 덱을 사용하면 일부 추가 연산을 편리하게 구현할 수 있다.</p> <p>Q: 스택에서 노드를 꺼낸 뒤 그 노드의 메모리를 해제해야 하는가?</p> <p>나중에 꺼낸 노드를 계속 사용해야 한다면 해제할 필요가 없다. 더 이상 사용하지 않는 경우, <code>Java</code>와 <code>Python</code> 같은 언어는 자동 가비지 컬렉션이 있으므로 직접 해제하지 않아도 된다. <code>C</code>와 <code>C++</code>에서는 직접 해제해야 한다.</p> <p>Q: 덱은 두 스택을 이어 붙인 것처럼 보이는데, 어디에 쓰이는가?</p> <p>덱은 스택과 큐를 합치거나 두 스택을 이어 붙인 것과 같다. 스택과 큐의 논리를 모두 지니므로 두 자료 구조의 모든 활용을 구현할 수 있고 더 유연하다.</p> <p>Q: 실행 취소(undo)와 다시 실행(redo)은 구체적으로 어떻게 구현하는가?</p> <p>스택 두 개를 사용한다. 스택 <code>A</code>는 실행 취소용, 스택 <code>B</code>는 다시 실행용이다.</p> <ol> <li>사용자가 작업을 수행할 때마다 그 작업을 스택 <code>A</code>에 넣고 스택 <code>B</code>를 비운다.</li> <li>‘실행 취소’를 수행하면 스택 <code>A</code>에서 가장 최근 작업을 꺼내 스택 <code>B</code>에 넣는다.</li> <li>‘다시 실행’을 수행하면 스택 <code>B</code>에서 가장 최근 작업을 꺼내 스택 <code>A</code>에 넣는다.</li> </ol>","path":["제 5 장   스택과 큐","5.4   요약"],"tags":[]},{"location":"chapter_tree/","level":1,"title":"제 7 장   트리","text":"<p>이 장을 시작하며</p> <p>하늘 높이 뻗은 나무는 깊은 뿌리와 무성한 잎, 넓게 펼쳐진 가지에 생명력이 가득하다.</p> <p>나무는 데이터를 나누어 다루는 모습을 생생하게 보여 준다.</p>","path":["제 7 장   트리"],"tags":[]},{"location":"chapter_tree/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>7.1   이진 트리</li> <li>7.2   이진 트리 순회</li> <li>7.3   이진 트리의 배열 표현</li> <li>7.4   이진 탐색 트리</li> <li>7.5   AVL 트리 *</li> <li>7.6   요약</li> <li>7.7   연습 문제</li> </ul>","path":["제 7 장   트리"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/","level":1,"title":"7.3   이진 트리의 배열 표현","text":"<p>연결 리스트로 표현할 때 이진 트리의 저장 단위는 노드 <code>TreeNode</code>이며, 노드들은 포인터로 연결된다. 앞 절에서는 이 표현을 사용하는 이진 트리의 기본 연산들을 살펴보았다.</p> <p>그렇다면 배열로도 이진 트리를 표현할 수 있을까? 가능하다.</p>","path":["제 7 장   트리","7.3   이진 트리의 배열 표현"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/#731","level":2,"title":"7.3.1   포화 이진 트리 표현","text":"<p>먼저 간단한 경우를 살펴보자. 포화 이진 트리의 모든 노드를 레벨 순서 순회 순서대로 배열에 저장하면 각 노드는 고유한 배열 인덱스에 대응한다.</p> <p>레벨 순서 순회의 특성에서 부모와 자식 인덱스 사이의 '매핑 공식'을 도출할 수 있다. 노드의 인덱스가 \\(i\\)이면 왼쪽 자식의 인덱스는 \\(2i + 1\\), 오른쪽 자식의 인덱스는 \\(2i + 2\\)다. 아래 그림(7-12)은 노드 인덱스 사이의 매핑 관계를 보여 준다.</p> <p></p> <p> 그림 7-12   포화 이진 트리의 배열 표현 </p> <p>매핑 공식은 연결 리스트의 노드 참조(포인터)와 같은 역할을 한다. 배열의 임의의 노드가 주어지면 이 공식으로 왼쪽(오른쪽) 자식에 접근할 수 있다.</p>","path":["제 7 장   트리","7.3   이진 트리의 배열 표현"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/#732","level":2,"title":"7.3.2   임의의 이진 트리 표현","text":"<p>포화 이진 트리는 특수한 경우이며, 일반적인 이진 트리의 중간 레벨에는 <code>None</code>이 많이 존재한다. 레벨 순서 순회 결과에는 이 <code>None</code>들이 포함되지 않으므로 결과만으로는 개수와 위치를 알 수 없다. 즉, 같은 레벨 순서 순회 결과에 여러 이진 트리 구조가 대응할 수 있다.</p> <p>아래 그림(7-13)처럼 포화 이진 트리가 아니면 앞의 배열 표현 방법은 더 이상 유효하지 않다.</p> <p></p> <p> 그림 7-13   하나의 레벨 순서 순회 결과에 대응하는 여러 이진 트리 </p> <p>이를 해결하려면 레벨 순서 순회 결과에 모든 <code>None</code>을 명시적으로 넣을 수 있다. 아래 그림(7-14)처럼 처리하면 순회 결과가 이진 트리를 유일하게 나타낸다. 예시 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># 이진 트리의 배열 표현\n# None으로 빈자리 표시\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// int의 최댓값 INT_MAX로 빈자리 표시\nvector&lt;int&gt; tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// int의 래퍼 클래스 Integer를 사용하면 null로 빈자리를 표시할 수 있다\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// 널 허용 타입 int?를 사용하면 null로 빈자리를 표시할 수 있다\nint?[] tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// any 타입의 슬라이스를 사용하면 nil로 빈자리를 표시할 수 있다\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// 널 허용 타입 Int?를 사용하면 nil로 빈자리를 표시할 수 있다\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// null로 빈자리 표시\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// null로 빈자리 표시\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// 널 허용 타입 int?를 사용하면 null로 빈자리를 표시할 수 있다\nList&lt;int?&gt; tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// None으로 빈자리 표시\nlet tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)];\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// int의 최댓값으로 빈자리를 표시하므로 노드 값은 INT_MAX이면 안 된다\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// null로 빈자리 표시\nval tree = arrayOf( 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 )\n</code></pre> <pre><code>### 이진 트리의 배열 표현 ###\n# nil로 빈자리 표시\ntree = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <p></p> <p> 그림 7-14   임의의 이진 트리의 배열 표현 </p> <p>완전 이진 트리는 배열로 표현하기에 매우 적합하다. 정의를 떠올려 보면 <code>None</code>은 맨 아래 레벨의 오른쪽에만 있으므로 레벨 순서 순회 결과의 끝에만 나타난다.</p> <p>따라서 완전 이진 트리를 배열로 표현할 때는 모든 <code>None</code>의 저장을 생략할 수 있어 매우 편리하다. 아래 그림(7-15)은 그 예다.</p> <p></p> <p> 그림 7-15   완전 이진 트리의 배열 표현 </p> <p>다음 코드는 배열로 표현한 이진 트리를 구현하며 아래 연산들을 포함한다.</p> <ul> <li>주어진 노드의 값, 왼쪽(오른쪽) 자식, 부모 노드를 구한다.</li> <li>전위, 중위, 후위, 레벨 순서 순회 결과를 구한다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_binary_tree.py<pre><code>class ArrayBinaryTree:\n    \"\"\"배열로 표현한 이진 트리 클래스\"\"\"\n\n    def __init__(self, arr: list[int | None]):\n        \"\"\"생성자\"\"\"\n        self._tree = list(arr)\n\n    def size(self):\n        \"\"\"리스트 용량\"\"\"\n        return len(self._tree)\n\n    def val(self, i: int) -&gt; int | None:\n        \"\"\"인덱스가 i인 노드의 값 반환\"\"\"\n        # 인덱스가 범위를 벗어나면 빈칸을 나타내는 None을 반환한다\n        if i &lt; 0 or i &gt;= self.size():\n            return None\n        return self._tree[i]\n\n    def left(self, i: int) -&gt; int | None:\n        \"\"\"인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환\"\"\"\n        return 2 * i + 1\n\n    def right(self, i: int) -&gt; int | None:\n        \"\"\"인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환\"\"\"\n        return 2 * i + 2\n\n    def parent(self, i: int) -&gt; int | None:\n        \"\"\"인덱스가 i인 노드의 부모 노드 인덱스 반환\"\"\"\n        return (i - 1) // 2\n\n    def level_order(self) -&gt; list[int]:\n        \"\"\"레벨 순서 순회\"\"\"\n        self.res = []\n        # 배열을 직접 순회한다\n        for i in range(self.size()):\n            if self.val(i) is not None:\n                self.res.append(self.val(i))\n        return self.res\n\n    def dfs(self, i: int, order: str):\n        \"\"\"깊이 우선 순회\"\"\"\n        if self.val(i) is None:\n            return\n        # 전위 순회\n        if order == \"pre\":\n            self.res.append(self.val(i))\n        self.dfs(self.left(i), order)\n        # 중위 순회\n        if order == \"in\":\n            self.res.append(self.val(i))\n        self.dfs(self.right(i), order)\n        # 후위 순회\n        if order == \"post\":\n            self.res.append(self.val(i))\n\n    def pre_order(self) -&gt; list[int]:\n        \"\"\"전위 순회\"\"\"\n        self.res = []\n        self.dfs(0, order=\"pre\")\n        return self.res\n\n    def in_order(self) -&gt; list[int]:\n        \"\"\"중위 순회\"\"\"\n        self.res = []\n        self.dfs(0, order=\"in\")\n        return self.res\n\n    def post_order(self) -&gt; list[int]:\n        \"\"\"후위 순회\"\"\"\n        self.res = []\n        self.dfs(0, order=\"post\")\n        return self.res\n</code></pre> array_binary_tree.cpp<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree {\n  public:\n    /* 생성자 */\n    ArrayBinaryTree(vector&lt;int&gt; arr) {\n        tree = arr;\n    }\n\n    /* 리스트 용량 */\n    int size() {\n        return tree.size();\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    int val(int i) {\n        // 인덱스가 범위를 벗어나면 빈자리를 나타내는 INT_MAX를 반환한다\n        if (i &lt; 0 || i &gt;= size())\n            return INT_MAX;\n        return tree[i];\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    int left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    int right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    int parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* 레벨 순서 순회 */\n    vector&lt;int&gt; levelOrder() {\n        vector&lt;int&gt; res;\n        // 배열을 직접 순회한다\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != INT_MAX)\n                res.push_back(val(i));\n        }\n        return res;\n    }\n\n    /* 전위 순회 */\n    vector&lt;int&gt; preOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* 중위 순회 */\n    vector&lt;int&gt; inOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* 후위 순회 */\n    vector&lt;int&gt; postOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"post\", res);\n        return res;\n    }\n\n  private:\n    vector&lt;int&gt; tree;\n\n    /* 깊이 우선 순회 */\n    void dfs(int i, string order, vector&lt;int&gt; &amp;res) {\n        // 빈칸이면 반환한다\n        if (val(i) == INT_MAX)\n            return;\n        // 전위 순회\n        if (order == \"pre\")\n            res.push_back(val(i));\n        dfs(left(i), order, res);\n        // 중위 순회\n        if (order == \"in\")\n            res.push_back(val(i));\n        dfs(right(i), order, res);\n        // 후위 순회\n        if (order == \"post\")\n            res.push_back(val(i));\n    }\n};\n</code></pre> array_binary_tree.java<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree {\n    private List&lt;Integer&gt; tree;\n\n    /* 생성자 */\n    public ArrayBinaryTree(List&lt;Integer&gt; arr) {\n        tree = new ArrayList&lt;&gt;(arr);\n    }\n\n    /* 리스트 용량 */\n    public int size() {\n        return tree.size();\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    public Integer val(int i) {\n        // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n        if (i &lt; 0 || i &gt;= size())\n            return null;\n        return tree.get(i);\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    public Integer left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    public Integer right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    public Integer parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* 레벨 순서 순회 */\n    public List&lt;Integer&gt; levelOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        // 배열을 직접 순회한다\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != null)\n                res.add(val(i));\n        }\n        return res;\n    }\n\n    /* 깊이 우선 순회 */\n    private void dfs(Integer i, String order, List&lt;Integer&gt; res) {\n        // 빈칸이면 반환한다\n        if (val(i) == null)\n            return;\n        // 전위 순회\n        if (\"pre\".equals(order))\n            res.add(val(i));\n        dfs(left(i), order, res);\n        // 중위 순회\n        if (\"in\".equals(order))\n            res.add(val(i));\n        dfs(right(i), order, res);\n        // 후위 순회\n        if (\"post\".equals(order))\n            res.add(val(i));\n    }\n\n    /* 전위 순회 */\n    public List&lt;Integer&gt; preOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* 중위 순회 */\n    public List&lt;Integer&gt; inOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* 후위 순회 */\n    public List&lt;Integer&gt; postOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.cs<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree(List&lt;int?&gt; arr) {\n    List&lt;int?&gt; tree = new(arr);\n\n    /* 리스트 용량 */\n    public int Size() {\n        return tree.Count;\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    public int? Val(int i) {\n        // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n        if (i &lt; 0 || i &gt;= Size())\n            return null;\n        return tree[i];\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    public int Left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    public int Right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    public int Parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* 레벨 순서 순회 */\n    public List&lt;int&gt; LevelOrder() {\n        List&lt;int&gt; res = [];\n        // 배열을 직접 순회한다\n        for (int i = 0; i &lt; Size(); i++) {\n            if (Val(i).HasValue)\n                res.Add(Val(i)!.Value);\n        }\n        return res;\n    }\n\n    /* 깊이 우선 순회 */\n    void DFS(int i, string order, List&lt;int&gt; res) {\n        // 빈칸이면 반환한다\n        if (!Val(i).HasValue)\n            return;\n        // 전위 순회\n        if (order == \"pre\")\n            res.Add(Val(i)!.Value);\n        DFS(Left(i), order, res);\n        // 중위 순회\n        if (order == \"in\")\n            res.Add(Val(i)!.Value);\n        DFS(Right(i), order, res);\n        // 후위 순회\n        if (order == \"post\")\n            res.Add(Val(i)!.Value);\n    }\n\n    /* 전위 순회 */\n    public List&lt;int&gt; PreOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"pre\", res);\n        return res;\n    }\n\n    /* 중위 순회 */\n    public List&lt;int&gt; InOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"in\", res);\n        return res;\n    }\n\n    /* 후위 순회 */\n    public List&lt;int&gt; PostOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.go<pre><code>/* 배열로 표현한 이진 트리 클래스 */\ntype arrayBinaryTree struct {\n    tree []any\n}\n\n/* 생성자 */\nfunc newArrayBinaryTree(arr []any) *arrayBinaryTree {\n    return &amp;arrayBinaryTree{\n        tree: arr,\n    }\n}\n\n/* 리스트 용량 */\nfunc (abt *arrayBinaryTree) size() int {\n    return len(abt.tree)\n}\n\n/* 인덱스가 i인 노드의 값 반환 */\nfunc (abt *arrayBinaryTree) val(i int) any {\n    // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n    if i &lt; 0 || i &gt;= abt.size() {\n        return nil\n    }\n    return abt.tree[i]\n}\n\n/* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\nfunc (abt *arrayBinaryTree) left(i int) int {\n    return 2*i + 1\n}\n\n/* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\nfunc (abt *arrayBinaryTree) right(i int) int {\n    return 2*i + 2\n}\n\n/* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\nfunc (abt *arrayBinaryTree) parent(i int) int {\n    return (i - 1) / 2\n}\n\n/* 레벨 순서 순회 */\nfunc (abt *arrayBinaryTree) levelOrder() []any {\n    var res []any\n    // 배열을 직접 순회한다\n    for i := 0; i &lt; abt.size(); i++ {\n        if abt.val(i) != nil {\n            res = append(res, abt.val(i))\n        }\n    }\n    return res\n}\n\n/* 깊이 우선 순회 */\nfunc (abt *arrayBinaryTree) dfs(i int, order string, res *[]any) {\n    // 빈칸이면 반환한다\n    if abt.val(i) == nil {\n        return\n    }\n    // 전위 순회\n    if order == \"pre\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.left(i), order, res)\n    // 중위 순회\n    if order == \"in\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.right(i), order, res)\n    // 후위 순회\n    if order == \"post\" {\n        *res = append(*res, abt.val(i))\n    }\n}\n\n/* 전위 순회 */\nfunc (abt *arrayBinaryTree) preOrder() []any {\n    var res []any\n    abt.dfs(0, \"pre\", &amp;res)\n    return res\n}\n\n/* 중위 순회 */\nfunc (abt *arrayBinaryTree) inOrder() []any {\n    var res []any\n    abt.dfs(0, \"in\", &amp;res)\n    return res\n}\n\n/* 후위 순회 */\nfunc (abt *arrayBinaryTree) postOrder() []any {\n    var res []any\n    abt.dfs(0, \"post\", &amp;res)\n    return res\n}\n</code></pre> array_binary_tree.swift<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree {\n    private var tree: [Int?]\n\n    /* 생성자 */\n    init(arr: [Int?]) {\n        tree = arr\n    }\n\n    /* 리스트 용량 */\n    func size() -&gt; Int {\n        tree.count\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    func val(i: Int) -&gt; Int? {\n        // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n        if i &lt; 0 || i &gt;= size() {\n            return nil\n        }\n        return tree[i]\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    func left(i: Int) -&gt; Int {\n        2 * i + 1\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    func right(i: Int) -&gt; Int {\n        2 * i + 2\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    func parent(i: Int) -&gt; Int {\n        (i - 1) / 2\n    }\n\n    /* 레벨 순서 순회 */\n    func levelOrder() -&gt; [Int] {\n        var res: [Int] = []\n        // 배열을 직접 순회한다\n        for i in 0 ..&lt; size() {\n            if let val = val(i: i) {\n                res.append(val)\n            }\n        }\n        return res\n    }\n\n    /* 깊이 우선 순회 */\n    private func dfs(i: Int, order: String, res: inout [Int]) {\n        // 빈칸이면 반환한다\n        guard let val = val(i: i) else {\n            return\n        }\n        // 전위 순회\n        if order == \"pre\" {\n            res.append(val)\n        }\n        dfs(i: left(i: i), order: order, res: &amp;res)\n        // 중위 순회\n        if order == \"in\" {\n            res.append(val)\n        }\n        dfs(i: right(i: i), order: order, res: &amp;res)\n        // 후위 순회\n        if order == \"post\" {\n            res.append(val)\n        }\n    }\n\n    /* 전위 순회 */\n    func preOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"pre\", res: &amp;res)\n        return res\n    }\n\n    /* 중위 순회 */\n    func inOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"in\", res: &amp;res)\n        return res\n    }\n\n    /* 후위 순회 */\n    func postOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"post\", res: &amp;res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.js<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree {\n    #tree;\n\n    /* 생성자 */\n    constructor(arr) {\n        this.#tree = arr;\n    }\n\n    /* 리스트 용량 */\n    size() {\n        return this.#tree.length;\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    val(i) {\n        // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    left(i) {\n        return 2 * i + 1;\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    right(i) {\n        return 2 * i + 2;\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    parent(i) {\n        return Math.floor((i - 1) / 2); // 나눗셈의 결과를 내림한다\n    }\n\n    /* 레벨 순서 순회 */\n    levelOrder() {\n        let res = [];\n        // 배열을 직접 순회한다\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* 깊이 우선 순회 */\n    #dfs(i, order, res) {\n        // 빈칸이면 반환한다\n        if (this.val(i) === null) return;\n        // 전위 순회\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // 중위 순회\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // 후위 순회\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* 전위 순회 */\n    preOrder() {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* 중위 순회 */\n    inOrder() {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* 후위 순회 */\n    postOrder() {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.ts<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree {\n    #tree: (number | null)[];\n\n    /* 생성자 */\n    constructor(arr: (number | null)[]) {\n        this.#tree = arr;\n    }\n\n    /* 리스트 용량 */\n    size(): number {\n        return this.#tree.length;\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    val(i: number): number | null {\n        // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    left(i: number): number {\n        return 2 * i + 1;\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    right(i: number): number {\n        return 2 * i + 2;\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    parent(i: number): number {\n        return Math.floor((i - 1) / 2); // 나눗셈의 결과를 내림한다\n    }\n\n    /* 레벨 순서 순회 */\n    levelOrder(): number[] {\n        let res = [];\n        // 배열을 직접 순회한다\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* 깊이 우선 순회 */\n    #dfs(i: number, order: Order, res: (number | null)[]): void {\n        // 빈칸이면 반환한다\n        if (this.val(i) === null) return;\n        // 전위 순회\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // 중위 순회\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // 후위 순회\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* 전위 순회 */\n    preOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* 중위 순회 */\n    inOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* 후위 순회 */\n    postOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.dart<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree {\n  late List&lt;int?&gt; _tree;\n\n  /* 생성자 */\n  ArrayBinaryTree(this._tree);\n\n  /* 리스트 용량 */\n  int size() {\n    return _tree.length;\n  }\n\n  /* 인덱스가 i인 노드의 값 반환 */\n  int? val(int i) {\n    // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n    if (i &lt; 0 || i &gt;= size()) {\n      return null;\n    }\n    return _tree[i];\n  }\n\n  /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n  int? left(int i) {\n    return 2 * i + 1;\n  }\n\n  /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n  int? right(int i) {\n    return 2 * i + 2;\n  }\n\n  /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n  int? parent(int i) {\n    return (i - 1) ~/ 2;\n  }\n\n  /* 레벨 순서 순회 */\n  List&lt;int&gt; levelOrder() {\n    List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; size(); i++) {\n      if (val(i) != null) {\n        res.add(val(i)!);\n      }\n    }\n    return res;\n  }\n\n  /* 깊이 우선 순회 */\n  void dfs(int i, String order, List&lt;int?&gt; res) {\n    // 빈칸이면 반환한다\n    if (val(i) == null) {\n      return;\n    }\n    // 전위 순회\n    if (order == 'pre') {\n      res.add(val(i));\n    }\n    dfs(left(i)!, order, res);\n    // 중위 순회\n    if (order == 'in') {\n      res.add(val(i));\n    }\n    dfs(right(i)!, order, res);\n    // 후위 순회\n    if (order == 'post') {\n      res.add(val(i));\n    }\n  }\n\n  /* 전위 순회 */\n  List&lt;int?&gt; preOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'pre', res);\n    return res;\n  }\n\n  /* 중위 순회 */\n  List&lt;int?&gt; inOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'in', res);\n    return res;\n  }\n\n  /* 후위 순회 */\n  List&lt;int?&gt; postOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'post', res);\n    return res;\n  }\n}\n</code></pre> array_binary_tree.rs<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nstruct ArrayBinaryTree {\n    tree: Vec&lt;Option&lt;i32&gt;&gt;,\n}\n\nimpl ArrayBinaryTree {\n    /* 생성자 */\n    fn new(arr: Vec&lt;Option&lt;i32&gt;&gt;) -&gt; Self {\n        Self { tree: arr }\n    }\n\n    /* 리스트 용량 */\n    fn size(&amp;self) -&gt; i32 {\n        self.tree.len() as i32\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    fn val(&amp;self, i: i32) -&gt; Option&lt;i32&gt; {\n        // 인덱스가 범위를 벗어나면 빈칸을 나타내는 None을 반환한다\n        if i &lt; 0 || i &gt;= self.size() {\n            None\n        } else {\n            self.tree[i as usize]\n        }\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    fn left(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 1\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    fn right(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 2\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    fn parent(&amp;self, i: i32) -&gt; i32 {\n        (i - 1) / 2\n    }\n\n    /* 레벨 순서 순회 */\n    fn level_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        self.tree.iter().filter_map(|&amp;x| x).collect()\n    }\n\n    /* 깊이 우선 순회 */\n    fn dfs(&amp;self, i: i32, order: &amp;'static str, res: &amp;mut Vec&lt;i32&gt;) {\n        if self.val(i).is_none() {\n            return;\n        }\n        let val = self.val(i).unwrap();\n        // 전위 순회\n        if order == \"pre\" {\n            res.push(val);\n        }\n        self.dfs(self.left(i), order, res);\n        // 중위 순회\n        if order == \"in\" {\n            res.push(val);\n        }\n        self.dfs(self.right(i), order, res);\n        // 후위 순회\n        if order == \"post\" {\n            res.push(val);\n        }\n    }\n\n    /* 전위 순회 */\n    fn pre_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"pre\", &amp;mut res);\n        res\n    }\n\n    /* 중위 순회 */\n    fn in_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"in\", &amp;mut res);\n        res\n    }\n\n    /* 후위 순회 */\n    fn post_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"post\", &amp;mut res);\n        res\n    }\n}\n</code></pre> array_binary_tree.c<pre><code>/* 배열로 표현한 이진 트리 구조체 */\ntypedef struct {\n    int *tree;\n    int size;\n} ArrayBinaryTree;\n\n/* 생성자 */\nArrayBinaryTree *newArrayBinaryTree(int *arr, int arrSize) {\n    ArrayBinaryTree *abt = (ArrayBinaryTree *)malloc(sizeof(ArrayBinaryTree));\n    abt-&gt;tree = malloc(sizeof(int) * arrSize);\n    memcpy(abt-&gt;tree, arr, sizeof(int) * arrSize);\n    abt-&gt;size = arrSize;\n    return abt;\n}\n\n/* 소멸자 */\nvoid delArrayBinaryTree(ArrayBinaryTree *abt) {\n    free(abt-&gt;tree);\n    free(abt);\n}\n\n/* 리스트 용량 */\nint size(ArrayBinaryTree *abt) {\n    return abt-&gt;size;\n}\n\n/* 인덱스가 i인 노드의 값 반환 */\nint val(ArrayBinaryTree *abt, int i) {\n    // 인덱스가 범위를 벗어나면 빈자리를 나타내는 INT_MAX를 반환한다\n    if (i &lt; 0 || i &gt;= size(abt))\n        return INT_MAX;\n    return abt-&gt;tree[i];\n}\n\n/* 레벨 순서 순회 */\nint *levelOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    // 배열을 직접 순회한다\n    for (int i = 0; i &lt; size(abt); i++) {\n        if (val(abt, i) != INT_MAX)\n            res[index++] = val(abt, i);\n    }\n    *returnSize = index;\n    return res;\n}\n\n/* 깊이 우선 순회 */\nvoid dfs(ArrayBinaryTree *abt, int i, char *order, int *res, int *index) {\n    // 빈칸이면 반환한다\n    if (val(abt, i) == INT_MAX)\n        return;\n    // 전위 순회\n    if (strcmp(order, \"pre\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, left(i), order, res, index);\n    // 중위 순회\n    if (strcmp(order, \"in\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, right(i), order, res, index);\n    // 후위 순회\n    if (strcmp(order, \"post\") == 0)\n        res[(*index)++] = val(abt, i);\n}\n\n/* 전위 순회 */\nint *preOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"pre\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* 중위 순회 */\nint *inOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"in\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* 후위 순회 */\nint *postOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"post\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n</code></pre> array_binary_tree.kt<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree(val tree: MutableList&lt;Int?&gt;) {\n    /* 리스트 용량 */\n    fun size(): Int {\n        return tree.size\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    fun _val(i: Int): Int? {\n        // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n        if (i &lt; 0 || i &gt;= size()) return null\n        return tree[i]\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    fun parent(i: Int): Int {\n        return (i - 1) / 2\n    }\n\n    /* 레벨 순서 순회 */\n    fun levelOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        // 배열을 직접 순회한다\n        for (i in 0..&lt;size()) {\n            if (_val(i) != null)\n                res.add(_val(i))\n        }\n        return res\n    }\n\n    /* 깊이 우선 순회 */\n    fun dfs(i: Int, order: String, res: MutableList&lt;Int?&gt;) {\n        // 빈칸이면 반환한다\n        if (_val(i) == null)\n            return\n        // 전위 순회\n        if (\"pre\" == order)\n            res.add(_val(i))\n        dfs(left(i), order, res)\n        // 중위 순회\n        if (\"in\" == order)\n            res.add(_val(i))\n        dfs(right(i), order, res)\n        // 후위 순회\n        if (\"post\" == order)\n            res.add(_val(i))\n    }\n\n    /* 전위 순회 */\n    fun preOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"pre\", res)\n        return res\n    }\n\n    /* 중위 순회 */\n    fun inOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"in\", res)\n        return res\n    }\n\n    /* 후위 순회 */\n    fun postOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"post\", res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.rb<pre><code>### 배열로 표현한 이진 트리 클래스 ###\nclass ArrayBinaryTree\n  ### 생성자 ###\n  def initialize(arr)\n    @tree = arr.to_a\n  end\n\n  ### 리스트 용량 ###\n  def size\n    @tree.length\n  end\n\n  ### 인덱스가 i인 노드의 값 반환 ###\n  def val(i)\n    # 인덱스가 범위를 벗어나면 빈칸을 나타내는 nil을 반환한다\n    return if i &lt; 0 || i &gt;= size\n\n    @tree[i]\n  end\n\n  ### 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 ###\n  def left(i)\n    2 * i + 1\n  end\n\n  ### 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 ###\n  def right(i)\n    2 * i + 2\n  end\n\n  ### 인덱스가 i인 노드의 부모 노드 인덱스 반환 ###\n  def parent(i)\n    (i - 1) / 2\n  end\n\n  ### 레벨 순서 순회 ###\n  def level_order\n    @res = []\n\n    # 배열을 직접 순회한다\n    for i in 0...size\n      @res &lt;&lt; val(i) unless val(i).nil?\n    end\n\n    @res\n  end\n\n  ### 깊이 우선 순회 ###\n  def dfs(i, order)\n    return if val(i).nil?\n    # 전위 순회\n    @res &lt;&lt; val(i) if order == :pre\n    dfs(left(i), order)\n    # 중위 순회\n    @res &lt;&lt; val(i) if order == :in\n    dfs(right(i), order)\n    # 후위 순회\n    @res &lt;&lt; val(i) if order == :post\n  end\n\n  ### 전위 순회 ###\n  def pre_order\n    @res = []\n    dfs(0, :pre)\n    @res\n  end\n\n  ### 중위 순회 ###\n  def in_order\n    @res = []\n    dfs(0, :in)\n    @res\n  end\n\n  ### 후위 순회 ###\n  def post_order\n    @res = []\n    dfs(0, :post)\n    @res\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 7 장   트리","7.3   이진 트리의 배열 표현"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/#733","level":2,"title":"7.3.3   장점과 한계","text":"<p>이진 트리의 배열 표현에는 다음 장점이 있다.</p> <ul> <li>배열은 연속된 메모리에 저장되어 캐시 친화적이며, 접근과 순회가 빠르다.</li> <li>포인터를 저장할 필요가 없어 공간을 절약한다.</li> <li>노드에 임의 접근할 수 있다.</li> </ul> <p>하지만 다음과 같은 한계도 있다.</p> <ul> <li>연속된 메모리가 필요하므로 데이터가 지나치게 많은 트리를 저장하기에는 적합하지 않다.</li> <li>노드를 추가하거나 삭제하려면 배열의 삽입과 삭제 연산을 수행해야 하므로 효율이 낮다.</li> <li>트리에 <code>None</code>이 많으면 배열에서 실제 노드 데이터가 차지하는 비율이 낮아 공간 활용률이 떨어진다.</li> </ul>","path":["제 7 장   트리","7.3   이진 트리의 배열 표현"],"tags":[]},{"location":"chapter_tree/avl_tree/","level":1,"title":"7.5   AVL 트리 *","text":"<p>'이진 탐색 트리' 절에서 삽입과 삭제를 여러 번 수행하면 트리가 연결 리스트로 퇴화할 수 있다고 설명했다. 이 경우 모든 연산의 시간 복잡도가 \\(O(\\log n)\\)에서 \\(O(n)\\)으로 저하된다.</p> <p>아래 그림(7-24)의 이진 탐색 트리는 노드를 두 번 삭제하면 연결 리스트로 퇴화한다.</p> <p></p> <p> 그림 7-24   노드 삭제 후 퇴화한 AVL 트리 </p> <p>또한 아래 그림(7-25)의 포화 이진 트리에 노드 두 개를 삽입하면 왼쪽으로 심하게 기울어 탐색의 시간 복잡도도 저하된다.</p> <p></p> <p> 그림 7-25   노드 삽입 후 퇴화한 AVL 트리 </p> <p>1962년 G. M. Adelson-Velsky와 E. M. Landis는 논문 'An algorithm for the organization of information'에서 AVL 트리를 제안했다. 논문은 노드를 계속 추가하거나 삭제해도 트리가 퇴화하지 않게 하는 일련의 연산을 상세히 설명하여, 각 연산의 시간 복잡도를 \\(O(\\log n)\\)으로 유지하도록 했다. 즉, 추가, 삭제, 조회, 수정이 잦은 상황에서도 효율적인 데이터 연산 성능을 유지하므로 활용 가치가 높다.</p>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#751-avl","level":2,"title":"7.5.1   AVL 트리의 주요 용어","text":"<p>AVL 트리는 이진 탐색 트리이면서 균형 이진 트리다. 두 종류의 모든 성질을 함께 만족하므로 균형 이진 탐색 트리(balanced binary search tree)다.</p>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#1","level":3,"title":"1.   노드 높이","text":"<p>AVL 트리 연산에는 노드 높이가 필요하므로 노드 클래스에 <code>height</code> 변수를 추가해야 한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class TreeNode:\n    \"\"\"AVL 트리 노드 클래스\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                 # 노드 값\n        self.height: int = 0                # 노드 높이\n        self.left: TreeNode | None = None   # 왼쪽 자식 노드 참조\n        self.right: TreeNode | None = None  # 오른쪽 자식 노드 참조\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nstruct TreeNode {\n    int val{};          // 노드 값\n    int height = 0;     // 노드 높이\n    TreeNode *left{};   // 왼쪽 자식 노드\n    TreeNode *right{};  // 오른쪽 자식 노드\n    TreeNode() = default;\n    explicit TreeNode(int x) : val(x){}\n};\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nclass TreeNode {\n    public int val;        // 노드 값\n    public int height;     // 노드 높이\n    public TreeNode left;  // 왼쪽 자식 노드\n    public TreeNode right; // 오른쪽 자식 노드\n    public TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nclass TreeNode(int? x) {\n    public int? val = x;    // 노드 값\n    public int height;      // 노드 높이\n    public TreeNode? left;  // 왼쪽 자식 노드 참조\n    public TreeNode? right; // 오른쪽 자식 노드 참조\n}\n</code></pre> <pre><code>/* AVL 트리 노드 구조체 */\ntype TreeNode struct {\n    Val    int       // 노드 값\n    Height int       // 노드 높이\n    Left   *TreeNode // 왼쪽 자식 노드 참조\n    Right  *TreeNode // 오른쪽 자식 노드 참조\n}\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nclass TreeNode {\n    var val: Int // 노드 값\n    var height: Int // 노드 높이\n    var left: TreeNode? // 왼쪽 자식 노드\n    var right: TreeNode? // 오른쪽 자식 노드\n\n    init(x: Int) {\n        val = x\n        height = 0\n    }\n}\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nclass TreeNode {\n    val; // 노드 값\n    height; //노드 높이\n    left; // 왼쪽 자식 노드 포인터\n    right; // 오른쪽 자식 노드 포인터\n    constructor(val, left, right, height) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nclass TreeNode {\n    val: number;            // 노드 값\n    height: number;         // 노드 높이\n    left: TreeNode | null;  // 왼쪽 자식 노드 포인터\n    right: TreeNode | null; // 오른쪽 자식 노드 포인터\n    constructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nclass TreeNode {\n  int val;         // 노드 값\n  int height;      // 노드 높이\n  TreeNode? left;  // 왼쪽 자식 노드\n  TreeNode? right; // 오른쪽 자식 노드\n  TreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* AVL 트리 노드 구조체 */\nstruct TreeNode {\n    val: i32,                               // 노드 값\n    height: i32,                            // 노드 높이\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // 왼쪽 자식 노드\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // 오른쪽 자식 노드\n}\n\nimpl TreeNode {\n    /* 생성자 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            height: 0,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* AVL 트리 노드 구조체 */\ntypedef struct TreeNode {\n    int val;\n    int height;\n    struct TreeNode *left;\n    struct TreeNode *right;\n} TreeNode;\n\n/* 생성자 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nclass TreeNode(val _val: Int) {  // 노드 값\n    var height: Int = 0          // 노드 높이\n    var left: TreeNode? = null   // 왼쪽 자식 노드\n    var right: TreeNode? = null  // 오른쪽 자식 노드\n}\n</code></pre> <pre><code>### AVL 트리 노드 클래스 ###\nclass TreeNode\n  attr_accessor :val    # 노드 값\n  attr_accessor :height # 노드 높이\n  attr_accessor :left   # 왼쪽 자식 노드 참조\n  attr_accessor :right  # 오른쪽 자식 노드 참조\n\n  def initialize(val)\n    @val = val\n    @height = 0\n  end\nend\n</code></pre> <p>'노드 높이'는 해당 노드에서 가장 먼 리프까지의 거리, 즉 거치는 '간선' 수다. 특히 리프의 높이는 \\(0\\), 빈 노드의 높이는 \\(-1\\)임에 주의하자. 높이를 조회하고 갱신하는 두 보조 함수를 만든다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def height(self, node: TreeNode | None) -&gt; int:\n    \"\"\"노드 높이 반환\"\"\"\n    # 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    if node is not None:\n        return node.height\n    return -1\n\ndef update_height(self, node: TreeNode | None):\n    \"\"\"노드 높이를 갱신한다\"\"\"\n    # 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node.height = max([self.height(node.left), self.height(node.right)]) + 1\n</code></pre> avl_tree.cpp<pre><code>/* 노드 높이 반환 */\nint height(TreeNode *node) {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    return node == nullptr ? -1 : node-&gt;height;\n}\n\n/* 노드 높이를 갱신한다 */\nvoid updateHeight(TreeNode *node) {\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node-&gt;height = max(height(node-&gt;left), height(node-&gt;right)) + 1;\n}\n</code></pre> avl_tree.java<pre><code>/* 노드 높이 반환 */\nint height(TreeNode node) {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    return node == null ? -1 : node.height;\n}\n\n/* 노드 높이를 갱신한다 */\nvoid updateHeight(TreeNode node) {\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.cs<pre><code>/* 노드 높이 반환 */\nint Height(TreeNode? node) {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    return node == null ? -1 : node.height;\n}\n\n/* 노드 높이를 갱신한다 */\nvoid UpdateHeight(TreeNode node) {\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node.height = Math.Max(Height(node.left), Height(node.right)) + 1;\n}\n</code></pre> avl_tree.go<pre><code>/* 노드 높이 반환 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    if node != nil {\n        return node.Height\n    }\n    return -1\n}\n\n/* 노드 높이를 갱신한다 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\n    lh := t.height(node.Left)\n    rh := t.height(node.Right)\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    if lh &gt; rh {\n        node.Height = lh + 1\n    } else {\n        node.Height = rh + 1\n    }\n}\n</code></pre> avl_tree.swift<pre><code>/* 노드 높이 반환 */\nfunc height(node: TreeNode?) -&gt; Int {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    node?.height ?? -1\n}\n\n/* 노드 높이를 갱신한다 */\nfunc updateHeight(node: TreeNode?) {\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n</code></pre> avl_tree.js<pre><code>/* 노드 높이 반환 */\nheight(node) {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    return node === null ? -1 : node.height;\n}\n\n/* 노드 높이를 갱신한다 */\n#updateHeight(node) {\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.ts<pre><code>/* 노드 높이 반환 */\nheight(node: TreeNode): number {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    return node === null ? -1 : node.height;\n}\n\n/* 노드 높이를 갱신한다 */\nupdateHeight(node: TreeNode): void {\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.dart<pre><code>/* 노드 높이 반환 */\nint height(TreeNode? node) {\n  // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n  return node == null ? -1 : node.height;\n}\n\n/* 노드 높이를 갱신한다 */\nvoid updateHeight(TreeNode? node) {\n  // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n  node!.height = max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.rs<pre><code>/* 노드 높이 반환 */\nfn height(node: OptionTreeNodeRc) -&gt; i32 {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    match node {\n        Some(node) =&gt; node.borrow().height,\n        None =&gt; -1,\n    }\n}\n\n/* 노드 높이를 갱신한다 */\nfn update_height(node: OptionTreeNodeRc) {\n    if let Some(node) = node {\n        let left = node.borrow().left.clone();\n        let right = node.borrow().right.clone();\n        // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n        node.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* 노드 높이 반환 */\nint height(TreeNode *node) {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    if (node != NULL) {\n        return node-&gt;height;\n    }\n    return -1;\n}\n\n/* 노드 높이를 갱신한다 */\nvoid updateHeight(TreeNode *node) {\n    int lh = height(node-&gt;left);\n    int rh = height(node-&gt;right);\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    if (lh &gt; rh) {\n        node-&gt;height = lh + 1;\n    } else {\n        node-&gt;height = rh + 1;\n    }\n}\n</code></pre> avl_tree.kt<pre><code>/* 노드 높이 반환 */\nfun height(node: TreeNode?): Int {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    return node?.height ?: -1\n}\n\n/* 노드 높이를 갱신한다 */\nfun updateHeight(node: TreeNode?) {\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node?.height = max(height(node?.left), height(node?.right)) + 1\n}\n</code></pre> avl_tree.rb<pre><code>### 노드 높이 반환 ###\ndef height(node)\n  # 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n  return node.height unless node.nil?\n\n  -1\nend\n\n### 노드 높이를 갱신한다 ###\ndef update_height(node)\n  # 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n  node.height = [height(node.left), height(node.right)].max + 1\nend\n</code></pre>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#2","level":3,"title":"2.   노드의 균형 인수","text":"<p>노드의 균형 인수(balance factor)는 왼쪽 서브트리 높이에서 오른쪽 서브트리 높이를 뺀 값이며, 빈 노드의 균형 인수는 \\(0\\)으로 정한다. 이후 편리하게 쓰도록 균형 인수를 구하는 기능도 함수로 만든다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def balance_factor(self, node: TreeNode | None) -&gt; int:\n    \"\"\"균형 인수 반환\"\"\"\n    # 빈 노드의 균형 인수는 0이다\n    if node is None:\n        return 0\n    # 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return self.height(node.left) - self.height(node.right)\n</code></pre> avl_tree.cpp<pre><code>/* 균형 인수 반환 */\nint balanceFactor(TreeNode *node) {\n    // 빈 노드의 균형 인수는 0이다\n    if (node == nullptr)\n        return 0;\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.java<pre><code>/* 균형 인수 반환 */\nint balanceFactor(TreeNode node) {\n    // 빈 노드의 균형 인수는 0이다\n    if (node == null)\n        return 0;\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.cs<pre><code>/* 균형 인수 반환 */\nint BalanceFactor(TreeNode? node) {\n    // 빈 노드의 균형 인수는 0이다\n    if (node == null) return 0;\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return Height(node.left) - Height(node.right);\n}\n</code></pre> avl_tree.go<pre><code>/* 균형 인수 반환 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n    // 빈 노드의 균형 인수는 0이다\n    if node == nil {\n        return 0\n    }\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return t.height(node.Left) - t.height(node.Right)\n}\n</code></pre> avl_tree.swift<pre><code>/* 균형 인수 반환 */\nfunc balanceFactor(node: TreeNode?) -&gt; Int {\n    // 빈 노드의 균형 인수는 0이다\n    guard let node = node else { return 0 }\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return height(node: node.left) - height(node: node.right)\n}\n</code></pre> avl_tree.js<pre><code>/* 균형 인수 반환 */\nbalanceFactor(node) {\n    // 빈 노드의 균형 인수는 0이다\n    if (node === null) return 0;\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.ts<pre><code>/* 균형 인수 반환 */\nbalanceFactor(node: TreeNode): number {\n    // 빈 노드의 균형 인수는 0이다\n    if (node === null) return 0;\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.dart<pre><code>/* 균형 인수 반환 */\nint balanceFactor(TreeNode? node) {\n  // 빈 노드의 균형 인수는 0이다\n  if (node == null) return 0;\n  // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n  return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.rs<pre><code>/* 균형 인수 반환 */\nfn balance_factor(node: OptionTreeNodeRc) -&gt; i32 {\n    match node {\n        // 빈 노드의 균형 인수는 0이다\n        None =&gt; 0,\n        // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n        Some(node) =&gt; {\n            Self::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())\n        }\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* 균형 인수 반환 */\nint balanceFactor(TreeNode *node) {\n    // 빈 노드의 균형 인수는 0이다\n    if (node == NULL) {\n        return 0;\n    }\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.kt<pre><code>/* 균형 인수 반환 */\nfun balanceFactor(node: TreeNode?): Int {\n    // 빈 노드의 균형 인수는 0이다\n    if (node == null) return 0\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return height(node.left) - height(node.right)\n}\n</code></pre> avl_tree.rb<pre><code>### 균형 인수 반환 ###\ndef balance_factor(node)\n  # 빈 노드의 균형 인수는 0이다\n  return 0 if node.nil?\n\n  # 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n  height(node.left) - height(node.right)\nend\n</code></pre> <p>참고</p> <p>균형 인수를 \\(f\\)라고 하면 AVL 트리의 모든 노드는 \\(-1 \\le f \\le 1\\)을 만족한다.</p>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#752-avl","level":2,"title":"7.5.2   AVL 트리의 회전","text":"<p>AVL 트리의 특징은 '회전' 연산이다. 이진 트리의 중위 순회 결과를 바꾸지 않으면서 불균형 노드의 균형을 회복한다. 즉, 회전은 '이진 탐색 트리'의 성질을 유지하면서 트리를 다시 '균형 이진 트리'로 만든다.</p> <p>균형 인수의 절댓값이 \\(&gt; 1\\)인 노드를 '불균형 노드'라고 한다. 불균형 상황에 따라 오른쪽 회전, 왼쪽 회전, 오른쪽 후 왼쪽 회전, 왼쪽 후 오른쪽 회전의 네 종류로 나뉜다. 각각 자세히 알아보자.</p>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#1_1","level":3,"title":"1.   오른쪽 회전","text":"<p>아래 그림(7-26)에서 노드 아래의 수는 균형 인수다. 아래에서 위로 볼 때 첫 불균형 노드는 '노드 3'이다. 이 노드를 루트로 하는 서브트리에 주목하여 불균형 노드를 <code>node</code>, 왼쪽 자식을 <code>child</code>라고 하고 '오른쪽 회전'을 수행한다. 회전 후 서브트리는 균형을 되찾으며 이진 탐색 트리의 성질도 유지한다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> 그림 7-26   오른쪽 회전 단계 </p> <p>아래 그림(7-27)처럼 <code>child</code>에 오른쪽 자식 <code>grand_child</code>가 있으면, 오른쪽 회전 중 <code>grand_child</code>를 <code>node</code>의 왼쪽 자식으로 연결하는 단계를 추가해야 한다.</p> <p></p> <p> 그림 7-27   grand_child가 있는 오른쪽 회전 </p> <p>'오른쪽으로 회전한다'는 시각적인 표현이며, 실제로는 노드 포인터를 수정하여 구현한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def right_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"오른쪽 회전 연산\"\"\"\n    child = node.left\n    grand_child = child.right\n    # child를 중심으로 node를 오른쪽으로 회전한다\n    child.right = node\n    node.left = grand_child\n    # 노드 높이를 갱신한다\n    self.update_height(node)\n    self.update_height(child)\n    # 회전한 서브트리의 루트 노드를 반환한다\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* 오른쪽 회전 연산 */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;left;\n    TreeNode *grandChild = child-&gt;right;\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* 오른쪽 회전 연산 */\nTreeNode rightRotate(TreeNode node) {\n    TreeNode child = node.left;\n    TreeNode grandChild = child.right;\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child.right = node;\n    node.left = grandChild;\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* 오른쪽 회전 연산 */\nTreeNode? RightRotate(TreeNode? node) {\n    TreeNode? child = node?.left;\n    TreeNode? grandChild = child?.right;\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child.right = node;\n    node.left = grandChild;\n    // 노드 높이를 갱신한다\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* 오른쪽 회전 연산 */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\n    child := node.Left\n    grandChild := child.Right\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child.Right = node\n    node.Left = grandChild\n    // 노드 높이를 갱신한다\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* 오른쪽 회전 연산 */\nfunc rightRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.left\n    let grandChild = child?.right\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child?.right = node\n    node?.left = grandChild\n    // 노드 높이를 갱신한다\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* 오른쪽 회전 연산 */\n#rightRotate(node) {\n    const child = node.left;\n    const grandChild = child.right;\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child.right = node;\n    node.left = grandChild;\n    // 노드 높이를 갱신한다\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* 오른쪽 회전 연산 */\nrightRotate(node: TreeNode): TreeNode {\n    const child = node.left;\n    const grandChild = child.right;\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child.right = node;\n    node.left = grandChild;\n    // 노드 높이를 갱신한다\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* 오른쪽 회전 연산 */\nTreeNode? rightRotate(TreeNode? node) {\n  TreeNode? child = node!.left;\n  TreeNode? grandChild = child!.right;\n  // child를 중심으로 node를 오른쪽으로 회전한다\n  child.right = node;\n  node.left = grandChild;\n  // 노드 높이를 갱신한다\n  updateHeight(node);\n  updateHeight(child);\n  // 회전한 서브트리의 루트 노드를 반환한다\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* 오른쪽 회전 연산 */\nfn right_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().left.clone().unwrap();\n            let grand_child = child.borrow().right.clone();\n            // child를 중심으로 node를 오른쪽으로 회전한다\n            child.borrow_mut().right = Some(node.clone());\n            node.borrow_mut().left = grand_child;\n            // 노드 높이를 갱신한다\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // 회전한 서브트리의 루트 노드를 반환한다\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* 오른쪽 회전 연산 */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;left;\n    grandChild = child-&gt;right;\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* 오른쪽 회전 연산 */\nfun rightRotate(node: TreeNode?): TreeNode {\n    val child = node!!.left\n    val grandChild = child!!.right\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child.right = node\n    node.left = grandChild\n    // 노드 높이를 갱신한다\n    updateHeight(node)\n    updateHeight(child)\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>### 오른쪽 회전 연산 ###\ndef right_rotate(node)\n  child = node.left\n  grand_child = child.right\n  # child를 중심으로 node를 오른쪽으로 회전한다\n  child.right = node\n  node.left = grand_child\n  # 노드 높이를 갱신한다\n  update_height(node)\n  update_height(child)\n  # 회전한 서브트리의 루트 노드를 반환한다\n  child\nend\n</code></pre>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#2_1","level":3,"title":"2.   왼쪽 회전","text":"<p>앞의 불균형 이진 트리를 '거울에 비춘' 경우에는 아래 그림(7-28)과 같이 '왼쪽 회전'이 필요하다.</p> <p></p> <p> 그림 7-28   왼쪽 회전 </p> <p>마찬가지로 아래 그림(7-29)처럼 <code>child</code>에 왼쪽 자식 <code>grand_child</code>가 있으면, 왼쪽 회전 중 <code>grand_child</code>를 <code>node</code>의 오른쪽 자식으로 연결하는 단계를 추가해야 한다.</p> <p></p> <p> 그림 7-29   grand_child가 있는 왼쪽 회전 </p> <p>오른쪽 회전과 왼쪽 회전은 논리적으로 거울 대칭이며, 각각 해결하는 불균형 상황도 대칭이다. 따라서 오른쪽 회전 코드의 모든 <code>left</code>를 <code>right</code>로, 모든 <code>right</code>를 <code>left</code>로 바꾸면 왼쪽 회전 코드를 얻는다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def left_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"왼쪽 회전 연산\"\"\"\n    child = node.right\n    grand_child = child.left\n    # child를 중심으로 node를 왼쪽으로 회전한다\n    child.left = node\n    node.right = grand_child\n    # 노드 높이를 갱신한다\n    self.update_height(node)\n    self.update_height(child)\n    # 회전한 서브트리의 루트 노드를 반환한다\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* 왼쪽 회전 연산 */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;right;\n    TreeNode *grandChild = child-&gt;left;\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* 왼쪽 회전 연산 */\nTreeNode leftRotate(TreeNode node) {\n    TreeNode child = node.right;\n    TreeNode grandChild = child.left;\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child.left = node;\n    node.right = grandChild;\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* 왼쪽 회전 연산 */\nTreeNode? LeftRotate(TreeNode? node) {\n    TreeNode? child = node?.right;\n    TreeNode? grandChild = child?.left;\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child.left = node;\n    node.right = grandChild;\n    // 노드 높이를 갱신한다\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* 왼쪽 회전 연산 */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\n    child := node.Right\n    grandChild := child.Left\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child.Left = node\n    node.Right = grandChild\n    // 노드 높이를 갱신한다\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* 왼쪽 회전 연산 */\nfunc leftRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.right\n    let grandChild = child?.left\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child?.left = node\n    node?.right = grandChild\n    // 노드 높이를 갱신한다\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* 왼쪽 회전 연산 */\n#leftRotate(node) {\n    const child = node.right;\n    const grandChild = child.left;\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child.left = node;\n    node.right = grandChild;\n    // 노드 높이를 갱신한다\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* 왼쪽 회전 연산 */\nleftRotate(node: TreeNode): TreeNode {\n    const child = node.right;\n    const grandChild = child.left;\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child.left = node;\n    node.right = grandChild;\n    // 노드 높이를 갱신한다\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* 왼쪽 회전 연산 */\nTreeNode? leftRotate(TreeNode? node) {\n  TreeNode? child = node!.right;\n  TreeNode? grandChild = child!.left;\n  // child를 중심으로 node를 왼쪽으로 회전한다\n  child.left = node;\n  node.right = grandChild;\n  // 노드 높이를 갱신한다\n  updateHeight(node);\n  updateHeight(child);\n  // 회전한 서브트리의 루트 노드를 반환한다\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* 왼쪽 회전 연산 */\nfn left_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().right.clone().unwrap();\n            let grand_child = child.borrow().left.clone();\n            // child를 중심으로 node를 왼쪽으로 회전한다\n            child.borrow_mut().left = Some(node.clone());\n            node.borrow_mut().right = grand_child;\n            // 노드 높이를 갱신한다\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // 회전한 서브트리의 루트 노드를 반환한다\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* 왼쪽 회전 연산 */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;right;\n    grandChild = child-&gt;left;\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* 왼쪽 회전 연산 */\nfun leftRotate(node: TreeNode?): TreeNode {\n    val child = node!!.right\n    val grandChild = child!!.left\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child.left = node\n    node.right = grandChild\n    // 노드 높이를 갱신한다\n    updateHeight(node)\n    updateHeight(child)\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>### 왼쪽 회전 연산 ###\ndef left_rotate(node)\n  child = node.right\n  grand_child = child.left\n  # child를 중심으로 node를 왼쪽으로 회전한다\n  child.left = node\n  node.right = grand_child\n  # 노드 높이를 갱신한다\n  update_height(node)\n  update_height(child)\n  # 회전한 서브트리의 루트 노드를 반환한다\n  child\nend\n</code></pre>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#3","level":3,"title":"3.   왼쪽 회전 후 오른쪽 회전","text":"<p>아래 그림(7-30)의 불균형 노드 3은 왼쪽 회전이나 오른쪽 회전 한 번만으로 균형을 회복할 수 없다. 먼저 <code>child</code>를 '왼쪽 회전'한 뒤 <code>node</code>를 '오른쪽 회전'해야 한다.</p> <p></p> <p> 그림 7-30   왼쪽 회전 후 오른쪽 회전 </p>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#4","level":3,"title":"4.   오른쪽 회전 후 왼쪽 회전","text":"<p>아래 그림(7-31)처럼 앞 상황의 거울 대칭인 경우에는 먼저 <code>child</code>를 '오른쪽 회전'한 뒤 <code>node</code>를 '왼쪽 회전'해야 한다.</p> <p></p> <p> 그림 7-31   오른쪽 회전 후 왼쪽 회전 </p>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#5","level":3,"title":"5.   회전 선택","text":"<p>아래 그림(7-32)의 네 불균형 상황은 앞의 예시들과 각각 대응한다. 순서대로 오른쪽 회전, 왼쪽 후 오른쪽 회전, 오른쪽 후 왼쪽 회전, 왼쪽 회전이 필요하다.</p> <p></p> <p> 그림 7-32   AVL 트리의 네 가지 회전 상황 </p> <p>아래 표(7-3)처럼 불균형 노드의 균형 인수와 더 높은 쪽 자식의 균형 인수 부호를 확인하여 어느 상황에 해당하는지 판단한다.</p> <p> 표 7-3   네 가지 회전의 선택 조건 </p> 불균형 노드의 균형 인수 자식 노드의 균형 인수 사용할 회전 \\(&gt; 1\\) (왼쪽으로 치우친 트리) \\(\\geq 0\\) 오른쪽 회전 \\(&gt; 1\\) (왼쪽으로 치우친 트리) \\(&lt;0\\) 왼쪽 회전 후 오른쪽 회전 \\(&lt; -1\\) (오른쪽으로 치우친 트리) \\(\\leq 0\\) 왼쪽 회전 \\(&lt; -1\\) (오른쪽으로 치우친 트리) \\(&gt;0\\) 오른쪽 회전 후 왼쪽 회전 <p>편리하게 사용하도록 회전 연산을 함수 하나로 묶는다. 이 함수를 사용하면 여러 불균형 상황에 맞게 회전하여 노드의 균형을 복구할 수 있다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"회전 연산을 수행하여 해당 서브트리의 균형 복원\"\"\"\n    # 노드 node의 균형 인수를 구한다\n    balance_factor = self.balance_factor(node)\n    # 왼쪽으로 치우친 트리\n    if balance_factor &gt; 1:\n        if self.balance_factor(node.left) &gt;= 0:\n            # 오른쪽 회전\n            return self.right_rotate(node)\n        else:\n            # 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node.left = self.left_rotate(node.left)\n            return self.right_rotate(node)\n    # 오른쪽으로 치우친 트리\n    elif balance_factor &lt; -1:\n        if self.balance_factor(node.right) &lt;= 0:\n            # 왼쪽 회전\n            return self.left_rotate(node)\n        else:\n            # 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node.right = self.right_rotate(node.right)\n            return self.left_rotate(node)\n    # 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node\n</code></pre> avl_tree.cpp<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nTreeNode *rotate(TreeNode *node) {\n    // 노드 node의 균형 인수를 구한다\n    int _balanceFactor = balanceFactor(node);\n    // 왼쪽으로 치우친 트리\n    if (_balanceFactor &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // 오른쪽 회전\n            return rightRotate(node);\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if (_balanceFactor &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // 왼쪽 회전\n            return leftRotate(node);\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nTreeNode rotate(TreeNode node) {\n    // 노드 node의 균형 인수를 구한다\n    int balanceFactor = balanceFactor(node);\n    // 왼쪽으로 치우친 트리\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // 오른쪽 회전\n            return rightRotate(node);\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node.left = leftRotate(node.left);\n            return rightRotate(node);\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // 왼쪽 회전\n            return leftRotate(node);\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node.right = rightRotate(node.right);\n            return leftRotate(node);\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nTreeNode? Rotate(TreeNode? node) {\n    // 노드 node의 균형 인수를 구한다\n    int balanceFactorInt = BalanceFactor(node);\n    // 왼쪽으로 치우친 트리\n    if (balanceFactorInt &gt; 1) {\n        if (BalanceFactor(node?.left) &gt;= 0) {\n            // 오른쪽 회전\n            return RightRotate(node);\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node!.left = LeftRotate(node!.left);\n            return RightRotate(node);\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if (balanceFactorInt &lt; -1) {\n        if (BalanceFactor(node?.right) &lt;= 0) {\n            // 왼쪽 회전\n            return LeftRotate(node);\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node!.right = RightRotate(node!.right);\n            return LeftRotate(node);\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n    // 노드 node의 균형 인수를 구한다\n    // Go에서는 짧은 변수명을 권장한다. 여기서 bf는 t.balanceFactor를 나타낸다\n    bf := t.balanceFactor(node)\n    // 왼쪽으로 치우친 트리\n    if bf &gt; 1 {\n        if t.balanceFactor(node.Left) &gt;= 0 {\n            // 오른쪽 회전\n            return t.rightRotate(node)\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node.Left = t.leftRotate(node.Left)\n            return t.rightRotate(node)\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if bf &lt; -1 {\n        if t.balanceFactor(node.Right) &lt;= 0 {\n            // 왼쪽 회전\n            return t.leftRotate(node)\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node.Right = t.rightRotate(node.Right)\n            return t.leftRotate(node)\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nfunc rotate(node: TreeNode?) -&gt; TreeNode? {\n    // 노드 node의 균형 인수를 구한다\n    let balanceFactor = balanceFactor(node: node)\n    // 왼쪽으로 치우친 트리\n    if balanceFactor &gt; 1 {\n        if self.balanceFactor(node: node?.left) &gt;= 0 {\n            // 오른쪽 회전\n            return rightRotate(node: node)\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node?.left = leftRotate(node: node?.left)\n            return rightRotate(node: node)\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if balanceFactor &lt; -1 {\n        if self.balanceFactor(node: node?.right) &lt;= 0 {\n            // 왼쪽 회전\n            return leftRotate(node: node)\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node?.right = rightRotate(node: node?.right)\n            return leftRotate(node: node)\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\n#rotate(node) {\n    // 노드 node의 균형 인수를 구한다\n    const balanceFactor = this.balanceFactor(node);\n    // 왼쪽으로 치우친 트리\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // 오른쪽 회전\n            return this.#rightRotate(node);\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node.left = this.#leftRotate(node.left);\n            return this.#rightRotate(node);\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // 왼쪽 회전\n            return this.#leftRotate(node);\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node.right = this.#rightRotate(node.right);\n            return this.#leftRotate(node);\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nrotate(node: TreeNode): TreeNode {\n    // 노드 node의 균형 인수를 구한다\n    const balanceFactor = this.balanceFactor(node);\n    // 왼쪽으로 치우친 트리\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // 오른쪽 회전\n            return this.rightRotate(node);\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node.left = this.leftRotate(node.left);\n            return this.rightRotate(node);\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // 왼쪽 회전\n            return this.leftRotate(node);\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node.right = this.rightRotate(node.right);\n            return this.leftRotate(node);\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nTreeNode? rotate(TreeNode? node) {\n  // 노드 node의 균형 인수를 구한다\n  int factor = balanceFactor(node);\n  // 왼쪽으로 치우친 트리\n  if (factor &gt; 1) {\n    if (balanceFactor(node!.left) &gt;= 0) {\n      // 오른쪽 회전\n      return rightRotate(node);\n    } else {\n      // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n      node.left = leftRotate(node.left);\n      return rightRotate(node);\n    }\n  }\n  // 오른쪽으로 치우친 트리\n  if (factor &lt; -1) {\n    if (balanceFactor(node!.right) &lt;= 0) {\n      // 왼쪽 회전\n      return leftRotate(node);\n    } else {\n      // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n      node.right = rightRotate(node.right);\n      return leftRotate(node);\n    }\n  }\n  // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nfn rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    // 노드 node의 균형 인수를 구한다\n    let balance_factor = Self::balance_factor(node.clone());\n    // 왼쪽으로 치우친 트리\n    if balance_factor &gt; 1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().left.clone()) &gt;= 0 {\n            // 오른쪽 회전\n            Self::right_rotate(Some(node))\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            let left = node.borrow().left.clone();\n            node.borrow_mut().left = Self::left_rotate(left);\n            Self::right_rotate(Some(node))\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    else if balance_factor &lt; -1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().right.clone()) &lt;= 0 {\n            // 왼쪽 회전\n            Self::left_rotate(Some(node))\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            let right = node.borrow().right.clone();\n            node.borrow_mut().right = Self::right_rotate(right);\n            Self::left_rotate(Some(node))\n        }\n    } else {\n        // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n        node\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nTreeNode *rotate(TreeNode *node) {\n    // 노드 node의 균형 인수를 구한다\n    int bf = balanceFactor(node);\n    // 왼쪽으로 치우친 트리\n    if (bf &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // 오른쪽 회전\n            return rightRotate(node);\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if (bf &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // 왼쪽 회전\n            return leftRotate(node);\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nfun rotate(node: TreeNode): TreeNode {\n    // 노드 node의 균형 인수를 구한다\n    val balanceFactor = balanceFactor(node)\n    // 왼쪽으로 치우친 트리\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // 오른쪽 회전\n            return rightRotate(node)\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node.left = leftRotate(node.left)\n            return rightRotate(node)\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // 왼쪽 회전\n            return leftRotate(node)\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node.right = rightRotate(node.right)\n            return leftRotate(node)\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### 회전 연산을 수행하여 해당 서브트리의 균형 복원 ###\ndef rotate(node)\n  # 노드 node의 균형 인수를 구한다\n  balance_factor = balance_factor(node)\n  # 왼쪽으로 치우친 트리\n  if balance_factor &gt; 1\n    if balance_factor(node.left) &gt;= 0\n      # 오른쪽 회전\n      return right_rotate(node)\n    else\n      # 왼쪽으로 회전한 후 오른쪽으로 회전한다\n      node.left = left_rotate(node.left)\n      return right_rotate(node)\n    end\n  # 오른쪽으로 치우친 트리\n  elsif balance_factor &lt; -1\n    if balance_factor(node.right) &lt;= 0\n      # 왼쪽 회전\n      return left_rotate(node)\n    else\n      # 오른쪽으로 회전한 후 왼쪽으로 회전한다\n      node.right = right_rotate(node.right)\n      return left_rotate(node)\n    end\n  end\n  # 균형 트리이므로 회전할 필요 없이 바로 반환한다\n  node\nend\n</code></pre>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#753-avl","level":2,"title":"7.5.3   AVL 트리의 주요 연산","text":"","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#1_2","level":3,"title":"1.   노드 삽입","text":"<p>AVL 트리의 삽입은 기본적으로 이진 탐색 트리와 비슷하다. 다만 삽입한 노드에서 루트까지의 경로에 불균형 노드가 여러 개 생길 수 있다는 차이가 있다. 따라서 삽입한 노드부터 아래에서 위로 회전하여 모든 불균형 노드의 균형을 회복해야 한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def insert(self, val):\n    \"\"\"노드 삽입\"\"\"\n    self._root = self.insert_helper(self._root, val)\n\ndef insert_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode:\n    \"\"\"노드의 재귀적 삽입(보조 메서드)\"\"\"\n    if node is None:\n        return TreeNode(val)\n    # 1. 삽입 위치를 찾아 노드를 삽입한다\n    if val &lt; node.val:\n        node.left = self.insert_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.insert_helper(node.right, val)\n    else:\n        # 중복 노드는 삽입하지 않고 바로 반환한다\n        return node\n    # 노드 높이를 갱신한다\n    self.update_height(node)\n    # 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* 노드 삽입 */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return new TreeNode(val);\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    else\n        return node;    // 중복 노드는 삽입하지 않고 바로 반환한다\n    updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* 노드 삽입 */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\nTreeNode insertHelper(TreeNode node, int val) {\n    if (node == null)\n        return new TreeNode(val);\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if (val &lt; node.val)\n        node.left = insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = insertHelper(node.right, val);\n    else\n        return node; // 중복 노드는 삽입하지 않고 바로 반환한다\n    updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* 노드 삽입 */\nvoid Insert(int val) {\n    root = InsertHelper(root, val);\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\nTreeNode? InsertHelper(TreeNode? node, int val) {\n    if (node == null) return new TreeNode(val);\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if (val &lt; node.val)\n        node.left = InsertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = InsertHelper(node.right, val);\n    else\n        return node;     // 중복 노드는 삽입하지 않고 바로 반환한다\n    UpdateHeight(node);  // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = Rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* 노드 삽입 */\nfunc (t *aVLTree) insert(val int) {\n    t.root = t.insertHelper(t.root, val)\n}\n\n/* 노드의 재귀적 삽입(보조 함수) */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return NewTreeNode(val)\n    }\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if val &lt; node.Val.(int) {\n        node.Left = t.insertHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.insertHelper(node.Right, val)\n    } else {\n        // 중복 노드는 삽입하지 않고 바로 반환한다\n        return node\n    }\n    // 노드 높이를 갱신한다\n    t.updateHeight(node)\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = t.rotate(node)\n    // 서브트리의 루트 노드를 반환한다\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* 노드 삽입 */\nfunc insert(val: Int) {\n    root = insertHelper(node: root, val: val)\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\nfunc insertHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return TreeNode(x: val)\n    }\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if val &lt; node!.val {\n        node?.left = insertHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = insertHelper(node: node?.right, val: val)\n    } else {\n        return node // 중복 노드는 삽입하지 않고 바로 반환한다\n    }\n    updateHeight(node: node) // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node: node)\n    // 서브트리의 루트 노드를 반환한다\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* 노드 삽입 */\ninsert(val) {\n    this.root = this.#insertHelper(this.root, val);\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\n#insertHelper(node, val) {\n    if (node === null) return new TreeNode(val);\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if (val &lt; node.val) node.left = this.#insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#insertHelper(node.right, val);\n    else return node; // 중복 노드는 삽입하지 않고 바로 반환한다\n    this.#updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = this.#rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* 노드 삽입 */\ninsert(val: number): void {\n    this.root = this.insertHelper(this.root, val);\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\ninsertHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return new TreeNode(val);\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if (val &lt; node.val) {\n        node.left = this.insertHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.insertHelper(node.right, val);\n    } else {\n        return node; // 중복 노드는 삽입하지 않고 바로 반환한다\n    }\n    this.updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = this.rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* 노드 삽입 */\nvoid insert(int val) {\n  root = insertHelper(root, val);\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\nTreeNode? insertHelper(TreeNode? node, int val) {\n  if (node == null) return TreeNode(val);\n  /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n  if (val &lt; node.val)\n    node.left = insertHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = insertHelper(node.right, val);\n  else\n    return node; // 중복 노드는 삽입하지 않고 바로 반환한다\n  updateHeight(node); // 노드 높이를 갱신한다\n  /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n  node = rotate(node);\n  // 서브트리의 루트 노드를 반환한다\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* 노드 삽입 */\nfn insert(&amp;mut self, val: i32) {\n    self.root = Self::insert_helper(self.root.clone(), val);\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\nfn insert_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n            match {\n                let node_val = node.borrow().val;\n                node_val\n            }\n            .cmp(&amp;val)\n            {\n                Ordering::Greater =&gt; {\n                    let left = node.borrow().left.clone();\n                    node.borrow_mut().left = Self::insert_helper(left, val);\n                }\n                Ordering::Less =&gt; {\n                    let right = node.borrow().right.clone();\n                    node.borrow_mut().right = Self::insert_helper(right, val);\n                }\n                Ordering::Equal =&gt; {\n                    return Some(node); // 중복 노드는 삽입하지 않고 바로 반환한다\n                }\n            }\n            Self::update_height(Some(node.clone())); // 노드 높이를 갱신한다\n\n            /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n            node = Self::rotate(Some(node)).unwrap();\n            // 서브트리의 루트 노드를 반환한다\n            Some(node)\n        }\n        None =&gt; Some(TreeNode::new(val)),\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* 노드 삽입 */\nvoid insert(AVLTree *tree, int val) {\n    tree-&gt;root = insertHelper(tree-&gt;root, val);\n}\n\n/* 노드의 재귀적 삽입(보조 함수) */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == NULL) {\n        return newTreeNode(val);\n    }\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    } else {\n        // 중복 노드는 삽입하지 않고 바로 반환한다\n        return node;\n    }\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* 노드 삽입 */\nfun insert(_val: Int) {\n    root = insertHelper(root, _val)\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\nfun insertHelper(n: TreeNode?, _val: Int): TreeNode {\n    if (n == null)\n        return TreeNode(_val)\n    var node = n\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if (_val &lt; node._val)\n        node.left = insertHelper(node.left, _val)\n    else if (_val &gt; node._val)\n        node.right = insertHelper(node.right, _val)\n    else\n        return node // 중복 노드는 삽입하지 않고 바로 반환한다\n    updateHeight(node) // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node)\n    // 서브트리의 루트 노드를 반환한다\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### 노드 삽입 ###\ndef insert(val)\n  @root = insert_helper(@root, val)\nend\n\n### 노드의 재귀적 삽입(보조 메서드)###\ndef insert_helper(node, val)\n  return TreeNode.new(val) if node.nil?\n  # 1. 삽입 위치를 찾아 노드를 삽입한다\n  if val &lt; node.val\n    node.left = insert_helper(node.left, val)\n  elsif val &gt; node.val\n    node.right = insert_helper(node.right, val)\n  else\n    # 중복 노드는 삽입하지 않고 바로 반환한다\n    return node\n  end\n  # 노드 높이를 갱신한다\n  update_height(node)\n  # 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다\n  rotate(node)\nend\n</code></pre>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#2_2","level":3,"title":"2.   노드 삭제","text":"<p>마찬가지로 이진 탐색 트리의 삭제 방법에 더하여 아래에서 위로 회전하며 모든 불균형 노드의 균형을 회복한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def remove(self, val: int):\n    \"\"\"노드 삭제\"\"\"\n    self._root = self.remove_helper(self._root, val)\n\ndef remove_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode | None:\n    \"\"\"노드의 재귀적 삭제(보조 메서드)\"\"\"\n    if node is None:\n        return None\n    # 1. 노드를 찾아 삭제한다\n    if val &lt; node.val:\n        node.left = self.remove_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.remove_helper(node.right, val)\n    else:\n        if node.left is None or node.right is None:\n            child = node.left or node.right\n            # 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if child is None:\n                return None\n            # 자식 노드 수 = 1이면 node를 바로 삭제한다\n            else:\n                node = child\n        else:\n            # 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            temp = node.right\n            while temp.left is not None:\n                temp = temp.left\n            node.right = self.remove_helper(node.right, temp.val)\n            node.val = temp.val\n    # 노드 높이를 갱신한다\n    self.update_height(node)\n    # 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* 노드 삭제 */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return nullptr;\n    /* 1. 노드를 찾아 삭제한다 */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    else {\n        if (node-&gt;left == nullptr || node-&gt;right == nullptr) {\n            TreeNode *child = node-&gt;left != nullptr ? node-&gt;left : node-&gt;right;\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if (child == nullptr) {\n                delete node;\n                return nullptr;\n            }\n            // 자식 노드 수 = 1이면 node를 바로 삭제한다\n            else {\n                delete node;\n                node = child;\n            }\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != nullptr) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* 노드 삭제 */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nTreeNode removeHelper(TreeNode node, int val) {\n    if (node == null)\n        return null;\n    /* 1. 노드를 찾아 삭제한다 */\n    if (val &lt; node.val)\n        node.left = removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = removeHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode child = node.left != null ? node.left : node.right;\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if (child == null)\n                return null;\n            // 자식 노드 수 = 1이면 node를 바로 삭제한다\n            else\n                node = child;\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            TreeNode temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* 노드 삭제 */\nvoid Remove(int val) {\n    root = RemoveHelper(root, val);\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nTreeNode? RemoveHelper(TreeNode? node, int val) {\n    if (node == null) return null;\n    /* 1. 노드를 찾아 삭제한다 */\n    if (val &lt; node.val)\n        node.left = RemoveHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = RemoveHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode? child = node.left ?? node.right;\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if (child == null)\n                return null;\n            // 자식 노드 수 = 1이면 node를 바로 삭제한다\n            else\n                node = child;\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            TreeNode? temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = RemoveHelper(node.right, temp.val!.Value);\n            node.val = temp.val;\n        }\n    }\n    UpdateHeight(node);  // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = Rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* 노드 삭제 */\nfunc (t *aVLTree) remove(val int) {\n    t.root = t.removeHelper(t.root, val)\n}\n\n/* 노드의 재귀적 삭제(보조 함수) */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return nil\n    }\n    /* 1. 노드를 찾아 삭제한다 */\n    if val &lt; node.Val.(int) {\n        node.Left = t.removeHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.removeHelper(node.Right, val)\n    } else {\n        if node.Left == nil || node.Right == nil {\n            child := node.Left\n            if node.Right != nil {\n                child = node.Right\n            }\n            if child == nil {\n                // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n                return nil\n            } else {\n                // 자식 노드 수 = 1이면 node를 바로 삭제한다\n                node = child\n            }\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            temp := node.Right\n            for temp.Left != nil {\n                temp = temp.Left\n            }\n            node.Right = t.removeHelper(node.Right, temp.Val.(int))\n            node.Val = temp.Val\n        }\n    }\n    // 노드 높이를 갱신한다\n    t.updateHeight(node)\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = t.rotate(node)\n    // 서브트리의 루트 노드를 반환한다\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* 노드 삭제 */\nfunc remove(val: Int) {\n    root = removeHelper(node: root, val: val)\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nfunc removeHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return nil\n    }\n    /* 1. 노드를 찾아 삭제한다 */\n    if val &lt; node!.val {\n        node?.left = removeHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = removeHelper(node: node?.right, val: val)\n    } else {\n        if node?.left == nil || node?.right == nil {\n            let child = node?.left ?? node?.right\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if child == nil {\n                return nil\n            }\n            // 자식 노드 수 = 1이면 node를 바로 삭제한다\n            else {\n                node = child\n            }\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            var temp = node?.right\n            while temp?.left != nil {\n                temp = temp?.left\n            }\n            node?.right = removeHelper(node: node?.right, val: temp!.val)\n            node?.val = temp!.val\n        }\n    }\n    updateHeight(node: node) // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node: node)\n    // 서브트리의 루트 노드를 반환한다\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* 노드 삭제 */\nremove(val) {\n    this.root = this.#removeHelper(this.root, val);\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\n#removeHelper(node, val) {\n    if (node === null) return null;\n    /* 1. 노드를 찾아 삭제한다 */\n    if (val &lt; node.val) node.left = this.#removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#removeHelper(node.right, val);\n    else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if (child === null) return null;\n            // 자식 노드 수 = 1이면 node를 바로 삭제한다\n            else node = child;\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.#removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.#updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = this.#rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* 노드 삭제 */\nremove(val: number): void {\n    this.root = this.removeHelper(this.root, val);\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nremoveHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return null;\n    /* 1. 노드를 찾아 삭제한다 */\n    if (val &lt; node.val) {\n        node.left = this.removeHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.removeHelper(node.right, val);\n    } else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if (child === null) {\n                return null;\n            } else {\n                // 자식 노드 수 = 1이면 node를 바로 삭제한다\n                node = child;\n            }\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = this.rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* 노드 삭제 */\nvoid remove(int val) {\n  root = removeHelper(root, val);\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nTreeNode? removeHelper(TreeNode? node, int val) {\n  if (node == null) return null;\n  /* 1. 노드를 찾아 삭제한다 */\n  if (val &lt; node.val)\n    node.left = removeHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = removeHelper(node.right, val);\n  else {\n    if (node.left == null || node.right == null) {\n      TreeNode? child = node.left ?? node.right;\n      // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n      if (child == null)\n        return null;\n      // 자식 노드 수 = 1이면 node를 바로 삭제한다\n      else\n        node = child;\n    } else {\n      // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n      TreeNode? temp = node.right;\n      while (temp!.left != null) {\n        temp = temp.left;\n      }\n      node.right = removeHelper(node.right, temp.val);\n      node.val = temp.val;\n    }\n  }\n  updateHeight(node); // 노드 높이를 갱신한다\n  /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n  node = rotate(node);\n  // 서브트리의 루트 노드를 반환한다\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* 노드 삭제 */\nfn remove(&amp;self, val: i32) {\n    Self::remove_helper(self.root.clone(), val);\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nfn remove_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. 노드를 찾아 삭제한다 */\n            if val &lt; node.borrow().val {\n                let left = node.borrow().left.clone();\n                node.borrow_mut().left = Self::remove_helper(left, val);\n            } else if val &gt; node.borrow().val {\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, val);\n            } else if node.borrow().left.is_none() || node.borrow().right.is_none() {\n                let child = if node.borrow().left.is_some() {\n                    node.borrow().left.clone()\n                } else {\n                    node.borrow().right.clone()\n                };\n                match child {\n                    // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n                    None =&gt; {\n                        return None;\n                    }\n                    // 자식 노드 수 = 1이면 node를 바로 삭제한다\n                    Some(child) =&gt; node = child,\n                }\n            } else {\n                // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n                let mut temp = node.borrow().right.clone().unwrap();\n                loop {\n                    let temp_left = temp.borrow().left.clone();\n                    if temp_left.is_none() {\n                        break;\n                    }\n                    temp = temp_left.unwrap();\n                }\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);\n                node.borrow_mut().val = temp.borrow().val;\n            }\n            Self::update_height(Some(node.clone())); // 노드 높이를 갱신한다\n\n            /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n            node = Self::rotate(Some(node)).unwrap();\n            // 서브트리의 루트 노드를 반환한다\n            Some(node)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* 노드 삭제 */\n// stdio.h를 포함했으므로 여기서는 remove라는 이름을 사용할 수 없다\nvoid removeItem(AVLTree *tree, int val) {\n    TreeNode *root = removeHelper(tree-&gt;root, val);\n}\n\n/* 노드의 재귀적 삭제(보조 함수) */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    TreeNode *child, *grandChild;\n    if (node == NULL) {\n        return NULL;\n    }\n    /* 1. 노드를 찾아 삭제한다 */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    } else {\n        if (node-&gt;left == NULL || node-&gt;right == NULL) {\n            child = node-&gt;left;\n            if (node-&gt;right != NULL) {\n                child = node-&gt;right;\n            }\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if (child == NULL) {\n                return NULL;\n            } else {\n                // 자식 노드 수 = 1이면 node를 바로 삭제한다\n                node = child;\n            }\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != NULL) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* 노드 삭제 */\nfun remove(_val: Int) {\n    root = removeHelper(root, _val)\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nfun removeHelper(n: TreeNode?, _val: Int): TreeNode? {\n    var node = n ?: return null\n    /* 1. 노드를 찾아 삭제한다 */\n    if (_val &lt; node._val)\n        node.left = removeHelper(node.left, _val)\n    else if (_val &gt; node._val)\n        node.right = removeHelper(node.right, _val)\n    else {\n        if (node.left == null || node.right == null) {\n            val child = if (node.left != null)\n                node.left\n            else\n                node.right\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if (child == null)\n                return null\n            // 자식 노드 수 = 1이면 node를 바로 삭제한다\n            else\n                node = child\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            var temp = node.right\n            while (temp!!.left != null) {\n                temp = temp.left\n            }\n            node.right = removeHelper(node.right, temp._val)\n            node._val = temp._val\n        }\n    }\n    updateHeight(node) // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node)\n    // 서브트리의 루트 노드를 반환한다\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### 노드 삭제 ###\ndef remove(val)\n  @root = remove_helper(@root, val)\nend\n\n### 노드의 재귀적 삭제(보조 메서드)###\ndef remove_helper(node, val)\n  return if node.nil?\n  # 1. 노드를 찾아 삭제한다\n  if val &lt; node.val\n    node.left = remove_helper(node.left, val)\n  elsif val &gt; node.val\n    node.right = remove_helper(node.right, val)\n  else\n    if node.left.nil? || node.right.nil?\n      child = node.left || node.right\n      # 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n      return if child.nil?\n      # 자식 노드 수 = 1이면 node를 바로 삭제한다\n      node = child\n    else\n      # 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n      temp = node.right\n      while !temp.left.nil?\n        temp = temp.left\n      end\n      node.right = remove_helper(node.right, temp.val)\n      node.val = temp.val\n    end\n  end\n  # 노드 높이를 갱신한다\n  update_height(node)\n  # 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다\n  rotate(node)\nend\n</code></pre>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#3_1","level":3,"title":"3.   노드 탐색","text":"<p>AVL 트리의 노드 탐색은 이진 탐색 트리와 같으므로 다시 설명하지 않는다.</p>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#754-avl","level":2,"title":"7.5.4   AVL 트리의 대표적인 응용","text":"<ul> <li>대규모 데이터를 구성하고 저장하며, 탐색이 잦고 추가와 삭제가 드문 상황에 적합하다.</li> <li>데이터베이스의 인덱스 시스템을 구축하는 데 사용한다.</li> <li>레드-블랙 트리도 대표적인 균형 이진 탐색 트리다. AVL 트리보다 균형 조건이 느슨하여 삽입과 삭제에 필요한 회전 수가 적고, 노드 추가와 삭제의 평균 효율이 더 높다.</li> </ul>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/binary_search_tree/","level":1,"title":"7.4   이진 탐색 트리","text":"<p>아래 그림(7-16)의 이진 탐색 트리(binary search tree)는 다음 조건을 만족한다.</p> <ol> <li>루트에 대해 왼쪽 서브트리의 모든 노드 값 \\(&lt;\\) 루트 값 \\(&lt;\\) 오른쪽 서브트리의 모든 노드 값이다.</li> <li>임의의 노드의 왼쪽과 오른쪽 서브트리도 이진 탐색 트리로, 조건 <code>1.</code>을 만족한다.</li> </ol> <p></p> <p> 그림 7-16   이진 탐색 트리 </p>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#741","level":2,"title":"7.4.1   이진 탐색 트리의 연산","text":"<p>이진 탐색 트리를 <code>BinarySearchTree</code> 클래스로 묶고, 루트 노드를 가리키는 멤버 변수 <code>root</code>를 정의한다.</p>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#1","level":3,"title":"1.   노드 탐색","text":"<p>목표 노드의 값 <code>num</code>이 주어지면 이진 탐색 트리의 성질로 찾을 수 있다. 아래 그림(7-17)과 같이 노드 <code>cur</code>를 두고 루트 <code>root</code>에서 출발하여 <code>cur.val</code>과 <code>num</code>의 대소 관계를 반복해서 비교한다.</p> <ul> <li><code>cur.val &lt; num</code>이면 목표 노드는 <code>cur</code>의 오른쪽 서브트리에 있으므로 <code>cur = cur.right</code>를 수행한다.</li> <li><code>cur.val &gt; num</code>이면 목표 노드는 <code>cur</code>의 왼쪽 서브트리에 있으므로 <code>cur = cur.left</code>를 수행한다.</li> <li><code>cur.val = num</code>이면 목표 노드를 찾았으므로 반복문을 빠져나와 해당 노드를 반환한다.</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> 그림 7-17   이진 탐색 트리의 노드 탐색 예시 </p> <p>이진 탐색 트리의 탐색 원리는 이진 탐색 알고리즘과 같으며, 매번 경우의 절반을 제외한다. 반복 횟수는 최대 트리 높이이므로 트리가 균형을 이루면 \\(O(\\log n)\\) 시간이 걸린다. 예시 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_tree.py<pre><code>def search(self, num: int) -&gt; TreeNode | None:\n    \"\"\"노드 탐색\"\"\"\n    cur = self._root\n    # 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while cur is not None:\n        # 목표 노드는 cur의 오른쪽 서브트리에 있다\n        if cur.val &lt; num:\n            cur = cur.right\n        # 목표 노드는 cur의 왼쪽 서브트리에 있다\n        elif cur.val &gt; num:\n            cur = cur.left\n        # 목표 노드를 찾았으므로 반복문을 종료한다\n        else:\n            break\n    return cur\n</code></pre> binary_search_tree.cpp<pre><code>/* 노드 탐색 */\nTreeNode *search(int num) {\n    TreeNode *cur = root;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != nullptr) {\n        // 목표 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // 목표 노드는 cur의 왼쪽 서브트리에 있다\n        else if (cur-&gt;val &gt; num)\n            cur = cur-&gt;left;\n        // 목표 노드를 찾았으므로 반복문을 종료한다\n        else\n            break;\n    }\n    // 목표 노드를 반환한다\n    return cur;\n}\n</code></pre> binary_search_tree.java<pre><code>/* 노드 탐색 */\nTreeNode search(int num) {\n    TreeNode cur = root;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 목표 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // 목표 노드는 cur의 왼쪽 서브트리에 있다\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // 목표 노드를 찾았으므로 반복문을 종료한다\n        else\n            break;\n    }\n    // 목표 노드를 반환한다\n    return cur;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* 노드 탐색 */\nTreeNode? Search(int num) {\n    TreeNode? cur = root;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 목표 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num) cur =\n            cur.right;\n        // 목표 노드는 cur의 왼쪽 서브트리에 있다\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // 목표 노드를 찾았으므로 반복문을 종료한다\n        else\n            break;\n    }\n    // 목표 노드를 반환한다\n    return cur;\n}\n</code></pre> binary_search_tree.go<pre><code>/* 노드 탐색 */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\n    node := bst.root\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    for node != nil {\n        if node.Val.(int) &lt; num {\n            // 목표 노드는 cur의 오른쪽 서브트리에 있다\n            node = node.Right\n        } else if node.Val.(int) &gt; num {\n            // 목표 노드는 cur의 왼쪽 서브트리에 있다\n            node = node.Left\n        } else {\n            // 목표 노드를 찾았으므로 반복문을 종료한다\n            break\n        }\n    }\n    // 목표 노드를 반환한다\n    return node\n}\n</code></pre> binary_search_tree.swift<pre><code>/* 노드 탐색 */\nfunc search(num: Int) -&gt; TreeNode? {\n    var cur = root\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while cur != nil {\n        // 목표 노드는 cur의 오른쪽 서브트리에 있다\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // 목표 노드는 cur의 왼쪽 서브트리에 있다\n        else if cur!.val &gt; num {\n            cur = cur?.left\n        }\n        // 목표 노드를 찾았으므로 반복문을 종료한다\n        else {\n            break\n        }\n    }\n    // 목표 노드를 반환한다\n    return cur\n}\n</code></pre> binary_search_tree.js<pre><code>/* 노드 탐색 */\nsearch(num) {\n    let cur = this.root;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur !== null) {\n        // 목표 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num) cur = cur.right;\n        // 목표 노드는 cur의 왼쪽 서브트리에 있다\n        else if (cur.val &gt; num) cur = cur.left;\n        // 목표 노드를 찾았으므로 반복문을 종료한다\n        else break;\n    }\n    // 목표 노드를 반환한다\n    return cur;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* 노드 탐색 */\nsearch(num: number): TreeNode | null {\n    let cur = this.root;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur !== null) {\n        // 목표 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num) cur = cur.right;\n        // 목표 노드는 cur의 왼쪽 서브트리에 있다\n        else if (cur.val &gt; num) cur = cur.left;\n        // 목표 노드를 찾았으므로 반복문을 종료한다\n        else break;\n    }\n    // 목표 노드를 반환한다\n    return cur;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* 노드 탐색 */\nTreeNode? search(int _num) {\n  TreeNode? cur = _root;\n  // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n  while (cur != null) {\n    // 목표 노드는 cur의 오른쪽 서브트리에 있다\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // 목표 노드는 cur의 왼쪽 서브트리에 있다\n    else if (cur.val &gt; _num)\n      cur = cur.left;\n    // 목표 노드를 찾았으므로 반복문을 종료한다\n    else\n      break;\n  }\n  // 목표 노드를 반환한다\n  return cur;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* 노드 탐색 */\npub fn search(&amp;self, num: i32) -&gt; OptionTreeNodeRc {\n    let mut cur = self.root.clone();\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // 목표 노드는 cur의 오른쪽 서브트리에 있다\n            Ordering::Greater =&gt; cur = node.borrow().right.clone(),\n            // 목표 노드는 cur의 왼쪽 서브트리에 있다\n            Ordering::Less =&gt; cur = node.borrow().left.clone(),\n            // 목표 노드를 찾았으므로 반복문을 종료한다\n            Ordering::Equal =&gt; break,\n        }\n    }\n\n    // 목표 노드를 반환한다\n    cur\n}\n</code></pre> binary_search_tree.c<pre><code>/* 노드 탐색 */\nTreeNode *search(BinarySearchTree *bst, int num) {\n    TreeNode *cur = bst-&gt;root;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != NULL) {\n        if (cur-&gt;val &lt; num) {\n            // 목표 노드는 cur의 오른쪽 서브트리에 있다\n            cur = cur-&gt;right;\n        } else if (cur-&gt;val &gt; num) {\n            // 목표 노드는 cur의 왼쪽 서브트리에 있다\n            cur = cur-&gt;left;\n        } else {\n            // 목표 노드를 찾았으므로 반복문을 종료한다\n            break;\n        }\n    }\n    // 목표 노드를 반환한다\n    return cur;\n}\n</code></pre> binary_search_tree.kt<pre><code>/* 노드 탐색 */\nfun search(num: Int): TreeNode? {\n    var cur = root\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 목표 노드는 cur의 오른쪽 서브트리에 있다\n        cur = if (cur._val &lt; num)\n            cur.right\n        // 목표 노드는 cur의 왼쪽 서브트리에 있다\n        else if (cur._val &gt; num)\n            cur.left\n        // 목표 노드를 찾았으므로 반복문을 종료한다\n        else\n            break\n    }\n    // 목표 노드를 반환한다\n    return cur\n}\n</code></pre> binary_search_tree.rb<pre><code>### 노드 탐색 ###\ndef search(num)\n  cur = @root\n\n  # 반복하여 탐색하다 리프 노드를 지나면 종료한다\n  while !cur.nil?\n    # 목표 노드는 cur의 오른쪽 서브트리에 있다\n    if cur.val &lt; num\n      cur = cur.right\n    # 목표 노드는 cur의 왼쪽 서브트리에 있다\n    elsif cur.val &gt; num\n      cur = cur.left\n    # 목표 노드를 찾았으므로 반복문을 종료한다\n    else\n      break\n    end\n  end\n\n  cur\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#2","level":3,"title":"2.   노드 삽입","text":"<p>삽입할 원소 <code>num</code>이 주어졌을 때 '왼쪽 서브트리 &lt; 루트 &lt; 오른쪽 서브트리'라는 성질을 유지하려면 아래 그림(7-18)과 같이 삽입한다.</p> <ol> <li>삽입 위치 찾기: 탐색과 비슷하게 루트에서 출발하여 현재 노드 값과 <code>num</code>의 대소 관계에 따라 아래로 탐색한다. 리프를 지나 <code>None</code>에 도달하면 반복문을 종료한다.</li> <li>그 위치에 노드 삽입: 값이 <code>num</code>인 노드를 초기화하여 <code>None</code> 자리에 놓는다.</li> </ol> <p></p> <p> 그림 7-18   이진 탐색 트리에 노드 삽입 </p> <p>구현 시 다음 두 가지에 주의하자.</p> <ul> <li>중복 노드가 있으면 정의에 어긋나므로 이진 탐색 트리는 중복을 허용하지 않는다. 삽입할 노드가 이미 있으면 삽입하지 않고 바로 반환한다.</li> <li>삽입을 위해 노드 <code>pre</code>에 직전 반복의 노드를 저장해야 한다. 그러면 <code>None</code>에 도달했을 때 그 부모를 얻어 삽입을 완료할 수 있다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_tree.py<pre><code>def insert(self, num: int):\n    \"\"\"노드 삽입\"\"\"\n    # 트리가 비어 있으면 루트 노드를 초기화한다\n    if self._root is None:\n        self._root = TreeNode(num)\n        return\n    # 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    cur, pre = self._root, None\n    while cur is not None:\n        # 중복 노드를 찾으면 바로 반환한다\n        if cur.val == num:\n            return\n        pre = cur\n        # 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        if cur.val &lt; num:\n            cur = cur.right\n        # 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else:\n            cur = cur.left\n    # 노드 삽입\n    node = TreeNode(num)\n    if pre.val &lt; num:\n        pre.right = node\n    else:\n        pre.left = node\n</code></pre> binary_search_tree.cpp<pre><code>/* 노드 삽입 */\nvoid insert(int num) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if (root == nullptr) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode *cur = root, *pre = nullptr;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != nullptr) {\n        // 중복 노드를 찾으면 바로 반환한다\n        if (cur-&gt;val == num)\n            return;\n        pre = cur;\n        // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else\n            cur = cur-&gt;left;\n    }\n    // 노드 삽입\n    TreeNode *node = new TreeNode(num);\n    if (pre-&gt;val &lt; num)\n        pre-&gt;right = node;\n    else\n        pre-&gt;left = node;\n}\n</code></pre> binary_search_tree.java<pre><code>/* 노드 삽입 */\nvoid insert(int num) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode cur = root, pre = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 중복 노드를 찾으면 바로 반환한다\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else\n            cur = cur.left;\n    }\n    // 노드 삽입\n    TreeNode node = new TreeNode(num);\n    if (pre.val &lt; num)\n        pre.right = node;\n    else\n        pre.left = node;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* 노드 삽입 */\nvoid Insert(int num) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode? cur = root, pre = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 중복 노드를 찾으면 바로 반환한다\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else\n            cur = cur.left;\n    }\n\n    // 노드 삽입\n    TreeNode node = new(num);\n    if (pre != null) {\n        if (pre.val &lt; num)\n            pre.right = node;\n        else\n            pre.left = node;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* 노드 삽입 */\nfunc (bst *binarySearchTree) insert(num int) {\n    cur := bst.root\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if cur == nil {\n        bst.root = NewTreeNode(num)\n        return\n    }\n    // 삽입할 노드의 부모가 될 노드 위치\n    var pre *TreeNode = nil\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    for cur != nil {\n        if cur.Val == num {\n            return\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            cur = cur.Right\n        } else {\n            cur = cur.Left\n        }\n    }\n    // 노드 삽입\n    node := NewTreeNode(num)\n    if pre.Val.(int) &lt; num {\n        pre.Right = node\n    } else {\n        pre.Left = node\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* 노드 삽입 */\nfunc insert(num: Int) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if root == nil {\n        root = TreeNode(x: num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while cur != nil {\n        // 중복 노드를 찾으면 바로 반환한다\n        if cur!.val == num {\n            return\n        }\n        pre = cur\n        // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else {\n            cur = cur?.left\n        }\n    }\n    // 노드 삽입\n    let node = TreeNode(x: num)\n    if pre!.val &lt; num {\n        pre?.right = node\n    } else {\n        pre?.left = node\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* 노드 삽입 */\ninsert(num) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur = this.root,\n        pre = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur !== null) {\n        // 중복 노드를 찾으면 바로 반환한다\n        if (cur.val === num) return;\n        pre = cur;\n        // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num) cur = cur.right;\n        // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else cur = cur.left;\n    }\n    // 노드 삽입\n    const node = new TreeNode(num);\n    if (pre.val &lt; num) pre.right = node;\n    else pre.left = node;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* 노드 삽입 */\ninsert(num: number): void {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur !== null) {\n        // 중복 노드를 찾으면 바로 반환한다\n        if (cur.val === num) return;\n        pre = cur;\n        // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num) cur = cur.right;\n        // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else cur = cur.left;\n    }\n    // 노드 삽입\n    const node = new TreeNode(num);\n    if (pre!.val &lt; num) pre!.right = node;\n    else pre!.left = node;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* 노드 삽입 */\nvoid insert(int _num) {\n  // 트리가 비어 있으면 루트 노드를 초기화한다\n  if (_root == null) {\n    _root = TreeNode(_num);\n    return;\n  }\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n  while (cur != null) {\n    // 중복 노드를 찾으면 바로 반환한다\n    if (cur.val == _num) return;\n    pre = cur;\n    // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n    else\n      cur = cur.left;\n  }\n  // 노드 삽입\n  TreeNode? node = TreeNode(_num);\n  if (pre!.val &lt; _num)\n    pre.right = node;\n  else\n    pre.left = node;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* 노드 삽입 */\npub fn insert(&amp;mut self, num: i32) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if self.root.is_none() {\n        self.root = Some(TreeNode::new(num));\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // 중복 노드를 찾으면 바로 반환한다\n            Ordering::Equal =&gt; return,\n            // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // 노드 삽입\n    let pre = pre.unwrap();\n    let node = Some(TreeNode::new(num));\n    if num &gt; pre.borrow().val {\n        pre.borrow_mut().right = node;\n    } else {\n        pre.borrow_mut().left = node;\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* 노드 삽입 */\nvoid insert(BinarySearchTree *bst, int num) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if (bst-&gt;root == NULL) {\n        bst-&gt;root = newTreeNode(num);\n        return;\n    }\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != NULL) {\n        // 중복 노드를 찾으면 바로 반환한다\n        if (cur-&gt;val == num) {\n            return;\n        }\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n            cur = cur-&gt;right;\n        } else {\n            // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n            cur = cur-&gt;left;\n        }\n    }\n    // 노드 삽입\n    TreeNode *node = newTreeNode(num);\n    if (pre-&gt;val &lt; num) {\n        pre-&gt;right = node;\n    } else {\n        pre-&gt;left = node;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* 노드 삽입 */\nfun insert(num: Int) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if (root == null) {\n        root = TreeNode(num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode? = null\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 중복 노드를 찾으면 바로 반환한다\n        if (cur._val == num)\n            return\n        pre = cur\n        // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        cur = if (cur._val &lt; num)\n            cur.right\n        // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else\n            cur.left\n    }\n    // 노드 삽입\n    val node = TreeNode(num)\n    if (pre?._val!! &lt; num)\n        pre.right = node\n    else\n        pre.left = node\n}\n</code></pre> binary_search_tree.rb<pre><code>### 노드 삽입 ###\ndef insert(num)\n  # 트리가 비어 있으면 루트 노드를 초기화한다\n  if @root.nil?\n    @root = TreeNode.new(num)\n    return\n  end\n\n  # 반복하여 탐색하다 리프 노드를 지나면 종료한다\n  cur, pre = @root, nil\n  while !cur.nil?\n    # 중복 노드를 찾으면 바로 반환한다\n    return if cur.val == num\n\n    pre = cur\n    # 삽입 위치는 cur의 오른쪽 서브트리에 있다\n    if cur.val &lt; num\n      cur = cur.right\n    # 삽입 위치는 cur의 왼쪽 서브트리에 있다\n    else\n      cur = cur.left\n    end\n  end\n\n  # 노드 삽입\n  node = TreeNode.new(num)\n  if pre.val &lt; num\n    pre.right = node\n  else\n    pre.left = node\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>노드 탐색과 마찬가지로 삽입에는 \\(O(\\log n)\\) 시간이 걸린다.</p>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#3","level":3,"title":"3.   노드 삭제","text":"<p>먼저 트리에서 목표 노드를 찾은 뒤 삭제한다. 삽입과 마찬가지로 삭제 후에도 '왼쪽 서브트리 &lt; 루트 &lt; 오른쪽 서브트리'라는 성질을 유지해야 한다. 따라서 목표 노드의 자식 수가 0, 1, 2인 세 경우로 나누어 삭제한다.</p> <p>아래 그림(7-19)과 같이 삭제할 노드의 차수가 \\(0\\)이면 리프 노드이므로 바로 삭제할 수 있다.</p> <p></p> <p> 그림 7-19   이진 탐색 트리의 노드 삭제: 차수 0 </p> <p>아래 그림(7-20)과 같이 차수가 \\(1\\)이면 삭제할 노드를 그 자식으로 바꾸면 된다.</p> <p></p> <p> 그림 7-20   이진 탐색 트리의 노드 삭제: 차수 1 </p> <p>차수가 \\(2\\)이면 바로 삭제할 수 없어 다른 노드로 대체해야 한다. '왼쪽 서브트리 \\(&lt;\\) 루트 \\(&lt;\\) 오른쪽 서브트리' 성질을 유지하려면 오른쪽 서브트리의 최소 노드 또는 왼쪽 서브트리의 최대 노드로 대체할 수 있다.</p> <p>오른쪽 서브트리의 최소 노드(중위 순회에서 다음 노드)를 선택하면 삭제 과정은 아래 그림(7-21)과 같다.</p> <ol> <li>삭제할 노드의 '중위 순회 순서'상 다음 노드를 찾아 <code>tmp</code>라고 한다.</li> <li>삭제할 노드의 값을 <code>tmp</code>의 값으로 덮어쓰고, 트리에서 노드 <code>tmp</code>를 재귀적으로 삭제한다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> 그림 7-21   이진 탐색 트리의 노드 삭제: 차수 2 </p> <p>노드 삭제에도 \\(O(\\log n)\\) 시간이 걸린다. 삭제할 노드를 찾는 데 \\(O(\\log n)\\), 중위 순회의 후속 노드를 구하는 데 \\(O(\\log n)\\) 시간이 든다. 예시 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_tree.py<pre><code>def remove(self, num: int):\n    \"\"\"노드 삭제\"\"\"\n    # 트리가 비어 있으면 바로 반환하여 종료한다\n    if self._root is None:\n        return\n    # 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    cur, pre = self._root, None\n    while cur is not None:\n        # 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if cur.val == num:\n            break\n        pre = cur\n        # 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        if cur.val &lt; num:\n            cur = cur.right\n        # 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else:\n            cur = cur.left\n    # 삭제할 노드가 없으면 바로 반환한다\n    if cur is None:\n        return\n\n    # 자식 노드 수 = 0 또는 1\n    if cur.left is None or cur.right is None:\n        # 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n        child = cur.left or cur.right\n        # 노드 cur를 삭제한다\n        if cur != self._root:\n            if pre.left == cur:\n                pre.left = child\n            else:\n                pre.right = child\n        else:\n            # 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            self._root = child\n    # 자식 노드 수 = 2\n    else:\n        # 중위 순회에서 cur의 다음 노드를 구한다\n        tmp: TreeNode = cur.right\n        while tmp.left is not None:\n            tmp = tmp.left\n        # 노드 tmp를 재귀적으로 삭제한다\n        self.remove(tmp.val)\n        # tmp로 cur를 덮어쓴다\n        cur.val = tmp.val\n</code></pre> binary_search_tree.cpp<pre><code>/* 노드 삭제 */\nvoid remove(int num) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if (root == nullptr)\n        return;\n    TreeNode *cur = root, *pre = nullptr;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != nullptr) {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else\n            cur = cur-&gt;left;\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if (cur == nullptr)\n        return;\n    // 자식 노드 수 = 0 또는 1\n    if (cur-&gt;left == nullptr || cur-&gt;right == nullptr) {\n        // 자식 노드 수가 0 / 1이면 child = nullptr / 해당 자식 노드다\n        TreeNode *child = cur-&gt;left != nullptr ? cur-&gt;left : cur-&gt;right;\n        // 노드 cur를 삭제한다\n        if (cur != root) {\n            if (pre-&gt;left == cur)\n                pre-&gt;left = child;\n            else\n                pre-&gt;right = child;\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            root = child;\n        }\n        // 메모리 해제\n        delete cur;\n    }\n    // 자식 노드 수 = 2\n    else {\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != nullptr) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // 노드 tmp를 재귀적으로 삭제한다\n        remove(tmp-&gt;val);\n        // tmp로 cur를 덮어쓴다\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.java<pre><code>/* 노드 삭제 */\nvoid remove(int num) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if (root == null)\n        return;\n    TreeNode cur = root, pre = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else\n            cur = cur.left;\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if (cur == null)\n        return;\n    // 자식 노드 수 = 0 또는 1\n    if (cur.left == null || cur.right == null) {\n        // 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n        TreeNode child = cur.left != null ? cur.left : cur.right;\n        // 노드 cur를 삭제한다\n        if (cur != root) {\n            if (pre.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            root = child;\n        }\n    }\n    // 자식 노드 수 = 2\n    else {\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        TreeNode tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // 노드 tmp를 재귀적으로 삭제한다\n        remove(tmp.val);\n        // tmp로 cur를 덮어쓴다\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.cs<pre><code>/* 노드 삭제 */\nvoid Remove(int num) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if (root == null)\n        return;\n    TreeNode? cur = root, pre = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else\n            cur = cur.left;\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if (cur == null)\n        return;\n    // 자식 노드 수 = 0 또는 1\n    if (cur.left == null || cur.right == null) {\n        // 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n        TreeNode? child = cur.left ?? cur.right;\n        // 노드 cur를 삭제한다\n        if (cur != root) {\n            if (pre!.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            root = child;\n        }\n    }\n    // 자식 노드 수 = 2\n    else {\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        TreeNode? tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // 노드 tmp를 재귀적으로 삭제한다\n        Remove(tmp.val!.Value);\n        // tmp로 cur를 덮어쓴다\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* 노드 삭제 */\nfunc (bst *binarySearchTree) remove(num int) {\n    cur := bst.root\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if cur == nil {\n        return\n    }\n    // 삭제할 노드의 부모 노드 위치\n    var pre *TreeNode = nil\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    for cur != nil {\n        if cur.Val == num {\n            break\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            // 삭제할 노드는 오른쪽 서브트리에 있다\n            cur = cur.Right\n        } else {\n            // 삭제할 노드는 왼쪽 서브트리에 있다\n            cur = cur.Left\n        }\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if cur == nil {\n        return\n    }\n    // 자식 노드 수가 0 또는 1\n    if cur.Left == nil || cur.Right == nil {\n        var child *TreeNode = nil\n        // 삭제할 노드의 자식 노드를 구한다\n        if cur.Left != nil {\n            child = cur.Left\n        } else {\n            child = cur.Right\n        }\n        // 노드 cur를 삭제한다\n        if cur != bst.root {\n            if pre.Left == cur {\n                pre.Left = child\n            } else {\n                pre.Right = child\n            }\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            bst.root = child\n        }\n        // 자식 노드 수가 2\n    } else {\n        // 중위 순회에서 삭제할 노드 cur의 다음 노드를 구한다\n        tmp := cur.Right\n        for tmp.Left != nil {\n            tmp = tmp.Left\n        }\n        // 노드 tmp를 재귀적으로 삭제한다\n        bst.remove(tmp.Val.(int))\n        // tmp로 cur를 덮어쓴다\n        cur.Val = tmp.Val\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* 노드 삭제 */\nfunc remove(num: Int) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if root == nil {\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while cur != nil {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if cur!.val == num {\n            break\n        }\n        pre = cur\n        // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else {\n            cur = cur?.left\n        }\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if cur == nil {\n        return\n    }\n    // 자식 노드 수 = 0 또는 1\n    if cur?.left == nil || cur?.right == nil {\n        // 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n        let child = cur?.left ?? cur?.right\n        // 노드 cur를 삭제한다\n        if cur !== root {\n            if pre?.left === cur {\n                pre?.left = child\n            } else {\n                pre?.right = child\n            }\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            root = child\n        }\n    }\n    // 자식 노드 수 = 2\n    else {\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        var tmp = cur?.right\n        while tmp?.left != nil {\n            tmp = tmp?.left\n        }\n        // 노드 tmp를 재귀적으로 삭제한다\n        remove(num: tmp!.val)\n        // tmp로 cur를 덮어쓴다\n        cur?.val = tmp!.val\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* 노드 삭제 */\nremove(num) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if (this.root === null) return;\n    let cur = this.root,\n        pre = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur !== null) {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if (cur.val === num) break;\n        pre = cur;\n        // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num) cur = cur.right;\n        // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else cur = cur.left;\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if (cur === null) return;\n    // 자식 노드 수 = 0 또는 1\n    if (cur.left === null || cur.right === null) {\n        // 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n        const child = cur.left !== null ? cur.left : cur.right;\n        // 노드 cur를 삭제한다\n        if (cur !== this.root) {\n            if (pre.left === cur) pre.left = child;\n            else pre.right = child;\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            this.root = child;\n        }\n    }\n    // 자식 노드 수 = 2\n    else {\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        let tmp = cur.right;\n        while (tmp.left !== null) {\n            tmp = tmp.left;\n        }\n        // 노드 tmp를 재귀적으로 삭제한다\n        this.remove(tmp.val);\n        // tmp로 cur를 덮어쓴다\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.ts<pre><code>/* 노드 삭제 */\nremove(num: number): void {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if (this.root === null) return;\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur !== null) {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if (cur.val === num) break;\n        pre = cur;\n        // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num) cur = cur.right;\n        // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else cur = cur.left;\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if (cur === null) return;\n    // 자식 노드 수 = 0 또는 1\n    if (cur.left === null || cur.right === null) {\n        // 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n        const child: TreeNode | null =\n            cur.left !== null ? cur.left : cur.right;\n        // 노드 cur를 삭제한다\n        if (cur !== this.root) {\n            if (pre!.left === cur) pre!.left = child;\n            else pre!.right = child;\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            this.root = child;\n        }\n    }\n    // 자식 노드 수 = 2\n    else {\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        let tmp: TreeNode | null = cur.right;\n        while (tmp!.left !== null) {\n            tmp = tmp!.left;\n        }\n        // 노드 tmp를 재귀적으로 삭제한다\n        this.remove(tmp!.val);\n        // tmp로 cur를 덮어쓴다\n        cur.val = tmp!.val;\n    }\n}\n</code></pre> binary_search_tree.dart<pre><code>/* 노드 삭제 */\nvoid remove(int _num) {\n  // 트리가 비어 있으면 바로 반환하여 종료한다\n  if (_root == null) return;\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n  while (cur != null) {\n    // 삭제할 노드를 찾았으므로 반복문을 종료한다\n    if (cur.val == _num) break;\n    pre = cur;\n    // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n    else\n      cur = cur.left;\n  }\n  // 삭제할 노드가 없으면 바로 반환한다\n  if (cur == null) return;\n  // 자식 노드 수 = 0 또는 1\n  if (cur.left == null || cur.right == null) {\n    // 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n    TreeNode? child = cur.left ?? cur.right;\n    // 노드 cur를 삭제한다\n    if (cur != _root) {\n      if (pre!.left == cur)\n        pre.left = child;\n      else\n        pre.right = child;\n    } else {\n      // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n      _root = child;\n    }\n  } else {\n    // 자식 노드 수 = 2\n    // 중위 순회에서 cur의 다음 노드를 구한다\n    TreeNode? tmp = cur.right;\n    while (tmp!.left != null) {\n      tmp = tmp.left;\n    }\n    // 노드 tmp를 재귀적으로 삭제한다\n    remove(tmp.val);\n    // tmp로 cur를 덮어쓴다\n    cur.val = tmp.val;\n  }\n}\n</code></pre> binary_search_tree.rs<pre><code>/* 노드 삭제 */\npub fn remove(&amp;mut self, num: i32) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if self.root.is_none() {\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // 삭제할 노드를 찾았으므로 반복문을 종료한다\n            Ordering::Equal =&gt; break,\n            // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if cur.is_none() {\n        return;\n    }\n    let cur = cur.unwrap();\n    let (left_child, right_child) = (cur.borrow().left.clone(), cur.borrow().right.clone());\n    match (left_child.clone(), right_child.clone()) {\n        // 자식 노드 수 = 0 또는 1\n        (None, None) | (Some(_), None) | (None, Some(_)) =&gt; {\n            // 자식 노드 수가 0 / 1이면 child = nullptr / 해당 자식 노드다\n            let child = left_child.or(right_child);\n            let pre = pre.unwrap();\n            // 노드 cur를 삭제한다\n            if !Rc::ptr_eq(&amp;cur, self.root.as_ref().unwrap()) {\n                let left = pre.borrow().left.clone();\n                if left.is_some() &amp;&amp; Rc::ptr_eq(left.as_ref().unwrap(), &amp;cur) {\n                    pre.borrow_mut().left = child;\n                } else {\n                    pre.borrow_mut().right = child;\n                }\n            } else {\n                // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n                self.root = child;\n            }\n        }\n        // 자식 노드 수 = 2\n        (Some(_), Some(_)) =&gt; {\n            // 중위 순회에서 cur의 다음 노드를 구한다\n            let mut tmp = cur.borrow().right.clone();\n            while let Some(node) = tmp.clone() {\n                if node.borrow().left.is_some() {\n                    tmp = node.borrow().left.clone();\n                } else {\n                    break;\n                }\n            }\n            let tmp_val = tmp.unwrap().borrow().val;\n            // 노드 tmp를 재귀적으로 삭제한다\n            self.remove(tmp_val);\n            // tmp로 cur를 덮어쓴다\n            cur.borrow_mut().val = tmp_val;\n        }\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* 노드 삭제 */\n// stdio.h를 포함했으므로 여기서는 remove라는 이름을 사용할 수 없다\nvoid removeItem(BinarySearchTree *bst, int num) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if (bst-&gt;root == NULL)\n        return;\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != NULL) {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // 삭제할 노드가 root의 오른쪽 서브트리에 있다\n            cur = cur-&gt;right;\n        } else {\n            // 삭제할 노드가 root의 왼쪽 서브트리에 있다\n            cur = cur-&gt;left;\n        }\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if (cur == NULL)\n        return;\n    // 삭제할 노드에 자식 노드가 있는지 확인한다\n    if (cur-&gt;left == NULL || cur-&gt;right == NULL) {\n        /* 자식 노드 수 = 0 또는 1 */\n        // 자식 노드 수가 0 / 1이면 child = nullptr / 해당 자식 노드다\n        TreeNode *child = cur-&gt;left != NULL ? cur-&gt;left : cur-&gt;right;\n        // 노드 cur를 삭제한다\n        if (pre-&gt;left == cur) {\n            pre-&gt;left = child;\n        } else {\n            pre-&gt;right = child;\n        }\n        // 메모리 해제\n        free(cur);\n    } else {\n        /* 자식 노드 수 = 2 */\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != NULL) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // 노드 tmp를 재귀적으로 삭제한다\n        removeItem(bst, tmp-&gt;val);\n        // tmp로 cur를 덮어쓴다\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* 노드 삭제 */\nfun remove(num: Int) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if (root == null)\n        return\n    var cur = root\n    var pre: TreeNode? = null\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if (cur._val == num)\n            break\n        pre = cur\n        // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        cur = if (cur._val &lt; num)\n            cur.right\n        // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else\n            cur.left\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if (cur == null)\n        return\n    // 자식 노드 수 = 0 또는 1\n    if (cur.left == null || cur.right == null) {\n        // 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n        val child = if (cur.left != null)\n            cur.left\n        else\n            cur.right\n        // 노드 cur를 삭제한다\n        if (cur != root) {\n            if (pre!!.left == cur)\n                pre.left = child\n            else\n                pre.right = child\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            root = child\n        }\n        // 자식 노드 수 = 2\n    } else {\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        var tmp = cur.right\n        while (tmp!!.left != null) {\n            tmp = tmp.left\n        }\n        // 노드 tmp를 재귀적으로 삭제한다\n        remove(tmp._val)\n        // tmp로 cur를 덮어쓴다\n        cur._val = tmp._val\n    }\n}\n</code></pre> binary_search_tree.rb<pre><code>### 노드 삭제 ###\ndef remove(num)\n  # 트리가 비어 있으면 바로 반환하여 종료한다\n  return if @root.nil?\n\n  # 반복하여 탐색하다 리프 노드를 지나면 종료한다\n  cur, pre = @root, nil\n  while !cur.nil?\n    # 삭제할 노드를 찾았으므로 반복문을 종료한다\n    break if cur.val == num\n\n    pre = cur\n    # 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n    if cur.val &lt; num\n      cur = cur.right\n    # 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n    else\n      cur = cur.left\n    end\n  end\n  # 삭제할 노드가 없으면 바로 반환한다\n  return if cur.nil?\n\n  # 자식 노드 수 = 0 또는 1\n  if cur.left.nil? || cur.right.nil?\n    # 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n    child = cur.left || cur.right\n    # 노드 cur를 삭제한다\n    if cur != @root\n      if pre.left == cur\n        pre.left = child\n      else\n        pre.right = child\n      end\n    else\n      # 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n      @root = child\n    end\n  # 자식 노드 수 = 2\n  else\n    # 중위 순회에서 cur의 다음 노드를 구한다\n    tmp = cur.right\n    while !tmp.left.nil?\n      tmp = tmp.left\n    end\n    # 노드 tmp를 재귀적으로 삭제한다\n    remove(tmp.val)\n    # tmp로 cur를 덮어쓴다\n    cur.val = tmp.val\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#4","level":3,"title":"4.   중위 순회 결과의 정렬 성질","text":"<p>아래 그림(7-22)과 같이 이진 트리의 중위 순회는 '왼쪽 \\(\\rightarrow\\) 루트 \\(\\rightarrow\\) 오른쪽' 순서이며, 이진 탐색 트리는 '왼쪽 자식 \\(&lt;\\) 루트 \\(&lt;\\) 오른쪽 자식'의 대소 관계를 만족한다.</p> <p>따라서 중위 순회에서는 항상 다음으로 작은 노드를 먼저 방문한다. 여기서 이진 탐색 트리의 중위 순회 결과는 오름차순이다라는 중요한 성질을 얻는다.</p> <p>이 성질을 이용하면 별도 정렬 없이 \\(O(n)\\) 시간만으로 정렬된 데이터를 얻을 수 있어 매우 효율적이다.</p> <p></p> <p> 그림 7-22   이진 탐색 트리의 중위 순회 결과 </p>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#742","level":2,"title":"7.4.2   이진 탐색 트리의 효율","text":"<p>데이터 집합을 배열이나 이진 탐색 트리에 저장한다고 하자. 아래 표(7-2)를 보면 이진 탐색 트리의 각 연산은 로그 시간 복잡도로 안정적이고 효율적이다. 데이터 추가는 잦고 탐색과 삭제는 드문 상황에서만 배열이 더 효율적이다.</p> <p> 표 7-2   배열과 탐색 트리의 효율 비교 </p> 정렬되지 않은 배열 이진 탐색 트리 원소 탐색 \\(O(n)\\) \\(O(\\log n)\\) 원소 삽입 \\(O(1)\\) \\(O(\\log n)\\) 원소 삭제 \\(O(n)\\) \\(O(\\log n)\\) <p>이상적인 경우 이진 탐색 트리는 '균형'을 이루므로 어떤 노드든 \\(\\log n\\) 회의 반복 안에 찾을 수 있다.</p> <p>하지만 삽입과 삭제를 계속하다 보면 아래 그림(7-23)의 연결 리스트로 퇴화할 수 있다. 이때 각 연산의 시간 복잡도도 \\(O(n)\\)으로 저하된다.</p> <p></p> <p> 그림 7-23   이진 탐색 트리의 퇴화 </p>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#743","level":2,"title":"7.4.3   이진 탐색 트리의 주요 응용","text":"<ul> <li>시스템의 다단계 인덱스로 사용하여 효율적인 탐색, 삽입, 삭제를 구현한다.</li> <li>일부 탐색 알고리즘의 기반 자료 구조로 사용한다.</li> <li>데이터 스트림을 저장하면서 정렬 상태를 유지한다.</li> </ul>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/","level":1,"title":"7.1   이진 트리","text":"<p>이진 트리(binary tree)는 '조상'과 '후손' 사이의 파생 관계를 나타내는 비선형 자료 구조로, '하나를 둘로 나누는' 분할 정복의 논리를 보여 준다. 연결 리스트처럼 기본 단위는 노드이며, 각 노드는 값과 왼쪽 자식 참조, 오른쪽 자식 참조를 포함한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class TreeNode:\n    \"\"\"이진 트리 노드 클래스\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # 노드 값\n        self.left: TreeNode | None = None  # 왼쪽 자식 노드 참조\n        self.right: TreeNode | None = None # 오른쪽 자식 노드 참조\n</code></pre> <pre><code>/* 이진 트리 노드 구조체 */\nstruct TreeNode {\n    int val;          // 노드 값\n    TreeNode *left;   // 왼쪽 자식 노드 포인터\n    TreeNode *right;  // 오른쪽 자식 노드 포인터\n    TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n</code></pre> <pre><code>/* 이진 트리 노드 클래스 */\nclass TreeNode {\n    int val;         // 노드 값\n    TreeNode left;   // 왼쪽 자식 노드 참조\n    TreeNode right;  // 오른쪽 자식 노드 참조\n    TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* 이진 트리 노드 클래스 */\nclass TreeNode(int? x) {\n    public int? val = x;    // 노드 값\n    public TreeNode? left;  // 왼쪽 자식 노드 참조\n    public TreeNode? right; // 오른쪽 자식 노드 참조\n}\n</code></pre> <pre><code>/* 이진 트리 노드 구조체 */\ntype TreeNode struct {\n    Val   int\n    Left  *TreeNode\n    Right *TreeNode\n}\n/* 생성자 */\nfunc NewTreeNode(v int) *TreeNode {\n    return &amp;TreeNode{\n        Left:  nil, // 왼쪽 자식 노드 포인터\n        Right: nil, // 오른쪽 자식 노드 포인터\n        Val:   v,   // 노드 값\n    }\n}\n</code></pre> <pre><code>/* 이진 트리 노드 클래스 */\nclass TreeNode {\n    var val: Int // 노드 값\n    var left: TreeNode? // 왼쪽 자식 노드 참조\n    var right: TreeNode? // 오른쪽 자식 노드 참조\n\n    init(x: Int) {\n        val = x\n    }\n}\n</code></pre> <pre><code>/* 이진 트리 노드 클래스 */\nclass TreeNode {\n    val; // 노드 값\n    left; // 왼쪽 자식 노드 포인터\n    right; // 오른쪽 자식 노드 포인터\n    constructor(val, left, right) {\n        this.val = val === undefined ? 0 : val;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* 이진 트리 노드 클래스 */\nclass TreeNode {\n    val: number;\n    left: TreeNode | null;\n    right: TreeNode | null;\n\n    constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val; // 노드 값\n        this.left = left === undefined ? null : left; // 왼쪽 자식 노드 참조\n        this.right = right === undefined ? null : right; // 오른쪽 자식 노드 참조\n    }\n}\n</code></pre> <pre><code>/* 이진 트리 노드 클래스 */\nclass TreeNode {\n  int val;         // 노드 값\n  TreeNode? left;  // 왼쪽 자식 노드 참조\n  TreeNode? right; // 오른쪽 자식 노드 참조\n  TreeNode(this.val, [this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* 이진 트리 노드 구조체 */\nstruct TreeNode {\n    val: i32,                               // 노드 값\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // 왼쪽 자식 노드 참조\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // 오른쪽 자식 노드 참조\n}\n\nimpl TreeNode {\n    /* 생성자 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* 이진 트리 노드 구조체 */\ntypedef struct TreeNode {\n    int val;                // 노드 값\n    int height;             // 노드 높이\n    struct TreeNode *left;  // 왼쪽 자식 노드 포인터\n    struct TreeNode *right; // 오른쪽 자식 노드 포인터\n} TreeNode;\n\n/* 생성자 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* 이진 트리 노드 클래스 */\nclass TreeNode(val _val: Int) {  // 노드 값\n    var left: TreeNode? = null   // 왼쪽 자식 노드 참조\n    var right: TreeNode? = null  // 오른쪽 자식 노드 참조\n}\n</code></pre> <pre><code>### 이진 트리 노드 클래스 ###\nclass TreeNode\n  attr_accessor :val    # 노드 값\n  attr_accessor :left   # 왼쪽 자식 노드 참조\n  attr_accessor :right  # 오른쪽 자식 노드 참조\n\n  def initialize(val)\n    @val = val\n  end\nend\n</code></pre> <p>각 노드는 두 참조(포인터)를 가지며, 각각 왼쪽 자식 노드(left-child node)와 오른쪽 자식 노드(right-child node)를 가리킨다. 해당 노드는 두 자식의 부모 노드(parent node)라고 한다. 어떤 노드의 왼쪽 자식과 그 아래 노드들이 이루는 트리를 그 노드의 왼쪽 서브트리(left subtree)라고 하며, 같은 방식으로 오른쪽 서브트리(right subtree)를 정의한다.</p> <p>이진 트리에서 리프 노드를 제외한 모든 노드에는 자식 노드와 비어 있지 않은 서브트리가 있다. 아래 그림(7-1)에서 '노드 2'를 부모로 보면 왼쪽과 오른쪽 자식은 각각 '노드 4', '노드 5'다. 왼쪽 서브트리는 '노드 4와 그 아래 노드들이 이루는 트리', 오른쪽 서브트리는 '노드 5와 그 아래 노드들이 이루는 트리'다.</p> <p></p> <p> 그림 7-1   부모 노드, 자식 노드, 서브트리 </p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#711","level":2,"title":"7.1.1   이진 트리의 주요 용어","text":"<p>이진 트리에서 자주 쓰는 용어는 아래 그림(7-2)과 같다.</p> <ul> <li>루트 노드(root node): 이진 트리의 최상위 노드로, 부모가 없다.</li> <li>리프 노드(leaf node): 자식이 없는 노드로, 두 포인터가 모두 <code>None</code>을 가리킨다.</li> <li>간선(edge): 두 노드를 잇는 선으로, 노드 참조(포인터)에 해당한다.</li> <li>노드의 레벨(level): 위에서 아래로 증가하며 루트의 레벨은 1이다.</li> <li>노드의 차수(degree): 자식 노드의 수다. 이진 트리에서는 0, 1, 2 중 하나다.</li> <li>이진 트리의 높이(height): 루트에서 가장 먼 리프까지 거치는 간선 수다.</li> <li>노드의 깊이(depth): 루트에서 해당 노드까지 거치는 간선 수다.</li> <li>노드의 높이(height): 해당 노드에서 가장 먼 리프와 해당 노드 사이에 있는 간선 수다.</li> </ul> <p></p> <p> 그림 7-2   이진 트리의 주요 용어 </p> <p>참고</p> <p>보통 '높이'와 '깊이'를 '거치는 간선 수'로 정의하지만, 일부 문제나 교재는 '거치는 노드 수'로 정의하기도 한다. 그런 경우에는 높이와 깊이에 각각 1을 더해야 한다.</p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#712","level":2,"title":"7.1.2   이진 트리의 기본 연산","text":"","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#1","level":3,"title":"1.   이진 트리 초기화","text":"<p>연결 리스트처럼 먼저 노드를 초기화하고 참조(포인터)를 연결한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree.py<pre><code># 이진 트리 초기화\n# 노드 초기화\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# 노드 사이의 참조(포인터) 연결\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.cpp<pre><code>/* 이진 트리 초기화 */\n// 노드 초기화\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// 노드 사이의 참조(포인터) 연결\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.java<pre><code>// 노드 초기화\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// 노드 사이의 참조(포인터) 연결\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.cs<pre><code>/* 이진 트리 초기화 */\n// 노드 초기화\nTreeNode n1 = new(1);\nTreeNode n2 = new(2);\nTreeNode n3 = new(3);\nTreeNode n4 = new(4);\nTreeNode n5 = new(5);\n// 노드 사이의 참조(포인터) 연결\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.go<pre><code>/* 이진 트리 초기화 */\n// 노드 초기화\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// 노드 사이의 참조(포인터) 연결\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n</code></pre> binary_tree.swift<pre><code>// 노드 초기화\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// 노드 사이의 참조(포인터) 연결\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.js<pre><code>/* 이진 트리 초기화 */\n// 노드 초기화\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// 노드 사이의 참조(포인터) 연결\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.ts<pre><code>/* 이진 트리 초기화 */\n// 노드 초기화\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// 노드 사이의 참조(포인터) 연결\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.dart<pre><code>/* 이진 트리 초기화 */\n// 노드 초기화\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// 노드 사이의 참조(포인터) 연결\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.rs<pre><code>// 노드 초기화\nlet n1 = TreeNode::new(1);\nlet n2 = TreeNode::new(2);\nlet n3 = TreeNode::new(3);\nlet n4 = TreeNode::new(4);\nlet n5 = TreeNode::new(5);\n// 노드 사이의 참조(포인터) 연결\nn1.borrow_mut().left = Some(n2.clone());\nn1.borrow_mut().right = Some(n3);\nn2.borrow_mut().left = Some(n4);\nn2.borrow_mut().right = Some(n5);\n</code></pre> binary_tree.c<pre><code>/* 이진 트리 초기화 */\n// 노드 초기화\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// 노드 사이의 참조(포인터) 연결\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.kt<pre><code>// 노드 초기화\nval n1 = TreeNode(1)\nval n2 = TreeNode(2)\nval n3 = TreeNode(3)\nval n4 = TreeNode(4)\nval n5 = TreeNode(5)\n// 노드 사이의 참조(포인터) 연결\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.rb<pre><code># 이진 트리 초기화\n# 노드 초기화\nn1 = TreeNode.new(1)\nn2 = TreeNode.new(2)\nn3 = TreeNode.new(3)\nn4 = TreeNode.new(4)\nn5 = TreeNode.new(5)\n# 노드 사이의 참조(포인터) 연결\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> 실행 시각화 <p> 전체 화면 &gt;</p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#2","level":3,"title":"2.   노드 삽입과 삭제","text":"<p>연결 리스트와 마찬가지로 이진 트리에서도 포인터를 수정하여 노드를 삽입하고 삭제할 수 있다. 아래 그림(7-3)은 그 예다.</p> <p></p> <p> 그림 7-3   이진 트리의 노드 삽입과 삭제 </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree.py<pre><code># 노드 삽입과 삭제\np = TreeNode(0)\n# n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = p\np.left = n2\n# 노드 P 삭제\nn1.left = n2\n</code></pre> binary_tree.cpp<pre><code>/* 노드 삽입과 삭제 */\nTreeNode* P = new TreeNode(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1-&gt;left = P;\nP-&gt;left = n2;\n// 노드 P 삭제\nn1-&gt;left = n2;\n// 메모리 해제\ndelete P;\n</code></pre> binary_tree.java<pre><code>TreeNode P = new TreeNode(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = P;\nP.left = n2;\n// 노드 P 삭제\nn1.left = n2;\n</code></pre> binary_tree.cs<pre><code>/* 노드 삽입과 삭제 */\nTreeNode P = new(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = P;\nP.left = n2;\n// 노드 P 삭제\nn1.left = n2;\n</code></pre> binary_tree.go<pre><code>/* 노드 삽입과 삭제 */\n// n1 -&gt; n2 사이에 노드 P 삽입\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// 노드 P 삭제\nn1.Left = n2\n</code></pre> binary_tree.swift<pre><code>let P = TreeNode(x: 0)\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = P\nP.left = n2\n// 노드 P 삭제\nn1.left = n2\n</code></pre> binary_tree.js<pre><code>/* 노드 삽입과 삭제 */\nlet P = new TreeNode(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = P;\nP.left = n2;\n// 노드 P 삭제\nn1.left = n2;\n</code></pre> binary_tree.ts<pre><code>/* 노드 삽입과 삭제 */\nconst P = new TreeNode(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = P;\nP.left = n2;\n// 노드 P 삭제\nn1.left = n2;\n</code></pre> binary_tree.dart<pre><code>/* 노드 삽입과 삭제 */\nTreeNode P = new TreeNode(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = P;\nP.left = n2;\n// 노드 P 삭제\nn1.left = n2;\n</code></pre> binary_tree.rs<pre><code>let p = TreeNode::new(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.borrow_mut().left = Some(p.clone());\np.borrow_mut().left = Some(n2.clone());\n// 노드 p 삭제\nn1.borrow_mut().left = Some(n2);\n</code></pre> binary_tree.c<pre><code>/* 노드 삽입과 삭제 */\nTreeNode *P = newTreeNode(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1-&gt;left = P;\nP-&gt;left = n2;\n// 노드 P 삭제\nn1-&gt;left = n2;\n// 메모리 해제\nfree(P);\n</code></pre> binary_tree.kt<pre><code>val P = TreeNode(0)\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = P\nP.left = n2\n// 노드 P 삭제\nn1.left = n2\n</code></pre> binary_tree.rb<pre><code># 노드 삽입과 삭제\n_p = TreeNode.new(0)\n# n1 -&gt; n2 사이에 노드 _p 삽입\nn1.left = _p\n_p.left = n2\n# 노드 삭제\nn1.left = n2\n</code></pre> 실행 시각화 <p> 전체 화면 &gt;</p> <p>참고</p> <p>노드 삽입은 이진 트리의 기존 논리 구조를 바꿀 수 있으며, 노드 삭제는 보통 해당 노드와 그 모든 서브트리를 삭제한다는 뜻이다. 따라서 의미 있는 작업을 수행하려면 이진 트리의 삽입과 삭제는 보통 여러 연산을 조합하여 이루어진다.</p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#713","level":2,"title":"7.1.3   주요 이진 트리 종류","text":"","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#1_1","level":3,"title":"1.   포화 이진 트리","text":"<p>아래 그림(7-4)의 포화 이진 트리(perfect binary tree)는 모든 레벨에 노드가 가득 차 있다. 리프 노드의 차수는 \\(0\\), 나머지 노드의 차수는 모두 \\(2\\)다. 높이가 \\(h\\)이면 전체 노드 수는 \\(2^{h+1} - 1\\)로 전형적인 지수 관계를 보이며, 자연에서 흔히 볼 수 있는 세포 분열 현상을 반영한다.</p> <p>참고</p> <p>중국어권에서는 포화 이진 트리를 흔히 满二叉树라고 부른다.</p> <p></p> <p> 그림 7-4   포화 이진 트리 </p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#2_1","level":3,"title":"2.   완전 이진 트리","text":"<p>아래 그림(7-5)의 완전 이진 트리(complete binary tree)는 맨 아래 레벨만 가득 차 있지 않아도 되며, 그 레벨의 노드는 왼쪽부터 오른쪽으로 빈틈없이 채워져야 한다. 포화 이진 트리도 완전 이진 트리다.</p> <p></p> <p> 그림 7-5   완전 이진 트리 </p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#3","level":3,"title":"3.   정 이진 트리","text":"<p>아래 그림(7-6)의 정 이진 트리(full binary tree)는 리프를 제외한 모든 노드가 자식 노드를 두 개씩 가진다.</p> <p></p> <p> 그림 7-6   정 이진 트리 </p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#4","level":3,"title":"4.   균형 이진 트리","text":"<p>아래 그림(7-7)의 균형 이진 트리(balanced binary tree)는 모든 노드에서 왼쪽과 오른쪽 서브트리의 높이 차의 절댓값이 1을 넘지 않는다.</p> <p></p> <p> 그림 7-7   균형 이진 트리 </p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#714","level":2,"title":"7.1.4   이진 트리의 퇴화","text":"<p>아래 그림(7-8)은 이진 트리의 이상적인 구조와 퇴화한 구조를 보여 준다. 모든 레벨이 가득 차면 '포화 이진 트리'가 되고, 모든 노드가 한쪽으로 치우치면 '연결 리스트'로 퇴화한다.</p> <ul> <li>포화 이진 트리는 이상적인 경우로, 이진 트리의 '분할 정복' 이점을 충분히 발휘할 수 있다.</li> <li>연결 리스트는 다른 극단으로, 모든 연산이 선형 연산이 되어 시간 복잡도가 \\(O(n)\\)으로 저하된다.</li> </ul> <p></p> <p> 그림 7-8   이진 트리의 최선 구조와 최악 구조 </p> <p>아래 표(7-1)와 같이 최선 및 최악 구조에서 이진 트리의 리프 수, 전체 노드 수, 높이 등이 최댓값이나 최솟값에 도달한다.</p> <p> 표 7-1   이진 트리의 최선 구조와 최악 구조 </p> 포화 이진 트리 연결 리스트 \\(i\\) 번째 레벨의 노드 수 \\(2^{i-1}\\) \\(1\\) 높이 \\(h\\)인 트리의 리프 수 \\(2^h\\) \\(1\\) 높이 \\(h\\)인 트리의 전체 노드 수 \\(2^{h+1} - 1\\) \\(h + 1\\) 전체 노드 수가 \\(n\\)인 트리의 높이 \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/","level":1,"title":"7.2   이진 트리 순회","text":"<p>물리적 구조의 관점에서 트리는 연결 리스트 기반 자료 구조이므로, 포인터로 노드를 하나씩 방문하여 순회한다. 하지만 비선형 자료 구조여서 연결 리스트보다 순회가 복잡하며 탐색 알고리즘이 필요하다.</p> <p>이진 트리의 주요 순회 방식은 레벨 순서 순회, 전위 순회, 중위 순회, 후위 순회 등이다.</p>","path":["제 7 장   트리","7.2   이진 트리 순회"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#721","level":2,"title":"7.2.1   레벨 순서 순회","text":"<p>아래 그림(7-9)과 같이 레벨 순서 순회(level-order traversal)는 위에서 아래로 한 레벨씩 순회하며, 각 레벨에서는 왼쪽부터 오른쪽 순서로 노드를 방문한다.</p> <p>레벨 순서 순회는 본질적으로 너비 우선 순회(breadth-first traversal), 즉 너비 우선 탐색(breadth-first search, BFS)에 속한다. '한 겹씩 바깥으로 확장'하는 층별 순회 방식이다.</p> <p></p> <p> 그림 7-9   이진 트리의 레벨 순서 순회 </p>","path":["제 7 장   트리","7.2   이진 트리 순회"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#1","level":3,"title":"1.   코드 구현","text":"<p>너비 우선 순회는 보통 '큐'로 구현한다. 큐는 '선입선출' 규칙을, 너비 우선 순회는 '한 레벨씩 진행'하는 규칙을 따르며, 바탕에 깔린 생각은 같다. 구현 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree_bfs.py<pre><code>def level_order(root: TreeNode | None) -&gt; list[int]:\n    \"\"\"레벨 순서 순회\"\"\"\n    # 큐를 초기화하고 루트 노드를 넣는다\n    queue: deque[TreeNode] = deque()\n    queue.append(root)\n    # 순회 순서를 저장할 리스트를 초기화한다\n    res = []\n    while queue:\n        node: TreeNode = queue.popleft()  # 큐에서 꺼낸다\n        res.append(node.val)  # 노드 값을 저장한다\n        if node.left is not None:\n            queue.append(node.left)  # 왼쪽 자식 노드를 큐에 삽입한다\n        if node.right is not None:\n            queue.append(node.right)  # 오른쪽 자식 노드를 큐에 삽입한다\n    return res\n</code></pre> binary_tree_bfs.cpp<pre><code>/* 레벨 순서 순회 */\nvector&lt;int&gt; levelOrder(TreeNode *root) {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    queue&lt;TreeNode *&gt; queue;\n    queue.push(root);\n    // 순회 순서를 저장할 리스트를 초기화한다\n    vector&lt;int&gt; vec;\n    while (!queue.empty()) {\n        TreeNode *node = queue.front();\n        queue.pop();              // 큐에서 꺼낸다\n        vec.push_back(node-&gt;val); // 노드 값을 저장한다\n        if (node-&gt;left != nullptr)\n            queue.push(node-&gt;left); // 왼쪽 자식 노드를 큐에 삽입한다\n        if (node-&gt;right != nullptr)\n            queue.push(node-&gt;right); // 오른쪽 자식 노드를 큐에 삽입한다\n    }\n    return vec;\n}\n</code></pre> binary_tree_bfs.java<pre><code>/* 레벨 순서 순회 */\nList&lt;Integer&gt; levelOrder(TreeNode root) {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    Queue&lt;TreeNode&gt; queue = new LinkedList&lt;&gt;();\n    queue.add(root);\n    // 순회 순서를 저장할 리스트를 초기화한다\n    List&lt;Integer&gt; list = new ArrayList&lt;&gt;();\n    while (!queue.isEmpty()) {\n        TreeNode node = queue.poll(); // 큐에서 꺼낸다\n        list.add(node.val);           // 노드 값을 저장한다\n        if (node.left != null)\n            queue.offer(node.left);   // 왼쪽 자식 노드를 큐에 삽입한다\n        if (node.right != null)\n            queue.offer(node.right);  // 오른쪽 자식 노드를 큐에 삽입한다\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.cs<pre><code>/* 레벨 순서 순회 */\nList&lt;int&gt; LevelOrder(TreeNode root) {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    Queue&lt;TreeNode&gt; queue = new();\n    queue.Enqueue(root);\n    // 순회 순서를 저장할 리스트를 초기화한다\n    List&lt;int&gt; list = [];\n    while (queue.Count != 0) {\n        TreeNode node = queue.Dequeue(); // 큐에서 꺼낸다\n        list.Add(node.val!.Value);       // 노드 값을 저장한다\n        if (node.left != null)\n            queue.Enqueue(node.left);    // 왼쪽 자식 노드를 큐에 삽입한다\n        if (node.right != null)\n            queue.Enqueue(node.right);   // 오른쪽 자식 노드를 큐에 삽입한다\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.go<pre><code>/* 레벨 순서 순회 */\nfunc levelOrder(root *TreeNode) []any {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    queue := list.New()\n    queue.PushBack(root)\n    // 순회 순서를 저장할 슬라이스를 초기화한다\n    nums := make([]any, 0)\n    for queue.Len() &gt; 0 {\n        // 큐에서 꺼낸다\n        node := queue.Remove(queue.Front()).(*TreeNode)\n        // 노드 값을 저장한다\n        nums = append(nums, node.Val)\n        if node.Left != nil {\n            // 왼쪽 자식 노드를 큐에 삽입한다\n            queue.PushBack(node.Left)\n        }\n        if node.Right != nil {\n            // 오른쪽 자식 노드를 큐에 삽입한다\n            queue.PushBack(node.Right)\n        }\n    }\n    return nums\n}\n</code></pre> binary_tree_bfs.swift<pre><code>/* 레벨 순서 순회 */\nfunc levelOrder(root: TreeNode) -&gt; [Int] {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    var queue: [TreeNode] = [root]\n    // 순회 순서를 저장할 리스트를 초기화한다\n    var list: [Int] = []\n    while !queue.isEmpty {\n        let node = queue.removeFirst() // 큐에서 꺼낸다\n        list.append(node.val) // 노드 값을 저장한다\n        if let left = node.left {\n            queue.append(left) // 왼쪽 자식 노드를 큐에 삽입한다\n        }\n        if let right = node.right {\n            queue.append(right) // 오른쪽 자식 노드를 큐에 삽입한다\n        }\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.js<pre><code>/* 레벨 순서 순회 */\nfunction levelOrder(root) {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    const queue = [root];\n    // 순회 순서를 저장할 리스트를 초기화한다\n    const list = [];\n    while (queue.length) {\n        let node = queue.shift(); // 큐에서 꺼낸다\n        list.push(node.val); // 노드 값을 저장한다\n        if (node.left) queue.push(node.left); // 왼쪽 자식 노드를 큐에 삽입한다\n        if (node.right) queue.push(node.right); // 오른쪽 자식 노드를 큐에 삽입한다\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.ts<pre><code>/* 레벨 순서 순회 */\nfunction levelOrder(root: TreeNode | null): number[] {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    const queue = [root];\n    // 순회 순서를 저장할 리스트를 초기화한다\n    const list: number[] = [];\n    while (queue.length) {\n        let node = queue.shift() as TreeNode; // 큐에서 꺼낸다\n        list.push(node.val); // 노드 값을 저장한다\n        if (node.left) {\n            queue.push(node.left); // 왼쪽 자식 노드를 큐에 삽입한다\n        }\n        if (node.right) {\n            queue.push(node.right); // 오른쪽 자식 노드를 큐에 삽입한다\n        }\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.dart<pre><code>/* 레벨 순서 순회 */\nList&lt;int&gt; levelOrder(TreeNode? root) {\n  // 큐를 초기화하고 루트 노드를 넣는다\n  Queue&lt;TreeNode?&gt; queue = Queue();\n  queue.add(root);\n  // 순회 순서를 저장할 리스트를 초기화한다\n  List&lt;int&gt; res = [];\n  while (queue.isNotEmpty) {\n    TreeNode? node = queue.removeFirst(); // 큐에서 꺼낸다\n    res.add(node!.val); // 노드 값을 저장한다\n    if (node.left != null) queue.add(node.left); // 왼쪽 자식 노드를 큐에 삽입한다\n    if (node.right != null) queue.add(node.right); // 오른쪽 자식 노드를 큐에 삽입한다\n  }\n  return res;\n}\n</code></pre> binary_tree_bfs.rs<pre><code>/* 레벨 순서 순회 */\nfn level_order(root: &amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    let mut que = VecDeque::new();\n    que.push_back(root.clone());\n    // 순회 순서를 저장할 리스트를 초기화한다\n    let mut vec = Vec::new();\n\n    while let Some(node) = que.pop_front() {\n        // 큐에서 꺼낸다\n        vec.push(node.borrow().val); // 노드 값을 저장한다\n        if let Some(left) = node.borrow().left.as_ref() {\n            que.push_back(left.clone()); // 왼쪽 자식 노드를 큐에 삽입한다\n        }\n        if let Some(right) = node.borrow().right.as_ref() {\n            que.push_back(right.clone()); // 오른쪽 자식 노드를 큐에 삽입한다\n        };\n    }\n    vec\n}\n</code></pre> binary_tree_bfs.c<pre><code>/* 레벨 순서 순회 */\nint *levelOrder(TreeNode *root, int *size) {\n    /* 보조 큐 */\n    int front, rear;\n    int index, *arr;\n    TreeNode *node;\n    TreeNode **queue;\n\n    /* 보조 큐 */\n    queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);\n    // 큐 포인터\n    front = 0, rear = 0;\n    // 루트 노드를 넣는다\n    queue[rear++] = root;\n    // 순회 순서를 저장할 리스트를 초기화한다\n    /* 보조 배열 */\n    arr = (int *)malloc(sizeof(int) * MAX_SIZE);\n    // 배열 포인터\n    index = 0;\n    while (front &lt; rear) {\n        // 큐에서 꺼낸다\n        node = queue[front++];\n        // 노드 값을 저장한다\n        arr[index++] = node-&gt;val;\n        if (node-&gt;left != NULL) {\n            // 왼쪽 자식 노드를 큐에 삽입한다\n            queue[rear++] = node-&gt;left;\n        }\n        if (node-&gt;right != NULL) {\n            // 오른쪽 자식 노드를 큐에 삽입한다\n            queue[rear++] = node-&gt;right;\n        }\n    }\n    // 배열 길이 값을 갱신한다\n    *size = index;\n    arr = realloc(arr, sizeof(int) * (*size));\n\n    // 보조 배열의 메모리 공간을 해제한다\n    free(queue);\n    return arr;\n}\n</code></pre> binary_tree_bfs.kt<pre><code>/* 레벨 순서 순회 */\nfun levelOrder(root: TreeNode?): MutableList&lt;Int&gt; {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    val queue = LinkedList&lt;TreeNode?&gt;()\n    queue.add(root)\n    // 순회 순서를 저장할 리스트를 초기화한다\n    val list = mutableListOf&lt;Int&gt;()\n    while (queue.isNotEmpty()) {\n        val node = queue.poll()      // 큐에서 꺼낸다\n        list.add(node?._val!!)       // 노드 값을 저장한다\n        if (node.left != null)\n            queue.offer(node.left)   // 왼쪽 자식 노드를 큐에 삽입한다\n        if (node.right != null)\n            queue.offer(node.right)  // 오른쪽 자식 노드를 큐에 삽입한다\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.rb<pre><code>### 레벨 순서 순회 ###\ndef level_order(root)\n  # 큐를 초기화하고 루트 노드를 넣는다\n  queue = [root]\n  # 순회 순서를 저장할 리스트를 초기화한다\n  res = []\n  while !queue.empty?\n    node = queue.shift # 큐에서 꺼낸다\n    res &lt;&lt; node.val # 노드 값을 저장한다\n    queue &lt;&lt; node.left unless node.left.nil? # 왼쪽 자식 노드를 큐에 삽입한다\n    queue &lt;&lt; node.right unless node.right.nil? # 오른쪽 자식 노드를 큐에 삽입한다\n  end\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 7 장   트리","7.2   이진 트리 순회"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#2","level":3,"title":"2.   복잡도 분석","text":"<ul> <li>시간 복잡도 \\(O(n)\\): 모든 노드를 한 번씩 방문하므로 \\(O(n)\\) 시간이 걸린다. \\(n\\)은 노드 수다.</li> <li>공간 복잡도 \\(O(n)\\): 최악의 경우인 포화 이진 트리에서는 맨 아래 레벨을 순회하기 직전에 큐에 최대 \\((n + 1) / 2\\) 개 노드가 동시에 들어 있어 \\(O(n)\\) 공간을 차지한다.</li> </ul>","path":["제 7 장   트리","7.2   이진 트리 순회"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#722","level":2,"title":"7.2.2   전위, 중위, 후위 순회","text":"<p>전위, 중위, 후위 순회는 모두 깊이 우선 순회(depth-first traversal), 즉 깊이 우선 탐색(depth-first search, DFS)에 속한다. '먼저 끝까지 간 뒤 돌아와 계속하는' 순회 방식이다.</p> <p>아래 그림(7-10)은 이진 트리의 깊이 우선 순회 원리를 보여 준다. 깊이 우선 순회는 트리 전체의 바깥을 한 바퀴 '도는' 것과 같다. 각 노드에서 세 위치를 만나며, 각각 전위, 중위, 후위 순회에 대응한다.</p> <p></p> <p> 그림 7-10   이진 탐색 트리의 전위, 중위, 후위 순회 </p>","path":["제 7 장   트리","7.2   이진 트리 순회"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#1_1","level":3,"title":"1.   코드 구현","text":"<p>깊이 우선 탐색은 보통 재귀로 구현한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree_dfs.py<pre><code>def pre_order(root: TreeNode | None):\n    \"\"\"전위 순회\"\"\"\n    if root is None:\n        return\n    # 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    res.append(root.val)\n    pre_order(root=root.left)\n    pre_order(root=root.right)\n\ndef in_order(root: TreeNode | None):\n    \"\"\"중위 순회\"\"\"\n    if root is None:\n        return\n    # 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    in_order(root=root.left)\n    res.append(root.val)\n    in_order(root=root.right)\n\ndef post_order(root: TreeNode | None):\n    \"\"\"후위 순회\"\"\"\n    if root is None:\n        return\n    # 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    post_order(root=root.left)\n    post_order(root=root.right)\n    res.append(root.val)\n</code></pre> binary_tree_dfs.cpp<pre><code>/* 전위 순회 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    vec.push_back(root-&gt;val);\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n\n/* 중위 순회 */\nvoid inOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(root-&gt;left);\n    vec.push_back(root-&gt;val);\n    inOrder(root-&gt;right);\n}\n\n/* 후위 순회 */\nvoid postOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(root-&gt;left);\n    postOrder(root-&gt;right);\n    vec.push_back(root-&gt;val);\n}\n</code></pre> binary_tree_dfs.java<pre><code>/* 전위 순회 */\nvoid preOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    list.add(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* 중위 순회 */\nvoid inOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(root.left);\n    list.add(root.val);\n    inOrder(root.right);\n}\n\n/* 후위 순회 */\nvoid postOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(root.left);\n    postOrder(root.right);\n    list.add(root.val);\n}\n</code></pre> binary_tree_dfs.cs<pre><code>/* 전위 순회 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) return;\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    list.Add(root.val!.Value);\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n\n/* 중위 순회 */\nvoid InOrder(TreeNode? root) {\n    if (root == null) return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    InOrder(root.left);\n    list.Add(root.val!.Value);\n    InOrder(root.right);\n}\n\n/* 후위 순회 */\nvoid PostOrder(TreeNode? root) {\n    if (root == null) return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    PostOrder(root.left);\n    PostOrder(root.right);\n    list.Add(root.val!.Value);\n}\n</code></pre> binary_tree_dfs.go<pre><code>/* 전위 순회 */\nfunc preOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    nums = append(nums, node.Val)\n    preOrder(node.Left)\n    preOrder(node.Right)\n}\n\n/* 중위 순회 */\nfunc inOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(node.Left)\n    nums = append(nums, node.Val)\n    inOrder(node.Right)\n}\n\n/* 후위 순회 */\nfunc postOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(node.Left)\n    postOrder(node.Right)\n    nums = append(nums, node.Val)\n}\n</code></pre> binary_tree_dfs.swift<pre><code>/* 전위 순회 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    list.append(root.val)\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n\n/* 중위 순회 */\nfunc inOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(root: root.left)\n    list.append(root.val)\n    inOrder(root: root.right)\n}\n\n/* 후위 순회 */\nfunc postOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(root: root.left)\n    postOrder(root: root.right)\n    list.append(root.val)\n}\n</code></pre> binary_tree_dfs.js<pre><code>/* 전위 순회 */\nfunction preOrder(root) {\n    if (root === null) return;\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* 중위 순회 */\nfunction inOrder(root) {\n    if (root === null) return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* 후위 순회 */\nfunction postOrder(root) {\n    if (root === null) return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.ts<pre><code>/* 전위 순회 */\nfunction preOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* 중위 순회 */\nfunction inOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* 후위 순회 */\nfunction postOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.dart<pre><code>/* 전위 순회 */\nvoid preOrder(TreeNode? node) {\n  if (node == null) return;\n  // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n  list.add(node.val);\n  preOrder(node.left);\n  preOrder(node.right);\n}\n\n/* 중위 순회 */\nvoid inOrder(TreeNode? node) {\n  if (node == null) return;\n  // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n  inOrder(node.left);\n  list.add(node.val);\n  inOrder(node.right);\n}\n\n/* 후위 순회 */\nvoid postOrder(TreeNode? node) {\n  if (node == null) return;\n  // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n  postOrder(node.left);\n  postOrder(node.right);\n  list.add(node.val);\n}\n</code></pre> binary_tree_dfs.rs<pre><code>/* 전위 순회 */\nfn pre_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n            let node = node.borrow();\n            res.push(node.val);\n            dfs(node.left.as_ref(), res);\n            dfs(node.right.as_ref(), res);\n        }\n    }\n    dfs(root, &amp;mut result);\n\n    result\n}\n\n/* 중위 순회 */\nfn in_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n            let node = node.borrow();\n            dfs(node.left.as_ref(), res);\n            res.push(node.val);\n            dfs(node.right.as_ref(), res);\n        }\n    }\n    dfs(root, &amp;mut result);\n\n    result\n}\n\n/* 후위 순회 */\nfn post_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n            let node = node.borrow();\n            dfs(node.left.as_ref(), res);\n            dfs(node.right.as_ref(), res);\n            res.push(node.val);\n        }\n    }\n\n    dfs(root, &amp;mut result);\n\n    result\n}\n</code></pre> binary_tree_dfs.c<pre><code>/* 전위 순회 */\nvoid preOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    arr[(*size)++] = root-&gt;val;\n    preOrder(root-&gt;left, size);\n    preOrder(root-&gt;right, size);\n}\n\n/* 중위 순회 */\nvoid inOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(root-&gt;left, size);\n    arr[(*size)++] = root-&gt;val;\n    inOrder(root-&gt;right, size);\n}\n\n/* 후위 순회 */\nvoid postOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(root-&gt;left, size);\n    postOrder(root-&gt;right, size);\n    arr[(*size)++] = root-&gt;val;\n}\n</code></pre> binary_tree_dfs.kt<pre><code>/* 전위 순회 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) return\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    list.add(root._val)\n    preOrder(root.left)\n    preOrder(root.right)\n}\n\n/* 중위 순회 */\nfun inOrder(root: TreeNode?) {\n    if (root == null) return\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(root.left)\n    list.add(root._val)\n    inOrder(root.right)\n}\n\n/* 후위 순회 */\nfun postOrder(root: TreeNode?) {\n    if (root == null) return\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(root.left)\n    postOrder(root.right)\n    list.add(root._val)\n}\n</code></pre> binary_tree_dfs.rb<pre><code>### 전위 순회 ###\ndef pre_order(root)\n  return if root.nil?\n\n  # 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n  $res &lt;&lt; root.val\n  pre_order(root.left)\n  pre_order(root.right)\nend\n\n### 중위 순회 ###\ndef in_order(root)\n  return if root.nil?\n\n  # 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n  in_order(root.left)\n  $res &lt;&lt; root.val\n  in_order(root.right)\nend\n\n### 후위 순회 ###\ndef post_order(root)\n  return if root.nil?\n\n  # 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n  post_order(root.left)\n  post_order(root.right)\n  $res &lt;&lt; root.val\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>참고</p> <p>깊이 우선 탐색은 반복으로도 구현할 수 있다. 관심 있는 독자는 직접 알아보자.</p> <p>아래 그림(7-11)은 이진 트리 전위 순회의 재귀 과정을 보여 준다. 서로 반대 방향인 '호출'과 '반환'으로 나뉜다.</p> <ol> <li>'호출'은 새 메서드를 시작하는 것으로, 프로그램은 이 과정에서 다음 노드를 방문한다.</li> <li>'반환'은 함수가 돌아오는 것으로, 현재 노드의 방문이 끝났음을 뜻한다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 7-11   전위 순회의 재귀 과정 </p>","path":["제 7 장   트리","7.2   이진 트리 순회"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#2_1","level":3,"title":"2.   복잡도 분석","text":"<ul> <li>시간 복잡도 \\(O(n)\\): 모든 노드를 한 번씩 방문하므로 \\(O(n)\\) 시간이 걸린다.</li> <li>공간 복잡도 \\(O(n)\\): 최악의 경우 트리가 연결 리스트로 퇴화하면 재귀 깊이가 \\(n\\)에 이르러 시스템이 \\(O(n)\\)의 스택 프레임 공간을 사용한다.</li> </ul>","path":["제 7 장   트리","7.2   이진 트리 순회"],"tags":[]},{"location":"chapter_tree/exercises/","level":1,"title":"7.7   연습 문제","text":"","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#771","level":2,"title":"7.7.1   개념 확인","text":"","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#1","level":3,"title":"1.   완전 이진 트리, 정 이진 트리, 포화 이진 트리","text":"<p>다음 두 배열은 이진 트리를 레벨 순서로 나타낸 것이다. <code>None</code>은 빈자리를 뜻한다.</p> <ul> <li>트리 A: <code>[1, 2, 3, 4, 5, 6]</code></li> <li>트리 B: <code>[1, 2, 3, None, None, 6, 7]</code></li> </ul> <ol> <li>어느 트리가 완전 이진 트리인가?</li> <li>어느 트리가 모든 비리프 노드에 자식이 두 개씩 있는 정 이진 트리인가?</li> <li>두 트리 중 포화 이진 트리가 있는가? 각각 이유를 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>트리 A는 완전 이진 트리다. 맨 아래 레벨만 가득 차지 않았고, 노드가 왼쪽부터 연속해서 배치되어 있다.     트리 B는 맨 아래 레벨의 왼쪽에 빈자리가 있는데 오른쪽에 노드가 있으므로 완전 이진 트리가 아니다.</p> </li> <li> <p>트리 B는 정 이진 트리다. 노드 1과 노드 3은 자식이 각각 두 개이고, 나머지는 모두 리프 노드다.     트리 A의 노드 3에는 왼쪽 자식 6만 있으므로 정 이진 트리가 아니다.</p> </li> <li> <p>두 트리 모두 맨 아래 레벨이 완전히 채워져 있지 않으므로 포화 이진 트리가 아니다.</p> </li> </ol>","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#2","level":3,"title":"2.   같은 트리의 세 가지 순회 순서","text":"<p>배열 <code>[1, 2, 3, 4, 5, 6, 7]</code>을 레벨 순서로 완전 이진 트리에 저장한다.</p> <ol> <li>이 트리를 그려 보자.</li> <li>전위, 중위, 후위 순회 결과를 각각 적어 보자.</li> <li>중위 순회 결과에서 루트 노드 1의 왼쪽과 오른쪽에 있는 두 수열은 각각 트리의 어느 부분에 대응하는가?</li> </ol> 참고 답안 <ol> <li> <p>트리는 다음과 같다.</p> <pre><code>      1\n    /   \\\n   2     3\n  / \\   / \\\n 4   5 6   7\n</code></pre> </li> <li> <p>전위 순회는 <code>1, 2, 4, 5, 3, 6, 7</code>이다.     중위 순회는 <code>4, 2, 5, 1, 6, 3, 7</code>이다.     후위 순회는 <code>4, 5, 2, 6, 7, 3, 1</code>이다.</p> </li> <li> <p>루트 노드 1의 왼쪽에 있는 <code>4, 2, 5</code>는 왼쪽 서브트리의 중위 순회 결과다.     오른쪽의 <code>6, 3, 7</code>은 오른쪽 서브트리의 중위 순회 결과다.</p> </li> </ol>","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#3","level":3,"title":"3.   두 이진 탐색 트리 비교하기","text":"<p>다음 두 수열을 각각 왼쪽부터 빈 이진 탐색 트리에 차례대로 삽입한다.</p> <ul> <li>수열 A: <code>[4, 2, 6, 1, 3, 5, 7]</code></li> <li>수열 B: <code>[1, 2, 3, 4, 5, 6, 7]</code></li> </ul> <ol> <li>숫자 7을 찾을 때 거치는 노드를 각각 적어 보자.</li> <li>높이를 루트에서 가장 먼 리프까지 지나는 간선 수로 계산할 때, 두 트리의 높이는 각각 얼마인가?</li> <li>앞선 두 질문을 바탕으로 두 트리에서 7을 찾는 효율성이 같은지 판단하라. 트리의 모양과 탐색 경로를 바탕으로 이유를 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>수열 A로 만든 트리의 탐색 경로는 <code>4 → 6 → 7</code>이다.     수열 B로 만든 트리의 탐색 경로는 <code>1 → 2 → 3 → 4 → 5 → 6 → 7</code>이다.</p> </li> <li> <p>첫 번째 트리는 모든 레벨이 가득 차 있고 높이는 2다. 두 번째 트리는 오른쪽 자식만 있으며 높이는 6이다.</p> </li> <li> <p>두 트리에서 7을 찾는 효율성은 다르다. 삽입 순서가 이진 탐색 트리의 모양과 높이를 바꾸기 때문이다. 첫 번째 트리는 노드 3개만 거치지만,     두 번째 트리는 노드 7개를 차례대로 거친다. 트리가 높을수록 최악의 경우 경로를 따라 비교해야 하는 노드가 많아진다.</p> </li> </ol>","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#772","level":2,"title":"7.7.2   프로그래밍 연습","text":"","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#1_1","level":3,"title":"1.   이진 트리의 최대 깊이","text":"<p>이진 트리의 루트 노드 <code>root</code>가 주어진다. 각 노드는 정숫값과 왼쪽 및 오른쪽 자식 노드에 대한 참조를 가진다.</p> <p>루트에서 가장 먼 리프까지 거치는 노드 수를 이진 트리의 최대 깊이라고 정의한다. 이 트리의 최대 깊이를 반환하라. 빈 트리의 최대 깊이는 0이다. 재귀를 사용하여 해결하라.</p> 풀이 힌트 <ol> <li>이 문제에서는 노드 수로 깊이를 계산한다. 루트 하나만 있는 트리의 최대 깊이는 1이다.</li> <li>재귀 함수가 현재 노드를 루트로 하는 서브트리의 최대 깊이를 반환하도록 한다.</li> <li>빈 노드는 0을 반환하고, 비어 있지 않은 노드는 max(depth(left), depth(right)) + 1을 반환한다.</li> </ol> <p>LeetCode</p>","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#2_1","level":3,"title":"2.   이진 트리를 레벨 순서로 순회하기","text":"<p>이진 트리의 루트 노드 <code>root</code>가 주어진다. 큐를 사용하여 위에서 아래로 한 레벨씩 모든 노드를 방문하라. 같은 레벨에서는 왼쪽부터 오른쪽으로 방문한다.</p> <p>2차원 배열을 반환하라. 첫 번째 하위 배열에는 루트가 있는 레벨의 노드 값을, 두 번째에는 다음 레벨의 노드 값을 저장하며 이 방식을 반복한다. 빈 트리라면 빈 배열을 반환하라.</p> 풀이 힌트 <ol> <li>레벨 순서 순회에서는 먼저 들어온 노드를 먼저 방문해야 하므로 큐를 사용한다.</li> <li>각 반복이 시작될 때 큐에 있는 노드들은 정확히 같은 레벨에 속한다.</li> <li>큐의 길이를 먼저 기록한 뒤 그 수만큼 노드를 꺼내고 각 노드의 자식을 넣는다.</li> </ol> <p>LeetCode</p>","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#3-k","level":3,"title":"3.   이진 탐색 트리에서 k번째로 작은 원소","text":"<p>이진 탐색 트리에 총 <code>n</code>개의 노드가 있으며, 노드 값은 모두 다르다. 모든 노드 값을 작은 것부터 나열하고 위치에 1부터 번호를 붙인다.</p> <p>루트 노드 <code>root</code>와 <code>1 &lt;= k &lt;= n</code>을 만족하는 정수 <code>k</code>가 주어질 때, <code>k</code>번째 노드 값을 반환하라. 모든 값을 미리 모으지 말고 중위 순회 도중에 바로 답을 찾아라.</p> 풀이 힌트 <ol> <li>이진 탐색 트리를 중위 순회하면 노드 값을 작은 것부터 방문한다.</li> <li>중위 순회는 왼쪽 서브트리, 현재 노드, 오른쪽 서브트리 순서로 처리한다. 현재 노드를 방문할 때 개수를 1 늘린다.</li> <li>개수가 처음 k가 되었을 때의 노드 값이 답이다. 이후에는 순회를 계속할 필요가 없다.</li> </ol> <p>LeetCode</p>","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/summary/","level":1,"title":"7.6   요약","text":"","path":["제 7 장   트리","7.6   요약"],"tags":[]},{"location":"chapter_tree/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>이진 트리는 '하나를 둘로 나누는' 분할 정복 논리를 보여 주는 비선형 자료 구조다. 각 노드는 값 하나와 왼쪽, 오른쪽 자식을 가리키는 포인터 두 개를 포함한다.</li> <li>어떤 노드의 왼쪽(오른쪽) 자식과 그 아래 노드들이 이루는 트리를 해당 노드의 왼쪽(오른쪽) 서브트리라고 한다.</li> <li>주요 용어에는 루트 노드, 리프 노드, 레벨, 차수, 간선, 높이, 깊이 등이 있다.</li> <li>이진 트리의 초기화, 노드 삽입과 삭제 방법은 연결 리스트와 비슷하다.</li> <li>주요 종류는 포화 이진 트리, 완전 이진 트리, 정 이진 트리, 균형 이진 트리다. 포화 이진 트리가 가장 이상적이며, 연결 리스트는 퇴화한 최악의 상태다.</li> <li>이진 트리는 노드 값과 빈자리를 레벨 순서대로 배열하여 배열로 표현할 수 있다. 부모와 자식의 인덱스 매핑 관계가 포인터를 대신한다.</li> <li>레벨 순서 순회는 '한 겹씩 바깥으로 확장'하는 너비 우선 탐색이며, 보통 큐로 구현한다.</li> <li>전위, 중위, 후위 순회는 '먼저 끝까지 간 뒤 돌아와 계속하는' 깊이 우선 탐색이며, 보통 재귀로 구현한다.</li> <li>이진 탐색 트리는 효율적인 원소 탐색 자료 구조로, 탐색, 삽입, 삭제의 시간 복잡도는 모두 \\(O(\\log n)\\)이다. 연결 리스트로 퇴화하면 각 복잡도가 \\(O(n)\\)으로 저하된다.</li> <li>AVL 트리는 균형 이진 탐색 트리라고도 하며, 회전 연산으로 삽입과 삭제가 반복되어도 균형을 유지한다.</li> <li>AVL 트리의 회전에는 오른쪽 회전, 왼쪽 회전, 오른쪽 후 왼쪽 회전, 왼쪽 후 오른쪽 회전이 있다. 삽입이나 삭제 후 아래에서 위로 회전하여 균형을 복구한다.</li> </ul>","path":["제 7 장   트리","7.6   요약"],"tags":[]},{"location":"chapter_tree/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 노드가 하나뿐인 이진 트리의 높이와 루트의 깊이는 모두 \\(0\\)인가?</p> <p>그렇다. 높이와 깊이는 보통 '거치는 간선 수'로 정의하기 때문이다.</p> <p>Q: 이진 트리의 삽입과 삭제를 보통 여러 연산의 조합으로 수행한다고 했는데, 여기서 '여러 연산'은 무엇인가? 해당 자원에 속한 자식 노드의 자원을 해제하는 것으로 이해해도 되는가?</p> <p>이진 탐색 트리를 예로 들면 삭제를 세 경우로 나누어 처리하며, 각 경우마다 여러 단계의 노드 연산이 필요하다.</p> <p>Q: 이진 트리의 DFS에는 왜 전위, 중위, 후위라는 세 순서가 있고 각각 어디에 쓰이는가?</p> <p>배열을 정방향이나 역방향으로 순회하듯이 전위, 중위, 후위는 이진 트리를 순회하는 세 방법이며 특정 순서의 결과를 얻는 데 사용한다. 예를 들어 이진 탐색 트리는 <code>왼쪽 자식 값 &lt; 루트 값 &lt; 오른쪽 자식 값</code>을 만족하므로 '왼쪽 \\(\\rightarrow\\) 루트 \\(\\rightarrow\\) 오른쪽' 우선순위로 순회하면 정렬된 노드 순서를 얻는다.</p> <p>Q: 오른쪽 회전은 불균형 노드 <code>node</code>, <code>child</code>, <code>grand_child</code> 사이의 관계를 처리하는데, <code>node</code>와 그 부모의 기존 연결은 관리하지 않아도 되는가? 회전 후 끊어지지 않는가?</p> <p>재귀의 관점에서 보아야 한다. <code>right_rotate(root)</code>는 서브트리의 루트를 전달받고, 마지막의 <code>return child</code>는 회전 후의 서브트리 루트를 반환한다. 새 서브트리 루트와 부모의 연결은 함수가 반환된 뒤 이루어지므로 오른쪽 회전 자체가 관리할 범위가 아니다.</p> <p>Q: C++에서 함수를 <code>private</code>와 <code>public</code>으로 나누는 기준은 무엇인가? <code>height()</code>는 <code>public</code>, <code>updateHeight()</code>는 <code>private</code>에 두는 이유는 무엇인가?</p> <p>주로 메서드의 사용 범위를 기준으로 한다. 클래스 내부에서만 쓰면 <code>private</code>로 설계한다. 예를 들어 <code>updateHeight()</code>는 삽입과 삭제의 한 단계일 뿐이라 사용자가 따로 호출할 의미가 없다. 반면 <code>height()</code>는 <code>vector.size()</code>처럼 노드 높이에 접근하는 기능이므로 편리하게 쓰도록 <code>public</code>으로 둔다.</p> <p>Q: 입력 데이터 집합에서 이진 탐색 트리를 어떻게 구성하는가? 루트 선택이 중요한가?</p> <p>그렇다. 구성 방법은 이진 탐색 트리 코드의 <code>build_tree()</code>에 제시되어 있다. 루트는 보통 입력을 정렬한 뒤 중간 원소로 정하고, 왼쪽과 오른쪽 서브트리를 재귀적으로 구성한다. 이렇게 하면 트리의 균형을 최대한 보장할 수 있다.</p> <p>Q: Java에서 문자열을 비교할 때 반드시 <code>equals()</code>를 사용해야 하는가?</p> <p>Java에서 기본 타입의 <code>==</code>는 두 변수의 값이 같은지 비교한다. 참조 타입에서는 두 방식의 원리가 다르다.</p> <ul> <li><code>==</code>: 두 변수가 같은 객체, 즉 메모리의 같은 위치를 가리키는지 비교한다.</li> <li><code>equals()</code>: 두 객체의 값이 같은지 비교한다.</li> </ul> <p>따라서 값을 비교하려면 <code>equals()</code>를 사용해야 한다. 다만 <code>String a = \"hi\"; String b = \"hi\";</code>로 초기화한 문자열들은 문자열 상수 풀에 저장되어 같은 객체를 가리키므로, 이 경우에는 <code>a == b</code>로 내용이 같은지 비교할 수도 있다.</p> <p>Q: 너비 우선 순회에서 맨 아래 레벨에 도달하기 전 큐의 노드 수는 \\(2^h\\)인가?</p> <p>그렇다. 예를 들어 높이 \\(h = 2\\)인 포화 이진 트리의 전체 노드 수는 \\(n = 7\\)이고, 맨 아래 레벨의 노드 수는 \\(4 = 2^h = (n + 1) / 2\\)다.</p>","path":["제 7 장   트리","7.6   요약"],"tags":[]}]}
\ No newline at end of file
+{"config":{"separator":"[\\s\\-_,:!=\\[\\]()\\\\\"`/]+|\\.(?!\\d)"},"items":[{"location":"chapter_appendix/","level":1,"title":"제 16 장   부록","text":"","path":["제 16 장   부록"],"tags":[]},{"location":"chapter_appendix/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>16.1   프로그래밍 환경 설치</li> <li>16.2   함께 만드는 책</li> <li>16.3   용어집</li> </ul>","path":["제 16 장   부록"],"tags":[]},{"location":"chapter_appendix/contribution/","level":1,"title":"16.2   함께 만드는 책","text":"<p>저자의 역량이 부족하여 책에 누락이나 오류가 있을 수 있다는 점을 양해 부탁드린다. 오탈자, 끊어진 링크, 빠진 내용, 모호한 표현, 불분명한 설명, 부자연스러운 글의 구성 등을 발견했다면 수정에 힘을 보태 주기 바란다. 독자에게 더 좋은 학습 자료를 제공하는 데 큰 도움이 된다.</p> <p>모든 집필 참여자의 GitHub ID는 저장소와 웹 버전의 첫 페이지에 표시된다. 이는 오픈 소스 커뮤니티에 아낌없이 기여한 분들에게 감사를 전하기 위한 것이다.</p> <p>오픈 소스의 매력</p> <p>종이책은 인쇄와 다음 인쇄 사이의 간격이 길어 내용을 갱신하기가 매우 불편하다.</p> <p>반면 이 오픈 소스 도서에서는 내용을 갱신하는 데 걸리는 시간이 며칠, 때로는 몇 시간으로 줄어든다.</p>","path":["제 16 장   부록","16.2   함께 만드는 책"],"tags":[]},{"location":"chapter_appendix/contribution/#1","level":3,"title":"1.   간단한 내용 수정","text":"<p>아래 그림(16-3)과 같이 각 페이지의 오른쪽 위에는 ‘편집 아이콘’이 있다. 다음 순서로 글이나 코드를 수정할 수 있다.</p> <ol> <li>‘편집 아이콘’을 누른다. 저장소를 포크해야 한다는 안내가 나오면 동의한다.</li> <li>Markdown 소스 파일을 수정하고 내용이 올바른지 확인한다. 서식은 가능한 한 일관되게 유지한다.</li> <li>페이지 아래쪽에 수정 내용을 설명하고 ‘Propose file change’ 버튼을 누른다. 이동한 페이지에서 ‘Create pull request’ 버튼을 누르면 풀 리퀘스트를 만들 수 있다.</li> </ol> <p></p> <p> 그림 16-3   페이지 편집 버튼 </p> <p>그림은 직접 수정할 수 없으므로 Issue를 만들거나 댓글로 문제를 설명해 주기 바란다. 확인 후 최대한 빨리 그림을 다시 제작하여 교체하겠다.</p>","path":["제 16 장   부록","16.2   함께 만드는 책"],"tags":[]},{"location":"chapter_appendix/contribution/#2","level":3,"title":"2.   내용 작성","text":"<p>코드를 다른 프로그래밍 언어로 옮기거나 글의 내용을 확장하는 등 이 오픈 소스 프로젝트에 참여하고 싶다면 다음 풀 리퀘스트 절차를 따르면 된다.</p> <ol> <li>GitHub에 로그인한 뒤 이 책의 코드 저장소를 개인 계정으로 포크한다.</li> <li>포크한 저장소의 웹페이지로 이동하여 <code>git clone</code> 명령으로 저장소를 로컬에 복제한다.</li> <li>로컬에서 내용을 작성하고 충분히 테스트하여 코드가 올바른지 검증한다.</li> <li>로컬 변경 사항을 커밋한 뒤 원격 저장소로 푸시한다.</li> <li>저장소 웹페이지를 새로 고침하고 ‘Create pull request’ 버튼을 눌러 풀 리퀘스트를 만든다.</li> </ol>","path":["제 16 장   부록","16.2   함께 만드는 책"],"tags":[]},{"location":"chapter_appendix/contribution/#3-docker","level":3,"title":"3.   Docker 배포","text":"<p><code>hello-algo</code> 루트 디렉터리에서 다음 Docker 명령을 실행하면 <code>http://localhost:8000/ko/</code>에서 한국어판을 열 수 있다.</p> <pre><code>docker-compose up -d\n</code></pre> <p>다음 명령으로 배포를 제거할 수 있다.</p> <pre><code>docker-compose down\n</code></pre>","path":["제 16 장   부록","16.2   함께 만드는 책"],"tags":[]},{"location":"chapter_appendix/installation/","level":1,"title":"16.1   프로그래밍 환경 설치","text":"","path":["제 16 장   부록","16.1   프로그래밍 환경 설치"],"tags":[]},{"location":"chapter_appendix/installation/#1611-ide","level":2,"title":"16.1.1   IDE 설치","text":"<p>로컬 통합 개발 환경(IDE)으로는 가벼운 오픈 소스 도구인 VS Code를 권장한다. VS Code 공식 웹사이트에서 운영체제에 맞는 버전을 내려받아 설치한다.</p> <p></p> <p> 그림 16-1   공식 웹사이트에서 VS Code 내려받기 </p> <p>VS Code는 풍부한 확장 생태계를 갖추고 있어 대부분의 프로그래밍 언어를 실행하고 디버깅할 수 있다. Python을 예로 들면, ‘Python Extension Pack’을 설치한 뒤 Python 코드를 디버깅할 수 있다. 설치 과정은 아래 그림(16-2)과 같다.</p> <p></p> <p> 그림 16-2   VS Code 확장 설치 </p>","path":["제 16 장   부록","16.1   프로그래밍 환경 설치"],"tags":[]},{"location":"chapter_appendix/installation/#1612","level":2,"title":"16.1.2   언어 실행 환경 설치","text":"","path":["제 16 장   부록","16.1   프로그래밍 환경 설치"],"tags":[]},{"location":"chapter_appendix/installation/#1-python","level":3,"title":"1.   Python 환경","text":"<ol> <li>Miniconda3를 내려받아 설치한다. Python 3.10 이상이 필요하다.</li> <li>VS Code 확장 마켓플레이스에서 <code>python</code>을 검색하여 Python Extension Pack을 설치한다.</li> <li>(선택 사항) 명령줄에 <code>pip install black</code>을 입력하여 코드 포매터를 설치한다.</li> </ol>","path":["제 16 장   부록","16.1   프로그래밍 환경 설치"],"tags":[]},{"location":"chapter_appendix/installation/#2-cc","level":3,"title":"2.   C/C++ 환경","text":"<ol> <li>Windows에서는 MinGW를 설치해야 한다(설정 안내). macOS에는 Clang이 기본으로 포함되어 있어 별도로 설치할 필요가 없다.</li> <li>VS Code 확장 마켓플레이스에서 <code>c++</code>를 검색하여 C/C++ Extension Pack을 설치한다.</li> <li>(선택 사항) Settings 페이지에서 코드 서식 옵션인 <code>Clang_format_fallback Style</code>을 검색하고 <code>{ BasedOnStyle: Microsoft, BreakBeforeBraces: Attach }</code>로 설정한다.</li> </ol>","path":["제 16 장   부록","16.1   프로그래밍 환경 설치"],"tags":[]},{"location":"chapter_appendix/installation/#3-java","level":3,"title":"3.   Java 환경","text":"<ol> <li>OpenJDK를 내려받아 설치한다(JDK 9보다 높은 버전이어야 한다).</li> <li>VS Code 확장 마켓플레이스에서 <code>java</code>를 검색하여 Extension Pack for Java를 설치한다.</li> </ol>","path":["제 16 장   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설치"],"tags":[]},{"location":"chapter_appendix/installation/#7-javascript","level":3,"title":"7.   JavaScript 환경","text":"<ol> <li>Node.js를 내려받아 설치한다.</li> <li>(선택 사항) VS Code 확장 마켓플레이스에서 <code>Prettier</code>를 검색하여 코드 포매터를 설치한다.</li> </ol>","path":["제 16 장   부록","16.1   프로그래밍 환경 설치"],"tags":[]},{"location":"chapter_appendix/installation/#8-typescript","level":3,"title":"8.   TypeScript 환경","text":"<ol> <li>JavaScript 환경과 같은 순서로 설치한다.</li> <li>TypeScript Execute (tsx)를 설치한다.</li> <li>VS Code 확장 마켓플레이스에서 <code>typescript</code>를 검색하여 Pretty TypeScript Errors를 설치한다.</li> </ol>","path":["제 16 장   부록","16.1   프로그래밍 환경 설치"],"tags":[]},{"location":"chapter_appendix/installation/#9-dart","level":3,"title":"9.   Dart 환경","text":"<ol> <li>Dart를 내려받아 설치한다.</li> <li>VS Code 확장 마켓플레이스에서 <code>dart</code>를 검색하여 Dart를 설치한다.</li> </ol>","path":["제 16 장   부록","16.1   프로그래밍 환경 설치"],"tags":[]},{"location":"chapter_appendix/installation/#10-rust","level":3,"title":"10.   Rust 환경","text":"<ol> <li>Rust를 내려받아 설치한다.</li> <li>VS Code 확장 마켓플레이스에서 <code>rust</code>를 검색하여 rust-analyzer를 설치한다.</li> </ol>","path":["제 16 장   부록","16.1   프로그래밍 환경 설치"],"tags":[]},{"location":"chapter_appendix/terminology/","level":1,"title":"16.3   용어집","text":"<p>아래 표(16-1)에는 이 책에 나오는 주요 용어를 정리했다. 영어 문헌을 읽을 때 도움이 되므로 각 용어의 영어 명칭을 기억해 두기를 권장한다.</p> <p> 표 16-1   자료 구조와 알고리즘의 주요 용어 </p> English 한국어 algorithm 알고리즘 data structure 자료 구조 code 코드 file 파일 function 함수 method 메서드 variable 변수 asymptotic complexity analysis 점근적 복잡도 분석 time complexity 시간 복잡도 space complexity 공간 복잡도 loop 반복문 iteration 반복 recursion 재귀 tail recursion 꼬리 재귀 recursion tree 재귀 트리 big-\\(O\\) notation 빅 \\(O\\) 표기법 asymptotic upper bound 점근적 상한 sign-magnitude 부호와 크기 표현 1’s complement 1의 보수 2’s complement 2의 보수 array 배열 index 인덱스 linked list 연결 리스트 linked list node, list node 연결 리스트 노드 head node 머리 노드 tail node 꼬리 노드 list 리스트 dynamic array 동적 배열 hard disk 하드 디스크 random-access memory (RAM) 메모리 cache memory 캐시 메모리 cache miss 캐시 미스 cache hit rate 캐시 적중률 stack 스택 top of the stack 스택의 맨 위 bottom of the stack 스택의 맨 아래 queue 큐 double-ended queue 덱 front of the queue 큐의 앞 rear of the queue 큐의 뒤 hash table 해시 테이블 hash set 해시 집합 bucket 버킷 hash function 해시 함수 hash collision 해시 충돌 load factor 적재율 separate chaining 체이닝 open addressing 개방 주소법 linear probing 선형 탐사 lazy deletion 지연 삭제 binary tree 이진 트리 tree node 트리 노드 left-child node 왼쪽 자식 노드 right-child node 오른쪽 자식 노드 parent node 부모 노드 left subtree 왼쪽 서브트리 right subtree 오른쪽 서브트리 root node 루트 노드 leaf node 리프 노드 edge 간선 level 레벨 degree 차수 height 높이 depth 깊이 perfect binary tree 포화 이진 트리 complete binary tree 완전 이진 트리 full binary tree 정 이진 트리 balanced binary tree 균형 이진 트리 binary search tree 이진 탐색 트리 AVL tree AVL 트리 red-black tree 레드-블랙 트리 level-order traversal 레벨 순서 순회 breadth-first traversal 너비 우선 순회 depth-first traversal 깊이 우선 순회 pre-order traversal 전위 순회 in-order traversal 중위 순회 post-order traversal 후위 순회 balanced binary search tree 균형 이진 탐색 트리 balance factor 균형 인수 heap 힙 max heap 최대 힙 min heap 최소 힙 priority queue 우선순위 큐 heapify 힙 재정렬 top-\\(k\\) problem Top-\\(k\\) 문제 graph 그래프 vertex 정점 undirected graph 무방향 그래프 directed graph 방향 그래프 connected graph 연결 그래프 disconnected graph 비연결 그래프 weighted graph 가중 그래프 adjacency 인접 path 경로 in-degree 진입 차수 out-degree 진출 차수 adjacency matrix 인접 행렬 adjacency list 인접 리스트 breadth-first search 너비 우선 탐색 depth-first search 깊이 우선 탐색 binary search 이진 탐색 searching algorithm 탐색 알고리즘 sorting algorithm 정렬 알고리즘 selection sort 선택 정렬 bubble sort 버블 정렬 insertion sort 삽입 정렬 quick sort 퀵 정렬 merge sort 병합 정렬 heap sort 힙 정렬 bucket sort 버킷 정렬 counting sort 계수 정렬 radix sort 기수 정렬 divide and conquer 분할 정복 hanota problem 하노이의 탑 문제 backtracking algorithm 백트래킹 알고리즘 constraint 제약 조건 solution 해 state 상태 pruning 가지치기 permutations problem 모든 순열 구하기 subset-sum problem 부분 집합의 합 문제 \\(n\\)-queens problem \\(n\\)-퀸 문제 dynamic programming 동적 계획법 initial state 초기 상태 state-transition equation 상태 전이 식 knapsack problem 배낭 문제 edit distance problem 편집 거리 문제 greedy algorithm 탐욕 알고리즘","path":["제 16 장   부록","16.3   용어집"],"tags":[]},{"location":"chapter_array_and_linkedlist/","level":1,"title":"제 4 장   배열과 연결 리스트","text":"<p>이 장을 시작하며</p> <p>자료 구조의 세계는 두꺼운 벽돌벽과 같다.</p> <p>배열의 벽돌은 가지런히 빈틈없이 붙어 있다. 연결 리스트의 벽돌은 곳곳에 흩어져 있으며, 이들을 잇는 덩굴이 벽돌 틈 사이를 자유롭게 누빈다.</p>","path":["제 4 장   배열과 연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>4.1   배열</li> <li>4.2   연결 리스트</li> <li>4.3   리스트</li> <li>4.4   메모리와 캐시 *</li> <li>4.5   요약</li> <li>4.6   연습 문제</li> </ul>","path":["제 4 장   배열과 연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/","level":1,"title":"4.1   배열","text":"<p>배열(array)은 같은 자료형의 원소를 연속된 메모리 공간에 저장하는 선형 자료 구조다. 배열에서 원소가 있는 위치를 해당 원소의 인덱스(index)라고 한다. 아래 그림(4-1)은 배열의 주요 개념과 저장 방식을 보여 준다.</p> <p></p> <p> 그림 4-1   배열의 정의와 저장 방식 </p>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#411","level":2,"title":"4.1.1   배열의 기본 연산","text":"","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#1","level":3,"title":"1.   배열 초기화","text":"<p>필요에 따라 초깃값을 지정하지 않거나 지정하는 두 가지 초기화 방식을 사용할 수 있다. 초깃값을 지정하지 않으면 대부분의 프로그래밍 언어는 배열 원소를 \\(0\\)으로 초기화한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code># 배열 초기화\narr: list[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> array.cpp<pre><code>/* 배열 초기화 */\n// 스택에 저장\nint arr[5] = {};\nint nums[5] = { 1, 3, 2, 5, 4 };\n// 힙에 저장(공간을 직접 해제해야 함)\nint* arr1 = new int[5] {};\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n</code></pre> array.java<pre><code>/* 배열 초기화 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n</code></pre> array.cs<pre><code>/* 배열 초기화 */\nint[] arr = new int[5]; // [ 0, 0, 0, 0, 0 ]\nint[] nums = [1, 3, 2, 5, 4];\n</code></pre> array.go<pre><code>/* 배열 초기화 */\nvar arr [5]int\n// Go에서 길이를 지정하면([5]int) 배열이고, 지정하지 않으면([]int) 슬라이스다\n// Go 배열의 길이는 컴파일 시 결정되도록 설계되었으므로 상수로만 길이를 지정할 수 있다\n// 용량 확장 메서드 extend()를 쉽게 구현하기 위해 아래에서는 슬라이스(Slice)를 배열(Array)로 간주한다\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> array.swift<pre><code>/* 배열 초기화 */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n</code></pre> array.js<pre><code>/* 배열 초기화 */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n</code></pre> array.ts<pre><code>/* 배열 초기화 */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> array.dart<pre><code>/* 배열 초기화 */\nList&lt;int&gt; arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> array.rs<pre><code>/* 배열 초기화 */\nlet arr: [i32; 5] = [0; 5]; // [0, 0, 0, 0, 0]\nlet slice: &amp;[i32] = &amp;[0; 5];\n// Rust에서 길이를 지정하면([i32; 5]) 배열이고, 지정하지 않으면(&amp;[i32]) 슬라이스다\n// Rust 배열의 길이는 컴파일 시 결정되도록 설계되었으므로 상수로만 길이를 지정할 수 있다\n// Vector는 Rust에서 일반적으로 동적 배열로 사용하는 자료형이다\n// 용량 확장 메서드 extend()를 쉽게 구현하기 위해 아래에서는 vector를 배열(array)로 간주한다\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> array.c<pre><code>/* 배열 초기화 */\nint arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n</code></pre> array.kt<pre><code>/* 배열 초기화 */\nvar arr = IntArray(5) // { 0, 0, 0, 0, 0 }\nvar nums = intArrayOf(1, 3, 2, 5, 4)\n</code></pre> array.rb<pre><code># 배열 초기화\narr = Array.new(5, 0)\nnums = [1, 3, 2, 5, 4]\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#2","level":3,"title":"2.   원소 접근","text":"<p>배열 원소는 연속된 메모리 공간에 저장되므로 메모리 주소를 매우 쉽게 계산할 수 있다. 배열의 메모리 주소(첫 원소의 메모리 주소)와 원소의 인덱스가 주어지면, 아래 그림(4-2)의 식으로 해당 원소의 메모리 주소를 구하여 직접 접근할 수 있다.</p> <p></p> <p> 그림 4-2   배열 원소의 메모리 주소 계산 </p> <p>위 그림(4-2)에서 배열의 첫 원소 인덱스는 \\(0\\)이다. \\(1\\)부터 세는 편이 더 자연스러워 보이므로 직관에 어긋날 수 있다. 그러나 주소 계산식의 관점에서 인덱스는 본질적으로 메모리 주소의 오프셋이다. 첫 원소의 주소 오프셋은 \\(0\\)이므로 인덱스가 \\(0\\)인 것은 합리적이다.</p> <p>배열의 원소 접근은 매우 효율적이며, 어떤 원소든 \\(O(1)\\) 시간에 임의 접근할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def random_access(nums: list[int]) -&gt; int:\n    \"\"\"원소에 임의 접근\"\"\"\n    # 구간 [0, len(nums)-1]에서 숫자 하나를 무작위로 뽑는다\n    random_index = random.randint(0, len(nums) - 1)\n    # 무작위 원소를 가져와 반환한다\n    random_num = nums[random_index]\n    return random_num\n</code></pre> array.cpp<pre><code>/* 원소에 임의 접근 */\nint randomAccess(int *nums, int size) {\n    // 구간 [0, size)에서 무작위로 수 하나 선택\n    int randomIndex = rand() % size;\n    // 무작위 원소를 가져와 반환한다\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.java<pre><code>/* 원소에 임의 접근 */\nint randomAccess(int[] nums) {\n    // 구간 [0, nums.length)에서 숫자 하나를 무작위로 뽑는다\n    int randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n    // 무작위 원소를 가져와 반환한다\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.cs<pre><code>/* 원소에 임의 접근 */\nint RandomAccess(int[] nums) {\n    Random random = new();\n    // 구간 [0, nums.Length)에서 무작위로 수 하나 선택\n    int randomIndex = random.Next(nums.Length);\n    // 무작위 원소를 가져와 반환한다\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.go<pre><code>/* 원소에 임의 접근 */\nfunc randomAccess(nums []int) (randomNum int) {\n    // 구간 [0, nums.length)에서 숫자 하나를 무작위로 뽑는다\n    randomIndex := rand.Intn(len(nums))\n    // 무작위 원소를 가져와 반환한다\n    randomNum = nums[randomIndex]\n    return\n}\n</code></pre> array.swift<pre><code>/* 원소에 임의 접근 */\nfunc randomAccess(nums: [Int]) -&gt; Int {\n    // 구간 [0, nums.count)에서 숫자 하나를 무작위로 뽑는다\n    let randomIndex = nums.indices.randomElement()!\n    // 무작위 원소를 가져와 반환한다\n    let randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.js<pre><code>/* 원소에 임의 접근 */\nfunction randomAccess(nums) {\n    // 구간 [0, nums.length)에서 숫자 하나를 무작위로 뽑는다\n    const random_index = Math.floor(Math.random() * nums.length);\n    // 무작위 원소를 가져와 반환한다\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.ts<pre><code>/* 원소에 임의 접근 */\nfunction randomAccess(nums: number[]): number {\n    // 구간 [0, nums.length)에서 숫자 하나를 무작위로 뽑는다\n    const random_index = Math.floor(Math.random() * nums.length);\n    // 무작위 원소를 가져와 반환한다\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.dart<pre><code>/* 원소에 임의 접근 */\nint randomAccess(List&lt;int&gt; nums) {\n  // 구간 [0, nums.length)에서 숫자 하나를 무작위로 뽑는다\n  int randomIndex = Random().nextInt(nums.length);\n  // 무작위 원소를 가져와 반환한다\n  int randomNum = nums[randomIndex];\n  return randomNum;\n}\n</code></pre> array.rs<pre><code>/* 원소에 임의 접근 */\nfn random_access(nums: &amp;[i32]) -&gt; i32 {\n    // 구간 [0, nums.len())에서 숫자 하나를 무작위로 뽑는다\n    let random_index = rand::thread_rng().gen_range(0..nums.len());\n    // 무작위 원소를 가져와 반환한다\n    let random_num = nums[random_index];\n    random_num\n}\n</code></pre> array.c<pre><code>/* 원소에 임의 접근 */\nint randomAccess(int *nums, int size) {\n    // 구간 [0, size)에서 무작위로 수 하나 선택\n    int randomIndex = rand() % size;\n    // 무작위 원소를 가져와 반환한다\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.kt<pre><code>/* 원소에 임의 접근 */\nfun randomAccess(nums: IntArray): Int {\n    // 구간 [0, nums.size)에서 숫자 하나를 무작위로 뽑는다\n    val randomIndex = ThreadLocalRandom.current().nextInt(0, nums.size)\n    // 무작위 원소를 가져와 반환한다\n    val randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.rb<pre><code>### 원소에 임의 접근 ###\ndef random_access(nums)\n  # 구간 [0, nums.length)에서 숫자 하나를 무작위로 뽑는다\n  random_index = Random.rand(0...nums.length)\n\n  # 무작위 원소를 가져와 반환한다\n  nums[random_index]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#3","level":3,"title":"3.   원소 삽입","text":"<p>배열 원소는 메모리에서 ‘빈틈없이 붙어 있어’ 사이에 다른 데이터를 넣을 공간이 없다. 아래 그림(4-3)과 같이 배열 중간에 원소를 삽입하려면 해당 위치 이후의 모든 원소를 한 칸씩 뒤로 옮긴 다음, 그 인덱스에 새 원소를 대입해야 한다.</p> <p></p> <p> 그림 4-3   배열에 원소를 삽입하는 예 </p> <p>배열의 길이는 고정되어 있으므로 원소를 하나 삽입하면 반드시 마지막 원소가 ‘사라진다’는 점에 주의하자. 이 문제의 해결책은 ‘리스트’ 절에서 다룬다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def insert(nums: list[int], num: int, index: int):\n    \"\"\"배열의 인덱스 index에 원소 num 삽입\"\"\"\n    # 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for i in range(len(nums) - 1, index, -1):\n        nums[i] = nums[i - 1]\n    # index 위치의 원소에 num을 대입한다\n    nums[index] = num\n</code></pre> array.cpp<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nvoid insert(int *nums, int size, int num, int index) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num;\n}\n</code></pre> array.java<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nvoid insert(int[] nums, int num, int index) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for (int i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num;\n}\n</code></pre> array.cs<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nvoid Insert(int[] nums, int num, int index) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for (int i = nums.Length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num;\n}\n</code></pre> array.go<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nfunc insert(nums []int, num int, index int) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for i := len(nums) - 1; i &gt; index; i-- {\n        nums[i] = nums[i-1]\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num\n}\n</code></pre> array.swift<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for i in stride(from: nums.count - 1, to: index, by: -1) {\n        nums[i] = nums[i - 1]\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num\n}\n</code></pre> array.js<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nfunction insert(nums, num, index) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num;\n}\n</code></pre> array.ts<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nfunction insert(nums: number[], num: number, index: number): void {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num;\n}\n</code></pre> array.dart<pre><code>/* 배열의 인덱스 index에 원소 _num 삽입 */\nvoid insert(List&lt;int&gt; nums, int _num, int index) {\n  // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n  for (var i = nums.length - 1; i &gt; index; i--) {\n    nums[i] = nums[i - 1];\n  }\n  // 인덱스 index의 원소에 _num을 대입한다\n  nums[index] = _num;\n}\n</code></pre> array.rs<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nfn insert(nums: &amp;mut [i32], num: i32, index: usize) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for i in (index + 1..nums.len()).rev() {\n        nums[i] = nums[i - 1];\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num;\n}\n</code></pre> array.c<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nvoid insert(int *nums, int size, int num, int index) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num;\n}\n</code></pre> array.kt<pre><code>/* 배열의 인덱스 index에 원소 num 삽입 */\nfun insert(nums: IntArray, num: Int, index: Int) {\n    // 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n    for (i in nums.size - 1 downTo index + 1) {\n        nums[i] = nums[i - 1]\n    }\n    // index 위치의 원소에 num을 대입한다\n    nums[index] = num\n}\n</code></pre> array.rb<pre><code>### 배열의 인덱스 index에 원소 num 삽입 ###\ndef insert(nums, num, index)\n  # 인덱스 index와 그 뒤에 있는 모든 원소를 한 칸씩 뒤로 이동한다\n  for i in (nums.length - 1).downto(index + 1)\n    nums[i] = nums[i - 1]\n  end\n\n  # index 위치의 원소에 num을 대입한다\n  nums[index] = num\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#4","level":3,"title":"4.   원소 삭제","text":"<p>마찬가지로 아래 그림(4-4)과 같이 인덱스 \\(i\\)의 원소를 삭제하려면 인덱스 \\(i\\) 뒤의 원소를 모두 한 칸씩 앞으로 옮겨야 한다.</p> <p></p> <p> 그림 4-4   배열에서 원소를 삭제하는 예 </p> <p>삭제가 끝난 뒤 원래의 마지막 원소는 ‘의미 없는’ 값이 되므로 따로 수정할 필요가 없다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def remove(nums: list[int], index: int):\n    \"\"\"인덱스 index에 있는 원소 삭제\"\"\"\n    # 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for i in range(index, len(nums) - 1):\n        nums[i] = nums[i + 1]\n</code></pre> array.cpp<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nvoid remove(int *nums, int size, int index) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.java<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nvoid remove(int[] nums, int index) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for (int i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.cs<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nvoid Remove(int[] nums, int index) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for (int i = index; i &lt; nums.Length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.go<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nfunc remove(nums []int, index int) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for i := index; i &lt; len(nums)-1; i++ {\n        nums[i] = nums[i+1]\n    }\n}\n</code></pre> array.swift<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nfunc remove(nums: inout [Int], index: Int) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for i in nums.indices.dropFirst(index).dropLast() {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.js<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nfunction remove(nums, index) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.ts<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nfunction remove(nums: number[], index: number): void {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.dart<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nvoid remove(List&lt;int&gt; nums, int index) {\n  // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n  for (var i = index; i &lt; nums.length - 1; i++) {\n    nums[i] = nums[i + 1];\n  }\n}\n</code></pre> array.rs<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nfn remove(nums: &amp;mut [i32], index: usize) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for i in index..nums.len() - 1 {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.c<pre><code>/* 인덱스 index에 있는 원소 삭제 */\n// 주의: stdio.h가 remove라는 이름을 사용한다\nvoid removeItem(int *nums, int size, int index) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.kt<pre><code>/* 인덱스 index에 있는 원소 삭제 */\nfun remove(nums: IntArray, index: Int) {\n    // 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n    for (i in index..&lt;nums.size - 1) {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.rb<pre><code>### 인덱스 index에 있는 원소 삭제 ###\ndef remove(nums, index)\n  # 인덱스 index 뒤에 있는 모든 원소를 한 칸씩 앞으로 이동한다\n  for i in index...(nums.length - 1)\n    nums[i] = nums[i + 1]\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>배열의 삽입과 삭제 연산에는 다음과 같은 단점이 있다.</p> <ul> <li>높은 시간 복잡도: 삽입과 삭제의 평균 시간 복잡도는 모두 \\(O(n)\\)이며, \\(n\\)은 배열 길이다.</li> <li>원소 손실: 배열의 길이는 바꿀 수 없으므로 삽입 후 길이 범위를 벗어나는 원소가 사라진다.</li> <li>메모리 낭비: 긴 배열을 초기화하고 앞부분만 사용하면 삽입 시 사라지는 마지막 원소는 모두 ‘의미 없는’ 값이 된다. 하지만 이 방식은 메모리 공간 일부를 낭비한다.</li> </ul>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#5","level":3,"title":"5.   배열 순회","text":"<p>대부분의 프로그래밍 언어에서는 인덱스로 배열을 순회할 수도 있고, 배열의 각 원소를 직접 얻으며 순회할 수도 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def traverse(nums: list[int]):\n    \"\"\"배열 순회\"\"\"\n    count = 0\n    # 인덱스를 통해 배열을 순회한다\n    for i in range(len(nums)):\n        count += nums[i]\n    # 배열의 원소를 직접 순회한다\n    for num in nums:\n        count += num\n    # 데이터의 인덱스와 원소를 함께 순회한다\n    for i, num in enumerate(nums):\n        count += nums[i]\n        count += num\n</code></pre> array.cpp<pre><code>/* 배열 순회 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // 인덱스를 통해 배열을 순회한다\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.java<pre><code>/* 배열 순회 */\nvoid traverse(int[] nums) {\n    int count = 0;\n    // 인덱스를 통해 배열을 순회한다\n    for (int i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // 배열의 원소를 직접 순회한다\n    for (int num : nums) {\n        count += num;\n    }\n}\n</code></pre> array.cs<pre><code>/* 배열 순회 */\nvoid Traverse(int[] nums) {\n    int count = 0;\n    // 인덱스를 통해 배열을 순회한다\n    for (int i = 0; i &lt; nums.Length; i++) {\n        count += nums[i];\n    }\n    // 배열의 원소를 직접 순회한다\n    foreach (int num in nums) {\n        count += num;\n    }\n}\n</code></pre> array.go<pre><code>/* 배열 순회 */\nfunc traverse(nums []int) {\n    count := 0\n    // 인덱스를 통해 배열을 순회한다\n    for i := 0; i &lt; len(nums); i++ {\n        count += nums[i]\n    }\n    count = 0\n    // 배열의 원소를 직접 순회한다\n    for _, num := range nums {\n        count += num\n    }\n    // 데이터의 인덱스와 원소를 함께 순회한다\n    for i, num := range nums {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.swift<pre><code>/* 배열 순회 */\nfunc traverse(nums: [Int]) {\n    var count = 0\n    // 인덱스를 통해 배열을 순회한다\n    for i in nums.indices {\n        count += nums[i]\n    }\n    // 배열의 원소를 직접 순회한다\n    for num in nums {\n        count += num\n    }\n    // 데이터의 인덱스와 원소를 함께 순회한다\n    for (i, num) in nums.enumerated() {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.js<pre><code>/* 배열 순회 */\nfunction traverse(nums) {\n    let count = 0;\n    // 인덱스를 통해 배열을 순회한다\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // 배열의 원소를 직접 순회한다\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.ts<pre><code>/* 배열 순회 */\nfunction traverse(nums: number[]): void {\n    let count = 0;\n    // 인덱스를 통해 배열을 순회한다\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // 배열의 원소를 직접 순회한다\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.dart<pre><code>/* 배열 원소 순회 */\nvoid traverse(List&lt;int&gt; nums) {\n  int count = 0;\n  // 인덱스를 통해 배열을 순회한다\n  for (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n  }\n  // 배열의 원소를 직접 순회한다\n  for (int _num in nums) {\n    count += _num;\n  }\n  // forEach 메서드로 배열을 순회한다\n  nums.forEach((_num) {\n    count += _num;\n  });\n}\n</code></pre> array.rs<pre><code>/* 배열 순회 */\nfn traverse(nums: &amp;[i32]) {\n    let mut _count = 0;\n    // 인덱스를 통해 배열을 순회한다\n    for i in 0..nums.len() {\n        _count += nums[i];\n    }\n    // 배열의 원소를 직접 순회한다\n    _count = 0;\n    for &amp;num in nums {\n        _count += num;\n    }\n}\n</code></pre> array.c<pre><code>/* 배열 순회 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // 인덱스를 통해 배열을 순회한다\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.kt<pre><code>/* 배열 순회 */\nfun traverse(nums: IntArray) {\n    var count = 0\n    // 인덱스를 통해 배열을 순회한다\n    for (i in nums.indices) {\n        count += nums[i]\n    }\n    // 배열의 원소를 직접 순회한다\n    for (j in nums) {\n        count += j\n    }\n}\n</code></pre> array.rb<pre><code>### 배열 순회 ###\ndef traverse(nums)\n  count = 0\n\n  # 인덱스를 통해 배열을 순회한다\n  for i in 0...nums.length\n    count += nums[i]\n  end\n\n  # 배열의 원소를 직접 순회한다\n  for num in nums\n    count += num\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#6","level":3,"title":"6.   원소 탐색","text":"<p>배열에서 특정 원소를 찾으려면 배열을 순회하면서 매번 값이 일치하는지 확인하고, 일치하면 해당 인덱스를 출력한다.</p> <p>배열은 선형 자료 구조이므로 이 탐색 연산을 ‘선형 탐색’이라고 한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def find(nums: list[int], target: int) -&gt; int:\n    \"\"\"배열에서 지정한 원소 탐색\"\"\"\n    for i in range(len(nums)):\n        if nums[i] == target:\n            return i\n    return -1\n</code></pre> array.cpp<pre><code>/* 배열에서 지정한 원소 탐색 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.java<pre><code>/* 배열에서 지정한 원소 탐색 */\nint find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.cs<pre><code>/* 배열에서 지정한 원소 탐색 */\nint Find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.go<pre><code>/* 배열에서 지정한 원소 탐색 */\nfunc find(nums []int, target int) (index int) {\n    index = -1\n    for i := 0; i &lt; len(nums); i++ {\n        if nums[i] == target {\n            index = i\n            break\n        }\n    }\n    return\n}\n</code></pre> array.swift<pre><code>/* 배열에서 지정한 원소 탐색 */\nfunc find(nums: [Int], target: Int) -&gt; Int {\n    for i in nums.indices {\n        if nums[i] == target {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> array.js<pre><code>/* 배열에서 지정한 원소 탐색 */\nfunction find(nums, target) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) return i;\n    }\n    return -1;\n}\n</code></pre> array.ts<pre><code>/* 배열에서 지정한 원소 탐색 */\nfunction find(nums: number[], target: number): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> array.dart<pre><code>/* 배열에서 지정한 원소 탐색 */\nint find(List&lt;int&gt; nums, int target) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    if (nums[i] == target) return i;\n  }\n  return -1;\n}\n</code></pre> array.rs<pre><code>/* 배열에서 지정한 원소 탐색 */\nfn find(nums: &amp;[i32], target: i32) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        if nums[i] == target {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> array.c<pre><code>/* 배열에서 지정한 원소 탐색 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.kt<pre><code>/* 배열에서 지정한 원소 탐색 */\nfun find(nums: IntArray, target: Int): Int {\n    for (i in nums.indices) {\n        if (nums[i] == target)\n            return i\n    }\n    return -1\n}\n</code></pre> array.rb<pre><code>### 배열에서 지정한 원소 탐색 ###\ndef find(nums, target)\n  for i in 0...nums.length\n    return i if nums[i] == target\n  end\n\n  -1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#7","level":3,"title":"7.   배열 용량 확장","text":"<p>복잡한 시스템 환경에서는 배열 바로 뒤의 메모리 공간을 사용할 수 있다고 보장하기 어려우므로, 배열 용량을 안전하게 늘릴 수 없다. 따라서 대부분의 프로그래밍 언어에서 배열 길이는 바꿀 수 없다.</p> <p>배열 용량을 늘리려면 더 큰 배열을 새로 만들고 기존 원소를 차례로 복사해야 한다. 이는 \\(O(n)\\) 연산으로, 배열이 매우 크면 시간이 많이 든다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array.py<pre><code>def extend(nums: list[int], enlarge: int) -&gt; list[int]:\n    \"\"\"배열 길이 확장\"\"\"\n    # 길이를 확장한 배열을 초기화한다\n    res = [0] * (len(nums) + enlarge)\n    # 원래 배열의 모든 원소를 새 배열에 복사한다\n    for i in range(len(nums)):\n        res[i] = nums[i]\n    # 확장한 새 배열을 반환한다\n    return res\n</code></pre> array.cpp<pre><code>/* 배열 길이 확장 */\nint *extend(int *nums, int size, int enlarge) {\n    // 길이를 확장한 배열을 초기화한다\n    int *res = new int[size + enlarge];\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // 메모리 해제\n    delete[] nums;\n    // 확장한 새 배열을 반환한다\n    return res;\n}\n</code></pre> array.java<pre><code>/* 배열 길이 확장 */\nint[] extend(int[] nums, int enlarge) {\n    // 길이를 확장한 배열을 초기화한다\n    int[] res = new int[nums.length + enlarge];\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for (int i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // 확장한 새 배열을 반환한다\n    return res;\n}\n</code></pre> array.cs<pre><code>/* 배열 길이 확장 */\nint[] Extend(int[] nums, int enlarge) {\n    // 길이를 확장한 배열을 초기화한다\n    int[] res = new int[nums.Length + enlarge];\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for (int i = 0; i &lt; nums.Length; i++) {\n        res[i] = nums[i];\n    }\n    // 확장한 새 배열을 반환한다\n    return res;\n}\n</code></pre> array.go<pre><code>/* 배열 길이 확장 */\nfunc extend(nums []int, enlarge int) []int {\n    // 길이를 확장한 배열을 초기화한다\n    res := make([]int, len(nums)+enlarge)\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for i, num := range nums {\n        res[i] = num\n    }\n    // 확장한 새 배열을 반환한다\n    return res\n}\n</code></pre> array.swift<pre><code>/* 배열 길이 확장 */\nfunc extend(nums: [Int], enlarge: Int) -&gt; [Int] {\n    // 길이를 확장한 배열을 초기화한다\n    var res = Array(repeating: 0, count: nums.count + enlarge)\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for i in nums.indices {\n        res[i] = nums[i]\n    }\n    // 확장한 새 배열을 반환한다\n    return res\n}\n</code></pre> array.js<pre><code>/* 배열 길이 확장 */\n// JavaScript의 Array는 동적 배열이므로 직접 확장할 수 있다\n// 학습의 편의를 위해 이 함수에서는 Array를 길이가 고정된 배열로 취급한다\nfunction extend(nums, enlarge) {\n    // 길이를 확장한 배열을 초기화한다\n    const res = new Array(nums.length + enlarge).fill(0);\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // 확장한 새 배열을 반환한다\n    return res;\n}\n</code></pre> array.ts<pre><code>/* 배열 길이 확장 */\n// TypeScript의 Array는 동적 배열이므로 직접 확장할 수 있다\n// 학습의 편의를 위해 이 함수에서는 Array를 길이가 고정된 배열로 취급한다\nfunction extend(nums: number[], enlarge: number): number[] {\n    // 길이를 확장한 배열을 초기화한다\n    const res = new Array(nums.length + enlarge).fill(0);\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // 확장한 새 배열을 반환한다\n    return res;\n}\n</code></pre> array.dart<pre><code>/* 배열 길이 확장 */\nList&lt;int&gt; extend(List&lt;int&gt; nums, int enlarge) {\n  // 길이를 확장한 배열을 초기화한다\n  List&lt;int&gt; res = List.filled(nums.length + enlarge, 0);\n  // 원래 배열의 모든 원소를 새 배열에 복사한다\n  for (var i = 0; i &lt; nums.length; i++) {\n    res[i] = nums[i];\n  }\n  // 확장한 새 배열을 반환한다\n  return res;\n}\n</code></pre> array.rs<pre><code>/* 배열 길이 확장 */\nfn extend(nums: &amp;[i32], enlarge: usize) -&gt; Vec&lt;i32&gt; {\n    // 길이를 확장한 배열을 초기화한다\n    let mut res: Vec&lt;i32&gt; = vec![0; nums.len() + enlarge];\n    // 원래 배열의 모든 원소를 새 배열로 복사한다\n    res[0..nums.len()].copy_from_slice(nums);\n\n    // 확장한 새 배열을 반환한다\n    res\n}\n</code></pre> array.c<pre><code>/* 배열 길이 확장 */\nint *extend(int *nums, int size, int enlarge) {\n    // 길이를 확장한 배열을 초기화한다\n    int *res = (int *)malloc(sizeof(int) * (size + enlarge));\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // 확장한 공간 초기화\n    for (int i = size; i &lt; size + enlarge; i++) {\n        res[i] = 0;\n    }\n    // 확장한 새 배열을 반환한다\n    return res;\n}\n</code></pre> array.kt<pre><code>/* 배열 길이 확장 */\nfun extend(nums: IntArray, enlarge: Int): IntArray {\n    // 길이를 확장한 배열을 초기화한다\n    val res = IntArray(nums.size + enlarge)\n    // 원래 배열의 모든 원소를 새 배열에 복사한다\n    for (i in nums.indices) {\n        res[i] = nums[i]\n    }\n    // 확장한 새 배열을 반환한다\n    return res\n}\n</code></pre> array.rb<pre><code>### 배열 길이 확장 ###\n# Ruby의 Array는 동적 배열이므로 직접 확장할 수 있다\n# 학습의 편의를 위해 이 함수에서는 Array를 길이가 고정된 배열로 취급한다\ndef extend(nums, enlarge)\n  # 길이를 확장한 배열을 초기화한다\n  res = Array.new(nums.length + enlarge, 0)\n\n  # 원래 배열의 모든 원소를 새 배열에 복사한다\n  for i in 0...nums.length\n    res[i] = nums[i]\n  end\n\n  # 확장한 새 배열을 반환한다\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#412","level":2,"title":"4.1.2   배열의 장점과 한계","text":"<p>배열은 연속된 메모리 공간에 저장되고 원소의 자료형이 모두 같다. 이러한 풍부한 사전 정보를 활용하면 시스템은 자료 구조의 연산 효율을 최적화할 수 있다.</p> <ul> <li>높은 공간 효율: 연속된 메모리 블록을 데이터에 할당하므로 추가적인 구조 비용이 없다.</li> <li>임의 접근 지원: 어떤 원소든 \\(O(1)\\) 시간에 접근할 수 있다.</li> <li>캐시 지역성: 배열 원소에 접근할 때 컴퓨터는 그 원소뿐 아니라 주변 데이터도 캐시에 저장한다. 이를 통해 이후 연산의 실행 속도를 높일 수 있다.</li> </ul> <p>연속 공간 저장은 양날의 검으로, 다음과 같은 한계가 있다.</p> <ul> <li>낮은 삽입 및 삭제 효율: 원소가 많으면 삽입이나 삭제 시 많은 원소를 이동해야 한다.</li> <li>고정된 길이: 초기화한 뒤에는 길이가 고정된다. 용량을 늘리려면 모든 데이터를 새 배열로 복사해야 하므로 비용이 크다.</li> <li>공간 낭비: 실제 필요한 크기보다 크게 할당하면 남는 공간이 낭비된다.</li> </ul>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/array/#413","level":2,"title":"4.1.3   배열의 대표적인 활용","text":"<p>배열은 기본적이면서도 흔히 쓰이는 자료 구조다. 다양한 알고리즘에 자주 사용되며 복잡한 자료 구조를 구현하는 데도 쓰인다.</p> <ul> <li>임의 접근: 일부 표본을 무작위로 추출하려면 데이터를 배열에 저장하고 무작위 수열을 생성하여 인덱스에 따라 표본을 뽑을 수 있다.</li> <li>정렬과 탐색: 배열은 정렬 및 탐색 알고리즘에서 가장 많이 쓰이는 자료 구조다. 퀵 정렬, 병합 정렬, 이진 탐색 등은 주로 배열에서 수행된다.</li> <li>조회 테이블: 원소나 대응 관계를 빠르게 찾고 싶을 때 배열을 조회 테이블로 사용할 수 있다. 예를 들어 문자와 ASCII 코드의 대응을 구현하려면 문자의 ASCII 코드 값을 인덱스로 삼고, 해당 원소를 배열의 대응 위치에 저장하면 된다.</li> <li>머신 러닝: 신경망에서는 벡터, 행렬, 텐서 사이의 선형 대수 연산을 많이 사용하며, 이 데이터들은 모두 배열 형태로 구성된다. 배열은 신경망 프로그래밍에서 가장 많이 쓰는 자료 구조다.</li> <li>자료 구조 구현: 배열로 스택, 큐, 해시 테이블, 힙, 그래프 등을 구현할 수 있다. 예를 들어 그래프의 인접 행렬 표현은 실제로 이차원 배열이다.</li> </ul>","path":["제 4 장   배열과 연결 리스트","4.1   배열"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/","level":1,"title":"4.6   연습 문제","text":"","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#461","level":2,"title":"4.6.1   개념 확인","text":"","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#1","level":3,"title":"1.   배열과 연결 리스트에서 원소 찾기","text":"<p>배열과 단일 연결 리스트에 모두 <code>[A, B, C, D, E]</code>가 순서대로 저장되어 있다. 네 번째 원소 <code>D</code>를 읽으려고 한다.</p> <ol> <li>배열에서는 어떤 인덱스를 바로 사용할 수 있는가?</li> <li>단일 연결 리스트에서는 머리 노드 <code>A</code>부터 <code>next</code>를 따라 어떤 노드를 차례대로 거쳐야 하는가?</li> <li>읽으려는 원소가 뒤쪽에 있을수록 두 구조에 필요한 단계 수는 어떻게 변하는가? 위치를 기준으로 반복해서 읽을 때는 어느 구조가 더 적합하며, 그 이유는 무엇인가?</li> </ol> 참고 답안 <ol> <li> <p>인덱스가 0부터 시작한다면 네 번째 원소의 인덱스는 3이므로, 배열에서는 <code>arr[3]</code>을 바로 읽을 수 있다.</p> </li> <li> <p>단일 연결 리스트는 맨 앞부터 시작해야 한다. 접근 경로는 <code>A → B → C → D</code>이며, <code>next</code>를 따라 3번 이동해야 한다.</p> </li> <li> <p>배열은 시작 주소와 인덱스로 원소의 위치를 바로 계산할 수 있으므로 위치를 기준으로 접근하는 시간 복잡도가 \\(O(1)\\)이다.     단일 연결 리스트에서 \\(k\\)번째 노드에 접근하려면 머리 노드부터 <code>next</code>를 따라 \\(k-1\\)번 이동해야 하므로,     최악의 경우 \\(O(n)\\) 시간이 필요하다.</p> <p>여기서는 위치를 기준으로 읽는 연산만 비교한 것이며, 연결 리스트가 모든 연산에서 더 느리다는 뜻은 아니다.</p> </li> </ol>","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#2","level":3,"title":"2.   배열과 연결 리스트에 원소 삽입하기","text":"<p>배열과 단일 연결 리스트에 모두 <code>A, B, C, D</code>가 저장되어 있다. <code>B</code> 뒤에 <code>X</code>를 삽입하려고 한다.</p> <ul> <li>배열의 용량은 5이고 현재 상태는 <code>[A, B, C, D, _]</code>다.</li> <li>연결 리스트는 <code>A → B → C → D</code>이며, 노드 <code>B</code>를 가리키는 참조를 이미 가지고 있다.</li> </ul> <ol> <li>배열에서는 어떤 원소를 이동해야 하는가? 삽입 후 배열을 적어 보자.</li> <li>연결 리스트에서는 <code>X.next</code>와 <code>B.next</code>를 어떤 순서로 수정해야 하는가? 삽입 후 연결 리스트를 적어 보자.</li> <li>삽입 효율성을 비교할 때 ‘노드 B의 참조를 이미 가지고 있다’는 조건을 강조하는 이유는 무엇인가?</li> </ol> 참고 답안 <ol> <li> <p>배열에서는 먼저 <code>D</code>를 오른쪽으로 한 칸 옮기고, 다음으로 <code>C</code>를 오른쪽으로 한 칸 옮긴 뒤, <code>X</code>를 인덱스 2에 넣는다.     결과는 <code>[A, B, X, C, D]</code>다.</p> </li> <li> <p><code>B.next</code>는 원래 <code>C</code>를 가리킨다. 먼저 <code>X.next = B.next</code>로 설정하여 <code>X</code>가 <code>C</code>를 가리키게 하고,     다음으로 <code>B.next = X</code>로 설정한다. 결과는 <code>A → B → X → C → D</code>다.     기존 연결을 저장하지 않은 채 <code>B.next</code>를 먼저 덮어쓰면 <code>C</code>를 찾지 못할 수 있다.</p> </li> <li> <p><code>B</code>의 위치를 알고 있다면 연결 리스트의 삽입은 연결 두 개만 수정하면 되므로 \\(O(1)\\) 시간에 완료할 수 있다.     맨 앞부터 <code>B</code>를 찾아야 한다면 탐색 자체에 \\(O(n)\\) 시간이 걸릴 수 있다.</p> </li> </ol>","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#3","level":3,"title":"3.   리스트의 용량은 어떻게 늘어나는가","text":"<p>배열로 구현한 리스트의 현재 내용은 <code>[A, B, C]</code>이고, 길이는 <code>size = 3</code>, 용량은 <code>capacity = 4</code>다. 용량이 부족해지면 새 배열의 용량을 기존의 2배로 늘리기로 한다.</p> <ol> <li><code>D</code>를 추가한 뒤 리스트의 길이와 용량은 각각 얼마인가? 용량을 늘려야 하는가?</li> <li>이어서 <code>E</code>를 추가하면 용량은 얼마가 되는가? 기존 원소를 몇 개 복사해야 하는가?</li> <li>내부 배열의 길이는 바꿀 수 없는데, 리스트의 용량은 어떻게 늘어나는 것처럼 보이는가?</li> </ol> 참고 답안 <ol> <li> <p><code>D</code>는 마지막 빈자리에 넣을 수 있다. 내용은 <code>[A, B, C, D]</code>가 되고,     <code>size = 4</code>, <code>capacity = 4</code>이며 용량 확장은 필요 없다.</p> </li> <li> <p>이어서 <code>E</code>를 추가할 때는 빈자리가 없으므로 용량이 8인 새 배열을 만들어야 한다.     기존 원소 <code>A, B, C, D</code> 네 개를 복사한 뒤 <code>E</code>를 넣는다.     이때 <code>size = 5</code>, <code>capacity = 8</code>이다.</p> </li> <li> <p>원래 배열 자체의 길이가 늘어난 것은 아니다. 더 큰 새 배열을 만들고 기존 원소를 복사한 뒤,     새 배열을 내부 저장 공간으로 사용하기 때문에 사용자에게는 용량이 늘어난 것으로 보인다.</p> </li> </ol>","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#462","level":2,"title":"4.6.2   프로그래밍 연습","text":"","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#1-1","level":3,"title":"1.   배열로 표현한 큰 정수에 1 더하기","text":"<p>배열 <code>digits</code>에는 음이 아닌 정수의 각 자릿수가 왼쪽부터 저장되어 있다. 예를 들어 <code>[3, 0, 8]</code>은 308을 나타낸다. 숫자 0은 <code>[0]</code>으로 나타내며, 그 밖의 입력은 첫 번째 자릿수가 0이 아니다.</p> <p>십진수의 세로셈 덧셈 과정을 따라 이 정수에 1을 더한 뒤, 결과를 같은 배열 형식으로 반환하라. <code>digits</code>를 직접 수정해도 된다. 맨 앞에 새로운 자리 올림이 생기면 더 긴 배열을 반환할 수 있다.</p> 풀이 힌트 <ol> <li>세로셈 덧셈처럼 배열의 마지막 자릿수부터 시작한다.</li> <li>현재 자릿수가 9보다 작으면 1을 더한 뒤 바로 반환할 수 있다.</li> <li>현재 자릿수가 9라면 0으로 바꾼다. 모든 자릿수가 9였다면 맨 앞에 1을 추가해야 한다.</li> </ol> <p>LeetCode</p>","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/exercises/#2_1","level":3,"title":"2.   단일 연결 리스트 뒤집기","text":"<p>단일 연결 리스트의 머리 노드 <code>head</code>가 주어진다. 각 노드에는 값과 다음 노드를 가리키는 <code>next</code>가 있다.</p> <p>반복 방식으로 모든 노드 사이의 연결 방향을 뒤집고, 뒤집힌 리스트의 머리 노드를 반환하라. 새로운 연결 리스트 노드를 만들어서는 안 된다.</p> 풀이 힌트 <ol> <li>먼저 종이에 연결된 노드 세 개와 prev, cur 두 포인터를 그려 보자.</li> <li>cur.next를 바꾸기 전에 nxt에 원래 다음 노드를 반드시 저장해야 한다.</li> <li>cur.next의 방향을 뒤집은 뒤 prev = cur, cur = nxt로 설정하고 원래 리스트의 다음 노드를 계속 처리한다.</li> </ol> <p>LeetCode</p>","path":["제 4 장   배열과 연결 리스트","4.6   연습 문제"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/","level":1,"title":"4.2   연결 리스트","text":"<p>메모리 공간은 모든 프로그램이 함께 사용하는 자원이다. 복잡한 시스템 실행 환경에서는 빈 메모리 공간이 곳곳에 흩어져 있을 수 있다. 배열을 저장할 메모리는 연속적이어야 하므로, 배열이 매우 크면 그만큼 큰 연속 공간을 확보하지 못할 수도 있다. 이때 연결 리스트의 유연성이 장점으로 드러난다.</p> <p>연결 리스트(linked list)는 각 원소가 노드 객체이며, 노드들이 ‘참조’로 연결된 선형 자료 구조다. 참조는 다음 노드의 메모리 주소를 기록하므로, 이를 통해 현재 노드에서 다음 노드에 접근할 수 있다.</p> <p>이러한 설계 덕분에 각 노드는 메모리 곳곳에 분산되어 저장될 수 있으며, 메모리 주소가 연속적일 필요가 없다.</p> <p></p> <p> 그림 4-5   연결 리스트의 정의와 저장 방식 </p> <p>위 그림(4-5)에서 연결 리스트를 구성하는 단위는 노드(node) 객체다. 각 노드는 노드의 ‘값’과 다음 노드를 가리키는 ‘참조’라는 두 데이터를 포함한다.</p> <ul> <li>연결 리스트의 첫 노드를 ‘머리 노드’, 마지막 노드를 ‘꼬리 노드’라고 한다.</li> <li>꼬리 노드는 ‘없음’을 가리키며, Java, C++, Python에서는 이를 각각 <code>null</code>, <code>nullptr</code>, <code>None</code>으로 표기한다.</li> <li>C, C++, Go, Rust처럼 포인터를 지원하는 언어에서는 위의 ‘참조’를 ‘포인터’로 바꾸어 이해하면 된다.</li> </ul> <p>다음 코드와 같이 연결 리스트 노드 <code>ListNode</code>는 값 외에도 참조(포인터)를 하나 더 저장해야 한다. 따라서 같은 양의 데이터를 저장할 때 연결 리스트는 배열보다 메모리 공간을 더 많이 차지한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class ListNode:\n    \"\"\"연결 리스트 노드 클래스\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val               # 노드 값\n        self.next: ListNode | None = None # 다음 노드를 가리키는 참조\n</code></pre> <pre><code>/* 연결 리스트 노드 구조체 */\nstruct ListNode {\n    int val;         // 노드 값\n    ListNode *next;  // 다음 노드를 가리키는 포인터\n    ListNode(int x) : val(x), next(nullptr) {}  // 생성자\n};\n</code></pre> <pre><code>/* 연결 리스트 노드 클래스 */\nclass ListNode {\n    int val;        // 노드 값\n    ListNode next;  // 다음 노드를 가리키는 참조\n    ListNode(int x) { val = x; }  // 생성자\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 클래스 */\nclass ListNode(int x) {  //생성자\n    int val = x;         // 노드 값\n    ListNode? next;      // 다음 노드를 가리키는 참조\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 구조체 */\ntype ListNode struct {\n    Val  int       // 노드 값\n    Next *ListNode // 다음 노드를 가리키는 포인터\n}\n\n// NewListNode 생성자: 새 연결 리스트 생성\nfunc NewListNode(val int) *ListNode {\n    return &amp;ListNode{\n        Val:  val,\n        Next: nil,\n    }\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 클래스 */\nclass ListNode {\n    var val: Int // 노드 값\n    var next: ListNode? // 다음 노드를 가리키는 참조\n\n    init(x: Int) { // 생성자\n        val = x\n    }\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 클래스 */\nclass ListNode {\n    constructor(val, next) {\n        this.val = (val === undefined ? 0 : val);       // 노드 값\n        this.next = (next === undefined ? null : next); // 다음 노드를 가리키는 참조\n    }\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 클래스 */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    constructor(val?: number, next?: ListNode | null) {\n        this.val = val === undefined ? 0 : val;        // 노드 값\n        this.next = next === undefined ? null : next;  // 다음 노드를 가리키는 참조\n    }\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 클래스 */\nclass ListNode {\n  int val; // 노드 값\n  ListNode? next; // 다음 노드를 가리키는 참조\n  ListNode(this.val, [this.next]); // 생성자\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* 연결 리스트 노드 클래스 */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // 노드 값\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // 다음 노드를 가리키는 포인터\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 구조체 */\ntypedef struct ListNode {\n    int val;               // 노드 값\n    struct ListNode *next; // 다음 노드를 가리키는 포인터\n} ListNode;\n\n/* 생성자 */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* 연결 리스트 노드 클래스 */\n// 생성자\nclass ListNode(x: Int) {\n    val _val: Int = x          // 노드 값\n    var next: ListNode? = null // 다음 노드를 가리키는 참조\n}\n</code></pre> <pre><code># 연결 리스트 노드 클래스\nclass ListNode\n  attr_accessor :val  # 노드 값\n  attr_accessor :next # 다음 노드를 가리키는 참조\n\n  def initialize(val=0, next_node=nil)\n    @val = val\n    @next = next_node\n  end\nend\n</code></pre>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#421","level":2,"title":"4.2.1   연결 리스트의 기본 연산","text":"","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#1","level":3,"title":"1.   연결 리스트 초기화","text":"<p>연결 리스트를 만드는 과정은 두 단계다. 먼저 각 노드 객체를 초기화한 다음, 노드 사이의 참조 관계를 설정한다. 초기화가 끝나면 머리 노드부터 시작하여 <code>next</code> 참조를 따라 모든 노드에 차례로 접근할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code># 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화\n# 각 노드 초기화\nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# 노드 사이의 참조 설정\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.cpp<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// 노드 사이의 참조 설정\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.java<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// 노드 사이의 참조 설정\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.cs<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nListNode n0 = new(1);\nListNode n1 = new(3);\nListNode n2 = new(2);\nListNode n3 = new(5);\nListNode n4 = new(4);\n// 노드 사이의 참조 설정\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.go<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// 노드 사이의 참조 설정\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n</code></pre> linked_list.swift<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// 노드 사이의 참조 설정\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.js<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// 노드 사이의 참조 설정\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.ts<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// 노드 사이의 참조 설정\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.dart<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\\\n// 각 노드 초기화\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// 노드 사이의 참조 설정\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rs<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nlet n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None }));\nlet n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None }));\nlet n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None }));\nlet n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None }));\nlet n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None }));\n\n// 노드 사이의 참조 설정\nn0.borrow_mut().next = Some(n1.clone());\nn1.borrow_mut().next = Some(n2.clone());\nn2.borrow_mut().next = Some(n3.clone());\nn3.borrow_mut().next = Some(n4.clone());\n</code></pre> linked_list.c<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// 노드 사이의 참조 설정\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.kt<pre><code>/* 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화 */\n// 각 노드 초기화\nval n0 = ListNode(1)\nval n1 = ListNode(3)\nval n2 = ListNode(2)\nval n3 = ListNode(5)\nval n4 = ListNode(4)\n// 노드 사이의 참조 설정\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rb<pre><code># 연결 리스트 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 초기화\n# 각 노드 초기화\nn0 = ListNode.new(1)\nn1 = ListNode.new(3)\nn2 = ListNode.new(2)\nn3 = ListNode.new(5)\nn4 = ListNode.new(4)\n# 노드 사이의 참조 설정\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>배열은 전체가 하나의 변수다. 예를 들어 배열 <code>nums</code>에는 <code>nums[0]</code>, <code>nums[1]</code> 등의 원소가 들어 있다. 반면 연결 리스트는 독립된 여러 노드 객체로 이루어진다. 보통 머리 노드로 연결 리스트 전체를 지칭한다. 예를 들어 위 코드의 연결 리스트는 연결 리스트 <code>n0</code>라고 부를 수 있다.</p>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#2","level":3,"title":"2.   노드 삽입","text":"<p>연결 리스트에 노드를 삽입하기는 매우 쉽다. 아래 그림(4-6)과 같이 인접한 노드 <code>n0</code>와 <code>n1</code> 사이에 새 노드 <code>P</code>를 삽입하려면 노드 참조(포인터) 두 개만 바꾸면 된다. 시간 복잡도는 \\(O(1)\\)이다.</p> <p>반면 배열의 원소 삽입은 시간 복잡도가 \\(O(n)\\)이므로 데이터가 많을 때 효율이 낮다.</p> <p></p> <p> 그림 4-6   연결 리스트에 노드를 삽입하는 예 </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def insert(n0: ListNode, P: ListNode):\n    \"\"\"연결 리스트의 노드 n0 뒤에 노드 P 삽입\"\"\"\n    n1 = n0.next\n    P.next = n1\n    n0.next = P\n</code></pre> linked_list.cpp<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.java<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nvoid insert(ListNode n0, ListNode P) {\n    ListNode n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.cs<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nvoid Insert(ListNode n0, ListNode P) {\n    ListNode? n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.go<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nfunc insertNode(n0 *ListNode, P *ListNode) {\n    n1 := n0.Next\n    P.Next = n1\n    n0.Next = P\n}\n</code></pre> linked_list.swift<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nfunc insert(n0: ListNode, P: ListNode) {\n    let n1 = n0.next\n    P.next = n1\n    n0.next = P\n}\n</code></pre> linked_list.js<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nfunction insert(n0, P) {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.ts<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nfunction insert(n0: ListNode, P: ListNode): void {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.dart<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nvoid insert(ListNode n0, ListNode P) {\n  ListNode? n1 = n0.next;\n  P.next = n1;\n  n0.next = P;\n}\n</code></pre> linked_list.rs<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\n#[allow(non_snake_case)]\npub fn insert&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, P: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    let n1 = n0.borrow_mut().next.take();\n    P.borrow_mut().next = n1;\n    n0.borrow_mut().next = Some(P);\n}\n</code></pre> linked_list.c<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.kt<pre><code>/* 연결 리스트의 노드 n0 뒤에 노드 P 삽입 */\nfun insert(n0: ListNode?, p: ListNode?) {\n    val n1 = n0?.next\n    p?.next = n1\n    n0?.next = p\n}\n</code></pre> linked_list.rb<pre><code>### 연결 리스트의 노드 n0 뒤에 노드 _p 삽입 ###\n# Ruby에서 `p`는 내장 함수이고 `P`는 상수이므로, 대신 `_p`를 사용할 수 있다\ndef insert(n0, _p)\n  n1 = n0.next\n  _p.next = n1\n  n0.next = _p\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#3","level":3,"title":"3.   노드 삭제","text":"<p>아래 그림(4-7)과 같이 연결 리스트의 노드 삭제도 매우 간단하다. 노드 하나의 참조(포인터)만 바꾸면 된다.</p> <p>삭제 후에도 노드 <code>P</code>는 여전히 <code>n1</code>을 가리키지만, 연결 리스트를 순회해서는 더 이상 <code>P</code>에 접근할 수 없다. 따라서 <code>P</code>는 이미 이 연결 리스트에 속하지 않는다.</p> <p></p> <p> 그림 4-7   연결 리스트의 노드 삭제 </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def remove(n0: ListNode):\n    \"\"\"연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제\"\"\"\n    if not n0.next:\n        return\n    # n0 -&gt; P -&gt; n1\n    P = n0.next\n    n1 = P.next\n    n0.next = n1\n</code></pre> linked_list.cpp<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nvoid remove(ListNode *n0) {\n    if (n0-&gt;next == nullptr)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // 메모리 해제\n    delete P;\n}\n</code></pre> linked_list.java<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nvoid remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.cs<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nvoid Remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode? n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.go<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nfunc removeItem(n0 *ListNode) {\n    if n0.Next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    P := n0.Next\n    n1 := P.Next\n    n0.Next = n1\n}\n</code></pre> linked_list.swift<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nfunc remove(n0: ListNode) {\n    if n0.next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    let P = n0.next\n    let n1 = P?.next\n    n0.next = n1\n}\n</code></pre> linked_list.js<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nfunction remove(n0) {\n    if (!n0.next) return;\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.ts<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nfunction remove(n0: ListNode): void {\n    if (!n0.next) {\n        return;\n    }\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.dart<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nvoid remove(ListNode n0) {\n  if (n0.next == null) return;\n  // n0 -&gt; P -&gt; n1\n  ListNode P = n0.next!;\n  ListNode? n1 = P.next;\n  n0.next = n1;\n}\n</code></pre> linked_list.rs<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\n#[allow(non_snake_case)]\npub fn remove&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    // n0 -&gt; P -&gt; n1\n    let P = n0.borrow_mut().next.take();\n    if let Some(node) = P {\n        let n1 = node.borrow_mut().next.take();\n        n0.borrow_mut().next = n1;\n    }\n}\n</code></pre> linked_list.c<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\n// 주의: stdio.h가 remove라는 이름을 사용한다\nvoid removeItem(ListNode *n0) {\n    if (!n0-&gt;next)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // 메모리 해제\n    free(P);\n}\n</code></pre> linked_list.kt<pre><code>/* 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 */\nfun remove(n0: ListNode?) {\n    if (n0?.next == null)\n        return\n    // n0 -&gt; P -&gt; n1\n    val p = n0.next\n    val n1 = p?.next\n    n0.next = n1\n}\n</code></pre> linked_list.rb<pre><code>### 연결 리스트의 노드 n0 바로 뒤에 있는 노드 삭제 ###\ndef remove(n0)\n  return if n0.next.nil?\n\n  # n0 -&gt; remove_node -&gt; n1\n  remove_node = n0.next\n  n1 = remove_node.next\n  n0.next = n1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#4","level":3,"title":"4.   노드 접근","text":"<p>연결 리스트의 노드 접근 효율은 낮다. 앞 절에서 설명했듯 배열은 어떤 원소든 \\(O(1)\\) 시간에 접근할 수 있다. 그러나 연결 리스트에서는 머리 노드부터 하나씩 뒤로 순회하여 목표 노드를 찾아야 한다. 즉, \\(i\\)번째 노드에 접근하려면 \\(i - 1\\)번 반복해야 하므로 시간 복잡도는 \\(O(n)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def access(head: ListNode, index: int) -&gt; ListNode | None:\n    \"\"\"연결 리스트에서 인덱스가 index인 노드에 접근\"\"\"\n    for _ in range(index):\n        if not head:\n            return None\n        head = head.next\n    return head\n</code></pre> linked_list.cpp<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == nullptr)\n            return nullptr;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.java<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nListNode access(ListNode head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.cs<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nListNode? Access(ListNode? head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.go<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nfunc access(head *ListNode, index int) *ListNode {\n    for i := 0; i &lt; index; i++ {\n        if head == nil {\n            return nil\n        }\n        head = head.Next\n    }\n    return head\n}\n</code></pre> linked_list.swift<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nfunc access(head: ListNode, index: Int) -&gt; ListNode? {\n    var head: ListNode? = head\n    for _ in 0 ..&lt; index {\n        if head == nil {\n            return nil\n        }\n        head = head?.next\n    }\n    return head\n}\n</code></pre> linked_list.js<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nfunction access(head, index) {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.ts<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nfunction access(head: ListNode | null, index: number): ListNode | null {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.dart<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nListNode? access(ListNode? head, int index) {\n  for (var i = 0; i &lt; index; i++) {\n    if (head == null) return null;\n    head = head.next;\n  }\n  return head;\n}\n</code></pre> linked_list.rs<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\npub fn access&lt;T&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, index: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n    fn dfs&lt;T&gt;(\n        head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,\n        index: i32,\n    ) -&gt; Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        if index &lt;= 0 {\n            return head.cloned();\n        }\n\n        if let Some(node) = head {\n            dfs(node.borrow().next.as_ref(), index - 1)\n        } else {\n            None\n        }\n    }\n\n    dfs(Some(head).as_ref(), index)\n}\n</code></pre> linked_list.c<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == NULL)\n            return NULL;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.kt<pre><code>/* 연결 리스트에서 인덱스가 index인 노드에 접근 */\nfun access(head: ListNode?, index: Int): ListNode? {\n    var h = head\n    for (i in 0..&lt;index) {\n        if (h == null)\n            return null\n        h = h.next\n    }\n    return h\n}\n</code></pre> linked_list.rb<pre><code>### 연결 리스트에서 인덱스가 index인 노드에 접근 ###\ndef access(head, index)\n  for i in 0...index\n    return nil if head.nil?\n    head = head.next\n  end\n\n  head\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#5","level":3,"title":"5.   노드 탐색","text":"<p>연결 리스트를 순회하여 값이 <code>target</code>인 노드를 찾고, 그 노드의 인덱스를 출력한다. 이 과정도 선형 탐색에 해당한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linked_list.py<pre><code>def find(head: ListNode, target: int) -&gt; int:\n    \"\"\"연결 리스트에서 값이 target인 첫 번째 노드 탐색\"\"\"\n    index = 0\n    while head:\n        if head.val == target:\n            return index\n        head = head.next\n        index += 1\n    return -1\n</code></pre> linked_list.cpp<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head != nullptr) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.java<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nint find(ListNode head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.cs<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nint Find(ListNode? head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.go<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nfunc findNode(head *ListNode, target int) int {\n    index := 0\n    for head != nil {\n        if head.Val == target {\n            return index\n        }\n        head = head.Next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.swift<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nfunc find(head: ListNode, target: Int) -&gt; Int {\n    var head: ListNode? = head\n    var index = 0\n    while head != nil {\n        if head?.val == target {\n            return index\n        }\n        head = head?.next\n        index += 1\n    }\n    return -1\n}\n</code></pre> linked_list.js<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nfunction find(head, target) {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.ts<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nfunction find(head: ListNode | null, target: number): number {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.dart<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nint find(ListNode? head, int target) {\n  int index = 0;\n  while (head != null) {\n    if (head.val == target) {\n      return index;\n    }\n    head = head.next;\n    index++;\n  }\n  return -1;\n}\n</code></pre> linked_list.rs<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\npub fn find&lt;T: PartialEq&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, target: T) -&gt; i32 {\n    fn find&lt;T: PartialEq&gt;(head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, target: T, idx: i32) -&gt; i32 {\n        if let Some(node) = head {\n            if node.borrow().val == target {\n                return idx;\n            }\n            return find(node.borrow().next.as_ref(), target, idx + 1);\n        } else {\n            -1\n        }\n    }\n\n    find(Some(head).as_ref(), target, 0)\n}\n</code></pre> linked_list.c<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.kt<pre><code>/* 연결 리스트에서 값이 target인 첫 번째 노드 탐색 */\nfun find(head: ListNode?, target: Int): Int {\n    var index = 0\n    var h = head\n    while (h != null) {\n        if (h._val == target)\n            return index\n        h = h.next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.rb<pre><code>### 연결 리스트에서 값이 target인 첫 번째 노드 탐색 ###\ndef find(head, target)\n  index = 0\n  while head\n    return index if head.val == target\n    head = head.next\n    index += 1\n  end\n\n  -1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#422-vs","level":2,"title":"4.2.2   배열 vs. 연결 리스트","text":"<p>아래 표(4-1)는 배열과 연결 리스트의 특징과 연산 효율을 비교한다. 둘은 서로 반대되는 저장 전략을 사용하므로 성질과 연산 효율도 대조적이다.</p> <p> 표 4-1   배열과 연결 리스트의 효율 비교 </p> 배열 연결 리스트 저장 방식 연속된 메모리 공간 분산된 메모리 공간 용량 확장 길이를 바꿀 수 없음 유연하게 확장 가능 메모리 효율 원소의 메모리 사용량은 적지만 공간을 낭비할 수 있음 원소의 메모리 사용량이 많음 원소 접근 \\(O(1)\\) \\(O(n)\\) 원소 추가 \\(O(n)\\) \\(O(1)\\) 원소 삭제 \\(O(n)\\) \\(O(1)\\)","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#423","level":2,"title":"4.2.3   대표적인 연결 리스트 종류","text":"<p>아래 그림(4-8)과 같이 대표적인 연결 리스트에는 세 종류가 있다.</p> <ul> <li>단일 연결 리스트: 앞에서 소개한 일반적인 연결 리스트다. 각 노드는 값과 다음 노드를 가리키는 참조라는 두 데이터를 포함한다. 첫 노드는 머리 노드, 마지막 노드는 꼬리 노드이며, 꼬리 노드는 <code>None</code>을 가리킨다.</li> <li>원형 연결 리스트: 단일 연결 리스트의 꼬리 노드가 머리 노드를 가리키도록 앞뒤를 이으면 원형 연결 리스트가 된다. 원형 연결 리스트에서는 어느 노드든 머리 노드로 볼 수 있다.</li> <li>이중 연결 리스트: 단일 연결 리스트와 달리 양방향 참조를 기록한다. 각 노드는 후속 노드(다음 노드)와 선행 노드(이전 노드)를 가리키는 참조(포인터)를 모두 포함한다. 양방향으로 순회할 수 있어 단일 연결 리스트보다 유연하지만, 그만큼 메모리 공간도 더 많이 차지한다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class ListNode:\n    \"\"\"이중 연결 리스트 노드 클래스\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # 노드 값\n        self.next: ListNode | None = None  # 후속 노드를 가리키는 참조\n        self.prev: ListNode | None = None  # 선행 노드를 가리키는 참조\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 구조체 */\nstruct ListNode {\n    int val;         // 노드 값\n    ListNode *next;  // 후속 노드를 가리키는 포인터\n    ListNode *prev;  // 선행 노드를 가리키는 포인터\n    ListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // 생성자\n};\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 클래스 */\nclass ListNode {\n    int val;        // 노드 값\n    ListNode next;  // 후속 노드를 가리키는 참조\n    ListNode prev;  // 선행 노드를 가리키는 참조\n    ListNode(int x) { val = x; }  // 생성자\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 클래스 */\nclass ListNode(int x) {  // 생성자\n    int val = x;    // 노드 값\n    ListNode next;  // 후속 노드를 가리키는 참조\n    ListNode prev;  // 선행 노드를 가리키는 참조\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 구조체 */\ntype DoublyListNode struct {\n    Val  int             // 노드 값\n    Next *DoublyListNode // 후속 노드를 가리키는 포인터\n    Prev *DoublyListNode // 선행 노드를 가리키는 포인터\n}\n\n// NewDoublyListNode 초기화\nfunc NewDoublyListNode(val int) *DoublyListNode {\n    return &amp;DoublyListNode{\n        Val:  val,\n        Next: nil,\n        Prev: nil,\n    }\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 클래스 */\nclass ListNode {\n    var val: Int // 노드 값\n    var next: ListNode? // 후속 노드를 가리키는 참조\n    var prev: ListNode? // 선행 노드를 가리키는 참조\n\n    init(x: Int) { // 생성자\n        val = x\n    }\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 클래스 */\nclass ListNode {\n    constructor(val, next, prev) {\n        this.val = val  ===  undefined ? 0 : val;        // 노드 값\n        this.next = next  ===  undefined ? null : next;  // 후속 노드를 가리키는 참조\n        this.prev = prev  ===  undefined ? null : prev;  // 선행 노드를 가리키는 참조\n    }\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 클래스 */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    prev: ListNode | null;\n    constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\n        this.val = val  ===  undefined ? 0 : val;        // 노드 값\n        this.next = next  ===  undefined ? null : next;  // 후속 노드를 가리키는 참조\n        this.prev = prev  ===  undefined ? null : prev;  // 선행 노드를 가리키는 참조\n    }\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 클래스 */\nclass ListNode {\n    int val;        // 노드 값\n    ListNode? next;  // 후속 노드를 가리키는 참조\n    ListNode? prev;  // 선행 노드를 가리키는 참조\n    ListNode(this.val, [this.next, this.prev]);  // 생성자\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* 이중 연결 리스트 노드 자료형 */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // 노드 값\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // 후속 노드를 가리키는 포인터\n    prev: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // 선행 노드를 가리키는 포인터\n}\n\n/* 생성자 */\nimpl ListNode {\n    fn new(val: i32) -&gt; Self {\n        ListNode {\n            val,\n            next: None,\n            prev: None,\n        }\n    }\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 구조체 */\ntypedef struct ListNode {\n    int val;               // 노드 값\n    struct ListNode *next; // 후속 노드를 가리키는 포인터\n    struct ListNode *prev; // 선행 노드를 가리키는 포인터\n} ListNode;\n\n/* 생성자 */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    node-&gt;prev = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* 이중 연결 리스트 노드 클래스 */\n// 생성자\nclass ListNode(x: Int) {\n    val _val: Int = x           // 노드 값\n    var next: ListNode? = null  // 후속 노드를 가리키는 참조\n    var prev: ListNode? = null  // 선행 노드를 가리키는 참조\n}\n</code></pre> <pre><code># 이중 연결 리스트 노드 클래스\nclass ListNode\n  attr_accessor :val    # 노드 값\n  attr_accessor :next   # 후속 노드를 가리키는 참조\n  attr_accessor :prev   # 선행 노드를 가리키는 참조\n\n  def initialize(val=0, next_node=nil, prev_node=nil)\n    @val = val\n    @next = next_node\n    @prev = prev_node\n  end\nend\n</code></pre> <p></p> <p> 그림 4-8   대표적인 연결 리스트 종류 </p>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/linked_list/#424","level":2,"title":"4.2.4   연결 리스트의 대표적인 활용","text":"<p>단일 연결 리스트는 보통 스택, 큐, 해시 테이블, 그래프 등의 자료 구조를 구현하는 데 사용된다.</p> <ul> <li>스택과 큐: 삽입과 삭제를 모두 연결 리스트의 같은 끝에서 수행하면 후입선출 특성이 나타나 스택에 해당한다. 한쪽 끝에서 삽입하고 반대쪽 끝에서 삭제하면 선입선출 특성이 나타나 큐에 해당한다.</li> <li>해시 테이블: 체이닝은 해시 충돌을 해결하는 대표적인 방법 중 하나로, 충돌하는 모든 원소를 하나의 연결 리스트에 넣는다.</li> <li>그래프: 인접 리스트는 그래프를 표현하는 흔한 방법이다. 각 정점에 연결 리스트 하나를 연결하고, 그 리스트의 각 원소로 해당 정점과 연결된 다른 정점을 나타낸다.</li> </ul> <p>이중 연결 리스트는 이전 원소와 다음 원소를 빠르게 찾아야 하는 상황에 자주 사용된다.</p> <ul> <li>고급 자료 구조: 레드-블랙 트리나 B 트리에서는 노드의 부모 노드에 접근해야 한다. 이중 연결 리스트처럼 노드에 부모를 가리키는 참조를 저장하여 구현할 수 있다.</li> <li>브라우저 방문 기록: 사용자가 웹 브라우저의 앞으로 또는 뒤로 버튼을 누르면 이전이나 다음에 방문했던 페이지를 알아야 한다. 이중 연결 리스트를 사용하면 이러한 연산이 간단해진다.</li> <li>LRU 알고리즘: 캐시 교체 알고리즘인 LRU에서는 가장 오랫동안 사용하지 않은 데이터를 빠르게 찾고 노드도 빠르게 추가하고 삭제해야 한다. 이때 이중 연결 리스트가 매우 적합하다.</li> </ul> <p>원형 연결 리스트는 운영체제의 자원 스케줄링처럼 주기적인 연산이 필요한 상황에 자주 사용된다.</p> <ul> <li>라운드 로빈 스케줄링 알고리즘: 운영체제의 대표적인 CPU 스케줄링 알고리즘으로, 여러 프로세스를 순환한다. 각 프로세스에 타임 슬라이스를 할당하고, 시간이 다 되면 CPU가 다음 프로세스로 전환한다. 이러한 순환은 원형 연결 리스트로 구현할 수 있다.</li> <li>데이터 버퍼: 일부 데이터 버퍼도 원형 연결 리스트로 구현할 수 있다. 예를 들어 오디오나 동영상 플레이어에서 데이터 스트림을 여러 버퍼 블록으로 나누어 원형 연결 리스트에 넣으면 끊김 없이 재생할 수 있다.</li> </ul>","path":["제 4 장   배열과 연결 리스트","4.2   연결 리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/","level":1,"title":"4.3   리스트","text":"<p>리스트(list)는 순서가 있는 원소의 모음을 나타내는 추상적인 자료 구조 개념이다. 원소 접근, 수정, 추가, 삭제, 순회 등의 연산을 지원하며, 사용자가 용량 제한을 신경 쓰지 않아도 된다. 리스트는 연결 리스트나 배열로 구현할 수 있다.</p> <ul> <li>연결 리스트는 원소의 삽입, 삭제, 탐색, 갱신을 지원하고 유연하게 용량을 늘릴 수 있으므로 자연스럽게 리스트로 볼 수 있다.</li> <li>배열도 원소의 삽입, 삭제, 탐색, 갱신을 지원하지만 길이를 바꿀 수 없으므로 길이가 제한된 리스트로 볼 수 있다.</li> </ul> <p>배열로 리스트를 구현하면 길이를 바꿀 수 없다는 특성 때문에 실용성이 낮아진다. 보통 저장할 데이터 양을 미리 알 수 없어 적절한 길이를 정하기 어렵기 때문이다. 길이가 너무 짧으면 필요를 충족하지 못하고, 너무 길면 메모리 공간을 낭비한다.</p> <p>이 문제는 동적 배열(dynamic array)로 리스트를 구현하여 해결할 수 있다. 동적 배열은 배열의 장점을 이어받으면서도 프로그램 실행 중에 용량을 동적으로 늘릴 수 있다.</p> <p>실제로 많은 프로그래밍 언어의 표준 라이브러리가 제공하는 리스트는 동적 배열로 구현된다. Python의 <code>list</code>, Java의 <code>ArrayList</code>, C++의 <code>vector</code>, C#의 <code>List</code> 등이 그 예다. 이후 설명에서는 ‘리스트’와 ‘동적 배열’을 같은 개념으로 다룬다.</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#431","level":2,"title":"4.3.1   리스트의 기본 연산","text":"","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#1","level":3,"title":"1.   리스트 초기화","text":"<p>보통 ‘초깃값 없음’과 ‘초깃값 지정’의 두 가지 초기화 방식을 사용한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># 리스트 초기화\n# 초깃값 없음\nnums1: list[int] = []\n# 초깃값 지정\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> list.cpp<pre><code>/* 리스트 초기화 */\n// C++의 vector가 이 글에서 설명하는 nums에 해당한다\n// 초깃값 없음\nvector&lt;int&gt; nums1;\n// 초깃값 지정\nvector&lt;int&gt; nums = { 1, 3, 2, 5, 4 };\n</code></pre> list.java<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;();\n// 초깃값 지정(배열은 int[]의 래퍼형인 Integer[]여야 함)\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList&lt;Integer&gt; nums = new ArrayList&lt;&gt;(Arrays.asList(numbers));\n</code></pre> list.cs<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nList&lt;int&gt; nums1 = [];\n// 초깃값 지정\nint[] numbers = [1, 3, 2, 5, 4];\nList&lt;int&gt; nums = [.. numbers];\n</code></pre> list_test.go<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nnums1 := []int{}\n// 초깃값 지정\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> list.swift<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nlet nums1: [Int] = []\n// 초깃값 지정\nvar nums = [1, 3, 2, 5, 4]\n</code></pre> list.js<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nconst nums1 = [];\n// 초깃값 지정\nconst nums = [1, 3, 2, 5, 4];\n</code></pre> list.ts<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nconst nums1: number[] = [];\n// 초깃값 지정\nconst nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> list.dart<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nList&lt;int&gt; nums1 = [];\n// 초깃값 지정\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> list.rs<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nlet nums1: Vec&lt;i32&gt; = Vec::new();\n// 초깃값 지정\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> list.c<pre><code>// C는 내장 동적 배열을 제공하지 않는다\n</code></pre> list.kt<pre><code>/* 리스트 초기화 */\n// 초깃값 없음\nvar nums1 = listOf&lt;Int&gt;()\n// 초깃값 지정\nvar numbers = arrayOf(1, 3, 2, 5, 4)\nvar nums = numbers.toMutableList()\n</code></pre> list.rb<pre><code># 리스트 초기화\n# 초깃값 없음\nnums1 = []\n# 초깃값 지정\nnums = [1, 3, 2, 5, 4]\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#2","level":3,"title":"2.   원소 접근","text":"<p>리스트는 본질적으로 배열이므로 원소에 접근하고 갱신하는 데 \\(O(1)\\) 시간이 걸려 매우 효율적이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># 원소 접근\nnum: int = nums[1]  # 인덱스 1의 원소에 접근\n\n# 원소 갱신\nnums[1] = 0    # 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.cpp<pre><code>/* 원소 접근 */\nint num = nums[1];  // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums[1] = 0;  // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.java<pre><code>/* 원소 접근 */\nint num = nums.get(1);  // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums.set(1, 0);  // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.cs<pre><code>/* 원소 접근 */\nint num = nums[1];  // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums[1] = 0;  // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list_test.go<pre><code>/* 원소 접근 */\nnum := nums[1]  // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums[1] = 0     // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.swift<pre><code>/* 원소 접근 */\nlet num = nums[1] // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums[1] = 0 // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.js<pre><code>/* 원소 접근 */\nconst num = nums[1];  // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums[1] = 0;  // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.ts<pre><code>/* 원소 접근 */\nconst num: number = nums[1];  // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums[1] = 0;  // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.dart<pre><code>/* 원소 접근 */\nint num = nums[1];  // 인덱스 1의 원소에 접근\n\n/* 원소 갱신 */\nnums[1] = 0;  // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.rs<pre><code>/* 원소 접근 */\nlet num: i32 = nums[1];  // 인덱스 1의 원소에 접근\n/* 원소 갱신 */\nnums[1] = 0;             // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.c<pre><code>// C는 내장 동적 배열을 제공하지 않는다\n</code></pre> list.kt<pre><code>/* 원소 접근 */\nval num = nums[1]       // 인덱스 1의 원소에 접근\n/* 원소 갱신 */\nnums[1] = 0             // 인덱스 1의 원소를 0으로 갱신\n</code></pre> list.rb<pre><code># 원소 접근\nnum = nums[1] # 인덱스 1의 원소에 접근\n# 원소 갱신\nnums[1] = 0 # 인덱스 1의 원소를 0으로 갱신\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#3","level":3,"title":"3.   원소 삽입과 삭제","text":"<p>배열과 달리 리스트는 원소를 자유롭게 추가하고 삭제할 수 있다. 끝에 원소를 추가하는 시간 복잡도는 \\(O(1)\\)이지만, 삽입과 삭제의 효율은 배열과 같아 시간 복잡도가 \\(O(n)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># 리스트 비우기\nnums.clear()\n\n# 끝에 원소 추가\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n# 중간에 원소 삽입\nnums.insert(3, 6)  # 인덱스 3에 숫자 6 삽입\n\n# 원소 삭제\nnums.pop(3)        # 인덱스 3의 원소 삭제\n</code></pre> list.cpp<pre><code>/* 리스트 비우기 */\nnums.clear();\n\n/* 끝에 원소 추가 */\nnums.push_back(1);\nnums.push_back(3);\nnums.push_back(2);\nnums.push_back(5);\nnums.push_back(4);\n\n/* 중간에 원소 삽입 */\nnums.insert(nums.begin() + 3, 6);  // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.erase(nums.begin() + 3);      // 인덱스 3의 원소 삭제\n</code></pre> list.java<pre><code>/* 리스트 비우기 */\nnums.clear();\n\n/* 끝에 원소 추가 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* 중간에 원소 삽입 */\nnums.add(3, 6);  // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.remove(3);  // 인덱스 3의 원소 삭제\n</code></pre> list.cs<pre><code>/* 리스트 비우기 */\nnums.Clear();\n\n/* 끝에 원소 추가 */\nnums.Add(1);\nnums.Add(3);\nnums.Add(2);\nnums.Add(5);\nnums.Add(4);\n\n/* 중간에 원소 삽입 */\nnums.Insert(3, 6);  // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.RemoveAt(3);  // 인덱스 3의 원소 삭제\n</code></pre> list_test.go<pre><code>/* 리스트 비우기 */\nnums = nil\n\n/* 끝에 원소 추가 */\nnums = append(nums, 1)\nnums = append(nums, 3)\nnums = append(nums, 2)\nnums = append(nums, 5)\nnums = append(nums, 4)\n\n/* 중간에 원소 삽입 */\nnums = append(nums[:3], append([]int{6}, nums[3:]...)...) // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums = append(nums[:3], nums[4:]...) // 인덱스 3의 원소 삭제\n</code></pre> list.swift<pre><code>/* 리스트 비우기 */\nnums.removeAll()\n\n/* 끝에 원소 추가 */\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n/* 중간에 원소 삽입 */\nnums.insert(6, at: 3) // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.remove(at: 3) // 인덱스 3의 원소 삭제\n</code></pre> list.js<pre><code>/* 리스트 비우기 */\nnums.length = 0;\n\n/* 끝에 원소 추가 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* 중간에 원소 삽입 */\nnums.splice(3, 0, 6); // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.splice(3, 1);  // 인덱스 3의 원소 삭제\n</code></pre> list.ts<pre><code>/* 리스트 비우기 */\nnums.length = 0;\n\n/* 끝에 원소 추가 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* 중간에 원소 삽입 */\nnums.splice(3, 0, 6); // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.splice(3, 1);  // 인덱스 3의 원소 삭제\n</code></pre> list.dart<pre><code>/* 리스트 비우기 */\nnums.clear();\n\n/* 끝에 원소 추가 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* 중간에 원소 삽입 */\nnums.insert(3, 6); // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.removeAt(3); // 인덱스 3의 원소 삭제\n</code></pre> list.rs<pre><code>/* 리스트 비우기 */\nnums.clear();\n\n/* 끝에 원소 추가 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* 중간에 원소 삽입 */\nnums.insert(3, 6);  // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.remove(3);    // 인덱스 3의 원소 삭제\n</code></pre> list.c<pre><code>// C는 내장 동적 배열을 제공하지 않는다\n</code></pre> list.kt<pre><code>/* 리스트 비우기 */\nnums.clear();\n\n/* 끝에 원소 추가 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* 중간에 원소 삽입 */\nnums.add(3, 6);  // 인덱스 3에 숫자 6 삽입\n\n/* 원소 삭제 */\nnums.remove(3);  // 인덱스 3의 원소 삭제\n</code></pre> list.rb<pre><code># 리스트 비우기\nnums.clear\n\n# 끝에 원소 추가\nnums &lt;&lt; 1\nnums &lt;&lt; 3\nnums &lt;&lt; 2\nnums &lt;&lt; 5\nnums &lt;&lt; 4\n\n# 중간에 원소 삽입\nnums.insert(3, 6) # 인덱스 3에 숫자 6 삽입\n\n# 원소 삭제\nnums.delete_at(3) # 인덱스 3의 원소 삭제\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#4","level":3,"title":"4.   리스트 순회","text":"<p>배열처럼 리스트도 인덱스로 순회하거나 각 원소를 직접 순회할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># 인덱스로 리스트 순회\ncount = 0\nfor i in range(len(nums)):\n    count += nums[i]\n\n# 리스트 원소를 직접 순회\nfor num in nums:\n    count += num\n</code></pre> list.cpp<pre><code>/* 인덱스로 리스트 순회 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums[i];\n}\n\n/* 리스트 원소를 직접 순회 */\ncount = 0;\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.java<pre><code>/* 인덱스로 리스트 순회 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums.get(i);\n}\n\n/* 리스트 원소를 직접 순회 */\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.cs<pre><code>/* 인덱스로 리스트 순회 */\nint count = 0;\nfor (int i = 0; i &lt; nums.Count; i++) {\n    count += nums[i];\n}\n\n/* 리스트 원소를 직접 순회 */\ncount = 0;\nforeach (int num in nums) {\n    count += num;\n}\n</code></pre> list_test.go<pre><code>/* 인덱스로 리스트 순회 */\ncount := 0\nfor i := 0; i &lt; len(nums); i++ {\n    count += nums[i]\n}\n\n/* 리스트 원소를 직접 순회 */\ncount = 0\nfor _, num := range nums {\n    count += num\n}\n</code></pre> list.swift<pre><code>/* 인덱스로 리스트 순회 */\nvar count = 0\nfor i in nums.indices {\n    count += nums[i]\n}\n\n/* 리스트 원소를 직접 순회 */\ncount = 0\nfor num in nums {\n    count += num\n}\n</code></pre> list.js<pre><code>/* 인덱스로 리스트 순회 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* 리스트 원소를 직접 순회 */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.ts<pre><code>/* 인덱스로 리스트 순회 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* 리스트 원소를 직접 순회 */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.dart<pre><code>/* 인덱스로 리스트 순회 */\nint count = 0;\nfor (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* 리스트 원소를 직접 순회 */\ncount = 0;\nfor (var num in nums) {\n    count += num;\n}\n</code></pre> list.rs<pre><code>// 인덱스로 리스트 순회\nlet mut _count = 0;\nfor i in 0..nums.len() {\n    _count += nums[i];\n}\n\n// 리스트 원소를 직접 순회\n_count = 0;\nfor num in &amp;nums {\n    _count += num;\n}\n</code></pre> list.c<pre><code>// C는 내장 동적 배열을 제공하지 않는다\n</code></pre> list.kt<pre><code>/* 인덱스로 리스트 순회 */\nvar count = 0\nfor (i in nums.indices) {\n    count += nums[i]\n}\n\n/* 리스트 원소를 직접 순회 */\nfor (num in nums) {\n    count += num\n}\n</code></pre> list.rb<pre><code># 인덱스로 리스트 순회\ncount = 0\nfor i in 0...nums.length\n    count += nums[i]\nend\n\n# 리스트 원소를 직접 순회\ncount = 0\nfor num in nums\n    count += num\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#5","level":3,"title":"5.   리스트 연결","text":"<p>새 리스트 <code>nums1</code>이 주어지면 기존 리스트의 끝에 이어 붙일 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># 두 리스트 연결\nnums1: list[int] = [6, 8, 7, 10, 9]\nnums += nums1  # 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.cpp<pre><code>/* 두 리스트 연결 */\nvector&lt;int&gt; nums1 = { 6, 8, 7, 10, 9 };\n// 리스트 nums1을 nums 뒤에 연결\nnums.insert(nums.end(), nums1.begin(), nums1.end());\n</code></pre> list.java<pre><code>/* 두 리스트 연결 */\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nnums.addAll(nums1);  // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.cs<pre><code>/* 두 리스트 연결 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.AddRange(nums1);  // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list_test.go<pre><code>/* 두 리스트 연결 */\nnums1 := []int{6, 8, 7, 10, 9}\nnums = append(nums, nums1...)  // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.swift<pre><code>/* 두 리스트 연결 */\nlet nums1 = [6, 8, 7, 10, 9]\nnums.append(contentsOf: nums1) // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.js<pre><code>/* 두 리스트 연결 */\nconst nums1 = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.ts<pre><code>/* 두 리스트 연결 */\nconst nums1: number[] = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.dart<pre><code>/* 두 리스트 연결 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.addAll(nums1);  // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.rs<pre><code>/* 두 리스트 연결 */\nlet nums1: Vec&lt;i32&gt; = vec![6, 8, 7, 10, 9];\nnums.extend(nums1);\n</code></pre> list.c<pre><code>// C는 내장 동적 배열을 제공하지 않는다\n</code></pre> list.kt<pre><code>/* 두 리스트 연결 */\nval nums1 = intArrayOf(6, 8, 7, 10, 9).toMutableList()\nnums.addAll(nums1)  // 리스트 nums1을 nums 뒤에 연결\n</code></pre> list.rb<pre><code># 두 리스트 연결\nnums1 = [6, 8, 7, 10, 9]\nnums += nums1\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#6","level":3,"title":"6.   리스트 정렬","text":"<p>리스트를 정렬하면 배열 관련 알고리즘 문제에 자주 등장하는 ‘이진 탐색’과 ‘투 포인터’ 알고리즘을 사용할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby list.py<pre><code># 리스트 정렬\nnums.sort()  # 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.cpp<pre><code>/* 리스트 정렬 */\nsort(nums.begin(), nums.end());  // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.java<pre><code>/* 리스트 정렬 */\nCollections.sort(nums);  // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.cs<pre><code>/* 리스트 정렬 */\nnums.Sort(); // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list_test.go<pre><code>/* 리스트 정렬 */\nsort.Ints(nums)  // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.swift<pre><code>/* 리스트 정렬 */\nnums.sort() // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.js<pre><code>/* 리스트 정렬 */\nnums.sort((a, b) =&gt; a - b);  // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.ts<pre><code>/* 리스트 정렬 */\nnums.sort((a, b) =&gt; a - b);  // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.dart<pre><code>/* 리스트 정렬 */\nnums.sort(); // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.rs<pre><code>/* 리스트 정렬 */\nnums.sort(); // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.c<pre><code>// C는 내장 동적 배열을 제공하지 않는다\n</code></pre> list.kt<pre><code>/* 리스트 정렬 */\nnums.sort() // 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> list.rb<pre><code># 리스트 정렬\nnums = nums.sort { |a, b| a &lt;=&gt; b } # 정렬 후 리스트 원소가 오름차순으로 배치됨\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/list/#432","level":2,"title":"4.3.2   리스트 구현","text":"<p>Java, C++, Python 등 많은 프로그래밍 언어는 리스트를 내장하고 있다. 이들의 구현은 비교적 복잡하며 초기 용량, 확장 배수 등의 매개변수도 정교하게 설정되어 있다. 관심 있는 독자는 소스 코드를 살펴보며 학습할 수 있다.</p> <p>리스트의 작동 원리를 더 깊이 이해하기 위해 간단한 리스트를 구현해 보자. 핵심 설계는 다음 세 가지다.</p> <ul> <li>초기 용량: 배열의 초기 용량을 적절히 선택한다. 이 예에서는 10을 초기 용량으로 정한다.</li> <li>개수 기록: 현재 원소 수를 기록하는 변수 <code>size</code>를 선언하고, 삽입과 삭제에 따라 즉시 갱신한다. 이 변수를 통해 리스트의 끝을 찾고 용량 확장이 필요한지 판단할 수 있다.</li> <li>용량 확장 방식: 원소를 삽입할 때 리스트가 가득 찼으면 용량을 늘려야 한다. 확장 배수에 따라 더 큰 배열을 만든 뒤 기존 원소를 모두 차례로 옮긴다. 이 예에서는 매번 배열 용량을 이전의 2배로 늘린다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_list.py<pre><code>class MyList:\n    \"\"\"리스트 클래스\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        self._capacity: int = 10  # 리스트 용량\n        self._arr: list[int] = [0] * self._capacity  # 배열(리스트 원소 저장)\n        self._size: int = 0  # 리스트 길이(현재 원소 개수)\n        self._extend_ratio: int = 2  # 리스트의 용량을 확장할 때마다 적용하는 배수\n\n    def size(self) -&gt; int:\n        \"\"\"리스트 길이(현재 원소 개수) 반환\"\"\"\n        return self._size\n\n    def capacity(self) -&gt; int:\n        \"\"\"리스트 용량 반환\"\"\"\n        return self._capacity\n\n    def get(self, index: int) -&gt; int:\n        \"\"\"원소 접근\"\"\"\n        # 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"인덱스가 범위를 벗어났다\")\n        return self._arr[index]\n\n    def set(self, num: int, index: int):\n        \"\"\"원소 갱신\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"인덱스가 범위를 벗어났다\")\n        self._arr[index] = num\n\n    def add(self, num: int):\n        \"\"\"맨 뒤에 원소 추가\"\"\"\n        # 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if self.size() == self.capacity():\n            self.extend_capacity()\n        self._arr[self._size] = num\n        self._size += 1\n\n    def insert(self, num: int, index: int):\n        \"\"\"중간에 원소 삽입\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"인덱스가 범위를 벗어났다\")\n        # 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if self._size == self.capacity():\n            self.extend_capacity()\n        # 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for j in range(self._size - 1, index - 1, -1):\n            self._arr[j + 1] = self._arr[j]\n        self._arr[index] = num\n        # 원소 개수를 갱신한다\n        self._size += 1\n\n    def remove(self, index: int) -&gt; int:\n        \"\"\"원소 삭제\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"인덱스가 범위를 벗어났다\")\n        num = self._arr[index]\n        # 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for j in range(index, self._size - 1):\n            self._arr[j] = self._arr[j + 1]\n        # 원소 개수를 갱신한다\n        self._size -= 1\n        # 삭제한 원소를 반환한다\n        return num\n\n    def extend_capacity(self):\n        \"\"\"리스트 용량 확장\"\"\"\n        # 원래 배열 길이의 _extend_ratio 배인 새 배열을 만들고 원래 배열을 새 배열에 복사한다\n        self._arr = self._arr + [0] * self.capacity() * (self._extend_ratio - 1)\n        # 리스트 용량을 갱신한다\n        self._capacity = len(self._arr)\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"유효한 길이만큼의 리스트 반환\"\"\"\n        return self._arr[: self._size]\n</code></pre> my_list.cpp<pre><code>/* 리스트 클래스 */\nclass MyList {\n  private:\n    int *arr;             // 배열(리스트 원소 저장)\n    int arrCapacity = 10; // 리스트 용량\n    int arrSize = 0;      // 리스트 길이(현재 원소 개수)\n    int extendRatio = 2;   // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n  public:\n    /* 생성자 */\n    MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* 소멸자 */\n    ~MyList() {\n        delete[] arr;\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환*/\n    int size() {\n        return arrSize;\n    }\n\n    /* 리스트 용량 반환 */\n    int capacity() {\n        return arrCapacity;\n    }\n\n    /* 원소 접근 */\n    int get(int index) {\n        // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"인덱스가 범위를 벗어났다\");\n        return arr[index];\n    }\n\n    /* 원소 갱신 */\n    void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"인덱스가 범위를 벗어났다\");\n        arr[index] = num;\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    void add(int num) {\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (size() == capacity())\n            extendCapacity();\n        arr[size()] = num;\n        // 원소 개수를 갱신한다\n        arrSize++;\n    }\n\n    /* 중간에 원소 삽입 */\n    void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"인덱스가 범위를 벗어났다\");\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (size() == capacity())\n            extendCapacity();\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for (int j = size() - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // 원소 개수를 갱신한다\n        arrSize++;\n    }\n\n    /* 원소 삭제 */\n    int remove(int index) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"인덱스가 범위를 벗어났다\");\n        int num = arr[index];\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for (int j = index; j &lt; size() - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // 원소 개수를 갱신한다\n        arrSize--;\n        // 삭제한 원소를 반환한다\n        return num;\n    }\n\n    /* 리스트 용량 확장 */\n    void extendCapacity() {\n        // 길이가 기존 배열의 extendRatio배인 새 배열 생성\n        int newCapacity = capacity() * extendRatio;\n        int *tmp = arr;\n        arr = new int[newCapacity];\n        // 원래 배열의 모든 원소를 새 배열에 복사한다\n        for (int i = 0; i &lt; size(); i++) {\n            arr[i] = tmp[i];\n        }\n        // 메모리 해제\n        delete[] tmp;\n        arrCapacity = newCapacity;\n    }\n\n    /* 출력을 위해 리스트를 Vector로 변환 */\n    vector&lt;int&gt; toVector() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        vector&lt;int&gt; vec(size());\n        for (int i = 0; i &lt; size(); i++) {\n            vec[i] = arr[i];\n        }\n        return vec;\n    }\n};\n</code></pre> my_list.java<pre><code>/* 리스트 클래스 */\nclass MyList {\n    private int[] arr; // 배열(리스트 원소 저장)\n    private int capacity = 10; // 리스트 용량\n    private int size = 0; // 리스트 길이(현재 원소 개수)\n    private int extendRatio = 2; // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n    /* 생성자 */\n    public MyList() {\n        arr = new int[capacity];\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환 */\n    public int size() {\n        return size;\n    }\n\n    /* 리스트 용량 반환 */\n    public int capacity() {\n        return capacity;\n    }\n\n    /* 원소 접근 */\n    public int get(int index) {\n        // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\");\n        return arr[index];\n    }\n\n    /* 원소 갱신 */\n    public void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\");\n        arr[index] = num;\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    public void add(int num) {\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (size == capacity())\n            extendCapacity();\n        arr[size] = num;\n        // 원소 개수를 갱신한다\n        size++;\n    }\n\n    /* 중간에 원소 삽입 */\n    public void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\");\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (size == capacity())\n            extendCapacity();\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for (int j = size - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // 원소 개수를 갱신한다\n        size++;\n    }\n\n    /* 원소 삭제 */\n    public int remove(int index) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\");\n        int num = arr[index];\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for (int j = index; j &lt; size - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // 원소 개수를 갱신한다\n        size--;\n        // 삭제한 원소를 반환한다\n        return num;\n    }\n\n    /* 리스트 용량 확장 */\n    public void extendCapacity() {\n        // 길이가 원래 배열의 extendRatio배인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n        arr = Arrays.copyOf(arr, capacity() * extendRatio);\n        // 리스트 용량을 갱신한다\n        capacity = arr.length;\n    }\n\n    /* 리스트를 배열로 변환 */\n    public int[] toArray() {\n        int size = size();\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        int[] arr = new int[size];\n        for (int i = 0; i &lt; size; i++) {\n            arr[i] = get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.cs<pre><code>/* 리스트 클래스 */\nclass MyList {\n    private int[] arr;           // 배열(리스트 원소 저장)\n    private int arrCapacity = 10;    // 리스트 용량\n    private int arrSize = 0;         // 리스트 길이(현재 원소 개수)\n    private readonly int extendRatio = 2;  // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n    /* 생성자 */\n    public MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환*/\n    public int Size() {\n        return arrSize;\n    }\n\n    /* 리스트 용량 반환 */\n    public int Capacity() {\n        return arrCapacity;\n    }\n\n    /* 원소 접근 */\n    public int Get(int index) {\n        // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"인덱스가 범위를 벗어났다\");\n        return arr[index];\n    }\n\n    /* 원소 갱신 */\n    public void Set(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"인덱스가 범위를 벗어났다\");\n        arr[index] = num;\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    public void Add(int num) {\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        arr[arrSize] = num;\n        // 원소 개수를 갱신한다\n        arrSize++;\n    }\n\n    /* 중간에 원소 삽입 */\n    public void Insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"인덱스가 범위를 벗어났다\");\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for (int j = arrSize - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // 원소 개수를 갱신한다\n        arrSize++;\n    }\n\n    /* 원소 삭제 */\n    public int Remove(int index) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"인덱스가 범위를 벗어났다\");\n        int num = arr[index];\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for (int j = index; j &lt; arrSize - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // 원소 개수를 갱신한다\n        arrSize--;\n        // 삭제한 원소를 반환한다\n        return num;\n    }\n\n    /* 리스트 용량 확장 */\n    public void ExtendCapacity() {\n        // 길이가 arrCapacity * extendRatio인 배열을 만들고 기존 배열을 새 배열에 복사\n        Array.Resize(ref arr, arrCapacity * extendRatio);\n        // 리스트 용량을 갱신한다\n        arrCapacity = arr.Length;\n    }\n\n    /* 리스트를 배열로 변환 */\n    public int[] ToArray() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        int[] arr = new int[arrSize];\n        for (int i = 0; i &lt; arrSize; i++) {\n            arr[i] = Get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.go<pre><code>/* 리스트 클래스 */\ntype myList struct {\n    arrCapacity int\n    arr         []int\n    arrSize     int\n    extendRatio int\n}\n\n/* 생성자 */\nfunc newMyList() *myList {\n    return &amp;myList{\n        arrCapacity: 10,              // 리스트 용량\n        arr:         make([]int, 10), // 배열(리스트 원소 저장)\n        arrSize:     0,               // 리스트 길이(현재 원소 개수)\n        extendRatio: 2,               // 리스트의 용량을 확장할 때마다 적용하는 배수\n    }\n}\n\n/* 리스트 길이(현재 원소 개수) 반환 */\nfunc (l *myList) size() int {\n    return l.arrSize\n}\n\n/*  리스트 용량 반환 */\nfunc (l *myList) capacity() int {\n    return l.arrCapacity\n}\n\n/* 원소 접근 */\nfunc (l *myList) get(index int) int {\n    // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"인덱스가 범위를 벗어났다\")\n    }\n    return l.arr[index]\n}\n\n/* 원소 갱신 */\nfunc (l *myList) set(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"인덱스가 범위를 벗어났다\")\n    }\n    l.arr[index] = num\n}\n\n/* 맨 뒤에 원소 추가 */\nfunc (l *myList) add(num int) {\n    // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    l.arr[l.arrSize] = num\n    // 원소 개수를 갱신한다\n    l.arrSize++\n}\n\n/* 중간에 원소 삽입 */\nfunc (l *myList) insert(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"인덱스가 범위를 벗어났다\")\n    }\n    // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n    for j := l.arrSize - 1; j &gt;= index; j-- {\n        l.arr[j+1] = l.arr[j]\n    }\n    l.arr[index] = num\n    // 원소 개수를 갱신한다\n    l.arrSize++\n}\n\n/* 원소 삭제 */\nfunc (l *myList) remove(index int) int {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"인덱스가 범위를 벗어났다\")\n    }\n    num := l.arr[index]\n    // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n    for j := index; j &lt; l.arrSize-1; j++ {\n        l.arr[j] = l.arr[j+1]\n    }\n    // 원소 개수를 갱신한다\n    l.arrSize--\n    // 삭제한 원소를 반환한다\n    return num\n}\n\n/* 리스트 용량 확장 */\nfunc (l *myList) extendCapacity() {\n    // 길이가 원래 배열의 extendRatio배인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n    l.arr = append(l.arr, make([]int, l.arrCapacity*(l.extendRatio-1))...)\n    // 리스트 용량을 갱신한다\n    l.arrCapacity = len(l.arr)\n}\n\n/* 유효한 길이만큼의 리스트 반환 */\nfunc (l *myList) toArray() []int {\n    // 유효한 길이 범위 안의 리스트 원소만 변환한다\n    return l.arr[:l.arrSize]\n}\n</code></pre> my_list.swift<pre><code>/* 리스트 클래스 */\nclass MyList {\n    private var arr: [Int] // 배열(리스트 원소 저장)\n    private var _capacity: Int // 리스트 용량\n    private var _size: Int // 리스트 길이(현재 원소 개수)\n    private let extendRatio: Int // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n    /* 생성자 */\n    init() {\n        _capacity = 10\n        _size = 0\n        extendRatio = 2\n        arr = Array(repeating: 0, count: _capacity)\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환*/\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* 리스트 용량 반환 */\n    func capacity() -&gt; Int {\n        _capacity\n    }\n\n    /* 원소 접근 */\n    func get(index: Int) -&gt; Int {\n        // 인덱스가 범위를 벗어나면 오류를 발생시킨다. 아래에서도 동일하다\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"인덱스가 범위를 벗어났다\")\n        }\n        return arr[index]\n    }\n\n    /* 원소 갱신 */\n    func set(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"인덱스가 범위를 벗어났다\")\n        }\n        arr[index] = num\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    func add(num: Int) {\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if size() == capacity() {\n            extendCapacity()\n        }\n        arr[size()] = num\n        // 원소 개수를 갱신한다\n        _size += 1\n    }\n\n    /* 중간에 원소 삽입 */\n    func insert(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"인덱스가 범위를 벗어났다\")\n        }\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if size() == capacity() {\n            extendCapacity()\n        }\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for j in (index ..&lt; size()).reversed() {\n            arr[j + 1] = arr[j]\n        }\n        arr[index] = num\n        // 원소 개수를 갱신한다\n        _size += 1\n    }\n\n    /* 원소 삭제 */\n    @discardableResult\n    func remove(index: Int) -&gt; Int {\n        if index &lt; 0 || index &gt;= size() {\n            fatalError(\"인덱스가 범위를 벗어났다\")\n        }\n        let num = arr[index]\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for j in index ..&lt; (size() - 1) {\n            arr[j] = arr[j + 1]\n        }\n        // 원소 개수를 갱신한다\n        _size -= 1\n        // 삭제한 원소를 반환한다\n        return num\n    }\n\n    /* 리스트 용량 확장 */\n    func extendCapacity() {\n        // 길이가 원래 배열의 extendRatio배인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n        arr = arr + Array(repeating: 0, count: capacity() * (extendRatio - 1))\n        // 리스트 용량을 갱신한다\n        _capacity = arr.count\n    }\n\n    /* 리스트를 배열로 변환 */\n    func toArray() -&gt; [Int] {\n        Array(arr.prefix(size()))\n    }\n}\n</code></pre> my_list.js<pre><code>/* 리스트 클래스 */\nclass MyList {\n    #arr = new Array(); // 배열(리스트 원소 저장)\n    #capacity = 10; // 리스트 용량\n    #size = 0; // 리스트 길이(현재 원소 개수)\n    #extendRatio = 2; // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n    /* 생성자 */\n    constructor() {\n        this.#arr = new Array(this.#capacity);\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환*/\n    size() {\n        return this.#size;\n    }\n\n    /* 리스트 용량 반환 */\n    capacity() {\n        return this.#capacity;\n    }\n\n    /* 원소 접근 */\n    get(index) {\n        // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('인덱스가 범위를 벗어났다');\n        return this.#arr[index];\n    }\n\n    /* 원소 갱신 */\n    set(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('인덱스가 범위를 벗어났다');\n        this.#arr[index] = num;\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    add(num) {\n        // 길이가 용량과 같으면 용량을 확장해야 한다\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // 새 원소를 리스트 맨 뒤에 추가한다\n        this.#arr[this.#size] = num;\n        this.#size++;\n    }\n\n    /* 중간에 원소 삽입 */\n    insert(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('인덱스가 범위를 벗어났다');\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for (let j = this.#size - 1; j &gt;= index; j--) {\n            this.#arr[j + 1] = this.#arr[j];\n        }\n        // 원소 개수를 갱신한다\n        this.#arr[index] = num;\n        this.#size++;\n    }\n\n    /* 원소 삭제 */\n    remove(index) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('인덱스가 범위를 벗어났다');\n        let num = this.#arr[index];\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for (let j = index; j &lt; this.#size - 1; j++) {\n            this.#arr[j] = this.#arr[j + 1];\n        }\n        // 원소 개수를 갱신한다\n        this.#size--;\n        // 삭제한 원소를 반환한다\n        return num;\n    }\n\n    /* 리스트 용량 확장 */\n    extendCapacity() {\n        // 길이가 원래 배열의 extendRatio배인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n        this.#arr = this.#arr.concat(\n            new Array(this.capacity() * (this.#extendRatio - 1))\n        );\n        // 리스트 용량을 갱신한다\n        this.#capacity = this.#arr.length;\n    }\n\n    /* 리스트를 배열로 변환 */\n    toArray() {\n        let size = this.size();\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.ts<pre><code>/* 리스트 클래스 */\nclass MyList {\n    private arr: Array&lt;number&gt;; // 배열(리스트 원소 저장)\n    private _capacity: number = 10; // 리스트 용량\n    private _size: number = 0; // 리스트 길이(현재 원소 개수)\n    private extendRatio: number = 2; // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n    /* 생성자 */\n    constructor() {\n        this.arr = new Array(this._capacity);\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환*/\n    public size(): number {\n        return this._size;\n    }\n\n    /* 리스트 용량 반환 */\n    public capacity(): number {\n        return this._capacity;\n    }\n\n    /* 원소 접근 */\n    public get(index: number): number {\n        // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('인덱스가 범위를 벗어났다');\n        return this.arr[index];\n    }\n\n    /* 원소 갱신 */\n    public set(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('인덱스가 범위를 벗어났다');\n        this.arr[index] = num;\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    public add(num: number): void {\n        // 길이가 용량과 같으면 용량을 확장해야 한다\n        if (this._size === this._capacity) this.extendCapacity();\n        // 새 원소를 리스트 맨 뒤에 추가한다\n        this.arr[this._size] = num;\n        this._size++;\n    }\n\n    /* 중간에 원소 삽입 */\n    public insert(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('인덱스가 범위를 벗어났다');\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (this._size === this._capacity) {\n            this.extendCapacity();\n        }\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for (let j = this._size - 1; j &gt;= index; j--) {\n            this.arr[j + 1] = this.arr[j];\n        }\n        // 원소 개수를 갱신한다\n        this.arr[index] = num;\n        this._size++;\n    }\n\n    /* 원소 삭제 */\n    public remove(index: number): number {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('인덱스가 범위를 벗어났다');\n        let num = this.arr[index];\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for (let j = index; j &lt; this._size - 1; j++) {\n            this.arr[j] = this.arr[j + 1];\n        }\n        // 원소 개수를 갱신한다\n        this._size--;\n        // 삭제한 원소를 반환한다\n        return num;\n    }\n\n    /* 리스트 용량 확장 */\n    public extendCapacity(): void {\n        // 길이가 size인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n        this.arr = this.arr.concat(\n            new Array(this.capacity() * (this.extendRatio - 1))\n        );\n        // 리스트 용량을 갱신한다\n        this._capacity = this.arr.length;\n    }\n\n    /* 리스트를 배열로 변환 */\n    public toArray(): number[] {\n        let size = this.size();\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.dart<pre><code>/* 리스트 클래스 */\nclass MyList {\n  late List&lt;int&gt; _arr; // 배열(리스트 원소 저장)\n  int _capacity = 10; // 리스트 용량\n  int _size = 0; // 리스트 길이(현재 원소 개수)\n  int _extendRatio = 2; // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n  /* 생성자 */\n  MyList() {\n    _arr = List.filled(_capacity, 0);\n  }\n\n  /* 리스트 길이(현재 원소 개수) 반환*/\n  int size() =&gt; _size;\n\n  /* 리스트 용량 반환 */\n  int capacity() =&gt; _capacity;\n\n  /* 원소 접근 */\n  int get(int index) {\n    if (index &gt;= _size) throw RangeError('인덱스가 범위를 벗어났다');\n    return _arr[index];\n  }\n\n  /* 원소 갱신 */\n  void set(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('인덱스가 범위를 벗어났다');\n    _arr[index] = _num;\n  }\n\n  /* 맨 뒤에 원소 추가 */\n  void add(int _num) {\n    // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n    if (_size == _capacity) extendCapacity();\n    _arr[_size] = _num;\n    // 원소 개수를 갱신한다\n    _size++;\n  }\n\n  /* 중간에 원소 삽입 */\n  void insert(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('인덱스가 범위를 벗어났다');\n    // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n    if (_size == _capacity) extendCapacity();\n    // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n    for (var j = _size - 1; j &gt;= index; j--) {\n      _arr[j + 1] = _arr[j];\n    }\n    _arr[index] = _num;\n    // 원소 개수를 갱신한다\n    _size++;\n  }\n\n  /* 원소 삭제 */\n  int remove(int index) {\n    if (index &gt;= _size) throw RangeError('인덱스가 범위를 벗어났다');\n    int _num = _arr[index];\n    // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n    for (var j = index; j &lt; _size - 1; j++) {\n      _arr[j] = _arr[j + 1];\n    }\n    // 원소 개수를 갱신한다\n    _size--;\n    // 삭제한 원소를 반환한다\n    return _num;\n  }\n\n  /* 리스트 용량 확장 */\n  void extendCapacity() {\n    // 길이가 원래 배열의 _extendRatio배인 새 배열을 생성한다\n    final _newNums = List.filled(_capacity * _extendRatio, 0);\n    // 원래 배열을 새 배열에 복사한다\n    List.copyRange(_newNums, 0, _arr);\n    // _arr의 참조를 갱신한다\n    _arr = _newNums;\n    // 리스트 용량을 갱신한다\n    _capacity = _arr.length;\n  }\n\n  /* 리스트를 배열로 변환 */\n  List&lt;int&gt; toArray() {\n    List&lt;int&gt; arr = [];\n    for (var i = 0; i &lt; _size; i++) {\n      arr.add(get(i));\n    }\n    return arr;\n  }\n}\n</code></pre> my_list.rs<pre><code>/* 리스트 클래스 */\n#[allow(dead_code)]\nstruct MyList {\n    arr: Vec&lt;i32&gt;,       // 배열(리스트 원소 저장)\n    capacity: usize,     // 리스트 용량\n    size: usize,         // 리스트 길이(현재 원소 개수)\n    extend_ratio: usize, // 리스트의 용량을 확장할 때마다 적용하는 배수\n}\n\n#[allow(unused, unused_comparisons)]\nimpl MyList {\n    /* 생성자 */\n    pub fn new(capacity: usize) -&gt; Self {\n        let mut vec = vec![0; capacity];\n        Self {\n            arr: vec,\n            capacity,\n            size: 0,\n            extend_ratio: 2,\n        }\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환*/\n    pub fn size(&amp;self) -&gt; usize {\n        return self.size;\n    }\n\n    /* 리스트 용량 반환 */\n    pub fn capacity(&amp;self) -&gt; usize {\n        return self.capacity;\n    }\n\n    /* 원소 접근 */\n    pub fn get(&amp;self, index: usize) -&gt; i32 {\n        // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if index &gt;= self.size {\n            panic!(\"인덱스가 범위를 벗어났다\")\n        };\n        return self.arr[index];\n    }\n\n    /* 원소 갱신 */\n    pub fn set(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size {\n            panic!(\"인덱스가 범위를 벗어났다\")\n        };\n        self.arr[index] = num;\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    pub fn add(&amp;mut self, num: i32) {\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        self.arr[self.size] = num;\n        // 원소 개수를 갱신한다\n        self.size += 1;\n    }\n\n    /* 중간에 원소 삽입 */\n    pub fn insert(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size() {\n            panic!(\"인덱스가 범위를 벗어났다\")\n        };\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for j in (index..self.size).rev() {\n            self.arr[j + 1] = self.arr[j];\n        }\n        self.arr[index] = num;\n        // 원소 개수를 갱신한다\n        self.size += 1;\n    }\n\n    /* 원소 삭제 */\n    pub fn remove(&amp;mut self, index: usize) -&gt; i32 {\n        if index &gt;= self.size() {\n            panic!(\"인덱스가 범위를 벗어났다\")\n        };\n        let num = self.arr[index];\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for j in index..self.size - 1 {\n            self.arr[j] = self.arr[j + 1];\n        }\n        // 원소 개수를 갱신한다\n        self.size -= 1;\n        // 삭제한 원소를 반환한다\n        return num;\n    }\n\n    /* 리스트 용량 확장 */\n    pub fn extend_capacity(&amp;mut self) {\n        // 길이가 원래 배열의 extend_ratio배인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n        let new_capacity = self.capacity * self.extend_ratio;\n        self.arr.resize(new_capacity, 0);\n        // 리스트 용량을 갱신한다\n        self.capacity = new_capacity;\n    }\n\n    /* 리스트를 배열로 변환 */\n    pub fn to_array(&amp;self) -&gt; Vec&lt;i32&gt; {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        let mut arr = Vec::new();\n        for i in 0..self.size {\n            arr.push(self.get(i));\n        }\n        arr\n    }\n}\n</code></pre> my_list.c<pre><code>/* 리스트 클래스 */\ntypedef struct {\n    int *arr;        // 배열(리스트 원소 저장)\n    int capacity;    // 리스트 용량\n    int size;        // 리스트 크기\n    int extendRatio; // 리스트 용량을 확장할 때마다 적용하는 배수\n} MyList;\n\n/* 생성자 */\nMyList *newMyList() {\n    MyList *nums = malloc(sizeof(MyList));\n    nums-&gt;capacity = 10;\n    nums-&gt;arr = malloc(sizeof(int) * nums-&gt;capacity);\n    nums-&gt;size = 0;\n    nums-&gt;extendRatio = 2;\n    return nums;\n}\n\n/* 소멸자 */\nvoid delMyList(MyList *nums) {\n    free(nums-&gt;arr);\n    free(nums);\n}\n\n/* 리스트 길이 구하기 */\nint size(MyList *nums) {\n    return nums-&gt;size;\n}\n\n/* 리스트 용량 반환 */\nint capacity(MyList *nums) {\n    return nums-&gt;capacity;\n}\n\n/* 원소 접근 */\nint get(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    return nums-&gt;arr[index];\n}\n\n/* 원소 갱신 */\nvoid set(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    nums-&gt;arr[index] = num;\n}\n\n/* 맨 뒤에 원소 추가 */\nvoid add(MyList *nums, int num) {\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // 용량 확장\n    }\n    nums-&gt;arr[size(nums)] = num;\n    nums-&gt;size++;\n}\n\n/* 중간에 원소 삽입 */\nvoid insert(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // 용량 확장\n    }\n    for (int i = size(nums); i &gt; index; --i) {\n        nums-&gt;arr[i] = nums-&gt;arr[i - 1];\n    }\n    nums-&gt;arr[index] = num;\n    nums-&gt;size++;\n}\n\n/* 원소 삭제 */\n// 주의: stdio.h가 remove라는 이름을 사용한다\nint removeItem(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    int num = nums-&gt;arr[index];\n    for (int i = index; i &lt; size(nums) - 1; i++) {\n        nums-&gt;arr[i] = nums-&gt;arr[i + 1];\n    }\n    nums-&gt;size--;\n    return num;\n}\n\n/* 리스트 용량 확장 */\nvoid extendCapacity(MyList *nums) {\n    // 먼저 공간 할당\n    int newCapacity = capacity(nums) * nums-&gt;extendRatio;\n    int *extend = (int *)malloc(sizeof(int) * newCapacity);\n    int *temp = nums-&gt;arr;\n\n    // 기존 데이터를 새 공간에 복사\n    for (int i = 0; i &lt; size(nums); i++)\n        extend[i] = nums-&gt;arr[i];\n\n    // 기존 데이터 공간 해제\n    free(temp);\n\n    // 새 데이터로 갱신\n    nums-&gt;arr = extend;\n    nums-&gt;capacity = newCapacity;\n}\n\n/* 출력을 위해 리스트를 Array로 변환 */\nint *toArray(MyList *nums) {\n    return nums-&gt;arr;\n}\n</code></pre> my_list.kt<pre><code>/* 리스트 클래스 */\nclass MyList {\n    private var arr: IntArray = intArrayOf() // 배열(리스트 원소 저장)\n    private var capacity: Int = 10 // 리스트 용량\n    private var size: Int = 0 // 리스트 길이(현재 원소 개수)\n    private var extendRatio: Int = 2 // 리스트의 용량을 확장할 때마다 적용하는 배수\n\n    /* 생성자 */\n    init {\n        arr = IntArray(capacity)\n    }\n\n    /* 리스트 길이(현재 원소 개수) 반환 */\n    fun size(): Int {\n        return size\n    }\n\n    /* 리스트 용량 반환 */\n    fun capacity(): Int {\n        return capacity\n    }\n\n    /* 원소 접근 */\n    fun get(index: Int): Int {\n        // 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\")\n        return arr[index]\n    }\n\n    /* 원소 갱신 */\n    fun set(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\")\n        arr[index] = num\n    }\n\n    /* 맨 뒤에 원소 추가 */\n    fun add(num: Int) {\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (size == capacity())\n            extendCapacity()\n        arr[size] = num\n        // 원소 개수를 갱신한다\n        size++\n    }\n\n    /* 중간에 원소 삽입 */\n    fun insert(index: Int, num: Int) {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\")\n        // 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n        if (size == capacity())\n            extendCapacity()\n        // 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n        for (j in size - 1 downTo index)\n            arr[j + 1] = arr[j]\n        arr[index] = num\n        // 원소 개수를 갱신한다\n        size++\n    }\n\n    /* 원소 삭제 */\n    fun remove(index: Int): Int {\n        if (index &lt; 0 || index &gt;= size)\n            throw IndexOutOfBoundsException(\"인덱스가 범위를 벗어났다\")\n        val num = arr[index]\n        // 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n        for (j in index..&lt;size - 1)\n            arr[j] = arr[j + 1]\n        // 원소 개수를 갱신한다\n        size--\n        // 삭제한 원소를 반환한다\n        return num\n    }\n\n    /* 리스트 용량 확장 */\n    fun extendCapacity() {\n        // 길이가 원래 배열의 extendRatio배인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n        arr = arr.copyOf(capacity() * extendRatio)\n        // 리스트 용량을 갱신한다\n        capacity = arr.size\n    }\n\n    /* 리스트를 배열로 변환 */\n    fun toArray(): IntArray {\n        val size = size()\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        val arr = IntArray(size)\n        for (i in 0..&lt;size) {\n            arr[i] = get(i)\n        }\n        return arr\n    }\n}\n</code></pre> my_list.rb<pre><code>### 리스트 클래스 ###\nclass MyList\n  attr_reader :size       # 리스트 길이(현재 원소 개수) 반환\n  attr_reader :capacity   # 리스트 용량 반환\n\n  ### 생성자 ###\n  def initialize\n    @capacity = 10\n    @size = 0\n    @extend_ratio = 2\n    @arr = Array.new(capacity)\n  end\n\n  ### 원소 접근 ###\n  def get(index)\n    # 인덱스가 범위를 벗어나면 예외를 발생시킨다. 아래에서도 동일하다\n    raise IndexError, \"인덱스가 범위를 벗어났다\" if index &lt; 0 || index &gt;= size\n    @arr[index]\n  end\n\n  ### 원소 접근 ###\n  def set(index, num)\n    raise IndexError, \"인덱스가 범위를 벗어났다\" if index &lt; 0 || index &gt;= size\n    @arr[index] = num\n  end\n\n  ### 맨 뒤에 원소 추가 ###\n  def add(num)\n    # 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n    extend_capacity if size == capacity\n    @arr[size] = num\n\n    # 원소 개수를 갱신한다\n    @size += 1\n  end\n\n  ### 중간에 원소 삽입 ###\n  def insert(index, num)\n    raise IndexError, \"인덱스가 범위를 벗어났다\" if index &lt; 0 || index &gt;= size\n\n    # 원소 개수가 용량을 초과하면 용량 확장을 수행한다\n    extend_capacity if size == capacity\n\n    # 인덱스 index와 그 뒤의 원소를 모두 한 칸씩 뒤로 이동한다\n    for j in (size - 1).downto(index)\n      @arr[j + 1] = @arr[j]\n    end\n    @arr[index] = num\n\n    # 원소 개수를 갱신한다\n    @size += 1\n  end\n\n  ### 원소 삭제 ###\n  def remove(index)\n    raise IndexError, \"인덱스가 범위를 벗어났다\" if index &lt; 0 || index &gt;= size\n    num = @arr[index]\n\n    # 인덱스 index 뒤의 원소를 모두 한 칸씩 앞으로 이동한다\n    for j in index...size\n      @arr[j] = @arr[j + 1]\n    end\n\n    # 원소 개수를 갱신한다\n    @size -= 1\n\n    # 삭제한 원소를 반환한다\n    num\n  end\n\n  ### 리스트 용량 확장 ###\n  def extend_capacity\n    # 길이가 원래 배열의 extend_ratio배인 새 배열을 생성하고, 원래 배열을 새 배열에 복사한다\n    arr = @arr.dup + Array.new(capacity * (@extend_ratio - 1))\n    # 리스트 용량을 갱신한다\n    @capacity = arr.length\n  end\n\n  ### 리스트를 배열로 변환 ###\n  def to_array\n    sz = size\n    # 유효한 길이 범위 안의 리스트 원소만 변환한다\n    arr = Array.new(sz)\n    for i in 0...sz\n      arr[i] = get(i)\n    end\n    arr\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 4 장   배열과 연결 리스트","4.3   리스트"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/","level":1,"title":"4.4   메모리와 캐시 *","text":"<p>이 장의 앞선 두 절에서는 기본적이고 중요한 두 자료 구조인 배열과 연결 리스트를 살펴보았다. 이들은 각각 ‘연속 저장’과 ‘분산 저장’이라는 물리적 구조를 대표한다.</p> <p>실제로 물리적 구조는 프로그램의 메모리 및 캐시 사용 효율을 크게 좌우하며, 나아가 알고리즘 프로그램의 전체 성능에 영향을 준다.</p>","path":["제 4 장   배열과 연결 리스트","4.4   메모리와 캐시 *"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/#441","level":2,"title":"4.4.1   컴퓨터의 저장 장치","text":"<p>컴퓨터에는 하드 디스크(hard disk), 메모리(random-access memory, RAM), 캐시(cache memory)라는 세 종류의 저장 장치가 있다. 아래 표(4-2)는 컴퓨터 시스템에서 각 장치가 맡는 역할과 성능 특징을 보여 준다.</p> <p> 표 4-2   컴퓨터의 저장 장치 </p> 하드 디스크 메모리 캐시 용도 운영체제, 프로그램, 파일 등의 데이터를 장기간 저장 실행 중인 프로그램과 처리 중인 데이터를 임시 저장 자주 접근하는 데이터와 명령어를 저장하여 CPU의 메모리 접근 횟수를 줄임 휘발성 전원이 꺼져도 데이터가 사라지지 않음 전원이 꺼지면 데이터가 사라짐 전원이 꺼지면 데이터가 사라짐 용량 큼, TB 단위 작음, GB 단위 매우 작음, MB 단위 속도 느림, 수백~수천 MB/s 빠름, 수십 GB/s 매우 빠름, 수십~수백 GB/s 가격(중국 위안) 저렴함, GB당 수십 펀~수 위안 비쌈, GB당 수십~수백 위안 매우 비쌈, CPU 가격에 포함 <p>컴퓨터의 저장 시스템은 아래 그림(4-9)과 같은 피라미드 구조로 생각할 수 있다. 꼭대기에 가까울수록 속도는 빠르고, 용량은 작으며, 비용은 높다. 이러한 계층적 설계는 우연이 아니라 컴퓨터 과학자와 엔지니어들이 깊이 고민한 결과다.</p> <ul> <li>메모리로 하드 디스크를 대체하기는 어렵다. 우선 메모리의 데이터는 전원이 꺼지면 사라지므로 장기 저장에 적합하지 않다. 또한 메모리 가격은 하드 디스크의 수십 배여서 소비자 시장에 보편적으로 적용하기 어렵다.</li> <li>캐시에서 큰 용량과 빠른 속도를 동시에 얻기는 어렵다. L1, L2, L3 캐시의 용량이 커질수록 물리적 크기도 커지고 CPU 코어와의 거리도 멀어진다. 그 결과 데이터 전송 시간이 늘고 원소 접근 지연도 커진다. 현재 기술에서 다단계 캐시 구조는 용량, 속도, 비용 사이의 최적 절충점이다.</li> </ul> <p></p> <p> 그림 4-9   컴퓨터 저장 시스템 </p> <p>참고</p> <p>컴퓨터의 저장 계층 구조는 속도, 용량, 비용 사이의 정교한 균형을 보여 준다. 이러한 절충은 모든 산업 분야에 널리 존재하며, 서로 다른 장점과 제약 사이에서 최적의 균형을 찾아야 한다.</p> <p>전체적으로 하드 디스크는 많은 데이터를 장기간 저장하고, 메모리는 실행 중인 프로그램이 처리하는 데이터를 임시 저장하며, 캐시는 자주 접근하는 데이터와 명령어를 저장하여 실행 효율을 높인다. 세 장치는 서로 협력하여 컴퓨터 시스템이 효율적으로 작동하게 한다.</p> <p>아래 그림(4-10)과 같이 프로그램이 실행되면 하드 디스크의 데이터를 메모리로 읽어 와 CPU가 계산에 사용한다. 캐시는 CPU의 일부로 볼 수 있으며, 메모리에서 데이터를 지능적으로 불러와 CPU에 빠른 데이터 읽기를 제공한다. 이를 통해 실행 효율을 크게 높이고 더 느린 메모리에 대한 의존을 줄인다.</p> <p></p> <p> 그림 4-10   하드 디스크, 메모리, 캐시 사이의 데이터 흐름 </p>","path":["제 4 장   배열과 연결 리스트","4.4   메모리와 캐시 *"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/#442","level":2,"title":"4.4.2   자료 구조의 메모리 효율","text":"<p>메모리 공간 활용 측면에서 배열과 연결 리스트에는 각각 장점과 한계가 있다.</p> <p>한편으로 메모리는 한정되어 있으며 같은 메모리 영역을 여러 프로그램이 공유할 수 없으므로, 자료 구조가 공간을 최대한 효율적으로 사용하기를 바란다. 배열 원소는 빈틈없이 붙어 있어 연결 리스트 노드 사이의 참조(포인터)를 저장할 추가 공간이 필요 없으므로 공간 효율이 더 높다. 하지만 배열은 충분한 연속 메모리를 한꺼번에 할당해야 하므로 공간 낭비가 발생할 수 있으며, 용량 확장에도 시간과 공간이 추가로 든다. 반면 연결 리스트는 ‘노드’ 단위로 메모리를 동적으로 할당하고 회수하므로 더 유연하다.</p> <p>다른 한편으로 프로그램이 실행되면서 메모리 할당과 해제가 반복되면 빈 메모리의 단편화가 점점 심해져 메모리 활용 효율이 낮아진다. 배열은 연속 저장 방식을 사용하므로 상대적으로 단편화를 덜 일으킨다. 반면 연결 리스트의 원소는 분산되어 있으므로 삽입과 삭제가 잦으면 단편화가 더 쉽게 발생한다.</p>","path":["제 4 장   배열과 연결 리스트","4.4   메모리와 캐시 *"],"tags":[]},{"location":"chapter_array_and_linkedlist/ram_and_cache/#443","level":2,"title":"4.4.3   자료 구조의 캐시 효율","text":"<p>캐시는 메모리보다 용량은 훨씬 작지만 속도는 훨씬 빨라 프로그램 실행 속도에 매우 중요한 역할을 한다. 용량이 제한되어 있어 자주 사용하는 데이터 일부만 저장할 수 있으므로, CPU가 접근하려는 데이터가 캐시에 없으면 캐시 미스(cache miss)가 발생한다. 이때 CPU는 더 느린 메모리에서 필요한 데이터를 불러와야 한다.</p> <p>따라서 ‘캐시 미스’가 적을수록 CPU의 데이터 읽기 및 쓰기 효율이 높아지고, 프로그램 성능도 좋아진다. CPU가 캐시에서 데이터를 성공적으로 가져오는 비율을 캐시 적중률(cache hit rate)이라고 하며, 캐시 효율을 측정하는 지표로 자주 사용한다.</p> <p>캐시는 효율을 최대한 높이기 위해 다음과 같은 데이터 로딩 방식을 사용한다.</p> <ul> <li>캐시 라인: 캐시는 데이터를 한 바이트씩 저장하고 불러오는 대신 캐시 라인 단위로 처리한다. 캐시 라인 단위 전송은 바이트 단위 전송보다 효율적이다.</li> <li>프리페치: 프로세서는 순차 접근이나 고정 간격 접근 같은 데이터 접근 패턴을 예측하고, 그 패턴에 따라 데이터를 미리 캐시에 불러와 적중률을 높인다.</li> <li>공간 지역성: 어떤 데이터에 접근하면 가까운 시일 안에 주변 데이터에도 접근할 가능성이 있다. 따라서 캐시는 데이터를 불러올 때 주변 데이터도 함께 불러와 적중률을 높인다.</li> <li>시간 지역성: 어떤 데이터에 접근하면 머지않아 다시 접근할 가능성이 높다. 캐시는 이 원리를 이용해 최근 접근한 데이터를 유지함으로써 적중률을 높인다.</li> </ul> <p>실제로 배열과 연결 리스트의 캐시 활용 효율은 서로 다르며, 주로 다음과 같은 차이가 있다.</p> <ul> <li>공간 사용량: 연결 리스트 원소는 배열 원소보다 더 많은 공간을 차지하므로 캐시에 담을 수 있는 유효 데이터의 양이 적다.</li> <li>캐시 라인: 연결 리스트의 데이터는 메모리 곳곳에 흩어져 있지만 캐시는 ‘라인 단위’로 불러오므로, 불필요한 데이터를 가져오는 비율이 높다.</li> <li>프리페치: 배열의 데이터 접근 패턴은 연결 리스트보다 ‘예측 가능성’이 높다. 즉, 시스템이 다음에 불러올 데이터를 더 쉽게 예상할 수 있다.</li> <li>공간 지역성: 배열은 한곳에 모인 메모리 공간에 저장되므로, 불러온 데이터 주변의 데이터에도 곧 접근할 가능성이 더 높다.</li> </ul> <p>대체로 배열은 캐시 적중률이 더 높아 연산 효율에서도 연결 리스트보다 유리한 경우가 많다. 이 때문에 알고리즘 문제를 풀 때는 배열 기반 자료 구조를 더 선호하는 경우가 많다.</p> <p>다만 캐시 효율이 높다고 해서 모든 상황에서 배열이 연결 리스트보다 좋은 것은 아니다. 실제로 어떤 자료 구조를 쓸지는 구체적인 요구에 따라 결정해야 한다. 예를 들어 배열과 연결 리스트 모두 ‘스택’을 구현할 수 있지만(다음 장에서 자세히 설명한다), 적합한 상황은 다르다.</p> <ul> <li>알고리즘 문제를 풀 때는 배열 기반 스택을 선호한다. 배열에 일정한 메모리 공간을 미리 할당하기만 하면, 높은 연산 효율과 임의 접근 기능을 얻을 수 있기 때문이다.</li> <li>데이터가 매우 많고 변화가 잦으며 스택의 예상 크기를 가늠하기 어렵다면 연결 리스트 기반 스택이 더 적합하다. 연결 리스트는 많은 데이터를 메모리 여러 곳에 분산 저장할 수 있으며, 배열 용량 확장에 따른 추가 비용도 피할 수 있다.</li> </ul>","path":["제 4 장   배열과 연결 리스트","4.4   메모리와 캐시 *"],"tags":[]},{"location":"chapter_array_and_linkedlist/summary/","level":1,"title":"4.5   요약","text":"","path":["제 4 장   배열과 연결 리스트","4.5   요약"],"tags":[]},{"location":"chapter_array_and_linkedlist/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>배열과 연결 리스트는 두 가지 기본 자료 구조로, 컴퓨터 메모리에 데이터를 저장하는 연속 공간 저장과 분산 공간 저장을 각각 대표한다. 둘의 특징은 서로 보완적이다.</li> <li>배열은 임의 접근을 지원하고 메모리 사용량이 적다. 그러나 원소 삽입과 삭제의 효율이 낮으며 초기화한 뒤 길이를 바꿀 수 없다.</li> <li>연결 리스트는 참조(포인터)를 바꾸어 노드를 효율적으로 삽입하고 삭제하며, 길이도 유연하게 조절할 수 있다. 하지만 노드 접근 효율이 낮고 메모리를 더 많이 사용한다. 대표적인 종류에는 단일 연결 리스트, 원형 연결 리스트, 이중 연결 리스트가 있다.</li> <li>리스트는 삽입, 삭제, 탐색, 갱신을 지원하는 순서 있는 원소의 모음으로, 보통 동적 배열로 구현한다. 배열의 장점을 유지하면서 길이를 유연하게 조절할 수 있다.</li> <li>리스트의 등장으로 배열의 실용성이 크게 높아졌지만, 메모리 공간 일부가 낭비될 수 있다.</li> <li>프로그램이 실행될 때 데이터는 주로 메모리에 저장된다. 배열은 메모리 공간 효율이 더 높고, 연결 리스트는 메모리를 더 유연하게 사용한다.</li> <li>캐시는 캐시 라인, 프리페치, 공간 지역성, 시간 지역성 등의 로딩 방식을 통해 CPU에 빠른 데이터 접근을 제공하여 프로그램 실행 효율을 크게 높인다.</li> <li>배열은 캐시 적중률이 더 높으므로 보통 연결 리스트보다 효율적이다. 자료 구조를 선택할 때는 구체적인 요구와 상황에 맞춰야 한다.</li> </ul>","path":["제 4 장   배열과 연결 리스트","4.5   요약"],"tags":[]},{"location":"chapter_array_and_linkedlist/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 배열을 스택에 저장하는지 힙에 저장하는지에 따라 시간 효율과 공간 효율이 달라지는가?</p> <p>스택에 저장한 배열도 힙에 저장한 배열도 연속된 메모리 공간에 있으므로, 데이터 연산 효율은 대체로 같다. 다만 스택과 힙의 특성 때문에 다음과 같은 차이가 생긴다.</p> <ol> <li>할당 및 해제 효율: 스택은 비교적 작은 메모리 영역으로 컴파일러가 자동으로 할당한다. 힙은 더 크며 코드에서 동적으로 할당할 수 있고 단편화도 더 쉽게 발생한다. 따라서 힙의 할당 및 해제 연산은 보통 스택보다 느리다.</li> <li>크기 제한: 스택 메모리는 비교적 작고 힙의 크기는 대개 사용 가능한 메모리에 의해 제한된다. 따라서 큰 배열을 저장하기에는 힙이 더 적합하다.</li> <li>유연성: 스택에 저장하는 배열의 크기는 컴파일 시 정해야 하지만, 힙에 저장하는 배열의 크기는 실행 중에 동적으로 정할 수 있다.</li> </ol> <p>Q: 배열은 원소의 자료형이 같아야 한다고 하는데, 연결 리스트에서는 왜 이를 강조하지 않는가?</p> <p>연결 리스트는 참조(포인터)로 연결된 노드들로 이루어진다. 각 노드에는 <code>int</code>, <code>double</code>, <code>string</code>, <code>object</code> 등 서로 다른 자료형의 데이터를 저장할 수 있다.</p> <p>반면 배열 원소는 자료형이 같아야 오프셋을 계산하여 해당 원소의 위치를 구할 수 있다. 예를 들어 배열에 <code>int</code>와 <code>long</code>이 함께 있으면 원소 하나가 각각 4바이트와 8바이트를 차지한다. ‘원소 길이’가 두 가지이므로 다음 식으로 오프셋을 계산할 수 없다.</p> <pre><code># 원소 메모리 주소 = 배열 메모리 주소(첫 원소의 메모리 주소) + 원소 길이 * 원소 인덱스\n</code></pre> <p>Q: 노드 <code>P</code>를 삭제한 뒤 <code>P.next</code>를 <code>None</code>으로 설정해야 하는가?</p> <p><code>P.next</code>를 바꾸지 않아도 된다. 해당 연결 리스트에서는 머리 노드부터 꼬리 노드까지 순회해도 더 이상 <code>P</code>를 만나지 않는다. 즉, <code>P</code>는 이미 연결 리스트에서 삭제되었으므로 어디를 가리키든 이 연결 리스트에 영향을 주지 않는다.</p> <p>자료 구조와 알고리즘 문제를 푸는 관점에서는 프로그램 논리가 올바르기만 하면 연결을 끊지 않아도 된다. 표준 라이브러리 관점에서는 연결을 끊는 편이 더 안전하고 논리도 명확하다. 연결을 끊지 않은 상태에서 삭제한 노드가 정상적으로 회수되지 않으면, 뒤따르는 노드의 메모리 회수에도 영향을 줄 수 있다.</p> <p>Q: 연결 리스트의 삽입과 삭제는 시간 복잡도가 \\(O(1)\\)이라고 한다. 하지만 먼저 원소를 찾는 데 \\(O(n)\\)이 필요한데, 전체 시간 복잡도는 왜 \\(O(n)\\)이 아닌가?</p> <p>원소를 먼저 찾고 삭제한다면 시간 복잡도는 실제로 \\(O(n)\\)이다. 다만 연결 리스트의 \\(O(1)\\) 삽입 및 삭제는 다른 응용에서 장점을 발휘한다. 예를 들어 덱은 연결 리스트로 구현하기에 적합하다. 머리 노드와 꼬리 노드를 항상 가리키는 포인터 변수를 유지하면 매번 삽입과 삭제를 \\(O(1)\\)에 수행할 수 있다.</p> <p>Q: ‘연결 리스트의 정의와 저장 방식’ 그림에서 노드 포인터를 저장하는 연한 파란색 부분은 하나의 메모리 주소를 차지하는가? 아니면 노드 값과 반씩 차지하는가?</p> <p>그 그림은 정성적인 설명이다. 정확한 크기는 구체적인 상황에 따라 분석해야 한다.</p> <ul> <li>노드 값이 차지하는 공간은 자료형에 따라 다르다. 예를 들어 <code>int</code>, <code>long</code>, <code>double</code>, 인스턴스 객체 등이 있다.</li> <li>포인터 변수가 차지하는 메모리 크기는 운영체제와 컴파일 환경에 따라 정해지며, 대개 8바이트 또는 4바이트다.</li> </ul> <p>Q: 리스트 끝에 원소를 추가하는 연산은 항상 \\(O(1)\\)인가?</p> <p>원소를 추가할 때 리스트의 길이를 넘으면 먼저 용량을 늘려야 한다. 시스템이 새 메모리 영역을 요청하고 기존 원소 전체를 옮기므로, 이때의 시간 복잡도는 \\(O(n)\\)이다.</p> <p>Q: ‘리스트는 배열의 실용성을 크게 높였지만 메모리 공간 일부를 낭비할 수 있다’는 설명에서, 공간 낭비는 용량, 길이, 확장 배수 같은 추가 변수의 메모리를 뜻하는가?</p> <p>여기서 공간 낭비는 주로 두 가지를 뜻한다. 하나는 리스트에 초기 길이를 설정하지만 그만큼 필요하지 않을 수 있다는 것이다. 다른 하나는 잦은 확장을 피하려고 보통 \\(\\times 1.5\\)처럼 일정한 계수를 곱해 용량을 늘린다는 것이다. 이때도 빈자리가 많이 생기며, 보통 이를 전부 채우지는 못한다.</p> <p>Q: Python에서 <code>n = [1, 2, 3]</code>을 초기화하면 세 원소의 주소가 이어져 있다. 하지만 <code>m = [2, 1, 3]</code>을 초기화하면 원소들의 id가 연속적이지 않고 <code>n</code>의 같은 값과 각각 일치한다. 주소가 연속적이지 않아도 <code>m</code>은 배열인가?</p> <p>리스트 원소를 연결 리스트 노드로 바꾸어 <code>n = [n1, n2, n3, n4, n5]</code>라고 하면, 이 다섯 노드 객체도 보통 메모리 곳곳에 흩어져 있다. 그래도 리스트 인덱스가 주어지면 \\(O(1)\\) 시간에 노드의 메모리 주소를 얻어 해당 노드에 접근할 수 있다. 배열에 저장된 것은 노드 자체가 아니라 노드의 참조이기 때문이다.</p> <p>많은 언어와 달리 Python에서는 숫자도 객체로 감싸져 있으며, 리스트에는 숫자 자체가 아닌 숫자의 참조가 저장된다. 따라서 두 배열의 같은 숫자가 같은 id를 가지며, 숫자들의 메모리 주소는 연속적일 필요가 없다.</p> <p>Q: C++ STL의 <code>std::list</code>는 이중 연결 리스트를 구현하는데, 일부 알고리즘 책에서는 거의 직접 쓰지 않는 것 같다. 어떤 한계가 있기 때문인가?</p> <p>한편으로는 보통 배열로 알고리즘을 구현하는 것을 선호하며, 꼭 필요할 때만 연결 리스트를 사용한다. 주된 이유는 두 가지다.</p> <ul> <li>공간 비용: 원소마다 이전 원소와 다음 원소를 위한 포인터가 두 개 더 필요하므로 <code>std::list</code>는 보통 <code>std::vector</code>보다 공간을 많이 차지한다.</li> <li>캐시에 불리함: 데이터가 연속적으로 저장되지 않아 <code>std::list</code>의 캐시 활용률이 낮다. 일반적으로는 <code>std::vector</code>의 성능이 더 좋다.</li> </ul> <p>다른 한편으로 연결 리스트가 필요한 경우는 주로 이진 트리와 그래프다. 스택과 큐는 연결 리스트 대신 프로그래밍 언어가 제공하는 <code>stack</code>과 <code>queue</code>를 쓰는 경우가 많다.</p> <p>Q: <code>res = [[0]] * n</code>으로 만든 이차원 리스트에서 각 <code>[0]</code>은 서로 독립적인가?</p> <p>독립적이지 않다. 이 이차원 리스트의 모든 <code>[0]</code>은 실제로 같은 객체를 참조한다. 한 원소를 수정하면 대응하는 원소가 모두 함께 바뀐다.</p> <p>각 <code>[0]</code>을 독립적으로 만들려면 <code>res = [[0] for _ in range(n)]</code>을 사용하면 된다. 이 방식은 서로 독립적인 <code>[0]</code> 리스트 객체 \\(n\\)개를 초기화한다.</p> <p>Q: <code>res = [0] * n</code>으로 만든 리스트에서 각 정수 0은 서로 독립적인가?</p> <p>이 리스트의 모든 정수 0은 같은 객체를 참조한다. Python은 객체 재사용을 극대화하고 성능을 높이기 위해 작은 정수(보통 -5부터 256까지)를 캐시 풀로 관리하기 때문이다.</p> <p>모두 같은 객체를 가리켜도 리스트의 각 원소는 독립적으로 바꿀 수 있다. Python의 정수는 ‘불변 객체’이기 때문이다. 원소를 바꾸면 기존 객체 자체가 바뀌는 것이 아니라 다른 객체를 참조하게 된다.</p> <p>그러나 리스트 원소가 리스트, 딕셔너리, 클래스 인스턴스 같은 ‘가변 객체’라면, 원소의 내용을 수정할 때 객체 자체가 바뀌므로 이를 참조하는 모든 원소에서 같은 변화가 나타난다.</p>","path":["제 4 장   배열과 연결 리스트","4.5   요약"],"tags":[]},{"location":"chapter_backtracking/","level":1,"title":"제 13 장   백트래킹","text":"<p>이 장을 시작하며</p> <p>우리는 미로 속 탐험가처럼 앞으로 나아가는 길에서 어려움을 만날 수 있다.</p> <p>백트래킹의 힘은 다시 출발하고 끊임없이 시도하여, 마침내 빛으로 향하는 출구를 찾게 해 준다.</p>","path":["제 13 장   백트래킹"],"tags":[]},{"location":"chapter_backtracking/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>13.1   백트래킹 알고리즘</li> <li>13.2   모든 순열 문제</li> <li>13.3   부분 집합 합 문제</li> <li>13.4   n-퀸 문제</li> <li>13.5   요약</li> <li>13.6   연습 문제</li> </ul>","path":["제 13 장   백트래킹"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/","level":1,"title":"13.1   백트래킹 알고리즘","text":"<p>백트래킹 알고리즘(backtracking algorithm)은 모든 경우를 열거하여 문제를 푸는 방법이다. 초기 상태에서 출발하여 가능한 모든 해결 방법을 완전 탐색하고, 올바른 해를 만나면 기록한다. 해를 찾거나 가능한 선택을 모두 시도해도 찾을 수 없을 때까지 진행하는 것이 핵심이다.</p> <p>백트래킹은 보통 '깊이 우선 탐색'으로 해 공간을 순회한다. '이진 트리' 장에서 전위, 중위, 후위 순회가 모두 깊이 우선 탐색이라고 설명했다. 이제 전위 순회를 이용한 백트래킹 문제를 만들어 작동 원리를 차근차근 알아보자.</p> <p>예제 1</p> <p>이진 트리가 주어졌을 때 값이 \\(7\\)인 모든 노드를 찾아 기록하고, 노드 리스트를 반환하라.</p> <p>이 문제에서는 트리를 전위 순회하며 현재 노드의 값이 \\(7\\)인지 확인하고, 맞으면 해당 노드의 값을 결과 리스트 <code>res</code>에 추가한다. 과정은 아래 그림(13-1)과 코드에 나타나 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_i_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"전위 순회: 예제 1\"\"\"\n    if root is None:\n        return\n    if root.val == 7:\n        # 해를 기록한다\n        res.append(root)\n    pre_order(root.left)\n    pre_order(root.right)\n</code></pre> preorder_traversal_i_compact.cpp<pre><code>/* 전위 순회: 예제 1 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // 해를 기록한다\n        res.push_back(root);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.java<pre><code>/* 전위 순회: 예제 1 */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // 해를 기록한다\n        res.add(root);\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.cs<pre><code>/* 전위 순회: 예제 1 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // 해를 기록한다\n        res.Add(root);\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.go<pre><code>/* 전위 순회: 예제 1 */\nfunc preOrderI(root *TreeNode, res *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    if (root.Val).(int) == 7 {\n        // 해를 기록한다\n        *res = append(*res, root)\n    }\n    preOrderI(root.Left, res)\n    preOrderI(root.Right, res)\n}\n</code></pre> preorder_traversal_i_compact.swift<pre><code>/* 전위 순회: 예제 1 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    if root.val == 7 {\n        // 해를 기록한다\n        res.append(root)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n</code></pre> preorder_traversal_i_compact.js<pre><code>/* 전위 순회: 예제 1 */\nfunction preOrder(root, res) {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // 해를 기록한다\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.ts<pre><code>/* 전위 순회: 예제 1 */\nfunction preOrder(root: TreeNode | null, res: TreeNode[]): void {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // 해를 기록한다\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.dart<pre><code>/* 전위 순회: 예제 1 */\nvoid preOrder(TreeNode? root, List&lt;TreeNode&gt; res) {\n  if (root == null) {\n    return;\n  }\n  if (root.val == 7) {\n    // 해를 기록한다\n    res.add(root);\n  }\n  preOrder(root.left, res);\n  preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.rs<pre><code>/* 전위 순회: 예제 1 */\nfn pre_order(res: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        if node.borrow().val == 7 {\n            // 해를 기록한다\n            res.push(node.clone());\n        }\n        pre_order(res, node.borrow().left.as_ref());\n        pre_order(res, node.borrow().right.as_ref());\n    }\n}\n</code></pre> preorder_traversal_i_compact.c<pre><code>/* 전위 순회: 예제 1 */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // 해를 기록한다\n        res[resSize++] = root;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.kt<pre><code>/* 전위 순회: 예제 1 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    if (root._val == 7) {\n        // 해를 기록한다\n        res!!.add(root)\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n}\n</code></pre> preorder_traversal_i_compact.rb<pre><code>### 전위 순회: 예제 1 ###\ndef pre_order(root)\n  return unless root\n\n  # 해를 기록한다\n  $res &lt;&lt; root if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 13-1   전위 순회로 노드 탐색 </p>","path":["제 13 장   백트래킹","13.1   백트래킹 알고리즘"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1311","level":2,"title":"13.1.1   시도와 되돌리기","text":"<p>백트래킹이라고 부르는 이유는 해 공간을 탐색할 때 '시도'와 '되돌리기' 전략을 사용하기 때문이다. 탐색 중 더 진행할 수 없거나 조건을 만족하는 해를 얻을 수 없는 상태를 만나면, 직전 선택을 취소하고 이전 상태로 돌아가 다른 선택을 시도한다.</p> <p>예제 1에서는 각 노드의 방문이 한 번의 '시도'이며, 리프 노드를 지나거나 부모 노드로 돌아가는 <code>return</code>이 '되돌리기'에 해당한다.</p> <p>되돌리기에는 함수 반환만 포함되는 것이 아니다. 이를 설명하기 위해 예제 1을 조금 확장해 보자.</p> <p>예제 2</p> <p>이진 트리에서 값이 \\(7\\)인 모든 노드를 찾아 루트 노드에서 각 노드까지의 경로를 반환하라.</p> <p>예제 1의 코드에 리스트 <code>path</code>를 추가하여 방문한 노드의 경로를 기록해야 한다. 값이 \\(7\\)인 노드를 방문하면 <code>path</code>를 복사하여 결과 리스트 <code>res</code>에 추가한다. 순회를 마치면 <code>res</code>에 모든 해가 저장된다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_ii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"전위 순회: 예제 2\"\"\"\n    if root is None:\n        return\n    # 시도\n    path.append(root)\n    if root.val == 7:\n        # 해를 기록한다\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # 되돌리기\n    path.pop()\n</code></pre> preorder_traversal_ii_compact.cpp<pre><code>/* 전위 순회: 예제 2 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    // 시도\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // 해를 기록한다\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // 되돌리기\n    path.pop_back();\n}\n</code></pre> preorder_traversal_ii_compact.java<pre><code>/* 전위 순회: 예제 2 */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    // 시도\n    path.add(root);\n    if (root.val == 7) {\n        // 해를 기록한다\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // 되돌리기\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_ii_compact.cs<pre><code>/* 전위 순회: 예제 2 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    // 시도\n    path.Add(root);\n    if (root.val == 7) {\n        // 해를 기록한다\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // 되돌리기\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_ii_compact.go<pre><code>/* 전위 순회: 예제 2 */\nfunc preOrderII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    // 시도\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // 해를 기록한다\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderII(root.Left, res, path)\n    preOrderII(root.Right, res, path)\n    // 되돌리기\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_ii_compact.swift<pre><code>/* 전위 순회: 예제 2 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // 시도\n    path.append(root)\n    if root.val == 7 {\n        // 해를 기록한다\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // 되돌리기\n    path.removeLast()\n}\n</code></pre> preorder_traversal_ii_compact.js<pre><code>/* 전위 순회: 예제 2 */\nfunction preOrder(root, path, res) {\n    if (root === null) {\n        return;\n    }\n    // 시도\n    path.push(root);\n    if (root.val === 7) {\n        // 해를 기록한다\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // 되돌리기\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.ts<pre><code>/* 전위 순회: 예제 2 */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    if (root === null) {\n        return;\n    }\n    // 시도\n    path.push(root);\n    if (root.val === 7) {\n        // 해를 기록한다\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // 되돌리기\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.dart<pre><code>/* 전위 순회: 예제 2 */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null) {\n    return;\n  }\n\n  // 시도\n  path.add(root);\n  if (root.val == 7) {\n    // 해를 기록한다\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // 되돌리기\n  path.removeLast();\n}\n</code></pre> preorder_traversal_ii_compact.rs<pre><code>/* 전위 순회: 예제 2 */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        // 시도\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // 해를 기록한다\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.as_ref());\n        pre_order(res, path, node.borrow().right.as_ref());\n        // 되돌리기\n        path.pop();\n    }\n}\n</code></pre> preorder_traversal_ii_compact.c<pre><code>/* 전위 순회: 예제 2 */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    // 시도\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // 해를 기록한다\n        for (int i = 0; i &lt; pathSize; ++i) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // 되돌리기\n    pathSize--;\n}\n</code></pre> preorder_traversal_ii_compact.kt<pre><code>/* 전위 순회: 예제 2 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) {\n        return\n    }\n    // 시도\n    path!!.add(root)\n    if (root._val == 7) {\n        // 해를 기록한다\n        res!!.add(path!!.toMutableList())\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // 되돌리기\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_ii_compact.rb<pre><code>### 전위 순회: 예제 2 ###\ndef pre_order(root)\n  return unless root\n\n  # 시도\n  $path &lt;&lt; root\n\n  # 해를 기록한다\n  $res &lt;&lt; $path.dup if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\n\n  # 되돌리기\n  $path.pop\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>매번 '시도'할 때는 현재 노드를 <code>path</code>에 추가하여 경로를 기록한다. '되돌리기' 전에는 이 노드를 <code>path</code>에서 꺼내어 이번 시도 이전의 상태를 복원해야 한다.</p> <p>아래 그림(13-2)의 과정을 보면 시도와 되돌리기를 각각 '전진'과 '취소'로 이해할 수 있으며, 두 연산은 서로 반대 방향이다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 13-2   시도와 되돌리기 </p>","path":["제 13 장   백트래킹","13.1   백트래킹 알고리즘"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1312","level":2,"title":"13.1.2   가지치기","text":"<p>복잡한 백트래킹 문제에는 보통 하나 이상의 제약 조건이 있으며, 이 제약 조건을 '가지치기'에 활용할 수 있다.</p> <p>예제 3</p> <p>이진 트리에서 값이 \\(7\\)인 모든 노드를 찾아 루트 노드에서 각 노드까지의 경로를 반환하라. 단, 경로에 값이 \\(3\\)인 노드가 포함되면 안 된다.</p> <p>이 제약을 만족하려면 가지치기 연산을 추가해야 한다. 탐색 중 값이 \\(3\\)인 노드를 만나면 더 탐색하지 않고 조기에 반환한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_iii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"전위 순회: 예제 3\"\"\"\n    # 가지치기\n    if root is None or root.val == 3:\n        return\n    # 시도\n    path.append(root)\n    if root.val == 7:\n        # 해를 기록한다\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # 되돌리기\n    path.pop()\n</code></pre> preorder_traversal_iii_compact.cpp<pre><code>/* 전위 순회: 예제 3 */\nvoid preOrder(TreeNode *root) {\n    // 가지치기\n    if (root == nullptr || root-&gt;val == 3) {\n        return;\n    }\n    // 시도\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // 해를 기록한다\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // 되돌리기\n    path.pop_back();\n}\n</code></pre> preorder_traversal_iii_compact.java<pre><code>/* 전위 순회: 예제 3 */\nvoid preOrder(TreeNode root) {\n    // 가지치기\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // 시도\n    path.add(root);\n    if (root.val == 7) {\n        // 해를 기록한다\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // 되돌리기\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_iii_compact.cs<pre><code>/* 전위 순회: 예제 3 */\nvoid PreOrder(TreeNode? root) {\n    // 가지치기\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // 시도\n    path.Add(root);\n    if (root.val == 7) {\n        // 해를 기록한다\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // 되돌리기\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_iii_compact.go<pre><code>/* 전위 순회: 예제 3 */\nfunc preOrderIII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    // 가지치기\n    if root == nil || root.Val == 3 {\n        return\n    }\n    // 시도\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // 해를 기록한다\n        *res = append(*res, append([]*TreeNode{}, *path...))\n    }\n    preOrderIII(root.Left, res, path)\n    preOrderIII(root.Right, res, path)\n    // 되돌리기\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_iii_compact.swift<pre><code>/* 전위 순회: 예제 3 */\nfunc preOrder(root: TreeNode?) {\n    // 가지치기\n    guard let root = root, root.val != 3 else {\n        return\n    }\n    // 시도\n    path.append(root)\n    if root.val == 7 {\n        // 해를 기록한다\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // 되돌리기\n    path.removeLast()\n}\n</code></pre> preorder_traversal_iii_compact.js<pre><code>/* 전위 순회: 예제 3 */\nfunction preOrder(root, path, res) {\n    // 가지치기\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // 시도\n    path.push(root);\n    if (root.val === 7) {\n        // 해를 기록한다\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // 되돌리기\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.ts<pre><code>/* 전위 순회: 예제 3 */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // 가지치기\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // 시도\n    path.push(root);\n    if (root.val === 7) {\n        // 해를 기록한다\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // 되돌리기\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.dart<pre><code>/* 전위 순회: 예제 3 */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null || root.val == 3) {\n    return;\n  }\n\n  // 시도\n  path.add(root);\n  if (root.val == 7) {\n    // 해를 기록한다\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // 되돌리기\n  path.removeLast();\n}\n</code></pre> preorder_traversal_iii_compact.rs<pre><code>/* 전위 순회: 예제 3 */\nfn pre_order(\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n) {\n    // 가지치기\n    if root.is_none() || root.as_ref().unwrap().borrow().val == 3 {\n        return;\n    }\n    if let Some(node) = root {\n        // 시도\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // 해를 기록한다\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.as_ref());\n        pre_order(res, path, node.borrow().right.as_ref());\n        // 되돌리기\n        path.pop();\n    }\n}\n</code></pre> preorder_traversal_iii_compact.c<pre><code>/* 전위 순회: 예제 3 */\nvoid preOrder(TreeNode *root) {\n    // 가지치기\n    if (root == NULL || root-&gt;val == 3) {\n        return;\n    }\n    // 시도\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // 해를 기록한다\n        for (int i = 0; i &lt; pathSize; i++) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // 되돌리기\n    pathSize--;\n}\n</code></pre> preorder_traversal_iii_compact.kt<pre><code>/* 전위 순회: 예제 3 */\nfun preOrder(root: TreeNode?) {\n    // 가지치기\n    if (root == null || root._val == 3) {\n        return\n    }\n    // 시도\n    path!!.add(root)\n    if (root._val == 7) {\n        // 해를 기록한다\n        res!!.add(path!!.toMutableList())\n    }\n    preOrder(root.left)\n    preOrder(root.right)\n    // 되돌리기\n    path!!.removeAt(path!!.size - 1)\n}\n</code></pre> preorder_traversal_iii_compact.rb<pre><code>### 전위 순회: 예제 3 ###\ndef pre_order(root)\n  # 가지치기\n  return if !root || root.val == 3\n\n  # 시도\n  $path.append(root)\n\n  # 해를 기록한다\n  $res &lt;&lt; $path.dup if root.val == 7\n\n  pre_order(root.left)\n  pre_order(root.right)\n\n  # 되돌리기\n  $path.pop\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>'가지치기'는 매우 직관적인 이름이다. 아래 그림(13-3)과 같이 탐색 중 제약 조건을 만족하지 않는 탐색 가지를 '잘라 내어' 의미 없는 시도를 많이 피하고 탐색 효율을 높인다.</p> <p></p> <p> 그림 13-3   제약 조건에 따른 가지치기 </p>","path":["제 13 장   백트래킹","13.1   백트래킹 알고리즘"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1313","level":2,"title":"13.1.3   기본 틀 코드","text":"<p>이제 백트래킹의 '시도, 되돌리기, 가지치기'라는 기본 틀을 추출하여 코드의 범용성을 높여 보자.</p> <p>아래 기본 틀 코드에서 <code>state</code>는 문제의 현재 상태를, <code>choices</code>는 그 상태에서 가능한 선택들을 나타낸다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def backtrack(state: State, choices: list[choice], res: list[state]):\n    \"\"\"백트래킹 알고리즘의 기본 틀\"\"\"\n    # 해인지 확인\n    if is_solution(state):\n        # 해 기록\n        record_solution(state, res)\n        # 더 이상 탐색하지 않는다\n        return\n    # 모든 선택지 순회\n    for choice in choices:\n        # 가지치기: 선택이 유효한지 확인\n        if is_valid(state, choice):\n            # 시도: 선택하고 상태 갱신\n            make_choice(state, choice)\n            backtrack(state, choices, res)\n            # 되돌리기: 선택을 취소하고 이전 상태 복원\n            undo_choice(state, choice)\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nvoid backtrack(State *state, vector&lt;Choice *&gt; &amp;choices, vector&lt;State *&gt; &amp;res) {\n    // 해인지 확인\n    if (isSolution(state)) {\n        // 해 기록\n        recordSolution(state, res);\n        // 더 이상 탐색하지 않는다\n        return;\n    }\n    // 모든 선택지 순회\n    for (Choice choice : choices) {\n        // 가지치기: 선택이 유효한지 확인\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nvoid backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // 해인지 확인\n    if (isSolution(state)) {\n        // 해 기록\n        recordSolution(state, res);\n        // 더 이상 탐색하지 않는다\n        return;\n    }\n    // 모든 선택지 순회\n    for (Choice choice : choices) {\n        // 가지치기: 선택이 유효한지 확인\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nvoid Backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // 해인지 확인\n    if (IsSolution(state)) {\n        // 해 기록\n        RecordSolution(state, res);\n        // 더 이상 탐색하지 않는다\n        return;\n    }\n    // 모든 선택지 순회\n    foreach (Choice choice in choices) {\n        // 가지치기: 선택이 유효한지 확인\n        if (IsValid(state, choice)) {\n            // 시도: 선택하고 상태 갱신\n            MakeChoice(state, choice);\n            Backtrack(state, choices, res);\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nfunc backtrack(state *State, choices []Choice, res *[]State) {\n    // 해인지 확인\n    if isSolution(state) {\n        // 해 기록\n        recordSolution(state, res)\n        // 더 이상 탐색하지 않는다\n        return\n    }\n    // 모든 선택지 순회\n    for _, choice := range choices {\n        // 가지치기: 선택이 유효한지 확인\n        if isValid(state, choice) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nfunc backtrack(state: inout State, choices: [Choice], res: inout [State]) {\n    // 해인지 확인\n    if isSolution(state: state) {\n        // 해 기록\n        recordSolution(state: state, res: &amp;res)\n        // 더 이상 탐색하지 않는다\n        return\n    }\n    // 모든 선택지 순회\n    for choice in choices {\n        // 가지치기: 선택이 유효한지 확인\n        if isValid(state: state, choice: choice) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state: &amp;state, choice: choice)\n            backtrack(state: &amp;state, choices: choices, res: &amp;res)\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nfunction backtrack(state, choices, res) {\n    // 해인지 확인\n    if (isSolution(state)) {\n        // 해 기록\n        recordSolution(state, res);\n        // 더 이상 탐색하지 않는다\n        return;\n    }\n    // 모든 선택지 순회\n    for (let choice of choices) {\n        // 가지치기: 선택이 유효한지 확인\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nfunction backtrack(state: State, choices: Choice[], res: State[]): void {\n    // 해인지 확인\n    if (isSolution(state)) {\n        // 해 기록\n        recordSolution(state, res);\n        // 더 이상 탐색하지 않는다\n        return;\n    }\n    // 모든 선택지 순회\n    for (let choice of choices) {\n        // 가지치기: 선택이 유효한지 확인\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nvoid backtrack(State state, List&lt;Choice&gt;, List&lt;State&gt; res) {\n  // 해인지 확인\n  if (isSolution(state)) {\n    // 해 기록\n    recordSolution(state, res);\n    // 더 이상 탐색하지 않는다\n    return;\n  }\n  // 모든 선택지 순회\n  for (Choice choice in choices) {\n    // 가지치기: 선택이 유효한지 확인\n    if (isValid(state, choice)) {\n      // 시도: 선택하고 상태 갱신\n      makeChoice(state, choice);\n      backtrack(state, choices, res);\n      // 되돌리기: 선택을 취소하고 이전 상태 복원\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nfn backtrack(state: &amp;mut State, choices: &amp;Vec&lt;Choice&gt;, res: &amp;mut Vec&lt;State&gt;) {\n    // 해인지 확인\n    if is_solution(state) {\n        // 해 기록\n        record_solution(state, res);\n        // 더 이상 탐색하지 않는다\n        return;\n    }\n    // 모든 선택지 순회\n    for choice in choices {\n        // 가지치기: 선택이 유효한지 확인\n        if is_valid(state, choice) {\n            // 시도: 선택하고 상태 갱신\n            make_choice(state, choice);\n            backtrack(state, choices, res);\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undo_choice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nvoid backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) {\n    // 해인지 확인\n    if (isSolution(state)) {\n        // 해 기록\n        recordSolution(state, res, numRes);\n        // 더 이상 탐색하지 않는다\n        return;\n    }\n    // 모든 선택지 순회\n    for (int i = 0; i &lt; numChoices; i++) {\n        // 가지치기: 선택이 유효한지 확인\n        if (isValid(state, &amp;choices[i])) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state, &amp;choices[i]);\n            backtrack(state, choices, numChoices, res, numRes);\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state, &amp;choices[i]);\n        }\n    }\n}\n</code></pre> <pre><code>/* 백트래킹 알고리즘의 기본 틀 */\nfun backtrack(state: State?, choices: List&lt;Choice?&gt;, res: List&lt;State?&gt;?) {\n    // 해인지 확인\n    if (isSolution(state)) {\n        // 해 기록\n        recordSolution(state, res)\n        // 더 이상 탐색하지 않는다\n        return\n    }\n    // 모든 선택지 순회\n    for (choice in choices) {\n        // 가지치기: 선택이 유효한지 확인\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태 갱신\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // 되돌리기: 선택을 취소하고 이전 상태 복원\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>### 백트래킹 알고리즘의 기본 틀 ###\ndef backtrack(state, choices, res)\n    # 해인지 확인\n    if is_solution?(state)\n        # 해 기록\n        record_solution(state, res)\n        return\n    end\n\n    # 모든 선택지 순회\n    for choice in choices\n        # 가지치기: 선택이 유효한지 확인\n        if is_valid?(state, choice)\n            # 시도: 선택하고 상태 갱신\n            make_choice(state, choice)\n            backtrack(state, choices, res)\n            # 되돌리기: 선택을 취소하고 이전 상태 복원\n            undo_choice(state, choice)\n        end\n    end\nend\n</code></pre> <p>이제 기본 틀 코드로 예제 3을 해결해 보자. 상태 <code>state</code>는 노드 순회 경로, 선택지 <code>choices</code>는 현재 노드의 왼쪽과 오른쪽 자식 노드, 결과 <code>res</code>는 경로들의 리스트다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby preorder_traversal_iii_template.py<pre><code>def is_solution(state: list[TreeNode]) -&gt; bool:\n    \"\"\"현재 상태가 해인지 판단\"\"\"\n    return state and state[-1].val == 7\n\ndef record_solution(state: list[TreeNode], res: list[list[TreeNode]]):\n    \"\"\"해를 기록한다\"\"\"\n    res.append(list(state))\n\ndef is_valid(state: list[TreeNode], choice: TreeNode) -&gt; bool:\n    \"\"\"현재 상태에서 해당 선택이 유효한지 판단\"\"\"\n    return choice is not None and choice.val != 3\n\ndef make_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"상태 갱신\"\"\"\n    state.append(choice)\n\ndef undo_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"상태 복원\"\"\"\n    state.pop()\n\ndef backtrack(\n    state: list[TreeNode], choices: list[TreeNode], res: list[list[TreeNode]]\n):\n    \"\"\"백트래킹 알고리즘: 예제 3\"\"\"\n    # 해인지 확인한다\n    if is_solution(state):\n        # 해를 기록한다\n        record_solution(state, res)\n    # 모든 선택지를 순회한다\n    for choice in choices:\n        # 가지치기: 선택이 유효한지 확인한다\n        if is_valid(state, choice):\n            # 시도: 선택하고 상태를 갱신한다\n            make_choice(state, choice)\n            # 다음 선택을 진행한다\n            backtrack(state, [choice.left, choice.right], res)\n            # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undo_choice(state, choice)\n</code></pre> preorder_traversal_iii_template.cpp<pre><code>/* 현재 상태가 해인지 판단 */\nbool isSolution(vector&lt;TreeNode *&gt; &amp;state) {\n    return !state.empty() &amp;&amp; state.back()-&gt;val == 7;\n}\n\n/* 해를 기록한다 */\nvoid recordSolution(vector&lt;TreeNode *&gt; &amp;state, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    res.push_back(state);\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nbool isValid(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    return choice != nullptr &amp;&amp; choice-&gt;val != 3;\n}\n\n/* 상태 갱신 */\nvoid makeChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.push_back(choice);\n}\n\n/* 상태 복원 */\nvoid undoChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.pop_back();\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nvoid backtrack(vector&lt;TreeNode *&gt; &amp;state, vector&lt;TreeNode *&gt; &amp;choices, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    // 해인지 확인한다\n    if (isSolution(state)) {\n        // 해를 기록한다\n        recordSolution(state, res);\n    }\n    // 모든 선택지를 순회한다\n    for (TreeNode *choice : choices) {\n        // 가지치기: 선택이 유효한지 확인한다\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(state, choice);\n            // 다음 선택을 진행한다\n            vector&lt;TreeNode *&gt; nextChoices{choice-&gt;left, choice-&gt;right};\n            backtrack(state, nextChoices, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.java<pre><code>/* 현재 상태가 해인지 판단 */\nboolean isSolution(List&lt;TreeNode&gt; state) {\n    return !state.isEmpty() &amp;&amp; state.get(state.size() - 1).val == 7;\n}\n\n/* 해를 기록한다 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.add(new ArrayList&lt;&gt;(state));\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nboolean isValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* 상태 갱신 */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.add(choice);\n}\n\n/* 상태 복원 */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.remove(state.size() - 1);\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nvoid backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // 해인지 확인한다\n    if (isSolution(state)) {\n        // 해를 기록한다\n        recordSolution(state, res);\n    }\n    // 모든 선택지를 순회한다\n    for (TreeNode choice : choices) {\n        // 가지치기: 선택이 유효한지 확인한다\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(state, choice);\n            // 다음 선택을 진행한다\n            backtrack(state, Arrays.asList(choice.left, choice.right), res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.cs<pre><code>/* 현재 상태가 해인지 판단 */\nbool IsSolution(List&lt;TreeNode&gt; state) {\n    return state.Count != 0 &amp;&amp; state[^1].val == 7;\n}\n\n/* 해를 기록한다 */\nvoid RecordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.Add(new List&lt;TreeNode&gt;(state));\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nbool IsValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* 상태 갱신 */\nvoid MakeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.Add(choice);\n}\n\n/* 상태 복원 */\nvoid UndoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.RemoveAt(state.Count - 1);\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nvoid Backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // 해인지 확인한다\n    if (IsSolution(state)) {\n        // 해를 기록한다\n        RecordSolution(state, res);\n    }\n    // 모든 선택지를 순회한다\n    foreach (TreeNode choice in choices) {\n        // 가지치기: 선택이 유효한지 확인한다\n        if (IsValid(state, choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            MakeChoice(state, choice);\n            // 다음 선택을 진행한다\n            Backtrack(state, [choice.left!, choice.right!], res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.go<pre><code>/* 현재 상태가 해인지 판단 */\nfunc isSolution(state *[]*TreeNode) bool {\n    return len(*state) != 0 &amp;&amp; (*state)[len(*state)-1].Val == 7\n}\n\n/* 해를 기록한다 */\nfunc recordSolution(state *[]*TreeNode, res *[][]*TreeNode) {\n    *res = append(*res, append([]*TreeNode{}, *state...))\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nfunc isValid(state *[]*TreeNode, choice *TreeNode) bool {\n    return choice != nil &amp;&amp; choice.Val != 3\n}\n\n/* 상태 갱신 */\nfunc makeChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = append(*state, choice)\n}\n\n/* 상태 복원 */\nfunc undoChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = (*state)[:len(*state)-1]\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nfunc backtrackIII(state *[]*TreeNode, choices *[]*TreeNode, res *[][]*TreeNode) {\n    // 해인지 확인한다\n    if isSolution(state) {\n        // 해를 기록한다\n        recordSolution(state, res)\n    }\n    // 모든 선택지를 순회한다\n    for _, choice := range *choices {\n        // 가지치기: 선택이 유효한지 확인한다\n        if isValid(state, choice) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(state, choice)\n            // 다음 선택을 진행한다\n            temp := make([]*TreeNode, 0)\n            temp = append(temp, choice.Left, choice.Right)\n            backtrackIII(state, &amp;temp, res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.swift<pre><code>/* 현재 상태가 해인지 판단 */\nfunc isSolution(state: [TreeNode]) -&gt; Bool {\n    !state.isEmpty &amp;&amp; state.last!.val == 7\n}\n\n/* 해를 기록한다 */\nfunc recordSolution(state: [TreeNode], res: inout [[TreeNode]]) {\n    res.append(state)\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nfunc isValid(state: [TreeNode], choice: TreeNode?) -&gt; Bool {\n    choice != nil &amp;&amp; choice!.val != 3\n}\n\n/* 상태 갱신 */\nfunc makeChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.append(choice)\n}\n\n/* 상태 복원 */\nfunc undoChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.removeLast()\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nfunc backtrack(state: inout [TreeNode], choices: [TreeNode], res: inout [[TreeNode]]) {\n    // 해인지 확인한다\n    if isSolution(state: state) {\n        recordSolution(state: state, res: &amp;res)\n    }\n    // 모든 선택지를 순회한다\n    for choice in choices {\n        // 가지치기: 선택이 유효한지 확인한다\n        if isValid(state: state, choice: choice) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(state: &amp;state, choice: choice)\n            // 다음 선택을 진행한다\n            backtrack(state: &amp;state, choices: [choice.left, choice.right].compactMap { $0 }, res: &amp;res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.js<pre><code>/* 현재 상태가 해인지 판단 */\nfunction isSolution(state) {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* 해를 기록한다 */\nfunction recordSolution(state, res) {\n    res.push([...state]);\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nfunction isValid(state, choice) {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* 상태 갱신 */\nfunction makeChoice(state, choice) {\n    state.push(choice);\n}\n\n/* 상태 복원 */\nfunction undoChoice(state) {\n    state.pop();\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nfunction backtrack(state, choices, res) {\n    // 해인지 확인한다\n    if (isSolution(state)) {\n        // 해를 기록한다\n        recordSolution(state, res);\n    }\n    // 모든 선택지를 순회한다\n    for (const choice of choices) {\n        // 가지치기: 선택이 유효한지 확인한다\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(state, choice);\n            // 다음 선택을 진행한다\n            backtrack(state, [choice.left, choice.right], res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.ts<pre><code>/* 현재 상태가 해인지 판단 */\nfunction isSolution(state: TreeNode[]): boolean {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* 해를 기록한다 */\nfunction recordSolution(state: TreeNode[], res: TreeNode[][]): void {\n    res.push([...state]);\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nfunction isValid(state: TreeNode[], choice: TreeNode): boolean {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* 상태 갱신 */\nfunction makeChoice(state: TreeNode[], choice: TreeNode): void {\n    state.push(choice);\n}\n\n/* 상태 복원 */\nfunction undoChoice(state: TreeNode[]): void {\n    state.pop();\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nfunction backtrack(\n    state: TreeNode[],\n    choices: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // 해인지 확인한다\n    if (isSolution(state)) {\n        // 해를 기록한다\n        recordSolution(state, res);\n    }\n    // 모든 선택지를 순회한다\n    for (const choice of choices) {\n        // 가지치기: 선택이 유효한지 확인한다\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(state, choice);\n            // 다음 선택을 진행한다\n            backtrack(state, [choice.left, choice.right], res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.dart<pre><code>/* 현재 상태가 해인지 판단 */\nbool isSolution(List&lt;TreeNode&gt; state) {\n  return state.isNotEmpty &amp;&amp; state.last.val == 7;\n}\n\n/* 해를 기록한다 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n  res.add(List.from(state));\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nbool isValid(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* 상태 갱신 */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.add(choice!);\n}\n\n/* 상태 복원 */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.removeLast();\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nvoid backtrack(\n  List&lt;TreeNode&gt; state,\n  List&lt;TreeNode?&gt; choices,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  // 해인지 확인한다\n  if (isSolution(state)) {\n    // 해를 기록한다\n    recordSolution(state, res);\n  }\n  // 모든 선택지를 순회한다\n  for (TreeNode? choice in choices) {\n    // 가지치기: 선택이 유효한지 확인한다\n    if (isValid(state, choice)) {\n      // 시도: 선택하고 상태를 갱신한다\n      makeChoice(state, choice);\n      // 다음 선택을 진행한다\n      backtrack(state, [choice!.left, choice.right], res);\n      // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> preorder_traversal_iii_template.rs<pre><code>/* 현재 상태가 해인지 판단 */\nfn is_solution(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\n    return !state.is_empty() &amp;&amp; state.last().unwrap().borrow().val == 7;\n}\n\n/* 해를 기록한다 */\nfn record_solution(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    res.push(state.clone());\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nfn is_valid(_: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\n    return choice.is_some() &amp;&amp; choice.unwrap().borrow().val != 3;\n}\n\n/* 상태 갱신 */\nfn make_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.push(choice);\n}\n\n/* 상태 복원 */\nfn undo_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, _: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.pop();\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nfn backtrack(\n    state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,\n    choices: &amp;Vec&lt;Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;,\n) {\n    // 해인지 확인한다\n    if is_solution(state) {\n        // 해를 기록한다\n        record_solution(state, res);\n    }\n    // 모든 선택지를 순회한다\n    for &amp;choice in choices.iter() {\n        // 가지치기: 선택이 유효한지 확인한다\n        if is_valid(state, choice) {\n            // 시도: 선택하고 상태를 갱신한다\n            make_choice(state, choice.unwrap().clone());\n            // 다음 선택을 진행한다\n            backtrack(\n                state,\n                &amp;vec![\n                    choice.unwrap().borrow().left.as_ref(),\n                    choice.unwrap().borrow().right.as_ref(),\n                ],\n                res,\n            );\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undo_choice(state, choice.unwrap().clone());\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.c<pre><code>/* 현재 상태가 해인지 판단 */\nbool isSolution(void) {\n    return pathSize &gt; 0 &amp;&amp; path[pathSize - 1]-&gt;val == 7;\n}\n\n/* 해를 기록한다 */\nvoid recordSolution(void) {\n    for (int i = 0; i &lt; pathSize; i++) {\n        res[resSize][i] = path[i];\n    }\n    resSize++;\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nbool isValid(TreeNode *choice) {\n    return choice != NULL &amp;&amp; choice-&gt;val != 3;\n}\n\n/* 상태 갱신 */\nvoid makeChoice(TreeNode *choice) {\n    path[pathSize++] = choice;\n}\n\n/* 상태 복원 */\nvoid undoChoice(void) {\n    pathSize--;\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nvoid backtrack(TreeNode *choices[2]) {\n    // 해인지 확인한다\n    if (isSolution()) {\n        // 해를 기록한다\n        recordSolution();\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; 2; i++) {\n        TreeNode *choice = choices[i];\n        // 가지치기: 선택이 유효한지 확인한다\n        if (isValid(choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(choice);\n            // 다음 선택을 진행한다\n            TreeNode *nextChoices[2] = {choice-&gt;left, choice-&gt;right};\n            backtrack(nextChoices);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice();\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.kt<pre><code>/* 현재 상태가 해인지 판단 */\nfun isSolution(state: MutableList&lt;TreeNode?&gt;): Boolean {\n    return state.isNotEmpty() &amp;&amp; state[state.size - 1]?._val == 7\n}\n\n/* 해를 기록한다 */\nfun recordSolution(state: MutableList&lt;TreeNode?&gt;?, res: MutableList&lt;MutableList&lt;TreeNode?&gt;?&gt;) {\n    res.add(state!!.toMutableList())\n}\n\n/* 현재 상태에서 해당 선택이 유효한지 판단 */\nfun isValid(state: MutableList&lt;TreeNode?&gt;?, choice: TreeNode?): Boolean {\n    return choice != null &amp;&amp; choice._val != 3\n}\n\n/* 상태 갱신 */\nfun makeChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.add(choice)\n}\n\n/* 상태 복원 */\nfun undoChoice(state: MutableList&lt;TreeNode?&gt;, choice: TreeNode?) {\n    state.removeLast()\n}\n\n/* 백트래킹 알고리즘: 예제 3 */\nfun backtrack(\n    state: MutableList&lt;TreeNode?&gt;,\n    choices: MutableList&lt;TreeNode?&gt;,\n    res: MutableList&lt;MutableList&lt;TreeNode?&gt;?&gt;\n) {\n    // 해인지 확인한다\n    if (isSolution(state)) {\n        // 해를 기록한다\n        recordSolution(state, res)\n    }\n    // 모든 선택지를 순회한다\n    for (choice in choices) {\n        // 가지치기: 선택이 유효한지 확인한다\n        if (isValid(state, choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            makeChoice(state, choice)\n            // 다음 선택을 진행한다\n            backtrack(state, mutableListOf(choice!!.left, choice.right), res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.rb<pre><code>### 현재 상태가 해인지 판단 ###\ndef is_solution?(state)\n  !state.empty? &amp;&amp; state.last.val == 7\nend\n\n### 해를 기록한다 ###\ndef record_solution(state, res)\n  res &lt;&lt; state.dup\nend\n\n### 현재 상태에서 해당 선택이 유효한지 판단 ###\ndef is_valid?(state, choice)\n  choice &amp;&amp; choice.val != 3\nend\n\n### 상태 갱신 ###\ndef make_choice(state, choice)\n  state &lt;&lt; choice\nend\n\n### 상태 복원 ###\ndef undo_choice(state, choice)\n  state.pop\nend\n\n### 백트래킹 알고리즘: 예제 3 ###\ndef backtrack(state, choices, res)\n  # 해인지 확인한다\n  record_solution(state, res) if is_solution?(state)\n\n  # 모든 선택지를 순회한다\n  for choice in choices\n    # 가지치기: 선택이 유효한지 확인한다\n    if is_valid?(state, choice)\n      # 시도: 선택하고 상태를 갱신한다\n      make_choice(state, choice)\n      # 다음 선택을 진행한다\n      backtrack(state, [choice.left, choice.right], res)\n      # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n      undo_choice(state, choice)\n    end\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>문제의 요구에 따르면 값이 \\(7\\)인 노드를 찾은 뒤에도 탐색을 계속해야 한다. 따라서 해를 기록한 뒤의 <code>return</code> 문을 삭제해야 한다. 아래 그림(13-4)은 <code>return</code> 문을 유지하거나 삭제했을 때의 탐색 과정을 비교한다.</p> <p></p> <p> 그림 13-4   return 유지 여부에 따른 탐색 과정 비교 </p> <p>전위 순회 기반 코드에 비해 백트래킹 기본 틀을 사용한 코드는 다소 장황하지만 범용성이 더 좋다. 실제로 많은 백트래킹 문제를 이 틀 안에서 해결할 수 있다. 구체적인 문제에 맞게 <code>state</code>와 <code>choices</code>를 정의하고, 틀 안의 각 메서드만 구현하면 된다.</p>","path":["제 13 장   백트래킹","13.1   백트래킹 알고리즘"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1314","level":2,"title":"13.1.4   주요 용어","text":"<p>알고리즘 문제를 더 명확히 분석하기 위해 백트래킹에서 자주 쓰는 용어를 정리하고, 예제 3에 대응하는 사례를 아래 표(13-1)에 나타냈다.</p> <p> 표 13-1   백트래킹 알고리즘의 주요 용어 </p> 용어 정의 예제 3 해(solution) 문제의 특정 조건을 만족하는 답으로, 하나 또는 여러 개일 수 있다 루트 노드에서 노드 \\(7\\)까지 제약 조건을 만족하는 모든 경로 제약 조건(constraint) 해의 유효성을 제한하는 조건으로, 보통 가지치기에 사용한다 경로에 노드 \\(3\\)이 포함되지 않는다 상태(state) 이미 내린 선택을 포함하여 특정 시점의 문제 상황을 나타낸다 현재까지 방문한 노드의 경로, 즉 노드 리스트 <code>path</code> 시도(attempt) 가능한 선택으로 해 공간을 탐색하는 과정으로, 선택, 상태 갱신, 해인지 확인하는 작업을 포함한다 왼쪽(오른쪽) 자식 노드를 재귀적으로 방문하고 <code>path</code>에 추가한 뒤 값이 \\(7\\)인지 확인한다 되돌리기(backtracking) 제약 조건을 만족하지 않는 상태를 만나면 이전 선택을 취소하고 직전 상태로 돌아간다 리프 노드를 지나거나 노드 방문을 마치거나 값이 \\(3\\)인 노드를 만나면 탐색을 멈추고 함수가 반환된다 가지치기(pruning) 문제의 특성과 제약 조건에 따라 의미 없는 탐색 경로를 피하여 효율을 높이는 방법이다 값이 \\(3\\)인 노드를 만나면 탐색을 더 진행하지 않는다 <p>참고</p> <p>문제, 해, 상태 등의 개념은 분할 정복, 백트래킹, 동적 계획법, 탐욕 알고리즘 등에서 공통으로 사용된다.</p>","path":["제 13 장   백트래킹","13.1   백트래킹 알고리즘"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1315","level":2,"title":"13.1.5   장점과 한계","text":"<p>백트래킹은 본질적으로 깊이 우선 탐색 알고리즘으로, 조건을 만족하는 해를 찾을 때까지 가능한 모든 해결 방법을 시도한다. 모든 가능한 해를 찾을 수 있으며, 적절히 가지치기하면 매우 효율적이라는 장점이 있다.</p> <p>하지만 규모가 크거나 복잡한 문제에서는 백트래킹의 실행 효율을 받아들이기 어려울 수 있다.</p> <ul> <li>시간: 상태 공간의 모든 가능성을 순회해야 하는 경우가 많아 시간 복잡도가 지수 또는 팩토리얼 수준에 이를 수 있다.</li> <li>공간: 재귀 호출 중 현재 상태(경로, 가지치기용 보조 변수 등)를 저장해야 하므로 깊이가 크면 공간 요구량도 매우 커질 수 있다.</li> </ul> <p>그렇더라도 일부 탐색 문제와 제약 만족 문제에서는 백트래킹이 여전히 최선의 해결 방법이다. 어떤 선택이 유효한 해를 만들지 예측할 수 없으므로 가능한 모든 선택을 순회해야 하기 때문이다. 이때는 효율을 어떻게 최적화하느냐가 핵심이며, 대표적인 방법은 두 가지다.</p> <ul> <li>가지치기: 해가 나올 수 없는 경로를 탐색하지 않아 시간과 공간을 절약한다.</li> <li>휴리스틱 탐색: 탐색에 전략이나 추정값을 도입하여 유효한 해가 나올 가능성이 높은 경로를 먼저 탐색한다.</li> </ul>","path":["제 13 장   백트래킹","13.1   백트래킹 알고리즘"],"tags":[]},{"location":"chapter_backtracking/backtracking_algorithm/#1316","level":2,"title":"13.1.6   대표적인 백트래킹 문제","text":"<p>백트래킹은 많은 탐색 문제, 제약 만족 문제, 조합 최적화 문제를 해결하는 데 사용할 수 있다.</p> <p>탐색 문제: 특정 조건을 만족하는 해결 방법을 찾는 것이 목표다.</p> <ul> <li>모든 순열 문제: 집합이 주어졌을 때 가능한 모든 배열 순서를 구한다.</li> <li>부분 집합 합 문제: 집합과 목표 합이 주어졌을 때 원소의 합이 목표 합인 모든 부분 집합을 찾는다.</li> <li>하노이의 탑 문제: 기둥 세 개와 크기가 다른 원판들이 주어졌을 때 모든 원판을 한 기둥에서 다른 기둥으로 옮긴다. 한 번에 하나만 옮길 수 있고 큰 원판을 작은 원판 위에 놓으면 안 된다.</li> </ul> <p>제약 만족 문제: 모든 제약 조건을 만족하는 해를 찾는 것이 목표다.</p> <ul> <li>\\(n\\)-퀸: \\(n \\times n\\) 체스판에 퀸 \\(n\\) 개를 서로 공격하지 않도록 놓는다.</li> <li>스도쿠: \\(9 \\times 9\\) 격자에 \\(1\\) ~ \\(9\\)를 넣되, 각 행과 열 및 각 \\(3 \\times 3\\) 부분 격자에서 숫자가 중복되지 않게 한다.</li> <li>그래프 색칠 문제: 무방향 그래프의 각 정점을 가능한 한 적은 색으로 칠하되, 인접한 정점은 서로 다른 색을 갖게 한다.</li> </ul> <p>조합 최적화 문제: 조합 공간에서 특정 조건을 만족하는 최적해를 찾는 것이 목표다.</p> <ul> <li>0-1 배낭 문제: 가치와 무게가 있는 물건들과 배낭이 주어졌을 때, 용량 제한 안에서 총가치가 최대가 되도록 물건을 고른다.</li> <li>외판원 문제: 그래프의 한 점에서 출발하여 다른 모든 점을 정확히 한 번 방문한 뒤 출발점으로 돌아오는 최단 경로를 구한다.</li> <li>최대 클리크 문제: 무방향 그래프에서 가장 큰 완전 부분 그래프, 즉 임의의 두 정점 사이에 간선이 있는 부분 그래프를 찾는다.</li> </ul> <p>많은 조합 최적화 문제에서 백트래킹이 최적의 해결 방법은 아니라는 점에 주의하자.</p> <ul> <li>0-1 배낭 문제는 더 높은 시간 효율을 위해 보통 동적 계획법으로 해결한다.</li> <li>외판원 문제는 유명한 NP-Hard 문제이며, 유전 알고리즘과 개미 군집 알고리즘 등이 자주 쓰인다.</li> <li>최대 클리크 문제는 그래프 이론의 대표적인 문제로, 탐욕 알고리즘 같은 휴리스틱 알고리즘을 사용할 수 있다.</li> </ul>","path":["제 13 장   백트래킹","13.1   백트래킹 알고리즘"],"tags":[]},{"location":"chapter_backtracking/exercises/","level":1,"title":"13.6   연습 문제","text":"","path":["제 13 장   백트래킹","13.6   연습 문제"],"tags":[]},{"location":"chapter_backtracking/exercises/#1361","level":2,"title":"13.6.1   개념 확인","text":"","path":["제 13 장   백트래킹","13.6   연습 문제"],"tags":[]},{"location":"chapter_backtracking/exercises/#1","level":3,"title":"1.   이 순열 알고리즘은 결과를 빠뜨리는가","text":"<p>한 백트래킹 알고리즘이 <code>1, 2, 3</code> 순서로 숫자를 시도하며 모든 순열을 생성한다. 숫자 <code>x</code>를 선택할 때마다 다음을 수행한다.</p> <ol> <li>현재 경로의 끝에 <code>x</code>를 추가한다.</li> <li><code>x</code>를 ‘사용함’으로 표시한다.</li> <li>재귀 호출로 다음 위치를 채운다.</li> </ol> <p>재귀 호출이 반환된 뒤, 학생은 경로 끝의 <code>x</code>만 삭제하고 다음 숫자를 계속 시도했다.</p> <ol> <li>알고리즘이 처음 얻는 순열은 무엇인가? 순열 6개를 모두 얻을 수 있는가?</li> <li>재귀 호출에서 이전 단계로 돌아가기 전에 경로 끝의 숫자를 삭제하는 것만으로 충분한가? 충분하지 않다면 무엇을 더 해야 하는지 이유와 함께 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>처음 얻는 순열은 <code>[1, 2, 3]</code>이지만, 모든 순열을 얻을 수는 없다. 반환하면서 경로는 짧아져도     숫자 1, 2, 3은 여전히 모두 ‘사용함’으로 표시되어 있어 이후 분기에서 선택할 숫자가 없기 때문이다.</p> </li> <li> <p>충분하지 않다. 경로 끝의 <code>x</code>를 삭제한 뒤에는 <code>x</code>를 다시 ‘사용하지 않음’으로 표시해야 한다.     현재 경로와 사용 여부 표시가 함께 탐색 상태를 나타낸다. 선택할 때 두 곳을 수정했으므로 되돌아갈 때도 두 곳을 모두 복원해야     다른 분기에서 <code>x</code>를 다시 선택할 수 있다.</p> </li> </ol>","path":["제 13 장   백트래킹","13.6   연습 문제"],"tags":[]},{"location":"chapter_backtracking/exercises/#2","level":3,"title":"2.   숫자를 선택하는 순서가 중요한가","text":"<p>정렬된 배열 <code>[2, 3, 5]</code>와 목표 값 5가 주어지며, 각 숫자는 반복해서 선택할 수 있다. 알고리즘에서는 각 탐색 경로의 숫자가 작은 것부터 큰 것 순으로만 나타나도록 제한한다.</p> <ol> <li>서로 다른 어떤 조합을 얻을 수 있는가?</li> <li>같은 숫자들을 다른 순서로 반복해서 탐색할 필요가 없는 이유는 무엇인가? ‘작은 것부터 큰 것 순’이라는 제한은 어떤 역할을 하는가?</li> <li>현재 경로가 <code>[3]</code>이고 2가 더 필요한 상황에서 다음 후보 숫자가 3이다. 이때 이 단계의 나머지 후보를 모두 확인하지 않고 멈출 수 있는 이유는 무엇인가?</li> </ol> 참고 답안 <ol> <li> <p>서로 다른 조합은 <code>[2, 3]</code>과 <code>[5]</code>다.</p> </li> <li> <p>이 문제에서는 <code>[2, 3]</code>과 <code>[3, 2]</code>를 같은 조합으로 취급하므로 숫자를 선택하는 순서는 답에 반영하지 않는다.     경로의 숫자가 작은 것부터 큰 것 순으로 나타나도록 정하면 탐색 중에 <code>[3, 2]</code> 같은 중복 조합을 바로 건너뛸 수 있다.</p> </li> <li> <p>현재 2가 더 필요한데 후보 숫자 3은 이미 2보다 크다. 배열이 정렬되어 있으므로     3 뒤의 후보는 더 크기만 할 뿐이고, 모두 현재 조합에 추가할 수 없다. 따라서 이 단계의 확인을 바로 끝낼 수 있다.</p> </li> </ol>","path":["제 13 장   백트래킹","13.6   연습 문제"],"tags":[]},{"location":"chapter_backtracking/exercises/#3","level":3,"title":"3.   다음 퀸은 어디에 놓을 수 있는가","text":"<p><code>4 × 4</code> 체스판에 행별로 퀸을 놓으며, 행과 열의 인덱스는 모두 0부터 시작한다. 현재 <code>(0, 1)</code>과 <code>(1, 3)</code>에 퀸을 놓았고, 이제 인덱스 2인 행에 다음 퀸을 놓으려 한다.</p> <ol> <li>‘같은 열’이라는 이유로 제외되는 열은 어디인가?</li> <li>남은 열 가운데 ‘같은 대각선’이라는 이유로 제외되는 위치는 어디인가?</li> <li>인덱스 2인 행에서 시도해 볼 수 있는 위치는 어디인가?</li> </ol> 참고 답안 <ol> <li> <p>인덱스 1과 3인 열에 이미 퀸이 있으므로 <code>(2, 1)</code>과 <code>(2, 3)</code>은 제외된다.</p> </li> <li> <p>남은 위치 중 <code>(2, 2)</code>는 <code>(1, 3)</code>과 같은 대각선에 있으므로 역시 제외된다.     <code>(2, 0)</code>은 이미 놓인 두 퀸 중 어느 것과도 같은 열이나 같은 대각선에 있지 않다.</p> </li> <li> <p>인덱스 2인 행에서 시도할 수 있는 위치는 <code>(2, 0)</code>뿐이다.</p> <p>이 단계에서는 현재 배치가 조건에 맞는다는 것만 확인한 셈이다. 이후 체스판을 완성할 수 없다면 여전히 되돌아가 더 앞 단계의 다른 선택을 시도해야 한다.</p> </li> </ol>","path":["제 13 장   백트래킹","13.6   연습 문제"],"tags":[]},{"location":"chapter_backtracking/exercises/#1362","level":2,"title":"13.6.2   프로그래밍 연습","text":"","path":["제 13 장   백트래킹","13.6   연습 문제"],"tags":[]},{"location":"chapter_backtracking/exercises/#1_1","level":3,"title":"1.   중복 원소가 없는 모든 순열","text":"<p>정수 배열 <code>nums</code>에는 원소가 적어도 하나 있으며, 모든 원소가 서로 다르다. 각 원소를 한 번씩 사용하여 만들 수 있는 모든 순서를 나열하고, 각 순서를 배열로 반환하라. 결과에서 순열들이 나오는 순서는 상관없다. 백트래킹을 사용하고, 각 위치의 원소가 현재 순열에 이미 선택되었는지 불리언 배열로 기록하라.</p> 풀이 힌트 <ol> <li>재귀 깊이는 순열의 몇 번째 위치를 채우는 중인지를 나타낸다.</li> <li>각 단계에서는 아직 사용하지 않은 원소만 시도한다.</li> <li>경로 길이가 <code>nums</code>의 길이에 도달하면 경로의 복사본을 답에 추가한다.</li> </ol> <p>LeetCode</p>","path":["제 13 장   백트래킹","13.6   연습 문제"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/","level":1,"title":"13.4   n-퀸 문제","text":"<p>문제</p> <p>체스 규칙에서 퀸은 같은 행, 열 또는 대각선에 있는 말을 공격할 수 있다. 퀸 \\(n\\) 개와 \\(n \\times n\\) 크기의 체스판이 주어졌을 때, 퀸들이 서로 공격할 수 없도록 배치하는 방법을 찾아라.</p> <p>아래 그림(13-15)과 같이 \\(n = 4\\)이면 해는 두 개다. 백트래킹 관점에서 \\(n \\times n\\) 체스판의 \\(n^2\\) 개 칸은 모든 선택지 <code>choices</code>를 제공한다. 퀸을 하나씩 놓는 동안 체스판은 계속 바뀌며, 각 시점의 체스판이 상태 <code>state</code>다.</p> <p></p> <p> 그림 13-15   4-퀸 문제의 해 </p> <p>아래 그림(13-16)은 세 가지 제약 조건을 보여 준다. 여러 퀸이 같은 행, 같은 열, 같은 대각선에 있으면 안 된다. 대각선은 주대각선 방향 <code>\\</code>과 부대각선 방향 <code>/</code>의 두 종류로 나뉜다.</p> <p></p> <p> 그림 13-16   n-퀸 문제의 제약 조건 </p>","path":["제 13 장   백트래킹","13.4   n-퀸 문제"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/#1","level":3,"title":"1.   한 행씩 배치하는 전략","text":"<p>퀸 수와 체스판의 행 수가 모두 \\(n\\)이므로 각 행에는 퀸을 정확히 하나씩 놓아야 한다는 사실을 쉽게 도출할 수 있다.</p> <p>따라서 첫 행부터 마지막 행까지 매 행에 퀸 하나씩 놓는 전략을 사용할 수 있다.</p> <p>아래 그림(13-17)은 4-퀸 문제에서 한 행씩 배치하는 과정이다. 공간 제약 때문에 첫 행의 탐색 가지 중 하나만 펼쳤으며, 열과 대각선 제약을 만족하지 않는 배치는 모두 가지치기했다.</p> <p></p> <p> 그림 13-17   한 행씩 배치하는 전략 </p> <p>본질적으로 행별 배치 전략 자체가 가지치기 역할을 한다. 같은 행에 여러 퀸이 놓이는 모든 탐색 가지를 피하기 때문이다.</p>","path":["제 13 장   백트래킹","13.4   n-퀸 문제"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/#2","level":3,"title":"2.   열과 대각선 가지치기","text":"<p>열 제약을 만족하기 위해 길이 \\(n\\)의 불리언 배열 <code>cols</code>에 각 열의 퀸 존재 여부를 기록할 수 있다. 배치를 결정하기 전에 <code>cols</code>로 이미 퀸이 있는 열을 가지치기하고, 백트래킹 중 <code>cols</code>의 상태를 동적으로 갱신한다.</p> <p>참고</p> <p>행렬의 시작점은 왼쪽 위이며, 행 인덱스는 아래로 갈수록, 열 인덱스는 오른쪽으로 갈수록 커진다.</p> <p>대각선 제약은 어떻게 처리할까? 한 칸의 행과 열 인덱스를 \\((row, col)\\)이라고 하자. 주대각선 방향의 대각선 하나를 고르면, 그 위의 모든 칸에서 행 인덱스와 열 인덱스의 차가 같다. 즉, 같은 주대각선 방향 대각선에서는 \\(row - col\\)이 일정하다.</p> <p>따라서 두 칸이 \\(row_1 - col_1 = row_2 - col_2\\)를 만족하면 반드시 같은 주대각선 방향 대각선에 있다. 이 규칙을 이용하여 아래 그림(13-18)의 배열 <code>diags1</code>에 각 대각선의 퀸 존재 여부를 기록할 수 있다.</p> <p>마찬가지로 같은 부대각선 방향 대각선의 모든 칸에서는 \\(row + col\\)이 일정하다. 배열 <code>diags2</code>를 이용하여 이 방향의 제약도 처리할 수 있다.</p> <p></p> <p> 그림 13-18   열 제약과 대각선 제약 처리 </p>","path":["제 13 장   백트래킹","13.4   n-퀸 문제"],"tags":[]},{"location":"chapter_backtracking/n_queens_problem/#3","level":3,"title":"3.   코드 구현","text":"<p>\\(n\\) 차 정방행렬에서 \\(row - col\\)의 범위는 \\([-n + 1, n - 1]\\), \\(row + col\\)의 범위는 \\([0, 2n - 2]\\)이다. 따라서 두 방향의 대각선 수는 각각 \\(2n - 1\\) 개이며, 배열 <code>diags1</code>과 <code>diags2</code>의 길이도 모두 \\(2n - 1\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby n_queens.py<pre><code>def backtrack(\n    row: int,\n    n: int,\n    state: list[list[str]],\n    res: list[list[list[str]]],\n    cols: list[bool],\n    diags1: list[bool],\n    diags2: list[bool],\n):\n    \"\"\"백트래킹 알고리즘: n-퀸\"\"\"\n    # 모든 행에 배치했으면 해를 기록한다\n    if row == n:\n        res.append([list(row) for row in state])\n        return\n    # 모든 열을 순회한다\n    for col in range(n):\n        # 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        diag1 = row - col + n - 1\n        diag2 = row + col\n        # 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if not cols[col] and not diags1[diag1] and not diags2[diag2]:\n            # 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = \"Q\"\n            cols[col] = diags1[diag1] = diags2[diag2] = True\n            # 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            # 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = \"#\"\n            cols[col] = diags1[diag1] = diags2[diag2] = False\n\ndef n_queens(n: int) -&gt; list[list[list[str]]]:\n    \"\"\"n-퀸 문제 풀기\"\"\"\n    # n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    state = [[\"#\" for _ in range(n)] for _ in range(n)]\n    cols = [False] * n  # 열에 퀸이 있는지 기록한다\n    diags1 = [False] * (2 * n - 1)  # 주대각선에 퀸이 있는지 기록한다\n    diags2 = [False] * (2 * n - 1)  # 부대각선에 퀸이 있는지 기록한다\n    res = []\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n</code></pre> n_queens.cpp<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nvoid backtrack(int row, int n, vector&lt;vector&lt;string&gt;&gt; &amp;state, vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; &amp;res, vector&lt;bool&gt; &amp;cols,\n               vector&lt;bool&gt; &amp;diags1, vector&lt;bool&gt; &amp;diags2) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if (row == n) {\n        res.push_back(state);\n        return;\n    }\n    // 모든 열을 순회한다\n    for (int col = 0; col &lt; n; col++) {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nvector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; nQueens(int n) {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    vector&lt;vector&lt;string&gt;&gt; state(n, vector&lt;string&gt;(n, \"#\"));\n    vector&lt;bool&gt; cols(n, false);           // 열에 퀸이 있는지 기록한다\n    vector&lt;bool&gt; diags1(2 * n - 1, false); // 주대각선에 퀸이 있는지 기록한다\n    vector&lt;bool&gt; diags2(2 * n - 1, false); // 부대각선에 퀸이 있는지 기록한다\n    vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; res;\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.java<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nvoid backtrack(int row, int n, List&lt;List&lt;String&gt;&gt; state, List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n        boolean[] cols, boolean[] diags1, boolean[] diags2) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if (row == n) {\n        List&lt;List&lt;String&gt;&gt; copyState = new ArrayList&lt;&gt;();\n        for (List&lt;String&gt; sRow : state) {\n            copyState.add(new ArrayList&lt;&gt;(sRow));\n        }\n        res.add(copyState);\n        return;\n    }\n    // 모든 열을 순회한다\n    for (int col = 0; col &lt; n; col++) {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state.get(row).set(col, \"Q\");\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state.get(row).set(col, \"#\");\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    List&lt;List&lt;String&gt;&gt; state = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;String&gt; row = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            row.add(\"#\");\n        }\n        state.add(row);\n    }\n    boolean[] cols = new boolean[n]; // 열에 퀸이 있는지 기록한다\n    boolean[] diags1 = new boolean[2 * n - 1]; // 주대각선에 퀸이 있는지 기록한다\n    boolean[] diags2 = new boolean[2 * n - 1]; // 부대각선에 퀸이 있는지 기록한다\n    List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = new ArrayList&lt;&gt;();\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.cs<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nvoid Backtrack(int row, int n, List&lt;List&lt;string&gt;&gt; state, List&lt;List&lt;List&lt;string&gt;&gt;&gt; res,\n        bool[] cols, bool[] diags1, bool[] diags2) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if (row == n) {\n        List&lt;List&lt;string&gt;&gt; copyState = [];\n        foreach (List&lt;string&gt; sRow in state) {\n            copyState.Add(new List&lt;string&gt;(sRow));\n        }\n        res.Add(copyState);\n        return;\n    }\n    // 모든 열을 순회한다\n    for (int col = 0; col &lt; n; col++) {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // 다음 행에 배치한다\n            Backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nList&lt;List&lt;List&lt;string&gt;&gt;&gt; NQueens(int n) {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    List&lt;List&lt;string&gt;&gt; state = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;string&gt; row = [];\n        for (int j = 0; j &lt; n; j++) {\n            row.Add(\"#\");\n        }\n        state.Add(row);\n    }\n    bool[] cols = new bool[n]; // 열에 퀸이 있는지 기록한다\n    bool[] diags1 = new bool[2 * n - 1]; // 주대각선에 퀸이 있는지 기록한다\n    bool[] diags2 = new bool[2 * n - 1]; // 부대각선에 퀸이 있는지 기록한다\n    List&lt;List&lt;List&lt;string&gt;&gt;&gt; res = [];\n\n    Backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.go<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if row == n {\n        newState := make([][]string, len(*state))\n        for i, _ := range newState {\n            newState[i] = make([]string, len((*state)[0]))\n            copy(newState[i], (*state)[i])\n\n        }\n        *res = append(*res, newState)\n        return\n    }\n    // 모든 열을 순회한다\n    for col := 0; col &lt; n; col++ {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        diag1 := row - col + n - 1\n        diag2 := row + col\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if !(*cols)[col] &amp;&amp; !(*diags1)[diag1] &amp;&amp; !(*diags2)[diag2] {\n            // 시도: 해당 칸에 퀸을 놓는다\n            (*state)[row][col] = \"Q\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n            // 다음 행에 배치한다\n            backtrack(row+1, n, state, res, cols, diags1, diags2)\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            (*state)[row][col] = \"#\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nfunc nQueens(n int) [][][]string {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    state := make([][]string, n)\n    for i := 0; i &lt; n; i++ {\n        row := make([]string, n)\n        for i := 0; i &lt; n; i++ {\n            row[i] = \"#\"\n        }\n        state[i] = row\n    }\n    // 열에 퀸이 있는지 기록한다\n    cols := make([]bool, n)\n    diags1 := make([]bool, 2*n-1)\n    diags2 := make([]bool, 2*n-1)\n    res := make([][][]string, 0)\n    backtrack(0, n, &amp;state, &amp;res, &amp;cols, &amp;diags1, &amp;diags2)\n    return res\n}\n</code></pre> n_queens.swift<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nfunc backtrack(row: Int, n: Int, state: inout [[String]], res: inout [[[String]]], cols: inout [Bool], diags1: inout [Bool], diags2: inout [Bool]) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if row == n {\n        res.append(state)\n        return\n    }\n    // 모든 열을 순회한다\n    for col in 0 ..&lt; n {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        let diag1 = row - col + n - 1\n        let diag2 = row + col\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = \"Q\"\n            cols[col] = true\n            diags1[diag1] = true\n            diags2[diag2] = true\n            // 다음 행에 배치한다\n            backtrack(row: row + 1, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = \"#\"\n            cols[col] = false\n            diags1[diag1] = false\n            diags2[diag2] = false\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nfunc nQueens(n: Int) -&gt; [[[String]]] {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    var state = Array(repeating: Array(repeating: \"#\", count: n), count: n)\n    var cols = Array(repeating: false, count: n) // 열에 퀸이 있는지 기록한다\n    var diags1 = Array(repeating: false, count: 2 * n - 1) // 주대각선에 퀸이 있는지 기록한다\n    var diags2 = Array(repeating: false, count: 2 * n - 1) // 부대각선에 퀸이 있는지 기록한다\n    var res: [[[String]]] = []\n\n    backtrack(row: 0, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n\n    return res\n}\n</code></pre> n_queens.js<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nfunction backtrack(row, n, state, res, cols, diags1, diags2) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // 모든 열을 순회한다\n    for (let col = 0; col &lt; n; col++) {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nfunction nQueens(n) {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // 열에 퀸이 있는지 기록한다\n    const diags1 = Array(2 * n - 1).fill(false); // 주대각선에 퀸이 있는지 기록한다\n    const diags2 = Array(2 * n - 1).fill(false); // 부대각선에 퀸이 있는지 기록한다\n    const res = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.ts<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nfunction backtrack(\n    row: number,\n    n: number,\n    state: string[][],\n    res: string[][][],\n    cols: boolean[],\n    diags1: boolean[],\n    diags2: boolean[]\n): void {\n    // 모든 행에 배치했으면 해를 기록한다\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // 모든 열을 순회한다\n    for (let col = 0; col &lt; n; col++) {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nfunction nQueens(n: number): string[][][] {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // 열에 퀸이 있는지 기록한다\n    const diags1 = Array(2 * n - 1).fill(false); // 주대각선에 퀸이 있는지 기록한다\n    const diags2 = Array(2 * n - 1).fill(false); // 부대각선에 퀸이 있는지 기록한다\n    const res: string[][][] = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.dart<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nvoid backtrack(\n  int row,\n  int n,\n  List&lt;List&lt;String&gt;&gt; state,\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n  List&lt;bool&gt; cols,\n  List&lt;bool&gt; diags1,\n  List&lt;bool&gt; diags2,\n) {\n  // 모든 행에 배치했으면 해를 기록한다\n  if (row == n) {\n    List&lt;List&lt;String&gt;&gt; copyState = [];\n    for (List&lt;String&gt; sRow in state) {\n      copyState.add(List.from(sRow));\n    }\n    res.add(copyState);\n    return;\n  }\n  // 모든 열을 순회한다\n  for (int col = 0; col &lt; n; col++) {\n    // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n    int diag1 = row - col + n - 1;\n    int diag2 = row + col;\n    // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n    if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n      // 시도: 해당 칸에 퀸을 놓는다\n      state[row][col] = \"Q\";\n      cols[col] = true;\n      diags1[diag1] = true;\n      diags2[diag2] = true;\n      // 다음 행에 배치한다\n      backtrack(row + 1, n, state, res, cols, diags1, diags2);\n      // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n      state[row][col] = \"#\";\n      cols[col] = false;\n      diags1[diag1] = false;\n      diags2[diag2] = false;\n    }\n  }\n}\n\n/* n-퀸 문제 풀기 */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n  // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n  List&lt;List&lt;String&gt;&gt; state = List.generate(n, (index) =&gt; List.filled(n, \"#\"));\n  List&lt;bool&gt; cols = List.filled(n, false); // 열에 퀸이 있는지 기록한다\n  List&lt;bool&gt; diags1 = List.filled(2 * n - 1, false); // 주대각선에 퀸이 있는지 기록한다\n  List&lt;bool&gt; diags2 = List.filled(2 * n - 1, false); // 부대각선에 퀸이 있는지 기록한다\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = [];\n\n  backtrack(0, n, state, res, cols, diags1, diags2);\n\n  return res;\n}\n</code></pre> n_queens.rs<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nfn backtrack(\n    row: usize,\n    n: usize,\n    state: &amp;mut Vec&lt;Vec&lt;String&gt;&gt;,\n    res: &amp;mut Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt;,\n    cols: &amp;mut [bool],\n    diags1: &amp;mut [bool],\n    diags2: &amp;mut [bool],\n) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if row == n {\n        res.push(state.clone());\n        return;\n    }\n    // 모든 열을 순회한다\n    for col in 0..n {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        let diag1 = row + n - 1 - col;\n        let diag2 = row + col;\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = \"Q\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (true, true, true);\n            // 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = \"#\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (false, false, false);\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nfn n_queens(n: usize) -&gt; Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    let mut state: Vec&lt;Vec&lt;String&gt;&gt; = vec![vec![\"#\".to_string(); n]; n];\n    let mut cols = vec![false; n]; // 열에 퀸이 있는지 기록한다\n    let mut diags1 = vec![false; 2 * n - 1]; // 주대각선에 퀸이 있는지 기록한다\n    let mut diags2 = vec![false; 2 * n - 1]; // 부대각선에 퀸이 있는지 기록한다\n    let mut res: Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; = Vec::new();\n\n    backtrack(\n        0,\n        n,\n        &amp;mut state,\n        &amp;mut res,\n        &amp;mut cols,\n        &amp;mut diags1,\n        &amp;mut diags2,\n    );\n\n    res\n}\n</code></pre> n_queens.c<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nvoid backtrack(int row, int n, char state[MAX_SIZE][MAX_SIZE], char ***res, int *resSize, bool cols[MAX_SIZE],\n               bool diags1[2 * MAX_SIZE - 1], bool diags2[2 * MAX_SIZE - 1]) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if (row == n) {\n        res[*resSize] = (char **)malloc(sizeof(char *) * n);\n        for (int i = 0; i &lt; n; ++i) {\n            res[*resSize][i] = (char *)malloc(sizeof(char) * (n + 1));\n            strcpy(res[*resSize][i], state[i]);\n        }\n        (*resSize)++;\n        return;\n    }\n    // 모든 열을 순회한다\n    for (int col = 0; col &lt; n; col++) {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, resSize, cols, diags1, diags2);\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nchar ***nQueens(int n, int *returnSize) {\n    char state[MAX_SIZE][MAX_SIZE];\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    for (int i = 0; i &lt; n; ++i) {\n        for (int j = 0; j &lt; n; ++j) {\n            state[i][j] = '#';\n        }\n        state[i][n] = '\\0';\n    }\n    bool cols[MAX_SIZE] = {false};           // 열에 퀸이 있는지 기록한다\n    bool diags1[2 * MAX_SIZE - 1] = {false}; // 주대각선에 퀸이 있는지 기록한다\n    bool diags2[2 * MAX_SIZE - 1] = {false}; // 부대각선에 퀸이 있는지 기록한다\n\n    char ***res = (char ***)malloc(sizeof(char **) * MAX_SIZE);\n    *returnSize = 0;\n    backtrack(0, n, state, res, returnSize, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.kt<pre><code>/* 백트래킹 알고리즘: n-퀸 */\nfun backtrack(\n    row: Int,\n    n: Int,\n    state: MutableList&lt;MutableList&lt;String&gt;&gt;,\n    res: MutableList&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt;,\n    cols: BooleanArray,\n    diags1: BooleanArray,\n    diags2: BooleanArray\n) {\n    // 모든 행에 배치했으면 해를 기록한다\n    if (row == n) {\n        val copyState = mutableListOf&lt;MutableList&lt;String&gt;&gt;()\n        for (sRow in state) {\n            copyState.add(sRow.toMutableList())\n        }\n        res.add(copyState)\n        return\n    }\n    // 모든 열을 순회한다\n    for (col in 0..&lt;n) {\n        // 해당 칸이 속한 주대각선과 부대각선을 계산한다\n        val diag1 = row - col + n - 1\n        val diag2 = row + col\n        // 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // 시도: 해당 칸에 퀸을 놓는다\n            state[row][col] = \"Q\"\n            diags2[diag2] = true\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n            // 다음 행에 배치한다\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            // 되돌리기: 해당 칸을 빈칸으로 되돌린다\n            state[row][col] = \"#\"\n            diags2[diag2] = false\n            diags1[diag1] = diags2[diag2]\n            cols[col] = diags1[diag1]\n        }\n    }\n}\n\n/* n-퀸 문제 풀기 */\nfun nQueens(n: Int): MutableList&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt; {\n    // n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n    val state = mutableListOf&lt;MutableList&lt;String&gt;&gt;()\n    for (i in 0..&lt;n) {\n        val row = mutableListOf&lt;String&gt;()\n        for (j in 0..&lt;n) {\n            row.add(\"#\")\n        }\n        state.add(row)\n    }\n    val cols = BooleanArray(n) // 열에 퀸이 있는지 기록한다\n    val diags1 = BooleanArray(2 * n - 1) // 주대각선에 퀸이 있는지 기록한다\n    val diags2 = BooleanArray(2 * n - 1) // 부대각선에 퀸이 있는지 기록한다\n    val res = mutableListOf&lt;MutableList&lt;MutableList&lt;String&gt;&gt;?&gt;()\n\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n}\n</code></pre> n_queens.rb<pre><code>### 백트래킹 알고리즘: n-퀸 ###\ndef backtrack(row, n, state, res, cols, diags1, diags2)\n  # 모든 행에 배치했으면 해를 기록한다\n  if row == n\n    res &lt;&lt; state.map { |row| row.dup }\n    return\n  end\n\n  # 모든 열을 순회한다\n  for col in 0...n\n    # 해당 칸이 속한 주대각선과 부대각선을 계산한다\n    diag1 = row - col + n - 1\n    diag2 = row + col\n    # 가지치기: 해당 칸이 속한 열, 주대각선, 부대각선에 퀸이 있으면 안 된다\n    if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]\n      # 시도: 해당 칸에 퀸을 놓는다\n      state[row][col] = \"Q\"\n      cols[col] = diags1[diag1] = diags2[diag2] = true\n      # 다음 행에 배치한다\n      backtrack(row + 1, n, state, res, cols, diags1, diags2)\n      # 되돌리기: 해당 칸을 빈칸으로 되돌린다\n      state[row][col] = \"#\"\n      cols[col] = diags1[diag1] = diags2[diag2] = false\n    end\n  end\nend\n\n### n-퀸 문제 풀기 ###\ndef n_queens(n)\n  # n*n 크기의 체스판을 초기화한다. 'Q'는 퀸, '#'은 빈칸을 나타낸다\n  state = Array.new(n) { Array.new(n, \"#\") }\n  cols = Array.new(n, false) # 열에 퀸이 있는지 기록한다\n  diags1 = Array.new(2 * n - 1, false) # 주대각선에 퀸이 있는지 기록한다\n  diags2 = Array.new(2 * n - 1, false) # 부대각선에 퀸이 있는지 기록한다\n  res = []\n  backtrack(0, n, state, res, cols, diags1, diags2)\n\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>행별로 \\(n\\) 번 배치할 때 열 제약을 고려하면 첫 행부터 마지막 행까지 선택지 수가 \\(n\\), \\(n-1\\), \\(\\dots\\), \\(2\\), \\(1\\)이므로 \\(O(n!)\\) 시간이 걸린다. 해를 기록할 때 행렬 <code>state</code>를 복사하여 <code>res</code>에 추가해야 하며, 복사에는 \\(O(n^2)\\) 시간이 든다. 따라서 전체 시간 복잡도는 \\(O(n! \\cdot n^2)\\)이다. 실제로는 대각선 제약에 따른 가지치기도 탐색 공간을 크게 줄이므로 위 복잡도보다 탐색 효율이 좋은 경우가 많다.</p> <p>배열 <code>state</code>는 \\(O(n^2)\\) 공간을, <code>cols</code>, <code>diags1</code>, <code>diags2</code>는 각각 \\(O(n)\\) 공간을 사용한다. 최대 재귀 깊이가 \\(n\\)이므로 스택 프레임 공간은 \\(O(n)\\)이다. 따라서 공간 복잡도는 \\(O(n^2)\\)이다.</p>","path":["제 13 장   백트래킹","13.4   n-퀸 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/","level":1,"title":"13.2   모든 순열 문제","text":"<p>모든 순열 문제는 백트래킹 알고리즘의 대표적인 응용이다. 배열이나 문자열 같은 집합이 주어졌을 때 원소들의 가능한 모든 순열을 찾는 문제로 정의된다.</p> <p>아래 표(13-2)에는 입력 배열과 그에 대응하는 모든 순열의 예시가 나와 있다.</p> <p> 표 13-2   모든 순열의 예시 </p> 입력 배열 모든 순열 \\([1]\\) \\([1]\\) \\([1, 2]\\) \\([1, 2], [2, 1]\\) \\([1, 2, 3]\\) \\([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\\)","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1321","level":2,"title":"13.2.1   값이 같은 원소가 없는 경우","text":"<p>문제</p> <p>중복 원소가 없는 정수 배열을 입력받아 가능한 모든 순열을 반환하라.</p> <p>백트래킹 관점에서는 순열을 만드는 과정을 연속된 선택의 결과로 볼 수 있다. 입력 배열이 \\([1, 2, 3]\\)이고 \\(1\\), \\(3\\), \\(2\\) 순서로 선택하면 순열 \\([1, 3, 2]\\)를 얻는다. 되돌리기는 선택 하나를 취소한 뒤 다른 선택을 계속 시도한다는 뜻이다.</p> <p>백트래킹 코드에서 후보 집합 <code>choices</code>는 입력 배열의 모든 원소이고, 상태 <code>state</code>는 지금까지 선택한 원소들이다. 각 원소는 한 번만 선택할 수 있으므로 <code>state</code> 안의 원소는 모두 중복되지 않아야 한다.</p> <p>아래 그림(13-5)과 같이 탐색 과정을 재귀 트리로 펼칠 수 있다. 각 노드는 현재 상태 <code>state</code>를 나타낸다. 루트에서 시작해 세 차례 선택하면 리프에 도달하며, 각 리프 노드는 순열 하나에 대응한다.</p> <p></p> <p> 그림 13-5   모든 순열의 재귀 트리 </p>","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1","level":3,"title":"1.   중복 선택 가지치기","text":"<p>각 원소를 한 번만 선택하도록 불리언 배열 <code>selected</code>를 도입하자. <code>selected[i]</code>는 <code>choices[i]</code>를 이미 선택했는지 나타내며, 이를 바탕으로 다음과 같이 가지치기한다.</p> <ul> <li><code>choices[i]</code>를 선택하면 <code>selected[i]</code>를 \\(\\text{True}\\)로 설정하여 이미 선택했음을 표시한다.</li> <li>선택지 리스트 <code>choices</code>를 순회할 때 이미 선택한 모든 노드를 건너뛰어 가지치기한다.</li> </ul> <p>아래 그림(13-6)처럼 첫 번째에 1, 두 번째에 3, 세 번째에 2를 선택한다면, 두 번째에서는 원소 1의 가지를, 세 번째에서는 원소 1과 3의 가지를 잘라 내야 한다.</p> <p></p> <p> 그림 13-6   모든 순열의 가지치기 예시 </p> <p>위 그림(13-6)에서 이 가지치기가 탐색 공간의 크기를 \\(O(n^n)\\)에서 \\(O(n!)\\)로 줄이는 것을 볼 수 있다.</p>","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#2","level":3,"title":"2.   코드 구현","text":"<p>위 내용을 정리했으니 기본 틀 코드의 '빈칸을 채울' 수 있다. 전체 코드를 짧게 만들기 위해 각 함수를 따로 구현하지 않고 <code>backtrack()</code> 함수 안에 직접 작성한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby permutations_i.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"백트래킹 알고리즘: 모든 순열 I\"\"\"\n    # 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # 모든 선택지를 순회한다\n    for i, choice in enumerate(choices):\n        # 가지치기: 원소를 중복하여 선택할 수 없다\n        if not selected[i]:\n            # 시도: 선택하고 상태를 갱신한다\n            selected[i] = True\n            state.append(choice)\n            # 다음 선택을 진행한다\n            backtrack(state, choices, selected, res)\n            # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = False\n            state.pop()\n\ndef permutations_i(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"모든 순열 I\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_i.cpp<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if (!selected[i]) {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true;\n            state.push_back(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* 모든 순열 I */\nvector&lt;vector&lt;int&gt;&gt; permutationsI(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_i.java<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if (!selected[i]) {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true;\n            state.add(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* 모든 순열 I */\nList&lt;List&lt;Integer&gt;&gt; permutationsI(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_i.cs<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if (!selected[i]) {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true;\n            state.Add(choice);\n            // 다음 선택을 진행한다\n            Backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* 모든 순열 I */\nList&lt;List&lt;int&gt;&gt; PermutationsI(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_i.go<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nfunc backtrackI(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // 모든 선택지를 순회한다\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if !(*selected)[i] {\n            // 시도: 선택하고 상태를 갱신한다\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // 다음 선택을 진행한다\n            backtrackI(state, choices, selected, res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* 모든 순열 I */\nfunc permutationsI(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackI(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_i.swift<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // 모든 선택지를 순회한다\n    for (i, choice) in choices.enumerated() {\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if !selected[i] {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true\n            state.append(choice)\n            // 다음 선택을 진행한다\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* 모든 순열 I */\nfunc permutationsI(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_i.js<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nfunction backtrack(state, choices, selected, res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    choices.forEach((choice, i) =&gt; {\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if (!selected[i]) {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true;\n            state.push(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* 모든 순열 I */\nfunction permutationsI(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.ts<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    choices.forEach((choice, i) =&gt; {\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if (!selected[i]) {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true;\n            state.push(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* 모든 순열 I */\nfunction permutationsI(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.dart<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // 모든 선택지를 순회한다\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // 가지치기: 원소를 중복하여 선택할 수 없다\n    if (!selected[i]) {\n      // 시도: 선택하고 상태를 갱신한다\n      selected[i] = true;\n      state.add(choice);\n      // 다음 선택을 진행한다\n      backtrack(state, choices, selected, res);\n      // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* 모든 순열 I */\nList&lt;List&lt;int&gt;&gt; permutationsI(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_i.rs<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if !selected[i] {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true;\n            state.push(choice);\n            // 다음 선택을 진행한다\n            backtrack(state.clone(), choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop();\n        }\n    }\n}\n\n/* 모든 순열 I */\nfn permutations_i(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new(); // 상태(부분 집합)\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_i.c<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if (!selected[i]) {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true;\n            state[stateSize] = choice;\n            // 다음 선택을 진행한다\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n        }\n    }\n}\n\n/* 모든 순열 I */\nint **permutationsI(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_i.kt<pre><code>/* 백트래킹 알고리즘: 모든 순열 I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.size == choices.size) {\n        res.add(state.toMutableList())\n        return\n    }\n    // 모든 선택지를 순회한다\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // 가지치기: 원소를 중복하여 선택할 수 없다\n        if (!selected[i]) {\n            // 시도: 선택하고 상태를 갱신한다\n            selected[i] = true\n            state.add(choice)\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* 모든 순열 I */\nfun permutationsI(nums: IntArray): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;()\n    backtrack(mutableListOf(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_i.rb<pre><code>### 백트래킹 알고리즘: 모든 순열 I ###\ndef backtrack(state, choices, selected, res)\n  # 상태의 길이가 원소 개수와 같으면 해를 기록한다\n  if state.length == choices.length\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # 모든 선택지를 순회한다\n  choices.each_with_index do |choice, i|\n    # 가지치기: 원소를 중복하여 선택할 수 없다\n    unless selected[i]\n      # 시도: 선택하고 상태를 갱신한다\n      selected[i] = true\n      state &lt;&lt; choice\n      # 다음 선택을 진행한다\n      backtrack(state, choices, selected, res)\n      # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n      selected[i] = false\n      state.pop\n    end\n  end\nend\n\n### 모든 순열 I ###\ndef permutations_i(nums)\n  res = []\n  backtrack([], nums, Array.new(nums.length, false), res)\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1322","level":2,"title":"13.2.2   값이 같은 원소가 있는 경우","text":"<p>문제</p> <p>중복 원소가 있을 수 있는 정수 배열을 입력받아 중복되지 않는 모든 순열을 반환하라.</p> <p>입력 배열이 \\([1, 1, 2]\\)라고 하자. 두 원소 \\(1\\)을 쉽게 구분하기 위해 두 번째 \\(1\\)을 \\(\\hat{1}\\)로 표시한다.</p> <p>아래 그림(13-7)과 같이 앞의 방법으로 만든 순열의 절반은 중복된다.</p> <p></p> <p> 그림 13-7   중복된 순열 </p> <p>중복된 순열은 어떻게 제거할까? 가장 직접적으로는 해시 집합을 이용해 결과에서 중복을 제거할 수 있다. 하지만 이 방법은 충분히 간결하지 않다. 중복된 순열을 만드는 탐색 가지는 불필요하므로 미리 알아내어 잘라 내야 하며, 그러면 알고리즘의 효율을 더 높일 수 있다.</p>","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#1_1","level":3,"title":"1.   같은 값의 원소 가지치기","text":"<p>아래 그림(13-8)에서 첫 번째 선택으로 \\(1\\)을 고르든 \\(\\hat{1}\\)을 고르든 같다. 두 선택 아래에서 만들어지는 모든 순열이 중복되므로 \\(\\hat{1}\\)을 가지치기해야 한다.</p> <p>마찬가지로 첫 번째에 \\(2\\)를 선택한 뒤에도 두 번째의 \\(1\\)과 \\(\\hat{1}\\)이 중복 가지를 만들므로 두 번째의 \\(\\hat{1}\\)도 잘라 내야 한다.</p> <p>본질적으로 한 차례의 선택에서 값이 같은 여러 원소를 한 번만 선택하도록 보장하는 것이 목표다.</p> <p></p> <p> 그림 13-8   중복된 순열 가지치기 </p>","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#2_1","level":3,"title":"2.   코드 구현","text":"<p>앞 문제의 코드를 바탕으로 매 차례의 선택에서 해시 집합 <code>duplicated</code>를 만들어 이미 시도한 원소를 기록하고 중복 원소를 가지치기한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby permutations_ii.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"백트래킹 알고리즘: 모든 순열 II\"\"\"\n    # 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # 모든 선택지를 순회한다\n    duplicated = set[int]()\n    for i, choice in enumerate(choices):\n        # 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if not selected[i] and choice not in duplicated:\n            # 시도: 선택하고 상태를 갱신한다\n            duplicated.add(choice)  # 선택한 원소의 값을 기록한다\n            selected[i] = True\n            state.append(choice)\n            # 다음 선택을 진행한다\n            backtrack(state, choices, selected, res)\n            # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = False\n            state.pop()\n\ndef permutations_ii(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"모든 순열 II\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_ii.cpp<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    unordered_set&lt;int&gt; duplicated;\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if (!selected[i] &amp;&amp; duplicated.find(choice) == duplicated.end()) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.emplace(choice); // 선택한 원소의 값을 기록한다\n            selected[i] = true;\n            state.push_back(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* 모든 순열 II */\nvector&lt;vector&lt;int&gt;&gt; permutationsII(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_ii.java<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    Set&lt;Integer&gt; duplicated = new HashSet&lt;Integer&gt;();\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.add(choice); // 선택한 원소의 값을 기록한다\n            selected[i] = true;\n            state.add(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* 모든 순열 II */\nList&lt;List&lt;Integer&gt;&gt; permutationsII(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_ii.cs<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    HashSet&lt;int&gt; duplicated = [];\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if (!selected[i] &amp;&amp; !duplicated.Contains(choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.Add(choice); // 선택한 원소의 값을 기록한다\n            selected[i] = true;\n            state.Add(choice);\n            // 다음 선택을 진행한다\n            Backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* 모든 순열 II */\nList&lt;List&lt;int&gt;&gt; PermutationsII(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_ii.go<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nfunc backtrackII(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // 모든 선택지를 순회한다\n    duplicated := make(map[int]struct{}, 0)\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if _, ok := duplicated[choice]; !ok &amp;&amp; !(*selected)[i] {\n            // 시도: 선택하고 상태를 갱신한다\n            // 선택한 원소의 값을 기록한다\n            duplicated[choice] = struct{}{}\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // 다음 선택을 진행한다\n            backtrackII(state, choices, selected, res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* 모든 순열 II */\nfunc permutationsII(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackII(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_ii.swift<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // 모든 선택지를 순회한다\n    var duplicated: Set&lt;Int&gt; = []\n    for (i, choice) in choices.enumerated() {\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if !selected[i], !duplicated.contains(choice) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.insert(choice) // 선택한 원소의 값을 기록한다\n            selected[i] = true\n            state.append(choice)\n            // 다음 선택을 진행한다\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* 모든 순열 II */\nfunc permutationsII(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_ii.js<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nfunction backtrack(state, choices, selected, res) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.add(choice); // 선택한 원소의 값을 기록한다\n            selected[i] = true;\n            state.push(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* 모든 순열 II */\nfunction permutationsII(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.ts<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.add(choice); // 선택한 원소의 값을 기록한다\n            selected[i] = true;\n            state.push(choice);\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* 모든 순열 II */\nfunction permutationsII(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.dart<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // 모든 선택지를 순회한다\n  Set&lt;int&gt; duplicated = {};\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n    if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n      // 시도: 선택하고 상태를 갱신한다\n      duplicated.add(choice); // 선택한 원소의 값을 기록한다\n      selected[i] = true;\n      state.add(choice);\n      // 다음 선택을 진행한다\n      backtrack(state, choices, selected, res);\n      // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* 모든 순열 II */\nList&lt;List&lt;int&gt;&gt; permutationsII(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_ii.rs<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    let mut duplicated = HashSet::&lt;i32&gt;::new();\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if !selected[i] &amp;&amp; !duplicated.contains(&amp;choice) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.insert(choice); // 선택한 원소의 값을 기록한다\n            selected[i] = true;\n            state.push(choice);\n            // 다음 선택을 진행한다\n            backtrack(state.clone(), choices, selected, res);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n            state.pop();\n        }\n    }\n}\n\n/* 모든 순열 II */\nfn permutations_ii(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new();\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_ii.c<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // 모든 선택지를 순회한다\n    bool duplicated[MAX_SIZE] = {false};\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if (!selected[i] &amp;&amp; !duplicated[choice]) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated[choice] = true; // 선택한 원소의 값을 기록한다\n            selected[i] = true;\n            state[stateSize] = choice;\n            // 다음 선택을 진행한다\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false;\n        }\n    }\n}\n\n/* 모든 순열 II */\nint **permutationsII(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_ii.kt<pre><code>/* 백트래킹 알고리즘: 모든 순열 II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    choices: IntArray,\n    selected: BooleanArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // 상태의 길이가 원소 개수와 같으면 해를 기록한다\n    if (state.size == choices.size) {\n        res.add(state.toMutableList())\n        return\n    }\n    // 모든 선택지를 순회한다\n    val duplicated = HashSet&lt;Int&gt;()\n    for (i in choices.indices) {\n        val choice = choices[i]\n        // 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // 시도: 선택하고 상태를 갱신한다\n            duplicated.add(choice) // 선택한 원소의 값을 기록한다\n            selected[i] = true\n            state.add(choice)\n            // 다음 선택을 진행한다\n            backtrack(state, choices, selected, res)\n            // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n            selected[i] = false\n            state.removeAt(state.size - 1)\n        }\n    }\n}\n\n/* 모든 순열 II */\nfun permutationsII(nums: IntArray): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;()\n    backtrack(mutableListOf(), nums, BooleanArray(nums.size), res)\n    return res\n}\n</code></pre> permutations_ii.rb<pre><code>### 백트래킹 알고리즘: 모든 순열 II ###\ndef backtrack(state, choices, selected, res)\n  # 상태의 길이가 원소 개수와 같으면 해를 기록한다\n  if state.length == choices.length\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # 모든 선택지를 순회한다\n  duplicated = Set.new\n  choices.each_with_index do |choice, i|\n    # 가지치기: 원소를 중복하여 선택할 수 없으며, 값이 같은 원소도 중복하여 선택할 수 없다\n    if !selected[i] &amp;&amp; !duplicated.include?(choice)\n      # 시도: 선택하고 상태를 갱신한다\n      duplicated.add(choice)\n      selected[i] = true\n      state &lt;&lt; choice\n      # 다음 선택을 진행한다\n      backtrack(state, choices, selected, res)\n      # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n      selected[i] = false\n      state.pop\n    end\n  end\nend\n\n### 모든 순열 II ###\ndef permutations_ii(nums)\n  res = []\n  backtrack([], nums, Array.new(nums.length, false), res)\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>모든 원소가 서로 다르다면 \\(n\\) 개 원소의 순열은 \\(n!\\) 가지(팩토리얼)다. 결과를 기록할 때마다 길이 \\(n\\)의 리스트를 복사하는 데 \\(O(n)\\) 시간이 걸린다. 따라서 시간 복잡도는 \\(O(n!n)\\)이다.</p> <p>최대 재귀 깊이는 \\(n\\)이므로 \\(O(n)\\)의 스택 프레임 공간을 사용한다. <code>selected</code>는 \\(O(n)\\) 공간을 사용한다. 동시에 최대 \\(n\\) 개의 <code>duplicated</code>가 존재하여 \\(O(n^2)\\) 공간을 사용한다. 따라서 공간 복잡도는 \\(O(n^2)\\)이다.</p>","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/permutations_problem/#3","level":3,"title":"3.   두 가지 가지치기 비교","text":"<p><code>selected</code>와 <code>duplicated</code>는 모두 가지치기에 쓰이지만 목적이 다르다.</p> <ul> <li>중복 선택 가지치기: 전체 탐색 과정에 <code>selected</code>는 하나만 존재한다. 현재 상태에 어떤 원소가 포함되어 있는지를 기록하여 같은 원소가 <code>state</code>에 중복 등장하지 않게 한다.</li> <li>같은 값의 원소 가지치기: 매 차례의 선택(호출된 각 <code>backtrack</code> 함수)마다 <code>duplicated</code>가 하나씩 있다. 이번 순회(<code>for</code> 반복문)에서 이미 선택한 원소를 기록하여 같은 값의 원소를 한 번만 선택하게 한다.</li> </ul> <p>아래 그림(13-9)은 두 가지 가지치기 조건이 적용되는 범위를 보여 준다. 각 트리 노드는 선택 하나를 나타내며, 루트에서 리프까지의 경로에 있는 노드들이 하나의 순열을 이룬다.</p> <p></p> <p> 그림 13-9   두 가지 가지치기 조건의 적용 범위 </p>","path":["제 13 장   백트래킹","13.2   모든 순열 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/","level":1,"title":"13.3   부분 집합 합 문제","text":"","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1331","level":2,"title":"13.3.1   중복 원소가 없는 경우","text":"<p>문제</p> <p>양의 정수 배열 <code>nums</code>와 목표 양의 정수 <code>target</code>이 주어진다. 원소의 합이 <code>target</code>인 모든 조합을 찾아라. 배열에는 중복 원소가 없고, 각 원소는 여러 번 선택할 수 있다. 조합들을 리스트로 반환하되 중복 조합이 없어야 한다.</p> <p>예를 들어 입력 집합이 \\(\\{3, 4, 5\\}\\)이고 목표 정수가 \\(9\\)이면 해는 \\(\\{3, 3, 3\\}, \\{4, 5\\}\\)다. 다음 두 가지에 주의하자.</p> <ul> <li>입력 집합의 원소는 횟수 제한 없이 반복 선택할 수 있다.</li> <li>부분 집합에서는 원소의 순서를 구분하지 않는다. 예를 들어 \\(\\{4, 5\\}\\)와 \\(\\{5, 4\\}\\)는 같은 부분 집합이다.</li> </ul>","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1","level":3,"title":"1.   모든 순열 풀이 참고","text":"<p>모든 순열 문제처럼 부분 집합을 만드는 과정을 연속된 선택의 결과로 볼 수 있다. 선택할 때마다 '원소의 합'을 갱신하고, 합이 <code>target</code>이 되면 해당 부분 집합을 결과 리스트에 기록한다.</p> <p>모든 순열 문제와 달리 이 문제에서는 집합의 원소를 횟수 제한 없이 선택할 수 있다. 따라서 선택 여부를 기록하는 불리언 리스트 <code>selected</code>가 필요하지 않다. 모든 순열 코드를 조금 수정하면 다음과 같은 첫 풀이를 얻는다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby subset_sum_i_naive.py<pre><code>def backtrack(\n    state: list[int],\n    target: int,\n    total: int,\n    choices: list[int],\n    res: list[list[int]],\n):\n    \"\"\"백트래킹 알고리즘: 부분 집합 합 I\"\"\"\n    # 부분 집합의 합이 target과 같으면 해를 기록한다\n    if total == target:\n        res.append(list(state))\n        return\n    # 모든 선택지를 순회한다\n    for i in range(len(choices)):\n        # 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if total + choices[i] &gt; target:\n            continue\n        # 시도: 선택하고 원소의 합 total을 갱신한다\n        state.append(choices[i])\n        # 다음 선택을 진행한다\n        backtrack(state, target, total + choices[i], choices, res)\n        # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop()\n\ndef subset_sum_i_naive(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"부분 집합 합 I 풀기(중복 부분 집합 포함)\"\"\"\n    state = []  # 상태(부분 집합)\n    total = 0  # 부분 집합의 합\n    res = []  # 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, total, nums, res)\n    return res\n</code></pre> subset_sum_i_naive.cpp<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, int total, vector&lt;int&gt; &amp;choices, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (total == target) {\n        res.push_back(state);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (size_t i = 0; i &lt; choices.size(); i++) {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.push_back(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target, total + choices[i], choices, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop_back();\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nvector&lt;vector&lt;int&gt;&gt; subsetSumINaive(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;       // 상태(부분 집합)\n    int total = 0;           // 부분 집합의 합\n    vector&lt;vector&lt;int&gt;&gt; res; // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.java<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int total, int[] choices, List&lt;List&lt;Integer&gt;&gt; res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (total == target) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; choices.length; i++) {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.add(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target, total + choices[i], choices, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.remove(state.size() - 1);\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nList&lt;List&lt;Integer&gt;&gt; subsetSumINaive(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // 상태(부분 집합)\n    int total = 0; // 부분 집합의 합\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.cs<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid Backtrack(List&lt;int&gt; state, int target, int total, int[] choices, List&lt;List&lt;int&gt;&gt; res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (total == target) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; choices.Length; i++) {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.Add(choices[i]);\n        // 다음 선택을 진행한다\n        Backtrack(state, target, total + choices[i], choices, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nList&lt;List&lt;int&gt;&gt; SubsetSumINaive(int[] nums, int target) {\n    List&lt;int&gt; state = []; // 상태(부분 집합)\n    int total = 0; // 부분 집합의 합\n    List&lt;List&lt;int&gt;&gt; res = []; // 결과 리스트(부분 집합 리스트)\n    Backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.go<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunc backtrackSubsetSumINaive(total, target int, state, choices *[]int, res *[][]int) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == total {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // 모든 선택지를 순회한다\n    for i := 0; i &lt; len(*choices); i++ {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if total+(*choices)[i] &gt; target {\n            continue\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        *state = append(*state, (*choices)[i])\n        // 다음 선택을 진행한다\n        backtrackSubsetSumINaive(total+(*choices)[i], target, state, choices, res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nfunc subsetSumINaive(nums []int, target int) [][]int {\n    state := make([]int, 0) // 상태(부분 집합)\n    total := 0              // 부분 집합의 합\n    res := make([][]int, 0) // 결과 리스트(부분 집합 리스트)\n    backtrackSubsetSumINaive(total, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.swift<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunc backtrack(state: inout [Int], target: Int, total: Int, choices: [Int], res: inout [[Int]]) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if total == target {\n        res.append(state)\n        return\n    }\n    // 모든 선택지를 순회한다\n    for i in choices.indices {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if total + choices[i] &gt; target {\n            continue\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.append(choices[i])\n        // 다음 선택을 진행한다\n        backtrack(state: &amp;state, target: target, total: total + choices[i], choices: choices, res: &amp;res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.removeLast()\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nfunc subsetSumINaive(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // 상태(부분 집합)\n    let total = 0 // 부분 집합의 합\n    var res: [[Int]] = [] // 결과 리스트(부분 집합 리스트)\n    backtrack(state: &amp;state, target: target, total: total, choices: nums, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.js<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunction backtrack(state, target, total, choices, res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (let i = 0; i &lt; choices.length; i++) {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target, total + choices[i], choices, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nfunction subsetSumINaive(nums, target) {\n    const state = []; // 상태(부분 집합)\n    const total = 0; // 부분 집합의 합\n    const res = []; // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.ts<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    total: number,\n    choices: number[],\n    res: number[][]\n): void {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (let i = 0; i &lt; choices.length; i++) {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target, total + choices[i], choices, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nfunction subsetSumINaive(nums: number[], target: number): number[][] {\n    const state = []; // 상태(부분 집합)\n    const total = 0; // 부분 집합의 합\n    const res = []; // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.dart<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  int total,\n  List&lt;int&gt; choices,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // 부분 집합의 합이 target과 같으면 해를 기록한다\n  if (total == target) {\n    res.add(List.from(state));\n    return;\n  }\n  // 모든 선택지를 순회한다\n  for (int i = 0; i &lt; choices.length; i++) {\n    // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n    if (total + choices[i] &gt; target) {\n      continue;\n    }\n    // 시도: 선택하고 원소의 합 total을 갱신한다\n    state.add(choices[i]);\n    // 다음 선택을 진행한다\n    backtrack(state, target, total + choices[i], choices, res);\n    // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n    state.removeLast();\n  }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nList&lt;List&lt;int&gt;&gt; subsetSumINaive(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // 상태(부분 집합)\n  int total = 0; // 원소의 합\n  List&lt;List&lt;int&gt;&gt; res = []; // 결과 리스트(부분 집합 리스트)\n  backtrack(state, target, total, nums, res);\n  return res;\n}\n</code></pre> subset_sum_i_naive.rs<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    total: i32,\n    choices: &amp;[i32],\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if total == target {\n        res.push(state.clone());\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for i in 0..choices.len() {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if total + choices[i] &gt; target {\n            continue;\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target, total + choices[i], choices, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nfn subset_sum_i_naive(nums: &amp;[i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // 상태(부분 집합)\n    let total = 0; // 부분 집합의 합\n    let mut res = Vec::new(); // 결과 리스트(부분 집합 리스트)\n    backtrack(&amp;mut state, target, total, nums, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i_naive.c<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(int target, int total, int *choices, int choicesSize) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (total == target) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; choicesSize; i++) {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state[stateSize++] = choices[i];\n        // 다음 선택을 진행한다\n        backtrack(target, total + choices[i], choices, choicesSize);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        stateSize--;\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nvoid subsetSumINaive(int *nums, int numsSize, int target) {\n    resSize = 0; // 해의 개수를 0으로 초기화\n    backtrack(target, 0, nums, numsSize);\n}\n</code></pre> subset_sum_i_naive.kt<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    total: Int,\n    choices: IntArray,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (total == target) {\n        res.add(state.toMutableList())\n        return\n    }\n    // 모든 선택지를 순회한다\n    for (i in choices.indices) {\n        // 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n        if (total + choices[i] &gt; target) {\n            continue\n        }\n        // 시도: 선택하고 원소의 합 total을 갱신한다\n        state.add(choices[i])\n        // 다음 선택을 진행한다\n        backtrack(state, target, total + choices[i], choices, res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* 부분 집합 합 I 풀기(중복 부분 집합 포함) */\nfun subsetSumINaive(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // 상태(부분 집합)\n    val total = 0 // 부분 집합의 합\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, total, nums, res)\n    return res\n}\n</code></pre> subset_sum_i_naive.rb<pre><code>### 백트래킹 알고리즘: 부분 집합 합 I ###\ndef backtrack(state, target, total, choices, res)\n  # 부분 집합의 합이 target과 같으면 해를 기록한다\n  if total == target\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # 모든 선택지를 순회한다\n  for i in 0...choices.length\n    # 가지치기: 부분 집합의 합이 target을 초과하면 해당 선택을 건너뛴다\n    next if total + choices[i] &gt; target\n    # 시도: 선택하고 원소의 합 total을 갱신한다\n    state &lt;&lt; choices[i]\n    # 다음 선택을 진행한다\n    backtrack(state, target, total + choices[i], choices, res)\n    # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n    state.pop\n  end\nend\n\n### 부분 집합 합 I 풀기(중복 부분 집합 포함)###\ndef subset_sum_i_naive(nums, target)\n  state = [] # 상태(부분 집합)\n  total = 0 # 부분 집합의 합\n  res = [] # 결과 리스트(부분 집합 리스트)\n  backtrack(state, target, total, nums, res)\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>위 코드에 배열 \\([3, 4, 5]\\)와 목표 원소 \\(9\\)를 넣으면 \\([3, 3, 3], [4, 5], [5, 4]\\)가 나온다. 합이 \\(9\\)인 모든 부분 집합을 찾았지만 \\([4, 5]\\)와 \\([5, 4]\\)는 중복이다.</p> <p>탐색 과정에서는 선택 순서를 구분하지만 부분 집합에서는 구분하지 않기 때문이다. 아래 그림(13-10)에서 \\(4\\) 다음 \\(5\\)를 고르는 것과 \\(5\\) 다음 \\(4\\)를 고르는 것은 서로 다른 가지이지만 같은 부분 집합에 대응한다.</p> <p></p> <p> 그림 13-10   부분 집합 탐색과 범위 초과 가지치기 </p> <p>중복 부분 집합을 제거하는 직접적인 방법은 결과 리스트에서 중복을 제거하는 것이다. 하지만 다음 두 이유로 매우 비효율적이다.</p> <ul> <li>배열 원소가 많고 특히 <code>target</code>이 크면 탐색 과정에서 중복 부분 집합이 매우 많이 생긴다.</li> <li>부분 집합(배열)이 같은지 비교하는 데 시간이 많이 든다. 먼저 배열을 정렬한 뒤 원소를 하나씩 비교해야 한다.</li> </ul>","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#2","level":3,"title":"2.   중복 부분 집합 가지치기","text":"<p>탐색 중 가지치기로 중복을 제거해 보자. 아래 그림(13-11)에서 중복 부분 집합은 배열 원소를 서로 다른 순서로 선택할 때 생긴다. 예를 들면 다음과 같다.</p> <ol> <li>첫 번째와 두 번째에 각각 \\(3\\)과 \\(4\\)를 선택하면 이 두 원소를 포함하는 모든 부분 집합 \\([3, 4, \\dots]\\)가 만들어진다.</li> <li>이후 첫 번째에 \\(4\\)를 선택하면 두 번째에서는 \\(3\\)을 건너뛰어야 한다. 이 선택으로 만드는 \\([4, 3, \\dots]\\)는 <code>1.</code> 단계에서 만든 부분 집합과 완전히 중복되기 때문이다.</li> </ol> <p>탐색에서 각 레벨의 선택은 왼쪽부터 오른쪽 순서로 시도하므로, 오른쪽으로 갈수록 잘라 내는 가지가 많아진다.</p> <ol> <li>첫 두 차례에 \\(3\\)과 \\(5\\)를 선택하여 부분 집합 \\([3, 5, \\dots]\\)를 만든다.</li> <li>첫 두 차례에 \\(4\\)와 \\(5\\)를 선택하여 부분 집합 \\([4, 5, \\dots]\\)를 만든다.</li> <li>첫 번째에 \\(5\\)를 선택한다면 두 번째에서는 \\(3\\)과 \\(4\\)를 건너뛰어야 한다. \\([5, 3, \\dots]\\)와 \\([5, 4, \\dots]\\)는 <code>1.</code> 단계와 <code>2.</code> 단계의 부분 집합과 완전히 중복되기 때문이다.</li> </ol> <p></p> <p> 그림 13-11   선택 순서의 차이로 생기는 중복 부분 집합 </p> <p>정리하면 입력 배열이 \\([x_1, x_2, \\dots, x_n]\\)일 때 선택 순서 \\([x_{i_1}, x_{i_2}, \\dots, x_{i_m}]\\)는 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\)을 만족해야 한다. 이 조건을 만족하지 않는 선택 순서는 중복을 만들므로 가지치기해야 한다.</p>","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#3","level":3,"title":"3.   코드 구현","text":"<p>이 가지치기를 위해 순회 시작점을 나타내는 변수 <code>start</code>를 초기화한다. \\(x_{i}\\)를 선택한 뒤 다음 차례는 인덱스 \\(i\\)부터 순회하도록 설정한다. 그러면 선택 순서가 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\)을 만족하여 부분 집합의 유일성을 보장한다.</p> <p>이 밖에도 다음 두 가지를 최적화했다.</p> <ul> <li>탐색을 시작하기 전에 배열 <code>nums</code>를 정렬한다. 선택지를 순회하다가 부분 집합의 합이 <code>target</code>을 넘으면 반복문을 바로 종료한다. 뒤의 원소들은 더 크므로 합도 반드시 <code>target</code>을 넘기 때문이다.</li> <li>원소 합 변수 <code>total</code>을 없애고 <code>target</code>에서 값을 빼는 방식으로 합을 추적한다. <code>target</code>이 \\(0\\)이 되면 해를 기록한다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby subset_sum_i.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"백트래킹 알고리즘: 부분 집합 합 I\"\"\"\n    # 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0:\n        res.append(list(state))\n        return\n    # 모든 선택지를 순회한다\n    # 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for i in range(start, len(choices)):\n        # 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        # 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target - choices[i] &lt; 0:\n            break\n        # 시도: 선택하고 target, start를 갱신한다\n        state.append(choices[i])\n        # 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i, res)\n        # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop()\n\ndef subset_sum_i(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"부분 집합 합 I 풀기\"\"\"\n    state = []  # 상태(부분 집합)\n    nums.sort()  # nums를 정렬한다\n    start = 0  # 순회 시작 위치\n    res = []  # 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_i.cpp<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for (int i = start; i &lt; choices.size(); i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push_back(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop_back();\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nvector&lt;vector&lt;int&gt;&gt; subsetSumI(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // 상태(부분 집합)\n    sort(nums.begin(), nums.end()); // nums를 정렬한다\n    int start = 0;                  // 순회 시작 위치\n    vector&lt;vector&lt;int&gt;&gt; res;        // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.java<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for (int i = start; i &lt; choices.length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.add(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.remove(state.size() - 1);\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nList&lt;List&lt;Integer&gt;&gt; subsetSumI(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // 상태(부분 집합)\n    Arrays.sort(nums); // nums를 정렬한다\n    int start = 0; // 순회 시작 위치\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.cs<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for (int i = start; i &lt; choices.Length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.Add(choices[i]);\n        // 다음 선택을 진행한다\n        Backtrack(state, target - choices[i], choices, i, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nList&lt;List&lt;int&gt;&gt; SubsetSumI(int[] nums, int target) {\n    List&lt;int&gt; state = []; // 상태(부분 집합)\n    Array.Sort(nums); // nums를 정렬한다\n    int start = 0; // 순회 시작 위치\n    List&lt;List&lt;int&gt;&gt; res = []; // 결과 리스트(부분 집합 리스트)\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.go<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunc backtrackSubsetSumI(start, target int, state, choices *[]int, res *[][]int) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for i := start; i &lt; len(*choices); i++ {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        *state = append(*state, (*choices)[i])\n        // 다음 선택을 진행한다\n        backtrackSubsetSumI(i, target-(*choices)[i], state, choices, res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nfunc subsetSumI(nums []int, target int) [][]int {\n    state := make([]int, 0) // 상태(부분 집합)\n    sort.Ints(nums)         // nums를 정렬한다\n    start := 0              // 순회 시작 위치\n    res := make([][]int, 0) // 결과 리스트(부분 집합 리스트)\n    backtrackSubsetSumI(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.swift<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for i in choices.indices.dropFirst(start) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.append(choices[i])\n        // 다음 선택을 진행한다\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i, res: &amp;res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.removeLast()\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nfunc subsetSumI(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // 상태(부분 집합)\n    let nums = nums.sorted() // nums를 정렬한다\n    let start = 0 // 순회 시작 위치\n    var res: [[Int]] = [] // 결과 리스트(부분 집합 리스트)\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.js<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunction backtrack(state, target, choices, start, res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for (let i = start; i &lt; choices.length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nfunction subsetSumI(nums, target) {\n    const state = []; // 상태(부분 집합)\n    nums.sort((a, b) =&gt; a - b); // nums를 정렬한다\n    const start = 0; // 순회 시작 위치\n    const res = []; // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.ts<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for (let i = start; i &lt; choices.length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nfunction subsetSumI(nums: number[], target: number): number[][] {\n    const state = []; // 상태(부분 집합)\n    nums.sort((a, b) =&gt; a - b); // nums를 정렬한다\n    const start = 0; // 순회 시작 위치\n    const res = []; // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.dart<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // 부분 집합의 합이 target과 같으면 해를 기록한다\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // 모든 선택지를 순회한다\n  // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n  for (int i = start; i &lt; choices.length; i++) {\n    // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n    // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // 시도: 선택하고 target, start를 갱신한다\n    state.add(choices[i]);\n    // 다음 선택을 진행한다\n    backtrack(state, target - choices[i], choices, i, res);\n    // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n    state.removeLast();\n  }\n}\n\n/* 부분 집합 합 I 풀기 */\nList&lt;List&lt;int&gt;&gt; subsetSumI(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // 상태(부분 집합)\n  nums.sort(); // nums를 정렬한다\n  int start = 0; // 순회 시작 위치\n  List&lt;List&lt;int&gt;&gt; res = []; // 결과 리스트(부분 집합 리스트)\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_i.rs<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0 {\n        res.push(state.clone());\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for i in start..choices.len() {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nfn subset_sum_i(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // 상태(부분 집합)\n    nums.sort(); // nums를 정렬한다\n    let start = 0; // 순회 시작 위치\n    let mut res = Vec::new(); // 결과 리스트(부분 집합 리스트)\n    backtrack(&amp;mut state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i.c<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; ++i) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for (int i = start; i &lt; choicesSize; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state[stateSize] = choices[i];\n        stateSize++;\n        // 다음 선택을 진행한다\n        backtrack(target - choices[i], choices, choicesSize, i);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        stateSize--;\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nvoid subsetSumI(int *nums, int numsSize, int target) {\n    qsort(nums, numsSize, sizeof(int), cmp); // nums를 정렬한다\n    int start = 0;                           // 순회 시작 위치\n    backtrack(target, nums, numsSize, start);\n}\n</code></pre> subset_sum_i.kt<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 I */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.add(state.toMutableList())\n        return\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    for (i in start..&lt;choices.size) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.add(choices[i])\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i, res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* 부분 집합 합 I 풀기 */\nfun subsetSumI(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // 상태(부분 집합)\n    nums.sort() // nums를 정렬한다\n    val start = 0 // 순회 시작 위치\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_i.rb<pre><code>### 백트래킹 알고리즘: 부분 집합 합 I ###\ndef backtrack(state, target, choices, start, res)\n  # 부분 집합의 합이 target과 같으면 해를 기록한다\n  if target.zero?\n    res &lt;&lt; state.dup\n    return\n  end\n  # 모든 선택지를 순회한다\n  # 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n  for i in start...choices.length\n    # 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n    # 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n    break if target - choices[i] &lt; 0\n    # 시도: 선택하고 target, start를 갱신한다\n    state &lt;&lt; choices[i]\n    # 다음 선택을 진행한다\n    backtrack(state, target - choices[i], choices, i, res)\n    # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n    state.pop\n  end\nend\n\n### 부분 집합 합 I 풀기 ###\ndef subset_sum_i(nums, target)\n  state = [] # 상태(부분 집합)\n  nums.sort! # nums를 정렬한다\n  start = 0 # 순회 시작 위치\n  res = [] # 결과 리스트(부분 집합 리스트)\n  backtrack(state, target, nums, start, res)\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(13-12)은 배열 \\([3, 4, 5]\\)와 목표 원소 \\(9\\)를 위 코드에 넣었을 때의 전체 백트래킹 과정이다.</p> <p></p> <p> 그림 13-12   부분 집합 합 I의 백트래킹 과정 </p>","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1332","level":2,"title":"13.3.2   중복 원소가 있는 경우","text":"<p>문제</p> <p>양의 정수 배열 <code>nums</code>와 목표 양의 정수 <code>target</code>이 주어진다. 원소의 합이 <code>target</code>인 모든 조합을 찾아라. 배열에는 중복 원소가 있을 수 있고, 각 원소는 한 번만 선택할 수 있다. 조합들을 리스트로 반환하되 중복 조합이 없어야 한다.</p> <p>앞 문제와 달리 입력 배열에 중복 원소가 있을 수 있어 새로운 문제가 생긴다. 예를 들어 배열 \\([4, \\hat{4}, 5]\\)와 목표 원소 \\(9\\)를 기존 코드에 넣으면 \\([4, 5], [\\hat{4}, 5]\\)가 나와 부분 집합이 중복된다.</p> <p>중복의 원인은 값이 같은 원소를 한 차례의 선택에서 여러 번 고르기 때문이다. 아래 그림(13-13)에서 첫 차례의 세 선택지 중 두 개가 \\(4\\)여서 중복된 두 탐색 가지가 만들어지고, 같은 부분 집합을 출력한다. 마찬가지로 두 번째 차례의 두 \\(4\\)도 중복을 만든다.</p> <p></p> <p> 그림 13-13   같은 값의 원소로 생기는 중복 부분 집합 </p>","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#1_1","level":3,"title":"1.   같은 값의 원소 가지치기","text":"<p>이를 해결하려면 각 차례에서 같은 값의 원소를 한 번만 선택하도록 제한해야 한다. 구현 방법은 간단하면서도 기발하다. 배열이 정렬되어 있으므로 같은 원소끼리는 인접한다. 따라서 한 차례의 선택에서 현재 원소가 왼쪽 원소와 같다면 이미 선택한 값이라는 뜻이므로 바로 건너뛴다.</p> <p>또한 이 문제에서는 각 배열 원소를 한 번만 선택할 수 있다. 다행히 <code>start</code>로 이 제약도 만족시킬 수 있다. \\(x_{i}\\)를 선택한 뒤 다음 차례는 인덱스 \\(i + 1\\)부터 배열의 끝을 향해 순회하도록 설정한다. 그러면 중복 부분 집합과 같은 원소의 반복 선택을 모두 막을 수 있다.</p>","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/subset_sum_problem/#2_1","level":3,"title":"2.   코드 구현","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby subset_sum_ii.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"백트래킹 알고리즘: 부분 집합 합 II\"\"\"\n    # 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0:\n        res.append(list(state))\n        return\n    # 모든 선택지를 순회한다\n    # 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    # 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for i in range(start, len(choices)):\n        # 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        # 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target - choices[i] &lt; 0:\n            break\n        # 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if i &gt; start and choices[i] == choices[i - 1]:\n            continue\n        # 시도: 선택하고 target, start를 갱신한다\n        state.append(choices[i])\n        # 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop()\n\ndef subset_sum_ii(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"부분 집합 합 II 풀기\"\"\"\n    state = []  # 상태(부분 집합)\n    nums.sort()  # nums를 정렬한다\n    start = 0  # 순회 시작 위치\n    res = []  # 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_ii.cpp<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for (int i = start; i &lt; choices.size(); i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push_back(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop_back();\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nvector&lt;vector&lt;int&gt;&gt; subsetSumII(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // 상태(부분 집합)\n    sort(nums.begin(), nums.end()); // nums를 정렬한다\n    int start = 0;                  // 순회 시작 위치\n    vector&lt;vector&lt;int&gt;&gt; res;        // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.java<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for (int i = start; i &lt; choices.length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.add(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.remove(state.size() - 1);\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nList&lt;List&lt;Integer&gt;&gt; subsetSumII(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // 상태(부분 집합)\n    Arrays.sort(nums); // nums를 정렬한다\n    int start = 0; // 순회 시작 위치\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.cs<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for (int i = start; i &lt; choices.Length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.Add(choices[i]);\n        // 다음 선택을 진행한다\n        Backtrack(state, target - choices[i], choices, i + 1, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nList&lt;List&lt;int&gt;&gt; SubsetSumII(int[] nums, int target) {\n    List&lt;int&gt; state = []; // 상태(부분 집합)\n    Array.Sort(nums); // nums를 정렬한다\n    int start = 0; // 순회 시작 위치\n    List&lt;List&lt;int&gt;&gt; res = []; // 결과 리스트(부분 집합 리스트)\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.go<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nfunc backtrackSubsetSumII(start, target int, state, choices *[]int, res *[][]int) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for i := start; i &lt; len(*choices); i++ {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if i &gt; start &amp;&amp; (*choices)[i] == (*choices)[i-1] {\n            continue\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        *state = append(*state, (*choices)[i])\n        // 다음 선택을 진행한다\n        backtrackSubsetSumII(i+1, target-(*choices)[i], state, choices, res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nfunc subsetSumII(nums []int, target int) [][]int {\n    state := make([]int, 0) // 상태(부분 집합)\n    sort.Ints(nums)         // nums를 정렬한다\n    start := 0              // 순회 시작 위치\n    res := make([][]int, 0) // 결과 리스트(부분 집합 리스트)\n    backtrackSubsetSumII(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.swift<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for i in choices.indices.dropFirst(start) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if i &gt; start, choices[i] == choices[i - 1] {\n            continue\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.append(choices[i])\n        // 다음 선택을 진행한다\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i + 1, res: &amp;res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.removeLast()\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nfunc subsetSumII(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // 상태(부분 집합)\n    let nums = nums.sorted() // nums를 정렬한다\n    let start = 0 // 순회 시작 위치\n    var res: [[Int]] = [] // 결과 리스트(부분 집합 리스트)\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.js<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nfunction backtrack(state, target, choices, start, res) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for (let i = start; i &lt; choices.length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nfunction subsetSumII(nums, target) {\n    const state = []; // 상태(부분 집합)\n    nums.sort((a, b) =&gt; a - b); // nums를 정렬한다\n    const start = 0; // 순회 시작 위치\n    const res = []; // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.ts<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for (let i = start; i &lt; choices.length; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nfunction subsetSumII(nums: number[], target: number): number[][] {\n    const state = []; // 상태(부분 집합)\n    nums.sort((a, b) =&gt; a - b); // nums를 정렬한다\n    const start = 0; // 순회 시작 위치\n    const res = []; // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.dart<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // 부분 집합의 합이 target과 같으면 해를 기록한다\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // 모든 선택지를 순회한다\n  // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n  // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n  for (int i = start; i &lt; choices.length; i++) {\n    // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n    // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n    if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n      continue;\n    }\n    // 시도: 선택하고 target, start를 갱신한다\n    state.add(choices[i]);\n    // 다음 선택을 진행한다\n    backtrack(state, target - choices[i], choices, i + 1, res);\n    // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n    state.removeLast();\n  }\n}\n\n/* 부분 집합 합 II 풀기 */\nList&lt;List&lt;int&gt;&gt; subsetSumII(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // 상태(부분 집합)\n  nums.sort(); // nums를 정렬한다\n  int start = 0; // 순회 시작 위치\n  List&lt;List&lt;int&gt;&gt; res = []; // 결과 리스트(부분 집합 리스트)\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_ii.rs<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nfn backtrack(\n    state: &amp;mut Vec&lt;i32&gt;,\n    target: i32,\n    choices: &amp;[i32],\n    start: usize,\n    res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;,\n) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if target == 0 {\n        res.push(state.clone());\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for i in start..choices.len() {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if i &gt; start &amp;&amp; choices[i] == choices[i - 1] {\n            continue;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.push(choices[i]);\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.pop();\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nfn subset_sum_ii(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut state = Vec::new(); // 상태(부분 집합)\n    nums.sort(); // nums를 정렬한다\n    let start = 0; // 순회 시작 위치\n    let mut res = Vec::new(); // 결과 리스트(부분 집합 리스트)\n    backtrack(&amp;mut state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_ii.c<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for (int i = start; i &lt; choicesSize; i++) {\n        // 가지치기 1: 부분 집합의 합이 target을 넘으면 바로 건너뛴다\n        if (target - choices[i] &lt; 0) {\n            continue;\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state[stateSize] = choices[i];\n        stateSize++;\n        // 다음 선택을 진행한다\n        backtrack(target - choices[i], choices, choicesSize, i + 1);\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        stateSize--;\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nvoid subsetSumII(int *nums, int numsSize, int target) {\n    // nums를 정렬한다\n    qsort(nums, numsSize, sizeof(int), cmp);\n    // 백트래킹 시작\n    backtrack(target, nums, numsSize, 0);\n}\n</code></pre> subset_sum_ii.kt<pre><code>/* 백트래킹 알고리즘: 부분 집합 합 II */\nfun backtrack(\n    state: MutableList&lt;Int&gt;,\n    target: Int,\n    choices: IntArray,\n    start: Int,\n    res: MutableList&lt;MutableList&lt;Int&gt;?&gt;\n) {\n    // 부분 집합의 합이 target과 같으면 해를 기록한다\n    if (target == 0) {\n        res.add(state.toMutableList())\n        return\n    }\n    // 모든 선택지를 순회한다\n    // 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n    // 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n    for (i in start..&lt;choices.size) {\n        // 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n        // 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n        if (target - choices[i] &lt; 0) {\n            break\n        }\n        // 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue\n        }\n        // 시도: 선택하고 target, start를 갱신한다\n        state.add(choices[i])\n        // 다음 선택을 진행한다\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        // 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n        state.removeAt(state.size - 1)\n    }\n}\n\n/* 부분 집합 합 II 풀기 */\nfun subsetSumII(nums: IntArray, target: Int): MutableList&lt;MutableList&lt;Int&gt;?&gt; {\n    val state = mutableListOf&lt;Int&gt;() // 상태(부분 집합)\n    nums.sort() // nums를 정렬한다\n    val start = 0 // 순회 시작 위치\n    val res = mutableListOf&lt;MutableList&lt;Int&gt;?&gt;() // 결과 리스트(부분 집합 리스트)\n    backtrack(state, target, nums, start, res)\n    return res\n}\n</code></pre> subset_sum_ii.rb<pre><code>### 백트래킹 알고리즘: 부분 집합 합 II ###\ndef backtrack(state, target, choices, start, res)\n  # 부분 집합의 합이 target과 같으면 해를 기록한다\n  if target.zero?\n    res &lt;&lt; state.dup\n    return\n  end\n\n  # 모든 선택지를 순회한다\n  # 가지치기 2: start부터 순회하여 중복 부분 집합 생성을 방지한다\n  # 가지치기 3: start부터 순회하여 같은 원소의 중복 선택을 방지한다\n  for i in start...choices.length\n    # 가지치기 1: 부분 집합의 합이 target을 초과하면 즉시 반복문을 종료한다\n    # 배열이 정렬되어 있어 뒤의 원소가 더 크므로, 부분 집합의 합이 반드시 target을 초과하기 때문이다\n    break if target - choices[i] &lt; 0\n    # 가지치기 4: 해당 원소가 왼쪽 원소와 같으면 탐색 분기가 중복되므로 바로 건너뛴다\n    next if i &gt; start &amp;&amp; choices[i] == choices[i - 1]\n    # 시도: 선택하고 target, start를 갱신한다\n    state &lt;&lt; choices[i]\n    # 다음 선택을 진행한다\n    backtrack(state, target - choices[i], choices, i + 1, res)\n    # 되돌리기: 선택을 취소하고 이전 상태로 되돌린다\n    state.pop\n  end\nend\n\n### 부분 집합 합 II 풀기 ###\ndef subset_sum_ii(nums, target)\n  state = [] # 상태(부분 집합)\n  nums.sort! # nums를 정렬한다\n  start = 0 # 순회 시작 위치\n  res = [] # 결과 리스트(부분 집합 리스트)\n  backtrack(state, target, nums, start, res)\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(13-14)은 배열 \\([4, 4, 5]\\)와 목표 원소 \\(9\\)의 백트래킹 과정으로, 네 가지 가지치기를 포함한다. 그림과 코드 주석을 함께 보면서 전체 탐색 과정과 각 가지치기가 어떻게 작동하는지 이해해 보자.</p> <p></p> <p> 그림 13-14   부분 집합 합 II의 백트래킹 과정 </p>","path":["제 13 장   백트래킹","13.3   부분 집합 합 문제"],"tags":[]},{"location":"chapter_backtracking/summary/","level":1,"title":"13.5   요약","text":"","path":["제 13 장   백트래킹","13.5   요약"],"tags":[]},{"location":"chapter_backtracking/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>백트래킹은 본질적으로 모든 경우를 열거하는 방법이다. 해 공간을 깊이 우선으로 순회하여 조건에 맞는 해를 찾고 기록하며, 모든 해를 찾거나 순회가 끝나면 종료한다.</li> <li>백트래킹의 탐색 과정은 시도와 되돌리기로 이루어진다. 깊이 우선 탐색으로 여러 선택을 시도하고, 제약 조건을 만족하지 않으면 직전 선택을 취소하여 이전 상태로 돌아간 뒤 다른 선택을 계속 시도한다. 시도와 되돌리기는 서로 반대 방향의 연산이다.</li> <li>백트래킹 문제에는 보통 여러 제약 조건이 있고, 이를 가지치기에 사용할 수 있다. 가지치기는 불필요한 탐색 가지를 일찍 종료하여 효율을 크게 높인다.</li> <li>백트래킹은 주로 탐색 문제와 제약 만족 문제에 사용한다. 조합 최적화 문제도 해결할 수 있지만, 더 효율적이거나 더 좋은 결과를 내는 풀이가 있는 경우가 많다.</li> <li>모든 순열 문제는 주어진 집합 원소들의 가능한 모든 순열을 찾는 것이 목표다. 배열로 각 원소의 선택 여부를 기록하고 동일 원소를 중복 선택하는 가지를 잘라 내어 각 원소를 한 번만 선택하도록 한다.</li> <li>모든 순열 문제에서 집합에 중복 원소가 있으면 결과에 중복된 순열이 생긴다. 각 차례에서 같은 값의 원소를 한 번만 선택하도록 제한해야 하며, 보통 해시 집합으로 구현한다.</li> <li>부분 집합 합 문제는 주어진 집합에서 합이 목표 값인 모든 부분 집합을 찾는다. 집합은 원소 순서를 구분하지 않지만 탐색은 모든 순서의 결과를 출력하므로 중복이 생긴다. 백트래킹 전에 데이터를 정렬하고 각 차례의 순회 시작점을 변수로 지정하여 중복 부분 집합을 만드는 가지를 잘라 낸다.</li> <li>부분 집합 합 문제에서는 배열의 같은 값 원소가 중복 집합을 만들 수 있다. 배열이 정렬되어 있다는 전제를 이용해 인접 원소의 값이 같은지 확인하여 가지치기하고, 같은 값을 매 차례 한 번만 선택하도록 한다.</li> <li>\\(n\\)-퀸 문제는 \\(n \\times n\\) 체스판에 퀸 \\(n\\) 개를 서로 공격하지 않게 배치하는 문제다. 행, 열, 주대각선 방향, 부대각선 방향의 제약이 있다. 행 제약을 만족하려고 행별 배치 전략을 사용하여 각 행에 퀸 하나를 놓는다.</li> <li>열 제약과 대각선 제약은 비슷하게 처리한다. 열 제약은 배열에 각 열의 퀸 존재 여부를 기록하여 선택한 칸이 유효한지 판단한다. 대각선 제약은 두 배열로 각 방향 대각선의 퀸 존재 여부를 기록한다. 핵심은 같은 주대각선(부대각선) 방향 대각선의 칸들이 만족하는 행과 열 인덱스의 규칙을 찾는 것이다.</li> </ul>","path":["제 13 장   백트래킹","13.5   요약"],"tags":[]},{"location":"chapter_backtracking/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 백트래킹과 재귀의 관계는 어떻게 이해하면 되는가?</p> <p>백트래킹은 '알고리즘 전략'이고, 재귀는 '도구'에 더 가깝다.</p> <ul> <li>백트래킹은 보통 재귀로 구현한다. 백트래킹은 재귀의 활용 사례 중 하나이며, 탐색 문제에 재귀를 적용한 것이다.</li> <li>재귀의 구조는 '부분 문제로 분해'하는 풀이 패턴을 보여 주며, 분할 정복, 백트래킹, 동적 계획법(메모이제이션을 이용한 재귀) 등의 문제에 자주 사용한다.</li> </ul>","path":["제 13 장   백트래킹","13.5   요약"],"tags":[]},{"location":"chapter_computational_complexity/","level":1,"title":"제 2 장   복잡도 분석","text":"<p>이 장을 시작하며</p> <p>복잡도 분석은 광활한 알고리즘 우주에서 시간과 공간을 안내하는 길잡이와 같다.</p> <p>시간과 공간이라는 두 차원을 깊이 탐구하며 더 우아한 해법을 찾도록 이끌어 준다.</p>","path":["제 2 장   복잡도 분석"],"tags":[]},{"location":"chapter_computational_complexity/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>2.1   알고리즘 효율 평가</li> <li>2.2   반복과 재귀</li> <li>2.3   시간 복잡도</li> <li>2.4   공간 복잡도</li> <li>2.5   요약</li> <li>2.6   연습 문제</li> </ul>","path":["제 2 장   복잡도 분석"],"tags":[]},{"location":"chapter_computational_complexity/exercises/","level":1,"title":"2.6   연습 문제","text":"","path":["제 2 장   복잡도 분석","2.6   연습 문제"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#261","level":2,"title":"2.6.1   개념 확인","text":"","path":["제 2 장   복잡도 분석","2.6   연습 문제"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#1","level":3,"title":"1.   반복과 재귀의 시간과 공간","text":"<p>다음 두 코드는 모두 \\(1 + 2 + \\dots + n\\)을 계산한다(\\(n \\ge 1\\)로 가정한다). <code>n</code>을 4로 설정하고, 프로그램이 실제로 실행되는 순서에 따라 질문에 답한 뒤 두 방식의 효율성을 비교해 보자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby complexity_exercises.py<pre><code>def sum_iter(n: int) -&gt; int:\n    \"\"\"반복으로 합 구하기\"\"\"\n    res = 0\n    for i in range(1, n + 1):\n        res += i\n    return res\n\ndef sum_recur(n: int) -&gt; int:\n    \"\"\"재귀로 합 구하기\"\"\"\n    if n == 1:\n        return 1\n    return n + sum_recur(n - 1)\n</code></pre> complexity_exercises.cpp<pre><code>/* 반복으로 합 구하기 */\nint sumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; ++i) {\n        res += i;\n    }\n    return res;\n}\n\n/* 재귀로 합 구하기 */\nint sumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.java<pre><code>/* 반복으로 합 구하기 */\nint sumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 재귀로 합 구하기 */\nint sumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.cs<pre><code>/* 반복으로 합 구하기 */\nint SumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 재귀로 합 구하기 */\nint SumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + SumRecur(n - 1);\n}\n</code></pre> complexity_exercises.go<pre><code>/* 반복으로 합 구하기 */\nfunc sumIter(n int) int {\n    res := 0\n    for i := 1; i &lt;= n; i++ {\n        res += i\n    }\n    return res\n}\n\n/* 재귀로 합 구하기 */\nfunc sumRecur(n int) int {\n    if n == 1 {\n        return 1\n    }\n    return n + sumRecur(n-1)\n}\n</code></pre> complexity_exercises.swift<pre><code>/* 반복으로 합 구하기 */\nfunc sumIter(n: Int) -&gt; Int {\n    var res = 0\n    for i in 1 ... n {\n        res += i\n    }\n    return res\n}\n\n/* 재귀로 합 구하기 */\nfunc sumRecur(n: Int) -&gt; Int {\n    if n == 1 {\n        return 1\n    }\n    return n + sumRecur(n: n - 1)\n}\n</code></pre> complexity_exercises.js<pre><code>/* 반복으로 합 구하기 */\nfunction sumIter(n) {\n    let res = 0;\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 재귀로 합 구하기 */\nfunction sumRecur(n) {\n    if (n === 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.ts<pre><code>/* 반복으로 합 구하기 */\nfunction sumIter(n: number): number {\n    let res = 0;\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 재귀로 합 구하기 */\nfunction sumRecur(n: number): number {\n    if (n === 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.dart<pre><code>/* 반복으로 합 구하기 */\nint sumIter(int n) {\n  int res = 0;\n  for (int i = 1; i &lt;= n; i++) {\n    res += i;\n  }\n  return res;\n}\n\n/* 재귀로 합 구하기 */\nint sumRecur(int n) {\n  if (n == 1) {\n    return 1;\n  }\n  return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.rs<pre><code>/* 반복으로 합 구하기 */\nfn sum_iter(n: i32) -&gt; i32 {\n    let mut res = 0;\n    for i in 1..=n {\n        res += i;\n    }\n    res\n}\n\n/* 재귀로 합 구하기 */\nfn sum_recur(n: i32) -&gt; i32 {\n    if n == 1 {\n        return 1;\n    }\n    n + sum_recur(n - 1)\n}\n</code></pre> complexity_exercises.c<pre><code>/* 반복으로 합 구하기 */\nint sumIter(int n) {\n    int res = 0;\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 재귀로 합 구하기 */\nint sumRecur(int n) {\n    if (n == 1) {\n        return 1;\n    }\n    return n + sumRecur(n - 1);\n}\n</code></pre> complexity_exercises.kt<pre><code>/* 반복으로 합 구하기 */\nfun sumIter(n: Int): Int {\n    var res = 0\n    for (i in 1..n) {\n        res += i\n    }\n    return res\n}\n\n/* 재귀로 합 구하기 */\nfun sumRecur(n: Int): Int {\n    if (n == 1) {\n        return 1\n    }\n    return n + sumRecur(n - 1)\n}\n</code></pre> complexity_exercises.rb<pre><code>### 반복으로 합 구하기 ###\ndef sum_iter(n)\n  res = 0\n  for i in 1..n\n    res += i\n  end\n  res\nend\n\n### 재귀로 합 구하기 ###\ndef sum_recur(n)\n  return 1 if n == 1\n\n  n + sum_recur(n - 1)\nend\n</code></pre> <ol> <li><code>n = 4</code>를 입력하여 반복 함수를 실행할 때, 각 반복이 끝난 뒤 누적 변수 <code>res</code>의 값은 각각 얼마인가?</li> <li><code>n = 4</code>를 입력하여 재귀 함수를 실행할 때, 매개변수 <code>n</code>은 어떤 값을 순서대로 갖는가? 가장 깊은 호출부터 반환할 때 결과는 어떻게 구해지는가?</li> <li>두 방식의 시간 복잡도와 공간 복잡도는 각각 얼마인가? 1번과 2번의 실행 과정을 바탕으로 이유를 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>반복 변수 <code>i</code>는 차례대로 <code>1, 2, 3, 4</code>가 되고, 각 반복이 끝날 때 <code>res</code>는     <code>1, 3, 6, 10</code>으로 변한다. 따라서 반복 함수는 10을 반환한다.</p> </li> <li> <p>매개변수 <code>n</code>은 <code>4 → 3 → 2 → 1</code> 순서로 변한다.     가장 깊은 호출이 1을 반환한 뒤, 각 호출은 차례대로 <code>2 + 1 = 3</code>, <code>3 + 3 = 6</code>, <code>4 + 6 = 10</code>을 얻는다.     가장 깊은 지점에서는 4번의 함수 호출이 모두 아직 끝나지 않은 상태다.</p> </li> <li> <p>두 코드 모두 \\(n\\)에 비례하는 횟수만큼 반복하거나 호출하므로 시간 복잡도는 모두 \\(O(n)\\)이다.     공간 복잡도는 다르다. 반복 방식은 상수 개의 변수만 사용하므로 \\(O(1)\\)이다.     재귀 방식은 종료 조건에 도달할 때까지 앞선 호출들이 반환값을 기다려야 하므로 호출 스택에 최대 \\(n\\)번의 호출이 동시에 저장된다.     따라서 공간 복잡도는 \\(O(n)\\)이다.</p> <p>공간 복잡도를 분석할 때는 코드에 선언된 변수뿐 아니라 재귀 호출이 사용하는 공간도 고려해야 한다.</p> </li> </ol>","path":["제 2 장   복잡도 분석","2.6   연습 문제"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#2","level":3,"title":"2.   세 코드의 시간 복잡도","text":"<p>다음 세 코드 조각은 모두 양의 정수 \\(n\\)을 입력받는다. 시간 복잡도가 낮은 순서대로 나열하고 각각의 복잡도를 적어 보자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby complexity_exercises.py<pre><code>def linear_loop(n: int) -&gt; int:\n    \"\"\"선형 차수 반복문\"\"\"\n    res = 0\n    for i in range(n):\n        res += i\n    return res\n\ndef quadratic_loop(n: int) -&gt; int:\n    \"\"\"제곱 차수 반복문\"\"\"\n    res = 0\n    for i in range(n):\n        for j in range(i, n):\n            res += j\n    return res\n\ndef logarithmic_loop(n: int) -&gt; int:\n    \"\"\"로그 차수 반복문\"\"\"\n    while n &gt; 1:\n        n //= 2\n    return n\n</code></pre> complexity_exercises.cpp<pre><code>/* 선형 차수 반복문 */\nint linearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; ++i) {\n        res += i;\n    }\n    return res;\n}\n\n/* 제곱 차수 반복문 */\nint quadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; ++i) {\n        for (int j = i; j &lt; n; ++j) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* 로그 차수 반복문 */\nint logarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.java<pre><code>/* 선형 차수 반복문 */\nint linearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 제곱 차수 반복문 */\nint quadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* 로그 차수 반복문 */\nint logarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.cs<pre><code>/* 선형 차수 반복문 */\nint LinearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 제곱 차수 반복문 */\nint QuadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* 로그 차수 반복문 */\nint LogarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.go<pre><code>/* 선형 차수 반복문 */\nfunc linearLoop(n int) int {\n    res := 0\n    for i := 0; i &lt; n; i++ {\n        res += i\n    }\n    return res\n}\n\n/* 제곱 차수 반복문 */\nfunc quadraticLoop(n int) int {\n    res := 0\n    for i := 0; i &lt; n; i++ {\n        for j := i; j &lt; n; j++ {\n            res += j\n        }\n    }\n    return res\n}\n\n/* 로그 차수 반복문 */\nfunc logarithmicLoop(n int) int {\n    for n &gt; 1 {\n        n /= 2\n    }\n    return n\n}\n</code></pre> complexity_exercises.swift<pre><code>/* 선형 차수 반복문 */\nfunc linearLoop(n: Int) -&gt; Int {\n    var res = 0\n    for i in 0 ..&lt; n {\n        res += i\n    }\n    return res\n}\n\n/* 제곱 차수 반복문 */\nfunc quadraticLoop(n: Int) -&gt; Int {\n    var res = 0\n    for i in 0 ..&lt; n {\n        for j in i ..&lt; n {\n            res += j\n        }\n    }\n    return res\n}\n\n/* 로그 차수 반복문 */\nfunc logarithmicLoop(n: Int) -&gt; Int {\n    var n = n\n    while n &gt; 1 {\n        n /= 2\n    }\n    return n\n}\n</code></pre> complexity_exercises.js<pre><code>/* 선형 차수 반복문 */\nfunction linearLoop(n) {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 제곱 차수 반복문 */\nfunction quadraticLoop(n) {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* 로그 차수 반복문 */\nfunction logarithmicLoop(n) {\n    while (n &gt; 1) {\n        n = Math.floor(n / 2);\n    }\n    return n;\n}\n</code></pre> complexity_exercises.ts<pre><code>/* 선형 차수 반복문 */\nfunction linearLoop(n: number): number {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 제곱 차수 반복문 */\nfunction quadraticLoop(n: number): number {\n    let res = 0;\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* 로그 차수 반복문 */\nfunction logarithmicLoop(n: number): number {\n    while (n &gt; 1) {\n        n = Math.floor(n / 2);\n    }\n    return n;\n}\n</code></pre> complexity_exercises.dart<pre><code>/* 선형 차수 반복문 */\nint linearLoop(int n) {\n  int res = 0;\n  for (int i = 0; i &lt; n; i++) {\n    res += i;\n  }\n  return res;\n}\n\n/* 제곱 차수 반복문 */\nint quadraticLoop(int n) {\n  int res = 0;\n  for (int i = 0; i &lt; n; i++) {\n    for (int j = i; j &lt; n; j++) {\n      res += j;\n    }\n  }\n  return res;\n}\n\n/* 로그 차수 반복문 */\nint logarithmicLoop(int n) {\n  while (n &gt; 1) {\n    n ~/= 2;\n  }\n  return n;\n}\n</code></pre> complexity_exercises.rs<pre><code>/* 선형 차수 반복문 */\nfn linear_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    for i in 0..n {\n        res += i;\n    }\n    res\n}\n\n/* 제곱 차수 반복문 */\nfn quadratic_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    for i in 0..n {\n        for j in i..n {\n            res += j;\n        }\n    }\n    res\n}\n\n/* 로그 차수 반복문 */\nfn logarithmic_loop(mut n: i32) -&gt; i32 {\n    while n &gt; 1 {\n        n /= 2;\n    }\n    n\n}\n</code></pre> complexity_exercises.c<pre><code>/* 선형 차수 반복문 */\nint linearLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        res += i;\n    }\n    return res;\n}\n\n/* 제곱 차수 반복문 */\nint quadraticLoop(int n) {\n    int res = 0;\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = i; j &lt; n; j++) {\n            res += j;\n        }\n    }\n    return res;\n}\n\n/* 로그 차수 반복문 */\nint logarithmicLoop(int n) {\n    while (n &gt; 1) {\n        n /= 2;\n    }\n    return n;\n}\n</code></pre> complexity_exercises.kt<pre><code>/* 선형 차수 반복문 */\nfun linearLoop(n: Int): Int {\n    var res = 0\n    for (i in 0 until n) {\n        res += i\n    }\n    return res\n}\n\n/* 제곱 차수 반복문 */\nfun quadraticLoop(n: Int): Int {\n    var res = 0\n    for (i in 0 until n) {\n        for (j in i until n) {\n            res += j\n        }\n    }\n    return res\n}\n\n/* 로그 차수 반복문 */\nfun logarithmicLoop(n: Int): Int {\n    var value = n\n    while (value &gt; 1) {\n        value /= 2\n    }\n    return value\n}\n</code></pre> complexity_exercises.rb<pre><code>### 선형 차수 반복문 ###\ndef linear_loop(n)\n  res = 0\n  for i in 0...n\n    res += i\n  end\n  res\nend\n\n### 제곱 차수 반복문 ###\ndef quadratic_loop(n)\n  res = 0\n  for i in 0...n\n    for j in i...n\n      res += j\n    end\n  end\n  res\nend\n\n### 로그 차수 반복문 ###\ndef logarithmic_loop(n)\n  n /= 2 while n &gt; 1\n  n\nend\n</code></pre> 참고 답안 <p>낮은 순서대로 코드 3은 \\(O(\\log n)\\), 코드 1은 \\(O(n)\\), 코드 2는 \\(O(n^2)\\)이다. 코드 3은 매번 \\(n\\)을 절반으로 줄이므로 반복 횟수는 약 \\(\\log_2 n\\)번이다. 코드 1의 반복문은 정확히 \\(n\\)번 실행된다. 코드 2의 안쪽 반복 횟수는 차례대로 \\(n,n-1,\\dots,1\\)이므로 총 \\(n(n+1)/2\\)번이며, 제곱 차수에 해당한다.</p>","path":["제 2 장   복잡도 분석","2.6   연습 문제"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#3","level":3,"title":"3.   어느 뒤집기 방식이 공간을 더 절약하는가","text":"<p>배열 <code>nums</code>의 모든 원소를 역순으로 뒤집는 두 가지 방법이 있다.</p> <ol> <li>길이가 같은 새 배열 <code>res</code>를 만들고 원소를 역순으로 복사한 뒤 반환한다.</li> <li> <p>두 인덱스 <code>i</code>와 <code>j</code>를 각각 처음과 끝에서 가운데로 이동시키며 <code>nums[i]</code>와 <code>nums[j]</code>를 쌍으로 교환한다.</p> <p>두 방법의 공간 복잡도는 각각 얼마인가? 어느 방법이 ‘제자리’ 연산에 해당하는가?</p> </li> </ol> 참고 답안 <ol> <li> <p>입력과 길이가 같은 보조 배열이 필요하므로 공간 복잡도는 \\(O(n)\\)이다.</p> </li> <li> <p>인덱스 변수 두 개만 사용하므로     공간 복잡도는 \\(O(1)\\)이며, 제자리 연산이다.</p> <p>주의할 점은 제자리 뒤집기가 입력 배열을 변경한다는 것이다. 입력을 변경해도 되는 경우에만 우선적으로 선택해야 한다. 원본 배열을 보존해야 한다면 첫 번째 방법의 복사 비용은 피할 수 없다.</p> </li> </ol>","path":["제 2 장   복잡도 분석","2.6   연습 문제"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#262","level":2,"title":"2.6.2   프로그래밍 연습","text":"","path":["제 2 장   복잡도 분석","2.6   연습 문제"],"tags":[]},{"location":"chapter_computational_complexity/exercises/#1_1","level":3,"title":"1.   피보나치 수","text":"<p>피보나치 수열은 \\(F(0)=0\\), \\(F(1)=1\\)이며, \\(n\\ge2\\)일 때 \\(F(n)=F(n-1)+F(n-2)\\)를 만족한다.</p> <p>음이 아닌 정수 <code>n</code>이 주어질 때, 재귀를 사용하지 않고 반복문으로 \\(F(n)\\)을 계산하여 반환하라.</p> 풀이 힌트 <ol> <li>n이 0 또는 1인 경우를 먼저 따로 처리한다.</li> <li>다음 항을 계산할 때는 앞선 두 항만 필요하므로 수열 전체를 저장할 필요가 없다.</li> <li>두 변수를 갱신할 때 아직 사용해야 하는 이전 값을 너무 일찍 덮어쓰지 않도록 주의한다.</li> </ol> <p>LeetCode</p>","path":["제 2 장   복잡도 분석","2.6   연습 문제"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/","level":1,"title":"2.2   반복과 재귀","text":"<p>알고리즘에서는 특정 작업을 반복해서 실행하는 일이 흔하며, 이는 복잡도 분석과 밀접하게 연결되어 있다. 따라서 시간 복잡도와 공간 복잡도를 소개하기 전에, 프로그램에서 작업을 반복 실행하는 두 가지 기본 제어 구조인 반복과 재귀부터 알아보자.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#221","level":2,"title":"2.2.1   반복","text":"<p>반복(iteration)은 특정 작업을 거듭 실행하는 제어 구조다. 반복에서는 일정한 조건을 만족하는 동안 코드 일부를 계속 실행하다가, 조건을 더 이상 만족하지 않으면 멈춘다.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for","level":3,"title":"1.   for 반복문","text":"<p><code>for</code> 반복문은 가장 흔한 반복 형식 중 하나로, 반복 횟수를 미리 알고 있을 때 적합하다.</p> <p>다음 함수는 <code>for</code> 반복문으로 \\(1 + 2 + \\dots + n\\)을 계산하고, 그 합을 변수 <code>res</code>에 기록한다. Python의 <code>range(a, b)</code>는 왼쪽 끝을 포함하고 오른쪽 끝은 포함하지 않는 구간이므로, 순회 범위는 \\(a, a + 1, \\dots, b-1\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def for_loop(n: int) -&gt; int:\n    \"\"\"for 반복문\"\"\"\n    res = 0\n    # 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for i in range(1, n + 1):\n        res += i\n    return res\n</code></pre> iteration.cpp<pre><code>/* for 반복문 */\nint forLoop(int n) {\n    int res = 0;\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for (int i = 1; i &lt;= n; ++i) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* for 반복문 */\nint forLoop(int n) {\n    int res = 0;\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* for 반복문 */\nint ForLoop(int n) {\n    int res = 0;\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* for 반복문 */\nfunc forLoop(n int) int {\n    res := 0\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for i := 1; i &lt;= n; i++ {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* for 반복문 */\nfunc forLoop(n: Int) -&gt; Int {\n    var res = 0\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for i in 1 ... n {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* for 반복문 */\nfunction forLoop(n) {\n    let res = 0;\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* for 반복문 */\nfunction forLoop(n: number): number {\n    let res = 0;\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* for 반복문 */\nint forLoop(int n) {\n  int res = 0;\n  // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n  for (int i = 1; i &lt;= n; i++) {\n    res += i;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* for 반복문 */\nfn for_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for i in 1..=n {\n        res += i;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* for 반복문 */\nint forLoop(int n) {\n    int res = 0;\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* for 반복문 */\nfun forLoop(n: Int): Int {\n    var res = 0\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for (i in 1..n) {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### for 반복문 ###\ndef for_loop(n)\n  res = 0\n\n  # 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n  for i in 1..n\n    res += i\n  end\n\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(2-1)은 이 합계 함수의 순서도다.</p> <p></p> <p> 그림 2-1   합계 함수의 순서도 </p> <p>이 합계 함수의 연산 횟수는 입력 데이터 크기 \\(n\\)에 비례한다. 이를 ‘선형 관계’라고도 한다. 실제로 시간 복잡도가 설명하는 것이 바로 이 ‘선형 관계’다. 관련 내용은 다음 절에서 자세히 소개한다.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while","level":3,"title":"2.   while 반복문","text":"<p><code>for</code> 반복문과 마찬가지로 <code>while</code> 반복문도 반복을 구현하는 방법이다. <code>while</code> 반복문은 매번 먼저 조건을 검사하여 참이면 실행을 계속하고, 그렇지 않으면 반복을 끝낸다.</p> <p>다음은 <code>while</code> 반복문으로 \\(1 + 2 + \\dots + n\\)을 계산하는 코드다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def while_loop(n: int) -&gt; int:\n    \"\"\"while 반복문\"\"\"\n    res = 0\n    i = 1  # 조건 변수를 초기화한다\n    # 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while i &lt;= n:\n        res += i\n        i += 1  # 조건 변수를 갱신한다\n    return res\n</code></pre> iteration.cpp<pre><code>/* while 반복문 */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        i++; // 조건 변수를 갱신한다\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while 반복문 */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        i++; // 조건 변수를 갱신한다\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while 반복문 */\nint WhileLoop(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        i += 1; // 조건 변수를 갱신한다\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while 반복문 */\nfunc whileLoop(n int) int {\n    res := 0\n    // 조건 변수를 초기화한다\n    i := 1\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    for i &lt;= n {\n        res += i\n        // 조건 변수를 갱신한다\n        i++\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while 반복문 */\nfunc whileLoop(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while i &lt;= n {\n        res += i\n        i += 1 // 조건 변수를 갱신한다\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while 반복문 */\nfunction whileLoop(n) {\n    let res = 0;\n    let i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        i++; // 조건 변수를 갱신한다\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while 반복문 */\nfunction whileLoop(n: number): number {\n    let res = 0;\n    let i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        i++; // 조건 변수를 갱신한다\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while 반복문 */\nint whileLoop(int n) {\n  int res = 0;\n  int i = 1; // 조건 변수를 초기화한다\n  // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n  while (i &lt;= n) {\n    res += i;\n    i++; // 조건 변수를 갱신한다\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while 반복문 */\nfn while_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // 조건 변수를 초기화한다\n\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while i &lt;= n {\n        res += i;\n        i += 1; // 조건 변수를 갱신한다\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while 반복문 */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        i++; // 조건 변수를 갱신한다\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while 반복문 */\nfun whileLoop(n: Int): Int {\n    var res = 0\n    var i = 1 // 조건 변수를 초기화한다\n    // 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n    while (i &lt;= n) {\n        res += i\n        i++ // 조건 변수를 갱신한다\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### while 반복문 ###\ndef while_loop(n)\n  res = 0\n  i = 1 # 조건 변수를 초기화한다\n\n  # 반복문으로 1, 2, ..., n-1, n의 합을 구한다\n  while i &lt;= n\n    res += i\n    i += 1 # 조건 변수를 갱신한다\n  end\n\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p><code>while</code> 반복문은 <code>for</code> 반복문보다 자유도가 높다. <code>while</code> 반복문에서는 조건 변수의 초기화와 갱신 단계를 자유롭게 설계할 수 있다.</p> <p>예를 들어 다음 코드에서는 조건 변수 \\(i\\)를 매번 두 차례 갱신한다. 이런 경우에는 <code>for</code> 반복문으로 구현하기가 다소 불편하다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def while_loop_ii(n: int) -&gt; int:\n    \"\"\"while 반복문(두 번 갱신)\"\"\"\n    res = 0\n    i = 1  # 조건 변수를 초기화한다\n    # 반복문으로 1, 4, 10, ...의 합을 구한다\n    while i &lt;= n:\n        res += i\n        # 조건 변수를 갱신한다\n        i += 1\n        i *= 2\n    return res\n</code></pre> iteration.cpp<pre><code>/* while 반복문(두 번 갱신) */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        // 조건 변수를 갱신한다\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while 반복문(두 번 갱신) */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        // 조건 변수를 갱신한다\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while 반복문(두 번 갱신) */\nint WhileLoopII(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        // 조건 변수를 갱신한다\n        i += 1; \n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while 반복문(두 번 갱신) */\nfunc whileLoopII(n int) int {\n    res := 0\n    // 조건 변수를 초기화한다\n    i := 1\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    for i &lt;= n {\n        res += i\n        // 조건 변수를 갱신한다\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while 반복문(두 번 갱신) */\nfunc whileLoopII(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while i &lt;= n {\n        res += i\n        // 조건 변수를 갱신한다\n        i += 1\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while 반복문(두 번 갱신) */\nfunction whileLoopII(n) {\n    let res = 0;\n    let i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        // 조건 변수를 갱신한다\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while 반복문(두 번 갱신) */\nfunction whileLoopII(n: number): number {\n    let res = 0;\n    let i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        // 조건 변수를 갱신한다\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while 반복문(두 번 갱신) */\nint whileLoopII(int n) {\n  int res = 0;\n  int i = 1; // 조건 변수를 초기화한다\n  // 반복문으로 1, 4, 10, ...의 합을 구한다\n  while (i &lt;= n) {\n    res += i;\n    // 조건 변수를 갱신한다\n    i++;\n    i *= 2;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while 반복문(두 번 갱신) */\nfn while_loop_ii(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // 조건 변수를 초기화한다\n\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while i &lt;= n {\n        res += i;\n        // 조건 변수를 갱신한다\n        i += 1;\n        i *= 2;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while 반복문(두 번 갱신) */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while (i &lt;= n) {\n        res += i;\n        // 조건 변수를 갱신한다\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* while 반복문(두 번 갱신) */\nfun whileLoopII(n: Int): Int {\n    var res = 0\n    var i = 1 // 조건 변수를 초기화한다\n    // 반복문으로 1, 4, 10, ...의 합을 구한다\n    while (i &lt;= n) {\n        res += i\n        // 조건 변수를 갱신한다\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.rb<pre><code>### while 반복문(두 번 갱신)###\ndef while_loop_ii(n)\n  res = 0\n  i = 1 # 조건 변수를 초기화한다\n\n  # 반복문으로 1, 4, 10, ...의 합을 구한다\n  while i &lt;= n\n    res += i\n    # 조건 변수를 갱신한다\n    i += 1\n    i *= 2\n  end\n\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>전반적으로 <code>for</code> 반복문은 코드가 더 간결하고, <code>while</code> 반복문은 더 유연하다. 둘 다 반복 구조를 구현할 수 있으며, 어떤 것을 쓸지는 문제의 구체적인 요구에 따라 결정해야 한다.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#3","level":3,"title":"3.   중첩 반복문","text":"<p>하나의 반복 구조 안에 다른 반복 구조를 중첩할 수 있다. 다음은 <code>for</code> 반복문을 사용한 예다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby iteration.py<pre><code>def nested_for_loop(n: int) -&gt; str:\n    \"\"\"이중 for 반복문\"\"\"\n    res = \"\"\n    # i = 1, 2, ..., n-1, n을 반복한다\n    for i in range(1, n + 1):\n        # j = 1, 2, ..., n-1, n을 반복한다\n        for j in range(1, n + 1):\n            res += f\"({i}, {j}), \"\n    return res\n</code></pre> iteration.cpp<pre><code>/* 이중 for 반복문 */\nstring nestedForLoop(int n) {\n    ostringstream res;\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for (int i = 1; i &lt;= n; ++i) {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for (int j = 1; j &lt;= n; ++j) {\n            res &lt;&lt; \"(\" &lt;&lt; i &lt;&lt; \", \" &lt;&lt; j &lt;&lt; \"), \";\n        }\n    }\n    return res.str();\n}\n</code></pre> iteration.java<pre><code>/* 이중 for 반복문 */\nString nestedForLoop(int n) {\n    StringBuilder res = new StringBuilder();\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for (int i = 1; i &lt;= n; i++) {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for (int j = 1; j &lt;= n; j++) {\n            res.append(\"(\" + i + \", \" + j + \"), \");\n        }\n    }\n    return res.toString();\n}\n</code></pre> iteration.cs<pre><code>/* 이중 for 반복문 */\nstring NestedForLoop(int n) {\n    StringBuilder res = new();\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for (int i = 1; i &lt;= n; i++) {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for (int j = 1; j &lt;= n; j++) {\n            res.Append($\"({i}, {j}), \");\n        }\n    }\n    return res.ToString();\n}\n</code></pre> iteration.go<pre><code>/* 이중 for 반복문 */\nfunc nestedForLoop(n int) string {\n    res := \"\"\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= n; j++ {\n            // j = 1, 2, ..., n-1, n을 반복한다\n            res += fmt.Sprintf(\"(%d, %d), \", i, j)\n        }\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* 이중 for 반복문 */\nfunc nestedForLoop(n: Int) -&gt; String {\n    var res = \"\"\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for i in 1 ... n {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for j in 1 ... n {\n            res.append(\"(\\(i), \\(j)), \")\n        }\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* 이중 for 반복문 */\nfunction nestedForLoop(n) {\n    let res = '';\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for (let i = 1; i &lt;= n; i++) {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* 이중 for 반복문 */\nfunction nestedForLoop(n: number): string {\n    let res = '';\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for (let i = 1; i &lt;= n; i++) {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* 이중 for 반복문 */\nString nestedForLoop(int n) {\n  String res = \"\";\n  // i = 1, 2, ..., n-1, n을 반복한다\n  for (int i = 1; i &lt;= n; i++) {\n    // j = 1, 2, ..., n-1, n을 반복한다\n    for (int j = 1; j &lt;= n; j++) {\n      res += \"($i, $j), \";\n    }\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* 이중 for 반복문 */\nfn nested_for_loop(n: i32) -&gt; String {\n    let mut res = vec![];\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for i in 1..=n {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for j in 1..=n {\n            res.push(format!(\"({}, {}), \", i, j));\n        }\n    }\n    res.join(\"\")\n}\n</code></pre> iteration.c<pre><code>/* 이중 for 반복문 */\nchar *nestedForLoop(int n) {\n    // n * n은 해당 점의 개수이며, \"(i, j), \"의 최대 문자열 길이는 6+10*2이다. 마지막 널 문자 \\0을 위한 공간도 추가한다\n    int size = n * n * 26 + 1;\n    char *res = malloc(size * sizeof(char));\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for (int i = 1; i &lt;= n; i++) {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for (int j = 1; j &lt;= n; j++) {\n            char tmp[26];\n            snprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\n            strncat(res, tmp, size - strlen(res) - 1);\n        }\n    }\n    return res;\n}\n</code></pre> iteration.kt<pre><code>/* 이중 for 반복문 */\nfun nestedForLoop(n: Int): String {\n    val res = StringBuilder()\n    // i = 1, 2, ..., n-1, n을 반복한다\n    for (i in 1..n) {\n        // j = 1, 2, ..., n-1, n을 반복한다\n        for (j in 1..n) {\n            res.append(\" ($i, $j), \")\n        }\n    }\n    return res.toString()\n}\n</code></pre> iteration.rb<pre><code>### 이중 for 반복문 ###\ndef nested_for_loop(n)\n  res = \"\"\n\n  # i = 1, 2, ..., n-1, n을 반복한다\n  for i in 1..n\n    # j = 1, 2, ..., n-1, n을 반복한다\n    for j in 1..n\n      res += \"(#{i}, #{j}), \"\n    end\n  end\n\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(2-2)은 이 중첩 반복문의 순서도다.</p> <p></p> <p> 그림 2-2   중첩 반복문의 순서도 </p> <p>이 경우 함수의 연산 횟수는 \\(n^2\\)에 비례한다. 즉, 알고리즘 실행 시간과 입력 데이터 크기 \\(n\\) 사이에는 ‘제곱 관계’가 있다.</p> <p>반복문을 계속 중첩할 수도 있다. 한 번 중첩할 때마다 ‘차원이 하나씩 늘어나’ 시간 복잡도가 ‘세제곱 관계’, ‘네제곱 관계’ 등으로 높아진다.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#222","level":2,"title":"2.2.2   재귀","text":"<p>재귀(recursion)는 함수가 자기 자신을 호출하여 문제를 해결하는 알고리즘 전략이다. 크게 두 단계로 이루어진다.</p> <ol> <li>내려가기: 프로그램이 자기 자신을 계속 더 깊이 호출한다. 보통 더 작거나 단순한 인수를 전달하며, ‘종료 조건’에 도달할 때까지 이어진다.</li> <li>돌아오기: ‘종료 조건’에 도달하면 가장 깊은 재귀 함수부터 한 단계씩 반환하며, 각 단계의 결과를 모은다.</li> </ol> <p>구현 관점에서 보면 재귀 코드는 크게 세 요소로 이루어진다.</p> <ol> <li>종료 조건: ‘내려가기’에서 ‘돌아오기’로 전환할 시점을 결정한다.</li> <li>재귀 호출: ‘내려가기’에 해당한다. 함수가 자기 자신을 호출하며, 보통 더 작거나 단순한 인수를 입력한다.</li> <li>결과 반환: ‘돌아오기’에 해당한다. 현재 재귀 단계의 결과를 이전 단계로 반환한다.</li> </ol> <p>다음 코드를 보면, 함수 <code>recur(n)</code>을 호출하기만 하면 \\(1 + 2 + \\dots + n\\)을 계산할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def recur(n: int) -&gt; int:\n    \"\"\"재귀\"\"\"\n    # 종료 조건\n    if n == 1:\n        return 1\n    # 내려가기: 재귀 호출\n    res = recur(n - 1)\n    # 돌아오기: 결과 반환\n    return n + res\n</code></pre> recursion.cpp<pre><code>/* 재귀 */\nint recur(int n) {\n    // 종료 조건\n    if (n == 1)\n        return 1;\n    // 내려가기: 재귀 호출\n    int res = recur(n - 1);\n    // 돌아오기: 결과 반환\n    return n + res;\n}\n</code></pre> recursion.java<pre><code>/* 재귀 */\nint recur(int n) {\n    // 종료 조건\n    if (n == 1)\n        return 1;\n    // 내려가기: 재귀 호출\n    int res = recur(n - 1);\n    // 돌아오기: 결과 반환\n    return n + res;\n}\n</code></pre> recursion.cs<pre><code>/* 재귀 */\nint Recur(int n) {\n    // 종료 조건\n    if (n == 1)\n        return 1;\n    // 내려가기: 재귀 호출\n    int res = Recur(n - 1);\n    // 돌아오기: 결과 반환\n    return n + res;\n}\n</code></pre> recursion.go<pre><code>/* 재귀 */\nfunc recur(n int) int {\n    // 종료 조건\n    if n == 1 {\n        return 1\n    }\n    // 내려가기: 재귀 호출\n    res := recur(n - 1)\n    // 돌아오기: 결과 반환\n    return n + res\n}\n</code></pre> recursion.swift<pre><code>/* 재귀 */\nfunc recur(n: Int) -&gt; Int {\n    // 종료 조건\n    if n == 1 {\n        return 1\n    }\n    // 내려가기: 재귀 호출\n    let res = recur(n: n - 1)\n    // 돌아오기: 결과 반환\n    return n + res\n}\n</code></pre> recursion.js<pre><code>/* 재귀 */\nfunction recur(n) {\n    // 종료 조건\n    if (n === 1) return 1;\n    // 내려가기: 재귀 호출\n    const res = recur(n - 1);\n    // 돌아오기: 결과 반환\n    return n + res;\n}\n</code></pre> recursion.ts<pre><code>/* 재귀 */\nfunction recur(n: number): number {\n    // 종료 조건\n    if (n === 1) return 1;\n    // 내려가기: 재귀 호출\n    const res = recur(n - 1);\n    // 돌아오기: 결과 반환\n    return n + res;\n}\n</code></pre> recursion.dart<pre><code>/* 재귀 */\nint recur(int n) {\n  // 종료 조건\n  if (n == 1) return 1;\n  // 내려가기: 재귀 호출\n  int res = recur(n - 1);\n  // 돌아오기: 결과 반환\n  return n + res;\n}\n</code></pre> recursion.rs<pre><code>/* 재귀 */\nfn recur(n: i32) -&gt; i32 {\n    // 종료 조건\n    if n == 1 {\n        return 1;\n    }\n    // 내려가기: 재귀 호출\n    let res = recur(n - 1);\n    // 돌아오기: 결과 반환\n    n + res\n}\n</code></pre> recursion.c<pre><code>/* 재귀 */\nint recur(int n) {\n    // 종료 조건\n    if (n == 1)\n        return 1;\n    // 내려가기: 재귀 호출\n    int res = recur(n - 1);\n    // 돌아오기: 결과 반환\n    return n + res;\n}\n</code></pre> recursion.kt<pre><code>/* 재귀 */\nfun recur(n: Int): Int {\n    // 종료 조건\n    if (n == 1)\n        return 1\n    // 내려가기: 재귀 호출\n    val res = recur(n - 1)\n    // 돌아오기: 결과 반환\n    return n + res\n}\n</code></pre> recursion.rb<pre><code>### 재귀 ###\ndef recur(n)\n  # 종료 조건\n  return 1 if n == 1\n  # 내려가기: 재귀 호출\n  res = recur(n - 1)\n  # 돌아오기: 결과 반환\n  n + res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(2-3)은 이 함수의 재귀 과정을 보여 준다.</p> <p></p> <p> 그림 2-3   합계 함수의 재귀 과정 </p> <p>반복과 재귀는 계산 결과가 같을 수 있지만, 문제를 생각하고 해결하는 방식은 완전히 다르다.</p> <ul> <li>반복: ‘상향식’으로 문제를 해결한다. 가장 기본적인 단계에서 시작해 그 단계를 계속 반복하거나 누적하여 작업을 완료한다.</li> <li>재귀: ‘하향식’으로 문제를 해결한다. 원래 문제를 같은 형태의 더 작은 부분 문제로 나눈다. 그런 다음 부분 문제를 더 작은 부분 문제로 계속 나누다가 기저 사례에 도달하면 멈춘다(기저 사례의 해는 이미 알고 있다).</li> </ul> <p>앞의 합계 함수를 예로 들어, 문제를 \\(f(n) = 1 + 2 + \\dots + n\\)이라고 하자.</p> <ul> <li>반복: 반복문에서 합을 구하는 과정을 재현한다. \\(1\\)부터 \\(n\\)까지 순회하면서 매번 더하면 \\(f(n)\\)을 구할 수 있다.</li> <li>재귀: 문제를 부분 문제 \\(f(n) = n + f(n-1)\\)로 나누고, 계속 재귀적으로 나누다가 기저 사례 \\(f(1) = 1\\)에 도달하면 멈춘다.</li> </ul>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#1","level":3,"title":"1.   호출 스택","text":"<p>재귀 함수가 자기 자신을 호출할 때마다 시스템은 새로 실행되는 함수를 위해 메모리를 할당하고 지역 변수, 호출 주소 등의 정보를 저장한다. 이로 인해 두 가지 결과가 나타난다.</p> <ul> <li>함수의 문맥 데이터는 ‘스택 프레임 공간’이라는 메모리 영역에 저장되며, 함수가 반환된 뒤에야 해제된다. 따라서 재귀는 보통 반복보다 더 많은 메모리 공간을 사용한다.</li> <li>재귀 함수 호출에는 추가 비용이 발생한다. 따라서 재귀는 보통 반복문보다 시간 효율이 낮다.</li> </ul> <p>아래 그림(2-4)과 같이 종료 조건에 도달하기 전에는 아직 반환되지 않은 재귀 함수가 \\(n\\)개 동시에 존재하므로, 재귀 깊이는 \\(n\\)이다.</p> <p></p> <p> 그림 2-4   재귀 호출 깊이 </p> <p>실제로 프로그래밍 언어가 허용하는 재귀 깊이는 보통 제한되어 있으며, 재귀가 너무 깊어지면 스택 오버플로 오류가 발생할 수 있다.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#2","level":3,"title":"2.   꼬리 재귀","text":"<p>흥미롭게도 함수가 반환하기 직전의 마지막 단계에서 재귀 호출을 수행하면, 컴파일러나 인터프리터가 이를 최적화하여 반복과 비슷한 공간 효율을 얻을 수 있다. 이런 경우를 꼬리 재귀(tail recursion)라고 한다.</p> <ul> <li>일반 재귀: 함수가 이전 단계의 함수로 반환된 뒤에도 실행할 코드가 남아 있으므로, 시스템은 이전 호출의 문맥을 저장해야 한다.</li> <li>꼬리 재귀: 재귀 호출이 함수 반환 전의 마지막 연산이다. 따라서 이전 단계로 반환된 뒤에 다른 연산을 수행할 필요가 없으므로, 시스템은 이전 함수의 문맥을 저장하지 않아도 된다.</li> </ul> <p>\\(1 + 2 + \\dots + n\\)을 계산하는 예에서 결과 변수 <code>res</code>를 함수 매개변수로 두면 꼬리 재귀를 구현할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def tail_recur(n, res):\n    \"\"\"꼬리 재귀\"\"\"\n    # 종료 조건\n    if n == 0:\n        return res\n    # 꼬리 재귀 호출\n    return tail_recur(n - 1, res + n)\n</code></pre> recursion.cpp<pre><code>/* 꼬리 재귀 */\nint tailRecur(int n, int res) {\n    // 종료 조건\n    if (n == 0)\n        return res;\n    // 꼬리 재귀 호출\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.java<pre><code>/* 꼬리 재귀 */\nint tailRecur(int n, int res) {\n    // 종료 조건\n    if (n == 0)\n        return res;\n    // 꼬리 재귀 호출\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.cs<pre><code>/* 꼬리 재귀 */\nint TailRecur(int n, int res) {\n    // 종료 조건\n    if (n == 0)\n        return res;\n    // 꼬리 재귀 호출\n    return TailRecur(n - 1, res + n);\n}\n</code></pre> recursion.go<pre><code>/* 꼬리 재귀 */\nfunc tailRecur(n int, res int) int {\n    // 종료 조건\n    if n == 0 {\n        return res\n    }\n    // 꼬리 재귀 호출\n    return tailRecur(n-1, res+n)\n}\n</code></pre> recursion.swift<pre><code>/* 꼬리 재귀 */\nfunc tailRecur(n: Int, res: Int) -&gt; Int {\n    // 종료 조건\n    if n == 0 {\n        return res\n    }\n    // 꼬리 재귀 호출\n    return tailRecur(n: n - 1, res: res + n)\n}\n</code></pre> recursion.js<pre><code>/* 꼬리 재귀 */\nfunction tailRecur(n, res) {\n    // 종료 조건\n    if (n === 0) return res;\n    // 꼬리 재귀 호출\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.ts<pre><code>/* 꼬리 재귀 */\nfunction tailRecur(n: number, res: number): number {\n    // 종료 조건\n    if (n === 0) return res;\n    // 꼬리 재귀 호출\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.dart<pre><code>/* 꼬리 재귀 */\nint tailRecur(int n, int res) {\n  // 종료 조건\n  if (n == 0) return res;\n  // 꼬리 재귀 호출\n  return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.rs<pre><code>/* 꼬리 재귀 */\nfn tail_recur(n: i32, res: i32) -&gt; i32 {\n    // 종료 조건\n    if n == 0 {\n        return res;\n    }\n    // 꼬리 재귀 호출\n    tail_recur(n - 1, res + n)\n}\n</code></pre> recursion.c<pre><code>/* 꼬리 재귀 */\nint tailRecur(int n, int res) {\n    // 종료 조건\n    if (n == 0)\n        return res;\n    // 꼬리 재귀 호출\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.kt<pre><code>/* 꼬리 재귀 */\ntailrec fun tailRecur(n: Int, res: Int): Int {\n    // tailrec 키워드를 추가하여 꼬리 재귀 최적화를 활성화한다\n    // 종료 조건\n    if (n == 0)\n        return res\n    // 꼬리 재귀 호출\n    return tailRecur(n - 1, res + n)\n}\n</code></pre> recursion.rb<pre><code>### 꼬리 재귀 ###\ndef tail_recur(n, res)\n  # 종료 조건\n  return res if n == 0\n  # 꼬리 재귀 호출\n  tail_recur(n - 1, res + n)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>꼬리 재귀의 실행 과정은 아래 그림(2-5)과 같다. 일반 재귀와 꼬리 재귀를 비교하면 합을 구하는 시점이 다르다.</p> <ul> <li>일반 재귀: ‘돌아오기’ 과정에서 합을 구한다. 각 단계가 반환된 뒤에 덧셈을 한 번씩 수행한다.</li> <li>꼬리 재귀: ‘내려가기’ 과정에서 합을 구한다. ‘돌아오기’ 과정에서는 단계별로 반환만 하면 된다.</li> </ul> <p></p> <p> 그림 2-5   꼬리 재귀 과정 </p> <p>참고</p> <p>많은 컴파일러와 인터프리터가 꼬리 재귀 최적화를 지원하지 않는다는 점에 주의하자. 예를 들어 Python은 기본적으로 꼬리 재귀 최적화를 지원하지 않으므로, 함수가 꼬리 재귀 형태여도 스택 오버플로 문제가 발생할 수 있다.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#3_1","level":3,"title":"3.   재귀 트리","text":"<p>‘분할 정복’과 관련된 알고리즘 문제에서는 재귀가 반복보다 생각하기 쉽고 코드도 읽기 쉬운 경우가 많다. ‘피보나치 수열’을 예로 살펴보자.</p> <p>문제</p> <p>피보나치 수열 \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\)가 주어졌을 때, 이 수열의 \\(n\\)번째 수를 구하라.</p> <p>피보나치 수열의 \\(n\\)번째 수를 \\(f(n)\\)이라고 하면 다음 두 가지 사실을 쉽게 알 수 있다.</p> <ul> <li>수열의 첫 두 수는 \\(f(1) = 0\\)과 \\(f(2) = 1\\)이다.</li> <li>수열의 각 수는 앞의 두 수의 합이다. 즉, \\(f(n) = f(n - 1) + f(n - 2)\\)이다.</li> </ul> <p>점화 관계에 따라 재귀 호출을 하고 첫 두 수를 종료 조건으로 삼으면 재귀 코드를 작성할 수 있다. <code>fib(n)</code>을 호출하면 피보나치 수열의 \\(n\\)번째 수를 얻는다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def fib(n: int) -&gt; int:\n    \"\"\"피보나치 수열: 재귀\"\"\"\n    # 종료 조건 f(1) = 0, f(2) = 1\n    if n == 1 or n == 2:\n        return n - 1\n    # 재귀 호출 f(n) = f(n-1) + f(n-2)\n    res = fib(n - 1) + fib(n - 2)\n    # 결과 f(n)을 반환한다\n    return res\n</code></pre> recursion.cpp<pre><code>/* 피보나치 수열: 재귀 */\nint fib(int n) {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // 결과 f(n)을 반환한다\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* 피보나치 수열: 재귀 */\nint fib(int n) {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // 결과 f(n)을 반환한다\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* 피보나치 수열: 재귀 */\nint Fib(int n) {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    int res = Fib(n - 1) + Fib(n - 2);\n    // 결과 f(n)을 반환한다\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* 피보나치 수열: 재귀 */\nfunc fib(n int) int {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    res := fib(n-1) + fib(n-2)\n    // 결과 f(n)을 반환한다\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* 피보나치 수열: 재귀 */\nfunc fib(n: Int) -&gt; Int {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    let res = fib(n: n - 1) + fib(n: n - 2)\n    // 결과 f(n)을 반환한다\n    return res\n}\n</code></pre> recursion.js<pre><code>/* 피보나치 수열: 재귀 */\nfunction fib(n) {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // 결과 f(n)을 반환한다\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* 피보나치 수열: 재귀 */\nfunction fib(n: number): number {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // 결과 f(n)을 반환한다\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* 피보나치 수열: 재귀 */\nint fib(int n) {\n  // 종료 조건 f(1) = 0, f(2) = 1\n  if (n == 1 || n == 2) return n - 1;\n  // 재귀 호출 f(n) = f(n-1) + f(n-2)\n  int res = fib(n - 1) + fib(n - 2);\n  // 결과 f(n)을 반환한다\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* 피보나치 수열: 재귀 */\nfn fib(n: i32) -&gt; i32 {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1;\n    }\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    let res = fib(n - 1) + fib(n - 2);\n    // 결과를 반환한다\n    res\n}\n</code></pre> recursion.c<pre><code>/* 피보나치 수열: 재귀 */\nint fib(int n) {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // 결과 f(n)을 반환한다\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* 피보나치 수열: 재귀 */\nfun fib(n: Int): Int {\n    // 종료 조건 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1\n    // 재귀 호출 f(n) = f(n-1) + f(n-2)\n    val res = fib(n - 1) + fib(n - 2)\n    // 결과 f(n)을 반환한다\n    return res\n}\n</code></pre> recursion.rb<pre><code>### 피보나치 수열: 재귀 ###\ndef fib(n)\n  # 종료 조건 f(1) = 0, f(2) = 1\n  return n - 1 if n == 1 || n == 2\n  # 재귀 호출 f(n) = f(n-1) + f(n-2)\n  res = fib(n - 1) + fib(n - 2)\n  # 결과 f(n)을 반환한다\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>위 코드는 함수 안에서 두 번의 재귀 호출을 수행한다. 이는 하나의 호출에서 두 개의 호출 분기가 생긴다는 뜻이다. 아래 그림(2-6)과 같이 재귀 호출을 계속하면 최종적으로 \\(n\\)개 층으로 이루어진 재귀 트리(recursion tree)가 만들어진다.</p> <p></p> <p> 그림 2-6   피보나치 수열의 재귀 트리 </p> <p>본질적으로 재귀는 ‘문제를 더 작은 부분 문제로 나누는’ 사고방식을 보여 준다. 이러한 분할 정복 전략은 매우 중요하다.</p> <ul> <li>알고리즘 관점에서 탐색, 정렬, 백트래킹, 분할 정복, 동적 계획법 등 중요한 알고리즘 전략 다수가 이 사고방식을 직간접적으로 활용한다.</li> <li>자료 구조 관점에서 재귀는 연결 리스트, 트리, 그래프와 관련된 문제를 다루는 데 자연스럽게 어울린다. 이들은 분할 정복 관점으로 분석하기에 매우 적합하기 때문이다.</li> </ul>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/iteration_and_recursion/#223","level":2,"title":"2.2.3   두 방식의 비교","text":"<p>앞의 내용을 정리하면, 아래 표(2-1)와 같이 반복과 재귀는 구현, 성능, 적용 범위에서 차이가 있다.</p> <p> 표 2-1   반복과 재귀의 특징 비교 </p> 반복 재귀 구현 방식 반복 구조 함수가 자기 자신을 호출 시간 효율 보통 효율이 높고 함수 호출 비용이 없음 함수 호출마다 비용이 발생함 메모리 사용 보통 고정 크기의 메모리 공간을 사용함 함수 호출이 누적되면 스택 프레임 공간을 많이 사용할 수 있음 적합한 문제 간단한 반복 작업에 적합하며 코드가 직관적이고 읽기 쉬움 트리, 그래프, 분할 정복, 백트래킹 등 부분 문제로 나누는 문제에 적합하며 코드 구조가 간결하고 명확함 <p>참고</p> <p>다음 내용이 이해하기 어렵다면 ‘스택’ 장을 읽은 뒤에 다시 복습해도 좋다.</p> <p>그렇다면 반복과 재귀는 내부적으로 어떤 관계가 있을까? 앞의 재귀 함수를 예로 들면, 합을 구하는 연산은 재귀의 ‘돌아오기’ 단계에서 수행된다. 즉, 가장 먼저 호출된 함수가 실제로는 가장 나중에 자신의 덧셈을 마친다. 이 작동 방식은 스택의 ‘후입선출’ 원리와 통한다.</p> <p>실제로 ‘호출 스택’, ‘스택 프레임 공간’ 같은 재귀 관련 용어에도 재귀와 스택의 밀접한 관계가 드러난다.</p> <ol> <li>내려가기: 함수가 호출되면 시스템은 ‘호출 스택’에 새로운 스택 프레임을 할당하여 함수의 지역 변수, 매개변수, 반환 주소 등의 데이터를 저장한다.</li> <li>돌아오기: 함수가 실행을 마치고 반환되면 해당 스택 프레임이 ‘호출 스택’에서 제거되고, 이전 함수의 실행 환경이 복원된다.</li> </ol> <p>따라서 명시적인 스택으로 호출 스택의 동작을 재현하면, 재귀를 반복 형태로 바꿀 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby recursion.py<pre><code>def for_loop_recur(n: int) -&gt; int:\n    \"\"\"반복으로 재귀 구현\"\"\"\n    # 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    stack = []\n    res = 0\n    # 내려가기: 재귀 호출\n    for i in range(n, 0, -1):\n        # 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.append(i)\n    # 돌아오기: 결과 반환\n    while stack:\n        # 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.pop()\n    # res = 1+2+3+...+n\n    return res\n</code></pre> recursion.cpp<pre><code>/* 반복으로 재귀 구현 */\nint forLoopRecur(int n) {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    stack&lt;int&gt; stack;\n    int res = 0;\n    // 내려가기: 재귀 호출\n    for (int i = n; i &gt; 0; i--) {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.push(i);\n    }\n    // 돌아오기: 결과 반환\n    while (!stack.empty()) {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.top();\n        stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* 반복으로 재귀 구현 */\nint forLoopRecur(int n) {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    Stack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n    int res = 0;\n    // 내려가기: 재귀 호출\n    for (int i = n; i &gt; 0; i--) {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.push(i);\n    }\n    // 돌아오기: 결과 반환\n    while (!stack.isEmpty()) {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* 반복으로 재귀 구현 */\nint ForLoopRecur(int n) {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    Stack&lt;int&gt; stack = new();\n    int res = 0;\n    // 내려가기: 재귀 호출\n    for (int i = n; i &gt; 0; i--) {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.Push(i);\n    }\n    // 돌아오기: 결과 반환\n    while (stack.Count &gt; 0) {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.Pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* 반복으로 재귀 구현 */\nfunc forLoopRecur(n int) int {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    stack := list.New()\n    res := 0\n    // 내려가기: 재귀 호출\n    for i := n; i &gt; 0; i-- {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.PushBack(i)\n    }\n    // 돌아오기: 결과 반환\n    for stack.Len() != 0 {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.Back().Value.(int)\n        stack.Remove(stack.Back())\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* 반복으로 재귀 구현 */\nfunc forLoopRecur(n: Int) -&gt; Int {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    var stack: [Int] = []\n    var res = 0\n    // 내려가기: 재귀 호출\n    for i in (1 ... n).reversed() {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.append(i)\n    }\n    // 돌아오기: 결과 반환\n    while !stack.isEmpty {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.removeLast()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.js<pre><code>/* 반복으로 재귀 구현 */\nfunction forLoopRecur(n) {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    const stack = [];\n    let res = 0;\n    // 내려가기: 재귀 호출\n    for (let i = n; i &gt; 0; i--) {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.push(i);\n    }\n    // 돌아오기: 결과 반환\n    while (stack.length) {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* 반복으로 재귀 구현 */\nfunction forLoopRecur(n: number): number {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다 \n    const stack: number[] = [];\n    let res: number = 0;\n    // 내려가기: 재귀 호출\n    for (let i = n; i &gt; 0; i--) {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.push(i);\n    }\n    // 돌아오기: 결과 반환\n    while (stack.length) {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* 반복으로 재귀 구현 */\nint forLoopRecur(int n) {\n  // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n  List&lt;int&gt; stack = [];\n  int res = 0;\n  // 내려가기: 재귀 호출\n  for (int i = n; i &gt; 0; i--) {\n    // 스택에 삽입하여 내려가는 과정을 모방한다\n    stack.add(i);\n  }\n  // 돌아오기: 결과 반환\n  while (!stack.isEmpty) {\n    // 스택에서 꺼내어 돌아오는 과정을 모방한다\n    res += stack.removeLast();\n  }\n  // res = 1+2+3+...+n\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* 반복으로 재귀 구현 */\nfn for_loop_recur(n: i32) -&gt; i32 {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    let mut stack = Vec::new();\n    let mut res = 0;\n    // 내려가기: 재귀 호출\n    for i in (1..=n).rev() {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.push(i);\n    }\n    // 돌아오기: 결과 반환\n    while !stack.is_empty() {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.pop().unwrap();\n    }\n    // res = 1+2+3+...+n\n    res\n}\n</code></pre> recursion.c<pre><code>/* 반복으로 재귀 구현 */\nint forLoopRecur(int n) {\n    int stack[1000]; // 큰 배열을 사용하여 스택을 구현한다\n    int top = -1;    // 스택의 맨 위 인덱스\n    int res = 0;\n    // 내려가기: 재귀 호출\n    for (int i = n; i &gt; 0; i--) {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack[1 + top++] = i;\n    }\n    // 돌아오기: 결과 반환\n    while (top &gt;= 0) {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack[top--];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.kt<pre><code>/* 반복으로 재귀 구현 */\nfun forLoopRecur(n: Int): Int {\n    // 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n    val stack = Stack&lt;Int&gt;()\n    var res = 0\n    // 내려가기: 재귀 호출\n    for (i in n downTo 0) {\n        // 스택에 삽입하여 내려가는 과정을 모방한다\n        stack.push(i)\n    }\n    // 돌아오기: 결과 반환\n    while (stack.isNotEmpty()) {\n        // 스택에서 꺼내어 돌아오는 과정을 모방한다\n        res += stack.pop()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.rb<pre><code>### 반복으로 재귀 구현 ###\ndef for_loop_recur(n)\n  # 명시적인 스택을 사용하여 시스템 호출 스택을 모방한다\n  stack = []\n  res = 0\n\n  # 내려가기: 재귀 호출\n  for i in n.downto(0)\n    # 스택에 삽입하여 내려가는 과정을 모방한다\n    stack &lt;&lt; i\n  end\n  # 돌아오기: 결과 반환\n  while !stack.empty?\n    res += stack.pop\n  end\n\n  # res = 1+2+3+...+n\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>위 코드를 보면 재귀를 반복으로 바꾸자 코드가 더 복잡해졌다. 반복과 재귀는 많은 경우 서로 변환할 수 있지만, 다음 두 가지 이유로 항상 그럴 가치가 있는 것은 아니다.</p> <ul> <li>변환한 코드가 더 이해하기 어렵고 가독성이 떨어질 수 있다.</li> <li>복잡한 문제에서는 시스템 호출 스택의 동작을 재현하기가 매우 어려울 수 있다.</li> </ul> <p>결국 반복과 재귀 중 어떤 것을 선택할지는 문제의 구체적인 성격에 달려 있다. 실제 프로그래밍에서는 둘의 장단점을 비교하고 상황에 맞는 방법을 선택하는 것이 중요하다.</p>","path":["제 2 장   복잡도 분석","2.2   반복과 재귀"],"tags":[]},{"location":"chapter_computational_complexity/performance_evaluation/","level":1,"title":"2.1   알고리즘 효율 평가","text":"<p>알고리즘을 설계할 때는 다음 두 단계의 목표를 차례로 추구한다.</p> <ol> <li>문제의 해법 찾기: 알고리즘은 정해진 입력 범위 안에서 문제의 올바른 해를 안정적으로 구해야 한다.</li> <li>최적의 해법 찾기: 같은 문제에도 여러 해법이 있을 수 있으므로, 가능한 한 효율적인 알고리즘을 찾고자 한다.</li> </ol> <p>즉, 문제를 해결할 수 있다는 전제하에 알고리즘 효율은 알고리즘의 우수성을 평가하는 주요 지표가 된다. 여기에는 다음 두 가지 측면이 있다.</p> <ul> <li>시간 효율: 알고리즘의 실행 시간이 얼마나 긴가.</li> <li>공간 효율: 알고리즘이 차지하는 메모리 공간이 얼마나 큰가.</li> </ul> <p>간단히 말해, 우리의 목표는 ‘빠르면서도 메모리를 적게 사용하는’ 자료 구조와 알고리즘을 설계하는 것이다. 알고리즘 효율을 효과적으로 평가하는 일도 매우 중요하다. 그래야 여러 알고리즘을 비교하고, 이를 바탕으로 알고리즘을 설계하고 최적화할 수 있기 때문이다.</p> <p>효율 평가 방법은 크게 실제 측정과 이론적 추정으로 나뉜다.</p>","path":["제 2 장   복잡도 분석","2.1   알고리즘 효율 평가"],"tags":[]},{"location":"chapter_computational_complexity/performance_evaluation/#211","level":2,"title":"2.1.1   실제 측정","text":"<p>같은 문제를 해결할 수 있는 알고리즘 <code>A</code>와 알고리즘 <code>B</code>가 있으며, 두 알고리즘의 효율을 비교해야 한다고 하자. 가장 직접적인 방법은 컴퓨터에서 두 알고리즘을 실행하고 실행 시간과 메모리 사용량을 관찰해 기록하는 것이다. 이러한 평가 방식은 실제 상황을 반영할 수 있지만 한계도 크다.</p> <p>한편으로는 측정 환경에 따른 영향을 배제하기 어렵다. 하드웨어 구성은 알고리즘 성능에 영향을 준다. 예를 들어 병렬성이 높은 알고리즘은 멀티코어 CPU에서 실행하기에 더 적합하고, 메모리 연산이 집중된 알고리즘은 고성능 메모리에서 더 좋은 성능을 보인다. 즉, 같은 알고리즘도 컴퓨터에 따라 측정 결과가 다를 수 있다. 따라서 다양한 컴퓨터에서 측정해 평균 효율을 구해야 하지만, 이는 현실적이지 않다.</p> <p>다른 한편으로는 충분한 측정을 수행하는 데 매우 많은 자원이 든다. 입력 데이터의 양이 달라지면 알고리즘의 효율도 달라진다. 예를 들어 입력 데이터가 적을 때는 알고리즘 <code>A</code>의 실행 시간이 알고리즘 <code>B</code>보다 짧지만, 입력 데이터가 많을 때는 결과가 정반대일 수 있다. 따라서 설득력 있는 결론을 얻으려면 다양한 크기의 입력 데이터를 측정해야 하며, 여기에는 많은 컴퓨팅 자원이 필요하다.</p>","path":["제 2 장   복잡도 분석","2.1   알고리즘 효율 평가"],"tags":[]},{"location":"chapter_computational_complexity/performance_evaluation/#212","level":2,"title":"2.1.2   이론적 추정","text":"<p>실제 측정에는 큰 한계가 있으므로, 계산만으로 알고리즘 효율을 평가하는 방법을 생각해 볼 수 있다. 이러한 추정 방법을 점근적 복잡도 분석(asymptotic complexity analysis), 줄여서 복잡도 분석이라고 한다.</p> <p>복잡도 분석은 알고리즘 실행에 필요한 시간 및 공간 자원과 입력 데이터 크기의 관계를 나타낸다. 입력 데이터 크기가 증가할 때 알고리즘 실행에 필요한 시간과 공간이 증가하는 추세를 설명한다. 이 정의는 다소 이해하기 어려우므로 세 가지 핵심으로 나누어 살펴보자.</p> <ul> <li>‘시간 및 공간 자원’은 각각 시간 복잡도(time complexity)와 공간 복잡도(space complexity)에 해당한다.</li> <li>‘입력 데이터 크기가 증가할 때’는 복잡도가 알고리즘의 실행 효율과 입력 데이터 크기 사이의 관계를 반영한다는 뜻이다.</li> <li>‘시간과 공간이 증가하는 추세’는 복잡도 분석이 실행 시간이나 사용 공간의 구체적인 값보다 시간 또는 공간이 얼마나 빠르게 증가하는지에 주목한다는 뜻이다.</li> </ul> <p>복잡도 분석은 실제 측정 방법의 단점을 극복한다. 구체적으로 다음과 같은 장점이 있다.</p> <ul> <li>실제로 코드를 실행하지 않아도 되므로 에너지를 절약하고 환경 부담을 줄인다.</li> <li>측정 환경에 의존하지 않아 분석 결과를 모든 실행 플랫폼에 적용할 수 있다.</li> <li>데이터 양에 따른 알고리즘 효율, 특히 데이터가 많을 때의 알고리즘 성능을 보여 준다.</li> </ul> <p>참고</p> <p>복잡도 개념이 아직 혼란스러워도 걱정할 필요 없다. 이후 절에서 자세히 소개한다.</p> <p>복잡도 분석은 알고리즘 효율을 평가하는 ‘자’와 같다. 이를 통해 알고리즘 실행에 필요한 시간과 공간 자원을 가늠하고 여러 알고리즘의 효율을 비교할 수 있다.</p> <p>복잡도는 수학적 개념이므로 초보자에게는 추상적으로 느껴지고 배우기도 다소 어려울 수 있다. 이런 관점에서는 복잡도 분석이 맨 처음 소개하기에 적절하지 않을 수도 있다. 그러나 어떤 자료 구조나 알고리즘의 특징을 논할 때는 실행 속도와 공간 사용량에 대한 분석을 피하기 어렵다.</p> <p>따라서 자료 구조와 알고리즘을 깊이 학습하기 전에, 복잡도 분석을 기초적으로 이해하여 간단한 알고리즘의 복잡도를 분석할 수 있도록 하는 것을 권한다.</p>","path":["제 2 장   복잡도 분석","2.1   알고리즘 효율 평가"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/","level":1,"title":"2.4   공간 복잡도","text":"<p>공간 복잡도(space complexity)는 데이터 양이 많아질 때 알고리즘이 차지하는 메모리 공간이 증가하는 추세를 측정한다. 시간 복잡도와 매우 비슷한 개념으로, ‘실행 시간’을 ‘차지하는 메모리 공간’으로 바꾸면 된다.</p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#241","level":2,"title":"2.4.1   알고리즘과 관련된 공간","text":"<p>알고리즘이 실행 중 사용하는 메모리 공간에는 주로 다음과 같은 종류가 있다.</p> <ul> <li>입력 공간: 알고리즘의 입력 데이터를 저장한다.</li> <li>임시 공간: 실행 중의 변수, 객체, 함수 문맥 등의 데이터를 저장한다.</li> <li>출력 공간: 알고리즘의 출력 데이터를 저장한다.</li> </ul> <p>일반적으로 공간 복잡도에 포함하는 범위는 ‘임시 공간’과 ‘출력 공간’의 합이다.</p> <p>임시 공간은 다시 세 부분으로 나눌 수 있다.</p> <ul> <li>임시 데이터: 실행 중의 각종 상수, 변수, 객체 등을 저장한다.</li> <li>스택 프레임 공간: 호출한 함수의 문맥 데이터를 저장한다. 시스템은 함수가 호출될 때마다 스택 맨 위에 스택 프레임을 만들며, 함수가 반환되면 이 공간을 해제한다.</li> <li>명령어 공간: 컴파일된 프로그램 명령어를 저장하며, 실제 계산에서는 보통 무시한다.</li> </ul> <p>프로그램의 공간 복잡도를 분석할 때는 아래 그림(2-15)과 같이 보통 임시 데이터, 스택 프레임 공간, 출력 데이터의 세 부분을 계산한다.</p> <p></p> <p> 그림 2-15   알고리즘이 사용하는 공간 </p> <p>관련 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class Node:\n    \"\"\"클래스\"\"\"\n    def __init__(self, x: int):\n        self.val: int = x              # 노드 값\n        self.next: Node | None = None  # 다음 노드를 가리키는 참조\n\ndef function() -&gt; int:\n    \"\"\"함수\"\"\"\n    # 몇 가지 연산 수행...\n    return 0\n\ndef algorithm(n) -&gt; int:  # 입력 데이터\n    A = 0                 # 임시 데이터(상수, 보통 대문자로 표기)\n    b = 0                 # 임시 데이터(변수)\n    node = Node(0)        # 임시 데이터(객체)\n    c = function()        # 스택 프레임 공간(함수 호출)\n    return A + b + c      # 출력 데이터\n</code></pre> <pre><code>/* 구조체 */\nstruct Node {\n    int val;\n    Node *next;\n    Node(int x) : val(x), next(nullptr) {}\n};\n\n/* 함수 */\nint func() {\n    // 몇 가지 연산 수행...\n    return 0;\n}\n\nint algorithm(int n) {        // 입력 데이터\n    const int a = 0;          // 임시 데이터(상수)\n    int b = 0;                // 임시 데이터(변수)\n    Node* node = new Node(0); // 임시 데이터(객체)\n    int c = func();           // 스택 프레임 공간(함수 호출)\n    return a + b + c;         // 출력 데이터\n}\n</code></pre> <pre><code>/* 클래스 */\nclass Node {\n    int val;\n    Node next;\n    Node(int x) { val = x; }\n}\n\n/* 함수 */\nint function() {\n    // 몇 가지 연산 수행...\n    return 0;\n}\n\nint algorithm(int n) {        // 입력 데이터\n    final int a = 0;          // 임시 데이터(상수)\n    int b = 0;                // 임시 데이터(변수)\n    Node node = new Node(0);  // 임시 데이터(객체)\n    int c = function();       // 스택 프레임 공간(함수 호출)\n    return a + b + c;         // 출력 데이터\n}\n</code></pre> <pre><code>/* 클래스 */\nclass Node(int x) {\n    int val = x;\n    Node next;\n}\n\n/* 함수 */\nint Function() {\n    // 몇 가지 연산 수행...\n    return 0;\n}\n\nint Algorithm(int n) {        // 입력 데이터\n    const int a = 0;          // 임시 데이터(상수)\n    int b = 0;                // 임시 데이터(변수)\n    Node node = new(0);       // 임시 데이터(객체)\n    int c = Function();       // 스택 프레임 공간(함수 호출)\n    return a + b + c;         // 출력 데이터\n}\n</code></pre> <pre><code>/* 구조체 */\ntype node struct {\n    val  int\n    next *node\n}\n\n/* node 구조체 생성  */\nfunc newNode(val int) *node {\n    return &amp;node{val: val}\n}\n\n/* 함수 */\nfunc function() int {\n    // 몇 가지 연산 수행...\n    return 0\n}\n\nfunc algorithm(n int) int { // 입력 데이터\n    const a = 0             // 임시 데이터(상수)\n    b := 0                  // 임시 데이터(변수)\n    newNode(0)              // 임시 데이터(객체)\n    c := function()         // 스택 프레임 공간(함수 호출)\n    return a + b + c        // 출력 데이터\n}\n</code></pre> <pre><code>/* 클래스 */\nclass Node {\n    var val: Int\n    var next: Node?\n\n    init(x: Int) {\n        val = x\n    }\n}\n\n/* 함수 */\nfunc function() -&gt; Int {\n    // 몇 가지 연산 수행...\n    return 0\n}\n\nfunc algorithm(n: Int) -&gt; Int { // 입력 데이터\n    let a = 0             // 임시 데이터(상수)\n    var b = 0             // 임시 데이터(변수)\n    let node = Node(x: 0) // 임시 데이터(객체)\n    let c = function()    // 스택 프레임 공간(함수 호출)\n    return a + b + c      // 출력 데이터\n}\n</code></pre> <pre><code>/* 클래스 */\nclass Node {\n    val;\n    next;\n    constructor(val) {\n        this.val = val === undefined ? 0 : val; // 노드 값\n        this.next = null;                       // 다음 노드를 가리키는 참조\n    }\n}\n\n/* 함수 */\nfunction constFunc() {\n    // 몇 가지 연산 수행\n    return 0;\n}\n\nfunction algorithm(n) {       // 입력 데이터\n    const a = 0;              // 임시 데이터(상수)\n    let b = 0;                // 임시 데이터(변수)\n    const node = new Node(0); // 임시 데이터(객체)\n    const c = constFunc();    // 스택 프레임 공간(함수 호출)\n    return a + b + c;         // 출력 데이터\n}\n</code></pre> <pre><code>/* 클래스 */\nclass Node {\n    val: number;\n    next: Node | null;\n    constructor(val?: number) {\n        this.val = val === undefined ? 0 : val; // 노드 값\n        this.next = null;                       // 다음 노드를 가리키는 참조\n    }\n}\n\n/* 함수 */\nfunction constFunc(): number {\n    // 몇 가지 연산 수행\n    return 0;\n}\n\nfunction algorithm(n: number): number { // 입력 데이터\n    const a = 0;                        // 임시 데이터(상수)\n    let b = 0;                          // 임시 데이터(변수)\n    const node = new Node(0);           // 임시 데이터(객체)\n    const c = constFunc();              // 스택 프레임 공간(함수 호출)\n    return a + b + c;                   // 출력 데이터\n}\n</code></pre> <pre><code>/* 클래스 */\nclass Node {\n  int val;\n  Node next;\n  Node(this.val, [this.next]);\n}\n\n/* 함수 */\nint function() {\n  // 몇 가지 연산 수행...\n  return 0;\n}\n\nint algorithm(int n) {  // 입력 데이터\n  const int a = 0;      // 임시 데이터(상수)\n  int b = 0;            // 임시 데이터(변수)\n  Node node = Node(0);  // 임시 데이터(객체)\n  int c = function();   // 스택 프레임 공간(함수 호출)\n  return a + b + c;     // 출력 데이터\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* 구조체 */\nstruct Node {\n    val: i32,\n    next: Option&lt;Rc&lt;RefCell&lt;Node&gt;&gt;&gt;,\n}\n\n/* Node 구조체 생성 */\nimpl Node {\n    fn new(val: i32) -&gt; Self {\n        Self { val: val, next: None }\n    }\n}\n\n/* 함수 */\nfn function() -&gt; i32 {      \n    // 몇 가지 연산 수행...\n    return 0;\n}\n\nfn algorithm(n: i32) -&gt; i32 {       // 입력 데이터\n    const a: i32 = 0;               // 임시 데이터(상수)\n    let mut b = 0;                  // 임시 데이터(변수)\n    let node = Node::new(0);        // 임시 데이터(객체)\n    let c = function();             // 스택 프레임 공간(함수 호출)\n    return a + b + c;               // 출력 데이터\n}\n</code></pre> <pre><code>/* 함수 */\nint func() {\n    // 몇 가지 연산 수행...\n    return 0;\n}\n\nint algorithm(int n) { // 입력 데이터\n    const int a = 0;   // 임시 데이터(상수)\n    int b = 0;         // 임시 데이터(변수)\n    int c = func();    // 스택 프레임 공간(함수 호출)\n    return a + b + c;  // 출력 데이터\n}\n</code></pre> <pre><code>/* 클래스 */\nclass Node(var _val: Int) {\n    var next: Node? = null\n}\n\n/* 함수 */\nfun function(): Int {\n    // 몇 가지 연산 수행...\n    return 0\n}\n\nfun algorithm(n: Int): Int { // 입력 데이터\n    val a = 0                // 임시 데이터(상수)\n    var b = 0                // 임시 데이터(변수)\n    val node = Node(0)       // 임시 데이터(객체)\n    val c = function()       // 스택 프레임 공간(함수 호출)\n    return a + b + c         // 출력 데이터\n}\n</code></pre> <pre><code>### 클래스 ###\nclass Node\n    attr_accessor :val      # 노드 값\n    attr_accessor :next     # 다음 노드를 가리키는 참조\n\n    def initialize(x)\n        @val = x\n    end\nend\n\n### 함수 ###\ndef function\n    # 몇 가지 연산 수행...\n    0\nend\n\n### 알고리즘 ###\ndef algorithm(n)        # 입력 데이터\n    a = 0               # 임시 데이터(상수)\n    b = 0               # 임시 데이터(변수)\n    node = Node.new(0)  # 임시 데이터(객체)\n    c = function        # 스택 프레임 공간(함수 호출)\n    a + b + c           # 출력 데이터\nend\n</code></pre>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#242","level":2,"title":"2.4.2   계산 방법","text":"<p>공간 복잡도를 구하는 방법은 시간 복잡도와 대체로 같다. 계산 대상을 ‘연산 횟수’에서 ‘사용 공간의 크기’로 바꾸면 된다.</p> <p>시간 복잡도와 달리 보통 최악 공간 복잡도에만 관심을 둔다. 메모리 공간은 반드시 충족해야 하는 요구 사항이므로, 어떤 입력 데이터에서도 충분한 메모리 공간이 확보되어 있어야 하기 때문이다.</p> <p>다음 코드를 보면 최악 공간 복잡도에서 ‘최악’은 두 가지 의미를 지닌다.</p> <ol> <li>가장 불리한 입력 데이터를 기준으로 한다: \\(n &lt; 10\\)이면 공간 복잡도는 \\(O(1)\\)이다. 그러나 \\(n &gt; 10\\)이면 초기화한 배열 <code>nums</code>가 \\(O(n)\\) 공간을 차지하므로 최악 공간 복잡도는 \\(O(n)\\)이다.</li> <li>알고리즘 실행 중 최대 메모리 사용량을 기준으로 한다: 예를 들어 프로그램이 마지막 줄을 실행하기 전에는 \\(O(1)\\) 공간을 차지하지만, 배열 <code>nums</code>를 초기화하면 \\(O(n)\\) 공간을 차지한다. 따라서 최악 공간 복잡도는 \\(O(n)\\)이다.</li> </ol> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def algorithm(n: int):\n    a = 0               # O(1)\n    b = [0] * 10000     # O(1)\n    if n &gt; 10:\n        nums = [0] * n  # O(n)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    vector&lt;int&gt; b(10000);    // O(1)\n    if (n &gt; 10)\n        vector&lt;int&gt; nums(n); // O(n)\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10)\n        int[] nums = new int[n]; // O(n)\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10) {\n        int[] nums = new int[n]; // O(n)\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 0                      // O(1)\n    b := make([]int, 10000)     // O(1)\n    var nums []int\n    if n &gt; 10 {\n        nums := make([]int, n)  // O(n)\n    }\n    fmt.Println(a, b, nums)\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    let a = 0 // O(1)\n    let b = Array(repeating: 0, count: 10000) // O(1)\n    if n &gt; 10 {\n        let nums = Array(repeating: 0, count: n) // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 0;                            // O(1)\n  List&lt;int&gt; b = List.filled(10000, 0);  // O(1)\n  if (n &gt; 10) {\n    List&lt;int&gt; nums = List.filled(n, 0); // O(n)\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let a = 0;                              // O(1)\n    let b = [0; 10000];                     // O(1)\n    if n &gt; 10 {\n        let nums = vec![0; n as usize];     // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    int b[10000];            // O(1)\n    if (n &gt; 10)\n        int nums[n] = {0};   // O(n)\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    val a = 0                    // O(1)\n    val b = IntArray(10000)      // O(1)\n    if (n &gt; 10) {\n        val nums = IntArray(n)   // O(n)\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 0                           # O(1)\n    b = Array.new(10000)            # O(1)\n    nums = Array.new(n) if n &gt; 10   # O(n)\nend\n</code></pre> <p>재귀 함수에서는 스택 프레임 공간을 계산하는 데 주의해야 한다. 다음 코드를 살펴보자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def function() -&gt; int:\n    # 몇 가지 연산 수행\n    return 0\n\ndef loop(n: int):\n    \"\"\"반복문의 공간 복잡도는 O(1)\"\"\"\n    for _ in range(n):\n        function()\n\ndef recur(n: int):\n    \"\"\"재귀의 공간 복잡도는 O(n)\"\"\"\n    if n == 1:\n        return\n    return recur(n - 1)\n</code></pre> <pre><code>int func() {\n    // 몇 가지 연산 수행\n    return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n    // 몇 가지 연산 수행\n    return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    recur(n - 1);\n}\n</code></pre> <pre><code>int Function() {\n    // 몇 가지 연산 수행\n    return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nvoid Loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nint Recur(int n) {\n    if (n == 1) return 1;\n    return Recur(n - 1);\n}\n</code></pre> <pre><code>func function() int {\n    // 몇 가지 연산 수행\n    return 0\n}\n\n/* 반복문의 공간 복잡도는 O(1) */\nfunc loop(n int) {\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n}\n\n/* 재귀의 공간 복잡도는 O(n) */\nfunc recur(n int) {\n    if n == 1 {\n        return\n    }\n    recur(n - 1)\n}\n</code></pre> <pre><code>@discardableResult\nfunc function() -&gt; Int {\n    // 몇 가지 연산 수행\n    return 0\n}\n\n/* 반복문의 공간 복잡도는 O(1) */\nfunc loop(n: Int) {\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n\n/* 재귀의 공간 복잡도는 O(n) */\nfunc recur(n: Int) {\n    if n == 1 {\n        return\n    }\n    recur(n: n - 1)\n}\n</code></pre> <pre><code>function constFunc() {\n    // 몇 가지 연산 수행\n    return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nfunction loop(n) {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nfunction recur(n) {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>function constFunc(): number {\n    // 몇 가지 연산 수행\n    return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nfunction loop(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nfunction recur(n: number): void {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n  // 몇 가지 연산 수행\n  return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nvoid loop(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nvoid recur(int n) {\n  if (n == 1) return;\n  recur(n - 1);\n}\n</code></pre> <pre><code>fn function() -&gt; i32 {\n    // 몇 가지 연산 수행\n    return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nfn loop(n: i32) {\n    for i in 0..n {\n        function();\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nfn recur(n: i32) {\n    if n == 1 {\n        return;\n    }\n    recur(n - 1);\n}\n</code></pre> <pre><code>int func() {\n    // 몇 가지 연산 수행\n    return 0;\n}\n/* 반복문의 공간 복잡도는 O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    recur(n - 1);\n}\n</code></pre> <pre><code>fun function(): Int {\n    // 몇 가지 연산 수행\n    return 0\n}\n/* 반복문의 공간 복잡도는 O(1) */\nfun loop(n: Int) {\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n/* 재귀의 공간 복잡도는 O(n) */\nfun recur(n: Int) {\n    if (n == 1) return\n    return recur(n - 1)\n}\n</code></pre> <pre><code>def function\n    # 몇 가지 연산 수행\n    0\nend\n\n### 반복문의 공간 복잡도는 O(1) ###\ndef loop(n)\n    (0...n).each { function }\nend\n\n### 재귀의 공간 복잡도는 O(n) ###\ndef recur(n)\n    return if n == 1\n    recur(n - 1)\nend\n</code></pre> <p>함수 <code>loop()</code>와 <code>recur()</code>의 시간 복잡도는 모두 \\(O(n)\\)이지만 공간 복잡도는 다르다.</p> <ul> <li>함수 <code>loop()</code>는 반복문에서 <code>function()</code>을 \\(n\\)번 호출한다. 매번 <code>function()</code>이 반환되어 스택 프레임 공간이 해제되므로, 공간 복잡도는 여전히 \\(O(1)\\)이다.</li> <li>재귀 함수 <code>recur()</code>의 실행 중에는 아직 반환되지 않은 <code>recur()</code>가 \\(n\\)개 동시에 존재하므로, \\(O(n)\\)의 스택 프레임 공간을 차지한다.</li> </ul>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#243","level":2,"title":"2.4.3   대표적인 유형","text":"<p>입력 데이터 크기를 \\(n\\)이라고 할 때, 아래 그림(2-16)은 대표적인 공간 복잡도 유형을 낮은 순서대로 보여 준다.</p> \\[ \\begin{aligned} &amp; O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \\newline &amp; \\text{상수} &lt; \\text{로그} &lt; \\text{선형} &lt; \\text{제곱} &lt; \\text{지수} \\end{aligned} \\] <p></p> <p> 그림 2-16   대표적인 공간 복잡도 유형 </p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#1-o1","level":3,"title":"1.   상수 공간 \\(O(1)\\)","text":"<p>상수 공간은 개수가 입력 데이터 크기 \\(n\\)과 무관한 상수, 변수, 객체에서 흔히 나타난다.</p> <p>반복문에서 변수를 초기화하거나 함수를 호출할 때 사용하는 메모리는 다음 반복에 들어가면 해제되므로 공간이 누적되지 않는다. 따라서 공간 복잡도는 여전히 \\(O(1)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def function() -&gt; int:\n    \"\"\"함수\"\"\"\n    # 몇 가지 연산을 수행한다\n    return 0\n\ndef constant(n: int):\n    \"\"\"상수 차수\"\"\"\n    # 상수, 변수, 객체는 O(1) 공간을 차지한다\n    a = 0\n    nums = [0] * 10000\n    node = ListNode(0)\n    # 반복문 안의 변수는 O(1) 공간을 차지한다\n    for _ in range(n):\n        c = 0\n    # 반복문 안의 함수는 O(1) 공간을 차지한다\n    for _ in range(n):\n        function()\n</code></pre> space_complexity.cpp<pre><code>/* 함수 */\nint func() {\n    // 몇 가지 연산을 수행한다\n    return 0;\n}\n\n/* 상수 차수 */\nvoid constant(int n) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    const int a = 0;\n    int b = 0;\n    vector&lt;int&gt; nums(10000);\n    ListNode node(0);\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* 함수 */\nint function() {\n    // 몇 가지 연산을 수행한다\n    return 0;\n}\n\n/* 상수 차수 */\nvoid constant(int n) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    final int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new ListNode(0);\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* 함수 */\nint Function() {\n    // 몇 가지 연산을 수행한다\n    return 0;\n}\n\n/* 상수 차수 */\nvoid Constant(int n) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new(0);\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* 함수 */\nfunc function() int {\n    // 몇 가지 연산을 수행한다...\n    return 0\n}\n\n/* 상수 차수 */\nfunc spaceConstant(n int) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    const a = 0\n    b := 0\n    nums := make([]int, 10000)\n    node := newNode(0)\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    var c int\n    for i := 0; i &lt; n; i++ {\n        c = 0\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n    b += 0\n    c += 0\n    nums[0] = 0\n    node.val = 0\n}\n</code></pre> space_complexity.swift<pre><code>/* 함수 */\n@discardableResult\nfunc function() -&gt; Int {\n    // 몇 가지 연산을 수행한다\n    return 0\n}\n\n/* 상수 차수 */\nfunc constant(n: Int) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    let a = 0\n    var b = 0\n    let nums = Array(repeating: 0, count: 10000)\n    let node = ListNode(x: 0)\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for _ in 0 ..&lt; n {\n        let c = 0\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n</code></pre> space_complexity.js<pre><code>/* 함수 */\nfunction constFunc() {\n    // 몇 가지 연산을 수행한다\n    return 0;\n}\n\n/* 상수 차수 */\nfunction constant(n) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* 함수 */\nfunction constFunc(): number {\n    // 몇 가지 연산을 수행한다\n    return 0;\n}\n\n/* 상수 차수 */\nfunction constant(n: number): void {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* 함수 */\nint function() {\n  // 몇 가지 연산을 수행한다\n  return 0;\n}\n\n/* 상수 차수 */\nvoid constant(int n) {\n  // 상수, 변수, 객체는 O(1) 공간을 차지한다\n  final int a = 0;\n  int b = 0;\n  List&lt;int&gt; nums = List.filled(10000, 0);\n  ListNode node = ListNode(0);\n  // 반복문 안의 변수는 O(1) 공간을 차지한다\n  for (var i = 0; i &lt; n; i++) {\n    int c = 0;\n  }\n  // 반복문 안의 함수는 O(1) 공간을 차지한다\n  for (var i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* 함수 */\nfn function() -&gt; i32 {\n    // 몇 가지 연산을 수행한다\n    return 0;\n}\n\n/* 상수 차수 */\n#[allow(unused)]\nfn constant(n: i32) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    const A: i32 = 0;\n    let b = 0;\n    let nums = vec![0; 10000];\n    let node = ListNode::new(0);\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for i in 0..n {\n        let c = 0;\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for i in 0..n {\n        function();\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* 함수 */\nint func() {\n    // 몇 가지 연산을 수행한다\n    return 0;\n}\n\n/* 상수 차수 */\nvoid constant(int n) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    const int a = 0;\n    int b = 0;\n    int nums[1000];\n    ListNode *node = newListNode(0);\n    free(node);\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* 함수 */\nfun function(): Int {\n    // 몇 가지 연산을 수행한다\n    return 0\n}\n\n/* 상수 차수 */\nfun constant(n: Int) {\n    // 상수, 변수, 객체는 O(1) 공간을 차지한다\n    val a = 0\n    var b = 0\n    val nums = Array(10000) { 0 }\n    val node = ListNode(0)\n    // 반복문 안의 변수는 O(1) 공간을 차지한다\n    for (i in 0..&lt;n) {\n        val c = 0\n    }\n    // 반복문 안의 함수는 O(1) 공간을 차지한다\n    for (i in 0..&lt;n) {\n        function()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### 함수 ###\ndef function\n  # 몇 가지 연산을 수행한다\n  0\nend\n\n### 상수 차수 ###\ndef constant(n)\n  # 상수, 변수, 객체는 O(1) 공간을 차지한다\n  a = 0\n  nums = [0] * 10000\n  node = ListNode.new\n\n  # 반복문 안의 변수는 O(1) 공간을 차지한다\n  (0...n).each { c = 0 }\n  # 반복문 안의 함수는 O(1) 공간을 차지한다\n  (0...n).each { function }\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#2-on","level":3,"title":"2.   선형 공간 \\(O(n)\\)","text":"<p>선형 공간은 원소 수가 \\(n\\)에 비례하는 배열, 연결 리스트, 스택, 큐 등에 흔히 나타난다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def linear(n: int):\n    \"\"\"선형 차수\"\"\"\n    # 길이가 n인 리스트는 O(n) 공간을 차지한다\n    nums = [0] * n\n    # 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    hmap = dict[int, str]()\n    for i in range(n):\n        hmap[i] = str(i)\n</code></pre> space_complexity.cpp<pre><code>/* 선형 차수 */\nvoid linear(int n) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    vector&lt;int&gt; nums(n);\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    vector&lt;ListNode&gt; nodes;\n    for (int i = 0; i &lt; n; i++) {\n        nodes.push_back(ListNode(i));\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    unordered_map&lt;int, string&gt; map;\n    for (int i = 0; i &lt; n; i++) {\n        map[i] = to_string(i);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* 선형 차수 */\nvoid linear(int n) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    int[] nums = new int[n];\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    List&lt;ListNode&gt; nodes = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        nodes.add(new ListNode(i));\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    Map&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        map.put(i, String.valueOf(i));\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* 선형 차수 */\nvoid Linear(int n) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    int[] nums = new int[n];\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    List&lt;ListNode&gt; nodes = [];\n    for (int i = 0; i &lt; n; i++) {\n        nodes.Add(new ListNode(i));\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    Dictionary&lt;int, string&gt; map = [];\n    for (int i = 0; i &lt; n; i++) {\n        map.Add(i, i.ToString());\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* 선형 차수 */\nfunc spaceLinear(n int) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    _ = make([]int, n)\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    var nodes []*node\n    for i := 0; i &lt; n; i++ {\n        nodes = append(nodes, newNode(i))\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    m := make(map[int]string, n)\n    for i := 0; i &lt; n; i++ {\n        m[i] = strconv.Itoa(i)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* 선형 차수 */\nfunc linear(n: Int) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    let nums = Array(repeating: 0, count: n)\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    let nodes = (0 ..&lt; n).map { ListNode(x: $0) }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    let map = Dictionary(uniqueKeysWithValues: (0 ..&lt; n).map { ($0, \"\\($0)\") })\n}\n</code></pre> space_complexity.js<pre><code>/* 선형 차수 */\nfunction linear(n) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    const nums = new Array(n);\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    const nodes = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* 선형 차수 */\nfunction linear(n: number): void {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    const nums = new Array(n);\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    const nodes: ListNode[] = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* 선형 차수 */\nvoid linear(int n) {\n  // 길이가 n인 배열은 O(n) 공간을 차지한다\n  List&lt;int&gt; nums = List.filled(n, 0);\n  // 길이가 n인 리스트는 O(n) 공간을 차지한다\n  List&lt;ListNode&gt; nodes = [];\n  for (var i = 0; i &lt; n; i++) {\n    nodes.add(ListNode(i));\n  }\n  // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n  Map&lt;int, String&gt; map = HashMap();\n  for (var i = 0; i &lt; n; i++) {\n    map.putIfAbsent(i, () =&gt; i.toString());\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* 선형 차수 */\n#[allow(unused)]\nfn linear(n: i32) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    let mut nums = vec![0; n as usize];\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    let mut nodes = Vec::new();\n    for i in 0..n {\n        nodes.push(ListNode::new(i))\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    let mut map = HashMap::new();\n    for i in 0..n {\n        map.insert(i, i.to_string());\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* 해시 테이블 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // uthash.h를 기반으로 구현한다\n} HashTable;\n\n/* 선형 차수 */\nvoid linear(int n) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    int *nums = malloc(sizeof(int) * n);\n    free(nums);\n\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    ListNode **nodes = malloc(sizeof(ListNode *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        nodes[i] = newListNode(i);\n    }\n    // 메모리 해제\n    for (int i = 0; i &lt; n; i++) {\n        free(nodes[i]);\n    }\n    free(nodes);\n\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    HashTable *h = NULL;\n    for (int i = 0; i &lt; n; i++) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = i;\n        tmp-&gt;val = i;\n        HASH_ADD_INT(h, key, tmp);\n    }\n\n    // 메모리 해제\n    HashTable *curr, *tmp;\n    HASH_ITER(hh, h, curr, tmp) {\n        HASH_DEL(h, curr);\n        free(curr);\n    }\n}\n</code></pre> space_complexity.kt<pre><code>/* 선형 차수 */\nfun linear(n: Int) {\n    // 길이가 n인 배열은 O(n) 공간을 차지한다\n    val nums = Array(n) { 0 }\n    // 길이가 n인 리스트는 O(n) 공간을 차지한다\n    val nodes = mutableListOf&lt;ListNode&gt;()\n    for (i in 0..&lt;n) {\n        nodes.add(ListNode(i))\n    }\n    // 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n    val map = mutableMapOf&lt;Int, String&gt;()\n    for (i in 0..&lt;n) {\n        map[i] = i.toString()\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### 선형 차수 ###\ndef linear(n)\n  # 길이가 n인 리스트는 O(n) 공간을 차지한다\n  nums = Array.new(n, 0)\n\n  # 길이가 n인 해시 테이블은 O(n) 공간을 차지한다\n  hmap = {}\n  for i in 0...n\n    hmap[i] = i.to_s\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(2-17)과 같이 이 함수의 재귀 깊이는 \\(n\\)이다. 즉, 아직 반환되지 않은 <code>linear_recur()</code> 함수가 \\(n\\)개 동시에 존재하며 \\(O(n)\\) 크기의 스택 프레임 공간을 사용한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def linear_recur(n: int):\n    \"\"\"선형 차수(재귀 구현)\"\"\"\n    print(\"재귀 n =\", n)\n    if n == 1:\n        return\n    linear_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* 선형 차수(재귀 구현) */\nvoid linearRecur(int n) {\n    cout &lt;&lt; \"재귀 n = \" &lt;&lt; n &lt;&lt; endl;\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* 선형 차수(재귀 구현) */\nvoid linearRecur(int n) {\n    System.out.println(\"재귀 n = \" + n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* 선형 차수(재귀 구현) */\nvoid LinearRecur(int n) {\n    Console.WriteLine(\"재귀 n = \" + n);\n    if (n == 1) return;\n    LinearRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* 선형 차수(재귀 구현) */\nfunc spaceLinearRecur(n int) {\n    fmt.Println(\"재귀 n =\", n)\n    if n == 1 {\n        return\n    }\n    spaceLinearRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* 선형 차수(재귀 구현) */\nfunc linearRecur(n: Int) {\n    print(\"재귀 n = \\(n)\")\n    if n == 1 {\n        return\n    }\n    linearRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* 선형 차수(재귀 구현) */\nfunction linearRecur(n) {\n    console.log(`재귀 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* 선형 차수(재귀 구현) */\nfunction linearRecur(n: number): void {\n    console.log(`재귀 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* 선형 차수(재귀 구현) */\nvoid linearRecur(int n) {\n  print('재귀 n = $n');\n  if (n == 1) return;\n  linearRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* 선형 차수(재귀 구현) */\nfn linear_recur(n: i32) {\n    println!(\"재귀 n = {}\", n);\n    if n == 1 {\n        return;\n    };\n    linear_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* 선형 차수(재귀 구현) */\nvoid linearRecur(int n) {\n    printf(\"재귀 n = %d\\r\\n\", n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.kt<pre><code>/* 선형 차수(재귀 구현) */\nfun linearRecur(n: Int) {\n    println(\"재귀 n = $n\")\n    if (n == 1)\n        return\n    linearRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>### 선형 차수(재귀 구현)###\ndef linear_recur(n)\n  puts \"재귀 n = #{n}\"\n  return if n == 1\n  linear_recur(n - 1)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 2-17   재귀 함수에서 발생하는 선형 공간 복잡도 </p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#3-on2","level":3,"title":"3.   제곱 공간 \\(O(n^2)\\)","text":"<p>제곱 공간은 원소 수가 \\(n\\)의 제곱에 비례하는 행렬과 그래프에 흔히 나타난다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def quadratic(n: int):\n    \"\"\"제곱 차수\"\"\"\n    # 이차원 리스트는 O(n^2) 공간을 차지한다\n    num_matrix = [[0] * n for _ in range(n)]\n</code></pre> space_complexity.cpp<pre><code>/* 제곱 차수 */\nvoid quadratic(int n) {\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    vector&lt;vector&lt;int&gt;&gt; numMatrix;\n    for (int i = 0; i &lt; n; i++) {\n        vector&lt;int&gt; tmp;\n        for (int j = 0; j &lt; n; j++) {\n            tmp.push_back(0);\n        }\n        numMatrix.push_back(tmp);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* 제곱 차수 */\nvoid quadratic(int n) {\n    // 행렬은 O(n^2) 공간을 차지한다\n    int[][] numMatrix = new int[n][n];\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    List&lt;List&lt;Integer&gt;&gt; numList = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            tmp.add(0);\n        }\n        numList.add(tmp);\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* 제곱 차수 */\nvoid Quadratic(int n) {\n    // 행렬은 O(n^2) 공간을 차지한다\n    int[,] numMatrix = new int[n, n];\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    List&lt;List&lt;int&gt;&gt; numList = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;int&gt; tmp = [];\n        for (int j = 0; j &lt; n; j++) {\n            tmp.Add(0);\n        }\n        numList.Add(tmp);\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* 제곱 차수 */\nfunc spaceQuadratic(n int) {\n    // 행렬은 O(n^2) 공간을 차지한다\n    numMatrix := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        numMatrix[i] = make([]int, n)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* 제곱 차수 */\nfunc quadratic(n: Int) {\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    let numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n</code></pre> space_complexity.js<pre><code>/* 제곱 차수 */\nfunction quadratic(n) {\n    // 행렬은 O(n^2) 공간을 차지한다\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* 제곱 차수 */\nfunction quadratic(n: number): void {\n    // 행렬은 O(n^2) 공간을 차지한다\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* 제곱 차수 */\nvoid quadratic(int n) {\n  // 행렬은 O(n^2) 공간을 차지한다\n  List&lt;List&lt;int&gt;&gt; numMatrix = List.generate(n, (_) =&gt; List.filled(n, 0));\n  // 이차원 리스트는 O(n^2) 공간을 차지한다\n  List&lt;List&lt;int&gt;&gt; numList = [];\n  for (var i = 0; i &lt; n; i++) {\n    List&lt;int&gt; tmp = [];\n    for (int j = 0; j &lt; n; j++) {\n      tmp.add(0);\n    }\n    numList.add(tmp);\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* 제곱 차수 */\n#[allow(unused)]\nfn quadratic(n: i32) {\n    // 행렬은 O(n^2) 공간을 차지한다\n    let num_matrix = vec![vec![0; n as usize]; n as usize];\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    let mut num_list = Vec::new();\n    for i in 0..n {\n        let mut tmp = Vec::new();\n        for j in 0..n {\n            tmp.push(0);\n        }\n        num_list.push(tmp);\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* 제곱 차수 */\nvoid quadratic(int n) {\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    int **numMatrix = malloc(sizeof(int *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        int *tmp = malloc(sizeof(int) * n);\n        for (int j = 0; j &lt; n; j++) {\n            tmp[j] = 0;\n        }\n        numMatrix[i] = tmp;\n    }\n\n    // 메모리 해제\n    for (int i = 0; i &lt; n; i++) {\n        free(numMatrix[i]);\n    }\n    free(numMatrix);\n}\n</code></pre> space_complexity.kt<pre><code>/* 제곱 차수 */\nfun quadratic(n: Int) {\n    // 행렬은 O(n^2) 공간을 차지한다\n    val numMatrix = arrayOfNulls&lt;Array&lt;Int&gt;?&gt;(n)\n    // 이차원 리스트는 O(n^2) 공간을 차지한다\n    val numList = mutableListOf&lt;MutableList&lt;Int&gt;&gt;()\n    for (i in 0..&lt;n) {\n        val tmp = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            tmp.add(0)\n        }\n        numList.add(tmp)\n    }\n}\n</code></pre> space_complexity.rb<pre><code>### 제곱 차수 ###\ndef quadratic(n)\n  # 이차원 리스트는 O(n^2) 공간을 차지한다\n  Array.new(n) { Array.new(n, 0) }\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(2-18)과 같이 이 함수의 재귀 깊이는 \\(n\\)이고, 각 재귀 함수에서 길이가 각각 \\(n\\), \\(n-1\\), \\(\\dots\\), \\(2\\), \\(1\\)인 배열을 초기화한다. 평균 길이는 \\(n / 2\\)이므로 전체적으로 \\(O(n^2)\\) 공간을 차지한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def quadratic_recur(n: int) -&gt; int:\n    \"\"\"제곱 차수(재귀 구현)\"\"\"\n    if n &lt;= 0:\n        return 0\n    # 배열 nums의 길이는 n, n-1, ..., 2, 1이다\n    nums = [0] * n\n    return quadratic_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* 제곱 차수(재귀 구현) */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    vector&lt;int&gt; nums(n);\n    cout &lt;&lt; \"재귀 n = \" &lt;&lt; n &lt;&lt; \" 일 때 nums의 길이 = \" &lt;&lt; nums.size() &lt;&lt; endl;\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* 제곱 차수(재귀 구현) */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    // 배열 nums의 길이는 n, n-1, ..., 2, 1이다\n    int[] nums = new int[n];\n    System.out.println(\"재귀 n = \" + n + \" 일 때 nums의 길이 = \" + nums.length);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* 제곱 차수(재귀 구현) */\nint QuadraticRecur(int n) {\n    if (n &lt;= 0) return 0;\n    int[] nums = new int[n];\n    Console.WriteLine(\"재귀 n = \" + n + \" 일 때 nums의 길이 = \" + nums.Length);\n    return QuadraticRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* 제곱 차수(재귀 구현) */\nfunc spaceQuadraticRecur(n int) int {\n    if n &lt;= 0 {\n        return 0\n    }\n    nums := make([]int, n)\n    fmt.Printf(\"재귀 n = %d에서 nums의 길이 = %d \\n\", n, len(nums))\n    return spaceQuadraticRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* 제곱 차수(재귀 구현) */\n@discardableResult\nfunc quadraticRecur(n: Int) -&gt; Int {\n    if n &lt;= 0 {\n        return 0\n    }\n    // 배열 nums의 길이는 n, n-1, ..., 2, 1이다\n    let nums = Array(repeating: 0, count: n)\n    print(\"재귀(n = \\(n))의 nums 길이 = \\(nums.count)\")\n    return quadraticRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* 제곱 차수(재귀 구현) */\nfunction quadraticRecur(n) {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`재귀 n = ${n}일 때 nums의 길이 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* 제곱 차수(재귀 구현) */\nfunction quadraticRecur(n: number): number {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`재귀 n = ${n}일 때 nums의 길이 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* 제곱 차수(재귀 구현) */\nint quadraticRecur(int n) {\n  if (n &lt;= 0) return 0;\n  List&lt;int&gt; nums = List.filled(n, 0);\n  print('재귀(n = $n)의 nums 길이 = ${nums.length}');\n  return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* 제곱 차수(재귀 구현) */\nfn quadratic_recur(n: i32) -&gt; i32 {\n    if n &lt;= 0 {\n        return 0;\n    };\n    // 배열 nums의 길이는 n, n-1, ..., 2, 1이다\n    let nums = vec![0; n as usize];\n    println!(\"재귀 n = {}에서 nums의 길이 = {}\", n, nums.len());\n    return quadratic_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* 제곱 차수(재귀 구현) */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    int *nums = malloc(sizeof(int) * n);\n    printf(\"재귀 n = %d에서 nums의 길이 = %d\\r\\n\", n, n);\n    int res = quadraticRecur(n - 1);\n    free(nums);\n    return res;\n}\n</code></pre> space_complexity.kt<pre><code>/* 제곱 차수(재귀 구현) */\ntailrec fun quadraticRecur(n: Int): Int {\n    if (n &lt;= 0)\n        return 0\n    // 배열 nums의 길이는 n, n-1, ..., 2, 1이다\n    val nums = Array(n) { 0 }\n    println(\"재귀(n = $n)의 nums 길이 = ${nums.size}\")\n    return quadraticRecur(n - 1)\n}\n</code></pre> space_complexity.rb<pre><code>### 제곱 차수(재귀 구현)###\ndef quadratic_recur(n)\n  return 0 unless n &gt; 0\n\n  # 배열 nums의 길이는 n, n-1, ..., 2, 1이다\n  nums = Array.new(n, 0)\n  quadratic_recur(n - 1)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 2-18   재귀 함수에서 발생하는 제곱 공간 복잡도 </p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#4-o2n","level":3,"title":"4.   지수 공간 \\(O(2^n)\\)","text":"<p>지수 공간은 이진 트리에 흔히 나타난다. 아래 그림(2-19)에서 층수가 \\(n\\)인 ‘포화 이진 트리’의 노드 수는 \\(2^n - 1\\)이므로 \\(O(2^n)\\) 공간을 차지한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby space_complexity.py<pre><code>def build_tree(n: int) -&gt; TreeNode | None:\n    \"\"\"지수 차수(포화 이진 트리 구성)\"\"\"\n    if n == 0:\n        return None\n    root = TreeNode(0)\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n    return root\n</code></pre> space_complexity.cpp<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return nullptr;\n    TreeNode *root = new TreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.java<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nTreeNode buildTree(int n) {\n    if (n == 0)\n        return null;\n    TreeNode root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.cs<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nTreeNode? BuildTree(int n) {\n    if (n == 0) return null;\n    TreeNode root = new(0) {\n        left = BuildTree(n - 1),\n        right = BuildTree(n - 1)\n    };\n    return root;\n}\n</code></pre> space_complexity.go<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nfunc buildTree(n int) *TreeNode {\n    if n == 0 {\n        return nil\n    }\n    root := NewTreeNode(0)\n    root.Left = buildTree(n - 1)\n    root.Right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.swift<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nfunc buildTree(n: Int) -&gt; TreeNode? {\n    if n == 0 {\n        return nil\n    }\n    let root = TreeNode(x: 0)\n    root.left = buildTree(n: n - 1)\n    root.right = buildTree(n: n - 1)\n    return root\n}\n</code></pre> space_complexity.js<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nfunction buildTree(n) {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.ts<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nfunction buildTree(n: number): TreeNode | null {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.dart<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nTreeNode? buildTree(int n) {\n  if (n == 0) return null;\n  TreeNode root = TreeNode(0);\n  root.left = buildTree(n - 1);\n  root.right = buildTree(n - 1);\n  return root;\n}\n</code></pre> space_complexity.rs<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nfn build_tree(n: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    if n == 0 {\n        return None;\n    };\n    let root = TreeNode::new(0);\n    root.borrow_mut().left = build_tree(n - 1);\n    root.borrow_mut().right = build_tree(n - 1);\n    return Some(root);\n}\n</code></pre> space_complexity.c<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return NULL;\n    TreeNode *root = newTreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.kt<pre><code>/* 지수 차수(포화 이진 트리 구성) */\nfun buildTree(n: Int): TreeNode? {\n    if (n == 0)\n        return null\n    val root = TreeNode(0)\n    root.left = buildTree(n - 1)\n    root.right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.rb<pre><code>### 지수 차수(포화 이진 트리 구성)###\ndef build_tree(n)\n  return if n == 0\n\n  TreeNode.new.tap do |root|\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 2-19   포화 이진 트리에서 발생하는 지수 공간 복잡도 </p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#5-olog-n","level":3,"title":"5.   로그 공간 \\(O(\\log n)\\)","text":"<p>로그 공간은 분할 정복 알고리즘에 흔히 나타난다. 예를 들어 병합 정렬은 길이가 \\(n\\)인 배열을 입력받아 재귀 호출마다 중간 지점에서 반으로 나눈다. 높이가 \\(\\log n\\)인 재귀 트리가 만들어지므로 \\(O(\\log n)\\)의 스택 프레임 공간을 사용한다.</p> <p>또 다른 예로 수를 문자열로 변환하는 경우를 보자. 양의 정수 \\(n\\)을 입력받으면 자릿수는 \\(\\lfloor \\log_{10} n \\rfloor + 1\\)이며, 대응하는 문자열 길이도 \\(\\lfloor \\log_{10} n \\rfloor + 1\\)이다. 따라서 공간 복잡도는 \\(O(\\log_{10} n + 1) = O(\\log n)\\)이다.</p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/space_complexity/#244","level":2,"title":"2.4.4   시간과 공간의 절충","text":"<p>이상적으로는 알고리즘의 시간 복잡도와 공간 복잡도가 모두 최적이기를 바란다. 그러나 실제로 두 복잡도를 동시에 최적화하기는 대개 매우 어렵다.</p> <p>시간 복잡도를 낮추려면 보통 공간 복잡도가 높아지는 대가를 치러야 하며, 그 반대도 마찬가지다. 메모리 공간을 더 사용하여 실행 속도를 높이는 방식을 ‘공간으로 시간을 절약하기’, 반대의 방식을 ‘시간으로 공간을 절약하기’라고 한다.</p> <p>어떤 방식을 선택할지는 어느 쪽을 더 중시하는지에 달려 있다. 대부분의 경우 시간이 공간보다 귀하므로 ‘공간으로 시간을 절약하기’가 더 널리 쓰인다. 물론 데이터 양이 매우 많다면 공간 복잡도를 관리하는 것 역시 중요하다.</p>","path":["제 2 장   복잡도 분석","2.4   공간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/summary/","level":1,"title":"2.5   요약","text":"","path":["제 2 장   복잡도 분석","2.5   요약"],"tags":[]},{"location":"chapter_computational_complexity/summary/#1","level":3,"title":"1.   핵심 정리","text":"<p>알고리즘 효율 평가</p> <ul> <li>시간 효율과 공간 효율은 알고리즘의 우수성을 평가하는 두 가지 주요 지표다.</li> <li>실제 측정으로 알고리즘 효율을 평가할 수 있지만, 측정 환경의 영향을 없애기 어렵고 많은 컴퓨팅 자원이 든다.</li> <li>복잡도 분석은 실제 측정의 단점을 극복한다. 분석 결과는 모든 실행 플랫폼에 적용할 수 있으며, 데이터 크기에 따른 알고리즘 효율을 보여 준다.</li> </ul> <p>시간 복잡도</p> <ul> <li>시간 복잡도는 데이터 양이 증가할 때 알고리즘 실행 시간이 증가하는 추세를 측정하여 효율을 효과적으로 평가한다. 다만 입력 데이터가 적거나 시간 복잡도가 같은 경우처럼, 알고리즘 효율의 우열을 정확히 비교할 수 없는 상황도 있다.</li> <li>최악 시간 복잡도는 빅 \\(O\\) 표기법으로 나타내며 함수의 점근적 상한에 해당한다. \\(n\\)이 양의 무한대로 향할 때 연산 횟수 \\(T(n)\\)의 증가 차수를 반영한다.</li> <li>시간 복잡도를 구하는 과정은 두 단계다. 먼저 연산 횟수를 세고, 그다음 점근적 상한을 판단한다.</li> <li>대표적인 시간 복잡도를 낮은 순서대로 나열하면 \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n \\log n)\\), \\(O(n^2)\\), \\(O(2^n)\\), \\(O(n!)\\) 등이 있다.</li> <li>일부 알고리즘의 시간 복잡도는 고정되어 있지 않고 입력 데이터의 분포에 따라 달라진다. 시간 복잡도는 최악, 최선, 평균 시간 복잡도로 나뉜다. 최선의 경우에 도달하려면 대개 입력 데이터가 엄격한 조건을 만족해야 하므로 최선 시간 복잡도는 거의 쓰지 않는다.</li> <li>평균 시간 복잡도는 무작위 입력 데이터에서의 실행 효율을 반영하므로 실제 응용에서의 알고리즘 성능에 가장 가깝다. 이를 계산하려면 입력 데이터의 분포와 이를 종합한 기댓값을 구해야 한다.</li> </ul> <p>공간 복잡도</p> <ul> <li>공간 복잡도는 시간 복잡도와 비슷하게, 데이터 양이 증가할 때 알고리즘이 차지하는 메모리 공간이 증가하는 추세를 측정한다.</li> <li>알고리즘 실행에 쓰이는 메모리 공간은 입력 공간, 임시 공간, 출력 공간으로 나뉜다. 보통 입력 공간은 공간 복잡도 계산에 포함하지 않는다. 임시 공간은 임시 데이터, 스택 프레임 공간, 명령어 공간으로 나뉘며, 스택 프레임 공간은 보통 재귀 함수에서만 공간 복잡도에 영향을 준다.</li> <li>보통 최악 공간 복잡도에만 관심을 둔다. 즉, 가장 불리한 입력 데이터와 실행 시점을 기준으로 공간 복잡도를 계산한다.</li> <li>대표적인 공간 복잡도를 낮은 순서대로 나열하면 \\(O(1)\\), \\(O(\\log n)\\), \\(O(n)\\), \\(O(n^2)\\), \\(O(2^n)\\) 등이 있다.</li> </ul>","path":["제 2 장   복잡도 분석","2.5   요약"],"tags":[]},{"location":"chapter_computational_complexity/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 꼬리 재귀의 공간 복잡도는 \\(O(1)\\)인가?</p> <p>이론적으로 꼬리 재귀 함수의 공간 복잡도는 \\(O(1)\\)로 최적화할 수 있다. 그러나 Java, Python, C++, Go, C# 등 대다수 프로그래밍 언어는 자동 꼬리 재귀 최적화를 지원하지 않으므로, 보통 공간 복잡도를 \\(O(n)\\)으로 본다.</p> <p>Q: 함수와 메서드라는 두 용어는 어떻게 다른가?</p> <p>함수(function)는 독립적으로 실행할 수 있으며, 모든 인수가 명시적으로 전달된다. 메서드(method)는 객체와 연결되어 있고, 자신을 호출하는 객체에 암묵적으로 전달되며, 클래스 인스턴스에 포함된 데이터를 조작할 수 있다.</p> <p>몇 가지 흔히 쓰는 프로그래밍 언어를 예로 살펴보자.</p> <ul> <li>C는 절차적 프로그래밍 언어로 객체 지향 개념이 없으므로 함수만 있다. 다만 구조체(struct)를 만들어 객체 지향 프로그래밍을 흉내 낼 수 있으며, 구조체에 연결된 함수는 다른 언어의 메서드에 해당한다.</li> <li>Java와 C#은 객체 지향 프로그래밍 언어로, 코드 블록(메서드)은 보통 클래스의 일부다. 정적 메서드는 클래스에 묶여 있고 특정 인스턴스 변수에 접근할 수 없으므로 함수와 비슷하게 동작한다.</li> <li>C++와 Python은 절차적 프로그래밍(함수)과 객체 지향 프로그래밍(메서드)을 모두 지원한다.</li> </ul> <p>Q: ‘대표적인 공간 복잡도 유형’ 그림은 사용 공간의 절대적인 크기를 나타내는가?</p> <p>아니다. 이 그림은 공간 복잡도를 보여 주므로, 사용 공간의 절대적인 크기가 아니라 증가 추세를 나타낸다.</p> <p>\\(n = 8\\)이라고 놓으면 각 곡선의 값이 해당 함수와 일치하지 않을 수 있다. 이는 각 곡선에 상수항을 넣어 값의 범위를 보기 편한 크기로 줄였기 때문이다.</p> <p>실제로는 각 방법의 ‘상수항’에 해당하는 비용을 보통 알 수 없으므로, \\(n = 8\\)일 때는 복잡도만으로 최적의 해법을 고르기 어렵다. 반면 \\(n = 8^5\\)일 때는 증가 추세가 지배적인 영향을 주므로 훨씬 쉽게 선택할 수 있다.</p> <p>Q 실제 사용 상황에 따라 시간이나 공간을 희생하도록 알고리즘을 설계하기도 하는가?</p> <p>실제 응용에서는 대부분 공간을 더 사용하여 시간을 줄이는 쪽을 선택한다. 예를 들어 데이터베이스 인덱스에서는 보통 B+ 트리나 해시 인덱스를 구축한다. 메모리 공간을 많이 사용하는 대신 \\(O(\\log n)\\), 나아가 \\(O(1)\\)의 효율적인 조회가 가능하다.</p> <p>공간 자원이 귀한 상황에서는 시간을 더 들여 공간을 절약하기도 한다. 예를 들어 임베디드 개발에서는 장치 메모리가 귀하므로, 해시 테이블 대신 배열에서 순차 탐색을 수행할 수 있다. 메모리 사용량을 줄이는 대가로 탐색 속도가 느려진다.</p>","path":["제 2 장   복잡도 분석","2.5   요약"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/","level":1,"title":"2.3   시간 복잡도","text":"<p>실행 시간은 알고리즘 효율을 직관적이고 정확하게 보여 준다. 코드의 실행 시간을 정확히 추정하려면 어떻게 해야 할까?</p> <ol> <li>실행 플랫폼을 정한다. 하드웨어 구성, 프로그래밍 언어, 시스템 환경 등이 포함되며, 모두 코드의 실행 효율에 영향을 준다.</li> <li>각 연산에 필요한 실행 시간을 평가한다. 예를 들어 덧셈 연산 <code>+</code>는 1 ns, 곱셈 연산 <code>*</code>는 10 ns, 출력 연산 <code>print()</code>는 5 ns가 걸린다고 하자.</li> <li>코드에 있는 모든 연산을 센다. 그런 다음 각 연산의 실행 시간을 모두 더해 전체 실행 시간을 구한다.</li> </ol> <p>예를 들어 다음 코드에서 입력 데이터 크기는 \\(n\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># 어떤 실행 플랫폼에서\ndef algorithm(n: int):\n    a = 2      # 1 ns\n    a = a + 1  # 1 ns\n    a = a * 2  # 10 ns\n    # n번 반복\n    for _ in range(n):  # 1 ns\n        print(0)        # 5 ns\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        cout &lt;&lt; 0 &lt;&lt; endl;         // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        System.out.println(0);     // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nvoid Algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) {  // 1 ns\n        Console.WriteLine(0);      // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nfunc algorithm(n int) {\n    a := 2     // 1 ns\n    a = a + 1  // 1 ns\n    a = a * 2  // 10 ns\n    // n번 반복\n    for i := 0; i &lt; n; i++ {  // 1 ns\n        fmt.Println(a)        // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nfunc algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // n번 반복\n    for _ in 0 ..&lt; n { // 1 ns\n        print(0) // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nfunction algorithm(n) {\n    var a = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // n번 반복\n    for(let i = 0; i &lt; n; i++) { // 1 ns\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nfunction algorithm(n: number): void {\n    var a: number = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // n번 반복\n    for(let i = 0; i &lt; n; i++) { // 1 ns\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nvoid algorithm(int n) {\n  int a = 2; // 1 ns\n  a = a + 1; // 1 ns\n  a = a * 2; // 10 ns\n  // n번 반복\n  for (int i = 0; i &lt; n; i++) { // 1 ns\n    print(0); // 5 ns\n  }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nfn algorithm(n: i32) {\n    let mut a = 2;      // 1 ns\n    a = a + 1;          // 1 ns\n    a = a * 2;          // 10 ns\n    // n번 반복\n    for _ in 0..n {     // 1 ns\n        println!(\"{}\", 0);  // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) {   // 1 ns\n        printf(\"%d\", 0);            // 5 ns\n    }\n}\n</code></pre> <pre><code>// 어떤 실행 플랫폼에서\nfun algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // n번 반복\n    for (i in 0..&lt;n) {  // 1 ns\n        println(0)      // 5 ns\n    }\n}\n</code></pre> <pre><code># 어떤 실행 플랫폼에서\ndef algorithm(n)\n    a = 2       # 1 ns\n    a = a + 1   # 1 ns\n    a = a * 2   # 10 ns\n    # n번 반복\n    (0...n).each do # 1 ns\n        puts 0      # 5 ns\n    end\nend\n</code></pre> <p>위 방법으로 구한 알고리즘의 실행 시간은 \\((6n + 12)\\) ns다.</p> \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] <p>그러나 실제로 알고리즘의 실행 시간을 이렇게 계산하는 것은 합리적이지도 현실적이지도 않다. 우선 알고리즘은 여러 플랫폼에서 실행되어야 하므로 추정 시간을 특정 실행 플랫폼에 묶고 싶지 않다. 또한 각 연산의 실행 시간을 알아내기 어려워 추정 과정이 매우 복잡해진다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#231","level":2,"title":"2.3.1   시간의 증가 추세 계산","text":"<p>시간 복잡도 분석이 계산하는 것은 알고리즘 실행 시간 자체가 아니라, 데이터 양이 많아질 때 알고리즘 실행 시간이 증가하는 추세다.</p> <p>‘시간의 증가 추세’는 추상적인 개념이므로 예를 통해 이해해 보자. 입력 데이터 크기가 \\(n\\)이고, 세 알고리즘 <code>A</code>, <code>B</code>, <code>C</code>가 주어졌다고 하자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># 알고리즘 A의 시간 복잡도: 상수 시간\ndef algorithm_A(n: int):\n    print(0)\n# 알고리즘 B의 시간 복잡도: 선형 시간\ndef algorithm_B(n: int):\n    for _ in range(n):\n        print(0)\n# 알고리즘 C의 시간 복잡도: 상수 시간\ndef algorithm_C(n: int):\n    for _ in range(1000000):\n        print(0)\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nvoid algorithm_A(int n) {\n    cout &lt;&lt; 0 &lt;&lt; endl;\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nvoid algorithm_A(int n) {\n    System.out.println(0);\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        System.out.println(0);\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        System.out.println(0);\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nvoid AlgorithmA(int n) {\n    Console.WriteLine(0);\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nvoid AlgorithmB(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Console.WriteLine(0);\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nvoid AlgorithmC(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        Console.WriteLine(0);\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nfunc algorithm_A(n int) {\n    fmt.Println(0)\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nfunc algorithm_B(n int) {\n    for i := 0; i &lt; n; i++ {\n        fmt.Println(0)\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nfunc algorithm_C(n int) {\n    for i := 0; i &lt; 1000000; i++ {\n        fmt.Println(0)\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nfunc algorithmA(n: Int) {\n    print(0)\n}\n\n// 알고리즘 B의 시간 복잡도: 선형 시간\nfunc algorithmB(n: Int) {\n    for _ in 0 ..&lt; n {\n        print(0)\n    }\n}\n\n// 알고리즘 C의 시간 복잡도: 상수 시간\nfunc algorithmC(n: Int) {\n    for _ in 0 ..&lt; 1_000_000 {\n        print(0)\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nfunction algorithm_A(n) {\n    console.log(0);\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nfunction algorithm_B(n) {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nfunction algorithm_C(n) {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nfunction algorithm_A(n: number): void {\n    console.log(0);\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nfunction algorithm_B(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nfunction algorithm_C(n: number): void {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nvoid algorithmA(int n) {\n  print(0);\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nvoid algorithmB(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    print(0);\n  }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nvoid algorithmC(int n) {\n  for (int i = 0; i &lt; 1000000; i++) {\n    print(0);\n  }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nfn algorithm_A(n: i32) {\n    println!(\"{}\", 0);\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nfn algorithm_B(n: i32) {\n    for _ in 0..n {\n        println!(\"{}\", 0);\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nfn algorithm_C(n: i32) {\n    for _ in 0..1000000 {\n        println!(\"{}\", 0);\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nvoid algorithm_A(int n) {\n    printf(\"%d\", 0);\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        printf(\"%d\", 0);\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        printf(\"%d\", 0);\n    }\n}\n</code></pre> <pre><code>// 알고리즘 A의 시간 복잡도: 상수 시간\nfun algoritm_A(n: Int) {\n    println(0)\n}\n// 알고리즘 B의 시간 복잡도: 선형 시간\nfun algorithm_B(n: Int) {\n    for (i in 0..&lt;n){\n        println(0)\n    }\n}\n// 알고리즘 C의 시간 복잡도: 상수 시간\nfun algorithm_C(n: Int) {\n    for (i in 0..&lt;1000000) {\n        println(0)\n    }\n}\n</code></pre> <pre><code># 알고리즘 A의 시간 복잡도: 상수 시간\ndef algorithm_A(n)\n    puts 0\nend\n\n# 알고리즘 B의 시간 복잡도: 선형 시간\ndef algorithm_B(n)\n    (0...n).each { puts 0 }\nend\n\n# 알고리즘 C의 시간 복잡도: 상수 시간\ndef algorithm_C(n)\n    (0...1_000_000).each { puts 0 }\nend\n</code></pre> <p>아래 그림(2-7)은 이 세 알고리즘 함수의 시간 복잡도를 보여 준다.</p> <ul> <li>알고리즘 <code>A</code>에는 출력 연산이 \\(1\\)번뿐이므로, \\(n\\)이 커져도 실행 시간이 늘어나지 않는다. 이런 알고리즘의 시간 복잡도를 ‘상수 시간’이라고 한다.</li> <li>알고리즘 <code>B</code>의 출력 연산은 \\(n\\)번 반복되므로, \\(n\\)이 커짐에 따라 실행 시간이 선형으로 증가한다. 이런 알고리즘의 시간 복잡도를 ‘선형 시간’이라고 한다.</li> <li>알고리즘 <code>C</code>의 출력 연산은 \\(1000000\\)번 반복된다. 실행 시간은 길지만 입력 데이터 크기 \\(n\\)과는 무관하다. 따라서 <code>C</code>의 시간 복잡도는 <code>A</code>와 같은 ‘상수 시간’이다.</li> </ul> <p></p> <p> 그림 2-7   알고리즘 A, B, C의 시간 증가 추세 </p> <p>알고리즘의 실행 시간을 직접 계산하는 방식과 비교하면 시간 복잡도 분석에는 어떤 특징이 있을까?</p> <ul> <li>시간 복잡도는 알고리즘 효율을 효과적으로 평가할 수 있다. 예를 들어 알고리즘 <code>B</code>의 실행 시간은 선형으로 증가하므로 \\(n &gt; 1\\)이면 알고리즘 <code>A</code>보다 느리고, \\(n &gt; 1000000\\)이면 알고리즘 <code>C</code>보다 느리다. 실제로 입력 데이터 크기 \\(n\\)이 충분히 크면 ‘상수 시간’ 알고리즘은 반드시 ‘선형 시간’ 알고리즘보다 유리하다. 이것이 바로 시간의 증가 추세가 뜻하는 바다.</li> <li>시간 복잡도를 구하는 방법은 더 간단하다. 실행 플랫폼과 연산 종류는 알고리즘 실행 시간의 증가 추세와 무관하다. 따라서 시간 복잡도 분석에서는 모든 연산의 실행 시간을 같은 ‘단위 시간’으로 볼 수 있다. 그러면 ‘연산 실행 시간 계산’을 ‘연산 횟수 계산’으로 단순화할 수 있어 추정이 훨씬 쉬워진다.</li> <li>시간 복잡도에도 한계는 있다. 예를 들어 알고리즘 <code>A</code>와 <code>C</code>의 시간 복잡도는 같지만 실제 실행 시간에는 큰 차이가 있다. 마찬가지로 알고리즘 <code>B</code>의 시간 복잡도는 <code>C</code>보다 높지만, 입력 데이터 크기 \\(n\\)이 작으면 <code>B</code>가 <code>C</code>보다 확실히 유리하다. 이런 경우에는 시간 복잡도만으로 효율의 우열을 판단하기 어렵다. 물론 이러한 문제에도 불구하고 복잡도 분석은 여전히 알고리즘 효율을 평가하는 가장 효과적이고 널리 쓰이는 방법이다.</li> </ul>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#232","level":2,"title":"2.3.2   함수의 점근적 상한","text":"<p>입력 크기가 \\(n\\)인 다음 함수를 살펴보자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def algorithm(n: int):\n    a = 1      # +1\n    a = a + 1  # +1\n    a = a * 2  # +1\n    # n번 반복\n    for i in range(n):  # +1\n        print(0)        # +1\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) { // +1(매번 i ++ 실행)\n        cout &lt;&lt; 0 &lt;&lt; endl;    // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) { // +1(매번 i ++ 실행)\n        System.out.println(0);    // +1\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) {   // +1(매번 i ++ 실행)\n        Console.WriteLine(0);   // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1      // +1\n    a = a + 1   // +1\n    a = a * 2   // +1\n    // n번 반복\n    for i := 0; i &lt; n; i++ {   // +1\n        fmt.Println(a)         // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // n번 반복\n    for _ in 0 ..&lt; n { // +1\n        print(0) // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    var a = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // n번 반복\n    for(let i = 0; i &lt; n; i++){ // +1(매번 i ++ 실행)\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void{\n    var a: number = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // n번 반복\n    for(let i = 0; i &lt; n; i++){ // +1(매번 i ++ 실행)\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +1\n  a = a + 1; // +1\n  a = a * 2; // +1\n  // n번 반복\n  for (int i = 0; i &lt; n; i++) { // +1(매번 i ++ 실행)\n    print(0); // +1\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;   // +1\n    a = a + 1;      // +1\n    a = a * 2;      // +1\n\n    // n번 반복\n    for _ in 0..n { // +1(매번 i ++ 실행)\n        println!(\"{}\", 0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // n번 반복\n    for (int i = 0; i &lt; n; i++) {   // +1(매번 i ++ 실행)\n        printf(\"%d\", 0);            // +1\n    }\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // n번 반복\n    for (i in 0..&lt;n) { // +1(매번 i ++ 실행)\n        println(0) // +1\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 1       # +1\n    a = a + 1   # +1\n    a = a * 2   # +1\n    # n번 반복\n    (0...n).each do # +1\n        puts 0      # +1\n    end\nend\n</code></pre> <p>알고리즘의 연산 횟수를 입력 데이터 크기 \\(n\\)에 대한 함수 \\(T(n)\\)으로 나타내면, 위 함수의 연산 횟수는 다음과 같다.</p> \\[ T(n) = 3 + 2n \\] <p>\\(T(n)\\)은 일차 함수이므로 실행 시간이 선형으로 증가한다. 따라서 시간 복잡도는 선형 시간이다.</p> <p>선형 시간 복잡도를 \\(O(n)\\)으로 표기한다. 이 수학 기호를 빅 \\(O\\) 표기법(big-\\(O\\) notation)이라고 하며, 함수 \\(T(n)\\)의 점근적 상한(asymptotic upper bound)을 나타낸다.</p> <p>시간 복잡도 분석은 본질적으로 ‘연산 횟수 \\(T(n)\\)’의 점근적 상한을 계산하는 것으로, 명확한 수학적 정의가 있다.</p> <p>함수의 점근적 상한</p> <p>양의 실수 \\(c\\)와 실수 \\(n_0\\)가 존재하여 모든 \\(n &gt; n_0\\)에 대해 \\(T(n) \\leq c \\cdot f(n)\\)이면, \\(f(n)\\)은 \\(T(n)\\)의 점근적 상한을 제공한다고 하며 \\(T(n) = O(f(n))\\)으로 표기한다.</p> <p>아래 그림(2-8)과 같이 점근적 상한을 계산한다는 것은 함수 \\(f(n)\\)을 찾는 것이다. 이 함수는 \\(n\\)이 무한대로 향할 때 \\(T(n)\\)과 \\(f(n)\\)의 증가 차수가 같고 상수 계수 \\(c\\)만큼만 차이가 나도록 해야 한다.</p> <p></p> <p> 그림 2-8   함수의 점근적 상한 </p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#233","level":2,"title":"2.3.3   계산 방법","text":"<p>점근적 상한은 수학적인 느낌이 강하므로, 완전히 이해되지 않아도 걱정할 필요 없다. 우선 계산 방법을 익힌 뒤 반복해서 연습하면 그 수학적 의미도 점차 이해할 수 있다.</p> <p>정의에 따르면 \\(f(n)\\)을 정하면 시간 복잡도 \\(O(f(n))\\)을 얻는다. 그렇다면 점근적 상한 \\(f(n)\\)은 어떻게 정할까? 크게 두 단계로 나뉜다. 먼저 연산 횟수를 세고, 그다음 점근적 상한을 판단한다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#1-1","level":3,"title":"1.   1단계: 연산 횟수 세기","text":"<p>코드를 위에서 아래로 한 줄씩 살펴보며 계산하면 된다. 다만 앞서 본 \\(c \\cdot f(n)\\)의 상수 계수 \\(c\\)는 어떤 크기든 가능하므로, 연산 횟수 \\(T(n)\\)의 각종 계수와 상수항은 무시해도 된다. 이 원리를 바탕으로 계산을 단순화하는 다음 요령을 얻을 수 있다.</p> <ol> <li>\\(T(n)\\)의 상수를 무시한다. 상수는 \\(n\\)과 무관하므로 시간 복잡도에 영향을 주지 않는다.</li> <li>모든 계수를 생략한다. 예를 들어 \\(2n\\)번, \\(5n + 1\\)번 등의 반복은 모두 \\(n\\)번으로 단순화할 수 있다. \\(n\\) 앞의 계수는 시간 복잡도에 영향을 주지 않기 때문이다.</li> <li>중첩 반복문에는 곱셈을 사용한다. 전체 연산 횟수는 바깥쪽 반복문과 안쪽 반복문의 연산 횟수의 곱이며, 각 반복문에는 여전히 <code>1.</code>과 <code>2.</code>의 요령을 따로 적용할 수 있다.</li> </ol> <p>다음 함수에 위 요령을 적용하여 연산 횟수를 세어 보자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>def algorithm(n: int):\n    a = 1      # +0(요령 1)\n    a = a + n  # +0(요령 1)\n    # +n(요령 2)\n    for i in range(5 * n + 1):\n        print(0)\n    # +n*n(요령 3)\n    for i in range(2 * n):\n        for j in range(n + 1):\n            print(0)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0(요령 1)\n    a = a + n;  // +0(요령 1)\n    // +n(요령 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n    // +n*n(요령 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            cout &lt;&lt; 0 &lt;&lt; endl;\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0(요령 1)\n    a = a + n;  // +0(요령 1)\n    // +n(요령 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        System.out.println(0);\n    }\n    // +n*n(요령 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            System.out.println(0);\n        }\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +0(요령 1)\n    a = a + n;  // +0(요령 1)\n    // +n(요령 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        Console.WriteLine(0);\n    }\n    // +n*n(요령 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            Console.WriteLine(0);\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1     // +0(요령 1)\n    a = a + n  // +0(요령 1)\n    // +n(요령 2)\n    for i := 0; i &lt; 5 * n + 1; i++ {\n        fmt.Println(0)\n    }\n    // +n*n(요령 3)\n    for i := 0; i &lt; 2 * n; i++ {\n        for j := 0; j &lt; n + 1; j++ {\n            fmt.Println(0)\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +0(요령 1)\n    a = a + n // +0(요령 1)\n    // +n(요령 2)\n    for _ in 0 ..&lt; (5 * n + 1) {\n        print(0)\n    }\n    // +n*n(요령 3)\n    for _ in 0 ..&lt; (2 * n) {\n        for _ in 0 ..&lt; (n + 1) {\n            print(0)\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    let a = 1;  // +0(요령 1)\n    a = a + n;  // +0(요령 1)\n    // +n(요령 2)\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n(요령 3)\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    let a = 1;  // +0(요령 1)\n    a = a + n;  // +0(요령 1)\n    // +n(요령 2)\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n(요령 3)\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +0(요령 1)\n  a = a + n; // +0(요령 1)\n  // +n(요령 2)\n  for (int i = 0; i &lt; 5 * n + 1; i++) {\n    print(0);\n  }\n  // +n*n(요령 3)\n  for (int i = 0; i &lt; 2 * n; i++) {\n    for (int j = 0; j &lt; n + 1; j++) {\n      print(0);\n    }\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;     // +0(요령 1)\n    a = a + n;        // +0(요령 1)\n\n    // +n(요령 2)\n    for i in 0..(5 * n + 1) {\n        println!(\"{}\", 0);\n    }\n\n    // +n*n(요령 3)\n    for i in 0..(2 * n) {\n        for j in 0..(n + 1) {\n            println!(\"{}\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0(요령 1)\n    a = a + n;  // +0(요령 1)\n    // +n(요령 2)\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        printf(\"%d\", 0);\n    }\n    // +n*n(요령 3)\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            printf(\"%d\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>fun algorithm(n: Int) {\n    var a = 1   // +0(요령 1)\n    a = a + n   // +0(요령 1)\n    // +n(요령 2)\n    for (i in 0..&lt;5 * n + 1) {\n        println(0)\n    }\n    // +n*n(요령 3)\n    for (i in 0..&lt;2 * n) {\n        for (j in 0..&lt;n + 1) {\n            println(0)\n        }\n    }\n}\n</code></pre> <pre><code>def algorithm(n)\n    a = 1       # +0(요령 1)\n    a = a + n   # +0(요령 1)\n    # +n(요령 2)\n    (0...(5 * n + 1)).each do { puts 0 }\n    # +n*n(요령 3)\n    (0...(2 * n)).each do\n        (0...(n + 1)).each do { puts 0 }\n    end\nend\n</code></pre> <p>아래 식은 위 요령을 적용하기 전과 후의 계산 결과를 보여 준다. 두 결과에서 구한 시간 복잡도는 모두 \\(O(n^2)\\)이다.</p> \\[ \\begin{aligned} T(n) &amp; = 2n(n + 1) + (5n + 1) + 2 &amp; \\text{빠짐없이 계산 (-.-|||)} \\newline &amp; = 2n^2 + 7n + 3 \\newline T(n) &amp; = n^2 + n &amp; \\text{간단히 계산 (o.O)} \\end{aligned} \\]","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#2-2","level":3,"title":"2.   2단계: 점근적 상한 판단","text":"<p>시간 복잡도는 \\(T(n)\\)에서 가장 높은 차수의 항으로 결정된다. \\(n\\)이 무한대로 향하면 가장 높은 차수의 항이 지배적인 영향을 주므로 나머지 항의 영향은 무시할 수 있기 때문이다.</p> <p>아래 표(2-2)는 몇 가지 예를 보여 준다. 일부 값이 과장되어 있는 이유는 ‘계수로는 차수를 뒤집을 수 없다’는 결론을 강조하기 위해서다. \\(n\\)이 무한대로 향하면 이러한 상수는 중요하지 않게 된다.</p> <p> 표 2-2   연산 횟수에 따른 시간 복잡도 </p> 연산 횟수 \\(T(n)\\) 시간 복잡도 \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#234","level":2,"title":"2.3.4   대표적인 유형","text":"<p>입력 데이터 크기를 \\(n\\)이라고 할 때, 대표적인 시간 복잡도 유형은 아래 그림(2-9)과 같다(낮은 순서대로 나열).</p> \\[ \\begin{aligned} &amp; O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n \\log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!) \\newline &amp; \\text{상수} &lt; \\text{로그} &lt; \\text{선형} &lt; \\text{선형 로그} &lt; \\text{제곱} &lt; \\text{지수} &lt; \\text{팩토리얼} \\end{aligned} \\] <p></p> <p> 그림 2-9   대표적인 시간 복잡도 유형 </p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#1-o1","level":3,"title":"1.   상수 시간 \\(O(1)\\)","text":"<p>상수 시간의 연산 횟수는 입력 데이터 크기 \\(n\\)과 무관하며, \\(n\\)이 변해도 달라지지 않는다.</p> <p>다음 함수에서 연산 횟수 <code>size</code>는 매우 클 수 있지만 입력 데이터 크기 \\(n\\)과 무관하므로, 시간 복잡도는 여전히 \\(O(1)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def constant(n: int) -&gt; int:\n    \"\"\"상수 차수\"\"\"\n    count = 0\n    size = 100000\n    for _ in range(size):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 상수 차수 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 상수 차수 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 상수 차수 */\nint Constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 상수 차수 */\nfunc constant(n int) int {\n    count := 0\n    size := 100000\n    for i := 0; i &lt; size; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 상수 차수 */\nfunc constant(n: Int) -&gt; Int {\n    var count = 0\n    let size = 100_000\n    for _ in 0 ..&lt; size {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 상수 차수 */\nfunction constant(n) {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 상수 차수 */\nfunction constant(n: number): number {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 상수 차수 */\nint constant(int n) {\n  int count = 0;\n  int size = 100000;\n  for (var i = 0; i &lt; size; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 상수 차수 */\nfn constant(n: i32) -&gt; i32 {\n    _ = n;\n    let mut count = 0;\n    let size = 100_000;\n    for _ in 0..size {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 상수 차수 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 상수 차수 */\nfun constant(n: Int): Int {\n    var count = 0\n    val size = 100000\n    for (i in 0..&lt;size)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 상수 차수 ###\ndef constant(n)\n  count = 0\n  size = 100000\n\n  (0...size).each { count += 1 }\n\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#2-on","level":3,"title":"2.   선형 시간 \\(O(n)\\)","text":"<p>선형 시간의 연산 횟수는 입력 데이터 크기 \\(n\\)에 대해 선형으로 증가한다. 선형 시간은 보통 단일 반복문에서 나타난다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def linear(n: int) -&gt; int:\n    \"\"\"선형 차수\"\"\"\n    count = 0\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 선형 차수 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 선형 차수 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 선형 차수 */\nint Linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 선형 차수 */\nfunc linear(n int) int {\n    count := 0\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 선형 차수 */\nfunc linear(n: Int) -&gt; Int {\n    var count = 0\n    for _ in 0 ..&lt; n {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 선형 차수 */\nfunction linear(n) {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 선형 차수 */\nfunction linear(n: number): number {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 선형 차수 */\nint linear(int n) {\n  int count = 0;\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 선형 차수 */\nfn linear(n: i32) -&gt; i32 {\n    let mut count = 0;\n    for _ in 0..n {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 선형 차수 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 선형 차수 */\nfun linear(n: Int): Int {\n    var count = 0\n    for (i in 0..&lt;n)\n        count++\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 선형 차수 ###\ndef linear(n)\n  count = 0\n  (0...n).each { count += 1 }\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>배열 순회와 연결 리스트 순회 등의 연산은 시간 복잡도가 모두 \\(O(n)\\)이다. 여기서 \\(n\\)은 배열이나 연결 리스트의 길이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def array_traversal(nums: list[int]) -&gt; int:\n    \"\"\"선형 차수(배열 순회)\"\"\"\n    count = 0\n    # 반복 횟수는 배열 길이에 비례한다\n    for num in nums:\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 선형 차수(배열 순회) */\nint arrayTraversal(vector&lt;int&gt; &amp;nums) {\n    int count = 0;\n    // 반복 횟수는 배열 길이에 비례한다\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 선형 차수(배열 순회) */\nint arrayTraversal(int[] nums) {\n    int count = 0;\n    // 반복 횟수는 배열 길이에 비례한다\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 선형 차수(배열 순회) */\nint ArrayTraversal(int[] nums) {\n    int count = 0;\n    // 반복 횟수는 배열 길이에 비례한다\n    foreach (int num in nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 선형 차수(배열 순회) */\nfunc arrayTraversal(nums []int) int {\n    count := 0\n    // 반복 횟수는 배열 길이에 비례한다\n    for range nums {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 선형 차수(배열 순회) */\nfunc arrayTraversal(nums: [Int]) -&gt; Int {\n    var count = 0\n    // 반복 횟수는 배열 길이에 비례한다\n    for _ in nums {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 선형 차수(배열 순회) */\nfunction arrayTraversal(nums) {\n    let count = 0;\n    // 반복 횟수는 배열 길이에 비례한다\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 선형 차수(배열 순회) */\nfunction arrayTraversal(nums: number[]): number {\n    let count = 0;\n    // 반복 횟수는 배열 길이에 비례한다\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 선형 차수(배열 순회) */\nint arrayTraversal(List&lt;int&gt; nums) {\n  int count = 0;\n  // 반복 횟수는 배열 길이에 비례한다\n  for (var _num in nums) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 선형 차수(배열 순회) */\nfn array_traversal(nums: &amp;[i32]) -&gt; i32 {\n    let mut count = 0;\n    // 반복 횟수는 배열 길이에 비례한다\n    for _ in nums {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 선형 차수(배열 순회) */\nint arrayTraversal(int *nums, int n) {\n    int count = 0;\n    // 반복 횟수는 배열 길이에 비례한다\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 선형 차수(배열 순회) */\nfun arrayTraversal(nums: IntArray): Int {\n    var count = 0\n    // 반복 횟수는 배열 길이에 비례한다\n    for (num in nums) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 선형 차수(배열 순회)###\ndef array_traversal(nums)\n  count = 0\n\n  # 반복 횟수는 배열 길이에 비례한다\n  for num in nums\n    count += 1\n  end\n\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>입력 데이터 크기 \\(n\\)은 입력 데이터의 종류에 따라 구체적으로 정해야 한다는 점에 주의하자. 첫 번째 예에서는 변수 \\(n\\)이 입력 데이터 크기이고, 두 번째 예에서는 배열의 길이 \\(n\\)이 데이터 크기다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#3-on2","level":3,"title":"3.   제곱 시간 \\(O(n^2)\\)","text":"<p>제곱 시간의 연산 횟수는 입력 데이터 크기 \\(n\\)의 제곱에 비례하여 증가한다. 보통 중첩 반복문에서 나타나며, 바깥쪽과 안쪽 반복문의 시간 복잡도가 각각 \\(O(n)\\)이므로 전체 시간 복잡도는 \\(O(n^2)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def quadratic(n: int) -&gt; int:\n    \"\"\"제곱 차수\"\"\"\n    count = 0\n    # 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for i in range(n):\n        for j in range(n):\n            count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 제곱 차수 */\nint quadratic(int n) {\n    int count = 0;\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 제곱 차수 */\nint quadratic(int n) {\n    int count = 0;\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 제곱 차수 */\nint Quadratic(int n) {\n    int count = 0;\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 제곱 차수 */\nfunc quadratic(n int) int {\n    count := 0\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; n; j++ {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 제곱 차수 */\nfunc quadratic(n: Int) -&gt; Int {\n    var count = 0\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; n {\n            count += 1\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 제곱 차수 */\nfunction quadratic(n) {\n    let count = 0;\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 제곱 차수 */\nfunction quadratic(n: number): number {\n    let count = 0;\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 제곱 차수 */\nint quadratic(int n) {\n  int count = 0;\n  // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n  for (int i = 0; i &lt; n; i++) {\n    for (int j = 0; j &lt; n; j++) {\n      count++;\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 제곱 차수 */\nfn quadratic(n: i32) -&gt; i32 {\n    let mut count = 0;\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for _ in 0..n {\n        for _ in 0..n {\n            count += 1;\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 제곱 차수 */\nint quadratic(int n) {\n    int count = 0;\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 제곱 차수 */\nfun quadratic(n: Int): Int {\n    var count = 0\n    // 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;n) {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 제곱 차수 ###\ndef quadratic(n)\n  count = 0\n\n  # 반복 횟수는 데이터 크기 n의 제곱에 비례한다\n  for i in 0...n\n    for j in 0...n\n      count += 1\n    end\n  end\n\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(2-10)은 상수 시간, 선형 시간, 제곱 시간의 세 가지 시간 복잡도를 비교한다.</p> <p></p> <p> 그림 2-10   상수 시간, 선형 시간, 제곱 시간의 시간 복잡도 </p> <p>버블 정렬을 예로 들면, 바깥쪽 반복문은 \\(n - 1\\)번 실행되고 안쪽 반복문은 \\(n-1\\), \\(n-2\\), \\(\\dots\\), \\(2\\), \\(1\\)번, 평균 \\(n / 2\\)번 실행된다. 따라서 시간 복잡도는 \\(O((n - 1) n / 2) = O(n^2)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def bubble_sort(nums: list[int]) -&gt; int:\n    \"\"\"제곱 차수(버블 정렬)\"\"\"\n    count = 0  # 카운터\n    # 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in range(len(nums) - 1, 0, -1):\n        # 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # nums[j]와 nums[j + 1]을 교환한다\n                tmp: int = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3  # 원소 교환에는 단위 연산 3회가 포함된다\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 제곱 차수(버블 정렬) */\nint bubbleSort(vector&lt;int&gt; &amp;nums) {\n    int count = 0; // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 제곱 차수(버블 정렬) */\nint bubbleSort(int[] nums) {\n    int count = 0; // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 제곱 차수(버블 정렬) */\nint BubbleSort(int[] nums) {\n    int count = 0;  // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                count += 3;  // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 제곱 차수(버블 정렬) */\nfunc bubbleSort(nums []int) int {\n    count := 0 // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                tmp := nums[j]\n                nums[j] = nums[j+1]\n                nums[j+1] = tmp\n                count += 3 // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 제곱 차수(버블 정렬) */\nfunc bubbleSort(nums: inout [Int]) -&gt; Int {\n    var count = 0 // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in nums.indices.dropFirst().reversed() {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3 // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 제곱 차수(버블 정렬) */\nfunction bubbleSort(nums) {\n    let count = 0; // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 제곱 차수(버블 정렬) */\nfunction bubbleSort(nums: number[]): number {\n    let count = 0; // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 제곱 차수(버블 정렬) */\nint bubbleSort(List&lt;int&gt; nums) {\n  int count = 0; // 카운터\n  // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n  for (var i = nums.length - 1; i &gt; 0; i--) {\n    // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n    for (var j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // nums[j]와 nums[j + 1]을 교환한다\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        count += 3; // 원소 교환에는 단위 연산 3회가 포함된다\n      }\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 제곱 차수(버블 정렬) */\nfn bubble_sort(nums: &amp;mut [i32]) -&gt; i32 {\n    let mut count = 0; // 카운터\n\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in (1..nums.len()).rev() {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 제곱 차수(버블 정렬) */\nint bubbleSort(int *nums, int n) {\n    int count = 0; // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = n - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 제곱 차수(버블 정렬) */\nfun bubbleSort(nums: IntArray): Int {\n    var count = 0 // 카운터\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (i in nums.size - 1 downTo 1) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n                count += 3 // 원소 교환에는 단위 연산 3회가 포함된다\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 제곱 차수(버블 정렬)###\ndef bubble_sort(nums)\n  count = 0  # 카운터\n\n  # 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n  for i in (nums.length - 1).downto(0)\n    # 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # nums[j]와 nums[j + 1]을 교환한다\n        tmp = nums[j]\n        nums[j] = nums[j + 1]\n        nums[j + 1] = tmp\n        count += 3 # 원소 교환에는 단위 연산 3회가 포함된다\n      end\n    end\n  end\n\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#4-o2n","level":3,"title":"4.   지수 시간 \\(O(2^n)\\)","text":"<p>생물학의 ‘세포 분열’은 지수적 증가의 대표적인 예다. 처음에는 세포가 \\(1\\)개이고, 한 차례 분열하면 \\(2\\)개, 두 차례 분열하면 \\(4\\)개가 된다. 이처럼 \\(n\\)차례 분열하면 세포는 \\(2^n\\)개가 된다.</p> <p>아래 그림(2-11)과 코드는 세포 분열 과정을 재현하며, 시간 복잡도는 \\(O(2^n)\\)이다. 입력 \\(n\\)은 분열이 진행되는 라운드 수이고, 반환값 <code>count</code>는 전체 분열 횟수라는 점에 주의하자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def exponential(n: int) -&gt; int:\n    \"\"\"지수 차수(반복문 구현)\"\"\"\n    count = 0\n    base = 1\n    # 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for _ in range(n):\n        for _ in range(base):\n            count += 1\n        base *= 2\n    # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 지수 차수(반복문 구현) */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 지수 차수(반복문 구현) */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 지수 차수(반복문 구현) */\nint Exponential(int n) {\n    int count = 0, bas = 1;\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 지수 차수(반복문 구현)*/\nfunc exponential(n int) int {\n    count, base := 0, 1\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; base; j++ {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 지수 차수(반복문 구현) */\nfunc exponential(n: Int) -&gt; Int {\n    var count = 0\n    var base = 1\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; base {\n            count += 1\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 지수 차수(반복문 구현) */\nfunction exponential(n) {\n    let count = 0,\n        base = 1;\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 지수 차수(반복문 구현) */\nfunction exponential(n: number): number {\n    let count = 0,\n        base = 1;\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 지수 차수(반복문 구현) */\nint exponential(int n) {\n  int count = 0, base = 1;\n  // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n  for (var i = 0; i &lt; n; i++) {\n    for (var j = 0; j &lt; base; j++) {\n      count++;\n    }\n    base *= 2;\n  }\n  // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 지수 차수(반복문 구현) */\nfn exponential(n: i32) -&gt; i32 {\n    let mut count = 0;\n    let mut base = 1;\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for _ in 0..n {\n        for _ in 0..base {\n            count += 1\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 지수 차수(반복문 구현) */\nint exponential(int n) {\n    int count = 0;\n    int bas = 1;\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 지수 차수(반복문 구현) */\nfun exponential(n: Int): Int {\n    var count = 0\n    var base = 1\n    // 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n    for (i in 0..&lt;n) {\n        for (j in 0..&lt;base) {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 지수 차수(반복문 구현)###\ndef exponential(n)\n  count, base = 0, 1\n\n  # 세포가 매번 둘로 나뉘어 수열 1, 2, 4, 8, ..., 2^(n-1)을 이룬다\n  (0...n).each do\n    (0...base).each { count += 1 }\n    base *= 2\n  end\n\n  # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 2-11   지수 시간 복잡도 </p> <p>실제 알고리즘에서 지수 시간은 재귀 함수에 자주 나타난다. 예를 들어 다음 코드는 재귀적으로 둘로 나뉘며, \\(n\\)번 나뉜 뒤 멈춘다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def exp_recur(n: int) -&gt; int:\n    \"\"\"지수 차수(재귀 구현)\"\"\"\n    if n == 1:\n        return 1\n    return exp_recur(n - 1) + exp_recur(n - 1) + 1\n</code></pre> time_complexity.cpp<pre><code>/* 지수 차수(재귀 구현) */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* 지수 차수(재귀 구현) */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* 지수 차수(재귀 구현) */\nint ExpRecur(int n) {\n    if (n == 1) return 1;\n    return ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* 지수 차수(재귀 구현)*/\nfunc expRecur(n int) int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n-1) + expRecur(n-1) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* 지수 차수(재귀 구현) */\nfunc expRecur(n: Int) -&gt; Int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* 지수 차수(재귀 구현) */\nfunction expRecur(n) {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* 지수 차수(재귀 구현) */\nfunction expRecur(n: number): number {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* 지수 차수(재귀 구현) */\nint expRecur(int n) {\n  if (n == 1) return 1;\n  return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* 지수 차수(재귀 구현) */\nfn exp_recur(n: i32) -&gt; i32 {\n    if n == 1 {\n        return 1;\n    }\n    exp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* 지수 차수(재귀 구현) */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* 지수 차수(재귀 구현) */\nfun expRecur(n: Int): Int {\n    if (n == 1) {\n        return 1\n    }\n    return expRecur(n - 1) + expRecur(n - 1) + 1\n}\n</code></pre> time_complexity.rb<pre><code>### 지수 차수(재귀 구현)###\ndef exp_recur(n)\n  return 1 if n == 1\n  exp_recur(n - 1) + exp_recur(n - 1) + 1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>지수 시간은 매우 빠르게 증가하며 완전 열거(완전 탐색, 백트래킹 등)에 흔히 나타난다. 데이터가 많은 문제에서는 지수 시간을 감당하기 어려우므로, 보통 동적 계획법이나 탐욕 알고리즘 등으로 해결해야 한다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#5-olog-n","level":3,"title":"5.   로그 시간 \\(O(\\log n)\\)","text":"<p>지수 시간과 반대로 로그 시간은 ‘매번 절반으로 줄어드는’ 상황을 나타낸다. 입력 데이터 크기가 \\(n\\)이고 매번 절반으로 줄어들므로, 반복 횟수는 \\(2^n\\)의 역함수인 \\(\\log_2 n\\)이다.</p> <p>아래 그림(2-12)과 코드는 ‘매번 절반으로 줄어드는’ 과정을 재현한다. 시간 복잡도는 \\(O(\\log_2 n)\\)이며, 줄여서 \\(O(\\log n)\\)으로 쓴다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def logarithmic(n: int) -&gt; int:\n    \"\"\"로그 차수(반복문 구현)\"\"\"\n    count = 0\n    while n &gt; 1:\n        n = n / 2\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 로그 차수(반복문 구현) */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 로그 차수(반복문 구현) */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 로그 차수(반복문 구현) */\nint Logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n /= 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 로그 차수(반복문 구현)*/\nfunc logarithmic(n int) int {\n    count := 0\n    for n &gt; 1 {\n        n = n / 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 로그 차수(반복문 구현) */\nfunc logarithmic(n: Int) -&gt; Int {\n    var count = 0\n    var n = n\n    while n &gt; 1 {\n        n = n / 2\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 로그 차수(반복문 구현) */\nfunction logarithmic(n) {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 로그 차수(반복문 구현) */\nfunction logarithmic(n: number): number {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 로그 차수(반복문 구현) */\nint logarithmic(int n) {\n  int count = 0;\n  while (n &gt; 1) {\n    n = n ~/ 2;\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 로그 차수(반복문 구현) */\nfn logarithmic(mut n: i32) -&gt; i32 {\n    let mut count = 0;\n    while n &gt; 1 {\n        n = n / 2;\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 로그 차수(반복문 구현) */\nint logarithmic(int n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 로그 차수(반복문 구현) */\nfun logarithmic(n: Int): Int {\n    var n1 = n\n    var count = 0\n    while (n1 &gt; 1) {\n        n1 /= 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 로그 차수(반복문 구현)###\ndef logarithmic(n)\n  count = 0\n\n  while n &gt; 1\n    n /= 2\n    count += 1\n  end\n\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 2-12   로그 시간 복잡도 </p> <p>지수 시간처럼 로그 시간도 재귀 함수에 자주 나타난다. 다음 코드는 높이가 \\(\\log_2 n\\)인 재귀 트리를 만든다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def log_recur(n: int) -&gt; int:\n    \"\"\"로그 차수(재귀 구현)\"\"\"\n    if n &lt;= 1:\n        return 0\n    return log_recur(n / 2) + 1\n</code></pre> time_complexity.cpp<pre><code>/* 로그 차수(재귀 구현) */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* 로그 차수(재귀 구현) */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* 로그 차수(재귀 구현) */\nint LogRecur(int n) {\n    if (n &lt;= 1) return 0;\n    return LogRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* 로그 차수(재귀 구현)*/\nfunc logRecur(n int) int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n/2) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* 로그 차수(재귀 구현) */\nfunc logRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n: n / 2) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* 로그 차수(재귀 구현) */\nfunction logRecur(n) {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* 로그 차수(재귀 구현) */\nfunction logRecur(n: number): number {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* 로그 차수(재귀 구현) */\nint logRecur(int n) {\n  if (n &lt;= 1) return 0;\n  return logRecur(n ~/ 2) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* 로그 차수(재귀 구현) */\nfn log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 0;\n    }\n    log_recur(n / 2) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* 로그 차수(재귀 구현) */\nint logRecur(int n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.kt<pre><code>/* 로그 차수(재귀 구현) */\nfun logRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 0\n    return logRecur(n / 2) + 1\n}\n</code></pre> time_complexity.rb<pre><code>### 로그 차수(재귀 구현)###\ndef log_recur(n)\n  return 0 unless n &gt; 1\n  log_recur(n / 2) + 1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>로그 시간은 분할 정복 전략에 기반한 알고리즘에 자주 나타나며, ‘하나를 여러 개로 나누기’, ‘복잡한 것을 단순하게 만들기’라는 알고리즘 사고방식을 보여 준다. 증가 속도가 느려 상수 시간 다음으로 이상적인 시간 복잡도다.</p> <p>\\(O(\\log n)\\)의 밑은 무엇인가?</p> <p>정확히 말하면, ‘하나를 \\(m\\)개로 나누기’에 해당하는 시간 복잡도는 \\(O(\\log_m n)\\)이다. 로그의 밑 변환 공식을 적용하면 밑이 달라도 같은 시간 복잡도를 얻는다.</p> \\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\] <p>즉, 밑 \\(m\\)은 복잡도에 영향을 주지 않고 바꿀 수 있다. 따라서 보통 밑 \\(m\\)을 생략하고 로그 시간을 \\(O(\\log n)\\)으로 표기한다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#6-on-log-n","level":3,"title":"6.   선형 로그 시간 \\(O(n \\log n)\\)","text":"<p>선형 로그 시간은 중첩 반복문에 자주 나타나며, 두 반복문의 시간 복잡도는 각각 \\(O(\\log n)\\)과 \\(O(n)\\)이다. 관련 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def linear_log_recur(n: int) -&gt; int:\n    \"\"\"선형 로그 차수\"\"\"\n    if n &lt;= 1:\n        return 1\n    # 둘로 나누어 부분 문제의 크기를 절반으로 줄인다\n    count = linear_log_recur(n // 2) + linear_log_recur(n // 2)\n    # 현재 부분 문제에는 n회의 연산이 포함된다\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 선형 로그 차수 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 선형 로그 차수 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 선형 로그 차수 */\nint LinearLogRecur(int n) {\n    if (n &lt;= 1) return 1;\n    int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 선형 로그 차수 */\nfunc linearLogRecur(n int) int {\n    if n &lt;= 1 {\n        return 1\n    }\n    count := linearLogRecur(n/2) + linearLogRecur(n/2)\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 선형 로그 차수 */\nfunc linearLogRecur(n: Int) -&gt; Int {\n    if n &lt;= 1 {\n        return 1\n    }\n    var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\n    for _ in stride(from: 0, to: n, by: 1) {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 선형 로그 차수 */\nfunction linearLogRecur(n) {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 선형 로그 차수 */\nfunction linearLogRecur(n: number): number {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 선형 로그 차수 */\nint linearLogRecur(int n) {\n  if (n &lt;= 1) return 1;\n  int count = linearLogRecur(n ~/ 2) + linearLogRecur(n ~/ 2);\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 선형 로그 차수 */\nfn linear_log_recur(n: i32) -&gt; i32 {\n    if n &lt;= 1 {\n        return 1;\n    }\n    let mut count = linear_log_recur(n / 2) + linear_log_recur(n / 2);\n    for _ in 0..n {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> time_complexity.c<pre><code>/* 선형 로그 차수 */\nint linearLogRecur(int n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 선형 로그 차수 */\nfun linearLogRecur(n: Int): Int {\n    if (n &lt;= 1)\n        return 1\n    var count = linearLogRecur(n / 2) + linearLogRecur(n / 2)\n    for (i in 0..&lt;n) {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 선형 로그 차수 ###\ndef linear_log_recur(n)\n  return 1 unless n &gt; 1\n\n  count = linear_log_recur(n / 2) + linear_log_recur(n / 2)\n  (0...n).each { count += 1 }\n\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(2-13)은 선형 로그 시간이 만들어지는 방식을 보여 준다. 이진 트리의 각 층에서 수행하는 전체 연산 횟수는 \\(n\\)이고, 층수는 \\(\\log_2 n + 1\\)이므로 시간 복잡도는 \\(O(n \\log n)\\)이다.</p> <p></p> <p> 그림 2-13   선형 로그 시간 복잡도 </p> <p>퀵 정렬, 병합 정렬, 힙 정렬 등 널리 쓰이는 정렬 알고리즘의 시간 복잡도는 보통 \\(O(n \\log n)\\)이다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#7-on","level":3,"title":"7.   팩토리얼 시간 \\(O(n!)\\)","text":"<p>팩토리얼 시간은 수학의 ‘모든 순열 구하기’ 문제에 해당한다. 서로 다른 원소 \\(n\\)개가 주어졌을 때 가능한 모든 순열을 구하면, 그 개수는 다음과 같다.</p> \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\] <p>팩토리얼은 보통 재귀로 구현한다. 아래 그림(2-14)과 코드에서 첫 번째 층은 \\(n\\)개로, 두 번째 층은 \\(n - 1\\)개로 분기하며, 이를 반복하다가 \\(n\\)번째 층에 도달하면 분기를 멈춘다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby time_complexity.py<pre><code>def factorial_recur(n: int) -&gt; int:\n    \"\"\"팩토리얼 차수(재귀 구현)\"\"\"\n    if n == 0:\n        return 1\n    count = 0\n    # 하나에서 n개로 분열한다\n    for _ in range(n):\n        count += factorial_recur(n - 1)\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* 팩토리얼 차수(재귀 구현) */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // 하나에서 n개로 분열한다\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* 팩토리얼 차수(재귀 구현) */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // 하나에서 n개로 분열한다\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* 팩토리얼 차수(재귀 구현) */\nint FactorialRecur(int n) {\n    if (n == 0) return 1;\n    int count = 0;\n    // 하나에서 n개로 분열한다\n    for (int i = 0; i &lt; n; i++) {\n        count += FactorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* 팩토리얼 차수(재귀 구현) */\nfunc factorialRecur(n int) int {\n    if n == 0 {\n        return 1\n    }\n    count := 0\n    // 하나에서 n개로 분열한다\n    for i := 0; i &lt; n; i++ {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* 팩토리얼 차수(재귀 구현) */\nfunc factorialRecur(n: Int) -&gt; Int {\n    if n == 0 {\n        return 1\n    }\n    var count = 0\n    // 하나에서 n개로 분열한다\n    for _ in 0 ..&lt; n {\n        count += factorialRecur(n: n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* 팩토리얼 차수(재귀 구현) */\nfunction factorialRecur(n) {\n    if (n === 0) return 1;\n    let count = 0;\n    // 하나에서 n개로 분열한다\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* 팩토리얼 차수(재귀 구현) */\nfunction factorialRecur(n: number): number {\n    if (n === 0) return 1;\n    let count = 0;\n    // 하나에서 n개로 분열한다\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* 팩토리얼 차수(재귀 구현) */\nint factorialRecur(int n) {\n  if (n == 0) return 1;\n  int count = 0;\n  // 하나에서 n개로 분열한다\n  for (var i = 0; i &lt; n; i++) {\n    count += factorialRecur(n - 1);\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* 팩토리얼 차수(재귀 구현) */\nfn factorial_recur(n: i32) -&gt; i32 {\n    if n == 0 {\n        return 1;\n    }\n    let mut count = 0;\n    // 하나에서 n개로 분열한다\n    for _ in 0..n {\n        count += factorial_recur(n - 1);\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* 팩토리얼 차수(재귀 구현) */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.kt<pre><code>/* 팩토리얼 차수(재귀 구현) */\nfun factorialRecur(n: Int): Int {\n    if (n == 0)\n        return 1\n    var count = 0\n    // 하나에서 n개로 분열한다\n    for (i in 0..&lt;n) {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.rb<pre><code>### 팩토리얼 차수(재귀 구현)###\ndef factorial_recur(n)\n  return 1 if n == 0\n\n  count = 0\n  # 하나에서 n개로 분열한다\n  (0...n).each { count += factorial_recur(n - 1) }\n\n  count\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 2-14   팩토리얼 시간 복잡도 </p> <p>\\(n \\geq 4\\)이면 항상 \\(n! &gt; 2^n\\)이므로 팩토리얼 시간은 지수 시간보다 더 빠르게 증가한다. 따라서 \\(n\\)이 클 때는 이 역시 감당하기 어렵다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_computational_complexity/time_complexity/#235","level":2,"title":"2.3.5   최악, 최선, 평균 시간 복잡도","text":"<p>알고리즘의 시간 효율은 고정되어 있지 않고 입력 데이터의 분포에 따라 달라지는 경우가 많다. 길이가 \\(n\\)인 배열 <code>nums</code>를 입력받는다고 하자. <code>nums</code>에는 \\(1\\)부터 \\(n\\)까지의 수가 한 번씩 들어 있으며, 원소 순서는 무작위로 섞여 있다. 목표는 원소 \\(1\\)의 인덱스를 반환하는 것이다. 여기서 다음과 같은 결론을 얻는다.</p> <ul> <li><code>nums = [?, ?, ..., 1]</code>, 즉 마지막 원소가 \\(1\\)이면 배열 전체를 순회해야 하므로 최악 시간 복잡도 \\(O(n)\\)에 도달한다.</li> <li><code>nums = [1, ?, ?, ...]</code>, 즉 첫 번째 원소가 \\(1\\)이면 배열 길이에 관계없이 더 순회할 필요가 없으므로 최선 시간 복잡도 \\(\\Omega(1)\\)에 도달한다.</li> </ul> <p>‘최악 시간 복잡도’는 함수의 점근적 상한에 해당하며 빅 \\(O\\) 표기법으로 나타낸다. 이에 대응하여 ‘최선 시간 복잡도’는 함수의 점근적 하한에 해당하며 \\(\\Omega\\) 표기법으로 나타낸다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby worst_best_time_complexity.py<pre><code>def random_numbers(n: int) -&gt; list[int]:\n    \"\"\"원소가 1, 2, ..., n이며 순서가 섞인 배열 생성\"\"\"\n    # 배열 nums =: 1, 2, 3, ..., n을 생성한다\n    nums = [i for i in range(1, n + 1)]\n    # 배열 원소를 무작위로 섞는다\n    random.shuffle(nums)\n    return nums\n\ndef find_one(nums: list[int]) -&gt; int:\n    \"\"\"배열 nums에서 숫자 1이 있는 인덱스 탐색\"\"\"\n    for i in range(len(nums)):\n        # 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        # 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if nums[i] == 1:\n            return i\n    return -1\n</code></pre> worst_best_time_complexity.cpp<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nvector&lt;int&gt; randomNumbers(int n) {\n    vector&lt;int&gt; nums(n);\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // 시스템 시간으로 난수 시드를 생성한다\n    unsigned seed = chrono::system_clock::now().time_since_epoch().count();\n    // 배열 원소를 무작위로 섞는다\n    shuffle(nums.begin(), nums.end(), default_random_engine(seed));\n    return nums;\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nint findOne(vector&lt;int&gt; &amp;nums) {\n    for (int i = 0; i &lt; nums.size(); i++) {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.java<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nint[] randomNumbers(int n) {\n    Integer[] nums = new Integer[n];\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // 배열 원소를 무작위로 섞는다\n    Collections.shuffle(Arrays.asList(nums));\n    // Integer[] -&gt; int[]\n    int[] res = new int[n];\n    for (int i = 0; i &lt; n; i++) {\n        res[i] = nums[i];\n    }\n    return res;\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nint findOne(int[] nums) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.cs<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nint[] RandomNumbers(int n) {\n    int[] nums = new int[n];\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n\n    // 배열 원소를 무작위로 섞는다\n    for (int i = 0; i &lt; nums.Length; i++) {\n        int index = new Random().Next(i, nums.Length);\n        (nums[i], nums[index]) = (nums[index], nums[i]);\n    }\n    return nums;\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nint FindOne(int[] nums) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.go<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nfunc randomNumbers(n int) []int {\n    nums := make([]int, n)\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for i := 0; i &lt; n; i++ {\n        nums[i] = i + 1\n    }\n    // 배열 원소를 무작위로 섞는다\n    rand.Shuffle(len(nums), func(i, j int) {\n        nums[i], nums[j] = nums[j], nums[i]\n    })\n    return nums\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nfunc findOne(nums []int) int {\n    for i := 0; i &lt; len(nums); i++ {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.swift<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nfunc randomNumbers(n: Int) -&gt; [Int] {\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    var nums = Array(1 ... n)\n    // 배열 원소를 무작위로 섞는다\n    nums.shuffle()\n    return nums\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nfunc findOne(nums: [Int]) -&gt; Int {\n    for i in nums.indices {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.js<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nfunction randomNumbers(n) {\n    const nums = Array(n);\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // 배열 원소를 무작위로 섞는다\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nfunction findOne(nums) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.ts<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nfunction randomNumbers(n: number): number[] {\n    const nums = Array(n);\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // 배열 원소를 무작위로 섞는다\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nfunction findOne(nums: number[]): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.dart<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nList&lt;int&gt; randomNumbers(int n) {\n  final nums = List.filled(n, 0);\n  // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n  for (var i = 0; i &lt; n; i++) {\n    nums[i] = i + 1;\n  }\n  // 배열 원소를 무작위로 섞는다\n  nums.shuffle();\n\n  return nums;\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nint findOne(List&lt;int&gt; nums) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n    // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n    if (nums[i] == 1) return i;\n  }\n\n  return -1;\n}\n</code></pre> worst_best_time_complexity.rs<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nfn random_numbers(n: i32) -&gt; Vec&lt;i32&gt; {\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    let mut nums = (1..=n).collect::&lt;Vec&lt;i32&gt;&gt;();\n    // 배열 원소를 무작위로 섞는다\n    nums.shuffle(&amp;mut thread_rng());\n    nums\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nfn find_one(nums: &amp;[i32]) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if nums[i] == 1 {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> worst_best_time_complexity.c<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nint *randomNumbers(int n) {\n    // 힙 메모리를 할당한다(원소 개수가 n이고 원소 타입이 int인 1차원 가변 길이 배열을 생성한다)\n    int *nums = (int *)malloc(n * sizeof(int));\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // 배열 원소를 무작위로 섞는다\n    for (int i = n - 1; i &gt; 0; i--) {\n        int j = rand() % (i + 1);\n        int temp = nums[i];\n        nums[i] = nums[j];\n        nums[j] = temp;\n    }\n    return nums;\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nint findOne(int *nums, int n) {\n    for (int i = 0; i &lt; n; i++) {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.kt<pre><code>/* 원소가 { 1, 2, ..., n }이며 순서가 섞인 배열 생성 */\nfun randomNumbers(n: Int): Array&lt;Int?&gt; {\n    val nums = IntArray(n)\n    // 배열 nums = { 1, 2, 3, ..., n }을 생성한다\n    for (i in 0..&lt;n) {\n        nums[i] = i + 1\n    }\n    // 배열 원소를 무작위로 섞는다\n    nums.shuffle()\n    val res = arrayOfNulls&lt;Int&gt;(n)\n    for (i in 0..&lt;n) {\n        res[i] = nums[i]\n    }\n    return res\n}\n\n/* 배열 nums에서 숫자 1이 있는 인덱스 탐색 */\nfun findOne(nums: Array&lt;Int?&gt;): Int {\n    for (i in nums.indices) {\n        // 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n        // 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n        if (nums[i] == 1)\n            return i\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.rb<pre><code>### 원소가 1, 2, ..., n이며 순서가 섞인 배열 생성 ###\ndef random_numbers(n)\n  # 배열 nums =: 1, 2, 3, ..., n을 생성한다\n  nums = Array.new(n) { |i| i + 1 }\n  # 배열 원소를 무작위로 섞는다\n  nums.shuffle!\nend\n\n### 배열 nums에서 숫자 1이 있는 인덱스 탐색 ###\ndef find_one(nums)\n  for i in 0...nums.length\n    # 원소 1이 배열 맨 앞에 있으면 최선 시간 복잡도 O(1)에 도달한다\n    # 원소 1이 배열 맨 뒤에 있으면 최악 시간 복잡도 O(n)에 도달한다\n    return i if nums[i] == 1\n  end\n\n  -1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>실제로 최선 시간 복잡도는 거의 사용하지 않는다. 보통 최선의 경우가 발생할 확률이 매우 낮아 오해를 줄 수 있기 때문이다. 최악 시간 복잡도는 효율의 안전한 기준을 제시하므로 더 실용적이며, 이를 통해 안심하고 알고리즘을 사용할 수 있다.</p> <p>앞의 예에서 최악 시간 복잡도와 최선 시간 복잡도는 ‘특수한 데이터 분포’에서만 나타난다. 이러한 경우는 발생 확률이 낮을 수 있어 실제 실행 효율을 제대로 반영하지 못한다. 반면 평균 시간 복잡도는 무작위 입력 데이터에서의 알고리즘 실행 효율을 나타내며, \\(\\Theta\\) 표기법으로 표시한다.</p> <p>일부 알고리즘은 무작위 데이터 분포에서의 평균 상황을 간단히 계산할 수 있다. 앞의 예에서는 입력 배열이 섞여 있으므로 원소 \\(1\\)이 어느 인덱스에 있을 확률도 같다. 따라서 평균 반복 횟수는 배열 길이의 절반인 \\(n / 2\\)이며, 평균 시간 복잡도는 \\(\\Theta(n / 2) = \\Theta(n)\\)이다.</p> <p>하지만 더 복잡한 알고리즘에서는 데이터 분포에 따른 전체 기댓값을 분석하기 어려워 평균 시간 복잡도를 구하기 힘든 경우가 많다. 이럴 때는 보통 최악 시간 복잡도를 알고리즘 효율의 평가 기준으로 삼는다.</p> <p>\\(\\Theta\\) 기호는 왜 보기 드문가?</p> <p>\\(O\\) 기호가 워낙 익숙해서인지 평균 시간 복잡도에도 이를 자주 사용한다. 그러나 엄밀하게는 올바른 표기 방식이 아니다. 이 책이나 다른 자료에서 ‘평균 시간 복잡도 \\(O(n)\\)’ 같은 표현을 만나면 \\(\\Theta(n)\\)으로 이해하면 된다.</p>","path":["제 2 장   복잡도 분석","2.3   시간 복잡도"],"tags":[]},{"location":"chapter_data_structure/","level":1,"title":"제 3 장   자료 구조","text":"<p>이 장을 시작하며</p> <p>자료 구조는 튼튼하면서도 다양한 형태를 지닌 뼈대와 같다.</p> <p>데이터를 질서 있게 조직하는 설계도를 제공하며, 알고리즘은 그 토대 위에서 생동감을 얻는다.</p>","path":["제 3 장   자료 구조"],"tags":[]},{"location":"chapter_data_structure/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>3.1   자료 구조의 분류</li> <li>3.2   기본 자료형</li> <li>3.3   수의 인코딩 *</li> <li>3.4   문자 인코딩 *</li> <li>3.5   요약</li> <li>3.6   연습 문제</li> </ul>","path":["제 3 장   자료 구조"],"tags":[]},{"location":"chapter_data_structure/basic_data_types/","level":1,"title":"3.2   기본 자료형","text":"<p>컴퓨터의 데이터라고 하면 텍스트, 이미지, 동영상, 음성, 3D 모델 등 다양한 형태가 떠오른다. 이 데이터들은 조직된 형태는 서로 다르지만, 모두 여러 기본 자료형으로 이루어져 있다.</p> <p>기본 자료형은 CPU가 직접 연산할 수 있는 자료형으로, 알고리즘에서 직접 사용된다. 주로 다음과 같은 종류가 있다.</p> <ul> <li>정수형 <code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>.</li> <li>소수를 표현하는 부동 소수점형 <code>float</code>, <code>double</code>.</li> <li>여러 언어의 글자, 문장 부호, 나아가 이모지 등을 표현하는 문자형 <code>char</code>.</li> <li>‘예’와 ‘아니요’를 판단하는 불리언형 <code>bool</code>.</li> </ul> <p>기본 자료형은 컴퓨터에 이진수 형태로 저장된다. 이진수 한 자리가 \\(1\\)비트다. 현대 운영체제의 대다수에서 \\(1\\)바이트(byte)는 \\(8\\)비트(bit)로 이루어진다.</p> <p>기본 자료형의 값 범위는 차지하는 공간의 크기에 따라 결정된다. Java를 예로 살펴보자.</p> <ul> <li>정수형 <code>byte</code>는 \\(1\\)바이트 = \\(8\\)비트를 차지하며, \\(2^{8}\\)개의 수를 표현할 수 있다.</li> <li>정수형 <code>int</code>는 \\(4\\)바이트 = \\(32\\)비트를 차지하며, \\(2^{32}\\)개의 수를 표현할 수 있다.</li> </ul> <p>아래 표(3-1)는 Java의 여러 기본 자료형이 차지하는 공간, 값 범위, 기본값을 나열한다. 표를 외울 필요는 없으며, 대략 이해한 뒤 필요할 때 찾아보면 된다.</p> <p> 표 3-1   기본 자료형의 공간 사용량과 값 범위 </p> 자료형 기호 공간 사용량 최솟값 최댓값 기본값 정수 <code>byte</code> 1바이트 \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) \\(0\\) <code>short</code> 2바이트 \\(-2^{15}\\) \\(2^{15} - 1\\) \\(0\\) <code>int</code> 4바이트 \\(-2^{31}\\) \\(2^{31} - 1\\) \\(0\\) <code>long</code> 8바이트 \\(-2^{63}\\) \\(2^{63} - 1\\) \\(0\\) 부동 소수점 <code>float</code> 4바이트 \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0\\text{f}\\) <code>double</code> 8바이트 \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) \\(0.0\\) 문자 <code>char</code> 2바이트 \\(0\\) \\(2^{16} - 1\\) \\(0\\) 불리언 <code>bool</code> 1바이트 \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\) <p>위 표(3-1)는 Java의 기본 자료형을 기준으로 한다. 프로그래밍 언어마다 자료형을 정의하는 방식이 있으므로, 공간 사용량과 값 범위, 기본값은 다를 수 있다.</p> <ul> <li>Python의 정수형 <code>int</code>는 사용 가능한 메모리만 충분하면 크기에 제한이 없다. 부동 소수점형 <code>float</code>는 64비트 배정밀도다. <code>char</code> 자료형은 없으며, 문자 하나는 실제로 길이가 1인 문자열 <code>str</code>이다.</li> <li>C와 C++는 기본 자료형의 크기를 명확히 규정하지 않으므로 구현과 플랫폼에 따라 다르다. 위 표(3-1)는 Linux, macOS 등의 64비트 Unix 운영체제에서 사용하는 LP64 데이터 모델을 따른다.</li> <li>문자형 <code>char</code>의 크기는 C와 C++에서 1바이트이며, 대부분의 프로그래밍 언어에서는 사용하는 문자 인코딩 방식에 따라 정해진다. 자세한 내용은 ‘문자 인코딩’ 절에서 설명한다.</li> <li>불리언 값을 표현하는 데는 1비트(\\(0\\) 또는 \\(1\\))면 충분하지만, 메모리에서는 보통 1바이트로 저장한다. 현대 컴퓨터 CPU가 대개 1바이트를 주소 지정이 가능한 최소 메모리 단위로 사용하기 때문이다.</li> </ul> <p>그렇다면 기본 자료형과 자료 구조는 어떤 관계가 있을까? 자료 구조란 컴퓨터에서 데이터를 조직하고 저장하는 방식이다. 이 문장의 핵심은 ‘데이터’보다 ‘구조’에 있다.</p> <p>‘한 줄로 늘어선 수’를 표현하려면 자연스럽게 배열을 떠올린다. 배열의 선형 구조가 수 사이의 인접 관계와 순서 관계를 표현할 수 있기 때문이다. 그러나 저장할 내용이 정수 <code>int</code>인지, 소수 <code>float</code>인지, 문자 <code>char</code>인지는 ‘자료 구조’와 무관하다.</p> <p>즉, 기본 자료형은 데이터의 ‘내용 유형’을 제공하고, 자료 구조는 데이터의 ‘조직 방식’을 제공한다. 다음 코드는 같은 자료 구조인 배열을 사용하여 <code>int</code>, <code>float</code>, <code>char</code>, <code>bool</code> 등 서로 다른 기본 자료형을 저장하고 표현한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># 여러 기본 자료형으로 배열 초기화\nnumbers: list[int] = [0] * 5\ndecimals: list[float] = [0.0] * 5\n# Python의 문자는 실제로 길이가 1인 문자열이다\ncharacters: list[str] = ['0'] * 5\nbools: list[bool] = [False] * 5\n# Python의 리스트는 여러 기본 자료형과 객체 참조를 자유롭게 저장할 수 있다\ndata = [0, 0.0, 'a', False, ListNode(0)]\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool bools[5];\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] bools = new boolean[5];\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] bools = new bool[5];\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar bools = [5]bool{}\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nlet numbers = Array(repeating: 0, count: 5)\nlet decimals = Array(repeating: 0.0, count: 5)\nlet characters: [Character] = Array(repeating: \"a\", count: 5)\nlet bools = Array(repeating: false, count: 5)\n</code></pre> <pre><code>// JavaScript의 배열은 여러 기본 자료형과 객체를 자유롭게 저장할 수 있다\nconst array = [0, 0.0, 'a', false];\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst bools: boolean[] = [];\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nList&lt;int&gt; numbers = List.filled(5, 0);\nList&lt;double&gt; decimals = List.filled(5, 0.0);\nList&lt;String&gt; characters = List.filled(5, 'a');\nList&lt;bool&gt; bools = List.filled(5, false);\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nlet numbers: Vec&lt;i32&gt; = vec![0; 5];\nlet decimals: Vec&lt;f32&gt; = vec![0.0; 5];\nlet characters: Vec&lt;char&gt; = vec!['0'; 5];\nlet bools: Vec&lt;bool&gt; = vec![false; 5];\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool bools[10];\n</code></pre> <pre><code>// 여러 기본 자료형으로 배열 초기화\nval numbers = IntArray(5)\nval decinals = FloatArray(5)\nval characters = CharArray(5)\nval bools = BooleanArray(5)\n</code></pre> <pre><code># Ruby의 리스트는 여러 기본 자료형과 객체 참조를 자유롭게 저장할 수 있다\ndata = [0, 0.0, 'a', false, ListNode(0)]\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 3 장   자료 구조","3.2   기본 자료형"],"tags":[]},{"location":"chapter_data_structure/character_encoding/","level":1,"title":"3.4   문자 인코딩 *","text":"<p>컴퓨터에서는 모든 데이터를 이진수 형태로 저장하며, 문자 <code>char</code>도 예외가 아니다. 문자를 표현하려면 각 문자와 이진수 사이의 일대일 대응 관계를 정한 ‘문자 집합’이 필요하다. 문자 집합이 있으면 컴퓨터는 대응표를 찾아 이진수를 문자로 바꿀 수 있다.</p>","path":["제 3 장   자료 구조","3.4   문자 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#341-ascii","level":2,"title":"3.4.1   ASCII 문자 집합","text":"<p>ASCII 코드는 가장 초기에 등장한 문자 집합으로, 정식 명칭은 American Standard Code for Information Interchange(미국 정보 교환 표준 부호)다. 7비트 이진수(한 바이트의 하위 7비트)로 문자 하나를 표현하여 최대 128개의 서로 다른 문자를 나타낸다. 아래 그림(3-6)과 같이 영문 대소문자, 숫자 0 ~ 9, 일부 문장 부호와 제어 문자(줄바꿈, 탭 등)를 포함한다.</p> <p></p> <p> 그림 3-6   ASCII 코드 </p> <p>그러나 ASCII 코드는 영어만 표현할 수 있다. 컴퓨터가 전 세계로 보급되면서 더 많은 언어를 표현할 수 있는 EASCII 문자 집합이 등장했다. ASCII의 7비트를 8비트로 확장하여 서로 다른 문자 256개를 표현한다.</p> <p>이후 세계 여러 지역에 맞는 EASCII 문자 집합이 잇따라 등장했다. 이 문자 집합들은 앞의 128개 문자는 모두 ASCII 코드로 통일하고, 뒤의 128개 문자는 각 언어의 필요에 맞게 서로 다르게 정의했다.</p>","path":["제 3 장   자료 구조","3.4   문자 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#342-gbk","level":2,"title":"3.4.2   GBK 문자 집합","text":"<p>하지만 EASCII 코드도 여러 언어에 필요한 문자 수를 충족하지 못했다. 예를 들어 한자는 거의 10만 자에 이르며, 일상에서 사용하는 것만 해도 수천 자다. 중국 국가표준총국은 1980년에 한자 6763자를 수록한 GB2312 문자 집합을 발표하여 컴퓨터의 한자 처리에 필요한 기본적인 요구를 충족했다.</p> <p>그러나 GB2312는 일부 드문 한자와 번체자를 처리할 수 없었다. GBK 문자 집합은 GB2312를 확장한 것으로 총 21886자의 한자를 수록한다. GBK 인코딩에서는 ASCII 문자를 1바이트로, 한자를 2바이트로 표현한다.</p>","path":["제 3 장   자료 구조","3.4   문자 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#343-unicode","level":2,"title":"3.4.3   Unicode 문자 집합","text":"<p>컴퓨터 기술이 빠르게 발전하면서 문자 집합과 인코딩 표준이 다양하게 등장했지만, 이는 여러 문제를 낳았다. 한편으로는 문자 집합이 대개 특정 언어의 문자만 정의하여 다국어 환경에서 제대로 작동하지 못했다. 다른 한편으로는 같은 언어에도 여러 문자 집합 표준이 있어, 두 컴퓨터의 인코딩 표준이 다르면 정보를 주고받을 때 글자가 깨졌다.</p> <p>당시 연구자들은 이렇게 생각했다. 세계의 모든 언어와 기호를 담을 만큼 완전한 문자 집합을 만들면 다국어 환경과 글자 깨짐 문제를 해결할 수 있지 않을까? 이러한 생각에서 크고 포괄적인 문자 집합 Unicode가 탄생했다.</p> <p>Unicode는 중국어로 ‘통일 부호’라는 뜻의 이름으로 불리며, 이론적으로 100만 개 이상의 문자를 수용할 수 있다. 전 세계 문자를 하나의 문자 집합에 통합하고, 여러 언어의 문자를 처리하고 표시하는 공통 문자 집합을 제공하여 인코딩 표준의 차이로 발생하는 글자 깨짐을 줄이는 것이 목표다. 1991년 발표된 뒤로 새로운 언어와 문자가 계속 추가되었다. 2022년 9월 기준으로 Unicode에는 여러 언어의 문자, 기호, 이모지 등을 포함하여 149186개의 문자가 들어 있다.</p> <p>Unicode는 범용 문자 집합으로서 각 문자에 고유한 ‘코드 포인트’(문자 번호)를 부여한다. 범위는 U+0000부터 U+10FFFF까지이며, 하나로 통일된 문자 번호 공간을 이룬다. 그러나 Unicode는 이러한 문자 코드 포인트를 컴퓨터에 어떻게 저장할지는 정하지 않는다. 길이가 다른 Unicode 코드 포인트들이 한 텍스트에 함께 나타나면 시스템은 문자를 어떻게 해석할까? 예를 들어 길이가 2바이트인 인코딩이 주어졌을 때, 2바이트 문자 하나인지 1바이트 문자 두 개인지 어떻게 판단할까?</p> <p>이 문제의 직접적인 해결책은 모든 문자를 같은 길이로 인코딩하여 저장하는 것이다. 아래 그림(3-7)에서 ‘Hello’의 각 문자는 1바이트, ‘算法’의 각 문자는 2바이트를 차지한다. 상위 비트에 0을 채우면 ‘Hello 算法’의 모든 문자를 2바이트 길이로 인코딩할 수 있다. 그러면 시스템이 2바이트마다 문자 하나를 해석하여 원래 문구를 복원할 수 있다.</p> <p></p> <p> 그림 3-7   Unicode 인코딩 예 </p> <p>하지만 ASCII 코드에서 보았듯이 영어 문자를 인코딩하는 데는 1바이트면 충분하다. 위 방식을 사용하면 영어 텍스트가 ASCII 인코딩의 두 배 공간을 차지하여 메모리를 크게 낭비한다. 따라서 더 효율적인 Unicode 인코딩 방법이 필요하다.</p>","path":["제 3 장   자료 구조","3.4   문자 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#344-utf-8","level":2,"title":"3.4.4   UTF-8 인코딩","text":"<p>현재 UTF-8은 세계에서 가장 널리 쓰이는 Unicode 인코딩 방식이다. 가변 길이 인코딩으로, 문자의 복잡성에 따라 1~4바이트로 문자 하나를 표현한다. ASCII 문자는 1바이트, 라틴 문자와 그리스 문자는 2바이트, 자주 쓰는 중국어 문자는 3바이트, 일부 드문 문자는 4바이트가 필요하다.</p> <p>UTF-8 인코딩 규칙은 복잡하지 않으며, 다음 두 경우로 나뉜다.</p> <ul> <li>길이가 1바이트인 문자는 최상위 비트를 \\(0\\)으로 두고 나머지 7비트에 Unicode 코드 포인트를 넣는다. ASCII 문자는 Unicode 문자 집합의 처음 128개 코드 포인트를 차지한다. 즉, UTF-8 인코딩은 ASCII 코드와 하위 호환된다. 따라서 오래된 ASCII 텍스트도 UTF-8로 해석할 수 있다.</li> <li>길이가 \\(n\\)바이트인 문자(\\(n &gt; 1\\))는 첫 바이트의 상위 \\(n\\)비트를 모두 \\(1\\)로, \\(n + 1\\)번째 비트를 \\(0\\)으로 둔다. 두 번째 바이트부터는 각 바이트의 상위 2비트를 \\(10\\)으로 두고, 나머지 모든 비트에 문자의 Unicode 코드 포인트를 채운다.</li> </ul> <p>아래 그림(3-8)은 ‘Hello算法’의 UTF-8 인코딩을 보여 준다. 상위 \\(n\\)비트가 모두 \\(1\\)이므로 시스템은 맨 앞에서 연속된 \\(1\\)의 개수를 읽어 문자 길이가 \\(n\\)임을 알 수 있다.</p> <p>그렇다면 나머지 바이트의 상위 2비트를 모두 \\(10\\)으로 두는 이유는 무엇일까? 이 \\(10\\)은 검사 표식 역할을 한다. 시스템이 잘못된 바이트에서 텍스트를 해석하기 시작했을 때, 바이트 앞부분의 \\(10\\)을 통해 이상을 빠르게 감지할 수 있다.</p> <p>\\(10\\)을 검사 표식으로 쓰는 이유는 UTF-8 규칙상 문자 인코딩의 맨 앞 두 비트가 \\(10\\)일 수 없기 때문이다. 이는 귀류법으로 증명할 수 있다. 문자 인코딩의 맨 앞 두 비트가 \\(10\\)이라고 가정하면, 길이가 \\(1\\)바이트인 ASCII 문자라는 뜻이다. 하지만 ASCII 코드의 최상위 비트는 \\(0\\)이어야 하므로 가정과 모순된다.</p> <p></p> <p> 그림 3-8   UTF-8 인코딩 예 </p> <p>UTF-8 외에도 다음 두 인코딩 방식이 널리 쓰인다.</p> <ul> <li>UTF-16 인코딩: 2바이트 또는 4바이트로 문자 하나를 표현한다. 모든 ASCII 문자와 자주 쓰는 비영어권 문자는 2바이트로 표현하고, 일부 문자는 4바이트가 필요하다. 2바이트 문자는 UTF-16 인코딩 값이 Unicode 코드 포인트와 같다.</li> <li>UTF-32 인코딩: 모든 문자에 4바이트를 사용한다. 따라서 UTF-8이나 UTF-16보다 공간을 더 많이 차지하며, 특히 ASCII 문자의 비중이 큰 텍스트에서 차이가 크다.</li> </ul> <p>저장 공간 관점에서 UTF-8은 영어 문자를 1바이트만으로 표현하므로 매우 효율적이다. 반면 중국어 같은 일부 비영어권 문자는 UTF-16으로 인코딩하는 것이 더 효율적이다. UTF-16은 2바이트면 되지만 UTF-8은 3바이트가 필요할 수 있기 때문이다.</p> <p>호환성 관점에서는 UTF-8의 범용성이 가장 뛰어나며, 많은 도구와 라이브러리가 UTF-8을 우선 지원한다.</p>","path":["제 3 장   자료 구조","3.4   문자 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/character_encoding/#345","level":2,"title":"3.4.5   프로그래밍 언어의 문자 인코딩","text":"<p>과거의 대다수 프로그래밍 언어에서는 실행 중인 프로그램의 문자열에 UTF-16이나 UTF-32 같은 고정 길이 인코딩을 사용했다. 고정 길이 인코딩에서는 문자열을 배열처럼 다룰 수 있어 다음과 같은 장점이 있다.</p> <ul> <li>임의 접근: UTF-16 문자열은 임의 접근이 쉽다. UTF-8은 가변 길이 인코딩이므로 \\(i\\)번째 문자를 찾으려면 문자열의 처음부터 \\(i\\)번째 문자까지 순회해야 하며, \\(O(n)\\) 시간이 필요하다.</li> <li>문자 수 계산: 임의 접근과 마찬가지로 UTF-16 문자열의 길이 계산도 \\(O(1)\\) 연산이다. 반면 UTF-8 문자열의 길이를 계산하려면 문자열 전체를 순회해야 한다.</li> <li>문자열 연산: UTF-16 문자열에서는 분할, 연결, 삽입, 삭제 등 여러 문자열 연산을 더 쉽게 수행할 수 있다. UTF-8 문자열에서는 잘못된 UTF-8 인코딩이 생기지 않도록 보장하기 위해 보통 추가 계산이 필요하다.</li> </ul> <p>프로그래밍 언어의 문자 인코딩 설계는 여러 요소가 관련된 흥미로운 주제다.</p> <ul> <li>Java의 <code>String</code>은 UTF-16 인코딩을 사용하며 각 문자는 2바이트를 차지한다. Java를 처음 설계할 당시에는 16비트로 가능한 모든 문자를 표현할 수 있다고 생각했기 때문이다. 그러나 이는 잘못된 판단이었다. 이후 Unicode 규격이 16비트를 넘어 확장되면서, 현재 Java의 문자는 16비트 값 두 개로 이루어진 ‘서로게이트 쌍’으로 표현될 수 있다.</li> <li>JavaScript와 TypeScript의 문자열이 UTF-16을 사용하는 이유도 Java와 비슷하다. Netscape가 1995년에 JavaScript를 처음 발표했을 때 Unicode는 초기 발전 단계였으며, 당시에는 16비트 인코딩으로 모든 Unicode 문자를 표현할 수 있었다.</li> <li>C#이 UTF-16을 사용하는 주된 이유는 .NET 플랫폼을 Microsoft가 설계했으며, Windows 운영체제를 비롯한 Microsoft의 여러 기술에서 UTF-16을 널리 사용하기 때문이다.</li> </ul> <p>이 언어들은 문자 수를 과소평가했기 때문에 16비트를 넘는 Unicode 문자를 표현하려고 ‘서로게이트 쌍’을 도입할 수밖에 없었다. 이는 불가피한 선택이었다. 한편으로는 서로게이트 쌍이 포함된 문자열에서 문자 하나가 2바이트 또는 4바이트를 차지하여 고정 길이 인코딩의 장점을 잃는다. 다른 한편으로는 서로게이트 쌍을 처리하는 코드가 더 필요해 프로그래밍이 복잡해지고 디버깅도 어려워진다.</p> <p>이러한 이유로 일부 프로그래밍 언어는 다른 인코딩 방식을 채택했다.</p> <ul> <li>Python의 <code>str</code>은 Unicode 인코딩과 유연한 문자열 표현을 사용한다. 저장되는 문자 길이는 문자열에서 가장 큰 Unicode 코드 포인트에 따라 정해진다. 모든 문자가 ASCII이면 문자마다 1바이트를 사용한다. ASCII 범위를 넘는 문자가 있지만 모두 기본 다국어 평면(BMP)에 속하면 문자마다 2바이트를 사용한다. BMP를 벗어나는 문자가 있으면 문자마다 4바이트를 사용한다.</li> <li>Go의 <code>string</code>은 내부적으로 UTF-8 인코딩을 사용한다. Go는 Unicode 코드 포인트 하나를 나타내는 <code>rune</code> 자료형도 제공한다.</li> <li>Rust의 <code>str</code>과 <code>String</code>은 내부적으로 UTF-8 인코딩을 사용한다. Rust도 Unicode 코드 포인트 하나를 나타내는 <code>char</code> 자료형을 제공한다.</li> </ul> <p>위 내용은 프로그래밍 언어 안에서 문자열을 저장하는 방식에 관한 것이다. 이는 문자열을 파일에 저장하거나 네트워크로 전송하는 방식과는 별개의 문제다. 파일 저장이나 네트워크 전송에서는 보통 호환성과 공간 효율을 높이기 위해 문자열을 UTF-8 형식으로 인코딩한다.</p>","path":["제 3 장   자료 구조","3.4   문자 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/classification_of_data_structure/","level":1,"title":"3.1   자료 구조의 분류","text":"<p>대표적인 자료 구조로는 배열, 연결 리스트, 스택, 큐, 해시 테이블, 트리, 힙, 그래프가 있다. 이들은 ‘논리적 구조’와 ‘물리적 구조’라는 두 관점에서 분류할 수 있다.</p>","path":["제 3 장   자료 구조","3.1   자료 구조의 분류"],"tags":[]},{"location":"chapter_data_structure/classification_of_data_structure/#311","level":2,"title":"3.1.1   논리적 구조: 선형과 비선형","text":"<p>논리적 구조는 데이터 원소 사이의 논리적 관계를 드러낸다. 배열과 연결 리스트에서는 데이터가 일정한 순서로 배열되어 데이터 사이의 선형 관계를 나타낸다. 트리에서는 데이터가 위에서 아래로 계층을 이루어 ‘조상’과 ‘자손’ 사이의 파생 관계를 나타낸다. 그래프는 노드와 간선으로 이루어져 복잡한 네트워크 관계를 표현한다.</p> <p>아래 그림(3-1)과 같이 논리적 구조는 크게 ‘선형’과 ‘비선형’으로 나뉜다. 선형 구조는 데이터가 논리적으로 일렬로 배열되어 있어 비교적 직관적이다. 비선형 구조는 이와 달리 일렬로 배열되지 않는다.</p> <ul> <li>선형 자료 구조: 배열, 연결 리스트, 스택, 큐, 해시 테이블이 해당하며, 원소 사이에 일대일의 순서 관계가 있다.</li> <li>비선형 자료 구조: 트리, 힙, 그래프, 해시 테이블이 해당한다.</li> </ul> <p>비선형 자료 구조는 다시 트리 구조와 네트워크 구조로 나뉜다.</p> <ul> <li>트리 구조: 트리, 힙, 해시 테이블이 해당하며, 원소 사이에 일대다 관계가 있다.</li> <li>네트워크 구조: 그래프가 해당하며, 원소 사이에 다대다 관계가 있다.</li> </ul> <p></p> <p> 그림 3-1   선형 자료 구조와 비선형 자료 구조 </p>","path":["제 3 장   자료 구조","3.1   자료 구조의 분류"],"tags":[]},{"location":"chapter_data_structure/classification_of_data_structure/#312","level":2,"title":"3.1.2   물리적 구조: 연속과 분산","text":"<p>알고리즘 프로그램이 실행될 때 처리 중인 데이터는 주로 메모리에 저장된다. 아래 그림(3-2)은 컴퓨터 메모리 모듈을 보여 주며, 검은색 칩마다 메모리 공간이 들어 있다. 메모리는 각 셀에 일정한 크기의 데이터를 저장할 수 있는 거대한 Excel 표라고 생각할 수 있다.</p> <p>시스템은 메모리 주소를 통해 원하는 위치의 데이터에 접근한다. 아래 그림(3-2)과 같이 컴퓨터는 특정 규칙에 따라 표의 각 셀에 번호를 부여하여 모든 메모리 공간이 고유한 메모리 주소를 갖게 한다. 프로그램은 이 주소를 이용해 메모리의 데이터에 접근한다.</p> <p></p> <p> 그림 3-2   메모리 모듈, 메모리 공간, 메모리 주소 </p> <p>참고</p> <p>메모리를 Excel 표에 빗댄 것은 단순화한 비유다. 실제 메모리의 작동 방식은 더 복잡하며 주소 공간, 메모리 관리, 캐시, 가상 메모리, 물리 메모리 등의 개념이 관련된다.</p> <p>메모리는 모든 프로그램이 공유하는 자원이다. 어떤 메모리 영역을 한 프로그램이 사용하면 보통 다른 프로그램은 동시에 사용할 수 없다. 따라서 자료 구조와 알고리즘을 설계할 때 메모리 자원은 중요한 고려 사항이다. 예를 들어 알고리즘의 최대 메모리 사용량은 시스템의 남은 메모리를 넘지 않아야 한다. 연속된 큰 메모리 공간이 부족하다면 분산된 메모리 공간에 저장할 수 있는 자료 구조를 선택해야 한다.</p> <p>아래 그림(3-3)과 같이 물리적 구조는 데이터가 컴퓨터 메모리에 저장되는 방식을 나타내며, 연속 공간 저장(배열)과 분산 공간 저장(연결 리스트)으로 나뉜다. 물리적 구조는 데이터 접근, 갱신, 삽입, 삭제 등의 연산 방식을 근본적으로 결정한다. 두 물리적 구조는 시간 효율과 공간 효율에서 서로 보완적인 특징을 지닌다.</p> <p></p> <p> 그림 3-3   연속 공간 저장과 분산 공간 저장 </p> <p>모든 자료 구조는 배열, 연결 리스트 또는 둘의 조합으로 구현된다는 점도 알아 두자. 예를 들어 스택과 큐는 배열이나 연결 리스트로 구현할 수 있으며, 해시 테이블 구현에는 배열과 연결 리스트가 함께 포함될 수 있다.</p> <ul> <li>배열로 구현할 수 있는 것: 스택, 큐, 해시 테이블, 트리, 힙, 그래프, 행렬, 텐서(차원이 \\(\\geq 3\\)인 배열) 등.</li> <li>연결 리스트로 구현할 수 있는 것: 스택, 큐, 해시 테이블, 트리, 힙, 그래프 등.</li> </ul> <p>연결 리스트는 초기화한 뒤에도 프로그램 실행 중에 길이를 조정할 수 있으므로 ‘동적 자료 구조’라고도 한다. 배열은 초기화한 뒤 길이를 바꿀 수 없으므로 ‘정적 자료 구조’라고도 한다. 다만 배열도 메모리를 다시 할당하면 길이를 바꿀 수 있어 어느 정도 ‘동적 특성’을 가질 수 있다.</p> <p>참고</p> <p>물리적 구조를 이해하기 어렵다면 다음 장을 먼저 읽은 뒤 이 절을 복습하는 것을 권한다.</p>","path":["제 3 장   자료 구조","3.1   자료 구조의 분류"],"tags":[]},{"location":"chapter_data_structure/exercises/","level":1,"title":"3.6   연습 문제","text":"","path":["제 3 장   자료 구조","3.6   연습 문제"],"tags":[]},{"location":"chapter_data_structure/exercises/#361","level":2,"title":"3.6.1   개념 확인","text":"","path":["제 3 장   자료 구조","3.6   연습 문제"],"tags":[]},{"location":"chapter_data_structure/exercises/#1","level":3,"title":"1.   일상 속 데이터의 관계","text":"<p>데이터 사이의 관계를 바탕으로 다음 세 상황에 대해 ‘선형 구조, 트리 구조, 네트워크 구조’ 중 하나를 고르고 이유를 설명하라.</p> <ol> <li>학생들이 한 줄로 서 있고, 각자 바로 앞과 뒤에 있는 사람만 신경 쓴다.</li> <li>학교를 ‘학교 → 학년 → 학급’의 계층으로 나누어 관리한다.</li> <li>도시의 도로가 여러 교차로를 연결한다. 한 교차로에서 여러 다른 교차로로 갈 수 있으며 순환하는 경로도 생길 수 있다.</li> </ol> 참고 답안 <ol> <li> <p>선형 구조다. 맨 앞과 맨 뒤에 선 사람을 제외하면 각 사람은 앞뒤로 한 사람씩만 이웃하며, 관계가 하나의 선을 따라 이어진다.</p> </li> <li> <p>트리 구조다. 각 학급은 한 학년에 속하고 각 학년은 학교에 속하므로, 관계가 위에서 아래로 계층적으로 펼쳐진다.</p> </li> <li> <p>네트워크 구조다. 한 교차로가 여러 교차로와 연결될 수 있고 경로들이 순환을 이룰 수도 있으므로, 하나의 순서나 엄격한 계층으로 나열할 수 없다.</p> <p>구조를 판단할 때는 메모리에서 차지하는 공간보다 원소 사이의 관계를 먼저 살펴보아야 한다.</p> </li> </ol>","path":["제 3 장   자료 구조","3.6   연습 문제"],"tags":[]},{"location":"chapter_data_structure/exercises/#2","level":3,"title":"2.   논리적 순서를 메모리에 저장하는 방법","text":"<p>논리적 순서 <code>A → B → C</code>를 저장하기 위해 다음과 같이 단순화한 두 메모리 배치 방식을 생각해 보자.</p> <ul> <li>방법 가: <code>A, B, C</code>를 각각 번호가 <code>20, 21, 22</code>인 메모리 칸에 저장한다.</li> <li>방법 나: <code>A, B, C</code>를 각각 번호가 <code>20, 7, 31</code>인 메모리 칸에 저장하고, <code>A</code>에 <code>B</code>의 위치를, <code>B</code>에 <code>C</code>의 위치를 기록한다.</li> </ul> <ol> <li>어느 방법이 연속 공간 저장이고, 어느 방법이 분산 공간 저장인가?</li> <li>두 방법은 각각 배열과 연결 리스트 중 어느 쪽에 더 가까운가?</li> <li>방법 나는 메모리 칸의 번호가 크기순으로 정렬되어 있지 않은데, 어떻게 <code>A → B → C</code>의 논리적 순서를 나타낼 수 있는가?</li> </ol> 참고 답안 <ol> <li> <p>방법 가는 연속된 메모리 칸을 사용하므로 연속 공간 저장이다. 방법 나는 노드가 여러 위치에 흩어져 있으므로 분산 공간 저장이다.</p> </li> <li> <p>방법 가는 배열에, 방법 나는 연결 리스트에 더 가깝다.</p> </li> <li> <p>논리적 순서는 메모리 칸 번호의 크기가 아니라 노드 사이에 기록된 연결 관계로 결정된다.     <code>A</code>에 기록된 위치에서 <code>B</code>를 찾고, 다시 <code>B</code>에 기록된 위치에서 <code>C</code>를 찾을 수 있으므로 <code>A, B, C</code>에 차례대로 접근할 수 있다.</p> <p>이는 논리적 구조와 물리적 구조가 같은 데이터 집합을 바라보는 서로 다른 두 관점임을 보여 준다.</p> </li> </ol>","path":["제 3 장   자료 구조","3.6   연습 문제"],"tags":[]},{"location":"chapter_data_structure/exercises/#3","level":3,"title":"3.   과제 기록의 자료형과 구조","text":"<p>한 학습 모임에서 학생 4명의 과제 제출 여부를 좌석 순서대로 기록한 결과가 다음과 같다.</p> <p><code>[true, false, true, true]</code></p> <ol> <li>각 원소에는 어떤 기본 자료형이 적합한가?</li> <li>이 4개 원소를 좌석 순서대로 한 줄에 나열한 것은 어떤 논리적 구조인가?</li> <li>나중에 각 학생의 과제 점수 <code>[90, 0, 85, 100]</code>를 기록하도록 바꾼다면, 달라지는 것은 데이터의 ‘내용 유형’인가, ‘조직 방식’인가? 이유를 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>각 원소는 ‘예’ 또는 ‘아니요’만 나타내므로 불리언 자료형 <code>bool</code>이 적합하다.</p> </li> <li> <p>원소들이 좌석 순서대로 나열되어 선형 구조를 이루며, 배열에 저장할 수 있다.</p> </li> <li> <p>달라지는 것은 내용 유형이다. 원소가 불리언 값에서 정수로 바뀐다. 조직 방식은 바뀌지 않는다.     데이터는 여전히 좌석 순서대로 한 줄에 나열되어 있으므로 배열이라는 선형 구조를 사용할 수 있다.</p> <p>기본 자료형은 ‘무엇을 저장하는가’를, 자료 구조는 ‘데이터를 어떻게 구성하는가’를 설명한다.</p> </li> </ol>","path":["제 3 장   자료 구조","3.6   연습 문제"],"tags":[]},{"location":"chapter_data_structure/exercises/#362","level":2,"title":"3.6.2   프로그래밍 연습","text":"","path":["제 3 장   자료 구조","3.6   연습 문제"],"tags":[]},{"location":"chapter_data_structure/exercises/#1-1","level":3,"title":"1.   이진수 표현에서 1의 개수 세기","text":"<p>음이 아닌 정수 <code>n</code>이 주어질 때, 이진수 표현에 포함된 1의 개수를 구하라.</p> <p>비트 연산을 사용해야 하며, 이진수 표현을 문자열로 변환하거나 1의 개수를 바로 세어 주는 내장 함수를 사용해서는 안 된다.</p> 풀이 힌트 <ol> <li>n &amp; 1로 n의 가장 오른쪽 비트를 얻어 그 비트가 1인지 확인할 수 있다.</li> <li>오른쪽으로 한 비트 이동하면 현재 가장 오른쪽 비트가 제거된다. 대부분의 언어에서는 &gt;&gt; 연산자를 사용한다.</li> <li>비트를 하나씩 확인하고 오른쪽으로 이동하는 방법을 완성한 뒤, n &amp; (n - 1)이 n의 가장 오른쪽 1을 0으로 바꾼다는 점도 살펴보자.</li> </ol> <p>LeetCode</p>","path":["제 3 장   자료 구조","3.6   연습 문제"],"tags":[]},{"location":"chapter_data_structure/number_encoding/","level":1,"title":"3.3   수의 인코딩 *","text":"<p>참고</p> <p>이 책에서 제목에 * 기호가 있는 절은 선택 학습 내용이다. 시간이 부족하거나 이해하기 어렵다면 먼저 건너뛰고, 필수 내용을 학습한 뒤 따로 도전해도 좋다.</p>","path":["제 3 장   자료 구조","3.3   수의 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/number_encoding/#331-1-2","level":2,"title":"3.3.1   부호와 크기, 1의 보수, 2의 보수","text":"<p>앞 절의 표에서 모든 정수형은 표현할 수 있는 음수가 양수보다 하나 더 많았다. 예를 들어 <code>byte</code>의 값 범위는 \\([-128, 127]\\)이다. 직관과 어긋나는 이 현상의 이유를 이해하려면 부호와 크기, 1의 보수, 2의 보수에 대해 알아야 한다.</p> <p>먼저 수는 컴퓨터에 ‘2의 보수’ 형태로 저장된다는 점을 알아 두자. 그 이유를 분석하기 전에 세 가지 표현을 정의하겠다.</p> <ul> <li>부호와 크기: 이진수 표현의 최상위 비트를 부호 비트로 사용한다. \\(0\\)은 양수, \\(1\\)은 음수를 나타내며 나머지 비트는 수의 크기를 나타낸다.</li> <li>1의 보수: 양수의 1의 보수는 부호와 크기 표현과 같다. 음수의 1의 보수는 부호와 크기 표현에서 부호 비트를 제외한 모든 비트를 반전한 것이다.</li> <li>2의 보수: 양수의 2의 보수는 부호와 크기 표현과 같다. 음수의 2의 보수는 1의 보수에 \\(1\\)을 더한 것이다.</li> </ul> <p>아래 그림(3-4)은 부호와 크기, 1의 보수, 2의 보수를 서로 변환하는 방법을 보여 준다.</p> <p></p> <p> 그림 3-4   부호와 크기, 1의 보수, 2의 보수의 상호 변환 </p> <p>부호와 크기(sign-magnitude) 표현은 가장 직관적이지만 한계가 있다. 우선 음수의 부호와 크기 표현은 연산에 그대로 사용할 수 없다. 예를 들어 이 표현으로 \\(1 + (-2)\\)를 계산하면 결과가 \\(-3\\)이 되어 명백히 틀린다.</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 + 1000 \\; 0010 \\newline &amp; = 1000 \\; 0011 \\newline &amp; \\rightarrow -3 \\end{aligned} \\] <p>이 문제를 해결하기 위해 1의 보수(1's complement)가 도입되었다. 부호와 크기 표현을 먼저 1의 보수로 바꾸고 \\(1 + (-2)\\)를 계산한 뒤, 결과를 다시 부호와 크기 표현으로 바꾸면 올바른 결과인 \\(-1\\)을 얻는다.</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 \\; \\text{(부호와 크기)} + 1000 \\; 0010 \\; \\text{(부호와 크기)} \\newline &amp; = 0000 \\; 0001 \\; \\text{(1의 보수)} + 1111  \\; 1101 \\; \\text{(1의 보수)} \\newline &amp; = 1111 \\; 1110 \\; \\text{(1의 보수)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(부호와 크기)} \\newline &amp; \\rightarrow -1 \\end{aligned} \\] <p>또한 부호와 크기 표현에서는 0을 \\(+0\\)과 \\(-0\\)의 두 가지 방식으로 나타낸다. 즉, 0에 서로 다른 두 이진 인코딩이 대응하므로 모호함이 생길 수 있다. 예를 들어 조건을 판단할 때 양의 0과 음의 0을 구분하지 않으면 결과가 틀릴 수 있다. 이러한 모호함을 처리하려면 추가 판단 연산이 필요하므로 컴퓨터의 연산 효율이 낮아질 수 있다.</p> \\[ \\begin{aligned} +0 &amp; \\rightarrow 0000 \\; 0000 \\newline -0 &amp; \\rightarrow 1000 \\; 0000 \\end{aligned} \\] <p>1의 보수에도 부호와 크기 표현처럼 양의 0과 음의 0의 모호함이 있다. 그래서 2의 보수(2's complement)가 도입되었다. 먼저 음의 0을 부호와 크기, 1의 보수, 2의 보수로 변환하는 과정을 살펴보자.</p> \\[ \\begin{aligned} -0 \\rightarrow \\; &amp; 1000 \\; 0000 \\; \\text{(부호와 크기)} \\newline = \\; &amp; 1111 \\; 1111 \\; \\text{(1의 보수)} \\newline = 1 \\; &amp; 0000 \\; 0000 \\; \\text{(2의 보수)} \\newline \\end{aligned} \\] <p>음의 0의 1의 보수에 \\(1\\)을 더하면 자리 올림이 발생한다. 그러나 <code>byte</code>는 길이가 8비트뿐이므로 9번째 비트로 넘친 \\(1\\)은 버려진다. 즉, 음의 0의 2의 보수는 \\(0000 \\; 0000\\)으로, 양의 0의 2의 보수와 같다. 따라서 2의 보수 표현에는 0이 하나만 존재하며, 양의 0과 음의 0의 모호함이 해결된다.</p> <p>마지막 의문이 남는다. <code>byte</code>의 값 범위는 \\([-128, 127]\\)인데, 하나 더 있는 음수 \\(-128\\)은 어떻게 얻을까? 구간 \\([-127, +127]\\)의 모든 정수에는 대응하는 부호와 크기, 1의 보수, 2의 보수 표현이 있으며, 부호와 크기 표현과 2의 보수는 서로 변환할 수 있다.</p> <p>그러나 2의 보수 \\(1000 \\; 0000\\)은 예외로, 대응하는 부호와 크기 표현이 없다. 변환 방법을 적용하면 부호와 크기 표현으로 \\(0000 \\; 0000\\)을 얻는다. 하지만 이 표현은 수 \\(0\\)을 나타내며 2의 보수도 자기 자신이어야 하므로 모순이다. 컴퓨터에서는 이 특수한 2의 보수 \\(1000 \\; 0000\\)이 \\(-128\\)을 나타내도록 정한다. 실제로 \\((-1) + (-127)\\)을 2의 보수로 계산한 결과는 \\(-128\\)이다.</p> \\[ \\begin{aligned} &amp; (-127) + (-1) \\newline &amp; \\rightarrow 1111 \\; 1111 \\; \\text{(부호와 크기)} + 1000 \\; 0001 \\; \\text{(부호와 크기)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(1의 보수)} + 1111  \\; 1110 \\; \\text{(1의 보수)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(2의 보수)} + 1111  \\; 1111 \\; \\text{(2의 보수)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(2의 보수)} \\newline &amp; \\rightarrow -128 \\end{aligned} \\] <p>위의 계산이 모두 덧셈이라는 사실을 발견했을 것이다. 여기에는 중요한 사실이 담겨 있다. 컴퓨터 내부의 하드웨어 회로는 주로 덧셈 연산을 기반으로 설계된다. 덧셈은 곱셈, 나눗셈, 뺄셈 같은 다른 연산보다 하드웨어로 구현하기 쉽고, 병렬 처리도 쉬우며, 연산 속도도 빠르기 때문이다.</p> <p>그렇다고 컴퓨터가 덧셈만 할 수 있다는 뜻은 아니다. 덧셈과 몇 가지 기본 논리 연산을 결합하면 다양한 수학 연산을 구현할 수 있다. 예를 들어 뺄셈 \\(a - b\\)는 덧셈 \\(a + (-b)\\)로 바꿀 수 있고, 곱셈과 나눗셈도 여러 번의 덧셈이나 뺄셈으로 바꿀 수 있다.</p> <p>이제 컴퓨터가 2의 보수를 사용하는 이유를 정리할 수 있다. 2의 보수 표현을 쓰면 같은 회로와 연산으로 양수와 음수의 덧셈을 처리할 수 있다. 뺄셈을 위한 별도 하드웨어 회로를 설계할 필요가 없으며, 양의 0과 음의 0의 모호함도 따로 처리하지 않아도 된다. 이는 하드웨어 설계를 크게 단순화하고 연산 효율을 높인다.</p> <p>2의 보수는 매우 정교하게 설계된 방식이지만, 분량상 여기까지만 소개한다. 관심 있는 독자는 더 깊이 알아보기를 권한다.</p>","path":["제 3 장   자료 구조","3.3   수의 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/number_encoding/#332","level":2,"title":"3.3.2   부동 소수점 인코딩","text":"<p>자세히 살펴보면 이런 의문이 생긴다. <code>int</code>와 <code>float</code>는 모두 4바이트로 길이가 같은데, 왜 <code>float</code>의 값 범위가 <code>int</code>보다 훨씬 넓을까? <code>float</code>는 소수도 표현해야 하므로 오히려 값 범위가 줄어들 것 같아 직관과 어긋난다.</p> <p>실제로 이는 부동 소수점형 <code>float</code>가 다른 표현 방식을 사용하기 때문이다. 길이가 32비트인 이진수를 다음과 같이 쓰자.</p> \\[ b_{31} b_{30} b_{29} \\ldots b_2 b_1 b_0 \\] <p>IEEE 754 표준에 따르면 32비트 <code>float</code>는 다음 세 부분으로 이루어진다.</p> <ul> <li>부호 비트 \\(\\mathrm{S}\\): 1비트를 차지하며 \\(b_{31}\\)에 해당한다.</li> <li>지수 비트 \\(\\mathrm{E}\\): 8비트를 차지하며 \\(b_{30} b_{29} \\ldots b_{23}\\)에 해당한다.</li> <li>가수의 소수부 비트 \\(\\mathrm{N}\\): 23비트를 차지하며 \\(b_{22} b_{21} \\ldots b_0\\)에 해당한다.</li> </ul> <p>이진수 <code>float</code>가 나타내는 값을 계산하는 방법은 다음과 같다.</p> \\[ \\text {val} = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\] <p>이를 십진수로 바꾸면 다음 식을 얻는다.</p> \\[ \\text {val}=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\] <p>각 항의 값 범위는 다음과 같다.</p> \\[ \\begin{aligned} \\mathrm{S} \\in &amp; \\{ 0, 1\\}, \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = &amp; (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\] <p></p> <p> 그림 3-5   IEEE 754 표준에 따른 float 계산 예 </p> <p>위 그림(3-5)에서 예로 든 데이터가 \\(\\mathrm{S} = 0\\), \\(\\mathrm{E} = 124\\), \\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\)이면 다음과 같다.</p> \\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\] <p>이제 처음의 질문에 답할 수 있다. <code>float</code>의 표현 방식에는 지수 비트가 있으므로 값 범위가 <code>int</code>보다 훨씬 넓다. 위 계산에 따르면 <code>float</code>로 표현할 수 있는 가장 큰 양수는 \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\)이며, 부호 비트를 바꾸면 가장 작은 음수를 얻는다.</p> <p>부동 소수점형 <code>float</code>는 값 범위를 넓히는 대신 정밀도를 희생한다. 정수형 <code>int</code>는 32비트 전체로 수를 표현하므로 수들이 균일하게 분포한다. 반면 부동 소수점형 <code>float</code>는 지수 비트 때문에 값이 커질수록 인접한 두 수 사이의 간격도 커지는 경향이 있다.</p> <p>아래 표(3-2)와 같이 지수 비트 \\(\\mathrm{E} = 0\\)과 \\(\\mathrm{E} = 255\\)는 특별한 의미를 가지며, 0, 무한대, \\(\\mathrm{NaN}\\) 등을 표현하는 데 사용된다.</p> <p> 표 3-2   지수 비트의 의미 </p> 지수 비트 E 소수부 \\(\\mathrm{N} = 0\\) 소수부 \\(\\mathrm{N} \\ne 0\\) 계산식 \\(0\\) \\(\\pm 0\\) 비정규화 수 \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) 정규화 수 정규화 수 \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\) <p>비정규화 수는 부동 소수점의 정밀도를 크게 높인다는 점도 알아 두자. 가장 작은 양의 정규화 수는 \\(2^{-126}\\)이고, 가장 작은 양의 비정규화 수는 \\(2^{-126} \\times 2^{-23}\\)이다.</p> <p>배정밀도 <code>double</code>도 <code>float</code>와 비슷한 표현 방식을 사용하므로 여기서는 자세한 설명을 생략한다.</p>","path":["제 3 장   자료 구조","3.3   수의 인코딩 *"],"tags":[]},{"location":"chapter_data_structure/summary/","level":1,"title":"3.5   요약","text":"","path":["제 3 장   자료 구조","3.5   요약"],"tags":[]},{"location":"chapter_data_structure/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>자료 구조는 논리적 구조와 물리적 구조라는 두 관점에서 분류할 수 있다. 논리적 구조는 데이터 원소 사이의 논리적 관계를 설명하고, 물리적 구조는 컴퓨터 메모리에 데이터를 저장하는 방식을 설명한다.</li> <li>대표적인 논리적 구조에는 선형, 트리형, 네트워크형 등이 있다. 보통 논리적 구조에 따라 자료 구조를 선형(배열, 연결 리스트, 스택, 큐)과 비선형(트리, 그래프, 힙)으로 나눈다. 해시 테이블 구현에는 선형과 비선형 자료 구조가 함께 포함될 수 있다.</li> <li>프로그램이 실행되면 데이터는 컴퓨터 메모리에 저장된다. 각 메모리 공간에는 대응하는 메모리 주소가 있으며, 프로그램은 이 주소로 데이터에 접근한다.</li> <li>물리적 구조는 주로 연속 공간 저장(배열)과 분산 공간 저장(연결 리스트)으로 나뉜다. 모든 자료 구조는 배열, 연결 리스트 또는 둘의 조합으로 구현된다.</li> <li>컴퓨터의 기본 자료형에는 정수형 <code>byte</code>, <code>short</code>, <code>int</code>, <code>long</code>, 부동 소수점형 <code>float</code>, <code>double</code>, 문자형 <code>char</code>, 불리언형 <code>bool</code>이 있다. 값 범위는 차지하는 공간의 크기와 표현 방식에 따라 결정된다.</li> <li>부호와 크기, 1의 보수, 2의 보수는 컴퓨터에서 수를 인코딩하는 세 가지 방식으로, 서로 변환할 수 있다. 정수의 부호와 크기 표현에서 최상위 비트는 부호 비트이고, 나머지 비트는 수의 크기를 나타낸다.</li> <li>정수는 컴퓨터에 2의 보수 형태로 저장된다. 이 표현에서는 양수와 음수의 덧셈을 똑같이 처리할 수 있어 뺄셈만을 위한 별도 하드웨어 회로가 필요 없으며, 양의 0과 음의 0의 모호함도 없다.</li> <li>부동 소수점 인코딩은 부호 1비트, 지수 8비트, 가수의 소수부 23비트로 이루어진다. 지수 비트 덕분에 정수보다 훨씬 넓은 값 범위를 갖는 대신 정밀도를 희생한다.</li> <li>ASCII 코드는 가장 초기에 등장한 영어 문자 집합으로, 길이가 1바이트이며 128개의 문자를 수록한다. GBK는 널리 쓰이는 중국어 문자 집합으로 2만 자 이상의 한자를 수록한다. Unicode는 전 세계 여러 언어의 문자를 담은 완전한 문자 집합 표준을 제공하여 인코딩 방식의 차이로 발생하는 글자 깨짐을 해결하고자 한다.</li> <li>UTF-8은 가장 널리 쓰이는 Unicode 인코딩 방식으로 범용성이 매우 좋다. 확장성이 뛰어난 가변 길이 인코딩이며 저장 공간을 효율적으로 활용한다. UTF-16과 UTF-32는 고정 길이 인코딩 방식이다. 중국어를 인코딩할 때 UTF-16은 UTF-8보다 공간을 적게 차지한다. Java, C# 등의 언어는 기본적으로 UTF-16을 사용한다.</li> </ul>","path":["제 3 장   자료 구조","3.5   요약"],"tags":[]},{"location":"chapter_data_structure/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 해시 테이블에는 왜 선형 자료 구조와 비선형 자료 구조가 함께 포함되는가?</p> <p>해시 테이블의 기반은 배열이며, 해시 충돌을 해결하기 위해 ‘체이닝’을 사용할 수 있다(뒤의 ‘해시 충돌’ 절에서 다룬다). 배열의 각 버킷은 연결 리스트를 가리키며, 연결 리스트 길이가 일정 기준을 넘으면 트리(보통 레드-블랙 트리)로 바뀔 수 있다.</p> <p>저장 관점에서 해시 테이블의 기반은 배열이고, 각 버킷의 슬롯에는 값 하나, 연결 리스트 또는 트리가 들어갈 수 있다. 따라서 해시 테이블에는 선형 자료 구조(배열, 연결 리스트)와 비선형 자료 구조(트리)가 함께 포함될 수 있다.</p> <p>Q: <code>char</code> 자료형의 길이는 1바이트인가?</p> <p><code>char</code>의 길이는 프로그래밍 언어가 사용하는 인코딩 방식에 따라 정해진다. 예를 들어 Java, JavaScript, TypeScript, C#은 모두 UTF-16 인코딩으로 Unicode 코드 포인트를 저장하므로 <code>char</code>의 길이는 2바이트다.</p> <p>Q: 배열로 구현한 자료 구조를 ‘정적 자료 구조’라고 부르면 모호하지 않은가? 스택에서도 원소를 꺼내거나 넣을 수 있는데, 이런 연산은 모두 ‘동적’이다.</p> <p>스택은 동적으로 데이터를 조작할 수 있지만 자료 구조는 여전히 ‘정적’이다. 즉, 길이를 바꿀 수 없다. 배열 기반 자료 구조는 원소를 동적으로 추가하거나 삭제할 수 있지만 용량은 고정되어 있다. 데이터 양이 미리 할당한 크기를 넘으면 더 큰 배열을 새로 만들고 기존 내용을 복사해야 한다.</p> <p>Q: 스택이나 큐를 만들 때 크기를 지정하지 않았는데, 왜 ‘정적 자료 구조’인가?</p> <p>고급 프로그래밍 언어에서는 스택이나 큐의 초기 용량을 직접 지정하지 않아도 클래스 내부에서 자동으로 처리한다. 예를 들어 Java <code>ArrayList</code>의 초기 용량은 보통 10이다. 용량 확장도 자동으로 이루어진다. 자세한 내용은 뒤의 ‘리스트’ 절에서 설명한다.</p> <p>Q: 부호와 크기 표현을 2의 보수로 바꾸려면 ‘반전한 뒤 1을 더한다’. 그렇다면 반대 변환은 역연산인 ‘1을 뺀 뒤 반전하기’여야 할 텐데, ‘반전한 뒤 1을 더하기’로도 가능한 이유는 무엇인가?</p> <p>두 표현의 상호 변환은 실제로 ‘보수’를 구하는 과정이기 때문이다. 먼저 보수를 정의하자. \\(a + b = c\\)라면 \\(a\\)를 \\(c\\)에 대한 \\(b\\)의 보수라 하고, 반대로 \\(b\\)도 \\(c\\)에 대한 \\(a\\)의 보수라고 한다.</p> <p>\\(n = 4\\)비트 이진수 \\(0010\\)이 주어졌다고 하자. 이를 부호와 크기 표현으로 보면(부호 비트는 고려하지 않는다), 2의 보수는 ‘반전한 뒤 1을 더해’ 얻는다.</p> \\[ 0010 \\rightarrow 1101 \\rightarrow 1110 \\] <p>원래 표현과 2의 보수의 합은 \\(0010 + 1110 = 10000\\)이다. 즉, 2의 보수 \\(1110\\)은 \\(10000\\)에 대한 원래 수 \\(0010\\)의 ‘보수’다. 따라서 ‘반전한 뒤 1을 더하기’는 실제로 \\(10000\\)에 대한 보수를 계산하는 과정이다.</p> <p>그렇다면 \\(10000\\)에 대한 2의 보수 \\(1110\\)의 ‘보수’는 무엇일까? 똑같이 ‘반전한 뒤 1을 더해서’ 얻을 수 있다.</p> \\[ 1110 \\rightarrow 0001 \\rightarrow 0010 \\] <p>즉, 원래 표현과 2의 보수는 \\(10000\\)에 대해 서로의 ‘보수’다. 따라서 두 방향의 변환에 같은 연산인 ‘반전한 뒤 1을 더하기’를 사용할 수 있다.</p> <p>물론 역연산인 ‘1을 뺀 뒤 반전하기’로도 2의 보수 \\(1110\\)에서 원래 표현을 구할 수 있다.</p> \\[ 1110 \\rightarrow 1101 \\rightarrow 0010 \\] <p>정리하면 ‘반전한 뒤 1을 더하기’와 ‘1을 뺀 뒤 반전하기’는 모두 \\(10000\\)에 대한 보수를 구하는 연산이므로 서로 동등하다.</p> <p>본질적으로 ‘반전’은 \\(1111\\)에 대한 보수를 구하는 연산이다(항상 <code>부호와 크기 표현 + 1의 보수 = 1111</code>이 성립하기 때문이다). 여기에 1을 더해서 얻은 2의 보수는 \\(10000\\)에 대한 보수다.</p> <p>위 설명은 \\(n = 4\\)를 예로 들었지만, 자릿수가 임의의 값인 이진수에도 일반화할 수 있다.</p>","path":["제 3 장   자료 구조","3.5   요약"],"tags":[]},{"location":"chapter_divide_and_conquer/","level":1,"title":"제 12 장   분할 정복","text":"<p>이 장을 시작하며</p> <p>어려운 문제도 한 단계씩 나눌 때마다 더욱 단순해진다.</p> <p>분할 정복은 중요한 사실을 알려 준다. 쉬운 것부터 시작하면 모든 것이 더 이상 복잡하지 않다.</p>","path":["제 12 장   분할 정복"],"tags":[]},{"location":"chapter_divide_and_conquer/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>12.1   분할 정복 알고리즘</li> <li>12.2   분할 정복 탐색 전략</li> <li>12.3   이진 트리 구성 문제</li> <li>12.4   하노이의 탑 문제</li> <li>12.5   요약</li> <li>12.6   연습 문제</li> </ul>","path":["제 12 장   분할 정복"],"tags":[]},{"location":"chapter_divide_and_conquer/binary_search_recur/","level":1,"title":"12.2   분할 정복 탐색 전략","text":"<p>앞에서 배웠듯이 탐색 알고리즘은 크게 두 종류로 나뉜다.</p> <ul> <li>완전 탐색: 자료 구조를 순회하며, 시간 복잡도는 \\(O(n)\\)이다.</li> <li>적응형 탐색: 데이터 고유의 구성 방식이나 사전 정보를 이용하며, 시간 복잡도가 \\(O(\\log n)\\) 또는 \\(O(1)\\)에 이를 수 있다.</li> </ul> <p>실제로 시간 복잡도가 \\(O(\\log n)\\)인 탐색 알고리즘은 보통 분할 정복 전략을 기반으로 구현된다. 이진 탐색과 트리가 그 예다.</p> <ul> <li>이진 탐색은 각 단계에서 원래 문제(배열에서 목표 원소 찾기)를 더 작은 문제(배열의 절반에서 목표 원소 찾기)로 나눈다. 배열이 비거나 목표 원소를 찾을 때까지 계속한다.</li> <li>트리는 분할 정복 사고의 대표적인 예다. 이진 탐색 트리, AVL 트리, 힙 등의 자료 구조에서 여러 연산의 시간 복잡도는 \\(O(\\log n)\\)이다.</li> </ul> <p>이진 탐색의 분할 정복 전략은 다음과 같다.</p> <ul> <li>문제를 나눌 수 있다: 중간 원소와 목표 원소를 비교하여 원래 문제(배열에서 찾기)를 부분 문제(배열의 절반에서 찾기)로 재귀적으로 나눈다.</li> <li>부분 문제가 독립적이다: 이진 탐색은 매번 부분 문제 하나만 처리하며 다른 부분 문제의 영향을 받지 않는다.</li> <li>부분 문제의 해를 합칠 필요가 없다: 특정 원소 하나를 찾는 것이 목적이므로 부분 문제의 해를 합칠 필요가 없다. 부분 문제를 해결하면 원래 문제도 함께 해결된다.</li> </ul> <p>분할 정복이 탐색 효율을 높이는 근본적인 이유는 완전 탐색이 매번 선택지 하나만 제외하는 데 비해 분할 정복 탐색은 매번 선택지의 절반을 제외할 수 있기 때문이다.</p>","path":["제 12 장   분할 정복","12.2   분할 정복 탐색 전략"],"tags":[]},{"location":"chapter_divide_and_conquer/binary_search_recur/#1","level":3,"title":"1.   분할 정복으로 이진 탐색 구현","text":"<p>앞 장에서는 이진 탐색을 반복 방식으로 구현했다. 이제 분할 정복(재귀)으로 구현해 보자.</p> <p>문제</p> <p>길이가 \\(n\\)이고 모든 원소가 서로 다른 정렬된 배열 <code>nums</code>가 주어진다. 원소 <code>target</code>을 찾아라.</p> <p>분할 정복의 관점에서 탐색 구간 \\([i, j]\\)에 해당하는 부분 문제를 \\(f(i, j)\\)라고 하자.</p> <p>원래 문제 \\(f(0, n-1)\\)에서 시작하여 다음 단계로 이진 탐색을 수행한다.</p> <ol> <li>탐색 구간 \\([i, j]\\)의 중간 인덱스 \\(m\\)을 계산하고, 이를 기준으로 탐색 구간의 절반을 제외한다.</li> <li>크기가 절반으로 줄어든 부분 문제 \\(f(i, m-1)\\) 또는 \\(f(m+1, j)\\)를 재귀적으로 푼다.</li> <li><code>target</code>을 찾거나 구간이 빌 때까지 <code>1.</code> 단계와 <code>2.</code> 단계를 반복한 뒤 반환한다.</li> </ol> <p>아래 그림(12-4)은 배열에서 원소 \\(6\\)을 이진 탐색하는 분할 정복 과정을 보여 준다.</p> <p></p> <p> 그림 12-4   이진 탐색의 분할 정복 과정 </p> <p>구현 코드에서는 문제 \\(f(i, j)\\)를 푸는 재귀 함수 <code>dfs()</code>를 정의한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_recur.py<pre><code>def dfs(nums: list[int], target: int, i: int, j: int) -&gt; int:\n    \"\"\"이진 탐색: 문제 f(i, j)\"\"\"\n    # 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if i &gt; j:\n        return -1\n    # 중간 인덱스 m을 계산한다\n    m = (i + j) // 2\n    if nums[m] &lt; target:\n        # 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m + 1, j)\n    elif nums[m] &gt; target:\n        # 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m - 1)\n    else:\n        # 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m\n\ndef binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"이진 탐색\"\"\"\n    n = len(nums)\n    # 문제 f(0, n-1)을 푼다\n    return dfs(nums, target, 0, n - 1)\n</code></pre> binary_search_recur.cpp<pre><code>/* 이진 탐색: 문제 f(i, j) */\nint dfs(vector&lt;int&gt; &amp;nums, int target, int i, int j) {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if (i &gt; j) {\n        return -1;\n    }\n    // 중간 인덱스 m을 계산한다\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m;\n    }\n}\n\n/* 이진 탐색 */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    int n = nums.size();\n    // 문제 f(0, n-1)을 푼다\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.java<pre><code>/* 이진 탐색: 문제 f(i, j) */\nint dfs(int[] nums, int target, int i, int j) {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if (i &gt; j) {\n        return -1;\n    }\n    // 중간 인덱스 m을 계산한다\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m;\n    }\n}\n\n/* 이진 탐색 */\nint binarySearch(int[] nums, int target) {\n    int n = nums.length;\n    // 문제 f(0, n-1)을 푼다\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.cs<pre><code>/* 이진 탐색: 문제 f(i, j) */\nint DFS(int[] nums, int target, int i, int j) {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if (i &gt; j) {\n        return -1;\n    }\n    // 중간 인덱스 m을 계산한다\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return DFS(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return DFS(nums, target, i, m - 1);\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m;\n    }\n}\n\n/* 이진 탐색 */\nint BinarySearch(int[] nums, int target) {\n    int n = nums.Length;\n    // 문제 f(0, n-1)을 푼다\n    return DFS(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.go<pre><code>/* 이진 탐색: 문제 f(i, j) */\nfunc dfs(nums []int, target, i, j int) int {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if i &gt; j {\n        return -1\n    }\n    //    중간 인덱스를 계산한다\n    m := i + ((j - i) &gt;&gt; 1)\n    //중간 원소와 목표 원소의 크기를 비교한다\n    if nums[m] &lt; target {\n        // 중간 원소가 더 작으면 배열의 오른쪽 절반을 재귀적으로 탐색한다\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m+1, j)\n    } else if nums[m] &gt; target {\n        // 중간 원소가 더 크면 배열의 왼쪽 절반을 재귀적으로 탐색한다\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m-1)\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m\n    }\n}\n\n/* 이진 탐색 */\nfunc binarySearch(nums []int, target int) int {\n    n := len(nums)\n    return dfs(nums, target, 0, n-1)\n}\n</code></pre> binary_search_recur.swift<pre><code>/* 이진 탐색: 문제 f(i, j) */\nfunc dfs(nums: [Int], target: Int, i: Int, j: Int) -&gt; Int {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if i &gt; j {\n        return -1\n    }\n    // 중간 인덱스 m을 계산한다\n    let m = (i + j) / 2\n    if nums[m] &lt; target {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums: nums, target: target, i: m + 1, j: j)\n    } else if nums[m] &gt; target {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums: nums, target: target, i: i, j: m - 1)\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m\n    }\n}\n\n/* 이진 탐색 */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // 문제 f(0, n-1)을 푼다\n    dfs(nums: nums, target: target, i: nums.startIndex, j: nums.endIndex - 1)\n}\n</code></pre> binary_search_recur.js<pre><code>/* 이진 탐색: 문제 f(i, j) */\nfunction dfs(nums, target, i, j) {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if (i &gt; j) {\n        return -1;\n    }\n    // 중간 인덱스 m을 계산한다\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m;\n    }\n}\n\n/* 이진 탐색 */\nfunction binarySearch(nums, target) {\n    const n = nums.length;\n    // 문제 f(0, n-1)을 푼다\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.ts<pre><code>/* 이진 탐색: 문제 f(i, j) */\nfunction dfs(nums: number[], target: number, i: number, j: number): number {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if (i &gt; j) {\n        return -1;\n    }\n    // 중간 인덱스 m을 계산한다\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m;\n    }\n}\n\n/* 이진 탐색 */\nfunction binarySearch(nums: number[], target: number): number {\n    const n = nums.length;\n    // 문제 f(0, n-1)을 푼다\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.dart<pre><code>/* 이진 탐색: 문제 f(i, j) */\nint dfs(List&lt;int&gt; nums, int target, int i, int j) {\n  // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n  if (i &gt; j) {\n    return -1;\n  }\n  // 중간 인덱스 m을 계산한다\n  int m = (i + j) ~/ 2;\n  if (nums[m] &lt; target) {\n    // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n    return dfs(nums, target, m + 1, j);\n  } else if (nums[m] &gt; target) {\n    // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n    return dfs(nums, target, i, m - 1);\n  } else {\n    // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n    return m;\n  }\n}\n\n/* 이진 탐색 */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  int n = nums.length;\n  // 문제 f(0, n-1)을 푼다\n  return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.rs<pre><code>/* 이진 탐색: 문제 f(i, j) */\nfn dfs(nums: &amp;[i32], target: i32, i: i32, j: i32) -&gt; i32 {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if i &gt; j {\n        return -1;\n    }\n    let m: i32 = i + (j - i) / 2;\n    if nums[m as usize] &lt; target {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m + 1, j);\n    } else if nums[m as usize] &gt; target {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m;\n    }\n}\n\n/* 이진 탐색 */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let n = nums.len() as i32;\n    // 문제 f(0, n-1)을 푼다\n    dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.c<pre><code>/* 이진 탐색: 문제 f(i, j) */\nint dfs(int nums[], int target, int i, int j) {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if (i &gt; j) {\n        return -1;\n    }\n    // 중간 인덱스 m을 계산한다\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        return m;\n    }\n}\n\n/* 이진 탐색 */\nint binarySearch(int nums[], int target, int numsSize) {\n    int n = numsSize;\n    // 문제 f(0, n-1)을 푼다\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.kt<pre><code>/* 이진 탐색: 문제 f(i, j) */\nfun dfs(\n    nums: IntArray,\n    target: Int,\n    i: Int,\n    j: Int\n): Int {\n    // 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n    if (i &gt; j) {\n        return -1\n    }\n    // 중간 인덱스 m을 계산한다\n    val m = (i + j) / 2\n    return if (nums[m] &lt; target) {\n        // 부분 문제 f(m+1, j)를 재귀적으로 푼다\n        dfs(nums, target, m + 1, j)\n    } else if (nums[m] &gt; target) {\n        // 부분 문제 f(i, m-1)를 재귀적으로 푼다\n        dfs(nums, target, i, m - 1)\n    } else {\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        m\n    }\n}\n\n/* 이진 탐색 */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    val n = nums.size\n    // 문제 f(0, n-1)을 푼다\n    return dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.rb<pre><code>### 이진 탐색: 문제 f(i, j) ###\ndef dfs(nums, target, i, j)\n  # 구간이 비어 있으면 목표 원소가 없다는 뜻이므로 -1을 반환한다\n  return -1 if i &gt; j\n\n  # 중간 인덱스 m을 계산한다\n  m = (i + j) / 2\n\n  if nums[m] &lt; target\n    # 부분 문제 f(m+1, j)를 재귀적으로 푼다\n    return dfs(nums, target, m + 1, j)\n  elsif nums[m] &gt; target\n    # 부분 문제 f(i, m-1)를 재귀적으로 푼다\n    return dfs(nums, target, i, m - 1)\n  else\n    # 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n    return m\n  end\nend\n\n### 이진 탐색 ###\ndef binary_search(nums, target)\n  n = nums.length\n  # 문제 f(0, n-1)을 푼다\n  dfs(nums, target, 0, n - 1)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 12 장   분할 정복","12.2   분할 정복 탐색 전략"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/","level":1,"title":"12.3   이진 트리 구성 문제","text":"<p>문제</p> <p>이진 트리의 전위 순회 <code>preorder</code>와 중위 순회 <code>inorder</code>가 주어진다. 이를 이용해 이진 트리를 구성하고 루트 노드를 반환하라. 트리에는 값이 중복되는 노드가 없다고 가정한다(아래 그림(12-5) 참고).</p> <p></p> <p> 그림 12-5   이진 트리 구성 예시 데이터 </p>","path":["제 12 장   분할 정복","12.3   이진 트리 구성 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#1","level":3,"title":"1.   분할 정복 문제인지 판단","text":"<p>원래 문제는 <code>preorder</code>와 <code>inorder</code>로 이진 트리를 구성하는 것으로, 전형적인 분할 정복 문제다.</p> <ul> <li>문제를 나눌 수 있다: 원래 문제를 왼쪽 서브트리 구성과 오른쪽 서브트리 구성이라는 두 부분 문제로 나누고, 루트 노드 초기화 연산을 하나 더할 수 있다. 각 서브트리(부분 문제)에도 같은 방법을 적용해 더 작은 서브트리(부분 문제)로 나눌 수 있으며, 가장 작은 부분 문제인 빈 서브트리에 도달하면 종료한다.</li> <li>부분 문제가 독립적이다: 왼쪽 서브트리와 오른쪽 서브트리는 겹치는 부분 없이 서로 독립적이다. 왼쪽 서브트리를 구성할 때는 중위 순회와 전위 순회에서 왼쪽 서브트리에 해당하는 부분만 보면 된다. 오른쪽도 마찬가지다.</li> <li>부분 문제의 해를 합칠 수 있다: 왼쪽 서브트리와 오른쪽 서브트리(부분 문제의 해)를 얻으면 루트 노드에 연결하여 원래 문제의 해를 만들 수 있다.</li> </ul>","path":["제 12 장   분할 정복","12.3   이진 트리 구성 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#2","level":3,"title":"2.   서브트리를 나누는 방법","text":"<p>위 분석에 따르면 분할 정복으로 해결할 수 있다. 그런데 전위 순회 <code>preorder</code>와 중위 순회 <code>inorder</code>로 왼쪽과 오른쪽 서브트리를 어떻게 나눌까?</p> <p>정의에 따르면 <code>preorder</code>와 <code>inorder</code>는 모두 세 부분으로 나뉜다.</p> <ul> <li>전위 순회: <code>[ 루트 노드 | 왼쪽 서브트리 | 오른쪽 서브트리 ]</code>. 예를 들어 위 그림(12-5)의 트리는 <code>[ 3 | 9 | 2 1 7 ]</code>에 해당한다.</li> <li>중위 순회: <code>[ 왼쪽 서브트리 | 루트 노드 | 오른쪽 서브트리 ]</code>. 예를 들어 위 그림(12-5)의 트리는 <code>[ 9 | 3 | 1 2 7 ]</code>에 해당한다.</li> </ul> <p>위 그림(12-5)의 데이터를 예로 들면 다음 그림(12-6)의 단계로 나눌 수 있다.</p> <ol> <li>전위 순회의 첫 원소 3은 루트 노드의 값이다.</li> <li><code>inorder</code>에서 루트 노드 3의 인덱스를 찾아 이를 기준으로 <code>inorder</code>를 <code>[ 9 | 3 | 1 2 7 ]</code>로 나눈다.</li> <li><code>inorder</code>의 분할 결과로 왼쪽 서브트리의 노드 수는 1, 오른쪽은 3임을 알 수 있다. 따라서 <code>preorder</code>를 <code>[ 3 | 9 | 2 1 7 ]</code>로 나눌 수 있다.</li> </ol> <p></p> <p> 그림 12-6   전위 순회와 중위 순회에서 서브트리 나누기 </p>","path":["제 12 장   분할 정복","12.3   이진 트리 구성 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#3","level":3,"title":"3.   변수로 서브트리 구간 표현","text":"<p>위 방법으로 <code>preorder</code>와 <code>inorder</code>에서 루트 노드, 왼쪽 서브트리, 오른쪽 서브트리의 인덱스 구간을 구했다. 이 구간들을 표현하려면 몇 개의 포인터 변수가 필요하다.</p> <ul> <li>현재 트리의 루트 노드가 <code>preorder</code>에서 갖는 인덱스를 \\(i\\)라고 한다.</li> <li>현재 트리의 루트 노드가 <code>inorder</code>에서 갖는 인덱스를 \\(m\\)이라고 한다.</li> <li>현재 트리가 <code>inorder</code>에서 차지하는 인덱스 구간을 \\([l, r]\\)이라고 한다.</li> </ul> <p>아래 표(12-1)와 같이 이 변수들로 <code>preorder</code>에서 루트 노드의 인덱스와 <code>inorder</code>에서 서브트리의 인덱스 구간을 나타낼 수 있다.</p> <p> 표 12-1   전위 순회와 중위 순회에서 루트 노드와 서브트리의 인덱스 </p> <code>preorder</code>에서 루트 노드의 인덱스 <code>inorder</code>에서 서브트리의 인덱스 구간 현재 트리 \\(i\\) \\([l, r]\\) 왼쪽 서브트리 \\(i + 1\\) \\([l, m-1]\\) 오른쪽 서브트리 \\(i + 1 + (m - l)\\) \\([m+1, r]\\) <p>오른쪽 서브트리의 루트 인덱스에 있는 \\((m-l)\\)은 '왼쪽 서브트리의 노드 수'를 뜻한다. 아래 그림(12-7)과 함께 이해해 보자.</p> <p></p> <p> 그림 12-7   루트 노드와 양쪽 서브트리의 인덱스 구간 표현 </p>","path":["제 12 장   분할 정복","12.3   이진 트리 구성 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#4","level":3,"title":"4.   코드 구현","text":"<p>\\(m\\)을 더 효율적으로 조회하기 위해 해시 테이블 <code>hmap</code>에 배열 <code>inorder</code>의 원소와 인덱스 사이의 매핑을 저장한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby build_tree.py<pre><code>def dfs(\n    preorder: list[int],\n    inorder_map: dict[int, int],\n    i: int,\n    l: int,\n    r: int,\n) -&gt; TreeNode | None:\n    \"\"\"이진 트리 구성: 분할 정복\"\"\"\n    # 서브트리 구간이 비어 있으면 종료한다\n    if r - l &lt; 0:\n        return None\n    # 루트 노드를 초기화한다\n    root = TreeNode(preorder[i])\n    # m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    m = inorder_map[preorder[i]]\n    # 부분 문제: 왼쪽 서브트리 구성\n    root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n    # 부분 문제: 오른쪽 서브트리 구성\n    root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n    # 루트 노드를 반환한다\n    return root\n\ndef build_tree(preorder: list[int], inorder: list[int]) -&gt; TreeNode | None:\n    \"\"\"이진 트리 구성\"\"\"\n    # inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    inorder_map = {val: i for i, val in enumerate(inorder)}\n    root = dfs(preorder, inorder_map, 0, 0, len(inorder) - 1)\n    return root\n</code></pre> build_tree.cpp<pre><code>/* 이진 트리 구성: 분할 정복 */\nTreeNode *dfs(vector&lt;int&gt; &amp;preorder, unordered_map&lt;int, int&gt; &amp;inorderMap, int i, int l, int r) {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if (r - l &lt; 0)\n        return NULL;\n    // 루트 노드를 초기화한다\n    TreeNode *root = new TreeNode(preorder[i]);\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    int m = inorderMap[preorder[i]];\n    // 부분 문제: 왼쪽 서브트리 구성\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // 부분 문제: 오른쪽 서브트리 구성\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // 루트 노드를 반환한다\n    return root;\n}\n\n/* 이진 트리 구성 */\nTreeNode *buildTree(vector&lt;int&gt; &amp;preorder, vector&lt;int&gt; &amp;inorder) {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    unordered_map&lt;int, int&gt; inorderMap;\n    for (int i = 0; i &lt; inorder.size(); i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorder.size() - 1);\n    return root;\n}\n</code></pre> build_tree.java<pre><code>/* 이진 트리 구성: 분할 정복 */\nTreeNode dfs(int[] preorder, Map&lt;Integer, Integer&gt; inorderMap, int i, int l, int r) {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if (r - l &lt; 0)\n        return null;\n    // 루트 노드를 초기화한다\n    TreeNode root = new TreeNode(preorder[i]);\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    int m = inorderMap.get(preorder[i]);\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // 루트 노드를 반환한다\n    return root;\n}\n\n/* 이진 트리 구성 */\nTreeNode buildTree(int[] preorder, int[] inorder) {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    Map&lt;Integer, Integer&gt; inorderMap = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; inorder.length; i++) {\n        inorderMap.put(inorder[i], i);\n    }\n    TreeNode root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.cs<pre><code>/* 이진 트리 구성: 분할 정복 */\nTreeNode? DFS(int[] preorder, Dictionary&lt;int, int&gt; inorderMap, int i, int l, int r) {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if (r - l &lt; 0)\n        return null;\n    // 루트 노드를 초기화한다\n    TreeNode root = new(preorder[i]);\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    int m = inorderMap[preorder[i]];\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.left = DFS(preorder, inorderMap, i + 1, l, m - 1);\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.right = DFS(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // 루트 노드를 반환한다\n    return root;\n}\n\n/* 이진 트리 구성 */\nTreeNode? BuildTree(int[] preorder, int[] inorder) {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    Dictionary&lt;int, int&gt; inorderMap = [];\n    for (int i = 0; i &lt; inorder.Length; i++) {\n        inorderMap.TryAdd(inorder[i], i);\n    }\n    TreeNode? root = DFS(preorder, inorderMap, 0, 0, inorder.Length - 1);\n    return root;\n}\n</code></pre> build_tree.go<pre><code>/* 이진 트리 구성: 분할 정복 */\nfunc dfsBuildTree(preorder []int, inorderMap map[int]int, i, l, r int) *TreeNode {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if r-l &lt; 0 {\n        return nil\n    }\n    // 루트 노드를 초기화한다\n    root := NewTreeNode(preorder[i])\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    m := inorderMap[preorder[i]]\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.Left = dfsBuildTree(preorder, inorderMap, i+1, l, m-1)\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.Right = dfsBuildTree(preorder, inorderMap, i+1+m-l, m+1, r)\n    // 루트 노드를 반환한다\n    return root\n}\n\n/* 이진 트리 구성 */\nfunc buildTree(preorder, inorder []int) *TreeNode {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    inorderMap := make(map[int]int, len(inorder))\n    for i := 0; i &lt; len(inorder); i++ {\n        inorderMap[inorder[i]] = i\n    }\n\n    root := dfsBuildTree(preorder, inorderMap, 0, 0, len(inorder)-1)\n    return root\n}\n</code></pre> build_tree.swift<pre><code>/* 이진 트리 구성: 분할 정복 */\nfunc dfs(preorder: [Int], inorderMap: [Int: Int], i: Int, l: Int, r: Int) -&gt; TreeNode? {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if r - l &lt; 0 {\n        return nil\n    }\n    // 루트 노드를 초기화한다\n    let root = TreeNode(x: preorder[i])\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    let m = inorderMap[preorder[i]]!\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.left = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1, l: l, r: m - 1)\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.right = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1 + m - l, l: m + 1, r: r)\n    // 루트 노드를 반환한다\n    return root\n}\n\n/* 이진 트리 구성 */\nfunc buildTree(preorder: [Int], inorder: [Int]) -&gt; TreeNode? {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    let inorderMap = inorder.enumerated().reduce(into: [:]) { $0[$1.element] = $1.offset }\n    return dfs(preorder: preorder, inorderMap: inorderMap, i: inorder.startIndex, l: inorder.startIndex, r: inorder.endIndex - 1)\n}\n</code></pre> build_tree.js<pre><code>/* 이진 트리 구성: 분할 정복 */\nfunction dfs(preorder, inorderMap, i, l, r) {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if (r - l &lt; 0) return null;\n    // 루트 노드를 초기화한다\n    const root = new TreeNode(preorder[i]);\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    const m = inorderMap.get(preorder[i]);\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // 루트 노드를 반환한다\n    return root;\n}\n\n/* 이진 트리 구성 */\nfunction buildTree(preorder, inorder) {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    let inorderMap = new Map();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.ts<pre><code>/* 이진 트리 구성: 분할 정복 */\nfunction dfs(\n    preorder: number[],\n    inorderMap: Map&lt;number, number&gt;,\n    i: number,\n    l: number,\n    r: number\n): TreeNode | null {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if (r - l &lt; 0) return null;\n    // 루트 노드를 초기화한다\n    const root: TreeNode = new TreeNode(preorder[i]);\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    const m = inorderMap.get(preorder[i]);\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // 루트 노드를 반환한다\n    return root;\n}\n\n/* 이진 트리 구성 */\nfunction buildTree(preorder: number[], inorder: number[]): TreeNode | null {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    let inorderMap = new Map&lt;number, number&gt;();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.dart<pre><code>/* 이진 트리 구성: 분할 정복 */\nTreeNode? dfs(\n  List&lt;int&gt; preorder,\n  Map&lt;int, int&gt; inorderMap,\n  int i,\n  int l,\n  int r,\n) {\n  // 서브트리 구간이 비어 있으면 종료한다\n  if (r - l &lt; 0) {\n    return null;\n  }\n  // 루트 노드를 초기화한다\n  TreeNode? root = TreeNode(preorder[i]);\n  // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n  int m = inorderMap[preorder[i]]!;\n  // 부분 문제: 왼쪽 서브트리 구성\n  root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n  // 부분 문제: 오른쪽 서브트리 구성\n  root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n  // 루트 노드를 반환한다\n  return root;\n}\n\n/* 이진 트리 구성 */\nTreeNode? buildTree(List&lt;int&gt; preorder, List&lt;int&gt; inorder) {\n  // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n  Map&lt;int, int&gt; inorderMap = {};\n  for (int i = 0; i &lt; inorder.length; i++) {\n    inorderMap[inorder[i]] = i;\n  }\n  TreeNode? root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n  return root;\n}\n</code></pre> build_tree.rs<pre><code>/* 이진 트리 구성: 분할 정복 */\nfn dfs(\n    preorder: &amp;[i32],\n    inorder_map: &amp;HashMap&lt;i32, i32&gt;,\n    i: i32,\n    l: i32,\n    r: i32,\n) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if r - l &lt; 0 {\n        return None;\n    }\n    // 루트 노드를 초기화한다\n    let root = TreeNode::new(preorder[i as usize]);\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    let m = inorder_map.get(&amp;preorder[i as usize]).unwrap();\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.borrow_mut().left = dfs(preorder, inorder_map, i + 1, l, m - 1);\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.borrow_mut().right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r);\n    // 루트 노드를 반환한다\n    Some(root)\n}\n\n/* 이진 트리 구성 */\nfn build_tree(preorder: &amp;[i32], inorder: &amp;[i32]) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    let mut inorder_map: HashMap&lt;i32, i32&gt; = HashMap::new();\n    for i in 0..inorder.len() {\n        inorder_map.insert(inorder[i], i as i32);\n    }\n    let root = dfs(preorder, &amp;inorder_map, 0, 0, inorder.len() as i32 - 1);\n    root\n}\n</code></pre> build_tree.c<pre><code>/* 이진 트리 구성: 분할 정복 */\nTreeNode *dfs(int *preorder, int *inorderMap, int i, int l, int r, int size) {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if (r - l &lt; 0)\n        return NULL;\n    // 루트 노드를 초기화한다\n    TreeNode *root = (TreeNode *)malloc(sizeof(TreeNode));\n    root-&gt;val = preorder[i];\n    root-&gt;left = NULL;\n    root-&gt;right = NULL;\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    int m = inorderMap[preorder[i]];\n    // 부분 문제: 왼쪽 서브트리 구성\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1, size);\n    // 부분 문제: 오른쪽 서브트리 구성\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r, size);\n    // 루트 노드를 반환한다\n    return root;\n}\n\n/* 이진 트리 구성 */\nTreeNode *buildTree(int *preorder, int preorderSize, int *inorder, int inorderSize) {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    int *inorderMap = (int *)malloc(sizeof(int) * MAX_SIZE);\n    for (int i = 0; i &lt; inorderSize; i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorderSize - 1, inorderSize);\n    free(inorderMap);\n    return root;\n}\n</code></pre> build_tree.kt<pre><code>/* 이진 트리 구성: 분할 정복 */\nfun dfs(\n    preorder: IntArray,\n    inorderMap: Map&lt;Int?, Int?&gt;,\n    i: Int,\n    l: Int,\n    r: Int\n): TreeNode? {\n    // 서브트리 구간이 비어 있으면 종료한다\n    if (r - l &lt; 0) return null\n    // 루트 노드를 초기화한다\n    val root = TreeNode(preorder[i])\n    // m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n    val m = inorderMap[preorder[i]]!!\n    // 부분 문제: 왼쪽 서브트리 구성\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1)\n    // 부분 문제: 오른쪽 서브트리 구성\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r)\n    // 루트 노드를 반환한다\n    return root\n}\n\n/* 이진 트리 구성 */\nfun buildTree(preorder: IntArray, inorder: IntArray): TreeNode? {\n    // inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n    val inorderMap = HashMap&lt;Int?, Int?&gt;()\n    for (i in inorder.indices) {\n        inorderMap[inorder[i]] = i\n    }\n    val root = dfs(preorder, inorderMap, 0, 0, inorder.size - 1)\n    return root\n}\n</code></pre> build_tree.rb<pre><code>### 이진 트리 구성: 분할 정복 ###\ndef dfs(preorder, inorder_map, i, l, r)\n  # 서브트리 구간이 비어 있으면 종료한다\n  return if r - l &lt; 0\n\n  # 루트 노드를 초기화한다\n  root = TreeNode.new(preorder[i])\n  # m을 조회하여 왼쪽과 오른쪽 서브트리를 나눈다\n  m = inorder_map[preorder[i]]\n  # 부분 문제: 왼쪽 서브트리 구성\n  root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n  # 부분 문제: 오른쪽 서브트리 구성\n  root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n\n  # 루트 노드를 반환한다\n  root\nend\n\n### 이진 트리 구성 ###\ndef build_tree(preorder, inorder)\n  # inorder 원소와 인덱스의 대응 관계를 저장할 해시 테이블을 초기화한다\n  inorder_map = {}\n  inorder.each_with_index { |val, i| inorder_map[val] = i }\n  dfs(preorder, inorder_map, 0, 0, inorder.length - 1)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(12-8)은 이진 트리 구성의 재귀 과정을 보여 준다. 각 노드는 아래로 '호출'하는 과정에서 만들어지고, 간선(참조)은 위로 '반환'하는 과정에서 만들어진다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 12-8   이진 트리 구성의 재귀 과정 </p> <p>각 재귀 함수에서 전위 순회 <code>preorder</code>와 중위 순회 <code>inorder</code>를 나눈 결과는 아래 그림(12-9)과 같다.</p> <p></p> <p> 그림 12-9   각 재귀 함수의 분할 결과 </p> <p>트리의 노드 수가 \\(n\\)일 때 노드 하나를 초기화하는 데(재귀 함수 <code>dfs()</code> 한 번 실행) \\(O(1)\\) 시간이 걸린다. 따라서 전체 시간 복잡도는 \\(O(n)\\)이다.</p> <p>해시 테이블은 <code>inorder</code> 원소와 인덱스의 매핑을 저장하므로 공간 복잡도가 \\(O(n)\\)이다. 최악의 경우인 이진 트리가 연결 리스트로 퇴화한 경우에는 재귀 깊이가 \\(n\\)에 이르러 \\(O(n)\\)의 스택 프레임 공간을 사용한다. 따라서 전체 공간 복잡도는 \\(O(n)\\)이다.</p>","path":["제 12 장   분할 정복","12.3   이진 트리 구성 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/","level":1,"title":"12.1   분할 정복 알고리즘","text":"<p>분할 정복(divide and conquer)은 '나누어 해결한다'는 뜻으로, 매우 중요하고 흔히 쓰이는 알고리즘 전략이다. 보통 재귀로 구현하며 '분할'과 '정복'의 두 단계로 이루어진다.</p> <ol> <li>분할(나누는 단계): 원래 문제를 둘 이상의 부분 문제로 재귀적으로 나누고, 가장 작은 부분 문제에 도달하면 멈춘다.</li> <li>정복(합치는 단계): 해를 이미 아는 가장 작은 부분 문제부터 시작하여 아래에서 위로 부분 문제의 해를 합침으로써 원래 문제의 해를 구성한다.</li> </ol> <p>아래 그림(12-1)의 '병합 정렬'은 분할 정복 전략의 대표적인 응용이다.</p> <ol> <li>분할: 원래 배열(원래 문제)을 두 부분 배열(부분 문제)로 재귀적으로 나누고, 부분 배열에 원소가 하나만 남으면(가장 작은 부분 문제) 멈춘다.</li> <li>정복: 정렬된 부분 배열(부분 문제의 해)을 아래에서 위로 합쳐 정렬된 원래 배열(원래 문제의 해)을 얻는다.</li> </ol> <p></p> <p> 그림 12-1   병합 정렬의 분할 정복 전략 </p>","path":["제 12 장   분할 정복","12.1   분할 정복 알고리즘"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1211","level":2,"title":"12.1.1   분할 정복 문제를 판단하는 방법","text":"<p>문제가 분할 정복에 적합한지는 보통 다음 기준으로 판단할 수 있다.</p> <ol> <li>문제를 나눌 수 있다: 원래 문제를 크기가 더 작고 형태가 비슷한 부분 문제로 나눌 수 있으며, 같은 방식으로 재귀적으로 나눌 수 있다.</li> <li>부분 문제가 독립적이다: 부분 문제들이 겹치지 않고 서로 의존하지 않으며, 독립적으로 해결할 수 있다.</li> <li>부분 문제의 해를 합칠 수 있다: 부분 문제의 해를 합쳐 원래 문제의 해를 얻는다.</li> </ol> <p>병합 정렬은 위 세 기준을 모두 만족한다.</p> <ol> <li>문제를 나눌 수 있다: 배열(원래 문제)을 두 부분 배열(부분 문제)로 재귀적으로 나눈다.</li> <li>부분 문제가 독립적이다: 각 부분 배열을 독립적으로 정렬할 수 있다(부분 문제를 독립적으로 해결할 수 있다).</li> <li>부분 문제의 해를 합칠 수 있다: 정렬된 두 부분 배열(부분 문제의 해)을 하나의 정렬된 배열(원래 문제의 해)로 합칠 수 있다.</li> </ol>","path":["제 12 장   분할 정복","12.1   분할 정복 알고리즘"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1212","level":2,"title":"12.1.2   분할 정복으로 효율 높이기","text":"<p>분할 정복은 알고리즘 문제를 효과적으로 해결할 뿐 아니라 효율을 높이는 경우도 많다. 퀵 정렬, 병합 정렬, 힙 정렬이 선택 정렬, 버블 정렬, 삽입 정렬보다 빠른 이유도 분할 정복 전략을 사용하기 때문이다.</p> <p>여기서 궁금해진다. 분할 정복이 알고리즘의 효율을 높이는 근본적인 원리는 무엇일까? 큰 문제를 여러 부분 문제로 나누고, 부분 문제를 해결하고, 그 해를 합쳐 원래 문제의 해를 구하는 과정이 왜 원래 문제를 직접 푸는 것보다 효율적일까? 연산 수와 병렬 계산이라는 두 관점에서 살펴볼 수 있다.</p>","path":["제 12 장   분할 정복","12.1   분할 정복 알고리즘"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1","level":3,"title":"1.   연산 수 최적화","text":"<p>'버블 정렬'은 길이 \\(n\\)의 배열을 처리하는 데 \\(O(n^2)\\) 시간이 걸린다. 아래 그림(12-2)처럼 중간에서 두 부분 배열로 나눈다고 하자. 분할에는 \\(O(n)\\) 시간, 각 부분 배열의 정렬에는 \\(O((n / 2)^2)\\) 시간, 두 부분 배열의 병합에는 \\(O(n)\\) 시간이 걸리므로 전체 시간 복잡도는 다음과 같다.</p> \\[ O(n + (\\frac{n}{2})^2 \\times 2 + n) = O(\\frac{n^2}{2} + 2n) \\] <p></p> <p> 그림 12-2   배열 분할 전후의 버블 정렬 </p> <p>이어서 왼쪽은 분할 전, 오른쪽은 분할 후의 총연산 수를 나타내는 다음 부등식을 계산해 보자.</p> \\[ \\begin{aligned} n^2 &amp; &gt; \\frac{n^2}{2} + 2n \\newline n^2 - \\frac{n^2}{2} - 2n &amp; &gt; 0 \\newline n(n - 4) &amp; &gt; 0 \\end{aligned} \\] <p>이는 \\(n &gt; 4\\)일 때 분할 후의 연산 수가 더 적어 정렬 효율이 높아질 것임을 뜻한다. 분할 후에도 시간 복잡도는 여전히 이차인 \\(O(n^2)\\)이며, 상수항만 작아졌다는 점에 주의하자.</p> <p>여기서 더 나아가 부분 배열을 계속 중간에서 두 개로 나누고, 원소가 하나만 남을 때까지 반복하면 어떨까? 이 발상은 바로 '병합 정렬'이며 시간 복잡도는 \\(O(n \\log n)\\)이다.</p> <p>다시 생각해 보자. 분할 지점을 더 많이 두어 원래 배열을 \\(k\\) 개의 부분 배열로 고르게 나누면 어떨까? 이는 '버킷 정렬'과 매우 비슷하다. 대규모 데이터 정렬에 적합하며 이론적으로 시간 복잡도가 \\(O(n + k)\\)에 이를 수 있다.</p>","path":["제 12 장   분할 정복","12.1   분할 정복 알고리즘"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#2","level":3,"title":"2.   병렬 계산 최적화","text":"<p>분할 정복으로 만들어진 부분 문제는 서로 독립적이므로 보통 병렬로 해결할 수 있다. 즉, 분할 정복은 알고리즘의 시간 복잡도를 줄일 뿐 아니라 운영체제의 병렬 최적화에도 유리하다.</p> <p>병렬 최적화는 멀티코어나 다중 프로세서 환경에서 특히 효과적이다. 여러 부분 문제를 동시에 처리하여 계산 자원을 더 충분히 활용하고 전체 실행 시간을 크게 줄일 수 있기 때문이다.</p> <p>예를 들어 아래 그림(12-3)의 '버킷 정렬'에서 대규모 데이터를 각 버킷에 고르게 분배하면, 버킷별 정렬 작업을 여러 계산 장치에 나누어 맡긴 뒤 결과를 합칠 수 있다.</p> <p></p> <p> 그림 12-3   버킷 정렬의 병렬 계산 </p>","path":["제 12 장   분할 정복","12.1   분할 정복 알고리즘"],"tags":[]},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1213","level":2,"title":"12.1.3   분할 정복의 주요 응용","text":"<p>분할 정복은 여러 대표적인 알고리즘 문제를 해결할 수 있다.</p> <ul> <li>가장 가까운 점 쌍 찾기: 먼저 점 집합을 두 부분으로 나누고 각 부분에서 가장 가까운 점 쌍을 찾은 뒤, 마지막으로 두 부분에 걸쳐 있는 가장 가까운 점 쌍을 찾는다.</li> <li>큰 정수의 곱셈: Karatsuba 알고리즘은 큰 정수의 곱셈을 더 작은 정수들의 곱셈과 덧셈으로 나눈다.</li> <li>행렬 곱셈: Strassen 알고리즘은 큰 행렬의 곱셈을 작은 행렬들의 곱셈과 덧셈으로 나눈다.</li> <li>하노이의 탑 문제: 재귀로 해결할 수 있는 하노이의 탑은 분할 정복 전략의 대표적인 응용이다.</li> <li>역전 쌍 구하기: 수열에서 앞의 수가 뒤의 수보다 크면 두 수가 역전 쌍을 이룬다. 역전 쌍 문제는 분할 정복의 아이디어로 병합 정렬을 이용하여 해결할 수 있다.</li> </ul> <p>또한 분할 정복은 알고리즘과 자료 구조의 설계에도 매우 널리 쓰인다.</p> <ul> <li>이진 탐색: 정렬된 배열을 중간 인덱스에서 두 부분으로 나누고, 목표 값과 중간 원소를 비교한 결과에 따라 제외할 절반을 정한다. 남은 구간에서 같은 이진 탐색 연산을 수행한다.</li> <li>병합 정렬: 이 절의 앞부분에서 소개했으므로 다시 설명하지 않는다.</li> <li>퀵 정렬: 피벗을 하나 고르고 피벗보다 작은 원소와 큰 원소를 각각 두 부분 배열로 나눈다. 부분 배열에 원소가 하나만 남을 때까지 같은 분할을 반복한다.</li> <li>버킷 정렬: 데이터를 여러 버킷에 분산하고 각 버킷의 원소를 정렬한 뒤, 버킷별 원소를 차례로 꺼내 정렬된 배열을 얻는다.</li> <li>트리: 이진 탐색 트리, AVL 트리, 레드-블랙 트리, B 트리, B+ 트리 등의 탐색, 삽입, 삭제 연산은 모두 분할 정복 전략의 응용으로 볼 수 있다.</li> <li>힙: 특수한 완전 이진 트리인 힙의 삽입, 삭제, 재정렬 등에도 분할 정복의 아이디어가 숨어 있다.</li> <li>해시 테이블: 분할 정복을 직접 적용하지는 않지만 일부 해시 충돌 해결 방법에는 간접적으로 사용된다. 예를 들어 체이닝의 긴 연결 리스트를 레드-블랙 트리로 바꾸어 조회 효율을 높인다.</li> </ul> <p>이처럼 분할 정복은 조용히 스며들어 작용하는 알고리즘 사고방식이며, 다양한 알고리즘과 자료 구조 속에 숨어 있다.</p>","path":["제 12 장   분할 정복","12.1   분할 정복 알고리즘"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/","level":1,"title":"12.6   연습 문제","text":"","path":["제 12 장   분할 정복","12.6   연습 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1261","level":2,"title":"12.6.1   개념 확인","text":"","path":["제 12 장   분할 정복","12.6   연습 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1","level":3,"title":"1.   어떤 작업에 분할 정복이 적합한가","text":"<p>한 학생이 ‘먼저 둘로 나누고, 각각 해결한 뒤, 결과를 합치는’ 방식으로 다음 작업을 수행하려 한다. 각 작업이 ‘분할 정복에 적합함’, ‘분할 정복이 가능하지만 전체 작업량은 줄어들지 않음’, ‘두 부분을 독립적으로 해결할 수 없음’ 중 어디에 해당하는지 판단하고 이유를 설명하라.</p> <ol> <li>정렬되지 않은 배열을 정렬한다.</li> <li>배열의 최댓값을 구한다.</li> <li>일련의 스택 연산 <code>push(x)</code>, <code>pop()</code>을 순서대로 실행하고 각 <code>pop()</code>에서 얻은 원소를 출력한다.</li> </ol> 참고 답안 <ol> <li>적합함: 반으로 나눈 뒤 두 부분을 독립적으로 정렬하고, \\(O(n)\\) 시간에 병합한다. 이것이 바로 병합 정렬이다.</li> <li>분할 정복이 가능하지만 전체 작업량은 줄어들지 않음: 왼쪽과 오른쪽을 합치면 여전히 \\(n\\)개 원소를 모두 확인해야 하므로,     직접 순회하는 것과 마찬가지로 \\(O(n)\\) 시간이 든다.</li> <li>두 부분을 독립적으로 해결할 수 없음: 후반부를 시작할 때의 스택 내용은 전반부의 실행 결과에 따라 달라진다.     따라서 서로의 결과를 모르는 상태에서 두 부분을 독립적으로 완료할 수 없다.</li> </ol>","path":["제 12 장   분할 정복","12.6   연습 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#2","level":3,"title":"2.   빠른 거듭제곱은 계산을 어떻게 줄이는가","text":"<p>다음 재귀 함수는 분할 정복으로 \\(x^n\\)을 계산한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby fast_power.py<pre><code>def fast_pow(x: int, n: int) -&gt; int:\n    \"\"\"빠른 거듭제곱\"\"\"\n    if n == 0:\n        return 1\n    half = fast_pow(x, n // 2)\n    if n % 2 == 0:\n        return half * half\n    return half * half * x\n</code></pre> fast_power.cpp<pre><code>/* 빠른 거듭제곱 */\nint fastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = fastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.java<pre><code>/* 빠른 거듭제곱 */\nint fastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = fastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.cs<pre><code>/* 빠른 거듭제곱 */\nint FastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = FastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.go<pre><code>/* 빠른 거듭제곱 */\nfunc fastPow(x, n int) int {\n    if n == 0 {\n        return 1\n    }\n    half := fastPow(x, n/2)\n    if n%2 == 0 {\n        return half * half\n    }\n    return half * half * x\n}\n</code></pre> fast_power.swift<pre><code>/* 빠른 거듭제곱 */\nfunc fastPow(x: Int, n: Int) -&gt; Int {\n    if n == 0 {\n        return 1\n    }\n    let half = fastPow(x: x, n: n / 2)\n    if n % 2 == 0 {\n        return half * half\n    }\n    return half * half * x\n}\n</code></pre> fast_power.js<pre><code>/* 빠른 거듭제곱 */\nfunction fastPow(x, n) {\n    if (n === 0) {\n        return 1;\n    }\n    const half = fastPow(x, Math.floor(n / 2));\n    if (n % 2 === 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.ts<pre><code>/* 빠른 거듭제곱 */\nfunction fastPow(x: number, n: number): number {\n    if (n === 0) {\n        return 1;\n    }\n    const half = fastPow(x, Math.floor(n / 2));\n    if (n % 2 === 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.dart<pre><code>/* 빠른 거듭제곱 */\nint fastPow(int x, int n) {\n  if (n == 0) {\n    return 1;\n  }\n  int half = fastPow(x, n ~/ 2);\n  if (n % 2 == 0) {\n    return half * half;\n  }\n  return half * half * x;\n}\n</code></pre> fast_power.rs<pre><code>/* 빠른 거듭제곱 */\nfn fast_pow(x: i32, n: i32) -&gt; i32 {\n    if n == 0 {\n        return 1;\n    }\n    let half = fast_pow(x, n / 2);\n    if n % 2 == 0 {\n        return half * half;\n    }\n    half * half * x\n}\n</code></pre> fast_power.c<pre><code>/* 빠른 거듭제곱 */\nint fastPow(int x, int n) {\n    if (n == 0) {\n        return 1;\n    }\n    int half = fastPow(x, n / 2);\n    if (n % 2 == 0) {\n        return half * half;\n    }\n    return half * half * x;\n}\n</code></pre> fast_power.kt<pre><code>/* 빠른 거듭제곱 */\nfun fastPow(x: Int, n: Int): Int {\n    if (n == 0) {\n        return 1\n    }\n    val half = fastPow(x, n / 2)\n    if (n % 2 == 0) {\n        return half * half\n    }\n    return half * half * x\n}\n</code></pre> fast_power.rb<pre><code>### 빠른 거듭제곱 ###\ndef fast_pow(x, n)\n  return 1 if n == 0\n\n  half = fast_pow(x, n / 2)\n  return half * half if n.even?\n\n  half * half * x\nend\n</code></pre> <p><code>x = 3</code>, <code>n = 5</code>로 설정하고 이 함수를 실행한다.</p> <ol> <li>재귀 호출에서 매개변수 <code>n</code>은 차례대로 어떤 값이 되는가?</li> <li>가장 깊은 호출부터 반환할 때 각 호출은 차례대로 어떤 값을 반환하는가?</li> <li>곱셈 양쪽에서 같은 부분 문제를 한 번씩 호출하지 않고, 먼저 재귀 호출의 결과를 <code>half</code>에 저장하는 이유는 무엇인가?</li> </ol> 참고 답안 <ol> <li> <p>매개변수는 <code>5 → 2 → 1 → 0</code> 순으로 바뀐다. 종료 조건에 도달할 때까지 매번 지수를 절반으로 줄인다.</p> </li> <li> <p><code>n = 0</code>이면 1을 반환한다. <code>n = 1</code>이면 \\(1×1×3=3\\)을 반환한다.     <code>n = 2</code>이면 \\(3×3=9\\)를, <code>n = 5</code>이면 \\(9×9×3=243\\)을 반환한다.</p> </li> <li> <p>곱셈 양쪽에서 같은 부분 문제를 한 번씩 호출하면 두 재귀 호출은 완전히 같은 계산을 수행한다.     먼저 결과를 <code>half</code>에 저장하면 각 단계에서 재귀 호출을 한 번만 하면 되고, 재귀 깊이는 대략 \\(\\log n\\)이다.     두 번 호출하면 같은 계산이 대량으로 반복된다.</p> </li> </ol>","path":["제 12 장   분할 정복","12.6   연습 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#3","level":3,"title":"3.   순회 결과에서 왼쪽 서브트리와 오른쪽 서브트리 나누기","text":"<p>중복된 노드가 없는 이진 트리의 전위 순회와 중위 순회 결과는 다음과 같다.</p> <ul> <li>전위 순회: <code>[A, B, D, E, C]</code></li> <li>중위 순회: <code>[D, B, E, A, C]</code></li> </ul> <p>루트 노드 수준에서만 나누고, 더 깊이 재귀적으로 나누거나 트리 전체를 그릴 필요는 없다.</p> <ol> <li>루트 노드는 무엇인가?</li> <li>중위 순회에서 왼쪽 서브트리와 오른쪽 서브트리에 해당하는 구간은 각각 어디인가?</li> <li>전위 순회에서 왼쪽 서브트리와 오른쪽 서브트리에 해당하는 구간은 각각 어디인가? 루트 노드의 직접적인 자식은 누구인가?</li> </ol> 참고 답안 <ol> <li> <p>전위 순회의 첫 번째 노드가 루트 노드이므로 루트는 <code>A</code>다.</p> </li> <li> <p><code>A</code>를 기준으로 중위 순회가 두 부분으로 나뉜다. 왼쪽 서브트리는 <code>[D, B, E]</code>, 오른쪽 서브트리는 <code>[C]</code>다.</p> </li> <li> <p>왼쪽 서브트리에 노드가 3개 있으므로 전위 순회에서 루트 노드 <code>A</code> 다음의 원소 3개가 왼쪽 서브트리에 속한다.     즉 <code>[B, D, E]</code>가 왼쪽 서브트리이고, 나머지 <code>[C]</code>가 오른쪽 서브트리다.     따라서 루트 노드의 왼쪽 자식은 <code>B</code>, 오른쪽 자식은 <code>C</code>다.</p> </li> </ol>","path":["제 12 장   분할 정복","12.6   연습 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1262","level":2,"title":"12.6.2   프로그래밍 연습","text":"","path":["제 12 장   분할 정복","12.6   연습 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/exercises/#1_1","level":3,"title":"1.   빠른 거듭제곱","text":"<p>실수 <code>x</code>와 정수 <code>n</code>이 주어진다. 언어에서 제공하는 거듭제곱 함수를 호출하지 않고 \\(x^n\\)을 계산하라. 재귀적인 분할 정복을 사용하여 매번 지수를 절반으로 줄이고, 이미 계산한 부분 문제의 결과를 재사용해야 한다. 이 문제에서는 <code>x = 0</code>인 경우를 포함하여 \\(x^0=1\\)로 정한다. <code>n &lt; 0</code>일 때는 <code>x != 0</code>임이 보장되며, 답을 \\((1/x)^{-n}\\)으로 바꾸어 구할 수 있다.</p> 풀이 힌트 <ol> <li>n이 0이면 답은 1이다.</li> <li>x의 n // 2제곱을 계산하여 half에 저장한다. 같은 재귀 호출을 두 번 하지 마라.</li> <li>n &lt; 0이면 먼저 x를 1 / x로 바꾸고 n을 -n으로 바꾼다. C++이나 Java에서는 n을 먼저 64비트 정수로 변환하면 가장 작은 32비트 정수의 부호를 반대로 바꿀 때 발생하는 오버플로를 피할 수 있다.</li> </ol> <p>LeetCode</p>","path":["제 12 장   분할 정복","12.6   연습 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/","level":1,"title":"12.4   하노이의 탑 문제","text":"<p>병합 정렬과 이진 트리 구성에서는 원래 문제를 크기가 절반인 두 부분 문제로 나누었다. 하지만 하노이의 탑 문제에서는 다른 분할 전략을 사용한다.</p> <p>문제</p> <p>기둥 세 개를 <code>A</code>, <code>B</code>, <code>C</code>라고 하자. 처음에는 기둥 <code>A</code>에 원판 \\(n\\) 개가 위에서 아래로 작은 것부터 큰 것 순서로 꽂혀 있다. 목표는 원래 순서를 유지하면서 이 원판 \\(n\\) 개를 기둥 <code>C</code>로 옮기는 것이다(아래 그림(12-10) 참고). 원판을 옮길 때는 다음 규칙을 지켜야 한다.</p> <ol> <li>원판은 한 기둥의 맨 위에서만 꺼내어 다른 기둥의 맨 위에 놓을 수 있다.</li> <li>한 번에 원판 하나만 옮길 수 있다.</li> <li>작은 원판은 항상 큰 원판 위에 있어야 한다.</li> </ol> <p></p> <p> 그림 12-10   하노이의 탑 문제 예시 </p> <p>크기가 \\(i\\)인 하노이의 탑 문제를 \\(f(i)\\)라고 하자. 예를 들어 \\(f(3)\\)은 원판 \\(3\\) 개를 <code>A</code>에서 <code>C</code>로 옮기는 문제다.</p>","path":["제 12 장   분할 정복","12.4   하노이의 탑 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/#1","level":3,"title":"1.   기본 경우 살펴보기","text":"<p>아래 그림(12-11)과 같이 \\(f(1)\\), 즉 원판이 하나뿐이면 <code>A</code>에서 <code>C</code>로 바로 옮기면 된다.</p> &lt;1&gt;&lt;2&gt; <p></p> <p></p> <p> 그림 12-11   크기 1인 문제의 해 </p> <p>아래 그림(12-12)과 같이 \\(f(2)\\), 즉 원판이 두 개이면 작은 원판이 항상 큰 원판 위에 있어야 하므로 <code>B</code>를 이용해야 한다.</p> <ol> <li>위의 작은 원판을 <code>A</code>에서 <code>B</code>로 옮긴다.</li> <li>큰 원판을 <code>A</code>에서 <code>C</code>로 옮긴다.</li> <li>작은 원판을 <code>B</code>에서 <code>C</code>로 옮긴다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> 그림 12-12   크기 2인 문제의 해 </p> <p>\\(f(2)\\)의 해결 과정은 <code>B</code>를 이용하여 원판 두 개를 <code>A</code>에서 <code>C</code>로 옮기는 것으로 요약할 수 있다. 여기서 <code>C</code>를 목표 기둥, <code>B</code>를 보조 기둥이라고 한다.</p>","path":["제 12 장   분할 정복","12.4   하노이의 탑 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/#2","level":3,"title":"2.   부분 문제로 나누기","text":"<p>\\(f(3)\\), 즉 원판이 세 개이면 상황이 조금 더 복잡해진다.</p> <p>\\(f(1)\\)과 \\(f(2)\\)의 해를 이미 알고 있으므로 분할 정복의 관점에서 생각해 보자. <code>A</code> 위쪽의 원판 두 개를 하나의 묶음으로 보고 아래 그림(12-13)의 단계를 수행하면 세 원판을 <code>A</code>에서 <code>C</code>로 옮길 수 있다.</p> <ol> <li><code>B</code>를 목표 기둥, <code>C</code>를 보조 기둥으로 삼아 원판 두 개를 <code>A</code>에서 <code>B</code>로 옮긴다.</li> <li><code>A</code>에 남은 원판 하나를 <code>C</code>로 바로 옮긴다.</li> <li><code>C</code>를 목표 기둥, <code>A</code>를 보조 기둥으로 삼아 원판 두 개를 <code>B</code>에서 <code>C</code>로 옮긴다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> 그림 12-13   크기 3인 문제의 해 </p> <p>본질적으로 문제 \\(f(3)\\)을 부분 문제 \\(f(2)\\) 두 개와 \\(f(1)\\) 하나로 나눈 것이다. 세 부분 문제를 순서대로 해결하면 원래 문제도 해결된다. 이는 부분 문제들이 독립적이고 해를 합칠 수 있음을 보여 준다.</p> <p>이제 아래 그림(12-14)과 같이 하노이의 탑을 푸는 분할 정복 전략을 정리할 수 있다. 원래 문제 \\(f(n)\\)을 부분 문제 \\(f(n-1)\\) 두 개와 \\(f(1)\\) 하나로 나누고, 다음 순서로 해결한다.</p> <ol> <li><code>C</code>를 이용하여 원판 \\(n-1\\) 개를 <code>A</code>에서 <code>B</code>로 옮긴다.</li> <li>남은 원판 \\(1\\) 개를 <code>A</code>에서 <code>C</code>로 바로 옮긴다.</li> <li><code>A</code>를 이용하여 원판 \\(n-1\\) 개를 <code>B</code>에서 <code>C</code>로 옮긴다.</li> </ol> <p>두 부분 문제 \\(f(n-1)\\)도 같은 방식으로 재귀적으로 나눌 수 있으며, 가장 작은 부분 문제 \\(f(1)\\)에 도달하면 멈춘다. \\(f(1)\\)의 해는 이미 알고 있고, 한 번만 옮기면 된다.</p> <p></p> <p> 그림 12-14   하노이의 탑을 푸는 분할 정복 전략 </p>","path":["제 12 장   분할 정복","12.4   하노이의 탑 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/hanota_problem/#3","level":3,"title":"3.   코드 구현","text":"<p>코드에서는 재귀 함수 <code>dfs(i, src, buf, tar)</code>를 정의한다. 이 함수는 기둥 <code>src</code> 위쪽의 원판 \\(i\\) 개를 보조 기둥 <code>buf</code>를 이용하여 목표 기둥 <code>tar</code>로 옮긴다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hanota.py<pre><code>def move(src: list[int], tar: list[int]):\n    \"\"\"원판 하나 이동\"\"\"\n    # src의 맨 위에서 원판 하나를 꺼낸다\n    pan = src.pop()\n    # 원판을 tar의 맨 위에 놓는다\n    tar.append(pan)\n\ndef dfs(i: int, src: list[int], buf: list[int], tar: list[int]):\n    \"\"\"하노이의 탑 문제 f(i) 풀기\"\"\"\n    # src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if i == 1:\n        move(src, tar)\n        return\n    # 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, tar, buf)\n    # 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, tar)\n    # 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, src, tar)\n\ndef solve_hanota(A: list[int], B: list[int], C: list[int]):\n    \"\"\"하노이의 탑 문제 풀기\"\"\"\n    n = len(A)\n    # B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(n, A, B, C)\n</code></pre> hanota.cpp<pre><code>/* 원판 하나 이동 */\nvoid move(vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;tar) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    int pan = src.back();\n    src.pop_back();\n    // 원판을 tar의 맨 위에 놓는다\n    tar.push_back(pan);\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nvoid dfs(int i, vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;buf, vector&lt;int&gt; &amp;tar) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, tar, buf);\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, tar);\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, src, tar);\n}\n\n/* 하노이의 탑 문제 풀기 */\nvoid solveHanota(vector&lt;int&gt; &amp;A, vector&lt;int&gt; &amp;B, vector&lt;int&gt; &amp;C) {\n    int n = A.size();\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.java<pre><code>/* 원판 하나 이동 */\nvoid move(List&lt;Integer&gt; src, List&lt;Integer&gt; tar) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    Integer pan = src.remove(src.size() - 1);\n    // 원판을 tar의 맨 위에 놓는다\n    tar.add(pan);\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nvoid dfs(int i, List&lt;Integer&gt; src, List&lt;Integer&gt; buf, List&lt;Integer&gt; tar) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, tar, buf);\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, tar);\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, src, tar);\n}\n\n/* 하노이의 탑 문제 풀기 */\nvoid solveHanota(List&lt;Integer&gt; A, List&lt;Integer&gt; B, List&lt;Integer&gt; C) {\n    int n = A.size();\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.cs<pre><code>/* 원판 하나 이동 */\nvoid Move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    int pan = src[^1];\n    src.RemoveAt(src.Count - 1);\n    // 원판을 tar의 맨 위에 놓는다\n    tar.Add(pan);\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nvoid DFS(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if (i == 1) {\n        Move(src, tar);\n        return;\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    DFS(i - 1, src, tar, buf);\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    Move(src, tar);\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    DFS(i - 1, buf, src, tar);\n}\n\n/* 하노이의 탑 문제 풀기 */\nvoid SolveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n    int n = A.Count;\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    DFS(n, A, B, C);\n}\n</code></pre> hanota.go<pre><code>/* 원판 하나 이동 */\nfunc move(src, tar *list.List) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    pan := src.Back()\n    // 원판을 tar의 맨 위에 놓는다\n    tar.PushBack(pan.Value)\n    // src의 맨 위 원판을 제거한다\n    src.Remove(pan)\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nfunc dfsHanota(i int, src, buf, tar *list.List) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if i == 1 {\n        move(src, tar)\n        return\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfsHanota(i-1, src, tar, buf)\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, tar)\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfsHanota(i-1, buf, src, tar)\n}\n\n/* 하노이의 탑 문제 풀기 */\nfunc solveHanota(A, B, C *list.List) {\n    n := A.Len()\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfsHanota(n, A, B, C)\n}\n</code></pre> hanota.swift<pre><code>/* 원판 하나 이동 */\nfunc move(src: inout [Int], tar: inout [Int]) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    let pan = src.popLast()!\n    // 원판을 tar의 맨 위에 놓는다\n    tar.append(pan)\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nfunc dfs(i: Int, src: inout [Int], buf: inout [Int], tar: inout [Int]) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if i == 1 {\n        move(src: &amp;src, tar: &amp;tar)\n        return\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i: i - 1, src: &amp;src, buf: &amp;tar, tar: &amp;buf)\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src: &amp;src, tar: &amp;tar)\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i: i - 1, src: &amp;buf, buf: &amp;src, tar: &amp;tar)\n}\n\n/* 하노이의 탑 문제 풀기 */\nfunc solveHanota(A: inout [Int], B: inout [Int], C: inout [Int]) {\n    let n = A.count\n    // 리스트의 맨 뒤가 기둥의 맨 위이다\n    // B를 이용하여 src의 맨 위 원판 n개를 C로 옮긴다\n    dfs(i: n, src: &amp;A, buf: &amp;B, tar: &amp;C)\n}\n</code></pre> hanota.js<pre><code>/* 원판 하나 이동 */\nfunction move(src, tar) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    const pan = src.pop();\n    // 원판을 tar의 맨 위에 놓는다\n    tar.push(pan);\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nfunction dfs(i, src, buf, tar) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, tar, buf);\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, tar);\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, src, tar);\n}\n\n/* 하노이의 탑 문제 풀기 */\nfunction solveHanota(A, B, C) {\n    const n = A.length;\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.ts<pre><code>/* 원판 하나 이동 */\nfunction move(src: number[], tar: number[]): void {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    const pan = src.pop();\n    // 원판을 tar의 맨 위에 놓는다\n    tar.push(pan);\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nfunction dfs(i: number, src: number[], buf: number[], tar: number[]): void {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, tar, buf);\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, tar);\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, src, tar);\n}\n\n/* 하노이의 탑 문제 풀기 */\nfunction solveHanota(A: number[], B: number[], C: number[]): void {\n    const n = A.length;\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.dart<pre><code>/* 원판 하나 이동 */\nvoid move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n  // src의 맨 위에서 원판 하나를 꺼낸다\n  int pan = src.removeLast();\n  // 원판을 tar의 맨 위에 놓는다\n  tar.add(pan);\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nvoid dfs(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n  // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n  if (i == 1) {\n    move(src, tar);\n    return;\n  }\n  // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n  dfs(i - 1, src, tar, buf);\n  // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n  move(src, tar);\n  // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n  dfs(i - 1, buf, src, tar);\n}\n\n/* 하노이의 탑 문제 풀기 */\nvoid solveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n  int n = A.length;\n  // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n  dfs(n, A, B, C);\n}\n</code></pre> hanota.rs<pre><code>/* 원판 하나 이동 */\nfn move_pan(src: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    let pan = src.pop().unwrap();\n    // 원판을 tar의 맨 위에 놓는다\n    tar.push(pan);\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nfn dfs(i: i32, src: &amp;mut Vec&lt;i32&gt;, buf: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if i == 1 {\n        move_pan(src, tar);\n        return;\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, tar, buf);\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move_pan(src, tar);\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, src, tar);\n}\n\n/* 하노이의 탑 문제 풀기 */\nfn solve_hanota(A: &amp;mut Vec&lt;i32&gt;, B: &amp;mut Vec&lt;i32&gt;, C: &amp;mut Vec&lt;i32&gt;) {\n    let n = A.len() as i32;\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.c<pre><code>/* 원판 하나 이동 */\nvoid move(int *src, int *srcSize, int *tar, int *tarSize) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    int pan = src[*srcSize - 1];\n    src[*srcSize - 1] = 0;\n    (*srcSize)--;\n    // 원판을 tar의 맨 위에 놓는다\n    tar[*tarSize] = pan;\n    (*tarSize)++;\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nvoid dfs(int i, int *src, int *srcSize, int *buf, int *bufSize, int *tar, int *tarSize) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if (i == 1) {\n        move(src, srcSize, tar, tarSize);\n        return;\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, srcSize, tar, tarSize, buf, bufSize);\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, srcSize, tar, tarSize);\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, bufSize, src, srcSize, tar, tarSize);\n}\n\n/* 하노이의 탑 문제 풀기 */\nvoid solveHanota(int *A, int *ASize, int *B, int *BSize, int *C, int *CSize) {\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(*ASize, A, ASize, B, BSize, C, CSize);\n}\n</code></pre> hanota.kt<pre><code>/* 원판 하나 이동 */\nfun move(src: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // src의 맨 위에서 원판 하나를 꺼낸다\n    val pan = src.removeAt(src.size - 1)\n    // 원판을 tar의 맨 위에 놓는다\n    tar.add(pan)\n}\n\n/* 하노이의 탑 문제 f(i) 풀기 */\nfun dfs(i: Int, src: MutableList&lt;Int&gt;, buf: MutableList&lt;Int&gt;, tar: MutableList&lt;Int&gt;) {\n    // src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n    if (i == 1) {\n        move(src, tar)\n        return\n    }\n    // 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n    dfs(i - 1, src, tar, buf)\n    // 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n    move(src, tar)\n    // 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n    dfs(i - 1, buf, src, tar)\n}\n\n/* 하노이의 탑 문제 풀기 */\nfun solveHanota(A: MutableList&lt;Int&gt;, B: MutableList&lt;Int&gt;, C: MutableList&lt;Int&gt;) {\n    val n = A.size\n    // B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n    dfs(n, A, B, C)\n}\n</code></pre> hanota.rb<pre><code>### 원판 하나 이동 ###\ndef move(src, tar)\n  # src의 맨 위에서 원판 하나를 꺼낸다\n  pan = src.pop\n  # 원판을 tar의 맨 위에 놓는다\n  tar &lt;&lt; pan\nend\n\n### 하노이의 탑 문제 f(i) 풀기 ###\ndef dfs(i, src, buf, tar)\n  # src에 원판이 하나만 남아 있으면 바로 tar로 옮긴다\n  if i == 1\n    move(src, tar)\n    return\n  end\n\n  # 부분 문제 f(i-1): tar를 이용하여 src의 맨 위 원판 i-1개를 buf로 옮긴다\n  dfs(i - 1, src, tar, buf)\n  # 부분 문제 f(1): src에 남은 원판 하나를 tar로 옮긴다\n  move(src, tar)\n  # 부분 문제 f(i-1): src를 이용하여 buf의 맨 위 원판 i-1개를 tar로 옮긴다\n  dfs(i - 1, buf, src, tar)\nend\n\n### 하노이의 탑 문제 풀기 ###\ndef solve_hanota(_A, _B, _C)\n  n = _A.length\n  # B를 이용하여 A의 맨 위 원판 n개를 C로 옮긴다\n  dfs(n, _A, _B, _C)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(12-15)과 같이 하노이의 탑 문제는 높이가 \\(n\\)인 재귀 트리를 이룬다. 각 노드는 부분 문제 하나를 나타내며 호출된 <code>dfs()</code> 함수 하나에 대응한다. 따라서 시간 복잡도는 \\(O(2^n)\\), 공간 복잡도는 \\(O(n)\\)이다.</p> <p></p> <p> 그림 12-15   하노이의 탑 문제의 재귀 트리 </p> <p>하노이의 탑 전설</p> <p>하노이의 탑 문제에는 오래된 전설이 전해진다. 고대 인도의 한 사원에서 승려들이 세 개의 높은 다이아몬드 기둥과 크기가 서로 다른 금 원판 \\(64\\) 개를 가지고 있었다. 승려들은 원판을 계속 옮기며 마지막 원판이 올바른 위치에 놓이는 순간 세상이 끝난다고 믿었다.</p> <p>하지만 매초 원판을 하나씩 옮겨도 총 \\(2^{64} \\approx 1.84×10^{19}\\) 초, 약 \\(5850\\) 억 년이 필요하다. 이는 현재 추정하는 우주의 나이보다 훨씬 길다. 따라서 전설이 사실이라고 해도 세상의 종말을 걱정할 필요는 없을 것이다.</p>","path":["제 12 장   분할 정복","12.4   하노이의 탑 문제"],"tags":[]},{"location":"chapter_divide_and_conquer/summary/","level":1,"title":"12.5   요약","text":"","path":["제 12 장   분할 정복","12.5   요약"],"tags":[]},{"location":"chapter_divide_and_conquer/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>분할 정복은 흔히 쓰이는 알고리즘 설계 전략이다. 분할(나누기)과 정복(합치기)의 두 단계로 이루어지며 보통 재귀로 구현한다.</li> <li>분할 정복 문제를 판단하는 기준은 문제를 나눌 수 있는지, 부분 문제들이 독립적인지, 부분 문제의 해를 합칠 수 있는지다.</li> <li>병합 정렬은 분할 정복 전략의 대표적인 응용이다. 배열을 같은 길이의 두 부분 배열로 재귀적으로 나누고, 원소가 하나만 남으면 단계적으로 병합하여 정렬한다.</li> <li>분할 정복 전략을 도입하면 알고리즘 효율이 높아지는 경우가 많다. 연산 수를 줄일 수 있고, 시스템의 병렬 최적화에도 유리하기 때문이다.</li> <li>분할 정복은 여러 알고리즘 문제를 해결할 뿐 아니라 자료 구조와 알고리즘의 설계에도 널리 사용되어 곳곳에서 찾아볼 수 있다.</li> <li>적응형 탐색은 완전 탐색보다 효율적이다. 시간 복잡도가 \\(O(\\log n)\\)인 탐색 알고리즘은 보통 분할 정복 전략을 기반으로 구현된다.</li> <li>이진 탐색은 분할 정복 전략의 또 다른 대표적인 응용으로, 부분 문제의 해를 합치는 단계가 없다. 재귀적 분할 정복으로 이진 탐색을 구현할 수 있다.</li> <li>이진 트리 구성 문제에서는 트리를 구성하는 원래 문제를 왼쪽 서브트리와 오른쪽 서브트리를 구성하는 부분 문제로 나눌 수 있다. 전위 순회와 중위 순회의 인덱스 구간을 나누어 구현한다.</li> <li>하노이의 탑에서는 크기 \\(n\\)인 문제를 크기 \\(n-1\\)인 부분 문제 두 개와 크기 \\(1\\)인 부분 문제 하나로 나눌 수 있다. 세 부분 문제를 순서대로 해결하면 원래 문제도 해결된다.</li> </ul>","path":["제 12 장   분할 정복","12.5   요약"],"tags":[]},{"location":"chapter_dynamic_programming/","level":1,"title":"제 14 장   동적 계획법","text":"<p>이 장을 시작하며</p> <p>시냇물은 강으로 모이고, 강물은 바다로 흘러든다.</p> <p>동적 계획법은 작은 문제의 해를 모아 큰 문제의 답을 만들며, 한 걸음씩 문제 해결의 건너편으로 우리를 이끈다.</p>","path":["제 14 장   동적 계획법"],"tags":[]},{"location":"chapter_dynamic_programming/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>14.1   동적 계획법 첫걸음</li> <li>14.2   동적 계획법 문제의 특성</li> <li>14.3   동적 계획법 문제 풀이 방법</li> <li>14.4   0-1 배낭 문제</li> <li>14.5   무한 배낭 문제</li> <li>14.6   편집 거리 문제</li> <li>14.7   요약</li> <li>14.8   연습 문제</li> </ul>","path":["제 14 장   동적 계획법"],"tags":[]},{"location":"chapter_dynamic_programming/dp_problem_features/","level":1,"title":"14.2   동적 계획법 문제의 특성","text":"<p>앞 절에서는 부분 문제로 나누어 원래 문제를 푸는 동적 계획법을 배웠다. 부분 문제로 나누는 것은 일반적인 알고리즘 아이디어지만 분할 정복, 동적 계획법, 백트래킹에서는 강조점이 다르다.</p> <ul> <li>분할 정복은 원래 문제를 독립적인 여러 부분 문제로 재귀적으로 나누어 가장 작은 부분 문제까지 내려간다. 이후 되돌아오면서 해를 합쳐 원래 문제의 해를 얻는다.</li> <li>동적 계획법도 문제를 재귀적으로 나누지만, 분할 정복과 달리 부분 문제들이 서로 의존하며 분해 과정에서 중복 부분 문제가 많이 생긴다.</li> <li>백트래킹은 시도와 되돌리기로 가능한 모든 해를 열거하고 가지치기로 불필요한 탐색 가지를 피한다. 원래 문제의 해는 일련의 의사 결정 단계로 이루어지며, 각 결정 단계에 앞선 부분 수열을 부분 문제로 볼 수 있다.</li> </ul> <p>동적 계획법은 최적화 문제에 자주 쓰인다. 이러한 문제에는 중복 부분 문제뿐 아니라 최적 부분 구조와 무후효성이라는 두 가지 특성이 더 있다.</p>","path":["제 14 장   동적 계획법","14.2   동적 계획법 문제의 특성"],"tags":[]},{"location":"chapter_dynamic_programming/dp_problem_features/#1421","level":2,"title":"14.2.1   최적 부분 구조","text":"<p>최적 부분 구조의 개념을 더 잘 보여 주도록 계단 오르기 문제를 조금 바꿔 보자.</p> <p>계단 오르기의 최소 비용</p> <p>계단을 한 번에 \\(1\\) 칸 또는 \\(2\\) 칸 오를 수 있다. 각 칸에는 그곳에서 지불할 비용인 음이 아닌 정수가 적혀 있다. 음이 아닌 정수 배열 \\(cost\\)가 주어지며, \\(cost[i]\\)는 \\(i\\) 번째 칸의 비용이고 \\(cost[0]\\)은 바닥(출발점)에 해당한다. 맨 위에 도달하는 데 필요한 최소 비용을 구하라.</p> <p>아래 그림(14-6)과 같이 \\(1\\), \\(2\\), \\(3\\) 번째 칸의 비용이 각각 \\(1\\), \\(10\\), \\(1\\)이면 바닥에서 \\(3\\) 번째 칸까지 오르는 최소 비용은 \\(2\\)다.</p> <p></p> <p> 그림 14-6   3번째 칸까지 오르는 최소 비용 </p> <p>\\(i\\) 번째 칸까지 오르는 데 지불한 누적 비용을 \\(dp[i]\\)라고 하자. \\(i\\) 번째 칸에는 \\(i - 1\\) 번째 또는 \\(i - 2\\) 번째 칸에서만 올 수 있으므로 \\(dp[i]\\)는 \\(dp[i - 1] + cost[i]\\) 또는 \\(dp[i - 2] + cost[i]\\)다. 비용을 최소화하려면 둘 중 작은 값을 선택해야 한다.</p> \\[ dp[i] = \\min(dp[i-1], dp[i-2]) + cost[i] \\] <p>여기서 최적 부분 구조의 의미를 알 수 있다. 원래 문제의 최적해는 부분 문제의 최적해로 구성된다.</p> <p>이 문제는 분명히 최적 부분 구조를 가진다. 두 부분 문제의 최적해 \\(dp[i-1]\\)과 \\(dp[i-2]\\) 중 더 좋은 것을 골라 원래 문제 \\(dp[i]\\)의 최적해를 구성하기 때문이다.</p> <p>그렇다면 앞 절의 계단 오르기 문제도 최적 부분 구조를 가질까? 방법 수를 구하는 것이 목표이므로 단순한 계수 문제처럼 보인다. 하지만 '최대 방법 수를 구하라'고 바꿔 보면 문제의 의미는 같은데 최적 부분 구조가 드러난다. \\(n\\) 번째 칸의 최대 방법 수는 \\(n-1\\) 번째와 \\(n-2\\) 번째 칸의 최대 방법 수를 더한 값이다. 이처럼 최적 부분 구조의 해석은 유연하며 문제마다 다른 의미를 가질 수 있다.</p> <p>상태 전이 식과 초기 상태 \\(dp[1] = cost[1]\\), \\(dp[2] = cost[2]\\)를 이용하면 다음 동적 계획법 코드를 얻는다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp(cost: list[int]) -&gt; int:\n    \"\"\"계단 오르기의 최소 비용: 동적 계획법\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    # 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    dp = [0] * (n + 1)\n    # 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1], dp[2] = cost[1], cost[2]\n    # 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in range(3, n + 1):\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    return dp[n]\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nint minCostClimbingStairsDP(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    vector&lt;int&gt; dp(n + 1);\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nint minCostClimbingStairsDP(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int[] dp = new int[n + 1];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nint MinCostClimbingStairsDP(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int[] dp = new int[n + 1];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.Min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nfunc minCostClimbingStairsDP(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    dp := make([]int, n+1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = min(dp[i-1], dp[i-2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nfunc minCostClimbingStairsDP(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    var dp = Array(repeating: 0, count: n + 1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in 3 ... n {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nfunction minCostClimbingStairsDP(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    const dp = new Array(n + 1);\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nfunction minCostClimbingStairsDP(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    const dp = new Array(n + 1);\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nint minCostClimbingStairsDP(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n  dp[1] = cost[1];\n  dp[2] = cost[2];\n  // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n  }\n  return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nfn min_cost_climbing_stairs_dp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    let mut dp = vec![-1; n + 1];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in 3..=n {\n        dp[i] = cmp::min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nint minCostClimbingStairsDP(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int *dp = calloc(n + 1, sizeof(int));\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = myMin(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    int res = dp[n];\n    // 메모리 해제\n    free(dp);\n    return res;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* 계단 오르기의 최소 비용: 동적 계획법 */\nfun minCostClimbingStairsDP(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    val dp = IntArray(n + 1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (i in 3..n) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>### 계단 오르기의 최소 비용: 동적 계획법 ###\ndef min_cost_climbing_stairs_dp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  # 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n  dp = Array.new(n + 1, 0)\n  # 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n  dp[1], dp[2] = cost[1], cost[2]\n  # 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n  (3...(n + 1)).each { |i| dp[i] = [dp[i - 1], dp[i - 2]].min + cost[i] }\n  dp[n]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-7)은 위 코드의 동적 계획법 과정이다.</p> <p></p> <p> 그림 14-7   계단 오르기 최소 비용의 동적 계획법 과정 </p> <p>이 문제도 공간을 최적화하여 1차원을 0차원으로 압축하고, 공간 복잡도를 \\(O(n)\\)에서 \\(O(1)\\)로 줄일 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp_comp(cost: list[int]) -&gt; int:\n    \"\"\"계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    a, b = cost[1], cost[2]\n    for i in range(3, n + 1):\n        a, b = b, min(a, b) + cost[i]\n    return b\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nint minCostClimbingStairsDPComp(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nint minCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nint MinCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.Min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nfunc minCostClimbingStairsDPComp(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    a, b := cost[1], cost[2]\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i := 3; i &lt;= n; i++ {\n        tmp := b\n        b = min(a, tmp) + cost[i]\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nfunc minCostClimbingStairsDPComp(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    var (a, b) = (cost[1], cost[2])\n    for i in 3 ... n {\n        (a, b) = (b, min(a, b) + cost[i])\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nfunction minCostClimbingStairsDPComp(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nfunction minCostClimbingStairsDPComp(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nint minCostClimbingStairsDPComp(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  int a = cost[1], b = cost[2];\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = min(a, tmp) + cost[i];\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nfn min_cost_climbing_stairs_dp_comp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 {\n        return cost[n];\n    };\n    let (mut a, mut b) = (cost[1], cost[2]);\n    for i in 3..=n {\n        let tmp = b;\n        b = cmp::min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    b\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nint minCostClimbingStairsDPComp(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = myMin(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.kt<pre><code>/* 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법 */\nfun minCostClimbingStairsDPComp(cost: IntArray): Int {\n    val n = cost.size - 1\n    if (n == 1 || n == 2) return cost[n]\n    var a = cost[1]\n    var b = cost[2]\n    for (i in 3..n) {\n        val tmp = b\n        b = min(a, tmp) + cost[i]\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.rb<pre><code>### 계단 오르기의 최소 비용: 동적 계획법 ###\ndef min_cost_climbing_stairs_dp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  # 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n  dp = Array.new(n + 1, 0)\n  # 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n  dp[1], dp[2] = cost[1], cost[2]\n  # 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n  (3...(n + 1)).each { |i| dp[i] = [dp[i - 1], dp[i - 2]].min + cost[i] }\n  dp[n]\nend\n\n# 계단 오르기의 최소 비용: 공간을 최적화한 동적 계획법\ndef min_cost_climbing_stairs_dp_comp(cost)\n  n = cost.length - 1\n  return cost[n] if n == 1 || n == 2\n  a, b = cost[1], cost[2]\n  (3...(n + 1)).each { |i| a, b = b, [a, b].min + cost[i] }\n  b\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.2   동적 계획법 문제의 특성"],"tags":[]},{"location":"chapter_dynamic_programming/dp_problem_features/#1422","level":2,"title":"14.2.2   무후효성","text":"<p>무후효성은 동적 계획법이 문제를 효과적으로 해결할 수 있게 하는 중요한 특성이다. 어떤 상태가 주어졌을 때 이후의 전개가 현재 상태에만 의존하고, 과거에 거쳐 온 모든 상태와는 무관하다는 뜻이다.</p> <p>계단 오르기에서 상태 \\(i\\)는 \\(1\\) 칸 또는 \\(2\\) 칸 오르는 선택에 따라 상태 \\(i+1\\) 또는 \\(i+2\\)로 이어진다. 두 선택을 할 때 상태 \\(i\\) 이전의 상태를 고려할 필요는 없다. 과거 상태는 \\(i\\) 이후의 전개에 영향을 주지 않기 때문이다.</p> <p>하지만 계단 오르기에 제약 하나를 추가하면 상황이 달라진다.</p> <p>제약이 있는 계단 오르기</p> <p>총 \\(n\\) 칸의 계단을 한 번에 \\(1\\) 칸 또는 \\(2\\) 칸 오를 수 있다. 단, 연속 두 번 \\(1\\) 칸씩 오를 수는 없다. 맨 위까지 오르는 방법은 몇 가지인가?</p> <p>아래 그림(14-8)에서는 \\(3\\) 번째 칸까지 오르는 유효한 방법이 \\(2\\) 가지만 남는다. \\(1\\) 칸씩 세 번 연속 오르는 방법은 제약 조건을 만족하지 않아 제외된다.</p> <p></p> <p> 그림 14-8   제약이 있을 때 3번째 칸까지 오르는 방법 수 </p> <p>이 문제에서 직전에 \\(1\\) 칸 올라왔다면 다음에는 반드시 \\(2\\) 칸 올라야 한다. 즉, 다음 선택은 현재 상태(현재 칸 번호)만으로 정해지지 않고 이전 상태(직전에 있던 칸 번호)에도 의존한다.</p> <p>따라서 기존 상태 정의로는 무후효성을 만족하지 않으며, 상태 전이 식 \\(dp[i] = dp[i-1] + dp[i-2]\\)도 더 이상 유효하지 않다. \\(dp[i-1]\\)은 이번에 \\(1\\) 칸 오르는 경우인데, 그 안에는 '직전에도 \\(1\\) 칸 올라온' 방법이 많이 들어 있다. 제약을 만족하려면 \\(dp[i-1]\\)을 그대로 \\(dp[i]\\)에 더할 수 없다.</p> <p>이를 위해 상태 정의를 확장해야 한다. 상태 \\([i, j]\\)는 현재 \\(i\\) 번째 칸에 있고 직전에 \\(j\\) 칸 올라왔음을 나타내며, \\(j \\in \\{1, 2\\}\\)이다. 이렇게 정의하면 직전에 \\(1\\) 칸 왔는지 \\(2\\) 칸 왔는지를 구분하여 현재 상태의 이전 상태를 판단할 수 있다.</p> <ul> <li>직전에 \\(1\\) 칸 왔다면 그 앞에서는 \\(2\\) 칸만 올 수 있다. 즉, \\(dp[i, 1]\\)은 \\(dp[i-1, 2]\\)에서만 전이된다.</li> <li>직전에 \\(2\\) 칸 왔다면 그 앞에서는 \\(1\\) 칸이나 \\(2\\) 칸 모두 가능하다. 즉, \\(dp[i, 2]\\)는 \\(dp[i-2, 1]\\) 또는 \\(dp[i-2, 2]\\)에서 전이된다.</li> </ul> <p>아래 그림(14-9)과 같이 이 정의에서 \\(dp[i, j]\\)는 상태 \\([i, j]\\)의 방법 수를 나타낸다. 상태 전이 식은 다음과 같다.</p> \\[ \\begin{cases} dp[i, 1] = dp[i-1, 2] \\\\ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] \\end{cases} \\] <p></p> <p> 그림 14-9   제약을 고려한 점화 관계 </p> <p>마지막으로 \\(dp[n, 1] + dp[n, 2]\\)를 반환하면 된다. 두 값의 합은 \\(n\\) 번째 칸까지 오르는 전체 방법 수다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_constraint_dp.py<pre><code>def climbing_stairs_constraint_dp(n: int) -&gt; int:\n    \"\"\"제약 조건이 있는 계단 오르기: 동적 계획법\"\"\"\n    if n == 1 or n == 2:\n        return 1\n    # 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    dp = [[0] * 3 for _ in range(n + 1)]\n    # 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1], dp[1][2] = 1, 0\n    dp[2][1], dp[2][2] = 0, 1\n    # 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in range(3, n + 1):\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    return dp[n][1] + dp[n][2]\n</code></pre> climbing_stairs_constraint_dp.cpp<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(3, 0));\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.java<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int[][] dp = new int[n + 1][3];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.cs<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nint ClimbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int[,] dp = new int[n + 1, 3];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1, 1] = 1;\n    dp[1, 2] = 0;\n    dp[2, 1] = 0;\n    dp[2, 2] = 1;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i, 1] = dp[i - 1, 2];\n        dp[i, 2] = dp[i - 2, 1] + dp[i - 2, 2];\n    }\n    return dp[n, 1] + dp[n, 2];\n}\n</code></pre> climbing_stairs_constraint_dp.go<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nfunc climbingStairsConstraintDP(n int) int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    dp := make([][3]int, n+1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i := 3; i &lt;= n; i++ {\n        dp[i][1] = dp[i-1][2]\n        dp[i][2] = dp[i-2][1] + dp[i-2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.swift<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nfunc climbingStairsConstraintDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    var dp = Array(repeating: Array(repeating: 0, count: 3), count: n + 1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in 3 ... n {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.js<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nfunction climbingStairsConstraintDP(n) {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    const dp = Array.from(new Array(n + 1), () =&gt; new Array(3));\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.ts<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nfunction climbingStairsConstraintDP(n: number): number {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(3));\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.dart<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nint climbingStairsConstraintDP(int n) {\n  if (n == 1 || n == 2) {\n    return 1;\n  }\n  // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(3, 0));\n  // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n  dp[1][1] = 1;\n  dp[1][2] = 0;\n  dp[2][1] = 0;\n  dp[2][2] = 1;\n  // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i][1] = dp[i - 1][2];\n    dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n  }\n  return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.rs<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nfn climbing_stairs_constraint_dp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return 1;\n    };\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    let mut dp = vec![vec![-1; 3]; n + 1];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in 3..=n {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.c<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(3, sizeof(int));\n    }\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    int res = dp[n][1] + dp[n][2];\n    // 메모리 해제\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> climbing_stairs_constraint_dp.kt<pre><code>/* 제약 조건이 있는 계단 오르기: 동적 계획법 */\nfun climbingStairsConstraintDP(n: Int): Int {\n    if (n == 1 || n == 2) {\n        return 1\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    val dp = Array(n + 1) { IntArray(3) }\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (i in 3..n) {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.rb<pre><code>### 제약 조건이 있는 계단 오르기: 동적 계획법 ###\ndef climbing_stairs_constraint_dp(n)\n  return 1 if n == 1 || n == 2\n\n  # 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n  dp = Array.new(n + 1) { Array.new(3, 0) }\n  # 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n  dp[1][1], dp[1][2] = 1, 0\n  dp[2][1], dp[2][2] = 0, 1\n  # 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n  for i in 3...(n + 1)\n    dp[i][1] = dp[i - 1][2]\n    dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n  end\n\n  dp[n][1] + dp[n][2]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>위 예시는 이전 상태 하나만 추가로 고려하면 되므로 상태 정의를 확장하여 무후효성을 다시 만족시킬 수 있었다. 하지만 일부 문제는 '과거의 영향'이 매우 강하다.</p> <p>계단 오르기와 장애물 생성</p> <p>총 \\(n\\) 칸의 계단을 한 번에 \\(1\\) 칸 또는 \\(2\\) 칸 오를 수 있다. \\(i\\) 번째 칸에 도착하면 시스템이 자동으로 \\(2i\\) 번째 칸에 장애물을 놓으며, 이후의 모든 이동에서는 \\(2i\\) 번째 칸으로 갈 수 없다. 예를 들어 처음 두 번에 \\(2\\), \\(3\\) 번째 칸에 도착했다면 이후에는 \\(4\\), \\(6\\) 번째 칸에 갈 수 없다. 맨 위까지 오르는 방법은 몇 가지인가?</p> <p>이 문제에서 다음 이동은 과거의 모든 상태에 의존한다. 매번 이동할 때마다 더 높은 칸에 장애물이 생겨 이후 이동에 영향을 주기 때문이다. 이런 문제는 동적 계획법으로 해결하기 어려운 경우가 많다.</p> <p>외판원 문제 같은 복잡한 조합 최적화 문제 중에는 무후효성을 만족하지 않는 경우가 많다. 이런 문제에는 보통 휴리스틱 탐색, 유전 알고리즘, 강화 학습 등의 다른 방법을 사용하여 제한된 시간 안에 활용 가능한 국소 최적해를 구한다.</p>","path":["제 14 장   동적 계획법","14.2   동적 계획법 문제의 특성"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/","level":1,"title":"14.3   동적 계획법 문제 풀이 방법","text":"<p>앞의 두 절에서는 동적 계획법 문제의 주요 특성을 살펴보았다. 이제 더 실용적인 두 질문을 알아보자.</p> <ol> <li>어떤 문제가 동적 계획법 문제인지 어떻게 판단할까?</li> <li>동적 계획법 문제는 어디서 시작하며, 전체 풀이 단계는 어떻게 될까?</li> </ol>","path":["제 14 장   동적 계획법","14.3   동적 계획법 문제 풀이 방법"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1431","level":2,"title":"14.3.1   문제 판단","text":"<p>대체로 중복 부분 문제와 최적 부분 구조가 있고 무후효성을 만족하면 동적 계획법에 적합하다. 하지만 문제 설명에서 이러한 특성을 바로 끌어내기는 어렵다. 따라서 보통 조건을 완화하여 먼저 백트래킹(모든 경우 열거)으로 풀기에 적합한지 살펴본다.</p> <p>백트래킹에 적합한 문제는 보통 '의사 결정 트리 모형'을 따른다. 각 노드가 결정 하나를, 각 경로가 결정의 순서를 나타내는 트리 구조로 표현할 수 있는 문제다.</p> <p>즉, 명확한 결정 개념이 있고 연속된 결정을 통해 해가 만들어진다면 의사 결정 트리 모형을 만족하며, 보통 백트래킹으로 해결할 수 있다.</p> <p>여기에 더해 동적 계획법 문제로 판단할 때 '가산점'이 되는 요소도 있다.</p> <ul> <li>최댓값, 최솟값, 가장 많은 수, 가장 적은 수 등의 최적화 표현이 포함되어 있다.</li> <li>상태를 리스트, 다차원 행렬, 트리로 표현할 수 있고, 한 상태와 주변 상태들 사이에 점화 관계가 있다.</li> </ul> <p>반대로 '감점' 요소도 있다.</p> <ul> <li>최적해 하나가 아니라 가능한 모든 해결 방법을 찾는 것이 목표다.</li> <li>문제 설명에 순열이나 조합의 특성이 뚜렷하며, 구체적인 여러 방법을 반환해야 한다.</li> </ul> <p>의사 결정 트리 모형을 만족하면서 '가산점' 요소가 뚜렷하다면, 우선 동적 계획법 문제라고 가정하고 풀이 과정에서 확인해 볼 수 있다.</p>","path":["제 14 장   동적 계획법","14.3   동적 계획법 문제 풀이 방법"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1432","level":2,"title":"14.3.2   문제 해결 단계","text":"<p>동적 계획법의 풀이 흐름은 문제의 성격과 난이도에 따라 다르지만, 보통 결정 설명, 상태 정의, \\(dp\\) 테이블 구성, 상태 전이 식 도출, 경계 조건 결정 등의 단계를 따른다.</p> <p>단계를 더 구체적으로 보여 주기 위해 대표적인 '최소 경로 합' 문제를 예로 든다.</p> <p>문제</p> <p>\\(n \\times m\\)의 2차원 격자 <code>grid</code>가 주어진다. 각 칸에는 그 칸의 비용인 음이 아닌 정수가 들어 있다. 로봇은 왼쪽 위에서 출발하여 매번 아래 또는 오른쪽으로 한 칸씩만 이동하고, 오른쪽 아래에 도착한다. 왼쪽 위에서 오른쪽 아래까지의 최소 경로 합을 반환하라.</p> <p>아래 그림(14-10)의 격자에서는 최소 경로 합이 \\(13\\)이다.</p> <p></p> <p> 그림 14-10   최소 경로 합 예시 데이터 </p> <p>1단계: 매 차례의 결정을 생각하고 상태를 정의하여 \\(dp\\) 테이블을 얻는다</p> <p>이 문제에서 매 차례의 결정은 현재 칸에서 아래 또는 오른쪽으로 한 칸 이동하는 것이다. 현재 칸의 행과 열 인덱스가 \\([i, j]\\)라면 이동 후에는 \\([i+1, j]\\) 또는 \\([i, j+1]\\)이 된다. 따라서 상태는 행 인덱스와 열 인덱스라는 두 변수를 포함해야 하며 \\([i, j]\\)로 표시한다.</p> <p>상태 \\([i, j]\\)의 부분 문제는 출발점 \\([0, 0]\\)에서 \\([i, j]\\)까지의 최소 경로 합이고, 해를 \\(dp[i, j]\\)라고 한다.</p> <p>이로써 아래 그림(14-11)처럼 입력 격자 \\(grid\\)와 크기가 같은 2차원 \\(dp\\) 행렬을 얻는다.</p> <p></p> <p> 그림 14-11   상태 정의와 dp 테이블 </p> <p>참고</p> <p>동적 계획법과 백트래킹 과정은 결정의 순서로 표현할 수 있으며, 상태는 모든 결정 변수로 이루어진다. 상태에는 풀이의 진행 정도를 나타내는 모든 변수가 포함되어야 한다. 즉, 다음 상태를 도출하기에 충분한 정보가 있어야 한다.</p> <p>각 상태는 부분 문제 하나에 대응하며, 모든 부분 문제의 해를 저장하기 위해 \\(dp\\) 테이블을 정의한다. 상태의 독립 변수 각각이 \\(dp\\) 테이블의 한 차원이 된다. 본질적으로 \\(dp\\) 테이블은 상태와 부분 문제의 해 사이의 매핑이다.</p> <p>2단계: 최적 부분 구조를 찾아 상태 전이 식을 도출한다</p> <p>상태 \\([i, j]\\)는 위쪽 칸 \\([i-1, j]\\) 또는 왼쪽 칸 \\([i, j-1]\\)에서만 올 수 있다. 따라서 최적 부분 구조는 다음과 같다. \\([i, j]\\)까지의 최소 경로 합은 \\([i, j-1]\\)까지의 최소 경로 합과 \\([i-1, j]\\)까지의 최소 경로 합 중 작은 쪽에 의해 결정된다.</p> <p>위 분석으로 아래 그림(14-12)의 상태 전이 식을 얻는다.</p> \\[ dp[i, j] = \\min(dp[i-1, j], dp[i, j-1]) + grid[i, j] \\] <p></p> <p> 그림 14-12   최적 부분 구조와 상태 전이 식 </p> <p>참고</p> <p>정의한 \\(dp\\) 테이블을 바탕으로 원래 문제와 부분 문제의 관계를 생각하고, 부분 문제의 최적해로 원래 문제의 최적해를 만드는 방법, 즉 최적 부분 구조를 찾는다.</p> <p>최적 부분 구조를 찾으면 이를 이용해 상태 전이 식을 세울 수 있다.</p> <p>3단계: 경계 조건과 상태 전이 순서를 정한다</p> <p>이 문제에서 첫 행의 상태는 왼쪽 상태에서만, 첫 열의 상태는 위쪽 상태에서만 올 수 있다. 따라서 첫 행 \\(i = 0\\)과 첫 열 \\(j = 0\\)이 경계 조건이다.</p> <p>아래 그림(14-13)과 같이 각 칸은 왼쪽 칸과 위쪽 칸에서 전이되므로, 바깥 반복문은 행을, 안쪽 반복문은 열을 순회한다.</p> <p></p> <p> 그림 14-13   경계 조건과 상태 전이 순서 </p> <p>참고</p> <p>경계 조건은 동적 계획법에서 \\(dp\\) 테이블을 초기화하는 데, 탐색에서는 가지치기에 사용한다.</p> <p>상태 전이 순서의 핵심은 현재 문제의 해를 계산하기 전에 그 해가 의존하는 모든 작은 부분 문제의 해가 올바르게 계산되어 있도록 하는 것이다.</p> <p>이제 동적 계획법 코드를 바로 작성할 수 있다. 하지만 부분 문제로 나누는 사고는 위에서 아래로 진행하므로, '완전 탐색 \\(\\rightarrow\\) 메모이제이션을 이용한 탐색 \\(\\rightarrow\\) 동적 계획법' 순서로 구현하는 편이 사고 흐름에 더 자연스럽다.</p>","path":["제 14 장   동적 계획법","14.3   동적 계획법 문제 풀이 방법"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1-1","level":3,"title":"1.   방법 1: 완전 탐색","text":"<p>상태 \\([i, j]\\)에서 탐색을 시작하여 더 작은 상태 \\([i-1, j]\\)와 \\([i, j-1]\\)로 계속 나눈다. 재귀 함수에는 다음 요소가 포함된다.</p> <ul> <li>재귀 매개변수: 상태 \\([i, j]\\).</li> <li>반환값: \\([0, 0]\\)에서 \\([i, j]\\)까지의 최소 경로 합 \\(dp[i, j]\\).</li> <li>종료 조건: \\(i = 0\\)이고 \\(j = 0\\)이면 비용 \\(grid[0, 0]\\)을 반환한다.</li> <li>가지치기: \\(i &lt; 0\\) 또는 \\(j &lt; 0\\)이면 인덱스가 범위를 벗어나므로 불가능함을 나타내는 비용 \\(+\\infty\\)를 반환한다.</li> </ul> <p>구현 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dfs(grid: list[list[int]], i: int, j: int) -&gt; int:\n    \"\"\"최소 경로 합: 완전 탐색\"\"\"\n    # 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    up = min_path_sum_dfs(grid, i - 1, j)\n    left = min_path_sum_dfs(grid, i, j - 1)\n    # 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return min(left, up) + grid[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* 최소 경로 합: 완전 탐색 */\nint minPathSumDFS(vector&lt;vector&lt;int&gt;&gt; &amp;grid, int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.java<pre><code>/* 최소 경로 합: 완전 탐색 */\nint minPathSumDFS(int[][] grid, int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* 최소 경로 합: 완전 탐색 */\nint MinPathSumDFS(int[][] grid, int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    int up = MinPathSumDFS(grid, i - 1, j);\n    int left = MinPathSumDFS(grid, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return Math.Min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* 최소 경로 합: 완전 탐색 */\nfunc minPathSumDFS(grid [][]int, i, j int) int {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    up := minPathSumDFS(grid, i-1, j)\n    left := minPathSumDFS(grid, i, j-1)\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return int(math.Min(float64(left), float64(up))) + grid[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* 최소 경로 합: 완전 탐색 */\nfunc minPathSumDFS(grid: [[Int]], i: Int, j: Int) -&gt; Int {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    let up = minPathSumDFS(grid: grid, i: i - 1, j: j)\n    let left = minPathSumDFS(grid: grid, i: i, j: j - 1)\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* 최소 경로 합: 완전 탐색 */\nfunction minPathSumDFS(grid, i, j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* 최소 경로 합: 완전 탐색 */\nfunction minPathSumDFS(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i === 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* 최소 경로 합: 완전 탐색 */\nint minPathSumDFS(List&lt;List&lt;int&gt;&gt; grid, int i, int j) {\n  // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n  if (i &lt; 0 || j &lt; 0) {\n    // Dart의 int 타입은 범위가 고정된 정수이며, 무한대를 나타내는 값이 없다\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n  int up = minPathSumDFS(grid, i - 1, j);\n  int left = minPathSumDFS(grid, i, j - 1);\n  // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n  return min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* 최소 경로 합: 완전 탐색 */\nfn min_path_sum_dfs(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    let up = min_path_sum_dfs(grid, i - 1, j);\n    let left = min_path_sum_dfs(grid, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    std::cmp::min(left, up) + grid[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* 최소 경로 합: 완전 탐색 */\nint minPathSumDFS(int grid[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.kt<pre><code>/* 최소 경로 합: 완전 탐색 */\nfun minPathSumDFS(grid: Array&lt;IntArray&gt;, i: Int, j: Int): Int {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n    val up = minPathSumDFS(grid, i - 1, j)\n    val left = minPathSumDFS(grid, i, j - 1)\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n    return min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.rb<pre><code>### 최소 경로 합: 완전 탐색 ###\ndef min_path_sum_dfs(grid, i, j)\n  # 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n  return grid[i][j] if i == 0 &amp;&amp; j == 0\n  # 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n  return Float::INFINITY if i &lt; 0 || j &lt; 0\n  # 왼쪽 위 모서리에서 (i-1, j)와 (i, j-1)까지의 최소 경로 비용을 계산한다\n  up = min_path_sum_dfs(grid, i - 1, j)\n  left = min_path_sum_dfs(grid, i, j - 1)\n  # 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 반환한다\n  [left, up].min + grid[i][j]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-14)은 \\(dp[2, 1]\\)을 루트로 하는 재귀 트리다. 중복 부분 문제가 포함되어 있으며, 그 수는 격자 <code>grid</code>의 크기가 커질수록 급격히 늘어난다.</p> <p>중복 부분 문제가 생기는 근본적인 이유는 왼쪽 위에서 어떤 칸에 도달하는 경로가 여러 개 있기 때문이다.</p> <p></p> <p> 그림 14-14   완전 탐색의 재귀 트리 </p> <p>각 상태에는 아래와 오른쪽이라는 두 선택이 있고, 왼쪽 위에서 오른쪽 아래까지는 총 \\(m + n - 2\\) 번 이동해야 한다. 따라서 최악의 시간 복잡도는 \\(O(2^{m + n})\\)이며, \\(n\\)과 \\(m\\)은 각각 격자의 행 수와 열 수다. 이 계산은 경계 근처의 상황을 고려하지 않았다. 격자 경계에서는 선택이 하나뿐이므로 실제 경로 수는 이보다 적다.</p>","path":["제 14 장   동적 계획법","14.3   동적 계획법 문제 풀이 방법"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#2-2","level":3,"title":"2.   방법 2: 메모이제이션을 이용한 탐색","text":"<p>격자 <code>grid</code>와 같은 크기의 메모이제이션 리스트 <code>mem</code>을 도입하여 각 부분 문제의 해를 기록하고 중복 부분 문제를 가지치기한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dfs_mem(\n    grid: list[list[int]], mem: list[list[int]], i: int, j: int\n) -&gt; int:\n    \"\"\"최소 경로 합: 메모이제이션을 이용한 탐색\"\"\"\n    # 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # 기록이 이미 있으면 바로 반환한다\n    if mem[i][j] != -1:\n        return mem[i][j]\n    # 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n    left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n    # 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nint minPathSumDFSMem(vector&lt;vector&lt;int&gt;&gt; &amp;grid, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.java<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nint minPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nint MinPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    int up = MinPathSumDFSMem(grid, mem, i - 1, j);\n    int left = MinPathSumDFSMem(grid, mem, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = Math.Min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nfunc minPathSumDFSMem(grid, mem [][]int, i, j int) int {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    up := minPathSumDFSMem(grid, mem, i-1, j)\n    left := minPathSumDFSMem(grid, mem, i, j-1)\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = int(math.Min(float64(left), float64(up))) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nfunc minPathSumDFSMem(grid: [[Int]], mem: inout [[Int]], i: Int, j: Int) -&gt; Int {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    let up = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i - 1, j: j)\n    let left = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i, j: j - 1)\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nfunction minPathSumDFSMem(grid, mem, i, j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][j] !== -1) {\n        return mem[i][j];\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nfunction minPathSumDFSMem(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nint minPathSumDFSMem(List&lt;List&lt;int&gt;&gt; grid, List&lt;List&lt;int&gt;&gt; mem, int i, int j) {\n  // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n  if (i &lt; 0 || j &lt; 0) {\n    // Dart의 int 타입은 범위가 고정된 정수이며, 무한대를 나타내는 값이 없다\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // 기록이 이미 있으면 바로 반환한다\n  if (mem[i][j] != -1) {\n    return mem[i][j];\n  }\n  // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n  int up = minPathSumDFSMem(grid, mem, i - 1, j);\n  int left = minPathSumDFSMem(grid, mem, i, j - 1);\n  // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n  mem[i][j] = min(left, up) + grid[i][j];\n  return mem[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nfn min_path_sum_dfs_mem(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if mem[i as usize][j as usize] != -1 {\n        return mem[i as usize][j as usize];\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    let up = min_path_sum_dfs_mem(grid, mem, i - 1, j);\n    let left = min_path_sum_dfs_mem(grid, mem, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];\n    mem[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nint minPathSumDFSMem(int grid[MAX_SIZE][MAX_SIZE], int mem[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.kt<pre><code>/* 최소 경로 합: 메모이제이션을 이용한 탐색 */\nfun minPathSumDFSMem(\n    grid: Array&lt;IntArray&gt;,\n    mem: Array&lt;IntArray&gt;,\n    i: Int,\n    j: Int\n): Int {\n    // 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0]\n    }\n    // 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n    if (i &lt; 0 || j &lt; 0) {\n        return Int.MAX_VALUE\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][j] != -1) {\n        return mem[i][j]\n    }\n    // 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n    val up = minPathSumDFSMem(grid, mem, i - 1, j)\n    val left = minPathSumDFSMem(grid, mem, i, j - 1)\n    // 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.rb<pre><code>### 최소 경로 합: 메모이제이션을 이용한 탐색 ###\ndef min_path_sum_dfs_mem(grid, mem, i, j)\n  # 왼쪽 위 모서리의 칸이면 탐색을 종료한다\n  return grid[0][0] if i == 0 &amp;&amp; j == 0\n  # 행 또는 열 인덱스가 범위를 벗어나면 비용 +∞를 반환한다\n  return Float::INFINITY if i &lt; 0 || j &lt; 0\n  # 기록이 이미 있으면 바로 반환한다\n  return mem[i][j] if mem[i][j] != -1\n  # 왼쪽 칸과 위쪽 칸까지의 최소 경로 비용\n  up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n  left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n  # 왼쪽 위 모서리에서 (i, j)까지의 최소 경로 비용을 기록하고 반환한다\n  mem[i][j] = [left, up].min + grid[i][j]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-15)과 같이 메모이제이션을 도입하면 모든 부분 문제의 해를 한 번만 계산한다. 따라서 시간 복잡도는 전체 상태 수, 즉 격자 크기인 \\(O(nm)\\)에 의해 결정된다.</p> <p></p> <p> 그림 14-15   메모이제이션을 이용한 탐색의 재귀 트리 </p>","path":["제 14 장   동적 계획법","14.3   동적 계획법 문제 풀이 방법"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#3-3","level":3,"title":"3.   방법 3: 동적 계획법","text":"<p>반복을 이용한 동적 계획법 구현은 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dp(grid: list[list[int]]) -&gt; int:\n    \"\"\"최소 경로 합: 동적 계획법\"\"\"\n    n, m = len(grid), len(grid[0])\n    # dp 테이블을 초기화한다\n    dp = [[0] * m for _ in range(n)]\n    dp[0][0] = grid[0][0]\n    # 상태 전이: 첫 번째 행\n    for j in range(1, m):\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    # 상태 전이: 첫 번째 열\n    for i in range(1, n):\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    # 상태 전이: 나머지 행과 열\n    for i in range(1, n):\n        for j in range(1, m):\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n    return dp[n - 1][m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* 최소 경로 합: 동적 계획법 */\nint minPathSumDP(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // dp 테이블을 초기화한다\n    vector&lt;vector&lt;int&gt;&gt; dp(n, vector&lt;int&gt;(m));\n    dp[0][0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // 상태 전이: 첫 번째 열\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* 최소 경로 합: 동적 계획법 */\nint minPathSumDP(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // dp 테이블을 초기화한다\n    int[][] dp = new int[n][m];\n    dp[0][0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // 상태 전이: 첫 번째 열\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* 최소 경로 합: 동적 계획법 */\nint MinPathSumDP(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // dp 테이블을 초기화한다\n    int[,] dp = new int[n, m];\n    dp[0, 0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt; m; j++) {\n        dp[0, j] = dp[0, j - 1] + grid[0][j];\n    }\n    // 상태 전이: 첫 번째 열\n    for (int i = 1; i &lt; n; i++) {\n        dp[i, 0] = dp[i - 1, 0] + grid[i][0];\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i, j] = Math.Min(dp[i, j - 1], dp[i - 1, j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1, m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* 최소 경로 합: 동적 계획법 */\nfunc minPathSumDP(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // dp 테이블을 초기화한다\n    dp := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        dp[i] = make([]int, m)\n    }\n    dp[0][0] = grid[0][0]\n    // 상태 전이: 첫 번째 행\n    for j := 1; j &lt; m; j++ {\n        dp[0][j] = dp[0][j-1] + grid[0][j]\n    }\n    // 상태 전이: 첫 번째 열\n    for i := 1; i &lt; n; i++ {\n        dp[i][0] = dp[i-1][0] + grid[i][0]\n    }\n    // 상태 전이: 나머지 행과 열\n    for i := 1; i &lt; n; i++ {\n        for j := 1; j &lt; m; j++ {\n            dp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(dp[i-1][j]))) + grid[i][j]\n        }\n    }\n    return dp[n-1][m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* 최소 경로 합: 동적 계획법 */\nfunc minPathSumDP(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: Array(repeating: 0, count: m), count: n)\n    dp[0][0] = grid[0][0]\n    // 상태 전이: 첫 번째 행\n    for j in 1 ..&lt; m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // 상태 전이: 첫 번째 열\n    for i in 1 ..&lt; n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // 상태 전이: 나머지 행과 열\n    for i in 1 ..&lt; n {\n        for j in 1 ..&lt; m {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* 최소 경로 합: 동적 계획법 */\nfunction minPathSumDP(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // 상태 전이: 첫 번째 열\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // 상태 전이: 나머지 행과 열\n    for (let i = 1; i &lt; n; i++) {\n        for (let j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* 최소 경로 합: 동적 계획법 */\nfunction minPathSumDP(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // 상태 전이: 첫 번째 열\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // 상태 전이: 나머지 행과 열\n    for (let i = 1; i &lt; n; i++) {\n        for (let j: number = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* 최소 경로 합: 동적 계획법 */\nint minPathSumDP(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // dp 테이블을 초기화한다\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n, (i) =&gt; List.filled(m, 0));\n  dp[0][0] = grid[0][0];\n  // 상태 전이: 첫 번째 행\n  for (int j = 1; j &lt; m; j++) {\n    dp[0][j] = dp[0][j - 1] + grid[0][j];\n  }\n  // 상태 전이: 첫 번째 열\n  for (int i = 1; i &lt; n; i++) {\n    dp[i][0] = dp[i - 1][0] + grid[i][0];\n  }\n  // 상태 전이: 나머지 행과 열\n  for (int i = 1; i &lt; n; i++) {\n    for (int j = 1; j &lt; m; j++) {\n      dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n    }\n  }\n  return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* 최소 경로 합: 동적 계획법 */\nfn min_path_sum_dp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // dp 테이블을 초기화한다\n    let mut dp = vec![vec![0; m]; n];\n    dp[0][0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for j in 1..m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // 상태 전이: 첫 번째 열\n    for i in 1..n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // 상태 전이: 나머지 행과 열\n    for i in 1..n {\n        for j in 1..m {\n            dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* 최소 경로 합: 동적 계획법 */\nint minPathSumDP(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // dp 테이블을 초기화한다\n    int **dp = malloc(n * sizeof(int *));\n    for (int i = 0; i &lt; n; i++) {\n        dp[i] = calloc(m, sizeof(int));\n    }\n    dp[0][0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // 상태 전이: 첫 번째 열\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = myMin(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    int res = dp[n - 1][m - 1];\n    // 메모리 해제\n    for (int i = 0; i &lt; n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* 최소 경로 합: 동적 계획법 */\nfun minPathSumDP(grid: Array&lt;IntArray&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // dp 테이블을 초기화한다\n    val dp = Array(n) { IntArray(m) }\n    dp[0][0] = grid[0][0]\n    // 상태 전이: 첫 번째 행\n    for (j in 1..&lt;m) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // 상태 전이: 첫 번째 열\n    for (i in 1..&lt;n) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // 상태 전이: 나머지 행과 열\n    for (i in 1..&lt;n) {\n        for (j in 1..&lt;m) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>### 최소 경로 합: 동적 계획법 ###\ndef min_path_sum_dp(grid)\n  n, m = grid.length, grid.first.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(n) { Array.new(m, 0) }\n  dp[0][0] = grid[0][0]\n  # 상태 전이: 첫 번째 행\n  (1...m).each { |j| dp[0][j] = dp[0][j - 1] + grid[0][j] }\n  # 상태 전이: 첫 번째 열\n  (1...n).each { |i| dp[i][0] = dp[i - 1][0] + grid[i][0] }\n  # 상태 전이: 나머지 행과 열\n  for i in 1...n\n    for j in 1...m\n      dp[i][j] = [dp[i][j - 1], dp[i - 1][j]].min + grid[i][j]\n    end\n  end\n  dp[n -1][m -1]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-16)의 최소 경로 합 상태 전이 과정에서는 전체 격자를 순회하므로 시간 복잡도는 \\(O(nm)\\)이다.</p> <p>배열 <code>dp</code>의 크기는 \\(n \\times m\\)이므로 공간 복잡도는 \\(O(nm)\\)이다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 14-16   최소 경로 합의 동적 계획법 과정 </p>","path":["제 14 장   동적 계획법","14.3   동적 계획법 문제 풀이 방법"],"tags":[]},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#4","level":3,"title":"4.   공간 최적화","text":"<p>각 칸은 왼쪽 칸과 위쪽 칸에만 의존하므로 한 행짜리 배열만으로 \\(dp\\) 테이블을 구현할 수 있다.</p> <p>배열 <code>dp</code>가 한 행의 상태만 나타내므로 첫 열의 상태를 미리 초기화할 수는 없다. 대신 각 행을 순회할 때 갱신해야 한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby min_path_sum.py<pre><code>def min_path_sum_dp_comp(grid: list[list[int]]) -&gt; int:\n    \"\"\"최소 경로 합: 공간을 최적화한 동적 계획법\"\"\"\n    n, m = len(grid), len(grid[0])\n    # dp 테이블을 초기화한다\n    dp = [0] * m\n    # 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0]\n    for j in range(1, m):\n        dp[j] = dp[j - 1] + grid[0][j]\n    # 상태 전이: 나머지 행\n    for i in range(1, n):\n        # 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0]\n        # 상태 전이: 나머지 열\n        for j in range(1, m):\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n    return dp[m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nint minPathSumDPComp(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // dp 테이블을 초기화한다\n    vector&lt;int&gt; dp(m);\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt; n; i++) {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0];\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nint minPathSumDPComp(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[m];\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt; n; i++) {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0];\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nint MinPathSumDPComp(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[m];\n    dp[0] = grid[0][0];\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt; n; i++) {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0];\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.Min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nfunc minPathSumDPComp(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // dp 테이블을 초기화한다\n    dp := make([]int, m)\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0]\n    for j := 1; j &lt; m; j++ {\n        dp[j] = dp[j-1] + grid[0][j]\n    }\n    // 상태 전이: 나머지 행과 열\n    for i := 1; i &lt; n; i++ {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0]\n        // 상태 전이: 나머지 열\n        for j := 1; j &lt; m; j++ {\n            dp[j] = int(math.Min(float64(dp[j-1]), float64(dp[j]))) + grid[i][j]\n        }\n    }\n    return dp[m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nfunc minPathSumDPComp(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: 0, count: m)\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0]\n    for j in 1 ..&lt; m {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // 상태 전이: 나머지 행\n    for i in 1 ..&lt; n {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0]\n        // 상태 전이: 나머지 열\n        for j in 1 ..&lt; m {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nfunction minPathSumDPComp(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // dp 테이블을 초기화한다\n    const dp = new Array(m);\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // 상태 전이: 나머지 행\n    for (let i = 1; i &lt; n; i++) {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0];\n        // 상태 전이: 나머지 열\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nfunction minPathSumDPComp(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // dp 테이블을 초기화한다\n    const dp = new Array(m);\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // 상태 전이: 나머지 행\n    for (let i = 1; i &lt; n; i++) {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0];\n        // 상태 전이: 나머지 열\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nint minPathSumDPComp(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // dp 테이블을 초기화한다\n  List&lt;int&gt; dp = List.filled(m, 0);\n  dp[0] = grid[0][0];\n  for (int j = 1; j &lt; m; j++) {\n    dp[j] = dp[j - 1] + grid[0][j];\n  }\n  // 상태 전이: 나머지 행\n  for (int i = 1; i &lt; n; i++) {\n    // 상태 전이: 첫 번째 열\n    dp[0] = dp[0] + grid[i][0];\n    // 상태 전이: 나머지 열\n    for (int j = 1; j &lt; m; j++) {\n      dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n    }\n  }\n  return dp[m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nfn min_path_sum_dp_comp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // dp 테이블을 초기화한다\n    let mut dp = vec![0; m];\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0];\n    for j in 1..m {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // 상태 전이: 나머지 행\n    for i in 1..n {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0];\n        // 상태 전이: 나머지 열\n        for j in 1..m {\n            dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    dp[m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nint minPathSumDPComp(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // dp 테이블을 초기화한다\n    int *dp = calloc(m, sizeof(int));\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt; n; i++) {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0];\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = myMin(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    int res = dp[m - 1];\n    // 메모리 해제\n    free(dp);\n    return res;\n}\n</code></pre> min_path_sum.kt<pre><code>/* 최소 경로 합: 공간을 최적화한 동적 계획법 */\nfun minPathSumDPComp(grid: Array&lt;IntArray&gt;): Int {\n    val n = grid.size\n    val m = grid[0].size\n    // dp 테이블을 초기화한다\n    val dp = IntArray(m)\n    // 상태 전이: 첫 번째 행\n    dp[0] = grid[0][0]\n    for (j in 1..&lt;m) {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // 상태 전이: 나머지 행\n    for (i in 1..&lt;n) {\n        // 상태 전이: 첫 번째 열\n        dp[0] = dp[0] + grid[i][0]\n        // 상태 전이: 나머지 열\n        for (j in 1..&lt;m) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.rb<pre><code>### 최소 경로 합: 공간을 최적화한 동적 계획법 ###\ndef min_path_sum_dp_comp(grid)\n  n, m = grid.length, grid.first.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(m, 0)\n  # 상태 전이: 첫 번째 행\n  dp[0] = grid[0][0]\n  (1...m).each { |j| dp[j] = dp[j - 1] + grid[0][j] }\n  # 상태 전이: 나머지 행\n  for i in 1...n\n    # 상태 전이: 첫 번째 열\n    dp[0] = dp[0] + grid[i][0]\n    # 상태 전이: 나머지 열\n    (1...m).each { |j| dp[j] = [dp[j - 1], dp[j]].min + grid[i][j] }\n  end\n  dp[m - 1]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.3   동적 계획법 문제 풀이 방법"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/","level":1,"title":"14.6   편집 거리 문제","text":"<p>편집 거리 또는 Levenshtein 거리는 두 문자열을 서로 바꾸는 데 필요한 최소 수정 횟수다. 정보 검색과 자연어 처리에서 두 시퀀스의 유사도를 측정할 때 자주 사용한다.</p> <p>문제</p> <p>두 문자열 \\(s\\)와 \\(t\\)를 입력받아 \\(s\\)를 \\(t\\)로 바꾸는 데 필요한 최소 편집 횟수를 반환하라.</p> <p>문자열에 문자를 하나 삽입하거나, 하나 삭제하거나, 한 문자를 임의의 다른 문자로 바꾸는 세 연산을 할 수 있다.</p> <p>아래 그림(14-27)과 같이 <code>kitten</code>을 <code>sitting</code>으로 바꾸려면 교체 2회와 추가 1회로 총 3회 편집해야 한다. <code>hello</code>를 <code>algo</code>로 바꾸려면 교체 2회와 삭제 1회로 총 3회가 필요하다.</p> <p></p> <p> 그림 14-27   편집 거리 예시 데이터 </p> <p>편집 거리 문제는 의사 결정 트리 모형으로 자연스럽게 설명할 수 있다. 문자열은 트리 노드에, 한 번의 결정(편집 연산)은 간선 하나에 대응한다.</p> <p>아래 그림(14-28)과 같이 연산을 제한하지 않으면 각 노드에서 다양한 연산에 대응하는 많은 간선이 뻗어 나온다. 즉, <code>hello</code>에서 <code>algo</code>로 가는 경로는 매우 많다.</p> <p>의사 결정 트리의 관점에서 이 문제의 목표는 노드 <code>hello</code>와 노드 <code>algo</code> 사이의 최단 경로를 구하는 것이다.</p> <p></p> <p> 그림 14-28   의사 결정 트리로 표현한 편집 거리 문제 </p>","path":["제 14 장   동적 계획법","14.6   편집 거리 문제"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/#1","level":3,"title":"1.   동적 계획법 접근","text":"<p>1단계: 매 차례의 결정을 생각하고 상태를 정의하여 \\(dp\\) 테이블을 얻는다</p> <p>매 차례의 결정은 문자열 \\(s\\)에 편집 연산을 한 번 적용하는 것이다.</p> <p>부분 문제를 만들려면 편집하는 동안 문제의 크기가 점차 줄어들어야 한다. 문자열 \\(s\\)와 \\(t\\)의 길이를 각각 \\(n\\)과 \\(m\\)이라고 하고, 두 문자열의 마지막 문자 \\(s[n-1]\\)과 \\(t[m-1]\\)을 먼저 살펴보자.</p> <ul> <li>\\(s[n-1]\\)과 \\(t[m-1]\\)이 같으면 두 문자를 건너뛰고 \\(s[n-2]\\)와 \\(t[m-2]\\)를 바로 살펴볼 수 있다.</li> <li>\\(s[n-1]\\)과 \\(t[m-1]\\)이 다르면 \\(s\\)에 편집(삽입, 삭제, 교체)을 한 번 적용하여 끝 문자가 같아지게 하고, 이를 건너뛰어 더 작은 문제를 살펴본다.</li> </ul> <p>즉, 문자열 \\(s\\)에서 결정(편집 연산)을 할 때마다 \\(s\\)와 \\(t\\)에서 아직 맞추어야 하는 문자들이 달라진다. 따라서 현재 살펴보는 \\(s\\)의 \\(i\\) 번째 문자와 \\(t\\)의 \\(j\\) 번째 문자를 상태로 정의하고 \\([i, j]\\)라고 한다.</p> <p>상태 \\([i, j]\\)의 부분 문제는 \\(s\\)의 앞 \\(i\\) 개 문자를 \\(t\\)의 앞 \\(j\\) 개 문자로 바꾸는 최소 편집 횟수다.</p> <p>이로써 \\((n+1) \\times (m+1)\\) 크기의 2차원 \\(dp\\) 테이블을 얻는다.</p> <p>2단계: 최적 부분 구조를 찾아 상태 전이 식을 도출한다</p> <p>부분 문제 \\(dp[i, j]\\)에 해당하는 두 문자열의 끝 문자는 \\(s[i-1]\\)과 \\(t[j-1]\\)이다. 편집 연산에 따라 아래 그림(14-29)의 세 경우로 나뉜다.</p> <ol> <li>\\(s[i-1]\\) 뒤에 \\(t[j-1]\\)을 추가하면 부분 문제 \\(dp[i, j-1]\\)이 남는다.</li> <li>\\(s[i-1]\\)을 삭제하면 부분 문제 \\(dp[i-1, j]\\)가 남는다.</li> <li>\\(s[i-1]\\)을 \\(t[j-1]\\)로 바꾸면 부분 문제 \\(dp[i-1, j-1]\\)이 남는다.</li> </ol> <p></p> <p> 그림 14-29   편집 거리의 상태 전이 </p> <p>위 분석으로 최적 부분 구조를 얻는다. \\(dp[i, j]\\)의 최소 편집 횟수는 \\(dp[i, j-1]\\), \\(dp[i-1, j]\\), \\(dp[i-1, j-1]\\) 중 가장 작은 편집 횟수에 이번 편집 횟수 \\(1\\)을 더한 값이다. 상태 전이 식은 다음과 같다.</p> \\[ dp[i, j] = \\min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1 \\] <p>\\(s[i-1]\\)과 \\(t[j-1]\\)이 같으면 현재 문자를 편집할 필요가 없으며, 이때 상태 전이 식은 다음과 같다.</p> \\[ dp[i, j] = dp[i-1, j-1] \\] <p>3단계: 경계 조건과 상태 전이 순서를 정한다</p> <p>두 문자열이 모두 비어 있으면 편집 횟수는 \\(0\\)이므로 \\(dp[0, 0] = 0\\)이다. \\(s\\)만 비어 있고 \\(t\\)는 비어 있지 않으면 최소 편집 횟수는 \\(t\\)의 길이이므로 첫 행은 \\(dp[0, j] = j\\)다. 반대로 \\(s\\)는 비어 있지 않고 \\(t\\)가 비어 있으면 최소 편집 횟수는 \\(s\\)의 길이이므로 첫 열은 \\(dp[i, 0] = i\\)다.</p> <p>상태 전이 식에서 \\(dp[i, j]\\)는 왼쪽, 위쪽, 왼쪽 위의 해에 의존하므로 이중 반복문으로 전체 \\(dp\\) 테이블을 정방향으로 순회하면 된다.</p>","path":["제 14 장   동적 계획법","14.6   편집 거리 문제"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/#2","level":3,"title":"2.   코드 구현","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby edit_distance.py<pre><code>def edit_distance_dp(s: str, t: str) -&gt; int:\n    \"\"\"편집 거리: 동적 계획법\"\"\"\n    n, m = len(s), len(t)\n    dp = [[0] * (m + 1) for _ in range(n + 1)]\n    # 상태 전이: 첫 번째 행과 첫 번째 열\n    for i in range(1, n + 1):\n        dp[i][0] = i\n    for j in range(1, m + 1):\n        dp[0][j] = j\n    # 상태 전이: 나머지 행과 열\n    for i in range(1, n + 1):\n        for j in range(1, m + 1):\n            if s[i - 1] == t[j - 1]:\n                # 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1]\n            else:\n                # 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] = min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1\n    return dp[n][m]\n</code></pre> edit_distance.cpp<pre><code>/* 편집 거리: 동적 계획법 */\nint editDistanceDP(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(m + 1, 0));\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.java<pre><code>/* 편집 거리: 동적 계획법 */\nint editDistanceDP(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[][] dp = new int[n + 1][m + 1];\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] = Math.min(Math.min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.cs<pre><code>/* 편집 거리: 동적 계획법 */\nint EditDistanceDP(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[,] dp = new int[n + 1, m + 1];\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i, 0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0, j] = j;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i, j] = dp[i - 1, j - 1];\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i, j] = Math.Min(Math.Min(dp[i, j - 1], dp[i - 1, j]), dp[i - 1, j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n, m];\n}\n</code></pre> edit_distance.go<pre><code>/* 편집 거리: 동적 계획법 */\nfunc editDistanceDP(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, m+1)\n    }\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for i := 1; i &lt;= n; i++ {\n        dp[i][0] = i\n    }\n    for j := 1; j &lt;= m; j++ {\n        dp[0][j] = j\n    }\n    // 상태 전이: 나머지 행과 열\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= m; j++ {\n            if s[i-1] == t[j-1] {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i-1][j-1]\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] = MinInt(MinInt(dp[i][j-1], dp[i-1][j]), dp[i-1][j-1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.swift<pre><code>/* 편집 거리: 동적 계획법 */\nfunc editDistanceDP(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: Array(repeating: 0, count: m + 1), count: n + 1)\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for i in 1 ... n {\n        dp[i][0] = i\n    }\n    for j in 1 ... m {\n        dp[0][j] = j\n    }\n    // 상태 전이: 나머지 행과 열\n    for i in 1 ... n {\n        for j in 1 ... m {\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.js<pre><code>/* 편집 거리: 동적 계획법 */\nfunction editDistanceDP(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(m + 1).fill(0));\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.ts<pre><code>/* 편집 거리: 동적 계획법 */\nfunction editDistanceDP(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: m + 1 }, () =&gt; 0)\n    );\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.dart<pre><code>/* 편집 거리: 동적 계획법 */\nint editDistanceDP(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (_) =&gt; List.filled(m + 1, 0));\n  // 상태 전이: 첫 번째 행과 첫 번째 열\n  for (int i = 1; i &lt;= n; i++) {\n    dp[i][0] = i;\n  }\n  for (int j = 1; j &lt;= m; j++) {\n    dp[0][j] = j;\n  }\n  // 상태 전이: 나머지 행과 열\n  for (int i = 1; i &lt;= n; i++) {\n    for (int j = 1; j &lt;= m; j++) {\n      if (s[i - 1] == t[j - 1]) {\n        // 두 문자가 같으면 두 문자를 바로 건너뛴다\n        dp[i][j] = dp[i - 1][j - 1];\n      } else {\n        // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n        dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n      }\n    }\n  }\n  return dp[n][m];\n}\n</code></pre> edit_distance.rs<pre><code>/* 편집 거리: 동적 계획법 */\nfn edit_distance_dp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![vec![0; m + 1]; n + 1];\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for i in 1..=n {\n        dp[i][0] = i as i32;\n    }\n    for j in 1..m {\n        dp[0][j] = j as i32;\n    }\n    // 상태 전이: 나머지 행과 열\n    for i in 1..=n {\n        for j in 1..=m {\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] =\n                    std::cmp::min(std::cmp::min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    dp[n][m]\n}\n</code></pre> edit_distance.c<pre><code>/* 편집 거리: 동적 계획법 */\nint editDistanceDP(char *s, char *t, int n, int m) {\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(m + 1, sizeof(int));\n    }\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] = myMin(myMin(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    int res = dp[n][m];\n    // 메모리 해제\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* 편집 거리: 동적 계획법 */\nfun editDistanceDP(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = Array(n + 1) { IntArray(m + 1) }\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (i in 1..n) {\n        dp[i][0] = i\n    }\n    for (j in 1..m) {\n        dp[0][j] = j\n    }\n    // 상태 전이: 나머지 행과 열\n    for (i in 1..n) {\n        for (j in 1..m) {\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.rb<pre><code>### 편집 거리: 동적 계획법 ###\ndef edit_distance_dp(s, t)\n  n, m = s.length, t.length\n  dp = Array.new(n + 1) { Array.new(m + 1, 0) }\n  # 상태 전이: 첫 번째 행과 첫 번째 열\n  (1...(n + 1)).each { |i| dp[i][0] = i }\n  (1...(m + 1)).each { |j| dp[0][j] = j }\n  # 상태 전이: 나머지 행과 열\n  for i in 1...(n + 1)\n    for j in 1...(m +1)\n      if s[i - 1] == t[j - 1]\n        # 두 문자가 같으면 두 문자를 바로 건너뛴다\n        dp[i][j] = dp[i - 1][j - 1]\n      else\n        # 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n        dp[i][j] = [dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]].min + 1\n      end\n    end\n  end\n  dp[n][m]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-30)과 같이 편집 거리의 상태 전이는 배낭 문제와 매우 비슷하며, 모두 2차원 격자를 채우는 과정으로 볼 수 있다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 14-30   편집 거리의 동적 계획법 과정 </p>","path":["제 14 장   동적 계획법","14.6   편집 거리 문제"],"tags":[]},{"location":"chapter_dynamic_programming/edit_distance_problem/#3","level":3,"title":"3.   공간 최적화","text":"<p>\\(dp[i,j]\\)는 위쪽 \\(dp[i-1, j]\\), 왼쪽 \\(dp[i, j-1]\\), 왼쪽 위 \\(dp[i-1, j-1]\\)에서 전이된다. 그런데 정방향 순회는 왼쪽 위 값 \\(dp[i-1, j-1]\\)을 잃고, 역방향 순회는 \\(dp[i, j-1]\\)을 미리 계산하지 못하므로 두 순서 모두 그대로는 사용할 수 없다.</p> <p>이를 위해 변수 <code>leftup</code>에 왼쪽 위의 해 \\(dp[i-1, j-1]\\)을 잠시 저장하면 왼쪽과 위쪽의 해만 고려하면 된다. 이때는 무한 배낭 문제와 같은 상황이므로 정방향 순회를 사용할 수 있다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby edit_distance.py<pre><code>def edit_distance_dp_comp(s: str, t: str) -&gt; int:\n    \"\"\"편집 거리: 공간을 최적화한 동적 계획법\"\"\"\n    n, m = len(s), len(t)\n    dp = [0] * (m + 1)\n    # 상태 전이: 첫 번째 행\n    for j in range(1, m + 1):\n        dp[j] = j\n    # 상태 전이: 나머지 행\n    for i in range(1, n + 1):\n        # 상태 전이: 첫 번째 열\n        leftup = dp[0]  # dp[i-1, j-1]을 임시로 저장한다\n        dp[0] += 1\n        # 상태 전이: 나머지 열\n        for j in range(1, m + 1):\n            temp = dp[j]\n            if s[i - 1] == t[j - 1]:\n                # 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup\n            else:\n                # 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = min(dp[j - 1], dp[j], leftup) + 1\n            leftup = temp  # 다음 반복의 dp[i-1, j-1]로 갱신한다\n    return dp[m]\n</code></pre> edit_distance.cpp<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nint editDistanceDPComp(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;int&gt; dp(m + 1, 0);\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt;= n; i++) {\n        // 상태 전이: 첫 번째 열\n        int leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i;\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup;\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.java<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nint editDistanceDPComp(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[] dp = new int[m + 1];\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt;= n; i++) {\n        // 상태 전이: 첫 번째 열\n        int leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i;\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup;\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = Math.min(Math.min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.cs<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nint EditDistanceDPComp(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[] dp = new int[m + 1];\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt;= n; i++) {\n        // 상태 전이: 첫 번째 열\n        int leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i;\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup;\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = Math.Min(Math.Min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.go<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nfunc editDistanceDPComp(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([]int, m+1)\n    // 상태 전이: 첫 번째 행\n    for j := 1; j &lt;= m; j++ {\n        dp[j] = j\n    }\n    // 상태 전이: 나머지 행\n    for i := 1; i &lt;= n; i++ {\n        // 상태 전이: 첫 번째 열\n        leftUp := dp[0] // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i\n        // 상태 전이: 나머지 열\n        for j := 1; j &lt;= m; j++ {\n            temp := dp[j]\n            if s[i-1] == t[j-1] {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftUp\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = MinInt(MinInt(dp[j-1], dp[j]), leftUp) + 1\n            }\n            leftUp = temp // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.swift<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nfunc editDistanceDPComp(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: 0, count: m + 1)\n    // 상태 전이: 첫 번째 행\n    for j in 1 ... m {\n        dp[j] = j\n    }\n    // 상태 전이: 나머지 행\n    for i in 1 ... n {\n        // 상태 전이: 첫 번째 열\n        var leftup = dp[0] // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i\n        // 상태 전이: 나머지 열\n        for j in 1 ... m {\n            let temp = dp[j]\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n            }\n            leftup = temp // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.js<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nfunction editDistanceDPComp(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // 상태 전이: 첫 번째 행\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // 상태 전이: 나머지 행\n    for (let i = 1; i &lt;= n; i++) {\n        // 상태 전이: 첫 번째 열\n        let leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i;\n        // 상태 전이: 나머지 열\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup;\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.ts<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nfunction editDistanceDPComp(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // 상태 전이: 첫 번째 행\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // 상태 전이: 나머지 행\n    for (let i = 1; i &lt;= n; i++) {\n        // 상태 전이: 첫 번째 열\n        let leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i;\n        // 상태 전이: 나머지 열\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup;\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.dart<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nint editDistanceDPComp(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;int&gt; dp = List.filled(m + 1, 0);\n  // 상태 전이: 첫 번째 행\n  for (int j = 1; j &lt;= m; j++) {\n    dp[j] = j;\n  }\n  // 상태 전이: 나머지 행\n  for (int i = 1; i &lt;= n; i++) {\n    // 상태 전이: 첫 번째 열\n    int leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n    dp[0] = i;\n    // 상태 전이: 나머지 열\n    for (int j = 1; j &lt;= m; j++) {\n      int temp = dp[j];\n      if (s[i - 1] == t[j - 1]) {\n        // 두 문자가 같으면 두 문자를 바로 건너뛴다\n        dp[j] = leftup;\n      } else {\n        // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n        dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n      }\n      leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n    }\n  }\n  return dp[m];\n}\n</code></pre> edit_distance.rs<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nfn edit_distance_dp_comp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![0; m + 1];\n    // 상태 전이: 첫 번째 행\n    for j in 1..m {\n        dp[j] = j as i32;\n    }\n    // 상태 전이: 나머지 행\n    for i in 1..=n {\n        // 상태 전이: 첫 번째 열\n        let mut leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i as i32;\n        // 상태 전이: 나머지 열\n        for j in 1..=m {\n            let temp = dp[j];\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup;\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = std::cmp::min(std::cmp::min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    dp[m]\n}\n</code></pre> edit_distance.c<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nint editDistanceDPComp(char *s, char *t, int n, int m) {\n    int *dp = calloc(m + 1, sizeof(int));\n    // 상태 전이: 첫 번째 행\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // 상태 전이: 나머지 행\n    for (int i = 1; i &lt;= n; i++) {\n        // 상태 전이: 첫 번째 열\n        int leftup = dp[0]; // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i;\n        // 상태 전이: 나머지 열\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup;\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = myMin(myMin(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    int res = dp[m];\n    // 메모리 해제\n    free(dp);\n    return res;\n}\n</code></pre> edit_distance.kt<pre><code>/* 편집 거리: 공간을 최적화한 동적 계획법 */\nfun editDistanceDPComp(s: String, t: String): Int {\n    val n = s.length\n    val m = t.length\n    val dp = IntArray(m + 1)\n    // 상태 전이: 첫 번째 행\n    for (j in 1..m) {\n        dp[j] = j\n    }\n    // 상태 전이: 나머지 행\n    for (i in 1..n) {\n        // 상태 전이: 첫 번째 열\n        var leftup = dp[0] // dp[i-1, j-1]을 임시로 저장한다\n        dp[0] = i\n        // 상태 전이: 나머지 열\n        for (j in 1..m) {\n            val temp = dp[j]\n            if (s[i - 1] == t[j - 1]) {\n                // 두 문자가 같으면 두 문자를 바로 건너뛴다\n                dp[j] = leftup\n            } else {\n                // 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n            }\n            leftup = temp // 다음 반복의 dp[i-1, j-1]로 갱신한다\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.rb<pre><code>### 편집 거리: 공간을 최적화한 동적 계획법 ###\ndef edit_distance_dp_comp(s, t)\n  n, m = s.length, t.length\n  dp = Array.new(m + 1, 0)\n  # 상태 전이: 첫 번째 행\n  (1...(m + 1)).each { |j| dp[j] = j }\n  # 상태 전이: 나머지 행\n  for i in 1...(n + 1)\n    # 상태 전이: 첫 번째 열\n    leftup = dp.first # dp[i-1, j-1]을 임시로 저장한다\n    dp[0] += 1\n    # 상태 전이: 나머지 열\n    for j in 1...(m + 1)\n      temp = dp[j]\n      if s[i - 1] == t[j - 1]\n        # 두 문자가 같으면 두 문자를 바로 건너뛴다\n        dp[j] = leftup\n      else\n        # 최소 편집 횟수 = 삽입, 삭제, 교체 세 연산에 대한 최소 편집 횟수 + 1\n        dp[j] = [dp[j - 1], dp[j], leftup].min + 1\n      end\n      leftup = temp # 다음 반복의 dp[i-1, j-1]로 갱신한다\n    end\n  end\n  dp[m]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.6   편집 거리 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/","level":1,"title":"14.8   연습 문제","text":"","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1481","level":2,"title":"14.8.1   개념 확인","text":"","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1","level":3,"title":"1.   언제 동적 계획법을 사용하는 것이 적합한가","text":"<p>한 학생이 “점화식을 쓸 수만 있으면 동적 계획법을 사용해야 한다.”라고 말했다. 다음 세 작업에 동적 계획법, 백트래킹, 또는 <code>dp</code> 테이블을 만들지 않는 반복문이나 수학 공식 중 어느 방법이 더 적합한지 판단하고, 핵심 이유를 하나씩 적어 보자.</p> <ol> <li>액면가가 <code>[1, 3, 4]</code>인 동전으로 금액 6을 만들 때 필요한 동전의 최소 개수를 구한다. 각 동전은 반복해서 사용할 수 있다.</li> <li><code>[1, 2, 3]</code>의 모든 순열을 출력한다.</li> <li> <p>\\(1 + 2 + \\dots + n\\)을 계산한다.</p> <p>동적 계획법이 적합하다고 판단한 작업에서는 <code>dp[i]</code>가 무엇을 나타내는지도 설명하라.</p> </li> </ol> 참고 답안 <ol> <li> <p>동적 계획법이 적합하다. <code>dp[i]</code>를 금액 <code>i</code>를 만드는 데 필요한 동전의 최소 개수로 정의한다.     <code>i</code> 이하인 각 동전 <code>c</code>에 대해 <code>dp[i-c] + 1</code>을 후보 답으로 삼고,     이 후보들 중 최솟값을 선택하면 된다. 서로 다른 선택에서 같은 금액을 반복해서 만나며, 더 큰 금액의 최적해도 더 작은 금액의 최적해로 구성할 수 있다.     금액 6의 답은 2이며, <code>3 + 3</code>으로 만들 수 있다.</p> </li> <li> <p>백트래킹을 사용해야 한다. 문제에서는 순열 6개를 모두 하나씩 생성해야 한다. 백트래킹은 하나를 선택하고 탐색을 이어 간 뒤,     선택을 취소하고 다른 분기를 시도하는 과정을 체계적으로 수행할 수 있다. 어떤 방법을 쓰더라도 이 순열들을 실제로 출력하려면 열거 과정을 생략할 수 없다.</p> </li> <li> <p>반복문이나 등차수열의 합 공식을 사용하면 된다. <code>S(i) = S(i-1) + i</code>라는 점화식을 쓸 수는 있지만, <code>S(i)</code>를 계산할 때는 더 작은 값 <code>S(i-1)</code> 하나에만 의존한다.     각 부분합을 한 번만 계산하면 되고 중복되는 부분 문제가 없으므로 <code>dp</code> 테이블을 만들 필요가 없다. ‘점화식을 쓸 수 있다’고 해서 ‘동적 계획법이 필요하다’는 뜻은 아니다.</p> </li> </ol>","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#2","level":3,"title":"2.   배낭 테이블의 한 칸은 어떻게 계산하는가","text":"<p>0-1 배낭 문제에서 물건의 무게는 <code>wgt = [1, 2, 3]</code>, 가치는 <code>val = [5, 11, 15]</code>이며, 배낭 용량은 4다. <code>dp[i][c]</code>는 앞의 물건 \\(i\\)개만 고려하고 배낭 용량의 상한이 \\(c\\)일 때 얻을 수 있는 최대 가치를 나타낸다. 배낭을 반드시 가득 채울 필요는 없다.</p> <p>지금은 상태 <code>dp[3][4]</code>만 계산한다. <code>dp[2][4] = 16</code>, <code>dp[2][1] = 5</code>임을 알고 있다.</p> <ol> <li>세 번째 물건을 선택하지 않을 때 후보 가치는 얼마인가?</li> <li>세 번째 물건을 선택하면 용량이 얼마나 남으며, 후보 가치는 얼마인가?</li> <li><code>dp[3][4]</code>는 얼마로 정해야 하는가? 이는 어떤 물건들을 선택한 결과인가?</li> </ol> 참고 답안 <ol> <li> <p>세 번째 물건을 선택하지 않으면 앞의 두 물건에 대한 결과를 그대로 사용하므로 후보 가치는 <code>dp[2][4] = 16</code>이다.</p> </li> <li> <p>세 번째 물건의 무게는 3이므로 넣은 뒤 남는 용량은 \\(4-3=1\\)이다. 후보 가치는     <code>dp[2][1] + 15 = 5 + 15 = 20</code>이다.</p> </li> <li> <p>16과 20을 비교하면 <code>dp[3][4] = 20</code>을 얻는다. 이는 첫 번째와 세 번째 물건을 선택한 결과로,     총무게는 \\(1+3=4\\), 총가치는 \\(5+15=20\\)이다.</p> <p>이 상태의 계산은 0-1 배낭 문제에서 한 번의 ‘선택하거나 선택하지 않거나’ 비교를 보여 준다.</p> </li> </ol>","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#3","level":3,"title":"3.   배낭 용량은 어떤 순서로 갱신해야 하는가","text":"<p>0-1 배낭 문제에 물건이 하나뿐이며, 무게는 2, 가치는 5다. 배낭 용량은 4다. 각 물건은 최대 한 번만 선택할 수 있고, 초기 일차원 배열은 <code>dp = [0, 0, 0, 0, 0]</code>이다.</p> <p>한 학생이 이 물건을 처리하면서 용량을 2부터 4까지 갱신했다.</p> <ul> <li><code>dp[2]</code>를 갱신한 뒤 5를 얻었다.</li> <li><code>dp[3]</code>을 갱신한 뒤에도 5를 얻었다.</li> <li><code>dp[4]</code>를 갱신할 때 방금 구한 <code>dp[2]</code>를 다시 사용하여 <code>dp[4] = 10</code>을 얻었다.</li> </ul> <ol> <li><code>dp[4] = 10</code>이라는 결과가 올바른가? 이유는 무엇인가?</li> <li>‘각 물건은 최대 한 번만 선택할 수 있다’는 조건에 따르면 <code>dp[4]</code>는 얼마여야 하는가?</li> <li>각 물건을 처리할 때 용량을 큰 것부터 작은 것 순으로 갱신해야 하는가, 그 반대인가? 그렇게 하면 어떤 문제를 피할 수 있는가?</li> </ol> 참고 답안 <ol> <li> <p>올바르지 않다. 가치 10은 가치가 5인 이 물건을 두 번 넣은 것과 같으므로,     ‘각 물건은 최대 한 번만 선택할 수 있다’는 조건에 어긋난다.</p> </li> <li> <p>배낭에는 이 물건 하나만 넣을 수 있으므로 올바른 <code>dp[4]</code> 값은 5다.</p> </li> <li> <p>용량을 큰 것부터 작은 것 순으로, 즉 4, 3, 2 순으로 갱신해야 한다.     그러면 <code>dp[c]</code>를 계산할 때 읽는 <code>dp[c-2]</code>는 아직 현재 물건을 처리하기 전의 결과이므로,     같은 회전에서 현재 물건을 반복해서 사용하지 않게 된다.</p> </li> </ol>","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1482","level":2,"title":"14.8.2   프로그래밍 연습","text":"","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#1_1","level":3,"title":"1.   계단을 오르는 방법의 수","text":"<p>계단은 총 <code>n</code>칸이다. 한 번에 1칸 또는 2칸만 올라갈 수 있으며, 정확히 <code>n</code>번째 칸에 도착해야 한다. 서로 다른 방법이 모두 몇 가지인지 계산하라. <code>n &gt;= 1</code>이며, 각 걸음에서 1칸을 오르는지 2칸을 오르는지에 따라서만 방법을 구분한다. 일차원 동적 계획법 배열을 사용하라. 상태 두 개만 유지하는 공간 최적화 방식은 아직 사용하지 마라.</p> 풀이 힌트 <ol> <li>i번째 칸에 도착하는 마지막 걸음은 1칸 또는 2칸을 오르는 것뿐이다.</li> <li>따라서 dp[i] = dp[i-1] + dp[i-2]다.</li> <li>n이 1과 2인 경우를 먼저 처리한 뒤, 세 번째 칸부터 테이블을 채운다.</li> </ol> <p>LeetCode</p>","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/exercises/#2-0-1","level":3,"title":"2.   0-1 배낭 문제","text":"<p>길이가 같은 배열 <code>wgt</code>와 <code>val</code>이 주어진다. <code>i</code>번째 물건의 무게 <code>wgt[i]</code>는 양의 정수이고, 가치 <code>val[i]</code>는 음이 아닌 정수다. 배낭 용량 <code>cap</code>은 음이 아닌 정수다. 각 물건은 최대 한 번만 선택할 수 있다. 총무게가 <code>cap</code>을 넘지 않는 조건에서 배낭에 넣을 수 있는 최대 총가치를 구하라. 일차원 동적 계획법으로 구현하라.</p> 풀이 힌트 <ol> <li>길이가 cap + 1인 배열 dp를 초기화한다. dp[c]는 용량 상한이 c일 때의 최대 가치를 나타낸다.</li> <li>물건 i를 처리할 때 ‘선택하지 않는 경우’의 dp[c]와 ‘선택하는 경우’의 dp[c-wgt[i]] + val[i]를 비교한다.</li> <li>같은 회전에서 현재 물건을 반복해서 선택하지 않도록 용량을 반드시 큰 것부터 작은 것 순으로 갱신해야 한다.</li> </ol>","path":["제 14 장   동적 계획법","14.8   연습 문제"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/","level":1,"title":"14.1   동적 계획법 첫걸음","text":"<p>동적 계획법(dynamic programming)은 중요한 알고리즘 패러다임이다. 문제를 더 작은 부분 문제들로 나누고 그 해를 저장하여 중복 계산을 피함으로써 시간 효율을 크게 높인다.</p> <p>이 절에서는 대표적인 예제에서 출발하여 먼저 완전 탐색을 수행하는 백트래킹 풀이를 제시한다. 그 안의 중복 부분 문제를 살펴보고, 더 효율적인 동적 계획법 풀이를 단계적으로 도출한다.</p> <p>계단 오르기</p> <p>총 \\(n\\) 칸인 계단이 있고, 한 번에 \\(1\\) 칸 또는 \\(2\\) 칸을 오를 수 있다. 맨 위까지 오르는 방법은 몇 가지인가?</p> <p>아래 그림(14-1)과 같이 \\(3\\) 칸짜리 계단의 맨 위에 오르는 방법은 총 \\(3\\) 가지다.</p> <p></p> <p> 그림 14-1   3번째 칸까지 오르는 방법 수 </p> <p>이 문제는 방법의 수를 구하는 것이므로 백트래킹으로 모든 가능성을 열거할 수 있다. 계단 오르기를 여러 차례의 선택 과정으로 생각해 보자. 바닥에서 출발하여 매번 \\(1\\) 칸 또는 \\(2\\) 칸을 고른다. 맨 위에 도착할 때마다 방법 수를 \\(1\\) 늘리고, 맨 위를 넘어가면 가지치기한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_backtrack.py<pre><code>def backtrack(choices: list[int], state: int, n: int, res: list[int]) -&gt; int:\n    \"\"\"백트래킹\"\"\"\n    # n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if state == n:\n        res[0] += 1\n    # 모든 선택지를 순회한다\n    for choice in choices:\n        # 가지치기: n번째 계단을 넘어갈 수 없다\n        if state + choice &gt; n:\n            continue\n        # 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res)\n        # 되돌리기\n\ndef climbing_stairs_backtrack(n: int) -&gt; int:\n    \"\"\"계단 오르기: 백트래킹\"\"\"\n    choices = [1, 2]  # 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    state = 0  # 0번째 계단에서 오르기 시작한다\n    res = [0]  # res[0]에 방법의 수를 기록한다\n    backtrack(choices, state, n, res)\n    return res[0]\n</code></pre> climbing_stairs_backtrack.cpp<pre><code>/* 백트래킹 */\nvoid backtrack(vector&lt;int&gt; &amp;choices, int state, int n, vector&lt;int&gt; &amp;res) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if (state == n)\n        res[0]++;\n    // 모든 선택지를 순회한다\n    for (auto &amp;choice : choices) {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if (state + choice &gt; n)\n            continue;\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res);\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nint climbingStairsBacktrack(int n) {\n    vector&lt;int&gt; choices = {1, 2}; // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    int state = 0;                // 0번째 계단에서 오르기 시작한다\n    vector&lt;int&gt; res = {0};        // res[0]에 방법의 수를 기록한다\n    backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.java<pre><code>/* 백트래킹 */\nvoid backtrack(List&lt;Integer&gt; choices, int state, int n, List&lt;Integer&gt; res) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if (state == n)\n        res.set(0, res.get(0) + 1);\n    // 모든 선택지를 순회한다\n    for (Integer choice : choices) {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if (state + choice &gt; n)\n            continue;\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res);\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nint climbingStairsBacktrack(int n) {\n    List&lt;Integer&gt; choices = Arrays.asList(1, 2); // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    int state = 0; // 0번째 계단에서 오르기 시작한다\n    List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n    res.add(0); // res[0]에 방법의 수를 기록한다\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.cs<pre><code>/* 백트래킹 */\nvoid Backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if (state == n)\n        res[0]++;\n    // 모든 선택지를 순회한다\n    foreach (int choice in choices) {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if (state + choice &gt; n)\n            continue;\n        // 시도: 선택하고 상태를 갱신한다\n        Backtrack(choices, state + choice, n, res);\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nint ClimbingStairsBacktrack(int n) {\n    List&lt;int&gt; choices = [1, 2]; // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    int state = 0; // 0번째 계단에서 오르기 시작한다\n    List&lt;int&gt; res = [0]; // res[0]에 방법의 수를 기록한다\n    Backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.go<pre><code>/* 백트래킹 */\nfunc backtrack(choices []int, state, n int, res []int) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if state == n {\n        res[0] = res[0] + 1\n    }\n    // 모든 선택지를 순회한다\n    for _, choice := range choices {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if state+choice &gt; n {\n            continue\n        }\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state+choice, n, res)\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nfunc climbingStairsBacktrack(n int) int {\n    // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    choices := []int{1, 2}\n    // 0번째 계단에서 오르기 시작한다\n    state := 0\n    res := make([]int, 1)\n    // res[0]에 방법의 수를 기록한다\n    res[0] = 0\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.swift<pre><code>/* 백트래킹 */\nfunc backtrack(choices: [Int], state: Int, n: Int, res: inout [Int]) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if state == n {\n        res[0] += 1\n    }\n    // 모든 선택지를 순회한다\n    for choice in choices {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if state + choice &gt; n {\n            continue\n        }\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices: choices, state: state + choice, n: n, res: &amp;res)\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nfunc climbingStairsBacktrack(n: Int) -&gt; Int {\n    let choices = [1, 2] // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    let state = 0 // 0번째 계단에서 오르기 시작한다\n    var res: [Int] = []\n    res.append(0) // res[0]에 방법의 수를 기록한다\n    backtrack(choices: choices, state: state, n: n, res: &amp;res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.js<pre><code>/* 백트래킹 */\nfunction backtrack(choices, state, n, res) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if (state === n) res.set(0, res.get(0) + 1);\n    // 모든 선택지를 순회한다\n    for (const choice of choices) {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if (state + choice &gt; n) continue;\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res);\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nfunction climbingStairsBacktrack(n) {\n    const choices = [1, 2]; // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    const state = 0; // 0번째 계단에서 오르기 시작한다\n    const res = new Map();\n    res.set(0, 0); // res[0]에 방법의 수를 기록한다\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.ts<pre><code>/* 백트래킹 */\nfunction backtrack(\n    choices: number[],\n    state: number,\n    n: number,\n    res: Map&lt;0, any&gt;\n): void {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if (state === n) res.set(0, res.get(0) + 1);\n    // 모든 선택지를 순회한다\n    for (const choice of choices) {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if (state + choice &gt; n) continue;\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res);\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nfunction climbingStairsBacktrack(n: number): number {\n    const choices = [1, 2]; // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    const state = 0; // 0번째 계단에서 오르기 시작한다\n    const res = new Map();\n    res.set(0, 0); // res[0]에 방법의 수를 기록한다\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.dart<pre><code>/* 백트래킹 */\nvoid backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n  // n번째 계단에 도달하면 방법의 수에 1을 더한다\n  if (state == n) {\n    res[0]++;\n  }\n  // 모든 선택지를 순회한다\n  for (int choice in choices) {\n    // 가지치기: n번째 계단을 넘어갈 수 없다\n    if (state + choice &gt; n) continue;\n    // 시도: 선택하고 상태를 갱신한다\n    backtrack(choices, state + choice, n, res);\n    // 되돌리기\n  }\n}\n\n/* 계단 오르기: 백트래킹 */\nint climbingStairsBacktrack(int n) {\n  List&lt;int&gt; choices = [1, 2]; // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n  int state = 0; // 0번째 계단에서 오르기 시작한다\n  List&lt;int&gt; res = [];\n  res.add(0); // res[0]에 방법의 수를 기록한다\n  backtrack(choices, state, n, res);\n  return res[0];\n}\n</code></pre> climbing_stairs_backtrack.rs<pre><code>/* 백트래킹 */\nfn backtrack(choices: &amp;[i32], state: i32, n: i32, res: &amp;mut [i32]) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if state == n {\n        res[0] = res[0] + 1;\n    }\n    // 모든 선택지를 순회한다\n    for &amp;choice in choices {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if state + choice &gt; n {\n            continue;\n        }\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res);\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nfn climbing_stairs_backtrack(n: usize) -&gt; i32 {\n    let choices = vec![1, 2]; // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    let state = 0; // 0번째 계단에서 오르기 시작한다\n    let mut res = Vec::new();\n    res.push(0); // res[0]에 방법의 수를 기록한다\n    backtrack(&amp;choices, state, n as i32, &amp;mut res);\n    res[0]\n}\n</code></pre> climbing_stairs_backtrack.c<pre><code>/* 백트래킹 */\nvoid backtrack(int *choices, int state, int n, int *res, int len) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if (state == n)\n        res[0]++;\n    // 모든 선택지를 순회한다\n    for (int i = 0; i &lt; len; i++) {\n        int choice = choices[i];\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if (state + choice &gt; n)\n            continue;\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res, len);\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nint climbingStairsBacktrack(int n) {\n    int choices[2] = {1, 2}; // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    int state = 0;           // 0번째 계단에서 오르기 시작한다\n    int *res = (int *)malloc(sizeof(int));\n    *res = 0; // res[0]에 방법의 수를 기록한다\n    int len = sizeof(choices) / sizeof(int);\n    backtrack(choices, state, n, res, len);\n    int result = *res;\n    free(res);\n    return result;\n}\n</code></pre> climbing_stairs_backtrack.kt<pre><code>/* 백트래킹 */\nfun backtrack(\n    choices: MutableList&lt;Int&gt;,\n    state: Int,\n    n: Int,\n    res: MutableList&lt;Int&gt;\n) {\n    // n번째 계단에 도달하면 방법의 수에 1을 더한다\n    if (state == n)\n        res[0] = res[0] + 1\n    // 모든 선택지를 순회한다\n    for (choice in choices) {\n        // 가지치기: n번째 계단을 넘어갈 수 없다\n        if (state + choice &gt; n) continue\n        // 시도: 선택하고 상태를 갱신한다\n        backtrack(choices, state + choice, n, res)\n        // 되돌리기\n    }\n}\n\n/* 계단 오르기: 백트래킹 */\nfun climbingStairsBacktrack(n: Int): Int {\n    val choices = mutableListOf(1, 2) // 한 계단 또는 두 계단 오르기를 선택할 수 있다\n    val state = 0 // 0번째 계단에서 오르기 시작한다\n    val res = mutableListOf&lt;Int&gt;()\n    res.add(0) // res[0]에 방법의 수를 기록한다\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.rb<pre><code>### 백트래킹 ###\ndef backtrack(choices, state, n, res)\n  # n번째 계단에 도달하면 방법의 수에 1을 더한다\n  res[0] += 1 if state == n\n  # 모든 선택지를 순회한다\n  for choice in choices\n    # 가지치기: n번째 계단을 넘어갈 수 없다\n    next if state + choice &gt; n\n\n    # 시도: 선택하고 상태를 갱신한다\n    backtrack(choices, state + choice, n, res)\n  end\n  # 되돌리기\nend\n\n### 계단 오르기: 백트래킹 ###\ndef climbing_stairs_backtrack(n)\n  choices = [1, 2] # 한 계단 또는 두 계단 오르기를 선택할 수 있다\n  state = 0 # 0번째 계단에서 오르기 시작한다\n  res = [0] # res[0]에 방법의 수를 기록한다\n  backtrack(choices, state, n, res)\n  res.first\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.1   동적 계획법 첫걸음"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1411-1","level":2,"title":"14.1.1   방법 1: 완전 탐색","text":"<p>백트래킹은 보통 문제를 명시적으로 나누지 않고 일련의 의사 결정 단계로 바라보며, 시도와 가지치기를 통해 가능한 모든 해를 찾는다.</p> <p>이 문제를 부분 문제로 나누는 관점에서 분석해 보자. \\(i\\) 번째 칸에 오르는 방법 수를 \\(dp[i]\\)라고 하면 \\(dp[i]\\)가 원래 문제이고, 다음은 그 부분 문제들이다.</p> \\[ dp[i-1], dp[i-2], \\dots, dp[2], dp[1] \\] <p>한 번에 \\(1\\) 칸 또는 \\(2\\) 칸만 오를 수 있으므로, 현재 \\(i\\) 번째 칸에 있다면 직전에는 \\(i - 1\\) 번째 또는 \\(i - 2\\) 번째 칸에 있었어야 한다. 즉, \\(i -1\\) 번째 또는 \\(i - 2\\) 번째 칸에서만 \\(i\\) 번째 칸으로 올라올 수 있다.</p> <p>여기서 중요한 결론을 얻는다. \\(i - 1\\) 번째 칸에 오르는 방법 수와 \\(i - 2\\) 번째 칸에 오르는 방법 수를 더하면 \\(i\\) 번째 칸에 오르는 방법 수가 된다. 식으로 나타내면 다음과 같다.</p> \\[ dp[i] = dp[i-1] + dp[i-2] \\] <p>이는 계단 오르기 문제의 부분 문제들 사이에 점화 관계가 있으며, 부분 문제의 해로 원래 문제의 해를 구성할 수 있다는 뜻이다. 아래 그림(14-2)이 이 점화 관계를 보여 준다.</p> <p></p> <p> 그림 14-2   방법 수의 점화 관계 </p> <p>점화식으로 완전 탐색 풀이를 얻을 수 있다. \\(dp[n]\\)에서 출발해 큰 문제 하나를 더 작은 두 문제의 합으로 재귀적으로 나누고, 가장 작은 부분 문제 \\(dp[1]\\)과 \\(dp[2]\\)에 도달하면 반환한다. 이들의 해는 \\(dp[1] = 1\\), \\(dp[2] = 2\\)로 이미 알려져 있다. 즉, \\(1\\), \\(2\\) 번째 칸에 오르는 방법은 각각 \\(1\\), \\(2\\) 가지다.</p> <p>아래 코드는 일반적인 백트래킹 코드와 마찬가지로 깊이 우선 탐색이지만 더 간결하다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dfs.py<pre><code>def dfs(i: int) -&gt; int:\n    \"\"\"탐색\"\"\"\n    # dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 or i == 2:\n        return i\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1) + dfs(i - 2)\n    return count\n\ndef climbing_stairs_dfs(n: int) -&gt; int:\n    \"\"\"계단 오르기: 탐색\"\"\"\n    return dfs(n)\n</code></pre> climbing_stairs_dfs.cpp<pre><code>/* 탐색 */\nint dfs(int i) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* 계단 오르기: 탐색 */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.java<pre><code>/* 탐색 */\nint dfs(int i) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* 계단 오르기: 탐색 */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.cs<pre><code>/* 탐색 */\nint DFS(int i) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1) + DFS(i - 2);\n    return count;\n}\n\n/* 계단 오르기: 탐색 */\nint ClimbingStairsDFS(int n) {\n    return DFS(n);\n}\n</code></pre> climbing_stairs_dfs.go<pre><code>/* 탐색 */\nfunc dfs(i int) int {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfs(i-1) + dfs(i-2)\n    return count\n}\n\n/* 계단 오르기: 탐색 */\nfunc climbingStairsDFS(n int) int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.swift<pre><code>/* 탐색 */\nfunc dfs(i: Int) -&gt; Int {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1) + dfs(i: i - 2)\n    return count\n}\n\n/* 계단 오르기: 탐색 */\nfunc climbingStairsDFS(n: Int) -&gt; Int {\n    dfs(i: n)\n}\n</code></pre> climbing_stairs_dfs.js<pre><code>/* 탐색 */\nfunction dfs(i) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* 계단 오르기: 탐색 */\nfunction climbingStairsDFS(n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.ts<pre><code>/* 탐색 */\nfunction dfs(i: number): number {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* 계단 오르기: 탐색 */\nfunction climbingStairsDFS(n: number): number {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.dart<pre><code>/* 탐색 */\nint dfs(int i) {\n  // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n  if (i == 1 || i == 2) return i;\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1) + dfs(i - 2);\n  return count;\n}\n\n/* 계단 오르기: 탐색 */\nint climbingStairsDFS(int n) {\n  return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.rs<pre><code>/* 탐색 */\nfn dfs(i: usize) -&gt; i32 {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1) + dfs(i - 2);\n    count\n}\n\n/* 계단 오르기: 탐색 */\nfn climbing_stairs_dfs(n: usize) -&gt; i32 {\n    dfs(n)\n}\n</code></pre> climbing_stairs_dfs.c<pre><code>/* 탐색 */\nint dfs(int i) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* 계단 오르기: 탐색 */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.kt<pre><code>/* 탐색 */\nfun dfs(i: Int): Int {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2) return i\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1) + dfs(i - 2)\n    return count\n}\n\n/* 계단 오르기: 탐색 */\nfun climbingStairsDFS(n: Int): Int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.rb<pre><code>### 탐색 ###\ndef dfs(i)\n  # dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n  return i if i == 1 || i == 2\n  # dp[i] = dp[i-1] + dp[i-2]\n  dfs(i - 1) + dfs(i - 2)\nend\n\n### 계단 오르기: 탐색 ###\ndef climbing_stairs_dfs(n)\n  dfs(n)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-3)은 완전 탐색으로 만들어지는 재귀 트리다. 문제 \\(dp[n]\\)의 재귀 트리 깊이는 \\(n\\)이고 시간 복잡도는 \\(O(2^n)\\)이다. 지수적 증가는 폭발적이므로 비교적 큰 \\(n\\)을 입력하면 매우 오래 기다려야 한다.</p> <p></p> <p> 그림 14-3   계단 오르기의 재귀 트리 </p> <p>위 그림(14-3)에서 지수 시간 복잡도는 '중복 부분 문제' 때문에 생긴다. 예를 들어 \\(dp[9]\\)는 \\(dp[8]\\)과 \\(dp[7]\\)로, \\(dp[8]\\)은 \\(dp[7]\\)과 \\(dp[6]\\)으로 나뉘므로 두 경우 모두 부분 문제 \\(dp[7]\\)을 포함한다.</p> <p>그 아래의 부분 문제에도 더 작은 중복 부분 문제가 계속 생겨난다. 계산 자원의 대부분은 이러한 중복 부분 문제에 낭비된다.</p>","path":["제 14 장   동적 계획법","14.1   동적 계획법 첫걸음"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1412-2","level":2,"title":"14.1.2   방법 2: 메모이제이션을 이용한 탐색","text":"<p>효율을 높이려면 모든 중복 부분 문제를 한 번씩만 계산해야 한다. 이를 위해 각 부분 문제의 해를 기록하는 배열 <code>mem</code>을 만들고, 탐색 중 중복 부분 문제를 가지치기한다.</p> <ol> <li>\\(dp[i]\\)를 처음 계산하면 나중에 사용할 수 있도록 <code>mem[i]</code>에 기록한다.</li> <li>\\(dp[i]\\)가 다시 필요하면 <code>mem[i]</code>에서 결과를 바로 가져와 중복 계산을 피한다.</li> </ol> <p>코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dfs_mem.py<pre><code>def dfs(i: int, mem: list[int]) -&gt; int:\n    \"\"\"메모이제이션을 이용한 탐색\"\"\"\n    # dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 or i == 2:\n        return i\n    # dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if mem[i] != -1:\n        return mem[i]\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    # dp[i]를 기록한다\n    mem[i] = count\n    return count\n\ndef climbing_stairs_dfs_mem(n: int) -&gt; int:\n    \"\"\"계단 오르기: 메모이제이션을 이용한 탐색\"\"\"\n    # mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    mem = [-1] * (n + 1)\n    return dfs(n, mem)\n</code></pre> climbing_stairs_dfs_mem.cpp<pre><code>/* 메모이제이션을 이용한 탐색 */\nint dfs(int i, vector&lt;int&gt; &amp;mem) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // dp[i]를 기록한다\n    mem[i] = count;\n    return count;\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nint climbingStairsDFSMem(int n) {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    vector&lt;int&gt; mem(n + 1, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.java<pre><code>/* 메모이제이션을 이용한 탐색 */\nint dfs(int i, int[] mem) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // dp[i]를 기록한다\n    mem[i] = count;\n    return count;\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nint climbingStairsDFSMem(int n) {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    int[] mem = new int[n + 1];\n    Arrays.fill(mem, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.cs<pre><code>/* 메모이제이션을 이용한 탐색 */\nint DFS(int i, int[] mem) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1, mem) + DFS(i - 2, mem);\n    // dp[i]를 기록한다\n    mem[i] = count;\n    return count;\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nint ClimbingStairsDFSMem(int n) {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    int[] mem = new int[n + 1];\n    Array.Fill(mem, -1);\n    return DFS(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.go<pre><code>/* 메모이제이션을 이용한 탐색 */\nfunc dfsMem(i int, mem []int) int {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfsMem(i-1, mem) + dfsMem(i-2, mem)\n    // dp[i]를 기록한다\n    mem[i] = count\n    return count\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nfunc climbingStairsDFSMem(n int) int {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    mem := make([]int, n+1)\n    for i := range mem {\n        mem[i] = -1\n    }\n    return dfsMem(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.swift<pre><code>/* 메모이제이션을 이용한 탐색 */\nfunc dfs(i: Int, mem: inout [Int]) -&gt; Int {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1, mem: &amp;mem) + dfs(i: i - 2, mem: &amp;mem)\n    // dp[i]를 기록한다\n    mem[i] = count\n    return count\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nfunc climbingStairsDFSMem(n: Int) -&gt; Int {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    var mem = Array(repeating: -1, count: n + 1)\n    return dfs(i: n, mem: &amp;mem)\n}\n</code></pre> climbing_stairs_dfs_mem.js<pre><code>/* 메모이제이션을 이용한 탐색 */\nfunction dfs(i, mem) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i === 1 || i === 2) return i;\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // dp[i]를 기록한다\n    mem[i] = count;\n    return count;\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nfunction climbingStairsDFSMem(n) {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.ts<pre><code>/* 메모이제이션을 이용한 탐색 */\nfunction dfs(i: number, mem: number[]): number {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i === 1 || i === 2) return i;\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // dp[i]를 기록한다\n    mem[i] = count;\n    return count;\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nfunction climbingStairsDFSMem(n: number): number {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.dart<pre><code>/* 메모이제이션을 이용한 탐색 */\nint dfs(int i, List&lt;int&gt; mem) {\n  // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n  if (i == 1 || i == 2) return i;\n  // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n  if (mem[i] != -1) return mem[i];\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n  // dp[i]를 기록한다\n  mem[i] = count;\n  return count;\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nint climbingStairsDFSMem(int n) {\n  // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n  List&lt;int&gt; mem = List.filled(n + 1, -1);\n  return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.rs<pre><code>/* 메모이제이션을 이용한 탐색 */\nfn dfs(i: usize, mem: &amp;mut [i32]) -&gt; i32 {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if i == 1 || i == 2 {\n        return i as i32;\n    }\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if mem[i] != -1 {\n        return mem[i];\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // dp[i]를 기록한다\n    mem[i] = count;\n    count\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nfn climbing_stairs_dfs_mem(n: usize) -&gt; i32 {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    let mut mem = vec![-1; n + 1];\n    dfs(n, &amp;mut mem)\n}\n</code></pre> climbing_stairs_dfs_mem.c<pre><code>/* 메모이제이션을 이용한 탐색 */\nint dfs(int i, int *mem) {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // dp[i]를 기록한다\n    mem[i] = count;\n    return count;\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nint climbingStairsDFSMem(int n) {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    int *mem = (int *)malloc((n + 1) * sizeof(int));\n    for (int i = 0; i &lt;= n; i++) {\n        mem[i] = -1;\n    }\n    int result = dfs(n, mem);\n    free(mem);\n    return result;\n}\n</code></pre> climbing_stairs_dfs_mem.kt<pre><code>/* 메모이제이션을 이용한 탐색 */\nfun dfs(i: Int, mem: IntArray): Int {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if (i == 1 || i == 2) return i\n    // dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n    if (mem[i] != -1) return mem[i]\n    // dp[i] = dp[i-1] + dp[i-2]\n    val count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    // dp[i]를 기록한다\n    mem[i] = count\n    return count\n}\n\n/* 계단 오르기: 메모이제이션을 이용한 탐색 */\nfun climbingStairsDFSMem(n: Int): Int {\n    // mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n    val mem = IntArray(n + 1)\n    mem.fill(-1)\n    return dfs(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.rb<pre><code>### 메모이제이션을 이용한 탐색 ###\ndef dfs(i, mem)\n  # dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n  return i if i == 1 || i == 2\n  # dp[i]의 기록이 있으면 해당 값을 바로 반환한다\n  return mem[i] if mem[i] != -1\n\n  # dp[i] = dp[i-1] + dp[i-2]\n  count = dfs(i - 1, mem) + dfs(i - 2, mem)\n  # dp[i]를 기록한다\n  mem[i] = count\nend\n\n### 계단 오르기: 메모이제이션을 이용한 탐색 ###\ndef climbing_stairs_dfs_mem(n)\n  # mem[i]는 i번째 계단까지 오르는 전체 방법의 수를 기록하며, -1은 기록이 없음을 나타낸다\n  mem = Array.new(n + 1, -1)\n  dfs(n, mem)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-4)과 같이 메모이제이션을 적용하면 모든 중복 부분 문제를 한 번만 계산하므로 시간 복잡도가 \\(O(n)\\)으로 개선된다. 이는 매우 큰 발전이다.</p> <p></p> <p> 그림 14-4   메모이제이션을 이용한 탐색의 재귀 트리 </p>","path":["제 14 장   동적 계획법","14.1   동적 계획법 첫걸음"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1413-3","level":2,"title":"14.1.3   방법 3: 동적 계획법","text":"<p>메모이제이션을 이용한 탐색은 '위에서 아래로' 진행하는 방법이다. 원래 문제(루트 노드)에서 시작하여 큰 부분 문제를 작은 부분 문제로 재귀적으로 나누고, 해를 이미 아는 가장 작은 부분 문제(리프 노드)에 도달한다. 이후 되돌아오면서 각 단계의 해를 모아 원래 문제의 해를 구성한다.</p> <p>반대로 동적 계획법은 '아래에서 위로' 진행하는 방법이다. 가장 작은 부분 문제의 해에서 시작하여 반복적으로 더 큰 부분 문제의 해를 구성하고, 마침내 원래 문제의 해를 얻는다.</p> <p>동적 계획법에는 되돌아오는 과정이 없으므로 재귀 없이 반복문만으로 구현할 수 있다. 아래 코드에서는 부분 문제의 해를 저장하는 배열 <code>dp</code>를 초기화한다. 이는 메모이제이션을 이용한 탐색의 배열 <code>mem</code>과 같은 기록 역할을 한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dp.py<pre><code>def climbing_stairs_dp(n: int) -&gt; int:\n    \"\"\"계단 오르기: 동적 계획법\"\"\"\n    if n == 1 or n == 2:\n        return n\n    # 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    dp = [0] * (n + 1)\n    # 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1], dp[2] = 1, 2\n    # 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in range(3, n + 1):\n        dp[i] = dp[i - 1] + dp[i - 2]\n    return dp[n]\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* 계단 오르기: 동적 계획법 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    vector&lt;int&gt; dp(n + 1);\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1;\n    dp[2] = 2;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* 계단 오르기: 동적 계획법 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int[] dp = new int[n + 1];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1;\n    dp[2] = 2;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* 계단 오르기: 동적 계획법 */\nint ClimbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int[] dp = new int[n + 1];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1;\n    dp[2] = 2;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* 계단 오르기: 동적 계획법 */\nfunc climbingStairsDP(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    dp := make([]int, n+1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1\n    dp[2] = 2\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = dp[i-1] + dp[i-2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* 계단 오르기: 동적 계획법 */\nfunc climbingStairsDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    var dp = Array(repeating: 0, count: n + 1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1\n    dp[2] = 2\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in 3 ... n {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* 계단 오르기: 동적 계획법 */\nfunction climbingStairsDP(n) {\n    if (n === 1 || n === 2) return n;\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    const dp = new Array(n + 1).fill(-1);\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1;\n    dp[2] = 2;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* 계단 오르기: 동적 계획법 */\nfunction climbingStairsDP(n: number): number {\n    if (n === 1 || n === 2) return n;\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    const dp = new Array(n + 1).fill(-1);\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1;\n    dp[2] = 2;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* 계단 오르기: 동적 계획법 */\nint climbingStairsDP(int n) {\n  if (n == 1 || n == 2) return n;\n  // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n  dp[1] = 1;\n  dp[2] = 2;\n  // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = dp[i - 1] + dp[i - 2];\n  }\n  return dp[n];\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* 계단 오르기: 동적 계획법 */\nfn climbing_stairs_dp(n: usize) -&gt; i32 {\n    // dp[1]과 dp[2]는 이미 알고 있으므로 해당 값을 반환한다\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    let mut dp = vec![-1; n + 1];\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1;\n    dp[2] = 2;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i in 3..=n {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    dp[n]\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* 계단 오르기: 동적 계획법 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    int *dp = (int *)malloc((n + 1) * sizeof(int));\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1;\n    dp[2] = 2;\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    int result = dp[n];\n    free(dp);\n    return result;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* 계단 오르기: 동적 계획법 */\nfun climbingStairsDP(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    // 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n    val dp = IntArray(n + 1)\n    // 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n    dp[1] = 1\n    dp[2] = 2\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for (i in 3..n) {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>### 계단 오르기: 동적 계획법 ###\ndef climbing_stairs_dp(n)\n  return n  if n == 1 || n == 2\n\n  # 부분 문제의 해를 저장할 dp 테이블을 초기화한다\n  dp = Array.new(n + 1, 0)\n  # 초기 상태: 가장 작은 부분 문제의 해를 미리 설정한다\n  dp[1], dp[2] = 1, 2\n  # 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n  (3...(n + 1)).each { |i| dp[i] = dp[i - 1] + dp[i - 2] }\n\n  dp[n]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-5)은 위 코드의 실행 과정을 보여 준다.</p> <p></p> <p> 그림 14-5   계단 오르기의 동적 계획법 과정 </p> <p>백트래킹과 마찬가지로 동적 계획법도 '상태'로 문제 해결의 특정 단계를 나타낸다. 각 상태는 부분 문제 하나와 그에 해당하는 국소 최적해에 대응한다. 예를 들어 계단 오르기 문제의 상태는 현재 위치한 계단의 칸 번호 \\(i\\)로 정의한다.</p> <p>위 내용을 바탕으로 동적 계획법의 주요 용어를 정리할 수 있다.</p> <ul> <li>배열 <code>dp</code>를 dp 테이블이라고 하며, \\(dp[i]\\)는 상태 \\(i\\)에 대응하는 부분 문제의 해다.</li> <li>가장 작은 부분 문제에 해당하는 상태(계단의 \\(1\\), \\(2\\) 번째 칸)를 초기 상태라고 한다.</li> <li>점화식 \\(dp[i] = dp[i-1] + dp[i-2]\\)를 상태 전이 식이라고 한다.</li> </ul>","path":["제 14 장   동적 계획법","14.1   동적 계획법 첫걸음"],"tags":[]},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1414","level":2,"title":"14.1.4   공간 최적화","text":"<p>주의 깊게 읽었다면 \\(dp[i]\\)는 \\(dp[i-1]\\)과 \\(dp[i-2]\\)에만 의존하므로 배열 <code>dp</code>에 모든 부분 문제의 해를 저장할 필요가 없음을 알아차렸을 것이다. 변수 두 개를 차례로 갱신하며 진행하면 된다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby climbing_stairs_dp.py<pre><code>def climbing_stairs_dp_comp(n: int) -&gt; int:\n    \"\"\"계단 오르기: 공간을 최적화한 동적 계획법\"\"\"\n    if n == 1 or n == 2:\n        return n\n    a, b = 1, 2\n    for _ in range(3, n + 1):\n        a, b = b, a + b\n    return b\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nint ClimbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nfunc climbingStairsDPComp(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    a, b := 1, 2\n    // 상태 전이: 작은 부분 문제부터 점차 큰 부분 문제를 푼다\n    for i := 3; i &lt;= n; i++ {\n        a, b = b, a+b\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nfunc climbingStairsDPComp(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    var a = 1\n    var b = 2\n    for _ in 3 ... n {\n        (a, b) = (b, a + b)\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nfunction climbingStairsDPComp(n) {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nfunction climbingStairsDPComp(n: number): number {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nint climbingStairsDPComp(int n) {\n  if (n == 1 || n == 2) return n;\n  int a = 1, b = 2;\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = a + b;\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nfn climbing_stairs_dp_comp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 {\n        return n as i32;\n    }\n    let (mut a, mut b) = (1, 2);\n    for _ in 3..=n {\n        let tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    b\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.kt<pre><code>/* 계단 오르기: 공간을 최적화한 동적 계획법 */\nfun climbingStairsDPComp(n: Int): Int {\n    if (n == 1 || n == 2) return n\n    var a = 1\n    var b = 2\n    for (i in 3..n) {\n        val temp = b\n        b += a\n        a = temp\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.rb<pre><code>### 계단 오르기: 공간을 최적화한 동적 계획법 ###\ndef climbing_stairs_dp_comp(n)\n  return n if n == 1 || n == 2\n\n  a, b = 1, 2\n  (3...(n + 1)).each { a, b = b, a + b }\n\n  b\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>위 코드에서는 배열 <code>dp</code>의 공간을 없앴으므로 공간 복잡도가 \\(O(n)\\)에서 \\(O(1)\\)로 줄어든다.</p> <p>동적 계획법의 현재 상태는 앞선 몇 개의 상태에만 의존하는 경우가 많다. 이때 필요한 상태만 남겨 '차원을 줄이면' 메모리를 절약할 수 있다. 이 공간 최적화 기법을 '롤링 변수' 또는 '롤링 배열'이라고 한다.</p>","path":["제 14 장   동적 계획법","14.1   동적 계획법 첫걸음"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/","level":1,"title":"14.4   0-1 배낭 문제","text":"<p>배낭 문제는 동적 계획법 입문에 매우 좋은 예제이며 가장 흔한 문제 유형이다. 0-1 배낭, 무한 배낭, 개수 제한 배낭 등 여러 변형이 있다.</p> <p>이 절에서는 가장 흔한 0-1 배낭 문제부터 해결한다.</p> <p>문제</p> <p>물건 \\(n\\) 개와 용량이 \\(cap\\)인 배낭이 주어진다. \\(i\\) 번째 물건의 무게는 \\(wgt[i-1]\\), 가치는 \\(val[i-1]\\)이다. 각 물건을 한 번만 선택할 수 있을 때, 배낭 용량 제한 안에 담을 수 있는 물건의 최대 가치를 구하라.</p> <p>아래 그림(14-17)에서 물건 번호 \\(i\\)는 \\(1\\)부터, 배열 인덱스는 \\(0\\)부터 시작하므로 물건 \\(i\\)의 무게와 가치는 각각 \\(wgt[i-1]\\), \\(val[i-1]\\)이다.</p> <p></p> <p> 그림 14-17   0-1 배낭 문제 예시 데이터 </p> <p>0-1 배낭 문제는 \\(n\\) 차례의 결정으로 이루어진 과정으로 볼 수 있다. 각 물건에는 넣지 않기와 넣기라는 두 결정이 있으므로 의사 결정 트리 모형을 만족한다.</p> <p>목표가 '배낭 용량 제한 안에 담을 수 있는 최대 가치'이므로 동적 계획법 문제일 가능성이 높다.</p> <p>1단계: 매 차례의 결정을 생각하고 상태를 정의하여 \\(dp\\) 테이블을 얻는다</p> <p>물건을 넣지 않으면 배낭 용량은 그대로이고, 넣으면 용량이 줄어든다. 따라서 현재 물건 번호 \\(i\\)와 배낭 용량 \\(c\\)로 상태를 정의하고 \\([i, c]\\)라고 한다.</p> <p>상태 \\([i, c]\\)의 부분 문제는 앞의 \\(i\\) 개 물건을 용량 \\(c\\)인 배낭에 담았을 때의 최대 가치이며, 이를 \\(dp[i, c]\\)라고 한다.</p> <p>구할 값은 \\(dp[n, cap]\\)이므로 \\((n+1) \\times (cap+1)\\) 크기의 2차원 \\(dp\\) 테이블이 필요하다.</p> <p>2단계: 최적 부분 구조를 찾아 상태 전이 식을 도출한다</p> <p>물건 \\(i\\)에 대한 결정을 내리면 앞의 \\(i-1\\) 개 물건을 결정하는 부분 문제가 남는다. 다음 두 경우로 나뉜다.</p> <ul> <li>물건 \\(i\\)를 넣지 않음: 용량은 그대로이며 상태는 \\([i-1, c]\\)로 바뀐다.</li> <li>물건 \\(i\\)를 넣음: 용량이 \\(wgt[i-1]\\) 줄고 가치가 \\(val[i-1]\\) 늘며, 상태는 \\([i-1, c-wgt[i-1]]\\)로 바뀐다.</li> </ul> <p>이 분석에서 최적 부분 구조가 드러난다. 최대 가치 \\(dp[i, c]\\)는 물건 \\(i\\)를 넣지 않는 경우와 물건 \\(i\\)를 넣는 경우 중 가치가 더 큰 쪽이다. 상태 전이 식은 다음과 같다.</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1]) \\] <p>현재 물건의 무게 \\(wgt[i - 1]\\)가 남은 용량 \\(c\\)를 넘으면 넣지 않는 선택만 가능하다.</p> <p>3단계: 경계 조건과 상태 전이 순서를 정한다</p> <p>물건이 없거나 배낭 용량이 \\(0\\)이면 최대 가치는 \\(0\\)이다. 즉, 첫 열 \\(dp[i, 0]\\)과 첫 행 \\(dp[0, c]\\)는 모두 \\(0\\)이다.</p> <p>현재 상태 \\([i, c]\\)는 바로 위의 \\([i-1, c]\\)와 왼쪽 위의 \\([i-1, c-wgt[i-1]]\\)에서 전이되므로, 이중 반복문으로 전체 \\(dp\\) 테이블을 정방향으로 순회하면 된다.</p> <p>이 분석을 바탕으로 완전 탐색, 메모이제이션을 이용한 탐색, 동적 계획법 순서로 구현한다.</p>","path":["제 14 장   동적 계획법","14.4   0-1 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#1-1","level":3,"title":"1.   방법 1: 완전 탐색","text":"<p>탐색 코드에는 다음 요소가 포함된다.</p> <ul> <li>재귀 매개변수: 상태 \\([i, c]\\).</li> <li>반환값: 부분 문제의 해 \\(dp[i, c]\\).</li> <li>종료 조건: 물건 번호가 범위를 벗어나 \\(i = 0\\)이 되거나 남은 용량이 \\(0\\)이면 재귀를 종료하고 가치 \\(0\\)을 반환한다.</li> <li>가지치기: 현재 물건의 무게가 남은 배낭 용량보다 크면 넣지 않는 선택만 할 수 있다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dfs(wgt: list[int], val: list[int], i: int, c: int) -&gt; int:\n    \"\"\"0-1 배낭: 완전 탐색\"\"\"\n    # 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 or c == 0:\n        return 0\n    # 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs(wgt, val, i - 1, c)\n    # 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    no = knapsack_dfs(wgt, val, i - 1, c)\n    yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # 두 경우 중 더 큰 가치를 반환한다\n    return max(no, yes)\n</code></pre> knapsack.cpp<pre><code>/* 0-1 배낭: 완전 탐색 */\nint knapsackDFS(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 반환한다\n    return max(no, yes);\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 배낭: 완전 탐색 */\nint knapsackDFS(int[] wgt, int[] val, int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 반환한다\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 배낭: 완전 탐색 */\nint KnapsackDFS(int[] weight, int[] val, int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFS(weight, val, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = KnapsackDFS(weight, val, i - 1, c);\n    int yes = KnapsackDFS(weight, val, i - 1, c - weight[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 반환한다\n    return Math.Max(no, yes);\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 배낭: 완전 탐색 */\nfunc knapsackDFS(wgt, val []int, i, c int) int {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i-1] &gt; c {\n        return knapsackDFS(wgt, val, i-1, c)\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    no := knapsackDFS(wgt, val, i-1, c)\n    yes := knapsackDFS(wgt, val, i-1, c-wgt[i-1]) + val[i-1]\n    // 두 경우 중 더 큰 가치를 반환한다\n    return int(math.Max(float64(no), float64(yes)))\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 배낭: 완전 탐색 */\nfunc knapsackDFS(wgt: [Int], val: [Int], i: Int, c: Int) -&gt; Int {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i - 1] &gt; c {\n        return knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    let no = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    let yes = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // 두 경우 중 더 큰 가치를 반환한다\n    return max(no, yes)\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 배낭: 완전 탐색 */\nfunction knapsackDFS(wgt, val, i, c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 반환한다\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 배낭: 완전 탐색 */\nfunction knapsackDFS(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    i: number,\n    c: number\n): number {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 반환한다\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 배낭: 완전 탐색 */\nint knapsackDFS(List&lt;int&gt; wgt, List&lt;int&gt; val, int i, int c) {\n  // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFS(wgt, val, i - 1, c);\n  }\n  // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n  int no = knapsackDFS(wgt, val, i - 1, c);\n  int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // 두 경우 중 더 큰 가치를 반환한다\n  return max(no, yes);\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 배낭: 완전 탐색 */\nfn knapsack_dfs(wgt: &amp;[i32], val: &amp;[i32], i: usize, c: usize) -&gt; i32 {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs(wgt, val, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    let no = knapsack_dfs(wgt, val, i - 1, c);\n    let yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 반환한다\n    std::cmp::max(no, yes)\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 배낭: 완전 탐색 */\nint knapsackDFS(int wgt[], int val[], int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 반환한다\n    return myMax(no, yes);\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 배낭: 완전 탐색 */\nfun knapsackDFS(\n    wgt: IntArray,\n    _val: IntArray,\n    i: Int,\n    c: Int\n): Int {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, _val, i - 1, c)\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    val no = knapsackDFS(wgt, _val, i - 1, c)\n    val yes = knapsackDFS(wgt, _val, i - 1, c - wgt[i - 1]) + _val[i - 1]\n    // 두 경우 중 더 큰 가치를 반환한다\n    return max(no, yes)\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 배낭: 완전 탐색 ###\ndef knapsack_dfs(wgt, val, i, c)\n  # 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n  return 0 if i == 0 || c == 0\n  # 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n  return knapsack_dfs(wgt, val, i - 1, c) if wgt[i - 1] &gt; c\n  # 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n  no = knapsack_dfs(wgt, val, i - 1, c)\n  yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n  # 두 경우 중 더 큰 가치를 반환한다\n  [no, yes].max\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-18)과 같이 물건마다 선택하지 않기와 선택하기라는 두 탐색 가지가 생기므로 시간 복잡도는 \\(O(2^n)\\)이다.</p> <p>재귀 트리에는 \\(dp[1, 10]\\) 같은 중복 부분 문제가 있음을 쉽게 알 수 있다. 물건이 많고 용량이 크며, 특히 무게가 같은 물건이 많을수록 중복 부분 문제의 수가 크게 늘어난다.</p> <p></p> <p> 그림 14-18   0-1 배낭 문제의 완전 탐색 재귀 트리 </p>","path":["제 14 장   동적 계획법","14.4   0-1 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#2-2","level":3,"title":"2.   방법 2: 메모이제이션을 이용한 탐색","text":"<p>중복 부분 문제를 한 번만 계산하도록 메모이제이션 리스트 <code>mem</code>에 부분 문제의 해를 저장한다. <code>mem[i][c]</code>는 \\(dp[i, c]\\)에 대응한다.</p> <p>메모이제이션을 도입하면 시간 복잡도는 부분 문제의 수에 의해 결정되므로 \\(O(n \\times cap)\\)이다. 구현 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dfs_mem(\n    wgt: list[int], val: list[int], mem: list[list[int]], i: int, c: int\n) -&gt; int:\n    \"\"\"0-1 배낭: 메모이제이션을 이용한 탐색\"\"\"\n    # 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 or c == 0:\n        return 0\n    # 기록이 이미 있으면 바로 반환한다\n    if mem[i][c] != -1:\n        return mem[i][c]\n    # 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    # 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nint knapsackDFSMem(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nint knapsackDFSMem(int[] wgt, int[] val, int[][] mem, int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nint KnapsackDFSMem(int[] weight, int[] val, int[][] mem, int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFSMem(weight, val, mem, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = KnapsackDFSMem(weight, val, mem, i - 1, c);\n    int yes = KnapsackDFSMem(weight, val, mem, i - 1, c - weight[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = Math.Max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nfunc knapsackDFSMem(wgt, val []int, mem [][]int, i, c int) int {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i-1] &gt; c {\n        return knapsackDFSMem(wgt, val, mem, i-1, c)\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    no := knapsackDFSMem(wgt, val, mem, i-1, c)\n    yes := knapsackDFSMem(wgt, val, mem, i-1, c-wgt[i-1]) + val[i-1]\n    // 두 경우 중 더 큰 가치를 반환한다\n    mem[i][c] = int(math.Max(float64(no), float64(yes)))\n    return mem[i][c]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nfunc knapsackDFSMem(wgt: [Int], val: [Int], mem: inout [[Int]], i: Int, c: Int) -&gt; Int {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i - 1] &gt; c {\n        return knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    let no = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    let yes = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nfunction knapsackDFSMem(wgt, val, mem, i, c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nfunction knapsackDFSMem(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    c: number\n): number {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nint knapsackDFSMem(\n  List&lt;int&gt; wgt,\n  List&lt;int&gt; val,\n  List&lt;List&lt;int&gt;&gt; mem,\n  int i,\n  int c,\n) {\n  // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // 기록이 이미 있으면 바로 반환한다\n  if (mem[i][c] != -1) {\n    return mem[i][c];\n  }\n  // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFSMem(wgt, val, mem, i - 1, c);\n  }\n  // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n  int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n  int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // 두 경우 중 더 큰 가치를 기록하고 반환한다\n  mem[i][c] = max(no, yes);\n  return mem[i][c];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nfn knapsack_dfs_mem(wgt: &amp;[i32], val: &amp;[i32], mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: usize, c: usize) -&gt; i32 {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if mem[i][c] != -1 {\n        return mem[i][c];\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    let no = knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    let yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = std::cmp::max(no, yes);\n    mem[i][c]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nint knapsackDFSMem(int wgt[], int val[], int memCols, int **mem, int i, int c) {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    int no = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = myMax(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 배낭: 메모이제이션을 이용한 탐색 */\nfun knapsackDFSMem(\n    wgt: IntArray,\n    _val: IntArray,\n    mem: Array&lt;IntArray&gt;,\n    i: Int,\n    c: Int\n): Int {\n    // 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n    if (i == 0 || c == 0) {\n        return 0\n    }\n    // 기록이 이미 있으면 바로 반환한다\n    if (mem[i][c] != -1) {\n        return mem[i][c]\n    }\n    // 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, _val, mem, i - 1, c)\n    }\n    // 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n    val no = knapsackDFSMem(wgt, _val, mem, i - 1, c)\n    val yes = knapsackDFSMem(wgt, _val, mem, i - 1, c - wgt[i - 1]) + _val[i - 1]\n    // 두 경우 중 더 큰 가치를 기록하고 반환한다\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 배낭: 메모이제이션을 이용한 탐색 ###\ndef knapsack_dfs_mem(wgt, val, mem, i, c)\n  # 모든 물건의 선택을 마쳤거나 배낭에 남은 용량이 없으면 가치 0을 반환한다\n  return 0 if i == 0 || c == 0\n  # 기록이 이미 있으면 바로 반환한다\n  return mem[i][c] if mem[i][c] != -1\n  # 배낭 용량을 초과하면 배낭에 넣지 않는 선택만 가능하다\n  return knapsack_dfs_mem(wgt, val, mem, i - 1, c) if wgt[i - 1] &gt; c\n  # 물건 i를 넣지 않는 경우와 넣는 경우의 최대 가치를 계산한다\n  no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n  yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n  # 두 경우 중 더 큰 가치를 기록하고 반환한다\n  mem[i][c] = [no, yes].max\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-19)은 메모이제이션을 이용한 탐색에서 잘라 낸 가지들을 보여 준다.</p> <p></p> <p> 그림 14-19   0-1 배낭 문제의 메모이제이션 탐색 재귀 트리 </p>","path":["제 14 장   동적 계획법","14.4   0-1 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#3-3","level":3,"title":"3.   방법 3: 동적 계획법","text":"<p>동적 계획법은 본질적으로 상태 전이를 통해 \\(dp\\) 테이블을 채우는 과정이다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 배낭: 동적 계획법\"\"\"\n    n = len(wgt)\n    # dp 테이블을 초기화한다\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # 상태 전이\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 배낭: 동적 계획법 */\nint knapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // dp 테이블을 초기화한다\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 배낭: 동적 계획법 */\nint knapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // dp 테이블을 초기화한다\n    int[][] dp = new int[n + 1][cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 배낭: 동적 계획법 */\nint KnapsackDP(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // dp 테이블을 초기화한다\n    int[,] dp = new int[n + 1, cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (weight[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i, c] = Math.Max(dp[i - 1, c - weight[i - 1]] + val[i - 1], dp[i - 1, c]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 배낭: 동적 계획법 */\nfunc knapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // dp 테이블을 초기화한다\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // 상태 전이\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i-1][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 배낭: 동적 계획법 */\nfunc knapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // 상태 전이\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 배낭: 동적 계획법 */\nfunction knapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array(n + 1)\n        .fill(0)\n        .map(() =&gt; Array(cap + 1).fill(0));\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 배낭: 동적 계획법 */\nfunction knapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 배낭: 동적 계획법 */\nint knapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // dp 테이블을 초기화한다\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // 상태 전이\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 배낭: 동적 계획법 */\nfn knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // dp 테이블을 초기화한다\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // 상태 전이\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = std::cmp::max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1] as usize] + val[i - 1],\n                );\n            }\n        }\n    }\n    dp[n][cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 배낭: 동적 계획법 */\nint knapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // dp 테이블을 초기화한다\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = myMax(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // 메모리 해제\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 배낭: 동적 계획법 */\nfun knapsackDP(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // dp 테이블을 초기화한다\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // 상태 전이\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 배낭: 동적 계획법 ###\ndef knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # 상태 전이\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[i][c] = dp[i - 1][c]\n      else\n        # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[i][c] = [dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-20)과 같이 시간 복잡도와 공간 복잡도는 모두 배열 <code>dp</code>의 크기인 \\(O(n \\times cap)\\)이다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 14-20   0-1 배낭 문제의 동적 계획법 과정 </p>","path":["제 14 장   동적 계획법","14.4   0-1 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/knapsack_problem/#4","level":3,"title":"4.   공간 최적화","text":"<p>각 상태는 바로 윗행의 상태에만 의존하므로 두 배열을 번갈아 갱신하면 공간 복잡도를 \\(O(n \\times cap)\\)에서 \\(O(cap)\\)으로 줄일 수 있다.</p> <p>배열 하나만으로도 가능할까? 각 상태는 바로 위 또는 왼쪽 위 칸에서 전이된다. 배열 하나를 쓴다면 \\(i\\) 번째 행의 순회를 시작할 때 그 배열에는 아직 \\(i-1\\) 번째 행의 상태가 저장되어 있다.</p> <ul> <li>정방향으로 순회하면 \\(dp[i, j]\\)에 도달했을 때 왼쪽 위 \\(dp[i-1, 1]\\) ~ \\(dp[i-1, j-1]\\)의 값이 이미 덮어써졌을 수 있어 올바른 상태 전이를 할 수 없다.</li> <li>역방향으로 순회하면 값이 덮어써지는 문제가 없어 상태 전이를 올바르게 수행할 수 있다.</li> </ul> <p>아래 그림(14-21)은 배열 하나로 \\(i = 1\\) 행에서 \\(i = 2\\) 행으로 바꾸는 과정이다. 정방향과 역방향 순회의 차이를 생각해 보자.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 14-21   공간 최적화 후 0-1 배낭의 동적 계획법 과정 </p> <p>구현에서는 배열 <code>dp</code>의 첫 번째 차원 \\(i\\)를 없애고 안쪽 반복문을 역방향 순회로 바꾸기만 하면 된다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby knapsack.py<pre><code>def knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 배낭: 공간을 최적화한 동적 계획법\"\"\"\n    n = len(wgt)\n    # dp 테이블을 초기화한다\n    dp = [0] * (cap + 1)\n    # 상태 전이\n    for i in range(1, n + 1):\n        # 역방향 순회\n        for c in range(cap, 0, -1):\n            if wgt[i - 1] &gt; c:\n                # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c]\n            else:\n                # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nint knapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // dp 테이블을 초기화한다\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        // 역방향 순회\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nint knapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        // 역방향 순회\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nint KnapsackDPComp(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        // 역방향 순회\n        for (int c = cap; c &gt; 0; c--) {\n            if (weight[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.Max(dp[c], dp[c - weight[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nfunc knapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // dp 테이블을 초기화한다\n    dp := make([]int, cap+1)\n    // 상태 전이\n    for i := 1; i &lt;= n; i++ {\n        // 역방향 순회\n        for c := cap; c &gt;= 1; c-- {\n            if wgt[i-1] &lt;= c {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nfunc knapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: 0, count: cap + 1)\n    // 상태 전이\n    for i in 1 ... n {\n        // 역방향 순회\n        for c in (1 ... cap).reversed() {\n            if wgt[i - 1] &lt;= c {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nfunction knapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array(cap + 1).fill(0);\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        // 역방향 순회\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nfunction knapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array(cap + 1).fill(0);\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        // 역방향 순회\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nint knapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // dp 테이블을 초기화한다\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // 상태 전이\n  for (int i = 1; i &lt;= n; i++) {\n    // 역방향 순회\n    for (int c = cap; c &gt;= 1; c--) {\n      if (wgt[i - 1] &lt;= c) {\n        // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nfn knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // dp 테이블을 초기화한다\n    let mut dp = vec![0; cap + 1];\n    // 상태 전이\n    for i in 1..=n {\n        // 역방향 순회\n        for c in (1..=cap).rev() {\n            if wgt[i - 1] &lt;= c as i32 {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nint knapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // dp 테이블을 초기화한다\n    int *dp = calloc(cap + 1, sizeof(int));\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        // 역방향 순회\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // 메모리 해제\n    free(dp);\n    return res;\n}\n</code></pre> knapsack.kt<pre><code>/* 0-1 배낭: 공간을 최적화한 동적 계획법 */\nfun knapsackDPComp(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // dp 테이블을 초기화한다\n    val dp = IntArray(cap + 1)\n    // 상태 전이\n    for (i in 1..n) {\n        // 역방향 순회\n        for (c in cap downTo 1) {\n            if (wgt[i - 1] &lt;= c) {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.rb<pre><code>### 0-1 배낭: 공간을 최적화한 동적 계획법 ###\ndef knapsack_dp_comp(wgt, val, cap)\n  n = wgt.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(cap + 1, 0)\n  # 상태 전이\n  for i in 1...(n + 1)\n    # 역방향 순회\n    for c in cap.downto(1)\n      if wgt[i - 1] &gt; c\n        # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[c] = dp[c]\n      else\n        # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[c] = [dp[c], dp[c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[cap]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.4   0-1 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/summary/","level":1,"title":"14.7   요약","text":"","path":["제 14 장   동적 계획법","14.7   요약"],"tags":[]},{"location":"chapter_dynamic_programming/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>동적 계획법은 문제를 나누고 부분 문제의 해를 저장하여 중복 계산을 피함으로써 계산 효율을 높인다.</li> <li>시간을 고려하지 않는다면 모든 동적 계획법 문제는 백트래킹(완전 탐색)으로 풀 수 있다. 하지만 재귀 트리에 중복 부분 문제가 매우 많아 비효율적이다. 메모이제이션 리스트에 계산한 해를 저장하면 중복 부분 문제를 한 번만 계산하도록 보장할 수 있다.</li> <li>메모이제이션을 이용한 탐색은 위에서 아래로 진행하는 재귀 풀이이며, 이에 대응하는 동적 계획법은 '표를 채우듯' 아래에서 위로 진행하는 반복 풀이이다. 현재 상태는 일부 주변 상태에만 의존하므로 \\(dp\\) 테이블의 한 차원을 없애 공간 복잡도를 줄일 수 있다.</li> <li>부분 문제로 나누는 것은 일반적인 알고리즘 아이디어이며, 분할 정복, 동적 계획법, 백트래킹에서 서로 다른 특성을 가진다.</li> <li>동적 계획법 문제의 세 가지 주요 특성은 중복 부분 문제, 최적 부분 구조, 무후효성이다.</li> <li>원래 문제의 최적해를 부분 문제의 최적해로 구성할 수 있다면 최적 부분 구조를 가진다.</li> <li>무후효성이란 한 상태의 이후 전개가 현재 상태에만 의존하고 과거의 모든 상태와는 무관하다는 뜻이다. 많은 조합 최적화 문제는 무후효성을 갖지 않아 동적 계획법으로 빠르게 풀 수 없다.</li> </ul> <p>배낭 문제</p> <ul> <li>배낭 문제는 가장 대표적인 동적 계획법 문제 중 하나로, 0-1 배낭, 무한 배낭, 개수 제한 배낭 등의 변형이 있다.</li> <li>0-1 배낭의 상태는 앞 \\(i\\) 개 물건을 용량 \\(c\\)인 배낭에 담을 때의 최대 가치로 정의한다. 넣지 않기와 넣기라는 두 결정으로 최적 부분 구조와 상태 전이 식을 얻는다. 공간 최적화에서는 바로 위와 왼쪽 위 상태에 의존하므로, 왼쪽 위 값이 덮어써지지 않도록 리스트를 역순으로 순회해야 한다.</li> <li>무한 배낭은 각 물건의 선택 횟수에 제한이 없어 물건을 넣는 상태 전이가 0-1 배낭과 다르다. 바로 위와 바로 왼쪽 상태에 의존하므로 공간 최적화에서는 정방향 순회를 해야 한다.</li> <li>동전 교환은 무한 배낭의 변형이다. '최대' 가치 대신 '최소' 동전 수를 구하므로 상태 전이 식의 \\(\\max()\\)를 \\(\\min()\\)으로 바꾼다. 용량을 '넘지 않는' 것이 아니라 목표 금액을 '정확히' 만들어야 하므로, 만들 수 없는 유효하지 않은 해를 \\(amt + 1\\)로 나타낸다.</li> <li>동전 교환 II는 '최소 동전 수' 대신 '동전 조합 수'를 구하므로 상태 전이 식의 \\(\\min()\\)을 합 연산으로 바꾼다.</li> </ul> <p>편집 거리 문제</p> <ul> <li>편집 거리(Levenshtein 거리)는 두 문자열의 유사도를 측정하며, 한 문자열을 다른 문자열로 바꾸는 최소 편집 횟수로 정의된다. 편집 연산에는 추가, 삭제, 교체가 있다.</li> <li>편집 거리의 상태는 \\(s\\)의 앞 \\(i\\) 개 문자를 \\(t\\)의 앞 \\(j\\) 개 문자로 바꾸는 최소 편집 횟수로 정의한다. \\(s[i-1] \\ne t[j-1]\\)이면 추가, 삭제, 교체의 세 결정이 있고 각각 남는 부분 문제가 있다. 이를 바탕으로 최적 부분 구조와 상태 전이 식을 구한다. \\(s[i-1] = t[j-1]\\)이면 현재 문자를 편집할 필요가 없다.</li> <li>편집 거리의 상태는 바로 위, 바로 왼쪽, 왼쪽 위 상태에 의존하므로 공간 최적화 후에는 정방향이나 역방향 순회만으로 올바른 전이를 할 수 없다. 변수를 하나 두어 왼쪽 위 상태를 임시 저장하면 무한 배낭과 같은 상황이 되어 정방향 순회가 가능하다.</li> </ul>","path":["제 14 장   동적 계획법","14.7   요약"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/","level":1,"title":"14.5   무한 배낭 문제","text":"<p>이 절에서는 또 다른 대표적인 배낭 문제인 무한 배낭을 먼저 풀고, 그 특수한 경우인 동전 교환을 알아본다.</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1451","level":2,"title":"14.5.1   무한 배낭 문제","text":"<p>문제</p> <p>물건 \\(n\\) 개와 용량이 \\(cap\\)인 배낭이 주어진다. \\(i\\) 번째 물건의 무게는 \\(wgt[i-1]\\), 가치는 \\(val[i-1]\\)이다. 각 물건을 반복해서 선택할 수 있을 때, 용량 제한 안에 담을 수 있는 최대 가치를 구하라. 예시는 아래 그림(14-22)과 같다.</p> <p></p> <p> 그림 14-22   무한 배낭 문제 예시 데이터 </p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1","level":3,"title":"1.   동적 계획법 접근","text":"<p>무한 배낭 문제는 0-1 배낭 문제와 매우 비슷하며, 물건을 선택하는 횟수에 제한이 없다는 점만 다르다.</p> <ul> <li>0-1 배낭에서는 종류별 물건이 하나뿐이므로 물건 \\(i\\)를 넣은 뒤에는 앞의 \\(i-1\\) 개 물건에서만 선택할 수 있다.</li> <li>무한 배낭에서는 각 종류의 물건이 무한히 많으므로 물건 \\(i\\)를 넣은 뒤에도 앞의 \\(i\\) 개 물건에서 계속 선택할 수 있다.</li> </ul> <p>무한 배낭의 규칙에 따라 상태 \\([i, c]\\)의 변화는 두 경우로 나뉜다.</p> <ul> <li>물건 \\(i\\)를 넣지 않음: 0-1 배낭과 동일하게 \\([i-1, c]\\)로 전이한다.</li> <li>물건 \\(i\\)를 넣음: 0-1 배낭과 달리 \\([i, c-wgt[i-1]]\\)로 전이한다.</li> </ul> <p>따라서 상태 전이 식은 다음과 같이 바뀐다.</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1]) \\]","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2","level":3,"title":"2.   코드 구현","text":"<p>두 문제의 코드를 비교하면 상태 전이의 한 곳이 \\(i-1\\)에서 \\(i\\)로 바뀌었을 뿐 나머지는 완전히 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"무한 배낭: 동적 계획법\"\"\"\n    n = len(wgt)\n    # dp 테이블을 초기화한다\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # 상태 전이\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* 무한 배낭: 동적 계획법 */\nint unboundedKnapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // dp 테이블을 초기화한다\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* 무한 배낭: 동적 계획법 */\nint unboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // dp 테이블을 초기화한다\n    int[][] dp = new int[n + 1][cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* 무한 배낭: 동적 계획법 */\nint UnboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // dp 테이블을 초기화한다\n    int[,] dp = new int[n + 1, cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i, c] = Math.Max(dp[i - 1, c], dp[i, c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* 무한 배낭: 동적 계획법 */\nfunc unboundedKnapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // dp 테이블을 초기화한다\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // 상태 전이\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* 무한 배낭: 동적 계획법 */\nfunc unboundedKnapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // 상태 전이\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* 무한 배낭: 동적 계획법 */\nfunction unboundedKnapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* 무한 배낭: 동적 계획법 */\nfunction unboundedKnapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* 무한 배낭: 동적 계획법 */\nint unboundedKnapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // dp 테이블을 초기화한다\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // 상태 전이\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* 무한 배낭: 동적 계획법 */\nfn unbounded_knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // dp 테이블을 초기화한다\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // 상태 전이\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i][c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* 무한 배낭: 동적 계획법 */\nint unboundedKnapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // dp 테이블을 초기화한다\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = myMax(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // 메모리 해제\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* 무한 배낭: 동적 계획법 */\nfun unboundedKnapsackDP(wgt: IntArray, _val: IntArray, cap: Int): Int {\n    val n = wgt.size\n    // dp 테이블을 초기화한다\n    val dp = Array(n + 1) { IntArray(cap + 1) }\n    // 상태 전이\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>### 무한 배낭: 동적 계획법 ###\ndef unbounded_knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # 상태 전이\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[i][c] = dp[i - 1][c]\n      else\n        # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[i][c] = [dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3","level":3,"title":"3.   공간 최적화","text":"<p>현재 상태는 왼쪽과 위쪽 상태에서 전이되므로 공간 최적화 후에는 \\(dp\\) 테이블의 각 행을 정방향으로 순회해야 한다.</p> <p>이는 0-1 배낭의 순회 순서와 정반대다. 아래 그림(14-23)으로 차이를 이해해 보자.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 14-23   공간 최적화 후 무한 배낭의 동적 계획법 과정 </p> <p>구현은 간단하다. 배열 <code>dp</code>의 첫 번째 차원을 없애면 된다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"무한 배낭: 공간을 최적화한 동적 계획법\"\"\"\n    n = len(wgt)\n    # dp 테이블을 초기화한다\n    dp = [0] * (cap + 1)\n    # 상태 전이\n    for i in range(1, n + 1):\n        # 정방향 순회\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c]\n            else:\n                # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nint unboundedKnapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // dp 테이블을 초기화한다\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nint unboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nint UnboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[cap + 1];\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.Max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nfunc unboundedKnapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // dp 테이블을 초기화한다\n    dp := make([]int, cap+1)\n    // 상태 전이\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nfunc unboundedKnapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: 0, count: cap + 1)\n    // 상태 전이\n    for i in 1 ... n {\n        for c in 1 ... cap {\n            if wgt[i - 1] &gt; c {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nfunction unboundedKnapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nfunction unboundedKnapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nint unboundedKnapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // dp 테이블을 초기화한다\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // 상태 전이\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[c] = dp[c];\n      } else {\n        // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nfn unbounded_knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // dp 테이블을 초기화한다\n    let mut dp = vec![0; cap + 1];\n    // 상태 전이\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nint unboundedKnapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // dp 테이블을 초기화한다\n    int *dp = calloc(cap + 1, sizeof(int));\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c];\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // 메모리 해제\n    free(dp);\n    return res;\n}\n</code></pre> unbounded_knapsack.kt<pre><code>/* 무한 배낭: 공간을 최적화한 동적 계획법 */\nfun unboundedKnapsackDPComp(\n    wgt: IntArray,\n    _val: IntArray,\n    cap: Int\n): Int {\n    val n = wgt.size\n    // dp 테이블을 초기화한다\n    val dp = IntArray(cap + 1)\n    // 상태 전이\n    for (i in 1..n) {\n        for (c in 1..cap) {\n            if (wgt[i - 1] &gt; c) {\n                // 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n                dp[c] = dp[c]\n            } else {\n                // 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + _val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.rb<pre><code>### 무한 배낭: 동적 계획법 ###\ndef unbounded_knapsack_dp(wgt, val, cap)\n  n = wgt.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(n + 1) { Array.new(cap + 1, 0) }\n  # 상태 전이\n  for i in 1...(n + 1)\n    for c in 1...(cap + 1)\n      if wgt[i - 1] &gt; c\n        # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[i][c] = dp[i - 1][c]\n      else\n        # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[i][c] = [dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[n][cap]\nend\n\n### 무한 배낭: 공간을 최적화한 동적 계획법 ##3\ndef unbounded_knapsack_dp_comp(wgt, val, cap)\n  n = wgt.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(cap + 1, 0)\n  # 상태 전이\n  for i in 1...(n + 1)\n    # 정방향 순회\n    for c in 1...(cap + 1)\n      if wgt[i -1] &gt; c\n        # 배낭 용량을 초과하면 물건 i를 선택하지 않는다\n        dp[c] = dp[c]\n      else\n        # 물건 i를 선택하지 않는 경우와 선택하는 경우 중 더 큰 값\n        dp[c] = [dp[c], dp[c - wgt[i - 1]] + val[i - 1]].max\n      end\n    end\n  end\n  dp[cap]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1452","level":2,"title":"14.5.2   동전 교환 문제","text":"<p>배낭 문제는 동적 계획법의 큰 문제군을 대표하며, 동전 교환 같은 여러 변형이 있다.</p> <p>문제</p> <p>동전 \\(n\\) 종류가 주어지며 \\(i\\) 번째 동전의 액면가는 \\(coins[i - 1]\\)이고 목표 금액은 \\(amt\\)다. 각 종류의 동전을 반복해서 선택할 수 있을 때, 목표 금액을 만드는 최소 동전 수를 구하라. 목표 금액을 만들 수 없으면 \\(-1\\)을 반환하라. 예시는 아래 그림(14-24)과 같다.</p> <p></p> <p> 그림 14-24   동전 교환 문제 예시 데이터 </p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_1","level":3,"title":"1.   동적 계획법 접근","text":"<p>동전 교환은 무한 배낭 문제의 특수한 경우로 볼 수 있으며, 다음과 같은 공통점과 차이점이 있다.</p> <ul> <li>두 문제는 서로 변환할 수 있다. '물건'은 '동전', '물건 무게'는 '동전 액면가', '배낭 용량'은 '목표 금액'에 대응한다.</li> <li>최적화 목표가 반대다. 무한 배낭은 물건 가치를 최대화하고, 동전 교환은 동전 수를 최소화한다.</li> <li>무한 배낭은 용량을 '넘지 않는' 해를 구하지만, 동전 교환은 목표 금액을 '정확히' 만드는 해를 구한다.</li> </ul> <p>1단계: 매 차례의 결정을 생각하고 상태를 정의하여 \\(dp\\) 테이블을 얻는다</p> <p>상태 \\([i, a]\\)의 부분 문제는 앞의 \\(i\\) 종류 동전으로 금액 \\(a\\)를 만드는 최소 동전 수이며, 이를 \\(dp[i, a]\\)라고 한다.</p> <p>2차원 \\(dp\\) 테이블의 크기는 \\((n+1) \\times (amt+1)\\)이다.</p> <p>2단계: 최적 부분 구조를 찾아 상태 전이 식을 도출한다</p> <p>이 문제의 상태 전이 식은 무한 배낭과 다음 두 가지가 다르다.</p> <ul> <li>최솟값을 구하므로 연산자 \\(\\max()\\)를 \\(\\min()\\)으로 바꾼다.</li> <li>최적화 대상은 상품 가치가 아니라 동전 수이므로, 동전을 선택하면 \\(+1\\)만 수행한다.</li> </ul> \\[ dp[i, a] = \\min(dp[i-1, a], dp[i, a - coins[i-1]] + 1) \\] <p>3단계: 경계 조건과 상태 전이 순서를 정한다</p> <p>목표 금액이 \\(0\\)이면 최소 동전 수는 \\(0\\)이다. 즉, 첫 열의 모든 \\(dp[i, 0]\\)은 \\(0\\)이다.</p> <p>동전이 없으면 \\(&gt; 0\\)인 어떤 목표 금액도 만들 수 없으므로 유효하지 않은 해다. 상태 전이 식의 \\(\\min()\\)이 이를 알아보고 제외하도록 \\(+ \\infty\\)로 나타낸다. 즉, 첫 행의 모든 \\(dp[0, a]\\)를 \\(+ \\infty\\)로 둔다.</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_1","level":3,"title":"2.   코드 구현","text":"<p>대부분의 프로그래밍 언어는 \\(+ \\infty\\) 변수를 제공하지 않아 정수 타입 <code>int</code>의 최댓값으로 대신해야 한다. 하지만 그러면 상태 전이 식의 \\(+ 1\\) 연산에서 정수 오버플로가 발생할 수 있다.</p> <p>이를 피하기 위해 유효하지 않은 해를 \\(amt + 1\\)로 나타낸다. \\(amt\\)를 만드는 동전 수는 많아도 \\(amt\\) 개이기 때문이다. 마지막에 \\(dp[n, amt]\\)가 \\(amt + 1\\)인지 확인하고, 같으면 목표 금액을 만들 수 없다는 뜻으로 \\(-1\\)을 반환한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change.py<pre><code>def coin_change_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"동전 교환: 동적 계획법\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # dp 테이블을 초기화한다\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # 상태 전이: 첫 번째 행과 첫 번째 열\n    for a in range(1, amt + 1):\n        dp[0][a] = MAX\n    # 상태 전이: 나머지 행과 열\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n    return dp[n][amt] if dp[n][amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* 동전 교환: 동적 계획법 */\nint coinChangeDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* 동전 교환: 동적 계획법 */\nint coinChangeDP(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    int[][] dp = new int[n + 1][amt + 1];\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* 동전 교환: 동적 계획법 */\nint CoinChangeDP(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    int[,] dp = new int[n + 1, amt + 1];\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0, a] = MAX;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i, a] = Math.Min(dp[i - 1, a], dp[i, a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n, amt] != MAX ? dp[n, amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* 동전 교환: 동적 계획법 */\nfunc coinChangeDP(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // dp 테이블을 초기화한다\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for a := 1; a &lt;= amt; a++ {\n        dp[0][a] = max\n    }\n    // 상태 전이: 나머지 행과 열\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = int(math.Min(float64(dp[i-1][a]), float64(dp[i][a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* 동전 교환: 동적 계획법 */\nfunc coinChangeDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for a in 1 ... amt {\n        dp[0][a] = MAX\n    }\n    // 상태 전이: 나머지 행과 열\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* 동전 교환: 동적 계획법 */\nfunction coinChangeDP(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* 동전 교환: 동적 계획법 */\nfunction coinChangeDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* 동전 교환: 동적 계획법 */\nint coinChangeDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // dp 테이블을 초기화한다\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // 상태 전이: 첫 번째 행과 첫 번째 열\n  for (int a = 1; a &lt;= amt; a++) {\n    dp[0][a] = MAX;\n  }\n  // 상태 전이: 나머지 행과 열\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n        dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* 동전 교환: 동적 계획법 */\nfn coin_change_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // dp 테이블을 초기화한다\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for a in 1..=amt {\n        dp[0][a] = max;\n    }\n    // 상태 전이: 나머지 행과 열\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = std::cmp::min(dp[i - 1][a], dp[i][a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* 동전 교환: 동적 계획법 */\nint coinChangeDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // 상태 전이: 나머지 행과 열\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = myMin(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[n][amt] != MAX ? dp[n][amt] : -1;\n    // 메모리 해제\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* 동전 교환: 동적 계획법 */\nfun coinChangeDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // dp 테이블을 초기화한다\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // 상태 전이: 첫 번째 행과 첫 번째 열\n    for (a in 1..amt) {\n        dp[0][a] = MAX\n    }\n    // 상태 전이: 나머지 행과 열\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return if (dp[n][amt] != MAX) dp[n][amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>### 동전 교환: 동적 계획법 ###\ndef coin_change_dp(coins, amt)\n  n = coins.length\n  _MAX = amt + 1\n  # dp 테이블을 초기화한다\n  dp = Array.new(n + 1) { Array.new(amt + 1, 0) }\n  # 상태 전이: 첫 번째 행과 첫 번째 열\n  (1...(amt + 1)).each { |a| dp[0][a] = _MAX }\n  # 상태 전이: 나머지 행과 열\n  for i in 1...(n + 1)\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[i][a] = dp[i - 1][a]\n      else\n        # 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n        dp[i][a] = [dp[i - 1][a], dp[i][a - coins[i - 1]] + 1].min\n      end\n    end\n  end\n  dp[n][amt] != _MAX ? dp[n][amt] : -1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(14-25)의 동전 교환 동적 계획법 과정은 무한 배낭 문제와 매우 비슷하다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 14-25   동전 교환 문제의 동적 계획법 과정 </p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_1","level":3,"title":"3.   공간 최적화","text":"<p>동전 교환의 공간 최적화는 무한 배낭과 같은 방식으로 처리한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change.py<pre><code>def coin_change_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"동전 교환: 공간을 최적화한 동적 계획법\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # dp 테이블을 초기화한다\n    dp = [MAX] * (amt + 1)\n    dp[0] = 0\n    # 상태 전이\n    for i in range(1, n + 1):\n        # 정방향 순회\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            else:\n                # 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n    return dp[amt] if dp[amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nint coinChangeDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    vector&lt;int&gt; dp(amt + 1, MAX);\n    dp[0] = 0;\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nint coinChangeDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[amt + 1];\n    Arrays.fill(dp, MAX);\n    dp[0] = 0;\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nint CoinChangeDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[amt + 1];\n    Array.Fill(dp, MAX);\n    dp[0] = 0;\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = Math.Min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* 동전 교환: 동적 계획법 */\nfunc coinChangeDPComp(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // dp 테이블을 초기화한다\n    dp := make([]int, amt+1)\n    for i := 1; i &lt;= amt; i++ {\n        dp[i] = max\n    }\n    // 상태 전이\n    for i := 1; i &lt;= n; i++ {\n        // 정방향 순회\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = int(math.Min(float64(dp[a]), float64(dp[a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nfunc coinChangeDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: MAX, count: amt + 1)\n    dp[0] = 0\n    // 상태 전이\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nfunction coinChangeDPComp(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nfunction coinChangeDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nint coinChangeDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // dp 테이블을 초기화한다\n  List&lt;int&gt; dp = List.filled(amt + 1, MAX);\n  dp[0] = 0;\n  // 상태 전이\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[a] = dp[a];\n      } else {\n        // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n        dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nfn coin_change_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // dp 테이블을 초기화한다\n    let mut dp = vec![0; amt + 1];\n    dp.fill(max);\n    dp[0] = 0;\n    // 상태 전이\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = std::cmp::min(dp[a], dp[a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt] as i32;\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change.c<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nint coinChangeDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // dp 테이블을 초기화한다\n    int *dp = malloc((amt + 1) * sizeof(int));\n    for (int j = 1; j &lt;= amt; j++) {\n        dp[j] = MAX;\n    } \n    dp[0] = 0;\n\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = myMin(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[amt] != MAX ? dp[amt] : -1;\n    // 메모리 해제\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.kt<pre><code>/* 동전 교환: 공간을 최적화한 동적 계획법 */\nfun coinChangeDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    val MAX = amt + 1\n    // dp 테이블을 초기화한다\n    val dp = IntArray(amt + 1)\n    dp.fill(MAX)\n    dp[0] = 0\n    // 상태 전이\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return if (dp[amt] != MAX) dp[amt] else -1\n}\n</code></pre> coin_change.rb<pre><code>### 동전 교환: 공간을 최적화한 동적 계획법 ###\ndef coin_change_dp_comp(coins, amt)\n  n = coins.length\n  _MAX = amt + 1\n  # dp 테이블을 초기화한다\n  dp = Array.new(amt + 1, _MAX)\n  dp[0] = 0\n  # 상태 전이\n  for i in 1...(n + 1)\n    # 정방향 순회\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[a] = dp[a]\n      else\n        # 동전 i를 선택하지 않는 경우와 선택하는 경우 중 더 작은 값\n        dp[a] = [dp[a], dp[a - coins[i - 1]] + 1].min\n      end\n    end\n  end\n  dp[amt] != _MAX ? dp[amt] : -1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1453-ii","level":2,"title":"14.5.3   동전 교환 문제 II","text":"<p>문제</p> <p>동전 \\(n\\) 종류가 주어지며 \\(i\\) 번째 동전의 액면가는 \\(coins[i - 1]\\)이고 목표 금액은 \\(amt\\)다. 각 종류의 동전을 반복해서 선택할 수 있을 때, 목표 금액을 만드는 동전 조합 수를 구하라. 예시는 아래 그림(14-26)과 같다.</p> <p></p> <p> 그림 14-26   동전 교환 문제 II 예시 데이터 </p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_2","level":3,"title":"1.   동적 계획법 접근","text":"<p>앞 문제와 달리 이번에는 조합 수를 구하므로 부분 문제는 앞의 \\(i\\) 종류 동전으로 금액 \\(a\\)를 만드는 조합 수가 된다. \\(dp\\) 테이블은 여전히 \\((n+1) \\times (amt + 1)\\) 크기의 2차원 행렬이다.</p> <p>현재 상태의 조합 수는 현재 동전을 선택하지 않는 경우와 선택하는 경우의 조합 수를 더한 값이다. 상태 전이 식은 다음과 같다.</p> \\[ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]] \\] <p>목표 금액이 \\(0\\)이면 동전을 하나도 선택하지 않고도 만들 수 있으므로 첫 열의 모든 \\(dp[i, 0]\\)을 \\(1\\)로 초기화한다. 동전이 없으면 \\(&gt;0\\)인 목표 금액을 만들 수 없으므로 첫 행의 모든 \\(dp[0, a]\\)는 \\(0\\)이다.</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_2","level":3,"title":"2.   코드 구현","text":"PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change_ii.py<pre><code>def coin_change_ii_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"동전 교환 II: 동적 계획법\"\"\"\n    n = len(coins)\n    # dp 테이블을 초기화한다\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # 첫 번째 열을 초기화한다\n    for i in range(n + 1):\n        dp[i][0] = 1\n    # 상태 전이\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n    return dp[n][amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* 동전 교환 II: 동적 계획법 */\nint coinChangeIIDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // dp 테이블을 초기화한다\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // 첫 번째 열을 초기화한다\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* 동전 교환 II: 동적 계획법 */\nint coinChangeIIDP(int[] coins, int amt) {\n    int n = coins.length;\n    // dp 테이블을 초기화한다\n    int[][] dp = new int[n + 1][amt + 1];\n    // 첫 번째 열을 초기화한다\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* 동전 교환 II: 동적 계획법 */\nint CoinChangeIIDP(int[] coins, int amt) {\n    int n = coins.Length;\n    // dp 테이블을 초기화한다\n    int[,] dp = new int[n + 1, amt + 1];\n    // 첫 번째 열을 초기화한다\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i, 0] = 1;\n    }\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i, a] = dp[i - 1, a] + dp[i, a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n, amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* 동전 교환 II: 동적 계획법 */\nfunc coinChangeIIDP(coins []int, amt int) int {\n    n := len(coins)\n    // dp 테이블을 초기화한다\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // 첫 번째 열을 초기화한다\n    for i := 0; i &lt;= n; i++ {\n        dp[i][0] = 1\n    }\n    // 상태 전이: 나머지 행과 열\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i-1][a] + dp[i][a-coins[i-1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* 동전 교환 II: 동적 계획법 */\nfunc coinChangeIIDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // 첫 번째 열을 초기화한다\n    for i in 0 ... n {\n        dp[i][0] = 1\n    }\n    // 상태 전이\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* 동전 교환 II: 동적 계획법 */\nfunction coinChangeIIDP(coins, amt) {\n    const n = coins.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // 첫 번째 열을 초기화한다\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* 동전 교환 II: 동적 계획법 */\nfunction coinChangeIIDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // 첫 번째 열을 초기화한다\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* 동전 교환 II: 동적 계획법 */\nint coinChangeIIDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // dp 테이블을 초기화한다\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // 첫 번째 열을 초기화한다\n  for (int i = 0; i &lt;= n; i++) {\n    dp[i][0] = 1;\n  }\n  // 상태 전이\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[n][amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* 동전 교환 II: 동적 계획법 */\nfn coin_change_ii_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // dp 테이블을 초기화한다\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // 첫 번째 열을 초기화한다\n    for i in 0..=n {\n        dp[i][0] = 1;\n    }\n    // 상태 전이\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[n][amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* 동전 교환 II: 동적 계획법 */\nint coinChangeIIDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // dp 테이블을 초기화한다\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // 첫 번째 열을 초기화한다\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[n][amt];\n    // 메모리 해제\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* 동전 교환 II: 동적 계획법 */\nfun coinChangeIIDP(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // dp 테이블을 초기화한다\n    val dp = Array(n + 1) { IntArray(amt + 1) }\n    // 첫 번째 열을 초기화한다\n    for (i in 0..n) {\n        dp[i][0] = 1\n    }\n    // 상태 전이\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>### 동전 교환 II: 동적 계획법 ###\ndef coin_change_ii_dp(coins, amt)\n  n = coins.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(n + 1) { Array.new(amt + 1, 0) }\n  # 첫 번째 열을 초기화한다\n  (0...(n + 1)).each { |i| dp[i][0] = 1 }\n  # 상태 전이\n  for i in 1...(n + 1)\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[i][a] = dp[i - 1][a]\n      else\n        # 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n      end\n    end\n  end\n  dp[n][amt]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_2","level":3,"title":"3.   공간 최적화","text":"<p>공간 최적화 방법도 같으며, 동전 차원을 없애면 된다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change_ii.py<pre><code>def coin_change_ii_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"동전 교환 II: 공간을 최적화한 동적 계획법\"\"\"\n    n = len(coins)\n    # dp 테이블을 초기화한다\n    dp = [0] * (amt + 1)\n    dp[0] = 1\n    # 상태 전이\n    for i in range(1, n + 1):\n        # 정방향 순회\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            else:\n                # 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n    return dp[amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nint coinChangeIIDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // dp 테이블을 초기화한다\n    vector&lt;int&gt; dp(amt + 1, 0);\n    dp[0] = 1;\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nint coinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nint CoinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    // dp 테이블을 초기화한다\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nfunc coinChangeIIDPComp(coins []int, amt int) int {\n    n := len(coins)\n    // dp 테이블을 초기화한다\n    dp := make([]int, amt+1)\n    dp[0] = 1\n    // 상태 전이\n    for i := 1; i &lt;= n; i++ {\n        // 정방향 순회\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a-coins[i-1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nfunc coinChangeIIDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // dp 테이블을 초기화한다\n    var dp = Array(repeating: 0, count: amt + 1)\n    dp[0] = 1\n    // 상태 전이\n    for i in 1 ... n {\n        for a in 1 ... amt {\n            if coins[i - 1] &gt; a {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nfunction coinChangeIIDPComp(coins, amt) {\n    const n = coins.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nfunction coinChangeIIDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // dp 테이블을 초기화한다\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // 상태 전이\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nint coinChangeIIDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // dp 테이블을 초기화한다\n  List&lt;int&gt; dp = List.filled(amt + 1, 0);\n  dp[0] = 1;\n  // 상태 전이\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[a] = dp[a];\n      } else {\n        // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n        dp[a] = dp[a] + dp[a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nfn coin_change_ii_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // dp 테이블을 초기화한다\n    let mut dp = vec![0; amt + 1];\n    dp[0] = 1;\n    // 상태 전이\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nint coinChangeIIDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // dp 테이블을 초기화한다\n    int *dp = calloc(amt + 1, sizeof(int));\n    dp[0] = 1;\n    // 상태 전이\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a];\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[amt];\n    // 메모리 해제\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.kt<pre><code>/* 동전 교환 II: 공간을 최적화한 동적 계획법 */\nfun coinChangeIIDPComp(coins: IntArray, amt: Int): Int {\n    val n = coins.size\n    // dp 테이블을 초기화한다\n    val dp = IntArray(amt + 1)\n    dp[0] = 1\n    // 상태 전이\n    for (i in 1..n) {\n        for (a in 1..amt) {\n            if (coins[i - 1] &gt; a) {\n                // 목표 금액을 초과하면 동전 i를 선택하지 않는다\n                dp[a] = dp[a]\n            } else {\n                // 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.rb<pre><code>### 동전 교환 II: 공간을 최적화한 동적 계획법 ###\ndef coin_change_ii_dp_comp(coins, amt)\n  n = coins.length\n  # dp 테이블을 초기화한다\n  dp = Array.new(amt + 1, 0)\n  dp[0] = 1\n  # 상태 전이\n  for i in 1...(n + 1)\n    # 정방향 순회\n    for a in 1...(amt + 1)\n      if coins[i - 1] &gt; a\n        # 목표 금액을 초과하면 동전 i를 선택하지 않는다\n        dp[a] = dp[a]\n      else\n        # 동전 i를 선택하지 않는 경우와 선택하는 경우의 방법 수를 더한다\n        dp[a] = dp[a] + dp[a - coins[i - 1]]\n      end\n    end\n  end\n  dp[amt]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 14 장   동적 계획법","14.5   무한 배낭 문제"],"tags":[]},{"location":"chapter_graph/","level":1,"title":"제 9 장   그래프","text":"<p>이 장을 시작하며</p> <p>삶이라는 여정에서 우리는 하나하나의 노드처럼, 보이지 않는 수많은 간선으로 연결되어 있다.</p> <p>만남과 헤어짐은 매번 이 거대한 네트워크 그래프에 저마다의 흔적을 남긴다.</p>","path":["제 9 장   그래프"],"tags":[]},{"location":"chapter_graph/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>9.1   그래프</li> <li>9.2   그래프의 기본 연산</li> <li>9.3   그래프 순회</li> <li>9.4   요약</li> <li>9.5   연습 문제</li> </ul>","path":["제 9 장   그래프"],"tags":[]},{"location":"chapter_graph/exercises/","level":1,"title":"9.5   연습 문제","text":"","path":["제 9 장   그래프","9.5   연습 문제"],"tags":[]},{"location":"chapter_graph/exercises/#951","level":2,"title":"9.5.1   개념 확인","text":"","path":["제 9 장   그래프","9.5   연습 문제"],"tags":[]},{"location":"chapter_graph/exercises/#1","level":3,"title":"1.   같은 그래프를 두 가지 방식으로 표현하기","text":"<p>무방향 그래프의 정점은 <code>A, B, C, D</code> 4개이며, 간선은 <code>A-B, A-C, B-C, C-D</code>다.</p> <ol> <li>이 그래프의 인접 리스트를 적어 보자.</li> <li>0과 1만 들어가는 인접 행렬을 완성해 보자.</li> <li><code>A</code>와 <code>D</code>가 직접 연결되어 있는지 판단할 때, 저장된 항목 하나만 확인하면 되는 표현 방식은 무엇인가?</li> <li>정점이 많고 간선이 적다면 일반적으로 어느 표현 방식이 공간을 더 절약하는가?</li> </ol> 참고 답안 <ol> <li> <p>인접 리스트는 다음과 같다.</p> <pre><code>A: B, C\nB: A, C\nC: A, B, D\nD: C\n</code></pre> </li> <li> <p>인접 행렬은 다음과 같다.</p> A B C D A 0 1 1 0 B 1 0 1 0 C 1 1 0 1 D 0 0 1 0 </li> <li> <p>인접 행렬에서는 <code>A</code>행과 <code>D</code>열이 만나는 항목을 바로 확인할 수 있다. 따라서 임의의 두 정점이 직접 연결되어 있는지 판단하기에 적합하다.</p> </li> <li> <p>정점이 많고 간선이 적을 때 인접 리스트는 실제로 존재하는 간선만 기록한다. 따라서 모든 정점 쌍에 대해 자리를 마련하는 인접 행렬보다 일반적으로 공간을 더 절약한다.</p> </li> </ol>","path":["제 9 장   그래프","9.5   연습 문제"],"tags":[]},{"location":"chapter_graph/exercises/#2","level":3,"title":"2.   너비 우선 탐색과 깊이 우선 탐색의 방문 순서","text":"<p>무방향 그래프의 정점은 <code>A, B, C, D, E</code>이며, 간선은 <code>A-B, A-C, B-D, C-D, D-E</code>다.</p> <p>A에서 출발하고, 방문하지 않은 인접 정점이 여러 개면 알파벳순으로 선택한다고 하자.</p> <ol> <li>너비 우선 탐색(BFS)의 방문 순서를 적어 보자.</li> <li>재귀 깊이 우선 탐색(DFS)의 방문 순서를 적어 보자.</li> <li>두 탐색 모두 이미 방문한 정점을 기록해야 하는 이유는 무엇인가?</li> </ol> 참고 답안 <ol> <li> <p>BFS의 방문 순서는 <code>A, B, C, D, E</code>다. A에서 간선 하나로 도달하는 B와 C를 먼저 방문하고,     그다음 더 멀리 있는 D와 E를 방문한다.</p> </li> <li> <p>DFS의 방문 순서는 <code>A, B, D, C, E</code>다. 현재 정점에서 방문하지 않은 인접 정점으로 차례대로 들어가므로     먼저 <code>A → B → D → C</code>를 따라간다. C에서 새로 방문할 인접 정점이 없으면 D로 돌아와 E를 방문한다.</p> </li> <li> <p>그래프에는 <code>A-B-D-C-A</code>와 같은 사이클이 있다. 이미 방문한 정점을 기록하지 않으면     같은 정점들을 사이클을 따라 반복해서 방문하여 탐색이 정상적으로 끝나지 않을 수 있다.</p> </li> </ol>","path":["제 9 장   그래프","9.5   연습 문제"],"tags":[]},{"location":"chapter_graph/exercises/#3-bfs","level":3,"title":"3.   BFS 한 번으로 그래프 전체를 방문할 수 있는가","text":"<p>무방향 그래프의 정점은 <code>A, B, C, D, E, F</code>이며, 간선은 <code>A-B, B-C, D-E</code>뿐이다.</p> <ol> <li>A에서 BFS를 한 번 시작하면 어떤 정점을 방문할 수 있는가?</li> <li>1번의 결과를 바탕으로, 이 BFS가 그래프의 모든 정점을 방문했는지 판단하고 이유를 설명하라.</li> <li>모든 정점을 알파벳순으로 확인하면서 방문하지 않은 정점을 만날 때마다 BFS를 새로 시작한다면,     각 BFS의 시작 정점은 무엇인가? 이 그래프는 서로 연결되지 않은 부분인 연결 요소 몇 개로 이루어져 있는가?</li> </ol> 참고 답안 <ol> <li> <p>A에서 출발하면 <code>A, B, C</code>만 방문할 수 있다.</p> </li> <li> <p>모든 정점을 방문한 것은 아니다. <code>D, E</code>는 서로 연결된 별개의 부분을 이루고, F는 홀로 떨어진 정점이다.     이 정점들과 A 사이에는 경로가 없으므로 A에서 출발해서는 도달할 수 없다.</p> </li> <li> <p>세 번의 BFS는 차례대로 <code>A, D, F</code>에서 시작하여 각각     <code>{A, B, C}</code>, <code>{D, E}</code>, <code>{F}</code>를 방문한다. 따라서 이 그래프의 연결 요소는 3개다.</p> </li> </ol>","path":["제 9 장   그래프","9.5   연습 문제"],"tags":[]},{"location":"chapter_graph/exercises/#952","level":2,"title":"9.5.2   프로그래밍 연습","text":"","path":["제 9 장   그래프","9.5   연습 문제"],"tags":[]},{"location":"chapter_graph/exercises/#1_1","level":3,"title":"1.   무방향 그래프에 경로가 존재하는지 판단하기","text":"<p>정점이 \\(n\\)개인 무방향 그래프가 주어지며, 정점 번호는 \\(0\\)부터 \\(n-1\\)까지다. 배열 <code>edges</code>의 각 항목 <code>[u, v]</code>는 정점 <code>u</code>와 <code>v</code> 사이에 무방향 간선이 있음을 나타낸다.</p> <p>시작 정점 <code>source</code>와 도착 정점 <code>destination</code>도 주어진다. 먼저 <code>edges</code>로 인접 리스트를 만든 뒤 BFS 또는 DFS를 사용하여 <code>source</code>에서 <code>destination</code>까지의 경로가 있는지 판단하라. 있으면 <code>true</code>를, 없으면 <code>false</code>를 반환하라. 그래프에는 사이클이 있을 수 있으며, 모든 정점이 서로 연결되어 있지 않을 수도 있다.</p> 풀이 힌트 <ol> <li>무방향 간선은 양쪽 방향을 모두 추가해야 한다.</li> <li>그래프에는 사이클이 있을 수 있으므로 이미 방문한 노드를 반드시 기록해야 한다.</li> <li>source에서 출발하여 destination을 만나면 true를 반환한다. 탐색을 마칠 때까지 만나지 못하면 false를 반환한다.</li> </ol> <p>LeetCode</p>","path":["제 9 장   그래프","9.5   연습 문제"],"tags":[]},{"location":"chapter_graph/graph/","level":1,"title":"9.1   그래프","text":"<p>그래프(graph)는 정점(vertex)과 간선(edge)으로 이루어진 비선형 자료 구조다. 그래프 \\(G\\)는 정점 집합 \\(V\\)와 간선 집합 \\(E\\)로 추상적으로 표현할 수 있다. 다음 예시는 정점 5개와 간선 7개로 이루어진 그래프를 나타낸다.</p> \\[ \\begin{aligned} V &amp; = \\{ 1, 2, 3, 4, 5 \\} \\newline E &amp; = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G &amp; = \\{ V, E \\} \\newline \\end{aligned} \\] <p>정점을 노드로, 간선을 노드들을 연결하는 참조(포인터)로 보면 그래프는 연결 리스트를 확장한 자료 구조로 생각할 수 있다. 아래 그림(9-1)과 같이 선형 관계(연결 리스트)와 분할 정복 관계(트리)에 비해 네트워크 관계(그래프)는 자유도가 높아 더 복잡하다.</p> <p></p> <p> 그림 9-1   연결 리스트, 트리, 그래프 사이의 관계 </p>","path":["제 9 장   그래프","9.1   그래프"],"tags":[]},{"location":"chapter_graph/graph/#911","level":2,"title":"9.1.1   그래프의 주요 종류와 용어","text":"<p>간선에 방향이 있는지에 따라 아래 그림(9-2)과 같이 무방향 그래프(undirected graph)와 방향 그래프(directed graph)로 나뉜다.</p> <ul> <li>무방향 그래프에서 간선은 두 정점 사이의 '양방향' 연결 관계를 나타낸다. 위챗이나 QQ의 '친구 관계'가 그 예다.</li> <li>방향 그래프에서 간선은 방향을 가진다. 즉, \\(A \\rightarrow B\\)와 \\(A \\leftarrow B\\) 방향의 간선은 서로 독립적이다. 웨이보나 더우인의 '팔로우'와 '팔로워' 관계가 그 예다.</li> </ul> <p></p> <p> 그림 9-2   방향 그래프와 무방향 그래프 </p> <p>모든 정점이 연결되어 있는지에 따라 아래 그림(9-3)과 같이 연결 그래프(connected graph)와 비연결 그래프(disconnected graph)로 나뉜다.</p> <ul> <li>연결 그래프에서는 한 정점에서 출발하여 다른 모든 정점에 도달할 수 있다.</li> <li>비연결 그래프에서는 한 정점에서 출발했을 때 도달할 수 없는 정점이 하나 이상 존재한다.</li> </ul> <p></p> <p> 그림 9-3   연결 그래프와 비연결 그래프 </p> <p>간선에 '가중치' 변수를 추가하면 아래 그림(9-4)과 같은 가중 그래프(weighted graph)를 얻는다. 예를 들어 『왕자영요』 같은 모바일 게임에서는 함께 게임한 시간을 바탕으로 플레이어 간의 '친밀도'를 계산한다. 이러한 친밀도 네트워크는 가중 그래프로 표현할 수 있다.</p> <p></p> <p> 그림 9-4   가중 그래프와 비가중 그래프 </p> <p>그래프 자료 구조에서 자주 사용하는 용어는 다음과 같다.</p> <ul> <li>인접(adjacency): 두 정점 사이에 간선이 있으면 두 정점이 '인접한다'고 한다. 위 그림(9-4)에서 정점 1에 인접한 정점은 2, 3, 5다.</li> <li>경로(path): 정점 A에서 정점 B로 갈 때 거치는 간선의 순서를 A에서 B로 가는 '경로'라고 한다. 위 그림(9-4)에서 간선 순서 1-5-2-4는 정점 1에서 정점 4로 가는 경로다.</li> <li>차수(degree): 한 정점에 연결된 간선의 수다. 방향 그래프에서는 진입 차수(in-degree)가 해당 정점으로 들어오는 간선 수를, 진출 차수(out-degree)가 해당 정점에서 나가는 간선 수를 나타낸다.</li> </ul>","path":["제 9 장   그래프","9.1   그래프"],"tags":[]},{"location":"chapter_graph/graph/#912","level":2,"title":"9.1.2   그래프의 표현","text":"<p>그래프를 표현하는 대표적인 방법은 '인접 행렬'과 '인접 리스트'다. 아래에서는 무방향 그래프를 예로 든다.</p>","path":["제 9 장   그래프","9.1   그래프"],"tags":[]},{"location":"chapter_graph/graph/#1","level":3,"title":"1.   인접 행렬","text":"<p>그래프의 정점 수를 \\(n\\)이라고 하자. 인접 행렬(adjacency matrix)은 \\(n \\times n\\) 크기의 행렬로 그래프를 표현한다. 각 행(열)은 정점 하나를, 행렬의 원소는 간선을 나타내며, \\(1\\) 또는 \\(0\\)으로 두 정점 사이에 간선이 있는지 표시한다.</p> <p>아래 그림(9-5)과 같이 인접 행렬을 \\(M\\), 정점 리스트를 \\(V\\)라고 하면, 행렬 원소 \\(M[i, j] = 1\\)은 정점 \\(V[i]\\)와 정점 \\(V[j]\\) 사이에 간선이 있음을 뜻한다. 반대로 \\(M[i, j] = 0\\)은 두 정점 사이에 간선이 없음을 뜻한다.</p> <p></p> <p> 그림 9-5   그래프의 인접 행렬 표현 </p> <p>인접 행렬에는 다음 성질이 있다.</p> <ul> <li>단순 그래프에서는 정점이 자기 자신과 연결될 수 없으므로, 인접 행렬의 주대각선 원소는 의미가 없다.</li> <li>무방향 그래프에서는 두 방향의 간선이 같으므로, 인접 행렬은 주대각선에 대해 대칭이다.</li> <li>인접 행렬의 원소 \\(1\\)과 \\(0\\)을 가중치로 바꾸면 가중 그래프를 표현할 수 있다.</li> </ul> <p>인접 행렬로 그래프를 표현하면 행렬 원소에 직접 접근하여 간선을 얻을 수 있다. 따라서 추가, 삭제, 조회, 수정 연산이 매우 효율적이며 시간 복잡도는 모두 \\(O(1)\\)이다. 하지만 행렬의 공간 복잡도는 \\(O(n^2)\\)이므로 메모리를 많이 차지한다.</p>","path":["제 9 장   그래프","9.1   그래프"],"tags":[]},{"location":"chapter_graph/graph/#2","level":3,"title":"2.   인접 리스트","text":"<p>인접 리스트(adjacency list)는 \\(n\\) 개의 연결 리스트로 그래프를 표현하며, 연결 리스트의 노드는 정점을 나타낸다. \\(i\\) 번째 연결 리스트는 정점 \\(i\\)에 대응하고, 해당 정점에 인접한 모든 정점(간선으로 연결된 정점)을 저장한다. 아래 그림(9-6)은 인접 리스트로 그래프를 저장한 예다.</p> <p></p> <p> 그림 9-6   그래프의 인접 리스트 표현 </p> <p>인접 리스트는 실제로 존재하는 간선만 저장한다. 전체 간선 수는 보통 \\(n^2\\)보다 훨씬 작으므로 공간을 더 절약할 수 있다. 하지만 간선을 찾으려면 연결 리스트를 순회해야 하므로 시간 효율은 인접 행렬보다 낮다.</p> <p>위 그림(9-6)을 보면 인접 리스트의 구조는 해시 테이블의 '체이닝'과 매우 비슷하므로, 비슷한 방법으로 효율을 높일 수 있다. 예를 들어 연결 리스트가 길면 AVL 트리나 레드-블랙 트리로 바꾸어 시간 복잡도를 \\(O(n)\\)에서 \\(O(\\log n)\\)으로 개선할 수 있다. 연결 리스트를 해시 테이블로 바꾸면 시간 복잡도를 \\(O(1)\\)로 줄일 수도 있다.</p>","path":["제 9 장   그래프","9.1   그래프"],"tags":[]},{"location":"chapter_graph/graph/#913","level":2,"title":"9.1.3   그래프의 주요 응용","text":"<p>아래 표(9-1)와 같이 현실의 많은 시스템을 그래프로 모델링할 수 있으며, 관련 문제를 그래프 계산 문제로 환원할 수 있다.</p> <p> 표 9-1   일상에서 볼 수 있는 그래프 </p> 정점 간선 그래프 계산 문제 소셜 네트워크 사용자 친구 관계 잠재적 친구 추천 지하철 노선 역 역 사이의 연결 관계 최단 경로 추천 태양계 천체 천체 사이의 만유인력 행성 궤도 계산","path":["제 9 장   그래프","9.1   그래프"],"tags":[]},{"location":"chapter_graph/graph_operations/","level":1,"title":"9.2   그래프의 기본 연산","text":"<p>그래프의 기본 연산은 '간선'에 대한 연산과 '정점'에 대한 연산으로 나뉜다. '인접 행렬'과 '인접 리스트' 중 어느 표현을 사용하는지에 따라 구현 방법이 달라진다.</p>","path":["제 9 장   그래프","9.2   그래프의 기본 연산"],"tags":[]},{"location":"chapter_graph/graph_operations/#921","level":2,"title":"9.2.1   인접 행렬 기반 구현","text":"<p>정점 수가 \\(n\\)인 무방향 그래프가 주어졌을 때 각 연산은 아래 그림(9-7)과 같이 구현한다.</p> <ul> <li>간선 추가 또는 삭제: 인접 행렬에서 해당 간선을 직접 수정하면 되며, \\(O(1)\\) 시간이 걸린다. 무방향 그래프이므로 두 방향의 간선을 함께 갱신해야 한다.</li> <li>정점 추가: 인접 행렬의 끝에 행과 열을 하나씩 추가하고 모두 \\(0\\)으로 채운다. \\(O(n)\\) 시간이 걸린다.</li> <li>정점 삭제: 인접 행렬에서 행과 열을 하나씩 삭제한다. 첫 번째 행과 열을 삭제할 때가 최악의 경우로, \\((n-1)^2\\) 개 원소를 '왼쪽 위로 이동'해야 하므로 \\(O(n^2)\\) 시간이 걸린다.</li> <li>초기화: 정점 \\(n\\) 개를 전달받아 길이가 \\(n\\)인 정점 리스트 <code>vertices</code>를 초기화하는 데 \\(O(n)\\) 시간이 걸린다. \\(n \\times n\\) 크기의 인접 행렬 <code>adjMat</code>을 초기화하는 데는 \\(O(n^2)\\) 시간이 걸린다.</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 9-7   인접 행렬의 초기화, 간선 추가와 삭제, 정점 추가와 삭제 </p> <p>다음은 인접 행렬로 그래프를 표현한 구현 코드다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_adjacency_matrix.py<pre><code>class GraphAdjMat:\n    \"\"\"인접 행렬로 구현한 무방향 그래프 클래스\"\"\"\n\n    def __init__(self, vertices: list[int], edges: list[list[int]]):\n        \"\"\"생성자\"\"\"\n        # 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n        self.vertices: list[int] = []\n        # 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n        self.adj_mat: list[list[int]] = []\n        # 정점 추가\n        for val in vertices:\n            self.add_vertex(val)\n        # 간선 추가\n        # edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for e in edges:\n            self.add_edge(e[0], e[1])\n\n    def size(self) -&gt; int:\n        \"\"\"정점 개수 반환\"\"\"\n        return len(self.vertices)\n\n    def add_vertex(self, val: int):\n        \"\"\"정점 추가\"\"\"\n        n = self.size()\n        # 정점 리스트에 새 정점의 값을 추가한다\n        self.vertices.append(val)\n        # 인접 행렬에 행 하나를 추가한다\n        new_row = [0] * n\n        self.adj_mat.append(new_row)\n        # 인접 행렬에 열 하나를 추가한다\n        for row in self.adj_mat:\n            row.append(0)\n\n    def remove_vertex(self, index: int):\n        \"\"\"정점 삭제\"\"\"\n        if index &gt;= self.size():\n            raise IndexError()\n        # 정점 리스트에서 인덱스 index의 정점을 제거한다\n        self.vertices.pop(index)\n        # 인접 행렬에서 인덱스 index의 행을 삭제한다\n        self.adj_mat.pop(index)\n        # 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for row in self.adj_mat:\n            row.pop(index)\n\n    def add_edge(self, i: int, j: int):\n        \"\"\"간선 추가\"\"\"\n        # 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n        # 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        # 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n        self.adj_mat[i][j] = 1\n        self.adj_mat[j][i] = 1\n\n    def remove_edge(self, i: int, j: int):\n        \"\"\"간선 삭제\"\"\"\n        # 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n        # 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        self.adj_mat[i][j] = 0\n        self.adj_mat[j][i] = 0\n\n    def print(self):\n        \"\"\"인접 행렬 출력\"\"\"\n        print(\"정점 리스트 =\", self.vertices)\n        print(\"인접 행렬 =\")\n        print_matrix(self.adj_mat)\n</code></pre> graph_adjacency_matrix.cpp<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat {\n    vector&lt;int&gt; vertices;       // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    vector&lt;vector&lt;int&gt;&gt; adjMat; // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n  public:\n    /* 생성자 */\n    GraphAdjMat(const vector&lt;int&gt; &amp;vertices, const vector&lt;vector&lt;int&gt;&gt; &amp;edges) {\n        // 정점 추가\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for (const vector&lt;int&gt; &amp;edge : edges) {\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    int size() const {\n        return vertices.size();\n    }\n\n    /* 정점 추가 */\n    void addVertex(int val) {\n        int n = size();\n        // 정점 리스트에 새 정점의 값을 추가한다\n        vertices.push_back(val);\n        // 인접 행렬에 행 하나를 추가한다\n        adjMat.emplace_back(vector&lt;int&gt;(n, 0));\n        // 인접 행렬에 열 하나를 추가한다\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.push_back(0);\n        }\n    }\n\n    /* 정점 삭제 */\n    void removeVertex(int index) {\n        if (index &gt;= size()) {\n            throw out_of_range(\"정점이 존재하지 않는다\");\n        }\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        vertices.erase(vertices.begin() + index);\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        adjMat.erase(adjMat.begin() + index);\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.erase(row.begin() + index);\n        }\n    }\n\n    /* 간선 추가 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    void addEdge(int i, int j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"정점이 존재하지 않는다\");\n        }\n        // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    void removeEdge(int i, int j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"정점이 존재하지 않는다\");\n        }\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* 인접 행렬 출력 */\n    void print() {\n        cout &lt;&lt; \"정점 리스트 = \";\n        printVector(vertices);\n        cout &lt;&lt; \"인접 행렬 =\" &lt;&lt; endl;\n        printVectorMatrix(adjMat);\n    }\n};\n</code></pre> graph_adjacency_matrix.java<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat {\n    List&lt;Integer&gt; vertices; // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    List&lt;List&lt;Integer&gt;&gt; adjMat; // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n    /* 생성자 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = new ArrayList&lt;&gt;();\n        this.adjMat = new ArrayList&lt;&gt;();\n        // 정점 추가\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for (int[] e : edges) {\n            addEdge(e[0], e[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    public int size() {\n        return vertices.size();\n    }\n\n    /* 정점 추가 */\n    public void addVertex(int val) {\n        int n = size();\n        // 정점 리스트에 새 정점의 값을 추가한다\n        vertices.add(val);\n        // 인접 행렬에 행 하나를 추가한다\n        List&lt;Integer&gt; newRow = new ArrayList&lt;&gt;(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.add(0);\n        }\n        adjMat.add(newRow);\n        // 인접 행렬에 열 하나를 추가한다\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.add(0);\n        }\n    }\n\n    /* 정점 삭제 */\n    public void removeVertex(int index) {\n        if (index &gt;= size())\n            throw new IndexOutOfBoundsException();\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        vertices.remove(index);\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        adjMat.remove(index);\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.remove(index);\n        }\n    }\n\n    /* 간선 추가 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    public void addEdge(int i, int j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n        adjMat.get(i).set(j, 1);\n        adjMat.get(j).set(i, 1);\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    public void removeEdge(int i, int j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        adjMat.get(i).set(j, 0);\n        adjMat.get(j).set(i, 0);\n    }\n\n    /* 인접 행렬 출력 */\n    public void print() {\n        System.out.print(\"정점 리스트 = \");\n        System.out.println(vertices);\n        System.out.println(\"인접 행렬 =\");\n        PrintUtil.printMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.cs<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat {\n    List&lt;int&gt; vertices;     // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    List&lt;List&lt;int&gt;&gt; adjMat; // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n    /* 생성자 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // 정점 추가\n        foreach (int val in vertices) {\n            AddVertex(val);\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        foreach (int[] e in edges) {\n            AddEdge(e[0], e[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    int Size() {\n        return vertices.Count;\n    }\n\n    /* 정점 추가 */\n    public void AddVertex(int val) {\n        int n = Size();\n        // 정점 리스트에 새 정점의 값을 추가한다\n        vertices.Add(val);\n        // 인접 행렬에 행 하나를 추가한다\n        List&lt;int&gt; newRow = new(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.Add(0);\n        }\n        adjMat.Add(newRow);\n        // 인접 행렬에 열 하나를 추가한다\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.Add(0);\n        }\n    }\n\n    /* 정점 삭제 */\n    public void RemoveVertex(int index) {\n        if (index &gt;= Size())\n            throw new IndexOutOfRangeException();\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        vertices.RemoveAt(index);\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        adjMat.RemoveAt(index);\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.RemoveAt(index);\n        }\n    }\n\n    /* 간선 추가 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    public void AddEdge(int i, int j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    public void RemoveEdge(int i, int j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* 인접 행렬 출력 */\n    public void Print() {\n        Console.Write(\"정점 리스트 = \");\n        PrintUtil.PrintList(vertices);\n        Console.WriteLine(\"인접 행렬 =\");\n        PrintUtil.PrintMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.go<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\ntype graphAdjMat struct {\n    // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    vertices []int\n    // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n    adjMat [][]int\n}\n\n/* 생성자 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n    // 정점 추가\n    n := len(vertices)\n    adjMat := make([][]int, n)\n    for i := range adjMat {\n        adjMat[i] = make([]int, n)\n    }\n    // 그래프 초기화\n    g := &amp;graphAdjMat{\n        vertices: vertices,\n        adjMat:   adjMat,\n    }\n    // 간선 추가\n    // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n    for i := range edges {\n        g.addEdge(edges[i][0], edges[i][1])\n    }\n    return g\n}\n\n/* 정점 개수 반환 */\nfunc (g *graphAdjMat) size() int {\n    return len(g.vertices)\n}\n\n/* 정점 추가 */\nfunc (g *graphAdjMat) addVertex(val int) {\n    n := g.size()\n    // 정점 리스트에 새 정점의 값을 추가한다\n    g.vertices = append(g.vertices, val)\n    // 인접 행렬에 행 하나를 추가한다\n    newRow := make([]int, n)\n    g.adjMat = append(g.adjMat, newRow)\n    // 인접 행렬에 열 하나를 추가한다\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i], 0)\n    }\n}\n\n/* 정점 삭제 */\nfunc (g *graphAdjMat) removeVertex(index int) {\n    if index &gt;= g.size() {\n        return\n    }\n    // 정점 리스트에서 인덱스 index의 정점을 제거한다\n    g.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n    // 인접 행렬에서 인덱스 index의 행을 삭제한다\n    g.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n    // 인접 행렬에서 인덱스 index의 열을 삭제한다\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n    }\n}\n\n/* 간선 추가 */\n// 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\nfunc (g *graphAdjMat) addEdge(i, j int) {\n    // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"인덱스가 범위를 벗어났다\")\n    }\n    // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n    g.adjMat[i][j] = 1\n    g.adjMat[j][i] = 1\n}\n\n/* 간선 삭제 */\n// 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n    // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"인덱스가 범위를 벗어났다\")\n    }\n    g.adjMat[i][j] = 0\n    g.adjMat[j][i] = 0\n}\n\n/* 인접 행렬 출력 */\nfunc (g *graphAdjMat) print() {\n    fmt.Printf(\"\\t정점 리스트 = %v\\n\", g.vertices)\n    fmt.Printf(\"\\t인접 행렬 = \\n\")\n    for i := range g.adjMat {\n        fmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n    }\n}\n</code></pre> graph_adjacency_matrix.swift<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat {\n    private var vertices: [Int] // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    private var adjMat: [[Int]] // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n    /* 생성자 */\n    init(vertices: [Int], edges: [[Int]]) {\n        self.vertices = []\n        adjMat = []\n        // 정점 추가\n        for val in vertices {\n            addVertex(val: val)\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for e in edges {\n            addEdge(i: e[0], j: e[1])\n        }\n    }\n\n    /* 정점 개수 반환 */\n    func size() -&gt; Int {\n        vertices.count\n    }\n\n    /* 정점 추가 */\n    func addVertex(val: Int) {\n        let n = size()\n        // 정점 리스트에 새 정점의 값을 추가한다\n        vertices.append(val)\n        // 인접 행렬에 행 하나를 추가한다\n        let newRow = Array(repeating: 0, count: n)\n        adjMat.append(newRow)\n        // 인접 행렬에 열 하나를 추가한다\n        for i in adjMat.indices {\n            adjMat[i].append(0)\n        }\n    }\n\n    /* 정점 삭제 */\n    func removeVertex(index: Int) {\n        if index &gt;= size() {\n            fatalError(\"범위를 벗어남\")\n        }\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        vertices.remove(at: index)\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        adjMat.remove(at: index)\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for i in adjMat.indices {\n            adjMat[i].remove(at: index)\n        }\n    }\n\n    /* 간선 추가 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    func addEdge(i: Int, j: Int) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"범위를 벗어남\")\n        }\n        // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    func removeEdge(i: Int, j: Int) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"범위를 벗어남\")\n        }\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* 인접 행렬 출력 */\n    func print() {\n        Swift.print(\"정점 리스트 = \", terminator: \"\")\n        Swift.print(vertices)\n        Swift.print(\"인접 행렬 =\")\n        PrintUtil.printMatrix(matrix: adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.js<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat {\n    vertices; // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    adjMat; // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n    /* 생성자 */\n    constructor(vertices, edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // 정점 추가\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    size() {\n        return this.vertices.length;\n    }\n\n    /* 정점 추가 */\n    addVertex(val) {\n        const n = this.size();\n        // 정점 리스트에 새 정점의 값을 추가한다\n        this.vertices.push(val);\n        // 인접 행렬에 행 하나를 추가한다\n        const newRow = [];\n        for (let j = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // 인접 행렬에 열 하나를 추가한다\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* 정점 삭제 */\n    removeVertex(index) {\n        if (index &gt;= this.size()) {\n            throw new RangeError('인덱스가 범위를 벗어났다');\n        }\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        this.vertices.splice(index, 1);\n\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        this.adjMat.splice(index, 1);\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* 간선 추가 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    addEdge(i, j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('인덱스가 범위를 벗어났다');\n        }\n        // 무방향 그래프의 인접 행렬은 주대각선을 기준으로 대칭이므로 (i, j) === (j, i)를 만족한다\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    removeEdge(i, j) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('인덱스가 범위를 벗어났다');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* 인접 행렬 출력 */\n    print() {\n        console.log('정점 리스트 = ', this.vertices);\n        console.log('인접 행렬 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.ts<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat {\n    vertices: number[]; // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    adjMat: number[][]; // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n    /* 생성자 */\n    constructor(vertices: number[], edges: number[][]) {\n        this.vertices = [];\n        this.adjMat = [];\n        // 정점 추가\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    size(): number {\n        return this.vertices.length;\n    }\n\n    /* 정점 추가 */\n    addVertex(val: number): void {\n        const n: number = this.size();\n        // 정점 리스트에 새 정점의 값을 추가한다\n        this.vertices.push(val);\n        // 인접 행렬에 행 하나를 추가한다\n        const newRow: number[] = [];\n        for (let j: number = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // 인접 행렬에 열 하나를 추가한다\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* 정점 삭제 */\n    removeVertex(index: number): void {\n        if (index &gt;= this.size()) {\n            throw new RangeError('인덱스가 범위를 벗어났다');\n        }\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        this.vertices.splice(index, 1);\n\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        this.adjMat.splice(index, 1);\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* 간선 추가 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    addEdge(i: number, j: number): void {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('인덱스가 범위를 벗어났다');\n        }\n        // 무방향 그래프의 인접 행렬은 주대각선을 기준으로 대칭이므로 (i, j) === (j, i)를 만족한다\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    removeEdge(i: number, j: number): void {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('인덱스가 범위를 벗어났다');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* 인접 행렬 출력 */\n    print(): void {\n        console.log('정점 리스트 = ', this.vertices);\n        console.log('인접 행렬 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.dart<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat {\n  List&lt;int&gt; vertices = []; // 정점 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n  List&lt;List&lt;int&gt;&gt; adjMat = []; //인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n  /* 생성자 */\n  GraphAdjMat(List&lt;int&gt; vertices, List&lt;List&lt;int&gt;&gt; edges) {\n    this.vertices = [];\n    this.adjMat = [];\n    // 정점 추가\n    for (int val in vertices) {\n      addVertex(val);\n    }\n    // 간선 추가\n    // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n    for (List&lt;int&gt; e in edges) {\n      addEdge(e[0], e[1]);\n    }\n  }\n\n  /* 정점 개수 반환 */\n  int size() {\n    return vertices.length;\n  }\n\n  /* 정점 추가 */\n  void addVertex(int val) {\n    int n = size();\n    // 정점 리스트에 새 정점의 값을 추가한다\n    vertices.add(val);\n    // 인접 행렬에 행 하나를 추가한다\n    List&lt;int&gt; newRow = List.filled(n, 0, growable: true);\n    adjMat.add(newRow);\n    // 인접 행렬에 열 하나를 추가한다\n    for (List&lt;int&gt; row in adjMat) {\n      row.add(0);\n    }\n  }\n\n  /* 정점 삭제 */\n  void removeVertex(int index) {\n    if (index &gt;= size()) {\n      throw IndexError;\n    }\n    // 정점 리스트에서 인덱스 index의 정점을 제거한다\n    vertices.removeAt(index);\n    // 인접 행렬에서 인덱스 index의 행을 삭제한다\n    adjMat.removeAt(index);\n    // 인접 행렬에서 인덱스 index의 열을 삭제한다\n    for (List&lt;int&gt; row in adjMat) {\n      row.removeAt(index);\n    }\n  }\n\n  /* 간선 추가 */\n  // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n  void addEdge(int i, int j) {\n    // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n    adjMat[i][j] = 1;\n    adjMat[j][i] = 1;\n  }\n\n  /* 간선 삭제 */\n  // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n  void removeEdge(int i, int j) {\n    // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    adjMat[i][j] = 0;\n    adjMat[j][i] = 0;\n  }\n\n  /* 인접 행렬 출력 */\n  void printAdjMat() {\n    print(\"정점 리스트 = $vertices\");\n    print(\"인접 행렬 = \");\n    printMatrix(adjMat);\n  }\n}\n</code></pre> graph_adjacency_matrix.rs<pre><code>/* 인접 행렬로 구현한 무방향 그래프 자료형 */\npub struct GraphAdjMat {\n    // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    pub vertices: Vec&lt;i32&gt;,\n    // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n    pub adj_mat: Vec&lt;Vec&lt;i32&gt;&gt;,\n}\n\nimpl GraphAdjMat {\n    /* 생성자 */\n    pub fn new(vertices: Vec&lt;i32&gt;, edges: Vec&lt;[usize; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjMat {\n            vertices: vec![],\n            adj_mat: vec![],\n        };\n        // 정점 추가\n        for val in vertices {\n            graph.add_vertex(val);\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for edge in edges {\n            graph.add_edge(edge[0], edge[1])\n        }\n\n        graph\n    }\n\n    /* 정점 개수 반환 */\n    pub fn size(&amp;self) -&gt; usize {\n        self.vertices.len()\n    }\n\n    /* 정점 추가 */\n    pub fn add_vertex(&amp;mut self, val: i32) {\n        let n = self.size();\n        // 정점 리스트에 새 정점의 값을 추가한다\n        self.vertices.push(val);\n        // 인접 행렬에 행 하나를 추가한다\n        self.adj_mat.push(vec![0; n]);\n        // 인접 행렬에 열 하나를 추가한다\n        for row in self.adj_mat.iter_mut() {\n            row.push(0);\n        }\n    }\n\n    /* 정점 삭제 */\n    pub fn remove_vertex(&amp;mut self, index: usize) {\n        if index &gt;= self.size() {\n            panic!(\"인덱스 오류\")\n        }\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        self.vertices.remove(index);\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        self.adj_mat.remove(index);\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for row in self.adj_mat.iter_mut() {\n            row.remove(index);\n        }\n    }\n\n    /* 간선 추가 */\n    pub fn add_edge(&amp;mut self, i: usize, j: usize) {\n        // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"인덱스 오류\")\n        }\n        // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n        self.adj_mat[i][j] = 1;\n        self.adj_mat[j][i] = 1;\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    pub fn remove_edge(&amp;mut self, i: usize, j: usize) {\n        // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"인덱스 오류\")\n        }\n        self.adj_mat[i][j] = 0;\n        self.adj_mat[j][i] = 0;\n    }\n\n    /* 인접 행렬 출력 */\n    pub fn print(&amp;self) {\n        println!(\"정점 리스트 = {:?}\", self.vertices);\n        println!(\"인접 행렬 =\");\n        println!(\"[\");\n        for row in &amp;self.adj_mat {\n            println!(\"  {:?},\", row);\n        }\n        println!(\"]\")\n    }\n}\n</code></pre> graph_adjacency_matrix.c<pre><code>/* 인접 행렬로 구현한 무방향 그래프 구조체 */\ntypedef struct {\n    int vertices[MAX_SIZE];\n    int adjMat[MAX_SIZE][MAX_SIZE];\n    int size;\n} GraphAdjMat;\n\n/* 생성자 */\nGraphAdjMat *newGraphAdjMat() {\n    GraphAdjMat *graph = (GraphAdjMat *)malloc(sizeof(GraphAdjMat));\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        for (int j = 0; j &lt; MAX_SIZE; j++) {\n            graph-&gt;adjMat[i][j] = 0;\n        }\n    }\n    return graph;\n}\n\n/* 소멸자 */\nvoid delGraphAdjMat(GraphAdjMat *graph) {\n    free(graph);\n}\n\n/* 정점 추가 */\nvoid addVertex(GraphAdjMat *graph, int val) {\n    if (graph-&gt;size == MAX_SIZE) {\n        fprintf(stderr, \"그래프의 정점 개수가 최댓값에 도달했다\\n\");\n        return;\n    }\n    // n번째 정점을 추가하고 n번째 행과 열을 0으로 설정한다\n    int n = graph-&gt;size;\n    graph-&gt;vertices[n] = val;\n    for (int i = 0; i &lt;= n; i++) {\n        graph-&gt;adjMat[n][i] = graph-&gt;adjMat[i][n] = 0;\n    }\n    graph-&gt;size++;\n}\n\n/* 정점 삭제 */\nvoid removeVertex(GraphAdjMat *graph, int index) {\n    if (index &lt; 0 || index &gt;= graph-&gt;size) {\n        fprintf(stderr, \"정점 인덱스가 범위를 벗어났다\\n\");\n        return;\n    }\n    // 정점 리스트에서 인덱스 index의 정점을 제거한다\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        graph-&gt;vertices[i] = graph-&gt;vertices[i + 1];\n    }\n    // 인접 행렬에서 인덱스 index의 행을 삭제한다\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        for (int j = 0; j &lt; graph-&gt;size; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i + 1][j];\n        }\n    }\n    // 인접 행렬에서 인덱스 index의 열을 삭제한다\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        for (int j = index; j &lt; graph-&gt;size - 1; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i][j + 1];\n        }\n    }\n    graph-&gt;size--;\n}\n\n/* 간선 추가 */\n// 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\nvoid addEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"간선 인덱스가 범위를 벗어나거나 서로 같다\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 1;\n    graph-&gt;adjMat[j][i] = 1;\n}\n\n/* 간선 삭제 */\n// 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\nvoid removeEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"간선 인덱스가 범위를 벗어나거나 서로 같다\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 0;\n    graph-&gt;adjMat[j][i] = 0;\n}\n\n/* 인접 행렬 출력 */\nvoid printGraphAdjMat(GraphAdjMat *graph) {\n    printf(\"정점 리스트 = \");\n    printArray(graph-&gt;vertices, graph-&gt;size);\n    printf(\"인접 행렬 =\\n\");\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        printArray(graph-&gt;adjMat[i], graph-&gt;size);\n    }\n}\n</code></pre> graph_adjacency_matrix.kt<pre><code>/* 인접 행렬로 구현한 무방향 그래프 클래스 */\nclass GraphAdjMat(vertices: IntArray, edges: Array&lt;IntArray&gt;) {\n    val vertices = mutableListOf&lt;Int&gt;() // 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    val adjMat = mutableListOf&lt;MutableList&lt;Int&gt;&gt;() // 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n\n    /* 생성자 */\n    init {\n        // 정점 추가\n        for (vertex in vertices) {\n            addVertex(vertex)\n        }\n        // 간선 추가\n        // edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n        for (edge in edges) {\n            addEdge(edge[0], edge[1])\n        }\n    }\n\n    /* 정점 개수 반환 */\n    fun size(): Int {\n        return vertices.size\n    }\n\n    /* 정점 추가 */\n    fun addVertex(_val: Int) {\n        val n = size()\n        // 정점 리스트에 새 정점의 값을 추가한다\n        vertices.add(_val)\n        // 인접 행렬에 행 하나를 추가한다\n        val newRow = mutableListOf&lt;Int&gt;()\n        for (j in 0..&lt;n) {\n            newRow.add(0)\n        }\n        adjMat.add(newRow)\n        // 인접 행렬에 열 하나를 추가한다\n        for (row in adjMat) {\n            row.add(0)\n        }\n    }\n\n    /* 정점 삭제 */\n    fun removeVertex(index: Int) {\n        if (index &gt;= size())\n            throw IndexOutOfBoundsException()\n        // 정점 리스트에서 인덱스 index의 정점을 제거한다\n        vertices.removeAt(index)\n        // 인접 행렬에서 인덱스 index의 행을 삭제한다\n        adjMat.removeAt(index)\n        // 인접 행렬에서 인덱스 index의 열을 삭제한다\n        for (row in adjMat) {\n            row.removeAt(index)\n        }\n    }\n\n    /* 간선 추가 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    fun addEdge(i: Int, j: Int) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw IndexOutOfBoundsException()\n        // 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* 간선 삭제 */\n    // 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    fun removeEdge(i: Int, j: Int) {\n        // 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw IndexOutOfBoundsException()\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* 인접 행렬 출력 */\n    fun print() {\n        print(\"정점 리스트 = \")\n        println(vertices)\n        println(\"인접 행렬 =\")\n        printMatrix(adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.rb<pre><code>### 인접 행렬로 구현한 무방향 그래프 클래스 ###\nclass GraphAdjMat\n  def initialize(vertices, edges)\n    ### 생성자 ###\n    # 정점 리스트의 원소는 정점 값을, 인덱스는 정점 인덱스를 나타낸다\n    @vertices = []\n    # 인접 행렬의 행과 열 인덱스는 정점 인덱스에 해당한다\n    @adj_mat = []\n    # 정점 추가\n    vertices.each { |val| add_vertex(val) }\n    # 간선 추가\n    # edges의 원소는 정점 인덱스, 즉 vertices 원소의 인덱스를 나타낸다\n    edges.each { |e| add_edge(e[0], e[1]) }\n  end\n\n  ### 정점 개수 반환 ###\n  def size\n    @vertices.length\n  end\n\n  ### 정점 추가 ###\n  def add_vertex(val)\n    n = size\n    # 정점 리스트에 새 정점의 값을 추가한다\n    @vertices &lt;&lt; val\n    # 인접 행렬에 행 하나를 추가한다\n    new_row = Array.new(n, 0)\n    @adj_mat &lt;&lt; new_row\n    # 인접 행렬에 열 하나를 추가한다\n    @adj_mat.each { |row| row &lt;&lt; 0 }\n  end\n\n  ### 정점 삭제 ###\n  def remove_vertex(index)\n    raise IndexError if index &gt;= size\n\n    # 정점 리스트에서 인덱스 index의 정점을 제거한다\n    @vertices.delete_at(index)\n    # 인접 행렬에서 인덱스 index의 행을 삭제한다\n    @adj_mat.delete_at(index)\n    # 인접 행렬에서 인덱스 index의 열을 삭제한다\n    @adj_mat.each { |row| row.delete_at(index) }\n  end\n\n  ### 간선 추가 ###\n  def add_edge(i, j)\n    # 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    # 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n    if i &lt; 0 || j &lt; 0 || i &gt;= size || j &gt;= size || i == j\n      raise IndexError\n    end\n    # 무방향 그래프의 인접 행렬은 주대각선에 대해 대칭이며, (i, j) == (j, i)를 만족한다\n    @adj_mat[i][j] = 1\n    @adj_mat[j][i] = 1\n  end\n\n  ### 간선 삭제 ###\n  def remove_edge(i, j)\n    # 매개변수 i, j는 vertices 원소의 인덱스에 해당한다\n    # 인덱스가 범위를 벗어나거나 서로 같은 경우를 처리한다\n    if i &lt; 0 || j &lt; 0 || i &gt;= size || j &gt;= size || i == j\n      raise IndexError\n    end\n    @adj_mat[i][j] = 0\n    @adj_mat[j][i] = 0\n  end\n\n  ### 인접 행렬 출력 ###\n  def __print__\n    puts \"정점 리스트 = #{@vertices}\"\n    puts '인접 행렬 ='\n    print_matrix(@adj_mat)\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 9 장   그래프","9.2   그래프의 기본 연산"],"tags":[]},{"location":"chapter_graph/graph_operations/#922","level":2,"title":"9.2.2   인접 리스트 기반 구현","text":"<p>무방향 그래프의 전체 정점 수가 \\(n\\), 전체 간선 수가 \\(m\\)이라면 아래 그림(9-8)의 방법으로 각 연산을 구현할 수 있다.</p> <ul> <li>간선 추가: 해당 정점에 대응하는 연결 리스트 끝에 간선을 추가한다. \\(O(1)\\) 시간이 걸린다. 무방향 그래프이므로 두 방향의 간선을 함께 추가해야 한다.</li> <li>간선 삭제: 해당 정점에 대응하는 연결 리스트에서 지정한 간선을 찾아 삭제한다. \\(O(m)\\) 시간이 걸린다. 무방향 그래프에서는 두 방향의 간선을 함께 삭제해야 한다.</li> <li>정점 추가: 인접 리스트에 연결 리스트 하나를 추가하고 새 정점을 그 연결 리스트의 머리 노드로 둔다. \\(O(1)\\) 시간이 걸린다.</li> <li>정점 삭제: 인접 리스트 전체를 순회하며 지정한 정점이 포함된 모든 간선을 삭제한다. \\(O(n + m)\\) 시간이 걸린다.</li> <li>초기화: 인접 리스트에 정점 \\(n\\) 개와 간선 \\(2m\\) 개를 만든다. \\(O(n + m)\\) 시간이 걸린다.</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 9-8   인접 리스트의 초기화, 간선 추가와 삭제, 정점 추가와 삭제 </p> <p>다음은 인접 리스트의 구현 코드다. 위 그림(9-8)과 비교하면 실제 코드에는 다음 차이가 있다.</p> <ul> <li>정점을 편리하게 추가하고 삭제하며 코드를 간단하게 만들기 위해 연결 리스트 대신 리스트(동적 배열)를 사용한다.</li> <li>해시 테이블로 인접 리스트를 저장한다. <code>key</code>는 정점 인스턴스이고, <code>value</code>는 그 정점의 인접 정점 리스트(연결 리스트)다.</li> </ul> <p>또한 인접 리스트에서 정점을 나타내기 위해 <code>Vertex</code> 클래스를 사용한다. 인접 행렬처럼 리스트 인덱스로 정점을 구분한다면 인덱스가 \\(i\\)인 정점을 삭제할 때 인접 리스트 전체를 순회하면서 \\(i\\)보다 큰 모든 인덱스를 \\(1\\)씩 줄여야 하므로 매우 비효율적이기 때문이다. 반면 정점마다 고유한 <code>Vertex</code> 인스턴스를 사용하면 어느 정점을 삭제해도 다른 정점을 수정할 필요가 없다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_adjacency_list.py<pre><code>class GraphAdjList:\n    \"\"\"인접 리스트로 구현한 무방향 그래프 클래스\"\"\"\n\n    def __init__(self, edges: list[list[Vertex]]):\n        \"\"\"생성자\"\"\"\n        # 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n        self.adj_list = dict[Vertex, list[Vertex]]()\n        # 모든 정점과 간선을 추가한다\n        for edge in edges:\n            self.add_vertex(edge[0])\n            self.add_vertex(edge[1])\n            self.add_edge(edge[0], edge[1])\n\n    def size(self) -&gt; int:\n        \"\"\"정점 개수 반환\"\"\"\n        return len(self.adj_list)\n\n    def add_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"간선 추가\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # 간선 vet1 - vet2를 추가한다\n        self.adj_list[vet1].append(vet2)\n        self.adj_list[vet2].append(vet1)\n\n    def remove_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"간선 삭제\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # 간선 vet1 - vet2를 삭제한다\n        self.adj_list[vet1].remove(vet2)\n        self.adj_list[vet2].remove(vet1)\n\n    def add_vertex(self, vet: Vertex):\n        \"\"\"정점 추가\"\"\"\n        if vet in self.adj_list:\n            return\n        # 인접 리스트에 새 연결 리스트를 추가한다\n        self.adj_list[vet] = []\n\n    def remove_vertex(self, vet: Vertex):\n        \"\"\"정점 삭제\"\"\"\n        if vet not in self.adj_list:\n            raise ValueError()\n        # 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        self.adj_list.pop(vet)\n        # 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for vertex in self.adj_list:\n            if vet in self.adj_list[vertex]:\n                self.adj_list[vertex].remove(vet)\n\n    def print(self):\n        \"\"\"인접 리스트 출력\"\"\"\n        print(\"인접 리스트 =\")\n        for vertex in self.adj_list:\n            tmp = [v.val for v in self.adj_list[vertex]]\n            print(f\"{vertex.val}: {tmp},\")\n</code></pre> graph_adjacency_list.cpp<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList {\n  public:\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    unordered_map&lt;Vertex *, vector&lt;Vertex *&gt;&gt; adjList;\n\n    /* vector에서 지정한 노드 삭제 */\n    void remove(vector&lt;Vertex *&gt; &amp;vec, Vertex *vet) {\n        for (int i = 0; i &lt; vec.size(); i++) {\n            if (vec[i] == vet) {\n                vec.erase(vec.begin() + i);\n                break;\n            }\n        }\n    }\n\n    /* 생성자 */\n    GraphAdjList(const vector&lt;vector&lt;Vertex *&gt;&gt; &amp;edges) {\n        // 모든 정점과 간선을 추가한다\n        for (const vector&lt;Vertex *&gt; &amp;edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    int size() {\n        return adjList.size();\n    }\n\n    /* 간선 추가 */\n    void addEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"정점이 존재하지 않는다\");\n        // 간선 vet1 - vet2를 추가한다\n        adjList[vet1].push_back(vet2);\n        adjList[vet2].push_back(vet1);\n    }\n\n    /* 간선 삭제 */\n    void removeEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"정점이 존재하지 않는다\");\n        // 간선 vet1 - vet2를 삭제한다\n        remove(adjList[vet1], vet2);\n        remove(adjList[vet2], vet1);\n    }\n\n    /* 정점 추가 */\n    void addVertex(Vertex *vet) {\n        if (adjList.count(vet))\n            return;\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        adjList[vet] = vector&lt;Vertex *&gt;();\n    }\n\n    /* 정점 삭제 */\n    void removeVertex(Vertex *vet) {\n        if (!adjList.count(vet))\n            throw invalid_argument(\"정점이 존재하지 않는다\");\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        adjList.erase(vet);\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for (auto &amp;adj : adjList) {\n            remove(adj.second, vet);\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    void print() {\n        cout &lt;&lt; \"인접 리스트 =\" &lt;&lt; endl;\n        for (auto &amp;adj : adjList) {\n            const auto &amp;key = adj.first;\n            const auto &amp;vec = adj.second;\n            cout &lt;&lt; key-&gt;val &lt;&lt; \": \";\n            printVector(vetsToVals(vec));\n        }\n    }\n};\n</code></pre> graph_adjacency_list.java<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* 생성자 */\n    public GraphAdjList(Vertex[][] edges) {\n        this.adjList = new HashMap&lt;&gt;();\n        // 모든 정점과 간선을 추가한다\n        for (Vertex[] edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    public int size() {\n        return adjList.size();\n    }\n\n    /* 간선 추가 */\n    public void addEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // 간선 vet1 - vet2를 추가한다\n        adjList.get(vet1).add(vet2);\n        adjList.get(vet2).add(vet1);\n    }\n\n    /* 간선 삭제 */\n    public void removeEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // 간선 vet1 - vet2를 삭제한다\n        adjList.get(vet1).remove(vet2);\n        adjList.get(vet2).remove(vet1);\n    }\n\n    /* 정점 추가 */\n    public void addVertex(Vertex vet) {\n        if (adjList.containsKey(vet))\n            return;\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        adjList.put(vet, new ArrayList&lt;&gt;());\n    }\n\n    /* 정점 삭제 */\n    public void removeVertex(Vertex vet) {\n        if (!adjList.containsKey(vet))\n            throw new IllegalArgumentException();\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        adjList.remove(vet);\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for (List&lt;Vertex&gt; list : adjList.values()) {\n            list.remove(vet);\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    public void print() {\n        System.out.println(\"인접 리스트 =\");\n        for (Map.Entry&lt;Vertex, List&lt;Vertex&gt;&gt; pair : adjList.entrySet()) {\n            List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n            for (Vertex vertex : pair.getValue())\n                tmp.add(vertex.val);\n            System.out.println(pair.getKey().val + \": \" + tmp + \",\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.cs<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    public Dictionary&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* 생성자 */\n    public GraphAdjList(Vertex[][] edges) {\n        adjList = [];\n        // 모든 정점과 간선을 추가한다\n        foreach (Vertex[] edge in edges) {\n            AddVertex(edge[0]);\n            AddVertex(edge[1]);\n            AddEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    int Size() {\n        return adjList.Count;\n    }\n\n    /* 간선 추가 */\n    public void AddEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // 간선 vet1 - vet2를 추가한다\n        adjList[vet1].Add(vet2);\n        adjList[vet2].Add(vet1);\n    }\n\n    /* 간선 삭제 */\n    public void RemoveEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // 간선 vet1 - vet2를 삭제한다\n        adjList[vet1].Remove(vet2);\n        adjList[vet2].Remove(vet1);\n    }\n\n    /* 정점 추가 */\n    public void AddVertex(Vertex vet) {\n        if (adjList.ContainsKey(vet))\n            return;\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        adjList.Add(vet, []);\n    }\n\n    /* 정점 삭제 */\n    public void RemoveVertex(Vertex vet) {\n        if (!adjList.ContainsKey(vet))\n            throw new InvalidOperationException();\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        adjList.Remove(vet);\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        foreach (List&lt;Vertex&gt; list in adjList.Values) {\n            list.Remove(vet);\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    public void Print() {\n        Console.WriteLine(\"인접 리스트 =\");\n        foreach (KeyValuePair&lt;Vertex, List&lt;Vertex&gt;&gt; pair in adjList) {\n            List&lt;int&gt; tmp = [];\n            foreach (Vertex vertex in pair.Value)\n                tmp.Add(vertex.val);\n            Console.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.go<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\ntype graphAdjList struct {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    adjList map[Vertex][]Vertex\n}\n\n/* 생성자 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\n    g := &amp;graphAdjList{\n        adjList: make(map[Vertex][]Vertex),\n    }\n    // 모든 정점과 간선을 추가한다\n    for _, edge := range edges {\n        g.addVertex(edge[0])\n        g.addVertex(edge[1])\n        g.addEdge(edge[0], edge[1])\n    }\n    return g\n}\n\n/* 정점 개수 반환 */\nfunc (g *graphAdjList) size() int {\n    return len(g.adjList)\n}\n\n/* 간선 추가 */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"오류\")\n    }\n    // 간선 vet1 - vet2를 추가하고 익명 struct{}를 추가한다\n    g.adjList[vet1] = append(g.adjList[vet1], vet2)\n    g.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n\n/* 간선 삭제 */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"오류\")\n    }\n    // 간선 vet1 - vet2를 삭제한다\n    g.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2)\n    g.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1)\n}\n\n/* 정점 추가 */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if ok {\n        return\n    }\n    // 인접 리스트에 새 연결 리스트를 추가한다\n    g.adjList[vet] = make([]Vertex, 0)\n}\n\n/* 정점 삭제 */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if !ok {\n        panic(\"오류\")\n    }\n    // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n    delete(g.adjList, vet)\n    // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n    for v, list := range g.adjList {\n        g.adjList[v] = DeleteSliceElms(list, vet)\n    }\n}\n\n/* 인접 리스트 출력 */\nfunc (g *graphAdjList) print() {\n    var builder strings.Builder\n    fmt.Printf(\"인접 리스트 = \\n\")\n    for k, v := range g.adjList {\n        builder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\n        for _, vet := range v {\n            builder.WriteString(strconv.Itoa(vet.Val) + \" \")\n        }\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> graph_adjacency_list.swift<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    public private(set) var adjList: [Vertex: [Vertex]]\n\n    /* 생성자 */\n    public init(edges: [[Vertex]]) {\n        adjList = [:]\n        // 모든 정점과 간선을 추가한다\n        for edge in edges {\n            addVertex(vet: edge[0])\n            addVertex(vet: edge[1])\n            addEdge(vet1: edge[0], vet2: edge[1])\n        }\n    }\n\n    /* 정점 개수 반환 */\n    public func size() -&gt; Int {\n        adjList.count\n    }\n\n    /* 간선 추가 */\n    public func addEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"인수 오류\")\n        }\n        // 간선 vet1 - vet2를 추가한다\n        adjList[vet1]?.append(vet2)\n        adjList[vet2]?.append(vet1)\n    }\n\n    /* 간선 삭제 */\n    public func removeEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"인수 오류\")\n        }\n        // 간선 vet1 - vet2를 삭제한다\n        adjList[vet1]?.removeAll { $0 == vet2 }\n        adjList[vet2]?.removeAll { $0 == vet1 }\n    }\n\n    /* 정점 추가 */\n    public func addVertex(vet: Vertex) {\n        if adjList[vet] != nil {\n            return\n        }\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        adjList[vet] = []\n    }\n\n    /* 정점 삭제 */\n    public func removeVertex(vet: Vertex) {\n        if adjList[vet] == nil {\n            fatalError(\"인수 오류\")\n        }\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        adjList.removeValue(forKey: vet)\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for key in adjList.keys {\n            adjList[key]?.removeAll { $0 == vet }\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    public func print() {\n        Swift.print(\"인접 리스트 =\")\n        for (vertex, list) in adjList {\n            let list = list.map { $0.val }\n            Swift.print(\"\\(vertex.val): \\(list),\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.js<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    adjList;\n\n    /* 생성자 */\n    constructor(edges) {\n        this.adjList = new Map();\n        // 모든 정점과 간선을 추가한다\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    size() {\n        return this.adjList.size;\n    }\n\n    /* 간선 추가 */\n    addEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('잘못된 인수');\n        }\n        // 간선 vet1 - vet2를 추가한다\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* 간선 삭제 */\n    removeEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2 ||\n            this.adjList.get(vet1).indexOf(vet2) === -1\n        ) {\n            throw new Error('잘못된 인수');\n        }\n        // 간선 vet1 - vet2를 삭제한다\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* 정점 추가 */\n    addVertex(vet) {\n        if (this.adjList.has(vet)) return;\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        this.adjList.set(vet, []);\n    }\n\n    /* 정점 삭제 */\n    removeVertex(vet) {\n        if (!this.adjList.has(vet)) {\n            throw new Error('잘못된 인수');\n        }\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        this.adjList.delete(vet);\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for (const set of this.adjList.values()) {\n            const index = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    print() {\n        console.log('인접 리스트 =');\n        for (const [key, value] of this.adjList) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.ts<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    adjList: Map&lt;Vertex, Vertex[]&gt;;\n\n    /* 생성자 */\n    constructor(edges: Vertex[][]) {\n        this.adjList = new Map();\n        // 모든 정점과 간선을 추가한다\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* 정점 개수 반환 */\n    size(): number {\n        return this.adjList.size;\n    }\n\n    /* 간선 추가 */\n    addEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('잘못된 인수');\n        }\n        // 간선 vet1 - vet2를 추가한다\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* 간선 삭제 */\n    removeEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2 ||\n            this.adjList.get(vet1).indexOf(vet2) === -1\n        ) {\n            throw new Error('잘못된 인수');\n        }\n        // 간선 vet1 - vet2를 삭제한다\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* 정점 추가 */\n    addVertex(vet: Vertex): void {\n        if (this.adjList.has(vet)) return;\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        this.adjList.set(vet, []);\n    }\n\n    /* 정점 삭제 */\n    removeVertex(vet: Vertex): void {\n        if (!this.adjList.has(vet)) {\n            throw new Error('잘못된 인수');\n        }\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        this.adjList.delete(vet);\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for (const set of this.adjList.values()) {\n            const index: number = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    print(): void {\n        console.log('인접 리스트 =');\n        for (const [key, value] of this.adjList.entries()) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.dart<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList {\n  // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n  Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList = {};\n\n  /* 생성자 */\n  GraphAdjList(List&lt;List&lt;Vertex&gt;&gt; edges) {\n    for (List&lt;Vertex&gt; edge in edges) {\n      addVertex(edge[0]);\n      addVertex(edge[1]);\n      addEdge(edge[0], edge[1]);\n    }\n  }\n\n  /* 정점 개수 반환 */\n  int size() {\n    return adjList.length;\n  }\n\n  /* 간선 추가 */\n  void addEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // 간선 vet1 - vet2를 추가한다\n    adjList[vet1]!.add(vet2);\n    adjList[vet2]!.add(vet1);\n  }\n\n  /* 간선 삭제 */\n  void removeEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // 간선 vet1 - vet2를 삭제한다\n    adjList[vet1]!.remove(vet2);\n    adjList[vet2]!.remove(vet1);\n  }\n\n  /* 정점 추가 */\n  void addVertex(Vertex vet) {\n    if (adjList.containsKey(vet)) return;\n    // 인접 리스트에 새 연결 리스트를 추가한다\n    adjList[vet] = [];\n  }\n\n  /* 정점 삭제 */\n  void removeVertex(Vertex vet) {\n    if (!adjList.containsKey(vet)) {\n      throw ArgumentError;\n    }\n    // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n    adjList.remove(vet);\n    // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n    adjList.forEach((key, value) {\n      value.remove(vet);\n    });\n  }\n\n  /* 인접 리스트 출력 */\n  void printAdjList() {\n    print(\"인접 리스트 =\");\n    adjList.forEach((key, value) {\n      List&lt;int&gt; tmp = [];\n      for (Vertex vertex in value) {\n        tmp.add(vertex.val);\n      }\n      print(\"${key.val}: $tmp,\");\n    });\n  }\n}\n</code></pre> graph_adjacency_list.rs<pre><code>/* 인접 리스트로 구현한 무방향 그래프 자료형 */\npub struct GraphAdjList {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    pub adj_list: HashMap&lt;Vertex, Vec&lt;Vertex&gt;&gt;, // 값 부분에는 HashSet&lt;Vertex&gt;가 더 적합할까?\n}\n\nimpl GraphAdjList {\n    /* 생성자 */\n    pub fn new(edges: Vec&lt;[Vertex; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjList {\n            adj_list: HashMap::new(),\n        };\n        // 모든 정점과 간선을 추가한다\n        for edge in edges {\n            graph.add_vertex(edge[0]);\n            graph.add_vertex(edge[1]);\n            graph.add_edge(edge[0], edge[1]);\n        }\n\n        graph\n    }\n\n    /* 정점 개수 반환 */\n    #[allow(unused)]\n    pub fn size(&amp;self) -&gt; usize {\n        self.adj_list.len()\n    }\n\n    /* 간선 추가 */\n    pub fn add_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if vet1 == vet2 {\n            panic!(\"값 오류\");\n        }\n        // 간선 vet1 - vet2를 추가한다\n        self.adj_list.entry(vet1).or_default().push(vet2);\n        self.adj_list.entry(vet2).or_default().push(vet1);\n    }\n\n    /* 간선 삭제 */\n    #[allow(unused)]\n    pub fn remove_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if vet1 == vet2 {\n            panic!(\"값 오류\");\n        }\n        // 간선 vet1 - vet2를 삭제한다\n        self.adj_list\n            .entry(vet1)\n            .and_modify(|v| v.retain(|&amp;e| e != vet2));\n        self.adj_list\n            .entry(vet2)\n            .and_modify(|v| v.retain(|&amp;e| e != vet1));\n    }\n\n    /* 정점 추가 */\n    pub fn add_vertex(&amp;mut self, vet: Vertex) {\n        if self.adj_list.contains_key(&amp;vet) {\n            return;\n        }\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        self.adj_list.insert(vet, vec![]);\n    }\n\n    /* 정점 삭제 */\n    #[allow(unused)]\n    pub fn remove_vertex(&amp;mut self, vet: Vertex) {\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        self.adj_list.remove(&amp;vet);\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for list in self.adj_list.values_mut() {\n            list.retain(|&amp;v| v != vet);\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    pub fn print(&amp;self) {\n        println!(\"인접 리스트 =\");\n        for (vertex, list) in &amp;self.adj_list {\n            let list = list.iter().map(|vertex| vertex.val).collect::&lt;Vec&lt;i32&gt;&gt;();\n            println!(\"{}: {:?},\", vertex.val, list);\n        }\n    }\n}\n</code></pre> graph_adjacency_list.c<pre><code>/* 노드 구조체 */\ntypedef struct AdjListNode {\n    Vertex *vertex;           // 정점\n    struct AdjListNode *next; // 다음 노드\n} AdjListNode;\n\n/* 정점에 해당하는 노드 찾기 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* 간선 추가 보조 함수 */\nvoid addEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *node = (AdjListNode *)malloc(sizeof(AdjListNode));\n    node-&gt;vertex = vet;\n    // 맨 앞에 삽입한다\n    node-&gt;next = head-&gt;next;\n    head-&gt;next = node;\n}\n\n/* 간선 삭제 보조 함수 */\nvoid removeEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *pre = head;\n    AdjListNode *cur = head-&gt;next;\n    // 연결 리스트에서 vet에 해당하는 노드를 찾는다\n    while (cur != NULL &amp;&amp; cur-&gt;vertex != vet) {\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n    if (cur == NULL)\n        return;\n    // 연결 리스트에서 vet에 해당하는 노드를 삭제한다\n    pre-&gt;next = cur-&gt;next;\n    // 메모리 해제\n    free(cur);\n}\n\n/* 인접 리스트로 구현한 무방향 그래프 클래스 */\ntypedef struct {\n    AdjListNode *heads[MAX_SIZE]; // 노드 배열\n    int size;                     // 노드 개수\n} GraphAdjList;\n\n/* 생성자 */\nGraphAdjList *newGraphAdjList() {\n    GraphAdjList *graph = (GraphAdjList *)malloc(sizeof(GraphAdjList));\n    if (!graph) {\n        return NULL;\n    }\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        graph-&gt;heads[i] = NULL;\n    }\n    return graph;\n}\n\n/* 소멸자 */\nvoid delGraphAdjList(GraphAdjList *graph) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        AdjListNode *cur = graph-&gt;heads[i];\n        while (cur != NULL) {\n            AdjListNode *next = cur-&gt;next;\n            if (cur != graph-&gt;heads[i]) {\n                free(cur);\n            }\n            cur = next;\n        }\n        free(graph-&gt;heads[i]-&gt;vertex);\n        free(graph-&gt;heads[i]);\n    }\n    free(graph);\n}\n\n/* 정점에 해당하는 노드 찾기 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* 간선 추가 */\nvoid addEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL &amp;&amp; head1 != head2);\n    // 간선 vet1 - vet2를 추가한다\n    addEdgeHelper(head1, vet2);\n    addEdgeHelper(head2, vet1);\n}\n\n/* 간선 삭제 */\nvoid removeEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL);\n    // 간선 vet1 - vet2를 삭제한다\n    removeEdgeHelper(head1, head2-&gt;vertex);\n    removeEdgeHelper(head2, head1-&gt;vertex);\n}\n\n/* 정점 추가 */\nvoid addVertex(GraphAdjList *graph, Vertex *vet) {\n    assert(graph != NULL &amp;&amp; graph-&gt;size &lt; MAX_SIZE);\n    AdjListNode *head = (AdjListNode *)malloc(sizeof(AdjListNode));\n    head-&gt;vertex = vet;\n    head-&gt;next = NULL;\n    // 인접 리스트에 새 연결 리스트를 추가한다\n    graph-&gt;heads[graph-&gt;size++] = head;\n}\n\n/* 정점 삭제 */\nvoid removeVertex(GraphAdjList *graph, Vertex *vet) {\n    AdjListNode *node = findNode(graph, vet);\n    assert(node != NULL);\n    // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n    AdjListNode *cur = node, *pre = NULL;\n    while (cur) {\n        pre = cur;\n        cur = cur-&gt;next;\n        free(pre);\n    }\n    // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        cur = graph-&gt;heads[i];\n        pre = NULL;\n        while (cur) {\n            pre = cur;\n            cur = cur-&gt;next;\n            if (cur &amp;&amp; cur-&gt;vertex == vet) {\n                pre-&gt;next = cur-&gt;next;\n                free(cur);\n                break;\n            }\n        }\n    }\n    // 해당 정점 뒤의 정점을 앞으로 옮겨 빈자리를 채운다\n    int i;\n    for (i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i] == node)\n            break;\n    }\n    for (int j = i; j &lt; graph-&gt;size - 1; j++) {\n        graph-&gt;heads[j] = graph-&gt;heads[j + 1];\n    }\n    graph-&gt;size--;\n    free(vet);\n}\n</code></pre> graph_adjacency_list.kt<pre><code>/* 인접 리스트로 구현한 무방향 그래프 클래스 */\nclass GraphAdjList(edges: Array&lt;Array&lt;Vertex?&gt;&gt;) {\n    // 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    val adjList = HashMap&lt;Vertex, MutableList&lt;Vertex&gt;&gt;()\n\n    /* 생성자 */\n    init {\n        // 모든 정점과 간선을 추가한다\n        for (edge in edges) {\n            addVertex(edge[0]!!)\n            addVertex(edge[1]!!)\n            addEdge(edge[0]!!, edge[1]!!)\n        }\n    }\n\n    /* 정점 개수 반환 */\n    fun size(): Int {\n        return adjList.size\n    }\n\n    /* 간선 추가 */\n    fun addEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // 간선 vet1 - vet2를 추가한다\n        adjList[vet1]?.add(vet2)\n        adjList[vet2]?.add(vet1)\n    }\n\n    /* 간선 삭제 */\n    fun removeEdge(vet1: Vertex, vet2: Vertex) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw IllegalArgumentException()\n        // 간선 vet1 - vet2를 삭제한다\n        adjList[vet1]?.remove(vet2)\n        adjList[vet2]?.remove(vet1)\n    }\n\n    /* 정점 추가 */\n    fun addVertex(vet: Vertex) {\n        if (adjList.containsKey(vet))\n            return\n        // 인접 리스트에 새 연결 리스트를 추가한다\n        adjList[vet] = mutableListOf()\n    }\n\n    /* 정점 삭제 */\n    fun removeVertex(vet: Vertex) {\n        if (!adjList.containsKey(vet))\n            throw IllegalArgumentException()\n        // 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n        adjList.remove(vet)\n        // 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n        for (list in adjList.values) {\n            list.remove(vet)\n        }\n    }\n\n    /* 인접 리스트 출력 */\n    fun print() {\n        println(\"인접 리스트 =\")\n        for (pair in adjList.entries) {\n            val tmp = mutableListOf&lt;Int&gt;()\n            for (vertex in pair.value) {\n                tmp.add(vertex._val)\n            }\n            println(\"${pair.key._val}: $tmp,\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.rb<pre><code>### 인접 리스트로 구현한 무방향 그래프 클래스 ###\nclass GraphAdjList\n  attr_reader :adj_list\n\n  ### 생성자 ###\n  def initialize(edges)\n    # 인접 리스트의 key: 정점, value: 해당 정점의 모든 인접 정점\n    @adj_list = {}\n    # 모든 정점과 간선을 추가한다\n    for edge in edges\n      add_vertex(edge[0])\n      add_vertex(edge[1])\n      add_edge(edge[0], edge[1])\n    end\n  end\n\n  ### 정점 개수 반환 ###\n  def size\n    @adj_list.length\n  end\n\n  ### 간선 추가 ###\n  def add_edge(vet1, vet2)\n    raise ArgumentError if !@adj_list.include?(vet1) || !@adj_list.include?(vet2)\n\n    @adj_list[vet1] &lt;&lt; vet2\n    @adj_list[vet2] &lt;&lt; vet1\n  end\n\n  ### 간선 삭제 ###\n  def remove_edge(vet1, vet2)\n    raise ArgumentError if !@adj_list.include?(vet1) || !@adj_list.include?(vet2)\n\n    # 간선 vet1 - vet2를 삭제한다\n    @adj_list[vet1].delete(vet2)\n    @adj_list[vet2].delete(vet1)\n  end\n\n  ### 정점 추가 ###\n  def add_vertex(vet)\n    return if @adj_list.include?(vet)\n\n    # 인접 리스트에 새 연결 리스트를 추가한다\n    @adj_list[vet] = []\n  end\n\n  ### 정점 삭제 ###\n  def remove_vertex(vet)\n    raise ArgumentError unless @adj_list.include?(vet)\n\n    # 인접 리스트에서 정점 vet에 해당하는 연결 리스트를 삭제한다\n    @adj_list.delete(vet)\n    # 다른 정점의 연결 리스트를 순회하여 vet를 포함하는 모든 간선을 삭제한다\n    for vertex in @adj_list\n      @adj_list[vertex.first].delete(vet) if @adj_list[vertex.first].include?(vet)\n    end\n  end\n\n  ### 인접 리스트 출력 ###\n  def __print__\n    puts '인접 리스트 ='\n    for vertex in @adj_list\n      tmp = @adj_list[vertex.first].map { |v| v.val }\n      puts \"#{vertex.first.val}: #{tmp},\"\n    end\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 9 장   그래프","9.2   그래프의 기본 연산"],"tags":[]},{"location":"chapter_graph/graph_operations/#923","level":2,"title":"9.2.3   효율 비교","text":"<p>그래프에 정점 \\(n\\) 개와 간선 \\(m\\) 개가 있다고 하자. 아래 표(9-2)는 인접 행렬과 인접 리스트의 시간 효율과 공간 효율을 비교한다. 여기서 인접 리스트(연결 리스트)는 이 절의 구현에 해당하며, 인접 리스트(해시 테이블)는 모든 연결 리스트를 해시 테이블로 바꾼 구현을 뜻한다.</p> <p> 표 9-2   인접 행렬과 인접 리스트 비교 </p> 인접 행렬 인접 리스트(연결 리스트) 인접 리스트(해시 테이블) 인접 여부 확인 \\(O(1)\\) \\(O(n)\\) \\(O(1)\\) 간선 추가 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) 간선 삭제 \\(O(1)\\) \\(O(n)\\) \\(O(1)\\) 정점 추가 \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) 정점 삭제 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) 메모리 사용량 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\) <p>위 표(9-2)만 보면 인접 리스트(해시 테이블)의 시간 효율과 공간 효율이 가장 좋아 보인다. 하지만 실제로는 배열 접근이나 대입 한 번이면 되는 인접 행렬의 간선 연산이 더 효율적이다. 종합하면 인접 행렬은 '공간을 더 써서 시간을 절약하는' 원리를, 인접 리스트는 '시간을 더 써서 공간을 절약하는' 원리를 보여 준다.</p>","path":["제 9 장   그래프","9.2   그래프의 기본 연산"],"tags":[]},{"location":"chapter_graph/graph_traversal/","level":1,"title":"9.3   그래프 순회","text":"<p>트리는 '일대다' 관계를 나타내지만, 그래프는 자유도가 더 높아 임의의 '다대다' 관계를 표현할 수 있다. 따라서 트리는 그래프의 특수한 경우로 볼 수 있다. 트리 순회 역시 그래프 순회의 특수한 경우다.</p> <p>그래프와 트리 모두 탐색 알고리즘을 이용하여 순회한다. 그래프 순회도 너비 우선 순회와 깊이 우선 순회의 두 가지로 나뉜다.</p>","path":["제 9 장   그래프","9.3   그래프 순회"],"tags":[]},{"location":"chapter_graph/graph_traversal/#931","level":2,"title":"9.3.1   너비 우선 순회","text":"<p>너비 우선 순회는 가까운 곳에서 먼 곳으로 진행하는 순회 방식이다. 한 노드에서 출발하여 항상 거리가 가장 가까운 정점을 먼저 방문하고, 한 층씩 바깥으로 확장한다. 아래 그림(9-9)에서는 왼쪽 위 정점에서 출발하여 해당 정점의 모든 인접 정점을 먼저 순회한 다음, 다음 정점의 모든 인접 정점을 순회한다. 이를 모든 정점을 방문할 때까지 반복한다.</p> <p></p> <p> 그림 9-9   그래프의 너비 우선 순회 </p>","path":["제 9 장   그래프","9.3   그래프 순회"],"tags":[]},{"location":"chapter_graph/graph_traversal/#1","level":3,"title":"1.   알고리즘 구현","text":"<p>BFS는 보통 큐로 구현하며, 코드는 다음과 같다. 큐의 '선입선출' 성질은 BFS의 '가까운 곳부터 먼 곳으로'라는 발상과 잘 맞는다.</p> <ol> <li>순회의 시작 정점 <code>startVet</code>를 큐에 넣고 반복문을 시작한다.</li> <li>매 반복에서 큐의 맨 앞 정점을 꺼내 방문을 기록한 다음, 해당 정점의 모든 인접 정점을 큐의 끝에 넣는다.</li> <li>모든 정점을 방문할 때까지 <code>2.</code> 단계를 반복한 뒤 종료한다.</li> </ol> <p>같은 정점을 여러 번 순회하지 않도록 해시 집합 <code>visited</code>를 사용하여 방문한 노드를 기록한다.</p> <p>참고</p> <p>해시 집합은 <code>value</code> 없이 <code>key</code>만 저장하는 해시 테이블로 볼 수 있다. <code>key</code>의 추가, 삭제, 조회, 수정 연산을 \\(O(1)\\) 시간에 수행한다. <code>key</code>는 중복될 수 없으므로, 해시 집합은 데이터 중복 제거 등에 자주 사용된다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_bfs.py<pre><code>def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"너비 우선 순회\"\"\"\n    # 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\n    # 정점 순회 순서\n    res = []\n    # 방문한 정점을 기록하는 해시 집합\n    visited = set[Vertex]([start_vet])\n    # 큐를 사용하여 BFS를 구현한다\n    que = deque[Vertex]([start_vet])\n    # 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while len(que) &gt; 0:\n        vet = que.popleft()  # 큐 맨 앞의 정점을 꺼낸다\n        res.append(vet)  # 방문한 정점을 기록한다\n        # 해당 정점의 모든 인접 정점을 순회한다\n        for adj_vet in graph.adj_list[vet]:\n            if adj_vet in visited:\n                continue  # 이미 방문한 정점은 건너뛴다\n            que.append(adj_vet)  # 방문하지 않은 정점만 큐에 삽입한다\n            visited.add(adj_vet)  # 해당 정점을 방문했다고 표시한다\n    # 정점 순회 순서를 반환한다\n    return res\n</code></pre> graph_bfs.cpp<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nvector&lt;Vertex *&gt; graphBFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // 정점 순회 순서\n    vector&lt;Vertex *&gt; res;\n    // 방문한 정점을 기록하는 해시 집합\n    unordered_set&lt;Vertex *&gt; visited = {startVet};\n    // 큐를 사용하여 BFS를 구현한다\n    queue&lt;Vertex *&gt; que;\n    que.push(startVet);\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while (!que.empty()) {\n        Vertex *vet = que.front();\n        que.pop();          // 큐 맨 앞의 정점을 꺼낸다\n        res.push_back(vet); // 방문한 정점을 기록한다\n        // 해당 정점의 모든 인접 정점을 순회한다\n        for (auto adjVet : graph.adjList[vet]) {\n            if (visited.count(adjVet))\n                continue;            // 이미 방문한 정점은 건너뛴다\n            que.push(adjVet);        // 방문하지 않은 정점만 큐에 삽입한다\n            visited.emplace(adjVet); // 해당 정점을 방문했다고 표시한다\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    return res;\n}\n</code></pre> graph_bfs.java<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n    // 정점 순회 순서\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // 방문한 정점을 기록하는 해시 집합\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    visited.add(startVet);\n    // 큐를 사용하여 BFS를 구현한다\n    Queue&lt;Vertex&gt; que = new LinkedList&lt;&gt;();\n    que.offer(startVet);\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while (!que.isEmpty()) {\n        Vertex vet = que.poll(); // 큐 맨 앞의 정점을 꺼낸다\n        res.add(vet);            // 방문한 정점을 기록한다\n        // 해당 정점의 모든 인접 정점을 순회한다\n        for (Vertex adjVet : graph.adjList.get(vet)) {\n            if (visited.contains(adjVet))\n                continue;        // 이미 방문한 정점은 건너뛴다\n            que.offer(adjVet);   // 방문하지 않은 정점만 큐에 삽입한다\n            visited.add(adjVet); // 해당 정점을 방문했다고 표시한다\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    return res;\n}\n</code></pre> graph_bfs.cs<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nList&lt;Vertex&gt; GraphBFS(GraphAdjList graph, Vertex startVet) {\n    // 정점 순회 순서\n    List&lt;Vertex&gt; res = [];\n    // 방문한 정점을 기록하는 해시 집합\n    HashSet&lt;Vertex&gt; visited = [startVet];\n    // 큐를 사용하여 BFS를 구현한다\n    Queue&lt;Vertex&gt; que = new();\n    que.Enqueue(startVet);\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while (que.Count &gt; 0) {\n        Vertex vet = que.Dequeue(); // 큐 맨 앞의 정점을 꺼낸다\n        res.Add(vet);               // 방문한 정점을 기록한다\n        foreach (Vertex adjVet in graph.adjList[vet]) {\n            if (visited.Contains(adjVet)) {\n                continue;          // 이미 방문한 정점은 건너뛴다\n            }\n            que.Enqueue(adjVet);   // 방문하지 않은 정점만 큐에 삽입한다\n            visited.Add(adjVet);   // 해당 정점을 방문했다고 표시한다\n        }\n    }\n\n    // 정점 순회 순서를 반환한다\n    return res;\n}\n</code></pre> graph_bfs.go<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // 정점 순회 순서\n    res := make([]Vertex, 0)\n    // 방문한 정점을 기록하는 해시 집합\n    visited := make(map[Vertex]struct{})\n    visited[startVet] = struct{}{}\n    // 큐로 BFS를 구현하며, 슬라이스로 큐를 구현한다\n    queue := make([]Vertex, 0)\n    queue = append(queue, startVet)\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    for len(queue) &gt; 0 {\n        // 큐 맨 앞의 정점을 꺼낸다\n        vet := queue[0]\n        queue = queue[1:]\n        // 방문한 정점을 기록한다\n        res = append(res, vet)\n        // 해당 정점의 모든 인접 정점을 순회한다\n        for _, adjVet := range g.adjList[vet] {\n            _, isExist := visited[adjVet]\n            // 방문하지 않은 정점만 큐에 삽입한다\n            if !isExist {\n                queue = append(queue, adjVet)\n                visited[adjVet] = struct{}{}\n            }\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    return res\n}\n</code></pre> graph_bfs.swift<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // 정점 순회 순서\n    var res: [Vertex] = []\n    // 방문한 정점을 기록하는 해시 집합\n    var visited: Set&lt;Vertex&gt; = [startVet]\n    // 큐를 사용하여 BFS를 구현한다\n    var que: [Vertex] = [startVet]\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while !que.isEmpty {\n        let vet = que.removeFirst() // 큐 맨 앞의 정점을 꺼낸다\n        res.append(vet) // 방문한 정점을 기록한다\n        // 해당 정점의 모든 인접 정점을 순회한다\n        for adjVet in graph.adjList[vet] ?? [] {\n            if visited.contains(adjVet) {\n                continue // 이미 방문한 정점은 건너뛴다\n            }\n            que.append(adjVet) // 방문하지 않은 정점만 큐에 삽입한다\n            visited.insert(adjVet) // 해당 정점을 방문했다고 표시한다\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    return res\n}\n</code></pre> graph_bfs.js<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunction graphBFS(graph, startVet) {\n    // 정점 순회 순서\n    const res = [];\n    // 방문한 정점을 기록하는 해시 집합\n    const visited = new Set();\n    visited.add(startVet);\n    // 큐를 사용하여 BFS를 구현한다\n    const que = [startVet];\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while (que.length) {\n        const vet = que.shift(); // 큐 맨 앞의 정점을 꺼낸다\n        res.push(vet); // 방문한 정점을 기록한다\n        // 해당 정점의 모든 인접 정점을 순회한다\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // 이미 방문한 정점은 건너뛴다\n            }\n            que.push(adjVet); // 방문하지 않은 정점만 큐에 삽입한다\n            visited.add(adjVet); // 해당 정점을 방문했다고 표시한다\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    return res;\n}\n</code></pre> graph_bfs.ts<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // 정점 순회 순서\n    const res: Vertex[] = [];\n    // 방문한 정점을 기록하는 해시 집합\n    const visited: Set&lt;Vertex&gt; = new Set();\n    visited.add(startVet);\n    // 큐를 사용하여 BFS를 구현한다\n    const que = [startVet];\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while (que.length) {\n        const vet = que.shift(); // 큐 맨 앞의 정점을 꺼낸다\n        res.push(vet); // 방문한 정점을 기록한다\n        // 해당 정점의 모든 인접 정점을 순회한다\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // 이미 방문한 정점은 건너뛴다\n            }\n            que.push(adjVet); // 방문하지 않은 정점만 큐에 삽입한다\n            visited.add(adjVet); // 해당 정점을 방문했다고 표시한다\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    return res;\n}\n</code></pre> graph_bfs.dart<pre><code>/* 너비 우선 순회 */\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n  // 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\n  // 정점 순회 순서\n  List&lt;Vertex&gt; res = [];\n  // 방문한 정점을 기록하는 해시 집합\n  Set&lt;Vertex&gt; visited = {};\n  visited.add(startVet);\n  // 큐를 사용하여 BFS를 구현한다\n  Queue&lt;Vertex&gt; que = Queue();\n  que.add(startVet);\n  // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n  while (que.isNotEmpty) {\n    Vertex vet = que.removeFirst(); // 큐 맨 앞의 정점을 꺼낸다\n    res.add(vet); // 방문한 정점을 기록한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for (Vertex adjVet in graph.adjList[vet]!) {\n      if (visited.contains(adjVet)) {\n        continue; // 이미 방문한 정점은 건너뛴다\n      }\n      que.add(adjVet); // 방문하지 않은 정점만 큐에 삽입한다\n      visited.add(adjVet); // 해당 정점을 방문했다고 표시한다\n    }\n  }\n  // 정점 순회 순서를 반환한다\n  return res;\n}\n</code></pre> graph_bfs.rs<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfn graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // 정점 순회 순서\n    let mut res = vec![];\n    // 방문한 정점을 기록하는 해시 집합\n    let mut visited = HashSet::new();\n    visited.insert(start_vet);\n    // 큐를 사용하여 BFS를 구현한다\n    let mut que = VecDeque::new();\n    que.push_back(start_vet);\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while let Some(vet) = que.pop_front() {\n        res.push(vet); // 방문한 정점을 기록한다\n\n        // 해당 정점의 모든 인접 정점을 순회한다\n        if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n            for &amp;adj_vet in adj_vets {\n                if visited.contains(&amp;adj_vet) {\n                    continue; // 이미 방문한 정점은 건너뛴다\n                }\n                que.push_back(adj_vet); // 방문하지 않은 정점만 큐에 삽입한다\n                visited.insert(adj_vet); // 해당 정점을 방문했다고 표시한다\n            }\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    res\n}\n</code></pre> graph_bfs.c<pre><code>/* 노드 큐 구조체 */\ntypedef struct {\n    Vertex *vertices[MAX_SIZE];\n    int front, rear, size;\n} Queue;\n\n/* 생성자 */\nQueue *newQueue() {\n    Queue *q = (Queue *)malloc(sizeof(Queue));\n    q-&gt;front = q-&gt;rear = q-&gt;size = 0;\n    return q;\n}\n\n/* 큐가 비어 있는지 판단 */\nint isEmpty(Queue *q) {\n    return q-&gt;size == 0;\n}\n\n/* 큐 삽입 연산 */\nvoid enqueue(Queue *q, Vertex *vet) {\n    q-&gt;vertices[q-&gt;rear] = vet;\n    q-&gt;rear = (q-&gt;rear + 1) % MAX_SIZE;\n    q-&gt;size++;\n}\n\n/* 큐에서 꺼내는 연산 */\nVertex *dequeue(Queue *q) {\n    Vertex *vet = q-&gt;vertices[q-&gt;front];\n    q-&gt;front = (q-&gt;front + 1) % MAX_SIZE;\n    q-&gt;size--;\n    return vet;\n}\n\n/* 정점을 이미 방문했는지 확인 */\nint isVisited(Vertex **visited, int size, Vertex *vet) {\n    // O(n) 시간에 순회하여 노드를 찾는다\n    for (int i = 0; i &lt; size; i++) {\n        if (visited[i] == vet)\n            return 1;\n    }\n    return 0;\n}\n\n/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nvoid graphBFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize, Vertex **visited, int *visitedSize) {\n    // 큐를 사용하여 BFS를 구현한다\n    Queue *queue = newQueue();\n    enqueue(queue, startVet);\n    visited[(*visitedSize)++] = startVet;\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while (!isEmpty(queue)) {\n        Vertex *vet = dequeue(queue); // 큐 맨 앞의 정점을 꺼낸다\n        res[(*resSize)++] = vet;      // 방문한 정점을 기록한다\n        // 해당 정점의 모든 인접 정점을 순회한다\n        AdjListNode *node = findNode(graph, vet);\n        while (node != NULL) {\n            // 이미 방문한 정점은 건너뛴다\n            if (!isVisited(visited, *visitedSize, node-&gt;vertex)) {\n                enqueue(queue, node-&gt;vertex);             // 방문하지 않은 정점만 큐에 삽입한다\n                visited[(*visitedSize)++] = node-&gt;vertex; // 해당 정점을 방문했다고 표시한다\n            }\n            node = node-&gt;next;\n        }\n    }\n    // 메모리 해제\n    free(queue);\n}\n</code></pre> graph_bfs.kt<pre><code>/* 너비 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfun graphBFS(graph: GraphAdjList, startVet: Vertex): MutableList&lt;Vertex?&gt; {\n    // 정점 순회 순서\n    val res = mutableListOf&lt;Vertex?&gt;()\n    // 방문한 정점을 기록하는 해시 집합\n    val visited = HashSet&lt;Vertex&gt;()\n    visited.add(startVet)\n    // 큐를 사용하여 BFS를 구현한다\n    val que = LinkedList&lt;Vertex&gt;()\n    que.offer(startVet)\n    // 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n    while (!que.isEmpty()) {\n        val vet = que.poll() // 큐 맨 앞의 정점을 꺼낸다\n        res.add(vet)         // 방문한 정점을 기록한다\n        // 해당 정점의 모든 인접 정점을 순회한다\n        for (adjVet in graph.adjList[vet]!!) {\n            if (visited.contains(adjVet))\n                continue        // 이미 방문한 정점은 건너뛴다\n            que.offer(adjVet)   // 방문하지 않은 정점만 큐에 삽입한다\n            visited.add(adjVet) // 해당 정점을 방문했다고 표시한다\n        }\n    }\n    // 정점 순회 순서를 반환한다\n    return res\n}\n</code></pre> graph_bfs.rb<pre><code>### 너비 우선 순회 ###\ndef graph_bfs(graph, start_vet)\n  # 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\n  # 정점 순회 순서\n  res = []\n  # 방문한 정점을 기록하는 해시 집합\n  visited = Set.new([start_vet])\n  # 큐를 사용하여 BFS를 구현한다\n  que = [start_vet]\n  # 정점 vet에서 시작하여 모든 정점을 방문할 때까지 반복한다\n  while que.length &gt; 0\n    vet = que.shift # 큐 맨 앞의 정점을 꺼낸다\n    res &lt;&lt; vet # 방문한 정점을 기록한다\n    # 해당 정점의 모든 인접 정점을 순회한다\n    for adj_vet in graph.adj_list[vet]\n      next if visited.include?(adj_vet) # 이미 방문한 정점은 건너뛴다\n      que &lt;&lt; adj_vet # 방문하지 않은 정점만 큐에 삽입한다\n      visited.add(adj_vet) # 해당 정점을 방문했다고 표시한다\n    end\n  end\n  # 정점 순회 순서를 반환한다\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>코드만으로는 추상적일 수 있으므로 아래 그림(9-10)과 함께 살펴보며 이해를 깊이 다져 보자.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 9-10   그래프의 너비 우선 순회 단계 </p> <p>너비 우선 순회 순서는 유일한가?</p> <p>유일하지 않다. 너비 우선 순회는 '가까운 곳부터 먼 곳으로' 순회하기만 하면 되므로, 거리가 같은 여러 정점의 방문 순서는 자유롭게 바꿀 수 있다. 위 그림(9-10)에서는 정점 \\(1\\), \\(3\\)의 방문 순서를 바꿀 수 있고, 정점 \\(2\\), \\(4\\), \\(6\\)의 순서도 자유롭게 바꿀 수 있다.</p>","path":["제 9 장   그래프","9.3   그래프 순회"],"tags":[]},{"location":"chapter_graph/graph_traversal/#2","level":3,"title":"2.   복잡도 분석","text":"<p>시간 복잡도: 모든 정점은 큐에 한 번 들어갔다가 한 번 나오므로 \\(O(|V|)\\) 시간이 걸린다. 인접 정점을 순회할 때는 무방향 그래프이므로 모든 간선을 \\(2\\) 번씩 방문하여 \\(O(2|E|)\\) 시간이 걸린다. 전체 시간은 \\(O(|V| + |E|)\\)이다.</p> <p>공간 복잡도: 리스트 <code>res</code>, 해시 집합 <code>visited</code>, 큐 <code>que</code>에 저장되는 정점 수는 각각 최대 \\(|V|\\)이므로 \\(O(|V|)\\) 공간을 사용한다.</p>","path":["제 9 장   그래프","9.3   그래프 순회"],"tags":[]},{"location":"chapter_graph/graph_traversal/#932","level":2,"title":"9.3.2   깊이 우선 순회","text":"<p>깊이 우선 순회는 먼저 끝까지 나아가고, 더 갈 곳이 없으면 되돌아오는 순회 방식이다. 아래 그림(9-11)에서는 왼쪽 위 정점에서 출발하여 현재 정점의 인접 정점 하나를 방문한다. 끝에 도달하면 되돌아오고, 다시 끝까지 나아갔다가 돌아오는 과정을 모든 정점을 순회할 때까지 반복한다.</p> <p></p> <p> 그림 9-11   그래프의 깊이 우선 순회 </p>","path":["제 9 장   그래프","9.3   그래프 순회"],"tags":[]},{"location":"chapter_graph/graph_traversal/#1_1","level":3,"title":"1.   알고리즘 구현","text":"<p>'끝까지 갔다가 돌아오는' 알고리즘 패턴은 보통 재귀로 구현한다. 너비 우선 순회와 마찬가지로 깊이 우선 순회에서도 해시 집합 <code>visited</code>로 이미 방문한 정점을 기록하여 중복 방문을 방지한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby graph_dfs.py<pre><code>def dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n    \"\"\"깊이 우선 순회 보조 함수\"\"\"\n    res.append(vet)  # 방문한 정점을 기록한다\n    visited.add(vet)  # 해당 정점을 방문했다고 표시한다\n    # 해당 정점의 모든 인접 정점을 순회한다\n    for adjVet in graph.adj_list[vet]:\n        if adjVet in visited:\n            continue  # 이미 방문한 정점은 건너뛴다\n        # 인접 정점을 재귀적으로 방문한다\n        dfs(graph, visited, res, adjVet)\n\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"깊이 우선 순회\"\"\"\n    # 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\n    # 정점 순회 순서\n    res = []\n    # 방문한 정점을 기록하는 해시 집합\n    visited = set[Vertex]()\n    dfs(graph, visited, res, start_vet)\n    return res\n</code></pre> graph_dfs.cpp<pre><code>/* 깊이 우선 순회 보조 함수 */\nvoid dfs(GraphAdjList &amp;graph, unordered_set&lt;Vertex *&gt; &amp;visited, vector&lt;Vertex *&gt; &amp;res, Vertex *vet) {\n    res.push_back(vet);   // 방문한 정점을 기록한다\n    visited.emplace(vet); // 해당 정점을 방문했다고 표시한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for (Vertex *adjVet : graph.adjList[vet]) {\n        if (visited.count(adjVet))\n            continue; // 이미 방문한 정점은 건너뛴다\n        // 인접 정점을 재귀적으로 방문한다\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nvector&lt;Vertex *&gt; graphDFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // 정점 순회 순서\n    vector&lt;Vertex *&gt; res;\n    // 방문한 정점을 기록하는 해시 집합\n    unordered_set&lt;Vertex *&gt; visited;\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.java<pre><code>/* 깊이 우선 순회 보조 함수 */\nvoid dfs(GraphAdjList graph, Set&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.add(vet);     // 방문한 정점을 기록한다\n    visited.add(vet); // 해당 정점을 방문했다고 표시한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for (Vertex adjVet : graph.adjList.get(vet)) {\n        if (visited.contains(adjVet))\n            continue; // 이미 방문한 정점은 건너뛴다\n        // 인접 정점을 재귀적으로 방문한다\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n    // 정점 순회 순서\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // 방문한 정점을 기록하는 해시 집합\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.cs<pre><code>/* 깊이 우선 순회 보조 함수 */\nvoid DFS(GraphAdjList graph, HashSet&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.Add(vet);     // 방문한 정점을 기록한다\n    visited.Add(vet); // 해당 정점을 방문했다고 표시한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    foreach (Vertex adjVet in graph.adjList[vet]) {\n        if (visited.Contains(adjVet)) {\n            continue; // 이미 방문한 정점은 건너뛴다                             \n        }\n        // 인접 정점을 재귀적으로 방문한다\n        DFS(graph, visited, res, adjVet);\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nList&lt;Vertex&gt; GraphDFS(GraphAdjList graph, Vertex startVet) {\n    // 정점 순회 순서\n    List&lt;Vertex&gt; res = [];\n    // 방문한 정점을 기록하는 해시 집합\n    HashSet&lt;Vertex&gt; visited = [];\n    DFS(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.go<pre><code>/* 깊이 우선 순회 보조 함수 */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n    // append 연산은 새 참조를 반환하므로 원래 참조에 새 슬라이스의 참조를 다시 대입해야 한다\n    *res = append(*res, vet)\n    visited[vet] = struct{}{}\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for _, adjVet := range g.adjList[vet] {\n        _, isExist := visited[adjVet]\n        // 인접 정점을 재귀적으로 방문한다\n        if !isExist {\n            dfs(g, visited, res, adjVet)\n        }\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // 정점 순회 순서\n    res := make([]Vertex, 0)\n    // 방문한 정점을 기록하는 해시 집합\n    visited := make(map[Vertex]struct{})\n    dfs(g, visited, &amp;res, startVet)\n    // 정점 순회 순서를 반환한다\n    return res\n}\n</code></pre> graph_dfs.swift<pre><code>/* 깊이 우선 순회 보조 함수 */\nfunc dfs(graph: GraphAdjList, visited: inout Set&lt;Vertex&gt;, res: inout [Vertex], vet: Vertex) {\n    res.append(vet) // 방문한 정점을 기록한다\n    visited.insert(vet) // 해당 정점을 방문했다고 표시한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for adjVet in graph.adjList[vet] ?? [] {\n        if visited.contains(adjVet) {\n            continue // 이미 방문한 정점은 건너뛴다\n        }\n        // 인접 정점을 재귀적으로 방문한다\n        dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: adjVet)\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // 정점 순회 순서\n    var res: [Vertex] = []\n    // 방문한 정점을 기록하는 해시 집합\n    var visited: Set&lt;Vertex&gt; = []\n    dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: startVet)\n    return res\n}\n</code></pre> graph_dfs.js<pre><code>/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunction dfs(graph, visited, res, vet) {\n    res.push(vet); // 방문한 정점을 기록한다\n    visited.add(vet); // 해당 정점을 방문했다고 표시한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // 이미 방문한 정점은 건너뛴다\n        }\n        // 인접 정점을 재귀적으로 방문한다\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunction graphDFS(graph, startVet) {\n    // 정점 순회 순서\n    const res = [];\n    // 방문한 정점을 기록하는 해시 집합\n    const visited = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.ts<pre><code>/* 깊이 우선 순회 보조 함수 */\nfunction dfs(\n    graph: GraphAdjList,\n    visited: Set&lt;Vertex&gt;,\n    res: Vertex[],\n    vet: Vertex\n): void {\n    res.push(vet); // 방문한 정점을 기록한다\n    visited.add(vet); // 해당 정점을 방문했다고 표시한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // 이미 방문한 정점은 건너뛴다\n        }\n        // 인접 정점을 재귀적으로 방문한다\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // 정점 순회 순서\n    const res: Vertex[] = [];\n    // 방문한 정점을 기록하는 해시 집합\n    const visited: Set&lt;Vertex&gt; = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.dart<pre><code>/* 깊이 우선 순회 보조 함수 */\nvoid dfs(\n  GraphAdjList graph,\n  Set&lt;Vertex&gt; visited,\n  List&lt;Vertex&gt; res,\n  Vertex vet,\n) {\n  res.add(vet); // 방문한 정점을 기록한다\n  visited.add(vet); // 해당 정점을 방문했다고 표시한다\n  // 해당 정점의 모든 인접 정점을 순회한다\n  for (Vertex adjVet in graph.adjList[vet]!) {\n    if (visited.contains(adjVet)) {\n      continue; // 이미 방문한 정점은 건너뛴다\n    }\n    // 인접 정점을 재귀적으로 방문한다\n    dfs(graph, visited, res, adjVet);\n  }\n}\n\n/* 깊이 우선 순회 */\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n  // 정점 순회 순서\n  List&lt;Vertex&gt; res = [];\n  // 방문한 정점을 기록하는 해시 집합\n  Set&lt;Vertex&gt; visited = {};\n  dfs(graph, visited, res, startVet);\n  return res;\n}\n</code></pre> graph_dfs.rs<pre><code>/* 깊이 우선 순회 보조 함수 */\nfn dfs(graph: &amp;GraphAdjList, visited: &amp;mut HashSet&lt;Vertex&gt;, res: &amp;mut Vec&lt;Vertex&gt;, vet: Vertex) {\n    res.push(vet); // 방문한 정점을 기록한다\n    visited.insert(vet); // 해당 정점을 방문했다고 표시한다\n                         // 해당 정점의 모든 인접 정점을 순회한다\n    if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n        for &amp;adj_vet in adj_vets {\n            if visited.contains(&amp;adj_vet) {\n                continue; // 이미 방문한 정점은 건너뛴다\n            }\n            // 인접 정점을 재귀적으로 방문한다\n            dfs(graph, visited, res, adj_vet);\n        }\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfn graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // 정점 순회 순서\n    let mut res = vec![];\n    // 방문한 정점을 기록하는 해시 집합\n    let mut visited = HashSet::new();\n    dfs(&amp;graph, &amp;mut visited, &amp;mut res, start_vet);\n\n    res\n}\n</code></pre> graph_dfs.c<pre><code>/* 정점을 이미 방문했는지 확인 */\nint isVisited(Vertex **res, int size, Vertex *vet) {\n    // O(n) 시간에 순회하여 노드를 찾는다\n    for (int i = 0; i &lt; size; i++) {\n        if (res[i] == vet) {\n            return 1;\n        }\n    }\n    return 0;\n}\n\n/* 깊이 우선 순회 보조 함수 */\nvoid dfs(GraphAdjList *graph, Vertex **res, int *resSize, Vertex *vet) {\n    // 방문한 정점을 기록한다\n    res[(*resSize)++] = vet;\n    // 해당 정점의 모든 인접 정점을 순회한다\n    AdjListNode *node = findNode(graph, vet);\n    while (node != NULL) {\n        // 이미 방문한 정점은 건너뛴다\n        if (!isVisited(res, *resSize, node-&gt;vertex)) {\n            // 인접 정점을 재귀적으로 방문한다\n            dfs(graph, res, resSize, node-&gt;vertex);\n        }\n        node = node-&gt;next;\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nvoid graphDFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize) {\n    dfs(graph, res, resSize, startVet);\n}\n</code></pre> graph_dfs.kt<pre><code>/* 깊이 우선 순회 보조 함수 */\nfun dfs(\n    graph: GraphAdjList,\n    visited: MutableSet&lt;Vertex?&gt;,\n    res: MutableList&lt;Vertex?&gt;,\n    vet: Vertex?\n) {\n    res.add(vet)     // 방문한 정점을 기록한다\n    visited.add(vet) // 해당 정점을 방문했다고 표시한다\n    // 해당 정점의 모든 인접 정점을 순회한다\n    for (adjVet in graph.adjList[vet]!!) {\n        if (visited.contains(adjVet))\n            continue  // 이미 방문한 정점은 건너뛴다\n        // 인접 정점을 재귀적으로 방문한다\n        dfs(graph, visited, res, adjVet)\n    }\n}\n\n/* 깊이 우선 순회 */\n// 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\nfun graphDFS(graph: GraphAdjList, startVet: Vertex?): MutableList&lt;Vertex?&gt; {\n    // 정점 순회 순서\n    val res = mutableListOf&lt;Vertex?&gt;()\n    // 방문한 정점을 기록하는 해시 집합\n    val visited = HashSet&lt;Vertex?&gt;()\n    dfs(graph, visited, res, startVet)\n    return res\n}\n</code></pre> graph_dfs.rb<pre><code>### 깊이 우선 순회 보조 함수 ###\ndef dfs(graph, visited, res, vet)\n  res &lt;&lt; vet # 방문한 정점을 기록한다\n  visited.add(vet) # 해당 정점을 방문했다고 표시한다\n  # 해당 정점의 모든 인접 정점을 순회한다\n  for adj_vet in graph.adj_list[vet]\n    next if visited.include?(adj_vet) # 이미 방문한 정점은 건너뛴다\n    # 인접 정점을 재귀적으로 방문한다\n    dfs(graph, visited, res, adj_vet)\n  end\nend\n\n### 깊이 우선 순회 ###\ndef graph_dfs(graph, start_vet)\n  # 지정한 정점의 모든 인접 정점을 얻기 쉽도록 그래프를 인접 리스트로 표현한다\n  # 정점 순회 순서\n  res = []\n  # 방문한 정점을 기록하는 해시 집합\n  visited = Set.new\n  dfs(graph, visited, res, start_vet)\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>깊이 우선 순회의 진행 과정은 아래 그림(9-12)과 같다.</p> <ul> <li>직선 점선은 아래로 내려가는 재귀 호출을 나타내며, 새 정점을 방문하기 위해 새로운 재귀 메서드를 호출했다는 뜻이다.</li> <li>곡선 점선은 위로 되돌아가는 백트래킹을 나타내며, 재귀 메서드가 반환되어 이를 호출한 위치로 돌아왔다는 뜻이다.</li> </ul> <p>이해를 깊이 다지기 위해 아래 그림(9-12)과 코드를 함께 보면서, 각 재귀 메서드가 언제 호출되고 언제 반환되는지까지 포함하여 전체 DFS 과정을 머릿속으로 따라가 보자(또는 직접 그려 보자).</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 9-12   그래프의 깊이 우선 순회 단계 </p> <p>깊이 우선 순회 순서는 유일한가?</p> <p>너비 우선 순회와 마찬가지로 깊이 우선 순회 순서도 유일하지 않다. 한 정점에서 어느 방향을 먼저 탐색해도 되므로, 인접 정점의 순서를 자유롭게 바꾸어도 모두 깊이 우선 순회에 해당한다.</p> <p>트리 순회를 예로 들면 '루트 \\(\\rightarrow\\) 왼쪽 \\(\\rightarrow\\) 오른쪽', '왼쪽 \\(\\rightarrow\\) 루트 \\(\\rightarrow\\) 오른쪽', '왼쪽 \\(\\rightarrow\\) 오른쪽 \\(\\rightarrow\\) 루트'는 각각 전위, 중위, 후위 순회에 해당한다. 세 방식은 방문 우선순위가 다르지만 모두 깊이 우선 순회다.</p>","path":["제 9 장   그래프","9.3   그래프 순회"],"tags":[]},{"location":"chapter_graph/graph_traversal/#2_1","level":3,"title":"2.   복잡도 분석","text":"<p>시간 복잡도: 모든 정점을 \\(1\\) 번씩 방문하는 데 \\(O(|V|)\\) 시간이 걸리고, 모든 간선을 \\(2\\) 번씩 방문하는 데 \\(O(2|E|)\\) 시간이 걸린다. 전체 시간은 \\(O(|V| + |E|)\\)이다.</p> <p>공간 복잡도: 리스트 <code>res</code>와 해시 집합 <code>visited</code>의 정점 수는 각각 최대 \\(|V|\\)이고, 재귀 깊이도 최대 \\(|V|\\)이므로 \\(O(|V|)\\) 공간을 사용한다.</p>","path":["제 9 장   그래프","9.3   그래프 순회"],"tags":[]},{"location":"chapter_graph/summary/","level":1,"title":"9.4   요약","text":"","path":["제 9 장   그래프","9.4   요약"],"tags":[]},{"location":"chapter_graph/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>그래프는 정점과 간선으로 이루어지며, 정점 집합과 간선 집합으로 표현할 수 있다.</li> <li>선형 관계(연결 리스트)와 분할 정복 관계(트리)에 비해 네트워크 관계(그래프)는 자유도가 높아 더 복잡하다.</li> <li>방향 그래프의 간선에는 방향이 있고, 연결 그래프에서는 모든 정점에 도달할 수 있으며, 가중 그래프의 각 간선에는 가중치 변수가 있다.</li> <li>인접 행렬은 행렬로 그래프를 표현한다. 각 행(열)은 정점 하나를, 행렬 원소는 간선을 나타내며, \\(1\\) 또는 \\(0\\)으로 두 정점 사이에 간선이 있는지 표시한다. 인접 행렬은 추가, 삭제, 조회, 수정 연산이 매우 효율적이지만 공간을 많이 차지한다.</li> <li>인접 리스트는 여러 연결 리스트로 그래프를 표현한다. \\(i\\) 번째 연결 리스트는 정점 \\(i\\)에 대응하고, 해당 정점의 모든 인접 정점을 저장한다. 인접 행렬보다 공간을 절약하지만, 간선을 찾으려면 연결 리스트를 순회해야 하므로 시간 효율이 낮다.</li> <li>인접 리스트 안의 연결 리스트가 너무 길면 레드-블랙 트리나 해시 테이블로 바꾸어 조회 효율을 높일 수 있다.</li> <li>알고리즘의 관점에서 보면 인접 행렬은 '공간을 더 써서 시간을 절약'하고, 인접 리스트는 '시간을 더 써서 공간을 절약'한다.</li> <li>그래프는 소셜 네트워크나 지하철 노선 등 다양한 현실 시스템을 모델링하는 데 사용할 수 있다.</li> <li>트리는 그래프의 특수한 경우이며, 트리 순회도 그래프 순회의 특수한 경우다.</li> <li>그래프의 너비 우선 순회는 가까운 곳에서 먼 곳으로 한 층씩 확장하는 탐색 방식이며, 보통 큐로 구현한다.</li> <li>그래프의 깊이 우선 순회는 먼저 끝까지 나아가고 더 갈 곳이 없으면 되돌아오는 탐색 방식이며, 보통 재귀로 구현한다.</li> </ul>","path":["제 9 장   그래프","9.4   요약"],"tags":[]},{"location":"chapter_graph/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 경로는 정점의 순서로 정의하는가, 간선의 순서로 정의하는가?</p> <p>위키백과의 언어별 정의는 서로 다르다. 영어판에서는 '경로는 간선의 순서'라고 하고, 중국어판에서는 '경로는 정점의 순서'라고 한다. 영어판 원문을 옮기면 다음과 같다. '그래프 이론에서 그래프의 경로는 정점의 나열을 연결하는 간선의 유한하거나 무한한 나열이다.'</p> <p>이 절에서는 경로를 정점의 순서가 아니라 간선의 순서로 본다. 두 정점 사이에 여러 간선이 연결될 수 있으며, 이때 각 간선은 각각 하나의 경로에 해당하기 때문이다.</p> <p>Q: 비연결 그래프에는 순회할 수 없는 정점이 있는가?</p> <p>비연결 그래프에서는 한 정점에서 출발했을 때 도달할 수 없는 정점이 하나 이상 존재한다. 그래프의 모든 연결 요소를 순회하려면 시작점을 여러 개 설정해야 한다.</p> <p>Q: 인접 리스트에서 '해당 정점과 연결된 모든 정점'의 순서에 제약이 있는가?</p> <p>순서는 자유롭다. 하지만 실제 응용에서는 지정한 규칙에 따라 정렬해야 할 수도 있다. 예를 들어 정점이 추가된 순서나 정점 값의 크기순으로 정렬하면, '특정 기준에서 최댓값이나 최솟값을 가진' 정점을 빠르게 찾는 데 도움이 된다.</p>","path":["제 9 장   그래프","9.4   요약"],"tags":[]},{"location":"chapter_greedy/","level":1,"title":"제 15 장   탐욕","text":"<p>이 장을 시작하며</p> <p>해바라기는 태양을 따라 돌며 매 순간 자신의 성장 가능성을 최대한 추구한다.</p> <p>탐욕 전략은 단순한 선택을 거듭하면서 차츰 최선의 답으로 나아간다.</p>","path":["제 15 장   탐욕"],"tags":[]},{"location":"chapter_greedy/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>15.1   탐욕 알고리즘</li> <li>15.2   분할 가능한 배낭 문제</li> <li>15.3   최대 용량 문제</li> <li>15.4   정수 분할의 최대 곱 문제</li> <li>15.5   요약</li> <li>15.6   연습 문제</li> </ul>","path":["제 15 장   탐욕"],"tags":[]},{"location":"chapter_greedy/exercises/","level":1,"title":"15.6   연습 문제","text":"","path":["제 15 장   탐욕","15.6   연습 문제"],"tags":[]},{"location":"chapter_greedy/exercises/#1561","level":2,"title":"15.6.1   개념 확인","text":"","path":["제 15 장   탐욕","15.6   연습 문제"],"tags":[]},{"location":"chapter_greedy/exercises/#1","level":3,"title":"1.   매번 가장 큰 동전을 고르면 항상 최선인가","text":"<p>동전의 액면가는 <code>[1, 7, 10]</code>이며, 목표 금액은 14다.</p> <ol> <li>‘남은 금액을 넘지 않는 가장 큰 액면가의 동전을 매번 고른다’는 규칙을 실행하고 선택한 동전을 적어 보자.</li> <li>동전을 더 적게 사용하는 방법이 있는가? 있으면 하나를 적고, 없으면 이유를 설명하라.</li> <li>이 결과로 해당 탐욕 전략이 어떤 동전 액면가에서든 올바르다고 할 수 있는가?</li> </ol> 참고 답안 <ol> <li> <p>탐욕 방식은 <code>10 + 1 + 1 + 1 + 1</code>을 차례대로 선택하여 총 5개의 동전을 사용한다.</p> </li> <li> <p>더 적게 사용하는 방법이 있다. <code>7 + 7</code>이면 동전 2개만 필요하다.</p> </li> <li> <p>그럴 수 없다. 이 반례는 임의의 동전 액면가에서 현재 고를 수 있는 가장 큰 액면가를 매번 선택한다고 해서 반드시 최소 동전 개수를 얻는 것은 아님을 보여 준다.     당장 가장 큰 것을 선택하면 이후에 더 좋은 조합을 만들 기회를 잃을 수 있다.</p> </li> </ol>","path":["제 15 장   탐욕","15.6   연습 문제"],"tags":[]},{"location":"chapter_greedy/exercises/#2","level":3,"title":"2.   배낭에 어떤 물건을 먼저 넣어야 하는가","text":"<p>용량이 4킬로그램인 배낭에 다음 물건을 넣을 수 있다. 물건의 일부만 넣어도 되며, 얻는 가치는 넣은 무게에 비례한다.</p> <ul> <li>물건 A: 무게 4킬로그램, 가치 20</li> <li>물건 B: 무게 3킬로그램, 가치 18</li> </ul> <ol> <li>두 물건의 킬로그램당 가치는 각각 얼마인가? 어느 물건을 먼저 넣어야 하는가?</li> <li>분할 가능한 배낭 문제의 탐욕 전략으로 배낭을 가득 채우면 최종 가치는 얼마인가?</li> <li>물건을 나눌 수 있고 배낭이 총무게를 제한하는 상황에서는 물건을 고를 때 총가치와 킬로그램당 가치 중 무엇을 비교해야 하는가? 이유도 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>A의 킬로그램당 가치는 <code>20 ÷ 4 = 5</code>, B의 킬로그램당 가치는 <code>18 ÷ 3 = 6</code>이다.     따라서 단위 무게당 가치가 더 높은 B를 먼저 넣어야 한다.</p> </li> <li> <p>먼저 B를 전부 넣으면 3킬로그램을 차지하고 가치 18을 얻는다. 남은 용량 1킬로그램에는     A를 1킬로그램 넣어 가치 5를 얻는다. 최종 가치는 <code>18 + 5 = 23</code>이다.</p> </li> <li> <p>배낭은 총무게를 제한하고 물건을 나눌 수 있으므로 단위 무게당 가치를 비교해야 한다.     A는 총가치가 더 높지만 킬로그램당 가치는 B보다 낮다. A부터 넣어 배낭을 채우면 가치 20밖에 얻지 못한다.</p> </li> </ol>","path":["제 15 장   탐욕","15.6   연습 문제"],"tags":[]},{"location":"chapter_greedy/exercises/#3","level":3,"title":"3.   두 포인터를 다음에 어떻게 움직여야 하는가","text":"<p>칸막이 높이가 <code>[1, 8, 6, 2, 5]</code>일 때 양 끝에서 시작하는 두 포인터로 최대 용량을 찾는다. 용량은 ‘두 칸막이 중 더 낮은 높이 × 두 칸막이의 인덱스 차이’다.</p> <ol> <li>처음에는 왼쪽 포인터가 인덱스 0, 오른쪽 포인터가 인덱스 4에 있다. 현재 용량은 얼마인가? 다음에는 어느 포인터를 움직여야 하는가?</li> <li>1번에서 선택한 대로 한 번 움직이면 두 포인터는 각각 어느 인덱스에 있는가? 이때 용량은 얼마이며, 다음에는 어느 포인터를 움직여야 하는가?</li> <li>현재 두 칸막이에서 더 낮은 쪽의 포인터를 움직일 수도 있고, 더 높은 쪽의 포인터를 움직일 수도 있다. 어느 쪽을 움직여야 더 큰 용량을 얻을 가능성이 남는가? 이유는 무엇인가?</li> </ol> 참고 답안 <ol> <li> <p>현재 용량은 <code>min(1, 5) × (4 - 0) = 4</code>다. 왼쪽 칸막이가 더 낮으므로 왼쪽 포인터를 움직인다.</p> </li> <li> <p>왼쪽 포인터를 움직이면 두 포인터는 각각 인덱스 1과 4에 있다. 현재 용량은     <code>min(8, 5) × (4 - 1) = 15</code>다. 오른쪽 칸막이가 더 낮으므로 다음에는 오른쪽 포인터를 움직인다.</p> </li> <li> <p>더 낮은 칸막이 쪽의 포인터를 움직여야 더 큰 용량을 얻을 가능성이 남는다. 더 높은 칸막이를 움직이면 두 칸막이 사이의 거리는 반드시 줄어드는 반면, 용기의 높이는 여전히 움직이지 않은 낮은 칸막이에 의해 제한되므로     그대로이거나 더 낮아질 뿐이다. 따라서 용량은 움직이기 전보다 커질 수 없다. 낮은 칸막이를 움직여야만 더 높은 칸막이를 만날 기회가 생긴다.</p> </li> </ol>","path":["제 15 장   탐욕","15.6   연습 문제"],"tags":[]},{"location":"chapter_greedy/exercises/#1562","level":2,"title":"15.6.2   프로그래밍 연습","text":"","path":["제 15 장   탐욕","15.6   연습 문제"],"tags":[]},{"location":"chapter_greedy/exercises/#1_1","level":3,"title":"1.   분할 가능한 배낭 문제","text":"<p>길이가 같은 배열 <code>wgt</code>와 <code>val</code>이 주어진다. <code>wgt[i] &gt; 0</code>, <code>val[i] &gt;= 0</code>이며 배낭 용량은 <code>cap &gt;= 0</code>이다. 각 종류의 물건은 하나씩만 있지만, 그중 일부만 넣어도 된다. 얻는 가치는 넣은 무게가 해당 물건의 총무게에서 차지하는 비율로 계산한다. 탐욕 알고리즘을 사용하여 배낭에서 얻을 수 있는 최대 총가치를 실수로 반환하라.</p> 풀이 힌트 <ol> <li>먼저 각 물건의 단위 무게당 가치 val[i] / wgt[i]를 계산한다. 나눗셈 결과의 소수 부분을 유지해야 한다.</li> <li>단위 무게당 가치가 높은 물건부터 배낭에 넣는다.</li> <li>남은 용량이 현재 물건의 무게보다 작으면 배낭을 정확히 채우는 만큼만 넣고 끝낸다.</li> </ol>","path":["제 15 장   탐욕","15.6   연습 문제"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/","level":1,"title":"15.2   분할 가능한 배낭 문제","text":"<p>문제</p> <p>물건 \\(n\\) 개와 용량이 \\(cap\\)인 배낭이 주어진다. \\(i\\) 번째 물건의 무게는 \\(wgt[i-1]\\), 가치는 \\(val[i-1]\\)이다. 각 물건은 한 번만 선택할 수 있지만, 일부만 선택할 수 있으며 가치는 선택한 무게의 비율에 따라 계산한다. 용량 제한 안에서 배낭에 담을 수 있는 최대 가치를 구하라. 예시는 아래 그림(15-3)과 같다.</p> <p></p> <p> 그림 15-3   분할 가능한 배낭 문제 예시 데이터 </p> <p>분할 가능한 배낭 문제는 전체적으로 0-1 배낭 문제와 매우 비슷하다. 상태는 현재 물건 \\(i\\)와 용량 \\(c\\)를 포함하며, 목표는 용량 제한 안에서 최대 가치를 구하는 것이다.</p> <p>차이점은 물건의 일부만 선택할 수 있다는 점이다. 아래 그림(15-4)과 같이 물건을 원하는 대로 나누고 무게 비율에 따라 그 가치를 계산할 수 있다.</p> <ol> <li>물건 \\(i\\)의 단위 무게당 가치는 \\(val[i-1] / wgt[i-1]\\)이며, 줄여서 단위 가치라고 한다.</li> <li>물건 \\(i\\)의 일부를 무게 \\(w\\)만큼 넣으면 배낭의 가치는 \\(w \\times val[i-1] / wgt[i-1]\\)만큼 늘어난다.</li> </ol> <p></p> <p> 그림 15-4   물건의 단위 무게당 가치 </p>","path":["제 15 장   탐욕","15.2   분할 가능한 배낭 문제"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/#1","level":3,"title":"1.   탐욕 전략 결정","text":"<p>배낭 안의 총가치를 최대화하는 것은 본질적으로 단위 무게당 물건 가치를 최대화하는 것이다. 여기서 아래 그림(15-5)의 탐욕 전략을 도출할 수 있다.</p> <ol> <li>물건을 단위 가치가 높은 순서로 정렬한다.</li> <li>모든 물건을 순회하며 매번 단위 가치가 가장 높은 물건을 탐욕적으로 선택한다.</li> <li>남은 용량이 부족하면 현재 물건의 일부로 배낭을 가득 채운다.</li> </ol> <p></p> <p> 그림 15-5   분할 가능한 배낭 문제의 탐욕 전략 </p>","path":["제 15 장   탐욕","15.2   분할 가능한 배낭 문제"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/#2","level":3,"title":"2.   코드 구현","text":"<p>단위 가치순으로 정렬하기 쉽도록 물건 클래스 <code>Item</code>을 만든다. 탐욕적 선택을 반복하다가 배낭이 가득 차면 반복문을 빠져나와 해를 반환한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby fractional_knapsack.py<pre><code>class Item:\n    \"\"\"물건\"\"\"\n\n    def __init__(self, w: int, v: int):\n        self.w = w  # 물건의 무게\n        self.v = v  # 물건의 가치\n\ndef fractional_knapsack(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"분할 가능한 배낭: 탐욕\"\"\"\n    # 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    items = [Item(w, v) for w, v in zip(wgt, val)]\n    # 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    items.sort(key=lambda item: item.v / item.w, reverse=True)\n    # 탐욕적 선택을 반복한다\n    res = 0\n    for item in items:\n        if item.w &lt;= cap:\n            # 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v\n            cap -= item.w\n        else:\n            # 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += (item.v / item.w) * cap\n            # 남은 용량이 없으므로 반복문을 종료한다\n            break\n    return res\n</code></pre> fractional_knapsack.cpp<pre><code>/* 물건 */\nclass Item {\n  public:\n    int w; // 물건의 무게\n    int v; // 물건의 가치\n\n    Item(int w, int v) : w(w), v(v) {\n    }\n};\n\n/* 분할 가능한 배낭: 탐욕 */\ndouble fractionalKnapsack(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    vector&lt;Item&gt; items;\n    for (int i = 0; i &lt; wgt.size(); i++) {\n        items.push_back(Item(wgt[i], val[i]));\n    }\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    sort(items.begin(), items.end(), [](Item &amp;a, Item &amp;b) { return (double)a.v / a.w &gt; (double)b.v / b.w; });\n    // 탐욕적 선택을 반복한다\n    double res = 0;\n    for (auto &amp;item : items) {\n        if (item.w &lt;= cap) {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += (double)item.v / item.w * cap;\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.java<pre><code>/* 물건 */\nclass Item {\n    int w; // 물건의 무게\n    int v; // 물건의 가치\n\n    public Item(int w, int v) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* 분할 가능한 배낭: 탐욕 */\ndouble fractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    Item[] items = new Item[wgt.length];\n    for (int i = 0; i &lt; wgt.length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    Arrays.sort(items, Comparator.comparingDouble(item -&gt; -((double) item.v / item.w)));\n    // 탐욕적 선택을 반복한다\n    double res = 0;\n    for (Item item : items) {\n        if (item.w &lt;= cap) {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += (double) item.v / item.w * cap;\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.cs<pre><code>/* 물건 */\nclass Item(int w, int v) {\n    public int w = w; // 물건의 무게\n    public int v = v; // 물건의 가치\n}\n\n/* 분할 가능한 배낭: 탐욕 */\ndouble FractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    Item[] items = new Item[wgt.Length];\n    for (int i = 0; i &lt; wgt.Length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    Array.Sort(items, (x, y) =&gt; (y.v / y.w).CompareTo(x.v / x.w));\n    // 탐욕적 선택을 반복한다\n    double res = 0;\n    foreach (Item item in items) {\n        if (item.w &lt;= cap) {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += (double)item.v / item.w * cap;\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.go<pre><code>/* 물건 */\ntype Item struct {\n    w int // 물건의 무게\n    v int // 물건의 가치\n}\n\n/* 분할 가능한 배낭: 탐욕 */\nfunc fractionalKnapsack(wgt []int, val []int, cap int) float64 {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    items := make([]Item, len(wgt))\n    for i := 0; i &lt; len(wgt); i++ {\n        items[i] = Item{wgt[i], val[i]}\n    }\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    sort.Slice(items, func(i, j int) bool {\n        return float64(items[i].v)/float64(items[i].w) &gt; float64(items[j].v)/float64(items[j].w)\n    })\n    // 탐욕적 선택을 반복한다\n    res := 0.0\n    for _, item := range items {\n        if item.w &lt;= cap {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += float64(item.v)\n            cap -= item.w\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += float64(item.v) / float64(item.w) * float64(cap)\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.swift<pre><code>/* 물건 */\nclass Item {\n    var w: Int // 물건의 무게\n    var v: Int // 물건의 가치\n\n    init(w: Int, v: Int) {\n        self.w = w\n        self.v = v\n    }\n}\n\n/* 분할 가능한 배낭: 탐욕 */\nfunc fractionalKnapsack(wgt: [Int], val: [Int], cap: Int) -&gt; Double {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    var items = zip(wgt, val).map { Item(w: $0, v: $1) }\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    items.sort { -(Double($0.v) / Double($0.w)) &lt; -(Double($1.v) / Double($1.w)) }\n    // 탐욕적 선택을 반복한다\n    var res = 0.0\n    var cap = cap\n    for item in items {\n        if item.w &lt;= cap {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += Double(item.v)\n            cap -= item.w\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += Double(item.v) / Double(item.w) * Double(cap)\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.js<pre><code>/* 물건 */\nclass Item {\n    constructor(w, v) {\n        this.w = w; // 물건의 무게\n        this.v = v; // 물건의 가치\n    }\n}\n\n/* 분할 가능한 배낭: 탐욕 */\nfunction fractionalKnapsack(wgt, val, cap) {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    const items = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // 탐욕적 선택을 반복한다\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += (item.v / item.w) * cap;\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.ts<pre><code>/* 물건 */\nclass Item {\n    w: number; // 물건의 무게\n    v: number; // 물건의 가치\n\n    constructor(w: number, v: number) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* 분할 가능한 배낭: 탐욕 */\nfunction fractionalKnapsack(wgt: number[], val: number[], cap: number): number {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    const items: Item[] = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // 탐욕적 선택을 반복한다\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += (item.v / item.w) * cap;\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.dart<pre><code>/* 물건 */\nclass Item {\n  int w; // 물건의 무게\n  int v; // 물건의 가치\n\n  Item(this.w, this.v);\n}\n\n/* 분할 가능한 배낭: 탐욕 */\ndouble fractionalKnapsack(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n  List&lt;Item&gt; items = List.generate(wgt.length, (i) =&gt; Item(wgt[i], val[i]));\n  // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n  items.sort((a, b) =&gt; (b.v / b.w).compareTo(a.v / a.w));\n  // 탐욕적 선택을 반복한다\n  double res = 0;\n  for (Item item in items) {\n    if (item.w &lt;= cap) {\n      // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n      res += item.v;\n      cap -= item.w;\n    } else {\n      // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n      res += item.v / item.w * cap;\n      // 남은 용량이 없으므로 반복문을 종료한다\n      break;\n    }\n  }\n  return res;\n}\n</code></pre> fractional_knapsack.rs<pre><code>/* 물건 */\nstruct Item {\n    w: i32, // 물건의 무게\n    v: i32, // 물건의 가치\n}\n\nimpl Item {\n    fn new(w: i32, v: i32) -&gt; Self {\n        Self { w, v }\n    }\n}\n\n/* 분할 가능한 배낭: 탐욕 */\nfn fractional_knapsack(wgt: &amp;[i32], val: &amp;[i32], mut cap: i32) -&gt; f64 {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    let mut items = wgt\n        .iter()\n        .zip(val.iter())\n        .map(|(&amp;w, &amp;v)| Item::new(w, v))\n        .collect::&lt;Vec&lt;Item&gt;&gt;();\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    items.sort_by(|a, b| {\n        (b.v as f64 / b.w as f64)\n            .partial_cmp(&amp;(a.v as f64 / a.w as f64))\n            .unwrap()\n    });\n    // 탐욕적 선택을 반복한다\n    let mut res = 0.0;\n    for item in &amp;items {\n        if item.w &lt;= cap {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v as f64;\n            cap -= item.w;\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += item.v as f64 / item.w as f64 * cap as f64;\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break;\n        }\n    }\n    res\n}\n</code></pre> fractional_knapsack.c<pre><code>/* 물건 */\ntypedef struct {\n    int w; // 물건의 무게\n    int v; // 물건의 가치\n} Item;\n\n/* 분할 가능한 배낭: 탐욕 */\nfloat fractionalKnapsack(int wgt[], int val[], int itemCount, int cap) {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    Item *items = malloc(sizeof(Item) * itemCount);\n    for (int i = 0; i &lt; itemCount; i++) {\n        items[i] = (Item){.w = wgt[i], .v = val[i]};\n    }\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    qsort(items, (size_t)itemCount, sizeof(Item), sortByValueDensity);\n    // 탐욕적 선택을 반복한다\n    float res = 0.0;\n    for (int i = 0; i &lt; itemCount; i++) {\n        if (items[i].w &lt;= cap) {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += items[i].v;\n            cap -= items[i].w;\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += (float)cap / items[i].w * items[i].v;\n            cap = 0;\n            break;\n        }\n    }\n    free(items);\n    return res;\n}\n</code></pre> fractional_knapsack.kt<pre><code>/* 물건 */\nclass Item(\n    val w: Int, // 물건\n    val v: Int  // 물건의 가치\n)\n\n/* 분할 가능한 배낭: 탐욕 */\nfun fractionalKnapsack(wgt: IntArray, _val: IntArray, c: Int): Double {\n    // 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n    var cap = c\n    val items = arrayOfNulls&lt;Item&gt;(wgt.size)\n    for (i in wgt.indices) {\n        items[i] = Item(wgt[i], _val[i])\n    }\n    // 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n    items.sortBy { item: Item? -&gt; -(item!!.v.toDouble() / item.w) }\n    // 탐욕적 선택을 반복한다\n    var res = 0.0\n    for (item in items) {\n        if (item!!.w &lt;= cap) {\n            // 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n            res += item.v\n            cap -= item.w\n        } else {\n            // 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n            res += item.v.toDouble() / item.w * cap\n            // 남은 용량이 없으므로 반복문을 종료한다\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.rb<pre><code>### 물건 ###\nclass Item\n  attr_accessor :w # 물건의 무게\n  attr_accessor :v # 물건의 가치\n\n  def initialize(w, v)\n    @w = w\n    @v = v\n  end\nend\n\n### 분할 가능한 배낭: 탐욕 ###\ndef fractional_knapsack(wgt, val, cap)\n  # 무게와 가치 두 속성을 갖는 물건의 리스트를 생성한다\n  items = wgt.each_with_index.map { |w, i| Item.new(w, val[i]) }\n  # 단위 가치 item.v / item.w를 기준으로 내림차순 정렬한다\n  items.sort! { |a, b| (b.v.to_f / b.w) &lt;=&gt; (a.v.to_f / a.w) }\n  # 탐욕적 선택을 반복한다\n  res = 0\n  for item in items\n    if item.w &lt;= cap\n      # 남은 용량이 충분하면 현재 물건을 통째로 배낭에 넣는다\n      res += item.v\n      cap -= item.w\n    else\n      # 남은 용량이 부족하면 현재 물건의 일부를 배낭에 넣는다\n      res += (item.v.to_f / item.w) * cap\n      # 남은 용량이 없으므로 반복문을 종료한다\n      break\n    end\n  end\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>내장 정렬 알고리즘의 시간 복잡도는 보통 \\(O(n \\log n)\\)이고, 공간 복잡도는 언어의 구체적인 구현에 따라 대체로 \\(O(\\log n)\\) 또는 \\(O(n)\\)이다.</p> <p>정렬을 제외하면 최악의 경우 물건 리스트 전체를 순회해야 하므로 시간 복잡도는 \\(O(n)\\)이다. 여기서 \\(n\\)은 물건 수다.</p> <p><code>Item</code> 객체 리스트를 초기화하므로 공간 복잡도는 \\(O(n)\\)이다.</p>","path":["제 15 장   탐욕","15.2   분할 가능한 배낭 문제"],"tags":[]},{"location":"chapter_greedy/fractional_knapsack_problem/#3","level":3,"title":"3.   정확성 증명","text":"<p>귀류법을 사용하자. 물건 \\(x\\)의 단위 가치가 가장 높고, 어떤 알고리즘으로 구한 최대 가치가 <code>res</code>인데 그 해에 \\(x\\)가 없다고 가정한다.</p> <p>배낭에서 임의의 물건을 단위 무게만큼 꺼내고 같은 무게의 물건 \\(x\\)로 바꾼다. \\(x\\)의 단위 가치가 가장 높으므로 바꾼 뒤의 총가치는 반드시 <code>res</code>보다 커진다. 이는 <code>res</code>가 최적해라는 가정에 모순이므로 최적해에는 물건 \\(x\\)가 포함되어야 한다.</p> <p>그 해의 다른 물건에도 같은 모순을 구성할 수 있다. 즉, 단위 가치가 더 높은 물건이 항상 더 좋은 선택이며, 탐욕 전략이 유효함을 알 수 있다.</p> <p>아래 그림(15-6)과 같이 물건 무게를 가로축, 단위 가치를 세로축으로 보면, 분할 가능한 배낭 문제는 '제한된 가로축 구간에서 최대 넓이를 구하는' 문제로 바뀐다. 이 비유는 탐욕 전략의 유효성을 기하학적으로 이해하는 데 도움이 된다.</p> <p></p> <p> 그림 15-6   분할 가능한 배낭 문제의 기하학적 표현 </p>","path":["제 15 장   탐욕","15.2   분할 가능한 배낭 문제"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/","level":1,"title":"15.1   탐욕 알고리즘","text":"<p>탐욕 알고리즘(greedy algorithm)은 최적화 문제를 푸는 대표적인 알고리즘이다. 매 결정 단계에서 현재 가장 좋아 보이는 선택, 즉 국소적으로 최적인 결정을 탐욕적으로 내려 전역 최적해를 얻으려는 것이 기본 아이디어다. 간결하고 효율적이어서 많은 실제 문제에 널리 쓰인다.</p> <p>탐욕 알고리즘과 동적 계획법은 모두 최적화 문제에 자주 사용한다. 최적 부분 구조에 의존한다는 공통점 등이 있지만 작동 원리는 다르다.</p> <ul> <li>동적 계획법은 이전 단계의 모든 결정을 바탕으로 현재 결정을 고려하고, 과거 부분 문제의 해로 현재 부분 문제의 해를 구성한다.</li> <li>탐욕 알고리즘은 과거 결정을 고려하지 않고 계속 앞으로 나아가며 탐욕적으로 선택한다. 문제의 범위를 점차 줄여 해결될 때까지 진행한다.</li> </ul> <p>먼저 '동전 교환' 예제로 탐욕 알고리즘의 원리를 알아보자. '무한 배낭 문제' 절에서 소개한 문제라 익숙할 것이다.</p> <p>문제</p> <p>동전 \\(n\\) 종류가 주어지며 \\(i\\) 번째 동전의 액면가는 \\(coins[i - 1]\\)이고 목표 금액은 \\(amt\\)다. 각 종류의 동전을 반복해서 선택할 수 있을 때 목표 금액을 만드는 최소 동전 수를 구하라. 만들 수 없다면 \\(-1\\)을 반환하라.</p> <p>이 문제에서 사용하는 탐욕 전략은 아래 그림(15-1)과 같다. 주어진 목표 금액에 대해 그 금액을 넘지 않으면서 가장 가까운 액면가의 동전을 탐욕적으로 고르고, 목표 금액을 만들 때까지 반복한다.</p> <p></p> <p> 그림 15-1   동전 교환의 탐욕 전략 </p> <p>구현 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby coin_change_greedy.py<pre><code>def coin_change_greedy(coins: list[int], amt: int) -&gt; int:\n    \"\"\"동전 교환: 탐욕\"\"\"\n    # coins 리스트가 정렬되어 있다고 가정한다\n    i = len(coins) - 1\n    count = 0\n    # 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while amt &gt; 0:\n        # 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while i &gt; 0 and coins[i] &gt; amt:\n            i -= 1\n        # coins[i]를 선택한다\n        amt -= coins[i]\n        count += 1\n    # 가능한 방법을 찾지 못하면 -1을 반환한다\n    return count if amt == 0 else -1\n</code></pre> coin_change_greedy.cpp<pre><code>/* 동전 교환: 탐욕 */\nint coinChangeGreedy(vector&lt;int&gt; &amp;coins, int amt) {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    int i = coins.size() - 1;\n    int count = 0;\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while (amt &gt; 0) {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i];\n        count++;\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.java<pre><code>/* 동전 교환: 탐욕 */\nint coinChangeGreedy(int[] coins, int amt) {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    int i = coins.length - 1;\n    int count = 0;\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while (amt &gt; 0) {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i];\n        count++;\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.cs<pre><code>/* 동전 교환: 탐욕 */\nint CoinChangeGreedy(int[] coins, int amt) {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    int i = coins.Length - 1;\n    int count = 0;\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while (amt &gt; 0) {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i];\n        count++;\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.go<pre><code>/* 동전 교환: 탐욕 */\nfunc coinChangeGreedy(coins []int, amt int) int {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    i := len(coins) - 1\n    count := 0\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    for amt &gt; 0 {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        for i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i--\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i]\n        count++\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    if amt != 0 {\n        return -1\n    }\n    return count\n}\n</code></pre> coin_change_greedy.swift<pre><code>/* 동전 교환: 탐욕 */\nfunc coinChangeGreedy(coins: [Int], amt: Int) -&gt; Int {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    var i = coins.count - 1\n    var count = 0\n    var amt = amt\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while amt &gt; 0 {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i]\n        count += 1\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return amt == 0 ? count : -1\n}\n</code></pre> coin_change_greedy.js<pre><code>/* 동전 교환: 탐욕 */\nfunction coinChangeGreedy(coins, amt) {\n    // coins 배열이 정렬되어 있다고 가정한다\n    let i = coins.length - 1;\n    let count = 0;\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while (amt &gt; 0) {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i];\n        count++;\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.ts<pre><code>/* 동전 교환: 탐욕 */\nfunction coinChangeGreedy(coins: number[], amt: number): number {\n    // coins 배열이 정렬되어 있다고 가정한다\n    let i = coins.length - 1;\n    let count = 0;\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while (amt &gt; 0) {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i];\n        count++;\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.dart<pre><code>/* 동전 교환: 탐욕 */\nint coinChangeGreedy(List&lt;int&gt; coins, int amt) {\n  // coins 리스트가 정렬되어 있다고 가정한다\n  int i = coins.length - 1;\n  int count = 0;\n  // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n  while (amt &gt; 0) {\n    // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n    while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n      i--;\n    }\n    // coins[i]를 선택한다\n    amt -= coins[i];\n    count++;\n  }\n  // 가능한 방법을 찾지 못하면 -1을 반환한다\n  return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.rs<pre><code>/* 동전 교환: 탐욕 */\nfn coin_change_greedy(coins: &amp;[i32], mut amt: i32) -&gt; i32 {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    let mut i = coins.len() - 1;\n    let mut count = 0;\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while amt &gt; 0 {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1;\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i];\n        count += 1;\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    if amt == 0 {\n        count\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change_greedy.c<pre><code>/* 동전 교환: 탐욕 */\nint coinChangeGreedy(int *coins, int size, int amt) {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    int i = size - 1;\n    int count = 0;\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while (amt &gt; 0) {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // coins[i]를 선택한다\n        amt -= coins[i];\n        count++;\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.kt<pre><code>/* 동전 교환: 탐욕 */\nfun coinChangeGreedy(coins: IntArray, amt: Int): Int {\n    // coins 리스트가 정렬되어 있다고 가정한다\n    var am = amt\n    var i = coins.size - 1\n    var count = 0\n    // 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n    while (am &gt; 0) {\n        // 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; am) {\n            i--\n        }\n        // coins[i]를 선택한다\n        am -= coins[i]\n        count++\n    }\n    // 가능한 방법을 찾지 못하면 -1을 반환한다\n    return if (am == 0) count else -1\n}\n</code></pre> coin_change_greedy.rb<pre><code>### 동전 교환: 탐욕 ###\ndef coin_change_greedy(coins, amt)\n  # coins 리스트가 정렬되어 있다고 가정한다\n  i = coins.length - 1\n  count = 0\n  # 남은 금액이 없을 때까지 탐욕적 선택을 반복한다\n  while amt &gt; 0\n    # 남은 금액보다 작으면서 가장 가까운 금액의 동전을 찾는다\n    while i &gt; 0 &amp;&amp; coins[i] &gt; amt\n      i -= 1\n    end\n    # coins[i]를 선택한다\n    amt -= coins[i]\n    count += 1\n  end\n  # 가능한 방법을 찾지 못하면 -1을 반환한다\n  amt == 0 ? count : -1\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>'정말 간결하다!'라는 감탄이 나올 만하다. 탐욕 알고리즘은 약 열 줄의 코드만으로 동전 교환 문제를 해결했다.</p>","path":["제 15 장   탐욕","15.1   탐욕 알고리즘"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1511","level":2,"title":"15.1.1   탐욕 알고리즘의 장점과 한계","text":"<p>탐욕 알고리즘은 연산이 직접적이고 구현이 간단할 뿐 아니라 보통 효율도 매우 높다. 위 코드에서 최소 액면가를 \\(\\min(coins)\\)라고 하면 탐욕적 선택은 최대 \\(amt / \\min(coins)\\) 번 반복하므로 시간 복잡도는 \\(O(amt / \\min(coins))\\)이다. 동적 계획법의 \\(O(n \\times amt)\\)보다 한 차수 낮다.</p> <p>하지만 일부 동전 액면가 조합에서는 최적해를 찾지 못한다. 아래 그림(15-2)은 두 예를 보여 준다.</p> <ul> <li>성립 예 \\(coins = [1, 5, 10, 20, 50, 100]\\): 이 동전 조합에서는 어떤 \\(amt\\)에 대해서도 탐욕 알고리즘이 최적해를 찾는다.</li> <li>반례 \\(coins = [1, 20, 50]\\): \\(amt = 60\\)이면 탐욕 알고리즘은 \\(50 + 1 \\times 10\\)으로 동전 \\(11\\) 개를 고른다. 하지만 동적 계획법은 \\(20 + 20 + 20\\)이라는 최적해를 찾아 \\(3\\) 개만 사용한다.</li> <li>반례 \\(coins = [1, 49, 50]\\): \\(amt = 98\\)이면 탐욕 알고리즘은 \\(50 + 1 \\times 48\\)로 동전 \\(49\\) 개를 고른다. 하지만 동적 계획법은 \\(49 + 49\\)라는 최적해를 찾아 \\(2\\) 개만 사용한다.</li> </ul> <p></p> <p> 그림 15-2   탐욕 알고리즘이 최적해를 찾지 못하는 예 </p> <p>즉, 동전 교환에서 탐욕 알고리즘은 전역 최적해를 보장하지 못하고 매우 나쁜 해를 찾을 수도 있다. 이 문제는 동적 계획법으로 푸는 편이 더 적합하다.</p> <p>일반적으로 탐욕 알고리즘을 적용하는 상황은 다음 두 가지다.</p> <ol> <li>최적해를 보장할 수 있다: 보통 백트래킹이나 동적 계획법보다 효율적이므로 이런 경우에는 탐욕 알고리즘이 가장 좋은 선택인 경우가 많다.</li> <li>최적해에 가까운 해를 찾을 수 있다: 이 경우에도 탐욕 알고리즘을 쓸 수 있다. 복잡한 문제에서는 전역 최적해를 찾기 매우 어려우므로, 효율적으로 차선의 해를 찾는 것만으로도 유용하다.</li> </ol>","path":["제 15 장   탐욕","15.1   탐욕 알고리즘"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1512","level":2,"title":"15.1.2   탐욕 알고리즘의 특성","text":"<p>그렇다면 어떤 문제가 탐욕 알고리즘에 적합할까? 다시 말해 어떤 경우에 최적해를 보장할 수 있을까?</p> <p>탐욕 알고리즘은 동적 계획법보다 적용 조건이 엄격하며, 주로 다음 두 성질에 주목한다.</p> <ul> <li>탐욕적 선택 속성: 국소 최적 선택이 항상 전역 최적해로 이어져야만 탐욕 알고리즘이 최적해를 보장할 수 있다.</li> <li>최적 부분 구조: 원래 문제의 최적해가 부분 문제의 최적해를 포함한다.</li> </ul> <p>최적 부분 구조는 '동적 계획법' 장에서 설명했으므로 반복하지 않는다. 일부 문제는 최적 부분 구조가 뚜렷하지 않아도 탐욕 알고리즘으로 해결할 수 있다는 점을 알아 두자.</p> <p>여기서는 탐욕적 선택 속성을 판단하는 방법을 주로 살펴본다. 설명은 간단해 보이지만 많은 문제에서 이 속성을 증명하기는 쉽지 않다.</p> <p>예를 들어 동전 교환에서는 반례로 탐욕적 선택 속성을 부정하기는 쉽지만 증명하기는 어렵다. 어떤 조건을 만족하는 동전 조합을 탐욕 알고리즘으로 풀 수 있는가라는 질문에는 직관이나 사례를 바탕으로 애매한 답을 내놓기 쉽고, 엄밀한 수학적 증명을 제시하기는 어렵다.</p> <p>참고 문헌</p> <p>어떤 동전 조합에 대해 탐욕 알고리즘이 모든 금액의 최적해를 찾을 수 있는지 판단하는 \\(O(n^3)\\) 알고리즘을 제시한 논문이 있다.</p> <p>Pearson, D. A polynomial-time algorithm for the change-making problem[J]. Operations Research Letters, 2005, 33(3): 231-234.</p>","path":["제 15 장   탐욕","15.1   탐욕 알고리즘"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1513","level":2,"title":"15.1.3   탐욕 알고리즘의 문제 해결 단계","text":"<p>탐욕 문제의 풀이 과정은 대체로 다음 세 단계다.</p> <ol> <li>문제 분석: 상태 정의, 최적화 목표, 제약 조건 등 문제의 특성을 정리하고 이해한다. 백트래킹과 동적 계획법에서도 필요한 단계다.</li> <li>탐욕 전략 결정: 매 단계에서 어떻게 탐욕적으로 선택할지 정한다. 각 단계에서 문제 크기를 줄여 최종적으로 전체 문제를 해결할 수 있어야 한다.</li> <li>정확성 증명: 보통 탐욕적 선택 속성과 최적 부분 구조를 증명해야 한다. 귀납법이나 귀류법 같은 수학적 증명이 필요할 수 있다.</li> </ol> <p>탐욕 전략을 정하는 것이 핵심 단계지만, 다음 이유로 쉽지 않을 수 있다.</p> <ul> <li>문제마다 탐욕 전략이 크게 다르다. 많은 문제는 전략이 비교적 명확하여 대략적인 생각과 시도만으로 찾을 수 있다. 반면 복잡한 문제에서는 전략이 잘 드러나지 않아 문제 풀이 경험과 알고리즘 역량이 크게 요구된다.</li> <li>일부 탐욕 전략은 그럴듯해 보이지만 함정이 있다. 자신 있게 전략을 설계하고 코드를 작성해 제출해도 일부 테스트를 통과하지 못할 수 있다. 전략이 '부분적으로만 옳기' 때문이다. 앞에서 본 동전 교환이 대표적인 사례다.</li> </ul> <p>정확성을 보장하려면 탐욕 전략을 엄밀히 증명해야 하며, 보통 귀류법이나 수학적 귀납법을 사용한다.</p> <p>하지만 정확성 증명 자체도 쉽지 않을 수 있다. 실마리가 없으면 보통 테스트 사례로 코드를 디버깅하면서 탐욕 전략을 조금씩 수정하고 검증한다.</p>","path":["제 15 장   탐욕","15.1   탐욕 알고리즘"],"tags":[]},{"location":"chapter_greedy/greedy_algorithm/#1514","level":2,"title":"15.1.4   대표적인 탐욕 알고리즘 문제","text":"<p>탐욕 알고리즘은 탐욕적 선택 속성과 최적 부분 구조를 만족하는 최적화 문제에 자주 사용한다. 대표적인 예는 다음과 같다.</p> <ul> <li>동전 거스름돈 문제: 일부 동전 조합에서는 탐욕 알고리즘이 항상 최적해를 얻는다.</li> <li>구간 스케줄링 문제: 각 작업이 일정 시간 구간에 걸쳐 진행될 때 가능한 한 많은 작업을 완료하는 것이 목표다. 매번 가장 일찍 끝나는 작업을 고르면 최적해를 얻을 수 있다.</li> <li>분할 가능한 배낭 문제: 물건들과 적재 용량이 주어졌을 때 총무게가 용량을 넘지 않으면서 총가치가 최대가 되도록 물건을 고른다. 매번 무게 대비 가치의 비율(가치 / 무게)이 가장 높은 물건을 고르면 일부 경우에 최적해를 얻을 수 있다.</li> <li>주식 매매 문제: 주가의 과거 기록이 주어졌을 때 여러 번 매매하여 최대 이익을 구한다. 단, 이미 주식을 보유하고 있으면 팔기 전에는 다시 살 수 없다.</li> <li>허프먼 코딩: 무손실 데이터 압축에 쓰이는 탐욕 알고리즘이다. 매번 빈도가 가장 낮은 두 노드를 합쳐 허프먼 트리를 만들면, 최종 트리의 가중 경로 길이(부호 길이)가 최소가 된다.</li> <li>Dijkstra 알고리즘: 모든 간선 가중치가 음이 아닌 그래프에서 주어진 시작 정점부터 나머지 정점까지의 최단 경로를 구하는 탐욕 알고리즘이다.</li> </ul>","path":["제 15 장   탐욕","15.1   탐욕 알고리즘"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/","level":1,"title":"15.3   최대 용량 문제","text":"<p>문제</p> <p>배열 \\(ht\\)의 각 원소는 수직 칸막이의 높이를 나타낸다. 배열에서 임의의 두 칸막이와 그 사이의 공간으로 용기를 만들 수 있다.</p> <p>용량은 높이와 너비의 곱(넓이)이다. 높이는 더 낮은 칸막이로 결정되고, 너비는 두 칸막이의 배열 인덱스 차이다.</p> <p>용량이 최대가 되도록 두 칸막이를 골라 최대 용량을 반환하라. 예시는 아래 그림(15-7)과 같다.</p> <p></p> <p> 그림 15-7   최대 용량 문제 예시 데이터 </p> <p>임의의 두 칸막이로 용기를 만들므로 이 문제의 상태는 두 칸막이의 인덱스 \\([i, j]\\)다.</p> <p>문제에 따르면 용량은 높이와 너비의 곱이고, 높이는 낮은 칸막이로 결정되며 너비는 두 인덱스의 차다. 용량을 \\(cap[i, j]\\)라고 하면 다음 공식을 얻는다.</p> \\[ cap[i, j] = \\min(ht[i], ht[j]) \\times (j - i) \\] <p>배열 길이가 \\(n\\)이면 두 칸막이의 조합 수(전체 상태 수)는 \\(C_n^2 = \\frac{n(n - 1)}{2}\\)이다. 가장 직접적으로는 모든 상태를 열거하여 최대 용량을 구할 수 있으며, 시간 복잡도는 \\(O(n^2)\\)이다.</p>","path":["제 15 장   탐욕","15.3   최대 용량 문제"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/#1","level":3,"title":"1.   탐욕 전략 결정","text":"<p>더 효율적인 풀이도 있다. 아래 그림(15-8)처럼 인덱스 \\(i &lt; j\\)이고 높이 \\(ht[i] &lt; ht[j]\\)인 상태 \\([i, j]\\)를 고르자. 즉, \\(i\\)는 낮은 칸막이, \\(j\\)는 높은 칸막이다.</p> <p></p> <p> 그림 15-8   초기 상태 </p> <p>아래 그림(15-9)과 같이 높은 칸막이 \\(j\\)를 낮은 칸막이 \\(i\\) 쪽으로 옮기면 용량은 반드시 작아진다.</p> <p>\\(j\\)를 옮기면 너비 \\(j-i\\)는 확실히 줄어든다. 높이는 낮은 칸막이로 결정되므로 그대로이거나(\\(i\\)가 계속 더 낮음), 더 낮아질 수만 있다(이동한 \\(j\\)가 더 낮아짐).</p> <p></p> <p> 그림 15-9   높은 칸막이를 안쪽으로 옮긴 상태 </p> <p>반대로 생각하면 낮은 칸막이 \\(i\\)를 안쪽으로 옮겨야만 용량이 커질 가능성이 있다. 너비는 반드시 줄지만 높이는 커질 수도 있기 때문이다(이동한 \\(i\\)가 더 높을 수 있다). 아래 그림(15-10)에서는 낮은 칸막이를 옮긴 뒤 넓이가 커진다.</p> <p></p> <p> 그림 15-10   낮은 칸막이를 안쪽으로 옮긴 상태 </p> <p>따라서 두 포인터를 용기의 양 끝에 놓고, 매번 낮은 칸막이 쪽 포인터를 안쪽으로 옮겨 두 포인터가 만날 때까지 진행하는 탐욕 전략을 도출할 수 있다.</p> <p>아래 그림(15-11)은 탐욕 전략의 실행 과정이다.</p> <ol> <li>처음에는 포인터 \\(i\\)와 \\(j\\)를 배열의 양 끝에 둔다.</li> <li>현재 상태의 용량 \\(cap[i, j]\\)를 계산하고 최대 용량을 갱신한다.</li> <li>칸막이 \\(i\\)와 \\(j\\)의 높이를 비교하여 낮은 쪽을 안쪽으로 한 칸 옮긴다.</li> <li>\\(i\\)와 \\(j\\)가 만날 때까지 <code>2.</code> 단계와 <code>3.</code> 단계를 반복한다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 15-11   최대 용량 문제의 탐욕 과정 </p>","path":["제 15 장   탐욕","15.3   최대 용량 문제"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/#2","level":3,"title":"2.   코드 구현","text":"<p>반복문은 최대 \\(n\\) 회 실행되므로 시간 복잡도는 \\(O(n)\\)이다.</p> <p>변수 \\(i\\), \\(j\\), \\(res\\)는 상수 크기의 추가 공간을 사용하므로 공간 복잡도는 \\(O(1)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby max_capacity.py<pre><code>def max_capacity(ht: list[int]) -&gt; int:\n    \"\"\"최대 용량: 탐욕\"\"\"\n    # i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    i, j = 0, len(ht) - 1\n    # 최대 용량의 초깃값은 0이다\n    res = 0\n    # 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while i &lt; j:\n        # 최대 용량을 갱신한다\n        cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        # 낮은 칸막이를 안쪽으로 이동한다\n        if ht[i] &lt; ht[j]:\n            i += 1\n        else:\n            j -= 1\n    return res\n</code></pre> max_capacity.cpp<pre><code>/* 최대 용량: 탐욕 */\nint maxCapacity(vector&lt;int&gt; &amp;ht) {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    int i = 0, j = ht.size() - 1;\n    // 최대 용량의 초깃값은 0이다\n    int res = 0;\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while (i &lt; j) {\n        // 최대 용량을 갱신한다\n        int cap = min(ht[i], ht[j]) * (j - i);\n        res = max(res, cap);\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.java<pre><code>/* 최대 용량: 탐욕 */\nint maxCapacity(int[] ht) {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    int i = 0, j = ht.length - 1;\n    // 최대 용량의 초깃값은 0이다\n    int res = 0;\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while (i &lt; j) {\n        // 최대 용량을 갱신한다\n        int cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.cs<pre><code>/* 최대 용량: 탐욕 */\nint MaxCapacity(int[] ht) {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    int i = 0, j = ht.Length - 1;\n    // 최대 용량의 초깃값은 0이다\n    int res = 0;\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while (i &lt; j) {\n        // 최대 용량을 갱신한다\n        int cap = Math.Min(ht[i], ht[j]) * (j - i);\n        res = Math.Max(res, cap);\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.go<pre><code>/* 최대 용량: 탐욕 */\nfunc maxCapacity(ht []int) int {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    i, j := 0, len(ht)-1\n    // 최대 용량의 초깃값은 0이다\n    res := 0\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    for i &lt; j {\n        // 최대 용량을 갱신한다\n        capacity := int(math.Min(float64(ht[i]), float64(ht[j]))) * (j - i)\n        res = int(math.Max(float64(res), float64(capacity)))\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if ht[i] &lt; ht[j] {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.swift<pre><code>/* 최대 용량: 탐욕 */\nfunc maxCapacity(ht: [Int]) -&gt; Int {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    var i = ht.startIndex, j = ht.endIndex - 1\n    // 최대 용량의 초깃값은 0이다\n    var res = 0\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while i &lt; j {\n        // 최대 용량을 갱신한다\n        let cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if ht[i] &lt; ht[j] {\n            i += 1\n        } else {\n            j -= 1\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.js<pre><code>/* 최대 용량: 탐욕 */\nfunction maxCapacity(ht) {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    let i = 0,\n        j = ht.length - 1;\n    // 최대 용량의 초깃값은 0이다\n    let res = 0;\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while (i &lt; j) {\n        // 최대 용량을 갱신한다\n        const cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.ts<pre><code>/* 최대 용량: 탐욕 */\nfunction maxCapacity(ht: number[]): number {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    let i = 0,\n        j = ht.length - 1;\n    // 최대 용량의 초깃값은 0이다\n    let res = 0;\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while (i &lt; j) {\n        // 최대 용량을 갱신한다\n        const cap: number = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.dart<pre><code>/* 최대 용량: 탐욕 */\nint maxCapacity(List&lt;int&gt; ht) {\n  // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n  int i = 0, j = ht.length - 1;\n  // 최대 용량의 초깃값은 0이다\n  int res = 0;\n  // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n  while (i &lt; j) {\n    // 최대 용량을 갱신한다\n    int cap = min(ht[i], ht[j]) * (j - i);\n    res = max(res, cap);\n    // 낮은 칸막이를 안쪽으로 이동한다\n    if (ht[i] &lt; ht[j]) {\n      i++;\n    } else {\n      j--;\n    }\n  }\n  return res;\n}\n</code></pre> max_capacity.rs<pre><code>/* 최대 용량: 탐욕 */\nfn max_capacity(ht: &amp;[i32]) -&gt; i32 {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    let mut i = 0;\n    let mut j = ht.len() - 1;\n    // 최대 용량의 초깃값은 0이다\n    let mut res = 0;\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while i &lt; j {\n        // 최대 용량을 갱신한다\n        let cap = std::cmp::min(ht[i], ht[j]) * (j - i) as i32;\n        res = std::cmp::max(res, cap);\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if ht[i] &lt; ht[j] {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    res\n}\n</code></pre> max_capacity.c<pre><code>/* 최대 용량: 탐욕 */\nint maxCapacity(int ht[], int htLength) {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    int i = 0;\n    int j = htLength - 1;\n    // 최대 용량의 초깃값은 0이다\n    int res = 0;\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while (i &lt; j) {\n        // 최대 용량을 갱신한다\n        int capacity = myMin(ht[i], ht[j]) * (j - i);\n        res = myMax(res, capacity);\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.kt<pre><code>/* 최대 용량: 탐욕 */\nfun maxCapacity(ht: IntArray): Int {\n    // i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n    var i = 0\n    var j = ht.size - 1\n    // 최대 용량의 초깃값은 0이다\n    var res = 0\n    // 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n    while (i &lt; j) {\n        // 최대 용량을 갱신한다\n        val cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        // 낮은 칸막이를 안쪽으로 이동한다\n        if (ht[i] &lt; ht[j]) {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.rb<pre><code>### 최대 용량: 탐욕 ###\ndef max_capacity(ht)\n  # i와 j가 각각 배열의 양 끝을 가리키도록 초기화한다\n  i, j = 0, ht.length - 1\n  # 최대 용량의 초깃값은 0이다\n  res = 0\n\n  # 두 칸막이가 만날 때까지 탐욕적 선택을 반복한다\n  while i &lt; j\n    # 최대 용량을 갱신한다\n    cap = [ht[i], ht[j]].min * (j - i)\n    res = [res, cap].max\n    # 낮은 칸막이를 안쪽으로 이동한다\n    if ht[i] &lt; ht[j]\n      i += 1\n    else\n      j -= 1\n    end\n  end\n\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 15 장   탐욕","15.3   최대 용량 문제"],"tags":[]},{"location":"chapter_greedy/max_capacity_problem/#3","level":3,"title":"3.   정확성 증명","text":"<p>탐욕 알고리즘이 모든 경우 열거보다 빠른 이유는 매번 일부 상태를 '건너뛰기' 때문이다.</p> <p>예를 들어 상태 \\(cap[i, j]\\)에서 \\(i\\)가 낮은 칸막이, \\(j\\)가 높은 칸막이라고 하자. 낮은 \\(i\\)를 탐욕적으로 한 칸 안쪽으로 옮기면 아래 그림(15-12)의 상태들을 건너뛴다. 이는 이후에 그 상태들의 용량을 확인할 수 없다는 뜻이다.</p> \\[ cap[i, i+1], cap[i, i+2], \\dots, cap[i, j-2], cap[i, j-1] \\] <p></p> <p> 그림 15-12   낮은 칸막이를 옮겨 건너뛴 상태들 </p> <p>건너뛴 상태들은 사실 높은 칸막이 \\(j\\)를 안쪽으로 옮긴 모든 상태다. 앞에서 높은 칸막이를 안쪽으로 옮기면 용량이 반드시 줄어듦을 보였다. 따라서 건너뛴 상태는 최적해가 될 수 없으며, 이를 건너뛰어도 최적해를 놓치지 않는다.</p> <p>이 분석은 낮은 칸막이를 옮기는 연산이 '안전'하며 탐욕 전략이 유효함을 보여 준다.</p>","path":["제 15 장   탐욕","15.3   최대 용량 문제"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/","level":1,"title":"15.4   정수 분할의 최대 곱 문제","text":"<p>문제</p> <p>양의 정수 \\(n\\)을 두 개 이상의 양의 정수의 합으로 나누었을 때, 나눈 정수들의 곱이 최대 얼마가 되는지 구하라. 아래 그림(15-13)을 참고하자.</p> <p></p> <p> 그림 15-13   정수 분할의 최대 곱 문제 정의 </p> <p>\\(n\\)을 \\(m\\) 개의 정수 인수로 나누고 \\(i\\) 번째 인수를 \\(n_i\\)라고 하면 다음과 같다.</p> \\[ n = \\sum_{i=1}^{m}n_i \\] <p>목표는 모든 정수 인수의 곱을 최대화하는 것이다.</p> \\[ \\max(\\prod_{i=1}^{m}n_i) \\] <p>생각해야 할 것은 나누는 개수 \\(m\\)과 각 \\(n_i\\)의 값이다.</p>","path":["제 15 장   탐욕","15.4   정수 분할의 최대 곱 문제"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/#1","level":3,"title":"1.   탐욕 전략 결정","text":"<p>경험상 두 정수의 곱은 합보다 큰 경우가 많다. \\(n\\)에서 인수 \\(2\\)를 나누어 내면 곱은 \\(2(n-2)\\)다. 이를 \\(n\\)과 비교해 보자.</p> \\[ \\begin{aligned} 2(n-2) &amp; \\geq n \\newline 2n - n - 4 &amp; \\geq 0 \\newline n &amp; \\geq 4 \\end{aligned} \\] <p>아래 그림(15-14)과 같이 \\(n \\geq 4\\)이면 \\(2\\)를 나누어 낸 뒤 곱이 커진다. 이는 \\(4\\) 이상의 정수는 모두 나누어야 한다는 뜻이다.</p> <p>탐욕 전략 1: 분할에 \\(\\geq 4\\)인 인수가 있으면 계속 나누어야 한다. 최종 분할에는 \\(1\\), \\(2\\), \\(3\\)만 있어야 한다.</p> <p></p> <p> 그림 15-14   분할로 커지는 곱 </p> <p>이제 어느 인수가 가장 좋은지 생각해 보자. \\(1\\), \\(2\\), \\(3\\) 중 \\(1\\)이 가장 나쁘다는 것은 분명하다. \\(1 \\times (n-1) &lt; n\\)이 항상 성립하므로 \\(1\\)을 나누어 내면 오히려 곱이 줄어들기 때문이다.</p> <p>아래 그림(15-15)에서 \\(n = 6\\)일 때 \\(3 \\times 3 &gt; 2 \\times 2 \\times 2\\)다. 따라서 \\(2\\)보다 \\(3\\)으로 나누는 편이 더 좋다.</p> <p>탐욕 전략 2: 분할에는 \\(2\\)가 많아도 두 개만 있어야 한다. \\(2\\) 세 개는 언제나 \\(3\\) 두 개로 바꾸어 더 큰 곱을 얻을 수 있기 때문이다.</p> <p></p> <p> 그림 15-15   최적인 분할 인수 </p> <p>위 내용을 종합하면 다음 탐욕 전략을 얻는다.</p> <ol> <li>정수 \\(n\\)에서 인수 \\(3\\)을 계속 나누어 내고, 나머지가 \\(0\\), \\(1\\), \\(2\\)가 되면 멈춘다.</li> <li>나머지가 \\(0\\)이면 \\(n\\)이 \\(3\\)의 배수이므로 추가로 처리하지 않는다.</li> <li>나머지가 \\(2\\)이면 더 나누지 않고 그대로 둔다.</li> <li>나머지가 \\(1\\)이면 \\(2 \\times 2 &gt; 1 \\times 3\\)이므로 마지막 \\(3\\)과 나머지 \\(1\\)을 \\(2\\) 두 개로 바꾼다.</li> </ol>","path":["제 15 장   탐욕","15.4   정수 분할의 최대 곱 문제"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/#2","level":3,"title":"2.   코드 구현","text":"<p>아래 그림(15-16)과 같이 반복문으로 정수를 나눌 필요 없이 내림 정수 나눗셈으로 \\(3\\)의 개수 \\(a\\)를, 나머지 연산으로 나머지 \\(b\\)를 구할 수 있다. 이때 다음이 성립한다.</p> \\[ n = 3 a + b \\] <p>경계 경우인 \\(n \\leq 3\\)에서는 반드시 \\(1\\)을 하나 나누어 내야 하므로 곱은 \\(1 \\times (n - 1)\\)이다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby max_product_cutting.py<pre><code>def max_product_cutting(n: int) -&gt; int:\n    \"\"\"분할 후의 최대 곱: 탐욕\"\"\"\n    # n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if n &lt;= 3:\n        return 1 * (n - 1)\n    # 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    a, b = n // 3, n % 3\n    if b == 1:\n        # 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return int(math.pow(3, a - 1)) * 2 * 2\n    if b == 2:\n        # 나머지가 2이면 별도로 처리하지 않는다\n        return int(math.pow(3, a)) * 2\n    # 나머지가 0이면 별도로 처리하지 않는다\n    return int(math.pow(3, a))\n</code></pre> max_product_cutting.cpp<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nint maxProductCutting(int n) {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return (int)pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return (int)pow(3, a) * 2;\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return (int)pow(3, a);\n}\n</code></pre> max_product_cutting.java<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nint maxProductCutting(int n) {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return (int) Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return (int) Math.pow(3, a) * 2;\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return (int) Math.pow(3, a);\n}\n</code></pre> max_product_cutting.cs<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nint MaxProductCutting(int n) {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return (int)Math.Pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return (int)Math.Pow(3, a) * 2;\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return (int)Math.Pow(3, a);\n}\n</code></pre> max_product_cutting.go<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nfunc maxProductCutting(n int) int {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    a := n / 3\n    b := n % 3\n    if b == 1 {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return int(math.Pow(3, float64(a-1))) * 2 * 2\n    }\n    if b == 2 {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return int(math.Pow(3, float64(a))) * 2\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return int(math.Pow(3, float64(a)))\n}\n</code></pre> max_product_cutting.swift<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nfunc maxProductCutting(n: Int) -&gt; Int {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    let a = n / 3\n    let b = n % 3\n    if b == 1 {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return pow(3, a - 1) * 2 * 2\n    }\n    if b == 2 {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return pow(3, a) * 2\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return pow(3, a)\n}\n</code></pre> max_product_cutting.js<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nfunction maxProductCutting(n) {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    let a = Math.floor(n / 3);\n    let b = n % 3;\n    if (b === 1) {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return Math.pow(3, a) * 2;\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.ts<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nfunction maxProductCutting(n: number): number {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    let a: number = Math.floor(n / 3);\n    let b: number = n % 3;\n    if (b === 1) {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return Math.pow(3, a) * 2;\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.dart<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nint maxProductCutting(int n) {\n  // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n  if (n &lt;= 3) {\n    return 1 * (n - 1);\n  }\n  // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n  int a = n ~/ 3;\n  int b = n % 3;\n  if (b == 1) {\n    // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n    return (pow(3, a - 1) * 2 * 2).toInt();\n  }\n  if (b == 2) {\n    // 나머지가 2이면 별도로 처리하지 않는다\n    return (pow(3, a) * 2).toInt();\n  }\n  // 나머지가 0이면 별도로 처리하지 않는다\n  return pow(3, a).toInt();\n}\n</code></pre> max_product_cutting.rs<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nfn max_product_cutting(n: i32) -&gt; i32 {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if n &lt;= 3 {\n        return 1 * (n - 1);\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    let a = n / 3;\n    let b = n % 3;\n    if b == 1 {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        3_i32.pow(a as u32 - 1) * 2 * 2\n    } else if b == 2 {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        3_i32.pow(a as u32) * 2\n    } else {\n        // 나머지가 0이면 별도로 처리하지 않는다\n        3_i32.pow(a as u32)\n    }\n}\n</code></pre> max_product_cutting.c<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nint maxProductCutting(int n) {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return pow(3, a) * 2;\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return pow(3, a);\n}\n</code></pre> max_product_cutting.kt<pre><code>/* 분할 후의 최대 곱: 탐욕 */\nfun maxProductCutting(n: Int): Int {\n    // n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n    if (n &lt;= 3) {\n        return 1 * (n - 1)\n    }\n    // 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n    val a = n / 3\n    val b = n % 3\n    if (b == 1) {\n        // 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n        return 3.0.pow((a - 1)).toInt() * 2 * 2\n    }\n    if (b == 2) {\n        // 나머지가 2이면 별도로 처리하지 않는다\n        return 3.0.pow(a).toInt() * 2 * 2\n    }\n    // 나머지가 0이면 별도로 처리하지 않는다\n    return 3.0.pow(a).toInt()\n}\n</code></pre> max_product_cutting.rb<pre><code>### 분할 후의 최대 곱: 탐욕 ###\ndef max_product_cutting(n)\n  # n &lt;= 3이면 반드시 1 하나가 나오도록 분할해야 한다\n  return 1 * (n - 1) if n &lt;= 3\n  # 탐욕적으로 3을 떼어 낸다. a는 3의 개수이고 b는 나머지이다\n  a, b = n / 3, n % 3\n  # 나머지가 1이면 1 * 3 한 쌍을 2 * 2로 바꾼다\n  return (3.pow(a - 1) * 2 * 2).to_i if b == 1\n  # 나머지가 2이면 별도로 처리하지 않는다\n  return (3.pow(a) * 2).to_i if b == 2\n  # 나머지가 0이면 별도로 처리하지 않는다\n  3.pow(a).to_i\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p></p> <p> 그림 15-16   정수 분할 최대 곱의 계산 방법 </p> <p>시간 복잡도는 프로그래밍 언어의 거듭제곱 구현 방식에 달려 있다. Python에서 자주 쓰는 거듭제곱 계산 방법은 세 가지다.</p> <ul> <li>연산자 <code>**</code>와 함수 <code>pow()</code>의 시간 복잡도는 모두 \\(O(\\log⁡ a)\\)이다.</li> <li>함수 <code>math.pow()</code>는 내부적으로 C 라이브러리의 <code>pow()</code>를 호출하여 부동 소수점 거듭제곱을 계산하며, 시간 복잡도는 \\(O(1)\\)이다.</li> </ul> <p>변수 \\(a\\)와 \\(b\\)는 상수 크기의 추가 공간을 사용하므로 공간 복잡도는 \\(O(1)\\)이다.</p>","path":["제 15 장   탐욕","15.4   정수 분할의 최대 곱 문제"],"tags":[]},{"location":"chapter_greedy/max_product_cutting_problem/#3","level":3,"title":"3.   정확성 증명","text":"<p>귀류법을 사용하며 \\(n \\geq 4\\)인 경우만 분석한다.</p> <ol> <li>모든 인수는 \\(\\leq 3\\)이다: 최적 분할에 \\(\\geq 4\\)인 인수 \\(x\\)가 있다고 가정하자. 이를 \\(2(x-2)\\)로 더 나누면 곱이 커지거나 같아진다. 이는 가정과 모순이다.</li> <li>분할에 \\(1\\)이 없다: 최적 분할에 인수 \\(1\\)이 있다고 가정하자. 이를 다른 인수에 합치면 더 큰 곱을 얻으므로 가정과 모순이다.</li> <li>분할에 \\(2\\)는 최대 두 개다: 최적 분할에 \\(2\\)가 세 개 있다고 가정하자. \\(3\\) 두 개로 바꾸면 곱이 더 커지므로 가정과 모순이다.</li> </ol>","path":["제 15 장   탐욕","15.4   정수 분할의 최대 곱 문제"],"tags":[]},{"location":"chapter_greedy/summary/","level":1,"title":"15.5   요약","text":"","path":["제 15 장   탐욕","15.5   요약"],"tags":[]},{"location":"chapter_greedy/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>탐욕 알고리즘은 주로 최적화 문제에 사용하며, 매 결정 단계에서 국소 최적 선택을 내려 전역 최적해를 얻으려는 원리다.</li> <li>탐욕 알고리즘은 선택을 하나씩 반복하며 매번 문제를 더 작은 부분 문제로 바꾸고, 해결될 때까지 진행한다.</li> <li>탐욕 알고리즘은 구현이 간단하고 풀이 효율이 높다. 시간 복잡도는 보통 동적 계획법보다 낮다.</li> <li>동전 교환에서는 일부 동전 조합에 대해 최적해를 보장하지만, 다른 조합에서는 매우 나쁜 해를 찾을 수도 있다.</li> <li>탐욕 알고리즘에 적합한 문제는 탐욕적 선택 속성과 최적 부분 구조를 가진다. 탐욕적 선택 속성은 전략의 유효성을 나타낸다.</li> <li>복잡한 문제에서는 탐욕적 선택 속성의 증명이 쉽지 않다. 동전 교환처럼 반례로 부정하는 편이 상대적으로 쉬운 경우가 있다.</li> <li>탐욕 문제의 풀이는 문제 분석, 탐욕 전략 결정, 정확성 증명의 세 단계로 나뉜다. 전략 결정이 핵심이며, 정확성 증명이 어려운 경우가 많다.</li> <li>분할 가능한 배낭은 0-1 배낭에서 물건의 일부를 고를 수 있도록 바꾼 문제이므로 탐욕 알고리즘으로 풀 수 있다. 전략의 정확성은 귀류법으로 증명할 수 있다.</li> <li>최대 용량 문제는 모든 경우를 열거하면 시간 복잡도가 \\(O(n^2)\\)이다. 매번 낮은 칸막이를 안쪽으로 옮기는 탐욕 전략으로 \\(O(n)\\)까지 최적화할 수 있다.</li> <li>정수 분할의 최대 곱 문제에서는 \\(\\geq 4\\)인 정수를 계속 나누어야 하고 최적인 인수는 \\(3\\)이라는 두 탐욕 전략을 차례로 도출했다. 코드에는 거듭제곱이 포함되므로 시간 복잡도는 구현 방식에 따라 보통 \\(O(1)\\) 또는 \\(O(\\log n)\\)이다.</li> </ul>","path":["제 15 장   탐욕","15.5   요약"],"tags":[]},{"location":"chapter_hashing/","level":1,"title":"제 6 장   해시 테이블","text":"<p>이 장을 시작하며</p> <p>컴퓨터 세계에서 해시 테이블은 영리한 사서와 같다.</p> <p>청구 기호를 계산하는 방법을 알기에 원하는 책을 빠르게 찾아낸다.</p>","path":["제 6 장   해시 테이블"],"tags":[]},{"location":"chapter_hashing/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>6.1   해시 테이블</li> <li>6.2   해시 충돌</li> <li>6.3   해시 알고리즘</li> <li>6.4   요약</li> <li>6.5   연습 문제</li> </ul>","path":["제 6 장   해시 테이블"],"tags":[]},{"location":"chapter_hashing/exercises/","level":1,"title":"6.5   연습 문제","text":"","path":["제 6 장   해시 테이블","6.5   연습 문제"],"tags":[]},{"location":"chapter_hashing/exercises/#651","level":2,"title":"6.5.1   개념 확인","text":"","path":["제 6 장   해시 테이블","6.5   연습 문제"],"tags":[]},{"location":"chapter_hashing/exercises/#1","level":3,"title":"1.   해시 충돌이 일어난 뒤 탐색하는 방법","text":"<p>버킷이 5개인 해시 테이블의 해시 함수는 \\(h(x)=x \\bmod 5\\)다. 충돌이 나면 해당 버킷의 리스트에 원소를 순서대로 넣는다. <code>[1, 6, 11, 7]</code>을 차례대로 삽입한다.</p> <ol> <li>0~4번 버킷의 내용을 적어 보자.</li> <li>6을 찾을 때 어느 버킷으로 들어가고, 어떤 원소를 차례대로 확인하는가?</li> <li>1번에서 적은 버킷의 내용을 보면 나중에 삽입한 원소가 먼저 삽입한 원소를 덮어썼는가? 이 충돌 처리 방식과 연결하여 이유를 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>\\(1\\bmod5=6\\bmod5=11\\bmod5=1\\)이고 \\(7\\bmod5=2\\)이므로 각 버킷의 내용은 다음과 같다.</p> <pre><code>0: []\n1: [1, 6, 11]\n2: [7]\n3: []\n4: []\n</code></pre> </li> <li> <p>6을 찾을 때는 먼저 1번 버킷으로 들어가 1, 6을 차례대로 비교한다. 두 번째 비교에서 목표를 찾는다.</p> </li> <li> <p>해시값이 같다는 것은 같은 버킷에 들어간다는 뜻일 뿐, 원소들이 같다는 뜻은 아니다. 체이닝은 충돌한 원소를 모두 버킷에 저장하고,     탐색할 때 하나씩 비교한다. 따라서 1, 6, 11은 서로를 덮어쓰지 않는다.</p> </li> </ol>","path":["제 6 장   해시 테이블","6.5   연습 문제"],"tags":[]},{"location":"chapter_hashing/exercises/#2","level":3,"title":"2.   해시 테이블의 용량을 늘리면 원소는 어디로 가는가","text":"<p>체이닝을 사용하는 해시 테이블에 원래 5개의 버킷이 있고, 해시 함수는 \\(h(x)=x\\bmod5\\)다. 키 <code>[1, 6, 11]</code>은 모두 1번 버킷에 있다.</p> <p>이제 버킷 수를 7개로 늘리고 해시 함수도 \\(h(x)=x\\bmod7\\)로 바꾼다.</p> <ol> <li>1, 6, 11의 새 버킷 번호를 각각 계산하라.</li> <li>용량을 늘린 뒤 어느 버킷에 원소가 들어 있는가?</li> <li>기존 1번 버킷의 리스트를 새 1번 버킷에 그대로 복사해도 되는가? 1번과 2번의 결과를 바탕으로 이유를 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>새 버킷 번호는 각각 다음과 같다.</p> <ul> <li>\\(1\\bmod7=1\\)</li> <li>\\(6\\bmod7=6\\)</li> <li>\\(11\\bmod7=4\\)</li> </ul> </li> <li> <p>1번 버킷에 1, 4번 버킷에 11, 6번 버킷에 6이 저장된다. 세 키가 더 이상 하나의 버킷에 모여 있지 않다.</p> </li> <li> <p>그대로 복사해서는 안 된다. 버킷 번호는 키를 버킷 수로 나눈 나머지로 구한다. 버킷 수가 5에서 7로 바뀌면 같은 키의 버킷 번호도 달라질 수 있으므로,     모든 키의 위치를 다시 계산해야 한다. 이전 1번 버킷을 그대로 복사하면 새 식으로 6과 11을 찾을 때     각각 6번 버킷과 4번 버킷으로 가게 되어 찾지 못한다.</p> </li> </ol>","path":["제 6 장   해시 테이블","6.5   연습 문제"],"tags":[]},{"location":"chapter_hashing/exercises/#3-6-11","level":3,"title":"3.   6을 삭제한 뒤에도 11을 찾을 수 있는가","text":"<p>해시 테이블에 5개의 자리가 있고 인덱스는 <code>0～4</code>이며, 해시 함수는 \\(h(x)=x\\bmod5\\)다. 충돌이 나면 해시 함수로 계산한 인덱스부터 오른쪽으로 이동하며 처음 만나는 빈자리를 찾는다.</p> <p><code>[1, 6, 11]</code>을 차례대로 삽입한다.</p> <ol> <li>세 수는 최종적으로 각각 어느 인덱스에 저장되는가?</li> <li>11을 찾을 때 어떤 인덱스를 차례대로 확인하는가?</li> <li>6을 삭제하면서 그 자리를 ‘한 번도 사용하지 않은 빈자리’로 바꾸고, 탐색은 빈자리를 만나면 멈춘다고 하자.     다시 11을 찾으면 어떤 일이 일어나는가? 그 결과는 올바른가? 문제가 있다면 어떻게 피할 수 있는가?</li> </ol> 참고 답안 <ol> <li> <p>1은 인덱스 1에 저장된다. 6도 인덱스 1로 매핑되지만 충돌하므로 인덱스 2에 저장된다.     11 역시 인덱스 1부터 시작해 이미 사용 중인 인덱스 1과 2를 지나 인덱스 3에 저장된다.</p> </li> <li> <p>11을 찾을 때는 인덱스 <code>1, 2, 3</code>을 차례대로 확인하고 인덱스 3에서 찾는다.</p> </li> <li> <p>인덱스 2를 ‘한 번도 사용하지 않음’을 뜻하는 빈자리로 바꾸면, 11을 찾다가 인덱스 1을 확인한 뒤 인덱스 2에서 멈춘다.     따라서 11이 없다고 잘못 판단한다. 삭제할 때는 ‘삭제됨’ 표시를 남겨야 한다.     탐색 중 이 표시를 만나면 다음 인덱스를 계속 확인하고, 인덱스 4를 지나면 0으로 돌아간다. 이후 삽입할 때는 이 자리를 다시 사용할 수 있다.</p> </li> </ol>","path":["제 6 장   해시 테이블","6.5   연습 문제"],"tags":[]},{"location":"chapter_hashing/exercises/#652","level":2,"title":"6.5.2   프로그래밍 연습","text":"","path":["제 6 장   해시 테이블","6.5   연습 문제"],"tags":[]},{"location":"chapter_hashing/exercises/#1_1","level":3,"title":"1.   두 문자열의 문자 구성 비교하기","text":"<p>영어 소문자만 포함하는 두 문자열 <code>s</code>와 <code>t</code>가 주어진다. <code>s</code>의 문자 위치는 자유롭게 바꿀 수 있지만 문자를 추가하거나 삭제하거나 다른 문자로 바꿀 수는 없다.</p> <p>순서를 바꿔 <code>t</code>를 만들 수 있는지 판단하고, 가능하면 <code>true</code>, 그렇지 않으면 <code>false</code>를 반환하라. 문자열의 문자를 정렬하지 말고 해시 테이블로 각 문자의 등장 횟수를 기록하라.</p> 풀이 힌트 <ol> <li>두 문자열의 길이가 다르면 문자 구성도 반드시 다르다.</li> <li>해시 테이블에 각 문자의 개수를 기록한다. s를 순회할 때 해당 개수를 1씩 늘린다.</li> <li>t를 순회할 때 해당 개수를 1씩 줄인다. 마지막에 모든 개수가 0이어야 문자 구성이 같다.</li> </ol> <p>LeetCode</p>","path":["제 6 장   해시 테이블","6.5   연습 문제"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/","level":1,"title":"6.3   해시 알고리즘","text":"<p>앞의 두 절에서는 해시 테이블의 작동 원리와 해시 충돌 처리 방법을 알아보았다. 하지만 개방 주소법이든 체이닝이든 충돌이 발생해도 해시 테이블이 정상적으로 작동하게 할 뿐, 충돌 발생 자체를 줄이지는 못한다.</p> <p>충돌이 너무 자주 발생하면 성능이 급격히 나빠진다. 아래 그림(6-8)의 체이닝 해시 테이블에서 이상적인 경우는 키-값 쌍이 각 버킷에 고르게 분포하여 최고의 조회 효율을 얻는 것이다. 최악의 경우에는 모든 키-값 쌍이 같은 버킷에 들어가 시간 복잡도가 \\(O(n)\\)으로 나빠진다.</p> <p></p> <p> 그림 6-8   해시 충돌의 최선과 최악의 경우 </p> <p>키-값 쌍의 분포는 해시 함수가 결정한다. 해시 함수는 먼저 해시값을 계산하고 배열 길이로 나눈 나머지를 구한다.</p> <pre><code>index = hash(key) % capacity\n</code></pre> <p>위 식에서 해시 테이블 용량 <code>capacity</code>가 고정되면 해시 알고리즘 <code>hash()</code>가 출력값을 결정하며, 나아가 키-값 쌍의 분포를 결정한다.</p> <p>따라서 해시 충돌 확률을 낮추려면 해시 알고리즘 <code>hash()</code>의 설계에 집중해야 한다.</p>","path":["제 6 장   해시 테이블","6.3   해시 알고리즘"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#631","level":2,"title":"6.3.1   해시 알고리즘의 목표","text":"<p>‘빠르고 안정적인’ 해시 테이블을 구현하려면 해시 알고리즘이 다음 특징을 갖추어야 한다.</p> <ul> <li>결정성: 같은 입력에는 항상 같은 출력을 생성해야 한다. 그래야 해시 테이블을 신뢰할 수 있다.</li> <li>높은 효율: 해시값 계산이 충분히 빨라야 한다. 계산 비용이 작을수록 해시 테이블의 실용성이 높아진다.</li> <li>균등 분포: 키-값 쌍이 해시 테이블에 고르게 분포하도록 해야 한다. 분포가 고를수록 충돌 확률은 낮아진다.</li> </ul> <p>해시 알고리즘은 해시 테이블 외에도 여러 분야에서 널리 사용된다.</p> <ul> <li>비밀번호 저장: 사용자의 비밀번호를 보호하기 위해 시스템은 보통 평문 대신 해시값을 저장한다. 사용자가 비밀번호를 입력하면 해시값을 계산해 저장된 값과 비교하고, 두 값이 같으면 올바른 비밀번호로 판단한다.</li> <li>데이터 무결성 검사: 송신자는 데이터의 해시값을 계산해 함께 보낼 수 있다. 수신자는 받은 데이터의 해시값을 다시 계산하여 전달받은 해시값과 비교한다. 둘이 같으면 데이터가 온전하다고 판단한다.</li> </ul> <p>암호학 관련 응용에서는 해시값으로 원래 비밀번호를 추론하는 등의 역공학을 막기 위해 더 높은 수준의 보안 특성이 필요하다.</p> <ul> <li>일방향성: 해시값으로 입력 데이터에 관한 정보를 역으로 추론할 수 없어야 한다.</li> <li>충돌 저항성: 해시값이 같은 서로 다른 두 입력을 찾기가 극히 어려워야 한다.</li> <li>눈사태 효과: 입력의 작은 변화가 출력의 크고 예측할 수 없는 변화로 이어져야 한다.</li> </ul> <p>‘균등 분포’와 ‘충돌 저항성’은 서로 독립된 개념이다. 균등 분포를 만족한다고 충돌 저항성도 만족하는 것은 아니다. 예를 들어 <code>key</code>가 무작위로 입력되면 해시 함수 <code>key % 100</code>은 고르게 분포하는 출력을 만들 수 있다. 하지만 이 알고리즘은 너무 단순하여 마지막 두 자리가 같은 모든 <code>key</code>의 출력이 같다. 따라서 해시값에서 유효한 <code>key</code>를 쉽게 역추론하여 비밀번호를 깰 수 있다.</p>","path":["제 6 장   해시 테이블","6.3   해시 알고리즘"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#632","level":2,"title":"6.3.2   해시 알고리즘의 설계","text":"<p>해시 알고리즘 설계는 여러 요소를 고려해야 하는 복잡한 문제다. 다만 요구 수준이 높지 않은 상황에서는 간단한 알고리즘을 설계할 수 있다.</p> <ul> <li>덧셈 해싱: 입력의 각 문자에 해당하는 ASCII 코드를 모두 더한 값을 해시값으로 삼는다.</li> <li>곱셈 해싱: 곱셈의 비상관성을 이용하여 매번 상수를 곱하고, 각 문자의 ASCII 코드를 해시값에 누적한다.</li> <li>XOR 해싱: 입력 데이터의 각 원소를 XOR 연산으로 하나의 해시값에 누적한다.</li> <li>회전 해싱: 각 문자의 ASCII 코드를 해시값에 누적하되, 누적하기 전에 해시값에 회전 연산을 수행한다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby simple_hash.py<pre><code>def add_hash(key: str) -&gt; int:\n    \"\"\"덧셈 해시\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash += ord(c)\n    return hash % modulus\n\ndef mul_hash(key: str) -&gt; int:\n    \"\"\"곱셈 해시\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = 31 * hash + ord(c)\n    return hash % modulus\n\ndef xor_hash(key: str) -&gt; int:\n    \"\"\"XOR 해시\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash ^= ord(c)\n    return hash % modulus\n\ndef rot_hash(key: str) -&gt; int:\n    \"\"\"회전 해시\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ ord(c)\n    return hash % modulus\n</code></pre> simple_hash.cpp<pre><code>/* 덧셈 해시 */\nint addHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* 곱셈 해시 */\nint mulHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (31 * hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* XOR 해시 */\nint xorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash ^= (int)c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* 회전 해시 */\nint rotHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.java<pre><code>/* 덧셈 해시 */\nint addHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* 곱셈 해시 */\nint mulHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (31 * hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* XOR 해시 */\nint xorHash(String key) {\n    int hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash ^= (int) c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* 회전 해시 */\nint rotHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n</code></pre> simple_hash.cs<pre><code>/* 덧셈 해시 */\nint AddHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* 곱셈 해시 */\nint MulHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (31 * hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* XOR 해시 */\nint XorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash ^= c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* 회전 해시 */\nint RotHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.go<pre><code>/* 덧셈 해시 */\nfunc addHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* 곱셈 해시 */\nfunc mulHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (31*hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* XOR 해시 */\nfunc xorHash(key string) int {\n    hash := 0\n    modulus := 1000000007\n    for _, b := range []byte(key) {\n        fmt.Println(int(b))\n        hash ^= int(b)\n        hash = (31*hash + int(b)) % modulus\n    }\n    return hash &amp; modulus\n}\n\n/* 회전 해시 */\nfunc rotHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ int64(b)) % modulus\n    }\n    return int(hash)\n}\n</code></pre> simple_hash.swift<pre><code>/* 덧셈 해시 */\nfunc addHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* 곱셈 해시 */\nfunc mulHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (31 * hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* XOR 해시 */\nfunc xorHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash ^= Int(scalar.value)\n        }\n    }\n    return hash &amp; MODULUS\n}\n\n/* 회전 해시 */\nfunc rotHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n</code></pre> simple_hash.js<pre><code>/* 덧셈 해시 */\nfunction addHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* 곱셈 해시 */\nfunction mulHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* XOR 해시 */\nfunction xorHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash % MODULUS;\n}\n\n/* 회전 해시 */\nfunction rotHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.ts<pre><code>/* 덧셈 해시 */\nfunction addHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* 곱셈 해시 */\nfunction mulHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* XOR 해시 */\nfunction xorHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash % MODULUS;\n}\n\n/* 회전 해시 */\nfunction rotHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.dart<pre><code>/* 덧셈 해시 */\nint addHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* 곱셈 해시 */\nint mulHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (31 * hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* XOR 해시 */\nint xorHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash ^= key.codeUnitAt(i);\n  }\n  return hash &amp; MODULUS;\n}\n\n/* 회전 해시 */\nint rotHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n</code></pre> simple_hash.rs<pre><code>/* 덧셈 해시 */\nfn add_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* 곱셈 해시 */\nfn mul_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (31 * hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* XOR 해시 */\nfn xor_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash ^= c as i64;\n    }\n\n    (hash &amp; MODULUS) as i32\n}\n\n/* 회전 해시 */\nfn rot_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n</code></pre> simple_hash.c<pre><code>/* 덧셈 해시 */\nint addHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* 곱셈 해시 */\nint mulHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (31 * hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* XOR 해시 */\nint xorHash(char *key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash ^= (unsigned char)key[i];\n    }\n    return hash &amp; MODULUS;\n}\n\n/* 회전 해시 */\nint rotHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (unsigned char)key[i]) % MODULUS;\n    }\n\n    return (int)hash;\n}\n</code></pre> simple_hash.kt<pre><code>/* 덧셈 해시 */\nfun addHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = (hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* 곱셈 해시 */\nfun mulHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = (31 * hash + c.code) % MODULUS\n    }\n    return hash.toInt()\n}\n\n/* XOR 해시 */\nfun xorHash(key: String): Int {\n    var hash = 0\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = hash xor c.code\n    }\n    return hash and MODULUS\n}\n\n/* 회전 해시 */\nfun rotHash(key: String): Int {\n    var hash = 0L\n    val MODULUS = 1000000007\n    for (c in key.toCharArray()) {\n        hash = ((hash shl 4) xor (hash shr 28) xor c.code.toLong()) % MODULUS\n    }\n    return hash.toInt()\n}\n</code></pre> simple_hash.rb<pre><code>### 덧셈 해시 ###\ndef add_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash += c.ord }\n\n  hash % modulus\nend\n\n### 곱셈 해시 ###\ndef mul_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash = 31 * hash + c.ord }\n\n  hash % modulus\nend\n\n### XOR 해시 ###\ndef xor_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash ^= c.ord }\n\n  hash % modulus\nend\n\n### 회전 해시 ###\ndef rot_hash(key)\n  hash = 0\n  modulus = 1_000_000_007\n\n  key.each_char { |c| hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.ord }\n\n  hash % modulus\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>각 알고리즘의 마지막 단계에서는 큰 소수 \\(1000000007\\)로 나눈 나머지를 구하여 해시값이 적절한 범위에 있도록 한다. 왜 소수로 나누어야 할까? 합성수로 나누면 어떤 단점이 있을까? 생각해 볼 만한 흥미로운 문제다.</p> <p>결론부터 말하면 큰 소수를 모듈러 연산의 법으로 사용하면 해시값의 균등 분포를 최대한 보장할 수 있다. 소수는 다른 수와 공약수를 갖지 않으므로 나머지 연산에서 생기는 주기적 패턴을 줄여 충돌을 피할 수 있기 때문이다.</p> <p>예를 들어 합성수 \\(9\\)를 법으로 선택했다고 하자. 이 수는 \\(3\\)으로 나누어떨어지므로, \\(3\\)으로 나누어떨어지는 모든 <code>key</code>는 \\(0\\), \\(3\\), \\(6\\)의 세 해시값에 매핑된다.</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 9 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\dots \\} \\end{aligned} \\] <p>입력 <code>key</code>가 마침 이런 등차수열 분포를 따르면 해시값에 군집이 생겨 충돌이 심해진다. 이제 <code>modulus</code>를 소수 \\(13\\)으로 바꾸면 <code>key</code>와 <code>modulus</code> 사이에 공약수가 없으므로 출력 해시값의 균등성이 뚜렷하게 좋아진다.</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 13 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\dots \\} \\end{aligned} \\] <p><code>key</code>가 무작위로 균등하게 분포함을 보장할 수 있다면 소수와 합성수 중 어느 것을 법으로 써도 균등한 해시값을 얻을 수 있다. 반면 <code>key</code>의 분포에 주기성이 있으면 합성수로 나눌 때 군집이 생기기 쉽다.</p> <p>따라서 보통 소수를 법으로 선택하며, 가능한 한 큰 소수를 사용해 주기적 패턴을 줄이고 해시 알고리즘의 안정성을 높인다.</p>","path":["제 6 장   해시 테이블","6.3   해시 알고리즘"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#633","level":2,"title":"6.3.3   대표적인 해시 알고리즘","text":"<p>앞에서 소개한 간단한 해시 알고리즘은 비교적 ‘취약하여’ 설계 목표에 크게 못 미친다. 예를 들어 덧셈과 XOR는 교환법칙을 만족하므로, 이 두 해싱 방식은 내용은 같고 순서만 다른 문자열을 구분하지 못한다. 이는 해시 충돌을 늘리고 보안 문제를 일으킬 수 있다.</p> <p>실제로는 MD5, SHA-1, SHA-2, SHA-3 같은 표준 해시 알고리즘을 주로 사용한다. 이들은 길이가 임의인 입력 데이터를 고정 길이의 해시값으로 매핑한다.</p> <p>지난 한 세기 가까이 해시 알고리즘은 계속 개선되고 최적화되어 왔다. 일부 연구자는 성능을 높이는 데 힘쓰고, 다른 연구자와 해커는 보안 문제를 찾는 데 집중한다. 아래 표(6-2)는 실제 응용에서 흔히 쓰이는 해시 알고리즘을 보여 준다.</p> <ul> <li>MD5와 SHA-1은 여러 차례 공격에 성공한 사례가 있어 각종 보안 응용에서 사용이 중단되었다.</li> <li>SHA-2 계열의 SHA-256은 가장 안전한 해시 알고리즘 중 하나로, 아직 성공한 공격 사례가 없어 각종 보안 응용과 프로토콜에서 자주 사용된다.</li> <li>SHA-3는 SHA-2보다 구현 비용이 낮고 계산 효율이 높지만, 현재 보급 범위는 SHA-2 계열에 못 미친다.</li> </ul> <p> 표 6-2   대표적인 해시 알고리즘 </p> MD5 SHA-1 SHA-2 SHA-3 발표 시기 1992 1995 2002 2008 출력 길이 128 bit 160 bit 256/512 bit 224/256/384/512 bit 해시 충돌 비교적 많음 비교적 많음 매우 적음 매우 적음 보안 수준 낮음, 공격 성공 사례가 있음 낮음, 공격 성공 사례가 있음 높음 높음 활용 사용 중단, 데이터 무결성 검사에는 여전히 사용 사용 중단 암호화폐 거래 검증, 전자 서명 등 SHA-2를 대체할 수 있음","path":["제 6 장   해시 테이블","6.3   해시 알고리즘"],"tags":[]},{"location":"chapter_hashing/hash_algorithm/#634","level":2,"title":"6.3.4   자료 구조의 해시값","text":"<p>해시 테이블의 <code>key</code>는 정수, 소수, 문자열 등의 자료형일 수 있다. 프로그래밍 언어는 보통 이러한 자료형에 내장 해시 알고리즘을 제공하여 버킷 인덱스를 계산한다. 예를 들어 Python에서는 <code>hash()</code> 함수로 여러 자료형의 해시값을 구할 수 있다.</p> <ul> <li>정수와 불리언의 해시값은 그 값 자체다.</li> <li>부동 소수점 수와 문자열의 해시값 계산은 더 복잡하다. 관심 있는 독자는 따로 알아보자.</li> <li>튜플은 각 원소의 해시값을 계산한 뒤 이를 결합하여 하나의 해시값을 얻는다.</li> <li>객체의 해시값은 메모리 주소를 기반으로 생성된다. 객체의 해시 메서드를 재정의하면 내용을 기반으로 해시값을 만들 수도 있다.</li> </ul> <p>참고</p> <p>프로그래밍 언어마다 내장 해시값 계산 함수의 정의와 방식이 다르다는 점에 주의하자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby built_in_hash.py<pre><code>num = 3\nhash_num = hash(num)\n# 정수 3의 해시값은 3\n\nbol = True\nhash_bol = hash(bol)\n# 불리언 True의 해시값은 1\n\ndec = 3.14159\nhash_dec = hash(dec)\n# 소수 3.14159의 해시값은 326484311674566659\n\nstr = \"Hello 算法\"\nhash_str = hash(str)\n# 문자열 “Hello 算法”의 해시값은 4617003410720528961\n\ntup = (12836, \"小哈\")\nhash_tup = hash(tup)\n# 튜플 (12836, '小哈')의 해시값은 1029005403108185979\n\nobj = ListNode(0)\nhash_obj = hash(obj)\n# 노드 객체 &lt;ListNode object at 0x1058fd810&gt;의 해시값은 274267521\n</code></pre> built_in_hash.cpp<pre><code>int num = 3;\nsize_t hashNum = hash&lt;int&gt;()(num);\n// 정수 3의 해시값은 3\n\nbool bol = true;\nsize_t hashBol = hash&lt;bool&gt;()(bol);\n// 불리언 1의 해시값은 1\n\ndouble dec = 3.14159;\nsize_t hashDec = hash&lt;double&gt;()(dec);\n// 소수 3.14159의 해시값은 4614256650576692846\n\nstring str = \"Hello 算法\";\nsize_t hashStr = hash&lt;string&gt;()(str);\n// 문자열 “Hello 算法”의 해시값은 15466937326284535026\n\n// C++의 내장 std::hash()는 기본 자료형의 해시값 계산만 제공한다\n// 배열과 객체의 해시값 계산은 직접 구현해야 한다\n</code></pre> built_in_hash.java<pre><code>int num = 3;\nint hashNum = Integer.hashCode(num);\n// 정수 3의 해시값은 3\n\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// 불리언 true의 해시값은 1231\n\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// 소수 3.14159의 해시값은 -1340954729\n\nString str = \"Hello 算法\";\nint hashStr = str.hashCode();\n// 문자열 “Hello 算法”의 해시값은 -727081396\n\nObject[] arr = { 12836, \"小哈\" };\nint hashTup = Arrays.hashCode(arr);\n// 배열 [12836, 小哈]의 해시값은 1151158\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// 노드 객체 utils.ListNode@7dc5e7b4의 해시값은 2110121908\n</code></pre> built_in_hash.cs<pre><code>int num = 3;\nint hashNum = num.GetHashCode();\n// 정수 3의 해시값은 3;\n\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// 불리언 true의 해시값은 1;\n\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// 소수 3.14159의 해시값은 -1340954729;\n\nstring str = \"Hello 算法\";\nint hashStr = str.GetHashCode();\n// 문자열 “Hello 算法”의 해시값은 -586107568;\n\nobject[] arr = [12836, \"小哈\"];\nint hashTup = arr.GetHashCode();\n// 배열 [12836, 小哈]의 해시값은 42931033;\n\nListNode obj = new(0);\nint hashObj = obj.GetHashCode();\n// 노드 객체 0의 해시값은 39053774;\n</code></pre> built_in_hash.go<pre><code>// Go는 내장 hash code 함수를 제공하지 않는다\n</code></pre> built_in_hash.swift<pre><code>let num = 3\nlet hashNum = num.hashValue\n// 정수 3의 해시값은 9047044699613009734\n\nlet bol = true\nlet hashBol = bol.hashValue\n// 불리언 true의 해시값은 -4431640247352757451\n\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// 소수 3.14159의 해시값은 -2465384235396674631\n\nlet str = \"Hello 算法\"\nlet hashStr = str.hashValue\n// 문자열 “Hello 算法”의 해시값은 -7850626797806988787\n\nlet arr = [AnyHashable(12836), AnyHashable(\"小哈\")]\nlet hashTup = arr.hashValue\n// 배열 [AnyHashable(12836), AnyHashable(\"小哈\")]의 해시값은 -2308633508154532996\n\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// 노드 객체 utils.ListNode의 해시값은 -2434780518035996159\n</code></pre> built_in_hash.js<pre><code>// JavaScript는 내장 hash code 함수를 제공하지 않는다\n</code></pre> built_in_hash.ts<pre><code>// TypeScript는 내장 hash code 함수를 제공하지 않는다\n</code></pre> built_in_hash.dart<pre><code>int num = 3;\nint hashNum = num.hashCode;\n// 정수 3의 해시값은 34803\n\nbool bol = true;\nint hashBol = bol.hashCode;\n// 불리언 true의 해시값은 1231\n\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// 소수 3.14159의 해시값은 2570631074981783\n\nString str = \"Hello 算法\";\nint hashStr = str.hashCode;\n// 문자열 “Hello 算法”의 해시값은 468167534\n\nList arr = [12836, \"小哈\"];\nint hashArr = arr.hashCode;\n// 배열 [12836, 小哈]의 해시값은 976512528\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// 노드 객체 Instance of 'ListNode'의 해시값은 1033450432\n</code></pre> built_in_hash.rs<pre><code>use std::collections::hash_map::DefaultHasher;\nuse std::hash::{Hash, Hasher};\n\nlet num = 3;\nlet mut num_hasher = DefaultHasher::new();\nnum.hash(&amp;mut num_hasher);\nlet hash_num = num_hasher.finish();\n// 정수 3의 해시값은 568126464209439262\n\nlet bol = true;\nlet mut bol_hasher = DefaultHasher::new();\nbol.hash(&amp;mut bol_hasher);\nlet hash_bol = bol_hasher.finish();\n// 불리언 true의 해시값은 4952851536318644461\n\nlet dec: f32 = 3.14159;\nlet mut dec_hasher = DefaultHasher::new();\ndec.to_bits().hash(&amp;mut dec_hasher);\nlet hash_dec = dec_hasher.finish();\n// 소수 3.14159의 해시값은 2566941990314602357\n\nlet str = \"Hello 算法\";\nlet mut str_hasher = DefaultHasher::new();\nstr.hash(&amp;mut str_hasher);\nlet hash_str = str_hasher.finish();\n// 문자열 “Hello 算法”의 해시값은 16092673739211250988\n\nlet arr = (&amp;12836, &amp;\"小哈\");\nlet mut tup_hasher = DefaultHasher::new();\narr.hash(&amp;mut tup_hasher);\nlet hash_tup = tup_hasher.finish();\n// 튜플 (12836, \"小哈\")의 해시값은 1885128010422702749\n\nlet node = ListNode::new(42);\nlet mut hasher = DefaultHasher::new();\nnode.borrow().val.hash(&amp;mut hasher);\nlet hash = hasher.finish();\n// 노드 객체 RefCell { value: ListNode { val: 42, next: None } }의 해시값은 15387811073369036852\n</code></pre> built_in_hash.c<pre><code>// C는 내장 hash code 함수를 제공하지 않는다\n</code></pre> built_in_hash.kt<pre><code>val num = 3\nval hashNum = num.hashCode()\n// 정수 3의 해시값은 3\n\nval bol = true\nval hashBol = bol.hashCode()\n// 불리언 true의 해시값은 1231\n\nval dec = 3.14159\nval hashDec = dec.hashCode()\n// 소수 3.14159의 해시값은 -1340954729\n\nval str = \"Hello 算法\"\nval hashStr = str.hashCode()\n// 문자열 “Hello 算法”의 해시값은 -727081396\n\nval arr = arrayOf&lt;Any&gt;(12836, \"小哈\")\nval hashTup = arr.hashCode()\n// 배열 [12836, 小哈]의 해시값은 189568618\n\nval obj = ListNode(0)\nval hashObj = obj.hashCode()\n// 노드 객체 utils.ListNode@1d81eb93의 해시값은 495053715\n</code></pre> built_in_hash.rb<pre><code>num = 3\nhash_num = num.hash\n# 정수 3의 해시값은 -4385856518450339636\n\nbol = true\nhash_bol = bol.hash\n# 불리언 true의 해시값은 -1617938112149317027\n\ndec = 3.14159\nhash_dec = dec.hash\n# 소수 3.14159의 해시값은 -1479186995943067893\n\nstr = \"Hello 算法\"\nhash_str = str.hash\n# 문자열 “Hello 算法”의 해시값은 -4075943250025831763\n\ntup = [12836, '小哈']\nhash_tup = tup.hash\n# 튜플 (12836, '小哈')의 해시값은 1999544809202288822\n\nobj = ListNode.new(0)\nhash_obj = obj.hash\n# 노드 객체 #&lt;ListNode:0x000078133140ab70&gt;의 해시값은 4302940560806366381\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>많은 프로그래밍 언어에서는 불변 객체만 해시 테이블의 <code>key</code>로 사용할 수 있다. 리스트(동적 배열)를 <code>key</code>로 사용하면 내용이 바뀔 때 해시값도 달라져, 기존의 <code>value</code>를 조회할 수 없게 된다.</p> <p>연결 리스트 노드 같은 사용자 정의 객체는 멤버 변수가 바뀔 수 있어도 해시 가능하다. 객체의 해시값은 보통 메모리 주소를 기반으로 생성되기 때문이다. 내용이 바뀌어도 메모리 주소가 그대로면 해시값도 바뀌지 않는다.</p> <p>서로 다른 콘솔에서 실행하면 출력되는 해시값이 다르다는 점을 발견했을 것이다. Python 인터프리터가 시작될 때마다 문자열 해시 함수에 무작위 솔트(salt) 값을 추가하기 때문이다. 이 방식은 HashDoS 공격을 효과적으로 방지하여 해시 알고리즘의 보안성을 높인다.</p>","path":["제 6 장   해시 테이블","6.3   해시 알고리즘"],"tags":[]},{"location":"chapter_hashing/hash_collision/","level":1,"title":"6.2   해시 충돌","text":"<p>앞 절에서 설명했듯이 해시 함수의 입력 공간은 보통 출력 공간보다 훨씬 크므로, 이론적으로 해시 충돌은 피할 수 없다. 예를 들어 입력 공간이 모든 정수이고 출력 공간이 배열 용량만큼이라면, 여러 정수가 같은 버킷 인덱스에 매핑될 수밖에 없다.</p> <p>해시 충돌은 잘못된 조회 결과를 일으켜 해시 테이블의 사용 가능성을 크게 해친다. 이를 해결하려면 충돌이 생길 때마다 해시 테이블을 확장하여 충돌이 없어질 때까지 반복할 수 있다. 단순하고 효과적인 방법이지만, 확장할 때 많은 데이터를 옮기고 해시값을 계산해야 하므로 효율은 매우 낮다. 효율을 높이려면 다음 전략을 사용할 수 있다.</p> <ol> <li>해시 테이블의 자료 구조를 개선하여 해시 충돌이 발생해도 정상적으로 작동하게 한다.</li> <li>필요한 경우, 즉 해시 충돌이 심할 때만 용량을 확장한다.</li> </ol> <p>해시 테이블의 구조를 개선하는 대표적인 방법은 ‘체이닝’과 ‘개방 주소법’이다.</p>","path":["제 6 장   해시 테이블","6.2   해시 충돌"],"tags":[]},{"location":"chapter_hashing/hash_collision/#621","level":2,"title":"6.2.1   체이닝","text":"<p>기본 해시 테이블에서는 버킷 하나에 키-값 쌍 하나만 저장할 수 있다. 체이닝(separate chaining)은 단일 원소를 연결 리스트로 바꾸고, 키-값 쌍을 노드로 삼아 충돌한 키-값 쌍을 모두 같은 연결 리스트에 저장한다. 아래 그림(6-5)은 체이닝 해시 테이블의 예다.</p> <p></p> <p> 그림 6-5   체이닝 해시 테이블 </p> <p>체이닝 해시 테이블에서는 연산 방식이 다음과 같이 바뀐다.</p> <ul> <li>원소 조회: <code>key</code>를 입력하여 해시 함수로 버킷 인덱스를 구한 뒤 연결 리스트의 머리 노드에 접근한다. 이후 리스트를 순회하며 <code>key</code>를 비교하여 목표 키-값 쌍을 찾는다.</li> <li>원소 추가: 해시 함수로 연결 리스트의 머리 노드에 접근한 뒤, 노드(키-값 쌍)를 리스트에 추가한다.</li> <li>원소 삭제: 해시 함수의 결과로 연결 리스트의 맨 앞에 접근한 뒤, 순회하여 목표 노드를 찾아 삭제한다.</li> </ul> <p>체이닝에는 다음과 같은 한계가 있다.</p> <ul> <li>공간 사용량 증가: 연결 리스트에는 노드 포인터가 있으므로 배열보다 메모리를 더 많이 사용한다.</li> <li>조회 효율 감소: 해당 원소를 찾으려면 연결 리스트를 선형으로 순회해야 한다.</li> </ul> <p>다음 코드는 간단한 체이닝 해시 테이블을 구현한다. 두 가지에 주의하자.</p> <ul> <li>코드를 단순화하기 위해 연결 리스트 대신 리스트(동적 배열)를 사용한다. 따라서 해시 테이블(배열)은 여러 버킷을 포함하며, 각 버킷은 리스트다.</li> <li>아래 구현에는 용량 확장 메서드가 포함된다. 적재율이 \\(\\frac{2}{3}\\)를 넘으면 해시 테이블을 기존 크기의 \\(2\\)배로 늘린다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map_chaining.py<pre><code>class HashMapChaining:\n    \"\"\"체이닝 해시 테이블\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        self.size = 0  # 키-값 쌍 개수\n        self.capacity = 4  # 해시 테이블 용량\n        self.load_thres = 2.0 / 3.0  # 용량 확장을 수행하는 적재율 임계값\n        self.extend_ratio = 2  # 용량 확장 배수\n        self.buckets = [[] for _ in range(self.capacity)]  # 버킷 배열\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"해시 함수\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"적재율\"\"\"\n        return self.size / self.capacity\n\n    def get(self, key: int) -&gt; str | None:\n        \"\"\"조회 연산\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for pair in bucket:\n            if pair.key == key:\n                return pair.val\n        # key를 찾지 못하면 None을 반환한다\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"추가 연산\"\"\"\n        # 적재율이 임계값을 초과하면 용량을 확장한다\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for pair in bucket:\n            if pair.key == key:\n                pair.val = val\n                return\n        # 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        pair = Pair(key, val)\n        bucket.append(pair)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"삭제 연산\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # 버킷을 순회하며 키-값 쌍을 삭제한다\n        for pair in bucket:\n            if pair.key == key:\n                bucket.remove(pair)\n                self.size -= 1\n                break\n\n    def extend(self):\n        \"\"\"해시 테이블 용량 확장\"\"\"\n        # 원래 해시 테이블을 임시로 저장한다\n        buckets = self.buckets\n        # 용량을 확장한 새 해시 테이블을 초기화한다\n        self.capacity *= self.extend_ratio\n        self.buckets = [[] for _ in range(self.capacity)]\n        self.size = 0\n        # 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for bucket in buckets:\n            for pair in bucket:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"해시 테이블 출력\"\"\"\n        for bucket in self.buckets:\n            res = []\n            for pair in bucket:\n                res.append(str(pair.key) + \" -&gt; \" + pair.val)\n            print(res)\n</code></pre> hash_map_chaining.cpp<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n  private:\n    int size;                       // 키-값 쌍 개수\n    int capacity;                   // 해시 테이블 용량\n    double loadThres;               // 용량 확장을 수행하는 적재율 임계값\n    int extendRatio;                // 용량 확장 배수\n    vector&lt;vector&lt;Pair *&gt;&gt; buckets; // 버킷 배열\n\n  public:\n    /* 생성자 */\n    HashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3.0), extendRatio(2) {\n        buckets.resize(capacity);\n    }\n\n    /* 소멸자 */\n    ~HashMapChaining() {\n        for (auto &amp;bucket : buckets) {\n            for (Pair *pair : bucket) {\n                // 메모리 해제\n                delete pair;\n            }\n        }\n    }\n\n    /* 해시 함수 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* 적재율 */\n    double loadFactor() {\n        return (double)size / (double)capacity;\n    }\n\n    /* 조회 연산 */\n    string get(int key) {\n        int index = hashFunc(key);\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                return pair-&gt;val;\n            }\n        }\n        // key를 찾지 못하면 빈 문자열을 반환한다\n        return \"\";\n    }\n\n    /* 추가 연산 */\n    void put(int key, string val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                pair-&gt;val = val;\n                return;\n            }\n        }\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        buckets[index].push_back(new Pair(key, val));\n        size++;\n    }\n\n    /* 삭제 연산 */\n    void remove(int key) {\n        int index = hashFunc(key);\n        auto &amp;bucket = buckets[index];\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        for (int i = 0; i &lt; bucket.size(); i++) {\n            if (bucket[i]-&gt;key == key) {\n                Pair *tmp = bucket[i];\n                bucket.erase(bucket.begin() + i); // 해당 키-값 쌍을 삭제한다\n                delete tmp;                       // 메모리 해제\n                size--;\n                return;\n            }\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    void extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        vector&lt;vector&lt;Pair *&gt;&gt; bucketsTmp = buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio;\n        buckets.clear();\n        buckets.resize(capacity);\n        size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (auto &amp;bucket : bucketsTmp) {\n            for (Pair *pair : bucket) {\n                put(pair-&gt;key, pair-&gt;val);\n                // 메모리 해제\n                delete pair;\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    void print() {\n        for (auto &amp;bucket : buckets) {\n            cout &lt;&lt; \"[\";\n            for (Pair *pair : bucket) {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; \", \";\n            }\n            cout &lt;&lt; \"]\\n\";\n        }\n    }\n};\n</code></pre> hash_map_chaining.java<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n    int size; // 키-값 쌍 개수\n    int capacity; // 해시 테이블 용량\n    double loadThres; // 용량 확장을 수행하는 적재율 임계값\n    int extendRatio; // 용량 확장 배수\n    List&lt;List&lt;Pair&gt;&gt; buckets; // 버킷 배열\n\n    /* 생성자 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n    }\n\n    /* 해시 함수 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* 적재율 */\n    double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* 조회 연산 */\n    String get(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // key를 찾지 못하면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    void put(int key, String val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        Pair pair = new Pair(key, val);\n        bucket.add(pair);\n        size++;\n    }\n\n    /* 삭제 연산 */\n    void remove(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    void extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n        size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (List&lt;Pair&gt; bucket : bucketsTmp) {\n            for (Pair pair : bucket) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    void print() {\n        for (List&lt;Pair&gt; bucket : buckets) {\n            List&lt;String&gt; res = new ArrayList&lt;&gt;();\n            for (Pair pair : bucket) {\n                res.add(pair.key + \" -&gt; \" + pair.val);\n            }\n            System.out.println(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.cs<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n    int size; // 키-값 쌍 개수\n    int capacity; // 해시 테이블 용량\n    double loadThres; // 용량 확장을 수행하는 적재율 임계값\n    int extendRatio; // 용량 확장 배수\n    List&lt;List&lt;Pair&gt;&gt; buckets; // 버킷 배열\n\n    /* 생성자 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n    }\n\n    /* 해시 함수 */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* 적재율 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* 조회 연산 */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // key를 찾지 못하면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    public void Put(int key, string val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        int index = HashFunc(key);\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        buckets[index].Add(new Pair(key, val));\n        size++;\n    }\n\n    /* 삭제 연산 */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        foreach (Pair pair in buckets[index].ToList()) {\n            if (pair.key == key) {\n                buckets[index].Remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    void Extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n        size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        foreach (List&lt;Pair&gt; bucket in bucketsTmp) {\n            foreach (Pair pair in bucket) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    public void Print() {\n        foreach (List&lt;Pair&gt; bucket in buckets) {\n            List&lt;string&gt; res = [];\n            foreach (Pair pair in bucket) {\n                res.Add(pair.key + \" -&gt; \" + pair.val);\n            }\n            foreach (string kv in res) {\n                Console.WriteLine(kv);\n            }\n        }\n    }\n}\n</code></pre> hash_map_chaining.go<pre><code>/* 체이닝 해시 테이블 */\ntype hashMapChaining struct {\n    size        int      // 키-값 쌍 개수\n    capacity    int      // 해시 테이블 용량\n    loadThres   float64  // 용량 확장을 수행하는 적재율 임계값\n    extendRatio int      // 용량 확장 배수\n    buckets     [][]pair // 버킷 배열\n}\n\n/* 생성자 */\nfunc newHashMapChaining() *hashMapChaining {\n    buckets := make([][]pair, 4)\n    for i := 0; i &lt; 4; i++ {\n        buckets[i] = make([]pair, 0)\n    }\n    return &amp;hashMapChaining{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     buckets,\n    }\n}\n\n/* 해시 함수 */\nfunc (m *hashMapChaining) hashFunc(key int) int {\n    return key % m.capacity\n}\n\n/* 적재율 */\nfunc (m *hashMapChaining) loadFactor() float64 {\n    return float64(m.size) / float64(m.capacity)\n}\n\n/* 조회 연산 */\nfunc (m *hashMapChaining) get(key int) string {\n    idx := m.hashFunc(key)\n    bucket := m.buckets[idx]\n    // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n    for _, p := range bucket {\n        if p.key == key {\n            return p.val\n        }\n    }\n    // key를 찾지 못하면 빈 문자열을 반환한다\n    return \"\"\n}\n\n/* 추가 연산 */\nfunc (m *hashMapChaining) put(key int, val string) {\n    // 적재율이 임계값을 초과하면 용량을 확장한다\n    if m.loadFactor() &gt; m.loadThres {\n        m.extend()\n    }\n    idx := m.hashFunc(key)\n    // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n    for i := range m.buckets[idx] {\n        if m.buckets[idx][i].key == key {\n            m.buckets[idx][i].val = val\n            return\n        }\n    }\n    // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n    p := pair{\n        key: key,\n        val: val,\n    }\n    m.buckets[idx] = append(m.buckets[idx], p)\n    m.size += 1\n}\n\n/* 삭제 연산 */\nfunc (m *hashMapChaining) remove(key int) {\n    idx := m.hashFunc(key)\n    // 버킷을 순회하며 키-값 쌍을 삭제한다\n    for i, p := range m.buckets[idx] {\n        if p.key == key {\n            // 슬라이스에서 삭제한다\n            m.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\n            m.size -= 1\n            break\n        }\n    }\n}\n\n/* 해시 테이블 용량 확장 */\nfunc (m *hashMapChaining) extend() {\n    // 원래 해시 테이블을 임시로 저장한다\n    tmpBuckets := make([][]pair, len(m.buckets))\n    for i := 0; i &lt; len(m.buckets); i++ {\n        tmpBuckets[i] = make([]pair, len(m.buckets[i]))\n        copy(tmpBuckets[i], m.buckets[i])\n    }\n    // 용량을 확장한 새 해시 테이블을 초기화한다\n    m.capacity *= m.extendRatio\n    m.buckets = make([][]pair, m.capacity)\n    for i := 0; i &lt; m.capacity; i++ {\n        m.buckets[i] = make([]pair, 0)\n    }\n    m.size = 0\n    // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for _, bucket := range tmpBuckets {\n        for _, p := range bucket {\n            m.put(p.key, p.val)\n        }\n    }\n}\n\n/* 해시 테이블 출력 */\nfunc (m *hashMapChaining) print() {\n    var builder strings.Builder\n\n    for _, bucket := range m.buckets {\n        builder.WriteString(\"[\")\n        for _, p := range bucket {\n            builder.WriteString(strconv.Itoa(p.key) + \" -&gt; \" + p.val + \" \")\n        }\n        builder.WriteString(\"]\")\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> hash_map_chaining.swift<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n    var size: Int // 키-값 쌍 개수\n    var capacity: Int // 해시 테이블 용량\n    var loadThres: Double // 용량 확장을 수행하는 적재율 임계값\n    var extendRatio: Int // 용량 확장 배수\n    var buckets: [[Pair]] // 버킷 배열\n\n    /* 생성자 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: [], count: capacity)\n    }\n\n    /* 해시 함수 */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* 적재율 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* 조회 연산 */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for pair in bucket {\n            if pair.key == key {\n                return pair.val\n            }\n        }\n        // key를 찾지 못하면 nil을 반환한다\n        return nil\n    }\n\n    /* 추가 연산 */\n    func put(key: Int, val: String) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val\n                return\n            }\n        }\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        let pair = Pair(key: key, val: val)\n        buckets[index].append(pair)\n        size += 1\n    }\n\n    /* 삭제 연산 */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        for (pairIndex, pair) in bucket.enumerated() {\n            if pair.key == key {\n                buckets[index].remove(at: pairIndex)\n                size -= 1\n                break\n            }\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    func extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        let bucketsTmp = buckets\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio\n        buckets = Array(repeating: [], count: capacity)\n        size = 0\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for bucket in bucketsTmp {\n            for pair in bucket {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    func print() {\n        for bucket in buckets {\n            let res = bucket.map { \"\\($0.key) -&gt; \\($0.val)\" }\n            Swift.print(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.js<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n    #size; // 키-값 쌍 개수\n    #capacity; // 해시 테이블 용량\n    #loadThres; // 용량 확장을 수행하는 적재율 임계값\n    #extendRatio; // 용량 확장 배수\n    #buckets; // 버킷 배열\n\n    /* 생성자 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* 해시 함수 */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* 적재율 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* 조회 연산 */\n    get(key) {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // key를 찾지 못하면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    put(key, val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* 삭제 연산 */\n    remove(key) {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    #extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        const bucketsTmp = this.#buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    print() {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.ts<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n    #size: number; // 키-값 쌍 개수\n    #capacity: number; // 해시 테이블 용량\n    #loadThres: number; // 용량 확장을 수행하는 적재율 임계값\n    #extendRatio: number; // 용량 확장 배수\n    #buckets: Pair[][]; // 버킷 배열\n\n    /* 생성자 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* 해시 함수 */\n    #hashFunc(key: number): number {\n        return key % this.#capacity;\n    }\n\n    /* 적재율 */\n    #loadFactor(): number {\n        return this.#size / this.#capacity;\n    }\n\n    /* 조회 연산 */\n    get(key: number): string | null {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // key를 찾지 못하면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    put(key: number, val: string): void {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* 삭제 연산 */\n    remove(key: number): void {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    #extend(): void {\n        // 원래 해시 테이블을 임시로 저장한다\n        const bucketsTmp = this.#buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    print(): void {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.dart<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n  late int size; // 키-값 쌍 개수\n  late int capacity; // 해시 테이블 용량\n  late double loadThres; // 용량 확장을 수행하는 적재율 임계값\n  late int extendRatio; // 용량 확장 배수\n  late List&lt;List&lt;Pair&gt;&gt; buckets; // 버킷 배열\n\n  /* 생성자 */\n  HashMapChaining() {\n    size = 0;\n    capacity = 4;\n    loadThres = 2.0 / 3.0;\n    extendRatio = 2;\n    buckets = List.generate(capacity, (_) =&gt; []);\n  }\n\n  /* 해시 함수 */\n  int hashFunc(int key) {\n    return key % capacity;\n  }\n\n  /* 적재율 */\n  double loadFactor() {\n    return size / capacity;\n  }\n\n  /* 조회 연산 */\n  String? get(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        return pair.val;\n      }\n    }\n    // key를 찾지 못하면 null을 반환한다\n    return null;\n  }\n\n  /* 추가 연산 */\n  void put(int key, String val) {\n    // 적재율이 임계값을 초과하면 용량을 확장한다\n    if (loadFactor() &gt; loadThres) {\n      extend();\n    }\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        pair.val = val;\n        return;\n      }\n    }\n    // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n    Pair pair = Pair(key, val);\n    bucket.add(pair);\n    size++;\n  }\n\n  /* 삭제 연산 */\n  void remove(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // 버킷을 순회하며 키-값 쌍을 삭제한다\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        bucket.remove(pair);\n        size--;\n        break;\n      }\n    }\n  }\n\n  /* 해시 테이블 용량 확장 */\n  void extend() {\n    // 원래 해시 테이블을 임시로 저장한다\n    List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n    // 용량을 확장한 새 해시 테이블을 초기화한다\n    capacity *= extendRatio;\n    buckets = List.generate(capacity, (_) =&gt; []);\n    size = 0;\n    // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for (List&lt;Pair&gt; bucket in bucketsTmp) {\n      for (Pair pair in bucket) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* 해시 테이블 출력 */\n  void printHashMap() {\n    for (List&lt;Pair&gt; bucket in buckets) {\n      List&lt;String&gt; res = [];\n      for (Pair pair in bucket) {\n        res.add(\"${pair.key} -&gt; ${pair.val}\");\n      }\n      print(res);\n    }\n  }\n}\n</code></pre> hash_map_chaining.rs<pre><code>/* 체이닝 해시 테이블 */\nstruct HashMapChaining {\n    size: usize,\n    capacity: usize,\n    load_thres: f32,\n    extend_ratio: usize,\n    buckets: Vec&lt;Vec&lt;Pair&gt;&gt;,\n}\n\nimpl HashMapChaining {\n    /* 생성자 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![vec![]; 4],\n        }\n    }\n\n    /* 해시 함수 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % self.capacity\n    }\n\n    /* 적재율 */\n    fn load_factor(&amp;self) -&gt; f32 {\n        self.size as f32 / self.capacity as f32\n    }\n\n    /* 삭제 연산 */\n    fn remove(&amp;mut self, key: i32) -&gt; Option&lt;String&gt; {\n        let index = self.hash_func(key);\n\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        for (i, p) in self.buckets[index].iter_mut().enumerate() {\n            if p.key == key {\n                let pair = self.buckets[index].remove(i);\n                self.size -= 1;\n                return Some(pair.val);\n            }\n        }\n\n        // key를 찾지 못하면 None을 반환한다\n        None\n    }\n\n    /* 해시 테이블 용량 확장 */\n    fn extend(&amp;mut self) {\n        // 원래 해시 테이블을 임시로 저장한다\n        let buckets_tmp = std::mem::take(&amp;mut self.buckets);\n\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![Vec::new(); self.capacity as usize];\n        self.size = 0;\n\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for bucket in buckets_tmp {\n            for pair in bucket {\n                self.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    fn print(&amp;self) {\n        for bucket in &amp;self.buckets {\n            let mut res = Vec::new();\n            for pair in bucket {\n                res.push(format!(\"{} -&gt; {}\", pair.key, pair.val));\n            }\n            println!(\"{:?}\", res);\n        }\n    }\n\n    /* 추가 연산 */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n\n        let index = self.hash_func(key);\n\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for pair in self.buckets[index].iter_mut() {\n            if pair.key == key {\n                pair.val = val;\n                return;\n            }\n        }\n\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        let pair = Pair { key, val };\n        self.buckets[index].push(pair);\n        self.size += 1;\n    }\n\n    /* 조회 연산 */\n    fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        let index = self.hash_func(key);\n\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for pair in self.buckets[index].iter() {\n            if pair.key == key {\n                return Some(&amp;pair.val);\n            }\n        }\n\n        // key를 찾지 못하면 None을 반환한다\n        None\n    }\n}\n</code></pre> hash_map_chaining.c<pre><code>/* 연결 리스트 노드 */\ntypedef struct Node {\n    Pair *pair;\n    struct Node *next;\n} Node;\n\n/* 체이닝 해시 테이블 */\ntypedef struct {\n    int size;         // 키-값 쌍 개수\n    int capacity;     // 해시 테이블 용량\n    double loadThres; // 용량 확장을 수행하는 적재율 임계값\n    int extendRatio;  // 용량 확장 배수\n    Node **buckets;   // 버킷 배열\n} HashMapChaining;\n\n/* 생성자 */\nHashMapChaining *newHashMapChaining() {\n    HashMapChaining *hashMap = (HashMapChaining *)malloc(sizeof(HashMapChaining));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    return hashMap;\n}\n\n/* 소멸자 */\nvoid delHashMapChaining(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        while (cur) {\n            Node *tmp = cur;\n            cur = cur-&gt;next;\n            free(tmp-&gt;pair);\n            free(tmp);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap);\n}\n\n/* 해시 함수 */\nint hashFunc(HashMapChaining *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* 적재율 */\ndouble loadFactor(HashMapChaining *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* 조회 연산 */\nchar *get(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            return cur-&gt;pair-&gt;val;\n        }\n        cur = cur-&gt;next;\n    }\n    return \"\"; // key를 찾지 못하면 빈 문자열을 반환한다\n}\n\n/* 추가 연산 */\nvoid put(HashMapChaining *hashMap, int key, const char *val) {\n    // 적재율이 임계값을 초과하면 용량을 확장한다\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    int index = hashFunc(hashMap, key);\n    // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            strcpy(cur-&gt;pair-&gt;val, val); // 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n            return;\n        }\n        cur = cur-&gt;next;\n    }\n    // 해당 key가 없으면 키-값 쌍을 연결 리스트의 맨 앞에 추가한다\n    Pair *newPair = (Pair *)malloc(sizeof(Pair));\n    newPair-&gt;key = key;\n    strcpy(newPair-&gt;val, val);\n    Node *newNode = (Node *)malloc(sizeof(Node));\n    newNode-&gt;pair = newPair;\n    newNode-&gt;next = hashMap-&gt;buckets[index];\n    hashMap-&gt;buckets[index] = newNode;\n    hashMap-&gt;size++;\n}\n\n/* 해시 테이블 용량 확장 */\nvoid extend(HashMapChaining *hashMap) {\n    // 원래 해시 테이블을 임시로 저장한다\n    int oldCapacity = hashMap-&gt;capacity;\n    Node **oldBuckets = hashMap-&gt;buckets;\n    // 용량을 확장한 새 해시 테이블을 초기화한다\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    hashMap-&gt;size = 0;\n    // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Node *cur = oldBuckets[i];\n        while (cur) {\n            put(hashMap, cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            Node *temp = cur;\n            cur = cur-&gt;next;\n            // 메모리 해제\n            free(temp-&gt;pair);\n            free(temp);\n        }\n    }\n\n    free(oldBuckets);\n}\n\n/* 삭제 연산 */\nvoid removeItem(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    Node *cur = hashMap-&gt;buckets[index];\n    Node *pre = NULL;\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            // 해당 키-값 쌍을 삭제한다\n            if (pre) {\n                pre-&gt;next = cur-&gt;next;\n            } else {\n                hashMap-&gt;buckets[index] = cur-&gt;next;\n            }\n            // 메모리 해제\n            free(cur-&gt;pair);\n            free(cur);\n            hashMap-&gt;size--;\n            return;\n        }\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n}\n\n/* 해시 테이블 출력 */\nvoid print(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        printf(\"[\");\n        while (cur) {\n            printf(\"%d -&gt; %s, \", cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            cur = cur-&gt;next;\n        }\n        printf(\"]\\n\");\n    }\n}\n</code></pre> hash_map_chaining.kt<pre><code>/* 체이닝 해시 테이블 */\nclass HashMapChaining {\n    var size: Int // 키-값 쌍 개수\n    var capacity: Int // 해시 테이블 용량\n    val loadThres: Double // 용량 확장을 수행하는 적재율 임계값\n    val extendRatio: Int // 용량 확장 배수\n    var buckets: MutableList&lt;MutableList&lt;Pair&gt;&gt; // 버킷 배열\n\n    /* 생성자 */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n    }\n\n    /* 해시 함수 */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* 적재율 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* 조회 연산 */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n        for (pair in bucket) {\n            if (pair.key == key) return pair._val\n        }\n        // key를 찾지 못하면 null을 반환한다\n        return null\n    }\n\n    /* 추가 연산 */\n    fun put(key: Int, _val: String) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n        for (pair in bucket) {\n            if (pair.key == key) {\n                pair._val = _val\n                return\n            }\n        }\n        // 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n        val pair = Pair(key, _val)\n        bucket.add(pair)\n        size++\n    }\n\n    /* 삭제 연산 */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        val bucket = buckets[index]\n        // 버킷을 순회하며 키-값 쌍을 삭제한다\n        for (pair in bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair)\n                size--\n                break\n            }\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    fun extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        val bucketsTmp = buckets\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio\n        // mutablelist는 크기가 고정되어 있지 않다\n        buckets = mutableListOf()\n        for (i in 0..&lt;capacity) {\n            buckets.add(mutableListOf())\n        }\n        size = 0\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (bucket in bucketsTmp) {\n            for (pair in bucket) {\n                put(pair.key, pair._val)\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    fun print() {\n        for (bucket in buckets) {\n            val res = mutableListOf&lt;String&gt;()\n            for (pair in bucket) {\n                val k = pair.key\n                val v = pair._val\n                res.add(\"$k -&gt; $v\")\n            }\n            println(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.rb<pre><code>### 체이닝 해시 테이블 ###\nclass HashMapChaining\n  ### 생성자 ###\n  def initialize\n    @size = 0 # 키-값 쌍 개수\n    @capacity = 4 # 해시 테이블 용량\n    @load_thres = 2.0 / 3.0 # 용량 확장을 수행하는 적재율 임계값\n    @extend_ratio = 2 # 용량 확장 배수\n    @buckets = Array.new(@capacity) { [] } # 버킷 배열\n  end\n\n  ### 해시 함수 ###\n  def hash_func(key)\n    key % @capacity\n  end\n\n  ### 적재율 ###\n  def load_factor\n    @size / @capacity\n  end\n\n  ### 조회 연산 ###\n  def get(key)\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # 버킷을 순회하다 key를 찾으면 해당 val을 반환한다\n    for pair in bucket\n      return pair.val if pair.key == key\n    end\n    # key를 찾지 못하면 nil을 반환한다\n    nil\n  end\n\n  ### 추가 연산 ###\n  def put(key, val)\n    # 적재율이 임계값을 초과하면 용량을 확장한다\n    extend if load_factor &gt; @load_thres\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # 버킷을 순회하다 지정한 key를 만나면 해당 val을 갱신하고 반환한다\n    for pair in bucket\n      if pair.key == key\n        pair.val = val\n        return\n      end\n    end\n    # 해당 key가 없으면 키-값 쌍을 맨 뒤에 추가한다\n    pair = Pair.new(key, val)\n    bucket &lt;&lt; pair\n    @size += 1\n  end\n\n  ### 삭제 연산 ###\n  def remove(key)\n    index = hash_func(key)\n    bucket = @buckets[index]\n    # 버킷을 순회하며 키-값 쌍을 삭제한다\n    for pair in bucket\n      if pair.key == key\n        bucket.delete(pair)\n        @size -= 1\n        break\n      end\n    end\n  end\n\n  ### 해시 테이블 용량 확장 ###\n  def extend\n    # 원래 해시 테이블을 임시로 저장한다\n    buckets = @buckets\n    # 용량을 확장한 새 해시 테이블을 초기화한다\n    @capacity *= @extend_ratio\n    @buckets = Array.new(@capacity) { [] }\n    @size = 0\n    # 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for bucket in buckets\n      for pair in bucket\n        put(pair.key, pair.val)\n      end\n    end\n  end\n\n  ### 해시 테이블 출력 ###\n  def print\n    for bucket in @buckets\n      res = []\n      for pair in bucket\n        res &lt;&lt; \"#{pair.key} -&gt; #{pair.val}\"\n      end\n      pp res\n    end\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>연결 리스트가 매우 길어지면 \\(O(n)\\)의 조회 효율은 나빠진다. 이때 연결 리스트를 ‘AVL 트리’나 ‘레드-블랙 트리’로 바꾸면 조회의 시간 복잡도를 \\(O(\\log n)\\)으로 개선할 수 있다.</p>","path":["제 6 장   해시 테이블","6.2   해시 충돌"],"tags":[]},{"location":"chapter_hashing/hash_collision/#622","level":2,"title":"6.2.2   개방 주소법","text":"<p>개방 주소법(open addressing)은 추가 자료 구조를 도입하지 않고 ‘여러 차례 탐사’하여 해시 충돌을 처리한다. 대표적인 탐사 방식에는 선형 탐사, 이차 탐사, 다중 해싱 등이 있다.</p> <p>선형 탐사를 예로 개방 주소법 해시 테이블의 작동 원리를 알아보자.</p>","path":["제 6 장   해시 테이블","6.2   해시 충돌"],"tags":[]},{"location":"chapter_hashing/hash_collision/#1","level":3,"title":"1.   선형 탐사","text":"<p>선형 탐사는 고정 간격의 선형 탐색으로 탐사하며, 일반 해시 테이블과 연산 방식이 다르다.</p> <ul> <li>원소 삽입: 해시 함수로 버킷 인덱스를 구한다. 이미 원소가 있으면 충돌 위치부터 뒤로 일정한 간격(보통 \\(1\\))으로 순회하여 빈 버킷을 찾고 원소를 넣는다.</li> <li>원소 탐색: 해시 충돌이 있으면 같은 간격으로 뒤를 순회하다가 해당 원소를 찾으면 <code>value</code>를 반환한다. 빈 버킷을 만나면 목표 원소가 없다는 뜻이므로 <code>None</code>을 반환한다.</li> </ul> <p>아래 그림(6-6)은 개방 주소법(선형 탐사) 해시 테이블의 키-값 쌍 분포를 보여 준다. 이 해시 함수에서는 마지막 두 자리가 같은 <code>key</code>가 모두 같은 버킷에 매핑된다. 선형 탐사를 통해 이들은 해당 버킷과 그 뒤의 버킷에 차례로 저장된다.</p> <p></p> <p> 그림 6-6   개방 주소법(선형 탐사) 해시 테이블의 키-값 쌍 분포 </p> <p>하지만 선형 탐사에서는 ‘군집 현상’이 쉽게 발생한다. 배열에서 연속으로 채워진 구간이 길수록 그 구간에서 충돌할 가능성도 커진다. 그 결과 군집이 더욱 커지는 악순환이 생기고, 결국 삽입, 삭제, 탐색, 갱신의 효율이 낮아진다.</p> <p>개방 주소법 해시 테이블에서는 원소를 곧바로 삭제하면 안 된다는 점에 주의하자. 삭제하면 배열에 빈 버킷 <code>None</code>이 생긴다. 조회 중 선형 탐사가 이 버킷에 도달하면 바로 반환하므로 그 뒤의 원소에 접근할 수 없게 된다. 아래 그림(6-7)과 같이 프로그램이 그 원소들을 존재하지 않는다고 잘못 판단할 수 있다.</p> <p></p> <p> 그림 6-7   개방 주소법에서 원소 삭제로 발생하는 조회 문제 </p> <p>이 문제는 지연 삭제(lazy deletion)로 해결할 수 있다. 원소를 해시 테이블에서 곧바로 제거하지 않고 상수 <code>TOMBSTONE</code>으로 해당 버킷을 표시한다. <code>None</code>과 <code>TOMBSTONE</code>은 모두 빈 버킷을 나타내며 새 키-값 쌍을 넣을 수 있다. 하지만 <code>TOMBSTONE</code>을 만나면 그 뒤에도 키-값 쌍이 있을 수 있으므로 탐사를 계속해야 한다.</p> <p>그러나 지연 삭제는 해시 테이블의 성능 저하를 가속할 수 있다. 삭제할 때마다 삭제 표시가 생기므로 <code>TOMBSTONE</code>이 늘어날수록 탐색 시간도 길어진다. 목표 원소를 찾기 위해 여러 <code>TOMBSTONE</code>을 건너뛰어야 할 수 있기 때문이다.</p> <p>이를 위해 선형 탐사 중 처음 만난 <code>TOMBSTONE</code>의 인덱스를 기록하고, 찾은 목표 원소와 그 <code>TOMBSTONE</code>의 위치를 교환하는 방법을 생각할 수 있다. 조회하거나 추가할 때마다 원소가 이상적인 위치(탐사 시작점)에 더 가까운 버킷으로 옮겨지므로 조회 효율이 좋아진다.</p> <p>다음 코드는 지연 삭제를 포함한 개방 주소법(선형 탐사) 해시 테이블을 구현한다. 공간을 더 충분히 활용하기 위해 해시 테이블을 ‘원형 배열’로 보고, 끝을 넘으면 처음으로 돌아와 순회를 계속한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map_open_addressing.py<pre><code>class HashMapOpenAddressing:\n    \"\"\"개방 주소법 해시 테이블\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        self.size = 0  # 키-값 쌍 개수\n        self.capacity = 4  # 해시 테이블 용량\n        self.load_thres = 2.0 / 3.0  # 용량 확장을 수행하는 적재율 임계값\n        self.extend_ratio = 2  # 용량 확장 배수\n        self.buckets: list[Pair | None] = [None] * self.capacity  # 버킷 배열\n        self.TOMBSTONE = Pair(-1, \"-1\")  # 삭제 표시\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"해시 함수\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"적재율\"\"\"\n        return self.size / self.capacity\n\n    def find_bucket(self, key: int) -&gt; int:\n        \"\"\"key에 해당하는 버킷 인덱스 탐색\"\"\"\n        index = self.hash_func(key)\n        first_tombstone = -1\n        # 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while self.buckets[index] is not None:\n            # key를 만나면 해당 버킷 인덱스를 반환한다\n            if self.buckets[index].key == key:\n                # 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if first_tombstone != -1:\n                    self.buckets[first_tombstone] = self.buckets[index]\n                    self.buckets[index] = self.TOMBSTONE\n                    return first_tombstone  # 옮긴 후의 버킷 인덱스를 반환한다\n                return index  # 버킷 인덱스를 반환한다\n            # 처음 만난 삭제 표시를 기록한다\n            if first_tombstone == -1 and self.buckets[index] is self.TOMBSTONE:\n                first_tombstone = index\n            # 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % self.capacity\n        # key가 없으면 추가할 위치의 인덱스를 반환한다\n        return index if first_tombstone == -1 else first_tombstone\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"조회 연산\"\"\"\n        # key에 해당하는 버킷 인덱스 탐색\n        index = self.find_bucket(key)\n        # 키-값 쌍을 찾으면 해당 val을 반환한다\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            return self.buckets[index].val\n        # 키-값 쌍이 없으면 None을 반환한다\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"추가 연산\"\"\"\n        # 적재율이 임계값을 초과하면 용량을 확장한다\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        # key에 해당하는 버킷 인덱스 탐색\n        index = self.find_bucket(key)\n        # 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index].val = val\n            return\n        # 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        self.buckets[index] = Pair(key, val)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"삭제 연산\"\"\"\n        # key에 해당하는 버킷 인덱스 탐색\n        index = self.find_bucket(key)\n        # 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index] = self.TOMBSTONE\n            self.size -= 1\n\n    def extend(self):\n        \"\"\"해시 테이블 용량 확장\"\"\"\n        # 원래 해시 테이블을 임시로 저장한다\n        buckets_tmp = self.buckets\n        # 용량을 확장한 새 해시 테이블을 초기화한다\n        self.capacity *= self.extend_ratio\n        self.buckets = [None] * self.capacity\n        self.size = 0\n        # 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for pair in buckets_tmp:\n            if pair not in [None, self.TOMBSTONE]:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"해시 테이블 출력\"\"\"\n        for pair in self.buckets:\n            if pair is None:\n                print(\"None\")\n            elif pair is self.TOMBSTONE:\n                print(\"TOMBSTONE\")\n            else:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> hash_map_open_addressing.cpp<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n  private:\n    int size;                             // 키-값 쌍 개수\n    int capacity = 4;                     // 해시 테이블 용량\n    const double loadThres = 2.0 / 3.0;     // 용량 확장을 수행하는 적재율 임계값\n    const int extendRatio = 2;            // 용량 확장 배수\n    vector&lt;Pair *&gt; buckets;               // 버킷 배열\n    Pair *TOMBSTONE = new Pair(-1, \"-1\"); // 삭제 표시\n\n  public:\n    /* 생성자 */\n    HashMapOpenAddressing() : size(0), buckets(capacity, nullptr) {\n    }\n\n    /* 소멸자 */\n    ~HashMapOpenAddressing() {\n        for (Pair *pair : buckets) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                delete pair;\n            }\n        }\n        delete TOMBSTONE;\n    }\n\n    /* 해시 함수 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* 적재율 */\n    double loadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while (buckets[index] != nullptr) {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if (buckets[index]-&gt;key == key) {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index; // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % capacity;\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* 조회 연산 */\n    string get(int key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = findBucket(key);\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]-&gt;val;\n        }\n        // 키-값 쌍이 없으면 빈 문자열을 반환한다\n        return \"\";\n    }\n\n    /* 추가 연산 */\n    void put(int key, string val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = findBucket(key);\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]-&gt;val = val;\n            return;\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* 삭제 연산 */\n    void remove(int key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = findBucket(key);\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            delete buckets[index];\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    void extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        vector&lt;Pair *&gt; bucketsTmp = buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio;\n        buckets = vector&lt;Pair *&gt;(capacity, nullptr);\n        size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (Pair *pair : bucketsTmp) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                put(pair-&gt;key, pair-&gt;val);\n                delete pair;\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    void print() {\n        for (Pair *pair : buckets) {\n            if (pair == nullptr) {\n                cout &lt;&lt; \"nullptr\" &lt;&lt; endl;\n            } else if (pair == TOMBSTONE) {\n                cout &lt;&lt; \"TOMBSTONE\" &lt;&lt; endl;\n            } else {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; endl;\n            }\n        }\n    }\n};\n</code></pre> hash_map_open_addressing.java<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n    private int size; // 키-값 쌍 개수\n    private int capacity = 4; // 해시 테이블 용량\n    private final double loadThres = 2.0 / 3.0; // 용량 확장을 수행하는 적재율 임계값\n    private final int extendRatio = 2; // 용량 확장 배수\n    private Pair[] buckets; // 버킷 배열\n    private final Pair TOMBSTONE = new Pair(-1, \"-1\"); // 삭제 표시\n\n    /* 생성자 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* 해시 함수 */\n    private int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* 적재율 */\n    private double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    private int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while (buckets[index] != null) {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if (buckets[index].key == key) {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index; // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % capacity;\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* 조회 연산 */\n    public String get(int key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = findBucket(key);\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // 키-값 쌍이 없으면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    public void put(int key, String val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = findBucket(key);\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* 삭제 연산 */\n    public void remove(int key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = findBucket(key);\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    private void extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        Pair[] bucketsTmp = buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (Pair pair : bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    public void print() {\n        for (Pair pair : buckets) {\n            if (pair == null) {\n                System.out.println(\"null\");\n            } else if (pair == TOMBSTONE) {\n                System.out.println(\"TOMBSTONE\");\n            } else {\n                System.out.println(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.cs<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n    int size; // 키-값 쌍 개수\n    int capacity = 4; // 해시 테이블 용량\n    double loadThres = 2.0 / 3.0; // 용량 확장을 수행하는 적재율 임계값\n    int extendRatio = 2; // 용량 확장 배수\n    Pair[] buckets; // 버킷 배열\n    Pair TOMBSTONE = new(-1, \"-1\"); // 삭제 표시\n\n    /* 생성자 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* 해시 함수 */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* 적재율 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    int FindBucket(int key) {\n        int index = HashFunc(key);\n        int firstTombstone = -1;\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while (buckets[index] != null) {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if (buckets[index].key == key) {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index; // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % capacity;\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* 조회 연산 */\n    public string? Get(int key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = FindBucket(key);\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // 키-값 쌍이 없으면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    public void Put(int key, string val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = FindBucket(key);\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* 삭제 연산 */\n    public void Remove(int key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        int index = FindBucket(key);\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    void Extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        Pair[] bucketsTmp = buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        foreach (Pair pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    public void Print() {\n        foreach (Pair pair in buckets) {\n            if (pair == null) {\n                Console.WriteLine(\"null\");\n            } else if (pair == TOMBSTONE) {\n                Console.WriteLine(\"TOMBSTONE\");\n            } else {\n                Console.WriteLine(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.go<pre><code>/* 개방 주소법 해시 테이블 */\ntype hashMapOpenAddressing struct {\n    size        int     // 키-값 쌍 개수\n    capacity    int     // 해시 테이블 용량\n    loadThres   float64 // 용량 확장을 수행하는 적재율 임계값\n    extendRatio int     // 용량 확장 배수\n    buckets     []*pair // 버킷 배열\n    TOMBSTONE   *pair   // 삭제 표시\n}\n\n/* 생성자 */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\n    return &amp;hashMapOpenAddressing{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     make([]*pair, 4),\n        TOMBSTONE:   &amp;pair{-1, \"-1\"},\n    }\n}\n\n/* 해시 함수 */\nfunc (h *hashMapOpenAddressing) hashFunc(key int) int {\n    return key % h.capacity // 키를 바탕으로 해시값을 계산한다\n}\n\n/* 적재율 */\nfunc (h *hashMapOpenAddressing) loadFactor() float64 {\n    return float64(h.size) / float64(h.capacity) // 현재 적재율을 계산한다\n}\n\n/* key에 해당하는 버킷 인덱스 탐색 */\nfunc (h *hashMapOpenAddressing) findBucket(key int) int {\n    index := h.hashFunc(key) // 초기 인덱스를 구한다\n    firstTombstone := -1     // 처음 만난 TOMBSTONE의 위치를 기록한다\n    for h.buckets[index] != nil {\n        if h.buckets[index].key == key {\n            if firstTombstone != -1 {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                h.buckets[firstTombstone] = h.buckets[index]\n                h.buckets[index] = h.TOMBSTONE\n                return firstTombstone // 옮긴 후의 버킷 인덱스를 반환한다\n            }\n            return index // 찾은 인덱스를 반환한다\n        }\n        if firstTombstone == -1 &amp;&amp; h.buckets[index] == h.TOMBSTONE {\n            firstTombstone = index // 처음 만난 삭제 표시의 위치를 기록한다\n        }\n        index = (index + 1) % h.capacity // 선형 탐사를 수행하며, 끝을 넘어가면 처음으로 돌아온다\n    }\n    // key가 없으면 추가할 위치의 인덱스를 반환한다\n    if firstTombstone != -1 {\n        return firstTombstone\n    }\n    return index\n}\n\n/* 조회 연산 */\nfunc (h *hashMapOpenAddressing) get(key int) string {\n    index := h.findBucket(key) // key에 해당하는 버킷 인덱스 탐색\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        return h.buckets[index].val // 키-값 쌍을 찾으면 해당 val을 반환한다\n    }\n    return \"\" // 키-값 쌍이 없으면 \"\"를 반환한다\n}\n\n/* 추가 연산 */\nfunc (h *hashMapOpenAddressing) put(key int, val string) {\n    if h.loadFactor() &gt; h.loadThres {\n        h.extend() // 적재율이 임계값을 초과하면 용량을 확장한다\n    }\n    index := h.findBucket(key) // key에 해당하는 버킷 인덱스 탐색\n    if h.buckets[index] == nil || h.buckets[index] == h.TOMBSTONE {\n        h.buckets[index] = &amp;pair{key, val} // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        h.size++\n    } else {\n        h.buckets[index].val = val // 키-값 쌍을 찾으면 val을 덮어쓴다\n    }\n}\n\n/* 삭제 연산 */\nfunc (h *hashMapOpenAddressing) remove(key int) {\n    index := h.findBucket(key) // key에 해당하는 버킷 인덱스 탐색\n    if h.buckets[index] != nil &amp;&amp; h.buckets[index] != h.TOMBSTONE {\n        h.buckets[index] = h.TOMBSTONE // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        h.size--\n    }\n}\n\n/* 해시 테이블 용량 확장 */\nfunc (h *hashMapOpenAddressing) extend() {\n    oldBuckets := h.buckets               // 원래 해시 테이블을 임시로 저장한다\n    h.capacity *= h.extendRatio           // 용량을 갱신한다\n    h.buckets = make([]*pair, h.capacity) // 용량을 확장한 새 해시 테이블을 초기화한다\n    h.size = 0                            // 크기를 초기화한다\n    // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for _, pair := range oldBuckets {\n        if pair != nil &amp;&amp; pair != h.TOMBSTONE {\n            h.put(pair.key, pair.val)\n        }\n    }\n}\n\n/* 해시 테이블 출력 */\nfunc (h *hashMapOpenAddressing) print() {\n    for _, pair := range h.buckets {\n        if pair == nil {\n            fmt.Println(\"nil\")\n        } else if pair == h.TOMBSTONE {\n            fmt.Println(\"TOMBSTONE\")\n        } else {\n            fmt.Printf(\"%d -&gt; %s\\n\", pair.key, pair.val)\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.swift<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n    var size: Int // 키-값 쌍 개수\n    var capacity: Int // 해시 테이블 용량\n    var loadThres: Double // 용량 확장을 수행하는 적재율 임계값\n    var extendRatio: Int // 용량 확장 배수\n    var buckets: [Pair?] // 버킷 배열\n    var TOMBSTONE: Pair // 삭제 표시\n\n    /* 생성자 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: nil, count: capacity)\n        TOMBSTONE = Pair(key: -1, val: \"-1\")\n    }\n\n    /* 해시 함수 */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* 적재율 */\n    func loadFactor() -&gt; Double {\n        Double(size) / Double(capacity)\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    func findBucket(key: Int) -&gt; Int {\n        var index = hashFunc(key: key)\n        var firstTombstone = -1\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while buckets[index] != nil {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if buckets[index]!.key == key {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if firstTombstone != -1 {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE {\n                firstTombstone = index\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % capacity\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        return firstTombstone == -1 ? index : firstTombstone\n    }\n\n    /* 조회 연산 */\n    func get(key: Int) -&gt; String? {\n        // key에 해당하는 버킷 인덱스 탐색\n        let index = findBucket(key: key)\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            return buckets[index]!.val\n        }\n        // 키-값 쌍이 없으면 null을 반환한다\n        return nil\n    }\n\n    /* 추가 연산 */\n    func put(key: Int, val: String) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        let index = findBucket(key: key)\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index]!.val = val\n            return\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        buckets[index] = Pair(key: key, val: val)\n        size += 1\n    }\n\n    /* 삭제 연산 */\n    func remove(key: Int) {\n        // key에 해당하는 버킷 인덱스 탐색\n        let index = findBucket(key: key)\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index] = TOMBSTONE\n            size -= 1\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    func extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        let bucketsTmp = buckets\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio\n        buckets = Array(repeating: nil, count: capacity)\n        size = 0\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for pair in bucketsTmp {\n            if let pair, pair != TOMBSTONE {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    func print() {\n        for pair in buckets {\n            if pair == nil {\n                Swift.print(\"null\")\n            } else if pair == TOMBSTONE {\n                Swift.print(\"TOMBSTONE\")\n            } else {\n                Swift.print(\"\\(pair!.key) -&gt; \\(pair!.val)\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.js<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n    #size; // 키-값 쌍 개수\n    #capacity; // 해시 테이블 용량\n    #loadThres; // 용량 확장을 수행하는 적재율 임계값\n    #extendRatio; // 용량 확장 배수\n    #buckets; // 버킷 배열\n    #TOMBSTONE; // 삭제 표시\n\n    /* 생성자 */\n    constructor() {\n        this.#size = 0; // 키-값 쌍 개수\n        this.#capacity = 4; // 해시 테이블 용량\n        this.#loadThres = 2.0 / 3.0; // 용량 확장을 수행하는 적재율 임계값\n        this.#extendRatio = 2; // 용량 확장 배수\n        this.#buckets = Array(this.#capacity).fill(null); // 버킷 배열\n        this.#TOMBSTONE = new Pair(-1, '-1'); // 삭제 표시\n    }\n\n    /* 해시 함수 */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* 적재율 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    #findBucket(key) {\n        let index = this.#hashFunc(key);\n        let firstTombstone = -1;\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while (this.#buckets[index] !== null) {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if (this.#buckets[index].key === key) {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if (firstTombstone !== -1) {\n                    this.#buckets[firstTombstone] = this.#buckets[index];\n                    this.#buckets[index] = this.#TOMBSTONE;\n                    return firstTombstone; // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index; // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.#buckets[index] === this.#TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % this.#capacity;\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* 조회 연산 */\n    get(key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        const index = this.#findBucket(key);\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            return this.#buckets[index].val;\n        }\n        // 키-값 쌍이 없으면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    put(key, val) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        const index = this.#findBucket(key);\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index].val = val;\n            return;\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        this.#buckets[index] = new Pair(key, val);\n        this.#size++;\n    }\n\n    /* 삭제 연산 */\n    remove(key) {\n        // key에 해당하는 버킷 인덱스 탐색\n        const index = this.#findBucket(key);\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index] = this.#TOMBSTONE;\n            this.#size--;\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    #extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        const bucketsTmp = this.#buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = Array(this.#capacity).fill(null);\n        this.#size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.#TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    print() {\n        for (const pair of this.#buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.#TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.ts<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n    private size: number; // 키-값 쌍 개수\n    private capacity: number; // 해시 테이블 용량\n    private loadThres: number; // 용량 확장을 수행하는 적재율 임계값\n    private extendRatio: number; // 용량 확장 배수\n    private buckets: Array&lt;Pair | null&gt;; // 버킷 배열\n    private TOMBSTONE: Pair; // 삭제 표시\n\n    /* 생성자 */\n    constructor() {\n        this.size = 0; // 키-값 쌍 개수\n        this.capacity = 4; // 해시 테이블 용량\n        this.loadThres = 2.0 / 3.0; // 용량 확장을 수행하는 적재율 임계값\n        this.extendRatio = 2; // 용량 확장 배수\n        this.buckets = Array(this.capacity).fill(null); // 버킷 배열\n        this.TOMBSTONE = new Pair(-1, '-1'); // 삭제 표시\n    }\n\n    /* 해시 함수 */\n    private hashFunc(key: number): number {\n        return key % this.capacity;\n    }\n\n    /* 적재율 */\n    private loadFactor(): number {\n        return this.size / this.capacity;\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    private findBucket(key: number): number {\n        let index = this.hashFunc(key);\n        let firstTombstone = -1;\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while (this.buckets[index] !== null) {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if (this.buckets[index]!.key === key) {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if (firstTombstone !== -1) {\n                    this.buckets[firstTombstone] = this.buckets[index];\n                    this.buckets[index] = this.TOMBSTONE;\n                    return firstTombstone; // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index; // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.buckets[index] === this.TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % this.capacity;\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* 조회 연산 */\n    get(key: number): string | null {\n        // key에 해당하는 버킷 인덱스 탐색\n        const index = this.findBucket(key);\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            return this.buckets[index]!.val;\n        }\n        // 키-값 쌍이 없으면 null을 반환한다\n        return null;\n    }\n\n    /* 추가 연산 */\n    put(key: number, val: string): void {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (this.loadFactor() &gt; this.loadThres) {\n            this.extend();\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        const index = this.findBucket(key);\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index]!.val = val;\n            return;\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        this.buckets[index] = new Pair(key, val);\n        this.size++;\n    }\n\n    /* 삭제 연산 */\n    remove(key: number): void {\n        // key에 해당하는 버킷 인덱스 탐색\n        const index = this.findBucket(key);\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index] = this.TOMBSTONE;\n            this.size--;\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    private extend(): void {\n        // 원래 해시 테이블을 임시로 저장한다\n        const bucketsTmp = this.buckets;\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        this.capacity *= this.extendRatio;\n        this.buckets = Array(this.capacity).fill(null);\n        this.size = 0;\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    print(): void {\n        for (const pair of this.buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.dart<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n  late int _size; // 키-값 쌍 개수\n  int _capacity = 4; // 해시 테이블 용량\n  double _loadThres = 2.0 / 3.0; // 용량 확장을 수행하는 적재율 임계값\n  int _extendRatio = 2; // 용량 확장 배수\n  late List&lt;Pair?&gt; _buckets; // 버킷 배열\n  Pair _TOMBSTONE = Pair(-1, \"-1\"); // 삭제 표시\n\n  /* 생성자 */\n  HashMapOpenAddressing() {\n    _size = 0;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n  }\n\n  /* 해시 함수 */\n  int hashFunc(int key) {\n    return key % _capacity;\n  }\n\n  /* 적재율 */\n  double loadFactor() {\n    return _size / _capacity;\n  }\n\n  /* key에 해당하는 버킷 인덱스 탐색 */\n  int findBucket(int key) {\n    int index = hashFunc(key);\n    int firstTombstone = -1;\n    // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n    while (_buckets[index] != null) {\n      // key를 만나면 해당 버킷 인덱스를 반환한다\n      if (_buckets[index]!.key == key) {\n        // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n        if (firstTombstone != -1) {\n          _buckets[firstTombstone] = _buckets[index];\n          _buckets[index] = _TOMBSTONE;\n          return firstTombstone; // 옮긴 후의 버킷 인덱스를 반환한다\n        }\n        return index; // 버킷 인덱스를 반환한다\n      }\n      // 처음 만난 삭제 표시를 기록한다\n      if (firstTombstone == -1 &amp;&amp; _buckets[index] == _TOMBSTONE) {\n        firstTombstone = index;\n      }\n      // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n      index = (index + 1) % _capacity;\n    }\n    // key가 없으면 추가할 위치의 인덱스를 반환한다\n    return firstTombstone == -1 ? index : firstTombstone;\n  }\n\n  /* 조회 연산 */\n  String? get(int key) {\n    // key에 해당하는 버킷 인덱스 탐색\n    int index = findBucket(key);\n    // 키-값 쌍을 찾으면 해당 val을 반환한다\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      return _buckets[index]!.val;\n    }\n    // 키-값 쌍이 없으면 null을 반환한다\n    return null;\n  }\n\n  /* 추가 연산 */\n  void put(int key, String val) {\n    // 적재율이 임계값을 초과하면 용량을 확장한다\n    if (loadFactor() &gt; _loadThres) {\n      extend();\n    }\n    // key에 해당하는 버킷 인덱스 탐색\n    int index = findBucket(key);\n    // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index]!.val = val;\n      return;\n    }\n    // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n    _buckets[index] = new Pair(key, val);\n    _size++;\n  }\n\n  /* 삭제 연산 */\n  void remove(int key) {\n    // key에 해당하는 버킷 인덱스 탐색\n    int index = findBucket(key);\n    // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index] = _TOMBSTONE;\n      _size--;\n    }\n  }\n\n  /* 해시 테이블 용량 확장 */\n  void extend() {\n    // 원래 해시 테이블을 임시로 저장한다\n    List&lt;Pair?&gt; bucketsTmp = _buckets;\n    // 용량을 확장한 새 해시 테이블을 초기화한다\n    _capacity *= _extendRatio;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n    _size = 0;\n    // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for (Pair? pair in bucketsTmp) {\n      if (pair != null &amp;&amp; pair != _TOMBSTONE) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* 해시 테이블 출력 */\n  void printHashMap() {\n    for (Pair? pair in _buckets) {\n      if (pair == null) {\n        print(\"null\");\n      } else if (pair == _TOMBSTONE) {\n        print(\"TOMBSTONE\");\n      } else {\n        print(\"${pair.key} -&gt; ${pair.val}\");\n      }\n    }\n  }\n}\n</code></pre> hash_map_open_addressing.rs<pre><code>/* 개방 주소법 해시 테이블 */\nstruct HashMapOpenAddressing {\n    size: usize,                // 키-값 쌍 개수\n    capacity: usize,            // 해시 테이블 용량\n    load_thres: f64,            // 용량 확장을 수행하는 적재율 임계값\n    extend_ratio: usize,        // 용량 확장 배수\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;, // 버킷 배열\n    TOMBSTONE: Option&lt;Pair&gt;,    // 삭제 표시\n}\n\nimpl HashMapOpenAddressing {\n    /* 생성자 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![None; 4],\n            TOMBSTONE: Some(Pair {\n                key: -1,\n                val: \"-1\".to_string(),\n            }),\n        }\n    }\n\n    /* 해시 함수 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        (key % self.capacity as i32) as usize\n    }\n\n    /* 적재율 */\n    fn load_factor(&amp;self) -&gt; f64 {\n        self.size as f64 / self.capacity as f64\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    fn find_bucket(&amp;mut self, key: i32) -&gt; usize {\n        let mut index = self.hash_func(key);\n        let mut first_tombstone = -1;\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while self.buckets[index].is_some() {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if self.buckets[index].as_ref().unwrap().key == key {\n                // 앞서 삭제 표시를 만났다면 키-값 쌍을 해당 인덱스로 옮긴다\n                if first_tombstone != -1 {\n                    self.buckets[first_tombstone as usize] = self.buckets[index].take();\n                    self.buckets[index] = self.TOMBSTONE.clone();\n                    return first_tombstone as usize; // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index; // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if first_tombstone == -1 &amp;&amp; self.buckets[index] == self.TOMBSTONE {\n                first_tombstone = index as i32;\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % self.capacity;\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        if first_tombstone == -1 {\n            index\n        } else {\n            first_tombstone as usize\n        }\n    }\n\n    /* 조회 연산 */\n    fn get(&amp;mut self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        // key에 해당하는 버킷 인덱스 탐색\n        let index = self.find_bucket(key);\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            return self.buckets[index].as_ref().map(|pair| &amp;pair.val as &amp;str);\n        }\n        // 키-값 쌍이 없으면 null을 반환한다\n        None\n    }\n\n    /* 추가 연산 */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        let index = self.find_bucket(key);\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index].as_mut().unwrap().val = val;\n            return;\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        self.buckets[index] = Some(Pair { key, val });\n        self.size += 1;\n    }\n\n    /* 삭제 연산 */\n    fn remove(&amp;mut self, key: i32) {\n        // key에 해당하는 버킷 인덱스 탐색\n        let index = self.find_bucket(key);\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index] = self.TOMBSTONE.clone();\n            self.size -= 1;\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    fn extend(&amp;mut self) {\n        // 원래 해시 테이블을 임시로 저장한다\n        let buckets_tmp = self.buckets.clone();\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![None; self.capacity];\n        self.size = 0;\n\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for pair in buckets_tmp {\n            if pair.is_none() || pair == self.TOMBSTONE {\n                continue;\n            }\n            let pair = pair.unwrap();\n\n            self.put(pair.key, pair.val);\n        }\n    }\n    /* 해시 테이블 출력 */\n    fn print(&amp;self) {\n        for pair in &amp;self.buckets {\n            if pair.is_none() {\n                println!(\"null\");\n            } else if pair == &amp;self.TOMBSTONE {\n                println!(\"TOMBSTONE\");\n            } else {\n                let pair = pair.as_ref().unwrap();\n                println!(\"{} -&gt; {}\", pair.key, pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.c<pre><code>/* 개방 주소법 해시 테이블 */\ntypedef struct {\n    int size;         // 키-값 쌍 개수\n    int capacity;     // 해시 테이블 용량\n    double loadThres; // 용량 확장을 수행하는 적재율 임계값\n    int extendRatio;  // 용량 확장 배수\n    Pair **buckets;   // 버킷 배열\n    Pair *TOMBSTONE;  // 삭제 표시\n} HashMapOpenAddressing;\n\n/* 생성자 */\nHashMapOpenAddressing *newHashMapOpenAddressing() {\n    HashMapOpenAddressing *hashMap = (HashMapOpenAddressing *)malloc(sizeof(HashMapOpenAddressing));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Pair **)calloc(hashMap-&gt;capacity, sizeof(Pair *));\n    hashMap-&gt;TOMBSTONE = (Pair *)malloc(sizeof(Pair));\n    hashMap-&gt;TOMBSTONE-&gt;key = -1;\n    hashMap-&gt;TOMBSTONE-&gt;val = \"-1\";\n\n    return hashMap;\n}\n\n/* 소멸자 */\nvoid delHashMapOpenAddressing(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap-&gt;TOMBSTONE);\n    free(hashMap);\n}\n\n/* 해시 함수 */\nint hashFunc(HashMapOpenAddressing *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* 적재율 */\ndouble loadFactor(HashMapOpenAddressing *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* key에 해당하는 버킷 인덱스 탐색 */\nint findBucket(HashMapOpenAddressing *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    int firstTombstone = -1;\n    // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n    while (hashMap-&gt;buckets[index] != NULL) {\n        // key를 만나면 해당 버킷 인덱스를 반환한다\n        if (hashMap-&gt;buckets[index]-&gt;key == key) {\n            // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n            if (firstTombstone != -1) {\n                hashMap-&gt;buckets[firstTombstone] = hashMap-&gt;buckets[index];\n                hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n                return firstTombstone; // 옮긴 후의 버킷 인덱스를 반환한다\n            }\n            return index; // 버킷 인덱스를 반환한다\n        }\n        // 처음 만난 삭제 표시를 기록한다\n        if (firstTombstone == -1 &amp;&amp; hashMap-&gt;buckets[index] == hashMap-&gt;TOMBSTONE) {\n            firstTombstone = index;\n        }\n        // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n        index = (index + 1) % hashMap-&gt;capacity;\n    }\n    // key가 없으면 추가할 위치의 인덱스를 반환한다\n    return firstTombstone == -1 ? index : firstTombstone;\n}\n\n/* 조회 연산 */\nchar *get(HashMapOpenAddressing *hashMap, int key) {\n    // key에 해당하는 버킷 인덱스 탐색\n    int index = findBucket(hashMap, key);\n    // 키-값 쌍을 찾으면 해당 val을 반환한다\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        return hashMap-&gt;buckets[index]-&gt;val;\n    }\n    // 키-값 쌍이 없으면 빈 문자열을 반환한다\n    return \"\";\n}\n\n/* 추가 연산 */\nvoid put(HashMapOpenAddressing *hashMap, int key, char *val) {\n    // 적재율이 임계값을 초과하면 용량을 확장한다\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    // key에 해당하는 버킷 인덱스 탐색\n    int index = findBucket(hashMap, key);\n    // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        free(hashMap-&gt;buckets[index]-&gt;val);\n        hashMap-&gt;buckets[index]-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n        strcpy(hashMap-&gt;buckets[index]-&gt;val, val);\n        hashMap-&gt;buckets[index]-&gt;val[strlen(val)] = '\\0';\n        return;\n    }\n    // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n    Pair *pair = (Pair *)malloc(sizeof(Pair));\n    pair-&gt;key = key;\n    pair-&gt;val = (char *)malloc(sizeof(strlen(val) + 1));\n    strcpy(pair-&gt;val, val);\n    pair-&gt;val[strlen(val)] = '\\0';\n\n    hashMap-&gt;buckets[index] = pair;\n    hashMap-&gt;size++;\n}\n\n/* 삭제 연산 */\nvoid removeItem(HashMapOpenAddressing *hashMap, int key) {\n    // key에 해당하는 버킷 인덱스 탐색\n    int index = findBucket(hashMap, key);\n    // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        Pair *pair = hashMap-&gt;buckets[index];\n        free(pair-&gt;val);\n        free(pair);\n        hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n        hashMap-&gt;size--;\n    }\n}\n\n/* 해시 테이블 용량 확장 */\nvoid extend(HashMapOpenAddressing *hashMap) {\n    // 원래 해시 테이블을 임시로 저장한다\n    Pair **bucketsTmp = hashMap-&gt;buckets;\n    int oldCapacity = hashMap-&gt;capacity;\n    // 용량을 확장한 새 해시 테이블을 초기화한다\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Pair **)calloc(hashMap-&gt;capacity, sizeof(Pair *));\n    hashMap-&gt;size = 0;\n    // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Pair *pair = bucketsTmp[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            put(hashMap, pair-&gt;key, pair-&gt;val);\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(bucketsTmp);\n}\n\n/* 해시 테이블 출력 */\nvoid print(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair == NULL) {\n            printf(\"NULL\\n\");\n        } else if (pair == hashMap-&gt;TOMBSTONE) {\n            printf(\"TOMBSTONE\\n\");\n        } else {\n            printf(\"%d -&gt; %s\\n\", pair-&gt;key, pair-&gt;val);\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.kt<pre><code>/* 개방 주소법 해시 테이블 */\nclass HashMapOpenAddressing {\n    private var size: Int               // 키-값 쌍 개수\n    private var capacity: Int           // 해시 테이블 용량\n    private val loadThres: Double       // 용량 확장을 수행하는 적재율 임계값\n    private val extendRatio: Int        // 용량 확장 배수\n    private var buckets: Array&lt;Pair?&gt;   // 버킷 배열\n    private val TOMBSTONE: Pair         // 삭제 표시\n\n    /* 생성자 */\n    init {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = arrayOfNulls(capacity)\n        TOMBSTONE = Pair(-1, \"-1\")\n    }\n\n    /* 해시 함수 */\n    fun hashFunc(key: Int): Int {\n        return key % capacity\n    }\n\n    /* 적재율 */\n    fun loadFactor(): Double {\n        return (size / capacity).toDouble()\n    }\n\n    /* key에 해당하는 버킷 인덱스 탐색 */\n    fun findBucket(key: Int): Int {\n        var index = hashFunc(key)\n        var firstTombstone = -1\n        // 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n        while (buckets[index] != null) {\n            // key를 만나면 해당 버킷 인덱스를 반환한다\n            if (buckets[index]?.key == key) {\n                // 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // 옮긴 후의 버킷 인덱스를 반환한다\n                }\n                return index // 버킷 인덱스를 반환한다\n            }\n            // 처음 만난 삭제 표시를 기록한다\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index\n            }\n            // 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n            index = (index + 1) % capacity\n        }\n        // key가 없으면 추가할 위치의 인덱스를 반환한다\n        return if (firstTombstone == -1) index else firstTombstone\n    }\n\n    /* 조회 연산 */\n    fun get(key: Int): String? {\n        // key에 해당하는 버킷 인덱스 탐색\n        val index = findBucket(key)\n        // 키-값 쌍을 찾으면 해당 val을 반환한다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]?._val\n        }\n        // 키-값 쌍이 없으면 null을 반환한다\n        return null\n    }\n\n    /* 추가 연산 */\n    fun put(key: Int, _val: String) {\n        // 적재율이 임계값을 초과하면 용량을 확장한다\n        if (loadFactor() &gt; loadThres) {\n            extend()\n        }\n        // key에 해당하는 버킷 인덱스 탐색\n        val index = findBucket(key)\n        // 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]!!._val = _val\n            return\n        }\n        // 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n        buckets[index] = Pair(key, _val)\n        size++\n    }\n\n    /* 삭제 연산 */\n    fun remove(key: Int) {\n        // key에 해당하는 버킷 인덱스 탐색\n        val index = findBucket(key)\n        // 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE\n            size--\n        }\n    }\n\n    /* 해시 테이블 용량 확장 */\n    fun extend() {\n        // 원래 해시 테이블을 임시로 저장한다\n        val bucketsTmp = buckets\n        // 용량을 확장한 새 해시 테이블을 초기화한다\n        capacity *= extendRatio\n        buckets = arrayOfNulls(capacity)\n        size = 0\n        // 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n        for (pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair._val)\n            }\n        }\n    }\n\n    /* 해시 테이블 출력 */\n    fun print() {\n        for (pair in buckets) {\n            if (pair == null) {\n                println(\"null\")\n            } else if (pair == TOMBSTONE) {\n                println(\"TOMESTOME\")\n            } else {\n                println(\"${pair.key} -&gt; ${pair._val}\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.rb<pre><code>### 개방 주소법 해시 테이블 ###\nclass HashMapOpenAddressing\n  TOMBSTONE = Pair.new(-1, '-1') # 삭제 표시\n\n  ### 생성자 ###\n  def initialize\n    @size = 0 # 키-값 쌍 개수\n    @capacity = 4 # 해시 테이블 용량\n    @load_thres = 2.0 / 3.0 # 용량 확장을 수행하는 적재율 임계값\n    @extend_ratio = 2 # 용량 확장 배수\n    @buckets = Array.new(@capacity) # 버킷 배열\n  end\n\n  ### 해시 함수 ###\n  def hash_func(key)\n    key % @capacity\n  end\n\n  ### 적재율 ###\n  def load_factor\n    @size / @capacity\n  end\n\n  ### key에 해당하는 버킷 인덱스 탐색 ###\n  def find_bucket(key)\n    index = hash_func(key)\n    first_tombstone = -1\n    # 선형 탐사로 탐색하고, 빈 버킷을 만나면 반복문을 종료한다\n    while !@buckets[index].nil?\n      # key를 만나면 해당 버킷 인덱스를 반환한다\n      if @buckets[index].key == key\n        # 이전에 삭제 표시를 만났으면 키-값 쌍을 그 인덱스로 옮긴다\n        if first_tombstone != -1\n          @buckets[first_tombstone] = @buckets[index]\n          @buckets[index] = TOMBSTONE\n          return first_tombstone # 옮긴 후의 버킷 인덱스를 반환한다\n        end\n        return index # 버킷 인덱스를 반환한다\n      end\n      # 처음 만난 삭제 표시를 기록한다\n      first_tombstone = index if first_tombstone == -1 &amp;&amp; @buckets[index] == TOMBSTONE\n      # 버킷 인덱스를 계산하고, 끝을 넘어가면 처음으로 돌아온다\n      index = (index + 1) % @capacity\n    end\n    # key가 없으면 추가할 위치의 인덱스를 반환한다\n    first_tombstone == -1 ? index : first_tombstone\n  end\n\n  ### 조회 연산 ###\n  def get(key)\n    # key에 해당하는 버킷 인덱스 탐색\n    index = find_bucket(key)\n    # 키-값 쌍을 찾으면 해당 val을 반환한다\n    return @buckets[index].val unless [nil, TOMBSTONE].include?(@buckets[index])\n    # 키-값 쌍이 없으면 nil을 반환한다\n    nil\n  end\n\n  ### 추가 연산 ###\n  def put(key, val)\n    # 적재율이 임계값을 초과하면 용량을 확장한다\n    extend if load_factor &gt; @load_thres\n    # key에 해당하는 버킷 인덱스 탐색\n    index = find_bucket(key)\n    # 키-값 쌍을 찾으면 val을 덮어쓰고 반환한다\n    unless [nil, TOMBSTONE].include?(@buckets[index])\n      @buckets[index].val = val\n      return\n    end\n    # 키-값 쌍이 없으면 해당 키-값 쌍을 추가한다\n    @buckets[index] = Pair.new(key, val)\n    @size += 1\n  end\n\n  ### 삭제 연산 ###\n  def remove(key)\n    # key에 해당하는 버킷 인덱스 탐색\n    index = find_bucket(key)\n    # 키-값 쌍을 찾으면 삭제 표시로 덮어쓴다\n    unless [nil, TOMBSTONE].include?(@buckets[index])\n      @buckets[index] = TOMBSTONE\n      @size -= 1\n    end\n  end\n\n  ### 해시 테이블 용량 확장 ###\n  def extend\n    # 원래 해시 테이블을 임시로 저장한다\n    buckets_tmp = @buckets\n    # 용량을 확장한 새 해시 테이블을 초기화한다\n    @capacity *= @extend_ratio\n    @buckets = Array.new(@capacity)\n    @size = 0\n    # 원래 해시 테이블의 키-값 쌍을 새 해시 테이블로 옮긴다\n    for pair in buckets_tmp\n      put(pair.key, pair.val) unless [nil, TOMBSTONE].include?(pair)\n    end\n  end\n\n  ### 해시 테이블 출력 ###\n  def print\n    for pair in @buckets\n      if pair.nil?\n        puts \"Nil\"\n      elsif pair == TOMBSTONE\n        puts \"TOMBSTONE\"\n      else\n        puts \"#{pair.key} -&gt; #{pair.val}\"\n      end\n    end\n  end\nend\n</code></pre>","path":["제 6 장   해시 테이블","6.2   해시 충돌"],"tags":[]},{"location":"chapter_hashing/hash_collision/#2","level":3,"title":"2.   이차 탐사","text":"<p>이차 탐사는 선형 탐사처럼 개방 주소법의 대표적인 전략이다. 충돌이 발생하면 고정 간격으로 이동하는 대신 ‘탐사 횟수의 제곱’만큼, 즉 \\(1, 4, 9, \\dots\\)칸을 이동한다.</p> <p>이차 탐사의 주요 장점은 다음과 같다.</p> <ul> <li>탐사 횟수의 제곱만큼 떨어진 위치로 이동하여 선형 탐사의 군집 현상을 완화하려 한다.</li> <li>더 먼 거리로 건너뛰며 빈 위치를 찾으므로 데이터를 더 고르게 분포시키는 데 도움이 된다.</li> </ul> <p>하지만 이차 탐사도 완벽하지는 않다.</p> <ul> <li>일부 위치가 다른 위치보다 더 쉽게 채워지는 군집 현상이 여전히 존재한다.</li> <li>제곱의 증가 때문에 해시 테이블 전체를 탐사하지 못할 수 있다. 즉, 빈 버킷이 있어도 그곳에 도달하지 못할 수 있다.</li> </ul>","path":["제 6 장   해시 테이블","6.2   해시 충돌"],"tags":[]},{"location":"chapter_hashing/hash_collision/#3","level":3,"title":"3.   다중 해싱","text":"<p>이름 그대로 여러 해시 함수 \\(f_1(x)\\), \\(f_2(x)\\), \\(f_3(x)\\), \\(\\dots\\)를 사용하여 탐사하는 방법이다.</p> <ul> <li>원소 삽입: 해시 함수 \\(f_1(x)\\)에서 충돌하면 \\(f_2(x)\\)를 시도한다. 이런 식으로 빈 위치를 찾을 때까지 계속한 뒤 원소를 넣는다.</li> <li>원소 탐색: 같은 해시 함수 순서로 탐색하여 목표 원소를 찾으면 반환한다. 빈 위치를 만나거나 모든 함수를 시도했다면 원소가 없다는 뜻이므로 <code>None</code>을 반환한다.</li> </ul> <p>선형 탐사보다 군집 현상이 덜 발생하지만, 여러 해시 함수로 인해 추가 계산이 필요하다.</p> <p>참고</p> <p>개방 주소법의 선형 탐사, 이차 탐사, 다중 해싱 모두 ‘원소를 곧바로 삭제할 수 없다’는 문제가 있다는 점에 주의하자.</p>","path":["제 6 장   해시 테이블","6.2   해시 충돌"],"tags":[]},{"location":"chapter_hashing/hash_collision/#623","level":2,"title":"6.2.3   프로그래밍 언어의 선택","text":"<p>프로그래밍 언어마다 해시 테이블 구현 전략이 다르다. 몇 가지 예를 살펴보자.</p> <ul> <li>Python은 개방 주소법을 사용한다. 딕셔너리 <code>dict</code>는 의사 난수를 이용해 탐사한다.</li> <li>Java는 체이닝을 사용한다. JDK 1.8부터는 <code>HashMap</code> 내부 배열 길이가 64 이상이고 연결 리스트 길이가 8 이상이면, 탐색 성능을 높이기 위해 연결 리스트를 레드-블랙 트리로 바꾼다.</li> <li>Go 1.24부터 내장 <code>map</code>은 기본적으로 Swiss Table 기반의 개방 주소법을 사용한다. Go 1.23 이하에서는 체이닝을 사용하며, 버킷 하나에 키-값 쌍을 최대 8개 저장하고 용량을 넘으면 오버플로 버킷을 연결한다.</li> </ul>","path":["제 6 장   해시 테이블","6.2   해시 충돌"],"tags":[]},{"location":"chapter_hashing/hash_map/","level":1,"title":"6.1   해시 테이블","text":"<p>해시 테이블(hash table)은 해시 표라고도 하며, 키 <code>key</code>와 값 <code>value</code> 사이의 대응 관계를 만들어 원소를 효율적으로 조회한다. 구체적으로 키 <code>key</code>를 입력하면 해당 값 <code>value</code>를 \\(O(1)\\) 시간에 얻을 수 있다.</p> <p>아래 그림(6-1)과 같이 학생 \\(n\\)명에게 각각 ‘이름’과 ‘학번’이라는 두 데이터가 있다고 하자. ‘학번을 입력하면 해당 이름을 반환하는’ 조회 기능을 구현하려면 그림의 해시 테이블을 사용할 수 있다.</p> <p></p> <p> 그림 6-1   해시 테이블의 추상적 표현 </p> <p>해시 테이블 외에 배열과 연결 리스트로도 조회 기능을 구현할 수 있다. 효율 비교는 아래 표(6-1)와 같다.</p> <ul> <li>원소 추가: 배열(연결 리스트)의 끝에 원소를 추가하면 되므로 \\(O(1)\\) 시간이 걸린다.</li> <li>원소 조회: 배열(연결 리스트)이 정렬되어 있지 않으므로 모든 원소를 순회해야 하며 \\(O(n)\\) 시간이 걸린다.</li> <li>원소 삭제: 먼저 원소를 찾은 뒤 배열(연결 리스트)에서 삭제해야 하므로 \\(O(n)\\) 시간이 걸린다.</li> </ul> <p> 표 6-1   원소 조회 효율 비교 </p> 배열 연결 리스트 해시 테이블 원소 탐색 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) 원소 추가 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) 원소 삭제 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) <p>이처럼 해시 테이블에서는 삽입, 삭제, 탐색, 갱신의 시간 복잡도가 모두 \\(O(1)\\)로 매우 효율적이다.</p>","path":["제 6 장   해시 테이블","6.1   해시 테이블"],"tags":[]},{"location":"chapter_hashing/hash_map/#611","level":2,"title":"6.1.1   해시 테이블의 기본 연산","text":"<p>대표적인 연산에는 초기화, 조회, 키-값 쌍 추가와 삭제 등이 있다. 예제 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map.py<pre><code># 해시 테이블 초기화\nhmap: dict = {}\n\n# 추가 연산\n# 해시 테이블에 키-값 쌍 (key, value) 추가\nhmap[12836] = \"小哈\"\nhmap[15937] = \"小啰\"\nhmap[16750] = \"小算\"\nhmap[13276] = \"小法\"\nhmap[10583] = \"小鸭\"\n\n# 조회 연산\n# 해시 테이블에 키 key를 입력하여 값 value를 얻음\nname: str = hmap[15937]\n\n# 삭제 연산\n# 해시 테이블에서 키-값 쌍 (key, value) 삭제\nhmap.pop(10583)\n</code></pre> hash_map.cpp<pre><code>/* 해시 테이블 초기화 */\nunordered_map&lt;int, string&gt; map;\n\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap[12836] = \"小哈\";\nmap[15937] = \"小啰\";\nmap[16750] = \"小算\";\nmap[13276] = \"小法\";\nmap[10583] = \"小鸭\";\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nstring name = map[15937];\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.erase(10583);\n</code></pre> hash_map.java<pre><code>/* 해시 테이블 초기화 */\nMap&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap.put(12836, \"小哈\");\nmap.put(15937, \"小啰\");\nmap.put(16750, \"小算\");\nmap.put(13276, \"小法\");\nmap.put(10583, \"小鸭\");\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nString name = map.get(15937);\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.remove(10583);\n</code></pre> hash_map.cs<pre><code>/* 해시 테이블 초기화 */\nDictionary&lt;int, string&gt; map = new() {\n    /* 추가 연산 */\n    // 해시 테이블에 키-값 쌍 (key, value) 추가\n    { 12836, \"小哈\" },\n    { 15937, \"小啰\" },\n    { 16750, \"小算\" },\n    { 13276, \"小法\" },\n    { 10583, \"小鸭\" }\n};\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nstring name = map[15937];\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.Remove(10583);\n</code></pre> hash_map_test.go<pre><code>/* 해시 테이블 초기화 */\nhmap := make(map[int]string)\n\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nhmap[12836] = \"小哈\"\nhmap[15937] = \"小啰\"\nhmap[16750] = \"小算\"\nhmap[13276] = \"小法\"\nhmap[10583] = \"小鸭\"\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nname := hmap[15937]\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\ndelete(hmap, 10583)\n</code></pre> hash_map.swift<pre><code>/* 해시 테이블 초기화 */\nvar map: [Int: String] = [:]\n\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap[12836] = \"小哈\"\nmap[15937] = \"小啰\"\nmap[16750] = \"小算\"\nmap[13276] = \"小法\"\nmap[10583] = \"小鸭\"\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nlet name = map[15937]!\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.removeValue(forKey: 10583)\n</code></pre> hash_map.js<pre><code>/* 해시 테이블 초기화 */\nconst map = new Map();\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap.set(12836, '小哈');\nmap.set(15937, '小啰');\nmap.set(16750, '小算');\nmap.set(13276, '小法');\nmap.set(10583, '小鸭');\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nlet name = map.get(15937);\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.delete(10583);\n</code></pre> hash_map.ts<pre><code>/* 해시 테이블 초기화 */\nconst map = new Map&lt;number, string&gt;();\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap.set(12836, '小哈');\nmap.set(15937, '小啰');\nmap.set(16750, '小算');\nmap.set(13276, '小法');\nmap.set(10583, '小鸭');\nconsole.info('\\n추가한 뒤의 해시 테이블\\nKey -&gt; Value');\nconsole.info(map);\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nlet name = map.get(15937);\nconsole.info('\\n학번 15937을 입력하여 찾은 이름 ' + name);\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.delete(10583);\nconsole.info('\\n10583을 삭제한 뒤의 해시 테이블\\nKey -&gt; Value');\nconsole.info(map);\n</code></pre> hash_map.dart<pre><code>/* 해시 테이블 초기화 */\nMap&lt;int, String&gt; map = {};\n\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap[12836] = \"小哈\";\nmap[15937] = \"小啰\";\nmap[16750] = \"小算\";\nmap[13276] = \"小法\";\nmap[10583] = \"小鸭\";\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nString name = map[15937];\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.remove(10583);\n</code></pre> hash_map.rs<pre><code>use std::collections::HashMap;\n\n/* 해시 테이블 초기화 */\nlet mut map: HashMap&lt;i32, String&gt; = HashMap::new();\n\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap.insert(12836, \"小哈\".to_string());\nmap.insert(15937, \"小啰\".to_string());\nmap.insert(16750, \"小算\".to_string());\nmap.insert(13279, \"小法\".to_string());\nmap.insert(10583, \"小鸭\".to_string());\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nlet _name: Option&lt;&amp;String&gt; = map.get(&amp;15937);\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nlet _removed_value: Option&lt;String&gt; = map.remove(&amp;10583);\n</code></pre> hash_map.c<pre><code>// C는 내장 해시 테이블을 제공하지 않는다\n</code></pre> hash_map.kt<pre><code>/* 해시 테이블 초기화 */\nval map = HashMap&lt;Int,String&gt;()\n\n/* 추가 연산 */\n// 해시 테이블에 키-값 쌍 (key, value) 추가\nmap[12836] = \"小哈\"\nmap[15937] = \"小啰\"\nmap[16750] = \"小算\"\nmap[13276] = \"小法\"\nmap[10583] = \"小鸭\"\n\n/* 조회 연산 */\n// 해시 테이블에 키 key를 입력하여 값 value를 얻음\nval name = map[15937]\n\n/* 삭제 연산 */\n// 해시 테이블에서 키-값 쌍 (key, value) 삭제\nmap.remove(10583)\n</code></pre> hash_map.rb<pre><code># 해시 테이블 초기화\nhmap = {}\n\n# 추가 연산\n# 해시 테이블에 키-값 쌍 (key, value) 추가\nhmap[12836] = \"小哈\"\nhmap[15937] = \"小啰\"\nhmap[16750] = \"小算\"\nhmap[13276] = \"小法\"\nhmap[10583] = \"小鸭\"\n\n# 조회 연산\n# 해시 테이블에 키 key를 입력하여 값 value를 얻음\nname = hmap[15937]\n\n# 삭제 연산\n# 해시 테이블에서 키-값 쌍 (key, value) 삭제\nhmap.delete(10583)\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>해시 테이블을 순회하는 대표적인 방식은 키-값 쌍 순회, 키 순회, 값 순회의 세 가지다. 예제 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby hash_map.py<pre><code># 해시 테이블 순회\n# 키-값 쌍 key-&gt;value 순회\nfor key, value in hmap.items():\n    print(key, \"-&gt;\", value)\n# 키 key만 순회\nfor key in hmap.keys():\n    print(key)\n# 값 value만 순회\nfor value in hmap.values():\n    print(value)\n</code></pre> hash_map.cpp<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 key-&gt;value 순회\nfor (auto kv: map) {\n    cout &lt;&lt; kv.first &lt;&lt; \" -&gt; \" &lt;&lt; kv.second &lt;&lt; endl;\n}\n// 이터레이터로 key-&gt;value 순회\nfor (auto iter = map.begin(); iter != map.end(); iter++) {\n    cout &lt;&lt; iter-&gt;first &lt;&lt; \"-&gt;\" &lt;&lt; iter-&gt;second &lt;&lt; endl;\n}\n</code></pre> hash_map.java<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 key-&gt;value 순회\nfor (Map.Entry &lt;Integer, String&gt; kv: map.entrySet()) {\n    System.out.println(kv.getKey() + \" -&gt; \" + kv.getValue());\n}\n// 키 key만 순회\nfor (int key: map.keySet()) {\n    System.out.println(key);\n}\n// 값 value만 순회\nfor (String val: map.values()) {\n    System.out.println(val);\n}\n</code></pre> hash_map.cs<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 Key-&gt;Value 순회\nforeach (var kv in map) {\n    Console.WriteLine(kv.Key + \" -&gt; \" + kv.Value);\n}\n// 키 key만 순회\nforeach (int key in map.Keys) {\n    Console.WriteLine(key);\n}\n// 값 value만 순회\nforeach (string val in map.Values) {\n    Console.WriteLine(val);\n}\n</code></pre> hash_map_test.go<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 key-&gt;value 순회\nfor key, value := range hmap {\n    fmt.Println(key, \"-&gt;\", value)\n}\n// 키 key만 순회\nfor key := range hmap {\n    fmt.Println(key)\n}\n// 값 value만 순회\nfor _, value := range hmap {\n    fmt.Println(value)\n}\n</code></pre> hash_map.swift<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 Key-&gt;Value 순회\nfor (key, value) in map {\n    print(\"\\(key) -&gt; \\(value)\")\n}\n// 키 Key만 순회\nfor key in map.keys {\n    print(key)\n}\n// 값 Value만 순회\nfor value in map.values {\n    print(value)\n}\n</code></pre> hash_map.js<pre><code>/* 해시 테이블 순회 */\nconsole.info('\\n키-값 쌍 Key-&gt;Value 순회');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n키 Key만 순회');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\n값 Value만 순회');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.ts<pre><code>/* 해시 테이블 순회 */\nconsole.info('\\n키-값 쌍 Key-&gt;Value 순회');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n키 Key만 순회');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\n값 Value만 순회');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.dart<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 Key-&gt;Value 순회\nmap.forEach((key, value) {\n  print('$key -&gt; $value');\n});\n\n// 키 Key만 순회\nmap.keys.forEach((key) {\n  print(key);\n});\n\n// 값 Value만 순회\nmap.values.forEach((value) {\n  print(value);\n});\n</code></pre> hash_map.rs<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 Key-&gt;Value 순회\nfor (key, value) in &amp;map {\n    println!(\"{key} -&gt; {value}\");\n}\n\n// 키 Key만 순회\nfor key in map.keys() {\n    println!(\"{key}\");\n}\n\n// 값 Value만 순회\nfor value in map.values() {\n    println!(\"{value}\");\n}\n</code></pre> hash_map.c<pre><code>// C는 내장 해시 테이블을 제공하지 않는다\n</code></pre> hash_map.kt<pre><code>/* 해시 테이블 순회 */\n// 키-값 쌍 key-&gt;value 순회\nfor ((key, value) in map) {\n    println(\"$key -&gt; $value\")\n}\n// 키 key만 순회\nfor (key in map.keys) {\n    println(key)\n}\n// 값 value만 순회\nfor (_val in map.values) {\n    println(_val)\n}\n</code></pre> hash_map.rb<pre><code># 해시 테이블 순회\n# 키-값 쌍 key-&gt;value 순회\nhmap.entries.each { |key, value| puts \"#{key} -&gt; #{value}\" }\n\n# 키 key만 순회\nhmap.keys.each { |key| puts key }\n\n# 값 value만 순회\nhmap.values.each { |val| puts val }\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 6 장   해시 테이블","6.1   해시 테이블"],"tags":[]},{"location":"chapter_hashing/hash_map/#612","level":2,"title":"6.1.2   간단한 해시 테이블 구현","text":"<p>먼저 가장 간단한 경우로 배열 하나만 사용하여 해시 테이블을 구현해 보자. 해시 테이블에서는 배열의 각 자리를 버킷(bucket)이라고 하며, 버킷마다 키-값 쌍 하나를 저장할 수 있다. 따라서 조회 연산은 <code>key</code>에 해당하는 버킷을 찾고 그 안에서 <code>value</code>를 얻는 것이다.</p> <p>그렇다면 <code>key</code>로 해당 버킷을 어떻게 찾을까? 이를 위해 해시 함수(hash function)를 사용한다. 해시 함수는 큰 입력 공간을 작은 출력 공간에 매핑한다. 해시 테이블에서 입력 공간은 모든 <code>key</code>이고, 출력 공간은 모든 버킷(배열 인덱스)이다. 즉, <code>key</code>를 입력하면 해시 함수를 통해 그 키-값 쌍이 배열에 저장되는 위치를 얻을 수 있다.</p> <p><code>key</code>를 입력했을 때 해시 함수의 계산 과정은 다음 두 단계로 나뉜다.</p> <ol> <li>어떤 해시 알고리즘 <code>hash()</code>로 해시값을 계산한다.</li> <li>해시값을 버킷 수(배열 길이) <code>capacity</code>로 나눈 나머지를 구하여 <code>key</code>에 해당하는 버킷(배열 인덱스) <code>index</code>를 얻는다.</li> </ol> <pre><code>index = hash(key) % capacity\n</code></pre> <p>그런 다음 <code>index</code>로 해시 테이블의 해당 버킷에 접근하여 <code>value</code>를 얻는다.</p> <p>배열 길이가 <code>capacity = 100</code>이고 해시 알고리즘이 <code>hash(key) = key</code>이면, 해시 함수는 <code>key % 100</code>이다. 아래 그림(6-2)은 학번을 <code>key</code>, 이름을 <code>value</code>로 삼아 해시 함수의 작동 원리를 보여 준다.</p> <p></p> <p> 그림 6-2   해시 함수의 작동 원리 </p> <p>다음 코드는 간단한 해시 테이블을 구현한다. <code>key</code>와 <code>value</code>를 <code>Pair</code> 클래스로 묶어 키-값 쌍을 표현한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_hash_map.py<pre><code>class Pair:\n    \"\"\"키-값 쌍\"\"\"\n\n    def __init__(self, key: int, val: str):\n        self.key = key\n        self.val = val\n\nclass ArrayHashMap:\n    \"\"\"배열로 구현한 해시 테이블\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        # 버킷 100개를 포함하는 배열을 초기화한다\n        self.buckets: list[Pair | None] = [None] * 100\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"해시 함수\"\"\"\n        index = key % 100\n        return index\n\n    def get(self, key: int) -&gt; str | None:\n        \"\"\"조회 연산\"\"\"\n        index: int = self.hash_func(key)\n        pair: Pair = self.buckets[index]\n        if pair is None:\n            return None\n        return pair.val\n\n    def put(self, key: int, val: str):\n        \"\"\"추가 및 갱신 연산\"\"\"\n        pair = Pair(key, val)\n        index: int = self.hash_func(key)\n        self.buckets[index] = pair\n\n    def remove(self, key: int):\n        \"\"\"삭제 연산\"\"\"\n        index: int = self.hash_func(key)\n        # None으로 설정하여 삭제를 나타낸다\n        self.buckets[index] = None\n\n    def entry_set(self) -&gt; list[Pair]:\n        \"\"\"모든 키-값 쌍 반환\"\"\"\n        result: list[Pair] = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair)\n        return result\n\n    def key_set(self) -&gt; list[int]:\n        \"\"\"모든 키 반환\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.key)\n        return result\n\n    def value_set(self) -&gt; list[str]:\n        \"\"\"모든 값 반환\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.val)\n        return result\n\n    def print(self):\n        \"\"\"해시 테이블 출력\"\"\"\n        for pair in self.buckets:\n            if pair is not None:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> array_hash_map.cpp<pre><code>/* 키-값 쌍 */\nstruct Pair {\n  public:\n    int key;\n    string val;\n    Pair(int key, string val) {\n        this-&gt;key = key;\n        this-&gt;val = val;\n    }\n};\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n  private:\n    vector&lt;Pair *&gt; buckets;\n\n  public:\n    ArrayHashMap() {\n        // 버킷 100개를 포함하는 배열을 초기화한다\n        buckets = vector&lt;Pair *&gt;(100);\n    }\n\n    ~ArrayHashMap() {\n        // 메모리 해제\n        for (const auto &amp;bucket : buckets) {\n            delete bucket;\n        }\n        buckets.clear();\n    }\n\n    /* 해시 함수 */\n    int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* 조회 연산 */\n    string get(int key) {\n        int index = hashFunc(key);\n        Pair *pair = buckets[index];\n        if (pair == nullptr)\n            return \"\";\n        return pair-&gt;val;\n    }\n\n    /* 추가 연산 */\n    void put(int key, string val) {\n        Pair *pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* 삭제 연산 */\n    void remove(int key) {\n        int index = hashFunc(key);\n        // 메모리를 해제하고 nullptr로 설정한다\n        delete buckets[index];\n        buckets[index] = nullptr;\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    vector&lt;Pair *&gt; pairSet() {\n        vector&lt;Pair *&gt; pairSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                pairSet.push_back(pair);\n            }\n        }\n        return pairSet;\n    }\n\n    /* 모든 키 반환 */\n    vector&lt;int&gt; keySet() {\n        vector&lt;int&gt; keySet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                keySet.push_back(pair-&gt;key);\n            }\n        }\n        return keySet;\n    }\n\n    /* 모든 값 반환 */\n    vector&lt;string&gt; valueSet() {\n        vector&lt;string&gt; valueSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                valueSet.push_back(pair-&gt;val);\n            }\n        }\n        return valueSet;\n    }\n\n    /* 해시 테이블 출력 */\n    void print() {\n        for (Pair *kv : pairSet()) {\n            cout &lt;&lt; kv-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; kv-&gt;val &lt;&lt; endl;\n        }\n    }\n};\n</code></pre> array_hash_map.java<pre><code>/* 키-값 쌍 */\nclass Pair {\n    public int key;\n    public String val;\n\n    public Pair(int key, String val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n    private List&lt;Pair&gt; buckets;\n\n    public ArrayHashMap() {\n        // 버킷 100개를 포함하는 배열을 초기화한다\n        buckets = new ArrayList&lt;&gt;();\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.add(null);\n        }\n    }\n\n    /* 해시 함수 */\n    private int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* 조회 연산 */\n    public String get(int key) {\n        int index = hashFunc(key);\n        Pair pair = buckets.get(index);\n        if (pair == null)\n            return null;\n        return pair.val;\n    }\n\n    /* 추가 연산 */\n    public void put(int key, String val) {\n        Pair pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets.set(index, pair);\n    }\n\n    /* 삭제 연산 */\n    public void remove(int key) {\n        int index = hashFunc(key);\n        // null로 설정하여 삭제를 나타낸다\n        buckets.set(index, null);\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    public List&lt;Pair&gt; pairSet() {\n        List&lt;Pair&gt; pairSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                pairSet.add(pair);\n        }\n        return pairSet;\n    }\n\n    /* 모든 키 반환 */\n    public List&lt;Integer&gt; keySet() {\n        List&lt;Integer&gt; keySet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                keySet.add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* 모든 값 반환 */\n    public List&lt;String&gt; valueSet() {\n        List&lt;String&gt; valueSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                valueSet.add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* 해시 테이블 출력 */\n    public void print() {\n        for (Pair kv : pairSet()) {\n            System.out.println(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.cs<pre><code>/* 키-값 쌍 int-&gt;string */\nclass Pair(int key, string val) {\n    public int key = key;\n    public string val = val;\n}\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n    List&lt;Pair?&gt; buckets;\n    public ArrayHashMap() {\n        // 버킷 100개를 포함하는 배열을 초기화한다\n        buckets = [];\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.Add(null);\n        }\n    }\n\n    /* 해시 함수 */\n    int HashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* 조회 연산 */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        Pair? pair = buckets[index];\n        if (pair == null) return null;\n        return pair.val;\n    }\n\n    /* 추가 연산 */\n    public void Put(int key, string val) {\n        Pair pair = new(key, val);\n        int index = HashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* 삭제 연산 */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // null로 설정하여 삭제를 나타낸다\n        buckets[index] = null;\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    public List&lt;Pair&gt; PairSet() {\n        List&lt;Pair&gt; pairSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                pairSet.Add(pair);\n        }\n        return pairSet;\n    }\n\n    /* 모든 키 반환 */\n    public List&lt;int&gt; KeySet() {\n        List&lt;int&gt; keySet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                keySet.Add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* 모든 값 반환 */\n    public List&lt;string&gt; ValueSet() {\n        List&lt;string&gt; valueSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                valueSet.Add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* 해시 테이블 출력 */\n    public void Print() {\n        foreach (Pair kv in PairSet()) {\n            Console.WriteLine(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.go<pre><code>/* 키-값 쌍 */\ntype pair struct {\n    key int\n    val string\n}\n\n/* 배열로 구현한 해시 테이블 */\ntype arrayHashMap struct {\n    buckets []*pair\n}\n\n/* 해시 테이블 초기화 */\nfunc newArrayHashMap() *arrayHashMap {\n    // 버킷 100개를 포함하는 배열을 초기화한다\n    buckets := make([]*pair, 100)\n    return &amp;arrayHashMap{buckets: buckets}\n}\n\n/* 해시 함수 */\nfunc (a *arrayHashMap) hashFunc(key int) int {\n    index := key % 100\n    return index\n}\n\n/* 조회 연산 */\nfunc (a *arrayHashMap) get(key int) string {\n    index := a.hashFunc(key)\n    pair := a.buckets[index]\n    if pair == nil {\n        return \"찾을 수 없음\"\n    }\n    return pair.val\n}\n\n/* 추가 연산 */\nfunc (a *arrayHashMap) put(key int, val string) {\n    pair := &amp;pair{key: key, val: val}\n    index := a.hashFunc(key)\n    a.buckets[index] = pair\n}\n\n/* 삭제 연산 */\nfunc (a *arrayHashMap) remove(key int) {\n    index := a.hashFunc(key)\n    // nil로 설정하여 삭제를 나타낸다\n    a.buckets[index] = nil\n}\n\n/* 모든 키-값 쌍 반환 */\nfunc (a *arrayHashMap) pairSet() []*pair {\n    var pairs []*pair\n    for _, pair := range a.buckets {\n        if pair != nil {\n            pairs = append(pairs, pair)\n        }\n    }\n    return pairs\n}\n\n/* 모든 키 반환 */\nfunc (a *arrayHashMap) keySet() []int {\n    var keys []int\n    for _, pair := range a.buckets {\n        if pair != nil {\n            keys = append(keys, pair.key)\n        }\n    }\n    return keys\n}\n\n/* 모든 값 반환 */\nfunc (a *arrayHashMap) valueSet() []string {\n    var values []string\n    for _, pair := range a.buckets {\n        if pair != nil {\n            values = append(values, pair.val)\n        }\n    }\n    return values\n}\n\n/* 해시 테이블 출력 */\nfunc (a *arrayHashMap) print() {\n    for _, pair := range a.buckets {\n        if pair != nil {\n            fmt.Println(pair.key, \"-&gt;\", pair.val)\n        }\n    }\n}\n</code></pre> array_hash_map.swift<pre><code>/* 키-값 쌍 */\nclass Pair: Equatable {\n    public var key: Int\n    public var val: String\n\n    public init(key: Int, val: String) {\n        self.key = key\n        self.val = val\n    }\n\n    public static func == (lhs: Pair, rhs: Pair) -&gt; Bool {\n        lhs.key == rhs.key &amp;&amp; lhs.val == rhs.val\n    }\n}\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n    private var buckets: [Pair?]\n\n    init() {\n        // 버킷 100개를 포함하는 배열을 초기화한다\n        buckets = Array(repeating: nil, count: 100)\n    }\n\n    /* 해시 함수 */\n    private func hashFunc(key: Int) -&gt; Int {\n        let index = key % 100\n        return index\n    }\n\n    /* 조회 연산 */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let pair = buckets[index]\n        return pair?.val\n    }\n\n    /* 추가 연산 */\n    func put(key: Int, val: String) {\n        let pair = Pair(key: key, val: val)\n        let index = hashFunc(key: key)\n        buckets[index] = pair\n    }\n\n    /* 삭제 연산 */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        // nil로 설정하여 삭제를 나타낸다\n        buckets[index] = nil\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    func pairSet() -&gt; [Pair] {\n        buckets.compactMap { $0 }\n    }\n\n    /* 모든 키 반환 */\n    func keySet() -&gt; [Int] {\n        buckets.compactMap { $0?.key }\n    }\n\n    /* 모든 값 반환 */\n    func valueSet() -&gt; [String] {\n        buckets.compactMap { $0?.val }\n    }\n\n    /* 해시 테이블 출력 */\n    func print() {\n        for pair in pairSet() {\n            Swift.print(\"\\(pair.key) -&gt; \\(pair.val)\")\n        }\n    }\n}\n</code></pre> array_hash_map.js<pre><code>/* 키-값 쌍 Number -&gt; String */\nclass Pair {\n    constructor(key, val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n    #buckets;\n    constructor() {\n        // 버킷 100개를 포함하는 배열을 초기화한다\n        this.#buckets = new Array(100).fill(null);\n    }\n\n    /* 해시 함수 */\n    #hashFunc(key) {\n        return key % 100;\n    }\n\n    /* 조회 연산 */\n    get(key) {\n        let index = this.#hashFunc(key);\n        let pair = this.#buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* 추가 연산 */\n    set(key, val) {\n        let index = this.#hashFunc(key);\n        this.#buckets[index] = new Pair(key, val);\n    }\n\n    /* 삭제 연산 */\n    delete(key) {\n        let index = this.#hashFunc(key);\n        // null로 설정하여 삭제를 나타낸다\n        this.#buckets[index] = null;\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    entries() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* 모든 키 반환 */\n    keys() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* 모든 값 반환 */\n    values() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* 해시 테이블 출력 */\n    print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.ts<pre><code>/* 키-값 쌍 Number -&gt; String */\nclass Pair {\n    public key: number;\n    public val: string;\n\n    constructor(key: number, val: string) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n    private readonly buckets: (Pair | null)[];\n\n    constructor() {\n        // 버킷 100개를 포함하는 배열을 초기화한다\n        this.buckets = new Array(100).fill(null);\n    }\n\n    /* 해시 함수 */\n    private hashFunc(key: number): number {\n        return key % 100;\n    }\n\n    /* 조회 연산 */\n    public get(key: number): string | null {\n        let index = this.hashFunc(key);\n        let pair = this.buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* 추가 연산 */\n    public set(key: number, val: string) {\n        let index = this.hashFunc(key);\n        this.buckets[index] = new Pair(key, val);\n    }\n\n    /* 삭제 연산 */\n    public delete(key: number) {\n        let index = this.hashFunc(key);\n        // null로 설정하여 삭제를 나타낸다\n        this.buckets[index] = null;\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    public entries(): (Pair | null)[] {\n        let arr: (Pair | null)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* 모든 키 반환 */\n    public keys(): (number | undefined)[] {\n        let arr: (number | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* 모든 값 반환 */\n    public values(): (string | undefined)[] {\n        let arr: (string | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* 해시 테이블 출력 */\n    public print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.dart<pre><code>/* 키-값 쌍 */\nclass Pair {\n  int key;\n  String val;\n  Pair(this.key, this.val);\n}\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n  late List&lt;Pair?&gt; _buckets;\n\n  ArrayHashMap() {\n    // 버킷 100개를 포함하는 배열을 초기화한다\n    _buckets = List.filled(100, null);\n  }\n\n  /* 해시 함수 */\n  int _hashFunc(int key) {\n    final int index = key % 100;\n    return index;\n  }\n\n  /* 조회 연산 */\n  String? get(int key) {\n    final int index = _hashFunc(key);\n    final Pair? pair = _buckets[index];\n    if (pair == null) {\n      return null;\n    }\n    return pair.val;\n  }\n\n  /* 추가 연산 */\n  void put(int key, String val) {\n    final Pair pair = Pair(key, val);\n    final int index = _hashFunc(key);\n    _buckets[index] = pair;\n  }\n\n  /* 삭제 연산 */\n  void remove(int key) {\n    final int index = _hashFunc(key);\n    _buckets[index] = null;\n  }\n\n  /* 모든 키-값 쌍 반환 */\n  List&lt;Pair&gt; pairSet() {\n    List&lt;Pair&gt; pairSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        pairSet.add(pair);\n      }\n    }\n    return pairSet;\n  }\n\n  /* 모든 키 반환 */\n  List&lt;int&gt; keySet() {\n    List&lt;int&gt; keySet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        keySet.add(pair.key);\n      }\n    }\n    return keySet;\n  }\n\n  /* 모든 값 반환 */\n  List&lt;String&gt; values() {\n    List&lt;String&gt; valueSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        valueSet.add(pair.val);\n      }\n    }\n    return valueSet;\n  }\n\n  /* 해시 테이블 출력 */\n  void printHashMap() {\n    for (final Pair kv in pairSet()) {\n      print(\"${kv.key} -&gt; ${kv.val}\");\n    }\n  }\n}\n</code></pre> array_hash_map.rs<pre><code>/* 키-값 쌍 */\n#[derive(Debug, Clone, PartialEq)]\npub struct Pair {\n    pub key: i32,\n    pub val: String,\n}\n\n/* 배열로 구현한 해시 테이블 */\npub struct ArrayHashMap {\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;,\n}\n\nimpl ArrayHashMap {\n    pub fn new() -&gt; ArrayHashMap {\n        // 버킷 100개를 포함하는 배열을 초기화한다\n        Self {\n            buckets: vec![None; 100],\n        }\n    }\n\n    /* 해시 함수 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % 100\n    }\n\n    /* 조회 연산 */\n    pub fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;String&gt; {\n        let index = self.hash_func(key);\n        self.buckets[index].as_ref().map(|pair| &amp;pair.val)\n    }\n\n    /* 추가 연산 */\n    pub fn put(&amp;mut self, key: i32, val: &amp;str) {\n        let index = self.hash_func(key);\n        self.buckets[index] = Some(Pair {\n            key,\n            val: val.to_string(),\n        });\n    }\n\n    /* 삭제 연산 */\n    pub fn remove(&amp;mut self, key: i32) {\n        let index = self.hash_func(key);\n        // None으로 설정하여 삭제를 나타낸다\n        self.buckets[index] = None;\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    pub fn entry_set(&amp;self) -&gt; Vec&lt;&amp;Pair&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref())\n            .collect()\n    }\n\n    /* 모든 키 반환 */\n    pub fn key_set(&amp;self) -&gt; Vec&lt;&amp;i32&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.key))\n            .collect()\n    }\n\n    /* 모든 값 반환 */\n    pub fn value_set(&amp;self) -&gt; Vec&lt;&amp;String&gt; {\n        self.buckets\n            .iter()\n            .filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.val))\n            .collect()\n    }\n\n    /* 해시 테이블 출력 */\n    pub fn print(&amp;self) {\n        for pair in self.entry_set() {\n            println!(\"{} -&gt; {}\", pair.key, pair.val);\n        }\n    }\n}\n</code></pre> array_hash_map.c<pre><code>/* 키-값 쌍 int-&gt;string */\ntypedef struct {\n    int key;\n    char *val;\n} Pair;\n\n/* 배열로 구현한 해시 테이블 */\ntypedef struct {\n    Pair *buckets[MAX_SIZE];\n} ArrayHashMap;\n\n/* 생성자 */\nArrayHashMap *newArrayHashMap() {\n    ArrayHashMap *hmap = malloc(sizeof(ArrayHashMap));\n    for (int i=0; i &lt; MAX_SIZE; i++) {\n        hmap-&gt;buckets[i] = NULL;\n    }\n    return hmap;\n}\n\n/* 소멸자 */\nvoid delArrayHashMap(ArrayHashMap *hmap) {\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            free(hmap-&gt;buckets[i]-&gt;val);\n            free(hmap-&gt;buckets[i]);\n        }\n    }\n    free(hmap);\n}\n\n/* 추가 연산 */\nvoid put(ArrayHashMap *hmap, const int key, const char *val) {\n    Pair *Pair = malloc(sizeof(Pair));\n    Pair-&gt;key = key;\n    Pair-&gt;val = malloc(strlen(val) + 1);\n    strcpy(Pair-&gt;val, val);\n\n    int index = hashFunc(key);\n    hmap-&gt;buckets[index] = Pair;\n}\n\n/* 삭제 연산 */\nvoid removeItem(ArrayHashMap *hmap, const int key) {\n    int index = hashFunc(key);\n    free(hmap-&gt;buckets[index]-&gt;val);\n    free(hmap-&gt;buckets[index]);\n    hmap-&gt;buckets[index] = NULL;\n}\n\n/* 모든 키-값 쌍 반환 */\nvoid pairSet(ArrayHashMap *hmap, MapSet *set) {\n    Pair *entries;\n    int i = 0, index = 0;\n    int total = 0;\n    /* 유효한 키-값 쌍의 개수를 센다 */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    entries = malloc(sizeof(Pair) * total);\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            entries[index].key = hmap-&gt;buckets[i]-&gt;key;\n            entries[index].val = malloc(strlen(hmap-&gt;buckets[i]-&gt;val) + 1);\n            strcpy(entries[index].val, hmap-&gt;buckets[i]-&gt;val);\n            index++;\n        }\n    }\n    set-&gt;set = entries;\n    set-&gt;len = total;\n}\n\n/* 모든 키 반환 */\nvoid keySet(ArrayHashMap *hmap, MapSet *set) {\n    int *keys;\n    int i = 0, index = 0;\n    int total = 0;\n    /* 유효한 키-값 쌍의 개수를 센다 */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    keys = malloc(total * sizeof(int));\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            keys[index] = hmap-&gt;buckets[i]-&gt;key;\n            index++;\n        }\n    }\n    set-&gt;set = keys;\n    set-&gt;len = total;\n}\n\n/* 모든 값 반환 */\nvoid valueSet(ArrayHashMap *hmap, MapSet *set) {\n    char **vals;\n    int i = 0, index = 0;\n    int total = 0;\n    /* 유효한 키-값 쌍의 개수를 센다 */\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    vals = malloc(total * sizeof(char *));\n    for (i = 0; i &lt; MAX_SIZE; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            vals[index] = hmap-&gt;buckets[i]-&gt;val;\n            index++;\n        }\n    }\n    set-&gt;set = vals;\n    set-&gt;len = total;\n}\n\n/* 해시 테이블 출력 */\nvoid print(ArrayHashMap *hmap) {\n    int i;\n    MapSet set;\n    pairSet(hmap, &amp;set);\n    Pair *entries = (Pair *)set.set;\n    for (i = 0; i &lt; set.len; i++) {\n        printf(\"%d -&gt; %s\\n\", entries[i].key, entries[i].val);\n    }\n    free(set.set);\n}\n</code></pre> array_hash_map.kt<pre><code>/* 키-값 쌍 */\nclass Pair(\n    var key: Int,\n    var _val: String\n)\n\n/* 배열로 구현한 해시 테이블 */\nclass ArrayHashMap {\n    // 버킷 100개를 포함하는 배열을 초기화한다\n    private val buckets = arrayOfNulls&lt;Pair&gt;(100)\n\n    /* 해시 함수 */\n    fun hashFunc(key: Int): Int {\n        val index = key % 100\n        return index\n    }\n\n    /* 조회 연산 */\n    fun get(key: Int): String? {\n        val index = hashFunc(key)\n        val pair = buckets[index] ?: return null\n        return pair._val\n    }\n\n    /* 추가 연산 */\n    fun put(key: Int, _val: String) {\n        val pair = Pair(key, _val)\n        val index = hashFunc(key)\n        buckets[index] = pair\n    }\n\n    /* 삭제 연산 */\n    fun remove(key: Int) {\n        val index = hashFunc(key)\n        // null로 설정하여 삭제를 나타낸다\n        buckets[index] = null\n    }\n\n    /* 모든 키-값 쌍 반환 */\n    fun pairSet(): MutableList&lt;Pair&gt; {\n        val pairSet = mutableListOf&lt;Pair&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                pairSet.add(pair)\n        }\n        return pairSet\n    }\n\n    /* 모든 키 반환 */\n    fun keySet(): MutableList&lt;Int&gt; {\n        val keySet = mutableListOf&lt;Int&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                keySet.add(pair.key)\n        }\n        return keySet\n    }\n\n    /* 모든 값 반환 */\n    fun valueSet(): MutableList&lt;String&gt; {\n        val valueSet = mutableListOf&lt;String&gt;()\n        for (pair in buckets) {\n            if (pair != null)\n                valueSet.add(pair._val)\n        }\n        return valueSet\n    }\n\n    /* 해시 테이블 출력 */\n    fun print() {\n        for (kv in pairSet()) {\n            val key = kv.key\n            val _val = kv._val\n            println(\"$key -&gt; $_val\")\n        }\n    }\n}\n</code></pre> array_hash_map.rb<pre><code>### 키-값 쌍 ###\nclass Pair\n  attr_accessor :key, :val\n\n  def initialize(key, val)\n    @key = key\n    @val = val\n  end\nend\n\n### 배열로 구현한 해시 테이블 ###\nclass ArrayHashMap\n  ### 생성자 ###\n  def initialize\n    # 버킷 100개를 포함하는 배열을 초기화한다\n    @buckets = Array.new(100)\n  end\n\n  ### 해시 함수 ###\n  def hash_func(key)\n    index = key % 100\n  end\n\n  ### 조회 연산 ###\n  def get(key)\n    index = hash_func(key)\n    pair = @buckets[index]\n\n    return if pair.nil?\n    pair.val\n  end\n\n  ### 추가 연산 ###\n  def put(key, val)\n    pair = Pair.new(key, val)\n    index = hash_func(key)\n    @buckets[index] = pair\n  end\n\n  ### 삭제 연산 ###\n  def remove(key)\n    index = hash_func(key)\n    # nil로 설정하여 삭제를 나타낸다\n    @buckets[index] = nil\n  end\n\n  ### 모든 키-값 쌍 반환 ###\n  def entry_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair unless pair.nil? }\n    result\n  end\n\n  ### 모든 키 반환 ###\n  def key_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair.key unless pair.nil? }\n    result\n  end\n\n  ### 모든 값 반환 ###\n  def value_set\n    result = []\n    @buckets.each { |pair| result &lt;&lt; pair.val unless pair.nil? }\n    result\n  end\n\n  ### 해시 테이블 출력 ###\n  def print\n    @buckets.each { |pair| puts \"#{pair.key} -&gt; #{pair.val}\" unless pair.nil? }\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 6 장   해시 테이블","6.1   해시 테이블"],"tags":[]},{"location":"chapter_hashing/hash_map/#613","level":2,"title":"6.1.3   해시 충돌과 용량 확장","text":"<p>해시 함수는 본질적으로 모든 <code>key</code>로 이루어진 입력 공간을 배열의 모든 인덱스로 이루어진 출력 공간에 매핑한다. 입력 공간은 대개 출력 공간보다 훨씬 크므로 이론적으로 ‘여러 입력이 같은 출력에 대응하는’ 경우가 반드시 존재한다.</p> <p>앞의 해시 함수에서는 입력 <code>key</code>의 마지막 두 자리가 같으면 출력도 같다. 예를 들어 학번이 12836과 20336인 두 학생을 조회하면 다음과 같다.</p> <pre><code>12836 % 100 = 36\n20336 % 100 = 36\n</code></pre> <p>아래 그림(6-3)에서 두 학번이 같은 이름을 가리키는데, 이는 명백히 잘못되었다. 이처럼 여러 입력이 같은 출력에 대응하는 현상을 해시 충돌(hash collision)이라고 한다.</p> <p></p> <p> 그림 6-3   해시 충돌의 예 </p> <p>해시 테이블의 용량 \\(n\\)이 커질수록 여러 <code>key</code>가 같은 버킷에 들어갈 확률은 낮아지고 충돌도 줄어든다. 따라서 해시 테이블의 용량을 늘려 해시 충돌을 줄일 수 있다.</p> <p>아래 그림(6-4)에서 확장 전에는 키-값 쌍 <code>(136, A)</code>와 <code>(236, D)</code>가 충돌하지만 확장 후에는 충돌이 사라진다.</p> <p></p> <p> 그림 6-4   해시 테이블의 용량 확장 </p> <p>배열 확장과 마찬가지로 해시 테이블을 확장하려면 모든 키-값 쌍을 기존 테이블에서 새 테이블로 옮겨야 하므로 시간이 많이 든다. 또한 용량 <code>capacity</code>가 바뀌므로 해시 함수로 모든 키-값 쌍의 저장 위치를 다시 계산해야 하며, 이는 확장 비용을 더욱 늘린다. 따라서 프로그래밍 언어는 보통 충분히 큰 용량을 미리 확보하여 잦은 확장을 방지한다.</p> <p>적재율(load factor)은 해시 테이블의 원소 수를 버킷 수로 나눈 값으로, 해시 충돌의 심각도를 측정하는 중요한 개념이다. 용량 확장을 시작하는 조건으로도 자주 사용된다. 예를 들어 Java에서는 적재율이 \\(0.75\\)를 넘으면 해시 테이블의 용량을 기존의 \\(2\\)배로 늘린다.</p>","path":["제 6 장   해시 테이블","6.1   해시 테이블"],"tags":[]},{"location":"chapter_hashing/summary/","level":1,"title":"6.4   요약","text":"","path":["제 6 장   해시 테이블","6.4   요약"],"tags":[]},{"location":"chapter_hashing/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>해시 테이블은 <code>key</code>를 입력받아 \\(O(1)\\) 시간에 <code>value</code>를 조회하므로 매우 효율적이다.</li> <li>대표적인 연산에는 조회, 키-값 쌍 추가와 삭제, 해시 테이블 순회 등이 있다.</li> <li>해시 함수는 <code>key</code>를 배열 인덱스로 매핑하여 해당 버킷에 접근하고 <code>value</code>를 얻도록 한다.</li> <li>서로 다른 두 <code>key</code>가 해시 함수를 거쳐 같은 배열 인덱스를 얻으면 잘못된 조회 결과가 발생할 수 있다. 이를 해시 충돌이라고 한다.</li> <li>해시 테이블 용량이 클수록 충돌 확률이 낮으므로 용량 확장으로 충돌을 완화할 수 있다. 배열과 마찬가지로 해시 테이블의 용량 확장 비용도 크다.</li> <li>적재율은 원소 수를 버킷 수로 나눈 값이며, 해시 충돌의 심각도를 반영한다. 용량 확장의 시작 조건으로 자주 사용된다.</li> <li>체이닝은 단일 원소를 연결 리스트로 바꾸어 충돌하는 모든 원소를 같은 리스트에 저장한다. 리스트가 너무 길면 조회 효율이 낮아지므로, 레드-블랙 트리로 바꾸어 효율을 높일 수 있다.</li> <li>개방 주소법은 여러 차례 탐사하여 해시 충돌을 처리한다. 선형 탐사는 고정 간격을 사용하며, 원소를 삭제할 수 없고 군집이 생기기 쉽다는 단점이 있다. 다중 해싱은 여러 해시 함수로 탐사하므로 군집은 덜 생기지만 계산량이 늘어난다.</li> <li>프로그래밍 언어마다 해시 테이블 구현이 다르다. 예를 들어 Java의 <code>HashMap</code>은 체이닝을, Python의 <code>dict</code>는 개방 주소법을 사용한다.</li> <li>해시 테이블의 해시 알고리즘에는 결정성, 높은 효율, 균등 분포가 필요하다. 암호학에서는 충돌 저항성과 눈사태 효과도 갖추어야 한다.</li> <li>해시 알고리즘은 해시값의 균등 분포를 최대한 보장하고 충돌을 줄이기 위해 보통 큰 소수를 모듈러 연산의 법으로 사용한다.</li> <li>대표적인 해시 알고리즘으로 MD5, SHA-1, SHA-2, SHA-3 등이 있다. MD5는 파일 무결성 검사에, SHA-2는 보안 응용과 프로토콜에 자주 쓰인다.</li> <li>프로그래밍 언어는 보통 자료형별 내장 해시 알고리즘을 제공하여 버킷 인덱스를 계산한다. 일반적으로 불변 객체만 해시 가능하다.</li> </ul>","path":["제 6 장   해시 테이블","6.4   요약"],"tags":[]},{"location":"chapter_hashing/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 해시 테이블의 시간 복잡도가 \\(O(n)\\)이 되는 경우는 언제인가?</p> <p>해시 충돌이 심하면 시간 복잡도가 \\(O(n)\\)으로 나빠진다. 해시 함수를 잘 설계하고 용량을 적절히 정하며 충돌이 비교적 고르게 분포하면 \\(O(1)\\)이다. 프로그래밍 언어의 내장 해시 테이블을 사용할 때는 보통 시간 복잡도를 \\(O(1)\\)로 본다.</p> <p>Q: 해시 함수 \\(f(x) = x\\)를 사용하면 충돌이 없을 텐데, 왜 그렇게 하지 않는가?</p> <p>\\(f(x) = x\\)에서는 각 원소에 고유한 버킷 인덱스가 대응하므로 배열과 같다. 그러나 입력 공간은 보통 출력 공간(배열 길이)보다 훨씬 크므로, 해시 함수의 마지막 단계에서 대개 배열 길이로 나눈 나머지를 구한다. 즉, 해시 테이블의 목적은 큰 상태 공간을 작은 공간에 매핑하면서 \\(O(1)\\)의 조회 효율을 제공하는 것이다.</p> <p>Q: 해시 테이블도 배열, 연결 리스트, 이진 트리로 구현되는데 어떻게 이들보다 효율이 높을 수 있는가?</p> <p>첫째, 시간 효율은 높아지지만 공간 효율은 낮아진다. 해시 테이블에는 사용하지 않는 메모리가 상당 부분 있다.</p> <p>둘째, 특정 활용 상황에서만 시간 효율이 높아진다. 같은 기능을 배열이나 연결 리스트로 같은 시간 복잡도에 구현할 수 있다면 보통 그쪽이 더 빠르다. 해시 함수 계산에 비용이 들어 시간 복잡도의 상수항이 더 크기 때문이다.</p> <p>마지막으로 해시 테이블의 시간 복잡도도 나빠질 수 있다. 예를 들어 체이닝에서는 연결 리스트나 레드-블랙 트리에서 탐색하므로, 여전히 \\(O(n)\\) 시간으로 나빠질 위험이 있다.</p> <p>Q: 다중 해싱도 원소를 곧바로 삭제할 수 없는가? 삭제 표시를 한 공간은 다시 사용할 수 있는가?</p> <p>다중 해싱은 개방 주소법의 한 종류다. 개방 주소법은 모두 원소를 곧바로 삭제할 수 없으므로 삭제 표시를 해야 한다. 표시된 공간은 다시 사용할 수 있다. 새 원소를 삽입할 때 해시 함수로 삭제 표시가 있는 위치를 찾으면 그곳에 새 원소를 넣을 수 있다. 이렇게 하면 탐사 순서를 유지하면서 공간 활용률도 보장한다.</p> <p>Q: 선형 탐사에서는 원소를 찾을 때도 왜 해시 충돌이 발생하는가?</p> <p>해시 함수로 해당 버킷과 키-값 쌍을 찾았는데 <code>key</code>가 일치하지 않으면 해시 충돌이 있다는 뜻이다. 따라서 미리 정한 간격으로 뒤를 차례로 탐색하다가 올바른 키-값 쌍을 찾거나, 찾을 수 없다고 판단하면 종료한다.</p> <p>Q: 해시 테이블의 용량 확장은 왜 해시 충돌을 완화하는가?</p> <p>해시 함수는 마지막에 대개 배열 길이 \\(n\\)으로 나눈 나머지를 구하여 출력이 인덱스 범위에 들어가도록 한다. 용량을 늘리면 \\(n\\)이 바뀌고 <code>key</code>에 대응하는 인덱스도 달라질 수 있다. 원래 같은 버킷에 있던 여러 <code>key</code>가 여러 버킷으로 분산될 수 있으므로 충돌이 완화된다.</p> <p>Q: 효율적으로 저장하고 접근하려는 목적이라면 배열을 바로 사용하면 되지 않는가?</p> <p><code>key</code>가 좁은 범위의 연속된 정수라면 배열을 바로 쓰는 것이 간단하고 효율적이다. 하지만 문자열처럼 다른 자료형이면 해시 함수로 <code>key</code>를 배열 인덱스에 매핑하고 버킷 배열에 원소를 저장해야 한다. 이것이 바로 해시 테이블이다.</p>","path":["제 6 장   해시 테이블","6.4   요약"],"tags":[]},{"location":"chapter_heap/","level":1,"title":"제 8 장   힙","text":"<p>이 장을 시작하며</p> <p>힙은 겹겹이 이어진 산봉우리처럼 높낮이도 모습도 제각각이다.</p> <p>봉우리마다 높이가 다르지만, 가장 높은 봉우리는 언제나 가장 먼저 눈에 들어온다.</p>","path":["제 8 장   힙"],"tags":[]},{"location":"chapter_heap/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>8.1   힙</li> <li>8.2   힙 구성 연산</li> <li>8.3   Top-k 문제</li> <li>8.4   요약</li> <li>8.5   연습 문제</li> </ul>","path":["제 8 장   힙"],"tags":[]},{"location":"chapter_heap/build_heap/","level":1,"title":"8.2   힙 구성 연산","text":"<p>어떤 상황에서는 리스트의 모든 원소를 사용해 힙을 구성하고 싶을 수 있다. 이 과정을 '힙 구성 연산'이라고 한다.</p>","path":["제 8 장   힙","8.2   힙 구성 연산"],"tags":[]},{"location":"chapter_heap/build_heap/#821","level":2,"title":"8.2.1   힙 삽입 연산을 이용한 구현","text":"<p>먼저 빈 힙을 만든 다음, 리스트를 순회하면서 각 원소에 대해 차례로 '힙 삽입 연산'을 수행한다. 즉, 원소를 힙의 끝에 추가한 뒤 해당 원소부터 '아래에서 위로' 힙을 재정렬한다.</p> <p>원소 하나를 힙에 삽입할 때마다 힙의 길이가 1씩 늘어난다. 이진 트리에 위에서 아래 순서로 노드를 추가하므로 힙은 '위에서 아래로' 구성된다.</p> <p>원소의 개수를 \\(n\\)이라고 하자. 각 원소의 힙 삽입 연산에는 \\(O(\\log{n})\\) 시간이 걸리므로, 이 힙 구성 방법의 시간 복잡도는 \\(O(n \\log n)\\)이다.</p>","path":["제 8 장   힙","8.2   힙 구성 연산"],"tags":[]},{"location":"chapter_heap/build_heap/#822","level":2,"title":"8.2.2   순회하며 힙을 재정렬하는 구현","text":"<p>실제로는 다음 두 단계로 더 효율적인 힙 구성 방법을 구현할 수 있다.</p> <ol> <li>리스트의 모든 원소를 그대로 힙에 추가한다. 이 시점에서는 아직 힙의 성질을 만족하지 않는다.</li> <li>힙을 역순으로 순회하면서(레벨 순서 순회의 역순), 각 내부 노드에 대해 차례로 '위에서 아래로 힙 재정렬'을 수행한다.</li> </ol> <p>노드 하나를 재정렬할 때마다 그 노드를 루트로 하는 서브트리가 올바른 부분 힙이 된다. 역순으로 순회하므로 힙은 '아래에서 위로' 구성된다.</p> <p>역순으로 순회하는 이유는 현재 노드 아래의 서브트리들이 이미 올바른 부분 힙임을 보장하기 위해서다. 그래야 현재 노드를 재정렬하는 작업이 유효해진다.</p> <p>리프 노드는 자식 노드가 없으므로 그 자체로 올바른 부분 힙이며, 재정렬할 필요가 없다. 아래 코드와 같이 마지막 내부 노드는 마지막 노드의 부모 노드다. 여기서부터 역순으로 순회하며 힙 재정렬을 수행한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def __init__(self, nums: list[int]):\n    \"\"\"생성자: 입력 리스트를 바탕으로 힙 구성\"\"\"\n    # 리스트 원소를 그대로 힙에 추가한다\n    self.max_heap = nums\n    # 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for i in range(self.parent(self.size() - 1), -1, -1):\n        self.sift_down(i)\n</code></pre> my_heap.cpp<pre><code>/* 생성자: 입력 리스트를 바탕으로 힙 구성 */\nMaxHeap(vector&lt;int&gt; nums) {\n    // 리스트 원소를 그대로 힙에 추가한다\n    maxHeap = nums;\n    // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.java<pre><code>/* 생성자: 입력 리스트를 바탕으로 힙 구성 */\nMaxHeap(List&lt;Integer&gt; nums) {\n    // 리스트 원소를 그대로 힙에 추가한다\n    maxHeap = new ArrayList&lt;&gt;(nums);\n    // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* 생성자: 입력 리스트로 힙 구성 */\nMaxHeap(IEnumerable&lt;int&gt; nums) {\n    // 리스트 원소를 그대로 힙에 추가한다\n    maxHeap = new List&lt;int&gt;(nums);\n    // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    var size = Parent(this.Size() - 1);\n    for (int i = size; i &gt;= 0; i--) {\n        SiftDown(i);\n    }\n}\n</code></pre> my_heap.go<pre><code>/* 생성자: 슬라이스를 바탕으로 힙 구성 */\nfunc newMaxHeap(nums []any) *maxHeap {\n    // 리스트 원소를 그대로 힙에 추가한다\n    h := &amp;maxHeap{data: nums}\n    for i := h.parent(len(h.data) - 1); i &gt;= 0; i-- {\n        // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n        h.siftDown(i)\n    }\n    return h\n}\n</code></pre> my_heap.swift<pre><code>/* 생성자: 입력 리스트를 바탕으로 힙 구성 */\ninit(nums: [Int]) {\n    // 리스트 원소를 그대로 힙에 추가한다\n    maxHeap = nums\n    // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for i in (0 ... parent(i: size() - 1)).reversed() {\n        siftDown(i: i)\n    }\n}\n</code></pre> my_heap.js<pre><code>/* 생성자: 빈 힙을 만들거나 입력 리스트로 힙을 구성한다 */\nconstructor(nums) {\n    // 리스트 원소를 그대로 힙에 추가한다\n    this.#maxHeap = nums === undefined ? [] : [...nums];\n    // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (let i = this.#parent(this.size() - 1); i &gt;= 0; i--) {\n        this.#siftDown(i);\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* 생성자: 빈 힙을 만들거나 입력 리스트로 힙을 구성한다 */\nconstructor(nums?: number[]) {\n    // 리스트 원소를 그대로 힙에 추가한다\n    this.maxHeap = nums === undefined ? [] : [...nums];\n    // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (let i = this.parent(this.size() - 1); i &gt;= 0; i--) {\n        this.siftDown(i);\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* 생성자: 입력 리스트를 바탕으로 힙 구성 */\nMaxHeap(List&lt;int&gt; nums) {\n  // 리스트 원소를 그대로 힙에 추가한다\n  _maxHeap = nums;\n  // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n  for (int i = _parent(size() - 1); i &gt;= 0; i--) {\n    siftDown(i);\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* 생성자: 입력 리스트를 바탕으로 힙 구성 */\nfn new(nums: Vec&lt;i32&gt;) -&gt; Self {\n    // 리스트 원소를 그대로 힙에 추가한다\n    let mut heap = MaxHeap { max_heap: nums };\n    // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for i in (0..=Self::parent(heap.size() - 1)).rev() {\n        heap.sift_down(i);\n    }\n    heap\n}\n</code></pre> my_heap.c<pre><code>/* 생성자: 슬라이스를 바탕으로 힙 구성 */\nMaxHeap *newMaxHeap(int nums[], int size) {\n    // 모든 원소를 힙에 삽입한다\n    MaxHeap *maxHeap = (MaxHeap *)malloc(sizeof(MaxHeap));\n    maxHeap-&gt;size = size;\n    memcpy(maxHeap-&gt;data, nums, size * sizeof(int));\n    for (int i = parent(maxHeap, size - 1); i &gt;= 0; i--) {\n        // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n        siftDown(maxHeap, i);\n    }\n    return maxHeap;\n}\n</code></pre> my_heap.kt<pre><code>/* 최대 힙 */\nclass MaxHeap(nums: MutableList&lt;Int&gt;?) {\n    // 배열 대신 리스트를 사용하므로 용량 확장을 고려할 필요가 없다\n    private val maxHeap = mutableListOf&lt;Int&gt;()\n\n    /* 생성자: 입력 리스트를 바탕으로 힙 구성 */\n    init {\n        // 리스트 원소를 그대로 힙에 추가한다\n        maxHeap.addAll(nums!!)\n        // 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n        for (i in parent(size() - 1) downTo 0) {\n            siftDown(i)\n        }\n    }\n\n    /* 왼쪽 자식 노드의 인덱스 반환 */\n    private fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* 오른쪽 자식 노드의 인덱스 반환 */\n    private fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* 부모 노드의 인덱스 반환 */\n    private fun parent(i: Int): Int {\n        return (i - 1) / 2 // 나눗셈의 결과를 내림한다\n    }\n\n    /* 원소 교환 */\n    private fun swap(i: Int, j: Int) {\n        val temp = maxHeap[i]\n        maxHeap[i] = maxHeap[j]\n        maxHeap[j] = temp\n    }\n\n    /* 힙 크기 반환 */\n    fun size(): Int {\n        return maxHeap.size\n    }\n\n    /* 힙이 비어 있는지 판단 */\n    fun isEmpty(): Boolean {\n        /* 힙이 비어 있는지 판단 */\n        return size() == 0\n    }\n\n    /* 힙의 루트 원소 접근 */\n    fun peek(): Int {\n        return maxHeap[0]\n    }\n\n    /* 힙에 원소 삽입 */\n    fun push(_val: Int) {\n        // 노드 추가\n        maxHeap.add(_val)\n        // 아래에서 위로 힙을 재정렬한다\n        siftUp(size() - 1)\n    }\n\n    /* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\n    private fun siftUp(it: Int) {\n        // Kotlin의 함수 매개변수는 변경할 수 없으므로 임시 변수를 생성한다\n        var i = it\n        while (true) {\n            // 노드 i의 부모 노드를 구한다\n            val p = parent(i)\n            // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n            if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n            // 두 노드를 교환한다\n            swap(i, p)\n            // 위쪽으로 힙 재정렬을 반복한다\n            i = p\n        }\n    }\n\n    /* 힙에서 원소 꺼내기 */\n    fun pop(): Int {\n        // 비어 있는 경우를 처리한다\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        swap(0, size() - 1)\n        // 노드 삭제\n        val _val = maxHeap.removeAt(size() - 1)\n        // 위에서 아래로 힙을 재정렬한다\n        siftDown(0)\n        // 힙의 루트 원소를 반환한다\n        return _val\n    }\n\n    /* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\n    private fun siftDown(it: Int) {\n        // Kotlin의 함수 매개변수는 변경할 수 없으므로 임시 변수를 생성한다\n        var i = it\n        while (true) {\n            // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n            val l = left(i)\n            val r = right(i)\n            var ma = i\n            if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n            if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n            // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n            if (ma == i) break\n            // 두 노드를 교환한다\n            swap(i, ma)\n            // 아래쪽으로 힙 재정렬을 반복한다\n            i = ma\n        }\n    }\n\n    /* 힙 출력(이진 트리) */\n    fun print() {\n        val queue = PriorityQueue { a: Int, b: Int -&gt; b - a }\n        queue.addAll(maxHeap)\n        printHeap(queue)\n    }\n}\n</code></pre> my_heap.rb<pre><code>### 생성자: 입력 리스트를 바탕으로 힙 구성 ###\ndef initialize(nums)\n  # 리스트 원소를 그대로 힙에 추가한다\n  @max_heap = nums\n  # 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n  parent(size - 1).downto(0) do |i|\n    sift_down(i)\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 8 장   힙","8.2   힙 구성 연산"],"tags":[]},{"location":"chapter_heap/build_heap/#823","level":2,"title":"8.2.3   복잡도 분석","text":"<p>이제 두 번째 힙 구성 방법의 시간 복잡도를 계산해 보자.</p> <ul> <li>완전 이진 트리의 노드 수가 \\(n\\)이라면, 리프 노드의 수는 \\((n + 1) / 2\\)이다. 여기서 \\(/\\)는 내림 정수 나눗셈이다. 따라서 재정렬해야 하는 노드의 수는 \\(n / 2\\)이다.</li> <li>위에서 아래로 힙을 재정렬할 때 각 노드는 최대 리프 노드까지 내려갈 수 있으므로, 최대 반복 횟수는 이진 트리의 높이인 \\(\\log n\\)이다.</li> </ul> <p>위의 두 값을 곱하면 힙 구성 과정의 시간 복잡도가 \\(O(n \\log n)\\)이 된다. 하지만 이 추정은 정확하지 않다. 이진 트리에서는 아래쪽 노드의 수가 위쪽 노드보다 훨씬 많다는 성질을 고려하지 않았기 때문이다.</p> <p>이제 더 정확하게 계산해 보자. 계산을 간단히 하기 위해 노드 수가 \\(n\\)이고 높이가 \\(h\\)인 '포화 이진 트리'가 주어졌다고 가정한다. 이 가정은 계산 결과의 정확성에 영향을 주지 않는다.</p> <p></p> <p> 그림 8-5   포화 이진 트리의 레벨별 노드 수 </p> <p>위 그림(8-5)에서 노드를 '위에서 아래로 재정렬'하는 최대 반복 횟수는 그 노드에서 리프 노드까지의 거리와 같으며, 이 거리는 바로 '노드의 높이'다. 따라서 각 레벨의 '노드 수 \\(\\times\\) 노드 높이'를 더하면 모든 노드에 대한 힙 재정렬의 총반복 횟수를 구할 수 있다.</p> \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{(h-1)}\\times1 \\] <p>위 식을 정리하려면 중등 수학의 수열 지식이 필요하다. 먼저 \\(T(h)\\)에 \\(2\\)를 곱하면 다음과 같다.</p> \\[ \\begin{aligned} T(h) &amp; = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{h-1}\\times1 \\newline 2 T(h) &amp; = 2^1h + 2^2(h-1) + 2^3(h-2) + \\dots + 2^{h}\\times1 \\newline \\end{aligned} \\] <p>두 식의 항을 한 칸씩 어긋나게 놓고 아래 식 \\(2 T(h)\\)에서 위 식 \\(T(h)\\)를 빼면 다음을 얻는다.</p> \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\dots + 2^{h-1} + 2^h \\] <p>위 식에서 \\(T(h)\\)를 이루는 항들이 등비수열을 이루므로 합 공식을 바로 적용하여 시간 복잡도를 구할 수 있다.</p> \\[ \\begin{aligned} T(h) &amp; = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline &amp; = 2^{h+1} - h - 2 \\newline &amp; = O(2^h) \\end{aligned} \\] <p>또한 높이가 \\(h\\)인 포화 이진 트리의 노드 수는 \\(n = 2^{h+1} - 1\\)이므로, 복잡도가 \\(O(2^h) = O(n)\\)임을 쉽게 알 수 있다. 위 계산은 리스트를 입력받아 힙을 구성하는 시간 복잡도가 \\(O(n)\\)으로 매우 효율적임을 보여 준다.</p>","path":["제 8 장   힙","8.2   힙 구성 연산"],"tags":[]},{"location":"chapter_heap/exercises/","level":1,"title":"8.5   연습 문제","text":"","path":["제 8 장   힙","8.5   연습 문제"],"tags":[]},{"location":"chapter_heap/exercises/#851","level":2,"title":"8.5.1   개념 확인","text":"","path":["제 8 장   힙","8.5   연습 문제"],"tags":[]},{"location":"chapter_heap/exercises/#1-10","level":3,"title":"1.   10을 힙에 넣은 뒤 어떻게 조정하는가","text":"<p>배열 <code>[9, 7, 8, 3, 5]</code>는 최대 힙을 나타낸다. 여기에 숫자 10을 삽입한다.</p> <ol> <li>먼저 10을 배열 끝에 추가한다. 부모 노드의 값은 얼마인가?</li> <li>새 노드부터 위로 힙을 재정렬하며, 교환할 때마다 배열을 적어 보자.</li> <li>최종 루트 원소는 무엇이며, 총 몇 번 교환했는가?</li> </ol> 참고 답안 <ol> <li> <p>추가한 10의 인덱스는 5이고, 부모 인덱스는     \\(\\lfloor(5-1)/2\\rfloor=2\\)이므로 부모 노드의 값은 8이다.</p> </li> <li> <p>10이 8보다 크므로 첫 번째 교환 후에는 <code>[9, 7, 10, 3, 5, 8]</code>이 된다.     10은 부모 9보다도 크므로 두 번째 교환 후에는 <code>[10, 7, 9, 3, 5, 8]</code>이 된다.     10이 루트에 도달했으므로 힙 재정렬을 마친다.</p> </li> <li> <p>최종 루트 값은 10이며, 총 2번 교환했다.</p> </li> </ol>","path":["제 8 장   힙","8.5   연습 문제"],"tags":[]},{"location":"chapter_heap/exercises/#2","level":3,"title":"2.   최소 힙의 부모와 자식 관계 확인하기","text":"<p>배열 <code>[1, 4, 3, 7, 6, 2]</code>는 완전 이진 트리를 나타낸다. 최소 힙에서는 모든 부모의 값이 자식의 값보다 작거나 같아야 한다. 인덱스 \\(i\\)의 왼쪽과 오른쪽 자식 인덱스는 각각 \\(2i+1\\), \\(2i+2\\)다.</p> <ol> <li>인덱스 2의 자식 인덱스와 값은 각각 무엇인가?</li> <li>인덱스 2의 값은 3이다. 이 노드와 자식의 관계가 최소 힙 규칙에 어긋나는가? 그렇다면 어떤 두 원소를 교환해야 하는가?</li> <li>2번의 판단에 따라, 규칙에 어긋나면 교환 후 배열을 적고 그렇지 않으면 교환할 필요가 없음을 설명하라. 마지막으로 나머지 부모와 자식 관계도 확인하라.</li> </ol> 참고 답안 <ol> <li> <p>인덱스 2의 왼쪽 자식 인덱스는 5이고 값은 2다. 오른쪽 자식 인덱스는 6이지만 배열 길이가 6이므로 오른쪽 자식은 없다.</p> </li> <li> <p>부모의 값 3이 자식의 값 2보다 크므로 최소 힙 규칙에 어긋난다. 인덱스 2와 5의 원소를 교환해야 한다.</p> </li> <li> <p>교환 후 배열은 <code>[1, 4, 2, 7, 6, 3]</code>이다. 관계를 하나씩 확인하면     <code>1 ≤ 4</code>, <code>1 ≤ 2</code>, <code>4 ≤ 7</code>, <code>4 ≤ 6</code>, <code>2 ≤ 3</code>이다.     모든 부모의 값이 자식의 값보다 작거나 같으므로 최소 힙 규칙을 만족한다.</p> </li> </ol>","path":["제 8 장   힙","8.5   연습 문제"],"tags":[]},{"location":"chapter_heap/exercises/#3","level":3,"title":"3.   최소 힙으로 가장 큰 수 세 개 유지하기","text":"<p>데이터 스트림 <code>[4, 1, 7, 3, 8]</code>에서 가장 큰 수 3개를 유지하려면 크기가 3을 넘지 않는 최소 힙을 사용할 수 있다.</p> <p>먼저 앞의 3개 수를 차례대로 최소 힙에 넣는다. 힙이 가득 찬 뒤에는 새 수를 읽을 때마다 루트보다 크면 루트를 삭제하고 새 수를 넣는다. 그렇지 않으면 힙을 그대로 둔다.</p> <p>각 수를 읽은 뒤 힙에 남아 있는 수와 루트 값을 적어 보자. 남아 있는 수는 집합으로만 적으면 되며, 힙 배열에서의 순서는 적지 않아도 된다.</p> 참고 답안 <p>각 수를 읽은 뒤의 결과는 다음과 같다.</p> 읽은 수 남아 있는 수 루트 4 <code>{4}</code> 4 1 <code>{1, 4}</code> 1 7 <code>{1, 4, 7}</code> 1 3 <code>{3, 4, 7}</code> 3 8 <code>{4, 7, 8}</code> 4 <p>힙이 가득 차면 루트는 현재 남아 있는 수 중 가장 작은 수다. 새 수가 루트보다 클 때만 새 수로 루트를 대체한다. 최종적으로 남은 <code>{4, 7, 8}</code>이 바로 가장 큰 수 3개다.</p>","path":["제 8 장   힙","8.5   연습 문제"],"tags":[]},{"location":"chapter_heap/exercises/#852","level":2,"title":"8.5.2   프로그래밍 연습","text":"","path":["제 8 장   힙","8.5   연습 문제"],"tags":[]},{"location":"chapter_heap/exercises/#1-k","level":3,"title":"1.   배열에서 k번째로 큰 원소 찾기","text":"<p>정수 배열 <code>nums</code>와 정수 \\(k\\)가 주어진다. 여기서 \\(1 \\le k \\le n\\)이며 \\(n\\)은 배열 길이다. 배열을 큰 것부터 나열했을 때 \\(k\\)번째 위치의 원소를 반환하라.</p> <p>중복 원소도 각각 센다. 예를 들어 <code>[5, 5, 2]</code>에서 두 번째로 큰 원소 역시 5다. 크기가 \\(k\\)를 넘지 않는 최소 힙을 사용하여 해결하라.</p> 풀이 힌트 <ol> <li>k번째로 큰 원소는 가장 큰 k개 수 중 가장 작은 수다.</li> <li>각 수를 최소 힙에 넣고 크기가 k를 넘으면 최솟값을 꺼낸다.</li> <li>순회가 끝나면 힙에 가장 큰 k개 수가 남으며, 루트가 답이다.</li> </ol> <p>LeetCode</p>","path":["제 8 장   힙","8.5   연습 문제"],"tags":[]},{"location":"chapter_heap/heap/","level":1,"title":"8.1   힙","text":"<p>힙(heap)은 특정 조건을 만족하는 완전 이진 트리로, 아래 그림(8-1)과 같이 크게 두 종류로 나뉜다.</p> <ul> <li>최소 힙(min heap): 모든 노드의 값이 자식 노드의 값보다 작거나 같다(\\(\\leq\\)).</li> <li>최대 힙(max heap): 모든 노드의 값이 자식 노드의 값보다 크거나 같다(\\(\\geq\\)).</li> </ul> <p></p> <p> 그림 8-1   최소 힙과 최대 힙 </p> <p>힙은 완전 이진 트리의 특수한 형태이므로 다음 성질을 가진다.</p> <ul> <li>가장 아래 레벨은 왼쪽부터 노드가 채워지며, 나머지 레벨은 모두 가득 차 있다.</li> <li>이진 트리의 루트 노드를 '힙의 루트', 가장 아래 레벨에서 맨 오른쪽에 있는 노드를 '힙의 끝'이라고 한다.</li> <li>최대 힙(최소 힙)에서는 힙의 루트 원소(루트 노드)의 값이 가장 크다(작다).</li> </ul>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/heap/#811","level":2,"title":"8.1.1   힙의 주요 연산","text":"<p>많은 프로그래밍 언어에서 제공하는 것은 우선순위 큐(priority queue)라는 점을 알아 두자. 우선순위 큐는 우선순위에 따라 정렬되는 큐로 정의되는 추상 자료 구조다.</p> <p>실제로 힙은 보통 우선순위 큐를 구현하는 데 사용되며, 최대 힙은 원소를 큰 값부터 작은 값 순서로 꺼내는 우선순위 큐에 해당한다. 사용하는 입장에서는 '우선순위 큐'와 '힙'을 동등한 자료 구조로 볼 수 있다. 따라서 이 책에서는 둘을 특별히 구분하지 않고 모두 '힙'이라고 한다.</p> <p>힙의 주요 연산은 아래 표(8-1)와 같다. 메서드 이름은 프로그래밍 언어에 따라 달라진다.</p> <p> 표 8-1   힙 연산의 효율 </p> 메서드 이름 설명 시간 복잡도 <code>push()</code> 힙에 원소 삽입 \\(O(\\log n)\\) <code>pop()</code> 힙의 루트 원소 꺼내기 \\(O(\\log n)\\) <code>peek()</code> 힙의 루트 원소에 접근(최대 / 최소 힙에서는 각각 최댓값 / 최솟값) \\(O(1)\\) <code>size()</code> 힙의 원소 수 구하기 \\(O(1)\\) <code>isEmpty()</code> 힙이 비어 있는지 확인 \\(O(1)\\) <p>실제 응용에서는 프로그래밍 언어가 제공하는 힙 클래스(또는 우선순위 큐 클래스)를 바로 사용할 수 있다.</p> <p>정렬 알고리즘에서 '오름차순 정렬'과 '내림차순 정렬'을 선택하듯이, <code>flag</code>를 설정하거나 <code>Comparator</code>를 수정하여 '최소 힙'과 '최대 힙'을 전환할 수 있다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby heap.py<pre><code># 최소 힙 초기화\nmin_heap, flag = [], 1\n# 최대 힙 초기화\nmax_heap, flag = [], -1\n\n# Python의 heapq 모듈은 기본적으로 최소 힙을 구현한다\n# 원소의 부호를 바꾼 뒤 삽입하면 대소 관계가 뒤집히므로 최대 힙을 구현할 수 있다\n# 이 예제에서 flag = 1이면 최소 힙, flag = -1이면 최대 힙에 해당한다\n\n# 힙에 원소 삽입\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n\n# 힙의 루트 원소 구하기\npeek: int = flag * max_heap[0] # 5\n\n# 힙의 루트 원소 꺼내기\n# 꺼낸 원소들은 큰 값부터 작은 값 순서로 나열된다\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n\n# 힙의 크기 구하기\nsize: int = len(max_heap)\n\n# 힙이 비어 있는지 확인\nis_empty: bool = not max_heap\n\n# 리스트를 입력받아 힙 구성\nmin_heap: list[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n</code></pre> heap.cpp<pre><code>/* 힙 초기화 */\n// 최소 힙 초기화\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap;\n// 최대 힙 초기화\npriority_queue&lt;int, vector&lt;int&gt;, less&lt;int&gt;&gt; maxHeap;\n\n/* 힙에 원소 삽입 */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n\n/* 힙의 루트 원소 구하기 */\nint peek = maxHeap.top(); // 5\n\n/* 힙의 루트 원소 꺼내기 */\n// 꺼낸 원소들은 큰 값부터 작은 값 순서로 나열된다\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n\n/* 힙의 크기 구하기 */\nint size = maxHeap.size();\n\n/* 힙이 비어 있는지 확인 */\nbool isEmpty = maxHeap.empty();\n\n/* 리스트를 입력받아 힙 구성 */\nvector&lt;int&gt; input{1, 3, 2, 5, 4};\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap(input.begin(), input.end());\n</code></pre> heap.java<pre><code>/* 힙 초기화 */\n// 최소 힙 초기화\nQueue&lt;Integer&gt; minHeap = new PriorityQueue&lt;&gt;();\n// 최대 힙 초기화(lambda 식으로 Comparator를 수정하면 된다)\nQueue&lt;Integer&gt; maxHeap = new PriorityQueue&lt;&gt;((a, b) -&gt; b - a);\n\n/* 힙에 원소 삽입 */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n\n/* 힙의 루트 원소 구하기 */\nint peek = maxHeap.peek(); // 5\n\n/* 힙의 루트 원소 꺼내기 */\n// 꺼낸 원소들은 큰 값부터 작은 값 순서로 나열된다\npeek = maxHeap.poll(); // 5\npeek = maxHeap.poll(); // 4\npeek = maxHeap.poll(); // 3\npeek = maxHeap.poll(); // 2\npeek = maxHeap.poll(); // 1\n\n/* 힙의 크기 구하기 */\nint size = maxHeap.size();\n\n/* 힙이 비어 있는지 확인 */\nboolean isEmpty = maxHeap.isEmpty();\n\n/* 리스트를 입력받아 힙 구성 */\nminHeap = new PriorityQueue&lt;&gt;(Arrays.asList(1, 3, 2, 5, 4));\n</code></pre> heap.cs<pre><code>/* 힙 초기화 */\n// 최소 힙 초기화\nPriorityQueue&lt;int, int&gt; minHeap = new();\n// 최대 힙 초기화(lambda 식으로 Comparer를 수정하면 된다)\nPriorityQueue&lt;int, int&gt; maxHeap = new(Comparer&lt;int&gt;.Create((x, y) =&gt; y.CompareTo(x)));\n\n/* 힙에 원소 삽입 */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n\n/* 힙의 루트 원소 구하기 */\nint peek = maxHeap.Peek();//5\n\n/* 힙의 루트 원소 꺼내기 */\n// 꺼낸 원소들은 큰 값부터 작은 값 순서로 나열된다\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n\n/* 힙의 크기 구하기 */\nint size = maxHeap.Count;\n\n/* 힙이 비어 있는지 확인 */\nbool isEmpty = maxHeap.Count == 0;\n\n/* 리스트를 입력받아 힙 구성 */\nminHeap = new PriorityQueue&lt;int, int&gt;([(1, 1), (3, 3), (2, 2), (5, 5), (4, 4)]);\n</code></pre> heap.go<pre><code>// Go에서는 heap.Interface를 구현하여 정수 최대 힙을 만들 수 있다\n// heap.Interface를 구현하려면 sort.Interface도 구현해야 한다\ntype intHeap []any\n\n// Push는 heap.Interface의 메서드로, 힙에 원소를 삽입한다\nfunc (h *intHeap) Push(x any) {\n    // Push와 Pop은 포인터 리시버(pointer receiver)를 매개변수로 사용한다\n    // 슬라이스의 내용뿐 아니라 길이도 수정하기 때문이다.\n    *h = append(*h, x.(int))\n}\n\n// Pop은 heap.Interface의 메서드로, 힙의 루트 원소를 꺼낸다\nfunc (h *intHeap) Pop() any {\n    // 꺼낼 원소는 맨 끝에 저장되어 있다\n    last := (*h)[len(*h)-1]\n    *h = (*h)[:len(*h)-1]\n    return last\n}\n\n// Len은 sort.Interface의 메서드다\nfunc (h *intHeap) Len() int {\n    return len(*h)\n}\n\n// Less는 sort.Interface의 메서드다\nfunc (h *intHeap) Less(i, j int) bool {\n    // 최소 힙을 구현하려면 부등호를 &lt;로 바꿔야 한다\n    return (*h)[i].(int) &gt; (*h)[j].(int)\n}\n\n// Swap은 sort.Interface의 메서드다\nfunc (h *intHeap) Swap(i, j int) {\n    (*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n\n// Top은 힙의 루트 원소를 구한다\nfunc (h *intHeap) Top() any {\n    return (*h)[0]\n}\n\n/* 실행 코드 */\nfunc TestHeap(t *testing.T) {\n    /* 힙 초기화 */\n    // 최대 힙 초기화\n    maxHeap := &amp;intHeap{}\n    heap.Init(maxHeap)\n    /* 힙에 원소 삽입 */\n    // heap.Interface의 메서드를 호출하여 원소를 추가한다\n    heap.Push(maxHeap, 1)\n    heap.Push(maxHeap, 3)\n    heap.Push(maxHeap, 2)\n    heap.Push(maxHeap, 4)\n    heap.Push(maxHeap, 5)\n\n    /* 힙의 루트 원소 구하기 */\n    top := maxHeap.Top()\n    fmt.Printf(\"힙의 루트 원소: %d\\n\", top)\n\n    /* 힙의 루트 원소 꺼내기 */\n    // heap.Interface의 메서드를 호출하여 원소를 제거한다\n    heap.Pop(maxHeap) // 5\n    heap.Pop(maxHeap) // 4\n    heap.Pop(maxHeap) // 3\n    heap.Pop(maxHeap) // 2\n    heap.Pop(maxHeap) // 1\n\n    /* 힙의 크기 구하기 */\n    size := len(*maxHeap)\n    fmt.Printf(\"힙의 원소 수: %d\\n\", size)\n\n    /* 힙이 비어 있는지 확인 */\n    isEmpty := len(*maxHeap) == 0\n    fmt.Printf(\"힙이 비어 있는가: %t\\n\", isEmpty)\n}\n</code></pre> heap.swift<pre><code>/* 힙 초기화 */\n// Swift의 Heap 타입은 최대 힙과 최소 힙을 모두 지원하며, swift-collections를 가져와야 한다\nvar heap = Heap&lt;Int&gt;()\n\n/* 힙에 원소 삽입 */\nheap.insert(1)\nheap.insert(3)\nheap.insert(2)\nheap.insert(5)\nheap.insert(4)\n\n/* 힙의 루트 원소 구하기 */\nvar peek = heap.max()!\n\n/* 힙의 루트 원소 꺼내기 */\npeek = heap.removeMax() // 5\npeek = heap.removeMax() // 4\npeek = heap.removeMax() // 3\npeek = heap.removeMax() // 2\npeek = heap.removeMax() // 1\n\n/* 힙의 크기 구하기 */\nlet size = heap.count\n\n/* 힙이 비어 있는지 확인 */\nlet isEmpty = heap.isEmpty\n\n/* 리스트를 입력받아 힙 구성 */\nlet heap2 = Heap([1, 3, 2, 5, 4])\n</code></pre> heap.js<pre><code>// JavaScript는 내장 Heap 클래스를 제공하지 않는다\n</code></pre> heap.ts<pre><code>// TypeScript는 내장 Heap 클래스를 제공하지 않는다\n</code></pre> heap.dart<pre><code>// Dart는 내장 Heap 클래스를 제공하지 않는다\n</code></pre> heap.rs<pre><code>use std::collections::BinaryHeap;\nuse std::cmp::Reverse;\n\n/* 힙 초기화 */\n// 최소 힙 초기화\nlet mut min_heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n// 최대 힙 초기화\nlet mut max_heap = BinaryHeap::new();\n\n/* 힙에 원소 삽입 */\nmax_heap.push(1);\nmax_heap.push(3);\nmax_heap.push(2);\nmax_heap.push(5);\nmax_heap.push(4);\n\n/* 힙의 루트 원소 구하기 */\nlet peek = max_heap.peek().unwrap();  // 5\n\n/* 힙의 루트 원소 꺼내기 */\n// 꺼낸 원소들은 큰 값부터 작은 값 순서로 나열된다\nlet peek = max_heap.pop().unwrap();   // 5\nlet peek = max_heap.pop().unwrap();   // 4\nlet peek = max_heap.pop().unwrap();   // 3\nlet peek = max_heap.pop().unwrap();   // 2\nlet peek = max_heap.pop().unwrap();   // 1\n\n/* 힙의 크기 구하기 */\nlet size = max_heap.len();\n\n/* 힙이 비어 있는지 확인 */\nlet is_empty = max_heap.is_empty();\n\n/* 리스트를 입력받아 힙 구성 */\nlet min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]);\n</code></pre> heap.c<pre><code>// C는 내장 Heap 클래스를 제공하지 않는다\n</code></pre> heap.kt<pre><code>/* 힙 초기화 */\n// 최소 힙 초기화\nvar minHeap = PriorityQueue&lt;Int&gt;()\n// 최대 힙 초기화(lambda 식으로 Comparator를 수정하면 된다)\nval maxHeap = PriorityQueue { a: Int, b: Int -&gt; b - a }\n\n/* 힙에 원소 삽입 */\nmaxHeap.offer(1)\nmaxHeap.offer(3)\nmaxHeap.offer(2)\nmaxHeap.offer(5)\nmaxHeap.offer(4)\n\n/* 힙의 루트 원소 구하기 */\nvar peek = maxHeap.peek() // 5\n\n/* 힙의 루트 원소 꺼내기 */\n// 꺼낸 원소들은 큰 값부터 작은 값 순서로 나열된다\npeek = maxHeap.poll() // 5\npeek = maxHeap.poll() // 4\npeek = maxHeap.poll() // 3\npeek = maxHeap.poll() // 2\npeek = maxHeap.poll() // 1\n\n/* 힙의 크기 구하기 */\nval size = maxHeap.size\n\n/* 힙이 비어 있는지 확인 */\nval isEmpty = maxHeap.isEmpty()\n\n/* 리스트를 입력받아 힙 구성 */\nminHeap = PriorityQueue(mutableListOf(1, 3, 2, 5, 4))\n</code></pre> heap.rb<pre><code># Ruby는 내장 Heap 클래스를 제공하지 않는다\n</code></pre> 실행 시각화 <p> 전체 화면 &gt;</p>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/heap/#812","level":2,"title":"8.1.2   힙의 구현","text":"<p>아래에서는 최대 힙을 구현한다. 이를 최소 힙으로 바꾸려면 모든 대소 비교 조건을 반대로 바꾸면 된다(예를 들어 \\(\\geq\\)를 \\(\\leq\\)로 바꾼다). 관심 있는 독자는 직접 구현해 보자.</p>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/heap/#1","level":3,"title":"1.   힙의 저장과 표현","text":"<p>'이진 트리' 장에서 살펴보았듯이 완전 이진 트리는 배열로 표현하기에 매우 적합하다. 힙은 완전 이진 트리이므로 배열을 사용하여 힙을 저장한다.</p> <p>배열로 이진 트리를 표현할 때 원소는 노드의 값을, 인덱스는 이진 트리에서 노드의 위치를 나타낸다. 노드 포인터는 인덱스 매핑 공식으로 구현한다.</p> <p>아래 그림(8-2)과 같이 인덱스 \\(i\\)가 주어지면 왼쪽 자식 노드의 인덱스는 \\(2i + 1\\), 오른쪽 자식 노드의 인덱스는 \\(2i + 2\\), 부모 노드의 인덱스는 \\((i - 1) / 2\\)(내림 정수 나눗셈)이다. 인덱스가 범위를 벗어나면 빈 노드이거나 노드가 존재하지 않는다는 뜻이다.</p> <p></p> <p> 그림 8-2   힙의 표현과 저장 </p> <p>나중에 편리하게 사용할 수 있도록 인덱스 매핑 공식을 함수로 만들 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def left(self, i: int) -&gt; int:\n    \"\"\"왼쪽 자식 노드의 인덱스 반환\"\"\"\n    return 2 * i + 1\n\ndef right(self, i: int) -&gt; int:\n    \"\"\"오른쪽 자식 노드의 인덱스 반환\"\"\"\n    return 2 * i + 2\n\ndef parent(self, i: int) -&gt; int:\n    \"\"\"부모 노드의 인덱스 반환\"\"\"\n    return (i - 1) // 2  # 나눗셈의 결과를 내림한다\n</code></pre> my_heap.cpp<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* 부모 노드의 인덱스 반환 */\nint parent(int i) {\n    return (i - 1) / 2; // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.java<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* 부모 노드의 인덱스 반환 */\nint parent(int i) {\n    return (i - 1) / 2; // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.cs<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nint Left(int i) {\n    return 2 * i + 1;\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nint Right(int i) {\n    return 2 * i + 2;\n}\n\n/* 부모 노드의 인덱스 반환 */\nint Parent(int i) {\n    return (i - 1) / 2; // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.go<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nfunc (h *maxHeap) left(i int) int {\n    return 2*i + 1\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nfunc (h *maxHeap) right(i int) int {\n    return 2*i + 2\n}\n\n/* 부모 노드의 인덱스 반환 */\nfunc (h *maxHeap) parent(i int) int {\n    // 나눗셈의 결과를 내림한다\n    return (i - 1) / 2\n}\n</code></pre> my_heap.swift<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nfunc left(i: Int) -&gt; Int {\n    2 * i + 1\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nfunc right(i: Int) -&gt; Int {\n    2 * i + 2\n}\n\n/* 부모 노드의 인덱스 반환 */\nfunc parent(i: Int) -&gt; Int {\n    (i - 1) / 2 // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.js<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\n#left(i) {\n    return 2 * i + 1;\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\n#right(i) {\n    return 2 * i + 2;\n}\n\n/* 부모 노드의 인덱스 반환 */\n#parent(i) {\n    return Math.floor((i - 1) / 2); // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.ts<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nleft(i: number): number {\n    return 2 * i + 1;\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nright(i: number): number {\n    return 2 * i + 2;\n}\n\n/* 부모 노드의 인덱스 반환 */\nparent(i: number): number {\n    return Math.floor((i - 1) / 2); // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.dart<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nint _left(int i) {\n  return 2 * i + 1;\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nint _right(int i) {\n  return 2 * i + 2;\n}\n\n/* 부모 노드의 인덱스 반환 */\nint _parent(int i) {\n  return (i - 1) ~/ 2; // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.rs<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nfn left(i: usize) -&gt; usize {\n    2 * i + 1\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nfn right(i: usize) -&gt; usize {\n    2 * i + 2\n}\n\n/* 부모 노드의 인덱스 반환 */\nfn parent(i: usize) -&gt; usize {\n    (i - 1) / 2 // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.c<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nint left(MaxHeap *maxHeap, int i) {\n    return 2 * i + 1;\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nint right(MaxHeap *maxHeap, int i) {\n    return 2 * i + 2;\n}\n\n/* 부모 노드의 인덱스 반환 */\nint parent(MaxHeap *maxHeap, int i) {\n    return (i - 1) / 2; // 내림\n}\n</code></pre> my_heap.kt<pre><code>/* 왼쪽 자식 노드의 인덱스 반환 */\nfun left(i: Int): Int {\n    return 2 * i + 1\n}\n\n/* 오른쪽 자식 노드의 인덱스 반환 */\nfun right(i: Int): Int {\n    return 2 * i + 2\n}\n\n/* 부모 노드의 인덱스 반환 */\nfun parent(i: Int): Int {\n    return (i - 1) / 2 // 나눗셈의 결과를 내림한다\n}\n</code></pre> my_heap.rb<pre><code>### 왼쪽 자식 노드의 인덱스 반환 ###\ndef left(i)\n  2 * i + 1\nend\n\n### 오른쪽 자식 노드의 인덱스 반환 ###\ndef right(i)\n  2 * i + 2\nend\n\n### 부모 노드의 인덱스 반환 ###\ndef parent(i)\n  (i - 1) / 2     # 나눗셈의 결과를 내림한다\nend\n</code></pre>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/heap/#2","level":3,"title":"2.   힙의 루트 원소에 접근","text":"<p>힙의 루트 원소는 이진 트리의 루트 노드, 즉 리스트의 첫 번째 원소다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def peek(self) -&gt; int:\n    \"\"\"힙의 루트 원소 접근\"\"\"\n    return self.max_heap[0]\n</code></pre> my_heap.cpp<pre><code>/* 힙의 루트 원소 접근 */\nint peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.java<pre><code>/* 힙의 루트 원소 접근 */\nint peek() {\n    return maxHeap.get(0);\n}\n</code></pre> my_heap.cs<pre><code>/* 힙의 루트 원소 접근 */\nint Peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.go<pre><code>/* 힙의 루트 원소 접근 */\nfunc (h *maxHeap) peek() any {\n    return h.data[0]\n}\n</code></pre> my_heap.swift<pre><code>/* 힙의 루트 원소 접근 */\nfunc peek() -&gt; Int {\n    maxHeap[0]\n}\n</code></pre> my_heap.js<pre><code>/* 힙의 루트 원소 접근 */\npeek() {\n    return this.#maxHeap[0];\n}\n</code></pre> my_heap.ts<pre><code>/* 힙의 루트 원소 접근 */\npeek(): number {\n    return this.maxHeap[0];\n}\n</code></pre> my_heap.dart<pre><code>/* 힙의 루트 원소 접근 */\nint peek() {\n  return _maxHeap[0];\n}\n</code></pre> my_heap.rs<pre><code>/* 힙의 루트 원소 접근 */\nfn peek(&amp;self) -&gt; Option&lt;i32&gt; {\n    self.max_heap.first().copied()\n}\n</code></pre> my_heap.c<pre><code>/* 힙의 루트 원소 접근 */\nint peek(MaxHeap *maxHeap) {\n    return maxHeap-&gt;data[0];\n}\n</code></pre> my_heap.kt<pre><code>/* 힙의 루트 원소 접근 */\nfun peek(): Int {\n    return maxHeap[0]\n}\n</code></pre> my_heap.rb<pre><code>### 힙의 루트 원소 접근 ###\ndef peek\n  @max_heap[0]\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/heap/#3","level":3,"title":"3.   힙에 원소 삽입","text":"<p>원소 <code>val</code>이 주어지면 먼저 힙의 끝에 추가한다. 추가한 <code>val</code>이 힙의 다른 원소보다 클 수 있으므로, 힙의 성립 조건이 깨질 수 있다. 따라서 삽입한 노드에서 루트 노드까지의 경로에 있는 노드들을 조정해야 한다. 이 연산을 힙 재정렬(heapify)이라고 한다.</p> <p>삽입한 노드부터 아래에서 위로 힙을 재정렬해 보자. 아래 그림(8-3)과 같이 삽입한 노드의 값과 부모 노드의 값을 비교하여 삽입한 노드가 더 크면 서로 바꾼다. 이 작업을 반복하여 아래에서 위로 힙의 노드들을 조정하고, 루트 노드를 지나거나 바꿀 필요가 없는 노드를 만나면 종료한다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 8-3   힙에 원소를 삽입하는 단계 </p> <p>전체 노드 수가 \\(n\\)이면 트리의 높이는 \\(O(\\log n)\\)이다. 따라서 힙 재정렬의 반복 횟수는 최대 \\(O(\\log n)\\)이며, 힙에 원소를 삽입하는 연산의 시간 복잡도는 \\(O(\\log n)\\)이다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def push(self, val: int):\n    \"\"\"힙에 원소 삽입\"\"\"\n    # 노드 추가\n    self.max_heap.append(val)\n    # 아래에서 위로 힙을 재정렬한다\n    self.sift_up(self.size() - 1)\n\ndef sift_up(self, i: int):\n    \"\"\"노드 i에서 시작하여 아래에서 위로 힙 재정렬\"\"\"\n    while True:\n        # 노드 i의 부모 노드를 구한다\n        p = self.parent(i)\n        # 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if p &lt; 0 or self.max_heap[i] &lt;= self.max_heap[p]:\n            break\n        # 두 노드를 교환한다\n        self.swap(i, p)\n        # 위쪽으로 힙 재정렬을 반복한다\n        i = p\n</code></pre> my_heap.cpp<pre><code>/* 힙에 원소 삽입 */\nvoid push(int val) {\n    // 노드 추가\n    maxHeap.push_back(val);\n    // 아래에서 위로 힙을 재정렬한다\n    siftUp(size() - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nvoid siftUp(int i) {\n    while (true) {\n        // 노드 i의 부모 노드를 구한다\n        int p = parent(i);\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // 두 노드를 교환한다\n        swap(maxHeap[i], maxHeap[p]);\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* 힙에 원소 삽입 */\nvoid push(int val) {\n    // 노드 추가\n    maxHeap.add(val);\n    // 아래에서 위로 힙을 재정렬한다\n    siftUp(size() - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nvoid siftUp(int i) {\n    while (true) {\n        // 노드 i의 부모 노드를 구한다\n        int p = parent(i);\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if (p &lt; 0 || maxHeap.get(i) &lt;= maxHeap.get(p))\n            break;\n        // 두 노드를 교환한다\n        swap(i, p);\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* 힙에 원소 삽입 */\nvoid Push(int val) {\n    // 노드 추가\n    maxHeap.Add(val);\n    // 아래에서 위로 힙을 재정렬한다\n    SiftUp(Size() - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nvoid SiftUp(int i) {\n    while (true) {\n        // 노드 i의 부모 노드를 구한다\n        int p = Parent(i);\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 종료한다\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // 두 노드를 교환한다\n        Swap(i, p);\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* 힙에 원소 삽입 */\nfunc (h *maxHeap) push(val any) {\n    // 노드 추가\n    h.data = append(h.data, val)\n    // 아래에서 위로 힙을 재정렬한다\n    h.siftUp(len(h.data) - 1)\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nfunc (h *maxHeap) siftUp(i int) {\n    for true {\n        // 노드 i의 부모 노드를 구한다\n        p := h.parent(i)\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if p &lt; 0 || h.data[i].(int) &lt;= h.data[p].(int) {\n            break\n        }\n        // 두 노드를 교환한다\n        h.swap(i, p)\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* 힙에 원소 삽입 */\nfunc push(val: Int) {\n    // 노드 추가\n    maxHeap.append(val)\n    // 아래에서 위로 힙을 재정렬한다\n    siftUp(i: size() - 1)\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nfunc siftUp(i: Int) {\n    var i = i\n    while true {\n        // 노드 i의 부모 노드를 구한다\n        let p = parent(i: i)\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if p &lt; 0 || maxHeap[i] &lt;= maxHeap[p] {\n            break\n        }\n        // 두 노드를 교환한다\n        swap(i: i, j: p)\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p\n    }\n}\n</code></pre> my_heap.js<pre><code>/* 힙에 원소 삽입 */\npush(val) {\n    // 노드 추가\n    this.#maxHeap.push(val);\n    // 아래에서 위로 힙을 재정렬한다\n    this.#siftUp(this.size() - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\n#siftUp(i) {\n    while (true) {\n        // 노드 i의 부모 노드를 구한다\n        const p = this.#parent(i);\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if (p &lt; 0 || this.#maxHeap[i] &lt;= this.#maxHeap[p]) break;\n        // 두 노드를 교환한다\n        this.#swap(i, p);\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* 힙에 원소 삽입 */\npush(val: number): void {\n    // 노드 추가\n    this.maxHeap.push(val);\n    // 아래에서 위로 힙을 재정렬한다\n    this.siftUp(this.size() - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nsiftUp(i: number): void {\n    while (true) {\n        // 노드 i의 부모 노드를 구한다\n        const p = this.parent(i);\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if (p &lt; 0 || this.maxHeap[i] &lt;= this.maxHeap[p]) break;\n        // 두 노드를 교환한다\n        this.swap(i, p);\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* 힙에 원소 삽입 */\nvoid push(int val) {\n  // 노드 추가\n  _maxHeap.add(val);\n  // 아래에서 위로 힙을 재정렬한다\n  siftUp(size() - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nvoid siftUp(int i) {\n  while (true) {\n    // 노드 i의 부모 노드를 구한다\n    int p = _parent(i);\n    // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n    if (p &lt; 0 || _maxHeap[i] &lt;= _maxHeap[p]) {\n      break;\n    }\n    // 두 노드를 교환한다\n    _swap(i, p);\n    // 위쪽으로 힙 재정렬을 반복한다\n    i = p;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* 힙에 원소 삽입 */\nfn push(&amp;mut self, val: i32) {\n    // 노드 추가\n    self.max_heap.push(val);\n    // 아래에서 위로 힙을 재정렬한다\n    self.sift_up(self.size() - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nfn sift_up(&amp;mut self, mut i: usize) {\n    loop {\n        // 노드 i가 이미 힙의 루트이므로 힙 재정렬을 마친다\n        if i == 0 {\n            break;\n        }\n        // 노드 i의 부모 노드를 구한다\n        let p = Self::parent(i);\n        // 노드를 조정할 필요가 없으면 힙 재정렬을 마친다\n        if self.max_heap[i] &lt;= self.max_heap[p] {\n            break;\n        }\n        // 두 노드를 교환한다\n        self.swap(i, p);\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* 힙에 원소 삽입 */\nvoid push(MaxHeap *maxHeap, int val) {\n    // 기본적으로 이렇게 많은 노드를 추가해서는 안 된다\n    if (maxHeap-&gt;size == MAX_SIZE) {\n        printf(\"힙이 가득 찼다!\");\n        return;\n    }\n    // 노드 추가\n    maxHeap-&gt;data[maxHeap-&gt;size] = val;\n    maxHeap-&gt;size++;\n\n    // 아래에서 위로 힙을 재정렬한다\n    siftUp(maxHeap, maxHeap-&gt;size - 1);\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nvoid siftUp(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // 노드 i의 부모 노드를 구한다\n        int p = parent(maxHeap, i);\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if (p &lt; 0 || maxHeap-&gt;data[i] &lt;= maxHeap-&gt;data[p]) {\n            break;\n        }\n        // 두 노드를 교환한다\n        swap(maxHeap, i, p);\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* 힙에 원소 삽입 */\nfun push(_val: Int) {\n    // 노드 추가\n    maxHeap.add(_val)\n    // 아래에서 위로 힙을 재정렬한다\n    siftUp(size() - 1)\n}\n\n/* 노드 i에서 시작하여 아래에서 위로 힙 재정렬 */\nfun siftUp(it: Int) {\n    // Kotlin의 함수 매개변수는 변경할 수 없으므로 임시 변수를 생성한다\n    var i = it\n    while (true) {\n        // 노드 i의 부모 노드를 구한다\n        val p = parent(i)\n        // 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p]) break\n        // 두 노드를 교환한다\n        swap(i, p)\n        // 위쪽으로 힙 재정렬을 반복한다\n        i = p\n    }\n}\n</code></pre> my_heap.rb<pre><code>### 힙에 원소 삽입 ###\ndef push(val)\n  # 노드 추가\n  @max_heap &lt;&lt; val\n  # 아래에서 위로 힙을 재정렬한다\n  sift_up(size - 1)\nend\n\n### 노드 i에서 시작하여 아래에서 위로 힙 재정렬 ###\ndef sift_up(i)\n  loop do\n    # 노드 i의 부모 노드를 구한다\n    p = parent(i)\n    # 루트 노드를 넘어가거나 노드를 수정할 필요가 없으면 힙 재정렬을 마친다\n    break if p &lt; 0 || @max_heap[i] &lt;= @max_heap[p]\n    # 두 노드를 교환한다\n    swap(i, p)\n    # 위쪽으로 힙 재정렬을 반복한다\n    i = p\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/heap/#4","level":3,"title":"4.   힙의 루트 원소 꺼내기","text":"<p>힙의 루트 원소는 이진 트리의 루트 노드, 즉 리스트의 첫 번째 원소다. 리스트에서 첫 번째 원소를 바로 삭제하면 이진 트리의 모든 노드 인덱스가 바뀌어, 이후 힙 재정렬로 복구하기가 어려워진다. 원소 인덱스의 변화를 최소화하기 위해 다음 절차를 사용한다.</p> <ol> <li>힙의 루트 원소와 끝 원소를 교환한다(루트 노드와 맨 오른쪽 리프 노드를 교환한다).</li> <li>교환 후 리스트에서 힙의 끝 원소를 삭제한다(이미 교환했으므로 실제로 삭제되는 것은 원래의 루트 원소다).</li> <li>루트 노드부터 위에서 아래로 힙을 재정렬한다.</li> </ol> <p>아래 그림(8-4)과 같이 '위에서 아래로 힙 재정렬'은 '아래에서 위로 힙 재정렬'과 진행 방향이 반대다. 루트 노드의 값을 두 자식 노드의 값과 비교하여, 더 큰 값을 가진 자식 노드와 루트 노드를 교환한다. 이 작업을 반복하며 리프 노드를 지나거나 교환할 필요가 없는 노드를 만나면 종료한다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 8-4   힙의 루트 원소를 꺼내는 단계 </p> <p>원소 삽입과 마찬가지로 힙의 루트 원소를 꺼내는 연산의 시간 복잡도도 \\(O(\\log n)\\)이다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby my_heap.py<pre><code>def pop(self) -&gt; int:\n    \"\"\"힙에서 원소 꺼내기\"\"\"\n    # 비어 있는 경우를 처리한다\n    if self.is_empty():\n        raise IndexError(\"힙이 비어 있다\")\n    # 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    self.swap(0, self.size() - 1)\n    # 노드 삭제\n    val = self.max_heap.pop()\n    # 위에서 아래로 힙을 재정렬한다\n    self.sift_down(0)\n    # 힙의 루트 원소를 반환한다\n    return val\n\ndef sift_down(self, i: int):\n    \"\"\"노드 i에서 시작하여 위에서 아래로 힙 재정렬\"\"\"\n    while True:\n        # 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        l, r, ma = self.left(i), self.right(i), i\n        if l &lt; self.size() and self.max_heap[l] &gt; self.max_heap[ma]:\n            ma = l\n        if r &lt; self.size() and self.max_heap[r] &gt; self.max_heap[ma]:\n            ma = r\n        # 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if ma == i:\n            break\n        # 두 노드를 교환한다\n        self.swap(i, ma)\n        # 아래쪽으로 힙 재정렬을 반복한다\n        i = ma\n</code></pre> my_heap.cpp<pre><code>/* 힙에서 원소 꺼내기 */\nvoid pop() {\n    // 비어 있는 경우를 처리한다\n    if (isEmpty()) {\n        throw out_of_range(\"힙이 비어 있다\");\n    }\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    swap(maxHeap[0], maxHeap[size() - 1]);\n    // 노드 삭제\n    maxHeap.pop_back();\n    // 위에서 아래로 힙을 재정렬한다\n    siftDown(0);\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nvoid siftDown(int i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i)\n            break;\n        swap(maxHeap[i], maxHeap[ma]);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* 힙에서 원소 꺼내기 */\nint pop() {\n    // 비어 있는 경우를 처리한다\n    if (isEmpty())\n        throw new IndexOutOfBoundsException();\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    swap(0, size() - 1);\n    // 노드 삭제\n    int val = maxHeap.remove(size() - 1);\n    // 위에서 아래로 힙을 재정렬한다\n    siftDown(0);\n    // 힙의 루트 원소를 반환한다\n    return val;\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nvoid siftDown(int i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap.get(l) &gt; maxHeap.get(ma))\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap.get(r) &gt; maxHeap.get(ma))\n            ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i)\n            break;\n        // 두 노드를 교환한다\n        swap(i, ma);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* 힙에서 원소 꺼내기 */\nint Pop() {\n    // 비어 있는 경우를 처리한다\n    if (IsEmpty())\n        throw new IndexOutOfRangeException();\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    Swap(0, Size() - 1);\n    // 노드 삭제\n    int val = maxHeap.Last();\n    maxHeap.RemoveAt(Size() - 1);\n    // 위에서 아래로 힙을 재정렬한다\n    SiftDown(0);\n    // 힙의 루트 원소를 반환한다\n    return val;\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nvoid SiftDown(int i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        int l = Left(i), r = Right(i), ma = i;\n        if (l &lt; Size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; Size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // 노드 i의 값이 가장 크거나 리프 노드를 넘어가면 힙 재정렬을 종료한다\n        if (ma == i) break;\n        // 두 노드를 교환한다\n        Swap(i, ma);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* 힙에서 원소 꺼내기 */\nfunc (h *maxHeap) pop() any {\n    // 비어 있는 경우를 처리한다\n    if h.isEmpty() {\n        fmt.Println(\"오류\")\n        return nil\n    }\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    h.swap(0, h.size()-1)\n    // 노드 삭제\n    val := h.data[len(h.data)-1]\n    h.data = h.data[:len(h.data)-1]\n    // 위에서 아래로 힙을 재정렬한다\n    h.siftDown(0)\n\n    // 힙의 루트 원소를 반환한다\n    return val\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nfunc (h *maxHeap) siftDown(i int) {\n    for true {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 max로 표시한다\n        l, r, max := h.left(i), h.right(i), i\n        if l &lt; h.size() &amp;&amp; h.data[l].(int) &gt; h.data[max].(int) {\n            max = l\n        }\n        if r &lt; h.size() &amp;&amp; h.data[r].(int) &gt; h.data[max].(int) {\n            max = r\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if max == i {\n            break\n        }\n        // 두 노드를 교환한다\n        h.swap(i, max)\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = max\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* 힙에서 원소 꺼내기 */\nfunc pop() -&gt; Int {\n    // 비어 있는 경우를 처리한다\n    if isEmpty() {\n        fatalError(\"힙이 비어 있다\")\n    }\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    swap(i: 0, j: size() - 1)\n    // 노드 삭제\n    let val = maxHeap.remove(at: size() - 1)\n    // 위에서 아래로 힙을 재정렬한다\n    siftDown(i: 0)\n    // 힙의 루트 원소를 반환한다\n    return val\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nfunc siftDown(i: Int) {\n    var i = i\n    while true {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        let l = left(i: i)\n        let r = right(i: i)\n        var ma = i\n        if l &lt; size(), maxHeap[l] &gt; maxHeap[ma] {\n            ma = l\n        }\n        if r &lt; size(), maxHeap[r] &gt; maxHeap[ma] {\n            ma = r\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if ma == i {\n            break\n        }\n        // 두 노드를 교환한다\n        swap(i: i, j: ma)\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma\n    }\n}\n</code></pre> my_heap.js<pre><code>/* 힙에서 원소 꺼내기 */\npop() {\n    // 비어 있는 경우를 처리한다\n    if (this.isEmpty()) throw new Error('힙이 비어 있다');\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    this.#swap(0, this.size() - 1);\n    // 노드 삭제\n    const val = this.#maxHeap.pop();\n    // 위에서 아래로 힙을 재정렬한다\n    this.#siftDown(0);\n    // 힙의 루트 원소를 반환한다\n    return val;\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\n#siftDown(i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        const l = this.#left(i),\n            r = this.#right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.#maxHeap[l] &gt; this.#maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.#maxHeap[r] &gt; this.#maxHeap[ma]) ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma === i) break;\n        // 두 노드를 교환한다\n        this.#swap(i, ma);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* 힙에서 원소 꺼내기 */\npop(): number {\n    // 비어 있는 경우를 처리한다\n    if (this.isEmpty()) throw new RangeError('힙이 비어 있다.');\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    this.swap(0, this.size() - 1);\n    // 노드 삭제\n    const val = this.maxHeap.pop();\n    // 위에서 아래로 힙을 재정렬한다\n    this.siftDown(0);\n    // 힙의 루트 원소를 반환한다\n    return val;\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nsiftDown(i: number): void {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        const l = this.left(i),\n            r = this.right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.maxHeap[l] &gt; this.maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.maxHeap[r] &gt; this.maxHeap[ma]) ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma === i) break;\n        // 두 노드를 교환한다\n        this.swap(i, ma);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* 힙에서 원소 꺼내기 */\nint pop() {\n  // 비어 있는 경우를 처리한다\n  if (isEmpty()) throw Exception('힙이 비어 있다');\n  // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n  _swap(0, size() - 1);\n  // 노드 삭제\n  int val = _maxHeap.removeLast();\n  // 위에서 아래로 힙을 재정렬한다\n  siftDown(0);\n  // 힙의 루트 원소를 반환한다\n  return val;\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nvoid siftDown(int i) {\n  while (true) {\n    // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n    int l = _left(i);\n    int r = _right(i);\n    int ma = i;\n    if (l &lt; size() &amp;&amp; _maxHeap[l] &gt; _maxHeap[ma]) ma = l;\n    if (r &lt; size() &amp;&amp; _maxHeap[r] &gt; _maxHeap[ma]) ma = r;\n    // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n    if (ma == i) break;\n    // 두 노드를 교환한다\n    _swap(i, ma);\n    // 아래쪽으로 힙 재정렬을 반복한다\n    i = ma;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* 힙에서 원소 꺼내기 */\nfn pop(&amp;mut self) -&gt; i32 {\n    // 비어 있는 경우를 처리한다\n    if self.is_empty() {\n        panic!(\"인덱스가 범위를 벗어났다\");\n    }\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    self.swap(0, self.size() - 1);\n    // 노드 삭제\n    let val = self.max_heap.pop().unwrap();\n    // 위에서 아래로 힙을 재정렬한다\n    self.sift_down(0);\n    // 힙의 루트 원소를 반환한다\n    val\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nfn sift_down(&amp;mut self, mut i: usize) {\n    loop {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        let (l, r, mut ma) = (Self::left(i), Self::right(i), i);\n        if l &lt; self.size() &amp;&amp; self.max_heap[l] &gt; self.max_heap[ma] {\n            ma = l;\n        }\n        if r &lt; self.size() &amp;&amp; self.max_heap[r] &gt; self.max_heap[ma] {\n            ma = r;\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if ma == i {\n            break;\n        }\n        // 두 노드를 교환한다\n        self.swap(i, ma);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* 힙에서 원소 꺼내기 */\nint pop(MaxHeap *maxHeap) {\n    // 비어 있는 경우를 처리한다\n    if (isEmpty(maxHeap)) {\n        printf(\"힙이 비어 있다!\");\n        return INT_MAX;\n    }\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    swap(maxHeap, 0, size(maxHeap) - 1);\n    // 노드 삭제\n    int val = maxHeap-&gt;data[maxHeap-&gt;size - 1];\n    maxHeap-&gt;size--;\n    // 위에서 아래로 힙을 재정렬한다\n    siftDown(maxHeap, 0);\n\n    // 힙의 루트 원소를 반환한다\n    return val;\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nvoid siftDown(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 max로 표시한다\n        int l = left(maxHeap, i);\n        int r = right(maxHeap, i);\n        int max = i;\n        if (l &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[l] &gt; maxHeap-&gt;data[max]) {\n            max = l;\n        }\n        if (r &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[r] &gt; maxHeap-&gt;data[max]) {\n            max = r;\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (max == i) {\n            break;\n        }\n        // 두 노드를 교환한다\n        swap(maxHeap, i, max);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = max;\n    }\n}\n</code></pre> my_heap.kt<pre><code>/* 힙에서 원소 꺼내기 */\nfun pop(): Int {\n    // 비어 있는 경우를 처리한다\n    if (isEmpty()) throw IndexOutOfBoundsException()\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    swap(0, size() - 1)\n    // 노드 삭제\n    val _val = maxHeap.removeAt(size() - 1)\n    // 위에서 아래로 힙을 재정렬한다\n    siftDown(0)\n    // 힙의 루트 원소를 반환한다\n    return _val\n}\n\n/* 노드 i에서 시작하여 위에서 아래로 힙 재정렬 */\nfun siftDown(it: Int) {\n    // Kotlin의 함수 매개변수는 변경할 수 없으므로 임시 변수를 생성한다\n    var i = it\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        val l = left(i)\n        val r = right(i)\n        var ma = i\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma]) ma = l\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma]) ma = r\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i) break\n        // 두 노드를 교환한다\n        swap(i, ma)\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma\n    }\n}\n</code></pre> my_heap.rb<pre><code>### 힙에서 원소 꺼내기 ###\ndef pop\n  # 비어 있는 경우를 처리한다\n  raise IndexError, \"힙이 비어 있다\" if is_empty?\n  # 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n  swap(0, size - 1)\n  # 노드 삭제\n  val = @max_heap.pop\n  # 위에서 아래로 힙을 재정렬한다\n  sift_down(0)\n  # 힙의 루트 원소를 반환한다\n  val\nend\n\n### 노드 i에서 시작하여 위에서 아래로 힙 재정렬 ###\ndef sift_down(i)\n  loop do\n    # 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n    l, r, ma = left(i), right(i), i\n    ma = l if l &lt; size &amp;&amp; @max_heap[l] &gt; @max_heap[ma]\n    ma = r if r &lt; size &amp;&amp; @max_heap[r] &gt; @max_heap[ma]\n\n    # 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n    break if ma == i\n\n    # 두 노드를 교환한다\n    swap(i, ma)\n    # 아래쪽으로 힙 재정렬을 반복한다\n    i = ma\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/heap/#813","level":2,"title":"8.1.3   힙의 주요 응용","text":"<ul> <li>우선순위 큐: 힙은 우선순위 큐를 구현할 때 가장 먼저 고려하는 자료 구조다. 삽입과 꺼내기 연산의 시간 복잡도는 모두 \\(O(\\log n)\\)이고, 힙 구성 연산은 \\(O(n)\\)으로 모두 매우 효율적이다.</li> <li>힙 정렬: 데이터가 주어지면 이를 이용해 힙을 만든 뒤 원소를 계속 꺼내어 정렬된 데이터를 얻을 수 있다. 다만 힙 정렬은 보통 더 간결한 방법으로 구현한다. 자세한 내용은 '힙 정렬' 절을 참고하자.</li> <li>가장 큰 \\(k\\) 개 원소 구하기: 대표적인 알고리즘 문제이자 흔한 응용 사례다. 예를 들어 인기 상위 10개 뉴스를 웨이보의 인기 검색어로 선정하거나, 판매량 상위 10개 상품을 고를 때 사용할 수 있다.</li> </ul>","path":["제 8 장   힙","8.1   힙"],"tags":[]},{"location":"chapter_heap/summary/","level":1,"title":"8.4   요약","text":"","path":["제 8 장   힙","8.4   요약"],"tags":[]},{"location":"chapter_heap/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>힙은 완전 이진 트리이며, 만족하는 조건에 따라 최대 힙과 최소 힙으로 나뉜다. 최대(최소) 힙의 루트 원소는 가장 큰(작은) 값이다.</li> <li>우선순위 큐는 원소를 꺼내는 우선순위가 있는 큐로 정의되며, 보통 힙으로 구현한다.</li> <li>힙의 주요 연산과 시간 복잡도는 원소 삽입 \\(O(\\log n)\\), 루트 원소 꺼내기 \\(O(\\log n)\\), 루트 원소 접근 \\(O(1)\\) 등이다.</li> <li>완전 이진 트리는 배열로 표현하기에 매우 적합하므로, 힙은 보통 배열에 저장한다.</li> <li>힙 재정렬 연산은 힙의 성질을 유지하는 데 사용하며, 삽입과 꺼내기 연산 모두에서 필요하다.</li> <li>\\(n\\) 개의 원소를 입력받아 힙을 구성하는 시간 복잡도는 \\(O(n)\\)까지 최적화할 수 있어 매우 효율적이다.</li> <li>Top-k는 대표적인 알고리즘 문제다. 힙 자료 구조를 사용하면 \\(O(n \\log k)\\)의 시간 복잡도로 효율적으로 해결할 수 있다.</li> </ul>","path":["제 8 장   힙","8.4   요약"],"tags":[]},{"location":"chapter_heap/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 자료 구조의 '힙'과 메모리 관리의 '힙'은 같은 개념인가?</p> <p>두 개념은 서로 다르며, 우연히 이름만 '힙'으로 같을 뿐이다. 컴퓨터 시스템 메모리의 힙은 동적 메모리 할당에 사용되는 영역으로, 프로그램이 실행 중에 데이터를 저장하는 데 사용할 수 있다. 프로그램은 객체나 배열 같은 복잡한 구조를 저장하기 위해 일정량의 힙 메모리를 요청할 수 있다. 데이터가 더 이상 필요하지 않으면 메모리 누수를 막기 위해 해당 메모리를 해제해야 한다. 스택 메모리에 비해 힙 메모리는 관리와 사용에 더 주의해야 하며, 잘못 사용하면 메모리 누수나 유효하지 않은 포인터 등의 문제가 발생할 수 있다.</p>","path":["제 8 장   힙","8.4   요약"],"tags":[]},{"location":"chapter_heap/top_k/","level":1,"title":"8.3   Top-k 문제","text":"<p>문제</p> <p>길이가 \\(n\\)인 정렬되지 않은 배열 <code>nums</code>가 주어졌을 때, 배열에서 가장 큰 \\(k\\) 개의 원소를 반환하라.</p> <p>이 문제에 대해 먼저 발상이 비교적 직접적인 두 가지 풀이를 살펴보고, 이어서 더 효율적인 힙 기반 풀이를 소개한다.</p>","path":["제 8 장   힙","8.3   Top-k 문제"],"tags":[]},{"location":"chapter_heap/top_k/#831-1","level":2,"title":"8.3.1   방법 1: 순회하며 선택","text":"<p>아래 그림(8-6)과 같이 \\(k\\) 번 순회하면서 매 순회에서 각각 \\(1\\), \\(2\\), \\(\\dots\\), \\(k\\) 번째로 큰 원소를 뽑을 수 있다. 시간 복잡도는 \\(O(nk)\\)이다.</p> <p>이 방법은 \\(k \\ll n\\)인 경우에만 적합하다. \\(k\\)가 \\(n\\)에 가까워지면 시간 복잡도가 \\(O(n^2)\\)에 가까워져 매우 오래 걸리기 때문이다.</p> <p></p> <p> 그림 8-6   순회로 가장 큰 k개의 원소 찾기 </p> <p>참고</p> <p>\\(k = n\\)이면 전체가 정렬된 수열을 얻을 수 있으며, 이는 '선택 정렬' 알고리즘과 같다.</p>","path":["제 8 장   힙","8.3   Top-k 문제"],"tags":[]},{"location":"chapter_heap/top_k/#832-2","level":2,"title":"8.3.2   방법 2: 정렬","text":"<p>아래 그림(8-7)과 같이 배열 <code>nums</code>를 먼저 정렬한 뒤 맨 오른쪽의 \\(k\\) 개 원소를 반환할 수 있다. 시간 복잡도는 \\(O(n \\log n)\\)이다.</p> <p>이 방법은 필요한 것보다 더 많은 일을 한다. 가장 큰 \\(k\\) 개 원소만 찾으면 되며, 나머지 원소를 정렬할 필요는 없기 때문이다.</p> <p></p> <p> 그림 8-7   정렬로 가장 큰 k개의 원소 찾기 </p>","path":["제 8 장   힙","8.3   Top-k 문제"],"tags":[]},{"location":"chapter_heap/top_k/#833-3","level":2,"title":"8.3.3   방법 3: 힙","text":"<p>힙을 이용하면 Top-k 문제를 더 효율적으로 해결할 수 있다. 과정은 아래 그림(8-8)과 같다.</p> <ol> <li>루트 원소가 가장 작은 최소 힙을 초기화한다.</li> <li>먼저 배열의 앞 \\(k\\) 개 원소를 차례로 힙에 삽입한다.</li> <li>\\(k + 1\\) 번째 원소부터 시작하여, 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 삽입한다.</li> <li>순회를 마치면 힙에 남은 원소가 가장 큰 \\(k\\) 개 원소다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 8-8   힙으로 가장 큰 k개의 원소 찾기 </p> <p>예시 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby top_k.py<pre><code>def top_k_heap(nums: list[int], k: int) -&gt; list[int]:\n    \"\"\"힙을 이용하여 배열에서 가장 큰 k개 원소 탐색\"\"\"\n    # 최소 힙 초기화\n    heap = []\n    # 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for i in range(k):\n        heapq.heappush(heap, nums[i])\n    # k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for i in range(k, len(nums)):\n        # 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if nums[i] &gt; heap[0]:\n            heapq.heappop(heap)\n            heapq.heappush(heap, nums[i])\n    return heap\n</code></pre> top_k.cpp<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; topKHeap(vector&lt;int&gt; &amp;nums, int k) {\n    // 최소 힙 초기화\n    priority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; heap;\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for (int i = 0; i &lt; k; i++) {\n        heap.push(nums[i]);\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for (int i = k; i &lt; nums.size(); i++) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if (nums[i] &gt; heap.top()) {\n            heap.pop();\n            heap.push(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.java<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nQueue&lt;Integer&gt; topKHeap(int[] nums, int k) {\n    // 최소 힙 초기화\n    Queue&lt;Integer&gt; heap = new PriorityQueue&lt;Integer&gt;();\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for (int i = 0; i &lt; k; i++) {\n        heap.offer(nums[i]);\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for (int i = k; i &lt; nums.length; i++) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll();\n            heap.offer(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.cs<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nPriorityQueue&lt;int, int&gt; TopKHeap(int[] nums, int k) {\n    // 최소 힙 초기화\n    PriorityQueue&lt;int, int&gt; heap = new();\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for (int i = 0; i &lt; k; i++) {\n        heap.Enqueue(nums[i], nums[i]);\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for (int i = k; i &lt; nums.Length; i++) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if (nums[i] &gt; heap.Peek()) {\n            heap.Dequeue();\n            heap.Enqueue(nums[i], nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.go<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nfunc topKHeap(nums []int, k int) *minHeap {\n    // 최소 힙 초기화\n    h := &amp;minHeap{}\n    heap.Init(h)\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for i := 0; i &lt; k; i++ {\n        heap.Push(h, nums[i])\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for i := k; i &lt; len(nums); i++ {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if nums[i] &gt; h.Top().(int) {\n            heap.Pop(h)\n            heap.Push(h, nums[i])\n        }\n    }\n    return h\n}\n</code></pre> top_k.swift<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nfunc topKHeap(nums: [Int], k: Int) -&gt; [Int] {\n    // 최소 힙을 초기화하여 앞쪽 k개 원소로 힙을 구성한다\n    var heap = Heap(nums.prefix(k))\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for i in nums.indices.dropFirst(k) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if nums[i] &gt; heap.min()! {\n            _ = heap.removeMin()\n            heap.insert(nums[i])\n        }\n    }\n    return heap.unordered\n}\n</code></pre> top_k.js<pre><code>/* 힙에 원소 삽입 */\nfunction pushMinHeap(maxHeap, val) {\n    // 원소의 부호를 반전한다\n    maxHeap.push(-val);\n}\n\n/* 힙에서 원소 꺼내기 */\nfunction popMinHeap(maxHeap) {\n    // 원소의 부호를 반전한다\n    return -maxHeap.pop();\n}\n\n/* 힙의 루트 원소 접근 */\nfunction peekMinHeap(maxHeap) {\n    // 원소의 부호를 반전한다\n    return -maxHeap.peek();\n}\n\n/* 힙의 원소를 반환한다 */\nfunction getMinHeap(maxHeap) {\n    // 원소의 부호를 반전한다\n    return maxHeap.getMaxHeap().map((num) =&gt; -num);\n}\n\n/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nfunction topKHeap(nums, k) {\n    // 최소 힙 초기화\n    // 힙의 모든 원소의 부호를 반전하여 최대 힙으로 최소 힙을 구현한다는 점에 유의한다\n    const maxHeap = new MaxHeap([]);\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for (let i = k; i &lt; nums.length; i++) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // 힙의 원소를 반환한다\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.ts<pre><code>/* 힙에 원소 삽입 */\nfunction pushMinHeap(maxHeap: MaxHeap, val: number): void {\n    // 원소의 부호를 반전한다\n    maxHeap.push(-val);\n}\n\n/* 힙에서 원소 꺼내기 */\nfunction popMinHeap(maxHeap: MaxHeap): number {\n    // 원소의 부호를 반전한다\n    return -maxHeap.pop();\n}\n\n/* 힙의 루트 원소 접근 */\nfunction peekMinHeap(maxHeap: MaxHeap): number {\n    // 원소의 부호를 반전한다\n    return -maxHeap.peek();\n}\n\n/* 힙의 원소를 반환한다 */\nfunction getMinHeap(maxHeap: MaxHeap): number[] {\n    // 원소의 부호를 반전한다\n    return maxHeap.getMaxHeap().map((num: number) =&gt; -num);\n}\n\n/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nfunction topKHeap(nums: number[], k: number): number[] {\n    // 최소 힙 초기화\n    // 힙의 모든 원소의 부호를 반전하여 최대 힙으로 최소 힙을 구현한다는 점에 유의한다\n    const maxHeap = new MaxHeap([]);\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for (let i = k; i &lt; nums.length; i++) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // 힙의 원소를 반환한다\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.dart<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nMinHeap topKHeap(List&lt;int&gt; nums, int k) {\n  // 최소 힙을 초기화하고 배열의 앞쪽 k개 원소를 힙에 삽입한다\n  MinHeap heap = MinHeap(nums.sublist(0, k));\n  // k+1번째 원소부터 힙의 길이를 k로 유지한다\n  for (int i = k; i &lt; nums.length; i++) {\n    // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n    if (nums[i] &gt; heap.peek()) {\n      heap.pop();\n      heap.push(nums[i]);\n    }\n  }\n  return heap;\n}\n</code></pre> top_k.rs<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nfn top_k_heap(nums: Vec&lt;i32&gt;, k: usize) -&gt; BinaryHeap&lt;Reverse&lt;i32&gt;&gt; {\n    // BinaryHeap은 최대 힙이므로 Reverse로 원소의 순서를 뒤집어 최소 힙을 구현한다\n    let mut heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for &amp;num in nums.iter().take(k) {\n        heap.push(Reverse(num));\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for &amp;num in nums.iter().skip(k) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if num &gt; heap.peek().unwrap().0 {\n            heap.pop();\n            heap.push(Reverse(num));\n        }\n    }\n    heap\n}\n</code></pre> top_k.c<pre><code>/* 힙에 원소 삽입 */\nvoid pushMinHeap(MaxHeap *maxHeap, int val) {\n    // 원소의 부호를 반전한다\n    push(maxHeap, -val);\n}\n\n/* 힙에서 원소 꺼내기 */\nint popMinHeap(MaxHeap *maxHeap) {\n    // 원소의 부호를 반전한다\n    return -pop(maxHeap);\n}\n\n/* 힙의 루트 원소 접근 */\nint peekMinHeap(MaxHeap *maxHeap) {\n    // 원소의 부호를 반전한다\n    return -peek(maxHeap);\n}\n\n/* 힙의 원소를 반환한다 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // 힙의 모든 원소의 부호를 반전하여 res 배열에 저장한다\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n/* 힙의 원소를 반환한다 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // 힙의 모든 원소의 부호를 반전하여 res 배열에 저장한다\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n// 힙으로 배열에서 가장 큰 k개 원소를 찾는 함수\nint *topKHeap(int *nums, int sizeNums, int k) {\n    // 최소 힙 초기화\n    // 힙의 모든 원소의 부호를 반전하여 최대 힙으로 최소 힙을 구현한다는 점에 유의한다\n    int *empty = (int *)malloc(0);\n    MaxHeap *maxHeap = newMaxHeap(empty, 0);\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for (int i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for (int i = k; i &lt; sizeNums; i++) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    int *res = getMinHeap(maxHeap);\n    // 메모리 해제\n    delMaxHeap(maxHeap);\n    return res;\n}\n</code></pre> top_k.kt<pre><code>/* 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 */\nfun topKHeap(nums: IntArray, k: Int): Queue&lt;Int&gt; {\n    // 최소 힙 초기화\n    val heap = PriorityQueue&lt;Int&gt;()\n    // 배열의 앞쪽 k개 원소를 힙에 삽입한다\n    for (i in 0..&lt;k) {\n        heap.offer(nums[i])\n    }\n    // k+1번째 원소부터 힙의 길이를 k로 유지한다\n    for (i in k..&lt;nums.size) {\n        // 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll()\n            heap.offer(nums[i])\n        }\n    }\n    return heap\n}\n</code></pre> top_k.rb<pre><code>### 힙을 이용하여 배열에서 가장 큰 k개 원소 탐색 ###\ndef top_k_heap(nums, k)\n  # 최소 힙 초기화\n  # 힙의 모든 원소의 부호를 반전하여 최대 힙으로 최소 힙을 구현한다는 점에 유의한다\n  max_heap = MaxHeap.new([])\n\n  # 배열의 앞쪽 k개 원소를 힙에 삽입한다\n  for i in 0...k\n    push_min_heap(max_heap, nums[i])\n  end\n\n  # k+1번째 원소부터 힙의 길이를 k로 유지한다\n  for i in k...nums.length\n    # 현재 원소가 힙의 루트 원소보다 크면 루트 원소를 꺼내고 현재 원소를 힙에 삽입한다\n    if nums[i] &gt; peek_min_heap(max_heap)\n      pop_min_heap(max_heap)\n      push_min_heap(max_heap, nums[i])\n    end\n  end\n\n  get_min_heap(max_heap)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>삽입과 꺼내기를 총 \\(n\\) 차례 수행하고 힙의 최대 길이는 \\(k\\)이므로, 시간 복잡도는 \\(O(n \\log k)\\)이다. 이 방법은 매우 효율적이다. \\(k\\)가 작으면 시간 복잡도가 \\(O(n)\\)에 가까워지고, \\(k\\)가 커도 \\(O(n \\log n)\\)을 넘지 않는다.</p> <p>또한 이 방법은 동적으로 들어오는 데이터 스트림에도 적합하다. 데이터가 계속 추가될 때 힙 안의 원소를 지속적으로 관리하면 가장 큰 \\(k\\) 개 원소를 동적으로 갱신할 수 있다.</p>","path":["제 8 장   힙","8.3   Top-k 문제"],"tags":[]},{"location":"chapter_hello_algo/","level":1,"title":"머리말","text":"<p>몇 년 전, 나는 리트코드에 『剑指 Offer』(IT 대기업 코딩 면접 대비서)의 문제 풀이 시리즈를 공유했고 많은 독자의 격려와 지지를 받았다. 독자들과 이야기하면서 가장 자주 받은 질문은 “알고리즘 공부를 어떻게 시작하면 좋을까요?”였다. 점차 나는 이 질문에 깊은 관심을 갖게 되었다.</p> <p>아무것도 모르는 상태에서 무작정 문제를 푸는 방식이 가장 인기 있는 방법인 듯하다. 간단하고 직접적이며 효과도 있다. 하지만 문제 풀이는 ‘지뢰 찾기’ 게임과 비슷하다. 스스로 학습하는 능력이 뛰어난 사람은 지뢰를 하나씩 찾아 제거하며 앞으로 나아가지만, 기초가 부족한 사람은 지뢰를 거듭 밟으며 좌절하고, 결국 앞으로 나아가지 못한 채 주저앉기 쉽다. 알고리즘 교재를 처음부터 끝까지 읽는 것도 흔한 방법이다. 그러나 취업을 준비하는 사람에게는 졸업 논문 작성, 이력서 제출, 필기시험과 면접 준비만으로도 대부분의 에너지가 소모되므로, 두꺼운 책을 파고드는 일이 큰 도전이 되곤 한다.</p> <p>비슷한 어려움을 겪고 있다면, 다행히도 이 책이 여러분을 '찾아왔다'고 말할 수 있을 것이다. 이 책은 앞서 던진 질문에 대한 나의 답이다. 최선의 해답은 아닐지라도, 적어도 앞으로 나아가는 데 도움이 되는 시도라고 생각한다. 물론 이 책 한 권만으로 곧바로 취업에 성공할 수는 없을 것이다. 하지만 자료 구조와 알고리즘의 '지식 지도'를 탐험하도록 안내하고, 여러 ‘지뢰’의 모양과 크기, 분포를 이해하며 다양한 ‘지뢰 제거 방법’을 익힐 수 있도록 도울 것이다. 이를 바탕으로 알고리즘 문제를 풀거나 기술 문헌을 읽는 일이 한층 수월해지고, 차근차근 체계적인 지식을 쌓아 나갈 수 있으리라 믿는다.</p> <p>나는 파인만 교수의 말에 깊이 공감한다. “지식은 공짜가 아니다. 주의를 기울여야 한다.” 이런 의미에서 이 책은 완전히 ‘무료’인 것은 아니다. 여러분이 이 책에 쏟는 소중한 ‘주의와 관심’을 헛되게 하지 않도록, 나 역시 최대한의 ‘주의와 관심’을 기울여 집필에 최선을 다하고자 한다.</p> <p>내 학식과 재능이 부족함을 잘 알고 있다. 책의 내용을 한동안 다듬어 왔지만 여전히 많은 오류가 있을 것이다. 선생님과 학생 여러분의 비판과 지적을 부탁드린다.</p> <p></p> Hello, 알고리즘! <p>컴퓨터의 등장은 세상을 크게 바꾸었다. 빠른 계산 능력과 뛰어난 프로그래밍 가능성을 갖춘 컴퓨터는 알고리즘을 실행하고 데이터를 처리하는 이상적인 수단이 되었다. 비디오 게임의 사실적인 화면, 자율 주행의 지능적인 판단, AlphaGo의 멋진 대국, ChatGPT의 자연스러운 상호작용까지, 이 모든 응용은 컴퓨터 위에서 알고리즘이 정교하게 구현된 결과다.</p> <p>사실 컴퓨터가 등장하기 전부터 알고리즘과 자료 구조는 세상 곳곳에 존재했다. 초기의 알고리즘은 고대의 셈법이나 도구 제작 절차처럼 비교적 단순했다. 문명이 발전하면서 알고리즘도 점점 더 정교하고 복잡해졌다. 장인의 놀라운 솜씨에서 생산력을 높여 준 산업 제품에 이르기까지, 나아가 우주의 다양한 현상을 설명하는 과학 법칙에 이르기까지, 평범하거나 경이로운 거의 모든 것의 이면에는 정교한 알고리즘적 사고가 숨어 있다.</p> <p>자료 구조 역시 어디에나 있다. 크게는 사회 관계망부터 작게는 지하철 노선까지 많은 시스템을 ‘그래프’로 모델링할 수 있다. 국가부터 가정에 이르기까지 사회의 주요 조직 형태는 ‘트리’의 특성을 보인다. 겨울옷은 ‘스택’과 같아서 가장 먼저 입은 옷을 가장 나중에 벗는다. 셔틀콕 통은 한쪽으로 넣고 다른 쪽으로 꺼낸다는 점에서 ‘큐’와 닮았다. 사전은 원하는 표제어를 빠르게 찾을 수 있다는 점에서 ‘해시 테이블’과 비슷하다.</p> <p>이 책은 명확하고 이해하기 쉬운 애니메이션 그림과 실행 가능한 코드 예제를 통해 독자가 알고리즘과 자료 구조의 핵심 개념을 이해하고 직접 프로그래밍으로 구현할 수 있도록 돕는 것을 목표로 한다. 이를 바탕으로 복잡한 세상 속에서 알고리즘이 생생하게 드러나는 모습을 보여 주며 알고리즘의 아름다움을 전하고자 한다. 이 책이 여러분에게 도움이 되기를 바란다!</p>","path":["머리말"],"tags":[]},{"location":"chapter_introduction/","level":1,"title":"제 1 장   알고리즘과의 첫 만남","text":"<p>이 장을 시작하며</p> <p>한 소녀가 데이터와 어우러져 우아하게 춤을 춘다. 치맛자락에는 알고리즘의 선율이 나부낀다.</p> <p>소녀가 함께 춤추자고 손을 내민다. 그 발걸음을 따라 논리와 아름다움으로 가득한 알고리즘의 세계에 들어서 보자.</p>","path":["제 1 장   알고리즘과의 첫 만남"],"tags":[]},{"location":"chapter_introduction/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>1.1   어디에나 있는 알고리즘</li> <li>1.2   알고리즘이란 무엇인가</li> <li>1.3   요약</li> </ul>","path":["제 1 장   알고리즘과의 첫 만남"],"tags":[]},{"location":"chapter_introduction/algorithms_are_everywhere/","level":1,"title":"1.1   어디에나 있는 알고리즘","text":"<p>‘알고리즘’이라는 말을 들으면 자연스럽게 수학을 떠올리게 된다. 하지만 실제로 많은 알고리즘은 복잡한 수학보다는 기본적인 논리에 더 많이 의존하며, 이러한 논리는 일상생활 곳곳에서 찾아볼 수 있다.</p> <p>알고리즘을 본격적으로 살펴보기 전에 흥미로운 사실 하나를 소개하고자 한다. 여러분은 이미 자신도 모르게 많은 알고리즘을 익혔고, 일상생활에서 자연스럽게 사용하고 있다. 몇 가지 구체적인 예를 통해 이 사실을 확인해 보자.</p> <p>예 1: 사전에서 단어 찾기. 중국어 사전에서는 각 한자에 대응하는 병음이 있고, 표제어는 병음의 알파벳순으로 배열되어 있다. 병음의 첫 글자가 \\(r\\)인 한자를 찾는다고 가정하면, 보통 아래 그림(1-1)과 같은 방법을 사용한다.</p> <ol> <li>사전의 중간쯤을 펼쳐 그 페이지의 병음 첫 글자를 확인한다. 첫 글자가 \\(m\\)이라고 가정하자.</li> <li>병음의 알파벳순에서 \\(r\\)은 \\(m\\) 뒤에 있으므로 사전의 앞쪽 절반을 제외하고 탐색 범위를 뒤쪽 절반으로 좁힌다.</li> <li>병음의 첫 글자가 \\(r\\)인 페이지를 찾을 때까지 단계 <code>1.</code>과 단계 <code>2.</code>를 반복한다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 1-1   사전에서 단어를 찾는 단계 </p> <p>초등학생도 익히는 사전 찾기는 사실 유명한 ‘이진 탐색’ 알고리즘이다. 자료 구조의 관점에서 사전은 정렬된 ‘배열’로 볼 수 있고, 알고리즘의 관점에서 앞서 살펴본 사전 찾기의 일련의 동작은 ‘이진 탐색’으로 볼 수 있다.</p> <p>예 2: 카드 정리하기. 카드 게임을 할 때는 매번 손에 든 카드를 작은 수부터 큰 수 순서로 정리한다. 그 과정은 아래 그림(1-2)과 같다.</p> <ol> <li>카드를 ‘정렬된 부분’과 ‘정렬되지 않은 부분’으로 나누고, 처음에는 맨 왼쪽 카드 1장이 이미 정렬되어 있다고 가정한다.</li> <li>정렬되지 않은 부분에서 카드 한 장을 꺼내 정렬된 부분의 올바른 위치에 끼워 넣는다. 그러면 맨 왼쪽 카드 2장이 정렬된 상태가 된다.</li> <li>단계 <code>2.</code>를 반복하여 매번 정렬되지 않은 부분의 카드 한 장을 정렬된 부분에 끼워 넣는다. 모든 카드가 정렬될 때까지 계속한다.</li> </ol> <p></p> <p> 그림 1-2   카드를 정렬하는 단계 </p> <p>이러한 카드 정리 방법은 본질적으로 ‘삽입 정렬’ 알고리즘이며, 작은 데이터 집합을 처리할 때 매우 효율적이다. 많은 프로그래밍 언어의 정렬 라이브러리 함수에서도 삽입 정렬이 사용된다.</p> <p>예 3: 거스름돈 주기. 마트에서 \\(69\\)위안어치 물건을 사고 계산원에게 \\(100\\)위안을 냈다고 가정하자. 계산원은 \\(31\\)위안을 거슬러 주어야 하며, 자연스럽게 아래 그림(1-3)과 같은 과정을 떠올린다.</p> <ol> <li>선택할 수 있는 것은 액면가가 \\(31\\)위안보다 작은 화폐로, \\(1\\)위안, \\(5\\)위안, \\(10\\)위안, \\(20\\)위안이 있다.</li> <li>선택지 중 가장 큰 \\(20\\)위안을 꺼내면 \\(31 - 20 = 11\\)위안이 남는다.</li> <li>남은 선택지 중 가장 큰 \\(10\\)위안을 꺼내면 \\(11 - 10 = 1\\)위안이 남는다.</li> <li>남은 선택지 중 가장 큰 \\(1\\)위안을 꺼내면 \\(1 - 1 = 0\\)위안이 남는다.</li> <li>거스름돈을 모두 주었다. 그 구성은 \\(20 + 10 + 1 = 31\\)위안이다.</li> </ol> <p></p> <p> 그림 1-3   거스름돈을 주는 과정 </p> <p>위 과정에서는 각 단계마다 현재 가장 좋아 보이는 선택, 즉 가능한 한 액면가가 큰 화폐를 고른다. 그 결과 거스름돈을 줄 수 있는 방법을 얻는다. 자료 구조와 알고리즘의 관점에서 이 방법은 본질적으로 ‘탐욕’ 알고리즘이다.</p> <p>작게는 요리 한 가지를 만드는 일부터 크게는 성간 항해에 이르기까지, 거의 모든 문제의 해결에는 알고리즘이 필요하다. 컴퓨터의 등장으로 우리는 프로그래밍을 통해 자료 구조를 메모리에 저장하고, CPU와 GPU가 알고리즘을 실행하도록 코드를 작성할 수 있게 되었다. 이로써 현실의 문제를 컴퓨터로 옮겨 다양한 복잡한 문제를 더 효율적으로 해결할 수 있다.</p> <p>참고</p> <p>자료 구조, 알고리즘, 배열, 이진 탐색 등의 개념이 아직 막연하게 느껴져도 계속 읽어 나가면 된다. 이 책이 자료 구조와 알고리즘의 지식 세계로 안내할 것이다.</p>","path":["제 1 장   알고리즘과의 첫 만남","1.1   어디에나 있는 알고리즘"],"tags":[]},{"location":"chapter_introduction/summary/","level":1,"title":"1.3   요약","text":"","path":["제 1 장   알고리즘과의 첫 만남","1.3   요약"],"tags":[]},{"location":"chapter_introduction/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>알고리즘은 일상생활 어디에나 있으며, 손에 닿지 않는 어려운 지식이 아니다. 사실 우리는 이미 자신도 모르게 많은 알고리즘을 익혀 크고 작은 생활 속 문제를 해결하고 있다.</li> <li>사전에서 단어를 찾는 원리는 이진 탐색 알고리즘과 같다. 이진 탐색에는 분할 정복이라는 중요한 알고리즘적 사고가 담겨 있다.</li> <li>카드를 정리하는 과정은 삽입 정렬 알고리즘과 매우 비슷하다. 삽입 정렬은 작은 데이터 집합을 정렬하는 데 적합하다.</li> <li>거스름돈을 주는 과정은 본질적으로 탐욕 알고리즘이다. 각 단계에서 현재 가장 좋아 보이는 선택을 한다.</li> <li>알고리즘은 유한한 시간 안에 특정 문제를 해결하는 명령 또는 작업 단계의 집합이고, 자료 구조는 컴퓨터에서 데이터를 구성하고 저장하는 방식이다.</li> <li>자료 구조와 알고리즘은 긴밀하게 연결되어 있다. 자료 구조는 알고리즘의 토대이며, 알고리즘은 자료 구조에 생명력을 불어넣는다.</li> <li>자료 구조와 알고리즘은 블록 조립에 비유할 수 있다. 블록은 데이터를, 블록의 모양과 연결 방식 등은 자료 구조를, 블록을 조립하는 단계는 알고리즘을 나타낸다.</li> </ul>","path":["제 1 장   알고리즘과의 첫 만남","1.3   요약"],"tags":[]},{"location":"chapter_introduction/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 프로그래머로 일하면서 직접 알고리즘으로 문제를 해결해 본 적이 없다. 자주 쓰는 알고리즘은 프로그래밍 언어에서 이미 제공하므로 그대로 사용하면 된다. 그렇다면 업무에서 다루는 문제가 아직 알고리즘이 필요할 정도는 아니라는 뜻인가?</p> <p>구체적인 업무 기술을 무술의 ‘동작’에 비유한다면, 기초 과목은 ‘내공’에 더 가깝다.</p> <p>알고리즘을 비롯한 기초 과목을 공부하는 의미는 업무에서 이를 처음부터 직접 구현하는 데 있지 않다고 생각한다. 배운 지식을 바탕으로 문제에 전문적으로 대응하고 판단하여 업무의 전반적인 품질을 높이는 데 의미가 있다. 간단한 예로, 프로그래밍 언어마다 내장 정렬 함수를 제공한다.</p> <ul> <li>자료 구조와 알고리즘을 배우지 않았다면 어떤 데이터든 이 정렬 함수에 넘길 수 있다. 실행도 원활하고 성능도 괜찮으니 별문제가 없어 보인다.</li> <li>하지만 알고리즘을 배웠다면 내장 정렬 함수의 시간 복잡도가 \\(O(n \\log n)\\)임을 안다. 주어진 데이터가 학번처럼 자릿수가 고정된 정수라면 더 효율적인 ‘기수 정렬’을 사용하여 시간 복잡도를 \\(O(nk)\\)로 낮출 수 있다. 여기서 \\(k\\)는 자릿수다. 데이터의 양이 많을 때는 이렇게 절약한 실행 시간이 비용 절감이나 사용자 경험 개선 등 큰 가치를 만들어 낼 수 있다.</li> </ul> <p>공학 분야의 많은 문제는 최적해를 얻기 어렵고, 그저 ‘어느 정도’ 해결된 상태로 남아 있다. 문제의 난이도는 문제 자체의 성질뿐 아니라 그 문제를 바라보는 사람의 지식에도 달려 있다. 지식이 체계적이고 경험이 많을수록 문제를 더 깊이 분석하고 더 우아하게 해결할 수 있다.</p> <p>Q: 이제는 거의 모든 코드를 AI 에이전트가 작성하는데, 그래도 알고리즘을 배워야 하는가?</p> <p>리무(李沐)의 말을 인용하자면, “에이전트는 능력 증폭기다. 능력에 10을 곱하는 것이지, 10을 더하는 덧셈기가 아니다.”</p> <p>AI가 능력 증폭기라면 증폭의 바탕이 되는 ‘능력’도 중요하다. 특히 두 가지 능력을 쌓을 가치가 있다고 생각한다. 첫째는 표현 능력, 특히 AI와 협업하는 능력이다. 목표를 명확히 설명하고 배경을 충분히 제공하며 잘못된 방향은 제때 바로잡아야 한다. AI에게 문제를 설명하는 일은 스스로 문제를 명확히 이해하는 데도 도움이 된다. 둘째는 기초 지식이다. 세부 사항은 AI가 보완할 수 있지만, 핵심을 짚어 질문하고 옳고 그름을 판단하며 AI의 출력을 유용한 결과물로 만들려면 기본 개념은 스스로 이해해야 한다.</p> <p>AI의 특별한 점은 과거의 도구처럼 기능의 경계가 뚜렷하지 않다는 것이다. AI는 물과 비슷해서 누구의 손에 들어가느냐에 따라 다른 모습을 띤다. 사람의 능력이 뛰어나고 안목이 좋을수록 AI의 증폭 효과도 더욱 뚜렷해진다.</p>","path":["제 1 장   알고리즘과의 첫 만남","1.3   요약"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/","level":1,"title":"1.2   알고리즘이란 무엇인가","text":"","path":["제 1 장   알고리즘과의 첫 만남","1.2   알고리즘이란 무엇인가"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/#121","level":2,"title":"1.2.1   알고리즘의 정의","text":"<p>알고리즘(algorithm)은 유한한 시간 안에 특정 문제를 해결하는 명령 또는 작업 단계의 집합이며, 다음과 같은 특성을 갖는다.</p> <ul> <li>문제가 명확하고 입력과 출력이 분명하게 정의되어 있다.</li> <li>실행 가능하며, 유한한 단계와 시간, 메모리 공간 안에서 완료할 수 있다.</li> <li>각 단계의 의미가 명확하여 입력과 실행 조건이 같으면 항상 같은 출력을 얻는다.</li> </ul>","path":["제 1 장   알고리즘과의 첫 만남","1.2   알고리즘이란 무엇인가"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/#122","level":2,"title":"1.2.2   자료 구조의 정의","text":"<p>자료 구조(data structure)는 데이터를 구성하고 저장하는 방식으로, 데이터의 내용, 데이터 사이의 관계, 데이터를 조작하는 방법을 포함한다. 자료 구조는 다음과 같은 목표를 고려하여 설계한다.</p> <ul> <li>컴퓨터 메모리를 절약하도록 공간 사용량을 최대한 줄인다.</li> <li>데이터 접근, 추가, 삭제, 갱신 등의 연산을 최대한 빠르게 수행한다.</li> <li>알고리즘이 효율적으로 실행되도록 간결한 데이터 표현과 논리적 정보를 제공한다.</li> </ul> <p>자료 구조를 설계하는 과정에는 수많은 절충이 따른다. 어느 한 측면을 개선하려면 다른 측면을 양보해야 할 때가 많다. 두 가지 예를 살펴보자.</p> <ul> <li>연결 리스트는 배열보다 데이터를 추가하거나 삭제하기가 편리하지만, 데이터에 접근하는 속도는 느리다.</li> <li>그래프는 연결 리스트보다 더 풍부한 논리적 정보를 제공하지만, 더 많은 메모리 공간이 필요하다.</li> </ul>","path":["제 1 장   알고리즘과의 첫 만남","1.2   알고리즘이란 무엇인가"],"tags":[]},{"location":"chapter_introduction/what_is_dsa/#123","level":2,"title":"1.2.3   자료 구조와 알고리즘의 관계","text":"<p>아래 그림(1-4)과 같이 자료 구조와 알고리즘은 밀접하게 연결되어 있다. 그 관계는 다음 세 가지 측면에서 살펴볼 수 있다.</p> <ul> <li>자료 구조는 알고리즘의 토대다. 자료 구조는 알고리즘에 체계적으로 저장된 데이터와 이를 조작하는 방법을 제공한다.</li> <li>알고리즘은 자료 구조에 생명력을 불어넣는다. 자료 구조 자체는 데이터 정보만 저장하며, 알고리즘과 결합해야 특정 문제를 해결할 수 있다.</li> <li>같은 알고리즘도 서로 다른 자료 구조를 바탕으로 구현할 수 있지만, 실행 효율은 크게 달라질 수 있다. 따라서 적절한 자료 구조를 선택하는 것이 중요하다.</li> </ul> <p></p> <p> 그림 1-4   자료 구조와 알고리즘의 관계 </p> <p>자료 구조와 알고리즘은 아래 그림(1-5)의 조립 블록에 비유할 수 있다. 블록 한 세트에는 여러 부품뿐 아니라 자세한 조립 설명서도 들어 있다. 설명서를 따라 차례대로 조립하면 멋진 블록 모형을 만들 수 있다.</p> <p></p> <p> 그림 1-5   블록 조립 </p> <p>구체적인 대응 관계는 아래 표(1-1)와 같다.</p> <p> 표 1-1   자료 구조와 알고리즘을 블록 조립에 비유하기 </p> 자료 구조와 알고리즘 블록 조립 입력 데이터 조립하기 전의 블록 자료 구조 모양, 크기, 연결 방식 등 블록의 구성 형태 알고리즘 블록을 원하는 형태로 조립하는 일련의 작업 단계 출력 데이터 완성된 블록 모형 <p>자료 구조와 알고리즘은 프로그래밍 언어에 종속되지 않는다. 이 때문에 이 책에서는 여러 프로그래밍 언어로 작성한 구현을 제공할 수 있다.</p> <p>관례적으로 사용하는 줄임말</p> <p>실제로 이야기할 때는 ‘자료 구조와 알고리즘’을 흔히 ‘알고리즘’이라고 줄여 부른다. 예를 들어 널리 알려진 리트코드의 알고리즘 문제는 실제로 자료 구조와 알고리즘 두 분야의 지식을 함께 평가한다.</p>","path":["제 1 장   알고리즘과의 첫 만남","1.2   알고리즘이란 무엇인가"],"tags":[]},{"location":"chapter_preface/","level":1,"title":"제 0 장   들어가며","text":"<p>이 장을 시작하며</p> <p>알고리즘은 아름다운 교향곡과 같다. 코드 한 줄 한 줄이 리듬처럼 흐른다.</p> <p>이 책이 여러분의 머릿속에 잔잔히 울려 퍼져, 특별하고 깊은 선율을 남기기를 바란다.</p>","path":["제 0 장   들어가며"],"tags":[]},{"location":"chapter_preface/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>0.1   이 책에 대하여</li> <li>0.2   이 책을 활용하는 방법</li> <li>0.3   요약</li> </ul>","path":["제 0 장   들어가며"],"tags":[]},{"location":"chapter_preface/about_the_book/","level":1,"title":"0.1   이 책에 대하여","text":"<p>이 프로젝트는 오픈 소스로 공개되는 무료 자료 구조와 알고리즘 입문서를 만드는 것을 목표로 한다. 처음 배우는 사람도 쉽게 접근할 수 있는 책을 지향한다.</p> <ul> <li>책 전체에 애니메이션 그림을 사용하여 내용을 명확하고 이해하기 쉽게 설명한다. 난이도를 완만하게 높여 가며 초보자가 자료 구조와 알고리즘의 지식 지도를 탐험하도록 안내한다.</li> <li>소스 코드를 클릭 한 번으로 실행할 수 있다. 독자가 실습을 통해 프로그래밍 능력을 키우고 알고리즘의 동작 원리와 자료 구조의 내부 구현을 이해하도록 돕는다.</li> <li>독자들이 서로 도우며 공부하기를 권장한다. 댓글에서 질문하고 의견을 나누며 함께 성장할 수 있다.</li> </ul>","path":["제 0 장   들어가며","0.1   이 책에 대하여"],"tags":[]},{"location":"chapter_preface/about_the_book/#011","level":2,"title":"0.1.1   대상 독자","text":"<p>알고리즘을 처음 접하는 입문자이거나, 문제를 풀어 본 경험은 조금 있지만 자료 구조와 알고리즘에 대한 이해가 막연해 알 것 같다가도 모르겠다는 생각이 든다면, 이 책은 바로 여러분을 위한 책이다!</p> <p>이미 상당한 수의 문제를 풀었고 대부분의 문제 유형에 익숙하다면, 이 책을 통해 알고리즘 지식을 되짚고 체계적으로 정리할 수 있다. 저장소의 소스 코드는 ‘문제 풀이 도구 모음’이나 ‘알고리즘 사전’처럼 활용할 수 있다.</p> <p>알고리즘의 ‘고수’라면 소중한 의견을 보내 주거나 집필에 함께 참여해 주기를 기대한다.</p> <p>선수 지식</p> <p>적어도 한 가지 언어의 프로그래밍 기초를 알고, 간단한 코드를 읽고 작성할 수 있어야 한다.</p>","path":["제 0 장   들어가며","0.1   이 책에 대하여"],"tags":[]},{"location":"chapter_preface/about_the_book/#012","level":2,"title":"0.1.2   내용 구성","text":"<p>이 책의 주요 내용은 아래 그림(0-1)과 같다.</p> <ul> <li>복잡도 분석: 자료 구조와 알고리즘을 평가하는 기준과 방법을 다룬다. 시간 복잡도와 공간 복잡도의 계산 방법, 대표적인 유형과 예제 등을 살펴본다.</li> <li>자료 구조: 기본 자료형과 자료 구조의 분류 방법을 다룬다. 배열, 연결 리스트, 스택, 큐, 해시 테이블, 트리, 힙, 그래프 등의 정의, 장단점, 자주 쓰는 연산, 대표적인 유형과 응용, 구현 방법 등을 살펴본다.</li> <li>알고리즘: 탐색, 정렬, 분할 정복, 백트래킹, 동적 계획법, 탐욕 등의 정의, 장단점, 효율성, 활용 상황, 문제 해결 단계와 예제 문제 등을 살펴본다.</li> </ul> <p></p> <p> 그림 0-1   이 책의 주요 내용 </p>","path":["제 0 장   들어가며","0.1   이 책에 대하여"],"tags":[]},{"location":"chapter_preface/about_the_book/#013","level":2,"title":"0.1.3   감사의 말","text":"<p>이 책은 오픈 소스 커뮤니티의 수많은 기여자가 함께 노력한 덕분에 계속 발전하고 있다. 시간과 노력을 들여 준 모든 집필 참여자에게 감사드린다. 참여자는 다음과 같다(GitHub가 자동으로 생성한 순서): krahets, coderonion, Gonglja, nuomi1, Reanon, justin-tse, hpstory, danielsss, curtishd, night-cruise, S-N-O-R-L-A-X, rongyi, msk397, gvenusleo, khoaxuantu, rivertwilight, K3v123, gyt95, zhuoqinyue, yuelinxin, Zuoxun, mingXta, Phoenix0415, FangYuan33, GN-Yu, longsizhuo, pengchzn, QiLOL, Cathay-Chen, guowei-gong, xBLACKICEx, IsChristina, JoseHung, qualifier1024, hello-ikun, magentaqin, Guanngxu, thomasq0, sunshinesDL, L-Super, Transmigration-zhou, WSL0809, Slone123c, lhxsm, yuan0221, what-is-me, theNefelibatas, Shyam-Chen, sangxiaai, longranger2, codeberg-user, xiongsp, JeffersonHuang, prinpal, seven1240, Wonderdch, malone6, xiaomiusa87, gaofer, bluebean-cloud, a16su, SamJin98, hongyun-robot, nanlei, XiaChuerwu, yd-j, iron-irax, mgisr, steventimes, junminhong, heshuyue, danny900714, Nigh, Dr-XYZ, MolDuM, XC-Zero, reeswell, PXG-XPG, NI-SW, Horbin-Magician, Enlightenus, YangXuanyi, xjr7670, beatrix-chan, DullSword, qq909244296, iStig, boloboloda, hts0000, gledfish, fbigm, echo1937, jiaxianhua, wenjianmin, keshida, kilikilikid, lclc6, lwbaptx, linyejoe2, liuxjerry, szu17dmy, dshlstarr, Yucao-cy, coderlef, czruby, bongbongbakudan, beintentional, ZongYangL, ZhongYuuu, ZhongGuanbin, hezhizhen, linzeyan, ZJKung, JTCPOWI, KawaiiAsh, luluxia, xb534, ztkuaikuai, yw-1021, ElaBosak233, baagod, zhouLion, yishangzhang, yi427, yanedie, yabo083, weibk, wangwang105, th1nk3r-ing, tao363, 4yDX3906, syd168, sslmj2020, smilelsb, siqyka, selear, sdshaoda, Xi-Row, popozhu, nuquist19, noobcodemaker, XiaoK29, chadyi, lyl625760, lucaswangdev, llql1211, 0130w, shanghai-Jerry, EJackYang, Javesun99, eltociear, lipusheng, KNChiu, BlindTerran, ShiMaRing, lovelock, FreddieLi, FloranceYeh, fanchenggang, gltianwen, goerll, nedchu, curly210102, CuB3y0nd, KraHsu, CarrotDLaw, youshaoXG, bubble9um, Asashishi, Asa0oo0o0o, fanenr, eagleanurag, akshiterate, 52coder, foursevenlove, KorsChen, hopkings2008, yang-le, realwujing, Evilrabbit520, Umer-Jahangir, Turing-1024-Lee, Suremotoo, paoxiaomooo, Chieko-Seren, Senrian, Allen-Scai, 19santosh99, ymmmas, Risuntsy, Richard-Zhang1019, RafaelCaso, qingpeng9802, primexiao, Urbaner3, codetypess, nidhoggfgg, MwumLi, CreatorMetaSky, martinx, ZnYang2018, hugtyftg, logan-qiu, psychelzh, Kunchen-Luo, Keynman 및 KeiichiKasai.</p> <p>이 책의 코드 검토는 coderonion, curtishd, Gonglja, gvenusleo, hpstory, justin-tse, khoaxuantu, krahets, night-cruise, nuomi1, Reanon 및 rongyi가 맡았다(알파벳순). 시간과 노력을 들여 준 이들에게 감사드린다. 이들 덕분에 각 언어의 코드가 일관된 규칙과 형식을 유지할 수 있었다.</p> <p>영문판은 yuelinxin, K3v123, magentaqin, QiLOL, Phoenix0415, SamJin98, yanedie, RafaelCaso, pengchzn 및 thomasq0가 검토했다. 일본어판은 eltociear가, 러시아어판은 И. А. Шевкун과 Yuyan Huang이, 번체 중국어판은 Shyam-Chen과 Dr-XYZ가 검토했다. 이들의 기여 덕분에 더 많은 독자에게 이 책을 전할 수 있게 되었다. 모두에게 감사드린다.</p> <p>이 책의 ePub 전자책 생성 도구는 zhongfq가 개발했다. 독자에게 더 유연한 독서 방식을 제공해 준 그의 기여에 감사드린다.</p> <p>이 책을 쓰는 동안 많은 분의 도움을 받았다.</p> <ul> <li>회사에서 나의 멘토였던 리시(李汐) 박사에게 감사드린다. 긴 대화 중 “어서 행동에 옮겨 보라”고 격려해 준 덕분에 이 책을 쓰겠다는 결심을 굳힐 수 있었다.</li> <li>이 책의 첫 독자가 되어 준 여자 친구 파오파오(泡泡)에게 감사드린다. 알고리즘 입문자의 시각에서 많은 소중한 의견을 주어 초보자가 읽기 더 좋은 책이 되었다.</li> <li>텅바오(腾宝), 치바오(琦宝), 페이바오(飞宝)에게 감사드린다. 이 책에 창의적인 이름을 지어 주어 모두가 첫 코드 “Hello World!”를 작성하던 아름다운 기억을 떠올리게 했다.</li> <li>지식재산권에 관한 전문적인 도움을 준 샤오취안(校铨)에게 감사드린다. 이 도움은 오픈 소스 도서인 이 책을 완성해 가는 데 중요한 역할을 했다.</li> <li>아름다운 표지와 로고를 디자인해 준 쑤퉁(苏潼)에게 감사드린다. 나의 지나친 꼼꼼함 때문에 여러 번 수정을 요청했는데도 인내심 있게 작업해 주었다.</li> <li>편집과 조판에 관한 조언을 해 주고 오픈 소스 문서 테마 Material-for-MkDocs를 개발한 @squidfunk에게 감사드린다.</li> </ul> <p>집필하면서 자료 구조와 알고리즘에 관한 많은 교재와 글을 읽었다. 이 작품들은 훌륭한 본보기가 되어 이 책의 정확성과 품질을 뒷받침해 주었다. 모든 선생님과 선배의 뛰어난 기여에 감사드린다!</p> <p>이 책은 손으로 실습하고 머리로 생각하며 배우는 방식을 권장한다. 이 점에서는 『실습으로 배우는 딥러닝(Dive into Deep Learning)』에서 깊은 영감을 받았다. 독자 여러분에게 이 훌륭한 책을 적극 추천한다.</p> <p>부모님께 진심으로 감사드린다. 늘 보내 주신 지지와 격려 덕분에 이처럼 흥미로운 일을 할 기회를 얻었다.</p>","path":["제 0 장   들어가며","0.1   이 책에 대하여"],"tags":[]},{"location":"chapter_preface/suggestions/","level":1,"title":"0.2   이 책을 활용하는 방법","text":"<p>참고</p> <p>가장 좋은 독서 경험을 위해 이 절을 끝까지 읽어 보기를 권장한다.</p>","path":["제 0 장   들어가며","0.2   이 책을 활용하는 방법"],"tags":[]},{"location":"chapter_preface/suggestions/#021","level":2,"title":"0.2.1   글의 표현과 서식에 관한 약속","text":"<ul> <li>제목 뒤에 <code>*</code>가 붙은 절은 선택해서 읽는 심화 내용이다. 시간이 부족하면 우선 건너뛰어도 된다.</li> <li>웹 버전에서는 전문 용어를 밑줄로 표시한다. 예를 들면 배열(array)과 같다. 문헌을 읽을 때 도움이 되므로 용어를 기억해 두기를 권장한다.</li> <li>핵심 내용과 요약 문장은 굵게 표시한다. 이러한 문장은 특히 주의 깊게 읽을 필요가 있다.</li> <li>특정한 의미로 사용하는 단어나 문구는 모호함을 피하기 위해 ‘따옴표’로 표시한다.</li> <li>프로그래밍 언어마다 명칭이 다를 때는 Python을 기준으로 삼는다. 예를 들어 ‘비어 있음’을 나타낼 때 <code>None</code>을 사용한다.</li> <li>이 책은 내용을 더 간결하게 배치하기 위해 프로그래밍 언어의 주석 작성 규칙을 일부 따르지 않는다. 주석은 주로 제목 주석, 설명 주석, 여러 줄 주석의 세 유형으로 나뉜다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>\"\"\"함수, 클래스, 테스트 사례 등을 표시하는 제목 주석\"\"\"\n\n# 코드를 자세히 설명하는 설명 주석\n\n\"\"\"\n여러 줄\n주석\n\"\"\"\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n// 여러 줄\n// 주석\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>/* 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 */\n\n// 코드를 자세히 설명하는 설명 주석\n\n/**\n * 여러 줄\n * 주석\n */\n</code></pre> <pre><code>### 함수, 클래스, 테스트 사례 등을 표시하는 제목 주석 ###\n\n# 코드를 자세히 설명하는 설명 주석\n\n# 여러 줄\n# 주석\n</code></pre>","path":["제 0 장   들어가며","0.2   이 책을 활용하는 방법"],"tags":[]},{"location":"chapter_preface/suggestions/#022","level":2,"title":"0.2.2   애니메이션 그림으로 효율적으로 배우기","text":"<p>영상과 그림은 글보다 정보 밀도가 높고 구조가 뚜렷하여 이해하기 쉽다. 이 책에서는 핵심 내용과 어려운 개념을 주로 애니메이션 그림으로 보여 주고, 글은 설명과 보충을 담당한다.</p> <p>책을 읽다가 아래 그림(0-2)과 같은 애니메이션 설명을 만나면 그림을 중심으로 보고 글을 보조 자료로 활용하여 두 가지를 함께 이해하기 바란다.</p> <p></p> <p> 그림 0-2   애니메이션 그림 예시 </p>","path":["제 0 장   들어가며","0.2   이 책을 활용하는 방법"],"tags":[]},{"location":"chapter_preface/suggestions/#023","level":2,"title":"0.2.3   코드 실습으로 이해를 깊게 하기","text":"<p>이 책의 예제 코드는 GitHub 저장소에 있다. 아래 그림(0-3)과 같이 소스 코드에는 테스트 사례가 포함되어 있어 클릭 한 번으로 실행할 수 있다.</p> <p>시간이 허락한다면 예제 코드를 참고하여 직접 한 번 작성해 보기를 권장한다. 공부할 시간이 부족하더라도 적어도 모든 코드를 읽고 실행해 보자.</p> <p>코드를 읽는 것보다 직접 작성하는 과정에서 더 많은 것을 얻는 경우가 많다. 직접 해 보며 배우는 것이 진짜 배움이다.</p> <p></p> <p> 그림 0-3   코드 실행 예시 </p> <p>코드를 실행하기 위한 준비는 크게 세 단계로 나뉜다.</p> <p>1단계: 로컬 프로그래밍 환경 설치. 부록의 안내에 따라 설치한다. 이미 설치했다면 이 단계는 건너뛰어도 된다.</p> <p>2단계: 코드 저장소 복제 또는 내려받기. GitHub 저장소로 이동한다. Git이 설치되어 있다면 다음 명령으로 저장소를 복제할 수 있다.</p> <pre><code>git clone https://github.com/krahets/hello-algo.git\n</code></pre> <p>물론 아래 그림(0-4)에 표시된 위치에서 ‘Download ZIP’ 버튼을 눌러 코드 압축 파일을 바로 내려받은 뒤 로컬에서 압축을 풀어도 된다.</p> <p></p> <p> 그림 0-4   저장소 복제와 코드 내려받기 </p> <p>3단계: 소스 코드 실행. 아래 그림(0-5)처럼 위쪽에 파일명이 표시된 코드 블록은 저장소의 <code>ko/codes</code> 폴더에서 해당 한국어판 소스 파일을 찾을 수 있다. 소스 파일은 클릭 한 번으로 실행할 수 있으므로 불필요한 디버깅 시간을 줄이고 학습 내용에 집중할 수 있다.</p> <p></p> <p> 그림 0-5   코드 블록과 이에 대응하는 소스 파일 </p> <p>로컬에서 실행하는 것 외에 웹 버전에서는 Python 코드의 실행 과정을 시각화할 수도 있다(pythontutor 기반). 아래 그림(0-6)처럼 코드 블록 아래의 ‘코드 시각화’를 누르면 화면이 펼쳐져 알고리즘 코드의 실행 과정을 관찰할 수 있다. ‘전체 화면’을 누르면 더 넓은 화면에서 볼 수 있다.</p> <p></p> <p> 그림 0-6   Python 코드 실행 시각화 </p>","path":["제 0 장   들어가며","0.2   이 책을 활용하는 방법"],"tags":[]},{"location":"chapter_preface/suggestions/#024","level":2,"title":"0.2.4   질문과 토론으로 함께 성장하기","text":"<p>책을 읽다가 이해하지 못한 내용을 만나도 쉽게 건너뛰지 않기를 바란다. 댓글란에 질문을 남기면 나와 동료들이 성심껏 답변하겠다. 보통 이틀 안에 답변을 받을 수 있다.</p> <p>아래 그림(0-7)처럼 웹 버전의 각 장 아래에는 댓글란이 있다. 댓글의 내용에도 관심을 가져 주기 바란다. 다른 사람들이 겪은 문제를 살펴보면서 놓친 내용을 보완하고 더 깊이 생각할 수 있다. 또한 다른 독자의 질문에 답하고 자신의 생각을 나누어 서로의 성장을 도울 수 있기를 기대한다.</p> <p></p> <p> 그림 0-7   댓글란 예시 </p>","path":["제 0 장   들어가며","0.2   이 책을 활용하는 방법"],"tags":[]},{"location":"chapter_preface/suggestions/#025","level":2,"title":"0.2.5   알고리즘 학습 경로","text":"<p>자료 구조와 알고리즘을 배우는 과정은 크게 세 단계로 나눌 수 있다.</p> <ol> <li>1단계: 알고리즘 입문. 다양한 자료 구조의 특성과 사용법에 익숙해지고, 각 알고리즘의 원리, 절차, 용도, 효율성 등을 배운다.</li> <li>2단계: 알고리즘 문제 풀기. 많이 다루는 문제부터 시작하여 최소 100문제 정도를 풀고 주요 알고리즘 문제에 익숙해지기를 권장한다. 처음 문제를 풀 때는 배운 내용을 잊는 일이 어려움으로 느껴질 수 있지만, 이는 매우 자연스러운 일이므로 걱정하지 않아도 된다. ‘에빙하우스의 망각 곡선’에 맞추어 문제를 복습하면 보통 3~5번 반복한 뒤에는 확실하게 기억할 수 있다. 추천 문제 목록과 학습 계획은 이 GitHub 저장소를 참고하기 바란다.</li> <li>3단계: 지식 체계 세우기. 학습 측면에서는 알고리즘 전문 글, 문제 해결 틀, 알고리즘 교재를 읽으며 지식 체계를 꾸준히 넓힐 수 있다. 문제 풀이 측면에서는 주제별 분류, 한 문제를 여러 방법으로 풀기, 한 해법을 여러 문제에 적용하기 등의 심화 전략을 시도할 수 있다. 관련 경험과 조언은 여러 커뮤니티에서 찾아볼 수 있다.</li> </ol> <p>아래 그림(0-8)처럼 이 책은 주로 ‘1단계’를 다루며, 이후 2단계와 3단계의 학습을 더 효율적으로 진행하도록 돕는 것을 목표로 한다.</p> <p></p> <p> 그림 0-8   알고리즘 학습 경로 </p>","path":["제 0 장   들어가며","0.2   이 책을 활용하는 방법"],"tags":[]},{"location":"chapter_preface/summary/","level":1,"title":"0.3   요약","text":"","path":["제 0 장   들어가며","0.3   요약"],"tags":[]},{"location":"chapter_preface/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>이 책은 주로 알고리즘 입문자를 대상으로 한다. 이미 기초를 갖추었다면 알고리즘 지식을 체계적으로 복습하는 데 도움이 되며, 소스 코드는 ‘문제 풀이 도구 모음’으로 활용할 수 있다.</li> <li>내용은 크게 복잡도 분석, 자료 구조, 알고리즘의 세 부분으로 구성되어 있으며, 이 분야의 주요 주제를 폭넓게 다룬다.</li> <li>알고리즘 초보자에게는 처음 공부할 때 입문서를 읽는 일이 매우 중요하다. 불필요하게 돌아가는 길을 많이 줄일 수 있다.</li> <li>애니메이션 그림은 주로 핵심 내용과 어려운 개념을 설명하는 데 쓰인다. 책을 읽을 때 이러한 내용을 특히 주의 깊게 살펴보자.</li> <li>실습은 프로그래밍을 배우는 가장 좋은 방법이다. 소스 코드를 실행하고 직접 코드를 작성해 볼 것을 적극 권장한다.</li> <li>웹 버전의 각 장에는 댓글란이 있다. 궁금한 점이나 생각을 언제든 나누기 바란다.</li> </ul>","path":["제 0 장   들어가며","0.3   요약"],"tags":[]},{"location":"chapter_reference/","level":1,"title":"참고 문헌","text":"<p>[1] Thomas H. Cormen, et al. Introduction to Algorithms (3<sup>rd</sup> Edition).</p> <p>[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1<sup>st</sup> Edition).</p> <p>[3] Robert Sedgewick, et al. Algorithms (4<sup>th</sup> Edition).</p> <p>[4] 옌웨이민(严蔚敏). 『자료 구조(C 언어판)』(数据结构（C 语言版）).</p> <p>[5] 덩쥔후이(邓俊辉). 『자료 구조(C++ 언어판, 제3판)』(数据结构（C++ 语言版，第三版）).</p> <p>[6] Mark Allen Weiss 지음, 천웨(陈越) 옮김. 『자료 구조와 알고리즘 분석: Java 언어로 설명하기(제3판)』(数据结构与算法分析：Java语言描述（第三版）).</p> <p>[7] 청제(程杰). 『쉽게 풀어 쓴 자료 구조』(大话数据结构).</p> <p>[8] 왕정(王争). 『자료 구조와 알고리즘의 아름다움』(数据结构与算法之美).</p> <p>[9] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6<sup>th</sup> Edition).</p> <p>[10] Aston Zhang, et al. Dive into Deep Learning.</p>","path":["참고 문헌"],"tags":[]},{"location":"chapter_searching/","level":1,"title":"제 10 장   탐색","text":"<p>이 장을 시작하며</p> <p>탐색은 미지의 세계로 떠나는 모험이다. 신비로운 공간 구석구석을 모두 돌아봐야 할 수도, 목표를 빠르게 찾아낼 수도 있다.</p> <p>무언가를 찾아 나서는 이 여정에서는 매번 새로운 곳을 살필 때마다 뜻밖의 답을 만날 수 있다.</p>","path":["제 10 장   탐색"],"tags":[]},{"location":"chapter_searching/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>10.1   이진 탐색</li> <li>10.2   이진 탐색으로 삽입 위치 찾기</li> <li>10.3   이진 탐색으로 경계 찾기</li> <li>10.4   해시를 이용한 최적화 전략</li> <li>10.5   탐색 알고리즘 다시 보기</li> <li>10.6   요약</li> <li>10.7   연습 문제</li> </ul>","path":["제 10 장   탐색"],"tags":[]},{"location":"chapter_searching/binary_search/","level":1,"title":"10.1   이진 탐색","text":"<p>이진 탐색(binary search)은 분할 정복 전략에 기반한 효율적인 탐색 알고리즘이다. 데이터가 정렬되어 있다는 성질을 이용하여 매번 탐색 범위를 절반으로 줄이며, 목표 원소를 찾거나 탐색 구간이 빌 때까지 반복한다.</p> <p>문제</p> <p>길이가 \\(n\\)인 배열 <code>nums</code>가 주어진다. 원소는 오름차순으로 정렬되어 있고 중복되지 않는다. 이 배열에서 원소 <code>target</code>을 찾아 그 인덱스를 반환하라. 배열에 해당 원소가 없으면 \\(-1\\)을 반환하라. 예시는 아래 그림(10-1)과 같다.</p> <p></p> <p> 그림 10-1   이진 탐색 예시 데이터 </p> <p>아래 그림(10-2)과 같이 포인터 \\(i = 0\\)과 \\(j = n - 1\\)을 초기화하여 각각 배열의 첫 원소와 마지막 원소를 가리키게 한다. 이는 탐색 구간 \\([0, n - 1]\\)을 나타낸다. 대괄호는 경곗값 자체를 포함하는 닫힌 구간을 뜻한다.</p> <p>이어서 다음 두 단계를 반복한다.</p> <ol> <li>중간 인덱스 \\(m = \\lfloor {(i + j) / 2} \\rfloor\\)를 계산한다. 여기서 \\(\\lfloor \\: \\rfloor\\)는 내림 연산이다.</li> <li><code>nums[m]</code>과 <code>target</code>의 대소 관계를 비교하며, 다음 세 경우로 나뉜다.<ol> <li><code>nums[m] &lt; target</code>이면 <code>target</code>은 구간 \\([m + 1, j]\\)에 있으므로 \\(i = m + 1\\)을 수행한다.</li> <li><code>nums[m] &gt; target</code>이면 <code>target</code>은 구간 \\([i, m - 1]\\)에 있으므로 \\(j = m - 1\\)을 수행한다.</li> <li><code>nums[m] = target</code>이면 <code>target</code>을 찾았으므로 인덱스 \\(m\\)을 반환한다.</li> </ol> </li> </ol> <p>배열에 목표 원소가 없으면 탐색 구간은 결국 비게 된다. 이때 \\(-1\\)을 반환한다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 10-2   이진 탐색 과정 </p> <p>\\(i\\)와 \\(j\\)가 모두 <code>int</code> 타입이므로 \\(i + j\\)가 <code>int</code> 타입의 표현 범위를 벗어날 수 있다. 정수 오버플로를 피하기 위해 보통 \\(m = \\lfloor {i + (j - i) / 2} \\rfloor\\) 공식으로 중간 인덱스를 계산한다.</p> <p>코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search.py<pre><code>def binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"이진 탐색(양 끝을 포함하는 닫힌 구간)\"\"\"\n    # 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    i, j = 0, len(nums) - 1\n    # 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while i &lt;= j:\n        # 이론적으로 Python의 수는 메모리 크기에 따라 무한히 커질 수 있으므로 큰 수의 범위 초과를 고려할 필요가 없다\n        m = (i + j) // 2  # 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target:\n            i = m + 1  # 이 경우 target은 구간 [m+1, j]에 있다\n        elif nums[m] &gt; target:\n            j = m - 1  # 이 경우 target은 구간 [i, m-1]에 있다\n        else:\n            return m  # 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n    return -1  # 목표 원소를 찾지 못했으므로 -1을 반환한다\n</code></pre> binary_search.cpp<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    int i = 0, j = nums.size() - 1;\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target)    // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1;\n        else // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nint binarySearch(int[] nums, int target) {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    int i = 0, j = nums.length - 1;\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1;\n        else // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nint BinarySearch(int[] nums, int target) {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    int i = 0, j = nums.Length - 1;\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2;   // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target)      // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1;\n        else                       // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nfunc binarySearch(nums []int, target int) int {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    i, j := 0, len(nums)-1\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    for i &lt;= j {\n        m := i + (j-i)/2      // 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target { // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1\n        } else if nums[m] &gt; target { // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1\n        } else { // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m\n        }\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target { // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1\n        } else if nums[m] &gt; target { // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1\n        } else { // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m\n        }\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nfunction binarySearch(nums, target) {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    let i = 0,\n        j = nums.length - 1;\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while (i &lt;= j) {\n        // 중간 인덱스 m을 계산하며, parseInt()로 소수 부분을 버린다\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1;\n        else return m; // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nfunction binarySearch(nums: number[], target: number): number {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    let i = 0,\n        j = nums.length - 1;\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while (i &lt;= j) {\n        // 중간 인덱스 m을 계산한다\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1;\n        } else {\n            // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n        }\n    }\n    return -1; // 목표 원소를 찾지 못했으므로 -1을 반환한다\n}\n</code></pre> binary_search.dart<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n  int i = 0, j = nums.length - 1;\n  // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // 중간 인덱스 m을 계산한다\n    if (nums[m] &lt; target) {\n      // 이 경우 target은 구간 [m+1, j]에 있다\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // 이 경우 target은 구간 [i, m-1]에 있다\n      j = m - 1;\n    } else {\n      // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n      return m;\n    }\n  }\n  // 목표 원소를 찾지 못했으므로 -1을 반환한다\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    let mut i = 0;\n    let mut j = nums.len() as i32 - 1;\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if nums[m as usize] &lt; target {\n            // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1;\n        } else {\n            // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n        }\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nint binarySearch(int *nums, int len, int target) {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    int i = 0, j = len - 1;\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target)    // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1;\n        else // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* 이진 탐색(양 끝을 포함하는 닫힌 구간) */\nfun binarySearch(nums: IntArray, target: Int): Int {\n    // 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n    var i = 0\n    var j = nums.size - 1\n    // 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) // 이 경우 target은 구간 [m+1, j]에 있다\n            i = m + 1\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m-1]에 있다\n            j = m - 1\n        else  // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>### 이진 탐색(양 끝을 포함하는 닫힌 구간) ###\ndef binary_search(nums, target)\n  # 닫힌 구간 [0, n-1]을 초기화한다. i와 j는 각각 배열의 첫 원소와 마지막 원소를 가리킨다\n  i, j = 0, nums.length - 1\n\n  # 반복하다 탐색 구간이 비면 종료한다(i &gt; j이면 빈 구간이다)\n  while i &lt;= j\n    # Ruby의 정수는 이론상 메모리가 허용하는 만큼 커질 수 있으므로 큰 수의 오버플로를 고려할 필요가 없다\n    m = (i + j) / 2   # 중간 인덱스 m을 계산한다\n\n    if nums[m] &lt; target\n      i = m + 1 # 이 경우 target은 구간 [m+1, j]에 있다\n    elsif nums[m] &gt; target\n      j = m - 1 # 이 경우 target은 구간 [i, m-1]에 있다\n    else\n      return m  # 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n    end\n  end\n\n  -1  # 목표 원소를 찾지 못했으므로 -1을 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>시간 복잡도는 \\(O(\\log n)\\)이다. 이진 탐색의 반복문에서 구간이 매번 절반으로 줄어들므로 반복 횟수는 \\(\\log_2 n\\)이다.</p> <p>공간 복잡도는 \\(O(1)\\)이다. 포인터 \\(i\\)와 \\(j\\)는 상수 크기의 공간을 사용한다.</p>","path":["제 10 장   탐색","10.1   이진 탐색"],"tags":[]},{"location":"chapter_searching/binary_search/#1011","level":2,"title":"10.1.1   구간 표현 방법","text":"<p>위의 양끝이 닫힌 구간 외에 '왼쪽은 닫히고 오른쪽은 열린' 구간도 자주 사용한다. \\([0, n)\\)으로 정의하며 왼쪽 경곗값은 포함하고 오른쪽 경곗값은 포함하지 않는다. 이 표현에서 구간 \\([i, j)\\)는 \\(i = j\\)일 때 비어 있다.</p> <p>이 표현을 사용하여 같은 기능의 이진 탐색 알고리즘을 구현할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search.py<pre><code>def binary_search_lcro(nums: list[int], target: int) -&gt; int:\n    \"\"\"이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간)\"\"\"\n    # 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    i, j = 0, len(nums)\n    # 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while i &lt; j:\n        m = (i + j) // 2  # 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target:\n            i = m + 1  # 이 경우 target은 구간 [m+1, j)에 있다\n        elif nums[m] &gt; target:\n            j = m  # 이 경우 target은 구간 [i, m)에 있다\n        else:\n            return m  # 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n    return -1  # 목표 원소를 찾지 못했으므로 -1을 반환한다\n</code></pre> binary_search.cpp<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nint binarySearchLCRO(vector&lt;int&gt; &amp;nums, int target) {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    int i = 0, j = nums.size();\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target)    // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m)에 있다\n            j = m;\n        else // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nint binarySearchLCRO(int[] nums, int target) {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    int i = 0, j = nums.length;\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m)에 있다\n            j = m;\n        else // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nint BinarySearchLCRO(int[] nums, int target) {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    int i = 0, j = nums.Length;\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2;   // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target)      // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m)에 있다\n            j = m;\n        else                       // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nfunc binarySearchLCRO(nums []int, target int) int {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    i, j := 0, len(nums)\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    for i &lt; j {\n        m := i + (j-i)/2      // 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target { // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1\n        } else if nums[m] &gt; target { // 이 경우 target은 구간 [i, m)에 있다\n            j = m\n        } else { // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m\n        }\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nfunc binarySearchLCRO(nums: [Int], target: Int) -&gt; Int {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    var i = nums.startIndex\n    var j = nums.endIndex\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while i &lt; j {\n        let m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target { // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1\n        } else if nums[m] &gt; target { // 이 경우 target은 구간 [i, m)에 있다\n            j = m\n        } else { // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m\n        }\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nfunction binarySearchLCRO(nums, target) {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    let i = 0,\n        j = nums.length;\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while (i &lt; j) {\n        // 중간 인덱스 m을 계산하며, parseInt()로 소수 부분을 버린다\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // 이 경우 target은 구간 [i, m)에 있다\n            j = m;\n        // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n        else return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nfunction binarySearchLCRO(nums: number[], target: number): number {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    let i = 0,\n        j = nums.length;\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while (i &lt; j) {\n        // 중간 인덱스 m을 계산한다\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // 이 경우 target은 구간 [i, m)에 있다\n            j = m;\n        } else {\n            // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n        }\n    }\n    return -1; // 목표 원소를 찾지 못했으므로 -1을 반환한다\n}\n</code></pre> binary_search.dart<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nint binarySearchLCRO(List&lt;int&gt; nums, int target) {\n  // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n  int i = 0, j = nums.length;\n  // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n  while (i &lt; j) {\n    int m = i + (j - i) ~/ 2; // 중간 인덱스 m을 계산한다\n    if (nums[m] &lt; target) {\n      // 이 경우 target은 구간 [m+1, j)에 있다\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // 이 경우 target은 구간 [i, m)에 있다\n      j = m;\n    } else {\n      // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n      return m;\n    }\n  }\n  // 목표 원소를 찾지 못했으므로 -1을 반환한다\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nfn binary_search_lcro(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    let mut i = 0;\n    let mut j = nums.len() as i32;\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while i &lt; j {\n        let m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if nums[m as usize] &lt; target {\n            // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {\n            // 이 경우 target은 구간 [i, m)에 있다\n            j = m;\n        } else {\n            // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n        }\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nint binarySearchLCRO(int *nums, int len, int target) {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    int i = 0, j = len;\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target)    // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1;\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m)에 있다\n            j = m;\n        else // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m;\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1;\n}\n</code></pre> binary_search.kt<pre><code>/* 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) */\nfun binarySearchLCRO(nums: IntArray, target: Int): Int {\n    // 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n    var i = 0\n    var j = nums.size\n    // 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n    while (i &lt; j) {\n        val m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) // 이 경우 target은 구간 [m+1, j)에 있다\n            i = m + 1\n        else if (nums[m] &gt; target) // 이 경우 target은 구간 [i, m)에 있다\n            j = m\n        else  // 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n            return m\n    }\n    // 목표 원소를 찾지 못했으므로 -1을 반환한다\n    return -1\n}\n</code></pre> binary_search.rb<pre><code>### 이진 탐색(왼쪽 끝은 포함하고 오른쪽 끝은 제외하는 구간) ###\ndef binary_search_lcro(nums, target)\n  # 왼쪽 끝만 포함하는 구간 [0, n)을 초기화한다. i는 배열의 첫 원소를, j는 마지막 원소 바로 다음 위치를 가리킨다\n  i, j = 0, nums.length\n\n  # 반복하다 탐색 구간이 비면 종료한다(i = j이면 빈 구간이다)\n  while i &lt; j\n    # 중간 인덱스 m을 계산한다\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # 이 경우 target은 구간 [m+1, j)에 있다\n    elsif nums[m] &gt; target\n      j = m - 1 # 이 경우 target은 구간 [i, m)에 있다\n    else\n      return m  # 목표 원소를 찾았으므로 해당 인덱스를 반환한다\n    end\n  end\n\n  -1  # 목표 원소를 찾지 못했으므로 -1을 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>아래 그림(10-3)과 같이 구간을 표현하는 방식에 따라 이진 탐색 알고리즘의 초기화, 반복 조건, 구간 축소 연산이 모두 달라진다.</p> <p>'양끝이 닫힌 구간'에서는 왼쪽과 오른쪽 경계를 모두 포함하므로, 포인터 \\(i\\)와 \\(j\\)로 구간을 줄이는 연산도 대칭적이다. 실수할 가능성이 더 작으므로 일반적으로 '양끝이 닫힌 구간'으로 작성하기를 권장한다.</p> <p></p> <p> 그림 10-3   두 가지 구간 정의 </p>","path":["제 10 장   탐색","10.1   이진 탐색"],"tags":[]},{"location":"chapter_searching/binary_search/#1012","level":2,"title":"10.1.2   장점과 한계","text":"<p>이진 탐색은 시간과 공간 양쪽에서 좋은 성능을 보인다.</p> <ul> <li>이진 탐색은 시간 효율이 높다. 데이터가 많을수록 로그 시간 복잡도의 이점이 뚜렷하다. 예를 들어 데이터 크기가 \\(n = 2^{20}\\)이면 선형 탐색은 \\(2^{20} = 1048576\\) 번 반복해야 하지만, 이진 탐색은 \\(\\log_2 2^{20} = 20\\) 번이면 된다.</li> <li>이진 탐색은 추가 공간이 필요하지 않다. 해시 탐색처럼 추가 공간을 사용하는 탐색 알고리즘보다 공간을 절약한다.</li> </ul> <p>하지만 이진 탐색이 모든 상황에 적합한 것은 아니다. 주요 이유는 다음과 같다.</p> <ul> <li>이진 탐색은 정렬된 데이터에만 적용할 수 있다. 입력이 정렬되지 않았다면 이진 탐색만을 위해 정렬하는 것은 오히려 손해다. 정렬 알고리즘의 시간 복잡도는 보통 \\(O(n \\log n)\\)으로 선형 탐색과 이진 탐색보다 높기 때문이다. 원소를 자주 삽입하는 상황에서도 정렬 상태를 유지하려면 특정 위치에 삽입해야 하므로, \\(O(n)\\)의 큰 시간 비용이 든다.</li> <li>이진 탐색은 배열에만 적합하다. 원소를 건너뛰며(연속적이지 않게) 접근해야 하는데, 연결 리스트에서는 이런 접근이 비효율적이다. 따라서 연결 리스트나 연결 리스트 기반 자료 구조에는 적합하지 않다.</li> <li>데이터가 적을 때는 선형 탐색의 성능이 더 좋다. 선형 탐색은 매번 비교 1회만 필요하지만, 이진 탐색은 덧셈 1회, 나눗셈 1회, 비교 1 ~ 3회, 덧셈(뺄셈) 1회로 총 4 ~ 6회의 기본 연산이 필요하다. 따라서 데이터 수 \\(n\\)이 작으면 선형 탐색이 오히려 더 빠르다.</li> </ul>","path":["제 10 장   탐색","10.1   이진 탐색"],"tags":[]},{"location":"chapter_searching/binary_search_edge/","level":1,"title":"10.3   이진 탐색으로 경계 찾기","text":"","path":["제 10 장   탐색","10.3   이진 탐색으로 경계 찾기"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#1031","level":2,"title":"10.3.1   왼쪽 경계 찾기","text":"<p>문제</p> <p>중복 원소를 포함할 수 있는 길이 \\(n\\)의 정렬된 배열 <code>nums</code>가 주어진다. 배열에서 가장 왼쪽에 있는 원소 <code>target</code>의 인덱스를 반환하라. 해당 원소가 없으면 \\(-1\\)을 반환하라.</p> <p>이진 탐색으로 삽입 위치를 찾는 방법을 떠올려 보자. 탐색이 끝나면 \\(i\\)는 가장 왼쪽의 <code>target</code>을 가리킨다. 따라서 삽입 위치 찾기는 본질적으로 가장 왼쪽 <code>target</code>의 인덱스를 찾는 과정이다.</p> <p>삽입 위치를 찾는 함수를 이용하여 왼쪽 경계를 찾아 보자. 배열에 <code>target</code>이 없을 수도 있으며, 이때는 다음 두 결과 중 하나가 나올 수 있다.</p> <ul> <li>삽입 위치의 인덱스 \\(i\\)가 범위를 벗어난다.</li> <li>원소 <code>nums[i]</code>가 <code>target</code>과 다르다.</li> </ul> <p>위 두 경우에는 \\(-1\\)을 바로 반환하면 된다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_edge.py<pre><code>def binary_search_left_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"가장 왼쪽에 있는 target을 이진 탐색\"\"\"\n    # target의 삽입 위치를 찾는 것과 같다\n    i = binary_search_insertion(nums, target)\n    # target을 찾지 못했으므로 -1을 반환한다\n    if i == len(nums) or nums[i] != target:\n        return -1\n    # target을 찾았으므로 인덱스 i를 반환한다\n    return i\n</code></pre> binary_search_edge.cpp<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nint binarySearchLeftEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // target의 삽입 위치를 찾는 것과 같다\n    int i = binarySearchInsertion(nums, target);\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (i == nums.size() || nums[i] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_edge.java<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nint binarySearchLeftEdge(int[] nums, int target) {\n    // target의 삽입 위치를 찾는 것과 같다\n    int i = binary_search_insertion.binarySearchInsertion(nums, target);\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (i == nums.length || nums[i] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nint BinarySearchLeftEdge(int[] nums, int target) {\n    // target의 삽입 위치를 찾는 것과 같다\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target);\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (i == nums.Length || nums[i] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_edge.go<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nfunc binarySearchLeftEdge(nums []int, target int) int {\n    // target의 삽입 위치를 찾는 것과 같다\n    i := binarySearchInsertion(nums, target)\n    // target을 찾지 못했으므로 -1을 반환한다\n    if i == len(nums) || nums[i] != target {\n        return -1\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i\n}\n</code></pre> binary_search_edge.swift<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nfunc binarySearchLeftEdge(nums: [Int], target: Int) -&gt; Int {\n    // target의 삽입 위치를 찾는 것과 같다\n    let i = binarySearchInsertion(nums: nums, target: target)\n    // target을 찾지 못했으므로 -1을 반환한다\n    if i == nums.endIndex || nums[i] != target {\n        return -1\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i\n}\n</code></pre> binary_search_edge.js<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nfunction binarySearchLeftEdge(nums, target) {\n    // target의 삽입 위치를 찾는 것과 같다\n    const i = binarySearchInsertion(nums, target);\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nfunction binarySearchLeftEdge(nums: Array&lt;number&gt;, target: number): number {\n    // target의 삽입 위치를 찾는 것과 같다\n    const i = binarySearchInsertion(nums, target);\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nint binarySearchLeftEdge(List&lt;int&gt; nums, int target) {\n  // target의 삽입 위치를 찾는 것과 같다\n  int i = binarySearchInsertion(nums, target);\n  // target을 찾지 못했으므로 -1을 반환한다\n  if (i == nums.length || nums[i] != target) {\n    return -1;\n  }\n  // target을 찾았으므로 인덱스 i를 반환한다\n  return i;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nfn binary_search_left_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // target의 삽입 위치를 찾는 것과 같다\n    let i = binary_search_insertion(nums, target);\n    // target을 찾지 못했으므로 -1을 반환한다\n    if i == nums.len() as i32 || nums[i as usize] != target {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    i\n}\n</code></pre> binary_search_edge.c<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nint binarySearchLeftEdge(int *nums, int numSize, int target) {\n    // target의 삽입 위치를 찾는 것과 같다\n    int i = binarySearchInsertion(nums, numSize, target);\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (i == numSize || nums[i] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* 가장 왼쪽에 있는 target을 이진 탐색 */\nfun binarySearchLeftEdge(nums: IntArray, target: Int): Int {\n    // target의 삽입 위치를 찾는 것과 같다\n    val i = binarySearchInsertion(nums, target)\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (i == nums.size || nums[i] != target) {\n        return -1\n    }\n    // target을 찾았으므로 인덱스 i를 반환한다\n    return i\n}\n</code></pre> binary_search_edge.rb<pre><code>### 가장 왼쪽에 있는 target을 이진 탐색 ###\ndef binary_search_left_edge(nums, target)\n  # target의 삽입 위치를 찾는 것과 같다\n  i = binary_search_insertion(nums, target)\n\n  # target을 찾지 못했으므로 -1을 반환한다\n  return -1 if i == nums.length || nums[i] != target\n\n  i # target을 찾았으므로 인덱스 i를 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 10 장   탐색","10.3   이진 탐색으로 경계 찾기"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#1032","level":2,"title":"10.3.2   오른쪽 경계 찾기","text":"<p>가장 오른쪽에 있는 <code>target</code>은 어떻게 찾을까? 가장 직접적인 방법은 코드를 수정하여 <code>nums[m] == target</code>일 때 포인터로 구간을 줄이는 방식을 바꾸는 것이다. 코드는 생략하니, 관심 있는 독자는 직접 구현해 보자.</p> <p>아래에서는 더 기발한 두 가지 방법을 소개한다.</p>","path":["제 10 장   탐색","10.3   이진 탐색으로 경계 찾기"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#1","level":3,"title":"1.   왼쪽 경계 찾기 재활용","text":"<p>가장 왼쪽 원소를 찾는 함수로 가장 오른쪽 원소도 찾을 수 있다. 구체적으로는 가장 오른쪽 <code>target</code>을 찾는 문제를 가장 왼쪽 <code>target + 1</code>을 찾는 문제로 바꾼다.</p> <p>아래 그림(10-7)과 같이 탐색이 끝나면 포인터 \\(i\\)는 가장 왼쪽 <code>target + 1</code>(존재한다면)을 가리키고, \\(j\\)는 가장 오른쪽 <code>target</code>을 가리킨다. 따라서 \\(j\\)를 반환하면 된다.</p> <p></p> <p> 그림 10-7   오른쪽 경계 찾기를 왼쪽 경계 찾기로 변환 </p> <p>반환되는 삽입 위치는 \\(i\\)이므로, 여기서 \\(1\\)을 빼야 \\(j\\)를 얻을 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_edge.py<pre><code>def binary_search_right_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"가장 오른쪽에 있는 target을 이진 탐색\"\"\"\n    # 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    i = binary_search_insertion(nums, target + 1)\n    # j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    j = i - 1\n    # target을 찾지 못했으므로 -1을 반환한다\n    if j == -1 or nums[j] != target:\n        return -1\n    # target을 찾았으므로 인덱스 j를 반환한다\n    return j\n</code></pre> binary_search_edge.cpp<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nint binarySearchRightEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    int i = binarySearchInsertion(nums, target + 1);\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    int j = i - 1;\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j;\n}\n</code></pre> binary_search_edge.java<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nint binarySearchRightEdge(int[] nums, int target) {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    int i = binary_search_insertion.binarySearchInsertion(nums, target + 1);\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    int j = i - 1;\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nint BinarySearchRightEdge(int[] nums, int target) {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target + 1);\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    int j = i - 1;\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j;\n}\n</code></pre> binary_search_edge.go<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nfunc binarySearchRightEdge(nums []int, target int) int {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    i := binarySearchInsertion(nums, target+1)\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    j := i - 1\n    // target을 찾지 못했으므로 -1을 반환한다\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j\n}\n</code></pre> binary_search_edge.swift<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nfunc binarySearchRightEdge(nums: [Int], target: Int) -&gt; Int {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    let i = binarySearchInsertion(nums: nums, target: target + 1)\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    let j = i - 1\n    // target을 찾지 못했으므로 -1을 반환한다\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j\n}\n</code></pre> binary_search_edge.js<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nfunction binarySearchRightEdge(nums, target) {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    const i = binarySearchInsertion(nums, target + 1);\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    const j = i - 1;\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nfunction binarySearchRightEdge(nums: Array&lt;number&gt;, target: number): number {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    const i = binarySearchInsertion(nums, target + 1);\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    const j = i - 1;\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nint binarySearchRightEdge(List&lt;int&gt; nums, int target) {\n  // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n  int i = binarySearchInsertion(nums, target + 1);\n  // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n  int j = i - 1;\n  // target을 찾지 못했으므로 -1을 반환한다\n  if (j == -1 || nums[j] != target) {\n    return -1;\n  }\n  // target을 찾았으므로 인덱스 j를 반환한다\n  return j;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nfn binary_search_right_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    let i = binary_search_insertion(nums, target + 1);\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    let j = i - 1;\n    // target을 찾지 못했으므로 -1을 반환한다\n    if j == -1 || nums[j as usize] != target {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    j\n}\n</code></pre> binary_search_edge.c<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nint binarySearchRightEdge(int *nums, int numSize, int target) {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    int i = binarySearchInsertion(nums, numSize, target + 1);\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    int j = i - 1;\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j;\n}\n</code></pre> binary_search_edge.kt<pre><code>/* 가장 오른쪽에 있는 target을 이진 탐색 */\nfun binarySearchRightEdge(nums: IntArray, target: Int): Int {\n    // 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n    val i = binarySearchInsertion(nums, target + 1)\n    // j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n    val j = i - 1\n    // target을 찾지 못했으므로 -1을 반환한다\n    if (j == -1 || nums[j] != target) {\n        return -1\n    }\n    // target을 찾았으므로 인덱스 j를 반환한다\n    return j\n}\n</code></pre> binary_search_edge.rb<pre><code>### 가장 오른쪽에 있는 target을 이진 탐색 ###\ndef binary_search_right_edge(nums, target)\n  # 가장 왼쪽에 있는 target + 1을 찾는 문제로 바꾼다\n  i = binary_search_insertion(nums, target + 1)\n\n  # j는 가장 오른쪽의 target을 가리키고, i는 target보다 큰 첫 번째 원소를 가리킨다\n  j = i - 1\n\n  # target을 찾지 못했으므로 -1을 반환한다\n  return -1 if j == -1 || nums[j] != target\n\n  j # target을 찾았으므로 인덱스 j를 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 10 장   탐색","10.3   이진 탐색으로 경계 찾기"],"tags":[]},{"location":"chapter_searching/binary_search_edge/#2","level":3,"title":"2.   원소 찾기로 변환","text":"<p>배열에 <code>target</code>이 없으면 최종적으로 \\(i\\)는 <code>target</code>보다 큰 첫 번째 원소를, \\(j\\)는 <code>target</code>보다 작은 가장 오른쪽 원소를 가리킨다.</p> <p>따라서 아래 그림(10-8)과 같이 배열에 없는 원소를 만들어 왼쪽 경계와 오른쪽 경계를 찾을 수 있다.</p> <ul> <li>가장 왼쪽 <code>target</code> 찾기: <code>target - 0.5</code>를 탐색하고 포인터 \\(i\\)를 반환하는 것으로 바꿀 수 있다.</li> <li>가장 오른쪽 <code>target</code> 찾기: <code>target + 0.5</code>를 탐색하고 포인터 \\(j\\)를 반환하는 것으로 바꿀 수 있다.</li> </ul> <p></p> <p> 그림 10-8   경계 찾기를 원소 찾기로 변환 </p> <p>코드는 생략하며, 다음 두 가지에 주의하자.</p> <ul> <li>주어진 배열에는 소수가 없으므로 값이 같은 경우의 처리는 고려하지 않아도 된다.</li> <li>이 방법은 소수를 사용하므로 함수의 변수 <code>target</code>을 부동 소수점 타입으로 바꿔야 한다(Python은 수정할 필요가 없다).</li> </ul>","path":["제 10 장   탐색","10.3   이진 탐색으로 경계 찾기"],"tags":[]},{"location":"chapter_searching/binary_search_insertion/","level":1,"title":"10.2   이진 탐색으로 삽입 위치 찾기","text":"<p>이진 탐색은 목표 원소를 찾는 데만 쓰이는 것이 아니다. 목표 원소를 삽입할 위치를 찾는 등 여러 변형 문제에도 사용할 수 있다.</p>","path":["제 10 장   탐색","10.2   이진 탐색으로 삽입 위치 찾기"],"tags":[]},{"location":"chapter_searching/binary_search_insertion/#1021","level":2,"title":"10.2.1   중복 원소가 없는 경우","text":"<p>문제</p> <p>중복 원소가 없는 길이 \\(n\\)의 정렬된 배열 <code>nums</code>와 원소 <code>target</code>이 주어진다. 정렬 상태를 유지하면서 <code>target</code>을 배열 <code>nums</code>에 삽입하려고 한다. 이미 <code>target</code>이 있으면 그 왼쪽에 삽입한다. 삽입 후 <code>target</code>의 배열 인덱스를 반환하라. 예시는 아래 그림(10-4)과 같다.</p> <p></p> <p> 그림 10-4   이진 탐색으로 삽입 위치를 찾는 예시 데이터 </p> <p>앞 절의 이진 탐색 코드를 재사용하려면 다음 두 질문에 답해야 한다.</p> <p>질문 1: 배열에 <code>target</code>이 있을 때 삽입 위치의 인덱스는 해당 원소의 인덱스인가?</p> <p>문제에서는 값이 같은 원소의 왼쪽에 <code>target</code>을 삽입하라고 한다. 따라서 새로 삽입하는 <code>target</code>이 기존 <code>target</code>의 자리를 차지한다. 즉, 배열에 <code>target</code>이 있으면 삽입 위치의 인덱스는 그 <code>target</code>의 인덱스다.</p> <p>질문 2: 배열에 <code>target</code>이 없을 때 삽입 위치는 어느 원소의 인덱스인가?</p> <p>이진 탐색 과정을 더 살펴보자. <code>nums[m] &lt; target</code>이면 \\(i\\)가 이동하므로 포인터 \\(i\\)는 <code>target</code> 이상인 원소에 가까워진다. 마찬가지로 포인터 \\(j\\)는 항상 <code>target</code> 이하인 원소에 가까워진다.</p> <p>따라서 이진 탐색이 끝나면 \\(i\\)는 <code>target</code>보다 큰 첫 번째 원소를, \\(j\\)는 <code>target</code>보다 작은 가장 오른쪽 원소를 가리킨다. 배열에 <code>target</code>이 없을 때 삽입 인덱스가 \\(i\\)임을 쉽게 알 수 있다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_insertion.py<pre><code>def binary_search_insertion_simple(nums: list[int], target: int) -&gt; int:\n    \"\"\"삽입 위치의 이진 탐색(중복 원소 없음)\"\"\"\n    i, j = 0, len(nums) - 1  # 닫힌 구간 [0, n-1]을 초기화한다\n    while i &lt;= j:\n        m = (i + j) // 2  # 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target:\n            i = m + 1  # target은 구간 [m+1, j]에 있다\n        elif nums[m] &gt; target:\n            j = m - 1  # target은 구간 [i, m-1]에 있다\n        else:\n            return m  # target을 찾았으므로 삽입 위치 m을 반환한다\n    # target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nint binarySearchInsertionSimple(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            return m; // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nint binarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            return m; // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nint BinarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            return m; // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nfunc binarySearchInsertionSimple(nums []int, target int) int {\n    // 닫힌 구간 [0, n-1]을 초기화한다\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // 중간 인덱스 m을 계산한다\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target은 구간 [m+1, j]에 있다\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target은 구간 [i, m-1]에 있다\n            j = m - 1\n        } else {\n            // target을 찾았으므로 삽입 위치 m을 반환한다\n            return m\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nfunc binarySearchInsertionSimple(nums: [Int], target: Int) -&gt; Int {\n    // 닫힌 구간 [0, n-1]을 초기화한다\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target {\n            i = m + 1 // target은 구간 [m+1, j]에 있다\n        } else if nums[m] &gt; target {\n            j = m - 1 // target은 구간 [i, m-1]에 있다\n        } else {\n            return m // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nfunction binarySearchInsertionSimple(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // 중간 인덱스 m을 계산하며, Math.floor()로 내림한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            return m; // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nfunction binarySearchInsertionSimple(\n    nums: Array&lt;number&gt;,\n    target: number\n): number {\n    let i = 0,\n        j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // 중간 인덱스 m을 계산하며, Math.floor()로 내림한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            return m; // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nint binarySearchInsertionSimple(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // 중간 인덱스 m을 계산한다\n    if (nums[m] &lt; target) {\n      i = m + 1; // target은 구간 [m+1, j]에 있다\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target은 구간 [i, m-1]에 있다\n    } else {\n      return m; // target을 찾았으므로 삽입 위치 m을 반환한다\n    }\n  }\n  // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nfn binary_search_insertion_simple(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // 닫힌 구간 [0, n-1]을 초기화한다\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if nums[m as usize] &lt; target {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            return m;\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nint binarySearchInsertionSimple(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            return m; // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 없음) */\nfun binarySearchInsertionSimple(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1 // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // target은 구간 [i, m-1]에 있다\n        } else {\n            return m // target을 찾았으므로 삽입 위치 m을 반환한다\n        }\n    }\n    // target을 찾지 못했으므로 삽입 위치 i를 반환한다\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>### 삽입 위치의 이진 탐색(중복 원소 없음) ###\ndef binary_search_insertion_simple(nums, target)\n  # 닫힌 구간 [0, n-1]을 초기화한다\n  i, j = 0, nums.length - 1\n\n  while i &lt;= j\n    # 중간 인덱스 m을 계산한다\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # target은 구간 [m+1, j]에 있다\n    elsif nums[m] &gt; target\n      j = m - 1 # target은 구간 [i, m-1]에 있다\n    else\n      return m  # target을 찾았으므로 삽입 위치 m을 반환한다\n    end\n  end\n\n  i # target을 찾지 못했으므로 삽입 위치 i를 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 10 장   탐색","10.2   이진 탐색으로 삽입 위치 찾기"],"tags":[]},{"location":"chapter_searching/binary_search_insertion/#1022","level":2,"title":"10.2.2   중복 원소가 있는 경우","text":"<p>문제</p> <p>앞 문제에서 배열에 중복 원소가 있을 수 있다는 조건만 추가하고, 나머지 조건은 그대로 둔다.</p> <p>배열에 <code>target</code>이 여러 개 있다면 일반적인 이진 탐색으로는 그중 하나의 인덱스만 구할 수 있다. 그 원소의 왼쪽과 오른쪽에 <code>target</code>이 몇 개 더 있는지는 알 수 없다.</p> <p>문제에서는 목표 원소를 가장 왼쪽에 삽입하라고 한다. 따라서 배열에서 가장 왼쪽 <code>target</code>의 인덱스를 찾아야 한다. 먼저 아래 그림(10-5)과 같은 방법을 생각해 볼 수 있다.</p> <ol> <li>이진 탐색으로 임의의 <code>target</code>의 인덱스를 구하고 이를 \\(k\\)라고 한다.</li> <li>인덱스 \\(k\\)부터 왼쪽으로 선형 순회하여 가장 왼쪽 <code>target</code>을 찾으면 반환한다.</li> </ol> <p></p> <p> 그림 10-5   선형 탐색으로 중복 원소의 삽입 위치 찾기 </p> <p>이 방법도 가능하지만 선형 탐색이 포함되어 시간 복잡도가 \\(O(n)\\)이다. 배열에 중복된 <code>target</code>이 많으면 매우 비효율적이다.</p> <p>이제 이진 탐색 코드를 확장해 보자. 아래 그림(10-6)과 같이 전체 흐름은 그대로 유지한다. 매 반복에서 중간 인덱스 \\(m\\)을 계산하고, <code>target</code>과 <code>nums[m]</code>의 대소 관계에 따라 다음과 같이 처리한다.</p> <ul> <li><code>nums[m] &lt; target</code> 또는 <code>nums[m] &gt; target</code>이면 아직 <code>target</code>을 찾지 못했으므로, 일반적인 이진 탐색처럼 구간을 줄여 포인터 \\(i\\)와 \\(j\\)가 <code>target</code>에 가까워지게 한다.</li> <li><code>nums[m] == target</code>이면 <code>target</code>보다 작은 원소는 구간 \\([i, m - 1]\\)에 있으므로, \\(j = m - 1\\)로 구간을 줄여 포인터 \\(j\\)가 <code>target</code>보다 작은 원소에 가까워지게 한다.</li> </ul> <p>반복이 끝나면 \\(i\\)는 가장 왼쪽 <code>target</code>을, \\(j\\)는 <code>target</code>보다 작은 가장 오른쪽 원소를 가리킨다. 따라서 인덱스 \\(i\\)가 삽입 위치다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 10-6   이진 탐색으로 중복 원소의 삽입 위치를 찾는 단계 </p> <p>아래 코드를 보면 <code>nums[m] &gt; target</code>과 <code>nums[m] == target</code> 분기에서 수행하는 연산이 같으므로 두 분기를 합칠 수 있다.</p> <p>그렇더라도 조건을 나누어 두면 논리가 더 명확하고 가독성이 좋으므로 그대로 둘 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_insertion.py<pre><code>def binary_search_insertion(nums: list[int], target: int) -&gt; int:\n    \"\"\"삽입 위치의 이진 탐색(중복 원소 있음)\"\"\"\n    i, j = 0, len(nums) - 1  # 닫힌 구간 [0, n-1]을 초기화한다\n    while i &lt;= j:\n        m = (i + j) // 2  # 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target:\n            i = m + 1  # target은 구간 [m+1, j]에 있다\n        elif nums[m] &gt; target:\n            j = m - 1  # target은 구간 [i, m-1]에 있다\n        else:\n            j = m - 1  # target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n    # 삽입 위치 i를 반환한다\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nint binarySearchInsertion(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nint binarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nint BinarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nfunc binarySearchInsertion(nums []int, target int) int {\n    // 닫힌 구간 [0, n-1]을 초기화한다\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // 중간 인덱스 m을 계산한다\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target은 구간 [m+1, j]에 있다\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target은 구간 [i, m-1]에 있다\n            j = m - 1\n        } else {\n            // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n            j = m - 1\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nfunc binarySearchInsertion(nums: [Int], target: Int) -&gt; Int {\n    // 닫힌 구간 [0, n-1]을 초기화한다\n    var i = nums.startIndex\n    var j = nums.endIndex - 1\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if nums[m] &lt; target {\n            i = m + 1 // target은 구간 [m+1, j]에 있다\n        } else if nums[m] &gt; target {\n            j = m - 1 // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1 // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nfunction binarySearchInsertion(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // 중간 인덱스 m을 계산하며, Math.floor()로 내림한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nfunction binarySearchInsertion(nums: Array&lt;number&gt;, target: number): number {\n    let i = 0,\n        j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // 중간 인덱스 m을 계산하며, Math.floor()로 내림한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nint binarySearchInsertion(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // 중간 인덱스 m을 계산한다\n    if (nums[m] &lt; target) {\n      i = m + 1; // target은 구간 [m+1, j]에 있다\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target은 구간 [i, m-1]에 있다\n    } else {\n      j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n    }\n  }\n  // 삽입 위치 i를 반환한다\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\npub fn binary_search_insertion(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1); // 닫힌 구간 [0, n-1]을 초기화한다\n    while i &lt;= j {\n        let m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if nums[m as usize] &lt; target {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if nums[m as usize] &gt; target {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nint binarySearchInsertion(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1; // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1; // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i;\n}\n</code></pre> binary_search_insertion.kt<pre><code>/* 삽입 위치의 이진 탐색(중복 원소 있음) */\nfun binarySearchInsertion(nums: IntArray, target: Int): Int {\n    var i = 0\n    var j = nums.size - 1 // 닫힌 구간 [0, n-1]을 초기화한다\n    while (i &lt;= j) {\n        val m = i + (j - i) / 2 // 중간 인덱스 m을 계산한다\n        if (nums[m] &lt; target) {\n            i = m + 1 // target은 구간 [m+1, j]에 있다\n        } else if (nums[m] &gt; target) {\n            j = m - 1 // target은 구간 [i, m-1]에 있다\n        } else {\n            j = m - 1 // target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n        }\n    }\n    // 삽입 위치 i를 반환한다\n    return i\n}\n</code></pre> binary_search_insertion.rb<pre><code>### 삽입 위치의 이진 탐색(중복 원소 있음) ###\ndef binary_search_insertion(nums, target)\n  # 닫힌 구간 [0, n-1]을 초기화한다\n  i, j = 0, nums.length - 1\n\n  while i &lt;= j\n    # 중간 인덱스 m을 계산한다\n    m = (i + j) / 2\n\n    if nums[m] &lt; target\n      i = m + 1 # target은 구간 [m+1, j]에 있다\n    elsif nums[m] &gt; target\n      j = m - 1 # target은 구간 [i, m-1]에 있다\n    else\n      j = m - 1 # target보다 작은 가장 오른쪽 원소는 구간 [i, m-1]에 있다\n    end\n  end\n\n  i # 삽입 위치 i를 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>참고</p> <p>이 절의 코드는 모두 '양끝이 닫힌 구간'을 사용한다. 관심 있는 독자는 '왼쪽은 닫히고 오른쪽은 열린 구간'으로 직접 구현해 보자.</p> <p>종합하면 이진 탐색은 포인터 \\(i\\)와 \\(j\\)에 각각 탐색 목표를 정해 주는 과정이다. 목표는 특정 원소(예: <code>target</code>)일 수도 있고, 원소의 범위(예: <code>target</code>보다 작은 원소)일 수도 있다.</p> <p>구간을 계속 반으로 나누는 동안 포인터 \\(i\\)와 \\(j\\)는 미리 정한 목표에 점차 가까워진다. 결국 답을 찾거나 경계를 넘어선 뒤 멈춘다.</p>","path":["제 10 장   탐색","10.2   이진 탐색으로 삽입 위치 찾기"],"tags":[]},{"location":"chapter_searching/exercises/","level":1,"title":"10.7   연습 문제","text":"","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/exercises/#1071","level":2,"title":"10.7.1   개념 확인","text":"","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/exercises/#1","level":3,"title":"1.   이진 탐색은 탐색 구간을 어떻게 줄이는가","text":"<p>정렬된 배열 <code>[2, 5, 8, 12, 16, 23, 38]</code>에서 16을 찾는다. 양 끝을 포함하는 닫힌 구간 <code>[i, j]</code>를 사용하며, 중간 인덱스는 \\(m=i+(j-i)/2\\)다. 나눗셈 결과의 소수 부분은 버린다.</p> <p>목표 값을 찾을 때까지 매 단계의 <code>(i, j, m)</code>과 중간 원소, 다음 단계에서 구간을 어떻게 줄이는지 적어 보자.</p> 참고 답안 <p>각 단계의 탐색 과정은 다음과 같다.</p> 단계 <code>(i, j, m)</code> 중간 원소 다음 동작 1 <code>(0, 6, 3)</code> 12 <code>12 &lt; 16</code>이므로 <code>i = 4</code>로 설정 2 <code>(4, 6, 5)</code> 23 <code>23 &gt; 16</code>이므로 <code>j = 4</code>로 설정 3 <code>(4, 4, 4)</code> 16 목표 값을 찾았으므로 인덱스 4를 반환 <p>배열이 정렬되어 있으므로 중간 값이 목표 값보다 작으면 중간 원소와 그 왼쪽을 제외할 수 있다. 중간 값이 목표 값보다 크면 중간 원소와 그 오른쪽을 제외할 수 있다.</p>","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/exercises/#2","level":3,"title":"2.   중복 원소의 왼쪽 경계와 오른쪽 경계","text":"<p>배열 <code>[1, 2, 2, 2, 4, 6]</code>에서 숫자 2를 찾는다. 한 학생이 이진 탐색으로 인덱스 2에서 목표 값을 찾자마자 반환한 뒤, “인덱스 2가 숫자 2의 왼쪽 경계다.”라고 말했다.</p> <ol> <li>이 학생의 말이 맞는가? 숫자 2의 왼쪽 경계와 오른쪽 경계는 각각 어디인지 이유와 함께 설명하라.</li> <li>왼쪽 경계를 찾을 때 중간 원소가 목표 값과 같다면, 다음에는 어느 쪽을 계속 탐색해야 하는가?</li> <li>오른쪽 경계를 찾을 때는 어느 쪽을 계속 탐색해야 하는가? 탐색 과정 전체를 적지 말고 방향만 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>이 학생의 말은 틀렸다. 2를 하나 찾자마자 반환하면 어떤 2를 찾았다는 것만 보장할 수 있고, 가장 왼쪽이나 가장 오른쪽의 2를 찾았다고 보장할 수는 없다.     이 문제에서 왼쪽 경계는 인덱스 1, 오른쪽 경계는 인덱스 3이다.</p> </li> <li> <p>왼쪽 경계를 찾을 때는 중간 원소가 2와 같더라도 왼쪽을 계속 탐색해야 한다.     예를 들어 양 끝을 포함하는 닫힌 구간을 사용하는 경우 <code>j = m - 1</code>로 설정한다.</p> </li> <li> <p>오른쪽 경계를 찾을 때는 중간 원소가 2와 같으면 오른쪽을 계속 탐색해야 한다.     예를 들어 <code>i = m + 1</code>로 설정한다.</p> </li> </ol>","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/exercises/#3","level":3,"title":"3.   데이터에 따라 어떤 탐색 방법을 선택해야 하는가","text":"<p>다음 세 상황에 알맞은 방법을 ‘선형 탐색, 이진 탐색, 해시 테이블’ 중에서 고르고 이유를 설명하라.</p> <ol> <li>이미 정렬되어 있고 더 이상 변하지 않는 정수 \\(10^7\\)개에서 반복해서 값을 찾으며, 다른 자료 구조를 추가로 만들지 않는다.</li> <li>삽입과 삭제가 빈번한 데이터 집합에서 특정 키가 있는지 반복해서 확인한다. 정렬 상태를 유지할 필요가 없고 범위 탐색도 하지 않는다.</li> <li>정렬되지 않은 배열에서 특정 값을 한 번만 찾는다.</li> </ol> 참고 답안 <ol> <li>이진 탐색: 데이터가 정렬되어 있고 변하지 않으므로 \\(O(\\log n)\\) 시간에 탐색할 수 있으며, 추가 공간도 필요하지 않다.</li> <li>해시 테이블: 해시 함수가 키를 각 버킷에 비교적 고르게 분산한다면 삽입, 삭제, 키를 이용한 탐색의 평균 시간 복잡도가 모두 \\(O(1)\\)이다.</li> <li> <p>처음부터 끝까지 직접 순회: 한 번만 찾는 경우에는 정렬하거나 해시 테이블을 만드는 데에도 먼저 배열 전체를 처리해야 하므로,     이번 작업의 전체 작업량이 줄어들지 않는다.</p> <p>어떤 방법을 선택할지는 데이터의 정렬 여부, 추가 자료 구조의 허용 여부, 조회 횟수, 수행해야 하는 연산에 따라 달라진다.</p> </li> </ol>","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/exercises/#1072","level":2,"title":"10.7.2   프로그래밍 연습","text":"","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/exercises/#1_1","level":3,"title":"1.   정렬된 배열의 이진 탐색","text":"<p>엄격한 오름차순으로 정렬된 정수 배열 <code>nums</code>와 목표 값 <code>target</code>이 주어진다. 이진 탐색으로 <code>target</code>을 찾아라. 존재하면 배열 인덱스를 반환하고, 존재하지 않으면 -1을 반환하라.</p> 풀이 힌트 <ol> <li>초기 구간은 left = 0, right = n - 1이며, 구간이 비어 있지 않을 조건은 left &lt;= right다.</li> <li>mid = left + (right - left) // 2로 중간 인덱스를 계산한다.</li> <li><code>nums[mid]</code>가 <code>target</code>보다 작으면 왼쪽 경계를 <code>mid + 1</code>로 옮긴다. <code>nums[mid]</code>가 <code>target</code>보다 크면 오른쪽 경계를 <code>mid - 1</code>로 옮긴다. 같으면 즉시 반환한다.</li> </ol> <p>LeetCode</p>","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/exercises/#2_1","level":3,"title":"2.   정렬된 배열의 삽입 위치","text":"<p>엄격한 오름차순으로 정렬된 정수 배열 <code>nums</code>와 목표 값 <code>target</code>이 주어진다.</p> <p><code>target</code>이 이미 배열에 있으면 그 인덱스를 반환하라. 없으면 <code>target</code>을 삽입해도 엄격한 오름차순을 유지할 수 있는 위치를 반환하라. 답은 0일 수도 있고 배열 길이와 같을 수도 있다. 이진 탐색을 사용하라.</p> 풀이 힌트 <ol> <li>답은 0일 수도 있고 배열 길이 n일 수도 있다.</li> <li>양 끝을 포함하는 닫힌 구간을 사용할 때 <code>nums[mid]</code>가 <code>target</code>보다 크거나 같으면 <code>right = mid - 1</code>로 설정하여 더 왼쪽의 위치를 계속 확인한다. 그렇지 않으면 <code>left = mid + 1</code>로 설정한다.</li> <li>반복문이 끝나면 left가 삽입 위치다.</li> </ol> <p>LeetCode</p>","path":["제 10 장   탐색","10.7   연습 문제"],"tags":[]},{"location":"chapter_searching/replace_linear_by_hashing/","level":1,"title":"10.4   해시를 이용한 최적화 전략","text":"<p>알고리즘 문제에서는 선형 탐색을 해시 탐색으로 바꾸어 시간 복잡도를 줄이는 경우가 많다. 한 가지 문제를 통해 더 자세히 알아보자.</p> <p>문제</p> <p>정수 배열 <code>nums</code>와 목표 원소 <code>target</code>이 주어진다. 배열에서 '합'이 <code>target</code>인 두 원소를 찾아 그 인덱스를 반환하라. 답이 여러 개라면 아무거나 하나를 반환해도 된다.</p>","path":["제 10 장   탐색","10.4   해시를 이용한 최적화 전략"],"tags":[]},{"location":"chapter_searching/replace_linear_by_hashing/#1041","level":2,"title":"10.4.1   선형 탐색: 시간을 더 써서 공간 절약","text":"<p>가능한 모든 조합을 직접 순회해 보자. 아래 그림(10-9)과 같이 이중 반복문을 사용하여 매번 두 정수의 합이 <code>target</code>인지 확인하고, 맞으면 두 인덱스를 반환한다.</p> <p></p> <p> 그림 10-9   선형 탐색으로 두 수의 합 구하기 </p> <p>코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby two_sum.py<pre><code>def two_sum_brute_force(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"방법 1: 완전 열거\"\"\"\n    # 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for i in range(len(nums) - 1):\n        for j in range(i + 1, len(nums)):\n            if nums[i] + nums[j] == target:\n                return [i, j]\n    return []\n</code></pre> two_sum.cpp<pre><code>/* 방법 1: 완전 열거 */\nvector&lt;int&gt; twoSumBruteForce(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return {i, j};\n        }\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* 방법 1: 완전 열거 */\nint[] twoSumBruteForce(int[] nums, int target) {\n    int size = nums.length;\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return new int[] { i, j };\n        }\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* 방법 1: 완전 열거 */\nint[] TwoSumBruteForce(int[] nums, int target) {\n    int size = nums.Length;\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return [i, j];\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* 방법 1: 완전 열거 */\nfunc twoSumBruteForce(nums []int, target int) []int {\n    size := len(nums)\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for i := 0; i &lt; size-1; i++ {\n        for j := i + 1; j &lt; size; j++ {\n            if nums[i]+nums[j] == target {\n                return []int{i, j}\n            }\n        }\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* 방법 1: 완전 열거 */\nfunc twoSumBruteForce(nums: [Int], target: Int) -&gt; [Int] {\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for i in nums.indices.dropLast() {\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[i] + nums[j] == target {\n                return [i, j]\n            }\n        }\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* 방법 1: 완전 열거 */\nfunction twoSumBruteForce(nums, target) {\n    const n = nums.length;\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* 방법 1: 완전 열거 */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\n    const n = nums.length;\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* 방법 1: 완전 열거 */\nList&lt;int&gt; twoSumBruteForce(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n  for (var i = 0; i &lt; size - 1; i++) {\n    for (var j = i + 1; j &lt; size; j++) {\n      if (nums[i] + nums[j] == target) return [i, j];\n    }\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* 방법 1: 완전 열거 */\npub fn two_sum_brute_force(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    let size = nums.len();\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for i in 0..size - 1 {\n        for j in i + 1..size {\n            if nums[i] + nums[j] == target {\n                return Some(vec![i as i32, j as i32]);\n            }\n        }\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* 방법 1: 완전 열거 */\nint *twoSumBruteForce(int *nums, int numsSize, int target, int *returnSize) {\n    for (int i = 0; i &lt; numsSize; ++i) {\n        for (int j = i + 1; j &lt; numsSize; ++j) {\n            if (nums[i] + nums[j] == target) {\n                int *res = malloc(sizeof(int) * 2);\n                res[0] = i, res[1] = j;\n                *returnSize = 2;\n                return res;\n            }\n        }\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* 방법 1: 완전 열거 */\nfun twoSumBruteForce(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // 이중 반복문이며 시간 복잡도는 O(n^2)이다\n    for (i in 0..&lt;size - 1) {\n        for (j in i + 1..&lt;size) {\n            if (nums[i] + nums[j] == target) return intArrayOf(i, j)\n        }\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>### 방법 1: 완전 열거 ###\ndef two_sum_brute_force(nums, target)\n  # 이중 반복문이며 시간 복잡도는 O(n^2)이다\n  for i in 0...(nums.length - 1)\n    for j in (i + 1)...nums.length\n      return [i, j] if nums[i] + nums[j] == target\n    end\n  end\n\n  []\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>이 방법의 시간 복잡도는 \\(O(n^2)\\), 공간 복잡도는 \\(O(1)\\)이다. 데이터가 많으면 시간이 매우 오래 걸린다.</p>","path":["제 10 장   탐색","10.4   해시를 이용한 최적화 전략"],"tags":[]},{"location":"chapter_searching/replace_linear_by_hashing/#1042","level":2,"title":"10.4.2   해시 탐색: 공간을 더 써서 시간 절약","text":"<p>키와 값으로 각각 배열 원소와 그 인덱스를 저장하는 해시 테이블을 사용해 보자. 배열을 순회하면서 매번 아래 그림(10-10)의 단계를 수행한다.</p> <ol> <li>수 <code>target - nums[i]</code>가 해시 테이블에 있는지 확인하고, 있으면 두 원소의 인덱스를 바로 반환한다.</li> <li><code>nums[i]</code>와 인덱스 <code>i</code>로 이루어진 키-값 쌍을 해시 테이블에 추가한다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> 그림 10-10   보조 해시 테이블로 두 수의 합 구하기 </p> <p>구현 코드는 다음과 같으며, 단일 반복문만 있으면 된다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby two_sum.py<pre><code>def two_sum_hash_table(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"방법 2: 보조 해시 테이블\"\"\"\n    # 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    dic = {}\n    # 단일 반복문이며 시간 복잡도는 O(n)이다\n    for i in range(len(nums)):\n        if target - nums[i] in dic:\n            return [dic[target - nums[i]], i]\n        dic[nums[i]] = i\n    return []\n</code></pre> two_sum.cpp<pre><code>/* 방법 2: 보조 해시 테이블 */\nvector&lt;int&gt; twoSumHashTable(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    unordered_map&lt;int, int&gt; dic;\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.find(target - nums[i]) != dic.end()) {\n            return {dic[target - nums[i]], i};\n        }\n        dic.emplace(nums[i], i);\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* 방법 2: 보조 해시 테이블 */\nint[] twoSumHashTable(int[] nums, int target) {\n    int size = nums.length;\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    Map&lt;Integer, Integer&gt; dic = new HashMap&lt;&gt;();\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.containsKey(target - nums[i])) {\n            return new int[] { dic.get(target - nums[i]), i };\n        }\n        dic.put(nums[i], i);\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* 방법 2: 보조 해시 테이블 */\nint[] TwoSumHashTable(int[] nums, int target) {\n    int size = nums.Length;\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    Dictionary&lt;int, int&gt; dic = [];\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.ContainsKey(target - nums[i])) {\n            return [dic[target - nums[i]], i];\n        }\n        dic.Add(nums[i], i);\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* 방법 2: 보조 해시 테이블 */\nfunc twoSumHashTable(nums []int, target int) []int {\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    hashTable := map[int]int{}\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for idx, val := range nums {\n        if preIdx, ok := hashTable[target-val]; ok {\n            return []int{preIdx, idx}\n        }\n        hashTable[val] = idx\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* 방법 2: 보조 해시 테이블 */\nfunc twoSumHashTable(nums: [Int], target: Int) -&gt; [Int] {\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    var dic: [Int: Int] = [:]\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for i in nums.indices {\n        if let j = dic[target - nums[i]] {\n            return [j, i]\n        }\n        dic[nums[i]] = i\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* 방법 2: 보조 해시 테이블 */\nfunction twoSumHashTable(nums, target) {\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    let m = {};\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (m[target - nums[i]] !== undefined) {\n            return [m[target - nums[i]], i];\n        } else {\n            m[nums[i]] = i;\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* 방법 2: 보조 해시 테이블 */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    let m: Map&lt;number, number&gt; = new Map();\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for (let i = 0; i &lt; nums.length; i++) {\n        let index = m.get(target - nums[i]);\n        if (index !== undefined) {\n            return [index, i];\n        } else {\n            m.set(nums[i], i);\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* 방법 2: 보조 해시 테이블 */\nList&lt;int&gt; twoSumHashTable(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n  Map&lt;int, int&gt; dic = HashMap();\n  // 단일 반복문이며 시간 복잡도는 O(n)이다\n  for (var i = 0; i &lt; size; i++) {\n    if (dic.containsKey(target - nums[i])) {\n      return [dic[target - nums[i]]!, i];\n    }\n    dic.putIfAbsent(nums[i], () =&gt; i);\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* 방법 2: 보조 해시 테이블 */\npub fn two_sum_hash_table(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    let mut dic = HashMap::new();\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for (i, num) in nums.iter().enumerate() {\n        match dic.get(&amp;(target - num)) {\n            Some(v) =&gt; return Some(vec![*v as i32, i as i32]),\n            None =&gt; dic.insert(num, i as i32),\n        };\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* 해시 테이블 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // uthash.h를 기반으로 구현한다\n} HashTable;\n\n/* 해시 테이블 조회 */\nHashTable *find(HashTable *h, int key) {\n    HashTable *tmp;\n    HASH_FIND_INT(h, &amp;key, tmp);\n    return tmp;\n}\n\n/* 해시 테이블에 원소 삽입 */\nvoid insert(HashTable **h, int key, int val) {\n    HashTable *t = find(*h, key);\n    if (t == NULL) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = key, tmp-&gt;val = val;\n        HASH_ADD_INT(*h, key, tmp);\n    } else {\n        t-&gt;val = val;\n    }\n}\n\n/* 방법 2: 보조 해시 테이블 */\nint *twoSumHashTable(int *nums, int numsSize, int target, int *returnSize) {\n    HashTable *hashtable = NULL;\n    for (int i = 0; i &lt; numsSize; i++) {\n        HashTable *t = find(hashtable, target - nums[i]);\n        if (t != NULL) {\n            int *res = malloc(sizeof(int) * 2);\n            res[0] = t-&gt;val, res[1] = i;\n            *returnSize = 2;\n            return res;\n        }\n        insert(&amp;hashtable, nums[i], i);\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.kt<pre><code>/* 방법 2: 보조 해시 테이블 */\nfun twoSumHashTable(nums: IntArray, target: Int): IntArray {\n    val size = nums.size\n    // 보조 해시 테이블의 공간 복잡도는 O(n)이다\n    val dic = HashMap&lt;Int, Int&gt;()\n    // 단일 반복문이며 시간 복잡도는 O(n)이다\n    for (i in 0..&lt;size) {\n        if (dic.containsKey(target - nums[i])) {\n            return intArrayOf(dic[target - nums[i]]!!, i)\n        }\n        dic[nums[i]] = i\n    }\n    return IntArray(0)\n}\n</code></pre> two_sum.rb<pre><code>### 방법 2: 보조 해시 테이블 ###\ndef two_sum_hash_table(nums, target)\n  # 보조 해시 테이블의 공간 복잡도는 O(n)이다\n  dic = {}\n  # 단일 반복문이며 시간 복잡도는 O(n)이다\n  for i in 0...nums.length\n    return [dic[target - nums[i]], i] if dic.has_key?(target - nums[i])\n\n    dic[nums[i]] = i\n  end\n\n  []\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>이 방법은 해시 탐색으로 시간 복잡도를 \\(O(n^2)\\)에서 \\(O(n)\\)으로 줄여 실행 효율을 크게 높인다.</p> <p>해시 테이블을 추가로 유지해야 하므로 공간 복잡도는 \\(O(n)\\)이다. 그렇더라도 시간과 공간 효율의 균형이 더 좋으므로 이 문제의 최적 풀이에 해당한다.</p>","path":["제 10 장   탐색","10.4   해시를 이용한 최적화 전략"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/","level":1,"title":"10.5   탐색 알고리즘 다시 보기","text":"<p>탐색 알고리즘(searching algorithm)은 배열, 연결 리스트, 트리, 그래프 등의 자료 구조에서 특정 조건을 만족하는 원소 하나 또는 여러 개를 찾는 데 사용한다.</p> <p>탐색 알고리즘은 구현 아이디어에 따라 다음 두 종류로 나뉜다.</p> <ul> <li>배열, 연결 리스트, 트리, 그래프의 순회처럼 자료 구조를 순회하여 목표 원소를 찾는다.</li> <li>이진 탐색, 해시 탐색, 이진 탐색 트리의 탐색처럼 데이터의 구성 방식이나 데이터에 대한 사전 정보를 이용하여 원소를 효율적으로 찾는다.</li> </ul> <p>이 내용은 모두 앞 장들에서 이미 소개했으므로 탐색 알고리즘은 낯설지 않을 것이다. 이 절에서는 더 체계적인 관점에서 탐색 알고리즘을 다시 살펴본다.</p>","path":["제 10 장   탐색","10.5   탐색 알고리즘 다시 보기"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/#1051","level":2,"title":"10.5.1   완전 탐색","text":"<p>완전 탐색은 자료 구조의 각 원소를 순회하여 목표 원소를 찾는다.</p> <ul> <li>'선형 탐색'은 배열이나 연결 리스트 같은 선형 자료 구조에 적합하다. 자료 구조의 한쪽 끝에서 원소를 하나씩 방문하며, 목표 원소를 찾거나 찾지 못한 채 반대쪽 끝에 도달하면 종료한다.</li> <li>'너비 우선 탐색'과 '깊이 우선 탐색'은 그래프와 트리의 두 가지 순회 전략이다. 너비 우선 탐색은 시작 노드에서 한 층씩 탐색하며 가까운 노드부터 먼 노드 순서로 방문한다. 깊이 우선 탐색은 시작 노드에서 한 경로를 따라 끝까지 간 뒤 되돌아와 다른 경로를 시도하며, 전체 자료 구조를 순회할 때까지 반복한다.</li> </ul> <p>완전 탐색은 간단하고 범용성이 좋으며, 데이터를 전처리하거나 추가 자료 구조를 사용할 필요가 없다.</p> <p>하지만 이러한 알고리즘의 시간 복잡도는 \\(O(n)\\)이다. 여기서 \\(n\\)은 원소 수이며, 데이터가 많으면 성능이 떨어진다.</p>","path":["제 10 장   탐색","10.5   탐색 알고리즘 다시 보기"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/#1052","level":2,"title":"10.5.2   적응형 탐색","text":"<p>적응형 탐색은 정렬 여부 같은 데이터 고유의 성질을 이용하여 탐색 과정을 최적화하고, 목표 원소를 더 효율적으로 찾는다.</p> <ul> <li>'이진 탐색'은 데이터의 정렬 상태를 이용하여 효율적으로 찾으며, 배열에만 적합하다.</li> <li>'해시 탐색'은 해시 테이블로 탐색 데이터와 목표 데이터 사이에 키-값 매핑을 만들어 조회한다.</li> <li>'트리 탐색'은 이진 탐색 트리 같은 특정 트리 구조에서 노드 값을 비교하여 탐색 대상에서 빠르게 제외하고 목표 원소를 찾는다.</li> </ul> <p>이러한 알고리즘은 효율이 높아 시간 복잡도가 \\(O(\\log n)\\), 나아가 \\(O(1)\\)에 이를 수 있다.</p> <p>하지만 이런 알고리즘을 사용하려면 데이터를 전처리해야 하는 경우가 많다. 예를 들어 이진 탐색은 배열을 미리 정렬해야 하며, 해시 탐색과 트리 탐색은 추가 자료 구조가 필요하다. 이러한 자료 구조를 유지하는 데도 시간과 공간이 추가로 든다.</p> <p>참고</p> <p>적응형 탐색 알고리즘은 조회 알고리즘이라고도 하며, 주로 특정 자료 구조에서 목표 원소를 빠르게 조회하는 데 사용한다.</p>","path":["제 10 장   탐색","10.5   탐색 알고리즘 다시 보기"],"tags":[]},{"location":"chapter_searching/searching_algorithm_revisited/#1053","level":2,"title":"10.5.3   탐색 방법 선택","text":"<p>크기가 \\(n\\)인 데이터 집합이 주어지면 선형 탐색, 이진 탐색, 트리 탐색, 해시 탐색 등 다양한 방법으로 목표 원소를 찾을 수 있다. 각 방법의 동작 원리는 아래 그림(10-11)과 같다.</p> <p></p> <p> 그림 10-11   다양한 탐색 전략 </p> <p>위 방법들의 연산 효율과 특성은 아래 표(10-1)와 같다.</p> <p> 표 10-1   탐색 알고리즘 효율 비교 </p> 선형 탐색 이진 탐색 트리 탐색 해시 탐색 원소 탐색 \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) 원소 삽입 \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) 원소 삭제 \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) 추가 공간 \\(O(1)\\) \\(O(1)\\) \\(O(n)\\) \\(O(n)\\) 데이터 전처리 / 정렬 \\(O(n \\log n)\\) 트리 구성 \\(O(n \\log n)\\) 해시 테이블 구성 \\(O(n)\\) 데이터 정렬 여부 정렬되지 않음 정렬됨 정렬됨 정렬되지 않음 <p>탐색 알고리즘을 선택할 때는 데이터 규모, 요구되는 탐색 성능, 데이터 조회와 갱신 빈도 등도 고려해야 한다.</p> <p>선형 탐색</p> <ul> <li>범용성이 좋으며 데이터 전처리가 전혀 필요하지 않다. 데이터를 한 번만 조회한다면 나머지 세 방법은 전처리 시간만으로도 선형 탐색보다 오래 걸린다.</li> <li>데이터 규모가 작을 때 적합하다. 이때는 시간 복잡도가 효율에 미치는 영향이 작다.</li> <li>데이터를 추가로 관리할 필요가 없으므로 데이터 갱신이 잦은 상황에 적합하다.</li> </ul> <p>이진 탐색</p> <ul> <li>데이터가 많을 때 적합하며, 최악의 시간 복잡도가 \\(O(\\log n)\\)으로 성능이 안정적이다.</li> <li>배열을 저장하려면 연속된 메모리 공간이 필요하므로 데이터가 지나치게 많으면 안 된다.</li> <li>정렬된 배열을 유지하는 비용이 크므로 데이터 추가와 삭제가 잦은 상황에는 적합하지 않다.</li> </ul> <p>해시 탐색</p> <ul> <li>평균 시간 복잡도가 \\(O(1)\\)이므로 매우 높은 조회 성능이 필요한 상황에 적합하다.</li> <li>해시 테이블은 데이터의 정렬 상태를 유지할 수 없으므로 정렬된 데이터나 범위 탐색이 필요한 상황에는 적합하지 않다.</li> <li>해시 함수와 해시 충돌 처리 전략에 크게 의존하므로 성능이 저하될 위험이 비교적 크다.</li> <li>충돌을 최대한 줄여 좋은 조회 성능을 내려면 추가 공간이 필요하므로 데이터가 지나치게 많은 경우에는 적합하지 않다.</li> </ul> <p>트리 탐색</p> <ul> <li>트리 노드는 메모리에 분산 저장되므로 대규모 데이터에 적합하다.</li> <li>데이터의 정렬 상태를 유지하거나 범위 탐색을 해야 하는 상황에 적합하다.</li> <li>노드를 계속 추가하고 삭제하다 보면 이진 탐색 트리가 한쪽으로 치우쳐 시간 복잡도가 \\(O(n)\\)으로 저하될 수 있다.</li> <li>AVL 트리나 레드-블랙 트리를 사용하면 각 연산을 안정적으로 \\(O(\\log n)\\)에 수행할 수 있지만, 트리의 균형을 유지하는 데 추가 비용이 든다.</li> </ul>","path":["제 10 장   탐색","10.5   탐색 알고리즘 다시 보기"],"tags":[]},{"location":"chapter_searching/summary/","level":1,"title":"10.6   요약","text":"","path":["제 10 장   탐색","10.6   요약"],"tags":[]},{"location":"chapter_searching/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>이진 탐색은 데이터가 정렬되어 있다는 성질을 이용하여 반복할 때마다 탐색 구간을 절반씩 줄인다. 입력 데이터가 정렬되어 있어야 하며, 배열이나 배열 기반 자료 구조에만 적합하다.</li> <li>완전 탐색은 자료 구조를 순회하여 데이터를 찾는다. 선형 탐색은 배열과 연결 리스트에, 너비 우선 탐색과 깊이 우선 탐색은 그래프와 트리에 적합하다. 범용성이 좋고 전처리가 필요하지 않지만, 시간 복잡도가 \\(O(n)\\)으로 비교적 높다.</li> <li>해시 탐색, 트리 탐색, 이진 탐색은 특정 자료 구조에서 목표 원소를 빠르게 찾는 효율적인 탐색 방법이다. 시간 복잡도가 \\(O(\\log n)\\) 또는 \\(O(1)\\)에 이를 만큼 효율적이지만, 보통 추가 자료 구조가 필요하다.</li> <li>실제로는 데이터 규모, 요구되는 탐색 성능, 데이터 조회와 갱신 빈도 등을 구체적으로 분석하여 적합한 탐색 방법을 골라야 한다.</li> <li>선형 탐색은 작거나 자주 갱신되는 데이터에, 이진 탐색은 크고 정렬된 데이터에 적합하다. 해시 탐색은 높은 조회 효율이 필요하면서 범위 조회는 필요 없는 데이터에, 트리 탐색은 순서 유지와 범위 조회가 필요한 대규모 동적 데이터에 적합하다.</li> <li>선형 탐색을 해시 탐색으로 바꾸는 것은 실행 시간을 최적화하는 흔한 전략으로, 시간 복잡도를 \\(O(n)\\)에서 \\(O(1)\\)로 줄일 수 있다.</li> </ul>","path":["제 10 장   탐색","10.6   요약"],"tags":[]},{"location":"chapter_sorting/","level":1,"title":"제 11 장   정렬","text":"<p>이 장을 시작하며</p> <p>정렬은 혼란을 질서로 바꾸는 마법의 열쇠와 같아서, 데이터를 더 효율적으로 이해하고 처리하게 해 준다.</p> <p>단순한 오름차순이든 복잡한 분류와 배열이든, 정렬은 데이터가 지닌 조화로운 아름다움을 보여 준다.</p>","path":["제 11 장   정렬"],"tags":[]},{"location":"chapter_sorting/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>11.1   정렬 알고리즘</li> <li>11.2   선택 정렬</li> <li>11.3   버블 정렬</li> <li>11.4   삽입 정렬</li> <li>11.5   퀵 정렬</li> <li>11.6   병합 정렬</li> <li>11.7   힙 정렬</li> <li>11.8   버킷 정렬</li> <li>11.9   계수 정렬</li> <li>11.10   기수 정렬</li> <li>11.11   요약</li> <li>11.12   연습 문제</li> </ul>","path":["제 11 장   정렬"],"tags":[]},{"location":"chapter_sorting/bubble_sort/","level":1,"title":"11.3   버블 정렬","text":"<p>버블 정렬(bubble sort)은 인접한 원소를 계속 비교하고 교환하여 정렬한다. 이 과정이 거품이 아래에서 위로 올라오는 모습과 비슷하여 버블 정렬이라는 이름이 붙었다.</p> <p>아래 그림(11-4)과 같이 원소 교환으로 거품이 올라오는 과정을 흉내 낼 수 있다. 배열의 맨 왼쪽부터 오른쪽으로 순회하며 인접한 두 원소를 비교하고, '왼쪽 원소 &gt; 오른쪽 원소'이면 둘을 교환한다. 순회를 마치면 가장 큰 원소가 배열의 맨 오른쪽으로 이동한다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 11-4   원소 교환으로 거품의 이동 구현 </p>","path":["제 11 장   정렬","11.3   버블 정렬"],"tags":[]},{"location":"chapter_sorting/bubble_sort/#1131","level":2,"title":"11.3.1   알고리즘 과정","text":"<p>배열의 길이가 \\(n\\)일 때 버블 정렬의 단계는 아래 그림(11-5)과 같다.</p> <ol> <li>먼저 \\(n\\) 개 원소에 '버블 연산'을 수행하여 배열에서 가장 큰 원소를 올바른 위치로 옮긴다.</li> <li>이어서 남은 \\(n - 1\\) 개 원소에 '버블 연산'을 수행하여 두 번째로 큰 원소를 올바른 위치로 옮긴다.</li> <li>같은 과정을 반복하여 \\(n - 1\\) 회의 '버블 연산'을 수행하면 큰 순서로 \\(n - 1\\) 개 원소가 모두 올바른 위치에 놓인다.</li> <li>남은 원소 하나는 반드시 가장 작은 원소이므로 정렬할 필요가 없으며, 배열 정렬이 완료된다.</li> </ol> <p></p> <p> 그림 11-5   버블 정렬 과정 </p> <p>예시 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby bubble_sort.py<pre><code>def bubble_sort(nums: list[int]):\n    \"\"\"버블 정렬\"\"\"\n    n = len(nums)\n    # 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in range(n - 1, 0, -1):\n        # 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # nums[j]와 nums[j + 1]을 교환한다\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n</code></pre> bubble_sort.cpp<pre><code>/* 버블 정렬 */\nvoid bubbleSort(vector&lt;int&gt; &amp;nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                // 여기서는 std::swap() 함수를 사용한다\n                swap(nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* 버블 정렬 */\nvoid bubbleSort(int[] nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* 버블 정렬 */\nvoid BubbleSort(int[] nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* 버블 정렬 */\nfunc bubbleSort(nums []int) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* 버블 정렬 */\nfunc bubbleSort(nums: inout [Int]) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in nums.indices.dropFirst().reversed() {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                nums.swapAt(j, j + 1)\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* 버블 정렬 */\nfunction bubbleSort(nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* 버블 정렬 */\nfunction bubbleSort(nums: number[]): void {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* 버블 정렬 */\nvoid bubbleSort(List&lt;int&gt; nums) {\n  // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // nums[j]와 nums[j + 1]을 교환한다\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n      }\n    }\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* 버블 정렬 */\nfn bubble_sort(nums: &amp;mut [i32]) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in (1..nums.len()).rev() {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                nums.swap(j, j + 1);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* 버블 정렬 */\nvoid bubbleSort(int nums[], int size) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = size - 1; i &gt; 0; i--) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* 버블 정렬 */\nfun bubbleSort(nums: IntArray) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (i in nums.size - 1 downTo 1) {\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>### 버블 정렬 ###\ndef bubble_sort(nums)\n  n = nums.length\n  # 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n  for i in (n - 1).downto(1)\n    # 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # nums[j]와 nums[j + 1]을 교환한다\n        nums[j], nums[j + 1] = nums[j + 1], nums[j]\n      end\n    end\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.3   버블 정렬"],"tags":[]},{"location":"chapter_sorting/bubble_sort/#1132","level":2,"title":"11.3.2   효율 최적화","text":"<p>어느 회의 '버블 연산'에서 교환이 한 번도 일어나지 않았다면 배열은 이미 정렬되어 있으므로 결과를 바로 반환할 수 있다. 따라서 이 상황을 감지하는 플래그 <code>flag</code>를 추가하고, 발생하는 즉시 반환할 수 있다.</p> <p>최적화 후에도 버블 정렬의 최악 및 평균 시간 복잡도는 \\(O(n^2)\\)이다. 하지만 입력 배열이 완전히 정렬되어 있으면 최선 시간 복잡도인 \\(O(n)\\)에 도달한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby bubble_sort.py<pre><code>def bubble_sort_with_flag(nums: list[int]):\n    \"\"\"버블 정렬(플래그 최적화)\"\"\"\n    n = len(nums)\n    # 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in range(n - 1, 0, -1):\n        flag = False  # 플래그를 초기화한다\n        # 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # nums[j]와 nums[j + 1]을 교환한다\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n                flag = True  # 원소가 교환되었음을 기록한다\n        if not flag:\n            break  # 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n</code></pre> bubble_sort.cpp<pre><code>/* 버블 정렬(플래그 최적화)*/\nvoid bubbleSortWithFlag(vector&lt;int&gt; &amp;nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        bool flag = false; // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                // 여기서는 std::swap() 함수를 사용한다\n                swap(nums[j], nums[j + 1]);\n                flag = true; // 원소가 교환되었음을 기록한다\n            }\n        }\n        if (!flag)\n            break; // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* 버블 정렬(플래그 최적화) */\nvoid bubbleSortWithFlag(int[] nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        boolean flag = false; // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // 원소가 교환되었음을 기록한다\n            }\n        }\n        if (!flag)\n            break; // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* 버블 정렬(플래그 최적화)*/\nvoid BubbleSortWithFlag(int[] nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        bool flag = false; // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                flag = true;  // 원소가 교환되었음을 기록한다\n            }\n        }\n        if (!flag) break;     // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* 버블 정렬(플래그 최적화)*/\nfunc bubbleSortWithFlag(nums []int) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        flag := false // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n                flag = true // 원소가 교환되었음을 기록한다\n            }\n        }\n        if flag == false { // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* 버블 정렬(플래그 최적화)*/\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in nums.indices.dropFirst().reversed() {\n        var flag = false // 플래그를 초기화한다\n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                nums.swapAt(j, j + 1)\n                flag = true // 원소가 교환되었음을 기록한다\n            }\n        }\n        if !flag { // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* 버블 정렬(플래그 최적화)*/\nfunction bubbleSortWithFlag(nums) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // 원소가 교환되었음을 기록한다\n            }\n        }\n        if (!flag) break; // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* 버블 정렬(플래그 최적화)*/\nfunction bubbleSortWithFlag(nums: number[]): void {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // 원소가 교환되었음을 기록한다\n            }\n        }\n        if (!flag) break; // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* 버블 정렬(플래그 최적화)*/\nvoid bubbleSortWithFlag(List&lt;int&gt; nums) {\n  // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    bool flag = false; // 플래그를 초기화한다\n    // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // nums[j]와 nums[j + 1]을 교환한다\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        flag = true; // 원소가 교환되었음을 기록한다\n      }\n    }\n    if (!flag) break; // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* 버블 정렬(플래그 최적화) */\nfn bubble_sort_with_flag(nums: &amp;mut [i32]) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for i in (1..nums.len()).rev() {\n        let mut flag = false; // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // nums[j]와 nums[j + 1]을 교환한다\n                nums.swap(j, j + 1);\n                flag = true; // 원소가 교환되었음을 기록한다\n            }\n        }\n        if !flag {\n            break; // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n        };\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* 버블 정렬(플래그 최적화)*/\nvoid bubbleSortWithFlag(int nums[], int size) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (int i = size - 1; i &gt; 0; i--) {\n        bool flag = false;\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n                flag = true;\n            }\n        }\n        if (!flag)\n            break;\n    }\n}\n</code></pre> bubble_sort.kt<pre><code>/* 버블 정렬(플래그 최적화) */\nfun bubbleSortWithFlag(nums: IntArray) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n    for (i in nums.size - 1 downTo 1) {\n        var flag = false // 플래그를 초기화한다\n        // 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n        for (j in 0..&lt;i) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // nums[j]와 nums[j + 1]을 교환한다\n                val temp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = temp\n                flag = true // 원소가 교환되었음을 기록한다\n            }\n        }\n        if (!flag) break // 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n    }\n}\n</code></pre> bubble_sort.rb<pre><code>### 버블 정렬(플래그 최적화)###\ndef bubble_sort_with_flag(nums)\n  n = nums.length\n  # 바깥쪽 반복문: 정렬되지 않은 구간은 [0, i]이다\n  for i in (n - 1).downto(1)\n    flag = false # 플래그를 초기화한다\n\n    # 안쪽 반복문: 정렬되지 않은 구간 [0, i]의 최댓값을 교환하여 이 구간의 맨 오른쪽으로 옮긴다\n    for j in 0...i\n      if nums[j] &gt; nums[j + 1]\n        # nums[j]와 nums[j + 1]을 교환한다\n        nums[j], nums[j + 1] = nums[j + 1], nums[j]\n        flag = true # 원소가 교환되었음을 기록한다\n      end\n    end\n\n    break unless flag # 이번 버블 과정에서 원소 교환이 없었으므로 바로 종료한다\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.3   버블 정렬"],"tags":[]},{"location":"chapter_sorting/bubble_sort/#1133","level":2,"title":"11.3.3   알고리즘 특성","text":"<ul> <li>시간 복잡도 \\(O(n^2)\\), 적응형 정렬: 각 회의 '버블 연산'에서 순회하는 배열 길이는 차례로 \\(n - 1\\), \\(n - 2\\), \\(\\dots\\), \\(2\\), \\(1\\)이며, 합은 \\((n - 1) n / 2\\)이다. <code>flag</code>를 도입하여 최적화하면 최선 시간 복잡도가 \\(O(n)\\)이 된다.</li> <li>공간 복잡도 \\(O(1)\\), 제자리 정렬: 포인터 \\(i\\)와 \\(j\\)는 상수 크기의 추가 공간을 사용한다.</li> <li>안정 정렬: '버블 연산'에서 값이 같은 원소는 교환하지 않는다.</li> </ul>","path":["제 11 장   정렬","11.3   버블 정렬"],"tags":[]},{"location":"chapter_sorting/bucket_sort/","level":1,"title":"11.8   버킷 정렬","text":"<p>앞에서 살펴본 정렬 알고리즘은 모두 원소의 크기를 비교하여 정렬하는 '비교 기반 정렬 알고리즘'이다. 이 알고리즘들의 최악 시간 복잡도 하한은 \\(\\Omega(n \\log n)\\)이다. 이제 시간 복잡도가 선형에 이를 수 있는 몇 가지 '비교를 사용하지 않는 정렬 알고리즘'을 살펴보자.</p> <p>버킷 정렬(bucket sort)은 분할 정복 전략의 대표적인 응용이다. 크기순으로 배열된 여러 버킷을 만들고 각 버킷에 데이터 범위를 대응시킨 뒤, 데이터를 각 버킷에 고르게 분배한다. 이어서 버킷별로 내부 정렬을 수행하고, 마지막으로 버킷 순서대로 모든 데이터를 합친다.</p>","path":["제 11 장   정렬","11.8   버킷 정렬"],"tags":[]},{"location":"chapter_sorting/bucket_sort/#1181","level":2,"title":"11.8.1   알고리즘 과정","text":"<p>길이가 \\(n\\)이고 원소가 범위 \\([0, 1)\\)의 부동 소수점 수인 배열을 생각해 보자. 버킷 정렬의 과정은 아래 그림(11-13)과 같다.</p> <ol> <li>버킷 \\(k\\) 개를 초기화하고 원소 \\(n\\) 개를 이 \\(k\\) 개의 버킷에 분배한다.</li> <li>각 버킷을 따로 정렬한다(여기서는 프로그래밍 언어의 내장 정렬 함수를 사용한다).</li> <li>작은 버킷부터 큰 버킷 순서로 결과를 합친다.</li> </ol> <p></p> <p> 그림 11-13   버킷 정렬 알고리즘 과정 </p> <p>코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby bucket_sort.py<pre><code>def bucket_sort(nums: list[float]):\n    \"\"\"버킷 정렬\"\"\"\n    # 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    k = len(nums) // 2\n    buckets = [[] for _ in range(k)]\n    # 1. 배열 원소를 각 버킷에 배정한다\n    for num in nums:\n        # 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        i = int(num * k)\n        # num을 버킷 i에 추가한다\n        buckets[i].append(num)\n    # 2. 각 버킷을 정렬한다\n    for bucket in buckets:\n        # 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        bucket.sort()\n    # 3. 버킷을 순회하여 결과를 합친다\n    i = 0\n    for bucket in buckets:\n        for num in bucket:\n            nums[i] = num\n            i += 1\n</code></pre> bucket_sort.cpp<pre><code>/* 버킷 정렬 */\nvoid bucketSort(vector&lt;float&gt; &amp;nums) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    int k = nums.size() / 2;\n    vector&lt;vector&lt;float&gt;&gt; buckets(k);\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for (float num : nums) {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        int i = num * k;\n        // num을 버킷 bucket_idx에 추가한다\n        buckets[i].push_back(num);\n    }\n    // 2. 각 버킷을 정렬한다\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        sort(bucket.begin(), bucket.end());\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    int i = 0;\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.java<pre><code>/* 버킷 정렬 */\nvoid bucketSort(float[] nums) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    int k = nums.length / 2;\n    List&lt;List&lt;Float&gt;&gt; buckets = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; k; i++) {\n        buckets.add(new ArrayList&lt;&gt;());\n    }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for (float num : nums) {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        int i = (int) (num * k);\n        // num을 버킷 i에 추가한다\n        buckets.get(i).add(num);\n    }\n    // 2. 각 버킷을 정렬한다\n    for (List&lt;Float&gt; bucket : buckets) {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        Collections.sort(bucket);\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    int i = 0;\n    for (List&lt;Float&gt; bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.cs<pre><code>/* 버킷 정렬 */\nvoid BucketSort(float[] nums) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    int k = nums.Length / 2;\n    List&lt;List&lt;float&gt;&gt; buckets = [];\n    for (int i = 0; i &lt; k; i++) {\n        buckets.Add([]);\n    }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    foreach (float num in nums) {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        int i = (int)(num * k);\n        // num을 버킷 i에 추가한다\n        buckets[i].Add(num);\n    }\n    // 2. 각 버킷을 정렬한다\n    foreach (List&lt;float&gt; bucket in buckets) {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        bucket.Sort();\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    int j = 0;\n    foreach (List&lt;float&gt; bucket in buckets) {\n        foreach (float num in bucket) {\n            nums[j++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.go<pre><code>/* 버킷 정렬 */\nfunc bucketSort(nums []float64) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    k := len(nums) / 2\n    buckets := make([][]float64, k)\n    for i := 0; i &lt; k; i++ {\n        buckets[i] = make([]float64, 0)\n    }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for _, num := range nums {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        i := int(num * float64(k))\n        // num을 버킷 i에 추가한다\n        buckets[i] = append(buckets[i], num)\n    }\n    // 2. 각 버킷을 정렬한다\n    for i := 0; i &lt; k; i++ {\n        // 내장 슬라이스 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        sort.Float64s(buckets[i])\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    i := 0\n    for _, bucket := range buckets {\n        for _, num := range bucket {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> bucket_sort.swift<pre><code>/* 버킷 정렬 */\nfunc bucketSort(nums: inout [Double]) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    let k = nums.count / 2\n    var buckets = (0 ..&lt; k).map { _ in [Double]() }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for num in nums {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        let i = Int(num * Double(k))\n        // num을 버킷 i에 추가한다\n        buckets[i].append(num)\n    }\n    // 2. 각 버킷을 정렬한다\n    for i in buckets.indices {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        buckets[i].sort()\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    var i = nums.startIndex\n    for bucket in buckets {\n        for num in bucket {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> bucket_sort.js<pre><code>/* 버킷 정렬 */\nfunction bucketSort(nums) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    const k = nums.length / 2;\n    const buckets = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for (const num of nums) {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        const i = Math.floor(num * k);\n        // num을 버킷 i에 추가한다\n        buckets[i].push(num);\n    }\n    // 2. 각 버킷을 정렬한다\n    for (const bucket of buckets) {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.ts<pre><code>/* 버킷 정렬 */\nfunction bucketSort(nums: number[]): void {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    const k = nums.length / 2;\n    const buckets: number[][] = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for (const num of nums) {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        const i = Math.floor(num * k);\n        // num을 버킷 i에 추가한다\n        buckets[i].push(num);\n    }\n    // 2. 각 버킷을 정렬한다\n    for (const bucket of buckets) {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.dart<pre><code>/* 버킷 정렬 */\nvoid bucketSort(List&lt;double&gt; nums) {\n  // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n  int k = nums.length ~/ 2;\n  List&lt;List&lt;double&gt;&gt; buckets = List.generate(k, (index) =&gt; []);\n\n  // 1. 배열 원소를 각 버킷에 배정한다\n  for (double _num in nums) {\n    // 입력 데이터의 범위는 [0, 1)이며, _num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n    int i = (_num * k).toInt();\n    // _num을 버킷 bucket_idx에 추가한다\n    buckets[i].add(_num);\n  }\n  // 2. 각 버킷을 정렬한다\n  for (List&lt;double&gt; bucket in buckets) {\n    bucket.sort();\n  }\n  // 3. 버킷을 순회하여 결과를 합친다\n  int i = 0;\n  for (List&lt;double&gt; bucket in buckets) {\n    for (double _num in bucket) {\n      nums[i++] = _num;\n    }\n  }\n}\n</code></pre> bucket_sort.rs<pre><code>/* 버킷 정렬 */\nfn bucket_sort(nums: &amp;mut [f64]) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    let k = nums.len() / 2;\n    let mut buckets = vec![vec![]; k];\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for &amp;num in nums.iter() {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        let i = (num * k as f64) as usize;\n        // num을 버킷 i에 추가한다\n        buckets[i].push(num);\n    }\n    // 2. 각 버킷을 정렬한다\n    for bucket in &amp;mut buckets {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        bucket.sort_by(|a, b| a.partial_cmp(b).unwrap());\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    let mut i = 0;\n    for bucket in buckets.iter() {\n        for &amp;num in bucket.iter() {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> bucket_sort.c<pre><code>/* 버킷 정렬 */\nvoid bucketSort(float nums[], int n) {\n    int k = n / 2;                                 // k = n/2개의 버킷을 초기화한다\n    int *sizes = malloc(k * sizeof(int));          // 각 버킷의 크기를 기록한다\n    float **buckets = malloc(k * sizeof(float *)); // 동적 배열의 배열(버킷)\n    // 각 버킷에 충분한 공간을 미리 할당한다\n    for (int i = 0; i &lt; k; ++i) {\n        buckets[i] = (float *)malloc(n * sizeof(float));\n        sizes[i] = 0;\n    }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for (int i = 0; i &lt; n; ++i) {\n        int idx = (int)(nums[i] * k);\n        buckets[idx][sizes[idx]++] = nums[i];\n    }\n    // 2. 각 버킷을 정렬한다\n    for (int i = 0; i &lt; k; ++i) {\n        qsort(buckets[i], sizes[i], sizeof(float), compare);\n    }\n    // 3. 정렬한 버킷을 병합한다\n    int idx = 0;\n    for (int i = 0; i &lt; k; ++i) {\n        for (int j = 0; j &lt; sizes[i]; ++j) {\n            nums[idx++] = buckets[i][j];\n        }\n        // 메모리 해제\n        free(buckets[i]);\n    }\n}\n</code></pre> bucket_sort.kt<pre><code>/* 버킷 정렬 */\nfun bucketSort(nums: FloatArray) {\n    // 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n    val k = nums.size / 2\n    val buckets = mutableListOf&lt;MutableList&lt;Float&gt;&gt;()\n    for (i in 0..&lt;k) {\n        buckets.add(mutableListOf())\n    }\n    // 1. 배열 원소를 각 버킷에 배정한다\n    for (num in nums) {\n        // 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n        val i = (num * k).toInt()\n        // num을 버킷 i에 추가한다\n        buckets[i].add(num)\n    }\n    // 2. 각 버킷을 정렬한다\n    for (bucket in buckets) {\n        // 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n        bucket.sort()\n    }\n    // 3. 버킷을 순회하여 결과를 합친다\n    var i = 0\n    for (bucket in buckets) {\n        for (num in bucket) {\n            nums[i++] = num\n        }\n    }\n}\n</code></pre> bucket_sort.rb<pre><code>### 버킷 정렬 ###\ndef bucket_sort(nums)\n  # 버킷 k = n/2개를 초기화하며, 각 버킷에 원소 2개가 배정될 것으로 예상한다\n  k = nums.length / 2\n  buckets = Array.new(k) { [] }\n\n  # 1. 배열 원소를 각 버킷에 배정한다\n  nums.each do |num|\n    # 입력 데이터의 범위는 [0, 1)이며, num * k를 사용하여 인덱스 범위 [0, k-1]에 대응시킨다\n    i = (num * k).to_i\n    # num을 버킷 i에 추가한다\n    buckets[i] &lt;&lt; num\n  end\n\n  # 2. 각 버킷을 정렬한다\n  buckets.each do |bucket|\n    # 내장 정렬 함수를 사용하며, 다른 정렬 알고리즘으로 대체할 수도 있다\n    bucket.sort!\n  end\n\n  # 3. 버킷을 순회하여 결과를 합친다\n  i = 0\n  buckets.each do |bucket|\n    bucket.each do |num|\n      nums[i] = num\n      i += 1\n    end\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.8   버킷 정렬"],"tags":[]},{"location":"chapter_sorting/bucket_sort/#1182","level":2,"title":"11.8.2   알고리즘 특성","text":"<p>버킷 정렬은 매우 큰 데이터를 처리하는 데 적합하다. 예를 들어 입력에 원소가 100만 개 있고 공간 제약 때문에 시스템 메모리에 모든 데이터를 한 번에 올릴 수 없다고 하자. 이때 데이터를 1000개 버킷으로 나누어 각각 정렬한 다음 결과를 합칠 수 있다.</p> <ul> <li>시간 복잡도 \\(O(n + k)\\): 원소가 각 버킷에 고르게 분포한다면 버킷당 원소 수는 \\(\\frac{n}{k}\\)이다. 버킷 하나를 정렬하는 데 \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) 시간이 걸린다고 하면 모든 버킷의 정렬에는 \\(O(n \\log\\frac{n}{k})\\) 시간이 걸린다. 버킷 수 \\(k\\)가 충분히 크면 시간 복잡도는 \\(O(n)\\)에 가까워진다. 결과를 합칠 때는 모든 버킷과 원소를 순회하므로 \\(O(n + k)\\) 시간이 든다. 최악의 경우 모든 데이터가 하나의 버킷에 들어가며, 그 버킷을 정렬하는 데 \\(O(n^2)\\) 시간이 걸린다.</li> <li>공간 복잡도 \\(O(n + k)\\), 제자리 정렬이 아님: 버킷 \\(k\\) 개와 총 \\(n\\) 개의 원소를 위한 추가 공간이 필요하다.</li> <li>버킷 정렬의 안정성은 버킷 내부를 정렬하는 알고리즘의 안정성에 달려 있다.</li> </ul>","path":["제 11 장   정렬","11.8   버킷 정렬"],"tags":[]},{"location":"chapter_sorting/bucket_sort/#1183","level":2,"title":"11.8.3   고르게 분배하는 방법","text":"<p>버킷 정렬의 시간 복잡도는 이론적으로 \\(O(n)\\)에 이를 수 있다. 핵심은 원소를 각 버킷에 고르게 분배하는 것인데, 실제 데이터는 균일하게 분포하지 않는 경우가 많다. 예를 들어 타오바오의 모든 상품을 가격 범위에 따라 10개 버킷에 고르게 나누고 싶다고 하자. 상품 가격은 불균일하여 100위안 미만의 상품은 매우 많고 1000위안 초과 상품은 매우 적다. 가격 구간을 같은 폭으로 10등분하면 버킷별 상품 수에 큰 차이가 생긴다.</p> <p>고르게 분배하려면 먼저 대략적인 경계를 정해 데이터를 버킷 3개로 나눌 수 있다. 분배한 뒤 상품이 많은 버킷을 다시 3개로 나누고, 모든 버킷의 원소 수가 비슷해질 때까지 반복한다.</p> <p>아래 그림(11-14)과 같이 이 방법은 본질적으로 재귀 트리를 만드는 과정이며, 리프 노드의 값을 가능한 한 고르게 만드는 것이 목표다. 반드시 매번 버킷 3개로 나눌 필요는 없으며, 데이터 특성에 맞추어 분할 방식을 유연하게 선택할 수 있다.</p> <p></p> <p> 그림 11-14   버킷을 재귀적으로 분할 </p> <p>상품 가격의 확률 분포를 미리 알고 있다면 그 분포에 따라 각 버킷의 가격 경계를 설정할 수 있다. 데이터 분포를 반드시 따로 집계할 필요는 없으며, 데이터 특성에 따라 적절한 확률 모형으로 근사할 수도 있다.</p> <p>아래 그림(11-15)에서는 상품 가격이 정규 분포를 따른다고 가정한다. 이를 바탕으로 가격 구간을 적절히 정하면 상품을 각 버킷에 고르게 분배할 수 있다.</p> <p></p> <p> 그림 11-15   확률 분포에 따른 버킷 분할 </p>","path":["제 11 장   정렬","11.8   버킷 정렬"],"tags":[]},{"location":"chapter_sorting/counting_sort/","level":1,"title":"11.9   계수 정렬","text":"<p>계수 정렬(counting sort)은 원소의 개수를 세어 정렬하며, 주로 정수 배열에 사용한다.</p>","path":["제 11 장   정렬","11.9   계수 정렬"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1191","level":2,"title":"11.9.1   간단한 구현","text":"<p>먼저 간단한 예를 보자. 길이가 \\(n\\)인 배열 <code>nums</code>의 모든 원소가 '음이 아닌 정수'라고 하자. 계수 정렬의 전체 과정은 아래 그림(11-16)과 같다.</p> <ol> <li>배열을 순회하여 가장 큰 수 \\(m\\)을 찾고, 길이가 \\(m + 1\\)인 보조 배열 <code>counter</code>를 만든다.</li> <li><code>counter</code>를 이용하여 <code>nums</code>에서 각 수가 나타나는 횟수를 센다. <code>counter[num]</code>은 수 <code>num</code>의 등장 횟수다. 세는 방법은 간단하다. <code>nums</code>를 순회하면서 현재 수를 <code>num</code>이라고 할 때 <code>counter[num]</code>을 \\(1\\)씩 늘리면 된다.</li> <li><code>counter</code>의 인덱스는 본래 순서대로 배열되어 있으므로, 모든 수가 이미 정렬된 셈이다. 이어서 <code>counter</code>를 순회하며 각 수의 등장 횟수만큼 작은 수부터 큰 수 순서로 <code>nums</code>에 채우면 된다.</li> </ol> <p></p> <p> 그림 11-16   계수 정렬 과정 </p> <p>코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby counting_sort.py<pre><code>def counting_sort_naive(nums: list[int]):\n    \"\"\"계수 정렬\"\"\"\n    # 간단한 구현이며, 객체 정렬에는 사용할 수 없다\n    # 1. 배열의 최댓값 m을 구한다\n    m = max(nums)\n    # 2. 각 숫자의 등장 횟수를 센다\n    # counter[num]은 num의 등장 횟수를 나타낸다\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    i = 0\n    for num in range(m + 1):\n        for _ in range(counter[num]):\n            nums[i] = num\n            i += 1\n</code></pre> counting_sort.cpp<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nvoid countingSortNaive(vector&lt;int&gt; &amp;nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.java<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nvoid countingSortNaive(int[] nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nvoid CountingSortNaive(int[] nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nfunc countingSortNaive(nums []int) {\n    // 1. 배열의 최댓값 m을 구한다\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    for i, num := 0, 0; num &lt; m+1; num++ {\n        for j := 0; j &lt; counter[num]; j++ {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.swift<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nfunc countingSortNaive(nums: inout [Int]) {\n    // 1. 배열의 최댓값 m을 구한다\n    let m = nums.max()!\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    var i = 0\n    for num in 0 ..&lt; m + 1 {\n        for _ in 0 ..&lt; counter[num] {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nfunction countingSortNaive(nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    let m = Math.max(...nums);\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nfunction countingSortNaive(nums: number[]): void {\n    // 1. 배열의 최댓값 m을 구한다\n    let m: number = Math.max(...nums);\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nvoid countingSortNaive(List&lt;int&gt; nums) {\n  // 1. 배열의 최댓값 m을 구한다\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. 각 숫자의 등장 횟수를 센다\n  // counter[_num]은 _num의 등장 횟수를 나타낸다\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n  int i = 0;\n  for (int _num = 0; _num &lt; m + 1; _num++) {\n    for (int j = 0; j &lt; counter[_num]; j++, i++) {\n      nums[i] = _num;\n    }\n  }\n}\n</code></pre> counting_sort.rs<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nfn counting_sort_naive(nums: &amp;mut [i32]) {\n    // 1. 배열의 최댓값 m을 구한다\n    let m = *nums.iter().max().unwrap();\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    let mut counter = vec![0; m as usize + 1];\n    for &amp;num in nums.iter() {\n        counter[num as usize] += 1;\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    let mut i = 0;\n    for num in 0..m + 1 {\n        for _ in 0..counter[num as usize] {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> counting_sort.c<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nvoid countingSortNaive(int nums[], int size) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    int *counter = calloc(m + 1, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n    // 4. 메모리를 해제한다\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* 계수 정렬 */\n// 간단한 구현이며, 객체 정렬에는 사용할 수 없다\nfun countingSortNaive(nums: IntArray) {\n    // 1. 배열의 최댓값 m을 구한다\n    var m = 0\n    for (num in nums) {\n        m = max(m, num)\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n    var i = 0\n    for (num in 0..&lt;m + 1) {\n        var j = 0\n        while (j &lt; counter[num]) {\n            nums[i] = num\n            j++\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.rb<pre><code>### 계수 정렬 ###\ndef counting_sort_naive(nums)\n  # 간단한 구현이며, 객체 정렬에는 사용할 수 없다\n  # 1. 배열의 최댓값 m을 구한다\n  m = 0\n  nums.each { |num| m = [m, num].max }\n  # 2. 각 숫자의 등장 횟수를 센다\n  # counter[num]은 num의 등장 횟수를 나타낸다\n  counter = Array.new(m + 1, 0)\n  nums.each { |num| counter[num] += 1 }\n  # 3. counter를 순회하며 각 원소를 원래 배열 nums에 채운다\n  i = 0\n  for num in 0...(m + 1)\n    (0...counter[num]).each do\n      nums[i] = num\n      i += 1\n    end\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>계수 정렬과 버킷 정렬의 관계</p> <p>버킷 정렬의 관점에서는 계수 배열 <code>counter</code>의 각 인덱스를 하나의 버킷으로, 개수를 세는 과정을 원소를 해당 버킷에 분배하는 과정으로 볼 수 있다. 본질적으로 계수 정렬은 정수 데이터에 적용되는 버킷 정렬의 특수한 경우다.</p>","path":["제 11 장   정렬","11.9   계수 정렬"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1192","level":2,"title":"11.9.2   완전한 구현","text":"<p>주의 깊게 읽었다면 입력 데이터가 객체일 때 위의 <code>3.</code> 단계가 작동하지 않는다는 것을 알아차렸을 것이다. 입력이 상품 객체이고 상품 가격(클래스의 멤버 변수)에 따라 정렬하려 한다고 하자. 위 알고리즘으로는 가격의 정렬 결과만 얻을 수 있다.</p> <p>그렇다면 원래 데이터의 정렬 결과는 어떻게 얻을까? 먼저 <code>counter</code>의 '접두 합'을 계산한다. 이름 그대로 인덱스 <code>i</code>의 접두 합 <code>prefix[i]</code>는 배열의 앞 <code>i</code> 개 원소를 더한 값이다.</p> \\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\] <p>접두 합은 명확한 의미를 가진다. <code>prefix[num] - 1</code>은 결과 배열 <code>res</code>에서 원소 <code>num</code>이 마지막으로 나타나는 인덱스다. 이는 각 원소가 결과 배열의 어디에 놓여야 하는지 알려 주는 핵심 정보다. 이어서 원래 배열 <code>nums</code>의 각 원소 <code>num</code>을 역순으로 순회하며 매번 다음 두 단계를 수행한다.</p> <ol> <li>배열 <code>res</code>의 인덱스 <code>prefix[num] - 1</code>에 <code>num</code>을 넣는다.</li> <li>접두 합 <code>prefix[num]</code>을 \\(1\\) 줄여 다음에 <code>num</code>을 넣을 인덱스를 얻는다.</li> </ol> <p>순회를 마치면 배열 <code>res</code>에 정렬된 결과가 들어 있다. 마지막으로 <code>res</code>를 원래 배열 <code>nums</code>에 덮어쓰면 된다. 아래 그림(11-17)은 계수 정렬의 전체 과정을 보여 준다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 11-17   계수 정렬 단계 </p> <p>계수 정렬의 구현 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby counting_sort.py<pre><code>def counting_sort(nums: list[int]):\n    \"\"\"계수 정렬\"\"\"\n    # 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\n    # 1. 배열의 최댓값 m을 구한다\n    m = max(nums)\n    # 2. 각 숫자의 등장 횟수를 센다\n    # counter[num]은 num의 등장 횟수를 나타낸다\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    # 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for i in range(m):\n        counter[i + 1] += counter[i]\n    # 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    # 결과를 기록할 배열 res를 초기화한다\n    n = len(nums)\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        num = nums[i]\n        res[counter[num] - 1] = num  # num을 해당 인덱스에 배치한다\n        counter[num] -= 1  # 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    # 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    for i in range(n):\n        nums[i] = res[i]\n</code></pre> counting_sort.cpp<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nvoid countingSort(vector&lt;int&gt; &amp;nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    int n = nums.size();\n    vector&lt;int&gt; res(n);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // num을 해당 인덱스에 배치한다\n        counter[num]--;              // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    nums = res;\n}\n</code></pre> counting_sort.java<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nvoid countingSort(int[] nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    int n = nums.length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // num을 해당 인덱스에 배치한다\n        counter[num]--; // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nvoid CountingSort(int[] nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    int n = nums.Length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // num을 해당 인덱스에 배치한다\n        counter[num]--; // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nfunc countingSort(nums []int) {\n    // 1. 배열의 최댓값 m을 구한다\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for i := 0; i &lt; m; i++ {\n        counter[i+1] += counter[i]\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    n := len(nums)\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        num := nums[i]\n        // num을 해당 인덱스에 배치한다\n        res[counter[num]-1] = num\n        // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n        counter[num]--\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    copy(nums, res)\n}\n</code></pre> counting_sort.swift<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nfunc countingSort(nums: inout [Int]) {\n    // 1. 배열의 최댓값 m을 구한다\n    let m = nums.max()!\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for i in 0 ..&lt; m {\n        counter[i + 1] += counter[i]\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let num = nums[i]\n        res[counter[num] - 1] = num // num을 해당 인덱스에 배치한다\n        counter[num] -= 1 // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nfunction countingSort(nums) {\n    // 1. 배열의 최댓값 m을 구한다\n    let m = Math.max(...nums);\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    const n = nums.length;\n    const res = new Array(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // num을 해당 인덱스에 배치한다\n        counter[num]--; // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nfunction countingSort(nums: number[]): void {\n    // 1. 배열의 최댓값 m을 구한다\n    let m: number = Math.max(...nums);\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    const n = nums.length;\n    const res: number[] = new Array&lt;number&gt;(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // num을 해당 인덱스에 배치한다\n        counter[num]--; // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nvoid countingSort(List&lt;int&gt; nums) {\n  // 1. 배열의 최댓값 m을 구한다\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. 각 숫자의 등장 횟수를 센다\n  // counter[_num]은 _num의 등장 횟수를 나타낸다\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n  // 즉, counter[_num]-1은 res에서 _num이 마지막으로 등장하는 인덱스이다\n  for (int i = 0; i &lt; m; i++) {\n    counter[i + 1] += counter[i];\n  }\n  // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n  // 결과를 기록할 배열 res를 초기화한다\n  int n = nums.length;\n  List&lt;int&gt; res = List.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int _num = nums[i];\n    res[counter[_num] - 1] = _num; // _num을 해당 인덱스에 배치한다\n    counter[_num]--; // 접두 합을 1 감소시켜 다음에 _num을 배치할 인덱스를 구한다\n  }\n  // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n  nums.setAll(0, res);\n}\n</code></pre> counting_sort.rs<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nfn counting_sort(nums: &amp;mut [i32]) {\n    // 1. 배열의 최댓값 m을 구한다\n    let m = *nums.iter().max().unwrap() as usize;\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    let mut counter = vec![0; m + 1];\n    for &amp;num in nums.iter() {\n        counter[num as usize] += 1;\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for i in 0..m {\n        counter[i + 1] += counter[i];\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    let n = nums.len();\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let num = nums[i];\n        res[counter[num as usize] - 1] = num; // num을 해당 인덱스에 배치한다\n        counter[num as usize] -= 1; // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    nums.copy_from_slice(&amp;res)\n}\n</code></pre> counting_sort.c<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nvoid countingSort(int nums[], int size) {\n    // 1. 배열의 최댓값 m을 구한다\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    int *counter = calloc(m, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    int *res = malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // num을 해당 인덱스에 배치한다\n        counter[num]--;              // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    memcpy(nums, res, size * sizeof(int));\n    // 5. 메모리를 해제한다\n    free(res);\n    free(counter);\n}\n</code></pre> counting_sort.kt<pre><code>/* 계수 정렬 */\n// 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\nfun countingSort(nums: IntArray) {\n    // 1. 배열의 최댓값 m을 구한다\n    var m = 0\n    for (num in nums) {\n        m = max(m, num)\n    }\n    // 2. 각 숫자의 등장 횟수를 센다\n    // counter[num]은 num의 등장 횟수를 나타낸다\n    val counter = IntArray(m + 1)\n    for (num in nums) {\n        counter[num]++\n    }\n    // 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n    // 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n    for (i in 0..&lt;m) {\n        counter[i + 1] += counter[i]\n    }\n    // 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n    // 결과를 기록할 배열 res를 초기화한다\n    val n = nums.size\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val num = nums[i]\n        res[counter[num] - 1] = num // num을 해당 인덱스에 배치한다\n        counter[num]-- // 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n    }\n    // 결과 배열 res로 원래 배열 nums를 덮어쓴다\n    for (i in 0..&lt;n) {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.rb<pre><code>### 계수 정렬 ###\ndef counting_sort(nums)\n  # 완전한 구현이며, 객체를 정렬할 수 있고 안정 정렬이다\n  # 1. 배열의 최댓값 m을 구한다\n  m = nums.max\n  # 2. 각 숫자의 등장 횟수를 센다\n  # counter[num]은 num의 등장 횟수를 나타낸다\n  counter = Array.new(m + 1, 0)\n  nums.each { |num| counter[num] += 1 }\n  # 3. counter의 접두 합을 구하여 등장 횟수를 마지막 인덱스로 변환한다\n  # 즉, counter[num]-1은 res에서 num이 마지막으로 등장하는 인덱스이다\n  (0...m).each { |i| counter[i + 1] += counter[i] }\n  # 4. nums를 역순으로 순회하며 각 원소를 결과 배열 res에 채운다\n  # 결과를 기록할 배열 res를 초기화한다\n  n = nums.length\n  res = Array.new(n, 0)\n  (n - 1).downto(0).each do |i|\n    num = nums[i]\n    res[counter[num] - 1] = num # num을 해당 인덱스에 배치한다\n    counter[num] -= 1 # 접두 합을 1 감소시켜 다음에 num을 배치할 인덱스를 구한다\n  end\n  # 결과 배열 res로 원래 배열 nums를 덮어쓴다\n  (0...n).each { |i| nums[i] = res[i] }\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.9   계수 정렬"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1193","level":2,"title":"11.9.3   알고리즘 특성","text":"<ul> <li>시간 복잡도 \\(O(n + m)\\), 비적응형 정렬: <code>nums</code>와 <code>counter</code>를 순회하며, 각각 선형 시간이 걸린다. 보통 \\(n \\gg m\\)이므로 시간 복잡도는 \\(O(n)\\)에 가까워진다.</li> <li>공간 복잡도 \\(O(n + m)\\), 제자리 정렬이 아님: 길이가 각각 \\(n\\)과 \\(m\\)인 배열 <code>res</code>와 <code>counter</code>를 사용한다.</li> <li>안정 정렬: <code>res</code>에는 '오른쪽에서 왼쪽으로' 원소를 채우므로, <code>nums</code>를 역순으로 순회하면 값이 같은 원소의 상대적 위치가 바뀌지 않아 안정 정렬이 된다. 정방향으로 순회해도 정렬 결과 자체는 올바르지만 안정적이지 않다.</li> </ul>","path":["제 11 장   정렬","11.9   계수 정렬"],"tags":[]},{"location":"chapter_sorting/counting_sort/#1194","level":2,"title":"11.9.4   한계","text":"<p>여기까지 보면 계수 정렬이 매우 기발하다고 느낄 수 있다. 개수만 세어도 효율적으로 정렬할 수 있기 때문이다. 하지만 계수 정렬을 사용하기 위한 전제 조건은 비교적 엄격하다.</p> <p>계수 정렬은 음이 아닌 정수에만 적합하다. 다른 타입에 적용하려면 해당 데이터를 음이 아닌 정수로 바꿀 수 있어야 하며, 변환 과정에서 원소 간 대소 관계가 바뀌면 안 된다. 예를 들어 음수가 포함된 정수 배열은 모든 수에 일정한 상수를 더해 양수로 바꾼 뒤, 정렬 후 다시 원래 값으로 돌릴 수 있다.</p> <p>계수 정렬은 데이터 수는 많지만 값의 범위는 작은 경우에 적합하다. 위 예시에서 \\(m\\)이 너무 크면 공간을 지나치게 많이 차지한다. 또한 \\(n \\ll m\\)이면 계수 정렬에 \\(O(m)\\) 시간이 걸리므로 \\(O(n \\log n)\\) 정렬 알고리즘보다 느릴 수도 있다.</p>","path":["제 11 장   정렬","11.9   계수 정렬"],"tags":[]},{"location":"chapter_sorting/exercises/","level":1,"title":"11.12   연습 문제","text":"","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/exercises/#11121","level":2,"title":"11.12.1   개념 확인","text":"","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/exercises/#1","level":3,"title":"1.   선택 정렬과 버블 정렬의 처음 몇 회전","text":"<p>배열 <code>[4, 2, 5, 1, 3]</code>이 주어진다. 다음은 모두 오름차순으로 정렬한다.</p> <ol> <li>선택 정렬의 첫 두 회전을 따라가며 각 회전이 끝난 뒤의 배열을 적고, 어느 위치가 확정되었는지 설명하라.</li> <li>버블 정렬의 첫 회전을 따라가며 배열 상태와 교환 횟수를 적고, 어느 위치가 확정되었는지 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>첫 두 회전은 다음과 같다.</p> 회전 배열 상태 설명 1 <code>[1, 2, 5, 4, 3]</code> 가장 작은 원소 1을 첫 번째 원소와 교환 2 <code>[1, 2, 5, 4, 3]</code> 값 2가 이미 인덱스 1에 있으므로 교환할 필요 없음 <p>이때 앞의 두 위치는 확정되었으므로, 이후에는 <code>[5, 4, 3]</code>에서 계속 최솟값을 선택하면 된다.</p> </li> <li> <p>이웃한 원소를 차례대로 비교한다. 4와 2는 교환하고, 4와 5는 교환하지 않으며, 5와 1을 교환한 뒤 5와 3을 교환한다.     결과는 <code>[2, 4, 1, 3, 5]</code>이고 총 3번 교환한다. 가장 큰 원소 5가 배열 끝으로 이동했으므로 마지막 위치가 확정되었다.</p> </li> </ol>","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/exercises/#2","level":3,"title":"2.   값이 같은 원소의 순서가 달라지는가","text":"<p>배열 \\([2_a, 2_b, 1]\\)에서 \\(2_a\\)와 \\(2_b\\)는 값이 같지만, 아래 첨자로 원래 순서를 표시했다.</p> <ol> <li>선택 정렬의 첫 회전이 끝난 뒤의 배열을 적어 보자. \\(2_a\\)와 \\(2_b\\)의 상대적인 순서가 바뀌었는가?</li> <li>버블 정렬의 첫 회전이 끝난 뒤의 배열을 적어 보자. \\(2_a\\)와 \\(2_b\\)의 상대적인 순서가 바뀌었는가?</li> <li>앞의 두 결과를 바탕으로, 값이 같은 원소의 원래 순서를 유지하는 측면에서 두 정렬이 어떻게 다른지 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>선택 정렬의 첫 회전에서는 가장 작은 원소 1을 골라 첫 원소 \\(2_a\\)와 교환한다. 결과는     \\([1, 2_b, 2_a]\\)다. \\(2_a\\)가 \\(2_b\\) 뒤로 옮겨졌으므로 두 원소의 상대적인 순서가 바뀌었다.</p> </li> <li> <p>버블 정렬은 먼저 \\(2_a\\)와 \\(2_b\\)를 비교한다. 두 값이 같으므로 교환하지 않는다. 이어서 \\(2_b\\)와 1을 비교하여 교환한다.     첫 회전이 끝나면 \\([2_a, 1, 2_b]\\)가 된다. \\(2_a\\)가 여전히 \\(2_b\\) 앞에 있으므로 상대적인 순서는 바뀌지 않았다.</p> </li> <li> <p>이 예에서 선택 정렬은 값이 같은 원소의 원래 순서를 바꾼다. 버블 정렬은 왼쪽 원소가 오른쪽 원소보다 클 때만     이웃한 원소를 교환한다. 값이 같은 원소끼리는 교환하지 않으므로 원래의 앞뒤 순서를 유지할 수 있다.</p> </li> </ol>","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/exercises/#3","level":3,"title":"3.   계수 정렬과 기수 정렬 비교하기","text":"<p>학교에서 자릿수가 8자리로 고정된 학번을 대량으로 정렬하려 한다. 다음에 답하라.</p> <ol> <li>가장 낮은 자리부터 처리하는 기수 정렬은 총 몇 회전이 필요한가?</li> <li>학번을 그대로 정수로 취급하여 계수 정렬을 하면, 사용하지 않는 계수 배열의 자리를 많이 마련하게 되는 이유는 무엇인가?</li> <li>앞의 두 결과를 바탕으로, 8자리로 고정된 학번을 대량으로 처리하려면 계수 정렬과 기수 정렬 중 어느 것을 선택하겠는가? 이유도 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>학번은 8자리이므로 가장 낮은 자리부터 가장 높은 자리까지 총 8회전이 필요하다. 매 회전에서는 0~9에 따라 그룹을 나누기만 하면 된다.</p> </li> <li> <p>학번 전체의 출현 횟수를 직접 세려면 가능한 모든 8자리 값에 대해 자리를 마련해야 한다. 하지만 실제 학생이 사용하는 학번은 그중 일부에 불과하므로,     계수 배열의 대부분은 계속 0으로 남는다.</p> </li> <li> <p>기수 정렬을 선택해야 한다. ‘자릿수가 고정되어 있고, 각 자리에 올 수 있는 값은 10가지뿐’이라는 특징을 이용하면 안정적인 그룹 나누기를 8회 반복하는 것으로 충분하다.     계수 정렬에서 8자리 학번 전체를 정수 인덱스로 사용하면 실제로는 나타나지 않을 수많은 값에 대해서도 계수 배열의 자리를 마련해야 한다.</p> </li> </ol>","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/exercises/#11122","level":2,"title":"11.12.2   프로그래밍 연습","text":"","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/exercises/#1_1","level":3,"title":"1.   병합 정렬로 배열 정렬하기","text":"<p>정수 배열 <code>nums</code>가 주어진다. 병합 정렬을 직접 구현하여 원소를 비내림차순으로 정렬한 뒤 반환하라. 언어에서 제공하는 정렬 함수를 호출하지 마라.</p> 풀이 힌트 <ol> <li>구간 길이가 1 이하면 이미 정렬되어 있다.</li> <li>중간 지점에서 구간을 둘로 나누고 각각 재귀적으로 정렬한다.</li> <li>두 포인터로 정렬된 두 구간을 병합한 뒤, 결과를 원래 배열에 다시 쓴다.</li> </ol> <p>LeetCode</p>","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/exercises/#2_1","level":3,"title":"2.   계수 정렬로 정수 배열 정렬하기","text":"<p>정수 배열 <code>nums</code>와 음이 아닌 정수 \\(K\\)가 주어진다. 배열의 모든 원소는 \\(0\\) 이상 \\(K\\) 이하이다.</p> <p>계수 정렬을 구현하여 비내림차순으로 정렬한 결과를 <code>nums</code>에 다시 쓰고 <code>nums</code>를 반환하라. 원소의 대소 비교로 순서를 결정하거나 언어에서 제공하는 정렬 함수를 호출하지 마라.</p> 풀이 힌트 <ol> <li>모든 원소가 0 이상 K 이하이므로 원소 값을 계수 배열의 인덱스로 바로 사용할 수 있다.</li> <li>먼저 nums를 한 번 순회하면서 해당 위치의 계수를 1씩 늘린다.</li> <li>이어서 계수 배열을 0부터 K까지 순회한다. 값 x가 나타난 횟수만큼 nums에 x를 연속해서 쓴다.</li> </ol>","path":["제 11 장   정렬","11.12   연습 문제"],"tags":[]},{"location":"chapter_sorting/heap_sort/","level":1,"title":"11.7   힙 정렬","text":"<p>참고</p> <p>이 절을 읽기 전에 '힙' 장을 먼저 학습하자.</p> <p>힙 정렬(heap sort)은 힙 자료 구조를 이용한 효율적인 정렬 알고리즘이다. 이미 배운 '힙 구성 연산'과 '힙에서 원소를 꺼내는 연산'으로 구현할 수 있다.</p> <ol> <li>배열을 입력받아 최소 힙을 구성한다. 이때 가장 작은 원소가 힙의 루트에 놓인다.</li> <li>힙에서 원소를 계속 꺼내며 순서대로 기록하면 오름차순으로 정렬된 수열을 얻는다.</li> </ol> <p>위 방법도 가능하지만 꺼낸 원소를 저장할 추가 배열이 필요하여 공간을 낭비한다. 실제로는 보통 더 간결한 구현 방식을 사용한다.</p>","path":["제 11 장   정렬","11.7   힙 정렬"],"tags":[]},{"location":"chapter_sorting/heap_sort/#1171","level":2,"title":"11.7.1   알고리즘 과정","text":"<p>배열의 길이가 \\(n\\)일 때 힙 정렬의 과정은 아래 그림(11-12)과 같다.</p> <ol> <li>배열을 입력받아 최대 힙을 구성한다. 완료하면 가장 큰 원소가 힙의 루트에 놓인다.</li> <li>힙의 루트 원소(첫 번째 원소)와 끝 원소(마지막 원소)를 교환한다. 교환 후 힙의 길이는 \\(1\\) 줄고, 정렬된 원소 수는 \\(1\\) 늘어난다.</li> <li>힙의 루트 원소부터 위에서 아래로 힙 재정렬(sift down)을 수행한다. 재정렬을 마치면 힙의 성질이 복구된다.</li> <li><code>2.</code> 단계와 <code>3.</code> 단계를 반복한다. \\(n - 1\\) 회 반복하면 배열 정렬이 완료된다.</li> </ol> <p>참고</p> <p>힙에서 원소를 꺼내는 연산에도 <code>2.</code> 단계와 <code>3.</code> 단계가 포함되며, 실제로 원소를 꺼내는 단계가 하나 더 있을 뿐이다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 11-12   힙 정렬 단계 </p> <p>구현에서는 '힙' 장과 동일한 하향 힙 재정렬 함수 <code>sift_down()</code>을 사용한다. 가장 큰 원소를 꺼낼 때마다 힙의 길이가 줄어들므로, 현재 유효한 힙 길이를 지정하는 매개변수 \\(n\\)을 <code>sift_down()</code>에 추가해야 한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby heap_sort.py<pre><code>def sift_down(nums: list[int], n: int, i: int):\n    \"\"\"길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬\"\"\"\n    while True:\n        # 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        l = 2 * i + 1\n        r = 2 * i + 2\n        ma = i\n        if l &lt; n and nums[l] &gt; nums[ma]:\n            ma = l\n        if r &lt; n and nums[r] &gt; nums[ma]:\n            ma = r\n        # 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if ma == i:\n            break\n        # 두 노드를 교환한다\n        nums[i], nums[ma] = nums[ma], nums[i]\n        # 아래쪽으로 힙 재정렬을 반복한다\n        i = ma\n\ndef heap_sort(nums: list[int]):\n    \"\"\"힙 정렬\"\"\"\n    # 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for i in range(len(nums) // 2 - 1, -1, -1):\n        sift_down(nums, len(nums), i)\n    # 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for i in range(len(nums) - 1, 0, -1):\n        # 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        nums[0], nums[i] = nums[i], nums[0]\n        # 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        sift_down(nums, i, 0)\n</code></pre> heap_sort.cpp<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nvoid siftDown(vector&lt;int&gt; &amp;nums, int n, int i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i) {\n            break;\n        }\n        // 두 노드를 교환한다\n        swap(nums[i], nums[ma]);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n\n/* 힙 정렬 */\nvoid heapSort(vector&lt;int&gt; &amp;nums) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (int i = nums.size() / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, nums.size(), i);\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for (int i = nums.size() - 1; i &gt; 0; --i) {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        swap(nums[0], nums[i]);\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.java<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nvoid siftDown(int[] nums, int n, int i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i)\n            break;\n        // 두 노드를 교환한다\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n\n/* 힙 정렬 */\nvoid heapSort(int[] nums) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (int i = nums.length / 2 - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.cs<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nvoid SiftDown(int[] nums, int n, int i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i)\n            break;\n        // 두 노드를 교환한다\n        (nums[ma], nums[i]) = (nums[i], nums[ma]);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n\n/* 힙 정렬 */\nvoid HeapSort(int[] nums) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (int i = nums.Length / 2 - 1; i &gt;= 0; i--) {\n        SiftDown(nums, nums.Length, i);\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        (nums[i], nums[0]) = (nums[0], nums[i]);\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        SiftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.go<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nfunc siftDown(nums *[]int, n, i int) {\n    for true {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        l := 2*i + 1\n        r := 2*i + 2\n        ma := i\n        if l &lt; n &amp;&amp; (*nums)[l] &gt; (*nums)[ma] {\n            ma = l\n        }\n        if r &lt; n &amp;&amp; (*nums)[r] &gt; (*nums)[ma] {\n            ma = r\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if ma == i {\n            break\n        }\n        // 두 노드를 교환한다\n        (*nums)[i], (*nums)[ma] = (*nums)[ma], (*nums)[i]\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma\n    }\n}\n\n/* 힙 정렬 */\nfunc heapSort(nums *[]int) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for i := len(*nums)/2 - 1; i &gt;= 0; i-- {\n        siftDown(nums, len(*nums), i)\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for i := len(*nums) - 1; i &gt; 0; i-- {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        (*nums)[0], (*nums)[i] = (*nums)[i], (*nums)[0]\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.swift<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nfunc siftDown(nums: inout [Int], n: Int, i: Int) {\n    var i = i\n    while true {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        let l = 2 * i + 1\n        let r = 2 * i + 2\n        var ma = i\n        if l &lt; n, nums[l] &gt; nums[ma] {\n            ma = l\n        }\n        if r &lt; n, nums[r] &gt; nums[ma] {\n            ma = r\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if ma == i {\n            break\n        }\n        // 두 노드를 교환한다\n        nums.swapAt(i, ma)\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma\n    }\n}\n\n/* 힙 정렬 */\nfunc heapSort(nums: inout [Int]) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for i in stride(from: nums.count / 2 - 1, through: 0, by: -1) {\n        siftDown(nums: &amp;nums, n: nums.count, i: i)\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for i in nums.indices.dropFirst().reversed() {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        nums.swapAt(0, i)\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums: &amp;nums, n: i, i: 0)\n    }\n}\n</code></pre> heap_sort.js<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nfunction siftDown(nums, n, i) {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma === i) {\n            break;\n        }\n        // 두 노드를 교환한다\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n\n/* 힙 정렬 */\nfunction heapSort(nums) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.ts<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nfunction siftDown(nums: number[], n: number, i: number): void {\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma === i) {\n            break;\n        }\n        // 두 노드를 교환한다\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n\n/* 힙 정렬 */\nfunction heapSort(nums: number[]): void {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.dart<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nvoid siftDown(List&lt;int&gt; nums, int n, int i) {\n  while (true) {\n    // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n    int l = 2 * i + 1;\n    int r = 2 * i + 2;\n    int ma = i;\n    if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) ma = l;\n    if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) ma = r;\n    // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n    if (ma == i) break;\n    // 두 노드를 교환한다\n    int temp = nums[i];\n    nums[i] = nums[ma];\n    nums[ma] = temp;\n    // 아래쪽으로 힙 재정렬을 반복한다\n    i = ma;\n  }\n}\n\n/* 힙 정렬 */\nvoid heapSort(List&lt;int&gt; nums) {\n  // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n  for (int i = nums.length ~/ 2 - 1; i &gt;= 0; i--) {\n    siftDown(nums, nums.length, i);\n  }\n  // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    int tmp = nums[0];\n    nums[0] = nums[i];\n    nums[i] = tmp;\n    // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n    siftDown(nums, i, 0);\n  }\n}\n</code></pre> heap_sort.rs<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nfn sift_down(nums: &amp;mut [i32], n: usize, mut i: usize) {\n    loop {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let mut ma = i;\n        if l &lt; n &amp;&amp; nums[l] &gt; nums[ma] {\n            ma = l;\n        }\n        if r &lt; n &amp;&amp; nums[r] &gt; nums[ma] {\n            ma = r;\n        }\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if ma == i {\n            break;\n        }\n        // 두 노드를 교환한다\n        nums.swap(i, ma);\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n\n/* 힙 정렬 */\nfn heap_sort(nums: &amp;mut [i32]) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for i in (0..nums.len() / 2).rev() {\n        sift_down(nums, nums.len(), i);\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for i in (1..nums.len()).rev() {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        nums.swap(0, i);\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        sift_down(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.c<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nvoid siftDown(int nums[], int n, int i) {\n    while (1) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i) {\n            break;\n        }\n        // 두 노드를 교환한다\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma;\n    }\n}\n\n/* 힙 정렬 */\nvoid heapSort(int nums[], int n) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (int i = n / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, n, i);\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for (int i = n - 1; i &gt; 0; --i) {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.kt<pre><code>/* 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 */\nfun siftDown(nums: IntArray, n: Int, li: Int) {\n    var i = li\n    while (true) {\n        // 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n        val l = 2 * i + 1\n        val r = 2 * i + 2\n        var ma = i\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) \n            ma = l\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) \n            ma = r\n        // 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n        if (ma == i) \n            break\n        // 두 노드를 교환한다\n        val temp = nums[i]\n        nums[i] = nums[ma]\n        nums[ma] = temp\n        // 아래쪽으로 힙 재정렬을 반복한다\n        i = ma\n    }\n}\n\n/* 힙 정렬 */\nfun heapSort(nums: IntArray) {\n    // 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n    for (i in nums.size / 2 - 1 downTo 0) {\n        siftDown(nums, nums.size, i)\n    }\n    // 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n    for (i in nums.size - 1 downTo 1) {\n        // 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n        val temp = nums[0]\n        nums[0] = nums[i]\n        nums[i] = temp\n        // 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.rb<pre><code>### 길이가 n인 힙에서 노드 i부터 위에서 아래로 힙 재정렬 ###\ndef sift_down(nums, n, i)\n  while true\n    # 노드 i, l, r 중 값이 가장 큰 노드를 찾아 ma로 표시한다\n    l = 2 * i + 1\n    r = 2 * i + 2\n    ma = i\n    ma = l if l &lt; n &amp;&amp; nums[l] &gt; nums[ma]\n    ma = r if r &lt; n &amp;&amp; nums[r] &gt; nums[ma]\n    # 노드 i의 값이 가장 크거나 인덱스 l, r이 범위를 벗어나면 힙을 더 재정렬할 필요가 없으므로 종료한다\n    break if ma == i\n    # 두 노드를 교환한다\n    nums[i], nums[ma] = nums[ma], nums[i]\n    # 아래쪽으로 힙 재정렬을 반복한다\n    i = ma\n  end\nend\n\n### 힙 정렬 ###\ndef heap_sort(nums)\n  # 힙 구성 연산: 리프 노드를 제외한 모든 노드에 힙 재정렬을 수행한다\n  (nums.length / 2 - 1).downto(0) do |i|\n    sift_down(nums, nums.length, i)\n  end\n  # 힙에서 최댓값을 꺼내는 작업을 n-1회 반복한다\n  (nums.length - 1).downto(1) do |i|\n    # 루트 노드와 가장 오른쪽 리프 노드를 교환한다(첫 원소와 마지막 원소 교환)\n    nums[0], nums[i] = nums[i], nums[0]\n    # 루트 노드에서 시작하여 위에서 아래로 힙을 재정렬한다\n    sift_down(nums, i, 0)\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.7   힙 정렬"],"tags":[]},{"location":"chapter_sorting/heap_sort/#1172","level":2,"title":"11.7.2   알고리즘 특성","text":"<ul> <li>시간 복잡도 \\(O(n \\log n)\\), 비적응형 정렬: 힙 구성에는 \\(O(n)\\) 시간이 걸린다. 힙에서 최댓값을 꺼내는 시간 복잡도는 \\(O(\\log n)\\)이며, 총 \\(n - 1\\) 회 반복한다.</li> <li>공간 복잡도 \\(O(1)\\), 제자리 정렬: 몇 개의 포인터 변수가 \\(O(1)\\) 공간을 사용한다. 원소 교환과 힙 재정렬은 모두 원래 배열에서 수행한다.</li> <li>불안정 정렬: 힙의 루트 원소와 끝 원소를 교환할 때 값이 같은 원소들의 상대적 위치가 바뀔 수 있다.</li> </ul>","path":["제 11 장   정렬","11.7   힙 정렬"],"tags":[]},{"location":"chapter_sorting/insertion_sort/","level":1,"title":"11.4   삽입 정렬","text":"<p>삽입 정렬(insertion sort)은 간단한 정렬 알고리즘으로, 손으로 카드 한 벌을 정리하는 과정과 원리가 매우 비슷하다.</p> <p>구체적으로는 미정렬 구간에서 기준 원소 하나를 골라, 왼쪽의 정렬된 구간에 있는 원소들과 하나씩 크기를 비교한 뒤 올바른 위치에 삽입한다.</p> <p>아래 그림(11-6)은 배열에 원소를 삽입하는 과정을 보여 준다. 기준 원소를 <code>base</code>라고 하면 목표 인덱스부터 <code>base</code> 사이의 모든 원소를 오른쪽으로 한 칸씩 이동한 뒤, 목표 인덱스에 <code>base</code>를 대입한다.</p> <p></p> <p> 그림 11-6   한 번의 삽입 연산 </p>","path":["제 11 장   정렬","11.4   삽입 정렬"],"tags":[]},{"location":"chapter_sorting/insertion_sort/#1141","level":2,"title":"11.4.1   알고리즘 과정","text":"<p>삽입 정렬의 전체 과정은 아래 그림(11-7)과 같다.</p> <ol> <li>처음에는 배열의 첫 번째 원소가 정렬되어 있다.</li> <li>배열의 두 번째 원소를 <code>base</code>로 골라 올바른 위치에 삽입하면 배열의 앞 2개 원소가 정렬된다.</li> <li>세 번째 원소를 <code>base</code>로 골라 올바른 위치에 삽입하면 배열의 앞 3개 원소가 정렬된다.</li> <li>같은 과정을 반복한다. 마지막 회에서 마지막 원소를 <code>base</code>로 골라 올바른 위치에 삽입하면 모든 원소가 정렬된다.</li> </ol> <p></p> <p> 그림 11-7   삽입 정렬 과정 </p> <p>예시 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby insertion_sort.py<pre><code>def insertion_sort(nums: list[int]):\n    \"\"\"삽입 정렬\"\"\"\n    # 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for i in range(1, len(nums)):\n        base = nums[i]\n        j = i - 1\n        # 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while j &gt;= 0 and nums[j] &gt; base:\n            nums[j + 1] = nums[j]  # nums[j]를 오른쪽으로 한 칸 이동한다\n            j -= 1\n        nums[j + 1] = base  # 올바른 위치에 base를 대입한다\n</code></pre> insertion_sort.cpp<pre><code>/* 삽입 정렬 */\nvoid insertionSort(vector&lt;int&gt; &amp;nums) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for (int i = 1; i &lt; nums.size(); i++) {\n        int base = nums[i], j = i - 1;\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // nums[j]를 오른쪽으로 한 칸 이동한다\n            j--;\n        }\n        nums[j + 1] = base; // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.java<pre><code>/* 삽입 정렬 */\nvoid insertionSort(int[] nums) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for (int i = 1; i &lt; nums.length; i++) {\n        int base = nums[i], j = i - 1;\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // nums[j]를 오른쪽으로 한 칸 이동한다\n            j--;\n        }\n        nums[j + 1] = base;        // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.cs<pre><code>/* 삽입 정렬 */\nvoid InsertionSort(int[] nums) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for (int i = 1; i &lt; nums.Length; i++) {\n        int bas = nums[i], j = i - 1;\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; bas) {\n            nums[j + 1] = nums[j]; // nums[j]를 오른쪽으로 한 칸 이동한다\n            j--;\n        }\n        nums[j + 1] = bas;         // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.go<pre><code>/* 삽입 정렬 */\nfunc insertionSort(nums []int) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for i := 1; i &lt; len(nums); i++ {\n        base := nums[i]\n        j := i - 1\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        for j &gt;= 0 &amp;&amp; nums[j] &gt; base {\n            nums[j+1] = nums[j] // nums[j]를 오른쪽으로 한 칸 이동한다\n            j--\n        }\n        nums[j+1] = base // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.swift<pre><code>/* 삽입 정렬 */\nfunc insertionSort(nums: inout [Int]) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for i in nums.indices.dropFirst() {\n        let base = nums[i]\n        var j = i - 1\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while j &gt;= 0, nums[j] &gt; base {\n            nums[j + 1] = nums[j] // nums[j]를 오른쪽으로 한 칸 이동한다\n            j -= 1\n        }\n        nums[j + 1] = base // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.js<pre><code>/* 삽입 정렬 */\nfunction insertionSort(nums) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for (let i = 1; i &lt; nums.length; i++) {\n        let base = nums[i],\n            j = i - 1;\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // nums[j]를 오른쪽으로 한 칸 이동한다\n            j--;\n        }\n        nums[j + 1] = base; // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.ts<pre><code>/* 삽입 정렬 */\nfunction insertionSort(nums: number[]): void {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for (let i = 1; i &lt; nums.length; i++) {\n        const base = nums[i];\n        let j = i - 1;\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // nums[j]를 오른쪽으로 한 칸 이동한다\n            j--;\n        }\n        nums[j + 1] = base; // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.dart<pre><code>/* 삽입 정렬 */\nvoid insertionSort(List&lt;int&gt; nums) {\n  // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n  for (int i = 1; i &lt; nums.length; i++) {\n    int base = nums[i], j = i - 1;\n    // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n    while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n      nums[j + 1] = nums[j]; // nums[j]를 오른쪽으로 한 칸 이동한다\n      j--;\n    }\n    nums[j + 1] = base; // 올바른 위치에 base를 대입한다\n  }\n}\n</code></pre> insertion_sort.rs<pre><code>/* 삽입 정렬 */\nfn insertion_sort(nums: &amp;mut [i32]) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for i in 1..nums.len() {\n        let (base, mut j) = (nums[i], (i - 1) as i32);\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while j &gt;= 0 &amp;&amp; nums[j as usize] &gt; base {\n            nums[(j + 1) as usize] = nums[j as usize]; // nums[j]를 오른쪽으로 한 칸 이동한다\n            j -= 1;\n        }\n        nums[(j + 1) as usize] = base; // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.c<pre><code>/* 삽입 정렬 */\nvoid insertionSort(int nums[], int size) {\n    // 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n    for (int i = 1; i &lt; size; i++) {\n        int base = nums[i], j = i - 1;\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            // nums[j]를 오른쪽으로 한 칸 이동한다\n            nums[j + 1] = nums[j];\n            j--;\n        }\n        // 올바른 위치에 base를 대입한다\n        nums[j + 1] = base;\n    }\n}\n</code></pre> insertion_sort.kt<pre><code>/* 삽입 정렬 */\nfun insertionSort(nums: IntArray) {\n    //바깥쪽 반복문: 정렬된 원소의 수는 1, 2, ..., n이 된다\n    for (i in nums.indices) {\n        val base = nums[i]\n        var j = i - 1\n        // 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j] // nums[j]를 오른쪽으로 한 칸 이동한다\n            j--\n        }\n        nums[j + 1] = base        // 올바른 위치에 base를 대입한다\n    }\n}\n</code></pre> insertion_sort.rb<pre><code>### 삽입 정렬 ###\ndef insertion_sort(nums)\n  n = nums.length\n  # 바깥쪽 반복문: 정렬된 구간은 [0, i-1]이다\n  for i in 1...n\n    base = nums[i]\n    j = i - 1\n    # 안쪽 반복문: base를 정렬된 구간 [0, i-1]의 올바른 위치에 삽입한다\n    while j &gt;= 0 &amp;&amp; nums[j] &gt; base\n      nums[j + 1] = nums[j] # nums[j]를 오른쪽으로 한 칸 이동한다\n      j -= 1\n    end\n    nums[j + 1] = base # 올바른 위치에 base를 대입한다\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.4   삽입 정렬"],"tags":[]},{"location":"chapter_sorting/insertion_sort/#1142","level":2,"title":"11.4.2   알고리즘 특성","text":"<ul> <li>시간 복잡도 \\(O(n^2)\\), 적응형 정렬: 최악의 경우 각 삽입 연산에서 \\(n - 1\\), \\(n-2\\), \\(\\dots\\), \\(2\\), \\(1\\) 번 반복해야 한다. 합은 \\((n - 1) n / 2\\)이므로 시간 복잡도는 \\(O(n^2)\\)이다. 정렬된 데이터를 만나면 삽입 연산이 일찍 끝난다. 입력 배열이 완전히 정렬되어 있으면 최선 시간 복잡도 \\(O(n)\\)에 도달한다.</li> <li>공간 복잡도 \\(O(1)\\), 제자리 정렬: 포인터 \\(i\\)와 \\(j\\)는 상수 크기의 추가 공간을 사용한다.</li> <li>안정 정렬: 삽입할 때 값이 같은 원소의 오른쪽에 삽입하므로 상대적 순서가 바뀌지 않는다.</li> </ul>","path":["제 11 장   정렬","11.4   삽입 정렬"],"tags":[]},{"location":"chapter_sorting/insertion_sort/#1143","level":2,"title":"11.4.3   삽입 정렬의 장점","text":"<p>삽입 정렬의 시간 복잡도는 \\(O(n^2)\\)이고, 곧 배울 퀵 정렬의 시간 복잡도는 \\(O(n \\log n)\\)이다. 삽입 정렬의 시간 복잡도가 더 높지만 데이터가 적으면 보통 삽입 정렬이 더 빠르다.</p> <p>이는 선형 탐색과 이진 탐색의 적합한 사용 상황에서 내린 결론과 비슷하다. 퀵 정렬 같은 \\(O(n \\log n)\\) 알고리즘은 분할 정복 전략에 기반한 정렬 알고리즘으로, 기본 연산이 더 많은 경우가 많다. 데이터가 적으면 \\(n^2\\)과 \\(n \\log n\\)의 값이 비슷하여 시간 복잡도가 결정적인 요인이 되지 못하고, 매 반복에서 수행하는 기본 연산 수가 결정적인 역할을 한다.</p> <p>실제로 Java 등 여러 프로그래밍 언어의 내장 정렬 함수는 삽입 정렬을 활용한다. 대체로 긴 배열에는 퀵 정렬 같은 분할 정복 기반 정렬을 사용하고, 짧은 배열에는 삽입 정렬을 바로 적용한다.</p> <p>버블 정렬, 선택 정렬, 삽입 정렬의 시간 복잡도는 모두 \\(O(n^2)\\)이지만, 실제로는 삽입 정렬이 버블 정렬이나 선택 정렬보다 훨씬 자주 사용된다. 주된 이유는 다음과 같다.</p> <ul> <li>버블 정렬은 원소 교환으로 구현하므로 임시 변수가 필요하고 총 3회의 기본 연산을 수행한다. 삽입 정렬은 원소 대입으로 구현하므로 기본 연산이 1회만 필요하다. 따라서 버블 정렬의 계산 비용은 보통 삽입 정렬보다 크다.</li> <li>선택 정렬의 시간 복잡도는 어떤 경우에도 \\(O(n^2)\\)이다. 데이터가 부분적으로 정렬되어 있다면 삽입 정렬이 보통 선택 정렬보다 효율적이다.</li> <li>선택 정렬은 불안정하므로 여러 기준으로 순차 정렬하는 데 사용할 수 없다.</li> </ul>","path":["제 11 장   정렬","11.4   삽입 정렬"],"tags":[]},{"location":"chapter_sorting/merge_sort/","level":1,"title":"11.6   병합 정렬","text":"<p>병합 정렬(merge sort)은 분할 정복 전략에 기반한 정렬 알고리즘으로, 아래 그림(11-10)의 '분할' 단계와 '병합' 단계로 이루어진다.</p> <ol> <li>분할 단계: 재귀적으로 배열을 계속 중간에서 나누어, 긴 배열을 정렬하는 문제를 짧은 배열을 정렬하는 문제로 바꾼다.</li> <li>병합 단계: 부분 배열의 길이가 1이 되면 분할을 멈추고 병합을 시작한다. 정렬된 짧은 왼쪽 배열과 오른쪽 배열을 하나의 더 긴 정렬된 배열로 계속 합치며, 끝날 때까지 반복한다.</li> </ol> <p></p> <p> 그림 11-10   병합 정렬의 분할 단계와 병합 단계 </p>","path":["제 11 장   정렬","11.6   병합 정렬"],"tags":[]},{"location":"chapter_sorting/merge_sort/#1161","level":2,"title":"11.6.1   알고리즘 과정","text":"<p>아래 그림(11-11)과 같이 '분할 단계'에서는 위에서 아래로 재귀하며 배열을 중간에서 두 부분 배열로 나눈다.</p> <ol> <li>배열의 중간 인덱스 <code>mid</code>를 계산하고 왼쪽 부분 배열(구간 <code>[left, mid]</code>)과 오른쪽 부분 배열(구간 <code>[mid + 1, right]</code>)을 재귀적으로 나눈다.</li> <li>부분 배열 구간의 길이가 1이 될 때까지 <code>1.</code> 단계를 재귀적으로 반복한다.</li> </ol> <p>'병합 단계'에서는 아래에서 위로 왼쪽 부분 배열과 오른쪽 부분 배열을 하나의 정렬된 배열로 합친다. 길이가 1인 부분 배열부터 병합하므로, 병합 단계의 각 부분 배열은 이미 정렬되어 있다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 11-11   병합 정렬 단계 </p> <p>병합 정렬의 재귀 순서는 이진 트리의 후위 순회와 같다는 것을 알 수 있다.</p> <ul> <li>후위 순회: 왼쪽 서브트리에 재귀한 다음 오른쪽 서브트리에 재귀하고, 마지막으로 루트 노드를 처리한다.</li> <li>병합 정렬: 왼쪽 부분 배열에 재귀한 다음 오른쪽 부분 배열에 재귀하고, 마지막으로 병합한다.</li> </ul> <p>병합 정렬의 구현은 다음 코드와 같다. <code>nums</code>에서 병합할 구간은 <code>[left, right]</code>이고, 이에 대응하는 <code>tmp</code>의 구간은 <code>[0, right - left]</code>라는 점에 주의하자.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby merge_sort.py<pre><code>def merge(nums: list[int], left: int, mid: int, right: int):\n    \"\"\"왼쪽 부분 배열과 오른쪽 부분 배열 병합\"\"\"\n    # 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    # 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    tmp = [0] * (right - left + 1)\n    # 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    i, j, k = left, mid + 1, 0\n    # 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while i &lt;= mid and j &lt;= right:\n        if nums[i] &lt;= nums[j]:\n            tmp[k] = nums[i]\n            i += 1\n        else:\n            tmp[k] = nums[j]\n            j += 1\n        k += 1\n    # 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while i &lt;= mid:\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    while j &lt;= right:\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    # 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for k in range(0, len(tmp)):\n        nums[left + k] = tmp[k]\n\ndef merge_sort(nums: list[int], left: int, right: int):\n    \"\"\"병합 정렬\"\"\"\n    # 종료 조건\n    if left &gt;= right:\n        return  # 부분 배열의 길이가 1이면 재귀를 종료한다\n    # 분할 단계\n    mid = (left + right) // 2 # 중간 위치를 계산한다\n    merge_sort(nums, left, mid)  # 왼쪽 부분 배열에 재귀 호출한다\n    merge_sort(nums, mid + 1, right)  # 오른쪽 부분 배열에 재귀 호출한다\n    # 병합 단계\n    merge(nums, left, mid, right)\n</code></pre> merge_sort.cpp<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nvoid merge(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    vector&lt;int&gt; tmp(right - left + 1);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    int i = left, j = mid + 1, k = 0;\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for (k = 0; k &lt; tmp.size(); k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* 병합 정렬 */\nvoid mergeSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // 종료 조건\n    if (left &gt;= right)\n        return; // 부분 배열의 길이가 1이면 재귀를 종료한다\n    // 분할 단계\n    int mid = left + (right - left) / 2;    // 중간 위치를 계산한다\n    mergeSort(nums, left, mid);      // 왼쪽 부분 배열에 재귀 호출한다\n    mergeSort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.java<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nvoid merge(int[] nums, int left, int mid, int right) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    int[] tmp = new int[right - left + 1];\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    int i = left, j = mid + 1, k = 0;\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* 병합 정렬 */\nvoid mergeSort(int[] nums, int left, int right) {\n    // 종료 조건\n    if (left &gt;= right)\n        return; // 부분 배열의 길이가 1이면 재귀를 종료한다\n    // 분할 단계\n    int mid = left + (right - left) / 2; // 중간 위치를 계산한다\n    mergeSort(nums, left, mid); // 왼쪽 부분 배열에 재귀 호출한다\n    mergeSort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.cs<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nvoid Merge(int[] nums, int left, int mid, int right) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    int[] tmp = new int[right - left + 1];\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    int i = left, j = mid + 1, k = 0;\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for (k = 0; k &lt; tmp.Length; ++k) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* 병합 정렬 */\nvoid MergeSort(int[] nums, int left, int right) {\n    // 종료 조건\n    if (left &gt;= right) return;       // 부분 배열의 길이가 1이면 재귀를 종료한다\n    // 분할 단계\n    int mid = left + (right - left) / 2;    // 중간 위치를 계산한다\n    MergeSort(nums, left, mid);      // 왼쪽 부분 배열에 재귀 호출한다\n    MergeSort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    Merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.go<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nfunc merge(nums []int, left, mid, right int) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    tmp := make([]int, right-left+1)\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    i, j, k := left, mid+1, 0\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    for i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i++\n        } else {\n            tmp[k] = nums[j]\n            j++\n        }\n        k++\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    for i &lt;= mid {\n        tmp[k] = nums[i]\n        i++\n        k++\n    }\n    for j &lt;= right {\n        tmp[k] = nums[j]\n        j++\n        k++\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for k := 0; k &lt; len(tmp); k++ {\n        nums[left+k] = tmp[k]\n    }\n}\n\n/* 병합 정렬 */\nfunc mergeSort(nums []int, left, right int) {\n    // 종료 조건\n    if left &gt;= right {\n        return\n    }\n    // 분할 단계\n    mid := left + (right - left) / 2\n    mergeSort(nums, left, mid)\n    mergeSort(nums, mid+1, right)\n    // 병합 단계\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.swift<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    var tmp = Array(repeating: 0, count: right - left + 1)\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    var i = left, j = mid + 1, k = 0\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while i &lt;= mid, j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i += 1\n        } else {\n            tmp[k] = nums[j]\n            j += 1\n        }\n        k += 1\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while i &lt;= mid {\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for k in tmp.indices {\n        nums[left + k] = tmp[k]\n    }\n}\n\n/* 병합 정렬 */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n    // 종료 조건\n    if left &gt;= right { // 부분 배열의 길이가 1이면 재귀를 종료한다\n        return\n    }\n    // 분할 단계\n    let mid = left + (right - left) / 2 // 중간 위치를 계산한다\n    mergeSort(nums: &amp;nums, left: left, right: mid) // 왼쪽 부분 배열에 재귀 호출한다\n    mergeSort(nums: &amp;nums, left: mid + 1, right: right) // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    merge(nums: &amp;nums, left: left, mid: mid, right: right)\n}\n</code></pre> merge_sort.js<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nfunction merge(nums, left, mid, right) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    const tmp = new Array(right - left + 1);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* 병합 정렬 */\nfunction mergeSort(nums, left, right) {\n    // 종료 조건\n    if (left &gt;= right) return; // 부분 배열의 길이가 1이면 재귀를 종료한다\n    // 분할 단계\n    let mid = Math.floor(left + (right - left) / 2); // 중간 위치를 계산한다\n    mergeSort(nums, left, mid); // 왼쪽 부분 배열에 재귀 호출한다\n    mergeSort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.ts<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    const tmp = new Array(right - left + 1);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* 병합 정렬 */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n    // 종료 조건\n    if (left &gt;= right) return; // 부분 배열의 길이가 1이면 재귀를 종료한다\n    // 분할 단계\n    let mid = Math.floor(left + (right - left) / 2); // 중간 위치를 계산한다\n    mergeSort(nums, left, mid); // 왼쪽 부분 배열에 재귀 호출한다\n    mergeSort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.dart<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nvoid merge(List&lt;int&gt; nums, int left, int mid, int right) {\n  // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n  // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n  List&lt;int&gt; tmp = List.filled(right - left + 1, 0);\n  // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n  int i = left, j = mid + 1, k = 0;\n  // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n  while (i &lt;= mid &amp;&amp; j &lt;= right) {\n    if (nums[i] &lt;= nums[j])\n      tmp[k++] = nums[i++];\n    else\n      tmp[k++] = nums[j++];\n  }\n  // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n  while (i &lt;= mid) {\n    tmp[k++] = nums[i++];\n  }\n  while (j &lt;= right) {\n    tmp[k++] = nums[j++];\n  }\n  // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n  for (k = 0; k &lt; tmp.length; k++) {\n    nums[left + k] = tmp[k];\n  }\n}\n\n/* 병합 정렬 */\nvoid mergeSort(List&lt;int&gt; nums, int left, int right) {\n  // 종료 조건\n  if (left &gt;= right) return; // 부분 배열의 길이가 1이면 재귀를 종료한다\n  // 분할 단계\n  int mid = left + (right - left) ~/ 2; // 중간 위치를 계산한다\n  mergeSort(nums, left, mid); // 왼쪽 부분 배열에 재귀 호출한다\n  mergeSort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n  // 병합 단계\n  merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.rs<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nfn merge(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    let tmp_size = right - left + 1;\n    let mut tmp = vec![0; tmp_size];\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    let (mut i, mut j, mut k) = (left, mid + 1, 0);\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i];\n            i += 1;\n        } else {\n            tmp[k] = nums[j];\n            j += 1;\n        }\n        k += 1;\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while i &lt;= mid {\n        tmp[k] = nums[i];\n        k += 1;\n        i += 1;\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j];\n        k += 1;\n        j += 1;\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for k in 0..tmp_size {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* 병합 정렬 */\nfn merge_sort(nums: &amp;mut [i32], left: usize, right: usize) {\n    // 종료 조건\n    if left &gt;= right {\n        return; // 부분 배열의 길이가 1이면 재귀를 종료한다\n    }\n\n    // 분할 단계\n    let mid = left + (right - left) / 2; // 중간 위치를 계산한다\n    merge_sort(nums, left, mid); // 왼쪽 부분 배열에 재귀 호출한다\n    merge_sort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n\n    // 병합 단계\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.c<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nvoid merge(int *nums, int left, int mid, int right) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    int tmpSize = right - left + 1;\n    int *tmp = (int *)malloc(tmpSize * sizeof(int));\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    int i = left, j = mid + 1, k = 0;\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for (k = 0; k &lt; tmpSize; ++k) {\n        nums[left + k] = tmp[k];\n    }\n    // 메모리 해제\n    free(tmp);\n}\n\n/* 병합 정렬 */\nvoid mergeSort(int *nums, int left, int right) {\n    // 종료 조건\n    if (left &gt;= right)\n        return; // 부분 배열의 길이가 1이면 재귀를 종료한다\n    // 분할 단계\n    int mid = left + (right - left) / 2;    // 중간 위치를 계산한다\n    mergeSort(nums, left, mid);      // 왼쪽 부분 배열에 재귀 호출한다\n    mergeSort(nums, mid + 1, right); // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.kt<pre><code>/* 왼쪽 부분 배열과 오른쪽 부분 배열 병합 */\nfun merge(nums: IntArray, left: Int, mid: Int, right: Int) {\n    // 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n    // 병합 결과를 저장할 임시 배열 tmp를 생성한다\n    val tmp = IntArray(right - left + 1)\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n    var i = left\n    var j = mid + 1\n    var k = 0\n    // 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++]\n        else\n            tmp[k++] = nums[j++]\n    }\n    // 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++]\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++]\n    }\n    // 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n    for (l in tmp.indices) {\n        nums[left + l] = tmp[l]\n    }\n}\n\n/* 병합 정렬 */\nfun mergeSort(nums: IntArray, left: Int, right: Int) {\n    // 종료 조건\n    if (left &gt;= right) return  // 부분 배열의 길이가 1이면 재귀를 종료한다\n    // 분할 단계\n    val mid = left + (right - left) / 2 // 중간 위치를 계산한다\n    mergeSort(nums, left, mid) // 왼쪽 부분 배열에 재귀 호출한다\n    mergeSort(nums, mid + 1, right) // 오른쪽 부분 배열에 재귀 호출한다\n    // 병합 단계\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.rb<pre><code>### 왼쪽 부분 배열과 오른쪽 부분 배열 병합 ###\ndef merge(nums, left, mid, right)\n  # 왼쪽 부분 배열의 구간은 [left, mid]이고, 오른쪽 부분 배열의 구간은 [mid+1, right]이다\n  # 병합 결과를 저장할 임시 배열 tmp를 생성한다\n  tmp = Array.new(right - left + 1, 0)\n  # 왼쪽 부분 배열과 오른쪽 부분 배열의 시작 인덱스를 초기화한다\n  i, j, k = left, mid + 1, 0\n  # 양쪽 부분 배열 모두에 원소가 남아 있으면 비교하여 더 작은 원소를 임시 배열에 복사한다\n  while i &lt;= mid &amp;&amp; j &lt;= right\n    if nums[i] &lt;= nums[j]\n      tmp[k] = nums[i]\n      i += 1\n    else\n      tmp[k] = nums[j]\n      j += 1\n    end\n    k += 1\n  end\n  # 왼쪽 부분 배열과 오른쪽 부분 배열의 남은 원소를 임시 배열에 복사한다\n  while i &lt;= mid\n    tmp[k] = nums[i]\n    i += 1\n    k += 1\n  end\n  while j &lt;= right\n    tmp[k] = nums[j]\n    j += 1\n    k += 1\n  end\n  # 임시 배열 tmp의 원소를 원래 배열 nums의 해당 구간에 다시 복사한다\n  (0...tmp.length).each do |k|\n    nums[left + k] = tmp[k]\n  end\nend\n\n### 병합 정렬 ###\ndef merge_sort(nums, left, right)\n  # 종료 조건\n  # 부분 배열의 길이가 1이면 재귀를 종료한다\n  return if left &gt;= right\n  # 분할 단계\n  mid = left + (right - left) / 2 # 중간 위치를 계산한다\n  merge_sort(nums, left, mid) # 왼쪽 부분 배열에 재귀 호출한다\n  merge_sort(nums, mid + 1, right) # 오른쪽 부분 배열에 재귀 호출한다\n  # 병합 단계\n  merge(nums, left, mid, right)\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.6   병합 정렬"],"tags":[]},{"location":"chapter_sorting/merge_sort/#1162","level":2,"title":"11.6.2   알고리즘 특성","text":"<ul> <li>시간 복잡도 \\(O(n \\log n)\\), 비적응형 정렬: 분할 과정에서 높이 \\(\\log n\\)의 재귀 트리가 만들어지고, 각 레벨에서 병합에 필요한 총연산 수는 \\(n\\)이므로 전체 시간 복잡도는 \\(O(n \\log n)\\)이다.</li> <li>공간 복잡도 \\(O(n)\\), 제자리 정렬이 아님: 재귀 깊이가 \\(\\log n\\)이므로 \\(O(\\log n)\\) 크기의 스택 프레임 공간을 사용한다. 병합에는 보조 배열이 필요하여 \\(O(n)\\)의 추가 공간을 사용한다.</li> <li>안정 정렬: 병합 과정에서 값이 같은 원소들의 순서가 유지된다.</li> </ul>","path":["제 11 장   정렬","11.6   병합 정렬"],"tags":[]},{"location":"chapter_sorting/merge_sort/#1163","level":2,"title":"11.6.3   연결 리스트 정렬","text":"<p>연결 리스트에서는 병합 정렬이 다른 정렬 알고리즘보다 뚜렷한 장점을 가진다. 연결 리스트 정렬의 공간 복잡도를 \\(O(1)\\)까지 최적화할 수 있다.</p> <ul> <li>분할 단계: 연결 리스트를 나눌 때 '재귀' 대신 '반복'을 사용하면 재귀에 필요한 스택 프레임 공간을 없앨 수 있다.</li> <li>병합 단계: 연결 리스트에서는 참조(포인터)만 바꾸어 노드를 추가하거나 삭제할 수 있다. 따라서 정렬된 짧은 연결 리스트 두 개를 하나의 긴 연결 리스트로 합칠 때 추가 연결 리스트를 만들 필요가 없다.</li> </ul> <p>구체적인 구현은 다소 복잡하므로, 관심 있는 독자는 관련 자료를 찾아 학습해 보자.</p>","path":["제 11 장   정렬","11.6   병합 정렬"],"tags":[]},{"location":"chapter_sorting/quick_sort/","level":1,"title":"11.5   퀵 정렬","text":"<p>퀵 정렬(quick sort)은 분할 정복 전략에 기반한 정렬 알고리즘으로, 실행 효율이 높고 널리 사용된다.</p> <p>퀵 정렬의 핵심 연산은 '센티널 분할'이다. 배열의 한 원소를 '피벗'으로 고른 다음, 피벗보다 작은 원소는 모두 왼쪽으로, 큰 원소는 모두 오른쪽으로 옮기는 것이 목표다. 구체적인 과정은 아래 그림(11-8)과 같다.</p> <ol> <li>배열의 맨 왼쪽 원소를 피벗으로 고르고, 포인터 <code>i</code>와 <code>j</code>가 배열의 양 끝을 가리키도록 초기화한다.</li> <li>반복문을 실행하여 매번 <code>i</code>(<code>j</code>)로 피벗보다 큰(작은) 첫 번째 원소를 각각 찾은 뒤 두 원소를 교환한다.</li> <li><code>i</code>와 <code>j</code>가 만날 때까지 <code>2.</code> 단계를 반복한다. 마지막으로 피벗을 두 부분 배열의 경계로 옮긴다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 11-8   센티널 분할 단계 </p> <p>분할을 마치면 원래 배열은 왼쪽 부분 배열, 피벗, 오른쪽 부분 배열의 세 부분으로 나뉘며, '왼쪽 부분 배열의 모든 원소 \\(\\leq\\) 피벗 \\(\\leq\\) 오른쪽 부분 배열의 모든 원소'를 만족한다. 따라서 이후에는 두 부분 배열만 정렬하면 된다.</p> <p>퀵 정렬의 분할 정복 전략</p> <p>센티널 분할은 본질적으로 긴 배열 하나를 정렬하는 문제를 짧은 배열 두 개를 정렬하는 문제로 단순화한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"센티널 분할\"\"\"\n    # nums[left]를 피벗으로 삼는다\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        # 원소 교환\n        nums[i], nums[j] = nums[j], nums[i]\n    # 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # 피벗의 인덱스를 반환한다\n</code></pre> quick_sort.cpp<pre><code>/* 센티널 분할 */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;                // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;                // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        swap(nums[i], nums[j]); // 두 원소를 교환한다\n    }\n    swap(nums[i], nums[left]);  // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i;                   // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.java<pre><code>/* 원소 교환 */\nvoid swap(int[] nums, int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* 센티널 분할 */\nint partition(int[] nums, int left, int right) {\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        swap(nums, i, j); // 두 원소를 교환한다\n    }\n    swap(nums, i, left);  // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i;             // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.cs<pre><code>/* 원소 교환 */\nvoid Swap(int[] nums, int i, int j) {\n    (nums[j], nums[i]) = (nums[i], nums[j]);\n}\n\n/* 센티널 분할 */\nint Partition(int[] nums, int left, int right) {\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        Swap(nums, i, j); // 두 원소를 교환한다\n    }\n    Swap(nums, i, left);  // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i;             // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.go<pre><code>/* 센티널 분할 */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n    // nums[left]를 피벗으로 삼는다\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        // 원소 교환\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    nums[i], nums[left] = nums[left], nums[i]\n    return i // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.swift<pre><code>/* 센티널 분할 */\nfunc partition(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // nums[left]를 피벗으로 삼는다\n    var i = left\n    var j = right\n    while i &lt; j {\n        while i &lt; j, nums[j] &gt;= nums[left] {\n            j -= 1 // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        while i &lt; j, nums[i] &lt;= nums[left] {\n            i += 1 // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        nums.swapAt(i, j) // 두 원소를 교환한다\n    }\n    nums.swapAt(i, left) // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.js<pre><code>/* 원소 교환 */\nswap(nums, i, j) {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* 센티널 분할 */\npartition(nums, left, right) {\n    // nums[left]를 피벗으로 삼는다\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        // 원소 교환\n        this.swap(nums, i, j); // 두 원소를 교환한다\n    }\n    this.swap(nums, i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i; // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.ts<pre><code>/* 원소 교환 */\nswap(nums: number[], i: number, j: number): void {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* 센티널 분할 */\npartition(nums: number[], left: number, right: number): number {\n    // nums[left]를 피벗으로 삼는다\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        // 원소 교환\n        this.swap(nums, i, j); // 두 원소를 교환한다\n    }\n    this.swap(nums, i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i; // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.dart<pre><code>/* 원소 교환 */\nvoid _swap(List&lt;int&gt; nums, int i, int j) {\n  int tmp = nums[i];\n  nums[i] = nums[j];\n  nums[j] = tmp;\n}\n\n/* 센티널 분할 */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // nums[left]를 피벗으로 삼는다\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n    _swap(nums, i, j); // 두 원소를 교환한다\n  }\n  _swap(nums, i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n  return i; // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.rs<pre><code>/* 센티널 분할 */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // nums[left]를 피벗으로 삼는다\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        nums.swap(i, j); // 두 원소를 교환한다\n    }\n    nums.swap(i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    i // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.c<pre><code>/* 원소 교환 */\nvoid swap(int nums[], int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* 센티널 분할 */\nint partition(int nums[], int left, int right) {\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j--; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i++; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        // 두 원소를 교환한다\n        swap(nums, i, j);\n    }\n    // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    swap(nums, i, left);\n    // 피벗의 인덱스를 반환한다\n    return i;\n}\n</code></pre> quick_sort.kt<pre><code>/* 원소 교환 */\nfun swap(nums: IntArray, i: Int, j: Int) {\n    val temp = nums[i]\n    nums[i] = nums[j]\n    nums[j] = temp\n}\n\n/* 센티널 분할 */\nfun partition(nums: IntArray, left: Int, right: Int): Int {\n    // nums[left]를 피벗으로 삼는다\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--           // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++           // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        swap(nums, i, j)  // 두 원소를 교환한다\n    }\n    swap(nums, i, left)   // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i              // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.rb<pre><code>### 센티널 분할 ###\ndef partition(nums, left, right)\n  # nums[left]를 피벗으로 삼는다\n  i, j = left, right\n  while i &lt; j\n    while i &lt; j &amp;&amp; nums[j] &gt;= nums[left]\n      j -= 1 # 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n    end\n    while i &lt; j &amp;&amp; nums[i] &lt;= nums[left]\n      i += 1 # 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n    end\n    # 원소 교환\n    nums[i], nums[j] = nums[j], nums[i]\n  end\n  # 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n  nums[i], nums[left] = nums[left], nums[i]\n  i # 피벗의 인덱스를 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.5   퀵 정렬"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1151","level":2,"title":"11.5.1   알고리즘 과정","text":"<p>퀵 정렬의 전체 과정은 아래 그림(11-9)과 같다.</p> <ol> <li>먼저 원래 배열에 '센티널 분할'을 한 번 수행하여 정렬되지 않은 왼쪽 부분 배열과 오른쪽 부분 배열을 얻는다.</li> <li>이어서 왼쪽 부분 배열과 오른쪽 부분 배열에 각각 '센티널 분할'을 재귀적으로 수행한다.</li> <li>부분 배열의 길이가 1이 될 때까지 재귀를 계속하면 배열 전체의 정렬이 완료된다.</li> </ol> <p></p> <p> 그림 11-9   퀵 정렬 과정 </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"퀵 정렬\"\"\"\n    # 부분 배열의 길이가 1이면 재귀를 종료한다\n    if left &gt;= right:\n        return\n    # 센티널 분할\n    pivot = self.partition(nums, left, right)\n    # 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    self.quick_sort(nums, left, pivot - 1)\n    self.quick_sort(nums, pivot + 1, right)\n</code></pre> quick_sort.cpp<pre><code>/* 퀵 정렬 */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if (left &gt;= right)\n        return;\n    // 센티널 분할\n    int pivot = partition(nums, left, right);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.java<pre><code>/* 퀵 정렬 */\nvoid quickSort(int[] nums, int left, int right) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if (left &gt;= right)\n        return;\n    // 센티널 분할\n    int pivot = partition(nums, left, right);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.cs<pre><code>/* 퀵 정렬 */\nvoid QuickSort(int[] nums, int left, int right) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if (left &gt;= right)\n        return;\n    // 센티널 분할\n    int pivot = Partition(nums, left, right);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    QuickSort(nums, left, pivot - 1);\n    QuickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.go<pre><code>/* 퀵 정렬 */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if left &gt;= right {\n        return\n    }\n    // 센티널 분할\n    pivot := q.partition(nums, left, right)\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    q.quickSort(nums, left, pivot-1)\n    q.quickSort(nums, pivot+1, right)\n}\n</code></pre> quick_sort.swift<pre><code>/* 퀵 정렬 */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if left &gt;= right {\n        return\n    }\n    // 센티널 분할\n    let pivot = partition(nums: &amp;nums, left: left, right: right)\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    quickSort(nums: &amp;nums, left: left, right: pivot - 1)\n    quickSort(nums: &amp;nums, left: pivot + 1, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* 퀵 정렬 */\nquickSort(nums, left, right) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if (left &gt;= right) return;\n    // 센티널 분할\n    const pivot = this.partition(nums, left, right);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.ts<pre><code>/* 퀵 정렬 */\nquickSort(nums: number[], left: number, right: number): void {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if (left &gt;= right) {\n        return;\n    }\n    // 센티널 분할\n    const pivot = this.partition(nums, left, right);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.dart<pre><code>/* 퀵 정렬 */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // 부분 배열의 길이가 1이면 재귀를 종료한다\n  if (left &gt;= right) return;\n  // 센티널 분할\n  int pivot = _partition(nums, left, right);\n  // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n  quickSort(nums, left, pivot - 1);\n  quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.rs<pre><code>/* 퀵 정렬 */\npub fn quick_sort(left: i32, right: i32, nums: &amp;mut [i32]) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if left &gt;= right {\n        return;\n    }\n    // 센티널 분할\n    let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    Self::quick_sort(left, pivot - 1, nums);\n    Self::quick_sort(pivot + 1, right, nums);\n}\n</code></pre> quick_sort.c<pre><code>/* 퀵 정렬 */\nvoid quickSort(int nums[], int left, int right) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if (left &gt;= right) {\n        return;\n    }\n    // 센티널 분할\n    int pivot = partition(nums, left, right);\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.kt<pre><code>/* 퀵 정렬 */\nfun quickSort(nums: IntArray, left: Int, right: Int) {\n    // 부분 배열의 길이가 1이면 재귀를 종료한다\n    if (left &gt;= right) return\n    // 센티널 분할\n    val pivot = partition(nums, left, right)\n    // 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    quickSort(nums, left, pivot - 1)\n    quickSort(nums, pivot + 1, right)\n}\n</code></pre> quick_sort.rb<pre><code>### 퀵 정렬 클래스 ###\ndef quick_sort(nums, left, right)\n  # 부분 배열의 길이가 1이 아니면 재귀 호출한다\n  if left &lt; right\n    # 센티널 분할\n    pivot = partition(nums, left, right)\n    # 왼쪽 부분 배열과 오른쪽 부분 배열에 재귀 호출한다\n    quick_sort(nums, left, pivot - 1)\n    quick_sort(nums, pivot + 1, right)\n  end\n  nums\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.5   퀵 정렬"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1152","level":2,"title":"11.5.2   알고리즘 특성","text":"<ul> <li>시간 복잡도 \\(O(n \\log n)\\), 비적응형 정렬: 평균적으로 분할의 재귀 레벨 수는 \\(\\log n\\)이고 각 레벨의 총반복 횟수는 \\(n\\)이므로 전체 시간은 \\(O(n \\log n)\\)이다. 최악의 경우 매 분할에서 길이 \\(n\\)의 배열이 길이 \\(0\\)과 \\(n - 1\\)인 두 부분 배열로 나뉜다. 이때 재귀 레벨 수는 \\(n\\)에 이르고 각 레벨의 반복 횟수는 \\(n\\)이므로 전체 시간은 \\(O(n^2)\\)이다.</li> <li>공간 복잡도 \\(O(n)\\), 제자리 정렬: 입력 배열이 완전히 역순이면 재귀 깊이가 최악의 값인 \\(n\\)에 이르므로 \\(O(n)\\)의 스택 프레임 공간을 사용한다. 정렬 자체는 추가 배열 없이 원래 배열에서 수행한다.</li> <li>불안정 정렬: 분할의 마지막 단계에서 피벗이 값이 같은 원소의 오른쪽으로 교환될 수 있다.</li> </ul>","path":["제 11 장   정렬","11.5   퀵 정렬"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1153","level":2,"title":"11.5.3   퀵 정렬이 빠른 이유","text":"<p>이름에서도 알 수 있듯이 퀵 정렬은 효율 측면에서 장점이 있다. 평균 시간 복잡도는 '병합 정렬', '힙 정렬'과 같지만 보통 퀵 정렬이 더 효율적이다. 주요 이유는 다음과 같다.</p> <ul> <li>최악의 경우가 발생할 확률이 낮다: 퀵 정렬의 최악 시간 복잡도는 \\(O(n^2)\\)으로 병합 정렬만큼 안정적이지 않지만, 대부분의 경우 \\(O(n \\log n)\\) 시간에 실행된다.</li> <li>캐시 활용 효율이 높다: 분할을 수행할 때 시스템이 부분 배열 전체를 캐시에 올릴 수 있어 원소 접근이 효율적이다. 반면 '힙 정렬' 같은 알고리즘은 원소를 건너뛰며 접근하므로 이런 특성이 없다.</li> <li>복잡도의 상수 계수가 작다: 위 세 알고리즘 중 퀵 정렬은 비교, 대입, 교환 등의 총연산 수가 가장 적다. '삽입 정렬'이 '버블 정렬'보다 빠른 이유와 비슷하다.</li> </ul>","path":["제 11 장   정렬","11.5   퀵 정렬"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1154","level":2,"title":"11.5.4   피벗 선택 최적화","text":"<p>일부 입력에서는 퀵 정렬의 시간 효율이 떨어질 수 있다. 극단적인 예로 입력 배열이 완전히 역순이라고 하자. 맨 왼쪽 원소를 피벗으로 고르면 분할 후 피벗이 맨 오른쪽으로 이동하여 왼쪽 부분 배열의 길이는 \\(n - 1\\), 오른쪽 부분 배열의 길이는 \\(0\\)이 된다. 이런 식으로 재귀하면 매번 한쪽 부분 배열의 길이가 \\(0\\)이 되어 분할 정복 전략이 제 기능을 하지 못하고, 퀵 정렬은 '버블 정렬'과 비슷한 형태로 퇴화한다.</p> <p>이 상황을 최대한 피하려면 분할에 사용할 피벗의 선택 전략을 최적화할 수 있다. 예를 들어 원소 하나를 무작위로 골라 피벗으로 삼을 수 있다. 하지만 운이 나빠 매번 좋지 않은 피벗을 고르면 여전히 만족스러운 효율을 얻지 못한다.</p> <p>프로그래밍 언어가 일반적으로 생성하는 것은 '의사 난수'라는 점에 주의하자. 의사 난수 수열에 맞추어 특정 테스트 사례를 만들면 퀵 정렬의 효율이 여전히 저하될 수 있다.</p> <p>더 개선하려면 배열에서 후보 원소 세 개(보통 첫 원소, 마지막 원소, 중간 원소)를 고르고, 이 세 후보의 중앙값을 피벗으로 사용한다. 그러면 피벗이 '너무 작지도 크지도 않을' 확률이 크게 높아진다. 후보를 더 많이 골라 알고리즘의 견고함을 높일 수도 있다. 이 방법을 사용하면 시간 복잡도가 \\(O(n^2)\\)으로 저하될 확률이 크게 줄어든다.</p> <p>예시 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def median_three(self, nums: list[int], left: int, mid: int, right: int) -&gt; int:\n    \"\"\"세 후보 원소의 중앙값 선택\"\"\"\n    l, m, r = nums[left], nums[mid], nums[right]\n    if (l &lt;= m &lt;= r) or (r &lt;= m &lt;= l):\n        return mid  # m은 l과 r 사이에 있다\n    if (m &lt;= l &lt;= r) or (r &lt;= l &lt;= m):\n        return left  # l은 m과 r 사이에 있다\n    return right\n\ndef partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"센티널 분할(세 수의 중앙값 선택)\"\"\"\n    # nums[left]를 피벗으로 삼는다\n    med = self.median_three(nums, left, (left + right) // 2, right)\n    # 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    nums[left], nums[med] = nums[med], nums[left]\n    # nums[left]를 피벗으로 삼는다\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        # 원소 교환\n        nums[i], nums[j] = nums[j], nums[i]\n    # 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # 피벗의 인덱스를 반환한다\n</code></pre> quick_sort.cpp<pre><code>/* 세 후보 원소의 중앙값 선택 */\nint medianThree(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m은 l과 r 사이에 있다\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l은 m과 r 사이에 있다\n    return right;\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // 세 후보 원소의 중앙값 선택\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    swap(nums[left], nums[med]);\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;                // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;                // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        swap(nums[i], nums[j]); // 두 원소를 교환한다\n    }\n    swap(nums[i], nums[left]);  // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i;                   // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.java<pre><code>/* 세 후보 원소의 중앙값 선택 */\nint medianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m은 l과 r 사이에 있다\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l은 m과 r 사이에 있다\n    return right;\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nint partition(int[] nums, int left, int right) {\n    // 세 후보 원소의 중앙값 선택\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    swap(nums, left, med);\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        swap(nums, i, j); // 두 원소를 교환한다\n    }\n    swap(nums, i, left);  // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i;             // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.cs<pre><code>/* 세 후보 원소의 중앙값 선택 */\nint MedianThree(int[] nums, int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m은 l과 r 사이에 있다\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l은 m과 r 사이에 있다\n    return right;\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nint Partition(int[] nums, int left, int right) {\n    // 세 후보 원소의 중앙값 선택\n    int med = MedianThree(nums, left, (left + right) / 2, right);\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    Swap(nums, left, med);\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        Swap(nums, i, j); // 두 원소를 교환한다\n    }\n    Swap(nums, i, left);  // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i;             // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.go<pre><code>/* 세 후보 원소의 중앙값 선택 */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n    l, m, r := nums[left], nums[mid], nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m은 l과 r 사이에 있다\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l은 m과 r 사이에 있다\n    }\n    return right\n}\n\n/* 센티널 분할(세 수의 중앙값 선택)*/\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n    // nums[left]를 피벗으로 삼는다\n    med := q.medianThree(nums, left, (left+right)/2, right)\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    nums[left], nums[med] = nums[med], nums[left]\n    // nums[left]를 피벗으로 삼는다\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- //오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ //왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        //원소 교환\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    //피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    nums[i], nums[left] = nums[left], nums[i]\n    return i //피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.swift<pre><code>/* 세 후보 원소의 중앙값 선택 */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -&gt; Int {\n    let l = nums[left]\n    let m = nums[mid]\n    let r = nums[right]\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid // m은 l과 r 사이에 있다\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left // l은 m과 r 사이에 있다\n    }\n    return right\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // 세 후보 원소의 중앙값 선택\n    let med = medianThree(nums: nums, left: left, mid: left + (right - left) / 2, right: right)\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    nums.swapAt(left, med)\n    return partition(nums: &amp;nums, left: left, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* 세 후보 원소의 중앙값 선택 */\nmedianThree(nums, left, mid, right) {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m은 l과 r 사이에 있다\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l은 m과 r 사이에 있다\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\npartition(nums, left, right) {\n    // 세 후보 원소의 중앙값 선택\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    this.swap(nums, left, med);\n    // nums[left]를 피벗으로 삼는다\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        this.swap(nums, i, j); // 두 원소를 교환한다\n    }\n    this.swap(nums, i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i; // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.ts<pre><code>/* 세 후보 원소의 중앙값 선택 */\nmedianThree(\n    nums: number[],\n    left: number,\n    mid: number,\n    right: number\n): number {\n    let l = nums[left],\n        m = nums[mid],\n        r = nums[right];\n    // m은 l과 r 사이에 있다\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l)) return mid;\n    // l은 m과 r 사이에 있다\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m)) return left;\n    return right;\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\npartition(nums: number[], left: number, right: number): number {\n    // 세 후보 원소의 중앙값 선택\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    this.swap(nums, left, med);\n    // nums[left]를 피벗으로 삼는다\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j--; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i++; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        this.swap(nums, i, j); // 두 원소를 교환한다\n    }\n    this.swap(nums, i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i; // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.dart<pre><code>/* 세 후보 원소의 중앙값 선택 */\nint _medianThree(List&lt;int&gt; nums, int left, int mid, int right) {\n  int l = nums[left], m = nums[mid], r = nums[right];\n  if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n    return mid; // m은 l과 r 사이에 있다\n  if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n    return left; // l은 m과 r 사이에 있다\n  return right;\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // 세 후보 원소의 중앙값 선택\n  int med = _medianThree(nums, left, (left + right) ~/ 2, right);\n  // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n  _swap(nums, left, med);\n  // nums[left]를 피벗으로 삼는다\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n    _swap(nums, i, j); // 두 원소를 교환한다\n  }\n  _swap(nums, i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n  return i; // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.rs<pre><code>/* 세 후보 원소의 중앙값 선택 */\nfn median_three(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) -&gt; usize {\n    let (l, m, r) = (nums[left], nums[mid], nums[right]);\n    if (l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l) {\n        return mid; // m은 l과 r 사이에 있다\n    }\n    if (m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m) {\n        return left; // l은 m과 r 사이에 있다\n    }\n    right\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // 세 후보 원소의 중앙값 선택\n    let med = Self::median_three(nums, left, (left + right) / 2, right);\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    nums.swap(left, med);\n    // nums[left]를 피벗으로 삼는다\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1; // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        }\n        nums.swap(i, j); // 두 원소를 교환한다\n    }\n    nums.swap(i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    i // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.c<pre><code>/* 세 후보 원소의 중앙값 선택 */\nint medianThree(int nums[], int left, int mid, int right) {\n    int l = nums[left], m = nums[mid], r = nums[right];\n    if ((l &lt;= m &amp;&amp; m &lt;= r) || (r &lt;= m &amp;&amp; m &lt;= l))\n        return mid; // m은 l과 r 사이에 있다\n    if ((m &lt;= l &amp;&amp; l &lt;= r) || (r &lt;= l &amp;&amp; l &lt;= m))\n        return left; // l은 m과 r 사이에 있다\n    return right;\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nint partitionMedian(int nums[], int left, int right) {\n    // 세 후보 원소의 중앙값 선택\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    swap(nums, left, med);\n    // nums[left]를 피벗으로 삼는다\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--; // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        swap(nums, i, j); // 두 원소를 교환한다\n    }\n    swap(nums, i, left); // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i;            // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.kt<pre><code>/* 세 후보 원소의 중앙값 선택 */\nfun medianThree(nums: IntArray, left: Int, mid: Int, right: Int): Int {\n    val l = nums[left]\n    val m = nums[mid]\n    val r = nums[right]\n    if ((m in l..r) || (m in r..l))\n        return mid  // m은 l과 r 사이에 있다\n    if ((l in m..r) || (l in r..m))\n        return left // l은 m과 r 사이에 있다\n    return right\n}\n\n/* 센티널 분할(세 수의 중앙값 선택) */\nfun partitionMedian(nums: IntArray, left: Int, right: Int): Int {\n    // 세 후보 원소의 중앙값 선택\n    val med = medianThree(nums, left, (left + right) / 2, right)\n    // 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n    swap(nums, left, med)\n    // nums[left]를 피벗으로 삼는다\n    var i = left\n    var j = right\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--                      // 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++                      // 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n        swap(nums, i, j)             // 두 원소를 교환한다\n    }\n    swap(nums, i, left)              // 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n    return i                         // 피벗의 인덱스를 반환한다\n}\n</code></pre> quick_sort.rb<pre><code>### 세 후보 원소의 중앙값 선택 ###\ndef median_three(nums, left, mid, right)\n  # 세 후보 원소의 중앙값 선택\n  _l, _m, _r = nums[left], nums[mid], nums[right]\n  # m은 l과 r 사이에 있다\n  return mid if (_l &lt;= _m &amp;&amp; _m &lt;= _r) || (_r &lt;= _m &amp;&amp; _m &lt;= _l)\n  # l은 m과 r 사이에 있다\n  return left if (_m &lt;= _l &amp;&amp; _l &lt;= _r) || (_r &lt;= _l &amp;&amp; _l &lt;= _m)\n  return right\nend\n\n### 센티널 분할(세 수의 중앙값 선택)###\ndef partition(nums, left, right)\n  ### nums[left]를 피벗으로 삼는다\n  med = median_three(nums, left, (left + right) / 2, right)\n  # 중앙값을 배열 맨 왼쪽으로 교환하여 옮긴다\n  nums[left], nums[med] = nums[med], nums[left]\n  i, j = left, right\n  while i &lt; j\n    while i &lt; j &amp;&amp; nums[j] &gt;= nums[left]\n      j -= 1 # 오른쪽에서 왼쪽으로 피벗보다 작은 첫 번째 원소를 찾는다\n    end\n    while i &lt; j &amp;&amp; nums[i] &lt;= nums[left]\n      i += 1 # 왼쪽에서 오른쪽으로 피벗보다 큰 첫 번째 원소를 찾는다\n    end\n    # 원소 교환\n    nums[i], nums[j] = nums[j], nums[i]\n  end\n  # 피벗을 두 부분 배열의 경계로 교환하여 옮긴다\n  nums[i], nums[left] = nums[left], nums[i]\n  i # 피벗의 인덱스를 반환한다\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.5   퀵 정렬"],"tags":[]},{"location":"chapter_sorting/quick_sort/#1155","level":2,"title":"11.5.5   재귀 깊이 최적화","text":"<p>일부 입력에서는 퀵 정렬이 많은 공간을 차지할 수 있다. 완전히 정렬된 입력 배열을 예로 들어 보자. 재귀 중인 부분 배열의 길이가 \\(m\\)이라면 매 분할에서 길이 \\(0\\)인 왼쪽 부분 배열과 길이 \\(m - 1\\)인 오른쪽 부분 배열이 생긴다. 이는 재귀 한 단계마다 줄어드는 문제 크기가 매우 작다는 뜻이다(원소 하나만 줄어든다). 재귀 트리의 높이는 \\(n - 1\\)에 이르며, \\(O(n)\\) 크기의 스택 프레임 공간이 필요하다.</p> <p>스택 프레임 공간이 쌓이지 않도록 매번 분할을 마친 뒤 두 부분 배열의 길이를 비교하여 더 짧은 부분 배열에만 재귀를 수행할 수 있다. 짧은 쪽의 길이는 \\(n / 2\\)를 넘지 않으므로 재귀 깊이를 \\(\\log n\\) 이하로 유지할 수 있다. 이로써 최악의 공간 복잡도를 \\(O(\\log n)\\)으로 개선한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"퀵 정렬(재귀 깊이 최적화)\"\"\"\n    # 부분 배열의 길이가 1이면 종료한다\n    while left &lt; right:\n        # 센티널 분할 연산\n        pivot = self.partition(nums, left, right)\n        # 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if pivot - left &lt; right - pivot:\n            self.quick_sort(nums, left, pivot - 1)  # 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1  # 남은 미정렬 구간은 [pivot + 1, right]이다\n        else:\n            self.quick_sort(nums, pivot + 1, right)  # 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1  # 남은 미정렬 구간은 [left, pivot - 1]이다\n</code></pre> quick_sort.cpp<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // 부분 배열의 길이가 1이면 종료한다\n    while (left &lt; right) {\n        // 센티널 분할 연산\n        int pivot = partition(nums, left, right);\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1;                 // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            quickSort(nums, pivot + 1, right); // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1;                 // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.java<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nvoid quickSort(int[] nums, int left, int right) {\n    // 부분 배열의 길이가 1이면 종료한다\n    while (left &lt; right) {\n        // 센티널 분할 연산\n        int pivot = partition(nums, left, right);\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1; // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            quickSort(nums, pivot + 1, right); // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1; // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.cs<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nvoid QuickSort(int[] nums, int left, int right) {\n    // 부분 배열의 길이가 1이면 종료한다\n    while (left &lt; right) {\n        // 센티널 분할 연산\n        int pivot = Partition(nums, left, right);\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - left &lt; right - pivot) {\n            QuickSort(nums, left, pivot - 1);  // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1;  // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            QuickSort(nums, pivot + 1, right); // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1; // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.go<pre><code>/* 퀵 정렬(재귀 깊이 최적화)*/\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n    // 부분 배열의 길이가 1이면 종료한다\n    for left &lt; right {\n        // 센티널 분할 연산\n        pivot := q.partition(nums, left, right)\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if pivot-left &lt; right-pivot {\n            q.quickSort(nums, left, pivot-1) // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1                 // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            q.quickSort(nums, pivot+1, right) // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1                 // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.swift<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\n    var left = left\n    var right = right\n    // 부분 배열의 길이가 1이면 종료한다\n    while left &lt; right {\n        // 센티널 분할 연산\n        let pivot = partition(nums: &amp;nums, left: left, right: right)\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - left) &lt; (right - pivot) {\n            quickSortTailCall(nums: &amp;nums, left: left, right: pivot - 1) // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1 // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            quickSortTailCall(nums: &amp;nums, left: pivot + 1, right: right) // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1 // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.js<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nquickSort(nums, left, right) {\n    // 부분 배열의 길이가 1이면 종료한다\n    while (left &lt; right) {\n        // 센티널 분할 연산\n        let pivot = this.partition(nums, left, right);\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1; // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            this.quickSort(nums, pivot + 1, right); // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1; // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.ts<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nquickSort(nums: number[], left: number, right: number): void {\n    // 부분 배열의 길이가 1이면 종료한다\n    while (left &lt; right) {\n        // 센티널 분할 연산\n        let pivot = this.partition(nums, left, right);\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1; // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            this.quickSort(nums, pivot + 1, right); // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1; // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.dart<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // 부분 배열의 길이가 1이면 종료한다\n  while (left &lt; right) {\n    // 센티널 분할 연산\n    int pivot = _partition(nums, left, right);\n    // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n    if (pivot - left &lt; right - pivot) {\n      quickSort(nums, left, pivot - 1); // 왼쪽 부분 배열을 재귀적으로 정렬한다\n      left = pivot + 1; // 남은 미정렬 구간은 [pivot + 1, right]이다\n    } else {\n      quickSort(nums, pivot + 1, right); // 오른쪽 부분 배열을 재귀적으로 정렬한다\n      right = pivot - 1; // 남은 미정렬 구간은 [left, pivot - 1]이다\n    }\n  }\n}\n</code></pre> quick_sort.rs<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\npub fn quick_sort(mut left: i32, mut right: i32, nums: &amp;mut [i32]) {\n    // 부분 배열의 길이가 1이면 종료한다\n    while left &lt; right {\n        // 센티널 분할 연산\n        let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if pivot - left &lt; right - pivot {\n            Self::quick_sort(left, pivot - 1, nums); // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            left = pivot + 1; // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            Self::quick_sort(pivot + 1, right, nums); // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            right = pivot - 1; // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.c<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nvoid quickSortTailCall(int nums[], int left, int right) {\n    // 부분 배열의 길이가 1이면 종료한다\n    while (left &lt; right) {\n        // 센티널 분할 연산\n        int pivot = partition(nums, left, right);\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - left &lt; right - pivot) {\n            // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            quickSortTailCall(nums, left, pivot - 1);\n            // 남은 미정렬 구간은 [pivot + 1, right]이다\n            left = pivot + 1;\n        } else {\n            // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            quickSortTailCall(nums, pivot + 1, right);\n            // 남은 미정렬 구간은 [left, pivot - 1]이다\n            right = pivot - 1;\n        }\n    }\n}\n</code></pre> quick_sort.kt<pre><code>/* 퀵 정렬(재귀 깊이 최적화) */\nfun quickSortTailCall(nums: IntArray, left: Int, right: Int) {\n    // 부분 배열의 길이가 1이면 종료한다\n    var l = left\n    var r = right\n    while (l &lt; r) {\n        // 센티널 분할 연산\n        val pivot = partition(nums, l, r)\n        // 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n        if (pivot - l &lt; r - pivot) {\n            quickSort(nums, l, pivot - 1) // 왼쪽 부분 배열을 재귀적으로 정렬한다\n            l = pivot + 1 // 남은 미정렬 구간은 [pivot + 1, right]이다\n        } else {\n            quickSort(nums, pivot + 1, r) // 오른쪽 부분 배열을 재귀적으로 정렬한다\n            r = pivot - 1 // 남은 미정렬 구간은 [left, pivot - 1]이다\n        }\n    }\n}\n</code></pre> quick_sort.rb<pre><code>### 퀵 정렬(재귀 깊이 최적화)###\ndef quick_sort(nums, left, right)\n  # 부분 배열의 길이가 1이 아니면 재귀 호출한다\n  while left &lt; right\n    # 센티널 분할\n    pivot = partition(nums, left, right)\n    # 두 부분 배열 중 더 짧은 쪽에 퀵 정렬을 수행한다\n    if pivot - left &lt; right - pivot\n      quick_sort(nums, left, pivot - 1)\n      left = pivot + 1 # 남은 미정렬 구간은 [pivot + 1, right]이다\n    else\n      quick_sort(nums, pivot + 1, right)\n      right = pivot - 1 # 남은 미정렬 구간은 [left, pivot - 1]이다\n    end\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.5   퀵 정렬"],"tags":[]},{"location":"chapter_sorting/radix_sort/","level":1,"title":"11.10   기수 정렬","text":"<p>앞 절에서 배운 계수 정렬은 데이터 수 \\(n\\)은 많고 값의 범위 \\(m\\)은 작은 경우에 적합하다. 학번 \\(n = 10^6\\) 개를 정렬해야 하며, 학번이 \\(8\\) 자리 수라고 하자. 이때 값의 범위는 \\(m = 10^8\\)로 매우 커서 계수 정렬에는 많은 메모리가 필요하다. 기수 정렬을 사용하면 이 문제를 피할 수 있다.</p> <p>기수 정렬(radix sort)의 핵심 아이디어는 계수 정렬과 같으며, 개수를 세어 정렬한다. 여기에 자릿수 사이의 순서 관계를 이용하여 각 자릿수를 차례로 정렬함으로써 최종 정렬 결과를 얻는다.</p>","path":["제 11 장   정렬","11.10   기수 정렬"],"tags":[]},{"location":"chapter_sorting/radix_sort/#11101","level":2,"title":"11.10.1   알고리즘 과정","text":"<p>학번을 예로 들어 가장 낮은 자리를 \\(1\\) 번째, 가장 높은 자리를 \\(8\\) 번째라고 하자. 기수 정렬의 과정은 아래 그림(11-18)과 같다.</p> <ol> <li>자릿수 \\(k = 1\\)로 초기화한다.</li> <li>학번의 \\(k\\) 번째 자리를 기준으로 '계수 정렬'을 수행한다. 완료하면 데이터는 \\(k\\) 번째 자리의 값에 따라 오름차순으로 정렬된다.</li> <li>\\(k\\)를 \\(1\\) 늘리고 <code>2.</code> 단계로 돌아가 반복한다. 모든 자리를 정렬하면 종료한다.</li> </ol> <p></p> <p> 그림 11-18   기수 정렬 알고리즘 과정 </p> <p>이제 코드 구현을 살펴보자. \\(d\\) 진법으로 표현한 수 \\(x\\)의 \\(k\\) 번째 자리 \\(x_k\\)는 다음 공식으로 구할 수 있다.</p> \\[ x_k = \\lfloor\\frac{x}{d^{k-1}}\\rfloor \\bmod d \\] <p>여기서 \\(\\lfloor a \\rfloor\\)는 부동 소수점 수 \\(a\\)를 내림하는 연산이고, \\(\\bmod \\: d\\)는 \\(d\\)로 나눈 나머지를 구하는 연산이다. 학번 데이터에서는 \\(d = 10\\)이고 \\(k \\in [1, 8]\\)이다.</p> <p>또한 수의 \\(k\\) 번째 자리를 기준으로 정렬할 수 있도록 계수 정렬 코드를 조금 수정해야 한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby radix_sort.py<pre><code>def digit(num: int, exp: int) -&gt; int:\n    \"\"\"원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다\"\"\"\n    # k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return (num // exp) % 10\n\ndef counting_sort_digit(nums: list[int], exp: int):\n    \"\"\"계수 정렬(nums의 k번째 자릿값을 기준으로 정렬)\"\"\"\n    # 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    counter = [0] * 10\n    n = len(nums)\n    # 0~9 각 숫자의 등장 횟수를 센다\n    for i in range(n):\n        d = digit(nums[i], exp)  # nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d] += 1  # 숫자 d의 등장 횟수를 센다\n    # 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for i in range(1, 10):\n        counter[i] += counter[i - 1]\n    # 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        d = digit(nums[i], exp)\n        j = counter[d] - 1  # 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i]  # 현재 원소를 인덱스 j에 채운다\n        counter[d] -= 1  # d의 개수를 1 줄인다\n    # 결과로 원래 배열 nums를 덮어쓴다\n    for i in range(n):\n        nums[i] = res[i]\n\ndef radix_sort(nums: list[int]):\n    \"\"\"기수 정렬\"\"\"\n    # 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    m = max(nums)\n    # 낮은 자리에서 높은 자리 순서로 순회한다\n    exp = 1\n    while exp &lt;= m:\n        # 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        # k = 1 -&gt; exp = 1\n        # k = 2 -&gt; exp = 10\n        # 즉, exp = 10^(k-1)이다\n        counting_sort_digit(nums, exp)\n        exp *= 10\n</code></pre> radix_sort.cpp<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nint digit(int num, int exp) {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return (num / exp) % 10;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nvoid countingSortDigit(vector&lt;int&gt; &amp;nums, int exp) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    vector&lt;int&gt; counter(10, 0);\n    int n = nums.size();\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d]++;                // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    vector&lt;int&gt; res(n, 0);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i];       // 현재 원소를 인덱스 j에 채운다\n        counter[d]--;           // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* 기수 정렬 */\nvoid radixSort(vector&lt;int&gt; &amp;nums) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    int m = *max_element(nums.begin(), nums.end());\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for (int exp = 1; exp &lt;= m; exp *= 10)\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.java<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nint digit(int num, int exp) {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return (num / exp) % 10;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nvoid countingSortDigit(int[] nums, int exp) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    int[] counter = new int[10];\n    int n = nums.length;\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d]++;                // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i];       // 현재 원소를 인덱스 j에 채운다\n        counter[d]--;           // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* 기수 정렬 */\nvoid radixSort(int[] nums) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    int m = Integer.MIN_VALUE;\n    for (int num : nums)\n        if (num &gt; m)\n            m = num;\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for (int exp = 1; exp &lt;= m; exp *= 10) {\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.cs<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nint Digit(int num, int exp) {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return (num / exp) % 10;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nvoid CountingSortDigit(int[] nums, int exp) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    int[] counter = new int[10];\n    int n = nums.Length;\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for (int i = 0; i &lt; n; i++) {\n        int d = Digit(nums[i], exp); // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d]++;                // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = Digit(nums[i], exp);\n        int j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i];       // 현재 원소를 인덱스 j에 채운다\n        counter[d]--;           // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* 기수 정렬 */\nvoid RadixSort(int[] nums) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    int m = int.MinValue;\n    foreach (int num in nums) {\n        if (num &gt; m) m = num;\n    }\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for (int exp = 1; exp &lt;= m; exp *= 10) {\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        CountingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.go<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nfunc digit(num, exp int) int {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return (num / exp) % 10\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nfunc countingSortDigit(nums []int, exp int) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    counter := make([]int, 10)\n    n := len(nums)\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for i := 0; i &lt; n; i++ {\n        d := digit(nums[i], exp) // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d]++             // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for i := 1; i &lt; 10; i++ {\n        counter[i] += counter[i-1]\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        d := digit(nums[i], exp)\n        j := counter[d] - 1 // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i]    // 현재 원소를 인덱스 j에 채운다\n        counter[d]--        // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for i := 0; i &lt; n; i++ {\n        nums[i] = res[i]\n    }\n}\n\n/* 기수 정렬 */\nfunc radixSort(nums []int) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    max := math.MinInt\n    for _, num := range nums {\n        if num &gt; max {\n            max = num\n        }\n    }\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for exp := 1; max &gt;= exp; exp *= 10 {\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums, exp)\n    }\n}\n</code></pre> radix_sort.swift<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nfunc digit(num: Int, exp: Int) -&gt; Int {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    (num / exp) % 10\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nfunc countingSortDigit(nums: inout [Int], exp: Int) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    var counter = Array(repeating: 0, count: 10)\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for i in nums.indices {\n        let d = digit(num: nums[i], exp: exp) // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d] += 1 // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for i in 1 ..&lt; 10 {\n        counter[i] += counter[i - 1]\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    var res = Array(repeating: 0, count: nums.count)\n    for i in nums.indices.reversed() {\n        let d = digit(num: nums[i], exp: exp)\n        let j = counter[d] - 1 // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i] // 현재 원소를 인덱스 j에 채운다\n        counter[d] -= 1 // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n\n/* 기수 정렬 */\nfunc radixSort(nums: inout [Int]) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    var m = Int.min\n    for num in nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for exp in sequence(first: 1, next: { m &gt;= ($0 * 10) ? $0 * 10 : nil }) {\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums: &amp;nums, exp: exp)\n    }\n}\n</code></pre> radix_sort.js<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nfunction digit(num, exp) {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return Math.floor(num / exp) % 10;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nfunction countingSortDigit(nums, exp) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d]++; // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i]; // 현재 원소를 인덱스 j에 채운다\n        counter[d]--; // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* 기수 정렬 */\nfunction radixSort(nums) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    let m = Math.max(... nums);\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.ts<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nfunction digit(num: number, exp: number): number {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return Math.floor(num / exp) % 10;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nfunction countingSortDigit(nums: number[], exp: number): void {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d]++; // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i]; // 현재 원소를 인덱스 j에 채운다\n        counter[d]--; // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* 기수 정렬 */\nfunction radixSort(nums: number[]): void {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    let m: number = Math.max(... nums);\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.dart<pre><code>/* 원소 _num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nint digit(int _num, int exp) {\n  // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n  return (_num ~/ exp) % 10;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nvoid countingSortDigit(List&lt;int&gt; nums, int exp) {\n  // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n  List&lt;int&gt; counter = List&lt;int&gt;.filled(10, 0);\n  int n = nums.length;\n  // 0~9 각 숫자의 등장 횟수를 센다\n  for (int i = 0; i &lt; n; i++) {\n    int d = digit(nums[i], exp); // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n    counter[d]++; // 숫자 d의 등장 횟수를 센다\n  }\n  // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n  for (int i = 1; i &lt; 10; i++) {\n    counter[i] += counter[i - 1];\n  }\n  // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n  List&lt;int&gt; res = List&lt;int&gt;.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int d = digit(nums[i], exp);\n    int j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n    res[j] = nums[i]; // 현재 원소를 인덱스 j에 채운다\n    counter[d]--; // d의 개수를 1 줄인다\n  }\n  // 결과로 원래 배열 nums를 덮어쓴다\n  for (int i = 0; i &lt; n; i++) nums[i] = res[i];\n}\n\n/* 기수 정렬 */\nvoid radixSort(List&lt;int&gt; nums) {\n  // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n  // Dart에서 int의 길이는 64비트이다\n  int m = -1 &lt;&lt; 63;\n  for (int _num in nums) if (_num &gt; m) m = _num;\n  // 낮은 자리에서 높은 자리 순서로 순회한다\n  for (int exp = 1; exp &lt;= m; exp *= 10)\n    // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n    // k = 1 -&gt; exp = 1\n    // k = 2 -&gt; exp = 10\n    // 즉, exp = 10^(k-1)이다\n    countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.rs<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nfn digit(num: i32, exp: i32) -&gt; usize {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return ((num / exp) % 10) as usize;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nfn counting_sort_digit(nums: &amp;mut [i32], exp: i32) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    let mut counter = [0; 10];\n    let n = nums.len();\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for i in 0..n {\n        let d = digit(nums[i], exp); // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d] += 1; // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for i in 1..10 {\n        counter[i] += counter[i - 1];\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let d = digit(nums[i], exp);\n        let j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i]; // 현재 원소를 인덱스 j에 채운다\n        counter[d] -= 1; // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    nums.copy_from_slice(&amp;res);\n}\n\n/* 기수 정렬 */\nfn radix_sort(nums: &amp;mut [i32]) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    let m = *nums.into_iter().max().unwrap();\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    let mut exp = 1;\n    while exp &lt;= m {\n        counting_sort_digit(nums, exp);\n        exp *= 10;\n    }\n}\n</code></pre> radix_sort.c<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nint digit(int num, int exp) {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return (num / exp) % 10;\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nvoid countingSortDigit(int nums[], int size, int exp) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    int *counter = (int *)malloc((sizeof(int) * 10));\n    memset(counter, 0, sizeof(int) * 10); // 이후 메모리 해제를 지원하도록 0으로 초기화한다\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for (int i = 0; i &lt; size; i++) {\n        // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        int d = digit(nums[i], exp);\n        // 숫자 d의 등장 횟수를 센다\n        counter[d]++;\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    int *res = (int *)malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i];       // 현재 원소를 인덱스 j에 채운다\n        counter[d]--;           // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for (int i = 0; i &lt; size; i++) {\n        nums[i] = res[i];\n    }\n    // 메모리 해제\n    free(res);\n    free(counter);\n}\n\n/* 기수 정렬 */\nvoid radixSort(int nums[], int size) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    int max = INT32_MIN;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; max) {\n            max = nums[i];\n        }\n    }\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    for (int exp = 1; max &gt;= exp; exp *= 10)\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums, size, exp);\n}\n</code></pre> radix_sort.kt<pre><code>/* 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 */\nfun digit(num: Int, exp: Int): Int {\n    // k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n    return (num / exp) % 10\n}\n\n/* 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬) */\nfun countingSortDigit(nums: IntArray, exp: Int) {\n    // 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n    val counter = IntArray(10)\n    val n = nums.size\n    // 0~9 각 숫자의 등장 횟수를 센다\n    for (i in 0..&lt;n) {\n        val d = digit(nums[i], exp) // nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n        counter[d]++                // 숫자 d의 등장 횟수를 센다\n    }\n    // 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n    for (i in 1..9) {\n        counter[i] += counter[i - 1]\n    }\n    // 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n    val res = IntArray(n)\n    for (i in n - 1 downTo 0) {\n        val d = digit(nums[i], exp)\n        val j = counter[d] - 1 // 배열에서 d의 인덱스 j를 구한다\n        res[j] = nums[i]       // 현재 원소를 인덱스 j에 채운다\n        counter[d]--           // d의 개수를 1 줄인다\n    }\n    // 결과로 원래 배열 nums를 덮어쓴다\n    for (i in 0..&lt;n)\n        nums[i] = res[i]\n}\n\n/* 기수 정렬 */\nfun radixSort(nums: IntArray) {\n    // 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n    var m = Int.MIN_VALUE\n    for (num in nums) if (num &gt; m) m = num\n    var exp = 1\n    // 낮은 자리에서 높은 자리 순서로 순회한다\n    while (exp &lt;= m) {\n        // 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // 즉, exp = 10^(k-1)이다\n        countingSortDigit(nums, exp)\n        exp *= 10\n    }\n}\n</code></pre> radix_sort.rb<pre><code>### 원소 num의 k번째 자릿값 반환. 이때 exp = 10^(k-1)이다 ###\ndef digit(num, exp)\n  # k 대신 exp를 전달하면 여기서 비용이 큰 거듭제곱 계산을 반복하지 않아도 된다\n  (num / exp) % 10\nend\n\n### 계수 정렬(nums의 k번째 자릿값을 기준으로 정렬)###\ndef counting_sort_digit(nums, exp)\n  # 십진수의 각 자리는 0~9 범위이므로 길이가 10인 버킷 배열이 필요하다\n  counter = Array.new(10, 0)\n  n = nums.length\n  # 0~9 각 숫자의 등장 횟수를 센다\n  for i in 0...n\n    d = digit(nums[i], exp) # nums[i]의 k번째 자릿값을 구하여 d로 표시한다\n    counter[d] += 1 # 숫자 d의 등장 횟수를 센다\n  end\n  # 접두 합을 구하여 등장 개수를 배열 인덱스로 변환한다\n  (1...10).each { |i| counter[i] += counter[i - 1] }\n  # 역순으로 순회하며 버킷에 집계된 결과에 따라 각 원소를 res에 채운다\n  res = Array.new(n, 0)\n  for i in (n - 1).downto(0)\n    d = digit(nums[i], exp)\n    j = counter[d] - 1 # 배열에서 d의 인덱스 j를 구한다\n    res[j] = nums[i] # 현재 원소를 인덱스 j에 채운다\n    counter[d] -= 1 # d의 개수를 1 줄인다\n  end\n  # 결과로 원래 배열 nums를 덮어쓴다\n  (0...n).each { |i| nums[i] = res[i] }\nend\n\n### 기수 정렬 ###\ndef radix_sort(nums)\n  # 배열의 최댓값을 구하여 최대 자릿수를 판단한다\n  m = nums.max\n  # 낮은 자리에서 높은 자리 순서로 순회한다\n  exp = 1\n  while exp &lt;= m\n    # 배열 원소의 k번째 자릿값에 대해 계수 정렬을 수행한다\n    # k = 1 -&gt; exp = 1\n    # k = 2 -&gt; exp = 10\n    # 즉, exp = 10^(k-1)이다\n    counting_sort_digit(nums, exp)\n    exp *= 10\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>왜 가장 낮은 자리부터 정렬하는가?</p> <p>여러 차례 연속으로 정렬하면 뒤의 정렬 결과가 앞의 결과를 덮어쓴다. 예를 들어 첫 번째 정렬에서는 \\(a &lt; b\\)였지만 두 번째 정렬에서는 \\(a &gt; b\\)라면, 두 번째 결과가 첫 번째 결과를 대신한다. 높은 자릿수가 낮은 자릿수보다 우선하므로 낮은 자리부터 정렬한 뒤 높은 자리를 정렬해야 한다.</p>","path":["제 11 장   정렬","11.10   기수 정렬"],"tags":[]},{"location":"chapter_sorting/radix_sort/#11102","level":2,"title":"11.10.2   알고리즘 특성","text":"<p>기수 정렬은 계수 정렬보다 값의 범위가 큰 경우에 적합하다. 단, 데이터를 고정된 자릿수의 형식으로 표현할 수 있어야 하며 자릿수가 지나치게 크면 안 된다. 예를 들어 부동 소수점 수는 자릿수 \\(k\\)가 너무 커서 시간 복잡도가 \\(O(nk) \\gg O(n^2)\\)이 될 수 있으므로 기수 정렬에 적합하지 않다.</p> <ul> <li>시간 복잡도 \\(O(nk)\\), 비적응형 정렬: 데이터 수가 \\(n\\), 진법이 \\(d\\), 최대 자릿수가 \\(k\\)라면 한 자리를 계수 정렬하는 데 \\(O(n + d)\\) 시간이 걸리고, \\(k\\) 자리 전체를 정렬하는 데 \\(O((n + d)k)\\) 시간이 걸린다. 일반적으로 \\(d\\)와 \\(k\\)는 모두 비교적 작으므로 시간 복잡도는 \\(O(n)\\)에 가까워진다.</li> <li>공간 복잡도 \\(O(n + d)\\), 제자리 정렬이 아님: 계수 정렬과 마찬가지로 길이가 \\(n\\)과 \\(d\\)인 배열 <code>res</code>와 <code>counter</code>가 필요하다.</li> <li>안정 정렬: 계수 정렬이 안정적이면 기수 정렬도 안정적이다. 계수 정렬이 불안정하면 기수 정렬은 올바른 정렬 결과를 보장할 수 없다.</li> </ul>","path":["제 11 장   정렬","11.10   기수 정렬"],"tags":[]},{"location":"chapter_sorting/selection_sort/","level":1,"title":"11.2   선택 정렬","text":"<p>선택 정렬(selection sort)의 원리는 매우 간단하다. 반복문을 실행하여 매번 정렬되지 않은 구간에서 가장 작은 원소를 고른 뒤 정렬된 구간의 끝에 놓는다.</p> <p>배열의 길이가 \\(n\\)일 때 선택 정렬의 과정은 아래 그림(11-2)과 같다.</p> <ol> <li>처음에는 모든 원소가 정렬되지 않았으므로 미정렬 구간(인덱스)은 \\([0, n-1]\\)이다.</li> <li>구간 \\([0, n-1]\\)에서 가장 작은 원소를 골라 인덱스 \\(0\\)의 원소와 교환한다. 그러면 배열의 앞 1개 원소가 정렬된다.</li> <li>구간 \\([1, n-1]\\)에서 가장 작은 원소를 골라 인덱스 \\(1\\)의 원소와 교환한다. 그러면 배열의 앞 2개 원소가 정렬된다.</li> <li>같은 과정을 반복한다. \\(n - 1\\) 번 선택하고 교환하면 배열의 앞 \\(n - 1\\) 개 원소가 정렬된다.</li> <li>남은 원소 하나는 반드시 가장 큰 원소이므로 정렬할 필요가 없으며, 배열 정렬이 완료된다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 11-2   선택 정렬 단계 </p> <p>코드에서는 \\(k\\)로 미정렬 구간의 최소 원소를 기록한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby selection_sort.py<pre><code>def selection_sort(nums: list[int]):\n    \"\"\"선택 정렬\"\"\"\n    n = len(nums)\n    # 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for i in range(n - 1):\n        # 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        k = i\n        for j in range(i + 1, n):\n            if nums[j] &lt; nums[k]:\n                k = j  # 최솟값의 인덱스를 기록한다\n        # 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        nums[i], nums[k] = nums[k], nums[i]\n</code></pre> selection_sort.cpp<pre><code>/* 선택 정렬 */\nvoid selectionSort(vector&lt;int&gt; &amp;nums) {\n    int n = nums.size();\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for (int i = 0; i &lt; n - 1; i++) {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // 최솟값의 인덱스를 기록한다\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        swap(nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.java<pre><code>/* 선택 정렬 */\nvoid selectionSort(int[] nums) {\n    int n = nums.length;\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for (int i = 0; i &lt; n - 1; i++) {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // 최솟값의 인덱스를 기록한다\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.cs<pre><code>/* 선택 정렬 */\nvoid SelectionSort(int[] nums) {\n    int n = nums.Length;\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for (int i = 0; i &lt; n - 1; i++) {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // 최솟값의 인덱스를 기록한다\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        (nums[k], nums[i]) = (nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.go<pre><code>/* 선택 정렬 */\nfunc selectionSort(nums []int) {\n    n := len(nums)\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for i := 0; i &lt; n-1; i++ {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        k := i\n        for j := i + 1; j &lt; n; j++ {\n            if nums[j] &lt; nums[k] {\n                // 최솟값의 인덱스를 기록한다\n                k = j\n            }\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        nums[i], nums[k] = nums[k], nums[i]\n\n    }\n}\n</code></pre> selection_sort.swift<pre><code>/* 선택 정렬 */\nfunc selectionSort(nums: inout [Int]) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for i in nums.indices.dropLast() {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        var k = i\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[j] &lt; nums[k] {\n                k = j // 최솟값의 인덱스를 기록한다\n            }\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        nums.swapAt(i, k)\n    }\n}\n</code></pre> selection_sort.js<pre><code>/* 선택 정렬 */\nfunction selectionSort(nums) {\n    let n = nums.length;\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for (let i = 0; i &lt; n - 1; i++) {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // 최솟값의 인덱스를 기록한다\n            }\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.ts<pre><code>/* 선택 정렬 */\nfunction selectionSort(nums: number[]): void {\n    let n = nums.length;\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for (let i = 0; i &lt; n - 1; i++) {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // 최솟값의 인덱스를 기록한다\n            }\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.dart<pre><code>/* 선택 정렬 */\nvoid selectionSort(List&lt;int&gt; nums) {\n  int n = nums.length;\n  // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n  for (int i = 0; i &lt; n - 1; i++) {\n    // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n    int k = i;\n    for (int j = i + 1; j &lt; n; j++) {\n      if (nums[j] &lt; nums[k]) k = j; // 최솟값의 인덱스를 기록한다\n    }\n    // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n    int temp = nums[i];\n    nums[i] = nums[k];\n    nums[k] = temp;\n  }\n}\n</code></pre> selection_sort.rs<pre><code>/* 선택 정렬 */\nfn selection_sort(nums: &amp;mut [i32]) {\n    if nums.is_empty() {\n        return;\n    }\n    let n = nums.len();\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for i in 0..n - 1 {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        let mut k = i;\n        for j in i + 1..n {\n            if nums[j] &lt; nums[k] {\n                k = j; // 최솟값의 인덱스를 기록한다\n            }\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        nums.swap(i, k);\n    }\n}\n</code></pre> selection_sort.c<pre><code>/* 선택 정렬 */\nvoid selectionSort(int nums[], int n) {\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for (int i = 0; i &lt; n - 1; i++) {\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // 최솟값의 인덱스를 기록한다\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.kt<pre><code>/* 선택 정렬 */\nfun selectionSort(nums: IntArray) {\n    val n = nums.size\n    // 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n    for (i in 0..&lt;n - 1) {\n        var k = i\n        // 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n        for (j in i + 1..&lt;n) {\n            if (nums[j] &lt; nums[k])\n                k = j // 최솟값의 인덱스를 기록한다\n        }\n        // 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n        val temp = nums[i]\n        nums[i] = nums[k]\n        nums[k] = temp\n    }\n}\n</code></pre> selection_sort.rb<pre><code>### 선택 정렬 ###\ndef selection_sort(nums)\n  n = nums.length\n  # 바깥쪽 반복문: 정렬되지 않은 구간은 [i, n-1]이다\n  for i in 0...(n - 1)\n    # 안쪽 반복문: 정렬되지 않은 구간의 최솟값을 찾는다\n    k = i\n    for j in (i + 1)...n\n      if nums[j] &lt; nums[k]\n        k = j # 최솟값의 인덱스를 기록한다\n      end\n    end\n    # 이 최솟값과 정렬되지 않은 구간의 첫 원소를 교환한다\n    nums[i], nums[k] = nums[k], nums[i]\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 11 장   정렬","11.2   선택 정렬"],"tags":[]},{"location":"chapter_sorting/selection_sort/#1121","level":2,"title":"11.2.1   알고리즘 특성","text":"<ul> <li>시간 복잡도 \\(O(n^2)\\), 비적응형 정렬: 바깥 반복문은 \\(n - 1\\) 번 실행된다. 안쪽 반복문은 첫 회에 \\(n - 1\\) 번, 마지막 회에 \\(1\\) 번 실행된다. 즉, 각 회의 실행 횟수는 \\(n - 1\\), \\(n - 2\\), \\(\\dots\\), \\(2\\), \\(1\\)이고 합은 \\(\\frac{n(n - 1)}{2}\\)이다.</li> <li>공간 복잡도 \\(O(1)\\), 제자리 정렬: 포인터 \\(i\\)와 \\(j\\)는 상수 크기의 추가 공간을 사용한다.</li> <li>불안정 정렬: 아래 그림(11-3)과 같이 원소 <code>nums[i]</code>가 자신과 값이 같은 원소의 오른쪽으로 교환되어 두 원소의 상대적 순서가 바뀔 수 있다.</li> </ul> <p> 그림 11-3   선택 정렬의 불안정성 예시 </p>","path":["제 11 장   정렬","11.2   선택 정렬"],"tags":[]},{"location":"chapter_sorting/sorting_algorithm/","level":1,"title":"11.1   정렬 알고리즘","text":"<p>정렬 알고리즘(sorting algorithm)은 데이터 집합을 특정 순서로 나열하는 데 사용한다. 정렬된 데이터는 대체로 더 효율적으로 탐색, 분석, 처리할 수 있으므로 정렬 알고리즘은 널리 활용된다.</p> <p>아래 그림(11-1)과 같이 정렬할 데이터의 타입은 정수, 부동 소수점 수, 문자, 문자열 등 다양하다. 정렬 기준도 수의 크기, 문자의 ASCII 코드 순서, 사용자 정의 규칙 등 필요에 따라 정할 수 있다.</p> <p></p> <p> 그림 11-1   데이터 타입과 정렬 기준의 예 </p>","path":["제 11 장   정렬","11.1   정렬 알고리즘"],"tags":[]},{"location":"chapter_sorting/sorting_algorithm/#1111","level":2,"title":"11.1.1   평가 기준","text":"<p>실행 효율: 정렬 알고리즘은 시간 복잡도가 가능한 한 낮고, 전체 연산 수도 적기를 기대한다(시간 복잡도의 상수항이 작아야 한다). 데이터가 많을 때는 실행 효율이 특히 중요하다.</p> <p>제자리 여부: 이름 그대로 제자리 정렬은 추가 보조 배열 없이 원래 배열을 직접 조작하여 정렬하므로 메모리를 절약한다. 보통 데이터 이동 연산이 적어 실행 속도도 더 빠르다.</p> <p>안정성: 안정 정렬은 정렬 후에도 값이 같은 원소들의 배열 내 상대적 순서가 바뀌지 않는다.</p> <p>안정 정렬은 여러 기준으로 순차 정렬할 때 필요한 조건이다. 학생 정보를 저장한 표의 첫 번째 열이 이름, 두 번째 열이 나이라고 하자. 이때 불안정 정렬을 사용하면 입력 데이터가 기존에 정렬되어 있던 순서가 깨질 수 있다.</p> <pre><code># 입력 데이터는 이름순으로 정렬되어 있다\n# (name, age)\n  ('A', 19)\n  ('B', 18)\n  ('C', 21)\n  ('D', 19)\n  ('E', 23)\n\n# 불안정 정렬 알고리즘으로 리스트를 나이순으로 정렬하면,\n# 결과에서 ('D', 19)와 ('A', 19)의 상대적 위치가 바뀌어,\n# 입력 데이터의 이름순 정렬 상태가 사라질 수 있다\n  ('B', 18)\n  ('D', 19)\n  ('A', 19)\n  ('C', 21)\n  ('E', 23)\n</code></pre> <p>적응성: 적응형 정렬은 입력 데이터에 이미 존재하는 순서 정보를 이용해 계산량을 줄이고 시간 효율을 높일 수 있다. 적응형 정렬 알고리즘의 최선 시간 복잡도는 보통 평균 시간 복잡도보다 낮다.</p> <p>비교 기반 여부: 비교 기반 정렬은 비교 연산자(\\(&lt;\\), \\(=\\), \\(&gt;\\))로 원소의 상대적 순서를 판단하여 배열 전체를 정렬한다. 최악의 시간 복잡도 하한은 \\(\\Omega(n \\log n)\\)이다. 반면 비교를 사용하지 않는 정렬은 비교 연산자를 사용하지 않으며 시간 복잡도가 \\(O(n)\\)에 이를 수 있지만, 범용성은 상대적으로 낮다.</p>","path":["제 11 장   정렬","11.1   정렬 알고리즘"],"tags":[]},{"location":"chapter_sorting/sorting_algorithm/#1112","level":2,"title":"11.1.2   이상적인 정렬 알고리즘","text":"<p>빠른 실행, 제자리 정렬, 안정성, 적응성, 높은 범용성을 모두 갖추면 이상적이다. 하지만 지금까지 이러한 특성을 모두 갖춘 정렬 알고리즘은 발견되지 않았다. 따라서 데이터의 구체적인 특성과 문제의 요구에 맞추어 정렬 알고리즘을 선택해야 한다.</p> <p>이제 다양한 정렬 알고리즘을 배우고, 위 평가 기준에 따라 각각의 장단점을 분석해 보자.</p>","path":["제 11 장   정렬","11.1   정렬 알고리즘"],"tags":[]},{"location":"chapter_sorting/summary/","level":1,"title":"11.11   요약","text":"","path":["제 11 장   정렬","11.11   요약"],"tags":[]},{"location":"chapter_sorting/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>버블 정렬은 인접한 원소를 교환하여 정렬한다. 플래그를 추가해 조기에 반환하도록 하면 최선 시간 복잡도를 \\(O(n)\\)으로 최적화할 수 있다.</li> <li>삽입 정렬은 매번 미정렬 구간의 원소를 정렬된 구간의 올바른 위치에 삽입한다. 시간 복잡도는 \\(O(n^2)\\)이지만 기본 연산 수가 비교적 적어, 데이터가 적은 정렬 작업에서 매우 많이 사용된다.</li> <li>퀵 정렬은 센티널 분할 연산으로 정렬한다. 분할할 때마다 최악의 피벗을 고르면 시간 복잡도가 \\(O(n^2)\\)으로 저하될 수 있다. 중앙값이나 무작위 원소를 피벗으로 사용하면 이런 확률을 줄일 수 있다. 더 짧은 부분 구간에 먼저 재귀하면 재귀 깊이를 효과적으로 줄여 공간 복잡도를 \\(O(\\log n)\\)으로 최적화할 수 있다.</li> <li>병합 정렬은 분할과 병합의 두 단계로 이루어지며 분할 정복 전략을 전형적으로 보여 준다. 배열을 정렬할 때는 보조 배열이 필요하여 공간 복잡도가 \\(O(n)\\)이지만, 연결 리스트 정렬은 \\(O(1)\\)까지 최적화할 수 있다.</li> <li>버킷 정렬은 데이터를 버킷에 분배하고, 각 버킷 안에서 정렬하고, 결과를 합치는 세 단계로 이루어진다. 역시 분할 정복 전략을 보여 주며 매우 큰 데이터에 적합하다. 핵심은 데이터를 고르게 분배하는 것이다.</li> <li>계수 정렬은 버킷 정렬의 특수한 경우로, 데이터의 등장 횟수를 세어 정렬한다. 데이터 수는 많지만 값의 범위가 제한된 경우에 적합하며, 데이터를 양의 정수로 변환할 수 있어야 한다.</li> <li>기수 정렬은 자릿수별로 정렬하여 전체 데이터를 정렬하며, 데이터를 고정된 자릿수의 수로 표현할 수 있어야 한다.</li> <li>우리는 효율, 안정성, 제자리 동작, 적응성 등의 장점을 갖춘 정렬 알고리즘을 원한다. 하지만 다른 자료 구조와 알고리즘처럼 이 모든 조건을 동시에 만족하는 정렬 알고리즘은 없다. 실제 응용에서는 데이터의 특성에 따라 적합한 알고리즘을 선택해야 한다.</li> <li>아래 그림(11-19)은 주요 정렬 알고리즘의 효율, 안정성, 제자리 여부, 적응성 등을 비교한다.</li> </ul> <p> 그림 11-19   정렬 알고리즘 비교 </p>","path":["제 11 장   정렬","11.11   요약"],"tags":[]},{"location":"chapter_sorting/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 정렬 알고리즘의 안정성은 어떤 상황에서 필요한가?</p> <p>실제로 객체의 특정 속성을 기준으로 정렬하는 경우가 있다. 예를 들어 학생에게 이름과 키라는 두 속성이 있고, 여러 기준으로 순차 정렬하고 싶다고 하자. 먼저 이름순으로 정렬하여 <code>(A, 180) (B, 185) (C, 170) (D, 170)</code>을 얻은 뒤 키순으로 정렬한다. 정렬 알고리즘이 불안정하면 <code>(D, 170) (C, 170) (A, 180) (B, 185)</code>가 나올 수 있다.</p> <p>학생 D와 C의 위치가 바뀌어 이름순 정렬이 깨졌음을 알 수 있으며, 이는 원하는 결과가 아니다.</p> <p>Q: 센티널 분할에서 '오른쪽에서 왼쪽으로 찾기'와 '왼쪽에서 오른쪽으로 찾기'의 순서를 바꿔도 되는가?</p> <p>안 된다. 맨 왼쪽 원소를 피벗으로 선택하면 반드시 '오른쪽에서 왼쪽으로 찾기'를 먼저 수행하고, 그다음 '왼쪽에서 오른쪽으로 찾기'를 수행해야 한다. 직관과 다르게 느껴질 수 있으니 이유를 분석해 보자.</p> <p>분할 함수 <code>partition()</code>의 마지막 단계는 <code>nums[left]</code>와 <code>nums[i]</code>를 교환하는 것이다. 교환 후에는 피벗 왼쪽의 모든 원소에 대해 '원소 <code>&lt;=</code> 피벗'이 성립해야 하므로, 마지막 교환 전에 반드시 <code>nums[left] &gt;= nums[i]</code>가 성립해야 한다. 먼저 '왼쪽에서 오른쪽으로 찾기'를 수행했는데 피벗보다 큰 원소를 찾지 못한다면, <code>i == j</code>일 때 반복문을 빠져나오며 이때 <code>nums[j] == nums[i] &gt; nums[left]</code>일 수 있다. 즉, 마지막 교환이 피벗보다 큰 원소를 맨 왼쪽으로 옮겨 분할을 실패하게 할 수 있다.</p> <p>예를 들어 배열이 <code>[0, 0, 0, 0, 1]</code>일 때 '왼쪽에서 오른쪽으로 찾기'를 먼저 하면, 분할 결과가 <code>[1, 0, 0, 0, 0]</code>으로 잘못 나온다.</p> <p>더 생각해 보면 <code>nums[right]</code>를 피벗으로 선택했을 때는 반대로 '왼쪽에서 오른쪽으로 찾기'를 먼저 해야 한다.</p> <p>Q: 퀵 정렬의 재귀 깊이를 최적화할 때 짧은 배열을 선택하면 왜 깊이가 \\(\\log n\\)을 넘지 않는가?</p> <p>재귀 깊이는 현재 아직 반환되지 않은 재귀 메서드의 수다. 매 분할에서 원래 배열을 두 부분 배열로 나눈다. 재귀 깊이를 최적화하면 아래로 재귀하는 부분 배열의 길이는 최대 원래 길이의 절반이다. 최악의 경우 매번 정확히 절반이라고 해도 최종 재귀 깊이는 \\(\\log n\\)이다.</p> <p>원래 퀵 정렬에서는 더 긴 배열에 계속 재귀할 수 있다. 최악의 경우 길이가 \\(n\\), \\(n - 1\\), \\(\\dots\\), \\(2\\), \\(1\\)로 줄어 재귀 깊이가 \\(n\\)이 된다. 재귀 깊이 최적화는 이 상황을 방지한다.</p> <p>Q: 배열의 모든 원소가 같으면 퀵 정렬의 시간 복잡도는 \\(O(n^2)\\)인가? 이런 퇴화는 어떻게 처리하는가?</p> <p>그렇다. 이 경우 배열을 피벗보다 작은 부분, 같은 부분, 큰 부분의 세 부분으로 나누고 작은 부분과 큰 부분에만 재귀하는 방법을 고려할 수 있다. 이렇게 하면 모든 원소가 같은 입력 배열은 분할 한 번으로 정렬된다.</p> <p>Q: 버킷 정렬의 최악 시간 복잡도는 왜 \\(O(n^2)\\)인가?</p> <p>최악의 경우 모든 원소가 같은 버킷에 들어간다. 이 원소들을 \\(O(n^2)\\) 알고리즘으로 정렬한다면 시간 복잡도는 \\(O(n^2)\\)이다.</p>","path":["제 11 장   정렬","11.11   요약"],"tags":[]},{"location":"chapter_stack_and_queue/","level":1,"title":"제 5 장   스택과 큐","text":"<p>이 장을 시작하며</p> <p>스택은 고양이들이 포개져 있는 모습과 같고, 큐는 고양이들이 줄을 서 있는 모습과 같다.</p> <p>둘은 각각 후입선출과 선입선출의 논리적 관계를 나타낸다.</p>","path":["제 5 장   스택과 큐"],"tags":[]},{"location":"chapter_stack_and_queue/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>5.1   스택</li> <li>5.2   큐</li> <li>5.3   덱</li> <li>5.4   요약</li> <li>5.5   연습 문제</li> </ul>","path":["제 5 장   스택과 큐"],"tags":[]},{"location":"chapter_stack_and_queue/deque/","level":1,"title":"5.3   덱","text":"<p>큐에서는 맨 앞 원소를 삭제하거나 맨 뒤에 원소를 추가하는 것만 가능하다. 아래 그림(5-7)의 덱(double-ended queue)은 앞뒤 양쪽에서 원소를 추가하고 삭제할 수 있어 더 유연하다.</p> <p></p> <p> 그림 5-7   덱의 연산 </p>","path":["제 5 장   스택과 큐","5.3   덱"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#531","level":2,"title":"5.3.1   덱의 기본 연산","text":"<p>덱의 기본 연산은 아래 표(5-3)와 같다. 구체적인 메서드 이름은 사용하는 프로그래밍 언어에 따라 정해진다.</p> <p> 표 5-3   덱의 연산 효율 </p> 메서드 설명 시간 복잡도 <code>push_first()</code> 맨 앞에 원소 추가 \\(O(1)\\) <code>push_last()</code> 맨 뒤에 원소 추가 \\(O(1)\\) <code>pop_first()</code> 맨 앞 원소 삭제 \\(O(1)\\) <code>pop_last()</code> 맨 뒤 원소 삭제 \\(O(1)\\) <code>peek_first()</code> 맨 앞 원소에 접근 \\(O(1)\\) <code>peek_last()</code> 맨 뒤 원소에 접근 \\(O(1)\\) <p>마찬가지로 프로그래밍 언어가 제공하는 덱 클래스를 바로 사용할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby deque.py<pre><code>from collections import deque\n\n# 덱 초기화\ndeq: deque[int] = deque()\n\n# 덱에 원소 삽입\ndeq.append(2)      # 맨 뒤에 추가\ndeq.append(5)\ndeq.append(4)\ndeq.appendleft(3)  # 맨 앞에 추가\ndeq.appendleft(1)\n\n# 원소 접근\nfront: int = deq[0]  # 맨 앞 원소\nrear: int = deq[-1]  # 맨 뒤 원소\n\n# 덱에서 원소 꺼내기\npop_front: int = deq.popleft()  # 맨 앞 원소 꺼내기\npop_rear: int = deq.pop()       # 맨 뒤 원소 꺼내기\n\n# 덱 길이 구하기\nsize: int = len(deq)\n\n# 덱이 비어 있는지 확인\nis_empty: bool = len(deq) == 0\n</code></pre> deque.cpp<pre><code>/* 덱 초기화 */\ndeque&lt;int&gt; deque;\n\n/* 덱에 원소 삽입 */\ndeque.push_back(2);   // 맨 뒤에 추가\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // 맨 앞에 추가\ndeque.push_front(1);\n\n/* 원소 접근 */\nint front = deque.front(); // 맨 앞 원소\nint back = deque.back();   // 맨 뒤 원소\n\n/* 덱에서 원소 꺼내기 */\ndeque.pop_front();  // 맨 앞 원소 꺼내기\ndeque.pop_back();   // 맨 뒤 원소 꺼내기\n\n/* 덱 길이 구하기 */\nint size = deque.size();\n\n/* 덱이 비어 있는지 확인 */\nbool empty = deque.empty();\n</code></pre> deque.java<pre><code>/* 덱 초기화 */\nDeque&lt;Integer&gt; deque = new LinkedList&lt;&gt;();\n\n/* 덱에 원소 삽입 */\ndeque.offerLast(2);   // 맨 뒤에 추가\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // 맨 앞에 추가\ndeque.offerFirst(1);\n\n/* 원소 접근 */\nint peekFirst = deque.peekFirst();  // 맨 앞 원소\nint peekLast = deque.peekLast();    // 맨 뒤 원소\n\n/* 덱에서 원소 꺼내기 */\nint popFirst = deque.pollFirst();  // 맨 앞 원소 꺼내기\nint popLast = deque.pollLast();    // 맨 뒤 원소 꺼내기\n\n/* 덱 길이 구하기 */\nint size = deque.size();\n\n/* 덱이 비어 있는지 확인 */\nboolean isEmpty = deque.isEmpty();\n</code></pre> deque.cs<pre><code>/* 덱 초기화 */\n// C#에서는 연결 리스트 LinkedList를 덱으로 사용한다\nLinkedList&lt;int&gt; deque = new();\n\n/* 덱에 원소 삽입 */\ndeque.AddLast(2);   // 맨 뒤에 추가\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // 맨 앞에 추가\ndeque.AddFirst(1);\n\n/* 원소 접근 */\nint peekFirst = deque.First.Value;  // 맨 앞 원소\nint peekLast = deque.Last.Value;    // 맨 뒤 원소\n\n/* 덱에서 원소 꺼내기 */\ndeque.RemoveFirst();  // 맨 앞 원소 꺼내기\ndeque.RemoveLast();   // 맨 뒤 원소 꺼내기\n\n/* 덱 길이 구하기 */\nint size = deque.Count;\n\n/* 덱이 비어 있는지 확인 */\nbool isEmpty = deque.Count == 0;\n</code></pre> deque_test.go<pre><code>/* 덱 초기화 */\n// Go에서는 list를 덱으로 사용한다\ndeque := list.New()\n\n/* 덱에 원소 삽입 */\ndeque.PushBack(2)      // 맨 뒤에 추가\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // 맨 앞에 추가\ndeque.PushFront(1)\n\n/* 원소 접근 */\nfront := deque.Front() // 맨 앞 원소\nrear := deque.Back()   // 맨 뒤 원소\n\n/* 덱에서 원소 꺼내기 */\ndeque.Remove(front)    // 맨 앞 원소 꺼내기\ndeque.Remove(rear)     // 맨 뒤 원소 꺼내기\n\n/* 덱 길이 구하기 */\nsize := deque.Len()\n\n/* 덱이 비어 있는지 확인 */\nisEmpty := deque.Len() == 0\n</code></pre> deque.swift<pre><code>/* 덱 초기화 */\n// Swift에는 내장 덱 클래스가 없으므로 Array를 덱으로 사용할 수 있다\nvar deque: [Int] = []\n\n/* 덱에 원소 삽입 */\ndeque.append(2) // 맨 뒤에 추가\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // 맨 앞에 추가\ndeque.insert(1, at: 0)\n\n/* 원소 접근 */\nlet peekFirst = deque.first! // 맨 앞 원소\nlet peekLast = deque.last! // 맨 뒤 원소\n\n/* 덱에서 원소 꺼내기 */\n// Array로 구현하면 popFirst의 복잡도는 O(n)이다\nlet popFirst = deque.removeFirst() // 맨 앞 원소 꺼내기\nlet popLast = deque.removeLast() // 맨 뒤 원소 꺼내기\n\n/* 덱 길이 구하기 */\nlet size = deque.count\n\n/* 덱이 비어 있는지 확인 */\nlet isEmpty = deque.isEmpty\n</code></pre> deque.js<pre><code>/* 덱 초기화 */\n// JavaScript에는 내장 덱이 없으므로 Array를 덱으로 사용해야 한다\nconst deque = [];\n\n/* 덱에 원소 삽입 */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// 배열이므로 unshift() 메서드의 시간 복잡도는 O(n)임에 주의\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* 원소 접근 */\nconst peekFirst = deque[0];\nconst peekLast = deque[deque.length - 1];\n\n/* 덱에서 원소 꺼내기 */\n// 배열이므로 shift() 메서드의 시간 복잡도는 O(n)임에 주의\nconst popFront = deque.shift();\nconst popBack = deque.pop();\n\n/* 덱 길이 구하기 */\nconst size = deque.length;\n\n/* 덱이 비어 있는지 확인 */\nconst isEmpty = size === 0;\n</code></pre> deque.ts<pre><code>/* 덱 초기화 */\n// TypeScript에는 내장 덱이 없으므로 Array를 덱으로 사용해야 한다\nconst deque: number[] = [];\n\n/* 덱에 원소 삽입 */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// 배열이므로 unshift() 메서드의 시간 복잡도는 O(n)임에 주의\ndeque.unshift(3);\ndeque.unshift(1);\n\n/* 원소 접근 */\nconst peekFirst: number = deque[0];\nconst peekLast: number = deque[deque.length - 1];\n\n/* 덱에서 원소 꺼내기 */\n// 배열이므로 shift() 메서드의 시간 복잡도는 O(n)임에 주의\nconst popFront: number = deque.shift() as number;\nconst popBack: number = deque.pop() as number;\n\n/* 덱 길이 구하기 */\nconst size: number = deque.length;\n\n/* 덱이 비어 있는지 확인 */\nconst isEmpty: boolean = size === 0;\n</code></pre> deque.dart<pre><code>/* 덱 초기화 */\n// Dart의 Queue는 덱으로 정의되어 있다\nQueue&lt;int&gt; deque = Queue&lt;int&gt;();\n\n/* 덱에 원소 삽입 */\ndeque.addLast(2);  // 맨 뒤에 추가\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // 맨 앞에 추가\ndeque.addFirst(1);\n\n/* 원소 접근 */\nint peekFirst = deque.first; // 맨 앞 원소\nint peekLast = deque.last;   // 맨 뒤 원소\n\n/* 덱에서 원소 꺼내기 */\nint popFirst = deque.removeFirst(); // 맨 앞 원소 꺼내기\nint popLast = deque.removeLast();   // 맨 뒤 원소 꺼내기\n\n/* 덱 길이 구하기 */\nint size = deque.length;\n\n/* 덱이 비어 있는지 확인 */\nbool isEmpty = deque.isEmpty;\n</code></pre> deque.rs<pre><code>/* 덱 초기화 */\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* 덱에 원소 삽입 */\ndeque.push_back(2);  // 맨 뒤에 추가\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // 맨 앞에 추가\ndeque.push_front(1);\n\n/* 원소 접근 */\nif let Some(front) = deque.front() { // 맨 앞 원소\n}\nif let Some(rear) = deque.back() {   // 맨 뒤 원소\n}\n\n/* 덱에서 원소 꺼내기 */\nif let Some(pop_front) = deque.pop_front() { // 맨 앞 원소 꺼내기\n}\nif let Some(pop_rear) = deque.pop_back() {   // 맨 뒤 원소 꺼내기\n}\n\n/* 덱 길이 구하기 */\nlet size = deque.len();\n\n/* 덱이 비어 있는지 확인 */\nlet is_empty = deque.is_empty();\n</code></pre> deque.c<pre><code>// C는 내장 덱을 제공하지 않는다\n</code></pre> deque.kt<pre><code>/* 덱 초기화 */\nval deque = LinkedList&lt;Int&gt;()\n\n/* 덱에 원소 삽입 */\ndeque.offerLast(2)  // 맨 뒤에 추가\ndeque.offerLast(5)\ndeque.offerLast(4)\ndeque.offerFirst(3) // 맨 앞에 추가\ndeque.offerFirst(1)\n\n/* 원소 접근 */\nval peekFirst = deque.peekFirst() // 맨 앞 원소\nval peekLast = deque.peekLast()   // 맨 뒤 원소\n\n/* 덱에서 원소 꺼내기 */\nval popFirst = deque.pollFirst() // 맨 앞 원소 꺼내기\nval popLast = deque.pollLast()   // 맨 뒤 원소 꺼내기\n\n/* 덱 길이 구하기 */\nval size = deque.size\n\n/* 덱이 비어 있는지 확인 */\nval isEmpty = deque.isEmpty()\n</code></pre> deque.rb<pre><code># 덱 초기화\n# Ruby에는 내장 덱이 없으므로 Array를 덱으로 사용해야 한다\ndeque = []\n\n# 덱에 원소 삽입\ndeque &lt;&lt; 2\ndeque &lt;&lt; 5\ndeque &lt;&lt; 4\n# 배열이므로 Array#unshift 메서드의 시간 복잡도는 O(n)임에 주의\ndeque.unshift(3)\ndeque.unshift(1)\n\n# 원소 접근\npeek_first = deque.first\npeek_last = deque.last\n\n# 덱에서 원소 꺼내기\n# 배열이므로 Array#shift 메서드의 시간 복잡도는 O(n)임에 주의\npop_front = deque.shift\npop_back = deque.pop\n\n# 덱 길이 구하기\nsize = deque.length\n\n# 덱이 비어 있는지 확인\nis_empty = size.zero?\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 5 장   스택과 큐","5.3   덱"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#532","level":2,"title":"5.3.2   덱 구현 *","text":"<p>덱의 구현은 큐와 비슷하며, 기반 자료 구조로 연결 리스트나 배열을 선택할 수 있다.</p>","path":["제 5 장   스택과 큐","5.3   덱"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#1","level":3,"title":"1.   이중 연결 리스트 기반 구현","text":"<p>앞 절에서는 일반적인 단일 연결 리스트로 큐를 구현했다. 머리 노드를 삭제하는 연산(큐에서 꺼내기)과 꼬리 노드 뒤에 새 노드를 추가하는 연산(큐에 삽입하기)을 쉽게 수행할 수 있기 때문이다.</p> <p>덱에서는 앞과 뒤 모두 삽입과 꺼내기를 수행할 수 있다. 즉, 큐와 대칭되는 반대 방향의 연산도 구현해야 한다. 이를 위해 ‘이중 연결 리스트’를 덱의 기반 자료 구조로 사용한다.</p> <p>아래 그림(5-8)과 같이 이중 연결 리스트의 머리 노드와 꼬리 노드를 덱의 맨 앞과 맨 뒤로 보고, 양 끝에서 노드를 추가하고 삭제하는 기능을 구현한다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 5-8   연결 리스트 기반 덱의 삽입과 꺼내기 </p> <p>구현 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linkedlist_deque.py<pre><code>class ListNode:\n    \"\"\"이중 연결 리스트 노드\"\"\"\n\n    def __init__(self, val: int):\n        \"\"\"생성자\"\"\"\n        self.val: int = val\n        self.next: ListNode | None = None  # 다음 노드 참조\n        self.prev: ListNode | None = None  # 이전 노드 참조\n\nclass LinkedListDeque:\n    \"\"\"이중 연결 리스트로 구현한 덱\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        self._front: ListNode | None = None  # 머리 노드 front\n        self._rear: ListNode | None = None  # 꼬리 노드 rear\n        self._size: int = 0  # 덱의 길이\n\n    def size(self) -&gt; int:\n        \"\"\"덱의 길이 반환\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"덱이 비어 있는지 판단\"\"\"\n        return self._size == 0\n\n    def push(self, num: int, is_front: bool):\n        \"\"\"큐 삽입 연산\"\"\"\n        node = ListNode(num)\n        # 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if self.is_empty():\n            self._front = self._rear = node\n        # 맨 앞에 삽입하는 연산\n        elif is_front:\n            # node를 연결 리스트의 맨 앞에 추가한다\n            self._front.prev = node\n            node.next = self._front\n            self._front = node  # 머리 노드를 갱신한다\n        # 맨 뒤에 삽입하는 연산\n        else:\n            # node를 연결 리스트의 맨 뒤에 추가한다\n            self._rear.next = node\n            node.prev = self._rear\n            self._rear = node  # 꼬리 노드를 갱신한다\n        self._size += 1  # 큐 길이를 갱신한다\n\n    def push_first(self, num: int):\n        \"\"\"맨 앞에 삽입\"\"\"\n        self.push(num, True)\n\n    def push_last(self, num: int):\n        \"\"\"맨 뒤에 삽입\"\"\"\n        self.push(num, False)\n\n    def pop(self, is_front: bool) -&gt; int:\n        \"\"\"큐에서 꺼내는 연산\"\"\"\n        if self.is_empty():\n            raise IndexError(\"덱이 비어 있다\")\n        # 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if self._size == 1:\n            val: int = self._front.val\n            self._front = None\n            self._rear = None\n        # 맨 앞에서 꺼내는 연산\n        elif is_front:\n            val: int = self._front.val  # 머리 노드의 값을 임시로 저장한다\n            # 머리 노드를 삭제한다\n            fnext: ListNode | None = self._front.next\n            fnext.prev = None\n            self._front.next = None\n            self._front = fnext  # 머리 노드를 갱신한다\n        # 맨 뒤에서 꺼내는 연산\n        else:\n            val: int = self._rear.val  # 꼬리 노드의 값을 임시로 저장한다\n            # 꼬리 노드를 삭제한다\n            rprev: ListNode | None = self._rear.prev\n            rprev.next = None\n            self._rear.prev = None\n            self._rear = rprev  # 꼬리 노드를 갱신한다\n        self._size -= 1  # 큐 길이를 갱신한다\n        return val\n\n    def pop_first(self) -&gt; int:\n        \"\"\"맨 앞에서 꺼내기\"\"\"\n        return self.pop(True)\n\n    def pop_last(self) -&gt; int:\n        \"\"\"맨 뒤에서 꺼내기\"\"\"\n        return self.pop(False)\n\n    def peek_first(self) -&gt; int:\n        \"\"\"맨 앞 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"덱이 비어 있다\")\n        return self._front.val\n\n    def peek_last(self) -&gt; int:\n        \"\"\"맨 뒤 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"덱이 비어 있다\")\n        return self._rear.val\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"출력에 사용할 배열 반환\"\"\"\n        node = self._front\n        res = [0] * self.size()\n        for i in range(self.size()):\n            res[i] = node.val\n            node = node.next\n        return res\n</code></pre> linkedlist_deque.cpp<pre><code>/* 이중 연결 리스트 노드 */\nstruct DoublyListNode {\n    int val;              // 노드 값\n    DoublyListNode *next; // 다음 노드 포인터\n    DoublyListNode *prev; // 이전 노드 포인터\n    DoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n    }\n};\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n  private:\n    DoublyListNode *front, *rear; // 머리 노드 front, 꼬리 노드 rear\n    int queSize = 0;              // 덱의 길이\n\n  public:\n    /* 생성자 */\n    LinkedListDeque() : front(nullptr), rear(nullptr) {\n    }\n\n    /* 소멸자 */\n    ~LinkedListDeque() {\n        // 연결 리스트를 순회하며 노드를 삭제하여 메모리를 해제한다\n        DoublyListNode *pre, *cur = front;\n        while (cur != nullptr) {\n            pre = cur;\n            cur = cur-&gt;next;\n            delete pre;\n        }\n    }\n\n    /* 덱의 길이 반환 */\n    int size() {\n        return queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* 큐 삽입 연산 */\n    void push(int num, bool isFront) {\n        DoublyListNode *node = new DoublyListNode(num);\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (isEmpty())\n            front = rear = node;\n        // 맨 앞에 삽입하는 연산\n        else if (isFront) {\n            // node를 연결 리스트의 맨 앞에 추가한다\n            front-&gt;prev = node;\n            node-&gt;next = front;\n            front = node; // 머리 노드를 갱신한다\n        // 맨 뒤에 삽입하는 연산\n        } else {\n            // node를 연결 리스트의 맨 뒤에 추가한다\n            rear-&gt;next = node;\n            node-&gt;prev = rear;\n            rear = node; // 꼬리 노드를 갱신한다\n        }\n        queSize++; // 큐 길이를 갱신한다\n    }\n\n    /* 맨 앞에 삽입 */\n    void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* 맨 뒤에 삽입 */\n    void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* 큐에서 꺼내는 연산 */\n    int pop(bool isFront) {\n        if (isEmpty())\n            throw out_of_range(\"큐가 비어 있다\");\n        int val;\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (queSize == 1) {\n            val = front-&gt;val;\n            delete front; // 같은 노드를 가리키므로 하나만 해제하면 된다\n            front = nullptr;\n            rear = nullptr;\n        }\n        // 맨 앞에서 꺼내는 연산\n        else if (isFront) {\n            val = front-&gt;val; // 머리 노드의 값을 임시로 저장한다\n            // 머리 노드를 삭제한다\n            DoublyListNode *fNext = front-&gt;next;\n            fNext-&gt;prev = nullptr;\n            front-&gt;next = nullptr;\n            delete front;\n            front = fNext; // 머리 노드를 갱신한다\n        // 맨 뒤에서 꺼내는 연산\n        } else {\n            val = rear-&gt;val; // 꼬리 노드의 값을 임시로 저장한다\n            // 꼬리 노드를 삭제한다\n            DoublyListNode *rPrev = rear-&gt;prev;\n            rPrev-&gt;next = nullptr;\n            rear-&gt;prev = nullptr;\n            delete rear;\n            rear = rPrev; // 꼬리 노드를 갱신한다\n        }\n        queSize--; // 큐 길이를 갱신한다\n        return val;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    int popFirst() {\n        return pop(true);\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    int popLast() {\n        return pop(false);\n    }\n\n    /* 맨 앞 원소 접근 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"덱이 비어 있다\");\n        return front-&gt;val;\n    }\n\n    /* 맨 뒤 원소 접근 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"덱이 비어 있다\");\n        return rear-&gt;val;\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    vector&lt;int&gt; toVector() {\n        DoublyListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_deque.java<pre><code>/* 이중 연결 리스트 노드 */\nclass ListNode {\n    int val; // 노드 값\n    ListNode next; // 다음 노드 참조\n    ListNode prev; // 이전 노드 참조\n\n    ListNode(int val) {\n        this.val = val;\n        prev = next = null;\n    }\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n    private ListNode front, rear; // 머리 노드 front, 꼬리 노드 rear\n    private int queSize = 0; // 덱의 길이\n\n    public LinkedListDeque() {\n        front = rear = null;\n    }\n\n    /* 덱의 길이 반환 */\n    public int size() {\n        return queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* 큐 삽입 연산 */\n    private void push(int num, boolean isFront) {\n        ListNode node = new ListNode(num);\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (isEmpty())\n            front = rear = node;\n        // 맨 앞에 삽입하는 연산\n        else if (isFront) {\n            // node를 연결 리스트의 맨 앞에 추가한다\n            front.prev = node;\n            node.next = front;\n            front = node; // 머리 노드를 갱신한다\n        // 맨 뒤에 삽입하는 연산\n        } else {\n            // node를 연결 리스트의 맨 뒤에 추가한다\n            rear.next = node;\n            node.prev = rear;\n            rear = node; // 꼬리 노드를 갱신한다\n        }\n        queSize++; // 큐 길이를 갱신한다\n    }\n\n    /* 맨 앞에 삽입 */\n    public void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* 맨 뒤에 삽입 */\n    public void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* 큐에서 꺼내는 연산 */\n    private int pop(boolean isFront) {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        int val;\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (queSize == 1) {\n            val = front.val;\n            front = null;\n            rear = null;\n        }\n        // 맨 앞에서 꺼내는 연산\n        else if (isFront) {\n            val = front.val; // 머리 노드의 값을 임시로 저장한다\n            // 머리 노드를 삭제한다\n            ListNode fNext = front.next;\n            fNext.prev = null;\n            front.next = null;\n            front = fNext; // 머리 노드를 갱신한다\n        // 맨 뒤에서 꺼내는 연산\n        } else {\n            val = rear.val; // 꼬리 노드의 값을 임시로 저장한다\n            // 꼬리 노드를 삭제한다\n            ListNode rPrev = rear.prev;\n            rPrev.next = null;\n            rear.prev = null;\n            rear = rPrev; // 꼬리 노드를 갱신한다\n        }\n        queSize--; // 큐 길이를 갱신한다\n        return val;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    public int popFirst() {\n        return pop(true);\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    public int popLast() {\n        return pop(false);\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* 맨 뒤 원소 접근 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return rear.val;\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.cs<pre><code>/* 이중 연결 리스트 노드 */\nclass ListNode(int val) {\n    public int val = val;       // 노드 값\n    public ListNode? next = null; // 다음 노드 참조\n    public ListNode? prev = null; // 이전 노드 참조\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n    ListNode? front, rear; // 머리 노드 front, 꼬리 노드 rear\n    int queSize = 0;      // 덱의 길이\n\n    public LinkedListDeque() {\n        front = null;\n        rear = null;\n    }\n\n    /* 덱의 길이 반환 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* 큐 삽입 연산 */\n    void Push(int num, bool isFront) {\n        ListNode node = new(num);\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (IsEmpty()) {\n            front = node;\n            rear = node;\n        }\n        // 맨 앞에 삽입하는 연산\n        else if (isFront) {\n            // node를 연결 리스트의 맨 앞에 추가한다\n            front!.prev = node;\n            node.next = front;\n            front = node; // 머리 노드를 갱신한다                           \n        }\n        // 맨 뒤에 삽입하는 연산\n        else {\n            // node를 연결 리스트의 맨 뒤에 추가한다\n            rear!.next = node;\n            node.prev = rear;\n            rear = node;  // 꼬리 노드를 갱신한다\n        }\n\n        queSize++; // 큐 길이를 갱신한다\n    }\n\n    /* 맨 앞에 삽입 */\n    public void PushFirst(int num) {\n        Push(num, true);\n    }\n\n    /* 맨 뒤에 삽입 */\n    public void PushLast(int num) {\n        Push(num, false);\n    }\n\n    /* 큐에서 꺼내는 연산 */\n    int? Pop(bool isFront) {\n        if (IsEmpty())\n            throw new Exception();\n        int? val;\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (queSize == 1) {\n            val = front!.val;\n            front = null;\n            rear = null;\n        }\n        // 맨 앞에서 꺼내는 연산\n        else if (isFront) {\n            val = front!.val; // 머리 노드의 값을 임시로 저장한다\n            // 머리 노드를 삭제한다\n            ListNode? fNext = front?.next;\n            fNext!.prev = null;\n            front.next = null;\n            front = fNext;   // 머리 노드를 갱신한다\n        }\n        // 맨 뒤에서 꺼내는 연산\n        else {\n            val = rear!.val;  // 꼬리 노드의 값을 임시로 저장한다\n            // 꼬리 노드를 삭제한다\n            ListNode? rPrev = rear?.prev;\n            rPrev!.next = null;\n            rear.prev = null;\n            rear = rPrev;    // 꼬리 노드를 갱신한다\n        }\n\n        queSize--; // 큐 길이를 갱신한다\n        return val;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    public int? PopFirst() {\n        return Pop(true);\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    public int? PopLast() {\n        return Pop(false);\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int? PeekFirst() {\n        if (IsEmpty())\n            throw new Exception();\n        return front?.val;\n    }\n\n    /* 맨 뒤 원소 접근 */\n    public int? PeekLast() {\n        if (IsEmpty())\n            throw new Exception();\n        return rear?.val;\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    public int?[] ToArray() {\n        ListNode? node = front;\n        int?[] res = new int?[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node?.val;\n            node = node?.next;\n        }\n\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.go<pre><code>/* 이중 연결 리스트로 구현한 덱 */\ntype linkedListDeque struct {\n    // 내장 패키지 list를 사용한다\n    data *list.List\n}\n\n/* 덱 초기화 */\nfunc newLinkedListDeque() *linkedListDeque {\n    return &amp;linkedListDeque{\n        data: list.New(),\n    }\n}\n\n/* 맨 앞에 원소 삽입 */\nfunc (s *linkedListDeque) pushFirst(value any) {\n    s.data.PushFront(value)\n}\n\n/* 맨 뒤에 원소 삽입 */\nfunc (s *linkedListDeque) pushLast(value any) {\n    s.data.PushBack(value)\n}\n\n/* 맨 앞 원소를 꺼낸다 */\nfunc (s *linkedListDeque) popFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* 맨 뒤 원소를 꺼낸다 */\nfunc (s *linkedListDeque) popLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* 맨 앞 원소 접근 */\nfunc (s *linkedListDeque) peekFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* 맨 뒤 원소 접근 */\nfunc (s *linkedListDeque) peekLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* 큐의 길이 반환 */\nfunc (s *linkedListDeque) size() int {\n    return s.data.Len()\n}\n\n/* 큐가 비어 있는지 판단 */\nfunc (s *linkedListDeque) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* 출력에 사용할 List 반환 */\nfunc (s *linkedListDeque) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_deque.swift<pre><code>/* 이중 연결 리스트 노드 */\nclass ListNode {\n    var val: Int // 노드 값\n    var next: ListNode? // 다음 노드 참조\n    weak var prev: ListNode? // 이전 노드 참조\n\n    init(val: Int) {\n        self.val = val\n    }\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n    private var front: ListNode? // 머리 노드 front\n    private var rear: ListNode? // 꼬리 노드 rear\n    private var _size: Int // 덱의 길이\n\n    init() {\n        _size = 0\n    }\n\n    /* 덱의 길이 반환 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* 큐 삽입 연산 */\n    private func push(num: Int, isFront: Bool) {\n        let node = ListNode(val: num)\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if isEmpty() {\n            front = node\n            rear = node\n        }\n        // 맨 앞에 삽입하는 연산\n        else if isFront {\n            // node를 연결 리스트의 맨 앞에 추가한다\n            front?.prev = node\n            node.next = front\n            front = node // 머리 노드를 갱신한다\n        }\n        // 맨 뒤에 삽입하는 연산\n        else {\n            // node를 연결 리스트의 맨 뒤에 추가한다\n            rear?.next = node\n            node.prev = rear\n            rear = node // 꼬리 노드를 갱신한다\n        }\n        _size += 1 // 큐 길이를 갱신한다\n    }\n\n    /* 맨 앞에 삽입 */\n    func pushFirst(num: Int) {\n        push(num: num, isFront: true)\n    }\n\n    /* 맨 뒤에 삽입 */\n    func pushLast(num: Int) {\n        push(num: num, isFront: false)\n    }\n\n    /* 큐에서 꺼내는 연산 */\n    private func pop(isFront: Bool) -&gt; Int {\n        if isEmpty() {\n            fatalError(\"덱이 비어 있다\")\n        }\n        let val: Int\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if _size == 1 {\n            val = front!.val\n            front = nil\n            rear = nil\n        }\n        // 맨 앞에서 꺼내는 연산\n        else if isFront {\n            val = front!.val // 머리 노드의 값을 임시로 저장한다\n            // 머리 노드를 삭제한다\n            let fNext = front?.next\n            fNext?.prev = nil\n            front?.next = nil\n            front = fNext // 머리 노드를 갱신한다\n        }\n        // 맨 뒤에서 꺼내는 연산\n        else {\n            val = rear!.val // 꼬리 노드의 값을 임시로 저장한다\n            // 꼬리 노드를 삭제한다\n            let rPrev = rear?.prev\n            rPrev?.next = nil\n            rear?.prev = nil\n            rear = rPrev // 꼬리 노드를 갱신한다\n        }\n        _size -= 1 // 큐 길이를 갱신한다\n        return val\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    func popFirst() -&gt; Int {\n        pop(isFront: true)\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    func popLast() -&gt; Int {\n        pop(isFront: false)\n    }\n\n    /* 맨 앞 원소 접근 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"덱이 비어 있다\")\n        }\n        return front!.val\n    }\n\n    /* 맨 뒤 원소 접근 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"덱이 비어 있다\")\n        }\n        return rear!.val\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.js<pre><code>/* 이중 연결 리스트 노드 */\nclass ListNode {\n    prev; // 이전 노드 참조(포인터)\n    next; // 다음 노드 참조(포인터)\n    val; // 노드 값\n\n    constructor(val) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n    #front; // 머리 노드 front\n    #rear; // 꼬리 노드 rear\n    #queSize; // 덱의 길이\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n        this.#queSize = 0;\n    }\n\n    /* 맨 뒤에 삽입하는 연산 */\n    pushLast(val) {\n        const node = new ListNode(val);\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // node를 연결 리스트의 맨 뒤에 추가한다\n            this.#rear.next = node;\n            node.prev = this.#rear;\n            this.#rear = node; // 꼬리 노드를 갱신한다\n        }\n        this.#queSize++;\n    }\n\n    /* 맨 앞에 삽입하는 연산 */\n    pushFirst(val) {\n        const node = new ListNode(val);\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // node를 연결 리스트의 맨 앞에 추가한다\n            this.#front.prev = node;\n            node.next = this.#front;\n            this.#front = node; // 머리 노드를 갱신한다\n        }\n        this.#queSize++;\n    }\n\n    /* 맨 뒤에서 꺼내는 연산 */\n    popLast() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#rear.val; // 꼬리 노드의 값을 저장한다\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (this.#queSize === 1) {\n            this.#front = null;\n            this.#rear = null;\n        }\n        // 꼬리 노드를 삭제한다\n        else {\n            let temp = this.#rear.prev;\n            temp.next = null;\n            this.#rear.prev = null;\n            this.#rear = temp; // 꼬리 노드를 갱신한다\n        }\n        this.#queSize--;\n        return value;\n    }\n\n    /* 맨 앞에서 꺼내는 연산 */\n    popFirst() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#front.val; // 머리 노드의 값을 저장한다\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (this.#queSize === 1) {\n            this.#front = null;\n            this.#rear = null;\n        }\n        // 머리 노드를 삭제한다\n        else {\n            let temp = this.#front.next;\n            temp.prev = null;\n            this.#front.next = null;\n            this.#front = temp; // 머리 노드를 갱신한다\n        }\n        this.#queSize--;\n        return value;\n    }\n\n    /* 맨 뒤 원소 접근 */\n    peekLast() {\n        return this.#queSize === 0 ? null : this.#rear.val;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peekFirst() {\n        return this.#queSize === 0 ? null : this.#front.val;\n    }\n\n    /* 덱의 길이 반환 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* 덱을 출력한다 */\n    print() {\n        const arr = [];\n        let temp = this.#front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.ts<pre><code>/* 이중 연결 리스트 노드 */\nclass ListNode {\n    prev: ListNode; // 이전 노드 참조(포인터)\n    next: ListNode; // 다음 노드 참조(포인터)\n    val: number; // 노드 값\n\n    constructor(val: number) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n    private front: ListNode; // 머리 노드 front\n    private rear: ListNode; // 꼬리 노드 rear\n    private queSize: number; // 덱의 길이\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n        this.queSize = 0;\n    }\n\n    /* 맨 뒤에 삽입하는 연산 */\n    pushLast(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // node를 연결 리스트의 맨 뒤에 추가한다\n            this.rear.next = node;\n            node.prev = this.rear;\n            this.rear = node; // 꼬리 노드를 갱신한다\n        }\n        this.queSize++;\n    }\n\n    /* 맨 앞에 삽입하는 연산 */\n    pushFirst(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // node를 연결 리스트의 맨 앞에 추가한다\n            this.front.prev = node;\n            node.next = this.front;\n            this.front = node; // 머리 노드를 갱신한다\n        }\n        this.queSize++;\n    }\n\n    /* 맨 뒤에서 꺼내는 연산 */\n    popLast(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.rear!.val; // 꼬리 노드의 값을 저장한다\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (this.queSize === 1) {\n            this.front = null;\n            this.rear = null;\n        }\n        // 꼬리 노드를 삭제한다\n        else {\n            let temp: ListNode = this.rear!.prev;\n            temp!.next = null;\n            this.rear!.prev = null;\n            this.rear = temp; // 꼬리 노드를 갱신한다\n        }\n        this.queSize--;\n        return value;\n    }\n\n    /* 맨 앞에서 꺼내는 연산 */\n    popFirst(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.front.val; // 머리 노드의 값을 저장한다\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (this.queSize === 1) {\n            this.front = null;\n            this.rear = null;\n        }\n        // 머리 노드를 삭제한다\n        else {\n            let temp: ListNode = this.front!.next;\n            temp!.prev = null;\n            this.front!.next = null;\n            this.front = temp; // 머리 노드를 갱신한다\n        }\n        this.queSize--;\n        return value;\n    }\n\n    /* 맨 뒤 원소 접근 */\n    peekLast(): number {\n        return this.queSize === 0 ? null : this.rear.val;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peekFirst(): number {\n        return this.queSize === 0 ? null : this.front.val;\n    }\n\n    /* 덱의 길이 반환 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* 덱을 출력한다 */\n    print(): void {\n        const arr: number[] = [];\n        let temp: ListNode = this.front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.dart<pre><code>/* 이중 연결 리스트 노드 */\nclass ListNode {\n  int val; // 노드 값\n  ListNode? next; // 다음 노드 참조\n  ListNode? prev; // 이전 노드 참조\n\n  ListNode(this.val, {this.next, this.prev});\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n  late ListNode? _front; // 머리 노드 _front\n  late ListNode? _rear; // 꼬리 노드 _rear\n  int _queSize = 0; // 덱의 길이\n\n  LinkedListDeque() {\n    this._front = null;\n    this._rear = null;\n  }\n\n  /* 덱의 길이 반환 */\n  int size() {\n    return this._queSize;\n  }\n\n  /* 덱이 비어 있는지 판단 */\n  bool isEmpty() {\n    return size() == 0;\n  }\n\n  /* 큐 삽입 연산 */\n  void push(int _num, bool isFront) {\n    final ListNode node = ListNode(_num);\n    if (isEmpty()) {\n      // 연결 리스트가 비어 있으면 _front와 _rear가 모두 node를 가리키게 한다\n      _front = _rear = node;\n    } else if (isFront) {\n      // 맨 앞에 삽입하는 연산\n      // node를 연결 리스트의 맨 앞에 추가한다\n      _front!.prev = node;\n      node.next = _front;\n      _front = node; // 머리 노드를 갱신한다\n    } else {\n      // 맨 뒤에 삽입하는 연산\n      // node를 연결 리스트의 맨 뒤에 추가한다\n      _rear!.next = node;\n      node.prev = _rear;\n      _rear = node; // 꼬리 노드를 갱신한다\n    }\n    _queSize++; // 큐 길이를 갱신한다\n  }\n\n  /* 맨 앞에 삽입 */\n  void pushFirst(int _num) {\n    push(_num, true);\n  }\n\n  /* 맨 뒤에 삽입 */\n  void pushLast(int _num) {\n    push(_num, false);\n  }\n\n  /* 큐에서 꺼내는 연산 */\n  int? pop(bool isFront) {\n    // 큐가 비어 있으면 바로 null을 반환한다\n    if (isEmpty()) {\n      return null;\n    }\n    final int val;\n    // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n    if (_queSize == 1) {\n      val = _front!.val;\n      _front = null;\n      _rear = null;\n    }\n    // 맨 앞에서 꺼내는 연산\n    else if (isFront) {\n      val = _front!.val; // 머리 노드의 값을 임시로 저장한다\n      // 머리 노드를 삭제한다\n      ListNode? fNext = _front!.next;\n      fNext!.prev = null;\n      _front!.next = null;\n      _front = fNext; // 머리 노드를 갱신한다\n    // 맨 뒤에서 꺼내는 연산\n    } else {\n      val = _rear!.val; // 꼬리 노드의 값을 임시로 저장한다\n      // 꼬리 노드를 삭제한다\n      ListNode? rPrev = _rear!.prev;\n      rPrev!.next = null;\n      _rear!.prev = null;\n      _rear = rPrev; // 꼬리 노드를 갱신한다\n    }\n    _queSize--; // 큐 길이를 갱신한다\n    return val;\n  }\n\n  /* 맨 앞에서 꺼내기 */\n  int? popFirst() {\n    return pop(true);\n  }\n\n  /* 맨 뒤에서 꺼내기 */\n  int? popLast() {\n    return pop(false);\n  }\n\n  /* 맨 앞 원소 접근 */\n  int? peekFirst() {\n    return _front?.val;\n  }\n\n  /* 맨 뒤 원소 접근 */\n  int? peekLast() {\n    return _rear?.val;\n  }\n\n  /* 출력에 사용할 배열 반환 */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; _queSize; i++) {\n      res.add(node!.val);\n      node = node.next;\n    }\n    return res;\n  }\n}\n</code></pre> linkedlist_deque.rs<pre><code>/* 이중 연결 리스트 노드 */\npub struct ListNode&lt;T&gt; {\n    pub val: T,                                 // 노드 값\n    pub next: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // 다음 노드 포인터\n    pub prev: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // 이전 노드 포인터\n}\n\nimpl&lt;T&gt; ListNode&lt;T&gt; {\n    pub fn new(val: T) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n        Rc::new(RefCell::new(ListNode {\n            val,\n            next: None,\n            prev: None,\n        }))\n    }\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\n#[allow(dead_code)]\npub struct LinkedListDeque&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // 머리 노드 front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // 꼬리 노드 rear\n    que_size: usize,                         // 덱의 길이\n}\n\nimpl&lt;T: Copy&gt; LinkedListDeque&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* 덱의 길이 반환 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.que_size == 0;\n    }\n\n    /* 큐 삽입 연산 */\n    fn push(&amp;mut self, num: T, is_front: bool) {\n        let node = ListNode::new(num);\n        // 맨 앞에 삽입하는 연산\n        if is_front {\n            match self.front.take() {\n                // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n                None =&gt; {\n                    self.rear = Some(node.clone());\n                    self.front = Some(node);\n                }\n                // node를 연결 리스트의 맨 앞에 추가한다\n                Some(old_front) =&gt; {\n                    old_front.borrow_mut().prev = Some(node.clone());\n                    node.borrow_mut().next = Some(old_front);\n                    self.front = Some(node); // 머리 노드를 갱신한다\n                }\n            }\n        }\n        // 맨 뒤에 삽입하는 연산\n        else {\n            match self.rear.take() {\n                // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n                None =&gt; {\n                    self.front = Some(node.clone());\n                    self.rear = Some(node);\n                }\n                // node를 연결 리스트의 맨 뒤에 추가한다\n                Some(old_rear) =&gt; {\n                    old_rear.borrow_mut().next = Some(node.clone());\n                    node.borrow_mut().prev = Some(old_rear);\n                    self.rear = Some(node); // 꼬리 노드를 갱신한다\n                }\n            }\n        }\n        self.que_size += 1; // 큐 길이를 갱신한다\n    }\n\n    /* 맨 앞에 삽입 */\n    pub fn push_first(&amp;mut self, num: T) {\n        self.push(num, true);\n    }\n\n    /* 맨 뒤에 삽입 */\n    pub fn push_last(&amp;mut self, num: T) {\n        self.push(num, false);\n    }\n\n    /* 큐에서 꺼내는 연산 */\n    fn pop(&amp;mut self, is_front: bool) -&gt; Option&lt;T&gt; {\n        // 큐가 비어 있으면 바로 None을 반환한다\n        if self.is_empty() {\n            return None;\n        };\n        // 맨 앞에서 꺼내는 연산\n        if is_front {\n            self.front.take().map(|old_front| {\n                match old_front.borrow_mut().next.take() {\n                    Some(new_front) =&gt; {\n                        new_front.borrow_mut().prev.take();\n                        self.front = Some(new_front); // 머리 노드를 갱신한다\n                    }\n                    None =&gt; {\n                        self.rear.take();\n                    }\n                }\n                self.que_size -= 1; // 큐 길이를 갱신한다\n                old_front.borrow().val\n            })\n        }\n        // 맨 뒤에서 꺼내는 연산\n        else {\n            self.rear.take().map(|old_rear| {\n                match old_rear.borrow_mut().prev.take() {\n                    Some(new_rear) =&gt; {\n                        new_rear.borrow_mut().next.take();\n                        self.rear = Some(new_rear); // 꼬리 노드를 갱신한다\n                    }\n                    None =&gt; {\n                        self.front.take();\n                    }\n                }\n                self.que_size -= 1; // 큐 길이를 갱신한다\n                old_rear.borrow().val\n            })\n        }\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    pub fn pop_first(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(true);\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    pub fn pop_last(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(false);\n    }\n\n    /* 맨 앞 원소 접근 */\n    pub fn peek_first(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* 맨 뒤 원소 접근 */\n    pub fn peek_last(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.rear.as_ref()\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        let mut res: Vec&lt;T&gt; = Vec::new();\n        fn recur&lt;T: Copy&gt;(cur: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, res: &amp;mut Vec&lt;T&gt;) {\n            if let Some(cur) = cur {\n                res.push(cur.borrow().val);\n                recur(cur.borrow().next.as_ref(), res);\n            }\n        }\n\n        recur(head, &amp;mut res);\n        res\n    }\n}\n</code></pre> linkedlist_deque.c<pre><code>/* 이중 연결 리스트 노드 */\ntypedef struct DoublyListNode {\n    int val;                     // 노드 값\n    struct DoublyListNode *next; // 다음 노드\n    struct DoublyListNode *prev; // 이전 노드\n} DoublyListNode;\n\n/* 생성자 */\nDoublyListNode *newDoublyListNode(int num) {\n    DoublyListNode *new = (DoublyListNode *)malloc(sizeof(DoublyListNode));\n    new-&gt;val = num;\n    new-&gt;next = NULL;\n    new-&gt;prev = NULL;\n    return new;\n}\n\n/* 소멸자 */\nvoid delDoublyListNode(DoublyListNode *node) {\n    free(node);\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\ntypedef struct {\n    DoublyListNode *front, *rear; // 머리 노드 front, 꼬리 노드 rear\n    int queSize;                  // 덱의 길이\n} LinkedListDeque;\n\n/* 생성자 */\nLinkedListDeque *newLinkedListDeque() {\n    LinkedListDeque *deque = (LinkedListDeque *)malloc(sizeof(LinkedListDeque));\n    deque-&gt;front = NULL;\n    deque-&gt;rear = NULL;\n    deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* 소멸자 */\nvoid delLinkedListdeque(LinkedListDeque *deque) {\n    // 모든 노드를 해제한다\n    for (int i = 0; i &lt; deque-&gt;queSize &amp;&amp; deque-&gt;front != NULL; i++) {\n        DoublyListNode *tmp = deque-&gt;front;\n        deque-&gt;front = deque-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // deque 구조체를 해제한다\n    free(deque);\n}\n\n/* 큐의 길이 반환 */\nint size(LinkedListDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* 큐가 비어 있는지 판단 */\nbool empty(LinkedListDeque *deque) {\n    return (size(deque) == 0);\n}\n\n/* 큐에 삽입 */\nvoid push(LinkedListDeque *deque, int num, bool isFront) {\n    DoublyListNode *node = newDoublyListNode(num);\n    // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n    if (empty(deque)) {\n        deque-&gt;front = deque-&gt;rear = node;\n    }\n    // 맨 앞에 삽입하는 연산\n    else if (isFront) {\n        // node를 연결 리스트의 맨 앞에 추가한다\n        deque-&gt;front-&gt;prev = node;\n        node-&gt;next = deque-&gt;front;\n        deque-&gt;front = node; // 머리 노드를 갱신한다\n    }\n    // 맨 뒤에 삽입하는 연산\n    else {\n        // node를 연결 리스트의 맨 뒤에 추가한다\n        deque-&gt;rear-&gt;next = node;\n        node-&gt;prev = deque-&gt;rear;\n        deque-&gt;rear = node;\n    }\n    deque-&gt;queSize++; // 큐 길이를 갱신한다\n}\n\n/* 맨 앞에 삽입 */\nvoid pushFirst(LinkedListDeque *deque, int num) {\n    push(deque, num, true);\n}\n\n/* 맨 뒤에 삽입 */\nvoid pushLast(LinkedListDeque *deque, int num) {\n    push(deque, num, false);\n}\n\n/* 맨 앞 원소 접근 */\nint peekFirst(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;front);\n    return deque-&gt;front-&gt;val;\n}\n\n/* 맨 뒤 원소 접근 */\nint peekLast(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;rear);\n    return deque-&gt;rear-&gt;val;\n}\n\n/* 큐에서 꺼내기 */\nint pop(LinkedListDeque *deque, bool isFront) {\n    if (empty(deque))\n        return -1;\n    int val;\n    // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n    if (deque-&gt;queSize == 1) {\n        val = deque-&gt;front-&gt;val;\n        delDoublyListNode(deque-&gt;front); // 같은 노드를 가리키므로 하나만 해제하면 된다\n        deque-&gt;front = NULL;\n        deque-&gt;rear = NULL;\n    }\n    // 맨 앞에서 꺼내는 연산\n    else if (isFront) {\n        val = peekFirst(deque); // 머리 노드의 값을 임시로 저장한다\n        DoublyListNode *fNext = deque-&gt;front-&gt;next;\n        fNext-&gt;prev = NULL;\n        deque-&gt;front-&gt;next = NULL;\n        delDoublyListNode(deque-&gt;front);\n        deque-&gt;front = fNext; // 머리 노드를 갱신한다\n    }\n    // 맨 뒤에서 꺼내는 연산\n    else {\n        val = peekLast(deque); // 꼬리 노드의 값을 임시로 저장한다\n        DoublyListNode *rPrev = deque-&gt;rear-&gt;prev;\n        rPrev-&gt;next = NULL;\n        deque-&gt;rear-&gt;prev = NULL;\n        delDoublyListNode(deque-&gt;rear);\n        deque-&gt;rear = rPrev; // 꼬리 노드를 갱신한다\n    }\n    deque-&gt;queSize--; // 큐 길이를 갱신한다\n    return val;\n}\n\n/* 맨 앞에서 꺼내기 */\nint popFirst(LinkedListDeque *deque) {\n    return pop(deque, true);\n}\n\n/* 맨 뒤에서 꺼내기 */\nint popLast(LinkedListDeque *deque) {\n    return pop(deque, false);\n}\n\n/* 큐 출력 */\nvoid printLinkedListDeque(LinkedListDeque *deque) {\n    int *arr = malloc(sizeof(int) * deque-&gt;queSize);\n    // 연결 리스트의 데이터를 배열에 복사한다\n    int i;\n    DoublyListNode *node;\n    for (i = 0, node = deque-&gt;front; i &lt; deque-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, deque-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_deque.kt<pre><code>/* 이중 연결 리스트 노드 */\nclass ListNode(var _val: Int) {\n    // 노드 값\n    var next: ListNode? = null // 다음 노드 참조\n    var prev: ListNode? = null // 이전 노드 참조\n}\n\n/* 이중 연결 리스트로 구현한 덱 */\nclass LinkedListDeque {\n    private var front: ListNode? = null // 머리 노드 front\n    private var rear: ListNode? = null // 꼬리 노드 rear\n    private var queSize: Int = 0 // 덱의 길이\n\n    /* 덱의 길이 반환 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* 큐 삽입 연산 */\n    fun push(num: Int, isFront: Boolean) {\n        val node = ListNode(num)\n        // 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n        if (isEmpty()) {\n            rear = node\n            front = rear\n            // 맨 앞에 삽입하는 연산\n        } else if (isFront) {\n            // node를 연결 리스트의 맨 앞에 추가한다\n            front?.prev = node\n            node.next = front\n            front = node // 머리 노드를 갱신한다\n            // 맨 뒤에 삽입하는 연산\n        } else {\n            // node를 연결 리스트의 맨 뒤에 추가한다\n            rear?.next = node\n            node.prev = rear\n            rear = node // 꼬리 노드를 갱신한다\n        }\n        queSize++ // 큐 길이를 갱신한다\n    }\n\n    /* 맨 앞에 삽입 */\n    fun pushFirst(num: Int) {\n        push(num, true)\n    }\n\n    /* 맨 뒤에 삽입 */\n    fun pushLast(num: Int) {\n        push(num, false)\n    }\n\n    /* 큐에서 꺼내는 연산 */\n    fun pop(isFront: Boolean): Int {\n        if (isEmpty()) \n            throw IndexOutOfBoundsException()\n        val _val: Int\n        // 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n        if (queSize == 1) {\n            _val = front!!._val\n            front = null\n            rear = null\n        }\n        // 맨 앞에서 꺼내는 연산\n        else if (isFront) {\n            _val = front!!._val // 머리 노드의 값을 임시로 저장한다\n            // 머리 노드를 삭제한다\n            val fNext = front!!.next\n            fNext!!.prev = null\n            front!!.next = null\n            front = fNext // 머리 노드를 갱신한다\n            // 맨 뒤에서 꺼내는 연산\n        } else {\n            _val = rear!!._val // 꼬리 노드의 값을 임시로 저장한다\n            // 꼬리 노드를 삭제한다\n            val rPrev = rear!!.prev\n            rPrev!!.next = null\n            rear!!.prev = null\n            rear = rPrev // 꼬리 노드를 갱신한다\n        }\n        queSize-- // 큐 길이를 갱신한다\n        return _val\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    fun popFirst(): Int {\n        return pop(true)\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    fun popLast(): Int {\n        return pop(false)\n    }\n\n    /* 맨 앞 원소 접근 */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!._val\n    }\n\n    /* 맨 뒤 원소 접근 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return rear!!._val\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!._val\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.rb<pre><code>=begin\nFile: linkedlist_deque.rb\nCreated Time: 2024-04-06\nAuthor: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com)\n=end\n\n### 이중 연결 리스트 노드\nclass ListNode\n  attr_accessor :val\n  attr_accessor :next # 다음 노드 참조\n  attr_accessor :prev # 이전 노드 참조\n\n  ### 생성자 ###\n  def initialize(val)\n    @val = val\n  end\nend\n\n### 이중 연결 리스트로 구현한 덱 ###\nclass LinkedListDeque\n  ### 덱의 길이 반환 ###\n  attr_reader :size\n\n  ### 생성자 ###\n  def initialize\n    @front = nil  # 머리 노드 front\n    @rear = nil   # 꼬리 노드 rear\n    @size = 0     # 덱의 길이\n  end\n\n  ### 덱이 비어 있는지 판단 ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### 큐 삽입 연산 ###\n  def push(num, is_front)\n    node = ListNode.new(num)\n    # 연결 리스트가 비어 있으면 front와 rear가 모두 node를 가리키게 한다\n    if is_empty?\n      @front = @rear = node\n    # 맨 앞에 삽입하는 연산\n    elsif is_front\n      # node를 연결 리스트의 맨 앞에 추가한다\n      @front.prev = node\n      node.next = @front\n      @front = node # 머리 노드를 갱신한다\n    # 맨 뒤에 삽입하는 연산\n    else\n      # node를 연결 리스트의 맨 뒤에 추가한다\n      @rear.next = node\n      node.prev = @rear\n      @rear = node # 꼬리 노드를 갱신한다\n    end\n    @size += 1 # 큐 길이를 갱신한다\n  end\n\n  ### 맨 앞에 삽입 ###\n  def push_first(num)\n    push(num, true)\n  end\n\n  ### 맨 뒤에 삽입 ###\n  def push_last(num)\n    push(num, false)\n  end\n\n  ### 큐에서 꺼내는 연산 ###\n  def pop(is_front)\n    raise IndexError, '덱이 비어 있다' if is_empty?\n\n    # 큐의 길이가 1이면 머리와 꼬리가 같은 노드를 가리킨다\n    if @size == 1\n      val = @front.val\n      @front = nil\n      @rear = nil\n    # 맨 앞에서 꺼내는 연산\n    elsif is_front\n      val = @front.val # 머리 노드의 값을 임시로 저장한다\n      # 머리 노드를 삭제한다\n      fnext = @front.next\n      fnext.prev = nil\n      @front.next = nil\n      @front = fnext # 머리 노드를 갱신한다\n    # 맨 뒤에서 꺼내는 연산\n    else\n      val = @rear.val # 꼬리 노드의 값을 임시로 저장한다\n      # 꼬리 노드를 삭제한다\n      rprev = @rear.prev\n      rprev.next = nil\n      @rear.prev = nil\n      @rear = rprev # 꼬리 노드를 갱신한다\n    end\n    @size -= 1 # 큐 길이를 갱신한다\n\n    val\n  end\n\n  ### 맨 앞에서 꺼내기 ###\n  def pop_first\n    pop(true)\n  end\n\n  ### 맨 앞에서 꺼내기 ###\n  def pop_last\n    pop(false)\n  end\n\n  ### 맨 앞 원소 접근 ###\n  def peek_first\n    raise IndexError, '덱이 비어 있다' if is_empty?\n\n    @front.val\n  end\n\n  ### 맨 뒤 원소 접근 ###\n  def peek_last\n    raise IndexError, '덱이 비어 있다' if is_empty?\n\n    @rear.val\n  end\n\n  ### 출력에 사용할 배열 반환 ###\n  def to_array\n    node = @front\n    res = Array.new(size, 0)\n    for i in 0...size\n      res[i] = node.val\n      node = node.next\n    end\n    res\n  end\nend\n</code></pre>","path":["제 5 장   스택과 큐","5.3   덱"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#2","level":3,"title":"2.   배열 기반 구현","text":"<p>아래 그림(5-9)과 같이 배열 기반 큐와 마찬가지로 원형 배열을 사용하여 덱을 구현할 수 있다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 5-9   배열 기반 덱의 삽입과 꺼내기 </p> <p>큐 구현에 ‘맨 앞에 삽입하기’와 ‘맨 뒤에서 꺼내기’ 메서드만 추가하면 된다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_deque.py<pre><code>class ArrayDeque:\n    \"\"\"원형 배열로 구현한 덱\"\"\"\n\n    def __init__(self, capacity: int):\n        \"\"\"생성자\"\"\"\n        self._nums: list[int] = [0] * capacity\n        self._front: int = 0\n        self._size: int = 0\n\n    def capacity(self) -&gt; int:\n        \"\"\"덱의 용량 반환\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"덱의 길이 반환\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"덱이 비어 있는지 판단\"\"\"\n        return self._size == 0\n\n    def index(self, i: int) -&gt; int:\n        \"\"\"원형 배열 인덱스 계산\"\"\"\n        # 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        # i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        # i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        return (i + self.capacity()) % self.capacity()\n\n    def push_first(self, num: int):\n        \"\"\"맨 앞에 삽입\"\"\"\n        if self._size == self.capacity():\n            print(\"덱이 가득 찼다\")\n            return\n        # 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        # 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        self._front = self.index(self._front - 1)\n        # num을 맨 앞에 추가한다\n        self._nums[self._front] = num\n        self._size += 1\n\n    def push_last(self, num: int):\n        \"\"\"맨 뒤에 삽입\"\"\"\n        if self._size == self.capacity():\n            print(\"덱이 가득 찼다\")\n            return\n        # 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        rear = self.index(self._front + self._size)\n        # num을 맨 뒤에 추가한다\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop_first(self) -&gt; int:\n        \"\"\"맨 앞에서 꺼내기\"\"\"\n        num = self.peek_first()\n        # 맨 앞 포인터를 뒤로 한 칸 이동한다\n        self._front = self.index(self._front + 1)\n        self._size -= 1\n        return num\n\n    def pop_last(self) -&gt; int:\n        \"\"\"맨 뒤에서 꺼내기\"\"\"\n        num = self.peek_last()\n        self._size -= 1\n        return num\n\n    def peek_first(self) -&gt; int:\n        \"\"\"맨 앞 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"덱이 비어 있다\")\n        return self._nums[self._front]\n\n    def peek_last(self) -&gt; int:\n        \"\"\"맨 뒤 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"덱이 비어 있다\")\n        # 마지막 원소의 인덱스를 계산한다\n        last = self.index(self._front + self._size - 1)\n        return self._nums[last]\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"출력에 사용할 배열 반환\"\"\"\n        # 유효한 길이 범위 안의 리스트 원소만 변환한다\n        res = []\n        for i in range(self._size):\n            res.append(self._nums[self.index(self._front + i)])\n        return res\n</code></pre> array_deque.cpp<pre><code>/* 원형 배열로 구현한 덱 */\nclass ArrayDeque {\n  private:\n    vector&lt;int&gt; nums; // 덱의 원소를 저장하는 배열\n    int front;        // 맨 앞 원소를 가리키는 맨 앞 포인터\n    int queSize;      // 덱 길이\n\n  public:\n    /* 생성자 */\n    ArrayDeque(int capacity) {\n        nums.resize(capacity);\n        front = queSize = 0;\n    }\n\n    /* 덱의 용량 반환 */\n    int capacity() {\n        return nums.size();\n    }\n\n    /* 덱의 길이 반환 */\n    int size() {\n        return queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    int index(int i) {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        return (i + capacity()) % capacity();\n    }\n\n    /* 맨 앞에 삽입 */\n    void pushFirst(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"덱이 가득 찼다\" &lt;&lt; endl;\n            return;\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        front = index(front - 1);\n        // num을 맨 앞에 추가한다\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* 맨 뒤에 삽입 */\n    void pushLast(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"덱이 가득 찼다\" &lt;&lt; endl;\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        int rear = index(front + queSize);\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    int popFirst() {\n        int num = peekFirst();\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"덱이 비어 있다\");\n        return nums[front];\n    }\n\n    /* 맨 뒤 원소 접근 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"덱이 비어 있다\");\n        // 마지막 원소의 인덱스를 계산한다\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    vector&lt;int&gt; toVector() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        vector&lt;int&gt; res(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n};\n</code></pre> array_deque.java<pre><code>/* 원형 배열로 구현한 덱 */\nclass ArrayDeque {\n    private int[] nums; // 덱의 원소를 저장하는 배열\n    private int front; // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private int queSize; // 덱 길이\n\n    /* 생성자 */\n    public ArrayDeque(int capacity) {\n        this.nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* 덱의 용량 반환 */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* 덱의 길이 반환 */\n    public int size() {\n        return queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    private int index(int i) {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        return (i + capacity()) % capacity();\n    }\n\n    /* 맨 앞에 삽입 */\n    public void pushFirst(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"덱이 가득 찼다\");\n            return;\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        front = index(front - 1);\n        // num을 맨 앞에 추가한다\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* 맨 뒤에 삽입 */\n    public void pushLast(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"덱이 가득 찼다\");\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        int rear = index(front + queSize);\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    public int popFirst() {\n        int num = peekFirst();\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    public int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* 맨 뒤 원소 접근 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        // 마지막 원소의 인덱스를 계산한다\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    public int[] toArray() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.cs<pre><code>/* 원형 배열로 구현한 덱 */\nclass ArrayDeque {\n    int[] nums;  // 덱의 원소를 저장하는 배열\n    int front;   // 맨 앞 원소를 가리키는 맨 앞 포인터\n    int queSize; // 덱 길이\n\n    /* 생성자 */\n    public ArrayDeque(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* 덱의 용량 반환 */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* 덱의 길이 반환 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    int Index(int i) {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        return (i + Capacity()) % Capacity();\n    }\n\n    /* 맨 앞에 삽입 */\n    public void PushFirst(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"덱이 가득 찼다\");\n            return;\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        front = Index(front - 1);\n        // num을 맨 앞에 추가한다\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* 맨 뒤에 삽입 */\n    public void PushLast(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"덱이 가득 찼다\");\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        int rear = Index(front + queSize);\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    public int PopFirst() {\n        int num = PeekFirst();\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        front = Index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    public int PopLast() {\n        int num = PeekLast();\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int PeekFirst() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        return nums[front];\n    }\n\n    /* 맨 뒤 원소 접근 */\n    public int PeekLast() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        // 마지막 원소의 인덱스를 계산한다\n        int last = Index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    public int[] ToArray() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[Index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.go<pre><code>/* 원형 배열로 구현한 덱 */\ntype arrayDeque struct {\n    nums        []int // 덱의 원소를 저장하는 배열\n    front       int   // 맨 앞 원소를 가리키는 맨 앞 포인터\n    queSize     int   // 덱 길이\n    queCapacity int   // 큐 용량(저장할 수 있는 최대 원소 개수)\n}\n\n/* 큐 초기화 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\n    return &amp;arrayDeque{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* 덱의 길이 반환 */\nfunc (q *arrayDeque) size() int {\n    return q.queSize\n}\n\n/* 덱이 비어 있는지 판단 */\nfunc (q *arrayDeque) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* 원형 배열 인덱스 계산 */\nfunc (q *arrayDeque) index(i int) int {\n    // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n    // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n    // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n    return (i + q.queCapacity) % q.queCapacity\n}\n\n/* 맨 앞에 삽입 */\nfunc (q *arrayDeque) pushFirst(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"덱이 가득 찼다\")\n        return\n    }\n    // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n    // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n    q.front = q.index(q.front - 1)\n    // num을 맨 앞에 추가한다\n    q.nums[q.front] = num\n    q.queSize++\n}\n\n/* 맨 뒤에 삽입 */\nfunc (q *arrayDeque) pushLast(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"덱이 가득 찼다\")\n        return\n    }\n    // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    rear := q.index(q.front + q.queSize)\n    // num을 맨 뒤에 추가한다\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* 맨 앞에서 꺼내기 */\nfunc (q *arrayDeque) popFirst() any {\n    num := q.peekFirst()\n    if num == nil {\n        return nil\n    }\n    // 맨 앞 포인터를 뒤로 한 칸 이동한다\n    q.front = q.index(q.front + 1)\n    q.queSize--\n    return num\n}\n\n/* 맨 뒤에서 꺼내기 */\nfunc (q *arrayDeque) popLast() any {\n    num := q.peekLast()\n    if num == nil {\n        return nil\n    }\n    q.queSize--\n    return num\n}\n\n/* 맨 앞 원소 접근 */\nfunc (q *arrayDeque) peekFirst() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* 맨 뒤 원소 접근 */\nfunc (q *arrayDeque) peekLast() any {\n    if q.isEmpty() {\n        return nil\n    }\n    // 마지막 원소의 인덱스를 계산한다\n    last := q.index(q.front + q.queSize - 1)\n    return q.nums[last]\n}\n\n/* 출력에 사용할 Slice 반환 */\nfunc (q *arrayDeque) toSlice() []int {\n    // 유효한 길이 범위 안의 리스트 원소만 변환한다\n    res := make([]int, q.queSize)\n    for i, j := 0, q.front; i &lt; q.queSize; i++ {\n        res[i] = q.nums[q.index(j)]\n        j++\n    }\n    return res\n}\n</code></pre> array_deque.swift<pre><code>/* 원형 배열로 구현한 덱 */\nclass ArrayDeque {\n    private var nums: [Int] // 덱의 원소를 저장하는 배열\n    private var front: Int // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private var _size: Int // 덱 길이\n\n    /* 생성자 */\n    init(capacity: Int) {\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* 덱의 용량 반환 */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* 덱의 길이 반환 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    private func index(i: Int) -&gt; Int {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        (i + capacity()) % capacity()\n    }\n\n    /* 맨 앞에 삽입 */\n    func pushFirst(num: Int) {\n        if size() == capacity() {\n            print(\"덱이 가득 찼다\")\n            return\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        front = index(i: front - 1)\n        // num을 맨 앞에 추가한다\n        nums[front] = num\n        _size += 1\n    }\n\n    /* 맨 뒤에 삽입 */\n    func pushLast(num: Int) {\n        if size() == capacity() {\n            print(\"덱이 가득 찼다\")\n            return\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        let rear = index(i: front + size())\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    func popFirst() -&gt; Int {\n        let num = peekFirst()\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        front = index(i: front + 1)\n        _size -= 1\n        return num\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    func popLast() -&gt; Int {\n        let num = peekLast()\n        _size -= 1\n        return num\n    }\n\n    /* 맨 앞 원소 접근 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"덱이 비어 있다\")\n        }\n        return nums[front]\n    }\n\n    /* 맨 뒤 원소 접근 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"덱이 비어 있다\")\n        }\n        // 마지막 원소의 인덱스를 계산한다\n        let last = index(i: front + size() - 1)\n        return nums[last]\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    func toArray() -&gt; [Int] {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        (front ..&lt; front + size()).map { nums[index(i: $0)] }\n    }\n}\n</code></pre> array_deque.js<pre><code>/* 원형 배열로 구현한 덱 */\nclass ArrayDeque {\n    #nums; // 덱의 원소를 저장하는 배열\n    #front; // 맨 앞 원소를 가리키는 맨 앞 포인터\n    #queSize; // 덱 길이\n\n    /* 생성자 */\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n        this.#front = 0;\n        this.#queSize = 0;\n    }\n\n    /* 덱의 용량 반환 */\n    capacity() {\n        return this.#nums.length;\n    }\n\n    /* 덱의 길이 반환 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    index(i) {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* 맨 앞에 삽입 */\n    pushFirst(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('덱이 가득 찼다');\n            return;\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        this.#front = this.index(this.#front - 1);\n        // num을 맨 앞에 추가한다\n        this.#nums[this.#front] = num;\n        this.#queSize++;\n    }\n\n    /* 맨 뒤에 삽입 */\n    pushLast(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('덱이 가득 찼다');\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        const rear = this.index(this.#front + this.#queSize);\n        // num을 맨 뒤에 추가한다\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    popFirst() {\n        const num = this.peekFirst();\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        this.#front = this.index(this.#front + 1);\n        this.#queSize--;\n        return num;\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    popLast() {\n        const num = this.peekLast();\n        this.#queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peekFirst() {\n        if (this.isEmpty()) throw new Error('덱이 비어 있다.');\n        return this.#nums[this.#front];\n    }\n\n    /* 맨 뒤 원소 접근 */\n    peekLast() {\n        if (this.isEmpty()) throw new Error('덱이 비어 있다.');\n        // 마지막 원소의 인덱스를 계산한다\n        const last = this.index(this.#front + this.#queSize - 1);\n        return this.#nums[last];\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    toArray() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        const res = [];\n        for (let i = 0, j = this.#front; i &lt; this.#queSize; i++, j++) {\n            res[i] = this.#nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.ts<pre><code>/* 원형 배열로 구현한 덱 */\nclass ArrayDeque {\n    private nums: number[]; // 덱의 원소를 저장하는 배열\n    private front: number; // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private queSize: number; // 덱 길이\n\n    /* 생성자 */\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = 0;\n        this.queSize = 0;\n    }\n\n    /* 덱의 용량 반환 */\n    capacity(): number {\n        return this.nums.length;\n    }\n\n    /* 덱의 길이 반환 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    index(i: number): number {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* 맨 앞에 삽입 */\n    pushFirst(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('덱이 가득 찼다');\n            return;\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        this.front = this.index(this.front - 1);\n        // num을 맨 앞에 추가한다\n        this.nums[this.front] = num;\n        this.queSize++;\n    }\n\n    /* 맨 뒤에 삽입 */\n    pushLast(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('덱이 가득 찼다');\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        const rear: number = this.index(this.front + this.queSize);\n        // num을 맨 뒤에 추가한다\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    popFirst(): number {\n        const num: number = this.peekFirst();\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        this.front = this.index(this.front + 1);\n        this.queSize--;\n        return num;\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    popLast(): number {\n        const num: number = this.peekLast();\n        this.queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peekFirst(): number {\n        if (this.isEmpty()) throw new Error('덱이 비어 있다.');\n        return this.nums[this.front];\n    }\n\n    /* 맨 뒤 원소 접근 */\n    peekLast(): number {\n        if (this.isEmpty()) throw new Error('덱이 비어 있다.');\n        // 마지막 원소의 인덱스를 계산한다\n        const last = this.index(this.front + this.queSize - 1);\n        return this.nums[last];\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    toArray(): number[] {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        const res: number[] = [];\n        for (let i = 0, j = this.front; i &lt; this.queSize; i++, j++) {\n            res[i] = this.nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.dart<pre><code>/* 원형 배열로 구현한 덱 */\nclass ArrayDeque {\n  late List&lt;int&gt; _nums; // 덱의 원소를 저장하는 배열\n  late int _front; // 맨 앞 원소를 가리키는 맨 앞 포인터\n  late int _queSize; // 덱 길이\n\n  /* 생성자 */\n  ArrayDeque(int capacity) {\n    this._nums = List.filled(capacity, 0);\n    this._front = this._queSize = 0;\n  }\n\n  /* 덱의 용량 반환 */\n  int capacity() {\n    return _nums.length;\n  }\n\n  /* 덱의 길이 반환 */\n  int size() {\n    return _queSize;\n  }\n\n  /* 덱이 비어 있는지 판단 */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* 원형 배열 인덱스 계산 */\n  int index(int i) {\n    // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n    // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n    // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n    return (i + capacity()) % capacity();\n  }\n\n  /* 맨 앞에 삽입 */\n  void pushFirst(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"덱이 가득 찼다\");\n    }\n    // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n    // 나머지 연산을 사용하여 _front가 배열의 시작을 넘어가면 끝으로 돌아가게 한다\n    _front = index(_front - 1);\n    // _num을 맨 앞에 추가한다\n    _nums[_front] = _num;\n    _queSize++;\n  }\n\n  /* 맨 뒤에 삽입 */\n  void pushLast(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"덱이 가득 찼다\");\n    }\n    // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    int rear = index(_front + _queSize);\n    // _num을 맨 뒤에 추가한다\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* 맨 앞에서 꺼내기 */\n  int popFirst() {\n    int _num = peekFirst();\n    // 맨 앞 포인터를 오른쪽으로 한 칸 이동한다\n    _front = index(_front + 1);\n    _queSize--;\n    return _num;\n  }\n\n  /* 맨 뒤에서 꺼내기 */\n  int popLast() {\n    int _num = peekLast();\n    _queSize--;\n    return _num;\n  }\n\n  /* 맨 앞 원소 접근 */\n  int peekFirst() {\n    if (isEmpty()) {\n      throw Exception(\"덱이 비어 있다\");\n    }\n    return _nums[_front];\n  }\n\n  /* 맨 뒤 원소 접근 */\n  int peekLast() {\n    if (isEmpty()) {\n      throw Exception(\"덱이 비어 있다\");\n    }\n    // 마지막 원소의 인덱스를 계산한다\n    int last = index(_front + _queSize - 1);\n    return _nums[last];\n  }\n\n  /* 출력에 사용할 배열 반환 */\n  List&lt;int&gt; toArray() {\n    // 유효한 길이 범위 안의 리스트 원소만 변환한다\n    List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[index(j)];\n    }\n    return res;\n  }\n}\n</code></pre> array_deque.rs<pre><code>/* 원형 배열로 구현한 덱 */\nstruct ArrayDeque&lt;T&gt; {\n    nums: Vec&lt;T&gt;,    // 덱의 원소를 저장하는 배열\n    front: usize,    // 맨 앞 원소를 가리키는 맨 앞 포인터\n    que_size: usize, // 덱 길이\n}\n\nimpl&lt;T: Copy + Default&gt; ArrayDeque&lt;T&gt; {\n    /* 생성자 */\n    pub fn new(capacity: usize) -&gt; Self {\n        Self {\n            nums: vec![T::default(); capacity],\n            front: 0,\n            que_size: 0,\n        }\n    }\n\n    /* 덱의 용량 반환 */\n    pub fn capacity(&amp;self) -&gt; usize {\n        self.nums.len()\n    }\n\n    /* 덱의 길이 반환 */\n    pub fn size(&amp;self) -&gt; usize {\n        self.que_size\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    fn index(&amp;self, i: i32) -&gt; usize {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        ((i + self.capacity() as i32) % self.capacity() as i32) as usize\n    }\n\n    /* 맨 앞에 삽입 */\n    pub fn push_first(&amp;mut self, num: T) {\n        if self.que_size == self.capacity() {\n            println!(\"덱이 가득 찼다\");\n            return;\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        self.front = self.index(self.front as i32 - 1);\n        // num을 맨 앞에 추가한다\n        self.nums[self.front] = num;\n        self.que_size += 1;\n    }\n\n    /* 맨 뒤에 삽입 */\n    pub fn push_last(&amp;mut self, num: T) {\n        if self.que_size == self.capacity() {\n            println!(\"덱이 가득 찼다\");\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        let rear = self.index(self.front as i32 + self.que_size as i32);\n        // num을 맨 뒤에 추가한다\n        self.nums[rear] = num;\n        self.que_size += 1;\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    fn pop_first(&amp;mut self) -&gt; T {\n        let num = self.peek_first();\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        self.front = self.index(self.front as i32 + 1);\n        self.que_size -= 1;\n        num\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    fn pop_last(&amp;mut self) -&gt; T {\n        let num = self.peek_last();\n        self.que_size -= 1;\n        num\n    }\n\n    /* 맨 앞 원소 접근 */\n    fn peek_first(&amp;self) -&gt; T {\n        if self.is_empty() {\n            panic!(\"덱이 비어 있다\")\n        };\n        self.nums[self.front]\n    }\n\n    /* 맨 뒤 원소 접근 */\n    fn peek_last(&amp;self) -&gt; T {\n        if self.is_empty() {\n            panic!(\"덱이 비어 있다\")\n        };\n        // 마지막 원소의 인덱스를 계산한다\n        let last = self.index(self.front as i32 + self.que_size as i32 - 1);\n        self.nums[last]\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    fn to_array(&amp;self) -&gt; Vec&lt;T&gt; {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        let mut res = vec![T::default(); self.que_size];\n        let mut j = self.front;\n        for i in 0..self.que_size {\n            res[i] = self.nums[self.index(j as i32)];\n            j += 1;\n        }\n        res\n    }\n}\n</code></pre> array_deque.c<pre><code>/* 원형 배열로 구현한 덱 */\ntypedef struct {\n    int *nums;       // 큐의 원소를 저장하는 배열\n    int front;       // 맨 앞 원소를 가리키는 맨 앞 포인터\n    int queSize;     // 맨 뒤 포인터, 큐의 맨 뒤 + 1을 가리킨다\n    int queCapacity; // 큐 용량\n} ArrayDeque;\n\n/* 생성자 */\nArrayDeque *newArrayDeque(int capacity) {\n    ArrayDeque *deque = (ArrayDeque *)malloc(sizeof(ArrayDeque));\n    // 배열 초기화\n    deque-&gt;queCapacity = capacity;\n    deque-&gt;nums = (int *)malloc(sizeof(int) * deque-&gt;queCapacity);\n    deque-&gt;front = deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* 소멸자 */\nvoid delArrayDeque(ArrayDeque *deque) {\n    free(deque-&gt;nums);\n    free(deque);\n}\n\n/* 덱의 용량 반환 */\nint capacity(ArrayDeque *deque) {\n    return deque-&gt;queCapacity;\n}\n\n/* 덱의 길이 반환 */\nint size(ArrayDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* 덱이 비어 있는지 판단 */\nbool empty(ArrayDeque *deque) {\n    return deque-&gt;queSize == 0;\n}\n\n/* 원형 배열 인덱스 계산 */\nint dequeIndex(ArrayDeque *deque, int i) {\n    // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n    // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n    // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n    return ((i + capacity(deque)) % capacity(deque));\n}\n\n/* 맨 앞에 삽입 */\nvoid pushFirst(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"덱이 가득 찼다\\r\\n\");\n        return;\n    }\n    // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n    // 나머지 연산으로 front가 배열의 처음을 넘어가면 끝으로 돌아가게 한다\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front - 1);\n    // num을 큐의 맨 앞에 추가한다\n    deque-&gt;nums[deque-&gt;front] = num;\n    deque-&gt;queSize++;\n}\n\n/* 맨 뒤에 삽입 */\nvoid pushLast(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"덱이 가득 찼다\\r\\n\");\n        return;\n    }\n    // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    int rear = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize);\n    // num을 맨 뒤에 추가한다\n    deque-&gt;nums[rear] = num;\n    deque-&gt;queSize++;\n}\n\n/* 맨 앞 원소 접근 */\nint peekFirst(ArrayDeque *deque) {\n    // 접근 예외: 덱이 비어 있다\n    assert(empty(deque) == 0);\n    return deque-&gt;nums[deque-&gt;front];\n}\n\n/* 맨 뒤 원소 접근 */\nint peekLast(ArrayDeque *deque) {\n    // 접근 예외: 덱이 비어 있다\n    assert(empty(deque) == 0);\n    int last = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize - 1);\n    return deque-&gt;nums[last];\n}\n\n/* 맨 앞에서 꺼내기 */\nint popFirst(ArrayDeque *deque) {\n    int num = peekFirst(deque);\n    // 맨 앞 포인터를 뒤로 한 칸 이동한다\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front + 1);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* 맨 뒤에서 꺼내기 */\nint popLast(ArrayDeque *deque) {\n    int num = peekLast(deque);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* 출력에 사용할 배열 반환 */\nint *toArray(ArrayDeque *deque, int *queSize) {\n    *queSize = deque-&gt;queSize;\n    int *res = (int *)calloc(deque-&gt;queSize, sizeof(int));\n    int j = deque-&gt;front;\n    for (int i = 0; i &lt; deque-&gt;queSize; i++) {\n        res[i] = deque-&gt;nums[j % deque-&gt;queCapacity];\n        j++;\n    }\n    return res;\n}\n</code></pre> array_deque.kt<pre><code>/* 생성자 */\nclass ArrayDeque(capacity: Int) {\n    private var nums: IntArray = IntArray(capacity) // 덱의 원소를 저장하는 배열\n    private var front: Int = 0 // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private var queSize: Int = 0 // 덱 길이\n\n    /* 덱의 용량 반환 */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* 덱의 길이 반환 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* 덱이 비어 있는지 판단 */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* 원형 배열 인덱스 계산 */\n    private fun index(i: Int): Int {\n        // 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n        // i가 배열의 끝을 넘어가면 처음으로 돌아온다\n        // i가 배열의 처음을 넘어가면 끝으로 돌아온다\n        return (i + capacity()) % capacity()\n    }\n\n    /* 맨 앞에 삽입 */\n    fun pushFirst(num: Int) {\n        if (queSize == capacity()) {\n            println(\"덱이 가득 찼다\")\n            return\n        }\n        // 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n        // 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n        front = index(front - 1)\n        // num을 맨 앞에 추가한다\n        nums[front] = num\n        queSize++\n    }\n\n    /* 맨 뒤에 삽입 */\n    fun pushLast(num: Int) {\n        if (queSize == capacity()) {\n            println(\"덱이 가득 찼다\")\n            return\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        val rear = index(front + queSize)\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num\n        queSize++\n    }\n\n    /* 맨 앞에서 꺼내기 */\n    fun popFirst(): Int {\n        val num = peekFirst()\n        // 맨 앞 포인터를 뒤로 한 칸 이동한다\n        front = index(front + 1)\n        queSize--\n        return num\n    }\n\n    /* 맨 뒤에서 꺼내기 */\n    fun popLast(): Int {\n        val num = peekLast()\n        queSize--\n        return num\n    }\n\n    /* 맨 앞 원소 접근 */\n    fun peekFirst(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* 맨 뒤 원소 접근 */\n    fun peekLast(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        // 마지막 원소의 인덱스를 계산한다\n        val last = index(front + queSize - 1)\n        return nums[last]\n    }\n\n    /* 출력에 사용할 배열 반환 */\n    fun toArray(): IntArray {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[index(j)]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_deque.rb<pre><code>### 원형 배열로 구현한 덱 ###\nclass ArrayDeque\n  ### 덱의 길이 반환 ###\n  attr_reader :size\n\n  ### 생성자 ###\n  def initialize(capacity)\n    @nums = Array.new(capacity, 0)\n    @front = 0\n    @size = 0\n  end\n\n  ### 덱의 용량 반환 ###\n  def capacity\n    @nums.length\n  end\n\n  ### 덱이 비어 있는지 판단 ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### 맨 앞에 삽입 ###\n  def push_first(num)\n    if size == capacity\n      puts '덱이 가득 찼다'\n      return\n    end\n\n    # 맨 앞 포인터를 왼쪽으로 한 칸 이동한다\n    # 나머지 연산을 통해 front가 배열의 처음을 넘어가면 끝으로 돌아오게 한다\n    @front = index(@front - 1)\n    # num을 맨 앞에 추가한다\n    @nums[@front] = num\n    @size += 1\n  end\n\n  ### 맨 뒤에 삽입 ###\n  def push_last(num)\n    if size == capacity\n      puts '덱이 가득 찼다'\n      return\n    end\n\n    # 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    rear = index(@front + size)\n    # num을 맨 뒤에 추가한다\n    @nums[rear] = num\n    @size += 1\n  end\n\n  ### 맨 앞에서 꺼내기 ###\n  def pop_first\n    num = peek_first\n    # 맨 앞 포인터를 뒤로 한 칸 이동한다\n    @front = index(@front + 1)\n    @size -= 1\n    num\n  end\n\n  ### 맨 뒤에서 꺼내기 ###\n  def pop_last\n    num = peek_last\n    @size -= 1\n    num\n  end\n\n  ### 맨 앞 원소 접근 ###\n  def peek_first\n    raise IndexError, '덱이 비어 있다' if is_empty?\n\n    @nums[@front]\n  end\n\n  ### 맨 뒤 원소 접근 ###\n  def peek_last\n    raise IndexError, '덱이 비어 있다' if is_empty?\n\n    # 마지막 원소의 인덱스를 계산한다\n    last = index(@front + size - 1)\n    @nums[last]\n  end\n\n  ### 출력에 사용할 배열 반환 ###\n  def to_array\n    # 유효한 길이 범위 안의 리스트 원소만 변환한다\n    res = []\n    for i in 0...size\n      res &lt;&lt; @nums[index(@front + i)]\n    end\n    res\n  end\n\n  private\n\n  ### 원형 배열 인덱스 계산 ###\n  def index(i)\n    # 나머지 연산을 통해 배열의 처음과 끝을 연결한다\n    # i가 배열의 끝을 넘어가면 처음으로 돌아온다\n    # i가 배열의 처음을 넘어가면 끝으로 돌아온다\n    (i + capacity) % capacity\n  end\nend\n</code></pre>","path":["제 5 장   스택과 큐","5.3   덱"],"tags":[]},{"location":"chapter_stack_and_queue/deque/#533","level":2,"title":"5.3.3   덱의 활용","text":"<p>덱은 스택과 큐의 논리를 모두 갖추므로 둘의 모든 활용 상황에 적용할 수 있으며, 더 높은 자유도도 제공한다.</p> <p>소프트웨어의 ‘실행 취소’는 보통 스택으로 구현한다. 시스템이 각 변경 작업을 스택에 <code>push</code>하고, <code>pop</code>으로 취소한다. 다만 자원 제약 때문에 보통 취소 가능한 단계 수를 제한한다(예를 들어 \\(50\\)단계만 저장). 스택 길이가 \\(50\\)을 넘으면 바닥(덱의 맨 앞)에서 삭제해야 한다. 스택은 이 기능을 지원하지 않으므로 덱으로 대체해야 한다. ‘실행 취소’의 핵심 논리는 여전히 스택의 후입선출 원리를 따르며, 덱은 여기에 필요한 추가 논리를 더 유연하게 구현하게 해 준다.</p>","path":["제 5 장   스택과 큐","5.3   덱"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/","level":1,"title":"5.5   연습 문제","text":"","path":["제 5 장   스택과 큐","5.5   연습 문제"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#551","level":2,"title":"5.5.1   개념 확인","text":"","path":["제 5 장   스택과 큐","5.5   연습 문제"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#1","level":3,"title":"1.   스택과 큐에서 무엇이 먼저 나오는가","text":"<p>빈 스택 <code>S</code>와 빈 큐 <code>Q</code>를 준비하고, 각각에 같은 연산을 수행한다.</p> <p>1단계: <code>A</code>를 넣는다. 2단계: <code>B</code>를 넣는다. 3단계: 원소 하나를 꺼내 기록한다. 4단계: <code>C</code>를 넣는다. 5단계: 비어 있을 때까지 원소를 계속 꺼내 기록한다.</p> <p><code>S</code>와 <code>Q</code>에서 원소가 나오는 순서를 각각 적고, ‘후입선출’ 또는 ‘선입선출’로 차이를 설명하라.</p> 참고 답안 <p>스택 <code>S</code>에서 나오는 순서는 <code>B, C, A</code>다. <code>A, B</code>를 넣은 뒤 가장 최근에 넣은 <code>B</code>를 먼저 꺼내고 <code>C</code>를 넣는다. 이후 <code>C, A</code>를 차례대로 꺼내므로 ‘후입선출’의 특성을 보인다.</p> <p>큐 <code>Q</code>에서 나오는 순서는 <code>A, B, C</code>다. <code>A, B</code>를 넣은 뒤 가장 먼저 넣은 <code>A</code>를 꺼내고, <code>C</code>를 넣는다. 이후 <code>B, C</code>를 차례대로 꺼내므로 ‘선입선출’의 특성을 보인다.</p>","path":["제 5 장   스택과 큐","5.5   연습 문제"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#2","level":3,"title":"2.   큐의 뒤가 배열의 끝을 넘어가면 어떻게 되는가","text":"<p>길이가 5인 원형 배열로 큐를 구현하며, 배열의 인덱스는 <code>0～4</code>다. 현재 <code>front = 3</code>, <code>size = 2</code>이고 큐의 <code>A, B</code>는 각각 인덱스 3과 4에 있다.</p> <ol> <li>‘<code>C</code>를 큐에 넣기’를 수행하면 <code>C</code>를 어느 인덱스에 저장해야 하는가? 삽입 후 <code>size</code>는 얼마인가?</li> <li>이어서 큐에서 한 번 꺼내면 어떤 원소가 나오는가? 새로운 <code>front</code>와 <code>size</code>는 각각 얼마인가?</li> <li>이때 큐의 앞에서 뒤로 이어지는 논리적 순서는 무엇인가? 원소를 꺼낼 때 배열의 다른 원소도 이동해야 하는가? 이유를 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>새 원소의 위치는     <code>(front + size) % 5 = (3 + 2) % 5 = 0</code>이므로,     <code>C</code>는 인덱스 0에 저장된다. 삽입 후 <code>size = 3</code>이다.</p> </li> <li> <p>현재 큐의 맨 앞 원소인 <code>A</code>가 나온다. 새로운 맨 앞 인덱스는     <code>(3 + 1) % 5 = 4</code>이므로 <code>front = 4</code>, <code>size = 2</code>다.</p> </li> <li> <p>유효한 원소의 논리적 순서는 <code>B, C</code>이며, <code>B</code>는 인덱스 4에, <code>C</code>는 인덱스 0에 있다.     원소를 꺼낼 때는 <code>front</code>를 이동하고 <code>size</code>만 수정하면 된다. 원형 배열에서는 나머지 연산으로 인덱스가 처음으로 돌아가므로,     다른 원소를 모두 앞으로 옮길 필요가 없다.</p> </li> </ol>","path":["제 5 장   스택과 큐","5.5   연습 문제"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#3","level":3,"title":"3.   덱의 양 끝 연산","text":"<p>여기서는 <code>push_first</code>를 앞에 넣기, <code>push_last</code>를 뒤에 넣기, <code>pop_first</code>를 앞에서 꺼내기, <code>pop_last</code>를 뒤에서 꺼내기로 정의한다.</p> <p>빈 덱 <code>deq</code>에 다음 연산을 차례대로 수행한다.</p> <ol> <li><code>push_last(A)</code></li> <li><code>push_last(B)</code></li> <li><code>push_first(C)</code></li> <li><code>pop_last()</code></li> <li><code>push_last(D)</code></li> <li><code>pop_first()</code></li> </ol> <ol> <li>두 번 꺼낸 원소는 각각 무엇인가?</li> <li>모든 연산을 마친 뒤 앞에서 뒤까지 어떤 원소가 남는가?</li> <li>뒤에서만 넣고 앞에서만 삭제할 수 있는 큐로 이 6단계를 모두 수행할 수 있는가? 불가능하다면 수행할 수 없는 연산을 찾아라. 이어서 덱은 모두 수행할 수 있는지와 그 이유를 설명하라.</li> </ol> 참고 답안 <p>처음 세 단계를 마치면 덱의 앞에서 뒤까지는 <code>[C, A, B]</code>다.</p> <ol> <li> <p><code>pop_last()</code>로 <code>B</code>를 꺼낸다. <code>D</code>를 넣으면 <code>[C, A, D]</code>가 되고,     <code>pop_first()</code>로 <code>C</code>를 꺼낸다.</p> </li> <li> <p>마지막에는 <code>[A, D]</code>가 남는다.</p> </li> <li> <p>뒤에서만 넣고 앞에서만 삭제하는 큐로는 모든 연산을 수행할 수 없다.     3단계의 <code>push_first(C)</code>는 앞에 넣어야 하고, 4단계의 <code>pop_last()</code>는 뒤에서 삭제해야 하므로 이 큐가 지원하는 연산 범위를 벗어난다.     덱은 양 끝에서 모두 삽입과 삭제가 가능하므로 6단계를 전부 수행할 수 있다.</p> </li> </ol>","path":["제 5 장   스택과 큐","5.5   연습 문제"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#552","level":2,"title":"5.5.2   프로그래밍 연습","text":"","path":["제 5 장   스택과 큐","5.5   연습 문제"],"tags":[]},{"location":"chapter_stack_and_queue/exercises/#1_1","level":3,"title":"1.   괄호 문자열 검사하기","text":"<p>세 종류의 괄호 <code>()</code>, <code>[]</code>, <code>{}</code>만 포함하는 문자열 <code>s</code>가 주어진다. 스택을 사용하여 올바른 괄호 문자열인지 판단하라.</p> <p>올바른 문자열은 다음 두 조건을 모두 만족해야 한다. 닫는 괄호는 가장 최근에 등장한, 아직 짝이 없는 여는 괄호와 종류가 같아야 하며, 순회를 마친 뒤 짝이 없는 여는 괄호가 남아 있어서는 안 된다. 판단 결과를 불리언 값으로 반환하라.</p> 풀이 힌트 <ol> <li>‘닫는 괄호에서 대응하는 여는 괄호로의 매핑’을 만들 수 있다.</li> <li>닫는 괄호를 만나면 먼저 스택이 비었는지 확인한 뒤 맨 위 괄호가 짝이 맞는지 확인한다.</li> <li>순회가 끝난 뒤에도 스택이 비어 있어야 한다.</li> </ol> <p>LeetCode</p>","path":["제 5 장   스택과 큐","5.5   연습 문제"],"tags":[]},{"location":"chapter_stack_and_queue/queue/","level":1,"title":"5.2   큐","text":"<p>큐(queue)는 선입선출 규칙을 따르는 선형 자료 구조다. 이름 그대로 줄 서는 현상을 재현한다. 새로 온 사람은 계속 줄의 뒤에 서고, 맨 앞에 있는 사람부터 하나씩 떠난다.</p> <p>아래 그림(5-4)과 같이 큐의 앞쪽을 ‘큐의 맨 앞’, 뒤쪽을 ‘큐의 맨 뒤’라고 한다. 맨 뒤에 원소를 추가하는 연산을 ‘큐에 삽입하기’, 맨 앞 원소를 삭제하는 연산을 ‘큐에서 꺼내기’라고 한다.</p> <p></p> <p> 그림 5-4   큐의 선입선출 규칙 </p>","path":["제 5 장   스택과 큐","5.2   큐"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#521","level":2,"title":"5.2.1   큐의 기본 연산","text":"<p>큐의 기본 연산은 아래 표(5-2)와 같다. 메서드 이름은 프로그래밍 언어에 따라 다를 수 있으며, 여기서는 스택과 같은 이름을 사용한다.</p> <p> 표 5-2   큐의 연산 효율 </p> 메서드 설명 시간 복잡도 <code>push()</code> 큐에 원소 삽입, 즉 맨 뒤에 추가 \\(O(1)\\) <code>pop()</code> 큐의 맨 앞 원소를 꺼냄 \\(O(1)\\) <code>peek()</code> 큐의 맨 앞 원소에 접근 \\(O(1)\\) <p>프로그래밍 언어가 제공하는 큐 클래스를 바로 사용할 수 있다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby queue.py<pre><code>from collections import deque\n\n# 큐 초기화\n# Python에서는 보통 덱 클래스 deque를 큐로 사용한다\n# queue.Queue()는 전용 큐 클래스이지만 사용이 불편하므로 권하지 않는다\nque: deque[int] = deque()\n\n# 큐에 원소 삽입\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n\n# 큐의 맨 앞 원소에 접근\nfront: int = que[0]\n\n# 큐에서 원소 꺼내기\npop: int = que.popleft()\n\n# 큐 길이 구하기\nsize: int = len(que)\n\n# 큐가 비어 있는지 확인\nis_empty: bool = len(que) == 0\n</code></pre> queue.cpp<pre><code>/* 큐 초기화 */\nqueue&lt;int&gt; queue;\n\n/* 큐에 원소 삽입 */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* 큐의 맨 앞 원소에 접근 */\nint front = queue.front();\n\n/* 큐에서 원소 꺼내기 */\nqueue.pop();\n\n/* 큐 길이 구하기 */\nint size = queue.size();\n\n/* 큐가 비어 있는지 확인 */\nbool empty = queue.empty();\n</code></pre> queue.java<pre><code>/* 큐 초기화 */\nQueue&lt;Integer&gt; queue = new LinkedList&lt;&gt;();\n\n/* 큐에 원소 삽입 */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n\n/* 큐의 맨 앞 원소에 접근 */\nint peek = queue.peek();\n\n/* 큐에서 원소 꺼내기 */\nint pop = queue.poll();\n\n/* 큐 길이 구하기 */\nint size = queue.size();\n\n/* 큐가 비어 있는지 확인 */\nboolean isEmpty = queue.isEmpty();\n</code></pre> queue.cs<pre><code>/* 큐 초기화 */\nQueue&lt;int&gt; queue = new();\n\n/* 큐에 원소 삽입 */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n\n/* 큐의 맨 앞 원소에 접근 */\nint peek = queue.Peek();\n\n/* 큐에서 원소 꺼내기 */\nint pop = queue.Dequeue();\n\n/* 큐 길이 구하기 */\nint size = queue.Count;\n\n/* 큐가 비어 있는지 확인 */\nbool isEmpty = queue.Count == 0;\n</code></pre> queue_test.go<pre><code>/* 큐 초기화 */\n// Go에서는 list를 큐로 사용한다\nqueue := list.New()\n\n/* 큐에 원소 삽입 */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n\n/* 큐의 맨 앞 원소에 접근 */\npeek := queue.Front()\n\n/* 큐에서 원소 꺼내기 */\npop := queue.Front()\nqueue.Remove(pop)\n\n/* 큐 길이 구하기 */\nsize := queue.Len()\n\n/* 큐가 비어 있는지 확인 */\nisEmpty := queue.Len() == 0\n</code></pre> queue.swift<pre><code>/* 큐 초기화 */\n// Swift에는 내장 큐 클래스가 없으므로 Array를 큐로 사용할 수 있다\nvar queue: [Int] = []\n\n/* 큐에 원소 삽입 */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n\n/* 큐의 맨 앞 원소에 접근 */\nlet peek = queue.first!\n\n/* 큐에서 원소 꺼내기 */\n// 배열이므로 removeFirst의 복잡도는 O(n)이다\nlet pop = queue.removeFirst()\n\n/* 큐 길이 구하기 */\nlet size = queue.count\n\n/* 큐가 비어 있는지 확인 */\nlet isEmpty = queue.isEmpty\n</code></pre> queue.js<pre><code>/* 큐 초기화 */\n// JavaScript에는 내장 큐가 없으므로 Array를 큐로 사용할 수 있다\nconst queue = [];\n\n/* 큐에 원소 삽입 */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* 큐의 맨 앞 원소에 접근 */\nconst peek = queue[0];\n\n/* 큐에서 원소 꺼내기 */\n// 기반이 배열이므로 shift() 메서드의 시간 복잡도는 O(n)이다\nconst pop = queue.shift();\n\n/* 큐 길이 구하기 */\nconst size = queue.length;\n\n/* 큐가 비어 있는지 확인 */\nconst empty = queue.length === 0;\n</code></pre> queue.ts<pre><code>/* 큐 초기화 */\n// TypeScript에는 내장 큐가 없으므로 Array를 큐로 사용할 수 있다\nconst queue: number[] = [];\n\n/* 큐에 원소 삽입 */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* 큐의 맨 앞 원소에 접근 */\nconst peek = queue[0];\n\n/* 큐에서 원소 꺼내기 */\n// 기반이 배열이므로 shift() 메서드의 시간 복잡도는 O(n)이다\nconst pop = queue.shift();\n\n/* 큐 길이 구하기 */\nconst size = queue.length;\n\n/* 큐가 비어 있는지 확인 */\nconst empty = queue.length === 0;\n</code></pre> queue.dart<pre><code>/* 큐 초기화 */\n// Dart의 Queue 클래스는 덱이며 일반 큐로도 사용할 수 있다\nQueue&lt;int&gt; queue = Queue();\n\n/* 큐에 원소 삽입 */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n\n/* 큐의 맨 앞 원소에 접근 */\nint peek = queue.first;\n\n/* 큐에서 원소 꺼내기 */\nint pop = queue.removeFirst();\n\n/* 큐 길이 구하기 */\nint size = queue.length;\n\n/* 큐가 비어 있는지 확인 */\nbool isEmpty = queue.isEmpty;\n</code></pre> queue.rs<pre><code>/* 덱 초기화 */\n// Rust에서는 덱을 일반 큐로 사용한다\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* 큐에 원소 삽입 */\ndeque.push_back(1);\ndeque.push_back(3);\ndeque.push_back(2);\ndeque.push_back(5);\ndeque.push_back(4);\n\n/* 큐의 맨 앞 원소에 접근 */\nif let Some(front) = deque.front() {\n}\n\n/* 큐에서 원소 꺼내기 */\nif let Some(pop) = deque.pop_front() {\n}\n\n/* 큐 길이 구하기 */\nlet size = deque.len();\n\n/* 큐가 비어 있는지 확인 */\nlet is_empty = deque.is_empty();\n</code></pre> queue.c<pre><code>// C는 내장 큐를 제공하지 않는다\n</code></pre> queue.kt<pre><code>/* 큐 초기화 */\nval queue = LinkedList&lt;Int&gt;()\n\n/* 큐에 원소 삽입 */\nqueue.offer(1)\nqueue.offer(3)\nqueue.offer(2)\nqueue.offer(5)\nqueue.offer(4)\n\n/* 큐의 맨 앞 원소에 접근 */\nval peek = queue.peek()\n\n/* 큐에서 원소 꺼내기 */\nval pop = queue.poll()\n\n/* 큐 길이 구하기 */\nval size = queue.size\n\n/* 큐가 비어 있는지 확인 */\nval isEmpty = queue.isEmpty()\n</code></pre> queue.rb<pre><code># 큐 초기화\n# Ruby의 내장 큐(Thread::Queue)에는 peek와 순회 메서드가 없으므로 Array를 큐로 사용할 수 있다\nqueue = []\n\n# 큐에 원소 삽입\nqueue.push(1)\nqueue.push(3)\nqueue.push(2)\nqueue.push(5)\nqueue.push(4)\n\n# 큐 원소에 접근\npeek = queue.first\n\n# 큐에서 원소 꺼내기\n# 배열이므로 Array#shift 메서드의 시간 복잡도는 O(n)임에 주의\npop = queue.shift\n\n# 큐 길이 구하기\nsize = queue.length\n\n# 큐가 비어 있는지 확인\nis_empty = queue.empty?\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 5 장   스택과 큐","5.2   큐"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#522","level":2,"title":"5.2.2   큐 구현","text":"<p>큐를 구현하려면 한쪽 끝에서 원소를 추가하고 반대쪽 끝에서 삭제할 수 있는 자료 구조가 필요하다. 연결 리스트와 배열 모두 이 조건을 만족한다.</p>","path":["제 5 장   스택과 큐","5.2   큐"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#1","level":3,"title":"1.   연결 리스트 기반 구현","text":"<p>아래 그림(5-5)과 같이 연결 리스트의 ‘머리 노드’와 ‘꼬리 노드’를 각각 ‘큐의 맨 앞’과 ‘큐의 맨 뒤’로 볼 수 있다. 맨 뒤에서는 노드 추가만, 맨 앞에서는 노드 삭제만 허용한다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> 그림 5-5   연결 리스트 기반 큐의 삽입과 꺼내기 </p> <p>연결 리스트로 큐를 구현한 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linkedlist_queue.py<pre><code>class LinkedListQueue:\n    \"\"\"연결 리스트로 구현한 큐\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        self._front: ListNode | None = None  # 머리 노드 front\n        self._rear: ListNode | None = None  # 꼬리 노드 rear\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"큐의 길이 반환\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"큐가 비어 있는지 판단\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"큐에 삽입\"\"\"\n        # 꼬리 노드 뒤에 num을 추가한다\n        node = ListNode(num)\n        # 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if self._front is None:\n            self._front = node\n            self._rear = node\n        # 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        else:\n            self._rear.next = node\n            self._rear = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"큐에서 꺼내기\"\"\"\n        num = self.peek()\n        # 머리 노드를 삭제한다\n        self._front = self._front.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"맨 앞 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"큐가 비어 있다\")\n        return self._front.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"출력에 사용할 리스트로 변환\"\"\"\n        queue = []\n        temp = self._front\n        while temp:\n            queue.append(temp.val)\n            temp = temp.next\n        return queue\n</code></pre> linkedlist_queue.cpp<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue {\n  private:\n    ListNode *front, *rear; // 머리 노드 front, 꼬리 노드 rear\n    int queSize;\n\n  public:\n    LinkedListQueue() {\n        front = nullptr;\n        rear = nullptr;\n        queSize = 0;\n    }\n\n    ~LinkedListQueue() {\n        // 연결 리스트를 순회하며 노드를 삭제하여 메모리를 해제한다\n        freeMemoryLinkedList(front);\n    }\n\n    /* 큐의 길이 반환 */\n    int size() {\n        return queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* 큐에 삽입 */\n    void push(int num) {\n        // 꼬리 노드 뒤에 num을 추가한다\n        ListNode *node = new ListNode(num);\n        // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if (front == nullptr) {\n            front = node;\n            rear = node;\n        }\n        // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        else {\n            rear-&gt;next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    int pop() {\n        int num = peek();\n        // 머리 노드를 삭제한다\n        ListNode *tmp = front;\n        front = front-&gt;next;\n        // 메모리 해제\n        delete tmp;\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    int peek() {\n        if (size() == 0)\n            throw out_of_range(\"큐가 비어 있다\");\n        return front-&gt;val;\n    }\n\n    /* 연결 리스트를 Vector로 변환하여 반환 */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_queue.java<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue {\n    private ListNode front, rear; // 머리 노드 front, 꼬리 노드 rear\n    private int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* 큐의 길이 반환 */\n    public int size() {\n        return queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* 큐에 삽입 */\n    public void push(int num) {\n        // 꼬리 노드 뒤에 num을 추가한다\n        ListNode node = new ListNode(num);\n        // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if (front == null) {\n            front = node;\n            rear = node;\n        // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        } else {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    public int pop() {\n        int num = peek();\n        // 머리 노드를 삭제한다\n        front = front.next;\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.cs<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue {\n    ListNode? front, rear;  // 머리 노드 front, 꼬리 노드 rear \n    int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* 큐의 길이 반환 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* 큐에 삽입 */\n    public void Push(int num) {\n        // 꼬리 노드 뒤에 num을 추가한다\n        ListNode node = new(num);\n        // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if (front == null) {\n            front = node;\n            rear = node;\n            // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        } else if (rear != null) {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    public int Pop() {\n        int num = Peek();\n        // 머리 노드를 삭제한다\n        front = front?.next;\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return front!.val;\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    public int[] ToArray() {\n        if (front == null)\n            return [];\n\n        ListNode? node = front;\n        int[] res = new int[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.go<pre><code>/* 연결 리스트로 구현한 큐 */\ntype linkedListQueue struct {\n    // 내장 패키지 list로 큐를 구현한다\n    data *list.List\n}\n\n/* 큐 초기화 */\nfunc newLinkedListQueue() *linkedListQueue {\n    return &amp;linkedListQueue{\n        data: list.New(),\n    }\n}\n\n/* 큐에 삽입 */\nfunc (s *linkedListQueue) push(value any) {\n    s.data.PushBack(value)\n}\n\n/* 큐에서 꺼내기 */\nfunc (s *linkedListQueue) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* 맨 앞 원소 접근 */\nfunc (s *linkedListQueue) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* 큐의 길이 반환 */\nfunc (s *linkedListQueue) size() int {\n    return s.data.Len()\n}\n\n/* 큐가 비어 있는지 판단 */\nfunc (s *linkedListQueue) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* 출력에 사용할 List 반환 */\nfunc (s *linkedListQueue) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_queue.swift<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue {\n    private var front: ListNode? // 머리 노드\n    private var rear: ListNode? // 꼬리 노드\n    private var _size: Int\n\n    init() {\n        _size = 0\n    }\n\n    /* 큐의 길이 반환 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* 큐에 삽입 */\n    func push(num: Int) {\n        // 꼬리 노드 뒤에 num을 추가한다\n        let node = ListNode(x: num)\n        // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if front == nil {\n            front = node\n            rear = node\n        }\n        // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        else {\n            rear?.next = node\n            rear = node\n        }\n        _size += 1\n    }\n\n    /* 큐에서 꺼내기 */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // 머리 노드를 삭제한다\n        front = front?.next\n        _size -= 1\n        return num\n    }\n\n    /* 맨 앞 원소 접근 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"큐가 비어 있다\")\n        }\n        return front!.val\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.js<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue {\n    #front; // 머리 노드 #front\n    #rear; // 꼬리 노드 #rear\n    #queSize = 0;\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n    }\n\n    /* 큐의 길이 반환 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* 큐에 삽입 */\n    push(num) {\n        // 꼬리 노드 뒤에 num을 추가한다\n        const node = new ListNode(num);\n        // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if (!this.#front) {\n            this.#front = node;\n            this.#rear = node;\n            // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        } else {\n            this.#rear.next = node;\n            this.#rear = node;\n        }\n        this.#queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    pop() {\n        const num = this.peek();\n        // 머리 노드를 삭제한다\n        this.#front = this.#front.next;\n        this.#queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peek() {\n        if (this.size === 0) throw new Error('큐가 비어 있다');\n        return this.#front.val;\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    toArray() {\n        let node = this.#front;\n        const res = new Array(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.ts<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue {\n    private front: ListNode | null; // 머리 노드 front\n    private rear: ListNode | null; // 꼬리 노드 rear\n    private queSize: number = 0;\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n    }\n\n    /* 큐의 길이 반환 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* 큐에 삽입 */\n    push(num: number): void {\n        // 꼬리 노드 뒤에 num을 추가한다\n        const node = new ListNode(num);\n        // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if (!this.front) {\n            this.front = node;\n            this.rear = node;\n            // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        } else {\n            this.rear!.next = node;\n            this.rear = node;\n        }\n        this.queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    pop(): number {\n        const num = this.peek();\n        if (!this.front) throw new Error('큐가 비어 있다');\n        // 머리 노드를 삭제한다\n        this.front = this.front.next;\n        this.queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peek(): number {\n        if (this.size === 0) throw new Error('큐가 비어 있다');\n        return this.front!.val;\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    toArray(): number[] {\n        let node = this.front;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.dart<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue {\n  ListNode? _front; // 머리 노드 _front\n  ListNode? _rear; // 꼬리 노드 _rear\n  int _queSize = 0; // 큐 길이\n\n  LinkedListQueue() {\n    _front = null;\n    _rear = null;\n  }\n\n  /* 큐의 길이 반환 */\n  int size() {\n    return _queSize;\n  }\n\n  /* 큐가 비어 있는지 판단 */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* 큐에 삽입 */\n  void push(int _num) {\n    // 꼬리 노드 뒤에 _num을 추가한다\n    final node = ListNode(_num);\n    // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n    if (_front == null) {\n      _front = node;\n      _rear = node;\n    } else {\n      // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n      _rear!.next = node;\n      _rear = node;\n    }\n    _queSize++;\n  }\n\n  /* 큐에서 꺼내기 */\n  int pop() {\n    final int _num = peek();\n    // 머리 노드를 삭제한다\n    _front = _front!.next;\n    _queSize--;\n    return _num;\n  }\n\n  /* 맨 앞 원소 접근 */\n  int peek() {\n    if (_queSize == 0) {\n      throw Exception('큐가 비어 있다');\n    }\n    return _front!.val;\n  }\n\n  /* 연결 리스트를 Array로 변환하여 반환 */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; queue = [];\n    while (node != null) {\n      queue.add(node.val);\n      node = node.next;\n    }\n    return queue;\n  }\n}\n</code></pre> linkedlist_queue.rs<pre><code>/* 연결 리스트로 구현한 큐 */\n#[allow(dead_code)]\npub struct LinkedListQueue&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // 머리 노드 front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,  // 꼬리 노드 rear\n    que_size: usize,                         // 큐의 길이\n}\n\nimpl&lt;T: Copy&gt; LinkedListQueue&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0,\n        }\n    }\n\n    /* 큐의 길이 반환 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.que_size == 0;\n    }\n\n    /* 큐에 삽입 */\n    pub fn push(&amp;mut self, num: T) {\n        // 꼬리 노드 뒤에 num을 추가한다\n        let new_rear = ListNode::new(num);\n        match self.rear.take() {\n            // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n            Some(old_rear) =&gt; {\n                old_rear.borrow_mut().next = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n            // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n            None =&gt; {\n                self.front = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n        }\n        self.que_size += 1;\n    }\n\n    /* 큐에서 꺼내기 */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.front.take().map(|old_front| {\n            match old_front.borrow_mut().next.take() {\n                Some(new_front) =&gt; {\n                    self.front = Some(new_front);\n                }\n                None =&gt; {\n                    self.rear.take();\n                }\n            }\n            self.que_size -= 1;\n            old_front.borrow().val\n        })\n    }\n\n    /* 맨 앞 원소 접근 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        let mut res: Vec&lt;T&gt; = Vec::new();\n\n        fn recur&lt;T: Copy&gt;(cur: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, res: &amp;mut Vec&lt;T&gt;) {\n            if let Some(cur) = cur {\n                res.push(cur.borrow().val);\n                recur(cur.borrow().next.as_ref(), res);\n            }\n        }\n\n        recur(head, &amp;mut res);\n\n        res\n    }\n}\n</code></pre> linkedlist_queue.c<pre><code>/* 연결 리스트로 구현한 큐 */\ntypedef struct {\n    ListNode *front, *rear;\n    int queSize;\n} LinkedListQueue;\n\n/* 생성자 */\nLinkedListQueue *newLinkedListQueue() {\n    LinkedListQueue *queue = (LinkedListQueue *)malloc(sizeof(LinkedListQueue));\n    queue-&gt;front = NULL;\n    queue-&gt;rear = NULL;\n    queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* 소멸자 */\nvoid delLinkedListQueue(LinkedListQueue *queue) {\n    // 모든 노드를 해제한다\n    while (queue-&gt;front != NULL) {\n        ListNode *tmp = queue-&gt;front;\n        queue-&gt;front = queue-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // queue 구조체를 해제한다\n    free(queue);\n}\n\n/* 큐의 길이 반환 */\nint size(LinkedListQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* 큐가 비어 있는지 판단 */\nbool empty(LinkedListQueue *queue) {\n    return (size(queue) == 0);\n}\n\n/* 큐에 삽입 */\nvoid push(LinkedListQueue *queue, int num) {\n    // 꼬리 노드 위치에 node를 추가한다\n    ListNode *node = newListNode(num);\n    // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n    if (queue-&gt;front == NULL) {\n        queue-&gt;front = node;\n        queue-&gt;rear = node;\n    }\n    // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n    else {\n        queue-&gt;rear-&gt;next = node;\n        queue-&gt;rear = node;\n    }\n    queue-&gt;queSize++;\n}\n\n/* 맨 앞 원소 접근 */\nint peek(LinkedListQueue *queue) {\n    assert(size(queue) &amp;&amp; queue-&gt;front);\n    return queue-&gt;front-&gt;val;\n}\n\n/* 큐에서 꺼내기 */\nint pop(LinkedListQueue *queue) {\n    int num = peek(queue);\n    ListNode *tmp = queue-&gt;front;\n    queue-&gt;front = queue-&gt;front-&gt;next;\n    free(tmp);\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* 큐 출력 */\nvoid printLinkedListQueue(LinkedListQueue *queue) {\n    int *arr = malloc(sizeof(int) * queue-&gt;queSize);\n    // 연결 리스트의 데이터를 배열에 복사한다\n    int i;\n    ListNode *node;\n    for (i = 0, node = queue-&gt;front; i &lt; queue-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, queue-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_queue.kt<pre><code>/* 연결 리스트로 구현한 큐 */\nclass LinkedListQueue(\n    // 머리 노드 front, 꼬리 노드 rear\n    private var front: ListNode? = null,\n    private var rear: ListNode? = null,\n    private var queSize: Int = 0\n) {\n\n    /* 큐의 길이 반환 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* 큐에 삽입 */\n    fun push(num: Int) {\n        // 꼬리 노드 뒤에 num을 추가한다\n        val node = ListNode(num)\n        // 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n        if (front == null) {\n            front = node\n            rear = node\n            // 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n        } else {\n            rear?.next = node\n            rear = node\n        }\n        queSize++\n    }\n\n    /* 큐에서 꺼내기 */\n    fun pop(): Int {\n        val num = peek()\n        // 머리 노드를 삭제한다\n        front = front?.next\n        queSize--\n        return num\n    }\n\n    /* 맨 앞 원소 접근 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return front!!._val\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    fun toArray(): IntArray {\n        var node = front\n        val res = IntArray(size())\n        for (i in res.indices) {\n            res[i] = node!!._val\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.rb<pre><code>### 연결 리스트로 구현한 큐 ###\nclass LinkedListQueue\n  ### 큐의 길이 반환 ###\n  attr_reader :size\n\n  ### 생성자 ###\n  def initialize\n    @front = nil  # 머리 노드 front\n    @rear = nil   # 꼬리 노드 rear\n    @size = 0\n  end\n\n  ### 큐가 비어 있는지 판단 ###\n  def is_empty?\n    @front.nil?\n  end\n\n  ### 큐에 삽입 ###\n  def push(num)\n    # 꼬리 노드 뒤에 num을 추가한다\n    node = ListNode.new(num)\n\n    # 큐가 비어 있으면 머리와 꼬리 노드가 모두 해당 노드를 가리키게 한다\n    if @front.nil?\n      @front = node\n      @rear = node\n    # 큐가 비어 있지 않으면 해당 노드를 꼬리 노드 뒤에 추가한다\n    else\n      @rear.next = node\n      @rear = node\n    end\n\n    @size += 1\n  end\n\n  ### 큐에서 꺼내기 ###\n  def pop\n    num = peek\n    # 머리 노드를 삭제한다\n    @front = @front.next\n    @size -= 1\n    num\n  end\n\n  ### 맨 앞 원소 접근 ###\n  def peek\n    raise IndexError, '큐가 비어 있다' if is_empty?\n\n    @front.val\n  end\n\n  ### 연결 리스트를 Array로 변환하여 반환 ###\n  def to_array\n    queue = []\n    temp = @front\n    while temp\n      queue &lt;&lt; temp.val\n      temp = temp.next\n    end\n    queue\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 5 장   스택과 큐","5.2   큐"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#2","level":3,"title":"2.   배열 기반 구현","text":"<p>배열에서 첫 원소를 삭제하는 시간 복잡도는 \\(O(n)\\)이므로, 큐에서 원소를 꺼내는 효율이 낮아진다. 그러나 다음과 같은 방법으로 이 문제를 피할 수 있다.</p> <p>변수 <code>front</code>로 큐의 맨 앞 원소 인덱스를 가리키고, 변수 <code>size</code>로 큐의 길이를 기록한다. <code>rear = front + size</code>로 정의하면 <code>rear</code>는 큐의 맨 뒤 원소 바로 다음 위치를 가리킨다.</p> <p>이 설계에서는 배열에서 원소가 들어 있는 유효 구간이 <code>[front, rear - 1]</code>이며, 각 연산의 구현 방법은 아래 그림(5-6)과 같다.</p> <ul> <li>삽입: 인덱스 <code>rear</code>에 입력 원소를 대입하고 <code>size</code>를 1 늘린다.</li> <li>꺼내기: <code>front</code>를 1 늘리고 <code>size</code>를 1 줄이면 된다.</li> </ul> <p>두 연산 모두 한 번의 처리만 필요하므로 시간 복잡도는 \\(O(1)\\)이다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> 그림 5-6   배열 기반 큐의 삽입과 꺼내기 </p> <p>여기에는 한 가지 문제가 있다. 삽입과 꺼내기를 반복하면 <code>front</code>와 <code>rear</code>가 계속 오른쪽으로 이동하다가 배열의 끝에 도달하면 더 이상 움직일 수 없다. 이를 해결하려면 배열을 앞뒤가 이어진 ‘원형 배열’로 볼 수 있다.</p> <p>원형 배열에서는 <code>front</code>나 <code>rear</code>가 배열의 끝을 넘으면 곧바로 처음으로 돌아와 순회를 계속하게 해야 한다. 이 주기적 규칙은 ‘나머지 연산’으로 구현할 수 있다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_queue.py<pre><code>class ArrayQueue:\n    \"\"\"원형 배열로 구현한 큐\"\"\"\n\n    def __init__(self, size: int):\n        \"\"\"생성자\"\"\"\n        self._nums: list[int] = [0] * size  # 큐의 원소를 저장하는 배열\n        self._front: int = 0  # 맨 앞 원소를 가리키는 맨 앞 포인터\n        self._size: int = 0  # 큐 길이\n\n    def capacity(self) -&gt; int:\n        \"\"\"큐의 용량 반환\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"큐의 길이 반환\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"큐가 비어 있는지 판단\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"큐에 삽입\"\"\"\n        if self._size == self.capacity():\n            raise IndexError(\"큐가 가득 찼다\")\n        # 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        # 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        rear: int = (self._front + self._size) % self.capacity()\n        # num을 맨 뒤에 추가한다\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"큐에서 꺼내기\"\"\"\n        num: int = self.peek()\n        # 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        self._front = (self._front + 1) % self.capacity()\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"맨 앞 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"큐가 비어 있다\")\n        return self._nums[self._front]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"출력에 사용할 리스트 반환\"\"\"\n        res = [0] * self.size()\n        j: int = self._front\n        for i in range(self.size()):\n            res[i] = self._nums[(j % self.capacity())]\n            j += 1\n        return res\n</code></pre> array_queue.cpp<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue {\n  private:\n    int *nums;       // 큐의 원소를 저장하는 배열\n    int front;       // 맨 앞 원소를 가리키는 맨 앞 포인터\n    int queSize;     // 큐 길이\n    int queCapacity; // 큐 용량\n\n  public:\n    ArrayQueue(int capacity) {\n        // 배열 초기화\n        nums = new int[capacity];\n        queCapacity = capacity;\n        front = queSize = 0;\n    }\n\n    ~ArrayQueue() {\n        delete[] nums;\n    }\n\n    /* 큐의 용량 반환 */\n    int capacity() {\n        return queCapacity;\n    }\n\n    /* 큐의 길이 반환 */\n    int size() {\n        return queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* 큐에 삽입 */\n    void push(int num) {\n        if (queSize == queCapacity) {\n            cout &lt;&lt; \"큐가 가득 찼다\" &lt;&lt; endl;\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        int rear = (front + queSize) % queCapacity;\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    int pop() {\n        int num = peek();\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        front = (front + 1) % queCapacity;\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    int peek() {\n        if (isEmpty())\n            throw out_of_range(\"큐가 비어 있다\");\n        return nums[front];\n    }\n\n    /* 배열을 Vector로 변환하여 반환 */\n    vector&lt;int&gt; toVector() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        vector&lt;int&gt; arr(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            arr[i] = nums[j % queCapacity];\n        }\n        return arr;\n    }\n};\n</code></pre> array_queue.java<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue {\n    private int[] nums; // 큐의 원소를 저장하는 배열\n    private int front; // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private int queSize; // 큐 길이\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* 큐의 용량 반환 */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* 큐의 길이 반환 */\n    public int size() {\n        return queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* 큐에 삽입 */\n    public void push(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"큐가 가득 찼다\");\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        int rear = (front + queSize) % capacity();\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    public int pop() {\n        int num = peek();\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        front = (front + 1) % capacity();\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* 배열 반환 */\n    public int[] toArray() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.cs<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue {\n    int[] nums;  // 큐의 원소를 저장하는 배열\n    int front;   // 맨 앞 원소를 가리키는 맨 앞 포인터\n    int queSize; // 큐 길이\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* 큐의 용량 반환 */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* 큐의 길이 반환 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* 큐에 삽입 */\n    public void Push(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"큐가 가득 찼다\");\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        int rear = (front + queSize) % Capacity();\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    public int Pop() {\n        int num = Peek();\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        front = (front + 1) % Capacity();\n        queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return nums[front];\n    }\n\n    /* 배열 반환 */\n    public int[] ToArray() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % this.Capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.go<pre><code>/* 원형 배열로 구현한 큐 */\ntype arrayQueue struct {\n    nums        []int // 큐의 원소를 저장하는 배열\n    front       int   // 맨 앞 원소를 가리키는 맨 앞 포인터\n    queSize     int   // 큐 길이\n    queCapacity int   // 큐 용량(저장할 수 있는 최대 원소 개수)\n}\n\n/* 큐 초기화 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\n    return &amp;arrayQueue{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* 큐의 길이 반환 */\nfunc (q *arrayQueue) size() int {\n    return q.queSize\n}\n\n/* 큐가 비어 있는지 판단 */\nfunc (q *arrayQueue) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* 큐에 삽입 */\nfunc (q *arrayQueue) push(num int) {\n    // rear == queCapacity이면 큐가 가득 찼음을 나타낸다\n    if q.queSize == q.queCapacity {\n        return\n    }\n    // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n    rear := (q.front + q.queSize) % q.queCapacity\n    // num을 맨 뒤에 추가한다\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* 큐에서 꺼내기 */\nfunc (q *arrayQueue) pop() any {\n    num := q.peek()\n    if num == nil {\n        return nil\n    }\n\n    // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n    q.front = (q.front + 1) % q.queCapacity\n    q.queSize--\n    return num\n}\n\n/* 맨 앞 원소 접근 */\nfunc (q *arrayQueue) peek() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* 출력에 사용할 Slice 반환 */\nfunc (q *arrayQueue) toSlice() []int {\n    rear := (q.front + q.queSize)\n    if rear &gt;= q.queCapacity {\n        rear %= q.queCapacity\n        return append(q.nums[q.front:], q.nums[:rear]...)\n    }\n    return q.nums[q.front:rear]\n}\n</code></pre> array_queue.swift<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue {\n    private var nums: [Int] // 큐의 원소를 저장하는 배열\n    private var front: Int // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private var _size: Int // 큐 길이\n\n    init(capacity: Int) {\n        // 배열 초기화\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        _size = 0\n    }\n\n    /* 큐의 용량 반환 */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* 큐의 길이 반환 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* 큐에 삽입 */\n    func push(num: Int) {\n        if size() == capacity() {\n            print(\"큐가 가득 찼다\")\n            return\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        let rear = (front + size()) % capacity()\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num\n        _size += 1\n    }\n\n    /* 큐에서 꺼내기 */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        front = (front + 1) % capacity()\n        _size -= 1\n        return num\n    }\n\n    /* 맨 앞 원소 접근 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"큐가 비어 있다\")\n        }\n        return nums[front]\n    }\n\n    /* 배열 반환 */\n    func toArray() -&gt; [Int] {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        (front ..&lt; front + size()).map { nums[$0 % capacity()] }\n    }\n}\n</code></pre> array_queue.js<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue {\n    #nums; // 큐의 원소를 저장하는 배열\n    #front = 0; // 맨 앞 원소를 가리키는 맨 앞 포인터\n    #queSize = 0; // 큐 길이\n\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n    }\n\n    /* 큐의 용량 반환 */\n    get capacity() {\n        return this.#nums.length;\n    }\n\n    /* 큐의 길이 반환 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* 큐에 삽입 */\n    push(num) {\n        if (this.size === this.capacity) {\n            console.log('큐가 가득 찼다');\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        const rear = (this.#front + this.size) % this.capacity;\n        // num을 맨 뒤에 추가한다\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    pop() {\n        const num = this.peek();\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        this.#front = (this.#front + 1) % this.capacity;\n        this.#queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peek() {\n        if (this.isEmpty()) throw new Error('큐가 비어 있다');\n        return this.#nums[this.#front];\n    }\n\n    /* Array를 반환한다 */\n    toArray() {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.#front; i &lt; this.size; i++, j++) {\n            arr[i] = this.#nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.ts<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue {\n    private nums: number[]; // 큐의 원소를 저장하는 배열\n    private front: number; // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private queSize: number; // 큐 길이\n\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = this.queSize = 0;\n    }\n\n    /* 큐의 용량 반환 */\n    get capacity(): number {\n        return this.nums.length;\n    }\n\n    /* 큐의 길이 반환 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* 큐에 삽입 */\n    push(num: number): void {\n        if (this.size === this.capacity) {\n            console.log('큐가 가득 찼다');\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        const rear = (this.front + this.queSize) % this.capacity;\n        // num을 맨 뒤에 추가한다\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* 큐에서 꺼내기 */\n    pop(): number {\n        const num = this.peek();\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        this.front = (this.front + 1) % this.capacity;\n        this.queSize--;\n        return num;\n    }\n\n    /* 맨 앞 원소 접근 */\n    peek(): number {\n        if (this.isEmpty()) throw new Error('큐가 비어 있다');\n        return this.nums[this.front];\n    }\n\n    /* Array를 반환한다 */\n    toArray(): number[] {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.front; i &lt; this.size; i++, j++) {\n            arr[i] = this.nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.dart<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue {\n  late List&lt;int&gt; _nums; // 큐의 원소를 저장하는 배열\n  late int _front; // 맨 앞 원소를 가리키는 맨 앞 포인터\n  late int _queSize; // 큐 길이\n\n  ArrayQueue(int capacity) {\n    _nums = List.filled(capacity, 0);\n    _front = _queSize = 0;\n  }\n\n  /* 큐의 용량 반환 */\n  int capaCity() {\n    return _nums.length;\n  }\n\n  /* 큐의 길이 반환 */\n  int size() {\n    return _queSize;\n  }\n\n  /* 큐가 비어 있는지 판단 */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* 큐에 삽입 */\n  void push(int _num) {\n    if (_queSize == capaCity()) {\n      throw Exception(\"큐가 가득 찼다\");\n    }\n    // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n    int rear = (_front + _queSize) % capaCity();\n    // _num을 맨 뒤에 추가한다\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* 큐에서 꺼내기 */\n  int pop() {\n    int _num = peek();\n    // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n    _front = (_front + 1) % capaCity();\n    _queSize--;\n    return _num;\n  }\n\n  /* 맨 앞 원소 접근 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"큐가 비어 있다\");\n    }\n    return _nums[_front];\n  }\n\n  /* Array를 반환한다 */\n  List&lt;int&gt; toArray() {\n    // 유효한 길이 범위 안의 리스트 원소만 변환한다\n    final List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[j % capaCity()];\n    }\n    return res;\n  }\n}\n</code></pre> array_queue.rs<pre><code>/* 원형 배열로 구현한 큐 */\nstruct ArrayQueue&lt;T&gt; {\n    nums: Vec&lt;T&gt;,      // 큐의 원소를 저장하는 배열\n    front: i32,        // 맨 앞 원소를 가리키는 맨 앞 포인터\n    que_size: i32,     // 큐 길이\n    que_capacity: i32, // 큐 용량\n}\n\nimpl&lt;T: Copy + Default&gt; ArrayQueue&lt;T&gt; {\n    /* 생성자 */\n    fn new(capacity: i32) -&gt; ArrayQueue&lt;T&gt; {\n        ArrayQueue {\n            nums: vec![T::default(); capacity as usize],\n            front: 0,\n            que_size: 0,\n            que_capacity: capacity,\n        }\n    }\n\n    /* 큐의 용량 반환 */\n    fn capacity(&amp;self) -&gt; i32 {\n        self.que_capacity\n    }\n\n    /* 큐의 길이 반환 */\n    fn size(&amp;self) -&gt; i32 {\n        self.que_size\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* 큐에 삽입 */\n    fn push(&amp;mut self, num: T) {\n        if self.que_size == self.capacity() {\n            println!(\"큐가 가득 찼다\");\n            return;\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        let rear = (self.front + self.que_size) % self.que_capacity;\n        // num을 맨 뒤에 추가한다\n        self.nums[rear as usize] = num;\n        self.que_size += 1;\n    }\n\n    /* 큐에서 꺼내기 */\n    fn pop(&amp;mut self) -&gt; T {\n        let num = self.peek();\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        self.front = (self.front + 1) % self.que_capacity;\n        self.que_size -= 1;\n        num\n    }\n\n    /* 맨 앞 원소 접근 */\n    fn peek(&amp;self) -&gt; T {\n        if self.is_empty() {\n            panic!(\"인덱스가 범위를 벗어났다\");\n        }\n        self.nums[self.front as usize]\n    }\n\n    /* 배열 반환 */\n    fn to_vector(&amp;self) -&gt; Vec&lt;T&gt; {\n        let cap = self.que_capacity;\n        let mut j = self.front;\n        let mut arr = vec![T::default(); cap as usize];\n        for i in 0..self.que_size {\n            arr[i as usize] = self.nums[(j % cap) as usize];\n            j += 1;\n        }\n        arr\n    }\n}\n</code></pre> array_queue.c<pre><code>/* 원형 배열로 구현한 큐 */\ntypedef struct {\n    int *nums;       // 큐의 원소를 저장하는 배열\n    int front;       // 맨 앞 원소를 가리키는 맨 앞 포인터\n    int queSize;     // 현재 큐의 원소 개수\n    int queCapacity; // 큐 용량\n} ArrayQueue;\n\n/* 생성자 */\nArrayQueue *newArrayQueue(int capacity) {\n    ArrayQueue *queue = (ArrayQueue *)malloc(sizeof(ArrayQueue));\n    // 배열 초기화\n    queue-&gt;queCapacity = capacity;\n    queue-&gt;nums = (int *)malloc(sizeof(int) * queue-&gt;queCapacity);\n    queue-&gt;front = queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* 소멸자 */\nvoid delArrayQueue(ArrayQueue *queue) {\n    free(queue-&gt;nums);\n    free(queue);\n}\n\n/* 큐의 용량 반환 */\nint capacity(ArrayQueue *queue) {\n    return queue-&gt;queCapacity;\n}\n\n/* 큐의 길이 반환 */\nint size(ArrayQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* 큐가 비어 있는지 판단 */\nbool empty(ArrayQueue *queue) {\n    return queue-&gt;queSize == 0;\n}\n\n/* 맨 앞 원소 접근 */\nint peek(ArrayQueue *queue) {\n    assert(size(queue) != 0);\n    return queue-&gt;nums[queue-&gt;front];\n}\n\n/* 큐에 삽입 */\nvoid push(ArrayQueue *queue, int num) {\n    if (size(queue) == capacity(queue)) {\n        printf(\"큐가 가득 찼다\\r\\n\");\n        return;\n    }\n    // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n    int rear = (queue-&gt;front + queue-&gt;queSize) % queue-&gt;queCapacity;\n    // num을 맨 뒤에 추가한다\n    queue-&gt;nums[rear] = num;\n    queue-&gt;queSize++;\n}\n\n/* 큐에서 꺼내기 */\nint pop(ArrayQueue *queue) {\n    int num = peek(queue);\n    // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n    queue-&gt;front = (queue-&gt;front + 1) % queue-&gt;queCapacity;\n    queue-&gt;queSize--;\n    return num;\n}\n\n/* 출력에 사용할 배열 반환 */\nint *toArray(ArrayQueue *queue, int *queSize) {\n    *queSize = queue-&gt;queSize;\n    int *res = (int *)calloc(queue-&gt;queSize, sizeof(int));\n    int j = queue-&gt;front;\n    for (int i = 0; i &lt; queue-&gt;queSize; i++) {\n        res[i] = queue-&gt;nums[j % queue-&gt;queCapacity];\n        j++;\n    }\n    return res;\n}\n</code></pre> array_queue.kt<pre><code>/* 원형 배열로 구현한 큐 */\nclass ArrayQueue(capacity: Int) {\n    private val nums: IntArray = IntArray(capacity) // 큐의 원소를 저장하는 배열\n    private var front: Int = 0 // 맨 앞 원소를 가리키는 맨 앞 포인터\n    private var queSize: Int = 0 // 큐 길이\n\n    /* 큐의 용량 반환 */\n    fun capacity(): Int {\n        return nums.size\n    }\n\n    /* 큐의 길이 반환 */\n    fun size(): Int {\n        return queSize\n    }\n\n    /* 큐가 비어 있는지 판단 */\n    fun isEmpty(): Boolean {\n        return queSize == 0\n    }\n\n    /* 큐에 삽입 */\n    fun push(num: Int) {\n        if (queSize == capacity()) {\n            println(\"큐가 가득 찼다\")\n            return\n        }\n        // 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n        // 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n        val rear = (front + queSize) % capacity()\n        // num을 맨 뒤에 추가한다\n        nums[rear] = num\n        queSize++\n    }\n\n    /* 큐에서 꺼내기 */\n    fun pop(): Int {\n        val num = peek()\n        // 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n        front = (front + 1) % capacity()\n        queSize--\n        return num\n    }\n\n    /* 맨 앞 원소 접근 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return nums[front]\n    }\n\n    /* 배열 반환 */\n    fun toArray(): IntArray {\n        // 유효한 길이 범위 안의 리스트 원소만 변환한다\n        val res = IntArray(queSize)\n        var i = 0\n        var j = front\n        while (i &lt; queSize) {\n            res[i] = nums[j % capacity()]\n            i++\n            j++\n        }\n        return res\n    }\n}\n</code></pre> array_queue.rb<pre><code>### 원형 배열로 구현한 큐 ###\nclass ArrayQueue\n  ### 큐의 길이 반환 ###\n  attr_reader :size\n\n  ### 생성자 ###\n  def initialize(size)\n    @nums = Array.new(size, 0) # 큐의 원소를 저장하는 배열\n    @front = 0 # 맨 앞 원소를 가리키는 맨 앞 포인터\n    @size = 0 # 큐 길이\n  end\n\n  ### 큐의 용량 반환 ###\n  def capacity\n    @nums.length\n  end\n\n  ### 큐가 비어 있는지 판단 ###\n  def is_empty?\n    size.zero?\n  end\n\n  ### 큐에 삽입 ###\n  def push(num)\n    raise IndexError, '큐가 가득 찼다' if size == capacity\n\n    # 맨 뒤 인덱스 + 1을 가리키는 맨 뒤 포인터를 계산한다\n    # 나머지 연산을 통해 rear가 배열의 끝을 넘어가면 처음으로 돌아오게 한다\n    rear = (@front + size) % capacity\n    # num을 맨 뒤에 추가한다\n    @nums[rear] = num\n    @size += 1\n  end\n\n  ### 큐에서 꺼내기 ###\n  def pop\n    num = peek\n    # 맨 앞 포인터를 뒤로 한 칸 이동하고, 끝을 넘어가면 배열의 처음으로 돌아온다\n    @front = (@front + 1) % capacity\n    @size -= 1\n    num\n  end\n\n  ### 맨 앞 원소 접근 ###\n  def peek\n    raise IndexError, '큐가 비어 있다' if is_empty?\n\n    @nums[@front]\n  end\n\n  ### 출력에 사용할 리스트 반환 ###\n  def to_array\n    res = Array.new(size, 0)\n    j = @front\n\n    for i in 0...size\n      res[i] = @nums[j % capacity]\n      j += 1\n    end\n\n    res\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>위 큐 구현에는 여전히 길이를 바꿀 수 없다는 한계가 있다. 하지만 배열을 동적 배열로 바꾸어 용량 확장 방식을 도입하면 쉽게 해결할 수 있다. 관심 있는 독자는 직접 구현해 보자.</p> <p>두 구현의 비교 결과는 스택과 같으므로 여기서는 반복해서 설명하지 않는다.</p>","path":["제 5 장   스택과 큐","5.2   큐"],"tags":[]},{"location":"chapter_stack_and_queue/queue/#523","level":2,"title":"5.2.3   큐의 대표적인 활용","text":"<ul> <li>타오바오 주문. 구매자가 주문하면 큐에 들어가고, 시스템은 순서대로 처리한다. 광군제 기간에는 짧은 시간에 엄청난 주문이 몰리므로 높은 동시성을 처리하는 일이 엔지니어들의 핵심 과제가 된다.</li> <li>여러 종류의 대기 작업. 프린터 작업 대기열, 식당 음식 제공 대기열 등 ‘먼저 온 순서’를 지켜야 하는 상황에서 큐는 처리 순서를 효과적으로 유지한다.</li> </ul>","path":["제 5 장   스택과 큐","5.2   큐"],"tags":[]},{"location":"chapter_stack_and_queue/stack/","level":1,"title":"5.1   스택","text":"<p>스택(stack)은 후입선출 원리를 따르는 선형 자료 구조다.</p> <p>스택은 탁자 위에 쌓인 접시에 비유할 수 있다. 한 번에 접시 하나만 옮길 수 있다면 맨 아래 접시를 꺼내려면 위의 접시들을 차례로 치워야 한다. 접시를 정수, 문자, 객체 등 여러 자료형의 원소로 바꾸면 스택이라는 자료 구조가 된다.</p> <p>아래 그림(5-1)과 같이 원소가 쌓인 맨 위를 ‘스택의 꼭대기’, 맨 아래를 ‘스택의 바닥’이라고 한다. 맨 위에 원소를 추가하는 연산을 ‘스택에 삽입하기’, 맨 위 원소를 삭제하는 연산을 ‘스택에서 꺼내기’라고 한다.</p> <p></p> <p> 그림 5-1   스택의 후입선출 규칙 </p>","path":["제 5 장   스택과 큐","5.1   스택"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#511","level":2,"title":"5.1.1   스택의 기본 연산","text":"<p>스택의 기본 연산은 아래 표(5-1)와 같다. 구체적인 메서드 이름은 프로그래밍 언어에 따라 다르며, 여기서는 흔히 쓰이는 <code>push()</code>, <code>pop()</code>, <code>peek()</code>를 예로 든다.</p> <p> 표 5-1   스택의 연산 효율 </p> 메서드 설명 시간 복잡도 <code>push()</code> 원소를 스택에 삽입(맨 위에 추가) \\(O(1)\\) <code>pop()</code> 스택 맨 위 원소를 꺼냄 \\(O(1)\\) <code>peek()</code> 스택 맨 위 원소에 접근 \\(O(1)\\) <p>보통 프로그래밍 언어에 내장된 스택 클래스를 바로 사용할 수 있다. 전용 스택 클래스가 없는 언어에서는 ‘배열’이나 ‘연결 리스트’를 스택으로 사용하면서, 프로그램 논리상 스택과 무관한 연산은 사용하지 않으면 된다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby stack.py<pre><code># 스택 초기화\n# Python에는 내장 스택 클래스가 없으므로 list를 스택으로 사용할 수 있다\nstack: list[int] = []\n\n# 스택에 원소 삽입\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n# 스택 맨 위 원소에 접근\npeek: int = stack[-1]\n\n# 스택에서 원소 꺼내기\npop: int = stack.pop()\n\n# 스택 길이 구하기\nsize: int = len(stack)\n\n# 비어 있는지 확인\nis_empty: bool = len(stack) == 0\n</code></pre> stack.cpp<pre><code>/* 스택 초기화 */\nstack&lt;int&gt; stack;\n\n/* 스택에 원소 삽입 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* 스택 맨 위 원소에 접근 */\nint top = stack.top();\n\n/* 스택에서 원소 꺼내기 */\nstack.pop(); // 반환값 없음\n\n/* 스택 길이 구하기 */\nint size = stack.size();\n\n/* 비어 있는지 확인 */\nbool empty = stack.empty();\n</code></pre> stack.java<pre><code>/* 스택 초기화 */\nStack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n\n/* 스택에 원소 삽입 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* 스택 맨 위 원소에 접근 */\nint peek = stack.peek();\n\n/* 스택에서 원소 꺼내기 */\nint pop = stack.pop();\n\n/* 스택 길이 구하기 */\nint size = stack.size();\n\n/* 비어 있는지 확인 */\nboolean isEmpty = stack.isEmpty();\n</code></pre> stack.cs<pre><code>/* 스택 초기화 */\nStack&lt;int&gt; stack = new();\n\n/* 스택에 원소 삽입 */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n\n/* 스택 맨 위 원소에 접근 */\nint peek = stack.Peek();\n\n/* 스택에서 원소 꺼내기 */\nint pop = stack.Pop();\n\n/* 스택 길이 구하기 */\nint size = stack.Count;\n\n/* 비어 있는지 확인 */\nbool isEmpty = stack.Count == 0;\n</code></pre> stack_test.go<pre><code>/* 스택 초기화 */\n// Go에서는 Slice를 스택으로 사용하는 것을 권한다\nvar stack []int\n\n/* 스택에 원소 삽입 */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n\n/* 스택 맨 위 원소에 접근 */\npeek := stack[len(stack)-1]\n\n/* 스택에서 원소 꺼내기 */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n\n/* 스택 길이 구하기 */\nsize := len(stack)\n\n/* 비어 있는지 확인 */\nisEmpty := len(stack) == 0\n</code></pre> stack.swift<pre><code>/* 스택 초기화 */\n// Swift에는 내장 스택 클래스가 없으므로 Array를 스택으로 사용할 수 있다\nvar stack: [Int] = []\n\n/* 스택에 원소 삽입 */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n/* 스택 맨 위 원소에 접근 */\nlet peek = stack.last!\n\n/* 스택에서 원소 꺼내기 */\nlet pop = stack.removeLast()\n\n/* 스택 길이 구하기 */\nlet size = stack.count\n\n/* 비어 있는지 확인 */\nlet isEmpty = stack.isEmpty\n</code></pre> stack.js<pre><code>/* 스택 초기화 */\n// JavaScript에는 내장 스택 클래스가 없으므로 Array를 스택으로 사용할 수 있다\nconst stack = [];\n\n/* 스택에 원소 삽입 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* 스택 맨 위 원소에 접근 */\nconst peek = stack[stack.length-1];\n\n/* 스택에서 원소 꺼내기 */\nconst pop = stack.pop();\n\n/* 스택 길이 구하기 */\nconst size = stack.length;\n\n/* 비어 있는지 확인 */\nconst is_empty = stack.length === 0;\n</code></pre> stack.ts<pre><code>/* 스택 초기화 */\n// TypeScript에는 내장 스택 클래스가 없으므로 Array를 스택으로 사용할 수 있다\nconst stack: number[] = [];\n\n/* 스택에 원소 삽입 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* 스택 맨 위 원소에 접근 */\nconst peek = stack[stack.length - 1];\n\n/* 스택에서 원소 꺼내기 */\nconst pop = stack.pop();\n\n/* 스택 길이 구하기 */\nconst size = stack.length;\n\n/* 비어 있는지 확인 */\nconst is_empty = stack.length === 0;\n</code></pre> stack.dart<pre><code>/* 스택 초기화 */\n// Dart에는 내장 스택 클래스가 없으므로 List를 스택으로 사용할 수 있다\nList&lt;int&gt; stack = [];\n\n/* 스택에 원소 삽입 */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n\n/* 스택 맨 위 원소에 접근 */\nint peek = stack.last;\n\n/* 스택에서 원소 꺼내기 */\nint pop = stack.removeLast();\n\n/* 스택 길이 구하기 */\nint size = stack.length;\n\n/* 비어 있는지 확인 */\nbool isEmpty = stack.isEmpty;\n</code></pre> stack.rs<pre><code>/* 스택 초기화 */\n// Vec를 스택으로 사용\nlet mut stack: Vec&lt;i32&gt; = Vec::new();\n\n/* 스택에 원소 삽입 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* 스택 맨 위 원소에 접근 */\nlet top = stack.last().unwrap();\n\n/* 스택에서 원소 꺼내기 */\nlet pop = stack.pop().unwrap();\n\n/* 스택 길이 구하기 */\nlet size = stack.len();\n\n/* 비어 있는지 확인 */\nlet is_empty = stack.is_empty();\n</code></pre> stack.c<pre><code>// C는 내장 스택을 제공하지 않는다\n</code></pre> stack.kt<pre><code>/* 스택 초기화 */\nval stack = Stack&lt;Int&gt;()\n\n/* 스택에 원소 삽입 */\nstack.push(1)\nstack.push(3)\nstack.push(2)\nstack.push(5)\nstack.push(4)\n\n/* 스택 맨 위 원소에 접근 */\nval peek = stack.peek()\n\n/* 스택에서 원소 꺼내기 */\nval pop = stack.pop()\n\n/* 스택 길이 구하기 */\nval size = stack.size\n\n/* 비어 있는지 확인 */\nval isEmpty = stack.isEmpty()\n</code></pre> stack.rb<pre><code># 스택 초기화\n# Ruby에는 내장 스택 클래스가 없으므로 Array를 스택으로 사용할 수 있다\nstack = []\n\n# 스택에 원소 삽입\nstack &lt;&lt; 1\nstack &lt;&lt; 3\nstack &lt;&lt; 2\nstack &lt;&lt; 5\nstack &lt;&lt; 4\n\n# 스택 맨 위 원소에 접근\npeek = stack.last\n\n# 스택에서 원소 꺼내기\npop = stack.pop\n\n# 스택 길이 구하기\nsize = stack.length\n\n# 비어 있는지 확인\nis_empty = stack.empty?\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 5 장   스택과 큐","5.1   스택"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#512","level":2,"title":"5.1.2   스택 구현","text":"<p>스택의 작동 원리를 깊이 이해하기 위해 스택 클래스를 직접 구현해 보자.</p> <p>스택은 후입선출 원리를 따르므로 맨 위에서만 원소를 추가하거나 삭제할 수 있다. 반면 배열과 연결 리스트는 어느 위치에서든 추가와 삭제가 가능하므로, 스택은 제약을 가한 배열이나 연결 리스트로 볼 수 있다. 즉, 일부 무관한 연산을 ‘가려서’ 외부에 보이는 논리가 스택의 특성에 맞도록 만들 수 있다.</p>","path":["제 5 장   스택과 큐","5.1   스택"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#1","level":3,"title":"1.   연결 리스트 기반 구현","text":"<p>연결 리스트로 스택을 구현할 때는 머리 노드를 스택의 꼭대기로, 꼬리 노드를 바닥으로 볼 수 있다.</p> <p>아래 그림(5-2)과 같이 스택에 삽입할 때는 연결 리스트의 맨 앞에 원소를 넣으면 된다. 이 방식을 ‘머리 삽입’이라고 한다. 스택에서 꺼낼 때는 머리 노드를 삭제하면 된다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> 그림 5-2   연결 리스트 기반 스택의 삽입과 꺼내기 </p> <p>연결 리스트로 스택을 구현한 예제 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby linkedlist_stack.py<pre><code>class LinkedListStack:\n    \"\"\"연결 리스트로 구현한 스택\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        self._peek: ListNode | None = None\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"스택의 길이 반환\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"스택이 비어 있는지 판단\"\"\"\n        return self._size == 0\n\n    def push(self, val: int):\n        \"\"\"스택에 삽입\"\"\"\n        node = ListNode(val)\n        node.next = self._peek\n        self._peek = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"스택에서 꺼내기\"\"\"\n        num = self.peek()\n        self._peek = self._peek.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"스택 맨 위 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"스택이 비어 있다\")\n        return self._peek.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"출력에 사용할 리스트로 변환\"\"\"\n        arr = []\n        node = self._peek\n        while node:\n            arr.append(node.val)\n            node = node.next\n        arr.reverse()\n        return arr\n</code></pre> linkedlist_stack.cpp<pre><code>/* 연결 리스트로 구현한 스택 */\nclass LinkedListStack {\n  private:\n    ListNode *stackTop; // 머리 노드를 스택의 맨 위로 사용한다\n    int stkSize;        // 스택의 길이\n\n  public:\n    LinkedListStack() {\n        stackTop = nullptr;\n        stkSize = 0;\n    }\n\n    ~LinkedListStack() {\n        // 연결 리스트를 순회하며 노드를 삭제하여 메모리를 해제한다\n        freeMemoryLinkedList(stackTop);\n    }\n\n    /* 스택의 길이 반환 */\n    int size() {\n        return stkSize;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* 스택에 삽입 */\n    void push(int num) {\n        ListNode *node = new ListNode(num);\n        node-&gt;next = stackTop;\n        stackTop = node;\n        stkSize++;\n    }\n\n    /* 스택에서 꺼내기 */\n    int pop() {\n        int num = top();\n        ListNode *tmp = stackTop;\n        stackTop = stackTop-&gt;next;\n        // 메모리 해제\n        delete tmp;\n        stkSize--;\n        return num;\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"스택이 비어 있다\");\n        return stackTop-&gt;val;\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = stackTop;\n        vector&lt;int&gt; res(size());\n        for (int i = res.size() - 1; i &gt;= 0; i--) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_stack.java<pre><code>/* 연결 리스트로 구현한 스택 */\nclass LinkedListStack {\n    private ListNode stackPeek; // 머리 노드를 스택의 맨 위로 사용한다\n    private int stkSize = 0; // 스택의 길이\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* 스택의 길이 반환 */\n    public int size() {\n        return stkSize;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* 스택에 삽입 */\n    public void push(int num) {\n        ListNode node = new ListNode(num);\n        node.next = stackPeek;\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* 스택에서 꺼내기 */\n    public int pop() {\n        int num = peek();\n        stackPeek = stackPeek.next;\n        stkSize--;\n        return num;\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stackPeek.val;\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    public int[] toArray() {\n        ListNode node = stackPeek;\n        int[] res = new int[size()];\n        for (int i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.cs<pre><code>/* 연결 리스트로 구현한 스택 */\nclass LinkedListStack {\n    ListNode? stackPeek;  // 머리 노드를 스택의 맨 위로 사용한다\n    int stkSize = 0;   // 스택의 길이\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* 스택의 길이 반환 */\n    public int Size() {\n        return stkSize;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* 스택에 삽입 */\n    public void Push(int num) {\n        ListNode node = new(num) {\n            next = stackPeek\n        };\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* 스택에서 꺼내기 */\n    public int Pop() {\n        int num = Peek();\n        stackPeek = stackPeek!.next;\n        stkSize--;\n        return num;\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stackPeek!.val;\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    public int[] ToArray() {\n        if (stackPeek == null)\n            return [];\n\n        ListNode? node = stackPeek;\n        int[] res = new int[Size()];\n        for (int i = res.Length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.go<pre><code>/* 연결 리스트로 구현한 스택 */\ntype linkedListStack struct {\n    // 내장 패키지 list로 스택을 구현한다\n    data *list.List\n}\n\n/* 스택 초기화 */\nfunc newLinkedListStack() *linkedListStack {\n    return &amp;linkedListStack{\n        data: list.New(),\n    }\n}\n\n/* 스택에 삽입 */\nfunc (s *linkedListStack) push(value int) {\n    s.data.PushBack(value)\n}\n\n/* 스택에서 꺼내기 */\nfunc (s *linkedListStack) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* 스택 맨 위 원소 접근 */\nfunc (s *linkedListStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* 스택의 길이 반환 */\nfunc (s *linkedListStack) size() int {\n    return s.data.Len()\n}\n\n/* 스택이 비어 있는지 판단 */\nfunc (s *linkedListStack) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* 출력에 사용할 List 반환 */\nfunc (s *linkedListStack) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_stack.swift<pre><code>/* 연결 리스트로 구현한 스택 */\nclass LinkedListStack {\n    private var _peek: ListNode? // 머리 노드를 스택의 맨 위로 사용한다\n    private var _size: Int // 스택의 길이\n\n    init() {\n        _size = 0\n    }\n\n    /* 스택의 길이 반환 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* 스택에 삽입 */\n    func push(num: Int) {\n        let node = ListNode(x: num)\n        node.next = _peek\n        _peek = node\n        _size += 1\n    }\n\n    /* 스택에서 꺼내기 */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        _peek = _peek?.next\n        _size -= 1\n        return num\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"스택이 비어 있다\")\n        }\n        return _peek!.val\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    func toArray() -&gt; [Int] {\n        var node = _peek\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices.reversed() {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.js<pre><code>/* 연결 리스트로 구현한 스택 */\nclass LinkedListStack {\n    #stackPeek; // 머리 노드를 스택의 맨 위로 사용한다\n    #stkSize = 0; // 스택의 길이\n\n    constructor() {\n        this.#stackPeek = null;\n    }\n\n    /* 스택의 길이 반환 */\n    get size() {\n        return this.#stkSize;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* 스택에 삽입 */\n    push(num) {\n        const node = new ListNode(num);\n        node.next = this.#stackPeek;\n        this.#stackPeek = node;\n        this.#stkSize++;\n    }\n\n    /* 스택에서 꺼내기 */\n    pop() {\n        const num = this.peek();\n        this.#stackPeek = this.#stackPeek.next;\n        this.#stkSize--;\n        return num;\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    peek() {\n        if (!this.#stackPeek) throw new Error('스택이 비어 있다');\n        return this.#stackPeek.val;\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    toArray() {\n        let node = this.#stackPeek;\n        const res = new Array(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.ts<pre><code>/* 연결 리스트로 구현한 스택 */\nclass LinkedListStack {\n    private stackPeek: ListNode | null; // 머리 노드를 스택의 맨 위로 사용한다\n    private stkSize: number = 0; // 스택의 길이\n\n    constructor() {\n        this.stackPeek = null;\n    }\n\n    /* 스택의 길이 반환 */\n    get size(): number {\n        return this.stkSize;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* 스택에 삽입 */\n    push(num: number): void {\n        const node = new ListNode(num);\n        node.next = this.stackPeek;\n        this.stackPeek = node;\n        this.stkSize++;\n    }\n\n    /* 스택에서 꺼내기 */\n    pop(): number {\n        const num = this.peek();\n        if (!this.stackPeek) throw new Error('스택이 비어 있다');\n        this.stackPeek = this.stackPeek.next;\n        this.stkSize--;\n        return num;\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    peek(): number {\n        if (!this.stackPeek) throw new Error('스택이 비어 있다');\n        return this.stackPeek.val;\n    }\n\n    /* 연결 리스트를 Array로 변환하여 반환 */\n    toArray(): number[] {\n        let node = this.stackPeek;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.dart<pre><code>/* 연결 리스트 클래스로 구현한 스택 */\nclass LinkedListStack {\n  ListNode? _stackPeek; // 머리 노드를 스택의 맨 위로 사용한다\n  int _stkSize = 0; // 스택의 길이\n\n  LinkedListStack() {\n    _stackPeek = null;\n  }\n\n  /* 스택의 길이 반환 */\n  int size() {\n    return _stkSize;\n  }\n\n  /* 스택이 비어 있는지 판단 */\n  bool isEmpty() {\n    return _stkSize == 0;\n  }\n\n  /* 스택에 삽입 */\n  void push(int _num) {\n    final ListNode node = ListNode(_num);\n    node.next = _stackPeek;\n    _stackPeek = node;\n    _stkSize++;\n  }\n\n  /* 스택에서 꺼내기 */\n  int pop() {\n    final int _num = peek();\n    _stackPeek = _stackPeek!.next;\n    _stkSize--;\n    return _num;\n  }\n\n  /* 스택 맨 위 원소 접근 */\n  int peek() {\n    if (_stackPeek == null) {\n      throw Exception(\"스택이 비어 있다\");\n    }\n    return _stackPeek!.val;\n  }\n\n  /* 연결 리스트를 List로 변환하여 반환 */\n  List&lt;int&gt; toList() {\n    ListNode? node = _stackPeek;\n    List&lt;int&gt; list = [];\n    while (node != null) {\n      list.add(node.val);\n      node = node.next;\n    }\n    list = list.reversed.toList();\n    return list;\n  }\n}\n</code></pre> linkedlist_stack.rs<pre><code>/* 연결 리스트로 구현한 스택 */\n#[allow(dead_code)]\npub struct LinkedListStack&lt;T&gt; {\n    stack_peek: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // 머리 노드를 스택의 맨 위로 사용한다\n    stk_size: usize,                              // 스택의 길이\n}\n\nimpl&lt;T: Copy&gt; LinkedListStack&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            stack_peek: None,\n            stk_size: 0,\n        }\n    }\n\n    /* 스택의 길이 반환 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.stk_size;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* 스택에 삽입 */\n    pub fn push(&amp;mut self, num: T) {\n        let node = ListNode::new(num);\n        node.borrow_mut().next = self.stack_peek.take();\n        self.stack_peek = Some(node);\n        self.stk_size += 1;\n    }\n\n    /* 스택에서 꺼내기 */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack_peek.take().map(|old_head| {\n            self.stack_peek = old_head.borrow_mut().next.take();\n            self.stk_size -= 1;\n\n            old_head.borrow().val\n        })\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.stack_peek.as_ref()\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    pub fn to_array(&amp;self) -&gt; Vec&lt;T&gt; {\n        fn _to_array&lt;T: Sized + Copy&gt;(head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n            if let Some(node) = head {\n                let mut nums = _to_array(node.borrow().next.as_ref());\n                nums.push(node.borrow().val);\n                return nums;\n            }\n            return Vec::new();\n        }\n\n        _to_array(self.peek())\n    }\n}\n</code></pre> linkedlist_stack.c<pre><code>/* 연결 리스트로 구현한 스택 */\ntypedef struct {\n    ListNode *top; // 머리 노드를 스택의 맨 위로 사용한다\n    int size;      // 스택의 길이\n} LinkedListStack;\n\n/* 생성자 */\nLinkedListStack *newLinkedListStack() {\n    LinkedListStack *s = malloc(sizeof(LinkedListStack));\n    s-&gt;top = NULL;\n    s-&gt;size = 0;\n    return s;\n}\n\n/* 소멸자 */\nvoid delLinkedListStack(LinkedListStack *s) {\n    while (s-&gt;top) {\n        ListNode *n = s-&gt;top-&gt;next;\n        free(s-&gt;top);\n        s-&gt;top = n;\n    }\n    free(s);\n}\n\n/* 스택의 길이 반환 */\nint size(LinkedListStack *s) {\n    return s-&gt;size;\n}\n\n/* 스택이 비어 있는지 판단 */\nbool isEmpty(LinkedListStack *s) {\n    return size(s) == 0;\n}\n\n/* 스택에 삽입 */\nvoid push(LinkedListStack *s, int num) {\n    ListNode *node = (ListNode *)malloc(sizeof(ListNode));\n    node-&gt;next = s-&gt;top; // 새로 추가한 노드의 포인터 필드를 갱신한다\n    node-&gt;val = num;     // 새로 추가한 노드의 데이터 필드를 갱신한다\n    s-&gt;top = node;       // 스택의 맨 위를 갱신한다\n    s-&gt;size++;           // 스택 크기를 갱신한다\n}\n\n/* 스택 맨 위 원소 접근 */\nint peek(LinkedListStack *s) {\n    if (s-&gt;size == 0) {\n        printf(\"스택이 비어 있다\\n\");\n        return INT_MAX;\n    }\n    return s-&gt;top-&gt;val;\n}\n\n/* 스택에서 꺼내기 */\nint pop(LinkedListStack *s) {\n    int val = peek(s);\n    ListNode *tmp = s-&gt;top;\n    s-&gt;top = s-&gt;top-&gt;next;\n    // 메모리 해제\n    free(tmp);\n    s-&gt;size--;\n    return val;\n}\n</code></pre> linkedlist_stack.kt<pre><code>/* 연결 리스트로 구현한 스택 */\nclass LinkedListStack(\n    private var stackPeek: ListNode? = null, // 머리 노드를 스택의 맨 위로 사용한다\n    private var stkSize: Int = 0 // 스택의 길이\n) {\n\n    /* 스택의 길이 반환 */\n    fun size(): Int {\n        return stkSize\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* 스택에 삽입 */\n    fun push(num: Int) {\n        val node = ListNode(num)\n        node.next = stackPeek\n        stackPeek = node\n        stkSize++\n    }\n\n    /* 스택에서 꺼내기 */\n    fun pop(): Int? {\n        val num = peek()\n        stackPeek = stackPeek?.next\n        stkSize--\n        return num\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    fun peek(): Int? {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stackPeek?._val\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    fun toArray(): IntArray {\n        var node = stackPeek\n        val res = IntArray(size())\n        for (i in res.size - 1 downTo 0) {\n            res[i] = node?._val!!\n            node = node.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.rb<pre><code>### 연결 리스트로 구현한 스택 ###\nclass LinkedListStack\n  attr_reader :size\n\n  ### 생성자 ###\n  def initialize\n    @size = 0\n  end\n\n  ### 스택이 비어 있는지 판단 ###\n  def is_empty?\n    @peek.nil?\n  end\n\n  ### 스택에 삽입 ###\n  def push(val)\n    node = ListNode.new(val)\n    node.next = @peek\n    @peek = node\n    @size += 1\n  end\n\n  ### 스택에서 꺼내기 ###\n  def pop\n    num = peek\n    @peek = @peek.next\n    @size -= 1\n    num\n  end\n\n  ### 스택 맨 위 원소 접근 ###\n  def peek\n    raise IndexError, '스택이 비어 있다' if is_empty?\n\n    @peek.val\n  end\n\n  ### 연결 리스트를 Array로 변환하여 반환 ###\n  def to_array\n    arr = []\n    node = @peek\n    while node\n      arr &lt;&lt; node.val\n      node = node.next\n    end\n    arr.reverse\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 5 장   스택과 큐","5.1   스택"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#2","level":3,"title":"2.   배열 기반 구현","text":"<p>배열로 스택을 구현할 때는 배열의 끝을 스택의 꼭대기로 볼 수 있다. 아래 그림(5-3)과 같이 삽입과 꺼내기는 각각 배열 끝에 원소를 추가하고 삭제하는 연산에 해당하며, 시간 복잡도는 모두 \\(O(1)\\)이다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> 그림 5-3   배열 기반 스택의 삽입과 꺼내기 </p> <p>스택에 삽입되는 원소가 계속 늘어날 수 있으므로 동적 배열을 사용하면 배열의 용량 확장을 직접 처리하지 않아도 된다. 예제 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_stack.py<pre><code>class ArrayStack:\n    \"\"\"배열로 구현한 스택\"\"\"\n\n    def __init__(self):\n        \"\"\"생성자\"\"\"\n        self._stack: list[int] = []\n\n    def size(self) -&gt; int:\n        \"\"\"스택의 길이 반환\"\"\"\n        return len(self._stack)\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"스택이 비어 있는지 판단\"\"\"\n        return self.size() == 0\n\n    def push(self, item: int):\n        \"\"\"스택에 삽입\"\"\"\n        self._stack.append(item)\n\n    def pop(self) -&gt; int:\n        \"\"\"스택에서 꺼내기\"\"\"\n        if self.is_empty():\n            raise IndexError(\"스택이 비어 있다\")\n        return self._stack.pop()\n\n    def peek(self) -&gt; int:\n        \"\"\"스택 맨 위 원소 접근\"\"\"\n        if self.is_empty():\n            raise IndexError(\"스택이 비어 있다\")\n        return self._stack[-1]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"출력에 사용할 리스트 반환\"\"\"\n        return self._stack\n</code></pre> array_stack.cpp<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n  private:\n    vector&lt;int&gt; stack;\n\n  public:\n    /* 스택의 길이 반환 */\n    int size() {\n        return stack.size();\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    bool isEmpty() {\n        return stack.size() == 0;\n    }\n\n    /* 스택에 삽입 */\n    void push(int num) {\n        stack.push_back(num);\n    }\n\n    /* 스택에서 꺼내기 */\n    int pop() {\n        int num = top();\n        stack.pop_back();\n        return num;\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"스택이 비어 있다\");\n        return stack.back();\n    }\n\n    /* Vector 반환 */\n    vector&lt;int&gt; toVector() {\n        return stack;\n    }\n};\n</code></pre> array_stack.java<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n    private ArrayList&lt;Integer&gt; stack;\n\n    public ArrayStack() {\n        // 리스트(동적 배열)를 초기화한다\n        stack = new ArrayList&lt;&gt;();\n    }\n\n    /* 스택의 길이 반환 */\n    public int size() {\n        return stack.size();\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* 스택에 삽입 */\n    public void push(int num) {\n        stack.add(num);\n    }\n\n    /* 스택에서 꺼내기 */\n    public int pop() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.remove(size() - 1);\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.get(size() - 1);\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    public Object[] toArray() {\n        return stack.toArray();\n    }\n}\n</code></pre> array_stack.cs<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n    List&lt;int&gt; stack;\n    public ArrayStack() {\n        // 리스트(동적 배열)를 초기화한다\n        stack = [];\n    }\n\n    /* 스택의 길이 반환 */\n    public int Size() {\n        return stack.Count;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* 스택에 삽입 */\n    public void Push(int num) {\n        stack.Add(num);\n    }\n\n    /* 스택에서 꺼내기 */\n    public int Pop() {\n        if (IsEmpty())\n            throw new Exception();\n        var val = Peek();\n        stack.RemoveAt(Size() - 1);\n        return val;\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stack[Size() - 1];\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    public int[] ToArray() {\n        return [.. stack];\n    }\n}\n</code></pre> array_stack.go<pre><code>/* 배열로 구현한 스택 */\ntype arrayStack struct {\n    data []int // 데이터\n}\n\n/* 스택 초기화 */\nfunc newArrayStack() *arrayStack {\n    return &amp;arrayStack{\n        // 스택의 길이를 0, 용량을 16으로 설정한다\n        data: make([]int, 0, 16),\n    }\n}\n\n/* 스택의 길이 */\nfunc (s *arrayStack) size() int {\n    return len(s.data)\n}\n\n/* 스택이 비어 있는지 판단 */\nfunc (s *arrayStack) isEmpty() bool {\n    return s.size() == 0\n}\n\n/* 스택에 삽입 */\nfunc (s *arrayStack) push(v int) {\n    // 슬라이스는 용량이 자동으로 확장된다\n    s.data = append(s.data, v)\n}\n\n/* 스택에서 꺼내기 */\nfunc (s *arrayStack) pop() any {\n    val := s.peek()\n    s.data = s.data[:len(s.data)-1]\n    return val\n}\n\n/* 스택 맨 위 원소 반환 */\nfunc (s *arrayStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    val := s.data[len(s.data)-1]\n    return val\n}\n\n/* 출력에 사용할 Slice 반환 */\nfunc (s *arrayStack) toSlice() []int {\n    return s.data\n}\n</code></pre> array_stack.swift<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n    private var stack: [Int]\n\n    init() {\n        // 리스트(동적 배열)를 초기화한다\n        stack = []\n    }\n\n    /* 스택의 길이 반환 */\n    func size() -&gt; Int {\n        stack.count\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    func isEmpty() -&gt; Bool {\n        stack.isEmpty\n    }\n\n    /* 스택에 삽입 */\n    func push(num: Int) {\n        stack.append(num)\n    }\n\n    /* 스택에서 꺼내기 */\n    @discardableResult\n    func pop() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"스택이 비어 있다\")\n        }\n        return stack.removeLast()\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"스택이 비어 있다\")\n        }\n        return stack.last!\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    func toArray() -&gt; [Int] {\n        stack\n    }\n}\n</code></pre> array_stack.js<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n    #stack;\n    constructor() {\n        this.#stack = [];\n    }\n\n    /* 스택의 길이 반환 */\n    get size() {\n        return this.#stack.length;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    isEmpty() {\n        return this.#stack.length === 0;\n    }\n\n    /* 스택에 삽입 */\n    push(num) {\n        this.#stack.push(num);\n    }\n\n    /* 스택에서 꺼내기 */\n    pop() {\n        if (this.isEmpty()) throw new Error('스택이 비어 있다');\n        return this.#stack.pop();\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    top() {\n        if (this.isEmpty()) throw new Error('스택이 비어 있다');\n        return this.#stack[this.#stack.length - 1];\n    }\n\n    /* Array를 반환한다 */\n    toArray() {\n        return this.#stack;\n    }\n}\n</code></pre> array_stack.ts<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n    private stack: number[];\n    constructor() {\n        this.stack = [];\n    }\n\n    /* 스택의 길이 반환 */\n    get size(): number {\n        return this.stack.length;\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    isEmpty(): boolean {\n        return this.stack.length === 0;\n    }\n\n    /* 스택에 삽입 */\n    push(num: number): void {\n        this.stack.push(num);\n    }\n\n    /* 스택에서 꺼내기 */\n    pop(): number | undefined {\n        if (this.isEmpty()) throw new Error('스택이 비어 있다');\n        return this.stack.pop();\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    top(): number | undefined {\n        if (this.isEmpty()) throw new Error('스택이 비어 있다');\n        return this.stack[this.stack.length - 1];\n    }\n\n    /* Array를 반환한다 */\n    toArray() {\n        return this.stack;\n    }\n}\n</code></pre> array_stack.dart<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n  late List&lt;int&gt; _stack;\n  ArrayStack() {\n    _stack = [];\n  }\n\n  /* 스택의 길이 반환 */\n  int size() {\n    return _stack.length;\n  }\n\n  /* 스택이 비어 있는지 판단 */\n  bool isEmpty() {\n    return _stack.isEmpty;\n  }\n\n  /* 스택에 삽입 */\n  void push(int _num) {\n    _stack.add(_num);\n  }\n\n  /* 스택에서 꺼내기 */\n  int pop() {\n    if (isEmpty()) {\n      throw Exception(\"스택이 비어 있다\");\n    }\n    return _stack.removeLast();\n  }\n\n  /* 스택 맨 위 원소 접근 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"스택이 비어 있다\");\n    }\n    return _stack.last;\n  }\n\n  /* 스택을 Array로 변환하여 반환 */\n  List&lt;int&gt; toArray() =&gt; _stack;\n}\n</code></pre> array_stack.rs<pre><code>/* 배열로 구현한 스택 */\nstruct ArrayStack&lt;T&gt; {\n    stack: Vec&lt;T&gt;,\n}\n\nimpl&lt;T&gt; ArrayStack&lt;T&gt; {\n    /* 스택 초기화 */\n    fn new() -&gt; ArrayStack&lt;T&gt; {\n        ArrayStack::&lt;T&gt; {\n            stack: Vec::&lt;T&gt;::new(),\n        }\n    }\n\n    /* 스택의 길이 반환 */\n    fn size(&amp;self) -&gt; usize {\n        self.stack.len()\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.size() == 0\n    }\n\n    /* 스택에 삽입 */\n    fn push(&amp;mut self, num: T) {\n        self.stack.push(num);\n    }\n\n    /* 스택에서 꺼내기 */\n    fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack.pop()\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    fn peek(&amp;self) -&gt; Option&lt;&amp;T&gt; {\n        if self.is_empty() {\n            panic!(\"스택이 비어 있다\")\n        };\n        self.stack.last()\n    }\n\n    /* &amp;Vec를 반환한다 */\n    fn to_array(&amp;self) -&gt; &amp;Vec&lt;T&gt; {\n        &amp;self.stack\n    }\n}\n</code></pre> array_stack.c<pre><code>/* 배열로 구현한 스택 */\ntypedef struct {\n    int *data;\n    int size;\n} ArrayStack;\n\n/* 생성자 */\nArrayStack *newArrayStack() {\n    ArrayStack *stack = malloc(sizeof(ArrayStack));\n    // 용량 확장을 피하도록 큰 용량으로 초기화한다\n    stack-&gt;data = malloc(sizeof(int) * MAX_SIZE);\n    stack-&gt;size = 0;\n    return stack;\n}\n\n/* 소멸자 */\nvoid delArrayStack(ArrayStack *stack) {\n    free(stack-&gt;data);\n    free(stack);\n}\n\n/* 스택의 길이 반환 */\nint size(ArrayStack *stack) {\n    return stack-&gt;size;\n}\n\n/* 스택이 비어 있는지 판단 */\nbool isEmpty(ArrayStack *stack) {\n    return stack-&gt;size == 0;\n}\n\n/* 스택에 삽입 */\nvoid push(ArrayStack *stack, int num) {\n    if (stack-&gt;size == MAX_SIZE) {\n        printf(\"스택이 가득 찼다\\n\");\n        return;\n    }\n    stack-&gt;data[stack-&gt;size] = num;\n    stack-&gt;size++;\n}\n\n/* 스택 맨 위 원소 접근 */\nint peek(ArrayStack *stack) {\n    if (stack-&gt;size == 0) {\n        printf(\"스택이 비어 있다\\n\");\n        return INT_MAX;\n    }\n    return stack-&gt;data[stack-&gt;size - 1];\n}\n\n/* 스택에서 꺼내기 */\nint pop(ArrayStack *stack) {\n    int val = peek(stack);\n    stack-&gt;size--;\n    return val;\n}\n</code></pre> array_stack.kt<pre><code>/* 배열로 구현한 스택 */\nclass ArrayStack {\n    // 리스트(동적 배열)를 초기화한다\n    private val stack = mutableListOf&lt;Int&gt;()\n\n    /* 스택의 길이 반환 */\n    fun size(): Int {\n        return stack.size\n    }\n\n    /* 스택이 비어 있는지 판단 */\n    fun isEmpty(): Boolean {\n        return size() == 0\n    }\n\n    /* 스택에 삽입 */\n    fun push(num: Int) {\n        stack.add(num)\n    }\n\n    /* 스택에서 꺼내기 */\n    fun pop(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack.removeAt(size() - 1)\n    }\n\n    /* 스택 맨 위 원소 접근 */\n    fun peek(): Int {\n        if (isEmpty()) throw IndexOutOfBoundsException()\n        return stack[size() - 1]\n    }\n\n    /* List를 Array로 변환하여 반환 */\n    fun toArray(): Array&lt;Any&gt; {\n        return stack.toTypedArray()\n    }\n}\n</code></pre> array_stack.rb<pre><code>### 배열로 구현한 스택 ###\nclass ArrayStack\n  ### 생성자 ###\n  def initialize\n    @stack = []\n  end\n\n  ### 스택의 길이 반환 ###\n  def size\n    @stack.length\n  end\n\n  ### 스택이 비어 있는지 판단 ###\n  def is_empty?\n    @stack.empty?\n  end\n\n  ### 스택에 삽입 ###\n  def push(item)\n    @stack &lt;&lt; item\n  end\n\n  ### 스택에서 꺼내기 ###\n  def pop\n    raise IndexError, '스택이 비어 있다' if is_empty?\n\n    @stack.pop\n  end\n\n  ### 스택 맨 위 원소 접근 ###\n  def peek\n    raise IndexError, '스택이 비어 있다' if is_empty?\n\n    @stack.last\n  end\n\n  ### 출력에 사용할 리스트 반환 ###\n  def to_array\n    @stack\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 5 장   스택과 큐","5.1   스택"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#513","level":2,"title":"5.1.3   두 구현의 비교","text":"<p>지원 연산</p> <p>두 구현 모두 스택 정의에 포함된 모든 연산을 지원한다. 배열 기반 구현은 임의 접근도 지원하지만, 이는 스택의 정의를 벗어나므로 일반적으로 사용하지 않는다.</p> <p>시간 효율</p> <p>배열 기반 구현에서는 삽입과 꺼내기를 미리 할당한 연속 메모리에서 수행한다. 캐시 지역성이 좋아 효율이 높다. 다만 삽입 시 배열 용량을 넘으면 확장이 발생하여 그 삽입 연산의 시간 복잡도는 \\(O(n)\\)이 된다.</p> <p>연결 리스트 기반 구현은 크기를 유연하게 늘릴 수 있어 배열 확장에 따른 효율 저하가 없다. 하지만 삽입 시 노드 객체를 초기화하고 포인터를 수정해야 하므로 상대적으로 효율이 낮다. 삽입할 원소 자체가 노드 객체라면 초기화 단계를 생략하여 효율을 높일 수 있다.</p> <p>따라서 삽입하고 꺼내는 원소가 <code>int</code>나 <code>double</code> 같은 기본 자료형이라면 다음과 같이 정리할 수 있다.</p> <ul> <li>배열 기반 스택은 용량 확장이 발생할 때 효율이 낮아지지만, 확장은 드물게 일어나므로 평균 효율은 더 높다.</li> <li>연결 리스트 기반 스택은 더 안정적인 효율을 제공한다.</li> </ul> <p>공간 효율</p> <p>리스트를 초기화할 때 시스템은 ‘초기 용량’을 할당하며, 이 용량은 실제 필요보다 클 수 있다. 또한 확장 시 보통 2배처럼 일정한 비율로 늘리므로 확장 후 용량도 실제 필요보다 클 수 있다. 따라서 배열 기반 스택은 어느 정도 공간을 낭비할 수 있다.</p> <p>반면 연결 리스트 노드는 포인터를 추가로 저장해야 하므로 노드 하나가 차지하는 공간이 상대적으로 크다.</p> <p>따라서 어느 구현이 메모리를 더 절약하는지는 단정할 수 없으며 구체적인 상황에 따라 분석해야 한다.</p>","path":["제 5 장   스택과 큐","5.1   스택"],"tags":[]},{"location":"chapter_stack_and_queue/stack/#514","level":2,"title":"5.1.4   스택의 대표적인 활용","text":"<ul> <li>브라우저의 뒤로 및 앞으로, 소프트웨어의 실행 취소 및 다시 실행. 새 웹 페이지를 열 때마다 브라우저는 이전 페이지를 스택에 넣어 뒤로 가기로 돌아갈 수 있게 한다. 뒤로 가기는 실제로 스택에서 원소를 꺼내는 연산이다. 뒤로 가기와 앞으로 가기를 함께 지원하려면 스택 두 개를 사용해야 한다.</li> <li>프로그램 메모리 관리. 함수가 호출될 때마다 시스템은 스택 맨 위에 스택 프레임을 추가해 함수의 문맥 정보를 기록한다. 재귀 함수에서는 아래로 호출을 이어 가는 단계에서 스택에 계속 넣고, 위로 되돌아오는 단계에서 계속 꺼낸다.</li> </ul>","path":["제 5 장   스택과 큐","5.1   스택"],"tags":[]},{"location":"chapter_stack_and_queue/summary/","level":1,"title":"5.4   요약","text":"","path":["제 5 장   스택과 큐","5.4   요약"],"tags":[]},{"location":"chapter_stack_and_queue/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>스택은 후입선출 원리를 따르는 자료 구조로, 배열이나 연결 리스트로 구현할 수 있다.</li> <li>시간 효율 측면에서 배열 기반 스택의 평균 효율은 더 높지만, 용량 확장이 발생하면 한 번의 삽입 연산의 시간 복잡도가 \\(O(n)\\)으로 나빠진다. 반면 연결 리스트 기반 스택은 더 안정적인 효율을 보인다.</li> <li>공간 효율 측면에서 배열 기반 스택은 어느 정도 공간을 낭비할 수 있다. 다만 연결 리스트의 노드 하나는 배열 원소보다 더 많은 메모리를 차지한다.</li> <li>큐는 선입선출 원리를 따르는 자료 구조로, 역시 배열이나 연결 리스트로 구현할 수 있다. 시간 및 공간 효율의 비교 결과는 앞의 스택과 비슷하다.</li> <li>덱은 자유도가 더 높은 큐로, 양 끝에서 원소를 추가하고 삭제할 수 있다.</li> </ul>","path":["제 5 장   스택과 큐","5.4   요약"],"tags":[]},{"location":"chapter_stack_and_queue/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 브라우저의 앞으로 가기와 뒤로 가기는 이중 연결 리스트로 구현되는가?</p> <p>브라우저의 앞으로 가기와 뒤로 가기는 본질적으로 ‘스택’을 보여 준다. 사용자가 새 페이지를 방문하면 맨 위에 추가되고, 뒤로 버튼을 누르면 그 페이지가 맨 위에서 꺼내진다. ‘덱’ 절에서 설명했듯 덱을 사용하면 일부 추가 연산을 편리하게 구현할 수 있다.</p> <p>Q: 스택에서 노드를 꺼낸 뒤 그 노드의 메모리를 해제해야 하는가?</p> <p>나중에 꺼낸 노드를 계속 사용해야 한다면 해제할 필요가 없다. 더 이상 사용하지 않는 경우, <code>Java</code>와 <code>Python</code> 같은 언어는 자동 가비지 컬렉션이 있으므로 직접 해제하지 않아도 된다. <code>C</code>와 <code>C++</code>에서는 직접 해제해야 한다.</p> <p>Q: 덱은 두 스택을 이어 붙인 것처럼 보이는데, 어디에 쓰이는가?</p> <p>덱은 스택과 큐를 합치거나 두 스택을 이어 붙인 것과 같다. 스택과 큐의 논리를 모두 지니므로 두 자료 구조의 모든 활용을 구현할 수 있고 더 유연하다.</p> <p>Q: 실행 취소(undo)와 다시 실행(redo)은 구체적으로 어떻게 구현하는가?</p> <p>스택 두 개를 사용한다. 스택 <code>A</code>는 실행 취소용, 스택 <code>B</code>는 다시 실행용이다.</p> <ol> <li>사용자가 작업을 수행할 때마다 그 작업을 스택 <code>A</code>에 넣고 스택 <code>B</code>를 비운다.</li> <li>‘실행 취소’를 수행하면 스택 <code>A</code>에서 가장 최근 작업을 꺼내 스택 <code>B</code>에 넣는다.</li> <li>‘다시 실행’을 수행하면 스택 <code>B</code>에서 가장 최근 작업을 꺼내 스택 <code>A</code>에 넣는다.</li> </ol>","path":["제 5 장   스택과 큐","5.4   요약"],"tags":[]},{"location":"chapter_tree/","level":1,"title":"제 7 장   트리","text":"<p>이 장을 시작하며</p> <p>하늘 높이 뻗은 나무는 깊은 뿌리와 무성한 잎, 넓게 펼쳐진 가지에 생명력이 가득하다.</p> <p>나무는 데이터를 나누어 다루는 모습을 생생하게 보여 준다.</p>","path":["제 7 장   트리"],"tags":[]},{"location":"chapter_tree/#_1","level":2,"title":"이 장의 내용","text":"<ul> <li>7.1   이진 트리</li> <li>7.2   이진 트리 순회</li> <li>7.3   이진 트리의 배열 표현</li> <li>7.4   이진 탐색 트리</li> <li>7.5   AVL 트리 *</li> <li>7.6   요약</li> <li>7.7   연습 문제</li> </ul>","path":["제 7 장   트리"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/","level":1,"title":"7.3   이진 트리의 배열 표현","text":"<p>연결 리스트로 표현할 때 이진 트리의 저장 단위는 노드 <code>TreeNode</code>이며, 노드들은 포인터로 연결된다. 앞 절에서는 이 표현을 사용하는 이진 트리의 기본 연산들을 살펴보았다.</p> <p>그렇다면 배열로도 이진 트리를 표현할 수 있을까? 가능하다.</p>","path":["제 7 장   트리","7.3   이진 트리의 배열 표현"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/#731","level":2,"title":"7.3.1   포화 이진 트리 표현","text":"<p>먼저 간단한 경우를 살펴보자. 포화 이진 트리의 모든 노드를 레벨 순서 순회 순서대로 배열에 저장하면 각 노드는 고유한 배열 인덱스에 대응한다.</p> <p>레벨 순서 순회의 특성에서 부모와 자식 인덱스 사이의 '매핑 공식'을 도출할 수 있다. 노드의 인덱스가 \\(i\\)이면 왼쪽 자식의 인덱스는 \\(2i + 1\\), 오른쪽 자식의 인덱스는 \\(2i + 2\\)다. 아래 그림(7-12)은 노드 인덱스 사이의 매핑 관계를 보여 준다.</p> <p></p> <p> 그림 7-12   포화 이진 트리의 배열 표현 </p> <p>매핑 공식은 연결 리스트의 노드 참조(포인터)와 같은 역할을 한다. 배열의 임의의 노드가 주어지면 이 공식으로 왼쪽(오른쪽) 자식에 접근할 수 있다.</p>","path":["제 7 장   트리","7.3   이진 트리의 배열 표현"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/#732","level":2,"title":"7.3.2   임의의 이진 트리 표현","text":"<p>포화 이진 트리는 특수한 경우이며, 일반적인 이진 트리의 중간 레벨에는 <code>None</code>이 많이 존재한다. 레벨 순서 순회 결과에는 이 <code>None</code>들이 포함되지 않으므로 결과만으로는 개수와 위치를 알 수 없다. 즉, 같은 레벨 순서 순회 결과에 여러 이진 트리 구조가 대응할 수 있다.</p> <p>아래 그림(7-13)처럼 포화 이진 트리가 아니면 앞의 배열 표현 방법은 더 이상 유효하지 않다.</p> <p></p> <p> 그림 7-13   하나의 레벨 순서 순회 결과에 대응하는 여러 이진 트리 </p> <p>이를 해결하려면 레벨 순서 순회 결과에 모든 <code>None</code>을 명시적으로 넣을 수 있다. 아래 그림(7-14)처럼 처리하면 순회 결과가 이진 트리를 유일하게 나타낸다. 예시 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code># 이진 트리의 배열 표현\n# None으로 빈자리 표시\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// int의 최댓값 INT_MAX로 빈자리 표시\nvector&lt;int&gt; tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// int의 래퍼 클래스 Integer를 사용하면 null로 빈자리를 표시할 수 있다\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// 널 허용 타입 int?를 사용하면 null로 빈자리를 표시할 수 있다\nint?[] tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// any 타입의 슬라이스를 사용하면 nil로 빈자리를 표시할 수 있다\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// 널 허용 타입 Int?를 사용하면 nil로 빈자리를 표시할 수 있다\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// null로 빈자리 표시\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// null로 빈자리 표시\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// 널 허용 타입 int?를 사용하면 null로 빈자리를 표시할 수 있다\nList&lt;int?&gt; tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// None으로 빈자리 표시\nlet tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)];\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// int의 최댓값으로 빈자리를 표시하므로 노드 값은 INT_MAX이면 안 된다\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* 이진 트리의 배열 표현 */\n// null로 빈자리 표시\nval tree = arrayOf( 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 )\n</code></pre> <pre><code>### 이진 트리의 배열 표현 ###\n# nil로 빈자리 표시\ntree = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <p></p> <p> 그림 7-14   임의의 이진 트리의 배열 표현 </p> <p>완전 이진 트리는 배열로 표현하기에 매우 적합하다. 정의를 떠올려 보면 <code>None</code>은 맨 아래 레벨의 오른쪽에만 있으므로 레벨 순서 순회 결과의 끝에만 나타난다.</p> <p>따라서 완전 이진 트리를 배열로 표현할 때는 모든 <code>None</code>의 저장을 생략할 수 있어 매우 편리하다. 아래 그림(7-15)은 그 예다.</p> <p></p> <p> 그림 7-15   완전 이진 트리의 배열 표현 </p> <p>다음 코드는 배열로 표현한 이진 트리를 구현하며 아래 연산들을 포함한다.</p> <ul> <li>주어진 노드의 값, 왼쪽(오른쪽) 자식, 부모 노드를 구한다.</li> <li>전위, 중위, 후위, 레벨 순서 순회 결과를 구한다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby array_binary_tree.py<pre><code>class ArrayBinaryTree:\n    \"\"\"배열로 표현한 이진 트리 클래스\"\"\"\n\n    def __init__(self, arr: list[int | None]):\n        \"\"\"생성자\"\"\"\n        self._tree = list(arr)\n\n    def size(self):\n        \"\"\"리스트 용량\"\"\"\n        return len(self._tree)\n\n    def val(self, i: int) -&gt; int | None:\n        \"\"\"인덱스가 i인 노드의 값 반환\"\"\"\n        # 인덱스가 범위를 벗어나면 빈칸을 나타내는 None을 반환한다\n        if i &lt; 0 or i &gt;= self.size():\n            return None\n        return self._tree[i]\n\n    def left(self, i: int) -&gt; int | None:\n        \"\"\"인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환\"\"\"\n        return 2 * i + 1\n\n    def right(self, i: int) -&gt; int | None:\n        \"\"\"인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환\"\"\"\n        return 2 * i + 2\n\n    def parent(self, i: int) -&gt; int | None:\n        \"\"\"인덱스가 i인 노드의 부모 노드 인덱스 반환\"\"\"\n        return (i - 1) // 2\n\n    def level_order(self) -&gt; list[int]:\n        \"\"\"레벨 순서 순회\"\"\"\n        self.res = []\n        # 배열을 직접 순회한다\n        for i in range(self.size()):\n            if self.val(i) is not None:\n                self.res.append(self.val(i))\n        return self.res\n\n    def dfs(self, i: int, order: str):\n        \"\"\"깊이 우선 순회\"\"\"\n        if self.val(i) is None:\n            return\n        # 전위 순회\n        if order == \"pre\":\n            self.res.append(self.val(i))\n        self.dfs(self.left(i), order)\n        # 중위 순회\n        if order == \"in\":\n            self.res.append(self.val(i))\n        self.dfs(self.right(i), order)\n        # 후위 순회\n        if order == \"post\":\n            self.res.append(self.val(i))\n\n    def pre_order(self) -&gt; list[int]:\n        \"\"\"전위 순회\"\"\"\n        self.res = []\n        self.dfs(0, order=\"pre\")\n        return self.res\n\n    def in_order(self) -&gt; list[int]:\n        \"\"\"중위 순회\"\"\"\n        self.res = []\n        self.dfs(0, order=\"in\")\n        return self.res\n\n    def post_order(self) -&gt; list[int]:\n        \"\"\"후위 순회\"\"\"\n        self.res = []\n        self.dfs(0, order=\"post\")\n        return self.res\n</code></pre> array_binary_tree.cpp<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree {\n  public:\n    /* 생성자 */\n    ArrayBinaryTree(vector&lt;int&gt; arr) {\n        tree = arr;\n    }\n\n    /* 리스트 용량 */\n    int size() {\n        return tree.size();\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    int val(int i) {\n        // 인덱스가 범위를 벗어나면 빈자리를 나타내는 INT_MAX를 반환한다\n        if (i &lt; 0 || i &gt;= size())\n            return INT_MAX;\n        return tree[i];\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    int left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    int right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    int parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* 레벨 순서 순회 */\n    vector&lt;int&gt; levelOrder() {\n        vector&lt;int&gt; res;\n        // 배열을 직접 순회한다\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != INT_MAX)\n                res.push_back(val(i));\n        }\n        return res;\n    }\n\n    /* 전위 순회 */\n    vector&lt;int&gt; preOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* 중위 순회 */\n    vector&lt;int&gt; inOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* 후위 순회 */\n    vector&lt;int&gt; postOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"post\", res);\n        return res;\n    }\n\n  private:\n    vector&lt;int&gt; tree;\n\n    /* 깊이 우선 순회 */\n    void dfs(int i, string order, vector&lt;int&gt; &amp;res) {\n        // 빈칸이면 반환한다\n        if (val(i) == INT_MAX)\n            return;\n        // 전위 순회\n        if (order == \"pre\")\n            res.push_back(val(i));\n        dfs(left(i), order, res);\n        // 중위 순회\n        if (order == \"in\")\n            res.push_back(val(i));\n        dfs(right(i), order, res);\n        // 후위 순회\n        if (order == \"post\")\n            res.push_back(val(i));\n    }\n};\n</code></pre> array_binary_tree.java<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree {\n    private List&lt;Integer&gt; tree;\n\n    /* 생성자 */\n    public ArrayBinaryTree(List&lt;Integer&gt; arr) {\n        tree = new ArrayList&lt;&gt;(arr);\n    }\n\n    /* 리스트 용량 */\n    public int size() {\n        return tree.size();\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    public Integer val(int i) {\n        // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n        if (i &lt; 0 || i &gt;= size())\n            return null;\n        return tree.get(i);\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    public Integer left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    public Integer right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    public Integer parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* 레벨 순서 순회 */\n    public List&lt;Integer&gt; levelOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        // 배열을 직접 순회한다\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != null)\n                res.add(val(i));\n        }\n        return res;\n    }\n\n    /* 깊이 우선 순회 */\n    private void dfs(Integer i, String order, List&lt;Integer&gt; res) {\n        // 빈칸이면 반환한다\n        if (val(i) == null)\n            return;\n        // 전위 순회\n        if (\"pre\".equals(order))\n            res.add(val(i));\n        dfs(left(i), order, res);\n        // 중위 순회\n        if (\"in\".equals(order))\n            res.add(val(i));\n        dfs(right(i), order, res);\n        // 후위 순회\n        if (\"post\".equals(order))\n            res.add(val(i));\n    }\n\n    /* 전위 순회 */\n    public List&lt;Integer&gt; preOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* 중위 순회 */\n    public List&lt;Integer&gt; inOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* 후위 순회 */\n    public List&lt;Integer&gt; postOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.cs<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree(List&lt;int?&gt; arr) {\n    List&lt;int?&gt; tree = new(arr);\n\n    /* 리스트 용량 */\n    public int Size() {\n        return tree.Count;\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    public int? Val(int i) {\n        // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n        if (i &lt; 0 || i &gt;= Size())\n            return null;\n        return tree[i];\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    public int Left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    public int Right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    public int Parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* 레벨 순서 순회 */\n    public List&lt;int&gt; LevelOrder() {\n        List&lt;int&gt; res = [];\n        // 배열을 직접 순회한다\n        for (int i = 0; i &lt; Size(); i++) {\n            if (Val(i).HasValue)\n                res.Add(Val(i)!.Value);\n        }\n        return res;\n    }\n\n    /* 깊이 우선 순회 */\n    void DFS(int i, string order, List&lt;int&gt; res) {\n        // 빈칸이면 반환한다\n        if (!Val(i).HasValue)\n            return;\n        // 전위 순회\n        if (order == \"pre\")\n            res.Add(Val(i)!.Value);\n        DFS(Left(i), order, res);\n        // 중위 순회\n        if (order == \"in\")\n            res.Add(Val(i)!.Value);\n        DFS(Right(i), order, res);\n        // 후위 순회\n        if (order == \"post\")\n            res.Add(Val(i)!.Value);\n    }\n\n    /* 전위 순회 */\n    public List&lt;int&gt; PreOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"pre\", res);\n        return res;\n    }\n\n    /* 중위 순회 */\n    public List&lt;int&gt; InOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"in\", res);\n        return res;\n    }\n\n    /* 후위 순회 */\n    public List&lt;int&gt; PostOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.go<pre><code>/* 배열로 표현한 이진 트리 클래스 */\ntype arrayBinaryTree struct {\n    tree []any\n}\n\n/* 생성자 */\nfunc newArrayBinaryTree(arr []any) *arrayBinaryTree {\n    return &amp;arrayBinaryTree{\n        tree: arr,\n    }\n}\n\n/* 리스트 용량 */\nfunc (abt *arrayBinaryTree) size() int {\n    return len(abt.tree)\n}\n\n/* 인덱스가 i인 노드의 값 반환 */\nfunc (abt *arrayBinaryTree) val(i int) any {\n    // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n    if i &lt; 0 || i &gt;= abt.size() {\n        return nil\n    }\n    return abt.tree[i]\n}\n\n/* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\nfunc (abt *arrayBinaryTree) left(i int) int {\n    return 2*i + 1\n}\n\n/* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\nfunc (abt *arrayBinaryTree) right(i int) int {\n    return 2*i + 2\n}\n\n/* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\nfunc (abt *arrayBinaryTree) parent(i int) int {\n    return (i - 1) / 2\n}\n\n/* 레벨 순서 순회 */\nfunc (abt *arrayBinaryTree) levelOrder() []any {\n    var res []any\n    // 배열을 직접 순회한다\n    for i := 0; i &lt; abt.size(); i++ {\n        if abt.val(i) != nil {\n            res = append(res, abt.val(i))\n        }\n    }\n    return res\n}\n\n/* 깊이 우선 순회 */\nfunc (abt *arrayBinaryTree) dfs(i int, order string, res *[]any) {\n    // 빈칸이면 반환한다\n    if abt.val(i) == nil {\n        return\n    }\n    // 전위 순회\n    if order == \"pre\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.left(i), order, res)\n    // 중위 순회\n    if order == \"in\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.right(i), order, res)\n    // 후위 순회\n    if order == \"post\" {\n        *res = append(*res, abt.val(i))\n    }\n}\n\n/* 전위 순회 */\nfunc (abt *arrayBinaryTree) preOrder() []any {\n    var res []any\n    abt.dfs(0, \"pre\", &amp;res)\n    return res\n}\n\n/* 중위 순회 */\nfunc (abt *arrayBinaryTree) inOrder() []any {\n    var res []any\n    abt.dfs(0, \"in\", &amp;res)\n    return res\n}\n\n/* 후위 순회 */\nfunc (abt *arrayBinaryTree) postOrder() []any {\n    var res []any\n    abt.dfs(0, \"post\", &amp;res)\n    return res\n}\n</code></pre> array_binary_tree.swift<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree {\n    private var tree: [Int?]\n\n    /* 생성자 */\n    init(arr: [Int?]) {\n        tree = arr\n    }\n\n    /* 리스트 용량 */\n    func size() -&gt; Int {\n        tree.count\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    func val(i: Int) -&gt; Int? {\n        // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n        if i &lt; 0 || i &gt;= size() {\n            return nil\n        }\n        return tree[i]\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    func left(i: Int) -&gt; Int {\n        2 * i + 1\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    func right(i: Int) -&gt; Int {\n        2 * i + 2\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    func parent(i: Int) -&gt; Int {\n        (i - 1) / 2\n    }\n\n    /* 레벨 순서 순회 */\n    func levelOrder() -&gt; [Int] {\n        var res: [Int] = []\n        // 배열을 직접 순회한다\n        for i in 0 ..&lt; size() {\n            if let val = val(i: i) {\n                res.append(val)\n            }\n        }\n        return res\n    }\n\n    /* 깊이 우선 순회 */\n    private func dfs(i: Int, order: String, res: inout [Int]) {\n        // 빈칸이면 반환한다\n        guard let val = val(i: i) else {\n            return\n        }\n        // 전위 순회\n        if order == \"pre\" {\n            res.append(val)\n        }\n        dfs(i: left(i: i), order: order, res: &amp;res)\n        // 중위 순회\n        if order == \"in\" {\n            res.append(val)\n        }\n        dfs(i: right(i: i), order: order, res: &amp;res)\n        // 후위 순회\n        if order == \"post\" {\n            res.append(val)\n        }\n    }\n\n    /* 전위 순회 */\n    func preOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"pre\", res: &amp;res)\n        return res\n    }\n\n    /* 중위 순회 */\n    func inOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"in\", res: &amp;res)\n        return res\n    }\n\n    /* 후위 순회 */\n    func postOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"post\", res: &amp;res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.js<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree {\n    #tree;\n\n    /* 생성자 */\n    constructor(arr) {\n        this.#tree = arr;\n    }\n\n    /* 리스트 용량 */\n    size() {\n        return this.#tree.length;\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    val(i) {\n        // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    left(i) {\n        return 2 * i + 1;\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    right(i) {\n        return 2 * i + 2;\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    parent(i) {\n        return Math.floor((i - 1) / 2); // 나눗셈의 결과를 내림한다\n    }\n\n    /* 레벨 순서 순회 */\n    levelOrder() {\n        let res = [];\n        // 배열을 직접 순회한다\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* 깊이 우선 순회 */\n    #dfs(i, order, res) {\n        // 빈칸이면 반환한다\n        if (this.val(i) === null) return;\n        // 전위 순회\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // 중위 순회\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // 후위 순회\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* 전위 순회 */\n    preOrder() {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* 중위 순회 */\n    inOrder() {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* 후위 순회 */\n    postOrder() {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.ts<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree {\n    #tree: (number | null)[];\n\n    /* 생성자 */\n    constructor(arr: (number | null)[]) {\n        this.#tree = arr;\n    }\n\n    /* 리스트 용량 */\n    size(): number {\n        return this.#tree.length;\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    val(i: number): number | null {\n        // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    left(i: number): number {\n        return 2 * i + 1;\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    right(i: number): number {\n        return 2 * i + 2;\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    parent(i: number): number {\n        return Math.floor((i - 1) / 2); // 나눗셈의 결과를 내림한다\n    }\n\n    /* 레벨 순서 순회 */\n    levelOrder(): number[] {\n        let res = [];\n        // 배열을 직접 순회한다\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* 깊이 우선 순회 */\n    #dfs(i: number, order: Order, res: (number | null)[]): void {\n        // 빈칸이면 반환한다\n        if (this.val(i) === null) return;\n        // 전위 순회\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // 중위 순회\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // 후위 순회\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* 전위 순회 */\n    preOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* 중위 순회 */\n    inOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* 후위 순회 */\n    postOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.dart<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree {\n  late List&lt;int?&gt; _tree;\n\n  /* 생성자 */\n  ArrayBinaryTree(this._tree);\n\n  /* 리스트 용량 */\n  int size() {\n    return _tree.length;\n  }\n\n  /* 인덱스가 i인 노드의 값 반환 */\n  int? val(int i) {\n    // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n    if (i &lt; 0 || i &gt;= size()) {\n      return null;\n    }\n    return _tree[i];\n  }\n\n  /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n  int? left(int i) {\n    return 2 * i + 1;\n  }\n\n  /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n  int? right(int i) {\n    return 2 * i + 2;\n  }\n\n  /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n  int? parent(int i) {\n    return (i - 1) ~/ 2;\n  }\n\n  /* 레벨 순서 순회 */\n  List&lt;int&gt; levelOrder() {\n    List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; size(); i++) {\n      if (val(i) != null) {\n        res.add(val(i)!);\n      }\n    }\n    return res;\n  }\n\n  /* 깊이 우선 순회 */\n  void dfs(int i, String order, List&lt;int?&gt; res) {\n    // 빈칸이면 반환한다\n    if (val(i) == null) {\n      return;\n    }\n    // 전위 순회\n    if (order == 'pre') {\n      res.add(val(i));\n    }\n    dfs(left(i)!, order, res);\n    // 중위 순회\n    if (order == 'in') {\n      res.add(val(i));\n    }\n    dfs(right(i)!, order, res);\n    // 후위 순회\n    if (order == 'post') {\n      res.add(val(i));\n    }\n  }\n\n  /* 전위 순회 */\n  List&lt;int?&gt; preOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'pre', res);\n    return res;\n  }\n\n  /* 중위 순회 */\n  List&lt;int?&gt; inOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'in', res);\n    return res;\n  }\n\n  /* 후위 순회 */\n  List&lt;int?&gt; postOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'post', res);\n    return res;\n  }\n}\n</code></pre> array_binary_tree.rs<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nstruct ArrayBinaryTree {\n    tree: Vec&lt;Option&lt;i32&gt;&gt;,\n}\n\nimpl ArrayBinaryTree {\n    /* 생성자 */\n    fn new(arr: Vec&lt;Option&lt;i32&gt;&gt;) -&gt; Self {\n        Self { tree: arr }\n    }\n\n    /* 리스트 용량 */\n    fn size(&amp;self) -&gt; i32 {\n        self.tree.len() as i32\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    fn val(&amp;self, i: i32) -&gt; Option&lt;i32&gt; {\n        // 인덱스가 범위를 벗어나면 빈칸을 나타내는 None을 반환한다\n        if i &lt; 0 || i &gt;= self.size() {\n            None\n        } else {\n            self.tree[i as usize]\n        }\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    fn left(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 1\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    fn right(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 2\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    fn parent(&amp;self, i: i32) -&gt; i32 {\n        (i - 1) / 2\n    }\n\n    /* 레벨 순서 순회 */\n    fn level_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        self.tree.iter().filter_map(|&amp;x| x).collect()\n    }\n\n    /* 깊이 우선 순회 */\n    fn dfs(&amp;self, i: i32, order: &amp;'static str, res: &amp;mut Vec&lt;i32&gt;) {\n        if self.val(i).is_none() {\n            return;\n        }\n        let val = self.val(i).unwrap();\n        // 전위 순회\n        if order == \"pre\" {\n            res.push(val);\n        }\n        self.dfs(self.left(i), order, res);\n        // 중위 순회\n        if order == \"in\" {\n            res.push(val);\n        }\n        self.dfs(self.right(i), order, res);\n        // 후위 순회\n        if order == \"post\" {\n            res.push(val);\n        }\n    }\n\n    /* 전위 순회 */\n    fn pre_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"pre\", &amp;mut res);\n        res\n    }\n\n    /* 중위 순회 */\n    fn in_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"in\", &amp;mut res);\n        res\n    }\n\n    /* 후위 순회 */\n    fn post_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"post\", &amp;mut res);\n        res\n    }\n}\n</code></pre> array_binary_tree.c<pre><code>/* 배열로 표현한 이진 트리 구조체 */\ntypedef struct {\n    int *tree;\n    int size;\n} ArrayBinaryTree;\n\n/* 생성자 */\nArrayBinaryTree *newArrayBinaryTree(int *arr, int arrSize) {\n    ArrayBinaryTree *abt = (ArrayBinaryTree *)malloc(sizeof(ArrayBinaryTree));\n    abt-&gt;tree = malloc(sizeof(int) * arrSize);\n    memcpy(abt-&gt;tree, arr, sizeof(int) * arrSize);\n    abt-&gt;size = arrSize;\n    return abt;\n}\n\n/* 소멸자 */\nvoid delArrayBinaryTree(ArrayBinaryTree *abt) {\n    free(abt-&gt;tree);\n    free(abt);\n}\n\n/* 리스트 용량 */\nint size(ArrayBinaryTree *abt) {\n    return abt-&gt;size;\n}\n\n/* 인덱스가 i인 노드의 값 반환 */\nint val(ArrayBinaryTree *abt, int i) {\n    // 인덱스가 범위를 벗어나면 빈자리를 나타내는 INT_MAX를 반환한다\n    if (i &lt; 0 || i &gt;= size(abt))\n        return INT_MAX;\n    return abt-&gt;tree[i];\n}\n\n/* 레벨 순서 순회 */\nint *levelOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    // 배열을 직접 순회한다\n    for (int i = 0; i &lt; size(abt); i++) {\n        if (val(abt, i) != INT_MAX)\n            res[index++] = val(abt, i);\n    }\n    *returnSize = index;\n    return res;\n}\n\n/* 깊이 우선 순회 */\nvoid dfs(ArrayBinaryTree *abt, int i, char *order, int *res, int *index) {\n    // 빈칸이면 반환한다\n    if (val(abt, i) == INT_MAX)\n        return;\n    // 전위 순회\n    if (strcmp(order, \"pre\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, left(i), order, res, index);\n    // 중위 순회\n    if (strcmp(order, \"in\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, right(i), order, res, index);\n    // 후위 순회\n    if (strcmp(order, \"post\") == 0)\n        res[(*index)++] = val(abt, i);\n}\n\n/* 전위 순회 */\nint *preOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"pre\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* 중위 순회 */\nint *inOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"in\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* 후위 순회 */\nint *postOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"post\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n</code></pre> array_binary_tree.kt<pre><code>/* 배열로 표현한 이진 트리 클래스 */\nclass ArrayBinaryTree(val tree: MutableList&lt;Int?&gt;) {\n    /* 리스트 용량 */\n    fun size(): Int {\n        return tree.size\n    }\n\n    /* 인덱스가 i인 노드의 값 반환 */\n    fun _val(i: Int): Int? {\n        // 인덱스가 범위를 벗어나면 빈칸을 나타내는 null을 반환한다\n        if (i &lt; 0 || i &gt;= size()) return null\n        return tree[i]\n    }\n\n    /* 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 */\n    fun left(i: Int): Int {\n        return 2 * i + 1\n    }\n\n    /* 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 */\n    fun right(i: Int): Int {\n        return 2 * i + 2\n    }\n\n    /* 인덱스가 i인 노드의 부모 노드 인덱스 반환 */\n    fun parent(i: Int): Int {\n        return (i - 1) / 2\n    }\n\n    /* 레벨 순서 순회 */\n    fun levelOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        // 배열을 직접 순회한다\n        for (i in 0..&lt;size()) {\n            if (_val(i) != null)\n                res.add(_val(i))\n        }\n        return res\n    }\n\n    /* 깊이 우선 순회 */\n    fun dfs(i: Int, order: String, res: MutableList&lt;Int?&gt;) {\n        // 빈칸이면 반환한다\n        if (_val(i) == null)\n            return\n        // 전위 순회\n        if (\"pre\" == order)\n            res.add(_val(i))\n        dfs(left(i), order, res)\n        // 중위 순회\n        if (\"in\" == order)\n            res.add(_val(i))\n        dfs(right(i), order, res)\n        // 후위 순회\n        if (\"post\" == order)\n            res.add(_val(i))\n    }\n\n    /* 전위 순회 */\n    fun preOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"pre\", res)\n        return res\n    }\n\n    /* 중위 순회 */\n    fun inOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"in\", res)\n        return res\n    }\n\n    /* 후위 순회 */\n    fun postOrder(): MutableList&lt;Int?&gt; {\n        val res = mutableListOf&lt;Int?&gt;()\n        dfs(0, \"post\", res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.rb<pre><code>### 배열로 표현한 이진 트리 클래스 ###\nclass ArrayBinaryTree\n  ### 생성자 ###\n  def initialize(arr)\n    @tree = arr.to_a\n  end\n\n  ### 리스트 용량 ###\n  def size\n    @tree.length\n  end\n\n  ### 인덱스가 i인 노드의 값 반환 ###\n  def val(i)\n    # 인덱스가 범위를 벗어나면 빈칸을 나타내는 nil을 반환한다\n    return if i &lt; 0 || i &gt;= size\n\n    @tree[i]\n  end\n\n  ### 인덱스가 i인 노드의 왼쪽 자식 노드 인덱스 반환 ###\n  def left(i)\n    2 * i + 1\n  end\n\n  ### 인덱스가 i인 노드의 오른쪽 자식 노드 인덱스 반환 ###\n  def right(i)\n    2 * i + 2\n  end\n\n  ### 인덱스가 i인 노드의 부모 노드 인덱스 반환 ###\n  def parent(i)\n    (i - 1) / 2\n  end\n\n  ### 레벨 순서 순회 ###\n  def level_order\n    @res = []\n\n    # 배열을 직접 순회한다\n    for i in 0...size\n      @res &lt;&lt; val(i) unless val(i).nil?\n    end\n\n    @res\n  end\n\n  ### 깊이 우선 순회 ###\n  def dfs(i, order)\n    return if val(i).nil?\n    # 전위 순회\n    @res &lt;&lt; val(i) if order == :pre\n    dfs(left(i), order)\n    # 중위 순회\n    @res &lt;&lt; val(i) if order == :in\n    dfs(right(i), order)\n    # 후위 순회\n    @res &lt;&lt; val(i) if order == :post\n  end\n\n  ### 전위 순회 ###\n  def pre_order\n    @res = []\n    dfs(0, :pre)\n    @res\n  end\n\n  ### 중위 순회 ###\n  def in_order\n    @res = []\n    dfs(0, :in)\n    @res\n  end\n\n  ### 후위 순회 ###\n  def post_order\n    @res = []\n    dfs(0, :post)\n    @res\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 7 장   트리","7.3   이진 트리의 배열 표현"],"tags":[]},{"location":"chapter_tree/array_representation_of_tree/#733","level":2,"title":"7.3.3   장점과 한계","text":"<p>이진 트리의 배열 표현에는 다음 장점이 있다.</p> <ul> <li>배열은 연속된 메모리에 저장되어 캐시 친화적이며, 접근과 순회가 빠르다.</li> <li>포인터를 저장할 필요가 없어 공간을 절약한다.</li> <li>노드에 임의 접근할 수 있다.</li> </ul> <p>하지만 다음과 같은 한계도 있다.</p> <ul> <li>연속된 메모리가 필요하므로 데이터가 지나치게 많은 트리를 저장하기에는 적합하지 않다.</li> <li>노드를 추가하거나 삭제하려면 배열의 삽입과 삭제 연산을 수행해야 하므로 효율이 낮다.</li> <li>트리에 <code>None</code>이 많으면 배열에서 실제 노드 데이터가 차지하는 비율이 낮아 공간 활용률이 떨어진다.</li> </ul>","path":["제 7 장   트리","7.3   이진 트리의 배열 표현"],"tags":[]},{"location":"chapter_tree/avl_tree/","level":1,"title":"7.5   AVL 트리 *","text":"<p>'이진 탐색 트리' 절에서 삽입과 삭제를 여러 번 수행하면 트리가 연결 리스트로 퇴화할 수 있다고 설명했다. 이 경우 모든 연산의 시간 복잡도가 \\(O(\\log n)\\)에서 \\(O(n)\\)으로 저하된다.</p> <p>아래 그림(7-24)의 이진 탐색 트리는 노드를 두 번 삭제하면 연결 리스트로 퇴화한다.</p> <p></p> <p> 그림 7-24   노드 삭제 후 퇴화한 AVL 트리 </p> <p>또한 아래 그림(7-25)의 포화 이진 트리에 노드 두 개를 삽입하면 왼쪽으로 심하게 기울어 탐색의 시간 복잡도도 저하된다.</p> <p></p> <p> 그림 7-25   노드 삽입 후 퇴화한 AVL 트리 </p> <p>1962년 G. M. Adelson-Velsky와 E. M. Landis는 논문 'An algorithm for the organization of information'에서 AVL 트리를 제안했다. 논문은 노드를 계속 추가하거나 삭제해도 트리가 퇴화하지 않게 하는 일련의 연산을 상세히 설명하여, 각 연산의 시간 복잡도를 \\(O(\\log n)\\)으로 유지하도록 했다. 즉, 추가, 삭제, 조회, 수정이 잦은 상황에서도 효율적인 데이터 연산 성능을 유지하므로 활용 가치가 높다.</p>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#751-avl","level":2,"title":"7.5.1   AVL 트리의 주요 용어","text":"<p>AVL 트리는 이진 탐색 트리이면서 균형 이진 트리다. 두 종류의 모든 성질을 함께 만족하므로 균형 이진 탐색 트리(balanced binary search tree)다.</p>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#1","level":3,"title":"1.   노드 높이","text":"<p>AVL 트리 연산에는 노드 높이가 필요하므로 노드 클래스에 <code>height</code> 변수를 추가해야 한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class TreeNode:\n    \"\"\"AVL 트리 노드 클래스\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                 # 노드 값\n        self.height: int = 0                # 노드 높이\n        self.left: TreeNode | None = None   # 왼쪽 자식 노드 참조\n        self.right: TreeNode | None = None  # 오른쪽 자식 노드 참조\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nstruct TreeNode {\n    int val{};          // 노드 값\n    int height = 0;     // 노드 높이\n    TreeNode *left{};   // 왼쪽 자식 노드\n    TreeNode *right{};  // 오른쪽 자식 노드\n    TreeNode() = default;\n    explicit TreeNode(int x) : val(x){}\n};\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nclass TreeNode {\n    public int val;        // 노드 값\n    public int height;     // 노드 높이\n    public TreeNode left;  // 왼쪽 자식 노드\n    public TreeNode right; // 오른쪽 자식 노드\n    public TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nclass TreeNode(int? x) {\n    public int? val = x;    // 노드 값\n    public int height;      // 노드 높이\n    public TreeNode? left;  // 왼쪽 자식 노드 참조\n    public TreeNode? right; // 오른쪽 자식 노드 참조\n}\n</code></pre> <pre><code>/* AVL 트리 노드 구조체 */\ntype TreeNode struct {\n    Val    int       // 노드 값\n    Height int       // 노드 높이\n    Left   *TreeNode // 왼쪽 자식 노드 참조\n    Right  *TreeNode // 오른쪽 자식 노드 참조\n}\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nclass TreeNode {\n    var val: Int // 노드 값\n    var height: Int // 노드 높이\n    var left: TreeNode? // 왼쪽 자식 노드\n    var right: TreeNode? // 오른쪽 자식 노드\n\n    init(x: Int) {\n        val = x\n        height = 0\n    }\n}\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nclass TreeNode {\n    val; // 노드 값\n    height; //노드 높이\n    left; // 왼쪽 자식 노드 포인터\n    right; // 오른쪽 자식 노드 포인터\n    constructor(val, left, right, height) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nclass TreeNode {\n    val: number;            // 노드 값\n    height: number;         // 노드 높이\n    left: TreeNode | null;  // 왼쪽 자식 노드 포인터\n    right: TreeNode | null; // 오른쪽 자식 노드 포인터\n    constructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nclass TreeNode {\n  int val;         // 노드 값\n  int height;      // 노드 높이\n  TreeNode? left;  // 왼쪽 자식 노드\n  TreeNode? right; // 오른쪽 자식 노드\n  TreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* AVL 트리 노드 구조체 */\nstruct TreeNode {\n    val: i32,                               // 노드 값\n    height: i32,                            // 노드 높이\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // 왼쪽 자식 노드\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // 오른쪽 자식 노드\n}\n\nimpl TreeNode {\n    /* 생성자 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            height: 0,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* AVL 트리 노드 구조체 */\ntypedef struct TreeNode {\n    int val;\n    int height;\n    struct TreeNode *left;\n    struct TreeNode *right;\n} TreeNode;\n\n/* 생성자 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* AVL 트리 노드 클래스 */\nclass TreeNode(val _val: Int) {  // 노드 값\n    var height: Int = 0          // 노드 높이\n    var left: TreeNode? = null   // 왼쪽 자식 노드\n    var right: TreeNode? = null  // 오른쪽 자식 노드\n}\n</code></pre> <pre><code>### AVL 트리 노드 클래스 ###\nclass TreeNode\n  attr_accessor :val    # 노드 값\n  attr_accessor :height # 노드 높이\n  attr_accessor :left   # 왼쪽 자식 노드 참조\n  attr_accessor :right  # 오른쪽 자식 노드 참조\n\n  def initialize(val)\n    @val = val\n    @height = 0\n  end\nend\n</code></pre> <p>'노드 높이'는 해당 노드에서 가장 먼 리프까지의 거리, 즉 거치는 '간선' 수다. 특히 리프의 높이는 \\(0\\), 빈 노드의 높이는 \\(-1\\)임에 주의하자. 높이를 조회하고 갱신하는 두 보조 함수를 만든다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def height(self, node: TreeNode | None) -&gt; int:\n    \"\"\"노드 높이 반환\"\"\"\n    # 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    if node is not None:\n        return node.height\n    return -1\n\ndef update_height(self, node: TreeNode | None):\n    \"\"\"노드 높이를 갱신한다\"\"\"\n    # 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node.height = max([self.height(node.left), self.height(node.right)]) + 1\n</code></pre> avl_tree.cpp<pre><code>/* 노드 높이 반환 */\nint height(TreeNode *node) {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    return node == nullptr ? -1 : node-&gt;height;\n}\n\n/* 노드 높이를 갱신한다 */\nvoid updateHeight(TreeNode *node) {\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node-&gt;height = max(height(node-&gt;left), height(node-&gt;right)) + 1;\n}\n</code></pre> avl_tree.java<pre><code>/* 노드 높이 반환 */\nint height(TreeNode node) {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    return node == null ? -1 : node.height;\n}\n\n/* 노드 높이를 갱신한다 */\nvoid updateHeight(TreeNode node) {\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.cs<pre><code>/* 노드 높이 반환 */\nint Height(TreeNode? node) {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    return node == null ? -1 : node.height;\n}\n\n/* 노드 높이를 갱신한다 */\nvoid UpdateHeight(TreeNode node) {\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node.height = Math.Max(Height(node.left), Height(node.right)) + 1;\n}\n</code></pre> avl_tree.go<pre><code>/* 노드 높이 반환 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    if node != nil {\n        return node.Height\n    }\n    return -1\n}\n\n/* 노드 높이를 갱신한다 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\n    lh := t.height(node.Left)\n    rh := t.height(node.Right)\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    if lh &gt; rh {\n        node.Height = lh + 1\n    } else {\n        node.Height = rh + 1\n    }\n}\n</code></pre> avl_tree.swift<pre><code>/* 노드 높이 반환 */\nfunc height(node: TreeNode?) -&gt; Int {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    node?.height ?? -1\n}\n\n/* 노드 높이를 갱신한다 */\nfunc updateHeight(node: TreeNode?) {\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n</code></pre> avl_tree.js<pre><code>/* 노드 높이 반환 */\nheight(node) {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    return node === null ? -1 : node.height;\n}\n\n/* 노드 높이를 갱신한다 */\n#updateHeight(node) {\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.ts<pre><code>/* 노드 높이 반환 */\nheight(node: TreeNode): number {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    return node === null ? -1 : node.height;\n}\n\n/* 노드 높이를 갱신한다 */\nupdateHeight(node: TreeNode): void {\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.dart<pre><code>/* 노드 높이 반환 */\nint height(TreeNode? node) {\n  // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n  return node == null ? -1 : node.height;\n}\n\n/* 노드 높이를 갱신한다 */\nvoid updateHeight(TreeNode? node) {\n  // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n  node!.height = max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.rs<pre><code>/* 노드 높이 반환 */\nfn height(node: OptionTreeNodeRc) -&gt; i32 {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    match node {\n        Some(node) =&gt; node.borrow().height,\n        None =&gt; -1,\n    }\n}\n\n/* 노드 높이를 갱신한다 */\nfn update_height(node: OptionTreeNodeRc) {\n    if let Some(node) = node {\n        let left = node.borrow().left.clone();\n        let right = node.borrow().right.clone();\n        // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n        node.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* 노드 높이 반환 */\nint height(TreeNode *node) {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    if (node != NULL) {\n        return node-&gt;height;\n    }\n    return -1;\n}\n\n/* 노드 높이를 갱신한다 */\nvoid updateHeight(TreeNode *node) {\n    int lh = height(node-&gt;left);\n    int rh = height(node-&gt;right);\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    if (lh &gt; rh) {\n        node-&gt;height = lh + 1;\n    } else {\n        node-&gt;height = rh + 1;\n    }\n}\n</code></pre> avl_tree.kt<pre><code>/* 노드 높이 반환 */\nfun height(node: TreeNode?): Int {\n    // 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n    return node?.height ?: -1\n}\n\n/* 노드 높이를 갱신한다 */\nfun updateHeight(node: TreeNode?) {\n    // 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n    node?.height = max(height(node?.left), height(node?.right)) + 1\n}\n</code></pre> avl_tree.rb<pre><code>### 노드 높이 반환 ###\ndef height(node)\n  # 빈 노드의 높이는 -1이고, 리프 노드의 높이는 0이다\n  return node.height unless node.nil?\n\n  -1\nend\n\n### 노드 높이를 갱신한다 ###\ndef update_height(node)\n  # 노드 높이는 가장 높은 서브트리의 높이 + 1이다\n  node.height = [height(node.left), height(node.right)].max + 1\nend\n</code></pre>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#2","level":3,"title":"2.   노드의 균형 인수","text":"<p>노드의 균형 인수(balance factor)는 왼쪽 서브트리 높이에서 오른쪽 서브트리 높이를 뺀 값이며, 빈 노드의 균형 인수는 \\(0\\)으로 정한다. 이후 편리하게 쓰도록 균형 인수를 구하는 기능도 함수로 만든다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def balance_factor(self, node: TreeNode | None) -&gt; int:\n    \"\"\"균형 인수 반환\"\"\"\n    # 빈 노드의 균형 인수는 0이다\n    if node is None:\n        return 0\n    # 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return self.height(node.left) - self.height(node.right)\n</code></pre> avl_tree.cpp<pre><code>/* 균형 인수 반환 */\nint balanceFactor(TreeNode *node) {\n    // 빈 노드의 균형 인수는 0이다\n    if (node == nullptr)\n        return 0;\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.java<pre><code>/* 균형 인수 반환 */\nint balanceFactor(TreeNode node) {\n    // 빈 노드의 균형 인수는 0이다\n    if (node == null)\n        return 0;\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.cs<pre><code>/* 균형 인수 반환 */\nint BalanceFactor(TreeNode? node) {\n    // 빈 노드의 균형 인수는 0이다\n    if (node == null) return 0;\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return Height(node.left) - Height(node.right);\n}\n</code></pre> avl_tree.go<pre><code>/* 균형 인수 반환 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n    // 빈 노드의 균형 인수는 0이다\n    if node == nil {\n        return 0\n    }\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return t.height(node.Left) - t.height(node.Right)\n}\n</code></pre> avl_tree.swift<pre><code>/* 균형 인수 반환 */\nfunc balanceFactor(node: TreeNode?) -&gt; Int {\n    // 빈 노드의 균형 인수는 0이다\n    guard let node = node else { return 0 }\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return height(node: node.left) - height(node: node.right)\n}\n</code></pre> avl_tree.js<pre><code>/* 균형 인수 반환 */\nbalanceFactor(node) {\n    // 빈 노드의 균형 인수는 0이다\n    if (node === null) return 0;\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.ts<pre><code>/* 균형 인수 반환 */\nbalanceFactor(node: TreeNode): number {\n    // 빈 노드의 균형 인수는 0이다\n    if (node === null) return 0;\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.dart<pre><code>/* 균형 인수 반환 */\nint balanceFactor(TreeNode? node) {\n  // 빈 노드의 균형 인수는 0이다\n  if (node == null) return 0;\n  // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n  return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.rs<pre><code>/* 균형 인수 반환 */\nfn balance_factor(node: OptionTreeNodeRc) -&gt; i32 {\n    match node {\n        // 빈 노드의 균형 인수는 0이다\n        None =&gt; 0,\n        // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n        Some(node) =&gt; {\n            Self::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())\n        }\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* 균형 인수 반환 */\nint balanceFactor(TreeNode *node) {\n    // 빈 노드의 균형 인수는 0이다\n    if (node == NULL) {\n        return 0;\n    }\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.kt<pre><code>/* 균형 인수 반환 */\nfun balanceFactor(node: TreeNode?): Int {\n    // 빈 노드의 균형 인수는 0이다\n    if (node == null) return 0\n    // 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n    return height(node.left) - height(node.right)\n}\n</code></pre> avl_tree.rb<pre><code>### 균형 인수 반환 ###\ndef balance_factor(node)\n  # 빈 노드의 균형 인수는 0이다\n  return 0 if node.nil?\n\n  # 노드의 균형 인수 = 왼쪽 서브트리 높이 - 오른쪽 서브트리 높이\n  height(node.left) - height(node.right)\nend\n</code></pre> <p>참고</p> <p>균형 인수를 \\(f\\)라고 하면 AVL 트리의 모든 노드는 \\(-1 \\le f \\le 1\\)을 만족한다.</p>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#752-avl","level":2,"title":"7.5.2   AVL 트리의 회전","text":"<p>AVL 트리의 특징은 '회전' 연산이다. 이진 트리의 중위 순회 결과를 바꾸지 않으면서 불균형 노드의 균형을 회복한다. 즉, 회전은 '이진 탐색 트리'의 성질을 유지하면서 트리를 다시 '균형 이진 트리'로 만든다.</p> <p>균형 인수의 절댓값이 \\(&gt; 1\\)인 노드를 '불균형 노드'라고 한다. 불균형 상황에 따라 오른쪽 회전, 왼쪽 회전, 오른쪽 후 왼쪽 회전, 왼쪽 후 오른쪽 회전의 네 종류로 나뉜다. 각각 자세히 알아보자.</p>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#1_1","level":3,"title":"1.   오른쪽 회전","text":"<p>아래 그림(7-26)에서 노드 아래의 수는 균형 인수다. 아래에서 위로 볼 때 첫 불균형 노드는 '노드 3'이다. 이 노드를 루트로 하는 서브트리에 주목하여 불균형 노드를 <code>node</code>, 왼쪽 자식을 <code>child</code>라고 하고 '오른쪽 회전'을 수행한다. 회전 후 서브트리는 균형을 되찾으며 이진 탐색 트리의 성질도 유지한다.</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> 그림 7-26   오른쪽 회전 단계 </p> <p>아래 그림(7-27)처럼 <code>child</code>에 오른쪽 자식 <code>grand_child</code>가 있으면, 오른쪽 회전 중 <code>grand_child</code>를 <code>node</code>의 왼쪽 자식으로 연결하는 단계를 추가해야 한다.</p> <p></p> <p> 그림 7-27   grand_child가 있는 오른쪽 회전 </p> <p>'오른쪽으로 회전한다'는 시각적인 표현이며, 실제로는 노드 포인터를 수정하여 구현한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def right_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"오른쪽 회전 연산\"\"\"\n    child = node.left\n    grand_child = child.right\n    # child를 중심으로 node를 오른쪽으로 회전한다\n    child.right = node\n    node.left = grand_child\n    # 노드 높이를 갱신한다\n    self.update_height(node)\n    self.update_height(child)\n    # 회전한 서브트리의 루트 노드를 반환한다\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* 오른쪽 회전 연산 */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;left;\n    TreeNode *grandChild = child-&gt;right;\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* 오른쪽 회전 연산 */\nTreeNode rightRotate(TreeNode node) {\n    TreeNode child = node.left;\n    TreeNode grandChild = child.right;\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child.right = node;\n    node.left = grandChild;\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* 오른쪽 회전 연산 */\nTreeNode? RightRotate(TreeNode? node) {\n    TreeNode? child = node?.left;\n    TreeNode? grandChild = child?.right;\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child.right = node;\n    node.left = grandChild;\n    // 노드 높이를 갱신한다\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* 오른쪽 회전 연산 */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\n    child := node.Left\n    grandChild := child.Right\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child.Right = node\n    node.Left = grandChild\n    // 노드 높이를 갱신한다\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* 오른쪽 회전 연산 */\nfunc rightRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.left\n    let grandChild = child?.right\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child?.right = node\n    node?.left = grandChild\n    // 노드 높이를 갱신한다\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* 오른쪽 회전 연산 */\n#rightRotate(node) {\n    const child = node.left;\n    const grandChild = child.right;\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child.right = node;\n    node.left = grandChild;\n    // 노드 높이를 갱신한다\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* 오른쪽 회전 연산 */\nrightRotate(node: TreeNode): TreeNode {\n    const child = node.left;\n    const grandChild = child.right;\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child.right = node;\n    node.left = grandChild;\n    // 노드 높이를 갱신한다\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* 오른쪽 회전 연산 */\nTreeNode? rightRotate(TreeNode? node) {\n  TreeNode? child = node!.left;\n  TreeNode? grandChild = child!.right;\n  // child를 중심으로 node를 오른쪽으로 회전한다\n  child.right = node;\n  node.left = grandChild;\n  // 노드 높이를 갱신한다\n  updateHeight(node);\n  updateHeight(child);\n  // 회전한 서브트리의 루트 노드를 반환한다\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* 오른쪽 회전 연산 */\nfn right_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().left.clone().unwrap();\n            let grand_child = child.borrow().right.clone();\n            // child를 중심으로 node를 오른쪽으로 회전한다\n            child.borrow_mut().right = Some(node.clone());\n            node.borrow_mut().left = grand_child;\n            // 노드 높이를 갱신한다\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // 회전한 서브트리의 루트 노드를 반환한다\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* 오른쪽 회전 연산 */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;left;\n    grandChild = child-&gt;right;\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* 오른쪽 회전 연산 */\nfun rightRotate(node: TreeNode?): TreeNode {\n    val child = node!!.left\n    val grandChild = child!!.right\n    // child를 중심으로 node를 오른쪽으로 회전한다\n    child.right = node\n    node.left = grandChild\n    // 노드 높이를 갱신한다\n    updateHeight(node)\n    updateHeight(child)\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>### 오른쪽 회전 연산 ###\ndef right_rotate(node)\n  child = node.left\n  grand_child = child.right\n  # child를 중심으로 node를 오른쪽으로 회전한다\n  child.right = node\n  node.left = grand_child\n  # 노드 높이를 갱신한다\n  update_height(node)\n  update_height(child)\n  # 회전한 서브트리의 루트 노드를 반환한다\n  child\nend\n</code></pre>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#2_1","level":3,"title":"2.   왼쪽 회전","text":"<p>앞의 불균형 이진 트리를 '거울에 비춘' 경우에는 아래 그림(7-28)과 같이 '왼쪽 회전'이 필요하다.</p> <p></p> <p> 그림 7-28   왼쪽 회전 </p> <p>마찬가지로 아래 그림(7-29)처럼 <code>child</code>에 왼쪽 자식 <code>grand_child</code>가 있으면, 왼쪽 회전 중 <code>grand_child</code>를 <code>node</code>의 오른쪽 자식으로 연결하는 단계를 추가해야 한다.</p> <p></p> <p> 그림 7-29   grand_child가 있는 왼쪽 회전 </p> <p>오른쪽 회전과 왼쪽 회전은 논리적으로 거울 대칭이며, 각각 해결하는 불균형 상황도 대칭이다. 따라서 오른쪽 회전 코드의 모든 <code>left</code>를 <code>right</code>로, 모든 <code>right</code>를 <code>left</code>로 바꾸면 왼쪽 회전 코드를 얻는다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def left_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"왼쪽 회전 연산\"\"\"\n    child = node.right\n    grand_child = child.left\n    # child를 중심으로 node를 왼쪽으로 회전한다\n    child.left = node\n    node.right = grand_child\n    # 노드 높이를 갱신한다\n    self.update_height(node)\n    self.update_height(child)\n    # 회전한 서브트리의 루트 노드를 반환한다\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* 왼쪽 회전 연산 */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;right;\n    TreeNode *grandChild = child-&gt;left;\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* 왼쪽 회전 연산 */\nTreeNode leftRotate(TreeNode node) {\n    TreeNode child = node.right;\n    TreeNode grandChild = child.left;\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child.left = node;\n    node.right = grandChild;\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* 왼쪽 회전 연산 */\nTreeNode? LeftRotate(TreeNode? node) {\n    TreeNode? child = node?.right;\n    TreeNode? grandChild = child?.left;\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child.left = node;\n    node.right = grandChild;\n    // 노드 높이를 갱신한다\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* 왼쪽 회전 연산 */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\n    child := node.Right\n    grandChild := child.Left\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child.Left = node\n    node.Right = grandChild\n    // 노드 높이를 갱신한다\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* 왼쪽 회전 연산 */\nfunc leftRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.right\n    let grandChild = child?.left\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child?.left = node\n    node?.right = grandChild\n    // 노드 높이를 갱신한다\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* 왼쪽 회전 연산 */\n#leftRotate(node) {\n    const child = node.right;\n    const grandChild = child.left;\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child.left = node;\n    node.right = grandChild;\n    // 노드 높이를 갱신한다\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* 왼쪽 회전 연산 */\nleftRotate(node: TreeNode): TreeNode {\n    const child = node.right;\n    const grandChild = child.left;\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child.left = node;\n    node.right = grandChild;\n    // 노드 높이를 갱신한다\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* 왼쪽 회전 연산 */\nTreeNode? leftRotate(TreeNode? node) {\n  TreeNode? child = node!.right;\n  TreeNode? grandChild = child!.left;\n  // child를 중심으로 node를 왼쪽으로 회전한다\n  child.left = node;\n  node.right = grandChild;\n  // 노드 높이를 갱신한다\n  updateHeight(node);\n  updateHeight(child);\n  // 회전한 서브트리의 루트 노드를 반환한다\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* 왼쪽 회전 연산 */\nfn left_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().right.clone().unwrap();\n            let grand_child = child.borrow().left.clone();\n            // child를 중심으로 node를 왼쪽으로 회전한다\n            child.borrow_mut().left = Some(node.clone());\n            node.borrow_mut().right = grand_child;\n            // 노드 높이를 갱신한다\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // 회전한 서브트리의 루트 노드를 반환한다\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* 왼쪽 회전 연산 */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;right;\n    grandChild = child-&gt;left;\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    updateHeight(child);\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child;\n}\n</code></pre> avl_tree.kt<pre><code>/* 왼쪽 회전 연산 */\nfun leftRotate(node: TreeNode?): TreeNode {\n    val child = node!!.right\n    val grandChild = child!!.left\n    // child를 중심으로 node를 왼쪽으로 회전한다\n    child.left = node\n    node.right = grandChild\n    // 노드 높이를 갱신한다\n    updateHeight(node)\n    updateHeight(child)\n    // 회전한 서브트리의 루트 노드를 반환한다\n    return child\n}\n</code></pre> avl_tree.rb<pre><code>### 왼쪽 회전 연산 ###\ndef left_rotate(node)\n  child = node.right\n  grand_child = child.left\n  # child를 중심으로 node를 왼쪽으로 회전한다\n  child.left = node\n  node.right = grand_child\n  # 노드 높이를 갱신한다\n  update_height(node)\n  update_height(child)\n  # 회전한 서브트리의 루트 노드를 반환한다\n  child\nend\n</code></pre>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#3","level":3,"title":"3.   왼쪽 회전 후 오른쪽 회전","text":"<p>아래 그림(7-30)의 불균형 노드 3은 왼쪽 회전이나 오른쪽 회전 한 번만으로 균형을 회복할 수 없다. 먼저 <code>child</code>를 '왼쪽 회전'한 뒤 <code>node</code>를 '오른쪽 회전'해야 한다.</p> <p></p> <p> 그림 7-30   왼쪽 회전 후 오른쪽 회전 </p>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#4","level":3,"title":"4.   오른쪽 회전 후 왼쪽 회전","text":"<p>아래 그림(7-31)처럼 앞 상황의 거울 대칭인 경우에는 먼저 <code>child</code>를 '오른쪽 회전'한 뒤 <code>node</code>를 '왼쪽 회전'해야 한다.</p> <p></p> <p> 그림 7-31   오른쪽 회전 후 왼쪽 회전 </p>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#5","level":3,"title":"5.   회전 선택","text":"<p>아래 그림(7-32)의 네 불균형 상황은 앞의 예시들과 각각 대응한다. 순서대로 오른쪽 회전, 왼쪽 후 오른쪽 회전, 오른쪽 후 왼쪽 회전, 왼쪽 회전이 필요하다.</p> <p></p> <p> 그림 7-32   AVL 트리의 네 가지 회전 상황 </p> <p>아래 표(7-3)처럼 불균형 노드의 균형 인수와 더 높은 쪽 자식의 균형 인수 부호를 확인하여 어느 상황에 해당하는지 판단한다.</p> <p> 표 7-3   네 가지 회전의 선택 조건 </p> 불균형 노드의 균형 인수 자식 노드의 균형 인수 사용할 회전 \\(&gt; 1\\) (왼쪽으로 치우친 트리) \\(\\geq 0\\) 오른쪽 회전 \\(&gt; 1\\) (왼쪽으로 치우친 트리) \\(&lt;0\\) 왼쪽 회전 후 오른쪽 회전 \\(&lt; -1\\) (오른쪽으로 치우친 트리) \\(\\leq 0\\) 왼쪽 회전 \\(&lt; -1\\) (오른쪽으로 치우친 트리) \\(&gt;0\\) 오른쪽 회전 후 왼쪽 회전 <p>편리하게 사용하도록 회전 연산을 함수 하나로 묶는다. 이 함수를 사용하면 여러 불균형 상황에 맞게 회전하여 노드의 균형을 복구할 수 있다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"회전 연산을 수행하여 해당 서브트리의 균형 복원\"\"\"\n    # 노드 node의 균형 인수를 구한다\n    balance_factor = self.balance_factor(node)\n    # 왼쪽으로 치우친 트리\n    if balance_factor &gt; 1:\n        if self.balance_factor(node.left) &gt;= 0:\n            # 오른쪽 회전\n            return self.right_rotate(node)\n        else:\n            # 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node.left = self.left_rotate(node.left)\n            return self.right_rotate(node)\n    # 오른쪽으로 치우친 트리\n    elif balance_factor &lt; -1:\n        if self.balance_factor(node.right) &lt;= 0:\n            # 왼쪽 회전\n            return self.left_rotate(node)\n        else:\n            # 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node.right = self.right_rotate(node.right)\n            return self.left_rotate(node)\n    # 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node\n</code></pre> avl_tree.cpp<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nTreeNode *rotate(TreeNode *node) {\n    // 노드 node의 균형 인수를 구한다\n    int _balanceFactor = balanceFactor(node);\n    // 왼쪽으로 치우친 트리\n    if (_balanceFactor &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // 오른쪽 회전\n            return rightRotate(node);\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if (_balanceFactor &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // 왼쪽 회전\n            return leftRotate(node);\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nTreeNode rotate(TreeNode node) {\n    // 노드 node의 균형 인수를 구한다\n    int balanceFactor = balanceFactor(node);\n    // 왼쪽으로 치우친 트리\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // 오른쪽 회전\n            return rightRotate(node);\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node.left = leftRotate(node.left);\n            return rightRotate(node);\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // 왼쪽 회전\n            return leftRotate(node);\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node.right = rightRotate(node.right);\n            return leftRotate(node);\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nTreeNode? Rotate(TreeNode? node) {\n    // 노드 node의 균형 인수를 구한다\n    int balanceFactorInt = BalanceFactor(node);\n    // 왼쪽으로 치우친 트리\n    if (balanceFactorInt &gt; 1) {\n        if (BalanceFactor(node?.left) &gt;= 0) {\n            // 오른쪽 회전\n            return RightRotate(node);\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node!.left = LeftRotate(node!.left);\n            return RightRotate(node);\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if (balanceFactorInt &lt; -1) {\n        if (BalanceFactor(node?.right) &lt;= 0) {\n            // 왼쪽 회전\n            return LeftRotate(node);\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node!.right = RightRotate(node!.right);\n            return LeftRotate(node);\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n    // 노드 node의 균형 인수를 구한다\n    // Go에서는 짧은 변수명을 권장한다. 여기서 bf는 t.balanceFactor를 나타낸다\n    bf := t.balanceFactor(node)\n    // 왼쪽으로 치우친 트리\n    if bf &gt; 1 {\n        if t.balanceFactor(node.Left) &gt;= 0 {\n            // 오른쪽 회전\n            return t.rightRotate(node)\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node.Left = t.leftRotate(node.Left)\n            return t.rightRotate(node)\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if bf &lt; -1 {\n        if t.balanceFactor(node.Right) &lt;= 0 {\n            // 왼쪽 회전\n            return t.leftRotate(node)\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node.Right = t.rightRotate(node.Right)\n            return t.leftRotate(node)\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nfunc rotate(node: TreeNode?) -&gt; TreeNode? {\n    // 노드 node의 균형 인수를 구한다\n    let balanceFactor = balanceFactor(node: node)\n    // 왼쪽으로 치우친 트리\n    if balanceFactor &gt; 1 {\n        if self.balanceFactor(node: node?.left) &gt;= 0 {\n            // 오른쪽 회전\n            return rightRotate(node: node)\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node?.left = leftRotate(node: node?.left)\n            return rightRotate(node: node)\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if balanceFactor &lt; -1 {\n        if self.balanceFactor(node: node?.right) &lt;= 0 {\n            // 왼쪽 회전\n            return leftRotate(node: node)\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node?.right = rightRotate(node: node?.right)\n            return leftRotate(node: node)\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\n#rotate(node) {\n    // 노드 node의 균형 인수를 구한다\n    const balanceFactor = this.balanceFactor(node);\n    // 왼쪽으로 치우친 트리\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // 오른쪽 회전\n            return this.#rightRotate(node);\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node.left = this.#leftRotate(node.left);\n            return this.#rightRotate(node);\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // 왼쪽 회전\n            return this.#leftRotate(node);\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node.right = this.#rightRotate(node.right);\n            return this.#leftRotate(node);\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nrotate(node: TreeNode): TreeNode {\n    // 노드 node의 균형 인수를 구한다\n    const balanceFactor = this.balanceFactor(node);\n    // 왼쪽으로 치우친 트리\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // 오른쪽 회전\n            return this.rightRotate(node);\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node.left = this.leftRotate(node.left);\n            return this.rightRotate(node);\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // 왼쪽 회전\n            return this.leftRotate(node);\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node.right = this.rightRotate(node.right);\n            return this.leftRotate(node);\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nTreeNode? rotate(TreeNode? node) {\n  // 노드 node의 균형 인수를 구한다\n  int factor = balanceFactor(node);\n  // 왼쪽으로 치우친 트리\n  if (factor &gt; 1) {\n    if (balanceFactor(node!.left) &gt;= 0) {\n      // 오른쪽 회전\n      return rightRotate(node);\n    } else {\n      // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n      node.left = leftRotate(node.left);\n      return rightRotate(node);\n    }\n  }\n  // 오른쪽으로 치우친 트리\n  if (factor &lt; -1) {\n    if (balanceFactor(node!.right) &lt;= 0) {\n      // 왼쪽 회전\n      return leftRotate(node);\n    } else {\n      // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n      node.right = rightRotate(node.right);\n      return leftRotate(node);\n    }\n  }\n  // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nfn rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    // 노드 node의 균형 인수를 구한다\n    let balance_factor = Self::balance_factor(node.clone());\n    // 왼쪽으로 치우친 트리\n    if balance_factor &gt; 1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().left.clone()) &gt;= 0 {\n            // 오른쪽 회전\n            Self::right_rotate(Some(node))\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            let left = node.borrow().left.clone();\n            node.borrow_mut().left = Self::left_rotate(left);\n            Self::right_rotate(Some(node))\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    else if balance_factor &lt; -1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().right.clone()) &lt;= 0 {\n            // 왼쪽 회전\n            Self::left_rotate(Some(node))\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            let right = node.borrow().right.clone();\n            node.borrow_mut().right = Self::right_rotate(right);\n            Self::left_rotate(Some(node))\n        }\n    } else {\n        // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n        node\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nTreeNode *rotate(TreeNode *node) {\n    // 노드 node의 균형 인수를 구한다\n    int bf = balanceFactor(node);\n    // 왼쪽으로 치우친 트리\n    if (bf &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // 오른쪽 회전\n            return rightRotate(node);\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if (bf &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // 왼쪽 회전\n            return leftRotate(node);\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* 회전 연산을 수행하여 해당 서브트리의 균형 복원 */\nfun rotate(node: TreeNode): TreeNode {\n    // 노드 node의 균형 인수를 구한다\n    val balanceFactor = balanceFactor(node)\n    // 왼쪽으로 치우친 트리\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // 오른쪽 회전\n            return rightRotate(node)\n        } else {\n            // 왼쪽으로 회전한 후 오른쪽으로 회전한다\n            node.left = leftRotate(node.left)\n            return rightRotate(node)\n        }\n    }\n    // 오른쪽으로 치우친 트리\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // 왼쪽 회전\n            return leftRotate(node)\n        } else {\n            // 오른쪽으로 회전한 후 왼쪽으로 회전한다\n            node.right = rightRotate(node.right)\n            return leftRotate(node)\n        }\n    }\n    // 균형 트리이므로 회전할 필요 없이 바로 반환한다\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### 회전 연산을 수행하여 해당 서브트리의 균형 복원 ###\ndef rotate(node)\n  # 노드 node의 균형 인수를 구한다\n  balance_factor = balance_factor(node)\n  # 왼쪽으로 치우친 트리\n  if balance_factor &gt; 1\n    if balance_factor(node.left) &gt;= 0\n      # 오른쪽 회전\n      return right_rotate(node)\n    else\n      # 왼쪽으로 회전한 후 오른쪽으로 회전한다\n      node.left = left_rotate(node.left)\n      return right_rotate(node)\n    end\n  # 오른쪽으로 치우친 트리\n  elsif balance_factor &lt; -1\n    if balance_factor(node.right) &lt;= 0\n      # 왼쪽 회전\n      return left_rotate(node)\n    else\n      # 오른쪽으로 회전한 후 왼쪽으로 회전한다\n      node.right = right_rotate(node.right)\n      return left_rotate(node)\n    end\n  end\n  # 균형 트리이므로 회전할 필요 없이 바로 반환한다\n  node\nend\n</code></pre>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#753-avl","level":2,"title":"7.5.3   AVL 트리의 주요 연산","text":"","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#1_2","level":3,"title":"1.   노드 삽입","text":"<p>AVL 트리의 삽입은 기본적으로 이진 탐색 트리와 비슷하다. 다만 삽입한 노드에서 루트까지의 경로에 불균형 노드가 여러 개 생길 수 있다는 차이가 있다. 따라서 삽입한 노드부터 아래에서 위로 회전하여 모든 불균형 노드의 균형을 회복해야 한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def insert(self, val):\n    \"\"\"노드 삽입\"\"\"\n    self._root = self.insert_helper(self._root, val)\n\ndef insert_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode:\n    \"\"\"노드의 재귀적 삽입(보조 메서드)\"\"\"\n    if node is None:\n        return TreeNode(val)\n    # 1. 삽입 위치를 찾아 노드를 삽입한다\n    if val &lt; node.val:\n        node.left = self.insert_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.insert_helper(node.right, val)\n    else:\n        # 중복 노드는 삽입하지 않고 바로 반환한다\n        return node\n    # 노드 높이를 갱신한다\n    self.update_height(node)\n    # 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* 노드 삽입 */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return new TreeNode(val);\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    else\n        return node;    // 중복 노드는 삽입하지 않고 바로 반환한다\n    updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* 노드 삽입 */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\nTreeNode insertHelper(TreeNode node, int val) {\n    if (node == null)\n        return new TreeNode(val);\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if (val &lt; node.val)\n        node.left = insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = insertHelper(node.right, val);\n    else\n        return node; // 중복 노드는 삽입하지 않고 바로 반환한다\n    updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* 노드 삽입 */\nvoid Insert(int val) {\n    root = InsertHelper(root, val);\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\nTreeNode? InsertHelper(TreeNode? node, int val) {\n    if (node == null) return new TreeNode(val);\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if (val &lt; node.val)\n        node.left = InsertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = InsertHelper(node.right, val);\n    else\n        return node;     // 중복 노드는 삽입하지 않고 바로 반환한다\n    UpdateHeight(node);  // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = Rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* 노드 삽입 */\nfunc (t *aVLTree) insert(val int) {\n    t.root = t.insertHelper(t.root, val)\n}\n\n/* 노드의 재귀적 삽입(보조 함수) */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return NewTreeNode(val)\n    }\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if val &lt; node.Val.(int) {\n        node.Left = t.insertHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.insertHelper(node.Right, val)\n    } else {\n        // 중복 노드는 삽입하지 않고 바로 반환한다\n        return node\n    }\n    // 노드 높이를 갱신한다\n    t.updateHeight(node)\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = t.rotate(node)\n    // 서브트리의 루트 노드를 반환한다\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* 노드 삽입 */\nfunc insert(val: Int) {\n    root = insertHelper(node: root, val: val)\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\nfunc insertHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return TreeNode(x: val)\n    }\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if val &lt; node!.val {\n        node?.left = insertHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = insertHelper(node: node?.right, val: val)\n    } else {\n        return node // 중복 노드는 삽입하지 않고 바로 반환한다\n    }\n    updateHeight(node: node) // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node: node)\n    // 서브트리의 루트 노드를 반환한다\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* 노드 삽입 */\ninsert(val) {\n    this.root = this.#insertHelper(this.root, val);\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\n#insertHelper(node, val) {\n    if (node === null) return new TreeNode(val);\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if (val &lt; node.val) node.left = this.#insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#insertHelper(node.right, val);\n    else return node; // 중복 노드는 삽입하지 않고 바로 반환한다\n    this.#updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = this.#rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* 노드 삽입 */\ninsert(val: number): void {\n    this.root = this.insertHelper(this.root, val);\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\ninsertHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return new TreeNode(val);\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if (val &lt; node.val) {\n        node.left = this.insertHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.insertHelper(node.right, val);\n    } else {\n        return node; // 중복 노드는 삽입하지 않고 바로 반환한다\n    }\n    this.updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = this.rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* 노드 삽입 */\nvoid insert(int val) {\n  root = insertHelper(root, val);\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\nTreeNode? insertHelper(TreeNode? node, int val) {\n  if (node == null) return TreeNode(val);\n  /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n  if (val &lt; node.val)\n    node.left = insertHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = insertHelper(node.right, val);\n  else\n    return node; // 중복 노드는 삽입하지 않고 바로 반환한다\n  updateHeight(node); // 노드 높이를 갱신한다\n  /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n  node = rotate(node);\n  // 서브트리의 루트 노드를 반환한다\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* 노드 삽입 */\nfn insert(&amp;mut self, val: i32) {\n    self.root = Self::insert_helper(self.root.clone(), val);\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\nfn insert_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n            match {\n                let node_val = node.borrow().val;\n                node_val\n            }\n            .cmp(&amp;val)\n            {\n                Ordering::Greater =&gt; {\n                    let left = node.borrow().left.clone();\n                    node.borrow_mut().left = Self::insert_helper(left, val);\n                }\n                Ordering::Less =&gt; {\n                    let right = node.borrow().right.clone();\n                    node.borrow_mut().right = Self::insert_helper(right, val);\n                }\n                Ordering::Equal =&gt; {\n                    return Some(node); // 중복 노드는 삽입하지 않고 바로 반환한다\n                }\n            }\n            Self::update_height(Some(node.clone())); // 노드 높이를 갱신한다\n\n            /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n            node = Self::rotate(Some(node)).unwrap();\n            // 서브트리의 루트 노드를 반환한다\n            Some(node)\n        }\n        None =&gt; Some(TreeNode::new(val)),\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* 노드 삽입 */\nvoid insert(AVLTree *tree, int val) {\n    tree-&gt;root = insertHelper(tree-&gt;root, val);\n}\n\n/* 노드의 재귀적 삽입(보조 함수) */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == NULL) {\n        return newTreeNode(val);\n    }\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    } else {\n        // 중복 노드는 삽입하지 않고 바로 반환한다\n        return node;\n    }\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* 노드 삽입 */\nfun insert(_val: Int) {\n    root = insertHelper(root, _val)\n}\n\n/* 노드의 재귀적 삽입(보조 메서드) */\nfun insertHelper(n: TreeNode?, _val: Int): TreeNode {\n    if (n == null)\n        return TreeNode(_val)\n    var node = n\n    /* 1. 삽입 위치를 찾아 노드를 삽입한다 */\n    if (_val &lt; node._val)\n        node.left = insertHelper(node.left, _val)\n    else if (_val &gt; node._val)\n        node.right = insertHelper(node.right, _val)\n    else\n        return node // 중복 노드는 삽입하지 않고 바로 반환한다\n    updateHeight(node) // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node)\n    // 서브트리의 루트 노드를 반환한다\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### 노드 삽입 ###\ndef insert(val)\n  @root = insert_helper(@root, val)\nend\n\n### 노드의 재귀적 삽입(보조 메서드)###\ndef insert_helper(node, val)\n  return TreeNode.new(val) if node.nil?\n  # 1. 삽입 위치를 찾아 노드를 삽입한다\n  if val &lt; node.val\n    node.left = insert_helper(node.left, val)\n  elsif val &gt; node.val\n    node.right = insert_helper(node.right, val)\n  else\n    # 중복 노드는 삽입하지 않고 바로 반환한다\n    return node\n  end\n  # 노드 높이를 갱신한다\n  update_height(node)\n  # 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다\n  rotate(node)\nend\n</code></pre>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#2_2","level":3,"title":"2.   노드 삭제","text":"<p>마찬가지로 이진 탐색 트리의 삭제 방법에 더하여 아래에서 위로 회전하며 모든 불균형 노드의 균형을 회복한다. 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby avl_tree.py<pre><code>def remove(self, val: int):\n    \"\"\"노드 삭제\"\"\"\n    self._root = self.remove_helper(self._root, val)\n\ndef remove_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode | None:\n    \"\"\"노드의 재귀적 삭제(보조 메서드)\"\"\"\n    if node is None:\n        return None\n    # 1. 노드를 찾아 삭제한다\n    if val &lt; node.val:\n        node.left = self.remove_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.remove_helper(node.right, val)\n    else:\n        if node.left is None or node.right is None:\n            child = node.left or node.right\n            # 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if child is None:\n                return None\n            # 자식 노드 수 = 1이면 node를 바로 삭제한다\n            else:\n                node = child\n        else:\n            # 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            temp = node.right\n            while temp.left is not None:\n                temp = temp.left\n            node.right = self.remove_helper(node.right, temp.val)\n            node.val = temp.val\n    # 노드 높이를 갱신한다\n    self.update_height(node)\n    # 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* 노드 삭제 */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return nullptr;\n    /* 1. 노드를 찾아 삭제한다 */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    else {\n        if (node-&gt;left == nullptr || node-&gt;right == nullptr) {\n            TreeNode *child = node-&gt;left != nullptr ? node-&gt;left : node-&gt;right;\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if (child == nullptr) {\n                delete node;\n                return nullptr;\n            }\n            // 자식 노드 수 = 1이면 node를 바로 삭제한다\n            else {\n                delete node;\n                node = child;\n            }\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != nullptr) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* 노드 삭제 */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nTreeNode removeHelper(TreeNode node, int val) {\n    if (node == null)\n        return null;\n    /* 1. 노드를 찾아 삭제한다 */\n    if (val &lt; node.val)\n        node.left = removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = removeHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode child = node.left != null ? node.left : node.right;\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if (child == null)\n                return null;\n            // 자식 노드 수 = 1이면 node를 바로 삭제한다\n            else\n                node = child;\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            TreeNode temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* 노드 삭제 */\nvoid Remove(int val) {\n    root = RemoveHelper(root, val);\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nTreeNode? RemoveHelper(TreeNode? node, int val) {\n    if (node == null) return null;\n    /* 1. 노드를 찾아 삭제한다 */\n    if (val &lt; node.val)\n        node.left = RemoveHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = RemoveHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode? child = node.left ?? node.right;\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if (child == null)\n                return null;\n            // 자식 노드 수 = 1이면 node를 바로 삭제한다\n            else\n                node = child;\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            TreeNode? temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = RemoveHelper(node.right, temp.val!.Value);\n            node.val = temp.val;\n        }\n    }\n    UpdateHeight(node);  // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = Rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* 노드 삭제 */\nfunc (t *aVLTree) remove(val int) {\n    t.root = t.removeHelper(t.root, val)\n}\n\n/* 노드의 재귀적 삭제(보조 함수) */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return nil\n    }\n    /* 1. 노드를 찾아 삭제한다 */\n    if val &lt; node.Val.(int) {\n        node.Left = t.removeHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.removeHelper(node.Right, val)\n    } else {\n        if node.Left == nil || node.Right == nil {\n            child := node.Left\n            if node.Right != nil {\n                child = node.Right\n            }\n            if child == nil {\n                // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n                return nil\n            } else {\n                // 자식 노드 수 = 1이면 node를 바로 삭제한다\n                node = child\n            }\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            temp := node.Right\n            for temp.Left != nil {\n                temp = temp.Left\n            }\n            node.Right = t.removeHelper(node.Right, temp.Val.(int))\n            node.Val = temp.Val\n        }\n    }\n    // 노드 높이를 갱신한다\n    t.updateHeight(node)\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = t.rotate(node)\n    // 서브트리의 루트 노드를 반환한다\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* 노드 삭제 */\nfunc remove(val: Int) {\n    root = removeHelper(node: root, val: val)\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nfunc removeHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return nil\n    }\n    /* 1. 노드를 찾아 삭제한다 */\n    if val &lt; node!.val {\n        node?.left = removeHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = removeHelper(node: node?.right, val: val)\n    } else {\n        if node?.left == nil || node?.right == nil {\n            let child = node?.left ?? node?.right\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if child == nil {\n                return nil\n            }\n            // 자식 노드 수 = 1이면 node를 바로 삭제한다\n            else {\n                node = child\n            }\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            var temp = node?.right\n            while temp?.left != nil {\n                temp = temp?.left\n            }\n            node?.right = removeHelper(node: node?.right, val: temp!.val)\n            node?.val = temp!.val\n        }\n    }\n    updateHeight(node: node) // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node: node)\n    // 서브트리의 루트 노드를 반환한다\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* 노드 삭제 */\nremove(val) {\n    this.root = this.#removeHelper(this.root, val);\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\n#removeHelper(node, val) {\n    if (node === null) return null;\n    /* 1. 노드를 찾아 삭제한다 */\n    if (val &lt; node.val) node.left = this.#removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#removeHelper(node.right, val);\n    else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if (child === null) return null;\n            // 자식 노드 수 = 1이면 node를 바로 삭제한다\n            else node = child;\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.#removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.#updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = this.#rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* 노드 삭제 */\nremove(val: number): void {\n    this.root = this.removeHelper(this.root, val);\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nremoveHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return null;\n    /* 1. 노드를 찾아 삭제한다 */\n    if (val &lt; node.val) {\n        node.left = this.removeHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.removeHelper(node.right, val);\n    } else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if (child === null) {\n                return null;\n            } else {\n                // 자식 노드 수 = 1이면 node를 바로 삭제한다\n                node = child;\n            }\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.updateHeight(node); // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = this.rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* 노드 삭제 */\nvoid remove(int val) {\n  root = removeHelper(root, val);\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nTreeNode? removeHelper(TreeNode? node, int val) {\n  if (node == null) return null;\n  /* 1. 노드를 찾아 삭제한다 */\n  if (val &lt; node.val)\n    node.left = removeHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = removeHelper(node.right, val);\n  else {\n    if (node.left == null || node.right == null) {\n      TreeNode? child = node.left ?? node.right;\n      // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n      if (child == null)\n        return null;\n      // 자식 노드 수 = 1이면 node를 바로 삭제한다\n      else\n        node = child;\n    } else {\n      // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n      TreeNode? temp = node.right;\n      while (temp!.left != null) {\n        temp = temp.left;\n      }\n      node.right = removeHelper(node.right, temp.val);\n      node.val = temp.val;\n    }\n  }\n  updateHeight(node); // 노드 높이를 갱신한다\n  /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n  node = rotate(node);\n  // 서브트리의 루트 노드를 반환한다\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* 노드 삭제 */\nfn remove(&amp;self, val: i32) {\n    Self::remove_helper(self.root.clone(), val);\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nfn remove_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. 노드를 찾아 삭제한다 */\n            if val &lt; node.borrow().val {\n                let left = node.borrow().left.clone();\n                node.borrow_mut().left = Self::remove_helper(left, val);\n            } else if val &gt; node.borrow().val {\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, val);\n            } else if node.borrow().left.is_none() || node.borrow().right.is_none() {\n                let child = if node.borrow().left.is_some() {\n                    node.borrow().left.clone()\n                } else {\n                    node.borrow().right.clone()\n                };\n                match child {\n                    // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n                    None =&gt; {\n                        return None;\n                    }\n                    // 자식 노드 수 = 1이면 node를 바로 삭제한다\n                    Some(child) =&gt; node = child,\n                }\n            } else {\n                // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n                let mut temp = node.borrow().right.clone().unwrap();\n                loop {\n                    let temp_left = temp.borrow().left.clone();\n                    if temp_left.is_none() {\n                        break;\n                    }\n                    temp = temp_left.unwrap();\n                }\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);\n                node.borrow_mut().val = temp.borrow().val;\n            }\n            Self::update_height(Some(node.clone())); // 노드 높이를 갱신한다\n\n            /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n            node = Self::rotate(Some(node)).unwrap();\n            // 서브트리의 루트 노드를 반환한다\n            Some(node)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* 노드 삭제 */\n// stdio.h를 포함했으므로 여기서는 remove라는 이름을 사용할 수 없다\nvoid removeItem(AVLTree *tree, int val) {\n    TreeNode *root = removeHelper(tree-&gt;root, val);\n}\n\n/* 노드의 재귀적 삭제(보조 함수) */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    TreeNode *child, *grandChild;\n    if (node == NULL) {\n        return NULL;\n    }\n    /* 1. 노드를 찾아 삭제한다 */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    } else {\n        if (node-&gt;left == NULL || node-&gt;right == NULL) {\n            child = node-&gt;left;\n            if (node-&gt;right != NULL) {\n                child = node-&gt;right;\n            }\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if (child == NULL) {\n                return NULL;\n            } else {\n                // 자식 노드 수 = 1이면 node를 바로 삭제한다\n                node = child;\n            }\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != NULL) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    // 노드 높이를 갱신한다\n    updateHeight(node);\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node);\n    // 서브트리의 루트 노드를 반환한다\n    return node;\n}\n</code></pre> avl_tree.kt<pre><code>/* 노드 삭제 */\nfun remove(_val: Int) {\n    root = removeHelper(root, _val)\n}\n\n/* 노드의 재귀적 삭제(보조 메서드) */\nfun removeHelper(n: TreeNode?, _val: Int): TreeNode? {\n    var node = n ?: return null\n    /* 1. 노드를 찾아 삭제한다 */\n    if (_val &lt; node._val)\n        node.left = removeHelper(node.left, _val)\n    else if (_val &gt; node._val)\n        node.right = removeHelper(node.right, _val)\n    else {\n        if (node.left == null || node.right == null) {\n            val child = if (node.left != null)\n                node.left\n            else\n                node.right\n            // 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n            if (child == null)\n                return null\n            // 자식 노드 수 = 1이면 node를 바로 삭제한다\n            else\n                node = child\n        } else {\n            // 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n            var temp = node.right\n            while (temp!!.left != null) {\n                temp = temp.left\n            }\n            node.right = removeHelper(node.right, temp._val)\n            node._val = temp._val\n        }\n    }\n    updateHeight(node) // 노드 높이를 갱신한다\n    /* 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다 */\n    node = rotate(node)\n    // 서브트리의 루트 노드를 반환한다\n    return node\n}\n</code></pre> avl_tree.rb<pre><code>### 노드 삭제 ###\ndef remove(val)\n  @root = remove_helper(@root, val)\nend\n\n### 노드의 재귀적 삭제(보조 메서드)###\ndef remove_helper(node, val)\n  return if node.nil?\n  # 1. 노드를 찾아 삭제한다\n  if val &lt; node.val\n    node.left = remove_helper(node.left, val)\n  elsif val &gt; node.val\n    node.right = remove_helper(node.right, val)\n  else\n    if node.left.nil? || node.right.nil?\n      child = node.left || node.right\n      # 자식 노드 수 = 0이면 node를 바로 삭제하고 반환한다\n      return if child.nil?\n      # 자식 노드 수 = 1이면 node를 바로 삭제한다\n      node = child\n    else\n      # 자식 노드 수 = 2이면 중위 순회의 다음 노드를 삭제하고, 그 노드로 현재 노드를 대체한다\n      temp = node.right\n      while !temp.left.nil?\n        temp = temp.left\n      end\n      node.right = remove_helper(node.right, temp.val)\n      node.val = temp.val\n    end\n  end\n  # 노드 높이를 갱신한다\n  update_height(node)\n  # 2. 회전 연산을 수행하여 해당 서브트리의 균형을 복원한다\n  rotate(node)\nend\n</code></pre>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#3_1","level":3,"title":"3.   노드 탐색","text":"<p>AVL 트리의 노드 탐색은 이진 탐색 트리와 같으므로 다시 설명하지 않는다.</p>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/avl_tree/#754-avl","level":2,"title":"7.5.4   AVL 트리의 대표적인 응용","text":"<ul> <li>대규모 데이터를 구성하고 저장하며, 탐색이 잦고 추가와 삭제가 드문 상황에 적합하다.</li> <li>데이터베이스의 인덱스 시스템을 구축하는 데 사용한다.</li> <li>레드-블랙 트리도 대표적인 균형 이진 탐색 트리다. AVL 트리보다 균형 조건이 느슨하여 삽입과 삭제에 필요한 회전 수가 적고, 노드 추가와 삭제의 평균 효율이 더 높다.</li> </ul>","path":["제 7 장   트리","7.5   AVL 트리 *"],"tags":[]},{"location":"chapter_tree/binary_search_tree/","level":1,"title":"7.4   이진 탐색 트리","text":"<p>아래 그림(7-16)의 이진 탐색 트리(binary search tree)는 다음 조건을 만족한다.</p> <ol> <li>루트에 대해 왼쪽 서브트리의 모든 노드 값 \\(&lt;\\) 루트 값 \\(&lt;\\) 오른쪽 서브트리의 모든 노드 값이다.</li> <li>임의의 노드의 왼쪽과 오른쪽 서브트리도 이진 탐색 트리로, 조건 <code>1.</code>을 만족한다.</li> </ol> <p></p> <p> 그림 7-16   이진 탐색 트리 </p>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#741","level":2,"title":"7.4.1   이진 탐색 트리의 연산","text":"<p>이진 탐색 트리를 <code>BinarySearchTree</code> 클래스로 묶고, 루트 노드를 가리키는 멤버 변수 <code>root</code>를 정의한다.</p>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#1","level":3,"title":"1.   노드 탐색","text":"<p>목표 노드의 값 <code>num</code>이 주어지면 이진 탐색 트리의 성질로 찾을 수 있다. 아래 그림(7-17)과 같이 노드 <code>cur</code>를 두고 루트 <code>root</code>에서 출발하여 <code>cur.val</code>과 <code>num</code>의 대소 관계를 반복해서 비교한다.</p> <ul> <li><code>cur.val &lt; num</code>이면 목표 노드는 <code>cur</code>의 오른쪽 서브트리에 있으므로 <code>cur = cur.right</code>를 수행한다.</li> <li><code>cur.val &gt; num</code>이면 목표 노드는 <code>cur</code>의 왼쪽 서브트리에 있으므로 <code>cur = cur.left</code>를 수행한다.</li> <li><code>cur.val = num</code>이면 목표 노드를 찾았으므로 반복문을 빠져나와 해당 노드를 반환한다.</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> 그림 7-17   이진 탐색 트리의 노드 탐색 예시 </p> <p>이진 탐색 트리의 탐색 원리는 이진 탐색 알고리즘과 같으며, 매번 경우의 절반을 제외한다. 반복 횟수는 최대 트리 높이이므로 트리가 균형을 이루면 \\(O(\\log n)\\) 시간이 걸린다. 예시 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_tree.py<pre><code>def search(self, num: int) -&gt; TreeNode | None:\n    \"\"\"노드 탐색\"\"\"\n    cur = self._root\n    # 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while cur is not None:\n        # 목표 노드는 cur의 오른쪽 서브트리에 있다\n        if cur.val &lt; num:\n            cur = cur.right\n        # 목표 노드는 cur의 왼쪽 서브트리에 있다\n        elif cur.val &gt; num:\n            cur = cur.left\n        # 목표 노드를 찾았으므로 반복문을 종료한다\n        else:\n            break\n    return cur\n</code></pre> binary_search_tree.cpp<pre><code>/* 노드 탐색 */\nTreeNode *search(int num) {\n    TreeNode *cur = root;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != nullptr) {\n        // 목표 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // 목표 노드는 cur의 왼쪽 서브트리에 있다\n        else if (cur-&gt;val &gt; num)\n            cur = cur-&gt;left;\n        // 목표 노드를 찾았으므로 반복문을 종료한다\n        else\n            break;\n    }\n    // 목표 노드를 반환한다\n    return cur;\n}\n</code></pre> binary_search_tree.java<pre><code>/* 노드 탐색 */\nTreeNode search(int num) {\n    TreeNode cur = root;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 목표 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // 목표 노드는 cur의 왼쪽 서브트리에 있다\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // 목표 노드를 찾았으므로 반복문을 종료한다\n        else\n            break;\n    }\n    // 목표 노드를 반환한다\n    return cur;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* 노드 탐색 */\nTreeNode? Search(int num) {\n    TreeNode? cur = root;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 목표 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num) cur =\n            cur.right;\n        // 목표 노드는 cur의 왼쪽 서브트리에 있다\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // 목표 노드를 찾았으므로 반복문을 종료한다\n        else\n            break;\n    }\n    // 목표 노드를 반환한다\n    return cur;\n}\n</code></pre> binary_search_tree.go<pre><code>/* 노드 탐색 */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\n    node := bst.root\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    for node != nil {\n        if node.Val.(int) &lt; num {\n            // 목표 노드는 cur의 오른쪽 서브트리에 있다\n            node = node.Right\n        } else if node.Val.(int) &gt; num {\n            // 목표 노드는 cur의 왼쪽 서브트리에 있다\n            node = node.Left\n        } else {\n            // 목표 노드를 찾았으므로 반복문을 종료한다\n            break\n        }\n    }\n    // 목표 노드를 반환한다\n    return node\n}\n</code></pre> binary_search_tree.swift<pre><code>/* 노드 탐색 */\nfunc search(num: Int) -&gt; TreeNode? {\n    var cur = root\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while cur != nil {\n        // 목표 노드는 cur의 오른쪽 서브트리에 있다\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // 목표 노드는 cur의 왼쪽 서브트리에 있다\n        else if cur!.val &gt; num {\n            cur = cur?.left\n        }\n        // 목표 노드를 찾았으므로 반복문을 종료한다\n        else {\n            break\n        }\n    }\n    // 목표 노드를 반환한다\n    return cur\n}\n</code></pre> binary_search_tree.js<pre><code>/* 노드 탐색 */\nsearch(num) {\n    let cur = this.root;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur !== null) {\n        // 목표 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num) cur = cur.right;\n        // 목표 노드는 cur의 왼쪽 서브트리에 있다\n        else if (cur.val &gt; num) cur = cur.left;\n        // 목표 노드를 찾았으므로 반복문을 종료한다\n        else break;\n    }\n    // 목표 노드를 반환한다\n    return cur;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* 노드 탐색 */\nsearch(num: number): TreeNode | null {\n    let cur = this.root;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur !== null) {\n        // 목표 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num) cur = cur.right;\n        // 목표 노드는 cur의 왼쪽 서브트리에 있다\n        else if (cur.val &gt; num) cur = cur.left;\n        // 목표 노드를 찾았으므로 반복문을 종료한다\n        else break;\n    }\n    // 목표 노드를 반환한다\n    return cur;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* 노드 탐색 */\nTreeNode? search(int _num) {\n  TreeNode? cur = _root;\n  // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n  while (cur != null) {\n    // 목표 노드는 cur의 오른쪽 서브트리에 있다\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // 목표 노드는 cur의 왼쪽 서브트리에 있다\n    else if (cur.val &gt; _num)\n      cur = cur.left;\n    // 목표 노드를 찾았으므로 반복문을 종료한다\n    else\n      break;\n  }\n  // 목표 노드를 반환한다\n  return cur;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* 노드 탐색 */\npub fn search(&amp;self, num: i32) -&gt; OptionTreeNodeRc {\n    let mut cur = self.root.clone();\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // 목표 노드는 cur의 오른쪽 서브트리에 있다\n            Ordering::Greater =&gt; cur = node.borrow().right.clone(),\n            // 목표 노드는 cur의 왼쪽 서브트리에 있다\n            Ordering::Less =&gt; cur = node.borrow().left.clone(),\n            // 목표 노드를 찾았으므로 반복문을 종료한다\n            Ordering::Equal =&gt; break,\n        }\n    }\n\n    // 목표 노드를 반환한다\n    cur\n}\n</code></pre> binary_search_tree.c<pre><code>/* 노드 탐색 */\nTreeNode *search(BinarySearchTree *bst, int num) {\n    TreeNode *cur = bst-&gt;root;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != NULL) {\n        if (cur-&gt;val &lt; num) {\n            // 목표 노드는 cur의 오른쪽 서브트리에 있다\n            cur = cur-&gt;right;\n        } else if (cur-&gt;val &gt; num) {\n            // 목표 노드는 cur의 왼쪽 서브트리에 있다\n            cur = cur-&gt;left;\n        } else {\n            // 목표 노드를 찾았으므로 반복문을 종료한다\n            break;\n        }\n    }\n    // 목표 노드를 반환한다\n    return cur;\n}\n</code></pre> binary_search_tree.kt<pre><code>/* 노드 탐색 */\nfun search(num: Int): TreeNode? {\n    var cur = root\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 목표 노드는 cur의 오른쪽 서브트리에 있다\n        cur = if (cur._val &lt; num)\n            cur.right\n        // 목표 노드는 cur의 왼쪽 서브트리에 있다\n        else if (cur._val &gt; num)\n            cur.left\n        // 목표 노드를 찾았으므로 반복문을 종료한다\n        else\n            break\n    }\n    // 목표 노드를 반환한다\n    return cur\n}\n</code></pre> binary_search_tree.rb<pre><code>### 노드 탐색 ###\ndef search(num)\n  cur = @root\n\n  # 반복하여 탐색하다 리프 노드를 지나면 종료한다\n  while !cur.nil?\n    # 목표 노드는 cur의 오른쪽 서브트리에 있다\n    if cur.val &lt; num\n      cur = cur.right\n    # 목표 노드는 cur의 왼쪽 서브트리에 있다\n    elsif cur.val &gt; num\n      cur = cur.left\n    # 목표 노드를 찾았으므로 반복문을 종료한다\n    else\n      break\n    end\n  end\n\n  cur\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#2","level":3,"title":"2.   노드 삽입","text":"<p>삽입할 원소 <code>num</code>이 주어졌을 때 '왼쪽 서브트리 &lt; 루트 &lt; 오른쪽 서브트리'라는 성질을 유지하려면 아래 그림(7-18)과 같이 삽입한다.</p> <ol> <li>삽입 위치 찾기: 탐색과 비슷하게 루트에서 출발하여 현재 노드 값과 <code>num</code>의 대소 관계에 따라 아래로 탐색한다. 리프를 지나 <code>None</code>에 도달하면 반복문을 종료한다.</li> <li>그 위치에 노드 삽입: 값이 <code>num</code>인 노드를 초기화하여 <code>None</code> 자리에 놓는다.</li> </ol> <p></p> <p> 그림 7-18   이진 탐색 트리에 노드 삽입 </p> <p>구현 시 다음 두 가지에 주의하자.</p> <ul> <li>중복 노드가 있으면 정의에 어긋나므로 이진 탐색 트리는 중복을 허용하지 않는다. 삽입할 노드가 이미 있으면 삽입하지 않고 바로 반환한다.</li> <li>삽입을 위해 노드 <code>pre</code>에 직전 반복의 노드를 저장해야 한다. 그러면 <code>None</code>에 도달했을 때 그 부모를 얻어 삽입을 완료할 수 있다.</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_tree.py<pre><code>def insert(self, num: int):\n    \"\"\"노드 삽입\"\"\"\n    # 트리가 비어 있으면 루트 노드를 초기화한다\n    if self._root is None:\n        self._root = TreeNode(num)\n        return\n    # 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    cur, pre = self._root, None\n    while cur is not None:\n        # 중복 노드를 찾으면 바로 반환한다\n        if cur.val == num:\n            return\n        pre = cur\n        # 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        if cur.val &lt; num:\n            cur = cur.right\n        # 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else:\n            cur = cur.left\n    # 노드 삽입\n    node = TreeNode(num)\n    if pre.val &lt; num:\n        pre.right = node\n    else:\n        pre.left = node\n</code></pre> binary_search_tree.cpp<pre><code>/* 노드 삽입 */\nvoid insert(int num) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if (root == nullptr) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode *cur = root, *pre = nullptr;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != nullptr) {\n        // 중복 노드를 찾으면 바로 반환한다\n        if (cur-&gt;val == num)\n            return;\n        pre = cur;\n        // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else\n            cur = cur-&gt;left;\n    }\n    // 노드 삽입\n    TreeNode *node = new TreeNode(num);\n    if (pre-&gt;val &lt; num)\n        pre-&gt;right = node;\n    else\n        pre-&gt;left = node;\n}\n</code></pre> binary_search_tree.java<pre><code>/* 노드 삽입 */\nvoid insert(int num) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode cur = root, pre = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 중복 노드를 찾으면 바로 반환한다\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else\n            cur = cur.left;\n    }\n    // 노드 삽입\n    TreeNode node = new TreeNode(num);\n    if (pre.val &lt; num)\n        pre.right = node;\n    else\n        pre.left = node;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* 노드 삽입 */\nvoid Insert(int num) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode? cur = root, pre = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 중복 노드를 찾으면 바로 반환한다\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else\n            cur = cur.left;\n    }\n\n    // 노드 삽입\n    TreeNode node = new(num);\n    if (pre != null) {\n        if (pre.val &lt; num)\n            pre.right = node;\n        else\n            pre.left = node;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* 노드 삽입 */\nfunc (bst *binarySearchTree) insert(num int) {\n    cur := bst.root\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if cur == nil {\n        bst.root = NewTreeNode(num)\n        return\n    }\n    // 삽입할 노드의 부모가 될 노드 위치\n    var pre *TreeNode = nil\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    for cur != nil {\n        if cur.Val == num {\n            return\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            cur = cur.Right\n        } else {\n            cur = cur.Left\n        }\n    }\n    // 노드 삽입\n    node := NewTreeNode(num)\n    if pre.Val.(int) &lt; num {\n        pre.Right = node\n    } else {\n        pre.Left = node\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* 노드 삽입 */\nfunc insert(num: Int) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if root == nil {\n        root = TreeNode(x: num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while cur != nil {\n        // 중복 노드를 찾으면 바로 반환한다\n        if cur!.val == num {\n            return\n        }\n        pre = cur\n        // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else {\n            cur = cur?.left\n        }\n    }\n    // 노드 삽입\n    let node = TreeNode(x: num)\n    if pre!.val &lt; num {\n        pre?.right = node\n    } else {\n        pre?.left = node\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* 노드 삽입 */\ninsert(num) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur = this.root,\n        pre = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur !== null) {\n        // 중복 노드를 찾으면 바로 반환한다\n        if (cur.val === num) return;\n        pre = cur;\n        // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num) cur = cur.right;\n        // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else cur = cur.left;\n    }\n    // 노드 삽입\n    const node = new TreeNode(num);\n    if (pre.val &lt; num) pre.right = node;\n    else pre.left = node;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* 노드 삽입 */\ninsert(num: number): void {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur !== null) {\n        // 중복 노드를 찾으면 바로 반환한다\n        if (cur.val === num) return;\n        pre = cur;\n        // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num) cur = cur.right;\n        // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else cur = cur.left;\n    }\n    // 노드 삽입\n    const node = new TreeNode(num);\n    if (pre!.val &lt; num) pre!.right = node;\n    else pre!.left = node;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* 노드 삽입 */\nvoid insert(int _num) {\n  // 트리가 비어 있으면 루트 노드를 초기화한다\n  if (_root == null) {\n    _root = TreeNode(_num);\n    return;\n  }\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n  while (cur != null) {\n    // 중복 노드를 찾으면 바로 반환한다\n    if (cur.val == _num) return;\n    pre = cur;\n    // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n    else\n      cur = cur.left;\n  }\n  // 노드 삽입\n  TreeNode? node = TreeNode(_num);\n  if (pre!.val &lt; _num)\n    pre.right = node;\n  else\n    pre.left = node;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* 노드 삽입 */\npub fn insert(&amp;mut self, num: i32) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if self.root.is_none() {\n        self.root = Some(TreeNode::new(num));\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // 중복 노드를 찾으면 바로 반환한다\n            Ordering::Equal =&gt; return,\n            // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // 노드 삽입\n    let pre = pre.unwrap();\n    let node = Some(TreeNode::new(num));\n    if num &gt; pre.borrow().val {\n        pre.borrow_mut().right = node;\n    } else {\n        pre.borrow_mut().left = node;\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* 노드 삽입 */\nvoid insert(BinarySearchTree *bst, int num) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if (bst-&gt;root == NULL) {\n        bst-&gt;root = newTreeNode(num);\n        return;\n    }\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != NULL) {\n        // 중복 노드를 찾으면 바로 반환한다\n        if (cur-&gt;val == num) {\n            return;\n        }\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n            cur = cur-&gt;right;\n        } else {\n            // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n            cur = cur-&gt;left;\n        }\n    }\n    // 노드 삽입\n    TreeNode *node = newTreeNode(num);\n    if (pre-&gt;val &lt; num) {\n        pre-&gt;right = node;\n    } else {\n        pre-&gt;left = node;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* 노드 삽입 */\nfun insert(num: Int) {\n    // 트리가 비어 있으면 루트 노드를 초기화한다\n    if (root == null) {\n        root = TreeNode(num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode? = null\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 중복 노드를 찾으면 바로 반환한다\n        if (cur._val == num)\n            return\n        pre = cur\n        // 삽입 위치는 cur의 오른쪽 서브트리에 있다\n        cur = if (cur._val &lt; num)\n            cur.right\n        // 삽입 위치는 cur의 왼쪽 서브트리에 있다\n        else\n            cur.left\n    }\n    // 노드 삽입\n    val node = TreeNode(num)\n    if (pre?._val!! &lt; num)\n        pre.right = node\n    else\n        pre.left = node\n}\n</code></pre> binary_search_tree.rb<pre><code>### 노드 삽입 ###\ndef insert(num)\n  # 트리가 비어 있으면 루트 노드를 초기화한다\n  if @root.nil?\n    @root = TreeNode.new(num)\n    return\n  end\n\n  # 반복하여 탐색하다 리프 노드를 지나면 종료한다\n  cur, pre = @root, nil\n  while !cur.nil?\n    # 중복 노드를 찾으면 바로 반환한다\n    return if cur.val == num\n\n    pre = cur\n    # 삽입 위치는 cur의 오른쪽 서브트리에 있다\n    if cur.val &lt; num\n      cur = cur.right\n    # 삽입 위치는 cur의 왼쪽 서브트리에 있다\n    else\n      cur = cur.left\n    end\n  end\n\n  # 노드 삽입\n  node = TreeNode.new(num)\n  if pre.val &lt; num\n    pre.right = node\n  else\n    pre.left = node\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>노드 탐색과 마찬가지로 삽입에는 \\(O(\\log n)\\) 시간이 걸린다.</p>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#3","level":3,"title":"3.   노드 삭제","text":"<p>먼저 트리에서 목표 노드를 찾은 뒤 삭제한다. 삽입과 마찬가지로 삭제 후에도 '왼쪽 서브트리 &lt; 루트 &lt; 오른쪽 서브트리'라는 성질을 유지해야 한다. 따라서 목표 노드의 자식 수가 0, 1, 2인 세 경우로 나누어 삭제한다.</p> <p>아래 그림(7-19)과 같이 삭제할 노드의 차수가 \\(0\\)이면 리프 노드이므로 바로 삭제할 수 있다.</p> <p></p> <p> 그림 7-19   이진 탐색 트리의 노드 삭제: 차수 0 </p> <p>아래 그림(7-20)과 같이 차수가 \\(1\\)이면 삭제할 노드를 그 자식으로 바꾸면 된다.</p> <p></p> <p> 그림 7-20   이진 탐색 트리의 노드 삭제: 차수 1 </p> <p>차수가 \\(2\\)이면 바로 삭제할 수 없어 다른 노드로 대체해야 한다. '왼쪽 서브트리 \\(&lt;\\) 루트 \\(&lt;\\) 오른쪽 서브트리' 성질을 유지하려면 오른쪽 서브트리의 최소 노드 또는 왼쪽 서브트리의 최대 노드로 대체할 수 있다.</p> <p>오른쪽 서브트리의 최소 노드(중위 순회에서 다음 노드)를 선택하면 삭제 과정은 아래 그림(7-21)과 같다.</p> <ol> <li>삭제할 노드의 '중위 순회 순서'상 다음 노드를 찾아 <code>tmp</code>라고 한다.</li> <li>삭제할 노드의 값을 <code>tmp</code>의 값으로 덮어쓰고, 트리에서 노드 <code>tmp</code>를 재귀적으로 삭제한다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> 그림 7-21   이진 탐색 트리의 노드 삭제: 차수 2 </p> <p>노드 삭제에도 \\(O(\\log n)\\) 시간이 걸린다. 삭제할 노드를 찾는 데 \\(O(\\log n)\\), 중위 순회의 후속 노드를 구하는 데 \\(O(\\log n)\\) 시간이 든다. 예시 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_search_tree.py<pre><code>def remove(self, num: int):\n    \"\"\"노드 삭제\"\"\"\n    # 트리가 비어 있으면 바로 반환하여 종료한다\n    if self._root is None:\n        return\n    # 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    cur, pre = self._root, None\n    while cur is not None:\n        # 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if cur.val == num:\n            break\n        pre = cur\n        # 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        if cur.val &lt; num:\n            cur = cur.right\n        # 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else:\n            cur = cur.left\n    # 삭제할 노드가 없으면 바로 반환한다\n    if cur is None:\n        return\n\n    # 자식 노드 수 = 0 또는 1\n    if cur.left is None or cur.right is None:\n        # 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n        child = cur.left or cur.right\n        # 노드 cur를 삭제한다\n        if cur != self._root:\n            if pre.left == cur:\n                pre.left = child\n            else:\n                pre.right = child\n        else:\n            # 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            self._root = child\n    # 자식 노드 수 = 2\n    else:\n        # 중위 순회에서 cur의 다음 노드를 구한다\n        tmp: TreeNode = cur.right\n        while tmp.left is not None:\n            tmp = tmp.left\n        # 노드 tmp를 재귀적으로 삭제한다\n        self.remove(tmp.val)\n        # tmp로 cur를 덮어쓴다\n        cur.val = tmp.val\n</code></pre> binary_search_tree.cpp<pre><code>/* 노드 삭제 */\nvoid remove(int num) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if (root == nullptr)\n        return;\n    TreeNode *cur = root, *pre = nullptr;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != nullptr) {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else\n            cur = cur-&gt;left;\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if (cur == nullptr)\n        return;\n    // 자식 노드 수 = 0 또는 1\n    if (cur-&gt;left == nullptr || cur-&gt;right == nullptr) {\n        // 자식 노드 수가 0 / 1이면 child = nullptr / 해당 자식 노드다\n        TreeNode *child = cur-&gt;left != nullptr ? cur-&gt;left : cur-&gt;right;\n        // 노드 cur를 삭제한다\n        if (cur != root) {\n            if (pre-&gt;left == cur)\n                pre-&gt;left = child;\n            else\n                pre-&gt;right = child;\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            root = child;\n        }\n        // 메모리 해제\n        delete cur;\n    }\n    // 자식 노드 수 = 2\n    else {\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != nullptr) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // 노드 tmp를 재귀적으로 삭제한다\n        remove(tmp-&gt;val);\n        // tmp로 cur를 덮어쓴다\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.java<pre><code>/* 노드 삭제 */\nvoid remove(int num) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if (root == null)\n        return;\n    TreeNode cur = root, pre = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else\n            cur = cur.left;\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if (cur == null)\n        return;\n    // 자식 노드 수 = 0 또는 1\n    if (cur.left == null || cur.right == null) {\n        // 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n        TreeNode child = cur.left != null ? cur.left : cur.right;\n        // 노드 cur를 삭제한다\n        if (cur != root) {\n            if (pre.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            root = child;\n        }\n    }\n    // 자식 노드 수 = 2\n    else {\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        TreeNode tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // 노드 tmp를 재귀적으로 삭제한다\n        remove(tmp.val);\n        // tmp로 cur를 덮어쓴다\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.cs<pre><code>/* 노드 삭제 */\nvoid Remove(int num) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if (root == null)\n        return;\n    TreeNode? cur = root, pre = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else\n            cur = cur.left;\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if (cur == null)\n        return;\n    // 자식 노드 수 = 0 또는 1\n    if (cur.left == null || cur.right == null) {\n        // 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n        TreeNode? child = cur.left ?? cur.right;\n        // 노드 cur를 삭제한다\n        if (cur != root) {\n            if (pre!.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            root = child;\n        }\n    }\n    // 자식 노드 수 = 2\n    else {\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        TreeNode? tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // 노드 tmp를 재귀적으로 삭제한다\n        Remove(tmp.val!.Value);\n        // tmp로 cur를 덮어쓴다\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* 노드 삭제 */\nfunc (bst *binarySearchTree) remove(num int) {\n    cur := bst.root\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if cur == nil {\n        return\n    }\n    // 삭제할 노드의 부모 노드 위치\n    var pre *TreeNode = nil\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    for cur != nil {\n        if cur.Val == num {\n            break\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            // 삭제할 노드는 오른쪽 서브트리에 있다\n            cur = cur.Right\n        } else {\n            // 삭제할 노드는 왼쪽 서브트리에 있다\n            cur = cur.Left\n        }\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if cur == nil {\n        return\n    }\n    // 자식 노드 수가 0 또는 1\n    if cur.Left == nil || cur.Right == nil {\n        var child *TreeNode = nil\n        // 삭제할 노드의 자식 노드를 구한다\n        if cur.Left != nil {\n            child = cur.Left\n        } else {\n            child = cur.Right\n        }\n        // 노드 cur를 삭제한다\n        if cur != bst.root {\n            if pre.Left == cur {\n                pre.Left = child\n            } else {\n                pre.Right = child\n            }\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            bst.root = child\n        }\n        // 자식 노드 수가 2\n    } else {\n        // 중위 순회에서 삭제할 노드 cur의 다음 노드를 구한다\n        tmp := cur.Right\n        for tmp.Left != nil {\n            tmp = tmp.Left\n        }\n        // 노드 tmp를 재귀적으로 삭제한다\n        bst.remove(tmp.Val.(int))\n        // tmp로 cur를 덮어쓴다\n        cur.Val = tmp.Val\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* 노드 삭제 */\nfunc remove(num: Int) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if root == nil {\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while cur != nil {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if cur!.val == num {\n            break\n        }\n        pre = cur\n        // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else {\n            cur = cur?.left\n        }\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if cur == nil {\n        return\n    }\n    // 자식 노드 수 = 0 또는 1\n    if cur?.left == nil || cur?.right == nil {\n        // 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n        let child = cur?.left ?? cur?.right\n        // 노드 cur를 삭제한다\n        if cur !== root {\n            if pre?.left === cur {\n                pre?.left = child\n            } else {\n                pre?.right = child\n            }\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            root = child\n        }\n    }\n    // 자식 노드 수 = 2\n    else {\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        var tmp = cur?.right\n        while tmp?.left != nil {\n            tmp = tmp?.left\n        }\n        // 노드 tmp를 재귀적으로 삭제한다\n        remove(num: tmp!.val)\n        // tmp로 cur를 덮어쓴다\n        cur?.val = tmp!.val\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* 노드 삭제 */\nremove(num) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if (this.root === null) return;\n    let cur = this.root,\n        pre = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur !== null) {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if (cur.val === num) break;\n        pre = cur;\n        // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num) cur = cur.right;\n        // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else cur = cur.left;\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if (cur === null) return;\n    // 자식 노드 수 = 0 또는 1\n    if (cur.left === null || cur.right === null) {\n        // 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n        const child = cur.left !== null ? cur.left : cur.right;\n        // 노드 cur를 삭제한다\n        if (cur !== this.root) {\n            if (pre.left === cur) pre.left = child;\n            else pre.right = child;\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            this.root = child;\n        }\n    }\n    // 자식 노드 수 = 2\n    else {\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        let tmp = cur.right;\n        while (tmp.left !== null) {\n            tmp = tmp.left;\n        }\n        // 노드 tmp를 재귀적으로 삭제한다\n        this.remove(tmp.val);\n        // tmp로 cur를 덮어쓴다\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.ts<pre><code>/* 노드 삭제 */\nremove(num: number): void {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if (this.root === null) return;\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur !== null) {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if (cur.val === num) break;\n        pre = cur;\n        // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        if (cur.val &lt; num) cur = cur.right;\n        // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else cur = cur.left;\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if (cur === null) return;\n    // 자식 노드 수 = 0 또는 1\n    if (cur.left === null || cur.right === null) {\n        // 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n        const child: TreeNode | null =\n            cur.left !== null ? cur.left : cur.right;\n        // 노드 cur를 삭제한다\n        if (cur !== this.root) {\n            if (pre!.left === cur) pre!.left = child;\n            else pre!.right = child;\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            this.root = child;\n        }\n    }\n    // 자식 노드 수 = 2\n    else {\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        let tmp: TreeNode | null = cur.right;\n        while (tmp!.left !== null) {\n            tmp = tmp!.left;\n        }\n        // 노드 tmp를 재귀적으로 삭제한다\n        this.remove(tmp!.val);\n        // tmp로 cur를 덮어쓴다\n        cur.val = tmp!.val;\n    }\n}\n</code></pre> binary_search_tree.dart<pre><code>/* 노드 삭제 */\nvoid remove(int _num) {\n  // 트리가 비어 있으면 바로 반환하여 종료한다\n  if (_root == null) return;\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n  while (cur != null) {\n    // 삭제할 노드를 찾았으므로 반복문을 종료한다\n    if (cur.val == _num) break;\n    pre = cur;\n    // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n    else\n      cur = cur.left;\n  }\n  // 삭제할 노드가 없으면 바로 반환한다\n  if (cur == null) return;\n  // 자식 노드 수 = 0 또는 1\n  if (cur.left == null || cur.right == null) {\n    // 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n    TreeNode? child = cur.left ?? cur.right;\n    // 노드 cur를 삭제한다\n    if (cur != _root) {\n      if (pre!.left == cur)\n        pre.left = child;\n      else\n        pre.right = child;\n    } else {\n      // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n      _root = child;\n    }\n  } else {\n    // 자식 노드 수 = 2\n    // 중위 순회에서 cur의 다음 노드를 구한다\n    TreeNode? tmp = cur.right;\n    while (tmp!.left != null) {\n      tmp = tmp.left;\n    }\n    // 노드 tmp를 재귀적으로 삭제한다\n    remove(tmp.val);\n    // tmp로 cur를 덮어쓴다\n    cur.val = tmp.val;\n  }\n}\n</code></pre> binary_search_tree.rs<pre><code>/* 노드 삭제 */\npub fn remove(&amp;mut self, num: i32) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if self.root.is_none() {\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while let Some(node) = cur.clone() {\n        match num.cmp(&amp;node.borrow().val) {\n            // 삭제할 노드를 찾았으므로 반복문을 종료한다\n            Ordering::Equal =&gt; break,\n            // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n            Ordering::Greater =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().right.clone();\n            }\n            // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n            Ordering::Less =&gt; {\n                pre = cur.clone();\n                cur = node.borrow().left.clone();\n            }\n        }\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if cur.is_none() {\n        return;\n    }\n    let cur = cur.unwrap();\n    let (left_child, right_child) = (cur.borrow().left.clone(), cur.borrow().right.clone());\n    match (left_child.clone(), right_child.clone()) {\n        // 자식 노드 수 = 0 또는 1\n        (None, None) | (Some(_), None) | (None, Some(_)) =&gt; {\n            // 자식 노드 수가 0 / 1이면 child = nullptr / 해당 자식 노드다\n            let child = left_child.or(right_child);\n            let pre = pre.unwrap();\n            // 노드 cur를 삭제한다\n            if !Rc::ptr_eq(&amp;cur, self.root.as_ref().unwrap()) {\n                let left = pre.borrow().left.clone();\n                if left.is_some() &amp;&amp; Rc::ptr_eq(left.as_ref().unwrap(), &amp;cur) {\n                    pre.borrow_mut().left = child;\n                } else {\n                    pre.borrow_mut().right = child;\n                }\n            } else {\n                // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n                self.root = child;\n            }\n        }\n        // 자식 노드 수 = 2\n        (Some(_), Some(_)) =&gt; {\n            // 중위 순회에서 cur의 다음 노드를 구한다\n            let mut tmp = cur.borrow().right.clone();\n            while let Some(node) = tmp.clone() {\n                if node.borrow().left.is_some() {\n                    tmp = node.borrow().left.clone();\n                } else {\n                    break;\n                }\n            }\n            let tmp_val = tmp.unwrap().borrow().val;\n            // 노드 tmp를 재귀적으로 삭제한다\n            self.remove(tmp_val);\n            // tmp로 cur를 덮어쓴다\n            cur.borrow_mut().val = tmp_val;\n        }\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* 노드 삭제 */\n// stdio.h를 포함했으므로 여기서는 remove라는 이름을 사용할 수 없다\nvoid removeItem(BinarySearchTree *bst, int num) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if (bst-&gt;root == NULL)\n        return;\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != NULL) {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // 삭제할 노드가 root의 오른쪽 서브트리에 있다\n            cur = cur-&gt;right;\n        } else {\n            // 삭제할 노드가 root의 왼쪽 서브트리에 있다\n            cur = cur-&gt;left;\n        }\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if (cur == NULL)\n        return;\n    // 삭제할 노드에 자식 노드가 있는지 확인한다\n    if (cur-&gt;left == NULL || cur-&gt;right == NULL) {\n        /* 자식 노드 수 = 0 또는 1 */\n        // 자식 노드 수가 0 / 1이면 child = nullptr / 해당 자식 노드다\n        TreeNode *child = cur-&gt;left != NULL ? cur-&gt;left : cur-&gt;right;\n        // 노드 cur를 삭제한다\n        if (pre-&gt;left == cur) {\n            pre-&gt;left = child;\n        } else {\n            pre-&gt;right = child;\n        }\n        // 메모리 해제\n        free(cur);\n    } else {\n        /* 자식 노드 수 = 2 */\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != NULL) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // 노드 tmp를 재귀적으로 삭제한다\n        removeItem(bst, tmp-&gt;val);\n        // tmp로 cur를 덮어쓴다\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.kt<pre><code>/* 노드 삭제 */\nfun remove(num: Int) {\n    // 트리가 비어 있으면 바로 반환하여 종료한다\n    if (root == null)\n        return\n    var cur = root\n    var pre: TreeNode? = null\n    // 반복하여 탐색하다 리프 노드를 지나면 종료한다\n    while (cur != null) {\n        // 삭제할 노드를 찾았으므로 반복문을 종료한다\n        if (cur._val == num)\n            break\n        pre = cur\n        // 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n        cur = if (cur._val &lt; num)\n            cur.right\n        // 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n        else\n            cur.left\n    }\n    // 삭제할 노드가 없으면 바로 반환한다\n    if (cur == null)\n        return\n    // 자식 노드 수 = 0 또는 1\n    if (cur.left == null || cur.right == null) {\n        // 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n        val child = if (cur.left != null)\n            cur.left\n        else\n            cur.right\n        // 노드 cur를 삭제한다\n        if (cur != root) {\n            if (pre!!.left == cur)\n                pre.left = child\n            else\n                pre.right = child\n        } else {\n            // 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n            root = child\n        }\n        // 자식 노드 수 = 2\n    } else {\n        // 중위 순회에서 cur의 다음 노드를 구한다\n        var tmp = cur.right\n        while (tmp!!.left != null) {\n            tmp = tmp.left\n        }\n        // 노드 tmp를 재귀적으로 삭제한다\n        remove(tmp._val)\n        // tmp로 cur를 덮어쓴다\n        cur._val = tmp._val\n    }\n}\n</code></pre> binary_search_tree.rb<pre><code>### 노드 삭제 ###\ndef remove(num)\n  # 트리가 비어 있으면 바로 반환하여 종료한다\n  return if @root.nil?\n\n  # 반복하여 탐색하다 리프 노드를 지나면 종료한다\n  cur, pre = @root, nil\n  while !cur.nil?\n    # 삭제할 노드를 찾았으므로 반복문을 종료한다\n    break if cur.val == num\n\n    pre = cur\n    # 삭제할 노드는 cur의 오른쪽 서브트리에 있다\n    if cur.val &lt; num\n      cur = cur.right\n    # 삭제할 노드는 cur의 왼쪽 서브트리에 있다\n    else\n      cur = cur.left\n    end\n  end\n  # 삭제할 노드가 없으면 바로 반환한다\n  return if cur.nil?\n\n  # 자식 노드 수 = 0 또는 1\n  if cur.left.nil? || cur.right.nil?\n    # 자식 노드 수 = 0 / 1이면 child = null / 해당 자식 노드이다\n    child = cur.left || cur.right\n    # 노드 cur를 삭제한다\n    if cur != @root\n      if pre.left == cur\n        pre.left = child\n      else\n        pre.right = child\n      end\n    else\n      # 삭제할 노드가 루트 노드이면 루트 노드를 다시 지정한다\n      @root = child\n    end\n  # 자식 노드 수 = 2\n  else\n    # 중위 순회에서 cur의 다음 노드를 구한다\n    tmp = cur.right\n    while !tmp.left.nil?\n      tmp = tmp.left\n    end\n    # 노드 tmp를 재귀적으로 삭제한다\n    remove(tmp.val)\n    # tmp로 cur를 덮어쓴다\n    cur.val = tmp.val\n  end\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#4","level":3,"title":"4.   중위 순회 결과의 정렬 성질","text":"<p>아래 그림(7-22)과 같이 이진 트리의 중위 순회는 '왼쪽 \\(\\rightarrow\\) 루트 \\(\\rightarrow\\) 오른쪽' 순서이며, 이진 탐색 트리는 '왼쪽 자식 \\(&lt;\\) 루트 \\(&lt;\\) 오른쪽 자식'의 대소 관계를 만족한다.</p> <p>따라서 중위 순회에서는 항상 다음으로 작은 노드를 먼저 방문한다. 여기서 이진 탐색 트리의 중위 순회 결과는 오름차순이다라는 중요한 성질을 얻는다.</p> <p>이 성질을 이용하면 별도 정렬 없이 \\(O(n)\\) 시간만으로 정렬된 데이터를 얻을 수 있어 매우 효율적이다.</p> <p></p> <p> 그림 7-22   이진 탐색 트리의 중위 순회 결과 </p>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#742","level":2,"title":"7.4.2   이진 탐색 트리의 효율","text":"<p>데이터 집합을 배열이나 이진 탐색 트리에 저장한다고 하자. 아래 표(7-2)를 보면 이진 탐색 트리의 각 연산은 로그 시간 복잡도로 안정적이고 효율적이다. 데이터 추가는 잦고 탐색과 삭제는 드문 상황에서만 배열이 더 효율적이다.</p> <p> 표 7-2   배열과 탐색 트리의 효율 비교 </p> 정렬되지 않은 배열 이진 탐색 트리 원소 탐색 \\(O(n)\\) \\(O(\\log n)\\) 원소 삽입 \\(O(1)\\) \\(O(\\log n)\\) 원소 삭제 \\(O(n)\\) \\(O(\\log n)\\) <p>이상적인 경우 이진 탐색 트리는 '균형'을 이루므로 어떤 노드든 \\(\\log n\\) 회의 반복 안에 찾을 수 있다.</p> <p>하지만 삽입과 삭제를 계속하다 보면 아래 그림(7-23)의 연결 리스트로 퇴화할 수 있다. 이때 각 연산의 시간 복잡도도 \\(O(n)\\)으로 저하된다.</p> <p></p> <p> 그림 7-23   이진 탐색 트리의 퇴화 </p>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_search_tree/#743","level":2,"title":"7.4.3   이진 탐색 트리의 주요 응용","text":"<ul> <li>시스템의 다단계 인덱스로 사용하여 효율적인 탐색, 삽입, 삭제를 구현한다.</li> <li>일부 탐색 알고리즘의 기반 자료 구조로 사용한다.</li> <li>데이터 스트림을 저장하면서 정렬 상태를 유지한다.</li> </ul>","path":["제 7 장   트리","7.4   이진 탐색 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/","level":1,"title":"7.1   이진 트리","text":"<p>이진 트리(binary tree)는 '조상'과 '후손' 사이의 파생 관계를 나타내는 비선형 자료 구조로, '하나를 둘로 나누는' 분할 정복의 논리를 보여 준다. 연결 리스트처럼 기본 단위는 노드이며, 각 노드는 값과 왼쪽 자식 참조, 오른쪽 자식 참조를 포함한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby <pre><code>class TreeNode:\n    \"\"\"이진 트리 노드 클래스\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # 노드 값\n        self.left: TreeNode | None = None  # 왼쪽 자식 노드 참조\n        self.right: TreeNode | None = None # 오른쪽 자식 노드 참조\n</code></pre> <pre><code>/* 이진 트리 노드 구조체 */\nstruct TreeNode {\n    int val;          // 노드 값\n    TreeNode *left;   // 왼쪽 자식 노드 포인터\n    TreeNode *right;  // 오른쪽 자식 노드 포인터\n    TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n</code></pre> <pre><code>/* 이진 트리 노드 클래스 */\nclass TreeNode {\n    int val;         // 노드 값\n    TreeNode left;   // 왼쪽 자식 노드 참조\n    TreeNode right;  // 오른쪽 자식 노드 참조\n    TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* 이진 트리 노드 클래스 */\nclass TreeNode(int? x) {\n    public int? val = x;    // 노드 값\n    public TreeNode? left;  // 왼쪽 자식 노드 참조\n    public TreeNode? right; // 오른쪽 자식 노드 참조\n}\n</code></pre> <pre><code>/* 이진 트리 노드 구조체 */\ntype TreeNode struct {\n    Val   int\n    Left  *TreeNode\n    Right *TreeNode\n}\n/* 생성자 */\nfunc NewTreeNode(v int) *TreeNode {\n    return &amp;TreeNode{\n        Left:  nil, // 왼쪽 자식 노드 포인터\n        Right: nil, // 오른쪽 자식 노드 포인터\n        Val:   v,   // 노드 값\n    }\n}\n</code></pre> <pre><code>/* 이진 트리 노드 클래스 */\nclass TreeNode {\n    var val: Int // 노드 값\n    var left: TreeNode? // 왼쪽 자식 노드 참조\n    var right: TreeNode? // 오른쪽 자식 노드 참조\n\n    init(x: Int) {\n        val = x\n    }\n}\n</code></pre> <pre><code>/* 이진 트리 노드 클래스 */\nclass TreeNode {\n    val; // 노드 값\n    left; // 왼쪽 자식 노드 포인터\n    right; // 오른쪽 자식 노드 포인터\n    constructor(val, left, right) {\n        this.val = val === undefined ? 0 : val;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* 이진 트리 노드 클래스 */\nclass TreeNode {\n    val: number;\n    left: TreeNode | null;\n    right: TreeNode | null;\n\n    constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val; // 노드 값\n        this.left = left === undefined ? null : left; // 왼쪽 자식 노드 참조\n        this.right = right === undefined ? null : right; // 오른쪽 자식 노드 참조\n    }\n}\n</code></pre> <pre><code>/* 이진 트리 노드 클래스 */\nclass TreeNode {\n  int val;         // 노드 값\n  TreeNode? left;  // 왼쪽 자식 노드 참조\n  TreeNode? right; // 오른쪽 자식 노드 참조\n  TreeNode(this.val, [this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* 이진 트리 노드 구조체 */\nstruct TreeNode {\n    val: i32,                               // 노드 값\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // 왼쪽 자식 노드 참조\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // 오른쪽 자식 노드 참조\n}\n\nimpl TreeNode {\n    /* 생성자 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* 이진 트리 노드 구조체 */\ntypedef struct TreeNode {\n    int val;                // 노드 값\n    int height;             // 노드 높이\n    struct TreeNode *left;  // 왼쪽 자식 노드 포인터\n    struct TreeNode *right; // 오른쪽 자식 노드 포인터\n} TreeNode;\n\n/* 생성자 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>/* 이진 트리 노드 클래스 */\nclass TreeNode(val _val: Int) {  // 노드 값\n    var left: TreeNode? = null   // 왼쪽 자식 노드 참조\n    var right: TreeNode? = null  // 오른쪽 자식 노드 참조\n}\n</code></pre> <pre><code>### 이진 트리 노드 클래스 ###\nclass TreeNode\n  attr_accessor :val    # 노드 값\n  attr_accessor :left   # 왼쪽 자식 노드 참조\n  attr_accessor :right  # 오른쪽 자식 노드 참조\n\n  def initialize(val)\n    @val = val\n  end\nend\n</code></pre> <p>각 노드는 두 참조(포인터)를 가지며, 각각 왼쪽 자식 노드(left-child node)와 오른쪽 자식 노드(right-child node)를 가리킨다. 해당 노드는 두 자식의 부모 노드(parent node)라고 한다. 어떤 노드의 왼쪽 자식과 그 아래 노드들이 이루는 트리를 그 노드의 왼쪽 서브트리(left subtree)라고 하며, 같은 방식으로 오른쪽 서브트리(right subtree)를 정의한다.</p> <p>이진 트리에서 리프 노드를 제외한 모든 노드에는 자식 노드와 비어 있지 않은 서브트리가 있다. 아래 그림(7-1)에서 '노드 2'를 부모로 보면 왼쪽과 오른쪽 자식은 각각 '노드 4', '노드 5'다. 왼쪽 서브트리는 '노드 4와 그 아래 노드들이 이루는 트리', 오른쪽 서브트리는 '노드 5와 그 아래 노드들이 이루는 트리'다.</p> <p></p> <p> 그림 7-1   부모 노드, 자식 노드, 서브트리 </p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#711","level":2,"title":"7.1.1   이진 트리의 주요 용어","text":"<p>이진 트리에서 자주 쓰는 용어는 아래 그림(7-2)과 같다.</p> <ul> <li>루트 노드(root node): 이진 트리의 최상위 노드로, 부모가 없다.</li> <li>리프 노드(leaf node): 자식이 없는 노드로, 두 포인터가 모두 <code>None</code>을 가리킨다.</li> <li>간선(edge): 두 노드를 잇는 선으로, 노드 참조(포인터)에 해당한다.</li> <li>노드의 레벨(level): 위에서 아래로 증가하며 루트의 레벨은 1이다.</li> <li>노드의 차수(degree): 자식 노드의 수다. 이진 트리에서는 0, 1, 2 중 하나다.</li> <li>이진 트리의 높이(height): 루트에서 가장 먼 리프까지 거치는 간선 수다.</li> <li>노드의 깊이(depth): 루트에서 해당 노드까지 거치는 간선 수다.</li> <li>노드의 높이(height): 해당 노드에서 가장 먼 리프와 해당 노드 사이에 있는 간선 수다.</li> </ul> <p></p> <p> 그림 7-2   이진 트리의 주요 용어 </p> <p>참고</p> <p>보통 '높이'와 '깊이'를 '거치는 간선 수'로 정의하지만, 일부 문제나 교재는 '거치는 노드 수'로 정의하기도 한다. 그런 경우에는 높이와 깊이에 각각 1을 더해야 한다.</p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#712","level":2,"title":"7.1.2   이진 트리의 기본 연산","text":"","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#1","level":3,"title":"1.   이진 트리 초기화","text":"<p>연결 리스트처럼 먼저 노드를 초기화하고 참조(포인터)를 연결한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree.py<pre><code># 이진 트리 초기화\n# 노드 초기화\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# 노드 사이의 참조(포인터) 연결\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.cpp<pre><code>/* 이진 트리 초기화 */\n// 노드 초기화\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// 노드 사이의 참조(포인터) 연결\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.java<pre><code>// 노드 초기화\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// 노드 사이의 참조(포인터) 연결\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.cs<pre><code>/* 이진 트리 초기화 */\n// 노드 초기화\nTreeNode n1 = new(1);\nTreeNode n2 = new(2);\nTreeNode n3 = new(3);\nTreeNode n4 = new(4);\nTreeNode n5 = new(5);\n// 노드 사이의 참조(포인터) 연결\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.go<pre><code>/* 이진 트리 초기화 */\n// 노드 초기화\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// 노드 사이의 참조(포인터) 연결\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n</code></pre> binary_tree.swift<pre><code>// 노드 초기화\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// 노드 사이의 참조(포인터) 연결\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.js<pre><code>/* 이진 트리 초기화 */\n// 노드 초기화\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// 노드 사이의 참조(포인터) 연결\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.ts<pre><code>/* 이진 트리 초기화 */\n// 노드 초기화\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// 노드 사이의 참조(포인터) 연결\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.dart<pre><code>/* 이진 트리 초기화 */\n// 노드 초기화\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// 노드 사이의 참조(포인터) 연결\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.rs<pre><code>// 노드 초기화\nlet n1 = TreeNode::new(1);\nlet n2 = TreeNode::new(2);\nlet n3 = TreeNode::new(3);\nlet n4 = TreeNode::new(4);\nlet n5 = TreeNode::new(5);\n// 노드 사이의 참조(포인터) 연결\nn1.borrow_mut().left = Some(n2.clone());\nn1.borrow_mut().right = Some(n3);\nn2.borrow_mut().left = Some(n4);\nn2.borrow_mut().right = Some(n5);\n</code></pre> binary_tree.c<pre><code>/* 이진 트리 초기화 */\n// 노드 초기화\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// 노드 사이의 참조(포인터) 연결\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.kt<pre><code>// 노드 초기화\nval n1 = TreeNode(1)\nval n2 = TreeNode(2)\nval n3 = TreeNode(3)\nval n4 = TreeNode(4)\nval n5 = TreeNode(5)\n// 노드 사이의 참조(포인터) 연결\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.rb<pre><code># 이진 트리 초기화\n# 노드 초기화\nn1 = TreeNode.new(1)\nn2 = TreeNode.new(2)\nn3 = TreeNode.new(3)\nn4 = TreeNode.new(4)\nn5 = TreeNode.new(5)\n# 노드 사이의 참조(포인터) 연결\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> 실행 시각화 <p> 전체 화면 &gt;</p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#2","level":3,"title":"2.   노드 삽입과 삭제","text":"<p>연결 리스트와 마찬가지로 이진 트리에서도 포인터를 수정하여 노드를 삽입하고 삭제할 수 있다. 아래 그림(7-3)은 그 예다.</p> <p></p> <p> 그림 7-3   이진 트리의 노드 삽입과 삭제 </p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree.py<pre><code># 노드 삽입과 삭제\np = TreeNode(0)\n# n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = p\np.left = n2\n# 노드 P 삭제\nn1.left = n2\n</code></pre> binary_tree.cpp<pre><code>/* 노드 삽입과 삭제 */\nTreeNode* P = new TreeNode(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1-&gt;left = P;\nP-&gt;left = n2;\n// 노드 P 삭제\nn1-&gt;left = n2;\n// 메모리 해제\ndelete P;\n</code></pre> binary_tree.java<pre><code>TreeNode P = new TreeNode(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = P;\nP.left = n2;\n// 노드 P 삭제\nn1.left = n2;\n</code></pre> binary_tree.cs<pre><code>/* 노드 삽입과 삭제 */\nTreeNode P = new(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = P;\nP.left = n2;\n// 노드 P 삭제\nn1.left = n2;\n</code></pre> binary_tree.go<pre><code>/* 노드 삽입과 삭제 */\n// n1 -&gt; n2 사이에 노드 P 삽입\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// 노드 P 삭제\nn1.Left = n2\n</code></pre> binary_tree.swift<pre><code>let P = TreeNode(x: 0)\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = P\nP.left = n2\n// 노드 P 삭제\nn1.left = n2\n</code></pre> binary_tree.js<pre><code>/* 노드 삽입과 삭제 */\nlet P = new TreeNode(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = P;\nP.left = n2;\n// 노드 P 삭제\nn1.left = n2;\n</code></pre> binary_tree.ts<pre><code>/* 노드 삽입과 삭제 */\nconst P = new TreeNode(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = P;\nP.left = n2;\n// 노드 P 삭제\nn1.left = n2;\n</code></pre> binary_tree.dart<pre><code>/* 노드 삽입과 삭제 */\nTreeNode P = new TreeNode(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = P;\nP.left = n2;\n// 노드 P 삭제\nn1.left = n2;\n</code></pre> binary_tree.rs<pre><code>let p = TreeNode::new(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.borrow_mut().left = Some(p.clone());\np.borrow_mut().left = Some(n2.clone());\n// 노드 p 삭제\nn1.borrow_mut().left = Some(n2);\n</code></pre> binary_tree.c<pre><code>/* 노드 삽입과 삭제 */\nTreeNode *P = newTreeNode(0);\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1-&gt;left = P;\nP-&gt;left = n2;\n// 노드 P 삭제\nn1-&gt;left = n2;\n// 메모리 해제\nfree(P);\n</code></pre> binary_tree.kt<pre><code>val P = TreeNode(0)\n// n1 -&gt; n2 사이에 노드 P 삽입\nn1.left = P\nP.left = n2\n// 노드 P 삭제\nn1.left = n2\n</code></pre> binary_tree.rb<pre><code># 노드 삽입과 삭제\n_p = TreeNode.new(0)\n# n1 -&gt; n2 사이에 노드 _p 삽입\nn1.left = _p\n_p.left = n2\n# 노드 삭제\nn1.left = n2\n</code></pre> 실행 시각화 <p> 전체 화면 &gt;</p> <p>참고</p> <p>노드 삽입은 이진 트리의 기존 논리 구조를 바꿀 수 있으며, 노드 삭제는 보통 해당 노드와 그 모든 서브트리를 삭제한다는 뜻이다. 따라서 의미 있는 작업을 수행하려면 이진 트리의 삽입과 삭제는 보통 여러 연산을 조합하여 이루어진다.</p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#713","level":2,"title":"7.1.3   주요 이진 트리 종류","text":"","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#1_1","level":3,"title":"1.   포화 이진 트리","text":"<p>아래 그림(7-4)의 포화 이진 트리(perfect binary tree)는 모든 레벨에 노드가 가득 차 있다. 리프 노드의 차수는 \\(0\\), 나머지 노드의 차수는 모두 \\(2\\)다. 높이가 \\(h\\)이면 전체 노드 수는 \\(2^{h+1} - 1\\)로 전형적인 지수 관계를 보이며, 자연에서 흔히 볼 수 있는 세포 분열 현상을 반영한다.</p> <p>참고</p> <p>중국어권에서는 포화 이진 트리를 흔히 满二叉树라고 부른다.</p> <p></p> <p> 그림 7-4   포화 이진 트리 </p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#2_1","level":3,"title":"2.   완전 이진 트리","text":"<p>아래 그림(7-5)의 완전 이진 트리(complete binary tree)는 맨 아래 레벨만 가득 차 있지 않아도 되며, 그 레벨의 노드는 왼쪽부터 오른쪽으로 빈틈없이 채워져야 한다. 포화 이진 트리도 완전 이진 트리다.</p> <p></p> <p> 그림 7-5   완전 이진 트리 </p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#3","level":3,"title":"3.   정 이진 트리","text":"<p>아래 그림(7-6)의 정 이진 트리(full binary tree)는 리프를 제외한 모든 노드가 자식 노드를 두 개씩 가진다.</p> <p></p> <p> 그림 7-6   정 이진 트리 </p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#4","level":3,"title":"4.   균형 이진 트리","text":"<p>아래 그림(7-7)의 균형 이진 트리(balanced binary tree)는 모든 노드에서 왼쪽과 오른쪽 서브트리의 높이 차의 절댓값이 1을 넘지 않는다.</p> <p></p> <p> 그림 7-7   균형 이진 트리 </p>","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree/#714","level":2,"title":"7.1.4   이진 트리의 퇴화","text":"<p>아래 그림(7-8)은 이진 트리의 이상적인 구조와 퇴화한 구조를 보여 준다. 모든 레벨이 가득 차면 '포화 이진 트리'가 되고, 모든 노드가 한쪽으로 치우치면 '연결 리스트'로 퇴화한다.</p> <ul> <li>포화 이진 트리는 이상적인 경우로, 이진 트리의 '분할 정복' 이점을 충분히 발휘할 수 있다.</li> <li>연결 리스트는 다른 극단으로, 모든 연산이 선형 연산이 되어 시간 복잡도가 \\(O(n)\\)으로 저하된다.</li> </ul> <p></p> <p> 그림 7-8   이진 트리의 최선 구조와 최악 구조 </p> <p>아래 표(7-1)와 같이 최선 및 최악 구조에서 이진 트리의 리프 수, 전체 노드 수, 높이 등이 최댓값이나 최솟값에 도달한다.</p> <p> 표 7-1   이진 트리의 최선 구조와 최악 구조 </p> 포화 이진 트리 연결 리스트 \\(i\\) 번째 레벨의 노드 수 \\(2^{i-1}\\) \\(1\\) 높이 \\(h\\)인 트리의 리프 수 \\(2^h\\) \\(1\\) 높이 \\(h\\)인 트리의 전체 노드 수 \\(2^{h+1} - 1\\) \\(h + 1\\) 전체 노드 수가 \\(n\\)인 트리의 높이 \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)","path":["제 7 장   트리","7.1   이진 트리"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/","level":1,"title":"7.2   이진 트리 순회","text":"<p>물리적 구조의 관점에서 트리는 연결 리스트 기반 자료 구조이므로, 포인터로 노드를 하나씩 방문하여 순회한다. 하지만 비선형 자료 구조여서 연결 리스트보다 순회가 복잡하며 탐색 알고리즘이 필요하다.</p> <p>이진 트리의 주요 순회 방식은 레벨 순서 순회, 전위 순회, 중위 순회, 후위 순회 등이다.</p>","path":["제 7 장   트리","7.2   이진 트리 순회"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#721","level":2,"title":"7.2.1   레벨 순서 순회","text":"<p>아래 그림(7-9)과 같이 레벨 순서 순회(level-order traversal)는 위에서 아래로 한 레벨씩 순회하며, 각 레벨에서는 왼쪽부터 오른쪽 순서로 노드를 방문한다.</p> <p>레벨 순서 순회는 본질적으로 너비 우선 순회(breadth-first traversal), 즉 너비 우선 탐색(breadth-first search, BFS)에 속한다. '한 겹씩 바깥으로 확장'하는 층별 순회 방식이다.</p> <p></p> <p> 그림 7-9   이진 트리의 레벨 순서 순회 </p>","path":["제 7 장   트리","7.2   이진 트리 순회"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#1","level":3,"title":"1.   코드 구현","text":"<p>너비 우선 순회는 보통 '큐'로 구현한다. 큐는 '선입선출' 규칙을, 너비 우선 순회는 '한 레벨씩 진행'하는 규칙을 따르며, 바탕에 깔린 생각은 같다. 구현 코드는 다음과 같다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree_bfs.py<pre><code>def level_order(root: TreeNode | None) -&gt; list[int]:\n    \"\"\"레벨 순서 순회\"\"\"\n    # 큐를 초기화하고 루트 노드를 넣는다\n    queue: deque[TreeNode] = deque()\n    queue.append(root)\n    # 순회 순서를 저장할 리스트를 초기화한다\n    res = []\n    while queue:\n        node: TreeNode = queue.popleft()  # 큐에서 꺼낸다\n        res.append(node.val)  # 노드 값을 저장한다\n        if node.left is not None:\n            queue.append(node.left)  # 왼쪽 자식 노드를 큐에 삽입한다\n        if node.right is not None:\n            queue.append(node.right)  # 오른쪽 자식 노드를 큐에 삽입한다\n    return res\n</code></pre> binary_tree_bfs.cpp<pre><code>/* 레벨 순서 순회 */\nvector&lt;int&gt; levelOrder(TreeNode *root) {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    queue&lt;TreeNode *&gt; queue;\n    queue.push(root);\n    // 순회 순서를 저장할 리스트를 초기화한다\n    vector&lt;int&gt; vec;\n    while (!queue.empty()) {\n        TreeNode *node = queue.front();\n        queue.pop();              // 큐에서 꺼낸다\n        vec.push_back(node-&gt;val); // 노드 값을 저장한다\n        if (node-&gt;left != nullptr)\n            queue.push(node-&gt;left); // 왼쪽 자식 노드를 큐에 삽입한다\n        if (node-&gt;right != nullptr)\n            queue.push(node-&gt;right); // 오른쪽 자식 노드를 큐에 삽입한다\n    }\n    return vec;\n}\n</code></pre> binary_tree_bfs.java<pre><code>/* 레벨 순서 순회 */\nList&lt;Integer&gt; levelOrder(TreeNode root) {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    Queue&lt;TreeNode&gt; queue = new LinkedList&lt;&gt;();\n    queue.add(root);\n    // 순회 순서를 저장할 리스트를 초기화한다\n    List&lt;Integer&gt; list = new ArrayList&lt;&gt;();\n    while (!queue.isEmpty()) {\n        TreeNode node = queue.poll(); // 큐에서 꺼낸다\n        list.add(node.val);           // 노드 값을 저장한다\n        if (node.left != null)\n            queue.offer(node.left);   // 왼쪽 자식 노드를 큐에 삽입한다\n        if (node.right != null)\n            queue.offer(node.right);  // 오른쪽 자식 노드를 큐에 삽입한다\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.cs<pre><code>/* 레벨 순서 순회 */\nList&lt;int&gt; LevelOrder(TreeNode root) {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    Queue&lt;TreeNode&gt; queue = new();\n    queue.Enqueue(root);\n    // 순회 순서를 저장할 리스트를 초기화한다\n    List&lt;int&gt; list = [];\n    while (queue.Count != 0) {\n        TreeNode node = queue.Dequeue(); // 큐에서 꺼낸다\n        list.Add(node.val!.Value);       // 노드 값을 저장한다\n        if (node.left != null)\n            queue.Enqueue(node.left);    // 왼쪽 자식 노드를 큐에 삽입한다\n        if (node.right != null)\n            queue.Enqueue(node.right);   // 오른쪽 자식 노드를 큐에 삽입한다\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.go<pre><code>/* 레벨 순서 순회 */\nfunc levelOrder(root *TreeNode) []any {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    queue := list.New()\n    queue.PushBack(root)\n    // 순회 순서를 저장할 슬라이스를 초기화한다\n    nums := make([]any, 0)\n    for queue.Len() &gt; 0 {\n        // 큐에서 꺼낸다\n        node := queue.Remove(queue.Front()).(*TreeNode)\n        // 노드 값을 저장한다\n        nums = append(nums, node.Val)\n        if node.Left != nil {\n            // 왼쪽 자식 노드를 큐에 삽입한다\n            queue.PushBack(node.Left)\n        }\n        if node.Right != nil {\n            // 오른쪽 자식 노드를 큐에 삽입한다\n            queue.PushBack(node.Right)\n        }\n    }\n    return nums\n}\n</code></pre> binary_tree_bfs.swift<pre><code>/* 레벨 순서 순회 */\nfunc levelOrder(root: TreeNode) -&gt; [Int] {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    var queue: [TreeNode] = [root]\n    // 순회 순서를 저장할 리스트를 초기화한다\n    var list: [Int] = []\n    while !queue.isEmpty {\n        let node = queue.removeFirst() // 큐에서 꺼낸다\n        list.append(node.val) // 노드 값을 저장한다\n        if let left = node.left {\n            queue.append(left) // 왼쪽 자식 노드를 큐에 삽입한다\n        }\n        if let right = node.right {\n            queue.append(right) // 오른쪽 자식 노드를 큐에 삽입한다\n        }\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.js<pre><code>/* 레벨 순서 순회 */\nfunction levelOrder(root) {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    const queue = [root];\n    // 순회 순서를 저장할 리스트를 초기화한다\n    const list = [];\n    while (queue.length) {\n        let node = queue.shift(); // 큐에서 꺼낸다\n        list.push(node.val); // 노드 값을 저장한다\n        if (node.left) queue.push(node.left); // 왼쪽 자식 노드를 큐에 삽입한다\n        if (node.right) queue.push(node.right); // 오른쪽 자식 노드를 큐에 삽입한다\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.ts<pre><code>/* 레벨 순서 순회 */\nfunction levelOrder(root: TreeNode | null): number[] {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    const queue = [root];\n    // 순회 순서를 저장할 리스트를 초기화한다\n    const list: number[] = [];\n    while (queue.length) {\n        let node = queue.shift() as TreeNode; // 큐에서 꺼낸다\n        list.push(node.val); // 노드 값을 저장한다\n        if (node.left) {\n            queue.push(node.left); // 왼쪽 자식 노드를 큐에 삽입한다\n        }\n        if (node.right) {\n            queue.push(node.right); // 오른쪽 자식 노드를 큐에 삽입한다\n        }\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.dart<pre><code>/* 레벨 순서 순회 */\nList&lt;int&gt; levelOrder(TreeNode? root) {\n  // 큐를 초기화하고 루트 노드를 넣는다\n  Queue&lt;TreeNode?&gt; queue = Queue();\n  queue.add(root);\n  // 순회 순서를 저장할 리스트를 초기화한다\n  List&lt;int&gt; res = [];\n  while (queue.isNotEmpty) {\n    TreeNode? node = queue.removeFirst(); // 큐에서 꺼낸다\n    res.add(node!.val); // 노드 값을 저장한다\n    if (node.left != null) queue.add(node.left); // 왼쪽 자식 노드를 큐에 삽입한다\n    if (node.right != null) queue.add(node.right); // 오른쪽 자식 노드를 큐에 삽입한다\n  }\n  return res;\n}\n</code></pre> binary_tree_bfs.rs<pre><code>/* 레벨 순서 순회 */\nfn level_order(root: &amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    let mut que = VecDeque::new();\n    que.push_back(root.clone());\n    // 순회 순서를 저장할 리스트를 초기화한다\n    let mut vec = Vec::new();\n\n    while let Some(node) = que.pop_front() {\n        // 큐에서 꺼낸다\n        vec.push(node.borrow().val); // 노드 값을 저장한다\n        if let Some(left) = node.borrow().left.as_ref() {\n            que.push_back(left.clone()); // 왼쪽 자식 노드를 큐에 삽입한다\n        }\n        if let Some(right) = node.borrow().right.as_ref() {\n            que.push_back(right.clone()); // 오른쪽 자식 노드를 큐에 삽입한다\n        };\n    }\n    vec\n}\n</code></pre> binary_tree_bfs.c<pre><code>/* 레벨 순서 순회 */\nint *levelOrder(TreeNode *root, int *size) {\n    /* 보조 큐 */\n    int front, rear;\n    int index, *arr;\n    TreeNode *node;\n    TreeNode **queue;\n\n    /* 보조 큐 */\n    queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);\n    // 큐 포인터\n    front = 0, rear = 0;\n    // 루트 노드를 넣는다\n    queue[rear++] = root;\n    // 순회 순서를 저장할 리스트를 초기화한다\n    /* 보조 배열 */\n    arr = (int *)malloc(sizeof(int) * MAX_SIZE);\n    // 배열 포인터\n    index = 0;\n    while (front &lt; rear) {\n        // 큐에서 꺼낸다\n        node = queue[front++];\n        // 노드 값을 저장한다\n        arr[index++] = node-&gt;val;\n        if (node-&gt;left != NULL) {\n            // 왼쪽 자식 노드를 큐에 삽입한다\n            queue[rear++] = node-&gt;left;\n        }\n        if (node-&gt;right != NULL) {\n            // 오른쪽 자식 노드를 큐에 삽입한다\n            queue[rear++] = node-&gt;right;\n        }\n    }\n    // 배열 길이 값을 갱신한다\n    *size = index;\n    arr = realloc(arr, sizeof(int) * (*size));\n\n    // 보조 배열의 메모리 공간을 해제한다\n    free(queue);\n    return arr;\n}\n</code></pre> binary_tree_bfs.kt<pre><code>/* 레벨 순서 순회 */\nfun levelOrder(root: TreeNode?): MutableList&lt;Int&gt; {\n    // 큐를 초기화하고 루트 노드를 넣는다\n    val queue = LinkedList&lt;TreeNode?&gt;()\n    queue.add(root)\n    // 순회 순서를 저장할 리스트를 초기화한다\n    val list = mutableListOf&lt;Int&gt;()\n    while (queue.isNotEmpty()) {\n        val node = queue.poll()      // 큐에서 꺼낸다\n        list.add(node?._val!!)       // 노드 값을 저장한다\n        if (node.left != null)\n            queue.offer(node.left)   // 왼쪽 자식 노드를 큐에 삽입한다\n        if (node.right != null)\n            queue.offer(node.right)  // 오른쪽 자식 노드를 큐에 삽입한다\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.rb<pre><code>### 레벨 순서 순회 ###\ndef level_order(root)\n  # 큐를 초기화하고 루트 노드를 넣는다\n  queue = [root]\n  # 순회 순서를 저장할 리스트를 초기화한다\n  res = []\n  while !queue.empty?\n    node = queue.shift # 큐에서 꺼낸다\n    res &lt;&lt; node.val # 노드 값을 저장한다\n    queue &lt;&lt; node.left unless node.left.nil? # 왼쪽 자식 노드를 큐에 삽입한다\n    queue &lt;&lt; node.right unless node.right.nil? # 오른쪽 자식 노드를 큐에 삽입한다\n  end\n  res\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p>","path":["제 7 장   트리","7.2   이진 트리 순회"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#2","level":3,"title":"2.   복잡도 분석","text":"<ul> <li>시간 복잡도 \\(O(n)\\): 모든 노드를 한 번씩 방문하므로 \\(O(n)\\) 시간이 걸린다. \\(n\\)은 노드 수다.</li> <li>공간 복잡도 \\(O(n)\\): 최악의 경우인 포화 이진 트리에서는 맨 아래 레벨을 순회하기 직전에 큐에 최대 \\((n + 1) / 2\\) 개 노드가 동시에 들어 있어 \\(O(n)\\) 공간을 차지한다.</li> </ul>","path":["제 7 장   트리","7.2   이진 트리 순회"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#722","level":2,"title":"7.2.2   전위, 중위, 후위 순회","text":"<p>전위, 중위, 후위 순회는 모두 깊이 우선 순회(depth-first traversal), 즉 깊이 우선 탐색(depth-first search, DFS)에 속한다. '먼저 끝까지 간 뒤 돌아와 계속하는' 순회 방식이다.</p> <p>아래 그림(7-10)은 이진 트리의 깊이 우선 순회 원리를 보여 준다. 깊이 우선 순회는 트리 전체의 바깥을 한 바퀴 '도는' 것과 같다. 각 노드에서 세 위치를 만나며, 각각 전위, 중위, 후위 순회에 대응한다.</p> <p></p> <p> 그림 7-10   이진 탐색 트리의 전위, 중위, 후위 순회 </p>","path":["제 7 장   트리","7.2   이진 트리 순회"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#1_1","level":3,"title":"1.   코드 구현","text":"<p>깊이 우선 탐색은 보통 재귀로 구현한다.</p> PythonC++JavaC#GoSwiftJSTSDartRustCKotlinRuby binary_tree_dfs.py<pre><code>def pre_order(root: TreeNode | None):\n    \"\"\"전위 순회\"\"\"\n    if root is None:\n        return\n    # 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    res.append(root.val)\n    pre_order(root=root.left)\n    pre_order(root=root.right)\n\ndef in_order(root: TreeNode | None):\n    \"\"\"중위 순회\"\"\"\n    if root is None:\n        return\n    # 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    in_order(root=root.left)\n    res.append(root.val)\n    in_order(root=root.right)\n\ndef post_order(root: TreeNode | None):\n    \"\"\"후위 순회\"\"\"\n    if root is None:\n        return\n    # 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    post_order(root=root.left)\n    post_order(root=root.right)\n    res.append(root.val)\n</code></pre> binary_tree_dfs.cpp<pre><code>/* 전위 순회 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    vec.push_back(root-&gt;val);\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n\n/* 중위 순회 */\nvoid inOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(root-&gt;left);\n    vec.push_back(root-&gt;val);\n    inOrder(root-&gt;right);\n}\n\n/* 후위 순회 */\nvoid postOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(root-&gt;left);\n    postOrder(root-&gt;right);\n    vec.push_back(root-&gt;val);\n}\n</code></pre> binary_tree_dfs.java<pre><code>/* 전위 순회 */\nvoid preOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    list.add(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* 중위 순회 */\nvoid inOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(root.left);\n    list.add(root.val);\n    inOrder(root.right);\n}\n\n/* 후위 순회 */\nvoid postOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(root.left);\n    postOrder(root.right);\n    list.add(root.val);\n}\n</code></pre> binary_tree_dfs.cs<pre><code>/* 전위 순회 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) return;\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    list.Add(root.val!.Value);\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n\n/* 중위 순회 */\nvoid InOrder(TreeNode? root) {\n    if (root == null) return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    InOrder(root.left);\n    list.Add(root.val!.Value);\n    InOrder(root.right);\n}\n\n/* 후위 순회 */\nvoid PostOrder(TreeNode? root) {\n    if (root == null) return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    PostOrder(root.left);\n    PostOrder(root.right);\n    list.Add(root.val!.Value);\n}\n</code></pre> binary_tree_dfs.go<pre><code>/* 전위 순회 */\nfunc preOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    nums = append(nums, node.Val)\n    preOrder(node.Left)\n    preOrder(node.Right)\n}\n\n/* 중위 순회 */\nfunc inOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(node.Left)\n    nums = append(nums, node.Val)\n    inOrder(node.Right)\n}\n\n/* 후위 순회 */\nfunc postOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(node.Left)\n    postOrder(node.Right)\n    nums = append(nums, node.Val)\n}\n</code></pre> binary_tree_dfs.swift<pre><code>/* 전위 순회 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    list.append(root.val)\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n\n/* 중위 순회 */\nfunc inOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(root: root.left)\n    list.append(root.val)\n    inOrder(root: root.right)\n}\n\n/* 후위 순회 */\nfunc postOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(root: root.left)\n    postOrder(root: root.right)\n    list.append(root.val)\n}\n</code></pre> binary_tree_dfs.js<pre><code>/* 전위 순회 */\nfunction preOrder(root) {\n    if (root === null) return;\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* 중위 순회 */\nfunction inOrder(root) {\n    if (root === null) return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* 후위 순회 */\nfunction postOrder(root) {\n    if (root === null) return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.ts<pre><code>/* 전위 순회 */\nfunction preOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* 중위 순회 */\nfunction inOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* 후위 순회 */\nfunction postOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.dart<pre><code>/* 전위 순회 */\nvoid preOrder(TreeNode? node) {\n  if (node == null) return;\n  // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n  list.add(node.val);\n  preOrder(node.left);\n  preOrder(node.right);\n}\n\n/* 중위 순회 */\nvoid inOrder(TreeNode? node) {\n  if (node == null) return;\n  // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n  inOrder(node.left);\n  list.add(node.val);\n  inOrder(node.right);\n}\n\n/* 후위 순회 */\nvoid postOrder(TreeNode? node) {\n  if (node == null) return;\n  // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n  postOrder(node.left);\n  postOrder(node.right);\n  list.add(node.val);\n}\n</code></pre> binary_tree_dfs.rs<pre><code>/* 전위 순회 */\nfn pre_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n            let node = node.borrow();\n            res.push(node.val);\n            dfs(node.left.as_ref(), res);\n            dfs(node.right.as_ref(), res);\n        }\n    }\n    dfs(root, &amp;mut result);\n\n    result\n}\n\n/* 중위 순회 */\nfn in_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n            let node = node.borrow();\n            dfs(node.left.as_ref(), res);\n            res.push(node.val);\n            dfs(node.right.as_ref(), res);\n        }\n    }\n    dfs(root, &amp;mut result);\n\n    result\n}\n\n/* 후위 순회 */\nfn post_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    fn dfs(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;i32&gt;) {\n        if let Some(node) = root {\n            // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n            let node = node.borrow();\n            dfs(node.left.as_ref(), res);\n            dfs(node.right.as_ref(), res);\n            res.push(node.val);\n        }\n    }\n\n    dfs(root, &amp;mut result);\n\n    result\n}\n</code></pre> binary_tree_dfs.c<pre><code>/* 전위 순회 */\nvoid preOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    arr[(*size)++] = root-&gt;val;\n    preOrder(root-&gt;left, size);\n    preOrder(root-&gt;right, size);\n}\n\n/* 중위 순회 */\nvoid inOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(root-&gt;left, size);\n    arr[(*size)++] = root-&gt;val;\n    inOrder(root-&gt;right, size);\n}\n\n/* 후위 순회 */\nvoid postOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(root-&gt;left, size);\n    postOrder(root-&gt;right, size);\n    arr[(*size)++] = root-&gt;val;\n}\n</code></pre> binary_tree_dfs.kt<pre><code>/* 전위 순회 */\nfun preOrder(root: TreeNode?) {\n    if (root == null) return\n    // 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n    list.add(root._val)\n    preOrder(root.left)\n    preOrder(root.right)\n}\n\n/* 중위 순회 */\nfun inOrder(root: TreeNode?) {\n    if (root == null) return\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n    inOrder(root.left)\n    list.add(root._val)\n    inOrder(root.right)\n}\n\n/* 후위 순회 */\nfun postOrder(root: TreeNode?) {\n    if (root == null) return\n    // 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n    postOrder(root.left)\n    postOrder(root.right)\n    list.add(root._val)\n}\n</code></pre> binary_tree_dfs.rb<pre><code>### 전위 순회 ###\ndef pre_order(root)\n  return if root.nil?\n\n  # 방문 우선순위: 루트 노드 -&gt; 왼쪽 서브트리 -&gt; 오른쪽 서브트리\n  $res &lt;&lt; root.val\n  pre_order(root.left)\n  pre_order(root.right)\nend\n\n### 중위 순회 ###\ndef in_order(root)\n  return if root.nil?\n\n  # 방문 우선순위: 왼쪽 서브트리 -&gt; 루트 노드 -&gt; 오른쪽 서브트리\n  in_order(root.left)\n  $res &lt;&lt; root.val\n  in_order(root.right)\nend\n\n### 후위 순회 ###\ndef post_order(root)\n  return if root.nil?\n\n  # 방문 우선순위: 왼쪽 서브트리 -&gt; 오른쪽 서브트리 -&gt; 루트 노드\n  post_order(root.left)\n  post_order(root.right)\n  $res &lt;&lt; root.val\nend\n</code></pre> 코드 시각화 <p> 전체 화면 &gt;</p> <p>참고</p> <p>깊이 우선 탐색은 반복으로도 구현할 수 있다. 관심 있는 독자는 직접 알아보자.</p> <p>아래 그림(7-11)은 이진 트리 전위 순회의 재귀 과정을 보여 준다. 서로 반대 방향인 '호출'과 '반환'으로 나뉜다.</p> <ol> <li>'호출'은 새 메서드를 시작하는 것으로, 프로그램은 이 과정에서 다음 노드를 방문한다.</li> <li>'반환'은 함수가 돌아오는 것으로, 현재 노드의 방문이 끝났음을 뜻한다.</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> 그림 7-11   전위 순회의 재귀 과정 </p>","path":["제 7 장   트리","7.2   이진 트리 순회"],"tags":[]},{"location":"chapter_tree/binary_tree_traversal/#2_1","level":3,"title":"2.   복잡도 분석","text":"<ul> <li>시간 복잡도 \\(O(n)\\): 모든 노드를 한 번씩 방문하므로 \\(O(n)\\) 시간이 걸린다.</li> <li>공간 복잡도 \\(O(n)\\): 최악의 경우 트리가 연결 리스트로 퇴화하면 재귀 깊이가 \\(n\\)에 이르러 시스템이 \\(O(n)\\)의 스택 프레임 공간을 사용한다.</li> </ul>","path":["제 7 장   트리","7.2   이진 트리 순회"],"tags":[]},{"location":"chapter_tree/exercises/","level":1,"title":"7.7   연습 문제","text":"","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#771","level":2,"title":"7.7.1   개념 확인","text":"","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#1","level":3,"title":"1.   완전 이진 트리, 정 이진 트리, 포화 이진 트리","text":"<p>다음 두 배열은 이진 트리를 레벨 순서로 나타낸 것이다. <code>None</code>은 빈자리를 뜻한다.</p> <ul> <li>트리 A: <code>[1, 2, 3, 4, 5, 6]</code></li> <li>트리 B: <code>[1, 2, 3, None, None, 6, 7]</code></li> </ul> <ol> <li>어느 트리가 완전 이진 트리인가?</li> <li>어느 트리가 모든 비리프 노드에 자식이 두 개씩 있는 정 이진 트리인가?</li> <li>두 트리 중 포화 이진 트리가 있는가? 각각 이유를 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>트리 A는 완전 이진 트리다. 맨 아래 레벨만 가득 차지 않았고, 노드가 왼쪽부터 연속해서 배치되어 있다.     트리 B는 맨 아래 레벨의 왼쪽에 빈자리가 있는데 오른쪽에 노드가 있으므로 완전 이진 트리가 아니다.</p> </li> <li> <p>트리 B는 정 이진 트리다. 노드 1과 노드 3은 자식이 각각 두 개이고, 나머지는 모두 리프 노드다.     트리 A의 노드 3에는 왼쪽 자식 6만 있으므로 정 이진 트리가 아니다.</p> </li> <li> <p>두 트리 모두 맨 아래 레벨이 완전히 채워져 있지 않으므로 포화 이진 트리가 아니다.</p> </li> </ol>","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#2","level":3,"title":"2.   같은 트리의 세 가지 순회 순서","text":"<p>배열 <code>[1, 2, 3, 4, 5, 6, 7]</code>을 레벨 순서로 완전 이진 트리에 저장한다.</p> <ol> <li>이 트리를 그려 보자.</li> <li>전위, 중위, 후위 순회 결과를 각각 적어 보자.</li> <li>중위 순회 결과에서 루트 노드 1의 왼쪽과 오른쪽에 있는 두 수열은 각각 트리의 어느 부분에 대응하는가?</li> </ol> 참고 답안 <ol> <li> <p>트리는 다음과 같다.</p> <pre><code>      1\n    /   \\\n   2     3\n  / \\   / \\\n 4   5 6   7\n</code></pre> </li> <li> <p>전위 순회는 <code>1, 2, 4, 5, 3, 6, 7</code>이다.     중위 순회는 <code>4, 2, 5, 1, 6, 3, 7</code>이다.     후위 순회는 <code>4, 5, 2, 6, 7, 3, 1</code>이다.</p> </li> <li> <p>루트 노드 1의 왼쪽에 있는 <code>4, 2, 5</code>는 왼쪽 서브트리의 중위 순회 결과다.     오른쪽의 <code>6, 3, 7</code>은 오른쪽 서브트리의 중위 순회 결과다.</p> </li> </ol>","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#3","level":3,"title":"3.   두 이진 탐색 트리 비교하기","text":"<p>다음 두 수열을 각각 왼쪽부터 빈 이진 탐색 트리에 차례대로 삽입한다.</p> <ul> <li>수열 A: <code>[4, 2, 6, 1, 3, 5, 7]</code></li> <li>수열 B: <code>[1, 2, 3, 4, 5, 6, 7]</code></li> </ul> <ol> <li>숫자 7을 찾을 때 거치는 노드를 각각 적어 보자.</li> <li>높이를 루트에서 가장 먼 리프까지 지나는 간선 수로 계산할 때, 두 트리의 높이는 각각 얼마인가?</li> <li>앞선 두 질문을 바탕으로 두 트리에서 7을 찾는 효율성이 같은지 판단하라. 트리의 모양과 탐색 경로를 바탕으로 이유를 설명하라.</li> </ol> 참고 답안 <ol> <li> <p>수열 A로 만든 트리의 탐색 경로는 <code>4 → 6 → 7</code>이다.     수열 B로 만든 트리의 탐색 경로는 <code>1 → 2 → 3 → 4 → 5 → 6 → 7</code>이다.</p> </li> <li> <p>첫 번째 트리는 모든 레벨이 가득 차 있고 높이는 2다. 두 번째 트리는 오른쪽 자식만 있으며 높이는 6이다.</p> </li> <li> <p>두 트리에서 7을 찾는 효율성은 다르다. 삽입 순서가 이진 탐색 트리의 모양과 높이를 바꾸기 때문이다. 첫 번째 트리는 노드 3개만 거치지만,     두 번째 트리는 노드 7개를 차례대로 거친다. 트리가 높을수록 최악의 경우 경로를 따라 비교해야 하는 노드가 많아진다.</p> </li> </ol>","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#772","level":2,"title":"7.7.2   프로그래밍 연습","text":"","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#1_1","level":3,"title":"1.   이진 트리의 최대 깊이","text":"<p>이진 트리의 루트 노드 <code>root</code>가 주어진다. 각 노드는 정숫값과 왼쪽 및 오른쪽 자식 노드에 대한 참조를 가진다.</p> <p>루트에서 가장 먼 리프까지 거치는 노드 수를 이진 트리의 최대 깊이라고 정의한다. 이 트리의 최대 깊이를 반환하라. 빈 트리의 최대 깊이는 0이다. 재귀를 사용하여 해결하라.</p> 풀이 힌트 <ol> <li>이 문제에서는 노드 수로 깊이를 계산한다. 루트 하나만 있는 트리의 최대 깊이는 1이다.</li> <li>재귀 함수가 현재 노드를 루트로 하는 서브트리의 최대 깊이를 반환하도록 한다.</li> <li>빈 노드는 0을 반환하고, 비어 있지 않은 노드는 max(depth(left), depth(right)) + 1을 반환한다.</li> </ol> <p>LeetCode</p>","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#2_1","level":3,"title":"2.   이진 트리를 레벨 순서로 순회하기","text":"<p>이진 트리의 루트 노드 <code>root</code>가 주어진다. 큐를 사용하여 위에서 아래로 한 레벨씩 모든 노드를 방문하라. 같은 레벨에서는 왼쪽부터 오른쪽으로 방문한다.</p> <p>2차원 배열을 반환하라. 첫 번째 하위 배열에는 루트가 있는 레벨의 노드 값을, 두 번째에는 다음 레벨의 노드 값을 저장하며 이 방식을 반복한다. 빈 트리라면 빈 배열을 반환하라.</p> 풀이 힌트 <ol> <li>레벨 순서 순회에서는 먼저 들어온 노드를 먼저 방문해야 하므로 큐를 사용한다.</li> <li>각 반복이 시작될 때 큐에 있는 노드들은 정확히 같은 레벨에 속한다.</li> <li>큐의 길이를 먼저 기록한 뒤 그 수만큼 노드를 꺼내고 각 노드의 자식을 넣는다.</li> </ol> <p>LeetCode</p>","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/exercises/#3-k","level":3,"title":"3.   이진 탐색 트리에서 k번째로 작은 원소","text":"<p>이진 탐색 트리에 총 <code>n</code>개의 노드가 있으며, 노드 값은 모두 다르다. 모든 노드 값을 작은 것부터 나열하고 위치에 1부터 번호를 붙인다.</p> <p>루트 노드 <code>root</code>와 <code>1 &lt;= k &lt;= n</code>을 만족하는 정수 <code>k</code>가 주어질 때, <code>k</code>번째 노드 값을 반환하라. 모든 값을 미리 모으지 말고 중위 순회 도중에 바로 답을 찾아라.</p> 풀이 힌트 <ol> <li>이진 탐색 트리를 중위 순회하면 노드 값을 작은 것부터 방문한다.</li> <li>중위 순회는 왼쪽 서브트리, 현재 노드, 오른쪽 서브트리 순서로 처리한다. 현재 노드를 방문할 때 개수를 1 늘린다.</li> <li>개수가 처음 k가 되었을 때의 노드 값이 답이다. 이후에는 순회를 계속할 필요가 없다.</li> </ol> <p>LeetCode</p>","path":["제 7 장   트리","7.7   연습 문제"],"tags":[]},{"location":"chapter_tree/summary/","level":1,"title":"7.6   요약","text":"","path":["제 7 장   트리","7.6   요약"],"tags":[]},{"location":"chapter_tree/summary/#1","level":3,"title":"1.   핵심 정리","text":"<ul> <li>이진 트리는 '하나를 둘로 나누는' 분할 정복 논리를 보여 주는 비선형 자료 구조다. 각 노드는 값 하나와 왼쪽, 오른쪽 자식을 가리키는 포인터 두 개를 포함한다.</li> <li>어떤 노드의 왼쪽(오른쪽) 자식과 그 아래 노드들이 이루는 트리를 해당 노드의 왼쪽(오른쪽) 서브트리라고 한다.</li> <li>주요 용어에는 루트 노드, 리프 노드, 레벨, 차수, 간선, 높이, 깊이 등이 있다.</li> <li>이진 트리의 초기화, 노드 삽입과 삭제 방법은 연결 리스트와 비슷하다.</li> <li>주요 종류는 포화 이진 트리, 완전 이진 트리, 정 이진 트리, 균형 이진 트리다. 포화 이진 트리가 가장 이상적이며, 연결 리스트는 퇴화한 최악의 상태다.</li> <li>이진 트리는 노드 값과 빈자리를 레벨 순서대로 배열하여 배열로 표현할 수 있다. 부모와 자식의 인덱스 매핑 관계가 포인터를 대신한다.</li> <li>레벨 순서 순회는 '한 겹씩 바깥으로 확장'하는 너비 우선 탐색이며, 보통 큐로 구현한다.</li> <li>전위, 중위, 후위 순회는 '먼저 끝까지 간 뒤 돌아와 계속하는' 깊이 우선 탐색이며, 보통 재귀로 구현한다.</li> <li>이진 탐색 트리는 효율적인 원소 탐색 자료 구조로, 탐색, 삽입, 삭제의 시간 복잡도는 모두 \\(O(\\log n)\\)이다. 연결 리스트로 퇴화하면 각 복잡도가 \\(O(n)\\)으로 저하된다.</li> <li>AVL 트리는 균형 이진 탐색 트리라고도 하며, 회전 연산으로 삽입과 삭제가 반복되어도 균형을 유지한다.</li> <li>AVL 트리의 회전에는 오른쪽 회전, 왼쪽 회전, 오른쪽 후 왼쪽 회전, 왼쪽 후 오른쪽 회전이 있다. 삽입이나 삭제 후 아래에서 위로 회전하여 균형을 복구한다.</li> </ul>","path":["제 7 장   트리","7.6   요약"],"tags":[]},{"location":"chapter_tree/summary/#2-q-a","level":3,"title":"2.   Q &amp; A","text":"<p>Q: 노드가 하나뿐인 이진 트리의 높이와 루트의 깊이는 모두 \\(0\\)인가?</p> <p>그렇다. 높이와 깊이는 보통 '거치는 간선 수'로 정의하기 때문이다.</p> <p>Q: 이진 트리의 삽입과 삭제를 보통 여러 연산의 조합으로 수행한다고 했는데, 여기서 '여러 연산'은 무엇인가? 해당 자원에 속한 자식 노드의 자원을 해제하는 것으로 이해해도 되는가?</p> <p>이진 탐색 트리를 예로 들면 삭제를 세 경우로 나누어 처리하며, 각 경우마다 여러 단계의 노드 연산이 필요하다.</p> <p>Q: 이진 트리의 DFS에는 왜 전위, 중위, 후위라는 세 순서가 있고 각각 어디에 쓰이는가?</p> <p>배열을 정방향이나 역방향으로 순회하듯이 전위, 중위, 후위는 이진 트리를 순회하는 세 방법이며 특정 순서의 결과를 얻는 데 사용한다. 예를 들어 이진 탐색 트리는 <code>왼쪽 자식 값 &lt; 루트 값 &lt; 오른쪽 자식 값</code>을 만족하므로 '왼쪽 \\(\\rightarrow\\) 루트 \\(\\rightarrow\\) 오른쪽' 우선순위로 순회하면 정렬된 노드 순서를 얻는다.</p> <p>Q: 오른쪽 회전은 불균형 노드 <code>node</code>, <code>child</code>, <code>grand_child</code> 사이의 관계를 처리하는데, <code>node</code>와 그 부모의 기존 연결은 관리하지 않아도 되는가? 회전 후 끊어지지 않는가?</p> <p>재귀의 관점에서 보아야 한다. <code>right_rotate(root)</code>는 서브트리의 루트를 전달받고, 마지막의 <code>return child</code>는 회전 후의 서브트리 루트를 반환한다. 새 서브트리 루트와 부모의 연결은 함수가 반환된 뒤 이루어지므로 오른쪽 회전 자체가 관리할 범위가 아니다.</p> <p>Q: C++에서 함수를 <code>private</code>와 <code>public</code>으로 나누는 기준은 무엇인가? <code>height()</code>는 <code>public</code>, <code>updateHeight()</code>는 <code>private</code>에 두는 이유는 무엇인가?</p> <p>주로 메서드의 사용 범위를 기준으로 한다. 클래스 내부에서만 쓰면 <code>private</code>로 설계한다. 예를 들어 <code>updateHeight()</code>는 삽입과 삭제의 한 단계일 뿐이라 사용자가 따로 호출할 의미가 없다. 반면 <code>height()</code>는 <code>vector.size()</code>처럼 노드 높이에 접근하는 기능이므로 편리하게 쓰도록 <code>public</code>으로 둔다.</p> <p>Q: 입력 데이터 집합에서 이진 탐색 트리를 어떻게 구성하는가? 루트 선택이 중요한가?</p> <p>그렇다. 구성 방법은 이진 탐색 트리 코드의 <code>build_tree()</code>에 제시되어 있다. 루트는 보통 입력을 정렬한 뒤 중간 원소로 정하고, 왼쪽과 오른쪽 서브트리를 재귀적으로 구성한다. 이렇게 하면 트리의 균형을 최대한 보장할 수 있다.</p> <p>Q: Java에서 문자열을 비교할 때 반드시 <code>equals()</code>를 사용해야 하는가?</p> <p>Java에서 기본 타입의 <code>==</code>는 두 변수의 값이 같은지 비교한다. 참조 타입에서는 두 방식의 원리가 다르다.</p> <ul> <li><code>==</code>: 두 변수가 같은 객체, 즉 메모리의 같은 위치를 가리키는지 비교한다.</li> <li><code>equals()</code>: 두 객체의 값이 같은지 비교한다.</li> </ul> <p>따라서 값을 비교하려면 <code>equals()</code>를 사용해야 한다. 다만 <code>String a = \"hi\"; String b = \"hi\";</code>로 초기화한 문자열들은 문자열 상수 풀에 저장되어 같은 객체를 가리키므로, 이 경우에는 <code>a == b</code>로 내용이 같은지 비교할 수도 있다.</p> <p>Q: 너비 우선 순회에서 맨 아래 레벨에 도달하기 전 큐의 노드 수는 \\(2^h\\)인가?</p> <p>그렇다. 예를 들어 높이 \\(h = 2\\)인 포화 이진 트리의 전체 노드 수는 \\(n = 7\\)이고, 맨 아래 레벨의 노드 수는 \\(4 = 2^h = (n + 1) / 2\\)다.</p>","path":["제 7 장   트리","7.6   요약"],"tags":[]}]}
\ No newline at end of file
diff --git a/ko/stylesheets/animation_player.css b/ko/stylesheets/animation_player.css
index bc33d7f01..34c931ed4 100644
--- a/ko/stylesheets/animation_player.css
+++ b/ko/stylesheets/animation_player.css
@@ -176,4 +176,4 @@
     font-size: 0.7rem;
   }
 }
-/*! update cache: 20261029171440 */
+/*! update cache: 20261030193337 */
diff --git a/ko/stylesheets/extra.css b/ko/stylesheets/extra.css
index 1a7a31496..76fdc1eb7 100644
--- a/ko/stylesheets/extra.css
+++ b/ko/stylesheets/extra.css
@@ -477,10 +477,75 @@ html:has(body[data-md-color-scheme="default"]) {
   font-weight: normal;
 }
 
+/* Keep long English chapter labels inside the hero at tablet/desktop widths. */
 .hero-div--en {
+  width: min(112vh, 76vw);
+  height: min(84vh, 57vw);
   font-size: min(1.65vh, 2.3vw);
 }
 
+.hero-div--en > a > span {
+  z-index: 1;
+}
+
+/* The illustrated chapter map becomes readable touch targets on small screens. */
+@media screen and (max-width: 600px) {
+  .home-div:has(> .hero-div--en) {
+    display: block;
+    height: auto !important;
+    margin-top: 0 !important;
+    padding: 0.5rem 0 1rem !important;
+  }
+
+  .home-div > .hero-div--en {
+    display: grid;
+    grid-template-columns: repeat(2, minmax(0, 1fr));
+    gap: 0.35rem 0.5rem;
+    width: calc(100% - 1.5rem);
+    height: auto;
+    margin: 0.5rem auto 1rem;
+  }
+
+  .home-div > .hero-div--en > img {
+    display: none;
+  }
+
+  .home-div > .hero-div--en > a {
+    display: flex;
+    align-items: center;
+    gap: 0.4rem;
+    min-width: 0;
+    min-height: 48px;
+    padding: 0.3rem 0.15rem;
+    text-align: left;
+  }
+
+  .home-div > .hero-div--en > a > img {
+    position: static;
+    flex-shrink: 0;
+    width: 32px !important;
+    height: 32px !important;
+    object-fit: contain;
+  }
+
+  .home-div > .hero-div--en > a > span {
+    position: static;
+    min-width: 0;
+    transform: none;
+    white-space: normal;
+    font-size: 13px;
+    line-height: 1.3;
+  }
+
+  .home-div > .hero-div--en + .heading-div {
+    position: relative;
+    left: auto;
+    bottom: auto;
+    transform: none;
+    padding: 0 0.75rem;
+  }
+}
+
 .hero-div--ru {
   font-size: min(1.45vh, 2vw);
 }
@@ -931,4 +996,4 @@ a:hover .device-on-hover {
     max-width: 100%;
   }
 }
-/*! update cache: 20261029171440 */
+/*! update cache: 20261030193337 */
diff --git a/ko/stylesheets/giscus-dark.css b/ko/stylesheets/giscus-dark.css
index 0d6d424c8..9727cf830 100644
--- a/ko/stylesheets/giscus-dark.css
+++ b/ko/stylesheets/giscus-dark.css
@@ -122,4 +122,4 @@ main .gsc-loading-image {
 .gsc-reply-content::-webkit-scrollbar-track {
   background: transparent;
 }
-/*! update cache: 20261029171440 */
+/*! update cache: 20261030193337 */
diff --git a/ko/stylesheets/giscus-light.css b/ko/stylesheets/giscus-light.css
index 36fd855d4..6c7a2d321 100644
--- a/ko/stylesheets/giscus-light.css
+++ b/ko/stylesheets/giscus-light.css
@@ -153,4 +153,4 @@ main {
 .gsc-reply-content::-webkit-scrollbar-track {
   background: transparent;
 }
-/*! update cache: 20261029171440 */
+/*! update cache: 20261030193337 */
diff --git a/ru/javascripts/animation_player.js b/ru/javascripts/animation_player.js
index a1b0e149f..7ea0f97c4 100644
--- a/ru/javascripts/animation_player.js
+++ b/ru/javascripts/animation_player.js
@@ -301,4 +301,4 @@
     initAutoSlide();
   }
 })();
-/*! update cache: 20261029171410 */
+/*! update cache: 20261030193307 */
diff --git a/ru/javascripts/katex.js b/ru/javascripts/katex.js
index ddfc648c2..14f2d921e 100644
--- a/ru/javascripts/katex.js
+++ b/ru/javascripts/katex.js
@@ -8,4 +8,4 @@ document$.subscribe(({ body }) => {
     ],
   });
 });
-/*! update cache: 20261029171410 */
+/*! update cache: 20261030193307 */
diff --git a/ru/javascripts/mathjax.js b/ru/javascripts/mathjax.js
index 478f09386..ab0140850 100644
--- a/ru/javascripts/mathjax.js
+++ b/ru/javascripts/mathjax.js
@@ -15,4 +15,4 @@ window.MathJax = {
 document$.subscribe(() => {
   MathJax.typesetPromise();
 });
-/*! update cache: 20261029171410 */
+/*! update cache: 20261030193307 */
diff --git a/ru/javascripts/starfield.js b/ru/javascripts/starfield.js
index 4c3716924..9db5a22a7 100644
--- a/ru/javascripts/starfield.js
+++ b/ru/javascripts/starfield.js
@@ -469,4 +469,4 @@
 
   return Starfield;
 });
-/*! update cache: 20261029171410 */
+/*! update cache: 20261030193307 */
diff --git a/ru/stylesheets/animation_player.css b/ru/stylesheets/animation_player.css
index 175cf446d..e548b90dc 100644
--- a/ru/stylesheets/animation_player.css
+++ b/ru/stylesheets/animation_player.css
@@ -176,4 +176,4 @@
     font-size: 0.7rem;
   }
 }
-/*! update cache: 20261029171410 */
+/*! update cache: 20261030193307 */
diff --git a/ru/stylesheets/extra.css b/ru/stylesheets/extra.css
index d8acc3226..10f5c97ae 100644
--- a/ru/stylesheets/extra.css
+++ b/ru/stylesheets/extra.css
@@ -477,10 +477,75 @@ html:has(body[data-md-color-scheme="default"]) {
   font-weight: normal;
 }
 
+/* Keep long English chapter labels inside the hero at tablet/desktop widths. */
 .hero-div--en {
+  width: min(112vh, 76vw);
+  height: min(84vh, 57vw);
   font-size: min(1.65vh, 2.3vw);
 }
 
+.hero-div--en > a > span {
+  z-index: 1;
+}
+
+/* The illustrated chapter map becomes readable touch targets on small screens. */
+@media screen and (max-width: 600px) {
+  .home-div:has(> .hero-div--en) {
+    display: block;
+    height: auto !important;
+    margin-top: 0 !important;
+    padding: 0.5rem 0 1rem !important;
+  }
+
+  .home-div > .hero-div--en {
+    display: grid;
+    grid-template-columns: repeat(2, minmax(0, 1fr));
+    gap: 0.35rem 0.5rem;
+    width: calc(100% - 1.5rem);
+    height: auto;
+    margin: 0.5rem auto 1rem;
+  }
+
+  .home-div > .hero-div--en > img {
+    display: none;
+  }
+
+  .home-div > .hero-div--en > a {
+    display: flex;
+    align-items: center;
+    gap: 0.4rem;
+    min-width: 0;
+    min-height: 48px;
+    padding: 0.3rem 0.15rem;
+    text-align: left;
+  }
+
+  .home-div > .hero-div--en > a > img {
+    position: static;
+    flex-shrink: 0;
+    width: 32px !important;
+    height: 32px !important;
+    object-fit: contain;
+  }
+
+  .home-div > .hero-div--en > a > span {
+    position: static;
+    min-width: 0;
+    transform: none;
+    white-space: normal;
+    font-size: 13px;
+    line-height: 1.3;
+  }
+
+  .home-div > .hero-div--en + .heading-div {
+    position: relative;
+    left: auto;
+    bottom: auto;
+    transform: none;
+    padding: 0 0.75rem;
+  }
+}
+
 .hero-div--ru {
   font-size: min(1.45vh, 2vw);
 }
@@ -931,4 +996,4 @@ a:hover .device-on-hover {
     max-width: 100%;
   }
 }
-/*! update cache: 20261029171410 */
+/*! update cache: 20261030193307 */
diff --git a/ru/stylesheets/giscus-dark.css b/ru/stylesheets/giscus-dark.css
index 7a4480481..6286db69a 100644
--- a/ru/stylesheets/giscus-dark.css
+++ b/ru/stylesheets/giscus-dark.css
@@ -122,4 +122,4 @@ main .gsc-loading-image {
 .gsc-reply-content::-webkit-scrollbar-track {
   background: transparent;
 }
-/*! update cache: 20261029171410 */
+/*! update cache: 20261030193307 */
diff --git a/ru/stylesheets/giscus-light.css b/ru/stylesheets/giscus-light.css
index 1927bf0a0..9709948c9 100644
--- a/ru/stylesheets/giscus-light.css
+++ b/ru/stylesheets/giscus-light.css
@@ -153,4 +153,4 @@ main {
 .gsc-reply-content::-webkit-scrollbar-track {
   background: transparent;
 }
-/*! update cache: 20261029171410 */
+/*! update cache: 20261030193307 */
diff --git a/stylesheets/animation_player.css b/stylesheets/animation_player.css
index 991124223..f237dc61d 100644
--- a/stylesheets/animation_player.css
+++ b/stylesheets/animation_player.css
@@ -176,4 +176,4 @@
     font-size: 0.7rem;
   }
 }
-/*! update cache: 20261029171211 */
+/*! update cache: 20261030193105 */
diff --git a/stylesheets/extra.css b/stylesheets/extra.css
index 629c69139..31e056026 100644
--- a/stylesheets/extra.css
+++ b/stylesheets/extra.css
@@ -477,10 +477,75 @@ html:has(body[data-md-color-scheme="default"]) {
   font-weight: normal;
 }
 
+/* Keep long English chapter labels inside the hero at tablet/desktop widths. */
 .hero-div--en {
+  width: min(112vh, 76vw);
+  height: min(84vh, 57vw);
   font-size: min(1.65vh, 2.3vw);
 }
 
+.hero-div--en > a > span {
+  z-index: 1;
+}
+
+/* The illustrated chapter map becomes readable touch targets on small screens. */
+@media screen and (max-width: 600px) {
+  .home-div:has(> .hero-div--en) {
+    display: block;
+    height: auto !important;
+    margin-top: 0 !important;
+    padding: 0.5rem 0 1rem !important;
+  }
+
+  .home-div > .hero-div--en {
+    display: grid;
+    grid-template-columns: repeat(2, minmax(0, 1fr));
+    gap: 0.35rem 0.5rem;
+    width: calc(100% - 1.5rem);
+    height: auto;
+    margin: 0.5rem auto 1rem;
+  }
+
+  .home-div > .hero-div--en > img {
+    display: none;
+  }
+
+  .home-div > .hero-div--en > a {
+    display: flex;
+    align-items: center;
+    gap: 0.4rem;
+    min-width: 0;
+    min-height: 48px;
+    padding: 0.3rem 0.15rem;
+    text-align: left;
+  }
+
+  .home-div > .hero-div--en > a > img {
+    position: static;
+    flex-shrink: 0;
+    width: 32px !important;
+    height: 32px !important;
+    object-fit: contain;
+  }
+
+  .home-div > .hero-div--en > a > span {
+    position: static;
+    min-width: 0;
+    transform: none;
+    white-space: normal;
+    font-size: 13px;
+    line-height: 1.3;
+  }
+
+  .home-div > .hero-div--en + .heading-div {
+    position: relative;
+    left: auto;
+    bottom: auto;
+    transform: none;
+    padding: 0 0.75rem;
+  }
+}
+
 .hero-div--ru {
   font-size: min(1.45vh, 2vw);
 }
@@ -931,4 +996,4 @@ a:hover .device-on-hover {
     max-width: 100%;
   }
 }
-/*! update cache: 20261029171211 */
+/*! update cache: 20261030193105 */
diff --git a/stylesheets/giscus-dark.css b/stylesheets/giscus-dark.css
index f7c53bd2a..160d866fe 100644
--- a/stylesheets/giscus-dark.css
+++ b/stylesheets/giscus-dark.css
@@ -122,4 +122,4 @@ main .gsc-loading-image {
 .gsc-reply-content::-webkit-scrollbar-track {
   background: transparent;
 }
-/*! update cache: 20261029171211 */
+/*! update cache: 20261030193105 */
diff --git a/stylesheets/giscus-light.css b/stylesheets/giscus-light.css
index d6c7042fd..8d19d1d85 100644
--- a/stylesheets/giscus-light.css
+++ b/stylesheets/giscus-light.css
@@ -153,4 +153,4 @@ main {
 .gsc-reply-content::-webkit-scrollbar-track {
   background: transparent;
 }
-/*! update cache: 20261029171211 */
+/*! update cache: 20261030193105 */
diff --git a/zh-hant/javascripts/animation_player.js b/zh-hant/javascripts/animation_player.js
index 3ce1a5dc5..5d482c682 100644
--- a/zh-hant/javascripts/animation_player.js
+++ b/zh-hant/javascripts/animation_player.js
@@ -301,4 +301,4 @@
     initAutoSlide();
   }
 })();
-/*! update cache: 20261029171240 */
+/*! update cache: 20261030193135 */
diff --git a/zh-hant/javascripts/katex.js b/zh-hant/javascripts/katex.js
index bf554da49..d6ee12aff 100644
--- a/zh-hant/javascripts/katex.js
+++ b/zh-hant/javascripts/katex.js
@@ -8,4 +8,4 @@ document$.subscribe(({ body }) => {
     ],
   });
 });
-/*! update cache: 20261029171240 */
+/*! update cache: 20261030193135 */
diff --git a/zh-hant/javascripts/mathjax.js b/zh-hant/javascripts/mathjax.js
index 1cae7b249..41169a4d9 100644
--- a/zh-hant/javascripts/mathjax.js
+++ b/zh-hant/javascripts/mathjax.js
@@ -15,4 +15,4 @@ window.MathJax = {
 document$.subscribe(() => {
   MathJax.typesetPromise();
 });
-/*! update cache: 20261029171240 */
+/*! update cache: 20261030193135 */
diff --git a/zh-hant/javascripts/starfield.js b/zh-hant/javascripts/starfield.js
index b90c82524..e2872ff52 100644
--- a/zh-hant/javascripts/starfield.js
+++ b/zh-hant/javascripts/starfield.js
@@ -469,4 +469,4 @@
 
   return Starfield;
 });
-/*! update cache: 20261029171240 */
+/*! update cache: 20261030193135 */
diff --git a/zh-hant/stylesheets/animation_player.css b/zh-hant/stylesheets/animation_player.css
index 40e4cb8ba..b2a7aa697 100644
--- a/zh-hant/stylesheets/animation_player.css
+++ b/zh-hant/stylesheets/animation_player.css
@@ -176,4 +176,4 @@
     font-size: 0.7rem;
   }
 }
-/*! update cache: 20261029171240 */
+/*! update cache: 20261030193135 */
diff --git a/zh-hant/stylesheets/extra.css b/zh-hant/stylesheets/extra.css
index 33cf07d14..a0a4dbbac 100644
--- a/zh-hant/stylesheets/extra.css
+++ b/zh-hant/stylesheets/extra.css
@@ -477,10 +477,75 @@ html:has(body[data-md-color-scheme="default"]) {
   font-weight: normal;
 }
 
+/* Keep long English chapter labels inside the hero at tablet/desktop widths. */
 .hero-div--en {
+  width: min(112vh, 76vw);
+  height: min(84vh, 57vw);
   font-size: min(1.65vh, 2.3vw);
 }
 
+.hero-div--en > a > span {
+  z-index: 1;
+}
+
+/* The illustrated chapter map becomes readable touch targets on small screens. */
+@media screen and (max-width: 600px) {
+  .home-div:has(> .hero-div--en) {
+    display: block;
+    height: auto !important;
+    margin-top: 0 !important;
+    padding: 0.5rem 0 1rem !important;
+  }
+
+  .home-div > .hero-div--en {
+    display: grid;
+    grid-template-columns: repeat(2, minmax(0, 1fr));
+    gap: 0.35rem 0.5rem;
+    width: calc(100% - 1.5rem);
+    height: auto;
+    margin: 0.5rem auto 1rem;
+  }
+
+  .home-div > .hero-div--en > img {
+    display: none;
+  }
+
+  .home-div > .hero-div--en > a {
+    display: flex;
+    align-items: center;
+    gap: 0.4rem;
+    min-width: 0;
+    min-height: 48px;
+    padding: 0.3rem 0.15rem;
+    text-align: left;
+  }
+
+  .home-div > .hero-div--en > a > img {
+    position: static;
+    flex-shrink: 0;
+    width: 32px !important;
+    height: 32px !important;
+    object-fit: contain;
+  }
+
+  .home-div > .hero-div--en > a > span {
+    position: static;
+    min-width: 0;
+    transform: none;
+    white-space: normal;
+    font-size: 13px;
+    line-height: 1.3;
+  }
+
+  .home-div > .hero-div--en + .heading-div {
+    position: relative;
+    left: auto;
+    bottom: auto;
+    transform: none;
+    padding: 0 0.75rem;
+  }
+}
+
 .hero-div--ru {
   font-size: min(1.45vh, 2vw);
 }
@@ -931,4 +996,4 @@ a:hover .device-on-hover {
     max-width: 100%;
   }
 }
-/*! update cache: 20261029171240 */
+/*! update cache: 20261030193135 */
diff --git a/zh-hant/stylesheets/giscus-dark.css b/zh-hant/stylesheets/giscus-dark.css
index e4fcecdc9..3b37be952 100644
--- a/zh-hant/stylesheets/giscus-dark.css
+++ b/zh-hant/stylesheets/giscus-dark.css
@@ -122,4 +122,4 @@ main .gsc-loading-image {
 .gsc-reply-content::-webkit-scrollbar-track {
   background: transparent;
 }
-/*! update cache: 20261029171240 */
+/*! update cache: 20261030193135 */
diff --git a/zh-hant/stylesheets/giscus-light.css b/zh-hant/stylesheets/giscus-light.css
index 5b8d2d7cc..b77334411 100644
--- a/zh-hant/stylesheets/giscus-light.css
+++ b/zh-hant/stylesheets/giscus-light.css
@@ -153,4 +153,4 @@ main {
 .gsc-reply-content::-webkit-scrollbar-track {
   background: transparent;
 }
-/*! update cache: 20261029171240 */
+/*! update cache: 20261030193135 */